%------------------------------------------------------------------------------
% File : DT2H2X---1.9.5
% Problem : ITP211^3 : TPTP v9.2.1. Bugfixed v9.2.1.
% Transfm : none
% Format : tptp:raw
% Command : /export/starexec/sandbox/solver/bin/run_DT2H2X /export/starexec/sandbox/benchmark/theBenchmark.p 300
% Computer : n017.cluster.edu
% Model : x86_64 x86_64
% CPU : Intel(R) Xeon(R) CPU E5-2620 v4 2.10GHz
% Memory : 8042.1875MB
% OS : Linux 3.10.0-693.el7.x86_64
% CPULimit : 300s
% WCLimit : 300s
% DateTime : Sat Feb 28 09:04:16 AM UTC 2026
% Result : Theorem 129.49s 25.35s
% Output : Refutation 129.49s
% Verified :
% SZS Type : Refutation
% Derivation depth : 17
% Number of leaves : 17
% Syntax : Number of formulae : 101 ( 39 unt; 0 typ; 2 def)
% Number of atoms : 309 ( 155 equ; 0 cnn)
% Maximal formula atoms : 4 ( 3 avg)
% Number of connectives : 449 ( 71 ~; 52 |; 9 &; 295 @)
% ( 5 <=>; 16 =>; 0 <=; 1 <~>)
% Maximal formula depth : 7 ( 4 avg)
% Number of types : 160 ( 159 usr)
% Number of type conns : 18 ( 18 >; 0 *; 0 +; 0 <<)
% Number of symbols : 1788 (1785 usr; 87 con; 0-8 aty)
% Number of variables : 135 ( 0 ^; 109 !; 26 ?; 135 :)
% Comments :
%------------------------------------------------------------------------------
thf(type_def_5,type,
produc3658429121746597890et_nat: $tType ).
thf(type_def_7,type,
assn: $tType ).
thf(type_def_8,type,
produc6197004810343482825et_nat: $tType ).
thf(type_def_9,type,
set_nat: $tType ).
thf(type_def_10,type,
heap_e7401611519738050253t_unit: $tType ).
thf(type_def_11,type,
produc2732055786443039994et_nat: $tType ).
thf(type_def_12,type,
set_Pr5094982260447487303t_unit: $tType ).
thf(type_def_13,type,
set_Product_unit: $tType ).
thf(type_def_14,type,
int: $tType ).
thf(type_def_15,type,
product_prod_int_int: $tType ).
thf(type_def_16,type,
code_integer: $tType ).
thf(type_def_17,type,
nat: $tType ).
thf(type_def_18,type,
code_natural: $tType ).
thf(type_def_19,type,
rat: $tType ).
thf(type_def_20,type,
product_prod_nat_nat: $tType ).
thf(type_def_21,type,
set_Pr958786334691620121nt_int: $tType ).
thf(type_def_22,type,
set_Pr2560585780119916871nt_int: $tType ).
thf(type_def_23,type,
set_Pr1261947904930325089at_nat: $tType ).
thf(type_def_24,type,
set_Pr8693737435421807431at_nat: $tType ).
thf(type_def_25,type,
set_Pr2161125870931222855_nat_o: $tType ).
thf(type_def_26,type,
set_Product_prod_o_o: $tType ).
thf(type_def_27,type,
set_Pr4532377907799695533_nat_o: $tType ).
thf(type_def_28,type,
set_o: $tType ).
thf(type_def_29,type,
set_int: $tType ).
thf(type_def_30,type,
product_unit: $tType ).
thf(type_def_31,type,
produc3925858234332021118et_nat: $tType ).
thf(type_def_32,type,
set_he8644174446132254637t_unit: $tType ).
thf(type_def_33,type,
set_Pr3286484037609594932et_nat: $tType ).
thf(type_def_34,type,
set_Pr3948176798113811640et_nat: $tType ).
thf(type_def_35,type,
set_set_nat: $tType ).
thf(type_def_36,type,
num: $tType ).
thf(type_def_37,type,
produc8923325533196201883nteger: $tType ).
thf(type_def_38,type,
set_Pr6409761941060870310at_o_o: $tType ).
thf(type_def_39,type,
set_o_o: $tType ).
thf(type_def_40,type,
set_int_o: $tType ).
thf(type_def_41,type,
set_nat_o: $tType ).
thf(type_def_42,type,
set_Product_unit_o: $tType ).
thf(type_def_43,type,
set_filter_nat: $tType ).
thf(type_def_44,type,
filter_nat: $tType ).
thf(type_def_45,type,
set_se1353816476878259085_nat_o: $tType ).
thf(type_def_46,type,
set_set_o: $tType ).
thf(type_def_47,type,
set_set_int: $tType ).
thf(type_def_48,type,
set_se7855581050983116737at_nat: $tType ).
thf(type_def_49,type,
set_set_Product_unit: $tType ).
thf(type_def_50,type,
set_set_Code_integer: $tType ).
thf(type_def_51,type,
set_Code_integer: $tType ).
thf(type_def_52,type,
set_se7663487083892625191_nat_o: $tType ).
thf(type_def_53,type,
set_se491199571785101031od_o_o: $tType ).
thf(type_def_54,type,
set_se6260736226359567993nt_int: $tType ).
thf(type_def_55,type,
set_se8994758365886191399t_unit: $tType ).
thf(type_def_56,type,
filter1242075044329608583at_nat: $tType ).
thf(type_def_57,type,
filter_set_nat: $tType ).
thf(type_def_58,type,
set_assn: $tType ).
thf(type_def_59,type,
set_Pr4811707699266497531nteger: $tType ).
thf(type_def_60,type,
set_Pr8986988648414616686er_nat: $tType ).
thf(type_def_61,type,
set_Pr2406931829603198446nteger: $tType ).
thf(type_def_62,type,
set_rat: $tType ).
thf(type_def_63,type,
produc723880050079548563nteger: $tType ).
thf(type_def_64,type,
produc101387210554329899nteger: $tType ).
thf(type_def_65,type,
produc2221115339011613079nteger: $tType ).
thf(type_def_66,type,
produc1397831737260518551nteger: $tType ).
thf(type_def_67,type,
produc6433947425880249742er_nat: $tType ).
thf(type_def_68,type,
produc2591615631378489102nteger: $tType ).
thf(type_def_69,type,
typerep: $tType ).
thf(type_def_70,type,
list_nat: $tType ).
thf(type_def_71,type,
set_Pr3813038829465207111_nat_o: $tType ).
thf(type_def_72,type,
set_Pr4577759397028426247_set_o: $tType ).
thf(type_def_73,type,
set_Pr2522554150109002629et_int: $tType ).
thf(type_def_74,type,
set_Pr5488025237498180813et_nat: $tType ).
thf(type_def_75,type,
set_Pr3980508049012409671t_unit: $tType ).
thf(type_def_76,type,
set_Pr8536935166611901872et_nat: $tType ).
thf(type_def_77,type,
produc5030937667867769463nt_int: $tType ).
thf(type_def_78,type,
produc6128020648511768019nt_int: $tType ).
thf(type_def_79,type,
produc3574062574353363227at_nat: $tType ).
thf(type_def_80,type,
produc4671145554997361783at_nat: $tType ).
thf(type_def_81,type,
sum_sum_nat_nat: $tType ).
thf(type_def_82,type,
set_Pr4769669219625338370_o_nat: $tType ).
thf(type_def_83,type,
set_Pr2101469702781467981_o_nat: $tType ).
thf(type_def_84,type,
set_Pr3448869479623346877nt_nat: $tType ).
thf(type_def_85,type,
set_Pr1763845938948868674it_nat: $tType ).
thf(type_def_86,type,
set_set_set_nat: $tType ).
thf(type_def_87,type,
set_Co726186806682698383nteger: $tType ).
thf(type_def_88,type,
set_Code_natural: $tType ).
thf(type_def_89,type,
set_Pr448751882837621926eger_o: $tType ).
thf(type_def_90,type,
set_Pr6496905503482889930er_int: $tType ).
thf(type_def_91,type,
set_Pr4580871404036792859t_unit: $tType ).
thf(type_def_92,type,
produc4928098042776334183_nat_o: $tType ).
thf(type_def_93,type,
product_prod_o_o: $tType ).
thf(type_def_94,type,
product_prod_o_int: $tType ).
thf(type_def_95,type,
product_prod_o_nat: $tType ).
thf(type_def_96,type,
produc3788275437394581276t_unit: $tType ).
thf(type_def_97,type,
produc7822875418678951345atural: $tType ).
thf(type_def_98,type,
produc5835291356934675326atural: $tType ).
thf(type_def_99,type,
produc8126358835630522215t_unit: $tType ).
thf(type_def_100,type,
product_prod_int_o: $tType ).
thf(type_def_101,type,
product_prod_int_nat: $tType ).
thf(type_def_102,type,
produc6865911496949736918t_unit: $tType ).
thf(type_def_103,type,
product_prod_nat_o: $tType ).
thf(type_def_104,type,
product_prod_nat_int: $tType ).
thf(type_def_105,type,
produc526117512892780850t_unit: $tType ).
thf(type_def_106,type,
produc5657529347773406293et_nat: $tType ).
thf(type_def_107,type,
produc4962441903510821572_nat_o: $tType ).
thf(type_def_108,type,
produc6830853553727218525et_nat: $tType ).
thf(type_def_109,type,
produc4658369026151987169et_nat: $tType ).
thf(type_def_110,type,
produc6907119039986491712_nat_o: $tType ).
thf(type_def_111,type,
produc7389280635074931169et_nat: $tType ).
thf(type_def_112,type,
produc8111630337999740517et_nat: $tType ).
thf(type_def_113,type,
produc1219242969750017639nt_int: $tType ).
thf(type_def_114,type,
produc4806409095325742951nt_nat: $tType ).
thf(type_def_115,type,
produc6495656768158660455at_int: $tType ).
thf(type_def_116,type,
produc859450856879609959at_nat: $tType ).
thf(type_def_117,type,
produc7485711547916492274unit_o: $tType ).
thf(type_def_118,type,
produc5646078835734834920it_int: $tType ).
thf(type_def_119,type,
produc600557818389255820it_nat: $tType ).
thf(type_def_120,type,
produc8459935480633519975t_unit: $tType ).
thf(type_def_121,type,
produc6666532470309664615_nat_o: $tType ).
thf(type_def_122,type,
produc7369051934464679207_set_o: $tType ).
thf(type_def_123,type,
produc6491284506569428743nteger: $tType ).
thf(type_def_124,type,
produc2115011035271226405et_int: $tType ).
thf(type_def_125,type,
produc7348995623955488188t_unit: $tType ).
thf(type_def_126,type,
produc7819656566062154093et_nat: $tType ).
thf(type_def_127,type,
produc1357844946079266663t_unit: $tType ).
thf(type_def_128,type,
set_Pr8834758594704517033_o_int: $tType ).
thf(type_def_129,type,
set_Pr6671815714118170108t_unit: $tType ).
thf(type_def_130,type,
set_Pr903927857289325719_int_o: $tType ).
thf(type_def_131,type,
set_Pr5889264611543340172t_unit: $tType ).
thf(type_def_132,type,
set_Pr5623950326363488168_nat_o: $tType ).
thf(type_def_133,type,
set_Pr3149072824959771635_nat_o: $tType ).
thf(type_def_134,type,
set_Pr7995236796853374141at_int: $tType ).
thf(type_def_135,type,
set_Pr4334478416066269672t_unit: $tType ).
thf(type_def_136,type,
set_Pr5464989305201766866unit_o: $tType ).
thf(type_def_137,type,
produc7936767299998605306et_nat: $tType ).
thf(type_def_138,type,
produc5697189838369619070et_nat: $tType ).
thf(type_def_139,type,
produc6500295350249520386et_nat: $tType ).
thf(type_def_140,type,
produc2475246329834808232it_int: $tType ).
thf(type_def_141,type,
produc2771989624626745363et_nat: $tType ).
thf(type_def_142,type,
produc6653097349344004940it_nat: $tType ).
thf(type_def_143,type,
produc2400336064389900727et_nat: $tType ).
thf(type_def_144,type,
produc3776562785724475560_o_int: $tType ).
thf(type_def_145,type,
produc1049655590333717003et_nat: $tType ).
thf(type_def_146,type,
produc8065447618272157327et_nat: $tType ).
thf(type_def_147,type,
produc5966032636589952099nteger: $tType ).
thf(type_def_148,type,
produc6271795597528267376eger_o: $tType ).
thf(type_def_149,type,
produc2256096406371053034er_int: $tType ).
thf(type_def_150,type,
produc5496629617739085541t_unit: $tType ).
thf(type_def_151,type,
set_Pr8497134830871917726it_int: $tType ).
thf(type_def_152,type,
set_Pr7705059561495379578_nat_o: $tType ).
thf(type_def_153,type,
set_Pr4208978187861016310_nat_o: $tType ).
thf(type_def_154,type,
set_Pr4071100604818519410t_unit: $tType ).
thf(type_def_155,type,
set_Pr7928877670098842301et_nat: $tType ).
thf(type_def_156,type,
set_Pr3721596426557623617et_nat: $tType ).
thf(type_def_157,type,
set_Pr8807100098903490369et_nat: $tType ).
thf(type_def_158,type,
set_Pr719794911490849221et_nat: $tType ).
thf(type_def_159,type,
set_Pr1776176089890104647t_unit: $tType ).
thf(type_def_160,type,
char: $tType ).
thf(type_def_161,type,
set_Pr3444600963470892981et_nat: $tType ).
thf(type_def_162,type,
set_Pr5696620967126086215nt_nat: $tType ).
thf(type_def_163,type,
set_Pr5557702248415638087at_int: $tType ).
thf(type_def_164,type,
set_Pr7577011563204128103nteger: $tType ).
thf(func_def_0,type,
set_Pr3444600963470892981et_nat: $tType ).
thf(func_def_1,type,
produc5657529347773406293et_nat: $tType ).
thf(func_def_2,type,
set_Pr7928877670098842301et_nat: $tType ).
thf(func_def_3,type,
produc6830853553727218525et_nat: $tType ).
thf(func_def_4,type,
set_Pr7705059561495379578_nat_o: $tType ).
thf(func_def_5,type,
set_Pr8536935166611901872et_nat: $tType ).
thf(func_def_6,type,
set_Pr8807100098903490369et_nat: $tType ).
thf(func_def_7,type,
set_Pr3721596426557623617et_nat: $tType ).
thf(func_def_8,type,
produc4962441903510821572_nat_o: $tType ).
thf(func_def_9,type,
produc2732055786443039994et_nat: $tType ).
thf(func_def_10,type,
produc7389280635074931169et_nat: $tType ).
thf(func_def_11,type,
produc4658369026151987169et_nat: $tType ).
thf(func_def_12,type,
produc101387210554329899nteger: $tType ).
thf(func_def_13,type,
produc7936767299998605306et_nat: $tType ).
thf(func_def_14,type,
set_Pr3813038829465207111_nat_o: $tType ).
thf(func_def_15,type,
set_se7663487083892625191_nat_o: $tType ).
thf(func_def_16,type,
produc6666532470309664615_nat_o: $tType ).
thf(func_def_17,type,
produc1049655590333717003et_nat: $tType ).
thf(func_def_18,type,
set_Pr2161125870931222855_nat_o: $tType ).
thf(func_def_19,type,
set_Pr4208978187861016310_nat_o: $tType ).
thf(func_def_20,type,
set_Pr3286484037609594932et_nat: $tType ).
thf(func_def_21,type,
produc4928098042776334183_nat_o: $tType ).
thf(func_def_22,type,
set_Pr719794911490849221et_nat: $tType ).
thf(func_def_23,type,
produc6907119039986491712_nat_o: $tType ).
thf(func_def_24,type,
produc3925858234332021118et_nat: $tType ).
thf(func_def_25,type,
produc8111630337999740517et_nat: $tType ).
thf(func_def_26,type,
produc5966032636589952099nteger: $tType ).
thf(func_def_27,type,
produc5697189838369619070et_nat: $tType ).
thf(func_def_28,type,
set_Pr5623950326363488168_nat_o: $tType ).
thf(func_def_29,type,
set_Pr4769669219625338370_o_nat: $tType ).
thf(func_def_30,type,
produc6500295350249520386et_nat: $tType ).
thf(func_def_31,type,
produc3776562785724475560_o_int: $tType ).
thf(func_def_32,type,
produc723880050079548563nteger: $tType ).
thf(func_def_33,type,
set_se1353816476878259085_nat_o: $tType ).
thf(func_def_34,type,
set_Pr1776176089890104647t_unit: $tType ).
thf(func_def_35,type,
produc8065447618272157327et_nat: $tType ).
thf(func_def_36,type,
set_Pr6409761941060870310at_o_o: $tType ).
thf(func_def_37,type,
set_Pr8693737435421807431at_nat: $tType ).
thf(func_def_38,type,
set_Pr5557702248415638087at_int: $tType ).
thf(func_def_39,type,
set_Pr5696620967126086215nt_nat: $tType ).
thf(func_def_40,type,
set_Pr2560585780119916871nt_int: $tType ).
thf(func_def_41,type,
produc6197004810343482825et_nat: $tType ).
thf(func_def_42,type,
produc5835291356934675326atural: $tType ).
thf(func_def_43,type,
set_Pr4532377907799695533_nat_o: $tType ).
thf(func_def_44,type,
produc8126358835630522215t_unit: $tType ).
thf(func_def_45,type,
produc859450856879609959at_nat: $tType ).
thf(func_def_46,type,
produc6495656768158660455at_int: $tType ).
thf(func_def_47,type,
produc4806409095325742951nt_nat: $tType ).
thf(func_def_48,type,
produc1219242969750017639nt_int: $tType ).
thf(func_def_49,type,
set_Pr7577011563204128103nteger: $tType ).
thf(func_def_50,type,
set_Pr4071100604818519410t_unit: $tType ).
thf(func_def_51,type,
set_Pr3948176798113811640et_nat: $tType ).
thf(func_def_52,type,
set_Pr3980508049012409671t_unit: $tType ).
thf(func_def_53,type,
produc1397831737260518551nteger: $tType ).
thf(func_def_54,type,
produc2221115339011613079nteger: $tType ).
thf(func_def_55,type,
produc4671145554997361783at_nat: $tType ).
thf(func_def_56,type,
produc3574062574353363227at_nat: $tType ).
thf(func_def_57,type,
produc6128020648511768019nt_int: $tType ).
thf(func_def_58,type,
produc5030937667867769463nt_int: $tType ).
thf(func_def_59,type,
produc6491284506569428743nteger: $tType ).
thf(func_def_60,type,
produc7348995623955488188t_unit: $tType ).
thf(func_def_61,type,
produc3658429121746597890et_nat: $tType ).
thf(func_def_62,type,
set_se8994758365886191399t_unit: $tType ).
thf(func_def_63,type,
produc1357844946079266663t_unit: $tType ).
thf(func_def_64,type,
set_Pr5488025237498180813et_nat: $tType ).
thf(func_def_65,type,
set_Pr2522554150109002629et_int: $tType ).
thf(func_def_66,type,
set_Pr4811707699266497531nteger: $tType ).
thf(func_def_67,type,
set_Pr4580871404036792859t_unit: $tType ).
thf(func_def_68,type,
produc6653097349344004940it_nat: $tType ).
thf(func_def_69,type,
produc2475246329834808232it_int: $tType ).
thf(func_def_70,type,
set_Pr5094982260447487303t_unit: $tType ).
thf(func_def_71,type,
set_se7855581050983116737at_nat: $tType ).
thf(func_def_72,type,
set_se6260736226359567993nt_int: $tType ).
thf(func_def_73,type,
set_Pr2406931829603198446nteger: $tType ).
thf(func_def_74,type,
set_Pr8986988648414616686er_nat: $tType ).
thf(func_def_75,type,
set_Pr6496905503482889930er_int: $tType ).
thf(func_def_76,type,
produc7819656566062154093et_nat: $tType ).
thf(func_def_77,type,
produc2115011035271226405et_int: $tType ).
thf(func_def_78,type,
produc7822875418678951345atural: $tType ).
thf(func_def_79,type,
produc8923325533196201883nteger: $tType ).
thf(func_def_80,type,
set_Pr4577759397028426247_set_o: $tType ).
thf(func_def_81,type,
set_Pr1763845938948868674it_nat: $tType ).
thf(func_def_82,type,
set_Pr8497134830871917726it_int: $tType ).
thf(func_def_83,type,
set_Pr4334478416066269672t_unit: $tType ).
thf(func_def_84,type,
set_Pr5889264611543340172t_unit: $tType ).
thf(func_def_85,type,
produc5496629617739085541t_unit: $tType ).
thf(func_def_86,type,
set_Co726186806682698383nteger: $tType ).
thf(func_def_87,type,
produc8459935480633519975t_unit: $tType ).
thf(func_def_88,type,
set_Pr448751882837621926eger_o: $tType ).
thf(func_def_89,type,
filter1242075044329608583at_nat: $tType ).
thf(func_def_90,type,
set_Pr5464989305201766866unit_o: $tType ).
thf(func_def_91,type,
set_Pr6671815714118170108t_unit: $tType ).
thf(func_def_92,type,
set_he8644174446132254637t_unit: $tType ).
thf(func_def_93,type,
set_Pr1261947904930325089at_nat: $tType ).
thf(func_def_94,type,
set_Pr7995236796853374141at_int: $tType ).
thf(func_def_95,type,
set_Pr3448869479623346877nt_nat: $tType ).
thf(func_def_96,type,
set_Pr958786334691620121nt_int: $tType ).
thf(func_def_97,type,
produc2400336064389900727et_nat: $tType ).
thf(func_def_98,type,
produc2591615631378489102nteger: $tType ).
thf(func_def_99,type,
produc2771989624626745363et_nat: $tType ).
thf(func_def_100,type,
produc6433947425880249742er_nat: $tType ).
thf(func_def_101,type,
produc2256096406371053034er_int: $tType ).
thf(func_def_102,type,
set_se491199571785101031od_o_o: $tType ).
thf(func_def_103,type,
produc7369051934464679207_set_o: $tType ).
thf(func_def_104,type,
produc600557818389255820it_nat: $tType ).
thf(func_def_105,type,
produc5646078835734834920it_int: $tType ).
thf(func_def_106,type,
produc526117512892780850t_unit: $tType ).
thf(func_def_107,type,
produc6865911496949736918t_unit: $tType ).
thf(func_def_108,type,
set_Pr3149072824959771635_nat_o: $tType ).
thf(func_def_109,type,
set_Pr903927857289325719_int_o: $tType ).
thf(func_def_110,type,
set_Pr2101469702781467981_o_nat: $tType ).
thf(func_def_111,type,
set_Pr8834758594704517033_o_int: $tType ).
thf(func_def_112,type,
produc6271795597528267376eger_o: $tType ).
thf(func_def_113,type,
set_set_set_nat: $tType ).
thf(func_def_114,type,
set_set_Code_integer: $tType ).
thf(func_def_115,type,
produc7485711547916492274unit_o: $tType ).
thf(func_def_116,type,
produc3788275437394581276t_unit: $tType ).
thf(func_def_117,type,
set_set_Product_unit: $tType ).
thf(func_def_118,type,
heap_e7401611519738050253t_unit: $tType ).
thf(func_def_119,type,
product_prod_nat_nat: $tType ).
thf(func_def_120,type,
product_prod_nat_int: $tType ).
thf(func_def_121,type,
product_prod_int_nat: $tType ).
thf(func_def_122,type,
product_prod_int_int: $tType ).
thf(func_def_123,type,
set_Product_unit_o: $tType ).
thf(func_def_124,type,
set_Product_prod_o_o: $tType ).
thf(func_def_125,type,
sum_sum_nat_nat: $tType ).
thf(func_def_126,type,
set_filter_nat: $tType ).
thf(func_def_127,type,
filter_set_nat: $tType ).
thf(func_def_128,type,
product_prod_nat_o: $tType ).
thf(func_def_129,type,
product_prod_int_o: $tType ).
thf(func_def_130,type,
product_prod_o_nat: $tType ).
thf(func_def_131,type,
product_prod_o_int: $tType ).
thf(func_def_132,type,
set_set_nat: $tType ).
thf(func_def_133,type,
set_set_int: $tType ).
thf(func_def_134,type,
set_Code_natural: $tType ).
thf(func_def_135,type,
set_Code_integer: $tType ).
thf(func_def_136,type,
set_Product_unit: $tType ).
thf(func_def_137,type,
set_nat_o: $tType ).
thf(func_def_138,type,
set_int_o: $tType ).
thf(func_def_139,type,
product_prod_o_o: $tType ).
thf(func_def_140,type,
set_assn: $tType ).
thf(func_def_141,type,
set_set_o: $tType ).
thf(func_def_142,type,
filter_nat: $tType ).
thf(func_def_143,type,
set_o_o: $tType ).
thf(func_def_144,type,
list_nat: $tType ).
thf(func_def_145,type,
set_rat: $tType ).
thf(func_def_146,type,
set_nat: $tType ).
thf(func_def_147,type,
set_int: $tType ).
thf(func_def_148,type,
code_natural: $tType ).
thf(func_def_149,type,
code_integer: $tType ).
thf(func_def_150,type,
product_unit: $tType ).
thf(func_def_151,type,
typerep: $tType ).
thf(func_def_152,type,
assn: $tType ).
thf(func_def_153,type,
set_o: $tType ).
thf(func_def_154,type,
char: $tType ).
thf(func_def_155,type,
rat: $tType ).
thf(func_def_156,type,
num: $tType ).
thf(func_def_157,type,
nat: $tType ).
thf(func_def_158,type,
int: $tType ).
thf(func_def_159,type,
abs_assn: ( produc3658429121746597890et_nat > $o ) > assn ).
thf(func_def_160,type,
rep_assn: assn > produc3658429121746597890et_nat > $o ).
thf(func_def_161,type,
in_range: produc3658429121746597890et_nat > $o ).
thf(func_def_162,type,
in_range_rel: produc3658429121746597890et_nat > produc3658429121746597890et_nat > $o ).
thf(func_def_163,type,
is_pure_assn: assn > $o ).
thf(func_def_164,type,
one_assn_raw: produc3658429121746597890et_nat > $o ).
thf(func_def_165,type,
one_assn_raw_rel: produc3658429121746597890et_nat > produc3658429121746597890et_nat > $o ).
thf(func_def_166,type,
proper: ( produc3658429121746597890et_nat > $o ) > $o ).
thf(func_def_167,type,
pure_assn: $o > assn ).
thf(func_def_168,type,
pure_a825153325127701367it_nat: $o > produc3658429121746597890et_nat > $o ).
thf(func_def_169,type,
pure_a6022498039421069578it_nat: produc6197004810343482825et_nat > produc6197004810343482825et_nat > $o ).
thf(func_def_170,type,
relH: set_nat > heap_e7401611519738050253t_unit > heap_e7401611519738050253t_unit > $o ).
thf(func_def_171,type,
times_assn_raw: ( produc3658429121746597890et_nat > $o ) > ( produc3658429121746597890et_nat > $o ) > produc3658429121746597890et_nat > $o ).
thf(func_def_172,type,
times_assn_raw_rel: produc2732055786443039994et_nat > produc2732055786443039994et_nat > $o ).
thf(func_def_173,type,
wand_assn: assn > assn > assn ).
thf(func_def_174,type,
wand_raw: ( produc3658429121746597890et_nat > $o ) > ( produc3658429121746597890et_nat > $o ) > produc3658429121746597890et_nat > $o ).
thf(func_def_175,type,
wand_raw_rel: produc2732055786443039994et_nat > produc2732055786443039994et_nat > $o ).
thf(func_def_176,type,
bNF_Cardinal_cone: set_Pr5094982260447487303t_unit ).
thf(func_def_177,type,
bNF_Ca7083678892733797993t_unit: set_Product_unit > set_Pr5094982260447487303t_unit ).
thf(func_def_178,type,
bNF_fs3871268833404974121et_nat: ( heap_e7401611519738050253t_unit > set_nat > $o ) > ( set_nat > set_nat > $o ) > produc3658429121746597890et_nat > produc3658429121746597890et_nat ).
thf(func_def_179,type,
bNF_fs8167890477030535480nt_int: ( int > int > $o ) > ( int > int > $o ) > product_prod_int_int > product_prod_int_int ).
thf(func_def_180,type,
bNF_pi3450177011487049898et_nat: ( heap_e7401611519738050253t_unit > heap_e7401611519738050253t_unit > $o ) > ( heap_e7401611519738050253t_unit > set_nat > $o ) > heap_e7401611519738050253t_unit > set_nat > heap_e7401611519738050253t_unit ).
thf(func_def_181,type,
bNF_pi4042512755101277815et_nat: ( heap_e7401611519738050253t_unit > set_nat > $o ) > ( set_nat > set_nat > $o ) > heap_e7401611519738050253t_unit > set_nat > set_nat ).
thf(func_def_182,type,
bNF_pi8853482479757052266nt_int: ( int > int > $o ) > ( int > int > $o ) > int > int > int ).
thf(func_def_183,type,
bNF_re398004352372739002nteger: ( int > code_integer > $o ) > ( ( int > int ) > ( code_integer > code_integer ) > $o ) > ( int > int > int ) > ( code_integer > code_integer > code_integer ) > $o ).
thf(func_def_184,type,
bNF_re3379532845092657523nteger: ( int > code_integer > $o ) > ( int > code_integer > $o ) > ( int > int ) > ( code_integer > code_integer ) > $o ).
thf(func_def_185,type,
bNF_re711492959462206631nt_int: ( int > int > $o ) > ( ( int > int ) > ( int > int ) > $o ) > ( int > int > int ) > ( int > int > int ) > $o ).
thf(func_def_186,type,
bNF_re4712519889275205905nt_int: ( int > int > $o ) > ( int > int > $o ) > ( int > int ) > ( int > int ) > $o ).
thf(func_def_187,type,
bNF_re88643428490162567atural: ( nat > code_natural > $o ) > ( ( nat > nat ) > ( code_natural > code_natural ) > $o ) > ( nat > nat > nat ) > ( code_natural > code_natural > code_natural ) > $o ).
thf(func_def_188,type,
bNF_re3704215830270325841atural: ( nat > code_natural > $o ) > ( nat > code_natural > $o ) > ( nat > nat ) > ( code_natural > code_natural ) > $o ).
thf(func_def_189,type,
bNF_re1345281282404953727at_nat: ( nat > nat > $o ) > ( ( nat > nat ) > ( nat > nat ) > $o ) > ( nat > nat > nat ) > ( nat > nat > nat ) > $o ).
thf(func_def_190,type,
bNF_re5653821019739307937at_nat: ( nat > nat > $o ) > ( nat > nat > $o ) > ( nat > nat ) > ( nat > nat ) > $o ).
thf(func_def_191,type,
bNF_re5228765855967844073nt_int: ( product_prod_int_int > product_prod_int_int > $o ) > ( ( product_prod_int_int > product_prod_int_int ) > ( product_prod_int_int > product_prod_int_int ) > $o ) > ( product_prod_int_int > product_prod_int_int > product_prod_int_int ) > ( product_prod_int_int > product_prod_int_int > product_prod_int_int ) > $o ).
thf(func_def_192,type,
bNF_re8699439704749558557nt_o_o: ( product_prod_int_int > product_prod_int_int > $o ) > ( $o > $o > $o ) > ( product_prod_int_int > $o ) > ( product_prod_int_int > $o ) > $o ).
thf(func_def_193,type,
bNF_re7145576690424134365nt_int: ( product_prod_int_int > product_prod_int_int > $o ) > ( product_prod_int_int > product_prod_int_int > $o ) > ( product_prod_int_int > product_prod_int_int ) > ( product_prod_int_int > product_prod_int_int ) > $o ).
thf(func_def_194,type,
bNF_re7627151682743391978at_rat: ( product_prod_int_int > rat > $o ) > ( ( product_prod_int_int > product_prod_int_int ) > ( rat > rat ) > $o ) > ( product_prod_int_int > product_prod_int_int > product_prod_int_int ) > ( rat > rat > rat ) > $o ).
thf(func_def_195,type,
bNF_re1494630372529172596at_o_o: ( product_prod_int_int > rat > $o ) > ( $o > $o > $o ) > ( product_prod_int_int > $o ) > ( rat > $o ) > $o ).
thf(func_def_196,type,
bNF_re8279943556446156061nt_rat: ( product_prod_int_int > rat > $o ) > ( product_prod_int_int > rat > $o ) > ( product_prod_int_int > product_prod_int_int ) > ( rat > rat ) > $o ).
thf(func_def_197,type,
bNF_re7408651293131936558nt_int: ( product_prod_nat_nat > int > $o ) > ( ( product_prod_nat_nat > product_prod_nat_nat ) > ( int > int ) > $o ) > ( product_prod_nat_nat > product_prod_nat_nat > product_prod_nat_nat ) > ( int > int > int ) > $o ).
thf(func_def_198,type,
bNF_re7400052026677387805at_int: ( product_prod_nat_nat > int > $o ) > ( product_prod_nat_nat > int > $o ) > ( product_prod_nat_nat > product_prod_nat_nat ) > ( int > int ) > $o ).
thf(func_def_199,type,
bNF_re3099431351363272937at_nat: ( product_prod_nat_nat > product_prod_nat_nat > $o ) > ( ( product_prod_nat_nat > product_prod_nat_nat ) > ( product_prod_nat_nat > product_prod_nat_nat ) > $o ) > ( product_prod_nat_nat > product_prod_nat_nat > product_prod_nat_nat ) > ( product_prod_nat_nat > product_prod_nat_nat > product_prod_nat_nat ) > $o ).
thf(func_def_200,type,
bNF_re2241393799969408733at_nat: ( product_prod_nat_nat > product_prod_nat_nat > $o ) > ( product_prod_nat_nat > product_prod_nat_nat > $o ) > ( product_prod_nat_nat > product_prod_nat_nat ) > ( product_prod_nat_nat > product_prod_nat_nat ) > $o ).
thf(func_def_201,type,
bNF_sn4039337771197282234et_nat: ( heap_e7401611519738050253t_unit > heap_e7401611519738050253t_unit > $o ) > ( heap_e7401611519738050253t_unit > set_nat > $o ) > produc3658429121746597890et_nat > produc3658429121746597890et_nat ).
thf(func_def_202,type,
bNF_sn1062102010912252026nt_int: ( int > int > $o ) > ( int > int > $o ) > product_prod_int_int > product_prod_int_int ).
thf(func_def_203,type,
bNF_We1730399552001240015qr_int: set_Pr958786334691620121nt_int > set_Pr2560585780119916871nt_int ).
thf(func_def_204,type,
bNF_We1732890022510290291qr_nat: set_Pr1261947904930325089at_nat > set_Pr8693737435421807431at_nat ).
thf(func_def_205,type,
bNF_We740377018556551609_nat_o: set_Pr2161125870931222855_nat_o > $o ).
thf(func_def_206,type,
bNF_We6437023980930612612_rel_o: set_Product_prod_o_o > $o ).
thf(func_def_207,type,
bNF_We3815749466139970368el_int: set_Pr958786334691620121nt_int > $o ).
thf(func_def_208,type,
bNF_We3818239936649020644el_nat: set_Pr1261947904930325089at_nat > $o ).
thf(func_def_209,type,
bNF_We2091578341578594809t_unit: set_Pr5094982260447487303t_unit > $o ).
thf(func_def_210,type,
bNF_We5608535359669600131_nat_o: set_Pr2161125870931222855_nat_o > set_Pr4532377907799695533_nat_o > ( produc3658429121746597890et_nat > $o ) > $o ).
thf(func_def_211,type,
bNF_We4647864369647134286inim_o: set_Product_prod_o_o > set_o > $o > $o ).
thf(func_def_212,type,
bNF_We6511469253196982518im_int: set_Pr958786334691620121nt_int > set_int > int > $o ).
thf(func_def_213,type,
bNF_We6513959723706032794im_nat: set_Pr1261947904930325089at_nat > set_nat > nat > $o ).
thf(func_def_214,type,
bNF_We909621271159000003t_unit: set_Pr5094982260447487303t_unit > set_Product_unit > product_unit > $o ).
thf(func_def_215,type,
bNF_We6400040844883348763_nat_o: set_Pr2161125870931222855_nat_o > ( produc3658429121746597890et_nat > $o ) > ( produc3658429121746597890et_nat > $o ) > produc3658429121746597890et_nat > $o ).
thf(func_def_216,type,
bNF_We3104086342435940326max2_o: set_Product_prod_o_o > $o > $o > $o ).
thf(func_def_217,type,
bNF_We2378448595015613790x2_int: set_Pr958786334691620121nt_int > int > int > int ).
thf(func_def_218,type,
bNF_We2380939065524664066x2_nat: set_Pr1261947904930325089at_nat > nat > nat > nat ).
thf(func_def_219,type,
bNF_We7204996500647184731t_unit: set_Pr5094982260447487303t_unit > product_unit > product_unit > product_unit ).
thf(func_def_220,type,
bNF_We8236795211600914329_nat_o: set_Pr2161125870931222855_nat_o > set_Pr4532377907799695533_nat_o > produc3658429121746597890et_nat > $o ).
thf(func_def_221,type,
bNF_We537455116209922916inim_o: set_Product_prod_o_o > set_o > $o ).
thf(func_def_222,type,
bNF_We3272633801743244640im_int: set_Pr958786334691620121nt_int > set_int > int ).
thf(func_def_223,type,
bNF_We3275124272252294916im_nat: set_Pr1261947904930325089at_nat > set_nat > nat ).
thf(func_def_224,type,
bNF_We7185707985517244377t_unit: set_Pr5094982260447487303t_unit > set_Product_unit > product_unit ).
thf(func_def_225,type,
basic_5567952622663070111et_nat: ( heap_e7401611519738050253t_unit > nat ) > ( set_nat > nat ) > produc3658429121746597890et_nat > nat ).
thf(func_def_226,type,
basic_1872990034501187214nt_int: ( int > nat ) > ( int > nat ) > product_prod_int_int > nat ).
thf(func_def_227,type,
basic_3966130907827183364et_nat: produc2732055786443039994et_nat > set_Pr4532377907799695533_nat_o ).
thf(func_def_228,type,
basic_8654530971723826568et_nat: produc3925858234332021118et_nat > set_Pr4532377907799695533_nat_o ).
thf(func_def_229,type,
basic_7833836874476830220et_nat: produc3658429121746597890et_nat > set_he8644174446132254637t_unit ).
thf(func_def_230,type,
basic_fsts_int_int: product_prod_int_int > set_int ).
thf(func_def_231,type,
basic_fsts_nat_nat: product_prod_nat_nat > set_nat ).
thf(func_def_232,type,
basic_2768200282441914438et_nat: produc2732055786443039994et_nat > set_Pr3286484037609594932et_nat ).
thf(func_def_233,type,
basic_8479756702517612234et_nat: produc3925858234332021118et_nat > set_Pr3948176798113811640et_nat ).
thf(func_def_234,type,
basic_2698842327037663054et_nat: produc3658429121746597890et_nat > set_set_nat ).
thf(func_def_235,type,
basic_snds_int_int: product_prod_int_int > set_int ).
thf(func_def_236,type,
basic_snds_nat_nat: product_prod_nat_nat > set_nat ).
thf(func_def_237,type,
binomial: nat > nat > nat ).
thf(func_def_238,type,
gbinom8545251970709558553nteger: code_integer > nat > code_integer ).
thf(func_def_239,type,
gbinom7368847122466276068atural: code_natural > nat > code_natural ).
thf(func_def_240,type,
gbinomial_int: int > nat > int ).
thf(func_def_241,type,
gbinomial_nat: nat > nat > nat ).
thf(func_def_242,type,
gbinomial_rat: rat > nat > rat ).
thf(func_def_243,type,
bit_and_int_rel: product_prod_int_int > product_prod_int_int > $o ).
thf(func_def_244,type,
bit_concat_bit: nat > int > int > int ).
thf(func_def_245,type,
bit_ri7919022796975470100ot_int: int > int ).
thf(func_def_246,type,
bit_ri631733984087533419it_int: nat > int > int ).
thf(func_def_247,type,
bit_se725231765392027082nd_int: int > int > int ).
thf(func_def_248,type,
bit_se727722235901077358nd_nat: nat > nat > nat ).
thf(func_def_249,type,
bit_se1409905431419307370or_int: int > int > int ).
thf(func_def_250,type,
bit_se1412395901928357646or_nat: nat > nat > nat ).
thf(func_def_251,type,
bit_se545348938243370406it_int: nat > int > int ).
thf(func_def_252,type,
bit_se547839408752420682it_nat: nat > nat > nat ).
thf(func_def_253,type,
bit_se7879613467334960850it_int: nat > int > int ).
thf(func_def_254,type,
bit_se2923211474154528505it_int: nat > int > int ).
thf(func_def_255,type,
bit_se4203085406695923979it_int: nat > int > int ).
thf(func_def_256,type,
bit_se6526347334894502574or_int: int > int > int ).
thf(func_def_257,type,
bit_se6528837805403552850or_nat: nat > nat > nat ).
thf(func_def_258,type,
bit_se1146084159140164899it_int: int > nat > $o ).
thf(func_def_259,type,
code_Neg: num > code_integer ).
thf(func_def_260,type,
code_Pos: num > code_integer ).
thf(func_def_261,type,
code_divmod_abs: code_integer > code_integer > produc8923325533196201883nteger ).
thf(func_def_262,type,
code_divmod_integer: code_integer > code_integer > produc8923325533196201883nteger ).
thf(func_def_263,type,
code_int_of_integer: code_integer > int ).
thf(func_def_264,type,
code_integer_of_int: int > code_integer ).
thf(func_def_265,type,
code_nat_of_integer: code_integer > nat ).
thf(func_def_266,type,
code_nat_of_natural: code_natural > nat ).
thf(func_def_267,type,
code_natural_of_nat: nat > code_natural ).
thf(func_def_268,type,
code_pcr_integer: int > code_integer > $o ).
thf(func_def_269,type,
code_pcr_natural: nat > code_natural > $o ).
thf(func_def_270,type,
comple4605987646318907561at_o_o: set_Pr6409761941060870310at_o_o > ( produc3658429121746597890et_nat > $o ) > $o ).
thf(func_def_271,type,
complete_Inf_Inf_o_o: set_o_o > $o > $o ).
thf(func_def_272,type,
comple4393433247856107946_int_o: set_int_o > int > $o ).
thf(func_def_273,type,
comple6214475593288795910_nat_o: set_nat_o > nat > $o ).
thf(func_def_274,type,
comple90455603224345172_nat_o: set_Pr4532377907799695533_nat_o > produc3658429121746597890et_nat > $o ).
thf(func_def_275,type,
comple2092403528524206185unit_o: set_Product_unit_o > product_unit > $o ).
thf(func_def_276,type,
complete_Inf_Inf_o: set_o > $o ).
thf(func_def_277,type,
comple6612786467464916249er_nat: set_filter_nat > filter_nat ).
thf(func_def_278,type,
complete_Inf_Inf_int: set_int > int ).
thf(func_def_279,type,
complete_Inf_Inf_nat: set_nat > nat ).
thf(func_def_280,type,
comple2584293577114468500t_unit: set_Product_unit > product_unit ).
thf(func_def_281,type,
comple79274577958421812_nat_o: set_se1353816476878259085_nat_o > set_Pr4532377907799695533_nat_o ).
thf(func_def_282,type,
comple3063163877087187839_set_o: set_set_o > set_o ).
thf(func_def_283,type,
comple3628384868704368283et_int: set_set_int > set_int ).
thf(func_def_284,type,
comple7806235888213564991et_nat: set_set_nat > set_nat ).
thf(func_def_285,type,
comple1930127330929219688at_nat: set_se7855581050983116737at_nat > set_Pr1261947904930325089at_nat ).
thf(func_def_286,type,
comple7216939928495492468t_unit: set_set_Product_unit > set_Product_unit ).
thf(func_def_287,type,
comple3590721164184202819at_o_o: set_Pr6409761941060870310at_o_o > ( produc3658429121746597890et_nat > $o ) > $o ).
thf(func_def_288,type,
complete_Sup_Sup_o_o: set_o_o > $o > $o ).
thf(func_def_289,type,
comple6496622788309502864_int_o: set_int_o > int > $o ).
thf(func_def_290,type,
comple8317665133742190828_nat_o: set_nat_o > nat > $o ).
thf(func_def_291,type,
comple4225350640907012154_nat_o: set_Pr4532377907799695533_nat_o > produc3658429121746597890et_nat > $o ).
thf(func_def_292,type,
comple4773641153059738115unit_o: set_Product_unit_o > product_unit > $o ).
thf(func_def_293,type,
complete_Sup_Sup_o: set_o > $o ).
thf(func_def_294,type,
comple2454491259644266675er_nat: set_filter_nat > filter_nat ).
thf(func_def_295,type,
complete_Sup_Sup_int: set_int > int ).
thf(func_def_296,type,
complete_Sup_Sup_nat: set_nat > nat ).
thf(func_def_297,type,
comple4687483117567863418t_unit: set_Product_unit > product_unit ).
thf(func_def_298,type,
comple3382087578465172250_nat_o: set_se1353816476878259085_nat_o > set_Pr4532377907799695533_nat_o ).
thf(func_def_299,type,
comple90263536869209701_set_o: set_set_o > set_o ).
thf(func_def_300,type,
comple739944243200306918nteger: set_set_Code_integer > set_Code_integer ).
thf(func_def_301,type,
comple3221217463730067765et_int: set_set_int > set_int ).
thf(func_def_302,type,
comple7399068483239264473et_nat: set_set_nat > set_nat ).
thf(func_def_303,type,
comple8164645552537203636_nat_o: set_se7663487083892625191_nat_o > set_Pr2161125870931222855_nat_o ).
thf(func_def_304,type,
comple2567785453016644468od_o_o: set_se491199571785101031od_o_o > set_Product_prod_o_o ).
thf(func_def_305,type,
comple5382143125604098054nt_int: set_se6260736226359567993nt_int > set_Pr958786334691620121nt_int ).
thf(func_def_306,type,
comple5685304695842803022at_nat: set_se7855581050983116737at_nat > set_Pr1261947904930325089at_nat ).
thf(func_def_307,type,
comple2798140142350441396t_unit: set_se8994758365886191399t_unit > set_Pr5094982260447487303t_unit ).
thf(func_def_308,type,
comple3349607820504768346t_unit: set_set_Product_unit > set_Product_unit ).
thf(func_def_309,type,
condit4352657709669162033_nat_o: set_Pr4532377907799695533_nat_o > $o ).
thf(func_def_310,type,
condit5488710616941104124bove_o: set_o > $o ).
thf(func_def_311,type,
condit4029664723990530684er_nat: set_filter_nat > $o ).
thf(func_def_312,type,
condit2212336002400062152ve_int: set_int > $o ).
thf(func_def_313,type,
condit2214826472909112428ve_nat: set_nat > $o ).
thf(func_def_314,type,
condit2080475194861589617t_unit: set_Product_unit > $o ).
thf(func_def_315,type,
condit5477540289124974626et_nat: set_set_nat > $o ).
thf(func_def_316,type,
condit8399017556247434269_nat_o: set_Pr4532377907799695533_nat_o > $o ).
thf(func_def_317,type,
condit5413489452508810728elow_o: set_o > $o ).
thf(func_def_318,type,
condit3133331395780078992er_nat: set_filter_nat > $o ).
thf(func_def_319,type,
condit1735850657277959132ow_int: set_int > $o ).
thf(func_def_320,type,
condit1738341127787009408ow_nat: set_nat > $o ).
thf(func_def_321,type,
condit9161856309968323677t_unit: set_Product_unit > $o ).
thf(func_def_322,type,
condit68592940725977398et_nat: set_set_nat > $o ).
thf(func_def_323,type,
eucl_rel_int: int > int > product_prod_int_int > $o ).
thf(func_def_324,type,
unique5706413561485394159nteger: produc8923325533196201883nteger > $o ).
thf(func_def_325,type,
unique6319869463603278526ux_int: product_prod_int_int > $o ).
thf(func_def_326,type,
unique6322359934112328802ux_nat: product_prod_nat_nat > $o ).
thf(func_def_327,type,
unique4921790084139445826nteger: num > produc8923325533196201883nteger > produc8923325533196201883nteger ).
thf(func_def_328,type,
unique5024387138958732305ep_int: num > product_prod_int_int > product_prod_int_int ).
thf(func_def_329,type,
unique5026877609467782581ep_nat: num > product_prod_nat_nat > product_prod_nat_nat ).
thf(func_def_330,type,
equiv_6726761681920839555_nat_o: set_Pr4532377907799695533_nat_o > set_Pr2161125870931222855_nat_o > $o ).
thf(func_def_331,type,
equiv_equiv_o: set_o > set_Product_prod_o_o > $o ).
thf(func_def_332,type,
equiv_equiv_int: set_int > set_Pr958786334691620121nt_int > $o ).
thf(func_def_333,type,
equiv_equiv_nat: set_nat > set_Pr1261947904930325089at_nat > $o ).
thf(func_def_334,type,
equiv_4189356295871601603t_unit: set_Product_unit > set_Pr5094982260447487303t_unit > $o ).
thf(func_def_335,type,
equiv_3235788788526563786_nat_o: set_Pr2161125870931222855_nat_o > ( produc3658429121746597890et_nat > $o ) > set_Pr4532377907799695533_nat_o ).
thf(func_def_336,type,
equiv_proj_o_o: set_Product_prod_o_o > $o > set_o ).
thf(func_def_337,type,
equiv_proj_int_int: set_Pr958786334691620121nt_int > int > set_int ).
thf(func_def_338,type,
equiv_proj_nat_nat: set_Pr1261947904930325089at_nat > nat > set_nat ).
thf(func_def_339,type,
equiv_5763582929720877514t_unit: set_Pr5094982260447487303t_unit > product_unit > set_Product_unit ).
thf(func_def_340,type,
equiv_7899955228204526484_nat_o: set_Pr4532377907799695533_nat_o > set_Pr2161125870931222855_nat_o > set_se1353816476878259085_nat_o ).
thf(func_def_341,type,
equiv_quotient_o: set_o > set_Product_prod_o_o > set_set_o ).
thf(func_def_342,type,
equiv_quotient_int: set_int > set_Pr958786334691620121nt_int > set_set_int ).
thf(func_def_343,type,
equiv_quotient_nat: set_nat > set_Pr1261947904930325089at_nat > set_set_nat ).
thf(func_def_344,type,
equiv_5172566880711946708t_unit: set_Product_unit > set_Pr5094982260447487303t_unit > set_set_Product_unit ).
thf(func_def_345,type,
comm_s8582702949713902594nteger: code_integer > nat > code_integer ).
thf(func_def_346,type,
comm_s4660882817536571857er_int: int > nat > int ).
thf(func_def_347,type,
comm_s4028243227959126397er_rat: rat > nat > rat ).
thf(func_def_348,type,
semiri1408675320244567234ct_nat: nat > nat ).
thf(func_def_349,type,
inverse_inverse_rat: rat > rat ).
thf(func_def_350,type,
at_top_nat: filter_nat ).
thf(func_def_351,type,
cofinite_nat: filter_nat ).
thf(func_def_352,type,
eventually_nat: ( nat > $o ) > filter_nat > $o ).
thf(func_def_353,type,
eventu1038000079068216329at_nat: ( product_prod_nat_nat > $o ) > filter1242075044329608583at_nat > $o ).
thf(func_def_354,type,
filterlim_nat_nat: ( nat > nat ) > filter_nat > filter_nat > $o ).
thf(func_def_355,type,
filter3212408913953519116et_nat: ( nat > set_nat ) > filter_set_nat > filter_nat > $o ).
thf(func_def_356,type,
finite3254316476582989077op_nat: set_nat > filter_set_nat ).
thf(func_def_357,type,
frequently_nat: ( nat > $o ) > filter_nat > $o ).
thf(func_def_358,type,
prod_filter_nat_nat: filter_nat > filter_nat > filter1242075044329608583at_nat ).
thf(func_def_359,type,
finite1363419556375932405_nat_o: set_Pr4532377907799695533_nat_o > nat ).
thf(func_def_360,type,
finite_card_o: set_o > nat ).
thf(func_def_361,type,
finite4902975817058060853nteger: set_Code_integer > nat ).
thf(func_def_362,type,
finite_card_int: set_int > nat ).
thf(func_def_363,type,
finite_card_nat: set_nat > nat ).
thf(func_def_364,type,
finite410649719033368117t_unit: set_Product_unit > nat ).
thf(func_def_365,type,
finite4731484169140532437_nat_o: set_se1353816476878259085_nat_o > nat ).
thf(func_def_366,type,
finite_card_set_o: set_set_o > nat ).
thf(func_def_367,type,
finite1340570857867686763nteger: set_set_Code_integer > nat ).
thf(func_def_368,type,
finite_card_set_int: set_set_int > nat ).
thf(func_def_369,type,
finite_card_set_nat: set_set_nat > nat ).
thf(func_def_370,type,
finite22775419188893461t_unit: set_set_Product_unit > nat ).
thf(func_def_371,type,
finite3252695134891459830_nat_o: set_Pr4532377907799695533_nat_o > $o ).
thf(func_def_372,type,
finite_finite_o: set_o > $o ).
thf(func_def_373,type,
finite_finite_assn: set_assn > $o ).
thf(func_def_374,type,
finite6017078050557962740nteger: set_Code_integer > $o ).
thf(func_def_375,type,
finite2119507909894593271er_nat: set_filter_nat > $o ).
thf(func_def_376,type,
finite_finite_int: set_int > $o ).
thf(func_def_377,type,
finite_finite_nat: set_nat > $o ).
thf(func_def_378,type,
finite2154694888287295812nteger: set_Pr4811707699266497531nteger > $o ).
thf(func_def_379,type,
finite4996374594527322935er_nat: set_Pr8986988648414616686er_nat > $o ).
thf(func_def_380,type,
finite2998713641127702882nt_int: set_Pr958786334691620121nt_int > $o ).
thf(func_def_381,type,
finite1154042800025562295nteger: set_Pr2406931829603198446nteger > $o ).
thf(func_def_382,type,
finite6177210948735845034at_nat: set_Pr1261947904930325089at_nat > $o ).
thf(func_def_383,type,
finite4290736615968046902t_unit: set_Product_unit > $o ).
thf(func_def_384,type,
finite_finite_rat: set_rat > $o ).
thf(func_def_385,type,
finite6069017635265841110_nat_o: set_se1353816476878259085_nat_o > $o ).
thf(func_def_386,type,
finite_finite_set_o: set_set_o > $o ).
thf(func_def_387,type,
finite6931041176100689706nteger: set_set_Code_integer > $o ).
thf(func_def_388,type,
finite6197958912794628473et_int: set_set_int > $o ).
thf(func_def_389,type,
finite1152437895449049373et_nat: set_set_nat > $o ).
thf(func_def_390,type,
finite1772178364199683094t_unit: set_set_Product_unit > $o ).
thf(func_def_391,type,
finite_fold_o_o: ( $o > $o > $o ) > $o > set_o > $o ).
thf(func_def_392,type,
finite_fold_o_set_o: ( $o > set_o > set_o ) > set_o > set_o > set_o ).
thf(func_def_393,type,
finite1440822074540989594n_assn: ( assn > assn > assn ) > assn > set_assn > assn ).
thf(func_def_394,type,
finite6454954857057555716nteger: ( code_integer > code_integer > code_integer ) > code_integer > set_Code_integer > code_integer ).
thf(func_def_395,type,
finite2281032298429728806t_unit: ( code_integer > product_unit > product_unit ) > product_unit > set_Code_integer > product_unit ).
thf(func_def_396,type,
finite1928688952044918289_set_o: ( code_integer > set_o > set_o ) > set_o > set_Code_integer > set_o ).
thf(func_def_397,type,
finite7486036041278456378nteger: ( code_integer > set_Code_integer > set_Code_integer ) > set_Code_integer > set_Code_integer > set_Code_integer ).
thf(func_def_398,type,
finite6635835719294221449et_int: ( code_integer > set_int > set_int ) > set_int > set_Code_integer > set_int ).
thf(func_def_399,type,
finite1590314701948642349et_nat: ( code_integer > set_nat > set_nat ) > set_nat > set_Code_integer > set_nat ).
thf(func_def_400,type,
finite3687622995246107910t_unit: ( code_integer > set_Product_unit > set_Product_unit ) > set_Product_unit > set_Code_integer > set_Product_unit ).
thf(func_def_401,type,
finite_fold_int_int: ( int > int > int ) > int > set_int > int ).
thf(func_def_402,type,
finite1723285575846220376et_int: ( int > set_int > set_int ) > set_int > set_int > set_int ).
thf(func_def_403,type,
finite_fold_nat_nat: ( nat > nat > nat ) > nat > set_nat > nat ).
thf(func_def_404,type,
finite6602901031542678131t_unit: ( nat > product_unit > product_unit ) > product_unit > set_nat > product_unit ).
thf(func_def_405,type,
finite3217087857726763998_set_o: ( nat > set_o > set_o ) > set_o > set_nat > set_o ).
thf(func_def_406,type,
finite1351632015609375740et_int: ( nat > set_int > set_int ) > set_int > set_nat > set_int ).
thf(func_def_407,type,
finite5529483035118572448et_nat: ( nat > set_nat > set_nat ) > set_nat > set_nat > set_nat ).
thf(func_def_408,type,
finite9145781422887121747t_unit: ( nat > set_Product_unit > set_Product_unit ) > set_Product_unit > set_nat > set_Product_unit ).
thf(func_def_409,type,
finite8834431295148555784t_unit: ( product_unit > product_unit > product_unit ) > product_unit > set_Product_unit > product_unit ).
thf(func_def_410,type,
finite2740360166580943592t_unit: ( product_unit > set_Product_unit > set_Product_unit ) > set_Product_unit > set_Product_unit > set_Product_unit ).
thf(func_def_411,type,
finite8456856254070787592_nat_o: ( set_Pr4532377907799695533_nat_o > set_Pr4532377907799695533_nat_o > set_Pr4532377907799695533_nat_o ) > set_Pr4532377907799695533_nat_o > set_se1353816476878259085_nat_o > set_Pr4532377907799695533_nat_o ).
thf(func_def_412,type,
finite4337638375924247368_set_o: ( set_o > set_o > set_o ) > set_o > set_set_o > set_o ).
thf(func_def_413,type,
finite2508883933289561102et_int: ( set_int > set_int > set_int ) > set_int > set_set_int > set_int ).
thf(func_def_414,type,
finite677925301803182934et_nat: ( set_nat > set_nat > set_nat ) > set_nat > set_set_nat > set_nat ).
thf(func_def_415,type,
finite4243028863286625800t_unit: ( set_Product_unit > set_Product_unit > set_Product_unit ) > set_Product_unit > set_set_Product_unit > set_Product_unit ).
thf(func_def_416,type,
comp_C8797469213163452608nteger: ( ( code_integer > code_integer ) > produc8923325533196201883nteger > produc8923325533196201883nteger ) > ( code_integer > code_integer > code_integer ) > code_integer > produc8923325533196201883nteger > produc8923325533196201883nteger ).
thf(func_def_417,type,
comp_C5076291335850163704nteger: ( ( code_integer > code_integer ) > code_integer ) > ( code_integer > code_integer > code_integer ) > code_integer > code_integer ).
thf(func_def_418,type,
comp_P4885729792186999034nteger: ( ( produc8923325533196201883nteger > produc8923325533196201883nteger ) > set_Code_integer ) > ( code_integer > produc8923325533196201883nteger > produc8923325533196201883nteger ) > code_integer > set_Code_integer ).
thf(func_def_419,type,
comp_P8262219321096090349nteger: ( ( produc8923325533196201883nteger > produc8923325533196201883nteger ) > set_nat ) > ( code_integer > produc8923325533196201883nteger > produc8923325533196201883nteger ) > code_integer > set_nat ).
thf(func_def_420,type,
comp_s518380907012905326o_assn: ( ( set_o > assn ) > set_set_o > assn ) > ( ( $o > assn ) > set_o > assn ) > ( $o > assn ) > set_set_o > assn ).
thf(func_def_421,type,
comp_s6084864194472624150_o_nat: ( ( set_o > nat ) > set_set_o > nat ) > ( ( $o > nat ) > set_o > nat ) > ( $o > nat ) > set_set_o > nat ).
thf(func_def_422,type,
comp_s3531641609786562035r_assn: ( ( set_Code_integer > assn ) > set_set_Code_integer > assn ) > ( ( code_integer > assn ) > set_Code_integer > assn ) > ( code_integer > assn ) > set_set_Code_integer > assn ).
thf(func_def_423,type,
comp_s8759123648413916139er_nat: ( ( set_Code_integer > nat ) > set_set_Code_integer > nat ) > ( ( code_integer > nat ) > set_Code_integer > nat ) > ( code_integer > nat ) > set_set_Code_integer > nat ).
thf(func_def_424,type,
comp_s4595925057662634336t_assn: ( ( set_int > assn ) > set_set_int > assn ) > ( ( int > assn ) > set_int > assn ) > ( int > assn ) > set_set_int > assn ).
thf(func_def_425,type,
comp_s9098231955081793132nt_int: ( ( set_int > int ) > set_set_int > int ) > ( ( int > int ) > set_int > int ) > ( int > int ) > set_set_int > int ).
thf(func_def_426,type,
comp_s3237642481825588056nt_nat: ( ( set_int > nat ) > set_set_int > nat ) > ( ( int > nat ) > set_int > nat ) > ( int > nat ) > set_set_int > nat ).
thf(func_def_427,type,
comp_s2451231738238893644t_assn: ( ( set_nat > assn ) > set_set_nat > assn ) > ( ( nat > assn ) > set_nat > assn ) > ( nat > assn ) > set_set_nat > assn ).
thf(func_def_428,type,
comp_s5456809700045850692at_nat: ( ( set_nat > nat ) > set_set_nat > nat ) > ( ( nat > nat ) > set_nat > nat ) > ( nat > nat ) > set_set_nat > nat ).
thf(func_def_429,type,
comp_s8247530669214396195t_assn: ( ( set_Product_unit > assn ) > set_set_Product_unit > assn ) > ( ( product_unit > assn ) > set_Product_unit > assn ) > ( product_unit > assn ) > set_set_Product_unit > assn ).
thf(func_def_430,type,
comp_s8928436226432208331it_nat: ( ( set_Product_unit > nat ) > set_set_Product_unit > nat ) > ( ( product_unit > nat ) > set_Product_unit > nat ) > ( product_unit > nat ) > set_set_Product_unit > nat ).
thf(func_def_431,type,
comp_o8922910837708301817nteger: ( $o > produc8923325533196201883nteger > produc8923325533196201883nteger ) > ( code_integer > $o ) > code_integer > produc8923325533196201883nteger > produc8923325533196201883nteger ).
thf(func_def_432,type,
comp_o6961622296474427249t_unit: ( $o > produc8923325533196201883nteger > produc8923325533196201883nteger ) > ( product_unit > $o ) > product_unit > produc8923325533196201883nteger > produc8923325533196201883nteger ).
thf(func_def_433,type,
comp_o1085144040659434033nteger: ( $o > code_integer ) > ( code_integer > $o ) > code_integer > code_integer ).
thf(func_def_434,type,
comp_o780160699970281785t_unit: ( $o > code_integer ) > ( product_unit > $o ) > product_unit > code_integer ).
thf(func_def_435,type,
comp_o2643362028263920430t_unit: ( $o > code_natural ) > ( product_unit > $o ) > product_unit > code_natural ).
thf(func_def_436,type,
comp_o_int_int: ( $o > int ) > ( int > $o ) > int > int ).
thf(func_def_437,type,
comp_o3916225744453444672unit_o: ( $o > product_unit ) > ( $o > $o ) > $o > product_unit ).
thf(func_def_438,type,
comp_o6635994240230588853nteger: ( $o > product_unit ) > ( code_integer > $o ) > code_integer > product_unit ).
thf(func_def_439,type,
comp_o7049022088938261380it_int: ( $o > product_unit ) > ( int > $o ) > int > product_unit ).
thf(func_def_440,type,
comp_o7051512559447311656it_nat: ( $o > product_unit ) > ( nat > $o ) > nat > product_unit ).
thf(func_def_441,type,
comp_o7724735231917782709t_unit: ( $o > product_unit ) > ( product_unit > $o ) > product_unit > product_unit ).
thf(func_def_442,type,
comp_o6895741221744486154t_unit: ( $o > set_o ) > ( product_unit > $o ) > product_unit > set_o ).
thf(func_def_443,type,
comp_o7929101295636871983nteger: ( $o > set_Code_integer ) > ( ( produc8923325533196201883nteger > produc8923325533196201883nteger ) > $o ) > ( produc8923325533196201883nteger > produc8923325533196201883nteger ) > set_Code_integer ).
thf(func_def_444,type,
comp_o4291842030903409511nteger: ( $o > set_Code_integer ) > ( code_integer > $o ) > code_integer > set_Code_integer ).
thf(func_def_445,type,
comp_o1521574261646630388t_unit: ( $o > set_int ) > ( product_unit > $o ) > product_unit > set_int ).
thf(func_def_446,type,
comp_o1289984117834210338nteger: ( $o > set_nat ) > ( ( produc8923325533196201883nteger > produc8923325533196201883nteger ) > $o ) > ( produc8923325533196201883nteger > produc8923325533196201883nteger ) > set_nat ).
thf(func_def_447,type,
comp_o172754640513974618nteger: ( $o > set_nat ) > ( code_integer > $o ) > code_integer > set_nat ).
thf(func_def_448,type,
comp_o9190160103024335696t_unit: ( $o > set_nat ) > ( product_unit > $o ) > product_unit > set_nat ).
thf(func_def_449,type,
comp_o2086126948152544725t_unit: ( $o > set_Product_unit ) > ( product_unit > $o ) > product_unit > set_Product_unit ).
thf(func_def_450,type,
comp_C2931745410781319416nteger: ( code_integer > code_integer > code_integer ) > ( code_integer > code_integer ) > code_integer > code_integer > code_integer ).
thf(func_def_451,type,
comp_C95226695241045696nteger: ( code_integer > produc8923325533196201883nteger > produc8923325533196201883nteger ) > ( ( code_integer > code_integer ) > code_integer ) > ( code_integer > code_integer ) > produc8923325533196201883nteger > produc8923325533196201883nteger ).
thf(func_def_452,type,
comp_C5733264393235242609eger_o: ( code_integer > produc8923325533196201883nteger > produc8923325533196201883nteger ) > ( $o > code_integer ) > $o > produc8923325533196201883nteger > produc8923325533196201883nteger ).
thf(func_def_453,type,
comp_C1593894019821074884nteger: ( code_integer > produc8923325533196201883nteger > produc8923325533196201883nteger ) > ( code_integer > code_integer ) > code_integer > produc8923325533196201883nteger > produc8923325533196201883nteger ).
thf(func_def_454,type,
comp_C834242911430131603er_int: ( code_integer > produc8923325533196201883nteger > produc8923325533196201883nteger ) > ( int > code_integer ) > int > produc8923325533196201883nteger > produc8923325533196201883nteger ).
thf(func_def_455,type,
comp_C836733381939181879er_nat: ( code_integer > produc8923325533196201883nteger > produc8923325533196201883nteger ) > ( nat > code_integer ) > nat > produc8923325533196201883nteger > produc8923325533196201883nteger ).
thf(func_def_456,type,
comp_C5854136495726678886t_unit: ( code_integer > produc8923325533196201883nteger > produc8923325533196201883nteger ) > ( product_unit > code_integer ) > product_unit > produc8923325533196201883nteger > produc8923325533196201883nteger ).
thf(func_def_457,type,
comp_C3026223900822616519nteger: ( code_integer > assn ) > ( code_integer > code_integer ) > code_integer > assn ).
thf(func_def_458,type,
comp_C8462447471145125178sn_nat: ( code_integer > assn ) > ( nat > code_integer ) > nat > assn ).
thf(func_def_459,type,
comp_C7449957260575251196nteger: ( code_integer > code_integer ) > ( code_integer > code_integer ) > code_integer > code_integer ).
thf(func_def_460,type,
comp_C6273552412331968711atural: ( code_integer > code_integer ) > ( code_natural > code_integer ) > code_natural > code_integer ).
thf(func_def_461,type,
comp_C6974050283906349679er_nat: ( code_integer > code_integer ) > ( nat > code_integer ) > nat > code_integer ).
thf(func_def_462,type,
comp_C1043215000170864188nteger: ( code_integer > code_integer ) > ( produc8923325533196201883nteger > code_integer ) > produc8923325533196201883nteger > code_integer ).
thf(func_def_463,type,
comp_C2290379119555921966t_unit: ( code_integer > code_integer ) > ( product_unit > code_integer ) > product_unit > code_integer ).
thf(func_def_464,type,
comp_C1891057325546811591nteger: ( code_integer > code_natural ) > ( code_integer > code_integer ) > code_integer > code_natural ).
thf(func_def_465,type,
comp_C714652477303529106atural: ( code_integer > code_natural ) > ( code_natural > code_integer ) > code_natural > code_natural ).
thf(func_def_466,type,
comp_C4153580447849560611t_unit: ( code_integer > code_natural ) > ( product_unit > code_integer ) > product_unit > code_natural ).
thf(func_def_467,type,
comp_C7610259292922237131nteger: ( code_integer > int ) > ( code_integer > code_integer ) > code_integer > int ).
thf(func_def_468,type,
comp_C5048910307564104766nt_nat: ( code_integer > int ) > ( nat > code_integer ) > nat > int ).
thf(func_def_469,type,
comp_C7238605732685392495nteger: ( code_integer > nat ) > ( code_integer > code_integer ) > code_integer > nat ).
thf(func_def_470,type,
comp_C8146212659816229034nteger: ( code_integer > product_unit ) > ( code_integer > code_integer ) > code_integer > product_unit ).
thf(func_def_471,type,
comp_C6969807811572946549atural: ( code_integer > product_unit ) > ( code_natural > code_integer ) > code_natural > product_unit ).
thf(func_def_472,type,
comp_C2408090164605745152t_unit: ( code_integer > product_unit ) > ( product_unit > code_integer ) > product_unit > product_unit ).
thf(func_def_473,type,
comp_C6344129587964912378nteger: ( code_integer > set_Code_integer ) > ( ( produc8923325533196201883nteger > produc8923325533196201883nteger ) > code_integer ) > ( produc8923325533196201883nteger > produc8923325533196201883nteger ) > set_Code_integer ).
thf(func_def_474,type,
comp_C8186931911283765298nteger: ( code_integer > set_Code_integer ) > ( code_integer > code_integer ) > code_integer > set_Code_integer ).
thf(func_def_475,type,
comp_C3184339336801759213nteger: ( code_integer > set_nat ) > ( ( produc8923325533196201883nteger > produc8923325533196201883nteger ) > code_integer ) > ( produc8923325533196201883nteger > produc8923325533196201883nteger ) > set_nat ).
thf(func_def_476,type,
comp_C6537567860429791781nteger: ( code_integer > set_nat ) > ( code_integer > code_integer ) > code_integer > set_nat ).
thf(func_def_477,type,
comp_C456341964420721295nteger: ( code_natural > produc8923325533196201883nteger > produc8923325533196201883nteger ) > ( code_integer > code_natural ) > code_integer > produc8923325533196201883nteger > produc8923325533196201883nteger ).
thf(func_def_478,type,
comp_i2874689474467862163nteger: ( int > produc8923325533196201883nteger > produc8923325533196201883nteger ) > ( code_integer > int ) > code_integer > produc8923325533196201883nteger > produc8923325533196201883nteger ).
thf(func_def_479,type,
comp_int_o_o: ( int > $o ) > ( $o > int ) > $o > $o ).
thf(func_def_480,type,
comp_int_o_int: ( int > $o ) > ( int > int ) > int > $o ).
thf(func_def_481,type,
comp_int_o_nat: ( int > $o ) > ( nat > int ) > nat > $o ).
thf(func_def_482,type,
comp_i5286552343147518890eger_o: ( int > code_integer ) > ( $o > int ) > $o > code_integer ).
thf(func_def_483,type,
comp_i6014793192326455499nteger: ( int > code_integer ) > ( code_integer > int ) > code_integer > code_integer ).
thf(func_def_484,type,
comp_i1585864551200866970er_int: ( int > code_integer ) > ( int > int ) > int > code_integer ).
thf(func_def_485,type,
comp_i1588355021709917246er_nat: ( int > code_integer ) > ( nat > int ) > nat > code_integer ).
thf(func_def_486,type,
comp_i455893257298015894nteger: ( int > code_natural ) > ( code_integer > int ) > code_integer > code_natural ).
thf(func_def_487,type,
comp_int_int_o: ( int > int ) > ( $o > int ) > $o > int ).
thf(func_def_488,type,
comp_int_int_int: ( int > int ) > ( int > int ) > int > int ).
thf(func_def_489,type,
comp_int_int_nat: ( int > int ) > ( nat > int ) > nat > int ).
thf(func_def_490,type,
comp_i4330622015127193916nt_int: ( int > int ) > ( product_prod_int_int > int ) > product_prod_int_int > int ).
thf(func_def_491,type,
comp_int_nat_o: ( int > nat ) > ( $o > int ) > $o > nat ).
thf(func_def_492,type,
comp_int_nat_int: ( int > nat ) > ( int > int ) > int > nat ).
thf(func_def_493,type,
comp_int_nat_nat: ( int > nat ) > ( nat > int ) > nat > nat ).
thf(func_def_494,type,
comp_i8439899303502915483nteger: ( int > product_unit ) > ( code_integer > int ) > code_integer > product_unit ).
thf(func_def_495,type,
comp_i5479505385629679622nteger: ( int > set_o ) > ( code_integer > int ) > code_integer > set_o ).
thf(func_def_496,type,
comp_i1887841406321795152nteger: ( int > set_int ) > ( code_integer > int ) > code_integer > set_int ).
thf(func_def_497,type,
comp_i5103216804979976198nt_int: ( int > set_int ) > ( product_prod_int_int > int ) > product_prod_int_int > set_int ).
thf(func_def_498,type,
comp_i5102403792180996084nteger: ( int > set_nat ) > ( code_integer > int ) > code_integer > set_nat ).
thf(func_def_499,type,
comp_i6200299785623974754nt_int: ( int > set_nat ) > ( product_prod_int_int > int ) > product_prod_int_int > set_nat ).
thf(func_def_500,type,
comp_i3319814986458681467nteger: ( int > set_Product_unit ) > ( code_integer > int ) > code_integer > set_Product_unit ).
thf(func_def_501,type,
comp_nat_nat_nat_nat: ( nat > nat > nat ) > ( nat > nat ) > nat > nat > nat ).
thf(func_def_502,type,
comp_n2033352843805742135nteger: ( nat > produc8923325533196201883nteger > produc8923325533196201883nteger ) > ( code_integer > nat ) > code_integer > produc8923325533196201883nteger > produc8923325533196201883nteger ).
thf(func_def_503,type,
comp_n856947995562459650atural: ( nat > produc8923325533196201883nteger > produc8923325533196201883nteger ) > ( code_natural > nat ) > code_natural > produc8923325533196201883nteger > produc8923325533196201883nteger ).
thf(func_def_504,type,
comp_nat_assn_nat: ( nat > assn ) > ( nat > nat ) > nat > assn ).
thf(func_def_505,type,
comp_n6876073255676942959nteger: ( nat > code_integer ) > ( code_integer > nat ) > code_integer > code_integer ).
thf(func_def_506,type,
comp_n5699668407433660474atural: ( nat > code_integer ) > ( code_natural > nat ) > code_natural > code_integer ).
thf(func_def_507,type,
comp_n3898172953802868194er_nat: ( nat > code_integer ) > ( nat > nat ) > nat > code_integer ).
thf(func_def_508,type,
comp_n2857367870785791611ural_o: ( nat > code_natural ) > ( $o > nat ) > $o > code_natural ).
thf(func_def_509,type,
comp_n1317173320648503354nteger: ( nat > code_natural ) > ( code_integer > nat ) > code_integer > code_natural ).
thf(func_def_510,type,
comp_n140768472405220869atural: ( nat > code_natural ) > ( code_natural > nat ) > code_natural > code_natural ).
thf(func_def_511,type,
comp_n9160511493981425161al_int: ( nat > code_natural ) > ( int > nat ) > int > code_natural ).
thf(func_def_512,type,
comp_n9163001964490475437al_nat: ( nat > code_natural ) > ( nat > nat ) > nat > code_natural ).
thf(func_def_513,type,
comp_nat_int_nat: ( nat > int ) > ( nat > nat ) > nat > int ).
thf(func_def_514,type,
comp_n8360577435903631554atural: ( nat > product_unit ) > ( code_natural > nat ) > code_natural > product_unit ).
thf(func_def_515,type,
comp_n1572716621429000127atural: ( nat > set_int ) > ( code_natural > nat ) > code_natural > set_int ).
thf(func_def_516,type,
comp_n3623798160786426464nteger: ( nat > set_nat ) > ( ( produc8923325533196201883nteger > produc8923325533196201883nteger ) > nat ) > ( produc8923325533196201883nteger > produc8923325533196201883nteger ) > set_nat ).
thf(func_def_517,type,
comp_n5963683855531483544nteger: ( nat > set_nat ) > ( code_integer > nat ) > code_integer > set_nat ).
thf(func_def_518,type,
comp_n4787279007288201059atural: ( nat > set_nat ) > ( code_natural > nat ) > code_natural > set_nat ).
thf(func_def_519,type,
comp_P5544053077415136032nteger: ( produc723880050079548563nteger > produc101387210554329899nteger ) > ( produc8923325533196201883nteger > produc723880050079548563nteger ) > produc8923325533196201883nteger > produc101387210554329899nteger ).
thf(func_def_520,type,
comp_P1414880308642389412nteger: ( produc2221115339011613079nteger > produc101387210554329899nteger ) > ( produc8923325533196201883nteger > produc2221115339011613079nteger ) > produc8923325533196201883nteger > produc101387210554329899nteger ).
thf(func_def_521,type,
comp_P6312383820422015140nteger: ( produc1397831737260518551nteger > produc101387210554329899nteger ) > ( produc8923325533196201883nteger > produc1397831737260518551nteger ) > produc8923325533196201883nteger > produc101387210554329899nteger ).
thf(func_def_522,type,
comp_P7111796090210262324nteger: ( produc8923325533196201883nteger > code_integer ) > ( code_integer > produc8923325533196201883nteger ) > code_integer > code_integer ).
thf(func_def_523,type,
comp_P3428181540658402820nteger: ( produc8923325533196201883nteger > code_integer ) > ( produc8923325533196201883nteger > produc8923325533196201883nteger ) > produc8923325533196201883nteger > code_integer ).
thf(func_def_524,type,
comp_P659088277599099894t_unit: ( produc8923325533196201883nteger > code_integer ) > ( product_unit > produc8923325533196201883nteger ) > product_unit > code_integer ).
thf(func_def_525,type,
comp_P234064951390806312nteger: ( produc8923325533196201883nteger > produc101387210554329899nteger ) > ( produc8923325533196201883nteger > produc8923325533196201883nteger ) > produc8923325533196201883nteger > produc101387210554329899nteger ).
thf(func_def_526,type,
comp_P7101464685127059239nteger: ( produc6433947425880249742er_nat > code_integer ) > ( code_integer > produc6433947425880249742er_nat ) > code_integer > code_integer ).
thf(func_def_527,type,
comp_P7581484221039074243t_unit: ( produc6433947425880249742er_nat > code_integer ) > ( product_unit > produc6433947425880249742er_nat ) > product_unit > code_integer ).
thf(func_def_528,type,
comp_P5964733648194011802nteger: ( produc6433947425880249742er_nat > nat ) > ( code_integer > produc6433947425880249742er_nat ) > code_integer > nat ).
thf(func_def_529,type,
comp_P6821003776282389008t_unit: ( produc6433947425880249742er_nat > nat ) > ( product_unit > produc6433947425880249742er_nat ) > product_unit > nat ).
thf(func_def_530,type,
comp_P2930521774885306735et_nat: ( produc3658429121746597890et_nat > set_nat ) > ( produc3658429121746597890et_nat > produc3658429121746597890et_nat ) > produc3658429121746597890et_nat > set_nat ).
thf(func_def_531,type,
comp_P8606889026643445772nt_int: ( produc3658429121746597890et_nat > set_nat ) > ( product_prod_int_int > produc3658429121746597890et_nat ) > product_prod_int_int > set_nat ).
thf(func_def_532,type,
comp_P8308861049717174982et_nat: ( product_prod_int_int > int ) > ( produc3658429121746597890et_nat > product_prod_int_int ) > produc3658429121746597890et_nat > int ).
thf(func_def_533,type,
comp_P5356232565778667509nt_int: ( product_prod_int_int > int ) > ( product_prod_int_int > product_prod_int_int ) > product_prod_int_int > int ).
thf(func_def_534,type,
comp_P6796073005956588617t_unit: ( product_prod_int_int > int ) > ( product_unit > product_prod_int_int ) > product_unit > int ).
thf(func_def_535,type,
comp_P7177471617286032590eger_o: ( produc2591615631378489102nteger > code_integer ) > ( $o > produc2591615631378489102nteger ) > $o > code_integer ).
thf(func_def_536,type,
comp_P6122662217367900327nteger: ( produc2591615631378489102nteger > code_integer ) > ( code_integer > produc2591615631378489102nteger ) > code_integer > code_integer ).
thf(func_def_537,type,
comp_P7320554561093743171t_unit: ( produc2591615631378489102nteger > code_integer ) > ( product_unit > produc2591615631378489102nteger ) > product_unit > code_integer ).
thf(func_def_538,type,
comp_P7179991235626930843_nat_o: ( produc2591615631378489102nteger > nat ) > ( $o > produc2591615631378489102nteger ) > $o > nat ).
thf(func_def_539,type,
comp_P5725896914063526938nteger: ( produc2591615631378489102nteger > nat ) > ( code_integer > produc2591615631378489102nteger ) > code_integer > nat ).
thf(func_def_540,type,
comp_P7621350044865347728t_unit: ( produc2591615631378489102nteger > nat ) > ( product_unit > produc2591615631378489102nteger ) > product_unit > nat ).
thf(func_def_541,type,
comp_P4394204175247225320_nat_o: ( product_prod_nat_nat > nat ) > ( $o > product_prod_nat_nat ) > $o > nat ).
thf(func_def_542,type,
comp_P2980510830910040157t_unit: ( product_prod_nat_nat > nat ) > ( product_unit > product_prod_nat_nat ) > product_unit > nat ).
thf(func_def_543,type,
comp_P4511551582283981294nteger: ( product_unit > produc8923325533196201883nteger > produc8923325533196201883nteger ) > ( code_integer > product_unit ) > code_integer > produc8923325533196201883nteger > produc8923325533196201883nteger ).
thf(func_def_544,type,
comp_P4558236313141964142nteger: ( product_unit > product_unit > product_unit ) > ( code_integer > product_unit ) > code_integer > product_unit > product_unit ).
thf(func_def_545,type,
comp_P6204500334276302049it_nat: ( product_unit > product_unit > product_unit ) > ( nat > product_unit ) > nat > product_unit > product_unit ).
thf(func_def_546,type,
comp_P6790823133229852966nteger: ( product_unit > code_integer ) > ( code_integer > product_unit ) > code_integer > code_integer ).
thf(func_def_547,type,
comp_P4635212033913086756nteger: ( product_unit > set_Code_integer ) > ( ( produc8923325533196201883nteger > produc8923325533196201883nteger ) > product_unit ) > ( produc8923325533196201883nteger > produc8923325533196201883nteger ) > set_Code_integer ).
thf(func_def_548,type,
comp_P4767411901762164316nteger: ( product_unit > set_Code_integer ) > ( code_integer > product_unit ) > code_integer > set_Code_integer ).
thf(func_def_549,type,
comp_P6101996899264665623nteger: ( product_unit > set_nat ) > ( ( produc8923325533196201883nteger > produc8923325533196201883nteger ) > product_unit ) > ( produc8923325533196201883nteger > produc8923325533196201883nteger ) > set_nat ).
thf(func_def_550,type,
comp_P5878433733084393551nteger: ( product_unit > set_nat ) > ( code_integer > product_unit ) > code_integer > set_nat ).
thf(func_def_551,type,
comp_s90868338361139993nteger: ( set_o > set_o > set_o ) > ( code_integer > set_o ) > code_integer > set_o > set_o ).
thf(func_def_552,type,
comp_s4424444925706540684_o_nat: ( set_o > set_o > set_o ) > ( nat > set_o ) > nat > set_o > set_o ).
thf(func_def_553,type,
comp_s956550612328592744nteger: ( set_Code_integer > set_Code_integer ) > ( code_integer > set_Code_integer ) > code_integer > set_Code_integer ).
thf(func_def_554,type,
comp_s3634918272309710215nteger: ( set_int > set_int > set_int ) > ( code_integer > set_int ) > code_integer > set_int > set_int ).
thf(func_def_555,type,
comp_s6689493721604790522nt_nat: ( set_int > set_int > set_int ) > ( nat > set_int ) > nat > set_int > set_int ).
thf(func_def_556,type,
comp_s3169385134670675059nteger: ( set_nat > set_nat > set_nat ) > ( code_integer > set_nat ) > code_integer > set_nat > set_nat ).
thf(func_def_557,type,
comp_s2351873599094224870at_nat: ( set_nat > set_nat > set_nat ) > ( nat > set_nat ) > nat > set_nat > set_nat ).
thf(func_def_558,type,
comp_s1601710177542341017et_nat: ( set_nat > int ) > ( produc3658429121746597890et_nat > set_nat ) > produc3658429121746597890et_nat > int ).
thf(func_def_559,type,
comp_s8292494440407567195nteger: ( set_nat > set_Code_integer ) > ( code_integer > set_nat ) > code_integer > set_Code_integer ).
thf(func_def_560,type,
comp_s4605850120322998606nteger: ( set_nat > set_nat ) > ( code_integer > set_nat ) > code_integer > set_nat ).
thf(func_def_561,type,
comp_s5696514601250091763et_nat: ( set_nat > set_nat ) > ( produc3658429121746597890et_nat > set_nat ) > produc3658429121746597890et_nat > set_nat ).
thf(func_def_562,type,
comp_s3199726385246003790nteger: ( set_Product_unit > set_Product_unit > set_Product_unit ) > ( code_integer > set_Product_unit ) > code_integer > set_Product_unit > set_Product_unit ).
thf(func_def_563,type,
comp_s6943286182646368705it_nat: ( set_Product_unit > set_Product_unit > set_Product_unit ) > ( nat > set_Product_unit ) > nat > set_Product_unit > set_Product_unit ).
thf(func_def_564,type,
fun_upd_o_o: ( $o > $o ) > $o > $o > $o > $o ).
thf(func_def_565,type,
fun_upd_o_int: ( $o > int ) > $o > int > $o > int ).
thf(func_def_566,type,
fun_upd_o_nat: ( $o > nat ) > $o > nat > $o > nat ).
thf(func_def_567,type,
fun_up846054379331416818t_unit: ( $o > product_unit ) > $o > product_unit > $o > product_unit ).
thf(func_def_568,type,
fun_upd_o_set_o: ( $o > set_o ) > $o > set_o > $o > set_o ).
thf(func_def_569,type,
fun_upd_o_set_int: ( $o > set_int ) > $o > set_int > $o > set_int ).
thf(func_def_570,type,
fun_upd_o_set_nat: ( $o > set_nat ) > $o > set_nat > $o > set_nat ).
thf(func_def_571,type,
fun_up4718931970374059986t_unit: ( $o > set_Product_unit ) > $o > set_Product_unit > $o > set_Product_unit ).
thf(func_def_572,type,
fun_up8355007323143964473nteger: ( code_integer > code_integer ) > code_integer > code_integer > code_integer > code_integer ).
thf(func_def_573,type,
fun_upd_int_nat: ( int > nat ) > int > nat > int > nat ).
thf(func_def_574,type,
fun_upd_int_set_o: ( int > set_o ) > int > set_o > int > set_o ).
thf(func_def_575,type,
fun_up5054257004045450754t_unit: ( int > set_Product_unit ) > int > set_Product_unit > int > set_Product_unit ).
thf(func_def_576,type,
fun_up3904115110581602988nteger: ( nat > code_integer ) > nat > code_integer > nat > code_integer ).
thf(func_def_577,type,
fun_upd_nat_nat: ( nat > nat ) > nat > nat > nat > nat ).
thf(func_def_578,type,
fun_upd_nat_set_nat: ( nat > set_nat ) > nat > set_nat > nat > set_nat ).
thf(func_def_579,type,
fun_up5432024876597553480unit_o: ( product_unit > $o ) > product_unit > $o > product_unit > $o ).
thf(func_def_580,type,
fun_up6470560416331323516it_int: ( product_unit > int ) > product_unit > int > product_unit > int ).
thf(func_def_581,type,
fun_up6473050886840373792it_nat: ( product_unit > nat ) > product_unit > nat > product_unit > nat ).
thf(func_def_582,type,
fun_up6875000483049295805t_unit: ( product_unit > product_unit ) > product_unit > product_unit > product_unit > product_unit ).
thf(func_def_583,type,
fun_up1762901246852280360_set_o: ( product_unit > set_o ) > product_unit > set_o > product_unit > set_o ).
thf(func_def_584,type,
fun_up391983181201004082et_int: ( product_unit > set_int ) > product_unit > set_int > product_unit > set_int ).
thf(func_def_585,type,
fun_up4569834200710200790et_nat: ( product_unit > set_nat ) > product_unit > set_nat > product_unit > set_nat ).
thf(func_def_586,type,
fun_up488859312337516701t_unit: ( product_unit > set_Product_unit ) > product_unit > set_Product_unit > product_unit > set_Product_unit ).
thf(func_def_587,type,
id_o: $o > $o ).
thf(func_def_588,type,
inj_on5271586719803267564_o_nat: ( ( produc3658429121746597890et_nat > $o ) > nat ) > set_Pr4532377907799695533_nat_o > $o ).
thf(func_def_589,type,
inj_on_o_o: ( $o > $o ) > set_o > $o ).
thf(func_def_590,type,
inj_on_o_int: ( $o > int ) > set_o > $o ).
thf(func_def_591,type,
inj_on_o_nat: ( $o > nat ) > set_o > $o ).
thf(func_def_592,type,
inj_on4268752236855399398t_unit: ( $o > product_unit ) > set_o > $o ).
thf(func_def_593,type,
inj_on1158064722259180589nteger: ( code_integer > code_integer ) > set_Code_integer > $o ).
thf(func_def_594,type,
inj_on1981279154655846304er_nat: ( code_integer > nat ) > set_Code_integer > $o ).
thf(func_def_595,type,
inj_on_int_int: ( int > int ) > set_int > $o ).
thf(func_def_596,type,
inj_on_int_nat: ( int > nat ) > set_int > $o ).
thf(func_def_597,type,
inj_on7779444506608719434_nat_o: ( nat > produc3658429121746597890et_nat > $o ) > set_nat > $o ).
thf(func_def_598,type,
inj_on_nat_o: ( nat > $o ) > set_nat > $o ).
thf(func_def_599,type,
inj_on2245834603434889120nteger: ( nat > code_integer ) > set_nat > $o ).
thf(func_def_600,type,
inj_on_nat_int: ( nat > int ) > set_nat > $o ).
thf(func_def_601,type,
inj_on_nat_nat: ( nat > nat ) > set_nat > $o ).
thf(func_def_602,type,
inj_on7061601236592826506t_unit: ( nat > product_unit ) > set_nat > $o ).
thf(func_def_603,type,
inj_on_nat_set_nat: ( nat > set_nat ) > set_nat > $o ).
thf(func_def_604,type,
inj_on20428373856169009nteger: ( produc8923325533196201883nteger > produc8923325533196201883nteger ) > set_Pr4811707699266497531nteger > $o ).
thf(func_def_605,type,
inj_on8854722734121536060unit_o: ( product_unit > $o ) > set_Product_unit > $o ).
thf(func_def_606,type,
inj_on8427948621271784584it_int: ( product_unit > int ) > set_Product_unit > $o ).
thf(func_def_607,type,
inj_on8430439091780834860it_nat: ( product_unit > nat ) > set_Product_unit > $o ).
thf(func_def_608,type,
inj_on8151373323710067377t_unit: ( product_unit > product_unit ) > set_Product_unit > $o ).
thf(func_def_609,type,
inj_on_rat_rat: ( rat > rat ) > set_rat > $o ).
thf(func_def_610,type,
inj_on_set_nat_nat: ( set_nat > nat ) > set_set_nat > $o ).
thf(func_def_611,type,
map_fu8272188784021352819nteger: ( code_integer > int ) > ( ( int > int ) > code_integer > code_integer ) > ( int > int > int ) > code_integer > code_integer > code_integer ).
thf(func_def_612,type,
map_fu2599414010547811884nteger: ( code_integer > int ) > ( int > code_integer ) > ( int > int ) > code_integer > code_integer ).
thf(func_def_613,type,
map_fu6549440983881763648atural: ( code_natural > nat ) > ( ( nat > nat ) > code_natural > code_natural ) > ( nat > nat > nat ) > code_natural > code_natural > code_natural ).
thf(func_def_614,type,
map_fu1239815594074539274atural: ( code_natural > nat ) > ( nat > code_natural ) > ( nat > nat ) > code_natural > code_natural ).
thf(func_def_615,type,
map_fu4960017516451851995nt_int: ( int > product_prod_nat_nat ) > ( ( product_prod_nat_nat > product_prod_nat_nat ) > int > int ) > ( product_prod_nat_nat > product_prod_nat_nat > product_prod_nat_nat ) > int > int > int ).
thf(func_def_616,type,
map_fu3667384564859982768at_int: ( int > product_prod_nat_nat ) > ( product_prod_nat_nat > int ) > ( product_prod_nat_nat > product_prod_nat_nat ) > int > int ).
thf(func_def_617,type,
map_fu4333342158222067775at_rat: ( rat > product_prod_int_int ) > ( ( product_prod_int_int > product_prod_int_int ) > rat > rat ) > ( product_prod_int_int > product_prod_int_int > product_prod_int_int ) > rat > rat > rat ).
thf(func_def_618,type,
map_fu898904425404107465nt_o_o: ( rat > product_prod_int_int ) > ( $o > $o ) > ( product_prod_int_int > $o ) > rat > $o ).
thf(func_def_619,type,
map_fu5673905371560938248nt_rat: ( rat > product_prod_int_int ) > ( product_prod_int_int > rat ) > ( product_prod_int_int > product_prod_int_int ) > rat > rat ).
thf(func_def_620,type,
gcd_Gcd_int: set_int > int ).
thf(func_def_621,type,
gcd_Gcd_nat: set_nat > nat ).
thf(func_def_622,type,
gcd_Lcm_nat: set_nat > nat ).
thf(func_def_623,type,
bezw: nat > nat > product_prod_int_int ).
thf(func_def_624,type,
bezw_rel: product_prod_nat_nat > product_prod_nat_nat > $o ).
thf(func_def_625,type,
gcd_gcd_Code_integer: code_integer > code_integer > code_integer ).
thf(func_def_626,type,
gcd_gcd_int: int > int > int ).
thf(func_def_627,type,
gcd_gcd_nat: nat > nat > nat ).
thf(func_def_628,type,
gcd_lcm_Code_integer: code_integer > code_integer > code_integer ).
thf(func_def_629,type,
gcd_lcm_int: int > int > int ).
thf(func_def_630,type,
gcd_lcm_nat: nat > nat > nat ).
thf(func_def_631,type,
semiri4256215615220890538in_int: set_int > int ).
thf(func_def_632,type,
semiri4258706085729940814in_nat: set_nat > nat ).
thf(func_def_633,type,
abs_abs_Code_integer: code_integer > code_integer ).
thf(func_def_634,type,
abs_abs_int: int > int ).
thf(func_def_635,type,
minus_minus_assn: assn > assn > assn ).
thf(func_def_636,type,
minus_8373710615458151222nteger: code_integer > code_integer > code_integer ).
thf(func_def_637,type,
minus_minus_int: int > int > int ).
thf(func_def_638,type,
minus_minus_nat: nat > nat > nat ).
thf(func_def_639,type,
minus_minus_rat: rat > rat > rat ).
thf(func_def_640,type,
minus_1801376950450012436_nat_o: set_Pr4532377907799695533_nat_o > set_Pr4532377907799695533_nat_o > set_Pr4532377907799695533_nat_o ).
thf(func_def_641,type,
minus_minus_set_o: set_o > set_o > set_o ).
thf(func_def_642,type,
minus_minus_set_assn: set_assn > set_assn > set_assn ).
thf(func_def_643,type,
minus_2355218937544613996nteger: set_Code_integer > set_Code_integer > set_Code_integer ).
thf(func_def_644,type,
minus_minus_set_int: set_int > set_int > set_int ).
thf(func_def_645,type,
minus_minus_set_nat: set_nat > set_nat > set_nat ).
thf(func_def_646,type,
minus_9189903262233110958_nat_o: set_Pr2161125870931222855_nat_o > set_Pr2161125870931222855_nat_o > set_Pr2161125870931222855_nat_o ).
thf(func_def_647,type,
minus_7052513330764642158od_o_o: set_Product_prod_o_o > set_Product_prod_o_o > set_Product_prod_o_o ).
thf(func_def_648,type,
minus_1052850069191792384nt_int: set_Pr958786334691620121nt_int > set_Pr958786334691620121nt_int > set_Pr958786334691620121nt_int ).
thf(func_def_649,type,
minus_1356011639430497352at_nat: set_Pr1261947904930325089at_nat > set_Pr1261947904930325089at_nat > set_Pr1261947904930325089at_nat ).
thf(func_def_650,type,
minus_6985556733573110190t_unit: set_Pr5094982260447487303t_unit > set_Pr5094982260447487303t_unit > set_Pr5094982260447487303t_unit ).
thf(func_def_651,type,
minus_6452836326544984404t_unit: set_Product_unit > set_Product_unit > set_Product_unit ).
thf(func_def_652,type,
minus_2163939370556025621et_nat: set_set_nat > set_set_nat > set_set_nat ).
thf(func_def_653,type,
one_one_assn: assn ).
thf(func_def_654,type,
one_one_Code_integer: code_integer ).
thf(func_def_655,type,
one_one_Code_natural: code_natural ).
thf(func_def_656,type,
one_one_int: int ).
thf(func_def_657,type,
one_one_nat: nat ).
thf(func_def_658,type,
one_one_rat: rat ).
thf(func_def_659,type,
plus_p5714425477246183910nteger: code_integer > code_integer > code_integer ).
thf(func_def_660,type,
plus_p4538020629002901425atural: code_natural > code_natural > code_natural ).
thf(func_def_661,type,
plus_plus_int: int > int > int ).
thf(func_def_662,type,
plus_plus_nat: nat > nat > nat ).
thf(func_def_663,type,
plus_plus_num: num > num > num ).
thf(func_def_664,type,
plus_plus_rat: rat > rat > rat ).
thf(func_def_665,type,
sgn_sgn_Code_integer: code_integer > code_integer ).
thf(func_def_666,type,
sgn_sgn_int: int > int ).
thf(func_def_667,type,
sgn_sgn_rat: rat > rat ).
thf(func_def_668,type,
times_times_assn: assn > assn > assn ).
thf(func_def_669,type,
times_3573771949741848930nteger: code_integer > code_integer > code_integer ).
thf(func_def_670,type,
times_2397367101498566445atural: code_natural > code_natural > code_natural ).
thf(func_def_671,type,
times_times_int: int > int > int ).
thf(func_def_672,type,
times_times_nat: nat > nat > nat ).
thf(func_def_673,type,
times_times_num: num > num > num ).
thf(func_def_674,type,
times_times_rat: rat > rat > rat ).
thf(func_def_675,type,
uminus_uminus_assn: assn > assn ).
thf(func_def_676,type,
uminus1351360451143612070nteger: code_integer > code_integer ).
thf(func_def_677,type,
uminus_uminus_int: int > int ).
thf(func_def_678,type,
uminus2952777764628376836t_unit: product_unit > product_unit ).
thf(func_def_679,type,
uminus_uminus_rat: rat > rat ).
thf(func_def_680,type,
uminus5254974436814262692_nat_o: set_Pr4532377907799695533_nat_o > set_Pr4532377907799695533_nat_o ).
thf(func_def_681,type,
uminus_uminus_set_o: set_o > set_o ).
thf(func_def_682,type,
uminus1532241313380277803et_int: set_int > set_int ).
thf(func_def_683,type,
uminus5710092332889474511et_nat: set_nat > set_nat ).
thf(func_def_684,type,
uminus5944136376168626660t_unit: set_Product_unit > set_Product_unit ).
thf(func_def_685,type,
zero_z3403309356797280102nteger: code_integer ).
thf(func_def_686,type,
zero_z2226904508553997617atural: code_natural ).
thf(func_def_687,type,
zero_zero_int: int ).
thf(func_def_688,type,
zero_zero_nat: nat ).
thf(func_def_689,type,
zero_zero_rat: rat ).
thf(func_def_690,type,
groups6742167275697360483_o_nat: ( ( produc3658429121746597890et_nat > $o ) > nat ) > set_Pr4532377907799695533_nat_o > nat ).
thf(func_def_691,type,
groups4406642042086082107nteger: ( $o > code_integer ) > set_o > code_integer ).
thf(func_def_692,type,
groups8505340233167759370_o_int: ( $o > int ) > set_o > int ).
thf(func_def_693,type,
groups8507830703676809646_o_nat: ( $o > nat ) > set_o > nat ).
thf(func_def_694,type,
groups7872700643590313910_o_rat: ( $o > rat ) > set_o > rat ).
thf(func_def_695,type,
groups879477027807139574nteger: ( code_integer > code_integer ) > set_Code_integer > code_integer ).
thf(func_def_696,type,
groups7234854612051535045er_int: ( code_integer > int ) > set_Code_integer > int ).
thf(func_def_697,type,
groups7237345082560585321er_nat: ( code_integer > nat ) > set_Code_integer > nat ).
thf(func_def_698,type,
groups6602215022474089585er_rat: ( code_integer > rat ) > set_Code_integer > rat ).
thf(func_def_699,type,
groups7873554091576472773nteger: ( int > code_integer ) > set_int > code_integer ).
thf(func_def_700,type,
groups4538972089207619220nt_int: ( int > int ) > set_int > int ).
thf(func_def_701,type,
groups4541462559716669496nt_nat: ( int > nat ) > set_int > nat ).
thf(func_def_702,type,
groups3906332499630173760nt_rat: ( int > rat ) > set_int > rat ).
thf(func_def_703,type,
groups7501900531339628137nteger: ( nat > code_integer ) > set_nat > code_integer ).
thf(func_def_704,type,
groups3539618377306564664at_int: ( nat > int ) > set_nat > int ).
thf(func_def_705,type,
groups3542108847815614940at_nat: ( nat > nat ) > set_nat > nat ).
thf(func_def_706,type,
groups2906978787729119204at_rat: ( nat > rat ) > set_nat > rat ).
thf(func_def_707,type,
groups5393439102872367408nteger: ( product_unit > code_integer ) > set_Product_unit > code_integer ).
thf(func_def_708,type,
groups8764306707065906943it_int: ( product_unit > int ) > set_Product_unit > int ).
thf(func_def_709,type,
groups8766797177574957219it_nat: ( product_unit > nat ) > set_Product_unit > nat ).
thf(func_def_710,type,
groups8131667117488461483it_rat: ( product_unit > rat ) > set_Product_unit > rat ).
thf(func_def_711,type,
groups1256056575759317066nt_int: ( set_int > int ) > set_set_int > int ).
thf(func_def_712,type,
groups8294997508430121362at_nat: ( set_nat > nat ) > set_set_nat > nat ).
thf(func_def_713,type,
groups5301882518646026715o_assn: ( $o > assn ) > set_o > assn ).
thf(func_def_714,type,
groups7694694392188491536nteger: ( $o > code_integer ) > set_o > code_integer ).
thf(func_def_715,type,
groups6518289543945209051atural: ( $o > code_natural ) > set_o > code_natural ).
thf(func_def_716,type,
groups3502327434004483295_o_int: ( $o > int ) > set_o > int ).
thf(func_def_717,type,
groups3504817904513533571_o_nat: ( $o > nat ) > set_o > nat ).
thf(func_def_718,type,
groups2869687844427037835_o_rat: ( $o > rat ) > set_o > rat ).
thf(func_def_719,type,
groups1304777262505850412r_assn: ( code_integer > assn ) > set_Code_integer > assn ).
thf(func_def_720,type,
groups3674199335183972705nteger: ( code_integer > code_integer ) > set_Code_integer > code_integer ).
thf(func_def_721,type,
groups2497794486940690220atural: ( code_integer > code_natural ) > set_Code_integer > code_natural ).
thf(func_def_722,type,
groups3188404863801439024er_int: ( code_integer > int ) > set_Code_integer > int ).
thf(func_def_723,type,
groups3190895334310489300er_nat: ( code_integer > nat ) > set_Code_integer > nat ).
thf(func_def_724,type,
groups2555765274223993564er_rat: ( code_integer > rat ) > set_Code_integer > rat ).
thf(func_def_725,type,
groups7882442080178216443t_assn: ( int > assn ) > set_int > assn ).
thf(func_def_726,type,
groups3827104343326376752nteger: ( int > code_integer ) > set_int > code_integer ).
thf(func_def_727,type,
groups2650699495083094267atural: ( int > code_natural ) > set_int > code_natural ).
thf(func_def_728,type,
groups1705073143266064639nt_int: ( int > int ) > set_int > int ).
thf(func_def_729,type,
groups1707563613775114915nt_nat: ( int > nat ) > set_int > nat ).
thf(func_def_730,type,
groups1072433553688619179nt_rat: ( int > rat ) > set_int > rat ).
thf(func_def_731,type,
groups6906906614972039071t_assn: ( nat > assn ) > set_nat > assn ).
thf(func_def_732,type,
groups3455450783089532116nteger: ( nat > code_integer ) > set_nat > code_integer ).
thf(func_def_733,type,
groups2279045934846249631atural: ( nat > code_natural ) > set_nat > code_natural ).
thf(func_def_734,type,
groups705719431365010083at_int: ( nat > int ) > set_nat > int ).
thf(func_def_735,type,
groups708209901874060359at_nat: ( nat > nat ) > set_nat > nat ).
thf(func_def_736,type,
groups73079841787564623at_rat: ( nat > rat ) > set_nat > rat ).
thf(func_def_737,type,
groups5360233303959380816t_assn: ( product_unit > assn ) > set_Product_unit > assn ).
thf(func_def_738,type,
groups8185359811944078981nteger: ( product_unit > code_integer ) > set_Product_unit > code_integer ).
thf(func_def_739,type,
groups7008954963700796496atural: ( product_unit > code_natural ) > set_Product_unit > code_natural ).
thf(func_def_740,type,
groups393930778517121108it_int: ( product_unit > int ) > set_Product_unit > int ).
thf(func_def_741,type,
groups396421249026171384it_nat: ( product_unit > nat ) > set_Product_unit > nat ).
thf(func_def_742,type,
groups8984663225794451456it_rat: ( product_unit > rat ) > set_Product_unit > rat ).
thf(func_def_743,type,
groups6963733262650904763o_assn: ( set_o > assn ) > set_set_o > assn ).
thf(func_def_744,type,
groups4868773677410035043_o_nat: ( set_o > nat ) > set_set_o > nat ).
thf(func_def_745,type,
groups4245288774754739042r_assn: ( set_Code_integer > assn ) > set_set_Code_integer > assn ).
thf(func_def_746,type,
groups6044384099095135754er_nat: ( set_Code_integer > nat ) > set_set_Code_integer > nat ).
thf(func_def_747,type,
groups5950967880310095793t_assn: ( set_int > assn ) > set_set_int > assn ).
thf(func_def_748,type,
groups6432978864363996853nt_int: ( set_int > int ) > set_set_int > int ).
thf(func_def_749,type,
groups6435469334873047129nt_nat: ( set_int > nat ) > set_set_int > nat ).
thf(func_def_750,type,
groups1535173586962943829t_assn: ( set_nat > assn ) > set_set_nat > assn ).
thf(func_def_751,type,
groups4248547760180025341at_nat: ( set_nat > nat ) > set_set_nat > nat ).
thf(func_def_752,type,
groups2392624107559085104t_assn: ( set_Product_unit > assn ) > set_set_Product_unit > assn ).
thf(func_def_753,type,
groups8454329047868613336it_nat: ( set_Product_unit > nat ) > set_set_Product_unit > nat ).
thf(func_def_754,type,
defaul566961228789861419t_unit: product_unit ).
thf(func_def_755,type,
arrays_Product_unit: heap_e7401611519738050253t_unit > typerep > nat > list_nat ).
thf(func_def_756,type,
lim_Product_unit: heap_e7401611519738050253t_unit > nat ).
thf(func_def_757,type,
refs_Product_unit: heap_e7401611519738050253t_unit > typerep > nat > nat ).
thf(func_def_758,type,
if_assn: $o > assn > assn > assn ).
thf(func_def_759,type,
if_Code_integer: $o > code_integer > code_integer > code_integer ).
thf(func_def_760,type,
if_Code_natural: $o > code_natural > code_natural > code_natural ).
thf(func_def_761,type,
if_filter_nat: $o > filter_nat > filter_nat > filter_nat ).
thf(func_def_762,type,
if_int: $o > int > int > int ).
thf(func_def_763,type,
if_nat: $o > nat > nat > nat ).
thf(func_def_764,type,
if_Pro6119634080678213985nteger: $o > produc8923325533196201883nteger > produc8923325533196201883nteger > produc8923325533196201883nteger ).
thf(func_def_765,type,
if_Pro3027730157355071871nt_int: $o > product_prod_int_int > product_prod_int_int > product_prod_int_int ).
thf(func_def_766,type,
if_Pro6206227464963214023at_nat: $o > product_prod_nat_nat > product_prod_nat_nat > product_prod_nat_nat ).
thf(func_def_767,type,
if_rat: $o > rat > rat > rat ).
thf(func_def_768,type,
if_set_o: $o > set_o > set_o > set_o ).
thf(func_def_769,type,
if_set_int: $o > set_int > set_int > set_int ).
thf(func_def_770,type,
if_set_nat: $o > set_nat > set_nat > set_nat ).
thf(func_def_771,type,
if_set_Product_unit: $o > set_Product_unit > set_Product_unit > set_Product_unit ).
thf(func_def_772,type,
infini8530281810654367211te_nat: set_nat > nat > nat ).
thf(func_def_773,type,
infini7930543730640340914t_unit: set_Product_unit > nat > product_unit ).
thf(func_def_774,type,
abs_Integ: product_prod_nat_nat > int ).
thf(func_def_775,type,
neg: num > int ).
thf(func_def_776,type,
pos: num > int ).
thf(func_def_777,type,
rep_Integ: int > product_prod_nat_nat ).
thf(func_def_778,type,
intrel: product_prod_nat_nat > product_prod_nat_nat > $o ).
thf(func_def_779,type,
nat2: int > nat ).
thf(func_def_780,type,
pcr_int: product_prod_nat_nat > int > $o ).
thf(func_def_781,type,
ring_1_of_int_rat: int > rat ).
thf(func_def_782,type,
inf_in1318976480646536635_nat_o: ( produc3658429121746597890et_nat > $o ) > ( produc3658429121746597890et_nat > $o ) > produc3658429121746597890et_nat > $o ).
thf(func_def_783,type,
inf_inf_o: $o > $o > $o ).
thf(func_def_784,type,
inf_inf_assn: assn > assn > assn ).
thf(func_def_785,type,
inf_inf_filter_nat: filter_nat > filter_nat > filter_nat ).
thf(func_def_786,type,
inf_inf_int: int > int > int ).
thf(func_def_787,type,
inf_inf_nat: nat > nat > nat ).
thf(func_def_788,type,
inf_inf_Product_unit: product_unit > product_unit > product_unit ).
thf(func_def_789,type,
inf_in1906310914598751387_nat_o: set_Pr4532377907799695533_nat_o > set_Pr4532377907799695533_nat_o > set_Pr4532377907799695533_nat_o ).
thf(func_def_790,type,
inf_inf_set_o: set_o > set_o > set_o ).
thf(func_def_791,type,
inf_in1364745209274528805nteger: set_Code_integer > set_Code_integer > set_Code_integer ).
thf(func_def_792,type,
inf_in7395780696554941096er_nat: set_filter_nat > set_filter_nat > set_filter_nat ).
thf(func_def_793,type,
inf_inf_set_int: set_int > set_int > set_int ).
thf(func_def_794,type,
inf_inf_set_nat: set_nat > set_nat > set_nat ).
thf(func_def_795,type,
inf_in2269163501485487111nt_int: set_Pr958786334691620121nt_int > set_Pr958786334691620121nt_int > set_Pr958786334691620121nt_int ).
thf(func_def_796,type,
inf_in2572325071724192079at_nat: set_Pr1261947904930325089at_nat > set_Pr1261947904930325089at_nat > set_Pr1261947904930325089at_nat ).
thf(func_def_797,type,
inf_in4660618365625256667t_unit: set_Product_unit > set_Product_unit > set_Product_unit ).
thf(func_def_798,type,
inf_inf_set_set_nat: set_set_nat > set_set_nat > set_set_nat ).
thf(func_def_799,type,
sup_su1942872429929737346at_o_o: ( ( produc3658429121746597890et_nat > $o ) > ( produc3658429121746597890et_nat > $o ) > $o ) > ( ( produc3658429121746597890et_nat > $o ) > ( produc3658429121746597890et_nat > $o ) > $o ) > ( produc3658429121746597890et_nat > $o ) > ( produc3658429121746597890et_nat > $o ) > $o ).
thf(func_def_800,type,
sup_sup_o_o_o: ( $o > $o > $o ) > ( $o > $o > $o ) > $o > $o > $o ).
thf(func_def_801,type,
sup_sup_int_int_o: ( int > int > $o ) > ( int > int > $o ) > int > int > $o ).
thf(func_def_802,type,
sup_sup_nat_nat_o: ( nat > nat > $o ) > ( nat > nat > $o ) > nat > nat > $o ).
thf(func_def_803,type,
sup_su5453871518329203617_nat_o: ( produc3658429121746597890et_nat > $o ) > ( produc3658429121746597890et_nat > $o ) > produc3658429121746597890et_nat > $o ).
thf(func_def_804,type,
sup_su3411595859437804162unit_o: ( product_unit > product_unit > $o ) > ( product_unit > product_unit > $o ) > product_unit > product_unit > $o ).
thf(func_def_805,type,
sup_sup_o: $o > $o > $o ).
thf(func_def_806,type,
sup_sup_assn: assn > assn > assn ).
thf(func_def_807,type,
sup_sup_filter_nat: filter_nat > filter_nat > filter_nat ).
thf(func_def_808,type,
sup_sup_int: int > int > int ).
thf(func_def_809,type,
sup_sup_nat: nat > nat > nat ).
thf(func_def_810,type,
sup_sup_Product_unit: product_unit > product_unit > product_unit ).
thf(func_def_811,type,
sup_su5209123915105501825_nat_o: set_Pr4532377907799695533_nat_o > set_Pr4532377907799695533_nat_o > set_Pr4532377907799695533_nat_o ).
thf(func_def_812,type,
sup_sup_set_o: set_o > set_o > set_o ).
thf(func_def_813,type,
sup_sup_set_assn: set_assn > set_assn > set_assn ).
thf(func_def_814,type,
sup_su848401254843788991nteger: set_Code_integer > set_Code_integer > set_Code_integer ).
thf(func_def_815,type,
sup_sup_set_int: set_int > set_int > set_int ).
thf(func_def_816,type,
sup_sup_set_nat: set_nat > set_nat > set_nat ).
thf(func_def_817,type,
sup_su8328026650783933211_nat_o: set_Pr2161125870931222855_nat_o > set_Pr2161125870931222855_nat_o > set_Pr2161125870931222855_nat_o ).
thf(func_def_818,type,
sup_su5769328420594410459od_o_o: set_Product_prod_o_o > set_Product_prod_o_o > set_Product_prod_o_o ).
thf(func_def_819,type,
sup_su6024340866399070445nt_int: set_Pr958786334691620121nt_int > set_Pr958786334691620121nt_int > set_Pr958786334691620121nt_int ).
thf(func_def_820,type,
sup_su6327502436637775413at_nat: set_Pr1261947904930325089at_nat > set_Pr1261947904930325089at_nat > set_Pr1261947904930325089at_nat ).
thf(func_def_821,type,
sup_su6481247232241170203t_unit: set_Pr5094982260447487303t_unit > set_Pr5094982260447487303t_unit > set_Pr5094982260447487303t_unit ).
thf(func_def_822,type,
sup_su4863159966000489243_nat_o: set_Pr3813038829465207111_nat_o > set_Pr3813038829465207111_nat_o > set_Pr3813038829465207111_nat_o ).
thf(func_def_823,type,
sup_su7766833992073560027_set_o: set_Pr4577759397028426247_set_o > set_Pr4577759397028426247_set_o > set_Pr4577759397028426247_set_o ).
thf(func_def_824,type,
sup_su1286422330746572633et_int: set_Pr2522554150109002629et_int > set_Pr2522554150109002629et_int > set_Pr2522554150109002629et_int ).
thf(func_def_825,type,
sup_su4251893418135750817et_nat: set_Pr5488025237498180813et_nat > set_Pr5488025237498180813et_nat > set_Pr5488025237498180813et_nat ).
thf(func_def_826,type,
sup_su3573767163274163995t_unit: set_Pr3980508049012409671t_unit > set_Pr3980508049012409671t_unit > set_Pr3980508049012409671t_unit ).
thf(func_def_827,type,
sup_su793286257634532545t_unit: set_Product_unit > set_Product_unit > set_Product_unit ).
thf(func_def_828,type,
sup_sup_set_set_o: set_set_o > set_set_o > set_set_o ).
thf(func_def_829,type,
sup_sup_set_set_int: set_set_int > set_set_int > set_set_int ).
thf(func_def_830,type,
sup_sup_set_set_nat: set_set_nat > set_set_nat > set_set_nat ).
thf(func_def_831,type,
sup_su5975557051335651745t_unit: set_set_Product_unit > set_set_Product_unit > set_set_Product_unit ).
thf(func_def_832,type,
lattic1921953407002678535_Max_o: set_o > $o ).
thf(func_def_833,type,
lattic4901227151466704046nteger: set_Code_integer > code_integer ).
thf(func_def_834,type,
lattic8263393255366662781ax_int: set_int > int ).
thf(func_def_835,type,
lattic8265883725875713057ax_nat: set_nat > nat ).
thf(func_def_836,type,
lattic1210392890313158908t_unit: set_Product_unit > product_unit ).
thf(func_def_837,type,
lattic7630753665789217321ax_rat: set_rat > rat ).
thf(func_def_838,type,
lattic1973801136483472281_Min_o: set_o > $o ).
thf(func_def_839,type,
lattic1063845414844153500nteger: set_Code_integer > code_integer ).
thf(func_def_840,type,
lattic8718645017227715691in_int: set_int > int ).
thf(func_def_841,type,
lattic8721135487736765967in_nat: set_nat > nat ).
thf(func_def_842,type,
lattic8554435388794137742t_unit: set_Product_unit > product_unit ).
thf(func_def_843,type,
lattic8086005427650270231in_rat: set_rat > rat ).
thf(func_def_844,type,
lattic7708394068118247271o_assn: ( $o > assn ) > set_o > $o ).
thf(func_def_845,type,
lattic2775856028456453135_o_nat: ( $o > nat ) > set_o > $o ).
thf(func_def_846,type,
lattic2140725968369957399_o_rat: ( $o > rat ) > set_o > $o ).
thf(func_def_847,type,
lattic4682378690128378784r_assn: ( code_integer > assn ) > set_Code_integer > code_integer ).
thf(func_def_848,type,
lattic6074936555552774728er_nat: ( code_integer > nat ) > set_Code_integer > code_integer ).
thf(func_def_849,type,
lattic5439806495466278992er_rat: ( code_integer > rat ) > set_Code_integer > code_integer ).
thf(func_def_850,type,
lattic3162799611691135855t_assn: ( int > assn ) > set_int > int ).
thf(func_def_851,type,
lattic8446286672483414039nt_nat: ( int > nat ) > set_int > int ).
thf(func_def_852,type,
lattic7811156612396918303nt_rat: ( int > rat ) > set_int > int ).
thf(func_def_853,type,
lattic2187264146484958483t_assn: ( nat > assn ) > set_nat > nat ).
thf(func_def_854,type,
lattic7446932960582359483at_nat: ( nat > nat ) > set_nat > nat ).
thf(func_def_855,type,
lattic6811802900495863747at_rat: ( nat > rat ) > set_nat > nat ).
thf(func_def_856,type,
lattic4376901495776413404t_assn: ( product_unit > assn ) > set_Product_unit > product_unit ).
thf(func_def_857,type,
lattic1360832392136924548it_nat: ( product_unit > nat ) > set_Product_unit > product_unit ).
thf(func_def_858,type,
lattic725702332050428812it_rat: ( product_unit > rat ) > set_Product_unit > product_unit ).
thf(func_def_859,type,
lattic956194824204696298_nat_o: set_Pr4532377907799695533_nat_o > produc3658429121746597890et_nat > $o ).
thf(func_def_860,type,
lattic4107685809792843317_fin_o: set_o > $o ).
thf(func_def_861,type,
lattic47131356835913163n_assn: set_assn > assn ).
thf(func_def_862,type,
lattic9128708996678779395er_nat: set_filter_nat > filter_nat ).
thf(func_def_863,type,
lattic5235898064620869839in_int: set_int > int ).
thf(func_def_864,type,
lattic5238388535129920115in_nat: set_nat > nat ).
thf(func_def_865,type,
lattic1263872656861969706t_unit: set_Product_unit > product_unit ).
thf(func_def_866,type,
lattic4603258475043424379in_rat: set_rat > rat ).
thf(func_def_867,type,
lattic3014633134055518761et_nat: set_set_nat > set_nat ).
thf(func_def_868,type,
lattic7320199455484906628_nat_o: set_Pr4532377907799695533_nat_o > produc3658429121746597890et_nat > $o ).
thf(func_def_869,type,
lattic1508158080041050831_fin_o: set_o > $o ).
thf(func_def_870,type,
lattic2150320897289308081n_assn: set_assn > assn ).
thf(func_def_871,type,
lattic5930898082463196905er_nat: set_filter_nat > filter_nat ).
thf(func_def_872,type,
lattic1091506334969745077in_int: set_int > int ).
thf(func_def_873,type,
lattic1093996805478795353in_nat: set_nat > nat ).
thf(func_def_874,type,
lattic5294303975357428420t_unit: set_Product_unit > product_unit ).
thf(func_def_875,type,
lattic458866745392299617in_rat: set_rat > rat ).
thf(func_def_876,type,
lattic3835124923745554447et_nat: set_set_nat > set_nat ).
thf(func_def_877,type,
append_nat: list_nat > list_nat > list_nat ).
thf(func_def_878,type,
butlast_nat: list_nat > list_nat ).
thf(func_def_879,type,
filter_nat2: ( nat > $o ) > list_nat > list_nat ).
thf(func_def_880,type,
cons_nat: nat > list_nat > list_nat ).
thf(func_def_881,type,
nil_nat: list_nat ).
thf(func_def_882,type,
map_nat_nat: ( nat > nat ) > list_nat > list_nat ).
thf(func_def_883,type,
upt: nat > nat > list_nat ).
thf(func_def_884,type,
biject1468766312547416318et_nat: set_Pr8536935166611901872et_nat > $o ).
thf(func_def_885,type,
biject2615096655818420098et_nat: set_Pr3286484037609594932et_nat > $o ).
thf(func_def_886,type,
biject2219996528949639174et_nat: set_Pr3948176798113811640et_nat > $o ).
thf(func_def_887,type,
bijective_int_int: set_Pr958786334691620121nt_int > $o ).
thf(func_def_888,type,
bijective_nat_nat: set_Pr1261947904930325089at_nat > $o ).
thf(func_def_889,type,
fun_of384177504262425200et_nat: set_Pr8536935166611901872et_nat > ( produc3658429121746597890et_nat > $o ) > produc3925858234332021118et_nat ).
thf(func_def_890,type,
fun_of3673525222065131764et_nat: set_Pr3286484037609594932et_nat > ( produc3658429121746597890et_nat > $o ) > produc3658429121746597890et_nat ).
thf(func_def_891,type,
fun_of5878246503291251064et_nat: set_Pr3948176798113811640et_nat > heap_e7401611519738050253t_unit > set_nat ).
thf(func_def_892,type,
fun_of_rel_int_int: set_Pr958786334691620121nt_int > int > int ).
thf(func_def_893,type,
fun_of_rel_nat_nat: set_Pr1261947904930325089at_nat > nat > nat ).
thf(func_def_894,type,
inv_on_o_int: ( $o > int ) > set_o > int > $o ).
thf(func_def_895,type,
inv_on_o_nat: ( $o > nat ) > set_o > nat > $o ).
thf(func_def_896,type,
inv_on4869000451854154612nteger: ( code_integer > code_integer ) > set_Code_integer > code_integer > code_integer ).
thf(func_def_897,type,
inv_on_int_int: ( int > int ) > set_int > int > int ).
thf(func_def_898,type,
inv_on_int_nat: ( int > nat ) > set_int > nat > int ).
thf(func_def_899,type,
inv_on1369624593536199399nteger: ( nat > code_integer ) > set_nat > code_integer > nat ).
thf(func_def_900,type,
inv_on_nat_int: ( nat > int ) > set_nat > int > nat ).
thf(func_def_901,type,
inv_on_nat_nat: ( nat > nat ) > set_nat > nat > nat ).
thf(func_def_902,type,
inv_on_nat_set_nat: ( nat > set_nat ) > set_nat > set_nat > nat ).
thf(func_def_903,type,
inv_on7725125254116338689it_int: ( product_unit > int ) > set_Product_unit > int > product_unit ).
thf(func_def_904,type,
inv_on7727615724625388965it_nat: ( product_unit > nat ) > set_Product_unit > nat > product_unit ).
thf(func_def_905,type,
pairself_int_int: ( int > int ) > product_prod_int_int > product_prod_int_int ).
thf(func_def_906,type,
pairself_int_nat: ( int > nat ) > product_prod_int_int > product_prod_nat_nat ).
thf(func_def_907,type,
pairself_nat_int: ( nat > int ) > product_prod_nat_nat > product_prod_int_int ).
thf(func_def_908,type,
pairself_nat_nat: ( nat > nat ) > product_prod_nat_nat > product_prod_nat_nat ).
thf(func_def_909,type,
pairself_rel_int_int: produc5030937667867769463nt_int > produc5030937667867769463nt_int > $o ).
thf(func_def_910,type,
pairself_rel_int_nat: produc6128020648511768019nt_int > produc6128020648511768019nt_int > $o ).
thf(func_def_911,type,
pairself_rel_nat_int: produc3574062574353363227at_nat > produc3574062574353363227at_nat > $o ).
thf(func_def_912,type,
pairself_rel_nat_nat: produc4671145554997361783at_nat > produc4671145554997361783at_nat > $o ).
thf(func_def_913,type,
rel_re3331361160397388719_nat_o: set_Pr2161125870931222855_nat_o > set_Pr4532377907799695533_nat_o > set_Pr2161125870931222855_nat_o ).
thf(func_def_914,type,
rel_restrict_o: set_Product_prod_o_o > set_o > set_Product_prod_o_o ).
thf(func_def_915,type,
rel_restrict_int: set_Pr958786334691620121nt_int > set_int > set_Pr958786334691620121nt_int ).
thf(func_def_916,type,
rel_restrict_nat: set_Pr1261947904930325089at_nat > set_nat > set_Pr1261947904930325089at_nat ).
thf(func_def_917,type,
rel_re9207996470947987439t_unit: set_Pr5094982260447487303t_unit > set_Product_unit > set_Pr5094982260447487303t_unit ).
thf(func_def_918,type,
suc: nat > nat ).
thf(func_def_919,type,
semiri4939895301339042750nteger: nat > code_integer ).
thf(func_def_920,type,
semiri3763490453095760265atural: nat > code_natural ).
thf(func_def_921,type,
semiri1314217659103216013at_int: nat > int ).
thf(func_def_922,type,
semiri1316708129612266289at_nat: nat > nat ).
thf(func_def_923,type,
semiri681578069525770553at_rat: nat > rat ).
thf(func_def_924,type,
nat_set_decode: nat > set_nat ).
thf(func_def_925,type,
nat_set_encode: set_nat > nat ).
thf(func_def_926,type,
nat_sum_encode: sum_sum_nat_nat > nat ).
thf(func_def_927,type,
nat_triangle: nat > nat ).
thf(func_def_928,type,
bitM: num > num ).
thf(func_def_929,type,
inc: num > num ).
thf(func_def_930,type,
nat_of_num: num > nat ).
thf(func_def_931,type,
neg_nu7757733837767384882nteger: code_integer > code_integer ).
thf(func_def_932,type,
neg_nu3811975205180677377ec_int: int > int ).
thf(func_def_933,type,
neg_nu3179335615603231917ec_rat: rat > rat ).
thf(func_def_934,type,
neg_nu5831290666863070958nteger: code_integer > code_integer ).
thf(func_def_935,type,
neg_nu5851722552734809277nc_int: int > int ).
thf(func_def_936,type,
neg_nu5219082963157363817nc_rat: rat > rat ).
thf(func_def_937,type,
neg_numeral_sub_int: num > num > int ).
thf(func_def_938,type,
bit0: num > num ).
thf(func_def_939,type,
bit1: num > num ).
thf(func_def_940,type,
one: num ).
thf(func_def_941,type,
num_of_nat: nat > num ).
thf(func_def_942,type,
numera6620942414471956472nteger: num > code_integer ).
thf(func_def_943,type,
numera5444537566228673987atural: num > code_natural ).
thf(func_def_944,type,
numeral_numeral_int: num > int ).
thf(func_def_945,type,
numeral_numeral_nat: num > nat ).
thf(func_def_946,type,
pow: num > num > num ).
thf(func_def_947,type,
pred_numeral: num > nat ).
thf(func_def_948,type,
sqr: num > num ).
thf(func_def_949,type,
order_1145710701291701250_nat_o: set_Pr4532377907799695533_nat_o > set_Pr2161125870931222855_nat_o > $o ).
thf(func_def_950,type,
order_6238756238976269133r_on_o: set_o > set_Product_prod_o_o > $o ).
thf(func_def_951,type,
order_4471489696718655927on_int: set_int > set_Pr958786334691620121nt_int > $o ).
thf(func_def_952,type,
order_4473980167227706203on_nat: set_nat > set_Pr1261947904930325089at_nat > $o ).
thf(func_def_953,type,
order_888282469287196226t_unit: set_Product_unit > set_Pr5094982260447487303t_unit > $o ).
thf(func_def_954,type,
order_3690095022144090562_nat_o: set_Pr4532377907799695533_nat_o > set_Pr2161125870931222855_nat_o > $o ).
thf(func_def_955,type,
order_8538205470752950541r_on_o: set_o > set_Product_prod_o_o > $o ).
thf(func_def_956,type,
order_5248785102713058295on_int: set_int > set_Pr958786334691620121nt_int > $o ).
thf(func_def_957,type,
order_5251275573222108571on_nat: set_nat > set_Pr1261947904930325089at_nat > $o ).
thf(func_def_958,type,
order_2505591618835473410t_unit: set_Product_unit > set_Pr5094982260447487303t_unit > $o ).
thf(func_def_959,type,
order_174704566861754596_nat_o: set_Pr4532377907799695533_nat_o > set_Pr2161125870931222855_nat_o > $o ).
thf(func_def_960,type,
order_preorder_on_o: set_o > set_Product_prod_o_o > $o ).
thf(func_def_961,type,
order_4859164337913492053on_int: set_int > set_Pr958786334691620121nt_int > $o ).
thf(func_def_962,type,
order_4861654808422542329on_nat: set_nat > set_Pr1261947904930325089at_nat > $o ).
thf(func_def_963,type,
order_7534304569044981540t_unit: set_Product_unit > set_Pr5094982260447487303t_unit > $o ).
thf(func_def_964,type,
order_7262550146012401043_nat_o: set_Pr2161125870931222855_nat_o > ( produc3658429121746597890et_nat > $o ) > set_Pr4532377907799695533_nat_o ).
thf(func_def_965,type,
order_underS_o: set_Product_prod_o_o > $o > set_o ).
thf(func_def_966,type,
order_underS_int: set_Pr958786334691620121nt_int > int > set_int ).
thf(func_def_967,type,
order_underS_nat: set_Pr1261947904930325089at_nat > nat > set_nat ).
thf(func_def_968,type,
order_5809128134054296019t_unit: set_Pr5094982260447487303t_unit > product_unit > set_Product_unit ).
thf(func_def_969,type,
order_2885205318754725938_nat_o: set_Pr2161125870931222855_nat_o > ( produc3658429121746597890et_nat > $o ) > set_Pr4532377907799695533_nat_o ).
thf(func_def_970,type,
order_under_o: set_Product_prod_o_o > $o > set_o ).
thf(func_def_971,type,
order_under_int: set_Pr958786334691620121nt_int > int > set_int ).
thf(func_def_972,type,
order_under_nat: set_Pr1261947904930325089at_nat > nat > set_nat ).
thf(func_def_973,type,
order_6949322460238004850t_unit: set_Pr5094982260447487303t_unit > product_unit > set_Product_unit ).
thf(func_def_974,type,
order_2118566763991480939_nat_o: set_Pr4532377907799695533_nat_o > set_Pr2161125870931222855_nat_o > $o ).
thf(func_def_975,type,
order_2615775362358406454r_on_o: set_o > set_Product_prod_o_o > $o ).
thf(func_def_976,type,
order_2886507596567047182on_int: set_int > set_Pr958786334691620121nt_int > $o ).
thf(func_def_977,type,
order_2888998067076097458on_nat: set_nat > set_Pr1261947904930325089at_nat > $o ).
thf(func_def_978,type,
order_3164716736300062379t_unit: set_Product_unit > set_Pr5094982260447487303t_unit > $o ).
thf(func_def_979,type,
bot_bo7963750851167320836at_o_o: ( produc3658429121746597890et_nat > $o ) > $o ).
thf(func_def_980,type,
bot_bot_o_o: $o > $o ).
thf(func_def_981,type,
bot_bot_int_o: int > $o ).
thf(func_def_982,type,
bot_bot_nat_o: nat > $o ).
thf(func_def_983,type,
bot_bo6993851861578111033_nat_o: produc3658429121746597890et_nat > $o ).
thf(func_def_984,type,
bot_bo4642748612307482820unit_o: product_unit > $o ).
thf(func_def_986,type,
bot_bot_assn: assn ).
thf(func_def_987,type,
bot_bot_filter_nat: filter_nat ).
thf(func_def_988,type,
bot_bot_nat: nat ).
thf(func_def_989,type,
bot_bot_Product_unit: product_unit ).
thf(func_def_990,type,
bot_bo7824918357723582233_nat_o: set_Pr4532377907799695533_nat_o ).
thf(func_def_991,type,
bot_bot_set_o: set_o ).
thf(func_def_992,type,
bot_bot_set_assn: set_assn ).
thf(func_def_993,type,
bot_bo3990330152332043303nteger: set_Code_integer ).
thf(func_def_994,type,
bot_bo498966703094740906er_nat: set_filter_nat ).
thf(func_def_995,type,
bot_bo3686245532977126937t_unit: set_he8644174446132254637t_unit ).
thf(func_def_996,type,
bot_bot_set_int: set_int ).
thf(func_def_997,type,
bot_bot_set_nat: set_nat ).
thf(func_def_998,type,
bot_bo3657187498359500723_nat_o: set_Pr2161125870931222855_nat_o ).
thf(func_def_999,type,
bot_bo1481135142794719944et_nat: set_Pr3286484037609594932et_nat ).
thf(func_def_1000,type,
bot_bo7073875226086086771od_o_o: set_Product_prod_o_o ).
thf(func_def_1001,type,
bot_bo3771712512152092492et_nat: set_Pr3948176798113811640et_nat ).
thf(func_def_1002,type,
bot_bo1796632182523588997nt_int: set_Pr958786334691620121nt_int ).
thf(func_def_1003,type,
bot_bo2099793752762293965at_nat: set_Pr1261947904930325089at_nat ).
thf(func_def_1004,type,
bot_bo1654022762305692595t_unit: set_Pr5094982260447487303t_unit ).
thf(func_def_1005,type,
bot_bo7535334947429593011_nat_o: set_Pr3813038829465207111_nat_o ).
thf(func_def_1006,type,
bot_bo8143257439550059635_set_o: set_Pr4577759397028426247_set_o ).
thf(func_def_1007,type,
bot_bo1987282549146081265et_int: set_Pr2522554150109002629et_int ).
thf(func_def_1008,type,
bot_bo4952753636535259449et_nat: set_Pr5488025237498180813et_nat ).
thf(func_def_1009,type,
bot_bo5113747506523071411t_unit: set_Pr3980508049012409671t_unit ).
thf(func_def_1010,type,
bot_bo3957492148770167129t_unit: set_Product_unit ).
thf(func_def_1011,type,
bot_bot_set_rat: set_rat ).
thf(func_def_1012,type,
bot_bo3736544561982865913_nat_o: set_se1353816476878259085_nat_o ).
thf(func_def_1013,type,
bot_bot_set_set_o: set_set_o ).
thf(func_def_1014,type,
bot_bo9204850693367297629nteger: set_set_Code_integer ).
thf(func_def_1015,type,
bot_bot_set_set_int: set_set_int ).
thf(func_def_1016,type,
bot_bot_set_set_nat: set_set_nat ).
thf(func_def_1017,type,
bot_bo6284674557190688313t_unit: set_set_Product_unit ).
thf(func_def_1018,type,
ord_le8176553561544679545_nat_o: ( produc3658429121746597890et_nat > $o ) > ( produc3658429121746597890et_nat > $o ) > $o ).
thf(func_def_1019,type,
ord_less_o: $o > $o > $o ).
thf(func_def_1020,type,
ord_less_assn: assn > assn > $o ).
thf(func_def_1021,type,
ord_le6747313008572928689nteger: code_integer > code_integer > $o ).
thf(func_def_1022,type,
ord_le5570908160329646204atural: code_natural > code_natural > $o ).
thf(func_def_1023,type,
ord_less_filter_nat: filter_nat > filter_nat > $o ).
thf(func_def_1024,type,
ord_less_int: int > int > $o ).
thf(func_def_1025,type,
ord_less_nat: nat > nat > $o ).
thf(func_def_1026,type,
ord_le361264281704409273t_unit: product_unit > product_unit > $o ).
thf(func_def_1027,type,
ord_less_rat: rat > rat > $o ).
thf(func_def_1028,type,
ord_le2453136405763929_nat_o: set_Pr4532377907799695533_nat_o > set_Pr4532377907799695533_nat_o > $o ).
thf(func_def_1029,type,
ord_less_set_o: set_o > set_o > $o ).
thf(func_def_1030,type,
ord_le1307284697595431911nteger: set_Code_integer > set_Code_integer > $o ).
thf(func_def_1031,type,
ord_less_set_int: set_int > set_int > $o ).
thf(func_def_1032,type,
ord_less_set_nat: set_nat > set_nat > $o ).
thf(func_def_1033,type,
ord_le8056459307392131481t_unit: set_Product_unit > set_Product_unit > $o ).
thf(func_def_1034,type,
ord_le729326519192465773_nat_o: ( produc3658429121746597890et_nat > $o ) > ( produc3658429121746597890et_nat > $o ) > $o ).
thf(func_def_1035,type,
ord_less_eq_o: $o > $o > $o ).
thf(func_def_1036,type,
ord_less_eq_assn: assn > assn > $o ).
thf(func_def_1037,type,
ord_le3102999989581377725nteger: code_integer > code_integer > $o ).
thf(func_def_1038,type,
ord_le1926595141338095240atural: code_natural > code_natural > $o ).
thf(func_def_1039,type,
ord_le2510731241096832064er_nat: filter_nat > filter_nat > $o ).
thf(func_def_1040,type,
ord_less_eq_int: int > int > $o ).
thf(func_def_1041,type,
ord_less_eq_nat: nat > nat > $o ).
thf(func_def_1042,type,
ord_le3221252021190050221t_unit: product_unit > product_unit > $o ).
thf(func_def_1043,type,
ord_less_eq_rat: rat > rat > $o ).
thf(func_def_1044,type,
ord_le2965882846123202637_nat_o: set_Pr4532377907799695533_nat_o > set_Pr4532377907799695533_nat_o > $o ).
thf(func_def_1045,type,
ord_less_eq_set_o: set_o > set_o > $o ).
thf(func_def_1046,type,
ord_less_eq_set_assn: set_assn > set_assn > $o ).
thf(func_def_1047,type,
ord_le7084787975880047091nteger: set_Code_integer > set_Code_integer > $o ).
thf(func_def_1048,type,
ord_le2426478655948331894er_nat: set_filter_nat > set_filter_nat > $o ).
thf(func_def_1049,type,
ord_le4903435234951335757t_unit: set_he8644174446132254637t_unit > set_he8644174446132254637t_unit > $o ).
thf(func_def_1050,type,
ord_less_eq_set_int: set_int > set_int > $o ).
thf(func_def_1051,type,
ord_less_eq_set_nat: set_nat > set_nat > $o ).
thf(func_def_1052,type,
ord_le6659818918159358183_nat_o: set_Pr2161125870931222855_nat_o > set_Pr2161125870931222855_nat_o > $o ).
thf(func_def_1053,type,
ord_le4460016228128491106_o_nat: set_Pr4769669219625338370_o_nat > set_Pr4769669219625338370_o_nat > $o ).
thf(func_def_1054,type,
ord_le4763372923235995152et_nat: set_Pr8536935166611901872et_nat > set_Pr8536935166611901872et_nat > $o ).
thf(func_def_1055,type,
ord_le5966269811547037844et_nat: set_Pr3286484037609594932et_nat > set_Pr3286484037609594932et_nat > $o ).
thf(func_def_1056,type,
ord_le192195778054999463od_o_o: set_Product_prod_o_o > set_Product_prod_o_o > $o ).
thf(func_def_1057,type,
ord_le6472378536984071597_o_nat: set_Pr2101469702781467981_o_nat > set_Pr2101469702781467981_o_nat > $o ).
thf(func_def_1058,type,
ord_le8090042543978450200et_nat: set_Pr3948176798113811640et_nat > set_Pr3948176798113811640et_nat > $o ).
thf(func_def_1059,type,
ord_le2843351958646193337nt_int: set_Pr958786334691620121nt_int > set_Pr958786334691620121nt_int > $o ).
thf(func_def_1060,type,
ord_le5333435103577920093nt_nat: set_Pr3448869479623346877nt_nat > set_Pr3448869479623346877nt_nat > $o ).
thf(func_def_1061,type,
ord_le3146513528884898305at_nat: set_Pr1261947904930325089at_nat > set_Pr1261947904930325089at_nat > $o ).
thf(func_def_1062,type,
ord_le8121013740822022818it_nat: set_Pr1763845938948868674it_nat > set_Pr1763845938948868674it_nat > $o ).
thf(func_def_1063,type,
ord_le930853871177014503t_unit: set_Pr5094982260447487303t_unit > set_Pr5094982260447487303t_unit > $o ).
thf(func_def_1064,type,
ord_le8001614231985977575_nat_o: set_Pr3813038829465207111_nat_o > set_Pr3813038829465207111_nat_o > $o ).
thf(func_def_1065,type,
ord_le2885120122233735591_set_o: set_Pr4577759397028426247_set_o > set_Pr4577759397028426247_set_o > $o ).
thf(func_def_1066,type,
ord_le1436784781161120549et_int: set_Pr2522554150109002629et_int > set_Pr2522554150109002629et_int > $o ).
thf(func_def_1067,type,
ord_le4402255868550298733et_nat: set_Pr5488025237498180813et_nat > set_Pr5488025237498180813et_nat > $o ).
thf(func_def_1068,type,
ord_le8072594200992201959t_unit: set_Pr3980508049012409671t_unit > set_Pr3980508049012409671t_unit > $o ).
thf(func_def_1069,type,
ord_le3507040750410214029t_unit: set_Product_unit > set_Product_unit > $o ).
thf(func_def_1070,type,
ord_less_eq_set_rat: set_rat > set_rat > $o ).
thf(func_def_1071,type,
ord_le1914454125413604393nteger: set_set_Code_integer > set_set_Code_integer > $o ).
thf(func_def_1072,type,
ord_le6893508408891458716et_nat: set_set_nat > set_set_nat > $o ).
thf(func_def_1073,type,
ord_le9131159989063066194et_nat: set_set_set_nat > set_set_set_nat > $o ).
thf(func_def_1074,type,
ord_max_o: $o > $o > $o ).
thf(func_def_1075,type,
ord_max_assn: assn > assn > assn ).
thf(func_def_1076,type,
ord_max_Code_integer: code_integer > code_integer > code_integer ).
thf(func_def_1077,type,
ord_max_Code_natural: code_natural > code_natural > code_natural ).
thf(func_def_1078,type,
ord_max_filter_nat: filter_nat > filter_nat > filter_nat ).
thf(func_def_1079,type,
ord_max_int: int > int > int ).
thf(func_def_1080,type,
ord_max_nat: nat > nat > nat ).
thf(func_def_1081,type,
ord_max_Product_unit: product_unit > product_unit > product_unit ).
thf(func_def_1082,type,
ord_max_rat: rat > rat > rat ).
thf(func_def_1083,type,
ord_ma8206644353667601538_nat_o: set_Pr4532377907799695533_nat_o > set_Pr4532377907799695533_nat_o > set_Pr4532377907799695533_nat_o ).
thf(func_def_1084,type,
ord_max_set_o: set_o > set_o > set_o ).
thf(func_def_1085,type,
ord_max_set_int: set_int > set_int > set_int ).
thf(func_def_1086,type,
ord_max_set_nat: set_nat > set_nat > set_nat ).
thf(func_def_1087,type,
ord_ma508538259134630082t_unit: set_Product_unit > set_Product_unit > set_Product_unit ).
thf(func_def_1088,type,
ord_min_nat: nat > nat > nat ).
thf(func_def_1089,type,
order_mono_nat_nat: ( nat > nat ) > $o ).
thf(func_def_1090,type,
ordering_top_assn: ( assn > assn > $o ) > ( assn > assn > $o ) > assn > $o ).
thf(func_def_1091,type,
orderi2807725203697798511er_nat: ( filter_nat > filter_nat > $o ) > ( filter_nat > filter_nat > $o ) > filter_nat > $o ).
thf(func_def_1092,type,
ordering_top_set_o: ( set_o > set_o > $o ) > ( set_o > set_o > $o ) > set_o > $o ).
thf(func_def_1093,type,
ordering_top_set_int: ( set_int > set_int > $o ) > ( set_int > set_int > $o ) > set_int > $o ).
thf(func_def_1094,type,
ordering_top_set_nat: ( set_nat > set_nat > $o ) > ( set_nat > set_nat > $o ) > set_nat > $o ).
thf(func_def_1095,type,
orderi2935705153647960990t_unit: ( set_Product_unit > set_Product_unit > $o ) > ( set_Product_unit > set_Product_unit > $o ) > set_Product_unit > $o ).
thf(func_def_1096,type,
top_to2462227307520033056at_o_o: ( produc3658429121746597890et_nat > $o ) > $o ).
thf(func_def_1097,type,
top_top_o_o: $o > $o ).
thf(func_def_1098,type,
top_top_int_o: int > $o ).
thf(func_def_1099,type,
top_top_nat_o: nat > $o ).
thf(func_def_1100,type,
top_to5213382387105117981_nat_o: produc3658429121746597890et_nat > $o ).
thf(func_def_1101,type,
top_to2465898995584390880unit_o: product_unit > $o ).
thf(func_def_1103,type,
top_top_assn: assn ).
thf(func_def_1104,type,
top_top_filter_nat: filter_nat ).
thf(func_def_1105,type,
top_top_Product_unit: product_unit ).
thf(func_def_1106,type,
top_to3804862627464599871nteger: set_Co726186806682698383nteger ).
thf(func_def_1107,type,
top_to4869887016220417533_nat_o: set_Pr4532377907799695533_nat_o ).
thf(func_def_1108,type,
top_top_set_o: set_o ).
thf(func_def_1109,type,
top_top_set_assn: set_assn ).
thf(func_def_1110,type,
top_to4645266643341252675nteger: set_Code_integer ).
thf(func_def_1111,type,
top_to9103716463852441742atural: set_Code_natural ).
thf(func_def_1112,type,
top_top_set_int: set_int ).
thf(func_def_1113,type,
top_top_set_nat: set_nat ).
thf(func_def_1114,type,
top_to2658285827425157974eger_o: set_Pr448751882837621926eger_o ).
thf(func_def_1115,type,
top_to2363841527343591755nteger: set_Pr4811707699266497531nteger ).
thf(func_def_1116,type,
top_to6270328743839206426er_int: set_Pr6496905503482889930er_int ).
thf(func_def_1117,type,
top_to8760411888770933182er_nat: set_Pr8986988648414616686er_nat ).
thf(func_def_1118,type,
top_to3244471725241573579t_unit: set_Pr4580871404036792859t_unit ).
thf(func_def_1119,type,
top_to2524957550908328808et_nat: set_Pr3948176798113811640et_nat ).
thf(func_def_1120,type,
top_to4366644338036079209nt_int: set_Pr958786334691620121nt_int ).
thf(func_def_1121,type,
top_to1996260823553986621t_unit: set_Product_unit ).
thf(func_def_1122,type,
top_top_set_rat: set_rat ).
thf(func_def_1123,type,
top_to8515903933492314333_nat_o: set_se1353816476878259085_nat_o ).
thf(func_def_1124,type,
top_top_set_set_o: set_set_o ).
thf(func_def_1125,type,
top_top_set_set_int: set_set_int ).
thf(func_def_1126,type,
top_top_set_set_nat: set_set_nat ).
thf(func_def_1127,type,
top_to1767297665138865437t_unit: set_set_Product_unit ).
thf(func_def_1128,type,
power_power_assn: assn > nat > assn ).
thf(func_def_1129,type,
power_8256067586552552935nteger: code_integer > nat > code_integer ).
thf(func_def_1130,type,
power_7079662738309270450atural: code_natural > nat > code_natural ).
thf(func_def_1131,type,
power_power_int: int > nat > int ).
thf(func_def_1132,type,
power_power_nat: nat > nat > nat ).
thf(func_def_1133,type,
power_power_rat: rat > nat > rat ).
thf(func_def_1134,type,
produc1721739103881910569nt_int: ( int > int ) > product_prod_int_int > produc5030937667867769463nt_int ).
thf(func_def_1135,type,
produc2818822084525909125nt_int: ( int > nat ) > product_prod_int_int > produc6128020648511768019nt_int ).
thf(func_def_1136,type,
produc3140199747736941517at_nat: ( nat > int ) > product_prod_nat_nat > produc3574062574353363227at_nat ).
thf(func_def_1137,type,
produc4237282728380940073at_nat: ( nat > nat ) > product_prod_nat_nat > produc4671145554997361783at_nat ).
thf(func_def_1138,type,
produc7368190662567826135_nat_o: ( produc3658429121746597890et_nat > $o ) > ( produc3658429121746597890et_nat > $o ) > produc4928098042776334183_nat_o ).
thf(func_def_1139,type,
produc2245416461498447860et_nat: ( produc3658429121746597890et_nat > $o ) > produc3925858234332021118et_nat > produc2732055786443039994et_nat ).
thf(func_def_1140,type,
produc5001842942810119800et_nat: ( produc3658429121746597890et_nat > $o ) > produc3658429121746597890et_nat > produc3925858234332021118et_nat ).
thf(func_def_1141,type,
product_Pair_o_o: $o > $o > product_prod_o_o ).
thf(func_def_1142,type,
product_Pair_o_int: $o > int > product_prod_o_int ).
thf(func_def_1143,type,
product_Pair_o_nat: $o > nat > product_prod_o_nat ).
thf(func_def_1144,type,
produc3762353314782720579et_nat: $o > produc3658429121746597890et_nat > produc6197004810343482825et_nat ).
thf(func_def_1145,type,
produc3901946521647900940t_unit: $o > product_unit > produc3788275437394581276t_unit ).
thf(func_def_1146,type,
produc1086072967326762835nteger: code_integer > code_integer > produc8923325533196201883nteger ).
thf(func_def_1147,type,
produc3574140220909816553atural: code_natural > code_natural > produc7822875418678951345atural ).
thf(func_def_1148,type,
produc6639722614265839536atural: code_natural > produc7822875418678951345atural > produc5835291356934675326atural ).
thf(func_def_1149,type,
produc8256076736985659095t_unit: heap_e7401611519738050253t_unit > heap_e7401611519738050253t_unit > produc8126358835630522215t_unit ).
thf(func_def_1150,type,
produc7507926704131184380et_nat: heap_e7401611519738050253t_unit > set_nat > produc3658429121746597890et_nat ).
thf(func_def_1151,type,
product_Pair_int_o: int > $o > product_prod_int_o ).
thf(func_def_1152,type,
product_Pair_int_int: int > int > product_prod_int_int ).
thf(func_def_1153,type,
product_Pair_int_nat: int > nat > product_prod_int_nat ).
thf(func_def_1154,type,
produc550872421827702152t_unit: int > product_unit > produc6865911496949736918t_unit ).
thf(func_def_1155,type,
product_Pair_nat_o: nat > $o > product_prod_nat_o ).
thf(func_def_1156,type,
product_Pair_nat_int: nat > int > product_prod_nat_int ).
thf(func_def_1157,type,
product_Pair_nat_nat: nat > nat > product_prod_nat_nat ).
thf(func_def_1158,type,
produc3434450474625521892t_unit: nat > product_unit > produc526117512892780850t_unit ).
thf(func_def_1159,type,
produc1940133919992309389et_nat: produc2732055786443039994et_nat > produc2732055786443039994et_nat > produc5657529347773406293et_nat ).
thf(func_def_1160,type,
produc3674526432113299062_nat_o: produc3925858234332021118et_nat > ( produc3658429121746597890et_nat > $o ) > produc4962441903510821572_nat_o ).
thf(func_def_1161,type,
produc8599840265553166229et_nat: produc3925858234332021118et_nat > produc3925858234332021118et_nat > produc6830853553727218525et_nat ).
thf(func_def_1162,type,
produc5530789758353788441et_nat: produc3925858234332021118et_nat > produc3658429121746597890et_nat > produc4658369026151987169et_nat ).
thf(func_def_1163,type,
produc2829238748666232050_nat_o: produc3658429121746597890et_nat > ( produc3658429121746597890et_nat > $o ) > produc6907119039986491712_nat_o ).
thf(func_def_1164,type,
produc3909112513931304473et_nat: produc3658429121746597890et_nat > produc3925858234332021118et_nat > produc7389280635074931169et_nat ).
thf(func_def_1165,type,
produc8199053930788261021et_nat: produc3658429121746597890et_nat > produc3658429121746597890et_nat > produc8111630337999740517et_nat ).
thf(func_def_1166,type,
produc3646306378393792727nt_int: product_prod_int_int > product_prod_int_int > produc1219242969750017639nt_int ).
thf(func_def_1167,type,
produc8921240378546987991nt_nat: product_prod_int_nat > product_prod_int_nat > produc4806409095325742951nt_nat ).
thf(func_def_1168,type,
produc886916002739626967at_int: product_prod_nat_int > product_prod_nat_int > produc6495656768158660455at_int ).
thf(func_def_1169,type,
produc6161850002892822231at_nat: product_prod_nat_nat > product_prod_nat_nat > produc859450856879609959at_nat ).
thf(func_def_1170,type,
produc8487917018914037602unit_o: product_unit > $o > produc7485711547916492274unit_o ).
thf(func_def_1171,type,
produc4800797859304479970it_int: product_unit > int > produc5646078835734834920it_int ).
thf(func_def_1172,type,
produc4803288329813530246it_nat: product_unit > nat > produc600557818389255820it_nat ).
thf(func_def_1173,type,
produc5692694829384537815t_unit: product_unit > product_unit > produc8459935480633519975t_unit ).
thf(func_def_1174,type,
produc5149316574908490455_nat_o: set_Pr4532377907799695533_nat_o > set_Pr4532377907799695533_nat_o > produc6666532470309664615_nat_o ).
thf(func_def_1175,type,
produc5838405689764958487_set_o: set_o > set_o > produc7369051934464679207_set_o ).
thf(func_def_1176,type,
produc7443773368509356479nteger: set_Code_integer > set_Code_integer > produc6491284506569428743nteger ).
thf(func_def_1177,type,
produc6363374080413544029et_int: set_int > set_int > produc2115011035271226405et_int ).
thf(func_def_1178,type,
produc2471974422597015918t_unit: set_nat > heap_e7401611519738050253t_unit > produc7348995623955488188t_unit ).
thf(func_def_1179,type,
produc4532415448927165861et_nat: set_nat > set_nat > produc7819656566062154093et_nat ).
thf(func_def_1180,type,
produc7269801343787132631t_unit: set_Product_unit > set_Product_unit > produc1357844946079266663t_unit ).
thf(func_def_1181,type,
produc3142209312014666069_o_nat: set_Pr4532377907799695533_nat_o > ( ( produc3658429121746597890et_nat > $o ) > set_nat ) > set_Pr4769669219625338370_o_nat ).
thf(func_def_1182,type,
produc7016705354093686019et_nat: set_Pr4532377907799695533_nat_o > ( ( produc3658429121746597890et_nat > $o ) > set_Pr3286484037609594932et_nat ) > set_Pr8536935166611901872et_nat ).
thf(func_def_1183,type,
produc8052021730408606599et_nat: set_Pr4532377907799695533_nat_o > ( ( produc3658429121746597890et_nat > $o ) > set_Pr3948176798113811640et_nat ) > set_Pr3286484037609594932et_nat ).
thf(func_def_1184,type,
product_Sigma_o_o: set_o > ( $o > set_o ) > set_Product_prod_o_o ).
thf(func_def_1185,type,
product_Sigma_o_int: set_o > ( $o > set_int ) > set_Pr8834758594704517033_o_int ).
thf(func_def_1186,type,
product_Sigma_o_nat: set_o > ( $o > set_nat ) > set_Pr2101469702781467981_o_nat ).
thf(func_def_1187,type,
produc411009820408249917t_unit: set_o > ( $o > set_Product_unit ) > set_Pr6671815714118170108t_unit ).
thf(func_def_1188,type,
produc5260988393495149579et_nat: set_he8644174446132254637t_unit > ( heap_e7401611519738050253t_unit > set_set_nat ) > set_Pr3948176798113811640et_nat ).
thf(func_def_1189,type,
product_Sigma_int_o: set_int > ( int > set_o ) > set_Pr903927857289325719_int_o ).
thf(func_def_1190,type,
produc1453890548195736866nt_int: set_int > ( int > set_int ) > set_Pr958786334691620121nt_int ).
thf(func_def_1191,type,
produc1456381018704787142nt_nat: set_int > ( int > set_nat ) > set_Pr3448869479623346877nt_nat ).
thf(func_def_1192,type,
produc2622961094310184599t_unit: set_int > ( int > set_Product_unit ) > set_Pr5889264611543340172t_unit ).
thf(func_def_1193,type,
produc5650067098820117939_nat_o: set_nat > ( nat > set_Pr4532377907799695533_nat_o ) > set_Pr5623950326363488168_nat_o ).
thf(func_def_1194,type,
product_Sigma_nat_o: set_nat > ( nat > set_o ) > set_Pr3149072824959771635_nat_o ).
thf(func_def_1195,type,
produc454536836294682310at_int: set_nat > ( nat > set_int ) > set_Pr7995236796853374141at_int ).
thf(func_def_1196,type,
produc457027306803732586at_nat: set_nat > ( nat > set_nat ) > set_Pr1261947904930325089at_nat ).
thf(func_def_1197,type,
produc5506539147108004339t_unit: set_nat > ( nat > set_Product_unit ) > set_Pr4334478416066269672t_unit ).
thf(func_def_1198,type,
produc4996980317674386579unit_o: set_Product_unit > ( product_unit > set_o ) > set_Pr5464989305201766866unit_o ).
thf(func_def_1199,type,
produc6875377002296012693it_nat: set_Product_unit > ( product_unit > set_nat ) > set_Pr1763845938948868674it_nat ).
thf(func_def_1200,type,
produc6121607364029222536t_unit: set_Product_unit > ( product_unit > set_Product_unit ) > set_Pr5094982260447487303t_unit ).
thf(func_def_1201,type,
product_Unity: product_unit ).
thf(func_def_1202,type,
produc1442628905706551930et_nat: ( ( produc3658429121746597890et_nat > $o ) > produc3658429121746597890et_nat > $o ) > produc2732055786443039994et_nat > produc2732055786443039994et_nat ).
thf(func_def_1203,type,
produc3198230245617193214et_nat: ( ( produc3658429121746597890et_nat > $o ) > produc3658429121746597890et_nat > $o ) > produc3925858234332021118et_nat > produc3925858234332021118et_nat ).
thf(func_def_1204,type,
produc3195571438000033749nteger: ( code_integer > code_integer ) > produc8923325533196201883nteger > produc8923325533196201883nteger ).
thf(func_def_1205,type,
produc4273547285574235514et_nat: ( heap_e7401611519738050253t_unit > produc3658429121746597890et_nat > $o ) > produc7936767299998605306et_nat > produc2732055786443039994et_nat ).
thf(func_def_1206,type,
produc6095824665904122878et_nat: ( heap_e7401611519738050253t_unit > produc3658429121746597890et_nat > $o ) > produc5697189838369619070et_nat > produc3925858234332021118et_nat ).
thf(func_def_1207,type,
produc962520821680634754et_nat: ( heap_e7401611519738050253t_unit > produc3658429121746597890et_nat > $o ) > produc3658429121746597890et_nat > produc6500295350249520386et_nat ).
thf(func_def_1208,type,
produc4172003873565521026et_nat: ( heap_e7401611519738050253t_unit > heap_e7401611519738050253t_unit ) > produc3658429121746597890et_nat > produc3658429121746597890et_nat ).
thf(func_def_1209,type,
produc7422730317258219nt_int: ( heap_e7401611519738050253t_unit > int ) > produc2475246329834808232it_int > product_prod_int_int ).
thf(func_def_1210,type,
produc3346058198459025221et_nat: ( heap_e7401611519738050253t_unit > int ) > produc3658429121746597890et_nat > produc2771989624626745363et_nat ).
thf(func_def_1211,type,
produc8233931525780029747at_nat: ( heap_e7401611519738050253t_unit > nat ) > produc6653097349344004940it_nat > product_prod_nat_nat ).
thf(func_def_1212,type,
produc2974404638222180585et_nat: ( heap_e7401611519738050253t_unit > nat ) > produc3658429121746597890et_nat > produc2400336064389900727et_nat ).
thf(func_def_1213,type,
produc1933289243015553937_o_int: ( int > produc3658429121746597890et_nat > $o ) > product_prod_int_int > produc3776562785724475560_o_int ).
thf(func_def_1214,type,
produc3602029435013415139et_nat: ( int > produc3658429121746597890et_nat > $o ) > produc1049655590333717003et_nat > produc2732055786443039994et_nat ).
thf(func_def_1215,type,
produc5259063908904861543et_nat: ( int > produc3658429121746597890et_nat > $o ) > produc8065447618272157327et_nat > produc3925858234332021118et_nat ).
thf(func_def_1216,type,
produc1158806734383251089it_int: ( int > heap_e7401611519738050253t_unit ) > product_prod_int_int > produc2475246329834808232it_int ).
thf(func_def_1217,type,
produc7086134074927288299et_nat: ( int > heap_e7401611519738050253t_unit ) > produc2771989624626745363et_nat > produc3658429121746597890et_nat ).
thf(func_def_1218,type,
produc2355237562570658114nt_int: ( int > int ) > product_prod_int_int > product_prod_int_int ).
thf(func_def_1219,type,
produc1355883850669603558at_int: ( int > nat ) > product_prod_int_int > product_prod_nat_int ).
thf(func_def_1220,type,
produc1358374321178653834at_nat: ( int > nat ) > product_prod_int_nat > product_prod_nat_nat ).
thf(func_def_1221,type,
produc986720760941809198at_nat: ( nat > nat ) > product_prod_nat_nat > product_prod_nat_nat ).
thf(func_def_1222,type,
produc9020447175693601247nteger: ( code_integer > produc8923325533196201883nteger > produc8923325533196201883nteger ) > produc8923325533196201883nteger > produc5966032636589952099nteger ).
thf(func_def_1223,type,
produc6499014454317279255nteger: ( code_integer > code_integer ) > produc8923325533196201883nteger > produc8923325533196201883nteger ).
thf(func_def_1224,type,
produc4463282112584876420nt_int: ( int > int ) > product_prod_int_int > product_prod_int_int ).
thf(func_def_1225,type,
produc3463928400683821864at_int: ( int > nat ) > product_prod_int_int > product_prod_int_nat ).
thf(func_def_1226,type,
produc3466418871192872140at_nat: ( int > nat ) > product_prod_nat_int > product_prod_nat_nat ).
thf(func_def_1227,type,
produc1783401272518604195_nat_o: ( int > produc3925858234332021118et_nat ) > produc3776562785724475560_o_int > produc2732055786443039994et_nat ).
thf(func_def_1228,type,
produc2878475440847162070at_int: ( int > produc3925858234332021118et_nat ) > product_prod_int_int > produc1049655590333717003et_nat ).
thf(func_def_1229,type,
produc3447581998352395295_nat_o: ( int > produc3658429121746597890et_nat ) > produc3776562785724475560_o_int > produc3925858234332021118et_nat ).
thf(func_def_1230,type,
produc9121118195053341018at_int: ( int > produc3658429121746597890et_nat ) > product_prod_int_int > produc8065447618272157327et_nat ).
thf(func_def_1231,type,
produc2122153897363312795t_unit: ( int > set_nat ) > produc2475246329834808232it_int > produc3658429121746597890et_nat ).
thf(func_def_1232,type,
produc1253519502038700766at_int: ( int > set_nat ) > product_prod_int_int > produc2771989624626745363et_nat ).
thf(func_def_1233,type,
produc3094765310956027504at_nat: ( nat > nat ) > product_prod_nat_nat > product_prod_nat_nat ).
thf(func_def_1234,type,
produc969530845752564945_nat_o: ( produc3925858234332021118et_nat > produc3925858234332021118et_nat ) > produc2732055786443039994et_nat > produc2732055786443039994et_nat ).
thf(func_def_1235,type,
produc1515462096303866701_nat_o: ( produc3925858234332021118et_nat > produc3658429121746597890et_nat ) > produc2732055786443039994et_nat > produc3925858234332021118et_nat ).
thf(func_def_1236,type,
produc7714581247149323085_nat_o: ( produc3658429121746597890et_nat > produc3925858234332021118et_nat ) > produc3925858234332021118et_nat > produc2732055786443039994et_nat ).
thf(func_def_1237,type,
produc8750854537940449737_nat_o: ( produc3658429121746597890et_nat > produc3658429121746597890et_nat ) > produc3925858234332021118et_nat > produc3925858234332021118et_nat ).
thf(func_def_1238,type,
produc7345612820240450715t_unit: ( set_nat > int ) > produc3658429121746597890et_nat > produc2475246329834808232it_int ).
thf(func_def_1239,type,
produc2744032961877892830nt_int: ( set_nat > int ) > produc2771989624626745363et_nat > product_prod_int_int ).
thf(func_def_1240,type,
produc8442695800884449271t_unit: ( set_nat > nat ) > produc3658429121746597890et_nat > produc6653097349344004940it_nat ).
thf(func_def_1241,type,
produc1747169720485888550at_nat: ( set_nat > nat ) > produc2400336064389900727et_nat > product_prod_nat_nat ).
thf(func_def_1242,type,
produc3009152965630989513_nat_o: ( set_nat > produc3925858234332021118et_nat ) > produc6500295350249520386et_nat > produc2732055786443039994et_nat ).
thf(func_def_1243,type,
produc7176435795764513225t_unit: ( set_nat > produc3925858234332021118et_nat ) > produc3658429121746597890et_nat > produc7936767299998605306et_nat ).
thf(func_def_1244,type,
produc1718959742329256773_nat_o: ( set_nat > produc3658429121746597890et_nat ) > produc6500295350249520386et_nat > produc3925858234332021118et_nat ).
thf(func_def_1245,type,
produc7441988914112407621t_unit: ( set_nat > produc3658429121746597890et_nat ) > produc3658429121746597890et_nat > produc5697189838369619070et_nat ).
thf(func_def_1246,type,
produc3286488329990770625t_unit: ( set_nat > set_nat ) > produc3658429121746597890et_nat > produc3658429121746597890et_nat ).
thf(func_def_1247,type,
produc5101573711933517782_nat_o: ( produc2732055786443039994et_nat > $o ) > ( produc3658429121746597890et_nat > $o ) > produc3925858234332021118et_nat > $o ).
thf(func_def_1248,type,
produc6216949301066131538_nat_o: ( produc3925858234332021118et_nat > $o ) > ( produc3658429121746597890et_nat > $o ) > produc3658429121746597890et_nat > $o ).
thf(func_def_1249,type,
produc1807123492089482958_nat_o: ( produc3658429121746597890et_nat > $o ) > heap_e7401611519738050253t_unit > set_nat > $o ).
thf(func_def_1250,type,
produc175634133007206835_int_o: ( product_prod_int_int > $o ) > int > int > $o ).
thf(func_def_1251,type,
produc1310100445399344235_nat_o: ( product_prod_nat_nat > $o ) > nat > nat > $o ).
thf(func_def_1252,type,
produc4724772261320961782nteger: ( ( code_integer > code_integer ) > produc8923325533196201883nteger > produc8923325533196201883nteger ) > ( ( code_integer > code_integer ) > produc8923325533196201883nteger > produc8923325533196201883nteger ) > produc723880050079548563nteger > produc101387210554329899nteger ).
thf(func_def_1253,type,
produc6158254626097298938nteger: ( ( code_integer > code_integer ) > produc8923325533196201883nteger > produc8923325533196201883nteger ) > ( code_integer > produc8923325533196201883nteger > produc8923325533196201883nteger ) > produc2221115339011613079nteger > produc101387210554329899nteger ).
thf(func_def_1254,type,
produc2784369760062397435at_nat: ( ( produc3658429121746597890et_nat > $o ) > nat ) > ( produc3658429121746597890et_nat > nat ) > produc3925858234332021118et_nat > product_prod_nat_nat ).
thf(func_def_1255,type,
produc7818937854452556646nteger: ( code_integer > code_integer > code_integer ) > ( code_integer > code_integer > code_integer ) > produc8923325533196201883nteger > produc723880050079548563nteger ).
thf(func_def_1256,type,
produc3789967387199524714nteger: ( code_integer > code_integer > code_integer ) > ( code_integer > code_integer ) > produc8923325533196201883nteger > produc2221115339011613079nteger ).
thf(func_def_1257,type,
produc3090791198996863994nteger: ( code_integer > produc8923325533196201883nteger > produc8923325533196201883nteger ) > ( ( code_integer > code_integer ) > produc8923325533196201883nteger > produc8923325533196201883nteger ) > produc1397831737260518551nteger > produc101387210554329899nteger ).
thf(func_def_1258,type,
produc5081366107664765182nteger: ( code_integer > produc8923325533196201883nteger > produc8923325533196201883nteger ) > ( code_integer > produc8923325533196201883nteger > produc8923325533196201883nteger ) > produc8923325533196201883nteger > produc101387210554329899nteger ).
thf(func_def_1259,type,
produc1084426483498010290er_o_o: ( code_integer > code_integer ) > ( $o > $o ) > produc6271795597528267376eger_o > produc6271795597528267376eger_o ).
thf(func_def_1260,type,
produc7547902865030867346_o_int: ( code_integer > code_integer ) > ( $o > int ) > produc6271795597528267376eger_o > produc2256096406371053034er_int ).
thf(func_def_1261,type,
produc7550393335539917622_o_nat: ( code_integer > code_integer ) > ( $o > nat ) > produc6271795597528267376eger_o > produc6433947425880249742er_nat ).
thf(func_def_1262,type,
produc3530376873132787239t_unit: ( code_integer > code_integer ) > ( $o > product_unit ) > produc6271795597528267376eger_o > produc5496629617739085541t_unit ).
thf(func_def_1263,type,
produc6023007152014423402nteger: ( code_integer > code_integer ) > ( code_integer > code_integer > code_integer ) > produc8923325533196201883nteger > produc1397831737260518551nteger ).
thf(func_def_1264,type,
produc1802105962441033582nteger: ( code_integer > code_integer ) > ( code_integer > code_integer ) > produc8923325533196201883nteger > produc8923325533196201883nteger ).
thf(func_def_1265,type,
produc1622059938646302433nteger: ( code_integer > code_integer ) > ( nat > code_integer ) > produc6433947425880249742er_nat > produc8923325533196201883nteger ).
thf(func_def_1266,type,
produc8116347370398923901unit_o: ( code_integer > code_integer ) > ( product_unit > $o ) > produc5496629617739085541t_unit > produc6271795597528267376eger_o ).
thf(func_def_1267,type,
produc4019910415967566983it_int: ( code_integer > code_integer ) > ( product_unit > int ) > produc5496629617739085541t_unit > produc2256096406371053034er_int ).
thf(func_def_1268,type,
produc4022400886476617259it_nat: ( code_integer > code_integer ) > ( product_unit > nat ) > produc5496629617739085541t_unit > produc6433947425880249742er_nat ).
thf(func_def_1269,type,
produc3346700480963112818t_unit: ( code_integer > code_integer ) > ( product_unit > product_unit ) > produc5496629617739085541t_unit > produc5496629617739085541t_unit ).
thf(func_def_1270,type,
produc8234977478398262212et_nat: ( heap_e7401611519738050253t_unit > heap_e7401611519738050253t_unit ) > ( set_nat > set_nat ) > produc3658429121746597890et_nat > produc3658429121746597890et_nat ).
thf(func_def_1271,type,
produc5973434381925408567at_int: ( heap_e7401611519738050253t_unit > int ) > ( set_nat > int ) > produc3658429121746597890et_nat > product_prod_int_int ).
thf(func_def_1272,type,
produc8285742784527409791at_nat: ( heap_e7401611519738050253t_unit > nat ) > ( set_nat > nat ) > produc3658429121746597890et_nat > product_prod_nat_nat ).
thf(func_def_1273,type,
produc3152090231653401565et_nat: ( int > heap_e7401611519738050253t_unit ) > ( int > set_nat ) > product_prod_int_int > produc3658429121746597890et_nat ).
thf(func_def_1274,type,
produc6036585564866528938nt_int: ( int > int ) > ( int > int ) > product_prod_int_int > product_prod_int_int ).
thf(func_def_1275,type,
produc5667422475138734578nt_nat: ( int > nat ) > ( int > nat ) > product_prod_int_int > product_prod_nat_nat ).
thf(func_def_1276,type,
produc79747477514316377et_nat: ( nat > produc3658429121746597890et_nat > $o ) > ( nat > produc3658429121746597890et_nat ) > product_prod_nat_nat > produc3925858234332021118et_nat ).
thf(func_def_1277,type,
produc2064153442785501et_nat: ( nat > heap_e7401611519738050253t_unit ) > ( nat > set_nat ) > product_prod_nat_nat > produc3658429121746597890et_nat ).
thf(func_def_1278,type,
produc7347049785058425330at_int: ( nat > int ) > ( nat > int ) > product_prod_nat_nat > product_prod_int_int ).
thf(func_def_1279,type,
produc6977886695330630970at_nat: ( nat > nat ) > ( nat > nat ) > product_prod_nat_nat > product_prod_nat_nat ).
thf(func_def_1280,type,
produc1553301316500091796er_int: ( code_integer > code_integer > int ) > produc8923325533196201883nteger > int ).
thf(func_def_1281,type,
produc6916734918728496179nteger: ( code_integer > code_integer > produc8923325533196201883nteger ) > produc8923325533196201883nteger > produc8923325533196201883nteger ).
thf(func_def_1282,type,
produc4947309494688390418_int_o: ( int > int > $o ) > product_prod_int_int > $o ).
thf(func_def_1283,type,
produc4245557441103728435nt_int: ( int > int > product_prod_int_int ) > product_prod_int_int > product_prod_int_int ).
thf(func_def_1284,type,
produc27273713700761075at_nat: ( nat > nat > product_prod_nat_nat > product_prod_nat_nat ) > product_prod_nat_nat > product_prod_nat_nat > product_prod_nat_nat ).
thf(func_def_1285,type,
produc2626176000494625587at_nat: ( nat > nat > product_prod_nat_nat ) > product_prod_nat_nat > product_prod_nat_nat ).
thf(func_def_1286,type,
produc180342877477747464et_nat: produc2732055786443039994et_nat > produc3658429121746597890et_nat > $o ).
thf(func_def_1287,type,
produc995936583742144908et_nat: produc3925858234332021118et_nat > produc3658429121746597890et_nat > $o ).
thf(func_def_1288,type,
produc8508995932063986495nteger: produc8923325533196201883nteger > code_integer ).
thf(func_def_1289,type,
produc8632663045010267826er_nat: produc6433947425880249742er_nat > code_integer ).
thf(func_def_1290,type,
produc1824681642469235216et_nat: produc3658429121746597890et_nat > heap_e7401611519738050253t_unit ).
thf(func_def_1291,type,
product_fst_int_int: product_prod_int_int > int ).
thf(func_def_1292,type,
produc8897218493789310642nteger: produc2591615631378489102nteger > nat ).
thf(func_def_1293,type,
product_fst_nat_nat: product_prod_nat_nat > nat ).
thf(func_def_1294,type,
produc1609452848092598666_nat_o: produc4962441903510821572_nat_o > produc3925858234332021118et_nat ).
thf(func_def_1295,type,
produc8046704426453032966_nat_o: produc6907119039986491712_nat_o > produc3658429121746597890et_nat ).
thf(func_def_1296,type,
produc6012101397789842562t_unit: produc7348995623955488188t_unit > set_nat ).
thf(func_def_1297,type,
produc5374455773327741254et_nat: produc2732055786443039994et_nat > produc3925858234332021118et_nat ).
thf(func_def_1298,type,
produc4011572625026189258et_nat: produc3925858234332021118et_nat > produc3658429121746597890et_nat ).
thf(func_def_1299,type,
produc6174133586879617921nteger: produc8923325533196201883nteger > code_integer ).
thf(func_def_1300,type,
produc1517335558169710324er_nat: produc6433947425880249742er_nat > nat ).
thf(func_def_1301,type,
produc8586169260539613262et_nat: produc3658429121746597890et_nat > set_nat ).
thf(func_def_1302,type,
product_snd_int_int: product_prod_int_int > int ).
thf(func_def_1303,type,
produc1781891006948753140nteger: produc2591615631378489102nteger > code_integer ).
thf(func_def_1304,type,
product_snd_nat_nat: product_prod_nat_nat > nat ).
thf(func_def_1305,type,
produc6803565743942592456_nat_o: produc4962441903510821572_nat_o > produc3658429121746597890et_nat > $o ).
thf(func_def_1306,type,
produc1838968430882301508_nat_o: produc6907119039986491712_nat_o > produc3658429121746597890et_nat > $o ).
thf(func_def_1307,type,
produc3550216979005444800t_unit: produc7348995623955488188t_unit > heap_e7401611519738050253t_unit ).
thf(func_def_1308,type,
produc1151471490888544852et_nat: produc2732055786443039994et_nat > produc4962441903510821572_nat_o ).
thf(func_def_1309,type,
produc3251101204053596376et_nat: produc3925858234332021118et_nat > produc6907119039986491712_nat_o ).
thf(func_def_1310,type,
produc6300697153377358172et_nat: produc3658429121746597890et_nat > produc7348995623955488188t_unit ).
thf(func_def_1311,type,
product_swap_int_int: product_prod_int_int > product_prod_int_int ).
thf(func_def_1312,type,
product_swap_nat_nat: product_prod_nat_nat > product_prod_nat_nat ).
thf(func_def_1313,type,
produc2580581461503396054_nat_o: produc4962441903510821572_nat_o > produc2732055786443039994et_nat ).
thf(func_def_1314,type,
produc1078497009909708626_nat_o: produc6907119039986491712_nat_o > produc3925858234332021118et_nat ).
thf(func_def_1315,type,
produc1264744871843189710t_unit: produc7348995623955488188t_unit > produc3658429121746597890et_nat ).
thf(func_def_1316,type,
produc5538323210962509403atural: ( produc7822875418678951345atural > produc5835291356934675326atural ) > ( code_natural > produc7822875418678951345atural > produc5835291356934675326atural ) > produc7822875418678951345atural > produc5835291356934675326atural ).
thf(func_def_1317,type,
product_Abs_unit: $o > product_unit ).
thf(func_def_1318,type,
product_Rep_unit: product_unit > $o ).
thf(func_def_1319,type,
iterat8892046348760725948atural: code_natural > ( code_natural > produc7822875418678951345atural > produc5835291356934675326atural ) > code_natural > produc7822875418678951345atural > produc5835291356934675326atural ).
thf(func_def_1320,type,
log: code_natural > code_natural > code_natural ).
thf(func_def_1321,type,
minus_shift: code_natural > code_natural > code_natural > code_natural ).
thf(func_def_1322,type,
next: produc7822875418678951345atural > produc5835291356934675326atural ).
thf(func_def_1323,type,
range: code_natural > produc7822875418678951345atural > produc5835291356934675326atural ).
thf(func_def_1324,type,
abs_Rat: product_prod_int_int > rat ).
thf(func_def_1325,type,
fract: int > int > rat ).
thf(func_def_1326,type,
rep_Rat: rat > product_prod_int_int ).
thf(func_def_1327,type,
normalize: product_prod_int_int > product_prod_int_int ).
thf(func_def_1328,type,
pcr_rat: product_prod_int_int > rat > $o ).
thf(func_def_1329,type,
positive: rat > $o ).
thf(func_def_1330,type,
quotient_of: rat > product_prod_int_int ).
thf(func_def_1331,type,
ratrel: product_prod_int_int > product_prod_int_int > $o ).
thf(func_def_1332,type,
domain4881241903163524940_nat_o: set_Pr2161125870931222855_nat_o > set_Pr4532377907799695533_nat_o ).
thf(func_def_1333,type,
domain3618434998481624895et_nat: set_Pr8536935166611901872et_nat > set_Pr4532377907799695533_nat_o ).
thf(func_def_1334,type,
domain2358961613337345475et_nat: set_Pr3286484037609594932et_nat > set_Pr4532377907799695533_nat_o ).
thf(func_def_1335,type,
domain_o_o: set_Product_prod_o_o > set_o ).
thf(func_def_1336,type,
domain2007548326387822151et_nat: set_Pr3948176798113811640et_nat > set_he8644174446132254637t_unit ).
thf(func_def_1337,type,
domain_int_int: set_Pr958786334691620121nt_int > set_int ).
thf(func_def_1338,type,
domain_nat_nat: set_Pr1261947904930325089at_nat > set_nat ).
thf(func_def_1339,type,
domain4292024440186670924t_unit: set_Pr5094982260447487303t_unit > set_Product_unit ).
thf(func_def_1340,type,
field_81821745747897882_nat_o: set_Pr2161125870931222855_nat_o > set_Pr4532377907799695533_nat_o ).
thf(func_def_1341,type,
field_o: set_Product_prod_o_o > set_o ).
thf(func_def_1342,type,
field_Code_integer: set_Pr4811707699266497531nteger > set_Code_integer ).
thf(func_def_1343,type,
field_int: set_Pr958786334691620121nt_int > set_int ).
thf(func_def_1344,type,
field_nat: set_Pr1261947904930325089at_nat > set_nat ).
thf(func_def_1345,type,
field_Product_unit: set_Pr5094982260447487303t_unit > set_Product_unit ).
thf(func_def_1346,type,
id_Pro6775173067513470761_nat_o: set_Pr2161125870931222855_nat_o ).
thf(func_def_1347,type,
id_o2: set_Product_prod_o_o ).
thf(func_def_1348,type,
id_int: set_Pr958786334691620121nt_int ).
thf(func_def_1349,type,
id_nat: set_Pr1261947904930325089at_nat ).
thf(func_def_1350,type,
id_Product_unit: set_Pr5094982260447487303t_unit ).
thf(func_def_1351,type,
id_on_7438695062414216554_nat_o: set_Pr4532377907799695533_nat_o > set_Pr2161125870931222855_nat_o ).
thf(func_def_1352,type,
id_on_o: set_o > set_Product_prod_o_o ).
thf(func_def_1353,type,
id_on_int: set_int > set_Pr958786334691620121nt_int ).
thf(func_def_1354,type,
id_on_nat: set_nat > set_Pr1261947904930325089at_nat ).
thf(func_def_1355,type,
id_on_Product_unit: set_Product_unit > set_Pr5094982260447487303t_unit ).
thf(func_def_1356,type,
image_2232984160439939859_nat_o: set_Pr2161125870931222855_nat_o > set_Pr4532377907799695533_nat_o > set_Pr4532377907799695533_nat_o ).
thf(func_def_1357,type,
image_7240710759097644216et_nat: set_Pr8536935166611901872et_nat > set_Pr4532377907799695533_nat_o > set_Pr3286484037609594932et_nat ).
thf(func_def_1358,type,
image_1814916760386648892et_nat: set_Pr3286484037609594932et_nat > set_Pr4532377907799695533_nat_o > set_Pr3948176798113811640et_nat ).
thf(func_def_1359,type,
image_o_o: set_Product_prod_o_o > set_o > set_o ).
thf(func_def_1360,type,
image_o_int: set_Pr8834758594704517033_o_int > set_o > set_int ).
thf(func_def_1361,type,
image_o_nat: set_Pr2101469702781467981_o_nat > set_o > set_nat ).
thf(func_def_1362,type,
image_o_Product_unit: set_Pr6671815714118170108t_unit > set_o > set_Product_unit ).
thf(func_def_1363,type,
image_4147621952751189391nteger: set_Pr4811707699266497531nteger > set_Code_integer > set_Code_integer ).
thf(func_def_1364,type,
image_3942343239045813184et_nat: set_Pr3948176798113811640et_nat > set_he8644174446132254637t_unit > set_set_nat ).
thf(func_def_1365,type,
image_int_o: set_Pr903927857289325719_int_o > set_int > set_o ).
thf(func_def_1366,type,
image_int_int: set_Pr958786334691620121nt_int > set_int > set_int ).
thf(func_def_1367,type,
image_int_nat: set_Pr3448869479623346877nt_nat > set_int > set_nat ).
thf(func_def_1368,type,
image_1405326909533657676t_unit: set_Pr5889264611543340172t_unit > set_int > set_Product_unit ).
thf(func_def_1369,type,
image_nat_o: set_Pr3149072824959771635_nat_o > set_nat > set_o ).
thf(func_def_1370,type,
image_nat_int: set_Pr7995236796853374141at_int > set_nat > set_int ).
thf(func_def_1371,type,
image_nat_nat: set_Pr1261947904930325089at_nat > set_nat > set_nat ).
thf(func_def_1372,type,
image_4288904962331477416t_unit: set_Pr4334478416066269672t_unit > set_nat > set_Product_unit ).
thf(func_def_1373,type,
image_Product_unit_o: set_Pr5464989305201766866unit_o > set_Product_unit > set_o ).
thf(func_def_1374,type,
image_5655252347010435494it_int: set_Pr8497134830871917726it_int > set_Product_unit > set_int ).
thf(func_def_1375,type,
image_5657742817519485770it_nat: set_Pr1763845938948868674it_nat > set_Product_unit > set_nat ).
thf(func_def_1376,type,
image_8431784415008483091t_unit: set_Pr5094982260447487303t_unit > set_Product_unit > set_Product_unit ).
thf(func_def_1377,type,
range_5289318983786108341_nat_o: set_Pr2161125870931222855_nat_o > set_Pr4532377907799695533_nat_o ).
thf(func_def_1378,type,
range_7260058354244137878et_nat: set_Pr8536935166611901872et_nat > set_Pr3286484037609594932et_nat ).
thf(func_def_1379,type,
range_3398330936903821850et_nat: set_Pr3286484037609594932et_nat > set_Pr3948176798113811640et_nat ).
thf(func_def_1380,type,
range_o_o: set_Product_prod_o_o > set_o ).
thf(func_def_1381,type,
range_1907724884897625393nteger: set_Pr4811707699266497531nteger > set_Code_integer ).
thf(func_def_1382,type,
range_9132311542094983838et_nat: set_Pr3948176798113811640et_nat > set_set_nat ).
thf(func_def_1383,type,
range_int_int: set_Pr958786334691620121nt_int > set_int ).
thf(func_def_1384,type,
range_nat_nat: set_Pr1261947904930325089at_nat > set_nat ).
thf(func_def_1385,type,
range_6617724647562260917t_unit: set_Pr5094982260447487303t_unit > set_Product_unit ).
thf(func_def_1386,type,
antisy9190549550370998639_nat_o: set_Pr2161125870931222855_nat_o > $o ).
thf(func_def_1387,type,
antisym_o: set_Product_prod_o_o > $o ).
thf(func_def_1388,type,
antisym_int: set_Pr958786334691620121nt_int > $o ).
thf(func_def_1389,type,
antisym_nat: set_Pr1261947904930325089at_nat > $o ).
thf(func_def_1390,type,
antisym_Product_unit: set_Pr5094982260447487303t_unit > $o ).
thf(func_def_1391,type,
conver2764731894203590139_nat_o: set_Pr2161125870931222855_nat_o > set_Pr2161125870931222855_nat_o ).
thf(func_def_1392,type,
conver3577387295554834384et_nat: set_Pr8536935166611901872et_nat > set_Pr7705059561495379578_nat_o ).
thf(func_def_1393,type,
conver4003969491454187092et_nat: set_Pr3286484037609594932et_nat > set_Pr4208978187861016310_nat_o ).
thf(func_def_1394,type,
converse_o_o: set_Product_prod_o_o > set_Product_prod_o_o ).
thf(func_def_1395,type,
conver7272282955447398104et_nat: set_Pr3948176798113811640et_nat > set_Pr4071100604818519410t_unit ).
thf(func_def_1396,type,
converse_int_int: set_Pr958786334691620121nt_int > set_Pr958786334691620121nt_int ).
thf(func_def_1397,type,
converse_nat_nat: set_Pr1261947904930325089at_nat > set_Pr1261947904930325089at_nat ).
thf(func_def_1398,type,
conver5006497266169685586_nat_o: set_Pr7705059561495379578_nat_o > set_Pr8536935166611901872et_nat ).
thf(func_def_1399,type,
conver1831365297310299342_nat_o: set_Pr4208978187861016310_nat_o > set_Pr3286484037609594932et_nat ).
thf(func_def_1400,type,
conver2495244158383847931t_unit: set_Pr5094982260447487303t_unit > set_Pr5094982260447487303t_unit ).
thf(func_def_1401,type,
conver2236330673913229642t_unit: set_Pr4071100604818519410t_unit > set_Pr3948176798113811640et_nat ).
thf(func_def_1402,type,
inv_image_o_o: set_Product_prod_o_o > ( $o > $o ) > set_Product_prod_o_o ).
thf(func_def_1403,type,
inv_image_o_nat: set_Product_prod_o_o > ( nat > $o ) > set_Pr1261947904930325089at_nat ).
thf(func_def_1404,type,
inv_im4602722609573551687t_unit: set_Product_prod_o_o > ( product_unit > $o ) > set_Pr5094982260447487303t_unit ).
thf(func_def_1405,type,
inv_image_int_o: set_Pr958786334691620121nt_int > ( $o > int ) > set_Product_prod_o_o ).
thf(func_def_1406,type,
inv_image_int_int: set_Pr958786334691620121nt_int > ( int > int ) > set_Pr958786334691620121nt_int ).
thf(func_def_1407,type,
inv_image_int_nat: set_Pr958786334691620121nt_int > ( nat > int ) > set_Pr1261947904930325089at_nat ).
thf(func_def_1408,type,
inv_im4740476784543923469t_unit: set_Pr958786334691620121nt_int > ( product_unit > int ) > set_Pr5094982260447487303t_unit ).
thf(func_def_1409,type,
inv_image_nat_o: set_Pr1261947904930325089at_nat > ( $o > nat ) > set_Product_prod_o_o ).
thf(func_def_1410,type,
inv_image_nat_int: set_Pr1261947904930325089at_nat > ( int > nat ) > set_Pr958786334691620121nt_int ).
thf(func_def_1411,type,
inv_image_nat_nat: set_Pr1261947904930325089at_nat > ( nat > nat ) > set_Pr1261947904930325089at_nat ).
thf(func_def_1412,type,
inv_im7624054837341743209t_unit: set_Pr1261947904930325089at_nat > ( product_unit > nat ) > set_Pr5094982260447487303t_unit ).
thf(func_def_1413,type,
inv_im9188693106839688349unit_o: set_Pr5094982260447487303t_unit > ( $o > product_unit ) > set_Product_prod_o_o ).
thf(func_def_1414,type,
inv_im8992892692529751563it_nat: set_Pr5094982260447487303t_unit > ( nat > product_unit ) > set_Pr1261947904930325089at_nat ).
thf(func_def_1415,type,
inv_im8770536620514187154t_unit: set_Pr5094982260447487303t_unit > ( product_unit > product_unit ) > set_Pr5094982260447487303t_unit ).
thf(func_def_1416,type,
refl_o1075864025857821388_nat_o: set_Pr4532377907799695533_nat_o > set_Pr2161125870931222855_nat_o > $o ).
thf(func_def_1417,type,
refl_on_o: set_o > set_Product_prod_o_o > $o ).
thf(func_def_1418,type,
refl_on_int: set_int > set_Pr958786334691620121nt_int > $o ).
thf(func_def_1419,type,
refl_on_nat: set_nat > set_Pr1261947904930325089at_nat > $o ).
thf(func_def_1420,type,
refl_on_Product_unit: set_Product_unit > set_Pr5094982260447487303t_unit > $o ).
thf(func_def_1421,type,
relcom6251035627269349452_nat_o: set_Pr2161125870931222855_nat_o > set_Pr2161125870931222855_nat_o > set_Pr2161125870931222855_nat_o ).
thf(func_def_1422,type,
relcom2608444778321954367et_nat: set_Pr2161125870931222855_nat_o > set_Pr8536935166611901872et_nat > set_Pr8536935166611901872et_nat ).
thf(func_def_1423,type,
relcom6619595054755144899et_nat: set_Pr2161125870931222855_nat_o > set_Pr3286484037609594932et_nat > set_Pr3286484037609594932et_nat ).
thf(func_def_1424,type,
relcom8303140135939483018et_nat: set_Pr8536935166611901872et_nat > set_Pr7928877670098842301et_nat > set_Pr8536935166611901872et_nat ).
thf(func_def_1425,type,
relcom430014394250001422et_nat: set_Pr8536935166611901872et_nat > set_Pr3721596426557623617et_nat > set_Pr3286484037609594932et_nat ).
thf(func_def_1426,type,
relcom8031709186682293262et_nat: set_Pr3286484037609594932et_nat > set_Pr8807100098903490369et_nat > set_Pr8536935166611901872et_nat ).
thf(func_def_1427,type,
relcom5975814107203797650et_nat: set_Pr3286484037609594932et_nat > set_Pr719794911490849221et_nat > set_Pr3286484037609594932et_nat ).
thf(func_def_1428,type,
relcomp_o_o_o: set_Product_prod_o_o > set_Product_prod_o_o > set_Product_prod_o_o ).
thf(func_def_1429,type,
relcom4930750947919833671et_nat: set_Pr1776176089890104647t_unit > set_Pr3948176798113811640et_nat > set_Pr3948176798113811640et_nat ).
thf(func_def_1430,type,
relcom8881708762719574682et_nat: set_Pr3948176798113811640et_nat > set_Pr5488025237498180813et_nat > set_Pr3948176798113811640et_nat ).
thf(func_def_1431,type,
relcomp_int_int_int: set_Pr958786334691620121nt_int > set_Pr958786334691620121nt_int > set_Pr958786334691620121nt_int ).
thf(func_def_1432,type,
relcomp_nat_nat_nat: set_Pr1261947904930325089at_nat > set_Pr1261947904930325089at_nat > set_Pr1261947904930325089at_nat ).
thf(func_def_1433,type,
relcom848357998638694028t_unit: set_Pr5094982260447487303t_unit > set_Pr5094982260447487303t_unit > set_Pr5094982260447487303t_unit ).
thf(func_def_1434,type,
relcom7236094467932586796_nat_o: set_Pr3813038829465207111_nat_o > set_Pr3813038829465207111_nat_o > set_Pr3813038829465207111_nat_o ).
thf(func_def_1435,type,
relcom7534720040753365047_set_o: set_Pr4577759397028426247_set_o > set_Pr4577759397028426247_set_o > set_Pr4577759397028426247_set_o ).
thf(func_def_1436,type,
relcom4034950705519254697et_int: set_Pr2522554150109002629et_int > set_Pr2522554150109002629et_int > set_Pr2522554150109002629et_int ).
thf(func_def_1437,type,
relcom236561585145426837et_nat: set_Pr5488025237498180813et_nat > set_Pr5488025237498180813et_nat > set_Pr5488025237498180813et_nat ).
thf(func_def_1438,type,
relcom7108816970757792620t_unit: set_Pr3980508049012409671t_unit > set_Pr3980508049012409671t_unit > set_Pr3980508049012409671t_unit ).
thf(func_def_1439,type,
single261872189793494280et_nat: set_Pr8536935166611901872et_nat > $o ).
thf(func_def_1440,type,
single7715245794981374860et_nat: set_Pr3286484037609594932et_nat > $o ).
thf(func_def_1441,type,
single6620584844202787856et_nat: set_Pr3948176798113811640et_nat > $o ).
thf(func_def_1442,type,
single6503947681363884253nt_int: set_Pr958786334691620121nt_int > $o ).
thf(func_def_1443,type,
single5507084439971879973at_nat: set_Pr1261947904930325089at_nat > $o ).
thf(func_def_1444,type,
total_7429123086324524369_nat_o: set_Pr4532377907799695533_nat_o > set_Pr2161125870931222855_nat_o > $o ).
thf(func_def_1445,type,
total_on_o: set_o > set_Product_prod_o_o > $o ).
thf(func_def_1446,type,
total_on_int: set_int > set_Pr958786334691620121nt_int > $o ).
thf(func_def_1447,type,
total_on_nat: set_nat > set_Pr1261947904930325089at_nat > $o ).
thf(func_def_1448,type,
total_5665700600833547665t_unit: set_Product_unit > set_Pr5094982260447487303t_unit > $o ).
thf(func_def_1449,type,
trans_int: set_Pr958786334691620121nt_int > $o ).
thf(func_def_1450,type,
trans_nat: set_Pr1261947904930325089at_nat > $o ).
thf(func_def_1451,type,
algebr932160517623751201me_int: int > int > $o ).
thf(func_def_1452,type,
algebr934650988132801477me_nat: nat > nat > $o ).
thf(func_def_1453,type,
divide6298287555418463151nteger: code_integer > code_integer > code_integer ).
thf(func_def_1454,type,
divide5121882707175180666atural: code_natural > code_natural > code_natural ).
thf(func_def_1455,type,
divide_divide_int: int > int > int ).
thf(func_def_1456,type,
divide_divide_nat: nat > nat > nat ).
thf(func_def_1457,type,
divide_divide_rat: rat > rat > rat ).
thf(func_def_1458,type,
dvd_dvd_assn: assn > assn > $o ).
thf(func_def_1459,type,
dvd_dvd_Code_integer: code_integer > code_integer > $o ).
thf(func_def_1460,type,
dvd_dvd_Code_natural: code_natural > code_natural > $o ).
thf(func_def_1461,type,
dvd_dvd_int: int > int > $o ).
thf(func_def_1462,type,
dvd_dvd_nat: nat > nat > $o ).
thf(func_def_1463,type,
dvd_dvd_rat: rat > rat > $o ).
thf(func_def_1464,type,
modulo8411746178871703098atural: code_natural > code_natural > code_natural ).
thf(func_def_1465,type,
modulo_modulo_int: int > int > int ).
thf(func_def_1466,type,
modulo_modulo_nat: nat > nat > nat ).
thf(func_def_1467,type,
zero_n356916108424825756nteger: $o > code_integer ).
thf(func_def_1468,type,
zero_n2684676970156552555ol_int: $o > int ).
thf(func_def_1469,type,
zero_n2687167440665602831ol_nat: $o > nat ).
thf(func_def_1470,type,
collec939566748876313656_nat_o: ( ( produc3658429121746597890et_nat > $o ) > $o ) > set_Pr4532377907799695533_nat_o ).
thf(func_def_1471,type,
collect_o: ( $o > $o ) > set_o ).
thf(func_def_1472,type,
collect_int: ( int > $o ) > set_int ).
thf(func_def_1473,type,
collect_nat: ( nat > $o ) > set_nat ).
thf(func_def_1474,type,
collect_Product_unit: ( product_unit > $o ) > set_Product_unit ).
thf(func_def_1475,type,
pow_Pr7829143134171777386_nat_o: set_Pr4532377907799695533_nat_o > set_se1353816476878259085_nat_o ).
thf(func_def_1476,type,
pow_o: set_o > set_set_o ).
thf(func_def_1477,type,
pow_Code_integer: set_Code_integer > set_set_Code_integer ).
thf(func_def_1478,type,
pow_int: set_int > set_set_int ).
thf(func_def_1479,type,
pow_nat: set_nat > set_set_nat ).
thf(func_def_1480,type,
pow_Product_unit: set_Product_unit > set_set_Product_unit ).
thf(func_def_1481,type,
pow_set_nat: set_set_nat > set_set_set_nat ).
thf(func_def_1482,type,
disjnt7869448817988085652_nat_o: set_Pr4532377907799695533_nat_o > set_Pr4532377907799695533_nat_o > $o ).
thf(func_def_1483,type,
disjnt_o: set_o > set_o > $o ).
thf(func_def_1484,type,
disjnt_int: set_int > set_int > $o ).
thf(func_def_1485,type,
disjnt_nat: set_nat > set_nat > $o ).
thf(func_def_1486,type,
disjnt4476176636800211563_nat_o: set_Pr5623950326363488168_nat_o > set_Pr5623950326363488168_nat_o > $o ).
thf(func_def_1487,type,
disjnt5652777605541633334_nat_o: set_Pr3149072824959771635_nat_o > set_Pr3149072824959771635_nat_o > $o ).
thf(func_def_1488,type,
disjnt7759068009871657124at_int: set_Pr7995236796853374141at_int > set_Pr7995236796853374141at_int > $o ).
thf(func_def_1489,type,
disjnt2713546992526078024at_nat: set_Pr1261947904930325089at_nat > set_Pr1261947904930325089at_nat > $o ).
thf(func_def_1490,type,
disjnt595863415355979947t_unit: set_Pr4334478416066269672t_unit > set_Pr4334478416066269672t_unit > $o ).
thf(func_def_1491,type,
disjnt_Product_unit: set_Product_unit > set_Product_unit > $o ).
thf(func_def_1492,type,
image_8639947063239330806_nat_o: ( ( ( produc3658429121746597890et_nat > $o ) > $o ) > set_Pr4532377907799695533_nat_o ) > set_Pr6409761941060870310at_o_o > set_se1353816476878259085_nat_o ).
thf(func_def_1493,type,
image_o_o_set_o: ( ( $o > $o ) > set_o ) > set_o_o > set_set_o ).
thf(func_def_1494,type,
image_int_o_set_int: ( ( int > $o ) > set_int ) > set_int_o > set_set_int ).
thf(func_def_1495,type,
image_nat_o_set_nat: ( ( nat > $o ) > set_nat ) > set_nat_o > set_set_nat ).
thf(func_def_1496,type,
image_3399141053782081349_nat_o: ( ( produc3658429121746597890et_nat > $o ) > produc3658429121746597890et_nat > $o ) > set_Pr4532377907799695533_nat_o > set_Pr4532377907799695533_nat_o ).
thf(func_def_1497,type,
image_6451031022632655280o_assn: ( ( produc3658429121746597890et_nat > $o ) > assn ) > set_Pr4532377907799695533_nat_o > set_assn ).
thf(func_def_1498,type,
image_2174381888560928856_o_nat: ( ( produc3658429121746597890et_nat > $o ) > nat ) > set_Pr4532377907799695533_nat_o > set_nat ).
thf(func_def_1499,type,
image_8451000719036923781t_unit: ( ( produc3658429121746597890et_nat > $o ) > product_unit ) > set_Pr4532377907799695533_nat_o > set_Product_unit ).
thf(func_def_1500,type,
image_8657708916238040603nteger: ( ( produc3658429121746597890et_nat > $o ) > set_Code_integer ) > set_Pr4532377907799695533_nat_o > set_set_Code_integer ).
thf(func_def_1501,type,
image_6888641473154130958et_nat: ( ( produc3658429121746597890et_nat > $o ) > set_nat ) > set_Pr4532377907799695533_nat_o > set_set_nat ).
thf(func_def_1502,type,
image_5138372075374782070t_unit: ( ( product_unit > $o ) > set_Product_unit ) > set_Product_unit_o > set_set_Product_unit ).
thf(func_def_1503,type,
image_o_o2: ( $o > $o ) > set_o > set_o ).
thf(func_def_1504,type,
image_o_Code_integer: ( $o > code_integer ) > set_o > set_Code_integer ).
thf(func_def_1505,type,
image_o_filter_nat: ( $o > filter_nat ) > set_o > set_filter_nat ).
thf(func_def_1506,type,
image_o_int2: ( $o > int ) > set_o > set_int ).
thf(func_def_1507,type,
image_o_nat2: ( $o > nat ) > set_o > set_nat ).
thf(func_def_1508,type,
image_8407096251609565691nteger: ( $o > produc2591615631378489102nteger ) > set_o > set_Pr2406931829603198446nteger ).
thf(func_def_1509,type,
image_3855930084881510382at_nat: ( $o > product_prod_nat_nat ) > set_o > set_Pr1261947904930325089at_nat ).
thf(func_def_1510,type,
image_o_Product_unit2: ( $o > product_unit ) > set_o > set_Product_unit ).
thf(func_def_1511,type,
image_o_set_o: ( $o > set_o ) > set_o > set_set_o ).
thf(func_def_1512,type,
image_6491638074533506790nteger: ( $o > set_Code_integer ) > set_o > set_set_Code_integer ).
thf(func_def_1513,type,
image_o_set_int: ( $o > set_int ) > set_o > set_set_int ).
thf(func_def_1514,type,
image_o_set_nat: ( $o > set_nat ) > set_o > set_set_nat ).
thf(func_def_1515,type,
image_663195431845683034t_unit: ( $o > set_Product_unit ) > set_o > set_set_Product_unit ).
thf(func_def_1516,type,
image_5465535159827709278_nat_o: ( assn > produc3658429121746597890et_nat > $o ) > set_assn > set_Pr4532377907799695533_nat_o ).
thf(func_def_1517,type,
image_assn_assn: ( assn > assn ) > set_assn > set_assn ).
thf(func_def_1518,type,
image_2177946750938214461nteger: ( code_integer > code_integer > code_integer ) > set_Code_integer > set_Co726186806682698383nteger ).
thf(func_def_1519,type,
image_Code_integer_o: ( code_integer > $o ) > set_Code_integer > set_o ).
thf(func_def_1520,type,
image_4470545334726330049nteger: ( code_integer > code_integer ) > set_Code_integer > set_Code_integer ).
thf(func_def_1521,type,
image_948535463418740880er_int: ( code_integer > int ) > set_Code_integer > set_int ).
thf(func_def_1522,type,
image_951025933927791156er_nat: ( code_integer > nat ) > set_Code_integer > set_nat ).
thf(func_def_1523,type,
image_8049062792287394871nteger: ( code_integer > produc8923325533196201883nteger ) > set_Code_integer > set_Pr4811707699266497531nteger ).
thf(func_def_1524,type,
image_7250923904792228394er_nat: ( code_integer > produc6433947425880249742er_nat ) > set_Code_integer > set_Pr8986988648414616686er_nat ).
thf(func_def_1525,type,
image_3408592110290467754nteger: ( code_integer > produc2591615631378489102nteger ) > set_Code_integer > set_Pr2406931829603198446nteger ).
thf(func_def_1526,type,
image_7182656606676592809t_unit: ( code_integer > product_unit ) > set_Code_integer > set_Product_unit ).
thf(func_def_1527,type,
image_1867029960091478804_set_o: ( code_integer > set_o ) > set_Code_integer > set_set_o ).
thf(func_def_1528,type,
image_4651426196962995782et_int: ( code_integer > set_int ) > set_Code_integer > set_set_int ).
thf(func_def_1529,type,
image_8829277216472192490et_nat: ( code_integer > set_nat ) > set_Code_integer > set_set_nat ).
thf(func_def_1530,type,
image_5870443399019143561t_unit: ( code_integer > set_Product_unit ) > set_Code_integer > set_set_Product_unit ).
thf(func_def_1531,type,
image_6215854944615398399al_nat: ( code_natural > nat ) > set_Code_natural > set_nat ).
thf(func_def_1532,type,
image_int_o2: ( int > $o ) > set_int > set_o ).
thf(func_def_1533,type,
image_1587234942943678608nteger: ( int > code_integer ) > set_int > set_Code_integer ).
thf(func_def_1534,type,
image_int_filter_nat: ( int > filter_nat ) > set_int > set_filter_nat ).
thf(func_def_1535,type,
image_int_int2: ( int > int ) > set_int > set_int ).
thf(func_def_1536,type,
image_int_nat2: ( int > nat ) > set_int > set_nat ).
thf(func_def_1537,type,
image_5846526143423701914t_unit: ( int > product_unit ) > set_int > set_Product_unit ).
thf(func_def_1538,type,
image_int_set_o: ( int > set_o ) > set_int > set_set_o ).
thf(func_def_1539,type,
image_75573793354812870nteger: ( int > set_Code_integer ) > set_int > set_set_Code_integer ).
thf(func_def_1540,type,
image_int_set_nat: ( int > set_nat ) > set_int > set_set_nat ).
thf(func_def_1541,type,
image_8658272795163236474t_unit: ( int > set_Product_unit ) > set_int > set_set_Product_unit ).
thf(func_def_1542,type,
image_4682239675366380726_nat_o: ( nat > produc3658429121746597890et_nat > $o ) > set_nat > set_Pr4532377907799695533_nat_o ).
thf(func_def_1543,type,
image_nat_o2: ( nat > $o ) > set_nat > set_o ).
thf(func_def_1544,type,
image_1215581382706833972nteger: ( nat > code_integer ) > set_nat > set_Code_integer ).
thf(func_def_1545,type,
image_39176534463551487atural: ( nat > code_natural ) > set_nat > set_Code_natural ).
thf(func_def_1546,type,
image_nat_filter_nat: ( nat > filter_nat ) > set_nat > set_filter_nat ).
thf(func_def_1547,type,
image_nat_int2: ( nat > int ) > set_nat > set_int ).
thf(func_def_1548,type,
image_nat_nat2: ( nat > nat ) > set_nat > set_nat ).
thf(func_def_1549,type,
image_8730104196221521654t_unit: ( nat > product_unit ) > set_nat > set_Product_unit ).
thf(func_def_1550,type,
image_nat_set_o: ( nat > set_o ) > set_nat > set_set_o ).
thf(func_def_1551,type,
image_2385391725447763818nteger: ( nat > set_Code_integer ) > set_nat > set_set_Code_integer ).
thf(func_def_1552,type,
image_nat_set_int: ( nat > set_int ) > set_nat > set_set_int ).
thf(func_def_1553,type,
image_nat_set_nat: ( nat > set_nat ) > set_nat > set_set_nat ).
thf(func_def_1554,type,
image_7178329752028323786at_nat: ( nat > set_Pr1261947904930325089at_nat ) > set_nat > set_se7855581050983116737at_nat ).
thf(func_def_1555,type,
image_4824033694279209942t_unit: ( nat > set_Product_unit ) > set_nat > set_set_Product_unit ).
thf(func_def_1556,type,
image_2517677226577294348_nat_o: ( produc2732055786443039994et_nat > produc3658429121746597890et_nat > $o ) > set_Pr8536935166611901872et_nat > set_Pr4532377907799695533_nat_o ).
thf(func_def_1557,type,
image_4557310310409863807et_nat: ( produc2732055786443039994et_nat > produc3925858234332021118et_nat ) > set_Pr8536935166611901872et_nat > set_Pr3286484037609594932et_nat ).
thf(func_def_1558,type,
image_4071802983491761093_nat_o: ( produc2732055786443039994et_nat > produc4962441903510821572_nat_o ) > set_Pr8536935166611901872et_nat > set_Pr7705059561495379578_nat_o ).
thf(func_def_1559,type,
image_6802380912191114376_nat_o: ( produc3925858234332021118et_nat > produc3658429121746597890et_nat > $o ) > set_Pr3286484037609594932et_nat > set_Pr4532377907799695533_nat_o ).
thf(func_def_1560,type,
image_4390311411407041927et_nat: ( produc3925858234332021118et_nat > produc3658429121746597890et_nat ) > set_Pr3286484037609594932et_nat > set_Pr3948176798113811640et_nat ).
thf(func_def_1561,type,
image_7073998564157519676at_nat: ( produc3925858234332021118et_nat > product_prod_nat_nat ) > set_Pr3286484037609594932et_nat > set_Pr1261947904930325089at_nat ).
thf(func_def_1562,type,
image_6229134628169569989_nat_o: ( produc3925858234332021118et_nat > produc6907119039986491712_nat_o ) > set_Pr3286484037609594932et_nat > set_Pr4208978187861016310_nat_o ).
thf(func_def_1563,type,
image_227735840630799463eger_o: ( produc6271795597528267376eger_o > produc6271795597528267376eger_o ) > set_Pr448751882837621926eger_o > set_Pr448751882837621926eger_o ).
thf(func_def_1564,type,
image_6298138160576288691er_int: ( produc6271795597528267376eger_o > produc2256096406371053034er_int ) > set_Pr448751882837621926eger_o > set_Pr6496905503482889930er_int ).
thf(func_def_1565,type,
image_1252617143230709591er_nat: ( produc6271795597528267376eger_o > produc6433947425880249742er_nat ) > set_Pr448751882837621926eger_o > set_Pr8986988648414616686er_nat ).
thf(func_def_1566,type,
image_2425804817718344284t_unit: ( produc6271795597528267376eger_o > produc5496629617739085541t_unit ) > set_Pr448751882837621926eger_o > set_Pr4580871404036792859t_unit ).
thf(func_def_1567,type,
image_7082141833288935621nteger: ( produc8923325533196201883nteger > produc8923325533196201883nteger ) > set_Pr4811707699266497531nteger > set_Pr4811707699266497531nteger ).
thf(func_def_1568,type,
image_4921214744877245202nteger: ( produc6433947425880249742er_nat > produc8923325533196201883nteger ) > set_Pr8986988648414616686er_nat > set_Pr4811707699266497531nteger ).
thf(func_def_1569,type,
image_7060974388358999388eger_o: ( produc5496629617739085541t_unit > produc6271795597528267376eger_o ) > set_Pr4580871404036792859t_unit > set_Pr448751882837621926eger_o ).
thf(func_def_1570,type,
image_4058522669889801726er_int: ( produc5496629617739085541t_unit > produc2256096406371053034er_int ) > set_Pr4580871404036792859t_unit > set_Pr6496905503482889930er_int ).
thf(func_def_1571,type,
image_8236373689398998434er_nat: ( produc5496629617739085541t_unit > produc6433947425880249742er_nat ) > set_Pr4580871404036792859t_unit > set_Pr8986988648414616686er_nat ).
thf(func_def_1572,type,
image_3456600171786721233t_unit: ( produc5496629617739085541t_unit > produc5496629617739085541t_unit ) > set_Pr4580871404036792859t_unit > set_Pr4580871404036792859t_unit ).
thf(func_def_1573,type,
image_6824039242672219652t_unit: ( produc3658429121746597890et_nat > heap_e7401611519738050253t_unit ) > set_Pr3948176798113811640et_nat > set_he8644174446132254637t_unit ).
thf(func_def_1574,type,
image_7426123528223433739et_nat: ( produc3658429121746597890et_nat > produc3658429121746597890et_nat ) > set_Pr3948176798113811640et_nat > set_Pr3948176798113811640et_nat ).
thf(func_def_1575,type,
image_4579277974391593584nt_int: ( produc3658429121746597890et_nat > product_prod_int_int ) > set_Pr3948176798113811640et_nat > set_Pr958786334691620121nt_int ).
thf(func_def_1576,type,
image_7757775281999735736at_nat: ( produc3658429121746597890et_nat > product_prod_nat_nat ) > set_Pr3948176798113811640et_nat > set_Pr1261947904930325089at_nat ).
thf(func_def_1577,type,
image_1893317993577548229t_unit: ( produc3658429121746597890et_nat > produc7348995623955488188t_unit ) > set_Pr3948176798113811640et_nat > set_Pr4071100604818519410t_unit ).
thf(func_def_1578,type,
image_6088997261112373647et_nat: ( produc3658429121746597890et_nat > set_nat ) > set_Pr3948176798113811640et_nat > set_set_nat ).
thf(func_def_1579,type,
image_5042161079198086560nt_int: ( product_prod_int_int > int ) > set_Pr958786334691620121nt_int > set_int ).
thf(func_def_1580,type,
image_2127501072796355446et_nat: ( product_prod_int_int > produc3658429121746597890et_nat ) > set_Pr958786334691620121nt_int > set_Pr3948176798113811640et_nat ).
thf(func_def_1581,type,
image_2653370878348428101nt_int: ( product_prod_int_int > product_prod_int_int ) > set_Pr958786334691620121nt_int > set_Pr958786334691620121nt_int ).
thf(func_def_1582,type,
image_5831868185956570253at_nat: ( product_prod_int_int > product_prod_nat_nat ) > set_Pr958786334691620121nt_int > set_Pr1261947904930325089at_nat ).
thf(func_def_1583,type,
image_2486076414777270412at_nat: ( product_prod_nat_nat > nat ) > set_Pr1261947904930325089at_nat > set_nat ).
thf(func_def_1584,type,
image_5744643625575355962et_nat: ( product_prod_nat_nat > produc3925858234332021118et_nat ) > set_Pr1261947904930325089at_nat > set_Pr3286484037609594932et_nat ).
thf(func_def_1585,type,
image_9068448769188195518et_nat: ( product_prod_nat_nat > produc3658429121746597890et_nat ) > set_Pr1261947904930325089at_nat > set_Pr3948176798113811640et_nat ).
thf(func_def_1586,type,
image_1990417195239315453nt_int: ( product_prod_nat_nat > product_prod_int_int ) > set_Pr1261947904930325089at_nat > set_Pr958786334691620121nt_int ).
thf(func_def_1587,type,
image_5168914502847457605at_nat: ( product_prod_nat_nat > product_prod_nat_nat ) > set_Pr1261947904930325089at_nat > set_Pr1261947904930325089at_nat ).
thf(func_def_1588,type,
image_7255662321850427333et_nat: ( produc4962441903510821572_nat_o > produc2732055786443039994et_nat ) > set_Pr7705059561495379578_nat_o > set_Pr8536935166611901872et_nat ).
thf(func_def_1589,type,
image_4501728140019976901et_nat: ( produc6907119039986491712_nat_o > produc3925858234332021118et_nat ) > set_Pr4208978187861016310_nat_o > set_Pr3286484037609594932et_nat ).
thf(func_def_1590,type,
image_5838825082199883205et_nat: ( produc7348995623955488188t_unit > produc3658429121746597890et_nat ) > set_Pr4071100604818519410t_unit > set_Pr3948176798113811640et_nat ).
thf(func_def_1591,type,
image_Product_unit_o2: ( product_unit > $o ) > set_Product_unit > set_o ).
thf(func_def_1592,type,
image_3815118110082124069nteger: ( product_unit > code_integer ) > set_Product_unit > set_Code_integer ).
thf(func_def_1593,type,
image_1853478764106424488er_nat: ( product_unit > filter_nat ) > set_Product_unit > set_filter_nat ).
thf(func_def_1594,type,
image_873079544045703924it_int: ( product_unit > int ) > set_Product_unit > set_int ).
thf(func_def_1595,type,
image_875570014554754200it_nat: ( product_unit > nat ) > set_Product_unit > set_nat ).
thf(func_def_1596,type,
image_8089323762101378259nteger: ( product_unit > produc8923325533196201883nteger ) > set_Product_unit > set_Pr4811707699266497531nteger ).
thf(func_def_1597,type,
image_8789315437038609094er_nat: ( product_unit > produc6433947425880249742er_nat ) > set_Product_unit > set_Pr8986988648414616686er_nat ).
thf(func_def_1598,type,
image_4103462098156874993nt_int: ( product_unit > product_prod_int_int ) > set_Product_unit > set_Pr958786334691620121nt_int ).
thf(func_def_1599,type,
image_4946983642536848454nteger: ( product_unit > produc2591615631378489102nteger ) > set_Product_unit > set_Pr2406931829603198446nteger ).
thf(func_def_1600,type,
image_7281959405765017145at_nat: ( product_unit > product_prod_nat_nat ) > set_Product_unit > set_Pr1261947904930325089at_nat ).
thf(func_def_1601,type,
image_405062704495631173t_unit: ( product_unit > product_unit ) > set_Product_unit > set_Product_unit ).
thf(func_def_1602,type,
image_6930536745178679216_set_o: ( product_unit > set_o ) > set_Product_unit > set_set_o ).
thf(func_def_1603,type,
image_4571654851662262875nteger: ( product_unit > set_Code_integer ) > set_Product_unit > set_set_Code_integer ).
thf(func_def_1604,type,
image_3995998972318789802et_int: ( product_unit > set_int ) > set_Product_unit > set_set_int ).
thf(func_def_1605,type,
image_8173849991827986510et_nat: ( product_unit > set_nat ) > set_Product_unit > set_set_nat ).
thf(func_def_1606,type,
image_7778121751606633509t_unit: ( product_unit > set_Product_unit ) > set_Product_unit > set_set_Product_unit ).
thf(func_def_1607,type,
image_rat_rat: ( rat > rat ) > set_rat > set_rat ).
thf(func_def_1608,type,
image_8925030434418351941_nat_o: ( set_Pr4532377907799695533_nat_o > set_Pr4532377907799695533_nat_o ) > set_se1353816476878259085_nat_o > set_se1353816476878259085_nat_o ).
thf(func_def_1609,type,
image_set_o_set_o: ( set_o > set_o ) > set_set_o > set_set_o ).
thf(func_def_1610,type,
image_6270750266955655469nteger: ( set_Code_integer > set_Code_integer ) > set_set_Code_integer > set_set_Code_integer ).
thf(func_def_1611,type,
image_524474410958335435et_int: ( set_int > set_int ) > set_set_int > set_set_int ).
thf(func_def_1612,type,
image_set_nat_nat: ( set_nat > nat ) > set_set_nat > set_nat ).
thf(func_def_1613,type,
image_6916300684244607776nteger: ( set_nat > set_Code_integer ) > set_set_nat > set_set_Code_integer ).
thf(func_def_1614,type,
image_7916887816326733075et_nat: ( set_nat > set_nat ) > set_set_nat > set_set_nat ).
thf(func_def_1615,type,
image_6725021117256019401et_nat: ( set_nat > set_set_nat ) > set_set_nat > set_set_set_nat ).
thf(func_def_1616,type,
image_4019945301284682565t_unit: ( set_Product_unit > set_Product_unit ) > set_set_Product_unit > set_set_Product_unit ).
thf(func_def_1617,type,
insert5175938949040314269_nat_o: ( produc3658429121746597890et_nat > $o ) > set_Pr4532377907799695533_nat_o > set_Pr4532377907799695533_nat_o ).
thf(func_def_1618,type,
insert_o: $o > set_o > set_o ).
thf(func_def_1619,type,
insert_assn: assn > set_assn > set_assn ).
thf(func_def_1620,type,
insert_Code_integer: code_integer > set_Code_integer > set_Code_integer ).
thf(func_def_1621,type,
insert8563697753282147485t_unit: heap_e7401611519738050253t_unit > set_he8644174446132254637t_unit > set_he8644174446132254637t_unit ).
thf(func_def_1622,type,
insert_int: int > set_int > set_int ).
thf(func_def_1623,type,
insert_nat: nat > set_nat > set_nat ).
thf(func_def_1624,type,
insert5309882203591314743_nat_o: produc4928098042776334183_nat_o > set_Pr2161125870931222855_nat_o > set_Pr2161125870931222855_nat_o ).
thf(func_def_1625,type,
insert8297122186708525866et_nat: produc2732055786443039994et_nat > set_Pr8536935166611901872et_nat > set_Pr8536935166611901872et_nat ).
thf(func_def_1626,type,
insert3111075074545270702et_nat: produc3925858234332021118et_nat > set_Pr3286484037609594932et_nat > set_Pr3286484037609594932et_nat ).
thf(func_def_1627,type,
insert6201435330877294327od_o_o: product_prod_o_o > set_Product_prod_o_o > set_Product_prod_o_o ).
thf(func_def_1628,type,
insert1271470778893645362et_nat: produc3658429121746597890et_nat > set_Pr3948176798113811640et_nat > set_Pr3948176798113811640et_nat ).
thf(func_def_1629,type,
insert5033312907999012233nt_int: product_prod_int_int > set_Pr958786334691620121nt_int > set_Pr958786334691620121nt_int ).
thf(func_def_1630,type,
insert8211810215607154385at_nat: product_prod_nat_nat > set_Pr1261947904930325089at_nat > set_Pr1261947904930325089at_nat ).
thf(func_def_1631,type,
insert4616891024742550839t_unit: produc8459935480633519975t_unit > set_Pr5094982260447487303t_unit > set_Pr5094982260447487303t_unit ).
thf(func_def_1632,type,
insert3244108804717745463_nat_o: produc6666532470309664615_nat_o > set_Pr3813038829465207111_nat_o > set_Pr3813038829465207111_nat_o ).
thf(func_def_1633,type,
insert1494450971503307511_set_o: produc7369051934464679207_set_o > set_Pr4577759397028426247_set_o > set_Pr4577759397028426247_set_o ).
thf(func_def_1634,type,
insert7328952640415156725et_int: produc2115011035271226405et_int > set_Pr2522554150109002629et_int > set_Pr2522554150109002629et_int ).
thf(func_def_1635,type,
insert3810226134351308605et_nat: produc7819656566062154093et_nat > set_Pr5488025237498180813et_nat > set_Pr5488025237498180813et_nat ).
thf(func_def_1636,type,
insert6793119152790066487t_unit: produc1357844946079266663t_unit > set_Pr3980508049012409671t_unit > set_Pr3980508049012409671t_unit ).
thf(func_def_1637,type,
insert_Product_unit: product_unit > set_Product_unit > set_Product_unit ).
thf(func_def_1638,type,
insert_rat: rat > set_rat > set_rat ).
thf(func_def_1639,type,
insert2507855216428400253_nat_o: set_Pr4532377907799695533_nat_o > set_se1353816476878259085_nat_o > set_se1353816476878259085_nat_o ).
thf(func_def_1640,type,
insert_set_o: set_o > set_set_o > set_set_o ).
thf(func_def_1641,type,
insert_set_int: set_int > set_set_int > set_set_int ).
thf(func_def_1642,type,
insert_set_nat: set_nat > set_set_nat > set_set_nat ).
thf(func_def_1643,type,
insert8986028188087627965t_unit: set_Product_unit > set_set_Product_unit > set_set_Product_unit ).
thf(func_def_1644,type,
is_emp6650082560925436063_nat_o: set_Pr4532377907799695533_nat_o > $o ).
thf(func_def_1645,type,
is_empty_o: set_o > $o ).
thf(func_def_1646,type,
is_empty_int: set_int > $o ).
thf(func_def_1647,type,
is_empty_nat: set_nat > $o ).
thf(func_def_1648,type,
is_emp5529367068699929823t_unit: set_Product_unit > $o ).
thf(func_def_1649,type,
is_sin5180296473474724033_nat_o: set_Pr4532377907799695533_nat_o > $o ).
thf(func_def_1650,type,
is_singleton_o: set_o > $o ).
thf(func_def_1651,type,
is_singleton_int: set_int > $o ).
thf(func_def_1652,type,
is_singleton_nat: set_nat > $o ).
thf(func_def_1653,type,
is_sin2160648248035936513t_unit: set_Product_unit > $o ).
thf(func_def_1654,type,
pairwi2432143822833250616_nat_o: ( ( produc3658429121746597890et_nat > $o ) > ( produc3658429121746597890et_nat > $o ) > $o ) > set_Pr4532377907799695533_nat_o > $o ).
thf(func_def_1655,type,
pairwise_o: ( $o > $o > $o ) > set_o > $o ).
thf(func_def_1656,type,
pairwi4741737654671186738nteger: ( code_integer > code_integer > $o ) > set_Code_integer > $o ).
thf(func_def_1657,type,
pairwise_int: ( int > int > $o ) > set_int > $o ).
thf(func_def_1658,type,
pairwise_nat: ( nat > nat > $o ) > set_nat > $o ).
thf(func_def_1659,type,
pairwi7919127943033184120t_unit: ( product_unit > product_unit > $o ) > set_Product_unit > $o ).
thf(func_def_1660,type,
pairwise_set_nat: ( set_nat > set_nat > $o ) > set_set_nat > $o ).
thf(func_def_1661,type,
remove4651630035290841522_nat_o: ( produc3658429121746597890et_nat > $o ) > set_Pr4532377907799695533_nat_o > set_Pr4532377907799695533_nat_o ).
thf(func_def_1662,type,
remove_o: $o > set_o > set_o ).
thf(func_def_1663,type,
remove_int: int > set_int > set_int ).
thf(func_def_1664,type,
remove_nat: nat > set_nat > set_nat ).
thf(func_def_1665,type,
remove_Product_unit: product_unit > set_Product_unit > set_Product_unit ).
thf(func_def_1666,type,
the_el5959152443890766018_nat_o: set_Pr4532377907799695533_nat_o > produc3658429121746597890et_nat > $o ).
thf(func_def_1667,type,
the_elem_o: set_o > $o ).
thf(func_def_1668,type,
the_el4656377885804969768nteger: set_Code_integer > code_integer ).
thf(func_def_1669,type,
the_elem_int: set_int > int ).
thf(func_def_1670,type,
the_elem_nat: set_nat > nat ).
thf(func_def_1671,type,
the_el608902216710161154t_unit: set_Product_unit > product_unit ).
thf(func_def_1672,type,
the_elem_set_nat: set_set_nat > set_nat ).
thf(func_def_1673,type,
vimage_o_o: ( $o > $o ) > set_o > set_o ).
thf(func_def_1674,type,
vimage_o_int: ( $o > int ) > set_int > set_o ).
thf(func_def_1675,type,
vimage_o_nat: ( $o > nat ) > set_nat > set_o ).
thf(func_def_1676,type,
vimage8945963521958007626od_o_o: ( $o > product_prod_o_o ) > set_Product_prod_o_o > set_o ).
thf(func_def_1677,type,
vimage4304346610107161470_int_o: ( $o > product_prod_int_o ) > set_Pr903927857289325719_int_o > set_o ).
thf(func_def_1678,type,
vimage6125388955539849434_nat_o: ( $o > product_prod_nat_o ) > set_Pr3149072824959771635_nat_o > set_o ).
thf(func_def_1679,type,
vimage9156843197228287893unit_o: ( $o > produc7485711547916492274unit_o ) > set_Pr5464989305201766866unit_o > set_o ).
thf(func_def_1680,type,
vimage4222377557387472176t_unit: ( $o > product_unit ) > set_Product_unit > set_o ).
thf(func_def_1681,type,
vimage3005800680117336574eger_o: ( code_integer > $o ) > set_o > set_Code_integer ).
thf(func_def_1682,type,
vimage1318793397690208119nteger: ( code_integer > code_integer ) > set_Code_integer > set_Code_integer ).
thf(func_def_1683,type,
vimage5837348839036363078er_int: ( code_integer > int ) > set_int > set_Code_integer ).
thf(func_def_1684,type,
vimage5839839309545413354er_nat: ( code_integer > nat ) > set_nat > set_Code_integer ).
thf(func_def_1685,type,
vimage5651688891684837235t_unit: ( code_integer > product_unit ) > set_Product_unit > set_Code_integer ).
thf(func_def_1686,type,
vimage_int_o: ( int > $o ) > set_o > set_int ).
thf(func_def_1687,type,
vimage_int_int: ( int > int ) > set_int > set_int ).
thf(func_def_1688,type,
vimage_int_nat: ( int > nat ) > set_nat > set_int ).
thf(func_def_1689,type,
vimage7872673496510460816nt_int: ( int > product_prod_int_int ) > set_Pr958786334691620121nt_int > set_int ).
thf(func_def_1690,type,
vimage2000912565490760292t_unit: ( int > product_unit ) > set_Product_unit > set_int ).
thf(func_def_1691,type,
vimage8201027140005291904_nat_o: ( nat > produc3658429121746597890et_nat > $o ) > set_Pr4532377907799695533_nat_o > set_nat ).
thf(func_def_1692,type,
vimage_nat_o: ( nat > $o ) > set_o > set_nat ).
thf(func_def_1693,type,
vimage6104394758324456170nteger: ( nat > code_integer ) > set_Code_integer > set_nat ).
thf(func_def_1694,type,
vimage_nat_int: ( nat > int ) > set_int > set_nat ).
thf(func_def_1695,type,
vimage_nat_nat: ( nat > nat ) > set_nat > set_nat ).
thf(func_def_1696,type,
vimage5656379182745576100_o_nat: ( nat > product_prod_o_nat ) > set_Pr2101469702781467981_o_nat > set_nat ).
thf(func_def_1697,type,
vimage9012682431555523728nt_nat: ( nat > product_prod_int_nat ) > set_Pr3448869479623346877nt_nat > set_nat ).
thf(func_def_1698,type,
vimage8013328719654469172at_nat: ( nat > product_prod_nat_nat ) > set_Pr1261947904930325089at_nat > set_nat ).
thf(func_def_1699,type,
vimage4884490618288580032t_unit: ( nat > product_unit ) > set_Product_unit > set_nat ).
thf(func_def_1700,type,
vimage_nat_set_nat: ( nat > set_nat ) > set_set_nat > set_nat ).
thf(func_def_1701,type,
vimage7160910655466636085et_nat: ( produc3925858234332021118et_nat > produc2732055786443039994et_nat ) > set_Pr8536935166611901872et_nat > set_Pr3286484037609594932et_nat ).
thf(func_def_1702,type,
vimage1990136491886319165et_nat: ( produc3658429121746597890et_nat > produc3925858234332021118et_nat ) > set_Pr3286484037609594932et_nat > set_Pr3948176798113811640et_nat ).
thf(func_def_1703,type,
vimage8808348054653608838unit_o: ( product_unit > $o ) > set_o > set_Product_unit ).
thf(func_def_1704,type,
vimage6250838002967538110it_int: ( product_unit > int ) > set_int > set_Product_unit ).
thf(func_def_1705,type,
vimage6253328473476588386it_nat: ( product_unit > nat ) > set_nat > set_Product_unit ).
thf(func_def_1706,type,
vimage508909887551888138t_unit: ( product_unit > produc3788275437394581276t_unit ) > set_Pr6671815714118170108t_unit > set_Product_unit ).
thf(func_def_1707,type,
vimage4422279681566046568t_unit: ( product_unit > produc6865911496949736918t_unit ) > set_Pr5889264611543340172t_unit > set_Product_unit ).
thf(func_def_1708,type,
vimage7305857734363866308t_unit: ( product_unit > produc526117512892780850t_unit ) > set_Pr4334478416066269672t_unit > set_Product_unit ).
thf(func_def_1709,type,
vimage8114940889160087125t_unit: ( product_unit > produc8459935480633519975t_unit ) > set_Pr5094982260447487303t_unit > set_Product_unit ).
thf(func_def_1710,type,
vimage7995052115951654139t_unit: ( product_unit > product_unit ) > set_Product_unit > set_Product_unit ).
thf(func_def_1711,type,
vimage5412618565253405765et_nat: ( set_nat > produc3658429121746597890et_nat ) > set_Pr3948176798113811640et_nat > set_set_nat ).
thf(func_def_1712,type,
set_fo2584398358068434914at_nat: ( nat > nat > nat ) > nat > nat > nat > nat ).
thf(func_def_1713,type,
set_or189985376899183464nteger: code_integer > code_integer > set_Code_integer ).
thf(func_def_1714,type,
set_or1266510415728281911st_int: int > int > set_int ).
thf(func_def_1715,type,
set_or1269000886237332187st_nat: nat > nat > set_nat ).
thf(func_def_1716,type,
set_or8404916559141939852nteger: code_integer > code_integer > set_Code_integer ).
thf(func_def_1717,type,
set_or4665077453230672383an_nat: nat > nat > set_nat ).
thf(func_def_1718,type,
set_ord_atLeast_nat: nat > set_nat ).
thf(func_def_1719,type,
set_ord_atMost_nat: nat > set_nat ).
thf(func_def_1720,type,
set_or2715278749043346189nteger: code_integer > code_integer > set_Code_integer ).
thf(func_def_1721,type,
set_or4266950643985792945nteger: code_integer > code_integer > set_Code_integer ).
thf(func_def_1722,type,
set_or1210151606488870762an_nat: nat > set_nat ).
thf(func_def_1723,type,
set_ord_lessThan_nat: nat > set_nat ).
thf(func_def_1724,type,
char2: $o > $o > $o > $o > $o > $o > $o > $o > char ).
thf(func_def_1725,type,
integer_of_char: char > code_integer ).
thf(func_def_1726,type,
sum_ca6763686470577984908at_nat: ( nat > nat ) > ( nat > nat ) > sum_sum_nat_nat > nat ).
thf(func_def_1727,type,
syntax7398250324933576852n_assn: assn > assn > $o ).
thf(func_def_1728,type,
syntax5678989248478167196nt_int: int > int > $o ).
thf(func_def_1729,type,
syntax4682126007086162916at_nat: nat > nat > $o ).
thf(func_def_1730,type,
syntax3730441303064801268at_rat: rat > rat > $o ).
thf(func_def_1731,type,
transi3136567867383547549ic_int: set_Pr958786334691620121nt_int > $o ).
thf(func_def_1732,type,
transi3139058337892597825ic_nat: set_Pr1261947904930325089at_nat > $o ).
thf(func_def_1733,type,
transi712980445759334096_nat_o: set_Pr2161125870931222855_nat_o > set_Pr2161125870931222855_nat_o ).
thf(func_def_1734,type,
transitive_rtrancl_o: set_Product_prod_o_o > set_Product_prod_o_o ).
thf(func_def_1735,type,
transi3984306202085910426nteger: set_Pr4811707699266497531nteger > set_Pr4811707699266497531nteger ).
thf(func_def_1736,type,
transi2902850859426252137cl_int: set_Pr958786334691620121nt_int > set_Pr958786334691620121nt_int ).
thf(func_def_1737,type,
transi2905341329935302413cl_nat: set_Pr1261947904930325089at_nat > set_Pr1261947904930325089at_nat ).
thf(func_def_1738,type,
transi344104063936286263et_nat: set_Pr3444600963470892981et_nat > set_Pr3444600963470892981et_nat ).
thf(func_def_1739,type,
transi2414916928880341691et_nat: set_Pr7928877670098842301et_nat > set_Pr7928877670098842301et_nat ).
thf(func_def_1740,type,
transi6919426408697524543et_nat: set_Pr719794911490849221et_nat > set_Pr719794911490849221et_nat ).
thf(func_def_1741,type,
transi5430920176653612092nt_int: set_Pr2560585780119916871nt_int > set_Pr2560585780119916871nt_int ).
thf(func_def_1742,type,
transi8609417484261754244at_nat: set_Pr8693737435421807431at_nat > set_Pr8693737435421807431at_nat ).
thf(func_def_1743,type,
transi6481887417741755664t_unit: set_Pr5094982260447487303t_unit > set_Pr5094982260447487303t_unit ).
thf(func_def_1744,type,
transi4009562729418357614_nat_o: set_Pr2161125870931222855_nat_o > set_Pr2161125870931222855_nat_o ).
thf(func_def_1745,type,
transitive_trancl_o: set_Product_prod_o_o > set_Product_prod_o_o ).
thf(func_def_1746,type,
transi6261509568448316235cl_int: set_Pr958786334691620121nt_int > set_Pr958786334691620121nt_int ).
thf(func_def_1747,type,
transi6264000038957366511cl_nat: set_Pr1261947904930325089at_nat > set_Pr1261947904930325089at_nat ).
thf(func_def_1748,type,
transi5221092739591632921et_nat: set_Pr3444600963470892981et_nat > set_Pr3444600963470892981et_nat ).
thf(func_def_1749,type,
transi3145040225084697757et_nat: set_Pr7928877670098842301et_nat > set_Pr7928877670098842301et_nat ).
thf(func_def_1750,type,
transi4137310414928362273et_nat: set_Pr719794911490849221et_nat > set_Pr719794911490849221et_nat ).
thf(func_def_1751,type,
transi6288783178788033498nt_int: set_Pr2560585780119916871nt_int > set_Pr2560585780119916871nt_int ).
thf(func_def_1752,type,
transi243908449541399842at_nat: set_Pr8693737435421807431at_nat > set_Pr8693737435421807431at_nat ).
thf(func_def_1753,type,
transi430408012122777006t_unit: set_Pr5094982260447487303t_unit > set_Pr5094982260447487303t_unit ).
thf(func_def_1754,type,
type_d5515310227294319076nteger: ( ( code_integer > code_integer ) > code_integer ) > ( code_integer > code_integer > code_integer ) > set_Code_integer > $o ).
thf(func_def_1755,type,
type_d5661033022651575429eger_o: ( ( produc8923325533196201883nteger > produc8923325533196201883nteger ) > $o ) > ( $o > produc8923325533196201883nteger > produc8923325533196201883nteger ) > set_o > $o ).
thf(func_def_1756,type,
type_d6544512618915850416nteger: ( ( produc8923325533196201883nteger > produc8923325533196201883nteger ) > code_integer ) > ( code_integer > produc8923325533196201883nteger > produc8923325533196201883nteger ) > set_Code_integer > $o ).
thf(func_def_1757,type,
type_d6278479724763174947er_nat: ( ( produc8923325533196201883nteger > produc8923325533196201883nteger ) > nat ) > ( nat > produc8923325533196201883nteger > produc8923325533196201883nteger ) > set_nat > $o ).
thf(func_def_1758,type,
type_d6996789741066924922t_unit: ( ( produc8923325533196201883nteger > produc8923325533196201883nteger ) > product_unit ) > ( product_unit > produc8923325533196201883nteger > produc8923325533196201883nteger ) > set_Product_unit > $o ).
thf(func_def_1759,type,
type_d3909072315231072503_nat_o: ( assn > produc3658429121746597890et_nat > $o ) > ( ( produc3658429121746597890et_nat > $o ) > assn ) > set_Pr4532377907799695533_nat_o > $o ).
thf(func_def_1760,type,
type_d669523880323286244nteger: ( code_integer > code_integer > code_integer ) > ( ( code_integer > code_integer ) > code_integer ) > set_Co726186806682698383nteger > $o ).
thf(func_def_1761,type,
type_d6812736842639505741eger_o: ( code_integer > $o ) > ( $o > code_integer ) > set_o > $o ).
thf(func_def_1762,type,
type_d4796611474492816872nteger: ( code_integer > code_integer ) > ( code_integer > code_integer ) > set_Code_integer > $o ).
thf(func_def_1763,type,
type_d8366093980585677751er_int: ( code_integer > int ) > ( int > code_integer ) > set_int > $o ).
thf(func_def_1764,type,
type_d8368584451094728027er_nat: ( code_integer > nat ) > ( nat > code_integer ) > set_nat > $o ).
thf(func_def_1765,type,
type_d9006111508436661314t_unit: ( code_integer > product_unit ) > ( product_unit > code_integer ) > set_Product_unit > $o ).
thf(func_def_1766,type,
type_d4410041424927559462al_nat: ( code_natural > nat ) > ( nat > code_natural ) > set_nat > $o ).
thf(func_def_1767,type,
type_d8633139899873770843nteger: ( nat > code_integer ) > ( code_integer > nat ) > set_Code_integer > $o ).
thf(func_def_1768,type,
type_d6250493948777748686at_nat: ( nat > nat ) > ( nat > nat ) > set_nat > $o ).
thf(func_def_1769,type,
type_d3768499744764857476et_nat: ( nat > set_nat ) > ( set_nat > nat ) > set_set_nat > $o ).
thf(func_def_1770,type,
type_d6188575255521822967unit_o: ( product_unit > $o ) > ( $o > product_unit ) > set_o > $o ).
thf(func_def_1771,type,
type_d202864840109488260at_nat: ( set_nat > nat ) > ( nat > set_nat ) > set_nat > $o ).
thf(func_def_1772,type,
accp_P4507279754199951662nt_int: ( produc5030937667867769463nt_int > produc5030937667867769463nt_int > $o ) > produc5030937667867769463nt_int > $o ).
thf(func_def_1773,type,
accp_P5604362734843950218nt_int: ( produc6128020648511768019nt_int > produc6128020648511768019nt_int > $o ) > produc6128020648511768019nt_int > $o ).
thf(func_def_1774,type,
accp_P3050404660685545426at_nat: ( produc3574062574353363227at_nat > produc3574062574353363227at_nat > $o ) > produc3574062574353363227at_nat > $o ).
thf(func_def_1775,type,
accp_P4147487641329543982at_nat: ( produc4671145554997361783at_nat > produc4671145554997361783at_nat > $o ) > produc4671145554997361783at_nat > $o ).
thf(func_def_1776,type,
accp_P1862375125659990705et_nat: ( produc2732055786443039994et_nat > produc2732055786443039994et_nat > $o ) > produc2732055786443039994et_nat > $o ).
thf(func_def_1777,type,
accp_P8458817951426537472et_nat: ( produc6197004810343482825et_nat > produc6197004810343482825et_nat > $o ) > produc6197004810343482825et_nat > $o ).
thf(func_def_1778,type,
accp_P5801069581201407417et_nat: ( produc3658429121746597890et_nat > produc3658429121746597890et_nat > $o ) > produc3658429121746597890et_nat > $o ).
thf(func_def_1779,type,
accp_P1096762738010456898nt_int: ( product_prod_int_int > product_prod_int_int > $o ) > product_prod_int_int > $o ).
thf(func_def_1780,type,
accp_P4275260045618599050at_nat: ( product_prod_nat_nat > product_prod_nat_nat > $o ) > product_prod_nat_nat > $o ).
thf(func_def_1781,type,
lex_pr4722427456421979612et_nat: set_Pr2161125870931222855_nat_o > set_Pr7928877670098842301et_nat > set_Pr3444600963470892981et_nat ).
thf(func_def_1782,type,
lex_pr5340572901959109728et_nat: set_Pr2161125870931222855_nat_o > set_Pr719794911490849221et_nat > set_Pr7928877670098842301et_nat ).
thf(func_def_1783,type,
lex_pr538634069021852900et_nat: set_Pr1776176089890104647t_unit > set_Pr5488025237498180813et_nat > set_Pr719794911490849221et_nat ).
thf(func_def_1784,type,
lex_prod_int_int: set_Pr958786334691620121nt_int > set_Pr958786334691620121nt_int > set_Pr2560585780119916871nt_int ).
thf(func_def_1785,type,
lex_prod_int_nat: set_Pr958786334691620121nt_int > set_Pr1261947904930325089at_nat > set_Pr5696620967126086215nt_nat ).
thf(func_def_1786,type,
lex_prod_nat_int: set_Pr1261947904930325089at_nat > set_Pr958786334691620121nt_int > set_Pr5557702248415638087at_int ).
thf(func_def_1787,type,
lex_prod_nat_nat: set_Pr1261947904930325089at_nat > set_Pr1261947904930325089at_nat > set_Pr8693737435421807431at_nat ).
thf(func_def_1788,type,
max_ex5265246208886859912_nat_o: set_Pr2161125870931222855_nat_o > set_Pr3813038829465207111_nat_o ).
thf(func_def_1789,type,
max_ext_o: set_Product_prod_o_o > set_Pr4577759397028426247_set_o ).
thf(func_def_1790,type,
max_ext_Code_integer: set_Pr4811707699266497531nteger > set_Pr7577011563204128103nteger ).
thf(func_def_1791,type,
max_ext_int: set_Pr958786334691620121nt_int > set_Pr2522554150109002629et_int ).
thf(func_def_1792,type,
max_ext_nat: set_Pr1261947904930325089at_nat > set_Pr5488025237498180813et_nat ).
thf(func_def_1793,type,
max_ext_Product_unit: set_Pr5094982260447487303t_unit > set_Pr3980508049012409671t_unit ).
thf(func_def_1794,type,
measure_int: ( int > nat ) > set_Pr958786334691620121nt_int ).
thf(func_def_1795,type,
measure_nat: ( nat > nat ) > set_Pr1261947904930325089at_nat ).
thf(func_def_1796,type,
min_ex3730736537596016502_nat_o: set_Pr2161125870931222855_nat_o > set_Pr3813038829465207111_nat_o ).
thf(func_def_1797,type,
min_ext_o: set_Product_prod_o_o > set_Pr4577759397028426247_set_o ).
thf(func_def_1798,type,
min_ext_int: set_Pr958786334691620121nt_int > set_Pr2522554150109002629et_int ).
thf(func_def_1799,type,
min_ext_nat: set_Pr1261947904930325089at_nat > set_Pr5488025237498180813et_nat ).
thf(func_def_1800,type,
min_ext_Product_unit: set_Pr5094982260447487303t_unit > set_Pr3980508049012409671t_unit ).
thf(func_def_1801,type,
wf_Pro8787509627421085110_nat_o: set_Pr2161125870931222855_nat_o > $o ).
thf(func_def_1802,type,
wf_o: set_Product_prod_o_o > $o ).
thf(func_def_1803,type,
wf_int: set_Pr958786334691620121nt_int > $o ).
thf(func_def_1804,type,
wf_nat: set_Pr1261947904930325089at_nat > $o ).
thf(func_def_1805,type,
wf_Product_unit: set_Pr5094982260447487303t_unit > $o ).
thf(func_def_1806,type,
chains_int: set_Pr958786334691620121nt_int > set_set_int ).
thf(func_def_1807,type,
chains_nat: set_Pr1261947904930325089at_nat > set_set_nat ).
thf(func_def_1808,type,
chains_nat2: set_set_nat > set_set_set_nat ).
thf(func_def_1809,type,
pred_c177362287423414904_nat_o: set_Pr4532377907799695533_nat_o > ( ( produc3658429121746597890et_nat > $o ) > ( produc3658429121746597890et_nat > $o ) > $o ) > set_Pr4532377907799695533_nat_o > $o ).
thf(func_def_1810,type,
pred_chain_o: set_o > ( $o > $o > $o ) > set_o > $o ).
thf(func_def_1811,type,
pred_chain_int: set_int > ( int > int > $o ) > set_int > $o ).
thf(func_def_1812,type,
pred_chain_nat: set_nat > ( nat > nat > $o ) > set_nat > $o ).
thf(func_def_1813,type,
pred_c6915491785556283576t_unit: set_Product_unit > ( product_unit > product_unit > $o ) > set_Product_unit > $o ).
thf(func_def_1814,type,
pred_c5891848916688490280nteger: set_set_Code_integer > ( set_Code_integer > set_Code_integer > $o ) > set_set_Code_integer > $o ).
thf(func_def_1815,type,
pred_chain_set_nat: set_set_nat > ( set_nat > set_nat > $o ) > set_set_nat > $o ).
thf(func_def_1816,type,
member6576561426505652726_nat_o: ( produc3658429121746597890et_nat > $o ) > set_Pr4532377907799695533_nat_o > $o ).
thf(func_def_1817,type,
member_o: $o > set_o > $o ).
thf(func_def_1818,type,
member_assn: assn > set_assn > $o ).
thf(func_def_1819,type,
member_Code_integer: code_integer > set_Code_integer > $o ).
thf(func_def_1820,type,
member_filter_nat: filter_nat > set_filter_nat > $o ).
thf(func_def_1821,type,
member242389114213480694t_unit: heap_e7401611519738050253t_unit > set_he8644174446132254637t_unit > $o ).
thf(func_def_1822,type,
member_int: int > set_int > $o ).
thf(func_def_1823,type,
member_nat: nat > set_nat > $o ).
thf(func_def_1824,type,
member8781333585448626064_nat_o: produc4928098042776334183_nat_o > set_Pr2161125870931222855_nat_o > $o ).
thf(func_def_1825,type,
member6124377750444531601et_nat: produc2732055786443039994et_nat > set_Pr8536935166611901872et_nat > $o ).
thf(func_def_1826,type,
member1996754912294343701et_nat: produc3925858234332021118et_nat > set_Pr3286484037609594932et_nat > $o ).
thf(func_def_1827,type,
member7466972457876170832od_o_o: product_prod_o_o > set_Product_prod_o_o > $o ).
thf(func_def_1828,type,
member7847949116333733898_o_int: product_prod_o_int > set_Pr8834758594704517033_o_int > $o ).
thf(func_def_1829,type,
member2802428098988154798_o_nat: product_prod_o_nat > set_Pr2101469702781467981_o_nat > $o ).
thf(func_def_1830,type,
member9215336023991391429t_unit: produc3788275437394581276t_unit > set_Pr6671815714118170108t_unit > $o ).
thf(func_def_1831,type,
member157494554546826820nteger: produc8923325533196201883nteger > set_Pr4811707699266497531nteger > $o ).
thf(func_def_1832,type,
member5046970461208427408t_unit: produc8126358835630522215t_unit > set_Pr1776176089890104647t_unit > $o ).
thf(func_def_1833,type,
member6260224972018164377et_nat: produc3658429121746597890et_nat > set_Pr3948176798113811640et_nat > $o ).
thf(func_def_1834,type,
member4489920277610959864_int_o: product_prod_int_o > set_Pr903927857289325719_int_o > $o ).
thf(func_def_1835,type,
member5262025264175285858nt_int: product_prod_int_int > set_Pr958786334691620121nt_int > $o ).
thf(func_def_1836,type,
member216504246829706758nt_nat: product_prod_int_nat > set_Pr3448869479623346877nt_nat > $o ).
thf(func_def_1837,type,
member6688357403439981677t_unit: produc6865911496949736918t_unit > set_Pr5889264611543340172t_unit > $o ).
thf(func_def_1838,type,
member6310962623043647828_nat_o: product_prod_nat_o > set_Pr3149072824959771635_nat_o > $o ).
thf(func_def_1839,type,
member4262671552274231302at_int: product_prod_nat_int > set_Pr7995236796853374141at_int > $o ).
thf(func_def_1840,type,
member8440522571783428010at_nat: product_prod_nat_nat > set_Pr1261947904930325089at_nat > $o ).
thf(func_def_1841,type,
member348563419383025609t_unit: produc526117512892780850t_unit > set_Pr4334478416066269672t_unit > $o ).
thf(func_def_1842,type,
member6341495586645257982et_nat: produc5657529347773406293et_nat > set_Pr3444600963470892981et_nat > $o ).
thf(func_def_1843,type,
member8354763867512313179_nat_o: produc4962441903510821572_nat_o > set_Pr7705059561495379578_nat_o > $o ).
thf(func_def_1844,type,
member4763271486408492550et_nat: produc6830853553727218525et_nat > set_Pr7928877670098842301et_nat > $o ).
thf(func_def_1845,type,
member6613709072892554et_nat: produc4658369026151987169et_nat > set_Pr3721596426557623617et_nat > $o ).
thf(func_def_1846,type,
member4978015717948814295_nat_o: produc6907119039986491712_nat_o > set_Pr4208978187861016310_nat_o > $o ).
thf(func_def_1847,type,
member2737525317995836554et_nat: produc7389280635074931169et_nat > set_Pr8807100098903490369et_nat > $o ).
thf(func_def_1848,type,
member6099555550032318734et_nat: produc8111630337999740517et_nat > set_Pr719794911490849221et_nat > $o ).
thf(func_def_1849,type,
member8566619992076573584nt_int: produc1219242969750017639nt_int > set_Pr2560585780119916871nt_int > $o ).
thf(func_def_1850,type,
member2930414080797523088nt_nat: produc4806409095325742951nt_nat > set_Pr5696620967126086215nt_nat > $o ).
thf(func_def_1851,type,
member4619661753630440592at_int: produc6495656768158660455at_int > set_Pr5557702248415638087at_int > $o ).
thf(func_def_1852,type,
member8206827879206165904at_nat: produc859450856879609959at_nat > set_Pr8693737435421807431at_nat > $o ).
thf(func_def_1853,type,
member3689400097658526619unit_o: produc7485711547916492274unit_o > set_Pr5464989305201766866unit_o > $o ).
thf(func_def_1854,type,
member5468524742225079679it_int: produc5646078835734834920it_int > set_Pr8497134830871917726it_int > $o ).
thf(func_def_1855,type,
member423003724879500579it_nat: produc600557818389255820it_nat > set_Pr1763845938948868674it_nat > $o ).
thf(func_def_1856,type,
member7821138191923116944t_unit: produc8459935480633519975t_unit > set_Pr5094982260447487303t_unit > $o ).
thf(func_def_1857,type,
member6623172407746373520_nat_o: produc6666532470309664615_nat_o > set_Pr3813038829465207111_nat_o > $o ).
thf(func_def_1858,type,
member9116954335612470352_set_o: produc7369051934464679207_set_o > set_Pr4577759397028426247_set_o > $o ).
thf(func_def_1859,type,
member4307123515891402160nteger: produc6491284506569428743nteger > set_Pr7577011563204128103nteger > $o ).
thf(func_def_1860,type,
member2572552093476627150et_int: produc2115011035271226405et_int > set_Pr2522554150109002629et_int > $o ).
thf(func_def_1861,type,
member727419437372278867t_unit: produc7348995623955488188t_unit > set_Pr4071100604818519410t_unit > $o ).
thf(func_def_1862,type,
member8277197624267554838et_nat: produc7819656566062154093et_nat > set_Pr5488025237498180813et_nat > $o ).
thf(func_def_1863,type,
member4189953012975926160t_unit: produc1357844946079266663t_unit > set_Pr3980508049012409671t_unit > $o ).
thf(func_def_1864,type,
member_Product_unit: product_unit > set_Product_unit > $o ).
thf(func_def_1865,type,
member_rat: rat > set_rat > $o ).
thf(func_def_1866,type,
member1452989533377600726_nat_o: set_Pr4532377907799695533_nat_o > set_se1353816476878259085_nat_o > $o ).
thf(func_def_1867,type,
member_set_o: set_o > set_set_o > $o ).
thf(func_def_1868,type,
member1834298987299537450nteger: set_Code_integer > set_set_Code_integer > $o ).
thf(func_def_1869,type,
member_set_int: set_int > set_set_int > $o ).
thf(func_def_1870,type,
member_set_nat: set_nat > set_set_nat > $o ).
thf(func_def_1871,type,
member5475208728337331568_nat_o: set_Pr2161125870931222855_nat_o > set_se7663487083892625191_nat_o > $o ).
thf(func_def_1872,type,
member3157015407008874800od_o_o: set_Product_prod_o_o > set_se491199571785101031od_o_o > $o ).
thf(func_def_1873,type,
member2340774599025711042nt_int: set_Pr958786334691620121nt_int > set_se6260736226359567993nt_int > $o ).
thf(func_def_1874,type,
member2643936169264416010at_nat: set_Pr1261947904930325089at_nat > set_se7855581050983116737at_nat > $o ).
thf(func_def_1875,type,
member1248447801810324848t_unit: set_Pr5094982260447487303t_unit > set_se8994758365886191399t_unit > $o ).
thf(func_def_1876,type,
member5877623283571906838t_unit: set_Product_unit > set_set_Product_unit > $o ).
thf(func_def_1877,type,
member_set_set_nat: set_set_nat > set_set_set_nat > $o ).
thf(func_def_1878,type,
h: produc3658429121746597890et_nat ).
thf(func_def_1897,type,
sP0: set_nat > nat > set_nat > $o ).
thf(func_def_1898,type,
sP1: set_nat > nat > set_nat > $o ).
thf(func_def_1899,type,
sK2: set_nat > set_nat > nat ).
thf(func_def_1900,type,
sK3: produc3658429121746597890et_nat > $o > heap_e7401611519738050253t_unit ).
thf(func_def_1901,type,
sK4: produc3658429121746597890et_nat > $o > set_nat ).
thf(func_def_1902,type,
sK5: assn > produc3658429121746597890et_nat ).
thf(func_def_1903,type,
sK6: assn > produc3658429121746597890et_nat ).
thf(func_def_1904,type,
sK7: set_nat > nat ).
thf(func_def_1905,type,
sK8: set_nat > set_nat > set_nat > set_nat ).
thf(func_def_1906,type,
sK9: set_nat > set_nat > set_nat > set_nat ).
thf(func_def_1907,type,
sK10: produc3658429121746597890et_nat > set_nat ).
thf(func_def_1908,type,
sK11: produc3658429121746597890et_nat > heap_e7401611519738050253t_unit ).
thf(func_def_1909,type,
sK12: set_Pr4532377907799695533_nat_o > produc3658429121746597890et_nat > $o ).
thf(func_def_1910,type,
sK13: ( produc3658429121746597890et_nat > $o ) > assn > produc3658429121746597890et_nat ).
thf(func_def_1911,type,
sK14: set_nat > nat ).
thf(func_def_1912,type,
sK15: set_nat > set_nat > nat ).
thf(func_def_1913,type,
sK16: produc3658429121746597890et_nat > heap_e7401611519738050253t_unit ).
thf(func_def_1914,type,
sK17: produc3658429121746597890et_nat > set_nat ).
thf(func_def_1915,type,
sK18: produc3658429121746597890et_nat > heap_e7401611519738050253t_unit ).
thf(func_def_1916,type,
sK19: produc3658429121746597890et_nat > set_nat ).
thf(func_def_1917,type,
sK20: ( ( produc3658429121746597890et_nat > $o ) > $o ) > ( ( produc3658429121746597890et_nat > $o ) > $o ) > produc3658429121746597890et_nat > $o ).
thf(func_def_1918,type,
sK21: set_Pr4532377907799695533_nat_o > produc3658429121746597890et_nat > $o ).
thf(func_def_1919,type,
sK22: assn > ( produc3658429121746597890et_nat > $o ) > produc3658429121746597890et_nat ).
thf(func_def_1920,type,
sK23: ( produc3658429121746597890et_nat > $o ) > assn ).
thf(func_def_1921,type,
sK24: produc3658429121746597890et_nat > heap_e7401611519738050253t_unit ).
thf(func_def_1922,type,
sK25: produc3658429121746597890et_nat > set_nat ).
thf(func_def_1923,type,
sK26: ( ( produc3658429121746597890et_nat > $o ) > $o ) > assn ).
thf(func_def_1924,type,
sK27: set_nat > nat ).
thf(func_def_1925,type,
sK28: ( ( produc3658429121746597890et_nat > $o ) > $o ) > ( ( produc3658429121746597890et_nat > $o ) > $o ) > produc3658429121746597890et_nat > $o ).
thf(func_def_1926,type,
sK29: ( nat > $o ) > set_nat > nat ).
thf(func_def_1927,type,
sK30: ( nat > $o ) > set_nat > nat ).
thf(func_def_1928,type,
sK31: ( nat > $o ) > set_nat > set_nat > nat ).
thf(func_def_1929,type,
sK32: assn > produc3658429121746597890et_nat > assn > set_nat ).
thf(func_def_1930,type,
sK33: assn > produc3658429121746597890et_nat > assn > heap_e7401611519738050253t_unit ).
thf(func_def_1931,type,
sK34: assn > produc3658429121746597890et_nat > assn > set_nat ).
thf(func_def_1932,type,
sK35: set_Pr4532377907799695533_nat_o > produc3658429121746597890et_nat > $o ).
thf(func_def_1933,type,
sK36: ( produc3658429121746597890et_nat > $o ) > set_nat ).
thf(func_def_1934,type,
sK37: ( produc3658429121746597890et_nat > $o ) > heap_e7401611519738050253t_unit ).
thf(func_def_1935,type,
sK38: set_nat > ( nat > $o ) > nat ).
thf(func_def_1936,type,
sK39: ( nat > $o ) > set_nat > nat ).
thf(func_def_1937,type,
sK40: set_nat > ( nat > $o ) > set_nat > nat ).
thf(func_def_1938,type,
sK41: set_nat > produc3658429121746597890et_nat > heap_e7401611519738050253t_unit ).
thf(func_def_1939,type,
sK42: assn > assn > produc3658429121746597890et_nat ).
thf(func_def_1940,type,
sK43: assn > assn > produc3658429121746597890et_nat ).
thf(func_def_1941,type,
sK44: assn > produc3658429121746597890et_nat > $o ).
thf(func_def_1942,type,
sK45: set_nat > set_nat > nat ).
thf(func_def_1943,type,
sK46: ( ( produc3658429121746597890et_nat > $o ) > $o ) > produc3658429121746597890et_nat > $o ).
thf(func_def_1944,type,
sK47: produc3658429121746597890et_nat > set_nat ).
thf(func_def_1945,type,
sK48: produc3658429121746597890et_nat > heap_e7401611519738050253t_unit ).
thf(func_def_1946,type,
sK49: ( ( produc3658429121746597890et_nat > $o ) > $o ) > produc3658429121746597890et_nat > $o ).
thf(func_def_1947,type,
sK50: ( assn > $o ) > produc3658429121746597890et_nat > $o ).
thf(func_def_1948,type,
sK51: set_nat > set_nat > nat ).
thf(func_def_1949,type,
sK52: set_nat > set_nat > nat ).
thf(func_def_1950,type,
sK53: set_Pr4532377907799695533_nat_o > ( ( produc3658429121746597890et_nat > $o ) > $o ) > ( ( produc3658429121746597890et_nat > $o ) > $o ) > produc3658429121746597890et_nat > $o ).
thf(func_def_1951,type,
sK54: set_Pr4532377907799695533_nat_o > produc3658429121746597890et_nat > $o ).
thf(func_def_1952,type,
sK55: set_nat > nat ).
thf(func_def_1953,type,
sK56: ( ( produc3658429121746597890et_nat > $o ) > $o ) > ( ( produc3658429121746597890et_nat > $o ) > $o ) > produc3658429121746597890et_nat > $o ).
thf(func_def_1954,type,
sK57: set_Pr4532377907799695533_nat_o > ( ( produc3658429121746597890et_nat > $o ) > $o ) > produc3658429121746597890et_nat > $o ).
thf(func_def_1956,type,
ph59:
!>[X0: $tType] : X0 ).
thf(func_def_1957,type,
sK60: produc3658429121746597890et_nat ).
thf(func_def_1958,type,
sK61: assn > assn > produc3658429121746597890et_nat ).
thf(func_def_1959,type,
sK62: assn > assn > produc3658429121746597890et_nat ).
thf(func_def_1960,type,
sK63: assn > assn > produc3658429121746597890et_nat ).
thf(func_def_1961,type,
sK64: assn > produc3658429121746597890et_nat ).
thf(func_def_1962,type,
sK65: produc3658429121746597890et_nat ).
thf(f26173,plain,
$false,
inference(avatar_sat_refutation,[status(thm)],[f23972,f23975,f25697,f26172]) ).
thf(f26172,plain,
( ~ spl58_1
| spl58_2 ),
inference(avatar_contradiction_clause,[status(thm)],[f26171]) ).
thf(f26171,plain,
( $false
| ~ spl58_1
| spl58_2 ),
inference(subsumption_resolution,[status(thm)],[f26170,f26128]) ).
thf(f26128,plain,
( ( ( one_assn_raw @ h )
!= $true )
| spl58_2 ),
inference(superposition,[status(thm)],[f23970,f24618]) ).
thf(f24618,plain,
! [X1: produc3658429121746597890et_nat] :
( ( one_assn_raw @ X1 )
= ( rep_assn @ ( abs_assn @ one_assn_raw ) @ X1 ) ),
inference(argument_congruence,[status(thm)],[f24167]) ).
thf(f24167,plain,
( one_assn_raw
= ( rep_assn @ ( abs_assn @ one_assn_raw ) ) ),
inference(trivial_inequality_removal,[status(thm)],[f24165]) ).
thf(f24165,plain,
( ( one_assn_raw
= ( rep_assn @ ( abs_assn @ one_assn_raw ) ) )
| ( $true != $true ) ),
inference(superposition,[status(thm)],[f23977,f23559]) ).
thf(f23559,plain,
( ( proper @ one_assn_raw )
= $true ),
inference(cnf_transformation,[status(thm)],[f10142]) ).
thf(f10142,plain,
( ( proper @ one_assn_raw )
= $true ),
inference(fool_elimination,[status(thm)],[f10141]) ).
thf(f10141,plain,
proper @ one_assn_raw,
inference(rectify,[status(thm)],[f89]) ).
thf(f89,axiom,
proper @ one_assn_raw,
file('/export/starexec/sandbox/tmp/tmp.2EMJwA8PHy/DTF2THF_22696.p',fact_88_one__assn__proper) ).
thf(f23977,plain,
! [X0: produc3658429121746597890et_nat > $o] :
( ( ( proper @ X0 )
!= $true )
| ( ( rep_assn @ ( abs_assn @ X0 ) )
= X0 ) ),
inference(backward_demodulation,[status(thm)],[f23620,f23648]) ).
thf(f23648,plain,
! [X0: produc3658429121746597890et_nat > $o,X1: ( produc3658429121746597890et_nat > $o ) > $o] :
( ( X1 @ X0 )
= ( member6576561426505652726_nat_o @ X0 @ ( collec939566748876313656_nat_o @ X1 ) ) ),
inference(cnf_transformation,[status(thm)],[f12283]) ).
thf(f12283,plain,
! [X0: produc3658429121746597890et_nat > $o,X1: ( produc3658429121746597890et_nat > $o ) > $o] :
( ( X1 @ X0 )
= ( member6576561426505652726_nat_o @ X0 @ ( collec939566748876313656_nat_o @ X1 ) ) ),
inference(fool_elimination,[status(thm)],[f12282]) ).
thf(f12282,plain,
! [X0: produc3658429121746597890et_nat > $o,X1: ( produc3658429121746597890et_nat > $o ) > $o] :
( ( X1 @ X0 )
= ( member6576561426505652726_nat_o @ X0 @ ( collec939566748876313656_nat_o @ X1 ) ) ),
inference(rectify,[status(thm)],[f103]) ).
thf(f103,axiom,
! [X8: produc3658429121746597890et_nat > $o,X2: ( produc3658429121746597890et_nat > $o ) > $o] :
( ( member6576561426505652726_nat_o @ X8 @ ( collec939566748876313656_nat_o @ X2 ) )
= ( X2 @ X8 ) ),
file('/export/starexec/sandbox/tmp/tmp.2EMJwA8PHy/DTF2THF_22696.p',fact_102_mem__Collect__eq) ).
thf(f23620,plain,
! [X0: produc3658429121746597890et_nat > $o] :
( ( ( member6576561426505652726_nat_o @ X0 @ ( collec939566748876313656_nat_o @ proper ) )
!= $true )
| ( ( rep_assn @ ( abs_assn @ X0 ) )
= X0 ) ),
inference(cnf_transformation,[status(thm)],[f23198]) ).
thf(f23198,plain,
! [X0: produc3658429121746597890et_nat > $o] :
( ( ( member6576561426505652726_nat_o @ X0 @ ( collec939566748876313656_nat_o @ proper ) )
!= $true )
| ( ( rep_assn @ ( abs_assn @ X0 ) )
= X0 ) ),
inference(ennf_transformation,[status(thm)],[f11495]) ).
thf(f11495,plain,
! [X0: produc3658429121746597890et_nat > $o] :
( ( ( member6576561426505652726_nat_o @ X0 @ ( collec939566748876313656_nat_o @ proper ) )
= $true )
=> ( ( rep_assn @ ( abs_assn @ X0 ) )
= X0 ) ),
inference(fool_elimination,[status(thm)],[f11494]) ).
thf(f11494,plain,
! [X0: produc3658429121746597890et_nat > $o] :
( ( member6576561426505652726_nat_o @ X0 @ ( collec939566748876313656_nat_o @ proper ) )
=> ( ( rep_assn @ ( abs_assn @ X0 ) )
= X0 ) ),
inference(rectify,[status(thm)],[f78]) ).
thf(f78,axiom,
! [X1: produc3658429121746597890et_nat > $o] :
( ( member6576561426505652726_nat_o @ X1 @ ( collec939566748876313656_nat_o @ proper ) )
=> ( ( rep_assn @ ( abs_assn @ X1 ) )
= X1 ) ),
file('/export/starexec/sandbox/tmp/tmp.2EMJwA8PHy/DTF2THF_22696.p',fact_77_Abs__assn__inverse) ).
thf(f23970,plain,
( ( ( rep_assn @ ( abs_assn @ one_assn_raw ) @ h )
!= $true )
| spl58_2 ),
inference(avatar_component_clause,[status(thm)],[f23969]) ).
thf(f23969,definition,
( spl58_2
<=> ( ( rep_assn @ ( abs_assn @ one_assn_raw ) @ h )
= $true ) ),
introduced(definition,[new_symbols(naming,[spl58_2])],[avatar_definition]) ).
thf(f26170,plain,
( ( ( one_assn_raw @ h )
= $true )
| ~ spl58_1 ),
inference(equality_resolution,[status(thm)],[f25959]) ).
thf(f25959,plain,
( ! [X0: produc3658429121746597890et_nat] :
( ( ( sK47 @ h )
!= ( sK47 @ X0 ) )
| ( ( one_assn_raw @ X0 )
= $true ) )
| ~ spl58_1 ),
inference(forward_demodulation,[status(thm)],[f24604,f25886]) ).
thf(f25886,plain,
( ( bot_bot_set_nat
= ( sK47 @ h ) )
| ~ spl58_1 ),
inference(forward_demodulation,[status(thm)],[f23967,f24025]) ).
thf(f24025,plain,
! [X0: produc3658429121746597890et_nat] :
( ( produc8586169260539613262et_nat @ X0 )
= ( sK47 @ X0 ) ),
inference(superposition,[status(thm)],[f23857,f23649]) ).
thf(f23649,plain,
! [X0: produc3658429121746597890et_nat] :
( ( produc7507926704131184380et_nat @ ( sK48 @ X0 ) @ ( sK47 @ X0 ) )
= X0 ),
inference(cnf_transformation,[status(thm)],[f23336]) ).
thf(f23336,plain,
! [X0: produc3658429121746597890et_nat] :
( ( produc7507926704131184380et_nat @ ( sK48 @ X0 ) @ ( sK47 @ X0 ) )
= X0 ),
inference(skolemisation,[status(esa),new_symbols(skolem,[sK47,sK48])],[f23167,f23335]) ).
thf(f23335,plain,
! [X0: produc3658429121746597890et_nat] :
( ? [X1: set_nat,X2: heap_e7401611519738050253t_unit] :
( ( produc7507926704131184380et_nat @ X2 @ X1 )
= X0 )
=> ( ( produc7507926704131184380et_nat @ ( sK48 @ X0 ) @ ( sK47 @ X0 ) )
= X0 ) ),
introduced(definition,[],[choice_axiom]) ).
thf(f23167,plain,
! [X0: produc3658429121746597890et_nat] :
? [X1: set_nat,X2: heap_e7401611519738050253t_unit] :
( ( produc7507926704131184380et_nat @ X2 @ X1 )
= X0 ),
inference(ennf_transformation,[status(thm)],[f9906]) ).
thf(f9906,plain,
! [X0: produc3658429121746597890et_nat] :
~ ! [X1: set_nat,X2: heap_e7401611519738050253t_unit] :
( ( produc7507926704131184380et_nat @ X2 @ X1 )
!= X0 ),
inference(fool_elimination,[status(thm)],[f9905]) ).
thf(f9905,plain,
! [X0: produc3658429121746597890et_nat] :
~ ! [X1: set_nat,X2: heap_e7401611519738050253t_unit] :
( ( produc7507926704131184380et_nat @ X2 @ X1 )
!= X0 ),
inference(rectify,[status(thm)],[f161]) ).
thf(f161,axiom,
! [X1: produc3658429121746597890et_nat] :
~ ! [X21: set_nat,X20: heap_e7401611519738050253t_unit] :
( ( produc7507926704131184380et_nat @ X20 @ X21 )
!= X1 ),
file('/export/starexec/sandbox/tmp/tmp.2EMJwA8PHy/DTF2THF_22696.p',fact_160_old_Oprod_Oexhaust) ).
thf(f23857,plain,
! [X2: heap_e7401611519738050253t_unit,X1: set_nat] :
( ( produc8586169260539613262et_nat @ ( produc7507926704131184380et_nat @ X2 @ X1 ) )
= X1 ),
inference(equality_resolution,[status(thm)],[f23587]) ).
thf(f23587,plain,
! [X2: heap_e7401611519738050253t_unit,X0: produc3658429121746597890et_nat,X1: set_nat] :
( ( ( produc8586169260539613262et_nat @ X0 )
= X1 )
| ( ( produc7507926704131184380et_nat @ X2 @ X1 )
!= X0 ) ),
inference(cnf_transformation,[status(thm)],[f23314]) ).
thf(f23314,plain,
! [X0: produc3658429121746597890et_nat,X1: set_nat] :
( ( ( ( produc8586169260539613262et_nat @ X0 )
= X1 )
| ! [X2: heap_e7401611519738050253t_unit] :
( ( produc7507926704131184380et_nat @ X2 @ X1 )
!= X0 ) )
& ( ( ( produc7507926704131184380et_nat @ ( sK41 @ X1 @ X0 ) @ X1 )
= X0 )
| ( ( produc8586169260539613262et_nat @ X0 )
!= X1 ) ) ),
inference(skolemisation,[status(esa),new_symbols(skolem,[sK41])],[f23312,f23313]) ).
thf(f23313,plain,
! [X0: produc3658429121746597890et_nat,X1: set_nat] :
( ? [X3: heap_e7401611519738050253t_unit] :
( ( produc7507926704131184380et_nat @ X3 @ X1 )
= X0 )
=> ( ( produc7507926704131184380et_nat @ ( sK41 @ X1 @ X0 ) @ X1 )
= X0 ) ),
introduced(definition,[],[choice_axiom]) ).
thf(f23312,plain,
! [X0: produc3658429121746597890et_nat,X1: set_nat] :
( ( ( ( produc8586169260539613262et_nat @ X0 )
= X1 )
| ! [X2: heap_e7401611519738050253t_unit] :
( ( produc7507926704131184380et_nat @ X2 @ X1 )
!= X0 ) )
& ( ? [X3: heap_e7401611519738050253t_unit] :
( ( produc7507926704131184380et_nat @ X3 @ X1 )
= X0 )
| ( ( produc8586169260539613262et_nat @ X0 )
!= X1 ) ) ),
inference(rectify,[status(thm)],[f23311]) ).
thf(f23311,plain,
! [X0: produc3658429121746597890et_nat,X1: set_nat] :
( ( ( ( produc8586169260539613262et_nat @ X0 )
= X1 )
| ! [X2: heap_e7401611519738050253t_unit] :
( ( produc7507926704131184380et_nat @ X2 @ X1 )
!= X0 ) )
& ( ? [X2: heap_e7401611519738050253t_unit] :
( ( produc7507926704131184380et_nat @ X2 @ X1 )
= X0 )
| ( ( produc8586169260539613262et_nat @ X0 )
!= X1 ) ) ),
inference(nnf_transformation,[status(thm)],[f15946]) ).
thf(f15946,plain,
! [X0: produc3658429121746597890et_nat,X1: set_nat] :
( ( ( produc8586169260539613262et_nat @ X0 )
= X1 )
<=> ? [X2: heap_e7401611519738050253t_unit] :
( ( produc7507926704131184380et_nat @ X2 @ X1 )
= X0 ) ),
inference(fool_elimination,[status(thm)],[f15945]) ).
thf(f15945,plain,
! [X0: produc3658429121746597890et_nat,X1: set_nat] :
? [X2: heap_e7401611519738050253t_unit] :
( ( ( produc7507926704131184380et_nat @ X2 @ X1 )
= X0 )
= ( ( produc8586169260539613262et_nat @ X0 )
= X1 ) ),
inference(rectify,[status(thm)],[f152]) ).
thf(f152,axiom,
! [X30: produc3658429121746597890et_nat,X19: set_nat] :
? [X31: heap_e7401611519738050253t_unit] :
( ( ( produc7507926704131184380et_nat @ X31 @ X19 )
= X30 )
= ( ( produc8586169260539613262et_nat @ X30 )
= X19 ) ),
file('/export/starexec/sandbox/tmp/tmp.2EMJwA8PHy/DTF2THF_22696.p',fact_151_eq__snd__iff) ).
thf(f23967,plain,
( ( bot_bot_set_nat
= ( produc8586169260539613262et_nat @ h ) )
| ~ spl58_1 ),
inference(avatar_component_clause,[status(thm)],[f23965]) ).
thf(f23965,definition,
( spl58_1
<=> ( bot_bot_set_nat
= ( produc8586169260539613262et_nat @ h ) ) ),
introduced(definition,[new_symbols(naming,[spl58_1])],[avatar_definition]) ).
thf(f24604,plain,
! [X0: produc3658429121746597890et_nat] :
( ( bot_bot_set_nat
!= ( sK47 @ X0 ) )
| ( ( one_assn_raw @ X0 )
= $true ) ),
inference(duplicate_literal_removal,[status(thm)],[f24603]) ).
thf(f24603,plain,
! [X0: produc3658429121746597890et_nat] :
( ( bot_bot_set_nat
!= ( sK47 @ X0 ) )
| ( ( one_assn_raw @ X0 )
= $true )
| ( ( one_assn_raw @ X0 )
= $true ) ),
inference(superposition,[status(thm)],[f23451,f24197]) ).
thf(f24197,plain,
! [X0: produc3658429121746597890et_nat] :
( ( ( sK47 @ X0 )
= ( sK19 @ X0 ) )
| ( ( one_assn_raw @ X0 )
= $true ) ),
inference(superposition,[status(thm)],[f24036,f23450]) ).
thf(f23450,plain,
! [X0: produc3658429121746597890et_nat] :
( ( ( produc7507926704131184380et_nat @ ( sK18 @ X0 ) @ ( sK19 @ X0 ) )
= X0 )
| ( ( one_assn_raw @ X0 )
= $true ) ),
inference(cnf_transformation,[status(thm)],[f23249]) ).
thf(f23249,plain,
! [X0: produc3658429121746597890et_nat] :
( ( ( bot_bot_set_nat
!= ( sK19 @ X0 ) )
& ( ( produc7507926704131184380et_nat @ ( sK18 @ X0 ) @ ( sK19 @ X0 ) )
= X0 ) )
| ( ( one_assn_raw @ X0 )
= $true ) ),
inference(skolemisation,[status(esa),new_symbols(skolem,[sK18,sK19])],[f23203,f23248]) ).
thf(f23248,plain,
! [X0: produc3658429121746597890et_nat] :
( ? [X1: heap_e7401611519738050253t_unit,X2: set_nat] :
( ( bot_bot_set_nat != X2 )
& ( ( produc7507926704131184380et_nat @ X1 @ X2 )
= X0 ) )
=> ( ( bot_bot_set_nat
!= ( sK19 @ X0 ) )
& ( ( produc7507926704131184380et_nat @ ( sK18 @ X0 ) @ ( sK19 @ X0 ) )
= X0 ) ) ),
introduced(definition,[],[choice_axiom]) ).
thf(f23203,plain,
! [X0: produc3658429121746597890et_nat] :
( ? [X1: heap_e7401611519738050253t_unit,X2: set_nat] :
( ( bot_bot_set_nat != X2 )
& ( ( produc7507926704131184380et_nat @ X1 @ X2 )
= X0 ) )
| ( ( one_assn_raw @ X0 )
= $true ) ),
inference(ennf_transformation,[status(thm)],[f23085]) ).
thf(f23085,plain,
! [X0: produc3658429121746597890et_nat] :
( ( ( one_assn_raw @ X0 )
!= $true )
=> ~ ! [X1: heap_e7401611519738050253t_unit,X2: set_nat] :
( ( ( produc7507926704131184380et_nat @ X1 @ X2 )
= X0 )
=> ( bot_bot_set_nat = X2 ) ) ),
inference(flattening,[status(thm)],[f13534]) ).
thf(f13534,plain,
! [X0: produc3658429121746597890et_nat] :
( ( ( one_assn_raw @ X0 )
!= $true )
=> ~ ! [X1: heap_e7401611519738050253t_unit,X2: set_nat] :
( ( ( produc7507926704131184380et_nat @ X1 @ X2 )
= X0 )
=> ( bot_bot_set_nat = X2 ) ) ),
inference(fool_elimination,[status(thm)],[f13533]) ).
thf(f13533,plain,
! [X0: produc3658429121746597890et_nat] :
( ~ ( one_assn_raw @ X0 )
=> ~ ! [X1: heap_e7401611519738050253t_unit,X2: set_nat] :
( ( ( produc7507926704131184380et_nat @ X1 @ X2 )
= X0 )
=> ( bot_bot_set_nat = X2 ) ) ),
inference(rectify,[status(thm)],[f82]) ).
thf(f82,axiom,
! [X0: produc3658429121746597890et_nat] :
( ~ ( one_assn_raw @ X0 )
=> ~ ! [X4: heap_e7401611519738050253t_unit,X17: set_nat] :
( ( ( produc7507926704131184380et_nat @ X4 @ X17 )
= X0 )
=> ( bot_bot_set_nat = X17 ) ) ),
file('/export/starexec/sandbox/tmp/tmp.2EMJwA8PHy/DTF2THF_22696.p',fact_81_one__assn__raw_Oelims_I3_J) ).
thf(f24036,plain,
! [X0: heap_e7401611519738050253t_unit,X1: set_nat] :
( ( sK47 @ ( produc7507926704131184380et_nat @ X0 @ X1 ) )
= X1 ),
inference(superposition,[status(thm)],[f23857,f24025]) ).
thf(f23451,plain,
! [X0: produc3658429121746597890et_nat] :
( ( bot_bot_set_nat
!= ( sK19 @ X0 ) )
| ( ( one_assn_raw @ X0 )
= $true ) ),
inference(cnf_transformation,[status(thm)],[f23249]) ).
thf(f25697,plain,
( spl58_1
| ~ spl58_2 ),
inference(avatar_contradiction_clause,[status(thm)],[f25696]) ).
thf(f25696,plain,
( $false
| spl58_1
| ~ spl58_2 ),
inference(subsumption_resolution,[status(thm)],[f25695,f24032]) ).
thf(f24032,plain,
( ( bot_bot_set_nat
!= ( sK47 @ h ) )
| spl58_1 ),
inference(backward_demodulation,[status(thm)],[f24023,f24026]) ).
thf(f24026,plain,
! [X0: produc3658429121746597890et_nat] :
( ( sK47 @ X0 )
= ( sK25 @ X0 ) ),
inference(backward_demodulation,[status(thm)],[f24018,f24025]) ).
thf(f24018,plain,
! [X0: produc3658429121746597890et_nat] :
( ( produc8586169260539613262et_nat @ X0 )
= ( sK25 @ X0 ) ),
inference(superposition,[status(thm)],[f23857,f23493]) ).
thf(f23493,plain,
! [X0: produc3658429121746597890et_nat] :
( ( produc7507926704131184380et_nat @ ( sK24 @ X0 ) @ ( sK25 @ X0 ) )
= X0 ),
inference(cnf_transformation,[status(thm)],[f23268]) ).
thf(f23268,plain,
! [X0: produc3658429121746597890et_nat] :
( ( produc7507926704131184380et_nat @ ( sK24 @ X0 ) @ ( sK25 @ X0 ) )
= X0 ),
inference(skolemisation,[status(esa),new_symbols(skolem,[sK24,sK25])],[f23153,f23267]) ).
thf(f23267,plain,
! [X0: produc3658429121746597890et_nat] :
( ? [X1: heap_e7401611519738050253t_unit,X2: set_nat] :
( ( produc7507926704131184380et_nat @ X1 @ X2 )
= X0 )
=> ( ( produc7507926704131184380et_nat @ ( sK24 @ X0 ) @ ( sK25 @ X0 ) )
= X0 ) ),
introduced(definition,[],[choice_axiom]) ).
thf(f23153,plain,
! [X0: produc3658429121746597890et_nat] :
? [X1: heap_e7401611519738050253t_unit,X2: set_nat] :
( ( produc7507926704131184380et_nat @ X1 @ X2 )
= X0 ),
inference(ennf_transformation,[status(thm)],[f17946]) ).
thf(f17946,plain,
! [X0: produc3658429121746597890et_nat] :
~ ! [X1: heap_e7401611519738050253t_unit,X2: set_nat] :
( ( produc7507926704131184380et_nat @ X1 @ X2 )
!= X0 ),
inference(fool_elimination,[status(thm)],[f17945]) ).
thf(f17945,plain,
! [X0: produc3658429121746597890et_nat] :
~ ! [X1: heap_e7401611519738050253t_unit,X2: set_nat] :
( ( produc7507926704131184380et_nat @ X1 @ X2 )
!= X0 ),
inference(rectify,[status(thm)],[f96]) ).
thf(f96,axiom,
! [X0: produc3658429121746597890et_nat] :
~ ! [X4: heap_e7401611519738050253t_unit,X17: set_nat] :
( ( produc7507926704131184380et_nat @ X4 @ X17 )
!= X0 ),
file('/export/starexec/sandbox/tmp/tmp.2EMJwA8PHy/DTF2THF_22696.p',fact_95_one__assn__raw_Ocases) ).
thf(f24023,plain,
( ( bot_bot_set_nat
!= ( sK25 @ h ) )
| spl58_1 ),
inference(backward_demodulation,[status(thm)],[f24014,f24019]) ).
thf(f24019,plain,
! [X0: produc3658429121746597890et_nat] :
( ( sK25 @ X0 )
= ( sK10 @ X0 ) ),
inference(backward_demodulation,[status(thm)],[f24012,f24018]) ).
thf(f24012,plain,
! [X0: produc3658429121746597890et_nat] :
( ( produc8586169260539613262et_nat @ X0 )
= ( sK10 @ X0 ) ),
inference(superposition,[status(thm)],[f23857,f23416]) ).
thf(f23416,plain,
! [X0: produc3658429121746597890et_nat] :
( ( produc7507926704131184380et_nat @ ( sK11 @ X0 ) @ ( sK10 @ X0 ) )
= X0 ),
inference(cnf_transformation,[status(thm)],[f23231]) ).
thf(f23231,plain,
! [X0: produc3658429121746597890et_nat] :
( ( produc7507926704131184380et_nat @ ( sK11 @ X0 ) @ ( sK10 @ X0 ) )
= X0 ),
inference(skolemisation,[status(esa),new_symbols(skolem,[sK10,sK11])],[f18413,f23230]) ).
thf(f23230,plain,
! [X0: produc3658429121746597890et_nat] :
( ? [X1: set_nat,X2: heap_e7401611519738050253t_unit] :
( ( produc7507926704131184380et_nat @ X2 @ X1 )
= X0 )
=> ( ( produc7507926704131184380et_nat @ ( sK11 @ X0 ) @ ( sK10 @ X0 ) )
= X0 ) ),
introduced(definition,[],[choice_axiom]) ).
thf(f18413,plain,
! [X0: produc3658429121746597890et_nat] :
? [X1: set_nat,X2: heap_e7401611519738050253t_unit] :
( ( produc7507926704131184380et_nat @ X2 @ X1 )
= X0 ),
inference(fool_elimination,[status(thm)],[f18412]) ).
thf(f18412,plain,
! [X0: produc3658429121746597890et_nat] :
? [X1: set_nat,X2: heap_e7401611519738050253t_unit] :
( ( produc7507926704131184380et_nat @ X2 @ X1 )
= X0 ),
inference(rectify,[status(thm)],[f166]) ).
thf(f166,axiom,
! [X30: produc3658429121746597890et_nat] :
? [X9: set_nat,X11: heap_e7401611519738050253t_unit] :
( ( produc7507926704131184380et_nat @ X11 @ X9 )
= X30 ),
file('/export/starexec/sandbox/tmp/tmp.2EMJwA8PHy/DTF2THF_22696.p',fact_165_surj__pair) ).
thf(f24014,plain,
( ( bot_bot_set_nat
!= ( sK10 @ h ) )
| spl58_1 ),
inference(backward_demodulation,[status(thm)],[f23966,f24012]) ).
thf(f23966,plain,
( ( bot_bot_set_nat
!= ( produc8586169260539613262et_nat @ h ) )
| spl58_1 ),
inference(avatar_component_clause,[status(thm)],[f23965]) ).
thf(f25695,plain,
( ( bot_bot_set_nat
= ( sK47 @ h ) )
| ~ spl58_2 ),
inference(trivial_inequality_removal,[status(thm)],[f25683]) ).
thf(f25683,plain,
( ( bot_bot_set_nat
= ( sK47 @ h ) )
| ( $true != $true )
| ~ spl58_2 ),
inference(superposition,[status(thm)],[f24030,f25665]) ).
thf(f25665,plain,
( ( ( one_assn_raw @ h )
= $true )
| ~ spl58_2 ),
inference(superposition,[status(thm)],[f23971,f24618]) ).
thf(f23971,plain,
( ( ( rep_assn @ ( abs_assn @ one_assn_raw ) @ h )
= $true )
| ~ spl58_2 ),
inference(avatar_component_clause,[status(thm)],[f23969]) ).
thf(f24030,plain,
! [X0: produc3658429121746597890et_nat] :
( ( ( one_assn_raw @ X0 )
!= $true )
| ( bot_bot_set_nat
= ( sK47 @ X0 ) ) ),
inference(backward_demodulation,[status(thm)],[f24017,f24026]) ).
thf(f24017,plain,
! [X0: produc3658429121746597890et_nat] :
( ( bot_bot_set_nat
= ( sK25 @ X0 ) )
| ( ( one_assn_raw @ X0 )
!= $true ) ),
inference(superposition,[status(thm)],[f23721,f23493]) ).
thf(f23721,plain,
! [X0: set_nat,X1: heap_e7401611519738050253t_unit] :
( ( ( one_assn_raw @ ( produc7507926704131184380et_nat @ X1 @ X0 ) )
!= $true )
| ( bot_bot_set_nat = X0 ) ),
inference(cnf_transformation,[status(thm)],[f23376]) ).
thf(f23376,plain,
! [X0: set_nat,X1: heap_e7401611519738050253t_unit] :
( ( ( ( one_assn_raw @ ( produc7507926704131184380et_nat @ X1 @ X0 ) )
= $true )
| ( bot_bot_set_nat != X0 ) )
& ( ( bot_bot_set_nat = X0 )
| ( ( one_assn_raw @ ( produc7507926704131184380et_nat @ X1 @ X0 ) )
!= $true ) ) ),
inference(nnf_transformation,[status(thm)],[f13111]) ).
thf(f13111,plain,
! [X0: set_nat,X1: heap_e7401611519738050253t_unit] :
( ( ( one_assn_raw @ ( produc7507926704131184380et_nat @ X1 @ X0 ) )
= $true )
<=> ( bot_bot_set_nat = X0 ) ),
inference(fool_elimination,[status(thm)],[f13110]) ).
thf(f13110,plain,
! [X0: set_nat,X1: heap_e7401611519738050253t_unit] :
( ( bot_bot_set_nat = X0 )
= ( one_assn_raw @ ( produc7507926704131184380et_nat @ X1 @ X0 ) ) ),
inference(rectify,[status(thm)],[f79]) ).
thf(f79,axiom,
! [X16: set_nat,X12: heap_e7401611519738050253t_unit] :
( ( bot_bot_set_nat = X16 )
= ( one_assn_raw @ ( produc7507926704131184380et_nat @ X12 @ X16 ) ) ),
file('/export/starexec/sandbox/tmp/tmp.2EMJwA8PHy/DTF2THF_22696.p',fact_78_one__assn__raw_Osimps) ).
thf(f23975,plain,
( ~ spl58_2
| ~ spl58_1 ),
inference(avatar_split_clause,[status(thm)],[f23724,f23965,f23969]) ).
thf(f23724,plain,
( ( ( rep_assn @ ( abs_assn @ one_assn_raw ) @ h )
!= $true )
| ( bot_bot_set_nat
!= ( produc8586169260539613262et_nat @ h ) ) ),
inference(cnf_transformation,[status(thm)],[f23377]) ).
thf(f23377,plain,
( ( ( ( rep_assn @ ( abs_assn @ one_assn_raw ) @ h )
!= $true )
| ( bot_bot_set_nat
!= ( produc8586169260539613262et_nat @ h ) ) )
& ( ( ( rep_assn @ ( abs_assn @ one_assn_raw ) @ h )
= $true )
| ( bot_bot_set_nat
= ( produc8586169260539613262et_nat @ h ) ) ) ),
inference(nnf_transformation,[status(thm)],[f23199]) ).
thf(f23199,plain,
( ( bot_bot_set_nat
= ( produc8586169260539613262et_nat @ h ) )
<~> ( ( rep_assn @ ( abs_assn @ one_assn_raw ) @ h )
= $true ) ),
inference(ennf_transformation,[status(thm)],[f15878]) ).
thf(f15878,plain,
~ ( ( bot_bot_set_nat
= ( produc8586169260539613262et_nat @ h ) )
<=> ( ( rep_assn @ ( abs_assn @ one_assn_raw ) @ h )
= $true ) ),
inference(fool_elimination,[status(thm)],[f7827]) ).
thf(f7827,negated_conjecture,
( ( rep_assn @ ( abs_assn @ one_assn_raw ) @ h )
!= ( bot_bot_set_nat
= ( produc8586169260539613262et_nat @ h ) ) ),
inference(negated_conjecture,[status(cth)],[f7826]) ).
thf(f7826,conjecture,
( ( rep_assn @ ( abs_assn @ one_assn_raw ) @ h )
= ( bot_bot_set_nat
= ( produc8586169260539613262et_nat @ h ) ) ),
file('/export/starexec/sandbox/tmp/tmp.2EMJwA8PHy/DTF2THF_22696.p',conj_0) ).
thf(f23972,plain,
( spl58_1
| spl58_2 ),
inference(avatar_split_clause,[status(thm)],[f23723,f23969,f23965]) ).
thf(f23723,plain,
( ( ( rep_assn @ ( abs_assn @ one_assn_raw ) @ h )
= $true )
| ( bot_bot_set_nat
= ( produc8586169260539613262et_nat @ h ) ) ),
inference(cnf_transformation,[status(thm)],[f23377]) ).
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.00/0.11 % Problem : ITP211^3 : TPTP v9.2.1. Bugfixed v9.2.1.
% 0.12/0.13 % Command : /export/starexec/sandbox/solver/bin/run_DT2H2X /export/starexec/sandbox/benchmark/theBenchmark.p 300
% 0.15/0.34 % Computer : n017.cluster.edu
% 0.15/0.34 % Model : x86_64 x86_64
% 0.15/0.34 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.15/0.34 % Memory : 8042.1875MB
% 0.15/0.34 % OS : Linux 3.10.0-693.el7.x86_64
% 0.15/0.34 % CPULimit : 300
% 0.15/0.34 % WCLimit : 300
% 0.15/0.34 % DateTime : Thu Feb 26 02:54:26 EST 2026
% 0.15/0.34 % CPUTime :
% 0.15/0.34 Running /export/starexec/sandbox/solver/bin/run_DT2H2X /export/starexec/sandbox/benchmark/theBenchmark.p 300
% 0.34/0.49 ---- Original DTF file ---
% 0.34/0.49 thf(spec,logic,$$dhol).
% 0.34/0.49 %------------------------------------------------------------------------------
% 0.34/0.49 % File : ITP211^3 : TPTP v9.2.1. Bugfixed v9.2.1.
% 0.34/0.49 % Domain : Interactive Theorem Proving
% 0.34/0.49 % Problem : Sledgehammer problem Assertions 00607_017823
% 0.34/0.49 % Version : [Des22] axioms.
% 0.34/0.49 % English :
% 0.34/0.49
% 0.34/0.49 % Refs : [BH+15] Blanchette et al. (2015), Mining the Archive of Formal
% 0.34/0.49 % : [Des22] Desharnais (2022), Email to Geoff Sutcliffe
% 0.34/0.49 % Source : [Des22]
% 0.34/0.49 % Names : 0024_Assertions_00607_017823 [Des22]
% 0.34/0.49
% 0.34/0.49 % Status : Theorem
% 0.34/0.49 % Rating : ? v9.2.1
% 0.34/0.49 % Syntax : Number of formulae : 9705 (3210 unt;1879 typ; 0 def)
% 0.34/0.49 % Number of atoms : 24722 (9168 equ; 0 cnn)
% 0.34/0.49 % Maximal formula atoms : 17 ( 3 avg)
% 0.34/0.49 % Number of connectives : 84528 (3128 ~; 291 |;1450 &;67800 @)
% 0.34/0.49 % ( 0 <=>;11859 =>; 0 <=; 0 <~>)
% 0.34/0.49 % Maximal formula depth : 22 ( 7 avg)
% 0.34/0.49 % Number of types : 160 ( 159 usr)
% 0.34/0.49 % Number of type conns : 8578 (8578 >; 0 *; 0 +; 0 <<)
% 0.34/0.49 % Number of symbols : 1723 (1720 usr; 85 con; 0-8 aty)
% 0.34/0.49 % Number of variables : 22195 ( 759 ^;20824 !; 612 ?;22195 :)
% 0.34/0.49 % SPC : TH0_THM_EQU_NAR
% 0.34/0.49
% 0.34/0.49 % Comments : This file was generated by Isabelle (most likely Sledgehammer)
% 0.34/0.49 % from the van Emde Boas Trees session in the Archive of Formal
% 0.34/0.49 % proofs -
% 0.34/0.49 % www.isa-afp.org/browser_info/current/AFP/Van_Emde_Boas_Trees
% 0.34/0.49 % 2022-02-17 14:51:40.090
% 0.34/0.49 % Bugfixes : Added ()s around negated sides of equations.
% 0.34/0.49 %------------------------------------------------------------------------------
% 0.34/0.49 % Could-be-implicit typings (159)
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% 0.34/0.49 bNF_re398004352372739002nteger: ( int > code_integer > $o ) > ( ( int > int ) > ( code_integer > code_integer ) > $o ) > ( int > int > int ) > ( code_integer > code_integer > code_integer ) > $o ).
% 0.34/0.49
% 0.34/0.49 thf(sy_c_BNF__Def_Orel__fun_001t__Int__Oint_001t__Code____Numeral__Ointeger_001t__Int__Oint_001t__Code____Numeral__Ointeger,type,
% 0.34/0.49 bNF_re3379532845092657523nteger: ( int > code_integer > $o ) > ( int > code_integer > $o ) > ( int > int ) > ( code_integer > code_integer ) > $o ).
% 0.34/0.49
% 0.34/0.49 thf(sy_c_BNF__Def_Orel__fun_001t__Int__Oint_001t__Int__Oint_001_062_It__Int__Oint_Mt__Int__Oint_J_001_062_It__Int__Oint_Mt__Int__Oint_J,type,
% 0.34/0.49 bNF_re711492959462206631nt_int: ( int > int > $o ) > ( ( int > int ) > ( int > int ) > $o ) > ( int > int > int ) > ( int > int > int ) > $o ).
% 0.34/0.49
% 0.34/0.49 thf(sy_c_BNF__Def_Orel__fun_001t__Int__Oint_001t__Int__Oint_001t__Int__Oint_001t__Int__Oint,type,
% 0.34/0.49 bNF_re4712519889275205905nt_int: ( int > int > $o ) > ( int > int > $o ) > ( int > int ) > ( int > int ) > $o ).
% 0.34/0.49
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% 0.34/0.49
% 0.34/0.49 thf(sy_c_BNF__Def_Orel__fun_001t__Nat__Onat_001t__Code____Numeral__Onatural_001t__Nat__Onat_001t__Code____Numeral__Onatural,type,
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% 0.34/0.49
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% 0.34/0.49
% 0.34/0.49 thf(sy_c_BNF__Def_Orel__fun_001t__Nat__Onat_001t__Nat__Onat_001t__Nat__Onat_001t__Nat__Onat,type,
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% 0.34/0.49
% 0.34/0.49 thf(sy_c_BNF__Def_Orel__fun_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_001_062_It__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_Mt__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_J_001_062_It__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_Mt__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_J,type,
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% 0.34/0.49
% 0.34/0.49 thf(sy_c_BNF__Def_Orel__fun_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_001_Eo_001_Eo,type,
% 0.34/0.49 bNF_re8699439704749558557nt_o_o: ( product_prod_int_int > product_prod_int_int > $o ) > ( $o > $o > $o ) > ( product_prod_int_int > $o ) > ( product_prod_int_int > $o ) > $o ).
% 0.34/0.49
% 0.34/0.49 thf(sy_c_BNF__Def_Orel__fun_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J,type,
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% 0.34/0.49
% 0.34/0.49 thf(sy_c_BNF__Def_Orel__fun_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_001t__Rat__Orat_001_062_It__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_Mt__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_J_001_062_It__Rat__Orat_Mt__Rat__Orat_J,type,
% 0.34/0.49 bNF_re7627151682743391978at_rat: ( product_prod_int_int > rat > $o ) > ( ( product_prod_int_int > product_prod_int_int ) > ( rat > rat ) > $o ) > ( product_prod_int_int > product_prod_int_int > product_prod_int_int ) > ( rat > rat > rat ) > $o ).
% 0.34/0.49
% 0.34/0.49 thf(sy_c_BNF__Def_Orel__fun_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_001t__Rat__Orat_001_Eo_001_Eo,type,
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% 0.34/0.49
% 0.34/0.49 thf(sy_c_BNF__Def_Orel__fun_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_001t__Rat__Orat_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_001t__Rat__Orat,type,
% 0.34/0.49 bNF_re8279943556446156061nt_rat: ( product_prod_int_int > rat > $o ) > ( product_prod_int_int > rat > $o ) > ( product_prod_int_int > product_prod_int_int ) > ( rat > rat ) > $o ).
% 0.34/0.49
% 0.34/0.49 thf(sy_c_BNF__Def_Orel__fun_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_001t__Int__Oint_001_062_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_Mt__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_001_062_It__Int__Oint_Mt__Int__Oint_J,type,
% 0.34/0.49 bNF_re7408651293131936558nt_int: ( product_prod_nat_nat > int > $o ) > ( ( product_prod_nat_nat > product_prod_nat_nat ) > ( int > int ) > $o ) > ( product_prod_nat_nat > product_prod_nat_nat > product_prod_nat_nat ) > ( int > int > int ) > $o ).
% 0.34/0.49
% 0.34/0.49 thf(sy_c_BNF__Def_Orel__fun_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_001t__Int__Oint_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_001t__Int__Oint,type,
% 0.34/0.49 bNF_re7400052026677387805at_int: ( product_prod_nat_nat > int > $o ) > ( product_prod_nat_nat > int > $o ) > ( product_prod_nat_nat > product_prod_nat_nat ) > ( int > int ) > $o ).
% 0.34/0.49
% 0.34/0.49 thf(sy_c_BNF__Def_Orel__fun_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_001_062_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_Mt__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_001_062_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_Mt__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
% 0.34/0.49 bNF_re3099431351363272937at_nat: ( product_prod_nat_nat > product_prod_nat_nat > $o ) > ( ( product_prod_nat_nat > product_prod_nat_nat ) > ( product_prod_nat_nat > product_prod_nat_nat ) > $o ) > ( product_prod_nat_nat > product_prod_nat_nat > product_prod_nat_nat ) > ( product_prod_nat_nat > product_prod_nat_nat > product_prod_nat_nat ) > $o ).
% 0.34/0.49
% 0.34/0.49 thf(sy_c_BNF__Def_Orel__fun_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
% 0.34/0.49 bNF_re2241393799969408733at_nat: ( product_prod_nat_nat > product_prod_nat_nat > $o ) > ( product_prod_nat_nat > product_prod_nat_nat > $o ) > ( product_prod_nat_nat > product_prod_nat_nat ) > ( product_prod_nat_nat > product_prod_nat_nat ) > $o ).
% 0.34/0.49
% 0.34/0.49 thf(sy_c_BNF__Def_OsndOp_001t__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J_001t__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J_001t__Set__Oset_It__Nat__Onat_J,type,
% 0.34/0.49 bNF_sn4039337771197282234et_nat: ( heap_e7401611519738050253t_unit > heap_e7401611519738050253t_unit > $o ) > ( heap_e7401611519738050253t_unit > set_nat > $o ) > produc3658429121746597890et_nat > produc3658429121746597890et_nat ).
% 0.34/0.49
% 0.34/0.49 thf(sy_c_BNF__Def_OsndOp_001t__Int__Oint_001t__Int__Oint_001t__Int__Oint,type,
% 0.34/0.49 bNF_sn1062102010912252026nt_int: ( int > int > $o ) > ( int > int > $o ) > product_prod_int_int > product_prod_int_int ).
% 0.34/0.49
% 0.34/0.49 thf(sy_c_BNF__Wellorder__Constructions_Obsqr_001t__Int__Oint,type,
% 0.34/0.49 bNF_We1730399552001240015qr_int: set_Pr958786334691620121nt_int > set_Pr2560585780119916871nt_int ).
% 0.34/0.49
% 0.34/0.49 thf(sy_c_BNF__Wellorder__Constructions_Obsqr_001t__Nat__Onat,type,
% 0.34/0.49 bNF_We1732890022510290291qr_nat: set_Pr1261947904930325089at_nat > set_Pr8693737435421807431at_nat ).
% 0.34/0.49
% 0.34/0.49 thf(sy_c_BNF__Wellorder__Relation_Owo__rel_001_062_It__Product____Type__Oprod_It__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J_Mt__Set__Oset_It__Nat__Onat_J_J_M_Eo_J,type,
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% 0.34/0.49
% 0.34/0.49 thf(sy_c_BNF__Wellorder__Relation_Owo__rel_001_Eo,type,
% 0.34/0.49 bNF_We6437023980930612612_rel_o: set_Product_prod_o_o > $o ).
% 0.34/0.49
% 0.34/0.49 thf(sy_c_BNF__Wellorder__Relation_Owo__rel_001t__Int__Oint,type,
% 0.34/0.49 bNF_We3815749466139970368el_int: set_Pr958786334691620121nt_int > $o ).
% 0.34/0.49
% 0.34/0.49 thf(sy_c_BNF__Wellorder__Relation_Owo__rel_001t__Nat__Onat,type,
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% 0.34/0.49
% 0.34/0.49 thf(sy_c_BNF__Wellorder__Relation_Owo__rel_001t__Product____Type__Ounit,type,
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% 0.34/0.49
% 0.34/0.49 thf(sy_c_BNF__Wellorder__Relation_Owo__rel_OisMinim_001_062_It__Product____Type__Oprod_It__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J_Mt__Set__Oset_It__Nat__Onat_J_J_M_Eo_J,type,
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% 0.34/0.49
% 0.34/0.49 thf(sy_c_BNF__Wellorder__Relation_Owo__rel_OisMinim_001_Eo,type,
% 0.34/0.49 bNF_We4647864369647134286inim_o: set_Product_prod_o_o > set_o > $o > $o ).
% 0.34/0.49
% 0.34/0.49 thf(sy_c_BNF__Wellorder__Relation_Owo__rel_OisMinim_001t__Int__Oint,type,
% 0.34/0.49 bNF_We6511469253196982518im_int: set_Pr958786334691620121nt_int > set_int > int > $o ).
% 0.34/0.49
% 0.34/0.49 thf(sy_c_BNF__Wellorder__Relation_Owo__rel_OisMinim_001t__Nat__Onat,type,
% 0.34/0.49 bNF_We6513959723706032794im_nat: set_Pr1261947904930325089at_nat > set_nat > nat > $o ).
% 0.34/0.49
% 0.34/0.49 thf(sy_c_BNF__Wellorder__Relation_Owo__rel_OisMinim_001t__Product____Type__Ounit,type,
% 0.34/0.49 bNF_We909621271159000003t_unit: set_Pr5094982260447487303t_unit > set_Product_unit > product_unit > $o ).
% 0.34/0.49
% 0.34/0.49 thf(sy_c_BNF__Wellorder__Relation_Owo__rel_Omax2_001_062_It__Product____Type__Oprod_It__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J_Mt__Set__Oset_It__Nat__Onat_J_J_M_Eo_J,type,
% 0.34/0.49 bNF_We6400040844883348763_nat_o: set_Pr2161125870931222855_nat_o > ( produc3658429121746597890et_nat > $o ) > ( produc3658429121746597890et_nat > $o ) > produc3658429121746597890et_nat > $o ).
% 0.34/0.49
% 0.34/0.49 thf(sy_c_BNF__Wellorder__Relation_Owo__rel_Omax2_001_Eo,type,
% 0.34/0.49 bNF_We3104086342435940326max2_o: set_Product_prod_o_o > $o > $o > $o ).
% 0.34/0.49
% 0.34/0.49 thf(sy_c_BNF__Wellorder__Relation_Owo__rel_Omax2_001t__Int__Oint,type,
% 0.34/0.49 bNF_We2378448595015613790x2_int: set_Pr958786334691620121nt_int > int > int > int ).
% 0.34/0.49
% 0.34/0.49 thf(sy_c_BNF__Wellorder__Relation_Owo__rel_Omax2_001t__Nat__Onat,type,
% 0.34/0.49 bNF_We2380939065524664066x2_nat: set_Pr1261947904930325089at_nat > nat > nat > nat ).
% 0.34/0.49
% 0.34/0.49 thf(sy_c_BNF__Wellorder__Relation_Owo__rel_Omax2_001t__Product____Type__Ounit,type,
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% 0.34/0.49
% 0.34/0.49 thf(sy_c_BNF__Wellorder__Relation_Owo__rel_Ominim_001_Eo,type,
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% 0.34/0.49
% 0.34/0.49 thf(sy_c_BNF__Wellorder__Relation_Owo__rel_Ominim_001t__Int__Oint,type,
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% 0.34/0.49
% 0.34/0.49 thf(sy_c_BNF__Wellorder__Relation_Owo__rel_Ominim_001t__Nat__Onat,type,
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% 0.34/0.49
% 0.34/0.49 thf(sy_c_BNF__Wellorder__Relation_Owo__rel_Ominim_001t__Product____Type__Ounit,type,
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% 0.34/0.49
% 0.34/0.49 thf(sy_c_Basic__BNF__LFPs_Oprod_Osize__prod_001t__Int__Oint_001t__Int__Oint,type,
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% 0.34/0.49
% 0.34/0.49 thf(sy_c_Basic__BNFs_Ofsts_001t__Int__Oint_001t__Int__Oint,type,
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% 0.34/0.49
% 0.34/0.49 thf(sy_c_Basic__BNFs_Ofsts_001t__Nat__Onat_001t__Nat__Onat,type,
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% 0.34/0.49 finite3687622995246107910t_unit: ( code_integer > set_Product_unit > set_Product_unit ) > set_Product_unit > set_Code_integer > set_Product_unit ).
% 0.34/0.49
% 0.34/0.49 thf(sy_c_Finite__Set_Ofold_001t__Int__Oint_001t__Int__Oint,type,
% 0.34/0.49 finite_fold_int_int: ( int > int > int ) > int > set_int > int ).
% 0.34/0.49
% 0.34/0.49 thf(sy_c_Finite__Set_Ofold_001t__Int__Oint_001t__Set__Oset_It__Int__Oint_J,type,
% 0.34/0.49 finite1723285575846220376et_int: ( int > set_int > set_int ) > set_int > set_int > set_int ).
% 0.34/0.49
% 0.34/0.49 thf(sy_c_Finite__Set_Ofold_001t__Nat__Onat_001t__Nat__Onat,type,
% 0.34/0.49 finite_fold_nat_nat: ( nat > nat > nat ) > nat > set_nat > nat ).
% 0.34/0.49
% 0.34/0.49 thf(sy_c_Finite__Set_Ofold_001t__Nat__Onat_001t__Product____Type__Ounit,type,
% 0.34/0.49 finite6602901031542678131t_unit: ( nat > product_unit > product_unit ) > product_unit > set_nat > product_unit ).
% 0.34/0.49
% 0.34/0.49 thf(sy_c_Finite__Set_Ofold_001t__Nat__Onat_001t__Set__Oset_I_Eo_J,type,
% 0.34/0.49 finite3217087857726763998_set_o: ( nat > set_o > set_o ) > set_o > set_nat > set_o ).
% 0.34/0.49
% 0.34/0.49 thf(sy_c_Finite__Set_Ofold_001t__Nat__Onat_001t__Set__Oset_It__Int__Oint_J,type,
% 0.34/0.49 finite1351632015609375740et_int: ( nat > set_int > set_int ) > set_int > set_nat > set_int ).
% 0.34/0.49
% 0.34/0.49 thf(sy_c_Finite__Set_Ofold_001t__Nat__Onat_001t__Set__Oset_It__Nat__Onat_J,type,
% 0.34/0.49 finite5529483035118572448et_nat: ( nat > set_nat > set_nat ) > set_nat > set_nat > set_nat ).
% 0.34/0.49
% 0.34/0.49 thf(sy_c_Finite__Set_Ofold_001t__Nat__Onat_001t__Set__Oset_It__Product____Type__Ounit_J,type,
% 0.34/0.49 finite9145781422887121747t_unit: ( nat > set_Product_unit > set_Product_unit ) > set_Product_unit > set_nat > set_Product_unit ).
% 0.34/0.49
% 0.34/0.49 thf(sy_c_Finite__Set_Ofold_001t__Product____Type__Ounit_001t__Product____Type__Ounit,type,
% 0.34/0.49 finite8834431295148555784t_unit: ( product_unit > product_unit > product_unit ) > product_unit > set_Product_unit > product_unit ).
% 0.34/0.49
% 0.34/0.49 thf(sy_c_Finite__Set_Ofold_001t__Product____Type__Ounit_001t__Set__Oset_It__Product____Type__Ounit_J,type,
% 0.34/0.49 finite2740360166580943592t_unit: ( product_unit > set_Product_unit > set_Product_unit ) > set_Product_unit > set_Product_unit > set_Product_unit ).
% 0.34/0.49
% 0.34/0.49 thf(sy_c_Finite__Set_Ofold_001t__Set__Oset_I_062_It__Product____Type__Oprod_It__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J_Mt__Set__Oset_It__Nat__Onat_J_J_M_Eo_J_J_001t__Set__Oset_I_062_It__Product____Type__Oprod_It__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J_Mt__Set__Oset_It__Nat__Onat_J_J_M_Eo_J_J,type,
% 0.34/0.49 finite8456856254070787592_nat_o: ( set_Pr4532377907799695533_nat_o > set_Pr4532377907799695533_nat_o > set_Pr4532377907799695533_nat_o ) > set_Pr4532377907799695533_nat_o > set_se1353816476878259085_nat_o > set_Pr4532377907799695533_nat_o ).
% 0.34/0.49
% 0.34/0.49 thf(sy_c_Finite__Set_Ofold_001t__Set__Oset_I_Eo_J_001t__Set__Oset_I_Eo_J,type,
% 0.34/0.49 finite4337638375924247368_set_o: ( set_o > set_o > set_o ) > set_o > set_set_o > set_o ).
% 0.34/0.49
% 0.34/0.49 thf(sy_c_Finite__Set_Ofold_001t__Set__Oset_It__Int__Oint_J_001t__Set__Oset_It__Int__Oint_J,type,
% 0.34/0.49 finite2508883933289561102et_int: ( set_int > set_int > set_int ) > set_int > set_set_int > set_int ).
% 0.34/0.49
% 0.34/0.49 thf(sy_c_Finite__Set_Ofold_001t__Set__Oset_It__Nat__Onat_J_001t__Set__Oset_It__Nat__Onat_J,type,
% 0.34/0.49 finite677925301803182934et_nat: ( set_nat > set_nat > set_nat ) > set_nat > set_set_nat > set_nat ).
% 0.34/0.49
% 0.34/0.49 thf(sy_c_Finite__Set_Ofold_001t__Set__Oset_It__Product____Type__Ounit_J_001t__Set__Oset_It__Product____Type__Ounit_J,type,
% 0.34/0.49 finite4243028863286625800t_unit: ( set_Product_unit > set_Product_unit > set_Product_unit ) > set_Product_unit > set_set_Product_unit > set_Product_unit ).
% 0.34/0.49
% 0.34/0.49 thf(sy_c_Fun_Ocomp_001_062_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J_001_062_It__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J_Mt__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J_J_001t__Code____Numeral__Ointeger,type,
% 0.34/0.49 comp_C8797469213163452608nteger: ( ( code_integer > code_integer ) > produc8923325533196201883nteger > produc8923325533196201883nteger ) > ( code_integer > code_integer > code_integer ) > code_integer > produc8923325533196201883nteger > produc8923325533196201883nteger ).
% 0.34/0.49
% 0.34/0.49 thf(sy_c_Fun_Ocomp_001_062_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J_001t__Code____Numeral__Ointeger_001t__Code____Numeral__Ointeger,type,
% 0.34/0.49 comp_C5076291335850163704nteger: ( ( code_integer > code_integer ) > code_integer ) > ( code_integer > code_integer > code_integer ) > code_integer > code_integer ).
% 0.34/0.49
% 0.34/0.49 thf(sy_c_Fun_Ocomp_001_062_It__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J_Mt__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J_J_001t__Set__Oset_It__Code____Numeral__Ointeger_J_001t__Code____Numeral__Ointeger,type,
% 0.34/0.49 comp_P4885729792186999034nteger: ( ( produc8923325533196201883nteger > produc8923325533196201883nteger ) > set_Code_integer ) > ( code_integer > produc8923325533196201883nteger > produc8923325533196201883nteger ) > code_integer > set_Code_integer ).
% 0.34/0.49
% 0.34/0.49 thf(sy_c_Fun_Ocomp_001_062_It__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J_Mt__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J_J_001t__Set__Oset_It__Nat__Onat_J_001t__Code____Numeral__Ointeger,type,
% 0.34/0.49 comp_P8262219321096090349nteger: ( ( produc8923325533196201883nteger > produc8923325533196201883nteger ) > set_nat ) > ( code_integer > produc8923325533196201883nteger > produc8923325533196201883nteger ) > code_integer > set_nat ).
% 0.34/0.49
% 0.34/0.49 thf(sy_c_Fun_Ocomp_001_062_It__Set__Oset_I_Eo_J_Mt__Assertions__Oassn_J_001_062_It__Set__Oset_It__Set__Oset_I_Eo_J_J_Mt__Assertions__Oassn_J_001_062_I_Eo_Mt__Assertions__Oassn_J,type,
% 0.34/0.49 comp_s518380907012905326o_assn: ( ( set_o > assn ) > set_set_o > assn ) > ( ( $o > assn ) > set_o > assn ) > ( $o > assn ) > set_set_o > assn ).
% 0.34/0.49
% 0.34/0.49 thf(sy_c_Fun_Ocomp_001_062_It__Set__Oset_I_Eo_J_Mt__Nat__Onat_J_001_062_It__Set__Oset_It__Set__Oset_I_Eo_J_J_Mt__Nat__Onat_J_001_062_I_Eo_Mt__Nat__Onat_J,type,
% 0.34/0.49 comp_s6084864194472624150_o_nat: ( ( set_o > nat ) > set_set_o > nat ) > ( ( $o > nat ) > set_o > nat ) > ( $o > nat ) > set_set_o > nat ).
% 0.34/0.49
% 0.34/0.49 thf(sy_c_Fun_Ocomp_001_062_It__Set__Oset_It__Code____Numeral__Ointeger_J_Mt__Assertions__Oassn_J_001_062_It__Set__Oset_It__Set__Oset_It__Code____Numeral__Ointeger_J_J_Mt__Assertions__Oassn_J_001_062_It__Code____Numeral__Ointeger_Mt__Assertions__Oassn_J,type,
% 0.34/0.49 comp_s3531641609786562035r_assn: ( ( set_Code_integer > assn ) > set_set_Code_integer > assn ) > ( ( code_integer > assn ) > set_Code_integer > assn ) > ( code_integer > assn ) > set_set_Code_integer > assn ).
% 0.34/0.49
% 0.34/0.49 thf(sy_c_Fun_Ocomp_001_062_It__Set__Oset_It__Code____Numeral__Ointeger_J_Mt__Nat__Onat_J_001_062_It__Set__Oset_It__Set__Oset_It__Code____Numeral__Ointeger_J_J_Mt__Nat__Onat_J_001_062_It__Code____Numeral__Ointeger_Mt__Nat__Onat_J,type,
% 0.34/0.49 comp_s8759123648413916139er_nat: ( ( set_Code_integer > nat ) > set_set_Code_integer > nat ) > ( ( code_integer > nat ) > set_Code_integer > nat ) > ( code_integer > nat ) > set_set_Code_integer > nat ).
% 0.34/0.49
% 0.34/0.49 thf(sy_c_Fun_Ocomp_001_062_It__Set__Oset_It__Int__Oint_J_Mt__Assertions__Oassn_J_001_062_It__Set__Oset_It__Set__Oset_It__Int__Oint_J_J_Mt__Assertions__Oassn_J_001_062_It__Int__Oint_Mt__Assertions__Oassn_J,type,
% 0.34/0.49 comp_s4595925057662634336t_assn: ( ( set_int > assn ) > set_set_int > assn ) > ( ( int > assn ) > set_int > assn ) > ( int > assn ) > set_set_int > assn ).
% 0.34/0.49
% 0.34/0.49 thf(sy_c_Fun_Ocomp_001_062_It__Set__Oset_It__Int__Oint_J_Mt__Int__Oint_J_001_062_It__Set__Oset_It__Set__Oset_It__Int__Oint_J_J_Mt__Int__Oint_J_001_062_It__Int__Oint_Mt__Int__Oint_J,type,
% 0.34/0.49 comp_s9098231955081793132nt_int: ( ( set_int > int ) > set_set_int > int ) > ( ( int > int ) > set_int > int ) > ( int > int ) > set_set_int > int ).
% 0.34/0.49
% 0.34/0.49 thf(sy_c_Fun_Ocomp_001_062_It__Set__Oset_It__Int__Oint_J_Mt__Nat__Onat_J_001_062_It__Set__Oset_It__Set__Oset_It__Int__Oint_J_J_Mt__Nat__Onat_J_001_062_It__Int__Oint_Mt__Nat__Onat_J,type,
% 0.34/0.49 comp_s3237642481825588056nt_nat: ( ( set_int > nat ) > set_set_int > nat ) > ( ( int > nat ) > set_int > nat ) > ( int > nat ) > set_set_int > nat ).
% 0.34/0.49
% 0.34/0.49 thf(sy_c_Fun_Ocomp_001_062_It__Set__Oset_It__Nat__Onat_J_Mt__Assertions__Oassn_J_001_062_It__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J_Mt__Assertions__Oassn_J_001_062_It__Nat__Onat_Mt__Assertions__Oassn_J,type,
% 0.34/0.49 comp_s2451231738238893644t_assn: ( ( set_nat > assn ) > set_set_nat > assn ) > ( ( nat > assn ) > set_nat > assn ) > ( nat > assn ) > set_set_nat > assn ).
% 0.34/0.49
% 0.34/0.49 thf(sy_c_Fun_Ocomp_001_062_It__Set__Oset_It__Nat__Onat_J_Mt__Nat__Onat_J_001_062_It__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J_Mt__Nat__Onat_J_001_062_It__Nat__Onat_Mt__Nat__Onat_J,type,
% 0.34/0.49 comp_s5456809700045850692at_nat: ( ( set_nat > nat ) > set_set_nat > nat ) > ( ( nat > nat ) > set_nat > nat ) > ( nat > nat ) > set_set_nat > nat ).
% 0.34/0.49
% 0.34/0.49 thf(sy_c_Fun_Ocomp_001_062_It__Set__Oset_It__Product____Type__Ounit_J_Mt__Assertions__Oassn_J_001_062_It__Set__Oset_It__Set__Oset_It__Product____Type__Ounit_J_J_Mt__Assertions__Oassn_J_001_062_It__Product____Type__Ounit_Mt__Assertions__Oassn_J,type,
% 0.34/0.49 comp_s8247530669214396195t_assn: ( ( set_Product_unit > assn ) > set_set_Product_unit > assn ) > ( ( product_unit > assn ) > set_Product_unit > assn ) > ( product_unit > assn ) > set_set_Product_unit > assn ).
% 0.34/0.49
% 0.34/0.49 thf(sy_c_Fun_Ocomp_001_062_It__Set__Oset_It__Product____Type__Ounit_J_Mt__Nat__Onat_J_001_062_It__Set__Oset_It__Set__Oset_It__Product____Type__Ounit_J_J_Mt__Nat__Onat_J_001_062_It__Product____Type__Ounit_Mt__Nat__Onat_J,type,
% 0.34/0.49 comp_s8928436226432208331it_nat: ( ( set_Product_unit > nat ) > set_set_Product_unit > nat ) > ( ( product_unit > nat ) > set_Product_unit > nat ) > ( product_unit > nat ) > set_set_Product_unit > nat ).
% 0.34/0.49
% 0.34/0.49 thf(sy_c_Fun_Ocomp_001_Eo_001_062_It__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J_Mt__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J_J_001t__Code____Numeral__Ointeger,type,
% 0.34/0.49 comp_o8922910837708301817nteger: ( $o > produc8923325533196201883nteger > produc8923325533196201883nteger ) > ( code_integer > $o ) > code_integer > produc8923325533196201883nteger > produc8923325533196201883nteger ).
% 0.34/0.49
% 0.34/0.49 thf(sy_c_Fun_Ocomp_001_Eo_001_062_It__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J_Mt__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J_J_001t__Product____Type__Ounit,type,
% 0.34/0.49 comp_o6961622296474427249t_unit: ( $o > produc8923325533196201883nteger > produc8923325533196201883nteger ) > ( product_unit > $o ) > product_unit > produc8923325533196201883nteger > produc8923325533196201883nteger ).
% 0.34/0.49
% 0.34/0.49 thf(sy_c_Fun_Ocomp_001_Eo_001t__Code____Numeral__Ointeger_001t__Code____Numeral__Ointeger,type,
% 0.34/0.49 comp_o1085144040659434033nteger: ( $o > code_integer ) > ( code_integer > $o ) > code_integer > code_integer ).
% 0.34/0.49
% 0.34/0.49 thf(sy_c_Fun_Ocomp_001_Eo_001t__Code____Numeral__Ointeger_001t__Product____Type__Ounit,type,
% 0.34/0.49 comp_o780160699970281785t_unit: ( $o > code_integer ) > ( product_unit > $o ) > product_unit > code_integer ).
% 0.34/0.49
% 0.34/0.49 thf(sy_c_Fun_Ocomp_001_Eo_001t__Code____Numeral__Onatural_001t__Product____Type__Ounit,type,
% 0.34/0.49 comp_o2643362028263920430t_unit: ( $o > code_natural ) > ( product_unit > $o ) > product_unit > code_natural ).
% 0.34/0.49
% 0.34/0.49 thf(sy_c_Fun_Ocomp_001_Eo_001t__Int__Oint_001t__Int__Oint,type,
% 0.34/0.49 comp_o_int_int: ( $o > int ) > ( int > $o ) > int > int ).
% 0.34/0.49
% 0.34/0.49 thf(sy_c_Fun_Ocomp_001_Eo_001t__Product____Type__Ounit_001_Eo,type,
% 0.34/0.49 comp_o3916225744453444672unit_o: ( $o > product_unit ) > ( $o > $o ) > $o > product_unit ).
% 0.34/0.49
% 0.34/0.49 thf(sy_c_Fun_Ocomp_001_Eo_001t__Product____Type__Ounit_001t__Code____Numeral__Ointeger,type,
% 0.34/0.49 comp_o6635994240230588853nteger: ( $o > product_unit ) > ( code_integer > $o ) > code_integer > product_unit ).
% 0.34/0.49
% 0.34/0.49 thf(sy_c_Fun_Ocomp_001_Eo_001t__Product____Type__Ounit_001t__Int__Oint,type,
% 0.34/0.49 comp_o7049022088938261380it_int: ( $o > product_unit ) > ( int > $o ) > int > product_unit ).
% 0.34/0.49
% 0.34/0.49 thf(sy_c_Fun_Ocomp_001_Eo_001t__Product____Type__Ounit_001t__Nat__Onat,type,
% 0.34/0.49 comp_o7051512559447311656it_nat: ( $o > product_unit ) > ( nat > $o ) > nat > product_unit ).
% 0.34/0.49
% 0.34/0.49 thf(sy_c_Fun_Ocomp_001_Eo_001t__Product____Type__Ounit_001t__Product____Type__Ounit,type,
% 0.34/0.49 comp_o7724735231917782709t_unit: ( $o > product_unit ) > ( product_unit > $o ) > product_unit > product_unit ).
% 0.34/0.49
% 0.34/0.49 thf(sy_c_Fun_Ocomp_001_Eo_001t__Set__Oset_I_Eo_J_001t__Product____Type__Ounit,type,
% 0.34/0.49 comp_o6895741221744486154t_unit: ( $o > set_o ) > ( product_unit > $o ) > product_unit > set_o ).
% 0.34/0.49
% 0.34/0.49 thf(sy_c_Fun_Ocomp_001_Eo_001t__Set__Oset_It__Code____Numeral__Ointeger_J_001_062_It__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J_Mt__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J_J,type,
% 0.34/0.49 comp_o7929101295636871983nteger: ( $o > set_Code_integer ) > ( ( produc8923325533196201883nteger > produc8923325533196201883nteger ) > $o ) > ( produc8923325533196201883nteger > produc8923325533196201883nteger ) > set_Code_integer ).
% 0.34/0.49
% 0.34/0.49 thf(sy_c_Fun_Ocomp_001_Eo_001t__Set__Oset_It__Code____Numeral__Ointeger_J_001t__Code____Numeral__Ointeger,type,
% 0.34/0.49 comp_o4291842030903409511nteger: ( $o > set_Code_integer ) > ( code_integer > $o ) > code_integer > set_Code_integer ).
% 0.34/0.49
% 0.34/0.49 thf(sy_c_Fun_Ocomp_001_Eo_001t__Set__Oset_It__Int__Oint_J_001t__Product____Type__Ounit,type,
% 0.34/0.49 comp_o1521574261646630388t_unit: ( $o > set_int ) > ( product_unit > $o ) > product_unit > set_int ).
% 0.34/0.49
% 0.34/0.49 thf(sy_c_Fun_Ocomp_001_Eo_001t__Set__Oset_It__Nat__Onat_J_001_062_It__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J_Mt__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J_J,type,
% 0.34/0.49 comp_o1289984117834210338nteger: ( $o > set_nat ) > ( ( produc8923325533196201883nteger > produc8923325533196201883nteger ) > $o ) > ( produc8923325533196201883nteger > produc8923325533196201883nteger ) > set_nat ).
% 0.34/0.49
% 0.34/0.49 thf(sy_c_Fun_Ocomp_001_Eo_001t__Set__Oset_It__Nat__Onat_J_001t__Code____Numeral__Ointeger,type,
% 0.34/0.49 comp_o172754640513974618nteger: ( $o > set_nat ) > ( code_integer > $o ) > code_integer > set_nat ).
% 0.34/0.49
% 0.34/0.49 thf(sy_c_Fun_Ocomp_001_Eo_001t__Set__Oset_It__Nat__Onat_J_001t__Product____Type__Ounit,type,
% 0.34/0.49 comp_o9190160103024335696t_unit: ( $o > set_nat ) > ( product_unit > $o ) > product_unit > set_nat ).
% 0.34/0.49
% 0.34/0.49 thf(sy_c_Fun_Ocomp_001_Eo_001t__Set__Oset_It__Product____Type__Ounit_J_001t__Product____Type__Ounit,type,
% 0.34/0.49 comp_o2086126948152544725t_unit: ( $o > set_Product_unit ) > ( product_unit > $o ) > product_unit > set_Product_unit ).
% 0.34/0.49
% 0.34/0.49 thf(sy_c_Fun_Ocomp_001t__Code____Numeral__Ointeger_001_062_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J_001t__Code____Numeral__Ointeger,type,
% 0.34/0.49 comp_C2931745410781319416nteger: ( code_integer > code_integer > code_integer ) > ( code_integer > code_integer ) > code_integer > code_integer > code_integer ).
% 0.34/0.49
% 0.34/0.49 thf(sy_c_Fun_Ocomp_001t__Code____Numeral__Ointeger_001_062_It__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J_Mt__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J_J_001_062_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J,type,
% 0.34/0.49 comp_C95226695241045696nteger: ( code_integer > produc8923325533196201883nteger > produc8923325533196201883nteger ) > ( ( code_integer > code_integer ) > code_integer ) > ( code_integer > code_integer ) > produc8923325533196201883nteger > produc8923325533196201883nteger ).
% 0.34/0.49
% 0.34/0.49 thf(sy_c_Fun_Ocomp_001t__Code____Numeral__Ointeger_001_062_It__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J_Mt__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J_J_001_Eo,type,
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% 0.34/0.49
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% 0.34/0.49
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% 0.34/0.49
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% 0.34/0.49
% 0.34/0.49 thf(sy_c_Fun_Ocomp_001t__Code____Numeral__Ointeger_001t__Assertions__Oassn_001t__Code____Numeral__Ointeger,type,
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% 0.34/0.49
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% 0.34/0.50 thf(sy_c_Fun_Ocomp_001t__Code____Numeral__Ointeger_001t__Int__Oint_001t__Code____Numeral__Ointeger,type,
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% 0.34/0.50 thf(sy_c_Fun_Ocomp_001t__Code____Numeral__Ointeger_001t__Int__Oint_001t__Nat__Onat,type,
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% 0.34/0.50
% 0.34/0.50 thf(sy_c_Fun_Ocomp_001t__Code____Numeral__Ointeger_001t__Nat__Onat_001t__Code____Numeral__Ointeger,type,
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% 0.34/0.50 thf(sy_c_Fun_Ocomp_001t__Code____Numeral__Ointeger_001t__Product____Type__Ounit_001t__Code____Numeral__Ointeger,type,
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% 0.34/0.50
% 0.34/0.50 thf(sy_c_Fun_Ocomp_001t__Code____Numeral__Ointeger_001t__Product____Type__Ounit_001t__Code____Numeral__Onatural,type,
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% 0.34/0.50
% 0.34/0.50 thf(sy_c_Fun_Ocomp_001t__Code____Numeral__Ointeger_001t__Product____Type__Ounit_001t__Product____Type__Ounit,type,
% 0.34/0.50 comp_C2408090164605745152t_unit: ( code_integer > product_unit ) > ( product_unit > code_integer ) > product_unit > product_unit ).
% 0.34/0.50
% 0.34/0.50 thf(sy_c_Fun_Ocomp_001t__Code____Numeral__Ointeger_001t__Set__Oset_It__Code____Numeral__Ointeger_J_001_062_It__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J_Mt__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J_J,type,
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% 0.34/0.50
% 0.34/0.50 thf(sy_c_Fun_Ocomp_001t__Code____Numeral__Ointeger_001t__Set__Oset_It__Code____Numeral__Ointeger_J_001t__Code____Numeral__Ointeger,type,
% 0.34/0.50 comp_C8186931911283765298nteger: ( code_integer > set_Code_integer ) > ( code_integer > code_integer ) > code_integer > set_Code_integer ).
% 0.34/0.50
% 0.34/0.50 thf(sy_c_Fun_Ocomp_001t__Code____Numeral__Ointeger_001t__Set__Oset_It__Nat__Onat_J_001_062_It__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J_Mt__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J_J,type,
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% 0.34/0.50
% 0.34/0.50 thf(sy_c_Fun_Ocomp_001t__Code____Numeral__Ointeger_001t__Set__Oset_It__Nat__Onat_J_001t__Code____Numeral__Ointeger,type,
% 0.34/0.50 comp_C6537567860429791781nteger: ( code_integer > set_nat ) > ( code_integer > code_integer ) > code_integer > set_nat ).
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% 0.34/0.50 thf(sy_c_Fun_Ocomp_001t__Code____Numeral__Onatural_001_062_It__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J_Mt__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J_J_001t__Code____Numeral__Ointeger,type,
% 0.34/0.50 comp_C456341964420721295nteger: ( code_natural > produc8923325533196201883nteger > produc8923325533196201883nteger ) > ( code_integer > code_natural ) > code_integer > produc8923325533196201883nteger > produc8923325533196201883nteger ).
% 0.34/0.50
% 0.34/0.50 thf(sy_c_Fun_Ocomp_001t__Int__Oint_001_062_It__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J_Mt__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J_J_001t__Code____Numeral__Ointeger,type,
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% 0.34/0.50
% 0.34/0.50 thf(sy_c_Fun_Ocomp_001t__Int__Oint_001_Eo_001_Eo,type,
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% 0.34/0.50
% 0.34/0.50 thf(sy_c_Fun_Ocomp_001t__Int__Oint_001_Eo_001t__Int__Oint,type,
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% 0.34/0.50
% 0.34/0.50 thf(sy_c_Fun_Ocomp_001t__Int__Oint_001_Eo_001t__Nat__Onat,type,
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% 0.34/0.50
% 0.34/0.50 thf(sy_c_Fun_Ocomp_001t__Int__Oint_001t__Int__Oint_001_Eo,type,
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% 0.34/0.50
% 0.34/0.50 thf(sy_c_Fun_Ocomp_001t__Int__Oint_001t__Int__Oint_001t__Int__Oint,type,
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% 0.34/0.50
% 0.34/0.50 thf(sy_c_Fun_Ocomp_001t__Int__Oint_001t__Int__Oint_001t__Nat__Onat,type,
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% 0.34/0.50
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% 0.34/0.50
% 0.34/0.50 thf(sy_c_Fun_Ocomp_001t__Int__Oint_001t__Nat__Onat_001_Eo,type,
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% 0.34/0.50
% 0.34/0.50 thf(sy_c_Fun_Ocomp_001t__Int__Oint_001t__Nat__Onat_001t__Int__Oint,type,
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% 0.34/0.50
% 0.34/0.50 thf(sy_c_Fun_Ocomp_001t__Int__Oint_001t__Nat__Onat_001t__Nat__Onat,type,
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% 0.34/0.50
% 0.34/0.50 thf(sy_c_Fun_Ocomp_001t__Int__Oint_001t__Product____Type__Ounit_001t__Code____Numeral__Ointeger,type,
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% 0.34/0.50 thf(sy_c_Fun_Ocomp_001t__Int__Oint_001t__Set__Oset_I_Eo_J_001t__Code____Numeral__Ointeger,type,
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% 0.34/0.50 thf(sy_c_Fun_Ocomp_001t__Int__Oint_001t__Set__Oset_It__Int__Oint_J_001t__Code____Numeral__Ointeger,type,
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% 0.34/0.50 thf(sy_c_Fun_Ocomp_001t__Int__Oint_001t__Set__Oset_It__Int__Oint_J_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J,type,
% 0.34/0.50 comp_i5103216804979976198nt_int: ( int > set_int ) > ( product_prod_int_int > int ) > product_prod_int_int > set_int ).
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% 0.34/0.50 comp_i6200299785623974754nt_int: ( int > set_nat ) > ( product_prod_int_int > int ) > product_prod_int_int > set_nat ).
% 0.34/0.50
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% 0.34/0.50
% 0.34/0.50 thf(sy_c_Fun_Ocomp_001t__Nat__Onat_001_062_It__Nat__Onat_Mt__Nat__Onat_J_001t__Nat__Onat,type,
% 0.34/0.50 comp_nat_nat_nat_nat: ( nat > nat > nat ) > ( nat > nat ) > nat > nat > nat ).
% 0.34/0.50
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% 0.34/0.50
% 0.34/0.50 thf(sy_c_Fun_Ocomp_001t__Nat__Onat_001_062_It__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J_Mt__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J_J_001t__Code____Numeral__Onatural,type,
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% 0.34/0.50
% 0.34/0.50 thf(sy_c_Fun_Ocomp_001t__Nat__Onat_001t__Assertions__Oassn_001t__Nat__Onat,type,
% 0.34/0.50 comp_nat_assn_nat: ( nat > assn ) > ( nat > nat ) > nat > assn ).
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% 0.34/0.50 comp_n9163001964490475437al_nat: ( nat > code_natural ) > ( nat > nat ) > nat > code_natural ).
% 0.34/0.50
% 0.34/0.50 thf(sy_c_Fun_Ocomp_001t__Nat__Onat_001t__Int__Oint_001t__Nat__Onat,type,
% 0.34/0.50 comp_nat_int_nat: ( nat > int ) > ( nat > nat ) > nat > int ).
% 0.34/0.50
% 0.34/0.50 thf(sy_c_Fun_Ocomp_001t__Nat__Onat_001t__Product____Type__Ounit_001t__Code____Numeral__Onatural,type,
% 0.34/0.50 comp_n8360577435903631554atural: ( nat > product_unit ) > ( code_natural > nat ) > code_natural > product_unit ).
% 0.34/0.50
% 0.34/0.50 thf(sy_c_Fun_Ocomp_001t__Nat__Onat_001t__Set__Oset_It__Int__Oint_J_001t__Code____Numeral__Onatural,type,
% 0.34/0.50 comp_n1572716621429000127atural: ( nat > set_int ) > ( code_natural > nat ) > code_natural > set_int ).
% 0.34/0.50
% 0.34/0.50 thf(sy_c_Fun_Ocomp_001t__Nat__Onat_001t__Set__Oset_It__Nat__Onat_J_001_062_It__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J_Mt__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J_J,type,
% 0.34/0.50 comp_n3623798160786426464nteger: ( nat > set_nat ) > ( ( produc8923325533196201883nteger > produc8923325533196201883nteger ) > nat ) > ( produc8923325533196201883nteger > produc8923325533196201883nteger ) > set_nat ).
% 0.34/0.50
% 0.34/0.50 thf(sy_c_Fun_Ocomp_001t__Nat__Onat_001t__Set__Oset_It__Nat__Onat_J_001t__Code____Numeral__Ointeger,type,
% 0.34/0.50 comp_n5963683855531483544nteger: ( nat > set_nat ) > ( code_integer > nat ) > code_integer > set_nat ).
% 0.34/0.50
% 0.34/0.50 thf(sy_c_Fun_Ocomp_001t__Nat__Onat_001t__Set__Oset_It__Nat__Onat_J_001t__Code____Numeral__Onatural,type,
% 0.34/0.50 comp_n4787279007288201059atural: ( nat > set_nat ) > ( code_natural > nat ) > code_natural > set_nat ).
% 0.34/0.50
% 0.34/0.50 thf(sy_c_Fun_Ocomp_001t__Product____Type__Oprod_I_062_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J_M_062_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J_J_001t__Product____Type__Oprod_I_062_It__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J_Mt__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J_J_M_062_It__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J_Mt__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J_J_J_001t__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J,type,
% 0.34/0.50 comp_P5544053077415136032nteger: ( produc723880050079548563nteger > produc101387210554329899nteger ) > ( produc8923325533196201883nteger > produc723880050079548563nteger ) > produc8923325533196201883nteger > produc101387210554329899nteger ).
% 0.34/0.50
% 0.34/0.50 thf(sy_c_Fun_Ocomp_001t__Product____Type__Oprod_I_062_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J_Mt__Code____Numeral__Ointeger_J_001t__Product____Type__Oprod_I_062_It__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J_Mt__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J_J_M_062_It__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J_Mt__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J_J_J_001t__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J,type,
% 0.34/0.50 comp_P1414880308642389412nteger: ( produc2221115339011613079nteger > produc101387210554329899nteger ) > ( produc8923325533196201883nteger > produc2221115339011613079nteger ) > produc8923325533196201883nteger > produc101387210554329899nteger ).
% 0.34/0.50
% 0.34/0.50 thf(sy_c_Fun_Ocomp_001t__Product____Type__Oprod_It__Code____Numeral__Ointeger_M_062_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J_J_001t__Product____Type__Oprod_I_062_It__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J_Mt__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J_J_M_062_It__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J_Mt__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J_J_J_001t__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J,type,
% 0.34/0.50 comp_P6312383820422015140nteger: ( produc1397831737260518551nteger > produc101387210554329899nteger ) > ( produc8923325533196201883nteger > produc1397831737260518551nteger ) > produc8923325533196201883nteger > produc101387210554329899nteger ).
% 0.34/0.50
% 0.34/0.50 thf(sy_c_Fun_Ocomp_001t__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J_001t__Code____Numeral__Ointeger_001t__Code____Numeral__Ointeger,type,
% 0.34/0.50 comp_P7111796090210262324nteger: ( produc8923325533196201883nteger > code_integer ) > ( code_integer > produc8923325533196201883nteger ) > code_integer > code_integer ).
% 0.34/0.50
% 0.34/0.50 thf(sy_c_Fun_Ocomp_001t__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J_001t__Code____Numeral__Ointeger_001t__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J,type,
% 0.34/0.50 comp_P3428181540658402820nteger: ( produc8923325533196201883nteger > code_integer ) > ( produc8923325533196201883nteger > produc8923325533196201883nteger ) > produc8923325533196201883nteger > code_integer ).
% 0.34/0.50
% 0.34/0.50 thf(sy_c_Fun_Ocomp_001t__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J_001t__Code____Numeral__Ointeger_001t__Product____Type__Ounit,type,
% 0.34/0.50 comp_P659088277599099894t_unit: ( produc8923325533196201883nteger > code_integer ) > ( product_unit > produc8923325533196201883nteger ) > product_unit > code_integer ).
% 0.34/0.50
% 0.34/0.50 thf(sy_c_Fun_Ocomp_001t__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J_001t__Product____Type__Oprod_I_062_It__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J_Mt__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J_J_M_062_It__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J_Mt__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J_J_J_001t__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J,type,
% 0.34/0.50 comp_P234064951390806312nteger: ( produc8923325533196201883nteger > produc101387210554329899nteger ) > ( produc8923325533196201883nteger > produc8923325533196201883nteger ) > produc8923325533196201883nteger > produc101387210554329899nteger ).
% 0.34/0.50
% 0.34/0.50 thf(sy_c_Fun_Ocomp_001t__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__Nat__Onat_J_001t__Code____Numeral__Ointeger_001t__Code____Numeral__Ointeger,type,
% 0.34/0.50 comp_P7101464685127059239nteger: ( produc6433947425880249742er_nat > code_integer ) > ( code_integer > produc6433947425880249742er_nat ) > code_integer > code_integer ).
% 0.34/0.50
% 0.34/0.50 thf(sy_c_Fun_Ocomp_001t__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__Nat__Onat_J_001t__Code____Numeral__Ointeger_001t__Product____Type__Ounit,type,
% 0.34/0.50 comp_P7581484221039074243t_unit: ( produc6433947425880249742er_nat > code_integer ) > ( product_unit > produc6433947425880249742er_nat ) > product_unit > code_integer ).
% 0.34/0.50
% 0.34/0.50 thf(sy_c_Fun_Ocomp_001t__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__Nat__Onat_J_001t__Nat__Onat_001t__Code____Numeral__Ointeger,type,
% 0.34/0.50 comp_P5964733648194011802nteger: ( produc6433947425880249742er_nat > nat ) > ( code_integer > produc6433947425880249742er_nat ) > code_integer > nat ).
% 0.34/0.50
% 0.34/0.50 thf(sy_c_Fun_Ocomp_001t__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__Nat__Onat_J_001t__Nat__Onat_001t__Product____Type__Ounit,type,
% 0.34/0.50 comp_P6821003776282389008t_unit: ( produc6433947425880249742er_nat > nat ) > ( product_unit > produc6433947425880249742er_nat ) > product_unit > nat ).
% 0.34/0.50
% 0.34/0.50 thf(sy_c_Fun_Ocomp_001t__Product____Type__Oprod_It__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J_Mt__Set__Oset_It__Nat__Onat_J_J_001t__Set__Oset_It__Nat__Onat_J_001t__Product____Type__Oprod_It__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J_Mt__Set__Oset_It__Nat__Onat_J_J,type,
% 0.34/0.50 comp_P2930521774885306735et_nat: ( produc3658429121746597890et_nat > set_nat ) > ( produc3658429121746597890et_nat > produc3658429121746597890et_nat ) > produc3658429121746597890et_nat > set_nat ).
% 0.34/0.50
% 0.34/0.50 thf(sy_c_Fun_Ocomp_001t__Product____Type__Oprod_It__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J_Mt__Set__Oset_It__Nat__Onat_J_J_001t__Set__Oset_It__Nat__Onat_J_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J,type,
% 0.34/0.50 comp_P8606889026643445772nt_int: ( produc3658429121746597890et_nat > set_nat ) > ( product_prod_int_int > produc3658429121746597890et_nat ) > product_prod_int_int > set_nat ).
% 0.34/0.50
% 0.34/0.50 thf(sy_c_Fun_Ocomp_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_001t__Int__Oint_001t__Product____Type__Oprod_It__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J_Mt__Set__Oset_It__Nat__Onat_J_J,type,
% 0.34/0.50 comp_P8308861049717174982et_nat: ( product_prod_int_int > int ) > ( produc3658429121746597890et_nat > product_prod_int_int ) > produc3658429121746597890et_nat > int ).
% 0.34/0.50
% 0.34/0.50 thf(sy_c_Fun_Ocomp_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_001t__Int__Oint_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J,type,
% 0.34/0.50 comp_P5356232565778667509nt_int: ( product_prod_int_int > int ) > ( product_prod_int_int > product_prod_int_int ) > product_prod_int_int > int ).
% 0.34/0.50
% 0.34/0.50 thf(sy_c_Fun_Ocomp_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_001t__Int__Oint_001t__Product____Type__Ounit,type,
% 0.34/0.50 comp_P6796073005956588617t_unit: ( product_prod_int_int > int ) > ( product_unit > product_prod_int_int ) > product_unit > int ).
% 0.34/0.50
% 0.34/0.50 thf(sy_c_Fun_Ocomp_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Code____Numeral__Ointeger_J_001t__Code____Numeral__Ointeger_001_Eo,type,
% 0.34/0.50 comp_P7177471617286032590eger_o: ( produc2591615631378489102nteger > code_integer ) > ( $o > produc2591615631378489102nteger ) > $o > code_integer ).
% 0.34/0.50
% 0.34/0.50 thf(sy_c_Fun_Ocomp_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Code____Numeral__Ointeger_J_001t__Code____Numeral__Ointeger_001t__Code____Numeral__Ointeger,type,
% 0.34/0.50 comp_P6122662217367900327nteger: ( produc2591615631378489102nteger > code_integer ) > ( code_integer > produc2591615631378489102nteger ) > code_integer > code_integer ).
% 0.34/0.50
% 0.34/0.50 thf(sy_c_Fun_Ocomp_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Code____Numeral__Ointeger_J_001t__Code____Numeral__Ointeger_001t__Product____Type__Ounit,type,
% 0.34/0.50 comp_P7320554561093743171t_unit: ( produc2591615631378489102nteger > code_integer ) > ( product_unit > produc2591615631378489102nteger ) > product_unit > code_integer ).
% 0.34/0.50
% 0.34/0.50 thf(sy_c_Fun_Ocomp_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Code____Numeral__Ointeger_J_001t__Nat__Onat_001_Eo,type,
% 0.34/0.50 comp_P7179991235626930843_nat_o: ( produc2591615631378489102nteger > nat ) > ( $o > produc2591615631378489102nteger ) > $o > nat ).
% 0.34/0.50
% 0.34/0.50 thf(sy_c_Fun_Ocomp_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Code____Numeral__Ointeger_J_001t__Nat__Onat_001t__Code____Numeral__Ointeger,type,
% 0.34/0.50 comp_P5725896914063526938nteger: ( produc2591615631378489102nteger > nat ) > ( code_integer > produc2591615631378489102nteger ) > code_integer > nat ).
% 0.34/0.50
% 0.34/0.50 thf(sy_c_Fun_Ocomp_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Code____Numeral__Ointeger_J_001t__Nat__Onat_001t__Product____Type__Ounit,type,
% 0.34/0.50 comp_P7621350044865347728t_unit: ( produc2591615631378489102nteger > nat ) > ( product_unit > produc2591615631378489102nteger ) > product_unit > nat ).
% 0.34/0.50
% 0.34/0.50 thf(sy_c_Fun_Ocomp_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_001t__Nat__Onat_001_Eo,type,
% 0.34/0.50 comp_P4394204175247225320_nat_o: ( product_prod_nat_nat > nat ) > ( $o > product_prod_nat_nat ) > $o > nat ).
% 0.34/0.50
% 0.34/0.50 thf(sy_c_Fun_Ocomp_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_001t__Nat__Onat_001t__Product____Type__Ounit,type,
% 0.34/0.50 comp_P2980510830910040157t_unit: ( product_prod_nat_nat > nat ) > ( product_unit > product_prod_nat_nat ) > product_unit > nat ).
% 0.34/0.50
% 0.34/0.50 thf(sy_c_Fun_Ocomp_001t__Product____Type__Ounit_001_062_It__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J_Mt__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J_J_001t__Code____Numeral__Ointeger,type,
% 0.34/0.50 comp_P4511551582283981294nteger: ( product_unit > produc8923325533196201883nteger > produc8923325533196201883nteger ) > ( code_integer > product_unit ) > code_integer > produc8923325533196201883nteger > produc8923325533196201883nteger ).
% 0.34/0.50
% 0.34/0.50 thf(sy_c_Fun_Ocomp_001t__Product____Type__Ounit_001_062_It__Product____Type__Ounit_Mt__Product____Type__Ounit_J_001t__Code____Numeral__Ointeger,type,
% 0.34/0.50 comp_P4558236313141964142nteger: ( product_unit > product_unit > product_unit ) > ( code_integer > product_unit ) > code_integer > product_unit > product_unit ).
% 0.34/0.50
% 0.34/0.50 thf(sy_c_Fun_Ocomp_001t__Product____Type__Ounit_001_062_It__Product____Type__Ounit_Mt__Product____Type__Ounit_J_001t__Nat__Onat,type,
% 0.34/0.50 comp_P6204500334276302049it_nat: ( product_unit > product_unit > product_unit ) > ( nat > product_unit ) > nat > product_unit > product_unit ).
% 0.34/0.50
% 0.34/0.50 thf(sy_c_Fun_Ocomp_001t__Product____Type__Ounit_001t__Code____Numeral__Ointeger_001t__Code____Numeral__Ointeger,type,
% 0.34/0.50 comp_P6790823133229852966nteger: ( product_unit > code_integer ) > ( code_integer > product_unit ) > code_integer > code_integer ).
% 0.34/0.50
% 0.34/0.50 thf(sy_c_Fun_Ocomp_001t__Product____Type__Ounit_001t__Set__Oset_It__Code____Numeral__Ointeger_J_001_062_It__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J_Mt__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J_J,type,
% 0.34/0.50 comp_P4635212033913086756nteger: ( product_unit > set_Code_integer ) > ( ( produc8923325533196201883nteger > produc8923325533196201883nteger ) > product_unit ) > ( produc8923325533196201883nteger > produc8923325533196201883nteger ) > set_Code_integer ).
% 0.34/0.50
% 0.34/0.50 thf(sy_c_Fun_Ocomp_001t__Product____Type__Ounit_001t__Set__Oset_It__Code____Numeral__Ointeger_J_001t__Code____Numeral__Ointeger,type,
% 0.34/0.50 comp_P4767411901762164316nteger: ( product_unit > set_Code_integer ) > ( code_integer > product_unit ) > code_integer > set_Code_integer ).
% 0.34/0.50
% 0.34/0.50 thf(sy_c_Fun_Ocomp_001t__Product____Type__Ounit_001t__Set__Oset_It__Nat__Onat_J_001_062_It__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J_Mt__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J_J,type,
% 0.34/0.50 comp_P6101996899264665623nteger: ( product_unit > set_nat ) > ( ( produc8923325533196201883nteger > produc8923325533196201883nteger ) > product_unit ) > ( produc8923325533196201883nteger > produc8923325533196201883nteger ) > set_nat ).
% 0.34/0.50
% 0.34/0.50 thf(sy_c_Fun_Ocomp_001t__Product____Type__Ounit_001t__Set__Oset_It__Nat__Onat_J_001t__Code____Numeral__Ointeger,type,
% 0.34/0.50 comp_P5878433733084393551nteger: ( product_unit > set_nat ) > ( code_integer > product_unit ) > code_integer > set_nat ).
% 0.34/0.50
% 0.34/0.50 thf(sy_c_Fun_Ocomp_001t__Set__Oset_I_Eo_J_001_062_It__Set__Oset_I_Eo_J_Mt__Set__Oset_I_Eo_J_J_001t__Code____Numeral__Ointeger,type,
% 0.34/0.50 comp_s90868338361139993nteger: ( set_o > set_o > set_o ) > ( code_integer > set_o ) > code_integer > set_o > set_o ).
% 0.34/0.50
% 0.34/0.50 thf(sy_c_Fun_Ocomp_001t__Set__Oset_I_Eo_J_001_062_It__Set__Oset_I_Eo_J_Mt__Set__Oset_I_Eo_J_J_001t__Nat__Onat,type,
% 0.34/0.50 comp_s4424444925706540684_o_nat: ( set_o > set_o > set_o ) > ( nat > set_o ) > nat > set_o > set_o ).
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% 0.34/0.50
% 0.34/0.50 thf(sy_c_Fun_Ocomp_001t__Set__Oset_It__Int__Oint_J_001_062_It__Set__Oset_It__Int__Oint_J_Mt__Set__Oset_It__Int__Oint_J_J_001t__Code____Numeral__Ointeger,type,
% 0.34/0.50 comp_s3634918272309710215nteger: ( set_int > set_int > set_int ) > ( code_integer > set_int ) > code_integer > set_int > set_int ).
% 0.34/0.50
% 0.34/0.50 thf(sy_c_Fun_Ocomp_001t__Set__Oset_It__Int__Oint_J_001_062_It__Set__Oset_It__Int__Oint_J_Mt__Set__Oset_It__Int__Oint_J_J_001t__Nat__Onat,type,
% 0.34/0.50 comp_s6689493721604790522nt_nat: ( set_int > set_int > set_int ) > ( nat > set_int ) > nat > set_int > set_int ).
% 0.34/0.50
% 0.34/0.50 thf(sy_c_Fun_Ocomp_001t__Set__Oset_It__Nat__Onat_J_001_062_It__Set__Oset_It__Nat__Onat_J_Mt__Set__Oset_It__Nat__Onat_J_J_001t__Code____Numeral__Ointeger,type,
% 0.34/0.50 comp_s3169385134670675059nteger: ( set_nat > set_nat > set_nat ) > ( code_integer > set_nat ) > code_integer > set_nat > set_nat ).
% 0.34/0.50
% 0.34/0.50 thf(sy_c_Fun_Ocomp_001t__Set__Oset_It__Nat__Onat_J_001_062_It__Set__Oset_It__Nat__Onat_J_Mt__Set__Oset_It__Nat__Onat_J_J_001t__Nat__Onat,type,
% 0.34/0.50 comp_s2351873599094224870at_nat: ( set_nat > set_nat > set_nat ) > ( nat > set_nat ) > nat > set_nat > set_nat ).
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% 0.34/0.51 thf(sy_c_member_001t__Set__Oset_I_062_It__Product____Type__Oprod_It__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J_Mt__Set__Oset_It__Nat__Onat_J_J_M_Eo_J_J,type,
% 0.34/0.51 member1452989533377600726_nat_o: set_Pr4532377907799695533_nat_o > set_se1353816476878259085_nat_o > $o ).
% 0.34/0.51
% 0.34/0.51 thf(sy_c_member_001t__Set__Oset_I_Eo_J,type,
% 0.34/0.51 member_set_o: set_o > set_set_o > $o ).
% 0.34/0.51
% 0.34/0.51 thf(sy_c_member_001t__Set__Oset_It__Code____Numeral__Ointeger_J,type,
% 0.34/0.51 member1834298987299537450nteger: set_Code_integer > set_set_Code_integer > $o ).
% 0.34/0.51
% 0.34/0.51 thf(sy_c_member_001t__Set__Oset_It__Int__Oint_J,type,
% 0.34/0.51 member_set_int: set_int > set_set_int > $o ).
% 0.34/0.51
% 0.34/0.51 thf(sy_c_member_001t__Set__Oset_It__Nat__Onat_J,type,
% 0.34/0.51 member_set_nat: set_nat > set_set_nat > $o ).
% 0.34/0.51
% 0.34/0.51 thf(sy_c_member_001t__Set__Oset_It__Product____Type__Oprod_I_062_It__Product____Type__Oprod_It__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J_Mt__Set__Oset_It__Nat__Onat_J_J_M_Eo_J_M_062_It__Product____Type__Oprod_It__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J_Mt__Set__Oset_It__Nat__Onat_J_J_M_Eo_J_J_J,type,
% 0.34/0.51 member5475208728337331568_nat_o: set_Pr2161125870931222855_nat_o > set_se7663487083892625191_nat_o > $o ).
% 0.34/0.51
% 0.34/0.51 thf(sy_c_member_001t__Set__Oset_It__Product____Type__Oprod_I_Eo_M_Eo_J_J,type,
% 0.34/0.51 member3157015407008874800od_o_o: set_Product_prod_o_o > set_se491199571785101031od_o_o > $o ).
% 0.34/0.51
% 0.34/0.51 thf(sy_c_member_001t__Set__Oset_It__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_J,type,
% 0.34/0.51 member2340774599025711042nt_int: set_Pr958786334691620121nt_int > set_se6260736226359567993nt_int > $o ).
% 0.34/0.51
% 0.34/0.51 thf(sy_c_member_001t__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
% 0.34/0.51 member2643936169264416010at_nat: set_Pr1261947904930325089at_nat > set_se7855581050983116737at_nat > $o ).
% 0.34/0.51
% 0.34/0.51 thf(sy_c_member_001t__Set__Oset_It__Product____Type__Oprod_It__Product____Type__Ounit_Mt__Product____Type__Ounit_J_J,type,
% 0.34/0.51 member1248447801810324848t_unit: set_Pr5094982260447487303t_unit > set_se8994758365886191399t_unit > $o ).
% 0.34/0.51
% 0.34/0.51 thf(sy_c_member_001t__Set__Oset_It__Product____Type__Ounit_J,type,
% 0.34/0.51 member5877623283571906838t_unit: set_Product_unit > set_set_Product_unit > $o ).
% 0.34/0.51
% 0.34/0.51 thf(sy_c_member_001t__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J,type,
% 0.34/0.51 member_set_set_nat: set_set_nat > set_set_set_nat > $o ).
% 0.34/0.51
% 0.34/0.51 thf(sy_v_h,type,
% 0.34/0.51 h: produc3658429121746597890et_nat ).
% 0.34/0.51
% 0.34/0.51 % Relevant facts (7796)
% 0.34/0.51 thf(fact_0_Rep__assn__inject,axiom,
% 0.34/0.51 ! [X: assn,Y: assn] :
% 0.34/0.51 ( ( ( rep_assn @ X )
% 0.34/0.51 = ( rep_assn @ Y ) )
% 0.34/0.51 = ( X = Y ) ) ).
% 0.34/0.51
% 0.34/0.51 % Rep_assn_inject
% 0.34/0.51 thf(fact_1_Rep__assn__inverse,axiom,
% 0.34/0.51 ! [X: assn] :
% 0.34/0.51 ( ( abs_assn @ ( rep_assn @ X ) )
% 0.34/0.51 = X ) ).
% 0.34/0.51
% 0.34/0.51 % Rep_assn_inverse
% 0.34/0.51 thf(fact_2_Abs__assn__eqI_I2_J,axiom,
% 0.34/0.51 ! [P: produc3658429121746597890et_nat > $o,Pr: assn] :
% 0.34/0.51 ( ! [H: produc3658429121746597890et_nat] :
% 0.34/0.51 ( ( P @ H )
% 0.34/0.51 = ( rep_assn @ Pr @ H ) )
% 0.34/0.51 => ( Pr
% 0.34/0.51 = ( abs_assn @ P ) ) ) ).
% 0.34/0.51
% 0.34/0.51 % Abs_assn_eqI(2)
% 0.34/0.51 thf(fact_3_Abs__assn__eqI_I1_J,axiom,
% 0.34/0.51 ! [P: produc3658429121746597890et_nat > $o,Pr: assn] :
% 0.34/0.51 ( ! [H: produc3658429121746597890et_nat] :
% 0.34/0.51 ( ( P @ H )
% 0.34/0.51 = ( rep_assn @ Pr @ H ) )
% 0.34/0.51 => ( ( abs_assn @ P )
% 0.34/0.51 = Pr ) ) ).
% 0.34/0.51
% 0.34/0.51 % Abs_assn_eqI(1)
% 0.34/0.51 thf(fact_4_empty__iff,axiom,
% 0.34/0.51 ! [C: produc3658429121746597890et_nat > $o] :
% 0.34/0.51 ~ ( member6576561426505652726_nat_o @ C @ bot_bo7824918357723582233_nat_o ) ).
% 0.34/0.51
% 0.34/0.51 % empty_iff
% 0.34/0.51 thf(fact_5_empty__iff,axiom,
% 0.34/0.51 ! [C: product_unit] :
% 0.34/0.51 ~ ( member_Product_unit @ C @ bot_bo3957492148770167129t_unit ) ).
% 0.34/0.51
% 0.34/0.51 % empty_iff
% 0.34/0.51 thf(fact_6_empty__iff,axiom,
% 0.34/0.51 ! [C: $o] :
% 0.34/0.51 ~ ( member_o @ C @ bot_bot_set_o ) ).
% 0.34/0.51
% 0.34/0.51 % empty_iff
% 0.34/0.51 thf(fact_7_empty__iff,axiom,
% 0.34/0.51 ! [C: nat] :
% 0.34/0.51 ~ ( member_nat @ C @ bot_bot_set_nat ) ).
% 0.34/0.51
% 0.34/0.51 % empty_iff
% 0.34/0.51 thf(fact_8_empty__iff,axiom,
% 0.34/0.51 ! [C: int] :
% 0.34/0.51 ~ ( member_int @ C @ bot_bot_set_int ) ).
% 0.34/0.51
% 0.34/0.51 % empty_iff
% 0.34/0.51 thf(fact_9_all__not__in__conv,axiom,
% 0.34/0.51 ! [A: set_Pr4532377907799695533_nat_o] :
% 0.34/0.51 ( ( ! [X2: produc3658429121746597890et_nat > $o] :
% 0.34/0.51 ~ ( member6576561426505652726_nat_o @ X2 @ A ) )
% 0.34/0.51 = ( A = bot_bo7824918357723582233_nat_o ) ) ).
% 0.34/0.51
% 0.34/0.51 % all_not_in_conv
% 0.34/0.51 thf(fact_10_all__not__in__conv,axiom,
% 0.34/0.51 ! [A: set_Product_unit] :
% 0.34/0.51 ( ( ! [X2: product_unit] :
% 0.34/0.51 ~ ( member_Product_unit @ X2 @ A ) )
% 0.34/0.51 = ( A = bot_bo3957492148770167129t_unit ) ) ).
% 0.34/0.51
% 0.34/0.51 % all_not_in_conv
% 0.34/0.51 thf(fact_11_all__not__in__conv,axiom,
% 0.34/0.51 ! [A: set_o] :
% 0.34/0.51 ( ( ! [X2: $o] :
% 0.34/0.51 ~ ( member_o @ X2 @ A ) )
% 0.34/0.51 = ( A = bot_bot_set_o ) ) ).
% 0.34/0.51
% 0.34/0.51 % all_not_in_conv
% 0.34/0.51 thf(fact_12_all__not__in__conv,axiom,
% 0.34/0.51 ! [A: set_nat] :
% 0.34/0.51 ( ( ! [X2: nat] :
% 0.34/0.51 ~ ( member_nat @ X2 @ A ) )
% 0.34/0.51 = ( A = bot_bot_set_nat ) ) ).
% 0.34/0.51
% 0.34/0.51 % all_not_in_conv
% 0.34/0.51 thf(fact_13_all__not__in__conv,axiom,
% 0.34/0.51 ! [A: set_int] :
% 0.34/0.51 ( ( ! [X2: int] :
% 0.34/0.51 ~ ( member_int @ X2 @ A ) )
% 0.34/0.51 = ( A = bot_bot_set_int ) ) ).
% 0.34/0.51
% 0.34/0.51 % all_not_in_conv
% 0.34/0.51 thf(fact_14_Collect__empty__eq,axiom,
% 0.34/0.51 ! [P: ( produc3658429121746597890et_nat > $o ) > $o] :
% 0.34/0.51 ( ( ( collec939566748876313656_nat_o @ P )
% 0.34/0.51 = bot_bo7824918357723582233_nat_o )
% 0.34/0.51 = ( ! [X2: produc3658429121746597890et_nat > $o] :
% 0.34/0.51 ~ ( P @ X2 ) ) ) ).
% 0.34/0.51
% 0.34/0.51 % Collect_empty_eq
% 0.34/0.51 thf(fact_15_Collect__empty__eq,axiom,
% 0.34/0.51 ! [P: product_unit > $o] :
% 0.34/0.51 ( ( ( collect_Product_unit @ P )
% 0.34/0.51 = bot_bo3957492148770167129t_unit )
% 0.34/0.51 = ( ! [X2: product_unit] :
% 0.34/0.51 ~ ( P @ X2 ) ) ) ).
% 0.34/0.51
% 0.34/0.51 % Collect_empty_eq
% 0.34/0.51 thf(fact_16_Collect__empty__eq,axiom,
% 0.34/0.51 ! [P: $o > $o] :
% 0.34/0.51 ( ( ( collect_o @ P )
% 0.34/0.51 = bot_bot_set_o )
% 0.34/0.51 = ( ! [X2: $o] :
% 0.34/0.51 ~ ( P @ X2 ) ) ) ).
% 0.34/0.51
% 0.34/0.51 % Collect_empty_eq
% 0.34/0.51 thf(fact_17_Collect__empty__eq,axiom,
% 0.34/0.51 ! [P: nat > $o] :
% 0.34/0.51 ( ( ( collect_nat @ P )
% 0.34/0.51 = bot_bot_set_nat )
% 0.34/0.51 = ( ! [X2: nat] :
% 0.34/0.51 ~ ( P @ X2 ) ) ) ).
% 0.34/0.51
% 0.34/0.51 % Collect_empty_eq
% 0.34/0.51 thf(fact_18_Collect__empty__eq,axiom,
% 0.34/0.51 ! [P: int > $o] :
% 0.34/0.51 ( ( ( collect_int @ P )
% 0.34/0.51 = bot_bot_set_int )
% 0.34/0.51 = ( ! [X2: int] :
% 0.34/0.51 ~ ( P @ X2 ) ) ) ).
% 0.34/0.51
% 0.34/0.51 % Collect_empty_eq
% 0.34/0.51 thf(fact_19_empty__Collect__eq,axiom,
% 0.34/0.51 ! [P: ( produc3658429121746597890et_nat > $o ) > $o] :
% 0.34/0.51 ( ( bot_bo7824918357723582233_nat_o
% 0.34/0.51 = ( collec939566748876313656_nat_o @ P ) )
% 0.34/0.51 = ( ! [X2: produc3658429121746597890et_nat > $o] :
% 0.34/0.51 ~ ( P @ X2 ) ) ) ).
% 0.34/0.51
% 0.34/0.51 % empty_Collect_eq
% 0.34/0.51 thf(fact_20_empty__Collect__eq,axiom,
% 0.34/0.51 ! [P: product_unit > $o] :
% 0.34/0.51 ( ( bot_bo3957492148770167129t_unit
% 0.34/0.51 = ( collect_Product_unit @ P ) )
% 0.34/0.51 = ( ! [X2: product_unit] :
% 0.34/0.51 ~ ( P @ X2 ) ) ) ).
% 0.34/0.51
% 0.34/0.51 % empty_Collect_eq
% 0.34/0.51 thf(fact_21_empty__Collect__eq,axiom,
% 0.34/0.51 ! [P: $o > $o] :
% 0.34/0.51 ( ( bot_bot_set_o
% 0.34/0.51 = ( collect_o @ P ) )
% 0.34/0.51 = ( ! [X2: $o] :
% 0.34/0.51 ~ ( P @ X2 ) ) ) ).
% 0.34/0.51
% 0.34/0.51 % empty_Collect_eq
% 0.34/0.51 thf(fact_22_empty__Collect__eq,axiom,
% 0.34/0.51 ! [P: nat > $o] :
% 0.34/0.51 ( ( bot_bot_set_nat
% 0.34/0.51 = ( collect_nat @ P ) )
% 0.34/0.51 = ( ! [X2: nat] :
% 0.34/0.51 ~ ( P @ X2 ) ) ) ).
% 0.34/0.51
% 0.34/0.51 % empty_Collect_eq
% 0.34/0.51 thf(fact_23_empty__Collect__eq,axiom,
% 0.34/0.51 ! [P: int > $o] :
% 0.34/0.51 ( ( bot_bot_set_int
% 0.34/0.51 = ( collect_int @ P ) )
% 0.34/0.51 = ( ! [X2: int] :
% 0.34/0.51 ~ ( P @ X2 ) ) ) ).
% 0.34/0.51
% 0.34/0.51 % empty_Collect_eq
% 0.34/0.51 thf(fact_24_emptyE,axiom,
% 0.34/0.51 ! [A2: produc3658429121746597890et_nat > $o] :
% 0.34/0.51 ~ ( member6576561426505652726_nat_o @ A2 @ bot_bo7824918357723582233_nat_o ) ).
% 0.34/0.51
% 0.34/0.51 % emptyE
% 0.34/0.51 thf(fact_25_emptyE,axiom,
% 0.34/0.51 ! [A2: product_unit] :
% 0.34/0.51 ~ ( member_Product_unit @ A2 @ bot_bo3957492148770167129t_unit ) ).
% 0.34/0.51
% 0.34/0.51 % emptyE
% 0.34/0.51 thf(fact_26_emptyE,axiom,
% 0.34/0.51 ! [A2: $o] :
% 0.34/0.51 ~ ( member_o @ A2 @ bot_bot_set_o ) ).
% 0.34/0.51
% 0.34/0.51 % emptyE
% 0.34/0.51 thf(fact_27_emptyE,axiom,
% 0.34/0.51 ! [A2: nat] :
% 0.34/0.51 ~ ( member_nat @ A2 @ bot_bot_set_nat ) ).
% 0.34/0.51
% 0.34/0.51 % emptyE
% 0.34/0.51 thf(fact_28_emptyE,axiom,
% 0.34/0.51 ! [A2: int] :
% 0.34/0.51 ~ ( member_int @ A2 @ bot_bot_set_int ) ).
% 0.34/0.51
% 0.34/0.51 % emptyE
% 0.34/0.51 thf(fact_29_equals0D,axiom,
% 0.34/0.51 ! [A: set_Product_unit,A2: product_unit] :
% 0.34/0.51 ( ( A = bot_bo3957492148770167129t_unit )
% 0.34/0.51 => ~ ( member_Product_unit @ A2 @ A ) ) ).
% 0.34/0.51
% 0.34/0.51 % equals0D
% 0.34/0.51 thf(fact_30_equals0D,axiom,
% 0.34/0.51 ! [A: set_o,A2: $o] :
% 0.34/0.51 ( ( A = bot_bot_set_o )
% 0.34/0.51 => ~ ( member_o @ A2 @ A ) ) ).
% 0.34/0.51
% 0.34/0.51 % equals0D
% 0.34/0.51 thf(fact_31_equals0D,axiom,
% 0.34/0.51 ! [A: set_nat,A2: nat] :
% 0.34/0.51 ( ( A = bot_bot_set_nat )
% 0.34/0.51 => ~ ( member_nat @ A2 @ A ) ) ).
% 0.34/0.51
% 0.34/0.51 % equals0D
% 0.34/0.51 thf(fact_32_equals0D,axiom,
% 0.34/0.51 ! [A: set_int,A2: int] :
% 0.34/0.51 ( ( A = bot_bot_set_int )
% 0.34/0.51 => ~ ( member_int @ A2 @ A ) ) ).
% 0.34/0.51
% 0.34/0.51 % equals0D
% 0.34/0.51 thf(fact_33_equals0D,axiom,
% 0.34/0.51 ! [A: set_Pr4532377907799695533_nat_o,A2: produc3658429121746597890et_nat > $o] :
% 0.34/0.51 ( ( A = bot_bo7824918357723582233_nat_o )
% 0.34/0.51 => ~ ( member6576561426505652726_nat_o @ A2 @ A ) ) ).
% 0.34/0.51
% 0.34/0.51 % equals0D
% 0.34/0.51 thf(fact_34_equals0I,axiom,
% 0.34/0.51 ! [A: set_Product_unit] :
% 0.34/0.51 ( ! [Y2: product_unit] :
% 0.34/0.51 ~ ( member_Product_unit @ Y2 @ A )
% 0.34/0.51 => ( A = bot_bo3957492148770167129t_unit ) ) ).
% 0.34/0.51
% 0.34/0.51 % equals0I
% 0.34/0.51 thf(fact_35_equals0I,axiom,
% 0.34/0.51 ! [A: set_o] :
% 0.34/0.51 ( ! [Y2: $o] :
% 0.34/0.51 ~ ( member_o @ Y2 @ A )
% 0.34/0.51 => ( A = bot_bot_set_o ) ) ).
% 0.34/0.51
% 0.34/0.51 % equals0I
% 0.34/0.51 thf(fact_36_equals0I,axiom,
% 0.34/0.51 ! [A: set_nat] :
% 0.34/0.51 ( ! [Y2: nat] :
% 0.34/0.51 ~ ( member_nat @ Y2 @ A )
% 0.34/0.51 => ( A = bot_bot_set_nat ) ) ).
% 0.34/0.51
% 0.34/0.51 % equals0I
% 0.34/0.51 thf(fact_37_equals0I,axiom,
% 0.34/0.51 ! [A: set_int] :
% 0.34/0.51 ( ! [Y2: int] :
% 0.34/0.51 ~ ( member_int @ Y2 @ A )
% 0.34/0.51 => ( A = bot_bot_set_int ) ) ).
% 0.34/0.51
% 0.34/0.51 % equals0I
% 0.34/0.51 thf(fact_38_equals0I,axiom,
% 0.34/0.51 ! [A: set_Pr4532377907799695533_nat_o] :
% 0.34/0.51 ( ! [Y2: produc3658429121746597890et_nat > $o] :
% 0.34/0.51 ~ ( member6576561426505652726_nat_o @ Y2 @ A )
% 0.34/0.51 => ( A = bot_bo7824918357723582233_nat_o ) ) ).
% 0.34/0.51
% 0.34/0.51 % equals0I
% 0.34/0.51 thf(fact_39_ex__in__conv,axiom,
% 0.34/0.51 ! [A: set_Product_unit] :
% 0.34/0.51 ( ( ? [X2: product_unit] : ( member_Product_unit @ X2 @ A ) )
% 0.34/0.51 = ( A != bot_bo3957492148770167129t_unit ) ) ).
% 0.34/0.51
% 0.34/0.51 % ex_in_conv
% 0.34/0.51 thf(fact_40_ex__in__conv,axiom,
% 0.34/0.51 ! [A: set_o] :
% 0.34/0.51 ( ( ? [X2: $o] : ( member_o @ X2 @ A ) )
% 0.34/0.51 = ( A != bot_bot_set_o ) ) ).
% 0.34/0.51
% 0.34/0.51 % ex_in_conv
% 0.34/0.51 thf(fact_41_ex__in__conv,axiom,
% 0.34/0.51 ! [A: set_nat] :
% 0.34/0.51 ( ( ? [X2: nat] : ( member_nat @ X2 @ A ) )
% 0.34/0.51 = ( A != bot_bot_set_nat ) ) ).
% 0.34/0.51
% 0.34/0.51 % ex_in_conv
% 0.34/0.51 thf(fact_42_ex__in__conv,axiom,
% 0.34/0.51 ! [A: set_int] :
% 0.34/0.51 ( ( ? [X2: int] : ( member_int @ X2 @ A ) )
% 0.34/0.51 = ( A != bot_bot_set_int ) ) ).
% 0.34/0.51
% 0.34/0.51 % ex_in_conv
% 0.34/0.51 thf(fact_43_ex__in__conv,axiom,
% 0.34/0.51 ! [A: set_Pr4532377907799695533_nat_o] :
% 0.34/0.51 ( ( ? [X2: produc3658429121746597890et_nat > $o] : ( member6576561426505652726_nat_o @ X2 @ A ) )
% 0.34/0.51 = ( A != bot_bo7824918357723582233_nat_o ) ) ).
% 0.34/0.51
% 0.34/0.51 % ex_in_conv
% 0.34/0.51 thf(fact_44_set__notEmptyE,axiom,
% 0.34/0.51 ! [S: set_Product_unit] :
% 0.34/0.51 ( ( S != bot_bo3957492148770167129t_unit )
% 0.34/0.51 => ~ ! [X3: product_unit] :
% 0.34/0.51 ~ ( member_Product_unit @ X3 @ S ) ) ).
% 0.34/0.51
% 0.34/0.51 % set_notEmptyE
% 0.34/0.51 thf(fact_45_set__notEmptyE,axiom,
% 0.34/0.51 ! [S: set_o] :
% 0.34/0.51 ( ( S != bot_bot_set_o )
% 0.34/0.51 => ~ ! [X3: $o] :
% 0.34/0.51 ~ ( member_o @ X3 @ S ) ) ).
% 0.34/0.51
% 0.34/0.51 % set_notEmptyE
% 0.34/0.51 thf(fact_46_set__notEmptyE,axiom,
% 0.34/0.51 ! [S: set_nat] :
% 0.34/0.51 ( ( S != bot_bot_set_nat )
% 0.34/0.51 => ~ ! [X3: nat] :
% 0.34/0.51 ~ ( member_nat @ X3 @ S ) ) ).
% 0.34/0.51
% 0.34/0.51 % set_notEmptyE
% 0.34/0.51 thf(fact_47_set__notEmptyE,axiom,
% 0.34/0.51 ! [S: set_int] :
% 0.34/0.51 ( ( S != bot_bot_set_int )
% 0.34/0.51 => ~ ! [X3: int] :
% 0.34/0.51 ~ ( member_int @ X3 @ S ) ) ).
% 0.34/0.51
% 0.34/0.51 % set_notEmptyE
% 0.34/0.51 thf(fact_48_set__notEmptyE,axiom,
% 0.34/0.51 ! [S: set_Pr4532377907799695533_nat_o] :
% 0.34/0.51 ( ( S != bot_bo7824918357723582233_nat_o )
% 0.34/0.51 => ~ ! [X3: produc3658429121746597890et_nat > $o] :
% 0.34/0.51 ~ ( member6576561426505652726_nat_o @ X3 @ S ) ) ).
% 0.34/0.51
% 0.34/0.51 % set_notEmptyE
% 0.34/0.51 thf(fact_49_memb__imp__not__empty,axiom,
% 0.34/0.51 ! [X: product_unit,S: set_Product_unit] :
% 0.34/0.51 ( ( member_Product_unit @ X @ S )
% 0.34/0.51 => ( S != bot_bo3957492148770167129t_unit ) ) ).
% 0.34/0.51
% 0.34/0.51 % memb_imp_not_empty
% 0.34/0.51 thf(fact_50_memb__imp__not__empty,axiom,
% 0.34/0.51 ! [X: $o,S: set_o] :
% 0.34/0.51 ( ( member_o @ X @ S )
% 0.34/0.51 => ( S != bot_bot_set_o ) ) ).
% 0.34/0.51
% 0.34/0.51 % memb_imp_not_empty
% 0.34/0.51 thf(fact_51_memb__imp__not__empty,axiom,
% 0.34/0.51 ! [X: nat,S: set_nat] :
% 0.34/0.51 ( ( member_nat @ X @ S )
% 0.34/0.51 => ( S != bot_bot_set_nat ) ) ).
% 0.34/0.51
% 0.34/0.51 % memb_imp_not_empty
% 0.34/0.51 thf(fact_52_memb__imp__not__empty,axiom,
% 0.34/0.51 ! [X: int,S: set_int] :
% 0.34/0.51 ( ( member_int @ X @ S )
% 0.34/0.51 => ( S != bot_bot_set_int ) ) ).
% 0.34/0.51
% 0.34/0.51 % memb_imp_not_empty
% 0.34/0.51 thf(fact_53_memb__imp__not__empty,axiom,
% 0.34/0.51 ! [X: produc3658429121746597890et_nat > $o,S: set_Pr4532377907799695533_nat_o] :
% 0.34/0.51 ( ( member6576561426505652726_nat_o @ X @ S )
% 0.34/0.51 => ( S != bot_bo7824918357723582233_nat_o ) ) ).
% 0.34/0.51
% 0.34/0.51 % memb_imp_not_empty
% 0.34/0.51 thf(fact_54_bot__set__def,axiom,
% 0.34/0.51 ( bot_bo3957492148770167129t_unit
% 0.34/0.51 = ( collect_Product_unit @ bot_bo4642748612307482820unit_o ) ) ).
% 0.34/0.51
% 0.34/0.51 % bot_set_def
% 0.34/0.51 thf(fact_55_bot__set__def,axiom,
% 0.34/0.51 ( bot_bot_set_o
% 0.34/0.51 = ( collect_o @ bot_bot_o_o ) ) ).
% 0.34/0.51
% 0.34/0.51 % bot_set_def
% 0.34/0.51 thf(fact_56_bot__set__def,axiom,
% 0.34/0.51 ( bot_bot_set_nat
% 0.34/0.51 = ( collect_nat @ bot_bot_nat_o ) ) ).
% 0.34/0.51
% 0.34/0.51 % bot_set_def
% 0.34/0.51 thf(fact_57_bot__set__def,axiom,
% 0.34/0.51 ( bot_bot_set_int
% 0.34/0.51 = ( collect_int @ bot_bot_int_o ) ) ).
% 0.34/0.51
% 0.34/0.51 % bot_set_def
% 0.34/0.51 thf(fact_58_bot__set__def,axiom,
% 0.34/0.51 ( bot_bo7824918357723582233_nat_o
% 0.34/0.51 = ( collec939566748876313656_nat_o @ bot_bo7963750851167320836at_o_o ) ) ).
% 0.34/0.51
% 0.34/0.51 % bot_set_def
% 0.34/0.51 thf(fact_59_mod__false,axiom,
% 0.34/0.51 ! [H2: produc3658429121746597890et_nat] :
% 0.34/0.51 ~ ( rep_assn @ bot_bot_assn @ H2 ) ).
% 0.34/0.51
% 0.34/0.51 % mod_false
% 0.34/0.51 thf(fact_60_Set_Ois__empty__def,axiom,
% 0.34/0.51 ( is_emp5529367068699929823t_unit
% 0.34/0.51 = ( ^ [A3: set_Product_unit] : ( A3 = bot_bo3957492148770167129t_unit ) ) ) ).
% 0.34/0.51
% 0.34/0.51 % Set.is_empty_def
% 0.34/0.51 thf(fact_61_Set_Ois__empty__def,axiom,
% 0.34/0.51 ( is_empty_o
% 0.34/0.51 = ( ^ [A3: set_o] : ( A3 = bot_bot_set_o ) ) ) ).
% 0.34/0.51
% 0.34/0.51 % Set.is_empty_def
% 0.34/0.51 thf(fact_62_Set_Ois__empty__def,axiom,
% 0.34/0.51 ( is_empty_nat
% 0.34/0.51 = ( ^ [A3: set_nat] : ( A3 = bot_bot_set_nat ) ) ) ).
% 0.34/0.51
% 0.34/0.51 % Set.is_empty_def
% 0.34/0.51 thf(fact_63_Set_Ois__empty__def,axiom,
% 0.34/0.51 ( is_empty_int
% 0.34/0.51 = ( ^ [A3: set_int] : ( A3 = bot_bot_set_int ) ) ) ).
% 0.34/0.51
% 0.34/0.51 % Set.is_empty_def
% 0.34/0.51 thf(fact_64_Set_Ois__empty__def,axiom,
% 0.34/0.51 ( is_emp6650082560925436063_nat_o
% 0.34/0.51 = ( ^ [A3: set_Pr4532377907799695533_nat_o] : ( A3 = bot_bo7824918357723582233_nat_o ) ) ) ).
% 0.34/0.51
% 0.34/0.51 % Set.is_empty_def
% 0.34/0.51 thf(fact_65_Collect__empty__eq__bot,axiom,
% 0.34/0.51 ! [P: product_unit > $o] :
% 0.34/0.51 ( ( ( collect_Product_unit @ P )
% 0.34/0.51 = bot_bo3957492148770167129t_unit )
% 0.34/0.51 = ( P = bot_bo4642748612307482820unit_o ) ) ).
% 0.34/0.51
% 0.34/0.51 % Collect_empty_eq_bot
% 0.34/0.51 thf(fact_66_Collect__empty__eq__bot,axiom,
% 0.34/0.51 ! [P: $o > $o] :
% 0.34/0.51 ( ( ( collect_o @ P )
% 0.34/0.51 = bot_bot_set_o )
% 0.34/0.51 = ( P = bot_bot_o_o ) ) ).
% 0.34/0.51
% 0.34/0.51 % Collect_empty_eq_bot
% 0.34/0.51 thf(fact_67_Collect__empty__eq__bot,axiom,
% 0.34/0.51 ! [P: nat > $o] :
% 0.34/0.51 ( ( ( collect_nat @ P )
% 0.34/0.51 = bot_bot_set_nat )
% 0.34/0.51 = ( P = bot_bot_nat_o ) ) ).
% 0.34/0.51
% 0.34/0.51 % Collect_empty_eq_bot
% 0.34/0.51 thf(fact_68_Collect__empty__eq__bot,axiom,
% 0.34/0.51 ! [P: int > $o] :
% 0.34/0.51 ( ( ( collect_int @ P )
% 0.34/0.51 = bot_bot_set_int )
% 0.34/0.51 = ( P = bot_bot_int_o ) ) ).
% 0.34/0.51
% 0.34/0.51 % Collect_empty_eq_bot
% 0.34/0.51 thf(fact_69_Collect__empty__eq__bot,axiom,
% 0.34/0.51 ! [P: ( produc3658429121746597890et_nat > $o ) > $o] :
% 0.34/0.51 ( ( ( collec939566748876313656_nat_o @ P )
% 0.34/0.51 = bot_bo7824918357723582233_nat_o )
% 0.34/0.51 = ( P = bot_bo7963750851167320836at_o_o ) ) ).
% 0.34/0.51
% 0.34/0.51 % Collect_empty_eq_bot
% 0.34/0.51 thf(fact_70_bot__empty__eq,axiom,
% 0.34/0.51 ( bot_bo4642748612307482820unit_o
% 0.34/0.51 = ( ^ [X2: product_unit] : ( member_Product_unit @ X2 @ bot_bo3957492148770167129t_unit ) ) ) ).
% 0.34/0.51
% 0.34/0.51 % bot_empty_eq
% 0.34/0.51 thf(fact_71_bot__empty__eq,axiom,
% 0.34/0.51 ( bot_bot_o_o
% 0.34/0.51 = ( ^ [X2: $o] : ( member_o @ X2 @ bot_bot_set_o ) ) ) ).
% 0.34/0.51
% 0.34/0.51 % bot_empty_eq
% 0.34/0.51 thf(fact_72_bot__empty__eq,axiom,
% 0.34/0.51 ( bot_bot_nat_o
% 0.34/0.51 = ( ^ [X2: nat] : ( member_nat @ X2 @ bot_bot_set_nat ) ) ) ).
% 0.34/0.51
% 0.34/0.51 % bot_empty_eq
% 0.34/0.51 thf(fact_73_bot__empty__eq,axiom,
% 0.34/0.51 ( bot_bot_int_o
% 0.34/0.51 = ( ^ [X2: int] : ( member_int @ X2 @ bot_bot_set_int ) ) ) ).
% 0.34/0.51
% 0.34/0.51 % bot_empty_eq
% 0.34/0.51 thf(fact_74_bot__empty__eq,axiom,
% 0.34/0.51 ( bot_bo7963750851167320836at_o_o
% 0.34/0.51 = ( ^ [X2: produc3658429121746597890et_nat > $o] : ( member6576561426505652726_nat_o @ X2 @ bot_bo7824918357723582233_nat_o ) ) ) ).
% 0.34/0.51
% 0.34/0.51 % bot_empty_eq
% 0.34/0.51 thf(fact_75_wand__assn__def,axiom,
% 0.34/0.51 ( wand_assn
% 0.34/0.51 = ( ^ [P2: assn,Q: assn] : ( abs_assn @ ( wand_raw @ ( rep_assn @ P2 ) @ ( rep_assn @ Q ) ) ) ) ) ).
% 0.34/0.51
% 0.34/0.51 % wand_assn_def
% 0.34/0.51 thf(fact_76_one__assn__def,axiom,
% 0.34/0.51 ( one_one_assn
% 0.34/0.51 = ( abs_assn @ one_assn_raw ) ) ).
% 0.34/0.51
% 0.34/0.51 % one_assn_def
% 0.34/0.51 thf(fact_77_Abs__assn__inverse,axiom,
% 0.34/0.51 ! [Y: produc3658429121746597890et_nat > $o] :
% 0.34/0.51 ( ( member6576561426505652726_nat_o @ Y @ ( collec939566748876313656_nat_o @ proper ) )
% 0.34/0.51 => ( ( rep_assn @ ( abs_assn @ Y ) )
% 0.34/0.51 = Y ) ) ).
% 0.34/0.51
% 0.34/0.51 % Abs_assn_inverse
% 0.34/0.51 thf(fact_78_one__assn__raw_Osimps,axiom,
% 0.34/0.51 ! [H2: heap_e7401611519738050253t_unit,As: set_nat] :
% 0.34/0.51 ( ( one_assn_raw @ ( produc7507926704131184380et_nat @ H2 @ As ) )
% 0.34/0.51 = ( As = bot_bot_set_nat ) ) ).
% 0.34/0.51
% 0.34/0.51 % one_assn_raw.simps
% 0.34/0.51 thf(fact_79_one__assn__raw_Oelims_I1_J,axiom,
% 0.34/0.51 ! [X: produc3658429121746597890et_nat,Y: $o] :
% 0.34/0.51 ( ( ( one_assn_raw @ X )
% 0.34/0.51 = Y )
% 0.34/0.51 => ~ ! [H: heap_e7401611519738050253t_unit,As2: set_nat] :
% 0.34/0.51 ( ( X
% 0.34/0.51 = ( produc7507926704131184380et_nat @ H @ As2 ) )
% 0.34/0.51 => ( Y
% 0.34/0.51 = ( As2 != bot_bot_set_nat ) ) ) ) ).
% 0.34/0.51
% 0.34/0.51 % one_assn_raw.elims(1)
% 0.34/0.51 thf(fact_80_one__assn__raw_Oelims_I2_J,axiom,
% 0.34/0.51 ! [X: produc3658429121746597890et_nat] :
% 0.34/0.51 ( ( one_assn_raw @ X )
% 0.34/0.51 => ~ ! [H: heap_e7401611519738050253t_unit,As2: set_nat] :
% 0.34/0.51 ( ( X
% 0.34/0.51 = ( produc7507926704131184380et_nat @ H @ As2 ) )
% 0.34/0.51 => ( As2 != bot_bot_set_nat ) ) ) ).
% 0.34/0.51
% 0.34/0.51 % one_assn_raw.elims(2)
% 0.34/0.51 thf(fact_81_one__assn__raw_Oelims_I3_J,axiom,
% 0.34/0.51 ! [X: produc3658429121746597890et_nat] :
% 0.34/0.51 ( ~ ( one_assn_raw @ X )
% 0.34/0.51 => ~ ! [H: heap_e7401611519738050253t_unit,As2: set_nat] :
% 0.34/0.51 ( ( X
% 0.34/0.51 = ( produc7507926704131184380et_nat @ H @ As2 ) )
% 0.34/0.51 => ( As2 = bot_bot_set_nat ) ) ) ).
% 0.34/0.51
% 0.34/0.51 % one_assn_raw.elims(3)
% 0.34/0.51 thf(fact_82_is__singletonI_H,axiom,
% 0.34/0.51 ! [A: set_Product_unit] :
% 0.34/0.51 ( ( A != bot_bo3957492148770167129t_unit )
% 0.34/0.51 => ( ! [X3: product_unit,Y2: product_unit] :
% 0.34/0.51 ( ( member_Product_unit @ X3 @ A )
% 0.34/0.51 => ( ( member_Product_unit @ Y2 @ A )
% 0.34/0.51 => ( X3 = Y2 ) ) )
% 0.34/0.51 => ( is_sin2160648248035936513t_unit @ A ) ) ) ).
% 0.34/0.51
% 0.34/0.51 % is_singletonI'
% 0.34/0.51 thf(fact_83_is__singletonI_H,axiom,
% 0.34/0.51 ! [A: set_o] :
% 0.34/0.51 ( ( A != bot_bot_set_o )
% 0.34/0.51 => ( ! [X3: $o,Y2: $o] :
% 0.34/0.51 ( ( member_o @ X3 @ A )
% 0.34/0.51 => ( ( member_o @ Y2 @ A )
% 0.34/0.51 => ( X3 = Y2 ) ) )
% 0.34/0.51 => ( is_singleton_o @ A ) ) ) ).
% 0.34/0.51
% 0.34/0.51 % is_singletonI'
% 0.34/0.51 thf(fact_84_is__singletonI_H,axiom,
% 0.34/0.51 ! [A: set_nat] :
% 0.34/0.51 ( ( A != bot_bot_set_nat )
% 0.34/0.51 => ( ! [X3: nat,Y2: nat] :
% 0.34/0.51 ( ( member_nat @ X3 @ A )
% 0.34/0.51 => ( ( member_nat @ Y2 @ A )
% 0.34/0.51 => ( X3 = Y2 ) ) )
% 0.34/0.51 => ( is_singleton_nat @ A ) ) ) ).
% 0.34/0.51
% 0.34/0.51 % is_singletonI'
% 0.34/0.51 thf(fact_85_is__singletonI_H,axiom,
% 0.34/0.51 ! [A: set_int] :
% 0.34/0.51 ( ( A != bot_bot_set_int )
% 0.34/0.51 => ( ! [X3: int,Y2: int] :
% 0.34/0.51 ( ( member_int @ X3 @ A )
% 0.34/0.51 => ( ( member_int @ Y2 @ A )
% 0.34/0.51 => ( X3 = Y2 ) ) )
% 0.34/0.51 => ( is_singleton_int @ A ) ) ) ).
% 0.34/0.51
% 0.34/0.51 % is_singletonI'
% 0.34/0.51 thf(fact_86_is__singletonI_H,axiom,
% 0.34/0.51 ! [A: set_Pr4532377907799695533_nat_o] :
% 0.34/0.51 ( ( A != bot_bo7824918357723582233_nat_o )
% 0.34/0.51 => ( ! [X3: produc3658429121746597890et_nat > $o,Y2: produc3658429121746597890et_nat > $o] :
% 0.34/0.51 ( ( member6576561426505652726_nat_o @ X3 @ A )
% 0.34/0.51 => ( ( member6576561426505652726_nat_o @ Y2 @ A )
% 0.34/0.51 => ( X3 = Y2 ) ) )
% 0.34/0.51 => ( is_sin5180296473474724033_nat_o @ A ) ) ) ).
% 0.34/0.51
% 0.34/0.51 % is_singletonI'
% 0.34/0.51 thf(fact_87_wand__proper,axiom,
% 0.34/0.51 ! [P: produc3658429121746597890et_nat > $o,Q2: produc3658429121746597890et_nat > $o] : ( proper @ ( wand_raw @ P @ Q2 ) ) ).
% 0.34/0.51
% 0.34/0.51 % wand_proper
% 0.34/0.51 thf(fact_88_one__assn__proper,axiom,
% 0.34/0.51 proper @ one_assn_raw ).
% 0.34/0.51
% 0.34/0.51 % one_assn_proper
% 0.34/0.51 thf(fact_89_assn__basic__inequalities_I3_J,axiom,
% 0.34/0.51 bot_bot_assn != one_one_assn ).
% 0.34/0.51
% 0.34/0.51 % assn_basic_inequalities(3)
% 0.34/0.51 thf(fact_90_bex2I,axiom,
% 0.34/0.51 ! [A2: heap_e7401611519738050253t_unit,B: set_nat,S: set_Pr3948176798113811640et_nat,P: heap_e7401611519738050253t_unit > set_nat > $o] :
% 0.34/0.51 ( ( member6260224972018164377et_nat @ ( produc7507926704131184380et_nat @ A2 @ B ) @ S )
% 0.34/0.51 => ( ( ( member6260224972018164377et_nat @ ( produc7507926704131184380et_nat @ A2 @ B ) @ S )
% 0.34/0.51 => ( P @ A2 @ B ) )
% 0.34/0.51 => ? [A4: heap_e7401611519738050253t_unit,B2: set_nat] :
% 0.34/0.51 ( ( member6260224972018164377et_nat @ ( produc7507926704131184380et_nat @ A4 @ B2 ) @ S )
% 0.34/0.51 & ( P @ A4 @ B2 ) ) ) ) ).
% 0.34/0.51
% 0.34/0.51 % bex2I
% 0.34/0.51 thf(fact_91_bex2I,axiom,
% 0.34/0.51 ! [A2: nat,B: nat,S: set_Pr1261947904930325089at_nat,P: nat > nat > $o] :
% 0.34/0.51 ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A2 @ B ) @ S )
% 0.34/0.51 => ( ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A2 @ B ) @ S )
% 0.34/0.51 => ( P @ A2 @ B ) )
% 0.34/0.51 => ? [A4: nat,B2: nat] :
% 0.34/0.51 ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A4 @ B2 ) @ S )
% 0.34/0.51 & ( P @ A4 @ B2 ) ) ) ) ).
% 0.34/0.51
% 0.34/0.51 % bex2I
% 0.34/0.51 thf(fact_92_bex2I,axiom,
% 0.34/0.51 ! [A2: int,B: int,S: set_Pr958786334691620121nt_int,P: int > int > $o] :
% 0.34/0.51 ( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ A2 @ B ) @ S )
% 0.34/0.51 => ( ( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ A2 @ B ) @ S )
% 0.34/0.51 => ( P @ A2 @ B ) )
% 0.34/0.51 => ? [A4: int,B2: int] :
% 0.34/0.51 ( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ A4 @ B2 ) @ S )
% 0.34/0.51 & ( P @ A4 @ B2 ) ) ) ) ).
% 0.34/0.51
% 0.34/0.51 % bex2I
% 0.34/0.51 thf(fact_93_bex2I,axiom,
% 0.34/0.51 ! [A2: produc3658429121746597890et_nat > $o,B: produc3658429121746597890et_nat,S: set_Pr3286484037609594932et_nat,P: ( produc3658429121746597890et_nat > $o ) > produc3658429121746597890et_nat > $o] :
% 0.34/0.51 ( ( member1996754912294343701et_nat @ ( produc5001842942810119800et_nat @ A2 @ B ) @ S )
% 0.34/0.51 => ( ( ( member1996754912294343701et_nat @ ( produc5001842942810119800et_nat @ A2 @ B ) @ S )
% 0.34/0.51 => ( P @ A2 @ B ) )
% 0.34/0.51 => ? [A4: produc3658429121746597890et_nat > $o,B2: produc3658429121746597890et_nat] :
% 0.34/0.51 ( ( member1996754912294343701et_nat @ ( produc5001842942810119800et_nat @ A4 @ B2 ) @ S )
% 0.34/0.51 & ( P @ A4 @ B2 ) ) ) ) ).
% 0.34/0.51
% 0.34/0.51 % bex2I
% 0.34/0.51 thf(fact_94_bex2I,axiom,
% 0.34/0.51 ! [A2: produc3658429121746597890et_nat > $o,B: produc3925858234332021118et_nat,S: set_Pr8536935166611901872et_nat,P: ( produc3658429121746597890et_nat > $o ) > produc3925858234332021118et_nat > $o] :
% 0.34/0.51 ( ( member6124377750444531601et_nat @ ( produc2245416461498447860et_nat @ A2 @ B ) @ S )
% 0.34/0.51 => ( ( ( member6124377750444531601et_nat @ ( produc2245416461498447860et_nat @ A2 @ B ) @ S )
% 0.34/0.51 => ( P @ A2 @ B ) )
% 0.34/0.51 => ? [A4: produc3658429121746597890et_nat > $o,B2: produc3925858234332021118et_nat] :
% 0.34/0.51 ( ( member6124377750444531601et_nat @ ( produc2245416461498447860et_nat @ A4 @ B2 ) @ S )
% 0.34/0.51 & ( P @ A4 @ B2 ) ) ) ) ).
% 0.34/0.51
% 0.34/0.51 % bex2I
% 0.34/0.51 thf(fact_95_one__assn__raw_Ocases,axiom,
% 0.34/0.51 ! [X: produc3658429121746597890et_nat] :
% 0.34/0.51 ~ ! [H: heap_e7401611519738050253t_unit,As2: set_nat] :
% 0.34/0.51 ( X
% 0.34/0.51 != ( produc7507926704131184380et_nat @ H @ As2 ) ) ).
% 0.34/0.51
% 0.34/0.51 % one_assn_raw.cases
% 0.34/0.51 thf(fact_96_sndE,axiom,
% 0.34/0.51 ! [X: product_prod_nat_nat,A2: nat,B: nat,P: nat > $o] :
% 0.34/0.51 ( ( X
% 0.34/0.51 = ( product_Pair_nat_nat @ A2 @ B ) )
% 0.34/0.51 => ( ( P @ ( product_snd_nat_nat @ X ) )
% 0.34/0.51 => ( P @ B ) ) ) ).
% 0.34/0.51
% 0.34/0.51 % sndE
% 0.34/0.51 thf(fact_97_sndE,axiom,
% 0.34/0.51 ! [X: produc3925858234332021118et_nat,A2: produc3658429121746597890et_nat > $o,B: produc3658429121746597890et_nat,P: produc3658429121746597890et_nat > $o] :
% 0.34/0.51 ( ( X
% 0.34/0.51 = ( produc5001842942810119800et_nat @ A2 @ B ) )
% 0.34/0.51 => ( ( P @ ( produc4011572625026189258et_nat @ X ) )
% 0.34/0.51 => ( P @ B ) ) ) ).
% 0.34/0.51
% 0.34/0.51 % sndE
% 0.34/0.51 thf(fact_98_sndE,axiom,
% 0.34/0.51 ! [X: produc2732055786443039994et_nat,A2: produc3658429121746597890et_nat > $o,B: produc3925858234332021118et_nat,P: produc3925858234332021118et_nat > $o] :
% 0.34/0.51 ( ( X
% 0.34/0.51 = ( produc2245416461498447860et_nat @ A2 @ B ) )
% 0.34/0.51 => ( ( P @ ( produc5374455773327741254et_nat @ X ) )
% 0.34/0.51 => ( P @ B ) ) ) ).
% 0.34/0.51
% 0.34/0.51 % sndE
% 0.34/0.51 thf(fact_99_sndE,axiom,
% 0.34/0.51 ! [X: produc3658429121746597890et_nat,A2: heap_e7401611519738050253t_unit,B: set_nat,P: set_nat > $o] :
% 0.34/0.51 ( ( X
% 0.34/0.51 = ( produc7507926704131184380et_nat @ A2 @ B ) )
% 0.34/0.51 => ( ( P @ ( produc8586169260539613262et_nat @ X ) )
% 0.34/0.51 => ( P @ B ) ) ) ).
% 0.34/0.51
% 0.34/0.51 % sndE
% 0.34/0.51 thf(fact_100_sndE,axiom,
% 0.34/0.51 ! [X: product_prod_int_int,A2: int,B: int,P: int > $o] :
% 0.34/0.51 ( ( X
% 0.34/0.51 = ( product_Pair_int_int @ A2 @ B ) )
% 0.34/0.51 => ( ( P @ ( product_snd_int_int @ X ) )
% 0.34/0.51 => ( P @ B ) ) ) ).
% 0.34/0.51
% 0.34/0.51 % sndE
% 0.34/0.51 thf(fact_101_Rep__assn,axiom,
% 0.34/0.51 ! [X: assn] : ( member6576561426505652726_nat_o @ ( rep_assn @ X ) @ ( collec939566748876313656_nat_o @ proper ) ) ).
% 0.34/0.51
% 0.34/0.51 % Rep_assn
% 0.34/0.51 thf(fact_102_mem__Collect__eq,axiom,
% 0.34/0.51 ! [A2: produc3658429121746597890et_nat > $o,P: ( produc3658429121746597890et_nat > $o ) > $o] :
% 0.34/0.51 ( ( member6576561426505652726_nat_o @ A2 @ ( collec939566748876313656_nat_o @ P ) )
% 0.34/0.51 = ( P @ A2 ) ) ).
% 0.34/0.51
% 0.34/0.51 % mem_Collect_eq
% 0.34/0.51 thf(fact_103_mem__Collect__eq,axiom,
% 0.34/0.51 ! [A2: nat,P: nat > $o] :
% 0.34/0.51 ( ( member_nat @ A2 @ ( collect_nat @ P ) )
% 0.34/0.51 = ( P @ A2 ) ) ).
% 0.34/0.51
% 0.34/0.51 % mem_Collect_eq
% 0.34/0.51 thf(fact_104_mem__Collect__eq,axiom,
% 0.34/0.51 ! [A2: int,P: int > $o] :
% 0.34/0.51 ( ( member_int @ A2 @ ( collect_int @ P ) )
% 0.34/0.51 = ( P @ A2 ) ) ).
% 0.34/0.51
% 0.34/0.51 % mem_Collect_eq
% 0.34/0.51 thf(fact_105_mem__Collect__eq,axiom,
% 0.34/0.51 ! [A2: $o,P: $o > $o] :
% 0.34/0.51 ( ( member_o @ A2 @ ( collect_o @ P ) )
% 0.34/0.51 = ( P @ A2 ) ) ).
% 0.34/0.51
% 0.34/0.51 % mem_Collect_eq
% 0.34/0.51 thf(fact_106_mem__Collect__eq,axiom,
% 0.34/0.51 ! [A2: product_unit,P: product_unit > $o] :
% 0.34/0.51 ( ( member_Product_unit @ A2 @ ( collect_Product_unit @ P ) )
% 0.34/0.51 = ( P @ A2 ) ) ).
% 0.34/0.51
% 0.34/0.51 % mem_Collect_eq
% 0.34/0.51 thf(fact_107_Collect__mem__eq,axiom,
% 0.34/0.51 ! [A: set_Pr4532377907799695533_nat_o] :
% 0.34/0.51 ( ( collec939566748876313656_nat_o
% 0.34/0.51 @ ^ [X2: produc3658429121746597890et_nat > $o] : ( member6576561426505652726_nat_o @ X2 @ A ) )
% 0.34/0.51 = A ) ).
% 0.34/0.51
% 0.34/0.51 % Collect_mem_eq
% 0.34/0.51 thf(fact_108_Collect__mem__eq,axiom,
% 0.34/0.51 ! [A: set_nat] :
% 0.34/0.51 ( ( collect_nat
% 0.34/0.51 @ ^ [X2: nat] : ( member_nat @ X2 @ A ) )
% 0.34/0.51 = A ) ).
% 0.34/0.51
% 0.34/0.51 % Collect_mem_eq
% 0.34/0.51 thf(fact_109_Collect__mem__eq,axiom,
% 0.34/0.51 ! [A: set_int] :
% 0.34/0.51 ( ( collect_int
% 0.34/0.51 @ ^ [X2: int] : ( member_int @ X2 @ A ) )
% 0.34/0.51 = A ) ).
% 0.34/0.51
% 0.34/0.51 % Collect_mem_eq
% 0.34/0.51 thf(fact_110_Collect__mem__eq,axiom,
% 0.34/0.51 ! [A: set_o] :
% 0.34/0.51 ( ( collect_o
% 0.34/0.51 @ ^ [X2: $o] : ( member_o @ X2 @ A ) )
% 0.34/0.51 = A ) ).
% 0.34/0.51
% 0.34/0.51 % Collect_mem_eq
% 0.34/0.51 thf(fact_111_Collect__mem__eq,axiom,
% 0.34/0.51 ! [A: set_Product_unit] :
% 0.34/0.51 ( ( collect_Product_unit
% 0.34/0.51 @ ^ [X2: product_unit] : ( member_Product_unit @ X2 @ A ) )
% 0.34/0.51 = A ) ).
% 0.34/0.51
% 0.34/0.51 % Collect_mem_eq
% 0.34/0.51 thf(fact_112_Collect__cong,axiom,
% 0.34/0.51 ! [P: ( produc3658429121746597890et_nat > $o ) > $o,Q2: ( produc3658429121746597890et_nat > $o ) > $o] :
% 0.34/0.51 ( ! [X3: produc3658429121746597890et_nat > $o] :
% 0.34/0.51 ( ( P @ X3 )
% 0.34/0.51 = ( Q2 @ X3 ) )
% 0.34/0.51 => ( ( collec939566748876313656_nat_o @ P )
% 0.34/0.51 = ( collec939566748876313656_nat_o @ Q2 ) ) ) ).
% 0.34/0.51
% 0.34/0.51 % Collect_cong
% 0.34/0.51 thf(fact_113_Collect__cong,axiom,
% 0.34/0.51 ! [P: nat > $o,Q2: nat > $o] :
% 0.34/0.51 ( ! [X3: nat] :
% 0.34/0.51 ( ( P @ X3 )
% 0.34/0.51 = ( Q2 @ X3 ) )
% 0.34/0.51 => ( ( collect_nat @ P )
% 0.34/0.51 = ( collect_nat @ Q2 ) ) ) ).
% 0.34/0.51
% 0.34/0.51 % Collect_cong
% 0.34/0.51 thf(fact_114_Collect__cong,axiom,
% 0.34/0.51 ! [P: int > $o,Q2: int > $o] :
% 0.34/0.51 ( ! [X3: int] :
% 0.34/0.51 ( ( P @ X3 )
% 0.34/0.51 = ( Q2 @ X3 ) )
% 0.34/0.51 => ( ( collect_int @ P )
% 0.34/0.51 = ( collect_int @ Q2 ) ) ) ).
% 0.34/0.51
% 0.34/0.51 % Collect_cong
% 0.34/0.51 thf(fact_115_Collect__cong,axiom,
% 0.34/0.51 ! [P: $o > $o,Q2: $o > $o] :
% 0.34/0.51 ( ! [X3: $o] :
% 0.34/0.51 ( ( P @ X3 )
% 0.34/0.51 = ( Q2 @ X3 ) )
% 0.34/0.51 => ( ( collect_o @ P )
% 0.34/0.51 = ( collect_o @ Q2 ) ) ) ).
% 0.34/0.51
% 0.34/0.51 % Collect_cong
% 0.34/0.51 thf(fact_116_Collect__cong,axiom,
% 0.34/0.51 ! [P: product_unit > $o,Q2: product_unit > $o] :
% 0.34/0.51 ( ! [X3: product_unit] :
% 0.34/0.51 ( ( P @ X3 )
% 0.34/0.51 = ( Q2 @ X3 ) )
% 0.34/0.51 => ( ( collect_Product_unit @ P )
% 0.34/0.51 = ( collect_Product_unit @ Q2 ) ) ) ).
% 0.34/0.51
% 0.34/0.51 % Collect_cong
% 0.34/0.51 thf(fact_117_Rep__assn__cases,axiom,
% 0.34/0.51 ! [Y: produc3658429121746597890et_nat > $o] :
% 0.34/0.51 ( ( member6576561426505652726_nat_o @ Y @ ( collec939566748876313656_nat_o @ proper ) )
% 0.34/0.51 => ~ ! [X3: assn] :
% 0.34/0.51 ( Y
% 0.34/0.51 != ( rep_assn @ X3 ) ) ) ).
% 0.34/0.51
% 0.34/0.51 % Rep_assn_cases
% 0.34/0.51 thf(fact_118_Rep__assn__induct,axiom,
% 0.34/0.51 ! [Y: produc3658429121746597890et_nat > $o,P: ( produc3658429121746597890et_nat > $o ) > $o] :
% 0.34/0.51 ( ( member6576561426505652726_nat_o @ Y @ ( collec939566748876313656_nat_o @ proper ) )
% 0.34/0.51 => ( ! [X3: assn] : ( P @ ( rep_assn @ X3 ) )
% 0.34/0.51 => ( P @ Y ) ) ) ).
% 0.34/0.51
% 0.34/0.51 % Rep_assn_induct
% 0.34/0.51 thf(fact_119_Abs__assn__cases,axiom,
% 0.34/0.51 ! [X: assn] :
% 0.34/0.51 ~ ! [Y2: produc3658429121746597890et_nat > $o] :
% 0.34/0.51 ( ( X
% 0.34/0.51 = ( abs_assn @ Y2 ) )
% 0.34/0.51 => ~ ( member6576561426505652726_nat_o @ Y2 @ ( collec939566748876313656_nat_o @ proper ) ) ) ).
% 0.34/0.51
% 0.34/0.51 % Abs_assn_cases
% 0.34/0.51 thf(fact_120_Abs__assn__induct,axiom,
% 0.34/0.51 ! [P: assn > $o,X: assn] :
% 0.34/0.51 ( ! [Y2: produc3658429121746597890et_nat > $o] :
% 0.34/0.51 ( ( member6576561426505652726_nat_o @ Y2 @ ( collec939566748876313656_nat_o @ proper ) )
% 0.34/0.51 => ( P @ ( abs_assn @ Y2 ) ) )
% 0.34/0.51 => ( P @ X ) ) ).
% 0.34/0.51
% 0.34/0.51 % Abs_assn_induct
% 0.34/0.51 thf(fact_121_Abs__assn__inject,axiom,
% 0.34/0.51 ! [X: produc3658429121746597890et_nat > $o,Y: produc3658429121746597890et_nat > $o] :
% 0.34/0.51 ( ( member6576561426505652726_nat_o @ X @ ( collec939566748876313656_nat_o @ proper ) )
% 0.34/0.51 => ( ( member6576561426505652726_nat_o @ Y @ ( collec939566748876313656_nat_o @ proper ) )
% 0.34/0.51 => ( ( ( abs_assn @ X )
% 0.34/0.51 = ( abs_assn @ Y ) )
% 0.34/0.51 = ( X = Y ) ) ) ) ).
% 0.34/0.51
% 0.34/0.51 % Abs_assn_inject
% 0.34/0.51 thf(fact_122_mod__h__bot__indep,axiom,
% 0.34/0.51 ! [P: assn,H2: heap_e7401611519738050253t_unit,H3: heap_e7401611519738050253t_unit] :
% 0.34/0.51 ( ( rep_assn @ P @ ( produc7507926704131184380et_nat @ H2 @ bot_bot_set_nat ) )
% 0.34/0.51 = ( rep_assn @ P @ ( produc7507926704131184380et_nat @ H3 @ bot_bot_set_nat ) ) ) ).
% 0.34/0.51
% 0.34/0.51 % mod_h_bot_indep
% 0.34/0.51 thf(fact_123_old_Oprod_Oinject,axiom,
% 0.34/0.51 ! [A2: heap_e7401611519738050253t_unit,B: set_nat,A5: heap_e7401611519738050253t_unit,B3: set_nat] :
% 0.34/0.51 ( ( ( produc7507926704131184380et_nat @ A2 @ B )
% 0.34/0.51 = ( produc7507926704131184380et_nat @ A5 @ B3 ) )
% 0.34/0.51 = ( ( A2 = A5 )
% 0.34/0.51 & ( B = B3 ) ) ) ).
% 0.34/0.51
% 0.34/0.51 % old.prod.inject
% 0.34/0.51 thf(fact_124_old_Oprod_Oinject,axiom,
% 0.34/0.51 ! [A2: nat,B: nat,A5: nat,B3: nat] :
% 0.34/0.51 ( ( ( product_Pair_nat_nat @ A2 @ B )
% 0.34/0.51 = ( product_Pair_nat_nat @ A5 @ B3 ) )
% 0.34/0.51 = ( ( A2 = A5 )
% 0.34/0.51 & ( B = B3 ) ) ) ).
% 0.34/0.51
% 0.34/0.51 % old.prod.inject
% 0.34/0.51 thf(fact_125_old_Oprod_Oinject,axiom,
% 0.34/0.51 ! [A2: int,B: int,A5: int,B3: int] :
% 0.34/0.51 ( ( ( product_Pair_int_int @ A2 @ B )
% 0.34/0.51 = ( product_Pair_int_int @ A5 @ B3 ) )
% 0.34/0.51 = ( ( A2 = A5 )
% 0.34/0.51 & ( B = B3 ) ) ) ).
% 0.34/0.51
% 0.34/0.51 % old.prod.inject
% 0.34/0.51 thf(fact_126_old_Oprod_Oinject,axiom,
% 0.34/0.51 ! [A2: produc3658429121746597890et_nat > $o,B: produc3658429121746597890et_nat,A5: produc3658429121746597890et_nat > $o,B3: produc3658429121746597890et_nat] :
% 0.34/0.51 ( ( ( produc5001842942810119800et_nat @ A2 @ B )
% 0.34/0.51 = ( produc5001842942810119800et_nat @ A5 @ B3 ) )
% 0.34/0.51 = ( ( A2 = A5 )
% 0.34/0.51 & ( B = B3 ) ) ) ).
% 0.34/0.51
% 0.34/0.51 % old.prod.inject
% 0.34/0.51 thf(fact_127_old_Oprod_Oinject,axiom,
% 0.34/0.51 ! [A2: produc3658429121746597890et_nat > $o,B: produc3925858234332021118et_nat,A5: produc3658429121746597890et_nat > $o,B3: produc3925858234332021118et_nat] :
% 0.34/0.51 ( ( ( produc2245416461498447860et_nat @ A2 @ B )
% 0.34/0.51 = ( produc2245416461498447860et_nat @ A5 @ B3 ) )
% 0.34/0.51 = ( ( A2 = A5 )
% 0.34/0.51 & ( B = B3 ) ) ) ).
% 0.34/0.51
% 0.34/0.51 % old.prod.inject
% 0.34/0.51 thf(fact_128_prod_Oinject,axiom,
% 0.34/0.51 ! [X1: heap_e7401611519738050253t_unit,X22: set_nat,Y1: heap_e7401611519738050253t_unit,Y22: set_nat] :
% 0.34/0.51 ( ( ( produc7507926704131184380et_nat @ X1 @ X22 )
% 0.34/0.51 = ( produc7507926704131184380et_nat @ Y1 @ Y22 ) )
% 0.34/0.51 = ( ( X1 = Y1 )
% 0.34/0.51 & ( X22 = Y22 ) ) ) ).
% 0.34/0.51
% 0.34/0.51 % prod.inject
% 0.34/0.51 thf(fact_129_prod_Oinject,axiom,
% 0.34/0.51 ! [X1: nat,X22: nat,Y1: nat,Y22: nat] :
% 0.34/0.51 ( ( ( product_Pair_nat_nat @ X1 @ X22 )
% 0.34/0.51 = ( product_Pair_nat_nat @ Y1 @ Y22 ) )
% 0.34/0.51 = ( ( X1 = Y1 )
% 0.34/0.51 & ( X22 = Y22 ) ) ) ).
% 0.34/0.51
% 0.34/0.51 % prod.inject
% 0.34/0.51 thf(fact_130_prod_Oinject,axiom,
% 0.34/0.51 ! [X1: int,X22: int,Y1: int,Y22: int] :
% 0.34/0.51 ( ( ( product_Pair_int_int @ X1 @ X22 )
% 0.34/0.51 = ( product_Pair_int_int @ Y1 @ Y22 ) )
% 0.34/0.51 = ( ( X1 = Y1 )
% 0.34/0.51 & ( X22 = Y22 ) ) ) ).
% 0.34/0.51
% 0.34/0.51 % prod.inject
% 0.34/0.51 thf(fact_131_prod_Oinject,axiom,
% 0.34/0.51 ! [X1: produc3658429121746597890et_nat > $o,X22: produc3658429121746597890et_nat,Y1: produc3658429121746597890et_nat > $o,Y22: produc3658429121746597890et_nat] :
% 0.34/0.51 ( ( ( produc5001842942810119800et_nat @ X1 @ X22 )
% 0.34/0.51 = ( produc5001842942810119800et_nat @ Y1 @ Y22 ) )
% 0.34/0.51 = ( ( X1 = Y1 )
% 0.34/0.51 & ( X22 = Y22 ) ) ) ).
% 0.34/0.51
% 0.34/0.51 % prod.inject
% 0.34/0.51 thf(fact_132_prod_Oinject,axiom,
% 0.34/0.51 ! [X1: produc3658429121746597890et_nat > $o,X22: produc3925858234332021118et_nat,Y1: produc3658429121746597890et_nat > $o,Y22: produc3925858234332021118et_nat] :
% 0.34/0.51 ( ( ( produc2245416461498447860et_nat @ X1 @ X22 )
% 0.34/0.51 = ( produc2245416461498447860et_nat @ Y1 @ Y22 ) )
% 0.34/0.51 = ( ( X1 = Y1 )
% 0.34/0.51 & ( X22 = Y22 ) ) ) ).
% 0.34/0.51
% 0.34/0.51 % prod.inject
% 0.34/0.51 thf(fact_133_sndI,axiom,
% 0.34/0.51 ! [X: product_prod_nat_nat,Y: nat,Z: nat] :
% 0.34/0.51 ( ( X
% 0.34/0.51 = ( product_Pair_nat_nat @ Y @ Z ) )
% 0.34/0.51 => ( ( product_snd_nat_nat @ X )
% 0.34/0.51 = Z ) ) ).
% 0.34/0.51
% 0.34/0.51 % sndI
% 0.34/0.51 thf(fact_134_sndI,axiom,
% 0.34/0.51 ! [X: produc3925858234332021118et_nat,Y: produc3658429121746597890et_nat > $o,Z: produc3658429121746597890et_nat] :
% 0.34/0.51 ( ( X
% 0.34/0.51 = ( produc5001842942810119800et_nat @ Y @ Z ) )
% 0.34/0.51 => ( ( produc4011572625026189258et_nat @ X )
% 0.34/0.51 = Z ) ) ).
% 0.34/0.51
% 0.34/0.51 % sndI
% 0.34/0.51 thf(fact_135_sndI,axiom,
% 0.34/0.51 ! [X: produc2732055786443039994et_nat,Y: produc3658429121746597890et_nat > $o,Z: produc3925858234332021118et_nat] :
% 0.34/0.51 ( ( X
% 0.34/0.51 = ( produc2245416461498447860et_nat @ Y @ Z ) )
% 0.34/0.51 => ( ( produc5374455773327741254et_nat @ X )
% 0.34/0.51 = Z ) ) ).
% 0.34/0.51
% 0.34/0.51 % sndI
% 0.34/0.51 thf(fact_136_sndI,axiom,
% 0.34/0.51 ! [X: produc3658429121746597890et_nat,Y: heap_e7401611519738050253t_unit,Z: set_nat] :
% 0.34/0.51 ( ( X
% 0.34/0.51 = ( produc7507926704131184380et_nat @ Y @ Z ) )
% 0.34/0.51 => ( ( produc8586169260539613262et_nat @ X )
% 0.34/0.51 = Z ) ) ).
% 0.34/0.51
% 0.34/0.51 % sndI
% 0.34/0.51 thf(fact_137_sndI,axiom,
% 0.34/0.51 ! [X: product_prod_int_int,Y: int,Z: int] :
% 0.34/0.51 ( ( X
% 0.34/0.51 = ( product_Pair_int_int @ Y @ Z ) )
% 0.34/0.51 => ( ( product_snd_int_int @ X )
% 0.34/0.51 = Z ) ) ).
% 0.34/0.51
% 0.34/0.51 % sndI
% 0.34/0.51 thf(fact_138_snd__eqD,axiom,
% 0.34/0.51 ! [X: nat,Y: nat,A2: nat] :
% 0.34/0.51 ( ( ( product_snd_nat_nat @ ( product_Pair_nat_nat @ X @ Y ) )
% 0.34/0.51 = A2 )
% 0.34/0.51 => ( Y = A2 ) ) ).
% 0.34/0.51
% 0.34/0.51 % snd_eqD
% 0.34/0.51 thf(fact_139_snd__eqD,axiom,
% 0.34/0.51 ! [X: produc3658429121746597890et_nat > $o,Y: produc3658429121746597890et_nat,A2: produc3658429121746597890et_nat] :
% 0.34/0.51 ( ( ( produc4011572625026189258et_nat @ ( produc5001842942810119800et_nat @ X @ Y ) )
% 0.34/0.51 = A2 )
% 0.34/0.51 => ( Y = A2 ) ) ).
% 0.34/0.51
% 0.34/0.51 % snd_eqD
% 0.34/0.51 thf(fact_140_snd__eqD,axiom,
% 0.34/0.51 ! [X: produc3658429121746597890et_nat > $o,Y: produc3925858234332021118et_nat,A2: produc3925858234332021118et_nat] :
% 0.34/0.51 ( ( ( produc5374455773327741254et_nat @ ( produc2245416461498447860et_nat @ X @ Y ) )
% 0.34/0.51 = A2 )
% 0.34/0.51 => ( Y = A2 ) ) ).
% 0.34/0.51
% 0.34/0.51 % snd_eqD
% 0.34/0.51 thf(fact_141_snd__eqD,axiom,
% 0.34/0.51 ! [X: heap_e7401611519738050253t_unit,Y: set_nat,A2: set_nat] :
% 0.34/0.51 ( ( ( produc8586169260539613262et_nat @ ( produc7507926704131184380et_nat @ X @ Y ) )
% 0.34/0.51 = A2 )
% 0.34/0.51 => ( Y = A2 ) ) ).
% 0.34/0.51
% 0.34/0.51 % snd_eqD
% 0.34/0.51 thf(fact_142_snd__eqD,axiom,
% 0.34/0.51 ! [X: int,Y: int,A2: int] :
% 0.34/0.51 ( ( ( product_snd_int_int @ ( product_Pair_int_int @ X @ Y ) )
% 0.34/0.51 = A2 )
% 0.34/0.51 => ( Y = A2 ) ) ).
% 0.34/0.51
% 0.34/0.51 % snd_eqD
% 0.34/0.51 thf(fact_143_snd__conv,axiom,
% 0.34/0.51 ! [X1: nat,X22: nat] :
% 0.34/0.51 ( ( product_snd_nat_nat @ ( product_Pair_nat_nat @ X1 @ X22 ) )
% 0.34/0.51 = X22 ) ).
% 0.34/0.51
% 0.34/0.51 % snd_conv
% 0.34/0.51 thf(fact_144_snd__conv,axiom,
% 0.34/0.51 ! [X1: produc3658429121746597890et_nat > $o,X22: produc3658429121746597890et_nat] :
% 0.34/0.51 ( ( produc4011572625026189258et_nat @ ( produc5001842942810119800et_nat @ X1 @ X22 ) )
% 0.34/0.51 = X22 ) ).
% 0.34/0.51
% 0.34/0.51 % snd_conv
% 0.34/0.51 thf(fact_145_snd__conv,axiom,
% 0.34/0.51 ! [X1: produc3658429121746597890et_nat > $o,X22: produc3925858234332021118et_nat] :
% 0.34/0.51 ( ( produc5374455773327741254et_nat @ ( produc2245416461498447860et_nat @ X1 @ X22 ) )
% 0.34/0.51 = X22 ) ).
% 0.34/0.51
% 0.34/0.51 % snd_conv
% 0.34/0.51 thf(fact_146_snd__conv,axiom,
% 0.34/0.51 ! [X1: heap_e7401611519738050253t_unit,X22: set_nat] :
% 0.34/0.51 ( ( produc8586169260539613262et_nat @ ( produc7507926704131184380et_nat @ X1 @ X22 ) )
% 0.34/0.51 = X22 ) ).
% 0.34/0.51
% 0.34/0.51 % snd_conv
% 0.34/0.51 thf(fact_147_snd__conv,axiom,
% 0.34/0.51 ! [X1: int,X22: int] :
% 0.34/0.51 ( ( product_snd_int_int @ ( product_Pair_int_int @ X1 @ X22 ) )
% 0.34/0.51 = X22 ) ).
% 0.34/0.51
% 0.34/0.51 % snd_conv
% 0.34/0.51 thf(fact_148_eq__snd__iff,axiom,
% 0.34/0.51 ! [B: nat,P3: product_prod_nat_nat] :
% 0.34/0.51 ( ( B
% 0.34/0.51 = ( product_snd_nat_nat @ P3 ) )
% 0.34/0.51 = ( ? [A6: nat] :
% 0.34/0.51 ( P3
% 0.34/0.51 = ( product_Pair_nat_nat @ A6 @ B ) ) ) ) ).
% 0.34/0.51
% 0.34/0.51 % eq_snd_iff
% 0.34/0.51 thf(fact_149_eq__snd__iff,axiom,
% 0.34/0.51 ! [B: produc3658429121746597890et_nat,P3: produc3925858234332021118et_nat] :
% 0.34/0.51 ( ( B
% 0.34/0.51 = ( produc4011572625026189258et_nat @ P3 ) )
% 0.34/0.51 = ( ? [A6: produc3658429121746597890et_nat > $o] :
% 0.34/0.51 ( P3
% 0.34/0.51 = ( produc5001842942810119800et_nat @ A6 @ B ) ) ) ) ).
% 0.34/0.51
% 0.34/0.51 % eq_snd_iff
% 0.34/0.51 thf(fact_150_eq__snd__iff,axiom,
% 0.34/0.51 ! [B: produc3925858234332021118et_nat,P3: produc2732055786443039994et_nat] :
% 0.34/0.51 ( ( B
% 0.34/0.51 = ( produc5374455773327741254et_nat @ P3 ) )
% 0.34/0.51 = ( ? [A6: produc3658429121746597890et_nat > $o] :
% 0.34/0.51 ( P3
% 0.34/0.51 = ( produc2245416461498447860et_nat @ A6 @ B ) ) ) ) ).
% 0.34/0.51
% 0.34/0.51 % eq_snd_iff
% 0.34/0.51 thf(fact_151_eq__snd__iff,axiom,
% 0.34/0.51 ! [B: set_nat,P3: produc3658429121746597890et_nat] :
% 0.34/0.51 ( ( B
% 0.34/0.51 = ( produc8586169260539613262et_nat @ P3 ) )
% 0.34/0.51 = ( ? [A6: heap_e7401611519738050253t_unit] :
% 0.34/0.51 ( P3
% 0.34/0.51 = ( produc7507926704131184380et_nat @ A6 @ B ) ) ) ) ).
% 0.34/0.51
% 0.34/0.51 % eq_snd_iff
% 0.34/0.51 thf(fact_152_eq__snd__iff,axiom,
% 0.34/0.51 ! [B: int,P3: product_prod_int_int] :
% 0.34/0.51 ( ( B
% 0.34/0.51 = ( product_snd_int_int @ P3 ) )
% 0.34/0.51 = ( ? [A6: int] :
% 0.34/0.51 ( P3
% 0.34/0.51 = ( product_Pair_int_int @ A6 @ B ) ) ) ) ).
% 0.34/0.51
% 0.34/0.51 % eq_snd_iff
% 0.34/0.51 thf(fact_153_pure__assn__raw_Oelims_I3_J,axiom,
% 0.34/0.51 ! [X: $o,Xa: produc3658429121746597890et_nat] :
% 0.34/0.51 ( ~ ( pure_a825153325127701367it_nat @ X @ Xa )
% 0.34/0.51 => ~ ! [H: heap_e7401611519738050253t_unit,As2: set_nat] :
% 0.34/0.51 ( ( Xa
% 0.34/0.51 = ( produc7507926704131184380et_nat @ H @ As2 ) )
% 0.34/0.51 => ( ( As2 = bot_bot_set_nat )
% 0.34/0.51 & X ) ) ) ).
% 0.34/0.51
% 0.34/0.51 % pure_assn_raw.elims(3)
% 0.34/0.51 thf(fact_154_pure__assn__raw_Oelims_I2_J,axiom,
% 0.34/0.51 ! [X: $o,Xa: produc3658429121746597890et_nat] :
% 0.34/0.51 ( ( pure_a825153325127701367it_nat @ X @ Xa )
% 0.34/0.51 => ~ ! [H: heap_e7401611519738050253t_unit,As2: set_nat] :
% 0.34/0.51 ( ( Xa
% 0.34/0.51 = ( produc7507926704131184380et_nat @ H @ As2 ) )
% 0.34/0.51 => ~ ( ( As2 = bot_bot_set_nat )
% 0.34/0.51 & X ) ) ) ).
% 0.34/0.51
% 0.34/0.51 % pure_assn_raw.elims(2)
% 0.34/0.51 thf(fact_155_pure__assn__raw_Oelims_I1_J,axiom,
% 0.34/0.51 ! [X: $o,Xa: produc3658429121746597890et_nat,Y: $o] :
% 0.34/0.51 ( ( ( pure_a825153325127701367it_nat @ X @ Xa )
% 0.34/0.51 = Y )
% 0.34/0.51 => ~ ! [H: heap_e7401611519738050253t_unit,As2: set_nat] :
% 0.34/0.51 ( ( Xa
% 0.34/0.51 = ( produc7507926704131184380et_nat @ H @ As2 ) )
% 0.34/0.51 => ( Y
% 0.34/0.51 = ( ~ ( ( As2 = bot_bot_set_nat )
% 0.34/0.51 & X ) ) ) ) ) ).
% 0.34/0.51
% 0.34/0.51 % pure_assn_raw.elims(1)
% 0.34/0.51 thf(fact_156_pure__assn__raw_Osimps,axiom,
% 0.34/0.51 ! [B: $o,H2: heap_e7401611519738050253t_unit,As: set_nat] :
% 0.34/0.51 ( ( pure_a825153325127701367it_nat @ B @ ( produc7507926704131184380et_nat @ H2 @ As ) )
% 0.34/0.51 = ( ( As = bot_bot_set_nat )
% 0.34/0.51 & B ) ) ).
% 0.34/0.51
% 0.34/0.51 % pure_assn_raw.simps
% 0.34/0.51 thf(fact_157_pure__assn__proper,axiom,
% 0.34/0.51 ! [B: $o] : ( proper @ ( pure_a825153325127701367it_nat @ B ) ) ).
% 0.34/0.51
% 0.34/0.51 % pure_assn_proper
% 0.34/0.51 thf(fact_158_times__assn__raw_Ocases,axiom,
% 0.34/0.51 ! [X: produc2732055786443039994et_nat] :
% 0.34/0.51 ~ ! [P4: produc3658429121746597890et_nat > $o,Q3: produc3658429121746597890et_nat > $o,H: heap_e7401611519738050253t_unit,As2: set_nat] :
% 0.34/0.51 ( X
% 0.34/0.51 != ( produc2245416461498447860et_nat @ P4 @ ( produc5001842942810119800et_nat @ Q3 @ ( produc7507926704131184380et_nat @ H @ As2 ) ) ) ) ).
% 0.34/0.51
% 0.34/0.51 % times_assn_raw.cases
% 0.34/0.51 thf(fact_159_pure__assn__raw_Ocases,axiom,
% 0.34/0.51 ! [X: produc6197004810343482825et_nat] :
% 0.34/0.51 ~ ! [B2: $o,H: heap_e7401611519738050253t_unit,As2: set_nat] :
% 0.34/0.51 ( X
% 0.34/0.51 != ( produc3762353314782720579et_nat @ B2 @ ( produc7507926704131184380et_nat @ H @ As2 ) ) ) ).
% 0.34/0.51
% 0.34/0.51 % pure_assn_raw.cases
% 0.34/0.51 thf(fact_160_old_Oprod_Oexhaust,axiom,
% 0.34/0.51 ! [Y: produc3658429121746597890et_nat] :
% 0.34/0.51 ~ ! [A4: heap_e7401611519738050253t_unit,B2: set_nat] :
% 0.34/0.51 ( Y
% 0.34/0.51 != ( produc7507926704131184380et_nat @ A4 @ B2 ) ) ).
% 0.34/0.51
% 0.34/0.51 % old.prod.exhaust
% 0.34/0.51 thf(fact_161_old_Oprod_Oexhaust,axiom,
% 0.34/0.51 ! [Y: product_prod_nat_nat] :
% 0.34/0.51 ~ ! [A4: nat,B2: nat] :
% 0.34/0.51 ( Y
% 0.34/0.51 != ( product_Pair_nat_nat @ A4 @ B2 ) ) ).
% 0.34/0.51
% 0.34/0.51 % old.prod.exhaust
% 0.34/0.51 thf(fact_162_old_Oprod_Oexhaust,axiom,
% 0.34/0.51 ! [Y: product_prod_int_int] :
% 0.34/0.51 ~ ! [A4: int,B2: int] :
% 0.34/0.51 ( Y
% 0.34/0.51 != ( product_Pair_int_int @ A4 @ B2 ) ) ).
% 0.34/0.51
% 0.34/0.51 % old.prod.exhaust
% 0.34/0.51 thf(fact_163_old_Oprod_Oexhaust,axiom,
% 0.34/0.51 ! [Y: produc3925858234332021118et_nat] :
% 0.34/0.51 ~ ! [A4: produc3658429121746597890et_nat > $o,B2: produc3658429121746597890et_nat] :
% 0.34/0.51 ( Y
% 0.34/0.51 != ( produc5001842942810119800et_nat @ A4 @ B2 ) ) ).
% 0.34/0.51
% 0.34/0.51 % old.prod.exhaust
% 0.34/0.51 thf(fact_164_old_Oprod_Oexhaust,axiom,
% 0.34/0.51 ! [Y: produc2732055786443039994et_nat] :
% 0.34/0.51 ~ ! [A4: produc3658429121746597890et_nat > $o,B2: produc3925858234332021118et_nat] :
% 0.34/0.51 ( Y
% 0.34/0.51 != ( produc2245416461498447860et_nat @ A4 @ B2 ) ) ).
% 0.34/0.51
% 0.34/0.51 % old.prod.exhaust
% 0.34/0.51 thf(fact_165_surj__pair,axiom,
% 0.34/0.51 ! [P3: produc3658429121746597890et_nat] :
% 0.34/0.51 ? [X3: heap_e7401611519738050253t_unit,Y2: set_nat] :
% 0.34/0.51 ( P3
% 0.34/0.51 = ( produc7507926704131184380et_nat @ X3 @ Y2 ) ) ).
% 0.34/0.51
% 0.34/0.51 % surj_pair
% 0.34/0.51 thf(fact_166_surj__pair,axiom,
% 0.34/0.51 ! [P3: product_prod_nat_nat] :
% 0.34/0.51 ? [X3: nat,Y2: nat] :
% 0.34/0.51 ( P3
% 0.34/0.51 = ( product_Pair_nat_nat @ X3 @ Y2 ) ) ).
% 0.34/0.51
% 0.34/0.51 % surj_pair
% 0.34/0.51 thf(fact_167_surj__pair,axiom,
% 0.34/0.51 ! [P3: product_prod_int_int] :
% 0.34/0.51 ? [X3: int,Y2: int] :
% 0.34/0.51 ( P3
% 0.34/0.51 = ( product_Pair_int_int @ X3 @ Y2 ) ) ).
% 0.34/0.51
% 0.34/0.51 % surj_pair
% 0.34/0.51 thf(fact_168_surj__pair,axiom,
% 0.34/0.51 ! [P3: produc3925858234332021118et_nat] :
% 0.34/0.51 ? [X3: produc3658429121746597890et_nat > $o,Y2: produc3658429121746597890et_nat] :
% 0.34/0.51 ( P3
% 0.34/0.51 = ( produc5001842942810119800et_nat @ X3 @ Y2 ) ) ).
% 0.34/0.51
% 0.34/0.51 % surj_pair
% 0.34/0.51 thf(fact_169_surj__pair,axiom,
% 0.34/0.51 ! [P3: produc2732055786443039994et_nat] :
% 0.34/0.51 ? [X3: produc3658429121746597890et_nat > $o,Y2: produc3925858234332021118et_nat] :
% 0.34/0.51 ( P3
% 0.34/0.51 = ( produc2245416461498447860et_nat @ X3 @ Y2 ) ) ).
% 0.34/0.51
% 0.34/0.51 % surj_pair
% 0.34/0.51 thf(fact_170_prod__cases,axiom,
% 0.34/0.51 ! [P: produc3658429121746597890et_nat > $o,P3: produc3658429121746597890et_nat] :
% 0.34/0.51 ( ! [A4: heap_e7401611519738050253t_unit,B2: set_nat] : ( P @ ( produc7507926704131184380et_nat @ A4 @ B2 ) )
% 0.34/0.51 => ( P @ P3 ) ) ).
% 0.34/0.51
% 0.34/0.51 % prod_cases
% 0.34/0.51 thf(fact_171_prod__cases,axiom,
% 0.34/0.51 ! [P: product_prod_nat_nat > $o,P3: product_prod_nat_nat] :
% 0.34/0.51 ( ! [A4: nat,B2: nat] : ( P @ ( product_Pair_nat_nat @ A4 @ B2 ) )
% 0.34/0.51 => ( P @ P3 ) ) ).
% 0.34/0.51
% 0.34/0.51 % prod_cases
% 0.34/0.51 thf(fact_172_prod__cases,axiom,
% 0.34/0.51 ! [P: product_prod_int_int > $o,P3: product_prod_int_int] :
% 0.34/0.51 ( ! [A4: int,B2: int] : ( P @ ( product_Pair_int_int @ A4 @ B2 ) )
% 0.34/0.51 => ( P @ P3 ) ) ).
% 0.34/0.51
% 0.34/0.51 % prod_cases
% 0.34/0.51 thf(fact_173_prod__cases,axiom,
% 0.34/0.51 ! [P: produc3925858234332021118et_nat > $o,P3: produc3925858234332021118et_nat] :
% 0.34/0.51 ( ! [A4: produc3658429121746597890et_nat > $o,B2: produc3658429121746597890et_nat] : ( P @ ( produc5001842942810119800et_nat @ A4 @ B2 ) )
% 0.34/0.51 => ( P @ P3 ) ) ).
% 0.34/0.51
% 0.34/0.51 % prod_cases
% 0.34/0.51 thf(fact_174_prod__cases,axiom,
% 0.34/0.51 ! [P: produc2732055786443039994et_nat > $o,P3: produc2732055786443039994et_nat] :
% 0.34/0.51 ( ! [A4: produc3658429121746597890et_nat > $o,B2: produc3925858234332021118et_nat] : ( P @ ( produc2245416461498447860et_nat @ A4 @ B2 ) )
% 0.34/0.51 => ( P @ P3 ) ) ).
% 0.34/0.51
% 0.34/0.51 % prod_cases
% 0.34/0.51 thf(fact_175_Pair__inject,axiom,
% 0.34/0.51 ! [A2: heap_e7401611519738050253t_unit,B: set_nat,A5: heap_e7401611519738050253t_unit,B3: set_nat] :
% 0.34/0.51 ( ( ( produc7507926704131184380et_nat @ A2 @ B )
% 0.34/0.51 = ( produc7507926704131184380et_nat @ A5 @ B3 ) )
% 0.34/0.51 => ~ ( ( A2 = A5 )
% 0.34/0.51 => ( B != B3 ) ) ) ).
% 0.34/0.51
% 0.34/0.51 % Pair_inject
% 0.34/0.51 thf(fact_176_Pair__inject,axiom,
% 0.34/0.51 ! [A2: nat,B: nat,A5: nat,B3: nat] :
% 0.34/0.51 ( ( ( product_Pair_nat_nat @ A2 @ B )
% 0.34/0.51 = ( product_Pair_nat_nat @ A5 @ B3 ) )
% 0.34/0.51 => ~ ( ( A2 = A5 )
% 0.34/0.51 => ( B != B3 ) ) ) ).
% 0.34/0.51
% 0.34/0.51 % Pair_inject
% 0.34/0.51 thf(fact_177_Pair__inject,axiom,
% 0.34/0.51 ! [A2: int,B: int,A5: int,B3: int] :
% 0.34/0.51 ( ( ( product_Pair_int_int @ A2 @ B )
% 0.34/0.51 = ( product_Pair_int_int @ A5 @ B3 ) )
% 0.34/0.51 => ~ ( ( A2 = A5 )
% 0.34/0.51 => ( B != B3 ) ) ) ).
% 0.34/0.51
% 0.34/0.51 % Pair_inject
% 0.34/0.51 thf(fact_178_Pair__inject,axiom,
% 0.34/0.51 ! [A2: produc3658429121746597890et_nat > $o,B: produc3658429121746597890et_nat,A5: produc3658429121746597890et_nat > $o,B3: produc3658429121746597890et_nat] :
% 0.34/0.51 ( ( ( produc5001842942810119800et_nat @ A2 @ B )
% 0.34/0.51 = ( produc5001842942810119800et_nat @ A5 @ B3 ) )
% 0.34/0.51 => ~ ( ( A2 = A5 )
% 0.34/0.51 => ( B != B3 ) ) ) ).
% 0.34/0.51
% 0.34/0.51 % Pair_inject
% 0.34/0.51 thf(fact_179_Pair__inject,axiom,
% 0.34/0.51 ! [A2: produc3658429121746597890et_nat > $o,B: produc3925858234332021118et_nat,A5: produc3658429121746597890et_nat > $o,B3: produc3925858234332021118et_nat] :
% 0.34/0.51 ( ( ( produc2245416461498447860et_nat @ A2 @ B )
% 0.34/0.51 = ( produc2245416461498447860et_nat @ A5 @ B3 ) )
% 0.34/0.51 => ~ ( ( A2 = A5 )
% 0.34/0.51 => ( B != B3 ) ) ) ).
% 0.34/0.51
% 0.34/0.51 % Pair_inject
% 0.34/0.51 thf(fact_180_prod__cases3,axiom,
% 0.34/0.51 ! [Y: produc3925858234332021118et_nat] :
% 0.34/0.51 ~ ! [A4: produc3658429121746597890et_nat > $o,B2: heap_e7401611519738050253t_unit,C2: set_nat] :
% 0.34/0.51 ( Y
% 0.34/0.51 != ( produc5001842942810119800et_nat @ A4 @ ( produc7507926704131184380et_nat @ B2 @ C2 ) ) ) ).
% 0.34/0.51
% 0.34/0.51 % prod_cases3
% 0.34/0.51 thf(fact_181_prod__cases3,axiom,
% 0.34/0.51 ! [Y: produc2732055786443039994et_nat] :
% 0.34/0.51 ~ ! [A4: produc3658429121746597890et_nat > $o,B2: produc3658429121746597890et_nat > $o,C2: produc3658429121746597890et_nat] :
% 0.34/0.51 ( Y
% 0.34/0.51 != ( produc2245416461498447860et_nat @ A4 @ ( produc5001842942810119800et_nat @ B2 @ C2 ) ) ) ).
% 0.34/0.51
% 0.34/0.51 % prod_cases3
% 0.34/0.51 thf(fact_182_prod__cases4,axiom,
% 0.34/0.51 ! [Y: produc2732055786443039994et_nat] :
% 0.34/0.51 ~ ! [A4: produc3658429121746597890et_nat > $o,B2: produc3658429121746597890et_nat > $o,C2: heap_e7401611519738050253t_unit,D: set_nat] :
% 0.34/0.51 ( Y
% 0.34/0.51 != ( produc2245416461498447860et_nat @ A4 @ ( produc5001842942810119800et_nat @ B2 @ ( produc7507926704131184380et_nat @ C2 @ D ) ) ) ) ).
% 0.34/0.51
% 0.34/0.51 % prod_cases4
% 0.34/0.51 thf(fact_183_prod__induct3,axiom,
% 0.34/0.51 ! [P: produc3925858234332021118et_nat > $o,X: produc3925858234332021118et_nat] :
% 0.34/0.51 ( ! [A4: produc3658429121746597890et_nat > $o,B2: heap_e7401611519738050253t_unit,C2: set_nat] : ( P @ ( produc5001842942810119800et_nat @ A4 @ ( produc7507926704131184380et_nat @ B2 @ C2 ) ) )
% 0.34/0.51 => ( P @ X ) ) ).
% 0.34/0.51
% 0.34/0.51 % prod_induct3
% 0.34/0.51 thf(fact_184_prod__induct3,axiom,
% 0.34/0.51 ! [P: produc2732055786443039994et_nat > $o,X: produc2732055786443039994et_nat] :
% 0.34/0.51 ( ! [A4: produc3658429121746597890et_nat > $o,B2: produc3658429121746597890et_nat > $o,C2: produc3658429121746597890et_nat] : ( P @ ( produc2245416461498447860et_nat @ A4 @ ( produc5001842942810119800et_nat @ B2 @ C2 ) ) )
% 0.34/0.51 => ( P @ X ) ) ).
% 0.34/0.51
% 0.34/0.51 % prod_induct3
% 0.34/0.51 thf(fact_185_prod__induct4,axiom,
% 0.34/0.51 ! [P: produc2732055786443039994et_nat > $o,X: produc2732055786443039994et_nat] :
% 0.34/0.51 ( ! [A4: produc3658429121746597890et_nat > $o,B2: produc3658429121746597890et_nat > $o,C2: heap_e7401611519738050253t_unit,D: set_nat] : ( P @ ( produc2245416461498447860et_nat @ A4 @ ( produc5001842942810119800et_nat @ B2 @ ( produc7507926704131184380et_nat @ C2 @ D ) ) ) )
% 0.34/0.51 => ( P @ X ) ) ).
% 0.34/0.51
% 0.34/0.51 % prod_induct4
% 0.34/0.51 thf(fact_186_one__reorient,axiom,
% 0.34/0.51 ! [X: assn] :
% 0.34/0.51 ( ( one_one_assn = X )
% 0.34/0.51 = ( X = one_one_assn ) ) ).
% 0.34/0.51
% 0.34/0.51 % one_reorient
% 0.34/0.51 thf(fact_187_one__reorient,axiom,
% 0.34/0.51 ! [X: nat] :
% 0.34/0.51 ( ( one_one_nat = X )
% 0.34/0.51 = ( X = one_one_nat ) ) ).
% 0.34/0.51
% 0.34/0.51 % one_reorient
% 0.34/0.51 thf(fact_188_one__reorient,axiom,
% 0.34/0.51 ! [X: int] :
% 0.34/0.51 ( ( one_one_int = X )
% 0.34/0.51 = ( X = one_one_int ) ) ).
% 0.34/0.51
% 0.34/0.51 % one_reorient
% 0.34/0.51 thf(fact_189_one__reorient,axiom,
% 0.34/0.51 ! [X: code_integer] :
% 0.34/0.51 ( ( one_one_Code_integer = X )
% 0.34/0.51 = ( X = one_one_Code_integer ) ) ).
% 0.34/0.51
% 0.34/0.51 % one_reorient
% 0.34/0.51 thf(fact_190_one__reorient,axiom,
% 0.34/0.51 ! [X: code_natural] :
% 0.34/0.51 ( ( one_one_Code_natural = X )
% 0.34/0.51 = ( X = one_one_Code_natural ) ) ).
% 0.34/0.51
% 0.34/0.51 % one_reorient
% 0.34/0.51 thf(fact_191_type__definition__assn,axiom,
% 0.34/0.51 type_d3909072315231072503_nat_o @ rep_assn @ abs_assn @ ( collec939566748876313656_nat_o @ proper ) ).
% 0.34/0.51
% 0.34/0.51 % type_definition_assn
% 0.34/0.51 thf(fact_192_times__assn__proper,axiom,
% 0.34/0.51 ! [P: produc3658429121746597890et_nat > $o,Q2: produc3658429121746597890et_nat > $o] :
% 0.34/0.51 ( ( proper @ P )
% 0.34/0.51 => ( ( proper @ Q2 )
% 0.34/0.51 => ( proper @ ( times_assn_raw @ P @ Q2 ) ) ) ) ).
% 0.34/0.51
% 0.34/0.51 % times_assn_proper
% 0.34/0.51 thf(fact_193_is__pure__assn__basic__simps_I2_J,axiom,
% 0.34/0.51 is_pure_assn @ one_one_assn ).
% 0.34/0.51
% 0.34/0.51 % is_pure_assn_basic_simps(2)
% 0.34/0.51 thf(fact_194_bijective__def,axiom,
% 0.34/0.51 ( biject2219996528949639174et_nat
% 0.34/0.51 = ( ^ [R: set_Pr3948176798113811640et_nat] :
% 0.34/0.51 ( ! [X2: heap_e7401611519738050253t_unit,Y3: set_nat,Z2: set_nat] :
% 0.34/0.51 ( ( ( member6260224972018164377et_nat @ ( produc7507926704131184380et_nat @ X2 @ Y3 ) @ R )
% 0.34/0.51 & ( member6260224972018164377et_nat @ ( produc7507926704131184380et_nat @ X2 @ Z2 ) @ R ) )
% 0.34/0.51 => ( Y3 = Z2 ) )
% 0.34/0.51 & ! [X2: heap_e7401611519738050253t_unit,Y3: heap_e7401611519738050253t_unit,Z2: set_nat] :
% 0.34/0.51 ( ( ( member6260224972018164377et_nat @ ( produc7507926704131184380et_nat @ X2 @ Z2 ) @ R )
% 0.34/0.51 & ( member6260224972018164377et_nat @ ( produc7507926704131184380et_nat @ Y3 @ Z2 ) @ R ) )
% 0.34/0.51 => ( X2 = Y3 ) ) ) ) ) ).
% 0.34/0.51
% 0.34/0.51 % bijective_def
% 0.34/0.51 thf(fact_195_bijective__def,axiom,
% 0.34/0.51 ( bijective_nat_nat
% 0.34/0.51 = ( ^ [R: set_Pr1261947904930325089at_nat] :
% 0.34/0.51 ( ! [X2: nat,Y3: nat,Z2: nat] :
% 0.34/0.51 ( ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X2 @ Y3 ) @ R )
% 0.34/0.51 & ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X2 @ Z2 ) @ R ) )
% 0.34/0.51 => ( Y3 = Z2 ) )
% 0.34/0.51 & ! [X2: nat,Y3: nat,Z2: nat] :
% 0.34/0.51 ( ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X2 @ Z2 ) @ R )
% 0.34/0.51 & ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ Y3 @ Z2 ) @ R ) )
% 0.34/0.51 => ( X2 = Y3 ) ) ) ) ) ).
% 0.34/0.51
% 0.34/0.51 % bijective_def
% 0.34/0.51 thf(fact_196_bijective__def,axiom,
% 0.34/0.51 ( bijective_int_int
% 0.34/0.51 = ( ^ [R: set_Pr958786334691620121nt_int] :
% 0.34/0.51 ( ! [X2: int,Y3: int,Z2: int] :
% 0.34/0.51 ( ( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ X2 @ Y3 ) @ R )
% 0.34/0.51 & ( member5262025264175285858nt_int @ ( product_Pair_int_int @ X2 @ Z2 ) @ R ) )
% 0.34/0.51 => ( Y3 = Z2 ) )
% 0.34/0.51 & ! [X2: int,Y3: int,Z2: int] :
% 0.34/0.51 ( ( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ X2 @ Z2 ) @ R )
% 0.34/0.51 & ( member5262025264175285858nt_int @ ( product_Pair_int_int @ Y3 @ Z2 ) @ R ) )
% 0.34/0.51 => ( X2 = Y3 ) ) ) ) ) ).
% 0.34/0.51
% 0.34/0.51 % bijective_def
% 0.34/0.51 thf(fact_197_bijective__def,axiom,
% 0.34/0.51 ( biject2615096655818420098et_nat
% 0.34/0.51 = ( ^ [R: set_Pr3286484037609594932et_nat] :
% 0.34/0.51 ( ! [X2: produc3658429121746597890et_nat > $o,Y3: produc3658429121746597890et_nat,Z2: produc3658429121746597890et_nat] :
% 0.34/0.51 ( ( ( member1996754912294343701et_nat @ ( produc5001842942810119800et_nat @ X2 @ Y3 ) @ R )
% 0.34/0.51 & ( member1996754912294343701et_nat @ ( produc5001842942810119800et_nat @ X2 @ Z2 ) @ R ) )
% 0.34/0.51 => ( Y3 = Z2 ) )
% 0.34/0.51 & ! [X2: produc3658429121746597890et_nat > $o,Y3: produc3658429121746597890et_nat > $o,Z2: produc3658429121746597890et_nat] :
% 0.34/0.51 ( ( ( member1996754912294343701et_nat @ ( produc5001842942810119800et_nat @ X2 @ Z2 ) @ R )
% 0.34/0.51 & ( member1996754912294343701et_nat @ ( produc5001842942810119800et_nat @ Y3 @ Z2 ) @ R ) )
% 0.34/0.51 => ( X2 = Y3 ) ) ) ) ) ).
% 0.34/0.51
% 0.34/0.51 % bijective_def
% 0.34/0.51 thf(fact_198_bijective__def,axiom,
% 0.34/0.51 ( biject1468766312547416318et_nat
% 0.34/0.51 = ( ^ [R: set_Pr8536935166611901872et_nat] :
% 0.34/0.51 ( ! [X2: produc3658429121746597890et_nat > $o,Y3: produc3925858234332021118et_nat,Z2: produc3925858234332021118et_nat] :
% 0.34/0.51 ( ( ( member6124377750444531601et_nat @ ( produc2245416461498447860et_nat @ X2 @ Y3 ) @ R )
% 0.34/0.51 & ( member6124377750444531601et_nat @ ( produc2245416461498447860et_nat @ X2 @ Z2 ) @ R ) )
% 0.34/0.51 => ( Y3 = Z2 ) )
% 0.34/0.51 & ! [X2: produc3658429121746597890et_nat > $o,Y3: produc3658429121746597890et_nat > $o,Z2: produc3925858234332021118et_nat] :
% 0.34/0.51 ( ( ( member6124377750444531601et_nat @ ( produc2245416461498447860et_nat @ X2 @ Z2 ) @ R )
% 0.34/0.51 & ( member6124377750444531601et_nat @ ( produc2245416461498447860et_nat @ Y3 @ Z2 ) @ R ) )
% 0.34/0.51 => ( X2 = Y3 ) ) ) ) ) ).
% 0.34/0.51
% 0.34/0.51 % bijective_def
% 0.34/0.51 thf(fact_199_ssubst__Pair__rhs,axiom,
% 0.34/0.51 ! [R2: heap_e7401611519738050253t_unit,S2: set_nat,R3: set_Pr3948176798113811640et_nat,S3: set_nat] :
% 0.34/0.51 ( ( member6260224972018164377et_nat @ ( produc7507926704131184380et_nat @ R2 @ S2 ) @ R3 )
% 0.34/0.51 => ( ( S3 = S2 )
% 0.34/0.51 => ( member6260224972018164377et_nat @ ( produc7507926704131184380et_nat @ R2 @ S3 ) @ R3 ) ) ) ).
% 0.34/0.51
% 0.34/0.51 % ssubst_Pair_rhs
% 0.34/0.51 thf(fact_200_ssubst__Pair__rhs,axiom,
% 0.34/0.51 ! [R2: nat,S2: nat,R3: set_Pr1261947904930325089at_nat,S3: nat] :
% 0.34/0.51 ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ R2 @ S2 ) @ R3 )
% 0.34/0.51 => ( ( S3 = S2 )
% 0.34/0.51 => ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ R2 @ S3 ) @ R3 ) ) ) ).
% 0.34/0.51
% 0.34/0.51 % ssubst_Pair_rhs
% 0.34/0.51 thf(fact_201_ssubst__Pair__rhs,axiom,
% 0.34/0.51 ! [R2: int,S2: int,R3: set_Pr958786334691620121nt_int,S3: int] :
% 0.34/0.51 ( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ R2 @ S2 ) @ R3 )
% 0.34/0.51 => ( ( S3 = S2 )
% 0.34/0.51 => ( member5262025264175285858nt_int @ ( product_Pair_int_int @ R2 @ S3 ) @ R3 ) ) ) ).
% 0.34/0.51
% 0.34/0.51 % ssubst_Pair_rhs
% 0.34/0.51 thf(fact_202_ssubst__Pair__rhs,axiom,
% 0.34/0.51 ! [R2: produc3658429121746597890et_nat > $o,S2: produc3658429121746597890et_nat,R3: set_Pr3286484037609594932et_nat,S3: produc3658429121746597890et_nat] :
% 0.34/0.51 ( ( member1996754912294343701et_nat @ ( produc5001842942810119800et_nat @ R2 @ S2 ) @ R3 )
% 0.34/0.51 => ( ( S3 = S2 )
% 0.34/0.51 => ( member1996754912294343701et_nat @ ( produc5001842942810119800et_nat @ R2 @ S3 ) @ R3 ) ) ) ).
% 0.34/0.51
% 0.34/0.51 % ssubst_Pair_rhs
% 0.34/0.51 thf(fact_203_ssubst__Pair__rhs,axiom,
% 0.34/0.51 ! [R2: produc3658429121746597890et_nat > $o,S2: produc3925858234332021118et_nat,R3: set_Pr8536935166611901872et_nat,S3: produc3925858234332021118et_nat] :
% 0.34/0.51 ( ( member6124377750444531601et_nat @ ( produc2245416461498447860et_nat @ R2 @ S2 ) @ R3 )
% 0.34/0.51 => ( ( S3 = S2 )
% 0.34/0.51 => ( member6124377750444531601et_nat @ ( produc2245416461498447860et_nat @ R2 @ S3 ) @ R3 ) ) ) ).
% 0.34/0.51
% 0.34/0.51 % ssubst_Pair_rhs
% 0.34/0.51 thf(fact_204_is__singletonI,axiom,
% 0.34/0.51 ! [X: product_unit] : ( is_sin2160648248035936513t_unit @ ( insert_Product_unit @ X @ bot_bo3957492148770167129t_unit ) ) ).
% 0.34/0.51
% 0.34/0.51 % is_singletonI
% 0.34/0.51 thf(fact_205_is__singletonI,axiom,
% 0.34/0.51 ! [X: $o] : ( is_singleton_o @ ( insert_o @ X @ bot_bot_set_o ) ) ).
% 0.34/0.51
% 0.34/0.51 % is_singletonI
% 0.34/0.51 thf(fact_206_is__singletonI,axiom,
% 0.34/0.51 ! [X: nat] : ( is_singleton_nat @ ( insert_nat @ X @ bot_bot_set_nat ) ) ).
% 0.34/0.51
% 0.34/0.51 % is_singletonI
% 0.34/0.51 thf(fact_207_is__singletonI,axiom,
% 0.34/0.51 ! [X: int] : ( is_singleton_int @ ( insert_int @ X @ bot_bot_set_int ) ) ).
% 0.34/0.51
% 0.34/0.51 % is_singletonI
% 0.34/0.51 thf(fact_208_is__singletonI,axiom,
% 0.34/0.51 ! [X: produc3658429121746597890et_nat > $o] : ( is_sin5180296473474724033_nat_o @ ( insert5175938949040314269_nat_o @ X @ bot_bo7824918357723582233_nat_o ) ) ).
% 0.34/0.51
% 0.34/0.51 % is_singletonI
% 0.34/0.51 thf(fact_209_curryI,axiom,
% 0.34/0.51 ! [F: produc3658429121746597890et_nat > $o,A2: heap_e7401611519738050253t_unit,B: set_nat] :
% 0.34/0.51 ( ( F @ ( produc7507926704131184380et_nat @ A2 @ B ) )
% 0.34/0.51 => ( produc1807123492089482958_nat_o @ F @ A2 @ B ) ) ).
% 0.34/0.51
% 0.34/0.51 % curryI
% 0.34/0.51 thf(fact_210_curryI,axiom,
% 0.34/0.51 ! [F: product_prod_nat_nat > $o,A2: nat,B: nat] :
% 0.34/0.51 ( ( F @ ( product_Pair_nat_nat @ A2 @ B ) )
% 0.34/0.51 => ( produc1310100445399344235_nat_o @ F @ A2 @ B ) ) ).
% 0.34/0.51
% 0.34/0.51 % curryI
% 0.34/0.51 thf(fact_211_curryI,axiom,
% 0.34/0.51 ! [F: product_prod_int_int > $o,A2: int,B: int] :
% 0.34/0.51 ( ( F @ ( product_Pair_int_int @ A2 @ B ) )
% 0.34/0.51 => ( produc175634133007206835_int_o @ F @ A2 @ B ) ) ).
% 0.34/0.51
% 0.34/0.51 % curryI
% 0.34/0.51 thf(fact_212_curryI,axiom,
% 0.34/0.51 ! [F: produc3925858234332021118et_nat > $o,A2: produc3658429121746597890et_nat > $o,B: produc3658429121746597890et_nat] :
% 0.34/0.51 ( ( F @ ( produc5001842942810119800et_nat @ A2 @ B ) )
% 0.34/0.51 => ( produc6216949301066131538_nat_o @ F @ A2 @ B ) ) ).
% 0.34/0.51
% 0.34/0.51 % curryI
% 0.34/0.51 thf(fact_213_curryI,axiom,
% 0.34/0.51 ! [F: produc2732055786443039994et_nat > $o,A2: produc3658429121746597890et_nat > $o,B: produc3925858234332021118et_nat] :
% 0.34/0.51 ( ( F @ ( produc2245416461498447860et_nat @ A2 @ B ) )
% 0.34/0.51 => ( produc5101573711933517782_nat_o @ F @ A2 @ B ) ) ).
% 0.34/0.51
% 0.34/0.51 % curryI
% 0.34/0.51 thf(fact_214_insertCI,axiom,
% 0.34/0.51 ! [A2: produc3658429121746597890et_nat > $o,B4: set_Pr4532377907799695533_nat_o,B: produc3658429121746597890et_nat > $o] :
% 0.34/0.51 ( ( ~ ( member6576561426505652726_nat_o @ A2 @ B4 )
% 0.34/0.51 => ( A2 = B ) )
% 0.34/0.51 => ( member6576561426505652726_nat_o @ A2 @ ( insert5175938949040314269_nat_o @ B @ B4 ) ) ) ).
% 0.34/0.51
% 0.34/0.51 % insertCI
% 0.34/0.51 thf(fact_215_insertCI,axiom,
% 0.34/0.51 ! [A2: nat,B4: set_nat,B: nat] :
% 0.34/0.51 ( ( ~ ( member_nat @ A2 @ B4 )
% 0.34/0.51 => ( A2 = B ) )
% 0.34/0.51 => ( member_nat @ A2 @ ( insert_nat @ B @ B4 ) ) ) ).
% 0.34/0.51
% 0.34/0.51 % insertCI
% 0.34/0.51 thf(fact_216_insertCI,axiom,
% 0.34/0.51 ! [A2: int,B4: set_int,B: int] :
% 0.34/0.51 ( ( ~ ( member_int @ A2 @ B4 )
% 0.34/0.51 => ( A2 = B ) )
% 0.34/0.51 => ( member_int @ A2 @ ( insert_int @ B @ B4 ) ) ) ).
% 0.34/0.51
% 0.34/0.51 % insertCI
% 0.34/0.51 thf(fact_217_insertCI,axiom,
% 0.34/0.51 ! [A2: $o,B4: set_o,B: $o] :
% 0.34/0.51 ( ( ~ ( member_o @ A2 @ B4 )
% 0.34/0.51 => ( A2 = B ) )
% 0.34/0.51 => ( member_o @ A2 @ ( insert_o @ B @ B4 ) ) ) ).
% 0.34/0.51
% 0.34/0.51 % insertCI
% 0.34/0.51 thf(fact_218_insertCI,axiom,
% 0.34/0.51 ! [A2: product_unit,B4: set_Product_unit,B: product_unit] :
% 0.34/0.51 ( ( ~ ( member_Product_unit @ A2 @ B4 )
% 0.34/0.51 => ( A2 = B ) )
% 0.34/0.51 => ( member_Product_unit @ A2 @ ( insert_Product_unit @ B @ B4 ) ) ) ).
% 0.34/0.51
% 0.34/0.51 % insertCI
% 0.34/0.51 thf(fact_219_insert__iff,axiom,
% 0.34/0.51 ! [A2: produc3658429121746597890et_nat > $o,B: produc3658429121746597890et_nat > $o,A: set_Pr4532377907799695533_nat_o] :
% 0.34/0.51 ( ( member6576561426505652726_nat_o @ A2 @ ( insert5175938949040314269_nat_o @ B @ A ) )
% 0.34/0.51 = ( ( A2 = B )
% 0.34/0.51 | ( member6576561426505652726_nat_o @ A2 @ A ) ) ) ).
% 0.34/0.51
% 0.34/0.51 % insert_iff
% 0.34/0.51 thf(fact_220_insert__iff,axiom,
% 0.34/0.51 ! [A2: nat,B: nat,A: set_nat] :
% 0.34/0.51 ( ( member_nat @ A2 @ ( insert_nat @ B @ A ) )
% 0.34/0.51 = ( ( A2 = B )
% 0.34/0.51 | ( member_nat @ A2 @ A ) ) ) ).
% 0.34/0.51
% 0.34/0.51 % insert_iff
% 0.34/0.51 thf(fact_221_insert__iff,axiom,
% 0.34/0.51 ! [A2: int,B: int,A: set_int] :
% 0.34/0.51 ( ( member_int @ A2 @ ( insert_int @ B @ A ) )
% 0.34/0.51 = ( ( A2 = B )
% 0.34/0.51 | ( member_int @ A2 @ A ) ) ) ).
% 0.34/0.51
% 0.34/0.51 % insert_iff
% 0.34/0.51 thf(fact_222_insert__iff,axiom,
% 0.34/0.51 ! [A2: $o,B: $o,A: set_o] :
% 0.34/0.51 ( ( member_o @ A2 @ ( insert_o @ B @ A ) )
% 0.34/0.51 = ( ( A2 = B )
% 0.34/0.51 | ( member_o @ A2 @ A ) ) ) ).
% 0.34/0.51
% 0.34/0.51 % insert_iff
% 0.34/0.51 thf(fact_223_insert__iff,axiom,
% 0.34/0.51 ! [A2: product_unit,B: product_unit,A: set_Product_unit] :
% 0.34/0.51 ( ( member_Product_unit @ A2 @ ( insert_Product_unit @ B @ A ) )
% 0.34/0.51 = ( ( A2 = B )
% 0.34/0.51 | ( member_Product_unit @ A2 @ A ) ) ) ).
% 0.34/0.51
% 0.34/0.51 % insert_iff
% 0.34/0.51 thf(fact_224_insert__absorb2,axiom,
% 0.34/0.51 ! [X: nat,A: set_nat] :
% 0.34/0.51 ( ( insert_nat @ X @ ( insert_nat @ X @ A ) )
% 0.34/0.51 = ( insert_nat @ X @ A ) ) ).
% 0.34/0.51
% 0.34/0.51 % insert_absorb2
% 0.34/0.51 thf(fact_225_insert__absorb2,axiom,
% 0.34/0.51 ! [X: int,A: set_int] :
% 0.34/0.51 ( ( insert_int @ X @ ( insert_int @ X @ A ) )
% 0.34/0.51 = ( insert_int @ X @ A ) ) ).
% 0.34/0.51
% 0.34/0.51 % insert_absorb2
% 0.34/0.51 thf(fact_226_insert__absorb2,axiom,
% 0.34/0.51 ! [X: $o,A: set_o] :
% 0.34/0.51 ( ( insert_o @ X @ ( insert_o @ X @ A ) )
% 0.34/0.51 = ( insert_o @ X @ A ) ) ).
% 0.34/0.51
% 0.34/0.51 % insert_absorb2
% 0.34/0.51 thf(fact_227_insert__absorb2,axiom,
% 0.34/0.51 ! [X: product_unit,A: set_Product_unit] :
% 0.34/0.51 ( ( insert_Product_unit @ X @ ( insert_Product_unit @ X @ A ) )
% 0.34/0.51 = ( insert_Product_unit @ X @ A ) ) ).
% 0.34/0.51
% 0.34/0.51 % insert_absorb2
% 0.34/0.51 thf(fact_228_singletonI,axiom,
% 0.34/0.51 ! [A2: product_unit] : ( member_Product_unit @ A2 @ ( insert_Product_unit @ A2 @ bot_bo3957492148770167129t_unit ) ) ).
% 0.34/0.51
% 0.34/0.51 % singletonI
% 0.34/0.51 thf(fact_229_singletonI,axiom,
% 0.34/0.51 ! [A2: $o] : ( member_o @ A2 @ ( insert_o @ A2 @ bot_bot_set_o ) ) ).
% 0.34/0.51
% 0.34/0.51 % singletonI
% 0.34/0.51 thf(fact_230_singletonI,axiom,
% 0.34/0.51 ! [A2: nat] : ( member_nat @ A2 @ ( insert_nat @ A2 @ bot_bot_set_nat ) ) ).
% 0.34/0.51
% 0.34/0.51 % singletonI
% 0.34/0.51 thf(fact_231_singletonI,axiom,
% 0.34/0.51 ! [A2: int] : ( member_int @ A2 @ ( insert_int @ A2 @ bot_bot_set_int ) ) ).
% 0.34/0.51
% 0.34/0.51 % singletonI
% 0.34/0.51 thf(fact_232_singletonI,axiom,
% 0.34/0.51 ! [A2: produc3658429121746597890et_nat > $o] : ( member6576561426505652726_nat_o @ A2 @ ( insert5175938949040314269_nat_o @ A2 @ bot_bo7824918357723582233_nat_o ) ) ).
% 0.34/0.51
% 0.34/0.51 % singletonI
% 0.34/0.51 thf(fact_233_insertE,axiom,
% 0.34/0.51 ! [A2: produc3658429121746597890et_nat > $o,B: produc3658429121746597890et_nat > $o,A: set_Pr4532377907799695533_nat_o] :
% 0.34/0.51 ( ( member6576561426505652726_nat_o @ A2 @ ( insert5175938949040314269_nat_o @ B @ A ) )
% 0.34/0.51 => ( ( A2 != B )
% 0.34/0.51 => ( member6576561426505652726_nat_o @ A2 @ A ) ) ) ).
% 0.34/0.51
% 0.34/0.51 % insertE
% 0.34/0.51 thf(fact_234_insertE,axiom,
% 0.34/0.51 ! [A2: nat,B: nat,A: set_nat] :
% 0.34/0.51 ( ( member_nat @ A2 @ ( insert_nat @ B @ A ) )
% 0.34/0.51 => ( ( A2 != B )
% 0.34/0.51 => ( member_nat @ A2 @ A ) ) ) ).
% 0.34/0.51
% 0.34/0.51 % insertE
% 0.34/0.51 thf(fact_235_insertE,axiom,
% 0.34/0.51 ! [A2: int,B: int,A: set_int] :
% 0.34/0.51 ( ( member_int @ A2 @ ( insert_int @ B @ A ) )
% 0.34/0.51 => ( ( A2 != B )
% 0.34/0.51 => ( member_int @ A2 @ A ) ) ) ).
% 0.34/0.51
% 0.34/0.51 % insertE
% 0.34/0.51 thf(fact_236_insertE,axiom,
% 0.34/0.51 ! [A2: $o,B: $o,A: set_o] :
% 0.34/0.51 ( ( member_o @ A2 @ ( insert_o @ B @ A ) )
% 0.34/0.51 => ( ( A2 = ( ~ B ) )
% 0.34/0.51 => ( member_o @ A2 @ A ) ) ) ).
% 0.34/0.51
% 0.34/0.51 % insertE
% 0.34/0.51 thf(fact_237_insertE,axiom,
% 0.34/0.51 ! [A2: product_unit,B: product_unit,A: set_Product_unit] :
% 0.34/0.51 ( ( member_Product_unit @ A2 @ ( insert_Product_unit @ B @ A ) )
% 0.34/0.51 => ( ( A2 != B )
% 0.34/0.51 => ( member_Product_unit @ A2 @ A ) ) ) ).
% 0.34/0.51
% 0.34/0.51 % insertE
% 0.34/0.51 thf(fact_238_insertI1,axiom,
% 0.34/0.51 ! [A2: produc3658429121746597890et_nat > $o,B4: set_Pr4532377907799695533_nat_o] : ( member6576561426505652726_nat_o @ A2 @ ( insert5175938949040314269_nat_o @ A2 @ B4 ) ) ).
% 0.34/0.51
% 0.34/0.51 % insertI1
% 0.34/0.51 thf(fact_239_insertI1,axiom,
% 0.34/0.51 ! [A2: nat,B4: set_nat] : ( member_nat @ A2 @ ( insert_nat @ A2 @ B4 ) ) ).
% 0.34/0.51
% 0.34/0.51 % insertI1
% 0.34/0.51 thf(fact_240_insertI1,axiom,
% 0.34/0.51 ! [A2: int,B4: set_int] : ( member_int @ A2 @ ( insert_int @ A2 @ B4 ) ) ).
% 0.34/0.51
% 0.34/0.51 % insertI1
% 0.34/0.51 thf(fact_241_insertI1,axiom,
% 0.34/0.51 ! [A2: $o,B4: set_o] : ( member_o @ A2 @ ( insert_o @ A2 @ B4 ) ) ).
% 0.34/0.51
% 0.34/0.51 % insertI1
% 0.34/0.51 thf(fact_242_insertI1,axiom,
% 0.34/0.51 ! [A2: product_unit,B4: set_Product_unit] : ( member_Product_unit @ A2 @ ( insert_Product_unit @ A2 @ B4 ) ) ).
% 0.34/0.51
% 0.34/0.51 % insertI1
% 0.34/0.51 thf(fact_243_insertI2,axiom,
% 0.34/0.51 ! [A2: produc3658429121746597890et_nat > $o,B4: set_Pr4532377907799695533_nat_o,B: produc3658429121746597890et_nat > $o] :
% 0.34/0.51 ( ( member6576561426505652726_nat_o @ A2 @ B4 )
% 0.34/0.51 => ( member6576561426505652726_nat_o @ A2 @ ( insert5175938949040314269_nat_o @ B @ B4 ) ) ) ).
% 0.34/0.51
% 0.34/0.51 % insertI2
% 0.34/0.51 thf(fact_244_insertI2,axiom,
% 0.34/0.51 ! [A2: nat,B4: set_nat,B: nat] :
% 0.34/0.51 ( ( member_nat @ A2 @ B4 )
% 0.34/0.51 => ( member_nat @ A2 @ ( insert_nat @ B @ B4 ) ) ) ).
% 0.34/0.51
% 0.34/0.51 % insertI2
% 0.34/0.51 thf(fact_245_insertI2,axiom,
% 0.34/0.51 ! [A2: int,B4: set_int,B: int] :
% 0.34/0.51 ( ( member_int @ A2 @ B4 )
% 0.34/0.51 => ( member_int @ A2 @ ( insert_int @ B @ B4 ) ) ) ).
% 0.34/0.51
% 0.34/0.51 % insertI2
% 0.34/0.51 thf(fact_246_insertI2,axiom,
% 0.34/0.51 ! [A2: $o,B4: set_o,B: $o] :
% 0.34/0.51 ( ( member_o @ A2 @ B4 )
% 0.34/0.51 => ( member_o @ A2 @ ( insert_o @ B @ B4 ) ) ) ).
% 0.34/0.51
% 0.34/0.51 % insertI2
% 0.34/0.51 thf(fact_247_insertI2,axiom,
% 0.34/0.51 ! [A2: product_unit,B4: set_Product_unit,B: product_unit] :
% 0.34/0.51 ( ( member_Product_unit @ A2 @ B4 )
% 0.34/0.51 => ( member_Product_unit @ A2 @ ( insert_Product_unit @ B @ B4 ) ) ) ).
% 0.34/0.51
% 0.34/0.51 % insertI2
% 0.34/0.51 thf(fact_248_Set_Oset__insert,axiom,
% 0.34/0.51 ! [X: produc3658429121746597890et_nat > $o,A: set_Pr4532377907799695533_nat_o] :
% 0.34/0.51 ( ( member6576561426505652726_nat_o @ X @ A )
% 0.34/0.51 => ~ ! [B5: set_Pr4532377907799695533_nat_o] :
% 0.34/0.51 ( ( A
% 0.34/0.51 = ( insert5175938949040314269_nat_o @ X @ B5 ) )
% 0.34/0.51 => ( member6576561426505652726_nat_o @ X @ B5 ) ) ) ).
% 0.34/0.51
% 0.34/0.51 % Set.set_insert
% 0.34/0.51 thf(fact_249_Set_Oset__insert,axiom,
% 0.34/0.51 ! [X: nat,A: set_nat] :
% 0.34/0.51 ( ( member_nat @ X @ A )
% 0.34/0.51 => ~ ! [B5: set_nat] :
% 0.34/0.51 ( ( A
% 0.34/0.51 = ( insert_nat @ X @ B5 ) )
% 0.34/0.51 => ( member_nat @ X @ B5 ) ) ) ).
% 0.34/0.51
% 0.34/0.51 % Set.set_insert
% 0.34/0.51 thf(fact_250_Set_Oset__insert,axiom,
% 0.34/0.51 ! [X: int,A: set_int] :
% 0.34/0.51 ( ( member_int @ X @ A )
% 0.34/0.51 => ~ ! [B5: set_int] :
% 0.34/0.51 ( ( A
% 0.34/0.51 = ( insert_int @ X @ B5 ) )
% 0.34/0.51 => ( member_int @ X @ B5 ) ) ) ).
% 0.34/0.51
% 0.34/0.51 % Set.set_insert
% 0.34/0.51 thf(fact_251_Set_Oset__insert,axiom,
% 0.34/0.51 ! [X: $o,A: set_o] :
% 0.34/0.51 ( ( member_o @ X @ A )
% 0.34/0.51 => ~ ! [B5: set_o] :
% 0.34/0.51 ( ( A
% 0.34/0.51 = ( insert_o @ X @ B5 ) )
% 0.34/0.51 => ( member_o @ X @ B5 ) ) ) ).
% 0.34/0.51
% 0.34/0.51 % Set.set_insert
% 0.34/0.51 thf(fact_252_Set_Oset__insert,axiom,
% 0.34/0.51 ! [X: product_unit,A: set_Product_unit] :
% 0.34/0.51 ( ( member_Product_unit @ X @ A )
% 0.34/0.51 => ~ ! [B5: set_Product_unit] :
% 0.34/0.51 ( ( A
% 0.34/0.51 = ( insert_Product_unit @ X @ B5 ) )
% 0.34/0.51 => ( member_Product_unit @ X @ B5 ) ) ) ).
% 0.34/0.51
% 0.34/0.51 % Set.set_insert
% 0.34/0.51 thf(fact_253_insert__ident,axiom,
% 0.34/0.51 ! [X: produc3658429121746597890et_nat > $o,A: set_Pr4532377907799695533_nat_o,B4: set_Pr4532377907799695533_nat_o] :
% 0.34/0.51 ( ~ ( member6576561426505652726_nat_o @ X @ A )
% 0.34/0.51 => ( ~ ( member6576561426505652726_nat_o @ X @ B4 )
% 0.34/0.51 => ( ( ( insert5175938949040314269_nat_o @ X @ A )
% 0.34/0.51 = ( insert5175938949040314269_nat_o @ X @ B4 ) )
% 0.34/0.51 = ( A = B4 ) ) ) ) ).
% 0.34/0.51
% 0.34/0.51 % insert_ident
% 0.34/0.51 thf(fact_254_insert__ident,axiom,
% 0.34/0.51 ! [X: nat,A: set_nat,B4: set_nat] :
% 0.34/0.51 ( ~ ( member_nat @ X @ A )
% 0.34/0.51 => ( ~ ( member_nat @ X @ B4 )
% 0.34/0.51 => ( ( ( insert_nat @ X @ A )
% 0.34/0.51 = ( insert_nat @ X @ B4 ) )
% 0.34/0.51 = ( A = B4 ) ) ) ) ).
% 0.34/0.51
% 0.34/0.51 % insert_ident
% 0.34/0.51 thf(fact_255_insert__ident,axiom,
% 0.34/0.51 ! [X: int,A: set_int,B4: set_int] :
% 0.34/0.51 ( ~ ( member_int @ X @ A )
% 0.34/0.51 => ( ~ ( member_int @ X @ B4 )
% 0.34/0.51 => ( ( ( insert_int @ X @ A )
% 0.34/0.51 = ( insert_int @ X @ B4 ) )
% 0.34/0.51 = ( A = B4 ) ) ) ) ).
% 0.34/0.51
% 0.34/0.51 % insert_ident
% 0.34/0.51 thf(fact_256_insert__ident,axiom,
% 0.34/0.51 ! [X: $o,A: set_o,B4: set_o] :
% 0.34/0.51 ( ~ ( member_o @ X @ A )
% 0.34/0.51 => ( ~ ( member_o @ X @ B4 )
% 0.34/0.51 => ( ( ( insert_o @ X @ A )
% 0.34/0.51 = ( insert_o @ X @ B4 ) )
% 0.34/0.51 = ( A = B4 ) ) ) ) ).
% 0.34/0.51
% 0.34/0.51 % insert_ident
% 0.34/0.51 thf(fact_257_insert__ident,axiom,
% 0.34/0.51 ! [X: product_unit,A: set_Product_unit,B4: set_Product_unit] :
% 0.34/0.51 ( ~ ( member_Product_unit @ X @ A )
% 0.34/0.51 => ( ~ ( member_Product_unit @ X @ B4 )
% 0.34/0.51 => ( ( ( insert_Product_unit @ X @ A )
% 0.34/0.51 = ( insert_Product_unit @ X @ B4 ) )
% 0.34/0.51 = ( A = B4 ) ) ) ) ).
% 0.34/0.51
% 0.34/0.51 % insert_ident
% 0.34/0.51 thf(fact_258_insert__absorb,axiom,
% 0.34/0.51 ! [A2: produc3658429121746597890et_nat > $o,A: set_Pr4532377907799695533_nat_o] :
% 0.34/0.51 ( ( member6576561426505652726_nat_o @ A2 @ A )
% 0.34/0.51 => ( ( insert5175938949040314269_nat_o @ A2 @ A )
% 0.34/0.51 = A ) ) ).
% 0.34/0.51
% 0.34/0.51 % insert_absorb
% 0.34/0.51 thf(fact_259_insert__absorb,axiom,
% 0.34/0.51 ! [A2: nat,A: set_nat] :
% 0.34/0.51 ( ( member_nat @ A2 @ A )
% 0.34/0.51 => ( ( insert_nat @ A2 @ A )
% 0.34/0.51 = A ) ) ).
% 0.34/0.51
% 0.34/0.51 % insert_absorb
% 0.34/0.51 thf(fact_260_insert__absorb,axiom,
% 0.34/0.51 ! [A2: int,A: set_int] :
% 0.34/0.51 ( ( member_int @ A2 @ A )
% 0.34/0.51 => ( ( insert_int @ A2 @ A )
% 0.34/0.51 = A ) ) ).
% 0.34/0.51
% 0.34/0.51 % insert_absorb
% 0.34/0.51 thf(fact_261_insert__absorb,axiom,
% 0.34/0.51 ! [A2: $o,A: set_o] :
% 0.34/0.51 ( ( member_o @ A2 @ A )
% 0.34/0.51 => ( ( insert_o @ A2 @ A )
% 0.34/0.51 = A ) ) ).
% 0.34/0.51
% 0.34/0.51 % insert_absorb
% 0.34/0.51 thf(fact_262_insert__absorb,axiom,
% 0.34/0.51 ! [A2: product_unit,A: set_Product_unit] :
% 0.34/0.51 ( ( member_Product_unit @ A2 @ A )
% 0.34/0.51 => ( ( insert_Product_unit @ A2 @ A )
% 0.34/0.51 = A ) ) ).
% 0.34/0.51
% 0.34/0.51 % insert_absorb
% 0.34/0.51 thf(fact_263_insert__eq__iff,axiom,
% 0.34/0.51 ! [A2: produc3658429121746597890et_nat > $o,A: set_Pr4532377907799695533_nat_o,B: produc3658429121746597890et_nat > $o,B4: set_Pr4532377907799695533_nat_o] :
% 0.34/0.51 ( ~ ( member6576561426505652726_nat_o @ A2 @ A )
% 0.34/0.51 => ( ~ ( member6576561426505652726_nat_o @ B @ B4 )
% 0.34/0.51 => ( ( ( insert5175938949040314269_nat_o @ A2 @ A )
% 0.34/0.51 = ( insert5175938949040314269_nat_o @ B @ B4 ) )
% 0.34/0.51 = ( ( ( A2 = B )
% 0.34/0.51 => ( A = B4 ) )
% 0.34/0.51 & ( ( A2 != B )
% 0.34/0.51 => ? [C3: set_Pr4532377907799695533_nat_o] :
% 0.34/0.51 ( ( A
% 0.34/0.51 = ( insert5175938949040314269_nat_o @ B @ C3 ) )
% 0.34/0.51 & ~ ( member6576561426505652726_nat_o @ B @ C3 )
% 0.34/0.51 & ( B4
% 0.34/0.51 = ( insert5175938949040314269_nat_o @ A2 @ C3 ) )
% 0.34/0.51 & ~ ( member6576561426505652726_nat_o @ A2 @ C3 ) ) ) ) ) ) ) ).
% 0.34/0.51
% 0.34/0.51 % insert_eq_iff
% 0.34/0.51 thf(fact_264_insert__eq__iff,axiom,
% 0.34/0.51 ! [A2: nat,A: set_nat,B: nat,B4: set_nat] :
% 0.34/0.51 ( ~ ( member_nat @ A2 @ A )
% 0.34/0.51 => ( ~ ( member_nat @ B @ B4 )
% 0.34/0.51 => ( ( ( insert_nat @ A2 @ A )
% 0.34/0.51 = ( insert_nat @ B @ B4 ) )
% 0.34/0.51 = ( ( ( A2 = B )
% 0.34/0.51 => ( A = B4 ) )
% 0.34/0.51 & ( ( A2 != B )
% 0.34/0.51 => ? [C3: set_nat] :
% 0.34/0.51 ( ( A
% 0.34/0.51 = ( insert_nat @ B @ C3 ) )
% 0.34/0.51 & ~ ( member_nat @ B @ C3 )
% 0.34/0.51 & ( B4
% 0.34/0.51 = ( insert_nat @ A2 @ C3 ) )
% 0.34/0.51 & ~ ( member_nat @ A2 @ C3 ) ) ) ) ) ) ) ).
% 0.34/0.51
% 0.34/0.51 % insert_eq_iff
% 0.34/0.51 thf(fact_265_insert__eq__iff,axiom,
% 0.34/0.51 ! [A2: int,A: set_int,B: int,B4: set_int] :
% 0.34/0.51 ( ~ ( member_int @ A2 @ A )
% 0.34/0.51 => ( ~ ( member_int @ B @ B4 )
% 0.34/0.51 => ( ( ( insert_int @ A2 @ A )
% 0.34/0.51 = ( insert_int @ B @ B4 ) )
% 0.34/0.51 = ( ( ( A2 = B )
% 0.34/0.51 => ( A = B4 ) )
% 0.34/0.51 & ( ( A2 != B )
% 0.34/0.51 => ? [C3: set_int] :
% 0.34/0.51 ( ( A
% 0.34/0.51 = ( insert_int @ B @ C3 ) )
% 0.34/0.51 & ~ ( member_int @ B @ C3 )
% 0.34/0.51 & ( B4
% 0.34/0.51 = ( insert_int @ A2 @ C3 ) )
% 0.34/0.51 & ~ ( member_int @ A2 @ C3 ) ) ) ) ) ) ) ).
% 0.34/0.51
% 0.34/0.51 % insert_eq_iff
% 0.34/0.51 thf(fact_266_insert__eq__iff,axiom,
% 0.34/0.51 ! [A2: $o,A: set_o,B: $o,B4: set_o] :
% 0.34/0.51 ( ~ ( member_o @ A2 @ A )
% 0.34/0.51 => ( ~ ( member_o @ B @ B4 )
% 0.34/0.51 => ( ( ( insert_o @ A2 @ A )
% 0.34/0.51 = ( insert_o @ B @ B4 ) )
% 0.34/0.51 = ( ( ( A2 = B )
% 0.34/0.51 => ( A = B4 ) )
% 0.34/0.51 & ( ( A2 = ( ~ B ) )
% 0.34/0.51 => ? [C3: set_o] :
% 0.34/0.51 ( ( A
% 0.34/0.51 = ( insert_o @ B @ C3 ) )
% 0.34/0.51 & ~ ( member_o @ B @ C3 )
% 0.34/0.51 & ( B4
% 0.34/0.51 = ( insert_o @ A2 @ C3 ) )
% 0.34/0.51 & ~ ( member_o @ A2 @ C3 ) ) ) ) ) ) ) ).
% 0.34/0.51
% 0.34/0.51 % insert_eq_iff
% 0.34/0.51 thf(fact_267_insert__eq__iff,axiom,
% 0.34/0.51 ! [A2: product_unit,A: set_Product_unit,B: product_unit,B4: set_Product_unit] :
% 0.34/0.51 ( ~ ( member_Product_unit @ A2 @ A )
% 0.34/0.51 => ( ~ ( member_Product_unit @ B @ B4 )
% 0.34/0.51 => ( ( ( insert_Product_unit @ A2 @ A )
% 0.34/0.51 = ( insert_Product_unit @ B @ B4 ) )
% 0.34/0.51 = ( ( ( A2 = B )
% 0.34/0.51 => ( A = B4 ) )
% 0.34/0.51 & ( ( A2 != B )
% 0.34/0.51 => ? [C3: set_Product_unit] :
% 0.34/0.51 ( ( A
% 0.34/0.51 = ( insert_Product_unit @ B @ C3 ) )
% 0.34/0.51 & ~ ( member_Product_unit @ B @ C3 )
% 0.34/0.51 & ( B4
% 0.34/0.51 = ( insert_Product_unit @ A2 @ C3 ) )
% 0.34/0.51 & ~ ( member_Product_unit @ A2 @ C3 ) ) ) ) ) ) ) ).
% 0.34/0.51
% 0.34/0.51 % insert_eq_iff
% 0.34/0.51 thf(fact_268_insert__commute,axiom,
% 0.34/0.51 ! [X: nat,Y: nat,A: set_nat] :
% 0.34/0.51 ( ( insert_nat @ X @ ( insert_nat @ Y @ A ) )
% 0.34/0.51 = ( insert_nat @ Y @ ( insert_nat @ X @ A ) ) ) ).
% 0.34/0.51
% 0.34/0.51 % insert_commute
% 0.34/0.51 thf(fact_269_insert__commute,axiom,
% 0.34/0.51 ! [X: int,Y: int,A: set_int] :
% 0.34/0.51 ( ( insert_int @ X @ ( insert_int @ Y @ A ) )
% 0.34/0.51 = ( insert_int @ Y @ ( insert_int @ X @ A ) ) ) ).
% 0.34/0.51
% 0.34/0.51 % insert_commute
% 0.34/0.51 thf(fact_270_insert__commute,axiom,
% 0.34/0.51 ! [X: $o,Y: $o,A: set_o] :
% 0.34/0.51 ( ( insert_o @ X @ ( insert_o @ Y @ A ) )
% 0.34/0.51 = ( insert_o @ Y @ ( insert_o @ X @ A ) ) ) ).
% 0.34/0.51
% 0.34/0.51 % insert_commute
% 0.34/0.51 thf(fact_271_insert__commute,axiom,
% 0.34/0.51 ! [X: product_unit,Y: product_unit,A: set_Product_unit] :
% 0.34/0.51 ( ( insert_Product_unit @ X @ ( insert_Product_unit @ Y @ A ) )
% 0.34/0.51 = ( insert_Product_unit @ Y @ ( insert_Product_unit @ X @ A ) ) ) ).
% 0.34/0.51
% 0.34/0.51 % insert_commute
% 0.34/0.51 thf(fact_272_mk__disjoint__insert,axiom,
% 0.34/0.51 ! [A2: produc3658429121746597890et_nat > $o,A: set_Pr4532377907799695533_nat_o] :
% 0.34/0.51 ( ( member6576561426505652726_nat_o @ A2 @ A )
% 0.34/0.51 => ? [B5: set_Pr4532377907799695533_nat_o] :
% 0.34/0.51 ( ( A
% 0.34/0.51 = ( insert5175938949040314269_nat_o @ A2 @ B5 ) )
% 0.34/0.51 & ~ ( member6576561426505652726_nat_o @ A2 @ B5 ) ) ) ).
% 0.34/0.51
% 0.34/0.51 % mk_disjoint_insert
% 0.34/0.51 thf(fact_273_mk__disjoint__insert,axiom,
% 0.34/0.51 ! [A2: nat,A: set_nat] :
% 0.34/0.51 ( ( member_nat @ A2 @ A )
% 0.34/0.51 => ? [B5: set_nat] :
% 0.34/0.51 ( ( A
% 0.34/0.51 = ( insert_nat @ A2 @ B5 ) )
% 0.34/0.51 & ~ ( member_nat @ A2 @ B5 ) ) ) ).
% 0.34/0.51
% 0.34/0.51 % mk_disjoint_insert
% 0.34/0.51 thf(fact_274_mk__disjoint__insert,axiom,
% 0.34/0.51 ! [A2: int,A: set_int] :
% 0.34/0.51 ( ( member_int @ A2 @ A )
% 0.34/0.51 => ? [B5: set_int] :
% 0.34/0.51 ( ( A
% 0.34/0.51 = ( insert_int @ A2 @ B5 ) )
% 0.34/0.51 & ~ ( member_int @ A2 @ B5 ) ) ) ).
% 0.34/0.51
% 0.34/0.51 % mk_disjoint_insert
% 0.34/0.51 thf(fact_275_mk__disjoint__insert,axiom,
% 0.34/0.51 ! [A2: $o,A: set_o] :
% 0.34/0.51 ( ( member_o @ A2 @ A )
% 0.34/0.51 => ? [B5: set_o] :
% 0.34/0.51 ( ( A
% 0.34/0.51 = ( insert_o @ A2 @ B5 ) )
% 0.34/0.51 & ~ ( member_o @ A2 @ B5 ) ) ) ).
% 0.34/0.51
% 0.34/0.51 % mk_disjoint_insert
% 0.34/0.51 thf(fact_276_mk__disjoint__insert,axiom,
% 0.34/0.51 ! [A2: product_unit,A: set_Product_unit] :
% 0.34/0.51 ( ( member_Product_unit @ A2 @ A )
% 0.34/0.51 => ? [B5: set_Product_unit] :
% 0.34/0.51 ( ( A
% 0.34/0.51 = ( insert_Product_unit @ A2 @ B5 ) )
% 0.34/0.51 & ~ ( member_Product_unit @ A2 @ B5 ) ) ) ).
% 0.34/0.51
% 0.34/0.51 % mk_disjoint_insert
% 0.34/0.51 thf(fact_277_singletonD,axiom,
% 0.34/0.51 ! [B: product_unit,A2: product_unit] :
% 0.34/0.51 ( ( member_Product_unit @ B @ ( insert_Product_unit @ A2 @ bot_bo3957492148770167129t_unit ) )
% 0.34/0.51 => ( B = A2 ) ) ).
% 0.34/0.51
% 0.34/0.51 % singletonD
% 0.34/0.51 thf(fact_278_singletonD,axiom,
% 0.34/0.51 ! [B: $o,A2: $o] :
% 0.34/0.51 ( ( member_o @ B @ ( insert_o @ A2 @ bot_bot_set_o ) )
% 0.34/0.51 => ( B = A2 ) ) ).
% 0.34/0.51
% 0.34/0.51 % singletonD
% 0.34/0.51 thf(fact_279_singletonD,axiom,
% 0.34/0.51 ! [B: nat,A2: nat] :
% 0.34/0.51 ( ( member_nat @ B @ ( insert_nat @ A2 @ bot_bot_set_nat ) )
% 0.34/0.51 => ( B = A2 ) ) ).
% 0.34/0.51
% 0.34/0.51 % singletonD
% 0.34/0.51 thf(fact_280_singletonD,axiom,
% 0.34/0.51 ! [B: int,A2: int] :
% 0.34/0.51 ( ( member_int @ B @ ( insert_int @ A2 @ bot_bot_set_int ) )
% 0.34/0.51 => ( B = A2 ) ) ).
% 0.34/0.51
% 0.34/0.51 % singletonD
% 0.34/0.51 thf(fact_281_singletonD,axiom,
% 0.34/0.51 ! [B: produc3658429121746597890et_nat > $o,A2: produc3658429121746597890et_nat > $o] :
% 0.34/0.51 ( ( member6576561426505652726_nat_o @ B @ ( insert5175938949040314269_nat_o @ A2 @ bot_bo7824918357723582233_nat_o ) )
% 0.34/0.51 => ( B = A2 ) ) ).
% 0.34/0.51
% 0.34/0.51 % singletonD
% 0.34/0.51 thf(fact_282_singleton__iff,axiom,
% 0.34/0.51 ! [B: product_unit,A2: product_unit] :
% 0.34/0.51 ( ( member_Product_unit @ B @ ( insert_Product_unit @ A2 @ bot_bo3957492148770167129t_unit ) )
% 0.34/0.51 = ( B = A2 ) ) ).
% 0.34/0.51
% 0.34/0.51 % singleton_iff
% 0.34/0.51 thf(fact_283_singleton__iff,axiom,
% 0.34/0.51 ! [B: $o,A2: $o] :
% 0.34/0.51 ( ( member_o @ B @ ( insert_o @ A2 @ bot_bot_set_o ) )
% 0.34/0.51 = ( B = A2 ) ) ).
% 0.34/0.51
% 0.34/0.51 % singleton_iff
% 0.34/0.51 thf(fact_284_singleton__iff,axiom,
% 0.34/0.51 ! [B: nat,A2: nat] :
% 0.34/0.51 ( ( member_nat @ B @ ( insert_nat @ A2 @ bot_bot_set_nat ) )
% 0.34/0.51 = ( B = A2 ) ) ).
% 0.34/0.51
% 0.34/0.51 % singleton_iff
% 0.34/0.51 thf(fact_285_singleton__iff,axiom,
% 0.34/0.51 ! [B: int,A2: int] :
% 0.34/0.51 ( ( member_int @ B @ ( insert_int @ A2 @ bot_bot_set_int ) )
% 0.34/0.51 = ( B = A2 ) ) ).
% 0.34/0.51
% 0.34/0.51 % singleton_iff
% 0.34/0.51 thf(fact_286_singleton__iff,axiom,
% 0.34/0.52 ! [B: produc3658429121746597890et_nat > $o,A2: produc3658429121746597890et_nat > $o] :
% 0.34/0.52 ( ( member6576561426505652726_nat_o @ B @ ( insert5175938949040314269_nat_o @ A2 @ bot_bo7824918357723582233_nat_o ) )
% 0.34/0.52 = ( B = A2 ) ) ).
% 0.34/0.52
% 0.34/0.52 % singleton_iff
% 0.34/0.52 thf(fact_287_doubleton__eq__iff,axiom,
% 0.34/0.52 ! [A2: product_unit,B: product_unit,C: product_unit,D2: product_unit] :
% 0.34/0.52 ( ( ( insert_Product_unit @ A2 @ ( insert_Product_unit @ B @ bot_bo3957492148770167129t_unit ) )
% 0.34/0.52 = ( insert_Product_unit @ C @ ( insert_Product_unit @ D2 @ bot_bo3957492148770167129t_unit ) ) )
% 0.34/0.52 = ( ( ( A2 = C )
% 0.34/0.52 & ( B = D2 ) )
% 0.34/0.52 | ( ( A2 = D2 )
% 0.34/0.52 & ( B = C ) ) ) ) ).
% 0.34/0.52
% 0.34/0.52 % doubleton_eq_iff
% 0.34/0.52 thf(fact_288_doubleton__eq__iff,axiom,
% 0.34/0.52 ! [A2: $o,B: $o,C: $o,D2: $o] :
% 0.34/0.52 ( ( ( insert_o @ A2 @ ( insert_o @ B @ bot_bot_set_o ) )
% 0.34/0.52 = ( insert_o @ C @ ( insert_o @ D2 @ bot_bot_set_o ) ) )
% 0.34/0.52 = ( ( ( A2 = C )
% 0.34/0.52 & ( B = D2 ) )
% 0.34/0.52 | ( ( A2 = D2 )
% 0.34/0.52 & ( B = C ) ) ) ) ).
% 0.34/0.52
% 0.34/0.52 % doubleton_eq_iff
% 0.34/0.52 thf(fact_289_doubleton__eq__iff,axiom,
% 0.34/0.52 ! [A2: nat,B: nat,C: nat,D2: nat] :
% 0.34/0.52 ( ( ( insert_nat @ A2 @ ( insert_nat @ B @ bot_bot_set_nat ) )
% 0.34/0.52 = ( insert_nat @ C @ ( insert_nat @ D2 @ bot_bot_set_nat ) ) )
% 0.34/0.52 = ( ( ( A2 = C )
% 0.34/0.52 & ( B = D2 ) )
% 0.34/0.52 | ( ( A2 = D2 )
% 0.34/0.52 & ( B = C ) ) ) ) ).
% 0.34/0.52
% 0.34/0.52 % doubleton_eq_iff
% 0.34/0.52 thf(fact_290_doubleton__eq__iff,axiom,
% 0.34/0.52 ! [A2: int,B: int,C: int,D2: int] :
% 0.34/0.52 ( ( ( insert_int @ A2 @ ( insert_int @ B @ bot_bot_set_int ) )
% 0.34/0.52 = ( insert_int @ C @ ( insert_int @ D2 @ bot_bot_set_int ) ) )
% 0.34/0.52 = ( ( ( A2 = C )
% 0.34/0.52 & ( B = D2 ) )
% 0.34/0.52 | ( ( A2 = D2 )
% 0.34/0.52 & ( B = C ) ) ) ) ).
% 0.34/0.52
% 0.34/0.52 % doubleton_eq_iff
% 0.34/0.52 thf(fact_291_doubleton__eq__iff,axiom,
% 0.34/0.52 ! [A2: produc3658429121746597890et_nat > $o,B: produc3658429121746597890et_nat > $o,C: produc3658429121746597890et_nat > $o,D2: produc3658429121746597890et_nat > $o] :
% 0.34/0.52 ( ( ( insert5175938949040314269_nat_o @ A2 @ ( insert5175938949040314269_nat_o @ B @ bot_bo7824918357723582233_nat_o ) )
% 0.34/0.52 = ( insert5175938949040314269_nat_o @ C @ ( insert5175938949040314269_nat_o @ D2 @ bot_bo7824918357723582233_nat_o ) ) )
% 0.34/0.52 = ( ( ( A2 = C )
% 0.34/0.52 & ( B = D2 ) )
% 0.34/0.52 | ( ( A2 = D2 )
% 0.34/0.52 & ( B = C ) ) ) ) ).
% 0.34/0.52
% 0.34/0.52 % doubleton_eq_iff
% 0.34/0.52 thf(fact_292_insert__not__empty,axiom,
% 0.34/0.52 ! [A2: product_unit,A: set_Product_unit] :
% 0.34/0.52 ( ( insert_Product_unit @ A2 @ A )
% 0.34/0.52 != bot_bo3957492148770167129t_unit ) ).
% 0.34/0.52
% 0.34/0.52 % insert_not_empty
% 0.34/0.52 thf(fact_293_insert__not__empty,axiom,
% 0.34/0.52 ! [A2: $o,A: set_o] :
% 0.34/0.52 ( ( insert_o @ A2 @ A )
% 0.34/0.52 != bot_bot_set_o ) ).
% 0.34/0.52
% 0.34/0.52 % insert_not_empty
% 0.34/0.52 thf(fact_294_insert__not__empty,axiom,
% 0.34/0.52 ! [A2: nat,A: set_nat] :
% 0.34/0.52 ( ( insert_nat @ A2 @ A )
% 0.34/0.52 != bot_bot_set_nat ) ).
% 0.34/0.52
% 0.34/0.52 % insert_not_empty
% 0.34/0.52 thf(fact_295_insert__not__empty,axiom,
% 0.34/0.52 ! [A2: int,A: set_int] :
% 0.34/0.52 ( ( insert_int @ A2 @ A )
% 0.34/0.52 != bot_bot_set_int ) ).
% 0.34/0.52
% 0.34/0.52 % insert_not_empty
% 0.34/0.52 thf(fact_296_insert__not__empty,axiom,
% 0.34/0.52 ! [A2: produc3658429121746597890et_nat > $o,A: set_Pr4532377907799695533_nat_o] :
% 0.34/0.52 ( ( insert5175938949040314269_nat_o @ A2 @ A )
% 0.34/0.52 != bot_bo7824918357723582233_nat_o ) ).
% 0.34/0.52
% 0.34/0.52 % insert_not_empty
% 0.34/0.52 thf(fact_297_singleton__inject,axiom,
% 0.34/0.52 ! [A2: product_unit,B: product_unit] :
% 0.34/0.52 ( ( ( insert_Product_unit @ A2 @ bot_bo3957492148770167129t_unit )
% 0.34/0.52 = ( insert_Product_unit @ B @ bot_bo3957492148770167129t_unit ) )
% 0.34/0.52 => ( A2 = B ) ) ).
% 0.34/0.52
% 0.34/0.52 % singleton_inject
% 0.34/0.52 thf(fact_298_singleton__inject,axiom,
% 0.34/0.52 ! [A2: $o,B: $o] :
% 0.34/0.52 ( ( ( insert_o @ A2 @ bot_bot_set_o )
% 0.34/0.52 = ( insert_o @ B @ bot_bot_set_o ) )
% 0.34/0.52 => ( A2 = B ) ) ).
% 0.34/0.52
% 0.34/0.52 % singleton_inject
% 0.34/0.52 thf(fact_299_singleton__inject,axiom,
% 0.34/0.52 ! [A2: nat,B: nat] :
% 0.34/0.52 ( ( ( insert_nat @ A2 @ bot_bot_set_nat )
% 0.34/0.52 = ( insert_nat @ B @ bot_bot_set_nat ) )
% 0.34/0.52 => ( A2 = B ) ) ).
% 0.34/0.52
% 0.34/0.52 % singleton_inject
% 0.34/0.52 thf(fact_300_singleton__inject,axiom,
% 0.34/0.52 ! [A2: int,B: int] :
% 0.34/0.52 ( ( ( insert_int @ A2 @ bot_bot_set_int )
% 0.34/0.52 = ( insert_int @ B @ bot_bot_set_int ) )
% 0.34/0.52 => ( A2 = B ) ) ).
% 0.34/0.52
% 0.34/0.52 % singleton_inject
% 0.34/0.52 thf(fact_301_singleton__inject,axiom,
% 0.34/0.52 ! [A2: produc3658429121746597890et_nat > $o,B: produc3658429121746597890et_nat > $o] :
% 0.34/0.52 ( ( ( insert5175938949040314269_nat_o @ A2 @ bot_bo7824918357723582233_nat_o )
% 0.34/0.52 = ( insert5175938949040314269_nat_o @ B @ bot_bo7824918357723582233_nat_o ) )
% 0.34/0.52 => ( A2 = B ) ) ).
% 0.34/0.52
% 0.34/0.52 % singleton_inject
% 0.34/0.52 thf(fact_302_curryE,axiom,
% 0.34/0.52 ! [F: produc3658429121746597890et_nat > $o,A2: heap_e7401611519738050253t_unit,B: set_nat] :
% 0.34/0.52 ( ( produc1807123492089482958_nat_o @ F @ A2 @ B )
% 0.34/0.52 => ( F @ ( produc7507926704131184380et_nat @ A2 @ B ) ) ) ).
% 0.34/0.52
% 0.34/0.52 % curryE
% 0.34/0.52 thf(fact_303_curryE,axiom,
% 0.34/0.52 ! [F: product_prod_nat_nat > $o,A2: nat,B: nat] :
% 0.34/0.52 ( ( produc1310100445399344235_nat_o @ F @ A2 @ B )
% 0.34/0.52 => ( F @ ( product_Pair_nat_nat @ A2 @ B ) ) ) ).
% 0.34/0.52
% 0.34/0.52 % curryE
% 0.34/0.52 thf(fact_304_curryE,axiom,
% 0.34/0.52 ! [F: product_prod_int_int > $o,A2: int,B: int] :
% 0.34/0.52 ( ( produc175634133007206835_int_o @ F @ A2 @ B )
% 0.34/0.52 => ( F @ ( product_Pair_int_int @ A2 @ B ) ) ) ).
% 0.34/0.52
% 0.34/0.52 % curryE
% 0.34/0.52 thf(fact_305_curryE,axiom,
% 0.34/0.52 ! [F: produc3925858234332021118et_nat > $o,A2: produc3658429121746597890et_nat > $o,B: produc3658429121746597890et_nat] :
% 0.34/0.52 ( ( produc6216949301066131538_nat_o @ F @ A2 @ B )
% 0.34/0.52 => ( F @ ( produc5001842942810119800et_nat @ A2 @ B ) ) ) ).
% 0.34/0.52
% 0.34/0.52 % curryE
% 0.34/0.52 thf(fact_306_curryE,axiom,
% 0.34/0.52 ! [F: produc2732055786443039994et_nat > $o,A2: produc3658429121746597890et_nat > $o,B: produc3925858234332021118et_nat] :
% 0.34/0.52 ( ( produc5101573711933517782_nat_o @ F @ A2 @ B )
% 0.34/0.52 => ( F @ ( produc2245416461498447860et_nat @ A2 @ B ) ) ) ).
% 0.34/0.52
% 0.34/0.52 % curryE
% 0.34/0.52 thf(fact_307_curryD,axiom,
% 0.34/0.52 ! [F: produc3658429121746597890et_nat > $o,A2: heap_e7401611519738050253t_unit,B: set_nat] :
% 0.34/0.52 ( ( produc1807123492089482958_nat_o @ F @ A2 @ B )
% 0.34/0.52 => ( F @ ( produc7507926704131184380et_nat @ A2 @ B ) ) ) ).
% 0.34/0.52
% 0.34/0.52 % curryD
% 0.34/0.52 thf(fact_308_curryD,axiom,
% 0.34/0.52 ! [F: product_prod_nat_nat > $o,A2: nat,B: nat] :
% 0.34/0.52 ( ( produc1310100445399344235_nat_o @ F @ A2 @ B )
% 0.34/0.52 => ( F @ ( product_Pair_nat_nat @ A2 @ B ) ) ) ).
% 0.34/0.52
% 0.34/0.52 % curryD
% 0.34/0.52 thf(fact_309_curryD,axiom,
% 0.34/0.52 ! [F: product_prod_int_int > $o,A2: int,B: int] :
% 0.34/0.52 ( ( produc175634133007206835_int_o @ F @ A2 @ B )
% 0.34/0.52 => ( F @ ( product_Pair_int_int @ A2 @ B ) ) ) ).
% 0.34/0.52
% 0.34/0.52 % curryD
% 0.34/0.52 thf(fact_310_curryD,axiom,
% 0.34/0.52 ! [F: produc3925858234332021118et_nat > $o,A2: produc3658429121746597890et_nat > $o,B: produc3658429121746597890et_nat] :
% 0.34/0.52 ( ( produc6216949301066131538_nat_o @ F @ A2 @ B )
% 0.34/0.52 => ( F @ ( produc5001842942810119800et_nat @ A2 @ B ) ) ) ).
% 0.34/0.52
% 0.34/0.52 % curryD
% 0.34/0.52 thf(fact_311_curryD,axiom,
% 0.34/0.52 ! [F: produc2732055786443039994et_nat > $o,A2: produc3658429121746597890et_nat > $o,B: produc3925858234332021118et_nat] :
% 0.34/0.52 ( ( produc5101573711933517782_nat_o @ F @ A2 @ B )
% 0.34/0.52 => ( F @ ( produc2245416461498447860et_nat @ A2 @ B ) ) ) ).
% 0.34/0.52
% 0.34/0.52 % curryD
% 0.34/0.52 thf(fact_312_is__singleton__def,axiom,
% 0.34/0.52 ( is_sin2160648248035936513t_unit
% 0.34/0.52 = ( ^ [A3: set_Product_unit] :
% 0.34/0.52 ? [X2: product_unit] :
% 0.34/0.52 ( A3
% 0.34/0.52 = ( insert_Product_unit @ X2 @ bot_bo3957492148770167129t_unit ) ) ) ) ).
% 0.34/0.52
% 0.34/0.52 % is_singleton_def
% 0.34/0.52 thf(fact_313_is__singleton__def,axiom,
% 0.34/0.52 ( is_singleton_o
% 0.34/0.52 = ( ^ [A3: set_o] :
% 0.34/0.52 ? [X2: $o] :
% 0.34/0.52 ( A3
% 0.34/0.52 = ( insert_o @ X2 @ bot_bot_set_o ) ) ) ) ).
% 0.34/0.52
% 0.34/0.52 % is_singleton_def
% 0.34/0.52 thf(fact_314_is__singleton__def,axiom,
% 0.34/0.52 ( is_singleton_nat
% 0.34/0.52 = ( ^ [A3: set_nat] :
% 0.34/0.52 ? [X2: nat] :
% 0.34/0.52 ( A3
% 0.34/0.52 = ( insert_nat @ X2 @ bot_bot_set_nat ) ) ) ) ).
% 0.34/0.52
% 0.34/0.52 % is_singleton_def
% 0.34/0.52 thf(fact_315_is__singleton__def,axiom,
% 0.34/0.52 ( is_singleton_int
% 0.34/0.52 = ( ^ [A3: set_int] :
% 0.34/0.52 ? [X2: int] :
% 0.34/0.52 ( A3
% 0.34/0.52 = ( insert_int @ X2 @ bot_bot_set_int ) ) ) ) ).
% 0.34/0.52
% 0.34/0.52 % is_singleton_def
% 0.34/0.52 thf(fact_316_is__singleton__def,axiom,
% 0.34/0.52 ( is_sin5180296473474724033_nat_o
% 0.34/0.52 = ( ^ [A3: set_Pr4532377907799695533_nat_o] :
% 0.34/0.52 ? [X2: produc3658429121746597890et_nat > $o] :
% 0.34/0.52 ( A3
% 0.34/0.52 = ( insert5175938949040314269_nat_o @ X2 @ bot_bo7824918357723582233_nat_o ) ) ) ) ).
% 0.34/0.52
% 0.34/0.52 % is_singleton_def
% 0.34/0.52 thf(fact_317_is__singletonE,axiom,
% 0.34/0.52 ! [A: set_Product_unit] :
% 0.34/0.52 ( ( is_sin2160648248035936513t_unit @ A )
% 0.34/0.52 => ~ ! [X3: product_unit] :
% 0.34/0.52 ( A
% 0.34/0.52 != ( insert_Product_unit @ X3 @ bot_bo3957492148770167129t_unit ) ) ) ).
% 0.34/0.52
% 0.34/0.52 % is_singletonE
% 0.34/0.52 thf(fact_318_is__singletonE,axiom,
% 0.34/0.52 ! [A: set_o] :
% 0.34/0.52 ( ( is_singleton_o @ A )
% 0.34/0.52 => ~ ! [X3: $o] :
% 0.34/0.52 ( A
% 0.34/0.52 != ( insert_o @ X3 @ bot_bot_set_o ) ) ) ).
% 0.34/0.52
% 0.34/0.52 % is_singletonE
% 0.34/0.52 thf(fact_319_is__singletonE,axiom,
% 0.34/0.52 ! [A: set_nat] :
% 0.34/0.52 ( ( is_singleton_nat @ A )
% 0.34/0.52 => ~ ! [X3: nat] :
% 0.34/0.52 ( A
% 0.34/0.52 != ( insert_nat @ X3 @ bot_bot_set_nat ) ) ) ).
% 0.34/0.52
% 0.34/0.52 % is_singletonE
% 0.34/0.52 thf(fact_320_is__singletonE,axiom,
% 0.34/0.52 ! [A: set_int] :
% 0.34/0.52 ( ( is_singleton_int @ A )
% 0.34/0.52 => ~ ! [X3: int] :
% 0.34/0.52 ( A
% 0.34/0.52 != ( insert_int @ X3 @ bot_bot_set_int ) ) ) ).
% 0.34/0.52
% 0.34/0.52 % is_singletonE
% 0.34/0.52 thf(fact_321_is__singletonE,axiom,
% 0.34/0.52 ! [A: set_Pr4532377907799695533_nat_o] :
% 0.34/0.52 ( ( is_sin5180296473474724033_nat_o @ A )
% 0.34/0.52 => ~ ! [X3: produc3658429121746597890et_nat > $o] :
% 0.34/0.52 ( A
% 0.34/0.52 != ( insert5175938949040314269_nat_o @ X3 @ bot_bo7824918357723582233_nat_o ) ) ) ).
% 0.34/0.52
% 0.34/0.52 % is_singletonE
% 0.34/0.52 thf(fact_322_is__pure__assn__basic__simps_I1_J,axiom,
% 0.34/0.52 is_pure_assn @ bot_bot_assn ).
% 0.34/0.52
% 0.34/0.52 % is_pure_assn_basic_simps(1)
% 0.34/0.52 thf(fact_323_is__singleton__the__elem,axiom,
% 0.34/0.52 ( is_sin2160648248035936513t_unit
% 0.34/0.52 = ( ^ [A3: set_Product_unit] :
% 0.34/0.52 ( A3
% 0.34/0.52 = ( insert_Product_unit @ ( the_el608902216710161154t_unit @ A3 ) @ bot_bo3957492148770167129t_unit ) ) ) ) ).
% 0.34/0.52
% 0.34/0.52 % is_singleton_the_elem
% 0.34/0.52 thf(fact_324_is__singleton__the__elem,axiom,
% 0.34/0.52 ( is_singleton_o
% 0.34/0.52 = ( ^ [A3: set_o] :
% 0.34/0.52 ( A3
% 0.34/0.52 = ( insert_o @ ( the_elem_o @ A3 ) @ bot_bot_set_o ) ) ) ) ).
% 0.34/0.52
% 0.34/0.52 % is_singleton_the_elem
% 0.34/0.52 thf(fact_325_is__singleton__the__elem,axiom,
% 0.34/0.52 ( is_singleton_nat
% 0.34/0.52 = ( ^ [A3: set_nat] :
% 0.34/0.52 ( A3
% 0.34/0.52 = ( insert_nat @ ( the_elem_nat @ A3 ) @ bot_bot_set_nat ) ) ) ) ).
% 0.34/0.52
% 0.34/0.52 % is_singleton_the_elem
% 0.34/0.52 thf(fact_326_is__singleton__the__elem,axiom,
% 0.34/0.52 ( is_singleton_int
% 0.34/0.52 = ( ^ [A3: set_int] :
% 0.34/0.52 ( A3
% 0.34/0.52 = ( insert_int @ ( the_elem_int @ A3 ) @ bot_bot_set_int ) ) ) ) ).
% 0.34/0.52
% 0.34/0.52 % is_singleton_the_elem
% 0.34/0.52 thf(fact_327_is__singleton__the__elem,axiom,
% 0.34/0.52 ( is_sin5180296473474724033_nat_o
% 0.34/0.52 = ( ^ [A3: set_Pr4532377907799695533_nat_o] :
% 0.34/0.52 ( A3
% 0.34/0.52 = ( insert5175938949040314269_nat_o @ ( the_el5959152443890766018_nat_o @ A3 ) @ bot_bo7824918357723582233_nat_o ) ) ) ) ).
% 0.34/0.52
% 0.34/0.52 % is_singleton_the_elem
% 0.34/0.52 thf(fact_328_the__elem__eq,axiom,
% 0.34/0.52 ! [X: product_unit] :
% 0.34/0.52 ( ( the_el608902216710161154t_unit @ ( insert_Product_unit @ X @ bot_bo3957492148770167129t_unit ) )
% 0.34/0.52 = X ) ).
% 0.34/0.52
% 0.34/0.52 % the_elem_eq
% 0.34/0.52 thf(fact_329_the__elem__eq,axiom,
% 0.34/0.52 ! [X: $o] :
% 0.34/0.52 ( ( the_elem_o @ ( insert_o @ X @ bot_bot_set_o ) )
% 0.34/0.52 = X ) ).
% 0.34/0.52
% 0.34/0.52 % the_elem_eq
% 0.34/0.52 thf(fact_330_the__elem__eq,axiom,
% 0.34/0.52 ! [X: nat] :
% 0.34/0.52 ( ( the_elem_nat @ ( insert_nat @ X @ bot_bot_set_nat ) )
% 0.34/0.52 = X ) ).
% 0.34/0.52
% 0.34/0.52 % the_elem_eq
% 0.34/0.52 thf(fact_331_the__elem__eq,axiom,
% 0.34/0.52 ! [X: int] :
% 0.34/0.52 ( ( the_elem_int @ ( insert_int @ X @ bot_bot_set_int ) )
% 0.34/0.52 = X ) ).
% 0.34/0.52
% 0.34/0.52 % the_elem_eq
% 0.34/0.52 thf(fact_332_the__elem__eq,axiom,
% 0.34/0.52 ! [X: produc3658429121746597890et_nat > $o] :
% 0.34/0.52 ( ( the_el5959152443890766018_nat_o @ ( insert5175938949040314269_nat_o @ X @ bot_bo7824918357723582233_nat_o ) )
% 0.34/0.52 = X ) ).
% 0.34/0.52
% 0.34/0.52 % the_elem_eq
% 0.34/0.52 thf(fact_333_pure__assn__def,axiom,
% 0.34/0.52 ( pure_assn
% 0.34/0.52 = ( ^ [B6: $o] : ( abs_assn @ ( pure_a825153325127701367it_nat @ B6 ) ) ) ) ).
% 0.34/0.52
% 0.34/0.52 % pure_assn_def
% 0.34/0.52 thf(fact_334_times__assn__def,axiom,
% 0.34/0.52 ( times_times_assn
% 0.34/0.52 = ( ^ [P2: assn,Q: assn] : ( abs_assn @ ( times_assn_raw @ ( rep_assn @ P2 ) @ ( rep_assn @ Q ) ) ) ) ) ).
% 0.34/0.52
% 0.34/0.52 % times_assn_def
% 0.34/0.52 thf(fact_335_type__definition__def,axiom,
% 0.34/0.52 ( type_d3909072315231072503_nat_o
% 0.34/0.52 = ( ^ [Rep: assn > produc3658429121746597890et_nat > $o,Abs: ( produc3658429121746597890et_nat > $o ) > assn,A3: set_Pr4532377907799695533_nat_o] :
% 0.34/0.52 ( ! [X2: assn] : ( member6576561426505652726_nat_o @ ( Rep @ X2 ) @ A3 )
% 0.34/0.52 & ! [X2: assn] :
% 0.34/0.52 ( ( Abs @ ( Rep @ X2 ) )
% 0.34/0.52 = X2 )
% 0.34/0.52 & ! [Y3: produc3658429121746597890et_nat > $o] :
% 0.34/0.52 ( ( member6576561426505652726_nat_o @ Y3 @ A3 )
% 0.34/0.52 => ( ( Rep @ ( Abs @ Y3 ) )
% 0.34/0.52 = Y3 ) ) ) ) ) ).
% 0.34/0.52
% 0.34/0.52 % type_definition_def
% 0.34/0.52 thf(fact_336_type__definition__def,axiom,
% 0.34/0.52 ( type_d8366093980585677751er_int
% 0.34/0.52 = ( ^ [Rep: code_integer > int,Abs: int > code_integer,A3: set_int] :
% 0.34/0.52 ( ! [X2: code_integer] : ( member_int @ ( Rep @ X2 ) @ A3 )
% 0.34/0.52 & ! [X2: code_integer] :
% 0.34/0.52 ( ( Abs @ ( Rep @ X2 ) )
% 0.34/0.52 = X2 )
% 0.34/0.52 & ! [Y3: int] :
% 0.34/0.52 ( ( member_int @ Y3 @ A3 )
% 0.34/0.52 => ( ( Rep @ ( Abs @ Y3 ) )
% 0.34/0.52 = Y3 ) ) ) ) ) ).
% 0.34/0.52
% 0.34/0.52 % type_definition_def
% 0.34/0.52 thf(fact_337_type__definition__def,axiom,
% 0.34/0.52 ( type_d6188575255521822967unit_o
% 0.34/0.52 = ( ^ [Rep: product_unit > $o,Abs: $o > product_unit,A3: set_o] :
% 0.34/0.52 ( ! [X2: product_unit] : ( member_o @ ( Rep @ X2 ) @ A3 )
% 0.34/0.52 & ! [X2: product_unit] :
% 0.34/0.52 ( ( Abs @ ( Rep @ X2 ) )
% 0.34/0.52 = X2 )
% 0.34/0.52 & ! [Y3: $o] :
% 0.34/0.52 ( ( member_o @ Y3 @ A3 )
% 0.34/0.52 => ( ( Rep @ ( Abs @ Y3 ) )
% 0.34/0.52 = Y3 ) ) ) ) ) ).
% 0.34/0.52
% 0.34/0.52 % type_definition_def
% 0.34/0.52 thf(fact_338_type__definition__def,axiom,
% 0.34/0.52 ( type_d4410041424927559462al_nat
% 0.34/0.52 = ( ^ [Rep: code_natural > nat,Abs: nat > code_natural,A3: set_nat] :
% 0.34/0.52 ( ! [X2: code_natural] : ( member_nat @ ( Rep @ X2 ) @ A3 )
% 0.34/0.52 & ! [X2: code_natural] :
% 0.34/0.52 ( ( Abs @ ( Rep @ X2 ) )
% 0.34/0.52 = X2 )
% 0.34/0.52 & ! [Y3: nat] :
% 0.34/0.52 ( ( member_nat @ Y3 @ A3 )
% 0.34/0.52 => ( ( Rep @ ( Abs @ Y3 ) )
% 0.34/0.52 = Y3 ) ) ) ) ) ).
% 0.34/0.52
% 0.34/0.52 % type_definition_def
% 0.34/0.52 thf(fact_339_type__definition_ORep__inverse,axiom,
% 0.34/0.52 ! [Rep2: assn > produc3658429121746597890et_nat > $o,Abs2: ( produc3658429121746597890et_nat > $o ) > assn,A: set_Pr4532377907799695533_nat_o,X: assn] :
% 0.34/0.52 ( ( type_d3909072315231072503_nat_o @ Rep2 @ Abs2 @ A )
% 0.34/0.52 => ( ( Abs2 @ ( Rep2 @ X ) )
% 0.34/0.52 = X ) ) ).
% 0.34/0.52
% 0.34/0.52 % type_definition.Rep_inverse
% 0.34/0.52 thf(fact_340_type__definition_ORep__inverse,axiom,
% 0.34/0.52 ! [Rep2: code_integer > int,Abs2: int > code_integer,A: set_int,X: code_integer] :
% 0.34/0.52 ( ( type_d8366093980585677751er_int @ Rep2 @ Abs2 @ A )
% 0.34/0.52 => ( ( Abs2 @ ( Rep2 @ X ) )
% 0.34/0.52 = X ) ) ).
% 0.34/0.52
% 0.34/0.52 % type_definition.Rep_inverse
% 0.34/0.52 thf(fact_341_type__definition_ORep__inverse,axiom,
% 0.34/0.52 ! [Rep2: product_unit > $o,Abs2: $o > product_unit,A: set_o,X: product_unit] :
% 0.34/0.52 ( ( type_d6188575255521822967unit_o @ Rep2 @ Abs2 @ A )
% 0.34/0.52 => ( ( Abs2 @ ( Rep2 @ X ) )
% 0.34/0.52 = X ) ) ).
% 0.34/0.52
% 0.34/0.52 % type_definition.Rep_inverse
% 0.34/0.52 thf(fact_342_type__definition_ORep__inverse,axiom,
% 0.34/0.52 ! [Rep2: code_natural > nat,Abs2: nat > code_natural,A: set_nat,X: code_natural] :
% 0.34/0.52 ( ( type_d4410041424927559462al_nat @ Rep2 @ Abs2 @ A )
% 0.34/0.52 => ( ( Abs2 @ ( Rep2 @ X ) )
% 0.34/0.52 = X ) ) ).
% 0.34/0.52
% 0.34/0.52 % type_definition.Rep_inverse
% 0.34/0.52 thf(fact_343_type__definition_OAbs__inverse,axiom,
% 0.34/0.52 ! [Rep2: assn > produc3658429121746597890et_nat > $o,Abs2: ( produc3658429121746597890et_nat > $o ) > assn,A: set_Pr4532377907799695533_nat_o,Y: produc3658429121746597890et_nat > $o] :
% 0.34/0.52 ( ( type_d3909072315231072503_nat_o @ Rep2 @ Abs2 @ A )
% 0.34/0.52 => ( ( member6576561426505652726_nat_o @ Y @ A )
% 0.34/0.52 => ( ( Rep2 @ ( Abs2 @ Y ) )
% 0.34/0.52 = Y ) ) ) ).
% 0.34/0.52
% 0.34/0.52 % type_definition.Abs_inverse
% 0.34/0.52 thf(fact_344_type__definition_OAbs__inverse,axiom,
% 0.34/0.52 ! [Rep2: code_integer > int,Abs2: int > code_integer,A: set_int,Y: int] :
% 0.34/0.52 ( ( type_d8366093980585677751er_int @ Rep2 @ Abs2 @ A )
% 0.34/0.52 => ( ( member_int @ Y @ A )
% 0.34/0.52 => ( ( Rep2 @ ( Abs2 @ Y ) )
% 0.34/0.52 = Y ) ) ) ).
% 0.34/0.52
% 0.34/0.52 % type_definition.Abs_inverse
% 0.34/0.52 thf(fact_345_type__definition_OAbs__inverse,axiom,
% 0.34/0.52 ! [Rep2: product_unit > $o,Abs2: $o > product_unit,A: set_o,Y: $o] :
% 0.34/0.52 ( ( type_d6188575255521822967unit_o @ Rep2 @ Abs2 @ A )
% 0.34/0.52 => ( ( member_o @ Y @ A )
% 0.34/0.52 => ( ( Rep2 @ ( Abs2 @ Y ) )
% 0.34/0.52 = Y ) ) ) ).
% 0.34/0.52
% 0.34/0.52 % type_definition.Abs_inverse
% 0.34/0.52 thf(fact_346_type__definition_OAbs__inverse,axiom,
% 0.34/0.52 ! [Rep2: code_natural > nat,Abs2: nat > code_natural,A: set_nat,Y: nat] :
% 0.34/0.52 ( ( type_d4410041424927559462al_nat @ Rep2 @ Abs2 @ A )
% 0.34/0.52 => ( ( member_nat @ Y @ A )
% 0.34/0.52 => ( ( Rep2 @ ( Abs2 @ Y ) )
% 0.34/0.52 = Y ) ) ) ).
% 0.34/0.52
% 0.34/0.52 % type_definition.Abs_inverse
% 0.34/0.52 thf(fact_347_type__definition_ORep__inject,axiom,
% 0.34/0.52 ! [Rep2: assn > produc3658429121746597890et_nat > $o,Abs2: ( produc3658429121746597890et_nat > $o ) > assn,A: set_Pr4532377907799695533_nat_o,X: assn,Y: assn] :
% 0.34/0.52 ( ( type_d3909072315231072503_nat_o @ Rep2 @ Abs2 @ A )
% 0.34/0.52 => ( ( ( Rep2 @ X )
% 0.34/0.52 = ( Rep2 @ Y ) )
% 0.34/0.52 = ( X = Y ) ) ) ).
% 0.34/0.52
% 0.34/0.52 % type_definition.Rep_inject
% 0.34/0.52 thf(fact_348_type__definition_ORep__inject,axiom,
% 0.34/0.52 ! [Rep2: code_integer > int,Abs2: int > code_integer,A: set_int,X: code_integer,Y: code_integer] :
% 0.34/0.52 ( ( type_d8366093980585677751er_int @ Rep2 @ Abs2 @ A )
% 0.34/0.52 => ( ( ( Rep2 @ X )
% 0.34/0.52 = ( Rep2 @ Y ) )
% 0.34/0.52 = ( X = Y ) ) ) ).
% 0.34/0.52
% 0.34/0.52 % type_definition.Rep_inject
% 0.34/0.52 thf(fact_349_type__definition_ORep__inject,axiom,
% 0.34/0.52 ! [Rep2: product_unit > $o,Abs2: $o > product_unit,A: set_o,X: product_unit,Y: product_unit] :
% 0.34/0.52 ( ( type_d6188575255521822967unit_o @ Rep2 @ Abs2 @ A )
% 0.34/0.52 => ( ( ( Rep2 @ X )
% 0.34/0.52 = ( Rep2 @ Y ) )
% 0.34/0.52 = ( X = Y ) ) ) ).
% 0.34/0.52
% 0.34/0.52 % type_definition.Rep_inject
% 0.34/0.52 thf(fact_350_type__definition_ORep__inject,axiom,
% 0.34/0.52 ! [Rep2: code_natural > nat,Abs2: nat > code_natural,A: set_nat,X: code_natural,Y: code_natural] :
% 0.34/0.52 ( ( type_d4410041424927559462al_nat @ Rep2 @ Abs2 @ A )
% 0.34/0.52 => ( ( ( Rep2 @ X )
% 0.34/0.52 = ( Rep2 @ Y ) )
% 0.34/0.52 = ( X = Y ) ) ) ).
% 0.34/0.52
% 0.34/0.52 % type_definition.Rep_inject
% 0.34/0.52 thf(fact_351_type__definition_ORep__induct,axiom,
% 0.34/0.52 ! [Rep2: assn > produc3658429121746597890et_nat > $o,Abs2: ( produc3658429121746597890et_nat > $o ) > assn,A: set_Pr4532377907799695533_nat_o,Y: produc3658429121746597890et_nat > $o,P: ( produc3658429121746597890et_nat > $o ) > $o] :
% 0.34/0.52 ( ( type_d3909072315231072503_nat_o @ Rep2 @ Abs2 @ A )
% 0.34/0.52 => ( ( member6576561426505652726_nat_o @ Y @ A )
% 0.34/0.52 => ( ! [X3: assn] : ( P @ ( Rep2 @ X3 ) )
% 0.34/0.52 => ( P @ Y ) ) ) ) ).
% 0.34/0.52
% 0.34/0.52 % type_definition.Rep_induct
% 0.34/0.52 thf(fact_352_type__definition_ORep__induct,axiom,
% 0.34/0.52 ! [Rep2: code_integer > int,Abs2: int > code_integer,A: set_int,Y: int,P: int > $o] :
% 0.34/0.52 ( ( type_d8366093980585677751er_int @ Rep2 @ Abs2 @ A )
% 0.34/0.52 => ( ( member_int @ Y @ A )
% 0.34/0.52 => ( ! [X3: code_integer] : ( P @ ( Rep2 @ X3 ) )
% 0.34/0.52 => ( P @ Y ) ) ) ) ).
% 0.34/0.52
% 0.34/0.52 % type_definition.Rep_induct
% 0.34/0.52 thf(fact_353_type__definition_ORep__induct,axiom,
% 0.34/0.52 ! [Rep2: product_unit > $o,Abs2: $o > product_unit,A: set_o,Y: $o,P: $o > $o] :
% 0.34/0.52 ( ( type_d6188575255521822967unit_o @ Rep2 @ Abs2 @ A )
% 0.34/0.52 => ( ( member_o @ Y @ A )
% 0.34/0.52 => ( ! [X3: product_unit] : ( P @ ( Rep2 @ X3 ) )
% 0.34/0.52 => ( P @ Y ) ) ) ) ).
% 0.34/0.52
% 0.34/0.52 % type_definition.Rep_induct
% 0.34/0.52 thf(fact_354_type__definition_ORep__induct,axiom,
% 0.34/0.52 ! [Rep2: code_natural > nat,Abs2: nat > code_natural,A: set_nat,Y: nat,P: nat > $o] :
% 0.34/0.52 ( ( type_d4410041424927559462al_nat @ Rep2 @ Abs2 @ A )
% 0.34/0.52 => ( ( member_nat @ Y @ A )
% 0.34/0.52 => ( ! [X3: code_natural] : ( P @ ( Rep2 @ X3 ) )
% 0.34/0.52 => ( P @ Y ) ) ) ) ).
% 0.34/0.52
% 0.34/0.52 % type_definition.Rep_induct
% 0.34/0.52 thf(fact_355_type__definition_OAbs__inject,axiom,
% 0.34/0.52 ! [Rep2: assn > produc3658429121746597890et_nat > $o,Abs2: ( produc3658429121746597890et_nat > $o ) > assn,A: set_Pr4532377907799695533_nat_o,X: produc3658429121746597890et_nat > $o,Y: produc3658429121746597890et_nat > $o] :
% 0.34/0.52 ( ( type_d3909072315231072503_nat_o @ Rep2 @ Abs2 @ A )
% 0.34/0.52 => ( ( member6576561426505652726_nat_o @ X @ A )
% 0.34/0.52 => ( ( member6576561426505652726_nat_o @ Y @ A )
% 0.34/0.52 => ( ( ( Abs2 @ X )
% 0.34/0.52 = ( Abs2 @ Y ) )
% 0.34/0.52 = ( X = Y ) ) ) ) ) ).
% 0.34/0.52
% 0.34/0.52 % type_definition.Abs_inject
% 0.34/0.52 thf(fact_356_type__definition_OAbs__inject,axiom,
% 0.34/0.52 ! [Rep2: code_integer > int,Abs2: int > code_integer,A: set_int,X: int,Y: int] :
% 0.34/0.52 ( ( type_d8366093980585677751er_int @ Rep2 @ Abs2 @ A )
% 0.34/0.52 => ( ( member_int @ X @ A )
% 0.34/0.52 => ( ( member_int @ Y @ A )
% 0.34/0.52 => ( ( ( Abs2 @ X )
% 0.34/0.52 = ( Abs2 @ Y ) )
% 0.34/0.52 = ( X = Y ) ) ) ) ) ).
% 0.34/0.52
% 0.34/0.52 % type_definition.Abs_inject
% 0.34/0.52 thf(fact_357_type__definition_OAbs__inject,axiom,
% 0.34/0.52 ! [Rep2: product_unit > $o,Abs2: $o > product_unit,A: set_o,X: $o,Y: $o] :
% 0.34/0.52 ( ( type_d6188575255521822967unit_o @ Rep2 @ Abs2 @ A )
% 0.34/0.52 => ( ( member_o @ X @ A )
% 0.34/0.52 => ( ( member_o @ Y @ A )
% 0.34/0.52 => ( ( ( Abs2 @ X )
% 0.34/0.52 = ( Abs2 @ Y ) )
% 0.34/0.52 = ( X = Y ) ) ) ) ) ).
% 0.34/0.52
% 0.34/0.52 % type_definition.Abs_inject
% 0.34/0.52 thf(fact_358_type__definition_OAbs__inject,axiom,
% 0.34/0.52 ! [Rep2: code_natural > nat,Abs2: nat > code_natural,A: set_nat,X: nat,Y: nat] :
% 0.34/0.52 ( ( type_d4410041424927559462al_nat @ Rep2 @ Abs2 @ A )
% 0.34/0.52 => ( ( member_nat @ X @ A )
% 0.34/0.52 => ( ( member_nat @ Y @ A )
% 0.34/0.52 => ( ( ( Abs2 @ X )
% 0.34/0.52 = ( Abs2 @ Y ) )
% 0.34/0.52 = ( X = Y ) ) ) ) ) ).
% 0.34/0.52
% 0.34/0.52 % type_definition.Abs_inject
% 0.34/0.52 thf(fact_359_pure__assn__eq__conv,axiom,
% 0.34/0.52 ! [P: $o,Q2: $o] :
% 0.34/0.52 ( ( ( pure_assn @ P )
% 0.34/0.52 = ( pure_assn @ Q2 ) )
% 0.34/0.52 = ( P = Q2 ) ) ).
% 0.34/0.52
% 0.34/0.52 % pure_assn_eq_conv
% 0.34/0.52 thf(fact_360_mult_Oright__neutral,axiom,
% 0.34/0.52 ! [A2: code_integer] :
% 0.34/0.52 ( ( times_3573771949741848930nteger @ A2 @ one_one_Code_integer )
% 0.34/0.52 = A2 ) ).
% 0.34/0.52
% 0.34/0.52 % mult.right_neutral
% 0.34/0.52 thf(fact_361_mult_Oright__neutral,axiom,
% 0.34/0.52 ! [A2: code_natural] :
% 0.34/0.52 ( ( times_2397367101498566445atural @ A2 @ one_one_Code_natural )
% 0.34/0.52 = A2 ) ).
% 0.34/0.52
% 0.34/0.52 % mult.right_neutral
% 0.34/0.52 thf(fact_362_mult_Oright__neutral,axiom,
% 0.34/0.52 ! [A2: assn] :
% 0.34/0.52 ( ( times_times_assn @ A2 @ one_one_assn )
% 0.34/0.52 = A2 ) ).
% 0.34/0.52
% 0.34/0.52 % mult.right_neutral
% 0.34/0.52 thf(fact_363_mult_Oright__neutral,axiom,
% 0.34/0.52 ! [A2: rat] :
% 0.34/0.52 ( ( times_times_rat @ A2 @ one_one_rat )
% 0.34/0.52 = A2 ) ).
% 0.34/0.52
% 0.34/0.52 % mult.right_neutral
% 0.34/0.52 thf(fact_364_mult_Oright__neutral,axiom,
% 0.34/0.52 ! [A2: nat] :
% 0.34/0.52 ( ( times_times_nat @ A2 @ one_one_nat )
% 0.34/0.52 = A2 ) ).
% 0.34/0.52
% 0.34/0.52 % mult.right_neutral
% 0.34/0.52 thf(fact_365_mult_Oright__neutral,axiom,
% 0.34/0.52 ! [A2: int] :
% 0.34/0.52 ( ( times_times_int @ A2 @ one_one_int )
% 0.34/0.52 = A2 ) ).
% 0.34/0.52
% 0.34/0.52 % mult.right_neutral
% 0.34/0.52 thf(fact_366_mult__1,axiom,
% 0.34/0.52 ! [A2: code_integer] :
% 0.34/0.52 ( ( times_3573771949741848930nteger @ one_one_Code_integer @ A2 )
% 0.34/0.52 = A2 ) ).
% 0.34/0.52
% 0.34/0.52 % mult_1
% 0.34/0.52 thf(fact_367_mult__1,axiom,
% 0.34/0.52 ! [A2: code_natural] :
% 0.34/0.52 ( ( times_2397367101498566445atural @ one_one_Code_natural @ A2 )
% 0.34/0.52 = A2 ) ).
% 0.34/0.52
% 0.34/0.52 % mult_1
% 0.34/0.52 thf(fact_368_mult__1,axiom,
% 0.34/0.52 ! [A2: assn] :
% 0.34/0.52 ( ( times_times_assn @ one_one_assn @ A2 )
% 0.34/0.52 = A2 ) ).
% 0.34/0.52
% 0.34/0.52 % mult_1
% 0.34/0.52 thf(fact_369_mult__1,axiom,
% 0.34/0.52 ! [A2: rat] :
% 0.34/0.52 ( ( times_times_rat @ one_one_rat @ A2 )
% 0.34/0.52 = A2 ) ).
% 0.34/0.52
% 0.34/0.52 % mult_1
% 0.34/0.52 thf(fact_370_mult__1,axiom,
% 0.34/0.52 ! [A2: nat] :
% 0.34/0.52 ( ( times_times_nat @ one_one_nat @ A2 )
% 0.34/0.52 = A2 ) ).
% 0.34/0.52
% 0.34/0.52 % mult_1
% 0.34/0.52 thf(fact_371_mult__1,axiom,
% 0.34/0.52 ! [A2: int] :
% 0.34/0.52 ( ( times_times_int @ one_one_int @ A2 )
% 0.34/0.52 = A2 ) ).
% 0.34/0.52
% 0.34/0.52 % mult_1
% 0.34/0.52 thf(fact_372_merge__pure__star,axiom,
% 0.34/0.52 ! [A2: $o,B: $o] :
% 0.34/0.52 ( ( times_times_assn @ ( pure_assn @ A2 ) @ ( pure_assn @ B ) )
% 0.34/0.52 = ( pure_assn
% 0.34/0.52 @ ( A2
% 0.34/0.52 & B ) ) ) ).
% 0.34/0.52
% 0.34/0.52 % merge_pure_star
% 0.34/0.52 thf(fact_373_star__false__right,axiom,
% 0.34/0.52 ! [P: assn] :
% 0.34/0.52 ( ( times_times_assn @ P @ bot_bot_assn )
% 0.34/0.52 = bot_bot_assn ) ).
% 0.34/0.52
% 0.34/0.52 % star_false_right
% 0.34/0.52 thf(fact_374_star__false__left,axiom,
% 0.34/0.52 ! [P: assn] :
% 0.34/0.52 ( ( times_times_assn @ bot_bot_assn @ P )
% 0.34/0.52 = bot_bot_assn ) ).
% 0.34/0.52
% 0.34/0.52 % star_false_left
% 0.34/0.52 thf(fact_375_pure__false,axiom,
% 0.34/0.52 ( ( pure_assn @ $false )
% 0.34/0.52 = bot_bot_assn ) ).
% 0.34/0.52
% 0.34/0.52 % pure_false
% 0.34/0.52 thf(fact_376_pure__assn__eq__false__iff,axiom,
% 0.34/0.52 ! [P: $o] :
% 0.34/0.52 ( ( ( pure_assn @ P )
% 0.34/0.52 = bot_bot_assn )
% 0.34/0.52 = ( ~ P ) ) ).
% 0.34/0.52
% 0.34/0.52 % pure_assn_eq_false_iff
% 0.34/0.52 thf(fact_377_is__pure__assn__starI,axiom,
% 0.34/0.52 ! [A2: assn,B: assn] :
% 0.34/0.52 ( ( is_pure_assn @ A2 )
% 0.34/0.52 => ( ( is_pure_assn @ B )
% 0.34/0.52 => ( is_pure_assn @ ( times_times_assn @ A2 @ B ) ) ) ) ).
% 0.34/0.52
% 0.34/0.52 % is_pure_assn_starI
% 0.34/0.52 thf(fact_378_is__pure__assn__pure,axiom,
% 0.34/0.52 ! [P: $o] : ( is_pure_assn @ ( pure_assn @ P ) ) ).
% 0.34/0.52
% 0.34/0.52 % is_pure_assn_pure
% 0.34/0.52 thf(fact_379_pure__true,axiom,
% 0.34/0.52 ( ( pure_assn @ $true )
% 0.34/0.52 = one_one_assn ) ).
% 0.34/0.52
% 0.34/0.52 % pure_true
% 0.34/0.52 thf(fact_380_pure__assn__eq__emp__iff,axiom,
% 0.34/0.52 ! [P: $o] :
% 0.34/0.52 ( ( ( pure_assn @ P )
% 0.34/0.52 = one_one_assn )
% 0.34/0.52 = P ) ).
% 0.34/0.52
% 0.34/0.52 % pure_assn_eq_emp_iff
% 0.34/0.52 thf(fact_381_assn__times__comm,axiom,
% 0.34/0.52 ( times_times_assn
% 0.34/0.52 = ( ^ [P2: assn,Q: assn] : ( times_times_assn @ Q @ P2 ) ) ) ).
% 0.34/0.52
% 0.34/0.52 % assn_times_comm
% 0.34/0.52 thf(fact_382_assn__times__assoc,axiom,
% 0.34/0.52 ! [P: assn,Q2: assn,R3: assn] :
% 0.34/0.52 ( ( times_times_assn @ ( times_times_assn @ P @ Q2 ) @ R3 )
% 0.34/0.52 = ( times_times_assn @ P @ ( times_times_assn @ Q2 @ R3 ) ) ) ).
% 0.34/0.52
% 0.34/0.52 % assn_times_assoc
% 0.34/0.52 thf(fact_383_ab__semigroup__mult__class_Omult_Oleft__commute,axiom,
% 0.34/0.52 ! [B: assn,A2: assn,C: assn] :
% 0.34/0.52 ( ( times_times_assn @ B @ ( times_times_assn @ A2 @ C ) )
% 0.34/0.52 = ( times_times_assn @ A2 @ ( times_times_assn @ B @ C ) ) ) ).
% 0.34/0.52
% 0.34/0.52 % ab_semigroup_mult_class.mult.left_commute
% 0.34/0.52 thf(fact_384_ab__semigroup__mult__class_Omult_Oleft__commute,axiom,
% 0.34/0.52 ! [B: rat,A2: rat,C: rat] :
% 0.34/0.52 ( ( times_times_rat @ B @ ( times_times_rat @ A2 @ C ) )
% 0.34/0.52 = ( times_times_rat @ A2 @ ( times_times_rat @ B @ C ) ) ) ).
% 0.34/0.52
% 0.34/0.52 % ab_semigroup_mult_class.mult.left_commute
% 0.34/0.52 thf(fact_385_ab__semigroup__mult__class_Omult_Oleft__commute,axiom,
% 0.34/0.52 ! [B: nat,A2: nat,C: nat] :
% 0.34/0.52 ( ( times_times_nat @ B @ ( times_times_nat @ A2 @ C ) )
% 0.34/0.52 = ( times_times_nat @ A2 @ ( times_times_nat @ B @ C ) ) ) ).
% 0.34/0.52
% 0.34/0.52 % ab_semigroup_mult_class.mult.left_commute
% 0.34/0.52 thf(fact_386_ab__semigroup__mult__class_Omult_Oleft__commute,axiom,
% 0.34/0.52 ! [B: int,A2: int,C: int] :
% 0.34/0.52 ( ( times_times_int @ B @ ( times_times_int @ A2 @ C ) )
% 0.34/0.52 = ( times_times_int @ A2 @ ( times_times_int @ B @ C ) ) ) ).
% 0.34/0.52
% 0.34/0.52 % ab_semigroup_mult_class.mult.left_commute
% 0.34/0.52 thf(fact_387_ab__semigroup__mult__class_Omult_Ocommute,axiom,
% 0.34/0.52 ( times_times_assn
% 0.34/0.52 = ( ^ [A6: assn,B6: assn] : ( times_times_assn @ B6 @ A6 ) ) ) ).
% 0.34/0.52
% 0.34/0.52 % ab_semigroup_mult_class.mult.commute
% 0.34/0.52 thf(fact_388_ab__semigroup__mult__class_Omult_Ocommute,axiom,
% 0.34/0.52 ( times_times_rat
% 0.34/0.52 = ( ^ [A6: rat,B6: rat] : ( times_times_rat @ B6 @ A6 ) ) ) ).
% 0.34/0.52
% 0.34/0.52 % ab_semigroup_mult_class.mult.commute
% 0.34/0.52 thf(fact_389_ab__semigroup__mult__class_Omult_Ocommute,axiom,
% 0.34/0.52 ( times_times_nat
% 0.34/0.52 = ( ^ [A6: nat,B6: nat] : ( times_times_nat @ B6 @ A6 ) ) ) ).
% 0.34/0.52
% 0.34/0.52 % ab_semigroup_mult_class.mult.commute
% 0.34/0.52 thf(fact_390_ab__semigroup__mult__class_Omult_Ocommute,axiom,
% 0.34/0.52 ( times_times_int
% 0.34/0.52 = ( ^ [A6: int,B6: int] : ( times_times_int @ B6 @ A6 ) ) ) ).
% 0.34/0.52
% 0.34/0.52 % ab_semigroup_mult_class.mult.commute
% 0.34/0.52 thf(fact_391_mult_Oassoc,axiom,
% 0.34/0.52 ! [A2: assn,B: assn,C: assn] :
% 0.34/0.52 ( ( times_times_assn @ ( times_times_assn @ A2 @ B ) @ C )
% 0.34/0.52 = ( times_times_assn @ A2 @ ( times_times_assn @ B @ C ) ) ) ).
% 0.34/0.52
% 0.34/0.52 % mult.assoc
% 0.34/0.52 thf(fact_392_mult_Oassoc,axiom,
% 0.34/0.52 ! [A2: rat,B: rat,C: rat] :
% 0.34/0.52 ( ( times_times_rat @ ( times_times_rat @ A2 @ B ) @ C )
% 0.34/0.52 = ( times_times_rat @ A2 @ ( times_times_rat @ B @ C ) ) ) ).
% 0.34/0.52
% 0.34/0.52 % mult.assoc
% 0.34/0.52 thf(fact_393_mult_Oassoc,axiom,
% 0.34/0.52 ! [A2: nat,B: nat,C: nat] :
% 0.34/0.52 ( ( times_times_nat @ ( times_times_nat @ A2 @ B ) @ C )
% 0.34/0.52 = ( times_times_nat @ A2 @ ( times_times_nat @ B @ C ) ) ) ).
% 0.34/0.52
% 0.34/0.52 % mult.assoc
% 0.34/0.52 thf(fact_394_mult_Oassoc,axiom,
% 0.34/0.52 ! [A2: int,B: int,C: int] :
% 0.34/0.52 ( ( times_times_int @ ( times_times_int @ A2 @ B ) @ C )
% 0.34/0.52 = ( times_times_int @ A2 @ ( times_times_int @ B @ C ) ) ) ).
% 0.34/0.52
% 0.34/0.52 % mult.assoc
% 0.34/0.52 thf(fact_395_comm__monoid__mult__class_Omult__1,axiom,
% 0.34/0.52 ! [A2: code_integer] :
% 0.34/0.52 ( ( times_3573771949741848930nteger @ one_one_Code_integer @ A2 )
% 0.34/0.52 = A2 ) ).
% 0.34/0.52
% 0.34/0.52 % comm_monoid_mult_class.mult_1
% 0.34/0.52 thf(fact_396_comm__monoid__mult__class_Omult__1,axiom,
% 0.34/0.52 ! [A2: code_natural] :
% 0.34/0.52 ( ( times_2397367101498566445atural @ one_one_Code_natural @ A2 )
% 0.34/0.52 = A2 ) ).
% 0.34/0.52
% 0.34/0.52 % comm_monoid_mult_class.mult_1
% 0.34/0.52 thf(fact_397_comm__monoid__mult__class_Omult__1,axiom,
% 0.34/0.52 ! [A2: assn] :
% 0.34/0.52 ( ( times_times_assn @ one_one_assn @ A2 )
% 0.34/0.52 = A2 ) ).
% 0.34/0.52
% 0.34/0.52 % comm_monoid_mult_class.mult_1
% 0.34/0.52 thf(fact_398_comm__monoid__mult__class_Omult__1,axiom,
% 0.34/0.52 ! [A2: rat] :
% 0.34/0.52 ( ( times_times_rat @ one_one_rat @ A2 )
% 0.34/0.52 = A2 ) ).
% 0.34/0.52
% 0.34/0.52 % comm_monoid_mult_class.mult_1
% 0.34/0.52 thf(fact_399_comm__monoid__mult__class_Omult__1,axiom,
% 0.34/0.52 ! [A2: nat] :
% 0.34/0.52 ( ( times_times_nat @ one_one_nat @ A2 )
% 0.34/0.52 = A2 ) ).
% 0.34/0.52
% 0.34/0.52 % comm_monoid_mult_class.mult_1
% 0.34/0.52 thf(fact_400_comm__monoid__mult__class_Omult__1,axiom,
% 0.34/0.52 ! [A2: int] :
% 0.34/0.52 ( ( times_times_int @ one_one_int @ A2 )
% 0.34/0.52 = A2 ) ).
% 0.34/0.52
% 0.34/0.52 % comm_monoid_mult_class.mult_1
% 0.34/0.52 thf(fact_401_mult_Ocomm__neutral,axiom,
% 0.34/0.52 ! [A2: code_integer] :
% 0.34/0.52 ( ( times_3573771949741848930nteger @ A2 @ one_one_Code_integer )
% 0.34/0.52 = A2 ) ).
% 0.34/0.52
% 0.34/0.52 % mult.comm_neutral
% 0.34/0.52 thf(fact_402_mult_Ocomm__neutral,axiom,
% 0.34/0.52 ! [A2: code_natural] :
% 0.34/0.52 ( ( times_2397367101498566445atural @ A2 @ one_one_Code_natural )
% 0.34/0.52 = A2 ) ).
% 0.34/0.52
% 0.34/0.52 % mult.comm_neutral
% 0.34/0.52 thf(fact_403_mult_Ocomm__neutral,axiom,
% 0.34/0.52 ! [A2: assn] :
% 0.34/0.52 ( ( times_times_assn @ A2 @ one_one_assn )
% 0.34/0.52 = A2 ) ).
% 0.34/0.52
% 0.34/0.52 % mult.comm_neutral
% 0.34/0.52 thf(fact_404_mult_Ocomm__neutral,axiom,
% 0.34/0.52 ! [A2: rat] :
% 0.34/0.52 ( ( times_times_rat @ A2 @ one_one_rat )
% 0.34/0.52 = A2 ) ).
% 0.34/0.52
% 0.34/0.52 % mult.comm_neutral
% 0.34/0.52 thf(fact_405_mult_Ocomm__neutral,axiom,
% 0.34/0.52 ! [A2: nat] :
% 0.34/0.52 ( ( times_times_nat @ A2 @ one_one_nat )
% 0.34/0.52 = A2 ) ).
% 0.34/0.52
% 0.34/0.52 % mult.comm_neutral
% 0.34/0.52 thf(fact_406_mult_Ocomm__neutral,axiom,
% 0.34/0.52 ! [A2: int] :
% 0.34/0.52 ( ( times_times_int @ A2 @ one_one_int )
% 0.34/0.52 = A2 ) ).
% 0.34/0.52
% 0.34/0.52 % mult.comm_neutral
% 0.34/0.52 thf(fact_407_mod__starE,axiom,
% 0.34/0.52 ! [A2: assn,B: assn,H2: produc3658429121746597890et_nat] :
% 0.34/0.52 ( ( rep_assn @ ( times_times_assn @ A2 @ B ) @ H2 )
% 0.34/0.52 => ~ ( ? [X_1: produc3658429121746597890et_nat] : ( rep_assn @ A2 @ X_1 )
% 0.34/0.52 => ! [H_2: produc3658429121746597890et_nat] :
% 0.34/0.52 ~ ( rep_assn @ B @ H_2 ) ) ) ).
% 0.34/0.52
% 0.34/0.52 % mod_starE
% 0.34/0.52 thf(fact_408_mod__starD,axiom,
% 0.34/0.52 ! [A: assn,B4: assn,H2: produc3658429121746597890et_nat] :
% 0.34/0.52 ( ( rep_assn @ ( times_times_assn @ A @ B4 ) @ H2 )
% 0.34/0.52 => ? [H1: produc3658429121746597890et_nat,H22: produc3658429121746597890et_nat] :
% 0.34/0.52 ( ( rep_assn @ A @ H1 )
% 0.34/0.52 & ( rep_assn @ B4 @ H22 ) ) ) ).
% 0.34/0.52
% 0.34/0.52 % mod_starD
% 0.34/0.52 thf(fact_409_assn__one__left,axiom,
% 0.34/0.52 ! [P: assn] :
% 0.34/0.52 ( ( times_times_assn @ one_one_assn @ P )
% 0.34/0.52 = P ) ).
% 0.34/0.52
% 0.34/0.52 % assn_one_left
% 0.34/0.52 thf(fact_410_is__pure__assnE,axiom,
% 0.34/0.52 ! [A2: assn] :
% 0.34/0.52 ( ( is_pure_assn @ A2 )
% 0.34/0.52 => ~ ! [P4: $o] :
% 0.34/0.52 ( A2
% 0.34/0.52 != ( pure_assn @ P4 ) ) ) ).
% 0.34/0.52
% 0.34/0.52 % is_pure_assnE
% 0.34/0.52 thf(fact_411_is__pure__assn__def,axiom,
% 0.34/0.52 ( is_pure_assn
% 0.34/0.52 = ( ^ [A6: assn] :
% 0.34/0.52 ? [P2: $o] :
% 0.34/0.52 ( A6
% 0.34/0.52 = ( pure_assn @ P2 ) ) ) ) ).
% 0.34/0.52
% 0.34/0.52 % is_pure_assn_def
% 0.34/0.52 thf(fact_412_type__definition_ORep,axiom,
% 0.34/0.52 ! [Rep2: assn > produc3658429121746597890et_nat > $o,Abs2: ( produc3658429121746597890et_nat > $o ) > assn,A: set_Pr4532377907799695533_nat_o,X: assn] :
% 0.34/0.52 ( ( type_d3909072315231072503_nat_o @ Rep2 @ Abs2 @ A )
% 0.34/0.52 => ( member6576561426505652726_nat_o @ ( Rep2 @ X ) @ A ) ) ).
% 0.34/0.52
% 0.34/0.52 % type_definition.Rep
% 0.34/0.52 thf(fact_413_type__definition_ORep,axiom,
% 0.34/0.52 ! [Rep2: code_integer > int,Abs2: int > code_integer,A: set_int,X: code_integer] :
% 0.34/0.52 ( ( type_d8366093980585677751er_int @ Rep2 @ Abs2 @ A )
% 0.34/0.52 => ( member_int @ ( Rep2 @ X ) @ A ) ) ).
% 0.34/0.52
% 0.34/0.52 % type_definition.Rep
% 0.34/0.52 thf(fact_414_type__definition_ORep,axiom,
% 0.34/0.52 ! [Rep2: product_unit > $o,Abs2: $o > product_unit,A: set_o,X: product_unit] :
% 0.34/0.52 ( ( type_d6188575255521822967unit_o @ Rep2 @ Abs2 @ A )
% 0.34/0.52 => ( member_o @ ( Rep2 @ X ) @ A ) ) ).
% 0.34/0.52
% 0.34/0.52 % type_definition.Rep
% 0.34/0.52 thf(fact_415_type__definition_ORep,axiom,
% 0.34/0.52 ! [Rep2: code_natural > nat,Abs2: nat > code_natural,A: set_nat,X: code_natural] :
% 0.34/0.52 ( ( type_d4410041424927559462al_nat @ Rep2 @ Abs2 @ A )
% 0.34/0.52 => ( member_nat @ ( Rep2 @ X ) @ A ) ) ).
% 0.34/0.52
% 0.34/0.52 % type_definition.Rep
% 0.34/0.52 thf(fact_416_type__definition_Ointro,axiom,
% 0.34/0.52 ! [Rep2: assn > produc3658429121746597890et_nat > $o,A: set_Pr4532377907799695533_nat_o,Abs2: ( produc3658429121746597890et_nat > $o ) > assn] :
% 0.34/0.52 ( ! [X3: assn] : ( member6576561426505652726_nat_o @ ( Rep2 @ X3 ) @ A )
% 0.34/0.52 => ( ! [X3: assn] :
% 0.34/0.52 ( ( Abs2 @ ( Rep2 @ X3 ) )
% 0.34/0.52 = X3 )
% 0.34/0.52 => ( ! [Y2: produc3658429121746597890et_nat > $o] :
% 0.34/0.52 ( ( member6576561426505652726_nat_o @ Y2 @ A )
% 0.34/0.52 => ( ( Rep2 @ ( Abs2 @ Y2 ) )
% 0.34/0.52 = Y2 ) )
% 0.34/0.52 => ( type_d3909072315231072503_nat_o @ Rep2 @ Abs2 @ A ) ) ) ) ).
% 0.34/0.52
% 0.34/0.52 % type_definition.intro
% 0.34/0.52 thf(fact_417_type__definition_Ointro,axiom,
% 0.34/0.52 ! [Rep2: code_integer > int,A: set_int,Abs2: int > code_integer] :
% 0.34/0.52 ( ! [X3: code_integer] : ( member_int @ ( Rep2 @ X3 ) @ A )
% 0.34/0.52 => ( ! [X3: code_integer] :
% 0.34/0.52 ( ( Abs2 @ ( Rep2 @ X3 ) )
% 0.34/0.52 = X3 )
% 0.34/0.52 => ( ! [Y2: int] :
% 0.34/0.52 ( ( member_int @ Y2 @ A )
% 0.34/0.52 => ( ( Rep2 @ ( Abs2 @ Y2 ) )
% 0.34/0.52 = Y2 ) )
% 0.34/0.52 => ( type_d8366093980585677751er_int @ Rep2 @ Abs2 @ A ) ) ) ) ).
% 0.34/0.52
% 0.34/0.52 % type_definition.intro
% 0.34/0.52 thf(fact_418_type__definition_Ointro,axiom,
% 0.34/0.52 ! [Rep2: product_unit > $o,A: set_o,Abs2: $o > product_unit] :
% 0.34/0.52 ( ! [X3: product_unit] : ( member_o @ ( Rep2 @ X3 ) @ A )
% 0.34/0.52 => ( ! [X3: product_unit] :
% 0.34/0.52 ( ( Abs2 @ ( Rep2 @ X3 ) )
% 0.34/0.52 = X3 )
% 0.34/0.52 => ( ! [Y2: $o] :
% 0.34/0.52 ( ( member_o @ Y2 @ A )
% 0.34/0.52 => ( ( Rep2 @ ( Abs2 @ Y2 ) )
% 0.34/0.52 = Y2 ) )
% 0.34/0.52 => ( type_d6188575255521822967unit_o @ Rep2 @ Abs2 @ A ) ) ) ) ).
% 0.34/0.52
% 0.34/0.52 % type_definition.intro
% 0.34/0.52 thf(fact_419_type__definition_Ointro,axiom,
% 0.34/0.52 ! [Rep2: code_natural > nat,A: set_nat,Abs2: nat > code_natural] :
% 0.34/0.52 ( ! [X3: code_natural] : ( member_nat @ ( Rep2 @ X3 ) @ A )
% 0.34/0.52 => ( ! [X3: code_natural] :
% 0.34/0.52 ( ( Abs2 @ ( Rep2 @ X3 ) )
% 0.34/0.52 = X3 )
% 0.34/0.52 => ( ! [Y2: nat] :
% 0.34/0.52 ( ( member_nat @ Y2 @ A )
% 0.34/0.52 => ( ( Rep2 @ ( Abs2 @ Y2 ) )
% 0.34/0.52 = Y2 ) )
% 0.34/0.52 => ( type_d4410041424927559462al_nat @ Rep2 @ Abs2 @ A ) ) ) ) ).
% 0.34/0.52
% 0.34/0.52 % type_definition.intro
% 0.34/0.52 thf(fact_420_type__definition_OAbs__cases,axiom,
% 0.34/0.52 ! [Rep2: assn > produc3658429121746597890et_nat > $o,Abs2: ( produc3658429121746597890et_nat > $o ) > assn,A: set_Pr4532377907799695533_nat_o,X: assn] :
% 0.34/0.52 ( ( type_d3909072315231072503_nat_o @ Rep2 @ Abs2 @ A )
% 0.34/0.52 => ~ ! [Y2: produc3658429121746597890et_nat > $o] :
% 0.34/0.52 ( ( X
% 0.34/0.52 = ( Abs2 @ Y2 ) )
% 0.34/0.52 => ~ ( member6576561426505652726_nat_o @ Y2 @ A ) ) ) ).
% 0.34/0.52
% 0.34/0.52 % type_definition.Abs_cases
% 0.34/0.52 thf(fact_421_type__definition_OAbs__cases,axiom,
% 0.34/0.52 ! [Rep2: code_integer > int,Abs2: int > code_integer,A: set_int,X: code_integer] :
% 0.34/0.52 ( ( type_d8366093980585677751er_int @ Rep2 @ Abs2 @ A )
% 0.34/0.52 => ~ ! [Y2: int] :
% 0.34/0.52 ( ( X
% 0.34/0.52 = ( Abs2 @ Y2 ) )
% 0.34/0.52 => ~ ( member_int @ Y2 @ A ) ) ) ).
% 0.34/0.52
% 0.34/0.52 % type_definition.Abs_cases
% 0.34/0.52 thf(fact_422_type__definition_OAbs__cases,axiom,
% 0.34/0.52 ! [Rep2: product_unit > $o,Abs2: $o > product_unit,A: set_o,X: product_unit] :
% 0.34/0.52 ( ( type_d6188575255521822967unit_o @ Rep2 @ Abs2 @ A )
% 0.34/0.52 => ~ ! [Y2: $o] :
% 0.34/0.52 ( ( X
% 0.34/0.52 = ( Abs2 @ Y2 ) )
% 0.34/0.52 => ~ ( member_o @ Y2 @ A ) ) ) ).
% 0.34/0.52
% 0.34/0.52 % type_definition.Abs_cases
% 0.34/0.52 thf(fact_423_type__definition_OAbs__cases,axiom,
% 0.34/0.52 ! [Rep2: code_natural > nat,Abs2: nat > code_natural,A: set_nat,X: code_natural] :
% 0.34/0.52 ( ( type_d4410041424927559462al_nat @ Rep2 @ Abs2 @ A )
% 0.34/0.52 => ~ ! [Y2: nat] :
% 0.34/0.52 ( ( X
% 0.34/0.52 = ( Abs2 @ Y2 ) )
% 0.34/0.52 => ~ ( member_nat @ Y2 @ A ) ) ) ).
% 0.34/0.52
% 0.34/0.52 % type_definition.Abs_cases
% 0.34/0.52 thf(fact_424_type__definition_ORep__cases,axiom,
% 0.34/0.52 ! [Rep2: assn > produc3658429121746597890et_nat > $o,Abs2: ( produc3658429121746597890et_nat > $o ) > assn,A: set_Pr4532377907799695533_nat_o,Y: produc3658429121746597890et_nat > $o] :
% 0.34/0.52 ( ( type_d3909072315231072503_nat_o @ Rep2 @ Abs2 @ A )
% 0.34/0.52 => ( ( member6576561426505652726_nat_o @ Y @ A )
% 0.34/0.52 => ~ ! [X3: assn] :
% 0.34/0.52 ( Y
% 0.34/0.52 != ( Rep2 @ X3 ) ) ) ) ).
% 0.34/0.52
% 0.34/0.52 % type_definition.Rep_cases
% 0.34/0.52 thf(fact_425_type__definition_ORep__cases,axiom,
% 0.34/0.52 ! [Rep2: code_integer > int,Abs2: int > code_integer,A: set_int,Y: int] :
% 0.34/0.52 ( ( type_d8366093980585677751er_int @ Rep2 @ Abs2 @ A )
% 0.34/0.52 => ( ( member_int @ Y @ A )
% 0.34/0.52 => ~ ! [X3: code_integer] :
% 0.34/0.52 ( Y
% 0.34/0.52 != ( Rep2 @ X3 ) ) ) ) ).
% 0.34/0.52
% 0.34/0.52 % type_definition.Rep_cases
% 0.34/0.52 thf(fact_426_type__definition_ORep__cases,axiom,
% 0.34/0.52 ! [Rep2: product_unit > $o,Abs2: $o > product_unit,A: set_o,Y: $o] :
% 0.34/0.52 ( ( type_d6188575255521822967unit_o @ Rep2 @ Abs2 @ A )
% 0.34/0.52 => ( ( member_o @ Y @ A )
% 0.34/0.52 => ~ ! [X3: product_unit] :
% 0.34/0.52 ( Y
% 0.34/0.52 = ( ~ ( Rep2 @ X3 ) ) ) ) ) ).
% 0.34/0.52
% 0.34/0.52 % type_definition.Rep_cases
% 0.34/0.52 thf(fact_427_type__definition_ORep__cases,axiom,
% 0.34/0.52 ! [Rep2: code_natural > nat,Abs2: nat > code_natural,A: set_nat,Y: nat] :
% 0.34/0.52 ( ( type_d4410041424927559462al_nat @ Rep2 @ Abs2 @ A )
% 0.34/0.52 => ( ( member_nat @ Y @ A )
% 0.34/0.52 => ~ ! [X3: code_natural] :
% 0.34/0.52 ( Y
% 0.34/0.52 != ( Rep2 @ X3 ) ) ) ) ).
% 0.34/0.52
% 0.34/0.52 % type_definition.Rep_cases
% 0.34/0.52 thf(fact_428_type__definition_OAbs__induct,axiom,
% 0.34/0.52 ! [Rep2: assn > produc3658429121746597890et_nat > $o,Abs2: ( produc3658429121746597890et_nat > $o ) > assn,A: set_Pr4532377907799695533_nat_o,P: assn > $o,X: assn] :
% 0.34/0.52 ( ( type_d3909072315231072503_nat_o @ Rep2 @ Abs2 @ A )
% 0.34/0.52 => ( ! [Y2: produc3658429121746597890et_nat > $o] :
% 0.34/0.52 ( ( member6576561426505652726_nat_o @ Y2 @ A )
% 0.34/0.52 => ( P @ ( Abs2 @ Y2 ) ) )
% 0.34/0.52 => ( P @ X ) ) ) ).
% 0.34/0.52
% 0.34/0.52 % type_definition.Abs_induct
% 0.34/0.52 thf(fact_429_type__definition_OAbs__induct,axiom,
% 0.34/0.52 ! [Rep2: code_integer > int,Abs2: int > code_integer,A: set_int,P: code_integer > $o,X: code_integer] :
% 0.34/0.52 ( ( type_d8366093980585677751er_int @ Rep2 @ Abs2 @ A )
% 0.34/0.52 => ( ! [Y2: int] :
% 0.34/0.52 ( ( member_int @ Y2 @ A )
% 0.34/0.52 => ( P @ ( Abs2 @ Y2 ) ) )
% 0.34/0.52 => ( P @ X ) ) ) ).
% 0.34/0.52
% 0.34/0.52 % type_definition.Abs_induct
% 0.34/0.52 thf(fact_430_type__definition_OAbs__induct,axiom,
% 0.34/0.52 ! [Rep2: product_unit > $o,Abs2: $o > product_unit,A: set_o,P: product_unit > $o,X: product_unit] :
% 0.34/0.52 ( ( type_d6188575255521822967unit_o @ Rep2 @ Abs2 @ A )
% 0.34/0.52 => ( ! [Y2: $o] :
% 0.34/0.52 ( ( member_o @ Y2 @ A )
% 0.34/0.52 => ( P @ ( Abs2 @ Y2 ) ) )
% 0.34/0.52 => ( P @ X ) ) ) ).
% 0.34/0.52
% 0.34/0.52 % type_definition.Abs_induct
% 0.34/0.52 thf(fact_431_type__definition_OAbs__induct,axiom,
% 0.34/0.52 ! [Rep2: code_natural > nat,Abs2: nat > code_natural,A: set_nat,P: code_natural > $o,X: code_natural] :
% 0.34/0.52 ( ( type_d4410041424927559462al_nat @ Rep2 @ Abs2 @ A )
% 0.34/0.52 => ( ! [Y2: nat] :
% 0.34/0.52 ( ( member_nat @ Y2 @ A )
% 0.34/0.52 => ( P @ ( Abs2 @ Y2 ) ) )
% 0.34/0.52 => ( P @ X ) ) ) ).
% 0.34/0.52
% 0.34/0.52 % type_definition.Abs_induct
% 0.34/0.52 thf(fact_432_total__on__singleton,axiom,
% 0.34/0.52 ! [X: product_unit] : ( total_5665700600833547665t_unit @ ( insert_Product_unit @ X @ bot_bo3957492148770167129t_unit ) @ ( insert4616891024742550839t_unit @ ( produc5692694829384537815t_unit @ X @ X ) @ bot_bo1654022762305692595t_unit ) ) ).
% 0.34/0.52
% 0.34/0.52 % total_on_singleton
% 0.34/0.52 thf(fact_433_total__on__singleton,axiom,
% 0.34/0.52 ! [X: $o] : ( total_on_o @ ( insert_o @ X @ bot_bot_set_o ) @ ( insert6201435330877294327od_o_o @ ( product_Pair_o_o @ X @ X ) @ bot_bo7073875226086086771od_o_o ) ) ).
% 0.34/0.52
% 0.34/0.52 % total_on_singleton
% 0.34/0.52 thf(fact_434_total__on__singleton,axiom,
% 0.34/0.52 ! [X: nat] : ( total_on_nat @ ( insert_nat @ X @ bot_bot_set_nat ) @ ( insert8211810215607154385at_nat @ ( product_Pair_nat_nat @ X @ X ) @ bot_bo2099793752762293965at_nat ) ) ).
% 0.34/0.52
% 0.34/0.52 % total_on_singleton
% 0.34/0.52 thf(fact_435_total__on__singleton,axiom,
% 0.34/0.52 ! [X: int] : ( total_on_int @ ( insert_int @ X @ bot_bot_set_int ) @ ( insert5033312907999012233nt_int @ ( product_Pair_int_int @ X @ X ) @ bot_bo1796632182523588997nt_int ) ) ).
% 0.34/0.52
% 0.34/0.52 % total_on_singleton
% 0.34/0.52 thf(fact_436_total__on__singleton,axiom,
% 0.34/0.52 ! [X: produc3658429121746597890et_nat > $o] : ( total_7429123086324524369_nat_o @ ( insert5175938949040314269_nat_o @ X @ bot_bo7824918357723582233_nat_o ) @ ( insert5309882203591314743_nat_o @ ( produc7368190662567826135_nat_o @ X @ X ) @ bot_bo3657187498359500723_nat_o ) ) ).
% 0.34/0.52
% 0.34/0.52 % total_on_singleton
% 0.34/0.52 thf(fact_437_refl__on__singleton,axiom,
% 0.34/0.52 ! [X: product_unit] : ( refl_on_Product_unit @ ( insert_Product_unit @ X @ bot_bo3957492148770167129t_unit ) @ ( insert4616891024742550839t_unit @ ( produc5692694829384537815t_unit @ X @ X ) @ bot_bo1654022762305692595t_unit ) ) ).
% 0.34/0.52
% 0.34/0.52 % refl_on_singleton
% 0.34/0.52 thf(fact_438_refl__on__singleton,axiom,
% 0.34/0.52 ! [X: $o] : ( refl_on_o @ ( insert_o @ X @ bot_bot_set_o ) @ ( insert6201435330877294327od_o_o @ ( product_Pair_o_o @ X @ X ) @ bot_bo7073875226086086771od_o_o ) ) ).
% 0.34/0.52
% 0.34/0.52 % refl_on_singleton
% 0.34/0.52 thf(fact_439_refl__on__singleton,axiom,
% 0.34/0.52 ! [X: nat] : ( refl_on_nat @ ( insert_nat @ X @ bot_bot_set_nat ) @ ( insert8211810215607154385at_nat @ ( product_Pair_nat_nat @ X @ X ) @ bot_bo2099793752762293965at_nat ) ) ).
% 0.34/0.52
% 0.34/0.52 % refl_on_singleton
% 0.34/0.52 thf(fact_440_refl__on__singleton,axiom,
% 0.34/0.52 ! [X: int] : ( refl_on_int @ ( insert_int @ X @ bot_bot_set_int ) @ ( insert5033312907999012233nt_int @ ( product_Pair_int_int @ X @ X ) @ bot_bo1796632182523588997nt_int ) ) ).
% 0.34/0.52
% 0.34/0.52 % refl_on_singleton
% 0.34/0.52 thf(fact_441_refl__on__singleton,axiom,
% 0.34/0.52 ! [X: produc3658429121746597890et_nat > $o] : ( refl_o1075864025857821388_nat_o @ ( insert5175938949040314269_nat_o @ X @ bot_bo7824918357723582233_nat_o ) @ ( insert5309882203591314743_nat_o @ ( produc7368190662567826135_nat_o @ X @ X ) @ bot_bo3657187498359500723_nat_o ) ) ).
% 0.34/0.52
% 0.34/0.52 % refl_on_singleton
% 0.34/0.52 thf(fact_442_Id__on__empty,axiom,
% 0.34/0.52 ( ( id_on_Product_unit @ bot_bo3957492148770167129t_unit )
% 0.34/0.52 = bot_bo1654022762305692595t_unit ) ).
% 0.34/0.52
% 0.34/0.52 % Id_on_empty
% 0.34/0.52 thf(fact_443_Id__on__empty,axiom,
% 0.34/0.52 ( ( id_on_o @ bot_bot_set_o )
% 0.34/0.52 = bot_bo7073875226086086771od_o_o ) ).
% 0.34/0.52
% 0.34/0.52 % Id_on_empty
% 0.34/0.52 thf(fact_444_Id__on__empty,axiom,
% 0.34/0.52 ( ( id_on_nat @ bot_bot_set_nat )
% 0.34/0.52 = bot_bo2099793752762293965at_nat ) ).
% 0.34/0.52
% 0.34/0.52 % Id_on_empty
% 0.34/0.52 thf(fact_445_Id__on__empty,axiom,
% 0.34/0.52 ( ( id_on_int @ bot_bot_set_int )
% 0.34/0.52 = bot_bo1796632182523588997nt_int ) ).
% 0.34/0.52
% 0.34/0.52 % Id_on_empty
% 0.34/0.52 thf(fact_446_Id__on__empty,axiom,
% 0.34/0.52 ( ( id_on_7438695062414216554_nat_o @ bot_bo7824918357723582233_nat_o )
% 0.34/0.52 = bot_bo3657187498359500723_nat_o ) ).
% 0.34/0.52
% 0.34/0.52 % Id_on_empty
% 0.34/0.52 thf(fact_447_linear__order__on__singleton,axiom,
% 0.34/0.52 ! [X: product_unit] : ( order_888282469287196226t_unit @ ( insert_Product_unit @ X @ bot_bo3957492148770167129t_unit ) @ ( insert4616891024742550839t_unit @ ( produc5692694829384537815t_unit @ X @ X ) @ bot_bo1654022762305692595t_unit ) ) ).
% 0.34/0.52
% 0.34/0.52 % linear_order_on_singleton
% 0.34/0.52 thf(fact_448_linear__order__on__singleton,axiom,
% 0.34/0.52 ! [X: $o] : ( order_6238756238976269133r_on_o @ ( insert_o @ X @ bot_bot_set_o ) @ ( insert6201435330877294327od_o_o @ ( product_Pair_o_o @ X @ X ) @ bot_bo7073875226086086771od_o_o ) ) ).
% 0.34/0.52
% 0.34/0.52 % linear_order_on_singleton
% 0.34/0.52 thf(fact_449_linear__order__on__singleton,axiom,
% 0.34/0.52 ! [X: nat] : ( order_4473980167227706203on_nat @ ( insert_nat @ X @ bot_bot_set_nat ) @ ( insert8211810215607154385at_nat @ ( product_Pair_nat_nat @ X @ X ) @ bot_bo2099793752762293965at_nat ) ) ).
% 0.34/0.52
% 0.34/0.52 % linear_order_on_singleton
% 0.34/0.52 thf(fact_450_linear__order__on__singleton,axiom,
% 0.34/0.52 ! [X: int] : ( order_4471489696718655927on_int @ ( insert_int @ X @ bot_bot_set_int ) @ ( insert5033312907999012233nt_int @ ( product_Pair_int_int @ X @ X ) @ bot_bo1796632182523588997nt_int ) ) ).
% 0.34/0.52
% 0.34/0.52 % linear_order_on_singleton
% 0.34/0.52 thf(fact_451_linear__order__on__singleton,axiom,
% 0.34/0.52 ! [X: produc3658429121746597890et_nat > $o] : ( order_1145710701291701250_nat_o @ ( insert5175938949040314269_nat_o @ X @ bot_bo7824918357723582233_nat_o ) @ ( insert5309882203591314743_nat_o @ ( produc7368190662567826135_nat_o @ X @ X ) @ bot_bo3657187498359500723_nat_o ) ) ).
% 0.34/0.52
% 0.34/0.52 % linear_order_on_singleton
% 0.34/0.52 thf(fact_452_pure__assn__raw_Opelims_I1_J,axiom,
% 0.34/0.52 ! [X: $o,Xa: produc3658429121746597890et_nat,Y: $o] :
% 0.34/0.52 ( ( ( pure_a825153325127701367it_nat @ X @ Xa )
% 0.34/0.52 = Y )
% 0.34/0.52 => ( ( accp_P8458817951426537472et_nat @ pure_a6022498039421069578it_nat @ ( produc3762353314782720579et_nat @ X @ Xa ) )
% 0.34/0.52 => ~ ! [H: heap_e7401611519738050253t_unit,As2: set_nat] :
% 0.34/0.52 ( ( Xa
% 0.34/0.52 = ( produc7507926704131184380et_nat @ H @ As2 ) )
% 0.34/0.52 => ( ( Y
% 0.34/0.52 = ( ( As2 = bot_bot_set_nat )
% 0.34/0.52 & X ) )
% 0.34/0.52 => ~ ( accp_P8458817951426537472et_nat @ pure_a6022498039421069578it_nat @ ( produc3762353314782720579et_nat @ X @ ( produc7507926704131184380et_nat @ H @ As2 ) ) ) ) ) ) ) ).
% 0.34/0.52
% 0.34/0.52 % pure_assn_raw.pelims(1)
% 0.34/0.52 thf(fact_453_pure__assn__raw_Opelims_I2_J,axiom,
% 0.34/0.52 ! [X: $o,Xa: produc3658429121746597890et_nat] :
% 0.34/0.52 ( ( pure_a825153325127701367it_nat @ X @ Xa )
% 0.34/0.52 => ( ( accp_P8458817951426537472et_nat @ pure_a6022498039421069578it_nat @ ( produc3762353314782720579et_nat @ X @ Xa ) )
% 0.34/0.52 => ~ ! [H: heap_e7401611519738050253t_unit,As2: set_nat] :
% 0.34/0.52 ( ( Xa
% 0.34/0.52 = ( produc7507926704131184380et_nat @ H @ As2 ) )
% 0.34/0.52 => ( ( accp_P8458817951426537472et_nat @ pure_a6022498039421069578it_nat @ ( produc3762353314782720579et_nat @ X @ ( produc7507926704131184380et_nat @ H @ As2 ) ) )
% 0.34/0.52 => ~ ( ( As2 = bot_bot_set_nat )
% 0.34/0.52 & X ) ) ) ) ) ).
% 0.34/0.52
% 0.34/0.52 % pure_assn_raw.pelims(2)
% 0.34/0.52 thf(fact_454_pure__assn__raw_Opelims_I3_J,axiom,
% 0.34/0.52 ! [X: $o,Xa: produc3658429121746597890et_nat] :
% 0.34/0.52 ( ~ ( pure_a825153325127701367it_nat @ X @ Xa )
% 0.34/0.52 => ( ( accp_P8458817951426537472et_nat @ pure_a6022498039421069578it_nat @ ( produc3762353314782720579et_nat @ X @ Xa ) )
% 0.34/0.52 => ~ ! [H: heap_e7401611519738050253t_unit,As2: set_nat] :
% 0.34/0.52 ( ( Xa
% 0.34/0.52 = ( produc7507926704131184380et_nat @ H @ As2 ) )
% 0.34/0.52 => ( ( accp_P8458817951426537472et_nat @ pure_a6022498039421069578it_nat @ ( produc3762353314782720579et_nat @ X @ ( produc7507926704131184380et_nat @ H @ As2 ) ) )
% 0.34/0.52 => ( ( As2 = bot_bot_set_nat )
% 0.34/0.52 & X ) ) ) ) ) ).
% 0.34/0.52
% 0.34/0.52 % pure_assn_raw.pelims(3)
% 0.34/0.52 thf(fact_455_Image__singleton__iff,axiom,
% 0.34/0.52 ! [B: nat,R2: set_Pr1763845938948868674it_nat,A2: product_unit] :
% 0.34/0.52 ( ( member_nat @ B @ ( image_5657742817519485770it_nat @ R2 @ ( insert_Product_unit @ A2 @ bot_bo3957492148770167129t_unit ) ) )
% 0.34/0.52 = ( member423003724879500579it_nat @ ( produc4803288329813530246it_nat @ A2 @ B ) @ R2 ) ) ).
% 0.34/0.52
% 0.34/0.52 % Image_singleton_iff
% 0.34/0.52 thf(fact_456_Image__singleton__iff,axiom,
% 0.34/0.52 ! [B: int,R2: set_Pr8497134830871917726it_int,A2: product_unit] :
% 0.34/0.52 ( ( member_int @ B @ ( image_5655252347010435494it_int @ R2 @ ( insert_Product_unit @ A2 @ bot_bo3957492148770167129t_unit ) ) )
% 0.34/0.52 = ( member5468524742225079679it_int @ ( produc4800797859304479970it_int @ A2 @ B ) @ R2 ) ) ).
% 0.34/0.52
% 0.34/0.52 % Image_singleton_iff
% 0.34/0.52 thf(fact_457_Image__singleton__iff,axiom,
% 0.34/0.52 ! [B: $o,R2: set_Pr5464989305201766866unit_o,A2: product_unit] :
% 0.34/0.52 ( ( member_o @ B @ ( image_Product_unit_o @ R2 @ ( insert_Product_unit @ A2 @ bot_bo3957492148770167129t_unit ) ) )
% 0.34/0.52 = ( member3689400097658526619unit_o @ ( produc8487917018914037602unit_o @ A2 @ B ) @ R2 ) ) ).
% 0.34/0.52
% 0.34/0.52 % Image_singleton_iff
% 0.34/0.52 thf(fact_458_Image__singleton__iff,axiom,
% 0.34/0.52 ! [B: product_unit,R2: set_Pr5094982260447487303t_unit,A2: product_unit] :
% 0.34/0.52 ( ( member_Product_unit @ B @ ( image_8431784415008483091t_unit @ R2 @ ( insert_Product_unit @ A2 @ bot_bo3957492148770167129t_unit ) ) )
% 0.34/0.52 = ( member7821138191923116944t_unit @ ( produc5692694829384537815t_unit @ A2 @ B ) @ R2 ) ) ).
% 0.34/0.52
% 0.34/0.52 % Image_singleton_iff
% 0.34/0.52 thf(fact_459_Image__singleton__iff,axiom,
% 0.34/0.52 ! [B: nat,R2: set_Pr2101469702781467981_o_nat,A2: $o] :
% 0.34/0.52 ( ( member_nat @ B @ ( image_o_nat @ R2 @ ( insert_o @ A2 @ bot_bot_set_o ) ) )
% 0.34/0.52 = ( member2802428098988154798_o_nat @ ( product_Pair_o_nat @ A2 @ B ) @ R2 ) ) ).
% 0.34/0.52
% 0.34/0.52 % Image_singleton_iff
% 0.34/0.52 thf(fact_460_Image__singleton__iff,axiom,
% 0.34/0.52 ! [B: int,R2: set_Pr8834758594704517033_o_int,A2: $o] :
% 0.34/0.52 ( ( member_int @ B @ ( image_o_int @ R2 @ ( insert_o @ A2 @ bot_bot_set_o ) ) )
% 0.34/0.52 = ( member7847949116333733898_o_int @ ( product_Pair_o_int @ A2 @ B ) @ R2 ) ) ).
% 0.34/0.52
% 0.34/0.52 % Image_singleton_iff
% 0.34/0.52 thf(fact_461_Image__singleton__iff,axiom,
% 0.34/0.52 ! [B: $o,R2: set_Product_prod_o_o,A2: $o] :
% 0.34/0.52 ( ( member_o @ B @ ( image_o_o @ R2 @ ( insert_o @ A2 @ bot_bot_set_o ) ) )
% 0.34/0.52 = ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ A2 @ B ) @ R2 ) ) ).
% 0.34/0.52
% 0.34/0.52 % Image_singleton_iff
% 0.34/0.52 thf(fact_462_Image__singleton__iff,axiom,
% 0.34/0.52 ! [B: product_unit,R2: set_Pr6671815714118170108t_unit,A2: $o] :
% 0.34/0.52 ( ( member_Product_unit @ B @ ( image_o_Product_unit @ R2 @ ( insert_o @ A2 @ bot_bot_set_o ) ) )
% 0.34/0.52 = ( member9215336023991391429t_unit @ ( produc3901946521647900940t_unit @ A2 @ B ) @ R2 ) ) ).
% 0.34/0.52
% 0.34/0.52 % Image_singleton_iff
% 0.34/0.52 thf(fact_463_Image__singleton__iff,axiom,
% 0.34/0.52 ! [B: int,R2: set_Pr7995236796853374141at_int,A2: nat] :
% 0.34/0.52 ( ( member_int @ B @ ( image_nat_int @ R2 @ ( insert_nat @ A2 @ bot_bot_set_nat ) ) )
% 0.34/0.52 = ( member4262671552274231302at_int @ ( product_Pair_nat_int @ A2 @ B ) @ R2 ) ) ).
% 0.34/0.52
% 0.34/0.52 % Image_singleton_iff
% 0.34/0.52 thf(fact_464_Image__singleton__iff,axiom,
% 0.34/0.52 ! [B: $o,R2: set_Pr3149072824959771635_nat_o,A2: nat] :
% 0.34/0.52 ( ( member_o @ B @ ( image_nat_o @ R2 @ ( insert_nat @ A2 @ bot_bot_set_nat ) ) )
% 0.34/0.52 = ( member6310962623043647828_nat_o @ ( product_Pair_nat_o @ A2 @ B ) @ R2 ) ) ).
% 0.34/0.52
% 0.34/0.52 % Image_singleton_iff
% 0.34/0.52 thf(fact_465_mult_Oright__assoc,axiom,
% 0.34/0.52 ! [A2: assn,B: assn,C: assn] :
% 0.34/0.52 ( ( times_times_assn @ ( times_times_assn @ A2 @ B ) @ C )
% 0.34/0.52 = ( times_times_assn @ A2 @ ( times_times_assn @ B @ C ) ) ) ).
% 0.34/0.52
% 0.34/0.52 % mult.right_assoc
% 0.34/0.52 thf(fact_466_mult_Oright__assoc,axiom,
% 0.34/0.52 ! [A2: rat,B: rat,C: rat] :
% 0.34/0.52 ( ( times_times_rat @ ( times_times_rat @ A2 @ B ) @ C )
% 0.34/0.52 = ( times_times_rat @ A2 @ ( times_times_rat @ B @ C ) ) ) ).
% 0.34/0.52
% 0.34/0.52 % mult.right_assoc
% 0.34/0.52 thf(fact_467_mult_Oright__assoc,axiom,
% 0.34/0.52 ! [A2: nat,B: nat,C: nat] :
% 0.34/0.52 ( ( times_times_nat @ ( times_times_nat @ A2 @ B ) @ C )
% 0.34/0.52 = ( times_times_nat @ A2 @ ( times_times_nat @ B @ C ) ) ) ).
% 0.34/0.52
% 0.34/0.52 % mult.right_assoc
% 0.34/0.52 thf(fact_468_mult_Oright__assoc,axiom,
% 0.34/0.52 ! [A2: int,B: int,C: int] :
% 0.34/0.52 ( ( times_times_int @ ( times_times_int @ A2 @ B ) @ C )
% 0.34/0.52 = ( times_times_int @ A2 @ ( times_times_int @ B @ C ) ) ) ).
% 0.34/0.52
% 0.34/0.52 % mult.right_assoc
% 0.34/0.52 thf(fact_469_mult_Oright__commute,axiom,
% 0.34/0.52 ! [A2: assn,B: assn,C: assn] :
% 0.34/0.52 ( ( times_times_assn @ ( times_times_assn @ A2 @ B ) @ C )
% 0.34/0.52 = ( times_times_assn @ ( times_times_assn @ A2 @ C ) @ B ) ) ).
% 0.34/0.52
% 0.34/0.52 % mult.right_commute
% 0.34/0.52 thf(fact_470_mult_Oright__commute,axiom,
% 0.34/0.52 ! [A2: rat,B: rat,C: rat] :
% 0.34/0.52 ( ( times_times_rat @ ( times_times_rat @ A2 @ B ) @ C )
% 0.34/0.52 = ( times_times_rat @ ( times_times_rat @ A2 @ C ) @ B ) ) ).
% 0.34/0.52
% 0.34/0.52 % mult.right_commute
% 0.34/0.52 thf(fact_471_mult_Oright__commute,axiom,
% 0.34/0.52 ! [A2: nat,B: nat,C: nat] :
% 0.34/0.52 ( ( times_times_nat @ ( times_times_nat @ A2 @ B ) @ C )
% 0.34/0.52 = ( times_times_nat @ ( times_times_nat @ A2 @ C ) @ B ) ) ).
% 0.34/0.52
% 0.34/0.52 % mult.right_commute
% 0.34/0.52 thf(fact_472_mult_Oright__commute,axiom,
% 0.34/0.52 ! [A2: int,B: int,C: int] :
% 0.34/0.52 ( ( times_times_int @ ( times_times_int @ A2 @ B ) @ C )
% 0.34/0.52 = ( times_times_int @ ( times_times_int @ A2 @ C ) @ B ) ) ).
% 0.34/0.52
% 0.34/0.52 % mult.right_commute
% 0.34/0.52 thf(fact_473_in__range__empty,axiom,
% 0.34/0.52 ! [H2: heap_e7401611519738050253t_unit] : ( in_range @ ( produc7507926704131184380et_nat @ H2 @ bot_bot_set_nat ) ) ).
% 0.34/0.52
% 0.34/0.52 % in_range_empty
% 0.34/0.52 thf(fact_474_ImageI,axiom,
% 0.34/0.52 ! [A2: nat,B: int,R2: set_Pr7995236796853374141at_int,A: set_nat] :
% 0.34/0.52 ( ( member4262671552274231302at_int @ ( product_Pair_nat_int @ A2 @ B ) @ R2 )
% 0.34/0.52 => ( ( member_nat @ A2 @ A )
% 0.34/0.52 => ( member_int @ B @ ( image_nat_int @ R2 @ A ) ) ) ) ).
% 0.34/0.52
% 0.34/0.52 % ImageI
% 0.34/0.52 thf(fact_475_ImageI,axiom,
% 0.34/0.52 ! [A2: nat,B: $o,R2: set_Pr3149072824959771635_nat_o,A: set_nat] :
% 0.34/0.52 ( ( member6310962623043647828_nat_o @ ( product_Pair_nat_o @ A2 @ B ) @ R2 )
% 0.34/0.52 => ( ( member_nat @ A2 @ A )
% 0.34/0.52 => ( member_o @ B @ ( image_nat_o @ R2 @ A ) ) ) ) ).
% 0.34/0.52
% 0.34/0.52 % ImageI
% 0.34/0.52 thf(fact_476_ImageI,axiom,
% 0.34/0.52 ! [A2: nat,B: product_unit,R2: set_Pr4334478416066269672t_unit,A: set_nat] :
% 0.34/0.52 ( ( member348563419383025609t_unit @ ( produc3434450474625521892t_unit @ A2 @ B ) @ R2 )
% 0.34/0.52 => ( ( member_nat @ A2 @ A )
% 0.34/0.52 => ( member_Product_unit @ B @ ( image_4288904962331477416t_unit @ R2 @ A ) ) ) ) ).
% 0.34/0.52
% 0.34/0.52 % ImageI
% 0.34/0.52 thf(fact_477_ImageI,axiom,
% 0.34/0.52 ! [A2: int,B: nat,R2: set_Pr3448869479623346877nt_nat,A: set_int] :
% 0.34/0.52 ( ( member216504246829706758nt_nat @ ( product_Pair_int_nat @ A2 @ B ) @ R2 )
% 0.34/0.52 => ( ( member_int @ A2 @ A )
% 0.34/0.52 => ( member_nat @ B @ ( image_int_nat @ R2 @ A ) ) ) ) ).
% 0.34/0.52
% 0.34/0.52 % ImageI
% 0.34/0.52 thf(fact_478_ImageI,axiom,
% 0.34/0.52 ! [A2: int,B: $o,R2: set_Pr903927857289325719_int_o,A: set_int] :
% 0.34/0.52 ( ( member4489920277610959864_int_o @ ( product_Pair_int_o @ A2 @ B ) @ R2 )
% 0.34/0.52 => ( ( member_int @ A2 @ A )
% 0.34/0.52 => ( member_o @ B @ ( image_int_o @ R2 @ A ) ) ) ) ).
% 0.34/0.52
% 0.34/0.52 % ImageI
% 0.34/0.52 thf(fact_479_ImageI,axiom,
% 0.34/0.52 ! [A2: int,B: product_unit,R2: set_Pr5889264611543340172t_unit,A: set_int] :
% 0.34/0.52 ( ( member6688357403439981677t_unit @ ( produc550872421827702152t_unit @ A2 @ B ) @ R2 )
% 0.34/0.52 => ( ( member_int @ A2 @ A )
% 0.34/0.52 => ( member_Product_unit @ B @ ( image_1405326909533657676t_unit @ R2 @ A ) ) ) ) ).
% 0.34/0.52
% 0.34/0.52 % ImageI
% 0.34/0.52 thf(fact_480_ImageI,axiom,
% 0.34/0.52 ! [A2: $o,B: nat,R2: set_Pr2101469702781467981_o_nat,A: set_o] :
% 0.34/0.52 ( ( member2802428098988154798_o_nat @ ( product_Pair_o_nat @ A2 @ B ) @ R2 )
% 0.34/0.52 => ( ( member_o @ A2 @ A )
% 0.34/0.52 => ( member_nat @ B @ ( image_o_nat @ R2 @ A ) ) ) ) ).
% 0.34/0.52
% 0.34/0.52 % ImageI
% 0.34/0.52 thf(fact_481_ImageI,axiom,
% 0.34/0.52 ! [A2: $o,B: int,R2: set_Pr8834758594704517033_o_int,A: set_o] :
% 0.34/0.52 ( ( member7847949116333733898_o_int @ ( product_Pair_o_int @ A2 @ B ) @ R2 )
% 0.34/0.52 => ( ( member_o @ A2 @ A )
% 0.34/0.52 => ( member_int @ B @ ( image_o_int @ R2 @ A ) ) ) ) ).
% 0.34/0.52
% 0.34/0.52 % ImageI
% 0.34/0.52 thf(fact_482_ImageI,axiom,
% 0.34/0.52 ! [A2: $o,B: $o,R2: set_Product_prod_o_o,A: set_o] :
% 0.34/0.52 ( ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ A2 @ B ) @ R2 )
% 0.34/0.52 => ( ( member_o @ A2 @ A )
% 0.34/0.52 => ( member_o @ B @ ( image_o_o @ R2 @ A ) ) ) ) ).
% 0.34/0.52
% 0.34/0.52 % ImageI
% 0.34/0.52 thf(fact_483_ImageI,axiom,
% 0.34/0.52 ! [A2: $o,B: product_unit,R2: set_Pr6671815714118170108t_unit,A: set_o] :
% 0.34/0.52 ( ( member9215336023991391429t_unit @ ( produc3901946521647900940t_unit @ A2 @ B ) @ R2 )
% 0.34/0.52 => ( ( member_o @ A2 @ A )
% 0.34/0.52 => ( member_Product_unit @ B @ ( image_o_Product_unit @ R2 @ A ) ) ) ) ).
% 0.34/0.52
% 0.34/0.52 % ImageI
% 0.34/0.52 thf(fact_484_Image__empty2,axiom,
% 0.34/0.52 ! [R3: set_Pr5094982260447487303t_unit] :
% 0.34/0.52 ( ( image_8431784415008483091t_unit @ R3 @ bot_bo3957492148770167129t_unit )
% 0.34/0.52 = bot_bo3957492148770167129t_unit ) ).
% 0.34/0.52
% 0.34/0.52 % Image_empty2
% 0.34/0.52 thf(fact_485_Image__empty2,axiom,
% 0.34/0.52 ! [R3: set_Pr5464989305201766866unit_o] :
% 0.34/0.52 ( ( image_Product_unit_o @ R3 @ bot_bo3957492148770167129t_unit )
% 0.34/0.52 = bot_bot_set_o ) ).
% 0.34/0.52
% 0.34/0.52 % Image_empty2
% 0.34/0.52 thf(fact_486_Image__empty2,axiom,
% 0.34/0.52 ! [R3: set_Pr1763845938948868674it_nat] :
% 0.34/0.52 ( ( image_5657742817519485770it_nat @ R3 @ bot_bo3957492148770167129t_unit )
% 0.34/0.52 = bot_bot_set_nat ) ).
% 0.34/0.52
% 0.34/0.52 % Image_empty2
% 0.34/0.52 thf(fact_487_Image__empty2,axiom,
% 0.34/0.52 ! [R3: set_Pr8497134830871917726it_int] :
% 0.34/0.52 ( ( image_5655252347010435494it_int @ R3 @ bot_bo3957492148770167129t_unit )
% 0.34/0.52 = bot_bot_set_int ) ).
% 0.34/0.52
% 0.34/0.52 % Image_empty2
% 0.34/0.52 thf(fact_488_Image__empty2,axiom,
% 0.34/0.52 ! [R3: set_Pr6671815714118170108t_unit] :
% 0.34/0.52 ( ( image_o_Product_unit @ R3 @ bot_bot_set_o )
% 0.34/0.52 = bot_bo3957492148770167129t_unit ) ).
% 0.34/0.52
% 0.34/0.52 % Image_empty2
% 0.34/0.52 thf(fact_489_Image__empty2,axiom,
% 0.34/0.52 ! [R3: set_Product_prod_o_o] :
% 0.34/0.52 ( ( image_o_o @ R3 @ bot_bot_set_o )
% 0.34/0.52 = bot_bot_set_o ) ).
% 0.34/0.52
% 0.34/0.52 % Image_empty2
% 0.34/0.52 thf(fact_490_Image__empty2,axiom,
% 0.34/0.52 ! [R3: set_Pr2101469702781467981_o_nat] :
% 0.34/0.52 ( ( image_o_nat @ R3 @ bot_bot_set_o )
% 0.34/0.52 = bot_bot_set_nat ) ).
% 0.34/0.52
% 0.34/0.52 % Image_empty2
% 0.34/0.52 thf(fact_491_Image__empty2,axiom,
% 0.34/0.52 ! [R3: set_Pr8834758594704517033_o_int] :
% 0.34/0.52 ( ( image_o_int @ R3 @ bot_bot_set_o )
% 0.34/0.52 = bot_bot_set_int ) ).
% 0.34/0.52
% 0.34/0.52 % Image_empty2
% 0.34/0.52 thf(fact_492_Image__empty2,axiom,
% 0.34/0.52 ! [R3: set_Pr4334478416066269672t_unit] :
% 0.34/0.52 ( ( image_4288904962331477416t_unit @ R3 @ bot_bot_set_nat )
% 0.34/0.52 = bot_bo3957492148770167129t_unit ) ).
% 0.34/0.52
% 0.34/0.52 % Image_empty2
% 0.34/0.52 thf(fact_493_Image__empty2,axiom,
% 0.34/0.52 ! [R3: set_Pr3149072824959771635_nat_o] :
% 0.34/0.52 ( ( image_nat_o @ R3 @ bot_bot_set_nat )
% 0.34/0.52 = bot_bot_set_o ) ).
% 0.34/0.52
% 0.34/0.52 % Image_empty2
% 0.34/0.52 thf(fact_494_Id__onI,axiom,
% 0.34/0.52 ! [A2: produc3658429121746597890et_nat > $o,A: set_Pr4532377907799695533_nat_o] :
% 0.34/0.52 ( ( member6576561426505652726_nat_o @ A2 @ A )
% 0.34/0.52 => ( member8781333585448626064_nat_o @ ( produc7368190662567826135_nat_o @ A2 @ A2 ) @ ( id_on_7438695062414216554_nat_o @ A ) ) ) ).
% 0.34/0.52
% 0.34/0.52 % Id_onI
% 0.34/0.52 thf(fact_495_Id__onI,axiom,
% 0.34/0.52 ! [A2: $o,A: set_o] :
% 0.34/0.52 ( ( member_o @ A2 @ A )
% 0.34/0.52 => ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ A2 @ A2 ) @ ( id_on_o @ A ) ) ) ).
% 0.34/0.52
% 0.34/0.52 % Id_onI
% 0.34/0.52 thf(fact_496_Id__onI,axiom,
% 0.34/0.52 ! [A2: product_unit,A: set_Product_unit] :
% 0.34/0.52 ( ( member_Product_unit @ A2 @ A )
% 0.34/0.52 => ( member7821138191923116944t_unit @ ( produc5692694829384537815t_unit @ A2 @ A2 ) @ ( id_on_Product_unit @ A ) ) ) ).
% 0.34/0.52
% 0.34/0.52 % Id_onI
% 0.34/0.52 thf(fact_497_Id__onI,axiom,
% 0.34/0.52 ! [A2: nat,A: set_nat] :
% 0.34/0.52 ( ( member_nat @ A2 @ A )
% 0.34/0.52 => ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A2 @ A2 ) @ ( id_on_nat @ A ) ) ) ).
% 0.34/0.52
% 0.34/0.52 % Id_onI
% 0.34/0.52 thf(fact_498_Id__onI,axiom,
% 0.34/0.52 ! [A2: int,A: set_int] :
% 0.34/0.52 ( ( member_int @ A2 @ A )
% 0.34/0.52 => ( member5262025264175285858nt_int @ ( product_Pair_int_int @ A2 @ A2 ) @ ( id_on_int @ A ) ) ) ).
% 0.34/0.52
% 0.34/0.52 % Id_onI
% 0.34/0.52 thf(fact_499_bool__assn__proper_I1_J,axiom,
% 0.34/0.52 proper @ in_range ).
% 0.34/0.52
% 0.34/0.52 % bool_assn_proper(1)
% 0.34/0.52 thf(fact_500_refl__on__domain,axiom,
% 0.34/0.52 ! [A: set_Pr4532377907799695533_nat_o,R2: set_Pr2161125870931222855_nat_o,A2: produc3658429121746597890et_nat > $o,B: produc3658429121746597890et_nat > $o] :
% 0.34/0.52 ( ( refl_o1075864025857821388_nat_o @ A @ R2 )
% 0.34/0.52 => ( ( member8781333585448626064_nat_o @ ( produc7368190662567826135_nat_o @ A2 @ B ) @ R2 )
% 0.34/0.52 => ( ( member6576561426505652726_nat_o @ A2 @ A )
% 0.34/0.52 & ( member6576561426505652726_nat_o @ B @ A ) ) ) ) ).
% 0.34/0.52
% 0.34/0.52 % refl_on_domain
% 0.34/0.52 thf(fact_501_refl__on__domain,axiom,
% 0.34/0.52 ! [A: set_o,R2: set_Product_prod_o_o,A2: $o,B: $o] :
% 0.34/0.52 ( ( refl_on_o @ A @ R2 )
% 0.34/0.52 => ( ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ A2 @ B ) @ R2 )
% 0.34/0.52 => ( ( member_o @ A2 @ A )
% 0.34/0.52 & ( member_o @ B @ A ) ) ) ) ).
% 0.34/0.52
% 0.34/0.52 % refl_on_domain
% 0.34/0.52 thf(fact_502_refl__on__domain,axiom,
% 0.34/0.52 ! [A: set_Product_unit,R2: set_Pr5094982260447487303t_unit,A2: product_unit,B: product_unit] :
% 0.34/0.52 ( ( refl_on_Product_unit @ A @ R2 )
% 0.34/0.52 => ( ( member7821138191923116944t_unit @ ( produc5692694829384537815t_unit @ A2 @ B ) @ R2 )
% 0.34/0.52 => ( ( member_Product_unit @ A2 @ A )
% 0.34/0.52 & ( member_Product_unit @ B @ A ) ) ) ) ).
% 0.34/0.52
% 0.34/0.52 % refl_on_domain
% 0.34/0.52 thf(fact_503_refl__on__domain,axiom,
% 0.34/0.52 ! [A: set_nat,R2: set_Pr1261947904930325089at_nat,A2: nat,B: nat] :
% 0.34/0.52 ( ( refl_on_nat @ A @ R2 )
% 0.34/0.52 => ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A2 @ B ) @ R2 )
% 0.34/0.52 => ( ( member_nat @ A2 @ A )
% 0.34/0.52 & ( member_nat @ B @ A ) ) ) ) ).
% 0.34/0.52
% 0.34/0.52 % refl_on_domain
% 0.34/0.52 thf(fact_504_refl__on__domain,axiom,
% 0.34/0.52 ! [A: set_int,R2: set_Pr958786334691620121nt_int,A2: int,B: int] :
% 0.34/0.52 ( ( refl_on_int @ A @ R2 )
% 0.34/0.52 => ( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ A2 @ B ) @ R2 )
% 0.34/0.52 => ( ( member_int @ A2 @ A )
% 0.34/0.52 & ( member_int @ B @ A ) ) ) ) ).
% 0.34/0.52
% 0.34/0.52 % refl_on_domain
% 0.34/0.52 thf(fact_505_rev__ImageI,axiom,
% 0.34/0.52 ! [A2: nat,A: set_nat,B: int,R2: set_Pr7995236796853374141at_int] :
% 0.34/0.52 ( ( member_nat @ A2 @ A )
% 0.34/0.52 => ( ( member4262671552274231302at_int @ ( product_Pair_nat_int @ A2 @ B ) @ R2 )
% 0.34/0.52 => ( member_int @ B @ ( image_nat_int @ R2 @ A ) ) ) ) ).
% 0.34/0.52
% 0.34/0.52 % rev_ImageI
% 0.34/0.52 thf(fact_506_rev__ImageI,axiom,
% 0.34/0.52 ! [A2: nat,A: set_nat,B: $o,R2: set_Pr3149072824959771635_nat_o] :
% 0.34/0.52 ( ( member_nat @ A2 @ A )
% 0.34/0.52 => ( ( member6310962623043647828_nat_o @ ( product_Pair_nat_o @ A2 @ B ) @ R2 )
% 0.34/0.52 => ( member_o @ B @ ( image_nat_o @ R2 @ A ) ) ) ) ).
% 0.34/0.52
% 0.34/0.52 % rev_ImageI
% 0.34/0.52 thf(fact_507_rev__ImageI,axiom,
% 0.34/0.52 ! [A2: nat,A: set_nat,B: product_unit,R2: set_Pr4334478416066269672t_unit] :
% 0.34/0.52 ( ( member_nat @ A2 @ A )
% 0.34/0.52 => ( ( member348563419383025609t_unit @ ( produc3434450474625521892t_unit @ A2 @ B ) @ R2 )
% 0.34/0.52 => ( member_Product_unit @ B @ ( image_4288904962331477416t_unit @ R2 @ A ) ) ) ) ).
% 0.34/0.52
% 0.34/0.52 % rev_ImageI
% 0.34/0.52 thf(fact_508_rev__ImageI,axiom,
% 0.34/0.52 ! [A2: int,A: set_int,B: nat,R2: set_Pr3448869479623346877nt_nat] :
% 0.34/0.52 ( ( member_int @ A2 @ A )
% 0.34/0.52 => ( ( member216504246829706758nt_nat @ ( product_Pair_int_nat @ A2 @ B ) @ R2 )
% 0.34/0.52 => ( member_nat @ B @ ( image_int_nat @ R2 @ A ) ) ) ) ).
% 0.34/0.52
% 0.34/0.52 % rev_ImageI
% 0.34/0.52 thf(fact_509_rev__ImageI,axiom,
% 0.34/0.52 ! [A2: int,A: set_int,B: $o,R2: set_Pr903927857289325719_int_o] :
% 0.34/0.52 ( ( member_int @ A2 @ A )
% 0.34/0.52 => ( ( member4489920277610959864_int_o @ ( product_Pair_int_o @ A2 @ B ) @ R2 )
% 0.34/0.52 => ( member_o @ B @ ( image_int_o @ R2 @ A ) ) ) ) ).
% 0.34/0.52
% 0.34/0.52 % rev_ImageI
% 0.34/0.52 thf(fact_510_rev__ImageI,axiom,
% 0.34/0.52 ! [A2: int,A: set_int,B: product_unit,R2: set_Pr5889264611543340172t_unit] :
% 0.34/0.52 ( ( member_int @ A2 @ A )
% 0.34/0.52 => ( ( member6688357403439981677t_unit @ ( produc550872421827702152t_unit @ A2 @ B ) @ R2 )
% 0.34/0.52 => ( member_Product_unit @ B @ ( image_1405326909533657676t_unit @ R2 @ A ) ) ) ) ).
% 0.34/0.52
% 0.34/0.52 % rev_ImageI
% 0.34/0.52 thf(fact_511_rev__ImageI,axiom,
% 0.34/0.52 ! [A2: $o,A: set_o,B: nat,R2: set_Pr2101469702781467981_o_nat] :
% 0.34/0.52 ( ( member_o @ A2 @ A )
% 0.34/0.52 => ( ( member2802428098988154798_o_nat @ ( product_Pair_o_nat @ A2 @ B ) @ R2 )
% 0.34/0.52 => ( member_nat @ B @ ( image_o_nat @ R2 @ A ) ) ) ) ).
% 0.34/0.52
% 0.34/0.52 % rev_ImageI
% 0.34/0.52 thf(fact_512_rev__ImageI,axiom,
% 0.34/0.52 ! [A2: $o,A: set_o,B: int,R2: set_Pr8834758594704517033_o_int] :
% 0.34/0.52 ( ( member_o @ A2 @ A )
% 0.34/0.52 => ( ( member7847949116333733898_o_int @ ( product_Pair_o_int @ A2 @ B ) @ R2 )
% 0.34/0.52 => ( member_int @ B @ ( image_o_int @ R2 @ A ) ) ) ) ).
% 0.34/0.52
% 0.34/0.52 % rev_ImageI
% 0.34/0.52 thf(fact_513_rev__ImageI,axiom,
% 0.34/0.52 ! [A2: $o,A: set_o,B: $o,R2: set_Product_prod_o_o] :
% 0.34/0.52 ( ( member_o @ A2 @ A )
% 0.34/0.52 => ( ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ A2 @ B ) @ R2 )
% 0.34/0.52 => ( member_o @ B @ ( image_o_o @ R2 @ A ) ) ) ) ).
% 0.34/0.52
% 0.34/0.52 % rev_ImageI
% 0.34/0.52 thf(fact_514_rev__ImageI,axiom,
% 0.34/0.52 ! [A2: $o,A: set_o,B: product_unit,R2: set_Pr6671815714118170108t_unit] :
% 0.34/0.52 ( ( member_o @ A2 @ A )
% 0.34/0.52 => ( ( member9215336023991391429t_unit @ ( produc3901946521647900940t_unit @ A2 @ B ) @ R2 )
% 0.34/0.52 => ( member_Product_unit @ B @ ( image_o_Product_unit @ R2 @ A ) ) ) ) ).
% 0.34/0.52
% 0.34/0.52 % rev_ImageI
% 0.34/0.52 thf(fact_515_Image__iff,axiom,
% 0.34/0.52 ! [B: set_nat,R2: set_Pr3948176798113811640et_nat,A: set_he8644174446132254637t_unit] :
% 0.34/0.52 ( ( member_set_nat @ B @ ( image_3942343239045813184et_nat @ R2 @ A ) )
% 0.34/0.52 = ( ? [X2: heap_e7401611519738050253t_unit] :
% 0.34/0.52 ( ( member242389114213480694t_unit @ X2 @ A )
% 0.34/0.52 & ( member6260224972018164377et_nat @ ( produc7507926704131184380et_nat @ X2 @ B ) @ R2 ) ) ) ) ).
% 0.34/0.52
% 0.34/0.52 % Image_iff
% 0.34/0.52 thf(fact_516_Image__iff,axiom,
% 0.34/0.52 ! [B: nat,R2: set_Pr1261947904930325089at_nat,A: set_nat] :
% 0.34/0.52 ( ( member_nat @ B @ ( image_nat_nat @ R2 @ A ) )
% 0.34/0.52 = ( ? [X2: nat] :
% 0.34/0.52 ( ( member_nat @ X2 @ A )
% 0.34/0.52 & ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X2 @ B ) @ R2 ) ) ) ) ).
% 0.34/0.52
% 0.34/0.52 % Image_iff
% 0.34/0.52 thf(fact_517_Image__iff,axiom,
% 0.34/0.52 ! [B: int,R2: set_Pr958786334691620121nt_int,A: set_int] :
% 0.34/0.52 ( ( member_int @ B @ ( image_int_int @ R2 @ A ) )
% 0.34/0.52 = ( ? [X2: int] :
% 0.34/0.52 ( ( member_int @ X2 @ A )
% 0.34/0.52 & ( member5262025264175285858nt_int @ ( product_Pair_int_int @ X2 @ B ) @ R2 ) ) ) ) ).
% 0.34/0.52
% 0.34/0.52 % Image_iff
% 0.34/0.52 thf(fact_518_Image__iff,axiom,
% 0.34/0.52 ! [B: produc3658429121746597890et_nat,R2: set_Pr3286484037609594932et_nat,A: set_Pr4532377907799695533_nat_o] :
% 0.34/0.52 ( ( member6260224972018164377et_nat @ B @ ( image_1814916760386648892et_nat @ R2 @ A ) )
% 0.34/0.52 = ( ? [X2: produc3658429121746597890et_nat > $o] :
% 0.34/0.52 ( ( member6576561426505652726_nat_o @ X2 @ A )
% 0.34/0.52 & ( member1996754912294343701et_nat @ ( produc5001842942810119800et_nat @ X2 @ B ) @ R2 ) ) ) ) ).
% 0.34/0.52
% 0.34/0.52 % Image_iff
% 0.34/0.52 thf(fact_519_Image__iff,axiom,
% 0.34/0.52 ! [B: produc3925858234332021118et_nat,R2: set_Pr8536935166611901872et_nat,A: set_Pr4532377907799695533_nat_o] :
% 0.34/0.52 ( ( member1996754912294343701et_nat @ B @ ( image_7240710759097644216et_nat @ R2 @ A ) )
% 0.34/0.52 = ( ? [X2: produc3658429121746597890et_nat > $o] :
% 0.34/0.52 ( ( member6576561426505652726_nat_o @ X2 @ A )
% 0.34/0.52 & ( member6124377750444531601et_nat @ ( produc2245416461498447860et_nat @ X2 @ B ) @ R2 ) ) ) ) ).
% 0.34/0.52
% 0.34/0.52 % Image_iff
% 0.34/0.52 thf(fact_520_ImageE,axiom,
% 0.34/0.52 ! [B: nat,R2: set_Pr3448869479623346877nt_nat,A: set_int] :
% 0.34/0.52 ( ( member_nat @ B @ ( image_int_nat @ R2 @ A ) )
% 0.34/0.52 => ~ ! [X3: int] :
% 0.34/0.52 ( ( member216504246829706758nt_nat @ ( product_Pair_int_nat @ X3 @ B ) @ R2 )
% 0.34/0.52 => ~ ( member_int @ X3 @ A ) ) ) ).
% 0.34/0.52
% 0.34/0.52 % ImageE
% 0.34/0.52 thf(fact_521_ImageE,axiom,
% 0.34/0.52 ! [B: nat,R2: set_Pr2101469702781467981_o_nat,A: set_o] :
% 0.34/0.52 ( ( member_nat @ B @ ( image_o_nat @ R2 @ A ) )
% 0.34/0.52 => ~ ! [X3: $o] :
% 0.34/0.52 ( ( member2802428098988154798_o_nat @ ( product_Pair_o_nat @ X3 @ B ) @ R2 )
% 0.34/0.52 => ~ ( member_o @ X3 @ A ) ) ) ).
% 0.34/0.52
% 0.34/0.52 % ImageE
% 0.34/0.52 thf(fact_522_ImageE,axiom,
% 0.34/0.52 ! [B: nat,R2: set_Pr1763845938948868674it_nat,A: set_Product_unit] :
% 0.34/0.52 ( ( member_nat @ B @ ( image_5657742817519485770it_nat @ R2 @ A ) )
% 0.34/0.52 => ~ ! [X3: product_unit] :
% 0.34/0.52 ( ( member423003724879500579it_nat @ ( produc4803288329813530246it_nat @ X3 @ B ) @ R2 )
% 0.34/0.52 => ~ ( member_Product_unit @ X3 @ A ) ) ) ).
% 0.34/0.52
% 0.34/0.52 % ImageE
% 0.34/0.52 thf(fact_523_ImageE,axiom,
% 0.34/0.52 ! [B: int,R2: set_Pr7995236796853374141at_int,A: set_nat] :
% 0.34/0.52 ( ( member_int @ B @ ( image_nat_int @ R2 @ A ) )
% 0.34/0.52 => ~ ! [X3: nat] :
% 0.34/0.52 ( ( member4262671552274231302at_int @ ( product_Pair_nat_int @ X3 @ B ) @ R2 )
% 0.34/0.52 => ~ ( member_nat @ X3 @ A ) ) ) ).
% 0.34/0.52
% 0.34/0.52 % ImageE
% 0.34/0.52 thf(fact_524_ImageE,axiom,
% 0.34/0.52 ! [B: int,R2: set_Pr8834758594704517033_o_int,A: set_o] :
% 0.34/0.52 ( ( member_int @ B @ ( image_o_int @ R2 @ A ) )
% 0.34/0.52 => ~ ! [X3: $o] :
% 0.34/0.52 ( ( member7847949116333733898_o_int @ ( product_Pair_o_int @ X3 @ B ) @ R2 )
% 0.34/0.52 => ~ ( member_o @ X3 @ A ) ) ) ).
% 0.34/0.52
% 0.34/0.52 % ImageE
% 0.34/0.52 thf(fact_525_ImageE,axiom,
% 0.34/0.52 ! [B: int,R2: set_Pr8497134830871917726it_int,A: set_Product_unit] :
% 0.34/0.52 ( ( member_int @ B @ ( image_5655252347010435494it_int @ R2 @ A ) )
% 0.34/0.52 => ~ ! [X3: product_unit] :
% 0.34/0.52 ( ( member5468524742225079679it_int @ ( produc4800797859304479970it_int @ X3 @ B ) @ R2 )
% 0.34/0.52 => ~ ( member_Product_unit @ X3 @ A ) ) ) ).
% 0.34/0.52
% 0.34/0.52 % ImageE
% 0.34/0.52 thf(fact_526_ImageE,axiom,
% 0.34/0.52 ! [B: $o,R2: set_Pr3149072824959771635_nat_o,A: set_nat] :
% 0.34/0.52 ( ( member_o @ B @ ( image_nat_o @ R2 @ A ) )
% 0.34/0.52 => ~ ! [X3: nat] :
% 0.34/0.52 ( ( member6310962623043647828_nat_o @ ( product_Pair_nat_o @ X3 @ B ) @ R2 )
% 0.34/0.52 => ~ ( member_nat @ X3 @ A ) ) ) ).
% 0.34/0.52
% 0.34/0.52 % ImageE
% 0.34/0.52 thf(fact_527_ImageE,axiom,
% 0.34/0.52 ! [B: $o,R2: set_Pr903927857289325719_int_o,A: set_int] :
% 0.34/0.52 ( ( member_o @ B @ ( image_int_o @ R2 @ A ) )
% 0.34/0.52 => ~ ! [X3: int] :
% 0.34/0.52 ( ( member4489920277610959864_int_o @ ( product_Pair_int_o @ X3 @ B ) @ R2 )
% 0.34/0.52 => ~ ( member_int @ X3 @ A ) ) ) ).
% 0.34/0.52
% 0.34/0.52 % ImageE
% 0.34/0.52 thf(fact_528_ImageE,axiom,
% 0.34/0.52 ! [B: $o,R2: set_Product_prod_o_o,A: set_o] :
% 0.34/0.52 ( ( member_o @ B @ ( image_o_o @ R2 @ A ) )
% 0.34/0.52 => ~ ! [X3: $o] :
% 0.34/0.52 ( ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ X3 @ B ) @ R2 )
% 0.34/0.52 => ~ ( member_o @ X3 @ A ) ) ) ).
% 0.34/0.52
% 0.34/0.52 % ImageE
% 0.34/0.52 thf(fact_529_ImageE,axiom,
% 0.34/0.52 ! [B: $o,R2: set_Pr5464989305201766866unit_o,A: set_Product_unit] :
% 0.34/0.52 ( ( member_o @ B @ ( image_Product_unit_o @ R2 @ A ) )
% 0.34/0.52 => ~ ! [X3: product_unit] :
% 0.34/0.52 ( ( member3689400097658526619unit_o @ ( produc8487917018914037602unit_o @ X3 @ B ) @ R2 )
% 0.34/0.52 => ~ ( member_Product_unit @ X3 @ A ) ) ) ).
% 0.34/0.52
% 0.34/0.52 % ImageE
% 0.34/0.52 thf(fact_530_refl__onD2,axiom,
% 0.34/0.52 ! [A: set_Pr4532377907799695533_nat_o,R2: set_Pr2161125870931222855_nat_o,X: produc3658429121746597890et_nat > $o,Y: produc3658429121746597890et_nat > $o] :
% 0.34/0.52 ( ( refl_o1075864025857821388_nat_o @ A @ R2 )
% 0.34/0.52 => ( ( member8781333585448626064_nat_o @ ( produc7368190662567826135_nat_o @ X @ Y ) @ R2 )
% 0.34/0.52 => ( member6576561426505652726_nat_o @ Y @ A ) ) ) ).
% 0.34/0.52
% 0.34/0.52 % refl_onD2
% 0.34/0.52 thf(fact_531_refl__onD2,axiom,
% 0.34/0.52 ! [A: set_o,R2: set_Product_prod_o_o,X: $o,Y: $o] :
% 0.34/0.52 ( ( refl_on_o @ A @ R2 )
% 0.34/0.52 => ( ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ X @ Y ) @ R2 )
% 0.34/0.52 => ( member_o @ Y @ A ) ) ) ).
% 0.34/0.52
% 0.34/0.52 % refl_onD2
% 0.34/0.52 thf(fact_532_refl__onD2,axiom,
% 0.34/0.52 ! [A: set_Product_unit,R2: set_Pr5094982260447487303t_unit,X: product_unit,Y: product_unit] :
% 0.34/0.52 ( ( refl_on_Product_unit @ A @ R2 )
% 0.34/0.52 => ( ( member7821138191923116944t_unit @ ( produc5692694829384537815t_unit @ X @ Y ) @ R2 )
% 0.34/0.52 => ( member_Product_unit @ Y @ A ) ) ) ).
% 0.34/0.52
% 0.34/0.52 % refl_onD2
% 0.34/0.52 thf(fact_533_refl__onD2,axiom,
% 0.34/0.52 ! [A: set_nat,R2: set_Pr1261947904930325089at_nat,X: nat,Y: nat] :
% 0.34/0.52 ( ( refl_on_nat @ A @ R2 )
% 0.34/0.52 => ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ Y ) @ R2 )
% 0.34/0.52 => ( member_nat @ Y @ A ) ) ) ).
% 0.34/0.52
% 0.34/0.52 % refl_onD2
% 0.34/0.52 thf(fact_534_refl__onD2,axiom,
% 0.34/0.52 ! [A: set_int,R2: set_Pr958786334691620121nt_int,X: int,Y: int] :
% 0.34/0.52 ( ( refl_on_int @ A @ R2 )
% 0.34/0.52 => ( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ X @ Y ) @ R2 )
% 0.34/0.52 => ( member_int @ Y @ A ) ) ) ).
% 0.34/0.52
% 0.34/0.52 % refl_onD2
% 0.34/0.52 thf(fact_535_refl__onD1,axiom,
% 0.34/0.52 ! [A: set_Pr4532377907799695533_nat_o,R2: set_Pr2161125870931222855_nat_o,X: produc3658429121746597890et_nat > $o,Y: produc3658429121746597890et_nat > $o] :
% 0.34/0.52 ( ( refl_o1075864025857821388_nat_o @ A @ R2 )
% 0.34/0.52 => ( ( member8781333585448626064_nat_o @ ( produc7368190662567826135_nat_o @ X @ Y ) @ R2 )
% 0.34/0.52 => ( member6576561426505652726_nat_o @ X @ A ) ) ) ).
% 0.34/0.52
% 0.34/0.52 % refl_onD1
% 0.34/0.52 thf(fact_536_refl__onD1,axiom,
% 0.34/0.52 ! [A: set_o,R2: set_Product_prod_o_o,X: $o,Y: $o] :
% 0.34/0.52 ( ( refl_on_o @ A @ R2 )
% 0.34/0.52 => ( ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ X @ Y ) @ R2 )
% 0.34/0.52 => ( member_o @ X @ A ) ) ) ).
% 0.34/0.52
% 0.34/0.52 % refl_onD1
% 0.34/0.52 thf(fact_537_refl__onD1,axiom,
% 0.34/0.52 ! [A: set_Product_unit,R2: set_Pr5094982260447487303t_unit,X: product_unit,Y: product_unit] :
% 0.34/0.52 ( ( refl_on_Product_unit @ A @ R2 )
% 0.34/0.52 => ( ( member7821138191923116944t_unit @ ( produc5692694829384537815t_unit @ X @ Y ) @ R2 )
% 0.34/0.52 => ( member_Product_unit @ X @ A ) ) ) ).
% 0.34/0.52
% 0.34/0.52 % refl_onD1
% 0.34/0.52 thf(fact_538_refl__onD1,axiom,
% 0.34/0.52 ! [A: set_nat,R2: set_Pr1261947904930325089at_nat,X: nat,Y: nat] :
% 0.34/0.52 ( ( refl_on_nat @ A @ R2 )
% 0.34/0.52 => ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ Y ) @ R2 )
% 0.34/0.52 => ( member_nat @ X @ A ) ) ) ).
% 0.34/0.52
% 0.34/0.52 % refl_onD1
% 0.34/0.52 thf(fact_539_refl__onD1,axiom,
% 0.34/0.52 ! [A: set_int,R2: set_Pr958786334691620121nt_int,X: int,Y: int] :
% 0.34/0.52 ( ( refl_on_int @ A @ R2 )
% 0.34/0.52 => ( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ X @ Y ) @ R2 )
% 0.34/0.52 => ( member_int @ X @ A ) ) ) ).
% 0.34/0.52
% 0.34/0.52 % refl_onD1
% 0.34/0.52 thf(fact_540_refl__onD,axiom,
% 0.34/0.52 ! [A: set_Pr4532377907799695533_nat_o,R2: set_Pr2161125870931222855_nat_o,A2: produc3658429121746597890et_nat > $o] :
% 0.34/0.52 ( ( refl_o1075864025857821388_nat_o @ A @ R2 )
% 0.34/0.52 => ( ( member6576561426505652726_nat_o @ A2 @ A )
% 0.34/0.52 => ( member8781333585448626064_nat_o @ ( produc7368190662567826135_nat_o @ A2 @ A2 ) @ R2 ) ) ) ).
% 0.34/0.52
% 0.34/0.52 % refl_onD
% 0.34/0.52 thf(fact_541_refl__onD,axiom,
% 0.34/0.52 ! [A: set_o,R2: set_Product_prod_o_o,A2: $o] :
% 0.34/0.52 ( ( refl_on_o @ A @ R2 )
% 0.34/0.52 => ( ( member_o @ A2 @ A )
% 0.34/0.52 => ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ A2 @ A2 ) @ R2 ) ) ) ).
% 0.34/0.52
% 0.34/0.52 % refl_onD
% 0.34/0.52 thf(fact_542_refl__onD,axiom,
% 0.34/0.52 ! [A: set_Product_unit,R2: set_Pr5094982260447487303t_unit,A2: product_unit] :
% 0.34/0.52 ( ( refl_on_Product_unit @ A @ R2 )
% 0.34/0.52 => ( ( member_Product_unit @ A2 @ A )
% 0.34/0.52 => ( member7821138191923116944t_unit @ ( produc5692694829384537815t_unit @ A2 @ A2 ) @ R2 ) ) ) ).
% 0.34/0.52
% 0.34/0.52 % refl_onD
% 0.34/0.52 thf(fact_543_refl__onD,axiom,
% 0.34/0.52 ! [A: set_nat,R2: set_Pr1261947904930325089at_nat,A2: nat] :
% 0.34/0.52 ( ( refl_on_nat @ A @ R2 )
% 0.34/0.52 => ( ( member_nat @ A2 @ A )
% 0.34/0.52 => ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A2 @ A2 ) @ R2 ) ) ) ).
% 0.34/0.52
% 0.34/0.52 % refl_onD
% 0.34/0.52 thf(fact_544_refl__onD,axiom,
% 0.34/0.52 ! [A: set_int,R2: set_Pr958786334691620121nt_int,A2: int] :
% 0.34/0.52 ( ( refl_on_int @ A @ R2 )
% 0.34/0.52 => ( ( member_int @ A2 @ A )
% 0.34/0.52 => ( member5262025264175285858nt_int @ ( product_Pair_int_int @ A2 @ A2 ) @ R2 ) ) ) ).
% 0.34/0.52
% 0.34/0.52 % refl_onD
% 0.34/0.52 thf(fact_545_total__on__def,axiom,
% 0.34/0.52 ( total_on_nat
% 0.34/0.52 = ( ^ [A3: set_nat,R4: set_Pr1261947904930325089at_nat] :
% 0.34/0.52 ! [X2: nat] :
% 0.34/0.52 ( ( member_nat @ X2 @ A3 )
% 0.34/0.52 => ! [Y3: nat] :
% 0.34/0.52 ( ( member_nat @ Y3 @ A3 )
% 0.34/0.52 => ( ( X2 != Y3 )
% 0.34/0.52 => ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X2 @ Y3 ) @ R4 )
% 0.34/0.52 | ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ Y3 @ X2 ) @ R4 ) ) ) ) ) ) ) ).
% 0.34/0.52
% 0.34/0.52 % total_on_def
% 0.34/0.52 thf(fact_546_total__on__def,axiom,
% 0.34/0.52 ( total_on_int
% 0.34/0.52 = ( ^ [A3: set_int,R4: set_Pr958786334691620121nt_int] :
% 0.34/0.52 ! [X2: int] :
% 0.34/0.52 ( ( member_int @ X2 @ A3 )
% 0.34/0.52 => ! [Y3: int] :
% 0.34/0.52 ( ( member_int @ Y3 @ A3 )
% 0.34/0.52 => ( ( X2 != Y3 )
% 0.34/0.52 => ( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ X2 @ Y3 ) @ R4 )
% 0.34/0.52 | ( member5262025264175285858nt_int @ ( product_Pair_int_int @ Y3 @ X2 ) @ R4 ) ) ) ) ) ) ) ).
% 0.34/0.52
% 0.34/0.52 % total_on_def
% 0.34/0.52 thf(fact_547_total__onI,axiom,
% 0.34/0.52 ! [A: set_Pr4532377907799695533_nat_o,R2: set_Pr2161125870931222855_nat_o] :
% 0.34/0.52 ( ! [X3: produc3658429121746597890et_nat > $o,Y2: produc3658429121746597890et_nat > $o] :
% 0.34/0.52 ( ( member6576561426505652726_nat_o @ X3 @ A )
% 0.34/0.52 => ( ( member6576561426505652726_nat_o @ Y2 @ A )
% 0.34/0.52 => ( ( X3 != Y2 )
% 0.34/0.52 => ( ( member8781333585448626064_nat_o @ ( produc7368190662567826135_nat_o @ X3 @ Y2 ) @ R2 )
% 0.34/0.52 | ( member8781333585448626064_nat_o @ ( produc7368190662567826135_nat_o @ Y2 @ X3 ) @ R2 ) ) ) ) )
% 0.34/0.52 => ( total_7429123086324524369_nat_o @ A @ R2 ) ) ).
% 0.34/0.52
% 0.34/0.52 % total_onI
% 0.34/0.52 thf(fact_548_total__onI,axiom,
% 0.34/0.52 ! [A: set_o,R2: set_Product_prod_o_o] :
% 0.34/0.52 ( ! [X3: $o,Y2: $o] :
% 0.34/0.52 ( ( member_o @ X3 @ A )
% 0.34/0.52 => ( ( member_o @ Y2 @ A )
% 0.34/0.52 => ( ( X3 != Y2 )
% 0.34/0.52 => ( ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ X3 @ Y2 ) @ R2 )
% 0.34/0.52 | ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ Y2 @ X3 ) @ R2 ) ) ) ) )
% 0.34/0.52 => ( total_on_o @ A @ R2 ) ) ).
% 0.34/0.52
% 0.34/0.52 % total_onI
% 0.34/0.52 thf(fact_549_total__onI,axiom,
% 0.34/0.52 ! [A: set_Product_unit,R2: set_Pr5094982260447487303t_unit] :
% 0.34/0.52 ( ! [X3: product_unit,Y2: product_unit] :
% 0.34/0.52 ( ( member_Product_unit @ X3 @ A )
% 0.34/0.52 => ( ( member_Product_unit @ Y2 @ A )
% 0.34/0.52 => ( ( X3 != Y2 )
% 0.34/0.52 => ( ( member7821138191923116944t_unit @ ( produc5692694829384537815t_unit @ X3 @ Y2 ) @ R2 )
% 0.34/0.52 | ( member7821138191923116944t_unit @ ( produc5692694829384537815t_unit @ Y2 @ X3 ) @ R2 ) ) ) ) )
% 0.34/0.52 => ( total_5665700600833547665t_unit @ A @ R2 ) ) ).
% 0.34/0.52
% 0.34/0.52 % total_onI
% 0.34/0.52 thf(fact_550_total__onI,axiom,
% 0.34/0.52 ! [A: set_nat,R2: set_Pr1261947904930325089at_nat] :
% 0.34/0.52 ( ! [X3: nat,Y2: nat] :
% 0.34/0.52 ( ( member_nat @ X3 @ A )
% 0.34/0.52 => ( ( member_nat @ Y2 @ A )
% 0.34/0.52 => ( ( X3 != Y2 )
% 0.34/0.52 => ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X3 @ Y2 ) @ R2 )
% 0.34/0.52 | ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ Y2 @ X3 ) @ R2 ) ) ) ) )
% 0.34/0.52 => ( total_on_nat @ A @ R2 ) ) ).
% 0.34/0.52
% 0.34/0.52 % total_onI
% 0.34/0.52 thf(fact_551_total__onI,axiom,
% 0.34/0.52 ! [A: set_int,R2: set_Pr958786334691620121nt_int] :
% 0.34/0.52 ( ! [X3: int,Y2: int] :
% 0.34/0.52 ( ( member_int @ X3 @ A )
% 0.34/0.52 => ( ( member_int @ Y2 @ A )
% 0.34/0.52 => ( ( X3 != Y2 )
% 0.34/0.52 => ( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ X3 @ Y2 ) @ R2 )
% 0.34/0.52 | ( member5262025264175285858nt_int @ ( product_Pair_int_int @ Y2 @ X3 ) @ R2 ) ) ) ) )
% 0.34/0.52 => ( total_on_int @ A @ R2 ) ) ).
% 0.34/0.52
% 0.34/0.52 % total_onI
% 0.34/0.52 thf(fact_552_total__on__empty,axiom,
% 0.34/0.52 ! [R2: set_Pr5094982260447487303t_unit] : ( total_5665700600833547665t_unit @ bot_bo3957492148770167129t_unit @ R2 ) ).
% 0.34/0.52
% 0.34/0.52 % total_on_empty
% 0.34/0.52 thf(fact_553_total__on__empty,axiom,
% 0.34/0.52 ! [R2: set_Product_prod_o_o] : ( total_on_o @ bot_bot_set_o @ R2 ) ).
% 0.34/0.52
% 0.34/0.52 % total_on_empty
% 0.34/0.52 thf(fact_554_total__on__empty,axiom,
% 0.34/0.52 ! [R2: set_Pr1261947904930325089at_nat] : ( total_on_nat @ bot_bot_set_nat @ R2 ) ).
% 0.34/0.52
% 0.34/0.52 % total_on_empty
% 0.34/0.52 thf(fact_555_total__on__empty,axiom,
% 0.34/0.52 ! [R2: set_Pr958786334691620121nt_int] : ( total_on_int @ bot_bot_set_int @ R2 ) ).
% 0.34/0.52
% 0.34/0.52 % total_on_empty
% 0.34/0.52 thf(fact_556_total__on__empty,axiom,
% 0.34/0.52 ! [R2: set_Pr2161125870931222855_nat_o] : ( total_7429123086324524369_nat_o @ bot_bo7824918357723582233_nat_o @ R2 ) ).
% 0.34/0.52
% 0.34/0.52 % total_on_empty
% 0.34/0.52 thf(fact_557_models__in__range,axiom,
% 0.34/0.52 ! [P: assn,H2: produc3658429121746597890et_nat] :
% 0.34/0.52 ( ( rep_assn @ P @ H2 )
% 0.34/0.52 => ( in_range @ H2 ) ) ).
% 0.34/0.52
% 0.34/0.52 % models_in_range
% 0.34/0.52 thf(fact_558_Id__on__iff,axiom,
% 0.34/0.52 ! [X: produc3658429121746597890et_nat > $o,Y: produc3658429121746597890et_nat > $o,A: set_Pr4532377907799695533_nat_o] :
% 0.34/0.52 ( ( member8781333585448626064_nat_o @ ( produc7368190662567826135_nat_o @ X @ Y ) @ ( id_on_7438695062414216554_nat_o @ A ) )
% 0.34/0.52 = ( ( X = Y )
% 0.34/0.52 & ( member6576561426505652726_nat_o @ X @ A ) ) ) ).
% 0.34/0.52
% 0.34/0.52 % Id_on_iff
% 0.34/0.52 thf(fact_559_Id__on__iff,axiom,
% 0.34/0.52 ! [X: $o,Y: $o,A: set_o] :
% 0.34/0.52 ( ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ X @ Y ) @ ( id_on_o @ A ) )
% 0.34/0.52 = ( ( X = Y )
% 0.34/0.52 & ( member_o @ X @ A ) ) ) ).
% 0.34/0.52
% 0.34/0.52 % Id_on_iff
% 0.34/0.52 thf(fact_560_Id__on__iff,axiom,
% 0.34/0.52 ! [X: product_unit,Y: product_unit,A: set_Product_unit] :
% 0.34/0.52 ( ( member7821138191923116944t_unit @ ( produc5692694829384537815t_unit @ X @ Y ) @ ( id_on_Product_unit @ A ) )
% 0.34/0.52 = ( ( X = Y )
% 0.34/0.52 & ( member_Product_unit @ X @ A ) ) ) ).
% 0.34/0.52
% 0.34/0.52 % Id_on_iff
% 0.34/0.52 thf(fact_561_Id__on__iff,axiom,
% 0.34/0.52 ! [X: nat,Y: nat,A: set_nat] :
% 0.34/0.52 ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ Y ) @ ( id_on_nat @ A ) )
% 0.34/0.52 = ( ( X = Y )
% 0.34/0.52 & ( member_nat @ X @ A ) ) ) ).
% 0.34/0.52
% 0.34/0.52 % Id_on_iff
% 0.34/0.52 thf(fact_562_Id__on__iff,axiom,
% 0.34/0.52 ! [X: int,Y: int,A: set_int] :
% 0.34/0.52 ( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ X @ Y ) @ ( id_on_int @ A ) )
% 0.34/0.52 = ( ( X = Y )
% 0.34/0.52 & ( member_int @ X @ A ) ) ) ).
% 0.34/0.52
% 0.34/0.52 % Id_on_iff
% 0.34/0.52 thf(fact_563_Id__on__eqI,axiom,
% 0.34/0.52 ! [A2: produc3658429121746597890et_nat > $o,B: produc3658429121746597890et_nat > $o,A: set_Pr4532377907799695533_nat_o] :
% 0.34/0.52 ( ( A2 = B )
% 0.34/0.52 => ( ( member6576561426505652726_nat_o @ A2 @ A )
% 0.34/0.52 => ( member8781333585448626064_nat_o @ ( produc7368190662567826135_nat_o @ A2 @ B ) @ ( id_on_7438695062414216554_nat_o @ A ) ) ) ) ).
% 0.34/0.52
% 0.34/0.52 % Id_on_eqI
% 0.34/0.52 thf(fact_564_Id__on__eqI,axiom,
% 0.34/0.52 ! [A2: $o,B: $o,A: set_o] :
% 0.34/0.52 ( ( A2 = B )
% 0.34/0.52 => ( ( member_o @ A2 @ A )
% 0.34/0.52 => ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ A2 @ B ) @ ( id_on_o @ A ) ) ) ) ).
% 0.34/0.52
% 0.34/0.52 % Id_on_eqI
% 0.34/0.52 thf(fact_565_Id__on__eqI,axiom,
% 0.34/0.52 ! [A2: product_unit,B: product_unit,A: set_Product_unit] :
% 0.34/0.52 ( ( A2 = B )
% 0.34/0.52 => ( ( member_Product_unit @ A2 @ A )
% 0.34/0.52 => ( member7821138191923116944t_unit @ ( produc5692694829384537815t_unit @ A2 @ B ) @ ( id_on_Product_unit @ A ) ) ) ) ).
% 0.34/0.52
% 0.34/0.52 % Id_on_eqI
% 0.34/0.52 thf(fact_566_Id__on__eqI,axiom,
% 0.34/0.52 ! [A2: nat,B: nat,A: set_nat] :
% 0.34/0.52 ( ( A2 = B )
% 0.34/0.52 => ( ( member_nat @ A2 @ A )
% 0.34/0.52 => ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A2 @ B ) @ ( id_on_nat @ A ) ) ) ) ).
% 0.34/0.52
% 0.34/0.52 % Id_on_eqI
% 0.34/0.52 thf(fact_567_Id__on__eqI,axiom,
% 0.34/0.52 ! [A2: int,B: int,A: set_int] :
% 0.34/0.52 ( ( A2 = B )
% 0.34/0.52 => ( ( member_int @ A2 @ A )
% 0.34/0.52 => ( member5262025264175285858nt_int @ ( product_Pair_int_int @ A2 @ B ) @ ( id_on_int @ A ) ) ) ) ).
% 0.34/0.52
% 0.34/0.52 % Id_on_eqI
% 0.34/0.52 thf(fact_568_Id__onE,axiom,
% 0.34/0.52 ! [C: produc4928098042776334183_nat_o,A: set_Pr4532377907799695533_nat_o] :
% 0.34/0.52 ( ( member8781333585448626064_nat_o @ C @ ( id_on_7438695062414216554_nat_o @ A ) )
% 0.34/0.52 => ~ ! [X3: produc3658429121746597890et_nat > $o] :
% 0.34/0.52 ( ( member6576561426505652726_nat_o @ X3 @ A )
% 0.34/0.52 => ( C
% 0.34/0.52 != ( produc7368190662567826135_nat_o @ X3 @ X3 ) ) ) ) ).
% 0.34/0.52
% 0.34/0.52 % Id_onE
% 0.34/0.52 thf(fact_569_Id__onE,axiom,
% 0.34/0.52 ! [C: product_prod_o_o,A: set_o] :
% 0.34/0.52 ( ( member7466972457876170832od_o_o @ C @ ( id_on_o @ A ) )
% 0.34/0.52 => ~ ! [X3: $o] :
% 0.34/0.52 ( ( member_o @ X3 @ A )
% 0.34/0.52 => ( C
% 0.34/0.52 != ( product_Pair_o_o @ X3 @ X3 ) ) ) ) ).
% 0.34/0.52
% 0.34/0.52 % Id_onE
% 0.34/0.52 thf(fact_570_Id__onE,axiom,
% 0.34/0.52 ! [C: produc8459935480633519975t_unit,A: set_Product_unit] :
% 0.34/0.52 ( ( member7821138191923116944t_unit @ C @ ( id_on_Product_unit @ A ) )
% 0.34/0.52 => ~ ! [X3: product_unit] :
% 0.34/0.52 ( ( member_Product_unit @ X3 @ A )
% 0.34/0.52 => ( C
% 0.34/0.52 != ( produc5692694829384537815t_unit @ X3 @ X3 ) ) ) ) ).
% 0.34/0.52
% 0.34/0.52 % Id_onE
% 0.34/0.52 thf(fact_571_Id__onE,axiom,
% 0.34/0.52 ! [C: product_prod_nat_nat,A: set_nat] :
% 0.34/0.52 ( ( member8440522571783428010at_nat @ C @ ( id_on_nat @ A ) )
% 0.34/0.52 => ~ ! [X3: nat] :
% 0.34/0.52 ( ( member_nat @ X3 @ A )
% 0.34/0.52 => ( C
% 0.34/0.52 != ( product_Pair_nat_nat @ X3 @ X3 ) ) ) ) ).
% 0.34/0.52
% 0.34/0.52 % Id_onE
% 0.34/0.52 thf(fact_572_Id__onE,axiom,
% 0.34/0.52 ! [C: product_prod_int_int,A: set_int] :
% 0.34/0.52 ( ( member5262025264175285858nt_int @ C @ ( id_on_int @ A ) )
% 0.34/0.52 => ~ ! [X3: int] :
% 0.34/0.52 ( ( member_int @ X3 @ A )
% 0.34/0.52 => ( C
% 0.34/0.52 != ( product_Pair_int_int @ X3 @ X3 ) ) ) ) ).
% 0.34/0.52
% 0.34/0.52 % Id_onE
% 0.34/0.52 thf(fact_573_lnear__order__on__empty,axiom,
% 0.34/0.52 order_888282469287196226t_unit @ bot_bo3957492148770167129t_unit @ bot_bo1654022762305692595t_unit ).
% 0.34/0.52
% 0.34/0.52 % lnear_order_on_empty
% 0.34/0.52 thf(fact_574_lnear__order__on__empty,axiom,
% 0.34/0.52 order_6238756238976269133r_on_o @ bot_bot_set_o @ bot_bo7073875226086086771od_o_o ).
% 0.34/0.52
% 0.34/0.52 % lnear_order_on_empty
% 0.34/0.52 thf(fact_575_lnear__order__on__empty,axiom,
% 0.34/0.52 order_4473980167227706203on_nat @ bot_bot_set_nat @ bot_bo2099793752762293965at_nat ).
% 0.34/0.52
% 0.34/0.52 % lnear_order_on_empty
% 0.34/0.52 thf(fact_576_lnear__order__on__empty,axiom,
% 0.34/0.52 order_4471489696718655927on_int @ bot_bot_set_int @ bot_bo1796632182523588997nt_int ).
% 0.34/0.52
% 0.34/0.52 % lnear_order_on_empty
% 0.34/0.52 thf(fact_577_lnear__order__on__empty,axiom,
% 0.34/0.52 order_1145710701291701250_nat_o @ bot_bo7824918357723582233_nat_o @ bot_bo3657187498359500723_nat_o ).
% 0.34/0.52
% 0.34/0.52 % lnear_order_on_empty
% 0.34/0.52 thf(fact_578_refl__on__empty,axiom,
% 0.34/0.52 refl_on_Product_unit @ bot_bo3957492148770167129t_unit @ bot_bo1654022762305692595t_unit ).
% 0.34/0.52
% 0.34/0.52 % refl_on_empty
% 0.34/0.52 thf(fact_579_refl__on__empty,axiom,
% 0.34/0.52 refl_on_o @ bot_bot_set_o @ bot_bo7073875226086086771od_o_o ).
% 0.34/0.52
% 0.34/0.52 % refl_on_empty
% 0.34/0.52 thf(fact_580_refl__on__empty,axiom,
% 0.34/0.52 refl_on_nat @ bot_bot_set_nat @ bot_bo2099793752762293965at_nat ).
% 0.34/0.52
% 0.34/0.52 % refl_on_empty
% 0.34/0.52 thf(fact_581_refl__on__empty,axiom,
% 0.34/0.52 refl_on_int @ bot_bot_set_int @ bot_bo1796632182523588997nt_int ).
% 0.34/0.52
% 0.34/0.52 % refl_on_empty
% 0.34/0.52 thf(fact_582_refl__on__empty,axiom,
% 0.34/0.52 refl_o1075864025857821388_nat_o @ bot_bo7824918357723582233_nat_o @ bot_bo3657187498359500723_nat_o ).
% 0.34/0.52
% 0.34/0.52 % refl_on_empty
% 0.34/0.52 thf(fact_583_properD1,axiom,
% 0.34/0.52 ! [P: produc3658429121746597890et_nat > $o,H2: heap_e7401611519738050253t_unit,As: set_nat] :
% 0.34/0.52 ( ( proper @ P )
% 0.34/0.52 => ( ( P @ ( produc7507926704131184380et_nat @ H2 @ As ) )
% 0.34/0.52 => ( in_range @ ( produc7507926704131184380et_nat @ H2 @ As ) ) ) ) ).
% 0.34/0.52
% 0.34/0.52 % properD1
% 0.34/0.52 thf(fact_584_proper__iff,axiom,
% 0.34/0.52 ! [P: produc3658429121746597890et_nat > $o,As: set_nat,H2: heap_e7401611519738050253t_unit,H3: heap_e7401611519738050253t_unit] :
% 0.34/0.52 ( ( proper @ P )
% 0.34/0.52 => ( ( relH @ As @ H2 @ H3 )
% 0.34/0.52 => ( ( in_range @ ( produc7507926704131184380et_nat @ H3 @ As ) )
% 0.34/0.52 => ( ( P @ ( produc7507926704131184380et_nat @ H2 @ As ) )
% 0.34/0.52 = ( P @ ( produc7507926704131184380et_nat @ H3 @ As ) ) ) ) ) ) ).
% 0.34/0.52
% 0.34/0.52 % proper_iff
% 0.34/0.52 thf(fact_585_proper__def,axiom,
% 0.34/0.52 ( proper
% 0.34/0.52 = ( ^ [P2: produc3658429121746597890et_nat > $o] :
% 0.34/0.52 ! [H4: heap_e7401611519738050253t_unit,H5: heap_e7401611519738050253t_unit,As3: set_nat] :
% 0.34/0.52 ( ( ( P2 @ ( produc7507926704131184380et_nat @ H4 @ As3 ) )
% 0.34/0.52 => ( in_range @ ( produc7507926704131184380et_nat @ H4 @ As3 ) ) )
% 0.34/0.52 & ( ( ( P2 @ ( produc7507926704131184380et_nat @ H4 @ As3 ) )
% 0.34/0.52 & ( relH @ As3 @ H4 @ H5 )
% 0.34/0.52 & ( in_range @ ( produc7507926704131184380et_nat @ H5 @ As3 ) ) )
% 0.34/0.52 => ( P2 @ ( produc7507926704131184380et_nat @ H5 @ As3 ) ) ) ) ) ) ).
% 0.34/0.52
% 0.34/0.52 % proper_def
% 0.34/0.52 thf(fact_586_properD2,axiom,
% 0.34/0.52 ! [P: produc3658429121746597890et_nat > $o,H2: heap_e7401611519738050253t_unit,As: set_nat,H3: heap_e7401611519738050253t_unit] :
% 0.34/0.52 ( ( proper @ P )
% 0.34/0.52 => ( ( P @ ( produc7507926704131184380et_nat @ H2 @ As ) )
% 0.34/0.52 => ( ( relH @ As @ H2 @ H3 )
% 0.34/0.52 => ( ( in_range @ ( produc7507926704131184380et_nat @ H3 @ As ) )
% 0.34/0.52 => ( P @ ( produc7507926704131184380et_nat @ H3 @ As ) ) ) ) ) ) ).
% 0.34/0.52
% 0.34/0.52 % properD2
% 0.34/0.52 thf(fact_587_properI,axiom,
% 0.34/0.52 ! [P: produc3658429121746597890et_nat > $o] :
% 0.34/0.52 ( ! [As2: set_nat,H: heap_e7401611519738050253t_unit] :
% 0.34/0.52 ( ( P @ ( produc7507926704131184380et_nat @ H @ As2 ) )
% 0.34/0.52 => ( in_range @ ( produc7507926704131184380et_nat @ H @ As2 ) ) )
% 0.34/0.52 => ( ! [As2: set_nat,H: heap_e7401611519738050253t_unit,H6: heap_e7401611519738050253t_unit] :
% 0.34/0.52 ( ( P @ ( produc7507926704131184380et_nat @ H @ As2 ) )
% 0.34/0.52 => ( ( relH @ As2 @ H @ H6 )
% 0.34/0.52 => ( ( in_range @ ( produc7507926704131184380et_nat @ H6 @ As2 ) )
% 0.34/0.52 => ( P @ ( produc7507926704131184380et_nat @ H6 @ As2 ) ) ) ) )
% 0.34/0.52 => ( proper @ P ) ) ) ).
% 0.34/0.52
% 0.34/0.52 % properI
% 0.34/0.52 thf(fact_588_pair__vimage__is__Image,axiom,
% 0.34/0.52 ! [U: heap_e7401611519738050253t_unit,E: set_Pr3948176798113811640et_nat] :
% 0.34/0.52 ( ( vimage5412618565253405765et_nat @ ( produc7507926704131184380et_nat @ U ) @ E )
% 0.34/0.52 = ( image_3942343239045813184et_nat @ E @ ( insert8563697753282147485t_unit @ U @ bot_bo3686245532977126937t_unit ) ) ) ).
% 0.34/0.52
% 0.34/0.52 % pair_vimage_is_Image
% 0.34/0.52 thf(fact_589_pair__vimage__is__Image,axiom,
% 0.34/0.52 ! [U: nat,E: set_Pr1261947904930325089at_nat] :
% 0.34/0.52 ( ( vimage8013328719654469172at_nat @ ( product_Pair_nat_nat @ U ) @ E )
% 0.34/0.52 = ( image_nat_nat @ E @ ( insert_nat @ U @ bot_bot_set_nat ) ) ) ).
% 0.34/0.52
% 0.34/0.52 % pair_vimage_is_Image
% 0.34/0.52 thf(fact_590_pair__vimage__is__Image,axiom,
% 0.34/0.52 ! [U: int,E: set_Pr958786334691620121nt_int] :
% 0.34/0.52 ( ( vimage7872673496510460816nt_int @ ( product_Pair_int_int @ U ) @ E )
% 0.34/0.52 = ( image_int_int @ E @ ( insert_int @ U @ bot_bot_set_int ) ) ) ).
% 0.34/0.52
% 0.34/0.52 % pair_vimage_is_Image
% 0.34/0.52 thf(fact_591_pair__vimage__is__Image,axiom,
% 0.34/0.52 ! [U: produc3658429121746597890et_nat > $o,E: set_Pr3286484037609594932et_nat] :
% 0.34/0.52 ( ( vimage1990136491886319165et_nat @ ( produc5001842942810119800et_nat @ U ) @ E )
% 0.34/0.52 = ( image_1814916760386648892et_nat @ E @ ( insert5175938949040314269_nat_o @ U @ bot_bo7824918357723582233_nat_o ) ) ) ).
% 0.34/0.52
% 0.34/0.52 % pair_vimage_is_Image
% 0.34/0.52 thf(fact_592_pair__vimage__is__Image,axiom,
% 0.34/0.52 ! [U: produc3658429121746597890et_nat > $o,E: set_Pr8536935166611901872et_nat] :
% 0.34/0.52 ( ( vimage7160910655466636085et_nat @ ( produc2245416461498447860et_nat @ U ) @ E )
% 0.34/0.52 = ( image_7240710759097644216et_nat @ E @ ( insert5175938949040314269_nat_o @ U @ bot_bo7824918357723582233_nat_o ) ) ) ).
% 0.34/0.52
% 0.34/0.52 % pair_vimage_is_Image
% 0.34/0.52 thf(fact_593_Range__insert,axiom,
% 0.34/0.52 ! [A2: heap_e7401611519738050253t_unit,B: set_nat,R2: set_Pr3948176798113811640et_nat] :
% 0.34/0.52 ( ( range_9132311542094983838et_nat @ ( insert1271470778893645362et_nat @ ( produc7507926704131184380et_nat @ A2 @ B ) @ R2 ) )
% 0.34/0.52 = ( insert_set_nat @ B @ ( range_9132311542094983838et_nat @ R2 ) ) ) ).
% 0.34/0.52
% 0.34/0.52 % Range_insert
% 0.34/0.52 thf(fact_594_Range__insert,axiom,
% 0.34/0.52 ! [A2: nat,B: nat,R2: set_Pr1261947904930325089at_nat] :
% 0.34/0.52 ( ( range_nat_nat @ ( insert8211810215607154385at_nat @ ( product_Pair_nat_nat @ A2 @ B ) @ R2 ) )
% 0.34/0.52 = ( insert_nat @ B @ ( range_nat_nat @ R2 ) ) ) ).
% 0.34/0.52
% 0.34/0.52 % Range_insert
% 0.34/0.52 thf(fact_595_Range__insert,axiom,
% 0.34/0.52 ! [A2: int,B: int,R2: set_Pr958786334691620121nt_int] :
% 0.34/0.52 ( ( range_int_int @ ( insert5033312907999012233nt_int @ ( product_Pair_int_int @ A2 @ B ) @ R2 ) )
% 0.34/0.52 = ( insert_int @ B @ ( range_int_int @ R2 ) ) ) ).
% 0.34/0.52
% 0.34/0.52 % Range_insert
% 0.34/0.52 thf(fact_596_Range__insert,axiom,
% 0.34/0.52 ! [A2: produc3658429121746597890et_nat > $o,B: produc3658429121746597890et_nat,R2: set_Pr3286484037609594932et_nat] :
% 0.34/0.52 ( ( range_3398330936903821850et_nat @ ( insert3111075074545270702et_nat @ ( produc5001842942810119800et_nat @ A2 @ B ) @ R2 ) )
% 0.34/0.52 = ( insert1271470778893645362et_nat @ B @ ( range_3398330936903821850et_nat @ R2 ) ) ) ).
% 0.34/0.52
% 0.34/0.52 % Range_insert
% 0.34/0.52 thf(fact_597_Range__insert,axiom,
% 0.34/0.52 ! [A2: produc3658429121746597890et_nat > $o,B: produc3925858234332021118et_nat,R2: set_Pr8536935166611901872et_nat] :
% 0.34/0.52 ( ( range_7260058354244137878et_nat @ ( insert8297122186708525866et_nat @ ( produc2245416461498447860et_nat @ A2 @ B ) @ R2 ) )
% 0.34/0.52 = ( insert3111075074545270702et_nat @ B @ ( range_7260058354244137878et_nat @ R2 ) ) ) ).
% 0.34/0.52
% 0.34/0.52 % Range_insert
% 0.34/0.52 thf(fact_598_Domain__insert,axiom,
% 0.34/0.52 ! [A2: heap_e7401611519738050253t_unit,B: set_nat,R2: set_Pr3948176798113811640et_nat] :
% 0.34/0.52 ( ( domain2007548326387822151et_nat @ ( insert1271470778893645362et_nat @ ( produc7507926704131184380et_nat @ A2 @ B ) @ R2 ) )
% 0.34/0.52 = ( insert8563697753282147485t_unit @ A2 @ ( domain2007548326387822151et_nat @ R2 ) ) ) ).
% 0.34/0.52
% 0.34/0.52 % Domain_insert
% 0.34/0.52 thf(fact_599_Domain__insert,axiom,
% 0.34/0.52 ! [A2: nat,B: nat,R2: set_Pr1261947904930325089at_nat] :
% 0.34/0.52 ( ( domain_nat_nat @ ( insert8211810215607154385at_nat @ ( product_Pair_nat_nat @ A2 @ B ) @ R2 ) )
% 0.34/0.52 = ( insert_nat @ A2 @ ( domain_nat_nat @ R2 ) ) ) ).
% 0.34/0.52
% 0.34/0.52 % Domain_insert
% 0.34/0.52 thf(fact_600_Domain__insert,axiom,
% 0.34/0.52 ! [A2: int,B: int,R2: set_Pr958786334691620121nt_int] :
% 0.34/0.52 ( ( domain_int_int @ ( insert5033312907999012233nt_int @ ( product_Pair_int_int @ A2 @ B ) @ R2 ) )
% 0.34/0.52 = ( insert_int @ A2 @ ( domain_int_int @ R2 ) ) ) ).
% 0.34/0.52
% 0.34/0.52 % Domain_insert
% 0.34/0.52 thf(fact_601_Domain__insert,axiom,
% 0.34/0.52 ! [A2: produc3658429121746597890et_nat > $o,B: produc3658429121746597890et_nat,R2: set_Pr3286484037609594932et_nat] :
% 0.34/0.52 ( ( domain2358961613337345475et_nat @ ( insert3111075074545270702et_nat @ ( produc5001842942810119800et_nat @ A2 @ B ) @ R2 ) )
% 0.34/0.52 = ( insert5175938949040314269_nat_o @ A2 @ ( domain2358961613337345475et_nat @ R2 ) ) ) ).
% 0.34/0.52
% 0.34/0.52 % Domain_insert
% 0.34/0.52 thf(fact_602_Domain__insert,axiom,
% 0.34/0.52 ! [A2: produc3658429121746597890et_nat > $o,B: produc3925858234332021118et_nat,R2: set_Pr8536935166611901872et_nat] :
% 0.34/0.52 ( ( domain3618434998481624895et_nat @ ( insert8297122186708525866et_nat @ ( produc2245416461498447860et_nat @ A2 @ B ) @ R2 ) )
% 0.34/0.52 = ( insert5175938949040314269_nat_o @ A2 @ ( domain3618434998481624895et_nat @ R2 ) ) ) ).
% 0.34/0.52
% 0.34/0.52 % Domain_insert
% 0.34/0.52 thf(fact_603_preorder__on__empty,axiom,
% 0.34/0.52 order_7534304569044981540t_unit @ bot_bo3957492148770167129t_unit @ bot_bo1654022762305692595t_unit ).
% 0.34/0.52
% 0.34/0.52 % preorder_on_empty
% 0.34/0.52 thf(fact_604_preorder__on__empty,axiom,
% 0.34/0.52 order_preorder_on_o @ bot_bot_set_o @ bot_bo7073875226086086771od_o_o ).
% 0.34/0.52
% 0.34/0.52 % preorder_on_empty
% 0.34/0.52 thf(fact_605_preorder__on__empty,axiom,
% 0.34/0.52 order_4861654808422542329on_nat @ bot_bot_set_nat @ bot_bo2099793752762293965at_nat ).
% 0.34/0.52
% 0.34/0.52 % preorder_on_empty
% 0.34/0.52 thf(fact_606_preorder__on__empty,axiom,
% 0.34/0.52 order_4859164337913492053on_int @ bot_bot_set_int @ bot_bo1796632182523588997nt_int ).
% 0.34/0.52
% 0.34/0.52 % preorder_on_empty
% 0.34/0.52 thf(fact_607_preorder__on__empty,axiom,
% 0.34/0.52 order_174704566861754596_nat_o @ bot_bo7824918357723582233_nat_o @ bot_bo3657187498359500723_nat_o ).
% 0.34/0.52
% 0.34/0.52 % preorder_on_empty
% 0.34/0.52 thf(fact_608_trancl__single,axiom,
% 0.34/0.52 ! [A2: nat,B: nat] :
% 0.34/0.52 ( ( transi6264000038957366511cl_nat @ ( insert8211810215607154385at_nat @ ( product_Pair_nat_nat @ A2 @ B ) @ bot_bo2099793752762293965at_nat ) )
% 0.34/0.52 = ( insert8211810215607154385at_nat @ ( product_Pair_nat_nat @ A2 @ B ) @ bot_bo2099793752762293965at_nat ) ) ).
% 0.34/0.52
% 0.34/0.52 % trancl_single
% 0.34/0.52 thf(fact_609_trancl__single,axiom,
% 0.34/0.52 ! [A2: int,B: int] :
% 0.34/0.52 ( ( transi6261509568448316235cl_int @ ( insert5033312907999012233nt_int @ ( product_Pair_int_int @ A2 @ B ) @ bot_bo1796632182523588997nt_int ) )
% 0.34/0.52 = ( insert5033312907999012233nt_int @ ( product_Pair_int_int @ A2 @ B ) @ bot_bo1796632182523588997nt_int ) ) ).
% 0.34/0.52
% 0.34/0.52 % trancl_single
% 0.34/0.52 thf(fact_610_trans__singleton,axiom,
% 0.34/0.52 ! [A2: nat] : ( trans_nat @ ( insert8211810215607154385at_nat @ ( product_Pair_nat_nat @ A2 @ A2 ) @ bot_bo2099793752762293965at_nat ) ) ).
% 0.34/0.52
% 0.34/0.52 % trans_singleton
% 0.34/0.52 thf(fact_611_trans__singleton,axiom,
% 0.34/0.52 ! [A2: int] : ( trans_int @ ( insert5033312907999012233nt_int @ ( product_Pair_int_int @ A2 @ A2 ) @ bot_bo1796632182523588997nt_int ) ) ).
% 0.34/0.52
% 0.34/0.52 % trans_singleton
% 0.34/0.52 thf(fact_612_mult_Osafe__commute,axiom,
% 0.34/0.52 ! [X: assn,Y: assn,A2: assn,B: assn] :
% 0.34/0.52 ( ( syntax7398250324933576852n_assn @ ( times_times_assn @ X @ Y ) @ A2 )
% 0.34/0.52 => ( ( times_times_assn @ A2 @ B )
% 0.34/0.52 = ( times_times_assn @ B @ A2 ) ) ) ).
% 0.34/0.52
% 0.34/0.52 % mult.safe_commute
% 0.34/0.52 thf(fact_613_mult_Osafe__commute,axiom,
% 0.34/0.52 ! [X: rat,Y: rat,A2: rat,B: rat] :
% 0.34/0.52 ( ( syntax3730441303064801268at_rat @ ( times_times_rat @ X @ Y ) @ A2 )
% 0.34/0.52 => ( ( times_times_rat @ A2 @ B )
% 0.34/0.52 = ( times_times_rat @ B @ A2 ) ) ) ).
% 0.34/0.52
% 0.34/0.52 % mult.safe_commute
% 0.34/0.52 thf(fact_614_mult_Osafe__commute,axiom,
% 0.34/0.52 ! [X: nat,Y: nat,A2: nat,B: nat] :
% 0.34/0.52 ( ( syntax4682126007086162916at_nat @ ( times_times_nat @ X @ Y ) @ A2 )
% 0.34/0.52 => ( ( times_times_nat @ A2 @ B )
% 0.34/0.52 = ( times_times_nat @ B @ A2 ) ) ) ).
% 0.34/0.52
% 0.34/0.52 % mult.safe_commute
% 0.34/0.52 thf(fact_615_mult_Osafe__commute,axiom,
% 0.34/0.52 ! [X: int,Y: int,A2: int,B: int] :
% 0.34/0.52 ( ( syntax5678989248478167196nt_int @ ( times_times_int @ X @ Y ) @ A2 )
% 0.34/0.52 => ( ( times_times_int @ A2 @ B )
% 0.34/0.52 = ( times_times_int @ B @ A2 ) ) ) ).
% 0.34/0.52
% 0.34/0.52 % mult.safe_commute
% 0.34/0.52 thf(fact_616_vimage__eq,axiom,
% 0.34/0.52 ! [A2: nat,F: nat > nat,B4: set_nat] :
% 0.34/0.52 ( ( member_nat @ A2 @ ( vimage_nat_nat @ F @ B4 ) )
% 0.34/0.52 = ( member_nat @ ( F @ A2 ) @ B4 ) ) ).
% 0.34/0.52
% 0.34/0.52 % vimage_eq
% 0.34/0.52 thf(fact_617_vimage__eq,axiom,
% 0.34/0.52 ! [A2: nat,F: nat > int,B4: set_int] :
% 0.34/0.52 ( ( member_nat @ A2 @ ( vimage_nat_int @ F @ B4 ) )
% 0.34/0.52 = ( member_int @ ( F @ A2 ) @ B4 ) ) ).
% 0.34/0.52
% 0.34/0.52 % vimage_eq
% 0.34/0.52 thf(fact_618_vimage__eq,axiom,
% 0.34/0.52 ! [A2: nat,F: nat > $o,B4: set_o] :
% 0.34/0.52 ( ( member_nat @ A2 @ ( vimage_nat_o @ F @ B4 ) )
% 0.34/0.52 = ( member_o @ ( F @ A2 ) @ B4 ) ) ).
% 0.34/0.52
% 0.34/0.52 % vimage_eq
% 0.34/0.52 thf(fact_619_vimage__eq,axiom,
% 0.34/0.52 ! [A2: nat,F: nat > product_unit,B4: set_Product_unit] :
% 0.34/0.52 ( ( member_nat @ A2 @ ( vimage4884490618288580032t_unit @ F @ B4 ) )
% 0.34/0.52 = ( member_Product_unit @ ( F @ A2 ) @ B4 ) ) ).
% 0.34/0.52
% 0.34/0.52 % vimage_eq
% 0.34/0.52 thf(fact_620_vimage__eq,axiom,
% 0.34/0.52 ! [A2: int,F: int > nat,B4: set_nat] :
% 0.34/0.52 ( ( member_int @ A2 @ ( vimage_int_nat @ F @ B4 ) )
% 0.34/0.52 = ( member_nat @ ( F @ A2 ) @ B4 ) ) ).
% 0.34/0.52
% 0.34/0.52 % vimage_eq
% 0.34/0.52 thf(fact_621_vimage__eq,axiom,
% 0.34/0.52 ! [A2: int,F: int > int,B4: set_int] :
% 0.34/0.52 ( ( member_int @ A2 @ ( vimage_int_int @ F @ B4 ) )
% 0.34/0.52 = ( member_int @ ( F @ A2 ) @ B4 ) ) ).
% 0.34/0.52
% 0.34/0.52 % vimage_eq
% 0.34/0.52 thf(fact_622_vimage__eq,axiom,
% 0.34/0.52 ! [A2: int,F: int > $o,B4: set_o] :
% 0.34/0.52 ( ( member_int @ A2 @ ( vimage_int_o @ F @ B4 ) )
% 0.34/0.52 = ( member_o @ ( F @ A2 ) @ B4 ) ) ).
% 0.34/0.52
% 0.34/0.52 % vimage_eq
% 0.34/0.52 thf(fact_623_vimage__eq,axiom,
% 0.34/0.52 ! [A2: int,F: int > product_unit,B4: set_Product_unit] :
% 0.34/0.52 ( ( member_int @ A2 @ ( vimage2000912565490760292t_unit @ F @ B4 ) )
% 0.34/0.52 = ( member_Product_unit @ ( F @ A2 ) @ B4 ) ) ).
% 0.34/0.52
% 0.34/0.52 % vimage_eq
% 0.34/0.52 thf(fact_624_vimage__eq,axiom,
% 0.34/0.52 ! [A2: $o,F: $o > nat,B4: set_nat] :
% 0.34/0.52 ( ( member_o @ A2 @ ( vimage_o_nat @ F @ B4 ) )
% 0.34/0.52 = ( member_nat @ ( F @ A2 ) @ B4 ) ) ).
% 0.34/0.52
% 0.34/0.52 % vimage_eq
% 0.34/0.52 thf(fact_625_vimage__eq,axiom,
% 0.34/0.52 ! [A2: $o,F: $o > int,B4: set_int] :
% 0.34/0.52 ( ( member_o @ A2 @ ( vimage_o_int @ F @ B4 ) )
% 0.34/0.52 = ( member_int @ ( F @ A2 ) @ B4 ) ) ).
% 0.34/0.52
% 0.34/0.52 % vimage_eq
% 0.34/0.52 thf(fact_626_vimageI,axiom,
% 0.34/0.52 ! [F: nat > nat,A2: nat,B: nat,B4: set_nat] :
% 0.34/0.52 ( ( ( F @ A2 )
% 0.34/0.52 = B )
% 0.34/0.52 => ( ( member_nat @ B @ B4 )
% 0.34/0.52 => ( member_nat @ A2 @ ( vimage_nat_nat @ F @ B4 ) ) ) ) ).
% 0.34/0.52
% 0.34/0.52 % vimageI
% 0.34/0.52 thf(fact_627_vimageI,axiom,
% 0.34/0.52 ! [F: int > nat,A2: int,B: nat,B4: set_nat] :
% 0.34/0.52 ( ( ( F @ A2 )
% 0.34/0.52 = B )
% 0.34/0.52 => ( ( member_nat @ B @ B4 )
% 0.34/0.52 => ( member_int @ A2 @ ( vimage_int_nat @ F @ B4 ) ) ) ) ).
% 0.34/0.52
% 0.34/0.52 % vimageI
% 0.34/0.52 thf(fact_628_vimageI,axiom,
% 0.34/0.52 ! [F: $o > nat,A2: $o,B: nat,B4: set_nat] :
% 0.34/0.52 ( ( ( F @ A2 )
% 0.34/0.52 = B )
% 0.34/0.52 => ( ( member_nat @ B @ B4 )
% 0.34/0.52 => ( member_o @ A2 @ ( vimage_o_nat @ F @ B4 ) ) ) ) ).
% 0.34/0.52
% 0.34/0.52 % vimageI
% 0.34/0.52 thf(fact_629_vimageI,axiom,
% 0.34/0.52 ! [F: product_unit > nat,A2: product_unit,B: nat,B4: set_nat] :
% 0.34/0.52 ( ( ( F @ A2 )
% 0.34/0.52 = B )
% 0.34/0.52 => ( ( member_nat @ B @ B4 )
% 0.34/0.52 => ( member_Product_unit @ A2 @ ( vimage6253328473476588386it_nat @ F @ B4 ) ) ) ) ).
% 0.34/0.52
% 0.34/0.52 % vimageI
% 0.34/0.52 thf(fact_630_vimageI,axiom,
% 0.34/0.52 ! [F: nat > int,A2: nat,B: int,B4: set_int] :
% 0.34/0.52 ( ( ( F @ A2 )
% 0.34/0.52 = B )
% 0.34/0.52 => ( ( member_int @ B @ B4 )
% 0.34/0.52 => ( member_nat @ A2 @ ( vimage_nat_int @ F @ B4 ) ) ) ) ).
% 0.34/0.52
% 0.34/0.52 % vimageI
% 0.34/0.52 thf(fact_631_vimageI,axiom,
% 0.34/0.52 ! [F: int > int,A2: int,B: int,B4: set_int] :
% 0.34/0.52 ( ( ( F @ A2 )
% 0.34/0.52 = B )
% 0.34/0.52 => ( ( member_int @ B @ B4 )
% 0.34/0.52 => ( member_int @ A2 @ ( vimage_int_int @ F @ B4 ) ) ) ) ).
% 0.34/0.52
% 0.34/0.52 % vimageI
% 0.34/0.52 thf(fact_632_vimageI,axiom,
% 0.34/0.52 ! [F: $o > int,A2: $o,B: int,B4: set_int] :
% 0.34/0.52 ( ( ( F @ A2 )
% 0.34/0.52 = B )
% 0.34/0.52 => ( ( member_int @ B @ B4 )
% 0.34/0.52 => ( member_o @ A2 @ ( vimage_o_int @ F @ B4 ) ) ) ) ).
% 0.34/0.52
% 0.34/0.52 % vimageI
% 0.34/0.52 thf(fact_633_vimageI,axiom,
% 0.34/0.52 ! [F: product_unit > int,A2: product_unit,B: int,B4: set_int] :
% 0.34/0.52 ( ( ( F @ A2 )
% 0.34/0.52 = B )
% 0.34/0.52 => ( ( member_int @ B @ B4 )
% 0.34/0.52 => ( member_Product_unit @ A2 @ ( vimage6250838002967538110it_int @ F @ B4 ) ) ) ) ).
% 0.34/0.52
% 0.34/0.52 % vimageI
% 0.34/0.52 thf(fact_634_vimageI,axiom,
% 0.34/0.52 ! [F: nat > $o,A2: nat,B: $o,B4: set_o] :
% 0.34/0.52 ( ( ( F @ A2 )
% 0.34/0.52 = B )
% 0.34/0.52 => ( ( member_o @ B @ B4 )
% 0.34/0.52 => ( member_nat @ A2 @ ( vimage_nat_o @ F @ B4 ) ) ) ) ).
% 0.34/0.52
% 0.34/0.52 % vimageI
% 0.34/0.52 thf(fact_635_vimageI,axiom,
% 0.34/0.52 ! [F: int > $o,A2: int,B: $o,B4: set_o] :
% 0.34/0.52 ( ( ( F @ A2 )
% 0.34/0.52 = B )
% 0.34/0.52 => ( ( member_o @ B @ B4 )
% 0.34/0.52 => ( member_int @ A2 @ ( vimage_int_o @ F @ B4 ) ) ) ) ).
% 0.34/0.52
% 0.34/0.52 % vimageI
% 0.34/0.52 thf(fact_636_vimage__empty,axiom,
% 0.34/0.52 ! [F: product_unit > product_unit] :
% 0.34/0.52 ( ( vimage7995052115951654139t_unit @ F @ bot_bo3957492148770167129t_unit )
% 0.34/0.52 = bot_bo3957492148770167129t_unit ) ).
% 0.34/0.52
% 0.34/0.52 % vimage_empty
% 0.34/0.52 thf(fact_637_vimage__empty,axiom,
% 0.34/0.52 ! [F: $o > product_unit] :
% 0.34/0.52 ( ( vimage4222377557387472176t_unit @ F @ bot_bo3957492148770167129t_unit )
% 0.34/0.52 = bot_bot_set_o ) ).
% 0.34/0.52
% 0.34/0.52 % vimage_empty
% 0.34/0.52 thf(fact_638_vimage__empty,axiom,
% 0.34/0.52 ! [F: nat > product_unit] :
% 0.34/0.52 ( ( vimage4884490618288580032t_unit @ F @ bot_bo3957492148770167129t_unit )
% 0.34/0.52 = bot_bot_set_nat ) ).
% 0.34/0.52
% 0.34/0.52 % vimage_empty
% 0.34/0.52 thf(fact_639_vimage__empty,axiom,
% 0.34/0.52 ! [F: int > product_unit] :
% 0.34/0.52 ( ( vimage2000912565490760292t_unit @ F @ bot_bo3957492148770167129t_unit )
% 0.34/0.52 = bot_bot_set_int ) ).
% 0.34/0.52
% 0.34/0.52 % vimage_empty
% 0.34/0.52 thf(fact_640_vimage__empty,axiom,
% 0.34/0.52 ! [F: product_unit > $o] :
% 0.34/0.52 ( ( vimage8808348054653608838unit_o @ F @ bot_bot_set_o )
% 0.34/0.52 = bot_bo3957492148770167129t_unit ) ).
% 0.34/0.52
% 0.34/0.52 % vimage_empty
% 0.34/0.52 thf(fact_641_vimage__empty,axiom,
% 0.34/0.52 ! [F: $o > $o] :
% 0.34/0.52 ( ( vimage_o_o @ F @ bot_bot_set_o )
% 0.34/0.52 = bot_bot_set_o ) ).
% 0.34/0.52
% 0.34/0.52 % vimage_empty
% 0.34/0.52 thf(fact_642_vimage__empty,axiom,
% 0.34/0.52 ! [F: nat > $o] :
% 0.34/0.52 ( ( vimage_nat_o @ F @ bot_bot_set_o )
% 0.34/0.52 = bot_bot_set_nat ) ).
% 0.34/0.52
% 0.34/0.52 % vimage_empty
% 0.34/0.52 thf(fact_643_vimage__empty,axiom,
% 0.34/0.52 ! [F: int > $o] :
% 0.34/0.52 ( ( vimage_int_o @ F @ bot_bot_set_o )
% 0.34/0.52 = bot_bot_set_int ) ).
% 0.34/0.52
% 0.34/0.52 % vimage_empty
% 0.34/0.52 thf(fact_644_vimage__empty,axiom,
% 0.34/0.52 ! [F: product_unit > nat] :
% 0.34/0.52 ( ( vimage6253328473476588386it_nat @ F @ bot_bot_set_nat )
% 0.34/0.52 = bot_bo3957492148770167129t_unit ) ).
% 0.34/0.52
% 0.34/0.52 % vimage_empty
% 0.34/0.52 thf(fact_645_vimage__empty,axiom,
% 0.34/0.52 ! [F: $o > nat] :
% 0.34/0.52 ( ( vimage_o_nat @ F @ bot_bot_set_nat )
% 0.34/0.52 = bot_bot_set_o ) ).
% 0.34/0.52
% 0.34/0.52 % vimage_empty
% 0.34/0.52 thf(fact_646_relH__sym,axiom,
% 0.34/0.52 ! [As: set_nat,H2: heap_e7401611519738050253t_unit,H3: heap_e7401611519738050253t_unit] :
% 0.34/0.52 ( ( relH @ As @ H2 @ H3 )
% 0.34/0.52 => ( relH @ As @ H3 @ H2 ) ) ).
% 0.34/0.52
% 0.34/0.52 % relH_sym
% 0.34/0.52 thf(fact_647_relH__trans,axiom,
% 0.34/0.52 ! [As: set_nat,H12: heap_e7401611519738050253t_unit,H23: heap_e7401611519738050253t_unit,H32: heap_e7401611519738050253t_unit] :
% 0.34/0.52 ( ( relH @ As @ H12 @ H23 )
% 0.34/0.52 => ( ( relH @ As @ H23 @ H32 )
% 0.34/0.52 => ( relH @ As @ H12 @ H32 ) ) ) ).
% 0.34/0.52
% 0.34/0.52 % relH_trans
% 0.34/0.52 thf(fact_648_vimage__Collect,axiom,
% 0.34/0.52 ! [P: nat > $o,F: nat > nat,Q2: nat > $o] :
% 0.34/0.52 ( ! [X3: nat] :
% 0.34/0.52 ( ( P @ ( F @ X3 ) )
% 0.34/0.52 = ( Q2 @ X3 ) )
% 0.34/0.52 => ( ( vimage_nat_nat @ F @ ( collect_nat @ P ) )
% 0.34/0.52 = ( collect_nat @ Q2 ) ) ) ).
% 0.34/0.52
% 0.34/0.52 % vimage_Collect
% 0.34/0.52 thf(fact_649_vimage__Collect,axiom,
% 0.34/0.52 ! [P: nat > $o,F: int > nat,Q2: int > $o] :
% 0.34/0.52 ( ! [X3: int] :
% 0.34/0.52 ( ( P @ ( F @ X3 ) )
% 0.34/0.52 = ( Q2 @ X3 ) )
% 0.34/0.52 => ( ( vimage_int_nat @ F @ ( collect_nat @ P ) )
% 0.34/0.52 = ( collect_int @ Q2 ) ) ) ).
% 0.34/0.52
% 0.34/0.52 % vimage_Collect
% 0.34/0.52 thf(fact_650_vimage__Collect,axiom,
% 0.34/0.52 ! [P: nat > $o,F: $o > nat,Q2: $o > $o] :
% 0.34/0.52 ( ! [X3: $o] :
% 0.34/0.52 ( ( P @ ( F @ X3 ) )
% 0.34/0.52 = ( Q2 @ X3 ) )
% 0.34/0.52 => ( ( vimage_o_nat @ F @ ( collect_nat @ P ) )
% 0.34/0.52 = ( collect_o @ Q2 ) ) ) ).
% 0.34/0.52
% 0.34/0.52 % vimage_Collect
% 0.34/0.52 thf(fact_651_vimage__Collect,axiom,
% 0.34/0.52 ! [P: nat > $o,F: product_unit > nat,Q2: product_unit > $o] :
% 0.34/0.52 ( ! [X3: product_unit] :
% 0.34/0.52 ( ( P @ ( F @ X3 ) )
% 0.34/0.52 = ( Q2 @ X3 ) )
% 0.34/0.52 => ( ( vimage6253328473476588386it_nat @ F @ ( collect_nat @ P ) )
% 0.34/0.52 = ( collect_Product_unit @ Q2 ) ) ) ).
% 0.34/0.52
% 0.34/0.52 % vimage_Collect
% 0.34/0.52 thf(fact_652_vimage__Collect,axiom,
% 0.34/0.52 ! [P: int > $o,F: nat > int,Q2: nat > $o] :
% 0.34/0.52 ( ! [X3: nat] :
% 0.34/0.52 ( ( P @ ( F @ X3 ) )
% 0.34/0.52 = ( Q2 @ X3 ) )
% 0.34/0.52 => ( ( vimage_nat_int @ F @ ( collect_int @ P ) )
% 0.34/0.52 = ( collect_nat @ Q2 ) ) ) ).
% 0.34/0.52
% 0.34/0.52 % vimage_Collect
% 0.34/0.52 thf(fact_653_vimage__Collect,axiom,
% 0.34/0.52 ! [P: int > $o,F: int > int,Q2: int > $o] :
% 0.34/0.52 ( ! [X3: int] :
% 0.34/0.52 ( ( P @ ( F @ X3 ) )
% 0.34/0.52 = ( Q2 @ X3 ) )
% 0.34/0.52 => ( ( vimage_int_int @ F @ ( collect_int @ P ) )
% 0.34/0.52 = ( collect_int @ Q2 ) ) ) ).
% 0.34/0.52
% 0.34/0.52 % vimage_Collect
% 0.34/0.52 thf(fact_654_vimage__Collect,axiom,
% 0.34/0.52 ! [P: int > $o,F: $o > int,Q2: $o > $o] :
% 0.34/0.52 ( ! [X3: $o] :
% 0.34/0.52 ( ( P @ ( F @ X3 ) )
% 0.34/0.52 = ( Q2 @ X3 ) )
% 0.34/0.52 => ( ( vimage_o_int @ F @ ( collect_int @ P ) )
% 0.34/0.52 = ( collect_o @ Q2 ) ) ) ).
% 0.34/0.52
% 0.34/0.52 % vimage_Collect
% 0.34/0.52 thf(fact_655_vimage__Collect,axiom,
% 0.34/0.52 ! [P: int > $o,F: product_unit > int,Q2: product_unit > $o] :
% 0.34/0.52 ( ! [X3: product_unit] :
% 0.34/0.52 ( ( P @ ( F @ X3 ) )
% 0.34/0.52 = ( Q2 @ X3 ) )
% 0.34/0.52 => ( ( vimage6250838002967538110it_int @ F @ ( collect_int @ P ) )
% 0.34/0.52 = ( collect_Product_unit @ Q2 ) ) ) ).
% 0.34/0.52
% 0.34/0.52 % vimage_Collect
% 0.34/0.52 thf(fact_656_vimage__Collect,axiom,
% 0.34/0.52 ! [P: $o > $o,F: nat > $o,Q2: nat > $o] :
% 0.34/0.52 ( ! [X3: nat] :
% 0.34/0.52 ( ( P @ ( F @ X3 ) )
% 0.34/0.52 = ( Q2 @ X3 ) )
% 0.34/0.52 => ( ( vimage_nat_o @ F @ ( collect_o @ P ) )
% 0.34/0.52 = ( collect_nat @ Q2 ) ) ) ).
% 0.34/0.52
% 0.34/0.52 % vimage_Collect
% 0.34/0.52 thf(fact_657_vimage__Collect,axiom,
% 0.34/0.52 ! [P: $o > $o,F: int > $o,Q2: int > $o] :
% 0.34/0.52 ( ! [X3: int] :
% 0.34/0.52 ( ( P @ ( F @ X3 ) )
% 0.34/0.52 = ( Q2 @ X3 ) )
% 0.34/0.52 => ( ( vimage_int_o @ F @ ( collect_o @ P ) )
% 0.34/0.52 = ( collect_int @ Q2 ) ) ) ).
% 0.34/0.52
% 0.34/0.52 % vimage_Collect
% 0.34/0.52 thf(fact_658_vimageI2,axiom,
% 0.34/0.52 ! [F: nat > nat,A2: nat,A: set_nat] :
% 0.34/0.52 ( ( member_nat @ ( F @ A2 ) @ A )
% 0.34/0.52 => ( member_nat @ A2 @ ( vimage_nat_nat @ F @ A ) ) ) ).
% 0.34/0.52
% 0.34/0.52 % vimageI2
% 0.34/0.52 thf(fact_659_vimageI2,axiom,
% 0.34/0.52 ! [F: int > nat,A2: int,A: set_nat] :
% 0.34/0.52 ( ( member_nat @ ( F @ A2 ) @ A )
% 0.34/0.52 => ( member_int @ A2 @ ( vimage_int_nat @ F @ A ) ) ) ).
% 0.34/0.52
% 0.34/0.52 % vimageI2
% 0.34/0.52 thf(fact_660_vimageI2,axiom,
% 0.34/0.52 ! [F: $o > nat,A2: $o,A: set_nat] :
% 0.34/0.52 ( ( member_nat @ ( F @ A2 ) @ A )
% 0.34/0.52 => ( member_o @ A2 @ ( vimage_o_nat @ F @ A ) ) ) ).
% 0.34/0.52
% 0.34/0.52 % vimageI2
% 0.34/0.52 thf(fact_661_vimageI2,axiom,
% 0.34/0.52 ! [F: product_unit > nat,A2: product_unit,A: set_nat] :
% 0.34/0.52 ( ( member_nat @ ( F @ A2 ) @ A )
% 0.34/0.52 => ( member_Product_unit @ A2 @ ( vimage6253328473476588386it_nat @ F @ A ) ) ) ).
% 0.34/0.52
% 0.34/0.52 % vimageI2
% 0.34/0.52 thf(fact_662_vimageI2,axiom,
% 0.34/0.52 ! [F: nat > int,A2: nat,A: set_int] :
% 0.34/0.52 ( ( member_int @ ( F @ A2 ) @ A )
% 0.34/0.52 => ( member_nat @ A2 @ ( vimage_nat_int @ F @ A ) ) ) ).
% 0.34/0.52
% 0.34/0.52 % vimageI2
% 0.34/0.52 thf(fact_663_vimageI2,axiom,
% 0.34/0.52 ! [F: int > int,A2: int,A: set_int] :
% 0.34/0.52 ( ( member_int @ ( F @ A2 ) @ A )
% 0.34/0.52 => ( member_int @ A2 @ ( vimage_int_int @ F @ A ) ) ) ).
% 0.34/0.52
% 0.34/0.52 % vimageI2
% 0.34/0.52 thf(fact_664_vimageI2,axiom,
% 0.34/0.52 ! [F: $o > int,A2: $o,A: set_int] :
% 0.34/0.52 ( ( member_int @ ( F @ A2 ) @ A )
% 0.34/0.52 => ( member_o @ A2 @ ( vimage_o_int @ F @ A ) ) ) ).
% 0.34/0.52
% 0.34/0.52 % vimageI2
% 0.34/0.52 thf(fact_665_vimageI2,axiom,
% 0.34/0.52 ! [F: product_unit > int,A2: product_unit,A: set_int] :
% 0.34/0.52 ( ( member_int @ ( F @ A2 ) @ A )
% 0.34/0.52 => ( member_Product_unit @ A2 @ ( vimage6250838002967538110it_int @ F @ A ) ) ) ).
% 0.34/0.52
% 0.34/0.52 % vimageI2
% 0.34/0.52 thf(fact_666_vimageI2,axiom,
% 0.34/0.52 ! [F: nat > $o,A2: nat,A: set_o] :
% 0.34/0.52 ( ( member_o @ ( F @ A2 ) @ A )
% 0.34/0.52 => ( member_nat @ A2 @ ( vimage_nat_o @ F @ A ) ) ) ).
% 0.34/0.52
% 0.34/0.52 % vimageI2
% 0.34/0.52 thf(fact_667_vimageI2,axiom,
% 0.34/0.52 ! [F: int > $o,A2: int,A: set_o] :
% 0.34/0.52 ( ( member_o @ ( F @ A2 ) @ A )
% 0.34/0.52 => ( member_int @ A2 @ ( vimage_int_o @ F @ A ) ) ) ).
% 0.34/0.52
% 0.34/0.52 % vimageI2
% 0.34/0.52 thf(fact_668_vimageE,axiom,
% 0.34/0.52 ! [A2: nat,F: nat > nat,B4: set_nat] :
% 0.34/0.52 ( ( member_nat @ A2 @ ( vimage_nat_nat @ F @ B4 ) )
% 0.34/0.52 => ( member_nat @ ( F @ A2 ) @ B4 ) ) ).
% 0.34/0.52
% 0.34/0.52 % vimageE
% 0.34/0.52 thf(fact_669_vimageE,axiom,
% 0.34/0.52 ! [A2: nat,F: nat > int,B4: set_int] :
% 0.34/0.52 ( ( member_nat @ A2 @ ( vimage_nat_int @ F @ B4 ) )
% 0.34/0.52 => ( member_int @ ( F @ A2 ) @ B4 ) ) ).
% 0.34/0.52
% 0.34/0.52 % vimageE
% 0.34/0.52 thf(fact_670_vimageE,axiom,
% 0.34/0.52 ! [A2: nat,F: nat > $o,B4: set_o] :
% 0.34/0.52 ( ( member_nat @ A2 @ ( vimage_nat_o @ F @ B4 ) )
% 0.34/0.52 => ( member_o @ ( F @ A2 ) @ B4 ) ) ).
% 0.34/0.52
% 0.34/0.52 % vimageE
% 0.34/0.52 thf(fact_671_vimageE,axiom,
% 0.34/0.52 ! [A2: nat,F: nat > product_unit,B4: set_Product_unit] :
% 0.34/0.52 ( ( member_nat @ A2 @ ( vimage4884490618288580032t_unit @ F @ B4 ) )
% 0.34/0.52 => ( member_Product_unit @ ( F @ A2 ) @ B4 ) ) ).
% 0.34/0.52
% 0.34/0.52 % vimageE
% 0.34/0.52 thf(fact_672_vimageE,axiom,
% 0.34/0.52 ! [A2: int,F: int > nat,B4: set_nat] :
% 0.34/0.52 ( ( member_int @ A2 @ ( vimage_int_nat @ F @ B4 ) )
% 0.34/0.52 => ( member_nat @ ( F @ A2 ) @ B4 ) ) ).
% 0.34/0.52
% 0.34/0.52 % vimageE
% 0.34/0.52 thf(fact_673_vimageE,axiom,
% 0.34/0.52 ! [A2: int,F: int > int,B4: set_int] :
% 0.34/0.52 ( ( member_int @ A2 @ ( vimage_int_int @ F @ B4 ) )
% 0.34/0.52 => ( member_int @ ( F @ A2 ) @ B4 ) ) ).
% 0.34/0.52
% 0.34/0.52 % vimageE
% 0.34/0.52 thf(fact_674_vimageE,axiom,
% 0.34/0.52 ! [A2: int,F: int > $o,B4: set_o] :
% 0.34/0.52 ( ( member_int @ A2 @ ( vimage_int_o @ F @ B4 ) )
% 0.34/0.52 => ( member_o @ ( F @ A2 ) @ B4 ) ) ).
% 0.34/0.52
% 0.34/0.52 % vimageE
% 0.34/0.52 thf(fact_675_vimageE,axiom,
% 0.34/0.52 ! [A2: int,F: int > product_unit,B4: set_Product_unit] :
% 0.34/0.52 ( ( member_int @ A2 @ ( vimage2000912565490760292t_unit @ F @ B4 ) )
% 0.34/0.52 => ( member_Product_unit @ ( F @ A2 ) @ B4 ) ) ).
% 0.34/0.52
% 0.34/0.52 % vimageE
% 0.34/0.52 thf(fact_676_vimageE,axiom,
% 0.34/0.52 ! [A2: $o,F: $o > nat,B4: set_nat] :
% 0.34/0.52 ( ( member_o @ A2 @ ( vimage_o_nat @ F @ B4 ) )
% 0.34/0.52 => ( member_nat @ ( F @ A2 ) @ B4 ) ) ).
% 0.34/0.52
% 0.34/0.52 % vimageE
% 0.34/0.52 thf(fact_677_vimageE,axiom,
% 0.34/0.52 ! [A2: $o,F: $o > int,B4: set_int] :
% 0.34/0.52 ( ( member_o @ A2 @ ( vimage_o_int @ F @ B4 ) )
% 0.34/0.52 => ( member_int @ ( F @ A2 ) @ B4 ) ) ).
% 0.34/0.52
% 0.34/0.52 % vimageE
% 0.34/0.52 thf(fact_678_vimageD,axiom,
% 0.34/0.52 ! [A2: nat,F: nat > nat,A: set_nat] :
% 0.34/0.52 ( ( member_nat @ A2 @ ( vimage_nat_nat @ F @ A ) )
% 0.34/0.52 => ( member_nat @ ( F @ A2 ) @ A ) ) ).
% 0.34/0.52
% 0.34/0.52 % vimageD
% 0.34/0.52 thf(fact_679_vimageD,axiom,
% 0.34/0.52 ! [A2: nat,F: nat > int,A: set_int] :
% 0.34/0.52 ( ( member_nat @ A2 @ ( vimage_nat_int @ F @ A ) )
% 0.34/0.52 => ( member_int @ ( F @ A2 ) @ A ) ) ).
% 0.34/0.52
% 0.34/0.52 % vimageD
% 0.34/0.52 thf(fact_680_vimageD,axiom,
% 0.34/0.52 ! [A2: nat,F: nat > $o,A: set_o] :
% 0.34/0.52 ( ( member_nat @ A2 @ ( vimage_nat_o @ F @ A ) )
% 0.34/0.52 => ( member_o @ ( F @ A2 ) @ A ) ) ).
% 0.34/0.52
% 0.34/0.52 % vimageD
% 0.34/0.52 thf(fact_681_vimageD,axiom,
% 0.34/0.52 ! [A2: nat,F: nat > product_unit,A: set_Product_unit] :
% 0.34/0.52 ( ( member_nat @ A2 @ ( vimage4884490618288580032t_unit @ F @ A ) )
% 0.34/0.52 => ( member_Product_unit @ ( F @ A2 ) @ A ) ) ).
% 0.34/0.52
% 0.34/0.52 % vimageD
% 0.34/0.52 thf(fact_682_vimageD,axiom,
% 0.34/0.52 ! [A2: int,F: int > nat,A: set_nat] :
% 0.34/0.52 ( ( member_int @ A2 @ ( vimage_int_nat @ F @ A ) )
% 0.34/0.52 => ( member_nat @ ( F @ A2 ) @ A ) ) ).
% 0.34/0.52
% 0.34/0.52 % vimageD
% 0.34/0.52 thf(fact_683_vimageD,axiom,
% 0.34/0.52 ! [A2: int,F: int > int,A: set_int] :
% 0.34/0.52 ( ( member_int @ A2 @ ( vimage_int_int @ F @ A ) )
% 0.34/0.52 => ( member_int @ ( F @ A2 ) @ A ) ) ).
% 0.34/0.52
% 0.34/0.52 % vimageD
% 0.34/0.52 thf(fact_684_vimageD,axiom,
% 0.34/0.52 ! [A2: int,F: int > $o,A: set_o] :
% 0.34/0.52 ( ( member_int @ A2 @ ( vimage_int_o @ F @ A ) )
% 0.34/0.52 => ( member_o @ ( F @ A2 ) @ A ) ) ).
% 0.34/0.52
% 0.34/0.52 % vimageD
% 0.34/0.52 thf(fact_685_vimageD,axiom,
% 0.34/0.52 ! [A2: int,F: int > product_unit,A: set_Product_unit] :
% 0.34/0.52 ( ( member_int @ A2 @ ( vimage2000912565490760292t_unit @ F @ A ) )
% 0.34/0.52 => ( member_Product_unit @ ( F @ A2 ) @ A ) ) ).
% 0.34/0.52
% 0.34/0.52 % vimageD
% 0.34/0.52 thf(fact_686_vimageD,axiom,
% 0.34/0.52 ! [A2: $o,F: $o > nat,A: set_nat] :
% 0.34/0.52 ( ( member_o @ A2 @ ( vimage_o_nat @ F @ A ) )
% 0.34/0.52 => ( member_nat @ ( F @ A2 ) @ A ) ) ).
% 0.34/0.52
% 0.34/0.52 % vimageD
% 0.34/0.52 thf(fact_687_vimageD,axiom,
% 0.34/0.52 ! [A2: $o,F: $o > int,A: set_int] :
% 0.34/0.52 ( ( member_o @ A2 @ ( vimage_o_int @ F @ A ) )
% 0.34/0.52 => ( member_int @ ( F @ A2 ) @ A ) ) ).
% 0.34/0.52
% 0.34/0.52 % vimageD
% 0.34/0.52 thf(fact_688_transD,axiom,
% 0.34/0.52 ! [R2: set_Pr1261947904930325089at_nat,X: nat,Y: nat,Z: nat] :
% 0.34/0.52 ( ( trans_nat @ R2 )
% 0.34/0.52 => ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ Y ) @ R2 )
% 0.34/0.52 => ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ Y @ Z ) @ R2 )
% 0.34/0.52 => ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ Z ) @ R2 ) ) ) ) ).
% 0.34/0.52
% 0.34/0.52 % transD
% 0.34/0.52 thf(fact_689_transD,axiom,
% 0.34/0.52 ! [R2: set_Pr958786334691620121nt_int,X: int,Y: int,Z: int] :
% 0.34/0.52 ( ( trans_int @ R2 )
% 0.34/0.52 => ( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ X @ Y ) @ R2 )
% 0.34/0.52 => ( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ Y @ Z ) @ R2 )
% 0.34/0.52 => ( member5262025264175285858nt_int @ ( product_Pair_int_int @ X @ Z ) @ R2 ) ) ) ) ).
% 0.34/0.52
% 0.34/0.52 % transD
% 0.34/0.52 thf(fact_690_transE,axiom,
% 0.34/0.52 ! [R2: set_Pr1261947904930325089at_nat,X: nat,Y: nat,Z: nat] :
% 0.34/0.52 ( ( trans_nat @ R2 )
% 0.34/0.52 => ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ Y ) @ R2 )
% 0.34/0.52 => ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ Y @ Z ) @ R2 )
% 0.34/0.52 => ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ Z ) @ R2 ) ) ) ) ).
% 0.34/0.52
% 0.34/0.52 % transE
% 0.34/0.52 thf(fact_691_transE,axiom,
% 0.34/0.52 ! [R2: set_Pr958786334691620121nt_int,X: int,Y: int,Z: int] :
% 0.34/0.52 ( ( trans_int @ R2 )
% 0.34/0.52 => ( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ X @ Y ) @ R2 )
% 0.34/0.52 => ( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ Y @ Z ) @ R2 )
% 0.34/0.52 => ( member5262025264175285858nt_int @ ( product_Pair_int_int @ X @ Z ) @ R2 ) ) ) ) ).
% 0.34/0.52
% 0.34/0.52 % transE
% 0.34/0.52 thf(fact_692_transI,axiom,
% 0.34/0.52 ! [R2: set_Pr1261947904930325089at_nat] :
% 0.34/0.52 ( ! [X3: nat,Y2: nat,Z3: nat] :
% 0.34/0.52 ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X3 @ Y2 ) @ R2 )
% 0.34/0.52 => ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ Y2 @ Z3 ) @ R2 )
% 0.34/0.52 => ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X3 @ Z3 ) @ R2 ) ) )
% 0.34/0.52 => ( trans_nat @ R2 ) ) ).
% 0.34/0.52
% 0.34/0.52 % transI
% 0.34/0.52 thf(fact_693_transI,axiom,
% 0.34/0.52 ! [R2: set_Pr958786334691620121nt_int] :
% 0.34/0.52 ( ! [X3: int,Y2: int,Z3: int] :
% 0.34/0.52 ( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ X3 @ Y2 ) @ R2 )
% 0.34/0.52 => ( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ Y2 @ Z3 ) @ R2 )
% 0.34/0.52 => ( member5262025264175285858nt_int @ ( product_Pair_int_int @ X3 @ Z3 ) @ R2 ) ) )
% 0.34/0.52 => ( trans_int @ R2 ) ) ).
% 0.34/0.52
% 0.34/0.52 % transI
% 0.34/0.52 thf(fact_694_trans__def,axiom,
% 0.34/0.52 ( trans_nat
% 0.34/0.52 = ( ^ [R4: set_Pr1261947904930325089at_nat] :
% 0.34/0.52 ! [X2: nat,Y3: nat,Z2: nat] :
% 0.34/0.52 ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X2 @ Y3 ) @ R4 )
% 0.34/0.52 => ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ Y3 @ Z2 ) @ R4 )
% 0.34/0.52 => ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X2 @ Z2 ) @ R4 ) ) ) ) ) ).
% 0.34/0.52
% 0.34/0.52 % trans_def
% 0.34/0.52 thf(fact_695_trans__def,axiom,
% 0.34/0.52 ( trans_int
% 0.34/0.52 = ( ^ [R4: set_Pr958786334691620121nt_int] :
% 0.34/0.52 ! [X2: int,Y3: int,Z2: int] :
% 0.34/0.52 ( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ X2 @ Y3 ) @ R4 )
% 0.34/0.52 => ( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ Y3 @ Z2 ) @ R4 )
% 0.34/0.52 => ( member5262025264175285858nt_int @ ( product_Pair_int_int @ X2 @ Z2 ) @ R4 ) ) ) ) ) ).
% 0.34/0.52
% 0.34/0.52 % trans_def
% 0.34/0.52 thf(fact_696_Domain_Ocases,axiom,
% 0.34/0.52 ! [A2: heap_e7401611519738050253t_unit,R2: set_Pr3948176798113811640et_nat] :
% 0.34/0.52 ( ( member242389114213480694t_unit @ A2 @ ( domain2007548326387822151et_nat @ R2 ) )
% 0.34/0.52 => ~ ! [B2: set_nat] :
% 0.34/0.52 ~ ( member6260224972018164377et_nat @ ( produc7507926704131184380et_nat @ A2 @ B2 ) @ R2 ) ) ).
% 0.34/0.52
% 0.34/0.52 % Domain.cases
% 0.34/0.52 thf(fact_697_Domain_Ocases,axiom,
% 0.34/0.52 ! [A2: nat,R2: set_Pr1261947904930325089at_nat] :
% 0.34/0.52 ( ( member_nat @ A2 @ ( domain_nat_nat @ R2 ) )
% 0.34/0.52 => ~ ! [B2: nat] :
% 0.34/0.52 ~ ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A2 @ B2 ) @ R2 ) ) ).
% 0.34/0.52
% 0.34/0.52 % Domain.cases
% 0.34/0.52 thf(fact_698_Domain_Ocases,axiom,
% 0.34/0.52 ! [A2: int,R2: set_Pr958786334691620121nt_int] :
% 0.34/0.52 ( ( member_int @ A2 @ ( domain_int_int @ R2 ) )
% 0.34/0.52 => ~ ! [B2: int] :
% 0.34/0.52 ~ ( member5262025264175285858nt_int @ ( product_Pair_int_int @ A2 @ B2 ) @ R2 ) ) ).
% 0.34/0.52
% 0.34/0.52 % Domain.cases
% 0.34/0.52 thf(fact_699_Domain_Ocases,axiom,
% 0.34/0.52 ! [A2: produc3658429121746597890et_nat > $o,R2: set_Pr3286484037609594932et_nat] :
% 0.34/0.52 ( ( member6576561426505652726_nat_o @ A2 @ ( domain2358961613337345475et_nat @ R2 ) )
% 0.34/0.52 => ~ ! [B2: produc3658429121746597890et_nat] :
% 0.34/0.52 ~ ( member1996754912294343701et_nat @ ( produc5001842942810119800et_nat @ A2 @ B2 ) @ R2 ) ) ).
% 0.34/0.52
% 0.34/0.52 % Domain.cases
% 0.34/0.52 thf(fact_700_Domain_Ocases,axiom,
% 0.34/0.52 ! [A2: produc3658429121746597890et_nat > $o,R2: set_Pr8536935166611901872et_nat] :
% 0.34/0.52 ( ( member6576561426505652726_nat_o @ A2 @ ( domain3618434998481624895et_nat @ R2 ) )
% 0.34/0.52 => ~ ! [B2: produc3925858234332021118et_nat] :
% 0.34/0.52 ~ ( member6124377750444531601et_nat @ ( produc2245416461498447860et_nat @ A2 @ B2 ) @ R2 ) ) ).
% 0.34/0.52
% 0.34/0.52 % Domain.cases
% 0.34/0.52 thf(fact_701_Domain_Osimps,axiom,
% 0.34/0.52 ! [A2: heap_e7401611519738050253t_unit,R2: set_Pr3948176798113811640et_nat] :
% 0.34/0.52 ( ( member242389114213480694t_unit @ A2 @ ( domain2007548326387822151et_nat @ R2 ) )
% 0.34/0.52 = ( ? [A6: heap_e7401611519738050253t_unit,B6: set_nat] :
% 0.34/0.52 ( ( A2 = A6 )
% 0.34/0.52 & ( member6260224972018164377et_nat @ ( produc7507926704131184380et_nat @ A6 @ B6 ) @ R2 ) ) ) ) ).
% 0.34/0.52
% 0.34/0.52 % Domain.simps
% 0.34/0.52 thf(fact_702_Domain_Osimps,axiom,
% 0.34/0.52 ! [A2: nat,R2: set_Pr1261947904930325089at_nat] :
% 0.34/0.52 ( ( member_nat @ A2 @ ( domain_nat_nat @ R2 ) )
% 0.34/0.52 = ( ? [A6: nat,B6: nat] :
% 0.34/0.52 ( ( A2 = A6 )
% 0.34/0.52 & ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A6 @ B6 ) @ R2 ) ) ) ) ).
% 0.34/0.52
% 0.34/0.52 % Domain.simps
% 0.34/0.52 thf(fact_703_Domain_Osimps,axiom,
% 0.34/0.52 ! [A2: int,R2: set_Pr958786334691620121nt_int] :
% 0.34/0.52 ( ( member_int @ A2 @ ( domain_int_int @ R2 ) )
% 0.34/0.52 = ( ? [A6: int,B6: int] :
% 0.34/0.52 ( ( A2 = A6 )
% 0.34/0.52 & ( member5262025264175285858nt_int @ ( product_Pair_int_int @ A6 @ B6 ) @ R2 ) ) ) ) ).
% 0.34/0.52
% 0.34/0.52 % Domain.simps
% 0.34/0.52 thf(fact_704_Domain_Osimps,axiom,
% 0.34/0.52 ! [A2: produc3658429121746597890et_nat > $o,R2: set_Pr3286484037609594932et_nat] :
% 0.34/0.52 ( ( member6576561426505652726_nat_o @ A2 @ ( domain2358961613337345475et_nat @ R2 ) )
% 0.34/0.52 = ( ? [A6: produc3658429121746597890et_nat > $o,B6: produc3658429121746597890et_nat] :
% 0.34/0.52 ( ( A2 = A6 )
% 0.34/0.52 & ( member1996754912294343701et_nat @ ( produc5001842942810119800et_nat @ A6 @ B6 ) @ R2 ) ) ) ) ).
% 0.34/0.52
% 0.34/0.52 % Domain.simps
% 0.34/0.52 thf(fact_705_Domain_Osimps,axiom,
% 0.34/0.52 ! [A2: produc3658429121746597890et_nat > $o,R2: set_Pr8536935166611901872et_nat] :
% 0.34/0.52 ( ( member6576561426505652726_nat_o @ A2 @ ( domain3618434998481624895et_nat @ R2 ) )
% 0.34/0.52 = ( ? [A6: produc3658429121746597890et_nat > $o,B6: produc3925858234332021118et_nat] :
% 0.34/0.52 ( ( A2 = A6 )
% 0.34/0.52 & ( member6124377750444531601et_nat @ ( produc2245416461498447860et_nat @ A6 @ B6 ) @ R2 ) ) ) ) ).
% 0.34/0.52
% 0.34/0.52 % Domain.simps
% 0.34/0.52 thf(fact_706_Domain_ODomainI,axiom,
% 0.34/0.52 ! [A2: heap_e7401611519738050253t_unit,B: set_nat,R2: set_Pr3948176798113811640et_nat] :
% 0.34/0.52 ( ( member6260224972018164377et_nat @ ( produc7507926704131184380et_nat @ A2 @ B ) @ R2 )
% 0.34/0.52 => ( member242389114213480694t_unit @ A2 @ ( domain2007548326387822151et_nat @ R2 ) ) ) ).
% 0.34/0.52
% 0.34/0.52 % Domain.DomainI
% 0.34/0.52 thf(fact_707_Domain_ODomainI,axiom,
% 0.34/0.52 ! [A2: nat,B: nat,R2: set_Pr1261947904930325089at_nat] :
% 0.34/0.52 ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A2 @ B ) @ R2 )
% 0.34/0.52 => ( member_nat @ A2 @ ( domain_nat_nat @ R2 ) ) ) ).
% 0.34/0.52
% 0.34/0.52 % Domain.DomainI
% 0.34/0.52 thf(fact_708_Domain_ODomainI,axiom,
% 0.34/0.52 ! [A2: int,B: int,R2: set_Pr958786334691620121nt_int] :
% 0.34/0.52 ( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ A2 @ B ) @ R2 )
% 0.34/0.52 => ( member_int @ A2 @ ( domain_int_int @ R2 ) ) ) ).
% 0.34/0.52
% 0.34/0.52 % Domain.DomainI
% 0.34/0.52 thf(fact_709_Domain_ODomainI,axiom,
% 0.34/0.52 ! [A2: produc3658429121746597890et_nat > $o,B: produc3658429121746597890et_nat,R2: set_Pr3286484037609594932et_nat] :
% 0.34/0.52 ( ( member1996754912294343701et_nat @ ( produc5001842942810119800et_nat @ A2 @ B ) @ R2 )
% 0.34/0.52 => ( member6576561426505652726_nat_o @ A2 @ ( domain2358961613337345475et_nat @ R2 ) ) ) ).
% 0.34/0.52
% 0.34/0.52 % Domain.DomainI
% 0.34/0.52 thf(fact_710_Domain_ODomainI,axiom,
% 0.34/0.52 ! [A2: produc3658429121746597890et_nat > $o,B: produc3925858234332021118et_nat,R2: set_Pr8536935166611901872et_nat] :
% 0.34/0.52 ( ( member6124377750444531601et_nat @ ( produc2245416461498447860et_nat @ A2 @ B ) @ R2 )
% 0.34/0.52 => ( member6576561426505652726_nat_o @ A2 @ ( domain3618434998481624895et_nat @ R2 ) ) ) ).
% 0.34/0.52
% 0.34/0.52 % Domain.DomainI
% 0.34/0.52 thf(fact_711_DomainE,axiom,
% 0.34/0.52 ! [A2: heap_e7401611519738050253t_unit,R2: set_Pr3948176798113811640et_nat] :
% 0.34/0.52 ( ( member242389114213480694t_unit @ A2 @ ( domain2007548326387822151et_nat @ R2 ) )
% 0.34/0.52 => ~ ! [B2: set_nat] :
% 0.34/0.52 ~ ( member6260224972018164377et_nat @ ( produc7507926704131184380et_nat @ A2 @ B2 ) @ R2 ) ) ).
% 0.34/0.52
% 0.34/0.52 % DomainE
% 0.34/0.52 thf(fact_712_DomainE,axiom,
% 0.34/0.52 ! [A2: nat,R2: set_Pr1261947904930325089at_nat] :
% 0.34/0.52 ( ( member_nat @ A2 @ ( domain_nat_nat @ R2 ) )
% 0.34/0.52 => ~ ! [B2: nat] :
% 0.34/0.52 ~ ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A2 @ B2 ) @ R2 ) ) ).
% 0.34/0.52
% 0.34/0.52 % DomainE
% 0.34/0.52 thf(fact_713_DomainE,axiom,
% 0.34/0.52 ! [A2: int,R2: set_Pr958786334691620121nt_int] :
% 0.34/0.52 ( ( member_int @ A2 @ ( domain_int_int @ R2 ) )
% 0.34/0.52 => ~ ! [B2: int] :
% 0.34/0.52 ~ ( member5262025264175285858nt_int @ ( product_Pair_int_int @ A2 @ B2 ) @ R2 ) ) ).
% 0.34/0.52
% 0.34/0.52 % DomainE
% 0.34/0.52 thf(fact_714_DomainE,axiom,
% 0.34/0.52 ! [A2: produc3658429121746597890et_nat > $o,R2: set_Pr3286484037609594932et_nat] :
% 0.34/0.52 ( ( member6576561426505652726_nat_o @ A2 @ ( domain2358961613337345475et_nat @ R2 ) )
% 0.34/0.52 => ~ ! [B2: produc3658429121746597890et_nat] :
% 0.34/0.52 ~ ( member1996754912294343701et_nat @ ( produc5001842942810119800et_nat @ A2 @ B2 ) @ R2 ) ) ).
% 0.34/0.52
% 0.34/0.52 % DomainE
% 0.34/0.52 thf(fact_715_DomainE,axiom,
% 0.34/0.52 ! [A2: produc3658429121746597890et_nat > $o,R2: set_Pr8536935166611901872et_nat] :
% 0.34/0.52 ( ( member6576561426505652726_nat_o @ A2 @ ( domain3618434998481624895et_nat @ R2 ) )
% 0.34/0.52 => ~ ! [B2: produc3925858234332021118et_nat] :
% 0.34/0.52 ~ ( member6124377750444531601et_nat @ ( produc2245416461498447860et_nat @ A2 @ B2 ) @ R2 ) ) ).
% 0.34/0.52
% 0.34/0.52 % DomainE
% 0.34/0.52 thf(fact_716_Domain__iff,axiom,
% 0.34/0.52 ! [A2: heap_e7401611519738050253t_unit,R2: set_Pr3948176798113811640et_nat] :
% 0.34/0.52 ( ( member242389114213480694t_unit @ A2 @ ( domain2007548326387822151et_nat @ R2 ) )
% 0.34/0.52 = ( ? [Y3: set_nat] : ( member6260224972018164377et_nat @ ( produc7507926704131184380et_nat @ A2 @ Y3 ) @ R2 ) ) ) ).
% 0.34/0.52
% 0.34/0.52 % Domain_iff
% 0.34/0.52 thf(fact_717_Domain__iff,axiom,
% 0.34/0.52 ! [A2: nat,R2: set_Pr1261947904930325089at_nat] :
% 0.34/0.52 ( ( member_nat @ A2 @ ( domain_nat_nat @ R2 ) )
% 0.34/0.52 = ( ? [Y3: nat] : ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A2 @ Y3 ) @ R2 ) ) ) ).
% 0.34/0.52
% 0.34/0.52 % Domain_iff
% 0.34/0.52 thf(fact_718_Domain__iff,axiom,
% 0.34/0.52 ! [A2: int,R2: set_Pr958786334691620121nt_int] :
% 0.34/0.52 ( ( member_int @ A2 @ ( domain_int_int @ R2 ) )
% 0.34/0.52 = ( ? [Y3: int] : ( member5262025264175285858nt_int @ ( product_Pair_int_int @ A2 @ Y3 ) @ R2 ) ) ) ).
% 0.34/0.52
% 0.34/0.52 % Domain_iff
% 0.34/0.52 thf(fact_719_Domain__iff,axiom,
% 0.34/0.52 ! [A2: produc3658429121746597890et_nat > $o,R2: set_Pr3286484037609594932et_nat] :
% 0.34/0.52 ( ( member6576561426505652726_nat_o @ A2 @ ( domain2358961613337345475et_nat @ R2 ) )
% 0.34/0.52 = ( ? [Y3: produc3658429121746597890et_nat] : ( member1996754912294343701et_nat @ ( produc5001842942810119800et_nat @ A2 @ Y3 ) @ R2 ) ) ) ).
% 0.34/0.52
% 0.34/0.52 % Domain_iff
% 0.34/0.52 thf(fact_720_Domain__iff,axiom,
% 0.34/0.52 ! [A2: produc3658429121746597890et_nat > $o,R2: set_Pr8536935166611901872et_nat] :
% 0.34/0.52 ( ( member6576561426505652726_nat_o @ A2 @ ( domain3618434998481624895et_nat @ R2 ) )
% 0.34/0.52 = ( ? [Y3: produc3925858234332021118et_nat] : ( member6124377750444531601et_nat @ ( produc2245416461498447860et_nat @ A2 @ Y3 ) @ R2 ) ) ) ).
% 0.34/0.52
% 0.34/0.52 % Domain_iff
% 0.34/0.52 thf(fact_721_Range_Ocases,axiom,
% 0.34/0.52 ! [A2: set_nat,R2: set_Pr3948176798113811640et_nat] :
% 0.34/0.52 ( ( member_set_nat @ A2 @ ( range_9132311542094983838et_nat @ R2 ) )
% 0.34/0.52 => ~ ! [A4: heap_e7401611519738050253t_unit] :
% 0.34/0.52 ~ ( member6260224972018164377et_nat @ ( produc7507926704131184380et_nat @ A4 @ A2 ) @ R2 ) ) ).
% 0.34/0.52
% 0.34/0.52 % Range.cases
% 0.34/0.52 thf(fact_722_Range_Ocases,axiom,
% 0.34/0.52 ! [A2: nat,R2: set_Pr1261947904930325089at_nat] :
% 0.34/0.52 ( ( member_nat @ A2 @ ( range_nat_nat @ R2 ) )
% 0.34/0.52 => ~ ! [A4: nat] :
% 0.34/0.52 ~ ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A4 @ A2 ) @ R2 ) ) ).
% 0.34/0.52
% 0.34/0.52 % Range.cases
% 0.34/0.52 thf(fact_723_Range_Ocases,axiom,
% 0.34/0.52 ! [A2: int,R2: set_Pr958786334691620121nt_int] :
% 0.34/0.52 ( ( member_int @ A2 @ ( range_int_int @ R2 ) )
% 0.34/0.52 => ~ ! [A4: int] :
% 0.34/0.52 ~ ( member5262025264175285858nt_int @ ( product_Pair_int_int @ A4 @ A2 ) @ R2 ) ) ).
% 0.34/0.52
% 0.34/0.52 % Range.cases
% 0.34/0.52 thf(fact_724_Range_Ocases,axiom,
% 0.34/0.52 ! [A2: produc3658429121746597890et_nat,R2: set_Pr3286484037609594932et_nat] :
% 0.34/0.52 ( ( member6260224972018164377et_nat @ A2 @ ( range_3398330936903821850et_nat @ R2 ) )
% 0.34/0.52 => ~ ! [A4: produc3658429121746597890et_nat > $o] :
% 0.34/0.52 ~ ( member1996754912294343701et_nat @ ( produc5001842942810119800et_nat @ A4 @ A2 ) @ R2 ) ) ).
% 0.34/0.52
% 0.34/0.52 % Range.cases
% 0.34/0.52 thf(fact_725_Range_Ocases,axiom,
% 0.34/0.52 ! [A2: produc3925858234332021118et_nat,R2: set_Pr8536935166611901872et_nat] :
% 0.34/0.52 ( ( member1996754912294343701et_nat @ A2 @ ( range_7260058354244137878et_nat @ R2 ) )
% 0.34/0.52 => ~ ! [A4: produc3658429121746597890et_nat > $o] :
% 0.34/0.52 ~ ( member6124377750444531601et_nat @ ( produc2245416461498447860et_nat @ A4 @ A2 ) @ R2 ) ) ).
% 0.34/0.52
% 0.34/0.52 % Range.cases
% 0.34/0.52 thf(fact_726_Range_Osimps,axiom,
% 0.34/0.52 ! [A2: set_nat,R2: set_Pr3948176798113811640et_nat] :
% 0.34/0.52 ( ( member_set_nat @ A2 @ ( range_9132311542094983838et_nat @ R2 ) )
% 0.34/0.52 = ( ? [A6: heap_e7401611519738050253t_unit,B6: set_nat] :
% 0.34/0.52 ( ( A2 = B6 )
% 0.34/0.52 & ( member6260224972018164377et_nat @ ( produc7507926704131184380et_nat @ A6 @ B6 ) @ R2 ) ) ) ) ).
% 0.34/0.52
% 0.34/0.52 % Range.simps
% 0.34/0.52 thf(fact_727_Range_Osimps,axiom,
% 0.34/0.52 ! [A2: nat,R2: set_Pr1261947904930325089at_nat] :
% 0.34/0.52 ( ( member_nat @ A2 @ ( range_nat_nat @ R2 ) )
% 0.34/0.52 = ( ? [A6: nat,B6: nat] :
% 0.34/0.52 ( ( A2 = B6 )
% 0.34/0.52 & ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A6 @ B6 ) @ R2 ) ) ) ) ).
% 0.34/0.52
% 0.34/0.52 % Range.simps
% 0.34/0.52 thf(fact_728_Range_Osimps,axiom,
% 0.34/0.52 ! [A2: int,R2: set_Pr958786334691620121nt_int] :
% 0.34/0.52 ( ( member_int @ A2 @ ( range_int_int @ R2 ) )
% 0.34/0.52 = ( ? [A6: int,B6: int] :
% 0.34/0.52 ( ( A2 = B6 )
% 0.34/0.52 & ( member5262025264175285858nt_int @ ( product_Pair_int_int @ A6 @ B6 ) @ R2 ) ) ) ) ).
% 0.34/0.52
% 0.34/0.52 % Range.simps
% 0.34/0.52 thf(fact_729_Range_Osimps,axiom,
% 0.34/0.52 ! [A2: produc3658429121746597890et_nat,R2: set_Pr3286484037609594932et_nat] :
% 0.34/0.52 ( ( member6260224972018164377et_nat @ A2 @ ( range_3398330936903821850et_nat @ R2 ) )
% 0.34/0.52 = ( ? [A6: produc3658429121746597890et_nat > $o,B6: produc3658429121746597890et_nat] :
% 0.34/0.52 ( ( A2 = B6 )
% 0.34/0.52 & ( member1996754912294343701et_nat @ ( produc5001842942810119800et_nat @ A6 @ B6 ) @ R2 ) ) ) ) ).
% 0.34/0.52
% 0.34/0.52 % Range.simps
% 0.34/0.52 thf(fact_730_Range_Osimps,axiom,
% 0.34/0.52 ! [A2: produc3925858234332021118et_nat,R2: set_Pr8536935166611901872et_nat] :
% 0.34/0.52 ( ( member1996754912294343701et_nat @ A2 @ ( range_7260058354244137878et_nat @ R2 ) )
% 0.34/0.52 = ( ? [A6: produc3658429121746597890et_nat > $o,B6: produc3925858234332021118et_nat] :
% 0.34/0.52 ( ( A2 = B6 )
% 0.34/0.52 & ( member6124377750444531601et_nat @ ( produc2245416461498447860et_nat @ A6 @ B6 ) @ R2 ) ) ) ) ).
% 0.34/0.52
% 0.34/0.52 % Range.simps
% 0.34/0.52 thf(fact_731_Range_Ointros,axiom,
% 0.34/0.52 ! [A2: heap_e7401611519738050253t_unit,B: set_nat,R2: set_Pr3948176798113811640et_nat] :
% 0.34/0.52 ( ( member6260224972018164377et_nat @ ( produc7507926704131184380et_nat @ A2 @ B ) @ R2 )
% 0.34/0.52 => ( member_set_nat @ B @ ( range_9132311542094983838et_nat @ R2 ) ) ) ).
% 0.34/0.52
% 0.34/0.52 % Range.intros
% 0.34/0.52 thf(fact_732_Range_Ointros,axiom,
% 0.34/0.52 ! [A2: nat,B: nat,R2: set_Pr1261947904930325089at_nat] :
% 0.34/0.52 ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A2 @ B ) @ R2 )
% 0.34/0.52 => ( member_nat @ B @ ( range_nat_nat @ R2 ) ) ) ).
% 0.34/0.52
% 0.34/0.52 % Range.intros
% 0.34/0.52 thf(fact_733_Range_Ointros,axiom,
% 0.34/0.52 ! [A2: int,B: int,R2: set_Pr958786334691620121nt_int] :
% 0.34/0.52 ( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ A2 @ B ) @ R2 )
% 0.34/0.52 => ( member_int @ B @ ( range_int_int @ R2 ) ) ) ).
% 0.34/0.52
% 0.34/0.52 % Range.intros
% 0.34/0.52 thf(fact_734_Range_Ointros,axiom,
% 0.34/0.52 ! [A2: produc3658429121746597890et_nat > $o,B: produc3658429121746597890et_nat,R2: set_Pr3286484037609594932et_nat] :
% 0.34/0.52 ( ( member1996754912294343701et_nat @ ( produc5001842942810119800et_nat @ A2 @ B ) @ R2 )
% 0.34/0.52 => ( member6260224972018164377et_nat @ B @ ( range_3398330936903821850et_nat @ R2 ) ) ) ).
% 0.34/0.52
% 0.34/0.52 % Range.intros
% 0.34/0.52 thf(fact_735_Range_Ointros,axiom,
% 0.34/0.52 ! [A2: produc3658429121746597890et_nat > $o,B: produc3925858234332021118et_nat,R2: set_Pr8536935166611901872et_nat] :
% 0.34/0.52 ( ( member6124377750444531601et_nat @ ( produc2245416461498447860et_nat @ A2 @ B ) @ R2 )
% 0.34/0.52 => ( member1996754912294343701et_nat @ B @ ( range_7260058354244137878et_nat @ R2 ) ) ) ).
% 0.34/0.52
% 0.34/0.52 % Range.intros
% 0.34/0.52 thf(fact_736_RangeE,axiom,
% 0.34/0.52 ! [B: set_nat,R2: set_Pr3948176798113811640et_nat] :
% 0.34/0.52 ( ( member_set_nat @ B @ ( range_9132311542094983838et_nat @ R2 ) )
% 0.34/0.52 => ~ ! [A4: heap_e7401611519738050253t_unit] :
% 0.34/0.52 ~ ( member6260224972018164377et_nat @ ( produc7507926704131184380et_nat @ A4 @ B ) @ R2 ) ) ).
% 0.34/0.52
% 0.34/0.52 % RangeE
% 0.34/0.52 thf(fact_737_RangeE,axiom,
% 0.34/0.52 ! [B: nat,R2: set_Pr1261947904930325089at_nat] :
% 0.34/0.52 ( ( member_nat @ B @ ( range_nat_nat @ R2 ) )
% 0.34/0.52 => ~ ! [A4: nat] :
% 0.34/0.52 ~ ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A4 @ B ) @ R2 ) ) ).
% 0.34/0.52
% 0.34/0.52 % RangeE
% 0.34/0.52 thf(fact_738_RangeE,axiom,
% 0.34/0.52 ! [B: int,R2: set_Pr958786334691620121nt_int] :
% 0.34/0.52 ( ( member_int @ B @ ( range_int_int @ R2 ) )
% 0.34/0.52 => ~ ! [A4: int] :
% 0.34/0.52 ~ ( member5262025264175285858nt_int @ ( product_Pair_int_int @ A4 @ B ) @ R2 ) ) ).
% 0.34/0.52
% 0.34/0.52 % RangeE
% 0.34/0.52 thf(fact_739_RangeE,axiom,
% 0.34/0.52 ! [B: produc3658429121746597890et_nat,R2: set_Pr3286484037609594932et_nat] :
% 0.34/0.52 ( ( member6260224972018164377et_nat @ B @ ( range_3398330936903821850et_nat @ R2 ) )
% 0.34/0.52 => ~ ! [A4: produc3658429121746597890et_nat > $o] :
% 0.34/0.52 ~ ( member1996754912294343701et_nat @ ( produc5001842942810119800et_nat @ A4 @ B ) @ R2 ) ) ).
% 0.34/0.52
% 0.34/0.52 % RangeE
% 0.34/0.52 thf(fact_740_RangeE,axiom,
% 0.34/0.52 ! [B: produc3925858234332021118et_nat,R2: set_Pr8536935166611901872et_nat] :
% 0.34/0.52 ( ( member1996754912294343701et_nat @ B @ ( range_7260058354244137878et_nat @ R2 ) )
% 0.34/0.52 => ~ ! [A4: produc3658429121746597890et_nat > $o] :
% 0.34/0.52 ~ ( member6124377750444531601et_nat @ ( produc2245416461498447860et_nat @ A4 @ B ) @ R2 ) ) ).
% 0.34/0.52
% 0.34/0.52 % RangeE
% 0.34/0.52 thf(fact_741_Range__iff,axiom,
% 0.34/0.52 ! [A2: set_nat,R2: set_Pr3948176798113811640et_nat] :
% 0.34/0.52 ( ( member_set_nat @ A2 @ ( range_9132311542094983838et_nat @ R2 ) )
% 0.34/0.52 = ( ? [Y3: heap_e7401611519738050253t_unit] : ( member6260224972018164377et_nat @ ( produc7507926704131184380et_nat @ Y3 @ A2 ) @ R2 ) ) ) ).
% 0.34/0.52
% 0.34/0.52 % Range_iff
% 0.34/0.52 thf(fact_742_Range__iff,axiom,
% 0.34/0.52 ! [A2: nat,R2: set_Pr1261947904930325089at_nat] :
% 0.34/0.52 ( ( member_nat @ A2 @ ( range_nat_nat @ R2 ) )
% 0.34/0.52 = ( ? [Y3: nat] : ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ Y3 @ A2 ) @ R2 ) ) ) ).
% 0.34/0.52
% 0.34/0.52 % Range_iff
% 0.34/0.52 thf(fact_743_Range__iff,axiom,
% 0.34/0.52 ! [A2: int,R2: set_Pr958786334691620121nt_int] :
% 0.34/0.52 ( ( member_int @ A2 @ ( range_int_int @ R2 ) )
% 0.34/0.52 = ( ? [Y3: int] : ( member5262025264175285858nt_int @ ( product_Pair_int_int @ Y3 @ A2 ) @ R2 ) ) ) ).
% 0.34/0.52
% 0.34/0.52 % Range_iff
% 0.34/0.52 thf(fact_744_Range__iff,axiom,
% 0.34/0.52 ! [A2: produc3658429121746597890et_nat,R2: set_Pr3286484037609594932et_nat] :
% 0.34/0.52 ( ( member6260224972018164377et_nat @ A2 @ ( range_3398330936903821850et_nat @ R2 ) )
% 0.34/0.52 = ( ? [Y3: produc3658429121746597890et_nat > $o] : ( member1996754912294343701et_nat @ ( produc5001842942810119800et_nat @ Y3 @ A2 ) @ R2 ) ) ) ).
% 0.34/0.52
% 0.34/0.52 % Range_iff
% 0.34/0.52 thf(fact_745_Range__iff,axiom,
% 0.34/0.52 ! [A2: produc3925858234332021118et_nat,R2: set_Pr8536935166611901872et_nat] :
% 0.34/0.52 ( ( member1996754912294343701et_nat @ A2 @ ( range_7260058354244137878et_nat @ R2 ) )
% 0.34/0.52 = ( ? [Y3: produc3658429121746597890et_nat > $o] : ( member6124377750444531601et_nat @ ( produc2245416461498447860et_nat @ Y3 @ A2 ) @ R2 ) ) ) ).
% 0.34/0.52
% 0.34/0.52 % Range_iff
% 0.34/0.52 thf(fact_746_vimage__singleton__eq,axiom,
% 0.34/0.52 ! [A2: nat,F: nat > product_unit,B: product_unit] :
% 0.34/0.52 ( ( member_nat @ A2 @ ( vimage4884490618288580032t_unit @ F @ ( insert_Product_unit @ B @ bot_bo3957492148770167129t_unit ) ) )
% 0.34/0.52 = ( ( F @ A2 )
% 0.34/0.52 = B ) ) ).
% 0.34/0.52
% 0.34/0.52 % vimage_singleton_eq
% 0.34/0.52 thf(fact_747_vimage__singleton__eq,axiom,
% 0.34/0.52 ! [A2: int,F: int > product_unit,B: product_unit] :
% 0.34/0.52 ( ( member_int @ A2 @ ( vimage2000912565490760292t_unit @ F @ ( insert_Product_unit @ B @ bot_bo3957492148770167129t_unit ) ) )
% 0.34/0.52 = ( ( F @ A2 )
% 0.34/0.52 = B ) ) ).
% 0.34/0.52
% 0.34/0.52 % vimage_singleton_eq
% 0.34/0.52 thf(fact_748_vimage__singleton__eq,axiom,
% 0.34/0.52 ! [A2: $o,F: $o > product_unit,B: product_unit] :
% 0.34/0.52 ( ( member_o @ A2 @ ( vimage4222377557387472176t_unit @ F @ ( insert_Product_unit @ B @ bot_bo3957492148770167129t_unit ) ) )
% 0.34/0.52 = ( ( F @ A2 )
% 0.34/0.52 = B ) ) ).
% 0.34/0.52
% 0.34/0.52 % vimage_singleton_eq
% 0.34/0.52 thf(fact_749_vimage__singleton__eq,axiom,
% 0.34/0.52 ! [A2: product_unit,F: product_unit > product_unit,B: product_unit] :
% 0.34/0.52 ( ( member_Product_unit @ A2 @ ( vimage7995052115951654139t_unit @ F @ ( insert_Product_unit @ B @ bot_bo3957492148770167129t_unit ) ) )
% 0.34/0.52 = ( ( F @ A2 )
% 0.34/0.52 = B ) ) ).
% 0.34/0.52
% 0.34/0.52 % vimage_singleton_eq
% 0.34/0.52 thf(fact_750_vimage__singleton__eq,axiom,
% 0.34/0.52 ! [A2: nat,F: nat > $o,B: $o] :
% 0.34/0.52 ( ( member_nat @ A2 @ ( vimage_nat_o @ F @ ( insert_o @ B @ bot_bot_set_o ) ) )
% 0.34/0.52 = ( ( F @ A2 )
% 0.34/0.52 = B ) ) ).
% 0.34/0.52
% 0.34/0.52 % vimage_singleton_eq
% 0.34/0.52 thf(fact_751_vimage__singleton__eq,axiom,
% 0.34/0.52 ! [A2: int,F: int > $o,B: $o] :
% 0.34/0.52 ( ( member_int @ A2 @ ( vimage_int_o @ F @ ( insert_o @ B @ bot_bot_set_o ) ) )
% 0.34/0.52 = ( ( F @ A2 )
% 0.34/0.52 = B ) ) ).
% 0.34/0.52
% 0.34/0.52 % vimage_singleton_eq
% 0.34/0.52 thf(fact_752_vimage__singleton__eq,axiom,
% 0.34/0.52 ! [A2: $o,F: $o > $o,B: $o] :
% 0.34/0.52 ( ( member_o @ A2 @ ( vimage_o_o @ F @ ( insert_o @ B @ bot_bot_set_o ) ) )
% 0.34/0.52 = ( ( F @ A2 )
% 0.34/0.52 = B ) ) ).
% 0.34/0.52
% 0.34/0.52 % vimage_singleton_eq
% 0.34/0.52 thf(fact_753_vimage__singleton__eq,axiom,
% 0.34/0.52 ! [A2: product_unit,F: product_unit > $o,B: $o] :
% 0.34/0.52 ( ( member_Product_unit @ A2 @ ( vimage8808348054653608838unit_o @ F @ ( insert_o @ B @ bot_bot_set_o ) ) )
% 0.34/0.52 = ( ( F @ A2 )
% 0.34/0.52 = B ) ) ).
% 0.34/0.52
% 0.34/0.52 % vimage_singleton_eq
% 0.34/0.52 thf(fact_754_vimage__singleton__eq,axiom,
% 0.34/0.52 ! [A2: nat,F: nat > nat,B: nat] :
% 0.34/0.52 ( ( member_nat @ A2 @ ( vimage_nat_nat @ F @ ( insert_nat @ B @ bot_bot_set_nat ) ) )
% 0.34/0.52 = ( ( F @ A2 )
% 0.34/0.52 = B ) ) ).
% 0.34/0.52
% 0.34/0.52 % vimage_singleton_eq
% 0.34/0.52 thf(fact_755_vimage__singleton__eq,axiom,
% 0.34/0.52 ! [A2: int,F: int > nat,B: nat] :
% 0.34/0.52 ( ( member_int @ A2 @ ( vimage_int_nat @ F @ ( insert_nat @ B @ bot_bot_set_nat ) ) )
% 0.34/0.52 = ( ( F @ A2 )
% 0.34/0.52 = B ) ) ).
% 0.34/0.52
% 0.34/0.52 % vimage_singleton_eq
% 0.34/0.52 thf(fact_756_relH__in__rangeI_I2_J,axiom,
% 0.34/0.52 ! [As: set_nat,H2: heap_e7401611519738050253t_unit,H3: heap_e7401611519738050253t_unit] :
% 0.34/0.52 ( ( relH @ As @ H2 @ H3 )
% 0.34/0.52 => ( in_range @ ( produc7507926704131184380et_nat @ H3 @ As ) ) ) ).
% 0.34/0.52
% 0.34/0.52 % relH_in_rangeI(2)
% 0.34/0.52 thf(fact_757_relH__in__rangeI_I1_J,axiom,
% 0.34/0.52 ! [As: set_nat,H2: heap_e7401611519738050253t_unit,H3: heap_e7401611519738050253t_unit] :
% 0.34/0.52 ( ( relH @ As @ H2 @ H3 )
% 0.34/0.52 => ( in_range @ ( produc7507926704131184380et_nat @ H2 @ As ) ) ) ).
% 0.34/0.52
% 0.34/0.52 % relH_in_rangeI(1)
% 0.34/0.52 thf(fact_758_relH__refl,axiom,
% 0.34/0.52 ! [H2: heap_e7401611519738050253t_unit,As: set_nat] :
% 0.34/0.52 ( ( in_range @ ( produc7507926704131184380et_nat @ H2 @ As ) )
% 0.34/0.52 => ( relH @ As @ H2 @ H2 ) ) ).
% 0.34/0.52
% 0.34/0.52 % relH_refl
% 0.34/0.52 thf(fact_759_Image__empty__trancl__Image__empty,axiom,
% 0.34/0.52 ! [R3: set_Pr5094982260447487303t_unit,V: product_unit] :
% 0.34/0.52 ( ( ( image_8431784415008483091t_unit @ R3 @ ( insert_Product_unit @ V @ bot_bo3957492148770167129t_unit ) )
% 0.34/0.52 = bot_bo3957492148770167129t_unit )
% 0.34/0.52 => ( ( image_8431784415008483091t_unit @ ( transi430408012122777006t_unit @ R3 ) @ ( insert_Product_unit @ V @ bot_bo3957492148770167129t_unit ) )
% 0.34/0.52 = bot_bo3957492148770167129t_unit ) ) ).
% 0.34/0.52
% 0.34/0.52 % Image_empty_trancl_Image_empty
% 0.34/0.52 thf(fact_760_Image__empty__trancl__Image__empty,axiom,
% 0.34/0.53 ! [R3: set_Product_prod_o_o,V: $o] :
% 0.34/0.53 ( ( ( image_o_o @ R3 @ ( insert_o @ V @ bot_bot_set_o ) )
% 0.34/0.53 = bot_bot_set_o )
% 0.34/0.53 => ( ( image_o_o @ ( transitive_trancl_o @ R3 ) @ ( insert_o @ V @ bot_bot_set_o ) )
% 0.34/0.53 = bot_bot_set_o ) ) ).
% 0.34/0.53
% 0.34/0.53 % Image_empty_trancl_Image_empty
% 0.34/0.53 thf(fact_761_Image__empty__trancl__Image__empty,axiom,
% 0.34/0.53 ! [R3: set_Pr1261947904930325089at_nat,V: nat] :
% 0.34/0.53 ( ( ( image_nat_nat @ R3 @ ( insert_nat @ V @ bot_bot_set_nat ) )
% 0.34/0.53 = bot_bot_set_nat )
% 0.34/0.53 => ( ( image_nat_nat @ ( transi6264000038957366511cl_nat @ R3 ) @ ( insert_nat @ V @ bot_bot_set_nat ) )
% 0.34/0.53 = bot_bot_set_nat ) ) ).
% 0.34/0.53
% 0.34/0.53 % Image_empty_trancl_Image_empty
% 0.34/0.53 thf(fact_762_Image__empty__trancl__Image__empty,axiom,
% 0.34/0.53 ! [R3: set_Pr958786334691620121nt_int,V: int] :
% 0.34/0.53 ( ( ( image_int_int @ R3 @ ( insert_int @ V @ bot_bot_set_int ) )
% 0.34/0.53 = bot_bot_set_int )
% 0.34/0.53 => ( ( image_int_int @ ( transi6261509568448316235cl_int @ R3 ) @ ( insert_int @ V @ bot_bot_set_int ) )
% 0.34/0.53 = bot_bot_set_int ) ) ).
% 0.34/0.53
% 0.34/0.53 % Image_empty_trancl_Image_empty
% 0.34/0.53 thf(fact_763_Image__empty__trancl__Image__empty,axiom,
% 0.34/0.53 ! [R3: set_Pr2161125870931222855_nat_o,V: produc3658429121746597890et_nat > $o] :
% 0.34/0.53 ( ( ( image_2232984160439939859_nat_o @ R3 @ ( insert5175938949040314269_nat_o @ V @ bot_bo7824918357723582233_nat_o ) )
% 0.34/0.53 = bot_bo7824918357723582233_nat_o )
% 0.34/0.53 => ( ( image_2232984160439939859_nat_o @ ( transi4009562729418357614_nat_o @ R3 ) @ ( insert5175938949040314269_nat_o @ V @ bot_bo7824918357723582233_nat_o ) )
% 0.34/0.53 = bot_bo7824918357723582233_nat_o ) ) ).
% 0.34/0.53
% 0.34/0.53 % Image_empty_trancl_Image_empty
% 0.34/0.53 thf(fact_764_mod__relH,axiom,
% 0.34/0.53 ! [As: set_nat,H2: heap_e7401611519738050253t_unit,H3: heap_e7401611519738050253t_unit,P: assn] :
% 0.34/0.53 ( ( relH @ As @ H2 @ H3 )
% 0.34/0.53 => ( ( rep_assn @ P @ ( produc7507926704131184380et_nat @ H2 @ As ) )
% 0.34/0.53 = ( rep_assn @ P @ ( produc7507926704131184380et_nat @ H3 @ As ) ) ) ) ).
% 0.34/0.53
% 0.34/0.53 % mod_relH
% 0.34/0.53 thf(fact_765_for__in__RI,axiom,
% 0.34/0.53 ! [X: heap_e7401611519738050253t_unit,R3: set_Pr3948176798113811640et_nat] :
% 0.34/0.53 ( ( member242389114213480694t_unit @ X @ ( domain2007548326387822151et_nat @ R3 ) )
% 0.34/0.53 => ( member6260224972018164377et_nat @ ( produc7507926704131184380et_nat @ X @ ( fun_of5878246503291251064et_nat @ R3 @ X ) ) @ R3 ) ) ).
% 0.34/0.53
% 0.34/0.53 % for_in_RI
% 0.34/0.53 thf(fact_766_for__in__RI,axiom,
% 0.34/0.53 ! [X: nat,R3: set_Pr1261947904930325089at_nat] :
% 0.34/0.53 ( ( member_nat @ X @ ( domain_nat_nat @ R3 ) )
% 0.34/0.53 => ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ ( fun_of_rel_nat_nat @ R3 @ X ) ) @ R3 ) ) ).
% 0.34/0.53
% 0.34/0.53 % for_in_RI
% 0.34/0.53 thf(fact_767_for__in__RI,axiom,
% 0.34/0.53 ! [X: int,R3: set_Pr958786334691620121nt_int] :
% 0.34/0.53 ( ( member_int @ X @ ( domain_int_int @ R3 ) )
% 0.34/0.53 => ( member5262025264175285858nt_int @ ( product_Pair_int_int @ X @ ( fun_of_rel_int_int @ R3 @ X ) ) @ R3 ) ) ).
% 0.34/0.53
% 0.34/0.53 % for_in_RI
% 0.34/0.53 thf(fact_768_for__in__RI,axiom,
% 0.34/0.53 ! [X: produc3658429121746597890et_nat > $o,R3: set_Pr3286484037609594932et_nat] :
% 0.34/0.53 ( ( member6576561426505652726_nat_o @ X @ ( domain2358961613337345475et_nat @ R3 ) )
% 0.34/0.53 => ( member1996754912294343701et_nat @ ( produc5001842942810119800et_nat @ X @ ( fun_of3673525222065131764et_nat @ R3 @ X ) ) @ R3 ) ) ).
% 0.34/0.53
% 0.34/0.53 % for_in_RI
% 0.34/0.53 thf(fact_769_for__in__RI,axiom,
% 0.34/0.53 ! [X: produc3658429121746597890et_nat > $o,R3: set_Pr8536935166611901872et_nat] :
% 0.34/0.53 ( ( member6576561426505652726_nat_o @ X @ ( domain3618434998481624895et_nat @ R3 ) )
% 0.34/0.53 => ( member6124377750444531601et_nat @ ( produc2245416461498447860et_nat @ X @ ( fun_of384177504262425200et_nat @ R3 @ X ) ) @ R3 ) ) ).
% 0.34/0.53
% 0.34/0.53 % for_in_RI
% 0.34/0.53 thf(fact_770_trancl_Ocases,axiom,
% 0.34/0.53 ! [A1: nat,A22: nat,R2: set_Pr1261947904930325089at_nat] :
% 0.34/0.53 ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A1 @ A22 ) @ ( transi6264000038957366511cl_nat @ R2 ) )
% 0.34/0.53 => ( ~ ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A1 @ A22 ) @ R2 )
% 0.34/0.53 => ~ ! [B2: nat] :
% 0.34/0.53 ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A1 @ B2 ) @ ( transi6264000038957366511cl_nat @ R2 ) )
% 0.34/0.53 => ~ ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ B2 @ A22 ) @ R2 ) ) ) ) ).
% 0.34/0.53
% 0.34/0.53 % trancl.cases
% 0.34/0.53 thf(fact_771_trancl_Ocases,axiom,
% 0.34/0.53 ! [A1: int,A22: int,R2: set_Pr958786334691620121nt_int] :
% 0.34/0.53 ( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ A1 @ A22 ) @ ( transi6261509568448316235cl_int @ R2 ) )
% 0.34/0.53 => ( ~ ( member5262025264175285858nt_int @ ( product_Pair_int_int @ A1 @ A22 ) @ R2 )
% 0.34/0.53 => ~ ! [B2: int] :
% 0.34/0.53 ( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ A1 @ B2 ) @ ( transi6261509568448316235cl_int @ R2 ) )
% 0.34/0.53 => ~ ( member5262025264175285858nt_int @ ( product_Pair_int_int @ B2 @ A22 ) @ R2 ) ) ) ) ).
% 0.34/0.53
% 0.34/0.53 % trancl.cases
% 0.34/0.53 thf(fact_772_trancl_Osimps,axiom,
% 0.34/0.53 ! [A1: nat,A22: nat,R2: set_Pr1261947904930325089at_nat] :
% 0.34/0.53 ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A1 @ A22 ) @ ( transi6264000038957366511cl_nat @ R2 ) )
% 0.34/0.53 = ( ? [A6: nat,B6: nat] :
% 0.34/0.53 ( ( A1 = A6 )
% 0.34/0.53 & ( A22 = B6 )
% 0.34/0.53 & ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A6 @ B6 ) @ R2 ) )
% 0.34/0.53 | ? [A6: nat,B6: nat,C4: nat] :
% 0.34/0.53 ( ( A1 = A6 )
% 0.34/0.53 & ( A22 = C4 )
% 0.34/0.53 & ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A6 @ B6 ) @ ( transi6264000038957366511cl_nat @ R2 ) )
% 0.34/0.53 & ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ B6 @ C4 ) @ R2 ) ) ) ) ).
% 0.34/0.53
% 0.34/0.53 % trancl.simps
% 0.34/0.53 thf(fact_773_trancl_Osimps,axiom,
% 0.34/0.53 ! [A1: int,A22: int,R2: set_Pr958786334691620121nt_int] :
% 0.34/0.53 ( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ A1 @ A22 ) @ ( transi6261509568448316235cl_int @ R2 ) )
% 0.34/0.53 = ( ? [A6: int,B6: int] :
% 0.34/0.53 ( ( A1 = A6 )
% 0.34/0.53 & ( A22 = B6 )
% 0.34/0.53 & ( member5262025264175285858nt_int @ ( product_Pair_int_int @ A6 @ B6 ) @ R2 ) )
% 0.34/0.53 | ? [A6: int,B6: int,C4: int] :
% 0.34/0.53 ( ( A1 = A6 )
% 0.34/0.53 & ( A22 = C4 )
% 0.34/0.53 & ( member5262025264175285858nt_int @ ( product_Pair_int_int @ A6 @ B6 ) @ ( transi6261509568448316235cl_int @ R2 ) )
% 0.34/0.53 & ( member5262025264175285858nt_int @ ( product_Pair_int_int @ B6 @ C4 ) @ R2 ) ) ) ) ).
% 0.34/0.53
% 0.34/0.53 % trancl.simps
% 0.34/0.53 thf(fact_774_trancl_Or__into__trancl,axiom,
% 0.34/0.53 ! [A2: nat,B: nat,R2: set_Pr1261947904930325089at_nat] :
% 0.34/0.53 ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A2 @ B ) @ R2 )
% 0.34/0.53 => ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A2 @ B ) @ ( transi6264000038957366511cl_nat @ R2 ) ) ) ).
% 0.34/0.53
% 0.34/0.53 % trancl.r_into_trancl
% 0.34/0.53 thf(fact_775_trancl_Or__into__trancl,axiom,
% 0.34/0.53 ! [A2: int,B: int,R2: set_Pr958786334691620121nt_int] :
% 0.34/0.53 ( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ A2 @ B ) @ R2 )
% 0.34/0.53 => ( member5262025264175285858nt_int @ ( product_Pair_int_int @ A2 @ B ) @ ( transi6261509568448316235cl_int @ R2 ) ) ) ).
% 0.34/0.53
% 0.34/0.53 % trancl.r_into_trancl
% 0.34/0.53 thf(fact_776_tranclE,axiom,
% 0.34/0.53 ! [A2: nat,B: nat,R2: set_Pr1261947904930325089at_nat] :
% 0.34/0.53 ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A2 @ B ) @ ( transi6264000038957366511cl_nat @ R2 ) )
% 0.34/0.53 => ( ~ ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A2 @ B ) @ R2 )
% 0.34/0.53 => ~ ! [C2: nat] :
% 0.34/0.53 ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A2 @ C2 ) @ ( transi6264000038957366511cl_nat @ R2 ) )
% 0.34/0.53 => ~ ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ C2 @ B ) @ R2 ) ) ) ) ).
% 0.34/0.53
% 0.34/0.53 % tranclE
% 0.34/0.53 thf(fact_777_tranclE,axiom,
% 0.34/0.53 ! [A2: int,B: int,R2: set_Pr958786334691620121nt_int] :
% 0.34/0.53 ( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ A2 @ B ) @ ( transi6261509568448316235cl_int @ R2 ) )
% 0.34/0.53 => ( ~ ( member5262025264175285858nt_int @ ( product_Pair_int_int @ A2 @ B ) @ R2 )
% 0.34/0.53 => ~ ! [C2: int] :
% 0.34/0.53 ( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ A2 @ C2 ) @ ( transi6261509568448316235cl_int @ R2 ) )
% 0.34/0.53 => ~ ( member5262025264175285858nt_int @ ( product_Pair_int_int @ C2 @ B ) @ R2 ) ) ) ) ).
% 0.34/0.53
% 0.34/0.53 % tranclE
% 0.34/0.53 thf(fact_778_trancl__trans,axiom,
% 0.34/0.53 ! [X: nat,Y: nat,R2: set_Pr1261947904930325089at_nat,Z: nat] :
% 0.34/0.53 ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ Y ) @ ( transi6264000038957366511cl_nat @ R2 ) )
% 0.34/0.53 => ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ Y @ Z ) @ ( transi6264000038957366511cl_nat @ R2 ) )
% 0.34/0.53 => ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ Z ) @ ( transi6264000038957366511cl_nat @ R2 ) ) ) ) ).
% 0.34/0.53
% 0.34/0.53 % trancl_trans
% 0.34/0.53 thf(fact_779_trancl__trans,axiom,
% 0.34/0.53 ! [X: int,Y: int,R2: set_Pr958786334691620121nt_int,Z: int] :
% 0.34/0.53 ( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ X @ Y ) @ ( transi6261509568448316235cl_int @ R2 ) )
% 0.34/0.53 => ( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ Y @ Z ) @ ( transi6261509568448316235cl_int @ R2 ) )
% 0.34/0.53 => ( member5262025264175285858nt_int @ ( product_Pair_int_int @ X @ Z ) @ ( transi6261509568448316235cl_int @ R2 ) ) ) ) ).
% 0.34/0.53
% 0.34/0.53 % trancl_trans
% 0.34/0.53 thf(fact_780_trancl__induct,axiom,
% 0.34/0.53 ! [A2: nat,B: nat,R2: set_Pr1261947904930325089at_nat,P: nat > $o] :
% 0.34/0.53 ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A2 @ B ) @ ( transi6264000038957366511cl_nat @ R2 ) )
% 0.34/0.53 => ( ! [Y2: nat] :
% 0.34/0.53 ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A2 @ Y2 ) @ R2 )
% 0.34/0.53 => ( P @ Y2 ) )
% 0.34/0.53 => ( ! [Y2: nat,Z3: nat] :
% 0.34/0.53 ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A2 @ Y2 ) @ ( transi6264000038957366511cl_nat @ R2 ) )
% 0.34/0.53 => ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ Y2 @ Z3 ) @ R2 )
% 0.34/0.53 => ( ( P @ Y2 )
% 0.34/0.53 => ( P @ Z3 ) ) ) )
% 0.34/0.53 => ( P @ B ) ) ) ) ).
% 0.34/0.53
% 0.34/0.53 % trancl_induct
% 0.34/0.53 thf(fact_781_trancl__induct,axiom,
% 0.34/0.53 ! [A2: int,B: int,R2: set_Pr958786334691620121nt_int,P: int > $o] :
% 0.34/0.53 ( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ A2 @ B ) @ ( transi6261509568448316235cl_int @ R2 ) )
% 0.34/0.53 => ( ! [Y2: int] :
% 0.34/0.53 ( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ A2 @ Y2 ) @ R2 )
% 0.34/0.53 => ( P @ Y2 ) )
% 0.34/0.53 => ( ! [Y2: int,Z3: int] :
% 0.34/0.53 ( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ A2 @ Y2 ) @ ( transi6261509568448316235cl_int @ R2 ) )
% 0.34/0.53 => ( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ Y2 @ Z3 ) @ R2 )
% 0.34/0.53 => ( ( P @ Y2 )
% 0.34/0.53 => ( P @ Z3 ) ) ) )
% 0.34/0.53 => ( P @ B ) ) ) ) ).
% 0.34/0.53
% 0.34/0.53 % trancl_induct
% 0.34/0.53 thf(fact_782_r__r__into__trancl,axiom,
% 0.34/0.53 ! [A2: nat,B: nat,R3: set_Pr1261947904930325089at_nat,C: nat] :
% 0.34/0.53 ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A2 @ B ) @ R3 )
% 0.34/0.53 => ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ B @ C ) @ R3 )
% 0.34/0.53 => ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A2 @ C ) @ ( transi6264000038957366511cl_nat @ R3 ) ) ) ) ).
% 0.34/0.53
% 0.34/0.53 % r_r_into_trancl
% 0.34/0.53 thf(fact_783_r__r__into__trancl,axiom,
% 0.34/0.53 ! [A2: int,B: int,R3: set_Pr958786334691620121nt_int,C: int] :
% 0.34/0.53 ( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ A2 @ B ) @ R3 )
% 0.34/0.53 => ( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ B @ C ) @ R3 )
% 0.34/0.53 => ( member5262025264175285858nt_int @ ( product_Pair_int_int @ A2 @ C ) @ ( transi6261509568448316235cl_int @ R3 ) ) ) ) ).
% 0.34/0.53
% 0.34/0.53 % r_r_into_trancl
% 0.34/0.53 thf(fact_784_converse__tranclE,axiom,
% 0.34/0.53 ! [X: nat,Z: nat,R2: set_Pr1261947904930325089at_nat] :
% 0.34/0.53 ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ Z ) @ ( transi6264000038957366511cl_nat @ R2 ) )
% 0.34/0.53 => ( ~ ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ Z ) @ R2 )
% 0.34/0.53 => ~ ! [Y2: nat] :
% 0.34/0.53 ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ Y2 ) @ R2 )
% 0.34/0.53 => ~ ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ Y2 @ Z ) @ ( transi6264000038957366511cl_nat @ R2 ) ) ) ) ) ).
% 0.34/0.53
% 0.34/0.53 % converse_tranclE
% 0.34/0.53 thf(fact_785_converse__tranclE,axiom,
% 0.34/0.53 ! [X: int,Z: int,R2: set_Pr958786334691620121nt_int] :
% 0.34/0.53 ( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ X @ Z ) @ ( transi6261509568448316235cl_int @ R2 ) )
% 0.34/0.53 => ( ~ ( member5262025264175285858nt_int @ ( product_Pair_int_int @ X @ Z ) @ R2 )
% 0.34/0.53 => ~ ! [Y2: int] :
% 0.34/0.53 ( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ X @ Y2 ) @ R2 )
% 0.34/0.53 => ~ ( member5262025264175285858nt_int @ ( product_Pair_int_int @ Y2 @ Z ) @ ( transi6261509568448316235cl_int @ R2 ) ) ) ) ) ).
% 0.34/0.53
% 0.34/0.53 % converse_tranclE
% 0.34/0.53 thf(fact_786_trancl__induct2,axiom,
% 0.34/0.53 ! [Ax: heap_e7401611519738050253t_unit,Ay: set_nat,Bx: heap_e7401611519738050253t_unit,By: set_nat,R2: set_Pr719794911490849221et_nat,P: heap_e7401611519738050253t_unit > set_nat > $o] :
% 0.34/0.53 ( ( member6099555550032318734et_nat @ ( produc8199053930788261021et_nat @ ( produc7507926704131184380et_nat @ Ax @ Ay ) @ ( produc7507926704131184380et_nat @ Bx @ By ) ) @ ( transi4137310414928362273et_nat @ R2 ) )
% 0.34/0.53 => ( ! [A4: heap_e7401611519738050253t_unit,B2: set_nat] :
% 0.34/0.53 ( ( member6099555550032318734et_nat @ ( produc8199053930788261021et_nat @ ( produc7507926704131184380et_nat @ Ax @ Ay ) @ ( produc7507926704131184380et_nat @ A4 @ B2 ) ) @ R2 )
% 0.34/0.53 => ( P @ A4 @ B2 ) )
% 0.34/0.53 => ( ! [A4: heap_e7401611519738050253t_unit,B2: set_nat,Aa: heap_e7401611519738050253t_unit,Ba: set_nat] :
% 0.34/0.53 ( ( member6099555550032318734et_nat @ ( produc8199053930788261021et_nat @ ( produc7507926704131184380et_nat @ Ax @ Ay ) @ ( produc7507926704131184380et_nat @ A4 @ B2 ) ) @ ( transi4137310414928362273et_nat @ R2 ) )
% 0.34/0.53 => ( ( member6099555550032318734et_nat @ ( produc8199053930788261021et_nat @ ( produc7507926704131184380et_nat @ A4 @ B2 ) @ ( produc7507926704131184380et_nat @ Aa @ Ba ) ) @ R2 )
% 0.34/0.53 => ( ( P @ A4 @ B2 )
% 0.34/0.53 => ( P @ Aa @ Ba ) ) ) )
% 0.34/0.53 => ( P @ Bx @ By ) ) ) ) ).
% 0.34/0.53
% 0.34/0.53 % trancl_induct2
% 0.34/0.53 thf(fact_787_trancl__induct2,axiom,
% 0.34/0.53 ! [Ax: nat,Ay: nat,Bx: nat,By: nat,R2: set_Pr8693737435421807431at_nat,P: nat > nat > $o] :
% 0.34/0.53 ( ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ ( product_Pair_nat_nat @ Ax @ Ay ) @ ( product_Pair_nat_nat @ Bx @ By ) ) @ ( transi243908449541399842at_nat @ R2 ) )
% 0.34/0.53 => ( ! [A4: nat,B2: nat] :
% 0.34/0.53 ( ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ ( product_Pair_nat_nat @ Ax @ Ay ) @ ( product_Pair_nat_nat @ A4 @ B2 ) ) @ R2 )
% 0.34/0.53 => ( P @ A4 @ B2 ) )
% 0.34/0.53 => ( ! [A4: nat,B2: nat,Aa: nat,Ba: nat] :
% 0.34/0.53 ( ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ ( product_Pair_nat_nat @ Ax @ Ay ) @ ( product_Pair_nat_nat @ A4 @ B2 ) ) @ ( transi243908449541399842at_nat @ R2 ) )
% 0.34/0.53 => ( ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ ( product_Pair_nat_nat @ A4 @ B2 ) @ ( product_Pair_nat_nat @ Aa @ Ba ) ) @ R2 )
% 0.34/0.53 => ( ( P @ A4 @ B2 )
% 0.34/0.53 => ( P @ Aa @ Ba ) ) ) )
% 0.34/0.53 => ( P @ Bx @ By ) ) ) ) ).
% 0.34/0.53
% 0.34/0.53 % trancl_induct2
% 0.34/0.53 thf(fact_788_trancl__induct2,axiom,
% 0.34/0.53 ! [Ax: int,Ay: int,Bx: int,By: int,R2: set_Pr2560585780119916871nt_int,P: int > int > $o] :
% 0.34/0.53 ( ( member8566619992076573584nt_int @ ( produc3646306378393792727nt_int @ ( product_Pair_int_int @ Ax @ Ay ) @ ( product_Pair_int_int @ Bx @ By ) ) @ ( transi6288783178788033498nt_int @ R2 ) )
% 0.34/0.53 => ( ! [A4: int,B2: int] :
% 0.34/0.53 ( ( member8566619992076573584nt_int @ ( produc3646306378393792727nt_int @ ( product_Pair_int_int @ Ax @ Ay ) @ ( product_Pair_int_int @ A4 @ B2 ) ) @ R2 )
% 0.34/0.53 => ( P @ A4 @ B2 ) )
% 0.34/0.53 => ( ! [A4: int,B2: int,Aa: int,Ba: int] :
% 0.34/0.53 ( ( member8566619992076573584nt_int @ ( produc3646306378393792727nt_int @ ( product_Pair_int_int @ Ax @ Ay ) @ ( product_Pair_int_int @ A4 @ B2 ) ) @ ( transi6288783178788033498nt_int @ R2 ) )
% 0.34/0.53 => ( ( member8566619992076573584nt_int @ ( produc3646306378393792727nt_int @ ( product_Pair_int_int @ A4 @ B2 ) @ ( product_Pair_int_int @ Aa @ Ba ) ) @ R2 )
% 0.34/0.53 => ( ( P @ A4 @ B2 )
% 0.34/0.53 => ( P @ Aa @ Ba ) ) ) )
% 0.34/0.53 => ( P @ Bx @ By ) ) ) ) ).
% 0.34/0.53
% 0.34/0.53 % trancl_induct2
% 0.34/0.53 thf(fact_789_trancl__induct2,axiom,
% 0.34/0.53 ! [Ax: produc3658429121746597890et_nat > $o,Ay: produc3658429121746597890et_nat,Bx: produc3658429121746597890et_nat > $o,By: produc3658429121746597890et_nat,R2: set_Pr7928877670098842301et_nat,P: ( produc3658429121746597890et_nat > $o ) > produc3658429121746597890et_nat > $o] :
% 0.34/0.53 ( ( member4763271486408492550et_nat @ ( produc8599840265553166229et_nat @ ( produc5001842942810119800et_nat @ Ax @ Ay ) @ ( produc5001842942810119800et_nat @ Bx @ By ) ) @ ( transi3145040225084697757et_nat @ R2 ) )
% 0.34/0.53 => ( ! [A4: produc3658429121746597890et_nat > $o,B2: produc3658429121746597890et_nat] :
% 0.34/0.53 ( ( member4763271486408492550et_nat @ ( produc8599840265553166229et_nat @ ( produc5001842942810119800et_nat @ Ax @ Ay ) @ ( produc5001842942810119800et_nat @ A4 @ B2 ) ) @ R2 )
% 0.34/0.53 => ( P @ A4 @ B2 ) )
% 0.34/0.53 => ( ! [A4: produc3658429121746597890et_nat > $o,B2: produc3658429121746597890et_nat,Aa: produc3658429121746597890et_nat > $o,Ba: produc3658429121746597890et_nat] :
% 0.34/0.53 ( ( member4763271486408492550et_nat @ ( produc8599840265553166229et_nat @ ( produc5001842942810119800et_nat @ Ax @ Ay ) @ ( produc5001842942810119800et_nat @ A4 @ B2 ) ) @ ( transi3145040225084697757et_nat @ R2 ) )
% 0.34/0.53 => ( ( member4763271486408492550et_nat @ ( produc8599840265553166229et_nat @ ( produc5001842942810119800et_nat @ A4 @ B2 ) @ ( produc5001842942810119800et_nat @ Aa @ Ba ) ) @ R2 )
% 0.34/0.53 => ( ( P @ A4 @ B2 )
% 0.34/0.53 => ( P @ Aa @ Ba ) ) ) )
% 0.34/0.53 => ( P @ Bx @ By ) ) ) ) ).
% 0.34/0.53
% 0.34/0.53 % trancl_induct2
% 0.34/0.53 thf(fact_790_trancl__induct2,axiom,
% 0.34/0.53 ! [Ax: produc3658429121746597890et_nat > $o,Ay: produc3925858234332021118et_nat,Bx: produc3658429121746597890et_nat > $o,By: produc3925858234332021118et_nat,R2: set_Pr3444600963470892981et_nat,P: ( produc3658429121746597890et_nat > $o ) > produc3925858234332021118et_nat > $o] :
% 0.34/0.53 ( ( member6341495586645257982et_nat @ ( produc1940133919992309389et_nat @ ( produc2245416461498447860et_nat @ Ax @ Ay ) @ ( produc2245416461498447860et_nat @ Bx @ By ) ) @ ( transi5221092739591632921et_nat @ R2 ) )
% 0.34/0.53 => ( ! [A4: produc3658429121746597890et_nat > $o,B2: produc3925858234332021118et_nat] :
% 0.34/0.53 ( ( member6341495586645257982et_nat @ ( produc1940133919992309389et_nat @ ( produc2245416461498447860et_nat @ Ax @ Ay ) @ ( produc2245416461498447860et_nat @ A4 @ B2 ) ) @ R2 )
% 0.34/0.53 => ( P @ A4 @ B2 ) )
% 0.34/0.53 => ( ! [A4: produc3658429121746597890et_nat > $o,B2: produc3925858234332021118et_nat,Aa: produc3658429121746597890et_nat > $o,Ba: produc3925858234332021118et_nat] :
% 0.34/0.53 ( ( member6341495586645257982et_nat @ ( produc1940133919992309389et_nat @ ( produc2245416461498447860et_nat @ Ax @ Ay ) @ ( produc2245416461498447860et_nat @ A4 @ B2 ) ) @ ( transi5221092739591632921et_nat @ R2 ) )
% 0.34/0.53 => ( ( member6341495586645257982et_nat @ ( produc1940133919992309389et_nat @ ( produc2245416461498447860et_nat @ A4 @ B2 ) @ ( produc2245416461498447860et_nat @ Aa @ Ba ) ) @ R2 )
% 0.34/0.53 => ( ( P @ A4 @ B2 )
% 0.34/0.53 => ( P @ Aa @ Ba ) ) ) )
% 0.34/0.53 => ( P @ Bx @ By ) ) ) ) ).
% 0.34/0.53
% 0.34/0.53 % trancl_induct2
% 0.34/0.53 thf(fact_791_converse__trancl__induct,axiom,
% 0.34/0.53 ! [A2: nat,B: nat,R2: set_Pr1261947904930325089at_nat,P: nat > $o] :
% 0.34/0.53 ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A2 @ B ) @ ( transi6264000038957366511cl_nat @ R2 ) )
% 0.34/0.53 => ( ! [Y2: nat] :
% 0.34/0.53 ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ Y2 @ B ) @ R2 )
% 0.34/0.53 => ( P @ Y2 ) )
% 0.34/0.53 => ( ! [Y2: nat,Z3: nat] :
% 0.34/0.53 ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ Y2 @ Z3 ) @ R2 )
% 0.34/0.53 => ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ Z3 @ B ) @ ( transi6264000038957366511cl_nat @ R2 ) )
% 0.34/0.53 => ( ( P @ Z3 )
% 0.34/0.53 => ( P @ Y2 ) ) ) )
% 0.34/0.53 => ( P @ A2 ) ) ) ) ).
% 0.34/0.53
% 0.34/0.53 % converse_trancl_induct
% 0.34/0.53 thf(fact_792_converse__trancl__induct,axiom,
% 0.34/0.53 ! [A2: int,B: int,R2: set_Pr958786334691620121nt_int,P: int > $o] :
% 0.34/0.53 ( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ A2 @ B ) @ ( transi6261509568448316235cl_int @ R2 ) )
% 0.34/0.53 => ( ! [Y2: int] :
% 0.34/0.53 ( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ Y2 @ B ) @ R2 )
% 0.34/0.53 => ( P @ Y2 ) )
% 0.34/0.53 => ( ! [Y2: int,Z3: int] :
% 0.34/0.53 ( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ Y2 @ Z3 ) @ R2 )
% 0.34/0.53 => ( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ Z3 @ B ) @ ( transi6261509568448316235cl_int @ R2 ) )
% 0.34/0.53 => ( ( P @ Z3 )
% 0.34/0.53 => ( P @ Y2 ) ) ) )
% 0.34/0.53 => ( P @ A2 ) ) ) ) ).
% 0.34/0.53
% 0.34/0.53 % converse_trancl_induct
% 0.34/0.53 thf(fact_793_trancl__trans__induct,axiom,
% 0.34/0.53 ! [X: nat,Y: nat,R2: set_Pr1261947904930325089at_nat,P: nat > nat > $o] :
% 0.34/0.53 ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ Y ) @ ( transi6264000038957366511cl_nat @ R2 ) )
% 0.34/0.53 => ( ! [X3: nat,Y2: nat] :
% 0.34/0.53 ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X3 @ Y2 ) @ R2 )
% 0.34/0.53 => ( P @ X3 @ Y2 ) )
% 0.34/0.53 => ( ! [X3: nat,Y2: nat,Z3: nat] :
% 0.34/0.53 ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X3 @ Y2 ) @ ( transi6264000038957366511cl_nat @ R2 ) )
% 0.34/0.53 => ( ( P @ X3 @ Y2 )
% 0.34/0.53 => ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ Y2 @ Z3 ) @ ( transi6264000038957366511cl_nat @ R2 ) )
% 0.34/0.53 => ( ( P @ Y2 @ Z3 )
% 0.34/0.53 => ( P @ X3 @ Z3 ) ) ) ) )
% 0.34/0.53 => ( P @ X @ Y ) ) ) ) ).
% 0.34/0.53
% 0.34/0.53 % trancl_trans_induct
% 0.34/0.53 thf(fact_794_trancl__trans__induct,axiom,
% 0.34/0.53 ! [X: int,Y: int,R2: set_Pr958786334691620121nt_int,P: int > int > $o] :
% 0.34/0.53 ( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ X @ Y ) @ ( transi6261509568448316235cl_int @ R2 ) )
% 0.34/0.53 => ( ! [X3: int,Y2: int] :
% 0.34/0.53 ( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ X3 @ Y2 ) @ R2 )
% 0.34/0.53 => ( P @ X3 @ Y2 ) )
% 0.34/0.53 => ( ! [X3: int,Y2: int,Z3: int] :
% 0.34/0.53 ( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ X3 @ Y2 ) @ ( transi6261509568448316235cl_int @ R2 ) )
% 0.34/0.53 => ( ( P @ X3 @ Y2 )
% 0.34/0.53 => ( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ Y2 @ Z3 ) @ ( transi6261509568448316235cl_int @ R2 ) )
% 0.34/0.53 => ( ( P @ Y2 @ Z3 )
% 0.34/0.53 => ( P @ X3 @ Z3 ) ) ) ) )
% 0.34/0.53 => ( P @ X @ Y ) ) ) ) ).
% 0.34/0.53
% 0.34/0.53 % trancl_trans_induct
% 0.34/0.53 thf(fact_795_trancl__into__trancl2,axiom,
% 0.34/0.53 ! [A2: nat,B: nat,R2: set_Pr1261947904930325089at_nat,C: nat] :
% 0.34/0.53 ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A2 @ B ) @ R2 )
% 0.34/0.53 => ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ B @ C ) @ ( transi6264000038957366511cl_nat @ R2 ) )
% 0.34/0.53 => ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A2 @ C ) @ ( transi6264000038957366511cl_nat @ R2 ) ) ) ) ).
% 0.34/0.53
% 0.34/0.53 % trancl_into_trancl2
% 0.34/0.53 thf(fact_796_trancl__into__trancl2,axiom,
% 0.34/0.53 ! [A2: int,B: int,R2: set_Pr958786334691620121nt_int,C: int] :
% 0.34/0.53 ( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ A2 @ B ) @ R2 )
% 0.34/0.53 => ( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ B @ C ) @ ( transi6261509568448316235cl_int @ R2 ) )
% 0.34/0.53 => ( member5262025264175285858nt_int @ ( product_Pair_int_int @ A2 @ C ) @ ( transi6261509568448316235cl_int @ R2 ) ) ) ) ).
% 0.34/0.53
% 0.34/0.53 % trancl_into_trancl2
% 0.34/0.53 thf(fact_797_Transitive__Closure_Otrancl__into__trancl,axiom,
% 0.34/0.53 ! [A2: nat,B: nat,R2: set_Pr1261947904930325089at_nat,C: nat] :
% 0.34/0.53 ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A2 @ B ) @ ( transi6264000038957366511cl_nat @ R2 ) )
% 0.34/0.53 => ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ B @ C ) @ R2 )
% 0.34/0.53 => ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A2 @ C ) @ ( transi6264000038957366511cl_nat @ R2 ) ) ) ) ).
% 0.34/0.53
% 0.34/0.53 % Transitive_Closure.trancl_into_trancl
% 0.34/0.53 thf(fact_798_Transitive__Closure_Otrancl__into__trancl,axiom,
% 0.34/0.53 ! [A2: int,B: int,R2: set_Pr958786334691620121nt_int,C: int] :
% 0.34/0.53 ( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ A2 @ B ) @ ( transi6261509568448316235cl_int @ R2 ) )
% 0.34/0.53 => ( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ B @ C ) @ R2 )
% 0.34/0.53 => ( member5262025264175285858nt_int @ ( product_Pair_int_int @ A2 @ C ) @ ( transi6261509568448316235cl_int @ R2 ) ) ) ) ).
% 0.34/0.53
% 0.34/0.53 % Transitive_Closure.trancl_into_trancl
% 0.34/0.53 thf(fact_799_irrefl__trancl__rD,axiom,
% 0.34/0.53 ! [R2: set_Pr1261947904930325089at_nat,X: nat,Y: nat] :
% 0.34/0.53 ( ! [X3: nat] :
% 0.34/0.53 ~ ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X3 @ X3 ) @ ( transi6264000038957366511cl_nat @ R2 ) )
% 0.34/0.53 => ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ Y ) @ R2 )
% 0.34/0.53 => ( X != Y ) ) ) ).
% 0.34/0.53
% 0.34/0.53 % irrefl_trancl_rD
% 0.34/0.53 thf(fact_800_irrefl__trancl__rD,axiom,
% 0.34/0.53 ! [R2: set_Pr958786334691620121nt_int,X: int,Y: int] :
% 0.34/0.53 ( ! [X3: int] :
% 0.34/0.53 ~ ( member5262025264175285858nt_int @ ( product_Pair_int_int @ X3 @ X3 ) @ ( transi6261509568448316235cl_int @ R2 ) )
% 0.34/0.53 => ( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ X @ Y ) @ R2 )
% 0.34/0.53 => ( X != Y ) ) ) ).
% 0.34/0.53
% 0.34/0.53 % irrefl_trancl_rD
% 0.34/0.53 thf(fact_801_trancl__Image__advance__ss,axiom,
% 0.34/0.53 ! [U: product_unit,V: product_unit,E: set_Pr5094982260447487303t_unit] :
% 0.34/0.53 ( ( member7821138191923116944t_unit @ ( produc5692694829384537815t_unit @ U @ V ) @ E )
% 0.34/0.53 => ( ord_le3507040750410214029t_unit @ ( image_8431784415008483091t_unit @ ( transi430408012122777006t_unit @ E ) @ ( insert_Product_unit @ V @ bot_bo3957492148770167129t_unit ) ) @ ( image_8431784415008483091t_unit @ ( transi430408012122777006t_unit @ E ) @ ( insert_Product_unit @ U @ bot_bo3957492148770167129t_unit ) ) ) ) ).
% 0.34/0.53
% 0.34/0.53 % trancl_Image_advance_ss
% 0.34/0.53 thf(fact_802_trancl__Image__advance__ss,axiom,
% 0.34/0.53 ! [U: $o,V: $o,E: set_Product_prod_o_o] :
% 0.34/0.53 ( ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ U @ V ) @ E )
% 0.34/0.53 => ( ord_less_eq_set_o @ ( image_o_o @ ( transitive_trancl_o @ E ) @ ( insert_o @ V @ bot_bot_set_o ) ) @ ( image_o_o @ ( transitive_trancl_o @ E ) @ ( insert_o @ U @ bot_bot_set_o ) ) ) ) ).
% 0.34/0.53
% 0.34/0.53 % trancl_Image_advance_ss
% 0.34/0.53 thf(fact_803_trancl__Image__advance__ss,axiom,
% 0.34/0.53 ! [U: int,V: int,E: set_Pr958786334691620121nt_int] :
% 0.34/0.53 ( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ U @ V ) @ E )
% 0.34/0.53 => ( ord_less_eq_set_int @ ( image_int_int @ ( transi6261509568448316235cl_int @ E ) @ ( insert_int @ V @ bot_bot_set_int ) ) @ ( image_int_int @ ( transi6261509568448316235cl_int @ E ) @ ( insert_int @ U @ bot_bot_set_int ) ) ) ) ).
% 0.34/0.53
% 0.34/0.53 % trancl_Image_advance_ss
% 0.34/0.53 thf(fact_804_trancl__Image__advance__ss,axiom,
% 0.34/0.53 ! [U: produc3658429121746597890et_nat > $o,V: produc3658429121746597890et_nat > $o,E: set_Pr2161125870931222855_nat_o] :
% 0.34/0.53 ( ( member8781333585448626064_nat_o @ ( produc7368190662567826135_nat_o @ U @ V ) @ E )
% 0.34/0.53 => ( ord_le2965882846123202637_nat_o @ ( image_2232984160439939859_nat_o @ ( transi4009562729418357614_nat_o @ E ) @ ( insert5175938949040314269_nat_o @ V @ bot_bo7824918357723582233_nat_o ) ) @ ( image_2232984160439939859_nat_o @ ( transi4009562729418357614_nat_o @ E ) @ ( insert5175938949040314269_nat_o @ U @ bot_bo7824918357723582233_nat_o ) ) ) ) ).
% 0.34/0.53
% 0.34/0.53 % trancl_Image_advance_ss
% 0.34/0.53 thf(fact_805_trancl__Image__advance__ss,axiom,
% 0.34/0.53 ! [U: nat,V: nat,E: set_Pr1261947904930325089at_nat] :
% 0.34/0.53 ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ U @ V ) @ E )
% 0.34/0.53 => ( ord_less_eq_set_nat @ ( image_nat_nat @ ( transi6264000038957366511cl_nat @ E ) @ ( insert_nat @ V @ bot_bot_set_nat ) ) @ ( image_nat_nat @ ( transi6264000038957366511cl_nat @ E ) @ ( insert_nat @ U @ bot_bot_set_nat ) ) ) ) ).
% 0.34/0.53
% 0.34/0.53 % trancl_Image_advance_ss
% 0.34/0.53 thf(fact_806_wand__assnI,axiom,
% 0.34/0.53 ! [H2: heap_e7401611519738050253t_unit,As: set_nat,Q2: assn,R3: assn] :
% 0.34/0.53 ( ( in_range @ ( produc7507926704131184380et_nat @ H2 @ As ) )
% 0.34/0.53 => ( ! [H6: heap_e7401611519738050253t_unit,As4: set_nat] :
% 0.34/0.53 ( ( ( inf_inf_set_nat @ As @ As4 )
% 0.34/0.53 = bot_bot_set_nat )
% 0.34/0.53 => ( ( relH @ As @ H2 @ H6 )
% 0.34/0.53 => ( ( in_range @ ( produc7507926704131184380et_nat @ H6 @ As ) )
% 0.34/0.53 => ( ( rep_assn @ Q2 @ ( produc7507926704131184380et_nat @ H6 @ As4 ) )
% 0.34/0.53 => ( rep_assn @ R3 @ ( produc7507926704131184380et_nat @ H6 @ ( sup_sup_set_nat @ As @ As4 ) ) ) ) ) ) )
% 0.34/0.53 => ( rep_assn @ ( wand_assn @ Q2 @ R3 ) @ ( produc7507926704131184380et_nat @ H2 @ As ) ) ) ) ).
% 0.34/0.53
% 0.34/0.53 % wand_assnI
% 0.34/0.53 thf(fact_807_trancl__Image__in__Range,axiom,
% 0.34/0.53 ! [R3: set_Pr1261947904930325089at_nat,V2: set_nat] : ( ord_less_eq_set_nat @ ( image_nat_nat @ ( transi6264000038957366511cl_nat @ R3 ) @ V2 ) @ ( range_nat_nat @ R3 ) ) ).
% 0.34/0.53
% 0.34/0.53 % trancl_Image_in_Range
% 0.34/0.53 thf(fact_808_Pair__vimage__Sigma,axiom,
% 0.34/0.53 ! [X: nat,A: set_nat,F: nat > set_Product_unit] :
% 0.34/0.53 ( ( ( member_nat @ X @ A )
% 0.34/0.53 => ( ( vimage7305857734363866308t_unit @ ( produc3434450474625521892t_unit @ X ) @ ( produc5506539147108004339t_unit @ A @ F ) )
% 0.34/0.53 = ( F @ X ) ) )
% 0.34/0.53 & ( ~ ( member_nat @ X @ A )
% 0.34/0.53 => ( ( vimage7305857734363866308t_unit @ ( produc3434450474625521892t_unit @ X ) @ ( produc5506539147108004339t_unit @ A @ F ) )
% 0.34/0.53 = bot_bo3957492148770167129t_unit ) ) ) ).
% 0.34/0.53
% 0.34/0.53 % Pair_vimage_Sigma
% 0.34/0.53 thf(fact_809_Pair__vimage__Sigma,axiom,
% 0.34/0.53 ! [X: int,A: set_int,F: int > set_Product_unit] :
% 0.34/0.53 ( ( ( member_int @ X @ A )
% 0.34/0.53 => ( ( vimage4422279681566046568t_unit @ ( produc550872421827702152t_unit @ X ) @ ( produc2622961094310184599t_unit @ A @ F ) )
% 0.34/0.53 = ( F @ X ) ) )
% 0.34/0.53 & ( ~ ( member_int @ X @ A )
% 0.34/0.53 => ( ( vimage4422279681566046568t_unit @ ( produc550872421827702152t_unit @ X ) @ ( produc2622961094310184599t_unit @ A @ F ) )
% 0.34/0.53 = bot_bo3957492148770167129t_unit ) ) ) ).
% 0.34/0.53
% 0.34/0.53 % Pair_vimage_Sigma
% 0.34/0.53 thf(fact_810_Pair__vimage__Sigma,axiom,
% 0.34/0.53 ! [X: $o,A: set_o,F: $o > set_Product_unit] :
% 0.34/0.53 ( ( ( member_o @ X @ A )
% 0.34/0.53 => ( ( vimage508909887551888138t_unit @ ( produc3901946521647900940t_unit @ X ) @ ( produc411009820408249917t_unit @ A @ F ) )
% 0.34/0.53 = ( F @ X ) ) )
% 0.34/0.53 & ( ~ ( member_o @ X @ A )
% 0.34/0.53 => ( ( vimage508909887551888138t_unit @ ( produc3901946521647900940t_unit @ X ) @ ( produc411009820408249917t_unit @ A @ F ) )
% 0.34/0.53 = bot_bo3957492148770167129t_unit ) ) ) ).
% 0.34/0.53
% 0.34/0.53 % Pair_vimage_Sigma
% 0.34/0.53 thf(fact_811_Pair__vimage__Sigma,axiom,
% 0.34/0.53 ! [X: product_unit,A: set_Product_unit,F: product_unit > set_Product_unit] :
% 0.34/0.53 ( ( ( member_Product_unit @ X @ A )
% 0.34/0.53 => ( ( vimage8114940889160087125t_unit @ ( produc5692694829384537815t_unit @ X ) @ ( produc6121607364029222536t_unit @ A @ F ) )
% 0.34/0.53 = ( F @ X ) ) )
% 0.34/0.53 & ( ~ ( member_Product_unit @ X @ A )
% 0.34/0.53 => ( ( vimage8114940889160087125t_unit @ ( produc5692694829384537815t_unit @ X ) @ ( produc6121607364029222536t_unit @ A @ F ) )
% 0.34/0.53 = bot_bo3957492148770167129t_unit ) ) ) ).
% 0.34/0.53
% 0.34/0.53 % Pair_vimage_Sigma
% 0.34/0.53 thf(fact_812_Pair__vimage__Sigma,axiom,
% 0.34/0.53 ! [X: nat,A: set_nat,F: nat > set_o] :
% 0.34/0.53 ( ( ( member_nat @ X @ A )
% 0.34/0.53 => ( ( vimage6125388955539849434_nat_o @ ( product_Pair_nat_o @ X ) @ ( product_Sigma_nat_o @ A @ F ) )
% 0.34/0.53 = ( F @ X ) ) )
% 0.34/0.53 & ( ~ ( member_nat @ X @ A )
% 0.34/0.53 => ( ( vimage6125388955539849434_nat_o @ ( product_Pair_nat_o @ X ) @ ( product_Sigma_nat_o @ A @ F ) )
% 0.34/0.53 = bot_bot_set_o ) ) ) ).
% 0.34/0.53
% 0.34/0.53 % Pair_vimage_Sigma
% 0.34/0.53 thf(fact_813_Pair__vimage__Sigma,axiom,
% 0.34/0.53 ! [X: int,A: set_int,F: int > set_o] :
% 0.34/0.53 ( ( ( member_int @ X @ A )
% 0.34/0.53 => ( ( vimage4304346610107161470_int_o @ ( product_Pair_int_o @ X ) @ ( product_Sigma_int_o @ A @ F ) )
% 0.34/0.53 = ( F @ X ) ) )
% 0.34/0.53 & ( ~ ( member_int @ X @ A )
% 0.34/0.53 => ( ( vimage4304346610107161470_int_o @ ( product_Pair_int_o @ X ) @ ( product_Sigma_int_o @ A @ F ) )
% 0.34/0.53 = bot_bot_set_o ) ) ) ).
% 0.34/0.53
% 0.34/0.53 % Pair_vimage_Sigma
% 0.34/0.53 thf(fact_814_Pair__vimage__Sigma,axiom,
% 0.34/0.53 ! [X: $o,A: set_o,F: $o > set_o] :
% 0.34/0.53 ( ( ( member_o @ X @ A )
% 0.34/0.53 => ( ( vimage8945963521958007626od_o_o @ ( product_Pair_o_o @ X ) @ ( product_Sigma_o_o @ A @ F ) )
% 0.34/0.53 = ( F @ X ) ) )
% 0.34/0.53 & ( ~ ( member_o @ X @ A )
% 0.34/0.53 => ( ( vimage8945963521958007626od_o_o @ ( product_Pair_o_o @ X ) @ ( product_Sigma_o_o @ A @ F ) )
% 0.34/0.53 = bot_bot_set_o ) ) ) ).
% 0.34/0.53
% 0.34/0.53 % Pair_vimage_Sigma
% 0.34/0.53 thf(fact_815_Pair__vimage__Sigma,axiom,
% 0.34/0.53 ! [X: product_unit,A: set_Product_unit,F: product_unit > set_o] :
% 0.34/0.53 ( ( ( member_Product_unit @ X @ A )
% 0.34/0.53 => ( ( vimage9156843197228287893unit_o @ ( produc8487917018914037602unit_o @ X ) @ ( produc4996980317674386579unit_o @ A @ F ) )
% 0.34/0.53 = ( F @ X ) ) )
% 0.34/0.53 & ( ~ ( member_Product_unit @ X @ A )
% 0.34/0.53 => ( ( vimage9156843197228287893unit_o @ ( produc8487917018914037602unit_o @ X ) @ ( produc4996980317674386579unit_o @ A @ F ) )
% 0.34/0.53 = bot_bot_set_o ) ) ) ).
% 0.34/0.53
% 0.34/0.53 % Pair_vimage_Sigma
% 0.34/0.53 thf(fact_816_Pair__vimage__Sigma,axiom,
% 0.34/0.53 ! [X: int,A: set_int,F: int > set_nat] :
% 0.34/0.53 ( ( ( member_int @ X @ A )
% 0.34/0.53 => ( ( vimage9012682431555523728nt_nat @ ( product_Pair_int_nat @ X ) @ ( produc1456381018704787142nt_nat @ A @ F ) )
% 0.34/0.53 = ( F @ X ) ) )
% 0.34/0.53 & ( ~ ( member_int @ X @ A )
% 0.34/0.53 => ( ( vimage9012682431555523728nt_nat @ ( product_Pair_int_nat @ X ) @ ( produc1456381018704787142nt_nat @ A @ F ) )
% 0.34/0.53 = bot_bot_set_nat ) ) ) ).
% 0.34/0.53
% 0.34/0.53 % Pair_vimage_Sigma
% 0.34/0.53 thf(fact_817_Pair__vimage__Sigma,axiom,
% 0.34/0.53 ! [X: $o,A: set_o,F: $o > set_nat] :
% 0.34/0.53 ( ( ( member_o @ X @ A )
% 0.34/0.53 => ( ( vimage5656379182745576100_o_nat @ ( product_Pair_o_nat @ X ) @ ( product_Sigma_o_nat @ A @ F ) )
% 0.34/0.53 = ( F @ X ) ) )
% 0.34/0.53 & ( ~ ( member_o @ X @ A )
% 0.34/0.53 => ( ( vimage5656379182745576100_o_nat @ ( product_Pair_o_nat @ X ) @ ( product_Sigma_o_nat @ A @ F ) )
% 0.34/0.53 = bot_bot_set_nat ) ) ) ).
% 0.34/0.53
% 0.34/0.53 % Pair_vimage_Sigma
% 0.34/0.53 thf(fact_818_relH__def,axiom,
% 0.34/0.53 ( relH
% 0.34/0.53 = ( ^ [As3: set_nat,H4: heap_e7401611519738050253t_unit,H5: heap_e7401611519738050253t_unit] :
% 0.34/0.53 ( ( in_range @ ( produc7507926704131184380et_nat @ H4 @ As3 ) )
% 0.34/0.53 & ( in_range @ ( produc7507926704131184380et_nat @ H5 @ As3 ) )
% 0.34/0.53 & ! [T: typerep,X2: nat] :
% 0.34/0.53 ( ( member_nat @ X2 @ As3 )
% 0.34/0.53 => ( ( ( refs_Product_unit @ H4 @ T @ X2 )
% 0.34/0.53 = ( refs_Product_unit @ H5 @ T @ X2 ) )
% 0.34/0.53 & ( ( arrays_Product_unit @ H4 @ T @ X2 )
% 0.34/0.53 = ( arrays_Product_unit @ H5 @ T @ X2 ) ) ) ) ) ) ) ).
% 0.34/0.53
% 0.34/0.53 % relH_def
% 0.34/0.53 thf(fact_819_vimage__insert,axiom,
% 0.34/0.53 ! [F: nat > product_unit,A2: product_unit,B4: set_Product_unit] :
% 0.34/0.53 ( ( vimage4884490618288580032t_unit @ F @ ( insert_Product_unit @ A2 @ B4 ) )
% 0.34/0.53 = ( sup_sup_set_nat @ ( vimage4884490618288580032t_unit @ F @ ( insert_Product_unit @ A2 @ bot_bo3957492148770167129t_unit ) ) @ ( vimage4884490618288580032t_unit @ F @ B4 ) ) ) ).
% 0.34/0.53
% 0.34/0.53 % vimage_insert
% 0.34/0.53 thf(fact_820_vimage__insert,axiom,
% 0.34/0.53 ! [F: nat > $o,A2: $o,B4: set_o] :
% 0.34/0.53 ( ( vimage_nat_o @ F @ ( insert_o @ A2 @ B4 ) )
% 0.34/0.53 = ( sup_sup_set_nat @ ( vimage_nat_o @ F @ ( insert_o @ A2 @ bot_bot_set_o ) ) @ ( vimage_nat_o @ F @ B4 ) ) ) ).
% 0.34/0.53
% 0.34/0.53 % vimage_insert
% 0.34/0.53 thf(fact_821_vimage__insert,axiom,
% 0.34/0.53 ! [F: nat > nat,A2: nat,B4: set_nat] :
% 0.34/0.53 ( ( vimage_nat_nat @ F @ ( insert_nat @ A2 @ B4 ) )
% 0.34/0.53 = ( sup_sup_set_nat @ ( vimage_nat_nat @ F @ ( insert_nat @ A2 @ bot_bot_set_nat ) ) @ ( vimage_nat_nat @ F @ B4 ) ) ) ).
% 0.34/0.53
% 0.34/0.53 % vimage_insert
% 0.34/0.53 thf(fact_822_vimage__insert,axiom,
% 0.34/0.53 ! [F: nat > int,A2: int,B4: set_int] :
% 0.34/0.53 ( ( vimage_nat_int @ F @ ( insert_int @ A2 @ B4 ) )
% 0.34/0.53 = ( sup_sup_set_nat @ ( vimage_nat_int @ F @ ( insert_int @ A2 @ bot_bot_set_int ) ) @ ( vimage_nat_int @ F @ B4 ) ) ) ).
% 0.34/0.53
% 0.34/0.53 % vimage_insert
% 0.34/0.53 thf(fact_823_vimage__insert,axiom,
% 0.34/0.53 ! [F: nat > produc3658429121746597890et_nat > $o,A2: produc3658429121746597890et_nat > $o,B4: set_Pr4532377907799695533_nat_o] :
% 0.34/0.53 ( ( vimage8201027140005291904_nat_o @ F @ ( insert5175938949040314269_nat_o @ A2 @ B4 ) )
% 0.34/0.53 = ( sup_sup_set_nat @ ( vimage8201027140005291904_nat_o @ F @ ( insert5175938949040314269_nat_o @ A2 @ bot_bo7824918357723582233_nat_o ) ) @ ( vimage8201027140005291904_nat_o @ F @ B4 ) ) ) ).
% 0.34/0.53
% 0.34/0.53 % vimage_insert
% 0.34/0.53 thf(fact_824_trancl__over__edgeE,axiom,
% 0.34/0.53 ! [U: nat,W: nat,V1: nat,V22: nat,E: set_Pr1261947904930325089at_nat] :
% 0.34/0.53 ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ U @ W ) @ ( transi6264000038957366511cl_nat @ ( insert8211810215607154385at_nat @ ( product_Pair_nat_nat @ V1 @ V22 ) @ E ) ) )
% 0.34/0.53 => ( ~ ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ U @ W ) @ ( transi6264000038957366511cl_nat @ E ) )
% 0.34/0.53 => ~ ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ U @ V1 ) @ ( transi2905341329935302413cl_nat @ E ) )
% 0.34/0.53 => ~ ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ V22 @ W ) @ ( transi2905341329935302413cl_nat @ E ) ) ) ) ) ).
% 0.34/0.53
% 0.34/0.53 % trancl_over_edgeE
% 0.34/0.53 thf(fact_825_trancl__over__edgeE,axiom,
% 0.34/0.53 ! [U: int,W: int,V1: int,V22: int,E: set_Pr958786334691620121nt_int] :
% 0.34/0.53 ( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ U @ W ) @ ( transi6261509568448316235cl_int @ ( insert5033312907999012233nt_int @ ( product_Pair_int_int @ V1 @ V22 ) @ E ) ) )
% 0.34/0.53 => ( ~ ( member5262025264175285858nt_int @ ( product_Pair_int_int @ U @ W ) @ ( transi6261509568448316235cl_int @ E ) )
% 0.34/0.53 => ~ ( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ U @ V1 ) @ ( transi2902850859426252137cl_int @ E ) )
% 0.34/0.53 => ~ ( member5262025264175285858nt_int @ ( product_Pair_int_int @ V22 @ W ) @ ( transi2902850859426252137cl_int @ E ) ) ) ) ) ).
% 0.34/0.53
% 0.34/0.53 % trancl_over_edgeE
% 0.34/0.53 thf(fact_826_partial__order__on__empty,axiom,
% 0.34/0.53 order_2505591618835473410t_unit @ bot_bo3957492148770167129t_unit @ bot_bo1654022762305692595t_unit ).
% 0.34/0.53
% 0.34/0.53 % partial_order_on_empty
% 0.34/0.53 thf(fact_827_partial__order__on__empty,axiom,
% 0.34/0.53 order_8538205470752950541r_on_o @ bot_bot_set_o @ bot_bo7073875226086086771od_o_o ).
% 0.34/0.53
% 0.34/0.53 % partial_order_on_empty
% 0.34/0.53 thf(fact_828_partial__order__on__empty,axiom,
% 0.34/0.53 order_5251275573222108571on_nat @ bot_bot_set_nat @ bot_bo2099793752762293965at_nat ).
% 0.34/0.53
% 0.34/0.53 % partial_order_on_empty
% 0.34/0.53 thf(fact_829_partial__order__on__empty,axiom,
% 0.34/0.53 order_5248785102713058295on_int @ bot_bot_set_int @ bot_bo1796632182523588997nt_int ).
% 0.34/0.53
% 0.34/0.53 % partial_order_on_empty
% 0.34/0.53 thf(fact_830_partial__order__on__empty,axiom,
% 0.34/0.53 order_3690095022144090562_nat_o @ bot_bo7824918357723582233_nat_o @ bot_bo3657187498359500723_nat_o ).
% 0.34/0.53
% 0.34/0.53 % partial_order_on_empty
% 0.34/0.53 thf(fact_831_prod__set__simps_I2_J,axiom,
% 0.34/0.53 ! [X: heap_e7401611519738050253t_unit,Y: set_nat] :
% 0.34/0.53 ( ( basic_2698842327037663054et_nat @ ( produc7507926704131184380et_nat @ X @ Y ) )
% 0.34/0.53 = ( insert_set_nat @ Y @ bot_bot_set_set_nat ) ) ).
% 0.34/0.53
% 0.34/0.53 % prod_set_simps(2)
% 0.34/0.53 thf(fact_832_prod__set__simps_I2_J,axiom,
% 0.34/0.53 ! [X: produc3658429121746597890et_nat > $o,Y: produc3658429121746597890et_nat] :
% 0.34/0.53 ( ( basic_8479756702517612234et_nat @ ( produc5001842942810119800et_nat @ X @ Y ) )
% 0.34/0.53 = ( insert1271470778893645362et_nat @ Y @ bot_bo3771712512152092492et_nat ) ) ).
% 0.34/0.53
% 0.34/0.53 % prod_set_simps(2)
% 0.34/0.53 thf(fact_833_prod__set__simps_I2_J,axiom,
% 0.34/0.53 ! [X: produc3658429121746597890et_nat > $o,Y: produc3925858234332021118et_nat] :
% 0.34/0.53 ( ( basic_2768200282441914438et_nat @ ( produc2245416461498447860et_nat @ X @ Y ) )
% 0.34/0.53 = ( insert3111075074545270702et_nat @ Y @ bot_bo1481135142794719944et_nat ) ) ).
% 0.34/0.53
% 0.34/0.53 % prod_set_simps(2)
% 0.34/0.53 thf(fact_834_prod__set__simps_I2_J,axiom,
% 0.34/0.53 ! [X: nat,Y: nat] :
% 0.34/0.53 ( ( basic_snds_nat_nat @ ( product_Pair_nat_nat @ X @ Y ) )
% 0.34/0.53 = ( insert_nat @ Y @ bot_bot_set_nat ) ) ).
% 0.34/0.53
% 0.34/0.53 % prod_set_simps(2)
% 0.34/0.53 thf(fact_835_prod__set__simps_I2_J,axiom,
% 0.34/0.53 ! [X: int,Y: int] :
% 0.34/0.53 ( ( basic_snds_int_int @ ( product_Pair_int_int @ X @ Y ) )
% 0.34/0.53 = ( insert_int @ Y @ bot_bot_set_int ) ) ).
% 0.34/0.53
% 0.34/0.53 % prod_set_simps(2)
% 0.34/0.53 thf(fact_836_dual__order_Orefl,axiom,
% 0.34/0.53 ! [A2: filter_nat] : ( ord_le2510731241096832064er_nat @ A2 @ A2 ) ).
% 0.34/0.53
% 0.34/0.53 % dual_order.refl
% 0.34/0.53 thf(fact_837_dual__order_Orefl,axiom,
% 0.34/0.53 ! [A2: set_nat] : ( ord_less_eq_set_nat @ A2 @ A2 ) ).
% 0.34/0.53
% 0.34/0.53 % dual_order.refl
% 0.34/0.53 thf(fact_838_dual__order_Orefl,axiom,
% 0.34/0.53 ! [A2: rat] : ( ord_less_eq_rat @ A2 @ A2 ) ).
% 0.34/0.53
% 0.34/0.53 % dual_order.refl
% 0.34/0.53 thf(fact_839_dual__order_Orefl,axiom,
% 0.34/0.53 ! [A2: nat] : ( ord_less_eq_nat @ A2 @ A2 ) ).
% 0.34/0.53
% 0.34/0.53 % dual_order.refl
% 0.34/0.53 thf(fact_840_dual__order_Orefl,axiom,
% 0.34/0.53 ! [A2: int] : ( ord_less_eq_int @ A2 @ A2 ) ).
% 0.34/0.53
% 0.34/0.53 % dual_order.refl
% 0.34/0.53 thf(fact_841_order__refl,axiom,
% 0.34/0.53 ! [X: filter_nat] : ( ord_le2510731241096832064er_nat @ X @ X ) ).
% 0.34/0.53
% 0.34/0.53 % order_refl
% 0.34/0.53 thf(fact_842_order__refl,axiom,
% 0.34/0.53 ! [X: set_nat] : ( ord_less_eq_set_nat @ X @ X ) ).
% 0.34/0.53
% 0.34/0.53 % order_refl
% 0.34/0.53 thf(fact_843_order__refl,axiom,
% 0.34/0.53 ! [X: rat] : ( ord_less_eq_rat @ X @ X ) ).
% 0.34/0.53
% 0.34/0.53 % order_refl
% 0.34/0.53 thf(fact_844_order__refl,axiom,
% 0.34/0.53 ! [X: nat] : ( ord_less_eq_nat @ X @ X ) ).
% 0.34/0.53
% 0.34/0.53 % order_refl
% 0.34/0.53 thf(fact_845_order__refl,axiom,
% 0.34/0.53 ! [X: int] : ( ord_less_eq_int @ X @ X ) ).
% 0.34/0.53
% 0.34/0.53 % order_refl
% 0.34/0.53 thf(fact_846_subset__antisym,axiom,
% 0.34/0.53 ! [A: set_nat,B4: set_nat] :
% 0.34/0.53 ( ( ord_less_eq_set_nat @ A @ B4 )
% 0.34/0.53 => ( ( ord_less_eq_set_nat @ B4 @ A )
% 0.34/0.53 => ( A = B4 ) ) ) ).
% 0.34/0.53
% 0.34/0.53 % subset_antisym
% 0.34/0.53 thf(fact_847_subsetI,axiom,
% 0.34/0.53 ! [A: set_Pr4532377907799695533_nat_o,B4: set_Pr4532377907799695533_nat_o] :
% 0.34/0.53 ( ! [X3: produc3658429121746597890et_nat > $o] :
% 0.34/0.53 ( ( member6576561426505652726_nat_o @ X3 @ A )
% 0.34/0.53 => ( member6576561426505652726_nat_o @ X3 @ B4 ) )
% 0.34/0.53 => ( ord_le2965882846123202637_nat_o @ A @ B4 ) ) ).
% 0.34/0.53
% 0.34/0.53 % subsetI
% 0.34/0.53 thf(fact_848_subsetI,axiom,
% 0.34/0.53 ! [A: set_int,B4: set_int] :
% 0.34/0.53 ( ! [X3: int] :
% 0.34/0.53 ( ( member_int @ X3 @ A )
% 0.34/0.53 => ( member_int @ X3 @ B4 ) )
% 0.34/0.53 => ( ord_less_eq_set_int @ A @ B4 ) ) ).
% 0.34/0.53
% 0.34/0.53 % subsetI
% 0.34/0.53 thf(fact_849_subsetI,axiom,
% 0.34/0.53 ! [A: set_o,B4: set_o] :
% 0.34/0.53 ( ! [X3: $o] :
% 0.34/0.53 ( ( member_o @ X3 @ A )
% 0.34/0.53 => ( member_o @ X3 @ B4 ) )
% 0.34/0.53 => ( ord_less_eq_set_o @ A @ B4 ) ) ).
% 0.34/0.53
% 0.34/0.53 % subsetI
% 0.34/0.53 thf(fact_850_subsetI,axiom,
% 0.34/0.53 ! [A: set_Product_unit,B4: set_Product_unit] :
% 0.34/0.53 ( ! [X3: product_unit] :
% 0.34/0.53 ( ( member_Product_unit @ X3 @ A )
% 0.34/0.53 => ( member_Product_unit @ X3 @ B4 ) )
% 0.34/0.53 => ( ord_le3507040750410214029t_unit @ A @ B4 ) ) ).
% 0.34/0.53
% 0.34/0.53 % subsetI
% 0.34/0.53 thf(fact_851_subsetI,axiom,
% 0.34/0.53 ! [A: set_nat,B4: set_nat] :
% 0.34/0.53 ( ! [X3: nat] :
% 0.34/0.53 ( ( member_nat @ X3 @ A )
% 0.34/0.53 => ( member_nat @ X3 @ B4 ) )
% 0.34/0.53 => ( ord_less_eq_set_nat @ A @ B4 ) ) ).
% 0.34/0.53
% 0.34/0.53 % subsetI
% 0.34/0.53 thf(fact_852_Int__iff,axiom,
% 0.34/0.53 ! [C: produc3658429121746597890et_nat > $o,A: set_Pr4532377907799695533_nat_o,B4: set_Pr4532377907799695533_nat_o] :
% 0.34/0.53 ( ( member6576561426505652726_nat_o @ C @ ( inf_in1906310914598751387_nat_o @ A @ B4 ) )
% 0.34/0.53 = ( ( member6576561426505652726_nat_o @ C @ A )
% 0.34/0.53 & ( member6576561426505652726_nat_o @ C @ B4 ) ) ) ).
% 0.34/0.53
% 0.34/0.53 % Int_iff
% 0.34/0.53 thf(fact_853_Int__iff,axiom,
% 0.34/0.53 ! [C: int,A: set_int,B4: set_int] :
% 0.34/0.53 ( ( member_int @ C @ ( inf_inf_set_int @ A @ B4 ) )
% 0.34/0.53 = ( ( member_int @ C @ A )
% 0.34/0.53 & ( member_int @ C @ B4 ) ) ) ).
% 0.34/0.53
% 0.34/0.53 % Int_iff
% 0.34/0.53 thf(fact_854_Int__iff,axiom,
% 0.34/0.53 ! [C: $o,A: set_o,B4: set_o] :
% 0.34/0.53 ( ( member_o @ C @ ( inf_inf_set_o @ A @ B4 ) )
% 0.34/0.53 = ( ( member_o @ C @ A )
% 0.34/0.53 & ( member_o @ C @ B4 ) ) ) ).
% 0.34/0.53
% 0.34/0.53 % Int_iff
% 0.34/0.53 thf(fact_855_Int__iff,axiom,
% 0.34/0.53 ! [C: product_unit,A: set_Product_unit,B4: set_Product_unit] :
% 0.34/0.53 ( ( member_Product_unit @ C @ ( inf_in4660618365625256667t_unit @ A @ B4 ) )
% 0.34/0.53 = ( ( member_Product_unit @ C @ A )
% 0.34/0.53 & ( member_Product_unit @ C @ B4 ) ) ) ).
% 0.34/0.53
% 0.34/0.53 % Int_iff
% 0.34/0.53 thf(fact_856_Int__iff,axiom,
% 0.34/0.53 ! [C: nat,A: set_nat,B4: set_nat] :
% 0.34/0.53 ( ( member_nat @ C @ ( inf_inf_set_nat @ A @ B4 ) )
% 0.34/0.53 = ( ( member_nat @ C @ A )
% 0.34/0.53 & ( member_nat @ C @ B4 ) ) ) ).
% 0.34/0.53
% 0.34/0.53 % Int_iff
% 0.34/0.53 thf(fact_857_IntI,axiom,
% 0.34/0.53 ! [C: produc3658429121746597890et_nat > $o,A: set_Pr4532377907799695533_nat_o,B4: set_Pr4532377907799695533_nat_o] :
% 0.34/0.53 ( ( member6576561426505652726_nat_o @ C @ A )
% 0.34/0.53 => ( ( member6576561426505652726_nat_o @ C @ B4 )
% 0.34/0.53 => ( member6576561426505652726_nat_o @ C @ ( inf_in1906310914598751387_nat_o @ A @ B4 ) ) ) ) ).
% 0.34/0.53
% 0.34/0.53 % IntI
% 0.34/0.53 thf(fact_858_IntI,axiom,
% 0.34/0.53 ! [C: int,A: set_int,B4: set_int] :
% 0.34/0.53 ( ( member_int @ C @ A )
% 0.34/0.53 => ( ( member_int @ C @ B4 )
% 0.34/0.53 => ( member_int @ C @ ( inf_inf_set_int @ A @ B4 ) ) ) ) ).
% 0.34/0.53
% 0.34/0.53 % IntI
% 0.34/0.53 thf(fact_859_IntI,axiom,
% 0.34/0.53 ! [C: $o,A: set_o,B4: set_o] :
% 0.34/0.53 ( ( member_o @ C @ A )
% 0.34/0.53 => ( ( member_o @ C @ B4 )
% 0.34/0.53 => ( member_o @ C @ ( inf_inf_set_o @ A @ B4 ) ) ) ) ).
% 0.34/0.53
% 0.34/0.53 % IntI
% 0.34/0.53 thf(fact_860_IntI,axiom,
% 0.34/0.53 ! [C: product_unit,A: set_Product_unit,B4: set_Product_unit] :
% 0.34/0.53 ( ( member_Product_unit @ C @ A )
% 0.34/0.53 => ( ( member_Product_unit @ C @ B4 )
% 0.34/0.53 => ( member_Product_unit @ C @ ( inf_in4660618365625256667t_unit @ A @ B4 ) ) ) ) ).
% 0.34/0.53
% 0.34/0.53 % IntI
% 0.34/0.53 thf(fact_861_IntI,axiom,
% 0.34/0.53 ! [C: nat,A: set_nat,B4: set_nat] :
% 0.34/0.53 ( ( member_nat @ C @ A )
% 0.34/0.53 => ( ( member_nat @ C @ B4 )
% 0.34/0.53 => ( member_nat @ C @ ( inf_inf_set_nat @ A @ B4 ) ) ) ) ).
% 0.34/0.53
% 0.34/0.53 % IntI
% 0.34/0.53 thf(fact_862_Un__iff,axiom,
% 0.34/0.53 ! [C: produc3658429121746597890et_nat > $o,A: set_Pr4532377907799695533_nat_o,B4: set_Pr4532377907799695533_nat_o] :
% 0.34/0.53 ( ( member6576561426505652726_nat_o @ C @ ( sup_su5209123915105501825_nat_o @ A @ B4 ) )
% 0.34/0.53 = ( ( member6576561426505652726_nat_o @ C @ A )
% 0.34/0.53 | ( member6576561426505652726_nat_o @ C @ B4 ) ) ) ).
% 0.34/0.53
% 0.34/0.53 % Un_iff
% 0.34/0.53 thf(fact_863_Un__iff,axiom,
% 0.34/0.53 ! [C: int,A: set_int,B4: set_int] :
% 0.34/0.53 ( ( member_int @ C @ ( sup_sup_set_int @ A @ B4 ) )
% 0.34/0.53 = ( ( member_int @ C @ A )
% 0.34/0.53 | ( member_int @ C @ B4 ) ) ) ).
% 0.34/0.53
% 0.34/0.53 % Un_iff
% 0.34/0.53 thf(fact_864_Un__iff,axiom,
% 0.34/0.53 ! [C: $o,A: set_o,B4: set_o] :
% 0.34/0.53 ( ( member_o @ C @ ( sup_sup_set_o @ A @ B4 ) )
% 0.34/0.53 = ( ( member_o @ C @ A )
% 0.34/0.53 | ( member_o @ C @ B4 ) ) ) ).
% 0.34/0.53
% 0.34/0.53 % Un_iff
% 0.34/0.53 thf(fact_865_Un__iff,axiom,
% 0.34/0.53 ! [C: product_unit,A: set_Product_unit,B4: set_Product_unit] :
% 0.34/0.53 ( ( member_Product_unit @ C @ ( sup_su793286257634532545t_unit @ A @ B4 ) )
% 0.34/0.53 = ( ( member_Product_unit @ C @ A )
% 0.34/0.53 | ( member_Product_unit @ C @ B4 ) ) ) ).
% 0.34/0.53
% 0.34/0.53 % Un_iff
% 0.34/0.53 thf(fact_866_Un__iff,axiom,
% 0.34/0.53 ! [C: nat,A: set_nat,B4: set_nat] :
% 0.34/0.53 ( ( member_nat @ C @ ( sup_sup_set_nat @ A @ B4 ) )
% 0.34/0.53 = ( ( member_nat @ C @ A )
% 0.34/0.53 | ( member_nat @ C @ B4 ) ) ) ).
% 0.34/0.53
% 0.34/0.53 % Un_iff
% 0.34/0.53 thf(fact_867_UnCI,axiom,
% 0.34/0.53 ! [C: produc3658429121746597890et_nat > $o,B4: set_Pr4532377907799695533_nat_o,A: set_Pr4532377907799695533_nat_o] :
% 0.34/0.53 ( ( ~ ( member6576561426505652726_nat_o @ C @ B4 )
% 0.34/0.53 => ( member6576561426505652726_nat_o @ C @ A ) )
% 0.34/0.53 => ( member6576561426505652726_nat_o @ C @ ( sup_su5209123915105501825_nat_o @ A @ B4 ) ) ) ).
% 0.34/0.53
% 0.34/0.53 % UnCI
% 0.34/0.53 thf(fact_868_UnCI,axiom,
% 0.34/0.53 ! [C: int,B4: set_int,A: set_int] :
% 0.34/0.53 ( ( ~ ( member_int @ C @ B4 )
% 0.34/0.53 => ( member_int @ C @ A ) )
% 0.34/0.53 => ( member_int @ C @ ( sup_sup_set_int @ A @ B4 ) ) ) ).
% 0.34/0.53
% 0.34/0.53 % UnCI
% 0.34/0.53 thf(fact_869_UnCI,axiom,
% 0.34/0.53 ! [C: $o,B4: set_o,A: set_o] :
% 0.34/0.53 ( ( ~ ( member_o @ C @ B4 )
% 0.34/0.53 => ( member_o @ C @ A ) )
% 0.34/0.53 => ( member_o @ C @ ( sup_sup_set_o @ A @ B4 ) ) ) ).
% 0.34/0.53
% 0.34/0.53 % UnCI
% 0.34/0.53 thf(fact_870_UnCI,axiom,
% 0.34/0.53 ! [C: product_unit,B4: set_Product_unit,A: set_Product_unit] :
% 0.34/0.53 ( ( ~ ( member_Product_unit @ C @ B4 )
% 0.34/0.53 => ( member_Product_unit @ C @ A ) )
% 0.34/0.53 => ( member_Product_unit @ C @ ( sup_su793286257634532545t_unit @ A @ B4 ) ) ) ).
% 0.34/0.53
% 0.34/0.53 % UnCI
% 0.34/0.53 thf(fact_871_UnCI,axiom,
% 0.34/0.53 ! [C: nat,B4: set_nat,A: set_nat] :
% 0.34/0.53 ( ( ~ ( member_nat @ C @ B4 )
% 0.34/0.53 => ( member_nat @ C @ A ) )
% 0.34/0.53 => ( member_nat @ C @ ( sup_sup_set_nat @ A @ B4 ) ) ) ).
% 0.34/0.53
% 0.34/0.53 % UnCI
% 0.34/0.53 thf(fact_872_empty__subsetI,axiom,
% 0.34/0.53 ! [A: set_Product_unit] : ( ord_le3507040750410214029t_unit @ bot_bo3957492148770167129t_unit @ A ) ).
% 0.34/0.53
% 0.34/0.53 % empty_subsetI
% 0.34/0.53 thf(fact_873_empty__subsetI,axiom,
% 0.34/0.53 ! [A: set_o] : ( ord_less_eq_set_o @ bot_bot_set_o @ A ) ).
% 0.34/0.53
% 0.34/0.53 % empty_subsetI
% 0.34/0.53 thf(fact_874_empty__subsetI,axiom,
% 0.34/0.53 ! [A: set_int] : ( ord_less_eq_set_int @ bot_bot_set_int @ A ) ).
% 0.34/0.53
% 0.34/0.53 % empty_subsetI
% 0.34/0.53 thf(fact_875_empty__subsetI,axiom,
% 0.34/0.53 ! [A: set_Pr4532377907799695533_nat_o] : ( ord_le2965882846123202637_nat_o @ bot_bo7824918357723582233_nat_o @ A ) ).
% 0.34/0.53
% 0.34/0.53 % empty_subsetI
% 0.34/0.53 thf(fact_876_empty__subsetI,axiom,
% 0.34/0.53 ! [A: set_nat] : ( ord_less_eq_set_nat @ bot_bot_set_nat @ A ) ).
% 0.34/0.53
% 0.34/0.53 % empty_subsetI
% 0.34/0.53 thf(fact_877_subset__empty,axiom,
% 0.34/0.53 ! [A: set_Product_unit] :
% 0.34/0.53 ( ( ord_le3507040750410214029t_unit @ A @ bot_bo3957492148770167129t_unit )
% 0.34/0.53 = ( A = bot_bo3957492148770167129t_unit ) ) ).
% 0.34/0.53
% 0.34/0.53 % subset_empty
% 0.34/0.53 thf(fact_878_subset__empty,axiom,
% 0.34/0.53 ! [A: set_o] :
% 0.34/0.53 ( ( ord_less_eq_set_o @ A @ bot_bot_set_o )
% 0.34/0.53 = ( A = bot_bot_set_o ) ) ).
% 0.34/0.53
% 0.34/0.53 % subset_empty
% 0.34/0.53 thf(fact_879_subset__empty,axiom,
% 0.34/0.53 ! [A: set_int] :
% 0.34/0.53 ( ( ord_less_eq_set_int @ A @ bot_bot_set_int )
% 0.34/0.53 = ( A = bot_bot_set_int ) ) ).
% 0.34/0.53
% 0.34/0.53 % subset_empty
% 0.34/0.53 thf(fact_880_subset__empty,axiom,
% 0.34/0.53 ! [A: set_Pr4532377907799695533_nat_o] :
% 0.34/0.53 ( ( ord_le2965882846123202637_nat_o @ A @ bot_bo7824918357723582233_nat_o )
% 0.34/0.53 = ( A = bot_bo7824918357723582233_nat_o ) ) ).
% 0.34/0.53
% 0.34/0.53 % subset_empty
% 0.34/0.53 thf(fact_881_subset__empty,axiom,
% 0.34/0.53 ! [A: set_nat] :
% 0.34/0.53 ( ( ord_less_eq_set_nat @ A @ bot_bot_set_nat )
% 0.34/0.53 = ( A = bot_bot_set_nat ) ) ).
% 0.34/0.53
% 0.34/0.53 % subset_empty
% 0.34/0.53 thf(fact_882_insert__subset,axiom,
% 0.34/0.53 ! [X: produc3658429121746597890et_nat > $o,A: set_Pr4532377907799695533_nat_o,B4: set_Pr4532377907799695533_nat_o] :
% 0.34/0.53 ( ( ord_le2965882846123202637_nat_o @ ( insert5175938949040314269_nat_o @ X @ A ) @ B4 )
% 0.34/0.53 = ( ( member6576561426505652726_nat_o @ X @ B4 )
% 0.34/0.53 & ( ord_le2965882846123202637_nat_o @ A @ B4 ) ) ) ).
% 0.34/0.53
% 0.34/0.53 % insert_subset
% 0.34/0.53 thf(fact_883_insert__subset,axiom,
% 0.34/0.53 ! [X: int,A: set_int,B4: set_int] :
% 0.34/0.53 ( ( ord_less_eq_set_int @ ( insert_int @ X @ A ) @ B4 )
% 0.34/0.53 = ( ( member_int @ X @ B4 )
% 0.34/0.53 & ( ord_less_eq_set_int @ A @ B4 ) ) ) ).
% 0.34/0.53
% 0.34/0.53 % insert_subset
% 0.34/0.53 thf(fact_884_insert__subset,axiom,
% 0.34/0.53 ! [X: $o,A: set_o,B4: set_o] :
% 0.34/0.53 ( ( ord_less_eq_set_o @ ( insert_o @ X @ A ) @ B4 )
% 0.34/0.53 = ( ( member_o @ X @ B4 )
% 0.34/0.53 & ( ord_less_eq_set_o @ A @ B4 ) ) ) ).
% 0.34/0.53
% 0.34/0.53 % insert_subset
% 0.34/0.53 thf(fact_885_insert__subset,axiom,
% 0.34/0.53 ! [X: product_unit,A: set_Product_unit,B4: set_Product_unit] :
% 0.34/0.53 ( ( ord_le3507040750410214029t_unit @ ( insert_Product_unit @ X @ A ) @ B4 )
% 0.34/0.53 = ( ( member_Product_unit @ X @ B4 )
% 0.34/0.53 & ( ord_le3507040750410214029t_unit @ A @ B4 ) ) ) ).
% 0.34/0.53
% 0.34/0.53 % insert_subset
% 0.34/0.53 thf(fact_886_insert__subset,axiom,
% 0.34/0.53 ! [X: nat,A: set_nat,B4: set_nat] :
% 0.34/0.53 ( ( ord_less_eq_set_nat @ ( insert_nat @ X @ A ) @ B4 )
% 0.34/0.53 = ( ( member_nat @ X @ B4 )
% 0.34/0.53 & ( ord_less_eq_set_nat @ A @ B4 ) ) ) ).
% 0.34/0.53
% 0.34/0.53 % insert_subset
% 0.34/0.53 thf(fact_887_SigmaI,axiom,
% 0.34/0.53 ! [A2: nat,A: set_nat,B: int,B4: nat > set_int] :
% 0.34/0.53 ( ( member_nat @ A2 @ A )
% 0.34/0.53 => ( ( member_int @ B @ ( B4 @ A2 ) )
% 0.34/0.53 => ( member4262671552274231302at_int @ ( product_Pair_nat_int @ A2 @ B ) @ ( produc454536836294682310at_int @ A @ B4 ) ) ) ) ).
% 0.34/0.53
% 0.34/0.53 % SigmaI
% 0.34/0.53 thf(fact_888_SigmaI,axiom,
% 0.34/0.53 ! [A2: nat,A: set_nat,B: $o,B4: nat > set_o] :
% 0.34/0.53 ( ( member_nat @ A2 @ A )
% 0.34/0.53 => ( ( member_o @ B @ ( B4 @ A2 ) )
% 0.34/0.53 => ( member6310962623043647828_nat_o @ ( product_Pair_nat_o @ A2 @ B ) @ ( product_Sigma_nat_o @ A @ B4 ) ) ) ) ).
% 0.34/0.53
% 0.34/0.53 % SigmaI
% 0.34/0.53 thf(fact_889_SigmaI,axiom,
% 0.34/0.53 ! [A2: nat,A: set_nat,B: product_unit,B4: nat > set_Product_unit] :
% 0.34/0.53 ( ( member_nat @ A2 @ A )
% 0.34/0.53 => ( ( member_Product_unit @ B @ ( B4 @ A2 ) )
% 0.34/0.53 => ( member348563419383025609t_unit @ ( produc3434450474625521892t_unit @ A2 @ B ) @ ( produc5506539147108004339t_unit @ A @ B4 ) ) ) ) ).
% 0.34/0.53
% 0.34/0.53 % SigmaI
% 0.34/0.53 thf(fact_890_SigmaI,axiom,
% 0.34/0.53 ! [A2: int,A: set_int,B: nat,B4: int > set_nat] :
% 0.34/0.53 ( ( member_int @ A2 @ A )
% 0.34/0.53 => ( ( member_nat @ B @ ( B4 @ A2 ) )
% 0.34/0.53 => ( member216504246829706758nt_nat @ ( product_Pair_int_nat @ A2 @ B ) @ ( produc1456381018704787142nt_nat @ A @ B4 ) ) ) ) ).
% 0.34/0.53
% 0.34/0.53 % SigmaI
% 0.34/0.53 thf(fact_891_SigmaI,axiom,
% 0.34/0.53 ! [A2: int,A: set_int,B: $o,B4: int > set_o] :
% 0.34/0.53 ( ( member_int @ A2 @ A )
% 0.34/0.53 => ( ( member_o @ B @ ( B4 @ A2 ) )
% 0.34/0.53 => ( member4489920277610959864_int_o @ ( product_Pair_int_o @ A2 @ B ) @ ( product_Sigma_int_o @ A @ B4 ) ) ) ) ).
% 0.34/0.53
% 0.34/0.53 % SigmaI
% 0.34/0.53 thf(fact_892_SigmaI,axiom,
% 0.34/0.53 ! [A2: int,A: set_int,B: product_unit,B4: int > set_Product_unit] :
% 0.34/0.53 ( ( member_int @ A2 @ A )
% 0.34/0.53 => ( ( member_Product_unit @ B @ ( B4 @ A2 ) )
% 0.34/0.53 => ( member6688357403439981677t_unit @ ( produc550872421827702152t_unit @ A2 @ B ) @ ( produc2622961094310184599t_unit @ A @ B4 ) ) ) ) ).
% 0.34/0.53
% 0.34/0.53 % SigmaI
% 0.34/0.53 thf(fact_893_SigmaI,axiom,
% 0.34/0.53 ! [A2: $o,A: set_o,B: nat,B4: $o > set_nat] :
% 0.34/0.53 ( ( member_o @ A2 @ A )
% 0.34/0.53 => ( ( member_nat @ B @ ( B4 @ A2 ) )
% 0.34/0.53 => ( member2802428098988154798_o_nat @ ( product_Pair_o_nat @ A2 @ B ) @ ( product_Sigma_o_nat @ A @ B4 ) ) ) ) ).
% 0.34/0.53
% 0.34/0.53 % SigmaI
% 0.34/0.53 thf(fact_894_SigmaI,axiom,
% 0.34/0.53 ! [A2: $o,A: set_o,B: int,B4: $o > set_int] :
% 0.34/0.53 ( ( member_o @ A2 @ A )
% 0.34/0.53 => ( ( member_int @ B @ ( B4 @ A2 ) )
% 0.34/0.53 => ( member7847949116333733898_o_int @ ( product_Pair_o_int @ A2 @ B ) @ ( product_Sigma_o_int @ A @ B4 ) ) ) ) ).
% 0.34/0.53
% 0.34/0.53 % SigmaI
% 0.34/0.53 thf(fact_895_SigmaI,axiom,
% 0.34/0.53 ! [A2: $o,A: set_o,B: $o,B4: $o > set_o] :
% 0.34/0.53 ( ( member_o @ A2 @ A )
% 0.34/0.53 => ( ( member_o @ B @ ( B4 @ A2 ) )
% 0.34/0.53 => ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ A2 @ B ) @ ( product_Sigma_o_o @ A @ B4 ) ) ) ) ).
% 0.34/0.53
% 0.34/0.53 % SigmaI
% 0.34/0.53 thf(fact_896_SigmaI,axiom,
% 0.34/0.53 ! [A2: $o,A: set_o,B: product_unit,B4: $o > set_Product_unit] :
% 0.34/0.53 ( ( member_o @ A2 @ A )
% 0.34/0.53 => ( ( member_Product_unit @ B @ ( B4 @ A2 ) )
% 0.34/0.53 => ( member9215336023991391429t_unit @ ( produc3901946521647900940t_unit @ A2 @ B ) @ ( produc411009820408249917t_unit @ A @ B4 ) ) ) ) ).
% 0.34/0.53
% 0.34/0.53 % SigmaI
% 0.34/0.53 thf(fact_897_mem__Sigma__iff,axiom,
% 0.34/0.53 ! [A2: nat,B: int,A: set_nat,B4: nat > set_int] :
% 0.34/0.53 ( ( member4262671552274231302at_int @ ( product_Pair_nat_int @ A2 @ B ) @ ( produc454536836294682310at_int @ A @ B4 ) )
% 0.34/0.53 = ( ( member_nat @ A2 @ A )
% 0.34/0.53 & ( member_int @ B @ ( B4 @ A2 ) ) ) ) ).
% 0.34/0.53
% 0.34/0.53 % mem_Sigma_iff
% 0.34/0.53 thf(fact_898_mem__Sigma__iff,axiom,
% 0.34/0.53 ! [A2: nat,B: $o,A: set_nat,B4: nat > set_o] :
% 0.34/0.53 ( ( member6310962623043647828_nat_o @ ( product_Pair_nat_o @ A2 @ B ) @ ( product_Sigma_nat_o @ A @ B4 ) )
% 0.34/0.53 = ( ( member_nat @ A2 @ A )
% 0.34/0.53 & ( member_o @ B @ ( B4 @ A2 ) ) ) ) ).
% 0.34/0.53
% 0.34/0.53 % mem_Sigma_iff
% 0.34/0.53 thf(fact_899_mem__Sigma__iff,axiom,
% 0.34/0.53 ! [A2: nat,B: product_unit,A: set_nat,B4: nat > set_Product_unit] :
% 0.34/0.53 ( ( member348563419383025609t_unit @ ( produc3434450474625521892t_unit @ A2 @ B ) @ ( produc5506539147108004339t_unit @ A @ B4 ) )
% 0.34/0.53 = ( ( member_nat @ A2 @ A )
% 0.34/0.53 & ( member_Product_unit @ B @ ( B4 @ A2 ) ) ) ) ).
% 0.34/0.53
% 0.34/0.53 % mem_Sigma_iff
% 0.34/0.53 thf(fact_900_mem__Sigma__iff,axiom,
% 0.34/0.53 ! [A2: int,B: nat,A: set_int,B4: int > set_nat] :
% 0.34/0.53 ( ( member216504246829706758nt_nat @ ( product_Pair_int_nat @ A2 @ B ) @ ( produc1456381018704787142nt_nat @ A @ B4 ) )
% 0.34/0.53 = ( ( member_int @ A2 @ A )
% 0.34/0.53 & ( member_nat @ B @ ( B4 @ A2 ) ) ) ) ).
% 0.34/0.53
% 0.34/0.53 % mem_Sigma_iff
% 0.34/0.53 thf(fact_901_mem__Sigma__iff,axiom,
% 0.34/0.53 ! [A2: int,B: $o,A: set_int,B4: int > set_o] :
% 0.34/0.53 ( ( member4489920277610959864_int_o @ ( product_Pair_int_o @ A2 @ B ) @ ( product_Sigma_int_o @ A @ B4 ) )
% 0.34/0.53 = ( ( member_int @ A2 @ A )
% 0.34/0.53 & ( member_o @ B @ ( B4 @ A2 ) ) ) ) ).
% 0.34/0.53
% 0.34/0.53 % mem_Sigma_iff
% 0.34/0.53 thf(fact_902_mem__Sigma__iff,axiom,
% 0.34/0.53 ! [A2: int,B: product_unit,A: set_int,B4: int > set_Product_unit] :
% 0.34/0.53 ( ( member6688357403439981677t_unit @ ( produc550872421827702152t_unit @ A2 @ B ) @ ( produc2622961094310184599t_unit @ A @ B4 ) )
% 0.34/0.53 = ( ( member_int @ A2 @ A )
% 0.34/0.53 & ( member_Product_unit @ B @ ( B4 @ A2 ) ) ) ) ).
% 0.34/0.53
% 0.34/0.53 % mem_Sigma_iff
% 0.34/0.53 thf(fact_903_mem__Sigma__iff,axiom,
% 0.34/0.53 ! [A2: $o,B: nat,A: set_o,B4: $o > set_nat] :
% 0.34/0.53 ( ( member2802428098988154798_o_nat @ ( product_Pair_o_nat @ A2 @ B ) @ ( product_Sigma_o_nat @ A @ B4 ) )
% 0.34/0.53 = ( ( member_o @ A2 @ A )
% 0.34/0.53 & ( member_nat @ B @ ( B4 @ A2 ) ) ) ) ).
% 0.34/0.53
% 0.34/0.53 % mem_Sigma_iff
% 0.34/0.53 thf(fact_904_mem__Sigma__iff,axiom,
% 0.34/0.53 ! [A2: $o,B: int,A: set_o,B4: $o > set_int] :
% 0.34/0.53 ( ( member7847949116333733898_o_int @ ( product_Pair_o_int @ A2 @ B ) @ ( product_Sigma_o_int @ A @ B4 ) )
% 0.34/0.53 = ( ( member_o @ A2 @ A )
% 0.34/0.53 & ( member_int @ B @ ( B4 @ A2 ) ) ) ) ).
% 0.34/0.53
% 0.34/0.53 % mem_Sigma_iff
% 0.34/0.53 thf(fact_905_mem__Sigma__iff,axiom,
% 0.34/0.53 ! [A2: $o,B: $o,A: set_o,B4: $o > set_o] :
% 0.34/0.53 ( ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ A2 @ B ) @ ( product_Sigma_o_o @ A @ B4 ) )
% 0.34/0.53 = ( ( member_o @ A2 @ A )
% 0.34/0.53 & ( member_o @ B @ ( B4 @ A2 ) ) ) ) ).
% 0.34/0.53
% 0.34/0.53 % mem_Sigma_iff
% 0.34/0.53 thf(fact_906_mem__Sigma__iff,axiom,
% 0.34/0.53 ! [A2: $o,B: product_unit,A: set_o,B4: $o > set_Product_unit] :
% 0.34/0.53 ( ( member9215336023991391429t_unit @ ( produc3901946521647900940t_unit @ A2 @ B ) @ ( produc411009820408249917t_unit @ A @ B4 ) )
% 0.34/0.53 = ( ( member_o @ A2 @ A )
% 0.34/0.53 & ( member_Product_unit @ B @ ( B4 @ A2 ) ) ) ) ).
% 0.34/0.53
% 0.34/0.53 % mem_Sigma_iff
% 0.34/0.53 thf(fact_907_Un__empty,axiom,
% 0.34/0.53 ! [A: set_Product_unit,B4: set_Product_unit] :
% 0.34/0.53 ( ( ( sup_su793286257634532545t_unit @ A @ B4 )
% 0.34/0.53 = bot_bo3957492148770167129t_unit )
% 0.34/0.53 = ( ( A = bot_bo3957492148770167129t_unit )
% 0.34/0.53 & ( B4 = bot_bo3957492148770167129t_unit ) ) ) ).
% 0.34/0.53
% 0.34/0.53 % Un_empty
% 0.34/0.53 thf(fact_908_Un__empty,axiom,
% 0.34/0.53 ! [A: set_o,B4: set_o] :
% 0.34/0.53 ( ( ( sup_sup_set_o @ A @ B4 )
% 0.34/0.53 = bot_bot_set_o )
% 0.34/0.53 = ( ( A = bot_bot_set_o )
% 0.34/0.53 & ( B4 = bot_bot_set_o ) ) ) ).
% 0.34/0.53
% 0.34/0.53 % Un_empty
% 0.34/0.53 thf(fact_909_Un__empty,axiom,
% 0.34/0.53 ! [A: set_nat,B4: set_nat] :
% 0.34/0.53 ( ( ( sup_sup_set_nat @ A @ B4 )
% 0.34/0.53 = bot_bot_set_nat )
% 0.34/0.53 = ( ( A = bot_bot_set_nat )
% 0.34/0.53 & ( B4 = bot_bot_set_nat ) ) ) ).
% 0.34/0.53
% 0.34/0.53 % Un_empty
% 0.34/0.53 thf(fact_910_Un__empty,axiom,
% 0.34/0.53 ! [A: set_int,B4: set_int] :
% 0.34/0.53 ( ( ( sup_sup_set_int @ A @ B4 )
% 0.34/0.53 = bot_bot_set_int )
% 0.34/0.53 = ( ( A = bot_bot_set_int )
% 0.34/0.53 & ( B4 = bot_bot_set_int ) ) ) ).
% 0.34/0.53
% 0.34/0.53 % Un_empty
% 0.34/0.53 thf(fact_911_Un__empty,axiom,
% 0.34/0.53 ! [A: set_Pr4532377907799695533_nat_o,B4: set_Pr4532377907799695533_nat_o] :
% 0.34/0.53 ( ( ( sup_su5209123915105501825_nat_o @ A @ B4 )
% 0.34/0.53 = bot_bo7824918357723582233_nat_o )
% 0.34/0.53 = ( ( A = bot_bo7824918357723582233_nat_o )
% 0.34/0.53 & ( B4 = bot_bo7824918357723582233_nat_o ) ) ) ).
% 0.34/0.53
% 0.34/0.53 % Un_empty
% 0.34/0.53 thf(fact_912_Int__subset__iff,axiom,
% 0.34/0.53 ! [C5: set_nat,A: set_nat,B4: set_nat] :
% 0.34/0.53 ( ( ord_less_eq_set_nat @ C5 @ ( inf_inf_set_nat @ A @ B4 ) )
% 0.34/0.53 = ( ( ord_less_eq_set_nat @ C5 @ A )
% 0.34/0.53 & ( ord_less_eq_set_nat @ C5 @ B4 ) ) ) ).
% 0.34/0.53
% 0.34/0.53 % Int_subset_iff
% 0.34/0.53 thf(fact_913_Int__insert__right__if1,axiom,
% 0.34/0.53 ! [A2: produc3658429121746597890et_nat > $o,A: set_Pr4532377907799695533_nat_o,B4: set_Pr4532377907799695533_nat_o] :
% 0.34/0.53 ( ( member6576561426505652726_nat_o @ A2 @ A )
% 0.34/0.53 => ( ( inf_in1906310914598751387_nat_o @ A @ ( insert5175938949040314269_nat_o @ A2 @ B4 ) )
% 0.34/0.53 = ( insert5175938949040314269_nat_o @ A2 @ ( inf_in1906310914598751387_nat_o @ A @ B4 ) ) ) ) ).
% 0.34/0.53
% 0.34/0.53 % Int_insert_right_if1
% 0.34/0.53 thf(fact_914_Int__insert__right__if1,axiom,
% 0.34/0.53 ! [A2: int,A: set_int,B4: set_int] :
% 0.34/0.53 ( ( member_int @ A2 @ A )
% 0.34/0.53 => ( ( inf_inf_set_int @ A @ ( insert_int @ A2 @ B4 ) )
% 0.34/0.53 = ( insert_int @ A2 @ ( inf_inf_set_int @ A @ B4 ) ) ) ) ).
% 0.34/0.53
% 0.34/0.53 % Int_insert_right_if1
% 0.34/0.53 thf(fact_915_Int__insert__right__if1,axiom,
% 0.34/0.53 ! [A2: $o,A: set_o,B4: set_o] :
% 0.34/0.53 ( ( member_o @ A2 @ A )
% 0.34/0.53 => ( ( inf_inf_set_o @ A @ ( insert_o @ A2 @ B4 ) )
% 0.34/0.53 = ( insert_o @ A2 @ ( inf_inf_set_o @ A @ B4 ) ) ) ) ).
% 0.34/0.53
% 0.34/0.53 % Int_insert_right_if1
% 0.34/0.53 thf(fact_916_Int__insert__right__if1,axiom,
% 0.34/0.53 ! [A2: product_unit,A: set_Product_unit,B4: set_Product_unit] :
% 0.34/0.53 ( ( member_Product_unit @ A2 @ A )
% 0.34/0.53 => ( ( inf_in4660618365625256667t_unit @ A @ ( insert_Product_unit @ A2 @ B4 ) )
% 0.34/0.53 = ( insert_Product_unit @ A2 @ ( inf_in4660618365625256667t_unit @ A @ B4 ) ) ) ) ).
% 0.34/0.53
% 0.34/0.53 % Int_insert_right_if1
% 0.34/0.53 thf(fact_917_Int__insert__right__if1,axiom,
% 0.34/0.53 ! [A2: nat,A: set_nat,B4: set_nat] :
% 0.34/0.53 ( ( member_nat @ A2 @ A )
% 0.34/0.53 => ( ( inf_inf_set_nat @ A @ ( insert_nat @ A2 @ B4 ) )
% 0.34/0.53 = ( insert_nat @ A2 @ ( inf_inf_set_nat @ A @ B4 ) ) ) ) ).
% 0.34/0.53
% 0.34/0.53 % Int_insert_right_if1
% 0.34/0.53 thf(fact_918_Int__insert__right__if0,axiom,
% 0.34/0.53 ! [A2: produc3658429121746597890et_nat > $o,A: set_Pr4532377907799695533_nat_o,B4: set_Pr4532377907799695533_nat_o] :
% 0.34/0.53 ( ~ ( member6576561426505652726_nat_o @ A2 @ A )
% 0.34/0.53 => ( ( inf_in1906310914598751387_nat_o @ A @ ( insert5175938949040314269_nat_o @ A2 @ B4 ) )
% 0.34/0.53 = ( inf_in1906310914598751387_nat_o @ A @ B4 ) ) ) ).
% 0.34/0.53
% 0.34/0.53 % Int_insert_right_if0
% 0.34/0.53 thf(fact_919_Int__insert__right__if0,axiom,
% 0.34/0.53 ! [A2: int,A: set_int,B4: set_int] :
% 0.34/0.53 ( ~ ( member_int @ A2 @ A )
% 0.34/0.53 => ( ( inf_inf_set_int @ A @ ( insert_int @ A2 @ B4 ) )
% 0.34/0.53 = ( inf_inf_set_int @ A @ B4 ) ) ) ).
% 0.34/0.53
% 0.34/0.53 % Int_insert_right_if0
% 0.34/0.53 thf(fact_920_Int__insert__right__if0,axiom,
% 0.34/0.53 ! [A2: $o,A: set_o,B4: set_o] :
% 0.34/0.53 ( ~ ( member_o @ A2 @ A )
% 0.34/0.53 => ( ( inf_inf_set_o @ A @ ( insert_o @ A2 @ B4 ) )
% 0.34/0.53 = ( inf_inf_set_o @ A @ B4 ) ) ) ).
% 0.34/0.53
% 0.34/0.53 % Int_insert_right_if0
% 0.34/0.53 thf(fact_921_Int__insert__right__if0,axiom,
% 0.34/0.53 ! [A2: product_unit,A: set_Product_unit,B4: set_Product_unit] :
% 0.34/0.53 ( ~ ( member_Product_unit @ A2 @ A )
% 0.34/0.53 => ( ( inf_in4660618365625256667t_unit @ A @ ( insert_Product_unit @ A2 @ B4 ) )
% 0.34/0.53 = ( inf_in4660618365625256667t_unit @ A @ B4 ) ) ) ).
% 0.34/0.53
% 0.34/0.53 % Int_insert_right_if0
% 0.34/0.53 thf(fact_922_Int__insert__right__if0,axiom,
% 0.34/0.53 ! [A2: nat,A: set_nat,B4: set_nat] :
% 0.34/0.53 ( ~ ( member_nat @ A2 @ A )
% 0.34/0.53 => ( ( inf_inf_set_nat @ A @ ( insert_nat @ A2 @ B4 ) )
% 0.34/0.53 = ( inf_inf_set_nat @ A @ B4 ) ) ) ).
% 0.34/0.53
% 0.34/0.53 % Int_insert_right_if0
% 0.34/0.53 thf(fact_923_insert__inter__insert,axiom,
% 0.34/0.53 ! [A2: int,A: set_int,B4: set_int] :
% 0.34/0.53 ( ( inf_inf_set_int @ ( insert_int @ A2 @ A ) @ ( insert_int @ A2 @ B4 ) )
% 0.34/0.53 = ( insert_int @ A2 @ ( inf_inf_set_int @ A @ B4 ) ) ) ).
% 0.34/0.53
% 0.34/0.53 % insert_inter_insert
% 0.34/0.53 thf(fact_924_insert__inter__insert,axiom,
% 0.34/0.53 ! [A2: $o,A: set_o,B4: set_o] :
% 0.34/0.53 ( ( inf_inf_set_o @ ( insert_o @ A2 @ A ) @ ( insert_o @ A2 @ B4 ) )
% 0.34/0.53 = ( insert_o @ A2 @ ( inf_inf_set_o @ A @ B4 ) ) ) ).
% 0.34/0.53
% 0.34/0.53 % insert_inter_insert
% 0.34/0.53 thf(fact_925_insert__inter__insert,axiom,
% 0.34/0.53 ! [A2: product_unit,A: set_Product_unit,B4: set_Product_unit] :
% 0.34/0.53 ( ( inf_in4660618365625256667t_unit @ ( insert_Product_unit @ A2 @ A ) @ ( insert_Product_unit @ A2 @ B4 ) )
% 0.34/0.53 = ( insert_Product_unit @ A2 @ ( inf_in4660618365625256667t_unit @ A @ B4 ) ) ) ).
% 0.34/0.53
% 0.34/0.53 % insert_inter_insert
% 0.34/0.53 thf(fact_926_insert__inter__insert,axiom,
% 0.34/0.53 ! [A2: nat,A: set_nat,B4: set_nat] :
% 0.34/0.53 ( ( inf_inf_set_nat @ ( insert_nat @ A2 @ A ) @ ( insert_nat @ A2 @ B4 ) )
% 0.34/0.53 = ( insert_nat @ A2 @ ( inf_inf_set_nat @ A @ B4 ) ) ) ).
% 0.34/0.53
% 0.34/0.53 % insert_inter_insert
% 0.34/0.53 thf(fact_927_Int__insert__left__if1,axiom,
% 0.34/0.53 ! [A2: produc3658429121746597890et_nat > $o,C5: set_Pr4532377907799695533_nat_o,B4: set_Pr4532377907799695533_nat_o] :
% 0.34/0.53 ( ( member6576561426505652726_nat_o @ A2 @ C5 )
% 0.34/0.53 => ( ( inf_in1906310914598751387_nat_o @ ( insert5175938949040314269_nat_o @ A2 @ B4 ) @ C5 )
% 0.34/0.53 = ( insert5175938949040314269_nat_o @ A2 @ ( inf_in1906310914598751387_nat_o @ B4 @ C5 ) ) ) ) ).
% 0.34/0.53
% 0.34/0.53 % Int_insert_left_if1
% 0.34/0.53 thf(fact_928_Int__insert__left__if1,axiom,
% 0.34/0.53 ! [A2: int,C5: set_int,B4: set_int] :
% 0.34/0.53 ( ( member_int @ A2 @ C5 )
% 0.34/0.53 => ( ( inf_inf_set_int @ ( insert_int @ A2 @ B4 ) @ C5 )
% 0.34/0.53 = ( insert_int @ A2 @ ( inf_inf_set_int @ B4 @ C5 ) ) ) ) ).
% 0.34/0.53
% 0.34/0.53 % Int_insert_left_if1
% 0.34/0.53 thf(fact_929_Int__insert__left__if1,axiom,
% 0.34/0.53 ! [A2: $o,C5: set_o,B4: set_o] :
% 0.34/0.53 ( ( member_o @ A2 @ C5 )
% 0.34/0.53 => ( ( inf_inf_set_o @ ( insert_o @ A2 @ B4 ) @ C5 )
% 0.34/0.53 = ( insert_o @ A2 @ ( inf_inf_set_o @ B4 @ C5 ) ) ) ) ).
% 0.34/0.53
% 0.34/0.53 % Int_insert_left_if1
% 0.34/0.53 thf(fact_930_Int__insert__left__if1,axiom,
% 0.34/0.53 ! [A2: product_unit,C5: set_Product_unit,B4: set_Product_unit] :
% 0.34/0.53 ( ( member_Product_unit @ A2 @ C5 )
% 0.34/0.53 => ( ( inf_in4660618365625256667t_unit @ ( insert_Product_unit @ A2 @ B4 ) @ C5 )
% 0.34/0.53 = ( insert_Product_unit @ A2 @ ( inf_in4660618365625256667t_unit @ B4 @ C5 ) ) ) ) ).
% 0.34/0.53
% 0.34/0.53 % Int_insert_left_if1
% 0.34/0.53 thf(fact_931_Int__insert__left__if1,axiom,
% 0.34/0.53 ! [A2: nat,C5: set_nat,B4: set_nat] :
% 0.34/0.53 ( ( member_nat @ A2 @ C5 )
% 0.34/0.53 => ( ( inf_inf_set_nat @ ( insert_nat @ A2 @ B4 ) @ C5 )
% 0.34/0.53 = ( insert_nat @ A2 @ ( inf_inf_set_nat @ B4 @ C5 ) ) ) ) ).
% 0.34/0.53
% 0.34/0.53 % Int_insert_left_if1
% 0.34/0.53 thf(fact_932_Int__insert__left__if0,axiom,
% 0.34/0.53 ! [A2: produc3658429121746597890et_nat > $o,C5: set_Pr4532377907799695533_nat_o,B4: set_Pr4532377907799695533_nat_o] :
% 0.34/0.53 ( ~ ( member6576561426505652726_nat_o @ A2 @ C5 )
% 0.34/0.53 => ( ( inf_in1906310914598751387_nat_o @ ( insert5175938949040314269_nat_o @ A2 @ B4 ) @ C5 )
% 0.34/0.53 = ( inf_in1906310914598751387_nat_o @ B4 @ C5 ) ) ) ).
% 0.34/0.53
% 0.34/0.53 % Int_insert_left_if0
% 0.34/0.53 thf(fact_933_Int__insert__left__if0,axiom,
% 0.34/0.53 ! [A2: int,C5: set_int,B4: set_int] :
% 0.34/0.53 ( ~ ( member_int @ A2 @ C5 )
% 0.34/0.53 => ( ( inf_inf_set_int @ ( insert_int @ A2 @ B4 ) @ C5 )
% 0.34/0.53 = ( inf_inf_set_int @ B4 @ C5 ) ) ) ).
% 0.34/0.53
% 0.34/0.53 % Int_insert_left_if0
% 0.34/0.53 thf(fact_934_Int__insert__left__if0,axiom,
% 0.34/0.53 ! [A2: $o,C5: set_o,B4: set_o] :
% 0.34/0.53 ( ~ ( member_o @ A2 @ C5 )
% 0.34/0.53 => ( ( inf_inf_set_o @ ( insert_o @ A2 @ B4 ) @ C5 )
% 0.34/0.53 = ( inf_inf_set_o @ B4 @ C5 ) ) ) ).
% 0.34/0.53
% 0.34/0.53 % Int_insert_left_if0
% 0.34/0.53 thf(fact_935_Int__insert__left__if0,axiom,
% 0.34/0.53 ! [A2: product_unit,C5: set_Product_unit,B4: set_Product_unit] :
% 0.34/0.53 ( ~ ( member_Product_unit @ A2 @ C5 )
% 0.34/0.53 => ( ( inf_in4660618365625256667t_unit @ ( insert_Product_unit @ A2 @ B4 ) @ C5 )
% 0.34/0.53 = ( inf_in4660618365625256667t_unit @ B4 @ C5 ) ) ) ).
% 0.34/0.53
% 0.34/0.53 % Int_insert_left_if0
% 0.34/0.53 thf(fact_936_Int__insert__left__if0,axiom,
% 0.34/0.53 ! [A2: nat,C5: set_nat,B4: set_nat] :
% 0.34/0.53 ( ~ ( member_nat @ A2 @ C5 )
% 0.34/0.53 => ( ( inf_inf_set_nat @ ( insert_nat @ A2 @ B4 ) @ C5 )
% 0.34/0.53 = ( inf_inf_set_nat @ B4 @ C5 ) ) ) ).
% 0.34/0.53
% 0.34/0.53 % Int_insert_left_if0
% 0.34/0.53 thf(fact_937_Un__subset__iff,axiom,
% 0.34/0.53 ! [A: set_nat,B4: set_nat,C5: set_nat] :
% 0.34/0.53 ( ( ord_less_eq_set_nat @ ( sup_sup_set_nat @ A @ B4 ) @ C5 )
% 0.34/0.53 = ( ( ord_less_eq_set_nat @ A @ C5 )
% 0.34/0.53 & ( ord_less_eq_set_nat @ B4 @ C5 ) ) ) ).
% 0.34/0.53
% 0.34/0.53 % Un_subset_iff
% 0.34/0.53 thf(fact_938_Un__insert__right,axiom,
% 0.34/0.53 ! [A: set_int,A2: int,B4: set_int] :
% 0.34/0.53 ( ( sup_sup_set_int @ A @ ( insert_int @ A2 @ B4 ) )
% 0.34/0.53 = ( insert_int @ A2 @ ( sup_sup_set_int @ A @ B4 ) ) ) ).
% 0.34/0.53
% 0.34/0.53 % Un_insert_right
% 0.34/0.53 thf(fact_939_Un__insert__right,axiom,
% 0.34/0.53 ! [A: set_o,A2: $o,B4: set_o] :
% 0.34/0.53 ( ( sup_sup_set_o @ A @ ( insert_o @ A2 @ B4 ) )
% 0.34/0.53 = ( insert_o @ A2 @ ( sup_sup_set_o @ A @ B4 ) ) ) ).
% 0.34/0.53
% 0.34/0.53 % Un_insert_right
% 0.34/0.53 thf(fact_940_Un__insert__right,axiom,
% 0.34/0.53 ! [A: set_Product_unit,A2: product_unit,B4: set_Product_unit] :
% 0.34/0.53 ( ( sup_su793286257634532545t_unit @ A @ ( insert_Product_unit @ A2 @ B4 ) )
% 0.34/0.53 = ( insert_Product_unit @ A2 @ ( sup_su793286257634532545t_unit @ A @ B4 ) ) ) ).
% 0.34/0.53
% 0.34/0.53 % Un_insert_right
% 0.34/0.53 thf(fact_941_Un__insert__right,axiom,
% 0.34/0.53 ! [A: set_nat,A2: nat,B4: set_nat] :
% 0.34/0.53 ( ( sup_sup_set_nat @ A @ ( insert_nat @ A2 @ B4 ) )
% 0.34/0.53 = ( insert_nat @ A2 @ ( sup_sup_set_nat @ A @ B4 ) ) ) ).
% 0.34/0.53
% 0.34/0.53 % Un_insert_right
% 0.34/0.53 thf(fact_942_Un__insert__left,axiom,
% 0.34/0.53 ! [A2: int,B4: set_int,C5: set_int] :
% 0.34/0.53 ( ( sup_sup_set_int @ ( insert_int @ A2 @ B4 ) @ C5 )
% 0.34/0.53 = ( insert_int @ A2 @ ( sup_sup_set_int @ B4 @ C5 ) ) ) ).
% 0.34/0.53
% 0.34/0.53 % Un_insert_left
% 0.34/0.53 thf(fact_943_Un__insert__left,axiom,
% 0.34/0.53 ! [A2: $o,B4: set_o,C5: set_o] :
% 0.34/0.53 ( ( sup_sup_set_o @ ( insert_o @ A2 @ B4 ) @ C5 )
% 0.34/0.53 = ( insert_o @ A2 @ ( sup_sup_set_o @ B4 @ C5 ) ) ) ).
% 0.34/0.53
% 0.34/0.53 % Un_insert_left
% 0.34/0.53 thf(fact_944_Un__insert__left,axiom,
% 0.34/0.53 ! [A2: product_unit,B4: set_Product_unit,C5: set_Product_unit] :
% 0.34/0.53 ( ( sup_su793286257634532545t_unit @ ( insert_Product_unit @ A2 @ B4 ) @ C5 )
% 0.34/0.53 = ( insert_Product_unit @ A2 @ ( sup_su793286257634532545t_unit @ B4 @ C5 ) ) ) ).
% 0.34/0.53
% 0.34/0.53 % Un_insert_left
% 0.34/0.53 thf(fact_945_Un__insert__left,axiom,
% 0.34/0.53 ! [A2: nat,B4: set_nat,C5: set_nat] :
% 0.34/0.53 ( ( sup_sup_set_nat @ ( insert_nat @ A2 @ B4 ) @ C5 )
% 0.34/0.53 = ( insert_nat @ A2 @ ( sup_sup_set_nat @ B4 @ C5 ) ) ) ).
% 0.34/0.53
% 0.34/0.53 % Un_insert_left
% 0.34/0.53 thf(fact_946_Un__Int__eq_I1_J,axiom,
% 0.34/0.53 ! [S: set_nat,T2: set_nat] :
% 0.34/0.53 ( ( inf_inf_set_nat @ ( sup_sup_set_nat @ S @ T2 ) @ S )
% 0.34/0.53 = S ) ).
% 0.34/0.53
% 0.34/0.53 % Un_Int_eq(1)
% 0.34/0.53 thf(fact_947_Un__Int__eq_I2_J,axiom,
% 0.34/0.53 ! [S: set_nat,T2: set_nat] :
% 0.34/0.53 ( ( inf_inf_set_nat @ ( sup_sup_set_nat @ S @ T2 ) @ T2 )
% 0.34/0.53 = T2 ) ).
% 0.34/0.53
% 0.34/0.53 % Un_Int_eq(2)
% 0.34/0.53 thf(fact_948_Un__Int__eq_I3_J,axiom,
% 0.34/0.53 ! [S: set_nat,T2: set_nat] :
% 0.34/0.53 ( ( inf_inf_set_nat @ S @ ( sup_sup_set_nat @ S @ T2 ) )
% 0.34/0.53 = S ) ).
% 0.34/0.53
% 0.34/0.53 % Un_Int_eq(3)
% 0.34/0.53 thf(fact_949_Un__Int__eq_I4_J,axiom,
% 0.34/0.53 ! [T2: set_nat,S: set_nat] :
% 0.34/0.53 ( ( inf_inf_set_nat @ T2 @ ( sup_sup_set_nat @ S @ T2 ) )
% 0.34/0.53 = T2 ) ).
% 0.34/0.53
% 0.34/0.53 % Un_Int_eq(4)
% 0.34/0.53 thf(fact_950_Int__Un__eq_I1_J,axiom,
% 0.34/0.53 ! [S: set_nat,T2: set_nat] :
% 0.34/0.53 ( ( sup_sup_set_nat @ ( inf_inf_set_nat @ S @ T2 ) @ S )
% 0.34/0.53 = S ) ).
% 0.34/0.53
% 0.34/0.53 % Int_Un_eq(1)
% 0.34/0.53 thf(fact_951_Int__Un__eq_I2_J,axiom,
% 0.34/0.53 ! [S: set_nat,T2: set_nat] :
% 0.34/0.53 ( ( sup_sup_set_nat @ ( inf_inf_set_nat @ S @ T2 ) @ T2 )
% 0.34/0.53 = T2 ) ).
% 0.34/0.53
% 0.34/0.53 % Int_Un_eq(2)
% 0.34/0.53 thf(fact_952_Int__Un__eq_I3_J,axiom,
% 0.34/0.53 ! [S: set_nat,T2: set_nat] :
% 0.34/0.53 ( ( sup_sup_set_nat @ S @ ( inf_inf_set_nat @ S @ T2 ) )
% 0.34/0.53 = S ) ).
% 0.34/0.53
% 0.34/0.53 % Int_Un_eq(3)
% 0.34/0.53 thf(fact_953_Int__Un__eq_I4_J,axiom,
% 0.34/0.53 ! [T2: set_nat,S: set_nat] :
% 0.34/0.53 ( ( sup_sup_set_nat @ T2 @ ( inf_inf_set_nat @ S @ T2 ) )
% 0.34/0.53 = T2 ) ).
% 0.34/0.53
% 0.34/0.53 % Int_Un_eq(4)
% 0.34/0.53 thf(fact_954_vimage__Int,axiom,
% 0.34/0.53 ! [F: nat > nat,A: set_nat,B4: set_nat] :
% 0.34/0.53 ( ( vimage_nat_nat @ F @ ( inf_inf_set_nat @ A @ B4 ) )
% 0.34/0.53 = ( inf_inf_set_nat @ ( vimage_nat_nat @ F @ A ) @ ( vimage_nat_nat @ F @ B4 ) ) ) ).
% 0.34/0.53
% 0.34/0.53 % vimage_Int
% 0.34/0.53 thf(fact_955_vimage__Un,axiom,
% 0.34/0.53 ! [F: nat > nat,A: set_nat,B4: set_nat] :
% 0.34/0.53 ( ( vimage_nat_nat @ F @ ( sup_sup_set_nat @ A @ B4 ) )
% 0.34/0.53 = ( sup_sup_set_nat @ ( vimage_nat_nat @ F @ A ) @ ( vimage_nat_nat @ F @ B4 ) ) ) ).
% 0.34/0.53
% 0.34/0.53 % vimage_Un
% 0.34/0.53 thf(fact_956_relH__dist__union,axiom,
% 0.34/0.53 ! [As: set_nat,As5: set_nat,H2: heap_e7401611519738050253t_unit,H3: heap_e7401611519738050253t_unit] :
% 0.34/0.53 ( ( relH @ ( sup_sup_set_nat @ As @ As5 ) @ H2 @ H3 )
% 0.34/0.53 = ( ( relH @ As @ H2 @ H3 )
% 0.34/0.53 & ( relH @ As5 @ H2 @ H3 ) ) ) ).
% 0.34/0.53
% 0.34/0.53 % relH_dist_union
% 0.34/0.53 thf(fact_957_singleton__insert__inj__eq,axiom,
% 0.34/0.53 ! [B: product_unit,A2: product_unit,A: set_Product_unit] :
% 0.34/0.53 ( ( ( insert_Product_unit @ B @ bot_bo3957492148770167129t_unit )
% 0.34/0.53 = ( insert_Product_unit @ A2 @ A ) )
% 0.34/0.53 = ( ( A2 = B )
% 0.34/0.53 & ( ord_le3507040750410214029t_unit @ A @ ( insert_Product_unit @ B @ bot_bo3957492148770167129t_unit ) ) ) ) ).
% 0.34/0.53
% 0.34/0.53 % singleton_insert_inj_eq
% 0.34/0.53 thf(fact_958_singleton__insert__inj__eq,axiom,
% 0.34/0.53 ! [B: $o,A2: $o,A: set_o] :
% 0.34/0.53 ( ( ( insert_o @ B @ bot_bot_set_o )
% 0.34/0.53 = ( insert_o @ A2 @ A ) )
% 0.34/0.53 = ( ( A2 = B )
% 0.34/0.53 & ( ord_less_eq_set_o @ A @ ( insert_o @ B @ bot_bot_set_o ) ) ) ) ).
% 0.34/0.53
% 0.34/0.53 % singleton_insert_inj_eq
% 0.34/0.53 thf(fact_959_singleton__insert__inj__eq,axiom,
% 0.34/0.53 ! [B: int,A2: int,A: set_int] :
% 0.34/0.53 ( ( ( insert_int @ B @ bot_bot_set_int )
% 0.34/0.53 = ( insert_int @ A2 @ A ) )
% 0.34/0.53 = ( ( A2 = B )
% 0.34/0.53 & ( ord_less_eq_set_int @ A @ ( insert_int @ B @ bot_bot_set_int ) ) ) ) ).
% 0.34/0.53
% 0.34/0.53 % singleton_insert_inj_eq
% 0.34/0.53 thf(fact_960_singleton__insert__inj__eq,axiom,
% 0.34/0.53 ! [B: produc3658429121746597890et_nat > $o,A2: produc3658429121746597890et_nat > $o,A: set_Pr4532377907799695533_nat_o] :
% 0.34/0.53 ( ( ( insert5175938949040314269_nat_o @ B @ bot_bo7824918357723582233_nat_o )
% 0.34/0.53 = ( insert5175938949040314269_nat_o @ A2 @ A ) )
% 0.34/0.53 = ( ( A2 = B )
% 0.34/0.53 & ( ord_le2965882846123202637_nat_o @ A @ ( insert5175938949040314269_nat_o @ B @ bot_bo7824918357723582233_nat_o ) ) ) ) ).
% 0.34/0.53
% 0.34/0.53 % singleton_insert_inj_eq
% 0.34/0.53 thf(fact_961_singleton__insert__inj__eq,axiom,
% 0.34/0.53 ! [B: nat,A2: nat,A: set_nat] :
% 0.34/0.53 ( ( ( insert_nat @ B @ bot_bot_set_nat )
% 0.34/0.53 = ( insert_nat @ A2 @ A ) )
% 0.34/0.53 = ( ( A2 = B )
% 0.34/0.53 & ( ord_less_eq_set_nat @ A @ ( insert_nat @ B @ bot_bot_set_nat ) ) ) ) ).
% 0.34/0.53
% 0.34/0.53 % singleton_insert_inj_eq
% 0.34/0.53 thf(fact_962_singleton__insert__inj__eq_H,axiom,
% 0.34/0.53 ! [A2: product_unit,A: set_Product_unit,B: product_unit] :
% 0.34/0.53 ( ( ( insert_Product_unit @ A2 @ A )
% 0.34/0.53 = ( insert_Product_unit @ B @ bot_bo3957492148770167129t_unit ) )
% 0.34/0.53 = ( ( A2 = B )
% 0.34/0.53 & ( ord_le3507040750410214029t_unit @ A @ ( insert_Product_unit @ B @ bot_bo3957492148770167129t_unit ) ) ) ) ).
% 0.34/0.53
% 0.34/0.53 % singleton_insert_inj_eq'
% 0.34/0.53 thf(fact_963_singleton__insert__inj__eq_H,axiom,
% 0.34/0.53 ! [A2: $o,A: set_o,B: $o] :
% 0.34/0.53 ( ( ( insert_o @ A2 @ A )
% 0.34/0.53 = ( insert_o @ B @ bot_bot_set_o ) )
% 0.34/0.53 = ( ( A2 = B )
% 0.34/0.53 & ( ord_less_eq_set_o @ A @ ( insert_o @ B @ bot_bot_set_o ) ) ) ) ).
% 0.34/0.53
% 0.34/0.53 % singleton_insert_inj_eq'
% 0.34/0.53 thf(fact_964_singleton__insert__inj__eq_H,axiom,
% 0.34/0.53 ! [A2: int,A: set_int,B: int] :
% 0.34/0.53 ( ( ( insert_int @ A2 @ A )
% 0.34/0.53 = ( insert_int @ B @ bot_bot_set_int ) )
% 0.34/0.53 = ( ( A2 = B )
% 0.34/0.53 & ( ord_less_eq_set_int @ A @ ( insert_int @ B @ bot_bot_set_int ) ) ) ) ).
% 0.34/0.53
% 0.34/0.53 % singleton_insert_inj_eq'
% 0.34/0.53 thf(fact_965_singleton__insert__inj__eq_H,axiom,
% 0.34/0.53 ! [A2: produc3658429121746597890et_nat > $o,A: set_Pr4532377907799695533_nat_o,B: produc3658429121746597890et_nat > $o] :
% 0.34/0.53 ( ( ( insert5175938949040314269_nat_o @ A2 @ A )
% 0.34/0.53 = ( insert5175938949040314269_nat_o @ B @ bot_bo7824918357723582233_nat_o ) )
% 0.34/0.53 = ( ( A2 = B )
% 0.34/0.53 & ( ord_le2965882846123202637_nat_o @ A @ ( insert5175938949040314269_nat_o @ B @ bot_bo7824918357723582233_nat_o ) ) ) ) ).
% 0.34/0.53
% 0.34/0.53 % singleton_insert_inj_eq'
% 0.34/0.53 thf(fact_966_singleton__insert__inj__eq_H,axiom,
% 0.34/0.53 ! [A2: nat,A: set_nat,B: nat] :
% 0.34/0.53 ( ( ( insert_nat @ A2 @ A )
% 0.34/0.53 = ( insert_nat @ B @ bot_bot_set_nat ) )
% 0.34/0.53 = ( ( A2 = B )
% 0.34/0.53 & ( ord_less_eq_set_nat @ A @ ( insert_nat @ B @ bot_bot_set_nat ) ) ) ) ).
% 0.34/0.53
% 0.34/0.53 % singleton_insert_inj_eq'
% 0.34/0.53 thf(fact_967_disjoint__insert_I2_J,axiom,
% 0.34/0.53 ! [A: set_Product_unit,B: product_unit,B4: set_Product_unit] :
% 0.34/0.53 ( ( bot_bo3957492148770167129t_unit
% 0.34/0.53 = ( inf_in4660618365625256667t_unit @ A @ ( insert_Product_unit @ B @ B4 ) ) )
% 0.34/0.53 = ( ~ ( member_Product_unit @ B @ A )
% 0.34/0.53 & ( bot_bo3957492148770167129t_unit
% 0.34/0.53 = ( inf_in4660618365625256667t_unit @ A @ B4 ) ) ) ) ).
% 0.34/0.53
% 0.34/0.53 % disjoint_insert(2)
% 0.34/0.53 thf(fact_968_disjoint__insert_I2_J,axiom,
% 0.34/0.53 ! [A: set_o,B: $o,B4: set_o] :
% 0.34/0.53 ( ( bot_bot_set_o
% 0.34/0.53 = ( inf_inf_set_o @ A @ ( insert_o @ B @ B4 ) ) )
% 0.34/0.53 = ( ~ ( member_o @ B @ A )
% 0.34/0.53 & ( bot_bot_set_o
% 0.34/0.53 = ( inf_inf_set_o @ A @ B4 ) ) ) ) ).
% 0.34/0.53
% 0.34/0.53 % disjoint_insert(2)
% 0.34/0.53 thf(fact_969_disjoint__insert_I2_J,axiom,
% 0.34/0.53 ! [A: set_nat,B: nat,B4: set_nat] :
% 0.34/0.53 ( ( bot_bot_set_nat
% 0.34/0.53 = ( inf_inf_set_nat @ A @ ( insert_nat @ B @ B4 ) ) )
% 0.34/0.53 = ( ~ ( member_nat @ B @ A )
% 0.34/0.53 & ( bot_bot_set_nat
% 0.34/0.53 = ( inf_inf_set_nat @ A @ B4 ) ) ) ) ).
% 0.34/0.53
% 0.34/0.53 % disjoint_insert(2)
% 0.34/0.53 thf(fact_970_disjoint__insert_I2_J,axiom,
% 0.34/0.53 ! [A: set_int,B: int,B4: set_int] :
% 0.34/0.53 ( ( bot_bot_set_int
% 0.34/0.53 = ( inf_inf_set_int @ A @ ( insert_int @ B @ B4 ) ) )
% 0.34/0.53 = ( ~ ( member_int @ B @ A )
% 0.34/0.53 & ( bot_bot_set_int
% 0.34/0.53 = ( inf_inf_set_int @ A @ B4 ) ) ) ) ).
% 0.34/0.53
% 0.34/0.53 % disjoint_insert(2)
% 0.34/0.53 thf(fact_971_disjoint__insert_I2_J,axiom,
% 0.34/0.53 ! [A: set_Pr4532377907799695533_nat_o,B: produc3658429121746597890et_nat > $o,B4: set_Pr4532377907799695533_nat_o] :
% 0.34/0.53 ( ( bot_bo7824918357723582233_nat_o
% 0.34/0.53 = ( inf_in1906310914598751387_nat_o @ A @ ( insert5175938949040314269_nat_o @ B @ B4 ) ) )
% 0.34/0.53 = ( ~ ( member6576561426505652726_nat_o @ B @ A )
% 0.34/0.53 & ( bot_bo7824918357723582233_nat_o
% 0.34/0.53 = ( inf_in1906310914598751387_nat_o @ A @ B4 ) ) ) ) ).
% 0.34/0.53
% 0.34/0.53 % disjoint_insert(2)
% 0.34/0.53 thf(fact_972_disjoint__insert_I1_J,axiom,
% 0.34/0.53 ! [B4: set_Product_unit,A2: product_unit,A: set_Product_unit] :
% 0.34/0.53 ( ( ( inf_in4660618365625256667t_unit @ B4 @ ( insert_Product_unit @ A2 @ A ) )
% 0.34/0.53 = bot_bo3957492148770167129t_unit )
% 0.34/0.53 = ( ~ ( member_Product_unit @ A2 @ B4 )
% 0.34/0.53 & ( ( inf_in4660618365625256667t_unit @ B4 @ A )
% 0.34/0.53 = bot_bo3957492148770167129t_unit ) ) ) ).
% 0.34/0.53
% 0.34/0.53 % disjoint_insert(1)
% 0.34/0.53 thf(fact_973_disjoint__insert_I1_J,axiom,
% 0.34/0.53 ! [B4: set_o,A2: $o,A: set_o] :
% 0.34/0.53 ( ( ( inf_inf_set_o @ B4 @ ( insert_o @ A2 @ A ) )
% 0.34/0.53 = bot_bot_set_o )
% 0.34/0.53 = ( ~ ( member_o @ A2 @ B4 )
% 0.34/0.53 & ( ( inf_inf_set_o @ B4 @ A )
% 0.34/0.53 = bot_bot_set_o ) ) ) ).
% 0.34/0.53
% 0.34/0.53 % disjoint_insert(1)
% 0.34/0.53 thf(fact_974_disjoint__insert_I1_J,axiom,
% 0.34/0.53 ! [B4: set_nat,A2: nat,A: set_nat] :
% 0.34/0.53 ( ( ( inf_inf_set_nat @ B4 @ ( insert_nat @ A2 @ A ) )
% 0.34/0.53 = bot_bot_set_nat )
% 0.34/0.53 = ( ~ ( member_nat @ A2 @ B4 )
% 0.34/0.53 & ( ( inf_inf_set_nat @ B4 @ A )
% 0.34/0.53 = bot_bot_set_nat ) ) ) ).
% 0.34/0.53
% 0.34/0.53 % disjoint_insert(1)
% 0.34/0.53 thf(fact_975_disjoint__insert_I1_J,axiom,
% 0.34/0.53 ! [B4: set_int,A2: int,A: set_int] :
% 0.34/0.53 ( ( ( inf_inf_set_int @ B4 @ ( insert_int @ A2 @ A ) )
% 0.34/0.53 = bot_bot_set_int )
% 0.34/0.53 = ( ~ ( member_int @ A2 @ B4 )
% 0.34/0.53 & ( ( inf_inf_set_int @ B4 @ A )
% 0.34/0.53 = bot_bot_set_int ) ) ) ).
% 0.34/0.53
% 0.34/0.53 % disjoint_insert(1)
% 0.34/0.53 thf(fact_976_disjoint__insert_I1_J,axiom,
% 0.34/0.53 ! [B4: set_Pr4532377907799695533_nat_o,A2: produc3658429121746597890et_nat > $o,A: set_Pr4532377907799695533_nat_o] :
% 0.34/0.53 ( ( ( inf_in1906310914598751387_nat_o @ B4 @ ( insert5175938949040314269_nat_o @ A2 @ A ) )
% 0.34/0.53 = bot_bo7824918357723582233_nat_o )
% 0.34/0.53 = ( ~ ( member6576561426505652726_nat_o @ A2 @ B4 )
% 0.34/0.53 & ( ( inf_in1906310914598751387_nat_o @ B4 @ A )
% 0.34/0.53 = bot_bo7824918357723582233_nat_o ) ) ) ).
% 0.34/0.53
% 0.34/0.53 % disjoint_insert(1)
% 0.34/0.53 thf(fact_977_insert__disjoint_I2_J,axiom,
% 0.34/0.53 ! [A2: product_unit,A: set_Product_unit,B4: set_Product_unit] :
% 0.34/0.53 ( ( bot_bo3957492148770167129t_unit
% 0.34/0.53 = ( inf_in4660618365625256667t_unit @ ( insert_Product_unit @ A2 @ A ) @ B4 ) )
% 0.34/0.53 = ( ~ ( member_Product_unit @ A2 @ B4 )
% 0.34/0.53 & ( bot_bo3957492148770167129t_unit
% 0.34/0.53 = ( inf_in4660618365625256667t_unit @ A @ B4 ) ) ) ) ).
% 0.34/0.53
% 0.34/0.53 % insert_disjoint(2)
% 0.34/0.53 thf(fact_978_insert__disjoint_I2_J,axiom,
% 0.34/0.53 ! [A2: $o,A: set_o,B4: set_o] :
% 0.34/0.53 ( ( bot_bot_set_o
% 0.34/0.53 = ( inf_inf_set_o @ ( insert_o @ A2 @ A ) @ B4 ) )
% 0.34/0.53 = ( ~ ( member_o @ A2 @ B4 )
% 0.34/0.53 & ( bot_bot_set_o
% 0.34/0.53 = ( inf_inf_set_o @ A @ B4 ) ) ) ) ).
% 0.34/0.53
% 0.34/0.53 % insert_disjoint(2)
% 0.34/0.53 thf(fact_979_insert__disjoint_I2_J,axiom,
% 0.34/0.53 ! [A2: nat,A: set_nat,B4: set_nat] :
% 0.34/0.53 ( ( bot_bot_set_nat
% 0.34/0.53 = ( inf_inf_set_nat @ ( insert_nat @ A2 @ A ) @ B4 ) )
% 0.34/0.53 = ( ~ ( member_nat @ A2 @ B4 )
% 0.34/0.53 & ( bot_bot_set_nat
% 0.34/0.53 = ( inf_inf_set_nat @ A @ B4 ) ) ) ) ).
% 0.34/0.53
% 0.34/0.53 % insert_disjoint(2)
% 0.34/0.53 thf(fact_980_insert__disjoint_I2_J,axiom,
% 0.34/0.53 ! [A2: int,A: set_int,B4: set_int] :
% 0.34/0.53 ( ( bot_bot_set_int
% 0.34/0.53 = ( inf_inf_set_int @ ( insert_int @ A2 @ A ) @ B4 ) )
% 0.34/0.53 = ( ~ ( member_int @ A2 @ B4 )
% 0.34/0.53 & ( bot_bot_set_int
% 0.34/0.53 = ( inf_inf_set_int @ A @ B4 ) ) ) ) ).
% 0.34/0.53
% 0.34/0.53 % insert_disjoint(2)
% 0.34/0.53 thf(fact_981_insert__disjoint_I2_J,axiom,
% 0.34/0.53 ! [A2: produc3658429121746597890et_nat > $o,A: set_Pr4532377907799695533_nat_o,B4: set_Pr4532377907799695533_nat_o] :
% 0.34/0.53 ( ( bot_bo7824918357723582233_nat_o
% 0.34/0.53 = ( inf_in1906310914598751387_nat_o @ ( insert5175938949040314269_nat_o @ A2 @ A ) @ B4 ) )
% 0.34/0.53 = ( ~ ( member6576561426505652726_nat_o @ A2 @ B4 )
% 0.34/0.53 & ( bot_bo7824918357723582233_nat_o
% 0.34/0.53 = ( inf_in1906310914598751387_nat_o @ A @ B4 ) ) ) ) ).
% 0.34/0.53
% 0.34/0.53 % insert_disjoint(2)
% 0.34/0.53 thf(fact_982_insert__disjoint_I1_J,axiom,
% 0.34/0.53 ! [A2: product_unit,A: set_Product_unit,B4: set_Product_unit] :
% 0.34/0.53 ( ( ( inf_in4660618365625256667t_unit @ ( insert_Product_unit @ A2 @ A ) @ B4 )
% 0.34/0.53 = bot_bo3957492148770167129t_unit )
% 0.34/0.53 = ( ~ ( member_Product_unit @ A2 @ B4 )
% 0.34/0.53 & ( ( inf_in4660618365625256667t_unit @ A @ B4 )
% 0.34/0.53 = bot_bo3957492148770167129t_unit ) ) ) ).
% 0.34/0.53
% 0.34/0.53 % insert_disjoint(1)
% 0.34/0.53 thf(fact_983_insert__disjoint_I1_J,axiom,
% 0.34/0.53 ! [A2: $o,A: set_o,B4: set_o] :
% 0.34/0.53 ( ( ( inf_inf_set_o @ ( insert_o @ A2 @ A ) @ B4 )
% 0.34/0.53 = bot_bot_set_o )
% 0.34/0.53 = ( ~ ( member_o @ A2 @ B4 )
% 0.34/0.53 & ( ( inf_inf_set_o @ A @ B4 )
% 0.34/0.53 = bot_bot_set_o ) ) ) ).
% 0.34/0.53
% 0.34/0.53 % insert_disjoint(1)
% 0.34/0.53 thf(fact_984_insert__disjoint_I1_J,axiom,
% 0.34/0.53 ! [A2: nat,A: set_nat,B4: set_nat] :
% 0.34/0.53 ( ( ( inf_inf_set_nat @ ( insert_nat @ A2 @ A ) @ B4 )
% 0.34/0.53 = bot_bot_set_nat )
% 0.34/0.53 = ( ~ ( member_nat @ A2 @ B4 )
% 0.34/0.53 & ( ( inf_inf_set_nat @ A @ B4 )
% 0.34/0.53 = bot_bot_set_nat ) ) ) ).
% 0.34/0.53
% 0.34/0.53 % insert_disjoint(1)
% 0.34/0.53 thf(fact_985_insert__disjoint_I1_J,axiom,
% 0.34/0.53 ! [A2: int,A: set_int,B4: set_int] :
% 0.34/0.53 ( ( ( inf_inf_set_int @ ( insert_int @ A2 @ A ) @ B4 )
% 0.34/0.53 = bot_bot_set_int )
% 0.34/0.53 = ( ~ ( member_int @ A2 @ B4 )
% 0.34/0.53 & ( ( inf_inf_set_int @ A @ B4 )
% 0.34/0.53 = bot_bot_set_int ) ) ) ).
% 0.34/0.53
% 0.34/0.53 % insert_disjoint(1)
% 0.34/0.53 thf(fact_986_insert__disjoint_I1_J,axiom,
% 0.34/0.53 ! [A2: produc3658429121746597890et_nat > $o,A: set_Pr4532377907799695533_nat_o,B4: set_Pr4532377907799695533_nat_o] :
% 0.34/0.53 ( ( ( inf_in1906310914598751387_nat_o @ ( insert5175938949040314269_nat_o @ A2 @ A ) @ B4 )
% 0.34/0.53 = bot_bo7824918357723582233_nat_o )
% 0.34/0.53 = ( ~ ( member6576561426505652726_nat_o @ A2 @ B4 )
% 0.34/0.53 & ( ( inf_in1906310914598751387_nat_o @ A @ B4 )
% 0.34/0.53 = bot_bo7824918357723582233_nat_o ) ) ) ).
% 0.34/0.53
% 0.34/0.53 % insert_disjoint(1)
% 0.34/0.53 thf(fact_987_Image__Id__on,axiom,
% 0.34/0.53 ! [A: set_nat,B4: set_nat] :
% 0.34/0.53 ( ( image_nat_nat @ ( id_on_nat @ A ) @ B4 )
% 0.34/0.53 = ( inf_inf_set_nat @ A @ B4 ) ) ).
% 0.34/0.53
% 0.34/0.53 % Image_Id_on
% 0.34/0.53 thf(fact_988_in__range__dist__union,axiom,
% 0.34/0.53 ! [H2: heap_e7401611519738050253t_unit,As: set_nat,As5: set_nat] :
% 0.34/0.53 ( ( in_range @ ( produc7507926704131184380et_nat @ H2 @ ( sup_sup_set_nat @ As @ As5 ) ) )
% 0.34/0.53 = ( ( in_range @ ( produc7507926704131184380et_nat @ H2 @ As ) )
% 0.34/0.53 & ( in_range @ ( produc7507926704131184380et_nat @ H2 @ As5 ) ) ) ) ).
% 0.34/0.53
% 0.34/0.53 % in_range_dist_union
% 0.34/0.53 thf(fact_989_reachable__mono,axiom,
% 0.34/0.53 ! [R3: set_Pr1261947904930325089at_nat,R5: set_Pr1261947904930325089at_nat,X4: set_nat,X5: set_nat] :
% 0.34/0.53 ( ( ord_le3146513528884898305at_nat @ R3 @ R5 )
% 0.34/0.53 => ( ( ord_less_eq_set_nat @ X4 @ X5 )
% 0.34/0.53 => ( ord_less_eq_set_nat @ ( image_nat_nat @ ( transi2905341329935302413cl_nat @ R3 ) @ X4 ) @ ( image_nat_nat @ ( transi2905341329935302413cl_nat @ R5 ) @ X5 ) ) ) ) ).
% 0.34/0.53
% 0.34/0.53 % reachable_mono
% 0.34/0.53 thf(fact_990_order__antisym__conv,axiom,
% 0.34/0.53 ! [Y: filter_nat,X: filter_nat] :
% 0.34/0.53 ( ( ord_le2510731241096832064er_nat @ Y @ X )
% 0.34/0.53 => ( ( ord_le2510731241096832064er_nat @ X @ Y )
% 0.34/0.53 = ( X = Y ) ) ) ).
% 0.34/0.53
% 0.34/0.53 % order_antisym_conv
% 0.34/0.53 thf(fact_991_order__antisym__conv,axiom,
% 0.34/0.53 ! [Y: set_nat,X: set_nat] :
% 0.34/0.53 ( ( ord_less_eq_set_nat @ Y @ X )
% 0.34/0.53 => ( ( ord_less_eq_set_nat @ X @ Y )
% 0.34/0.53 = ( X = Y ) ) ) ).
% 0.34/0.53
% 0.34/0.53 % order_antisym_conv
% 0.34/0.53 thf(fact_992_order__antisym__conv,axiom,
% 0.34/0.53 ! [Y: rat,X: rat] :
% 0.34/0.53 ( ( ord_less_eq_rat @ Y @ X )
% 0.34/0.53 => ( ( ord_less_eq_rat @ X @ Y )
% 0.34/0.53 = ( X = Y ) ) ) ).
% 0.34/0.53
% 0.34/0.53 % order_antisym_conv
% 0.34/0.53 thf(fact_993_order__antisym__conv,axiom,
% 0.34/0.53 ! [Y: nat,X: nat] :
% 0.34/0.53 ( ( ord_less_eq_nat @ Y @ X )
% 0.34/0.53 => ( ( ord_less_eq_nat @ X @ Y )
% 0.34/0.53 = ( X = Y ) ) ) ).
% 0.34/0.53
% 0.34/0.53 % order_antisym_conv
% 0.34/0.53 thf(fact_994_order__antisym__conv,axiom,
% 0.34/0.53 ! [Y: int,X: int] :
% 0.34/0.53 ( ( ord_less_eq_int @ Y @ X )
% 0.34/0.53 => ( ( ord_less_eq_int @ X @ Y )
% 0.34/0.53 = ( X = Y ) ) ) ).
% 0.34/0.53
% 0.34/0.53 % order_antisym_conv
% 0.34/0.53 thf(fact_995_linorder__le__cases,axiom,
% 0.34/0.53 ! [X: rat,Y: rat] :
% 0.34/0.53 ( ~ ( ord_less_eq_rat @ X @ Y )
% 0.34/0.53 => ( ord_less_eq_rat @ Y @ X ) ) ).
% 0.34/0.53
% 0.34/0.53 % linorder_le_cases
% 0.34/0.53 thf(fact_996_linorder__le__cases,axiom,
% 0.34/0.53 ! [X: nat,Y: nat] :
% 0.34/0.53 ( ~ ( ord_less_eq_nat @ X @ Y )
% 0.34/0.53 => ( ord_less_eq_nat @ Y @ X ) ) ).
% 0.34/0.53
% 0.34/0.53 % linorder_le_cases
% 0.34/0.53 thf(fact_997_linorder__le__cases,axiom,
% 0.34/0.53 ! [X: int,Y: int] :
% 0.34/0.53 ( ~ ( ord_less_eq_int @ X @ Y )
% 0.34/0.53 => ( ord_less_eq_int @ Y @ X ) ) ).
% 0.34/0.53
% 0.34/0.53 % linorder_le_cases
% 0.34/0.53 thf(fact_998_ord__le__eq__subst,axiom,
% 0.34/0.53 ! [A2: rat,B: rat,F: rat > rat,C: rat] :
% 0.34/0.53 ( ( ord_less_eq_rat @ A2 @ B )
% 0.34/0.53 => ( ( ( F @ B )
% 0.34/0.53 = C )
% 0.34/0.53 => ( ! [X3: rat,Y2: rat] :
% 0.34/0.53 ( ( ord_less_eq_rat @ X3 @ Y2 )
% 0.34/0.53 => ( ord_less_eq_rat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
% 0.34/0.53 => ( ord_less_eq_rat @ ( F @ A2 ) @ C ) ) ) ) ).
% 0.34/0.53
% 0.34/0.53 % ord_le_eq_subst
% 0.34/0.53 thf(fact_999_ord__le__eq__subst,axiom,
% 0.34/0.53 ! [A2: rat,B: rat,F: rat > nat,C: nat] :
% 0.34/0.53 ( ( ord_less_eq_rat @ A2 @ B )
% 0.34/0.53 => ( ( ( F @ B )
% 0.34/0.53 = C )
% 0.34/0.53 => ( ! [X3: rat,Y2: rat] :
% 0.34/0.53 ( ( ord_less_eq_rat @ X3 @ Y2 )
% 0.34/0.53 => ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
% 0.34/0.53 => ( ord_less_eq_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% 0.34/0.53
% 0.34/0.53 % ord_le_eq_subst
% 0.34/0.53 thf(fact_1000_ord__le__eq__subst,axiom,
% 0.34/0.53 ! [A2: rat,B: rat,F: rat > int,C: int] :
% 0.34/0.53 ( ( ord_less_eq_rat @ A2 @ B )
% 0.34/0.53 => ( ( ( F @ B )
% 0.34/0.53 = C )
% 0.34/0.53 => ( ! [X3: rat,Y2: rat] :
% 0.34/0.53 ( ( ord_less_eq_rat @ X3 @ Y2 )
% 0.34/0.53 => ( ord_less_eq_int @ ( F @ X3 ) @ ( F @ Y2 ) ) )
% 0.34/0.53 => ( ord_less_eq_int @ ( F @ A2 ) @ C ) ) ) ) ).
% 0.34/0.53
% 0.34/0.53 % ord_le_eq_subst
% 0.34/0.53 thf(fact_1001_ord__le__eq__subst,axiom,
% 0.34/0.53 ! [A2: nat,B: nat,F: nat > rat,C: rat] :
% 0.34/0.53 ( ( ord_less_eq_nat @ A2 @ B )
% 0.34/0.53 => ( ( ( F @ B )
% 0.34/0.53 = C )
% 0.34/0.53 => ( ! [X3: nat,Y2: nat] :
% 0.34/0.53 ( ( ord_less_eq_nat @ X3 @ Y2 )
% 0.34/0.53 => ( ord_less_eq_rat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
% 0.34/0.53 => ( ord_less_eq_rat @ ( F @ A2 ) @ C ) ) ) ) ).
% 0.34/0.53
% 0.34/0.53 % ord_le_eq_subst
% 0.34/0.53 thf(fact_1002_ord__le__eq__subst,axiom,
% 0.34/0.53 ! [A2: nat,B: nat,F: nat > nat,C: nat] :
% 0.34/0.53 ( ( ord_less_eq_nat @ A2 @ B )
% 0.34/0.53 => ( ( ( F @ B )
% 0.34/0.53 = C )
% 0.34/0.53 => ( ! [X3: nat,Y2: nat] :
% 0.34/0.53 ( ( ord_less_eq_nat @ X3 @ Y2 )
% 0.34/0.53 => ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
% 0.34/0.53 => ( ord_less_eq_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% 0.34/0.53
% 0.34/0.53 % ord_le_eq_subst
% 0.34/0.53 thf(fact_1003_ord__le__eq__subst,axiom,
% 0.34/0.53 ! [A2: nat,B: nat,F: nat > int,C: int] :
% 0.34/0.53 ( ( ord_less_eq_nat @ A2 @ B )
% 0.34/0.53 => ( ( ( F @ B )
% 0.34/0.53 = C )
% 0.34/0.53 => ( ! [X3: nat,Y2: nat] :
% 0.34/0.53 ( ( ord_less_eq_nat @ X3 @ Y2 )
% 0.34/0.53 => ( ord_less_eq_int @ ( F @ X3 ) @ ( F @ Y2 ) ) )
% 0.34/0.53 => ( ord_less_eq_int @ ( F @ A2 ) @ C ) ) ) ) ).
% 0.34/0.53
% 0.34/0.53 % ord_le_eq_subst
% 0.34/0.53 thf(fact_1004_ord__le__eq__subst,axiom,
% 0.34/0.53 ! [A2: int,B: int,F: int > rat,C: rat] :
% 0.34/0.53 ( ( ord_less_eq_int @ A2 @ B )
% 0.34/0.53 => ( ( ( F @ B )
% 0.34/0.53 = C )
% 0.34/0.53 => ( ! [X3: int,Y2: int] :
% 0.34/0.53 ( ( ord_less_eq_int @ X3 @ Y2 )
% 0.34/0.53 => ( ord_less_eq_rat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
% 0.34/0.53 => ( ord_less_eq_rat @ ( F @ A2 ) @ C ) ) ) ) ).
% 0.34/0.53
% 0.34/0.53 % ord_le_eq_subst
% 0.34/0.53 thf(fact_1005_ord__le__eq__subst,axiom,
% 0.34/0.53 ! [A2: int,B: int,F: int > nat,C: nat] :
% 0.34/0.53 ( ( ord_less_eq_int @ A2 @ B )
% 0.34/0.53 => ( ( ( F @ B )
% 0.34/0.53 = C )
% 0.34/0.53 => ( ! [X3: int,Y2: int] :
% 0.34/0.53 ( ( ord_less_eq_int @ X3 @ Y2 )
% 0.34/0.53 => ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
% 0.34/0.53 => ( ord_less_eq_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% 0.34/0.53
% 0.34/0.53 % ord_le_eq_subst
% 0.34/0.53 thf(fact_1006_ord__le__eq__subst,axiom,
% 0.34/0.53 ! [A2: int,B: int,F: int > int,C: int] :
% 0.34/0.53 ( ( ord_less_eq_int @ A2 @ B )
% 0.34/0.53 => ( ( ( F @ B )
% 0.34/0.53 = C )
% 0.34/0.53 => ( ! [X3: int,Y2: int] :
% 0.34/0.53 ( ( ord_less_eq_int @ X3 @ Y2 )
% 0.34/0.53 => ( ord_less_eq_int @ ( F @ X3 ) @ ( F @ Y2 ) ) )
% 0.34/0.53 => ( ord_less_eq_int @ ( F @ A2 ) @ C ) ) ) ) ).
% 0.34/0.53
% 0.34/0.53 % ord_le_eq_subst
% 0.34/0.53 thf(fact_1007_ord__le__eq__subst,axiom,
% 0.34/0.53 ! [A2: filter_nat,B: filter_nat,F: filter_nat > rat,C: rat] :
% 0.34/0.53 ( ( ord_le2510731241096832064er_nat @ A2 @ B )
% 0.34/0.53 => ( ( ( F @ B )
% 0.34/0.53 = C )
% 0.34/0.53 => ( ! [X3: filter_nat,Y2: filter_nat] :
% 0.34/0.53 ( ( ord_le2510731241096832064er_nat @ X3 @ Y2 )
% 0.34/0.53 => ( ord_less_eq_rat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
% 0.34/0.53 => ( ord_less_eq_rat @ ( F @ A2 ) @ C ) ) ) ) ).
% 0.34/0.53
% 0.34/0.53 % ord_le_eq_subst
% 0.34/0.53 thf(fact_1008_ord__eq__le__subst,axiom,
% 0.34/0.53 ! [A2: rat,F: rat > rat,B: rat,C: rat] :
% 0.34/0.53 ( ( A2
% 0.34/0.53 = ( F @ B ) )
% 0.34/0.53 => ( ( ord_less_eq_rat @ B @ C )
% 0.34/0.53 => ( ! [X3: rat,Y2: rat] :
% 0.34/0.53 ( ( ord_less_eq_rat @ X3 @ Y2 )
% 0.34/0.53 => ( ord_less_eq_rat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
% 0.34/0.53 => ( ord_less_eq_rat @ A2 @ ( F @ C ) ) ) ) ) ).
% 0.34/0.53
% 0.34/0.53 % ord_eq_le_subst
% 0.34/0.53 thf(fact_1009_ord__eq__le__subst,axiom,
% 0.34/0.53 ! [A2: nat,F: rat > nat,B: rat,C: rat] :
% 0.34/0.53 ( ( A2
% 0.34/0.53 = ( F @ B ) )
% 0.34/0.53 => ( ( ord_less_eq_rat @ B @ C )
% 0.34/0.53 => ( ! [X3: rat,Y2: rat] :
% 0.34/0.53 ( ( ord_less_eq_rat @ X3 @ Y2 )
% 0.34/0.53 => ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
% 0.34/0.53 => ( ord_less_eq_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% 0.34/0.53
% 0.34/0.53 % ord_eq_le_subst
% 0.34/0.53 thf(fact_1010_ord__eq__le__subst,axiom,
% 0.34/0.53 ! [A2: int,F: rat > int,B: rat,C: rat] :
% 0.34/0.53 ( ( A2
% 0.34/0.53 = ( F @ B ) )
% 0.34/0.53 => ( ( ord_less_eq_rat @ B @ C )
% 0.34/0.53 => ( ! [X3: rat,Y2: rat] :
% 0.34/0.53 ( ( ord_less_eq_rat @ X3 @ Y2 )
% 0.34/0.53 => ( ord_less_eq_int @ ( F @ X3 ) @ ( F @ Y2 ) ) )
% 0.34/0.53 => ( ord_less_eq_int @ A2 @ ( F @ C ) ) ) ) ) ).
% 0.34/0.53
% 0.34/0.53 % ord_eq_le_subst
% 0.34/0.53 thf(fact_1011_ord__eq__le__subst,axiom,
% 0.34/0.53 ! [A2: rat,F: nat > rat,B: nat,C: nat] :
% 0.34/0.53 ( ( A2
% 0.34/0.53 = ( F @ B ) )
% 0.34/0.53 => ( ( ord_less_eq_nat @ B @ C )
% 0.34/0.53 => ( ! [X3: nat,Y2: nat] :
% 0.34/0.53 ( ( ord_less_eq_nat @ X3 @ Y2 )
% 0.34/0.53 => ( ord_less_eq_rat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
% 0.34/0.53 => ( ord_less_eq_rat @ A2 @ ( F @ C ) ) ) ) ) ).
% 0.34/0.53
% 0.34/0.53 % ord_eq_le_subst
% 0.34/0.53 thf(fact_1012_ord__eq__le__subst,axiom,
% 0.34/0.53 ! [A2: nat,F: nat > nat,B: nat,C: nat] :
% 0.34/0.53 ( ( A2
% 0.34/0.53 = ( F @ B ) )
% 0.34/0.53 => ( ( ord_less_eq_nat @ B @ C )
% 0.34/0.53 => ( ! [X3: nat,Y2: nat] :
% 0.34/0.53 ( ( ord_less_eq_nat @ X3 @ Y2 )
% 0.34/0.53 => ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
% 0.34/0.53 => ( ord_less_eq_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% 0.34/0.53
% 0.34/0.53 % ord_eq_le_subst
% 0.34/0.53 thf(fact_1013_ord__eq__le__subst,axiom,
% 0.34/0.53 ! [A2: int,F: nat > int,B: nat,C: nat] :
% 0.34/0.53 ( ( A2
% 0.34/0.53 = ( F @ B ) )
% 0.34/0.53 => ( ( ord_less_eq_nat @ B @ C )
% 0.34/0.53 => ( ! [X3: nat,Y2: nat] :
% 0.34/0.53 ( ( ord_less_eq_nat @ X3 @ Y2 )
% 0.34/0.53 => ( ord_less_eq_int @ ( F @ X3 ) @ ( F @ Y2 ) ) )
% 0.34/0.53 => ( ord_less_eq_int @ A2 @ ( F @ C ) ) ) ) ) ).
% 0.34/0.53
% 0.34/0.53 % ord_eq_le_subst
% 0.34/0.53 thf(fact_1014_ord__eq__le__subst,axiom,
% 0.34/0.53 ! [A2: rat,F: int > rat,B: int,C: int] :
% 0.34/0.53 ( ( A2
% 0.34/0.53 = ( F @ B ) )
% 0.34/0.53 => ( ( ord_less_eq_int @ B @ C )
% 0.34/0.53 => ( ! [X3: int,Y2: int] :
% 0.34/0.53 ( ( ord_less_eq_int @ X3 @ Y2 )
% 0.34/0.53 => ( ord_less_eq_rat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
% 0.34/0.53 => ( ord_less_eq_rat @ A2 @ ( F @ C ) ) ) ) ) ).
% 0.34/0.53
% 0.34/0.53 % ord_eq_le_subst
% 0.34/0.53 thf(fact_1015_ord__eq__le__subst,axiom,
% 0.34/0.53 ! [A2: nat,F: int > nat,B: int,C: int] :
% 0.34/0.53 ( ( A2
% 0.34/0.53 = ( F @ B ) )
% 0.34/0.53 => ( ( ord_less_eq_int @ B @ C )
% 0.34/0.53 => ( ! [X3: int,Y2: int] :
% 0.34/0.53 ( ( ord_less_eq_int @ X3 @ Y2 )
% 0.34/0.53 => ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
% 0.34/0.53 => ( ord_less_eq_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% 0.34/0.53
% 0.34/0.53 % ord_eq_le_subst
% 0.34/0.53 thf(fact_1016_ord__eq__le__subst,axiom,
% 0.34/0.53 ! [A2: int,F: int > int,B: int,C: int] :
% 0.34/0.53 ( ( A2
% 0.34/0.53 = ( F @ B ) )
% 0.34/0.53 => ( ( ord_less_eq_int @ B @ C )
% 0.34/0.53 => ( ! [X3: int,Y2: int] :
% 0.34/0.53 ( ( ord_less_eq_int @ X3 @ Y2 )
% 0.34/0.53 => ( ord_less_eq_int @ ( F @ X3 ) @ ( F @ Y2 ) ) )
% 0.34/0.53 => ( ord_less_eq_int @ A2 @ ( F @ C ) ) ) ) ) ).
% 0.34/0.53
% 0.34/0.53 % ord_eq_le_subst
% 0.34/0.53 thf(fact_1017_ord__eq__le__subst,axiom,
% 0.34/0.53 ! [A2: rat,F: filter_nat > rat,B: filter_nat,C: filter_nat] :
% 0.34/0.53 ( ( A2
% 0.34/0.53 = ( F @ B ) )
% 0.34/0.53 => ( ( ord_le2510731241096832064er_nat @ B @ C )
% 0.34/0.53 => ( ! [X3: filter_nat,Y2: filter_nat] :
% 0.34/0.53 ( ( ord_le2510731241096832064er_nat @ X3 @ Y2 )
% 0.34/0.53 => ( ord_less_eq_rat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
% 0.34/0.53 => ( ord_less_eq_rat @ A2 @ ( F @ C ) ) ) ) ) ).
% 0.34/0.53
% 0.34/0.53 % ord_eq_le_subst
% 0.34/0.53 thf(fact_1018_linorder__linear,axiom,
% 0.34/0.53 ! [X: rat,Y: rat] :
% 0.34/0.53 ( ( ord_less_eq_rat @ X @ Y )
% 0.34/0.53 | ( ord_less_eq_rat @ Y @ X ) ) ).
% 0.34/0.53
% 0.34/0.53 % linorder_linear
% 0.34/0.53 thf(fact_1019_linorder__linear,axiom,
% 0.34/0.53 ! [X: nat,Y: nat] :
% 0.34/0.53 ( ( ord_less_eq_nat @ X @ Y )
% 0.34/0.53 | ( ord_less_eq_nat @ Y @ X ) ) ).
% 0.34/0.53
% 0.34/0.53 % linorder_linear
% 0.34/0.53 thf(fact_1020_linorder__linear,axiom,
% 0.34/0.53 ! [X: int,Y: int] :
% 0.34/0.53 ( ( ord_less_eq_int @ X @ Y )
% 0.34/0.53 | ( ord_less_eq_int @ Y @ X ) ) ).
% 0.34/0.53
% 0.34/0.53 % linorder_linear
% 0.34/0.53 thf(fact_1021_subset__Collect__conv,axiom,
% 0.34/0.53 ! [S: set_Pr4532377907799695533_nat_o,P: ( produc3658429121746597890et_nat > $o ) > $o] :
% 0.34/0.53 ( ( ord_le2965882846123202637_nat_o @ S @ ( collec939566748876313656_nat_o @ P ) )
% 0.34/0.53 = ( ! [X2: produc3658429121746597890et_nat > $o] :
% 0.34/0.53 ( ( member6576561426505652726_nat_o @ X2 @ S )
% 0.34/0.53 => ( P @ X2 ) ) ) ) ).
% 0.34/0.53
% 0.34/0.53 % subset_Collect_conv
% 0.34/0.53 thf(fact_1022_subset__Collect__conv,axiom,
% 0.34/0.53 ! [S: set_int,P: int > $o] :
% 0.34/0.53 ( ( ord_less_eq_set_int @ S @ ( collect_int @ P ) )
% 0.34/0.53 = ( ! [X2: int] :
% 0.34/0.53 ( ( member_int @ X2 @ S )
% 0.34/0.53 => ( P @ X2 ) ) ) ) ).
% 0.34/0.53
% 0.34/0.53 % subset_Collect_conv
% 0.34/0.53 thf(fact_1023_subset__Collect__conv,axiom,
% 0.34/0.53 ! [S: set_o,P: $o > $o] :
% 0.34/0.53 ( ( ord_less_eq_set_o @ S @ ( collect_o @ P ) )
% 0.34/0.53 = ( ! [X2: $o] :
% 0.34/0.53 ( ( member_o @ X2 @ S )
% 0.34/0.53 => ( P @ X2 ) ) ) ) ).
% 0.34/0.53
% 0.34/0.53 % subset_Collect_conv
% 0.34/0.53 thf(fact_1024_subset__Collect__conv,axiom,
% 0.34/0.53 ! [S: set_Product_unit,P: product_unit > $o] :
% 0.34/0.53 ( ( ord_le3507040750410214029t_unit @ S @ ( collect_Product_unit @ P ) )
% 0.34/0.53 = ( ! [X2: product_unit] :
% 0.34/0.53 ( ( member_Product_unit @ X2 @ S )
% 0.34/0.53 => ( P @ X2 ) ) ) ) ).
% 0.34/0.53
% 0.34/0.53 % subset_Collect_conv
% 0.34/0.53 thf(fact_1025_subset__Collect__conv,axiom,
% 0.34/0.53 ! [S: set_nat,P: nat > $o] :
% 0.34/0.53 ( ( ord_less_eq_set_nat @ S @ ( collect_nat @ P ) )
% 0.34/0.53 = ( ! [X2: nat] :
% 0.34/0.53 ( ( member_nat @ X2 @ S )
% 0.34/0.53 => ( P @ X2 ) ) ) ) ).
% 0.34/0.53
% 0.34/0.53 % subset_Collect_conv
% 0.34/0.53 thf(fact_1026_order__eq__refl,axiom,
% 0.34/0.53 ! [X: filter_nat,Y: filter_nat] :
% 0.34/0.53 ( ( X = Y )
% 0.34/0.53 => ( ord_le2510731241096832064er_nat @ X @ Y ) ) ).
% 0.34/0.53
% 0.34/0.53 % order_eq_refl
% 0.34/0.53 thf(fact_1027_order__eq__refl,axiom,
% 0.34/0.53 ! [X: set_nat,Y: set_nat] :
% 0.34/0.53 ( ( X = Y )
% 0.34/0.53 => ( ord_less_eq_set_nat @ X @ Y ) ) ).
% 0.34/0.53
% 0.34/0.53 % order_eq_refl
% 0.34/0.53 thf(fact_1028_order__eq__refl,axiom,
% 0.34/0.53 ! [X: rat,Y: rat] :
% 0.34/0.53 ( ( X = Y )
% 0.34/0.53 => ( ord_less_eq_rat @ X @ Y ) ) ).
% 0.34/0.53
% 0.34/0.53 % order_eq_refl
% 0.34/0.53 thf(fact_1029_order__eq__refl,axiom,
% 0.34/0.53 ! [X: nat,Y: nat] :
% 0.34/0.53 ( ( X = Y )
% 0.34/0.53 => ( ord_less_eq_nat @ X @ Y ) ) ).
% 0.34/0.53
% 0.34/0.53 % order_eq_refl
% 0.34/0.53 thf(fact_1030_order__eq__refl,axiom,
% 0.34/0.53 ! [X: int,Y: int] :
% 0.34/0.53 ( ( X = Y )
% 0.34/0.53 => ( ord_less_eq_int @ X @ Y ) ) ).
% 0.34/0.53
% 0.34/0.53 % order_eq_refl
% 0.34/0.53 thf(fact_1031_ord__eq__le__eq__trans,axiom,
% 0.34/0.53 ! [A2: filter_nat,B: filter_nat,C: filter_nat,D2: filter_nat] :
% 0.34/0.53 ( ( A2 = B )
% 0.34/0.53 => ( ( ord_le2510731241096832064er_nat @ B @ C )
% 0.34/0.53 => ( ( C = D2 )
% 0.34/0.53 => ( ord_le2510731241096832064er_nat @ A2 @ D2 ) ) ) ) ).
% 0.34/0.53
% 0.34/0.53 % ord_eq_le_eq_trans
% 0.34/0.53 thf(fact_1032_ord__eq__le__eq__trans,axiom,
% 0.34/0.53 ! [A2: set_nat,B: set_nat,C: set_nat,D2: set_nat] :
% 0.34/0.53 ( ( A2 = B )
% 0.34/0.53 => ( ( ord_less_eq_set_nat @ B @ C )
% 0.34/0.53 => ( ( C = D2 )
% 0.34/0.53 => ( ord_less_eq_set_nat @ A2 @ D2 ) ) ) ) ).
% 0.34/0.53
% 0.34/0.53 % ord_eq_le_eq_trans
% 0.34/0.53 thf(fact_1033_ord__eq__le__eq__trans,axiom,
% 0.34/0.53 ! [A2: rat,B: rat,C: rat,D2: rat] :
% 0.34/0.53 ( ( A2 = B )
% 0.34/0.53 => ( ( ord_less_eq_rat @ B @ C )
% 0.34/0.53 => ( ( C = D2 )
% 0.34/0.53 => ( ord_less_eq_rat @ A2 @ D2 ) ) ) ) ).
% 0.34/0.53
% 0.34/0.53 % ord_eq_le_eq_trans
% 0.34/0.53 thf(fact_1034_ord__eq__le__eq__trans,axiom,
% 0.34/0.53 ! [A2: nat,B: nat,C: nat,D2: nat] :
% 0.34/0.53 ( ( A2 = B )
% 0.34/0.53 => ( ( ord_less_eq_nat @ B @ C )
% 0.34/0.53 => ( ( C = D2 )
% 0.34/0.53 => ( ord_less_eq_nat @ A2 @ D2 ) ) ) ) ).
% 0.34/0.53
% 0.34/0.53 % ord_eq_le_eq_trans
% 0.34/0.53 thf(fact_1035_ord__eq__le__eq__trans,axiom,
% 0.34/0.53 ! [A2: int,B: int,C: int,D2: int] :
% 0.34/0.53 ( ( A2 = B )
% 0.34/0.53 => ( ( ord_less_eq_int @ B @ C )
% 0.34/0.53 => ( ( C = D2 )
% 0.34/0.53 => ( ord_less_eq_int @ A2 @ D2 ) ) ) ) ).
% 0.34/0.53
% 0.34/0.53 % ord_eq_le_eq_trans
% 0.34/0.53 thf(fact_1036_order__subst2,axiom,
% 0.34/0.53 ! [A2: rat,B: rat,F: rat > rat,C: rat] :
% 0.34/0.53 ( ( ord_less_eq_rat @ A2 @ B )
% 0.34/0.53 => ( ( ord_less_eq_rat @ ( F @ B ) @ C )
% 0.34/0.53 => ( ! [X3: rat,Y2: rat] :
% 0.34/0.53 ( ( ord_less_eq_rat @ X3 @ Y2 )
% 0.34/0.53 => ( ord_less_eq_rat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
% 0.34/0.53 => ( ord_less_eq_rat @ ( F @ A2 ) @ C ) ) ) ) ).
% 0.34/0.53
% 0.34/0.53 % order_subst2
% 0.34/0.53 thf(fact_1037_order__subst2,axiom,
% 0.34/0.53 ! [A2: rat,B: rat,F: rat > nat,C: nat] :
% 0.34/0.53 ( ( ord_less_eq_rat @ A2 @ B )
% 0.34/0.53 => ( ( ord_less_eq_nat @ ( F @ B ) @ C )
% 0.34/0.53 => ( ! [X3: rat,Y2: rat] :
% 0.34/0.53 ( ( ord_less_eq_rat @ X3 @ Y2 )
% 0.34/0.53 => ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
% 0.34/0.53 => ( ord_less_eq_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% 0.34/0.53
% 0.34/0.53 % order_subst2
% 0.34/0.53 thf(fact_1038_order__subst2,axiom,
% 0.34/0.53 ! [A2: rat,B: rat,F: rat > int,C: int] :
% 0.34/0.53 ( ( ord_less_eq_rat @ A2 @ B )
% 0.34/0.53 => ( ( ord_less_eq_int @ ( F @ B ) @ C )
% 0.34/0.53 => ( ! [X3: rat,Y2: rat] :
% 0.34/0.53 ( ( ord_less_eq_rat @ X3 @ Y2 )
% 0.34/0.53 => ( ord_less_eq_int @ ( F @ X3 ) @ ( F @ Y2 ) ) )
% 0.34/0.53 => ( ord_less_eq_int @ ( F @ A2 ) @ C ) ) ) ) ).
% 0.34/0.53
% 0.34/0.53 % order_subst2
% 0.34/0.53 thf(fact_1039_order__subst2,axiom,
% 0.34/0.53 ! [A2: nat,B: nat,F: nat > rat,C: rat] :
% 0.34/0.53 ( ( ord_less_eq_nat @ A2 @ B )
% 0.34/0.53 => ( ( ord_less_eq_rat @ ( F @ B ) @ C )
% 0.34/0.53 => ( ! [X3: nat,Y2: nat] :
% 0.34/0.53 ( ( ord_less_eq_nat @ X3 @ Y2 )
% 0.34/0.53 => ( ord_less_eq_rat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
% 0.34/0.53 => ( ord_less_eq_rat @ ( F @ A2 ) @ C ) ) ) ) ).
% 0.34/0.53
% 0.34/0.53 % order_subst2
% 0.34/0.53 thf(fact_1040_order__subst2,axiom,
% 0.34/0.53 ! [A2: nat,B: nat,F: nat > nat,C: nat] :
% 0.34/0.53 ( ( ord_less_eq_nat @ A2 @ B )
% 0.34/0.53 => ( ( ord_less_eq_nat @ ( F @ B ) @ C )
% 0.34/0.53 => ( ! [X3: nat,Y2: nat] :
% 0.34/0.53 ( ( ord_less_eq_nat @ X3 @ Y2 )
% 0.34/0.53 => ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
% 0.34/0.53 => ( ord_less_eq_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% 0.34/0.53
% 0.34/0.53 % order_subst2
% 0.34/0.53 thf(fact_1041_order__subst2,axiom,
% 0.34/0.53 ! [A2: nat,B: nat,F: nat > int,C: int] :
% 0.34/0.53 ( ( ord_less_eq_nat @ A2 @ B )
% 0.34/0.53 => ( ( ord_less_eq_int @ ( F @ B ) @ C )
% 0.34/0.53 => ( ! [X3: nat,Y2: nat] :
% 0.34/0.53 ( ( ord_less_eq_nat @ X3 @ Y2 )
% 0.34/0.53 => ( ord_less_eq_int @ ( F @ X3 ) @ ( F @ Y2 ) ) )
% 0.34/0.53 => ( ord_less_eq_int @ ( F @ A2 ) @ C ) ) ) ) ).
% 0.34/0.53
% 0.34/0.53 % order_subst2
% 0.34/0.53 thf(fact_1042_order__subst2,axiom,
% 0.34/0.53 ! [A2: int,B: int,F: int > rat,C: rat] :
% 0.34/0.53 ( ( ord_less_eq_int @ A2 @ B )
% 0.34/0.53 => ( ( ord_less_eq_rat @ ( F @ B ) @ C )
% 0.34/0.53 => ( ! [X3: int,Y2: int] :
% 0.34/0.53 ( ( ord_less_eq_int @ X3 @ Y2 )
% 0.34/0.53 => ( ord_less_eq_rat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
% 0.34/0.53 => ( ord_less_eq_rat @ ( F @ A2 ) @ C ) ) ) ) ).
% 0.34/0.53
% 0.34/0.53 % order_subst2
% 0.34/0.53 thf(fact_1043_order__subst2,axiom,
% 0.34/0.53 ! [A2: int,B: int,F: int > nat,C: nat] :
% 0.34/0.53 ( ( ord_less_eq_int @ A2 @ B )
% 0.34/0.53 => ( ( ord_less_eq_nat @ ( F @ B ) @ C )
% 0.34/0.53 => ( ! [X3: int,Y2: int] :
% 0.34/0.53 ( ( ord_less_eq_int @ X3 @ Y2 )
% 0.34/0.53 => ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
% 0.34/0.53 => ( ord_less_eq_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% 0.34/0.53
% 0.34/0.53 % order_subst2
% 0.34/0.53 thf(fact_1044_order__subst2,axiom,
% 0.34/0.53 ! [A2: int,B: int,F: int > int,C: int] :
% 0.34/0.53 ( ( ord_less_eq_int @ A2 @ B )
% 0.34/0.53 => ( ( ord_less_eq_int @ ( F @ B ) @ C )
% 0.34/0.53 => ( ! [X3: int,Y2: int] :
% 0.34/0.53 ( ( ord_less_eq_int @ X3 @ Y2 )
% 0.34/0.53 => ( ord_less_eq_int @ ( F @ X3 ) @ ( F @ Y2 ) ) )
% 0.34/0.53 => ( ord_less_eq_int @ ( F @ A2 ) @ C ) ) ) ) ).
% 0.34/0.53
% 0.34/0.53 % order_subst2
% 0.34/0.53 thf(fact_1045_order__subst2,axiom,
% 0.34/0.53 ! [A2: filter_nat,B: filter_nat,F: filter_nat > rat,C: rat] :
% 0.34/0.53 ( ( ord_le2510731241096832064er_nat @ A2 @ B )
% 0.34/0.53 => ( ( ord_less_eq_rat @ ( F @ B ) @ C )
% 0.34/0.53 => ( ! [X3: filter_nat,Y2: filter_nat] :
% 0.34/0.53 ( ( ord_le2510731241096832064er_nat @ X3 @ Y2 )
% 0.34/0.53 => ( ord_less_eq_rat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
% 0.34/0.53 => ( ord_less_eq_rat @ ( F @ A2 ) @ C ) ) ) ) ).
% 0.34/0.53
% 0.34/0.53 % order_subst2
% 0.34/0.53 thf(fact_1046_order__subst1,axiom,
% 0.34/0.53 ! [A2: rat,F: rat > rat,B: rat,C: rat] :
% 0.34/0.53 ( ( ord_less_eq_rat @ A2 @ ( F @ B ) )
% 0.34/0.53 => ( ( ord_less_eq_rat @ B @ C )
% 0.34/0.53 => ( ! [X3: rat,Y2: rat] :
% 0.34/0.53 ( ( ord_less_eq_rat @ X3 @ Y2 )
% 0.34/0.53 => ( ord_less_eq_rat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
% 0.34/0.53 => ( ord_less_eq_rat @ A2 @ ( F @ C ) ) ) ) ) ).
% 0.34/0.53
% 0.34/0.53 % order_subst1
% 0.34/0.53 thf(fact_1047_order__subst1,axiom,
% 0.34/0.53 ! [A2: rat,F: nat > rat,B: nat,C: nat] :
% 0.34/0.53 ( ( ord_less_eq_rat @ A2 @ ( F @ B ) )
% 0.34/0.53 => ( ( ord_less_eq_nat @ B @ C )
% 0.34/0.53 => ( ! [X3: nat,Y2: nat] :
% 0.34/0.53 ( ( ord_less_eq_nat @ X3 @ Y2 )
% 0.34/0.53 => ( ord_less_eq_rat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
% 0.34/0.53 => ( ord_less_eq_rat @ A2 @ ( F @ C ) ) ) ) ) ).
% 0.34/0.53
% 0.34/0.53 % order_subst1
% 0.34/0.53 thf(fact_1048_order__subst1,axiom,
% 0.34/0.53 ! [A2: rat,F: int > rat,B: int,C: int] :
% 0.34/0.53 ( ( ord_less_eq_rat @ A2 @ ( F @ B ) )
% 0.34/0.53 => ( ( ord_less_eq_int @ B @ C )
% 0.34/0.53 => ( ! [X3: int,Y2: int] :
% 0.34/0.53 ( ( ord_less_eq_int @ X3 @ Y2 )
% 0.34/0.53 => ( ord_less_eq_rat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
% 0.34/0.53 => ( ord_less_eq_rat @ A2 @ ( F @ C ) ) ) ) ) ).
% 0.34/0.53
% 0.34/0.53 % order_subst1
% 0.34/0.53 thf(fact_1049_order__subst1,axiom,
% 0.34/0.53 ! [A2: nat,F: rat > nat,B: rat,C: rat] :
% 0.34/0.53 ( ( ord_less_eq_nat @ A2 @ ( F @ B ) )
% 0.34/0.53 => ( ( ord_less_eq_rat @ B @ C )
% 0.34/0.53 => ( ! [X3: rat,Y2: rat] :
% 0.34/0.53 ( ( ord_less_eq_rat @ X3 @ Y2 )
% 0.34/0.53 => ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
% 0.34/0.53 => ( ord_less_eq_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% 0.34/0.53
% 0.34/0.53 % order_subst1
% 0.34/0.53 thf(fact_1050_order__subst1,axiom,
% 0.34/0.53 ! [A2: nat,F: nat > nat,B: nat,C: nat] :
% 0.34/0.53 ( ( ord_less_eq_nat @ A2 @ ( F @ B ) )
% 0.34/0.53 => ( ( ord_less_eq_nat @ B @ C )
% 0.34/0.53 => ( ! [X3: nat,Y2: nat] :
% 0.34/0.53 ( ( ord_less_eq_nat @ X3 @ Y2 )
% 0.34/0.53 => ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
% 0.34/0.53 => ( ord_less_eq_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% 0.34/0.53
% 0.34/0.53 % order_subst1
% 0.34/0.53 thf(fact_1051_order__subst1,axiom,
% 0.34/0.53 ! [A2: nat,F: int > nat,B: int,C: int] :
% 0.34/0.53 ( ( ord_less_eq_nat @ A2 @ ( F @ B ) )
% 0.34/0.53 => ( ( ord_less_eq_int @ B @ C )
% 0.34/0.53 => ( ! [X3: int,Y2: int] :
% 0.34/0.53 ( ( ord_less_eq_int @ X3 @ Y2 )
% 0.34/0.53 => ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
% 0.34/0.53 => ( ord_less_eq_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% 0.34/0.53
% 0.34/0.53 % order_subst1
% 0.34/0.53 thf(fact_1052_order__subst1,axiom,
% 0.34/0.53 ! [A2: int,F: rat > int,B: rat,C: rat] :
% 0.34/0.53 ( ( ord_less_eq_int @ A2 @ ( F @ B ) )
% 0.34/0.53 => ( ( ord_less_eq_rat @ B @ C )
% 0.34/0.53 => ( ! [X3: rat,Y2: rat] :
% 0.34/0.53 ( ( ord_less_eq_rat @ X3 @ Y2 )
% 0.34/0.53 => ( ord_less_eq_int @ ( F @ X3 ) @ ( F @ Y2 ) ) )
% 0.34/0.53 => ( ord_less_eq_int @ A2 @ ( F @ C ) ) ) ) ) ).
% 0.34/0.53
% 0.34/0.53 % order_subst1
% 0.34/0.53 thf(fact_1053_order__subst1,axiom,
% 0.34/0.53 ! [A2: int,F: nat > int,B: nat,C: nat] :
% 0.34/0.53 ( ( ord_less_eq_int @ A2 @ ( F @ B ) )
% 0.34/0.53 => ( ( ord_less_eq_nat @ B @ C )
% 0.34/0.53 => ( ! [X3: nat,Y2: nat] :
% 0.34/0.53 ( ( ord_less_eq_nat @ X3 @ Y2 )
% 0.34/0.53 => ( ord_less_eq_int @ ( F @ X3 ) @ ( F @ Y2 ) ) )
% 0.34/0.53 => ( ord_less_eq_int @ A2 @ ( F @ C ) ) ) ) ) ).
% 0.34/0.53
% 0.34/0.53 % order_subst1
% 0.34/0.53 thf(fact_1054_order__subst1,axiom,
% 0.34/0.53 ! [A2: int,F: int > int,B: int,C: int] :
% 0.34/0.53 ( ( ord_less_eq_int @ A2 @ ( F @ B ) )
% 0.34/0.53 => ( ( ord_less_eq_int @ B @ C )
% 0.34/0.53 => ( ! [X3: int,Y2: int] :
% 0.34/0.53 ( ( ord_less_eq_int @ X3 @ Y2 )
% 0.34/0.53 => ( ord_less_eq_int @ ( F @ X3 ) @ ( F @ Y2 ) ) )
% 0.34/0.53 => ( ord_less_eq_int @ A2 @ ( F @ C ) ) ) ) ) ).
% 0.34/0.53
% 0.34/0.53 % order_subst1
% 0.34/0.53 thf(fact_1055_order__subst1,axiom,
% 0.34/0.53 ! [A2: filter_nat,F: rat > filter_nat,B: rat,C: rat] :
% 0.34/0.53 ( ( ord_le2510731241096832064er_nat @ A2 @ ( F @ B ) )
% 0.34/0.53 => ( ( ord_less_eq_rat @ B @ C )
% 0.34/0.53 => ( ! [X3: rat,Y2: rat] :
% 0.34/0.53 ( ( ord_less_eq_rat @ X3 @ Y2 )
% 0.34/0.53 => ( ord_le2510731241096832064er_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
% 0.34/0.53 => ( ord_le2510731241096832064er_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% 0.34/0.53
% 0.34/0.53 % order_subst1
% 0.34/0.53 thf(fact_1056_Orderings_Oorder__eq__iff,axiom,
% 0.34/0.53 ( ( ^ [Y4: filter_nat,Z4: filter_nat] : ( Y4 = Z4 ) )
% 0.34/0.53 = ( ^ [A6: filter_nat,B6: filter_nat] :
% 0.34/0.53 ( ( ord_le2510731241096832064er_nat @ A6 @ B6 )
% 0.34/0.53 & ( ord_le2510731241096832064er_nat @ B6 @ A6 ) ) ) ) ).
% 0.34/0.53
% 0.34/0.53 % Orderings.order_eq_iff
% 0.34/0.53 thf(fact_1057_Orderings_Oorder__eq__iff,axiom,
% 0.34/0.53 ( ( ^ [Y4: set_nat,Z4: set_nat] : ( Y4 = Z4 ) )
% 0.34/0.53 = ( ^ [A6: set_nat,B6: set_nat] :
% 0.34/0.53 ( ( ord_less_eq_set_nat @ A6 @ B6 )
% 0.34/0.53 & ( ord_less_eq_set_nat @ B6 @ A6 ) ) ) ) ).
% 0.34/0.53
% 0.34/0.53 % Orderings.order_eq_iff
% 0.34/0.53 thf(fact_1058_Orderings_Oorder__eq__iff,axiom,
% 0.34/0.53 ( ( ^ [Y4: rat,Z4: rat] : ( Y4 = Z4 ) )
% 0.34/0.53 = ( ^ [A6: rat,B6: rat] :
% 0.34/0.53 ( ( ord_less_eq_rat @ A6 @ B6 )
% 0.34/0.53 & ( ord_less_eq_rat @ B6 @ A6 ) ) ) ) ).
% 0.34/0.53
% 0.34/0.53 % Orderings.order_eq_iff
% 0.34/0.53 thf(fact_1059_Orderings_Oorder__eq__iff,axiom,
% 0.34/0.53 ( ( ^ [Y4: nat,Z4: nat] : ( Y4 = Z4 ) )
% 0.34/0.53 = ( ^ [A6: nat,B6: nat] :
% 0.34/0.53 ( ( ord_less_eq_nat @ A6 @ B6 )
% 0.34/0.53 & ( ord_less_eq_nat @ B6 @ A6 ) ) ) ) ).
% 0.34/0.53
% 0.34/0.53 % Orderings.order_eq_iff
% 0.34/0.53 thf(fact_1060_Orderings_Oorder__eq__iff,axiom,
% 0.34/0.53 ( ( ^ [Y4: int,Z4: int] : ( Y4 = Z4 ) )
% 0.34/0.53 = ( ^ [A6: int,B6: int] :
% 0.34/0.53 ( ( ord_less_eq_int @ A6 @ B6 )
% 0.34/0.53 & ( ord_less_eq_int @ B6 @ A6 ) ) ) ) ).
% 0.34/0.53
% 0.34/0.53 % Orderings.order_eq_iff
% 0.34/0.53 thf(fact_1061_inter__eq__subsetI,axiom,
% 0.34/0.53 ! [S: set_nat,S4: set_nat,A: set_nat,B4: set_nat] :
% 0.34/0.53 ( ( ord_less_eq_set_nat @ S @ S4 )
% 0.34/0.53 => ( ( ( inf_inf_set_nat @ A @ S4 )
% 0.34/0.53 = ( inf_inf_set_nat @ B4 @ S4 ) )
% 0.34/0.53 => ( ( inf_inf_set_nat @ A @ S )
% 0.34/0.53 = ( inf_inf_set_nat @ B4 @ S ) ) ) ) ).
% 0.34/0.53
% 0.34/0.53 % inter_eq_subsetI
% 0.34/0.53 thf(fact_1062_Int__left__commute,axiom,
% 0.34/0.53 ! [A: set_nat,B4: set_nat,C5: set_nat] :
% 0.34/0.53 ( ( inf_inf_set_nat @ A @ ( inf_inf_set_nat @ B4 @ C5 ) )
% 0.34/0.53 = ( inf_inf_set_nat @ B4 @ ( inf_inf_set_nat @ A @ C5 ) ) ) ).
% 0.34/0.53
% 0.34/0.53 % Int_left_commute
% 0.34/0.53 thf(fact_1063_Int__Collect__mono,axiom,
% 0.34/0.53 ! [A: set_Pr4532377907799695533_nat_o,B4: set_Pr4532377907799695533_nat_o,P: ( produc3658429121746597890et_nat > $o ) > $o,Q2: ( produc3658429121746597890et_nat > $o ) > $o] :
% 0.34/0.53 ( ( ord_le2965882846123202637_nat_o @ A @ B4 )
% 0.34/0.53 => ( ! [X3: produc3658429121746597890et_nat > $o] :
% 0.34/0.53 ( ( member6576561426505652726_nat_o @ X3 @ A )
% 0.34/0.53 => ( ( P @ X3 )
% 0.34/0.53 => ( Q2 @ X3 ) ) )
% 0.34/0.53 => ( ord_le2965882846123202637_nat_o @ ( inf_in1906310914598751387_nat_o @ A @ ( collec939566748876313656_nat_o @ P ) ) @ ( inf_in1906310914598751387_nat_o @ B4 @ ( collec939566748876313656_nat_o @ Q2 ) ) ) ) ) ).
% 0.34/0.53
% 0.34/0.53 % Int_Collect_mono
% 0.34/0.53 thf(fact_1064_Int__Collect__mono,axiom,
% 0.34/0.53 ! [A: set_int,B4: set_int,P: int > $o,Q2: int > $o] :
% 0.34/0.53 ( ( ord_less_eq_set_int @ A @ B4 )
% 0.34/0.53 => ( ! [X3: int] :
% 0.34/0.53 ( ( member_int @ X3 @ A )
% 0.34/0.53 => ( ( P @ X3 )
% 0.34/0.53 => ( Q2 @ X3 ) ) )
% 0.34/0.53 => ( ord_less_eq_set_int @ ( inf_inf_set_int @ A @ ( collect_int @ P ) ) @ ( inf_inf_set_int @ B4 @ ( collect_int @ Q2 ) ) ) ) ) ).
% 0.34/0.53
% 0.34/0.53 % Int_Collect_mono
% 0.34/0.53 thf(fact_1065_Int__Collect__mono,axiom,
% 0.34/0.53 ! [A: set_o,B4: set_o,P: $o > $o,Q2: $o > $o] :
% 0.34/0.53 ( ( ord_less_eq_set_o @ A @ B4 )
% 0.34/0.53 => ( ! [X3: $o] :
% 0.34/0.53 ( ( member_o @ X3 @ A )
% 0.34/0.53 => ( ( P @ X3 )
% 0.34/0.53 => ( Q2 @ X3 ) ) )
% 0.34/0.53 => ( ord_less_eq_set_o @ ( inf_inf_set_o @ A @ ( collect_o @ P ) ) @ ( inf_inf_set_o @ B4 @ ( collect_o @ Q2 ) ) ) ) ) ).
% 0.34/0.53
% 0.34/0.53 % Int_Collect_mono
% 0.34/0.53 thf(fact_1066_Int__Collect__mono,axiom,
% 0.34/0.53 ! [A: set_Product_unit,B4: set_Product_unit,P: product_unit > $o,Q2: product_unit > $o] :
% 0.34/0.53 ( ( ord_le3507040750410214029t_unit @ A @ B4 )
% 0.34/0.53 => ( ! [X3: product_unit] :
% 0.34/0.53 ( ( member_Product_unit @ X3 @ A )
% 0.34/0.53 => ( ( P @ X3 )
% 0.34/0.53 => ( Q2 @ X3 ) ) )
% 0.34/0.53 => ( ord_le3507040750410214029t_unit @ ( inf_in4660618365625256667t_unit @ A @ ( collect_Product_unit @ P ) ) @ ( inf_in4660618365625256667t_unit @ B4 @ ( collect_Product_unit @ Q2 ) ) ) ) ) ).
% 0.34/0.53
% 0.34/0.53 % Int_Collect_mono
% 0.34/0.53 thf(fact_1067_Int__Collect__mono,axiom,
% 0.34/0.53 ! [A: set_nat,B4: set_nat,P: nat > $o,Q2: nat > $o] :
% 0.34/0.53 ( ( ord_less_eq_set_nat @ A @ B4 )
% 0.34/0.53 => ( ! [X3: nat] :
% 0.34/0.53 ( ( member_nat @ X3 @ A )
% 0.34/0.53 => ( ( P @ X3 )
% 0.34/0.53 => ( Q2 @ X3 ) ) )
% 0.34/0.53 => ( ord_less_eq_set_nat @ ( inf_inf_set_nat @ A @ ( collect_nat @ P ) ) @ ( inf_inf_set_nat @ B4 @ ( collect_nat @ Q2 ) ) ) ) ) ).
% 0.34/0.53
% 0.34/0.53 % Int_Collect_mono
% 0.34/0.53 thf(fact_1068_Collect__mono__iff,axiom,
% 0.34/0.53 ! [P: ( produc3658429121746597890et_nat > $o ) > $o,Q2: ( produc3658429121746597890et_nat > $o ) > $o] :
% 0.34/0.53 ( ( ord_le2965882846123202637_nat_o @ ( collec939566748876313656_nat_o @ P ) @ ( collec939566748876313656_nat_o @ Q2 ) )
% 0.34/0.53 = ( ! [X2: produc3658429121746597890et_nat > $o] :
% 0.34/0.53 ( ( P @ X2 )
% 0.34/0.53 => ( Q2 @ X2 ) ) ) ) ).
% 0.34/0.53
% 0.34/0.53 % Collect_mono_iff
% 0.34/0.53 thf(fact_1069_Collect__mono__iff,axiom,
% 0.34/0.53 ! [P: int > $o,Q2: int > $o] :
% 0.34/0.53 ( ( ord_less_eq_set_int @ ( collect_int @ P ) @ ( collect_int @ Q2 ) )
% 0.34/0.53 = ( ! [X2: int] :
% 0.34/0.53 ( ( P @ X2 )
% 0.34/0.53 => ( Q2 @ X2 ) ) ) ) ).
% 0.34/0.53
% 0.34/0.53 % Collect_mono_iff
% 0.34/0.53 thf(fact_1070_Collect__mono__iff,axiom,
% 0.34/0.53 ! [P: $o > $o,Q2: $o > $o] :
% 0.34/0.53 ( ( ord_less_eq_set_o @ ( collect_o @ P ) @ ( collect_o @ Q2 ) )
% 0.34/0.53 = ( ! [X2: $o] :
% 0.34/0.53 ( ( P @ X2 )
% 0.34/0.53 => ( Q2 @ X2 ) ) ) ) ).
% 0.34/0.53
% 0.34/0.53 % Collect_mono_iff
% 0.34/0.53 thf(fact_1071_Collect__mono__iff,axiom,
% 0.34/0.53 ! [P: product_unit > $o,Q2: product_unit > $o] :
% 0.34/0.53 ( ( ord_le3507040750410214029t_unit @ ( collect_Product_unit @ P ) @ ( collect_Product_unit @ Q2 ) )
% 0.34/0.53 = ( ! [X2: product_unit] :
% 0.34/0.53 ( ( P @ X2 )
% 0.34/0.53 => ( Q2 @ X2 ) ) ) ) ).
% 0.34/0.53
% 0.34/0.53 % Collect_mono_iff
% 0.34/0.53 thf(fact_1072_Collect__mono__iff,axiom,
% 0.34/0.53 ! [P: nat > $o,Q2: nat > $o] :
% 0.34/0.53 ( ( ord_less_eq_set_nat @ ( collect_nat @ P ) @ ( collect_nat @ Q2 ) )
% 0.34/0.53 = ( ! [X2: nat] :
% 0.34/0.53 ( ( P @ X2 )
% 0.34/0.53 => ( Q2 @ X2 ) ) ) ) ).
% 0.34/0.53
% 0.34/0.53 % Collect_mono_iff
% 0.34/0.53 thf(fact_1073_Un__left__commute,axiom,
% 0.34/0.53 ! [A: set_nat,B4: set_nat,C5: set_nat] :
% 0.34/0.53 ( ( sup_sup_set_nat @ A @ ( sup_sup_set_nat @ B4 @ C5 ) )
% 0.34/0.53 = ( sup_sup_set_nat @ B4 @ ( sup_sup_set_nat @ A @ C5 ) ) ) ).
% 0.34/0.53
% 0.34/0.53 % Un_left_commute
% 0.34/0.53 thf(fact_1074_Un__Int__distrib2,axiom,
% 0.34/0.53 ! [B4: set_nat,C5: set_nat,A: set_nat] :
% 0.34/0.53 ( ( sup_sup_set_nat @ ( inf_inf_set_nat @ B4 @ C5 ) @ A )
% 0.34/0.53 = ( inf_inf_set_nat @ ( sup_sup_set_nat @ B4 @ A ) @ ( sup_sup_set_nat @ C5 @ A ) ) ) ).
% 0.34/0.53
% 0.34/0.53 % Un_Int_distrib2
% 0.34/0.53 thf(fact_1075_Un__Int__assoc__eq,axiom,
% 0.34/0.53 ! [A: set_nat,B4: set_nat,C5: set_nat] :
% 0.34/0.53 ( ( ( sup_sup_set_nat @ ( inf_inf_set_nat @ A @ B4 ) @ C5 )
% 0.34/0.53 = ( inf_inf_set_nat @ A @ ( sup_sup_set_nat @ B4 @ C5 ) ) )
% 0.34/0.53 = ( ord_less_eq_set_nat @ C5 @ A ) ) ).
% 0.34/0.53
% 0.34/0.53 % Un_Int_assoc_eq
% 0.34/0.53 thf(fact_1076_Int__left__absorb,axiom,
% 0.34/0.53 ! [A: set_nat,B4: set_nat] :
% 0.34/0.53 ( ( inf_inf_set_nat @ A @ ( inf_inf_set_nat @ A @ B4 ) )
% 0.34/0.53 = ( inf_inf_set_nat @ A @ B4 ) ) ).
% 0.34/0.53
% 0.34/0.53 % Int_left_absorb
% 0.34/0.53 thf(fact_1077_Int__Un__distrib2,axiom,
% 0.34/0.53 ! [B4: set_nat,C5: set_nat,A: set_nat] :
% 0.34/0.53 ( ( inf_inf_set_nat @ ( sup_sup_set_nat @ B4 @ C5 ) @ A )
% 0.34/0.53 = ( sup_sup_set_nat @ ( inf_inf_set_nat @ B4 @ A ) @ ( inf_inf_set_nat @ C5 @ A ) ) ) ).
% 0.34/0.53
% 0.34/0.53 % Int_Un_distrib2
% 0.34/0.53 thf(fact_1078_Un__left__absorb,axiom,
% 0.34/0.53 ! [A: set_nat,B4: set_nat] :
% 0.34/0.53 ( ( sup_sup_set_nat @ A @ ( sup_sup_set_nat @ A @ B4 ) )
% 0.34/0.53 = ( sup_sup_set_nat @ A @ B4 ) ) ).
% 0.34/0.53
% 0.34/0.53 % Un_left_absorb
% 0.34/0.53 thf(fact_1079_Un__Int__distrib,axiom,
% 0.34/0.53 ! [A: set_nat,B4: set_nat,C5: set_nat] :
% 0.34/0.53 ( ( sup_sup_set_nat @ A @ ( inf_inf_set_nat @ B4 @ C5 ) )
% 0.34/0.53 = ( inf_inf_set_nat @ ( sup_sup_set_nat @ A @ B4 ) @ ( sup_sup_set_nat @ A @ C5 ) ) ) ).
% 0.34/0.53
% 0.34/0.53 % Un_Int_distrib
% 0.34/0.53 thf(fact_1080_Int__Un__distrib,axiom,
% 0.34/0.53 ! [A: set_nat,B4: set_nat,C5: set_nat] :
% 0.34/0.53 ( ( inf_inf_set_nat @ A @ ( sup_sup_set_nat @ B4 @ C5 ) )
% 0.34/0.53 = ( sup_sup_set_nat @ ( inf_inf_set_nat @ A @ B4 ) @ ( inf_inf_set_nat @ A @ C5 ) ) ) ).
% 0.34/0.53
% 0.34/0.53 % Int_Un_distrib
% 0.34/0.53 thf(fact_1081_set__eq__subset,axiom,
% 0.34/0.53 ( ( ^ [Y4: set_nat,Z4: set_nat] : ( Y4 = Z4 ) )
% 0.34/0.53 = ( ^ [A3: set_nat,B7: set_nat] :
% 0.34/0.53 ( ( ord_less_eq_set_nat @ A3 @ B7 )
% 0.34/0.53 & ( ord_less_eq_set_nat @ B7 @ A3 ) ) ) ) ).
% 0.34/0.53
% 0.34/0.53 % set_eq_subset
% 0.34/0.53 thf(fact_1082_antisym,axiom,
% 0.34/0.53 ! [A2: filter_nat,B: filter_nat] :
% 0.34/0.53 ( ( ord_le2510731241096832064er_nat @ A2 @ B )
% 0.34/0.53 => ( ( ord_le2510731241096832064er_nat @ B @ A2 )
% 0.34/0.53 => ( A2 = B ) ) ) ).
% 0.34/0.53
% 0.34/0.53 % antisym
% 0.34/0.53 thf(fact_1083_antisym,axiom,
% 0.34/0.53 ! [A2: set_nat,B: set_nat] :
% 0.34/0.53 ( ( ord_less_eq_set_nat @ A2 @ B )
% 0.34/0.53 => ( ( ord_less_eq_set_nat @ B @ A2 )
% 0.34/0.53 => ( A2 = B ) ) ) ).
% 0.34/0.53
% 0.34/0.53 % antisym
% 0.34/0.53 thf(fact_1084_antisym,axiom,
% 0.34/0.53 ! [A2: rat,B: rat] :
% 0.34/0.53 ( ( ord_less_eq_rat @ A2 @ B )
% 0.34/0.53 => ( ( ord_less_eq_rat @ B @ A2 )
% 0.34/0.53 => ( A2 = B ) ) ) ).
% 0.34/0.53
% 0.34/0.53 % antisym
% 0.34/0.53 thf(fact_1085_antisym,axiom,
% 0.34/0.53 ! [A2: nat,B: nat] :
% 0.34/0.53 ( ( ord_less_eq_nat @ A2 @ B )
% 0.34/0.53 => ( ( ord_less_eq_nat @ B @ A2 )
% 0.34/0.53 => ( A2 = B ) ) ) ).
% 0.34/0.53
% 0.34/0.53 % antisym
% 0.34/0.53 thf(fact_1086_antisym,axiom,
% 0.34/0.53 ! [A2: int,B: int] :
% 0.34/0.53 ( ( ord_less_eq_int @ A2 @ B )
% 0.34/0.53 => ( ( ord_less_eq_int @ B @ A2 )
% 0.34/0.53 => ( A2 = B ) ) ) ).
% 0.34/0.53
% 0.34/0.53 % antisym
% 0.34/0.53 thf(fact_1087_subset__trans,axiom,
% 0.34/0.53 ! [A: set_nat,B4: set_nat,C5: set_nat] :
% 0.34/0.53 ( ( ord_less_eq_set_nat @ A @ B4 )
% 0.34/0.53 => ( ( ord_less_eq_set_nat @ B4 @ C5 )
% 0.34/0.53 => ( ord_less_eq_set_nat @ A @ C5 ) ) ) ).
% 0.34/0.53
% 0.34/0.53 % subset_trans
% 0.34/0.53 thf(fact_1088_subset__Un__eq,axiom,
% 0.34/0.53 ( ord_less_eq_set_nat
% 0.34/0.53 = ( ^ [A3: set_nat,B7: set_nat] :
% 0.34/0.53 ( ( sup_sup_set_nat @ A3 @ B7 )
% 0.34/0.53 = B7 ) ) ) ).
% 0.34/0.53
% 0.34/0.53 % subset_Un_eq
% 0.34/0.53 thf(fact_1089_Un__Int__crazy,axiom,
% 0.34/0.53 ! [A: set_nat,B4: set_nat,C5: set_nat] :
% 0.34/0.53 ( ( sup_sup_set_nat @ ( sup_sup_set_nat @ ( inf_inf_set_nat @ A @ B4 ) @ ( inf_inf_set_nat @ B4 @ C5 ) ) @ ( inf_inf_set_nat @ C5 @ A ) )
% 0.34/0.53 = ( inf_inf_set_nat @ ( inf_inf_set_nat @ ( sup_sup_set_nat @ A @ B4 ) @ ( sup_sup_set_nat @ B4 @ C5 ) ) @ ( sup_sup_set_nat @ C5 @ A ) ) ) ).
% 0.34/0.53
% 0.34/0.53 % Un_Int_crazy
% 0.34/0.53 thf(fact_1090_Int__greatest,axiom,
% 0.34/0.53 ! [C5: set_nat,A: set_nat,B4: set_nat] :
% 0.34/0.53 ( ( ord_less_eq_set_nat @ C5 @ A )
% 0.34/0.53 => ( ( ord_less_eq_set_nat @ C5 @ B4 )
% 0.34/0.53 => ( ord_less_eq_set_nat @ C5 @ ( inf_inf_set_nat @ A @ B4 ) ) ) ) ).
% 0.34/0.53
% 0.34/0.53 % Int_greatest
% 0.34/0.53 thf(fact_1091_Collect__mono,axiom,
% 0.34/0.53 ! [P: ( produc3658429121746597890et_nat > $o ) > $o,Q2: ( produc3658429121746597890et_nat > $o ) > $o] :
% 0.34/0.53 ( ! [X3: produc3658429121746597890et_nat > $o] :
% 0.34/0.53 ( ( P @ X3 )
% 0.34/0.53 => ( Q2 @ X3 ) )
% 0.34/0.53 => ( ord_le2965882846123202637_nat_o @ ( collec939566748876313656_nat_o @ P ) @ ( collec939566748876313656_nat_o @ Q2 ) ) ) ).
% 0.34/0.53
% 0.34/0.53 % Collect_mono
% 0.34/0.53 thf(fact_1092_Collect__mono,axiom,
% 0.34/0.53 ! [P: int > $o,Q2: int > $o] :
% 0.34/0.53 ( ! [X3: int] :
% 0.34/0.53 ( ( P @ X3 )
% 0.34/0.53 => ( Q2 @ X3 ) )
% 0.34/0.53 => ( ord_less_eq_set_int @ ( collect_int @ P ) @ ( collect_int @ Q2 ) ) ) ).
% 0.34/0.53
% 0.34/0.53 % Collect_mono
% 0.34/0.53 thf(fact_1093_Collect__mono,axiom,
% 0.34/0.53 ! [P: $o > $o,Q2: $o > $o] :
% 0.34/0.53 ( ! [X3: $o] :
% 0.34/0.53 ( ( P @ X3 )
% 0.34/0.53 => ( Q2 @ X3 ) )
% 0.34/0.53 => ( ord_less_eq_set_o @ ( collect_o @ P ) @ ( collect_o @ Q2 ) ) ) ).
% 0.34/0.53
% 0.34/0.53 % Collect_mono
% 0.34/0.53 thf(fact_1094_Collect__mono,axiom,
% 0.34/0.53 ! [P: product_unit > $o,Q2: product_unit > $o] :
% 0.34/0.53 ( ! [X3: product_unit] :
% 0.34/0.53 ( ( P @ X3 )
% 0.34/0.53 => ( Q2 @ X3 ) )
% 0.34/0.53 => ( ord_le3507040750410214029t_unit @ ( collect_Product_unit @ P ) @ ( collect_Product_unit @ Q2 ) ) ) ).
% 0.34/0.53
% 0.34/0.53 % Collect_mono
% 0.34/0.53 thf(fact_1095_Collect__mono,axiom,
% 0.34/0.53 ! [P: nat > $o,Q2: nat > $o] :
% 0.34/0.53 ( ! [X3: nat] :
% 0.34/0.53 ( ( P @ X3 )
% 0.34/0.53 => ( Q2 @ X3 ) )
% 0.34/0.53 => ( ord_less_eq_set_nat @ ( collect_nat @ P ) @ ( collect_nat @ Q2 ) ) ) ).
% 0.34/0.53
% 0.34/0.53 % Collect_mono
% 0.34/0.53 thf(fact_1096_subset__refl,axiom,
% 0.34/0.53 ! [A: set_nat] : ( ord_less_eq_set_nat @ A @ A ) ).
% 0.34/0.53
% 0.34/0.53 % subset_refl
% 0.34/0.53 thf(fact_1097_Int__commute,axiom,
% 0.34/0.53 ( inf_inf_set_nat
% 0.34/0.53 = ( ^ [A3: set_nat,B7: set_nat] : ( inf_inf_set_nat @ B7 @ A3 ) ) ) ).
% 0.34/0.53
% 0.34/0.53 % Int_commute
% 0.34/0.53 thf(fact_1098_Int__absorb2,axiom,
% 0.34/0.53 ! [A: set_nat,B4: set_nat] :
% 0.34/0.53 ( ( ord_less_eq_set_nat @ A @ B4 )
% 0.34/0.53 => ( ( inf_inf_set_nat @ A @ B4 )
% 0.34/0.53 = A ) ) ).
% 0.34/0.53
% 0.34/0.53 % Int_absorb2
% 0.34/0.53 thf(fact_1099_Int__absorb1,axiom,
% 0.34/0.53 ! [B4: set_nat,A: set_nat] :
% 0.34/0.53 ( ( ord_less_eq_set_nat @ B4 @ A )
% 0.34/0.53 => ( ( inf_inf_set_nat @ A @ B4 )
% 0.34/0.53 = B4 ) ) ).
% 0.34/0.53
% 0.34/0.53 % Int_absorb1
% 0.34/0.53 thf(fact_1100_subset__iff,axiom,
% 0.34/0.53 ( ord_le2965882846123202637_nat_o
% 0.34/0.53 = ( ^ [A3: set_Pr4532377907799695533_nat_o,B7: set_Pr4532377907799695533_nat_o] :
% 0.34/0.53 ! [T: produc3658429121746597890et_nat > $o] :
% 0.34/0.53 ( ( member6576561426505652726_nat_o @ T @ A3 )
% 0.34/0.53 => ( member6576561426505652726_nat_o @ T @ B7 ) ) ) ) ).
% 0.34/0.53
% 0.34/0.53 % subset_iff
% 0.34/0.53 thf(fact_1101_subset__iff,axiom,
% 0.34/0.53 ( ord_less_eq_set_int
% 0.34/0.53 = ( ^ [A3: set_int,B7: set_int] :
% 0.34/0.53 ! [T: int] :
% 0.34/0.53 ( ( member_int @ T @ A3 )
% 0.34/0.53 => ( member_int @ T @ B7 ) ) ) ) ).
% 0.34/0.53
% 0.34/0.53 % subset_iff
% 0.34/0.53 thf(fact_1102_subset__iff,axiom,
% 0.34/0.53 ( ord_less_eq_set_o
% 0.34/0.53 = ( ^ [A3: set_o,B7: set_o] :
% 0.34/0.53 ! [T: $o] :
% 0.34/0.53 ( ( member_o @ T @ A3 )
% 0.34/0.53 => ( member_o @ T @ B7 ) ) ) ) ).
% 0.34/0.53
% 0.34/0.53 % subset_iff
% 0.34/0.53 thf(fact_1103_subset__iff,axiom,
% 0.34/0.53 ( ord_le3507040750410214029t_unit
% 0.34/0.53 = ( ^ [A3: set_Product_unit,B7: set_Product_unit] :
% 0.34/0.53 ! [T: product_unit] :
% 0.34/0.53 ( ( member_Product_unit @ T @ A3 )
% 0.34/0.53 => ( member_Product_unit @ T @ B7 ) ) ) ) ).
% 0.34/0.53
% 0.34/0.53 % subset_iff
% 0.34/0.53 thf(fact_1104_subset__iff,axiom,
% 0.34/0.53 ( ord_less_eq_set_nat
% 0.34/0.53 = ( ^ [A3: set_nat,B7: set_nat] :
% 0.34/0.53 ! [T: nat] :
% 0.34/0.53 ( ( member_nat @ T @ A3 )
% 0.34/0.53 => ( member_nat @ T @ B7 ) ) ) ) ).
% 0.34/0.53
% 0.34/0.53 % subset_iff
% 0.34/0.53 thf(fact_1105_subset__UnE,axiom,
% 0.34/0.53 ! [C5: set_nat,A: set_nat,B4: set_nat] :
% 0.34/0.53 ( ( ord_less_eq_set_nat @ C5 @ ( sup_sup_set_nat @ A @ B4 ) )
% 0.34/0.53 => ~ ! [A7: set_nat] :
% 0.34/0.53 ( ( ord_less_eq_set_nat @ A7 @ A )
% 0.34/0.53 => ! [B8: set_nat] :
% 0.34/0.53 ( ( ord_less_eq_set_nat @ B8 @ B4 )
% 0.34/0.53 => ( C5
% 0.34/0.53 != ( sup_sup_set_nat @ A7 @ B8 ) ) ) ) ) ).
% 0.34/0.53
% 0.34/0.53 % subset_UnE
% 0.34/0.53 thf(fact_1106_equalityD2,axiom,
% 0.34/0.53 ! [A: set_nat,B4: set_nat] :
% 0.34/0.53 ( ( A = B4 )
% 0.34/0.53 => ( ord_less_eq_set_nat @ B4 @ A ) ) ).
% 0.34/0.53
% 0.34/0.53 % equalityD2
% 0.34/0.53 thf(fact_1107_equalityD1,axiom,
% 0.34/0.53 ! [A: set_nat,B4: set_nat] :
% 0.34/0.53 ( ( A = B4 )
% 0.34/0.53 => ( ord_less_eq_set_nat @ A @ B4 ) ) ).
% 0.34/0.53
% 0.34/0.53 % equalityD1
% 0.34/0.53 thf(fact_1108_Un__commute,axiom,
% 0.34/0.53 ( sup_sup_set_nat
% 0.34/0.53 = ( ^ [A3: set_nat,B7: set_nat] : ( sup_sup_set_nat @ B7 @ A3 ) ) ) ).
% 0.34/0.53
% 0.34/0.53 % Un_commute
% 0.34/0.53 thf(fact_1109_Un__absorb2,axiom,
% 0.34/0.53 ! [B4: set_nat,A: set_nat] :
% 0.34/0.53 ( ( ord_less_eq_set_nat @ B4 @ A )
% 0.34/0.53 => ( ( sup_sup_set_nat @ A @ B4 )
% 0.34/0.53 = A ) ) ).
% 0.34/0.53
% 0.34/0.53 % Un_absorb2
% 0.34/0.53 thf(fact_1110_Un__absorb1,axiom,
% 0.34/0.53 ! [A: set_nat,B4: set_nat] :
% 0.34/0.53 ( ( ord_less_eq_set_nat @ A @ B4 )
% 0.34/0.53 => ( ( sup_sup_set_nat @ A @ B4 )
% 0.34/0.53 = B4 ) ) ).
% 0.34/0.53
% 0.34/0.53 % Un_absorb1
% 0.34/0.53 thf(fact_1111_Int__lower2,axiom,
% 0.34/0.53 ! [A: set_nat,B4: set_nat] : ( ord_less_eq_set_nat @ ( inf_inf_set_nat @ A @ B4 ) @ B4 ) ).
% 0.34/0.53
% 0.34/0.53 % Int_lower2
% 0.34/0.53 thf(fact_1112_Int__lower1,axiom,
% 0.34/0.53 ! [A: set_nat,B4: set_nat] : ( ord_less_eq_set_nat @ ( inf_inf_set_nat @ A @ B4 ) @ A ) ).
% 0.34/0.53
% 0.34/0.53 % Int_lower1
% 0.34/0.53 thf(fact_1113_Int__absorb,axiom,
% 0.34/0.53 ! [A: set_nat] :
% 0.34/0.53 ( ( inf_inf_set_nat @ A @ A )
% 0.34/0.53 = A ) ).
% 0.34/0.53
% 0.34/0.53 % Int_absorb
% 0.34/0.53 thf(fact_1114_subset__eq,axiom,
% 0.34/0.53 ( ord_le2965882846123202637_nat_o
% 0.34/0.53 = ( ^ [A3: set_Pr4532377907799695533_nat_o,B7: set_Pr4532377907799695533_nat_o] :
% 0.34/0.53 ! [X2: produc3658429121746597890et_nat > $o] :
% 0.34/0.53 ( ( member6576561426505652726_nat_o @ X2 @ A3 )
% 0.34/0.53 => ( member6576561426505652726_nat_o @ X2 @ B7 ) ) ) ) ).
% 0.34/0.53
% 0.34/0.53 % subset_eq
% 0.34/0.53 thf(fact_1115_subset__eq,axiom,
% 0.34/0.53 ( ord_less_eq_set_int
% 0.34/0.53 = ( ^ [A3: set_int,B7: set_int] :
% 0.34/0.53 ! [X2: int] :
% 0.34/0.53 ( ( member_int @ X2 @ A3 )
% 0.34/0.53 => ( member_int @ X2 @ B7 ) ) ) ) ).
% 0.34/0.53
% 0.34/0.53 % subset_eq
% 0.34/0.53 thf(fact_1116_subset__eq,axiom,
% 0.34/0.53 ( ord_less_eq_set_o
% 0.34/0.53 = ( ^ [A3: set_o,B7: set_o] :
% 0.34/0.53 ! [X2: $o] :
% 0.34/0.53 ( ( member_o @ X2 @ A3 )
% 0.34/0.53 => ( member_o @ X2 @ B7 ) ) ) ) ).
% 0.34/0.53
% 0.34/0.53 % subset_eq
% 0.34/0.53 thf(fact_1117_subset__eq,axiom,
% 0.34/0.53 ( ord_le3507040750410214029t_unit
% 0.34/0.53 = ( ^ [A3: set_Product_unit,B7: set_Product_unit] :
% 0.34/0.53 ! [X2: product_unit] :
% 0.34/0.53 ( ( member_Product_unit @ X2 @ A3 )
% 0.34/0.53 => ( member_Product_unit @ X2 @ B7 ) ) ) ) ).
% 0.34/0.53
% 0.34/0.53 % subset_eq
% 0.34/0.53 thf(fact_1118_subset__eq,axiom,
% 0.34/0.53 ( ord_less_eq_set_nat
% 0.34/0.53 = ( ^ [A3: set_nat,B7: set_nat] :
% 0.34/0.53 ! [X2: nat] :
% 0.34/0.53 ( ( member_nat @ X2 @ A3 )
% 0.34/0.53 => ( member_nat @ X2 @ B7 ) ) ) ) ).
% 0.34/0.53
% 0.34/0.53 % subset_eq
% 0.34/0.53 thf(fact_1119_equalityE,axiom,
% 0.34/0.53 ! [A: set_nat,B4: set_nat] :
% 0.34/0.53 ( ( A = B4 )
% 0.34/0.53 => ~ ( ( ord_less_eq_set_nat @ A @ B4 )
% 0.34/0.53 => ~ ( ord_less_eq_set_nat @ B4 @ A ) ) ) ).
% 0.34/0.53
% 0.34/0.53 % equalityE
% 0.34/0.53 thf(fact_1120_Un__upper2,axiom,
% 0.34/0.53 ! [B4: set_nat,A: set_nat] : ( ord_less_eq_set_nat @ B4 @ ( sup_sup_set_nat @ A @ B4 ) ) ).
% 0.34/0.53
% 0.34/0.53 % Un_upper2
% 0.34/0.53 thf(fact_1121_Un__upper1,axiom,
% 0.34/0.53 ! [A: set_nat,B4: set_nat] : ( ord_less_eq_set_nat @ A @ ( sup_sup_set_nat @ A @ B4 ) ) ).
% 0.34/0.53
% 0.34/0.53 % Un_upper1
% 0.34/0.53 thf(fact_1122_Un__absorb,axiom,
% 0.34/0.53 ! [A: set_nat] :
% 0.34/0.53 ( ( sup_sup_set_nat @ A @ A )
% 0.34/0.53 = A ) ).
% 0.34/0.53
% 0.34/0.53 % Un_absorb
% 0.34/0.53 thf(fact_1123_Int__assoc,axiom,
% 0.34/0.53 ! [A: set_nat,B4: set_nat,C5: set_nat] :
% 0.34/0.53 ( ( inf_inf_set_nat @ ( inf_inf_set_nat @ A @ B4 ) @ C5 )
% 0.34/0.53 = ( inf_inf_set_nat @ A @ ( inf_inf_set_nat @ B4 @ C5 ) ) ) ).
% 0.34/0.53
% 0.34/0.53 % Int_assoc
% 0.34/0.53 thf(fact_1124_Un__least,axiom,
% 0.34/0.53 ! [A: set_nat,C5: set_nat,B4: set_nat] :
% 0.34/0.53 ( ( ord_less_eq_set_nat @ A @ C5 )
% 0.34/0.53 => ( ( ord_less_eq_set_nat @ B4 @ C5 )
% 0.34/0.53 => ( ord_less_eq_set_nat @ ( sup_sup_set_nat @ A @ B4 ) @ C5 ) ) ) ).
% 0.34/0.53
% 0.34/0.53 % Un_least
% 0.34/0.53 thf(fact_1125_Un__assoc,axiom,
% 0.34/0.53 ! [A: set_nat,B4: set_nat,C5: set_nat] :
% 0.34/0.53 ( ( sup_sup_set_nat @ ( sup_sup_set_nat @ A @ B4 ) @ C5 )
% 0.34/0.53 = ( sup_sup_set_nat @ A @ ( sup_sup_set_nat @ B4 @ C5 ) ) ) ).
% 0.34/0.53
% 0.34/0.53 % Un_assoc
% 0.34/0.53 thf(fact_1126_Int__mono,axiom,
% 0.34/0.53 ! [A: set_nat,C5: set_nat,B4: set_nat,D3: set_nat] :
% 0.34/0.53 ( ( ord_less_eq_set_nat @ A @ C5 )
% 0.34/0.53 => ( ( ord_less_eq_set_nat @ B4 @ D3 )
% 0.34/0.53 => ( ord_less_eq_set_nat @ ( inf_inf_set_nat @ A @ B4 ) @ ( inf_inf_set_nat @ C5 @ D3 ) ) ) ) ).
% 0.34/0.53
% 0.34/0.53 % Int_mono
% 0.34/0.53 thf(fact_1127_subsetD,axiom,
% 0.34/0.53 ! [A: set_Pr4532377907799695533_nat_o,B4: set_Pr4532377907799695533_nat_o,C: produc3658429121746597890et_nat > $o] :
% 0.34/0.53 ( ( ord_le2965882846123202637_nat_o @ A @ B4 )
% 0.34/0.53 => ( ( member6576561426505652726_nat_o @ C @ A )
% 0.34/0.53 => ( member6576561426505652726_nat_o @ C @ B4 ) ) ) ).
% 0.34/0.53
% 0.34/0.53 % subsetD
% 0.34/0.53 thf(fact_1128_subsetD,axiom,
% 0.34/0.53 ! [A: set_int,B4: set_int,C: int] :
% 0.34/0.53 ( ( ord_less_eq_set_int @ A @ B4 )
% 0.34/0.53 => ( ( member_int @ C @ A )
% 0.34/0.53 => ( member_int @ C @ B4 ) ) ) ).
% 0.34/0.53
% 0.34/0.53 % subsetD
% 0.34/0.53 thf(fact_1129_subsetD,axiom,
% 0.34/0.53 ! [A: set_o,B4: set_o,C: $o] :
% 0.34/0.53 ( ( ord_less_eq_set_o @ A @ B4 )
% 0.34/0.53 => ( ( member_o @ C @ A )
% 0.34/0.53 => ( member_o @ C @ B4 ) ) ) ).
% 0.34/0.53
% 0.34/0.53 % subsetD
% 0.34/0.53 thf(fact_1130_subsetD,axiom,
% 0.34/0.53 ! [A: set_Product_unit,B4: set_Product_unit,C: product_unit] :
% 0.34/0.53 ( ( ord_le3507040750410214029t_unit @ A @ B4 )
% 0.34/0.53 => ( ( member_Product_unit @ C @ A )
% 0.34/0.53 => ( member_Product_unit @ C @ B4 ) ) ) ).
% 0.34/0.53
% 0.34/0.53 % subsetD
% 0.34/0.53 thf(fact_1131_subsetD,axiom,
% 0.34/0.53 ! [A: set_nat,B4: set_nat,C: nat] :
% 0.34/0.53 ( ( ord_less_eq_set_nat @ A @ B4 )
% 0.34/0.53 => ( ( member_nat @ C @ A )
% 0.34/0.53 => ( member_nat @ C @ B4 ) ) ) ).
% 0.34/0.53
% 0.34/0.53 % subsetD
% 0.34/0.53 thf(fact_1132_in__mono,axiom,
% 0.34/0.53 ! [A: set_Pr4532377907799695533_nat_o,B4: set_Pr4532377907799695533_nat_o,X: produc3658429121746597890et_nat > $o] :
% 0.34/0.53 ( ( ord_le2965882846123202637_nat_o @ A @ B4 )
% 0.34/0.53 => ( ( member6576561426505652726_nat_o @ X @ A )
% 0.34/0.53 => ( member6576561426505652726_nat_o @ X @ B4 ) ) ) ).
% 0.34/0.53
% 0.34/0.53 % in_mono
% 0.34/0.53 thf(fact_1133_in__mono,axiom,
% 0.34/0.53 ! [A: set_int,B4: set_int,X: int] :
% 0.34/0.53 ( ( ord_less_eq_set_int @ A @ B4 )
% 0.34/0.53 => ( ( member_int @ X @ A )
% 0.34/0.53 => ( member_int @ X @ B4 ) ) ) ).
% 0.34/0.53
% 0.34/0.53 % in_mono
% 0.34/0.53 thf(fact_1134_in__mono,axiom,
% 0.34/0.53 ! [A: set_o,B4: set_o,X: $o] :
% 0.34/0.53 ( ( ord_less_eq_set_o @ A @ B4 )
% 0.34/0.53 => ( ( member_o @ X @ A )
% 0.34/0.53 => ( member_o @ X @ B4 ) ) ) ).
% 0.34/0.53
% 0.34/0.53 % in_mono
% 0.34/0.53 thf(fact_1135_in__mono,axiom,
% 0.34/0.53 ! [A: set_Product_unit,B4: set_Product_unit,X: product_unit] :
% 0.34/0.53 ( ( ord_le3507040750410214029t_unit @ A @ B4 )
% 0.34/0.53 => ( ( member_Product_unit @ X @ A )
% 0.34/0.53 => ( member_Product_unit @ X @ B4 ) ) ) ).
% 0.34/0.53
% 0.34/0.53 % in_mono
% 0.34/0.53 thf(fact_1136_in__mono,axiom,
% 0.34/0.53 ! [A: set_nat,B4: set_nat,X: nat] :
% 0.34/0.53 ( ( ord_less_eq_set_nat @ A @ B4 )
% 0.34/0.53 => ( ( member_nat @ X @ A )
% 0.34/0.53 => ( member_nat @ X @ B4 ) ) ) ).
% 0.34/0.53
% 0.34/0.53 % in_mono
% 0.34/0.53 thf(fact_1137_ball__Un,axiom,
% 0.34/0.53 ! [A: set_nat,B4: set_nat,P: nat > $o] :
% 0.34/0.53 ( ( ! [X2: nat] :
% 0.34/0.53 ( ( member_nat @ X2 @ ( sup_sup_set_nat @ A @ B4 ) )
% 0.34/0.53 => ( P @ X2 ) ) )
% 0.34/0.53 = ( ! [X2: nat] :
% 0.34/0.53 ( ( member_nat @ X2 @ A )
% 0.34/0.53 => ( P @ X2 ) )
% 0.34/0.53 & ! [X2: nat] :
% 0.34/0.53 ( ( member_nat @ X2 @ B4 )
% 0.34/0.53 => ( P @ X2 ) ) ) ) ).
% 0.34/0.53
% 0.34/0.53 % ball_Un
% 0.34/0.53 thf(fact_1138_Un__mono,axiom,
% 0.34/0.53 ! [A: set_nat,C5: set_nat,B4: set_nat,D3: set_nat] :
% 0.34/0.53 ( ( ord_less_eq_set_nat @ A @ C5 )
% 0.34/0.53 => ( ( ord_less_eq_set_nat @ B4 @ D3 )
% 0.34/0.53 => ( ord_less_eq_set_nat @ ( sup_sup_set_nat @ A @ B4 ) @ ( sup_sup_set_nat @ C5 @ D3 ) ) ) ) ).
% 0.34/0.53
% 0.34/0.53 % Un_mono
% 0.34/0.53 thf(fact_1139_bex__Un,axiom,
% 0.34/0.53 ! [A: set_nat,B4: set_nat,P: nat > $o] :
% 0.34/0.53 ( ( ? [X2: nat] :
% 0.34/0.53 ( ( member_nat @ X2 @ ( sup_sup_set_nat @ A @ B4 ) )
% 0.34/0.53 & ( P @ X2 ) ) )
% 0.34/0.53 = ( ? [X2: nat] :
% 0.34/0.53 ( ( member_nat @ X2 @ A )
% 0.34/0.53 & ( P @ X2 ) )
% 0.34/0.53 | ? [X2: nat] :
% 0.34/0.53 ( ( member_nat @ X2 @ B4 )
% 0.34/0.53 & ( P @ X2 ) ) ) ) ).
% 0.34/0.53
% 0.34/0.53 % bex_Un
% 0.34/0.53 thf(fact_1140_IntD2,axiom,
% 0.34/0.53 ! [C: produc3658429121746597890et_nat > $o,A: set_Pr4532377907799695533_nat_o,B4: set_Pr4532377907799695533_nat_o] :
% 0.34/0.53 ( ( member6576561426505652726_nat_o @ C @ ( inf_in1906310914598751387_nat_o @ A @ B4 ) )
% 0.34/0.53 => ( member6576561426505652726_nat_o @ C @ B4 ) ) ).
% 0.34/0.53
% 0.34/0.53 % IntD2
% 0.34/0.53 thf(fact_1141_IntD2,axiom,
% 0.34/0.53 ! [C: int,A: set_int,B4: set_int] :
% 0.34/0.53 ( ( member_int @ C @ ( inf_inf_set_int @ A @ B4 ) )
% 0.34/0.53 => ( member_int @ C @ B4 ) ) ).
% 0.34/0.53
% 0.34/0.53 % IntD2
% 0.34/0.53 thf(fact_1142_IntD2,axiom,
% 0.34/0.53 ! [C: $o,A: set_o,B4: set_o] :
% 0.34/0.53 ( ( member_o @ C @ ( inf_inf_set_o @ A @ B4 ) )
% 0.34/0.53 => ( member_o @ C @ B4 ) ) ).
% 0.34/0.53
% 0.34/0.53 % IntD2
% 0.34/0.53 thf(fact_1143_IntD2,axiom,
% 0.34/0.53 ! [C: product_unit,A: set_Product_unit,B4: set_Product_unit] :
% 0.34/0.53 ( ( member_Product_unit @ C @ ( inf_in4660618365625256667t_unit @ A @ B4 ) )
% 0.34/0.53 => ( member_Product_unit @ C @ B4 ) ) ).
% 0.34/0.53
% 0.34/0.53 % IntD2
% 0.34/0.53 thf(fact_1144_IntD2,axiom,
% 0.34/0.53 ! [C: nat,A: set_nat,B4: set_nat] :
% 0.34/0.53 ( ( member_nat @ C @ ( inf_inf_set_nat @ A @ B4 ) )
% 0.34/0.53 => ( member_nat @ C @ B4 ) ) ).
% 0.34/0.53
% 0.34/0.53 % IntD2
% 0.34/0.53 thf(fact_1145_IntD1,axiom,
% 0.34/0.53 ! [C: produc3658429121746597890et_nat > $o,A: set_Pr4532377907799695533_nat_o,B4: set_Pr4532377907799695533_nat_o] :
% 0.34/0.53 ( ( member6576561426505652726_nat_o @ C @ ( inf_in1906310914598751387_nat_o @ A @ B4 ) )
% 0.34/0.53 => ( member6576561426505652726_nat_o @ C @ A ) ) ).
% 0.34/0.53
% 0.34/0.53 % IntD1
% 0.34/0.53 thf(fact_1146_IntD1,axiom,
% 0.34/0.53 ! [C: int,A: set_int,B4: set_int] :
% 0.34/0.53 ( ( member_int @ C @ ( inf_inf_set_int @ A @ B4 ) )
% 0.34/0.53 => ( member_int @ C @ A ) ) ).
% 0.34/0.53
% 0.34/0.53 % IntD1
% 0.34/0.53 thf(fact_1147_IntD1,axiom,
% 0.34/0.53 ! [C: $o,A: set_o,B4: set_o] :
% 0.34/0.53 ( ( member_o @ C @ ( inf_inf_set_o @ A @ B4 ) )
% 0.34/0.53 => ( member_o @ C @ A ) ) ).
% 0.34/0.53
% 0.34/0.53 % IntD1
% 0.34/0.53 thf(fact_1148_IntD1,axiom,
% 0.34/0.53 ! [C: product_unit,A: set_Product_unit,B4: set_Product_unit] :
% 0.34/0.53 ( ( member_Product_unit @ C @ ( inf_in4660618365625256667t_unit @ A @ B4 ) )
% 0.34/0.53 => ( member_Product_unit @ C @ A ) ) ).
% 0.34/0.53
% 0.34/0.53 % IntD1
% 0.34/0.53 thf(fact_1149_IntD1,axiom,
% 0.34/0.53 ! [C: nat,A: set_nat,B4: set_nat] :
% 0.34/0.53 ( ( member_nat @ C @ ( inf_inf_set_nat @ A @ B4 ) )
% 0.34/0.53 => ( member_nat @ C @ A ) ) ).
% 0.34/0.53
% 0.34/0.53 % IntD1
% 0.34/0.53 thf(fact_1150_UnI2,axiom,
% 0.34/0.53 ! [C: produc3658429121746597890et_nat > $o,B4: set_Pr4532377907799695533_nat_o,A: set_Pr4532377907799695533_nat_o] :
% 0.34/0.53 ( ( member6576561426505652726_nat_o @ C @ B4 )
% 0.34/0.53 => ( member6576561426505652726_nat_o @ C @ ( sup_su5209123915105501825_nat_o @ A @ B4 ) ) ) ).
% 0.34/0.53
% 0.34/0.53 % UnI2
% 0.34/0.53 thf(fact_1151_UnI2,axiom,
% 0.34/0.53 ! [C: int,B4: set_int,A: set_int] :
% 0.34/0.53 ( ( member_int @ C @ B4 )
% 0.34/0.53 => ( member_int @ C @ ( sup_sup_set_int @ A @ B4 ) ) ) ).
% 0.34/0.53
% 0.34/0.53 % UnI2
% 0.34/0.53 thf(fact_1152_UnI2,axiom,
% 0.34/0.53 ! [C: $o,B4: set_o,A: set_o] :
% 0.34/0.53 ( ( member_o @ C @ B4 )
% 0.34/0.53 => ( member_o @ C @ ( sup_sup_set_o @ A @ B4 ) ) ) ).
% 0.34/0.53
% 0.34/0.53 % UnI2
% 0.34/0.53 thf(fact_1153_UnI2,axiom,
% 0.34/0.53 ! [C: product_unit,B4: set_Product_unit,A: set_Product_unit] :
% 0.34/0.53 ( ( member_Product_unit @ C @ B4 )
% 0.34/0.53 => ( member_Product_unit @ C @ ( sup_su793286257634532545t_unit @ A @ B4 ) ) ) ).
% 0.34/0.53
% 0.34/0.53 % UnI2
% 0.34/0.53 thf(fact_1154_UnI2,axiom,
% 0.34/0.53 ! [C: nat,B4: set_nat,A: set_nat] :
% 0.34/0.53 ( ( member_nat @ C @ B4 )
% 0.34/0.53 => ( member_nat @ C @ ( sup_sup_set_nat @ A @ B4 ) ) ) ).
% 0.34/0.53
% 0.34/0.53 % UnI2
% 0.34/0.53 thf(fact_1155_UnI1,axiom,
% 0.34/0.53 ! [C: produc3658429121746597890et_nat > $o,A: set_Pr4532377907799695533_nat_o,B4: set_Pr4532377907799695533_nat_o] :
% 0.34/0.53 ( ( member6576561426505652726_nat_o @ C @ A )
% 0.34/0.53 => ( member6576561426505652726_nat_o @ C @ ( sup_su5209123915105501825_nat_o @ A @ B4 ) ) ) ).
% 0.34/0.53
% 0.34/0.53 % UnI1
% 0.34/0.53 thf(fact_1156_UnI1,axiom,
% 0.34/0.53 ! [C: int,A: set_int,B4: set_int] :
% 0.34/0.53 ( ( member_int @ C @ A )
% 0.34/0.53 => ( member_int @ C @ ( sup_sup_set_int @ A @ B4 ) ) ) ).
% 0.34/0.53
% 0.34/0.53 % UnI1
% 0.34/0.53 thf(fact_1157_UnI1,axiom,
% 0.34/0.53 ! [C: $o,A: set_o,B4: set_o] :
% 0.34/0.53 ( ( member_o @ C @ A )
% 0.34/0.53 => ( member_o @ C @ ( sup_sup_set_o @ A @ B4 ) ) ) ).
% 0.34/0.53
% 0.34/0.53 % UnI1
% 0.34/0.53 thf(fact_1158_UnI1,axiom,
% 0.34/0.53 ! [C: product_unit,A: set_Product_unit,B4: set_Product_unit] :
% 0.34/0.53 ( ( member_Product_unit @ C @ A )
% 0.34/0.53 => ( member_Product_unit @ C @ ( sup_su793286257634532545t_unit @ A @ B4 ) ) ) ).
% 0.34/0.53
% 0.34/0.53 % UnI1
% 0.34/0.53 thf(fact_1159_UnI1,axiom,
% 0.34/0.53 ! [C: nat,A: set_nat,B4: set_nat] :
% 0.34/0.53 ( ( member_nat @ C @ A )
% 0.34/0.53 => ( member_nat @ C @ ( sup_sup_set_nat @ A @ B4 ) ) ) ).
% 0.34/0.53
% 0.34/0.53 % UnI1
% 0.34/0.53 thf(fact_1160_IntE,axiom,
% 0.34/0.53 ! [C: produc3658429121746597890et_nat > $o,A: set_Pr4532377907799695533_nat_o,B4: set_Pr4532377907799695533_nat_o] :
% 0.34/0.53 ( ( member6576561426505652726_nat_o @ C @ ( inf_in1906310914598751387_nat_o @ A @ B4 ) )
% 0.34/0.53 => ~ ( ( member6576561426505652726_nat_o @ C @ A )
% 0.34/0.53 => ~ ( member6576561426505652726_nat_o @ C @ B4 ) ) ) ).
% 0.34/0.53
% 0.34/0.53 % IntE
% 0.34/0.53 thf(fact_1161_IntE,axiom,
% 0.34/0.53 ! [C: int,A: set_int,B4: set_int] :
% 0.34/0.53 ( ( member_int @ C @ ( inf_inf_set_int @ A @ B4 ) )
% 0.34/0.53 => ~ ( ( member_int @ C @ A )
% 0.34/0.53 => ~ ( member_int @ C @ B4 ) ) ) ).
% 0.34/0.53
% 0.34/0.53 % IntE
% 0.34/0.53 thf(fact_1162_IntE,axiom,
% 0.34/0.53 ! [C: $o,A: set_o,B4: set_o] :
% 0.34/0.53 ( ( member_o @ C @ ( inf_inf_set_o @ A @ B4 ) )
% 0.34/0.53 => ~ ( ( member_o @ C @ A )
% 0.34/0.53 => ~ ( member_o @ C @ B4 ) ) ) ).
% 0.34/0.53
% 0.34/0.53 % IntE
% 0.34/0.53 thf(fact_1163_IntE,axiom,
% 0.34/0.53 ! [C: product_unit,A: set_Product_unit,B4: set_Product_unit] :
% 0.34/0.53 ( ( member_Product_unit @ C @ ( inf_in4660618365625256667t_unit @ A @ B4 ) )
% 0.34/0.53 => ~ ( ( member_Product_unit @ C @ A )
% 0.34/0.53 => ~ ( member_Product_unit @ C @ B4 ) ) ) ).
% 0.34/0.53
% 0.34/0.53 % IntE
% 0.34/0.53 thf(fact_1164_IntE,axiom,
% 0.34/0.53 ! [C: nat,A: set_nat,B4: set_nat] :
% 0.34/0.53 ( ( member_nat @ C @ ( inf_inf_set_nat @ A @ B4 ) )
% 0.34/0.53 => ~ ( ( member_nat @ C @ A )
% 0.34/0.53 => ~ ( member_nat @ C @ B4 ) ) ) ).
% 0.34/0.53
% 0.34/0.53 % IntE
% 0.34/0.53 thf(fact_1165_UnE,axiom,
% 0.34/0.53 ! [C: produc3658429121746597890et_nat > $o,A: set_Pr4532377907799695533_nat_o,B4: set_Pr4532377907799695533_nat_o] :
% 0.34/0.53 ( ( member6576561426505652726_nat_o @ C @ ( sup_su5209123915105501825_nat_o @ A @ B4 ) )
% 0.34/0.53 => ( ~ ( member6576561426505652726_nat_o @ C @ A )
% 0.34/0.53 => ( member6576561426505652726_nat_o @ C @ B4 ) ) ) ).
% 0.34/0.53
% 0.34/0.53 % UnE
% 0.34/0.53 thf(fact_1166_UnE,axiom,
% 0.34/0.53 ! [C: int,A: set_int,B4: set_int] :
% 0.34/0.53 ( ( member_int @ C @ ( sup_sup_set_int @ A @ B4 ) )
% 0.34/0.53 => ( ~ ( member_int @ C @ A )
% 0.34/0.53 => ( member_int @ C @ B4 ) ) ) ).
% 0.34/0.53
% 0.34/0.53 % UnE
% 0.34/0.53 thf(fact_1167_UnE,axiom,
% 0.34/0.53 ! [C: $o,A: set_o,B4: set_o] :
% 0.34/0.53 ( ( member_o @ C @ ( sup_sup_set_o @ A @ B4 ) )
% 0.34/0.53 => ( ~ ( member_o @ C @ A )
% 0.34/0.53 => ( member_o @ C @ B4 ) ) ) ).
% 0.34/0.53
% 0.34/0.53 % UnE
% 0.34/0.53 thf(fact_1168_UnE,axiom,
% 0.34/0.53 ! [C: product_unit,A: set_Product_unit,B4: set_Product_unit] :
% 0.34/0.53 ( ( member_Product_unit @ C @ ( sup_su793286257634532545t_unit @ A @ B4 ) )
% 0.34/0.53 => ( ~ ( member_Product_unit @ C @ A )
% 0.34/0.53 => ( member_Product_unit @ C @ B4 ) ) ) ).
% 0.34/0.53
% 0.34/0.53 % UnE
% 0.34/0.53 thf(fact_1169_UnE,axiom,
% 0.34/0.53 ! [C: nat,A: set_nat,B4: set_nat] :
% 0.34/0.53 ( ( member_nat @ C @ ( sup_sup_set_nat @ A @ B4 ) )
% 0.34/0.53 => ( ~ ( member_nat @ C @ A )
% 0.34/0.53 => ( member_nat @ C @ B4 ) ) ) ).
% 0.34/0.53
% 0.34/0.53 % UnE
% 0.34/0.53 thf(fact_1170_dual__order_Otrans,axiom,
% 0.34/0.53 ! [B: filter_nat,A2: filter_nat,C: filter_nat] :
% 0.34/0.53 ( ( ord_le2510731241096832064er_nat @ B @ A2 )
% 0.34/0.53 => ( ( ord_le2510731241096832064er_nat @ C @ B )
% 0.34/0.53 => ( ord_le2510731241096832064er_nat @ C @ A2 ) ) ) ).
% 0.34/0.53
% 0.34/0.53 % dual_order.trans
% 0.34/0.53 thf(fact_1171_dual__order_Otrans,axiom,
% 0.34/0.53 ! [B: set_nat,A2: set_nat,C: set_nat] :
% 0.34/0.53 ( ( ord_less_eq_set_nat @ B @ A2 )
% 0.34/0.53 => ( ( ord_less_eq_set_nat @ C @ B )
% 0.34/0.53 => ( ord_less_eq_set_nat @ C @ A2 ) ) ) ).
% 0.34/0.53
% 0.34/0.53 % dual_order.trans
% 0.34/0.53 thf(fact_1172_dual__order_Otrans,axiom,
% 0.34/0.53 ! [B: rat,A2: rat,C: rat] :
% 0.34/0.53 ( ( ord_less_eq_rat @ B @ A2 )
% 0.34/0.53 => ( ( ord_less_eq_rat @ C @ B )
% 0.34/0.53 => ( ord_less_eq_rat @ C @ A2 ) ) ) ).
% 0.34/0.53
% 0.34/0.53 % dual_order.trans
% 0.34/0.53 thf(fact_1173_dual__order_Otrans,axiom,
% 0.34/0.53 ! [B: nat,A2: nat,C: nat] :
% 0.34/0.53 ( ( ord_less_eq_nat @ B @ A2 )
% 0.34/0.53 => ( ( ord_less_eq_nat @ C @ B )
% 0.34/0.53 => ( ord_less_eq_nat @ C @ A2 ) ) ) ).
% 0.34/0.53
% 0.34/0.53 % dual_order.trans
% 0.34/0.53 thf(fact_1174_dual__order_Otrans,axiom,
% 0.34/0.53 ! [B: int,A2: int,C: int] :
% 0.34/0.53 ( ( ord_less_eq_int @ B @ A2 )
% 0.34/0.53 => ( ( ord_less_eq_int @ C @ B )
% 0.34/0.53 => ( ord_less_eq_int @ C @ A2 ) ) ) ).
% 0.34/0.53
% 0.34/0.53 % dual_order.trans
% 0.34/0.53 thf(fact_1175_dual__order_Oantisym,axiom,
% 0.34/0.53 ! [B: filter_nat,A2: filter_nat] :
% 0.34/0.53 ( ( ord_le2510731241096832064er_nat @ B @ A2 )
% 0.34/0.53 => ( ( ord_le2510731241096832064er_nat @ A2 @ B )
% 0.34/0.53 => ( A2 = B ) ) ) ).
% 0.34/0.53
% 0.34/0.53 % dual_order.antisym
% 0.34/0.53 thf(fact_1176_dual__order_Oantisym,axiom,
% 0.34/0.53 ! [B: set_nat,A2: set_nat] :
% 0.34/0.53 ( ( ord_less_eq_set_nat @ B @ A2 )
% 0.34/0.53 => ( ( ord_less_eq_set_nat @ A2 @ B )
% 0.34/0.53 => ( A2 = B ) ) ) ).
% 0.34/0.53
% 0.34/0.53 % dual_order.antisym
% 0.34/0.53 thf(fact_1177_dual__order_Oantisym,axiom,
% 0.34/0.53 ! [B: rat,A2: rat] :
% 0.34/0.53 ( ( ord_less_eq_rat @ B @ A2 )
% 0.34/0.53 => ( ( ord_less_eq_rat @ A2 @ B )
% 0.34/0.53 => ( A2 = B ) ) ) ).
% 0.34/0.53
% 0.34/0.53 % dual_order.antisym
% 0.34/0.53 thf(fact_1178_dual__order_Oantisym,axiom,
% 0.34/0.53 ! [B: nat,A2: nat] :
% 0.34/0.53 ( ( ord_less_eq_nat @ B @ A2 )
% 0.34/0.53 => ( ( ord_less_eq_nat @ A2 @ B )
% 0.34/0.53 => ( A2 = B ) ) ) ).
% 0.34/0.53
% 0.34/0.53 % dual_order.antisym
% 0.34/0.53 thf(fact_1179_dual__order_Oantisym,axiom,
% 0.34/0.53 ! [B: int,A2: int] :
% 0.34/0.53 ( ( ord_less_eq_int @ B @ A2 )
% 0.34/0.53 => ( ( ord_less_eq_int @ A2 @ B )
% 0.34/0.53 => ( A2 = B ) ) ) ).
% 0.34/0.53
% 0.34/0.53 % dual_order.antisym
% 0.34/0.53 thf(fact_1180_dual__order_Oeq__iff,axiom,
% 0.34/0.53 ( ( ^ [Y4: filter_nat,Z4: filter_nat] : ( Y4 = Z4 ) )
% 0.34/0.53 = ( ^ [A6: filter_nat,B6: filter_nat] :
% 0.34/0.53 ( ( ord_le2510731241096832064er_nat @ B6 @ A6 )
% 0.34/0.53 & ( ord_le2510731241096832064er_nat @ A6 @ B6 ) ) ) ) ).
% 0.34/0.53
% 0.34/0.53 % dual_order.eq_iff
% 0.34/0.53 thf(fact_1181_dual__order_Oeq__iff,axiom,
% 0.34/0.53 ( ( ^ [Y4: set_nat,Z4: set_nat] : ( Y4 = Z4 ) )
% 0.34/0.53 = ( ^ [A6: set_nat,B6: set_nat] :
% 0.34/0.53 ( ( ord_less_eq_set_nat @ B6 @ A6 )
% 0.34/0.53 & ( ord_less_eq_set_nat @ A6 @ B6 ) ) ) ) ).
% 0.34/0.53
% 0.34/0.53 % dual_order.eq_iff
% 0.34/0.53 thf(fact_1182_dual__order_Oeq__iff,axiom,
% 0.34/0.53 ( ( ^ [Y4: rat,Z4: rat] : ( Y4 = Z4 ) )
% 0.34/0.53 = ( ^ [A6: rat,B6: rat] :
% 0.34/0.53 ( ( ord_less_eq_rat @ B6 @ A6 )
% 0.34/0.53 & ( ord_less_eq_rat @ A6 @ B6 ) ) ) ) ).
% 0.34/0.53
% 0.34/0.53 % dual_order.eq_iff
% 0.34/0.53 thf(fact_1183_dual__order_Oeq__iff,axiom,
% 0.34/0.53 ( ( ^ [Y4: nat,Z4: nat] : ( Y4 = Z4 ) )
% 0.34/0.53 = ( ^ [A6: nat,B6: nat] :
% 0.34/0.53 ( ( ord_less_eq_nat @ B6 @ A6 )
% 0.34/0.53 & ( ord_less_eq_nat @ A6 @ B6 ) ) ) ) ).
% 0.34/0.53
% 0.34/0.53 % dual_order.eq_iff
% 0.34/0.53 thf(fact_1184_dual__order_Oeq__iff,axiom,
% 0.34/0.53 ( ( ^ [Y4: int,Z4: int] : ( Y4 = Z4 ) )
% 0.34/0.53 = ( ^ [A6: int,B6: int] :
% 0.34/0.53 ( ( ord_less_eq_int @ B6 @ A6 )
% 0.34/0.53 & ( ord_less_eq_int @ A6 @ B6 ) ) ) ) ).
% 0.34/0.53
% 0.34/0.53 % dual_order.eq_iff
% 0.34/0.53 thf(fact_1185_linorder__wlog,axiom,
% 0.34/0.53 ! [P: rat > rat > $o,A2: rat,B: rat] :
% 0.34/0.53 ( ! [A4: rat,B2: rat] :
% 0.34/0.53 ( ( ord_less_eq_rat @ A4 @ B2 )
% 0.34/0.53 => ( P @ A4 @ B2 ) )
% 0.34/0.53 => ( ! [A4: rat,B2: rat] :
% 0.34/0.53 ( ( P @ B2 @ A4 )
% 0.34/0.53 => ( P @ A4 @ B2 ) )
% 0.34/0.53 => ( P @ A2 @ B ) ) ) ).
% 0.34/0.53
% 0.34/0.53 % linorder_wlog
% 0.34/0.53 thf(fact_1186_linorder__wlog,axiom,
% 0.34/0.53 ! [P: nat > nat > $o,A2: nat,B: nat] :
% 0.34/0.53 ( ! [A4: nat,B2: nat] :
% 0.34/0.53 ( ( ord_less_eq_nat @ A4 @ B2 )
% 0.34/0.53 => ( P @ A4 @ B2 ) )
% 0.34/0.53 => ( ! [A4: nat,B2: nat] :
% 0.34/0.53 ( ( P @ B2 @ A4 )
% 0.34/0.53 => ( P @ A4 @ B2 ) )
% 0.34/0.53 => ( P @ A2 @ B ) ) ) ).
% 0.34/0.53
% 0.34/0.53 % linorder_wlog
% 0.34/0.53 thf(fact_1187_linorder__wlog,axiom,
% 0.34/0.53 ! [P: int > int > $o,A2: int,B: int] :
% 0.34/0.53 ( ! [A4: int,B2: int] :
% 0.34/0.53 ( ( ord_less_eq_int @ A4 @ B2 )
% 0.34/0.53 => ( P @ A4 @ B2 ) )
% 0.34/0.53 => ( ! [A4: int,B2: int] :
% 0.34/0.53 ( ( P @ B2 @ A4 )
% 0.34/0.53 => ( P @ A4 @ B2 ) )
% 0.34/0.53 => ( P @ A2 @ B ) ) ) ).
% 0.34/0.53
% 0.34/0.53 % linorder_wlog
% 0.34/0.53 thf(fact_1188_order__trans,axiom,
% 0.34/0.53 ! [X: filter_nat,Y: filter_nat,Z: filter_nat] :
% 0.34/0.53 ( ( ord_le2510731241096832064er_nat @ X @ Y )
% 0.34/0.53 => ( ( ord_le2510731241096832064er_nat @ Y @ Z )
% 0.34/0.53 => ( ord_le2510731241096832064er_nat @ X @ Z ) ) ) ).
% 0.34/0.53
% 0.34/0.53 % order_trans
% 0.34/0.53 thf(fact_1189_order__trans,axiom,
% 0.34/0.53 ! [X: set_nat,Y: set_nat,Z: set_nat] :
% 0.34/0.53 ( ( ord_less_eq_set_nat @ X @ Y )
% 0.34/0.53 => ( ( ord_less_eq_set_nat @ Y @ Z )
% 0.34/0.53 => ( ord_less_eq_set_nat @ X @ Z ) ) ) ).
% 0.34/0.53
% 0.34/0.53 % order_trans
% 0.34/0.53 thf(fact_1190_order__trans,axiom,
% 0.34/0.53 ! [X: rat,Y: rat,Z: rat] :
% 0.34/0.53 ( ( ord_less_eq_rat @ X @ Y )
% 0.34/0.53 => ( ( ord_less_eq_rat @ Y @ Z )
% 0.34/0.53 => ( ord_less_eq_rat @ X @ Z ) ) ) ).
% 0.34/0.53
% 0.34/0.53 % order_trans
% 0.34/0.53 thf(fact_1191_order__trans,axiom,
% 0.34/0.53 ! [X: nat,Y: nat,Z: nat] :
% 0.34/0.53 ( ( ord_less_eq_nat @ X @ Y )
% 0.34/0.53 => ( ( ord_less_eq_nat @ Y @ Z )
% 0.34/0.53 => ( ord_less_eq_nat @ X @ Z ) ) ) ).
% 0.34/0.53
% 0.34/0.53 % order_trans
% 0.34/0.53 thf(fact_1192_order__trans,axiom,
% 0.34/0.53 ! [X: int,Y: int,Z: int] :
% 0.34/0.53 ( ( ord_less_eq_int @ X @ Y )
% 0.34/0.53 => ( ( ord_less_eq_int @ Y @ Z )
% 0.34/0.53 => ( ord_less_eq_int @ X @ Z ) ) ) ).
% 0.34/0.53
% 0.34/0.53 % order_trans
% 0.34/0.53 thf(fact_1193_order_Otrans,axiom,
% 0.34/0.53 ! [A2: filter_nat,B: filter_nat,C: filter_nat] :
% 0.34/0.53 ( ( ord_le2510731241096832064er_nat @ A2 @ B )
% 0.34/0.53 => ( ( ord_le2510731241096832064er_nat @ B @ C )
% 0.34/0.53 => ( ord_le2510731241096832064er_nat @ A2 @ C ) ) ) ).
% 0.34/0.53
% 0.34/0.53 % order.trans
% 0.34/0.53 thf(fact_1194_order_Otrans,axiom,
% 0.34/0.53 ! [A2: set_nat,B: set_nat,C: set_nat] :
% 0.34/0.53 ( ( ord_less_eq_set_nat @ A2 @ B )
% 0.34/0.53 => ( ( ord_less_eq_set_nat @ B @ C )
% 0.34/0.53 => ( ord_less_eq_set_nat @ A2 @ C ) ) ) ).
% 0.34/0.53
% 0.34/0.53 % order.trans
% 0.34/0.54 thf(fact_1195_order_Otrans,axiom,
% 0.34/0.54 ! [A2: rat,B: rat,C: rat] :
% 0.34/0.54 ( ( ord_less_eq_rat @ A2 @ B )
% 0.34/0.54 => ( ( ord_less_eq_rat @ B @ C )
% 0.34/0.54 => ( ord_less_eq_rat @ A2 @ C ) ) ) ).
% 0.34/0.54
% 0.34/0.54 % order.trans
% 0.34/0.54 thf(fact_1196_order_Otrans,axiom,
% 0.34/0.54 ! [A2: nat,B: nat,C: nat] :
% 0.34/0.54 ( ( ord_less_eq_nat @ A2 @ B )
% 0.34/0.54 => ( ( ord_less_eq_nat @ B @ C )
% 0.34/0.54 => ( ord_less_eq_nat @ A2 @ C ) ) ) ).
% 0.34/0.54
% 0.34/0.54 % order.trans
% 0.34/0.54 thf(fact_1197_order_Otrans,axiom,
% 0.34/0.54 ! [A2: int,B: int,C: int] :
% 0.34/0.54 ( ( ord_less_eq_int @ A2 @ B )
% 0.34/0.54 => ( ( ord_less_eq_int @ B @ C )
% 0.34/0.54 => ( ord_less_eq_int @ A2 @ C ) ) ) ).
% 0.34/0.54
% 0.34/0.54 % order.trans
% 0.34/0.54 thf(fact_1198_order__antisym,axiom,
% 0.34/0.54 ! [X: filter_nat,Y: filter_nat] :
% 0.34/0.54 ( ( ord_le2510731241096832064er_nat @ X @ Y )
% 0.34/0.54 => ( ( ord_le2510731241096832064er_nat @ Y @ X )
% 0.34/0.54 => ( X = Y ) ) ) ).
% 0.34/0.54
% 0.34/0.54 % order_antisym
% 0.34/0.54 thf(fact_1199_order__antisym,axiom,
% 0.34/0.54 ! [X: set_nat,Y: set_nat] :
% 0.34/0.54 ( ( ord_less_eq_set_nat @ X @ Y )
% 0.34/0.54 => ( ( ord_less_eq_set_nat @ Y @ X )
% 0.34/0.54 => ( X = Y ) ) ) ).
% 0.34/0.54
% 0.34/0.54 % order_antisym
% 0.34/0.54 thf(fact_1200_order__antisym,axiom,
% 0.34/0.54 ! [X: rat,Y: rat] :
% 0.34/0.54 ( ( ord_less_eq_rat @ X @ Y )
% 0.34/0.54 => ( ( ord_less_eq_rat @ Y @ X )
% 0.34/0.54 => ( X = Y ) ) ) ).
% 0.34/0.54
% 0.34/0.54 % order_antisym
% 0.34/0.54 thf(fact_1201_order__antisym,axiom,
% 0.34/0.54 ! [X: nat,Y: nat] :
% 0.34/0.54 ( ( ord_less_eq_nat @ X @ Y )
% 0.34/0.54 => ( ( ord_less_eq_nat @ Y @ X )
% 0.34/0.54 => ( X = Y ) ) ) ).
% 0.34/0.54
% 0.34/0.54 % order_antisym
% 0.34/0.54 thf(fact_1202_order__antisym,axiom,
% 0.34/0.54 ! [X: int,Y: int] :
% 0.34/0.54 ( ( ord_less_eq_int @ X @ Y )
% 0.34/0.54 => ( ( ord_less_eq_int @ Y @ X )
% 0.34/0.54 => ( X = Y ) ) ) ).
% 0.34/0.54
% 0.34/0.54 % order_antisym
% 0.34/0.54 thf(fact_1203_ord__le__eq__trans,axiom,
% 0.34/0.54 ! [A2: filter_nat,B: filter_nat,C: filter_nat] :
% 0.34/0.54 ( ( ord_le2510731241096832064er_nat @ A2 @ B )
% 0.34/0.54 => ( ( B = C )
% 0.34/0.54 => ( ord_le2510731241096832064er_nat @ A2 @ C ) ) ) ).
% 0.34/0.54
% 0.34/0.54 % ord_le_eq_trans
% 0.34/0.54 thf(fact_1204_ord__le__eq__trans,axiom,
% 0.34/0.54 ! [A2: set_nat,B: set_nat,C: set_nat] :
% 0.34/0.54 ( ( ord_less_eq_set_nat @ A2 @ B )
% 0.34/0.54 => ( ( B = C )
% 0.34/0.54 => ( ord_less_eq_set_nat @ A2 @ C ) ) ) ).
% 0.34/0.54
% 0.34/0.54 % ord_le_eq_trans
% 0.34/0.54 thf(fact_1205_ord__le__eq__trans,axiom,
% 0.34/0.54 ! [A2: rat,B: rat,C: rat] :
% 0.34/0.54 ( ( ord_less_eq_rat @ A2 @ B )
% 0.34/0.54 => ( ( B = C )
% 0.34/0.54 => ( ord_less_eq_rat @ A2 @ C ) ) ) ).
% 0.34/0.54
% 0.34/0.54 % ord_le_eq_trans
% 0.34/0.54 thf(fact_1206_ord__le__eq__trans,axiom,
% 0.34/0.54 ! [A2: nat,B: nat,C: nat] :
% 0.34/0.54 ( ( ord_less_eq_nat @ A2 @ B )
% 0.34/0.54 => ( ( B = C )
% 0.34/0.54 => ( ord_less_eq_nat @ A2 @ C ) ) ) ).
% 0.34/0.54
% 0.34/0.54 % ord_le_eq_trans
% 0.34/0.54 thf(fact_1207_ord__le__eq__trans,axiom,
% 0.34/0.54 ! [A2: int,B: int,C: int] :
% 0.34/0.54 ( ( ord_less_eq_int @ A2 @ B )
% 0.34/0.54 => ( ( B = C )
% 0.34/0.54 => ( ord_less_eq_int @ A2 @ C ) ) ) ).
% 0.34/0.54
% 0.34/0.54 % ord_le_eq_trans
% 0.34/0.54 thf(fact_1208_ord__eq__le__trans,axiom,
% 0.34/0.54 ! [A2: filter_nat,B: filter_nat,C: filter_nat] :
% 0.34/0.54 ( ( A2 = B )
% 0.34/0.54 => ( ( ord_le2510731241096832064er_nat @ B @ C )
% 0.34/0.54 => ( ord_le2510731241096832064er_nat @ A2 @ C ) ) ) ).
% 0.34/0.54
% 0.34/0.54 % ord_eq_le_trans
% 0.34/0.54 thf(fact_1209_ord__eq__le__trans,axiom,
% 0.34/0.54 ! [A2: set_nat,B: set_nat,C: set_nat] :
% 0.34/0.54 ( ( A2 = B )
% 0.34/0.54 => ( ( ord_less_eq_set_nat @ B @ C )
% 0.34/0.54 => ( ord_less_eq_set_nat @ A2 @ C ) ) ) ).
% 0.34/0.54
% 0.34/0.54 % ord_eq_le_trans
% 0.34/0.54 thf(fact_1210_ord__eq__le__trans,axiom,
% 0.34/0.54 ! [A2: rat,B: rat,C: rat] :
% 0.34/0.54 ( ( A2 = B )
% 0.34/0.54 => ( ( ord_less_eq_rat @ B @ C )
% 0.34/0.54 => ( ord_less_eq_rat @ A2 @ C ) ) ) ).
% 0.34/0.54
% 0.34/0.54 % ord_eq_le_trans
% 0.34/0.54 thf(fact_1211_ord__eq__le__trans,axiom,
% 0.34/0.54 ! [A2: nat,B: nat,C: nat] :
% 0.34/0.54 ( ( A2 = B )
% 0.34/0.54 => ( ( ord_less_eq_nat @ B @ C )
% 0.34/0.54 => ( ord_less_eq_nat @ A2 @ C ) ) ) ).
% 0.34/0.54
% 0.34/0.54 % ord_eq_le_trans
% 0.34/0.54 thf(fact_1212_ord__eq__le__trans,axiom,
% 0.34/0.54 ! [A2: int,B: int,C: int] :
% 0.34/0.54 ( ( A2 = B )
% 0.34/0.54 => ( ( ord_less_eq_int @ B @ C )
% 0.34/0.54 => ( ord_less_eq_int @ A2 @ C ) ) ) ).
% 0.34/0.54
% 0.34/0.54 % ord_eq_le_trans
% 0.34/0.54 thf(fact_1213_order__class_Oorder__eq__iff,axiom,
% 0.34/0.54 ( ( ^ [Y4: filter_nat,Z4: filter_nat] : ( Y4 = Z4 ) )
% 0.34/0.54 = ( ^ [X2: filter_nat,Y3: filter_nat] :
% 0.34/0.54 ( ( ord_le2510731241096832064er_nat @ X2 @ Y3 )
% 0.34/0.54 & ( ord_le2510731241096832064er_nat @ Y3 @ X2 ) ) ) ) ).
% 0.34/0.54
% 0.34/0.54 % order_class.order_eq_iff
% 0.34/0.54 thf(fact_1214_order__class_Oorder__eq__iff,axiom,
% 0.34/0.54 ( ( ^ [Y4: set_nat,Z4: set_nat] : ( Y4 = Z4 ) )
% 0.34/0.54 = ( ^ [X2: set_nat,Y3: set_nat] :
% 0.34/0.54 ( ( ord_less_eq_set_nat @ X2 @ Y3 )
% 0.34/0.54 & ( ord_less_eq_set_nat @ Y3 @ X2 ) ) ) ) ).
% 0.34/0.54
% 0.34/0.54 % order_class.order_eq_iff
% 0.34/0.54 thf(fact_1215_order__class_Oorder__eq__iff,axiom,
% 0.34/0.54 ( ( ^ [Y4: rat,Z4: rat] : ( Y4 = Z4 ) )
% 0.34/0.54 = ( ^ [X2: rat,Y3: rat] :
% 0.34/0.54 ( ( ord_less_eq_rat @ X2 @ Y3 )
% 0.34/0.54 & ( ord_less_eq_rat @ Y3 @ X2 ) ) ) ) ).
% 0.34/0.54
% 0.34/0.54 % order_class.order_eq_iff
% 0.34/0.54 thf(fact_1216_order__class_Oorder__eq__iff,axiom,
% 0.34/0.54 ( ( ^ [Y4: nat,Z4: nat] : ( Y4 = Z4 ) )
% 0.34/0.54 = ( ^ [X2: nat,Y3: nat] :
% 0.34/0.54 ( ( ord_less_eq_nat @ X2 @ Y3 )
% 0.34/0.54 & ( ord_less_eq_nat @ Y3 @ X2 ) ) ) ) ).
% 0.34/0.54
% 0.34/0.54 % order_class.order_eq_iff
% 0.34/0.54 thf(fact_1217_order__class_Oorder__eq__iff,axiom,
% 0.34/0.54 ( ( ^ [Y4: int,Z4: int] : ( Y4 = Z4 ) )
% 0.34/0.54 = ( ^ [X2: int,Y3: int] :
% 0.34/0.54 ( ( ord_less_eq_int @ X2 @ Y3 )
% 0.34/0.54 & ( ord_less_eq_int @ Y3 @ X2 ) ) ) ) ).
% 0.34/0.54
% 0.34/0.54 % order_class.order_eq_iff
% 0.34/0.54 thf(fact_1218_le__cases3,axiom,
% 0.34/0.54 ! [X: rat,Y: rat,Z: rat] :
% 0.34/0.54 ( ( ( ord_less_eq_rat @ X @ Y )
% 0.34/0.54 => ~ ( ord_less_eq_rat @ Y @ Z ) )
% 0.34/0.54 => ( ( ( ord_less_eq_rat @ Y @ X )
% 0.34/0.54 => ~ ( ord_less_eq_rat @ X @ Z ) )
% 0.34/0.54 => ( ( ( ord_less_eq_rat @ X @ Z )
% 0.34/0.54 => ~ ( ord_less_eq_rat @ Z @ Y ) )
% 0.34/0.54 => ( ( ( ord_less_eq_rat @ Z @ Y )
% 0.34/0.54 => ~ ( ord_less_eq_rat @ Y @ X ) )
% 0.34/0.54 => ( ( ( ord_less_eq_rat @ Y @ Z )
% 0.34/0.54 => ~ ( ord_less_eq_rat @ Z @ X ) )
% 0.34/0.54 => ~ ( ( ord_less_eq_rat @ Z @ X )
% 0.34/0.54 => ~ ( ord_less_eq_rat @ X @ Y ) ) ) ) ) ) ) ).
% 0.34/0.54
% 0.34/0.54 % le_cases3
% 0.34/0.54 thf(fact_1219_le__cases3,axiom,
% 0.34/0.54 ! [X: nat,Y: nat,Z: nat] :
% 0.34/0.54 ( ( ( ord_less_eq_nat @ X @ Y )
% 0.34/0.54 => ~ ( ord_less_eq_nat @ Y @ Z ) )
% 0.34/0.54 => ( ( ( ord_less_eq_nat @ Y @ X )
% 0.34/0.54 => ~ ( ord_less_eq_nat @ X @ Z ) )
% 0.34/0.54 => ( ( ( ord_less_eq_nat @ X @ Z )
% 0.34/0.54 => ~ ( ord_less_eq_nat @ Z @ Y ) )
% 0.34/0.54 => ( ( ( ord_less_eq_nat @ Z @ Y )
% 0.34/0.54 => ~ ( ord_less_eq_nat @ Y @ X ) )
% 0.34/0.54 => ( ( ( ord_less_eq_nat @ Y @ Z )
% 0.34/0.54 => ~ ( ord_less_eq_nat @ Z @ X ) )
% 0.34/0.54 => ~ ( ( ord_less_eq_nat @ Z @ X )
% 0.34/0.54 => ~ ( ord_less_eq_nat @ X @ Y ) ) ) ) ) ) ) ).
% 0.34/0.54
% 0.34/0.54 % le_cases3
% 0.34/0.54 thf(fact_1220_le__cases3,axiom,
% 0.34/0.54 ! [X: int,Y: int,Z: int] :
% 0.34/0.54 ( ( ( ord_less_eq_int @ X @ Y )
% 0.34/0.54 => ~ ( ord_less_eq_int @ Y @ Z ) )
% 0.34/0.54 => ( ( ( ord_less_eq_int @ Y @ X )
% 0.34/0.54 => ~ ( ord_less_eq_int @ X @ Z ) )
% 0.34/0.54 => ( ( ( ord_less_eq_int @ X @ Z )
% 0.34/0.54 => ~ ( ord_less_eq_int @ Z @ Y ) )
% 0.34/0.54 => ( ( ( ord_less_eq_int @ Z @ Y )
% 0.34/0.54 => ~ ( ord_less_eq_int @ Y @ X ) )
% 0.34/0.54 => ( ( ( ord_less_eq_int @ Y @ Z )
% 0.34/0.54 => ~ ( ord_less_eq_int @ Z @ X ) )
% 0.34/0.54 => ~ ( ( ord_less_eq_int @ Z @ X )
% 0.34/0.54 => ~ ( ord_less_eq_int @ X @ Y ) ) ) ) ) ) ) ).
% 0.34/0.54
% 0.34/0.54 % le_cases3
% 0.34/0.54 thf(fact_1221_nle__le,axiom,
% 0.34/0.54 ! [A2: rat,B: rat] :
% 0.34/0.54 ( ( ~ ( ord_less_eq_rat @ A2 @ B ) )
% 0.34/0.54 = ( ( ord_less_eq_rat @ B @ A2 )
% 0.34/0.54 & ( B != A2 ) ) ) ).
% 0.34/0.54
% 0.34/0.54 % nle_le
% 0.34/0.54 thf(fact_1222_nle__le,axiom,
% 0.34/0.54 ! [A2: nat,B: nat] :
% 0.34/0.54 ( ( ~ ( ord_less_eq_nat @ A2 @ B ) )
% 0.34/0.54 = ( ( ord_less_eq_nat @ B @ A2 )
% 0.34/0.54 & ( B != A2 ) ) ) ).
% 0.34/0.54
% 0.34/0.54 % nle_le
% 0.34/0.54 thf(fact_1223_nle__le,axiom,
% 0.34/0.54 ! [A2: int,B: int] :
% 0.34/0.54 ( ( ~ ( ord_less_eq_int @ A2 @ B ) )
% 0.34/0.54 = ( ( ord_less_eq_int @ B @ A2 )
% 0.34/0.54 & ( B != A2 ) ) ) ).
% 0.34/0.54
% 0.34/0.54 % nle_le
% 0.34/0.54 thf(fact_1224_Sigma__mono,axiom,
% 0.34/0.54 ! [A: set_Pr4532377907799695533_nat_o,C5: set_Pr4532377907799695533_nat_o,B4: ( produc3658429121746597890et_nat > $o ) > set_nat,D3: ( produc3658429121746597890et_nat > $o ) > set_nat] :
% 0.34/0.54 ( ( ord_le2965882846123202637_nat_o @ A @ C5 )
% 0.34/0.54 => ( ! [X3: produc3658429121746597890et_nat > $o] :
% 0.34/0.54 ( ( member6576561426505652726_nat_o @ X3 @ A )
% 0.34/0.54 => ( ord_less_eq_set_nat @ ( B4 @ X3 ) @ ( D3 @ X3 ) ) )
% 0.34/0.54 => ( ord_le4460016228128491106_o_nat @ ( produc3142209312014666069_o_nat @ A @ B4 ) @ ( produc3142209312014666069_o_nat @ C5 @ D3 ) ) ) ) ).
% 0.34/0.54
% 0.34/0.54 % Sigma_mono
% 0.34/0.54 thf(fact_1225_Sigma__mono,axiom,
% 0.34/0.54 ! [A: set_int,C5: set_int,B4: int > set_nat,D3: int > set_nat] :
% 0.34/0.54 ( ( ord_less_eq_set_int @ A @ C5 )
% 0.34/0.54 => ( ! [X3: int] :
% 0.34/0.54 ( ( member_int @ X3 @ A )
% 0.34/0.54 => ( ord_less_eq_set_nat @ ( B4 @ X3 ) @ ( D3 @ X3 ) ) )
% 0.34/0.54 => ( ord_le5333435103577920093nt_nat @ ( produc1456381018704787142nt_nat @ A @ B4 ) @ ( produc1456381018704787142nt_nat @ C5 @ D3 ) ) ) ) ).
% 0.34/0.54
% 0.34/0.54 % Sigma_mono
% 0.34/0.54 thf(fact_1226_Sigma__mono,axiom,
% 0.34/0.54 ! [A: set_o,C5: set_o,B4: $o > set_nat,D3: $o > set_nat] :
% 0.34/0.54 ( ( ord_less_eq_set_o @ A @ C5 )
% 0.34/0.54 => ( ! [X3: $o] :
% 0.34/0.54 ( ( member_o @ X3 @ A )
% 0.34/0.54 => ( ord_less_eq_set_nat @ ( B4 @ X3 ) @ ( D3 @ X3 ) ) )
% 0.34/0.54 => ( ord_le6472378536984071597_o_nat @ ( product_Sigma_o_nat @ A @ B4 ) @ ( product_Sigma_o_nat @ C5 @ D3 ) ) ) ) ).
% 0.34/0.54
% 0.34/0.54 % Sigma_mono
% 0.34/0.54 thf(fact_1227_Sigma__mono,axiom,
% 0.34/0.54 ! [A: set_Product_unit,C5: set_Product_unit,B4: product_unit > set_nat,D3: product_unit > set_nat] :
% 0.34/0.54 ( ( ord_le3507040750410214029t_unit @ A @ C5 )
% 0.34/0.54 => ( ! [X3: product_unit] :
% 0.34/0.54 ( ( member_Product_unit @ X3 @ A )
% 0.34/0.54 => ( ord_less_eq_set_nat @ ( B4 @ X3 ) @ ( D3 @ X3 ) ) )
% 0.34/0.54 => ( ord_le8121013740822022818it_nat @ ( produc6875377002296012693it_nat @ A @ B4 ) @ ( produc6875377002296012693it_nat @ C5 @ D3 ) ) ) ) ).
% 0.34/0.54
% 0.34/0.54 % Sigma_mono
% 0.34/0.54 thf(fact_1228_Sigma__mono,axiom,
% 0.34/0.54 ! [A: set_nat,C5: set_nat,B4: nat > set_nat,D3: nat > set_nat] :
% 0.34/0.54 ( ( ord_less_eq_set_nat @ A @ C5 )
% 0.34/0.54 => ( ! [X3: nat] :
% 0.34/0.54 ( ( member_nat @ X3 @ A )
% 0.34/0.54 => ( ord_less_eq_set_nat @ ( B4 @ X3 ) @ ( D3 @ X3 ) ) )
% 0.34/0.54 => ( ord_le3146513528884898305at_nat @ ( produc457027306803732586at_nat @ A @ B4 ) @ ( produc457027306803732586at_nat @ C5 @ D3 ) ) ) ) ).
% 0.34/0.54
% 0.34/0.54 % Sigma_mono
% 0.34/0.54 thf(fact_1229_rtrancl__Un__separator__converseE,axiom,
% 0.34/0.54 ! [A2: nat,B: nat,P: set_Pr1261947904930325089at_nat,Q2: set_Pr1261947904930325089at_nat] :
% 0.34/0.54 ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A2 @ B ) @ ( transi2905341329935302413cl_nat @ ( sup_su6327502436637775413at_nat @ P @ Q2 ) ) )
% 0.34/0.54 => ( ! [X3: nat,Y2: nat] :
% 0.34/0.54 ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X3 @ B ) @ ( transi2905341329935302413cl_nat @ P ) )
% 0.34/0.54 => ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ Y2 @ X3 ) @ Q2 )
% 0.34/0.54 => ( Y2 = X3 ) ) )
% 0.34/0.54 => ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A2 @ B ) @ ( transi2905341329935302413cl_nat @ P ) ) ) ) ).
% 0.34/0.54
% 0.34/0.54 % rtrancl_Un_separator_converseE
% 0.34/0.54 thf(fact_1230_rtrancl__Un__separator__converseE,axiom,
% 0.34/0.54 ! [A2: int,B: int,P: set_Pr958786334691620121nt_int,Q2: set_Pr958786334691620121nt_int] :
% 0.34/0.54 ( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ A2 @ B ) @ ( transi2902850859426252137cl_int @ ( sup_su6024340866399070445nt_int @ P @ Q2 ) ) )
% 0.34/0.54 => ( ! [X3: int,Y2: int] :
% 0.34/0.54 ( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ X3 @ B ) @ ( transi2902850859426252137cl_int @ P ) )
% 0.34/0.54 => ( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ Y2 @ X3 ) @ Q2 )
% 0.34/0.54 => ( Y2 = X3 ) ) )
% 0.34/0.54 => ( member5262025264175285858nt_int @ ( product_Pair_int_int @ A2 @ B ) @ ( transi2902850859426252137cl_int @ P ) ) ) ) ).
% 0.34/0.54
% 0.34/0.54 % rtrancl_Un_separator_converseE
% 0.34/0.54 thf(fact_1231_rtrancl__Un__separatorE,axiom,
% 0.34/0.54 ! [A2: nat,B: nat,P: set_Pr1261947904930325089at_nat,Q2: set_Pr1261947904930325089at_nat] :
% 0.34/0.54 ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A2 @ B ) @ ( transi2905341329935302413cl_nat @ ( sup_su6327502436637775413at_nat @ P @ Q2 ) ) )
% 0.34/0.54 => ( ! [X3: nat,Y2: nat] :
% 0.34/0.54 ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A2 @ X3 ) @ ( transi2905341329935302413cl_nat @ P ) )
% 0.34/0.54 => ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X3 @ Y2 ) @ Q2 )
% 0.34/0.54 => ( X3 = Y2 ) ) )
% 0.34/0.54 => ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A2 @ B ) @ ( transi2905341329935302413cl_nat @ P ) ) ) ) ).
% 0.34/0.54
% 0.34/0.54 % rtrancl_Un_separatorE
% 0.34/0.54 thf(fact_1232_rtrancl__Un__separatorE,axiom,
% 0.34/0.54 ! [A2: int,B: int,P: set_Pr958786334691620121nt_int,Q2: set_Pr958786334691620121nt_int] :
% 0.34/0.54 ( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ A2 @ B ) @ ( transi2902850859426252137cl_int @ ( sup_su6024340866399070445nt_int @ P @ Q2 ) ) )
% 0.34/0.54 => ( ! [X3: int,Y2: int] :
% 0.34/0.54 ( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ A2 @ X3 ) @ ( transi2902850859426252137cl_int @ P ) )
% 0.34/0.54 => ( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ X3 @ Y2 ) @ Q2 )
% 0.34/0.54 => ( X3 = Y2 ) ) )
% 0.34/0.54 => ( member5262025264175285858nt_int @ ( product_Pair_int_int @ A2 @ B ) @ ( transi2902850859426252137cl_int @ P ) ) ) ) ).
% 0.34/0.54
% 0.34/0.54 % rtrancl_Un_separatorE
% 0.34/0.54 thf(fact_1233_disjoint__mono,axiom,
% 0.34/0.54 ! [A2: set_Product_unit,A5: set_Product_unit,B: set_Product_unit,B3: set_Product_unit] :
% 0.34/0.54 ( ( ord_le3507040750410214029t_unit @ A2 @ A5 )
% 0.34/0.54 => ( ( ord_le3507040750410214029t_unit @ B @ B3 )
% 0.34/0.54 => ( ( ( inf_in4660618365625256667t_unit @ A5 @ B3 )
% 0.34/0.54 = bot_bo3957492148770167129t_unit )
% 0.34/0.54 => ( ( inf_in4660618365625256667t_unit @ A2 @ B )
% 0.34/0.54 = bot_bo3957492148770167129t_unit ) ) ) ) ).
% 0.34/0.54
% 0.34/0.54 % disjoint_mono
% 0.34/0.54 thf(fact_1234_disjoint__mono,axiom,
% 0.34/0.54 ! [A2: set_o,A5: set_o,B: set_o,B3: set_o] :
% 0.34/0.54 ( ( ord_less_eq_set_o @ A2 @ A5 )
% 0.34/0.54 => ( ( ord_less_eq_set_o @ B @ B3 )
% 0.34/0.54 => ( ( ( inf_inf_set_o @ A5 @ B3 )
% 0.34/0.54 = bot_bot_set_o )
% 0.34/0.54 => ( ( inf_inf_set_o @ A2 @ B )
% 0.34/0.54 = bot_bot_set_o ) ) ) ) ).
% 0.34/0.54
% 0.34/0.54 % disjoint_mono
% 0.34/0.54 thf(fact_1235_disjoint__mono,axiom,
% 0.34/0.54 ! [A2: set_int,A5: set_int,B: set_int,B3: set_int] :
% 0.34/0.54 ( ( ord_less_eq_set_int @ A2 @ A5 )
% 0.34/0.54 => ( ( ord_less_eq_set_int @ B @ B3 )
% 0.34/0.54 => ( ( ( inf_inf_set_int @ A5 @ B3 )
% 0.34/0.54 = bot_bot_set_int )
% 0.34/0.54 => ( ( inf_inf_set_int @ A2 @ B )
% 0.34/0.54 = bot_bot_set_int ) ) ) ) ).
% 0.34/0.54
% 0.34/0.54 % disjoint_mono
% 0.34/0.54 thf(fact_1236_disjoint__mono,axiom,
% 0.34/0.54 ! [A2: set_Pr4532377907799695533_nat_o,A5: set_Pr4532377907799695533_nat_o,B: set_Pr4532377907799695533_nat_o,B3: set_Pr4532377907799695533_nat_o] :
% 0.34/0.54 ( ( ord_le2965882846123202637_nat_o @ A2 @ A5 )
% 0.34/0.54 => ( ( ord_le2965882846123202637_nat_o @ B @ B3 )
% 0.34/0.54 => ( ( ( inf_in1906310914598751387_nat_o @ A5 @ B3 )
% 0.34/0.54 = bot_bo7824918357723582233_nat_o )
% 0.34/0.54 => ( ( inf_in1906310914598751387_nat_o @ A2 @ B )
% 0.34/0.54 = bot_bo7824918357723582233_nat_o ) ) ) ) ).
% 0.34/0.54
% 0.34/0.54 % disjoint_mono
% 0.34/0.54 thf(fact_1237_disjoint__mono,axiom,
% 0.34/0.54 ! [A2: set_nat,A5: set_nat,B: set_nat,B3: set_nat] :
% 0.34/0.54 ( ( ord_less_eq_set_nat @ A2 @ A5 )
% 0.34/0.54 => ( ( ord_less_eq_set_nat @ B @ B3 )
% 0.34/0.54 => ( ( ( inf_inf_set_nat @ A5 @ B3 )
% 0.34/0.54 = bot_bot_set_nat )
% 0.34/0.54 => ( ( inf_inf_set_nat @ A2 @ B )
% 0.34/0.54 = bot_bot_set_nat ) ) ) ) ).
% 0.34/0.54
% 0.34/0.54 % disjoint_mono
% 0.34/0.54 thf(fact_1238_rtrancl__image__unfold__right,axiom,
% 0.34/0.54 ! [E: set_Pr1261947904930325089at_nat,V2: set_nat] : ( ord_less_eq_set_nat @ ( image_nat_nat @ E @ ( image_nat_nat @ ( transi2905341329935302413cl_nat @ E ) @ V2 ) ) @ ( image_nat_nat @ ( transi2905341329935302413cl_nat @ E ) @ V2 ) ) ).
% 0.34/0.54
% 0.34/0.54 % rtrancl_image_unfold_right
% 0.34/0.54 thf(fact_1239_rtrancl__reachable__induct,axiom,
% 0.34/0.54 ! [I: set_nat,INV: set_nat,E: set_Pr1261947904930325089at_nat] :
% 0.34/0.54 ( ( ord_less_eq_set_nat @ I @ INV )
% 0.34/0.54 => ( ( ord_less_eq_set_nat @ ( image_nat_nat @ E @ INV ) @ INV )
% 0.34/0.54 => ( ord_less_eq_set_nat @ ( image_nat_nat @ ( transi2905341329935302413cl_nat @ E ) @ I ) @ INV ) ) ) ).
% 0.34/0.54
% 0.34/0.54 % rtrancl_reachable_induct
% 0.34/0.54 thf(fact_1240_Image__Int__subset,axiom,
% 0.34/0.54 ! [R3: set_Pr1261947904930325089at_nat,A: set_nat,B4: set_nat] : ( ord_less_eq_set_nat @ ( image_nat_nat @ R3 @ ( inf_inf_set_nat @ A @ B4 ) ) @ ( inf_inf_set_nat @ ( image_nat_nat @ R3 @ A ) @ ( image_nat_nat @ R3 @ B4 ) ) ) ).
% 0.34/0.54
% 0.34/0.54 % Image_Int_subset
% 0.34/0.54 thf(fact_1241_Image__mono,axiom,
% 0.34/0.54 ! [R6: set_Pr1261947904930325089at_nat,R2: set_Pr1261947904930325089at_nat,A8: set_nat,A: set_nat] :
% 0.34/0.54 ( ( ord_le3146513528884898305at_nat @ R6 @ R2 )
% 0.34/0.54 => ( ( ord_less_eq_set_nat @ A8 @ A )
% 0.34/0.54 => ( ord_less_eq_set_nat @ ( image_nat_nat @ R6 @ A8 ) @ ( image_nat_nat @ R2 @ A ) ) ) ) ).
% 0.34/0.54
% 0.34/0.54 % Image_mono
% 0.34/0.54 thf(fact_1242_refl__on__Un,axiom,
% 0.34/0.54 ! [A: set_nat,R2: set_Pr1261947904930325089at_nat,B4: set_nat,S2: set_Pr1261947904930325089at_nat] :
% 0.34/0.54 ( ( refl_on_nat @ A @ R2 )
% 0.34/0.54 => ( ( refl_on_nat @ B4 @ S2 )
% 0.34/0.54 => ( refl_on_nat @ ( sup_sup_set_nat @ A @ B4 ) @ ( sup_su6327502436637775413at_nat @ R2 @ S2 ) ) ) ) ).
% 0.34/0.54
% 0.34/0.54 % refl_on_Un
% 0.34/0.54 thf(fact_1243_refl__on__Int,axiom,
% 0.34/0.54 ! [A: set_nat,R2: set_Pr1261947904930325089at_nat,B4: set_nat,S2: set_Pr1261947904930325089at_nat] :
% 0.34/0.54 ( ( refl_on_nat @ A @ R2 )
% 0.34/0.54 => ( ( refl_on_nat @ B4 @ S2 )
% 0.34/0.54 => ( refl_on_nat @ ( inf_inf_set_nat @ A @ B4 ) @ ( inf_in2572325071724192079at_nat @ R2 @ S2 ) ) ) ) ).
% 0.34/0.54
% 0.34/0.54 % refl_on_Int
% 0.34/0.54 thf(fact_1244_trancl__union__outside,axiom,
% 0.34/0.54 ! [V: nat,W: nat,E: set_Pr1261947904930325089at_nat,U2: set_Pr1261947904930325089at_nat] :
% 0.34/0.54 ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ V @ W ) @ ( transi6264000038957366511cl_nat @ ( sup_su6327502436637775413at_nat @ E @ U2 ) ) )
% 0.34/0.54 => ( ~ ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ V @ W ) @ ( transi6264000038957366511cl_nat @ E ) )
% 0.34/0.54 => ? [X3: nat,Y2: nat] :
% 0.34/0.54 ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ V @ X3 ) @ ( transi2905341329935302413cl_nat @ ( sup_su6327502436637775413at_nat @ E @ U2 ) ) )
% 0.34/0.54 & ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X3 @ Y2 ) @ U2 )
% 0.34/0.54 & ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ Y2 @ W ) @ ( transi2905341329935302413cl_nat @ ( sup_su6327502436637775413at_nat @ E @ U2 ) ) ) ) ) ) ).
% 0.34/0.54
% 0.34/0.54 % trancl_union_outside
% 0.34/0.54 thf(fact_1245_trancl__union__outside,axiom,
% 0.34/0.54 ! [V: int,W: int,E: set_Pr958786334691620121nt_int,U2: set_Pr958786334691620121nt_int] :
% 0.34/0.54 ( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ V @ W ) @ ( transi6261509568448316235cl_int @ ( sup_su6024340866399070445nt_int @ E @ U2 ) ) )
% 0.34/0.54 => ( ~ ( member5262025264175285858nt_int @ ( product_Pair_int_int @ V @ W ) @ ( transi6261509568448316235cl_int @ E ) )
% 0.34/0.54 => ? [X3: int,Y2: int] :
% 0.34/0.54 ( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ V @ X3 ) @ ( transi2902850859426252137cl_int @ ( sup_su6024340866399070445nt_int @ E @ U2 ) ) )
% 0.34/0.54 & ( member5262025264175285858nt_int @ ( product_Pair_int_int @ X3 @ Y2 ) @ U2 )
% 0.34/0.54 & ( member5262025264175285858nt_int @ ( product_Pair_int_int @ Y2 @ W ) @ ( transi2902850859426252137cl_int @ ( sup_su6024340866399070445nt_int @ E @ U2 ) ) ) ) ) ) ).
% 0.34/0.54
% 0.34/0.54 % trancl_union_outside
% 0.34/0.54 thf(fact_1246_SigmaE,axiom,
% 0.34/0.54 ! [C: product_prod_nat_int,A: set_nat,B4: nat > set_int] :
% 0.34/0.54 ( ( member4262671552274231302at_int @ C @ ( produc454536836294682310at_int @ A @ B4 ) )
% 0.34/0.54 => ~ ! [X3: nat] :
% 0.34/0.54 ( ( member_nat @ X3 @ A )
% 0.34/0.54 => ! [Y2: int] :
% 0.34/0.54 ( ( member_int @ Y2 @ ( B4 @ X3 ) )
% 0.34/0.54 => ( C
% 0.34/0.54 != ( product_Pair_nat_int @ X3 @ Y2 ) ) ) ) ) ).
% 0.34/0.54
% 0.34/0.54 % SigmaE
% 0.34/0.54 thf(fact_1247_SigmaE,axiom,
% 0.34/0.54 ! [C: product_prod_nat_o,A: set_nat,B4: nat > set_o] :
% 0.34/0.54 ( ( member6310962623043647828_nat_o @ C @ ( product_Sigma_nat_o @ A @ B4 ) )
% 0.34/0.54 => ~ ! [X3: nat] :
% 0.34/0.54 ( ( member_nat @ X3 @ A )
% 0.34/0.54 => ! [Y2: $o] :
% 0.34/0.54 ( ( member_o @ Y2 @ ( B4 @ X3 ) )
% 0.34/0.54 => ( C
% 0.34/0.54 != ( product_Pair_nat_o @ X3 @ Y2 ) ) ) ) ) ).
% 0.34/0.54
% 0.34/0.54 % SigmaE
% 0.34/0.54 thf(fact_1248_SigmaE,axiom,
% 0.34/0.54 ! [C: produc526117512892780850t_unit,A: set_nat,B4: nat > set_Product_unit] :
% 0.34/0.54 ( ( member348563419383025609t_unit @ C @ ( produc5506539147108004339t_unit @ A @ B4 ) )
% 0.34/0.54 => ~ ! [X3: nat] :
% 0.34/0.54 ( ( member_nat @ X3 @ A )
% 0.34/0.54 => ! [Y2: product_unit] :
% 0.34/0.54 ( ( member_Product_unit @ Y2 @ ( B4 @ X3 ) )
% 0.34/0.54 => ( C
% 0.34/0.54 != ( produc3434450474625521892t_unit @ X3 @ Y2 ) ) ) ) ) ).
% 0.34/0.54
% 0.34/0.54 % SigmaE
% 0.34/0.54 thf(fact_1249_SigmaE,axiom,
% 0.34/0.54 ! [C: product_prod_int_nat,A: set_int,B4: int > set_nat] :
% 0.34/0.54 ( ( member216504246829706758nt_nat @ C @ ( produc1456381018704787142nt_nat @ A @ B4 ) )
% 0.34/0.54 => ~ ! [X3: int] :
% 0.34/0.54 ( ( member_int @ X3 @ A )
% 0.34/0.54 => ! [Y2: nat] :
% 0.34/0.54 ( ( member_nat @ Y2 @ ( B4 @ X3 ) )
% 0.34/0.54 => ( C
% 0.34/0.54 != ( product_Pair_int_nat @ X3 @ Y2 ) ) ) ) ) ).
% 0.34/0.54
% 0.34/0.54 % SigmaE
% 0.34/0.54 thf(fact_1250_SigmaE,axiom,
% 0.34/0.54 ! [C: product_prod_int_o,A: set_int,B4: int > set_o] :
% 0.34/0.54 ( ( member4489920277610959864_int_o @ C @ ( product_Sigma_int_o @ A @ B4 ) )
% 0.34/0.54 => ~ ! [X3: int] :
% 0.34/0.54 ( ( member_int @ X3 @ A )
% 0.34/0.54 => ! [Y2: $o] :
% 0.34/0.54 ( ( member_o @ Y2 @ ( B4 @ X3 ) )
% 0.34/0.54 => ( C
% 0.34/0.54 != ( product_Pair_int_o @ X3 @ Y2 ) ) ) ) ) ).
% 0.34/0.54
% 0.34/0.54 % SigmaE
% 0.34/0.54 thf(fact_1251_SigmaE,axiom,
% 0.34/0.54 ! [C: produc6865911496949736918t_unit,A: set_int,B4: int > set_Product_unit] :
% 0.34/0.54 ( ( member6688357403439981677t_unit @ C @ ( produc2622961094310184599t_unit @ A @ B4 ) )
% 0.34/0.54 => ~ ! [X3: int] :
% 0.34/0.54 ( ( member_int @ X3 @ A )
% 0.34/0.54 => ! [Y2: product_unit] :
% 0.34/0.54 ( ( member_Product_unit @ Y2 @ ( B4 @ X3 ) )
% 0.34/0.54 => ( C
% 0.34/0.54 != ( produc550872421827702152t_unit @ X3 @ Y2 ) ) ) ) ) ).
% 0.34/0.54
% 0.34/0.54 % SigmaE
% 0.34/0.54 thf(fact_1252_SigmaE,axiom,
% 0.34/0.54 ! [C: product_prod_o_nat,A: set_o,B4: $o > set_nat] :
% 0.34/0.54 ( ( member2802428098988154798_o_nat @ C @ ( product_Sigma_o_nat @ A @ B4 ) )
% 0.34/0.54 => ~ ! [X3: $o] :
% 0.34/0.54 ( ( member_o @ X3 @ A )
% 0.34/0.54 => ! [Y2: nat] :
% 0.34/0.54 ( ( member_nat @ Y2 @ ( B4 @ X3 ) )
% 0.34/0.54 => ( C
% 0.34/0.54 != ( product_Pair_o_nat @ X3 @ Y2 ) ) ) ) ) ).
% 0.34/0.54
% 0.34/0.54 % SigmaE
% 0.34/0.54 thf(fact_1253_SigmaE,axiom,
% 0.34/0.54 ! [C: product_prod_o_int,A: set_o,B4: $o > set_int] :
% 0.34/0.54 ( ( member7847949116333733898_o_int @ C @ ( product_Sigma_o_int @ A @ B4 ) )
% 0.34/0.54 => ~ ! [X3: $o] :
% 0.34/0.54 ( ( member_o @ X3 @ A )
% 0.34/0.54 => ! [Y2: int] :
% 0.34/0.54 ( ( member_int @ Y2 @ ( B4 @ X3 ) )
% 0.34/0.54 => ( C
% 0.34/0.54 != ( product_Pair_o_int @ X3 @ Y2 ) ) ) ) ) ).
% 0.34/0.54
% 0.34/0.54 % SigmaE
% 0.34/0.54 thf(fact_1254_SigmaE,axiom,
% 0.34/0.54 ! [C: product_prod_o_o,A: set_o,B4: $o > set_o] :
% 0.34/0.54 ( ( member7466972457876170832od_o_o @ C @ ( product_Sigma_o_o @ A @ B4 ) )
% 0.34/0.54 => ~ ! [X3: $o] :
% 0.34/0.54 ( ( member_o @ X3 @ A )
% 0.34/0.54 => ! [Y2: $o] :
% 0.34/0.54 ( ( member_o @ Y2 @ ( B4 @ X3 ) )
% 0.34/0.54 => ( C
% 0.34/0.54 != ( product_Pair_o_o @ X3 @ Y2 ) ) ) ) ) ).
% 0.34/0.54
% 0.34/0.54 % SigmaE
% 0.34/0.54 thf(fact_1255_SigmaE,axiom,
% 0.34/0.54 ! [C: produc3788275437394581276t_unit,A: set_o,B4: $o > set_Product_unit] :
% 0.34/0.54 ( ( member9215336023991391429t_unit @ C @ ( produc411009820408249917t_unit @ A @ B4 ) )
% 0.34/0.54 => ~ ! [X3: $o] :
% 0.34/0.54 ( ( member_o @ X3 @ A )
% 0.34/0.54 => ! [Y2: product_unit] :
% 0.34/0.54 ( ( member_Product_unit @ Y2 @ ( B4 @ X3 ) )
% 0.34/0.54 => ( C
% 0.34/0.54 != ( produc3901946521647900940t_unit @ X3 @ Y2 ) ) ) ) ) ).
% 0.34/0.54
% 0.34/0.54 % SigmaE
% 0.34/0.54 thf(fact_1256_SigmaD1,axiom,
% 0.34/0.54 ! [A2: heap_e7401611519738050253t_unit,B: set_nat,A: set_he8644174446132254637t_unit,B4: heap_e7401611519738050253t_unit > set_set_nat] :
% 0.34/0.54 ( ( member6260224972018164377et_nat @ ( produc7507926704131184380et_nat @ A2 @ B ) @ ( produc5260988393495149579et_nat @ A @ B4 ) )
% 0.34/0.54 => ( member242389114213480694t_unit @ A2 @ A ) ) ).
% 0.34/0.54
% 0.34/0.54 % SigmaD1
% 0.34/0.54 thf(fact_1257_SigmaD1,axiom,
% 0.34/0.54 ! [A2: nat,B: nat,A: set_nat,B4: nat > set_nat] :
% 0.34/0.54 ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A2 @ B ) @ ( produc457027306803732586at_nat @ A @ B4 ) )
% 0.34/0.54 => ( member_nat @ A2 @ A ) ) ).
% 0.34/0.54
% 0.34/0.54 % SigmaD1
% 0.34/0.54 thf(fact_1258_SigmaD1,axiom,
% 0.34/0.54 ! [A2: int,B: int,A: set_int,B4: int > set_int] :
% 0.34/0.54 ( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ A2 @ B ) @ ( produc1453890548195736866nt_int @ A @ B4 ) )
% 0.34/0.54 => ( member_int @ A2 @ A ) ) ).
% 0.34/0.54
% 0.34/0.54 % SigmaD1
% 0.34/0.54 thf(fact_1259_SigmaD1,axiom,
% 0.34/0.54 ! [A2: produc3658429121746597890et_nat > $o,B: produc3658429121746597890et_nat,A: set_Pr4532377907799695533_nat_o,B4: ( produc3658429121746597890et_nat > $o ) > set_Pr3948176798113811640et_nat] :
% 0.34/0.54 ( ( member1996754912294343701et_nat @ ( produc5001842942810119800et_nat @ A2 @ B ) @ ( produc8052021730408606599et_nat @ A @ B4 ) )
% 0.34/0.54 => ( member6576561426505652726_nat_o @ A2 @ A ) ) ).
% 0.34/0.54
% 0.34/0.54 % SigmaD1
% 0.34/0.54 thf(fact_1260_SigmaD1,axiom,
% 0.34/0.54 ! [A2: produc3658429121746597890et_nat > $o,B: produc3925858234332021118et_nat,A: set_Pr4532377907799695533_nat_o,B4: ( produc3658429121746597890et_nat > $o ) > set_Pr3286484037609594932et_nat] :
% 0.34/0.54 ( ( member6124377750444531601et_nat @ ( produc2245416461498447860et_nat @ A2 @ B ) @ ( produc7016705354093686019et_nat @ A @ B4 ) )
% 0.34/0.54 => ( member6576561426505652726_nat_o @ A2 @ A ) ) ).
% 0.34/0.54
% 0.34/0.54 % SigmaD1
% 0.34/0.54 thf(fact_1261_SigmaD2,axiom,
% 0.34/0.54 ! [A2: heap_e7401611519738050253t_unit,B: set_nat,A: set_he8644174446132254637t_unit,B4: heap_e7401611519738050253t_unit > set_set_nat] :
% 0.34/0.54 ( ( member6260224972018164377et_nat @ ( produc7507926704131184380et_nat @ A2 @ B ) @ ( produc5260988393495149579et_nat @ A @ B4 ) )
% 0.34/0.54 => ( member_set_nat @ B @ ( B4 @ A2 ) ) ) ).
% 0.34/0.54
% 0.34/0.54 % SigmaD2
% 0.34/0.54 thf(fact_1262_SigmaD2,axiom,
% 0.34/0.54 ! [A2: nat,B: nat,A: set_nat,B4: nat > set_nat] :
% 0.34/0.54 ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A2 @ B ) @ ( produc457027306803732586at_nat @ A @ B4 ) )
% 0.34/0.54 => ( member_nat @ B @ ( B4 @ A2 ) ) ) ).
% 0.34/0.54
% 0.34/0.54 % SigmaD2
% 0.34/0.54 thf(fact_1263_SigmaD2,axiom,
% 0.34/0.54 ! [A2: int,B: int,A: set_int,B4: int > set_int] :
% 0.34/0.54 ( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ A2 @ B ) @ ( produc1453890548195736866nt_int @ A @ B4 ) )
% 0.34/0.54 => ( member_int @ B @ ( B4 @ A2 ) ) ) ).
% 0.34/0.54
% 0.34/0.54 % SigmaD2
% 0.34/0.54 thf(fact_1264_SigmaD2,axiom,
% 0.34/0.54 ! [A2: produc3658429121746597890et_nat > $o,B: produc3658429121746597890et_nat,A: set_Pr4532377907799695533_nat_o,B4: ( produc3658429121746597890et_nat > $o ) > set_Pr3948176798113811640et_nat] :
% 0.34/0.54 ( ( member1996754912294343701et_nat @ ( produc5001842942810119800et_nat @ A2 @ B ) @ ( produc8052021730408606599et_nat @ A @ B4 ) )
% 0.34/0.54 => ( member6260224972018164377et_nat @ B @ ( B4 @ A2 ) ) ) ).
% 0.34/0.54
% 0.34/0.54 % SigmaD2
% 0.34/0.54 thf(fact_1265_SigmaD2,axiom,
% 0.34/0.54 ! [A2: produc3658429121746597890et_nat > $o,B: produc3925858234332021118et_nat,A: set_Pr4532377907799695533_nat_o,B4: ( produc3658429121746597890et_nat > $o ) > set_Pr3286484037609594932et_nat] :
% 0.34/0.54 ( ( member6124377750444531601et_nat @ ( produc2245416461498447860et_nat @ A2 @ B ) @ ( produc7016705354093686019et_nat @ A @ B4 ) )
% 0.34/0.54 => ( member1996754912294343701et_nat @ B @ ( B4 @ A2 ) ) ) ).
% 0.34/0.54
% 0.34/0.54 % SigmaD2
% 0.34/0.54 thf(fact_1266_SigmaE2,axiom,
% 0.34/0.54 ! [A2: nat,B: int,A: set_nat,B4: nat > set_int] :
% 0.34/0.54 ( ( member4262671552274231302at_int @ ( product_Pair_nat_int @ A2 @ B ) @ ( produc454536836294682310at_int @ A @ B4 ) )
% 0.34/0.54 => ~ ( ( member_nat @ A2 @ A )
% 0.34/0.54 => ~ ( member_int @ B @ ( B4 @ A2 ) ) ) ) ).
% 0.34/0.54
% 0.34/0.54 % SigmaE2
% 0.34/0.54 thf(fact_1267_SigmaE2,axiom,
% 0.34/0.54 ! [A2: nat,B: $o,A: set_nat,B4: nat > set_o] :
% 0.34/0.54 ( ( member6310962623043647828_nat_o @ ( product_Pair_nat_o @ A2 @ B ) @ ( product_Sigma_nat_o @ A @ B4 ) )
% 0.34/0.54 => ~ ( ( member_nat @ A2 @ A )
% 0.34/0.54 => ~ ( member_o @ B @ ( B4 @ A2 ) ) ) ) ).
% 0.34/0.54
% 0.34/0.54 % SigmaE2
% 0.34/0.54 thf(fact_1268_SigmaE2,axiom,
% 0.34/0.54 ! [A2: nat,B: product_unit,A: set_nat,B4: nat > set_Product_unit] :
% 0.34/0.54 ( ( member348563419383025609t_unit @ ( produc3434450474625521892t_unit @ A2 @ B ) @ ( produc5506539147108004339t_unit @ A @ B4 ) )
% 0.34/0.54 => ~ ( ( member_nat @ A2 @ A )
% 0.34/0.54 => ~ ( member_Product_unit @ B @ ( B4 @ A2 ) ) ) ) ).
% 0.34/0.54
% 0.34/0.54 % SigmaE2
% 0.34/0.54 thf(fact_1269_SigmaE2,axiom,
% 0.34/0.54 ! [A2: int,B: nat,A: set_int,B4: int > set_nat] :
% 0.34/0.54 ( ( member216504246829706758nt_nat @ ( product_Pair_int_nat @ A2 @ B ) @ ( produc1456381018704787142nt_nat @ A @ B4 ) )
% 0.34/0.54 => ~ ( ( member_int @ A2 @ A )
% 0.34/0.54 => ~ ( member_nat @ B @ ( B4 @ A2 ) ) ) ) ).
% 0.34/0.54
% 0.34/0.54 % SigmaE2
% 0.34/0.54 thf(fact_1270_SigmaE2,axiom,
% 0.34/0.54 ! [A2: int,B: $o,A: set_int,B4: int > set_o] :
% 0.34/0.54 ( ( member4489920277610959864_int_o @ ( product_Pair_int_o @ A2 @ B ) @ ( product_Sigma_int_o @ A @ B4 ) )
% 0.34/0.54 => ~ ( ( member_int @ A2 @ A )
% 0.34/0.54 => ~ ( member_o @ B @ ( B4 @ A2 ) ) ) ) ).
% 0.34/0.54
% 0.34/0.54 % SigmaE2
% 0.34/0.54 thf(fact_1271_SigmaE2,axiom,
% 0.34/0.54 ! [A2: int,B: product_unit,A: set_int,B4: int > set_Product_unit] :
% 0.34/0.54 ( ( member6688357403439981677t_unit @ ( produc550872421827702152t_unit @ A2 @ B ) @ ( produc2622961094310184599t_unit @ A @ B4 ) )
% 0.34/0.54 => ~ ( ( member_int @ A2 @ A )
% 0.34/0.54 => ~ ( member_Product_unit @ B @ ( B4 @ A2 ) ) ) ) ).
% 0.34/0.54
% 0.34/0.54 % SigmaE2
% 0.34/0.54 thf(fact_1272_SigmaE2,axiom,
% 0.34/0.54 ! [A2: $o,B: nat,A: set_o,B4: $o > set_nat] :
% 0.34/0.54 ( ( member2802428098988154798_o_nat @ ( product_Pair_o_nat @ A2 @ B ) @ ( product_Sigma_o_nat @ A @ B4 ) )
% 0.34/0.54 => ~ ( ( member_o @ A2 @ A )
% 0.34/0.54 => ~ ( member_nat @ B @ ( B4 @ A2 ) ) ) ) ).
% 0.34/0.54
% 0.34/0.54 % SigmaE2
% 0.34/0.54 thf(fact_1273_SigmaE2,axiom,
% 0.34/0.54 ! [A2: $o,B: int,A: set_o,B4: $o > set_int] :
% 0.34/0.54 ( ( member7847949116333733898_o_int @ ( product_Pair_o_int @ A2 @ B ) @ ( product_Sigma_o_int @ A @ B4 ) )
% 0.34/0.54 => ~ ( ( member_o @ A2 @ A )
% 0.34/0.54 => ~ ( member_int @ B @ ( B4 @ A2 ) ) ) ) ).
% 0.34/0.54
% 0.34/0.54 % SigmaE2
% 0.34/0.54 thf(fact_1274_SigmaE2,axiom,
% 0.34/0.54 ! [A2: $o,B: $o,A: set_o,B4: $o > set_o] :
% 0.34/0.54 ( ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ A2 @ B ) @ ( product_Sigma_o_o @ A @ B4 ) )
% 0.34/0.54 => ~ ( ( member_o @ A2 @ A )
% 0.34/0.54 => ~ ( member_o @ B @ ( B4 @ A2 ) ) ) ) ).
% 0.34/0.54
% 0.34/0.54 % SigmaE2
% 0.34/0.54 thf(fact_1275_SigmaE2,axiom,
% 0.34/0.54 ! [A2: $o,B: product_unit,A: set_o,B4: $o > set_Product_unit] :
% 0.34/0.54 ( ( member9215336023991391429t_unit @ ( produc3901946521647900940t_unit @ A2 @ B ) @ ( produc411009820408249917t_unit @ A @ B4 ) )
% 0.34/0.54 => ~ ( ( member_o @ A2 @ A )
% 0.34/0.54 => ~ ( member_Product_unit @ B @ ( B4 @ A2 ) ) ) ) ).
% 0.34/0.54
% 0.34/0.54 % SigmaE2
% 0.34/0.54 thf(fact_1276_trancl__image__by__rtrancl,axiom,
% 0.34/0.54 ! [E: set_Pr1261947904930325089at_nat,Vi: set_nat] :
% 0.34/0.54 ( ( sup_sup_set_nat @ ( image_nat_nat @ ( transi6264000038957366511cl_nat @ E ) @ Vi ) @ Vi )
% 0.34/0.54 = ( image_nat_nat @ ( transi2905341329935302413cl_nat @ E ) @ Vi ) ) ).
% 0.34/0.54
% 0.34/0.54 % trancl_image_by_rtrancl
% 0.34/0.54 thf(fact_1277_converse__rtranclE_H,axiom,
% 0.34/0.54 ! [U: nat,V: nat,R3: set_Pr1261947904930325089at_nat] :
% 0.34/0.54 ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ U @ V ) @ ( transi2905341329935302413cl_nat @ R3 ) )
% 0.34/0.54 => ( ( U != V )
% 0.34/0.54 => ~ ! [Vh: nat] :
% 0.34/0.54 ( ( U != Vh )
% 0.34/0.54 => ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ U @ Vh ) @ R3 )
% 0.34/0.54 => ~ ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ Vh @ V ) @ ( transi2905341329935302413cl_nat @ R3 ) ) ) ) ) ) ).
% 0.34/0.54
% 0.34/0.54 % converse_rtranclE'
% 0.34/0.54 thf(fact_1278_converse__rtranclE_H,axiom,
% 0.34/0.54 ! [U: int,V: int,R3: set_Pr958786334691620121nt_int] :
% 0.34/0.54 ( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ U @ V ) @ ( transi2902850859426252137cl_int @ R3 ) )
% 0.34/0.54 => ( ( U != V )
% 0.34/0.54 => ~ ! [Vh: int] :
% 0.34/0.54 ( ( U != Vh )
% 0.34/0.54 => ( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ U @ Vh ) @ R3 )
% 0.34/0.54 => ~ ( member5262025264175285858nt_int @ ( product_Pair_int_int @ Vh @ V ) @ ( transi2902850859426252137cl_int @ R3 ) ) ) ) ) ) ).
% 0.34/0.54
% 0.34/0.54 % converse_rtranclE'
% 0.34/0.54 thf(fact_1279_rtrancl_Ocases,axiom,
% 0.34/0.54 ! [A1: nat,A22: nat,R2: set_Pr1261947904930325089at_nat] :
% 0.34/0.54 ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A1 @ A22 ) @ ( transi2905341329935302413cl_nat @ R2 ) )
% 0.34/0.54 => ( ( A22 != A1 )
% 0.34/0.54 => ~ ! [B2: nat] :
% 0.34/0.54 ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A1 @ B2 ) @ ( transi2905341329935302413cl_nat @ R2 ) )
% 0.34/0.54 => ~ ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ B2 @ A22 ) @ R2 ) ) ) ) ).
% 0.34/0.54
% 0.34/0.54 % rtrancl.cases
% 0.34/0.54 thf(fact_1280_rtrancl_Ocases,axiom,
% 0.34/0.54 ! [A1: int,A22: int,R2: set_Pr958786334691620121nt_int] :
% 0.34/0.54 ( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ A1 @ A22 ) @ ( transi2902850859426252137cl_int @ R2 ) )
% 0.34/0.54 => ( ( A22 != A1 )
% 0.34/0.54 => ~ ! [B2: int] :
% 0.34/0.54 ( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ A1 @ B2 ) @ ( transi2902850859426252137cl_int @ R2 ) )
% 0.34/0.54 => ~ ( member5262025264175285858nt_int @ ( product_Pair_int_int @ B2 @ A22 ) @ R2 ) ) ) ) ).
% 0.34/0.54
% 0.34/0.54 % rtrancl.cases
% 0.34/0.54 thf(fact_1281_rtrancl_Osimps,axiom,
% 0.34/0.54 ! [A1: nat,A22: nat,R2: set_Pr1261947904930325089at_nat] :
% 0.34/0.54 ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A1 @ A22 ) @ ( transi2905341329935302413cl_nat @ R2 ) )
% 0.34/0.54 = ( ? [A6: nat] :
% 0.34/0.54 ( ( A1 = A6 )
% 0.34/0.54 & ( A22 = A6 ) )
% 0.34/0.54 | ? [A6: nat,B6: nat,C4: nat] :
% 0.34/0.54 ( ( A1 = A6 )
% 0.34/0.54 & ( A22 = C4 )
% 0.34/0.54 & ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A6 @ B6 ) @ ( transi2905341329935302413cl_nat @ R2 ) )
% 0.34/0.54 & ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ B6 @ C4 ) @ R2 ) ) ) ) ).
% 0.34/0.54
% 0.34/0.54 % rtrancl.simps
% 0.34/0.54 thf(fact_1282_rtrancl_Osimps,axiom,
% 0.34/0.54 ! [A1: int,A22: int,R2: set_Pr958786334691620121nt_int] :
% 0.34/0.54 ( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ A1 @ A22 ) @ ( transi2902850859426252137cl_int @ R2 ) )
% 0.34/0.54 = ( ? [A6: int] :
% 0.34/0.54 ( ( A1 = A6 )
% 0.34/0.54 & ( A22 = A6 ) )
% 0.34/0.54 | ? [A6: int,B6: int,C4: int] :
% 0.34/0.54 ( ( A1 = A6 )
% 0.34/0.54 & ( A22 = C4 )
% 0.34/0.54 & ( member5262025264175285858nt_int @ ( product_Pair_int_int @ A6 @ B6 ) @ ( transi2902850859426252137cl_int @ R2 ) )
% 0.34/0.54 & ( member5262025264175285858nt_int @ ( product_Pair_int_int @ B6 @ C4 ) @ R2 ) ) ) ) ).
% 0.34/0.54
% 0.34/0.54 % rtrancl.simps
% 0.34/0.54 thf(fact_1283_rtrancl_Ortrancl__refl,axiom,
% 0.34/0.54 ! [A2: nat,R2: set_Pr1261947904930325089at_nat] : ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A2 @ A2 ) @ ( transi2905341329935302413cl_nat @ R2 ) ) ).
% 0.34/0.54
% 0.34/0.54 % rtrancl.rtrancl_refl
% 0.34/0.54 thf(fact_1284_rtrancl_Ortrancl__refl,axiom,
% 0.34/0.54 ! [A2: int,R2: set_Pr958786334691620121nt_int] : ( member5262025264175285858nt_int @ ( product_Pair_int_int @ A2 @ A2 ) @ ( transi2902850859426252137cl_int @ R2 ) ) ).
% 0.34/0.54
% 0.34/0.54 % rtrancl.rtrancl_refl
% 0.34/0.54 thf(fact_1285_rtrancl_Ortrancl__into__rtrancl,axiom,
% 0.34/0.54 ! [A2: nat,B: nat,R2: set_Pr1261947904930325089at_nat,C: nat] :
% 0.34/0.54 ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A2 @ B ) @ ( transi2905341329935302413cl_nat @ R2 ) )
% 0.34/0.54 => ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ B @ C ) @ R2 )
% 0.34/0.54 => ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A2 @ C ) @ ( transi2905341329935302413cl_nat @ R2 ) ) ) ) ).
% 0.34/0.54
% 0.34/0.54 % rtrancl.rtrancl_into_rtrancl
% 0.34/0.54 thf(fact_1286_rtrancl_Ortrancl__into__rtrancl,axiom,
% 0.34/0.54 ! [A2: int,B: int,R2: set_Pr958786334691620121nt_int,C: int] :
% 0.34/0.54 ( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ A2 @ B ) @ ( transi2902850859426252137cl_int @ R2 ) )
% 0.34/0.54 => ( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ B @ C ) @ R2 )
% 0.34/0.54 => ( member5262025264175285858nt_int @ ( product_Pair_int_int @ A2 @ C ) @ ( transi2902850859426252137cl_int @ R2 ) ) ) ) ).
% 0.34/0.54
% 0.34/0.54 % rtrancl.rtrancl_into_rtrancl
% 0.34/0.54 thf(fact_1287_rtranclE,axiom,
% 0.34/0.54 ! [A2: nat,B: nat,R2: set_Pr1261947904930325089at_nat] :
% 0.34/0.54 ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A2 @ B ) @ ( transi2905341329935302413cl_nat @ R2 ) )
% 0.34/0.54 => ( ( A2 != B )
% 0.34/0.54 => ~ ! [Y2: nat] :
% 0.34/0.54 ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A2 @ Y2 ) @ ( transi2905341329935302413cl_nat @ R2 ) )
% 0.34/0.54 => ~ ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ Y2 @ B ) @ R2 ) ) ) ) ).
% 0.34/0.54
% 0.34/0.54 % rtranclE
% 0.34/0.54 thf(fact_1288_rtranclE,axiom,
% 0.34/0.54 ! [A2: int,B: int,R2: set_Pr958786334691620121nt_int] :
% 0.34/0.54 ( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ A2 @ B ) @ ( transi2902850859426252137cl_int @ R2 ) )
% 0.34/0.54 => ( ( A2 != B )
% 0.34/0.54 => ~ ! [Y2: int] :
% 0.34/0.54 ( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ A2 @ Y2 ) @ ( transi2902850859426252137cl_int @ R2 ) )
% 0.34/0.54 => ~ ( member5262025264175285858nt_int @ ( product_Pair_int_int @ Y2 @ B ) @ R2 ) ) ) ) ).
% 0.34/0.54
% 0.34/0.54 % rtranclE
% 0.34/0.54 thf(fact_1289_rtrancl__trans,axiom,
% 0.34/0.54 ! [X: nat,Y: nat,R2: set_Pr1261947904930325089at_nat,Z: nat] :
% 0.34/0.54 ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ Y ) @ ( transi2905341329935302413cl_nat @ R2 ) )
% 0.34/0.54 => ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ Y @ Z ) @ ( transi2905341329935302413cl_nat @ R2 ) )
% 0.34/0.54 => ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ Z ) @ ( transi2905341329935302413cl_nat @ R2 ) ) ) ) ).
% 0.34/0.54
% 0.34/0.54 % rtrancl_trans
% 0.34/0.54 thf(fact_1290_rtrancl__trans,axiom,
% 0.34/0.54 ! [X: int,Y: int,R2: set_Pr958786334691620121nt_int,Z: int] :
% 0.34/0.54 ( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ X @ Y ) @ ( transi2902850859426252137cl_int @ R2 ) )
% 0.34/0.54 => ( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ Y @ Z ) @ ( transi2902850859426252137cl_int @ R2 ) )
% 0.34/0.54 => ( member5262025264175285858nt_int @ ( product_Pair_int_int @ X @ Z ) @ ( transi2902850859426252137cl_int @ R2 ) ) ) ) ).
% 0.34/0.54
% 0.34/0.54 % rtrancl_trans
% 0.34/0.54 thf(fact_1291_rtrancl__induct,axiom,
% 0.34/0.54 ! [A2: nat,B: nat,R2: set_Pr1261947904930325089at_nat,P: nat > $o] :
% 0.34/0.54 ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A2 @ B ) @ ( transi2905341329935302413cl_nat @ R2 ) )
% 0.34/0.54 => ( ( P @ A2 )
% 0.34/0.54 => ( ! [Y2: nat,Z3: nat] :
% 0.34/0.54 ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A2 @ Y2 ) @ ( transi2905341329935302413cl_nat @ R2 ) )
% 0.34/0.54 => ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ Y2 @ Z3 ) @ R2 )
% 0.34/0.54 => ( ( P @ Y2 )
% 0.34/0.54 => ( P @ Z3 ) ) ) )
% 0.34/0.54 => ( P @ B ) ) ) ) ).
% 0.34/0.54
% 0.34/0.54 % rtrancl_induct
% 0.34/0.54 thf(fact_1292_rtrancl__induct,axiom,
% 0.34/0.54 ! [A2: int,B: int,R2: set_Pr958786334691620121nt_int,P: int > $o] :
% 0.34/0.54 ( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ A2 @ B ) @ ( transi2902850859426252137cl_int @ R2 ) )
% 0.34/0.54 => ( ( P @ A2 )
% 0.34/0.54 => ( ! [Y2: int,Z3: int] :
% 0.34/0.54 ( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ A2 @ Y2 ) @ ( transi2902850859426252137cl_int @ R2 ) )
% 0.34/0.54 => ( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ Y2 @ Z3 ) @ R2 )
% 0.34/0.54 => ( ( P @ Y2 )
% 0.34/0.54 => ( P @ Z3 ) ) ) )
% 0.34/0.54 => ( P @ B ) ) ) ) ).
% 0.34/0.54
% 0.34/0.54 % rtrancl_induct
% 0.34/0.54 thf(fact_1293_converse__rtranclE,axiom,
% 0.34/0.54 ! [X: nat,Z: nat,R2: set_Pr1261947904930325089at_nat] :
% 0.34/0.54 ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ Z ) @ ( transi2905341329935302413cl_nat @ R2 ) )
% 0.34/0.54 => ( ( X != Z )
% 0.34/0.54 => ~ ! [Y2: nat] :
% 0.34/0.54 ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ Y2 ) @ R2 )
% 0.34/0.54 => ~ ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ Y2 @ Z ) @ ( transi2905341329935302413cl_nat @ R2 ) ) ) ) ) ).
% 0.34/0.54
% 0.34/0.54 % converse_rtranclE
% 0.34/0.54 thf(fact_1294_converse__rtranclE,axiom,
% 0.34/0.54 ! [X: int,Z: int,R2: set_Pr958786334691620121nt_int] :
% 0.34/0.54 ( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ X @ Z ) @ ( transi2902850859426252137cl_int @ R2 ) )
% 0.34/0.54 => ( ( X != Z )
% 0.34/0.54 => ~ ! [Y2: int] :
% 0.34/0.54 ( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ X @ Y2 ) @ R2 )
% 0.34/0.54 => ~ ( member5262025264175285858nt_int @ ( product_Pair_int_int @ Y2 @ Z ) @ ( transi2902850859426252137cl_int @ R2 ) ) ) ) ) ).
% 0.34/0.54
% 0.34/0.54 % converse_rtranclE
% 0.34/0.54 thf(fact_1295_converse__rtrancl__induct,axiom,
% 0.34/0.54 ! [A2: nat,B: nat,R2: set_Pr1261947904930325089at_nat,P: nat > $o] :
% 0.34/0.54 ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A2 @ B ) @ ( transi2905341329935302413cl_nat @ R2 ) )
% 0.34/0.54 => ( ( P @ B )
% 0.34/0.54 => ( ! [Y2: nat,Z3: nat] :
% 0.34/0.54 ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ Y2 @ Z3 ) @ R2 )
% 0.34/0.54 => ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ Z3 @ B ) @ ( transi2905341329935302413cl_nat @ R2 ) )
% 0.34/0.54 => ( ( P @ Z3 )
% 0.34/0.54 => ( P @ Y2 ) ) ) )
% 0.34/0.54 => ( P @ A2 ) ) ) ) ).
% 0.34/0.54
% 0.34/0.54 % converse_rtrancl_induct
% 0.34/0.54 thf(fact_1296_converse__rtrancl__induct,axiom,
% 0.34/0.54 ! [A2: int,B: int,R2: set_Pr958786334691620121nt_int,P: int > $o] :
% 0.34/0.54 ( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ A2 @ B ) @ ( transi2902850859426252137cl_int @ R2 ) )
% 0.34/0.54 => ( ( P @ B )
% 0.34/0.54 => ( ! [Y2: int,Z3: int] :
% 0.34/0.54 ( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ Y2 @ Z3 ) @ R2 )
% 0.34/0.54 => ( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ Z3 @ B ) @ ( transi2902850859426252137cl_int @ R2 ) )
% 0.34/0.54 => ( ( P @ Z3 )
% 0.34/0.54 => ( P @ Y2 ) ) ) )
% 0.34/0.54 => ( P @ A2 ) ) ) ) ).
% 0.34/0.54
% 0.34/0.54 % converse_rtrancl_induct
% 0.34/0.54 thf(fact_1297_converse__rtrancl__into__rtrancl,axiom,
% 0.34/0.54 ! [A2: nat,B: nat,R2: set_Pr1261947904930325089at_nat,C: nat] :
% 0.34/0.54 ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A2 @ B ) @ R2 )
% 0.34/0.54 => ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ B @ C ) @ ( transi2905341329935302413cl_nat @ R2 ) )
% 0.34/0.54 => ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A2 @ C ) @ ( transi2905341329935302413cl_nat @ R2 ) ) ) ) ).
% 0.34/0.54
% 0.34/0.54 % converse_rtrancl_into_rtrancl
% 0.34/0.54 thf(fact_1298_converse__rtrancl__into__rtrancl,axiom,
% 0.34/0.54 ! [A2: int,B: int,R2: set_Pr958786334691620121nt_int,C: int] :
% 0.34/0.54 ( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ A2 @ B ) @ R2 )
% 0.34/0.54 => ( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ B @ C ) @ ( transi2902850859426252137cl_int @ R2 ) )
% 0.34/0.54 => ( member5262025264175285858nt_int @ ( product_Pair_int_int @ A2 @ C ) @ ( transi2902850859426252137cl_int @ R2 ) ) ) ) ).
% 0.34/0.54
% 0.34/0.54 % converse_rtrancl_into_rtrancl
% 0.34/0.54 thf(fact_1299_rtrancl__induct2,axiom,
% 0.34/0.54 ! [Ax: heap_e7401611519738050253t_unit,Ay: set_nat,Bx: heap_e7401611519738050253t_unit,By: set_nat,R2: set_Pr719794911490849221et_nat,P: heap_e7401611519738050253t_unit > set_nat > $o] :
% 0.34/0.54 ( ( member6099555550032318734et_nat @ ( produc8199053930788261021et_nat @ ( produc7507926704131184380et_nat @ Ax @ Ay ) @ ( produc7507926704131184380et_nat @ Bx @ By ) ) @ ( transi6919426408697524543et_nat @ R2 ) )
% 0.34/0.54 => ( ( P @ Ax @ Ay )
% 0.34/0.54 => ( ! [A4: heap_e7401611519738050253t_unit,B2: set_nat,Aa: heap_e7401611519738050253t_unit,Ba: set_nat] :
% 0.34/0.54 ( ( member6099555550032318734et_nat @ ( produc8199053930788261021et_nat @ ( produc7507926704131184380et_nat @ Ax @ Ay ) @ ( produc7507926704131184380et_nat @ A4 @ B2 ) ) @ ( transi6919426408697524543et_nat @ R2 ) )
% 0.34/0.54 => ( ( member6099555550032318734et_nat @ ( produc8199053930788261021et_nat @ ( produc7507926704131184380et_nat @ A4 @ B2 ) @ ( produc7507926704131184380et_nat @ Aa @ Ba ) ) @ R2 )
% 0.34/0.54 => ( ( P @ A4 @ B2 )
% 0.34/0.54 => ( P @ Aa @ Ba ) ) ) )
% 0.34/0.54 => ( P @ Bx @ By ) ) ) ) ).
% 0.34/0.54
% 0.34/0.54 % rtrancl_induct2
% 0.34/0.54 thf(fact_1300_rtrancl__induct2,axiom,
% 0.34/0.54 ! [Ax: nat,Ay: nat,Bx: nat,By: nat,R2: set_Pr8693737435421807431at_nat,P: nat > nat > $o] :
% 0.34/0.54 ( ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ ( product_Pair_nat_nat @ Ax @ Ay ) @ ( product_Pair_nat_nat @ Bx @ By ) ) @ ( transi8609417484261754244at_nat @ R2 ) )
% 0.34/0.54 => ( ( P @ Ax @ Ay )
% 0.34/0.54 => ( ! [A4: nat,B2: nat,Aa: nat,Ba: nat] :
% 0.34/0.54 ( ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ ( product_Pair_nat_nat @ Ax @ Ay ) @ ( product_Pair_nat_nat @ A4 @ B2 ) ) @ ( transi8609417484261754244at_nat @ R2 ) )
% 0.34/0.54 => ( ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ ( product_Pair_nat_nat @ A4 @ B2 ) @ ( product_Pair_nat_nat @ Aa @ Ba ) ) @ R2 )
% 0.34/0.54 => ( ( P @ A4 @ B2 )
% 0.34/0.54 => ( P @ Aa @ Ba ) ) ) )
% 0.34/0.54 => ( P @ Bx @ By ) ) ) ) ).
% 0.34/0.54
% 0.34/0.54 % rtrancl_induct2
% 0.34/0.54 thf(fact_1301_rtrancl__induct2,axiom,
% 0.34/0.54 ! [Ax: int,Ay: int,Bx: int,By: int,R2: set_Pr2560585780119916871nt_int,P: int > int > $o] :
% 0.34/0.54 ( ( member8566619992076573584nt_int @ ( produc3646306378393792727nt_int @ ( product_Pair_int_int @ Ax @ Ay ) @ ( product_Pair_int_int @ Bx @ By ) ) @ ( transi5430920176653612092nt_int @ R2 ) )
% 0.34/0.54 => ( ( P @ Ax @ Ay )
% 0.34/0.54 => ( ! [A4: int,B2: int,Aa: int,Ba: int] :
% 0.34/0.54 ( ( member8566619992076573584nt_int @ ( produc3646306378393792727nt_int @ ( product_Pair_int_int @ Ax @ Ay ) @ ( product_Pair_int_int @ A4 @ B2 ) ) @ ( transi5430920176653612092nt_int @ R2 ) )
% 0.34/0.54 => ( ( member8566619992076573584nt_int @ ( produc3646306378393792727nt_int @ ( product_Pair_int_int @ A4 @ B2 ) @ ( product_Pair_int_int @ Aa @ Ba ) ) @ R2 )
% 0.34/0.54 => ( ( P @ A4 @ B2 )
% 0.34/0.54 => ( P @ Aa @ Ba ) ) ) )
% 0.34/0.54 => ( P @ Bx @ By ) ) ) ) ).
% 0.34/0.54
% 0.34/0.54 % rtrancl_induct2
% 0.34/0.54 thf(fact_1302_rtrancl__induct2,axiom,
% 0.34/0.54 ! [Ax: produc3658429121746597890et_nat > $o,Ay: produc3658429121746597890et_nat,Bx: produc3658429121746597890et_nat > $o,By: produc3658429121746597890et_nat,R2: set_Pr7928877670098842301et_nat,P: ( produc3658429121746597890et_nat > $o ) > produc3658429121746597890et_nat > $o] :
% 0.34/0.54 ( ( member4763271486408492550et_nat @ ( produc8599840265553166229et_nat @ ( produc5001842942810119800et_nat @ Ax @ Ay ) @ ( produc5001842942810119800et_nat @ Bx @ By ) ) @ ( transi2414916928880341691et_nat @ R2 ) )
% 0.34/0.54 => ( ( P @ Ax @ Ay )
% 0.34/0.54 => ( ! [A4: produc3658429121746597890et_nat > $o,B2: produc3658429121746597890et_nat,Aa: produc3658429121746597890et_nat > $o,Ba: produc3658429121746597890et_nat] :
% 0.34/0.54 ( ( member4763271486408492550et_nat @ ( produc8599840265553166229et_nat @ ( produc5001842942810119800et_nat @ Ax @ Ay ) @ ( produc5001842942810119800et_nat @ A4 @ B2 ) ) @ ( transi2414916928880341691et_nat @ R2 ) )
% 0.34/0.54 => ( ( member4763271486408492550et_nat @ ( produc8599840265553166229et_nat @ ( produc5001842942810119800et_nat @ A4 @ B2 ) @ ( produc5001842942810119800et_nat @ Aa @ Ba ) ) @ R2 )
% 0.34/0.54 => ( ( P @ A4 @ B2 )
% 0.34/0.54 => ( P @ Aa @ Ba ) ) ) )
% 0.34/0.54 => ( P @ Bx @ By ) ) ) ) ).
% 0.34/0.54
% 0.34/0.54 % rtrancl_induct2
% 0.34/0.54 thf(fact_1303_rtrancl__induct2,axiom,
% 0.34/0.54 ! [Ax: produc3658429121746597890et_nat > $o,Ay: produc3925858234332021118et_nat,Bx: produc3658429121746597890et_nat > $o,By: produc3925858234332021118et_nat,R2: set_Pr3444600963470892981et_nat,P: ( produc3658429121746597890et_nat > $o ) > produc3925858234332021118et_nat > $o] :
% 0.34/0.54 ( ( member6341495586645257982et_nat @ ( produc1940133919992309389et_nat @ ( produc2245416461498447860et_nat @ Ax @ Ay ) @ ( produc2245416461498447860et_nat @ Bx @ By ) ) @ ( transi344104063936286263et_nat @ R2 ) )
% 0.34/0.54 => ( ( P @ Ax @ Ay )
% 0.34/0.54 => ( ! [A4: produc3658429121746597890et_nat > $o,B2: produc3925858234332021118et_nat,Aa: produc3658429121746597890et_nat > $o,Ba: produc3925858234332021118et_nat] :
% 0.34/0.54 ( ( member6341495586645257982et_nat @ ( produc1940133919992309389et_nat @ ( produc2245416461498447860et_nat @ Ax @ Ay ) @ ( produc2245416461498447860et_nat @ A4 @ B2 ) ) @ ( transi344104063936286263et_nat @ R2 ) )
% 0.34/0.54 => ( ( member6341495586645257982et_nat @ ( produc1940133919992309389et_nat @ ( produc2245416461498447860et_nat @ A4 @ B2 ) @ ( produc2245416461498447860et_nat @ Aa @ Ba ) ) @ R2 )
% 0.34/0.54 => ( ( P @ A4 @ B2 )
% 0.34/0.54 => ( P @ Aa @ Ba ) ) ) )
% 0.34/0.54 => ( P @ Bx @ By ) ) ) ) ).
% 0.34/0.54
% 0.34/0.54 % rtrancl_induct2
% 0.34/0.54 thf(fact_1304_converse__rtranclE2,axiom,
% 0.34/0.54 ! [Xa: heap_e7401611519738050253t_unit,Xb: set_nat,Za: heap_e7401611519738050253t_unit,Zb: set_nat,R2: set_Pr719794911490849221et_nat] :
% 0.34/0.54 ( ( member6099555550032318734et_nat @ ( produc8199053930788261021et_nat @ ( produc7507926704131184380et_nat @ Xa @ Xb ) @ ( produc7507926704131184380et_nat @ Za @ Zb ) ) @ ( transi6919426408697524543et_nat @ R2 ) )
% 0.34/0.54 => ( ( ( produc7507926704131184380et_nat @ Xa @ Xb )
% 0.34/0.54 != ( produc7507926704131184380et_nat @ Za @ Zb ) )
% 0.34/0.54 => ~ ! [A4: heap_e7401611519738050253t_unit,B2: set_nat] :
% 0.34/0.54 ( ( member6099555550032318734et_nat @ ( produc8199053930788261021et_nat @ ( produc7507926704131184380et_nat @ Xa @ Xb ) @ ( produc7507926704131184380et_nat @ A4 @ B2 ) ) @ R2 )
% 0.34/0.54 => ~ ( member6099555550032318734et_nat @ ( produc8199053930788261021et_nat @ ( produc7507926704131184380et_nat @ A4 @ B2 ) @ ( produc7507926704131184380et_nat @ Za @ Zb ) ) @ ( transi6919426408697524543et_nat @ R2 ) ) ) ) ) ).
% 0.34/0.54
% 0.34/0.54 % converse_rtranclE2
% 0.34/0.54 thf(fact_1305_converse__rtranclE2,axiom,
% 0.34/0.54 ! [Xa: nat,Xb: nat,Za: nat,Zb: nat,R2: set_Pr8693737435421807431at_nat] :
% 0.34/0.54 ( ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ ( product_Pair_nat_nat @ Xa @ Xb ) @ ( product_Pair_nat_nat @ Za @ Zb ) ) @ ( transi8609417484261754244at_nat @ R2 ) )
% 0.34/0.54 => ( ( ( product_Pair_nat_nat @ Xa @ Xb )
% 0.34/0.54 != ( product_Pair_nat_nat @ Za @ Zb ) )
% 0.34/0.54 => ~ ! [A4: nat,B2: nat] :
% 0.34/0.54 ( ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ ( product_Pair_nat_nat @ Xa @ Xb ) @ ( product_Pair_nat_nat @ A4 @ B2 ) ) @ R2 )
% 0.34/0.54 => ~ ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ ( product_Pair_nat_nat @ A4 @ B2 ) @ ( product_Pair_nat_nat @ Za @ Zb ) ) @ ( transi8609417484261754244at_nat @ R2 ) ) ) ) ) ).
% 0.34/0.54
% 0.34/0.54 % converse_rtranclE2
% 0.34/0.54 thf(fact_1306_converse__rtranclE2,axiom,
% 0.34/0.54 ! [Xa: int,Xb: int,Za: int,Zb: int,R2: set_Pr2560585780119916871nt_int] :
% 0.34/0.54 ( ( member8566619992076573584nt_int @ ( produc3646306378393792727nt_int @ ( product_Pair_int_int @ Xa @ Xb ) @ ( product_Pair_int_int @ Za @ Zb ) ) @ ( transi5430920176653612092nt_int @ R2 ) )
% 0.34/0.54 => ( ( ( product_Pair_int_int @ Xa @ Xb )
% 0.34/0.54 != ( product_Pair_int_int @ Za @ Zb ) )
% 0.34/0.54 => ~ ! [A4: int,B2: int] :
% 0.34/0.54 ( ( member8566619992076573584nt_int @ ( produc3646306378393792727nt_int @ ( product_Pair_int_int @ Xa @ Xb ) @ ( product_Pair_int_int @ A4 @ B2 ) ) @ R2 )
% 0.34/0.54 => ~ ( member8566619992076573584nt_int @ ( produc3646306378393792727nt_int @ ( product_Pair_int_int @ A4 @ B2 ) @ ( product_Pair_int_int @ Za @ Zb ) ) @ ( transi5430920176653612092nt_int @ R2 ) ) ) ) ) ).
% 0.34/0.54
% 0.34/0.54 % converse_rtranclE2
% 0.34/0.54 thf(fact_1307_converse__rtranclE2,axiom,
% 0.34/0.54 ! [Xa: produc3658429121746597890et_nat > $o,Xb: produc3658429121746597890et_nat,Za: produc3658429121746597890et_nat > $o,Zb: produc3658429121746597890et_nat,R2: set_Pr7928877670098842301et_nat] :
% 0.34/0.54 ( ( member4763271486408492550et_nat @ ( produc8599840265553166229et_nat @ ( produc5001842942810119800et_nat @ Xa @ Xb ) @ ( produc5001842942810119800et_nat @ Za @ Zb ) ) @ ( transi2414916928880341691et_nat @ R2 ) )
% 0.34/0.54 => ( ( ( produc5001842942810119800et_nat @ Xa @ Xb )
% 0.34/0.54 != ( produc5001842942810119800et_nat @ Za @ Zb ) )
% 0.34/0.54 => ~ ! [A4: produc3658429121746597890et_nat > $o,B2: produc3658429121746597890et_nat] :
% 0.34/0.54 ( ( member4763271486408492550et_nat @ ( produc8599840265553166229et_nat @ ( produc5001842942810119800et_nat @ Xa @ Xb ) @ ( produc5001842942810119800et_nat @ A4 @ B2 ) ) @ R2 )
% 0.34/0.54 => ~ ( member4763271486408492550et_nat @ ( produc8599840265553166229et_nat @ ( produc5001842942810119800et_nat @ A4 @ B2 ) @ ( produc5001842942810119800et_nat @ Za @ Zb ) ) @ ( transi2414916928880341691et_nat @ R2 ) ) ) ) ) ).
% 0.34/0.54
% 0.34/0.54 % converse_rtranclE2
% 0.34/0.54 thf(fact_1308_converse__rtranclE2,axiom,
% 0.34/0.54 ! [Xa: produc3658429121746597890et_nat > $o,Xb: produc3925858234332021118et_nat,Za: produc3658429121746597890et_nat > $o,Zb: produc3925858234332021118et_nat,R2: set_Pr3444600963470892981et_nat] :
% 0.34/0.54 ( ( member6341495586645257982et_nat @ ( produc1940133919992309389et_nat @ ( produc2245416461498447860et_nat @ Xa @ Xb ) @ ( produc2245416461498447860et_nat @ Za @ Zb ) ) @ ( transi344104063936286263et_nat @ R2 ) )
% 0.34/0.54 => ( ( ( produc2245416461498447860et_nat @ Xa @ Xb )
% 0.34/0.54 != ( produc2245416461498447860et_nat @ Za @ Zb ) )
% 0.34/0.54 => ~ ! [A4: produc3658429121746597890et_nat > $o,B2: produc3925858234332021118et_nat] :
% 0.34/0.54 ( ( member6341495586645257982et_nat @ ( produc1940133919992309389et_nat @ ( produc2245416461498447860et_nat @ Xa @ Xb ) @ ( produc2245416461498447860et_nat @ A4 @ B2 ) ) @ R2 )
% 0.34/0.54 => ~ ( member6341495586645257982et_nat @ ( produc1940133919992309389et_nat @ ( produc2245416461498447860et_nat @ A4 @ B2 ) @ ( produc2245416461498447860et_nat @ Za @ Zb ) ) @ ( transi344104063936286263et_nat @ R2 ) ) ) ) ) ).
% 0.34/0.54
% 0.34/0.54 % converse_rtranclE2
% 0.34/0.54 thf(fact_1309_converse__rtrancl__induct2,axiom,
% 0.34/0.54 ! [Ax: heap_e7401611519738050253t_unit,Ay: set_nat,Bx: heap_e7401611519738050253t_unit,By: set_nat,R2: set_Pr719794911490849221et_nat,P: heap_e7401611519738050253t_unit > set_nat > $o] :
% 0.34/0.54 ( ( member6099555550032318734et_nat @ ( produc8199053930788261021et_nat @ ( produc7507926704131184380et_nat @ Ax @ Ay ) @ ( produc7507926704131184380et_nat @ Bx @ By ) ) @ ( transi6919426408697524543et_nat @ R2 ) )
% 0.34/0.54 => ( ( P @ Bx @ By )
% 0.34/0.54 => ( ! [A4: heap_e7401611519738050253t_unit,B2: set_nat,Aa: heap_e7401611519738050253t_unit,Ba: set_nat] :
% 0.34/0.54 ( ( member6099555550032318734et_nat @ ( produc8199053930788261021et_nat @ ( produc7507926704131184380et_nat @ A4 @ B2 ) @ ( produc7507926704131184380et_nat @ Aa @ Ba ) ) @ R2 )
% 0.34/0.54 => ( ( member6099555550032318734et_nat @ ( produc8199053930788261021et_nat @ ( produc7507926704131184380et_nat @ Aa @ Ba ) @ ( produc7507926704131184380et_nat @ Bx @ By ) ) @ ( transi6919426408697524543et_nat @ R2 ) )
% 0.34/0.54 => ( ( P @ Aa @ Ba )
% 0.34/0.54 => ( P @ A4 @ B2 ) ) ) )
% 0.34/0.54 => ( P @ Ax @ Ay ) ) ) ) ).
% 0.34/0.54
% 0.34/0.54 % converse_rtrancl_induct2
% 0.34/0.54 thf(fact_1310_converse__rtrancl__induct2,axiom,
% 0.34/0.54 ! [Ax: nat,Ay: nat,Bx: nat,By: nat,R2: set_Pr8693737435421807431at_nat,P: nat > nat > $o] :
% 0.34/0.54 ( ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ ( product_Pair_nat_nat @ Ax @ Ay ) @ ( product_Pair_nat_nat @ Bx @ By ) ) @ ( transi8609417484261754244at_nat @ R2 ) )
% 0.34/0.54 => ( ( P @ Bx @ By )
% 0.34/0.54 => ( ! [A4: nat,B2: nat,Aa: nat,Ba: nat] :
% 0.34/0.54 ( ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ ( product_Pair_nat_nat @ A4 @ B2 ) @ ( product_Pair_nat_nat @ Aa @ Ba ) ) @ R2 )
% 0.34/0.54 => ( ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ ( product_Pair_nat_nat @ Aa @ Ba ) @ ( product_Pair_nat_nat @ Bx @ By ) ) @ ( transi8609417484261754244at_nat @ R2 ) )
% 0.34/0.54 => ( ( P @ Aa @ Ba )
% 0.34/0.54 => ( P @ A4 @ B2 ) ) ) )
% 0.34/0.54 => ( P @ Ax @ Ay ) ) ) ) ).
% 0.34/0.54
% 0.34/0.54 % converse_rtrancl_induct2
% 0.34/0.54 thf(fact_1311_converse__rtrancl__induct2,axiom,
% 0.34/0.54 ! [Ax: int,Ay: int,Bx: int,By: int,R2: set_Pr2560585780119916871nt_int,P: int > int > $o] :
% 0.34/0.54 ( ( member8566619992076573584nt_int @ ( produc3646306378393792727nt_int @ ( product_Pair_int_int @ Ax @ Ay ) @ ( product_Pair_int_int @ Bx @ By ) ) @ ( transi5430920176653612092nt_int @ R2 ) )
% 0.34/0.54 => ( ( P @ Bx @ By )
% 0.34/0.54 => ( ! [A4: int,B2: int,Aa: int,Ba: int] :
% 0.34/0.54 ( ( member8566619992076573584nt_int @ ( produc3646306378393792727nt_int @ ( product_Pair_int_int @ A4 @ B2 ) @ ( product_Pair_int_int @ Aa @ Ba ) ) @ R2 )
% 0.34/0.54 => ( ( member8566619992076573584nt_int @ ( produc3646306378393792727nt_int @ ( product_Pair_int_int @ Aa @ Ba ) @ ( product_Pair_int_int @ Bx @ By ) ) @ ( transi5430920176653612092nt_int @ R2 ) )
% 0.34/0.54 => ( ( P @ Aa @ Ba )
% 0.34/0.54 => ( P @ A4 @ B2 ) ) ) )
% 0.34/0.54 => ( P @ Ax @ Ay ) ) ) ) ).
% 0.34/0.54
% 0.34/0.54 % converse_rtrancl_induct2
% 0.34/0.54 thf(fact_1312_converse__rtrancl__induct2,axiom,
% 0.34/0.54 ! [Ax: produc3658429121746597890et_nat > $o,Ay: produc3658429121746597890et_nat,Bx: produc3658429121746597890et_nat > $o,By: produc3658429121746597890et_nat,R2: set_Pr7928877670098842301et_nat,P: ( produc3658429121746597890et_nat > $o ) > produc3658429121746597890et_nat > $o] :
% 0.34/0.54 ( ( member4763271486408492550et_nat @ ( produc8599840265553166229et_nat @ ( produc5001842942810119800et_nat @ Ax @ Ay ) @ ( produc5001842942810119800et_nat @ Bx @ By ) ) @ ( transi2414916928880341691et_nat @ R2 ) )
% 0.34/0.54 => ( ( P @ Bx @ By )
% 0.34/0.54 => ( ! [A4: produc3658429121746597890et_nat > $o,B2: produc3658429121746597890et_nat,Aa: produc3658429121746597890et_nat > $o,Ba: produc3658429121746597890et_nat] :
% 0.34/0.54 ( ( member4763271486408492550et_nat @ ( produc8599840265553166229et_nat @ ( produc5001842942810119800et_nat @ A4 @ B2 ) @ ( produc5001842942810119800et_nat @ Aa @ Ba ) ) @ R2 )
% 0.34/0.54 => ( ( member4763271486408492550et_nat @ ( produc8599840265553166229et_nat @ ( produc5001842942810119800et_nat @ Aa @ Ba ) @ ( produc5001842942810119800et_nat @ Bx @ By ) ) @ ( transi2414916928880341691et_nat @ R2 ) )
% 0.34/0.54 => ( ( P @ Aa @ Ba )
% 0.34/0.54 => ( P @ A4 @ B2 ) ) ) )
% 0.34/0.54 => ( P @ Ax @ Ay ) ) ) ) ).
% 0.34/0.54
% 0.34/0.54 % converse_rtrancl_induct2
% 0.34/0.54 thf(fact_1313_converse__rtrancl__induct2,axiom,
% 0.34/0.54 ! [Ax: produc3658429121746597890et_nat > $o,Ay: produc3925858234332021118et_nat,Bx: produc3658429121746597890et_nat > $o,By: produc3925858234332021118et_nat,R2: set_Pr3444600963470892981et_nat,P: ( produc3658429121746597890et_nat > $o ) > produc3925858234332021118et_nat > $o] :
% 0.34/0.54 ( ( member6341495586645257982et_nat @ ( produc1940133919992309389et_nat @ ( produc2245416461498447860et_nat @ Ax @ Ay ) @ ( produc2245416461498447860et_nat @ Bx @ By ) ) @ ( transi344104063936286263et_nat @ R2 ) )
% 0.34/0.54 => ( ( P @ Bx @ By )
% 0.34/0.54 => ( ! [A4: produc3658429121746597890et_nat > $o,B2: produc3925858234332021118et_nat,Aa: produc3658429121746597890et_nat > $o,Ba: produc3925858234332021118et_nat] :
% 0.34/0.54 ( ( member6341495586645257982et_nat @ ( produc1940133919992309389et_nat @ ( produc2245416461498447860et_nat @ A4 @ B2 ) @ ( produc2245416461498447860et_nat @ Aa @ Ba ) ) @ R2 )
% 0.34/0.54 => ( ( member6341495586645257982et_nat @ ( produc1940133919992309389et_nat @ ( produc2245416461498447860et_nat @ Aa @ Ba ) @ ( produc2245416461498447860et_nat @ Bx @ By ) ) @ ( transi344104063936286263et_nat @ R2 ) )
% 0.34/0.54 => ( ( P @ Aa @ Ba )
% 0.34/0.54 => ( P @ A4 @ B2 ) ) ) )
% 0.34/0.54 => ( P @ Ax @ Ay ) ) ) ) ).
% 0.34/0.54
% 0.34/0.54 % converse_rtrancl_induct2
% 0.34/0.54 thf(fact_1314_Un__empty__right,axiom,
% 0.34/0.54 ! [A: set_Product_unit] :
% 0.34/0.54 ( ( sup_su793286257634532545t_unit @ A @ bot_bo3957492148770167129t_unit )
% 0.34/0.54 = A ) ).
% 0.34/0.54
% 0.34/0.54 % Un_empty_right
% 0.34/0.54 thf(fact_1315_Un__empty__right,axiom,
% 0.34/0.54 ! [A: set_o] :
% 0.34/0.54 ( ( sup_sup_set_o @ A @ bot_bot_set_o )
% 0.34/0.54 = A ) ).
% 0.34/0.54
% 0.34/0.54 % Un_empty_right
% 0.34/0.54 thf(fact_1316_Un__empty__right,axiom,
% 0.34/0.54 ! [A: set_nat] :
% 0.34/0.54 ( ( sup_sup_set_nat @ A @ bot_bot_set_nat )
% 0.34/0.54 = A ) ).
% 0.34/0.54
% 0.34/0.54 % Un_empty_right
% 0.34/0.54 thf(fact_1317_Un__empty__right,axiom,
% 0.34/0.54 ! [A: set_int] :
% 0.34/0.54 ( ( sup_sup_set_int @ A @ bot_bot_set_int )
% 0.34/0.54 = A ) ).
% 0.34/0.54
% 0.34/0.54 % Un_empty_right
% 0.34/0.54 thf(fact_1318_Un__empty__right,axiom,
% 0.34/0.54 ! [A: set_Pr4532377907799695533_nat_o] :
% 0.34/0.54 ( ( sup_su5209123915105501825_nat_o @ A @ bot_bo7824918357723582233_nat_o )
% 0.34/0.54 = A ) ).
% 0.34/0.54
% 0.34/0.54 % Un_empty_right
% 0.34/0.54 thf(fact_1319_Un__empty__left,axiom,
% 0.34/0.54 ! [B4: set_Product_unit] :
% 0.34/0.54 ( ( sup_su793286257634532545t_unit @ bot_bo3957492148770167129t_unit @ B4 )
% 0.34/0.54 = B4 ) ).
% 0.34/0.54
% 0.34/0.54 % Un_empty_left
% 0.34/0.54 thf(fact_1320_Un__empty__left,axiom,
% 0.34/0.54 ! [B4: set_o] :
% 0.34/0.54 ( ( sup_sup_set_o @ bot_bot_set_o @ B4 )
% 0.34/0.54 = B4 ) ).
% 0.34/0.54
% 0.34/0.54 % Un_empty_left
% 0.34/0.54 thf(fact_1321_Un__empty__left,axiom,
% 0.34/0.54 ! [B4: set_nat] :
% 0.34/0.54 ( ( sup_sup_set_nat @ bot_bot_set_nat @ B4 )
% 0.34/0.54 = B4 ) ).
% 0.34/0.54
% 0.34/0.54 % Un_empty_left
% 0.34/0.54 thf(fact_1322_Un__empty__left,axiom,
% 0.34/0.54 ! [B4: set_int] :
% 0.34/0.54 ( ( sup_sup_set_int @ bot_bot_set_int @ B4 )
% 0.34/0.54 = B4 ) ).
% 0.34/0.54
% 0.34/0.54 % Un_empty_left
% 0.34/0.54 thf(fact_1323_Un__empty__left,axiom,
% 0.34/0.54 ! [B4: set_Pr4532377907799695533_nat_o] :
% 0.34/0.54 ( ( sup_su5209123915105501825_nat_o @ bot_bo7824918357723582233_nat_o @ B4 )
% 0.34/0.54 = B4 ) ).
% 0.34/0.54
% 0.34/0.54 % Un_empty_left
% 0.34/0.54 thf(fact_1324_disjoint__iff__not__equal,axiom,
% 0.34/0.54 ! [A: set_Product_unit,B4: set_Product_unit] :
% 0.34/0.54 ( ( ( inf_in4660618365625256667t_unit @ A @ B4 )
% 0.34/0.54 = bot_bo3957492148770167129t_unit )
% 0.34/0.54 = ( ! [X2: product_unit] :
% 0.34/0.54 ( ( member_Product_unit @ X2 @ A )
% 0.34/0.54 => ! [Y3: product_unit] :
% 0.34/0.54 ( ( member_Product_unit @ Y3 @ B4 )
% 0.34/0.54 => ( X2 != Y3 ) ) ) ) ) ).
% 0.34/0.54
% 0.34/0.54 % disjoint_iff_not_equal
% 0.34/0.54 thf(fact_1325_disjoint__iff__not__equal,axiom,
% 0.34/0.54 ! [A: set_o,B4: set_o] :
% 0.34/0.54 ( ( ( inf_inf_set_o @ A @ B4 )
% 0.34/0.54 = bot_bot_set_o )
% 0.34/0.54 = ( ! [X2: $o] :
% 0.34/0.54 ( ( member_o @ X2 @ A )
% 0.34/0.54 => ! [Y3: $o] :
% 0.34/0.54 ( ( member_o @ Y3 @ B4 )
% 0.34/0.54 => ( X2 = ( ~ Y3 ) ) ) ) ) ) ).
% 0.34/0.54
% 0.34/0.54 % disjoint_iff_not_equal
% 0.34/0.54 thf(fact_1326_disjoint__iff__not__equal,axiom,
% 0.34/0.54 ! [A: set_nat,B4: set_nat] :
% 0.34/0.54 ( ( ( inf_inf_set_nat @ A @ B4 )
% 0.34/0.54 = bot_bot_set_nat )
% 0.34/0.54 = ( ! [X2: nat] :
% 0.34/0.54 ( ( member_nat @ X2 @ A )
% 0.34/0.54 => ! [Y3: nat] :
% 0.34/0.54 ( ( member_nat @ Y3 @ B4 )
% 0.34/0.54 => ( X2 != Y3 ) ) ) ) ) ).
% 0.34/0.54
% 0.34/0.54 % disjoint_iff_not_equal
% 0.34/0.54 thf(fact_1327_disjoint__iff__not__equal,axiom,
% 0.34/0.54 ! [A: set_int,B4: set_int] :
% 0.34/0.54 ( ( ( inf_inf_set_int @ A @ B4 )
% 0.34/0.54 = bot_bot_set_int )
% 0.34/0.54 = ( ! [X2: int] :
% 0.34/0.54 ( ( member_int @ X2 @ A )
% 0.34/0.54 => ! [Y3: int] :
% 0.34/0.54 ( ( member_int @ Y3 @ B4 )
% 0.34/0.54 => ( X2 != Y3 ) ) ) ) ) ).
% 0.34/0.54
% 0.34/0.54 % disjoint_iff_not_equal
% 0.34/0.54 thf(fact_1328_disjoint__iff__not__equal,axiom,
% 0.34/0.54 ! [A: set_Pr4532377907799695533_nat_o,B4: set_Pr4532377907799695533_nat_o] :
% 0.34/0.54 ( ( ( inf_in1906310914598751387_nat_o @ A @ B4 )
% 0.34/0.54 = bot_bo7824918357723582233_nat_o )
% 0.34/0.54 = ( ! [X2: produc3658429121746597890et_nat > $o] :
% 0.34/0.54 ( ( member6576561426505652726_nat_o @ X2 @ A )
% 0.34/0.54 => ! [Y3: produc3658429121746597890et_nat > $o] :
% 0.34/0.54 ( ( member6576561426505652726_nat_o @ Y3 @ B4 )
% 0.34/0.54 => ( X2 != Y3 ) ) ) ) ) ).
% 0.34/0.54
% 0.34/0.54 % disjoint_iff_not_equal
% 0.34/0.54 thf(fact_1329_Int__empty__right,axiom,
% 0.34/0.54 ! [A: set_Product_unit] :
% 0.34/0.54 ( ( inf_in4660618365625256667t_unit @ A @ bot_bo3957492148770167129t_unit )
% 0.34/0.54 = bot_bo3957492148770167129t_unit ) ).
% 0.34/0.54
% 0.34/0.54 % Int_empty_right
% 0.34/0.54 thf(fact_1330_Int__empty__right,axiom,
% 0.34/0.54 ! [A: set_o] :
% 0.34/0.54 ( ( inf_inf_set_o @ A @ bot_bot_set_o )
% 0.34/0.54 = bot_bot_set_o ) ).
% 0.34/0.54
% 0.34/0.54 % Int_empty_right
% 0.34/0.54 thf(fact_1331_Int__empty__right,axiom,
% 0.34/0.54 ! [A: set_nat] :
% 0.34/0.54 ( ( inf_inf_set_nat @ A @ bot_bot_set_nat )
% 0.34/0.54 = bot_bot_set_nat ) ).
% 0.34/0.54
% 0.34/0.54 % Int_empty_right
% 0.34/0.54 thf(fact_1332_Int__empty__right,axiom,
% 0.34/0.54 ! [A: set_int] :
% 0.34/0.54 ( ( inf_inf_set_int @ A @ bot_bot_set_int )
% 0.34/0.54 = bot_bot_set_int ) ).
% 0.34/0.54
% 0.34/0.54 % Int_empty_right
% 0.34/0.54 thf(fact_1333_Int__empty__right,axiom,
% 0.34/0.54 ! [A: set_Pr4532377907799695533_nat_o] :
% 0.34/0.54 ( ( inf_in1906310914598751387_nat_o @ A @ bot_bo7824918357723582233_nat_o )
% 0.34/0.54 = bot_bo7824918357723582233_nat_o ) ).
% 0.34/0.54
% 0.34/0.54 % Int_empty_right
% 0.34/0.54 thf(fact_1334_Int__empty__left,axiom,
% 0.34/0.54 ! [B4: set_Product_unit] :
% 0.34/0.54 ( ( inf_in4660618365625256667t_unit @ bot_bo3957492148770167129t_unit @ B4 )
% 0.34/0.54 = bot_bo3957492148770167129t_unit ) ).
% 0.34/0.54
% 0.34/0.54 % Int_empty_left
% 0.34/0.54 thf(fact_1335_Int__empty__left,axiom,
% 0.34/0.54 ! [B4: set_o] :
% 0.34/0.54 ( ( inf_inf_set_o @ bot_bot_set_o @ B4 )
% 0.34/0.54 = bot_bot_set_o ) ).
% 0.34/0.54
% 0.34/0.54 % Int_empty_left
% 0.34/0.54 thf(fact_1336_Int__empty__left,axiom,
% 0.34/0.54 ! [B4: set_nat] :
% 0.34/0.54 ( ( inf_inf_set_nat @ bot_bot_set_nat @ B4 )
% 0.34/0.54 = bot_bot_set_nat ) ).
% 0.34/0.54
% 0.34/0.54 % Int_empty_left
% 0.34/0.54 thf(fact_1337_Int__empty__left,axiom,
% 0.34/0.54 ! [B4: set_int] :
% 0.34/0.54 ( ( inf_inf_set_int @ bot_bot_set_int @ B4 )
% 0.34/0.54 = bot_bot_set_int ) ).
% 0.34/0.54
% 0.34/0.54 % Int_empty_left
% 0.34/0.54 thf(fact_1338_Int__empty__left,axiom,
% 0.34/0.54 ! [B4: set_Pr4532377907799695533_nat_o] :
% 0.34/0.54 ( ( inf_in1906310914598751387_nat_o @ bot_bo7824918357723582233_nat_o @ B4 )
% 0.34/0.54 = bot_bo7824918357723582233_nat_o ) ).
% 0.34/0.54
% 0.34/0.54 % Int_empty_left
% 0.34/0.54 thf(fact_1339_disjoint__iff,axiom,
% 0.34/0.54 ! [A: set_Product_unit,B4: set_Product_unit] :
% 0.34/0.54 ( ( ( inf_in4660618365625256667t_unit @ A @ B4 )
% 0.34/0.54 = bot_bo3957492148770167129t_unit )
% 0.34/0.54 = ( ! [X2: product_unit] :
% 0.34/0.54 ( ( member_Product_unit @ X2 @ A )
% 0.34/0.54 => ~ ( member_Product_unit @ X2 @ B4 ) ) ) ) ).
% 0.34/0.54
% 0.34/0.54 % disjoint_iff
% 0.34/0.54 thf(fact_1340_disjoint__iff,axiom,
% 0.34/0.54 ! [A: set_o,B4: set_o] :
% 0.34/0.54 ( ( ( inf_inf_set_o @ A @ B4 )
% 0.34/0.54 = bot_bot_set_o )
% 0.34/0.54 = ( ! [X2: $o] :
% 0.34/0.54 ( ( member_o @ X2 @ A )
% 0.34/0.54 => ~ ( member_o @ X2 @ B4 ) ) ) ) ).
% 0.34/0.54
% 0.34/0.54 % disjoint_iff
% 0.34/0.54 thf(fact_1341_disjoint__iff,axiom,
% 0.34/0.54 ! [A: set_nat,B4: set_nat] :
% 0.34/0.54 ( ( ( inf_inf_set_nat @ A @ B4 )
% 0.34/0.54 = bot_bot_set_nat )
% 0.34/0.54 = ( ! [X2: nat] :
% 0.34/0.54 ( ( member_nat @ X2 @ A )
% 0.34/0.54 => ~ ( member_nat @ X2 @ B4 ) ) ) ) ).
% 0.34/0.54
% 0.34/0.54 % disjoint_iff
% 0.34/0.54 thf(fact_1342_disjoint__iff,axiom,
% 0.34/0.54 ! [A: set_int,B4: set_int] :
% 0.34/0.54 ( ( ( inf_inf_set_int @ A @ B4 )
% 0.34/0.54 = bot_bot_set_int )
% 0.34/0.54 = ( ! [X2: int] :
% 0.34/0.54 ( ( member_int @ X2 @ A )
% 0.34/0.54 => ~ ( member_int @ X2 @ B4 ) ) ) ) ).
% 0.34/0.54
% 0.34/0.54 % disjoint_iff
% 0.34/0.54 thf(fact_1343_disjoint__iff,axiom,
% 0.34/0.54 ! [A: set_Pr4532377907799695533_nat_o,B4: set_Pr4532377907799695533_nat_o] :
% 0.34/0.54 ( ( ( inf_in1906310914598751387_nat_o @ A @ B4 )
% 0.34/0.54 = bot_bo7824918357723582233_nat_o )
% 0.34/0.54 = ( ! [X2: produc3658429121746597890et_nat > $o] :
% 0.34/0.54 ( ( member6576561426505652726_nat_o @ X2 @ A )
% 0.34/0.54 => ~ ( member6576561426505652726_nat_o @ X2 @ B4 ) ) ) ) ).
% 0.34/0.54
% 0.34/0.54 % disjoint_iff
% 0.34/0.54 thf(fact_1344_Int__emptyI,axiom,
% 0.34/0.54 ! [A: set_Product_unit,B4: set_Product_unit] :
% 0.34/0.54 ( ! [X3: product_unit] :
% 0.34/0.54 ( ( member_Product_unit @ X3 @ A )
% 0.34/0.54 => ~ ( member_Product_unit @ X3 @ B4 ) )
% 0.34/0.54 => ( ( inf_in4660618365625256667t_unit @ A @ B4 )
% 0.34/0.54 = bot_bo3957492148770167129t_unit ) ) ).
% 0.34/0.54
% 0.34/0.54 % Int_emptyI
% 0.34/0.54 thf(fact_1345_Int__emptyI,axiom,
% 0.34/0.54 ! [A: set_o,B4: set_o] :
% 0.34/0.54 ( ! [X3: $o] :
% 0.34/0.54 ( ( member_o @ X3 @ A )
% 0.34/0.54 => ~ ( member_o @ X3 @ B4 ) )
% 0.34/0.54 => ( ( inf_inf_set_o @ A @ B4 )
% 0.34/0.54 = bot_bot_set_o ) ) ).
% 0.34/0.54
% 0.34/0.54 % Int_emptyI
% 0.34/0.54 thf(fact_1346_Int__emptyI,axiom,
% 0.34/0.54 ! [A: set_nat,B4: set_nat] :
% 0.34/0.54 ( ! [X3: nat] :
% 0.34/0.54 ( ( member_nat @ X3 @ A )
% 0.34/0.54 => ~ ( member_nat @ X3 @ B4 ) )
% 0.34/0.54 => ( ( inf_inf_set_nat @ A @ B4 )
% 0.34/0.54 = bot_bot_set_nat ) ) ).
% 0.34/0.54
% 0.34/0.54 % Int_emptyI
% 0.34/0.54 thf(fact_1347_Int__emptyI,axiom,
% 0.34/0.54 ! [A: set_int,B4: set_int] :
% 0.34/0.54 ( ! [X3: int] :
% 0.34/0.54 ( ( member_int @ X3 @ A )
% 0.34/0.54 => ~ ( member_int @ X3 @ B4 ) )
% 0.34/0.54 => ( ( inf_inf_set_int @ A @ B4 )
% 0.34/0.54 = bot_bot_set_int ) ) ).
% 0.34/0.54
% 0.34/0.54 % Int_emptyI
% 0.34/0.54 thf(fact_1348_Int__emptyI,axiom,
% 0.34/0.54 ! [A: set_Pr4532377907799695533_nat_o,B4: set_Pr4532377907799695533_nat_o] :
% 0.34/0.54 ( ! [X3: produc3658429121746597890et_nat > $o] :
% 0.34/0.54 ( ( member6576561426505652726_nat_o @ X3 @ A )
% 0.34/0.54 => ~ ( member6576561426505652726_nat_o @ X3 @ B4 ) )
% 0.34/0.54 => ( ( inf_in1906310914598751387_nat_o @ A @ B4 )
% 0.34/0.54 = bot_bo7824918357723582233_nat_o ) ) ).
% 0.34/0.54
% 0.34/0.54 % Int_emptyI
% 0.34/0.54 thf(fact_1349_disjointI,axiom,
% 0.34/0.54 ! [A2: set_Product_unit,B: set_Product_unit] :
% 0.34/0.54 ( ! [X3: product_unit] :
% 0.34/0.54 ( ( member_Product_unit @ X3 @ A2 )
% 0.34/0.54 => ~ ( member_Product_unit @ X3 @ B ) )
% 0.34/0.54 => ( ( inf_in4660618365625256667t_unit @ A2 @ B )
% 0.34/0.54 = bot_bo3957492148770167129t_unit ) ) ).
% 0.34/0.54
% 0.34/0.54 % disjointI
% 0.34/0.54 thf(fact_1350_disjointI,axiom,
% 0.34/0.54 ! [A2: set_o,B: set_o] :
% 0.34/0.54 ( ! [X3: $o] :
% 0.34/0.54 ( ( member_o @ X3 @ A2 )
% 0.34/0.54 => ~ ( member_o @ X3 @ B ) )
% 0.34/0.54 => ( ( inf_inf_set_o @ A2 @ B )
% 0.34/0.54 = bot_bot_set_o ) ) ).
% 0.34/0.54
% 0.34/0.54 % disjointI
% 0.34/0.54 thf(fact_1351_disjointI,axiom,
% 0.34/0.54 ! [A2: set_nat,B: set_nat] :
% 0.34/0.54 ( ! [X3: nat] :
% 0.34/0.54 ( ( member_nat @ X3 @ A2 )
% 0.34/0.54 => ~ ( member_nat @ X3 @ B ) )
% 0.34/0.54 => ( ( inf_inf_set_nat @ A2 @ B )
% 0.34/0.54 = bot_bot_set_nat ) ) ).
% 0.34/0.54
% 0.34/0.54 % disjointI
% 0.34/0.54 thf(fact_1352_disjointI,axiom,
% 0.34/0.54 ! [A2: set_int,B: set_int] :
% 0.34/0.54 ( ! [X3: int] :
% 0.34/0.54 ( ( member_int @ X3 @ A2 )
% 0.34/0.54 => ~ ( member_int @ X3 @ B ) )
% 0.34/0.54 => ( ( inf_inf_set_int @ A2 @ B )
% 0.34/0.54 = bot_bot_set_int ) ) ).
% 0.34/0.54
% 0.34/0.54 % disjointI
% 0.34/0.54 thf(fact_1353_disjointI,axiom,
% 0.34/0.54 ! [A2: set_Pr4532377907799695533_nat_o,B: set_Pr4532377907799695533_nat_o] :
% 0.34/0.54 ( ! [X3: produc3658429121746597890et_nat > $o] :
% 0.34/0.54 ( ( member6576561426505652726_nat_o @ X3 @ A2 )
% 0.34/0.54 => ~ ( member6576561426505652726_nat_o @ X3 @ B ) )
% 0.34/0.54 => ( ( inf_in1906310914598751387_nat_o @ A2 @ B )
% 0.34/0.54 = bot_bo7824918357723582233_nat_o ) ) ).
% 0.34/0.54
% 0.34/0.54 % disjointI
% 0.34/0.54 thf(fact_1354_Int__insert__right,axiom,
% 0.34/0.54 ! [A2: produc3658429121746597890et_nat > $o,A: set_Pr4532377907799695533_nat_o,B4: set_Pr4532377907799695533_nat_o] :
% 0.34/0.54 ( ( ( member6576561426505652726_nat_o @ A2 @ A )
% 0.34/0.54 => ( ( inf_in1906310914598751387_nat_o @ A @ ( insert5175938949040314269_nat_o @ A2 @ B4 ) )
% 0.34/0.54 = ( insert5175938949040314269_nat_o @ A2 @ ( inf_in1906310914598751387_nat_o @ A @ B4 ) ) ) )
% 0.34/0.54 & ( ~ ( member6576561426505652726_nat_o @ A2 @ A )
% 0.34/0.54 => ( ( inf_in1906310914598751387_nat_o @ A @ ( insert5175938949040314269_nat_o @ A2 @ B4 ) )
% 0.34/0.54 = ( inf_in1906310914598751387_nat_o @ A @ B4 ) ) ) ) ).
% 0.34/0.54
% 0.34/0.54 % Int_insert_right
% 0.34/0.54 thf(fact_1355_Int__insert__right,axiom,
% 0.34/0.54 ! [A2: int,A: set_int,B4: set_int] :
% 0.34/0.54 ( ( ( member_int @ A2 @ A )
% 0.34/0.54 => ( ( inf_inf_set_int @ A @ ( insert_int @ A2 @ B4 ) )
% 0.34/0.54 = ( insert_int @ A2 @ ( inf_inf_set_int @ A @ B4 ) ) ) )
% 0.34/0.54 & ( ~ ( member_int @ A2 @ A )
% 0.34/0.54 => ( ( inf_inf_set_int @ A @ ( insert_int @ A2 @ B4 ) )
% 0.34/0.54 = ( inf_inf_set_int @ A @ B4 ) ) ) ) ).
% 0.34/0.54
% 0.34/0.54 % Int_insert_right
% 0.34/0.54 thf(fact_1356_Int__insert__right,axiom,
% 0.34/0.54 ! [A2: $o,A: set_o,B4: set_o] :
% 0.34/0.54 ( ( ( member_o @ A2 @ A )
% 0.34/0.54 => ( ( inf_inf_set_o @ A @ ( insert_o @ A2 @ B4 ) )
% 0.34/0.54 = ( insert_o @ A2 @ ( inf_inf_set_o @ A @ B4 ) ) ) )
% 0.34/0.54 & ( ~ ( member_o @ A2 @ A )
% 0.34/0.54 => ( ( inf_inf_set_o @ A @ ( insert_o @ A2 @ B4 ) )
% 0.34/0.54 = ( inf_inf_set_o @ A @ B4 ) ) ) ) ).
% 0.34/0.54
% 0.34/0.54 % Int_insert_right
% 0.34/0.54 thf(fact_1357_Int__insert__right,axiom,
% 0.34/0.54 ! [A2: product_unit,A: set_Product_unit,B4: set_Product_unit] :
% 0.34/0.54 ( ( ( member_Product_unit @ A2 @ A )
% 0.34/0.54 => ( ( inf_in4660618365625256667t_unit @ A @ ( insert_Product_unit @ A2 @ B4 ) )
% 0.34/0.54 = ( insert_Product_unit @ A2 @ ( inf_in4660618365625256667t_unit @ A @ B4 ) ) ) )
% 0.34/0.54 & ( ~ ( member_Product_unit @ A2 @ A )
% 0.34/0.54 => ( ( inf_in4660618365625256667t_unit @ A @ ( insert_Product_unit @ A2 @ B4 ) )
% 0.34/0.54 = ( inf_in4660618365625256667t_unit @ A @ B4 ) ) ) ) ).
% 0.34/0.54
% 0.34/0.54 % Int_insert_right
% 0.34/0.54 thf(fact_1358_Int__insert__right,axiom,
% 0.34/0.54 ! [A2: nat,A: set_nat,B4: set_nat] :
% 0.34/0.54 ( ( ( member_nat @ A2 @ A )
% 0.34/0.54 => ( ( inf_inf_set_nat @ A @ ( insert_nat @ A2 @ B4 ) )
% 0.34/0.54 = ( insert_nat @ A2 @ ( inf_inf_set_nat @ A @ B4 ) ) ) )
% 0.34/0.54 & ( ~ ( member_nat @ A2 @ A )
% 0.34/0.54 => ( ( inf_inf_set_nat @ A @ ( insert_nat @ A2 @ B4 ) )
% 0.34/0.54 = ( inf_inf_set_nat @ A @ B4 ) ) ) ) ).
% 0.34/0.54
% 0.34/0.54 % Int_insert_right
% 0.34/0.54 thf(fact_1359_Int__insert__left,axiom,
% 0.34/0.54 ! [A2: produc3658429121746597890et_nat > $o,C5: set_Pr4532377907799695533_nat_o,B4: set_Pr4532377907799695533_nat_o] :
% 0.34/0.54 ( ( ( member6576561426505652726_nat_o @ A2 @ C5 )
% 0.34/0.54 => ( ( inf_in1906310914598751387_nat_o @ ( insert5175938949040314269_nat_o @ A2 @ B4 ) @ C5 )
% 0.34/0.54 = ( insert5175938949040314269_nat_o @ A2 @ ( inf_in1906310914598751387_nat_o @ B4 @ C5 ) ) ) )
% 0.34/0.54 & ( ~ ( member6576561426505652726_nat_o @ A2 @ C5 )
% 0.34/0.54 => ( ( inf_in1906310914598751387_nat_o @ ( insert5175938949040314269_nat_o @ A2 @ B4 ) @ C5 )
% 0.34/0.54 = ( inf_in1906310914598751387_nat_o @ B4 @ C5 ) ) ) ) ).
% 0.34/0.54
% 0.34/0.54 % Int_insert_left
% 0.34/0.54 thf(fact_1360_Int__insert__left,axiom,
% 0.34/0.54 ! [A2: int,C5: set_int,B4: set_int] :
% 0.34/0.54 ( ( ( member_int @ A2 @ C5 )
% 0.34/0.54 => ( ( inf_inf_set_int @ ( insert_int @ A2 @ B4 ) @ C5 )
% 0.34/0.54 = ( insert_int @ A2 @ ( inf_inf_set_int @ B4 @ C5 ) ) ) )
% 0.34/0.54 & ( ~ ( member_int @ A2 @ C5 )
% 0.34/0.54 => ( ( inf_inf_set_int @ ( insert_int @ A2 @ B4 ) @ C5 )
% 0.34/0.54 = ( inf_inf_set_int @ B4 @ C5 ) ) ) ) ).
% 0.34/0.54
% 0.34/0.54 % Int_insert_left
% 0.34/0.54 thf(fact_1361_Int__insert__left,axiom,
% 0.34/0.54 ! [A2: $o,C5: set_o,B4: set_o] :
% 0.34/0.54 ( ( ( member_o @ A2 @ C5 )
% 0.34/0.54 => ( ( inf_inf_set_o @ ( insert_o @ A2 @ B4 ) @ C5 )
% 0.34/0.54 = ( insert_o @ A2 @ ( inf_inf_set_o @ B4 @ C5 ) ) ) )
% 0.34/0.54 & ( ~ ( member_o @ A2 @ C5 )
% 0.34/0.54 => ( ( inf_inf_set_o @ ( insert_o @ A2 @ B4 ) @ C5 )
% 0.34/0.54 = ( inf_inf_set_o @ B4 @ C5 ) ) ) ) ).
% 0.34/0.54
% 0.34/0.54 % Int_insert_left
% 0.34/0.54 thf(fact_1362_Int__insert__left,axiom,
% 0.34/0.54 ! [A2: product_unit,C5: set_Product_unit,B4: set_Product_unit] :
% 0.34/0.54 ( ( ( member_Product_unit @ A2 @ C5 )
% 0.34/0.54 => ( ( inf_in4660618365625256667t_unit @ ( insert_Product_unit @ A2 @ B4 ) @ C5 )
% 0.34/0.54 = ( insert_Product_unit @ A2 @ ( inf_in4660618365625256667t_unit @ B4 @ C5 ) ) ) )
% 0.34/0.54 & ( ~ ( member_Product_unit @ A2 @ C5 )
% 0.34/0.54 => ( ( inf_in4660618365625256667t_unit @ ( insert_Product_unit @ A2 @ B4 ) @ C5 )
% 0.34/0.54 = ( inf_in4660618365625256667t_unit @ B4 @ C5 ) ) ) ) ).
% 0.34/0.54
% 0.34/0.54 % Int_insert_left
% 0.34/0.54 thf(fact_1363_Int__insert__left,axiom,
% 0.34/0.54 ! [A2: nat,C5: set_nat,B4: set_nat] :
% 0.34/0.54 ( ( ( member_nat @ A2 @ C5 )
% 0.34/0.54 => ( ( inf_inf_set_nat @ ( insert_nat @ A2 @ B4 ) @ C5 )
% 0.34/0.54 = ( insert_nat @ A2 @ ( inf_inf_set_nat @ B4 @ C5 ) ) ) )
% 0.34/0.54 & ( ~ ( member_nat @ A2 @ C5 )
% 0.34/0.54 => ( ( inf_inf_set_nat @ ( insert_nat @ A2 @ B4 ) @ C5 )
% 0.34/0.54 = ( inf_inf_set_nat @ B4 @ C5 ) ) ) ) ).
% 0.34/0.54
% 0.34/0.54 % Int_insert_left
% 0.34/0.54 thf(fact_1364_subrelI,axiom,
% 0.34/0.54 ! [R2: set_Pr3948176798113811640et_nat,S2: set_Pr3948176798113811640et_nat] :
% 0.34/0.54 ( ! [X3: heap_e7401611519738050253t_unit,Y2: set_nat] :
% 0.34/0.54 ( ( member6260224972018164377et_nat @ ( produc7507926704131184380et_nat @ X3 @ Y2 ) @ R2 )
% 0.34/0.54 => ( member6260224972018164377et_nat @ ( produc7507926704131184380et_nat @ X3 @ Y2 ) @ S2 ) )
% 0.34/0.54 => ( ord_le8090042543978450200et_nat @ R2 @ S2 ) ) ).
% 0.34/0.54
% 0.34/0.54 % subrelI
% 0.34/0.54 thf(fact_1365_subrelI,axiom,
% 0.34/0.54 ! [R2: set_Pr1261947904930325089at_nat,S2: set_Pr1261947904930325089at_nat] :
% 0.34/0.54 ( ! [X3: nat,Y2: nat] :
% 0.34/0.54 ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X3 @ Y2 ) @ R2 )
% 0.34/0.54 => ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X3 @ Y2 ) @ S2 ) )
% 0.34/0.54 => ( ord_le3146513528884898305at_nat @ R2 @ S2 ) ) ).
% 0.34/0.54
% 0.34/0.54 % subrelI
% 0.34/0.54 thf(fact_1366_subrelI,axiom,
% 0.34/0.54 ! [R2: set_Pr958786334691620121nt_int,S2: set_Pr958786334691620121nt_int] :
% 0.34/0.54 ( ! [X3: int,Y2: int] :
% 0.34/0.54 ( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ X3 @ Y2 ) @ R2 )
% 0.34/0.54 => ( member5262025264175285858nt_int @ ( product_Pair_int_int @ X3 @ Y2 ) @ S2 ) )
% 0.34/0.54 => ( ord_le2843351958646193337nt_int @ R2 @ S2 ) ) ).
% 0.34/0.54
% 0.34/0.54 % subrelI
% 0.34/0.54 thf(fact_1367_subrelI,axiom,
% 0.34/0.54 ! [R2: set_Pr3286484037609594932et_nat,S2: set_Pr3286484037609594932et_nat] :
% 0.34/0.54 ( ! [X3: produc3658429121746597890et_nat > $o,Y2: produc3658429121746597890et_nat] :
% 0.34/0.54 ( ( member1996754912294343701et_nat @ ( produc5001842942810119800et_nat @ X3 @ Y2 ) @ R2 )
% 0.34/0.54 => ( member1996754912294343701et_nat @ ( produc5001842942810119800et_nat @ X3 @ Y2 ) @ S2 ) )
% 0.34/0.54 => ( ord_le5966269811547037844et_nat @ R2 @ S2 ) ) ).
% 0.34/0.54
% 0.34/0.54 % subrelI
% 0.34/0.54 thf(fact_1368_subrelI,axiom,
% 0.34/0.54 ! [R2: set_Pr8536935166611901872et_nat,S2: set_Pr8536935166611901872et_nat] :
% 0.34/0.54 ( ! [X3: produc3658429121746597890et_nat > $o,Y2: produc3925858234332021118et_nat] :
% 0.34/0.54 ( ( member6124377750444531601et_nat @ ( produc2245416461498447860et_nat @ X3 @ Y2 ) @ R2 )
% 0.34/0.54 => ( member6124377750444531601et_nat @ ( produc2245416461498447860et_nat @ X3 @ Y2 ) @ S2 ) )
% 0.34/0.54 => ( ord_le4763372923235995152et_nat @ R2 @ S2 ) ) ).
% 0.34/0.54
% 0.34/0.54 % subrelI
% 0.34/0.54 thf(fact_1369_bot_Oextremum__uniqueI,axiom,
% 0.34/0.54 ! [A2: set_Product_unit] :
% 0.34/0.54 ( ( ord_le3507040750410214029t_unit @ A2 @ bot_bo3957492148770167129t_unit )
% 0.34/0.54 => ( A2 = bot_bo3957492148770167129t_unit ) ) ).
% 0.34/0.54
% 0.34/0.54 % bot.extremum_uniqueI
% 0.34/0.54 thf(fact_1370_bot_Oextremum__uniqueI,axiom,
% 0.34/0.54 ! [A2: set_o] :
% 0.34/0.54 ( ( ord_less_eq_set_o @ A2 @ bot_bot_set_o )
% 0.34/0.54 => ( A2 = bot_bot_set_o ) ) ).
% 0.34/0.54
% 0.34/0.54 % bot.extremum_uniqueI
% 0.34/0.54 thf(fact_1371_bot_Oextremum__uniqueI,axiom,
% 0.34/0.54 ! [A2: set_int] :
% 0.34/0.54 ( ( ord_less_eq_set_int @ A2 @ bot_bot_set_int )
% 0.34/0.54 => ( A2 = bot_bot_set_int ) ) ).
% 0.34/0.54
% 0.34/0.54 % bot.extremum_uniqueI
% 0.34/0.54 thf(fact_1372_bot_Oextremum__uniqueI,axiom,
% 0.34/0.54 ! [A2: set_Pr4532377907799695533_nat_o] :
% 0.34/0.54 ( ( ord_le2965882846123202637_nat_o @ A2 @ bot_bo7824918357723582233_nat_o )
% 0.34/0.54 => ( A2 = bot_bo7824918357723582233_nat_o ) ) ).
% 0.34/0.54
% 0.34/0.54 % bot.extremum_uniqueI
% 0.34/0.54 thf(fact_1373_bot_Oextremum__uniqueI,axiom,
% 0.34/0.54 ! [A2: filter_nat] :
% 0.34/0.54 ( ( ord_le2510731241096832064er_nat @ A2 @ bot_bot_filter_nat )
% 0.34/0.54 => ( A2 = bot_bot_filter_nat ) ) ).
% 0.34/0.54
% 0.34/0.54 % bot.extremum_uniqueI
% 0.34/0.54 thf(fact_1374_bot_Oextremum__uniqueI,axiom,
% 0.34/0.54 ! [A2: set_nat] :
% 0.34/0.54 ( ( ord_less_eq_set_nat @ A2 @ bot_bot_set_nat )
% 0.34/0.54 => ( A2 = bot_bot_set_nat ) ) ).
% 0.34/0.54
% 0.34/0.54 % bot.extremum_uniqueI
% 0.34/0.54 thf(fact_1375_bot_Oextremum__uniqueI,axiom,
% 0.34/0.54 ! [A2: nat] :
% 0.34/0.54 ( ( ord_less_eq_nat @ A2 @ bot_bot_nat )
% 0.34/0.54 => ( A2 = bot_bot_nat ) ) ).
% 0.34/0.54
% 0.34/0.54 % bot.extremum_uniqueI
% 0.34/0.54 thf(fact_1376_bot_Oextremum__unique,axiom,
% 0.34/0.54 ! [A2: set_Product_unit] :
% 0.34/0.54 ( ( ord_le3507040750410214029t_unit @ A2 @ bot_bo3957492148770167129t_unit )
% 0.34/0.54 = ( A2 = bot_bo3957492148770167129t_unit ) ) ).
% 0.34/0.54
% 0.34/0.54 % bot.extremum_unique
% 0.34/0.54 thf(fact_1377_bot_Oextremum__unique,axiom,
% 0.34/0.54 ! [A2: set_o] :
% 0.34/0.54 ( ( ord_less_eq_set_o @ A2 @ bot_bot_set_o )
% 0.34/0.54 = ( A2 = bot_bot_set_o ) ) ).
% 0.34/0.54
% 0.34/0.54 % bot.extremum_unique
% 0.34/0.54 thf(fact_1378_bot_Oextremum__unique,axiom,
% 0.34/0.54 ! [A2: set_int] :
% 0.34/0.54 ( ( ord_less_eq_set_int @ A2 @ bot_bot_set_int )
% 0.34/0.54 = ( A2 = bot_bot_set_int ) ) ).
% 0.34/0.54
% 0.34/0.54 % bot.extremum_unique
% 0.34/0.54 thf(fact_1379_bot_Oextremum__unique,axiom,
% 0.34/0.54 ! [A2: set_Pr4532377907799695533_nat_o] :
% 0.34/0.54 ( ( ord_le2965882846123202637_nat_o @ A2 @ bot_bo7824918357723582233_nat_o )
% 0.34/0.54 = ( A2 = bot_bo7824918357723582233_nat_o ) ) ).
% 0.34/0.54
% 0.34/0.54 % bot.extremum_unique
% 0.34/0.54 thf(fact_1380_bot_Oextremum__unique,axiom,
% 0.34/0.54 ! [A2: filter_nat] :
% 0.34/0.54 ( ( ord_le2510731241096832064er_nat @ A2 @ bot_bot_filter_nat )
% 0.34/0.54 = ( A2 = bot_bot_filter_nat ) ) ).
% 0.34/0.54
% 0.34/0.54 % bot.extremum_unique
% 0.34/0.54 thf(fact_1381_bot_Oextremum__unique,axiom,
% 0.34/0.54 ! [A2: set_nat] :
% 0.34/0.54 ( ( ord_less_eq_set_nat @ A2 @ bot_bot_set_nat )
% 0.34/0.54 = ( A2 = bot_bot_set_nat ) ) ).
% 0.34/0.54
% 0.34/0.54 % bot.extremum_unique
% 0.34/0.54 thf(fact_1382_bot_Oextremum__unique,axiom,
% 0.34/0.54 ! [A2: nat] :
% 0.34/0.54 ( ( ord_less_eq_nat @ A2 @ bot_bot_nat )
% 0.34/0.54 = ( A2 = bot_bot_nat ) ) ).
% 0.34/0.54
% 0.34/0.54 % bot.extremum_unique
% 0.34/0.54 thf(fact_1383_bot_Oextremum,axiom,
% 0.34/0.54 ! [A2: set_Product_unit] : ( ord_le3507040750410214029t_unit @ bot_bo3957492148770167129t_unit @ A2 ) ).
% 0.34/0.54
% 0.34/0.54 % bot.extremum
% 0.34/0.54 thf(fact_1384_bot_Oextremum,axiom,
% 0.34/0.54 ! [A2: set_o] : ( ord_less_eq_set_o @ bot_bot_set_o @ A2 ) ).
% 0.34/0.54
% 0.34/0.54 % bot.extremum
% 0.34/0.54 thf(fact_1385_bot_Oextremum,axiom,
% 0.34/0.54 ! [A2: set_int] : ( ord_less_eq_set_int @ bot_bot_set_int @ A2 ) ).
% 0.34/0.54
% 0.34/0.54 % bot.extremum
% 0.34/0.54 thf(fact_1386_bot_Oextremum,axiom,
% 0.34/0.54 ! [A2: set_Pr4532377907799695533_nat_o] : ( ord_le2965882846123202637_nat_o @ bot_bo7824918357723582233_nat_o @ A2 ) ).
% 0.34/0.54
% 0.34/0.54 % bot.extremum
% 0.34/0.54 thf(fact_1387_bot_Oextremum,axiom,
% 0.34/0.54 ! [A2: filter_nat] : ( ord_le2510731241096832064er_nat @ bot_bot_filter_nat @ A2 ) ).
% 0.34/0.54
% 0.34/0.54 % bot.extremum
% 0.34/0.54 thf(fact_1388_bot_Oextremum,axiom,
% 0.34/0.54 ! [A2: set_nat] : ( ord_less_eq_set_nat @ bot_bot_set_nat @ A2 ) ).
% 0.34/0.54
% 0.34/0.54 % bot.extremum
% 0.34/0.54 thf(fact_1389_bot_Oextremum,axiom,
% 0.34/0.54 ! [A2: nat] : ( ord_less_eq_nat @ bot_bot_nat @ A2 ) ).
% 0.34/0.54
% 0.34/0.54 % bot.extremum
% 0.34/0.54 thf(fact_1390_subset__emptyI,axiom,
% 0.34/0.54 ! [A: set_Product_unit] :
% 0.34/0.54 ( ! [X3: product_unit] :
% 0.34/0.54 ~ ( member_Product_unit @ X3 @ A )
% 0.34/0.54 => ( ord_le3507040750410214029t_unit @ A @ bot_bo3957492148770167129t_unit ) ) ).
% 0.34/0.54
% 0.34/0.54 % subset_emptyI
% 0.34/0.54 thf(fact_1391_subset__emptyI,axiom,
% 0.34/0.54 ! [A: set_o] :
% 0.34/0.54 ( ! [X3: $o] :
% 0.34/0.54 ~ ( member_o @ X3 @ A )
% 0.34/0.54 => ( ord_less_eq_set_o @ A @ bot_bot_set_o ) ) ).
% 0.34/0.54
% 0.34/0.54 % subset_emptyI
% 0.34/0.54 thf(fact_1392_subset__emptyI,axiom,
% 0.34/0.54 ! [A: set_int] :
% 0.34/0.54 ( ! [X3: int] :
% 0.34/0.54 ~ ( member_int @ X3 @ A )
% 0.34/0.54 => ( ord_less_eq_set_int @ A @ bot_bot_set_int ) ) ).
% 0.34/0.54
% 0.34/0.54 % subset_emptyI
% 0.34/0.54 thf(fact_1393_subset__emptyI,axiom,
% 0.34/0.54 ! [A: set_Pr4532377907799695533_nat_o] :
% 0.34/0.54 ( ! [X3: produc3658429121746597890et_nat > $o] :
% 0.34/0.54 ~ ( member6576561426505652726_nat_o @ X3 @ A )
% 0.34/0.54 => ( ord_le2965882846123202637_nat_o @ A @ bot_bo7824918357723582233_nat_o ) ) ).
% 0.34/0.54
% 0.34/0.54 % subset_emptyI
% 0.34/0.54 thf(fact_1394_subset__emptyI,axiom,
% 0.34/0.54 ! [A: set_nat] :
% 0.34/0.54 ( ! [X3: nat] :
% 0.34/0.54 ~ ( member_nat @ X3 @ A )
% 0.34/0.54 => ( ord_less_eq_set_nat @ A @ bot_bot_set_nat ) ) ).
% 0.34/0.54
% 0.34/0.54 % subset_emptyI
% 0.34/0.54 thf(fact_1395_insert__subsetI,axiom,
% 0.34/0.54 ! [X: produc3658429121746597890et_nat > $o,A: set_Pr4532377907799695533_nat_o,X4: set_Pr4532377907799695533_nat_o] :
% 0.34/0.54 ( ( member6576561426505652726_nat_o @ X @ A )
% 0.34/0.54 => ( ( ord_le2965882846123202637_nat_o @ X4 @ A )
% 0.34/0.54 => ( ord_le2965882846123202637_nat_o @ ( insert5175938949040314269_nat_o @ X @ X4 ) @ A ) ) ) ).
% 0.34/0.54
% 0.34/0.54 % insert_subsetI
% 0.34/0.54 thf(fact_1396_insert__subsetI,axiom,
% 0.34/0.54 ! [X: int,A: set_int,X4: set_int] :
% 0.34/0.54 ( ( member_int @ X @ A )
% 0.34/0.54 => ( ( ord_less_eq_set_int @ X4 @ A )
% 0.34/0.54 => ( ord_less_eq_set_int @ ( insert_int @ X @ X4 ) @ A ) ) ) ).
% 0.34/0.54
% 0.34/0.54 % insert_subsetI
% 0.34/0.54 thf(fact_1397_insert__subsetI,axiom,
% 0.34/0.54 ! [X: $o,A: set_o,X4: set_o] :
% 0.34/0.54 ( ( member_o @ X @ A )
% 0.34/0.54 => ( ( ord_less_eq_set_o @ X4 @ A )
% 0.34/0.54 => ( ord_less_eq_set_o @ ( insert_o @ X @ X4 ) @ A ) ) ) ).
% 0.34/0.54
% 0.34/0.54 % insert_subsetI
% 0.34/0.54 thf(fact_1398_insert__subsetI,axiom,
% 0.34/0.54 ! [X: product_unit,A: set_Product_unit,X4: set_Product_unit] :
% 0.34/0.54 ( ( member_Product_unit @ X @ A )
% 0.34/0.54 => ( ( ord_le3507040750410214029t_unit @ X4 @ A )
% 0.34/0.54 => ( ord_le3507040750410214029t_unit @ ( insert_Product_unit @ X @ X4 ) @ A ) ) ) ).
% 0.34/0.54
% 0.34/0.54 % insert_subsetI
% 0.34/0.54 thf(fact_1399_insert__subsetI,axiom,
% 0.34/0.54 ! [X: nat,A: set_nat,X4: set_nat] :
% 0.34/0.54 ( ( member_nat @ X @ A )
% 0.34/0.54 => ( ( ord_less_eq_set_nat @ X4 @ A )
% 0.34/0.54 => ( ord_less_eq_set_nat @ ( insert_nat @ X @ X4 ) @ A ) ) ) ).
% 0.34/0.54
% 0.34/0.54 % insert_subsetI
% 0.34/0.54 thf(fact_1400_subset__insertI2,axiom,
% 0.34/0.54 ! [A: set_int,B4: set_int,B: int] :
% 0.34/0.54 ( ( ord_less_eq_set_int @ A @ B4 )
% 0.34/0.54 => ( ord_less_eq_set_int @ A @ ( insert_int @ B @ B4 ) ) ) ).
% 0.34/0.54
% 0.34/0.54 % subset_insertI2
% 0.34/0.54 thf(fact_1401_subset__insertI2,axiom,
% 0.34/0.54 ! [A: set_o,B4: set_o,B: $o] :
% 0.34/0.54 ( ( ord_less_eq_set_o @ A @ B4 )
% 0.34/0.54 => ( ord_less_eq_set_o @ A @ ( insert_o @ B @ B4 ) ) ) ).
% 0.34/0.54
% 0.34/0.54 % subset_insertI2
% 0.34/0.54 thf(fact_1402_subset__insertI2,axiom,
% 0.34/0.54 ! [A: set_Product_unit,B4: set_Product_unit,B: product_unit] :
% 0.34/0.54 ( ( ord_le3507040750410214029t_unit @ A @ B4 )
% 0.34/0.54 => ( ord_le3507040750410214029t_unit @ A @ ( insert_Product_unit @ B @ B4 ) ) ) ).
% 0.34/0.54
% 0.34/0.54 % subset_insertI2
% 0.34/0.54 thf(fact_1403_subset__insertI2,axiom,
% 0.34/0.54 ! [A: set_nat,B4: set_nat,B: nat] :
% 0.34/0.54 ( ( ord_less_eq_set_nat @ A @ B4 )
% 0.34/0.54 => ( ord_less_eq_set_nat @ A @ ( insert_nat @ B @ B4 ) ) ) ).
% 0.34/0.54
% 0.34/0.54 % subset_insertI2
% 0.34/0.54 thf(fact_1404_subset__insertI,axiom,
% 0.34/0.54 ! [B4: set_int,A2: int] : ( ord_less_eq_set_int @ B4 @ ( insert_int @ A2 @ B4 ) ) ).
% 0.34/0.54
% 0.34/0.54 % subset_insertI
% 0.34/0.54 thf(fact_1405_subset__insertI,axiom,
% 0.34/0.54 ! [B4: set_o,A2: $o] : ( ord_less_eq_set_o @ B4 @ ( insert_o @ A2 @ B4 ) ) ).
% 0.34/0.54
% 0.34/0.54 % subset_insertI
% 0.34/0.54 thf(fact_1406_subset__insertI,axiom,
% 0.34/0.54 ! [B4: set_Product_unit,A2: product_unit] : ( ord_le3507040750410214029t_unit @ B4 @ ( insert_Product_unit @ A2 @ B4 ) ) ).
% 0.34/0.54
% 0.34/0.54 % subset_insertI
% 0.34/0.54 thf(fact_1407_subset__insertI,axiom,
% 0.34/0.54 ! [B4: set_nat,A2: nat] : ( ord_less_eq_set_nat @ B4 @ ( insert_nat @ A2 @ B4 ) ) ).
% 0.34/0.54
% 0.34/0.54 % subset_insertI
% 0.34/0.54 thf(fact_1408_subset__insert,axiom,
% 0.34/0.54 ! [X: produc3658429121746597890et_nat > $o,A: set_Pr4532377907799695533_nat_o,B4: set_Pr4532377907799695533_nat_o] :
% 0.34/0.54 ( ~ ( member6576561426505652726_nat_o @ X @ A )
% 0.34/0.54 => ( ( ord_le2965882846123202637_nat_o @ A @ ( insert5175938949040314269_nat_o @ X @ B4 ) )
% 0.34/0.54 = ( ord_le2965882846123202637_nat_o @ A @ B4 ) ) ) ).
% 0.34/0.54
% 0.34/0.54 % subset_insert
% 0.34/0.54 thf(fact_1409_subset__insert,axiom,
% 0.34/0.54 ! [X: int,A: set_int,B4: set_int] :
% 0.34/0.54 ( ~ ( member_int @ X @ A )
% 0.34/0.54 => ( ( ord_less_eq_set_int @ A @ ( insert_int @ X @ B4 ) )
% 0.34/0.54 = ( ord_less_eq_set_int @ A @ B4 ) ) ) ).
% 0.34/0.54
% 0.34/0.54 % subset_insert
% 0.34/0.54 thf(fact_1410_subset__insert,axiom,
% 0.34/0.54 ! [X: $o,A: set_o,B4: set_o] :
% 0.34/0.54 ( ~ ( member_o @ X @ A )
% 0.34/0.54 => ( ( ord_less_eq_set_o @ A @ ( insert_o @ X @ B4 ) )
% 0.34/0.54 = ( ord_less_eq_set_o @ A @ B4 ) ) ) ).
% 0.34/0.54
% 0.34/0.54 % subset_insert
% 0.34/0.54 thf(fact_1411_subset__insert,axiom,
% 0.34/0.54 ! [X: product_unit,A: set_Product_unit,B4: set_Product_unit] :
% 0.34/0.54 ( ~ ( member_Product_unit @ X @ A )
% 0.34/0.54 => ( ( ord_le3507040750410214029t_unit @ A @ ( insert_Product_unit @ X @ B4 ) )
% 0.34/0.54 = ( ord_le3507040750410214029t_unit @ A @ B4 ) ) ) ).
% 0.34/0.54
% 0.34/0.54 % subset_insert
% 0.34/0.54 thf(fact_1412_subset__insert,axiom,
% 0.34/0.54 ! [X: nat,A: set_nat,B4: set_nat] :
% 0.34/0.54 ( ~ ( member_nat @ X @ A )
% 0.34/0.54 => ( ( ord_less_eq_set_nat @ A @ ( insert_nat @ X @ B4 ) )
% 0.34/0.54 = ( ord_less_eq_set_nat @ A @ B4 ) ) ) ).
% 0.34/0.54
% 0.34/0.54 % subset_insert
% 0.34/0.54 thf(fact_1413_insert__mono,axiom,
% 0.34/0.54 ! [C5: set_int,D3: set_int,A2: int] :
% 0.34/0.54 ( ( ord_less_eq_set_int @ C5 @ D3 )
% 0.34/0.54 => ( ord_less_eq_set_int @ ( insert_int @ A2 @ C5 ) @ ( insert_int @ A2 @ D3 ) ) ) ).
% 0.34/0.54
% 0.34/0.54 % insert_mono
% 0.34/0.54 thf(fact_1414_insert__mono,axiom,
% 0.34/0.54 ! [C5: set_o,D3: set_o,A2: $o] :
% 0.34/0.54 ( ( ord_less_eq_set_o @ C5 @ D3 )
% 0.34/0.54 => ( ord_less_eq_set_o @ ( insert_o @ A2 @ C5 ) @ ( insert_o @ A2 @ D3 ) ) ) ).
% 0.34/0.54
% 0.34/0.54 % insert_mono
% 0.34/0.54 thf(fact_1415_insert__mono,axiom,
% 0.34/0.54 ! [C5: set_Product_unit,D3: set_Product_unit,A2: product_unit] :
% 0.34/0.54 ( ( ord_le3507040750410214029t_unit @ C5 @ D3 )
% 0.34/0.54 => ( ord_le3507040750410214029t_unit @ ( insert_Product_unit @ A2 @ C5 ) @ ( insert_Product_unit @ A2 @ D3 ) ) ) ).
% 0.34/0.54
% 0.34/0.54 % insert_mono
% 0.34/0.54 thf(fact_1416_insert__mono,axiom,
% 0.34/0.54 ! [C5: set_nat,D3: set_nat,A2: nat] :
% 0.34/0.54 ( ( ord_less_eq_set_nat @ C5 @ D3 )
% 0.34/0.54 => ( ord_less_eq_set_nat @ ( insert_nat @ A2 @ C5 ) @ ( insert_nat @ A2 @ D3 ) ) ) ).
% 0.34/0.54
% 0.34/0.54 % insert_mono
% 0.34/0.54 thf(fact_1417_Image__Un,axiom,
% 0.34/0.54 ! [R3: set_Pr1261947904930325089at_nat,A: set_nat,B4: set_nat] :
% 0.34/0.54 ( ( image_nat_nat @ R3 @ ( sup_sup_set_nat @ A @ B4 ) )
% 0.34/0.54 = ( sup_sup_set_nat @ ( image_nat_nat @ R3 @ A ) @ ( image_nat_nat @ R3 @ B4 ) ) ) ).
% 0.34/0.54
% 0.34/0.54 % Image_Un
% 0.34/0.54 thf(fact_1418_subset__vimage__iff,axiom,
% 0.34/0.54 ! [A: set_nat,F: nat > produc3658429121746597890et_nat > $o,B4: set_Pr4532377907799695533_nat_o] :
% 0.34/0.54 ( ( ord_less_eq_set_nat @ A @ ( vimage8201027140005291904_nat_o @ F @ B4 ) )
% 0.34/0.54 = ( ! [X2: nat] :
% 0.34/0.54 ( ( member_nat @ X2 @ A )
% 0.34/0.54 => ( member6576561426505652726_nat_o @ ( F @ X2 ) @ B4 ) ) ) ) ).
% 0.34/0.54
% 0.34/0.54 % subset_vimage_iff
% 0.34/0.54 thf(fact_1419_subset__vimage__iff,axiom,
% 0.34/0.54 ! [A: set_nat,F: nat > nat,B4: set_nat] :
% 0.34/0.54 ( ( ord_less_eq_set_nat @ A @ ( vimage_nat_nat @ F @ B4 ) )
% 0.34/0.54 = ( ! [X2: nat] :
% 0.34/0.54 ( ( member_nat @ X2 @ A )
% 0.34/0.54 => ( member_nat @ ( F @ X2 ) @ B4 ) ) ) ) ).
% 0.34/0.54
% 0.34/0.54 % subset_vimage_iff
% 0.34/0.54 thf(fact_1420_subset__vimage__iff,axiom,
% 0.34/0.54 ! [A: set_nat,F: nat > int,B4: set_int] :
% 0.34/0.54 ( ( ord_less_eq_set_nat @ A @ ( vimage_nat_int @ F @ B4 ) )
% 0.34/0.54 = ( ! [X2: nat] :
% 0.34/0.54 ( ( member_nat @ X2 @ A )
% 0.34/0.54 => ( member_int @ ( F @ X2 ) @ B4 ) ) ) ) ).
% 0.34/0.54
% 0.34/0.54 % subset_vimage_iff
% 0.34/0.54 thf(fact_1421_subset__vimage__iff,axiom,
% 0.34/0.54 ! [A: set_nat,F: nat > $o,B4: set_o] :
% 0.34/0.54 ( ( ord_less_eq_set_nat @ A @ ( vimage_nat_o @ F @ B4 ) )
% 0.34/0.54 = ( ! [X2: nat] :
% 0.34/0.54 ( ( member_nat @ X2 @ A )
% 0.34/0.54 => ( member_o @ ( F @ X2 ) @ B4 ) ) ) ) ).
% 0.34/0.54
% 0.34/0.54 % subset_vimage_iff
% 0.34/0.54 thf(fact_1422_subset__vimage__iff,axiom,
% 0.34/0.54 ! [A: set_nat,F: nat > product_unit,B4: set_Product_unit] :
% 0.34/0.54 ( ( ord_less_eq_set_nat @ A @ ( vimage4884490618288580032t_unit @ F @ B4 ) )
% 0.34/0.54 = ( ! [X2: nat] :
% 0.34/0.54 ( ( member_nat @ X2 @ A )
% 0.34/0.54 => ( member_Product_unit @ ( F @ X2 ) @ B4 ) ) ) ) ).
% 0.34/0.54
% 0.34/0.54 % subset_vimage_iff
% 0.34/0.54 thf(fact_1423_vimage__mono,axiom,
% 0.34/0.54 ! [A: set_nat,B4: set_nat,F: nat > nat] :
% 0.34/0.54 ( ( ord_less_eq_set_nat @ A @ B4 )
% 0.34/0.54 => ( ord_less_eq_set_nat @ ( vimage_nat_nat @ F @ A ) @ ( vimage_nat_nat @ F @ B4 ) ) ) ).
% 0.34/0.54
% 0.34/0.54 % vimage_mono
% 0.34/0.54 thf(fact_1424_relH__subset,axiom,
% 0.34/0.54 ! [Bs: set_nat,H2: heap_e7401611519738050253t_unit,H3: heap_e7401611519738050253t_unit,As: set_nat] :
% 0.34/0.54 ( ( relH @ Bs @ H2 @ H3 )
% 0.34/0.54 => ( ( ord_less_eq_set_nat @ As @ Bs )
% 0.34/0.54 => ( relH @ As @ H2 @ H3 ) ) ) ).
% 0.34/0.54
% 0.34/0.54 % relH_subset
% 0.34/0.54 thf(fact_1425_rtrancl__apply__insert,axiom,
% 0.34/0.54 ! [R3: set_Pr5094982260447487303t_unit,X: product_unit,S: set_Product_unit] :
% 0.34/0.54 ( ( image_8431784415008483091t_unit @ ( transi6481887417741755664t_unit @ R3 ) @ ( insert_Product_unit @ X @ S ) )
% 0.34/0.54 = ( insert_Product_unit @ X @ ( image_8431784415008483091t_unit @ ( transi6481887417741755664t_unit @ R3 ) @ ( sup_su793286257634532545t_unit @ S @ ( image_8431784415008483091t_unit @ R3 @ ( insert_Product_unit @ X @ bot_bo3957492148770167129t_unit ) ) ) ) ) ) ).
% 0.34/0.54
% 0.34/0.54 % rtrancl_apply_insert
% 0.34/0.54 thf(fact_1426_rtrancl__apply__insert,axiom,
% 0.34/0.54 ! [R3: set_Product_prod_o_o,X: $o,S: set_o] :
% 0.34/0.54 ( ( image_o_o @ ( transitive_rtrancl_o @ R3 ) @ ( insert_o @ X @ S ) )
% 0.34/0.54 = ( insert_o @ X @ ( image_o_o @ ( transitive_rtrancl_o @ R3 ) @ ( sup_sup_set_o @ S @ ( image_o_o @ R3 @ ( insert_o @ X @ bot_bot_set_o ) ) ) ) ) ) ).
% 0.34/0.54
% 0.34/0.54 % rtrancl_apply_insert
% 0.34/0.54 thf(fact_1427_rtrancl__apply__insert,axiom,
% 0.34/0.54 ! [R3: set_Pr1261947904930325089at_nat,X: nat,S: set_nat] :
% 0.34/0.54 ( ( image_nat_nat @ ( transi2905341329935302413cl_nat @ R3 ) @ ( insert_nat @ X @ S ) )
% 0.34/0.54 = ( insert_nat @ X @ ( image_nat_nat @ ( transi2905341329935302413cl_nat @ R3 ) @ ( sup_sup_set_nat @ S @ ( image_nat_nat @ R3 @ ( insert_nat @ X @ bot_bot_set_nat ) ) ) ) ) ) ).
% 0.34/0.54
% 0.34/0.54 % rtrancl_apply_insert
% 0.34/0.54 thf(fact_1428_rtrancl__apply__insert,axiom,
% 0.34/0.54 ! [R3: set_Pr958786334691620121nt_int,X: int,S: set_int] :
% 0.34/0.54 ( ( image_int_int @ ( transi2902850859426252137cl_int @ R3 ) @ ( insert_int @ X @ S ) )
% 0.34/0.54 = ( insert_int @ X @ ( image_int_int @ ( transi2902850859426252137cl_int @ R3 ) @ ( sup_sup_set_int @ S @ ( image_int_int @ R3 @ ( insert_int @ X @ bot_bot_set_int ) ) ) ) ) ) ).
% 0.34/0.54
% 0.34/0.54 % rtrancl_apply_insert
% 0.34/0.54 thf(fact_1429_rtrancl__apply__insert,axiom,
% 0.34/0.54 ! [R3: set_Pr2161125870931222855_nat_o,X: produc3658429121746597890et_nat > $o,S: set_Pr4532377907799695533_nat_o] :
% 0.34/0.54 ( ( image_2232984160439939859_nat_o @ ( transi712980445759334096_nat_o @ R3 ) @ ( insert5175938949040314269_nat_o @ X @ S ) )
% 0.34/0.54 = ( insert5175938949040314269_nat_o @ X @ ( image_2232984160439939859_nat_o @ ( transi712980445759334096_nat_o @ R3 ) @ ( sup_su5209123915105501825_nat_o @ S @ ( image_2232984160439939859_nat_o @ R3 @ ( insert5175938949040314269_nat_o @ X @ bot_bo7824918357723582233_nat_o ) ) ) ) ) ) ).
% 0.34/0.54
% 0.34/0.54 % rtrancl_apply_insert
% 0.34/0.54 thf(fact_1430_antisymD,axiom,
% 0.34/0.54 ! [R2: set_Pr1261947904930325089at_nat,A2: nat,B: nat] :
% 0.34/0.54 ( ( antisym_nat @ R2 )
% 0.34/0.54 => ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A2 @ B ) @ R2 )
% 0.34/0.54 => ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ B @ A2 ) @ R2 )
% 0.34/0.54 => ( A2 = B ) ) ) ) ).
% 0.34/0.54
% 0.34/0.54 % antisymD
% 0.34/0.54 thf(fact_1431_antisymD,axiom,
% 0.34/0.54 ! [R2: set_Pr958786334691620121nt_int,A2: int,B: int] :
% 0.34/0.54 ( ( antisym_int @ R2 )
% 0.34/0.54 => ( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ A2 @ B ) @ R2 )
% 0.34/0.54 => ( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ B @ A2 ) @ R2 )
% 0.34/0.54 => ( A2 = B ) ) ) ) ).
% 0.34/0.54
% 0.34/0.54 % antisymD
% 0.34/0.54 thf(fact_1432_antisymI,axiom,
% 0.34/0.54 ! [R2: set_Pr1261947904930325089at_nat] :
% 0.34/0.54 ( ! [X3: nat,Y2: nat] :
% 0.34/0.54 ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X3 @ Y2 ) @ R2 )
% 0.34/0.54 => ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ Y2 @ X3 ) @ R2 )
% 0.34/0.54 => ( X3 = Y2 ) ) )
% 0.34/0.54 => ( antisym_nat @ R2 ) ) ).
% 0.34/0.54
% 0.34/0.54 % antisymI
% 0.34/0.54 thf(fact_1433_antisymI,axiom,
% 0.34/0.54 ! [R2: set_Pr958786334691620121nt_int] :
% 0.34/0.54 ( ! [X3: int,Y2: int] :
% 0.34/0.54 ( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ X3 @ Y2 ) @ R2 )
% 0.34/0.54 => ( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ Y2 @ X3 ) @ R2 )
% 0.34/0.54 => ( X3 = Y2 ) ) )
% 0.34/0.54 => ( antisym_int @ R2 ) ) ).
% 0.34/0.54
% 0.34/0.54 % antisymI
% 0.34/0.54 thf(fact_1434_antisym__def,axiom,
% 0.34/0.54 ( antisym_nat
% 0.34/0.54 = ( ^ [R4: set_Pr1261947904930325089at_nat] :
% 0.34/0.54 ! [X2: nat,Y3: nat] :
% 0.34/0.54 ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X2 @ Y3 ) @ R4 )
% 0.34/0.54 => ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ Y3 @ X2 ) @ R4 )
% 0.34/0.54 => ( X2 = Y3 ) ) ) ) ) ).
% 0.34/0.54
% 0.34/0.54 % antisym_def
% 0.34/0.54 thf(fact_1435_antisym__def,axiom,
% 0.34/0.54 ( antisym_int
% 0.34/0.54 = ( ^ [R4: set_Pr958786334691620121nt_int] :
% 0.34/0.54 ! [X2: int,Y3: int] :
% 0.34/0.54 ( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ X2 @ Y3 ) @ R4 )
% 0.34/0.54 => ( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ Y3 @ X2 ) @ R4 )
% 0.34/0.54 => ( X2 = Y3 ) ) ) ) ) ).
% 0.34/0.54
% 0.34/0.54 % antisym_def
% 0.34/0.54 thf(fact_1436_Image__absorb__rtrancl,axiom,
% 0.34/0.54 ! [A: set_Pr1261947904930325089at_nat,B4: set_nat,C5: set_nat] :
% 0.34/0.54 ( ( trans_nat @ A )
% 0.34/0.54 => ( ( refl_on_nat @ B4 @ A )
% 0.34/0.54 => ( ( ord_less_eq_set_nat @ C5 @ B4 )
% 0.34/0.54 => ( ( image_nat_nat @ ( transi2905341329935302413cl_nat @ A ) @ C5 )
% 0.34/0.54 = ( image_nat_nat @ A @ C5 ) ) ) ) ) ).
% 0.34/0.54
% 0.34/0.54 % Image_absorb_rtrancl
% 0.34/0.54 thf(fact_1437_times__assn__raw_Osimps,axiom,
% 0.34/0.54 ! [P: produc3658429121746597890et_nat > $o,Q2: produc3658429121746597890et_nat > $o,H2: heap_e7401611519738050253t_unit,As: set_nat] :
% 0.34/0.54 ( ( times_assn_raw @ P @ Q2 @ ( produc7507926704131184380et_nat @ H2 @ As ) )
% 0.34/0.54 = ( ? [As1: set_nat,As22: set_nat] :
% 0.34/0.54 ( ( As
% 0.34/0.54 = ( sup_sup_set_nat @ As1 @ As22 ) )
% 0.34/0.54 & ( ( inf_inf_set_nat @ As1 @ As22 )
% 0.34/0.54 = bot_bot_set_nat )
% 0.34/0.54 & ( P @ ( produc7507926704131184380et_nat @ H2 @ As1 ) )
% 0.34/0.54 & ( Q2 @ ( produc7507926704131184380et_nat @ H2 @ As22 ) ) ) ) ) ).
% 0.34/0.54
% 0.34/0.54 % times_assn_raw.simps
% 0.34/0.54 thf(fact_1438_times__assn__raw_Oelims_I1_J,axiom,
% 0.34/0.54 ! [X: produc3658429121746597890et_nat > $o,Xa: produc3658429121746597890et_nat > $o,Xb: produc3658429121746597890et_nat,Y: $o] :
% 0.34/0.54 ( ( ( times_assn_raw @ X @ Xa @ Xb )
% 0.34/0.54 = Y )
% 0.34/0.54 => ~ ! [H: heap_e7401611519738050253t_unit,As2: set_nat] :
% 0.34/0.54 ( ( Xb
% 0.34/0.54 = ( produc7507926704131184380et_nat @ H @ As2 ) )
% 0.34/0.54 => ( Y
% 0.34/0.54 = ( ~ ? [As1: set_nat,As22: set_nat] :
% 0.34/0.54 ( ( As2
% 0.34/0.54 = ( sup_sup_set_nat @ As1 @ As22 ) )
% 0.34/0.54 & ( ( inf_inf_set_nat @ As1 @ As22 )
% 0.34/0.54 = bot_bot_set_nat )
% 0.34/0.54 & ( X @ ( produc7507926704131184380et_nat @ H @ As1 ) )
% 0.34/0.54 & ( Xa @ ( produc7507926704131184380et_nat @ H @ As22 ) ) ) ) ) ) ) ).
% 0.34/0.54
% 0.34/0.54 % times_assn_raw.elims(1)
% 0.34/0.54 thf(fact_1439_times__assn__raw_Oelims_I2_J,axiom,
% 0.34/0.54 ! [X: produc3658429121746597890et_nat > $o,Xa: produc3658429121746597890et_nat > $o,Xb: produc3658429121746597890et_nat] :
% 0.34/0.54 ( ( times_assn_raw @ X @ Xa @ Xb )
% 0.34/0.54 => ~ ! [H: heap_e7401611519738050253t_unit,As2: set_nat] :
% 0.34/0.54 ( ( Xb
% 0.34/0.54 = ( produc7507926704131184380et_nat @ H @ As2 ) )
% 0.34/0.54 => ~ ? [As12: set_nat,As23: set_nat] :
% 0.34/0.54 ( ( As2
% 0.34/0.54 = ( sup_sup_set_nat @ As12 @ As23 ) )
% 0.34/0.54 & ( ( inf_inf_set_nat @ As12 @ As23 )
% 0.34/0.54 = bot_bot_set_nat )
% 0.34/0.54 & ( X @ ( produc7507926704131184380et_nat @ H @ As12 ) )
% 0.34/0.54 & ( Xa @ ( produc7507926704131184380et_nat @ H @ As23 ) ) ) ) ) ).
% 0.34/0.54
% 0.34/0.54 % times_assn_raw.elims(2)
% 0.34/0.54 thf(fact_1440_times__assn__raw_Oelims_I3_J,axiom,
% 0.34/0.54 ! [X: produc3658429121746597890et_nat > $o,Xa: produc3658429121746597890et_nat > $o,Xb: produc3658429121746597890et_nat] :
% 0.34/0.54 ( ~ ( times_assn_raw @ X @ Xa @ Xb )
% 0.34/0.54 => ~ ! [H: heap_e7401611519738050253t_unit,As2: set_nat] :
% 0.34/0.54 ( ( Xb
% 0.34/0.54 = ( produc7507926704131184380et_nat @ H @ As2 ) )
% 0.34/0.54 => ? [As13: set_nat,As24: set_nat] :
% 0.34/0.54 ( ( As2
% 0.34/0.54 = ( sup_sup_set_nat @ As13 @ As24 ) )
% 0.34/0.54 & ( ( inf_inf_set_nat @ As13 @ As24 )
% 0.34/0.54 = bot_bot_set_nat )
% 0.34/0.54 & ( X @ ( produc7507926704131184380et_nat @ H @ As13 ) )
% 0.34/0.54 & ( Xa @ ( produc7507926704131184380et_nat @ H @ As24 ) ) ) ) ) ).
% 0.34/0.54
% 0.34/0.54 % times_assn_raw.elims(3)
% 0.34/0.54 thf(fact_1441_rtrancl__Image__advance__ss,axiom,
% 0.34/0.54 ! [U: product_unit,V: product_unit,E: set_Pr5094982260447487303t_unit] :
% 0.34/0.54 ( ( member7821138191923116944t_unit @ ( produc5692694829384537815t_unit @ U @ V ) @ E )
% 0.34/0.54 => ( ord_le3507040750410214029t_unit @ ( image_8431784415008483091t_unit @ ( transi6481887417741755664t_unit @ E ) @ ( insert_Product_unit @ V @ bot_bo3957492148770167129t_unit ) ) @ ( image_8431784415008483091t_unit @ ( transi6481887417741755664t_unit @ E ) @ ( insert_Product_unit @ U @ bot_bo3957492148770167129t_unit ) ) ) ) ).
% 0.34/0.54
% 0.34/0.54 % rtrancl_Image_advance_ss
% 0.34/0.54 thf(fact_1442_rtrancl__Image__advance__ss,axiom,
% 0.34/0.54 ! [U: $o,V: $o,E: set_Product_prod_o_o] :
% 0.34/0.54 ( ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ U @ V ) @ E )
% 0.34/0.54 => ( ord_less_eq_set_o @ ( image_o_o @ ( transitive_rtrancl_o @ E ) @ ( insert_o @ V @ bot_bot_set_o ) ) @ ( image_o_o @ ( transitive_rtrancl_o @ E ) @ ( insert_o @ U @ bot_bot_set_o ) ) ) ) ).
% 0.34/0.54
% 0.34/0.54 % rtrancl_Image_advance_ss
% 0.34/0.54 thf(fact_1443_rtrancl__Image__advance__ss,axiom,
% 0.34/0.54 ! [U: int,V: int,E: set_Pr958786334691620121nt_int] :
% 0.34/0.54 ( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ U @ V ) @ E )
% 0.34/0.54 => ( ord_less_eq_set_int @ ( image_int_int @ ( transi2902850859426252137cl_int @ E ) @ ( insert_int @ V @ bot_bot_set_int ) ) @ ( image_int_int @ ( transi2902850859426252137cl_int @ E ) @ ( insert_int @ U @ bot_bot_set_int ) ) ) ) ).
% 0.34/0.54
% 0.34/0.54 % rtrancl_Image_advance_ss
% 0.34/0.54 thf(fact_1444_rtrancl__Image__advance__ss,axiom,
% 0.34/0.54 ! [U: produc3658429121746597890et_nat > $o,V: produc3658429121746597890et_nat > $o,E: set_Pr2161125870931222855_nat_o] :
% 0.34/0.54 ( ( member8781333585448626064_nat_o @ ( produc7368190662567826135_nat_o @ U @ V ) @ E )
% 0.34/0.54 => ( ord_le2965882846123202637_nat_o @ ( image_2232984160439939859_nat_o @ ( transi712980445759334096_nat_o @ E ) @ ( insert5175938949040314269_nat_o @ V @ bot_bo7824918357723582233_nat_o ) ) @ ( image_2232984160439939859_nat_o @ ( transi712980445759334096_nat_o @ E ) @ ( insert5175938949040314269_nat_o @ U @ bot_bo7824918357723582233_nat_o ) ) ) ) ).
% 0.34/0.54
% 0.34/0.54 % rtrancl_Image_advance_ss
% 0.34/0.54 thf(fact_1445_rtrancl__Image__advance__ss,axiom,
% 0.34/0.54 ! [U: nat,V: nat,E: set_Pr1261947904930325089at_nat] :
% 0.34/0.54 ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ U @ V ) @ E )
% 0.34/0.54 => ( ord_less_eq_set_nat @ ( image_nat_nat @ ( transi2905341329935302413cl_nat @ E ) @ ( insert_nat @ V @ bot_bot_set_nat ) ) @ ( image_nat_nat @ ( transi2905341329935302413cl_nat @ E ) @ ( insert_nat @ U @ bot_bot_set_nat ) ) ) ) ).
% 0.34/0.54
% 0.34/0.54 % rtrancl_Image_advance_ss
% 0.34/0.54 thf(fact_1446_rtrancl__image__advance__rtrancl,axiom,
% 0.34/0.54 ! [Q4: produc3658429121746597890et_nat > $o,R3: set_Pr2161125870931222855_nat_o,Q0: set_Pr4532377907799695533_nat_o,X: produc3658429121746597890et_nat > $o] :
% 0.34/0.54 ( ( member6576561426505652726_nat_o @ Q4 @ ( image_2232984160439939859_nat_o @ ( transi712980445759334096_nat_o @ R3 ) @ Q0 ) )
% 0.34/0.54 => ( ( member8781333585448626064_nat_o @ ( produc7368190662567826135_nat_o @ Q4 @ X ) @ ( transi712980445759334096_nat_o @ R3 ) )
% 0.34/0.54 => ( member6576561426505652726_nat_o @ X @ ( image_2232984160439939859_nat_o @ ( transi712980445759334096_nat_o @ R3 ) @ Q0 ) ) ) ) ).
% 0.34/0.54
% 0.34/0.54 % rtrancl_image_advance_rtrancl
% 0.34/0.54 thf(fact_1447_rtrancl__image__advance__rtrancl,axiom,
% 0.34/0.54 ! [Q4: $o,R3: set_Product_prod_o_o,Q0: set_o,X: $o] :
% 0.34/0.54 ( ( member_o @ Q4 @ ( image_o_o @ ( transitive_rtrancl_o @ R3 ) @ Q0 ) )
% 0.34/0.54 => ( ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ Q4 @ X ) @ ( transitive_rtrancl_o @ R3 ) )
% 0.34/0.54 => ( member_o @ X @ ( image_o_o @ ( transitive_rtrancl_o @ R3 ) @ Q0 ) ) ) ) ).
% 0.34/0.54
% 0.34/0.54 % rtrancl_image_advance_rtrancl
% 0.34/0.54 thf(fact_1448_rtrancl__image__advance__rtrancl,axiom,
% 0.34/0.54 ! [Q4: product_unit,R3: set_Pr5094982260447487303t_unit,Q0: set_Product_unit,X: product_unit] :
% 0.34/0.54 ( ( member_Product_unit @ Q4 @ ( image_8431784415008483091t_unit @ ( transi6481887417741755664t_unit @ R3 ) @ Q0 ) )
% 0.34/0.54 => ( ( member7821138191923116944t_unit @ ( produc5692694829384537815t_unit @ Q4 @ X ) @ ( transi6481887417741755664t_unit @ R3 ) )
% 0.34/0.54 => ( member_Product_unit @ X @ ( image_8431784415008483091t_unit @ ( transi6481887417741755664t_unit @ R3 ) @ Q0 ) ) ) ) ).
% 0.34/0.54
% 0.34/0.54 % rtrancl_image_advance_rtrancl
% 0.34/0.54 thf(fact_1449_rtrancl__image__advance__rtrancl,axiom,
% 0.34/0.54 ! [Q4: nat,R3: set_Pr1261947904930325089at_nat,Q0: set_nat,X: nat] :
% 0.34/0.54 ( ( member_nat @ Q4 @ ( image_nat_nat @ ( transi2905341329935302413cl_nat @ R3 ) @ Q0 ) )
% 0.34/0.54 => ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ Q4 @ X ) @ ( transi2905341329935302413cl_nat @ R3 ) )
% 0.34/0.54 => ( member_nat @ X @ ( image_nat_nat @ ( transi2905341329935302413cl_nat @ R3 ) @ Q0 ) ) ) ) ).
% 0.34/0.54
% 0.34/0.54 % rtrancl_image_advance_rtrancl
% 0.34/0.54 thf(fact_1450_rtrancl__image__advance__rtrancl,axiom,
% 0.34/0.54 ! [Q4: int,R3: set_Pr958786334691620121nt_int,Q0: set_int,X: int] :
% 0.34/0.54 ( ( member_int @ Q4 @ ( image_int_int @ ( transi2902850859426252137cl_int @ R3 ) @ Q0 ) )
% 0.34/0.54 => ( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ Q4 @ X ) @ ( transi2902850859426252137cl_int @ R3 ) )
% 0.34/0.54 => ( member_int @ X @ ( image_int_int @ ( transi2902850859426252137cl_int @ R3 ) @ Q0 ) ) ) ) ).
% 0.34/0.54
% 0.34/0.54 % rtrancl_image_advance_rtrancl
% 0.34/0.54 thf(fact_1451_rtrancl__image__advance,axiom,
% 0.34/0.54 ! [Q4: produc3658429121746597890et_nat > $o,R3: set_Pr2161125870931222855_nat_o,Q0: set_Pr4532377907799695533_nat_o,X: produc3658429121746597890et_nat > $o] :
% 0.34/0.54 ( ( member6576561426505652726_nat_o @ Q4 @ ( image_2232984160439939859_nat_o @ ( transi712980445759334096_nat_o @ R3 ) @ Q0 ) )
% 0.34/0.54 => ( ( member8781333585448626064_nat_o @ ( produc7368190662567826135_nat_o @ Q4 @ X ) @ R3 )
% 0.34/0.54 => ( member6576561426505652726_nat_o @ X @ ( image_2232984160439939859_nat_o @ ( transi712980445759334096_nat_o @ R3 ) @ Q0 ) ) ) ) ).
% 0.34/0.54
% 0.34/0.54 % rtrancl_image_advance
% 0.34/0.54 thf(fact_1452_rtrancl__image__advance,axiom,
% 0.34/0.54 ! [Q4: $o,R3: set_Product_prod_o_o,Q0: set_o,X: $o] :
% 0.34/0.54 ( ( member_o @ Q4 @ ( image_o_o @ ( transitive_rtrancl_o @ R3 ) @ Q0 ) )
% 0.34/0.54 => ( ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ Q4 @ X ) @ R3 )
% 0.34/0.54 => ( member_o @ X @ ( image_o_o @ ( transitive_rtrancl_o @ R3 ) @ Q0 ) ) ) ) ).
% 0.34/0.54
% 0.34/0.54 % rtrancl_image_advance
% 0.34/0.54 thf(fact_1453_rtrancl__image__advance,axiom,
% 0.34/0.54 ! [Q4: product_unit,R3: set_Pr5094982260447487303t_unit,Q0: set_Product_unit,X: product_unit] :
% 0.34/0.54 ( ( member_Product_unit @ Q4 @ ( image_8431784415008483091t_unit @ ( transi6481887417741755664t_unit @ R3 ) @ Q0 ) )
% 0.34/0.54 => ( ( member7821138191923116944t_unit @ ( produc5692694829384537815t_unit @ Q4 @ X ) @ R3 )
% 0.34/0.54 => ( member_Product_unit @ X @ ( image_8431784415008483091t_unit @ ( transi6481887417741755664t_unit @ R3 ) @ Q0 ) ) ) ) ).
% 0.34/0.54
% 0.34/0.54 % rtrancl_image_advance
% 0.34/0.54 thf(fact_1454_rtrancl__image__advance,axiom,
% 0.34/0.54 ! [Q4: nat,R3: set_Pr1261947904930325089at_nat,Q0: set_nat,X: nat] :
% 0.34/0.54 ( ( member_nat @ Q4 @ ( image_nat_nat @ ( transi2905341329935302413cl_nat @ R3 ) @ Q0 ) )
% 0.34/0.54 => ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ Q4 @ X ) @ R3 )
% 0.34/0.54 => ( member_nat @ X @ ( image_nat_nat @ ( transi2905341329935302413cl_nat @ R3 ) @ Q0 ) ) ) ) ).
% 0.34/0.54
% 0.34/0.54 % rtrancl_image_advance
% 0.34/0.54 thf(fact_1455_rtrancl__image__advance,axiom,
% 0.34/0.54 ! [Q4: int,R3: set_Pr958786334691620121nt_int,Q0: set_int,X: int] :
% 0.34/0.54 ( ( member_int @ Q4 @ ( image_int_int @ ( transi2902850859426252137cl_int @ R3 ) @ Q0 ) )
% 0.34/0.54 => ( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ Q4 @ X ) @ R3 )
% 0.34/0.54 => ( member_int @ X @ ( image_int_int @ ( transi2902850859426252137cl_int @ R3 ) @ Q0 ) ) ) ) ).
% 0.34/0.54
% 0.34/0.54 % rtrancl_image_advance
% 0.34/0.54 thf(fact_1456_trancl__rtrancl__trancl,axiom,
% 0.34/0.54 ! [A2: nat,B: nat,R2: set_Pr1261947904930325089at_nat,C: nat] :
% 0.34/0.54 ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A2 @ B ) @ ( transi6264000038957366511cl_nat @ R2 ) )
% 0.34/0.54 => ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ B @ C ) @ ( transi2905341329935302413cl_nat @ R2 ) )
% 0.34/0.54 => ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A2 @ C ) @ ( transi6264000038957366511cl_nat @ R2 ) ) ) ) ).
% 0.34/0.54
% 0.34/0.54 % trancl_rtrancl_trancl
% 0.34/0.54 thf(fact_1457_trancl__rtrancl__trancl,axiom,
% 0.34/0.54 ! [A2: int,B: int,R2: set_Pr958786334691620121nt_int,C: int] :
% 0.34/0.54 ( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ A2 @ B ) @ ( transi6261509568448316235cl_int @ R2 ) )
% 0.34/0.54 => ( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ B @ C ) @ ( transi2902850859426252137cl_int @ R2 ) )
% 0.34/0.54 => ( member5262025264175285858nt_int @ ( product_Pair_int_int @ A2 @ C ) @ ( transi6261509568448316235cl_int @ R2 ) ) ) ) ).
% 0.34/0.54
% 0.34/0.54 % trancl_rtrancl_trancl
% 0.34/0.54 thf(fact_1458_rtrancl__trancl__trancl,axiom,
% 0.34/0.54 ! [X: nat,Y: nat,R2: set_Pr1261947904930325089at_nat,Z: nat] :
% 0.34/0.54 ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ Y ) @ ( transi2905341329935302413cl_nat @ R2 ) )
% 0.34/0.54 => ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ Y @ Z ) @ ( transi6264000038957366511cl_nat @ R2 ) )
% 0.34/0.54 => ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ Z ) @ ( transi6264000038957366511cl_nat @ R2 ) ) ) ) ).
% 0.34/0.54
% 0.34/0.54 % rtrancl_trancl_trancl
% 0.34/0.54 thf(fact_1459_rtrancl__trancl__trancl,axiom,
% 0.34/0.54 ! [X: int,Y: int,R2: set_Pr958786334691620121nt_int,Z: int] :
% 0.34/0.54 ( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ X @ Y ) @ ( transi2902850859426252137cl_int @ R2 ) )
% 0.34/0.54 => ( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ Y @ Z ) @ ( transi6261509568448316235cl_int @ R2 ) )
% 0.34/0.54 => ( member5262025264175285858nt_int @ ( product_Pair_int_int @ X @ Z ) @ ( transi6261509568448316235cl_int @ R2 ) ) ) ) ).
% 0.34/0.54
% 0.34/0.54 % rtrancl_trancl_trancl
% 0.34/0.54 thf(fact_1460_rtrancl__into__trancl2,axiom,
% 0.34/0.54 ! [A2: nat,B: nat,R2: set_Pr1261947904930325089at_nat,C: nat] :
% 0.34/0.54 ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A2 @ B ) @ R2 )
% 0.34/0.54 => ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ B @ C ) @ ( transi2905341329935302413cl_nat @ R2 ) )
% 0.34/0.54 => ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A2 @ C ) @ ( transi6264000038957366511cl_nat @ R2 ) ) ) ) ).
% 0.34/0.54
% 0.34/0.54 % rtrancl_into_trancl2
% 0.34/0.54 thf(fact_1461_rtrancl__into__trancl2,axiom,
% 0.34/0.54 ! [A2: int,B: int,R2: set_Pr958786334691620121nt_int,C: int] :
% 0.34/0.54 ( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ A2 @ B ) @ R2 )
% 0.34/0.54 => ( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ B @ C ) @ ( transi2902850859426252137cl_int @ R2 ) )
% 0.34/0.54 => ( member5262025264175285858nt_int @ ( product_Pair_int_int @ A2 @ C ) @ ( transi6261509568448316235cl_int @ R2 ) ) ) ) ).
% 0.34/0.54
% 0.34/0.54 % rtrancl_into_trancl2
% 0.34/0.54 thf(fact_1462_rtrancl__into__trancl1,axiom,
% 0.34/0.54 ! [A2: nat,B: nat,R2: set_Pr1261947904930325089at_nat,C: nat] :
% 0.34/0.54 ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A2 @ B ) @ ( transi2905341329935302413cl_nat @ R2 ) )
% 0.34/0.54 => ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ B @ C ) @ R2 )
% 0.34/0.54 => ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A2 @ C ) @ ( transi6264000038957366511cl_nat @ R2 ) ) ) ) ).
% 0.34/0.54
% 0.34/0.54 % rtrancl_into_trancl1
% 0.34/0.54 thf(fact_1463_rtrancl__into__trancl1,axiom,
% 0.34/0.54 ! [A2: int,B: int,R2: set_Pr958786334691620121nt_int,C: int] :
% 0.34/0.54 ( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ A2 @ B ) @ ( transi2902850859426252137cl_int @ R2 ) )
% 0.34/0.54 => ( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ B @ C ) @ R2 )
% 0.34/0.54 => ( member5262025264175285858nt_int @ ( product_Pair_int_int @ A2 @ C ) @ ( transi6261509568448316235cl_int @ R2 ) ) ) ) ).
% 0.34/0.54
% 0.34/0.54 % rtrancl_into_trancl1
% 0.34/0.54 thf(fact_1464_rtrancl__eq__or__trancl,axiom,
% 0.34/0.54 ! [X: nat,Y: nat,R3: set_Pr1261947904930325089at_nat] :
% 0.34/0.54 ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ Y ) @ ( transi2905341329935302413cl_nat @ R3 ) )
% 0.34/0.54 = ( ( X = Y )
% 0.34/0.54 | ( ( X != Y )
% 0.34/0.54 & ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ Y ) @ ( transi6264000038957366511cl_nat @ R3 ) ) ) ) ) ).
% 0.34/0.54
% 0.34/0.54 % rtrancl_eq_or_trancl
% 0.34/0.54 thf(fact_1465_rtrancl__eq__or__trancl,axiom,
% 0.34/0.54 ! [X: int,Y: int,R3: set_Pr958786334691620121nt_int] :
% 0.34/0.54 ( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ X @ Y ) @ ( transi2902850859426252137cl_int @ R3 ) )
% 0.34/0.54 = ( ( X = Y )
% 0.34/0.54 | ( ( X != Y )
% 0.34/0.54 & ( member5262025264175285858nt_int @ ( product_Pair_int_int @ X @ Y ) @ ( transi6261509568448316235cl_int @ R3 ) ) ) ) ) ).
% 0.34/0.54
% 0.34/0.54 % rtrancl_eq_or_trancl
% 0.34/0.54 thf(fact_1466_trancl__into__rtrancl,axiom,
% 0.34/0.54 ! [A2: nat,B: nat,R2: set_Pr1261947904930325089at_nat] :
% 0.34/0.54 ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A2 @ B ) @ ( transi6264000038957366511cl_nat @ R2 ) )
% 0.34/0.54 => ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A2 @ B ) @ ( transi2905341329935302413cl_nat @ R2 ) ) ) ).
% 0.34/0.54
% 0.34/0.54 % trancl_into_rtrancl
% 0.34/0.54 thf(fact_1467_trancl__into__rtrancl,axiom,
% 0.34/0.54 ! [A2: int,B: int,R2: set_Pr958786334691620121nt_int] :
% 0.34/0.54 ( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ A2 @ B ) @ ( transi6261509568448316235cl_int @ R2 ) )
% 0.34/0.54 => ( member5262025264175285858nt_int @ ( product_Pair_int_int @ A2 @ B ) @ ( transi2902850859426252137cl_int @ R2 ) ) ) ).
% 0.34/0.54
% 0.34/0.54 % trancl_into_rtrancl
% 0.34/0.54 thf(fact_1468_tranclD2,axiom,
% 0.34/0.54 ! [X: nat,Y: nat,R3: set_Pr1261947904930325089at_nat] :
% 0.34/0.54 ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ Y ) @ ( transi6264000038957366511cl_nat @ R3 ) )
% 0.34/0.54 => ? [Z3: nat] :
% 0.34/0.54 ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ Z3 ) @ ( transi2905341329935302413cl_nat @ R3 ) )
% 0.34/0.54 & ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ Z3 @ Y ) @ R3 ) ) ) ).
% 0.34/0.54
% 0.34/0.54 % tranclD2
% 0.34/0.54 thf(fact_1469_tranclD2,axiom,
% 0.34/0.54 ! [X: int,Y: int,R3: set_Pr958786334691620121nt_int] :
% 0.34/0.54 ( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ X @ Y ) @ ( transi6261509568448316235cl_int @ R3 ) )
% 0.34/0.54 => ? [Z3: int] :
% 0.34/0.54 ( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ X @ Z3 ) @ ( transi2902850859426252137cl_int @ R3 ) )
% 0.34/0.54 & ( member5262025264175285858nt_int @ ( product_Pair_int_int @ Z3 @ Y ) @ R3 ) ) ) ).
% 0.34/0.54
% 0.34/0.54 % tranclD2
% 0.34/0.54 thf(fact_1470_rtranclD,axiom,
% 0.34/0.54 ! [A2: nat,B: nat,R3: set_Pr1261947904930325089at_nat] :
% 0.34/0.54 ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A2 @ B ) @ ( transi2905341329935302413cl_nat @ R3 ) )
% 0.34/0.54 => ( ( A2 = B )
% 0.34/0.54 | ( ( A2 != B )
% 0.34/0.54 & ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A2 @ B ) @ ( transi6264000038957366511cl_nat @ R3 ) ) ) ) ) ).
% 0.34/0.54
% 0.34/0.54 % rtranclD
% 0.34/0.54 thf(fact_1471_rtranclD,axiom,
% 0.34/0.54 ! [A2: int,B: int,R3: set_Pr958786334691620121nt_int] :
% 0.34/0.54 ( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ A2 @ B ) @ ( transi2902850859426252137cl_int @ R3 ) )
% 0.34/0.54 => ( ( A2 = B )
% 0.34/0.54 | ( ( A2 != B )
% 0.34/0.54 & ( member5262025264175285858nt_int @ ( product_Pair_int_int @ A2 @ B ) @ ( transi6261509568448316235cl_int @ R3 ) ) ) ) ) ).
% 0.34/0.54
% 0.34/0.54 % rtranclD
% 0.34/0.54 thf(fact_1472_tranclD,axiom,
% 0.34/0.54 ! [X: nat,Y: nat,R3: set_Pr1261947904930325089at_nat] :
% 0.34/0.54 ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ Y ) @ ( transi6264000038957366511cl_nat @ R3 ) )
% 0.34/0.54 => ? [Z3: nat] :
% 0.34/0.54 ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ Z3 ) @ R3 )
% 0.34/0.54 & ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ Z3 @ Y ) @ ( transi2905341329935302413cl_nat @ R3 ) ) ) ) ).
% 0.34/0.54
% 0.34/0.54 % tranclD
% 0.34/0.54 thf(fact_1473_tranclD,axiom,
% 0.34/0.54 ! [X: int,Y: int,R3: set_Pr958786334691620121nt_int] :
% 0.34/0.54 ( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ X @ Y ) @ ( transi6261509568448316235cl_int @ R3 ) )
% 0.34/0.54 => ? [Z3: int] :
% 0.34/0.54 ( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ X @ Z3 ) @ R3 )
% 0.34/0.54 & ( member5262025264175285858nt_int @ ( product_Pair_int_int @ Z3 @ Y ) @ ( transi2902850859426252137cl_int @ R3 ) ) ) ) ).
% 0.34/0.54
% 0.34/0.54 % tranclD
% 0.34/0.54 thf(fact_1474_insert__is__Un,axiom,
% 0.34/0.54 ( insert_Product_unit
% 0.34/0.54 = ( ^ [A6: product_unit] : ( sup_su793286257634532545t_unit @ ( insert_Product_unit @ A6 @ bot_bo3957492148770167129t_unit ) ) ) ) ).
% 0.34/0.54
% 0.34/0.54 % insert_is_Un
% 0.34/0.54 thf(fact_1475_insert__is__Un,axiom,
% 0.34/0.54 ( insert_o
% 0.34/0.54 = ( ^ [A6: $o] : ( sup_sup_set_o @ ( insert_o @ A6 @ bot_bot_set_o ) ) ) ) ).
% 0.34/0.54
% 0.34/0.54 % insert_is_Un
% 0.34/0.54 thf(fact_1476_insert__is__Un,axiom,
% 0.34/0.54 ( insert_nat
% 0.34/0.54 = ( ^ [A6: nat] : ( sup_sup_set_nat @ ( insert_nat @ A6 @ bot_bot_set_nat ) ) ) ) ).
% 0.34/0.54
% 0.34/0.54 % insert_is_Un
% 0.34/0.54 thf(fact_1477_insert__is__Un,axiom,
% 0.34/0.54 ( insert_int
% 0.34/0.54 = ( ^ [A6: int] : ( sup_sup_set_int @ ( insert_int @ A6 @ bot_bot_set_int ) ) ) ) ).
% 0.34/0.54
% 0.34/0.54 % insert_is_Un
% 0.34/0.54 thf(fact_1478_insert__is__Un,axiom,
% 0.34/0.54 ( insert5175938949040314269_nat_o
% 0.34/0.54 = ( ^ [A6: produc3658429121746597890et_nat > $o] : ( sup_su5209123915105501825_nat_o @ ( insert5175938949040314269_nat_o @ A6 @ bot_bo7824918357723582233_nat_o ) ) ) ) ).
% 0.34/0.54
% 0.34/0.54 % insert_is_Un
% 0.34/0.54 thf(fact_1479_Un__singleton__iff,axiom,
% 0.34/0.54 ! [A: set_Product_unit,B4: set_Product_unit,X: product_unit] :
% 0.34/0.54 ( ( ( sup_su793286257634532545t_unit @ A @ B4 )
% 0.34/0.54 = ( insert_Product_unit @ X @ bot_bo3957492148770167129t_unit ) )
% 0.34/0.54 = ( ( ( A = bot_bo3957492148770167129t_unit )
% 0.34/0.54 & ( B4
% 0.34/0.54 = ( insert_Product_unit @ X @ bot_bo3957492148770167129t_unit ) ) )
% 0.34/0.54 | ( ( A
% 0.34/0.54 = ( insert_Product_unit @ X @ bot_bo3957492148770167129t_unit ) )
% 0.34/0.54 & ( B4 = bot_bo3957492148770167129t_unit ) )
% 0.34/0.54 | ( ( A
% 0.34/0.54 = ( insert_Product_unit @ X @ bot_bo3957492148770167129t_unit ) )
% 0.34/0.54 & ( B4
% 0.34/0.54 = ( insert_Product_unit @ X @ bot_bo3957492148770167129t_unit ) ) ) ) ) ).
% 0.34/0.54
% 0.34/0.54 % Un_singleton_iff
% 0.34/0.54 thf(fact_1480_Un__singleton__iff,axiom,
% 0.34/0.54 ! [A: set_o,B4: set_o,X: $o] :
% 0.34/0.54 ( ( ( sup_sup_set_o @ A @ B4 )
% 0.34/0.54 = ( insert_o @ X @ bot_bot_set_o ) )
% 0.34/0.54 = ( ( ( A = bot_bot_set_o )
% 0.34/0.54 & ( B4
% 0.34/0.54 = ( insert_o @ X @ bot_bot_set_o ) ) )
% 0.34/0.54 | ( ( A
% 0.34/0.54 = ( insert_o @ X @ bot_bot_set_o ) )
% 0.34/0.54 & ( B4 = bot_bot_set_o ) )
% 0.34/0.54 | ( ( A
% 0.34/0.54 = ( insert_o @ X @ bot_bot_set_o ) )
% 0.34/0.54 & ( B4
% 0.34/0.54 = ( insert_o @ X @ bot_bot_set_o ) ) ) ) ) ).
% 0.34/0.54
% 0.34/0.54 % Un_singleton_iff
% 0.34/0.54 thf(fact_1481_Un__singleton__iff,axiom,
% 0.34/0.54 ! [A: set_nat,B4: set_nat,X: nat] :
% 0.34/0.54 ( ( ( sup_sup_set_nat @ A @ B4 )
% 0.34/0.54 = ( insert_nat @ X @ bot_bot_set_nat ) )
% 0.34/0.54 = ( ( ( A = bot_bot_set_nat )
% 0.34/0.54 & ( B4
% 0.34/0.54 = ( insert_nat @ X @ bot_bot_set_nat ) ) )
% 0.34/0.54 | ( ( A
% 0.34/0.54 = ( insert_nat @ X @ bot_bot_set_nat ) )
% 0.34/0.54 & ( B4 = bot_bot_set_nat ) )
% 0.34/0.54 | ( ( A
% 0.34/0.54 = ( insert_nat @ X @ bot_bot_set_nat ) )
% 0.34/0.54 & ( B4
% 0.34/0.54 = ( insert_nat @ X @ bot_bot_set_nat ) ) ) ) ) ).
% 0.34/0.54
% 0.34/0.54 % Un_singleton_iff
% 0.34/0.54 thf(fact_1482_Un__singleton__iff,axiom,
% 0.34/0.54 ! [A: set_int,B4: set_int,X: int] :
% 0.34/0.54 ( ( ( sup_sup_set_int @ A @ B4 )
% 0.34/0.54 = ( insert_int @ X @ bot_bot_set_int ) )
% 0.34/0.54 = ( ( ( A = bot_bot_set_int )
% 0.34/0.54 & ( B4
% 0.34/0.54 = ( insert_int @ X @ bot_bot_set_int ) ) )
% 0.34/0.54 | ( ( A
% 0.34/0.54 = ( insert_int @ X @ bot_bot_set_int ) )
% 0.34/0.54 & ( B4 = bot_bot_set_int ) )
% 0.34/0.54 | ( ( A
% 0.34/0.54 = ( insert_int @ X @ bot_bot_set_int ) )
% 0.34/0.54 & ( B4
% 0.34/0.54 = ( insert_int @ X @ bot_bot_set_int ) ) ) ) ) ).
% 0.34/0.54
% 0.34/0.54 % Un_singleton_iff
% 0.34/0.54 thf(fact_1483_Un__singleton__iff,axiom,
% 0.34/0.54 ! [A: set_Pr4532377907799695533_nat_o,B4: set_Pr4532377907799695533_nat_o,X: produc3658429121746597890et_nat > $o] :
% 0.34/0.54 ( ( ( sup_su5209123915105501825_nat_o @ A @ B4 )
% 0.34/0.54 = ( insert5175938949040314269_nat_o @ X @ bot_bo7824918357723582233_nat_o ) )
% 0.34/0.54 = ( ( ( A = bot_bo7824918357723582233_nat_o )
% 0.34/0.54 & ( B4
% 0.34/0.54 = ( insert5175938949040314269_nat_o @ X @ bot_bo7824918357723582233_nat_o ) ) )
% 0.34/0.54 | ( ( A
% 0.34/0.54 = ( insert5175938949040314269_nat_o @ X @ bot_bo7824918357723582233_nat_o ) )
% 0.34/0.54 & ( B4 = bot_bo7824918357723582233_nat_o ) )
% 0.34/0.54 | ( ( A
% 0.34/0.54 = ( insert5175938949040314269_nat_o @ X @ bot_bo7824918357723582233_nat_o ) )
% 0.34/0.54 & ( B4
% 0.34/0.54 = ( insert5175938949040314269_nat_o @ X @ bot_bo7824918357723582233_nat_o ) ) ) ) ) ).
% 0.34/0.54
% 0.34/0.54 % Un_singleton_iff
% 0.34/0.54 thf(fact_1484_singleton__Un__iff,axiom,
% 0.34/0.54 ! [X: product_unit,A: set_Product_unit,B4: set_Product_unit] :
% 0.34/0.54 ( ( ( insert_Product_unit @ X @ bot_bo3957492148770167129t_unit )
% 0.34/0.54 = ( sup_su793286257634532545t_unit @ A @ B4 ) )
% 0.34/0.54 = ( ( ( A = bot_bo3957492148770167129t_unit )
% 0.34/0.54 & ( B4
% 0.34/0.54 = ( insert_Product_unit @ X @ bot_bo3957492148770167129t_unit ) ) )
% 0.34/0.54 | ( ( A
% 0.34/0.54 = ( insert_Product_unit @ X @ bot_bo3957492148770167129t_unit ) )
% 0.34/0.54 & ( B4 = bot_bo3957492148770167129t_unit ) )
% 0.34/0.54 | ( ( A
% 0.34/0.54 = ( insert_Product_unit @ X @ bot_bo3957492148770167129t_unit ) )
% 0.34/0.54 & ( B4
% 0.34/0.54 = ( insert_Product_unit @ X @ bot_bo3957492148770167129t_unit ) ) ) ) ) ).
% 0.34/0.54
% 0.34/0.54 % singleton_Un_iff
% 0.34/0.54 thf(fact_1485_singleton__Un__iff,axiom,
% 0.34/0.54 ! [X: $o,A: set_o,B4: set_o] :
% 0.34/0.54 ( ( ( insert_o @ X @ bot_bot_set_o )
% 0.34/0.54 = ( sup_sup_set_o @ A @ B4 ) )
% 0.34/0.54 = ( ( ( A = bot_bot_set_o )
% 0.34/0.54 & ( B4
% 0.34/0.54 = ( insert_o @ X @ bot_bot_set_o ) ) )
% 0.34/0.54 | ( ( A
% 0.34/0.54 = ( insert_o @ X @ bot_bot_set_o ) )
% 0.34/0.54 & ( B4 = bot_bot_set_o ) )
% 0.34/0.54 | ( ( A
% 0.34/0.54 = ( insert_o @ X @ bot_bot_set_o ) )
% 0.34/0.54 & ( B4
% 0.34/0.54 = ( insert_o @ X @ bot_bot_set_o ) ) ) ) ) ).
% 0.34/0.54
% 0.34/0.54 % singleton_Un_iff
% 0.34/0.54 thf(fact_1486_singleton__Un__iff,axiom,
% 0.34/0.54 ! [X: nat,A: set_nat,B4: set_nat] :
% 0.34/0.54 ( ( ( insert_nat @ X @ bot_bot_set_nat )
% 0.34/0.54 = ( sup_sup_set_nat @ A @ B4 ) )
% 0.34/0.54 = ( ( ( A = bot_bot_set_nat )
% 0.34/0.54 & ( B4
% 0.34/0.54 = ( insert_nat @ X @ bot_bot_set_nat ) ) )
% 0.34/0.54 | ( ( A
% 0.34/0.54 = ( insert_nat @ X @ bot_bot_set_nat ) )
% 0.34/0.54 & ( B4 = bot_bot_set_nat ) )
% 0.34/0.54 | ( ( A
% 0.34/0.54 = ( insert_nat @ X @ bot_bot_set_nat ) )
% 0.34/0.54 & ( B4
% 0.34/0.54 = ( insert_nat @ X @ bot_bot_set_nat ) ) ) ) ) ).
% 0.34/0.54
% 0.34/0.54 % singleton_Un_iff
% 0.34/0.54 thf(fact_1487_singleton__Un__iff,axiom,
% 0.34/0.54 ! [X: int,A: set_int,B4: set_int] :
% 0.34/0.54 ( ( ( insert_int @ X @ bot_bot_set_int )
% 0.34/0.54 = ( sup_sup_set_int @ A @ B4 ) )
% 0.34/0.54 = ( ( ( A = bot_bot_set_int )
% 0.34/0.54 & ( B4
% 0.34/0.54 = ( insert_int @ X @ bot_bot_set_int ) ) )
% 0.34/0.54 | ( ( A
% 0.34/0.54 = ( insert_int @ X @ bot_bot_set_int ) )
% 0.34/0.54 & ( B4 = bot_bot_set_int ) )
% 0.34/0.54 | ( ( A
% 0.34/0.54 = ( insert_int @ X @ bot_bot_set_int ) )
% 0.34/0.54 & ( B4
% 0.34/0.54 = ( insert_int @ X @ bot_bot_set_int ) ) ) ) ) ).
% 0.34/0.54
% 0.34/0.54 % singleton_Un_iff
% 0.34/0.54 thf(fact_1488_singleton__Un__iff,axiom,
% 0.34/0.54 ! [X: produc3658429121746597890et_nat > $o,A: set_Pr4532377907799695533_nat_o,B4: set_Pr4532377907799695533_nat_o] :
% 0.34/0.54 ( ( ( insert5175938949040314269_nat_o @ X @ bot_bo7824918357723582233_nat_o )
% 0.34/0.54 = ( sup_su5209123915105501825_nat_o @ A @ B4 ) )
% 0.34/0.54 = ( ( ( A = bot_bo7824918357723582233_nat_o )
% 0.34/0.54 & ( B4
% 0.34/0.54 = ( insert5175938949040314269_nat_o @ X @ bot_bo7824918357723582233_nat_o ) ) )
% 0.34/0.54 | ( ( A
% 0.34/0.54 = ( insert5175938949040314269_nat_o @ X @ bot_bo7824918357723582233_nat_o ) )
% 0.34/0.54 & ( B4 = bot_bo7824918357723582233_nat_o ) )
% 0.34/0.54 | ( ( A
% 0.34/0.54 = ( insert5175938949040314269_nat_o @ X @ bot_bo7824918357723582233_nat_o ) )
% 0.34/0.54 & ( B4
% 0.34/0.54 = ( insert5175938949040314269_nat_o @ X @ bot_bo7824918357723582233_nat_o ) ) ) ) ) ).
% 0.34/0.54
% 0.34/0.54 % singleton_Un_iff
% 0.34/0.54 thf(fact_1489_Not__Domain__rtrancl,axiom,
% 0.34/0.54 ! [X: produc3658429121746597890et_nat > $o,R3: set_Pr2161125870931222855_nat_o,Y: produc3658429121746597890et_nat > $o] :
% 0.34/0.54 ( ~ ( member6576561426505652726_nat_o @ X @ ( domain4881241903163524940_nat_o @ R3 ) )
% 0.34/0.54 => ( ( member8781333585448626064_nat_o @ ( produc7368190662567826135_nat_o @ X @ Y ) @ ( transi712980445759334096_nat_o @ R3 ) )
% 0.34/0.54 = ( X = Y ) ) ) ).
% 0.34/0.54
% 0.34/0.54 % Not_Domain_rtrancl
% 0.34/0.54 thf(fact_1490_Not__Domain__rtrancl,axiom,
% 0.34/0.54 ! [X: $o,R3: set_Product_prod_o_o,Y: $o] :
% 0.34/0.54 ( ~ ( member_o @ X @ ( domain_o_o @ R3 ) )
% 0.34/0.54 => ( ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ X @ Y ) @ ( transitive_rtrancl_o @ R3 ) )
% 0.34/0.54 = ( X = Y ) ) ) ).
% 0.34/0.54
% 0.34/0.54 % Not_Domain_rtrancl
% 0.34/0.54 thf(fact_1491_Not__Domain__rtrancl,axiom,
% 0.34/0.54 ! [X: product_unit,R3: set_Pr5094982260447487303t_unit,Y: product_unit] :
% 0.34/0.54 ( ~ ( member_Product_unit @ X @ ( domain4292024440186670924t_unit @ R3 ) )
% 0.34/0.54 => ( ( member7821138191923116944t_unit @ ( produc5692694829384537815t_unit @ X @ Y ) @ ( transi6481887417741755664t_unit @ R3 ) )
% 0.34/0.54 = ( X = Y ) ) ) ).
% 0.34/0.54
% 0.34/0.54 % Not_Domain_rtrancl
% 0.34/0.54 thf(fact_1492_Not__Domain__rtrancl,axiom,
% 0.34/0.54 ! [X: nat,R3: set_Pr1261947904930325089at_nat,Y: nat] :
% 0.34/0.54 ( ~ ( member_nat @ X @ ( domain_nat_nat @ R3 ) )
% 0.34/0.54 => ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ Y ) @ ( transi2905341329935302413cl_nat @ R3 ) )
% 0.34/0.54 = ( X = Y ) ) ) ).
% 0.34/0.54
% 0.34/0.54 % Not_Domain_rtrancl
% 0.34/0.54 thf(fact_1493_Not__Domain__rtrancl,axiom,
% 0.34/0.54 ! [X: int,R3: set_Pr958786334691620121nt_int,Y: int] :
% 0.34/0.54 ( ~ ( member_int @ X @ ( domain_int_int @ R3 ) )
% 0.34/0.54 => ( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ X @ Y ) @ ( transi2902850859426252137cl_int @ R3 ) )
% 0.34/0.54 = ( X = Y ) ) ) ).
% 0.34/0.54
% 0.34/0.54 % Not_Domain_rtrancl
% 0.34/0.54 thf(fact_1494_subset__singletonD,axiom,
% 0.34/0.54 ! [A: set_Product_unit,X: product_unit] :
% 0.34/0.54 ( ( ord_le3507040750410214029t_unit @ A @ ( insert_Product_unit @ X @ bot_bo3957492148770167129t_unit ) )
% 0.34/0.54 => ( ( A = bot_bo3957492148770167129t_unit )
% 0.34/0.54 | ( A
% 0.34/0.54 = ( insert_Product_unit @ X @ bot_bo3957492148770167129t_unit ) ) ) ) ).
% 0.34/0.54
% 0.34/0.54 % subset_singletonD
% 0.34/0.54 thf(fact_1495_subset__singletonD,axiom,
% 0.34/0.54 ! [A: set_o,X: $o] :
% 0.34/0.54 ( ( ord_less_eq_set_o @ A @ ( insert_o @ X @ bot_bot_set_o ) )
% 0.34/0.54 => ( ( A = bot_bot_set_o )
% 0.34/0.54 | ( A
% 0.34/0.54 = ( insert_o @ X @ bot_bot_set_o ) ) ) ) ).
% 0.34/0.54
% 0.34/0.54 % subset_singletonD
% 0.34/0.54 thf(fact_1496_subset__singletonD,axiom,
% 0.34/0.54 ! [A: set_int,X: int] :
% 0.34/0.54 ( ( ord_less_eq_set_int @ A @ ( insert_int @ X @ bot_bot_set_int ) )
% 0.34/0.54 => ( ( A = bot_bot_set_int )
% 0.34/0.54 | ( A
% 0.34/0.54 = ( insert_int @ X @ bot_bot_set_int ) ) ) ) ).
% 0.34/0.54
% 0.34/0.54 % subset_singletonD
% 0.34/0.54 thf(fact_1497_subset__singletonD,axiom,
% 0.34/0.54 ! [A: set_Pr4532377907799695533_nat_o,X: produc3658429121746597890et_nat > $o] :
% 0.34/0.54 ( ( ord_le2965882846123202637_nat_o @ A @ ( insert5175938949040314269_nat_o @ X @ bot_bo7824918357723582233_nat_o ) )
% 0.34/0.54 => ( ( A = bot_bo7824918357723582233_nat_o )
% 0.34/0.54 | ( A
% 0.34/0.54 = ( insert5175938949040314269_nat_o @ X @ bot_bo7824918357723582233_nat_o ) ) ) ) ).
% 0.34/0.54
% 0.34/0.54 % subset_singletonD
% 0.34/0.54 thf(fact_1498_subset__singletonD,axiom,
% 0.34/0.54 ! [A: set_nat,X: nat] :
% 0.34/0.54 ( ( ord_less_eq_set_nat @ A @ ( insert_nat @ X @ bot_bot_set_nat ) )
% 0.34/0.54 => ( ( A = bot_bot_set_nat )
% 0.34/0.54 | ( A
% 0.34/0.54 = ( insert_nat @ X @ bot_bot_set_nat ) ) ) ) ).
% 0.34/0.54
% 0.34/0.54 % subset_singletonD
% 0.34/0.54 thf(fact_1499_subset__singleton__iff,axiom,
% 0.34/0.54 ! [X4: set_Product_unit,A2: product_unit] :
% 0.34/0.54 ( ( ord_le3507040750410214029t_unit @ X4 @ ( insert_Product_unit @ A2 @ bot_bo3957492148770167129t_unit ) )
% 0.34/0.54 = ( ( X4 = bot_bo3957492148770167129t_unit )
% 0.34/0.54 | ( X4
% 0.34/0.54 = ( insert_Product_unit @ A2 @ bot_bo3957492148770167129t_unit ) ) ) ) ).
% 0.34/0.54
% 0.34/0.54 % subset_singleton_iff
% 0.34/0.54 thf(fact_1500_subset__singleton__iff,axiom,
% 0.34/0.54 ! [X4: set_o,A2: $o] :
% 0.34/0.54 ( ( ord_less_eq_set_o @ X4 @ ( insert_o @ A2 @ bot_bot_set_o ) )
% 0.34/0.54 = ( ( X4 = bot_bot_set_o )
% 0.34/0.54 | ( X4
% 0.34/0.54 = ( insert_o @ A2 @ bot_bot_set_o ) ) ) ) ).
% 0.34/0.54
% 0.34/0.54 % subset_singleton_iff
% 0.34/0.54 thf(fact_1501_subset__singleton__iff,axiom,
% 0.34/0.54 ! [X4: set_int,A2: int] :
% 0.34/0.54 ( ( ord_less_eq_set_int @ X4 @ ( insert_int @ A2 @ bot_bot_set_int ) )
% 0.34/0.54 = ( ( X4 = bot_bot_set_int )
% 0.34/0.54 | ( X4
% 0.34/0.54 = ( insert_int @ A2 @ bot_bot_set_int ) ) ) ) ).
% 0.34/0.54
% 0.34/0.54 % subset_singleton_iff
% 0.34/0.54 thf(fact_1502_subset__singleton__iff,axiom,
% 0.34/0.54 ! [X4: set_Pr4532377907799695533_nat_o,A2: produc3658429121746597890et_nat > $o] :
% 0.34/0.54 ( ( ord_le2965882846123202637_nat_o @ X4 @ ( insert5175938949040314269_nat_o @ A2 @ bot_bo7824918357723582233_nat_o ) )
% 0.34/0.54 = ( ( X4 = bot_bo7824918357723582233_nat_o )
% 0.34/0.54 | ( X4
% 0.34/0.54 = ( insert5175938949040314269_nat_o @ A2 @ bot_bo7824918357723582233_nat_o ) ) ) ) ).
% 0.34/0.54
% 0.34/0.54 % subset_singleton_iff
% 0.34/0.54 thf(fact_1503_subset__singleton__iff,axiom,
% 0.34/0.54 ! [X4: set_nat,A2: nat] :
% 0.34/0.54 ( ( ord_less_eq_set_nat @ X4 @ ( insert_nat @ A2 @ bot_bot_set_nat ) )
% 0.34/0.54 = ( ( X4 = bot_bot_set_nat )
% 0.34/0.54 | ( X4
% 0.34/0.54 = ( insert_nat @ A2 @ bot_bot_set_nat ) ) ) ) ).
% 0.34/0.54
% 0.34/0.54 % subset_singleton_iff
% 0.34/0.54 thf(fact_1504_mod__star__conv,axiom,
% 0.34/0.54 ! [A: assn,B4: assn,H2: produc3658429121746597890et_nat] :
% 0.34/0.54 ( ( rep_assn @ ( times_times_assn @ A @ B4 ) @ H2 )
% 0.34/0.54 = ( ? [Hr: heap_e7401611519738050253t_unit,As1: set_nat,As22: set_nat] :
% 0.34/0.54 ( ( H2
% 0.34/0.54 = ( produc7507926704131184380et_nat @ Hr @ ( sup_sup_set_nat @ As1 @ As22 ) ) )
% 0.34/0.54 & ( ( inf_inf_set_nat @ As1 @ As22 )
% 0.34/0.54 = bot_bot_set_nat )
% 0.34/0.54 & ( rep_assn @ A @ ( produc7507926704131184380et_nat @ Hr @ As1 ) )
% 0.34/0.54 & ( rep_assn @ B4 @ ( produc7507926704131184380et_nat @ Hr @ As22 ) ) ) ) ) ).
% 0.34/0.54
% 0.34/0.54 % mod_star_conv
% 0.34/0.54 thf(fact_1505_star__assnI,axiom,
% 0.34/0.54 ! [P: assn,H2: heap_e7401611519738050253t_unit,As: set_nat,Q2: assn,As5: set_nat] :
% 0.34/0.54 ( ( rep_assn @ P @ ( produc7507926704131184380et_nat @ H2 @ As ) )
% 0.34/0.54 => ( ( rep_assn @ Q2 @ ( produc7507926704131184380et_nat @ H2 @ As5 ) )
% 0.34/0.54 => ( ( ( inf_inf_set_nat @ As @ As5 )
% 0.34/0.54 = bot_bot_set_nat )
% 0.34/0.54 => ( rep_assn @ ( times_times_assn @ P @ Q2 ) @ ( produc7507926704131184380et_nat @ H2 @ ( sup_sup_set_nat @ As @ As5 ) ) ) ) ) ) ).
% 0.34/0.54
% 0.34/0.54 % star_assnI
% 0.34/0.54 thf(fact_1506_in__range__subset,axiom,
% 0.34/0.54 ! [As: set_nat,As5: set_nat,H2: heap_e7401611519738050253t_unit] :
% 0.34/0.54 ( ( ord_less_eq_set_nat @ As @ As5 )
% 0.34/0.54 => ( ( in_range @ ( produc7507926704131184380et_nat @ H2 @ As5 ) )
% 0.34/0.54 => ( in_range @ ( produc7507926704131184380et_nat @ H2 @ As ) ) ) ) ).
% 0.34/0.54
% 0.34/0.54 % in_range_subset
% 0.34/0.54 thf(fact_1507_wand__raw_Osimps,axiom,
% 0.34/0.54 ! [P: produc3658429121746597890et_nat > $o,Q2: produc3658429121746597890et_nat > $o,H2: heap_e7401611519738050253t_unit,As: set_nat] :
% 0.34/0.54 ( ( wand_raw @ P @ Q2 @ ( produc7507926704131184380et_nat @ H2 @ As ) )
% 0.34/0.54 = ( ( in_range @ ( produc7507926704131184380et_nat @ H2 @ As ) )
% 0.34/0.54 & ! [H5: heap_e7401611519738050253t_unit,As6: set_nat] :
% 0.34/0.54 ( ( ( ( inf_inf_set_nat @ As @ As6 )
% 0.34/0.54 = bot_bot_set_nat )
% 0.34/0.54 & ( relH @ As @ H2 @ H5 )
% 0.34/0.54 & ( in_range @ ( produc7507926704131184380et_nat @ H5 @ As ) )
% 0.34/0.54 & ( P @ ( produc7507926704131184380et_nat @ H5 @ As6 ) ) )
% 0.34/0.54 => ( Q2 @ ( produc7507926704131184380et_nat @ H5 @ ( sup_sup_set_nat @ As @ As6 ) ) ) ) ) ) ).
% 0.34/0.54
% 0.34/0.54 % wand_raw.simps
% 0.34/0.54 thf(fact_1508_wand__raw_Oelims_I1_J,axiom,
% 0.34/0.54 ! [X: produc3658429121746597890et_nat > $o,Xa: produc3658429121746597890et_nat > $o,Xb: produc3658429121746597890et_nat,Y: $o] :
% 0.34/0.54 ( ( ( wand_raw @ X @ Xa @ Xb )
% 0.34/0.54 = Y )
% 0.34/0.54 => ~ ! [H: heap_e7401611519738050253t_unit,As2: set_nat] :
% 0.34/0.54 ( ( Xb
% 0.34/0.54 = ( produc7507926704131184380et_nat @ H @ As2 ) )
% 0.34/0.54 => ( Y
% 0.34/0.54 = ( ~ ( ( in_range @ ( produc7507926704131184380et_nat @ H @ As2 ) )
% 0.34/0.54 & ! [H5: heap_e7401611519738050253t_unit,As6: set_nat] :
% 0.34/0.54 ( ( ( ( inf_inf_set_nat @ As2 @ As6 )
% 0.34/0.54 = bot_bot_set_nat )
% 0.34/0.54 & ( relH @ As2 @ H @ H5 )
% 0.34/0.54 & ( in_range @ ( produc7507926704131184380et_nat @ H5 @ As2 ) )
% 0.34/0.54 & ( X @ ( produc7507926704131184380et_nat @ H5 @ As6 ) ) )
% 0.34/0.54 => ( Xa @ ( produc7507926704131184380et_nat @ H5 @ ( sup_sup_set_nat @ As2 @ As6 ) ) ) ) ) ) ) ) ) ).
% 0.34/0.54
% 0.34/0.54 % wand_raw.elims(1)
% 0.34/0.54 thf(fact_1509_wand__raw_Oelims_I2_J,axiom,
% 0.34/0.54 ! [X: produc3658429121746597890et_nat > $o,Xa: produc3658429121746597890et_nat > $o,Xb: produc3658429121746597890et_nat] :
% 0.34/0.54 ( ( wand_raw @ X @ Xa @ Xb )
% 0.34/0.54 => ~ ! [H: heap_e7401611519738050253t_unit,As2: set_nat] :
% 0.34/0.54 ( ( Xb
% 0.34/0.54 = ( produc7507926704131184380et_nat @ H @ As2 ) )
% 0.34/0.54 => ~ ( ( in_range @ ( produc7507926704131184380et_nat @ H @ As2 ) )
% 0.34/0.54 & ! [H7: heap_e7401611519738050253t_unit,As7: set_nat] :
% 0.34/0.54 ( ( ( ( inf_inf_set_nat @ As2 @ As7 )
% 0.34/0.54 = bot_bot_set_nat )
% 0.34/0.54 & ( relH @ As2 @ H @ H7 )
% 0.34/0.54 & ( in_range @ ( produc7507926704131184380et_nat @ H7 @ As2 ) )
% 0.34/0.54 & ( X @ ( produc7507926704131184380et_nat @ H7 @ As7 ) ) )
% 0.34/0.54 => ( Xa @ ( produc7507926704131184380et_nat @ H7 @ ( sup_sup_set_nat @ As2 @ As7 ) ) ) ) ) ) ) ).
% 0.34/0.54
% 0.34/0.54 % wand_raw.elims(2)
% 0.34/0.54 thf(fact_1510_wand__raw_Oelims_I3_J,axiom,
% 0.34/0.54 ! [X: produc3658429121746597890et_nat > $o,Xa: produc3658429121746597890et_nat > $o,Xb: produc3658429121746597890et_nat] :
% 0.34/0.54 ( ~ ( wand_raw @ X @ Xa @ Xb )
% 0.34/0.54 => ~ ! [H: heap_e7401611519738050253t_unit,As2: set_nat] :
% 0.34/0.54 ( ( Xb
% 0.34/0.54 = ( produc7507926704131184380et_nat @ H @ As2 ) )
% 0.34/0.54 => ( ( in_range @ ( produc7507926704131184380et_nat @ H @ As2 ) )
% 0.34/0.54 & ! [H6: heap_e7401611519738050253t_unit,As4: set_nat] :
% 0.34/0.54 ( ( ( ( inf_inf_set_nat @ As2 @ As4 )
% 0.34/0.54 = bot_bot_set_nat )
% 0.34/0.54 & ( relH @ As2 @ H @ H6 )
% 0.34/0.54 & ( in_range @ ( produc7507926704131184380et_nat @ H6 @ As2 ) )
% 0.34/0.54 & ( X @ ( produc7507926704131184380et_nat @ H6 @ As4 ) ) )
% 0.34/0.54 => ( Xa @ ( produc7507926704131184380et_nat @ H6 @ ( sup_sup_set_nat @ As2 @ As4 ) ) ) ) ) ) ) ).
% 0.34/0.54
% 0.34/0.54 % wand_raw.elims(3)
% 0.34/0.54 thf(fact_1511_Image__empty__rtrancl__Image__id,axiom,
% 0.34/0.54 ! [R3: set_Pr5094982260447487303t_unit,V: product_unit] :
% 0.34/0.54 ( ( ( image_8431784415008483091t_unit @ R3 @ ( insert_Product_unit @ V @ bot_bo3957492148770167129t_unit ) )
% 0.34/0.54 = bot_bo3957492148770167129t_unit )
% 0.34/0.54 => ( ( image_8431784415008483091t_unit @ ( transi6481887417741755664t_unit @ R3 ) @ ( insert_Product_unit @ V @ bot_bo3957492148770167129t_unit ) )
% 0.34/0.54 = ( insert_Product_unit @ V @ bot_bo3957492148770167129t_unit ) ) ) ).
% 0.34/0.54
% 0.34/0.54 % Image_empty_rtrancl_Image_id
% 0.34/0.54 thf(fact_1512_Image__empty__rtrancl__Image__id,axiom,
% 0.34/0.54 ! [R3: set_Product_prod_o_o,V: $o] :
% 0.34/0.54 ( ( ( image_o_o @ R3 @ ( insert_o @ V @ bot_bot_set_o ) )
% 0.34/0.54 = bot_bot_set_o )
% 0.34/0.54 => ( ( image_o_o @ ( transitive_rtrancl_o @ R3 ) @ ( insert_o @ V @ bot_bot_set_o ) )
% 0.34/0.54 = ( insert_o @ V @ bot_bot_set_o ) ) ) ).
% 0.34/0.54
% 0.34/0.54 % Image_empty_rtrancl_Image_id
% 0.34/0.54 thf(fact_1513_Image__empty__rtrancl__Image__id,axiom,
% 0.34/0.54 ! [R3: set_Pr1261947904930325089at_nat,V: nat] :
% 0.34/0.54 ( ( ( image_nat_nat @ R3 @ ( insert_nat @ V @ bot_bot_set_nat ) )
% 0.34/0.54 = bot_bot_set_nat )
% 0.34/0.54 => ( ( image_nat_nat @ ( transi2905341329935302413cl_nat @ R3 ) @ ( insert_nat @ V @ bot_bot_set_nat ) )
% 0.34/0.54 = ( insert_nat @ V @ bot_bot_set_nat ) ) ) ).
% 0.34/0.54
% 0.34/0.54 % Image_empty_rtrancl_Image_id
% 0.34/0.54 thf(fact_1514_Image__empty__rtrancl__Image__id,axiom,
% 0.34/0.54 ! [R3: set_Pr958786334691620121nt_int,V: int] :
% 0.34/0.54 ( ( ( image_int_int @ R3 @ ( insert_int @ V @ bot_bot_set_int ) )
% 0.34/0.54 = bot_bot_set_int )
% 0.34/0.54 => ( ( image_int_int @ ( transi2902850859426252137cl_int @ R3 ) @ ( insert_int @ V @ bot_bot_set_int ) )
% 0.34/0.54 = ( insert_int @ V @ bot_bot_set_int ) ) ) ).
% 0.34/0.54
% 0.34/0.54 % Image_empty_rtrancl_Image_id
% 0.34/0.54 thf(fact_1515_Image__empty__rtrancl__Image__id,axiom,
% 0.34/0.54 ! [R3: set_Pr2161125870931222855_nat_o,V: produc3658429121746597890et_nat > $o] :
% 0.34/0.54 ( ( ( image_2232984160439939859_nat_o @ R3 @ ( insert5175938949040314269_nat_o @ V @ bot_bo7824918357723582233_nat_o ) )
% 0.34/0.54 = bot_bo7824918357723582233_nat_o )
% 0.34/0.54 => ( ( image_2232984160439939859_nat_o @ ( transi712980445759334096_nat_o @ R3 ) @ ( insert5175938949040314269_nat_o @ V @ bot_bo7824918357723582233_nat_o ) )
% 0.34/0.54 = ( insert5175938949040314269_nat_o @ V @ bot_bo7824918357723582233_nat_o ) ) ) ).
% 0.34/0.54
% 0.34/0.54 % Image_empty_rtrancl_Image_id
% 0.34/0.54 thf(fact_1516_sup__bot_Oright__neutral,axiom,
% 0.34/0.54 ! [A2: assn] :
% 0.34/0.54 ( ( sup_sup_assn @ A2 @ bot_bot_assn )
% 0.34/0.54 = A2 ) ).
% 0.34/0.54
% 0.34/0.54 % sup_bot.right_neutral
% 0.34/0.54 thf(fact_1517_sup__bot_Oright__neutral,axiom,
% 0.34/0.54 ! [A2: product_unit] :
% 0.34/0.54 ( ( sup_sup_Product_unit @ A2 @ bot_bot_Product_unit )
% 0.34/0.54 = A2 ) ).
% 0.34/0.54
% 0.34/0.54 % sup_bot.right_neutral
% 0.34/0.54 thf(fact_1518_sup__bot_Oright__neutral,axiom,
% 0.34/0.54 ! [A2: set_Product_unit] :
% 0.34/0.54 ( ( sup_su793286257634532545t_unit @ A2 @ bot_bo3957492148770167129t_unit )
% 0.34/0.54 = A2 ) ).
% 0.34/0.54
% 0.34/0.54 % sup_bot.right_neutral
% 0.34/0.54 thf(fact_1519_sup__bot_Oright__neutral,axiom,
% 0.34/0.54 ! [A2: set_o] :
% 0.34/0.54 ( ( sup_sup_set_o @ A2 @ bot_bot_set_o )
% 0.34/0.54 = A2 ) ).
% 0.34/0.54
% 0.34/0.54 % sup_bot.right_neutral
% 0.34/0.54 thf(fact_1520_sup__bot_Oright__neutral,axiom,
% 0.34/0.54 ! [A2: set_nat] :
% 0.34/0.54 ( ( sup_sup_set_nat @ A2 @ bot_bot_set_nat )
% 0.34/0.54 = A2 ) ).
% 0.34/0.54
% 0.34/0.54 % sup_bot.right_neutral
% 0.34/0.54 thf(fact_1521_sup__bot_Oright__neutral,axiom,
% 0.34/0.54 ! [A2: set_int] :
% 0.34/0.54 ( ( sup_sup_set_int @ A2 @ bot_bot_set_int )
% 0.34/0.54 = A2 ) ).
% 0.34/0.54
% 0.34/0.54 % sup_bot.right_neutral
% 0.34/0.54 thf(fact_1522_sup__bot_Oright__neutral,axiom,
% 0.34/0.54 ! [A2: set_Pr4532377907799695533_nat_o] :
% 0.34/0.54 ( ( sup_su5209123915105501825_nat_o @ A2 @ bot_bo7824918357723582233_nat_o )
% 0.34/0.54 = A2 ) ).
% 0.34/0.54
% 0.34/0.54 % sup_bot.right_neutral
% 0.34/0.54 thf(fact_1523_sup__bot_Oneutr__eq__iff,axiom,
% 0.34/0.54 ! [A2: assn,B: assn] :
% 0.34/0.54 ( ( bot_bot_assn
% 0.34/0.54 = ( sup_sup_assn @ A2 @ B ) )
% 0.34/0.54 = ( ( A2 = bot_bot_assn )
% 0.34/0.54 & ( B = bot_bot_assn ) ) ) ).
% 0.34/0.54
% 0.34/0.54 % sup_bot.neutr_eq_iff
% 0.34/0.54 thf(fact_1524_sup__bot_Oneutr__eq__iff,axiom,
% 0.34/0.54 ! [A2: product_unit,B: product_unit] :
% 0.34/0.54 ( ( bot_bot_Product_unit
% 0.34/0.54 = ( sup_sup_Product_unit @ A2 @ B ) )
% 0.34/0.54 = ( ( A2 = bot_bot_Product_unit )
% 0.34/0.54 & ( B = bot_bot_Product_unit ) ) ) ).
% 0.34/0.54
% 0.34/0.54 % sup_bot.neutr_eq_iff
% 0.34/0.54 thf(fact_1525_sup__bot_Oneutr__eq__iff,axiom,
% 0.34/0.54 ! [A2: set_Product_unit,B: set_Product_unit] :
% 0.34/0.54 ( ( bot_bo3957492148770167129t_unit
% 0.34/0.54 = ( sup_su793286257634532545t_unit @ A2 @ B ) )
% 0.34/0.54 = ( ( A2 = bot_bo3957492148770167129t_unit )
% 0.34/0.54 & ( B = bot_bo3957492148770167129t_unit ) ) ) ).
% 0.34/0.54
% 0.34/0.54 % sup_bot.neutr_eq_iff
% 0.34/0.54 thf(fact_1526_sup__bot_Oneutr__eq__iff,axiom,
% 0.34/0.54 ! [A2: set_o,B: set_o] :
% 0.34/0.54 ( ( bot_bot_set_o
% 0.34/0.54 = ( sup_sup_set_o @ A2 @ B ) )
% 0.34/0.54 = ( ( A2 = bot_bot_set_o )
% 0.34/0.54 & ( B = bot_bot_set_o ) ) ) ).
% 0.34/0.54
% 0.34/0.54 % sup_bot.neutr_eq_iff
% 0.34/0.54 thf(fact_1527_sup__bot_Oneutr__eq__iff,axiom,
% 0.34/0.54 ! [A2: set_nat,B: set_nat] :
% 0.34/0.54 ( ( bot_bot_set_nat
% 0.34/0.54 = ( sup_sup_set_nat @ A2 @ B ) )
% 0.34/0.54 = ( ( A2 = bot_bot_set_nat )
% 0.34/0.54 & ( B = bot_bot_set_nat ) ) ) ).
% 0.34/0.54
% 0.34/0.54 % sup_bot.neutr_eq_iff
% 0.34/0.54 thf(fact_1528_sup__bot_Oneutr__eq__iff,axiom,
% 0.34/0.54 ! [A2: set_int,B: set_int] :
% 0.34/0.54 ( ( bot_bot_set_int
% 0.34/0.54 = ( sup_sup_set_int @ A2 @ B ) )
% 0.34/0.54 = ( ( A2 = bot_bot_set_int )
% 0.34/0.54 & ( B = bot_bot_set_int ) ) ) ).
% 0.34/0.54
% 0.34/0.54 % sup_bot.neutr_eq_iff
% 0.34/0.54 thf(fact_1529_sup__bot_Oneutr__eq__iff,axiom,
% 0.34/0.54 ! [A2: set_Pr4532377907799695533_nat_o,B: set_Pr4532377907799695533_nat_o] :
% 0.34/0.54 ( ( bot_bo7824918357723582233_nat_o
% 0.34/0.54 = ( sup_su5209123915105501825_nat_o @ A2 @ B ) )
% 0.34/0.54 = ( ( A2 = bot_bo7824918357723582233_nat_o )
% 0.34/0.54 & ( B = bot_bo7824918357723582233_nat_o ) ) ) ).
% 0.34/0.54
% 0.34/0.54 % sup_bot.neutr_eq_iff
% 0.34/0.54 thf(fact_1530_sup__bot_Oleft__neutral,axiom,
% 0.34/0.54 ! [A2: assn] :
% 0.34/0.54 ( ( sup_sup_assn @ bot_bot_assn @ A2 )
% 0.34/0.54 = A2 ) ).
% 0.34/0.54
% 0.34/0.54 % sup_bot.left_neutral
% 0.34/0.54 thf(fact_1531_sup__bot_Oleft__neutral,axiom,
% 0.34/0.54 ! [A2: product_unit] :
% 0.34/0.54 ( ( sup_sup_Product_unit @ bot_bot_Product_unit @ A2 )
% 0.34/0.54 = A2 ) ).
% 0.34/0.54
% 0.34/0.54 % sup_bot.left_neutral
% 0.34/0.54 thf(fact_1532_sup__bot_Oleft__neutral,axiom,
% 0.34/0.54 ! [A2: set_Product_unit] :
% 0.34/0.54 ( ( sup_su793286257634532545t_unit @ bot_bo3957492148770167129t_unit @ A2 )
% 0.34/0.54 = A2 ) ).
% 0.34/0.54
% 0.34/0.54 % sup_bot.left_neutral
% 0.34/0.54 thf(fact_1533_sup__bot_Oleft__neutral,axiom,
% 0.34/0.54 ! [A2: set_o] :
% 0.34/0.54 ( ( sup_sup_set_o @ bot_bot_set_o @ A2 )
% 0.34/0.54 = A2 ) ).
% 0.34/0.54
% 0.34/0.54 % sup_bot.left_neutral
% 0.34/0.54 thf(fact_1534_sup__bot_Oleft__neutral,axiom,
% 0.34/0.54 ! [A2: set_nat] :
% 0.34/0.54 ( ( sup_sup_set_nat @ bot_bot_set_nat @ A2 )
% 0.34/0.54 = A2 ) ).
% 0.34/0.54
% 0.34/0.54 % sup_bot.left_neutral
% 0.34/0.54 thf(fact_1535_sup__bot_Oleft__neutral,axiom,
% 0.34/0.54 ! [A2: set_int] :
% 0.34/0.54 ( ( sup_sup_set_int @ bot_bot_set_int @ A2 )
% 0.34/0.54 = A2 ) ).
% 0.34/0.54
% 0.34/0.54 % sup_bot.left_neutral
% 0.34/0.54 thf(fact_1536_sup__bot_Oleft__neutral,axiom,
% 0.34/0.54 ! [A2: set_Pr4532377907799695533_nat_o] :
% 0.34/0.54 ( ( sup_su5209123915105501825_nat_o @ bot_bo7824918357723582233_nat_o @ A2 )
% 0.34/0.54 = A2 ) ).
% 0.34/0.54
% 0.34/0.54 % sup_bot.left_neutral
% 0.34/0.54 thf(fact_1537_sup__bot_Oeq__neutr__iff,axiom,
% 0.34/0.54 ! [A2: assn,B: assn] :
% 0.34/0.54 ( ( ( sup_sup_assn @ A2 @ B )
% 0.34/0.54 = bot_bot_assn )
% 0.34/0.54 = ( ( A2 = bot_bot_assn )
% 0.34/0.54 & ( B = bot_bot_assn ) ) ) ).
% 0.34/0.54
% 0.34/0.54 % sup_bot.eq_neutr_iff
% 0.34/0.54 thf(fact_1538_sup__bot_Oeq__neutr__iff,axiom,
% 0.34/0.54 ! [A2: product_unit,B: product_unit] :
% 0.34/0.54 ( ( ( sup_sup_Product_unit @ A2 @ B )
% 0.34/0.54 = bot_bot_Product_unit )
% 0.34/0.54 = ( ( A2 = bot_bot_Product_unit )
% 0.34/0.54 & ( B = bot_bot_Product_unit ) ) ) ).
% 0.34/0.54
% 0.34/0.54 % sup_bot.eq_neutr_iff
% 0.34/0.54 thf(fact_1539_sup__bot_Oeq__neutr__iff,axiom,
% 0.34/0.54 ! [A2: set_Product_unit,B: set_Product_unit] :
% 0.34/0.54 ( ( ( sup_su793286257634532545t_unit @ A2 @ B )
% 0.34/0.54 = bot_bo3957492148770167129t_unit )
% 0.34/0.54 = ( ( A2 = bot_bo3957492148770167129t_unit )
% 0.34/0.54 & ( B = bot_bo3957492148770167129t_unit ) ) ) ).
% 0.34/0.54
% 0.34/0.54 % sup_bot.eq_neutr_iff
% 0.34/0.54 thf(fact_1540_sup__bot_Oeq__neutr__iff,axiom,
% 0.34/0.54 ! [A2: set_o,B: set_o] :
% 0.34/0.54 ( ( ( sup_sup_set_o @ A2 @ B )
% 0.34/0.54 = bot_bot_set_o )
% 0.34/0.54 = ( ( A2 = bot_bot_set_o )
% 0.34/0.54 & ( B = bot_bot_set_o ) ) ) ).
% 0.34/0.54
% 0.34/0.54 % sup_bot.eq_neutr_iff
% 0.34/0.54 thf(fact_1541_sup__bot_Oeq__neutr__iff,axiom,
% 0.34/0.54 ! [A2: set_nat,B: set_nat] :
% 0.34/0.54 ( ( ( sup_sup_set_nat @ A2 @ B )
% 0.34/0.54 = bot_bot_set_nat )
% 0.34/0.54 = ( ( A2 = bot_bot_set_nat )
% 0.34/0.54 & ( B = bot_bot_set_nat ) ) ) ).
% 0.34/0.54
% 0.34/0.54 % sup_bot.eq_neutr_iff
% 0.34/0.54 thf(fact_1542_sup__bot_Oeq__neutr__iff,axiom,
% 0.34/0.54 ! [A2: set_int,B: set_int] :
% 0.34/0.54 ( ( ( sup_sup_set_int @ A2 @ B )
% 0.34/0.54 = bot_bot_set_int )
% 0.34/0.54 = ( ( A2 = bot_bot_set_int )
% 0.34/0.54 & ( B = bot_bot_set_int ) ) ) ).
% 0.34/0.54
% 0.34/0.54 % sup_bot.eq_neutr_iff
% 0.34/0.54 thf(fact_1543_sup__bot_Oeq__neutr__iff,axiom,
% 0.34/0.54 ! [A2: set_Pr4532377907799695533_nat_o,B: set_Pr4532377907799695533_nat_o] :
% 0.34/0.54 ( ( ( sup_su5209123915105501825_nat_o @ A2 @ B )
% 0.34/0.54 = bot_bo7824918357723582233_nat_o )
% 0.34/0.54 = ( ( A2 = bot_bo7824918357723582233_nat_o )
% 0.34/0.54 & ( B = bot_bo7824918357723582233_nat_o ) ) ) ).
% 0.34/0.54
% 0.34/0.54 % sup_bot.eq_neutr_iff
% 0.34/0.54 thf(fact_1544_sup__eq__bot__iff,axiom,
% 0.34/0.54 ! [X: assn,Y: assn] :
% 0.34/0.54 ( ( ( sup_sup_assn @ X @ Y )
% 0.34/0.54 = bot_bot_assn )
% 0.34/0.54 = ( ( X = bot_bot_assn )
% 0.34/0.54 & ( Y = bot_bot_assn ) ) ) ).
% 0.34/0.54
% 0.34/0.54 % sup_eq_bot_iff
% 0.34/0.54 thf(fact_1545_sup__eq__bot__iff,axiom,
% 0.34/0.54 ! [X: product_unit,Y: product_unit] :
% 0.34/0.54 ( ( ( sup_sup_Product_unit @ X @ Y )
% 0.34/0.54 = bot_bot_Product_unit )
% 0.34/0.54 = ( ( X = bot_bot_Product_unit )
% 0.34/0.54 & ( Y = bot_bot_Product_unit ) ) ) ).
% 0.34/0.54
% 0.34/0.54 % sup_eq_bot_iff
% 0.34/0.54 thf(fact_1546_sup__eq__bot__iff,axiom,
% 0.34/0.54 ! [X: set_Product_unit,Y: set_Product_unit] :
% 0.34/0.54 ( ( ( sup_su793286257634532545t_unit @ X @ Y )
% 0.34/0.54 = bot_bo3957492148770167129t_unit )
% 0.34/0.54 = ( ( X = bot_bo3957492148770167129t_unit )
% 0.34/0.54 & ( Y = bot_bo3957492148770167129t_unit ) ) ) ).
% 0.34/0.54
% 0.34/0.54 % sup_eq_bot_iff
% 0.34/0.54 thf(fact_1547_sup__eq__bot__iff,axiom,
% 0.34/0.54 ! [X: set_o,Y: set_o] :
% 0.34/0.54 ( ( ( sup_sup_set_o @ X @ Y )
% 0.34/0.54 = bot_bot_set_o )
% 0.34/0.54 = ( ( X = bot_bot_set_o )
% 0.34/0.54 & ( Y = bot_bot_set_o ) ) ) ).
% 0.34/0.54
% 0.34/0.54 % sup_eq_bot_iff
% 0.34/0.54 thf(fact_1548_sup__eq__bot__iff,axiom,
% 0.34/0.54 ! [X: set_nat,Y: set_nat] :
% 0.34/0.54 ( ( ( sup_sup_set_nat @ X @ Y )
% 0.34/0.54 = bot_bot_set_nat )
% 0.34/0.54 = ( ( X = bot_bot_set_nat )
% 0.34/0.54 & ( Y = bot_bot_set_nat ) ) ) ).
% 0.34/0.54
% 0.34/0.54 % sup_eq_bot_iff
% 0.34/0.54 thf(fact_1549_sup__eq__bot__iff,axiom,
% 0.34/0.54 ! [X: set_int,Y: set_int] :
% 0.34/0.54 ( ( ( sup_sup_set_int @ X @ Y )
% 0.34/0.54 = bot_bot_set_int )
% 0.34/0.54 = ( ( X = bot_bot_set_int )
% 0.34/0.54 & ( Y = bot_bot_set_int ) ) ) ).
% 0.34/0.54
% 0.34/0.54 % sup_eq_bot_iff
% 0.34/0.54 thf(fact_1550_sup__eq__bot__iff,axiom,
% 0.34/0.54 ! [X: set_Pr4532377907799695533_nat_o,Y: set_Pr4532377907799695533_nat_o] :
% 0.34/0.54 ( ( ( sup_su5209123915105501825_nat_o @ X @ Y )
% 0.34/0.54 = bot_bo7824918357723582233_nat_o )
% 0.34/0.54 = ( ( X = bot_bo7824918357723582233_nat_o )
% 0.34/0.54 & ( Y = bot_bo7824918357723582233_nat_o ) ) ) ).
% 0.34/0.54
% 0.34/0.54 % sup_eq_bot_iff
% 0.34/0.54 thf(fact_1551_bot__eq__sup__iff,axiom,
% 0.34/0.54 ! [X: assn,Y: assn] :
% 0.34/0.54 ( ( bot_bot_assn
% 0.34/0.54 = ( sup_sup_assn @ X @ Y ) )
% 0.34/0.54 = ( ( X = bot_bot_assn )
% 0.34/0.54 & ( Y = bot_bot_assn ) ) ) ).
% 0.34/0.54
% 0.34/0.54 % bot_eq_sup_iff
% 0.34/0.54 thf(fact_1552_bot__eq__sup__iff,axiom,
% 0.34/0.54 ! [X: product_unit,Y: product_unit] :
% 0.34/0.54 ( ( bot_bot_Product_unit
% 0.34/0.54 = ( sup_sup_Product_unit @ X @ Y ) )
% 0.34/0.54 = ( ( X = bot_bot_Product_unit )
% 0.34/0.54 & ( Y = bot_bot_Product_unit ) ) ) ).
% 0.34/0.54
% 0.34/0.54 % bot_eq_sup_iff
% 0.34/0.54 thf(fact_1553_bot__eq__sup__iff,axiom,
% 0.34/0.54 ! [X: set_Product_unit,Y: set_Product_unit] :
% 0.34/0.54 ( ( bot_bo3957492148770167129t_unit
% 0.34/0.54 = ( sup_su793286257634532545t_unit @ X @ Y ) )
% 0.34/0.54 = ( ( X = bot_bo3957492148770167129t_unit )
% 0.34/0.54 & ( Y = bot_bo3957492148770167129t_unit ) ) ) ).
% 0.34/0.54
% 0.34/0.54 % bot_eq_sup_iff
% 0.34/0.54 thf(fact_1554_bot__eq__sup__iff,axiom,
% 0.34/0.54 ! [X: set_o,Y: set_o] :
% 0.34/0.54 ( ( bot_bot_set_o
% 0.34/0.54 = ( sup_sup_set_o @ X @ Y ) )
% 0.34/0.54 = ( ( X = bot_bot_set_o )
% 0.34/0.54 & ( Y = bot_bot_set_o ) ) ) ).
% 0.34/0.54
% 0.34/0.54 % bot_eq_sup_iff
% 0.34/0.54 thf(fact_1555_bot__eq__sup__iff,axiom,
% 0.34/0.54 ! [X: set_nat,Y: set_nat] :
% 0.34/0.54 ( ( bot_bot_set_nat
% 0.34/0.54 = ( sup_sup_set_nat @ X @ Y ) )
% 0.34/0.54 = ( ( X = bot_bot_set_nat )
% 0.34/0.54 & ( Y = bot_bot_set_nat ) ) ) ).
% 0.34/0.54
% 0.34/0.54 % bot_eq_sup_iff
% 0.34/0.54 thf(fact_1556_bot__eq__sup__iff,axiom,
% 0.34/0.54 ! [X: set_int,Y: set_int] :
% 0.34/0.54 ( ( bot_bot_set_int
% 0.34/0.54 = ( sup_sup_set_int @ X @ Y ) )
% 0.34/0.54 = ( ( X = bot_bot_set_int )
% 0.34/0.54 & ( Y = bot_bot_set_int ) ) ) ).
% 0.34/0.54
% 0.34/0.54 % bot_eq_sup_iff
% 0.34/0.54 thf(fact_1557_bot__eq__sup__iff,axiom,
% 0.34/0.54 ! [X: set_Pr4532377907799695533_nat_o,Y: set_Pr4532377907799695533_nat_o] :
% 0.34/0.54 ( ( bot_bo7824918357723582233_nat_o
% 0.34/0.54 = ( sup_su5209123915105501825_nat_o @ X @ Y ) )
% 0.34/0.54 = ( ( X = bot_bo7824918357723582233_nat_o )
% 0.34/0.54 & ( Y = bot_bo7824918357723582233_nat_o ) ) ) ).
% 0.34/0.54
% 0.34/0.54 % bot_eq_sup_iff
% 0.34/0.54 thf(fact_1558_sup__bot__right,axiom,
% 0.34/0.54 ! [X: assn] :
% 0.34/0.54 ( ( sup_sup_assn @ X @ bot_bot_assn )
% 0.34/0.54 = X ) ).
% 0.34/0.54
% 0.34/0.54 % sup_bot_right
% 0.34/0.54 thf(fact_1559_sup__bot__right,axiom,
% 0.34/0.54 ! [X: product_unit] :
% 0.34/0.54 ( ( sup_sup_Product_unit @ X @ bot_bot_Product_unit )
% 0.34/0.54 = X ) ).
% 0.34/0.54
% 0.34/0.54 % sup_bot_right
% 0.34/0.54 thf(fact_1560_sup__bot__right,axiom,
% 0.34/0.54 ! [X: set_Product_unit] :
% 0.34/0.54 ( ( sup_su793286257634532545t_unit @ X @ bot_bo3957492148770167129t_unit )
% 0.34/0.54 = X ) ).
% 0.34/0.54
% 0.34/0.54 % sup_bot_right
% 0.34/0.54 thf(fact_1561_sup__bot__right,axiom,
% 0.34/0.54 ! [X: set_o] :
% 0.34/0.54 ( ( sup_sup_set_o @ X @ bot_bot_set_o )
% 0.34/0.54 = X ) ).
% 0.34/0.54
% 0.34/0.54 % sup_bot_right
% 0.34/0.54 thf(fact_1562_sup__bot__right,axiom,
% 0.34/0.54 ! [X: set_nat] :
% 0.34/0.54 ( ( sup_sup_set_nat @ X @ bot_bot_set_nat )
% 0.34/0.54 = X ) ).
% 0.34/0.54
% 0.34/0.54 % sup_bot_right
% 0.34/0.54 thf(fact_1563_sup__bot__right,axiom,
% 0.34/0.54 ! [X: set_int] :
% 0.34/0.54 ( ( sup_sup_set_int @ X @ bot_bot_set_int )
% 0.34/0.54 = X ) ).
% 0.34/0.54
% 0.34/0.54 % sup_bot_right
% 0.34/0.54 thf(fact_1564_sup__bot__right,axiom,
% 0.34/0.54 ! [X: set_Pr4532377907799695533_nat_o] :
% 0.34/0.54 ( ( sup_su5209123915105501825_nat_o @ X @ bot_bo7824918357723582233_nat_o )
% 0.34/0.54 = X ) ).
% 0.34/0.54
% 0.34/0.54 % sup_bot_right
% 0.34/0.54 thf(fact_1565_sup__bot__left,axiom,
% 0.34/0.54 ! [X: assn] :
% 0.34/0.54 ( ( sup_sup_assn @ bot_bot_assn @ X )
% 0.34/0.54 = X ) ).
% 0.34/0.54
% 0.34/0.54 % sup_bot_left
% 0.34/0.54 thf(fact_1566_sup__bot__left,axiom,
% 0.34/0.54 ! [X: product_unit] :
% 0.34/0.54 ( ( sup_sup_Product_unit @ bot_bot_Product_unit @ X )
% 0.34/0.54 = X ) ).
% 0.34/0.54
% 0.34/0.54 % sup_bot_left
% 0.34/0.54 thf(fact_1567_sup__bot__left,axiom,
% 0.34/0.54 ! [X: set_Product_unit] :
% 0.34/0.54 ( ( sup_su793286257634532545t_unit @ bot_bo3957492148770167129t_unit @ X )
% 0.34/0.54 = X ) ).
% 0.34/0.54
% 0.34/0.54 % sup_bot_left
% 0.34/0.54 thf(fact_1568_sup__bot__left,axiom,
% 0.34/0.54 ! [X: set_o] :
% 0.34/0.54 ( ( sup_sup_set_o @ bot_bot_set_o @ X )
% 0.34/0.54 = X ) ).
% 0.34/0.54
% 0.34/0.54 % sup_bot_left
% 0.34/0.54 thf(fact_1569_sup__bot__left,axiom,
% 0.34/0.54 ! [X: set_nat] :
% 0.34/0.54 ( ( sup_sup_set_nat @ bot_bot_set_nat @ X )
% 0.34/0.54 = X ) ).
% 0.34/0.54
% 0.34/0.54 % sup_bot_left
% 0.34/0.54 thf(fact_1570_sup__bot__left,axiom,
% 0.34/0.54 ! [X: set_int] :
% 0.34/0.54 ( ( sup_sup_set_int @ bot_bot_set_int @ X )
% 0.34/0.54 = X ) ).
% 0.34/0.54
% 0.34/0.54 % sup_bot_left
% 0.34/0.54 thf(fact_1571_sup__bot__left,axiom,
% 0.34/0.54 ! [X: set_Pr4532377907799695533_nat_o] :
% 0.34/0.54 ( ( sup_su5209123915105501825_nat_o @ bot_bo7824918357723582233_nat_o @ X )
% 0.34/0.54 = X ) ).
% 0.34/0.54
% 0.34/0.54 % sup_bot_left
% 0.34/0.54 thf(fact_1572_boolean__algebra_Oconj__zero__right,axiom,
% 0.34/0.54 ! [X: assn] :
% 0.34/0.54 ( ( inf_inf_assn @ X @ bot_bot_assn )
% 0.34/0.54 = bot_bot_assn ) ).
% 0.34/0.54
% 0.34/0.54 % boolean_algebra.conj_zero_right
% 0.34/0.54 thf(fact_1573_boolean__algebra_Oconj__zero__right,axiom,
% 0.34/0.54 ! [X: product_unit] :
% 0.34/0.54 ( ( inf_inf_Product_unit @ X @ bot_bot_Product_unit )
% 0.34/0.54 = bot_bot_Product_unit ) ).
% 0.34/0.54
% 0.34/0.54 % boolean_algebra.conj_zero_right
% 0.34/0.54 thf(fact_1574_boolean__algebra_Oconj__zero__right,axiom,
% 0.34/0.54 ! [X: set_Product_unit] :
% 0.34/0.54 ( ( inf_in4660618365625256667t_unit @ X @ bot_bo3957492148770167129t_unit )
% 0.34/0.54 = bot_bo3957492148770167129t_unit ) ).
% 0.34/0.54
% 0.34/0.54 % boolean_algebra.conj_zero_right
% 0.34/0.54 thf(fact_1575_boolean__algebra_Oconj__zero__right,axiom,
% 0.34/0.54 ! [X: set_o] :
% 0.34/0.54 ( ( inf_inf_set_o @ X @ bot_bot_set_o )
% 0.34/0.54 = bot_bot_set_o ) ).
% 0.34/0.54
% 0.34/0.54 % boolean_algebra.conj_zero_right
% 0.34/0.54 thf(fact_1576_boolean__algebra_Oconj__zero__right,axiom,
% 0.34/0.54 ! [X: set_nat] :
% 0.34/0.54 ( ( inf_inf_set_nat @ X @ bot_bot_set_nat )
% 0.34/0.54 = bot_bot_set_nat ) ).
% 0.34/0.54
% 0.34/0.54 % boolean_algebra.conj_zero_right
% 0.34/0.54 thf(fact_1577_boolean__algebra_Oconj__zero__right,axiom,
% 0.34/0.54 ! [X: set_int] :
% 0.34/0.54 ( ( inf_inf_set_int @ X @ bot_bot_set_int )
% 0.34/0.54 = bot_bot_set_int ) ).
% 0.34/0.54
% 0.34/0.54 % boolean_algebra.conj_zero_right
% 0.34/0.54 thf(fact_1578_boolean__algebra_Oconj__zero__right,axiom,
% 0.34/0.54 ! [X: set_Pr4532377907799695533_nat_o] :
% 0.34/0.54 ( ( inf_in1906310914598751387_nat_o @ X @ bot_bo7824918357723582233_nat_o )
% 0.34/0.54 = bot_bo7824918357723582233_nat_o ) ).
% 0.34/0.54
% 0.34/0.54 % boolean_algebra.conj_zero_right
% 0.34/0.54 thf(fact_1579_boolean__algebra_Oconj__zero__left,axiom,
% 0.34/0.54 ! [X: assn] :
% 0.34/0.54 ( ( inf_inf_assn @ bot_bot_assn @ X )
% 0.34/0.54 = bot_bot_assn ) ).
% 0.34/0.54
% 0.34/0.54 % boolean_algebra.conj_zero_left
% 0.34/0.54 thf(fact_1580_boolean__algebra_Oconj__zero__left,axiom,
% 0.34/0.54 ! [X: product_unit] :
% 0.34/0.54 ( ( inf_inf_Product_unit @ bot_bot_Product_unit @ X )
% 0.34/0.54 = bot_bot_Product_unit ) ).
% 0.34/0.54
% 0.34/0.54 % boolean_algebra.conj_zero_left
% 0.34/0.54 thf(fact_1581_boolean__algebra_Oconj__zero__left,axiom,
% 0.34/0.54 ! [X: set_Product_unit] :
% 0.34/0.54 ( ( inf_in4660618365625256667t_unit @ bot_bo3957492148770167129t_unit @ X )
% 0.34/0.54 = bot_bo3957492148770167129t_unit ) ).
% 0.34/0.54
% 0.34/0.54 % boolean_algebra.conj_zero_left
% 0.34/0.54 thf(fact_1582_boolean__algebra_Oconj__zero__left,axiom,
% 0.34/0.54 ! [X: set_o] :
% 0.34/0.54 ( ( inf_inf_set_o @ bot_bot_set_o @ X )
% 0.34/0.54 = bot_bot_set_o ) ).
% 0.34/0.54
% 0.34/0.54 % boolean_algebra.conj_zero_left
% 0.34/0.54 thf(fact_1583_boolean__algebra_Oconj__zero__left,axiom,
% 0.34/0.54 ! [X: set_nat] :
% 0.34/0.54 ( ( inf_inf_set_nat @ bot_bot_set_nat @ X )
% 0.34/0.54 = bot_bot_set_nat ) ).
% 0.34/0.54
% 0.34/0.54 % boolean_algebra.conj_zero_left
% 0.34/0.54 thf(fact_1584_boolean__algebra_Oconj__zero__left,axiom,
% 0.34/0.54 ! [X: set_int] :
% 0.34/0.54 ( ( inf_inf_set_int @ bot_bot_set_int @ X )
% 0.34/0.54 = bot_bot_set_int ) ).
% 0.34/0.54
% 0.34/0.54 % boolean_algebra.conj_zero_left
% 0.34/0.54 thf(fact_1585_boolean__algebra_Oconj__zero__left,axiom,
% 0.34/0.54 ! [X: set_Pr4532377907799695533_nat_o] :
% 0.34/0.54 ( ( inf_in1906310914598751387_nat_o @ bot_bo7824918357723582233_nat_o @ X )
% 0.34/0.54 = bot_bo7824918357723582233_nat_o ) ).
% 0.34/0.54
% 0.34/0.54 % boolean_algebra.conj_zero_left
% 0.34/0.54 thf(fact_1586_inf__bot__right,axiom,
% 0.34/0.54 ! [X: assn] :
% 0.34/0.54 ( ( inf_inf_assn @ X @ bot_bot_assn )
% 0.34/0.54 = bot_bot_assn ) ).
% 0.34/0.54
% 0.34/0.54 % inf_bot_right
% 0.34/0.54 thf(fact_1587_inf__bot__right,axiom,
% 0.34/0.54 ! [X: product_unit] :
% 0.34/0.54 ( ( inf_inf_Product_unit @ X @ bot_bot_Product_unit )
% 0.34/0.54 = bot_bot_Product_unit ) ).
% 0.34/0.54
% 0.34/0.54 % inf_bot_right
% 0.34/0.54 thf(fact_1588_inf__bot__right,axiom,
% 0.34/0.54 ! [X: set_Product_unit] :
% 0.34/0.54 ( ( inf_in4660618365625256667t_unit @ X @ bot_bo3957492148770167129t_unit )
% 0.34/0.54 = bot_bo3957492148770167129t_unit ) ).
% 0.34/0.54
% 0.34/0.54 % inf_bot_right
% 0.34/0.54 thf(fact_1589_inf__bot__right,axiom,
% 0.34/0.54 ! [X: set_o] :
% 0.34/0.54 ( ( inf_inf_set_o @ X @ bot_bot_set_o )
% 0.34/0.54 = bot_bot_set_o ) ).
% 0.34/0.54
% 0.34/0.54 % inf_bot_right
% 0.34/0.54 thf(fact_1590_inf__bot__right,axiom,
% 0.34/0.54 ! [X: set_nat] :
% 0.34/0.54 ( ( inf_inf_set_nat @ X @ bot_bot_set_nat )
% 0.34/0.54 = bot_bot_set_nat ) ).
% 0.34/0.54
% 0.34/0.54 % inf_bot_right
% 0.34/0.54 thf(fact_1591_inf__bot__right,axiom,
% 0.34/0.54 ! [X: set_int] :
% 0.34/0.54 ( ( inf_inf_set_int @ X @ bot_bot_set_int )
% 0.34/0.54 = bot_bot_set_int ) ).
% 0.34/0.54
% 0.34/0.54 % inf_bot_right
% 0.34/0.54 thf(fact_1592_inf__bot__right,axiom,
% 0.34/0.54 ! [X: set_Pr4532377907799695533_nat_o] :
% 0.34/0.54 ( ( inf_in1906310914598751387_nat_o @ X @ bot_bo7824918357723582233_nat_o )
% 0.34/0.54 = bot_bo7824918357723582233_nat_o ) ).
% 0.34/0.54
% 0.34/0.54 % inf_bot_right
% 0.34/0.54 thf(fact_1593_merge__pure__or,axiom,
% 0.34/0.54 ! [A2: $o,B: $o] :
% 0.34/0.54 ( ( sup_sup_assn @ ( pure_assn @ A2 ) @ ( pure_assn @ B ) )
% 0.34/0.54 = ( pure_assn
% 0.34/0.54 @ ( A2
% 0.34/0.54 | B ) ) ) ).
% 0.34/0.54
% 0.34/0.54 % merge_pure_or
% 0.34/0.54 thf(fact_1594_merge__pure__and,axiom,
% 0.34/0.54 ! [A2: $o,B: $o] :
% 0.34/0.54 ( ( inf_inf_assn @ ( pure_assn @ A2 ) @ ( pure_assn @ B ) )
% 0.34/0.54 = ( pure_assn
% 0.34/0.54 @ ( A2
% 0.34/0.54 & B ) ) ) ).
% 0.34/0.54
% 0.34/0.54 % merge_pure_and
% 0.34/0.54 thf(fact_1595_inf__bot__left,axiom,
% 0.34/0.54 ! [X: assn] :
% 0.34/0.54 ( ( inf_inf_assn @ bot_bot_assn @ X )
% 0.34/0.54 = bot_bot_assn ) ).
% 0.34/0.55
% 0.34/0.55 % inf_bot_left
% 0.34/0.55 thf(fact_1596_inf__bot__left,axiom,
% 0.34/0.55 ! [X: product_unit] :
% 0.34/0.55 ( ( inf_inf_Product_unit @ bot_bot_Product_unit @ X )
% 0.34/0.55 = bot_bot_Product_unit ) ).
% 0.34/0.55
% 0.34/0.55 % inf_bot_left
% 0.34/0.55 thf(fact_1597_inf__bot__left,axiom,
% 0.34/0.55 ! [X: set_Product_unit] :
% 0.34/0.55 ( ( inf_in4660618365625256667t_unit @ bot_bo3957492148770167129t_unit @ X )
% 0.34/0.55 = bot_bo3957492148770167129t_unit ) ).
% 0.34/0.55
% 0.34/0.55 % inf_bot_left
% 0.34/0.55 thf(fact_1598_inf__bot__left,axiom,
% 0.34/0.55 ! [X: set_o] :
% 0.34/0.55 ( ( inf_inf_set_o @ bot_bot_set_o @ X )
% 0.34/0.55 = bot_bot_set_o ) ).
% 0.34/0.55
% 0.34/0.55 % inf_bot_left
% 0.34/0.55 thf(fact_1599_inf__bot__left,axiom,
% 0.34/0.55 ! [X: set_nat] :
% 0.34/0.55 ( ( inf_inf_set_nat @ bot_bot_set_nat @ X )
% 0.34/0.55 = bot_bot_set_nat ) ).
% 0.34/0.55
% 0.34/0.55 % inf_bot_left
% 0.34/0.55 thf(fact_1600_inf__bot__left,axiom,
% 0.34/0.55 ! [X: set_int] :
% 0.34/0.55 ( ( inf_inf_set_int @ bot_bot_set_int @ X )
% 0.34/0.55 = bot_bot_set_int ) ).
% 0.34/0.55
% 0.34/0.55 % inf_bot_left
% 0.34/0.55 thf(fact_1601_inf__bot__left,axiom,
% 0.34/0.55 ! [X: set_Pr4532377907799695533_nat_o] :
% 0.34/0.55 ( ( inf_in1906310914598751387_nat_o @ bot_bo7824918357723582233_nat_o @ X )
% 0.34/0.55 = bot_bo7824918357723582233_nat_o ) ).
% 0.34/0.55
% 0.34/0.55 % inf_bot_left
% 0.34/0.55 thf(fact_1602_boolean__algebra_Odisj__zero__right,axiom,
% 0.34/0.55 ! [X: assn] :
% 0.34/0.55 ( ( sup_sup_assn @ X @ bot_bot_assn )
% 0.34/0.55 = X ) ).
% 0.34/0.55
% 0.34/0.55 % boolean_algebra.disj_zero_right
% 0.34/0.55 thf(fact_1603_boolean__algebra_Odisj__zero__right,axiom,
% 0.34/0.55 ! [X: product_unit] :
% 0.34/0.55 ( ( sup_sup_Product_unit @ X @ bot_bot_Product_unit )
% 0.34/0.55 = X ) ).
% 0.34/0.55
% 0.34/0.55 % boolean_algebra.disj_zero_right
% 0.34/0.55 thf(fact_1604_boolean__algebra_Odisj__zero__right,axiom,
% 0.34/0.55 ! [X: set_Product_unit] :
% 0.34/0.55 ( ( sup_su793286257634532545t_unit @ X @ bot_bo3957492148770167129t_unit )
% 0.34/0.55 = X ) ).
% 0.34/0.55
% 0.34/0.55 % boolean_algebra.disj_zero_right
% 0.34/0.55 thf(fact_1605_boolean__algebra_Odisj__zero__right,axiom,
% 0.34/0.55 ! [X: set_o] :
% 0.34/0.55 ( ( sup_sup_set_o @ X @ bot_bot_set_o )
% 0.34/0.55 = X ) ).
% 0.34/0.55
% 0.34/0.55 % boolean_algebra.disj_zero_right
% 0.34/0.55 thf(fact_1606_boolean__algebra_Odisj__zero__right,axiom,
% 0.34/0.55 ! [X: set_nat] :
% 0.34/0.55 ( ( sup_sup_set_nat @ X @ bot_bot_set_nat )
% 0.34/0.55 = X ) ).
% 0.34/0.55
% 0.34/0.55 % boolean_algebra.disj_zero_right
% 0.34/0.55 thf(fact_1607_boolean__algebra_Odisj__zero__right,axiom,
% 0.34/0.55 ! [X: set_int] :
% 0.34/0.55 ( ( sup_sup_set_int @ X @ bot_bot_set_int )
% 0.34/0.55 = X ) ).
% 0.34/0.55
% 0.34/0.55 % boolean_algebra.disj_zero_right
% 0.34/0.55 thf(fact_1608_boolean__algebra_Odisj__zero__right,axiom,
% 0.34/0.55 ! [X: set_Pr4532377907799695533_nat_o] :
% 0.34/0.55 ( ( sup_su5209123915105501825_nat_o @ X @ bot_bo7824918357723582233_nat_o )
% 0.34/0.55 = X ) ).
% 0.34/0.55
% 0.34/0.55 % boolean_algebra.disj_zero_right
% 0.34/0.55 thf(fact_1609_prod__set__defs_I2_J,axiom,
% 0.34/0.55 ( basic_2698842327037663054et_nat
% 0.34/0.55 = ( ^ [P5: produc3658429121746597890et_nat] : ( insert_set_nat @ ( produc8586169260539613262et_nat @ P5 ) @ bot_bot_set_set_nat ) ) ) ).
% 0.34/0.55
% 0.34/0.55 % prod_set_defs(2)
% 0.34/0.55 thf(fact_1610_prod__set__defs_I2_J,axiom,
% 0.34/0.55 ( basic_snds_int_int
% 0.34/0.55 = ( ^ [P5: product_prod_int_int] : ( insert_int @ ( product_snd_int_int @ P5 ) @ bot_bot_set_int ) ) ) ).
% 0.34/0.55
% 0.34/0.55 % prod_set_defs(2)
% 0.34/0.55 thf(fact_1611_snds_Ocases,axiom,
% 0.34/0.55 ! [A2: set_nat,P3: produc3658429121746597890et_nat] :
% 0.34/0.55 ( ( member_set_nat @ A2 @ ( basic_2698842327037663054et_nat @ P3 ) )
% 0.34/0.55 => ( A2
% 0.34/0.55 = ( produc8586169260539613262et_nat @ P3 ) ) ) ).
% 0.34/0.55
% 0.34/0.55 % snds.cases
% 0.34/0.55 thf(fact_1612_snds_Ocases,axiom,
% 0.34/0.55 ! [A2: int,P3: product_prod_int_int] :
% 0.34/0.55 ( ( member_int @ A2 @ ( basic_snds_int_int @ P3 ) )
% 0.34/0.55 => ( A2
% 0.34/0.55 = ( product_snd_int_int @ P3 ) ) ) ).
% 0.34/0.55
% 0.34/0.55 % snds.cases
% 0.34/0.55 thf(fact_1613_snds_Osimps,axiom,
% 0.34/0.55 ! [A2: set_nat,P3: produc3658429121746597890et_nat] :
% 0.34/0.55 ( ( member_set_nat @ A2 @ ( basic_2698842327037663054et_nat @ P3 ) )
% 0.34/0.55 = ( A2
% 0.34/0.55 = ( produc8586169260539613262et_nat @ P3 ) ) ) ).
% 0.34/0.55
% 0.34/0.55 % snds.simps
% 0.34/0.55 thf(fact_1614_snds_Osimps,axiom,
% 0.34/0.55 ! [A2: int,P3: product_prod_int_int] :
% 0.34/0.55 ( ( member_int @ A2 @ ( basic_snds_int_int @ P3 ) )
% 0.34/0.55 = ( A2
% 0.34/0.55 = ( product_snd_int_int @ P3 ) ) ) ).
% 0.34/0.55
% 0.34/0.55 % snds.simps
% 0.34/0.55 thf(fact_1615_snds_Ointros,axiom,
% 0.34/0.55 ! [P3: produc3658429121746597890et_nat] : ( member_set_nat @ ( produc8586169260539613262et_nat @ P3 ) @ ( basic_2698842327037663054et_nat @ P3 ) ) ).
% 0.34/0.55
% 0.34/0.55 % snds.intros
% 0.34/0.55 thf(fact_1616_snds_Ointros,axiom,
% 0.34/0.55 ! [P3: product_prod_int_int] : ( member_int @ ( product_snd_int_int @ P3 ) @ ( basic_snds_int_int @ P3 ) ) ).
% 0.34/0.55
% 0.34/0.55 % snds.intros
% 0.34/0.55 thf(fact_1617_wand__raw_Opelims_I1_J,axiom,
% 0.34/0.55 ! [X: produc3658429121746597890et_nat > $o,Xa: produc3658429121746597890et_nat > $o,Xb: produc3658429121746597890et_nat,Y: $o] :
% 0.34/0.55 ( ( ( wand_raw @ X @ Xa @ Xb )
% 0.34/0.55 = Y )
% 0.34/0.55 => ( ( accp_P1862375125659990705et_nat @ wand_raw_rel @ ( produc2245416461498447860et_nat @ X @ ( produc5001842942810119800et_nat @ Xa @ Xb ) ) )
% 0.34/0.55 => ~ ! [H: heap_e7401611519738050253t_unit,As2: set_nat] :
% 0.34/0.55 ( ( Xb
% 0.34/0.55 = ( produc7507926704131184380et_nat @ H @ As2 ) )
% 0.34/0.55 => ( ( Y
% 0.34/0.55 = ( ( in_range @ ( produc7507926704131184380et_nat @ H @ As2 ) )
% 0.34/0.55 & ! [H5: heap_e7401611519738050253t_unit,As6: set_nat] :
% 0.34/0.55 ( ( ( ( inf_inf_set_nat @ As2 @ As6 )
% 0.34/0.55 = bot_bot_set_nat )
% 0.34/0.55 & ( relH @ As2 @ H @ H5 )
% 0.34/0.55 & ( in_range @ ( produc7507926704131184380et_nat @ H5 @ As2 ) )
% 0.34/0.55 & ( X @ ( produc7507926704131184380et_nat @ H5 @ As6 ) ) )
% 0.34/0.55 => ( Xa @ ( produc7507926704131184380et_nat @ H5 @ ( sup_sup_set_nat @ As2 @ As6 ) ) ) ) ) )
% 0.34/0.55 => ~ ( accp_P1862375125659990705et_nat @ wand_raw_rel @ ( produc2245416461498447860et_nat @ X @ ( produc5001842942810119800et_nat @ Xa @ ( produc7507926704131184380et_nat @ H @ As2 ) ) ) ) ) ) ) ) ).
% 0.34/0.55
% 0.34/0.55 % wand_raw.pelims(1)
% 0.34/0.55 thf(fact_1618_wand__raw_Opelims_I2_J,axiom,
% 0.34/0.55 ! [X: produc3658429121746597890et_nat > $o,Xa: produc3658429121746597890et_nat > $o,Xb: produc3658429121746597890et_nat] :
% 0.34/0.55 ( ( wand_raw @ X @ Xa @ Xb )
% 0.34/0.55 => ( ( accp_P1862375125659990705et_nat @ wand_raw_rel @ ( produc2245416461498447860et_nat @ X @ ( produc5001842942810119800et_nat @ Xa @ Xb ) ) )
% 0.34/0.55 => ~ ! [H: heap_e7401611519738050253t_unit,As2: set_nat] :
% 0.34/0.55 ( ( Xb
% 0.34/0.55 = ( produc7507926704131184380et_nat @ H @ As2 ) )
% 0.34/0.55 => ( ( accp_P1862375125659990705et_nat @ wand_raw_rel @ ( produc2245416461498447860et_nat @ X @ ( produc5001842942810119800et_nat @ Xa @ ( produc7507926704131184380et_nat @ H @ As2 ) ) ) )
% 0.34/0.55 => ~ ( ( in_range @ ( produc7507926704131184380et_nat @ H @ As2 ) )
% 0.34/0.55 & ! [H7: heap_e7401611519738050253t_unit,As7: set_nat] :
% 0.34/0.55 ( ( ( ( inf_inf_set_nat @ As2 @ As7 )
% 0.34/0.55 = bot_bot_set_nat )
% 0.34/0.55 & ( relH @ As2 @ H @ H7 )
% 0.34/0.55 & ( in_range @ ( produc7507926704131184380et_nat @ H7 @ As2 ) )
% 0.34/0.55 & ( X @ ( produc7507926704131184380et_nat @ H7 @ As7 ) ) )
% 0.34/0.55 => ( Xa @ ( produc7507926704131184380et_nat @ H7 @ ( sup_sup_set_nat @ As2 @ As7 ) ) ) ) ) ) ) ) ) ).
% 0.34/0.55
% 0.34/0.55 % wand_raw.pelims(2)
% 0.34/0.55 thf(fact_1619_wand__raw_Opelims_I3_J,axiom,
% 0.34/0.55 ! [X: produc3658429121746597890et_nat > $o,Xa: produc3658429121746597890et_nat > $o,Xb: produc3658429121746597890et_nat] :
% 0.34/0.55 ( ~ ( wand_raw @ X @ Xa @ Xb )
% 0.34/0.55 => ( ( accp_P1862375125659990705et_nat @ wand_raw_rel @ ( produc2245416461498447860et_nat @ X @ ( produc5001842942810119800et_nat @ Xa @ Xb ) ) )
% 0.34/0.55 => ~ ! [H: heap_e7401611519738050253t_unit,As2: set_nat] :
% 0.34/0.55 ( ( Xb
% 0.34/0.55 = ( produc7507926704131184380et_nat @ H @ As2 ) )
% 0.34/0.55 => ( ( accp_P1862375125659990705et_nat @ wand_raw_rel @ ( produc2245416461498447860et_nat @ X @ ( produc5001842942810119800et_nat @ Xa @ ( produc7507926704131184380et_nat @ H @ As2 ) ) ) )
% 0.34/0.55 => ( ( in_range @ ( produc7507926704131184380et_nat @ H @ As2 ) )
% 0.34/0.55 & ! [H6: heap_e7401611519738050253t_unit,As4: set_nat] :
% 0.34/0.55 ( ( ( ( inf_inf_set_nat @ As2 @ As4 )
% 0.34/0.55 = bot_bot_set_nat )
% 0.34/0.55 & ( relH @ As2 @ H @ H6 )
% 0.34/0.55 & ( in_range @ ( produc7507926704131184380et_nat @ H6 @ As2 ) )
% 0.34/0.55 & ( X @ ( produc7507926704131184380et_nat @ H6 @ As4 ) ) )
% 0.34/0.55 => ( Xa @ ( produc7507926704131184380et_nat @ H6 @ ( sup_sup_set_nat @ As2 @ As4 ) ) ) ) ) ) ) ) ) ).
% 0.34/0.55
% 0.34/0.55 % wand_raw.pelims(3)
% 0.34/0.55 thf(fact_1620_le__numeral__extra_I4_J,axiom,
% 0.34/0.55 ord_le3102999989581377725nteger @ one_one_Code_integer @ one_one_Code_integer ).
% 0.34/0.55
% 0.34/0.55 % le_numeral_extra(4)
% 0.34/0.55 thf(fact_1621_le__numeral__extra_I4_J,axiom,
% 0.34/0.55 ord_le1926595141338095240atural @ one_one_Code_natural @ one_one_Code_natural ).
% 0.34/0.55
% 0.34/0.55 % le_numeral_extra(4)
% 0.34/0.55 thf(fact_1622_le__numeral__extra_I4_J,axiom,
% 0.34/0.55 ord_less_eq_rat @ one_one_rat @ one_one_rat ).
% 0.34/0.55
% 0.34/0.55 % le_numeral_extra(4)
% 0.34/0.55 thf(fact_1623_le__numeral__extra_I4_J,axiom,
% 0.34/0.55 ord_less_eq_nat @ one_one_nat @ one_one_nat ).
% 0.34/0.55
% 0.34/0.55 % le_numeral_extra(4)
% 0.34/0.55 thf(fact_1624_le__numeral__extra_I4_J,axiom,
% 0.34/0.55 ord_less_eq_int @ one_one_int @ one_one_int ).
% 0.34/0.55
% 0.34/0.55 % le_numeral_extra(4)
% 0.34/0.55 thf(fact_1625_less__eq__assn__def,axiom,
% 0.34/0.55 ( ord_less_eq_assn
% 0.34/0.55 = ( ^ [A6: assn,B6: assn] :
% 0.34/0.55 ( A6
% 0.34/0.55 = ( inf_inf_assn @ A6 @ B6 ) ) ) ) ).
% 0.34/0.55
% 0.34/0.55 % less_eq_assn_def
% 0.34/0.55 thf(fact_1626_times__assn__raw_Opelims_I3_J,axiom,
% 0.34/0.55 ! [X: produc3658429121746597890et_nat > $o,Xa: produc3658429121746597890et_nat > $o,Xb: produc3658429121746597890et_nat] :
% 0.34/0.55 ( ~ ( times_assn_raw @ X @ Xa @ Xb )
% 0.34/0.55 => ( ( accp_P1862375125659990705et_nat @ times_assn_raw_rel @ ( produc2245416461498447860et_nat @ X @ ( produc5001842942810119800et_nat @ Xa @ Xb ) ) )
% 0.34/0.55 => ~ ! [H: heap_e7401611519738050253t_unit,As2: set_nat] :
% 0.34/0.55 ( ( Xb
% 0.34/0.55 = ( produc7507926704131184380et_nat @ H @ As2 ) )
% 0.34/0.55 => ( ( accp_P1862375125659990705et_nat @ times_assn_raw_rel @ ( produc2245416461498447860et_nat @ X @ ( produc5001842942810119800et_nat @ Xa @ ( produc7507926704131184380et_nat @ H @ As2 ) ) ) )
% 0.34/0.55 => ? [As13: set_nat,As24: set_nat] :
% 0.34/0.55 ( ( As2
% 0.34/0.55 = ( sup_sup_set_nat @ As13 @ As24 ) )
% 0.34/0.55 & ( ( inf_inf_set_nat @ As13 @ As24 )
% 0.34/0.55 = bot_bot_set_nat )
% 0.34/0.55 & ( X @ ( produc7507926704131184380et_nat @ H @ As13 ) )
% 0.34/0.55 & ( Xa @ ( produc7507926704131184380et_nat @ H @ As24 ) ) ) ) ) ) ) ).
% 0.34/0.55
% 0.34/0.55 % times_assn_raw.pelims(3)
% 0.34/0.55 thf(fact_1627_times__assn__raw_Opelims_I2_J,axiom,
% 0.34/0.55 ! [X: produc3658429121746597890et_nat > $o,Xa: produc3658429121746597890et_nat > $o,Xb: produc3658429121746597890et_nat] :
% 0.34/0.55 ( ( times_assn_raw @ X @ Xa @ Xb )
% 0.34/0.55 => ( ( accp_P1862375125659990705et_nat @ times_assn_raw_rel @ ( produc2245416461498447860et_nat @ X @ ( produc5001842942810119800et_nat @ Xa @ Xb ) ) )
% 0.34/0.55 => ~ ! [H: heap_e7401611519738050253t_unit,As2: set_nat] :
% 0.34/0.55 ( ( Xb
% 0.34/0.55 = ( produc7507926704131184380et_nat @ H @ As2 ) )
% 0.34/0.55 => ( ( accp_P1862375125659990705et_nat @ times_assn_raw_rel @ ( produc2245416461498447860et_nat @ X @ ( produc5001842942810119800et_nat @ Xa @ ( produc7507926704131184380et_nat @ H @ As2 ) ) ) )
% 0.34/0.55 => ~ ? [As12: set_nat,As23: set_nat] :
% 0.34/0.55 ( ( As2
% 0.34/0.55 = ( sup_sup_set_nat @ As12 @ As23 ) )
% 0.34/0.55 & ( ( inf_inf_set_nat @ As12 @ As23 )
% 0.34/0.55 = bot_bot_set_nat )
% 0.34/0.55 & ( X @ ( produc7507926704131184380et_nat @ H @ As12 ) )
% 0.34/0.55 & ( Xa @ ( produc7507926704131184380et_nat @ H @ As23 ) ) ) ) ) ) ) ).
% 0.34/0.55
% 0.34/0.55 % times_assn_raw.pelims(2)
% 0.34/0.55 thf(fact_1628_times__assn__raw_Opelims_I1_J,axiom,
% 0.34/0.55 ! [X: produc3658429121746597890et_nat > $o,Xa: produc3658429121746597890et_nat > $o,Xb: produc3658429121746597890et_nat,Y: $o] :
% 0.34/0.55 ( ( ( times_assn_raw @ X @ Xa @ Xb )
% 0.34/0.55 = Y )
% 0.34/0.55 => ( ( accp_P1862375125659990705et_nat @ times_assn_raw_rel @ ( produc2245416461498447860et_nat @ X @ ( produc5001842942810119800et_nat @ Xa @ Xb ) ) )
% 0.34/0.55 => ~ ! [H: heap_e7401611519738050253t_unit,As2: set_nat] :
% 0.34/0.55 ( ( Xb
% 0.34/0.55 = ( produc7507926704131184380et_nat @ H @ As2 ) )
% 0.34/0.55 => ( ( Y
% 0.34/0.55 = ( ? [As1: set_nat,As22: set_nat] :
% 0.34/0.55 ( ( As2
% 0.34/0.55 = ( sup_sup_set_nat @ As1 @ As22 ) )
% 0.34/0.55 & ( ( inf_inf_set_nat @ As1 @ As22 )
% 0.34/0.55 = bot_bot_set_nat )
% 0.34/0.55 & ( X @ ( produc7507926704131184380et_nat @ H @ As1 ) )
% 0.34/0.55 & ( Xa @ ( produc7507926704131184380et_nat @ H @ As22 ) ) ) ) )
% 0.34/0.55 => ~ ( accp_P1862375125659990705et_nat @ times_assn_raw_rel @ ( produc2245416461498447860et_nat @ X @ ( produc5001842942810119800et_nat @ Xa @ ( produc7507926704131184380et_nat @ H @ As2 ) ) ) ) ) ) ) ) ).
% 0.34/0.55
% 0.34/0.55 % times_assn_raw.pelims(1)
% 0.34/0.55 thf(fact_1629_subset__Image1__Image1__iff,axiom,
% 0.34/0.55 ! [R2: set_Pr5094982260447487303t_unit,A2: product_unit,B: product_unit] :
% 0.34/0.55 ( ( order_7534304569044981540t_unit @ ( field_Product_unit @ R2 ) @ R2 )
% 0.34/0.55 => ( ( member_Product_unit @ A2 @ ( field_Product_unit @ R2 ) )
% 0.34/0.55 => ( ( member_Product_unit @ B @ ( field_Product_unit @ R2 ) )
% 0.34/0.55 => ( ( ord_le3507040750410214029t_unit @ ( image_8431784415008483091t_unit @ R2 @ ( insert_Product_unit @ A2 @ bot_bo3957492148770167129t_unit ) ) @ ( image_8431784415008483091t_unit @ R2 @ ( insert_Product_unit @ B @ bot_bo3957492148770167129t_unit ) ) )
% 0.34/0.55 = ( member7821138191923116944t_unit @ ( produc5692694829384537815t_unit @ B @ A2 ) @ R2 ) ) ) ) ) ).
% 0.34/0.55
% 0.34/0.55 % subset_Image1_Image1_iff
% 0.34/0.55 thf(fact_1630_subset__Image1__Image1__iff,axiom,
% 0.34/0.55 ! [R2: set_Product_prod_o_o,A2: $o,B: $o] :
% 0.34/0.55 ( ( order_preorder_on_o @ ( field_o @ R2 ) @ R2 )
% 0.34/0.55 => ( ( member_o @ A2 @ ( field_o @ R2 ) )
% 0.34/0.55 => ( ( member_o @ B @ ( field_o @ R2 ) )
% 0.34/0.55 => ( ( ord_less_eq_set_o @ ( image_o_o @ R2 @ ( insert_o @ A2 @ bot_bot_set_o ) ) @ ( image_o_o @ R2 @ ( insert_o @ B @ bot_bot_set_o ) ) )
% 0.34/0.55 = ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ B @ A2 ) @ R2 ) ) ) ) ) ).
% 0.34/0.55
% 0.34/0.55 % subset_Image1_Image1_iff
% 0.34/0.55 thf(fact_1631_subset__Image1__Image1__iff,axiom,
% 0.34/0.55 ! [R2: set_Pr958786334691620121nt_int,A2: int,B: int] :
% 0.34/0.55 ( ( order_4859164337913492053on_int @ ( field_int @ R2 ) @ R2 )
% 0.34/0.55 => ( ( member_int @ A2 @ ( field_int @ R2 ) )
% 0.34/0.55 => ( ( member_int @ B @ ( field_int @ R2 ) )
% 0.34/0.55 => ( ( ord_less_eq_set_int @ ( image_int_int @ R2 @ ( insert_int @ A2 @ bot_bot_set_int ) ) @ ( image_int_int @ R2 @ ( insert_int @ B @ bot_bot_set_int ) ) )
% 0.34/0.55 = ( member5262025264175285858nt_int @ ( product_Pair_int_int @ B @ A2 ) @ R2 ) ) ) ) ) ).
% 0.34/0.55
% 0.34/0.55 % subset_Image1_Image1_iff
% 0.34/0.55 thf(fact_1632_subset__Image1__Image1__iff,axiom,
% 0.34/0.55 ! [R2: set_Pr2161125870931222855_nat_o,A2: produc3658429121746597890et_nat > $o,B: produc3658429121746597890et_nat > $o] :
% 0.34/0.55 ( ( order_174704566861754596_nat_o @ ( field_81821745747897882_nat_o @ R2 ) @ R2 )
% 0.34/0.55 => ( ( member6576561426505652726_nat_o @ A2 @ ( field_81821745747897882_nat_o @ R2 ) )
% 0.34/0.55 => ( ( member6576561426505652726_nat_o @ B @ ( field_81821745747897882_nat_o @ R2 ) )
% 0.34/0.55 => ( ( ord_le2965882846123202637_nat_o @ ( image_2232984160439939859_nat_o @ R2 @ ( insert5175938949040314269_nat_o @ A2 @ bot_bo7824918357723582233_nat_o ) ) @ ( image_2232984160439939859_nat_o @ R2 @ ( insert5175938949040314269_nat_o @ B @ bot_bo7824918357723582233_nat_o ) ) )
% 0.34/0.55 = ( member8781333585448626064_nat_o @ ( produc7368190662567826135_nat_o @ B @ A2 ) @ R2 ) ) ) ) ) ).
% 0.34/0.55
% 0.34/0.55 % subset_Image1_Image1_iff
% 0.34/0.55 thf(fact_1633_subset__Image1__Image1__iff,axiom,
% 0.34/0.55 ! [R2: set_Pr1261947904930325089at_nat,A2: nat,B: nat] :
% 0.34/0.55 ( ( order_4861654808422542329on_nat @ ( field_nat @ R2 ) @ R2 )
% 0.34/0.55 => ( ( member_nat @ A2 @ ( field_nat @ R2 ) )
% 0.34/0.55 => ( ( member_nat @ B @ ( field_nat @ R2 ) )
% 0.34/0.55 => ( ( ord_less_eq_set_nat @ ( image_nat_nat @ R2 @ ( insert_nat @ A2 @ bot_bot_set_nat ) ) @ ( image_nat_nat @ R2 @ ( insert_nat @ B @ bot_bot_set_nat ) ) )
% 0.34/0.55 = ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ B @ A2 ) @ R2 ) ) ) ) ) ).
% 0.34/0.55
% 0.34/0.55 % subset_Image1_Image1_iff
% 0.34/0.55 thf(fact_1634_Field__insert,axiom,
% 0.34/0.55 ! [A2: product_unit,B: product_unit,R2: set_Pr5094982260447487303t_unit] :
% 0.34/0.55 ( ( field_Product_unit @ ( insert4616891024742550839t_unit @ ( produc5692694829384537815t_unit @ A2 @ B ) @ R2 ) )
% 0.34/0.55 = ( sup_su793286257634532545t_unit @ ( insert_Product_unit @ A2 @ ( insert_Product_unit @ B @ bot_bo3957492148770167129t_unit ) ) @ ( field_Product_unit @ R2 ) ) ) ).
% 0.34/0.55
% 0.34/0.55 % Field_insert
% 0.34/0.55 thf(fact_1635_Field__insert,axiom,
% 0.34/0.55 ! [A2: $o,B: $o,R2: set_Product_prod_o_o] :
% 0.34/0.55 ( ( field_o @ ( insert6201435330877294327od_o_o @ ( product_Pair_o_o @ A2 @ B ) @ R2 ) )
% 0.34/0.55 = ( sup_sup_set_o @ ( insert_o @ A2 @ ( insert_o @ B @ bot_bot_set_o ) ) @ ( field_o @ R2 ) ) ) ).
% 0.34/0.55
% 0.34/0.55 % Field_insert
% 0.34/0.55 thf(fact_1636_Field__insert,axiom,
% 0.34/0.55 ! [A2: nat,B: nat,R2: set_Pr1261947904930325089at_nat] :
% 0.34/0.55 ( ( field_nat @ ( insert8211810215607154385at_nat @ ( product_Pair_nat_nat @ A2 @ B ) @ R2 ) )
% 0.34/0.55 = ( sup_sup_set_nat @ ( insert_nat @ A2 @ ( insert_nat @ B @ bot_bot_set_nat ) ) @ ( field_nat @ R2 ) ) ) ).
% 0.34/0.55
% 0.34/0.55 % Field_insert
% 0.34/0.55 thf(fact_1637_Field__insert,axiom,
% 0.34/0.55 ! [A2: int,B: int,R2: set_Pr958786334691620121nt_int] :
% 0.34/0.55 ( ( field_int @ ( insert5033312907999012233nt_int @ ( product_Pair_int_int @ A2 @ B ) @ R2 ) )
% 0.34/0.55 = ( sup_sup_set_int @ ( insert_int @ A2 @ ( insert_int @ B @ bot_bot_set_int ) ) @ ( field_int @ R2 ) ) ) ).
% 0.34/0.55
% 0.34/0.55 % Field_insert
% 0.34/0.55 thf(fact_1638_Field__insert,axiom,
% 0.34/0.55 ! [A2: produc3658429121746597890et_nat > $o,B: produc3658429121746597890et_nat > $o,R2: set_Pr2161125870931222855_nat_o] :
% 0.34/0.55 ( ( field_81821745747897882_nat_o @ ( insert5309882203591314743_nat_o @ ( produc7368190662567826135_nat_o @ A2 @ B ) @ R2 ) )
% 0.34/0.55 = ( sup_su5209123915105501825_nat_o @ ( insert5175938949040314269_nat_o @ A2 @ ( insert5175938949040314269_nat_o @ B @ bot_bo7824918357723582233_nat_o ) ) @ ( field_81821745747897882_nat_o @ R2 ) ) ) ).
% 0.34/0.55
% 0.34/0.55 % Field_insert
% 0.34/0.55 thf(fact_1639_dom__ran__disj__comp,axiom,
% 0.34/0.55 ! [R3: set_Pr5094982260447487303t_unit] :
% 0.34/0.55 ( ( ( inf_in4660618365625256667t_unit @ ( domain4292024440186670924t_unit @ R3 ) @ ( range_6617724647562260917t_unit @ R3 ) )
% 0.34/0.55 = bot_bo3957492148770167129t_unit )
% 0.34/0.55 => ( ( relcom848357998638694028t_unit @ R3 @ R3 )
% 0.34/0.55 = bot_bo1654022762305692595t_unit ) ) ).
% 0.34/0.55
% 0.34/0.55 % dom_ran_disj_comp
% 0.34/0.55 thf(fact_1640_dom__ran__disj__comp,axiom,
% 0.34/0.55 ! [R3: set_Product_prod_o_o] :
% 0.34/0.55 ( ( ( inf_inf_set_o @ ( domain_o_o @ R3 ) @ ( range_o_o @ R3 ) )
% 0.34/0.55 = bot_bot_set_o )
% 0.34/0.55 => ( ( relcomp_o_o_o @ R3 @ R3 )
% 0.34/0.55 = bot_bo7073875226086086771od_o_o ) ) ).
% 0.34/0.55
% 0.34/0.55 % dom_ran_disj_comp
% 0.34/0.55 thf(fact_1641_dom__ran__disj__comp,axiom,
% 0.34/0.55 ! [R3: set_Pr1261947904930325089at_nat] :
% 0.34/0.55 ( ( ( inf_inf_set_nat @ ( domain_nat_nat @ R3 ) @ ( range_nat_nat @ R3 ) )
% 0.34/0.55 = bot_bot_set_nat )
% 0.34/0.55 => ( ( relcomp_nat_nat_nat @ R3 @ R3 )
% 0.34/0.55 = bot_bo2099793752762293965at_nat ) ) ).
% 0.34/0.55
% 0.34/0.55 % dom_ran_disj_comp
% 0.34/0.55 thf(fact_1642_dom__ran__disj__comp,axiom,
% 0.34/0.55 ! [R3: set_Pr958786334691620121nt_int] :
% 0.34/0.55 ( ( ( inf_inf_set_int @ ( domain_int_int @ R3 ) @ ( range_int_int @ R3 ) )
% 0.34/0.55 = bot_bot_set_int )
% 0.34/0.55 => ( ( relcomp_int_int_int @ R3 @ R3 )
% 0.34/0.55 = bot_bo1796632182523588997nt_int ) ) ).
% 0.34/0.55
% 0.34/0.55 % dom_ran_disj_comp
% 0.34/0.55 thf(fact_1643_dom__ran__disj__comp,axiom,
% 0.34/0.55 ! [R3: set_Pr2161125870931222855_nat_o] :
% 0.34/0.55 ( ( ( inf_in1906310914598751387_nat_o @ ( domain4881241903163524940_nat_o @ R3 ) @ ( range_5289318983786108341_nat_o @ R3 ) )
% 0.34/0.55 = bot_bo7824918357723582233_nat_o )
% 0.34/0.55 => ( ( relcom6251035627269349452_nat_o @ R3 @ R3 )
% 0.34/0.55 = bot_bo3657187498359500723_nat_o ) ) ).
% 0.34/0.55
% 0.34/0.55 % dom_ran_disj_comp
% 0.34/0.55 thf(fact_1644_proj__iff,axiom,
% 0.34/0.55 ! [A: set_Product_unit,R2: set_Pr5094982260447487303t_unit,X: product_unit,Y: product_unit] :
% 0.34/0.55 ( ( equiv_4189356295871601603t_unit @ A @ R2 )
% 0.34/0.55 => ( ( ord_le3507040750410214029t_unit @ ( insert_Product_unit @ X @ ( insert_Product_unit @ Y @ bot_bo3957492148770167129t_unit ) ) @ A )
% 0.34/0.55 => ( ( ( equiv_5763582929720877514t_unit @ R2 @ X )
% 0.34/0.55 = ( equiv_5763582929720877514t_unit @ R2 @ Y ) )
% 0.34/0.55 = ( member7821138191923116944t_unit @ ( produc5692694829384537815t_unit @ X @ Y ) @ R2 ) ) ) ) ).
% 0.34/0.55
% 0.34/0.55 % proj_iff
% 0.34/0.55 thf(fact_1645_proj__iff,axiom,
% 0.34/0.55 ! [A: set_o,R2: set_Product_prod_o_o,X: $o,Y: $o] :
% 0.34/0.55 ( ( equiv_equiv_o @ A @ R2 )
% 0.34/0.55 => ( ( ord_less_eq_set_o @ ( insert_o @ X @ ( insert_o @ Y @ bot_bot_set_o ) ) @ A )
% 0.34/0.55 => ( ( ( equiv_proj_o_o @ R2 @ X )
% 0.34/0.55 = ( equiv_proj_o_o @ R2 @ Y ) )
% 0.34/0.55 = ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ X @ Y ) @ R2 ) ) ) ) ).
% 0.34/0.55
% 0.34/0.55 % proj_iff
% 0.34/0.55 thf(fact_1646_proj__iff,axiom,
% 0.34/0.55 ! [A: set_int,R2: set_Pr958786334691620121nt_int,X: int,Y: int] :
% 0.34/0.55 ( ( equiv_equiv_int @ A @ R2 )
% 0.34/0.55 => ( ( ord_less_eq_set_int @ ( insert_int @ X @ ( insert_int @ Y @ bot_bot_set_int ) ) @ A )
% 0.34/0.55 => ( ( ( equiv_proj_int_int @ R2 @ X )
% 0.34/0.55 = ( equiv_proj_int_int @ R2 @ Y ) )
% 0.34/0.55 = ( member5262025264175285858nt_int @ ( product_Pair_int_int @ X @ Y ) @ R2 ) ) ) ) ).
% 0.34/0.55
% 0.34/0.55 % proj_iff
% 0.34/0.55 thf(fact_1647_proj__iff,axiom,
% 0.34/0.55 ! [A: set_Pr4532377907799695533_nat_o,R2: set_Pr2161125870931222855_nat_o,X: produc3658429121746597890et_nat > $o,Y: produc3658429121746597890et_nat > $o] :
% 0.34/0.55 ( ( equiv_6726761681920839555_nat_o @ A @ R2 )
% 0.34/0.55 => ( ( ord_le2965882846123202637_nat_o @ ( insert5175938949040314269_nat_o @ X @ ( insert5175938949040314269_nat_o @ Y @ bot_bo7824918357723582233_nat_o ) ) @ A )
% 0.34/0.55 => ( ( ( equiv_3235788788526563786_nat_o @ R2 @ X )
% 0.34/0.55 = ( equiv_3235788788526563786_nat_o @ R2 @ Y ) )
% 0.34/0.55 = ( member8781333585448626064_nat_o @ ( produc7368190662567826135_nat_o @ X @ Y ) @ R2 ) ) ) ) ).
% 0.34/0.55
% 0.34/0.55 % proj_iff
% 0.34/0.55 thf(fact_1648_proj__iff,axiom,
% 0.34/0.55 ! [A: set_nat,R2: set_Pr1261947904930325089at_nat,X: nat,Y: nat] :
% 0.34/0.55 ( ( equiv_equiv_nat @ A @ R2 )
% 0.34/0.55 => ( ( ord_less_eq_set_nat @ ( insert_nat @ X @ ( insert_nat @ Y @ bot_bot_set_nat ) ) @ A )
% 0.34/0.55 => ( ( ( equiv_proj_nat_nat @ R2 @ X )
% 0.34/0.55 = ( equiv_proj_nat_nat @ R2 @ Y ) )
% 0.34/0.55 = ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ Y ) @ R2 ) ) ) ) ).
% 0.34/0.55
% 0.34/0.55 % proj_iff
% 0.34/0.55 thf(fact_1649_Refl__antisym__eq__Image1__Image1__iff,axiom,
% 0.34/0.55 ! [R2: set_Pr5094982260447487303t_unit,A2: product_unit,B: product_unit] :
% 0.34/0.55 ( ( refl_on_Product_unit @ ( field_Product_unit @ R2 ) @ R2 )
% 0.34/0.55 => ( ( antisym_Product_unit @ R2 )
% 0.34/0.55 => ( ( member_Product_unit @ A2 @ ( field_Product_unit @ R2 ) )
% 0.34/0.55 => ( ( member_Product_unit @ B @ ( field_Product_unit @ R2 ) )
% 0.34/0.55 => ( ( ( image_8431784415008483091t_unit @ R2 @ ( insert_Product_unit @ A2 @ bot_bo3957492148770167129t_unit ) )
% 0.34/0.55 = ( image_8431784415008483091t_unit @ R2 @ ( insert_Product_unit @ B @ bot_bo3957492148770167129t_unit ) ) )
% 0.34/0.55 = ( A2 = B ) ) ) ) ) ) ).
% 0.34/0.55
% 0.34/0.55 % Refl_antisym_eq_Image1_Image1_iff
% 0.34/0.55 thf(fact_1650_Refl__antisym__eq__Image1__Image1__iff,axiom,
% 0.34/0.55 ! [R2: set_Product_prod_o_o,A2: $o,B: $o] :
% 0.34/0.55 ( ( refl_on_o @ ( field_o @ R2 ) @ R2 )
% 0.34/0.55 => ( ( antisym_o @ R2 )
% 0.34/0.55 => ( ( member_o @ A2 @ ( field_o @ R2 ) )
% 0.34/0.55 => ( ( member_o @ B @ ( field_o @ R2 ) )
% 0.34/0.55 => ( ( ( image_o_o @ R2 @ ( insert_o @ A2 @ bot_bot_set_o ) )
% 0.34/0.55 = ( image_o_o @ R2 @ ( insert_o @ B @ bot_bot_set_o ) ) )
% 0.34/0.55 = ( A2 = B ) ) ) ) ) ) ).
% 0.34/0.55
% 0.34/0.55 % Refl_antisym_eq_Image1_Image1_iff
% 0.34/0.55 thf(fact_1651_Refl__antisym__eq__Image1__Image1__iff,axiom,
% 0.34/0.55 ! [R2: set_Pr1261947904930325089at_nat,A2: nat,B: nat] :
% 0.34/0.55 ( ( refl_on_nat @ ( field_nat @ R2 ) @ R2 )
% 0.34/0.55 => ( ( antisym_nat @ R2 )
% 0.34/0.55 => ( ( member_nat @ A2 @ ( field_nat @ R2 ) )
% 0.34/0.55 => ( ( member_nat @ B @ ( field_nat @ R2 ) )
% 0.34/0.55 => ( ( ( image_nat_nat @ R2 @ ( insert_nat @ A2 @ bot_bot_set_nat ) )
% 0.34/0.55 = ( image_nat_nat @ R2 @ ( insert_nat @ B @ bot_bot_set_nat ) ) )
% 0.34/0.55 = ( A2 = B ) ) ) ) ) ) ).
% 0.34/0.55
% 0.34/0.55 % Refl_antisym_eq_Image1_Image1_iff
% 0.34/0.55 thf(fact_1652_Refl__antisym__eq__Image1__Image1__iff,axiom,
% 0.34/0.55 ! [R2: set_Pr958786334691620121nt_int,A2: int,B: int] :
% 0.34/0.55 ( ( refl_on_int @ ( field_int @ R2 ) @ R2 )
% 0.34/0.55 => ( ( antisym_int @ R2 )
% 0.34/0.55 => ( ( member_int @ A2 @ ( field_int @ R2 ) )
% 0.34/0.55 => ( ( member_int @ B @ ( field_int @ R2 ) )
% 0.34/0.55 => ( ( ( image_int_int @ R2 @ ( insert_int @ A2 @ bot_bot_set_int ) )
% 0.34/0.55 = ( image_int_int @ R2 @ ( insert_int @ B @ bot_bot_set_int ) ) )
% 0.34/0.55 = ( A2 = B ) ) ) ) ) ) ).
% 0.34/0.55
% 0.34/0.55 % Refl_antisym_eq_Image1_Image1_iff
% 0.34/0.55 thf(fact_1653_Refl__antisym__eq__Image1__Image1__iff,axiom,
% 0.34/0.55 ! [R2: set_Pr2161125870931222855_nat_o,A2: produc3658429121746597890et_nat > $o,B: produc3658429121746597890et_nat > $o] :
% 0.34/0.55 ( ( refl_o1075864025857821388_nat_o @ ( field_81821745747897882_nat_o @ R2 ) @ R2 )
% 0.34/0.55 => ( ( antisy9190549550370998639_nat_o @ R2 )
% 0.34/0.55 => ( ( member6576561426505652726_nat_o @ A2 @ ( field_81821745747897882_nat_o @ R2 ) )
% 0.34/0.55 => ( ( member6576561426505652726_nat_o @ B @ ( field_81821745747897882_nat_o @ R2 ) )
% 0.34/0.55 => ( ( ( image_2232984160439939859_nat_o @ R2 @ ( insert5175938949040314269_nat_o @ A2 @ bot_bo7824918357723582233_nat_o ) )
% 0.34/0.55 = ( image_2232984160439939859_nat_o @ R2 @ ( insert5175938949040314269_nat_o @ B @ bot_bo7824918357723582233_nat_o ) ) )
% 0.34/0.55 = ( A2 = B ) ) ) ) ) ) ).
% 0.34/0.55
% 0.34/0.55 % Refl_antisym_eq_Image1_Image1_iff
% 0.34/0.55 thf(fact_1654_equiv__class__nondisjoint,axiom,
% 0.34/0.55 ! [A: set_Product_unit,R2: set_Pr5094982260447487303t_unit,X: product_unit,A2: product_unit,B: product_unit] :
% 0.34/0.55 ( ( equiv_4189356295871601603t_unit @ A @ R2 )
% 0.34/0.55 => ( ( member_Product_unit @ X @ ( inf_in4660618365625256667t_unit @ ( image_8431784415008483091t_unit @ R2 @ ( insert_Product_unit @ A2 @ bot_bo3957492148770167129t_unit ) ) @ ( image_8431784415008483091t_unit @ R2 @ ( insert_Product_unit @ B @ bot_bo3957492148770167129t_unit ) ) ) )
% 0.34/0.55 => ( member7821138191923116944t_unit @ ( produc5692694829384537815t_unit @ A2 @ B ) @ R2 ) ) ) ).
% 0.34/0.55
% 0.34/0.55 % equiv_class_nondisjoint
% 0.34/0.55 thf(fact_1655_equiv__class__nondisjoint,axiom,
% 0.34/0.55 ! [A: set_o,R2: set_Product_prod_o_o,X: $o,A2: $o,B: $o] :
% 0.34/0.55 ( ( equiv_equiv_o @ A @ R2 )
% 0.34/0.55 => ( ( member_o @ X @ ( inf_inf_set_o @ ( image_o_o @ R2 @ ( insert_o @ A2 @ bot_bot_set_o ) ) @ ( image_o_o @ R2 @ ( insert_o @ B @ bot_bot_set_o ) ) ) )
% 0.34/0.55 => ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ A2 @ B ) @ R2 ) ) ) ).
% 0.34/0.55
% 0.34/0.55 % equiv_class_nondisjoint
% 0.34/0.55 thf(fact_1656_equiv__class__nondisjoint,axiom,
% 0.34/0.55 ! [A: set_nat,R2: set_Pr1261947904930325089at_nat,X: nat,A2: nat,B: nat] :
% 0.34/0.55 ( ( equiv_equiv_nat @ A @ R2 )
% 0.34/0.55 => ( ( member_nat @ X @ ( inf_inf_set_nat @ ( image_nat_nat @ R2 @ ( insert_nat @ A2 @ bot_bot_set_nat ) ) @ ( image_nat_nat @ R2 @ ( insert_nat @ B @ bot_bot_set_nat ) ) ) )
% 0.34/0.55 => ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A2 @ B ) @ R2 ) ) ) ).
% 0.34/0.55
% 0.34/0.55 % equiv_class_nondisjoint
% 0.34/0.55 thf(fact_1657_equiv__class__nondisjoint,axiom,
% 0.34/0.55 ! [A: set_int,R2: set_Pr958786334691620121nt_int,X: int,A2: int,B: int] :
% 0.34/0.55 ( ( equiv_equiv_int @ A @ R2 )
% 0.34/0.55 => ( ( member_int @ X @ ( inf_inf_set_int @ ( image_int_int @ R2 @ ( insert_int @ A2 @ bot_bot_set_int ) ) @ ( image_int_int @ R2 @ ( insert_int @ B @ bot_bot_set_int ) ) ) )
% 0.34/0.55 => ( member5262025264175285858nt_int @ ( product_Pair_int_int @ A2 @ B ) @ R2 ) ) ) ).
% 0.34/0.55
% 0.34/0.55 % equiv_class_nondisjoint
% 0.34/0.55 thf(fact_1658_equiv__class__nondisjoint,axiom,
% 0.34/0.55 ! [A: set_Pr4532377907799695533_nat_o,R2: set_Pr2161125870931222855_nat_o,X: produc3658429121746597890et_nat > $o,A2: produc3658429121746597890et_nat > $o,B: produc3658429121746597890et_nat > $o] :
% 0.34/0.55 ( ( equiv_6726761681920839555_nat_o @ A @ R2 )
% 0.34/0.55 => ( ( member6576561426505652726_nat_o @ X @ ( inf_in1906310914598751387_nat_o @ ( image_2232984160439939859_nat_o @ R2 @ ( insert5175938949040314269_nat_o @ A2 @ bot_bo7824918357723582233_nat_o ) ) @ ( image_2232984160439939859_nat_o @ R2 @ ( insert5175938949040314269_nat_o @ B @ bot_bo7824918357723582233_nat_o ) ) ) )
% 0.34/0.55 => ( member8781333585448626064_nat_o @ ( produc7368190662567826135_nat_o @ A2 @ B ) @ R2 ) ) ) ).
% 0.34/0.55
% 0.34/0.55 % equiv_class_nondisjoint
% 0.34/0.55 thf(fact_1659_finite__reachable__advance,axiom,
% 0.34/0.55 ! [E: set_Pr4811707699266497531nteger,V0: code_integer,V: code_integer] :
% 0.34/0.55 ( ( finite6017078050557962740nteger @ ( image_4147621952751189391nteger @ ( transi3984306202085910426nteger @ E ) @ ( insert_Code_integer @ V0 @ bot_bo3990330152332043303nteger ) ) )
% 0.34/0.55 => ( ( member157494554546826820nteger @ ( produc1086072967326762835nteger @ V0 @ V ) @ ( transi3984306202085910426nteger @ E ) )
% 0.34/0.55 => ( finite6017078050557962740nteger @ ( image_4147621952751189391nteger @ ( transi3984306202085910426nteger @ E ) @ ( insert_Code_integer @ V @ bot_bo3990330152332043303nteger ) ) ) ) ) ).
% 0.34/0.55
% 0.34/0.55 % finite_reachable_advance
% 0.34/0.55 thf(fact_1660_finite__reachable__advance,axiom,
% 0.34/0.55 ! [E: set_Pr5094982260447487303t_unit,V0: product_unit,V: product_unit] :
% 0.34/0.55 ( ( finite4290736615968046902t_unit @ ( image_8431784415008483091t_unit @ ( transi6481887417741755664t_unit @ E ) @ ( insert_Product_unit @ V0 @ bot_bo3957492148770167129t_unit ) ) )
% 0.34/0.55 => ( ( member7821138191923116944t_unit @ ( produc5692694829384537815t_unit @ V0 @ V ) @ ( transi6481887417741755664t_unit @ E ) )
% 0.34/0.55 => ( finite4290736615968046902t_unit @ ( image_8431784415008483091t_unit @ ( transi6481887417741755664t_unit @ E ) @ ( insert_Product_unit @ V @ bot_bo3957492148770167129t_unit ) ) ) ) ) ).
% 0.34/0.55
% 0.34/0.55 % finite_reachable_advance
% 0.34/0.55 thf(fact_1661_finite__reachable__advance,axiom,
% 0.34/0.55 ! [E: set_Product_prod_o_o,V0: $o,V: $o] :
% 0.34/0.55 ( ( finite_finite_o @ ( image_o_o @ ( transitive_rtrancl_o @ E ) @ ( insert_o @ V0 @ bot_bot_set_o ) ) )
% 0.34/0.55 => ( ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ V0 @ V ) @ ( transitive_rtrancl_o @ E ) )
% 0.34/0.55 => ( finite_finite_o @ ( image_o_o @ ( transitive_rtrancl_o @ E ) @ ( insert_o @ V @ bot_bot_set_o ) ) ) ) ) ).
% 0.34/0.55
% 0.34/0.55 % finite_reachable_advance
% 0.34/0.55 thf(fact_1662_finite__reachable__advance,axiom,
% 0.34/0.55 ! [E: set_Pr1261947904930325089at_nat,V0: nat,V: nat] :
% 0.34/0.55 ( ( finite_finite_nat @ ( image_nat_nat @ ( transi2905341329935302413cl_nat @ E ) @ ( insert_nat @ V0 @ bot_bot_set_nat ) ) )
% 0.34/0.55 => ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ V0 @ V ) @ ( transi2905341329935302413cl_nat @ E ) )
% 0.34/0.55 => ( finite_finite_nat @ ( image_nat_nat @ ( transi2905341329935302413cl_nat @ E ) @ ( insert_nat @ V @ bot_bot_set_nat ) ) ) ) ) ).
% 0.34/0.55
% 0.34/0.55 % finite_reachable_advance
% 0.34/0.55 thf(fact_1663_finite__reachable__advance,axiom,
% 0.34/0.55 ! [E: set_Pr958786334691620121nt_int,V0: int,V: int] :
% 0.34/0.55 ( ( finite_finite_int @ ( image_int_int @ ( transi2902850859426252137cl_int @ E ) @ ( insert_int @ V0 @ bot_bot_set_int ) ) )
% 0.34/0.55 => ( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ V0 @ V ) @ ( transi2902850859426252137cl_int @ E ) )
% 0.34/0.55 => ( finite_finite_int @ ( image_int_int @ ( transi2902850859426252137cl_int @ E ) @ ( insert_int @ V @ bot_bot_set_int ) ) ) ) ) ).
% 0.34/0.55
% 0.34/0.55 % finite_reachable_advance
% 0.34/0.55 thf(fact_1664_finite__reachable__advance,axiom,
% 0.34/0.55 ! [E: set_Pr2161125870931222855_nat_o,V0: produc3658429121746597890et_nat > $o,V: produc3658429121746597890et_nat > $o] :
% 0.34/0.55 ( ( finite3252695134891459830_nat_o @ ( image_2232984160439939859_nat_o @ ( transi712980445759334096_nat_o @ E ) @ ( insert5175938949040314269_nat_o @ V0 @ bot_bo7824918357723582233_nat_o ) ) )
% 0.34/0.55 => ( ( member8781333585448626064_nat_o @ ( produc7368190662567826135_nat_o @ V0 @ V ) @ ( transi712980445759334096_nat_o @ E ) )
% 0.34/0.55 => ( finite3252695134891459830_nat_o @ ( image_2232984160439939859_nat_o @ ( transi712980445759334096_nat_o @ E ) @ ( insert5175938949040314269_nat_o @ V @ bot_bo7824918357723582233_nat_o ) ) ) ) ) ).
% 0.34/0.55
% 0.34/0.55 % finite_reachable_advance
% 0.34/0.55 thf(fact_1665_finite__Field__eq__finite,axiom,
% 0.34/0.55 ! [R3: set_Pr1261947904930325089at_nat] :
% 0.34/0.55 ( ( finite_finite_nat @ ( field_nat @ R3 ) )
% 0.34/0.55 = ( finite6177210948735845034at_nat @ R3 ) ) ).
% 0.34/0.55
% 0.34/0.55 % finite_Field_eq_finite
% 0.34/0.55 thf(fact_1666_finite__Field__eq__finite,axiom,
% 0.34/0.55 ! [R3: set_Pr4811707699266497531nteger] :
% 0.34/0.55 ( ( finite6017078050557962740nteger @ ( field_Code_integer @ R3 ) )
% 0.34/0.55 = ( finite2154694888287295812nteger @ R3 ) ) ).
% 0.34/0.55
% 0.34/0.55 % finite_Field_eq_finite
% 0.34/0.55 thf(fact_1667_Field__empty,axiom,
% 0.34/0.55 ( ( field_Product_unit @ bot_bo1654022762305692595t_unit )
% 0.34/0.55 = bot_bo3957492148770167129t_unit ) ).
% 0.34/0.55
% 0.34/0.55 % Field_empty
% 0.34/0.55 thf(fact_1668_Field__empty,axiom,
% 0.34/0.55 ( ( field_o @ bot_bo7073875226086086771od_o_o )
% 0.34/0.55 = bot_bot_set_o ) ).
% 0.34/0.55
% 0.34/0.55 % Field_empty
% 0.34/0.55 thf(fact_1669_Field__empty,axiom,
% 0.34/0.55 ( ( field_nat @ bot_bo2099793752762293965at_nat )
% 0.34/0.55 = bot_bot_set_nat ) ).
% 0.34/0.55
% 0.34/0.55 % Field_empty
% 0.34/0.55 thf(fact_1670_Field__empty,axiom,
% 0.34/0.55 ( ( field_int @ bot_bo1796632182523588997nt_int )
% 0.34/0.55 = bot_bot_set_int ) ).
% 0.34/0.55
% 0.34/0.55 % Field_empty
% 0.34/0.55 thf(fact_1671_Field__empty,axiom,
% 0.34/0.55 ( ( field_81821745747897882_nat_o @ bot_bo3657187498359500723_nat_o )
% 0.34/0.55 = bot_bo7824918357723582233_nat_o ) ).
% 0.34/0.55
% 0.34/0.55 % Field_empty
% 0.34/0.55 thf(fact_1672_Field__Un,axiom,
% 0.34/0.55 ! [R2: set_Pr1261947904930325089at_nat,S2: set_Pr1261947904930325089at_nat] :
% 0.34/0.55 ( ( field_nat @ ( sup_su6327502436637775413at_nat @ R2 @ S2 ) )
% 0.34/0.55 = ( sup_sup_set_nat @ ( field_nat @ R2 ) @ ( field_nat @ S2 ) ) ) ).
% 0.34/0.55
% 0.34/0.55 % Field_Un
% 0.34/0.55 thf(fact_1673_finite__Field,axiom,
% 0.34/0.55 ! [R2: set_Pr1261947904930325089at_nat] :
% 0.34/0.55 ( ( finite6177210948735845034at_nat @ R2 )
% 0.34/0.55 => ( finite_finite_nat @ ( field_nat @ R2 ) ) ) ).
% 0.34/0.55
% 0.34/0.55 % finite_Field
% 0.34/0.55 thf(fact_1674_finite__Field,axiom,
% 0.34/0.55 ! [R2: set_Pr4811707699266497531nteger] :
% 0.34/0.55 ( ( finite2154694888287295812nteger @ R2 )
% 0.34/0.55 => ( finite6017078050557962740nteger @ ( field_Code_integer @ R2 ) ) ) ).
% 0.34/0.55
% 0.34/0.55 % finite_Field
% 0.34/0.55 thf(fact_1675_relcompEpair,axiom,
% 0.34/0.55 ! [A2: heap_e7401611519738050253t_unit,C: set_nat,R2: set_Pr1776176089890104647t_unit,S2: set_Pr3948176798113811640et_nat] :
% 0.34/0.55 ( ( member6260224972018164377et_nat @ ( produc7507926704131184380et_nat @ A2 @ C ) @ ( relcom4930750947919833671et_nat @ R2 @ S2 ) )
% 0.34/0.55 => ~ ! [B2: heap_e7401611519738050253t_unit] :
% 0.34/0.55 ( ( member5046970461208427408t_unit @ ( produc8256076736985659095t_unit @ A2 @ B2 ) @ R2 )
% 0.34/0.55 => ~ ( member6260224972018164377et_nat @ ( produc7507926704131184380et_nat @ B2 @ C ) @ S2 ) ) ) ).
% 0.34/0.55
% 0.34/0.55 % relcompEpair
% 0.34/0.55 thf(fact_1676_relcompEpair,axiom,
% 0.34/0.55 ! [A2: heap_e7401611519738050253t_unit,C: set_nat,R2: set_Pr3948176798113811640et_nat,S2: set_Pr5488025237498180813et_nat] :
% 0.34/0.55 ( ( member6260224972018164377et_nat @ ( produc7507926704131184380et_nat @ A2 @ C ) @ ( relcom8881708762719574682et_nat @ R2 @ S2 ) )
% 0.34/0.55 => ~ ! [B2: set_nat] :
% 0.34/0.55 ( ( member6260224972018164377et_nat @ ( produc7507926704131184380et_nat @ A2 @ B2 ) @ R2 )
% 0.34/0.55 => ~ ( member8277197624267554838et_nat @ ( produc4532415448927165861et_nat @ B2 @ C ) @ S2 ) ) ) ).
% 0.34/0.55
% 0.34/0.55 % relcompEpair
% 0.34/0.55 thf(fact_1677_relcompEpair,axiom,
% 0.34/0.55 ! [A2: nat,C: nat,R2: set_Pr1261947904930325089at_nat,S2: set_Pr1261947904930325089at_nat] :
% 0.34/0.55 ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A2 @ C ) @ ( relcomp_nat_nat_nat @ R2 @ S2 ) )
% 0.34/0.55 => ~ ! [B2: nat] :
% 0.34/0.55 ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A2 @ B2 ) @ R2 )
% 0.34/0.55 => ~ ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ B2 @ C ) @ S2 ) ) ) ).
% 0.34/0.55
% 0.34/0.55 % relcompEpair
% 0.34/0.55 thf(fact_1678_relcompEpair,axiom,
% 0.34/0.55 ! [A2: int,C: int,R2: set_Pr958786334691620121nt_int,S2: set_Pr958786334691620121nt_int] :
% 0.34/0.55 ( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ A2 @ C ) @ ( relcomp_int_int_int @ R2 @ S2 ) )
% 0.34/0.55 => ~ ! [B2: int] :
% 0.34/0.55 ( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ A2 @ B2 ) @ R2 )
% 0.34/0.55 => ~ ( member5262025264175285858nt_int @ ( product_Pair_int_int @ B2 @ C ) @ S2 ) ) ) ).
% 0.34/0.55
% 0.34/0.55 % relcompEpair
% 0.34/0.55 thf(fact_1679_relcompEpair,axiom,
% 0.34/0.55 ! [A2: produc3658429121746597890et_nat > $o,C: produc3658429121746597890et_nat,R2: set_Pr2161125870931222855_nat_o,S2: set_Pr3286484037609594932et_nat] :
% 0.34/0.55 ( ( member1996754912294343701et_nat @ ( produc5001842942810119800et_nat @ A2 @ C ) @ ( relcom6619595054755144899et_nat @ R2 @ S2 ) )
% 0.34/0.55 => ~ ! [B2: produc3658429121746597890et_nat > $o] :
% 0.34/0.55 ( ( member8781333585448626064_nat_o @ ( produc7368190662567826135_nat_o @ A2 @ B2 ) @ R2 )
% 0.34/0.55 => ~ ( member1996754912294343701et_nat @ ( produc5001842942810119800et_nat @ B2 @ C ) @ S2 ) ) ) ).
% 0.34/0.55
% 0.34/0.55 % relcompEpair
% 0.34/0.55 thf(fact_1680_relcompEpair,axiom,
% 0.34/0.55 ! [A2: produc3658429121746597890et_nat > $o,C: produc3658429121746597890et_nat,R2: set_Pr3286484037609594932et_nat,S2: set_Pr719794911490849221et_nat] :
% 0.34/0.55 ( ( member1996754912294343701et_nat @ ( produc5001842942810119800et_nat @ A2 @ C ) @ ( relcom5975814107203797650et_nat @ R2 @ S2 ) )
% 0.34/0.55 => ~ ! [B2: produc3658429121746597890et_nat] :
% 0.34/0.55 ( ( member1996754912294343701et_nat @ ( produc5001842942810119800et_nat @ A2 @ B2 ) @ R2 )
% 0.34/0.55 => ~ ( member6099555550032318734et_nat @ ( produc8199053930788261021et_nat @ B2 @ C ) @ S2 ) ) ) ).
% 0.34/0.55
% 0.34/0.55 % relcompEpair
% 0.34/0.55 thf(fact_1681_relcompEpair,axiom,
% 0.34/0.55 ! [A2: produc3658429121746597890et_nat > $o,C: produc3658429121746597890et_nat,R2: set_Pr8536935166611901872et_nat,S2: set_Pr3721596426557623617et_nat] :
% 0.34/0.55 ( ( member1996754912294343701et_nat @ ( produc5001842942810119800et_nat @ A2 @ C ) @ ( relcom430014394250001422et_nat @ R2 @ S2 ) )
% 0.34/0.55 => ~ ! [B2: produc3925858234332021118et_nat] :
% 0.34/0.55 ( ( member6124377750444531601et_nat @ ( produc2245416461498447860et_nat @ A2 @ B2 ) @ R2 )
% 0.34/0.55 => ~ ( member6613709072892554et_nat @ ( produc5530789758353788441et_nat @ B2 @ C ) @ S2 ) ) ) ).
% 0.34/0.55
% 0.34/0.55 % relcompEpair
% 0.34/0.55 thf(fact_1682_relcompEpair,axiom,
% 0.34/0.55 ! [A2: produc3658429121746597890et_nat > $o,C: produc3925858234332021118et_nat,R2: set_Pr2161125870931222855_nat_o,S2: set_Pr8536935166611901872et_nat] :
% 0.34/0.55 ( ( member6124377750444531601et_nat @ ( produc2245416461498447860et_nat @ A2 @ C ) @ ( relcom2608444778321954367et_nat @ R2 @ S2 ) )
% 0.34/0.55 => ~ ! [B2: produc3658429121746597890et_nat > $o] :
% 0.34/0.55 ( ( member8781333585448626064_nat_o @ ( produc7368190662567826135_nat_o @ A2 @ B2 ) @ R2 )
% 0.34/0.55 => ~ ( member6124377750444531601et_nat @ ( produc2245416461498447860et_nat @ B2 @ C ) @ S2 ) ) ) ).
% 0.34/0.55
% 0.34/0.55 % relcompEpair
% 0.34/0.55 thf(fact_1683_relcompEpair,axiom,
% 0.34/0.55 ! [A2: produc3658429121746597890et_nat > $o,C: produc3925858234332021118et_nat,R2: set_Pr3286484037609594932et_nat,S2: set_Pr8807100098903490369et_nat] :
% 0.34/0.55 ( ( member6124377750444531601et_nat @ ( produc2245416461498447860et_nat @ A2 @ C ) @ ( relcom8031709186682293262et_nat @ R2 @ S2 ) )
% 0.34/0.55 => ~ ! [B2: produc3658429121746597890et_nat] :
% 0.34/0.55 ( ( member1996754912294343701et_nat @ ( produc5001842942810119800et_nat @ A2 @ B2 ) @ R2 )
% 0.34/0.55 => ~ ( member2737525317995836554et_nat @ ( produc3909112513931304473et_nat @ B2 @ C ) @ S2 ) ) ) ).
% 0.34/0.55
% 0.34/0.55 % relcompEpair
% 0.34/0.55 thf(fact_1684_relcompEpair,axiom,
% 0.34/0.55 ! [A2: produc3658429121746597890et_nat > $o,C: produc3925858234332021118et_nat,R2: set_Pr8536935166611901872et_nat,S2: set_Pr7928877670098842301et_nat] :
% 0.34/0.55 ( ( member6124377750444531601et_nat @ ( produc2245416461498447860et_nat @ A2 @ C ) @ ( relcom8303140135939483018et_nat @ R2 @ S2 ) )
% 0.34/0.55 => ~ ! [B2: produc3925858234332021118et_nat] :
% 0.34/0.55 ( ( member6124377750444531601et_nat @ ( produc2245416461498447860et_nat @ A2 @ B2 ) @ R2 )
% 0.34/0.55 => ~ ( member4763271486408492550et_nat @ ( produc8599840265553166229et_nat @ B2 @ C ) @ S2 ) ) ) ).
% 0.34/0.55
% 0.34/0.55 % relcompEpair
% 0.34/0.55 thf(fact_1685_relcompE,axiom,
% 0.34/0.55 ! [Xz: produc3658429121746597890et_nat,R2: set_Pr1776176089890104647t_unit,S2: set_Pr3948176798113811640et_nat] :
% 0.34/0.55 ( ( member6260224972018164377et_nat @ Xz @ ( relcom4930750947919833671et_nat @ R2 @ S2 ) )
% 0.34/0.55 => ~ ! [X3: heap_e7401611519738050253t_unit,Y2: heap_e7401611519738050253t_unit,Z3: set_nat] :
% 0.34/0.55 ( ( Xz
% 0.34/0.55 = ( produc7507926704131184380et_nat @ X3 @ Z3 ) )
% 0.34/0.55 => ( ( member5046970461208427408t_unit @ ( produc8256076736985659095t_unit @ X3 @ Y2 ) @ R2 )
% 0.34/0.55 => ~ ( member6260224972018164377et_nat @ ( produc7507926704131184380et_nat @ Y2 @ Z3 ) @ S2 ) ) ) ) ).
% 0.34/0.55
% 0.34/0.55 % relcompE
% 0.34/0.55 thf(fact_1686_relcompE,axiom,
% 0.34/0.55 ! [Xz: produc3658429121746597890et_nat,R2: set_Pr3948176798113811640et_nat,S2: set_Pr5488025237498180813et_nat] :
% 0.34/0.55 ( ( member6260224972018164377et_nat @ Xz @ ( relcom8881708762719574682et_nat @ R2 @ S2 ) )
% 0.34/0.55 => ~ ! [X3: heap_e7401611519738050253t_unit,Y2: set_nat,Z3: set_nat] :
% 0.34/0.55 ( ( Xz
% 0.34/0.55 = ( produc7507926704131184380et_nat @ X3 @ Z3 ) )
% 0.34/0.55 => ( ( member6260224972018164377et_nat @ ( produc7507926704131184380et_nat @ X3 @ Y2 ) @ R2 )
% 0.34/0.55 => ~ ( member8277197624267554838et_nat @ ( produc4532415448927165861et_nat @ Y2 @ Z3 ) @ S2 ) ) ) ) ).
% 0.34/0.55
% 0.34/0.55 % relcompE
% 0.34/0.55 thf(fact_1687_relcompE,axiom,
% 0.34/0.55 ! [Xz: product_prod_nat_nat,R2: set_Pr1261947904930325089at_nat,S2: set_Pr1261947904930325089at_nat] :
% 0.34/0.55 ( ( member8440522571783428010at_nat @ Xz @ ( relcomp_nat_nat_nat @ R2 @ S2 ) )
% 0.34/0.55 => ~ ! [X3: nat,Y2: nat,Z3: nat] :
% 0.34/0.55 ( ( Xz
% 0.34/0.55 = ( product_Pair_nat_nat @ X3 @ Z3 ) )
% 0.34/0.55 => ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X3 @ Y2 ) @ R2 )
% 0.34/0.55 => ~ ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ Y2 @ Z3 ) @ S2 ) ) ) ) ).
% 0.34/0.55
% 0.34/0.55 % relcompE
% 0.34/0.55 thf(fact_1688_relcompE,axiom,
% 0.34/0.55 ! [Xz: product_prod_int_int,R2: set_Pr958786334691620121nt_int,S2: set_Pr958786334691620121nt_int] :
% 0.34/0.55 ( ( member5262025264175285858nt_int @ Xz @ ( relcomp_int_int_int @ R2 @ S2 ) )
% 0.34/0.55 => ~ ! [X3: int,Y2: int,Z3: int] :
% 0.34/0.55 ( ( Xz
% 0.34/0.55 = ( product_Pair_int_int @ X3 @ Z3 ) )
% 0.34/0.55 => ( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ X3 @ Y2 ) @ R2 )
% 0.34/0.55 => ~ ( member5262025264175285858nt_int @ ( product_Pair_int_int @ Y2 @ Z3 ) @ S2 ) ) ) ) ).
% 0.34/0.55
% 0.34/0.55 % relcompE
% 0.34/0.55 thf(fact_1689_relcompE,axiom,
% 0.34/0.55 ! [Xz: produc3925858234332021118et_nat,R2: set_Pr2161125870931222855_nat_o,S2: set_Pr3286484037609594932et_nat] :
% 0.34/0.55 ( ( member1996754912294343701et_nat @ Xz @ ( relcom6619595054755144899et_nat @ R2 @ S2 ) )
% 0.34/0.55 => ~ ! [X3: produc3658429121746597890et_nat > $o,Y2: produc3658429121746597890et_nat > $o,Z3: produc3658429121746597890et_nat] :
% 0.34/0.55 ( ( Xz
% 0.34/0.55 = ( produc5001842942810119800et_nat @ X3 @ Z3 ) )
% 0.34/0.55 => ( ( member8781333585448626064_nat_o @ ( produc7368190662567826135_nat_o @ X3 @ Y2 ) @ R2 )
% 0.34/0.55 => ~ ( member1996754912294343701et_nat @ ( produc5001842942810119800et_nat @ Y2 @ Z3 ) @ S2 ) ) ) ) ).
% 0.34/0.55
% 0.34/0.55 % relcompE
% 0.34/0.55 thf(fact_1690_relcompE,axiom,
% 0.34/0.55 ! [Xz: produc3925858234332021118et_nat,R2: set_Pr3286484037609594932et_nat,S2: set_Pr719794911490849221et_nat] :
% 0.34/0.55 ( ( member1996754912294343701et_nat @ Xz @ ( relcom5975814107203797650et_nat @ R2 @ S2 ) )
% 0.34/0.55 => ~ ! [X3: produc3658429121746597890et_nat > $o,Y2: produc3658429121746597890et_nat,Z3: produc3658429121746597890et_nat] :
% 0.34/0.55 ( ( Xz
% 0.34/0.55 = ( produc5001842942810119800et_nat @ X3 @ Z3 ) )
% 0.34/0.55 => ( ( member1996754912294343701et_nat @ ( produc5001842942810119800et_nat @ X3 @ Y2 ) @ R2 )
% 0.34/0.55 => ~ ( member6099555550032318734et_nat @ ( produc8199053930788261021et_nat @ Y2 @ Z3 ) @ S2 ) ) ) ) ).
% 0.34/0.55
% 0.34/0.55 % relcompE
% 0.34/0.55 thf(fact_1691_relcompE,axiom,
% 0.34/0.55 ! [Xz: produc3925858234332021118et_nat,R2: set_Pr8536935166611901872et_nat,S2: set_Pr3721596426557623617et_nat] :
% 0.34/0.55 ( ( member1996754912294343701et_nat @ Xz @ ( relcom430014394250001422et_nat @ R2 @ S2 ) )
% 0.34/0.55 => ~ ! [X3: produc3658429121746597890et_nat > $o,Y2: produc3925858234332021118et_nat,Z3: produc3658429121746597890et_nat] :
% 0.34/0.55 ( ( Xz
% 0.34/0.55 = ( produc5001842942810119800et_nat @ X3 @ Z3 ) )
% 0.34/0.55 => ( ( member6124377750444531601et_nat @ ( produc2245416461498447860et_nat @ X3 @ Y2 ) @ R2 )
% 0.34/0.55 => ~ ( member6613709072892554et_nat @ ( produc5530789758353788441et_nat @ Y2 @ Z3 ) @ S2 ) ) ) ) ).
% 0.34/0.55
% 0.34/0.55 % relcompE
% 0.34/0.55 thf(fact_1692_relcompE,axiom,
% 0.34/0.55 ! [Xz: produc2732055786443039994et_nat,R2: set_Pr2161125870931222855_nat_o,S2: set_Pr8536935166611901872et_nat] :
% 0.34/0.55 ( ( member6124377750444531601et_nat @ Xz @ ( relcom2608444778321954367et_nat @ R2 @ S2 ) )
% 0.34/0.55 => ~ ! [X3: produc3658429121746597890et_nat > $o,Y2: produc3658429121746597890et_nat > $o,Z3: produc3925858234332021118et_nat] :
% 0.34/0.55 ( ( Xz
% 0.34/0.55 = ( produc2245416461498447860et_nat @ X3 @ Z3 ) )
% 0.34/0.55 => ( ( member8781333585448626064_nat_o @ ( produc7368190662567826135_nat_o @ X3 @ Y2 ) @ R2 )
% 0.34/0.55 => ~ ( member6124377750444531601et_nat @ ( produc2245416461498447860et_nat @ Y2 @ Z3 ) @ S2 ) ) ) ) ).
% 0.34/0.55
% 0.34/0.55 % relcompE
% 0.34/0.55 thf(fact_1693_relcompE,axiom,
% 0.34/0.55 ! [Xz: produc2732055786443039994et_nat,R2: set_Pr3286484037609594932et_nat,S2: set_Pr8807100098903490369et_nat] :
% 0.34/0.55 ( ( member6124377750444531601et_nat @ Xz @ ( relcom8031709186682293262et_nat @ R2 @ S2 ) )
% 0.34/0.55 => ~ ! [X3: produc3658429121746597890et_nat > $o,Y2: produc3658429121746597890et_nat,Z3: produc3925858234332021118et_nat] :
% 0.34/0.55 ( ( Xz
% 0.34/0.55 = ( produc2245416461498447860et_nat @ X3 @ Z3 ) )
% 0.34/0.55 => ( ( member1996754912294343701et_nat @ ( produc5001842942810119800et_nat @ X3 @ Y2 ) @ R2 )
% 0.34/0.55 => ~ ( member2737525317995836554et_nat @ ( produc3909112513931304473et_nat @ Y2 @ Z3 ) @ S2 ) ) ) ) ).
% 0.34/0.55
% 0.34/0.55 % relcompE
% 0.34/0.55 thf(fact_1694_relcompE,axiom,
% 0.34/0.55 ! [Xz: produc2732055786443039994et_nat,R2: set_Pr8536935166611901872et_nat,S2: set_Pr7928877670098842301et_nat] :
% 0.34/0.55 ( ( member6124377750444531601et_nat @ Xz @ ( relcom8303140135939483018et_nat @ R2 @ S2 ) )
% 0.34/0.55 => ~ ! [X3: produc3658429121746597890et_nat > $o,Y2: produc3925858234332021118et_nat,Z3: produc3925858234332021118et_nat] :
% 0.34/0.55 ( ( Xz
% 0.34/0.55 = ( produc2245416461498447860et_nat @ X3 @ Z3 ) )
% 0.34/0.55 => ( ( member6124377750444531601et_nat @ ( produc2245416461498447860et_nat @ X3 @ Y2 ) @ R2 )
% 0.34/0.55 => ~ ( member4763271486408492550et_nat @ ( produc8599840265553166229et_nat @ Y2 @ Z3 ) @ S2 ) ) ) ) ).
% 0.34/0.55
% 0.34/0.55 % relcompE
% 0.34/0.55 thf(fact_1695_relcomp_OrelcompI,axiom,
% 0.34/0.55 ! [A2: heap_e7401611519738050253t_unit,B: heap_e7401611519738050253t_unit,R2: set_Pr1776176089890104647t_unit,C: set_nat,S2: set_Pr3948176798113811640et_nat] :
% 0.34/0.55 ( ( member5046970461208427408t_unit @ ( produc8256076736985659095t_unit @ A2 @ B ) @ R2 )
% 0.34/0.55 => ( ( member6260224972018164377et_nat @ ( produc7507926704131184380et_nat @ B @ C ) @ S2 )
% 0.34/0.55 => ( member6260224972018164377et_nat @ ( produc7507926704131184380et_nat @ A2 @ C ) @ ( relcom4930750947919833671et_nat @ R2 @ S2 ) ) ) ) ).
% 0.34/0.55
% 0.34/0.55 % relcomp.relcompI
% 0.34/0.55 thf(fact_1696_relcomp_OrelcompI,axiom,
% 0.34/0.55 ! [A2: produc3658429121746597890et_nat > $o,B: produc3658429121746597890et_nat > $o,R2: set_Pr2161125870931222855_nat_o,C: produc3658429121746597890et_nat,S2: set_Pr3286484037609594932et_nat] :
% 0.34/0.55 ( ( member8781333585448626064_nat_o @ ( produc7368190662567826135_nat_o @ A2 @ B ) @ R2 )
% 0.34/0.55 => ( ( member1996754912294343701et_nat @ ( produc5001842942810119800et_nat @ B @ C ) @ S2 )
% 0.34/0.55 => ( member1996754912294343701et_nat @ ( produc5001842942810119800et_nat @ A2 @ C ) @ ( relcom6619595054755144899et_nat @ R2 @ S2 ) ) ) ) ).
% 0.34/0.55
% 0.34/0.55 % relcomp.relcompI
% 0.34/0.55 thf(fact_1697_relcomp_OrelcompI,axiom,
% 0.34/0.55 ! [A2: produc3658429121746597890et_nat > $o,B: produc3658429121746597890et_nat > $o,R2: set_Pr2161125870931222855_nat_o,C: produc3925858234332021118et_nat,S2: set_Pr8536935166611901872et_nat] :
% 0.34/0.55 ( ( member8781333585448626064_nat_o @ ( produc7368190662567826135_nat_o @ A2 @ B ) @ R2 )
% 0.34/0.55 => ( ( member6124377750444531601et_nat @ ( produc2245416461498447860et_nat @ B @ C ) @ S2 )
% 0.34/0.55 => ( member6124377750444531601et_nat @ ( produc2245416461498447860et_nat @ A2 @ C ) @ ( relcom2608444778321954367et_nat @ R2 @ S2 ) ) ) ) ).
% 0.34/0.55
% 0.34/0.55 % relcomp.relcompI
% 0.34/0.55 thf(fact_1698_relcomp_OrelcompI,axiom,
% 0.34/0.55 ! [A2: heap_e7401611519738050253t_unit,B: set_nat,R2: set_Pr3948176798113811640et_nat,C: set_nat,S2: set_Pr5488025237498180813et_nat] :
% 0.34/0.55 ( ( member6260224972018164377et_nat @ ( produc7507926704131184380et_nat @ A2 @ B ) @ R2 )
% 0.34/0.55 => ( ( member8277197624267554838et_nat @ ( produc4532415448927165861et_nat @ B @ C ) @ S2 )
% 0.34/0.55 => ( member6260224972018164377et_nat @ ( produc7507926704131184380et_nat @ A2 @ C ) @ ( relcom8881708762719574682et_nat @ R2 @ S2 ) ) ) ) ).
% 0.34/0.55
% 0.34/0.55 % relcomp.relcompI
% 0.34/0.55 thf(fact_1699_relcomp_OrelcompI,axiom,
% 0.34/0.55 ! [A2: nat,B: nat,R2: set_Pr1261947904930325089at_nat,C: nat,S2: set_Pr1261947904930325089at_nat] :
% 0.34/0.55 ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A2 @ B ) @ R2 )
% 0.34/0.55 => ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ B @ C ) @ S2 )
% 0.34/0.55 => ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A2 @ C ) @ ( relcomp_nat_nat_nat @ R2 @ S2 ) ) ) ) ).
% 0.34/0.55
% 0.34/0.55 % relcomp.relcompI
% 0.34/0.55 thf(fact_1700_relcomp_OrelcompI,axiom,
% 0.34/0.55 ! [A2: int,B: int,R2: set_Pr958786334691620121nt_int,C: int,S2: set_Pr958786334691620121nt_int] :
% 0.34/0.55 ( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ A2 @ B ) @ R2 )
% 0.34/0.55 => ( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ B @ C ) @ S2 )
% 0.34/0.55 => ( member5262025264175285858nt_int @ ( product_Pair_int_int @ A2 @ C ) @ ( relcomp_int_int_int @ R2 @ S2 ) ) ) ) ).
% 0.34/0.55
% 0.34/0.55 % relcomp.relcompI
% 0.34/0.55 thf(fact_1701_relcomp_OrelcompI,axiom,
% 0.34/0.55 ! [A2: produc3658429121746597890et_nat > $o,B: produc3658429121746597890et_nat,R2: set_Pr3286484037609594932et_nat,C: produc3658429121746597890et_nat,S2: set_Pr719794911490849221et_nat] :
% 0.34/0.55 ( ( member1996754912294343701et_nat @ ( produc5001842942810119800et_nat @ A2 @ B ) @ R2 )
% 0.34/0.55 => ( ( member6099555550032318734et_nat @ ( produc8199053930788261021et_nat @ B @ C ) @ S2 )
% 0.34/0.55 => ( member1996754912294343701et_nat @ ( produc5001842942810119800et_nat @ A2 @ C ) @ ( relcom5975814107203797650et_nat @ R2 @ S2 ) ) ) ) ).
% 0.34/0.55
% 0.34/0.55 % relcomp.relcompI
% 0.34/0.55 thf(fact_1702_relcomp_OrelcompI,axiom,
% 0.34/0.55 ! [A2: produc3658429121746597890et_nat > $o,B: produc3658429121746597890et_nat,R2: set_Pr3286484037609594932et_nat,C: produc3925858234332021118et_nat,S2: set_Pr8807100098903490369et_nat] :
% 0.34/0.55 ( ( member1996754912294343701et_nat @ ( produc5001842942810119800et_nat @ A2 @ B ) @ R2 )
% 0.34/0.55 => ( ( member2737525317995836554et_nat @ ( produc3909112513931304473et_nat @ B @ C ) @ S2 )
% 0.34/0.55 => ( member6124377750444531601et_nat @ ( produc2245416461498447860et_nat @ A2 @ C ) @ ( relcom8031709186682293262et_nat @ R2 @ S2 ) ) ) ) ).
% 0.34/0.55
% 0.34/0.55 % relcomp.relcompI
% 0.34/0.55 thf(fact_1703_relcomp_OrelcompI,axiom,
% 0.34/0.55 ! [A2: produc3658429121746597890et_nat > $o,B: produc3925858234332021118et_nat,R2: set_Pr8536935166611901872et_nat,C: produc3658429121746597890et_nat,S2: set_Pr3721596426557623617et_nat] :
% 0.34/0.55 ( ( member6124377750444531601et_nat @ ( produc2245416461498447860et_nat @ A2 @ B ) @ R2 )
% 0.34/0.55 => ( ( member6613709072892554et_nat @ ( produc5530789758353788441et_nat @ B @ C ) @ S2 )
% 0.34/0.55 => ( member1996754912294343701et_nat @ ( produc5001842942810119800et_nat @ A2 @ C ) @ ( relcom430014394250001422et_nat @ R2 @ S2 ) ) ) ) ).
% 0.34/0.55
% 0.34/0.55 % relcomp.relcompI
% 0.34/0.55 thf(fact_1704_relcomp_OrelcompI,axiom,
% 0.34/0.55 ! [A2: produc3658429121746597890et_nat > $o,B: produc3925858234332021118et_nat,R2: set_Pr8536935166611901872et_nat,C: produc3925858234332021118et_nat,S2: set_Pr7928877670098842301et_nat] :
% 0.34/0.55 ( ( member6124377750444531601et_nat @ ( produc2245416461498447860et_nat @ A2 @ B ) @ R2 )
% 0.34/0.55 => ( ( member4763271486408492550et_nat @ ( produc8599840265553166229et_nat @ B @ C ) @ S2 )
% 0.34/0.55 => ( member6124377750444531601et_nat @ ( produc2245416461498447860et_nat @ A2 @ C ) @ ( relcom8303140135939483018et_nat @ R2 @ S2 ) ) ) ) ).
% 0.34/0.55
% 0.34/0.55 % relcomp.relcompI
% 0.34/0.55 thf(fact_1705_relcomp_Osimps,axiom,
% 0.34/0.55 ! [A1: heap_e7401611519738050253t_unit,A22: set_nat,R2: set_Pr1776176089890104647t_unit,S2: set_Pr3948176798113811640et_nat] :
% 0.34/0.55 ( ( member6260224972018164377et_nat @ ( produc7507926704131184380et_nat @ A1 @ A22 ) @ ( relcom4930750947919833671et_nat @ R2 @ S2 ) )
% 0.34/0.55 = ( ? [A6: heap_e7401611519738050253t_unit,B6: heap_e7401611519738050253t_unit,C4: set_nat] :
% 0.34/0.55 ( ( A1 = A6 )
% 0.34/0.55 & ( A22 = C4 )
% 0.34/0.55 & ( member5046970461208427408t_unit @ ( produc8256076736985659095t_unit @ A6 @ B6 ) @ R2 )
% 0.34/0.55 & ( member6260224972018164377et_nat @ ( produc7507926704131184380et_nat @ B6 @ C4 ) @ S2 ) ) ) ) ).
% 0.34/0.55
% 0.34/0.55 % relcomp.simps
% 0.34/0.55 thf(fact_1706_relcomp_Osimps,axiom,
% 0.34/0.55 ! [A1: heap_e7401611519738050253t_unit,A22: set_nat,R2: set_Pr3948176798113811640et_nat,S2: set_Pr5488025237498180813et_nat] :
% 0.34/0.55 ( ( member6260224972018164377et_nat @ ( produc7507926704131184380et_nat @ A1 @ A22 ) @ ( relcom8881708762719574682et_nat @ R2 @ S2 ) )
% 0.34/0.55 = ( ? [A6: heap_e7401611519738050253t_unit,B6: set_nat,C4: set_nat] :
% 0.34/0.55 ( ( A1 = A6 )
% 0.34/0.55 & ( A22 = C4 )
% 0.34/0.55 & ( member6260224972018164377et_nat @ ( produc7507926704131184380et_nat @ A6 @ B6 ) @ R2 )
% 0.34/0.55 & ( member8277197624267554838et_nat @ ( produc4532415448927165861et_nat @ B6 @ C4 ) @ S2 ) ) ) ) ).
% 0.34/0.55
% 0.34/0.55 % relcomp.simps
% 0.34/0.55 thf(fact_1707_relcomp_Osimps,axiom,
% 0.34/0.55 ! [A1: nat,A22: nat,R2: set_Pr1261947904930325089at_nat,S2: set_Pr1261947904930325089at_nat] :
% 0.34/0.55 ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A1 @ A22 ) @ ( relcomp_nat_nat_nat @ R2 @ S2 ) )
% 0.34/0.55 = ( ? [A6: nat,B6: nat,C4: nat] :
% 0.34/0.55 ( ( A1 = A6 )
% 0.34/0.55 & ( A22 = C4 )
% 0.34/0.55 & ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A6 @ B6 ) @ R2 )
% 0.34/0.55 & ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ B6 @ C4 ) @ S2 ) ) ) ) ).
% 0.34/0.55
% 0.34/0.55 % relcomp.simps
% 0.34/0.55 thf(fact_1708_relcomp_Osimps,axiom,
% 0.34/0.55 ! [A1: int,A22: int,R2: set_Pr958786334691620121nt_int,S2: set_Pr958786334691620121nt_int] :
% 0.34/0.55 ( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ A1 @ A22 ) @ ( relcomp_int_int_int @ R2 @ S2 ) )
% 0.34/0.55 = ( ? [A6: int,B6: int,C4: int] :
% 0.34/0.55 ( ( A1 = A6 )
% 0.34/0.55 & ( A22 = C4 )
% 0.34/0.55 & ( member5262025264175285858nt_int @ ( product_Pair_int_int @ A6 @ B6 ) @ R2 )
% 0.34/0.55 & ( member5262025264175285858nt_int @ ( product_Pair_int_int @ B6 @ C4 ) @ S2 ) ) ) ) ).
% 0.34/0.55
% 0.34/0.55 % relcomp.simps
% 0.34/0.55 thf(fact_1709_relcomp_Osimps,axiom,
% 0.34/0.55 ! [A1: produc3658429121746597890et_nat > $o,A22: produc3658429121746597890et_nat,R2: set_Pr2161125870931222855_nat_o,S2: set_Pr3286484037609594932et_nat] :
% 0.34/0.55 ( ( member1996754912294343701et_nat @ ( produc5001842942810119800et_nat @ A1 @ A22 ) @ ( relcom6619595054755144899et_nat @ R2 @ S2 ) )
% 0.34/0.55 = ( ? [A6: produc3658429121746597890et_nat > $o,B6: produc3658429121746597890et_nat > $o,C4: produc3658429121746597890et_nat] :
% 0.34/0.55 ( ( A1 = A6 )
% 0.34/0.55 & ( A22 = C4 )
% 0.34/0.55 & ( member8781333585448626064_nat_o @ ( produc7368190662567826135_nat_o @ A6 @ B6 ) @ R2 )
% 0.34/0.55 & ( member1996754912294343701et_nat @ ( produc5001842942810119800et_nat @ B6 @ C4 ) @ S2 ) ) ) ) ).
% 0.34/0.55
% 0.34/0.55 % relcomp.simps
% 0.34/0.55 thf(fact_1710_relcomp_Osimps,axiom,
% 0.34/0.55 ! [A1: produc3658429121746597890et_nat > $o,A22: produc3658429121746597890et_nat,R2: set_Pr3286484037609594932et_nat,S2: set_Pr719794911490849221et_nat] :
% 0.34/0.55 ( ( member1996754912294343701et_nat @ ( produc5001842942810119800et_nat @ A1 @ A22 ) @ ( relcom5975814107203797650et_nat @ R2 @ S2 ) )
% 0.34/0.55 = ( ? [A6: produc3658429121746597890et_nat > $o,B6: produc3658429121746597890et_nat,C4: produc3658429121746597890et_nat] :
% 0.34/0.55 ( ( A1 = A6 )
% 0.34/0.55 & ( A22 = C4 )
% 0.34/0.55 & ( member1996754912294343701et_nat @ ( produc5001842942810119800et_nat @ A6 @ B6 ) @ R2 )
% 0.34/0.55 & ( member6099555550032318734et_nat @ ( produc8199053930788261021et_nat @ B6 @ C4 ) @ S2 ) ) ) ) ).
% 0.34/0.55
% 0.34/0.55 % relcomp.simps
% 0.34/0.55 thf(fact_1711_relcomp_Osimps,axiom,
% 0.34/0.55 ! [A1: produc3658429121746597890et_nat > $o,A22: produc3658429121746597890et_nat,R2: set_Pr8536935166611901872et_nat,S2: set_Pr3721596426557623617et_nat] :
% 0.34/0.55 ( ( member1996754912294343701et_nat @ ( produc5001842942810119800et_nat @ A1 @ A22 ) @ ( relcom430014394250001422et_nat @ R2 @ S2 ) )
% 0.34/0.55 = ( ? [A6: produc3658429121746597890et_nat > $o,B6: produc3925858234332021118et_nat,C4: produc3658429121746597890et_nat] :
% 0.34/0.55 ( ( A1 = A6 )
% 0.34/0.55 & ( A22 = C4 )
% 0.34/0.55 & ( member6124377750444531601et_nat @ ( produc2245416461498447860et_nat @ A6 @ B6 ) @ R2 )
% 0.34/0.55 & ( member6613709072892554et_nat @ ( produc5530789758353788441et_nat @ B6 @ C4 ) @ S2 ) ) ) ) ).
% 0.34/0.55
% 0.34/0.55 % relcomp.simps
% 0.34/0.55 thf(fact_1712_relcomp_Osimps,axiom,
% 0.34/0.55 ! [A1: produc3658429121746597890et_nat > $o,A22: produc3925858234332021118et_nat,R2: set_Pr2161125870931222855_nat_o,S2: set_Pr8536935166611901872et_nat] :
% 0.34/0.55 ( ( member6124377750444531601et_nat @ ( produc2245416461498447860et_nat @ A1 @ A22 ) @ ( relcom2608444778321954367et_nat @ R2 @ S2 ) )
% 0.34/0.55 = ( ? [A6: produc3658429121746597890et_nat > $o,B6: produc3658429121746597890et_nat > $o,C4: produc3925858234332021118et_nat] :
% 0.34/0.55 ( ( A1 = A6 )
% 0.34/0.55 & ( A22 = C4 )
% 0.34/0.55 & ( member8781333585448626064_nat_o @ ( produc7368190662567826135_nat_o @ A6 @ B6 ) @ R2 )
% 0.34/0.55 & ( member6124377750444531601et_nat @ ( produc2245416461498447860et_nat @ B6 @ C4 ) @ S2 ) ) ) ) ).
% 0.34/0.55
% 0.34/0.55 % relcomp.simps
% 0.34/0.55 thf(fact_1713_relcomp_Osimps,axiom,
% 0.34/0.55 ! [A1: produc3658429121746597890et_nat > $o,A22: produc3925858234332021118et_nat,R2: set_Pr3286484037609594932et_nat,S2: set_Pr8807100098903490369et_nat] :
% 0.34/0.55 ( ( member6124377750444531601et_nat @ ( produc2245416461498447860et_nat @ A1 @ A22 ) @ ( relcom8031709186682293262et_nat @ R2 @ S2 ) )
% 0.34/0.55 = ( ? [A6: produc3658429121746597890et_nat > $o,B6: produc3658429121746597890et_nat,C4: produc3925858234332021118et_nat] :
% 0.34/0.55 ( ( A1 = A6 )
% 0.34/0.55 & ( A22 = C4 )
% 0.34/0.55 & ( member1996754912294343701et_nat @ ( produc5001842942810119800et_nat @ A6 @ B6 ) @ R2 )
% 0.34/0.55 & ( member2737525317995836554et_nat @ ( produc3909112513931304473et_nat @ B6 @ C4 ) @ S2 ) ) ) ) ).
% 0.34/0.55
% 0.34/0.55 % relcomp.simps
% 0.34/0.55 thf(fact_1714_relcomp_Osimps,axiom,
% 0.34/0.55 ! [A1: produc3658429121746597890et_nat > $o,A22: produc3925858234332021118et_nat,R2: set_Pr8536935166611901872et_nat,S2: set_Pr7928877670098842301et_nat] :
% 0.34/0.55 ( ( member6124377750444531601et_nat @ ( produc2245416461498447860et_nat @ A1 @ A22 ) @ ( relcom8303140135939483018et_nat @ R2 @ S2 ) )
% 0.34/0.55 = ( ? [A6: produc3658429121746597890et_nat > $o,B6: produc3925858234332021118et_nat,C4: produc3925858234332021118et_nat] :
% 0.34/0.55 ( ( A1 = A6 )
% 0.34/0.55 & ( A22 = C4 )
% 0.34/0.55 & ( member6124377750444531601et_nat @ ( produc2245416461498447860et_nat @ A6 @ B6 ) @ R2 )
% 0.34/0.55 & ( member4763271486408492550et_nat @ ( produc8599840265553166229et_nat @ B6 @ C4 ) @ S2 ) ) ) ) ).
% 0.34/0.55
% 0.34/0.55 % relcomp.simps
% 0.34/0.55 thf(fact_1715_relcomp_Ocases,axiom,
% 0.34/0.55 ! [A1: heap_e7401611519738050253t_unit,A22: set_nat,R2: set_Pr1776176089890104647t_unit,S2: set_Pr3948176798113811640et_nat] :
% 0.34/0.55 ( ( member6260224972018164377et_nat @ ( produc7507926704131184380et_nat @ A1 @ A22 ) @ ( relcom4930750947919833671et_nat @ R2 @ S2 ) )
% 0.34/0.55 => ~ ! [B2: heap_e7401611519738050253t_unit] :
% 0.34/0.55 ( ( member5046970461208427408t_unit @ ( produc8256076736985659095t_unit @ A1 @ B2 ) @ R2 )
% 0.34/0.55 => ~ ( member6260224972018164377et_nat @ ( produc7507926704131184380et_nat @ B2 @ A22 ) @ S2 ) ) ) ).
% 0.34/0.55
% 0.34/0.55 % relcomp.cases
% 0.34/0.55 thf(fact_1716_relcomp_Ocases,axiom,
% 0.34/0.55 ! [A1: heap_e7401611519738050253t_unit,A22: set_nat,R2: set_Pr3948176798113811640et_nat,S2: set_Pr5488025237498180813et_nat] :
% 0.34/0.55 ( ( member6260224972018164377et_nat @ ( produc7507926704131184380et_nat @ A1 @ A22 ) @ ( relcom8881708762719574682et_nat @ R2 @ S2 ) )
% 0.34/0.55 => ~ ! [B2: set_nat] :
% 0.34/0.55 ( ( member6260224972018164377et_nat @ ( produc7507926704131184380et_nat @ A1 @ B2 ) @ R2 )
% 0.34/0.55 => ~ ( member8277197624267554838et_nat @ ( produc4532415448927165861et_nat @ B2 @ A22 ) @ S2 ) ) ) ).
% 0.34/0.55
% 0.34/0.55 % relcomp.cases
% 0.34/0.55 thf(fact_1717_relcomp_Ocases,axiom,
% 0.34/0.55 ! [A1: nat,A22: nat,R2: set_Pr1261947904930325089at_nat,S2: set_Pr1261947904930325089at_nat] :
% 0.34/0.55 ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A1 @ A22 ) @ ( relcomp_nat_nat_nat @ R2 @ S2 ) )
% 0.34/0.55 => ~ ! [B2: nat] :
% 0.34/0.55 ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A1 @ B2 ) @ R2 )
% 0.34/0.55 => ~ ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ B2 @ A22 ) @ S2 ) ) ) ).
% 0.34/0.55
% 0.34/0.55 % relcomp.cases
% 0.34/0.55 thf(fact_1718_relcomp_Ocases,axiom,
% 0.34/0.55 ! [A1: int,A22: int,R2: set_Pr958786334691620121nt_int,S2: set_Pr958786334691620121nt_int] :
% 0.34/0.55 ( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ A1 @ A22 ) @ ( relcomp_int_int_int @ R2 @ S2 ) )
% 0.34/0.55 => ~ ! [B2: int] :
% 0.34/0.55 ( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ A1 @ B2 ) @ R2 )
% 0.34/0.55 => ~ ( member5262025264175285858nt_int @ ( product_Pair_int_int @ B2 @ A22 ) @ S2 ) ) ) ).
% 0.34/0.55
% 0.34/0.55 % relcomp.cases
% 0.34/0.55 thf(fact_1719_relcomp_Ocases,axiom,
% 0.34/0.55 ! [A1: produc3658429121746597890et_nat > $o,A22: produc3658429121746597890et_nat,R2: set_Pr2161125870931222855_nat_o,S2: set_Pr3286484037609594932et_nat] :
% 0.34/0.55 ( ( member1996754912294343701et_nat @ ( produc5001842942810119800et_nat @ A1 @ A22 ) @ ( relcom6619595054755144899et_nat @ R2 @ S2 ) )
% 0.34/0.55 => ~ ! [B2: produc3658429121746597890et_nat > $o] :
% 0.34/0.55 ( ( member8781333585448626064_nat_o @ ( produc7368190662567826135_nat_o @ A1 @ B2 ) @ R2 )
% 0.34/0.55 => ~ ( member1996754912294343701et_nat @ ( produc5001842942810119800et_nat @ B2 @ A22 ) @ S2 ) ) ) ).
% 0.34/0.55
% 0.34/0.55 % relcomp.cases
% 0.34/0.55 thf(fact_1720_relcomp_Ocases,axiom,
% 0.34/0.55 ! [A1: produc3658429121746597890et_nat > $o,A22: produc3658429121746597890et_nat,R2: set_Pr3286484037609594932et_nat,S2: set_Pr719794911490849221et_nat] :
% 0.34/0.55 ( ( member1996754912294343701et_nat @ ( produc5001842942810119800et_nat @ A1 @ A22 ) @ ( relcom5975814107203797650et_nat @ R2 @ S2 ) )
% 0.34/0.55 => ~ ! [B2: produc3658429121746597890et_nat] :
% 0.34/0.55 ( ( member1996754912294343701et_nat @ ( produc5001842942810119800et_nat @ A1 @ B2 ) @ R2 )
% 0.34/0.55 => ~ ( member6099555550032318734et_nat @ ( produc8199053930788261021et_nat @ B2 @ A22 ) @ S2 ) ) ) ).
% 0.34/0.55
% 0.34/0.55 % relcomp.cases
% 0.34/0.55 thf(fact_1721_relcomp_Ocases,axiom,
% 0.34/0.55 ! [A1: produc3658429121746597890et_nat > $o,A22: produc3658429121746597890et_nat,R2: set_Pr8536935166611901872et_nat,S2: set_Pr3721596426557623617et_nat] :
% 0.34/0.55 ( ( member1996754912294343701et_nat @ ( produc5001842942810119800et_nat @ A1 @ A22 ) @ ( relcom430014394250001422et_nat @ R2 @ S2 ) )
% 0.34/0.55 => ~ ! [B2: produc3925858234332021118et_nat] :
% 0.34/0.55 ( ( member6124377750444531601et_nat @ ( produc2245416461498447860et_nat @ A1 @ B2 ) @ R2 )
% 0.34/0.55 => ~ ( member6613709072892554et_nat @ ( produc5530789758353788441et_nat @ B2 @ A22 ) @ S2 ) ) ) ).
% 0.34/0.55
% 0.34/0.55 % relcomp.cases
% 0.34/0.55 thf(fact_1722_relcomp_Ocases,axiom,
% 0.34/0.55 ! [A1: produc3658429121746597890et_nat > $o,A22: produc3925858234332021118et_nat,R2: set_Pr2161125870931222855_nat_o,S2: set_Pr8536935166611901872et_nat] :
% 0.34/0.55 ( ( member6124377750444531601et_nat @ ( produc2245416461498447860et_nat @ A1 @ A22 ) @ ( relcom2608444778321954367et_nat @ R2 @ S2 ) )
% 0.34/0.55 => ~ ! [B2: produc3658429121746597890et_nat > $o] :
% 0.34/0.55 ( ( member8781333585448626064_nat_o @ ( produc7368190662567826135_nat_o @ A1 @ B2 ) @ R2 )
% 0.34/0.55 => ~ ( member6124377750444531601et_nat @ ( produc2245416461498447860et_nat @ B2 @ A22 ) @ S2 ) ) ) ).
% 0.34/0.55
% 0.34/0.55 % relcomp.cases
% 0.34/0.55 thf(fact_1723_relcomp_Ocases,axiom,
% 0.34/0.55 ! [A1: produc3658429121746597890et_nat > $o,A22: produc3925858234332021118et_nat,R2: set_Pr3286484037609594932et_nat,S2: set_Pr8807100098903490369et_nat] :
% 0.34/0.55 ( ( member6124377750444531601et_nat @ ( produc2245416461498447860et_nat @ A1 @ A22 ) @ ( relcom8031709186682293262et_nat @ R2 @ S2 ) )
% 0.34/0.55 => ~ ! [B2: produc3658429121746597890et_nat] :
% 0.34/0.55 ( ( member1996754912294343701et_nat @ ( produc5001842942810119800et_nat @ A1 @ B2 ) @ R2 )
% 0.34/0.55 => ~ ( member2737525317995836554et_nat @ ( produc3909112513931304473et_nat @ B2 @ A22 ) @ S2 ) ) ) ).
% 0.34/0.55
% 0.34/0.55 % relcomp.cases
% 0.34/0.55 thf(fact_1724_relcomp_Ocases,axiom,
% 0.34/0.55 ! [A1: produc3658429121746597890et_nat > $o,A22: produc3925858234332021118et_nat,R2: set_Pr8536935166611901872et_nat,S2: set_Pr7928877670098842301et_nat] :
% 0.34/0.55 ( ( member6124377750444531601et_nat @ ( produc2245416461498447860et_nat @ A1 @ A22 ) @ ( relcom8303140135939483018et_nat @ R2 @ S2 ) )
% 0.34/0.55 => ~ ! [B2: produc3925858234332021118et_nat] :
% 0.34/0.55 ( ( member6124377750444531601et_nat @ ( produc2245416461498447860et_nat @ A1 @ B2 ) @ R2 )
% 0.34/0.55 => ~ ( member4763271486408492550et_nat @ ( produc8599840265553166229et_nat @ B2 @ A22 ) @ S2 ) ) ) ).
% 0.34/0.55
% 0.34/0.55 % relcomp.cases
% 0.34/0.55 thf(fact_1725_FieldI2,axiom,
% 0.34/0.55 ! [I2: produc3658429121746597890et_nat > $o,J: produc3658429121746597890et_nat > $o,R3: set_Pr2161125870931222855_nat_o] :
% 0.34/0.55 ( ( member8781333585448626064_nat_o @ ( produc7368190662567826135_nat_o @ I2 @ J ) @ R3 )
% 0.34/0.55 => ( member6576561426505652726_nat_o @ J @ ( field_81821745747897882_nat_o @ R3 ) ) ) ).
% 0.34/0.55
% 0.34/0.55 % FieldI2
% 0.34/0.55 thf(fact_1726_FieldI2,axiom,
% 0.34/0.55 ! [I2: $o,J: $o,R3: set_Product_prod_o_o] :
% 0.34/0.55 ( ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ I2 @ J ) @ R3 )
% 0.34/0.55 => ( member_o @ J @ ( field_o @ R3 ) ) ) ).
% 0.34/0.55
% 0.34/0.55 % FieldI2
% 0.34/0.55 thf(fact_1727_FieldI2,axiom,
% 0.34/0.55 ! [I2: product_unit,J: product_unit,R3: set_Pr5094982260447487303t_unit] :
% 0.34/0.55 ( ( member7821138191923116944t_unit @ ( produc5692694829384537815t_unit @ I2 @ J ) @ R3 )
% 0.34/0.55 => ( member_Product_unit @ J @ ( field_Product_unit @ R3 ) ) ) ).
% 0.34/0.55
% 0.34/0.55 % FieldI2
% 0.34/0.55 thf(fact_1728_FieldI2,axiom,
% 0.34/0.55 ! [I2: nat,J: nat,R3: set_Pr1261947904930325089at_nat] :
% 0.34/0.55 ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ I2 @ J ) @ R3 )
% 0.34/0.55 => ( member_nat @ J @ ( field_nat @ R3 ) ) ) ).
% 0.34/0.55
% 0.34/0.55 % FieldI2
% 0.34/0.55 thf(fact_1729_FieldI2,axiom,
% 0.34/0.55 ! [I2: int,J: int,R3: set_Pr958786334691620121nt_int] :
% 0.34/0.55 ( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ I2 @ J ) @ R3 )
% 0.34/0.55 => ( member_int @ J @ ( field_int @ R3 ) ) ) ).
% 0.34/0.55
% 0.34/0.55 % FieldI2
% 0.34/0.55 thf(fact_1730_FieldI1,axiom,
% 0.34/0.55 ! [I2: produc3658429121746597890et_nat > $o,J: produc3658429121746597890et_nat > $o,R3: set_Pr2161125870931222855_nat_o] :
% 0.34/0.55 ( ( member8781333585448626064_nat_o @ ( produc7368190662567826135_nat_o @ I2 @ J ) @ R3 )
% 0.34/0.55 => ( member6576561426505652726_nat_o @ I2 @ ( field_81821745747897882_nat_o @ R3 ) ) ) ).
% 0.34/0.55
% 0.34/0.55 % FieldI1
% 0.34/0.55 thf(fact_1731_FieldI1,axiom,
% 0.34/0.55 ! [I2: $o,J: $o,R3: set_Product_prod_o_o] :
% 0.34/0.55 ( ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ I2 @ J ) @ R3 )
% 0.34/0.55 => ( member_o @ I2 @ ( field_o @ R3 ) ) ) ).
% 0.34/0.55
% 0.34/0.55 % FieldI1
% 0.34/0.55 thf(fact_1732_FieldI1,axiom,
% 0.34/0.55 ! [I2: product_unit,J: product_unit,R3: set_Pr5094982260447487303t_unit] :
% 0.34/0.55 ( ( member7821138191923116944t_unit @ ( produc5692694829384537815t_unit @ I2 @ J ) @ R3 )
% 0.34/0.55 => ( member_Product_unit @ I2 @ ( field_Product_unit @ R3 ) ) ) ).
% 0.34/0.55
% 0.34/0.55 % FieldI1
% 0.34/0.55 thf(fact_1733_FieldI1,axiom,
% 0.34/0.55 ! [I2: nat,J: nat,R3: set_Pr1261947904930325089at_nat] :
% 0.34/0.55 ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ I2 @ J ) @ R3 )
% 0.34/0.55 => ( member_nat @ I2 @ ( field_nat @ R3 ) ) ) ).
% 0.34/0.55
% 0.34/0.55 % FieldI1
% 0.34/0.55 thf(fact_1734_FieldI1,axiom,
% 0.34/0.55 ! [I2: int,J: int,R3: set_Pr958786334691620121nt_int] :
% 0.34/0.55 ( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ I2 @ J ) @ R3 )
% 0.34/0.55 => ( member_int @ I2 @ ( field_int @ R3 ) ) ) ).
% 0.34/0.55
% 0.34/0.55 % FieldI1
% 0.34/0.55 thf(fact_1735_mono__Field,axiom,
% 0.34/0.55 ! [R2: set_Pr1261947904930325089at_nat,S2: set_Pr1261947904930325089at_nat] :
% 0.34/0.55 ( ( ord_le3146513528884898305at_nat @ R2 @ S2 )
% 0.34/0.55 => ( ord_less_eq_set_nat @ ( field_nat @ R2 ) @ ( field_nat @ S2 ) ) ) ).
% 0.34/0.55
% 0.34/0.55 % mono_Field
% 0.34/0.55 thf(fact_1736_equiv__class__self,axiom,
% 0.34/0.55 ! [A: set_Product_unit,R2: set_Pr5094982260447487303t_unit,A2: product_unit] :
% 0.34/0.55 ( ( equiv_4189356295871601603t_unit @ A @ R2 )
% 0.34/0.55 => ( ( member_Product_unit @ A2 @ A )
% 0.34/0.55 => ( member_Product_unit @ A2 @ ( image_8431784415008483091t_unit @ R2 @ ( insert_Product_unit @ A2 @ bot_bo3957492148770167129t_unit ) ) ) ) ) ).
% 0.34/0.55
% 0.34/0.55 % equiv_class_self
% 0.34/0.55 thf(fact_1737_equiv__class__self,axiom,
% 0.34/0.55 ! [A: set_o,R2: set_Product_prod_o_o,A2: $o] :
% 0.34/0.55 ( ( equiv_equiv_o @ A @ R2 )
% 0.34/0.55 => ( ( member_o @ A2 @ A )
% 0.34/0.55 => ( member_o @ A2 @ ( image_o_o @ R2 @ ( insert_o @ A2 @ bot_bot_set_o ) ) ) ) ) ).
% 0.34/0.55
% 0.34/0.55 % equiv_class_self
% 0.34/0.55 thf(fact_1738_equiv__class__self,axiom,
% 0.34/0.55 ! [A: set_nat,R2: set_Pr1261947904930325089at_nat,A2: nat] :
% 0.34/0.55 ( ( equiv_equiv_nat @ A @ R2 )
% 0.34/0.55 => ( ( member_nat @ A2 @ A )
% 0.34/0.55 => ( member_nat @ A2 @ ( image_nat_nat @ R2 @ ( insert_nat @ A2 @ bot_bot_set_nat ) ) ) ) ) ).
% 0.34/0.55
% 0.34/0.55 % equiv_class_self
% 0.34/0.55 thf(fact_1739_equiv__class__self,axiom,
% 0.34/0.55 ! [A: set_int,R2: set_Pr958786334691620121nt_int,A2: int] :
% 0.34/0.55 ( ( equiv_equiv_int @ A @ R2 )
% 0.34/0.55 => ( ( member_int @ A2 @ A )
% 0.34/0.55 => ( member_int @ A2 @ ( image_int_int @ R2 @ ( insert_int @ A2 @ bot_bot_set_int ) ) ) ) ) ).
% 0.34/0.55
% 0.34/0.55 % equiv_class_self
% 0.34/0.55 thf(fact_1740_equiv__class__self,axiom,
% 0.34/0.55 ! [A: set_Pr4532377907799695533_nat_o,R2: set_Pr2161125870931222855_nat_o,A2: produc3658429121746597890et_nat > $o] :
% 0.34/0.55 ( ( equiv_6726761681920839555_nat_o @ A @ R2 )
% 0.34/0.55 => ( ( member6576561426505652726_nat_o @ A2 @ A )
% 0.34/0.55 => ( member6576561426505652726_nat_o @ A2 @ ( image_2232984160439939859_nat_o @ R2 @ ( insert5175938949040314269_nat_o @ A2 @ bot_bo7824918357723582233_nat_o ) ) ) ) ) ).
% 0.34/0.55
% 0.34/0.55 % equiv_class_self
% 0.34/0.55 thf(fact_1741_Field__def,axiom,
% 0.34/0.55 ( field_nat
% 0.34/0.55 = ( ^ [R4: set_Pr1261947904930325089at_nat] : ( sup_sup_set_nat @ ( domain_nat_nat @ R4 ) @ ( range_nat_nat @ R4 ) ) ) ) ).
% 0.34/0.55
% 0.34/0.55 % Field_def
% 0.34/0.55 thf(fact_1742_equiv__class__eq__iff,axiom,
% 0.34/0.55 ! [A: set_Product_unit,R2: set_Pr5094982260447487303t_unit,X: product_unit,Y: product_unit] :
% 0.34/0.55 ( ( equiv_4189356295871601603t_unit @ A @ R2 )
% 0.34/0.55 => ( ( member7821138191923116944t_unit @ ( produc5692694829384537815t_unit @ X @ Y ) @ R2 )
% 0.34/0.55 = ( ( ( image_8431784415008483091t_unit @ R2 @ ( insert_Product_unit @ X @ bot_bo3957492148770167129t_unit ) )
% 0.34/0.55 = ( image_8431784415008483091t_unit @ R2 @ ( insert_Product_unit @ Y @ bot_bo3957492148770167129t_unit ) ) )
% 0.34/0.55 & ( member_Product_unit @ X @ A )
% 0.34/0.55 & ( member_Product_unit @ Y @ A ) ) ) ) ).
% 0.34/0.55
% 0.34/0.55 % equiv_class_eq_iff
% 0.34/0.55 thf(fact_1743_equiv__class__eq__iff,axiom,
% 0.34/0.55 ! [A: set_o,R2: set_Product_prod_o_o,X: $o,Y: $o] :
% 0.34/0.55 ( ( equiv_equiv_o @ A @ R2 )
% 0.34/0.55 => ( ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ X @ Y ) @ R2 )
% 0.34/0.55 = ( ( ( image_o_o @ R2 @ ( insert_o @ X @ bot_bot_set_o ) )
% 0.34/0.55 = ( image_o_o @ R2 @ ( insert_o @ Y @ bot_bot_set_o ) ) )
% 0.34/0.55 & ( member_o @ X @ A )
% 0.34/0.55 & ( member_o @ Y @ A ) ) ) ) ).
% 0.34/0.55
% 0.34/0.55 % equiv_class_eq_iff
% 0.34/0.55 thf(fact_1744_equiv__class__eq__iff,axiom,
% 0.34/0.55 ! [A: set_nat,R2: set_Pr1261947904930325089at_nat,X: nat,Y: nat] :
% 0.34/0.55 ( ( equiv_equiv_nat @ A @ R2 )
% 0.34/0.55 => ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ Y ) @ R2 )
% 0.34/0.55 = ( ( ( image_nat_nat @ R2 @ ( insert_nat @ X @ bot_bot_set_nat ) )
% 0.34/0.55 = ( image_nat_nat @ R2 @ ( insert_nat @ Y @ bot_bot_set_nat ) ) )
% 0.34/0.55 & ( member_nat @ X @ A )
% 0.34/0.55 & ( member_nat @ Y @ A ) ) ) ) ).
% 0.34/0.55
% 0.34/0.55 % equiv_class_eq_iff
% 0.34/0.55 thf(fact_1745_equiv__class__eq__iff,axiom,
% 0.34/0.55 ! [A: set_int,R2: set_Pr958786334691620121nt_int,X: int,Y: int] :
% 0.34/0.55 ( ( equiv_equiv_int @ A @ R2 )
% 0.34/0.55 => ( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ X @ Y ) @ R2 )
% 0.34/0.55 = ( ( ( image_int_int @ R2 @ ( insert_int @ X @ bot_bot_set_int ) )
% 0.34/0.55 = ( image_int_int @ R2 @ ( insert_int @ Y @ bot_bot_set_int ) ) )
% 0.34/0.55 & ( member_int @ X @ A )
% 0.34/0.55 & ( member_int @ Y @ A ) ) ) ) ).
% 0.34/0.55
% 0.34/0.55 % equiv_class_eq_iff
% 0.34/0.55 thf(fact_1746_equiv__class__eq__iff,axiom,
% 0.34/0.55 ! [A: set_Pr4532377907799695533_nat_o,R2: set_Pr2161125870931222855_nat_o,X: produc3658429121746597890et_nat > $o,Y: produc3658429121746597890et_nat > $o] :
% 0.34/0.55 ( ( equiv_6726761681920839555_nat_o @ A @ R2 )
% 0.34/0.55 => ( ( member8781333585448626064_nat_o @ ( produc7368190662567826135_nat_o @ X @ Y ) @ R2 )
% 0.34/0.55 = ( ( ( image_2232984160439939859_nat_o @ R2 @ ( insert5175938949040314269_nat_o @ X @ bot_bo7824918357723582233_nat_o ) )
% 0.34/0.55 = ( image_2232984160439939859_nat_o @ R2 @ ( insert5175938949040314269_nat_o @ Y @ bot_bo7824918357723582233_nat_o ) ) )
% 0.34/0.55 & ( member6576561426505652726_nat_o @ X @ A )
% 0.34/0.55 & ( member6576561426505652726_nat_o @ Y @ A ) ) ) ) ).
% 0.34/0.55
% 0.34/0.55 % equiv_class_eq_iff
% 0.34/0.55 thf(fact_1747_eq__equiv__class__iff,axiom,
% 0.34/0.55 ! [A: set_Product_unit,R2: set_Pr5094982260447487303t_unit,X: product_unit,Y: product_unit] :
% 0.34/0.55 ( ( equiv_4189356295871601603t_unit @ A @ R2 )
% 0.34/0.55 => ( ( member_Product_unit @ X @ A )
% 0.34/0.55 => ( ( member_Product_unit @ Y @ A )
% 0.34/0.55 => ( ( ( image_8431784415008483091t_unit @ R2 @ ( insert_Product_unit @ X @ bot_bo3957492148770167129t_unit ) )
% 0.34/0.55 = ( image_8431784415008483091t_unit @ R2 @ ( insert_Product_unit @ Y @ bot_bo3957492148770167129t_unit ) ) )
% 0.34/0.55 = ( member7821138191923116944t_unit @ ( produc5692694829384537815t_unit @ X @ Y ) @ R2 ) ) ) ) ) ).
% 0.34/0.55
% 0.34/0.55 % eq_equiv_class_iff
% 0.34/0.55 thf(fact_1748_eq__equiv__class__iff,axiom,
% 0.34/0.55 ! [A: set_o,R2: set_Product_prod_o_o,X: $o,Y: $o] :
% 0.34/0.55 ( ( equiv_equiv_o @ A @ R2 )
% 0.34/0.55 => ( ( member_o @ X @ A )
% 0.34/0.55 => ( ( member_o @ Y @ A )
% 0.34/0.55 => ( ( ( image_o_o @ R2 @ ( insert_o @ X @ bot_bot_set_o ) )
% 0.34/0.55 = ( image_o_o @ R2 @ ( insert_o @ Y @ bot_bot_set_o ) ) )
% 0.34/0.55 = ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ X @ Y ) @ R2 ) ) ) ) ) ).
% 0.34/0.55
% 0.34/0.55 % eq_equiv_class_iff
% 0.34/0.55 thf(fact_1749_eq__equiv__class__iff,axiom,
% 0.34/0.55 ! [A: set_nat,R2: set_Pr1261947904930325089at_nat,X: nat,Y: nat] :
% 0.34/0.55 ( ( equiv_equiv_nat @ A @ R2 )
% 0.34/0.55 => ( ( member_nat @ X @ A )
% 0.34/0.55 => ( ( member_nat @ Y @ A )
% 0.34/0.55 => ( ( ( image_nat_nat @ R2 @ ( insert_nat @ X @ bot_bot_set_nat ) )
% 0.34/0.55 = ( image_nat_nat @ R2 @ ( insert_nat @ Y @ bot_bot_set_nat ) ) )
% 0.34/0.55 = ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ Y ) @ R2 ) ) ) ) ) ).
% 0.34/0.55
% 0.34/0.55 % eq_equiv_class_iff
% 0.34/0.55 thf(fact_1750_eq__equiv__class__iff,axiom,
% 0.34/0.55 ! [A: set_int,R2: set_Pr958786334691620121nt_int,X: int,Y: int] :
% 0.34/0.55 ( ( equiv_equiv_int @ A @ R2 )
% 0.34/0.55 => ( ( member_int @ X @ A )
% 0.34/0.55 => ( ( member_int @ Y @ A )
% 0.34/0.55 => ( ( ( image_int_int @ R2 @ ( insert_int @ X @ bot_bot_set_int ) )
% 0.34/0.55 = ( image_int_int @ R2 @ ( insert_int @ Y @ bot_bot_set_int ) ) )
% 0.34/0.55 = ( member5262025264175285858nt_int @ ( product_Pair_int_int @ X @ Y ) @ R2 ) ) ) ) ) ).
% 0.34/0.55
% 0.34/0.55 % eq_equiv_class_iff
% 0.34/0.55 thf(fact_1751_eq__equiv__class__iff,axiom,
% 0.34/0.55 ! [A: set_Pr4532377907799695533_nat_o,R2: set_Pr2161125870931222855_nat_o,X: produc3658429121746597890et_nat > $o,Y: produc3658429121746597890et_nat > $o] :
% 0.34/0.55 ( ( equiv_6726761681920839555_nat_o @ A @ R2 )
% 0.34/0.55 => ( ( member6576561426505652726_nat_o @ X @ A )
% 0.34/0.55 => ( ( member6576561426505652726_nat_o @ Y @ A )
% 0.34/0.55 => ( ( ( image_2232984160439939859_nat_o @ R2 @ ( insert5175938949040314269_nat_o @ X @ bot_bo7824918357723582233_nat_o ) )
% 0.34/0.55 = ( image_2232984160439939859_nat_o @ R2 @ ( insert5175938949040314269_nat_o @ Y @ bot_bo7824918357723582233_nat_o ) ) )
% 0.34/0.55 = ( member8781333585448626064_nat_o @ ( produc7368190662567826135_nat_o @ X @ Y ) @ R2 ) ) ) ) ) ).
% 0.34/0.55
% 0.34/0.55 % eq_equiv_class_iff
% 0.34/0.55 thf(fact_1752_equiv__class__eq,axiom,
% 0.34/0.55 ! [A: set_Product_unit,R2: set_Pr5094982260447487303t_unit,A2: product_unit,B: product_unit] :
% 0.34/0.55 ( ( equiv_4189356295871601603t_unit @ A @ R2 )
% 0.34/0.55 => ( ( member7821138191923116944t_unit @ ( produc5692694829384537815t_unit @ A2 @ B ) @ R2 )
% 0.34/0.55 => ( ( image_8431784415008483091t_unit @ R2 @ ( insert_Product_unit @ A2 @ bot_bo3957492148770167129t_unit ) )
% 0.34/0.55 = ( image_8431784415008483091t_unit @ R2 @ ( insert_Product_unit @ B @ bot_bo3957492148770167129t_unit ) ) ) ) ) ).
% 0.34/0.55
% 0.34/0.55 % equiv_class_eq
% 0.34/0.55 thf(fact_1753_equiv__class__eq,axiom,
% 0.34/0.55 ! [A: set_o,R2: set_Product_prod_o_o,A2: $o,B: $o] :
% 0.34/0.55 ( ( equiv_equiv_o @ A @ R2 )
% 0.34/0.55 => ( ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ A2 @ B ) @ R2 )
% 0.34/0.55 => ( ( image_o_o @ R2 @ ( insert_o @ A2 @ bot_bot_set_o ) )
% 0.34/0.55 = ( image_o_o @ R2 @ ( insert_o @ B @ bot_bot_set_o ) ) ) ) ) ).
% 0.34/0.55
% 0.34/0.55 % equiv_class_eq
% 0.34/0.55 thf(fact_1754_equiv__class__eq,axiom,
% 0.34/0.55 ! [A: set_nat,R2: set_Pr1261947904930325089at_nat,A2: nat,B: nat] :
% 0.34/0.55 ( ( equiv_equiv_nat @ A @ R2 )
% 0.34/0.55 => ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A2 @ B ) @ R2 )
% 0.34/0.55 => ( ( image_nat_nat @ R2 @ ( insert_nat @ A2 @ bot_bot_set_nat ) )
% 0.34/0.55 = ( image_nat_nat @ R2 @ ( insert_nat @ B @ bot_bot_set_nat ) ) ) ) ) ).
% 0.34/0.55
% 0.34/0.55 % equiv_class_eq
% 0.34/0.55 thf(fact_1755_equiv__class__eq,axiom,
% 0.34/0.55 ! [A: set_int,R2: set_Pr958786334691620121nt_int,A2: int,B: int] :
% 0.34/0.55 ( ( equiv_equiv_int @ A @ R2 )
% 0.34/0.55 => ( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ A2 @ B ) @ R2 )
% 0.34/0.55 => ( ( image_int_int @ R2 @ ( insert_int @ A2 @ bot_bot_set_int ) )
% 0.34/0.55 = ( image_int_int @ R2 @ ( insert_int @ B @ bot_bot_set_int ) ) ) ) ) ).
% 0.34/0.55
% 0.34/0.55 % equiv_class_eq
% 0.34/0.55 thf(fact_1756_equiv__class__eq,axiom,
% 0.34/0.55 ! [A: set_Pr4532377907799695533_nat_o,R2: set_Pr2161125870931222855_nat_o,A2: produc3658429121746597890et_nat > $o,B: produc3658429121746597890et_nat > $o] :
% 0.34/0.55 ( ( equiv_6726761681920839555_nat_o @ A @ R2 )
% 0.34/0.55 => ( ( member8781333585448626064_nat_o @ ( produc7368190662567826135_nat_o @ A2 @ B ) @ R2 )
% 0.34/0.55 => ( ( image_2232984160439939859_nat_o @ R2 @ ( insert5175938949040314269_nat_o @ A2 @ bot_bo7824918357723582233_nat_o ) )
% 0.34/0.55 = ( image_2232984160439939859_nat_o @ R2 @ ( insert5175938949040314269_nat_o @ B @ bot_bo7824918357723582233_nat_o ) ) ) ) ) ).
% 0.34/0.55
% 0.34/0.55 % equiv_class_eq
% 0.34/0.55 thf(fact_1757_eq__equiv__class,axiom,
% 0.34/0.55 ! [R2: set_Pr5094982260447487303t_unit,A2: product_unit,B: product_unit,A: set_Product_unit] :
% 0.34/0.55 ( ( ( image_8431784415008483091t_unit @ R2 @ ( insert_Product_unit @ A2 @ bot_bo3957492148770167129t_unit ) )
% 0.34/0.55 = ( image_8431784415008483091t_unit @ R2 @ ( insert_Product_unit @ B @ bot_bo3957492148770167129t_unit ) ) )
% 0.34/0.55 => ( ( equiv_4189356295871601603t_unit @ A @ R2 )
% 0.34/0.55 => ( ( member_Product_unit @ B @ A )
% 0.34/0.55 => ( member7821138191923116944t_unit @ ( produc5692694829384537815t_unit @ A2 @ B ) @ R2 ) ) ) ) ).
% 0.34/0.55
% 0.34/0.55 % eq_equiv_class
% 0.34/0.55 thf(fact_1758_eq__equiv__class,axiom,
% 0.34/0.55 ! [R2: set_Product_prod_o_o,A2: $o,B: $o,A: set_o] :
% 0.34/0.55 ( ( ( image_o_o @ R2 @ ( insert_o @ A2 @ bot_bot_set_o ) )
% 0.34/0.55 = ( image_o_o @ R2 @ ( insert_o @ B @ bot_bot_set_o ) ) )
% 0.34/0.55 => ( ( equiv_equiv_o @ A @ R2 )
% 0.34/0.55 => ( ( member_o @ B @ A )
% 0.34/0.55 => ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ A2 @ B ) @ R2 ) ) ) ) ).
% 0.34/0.55
% 0.34/0.55 % eq_equiv_class
% 0.34/0.55 thf(fact_1759_eq__equiv__class,axiom,
% 0.34/0.55 ! [R2: set_Pr1261947904930325089at_nat,A2: nat,B: nat,A: set_nat] :
% 0.34/0.55 ( ( ( image_nat_nat @ R2 @ ( insert_nat @ A2 @ bot_bot_set_nat ) )
% 0.34/0.55 = ( image_nat_nat @ R2 @ ( insert_nat @ B @ bot_bot_set_nat ) ) )
% 0.34/0.55 => ( ( equiv_equiv_nat @ A @ R2 )
% 0.34/0.55 => ( ( member_nat @ B @ A )
% 0.34/0.55 => ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A2 @ B ) @ R2 ) ) ) ) ).
% 0.34/0.55
% 0.34/0.55 % eq_equiv_class
% 0.34/0.55 thf(fact_1760_eq__equiv__class,axiom,
% 0.34/0.55 ! [R2: set_Pr958786334691620121nt_int,A2: int,B: int,A: set_int] :
% 0.34/0.55 ( ( ( image_int_int @ R2 @ ( insert_int @ A2 @ bot_bot_set_int ) )
% 0.34/0.55 = ( image_int_int @ R2 @ ( insert_int @ B @ bot_bot_set_int ) ) )
% 0.34/0.55 => ( ( equiv_equiv_int @ A @ R2 )
% 0.34/0.55 => ( ( member_int @ B @ A )
% 0.34/0.55 => ( member5262025264175285858nt_int @ ( product_Pair_int_int @ A2 @ B ) @ R2 ) ) ) ) ).
% 0.34/0.55
% 0.34/0.55 % eq_equiv_class
% 0.34/0.55 thf(fact_1761_eq__equiv__class,axiom,
% 0.34/0.55 ! [R2: set_Pr2161125870931222855_nat_o,A2: produc3658429121746597890et_nat > $o,B: produc3658429121746597890et_nat > $o,A: set_Pr4532377907799695533_nat_o] :
% 0.34/0.55 ( ( ( image_2232984160439939859_nat_o @ R2 @ ( insert5175938949040314269_nat_o @ A2 @ bot_bo7824918357723582233_nat_o ) )
% 0.34/0.55 = ( image_2232984160439939859_nat_o @ R2 @ ( insert5175938949040314269_nat_o @ B @ bot_bo7824918357723582233_nat_o ) ) )
% 0.34/0.55 => ( ( equiv_6726761681920839555_nat_o @ A @ R2 )
% 0.34/0.55 => ( ( member6576561426505652726_nat_o @ B @ A )
% 0.34/0.55 => ( member8781333585448626064_nat_o @ ( produc7368190662567826135_nat_o @ A2 @ B ) @ R2 ) ) ) ) ).
% 0.34/0.55
% 0.34/0.55 % eq_equiv_class
% 0.34/0.55 thf(fact_1762_rtrancl__Image__in__Field,axiom,
% 0.34/0.55 ! [R3: set_Pr1261947904930325089at_nat,V2: set_nat] : ( ord_less_eq_set_nat @ ( image_nat_nat @ ( transi2905341329935302413cl_nat @ R3 ) @ V2 ) @ ( sup_sup_set_nat @ ( field_nat @ R3 ) @ V2 ) ) ).
% 0.34/0.55
% 0.34/0.55 % rtrancl_Image_in_Field
% 0.34/0.55 thf(fact_1763_refines__equiv__class__eq2,axiom,
% 0.34/0.55 ! [R3: set_Pr5094982260447487303t_unit,S: set_Pr5094982260447487303t_unit,A: set_Product_unit,A2: product_unit] :
% 0.34/0.55 ( ( ord_le930853871177014503t_unit @ R3 @ S )
% 0.34/0.55 => ( ( equiv_4189356295871601603t_unit @ A @ R3 )
% 0.34/0.55 => ( ( equiv_4189356295871601603t_unit @ A @ S )
% 0.34/0.55 => ( ( image_8431784415008483091t_unit @ S @ ( image_8431784415008483091t_unit @ R3 @ ( insert_Product_unit @ A2 @ bot_bo3957492148770167129t_unit ) ) )
% 0.34/0.55 = ( image_8431784415008483091t_unit @ S @ ( insert_Product_unit @ A2 @ bot_bo3957492148770167129t_unit ) ) ) ) ) ) ).
% 0.34/0.55
% 0.34/0.55 % refines_equiv_class_eq2
% 0.34/0.55 thf(fact_1764_refines__equiv__class__eq2,axiom,
% 0.34/0.55 ! [R3: set_Product_prod_o_o,S: set_Product_prod_o_o,A: set_o,A2: $o] :
% 0.34/0.55 ( ( ord_le192195778054999463od_o_o @ R3 @ S )
% 0.34/0.55 => ( ( equiv_equiv_o @ A @ R3 )
% 0.34/0.55 => ( ( equiv_equiv_o @ A @ S )
% 0.34/0.55 => ( ( image_o_o @ S @ ( image_o_o @ R3 @ ( insert_o @ A2 @ bot_bot_set_o ) ) )
% 0.34/0.55 = ( image_o_o @ S @ ( insert_o @ A2 @ bot_bot_set_o ) ) ) ) ) ) ).
% 0.34/0.55
% 0.34/0.55 % refines_equiv_class_eq2
% 0.34/0.55 thf(fact_1765_refines__equiv__class__eq2,axiom,
% 0.34/0.55 ! [R3: set_Pr1261947904930325089at_nat,S: set_Pr1261947904930325089at_nat,A: set_nat,A2: nat] :
% 0.34/0.55 ( ( ord_le3146513528884898305at_nat @ R3 @ S )
% 0.34/0.55 => ( ( equiv_equiv_nat @ A @ R3 )
% 0.34/0.55 => ( ( equiv_equiv_nat @ A @ S )
% 0.34/0.55 => ( ( image_nat_nat @ S @ ( image_nat_nat @ R3 @ ( insert_nat @ A2 @ bot_bot_set_nat ) ) )
% 0.34/0.55 = ( image_nat_nat @ S @ ( insert_nat @ A2 @ bot_bot_set_nat ) ) ) ) ) ) ).
% 0.34/0.55
% 0.34/0.55 % refines_equiv_class_eq2
% 0.34/0.55 thf(fact_1766_refines__equiv__class__eq2,axiom,
% 0.34/0.55 ! [R3: set_Pr958786334691620121nt_int,S: set_Pr958786334691620121nt_int,A: set_int,A2: int] :
% 0.34/0.55 ( ( ord_le2843351958646193337nt_int @ R3 @ S )
% 0.34/0.55 => ( ( equiv_equiv_int @ A @ R3 )
% 0.34/0.55 => ( ( equiv_equiv_int @ A @ S )
% 0.34/0.55 => ( ( image_int_int @ S @ ( image_int_int @ R3 @ ( insert_int @ A2 @ bot_bot_set_int ) ) )
% 0.34/0.55 = ( image_int_int @ S @ ( insert_int @ A2 @ bot_bot_set_int ) ) ) ) ) ) ).
% 0.34/0.55
% 0.34/0.55 % refines_equiv_class_eq2
% 0.34/0.55 thf(fact_1767_refines__equiv__class__eq2,axiom,
% 0.34/0.55 ! [R3: set_Pr2161125870931222855_nat_o,S: set_Pr2161125870931222855_nat_o,A: set_Pr4532377907799695533_nat_o,A2: produc3658429121746597890et_nat > $o] :
% 0.34/0.55 ( ( ord_le6659818918159358183_nat_o @ R3 @ S )
% 0.34/0.55 => ( ( equiv_6726761681920839555_nat_o @ A @ R3 )
% 0.34/0.55 => ( ( equiv_6726761681920839555_nat_o @ A @ S )
% 0.34/0.55 => ( ( image_2232984160439939859_nat_o @ S @ ( image_2232984160439939859_nat_o @ R3 @ ( insert5175938949040314269_nat_o @ A2 @ bot_bo7824918357723582233_nat_o ) ) )
% 0.34/0.55 = ( image_2232984160439939859_nat_o @ S @ ( insert5175938949040314269_nat_o @ A2 @ bot_bo7824918357723582233_nat_o ) ) ) ) ) ) ).
% 0.34/0.55
% 0.34/0.55 % refines_equiv_class_eq2
% 0.34/0.55 thf(fact_1768_refines__equiv__class__eq,axiom,
% 0.34/0.55 ! [R3: set_Pr5094982260447487303t_unit,S: set_Pr5094982260447487303t_unit,A: set_Product_unit,A2: product_unit] :
% 0.34/0.55 ( ( ord_le930853871177014503t_unit @ R3 @ S )
% 0.34/0.55 => ( ( equiv_4189356295871601603t_unit @ A @ R3 )
% 0.34/0.55 => ( ( equiv_4189356295871601603t_unit @ A @ S )
% 0.34/0.55 => ( ( image_8431784415008483091t_unit @ R3 @ ( image_8431784415008483091t_unit @ S @ ( insert_Product_unit @ A2 @ bot_bo3957492148770167129t_unit ) ) )
% 0.34/0.55 = ( image_8431784415008483091t_unit @ S @ ( insert_Product_unit @ A2 @ bot_bo3957492148770167129t_unit ) ) ) ) ) ) ).
% 0.34/0.55
% 0.34/0.55 % refines_equiv_class_eq
% 0.34/0.55 thf(fact_1769_refines__equiv__class__eq,axiom,
% 0.34/0.55 ! [R3: set_Product_prod_o_o,S: set_Product_prod_o_o,A: set_o,A2: $o] :
% 0.34/0.55 ( ( ord_le192195778054999463od_o_o @ R3 @ S )
% 0.34/0.55 => ( ( equiv_equiv_o @ A @ R3 )
% 0.34/0.55 => ( ( equiv_equiv_o @ A @ S )
% 0.34/0.55 => ( ( image_o_o @ R3 @ ( image_o_o @ S @ ( insert_o @ A2 @ bot_bot_set_o ) ) )
% 0.34/0.55 = ( image_o_o @ S @ ( insert_o @ A2 @ bot_bot_set_o ) ) ) ) ) ) ).
% 0.34/0.55
% 0.34/0.55 % refines_equiv_class_eq
% 0.34/0.55 thf(fact_1770_refines__equiv__class__eq,axiom,
% 0.34/0.55 ! [R3: set_Pr1261947904930325089at_nat,S: set_Pr1261947904930325089at_nat,A: set_nat,A2: nat] :
% 0.34/0.55 ( ( ord_le3146513528884898305at_nat @ R3 @ S )
% 0.34/0.55 => ( ( equiv_equiv_nat @ A @ R3 )
% 0.34/0.55 => ( ( equiv_equiv_nat @ A @ S )
% 0.34/0.55 => ( ( image_nat_nat @ R3 @ ( image_nat_nat @ S @ ( insert_nat @ A2 @ bot_bot_set_nat ) ) )
% 0.34/0.55 = ( image_nat_nat @ S @ ( insert_nat @ A2 @ bot_bot_set_nat ) ) ) ) ) ) ).
% 0.34/0.55
% 0.34/0.55 % refines_equiv_class_eq
% 0.34/0.55 thf(fact_1771_refines__equiv__class__eq,axiom,
% 0.34/0.55 ! [R3: set_Pr958786334691620121nt_int,S: set_Pr958786334691620121nt_int,A: set_int,A2: int] :
% 0.34/0.55 ( ( ord_le2843351958646193337nt_int @ R3 @ S )
% 0.34/0.55 => ( ( equiv_equiv_int @ A @ R3 )
% 0.34/0.55 => ( ( equiv_equiv_int @ A @ S )
% 0.34/0.55 => ( ( image_int_int @ R3 @ ( image_int_int @ S @ ( insert_int @ A2 @ bot_bot_set_int ) ) )
% 0.34/0.55 = ( image_int_int @ S @ ( insert_int @ A2 @ bot_bot_set_int ) ) ) ) ) ) ).
% 0.34/0.55
% 0.34/0.55 % refines_equiv_class_eq
% 0.34/0.55 thf(fact_1772_refines__equiv__class__eq,axiom,
% 0.34/0.55 ! [R3: set_Pr2161125870931222855_nat_o,S: set_Pr2161125870931222855_nat_o,A: set_Pr4532377907799695533_nat_o,A2: produc3658429121746597890et_nat > $o] :
% 0.34/0.55 ( ( ord_le6659818918159358183_nat_o @ R3 @ S )
% 0.34/0.55 => ( ( equiv_6726761681920839555_nat_o @ A @ R3 )
% 0.34/0.55 => ( ( equiv_6726761681920839555_nat_o @ A @ S )
% 0.34/0.55 => ( ( image_2232984160439939859_nat_o @ R3 @ ( image_2232984160439939859_nat_o @ S @ ( insert5175938949040314269_nat_o @ A2 @ bot_bo7824918357723582233_nat_o ) ) )
% 0.34/0.55 = ( image_2232984160439939859_nat_o @ S @ ( insert5175938949040314269_nat_o @ A2 @ bot_bo7824918357723582233_nat_o ) ) ) ) ) ) ).
% 0.34/0.55
% 0.34/0.55 % refines_equiv_class_eq
% 0.34/0.55 thf(fact_1773_Partial__order__eq__Image1__Image1__iff,axiom,
% 0.34/0.55 ! [R2: set_Pr5094982260447487303t_unit,A2: product_unit,B: product_unit] :
% 0.34/0.55 ( ( order_2505591618835473410t_unit @ ( field_Product_unit @ R2 ) @ R2 )
% 0.34/0.55 => ( ( member_Product_unit @ A2 @ ( field_Product_unit @ R2 ) )
% 0.34/0.55 => ( ( member_Product_unit @ B @ ( field_Product_unit @ R2 ) )
% 0.34/0.55 => ( ( ( image_8431784415008483091t_unit @ R2 @ ( insert_Product_unit @ A2 @ bot_bo3957492148770167129t_unit ) )
% 0.34/0.55 = ( image_8431784415008483091t_unit @ R2 @ ( insert_Product_unit @ B @ bot_bo3957492148770167129t_unit ) ) )
% 0.34/0.55 = ( A2 = B ) ) ) ) ) ).
% 0.34/0.55
% 0.34/0.55 % Partial_order_eq_Image1_Image1_iff
% 0.34/0.55 thf(fact_1774_Partial__order__eq__Image1__Image1__iff,axiom,
% 0.34/0.55 ! [R2: set_Product_prod_o_o,A2: $o,B: $o] :
% 0.34/0.55 ( ( order_8538205470752950541r_on_o @ ( field_o @ R2 ) @ R2 )
% 0.34/0.55 => ( ( member_o @ A2 @ ( field_o @ R2 ) )
% 0.34/0.55 => ( ( member_o @ B @ ( field_o @ R2 ) )
% 0.34/0.55 => ( ( ( image_o_o @ R2 @ ( insert_o @ A2 @ bot_bot_set_o ) )
% 0.34/0.55 = ( image_o_o @ R2 @ ( insert_o @ B @ bot_bot_set_o ) ) )
% 0.34/0.55 = ( A2 = B ) ) ) ) ) ).
% 0.34/0.55
% 0.34/0.55 % Partial_order_eq_Image1_Image1_iff
% 0.34/0.55 thf(fact_1775_Partial__order__eq__Image1__Image1__iff,axiom,
% 0.34/0.55 ! [R2: set_Pr1261947904930325089at_nat,A2: nat,B: nat] :
% 0.34/0.55 ( ( order_5251275573222108571on_nat @ ( field_nat @ R2 ) @ R2 )
% 0.34/0.55 => ( ( member_nat @ A2 @ ( field_nat @ R2 ) )
% 0.34/0.55 => ( ( member_nat @ B @ ( field_nat @ R2 ) )
% 0.34/0.55 => ( ( ( image_nat_nat @ R2 @ ( insert_nat @ A2 @ bot_bot_set_nat ) )
% 0.34/0.55 = ( image_nat_nat @ R2 @ ( insert_nat @ B @ bot_bot_set_nat ) ) )
% 0.34/0.55 = ( A2 = B ) ) ) ) ) ).
% 0.34/0.55
% 0.34/0.55 % Partial_order_eq_Image1_Image1_iff
% 0.34/0.55 thf(fact_1776_Partial__order__eq__Image1__Image1__iff,axiom,
% 0.34/0.55 ! [R2: set_Pr958786334691620121nt_int,A2: int,B: int] :
% 0.34/0.55 ( ( order_5248785102713058295on_int @ ( field_int @ R2 ) @ R2 )
% 0.34/0.55 => ( ( member_int @ A2 @ ( field_int @ R2 ) )
% 0.34/0.55 => ( ( member_int @ B @ ( field_int @ R2 ) )
% 0.34/0.55 => ( ( ( image_int_int @ R2 @ ( insert_int @ A2 @ bot_bot_set_int ) )
% 0.34/0.55 = ( image_int_int @ R2 @ ( insert_int @ B @ bot_bot_set_int ) ) )
% 0.34/0.55 = ( A2 = B ) ) ) ) ) ).
% 0.34/0.55
% 0.34/0.55 % Partial_order_eq_Image1_Image1_iff
% 0.34/0.55 thf(fact_1777_Partial__order__eq__Image1__Image1__iff,axiom,
% 0.34/0.55 ! [R2: set_Pr2161125870931222855_nat_o,A2: produc3658429121746597890et_nat > $o,B: produc3658429121746597890et_nat > $o] :
% 0.34/0.55 ( ( order_3690095022144090562_nat_o @ ( field_81821745747897882_nat_o @ R2 ) @ R2 )
% 0.34/0.55 => ( ( member6576561426505652726_nat_o @ A2 @ ( field_81821745747897882_nat_o @ R2 ) )
% 0.34/0.55 => ( ( member6576561426505652726_nat_o @ B @ ( field_81821745747897882_nat_o @ R2 ) )
% 0.34/0.55 => ( ( ( image_2232984160439939859_nat_o @ R2 @ ( insert5175938949040314269_nat_o @ A2 @ bot_bo7824918357723582233_nat_o ) )
% 0.34/0.55 = ( image_2232984160439939859_nat_o @ R2 @ ( insert5175938949040314269_nat_o @ B @ bot_bo7824918357723582233_nat_o ) ) )
% 0.34/0.55 = ( A2 = B ) ) ) ) ) ).
% 0.34/0.55
% 0.34/0.55 % Partial_order_eq_Image1_Image1_iff
% 0.34/0.55 thf(fact_1778_finite__reachable__restrictedI,axiom,
% 0.34/0.55 ! [Q2: set_Code_integer,I: set_Code_integer,E: set_Pr4811707699266497531nteger] :
% 0.34/0.55 ( ( finite6017078050557962740nteger @ Q2 )
% 0.34/0.55 => ( ( ord_le7084787975880047091nteger @ I @ Q2 )
% 0.34/0.55 => ( ( ord_le7084787975880047091nteger @ ( range_1907724884897625393nteger @ E ) @ Q2 )
% 0.34/0.55 => ( finite6017078050557962740nteger @ ( image_4147621952751189391nteger @ ( transi3984306202085910426nteger @ E ) @ I ) ) ) ) ) ).
% 0.34/0.55
% 0.34/0.55 % finite_reachable_restrictedI
% 0.34/0.55 thf(fact_1779_finite__reachable__restrictedI,axiom,
% 0.34/0.55 ! [Q2: set_nat,I: set_nat,E: set_Pr1261947904930325089at_nat] :
% 0.34/0.55 ( ( finite_finite_nat @ Q2 )
% 0.34/0.55 => ( ( ord_less_eq_set_nat @ I @ Q2 )
% 0.34/0.55 => ( ( ord_less_eq_set_nat @ ( range_nat_nat @ E ) @ Q2 )
% 0.34/0.55 => ( finite_finite_nat @ ( image_nat_nat @ ( transi2905341329935302413cl_nat @ E ) @ I ) ) ) ) ) ).
% 0.34/0.55
% 0.34/0.55 % finite_reachable_restrictedI
% 0.34/0.55 thf(fact_1780_subset__Image__Image__iff,axiom,
% 0.34/0.55 ! [R2: set_Pr958786334691620121nt_int,A: set_int,B4: set_int] :
% 0.34/0.55 ( ( order_4859164337913492053on_int @ ( field_int @ R2 ) @ R2 )
% 0.34/0.55 => ( ( ord_less_eq_set_int @ A @ ( field_int @ R2 ) )
% 0.34/0.55 => ( ( ord_less_eq_set_int @ B4 @ ( field_int @ R2 ) )
% 0.34/0.55 => ( ( ord_less_eq_set_int @ ( image_int_int @ R2 @ A ) @ ( image_int_int @ R2 @ B4 ) )
% 0.34/0.55 = ( ! [X2: int] :
% 0.34/0.55 ( ( member_int @ X2 @ A )
% 0.34/0.55 => ? [Y3: int] :
% 0.34/0.55 ( ( member_int @ Y3 @ B4 )
% 0.34/0.55 & ( member5262025264175285858nt_int @ ( product_Pair_int_int @ Y3 @ X2 ) @ R2 ) ) ) ) ) ) ) ) ).
% 0.34/0.55
% 0.34/0.55 % subset_Image_Image_iff
% 0.34/0.55 thf(fact_1781_subset__Image__Image__iff,axiom,
% 0.34/0.55 ! [R2: set_Pr1261947904930325089at_nat,A: set_nat,B4: set_nat] :
% 0.34/0.55 ( ( order_4861654808422542329on_nat @ ( field_nat @ R2 ) @ R2 )
% 0.34/0.55 => ( ( ord_less_eq_set_nat @ A @ ( field_nat @ R2 ) )
% 0.34/0.55 => ( ( ord_less_eq_set_nat @ B4 @ ( field_nat @ R2 ) )
% 0.34/0.55 => ( ( ord_less_eq_set_nat @ ( image_nat_nat @ R2 @ A ) @ ( image_nat_nat @ R2 @ B4 ) )
% 0.34/0.55 = ( ! [X2: nat] :
% 0.34/0.55 ( ( member_nat @ X2 @ A )
% 0.34/0.55 => ? [Y3: nat] :
% 0.34/0.55 ( ( member_nat @ Y3 @ B4 )
% 0.34/0.55 & ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ Y3 @ X2 ) @ R2 ) ) ) ) ) ) ) ) ).
% 0.34/0.55
% 0.34/0.55 % subset_Image_Image_iff
% 0.34/0.55 thf(fact_1782_equiv__class__subset,axiom,
% 0.34/0.55 ! [A: set_Product_unit,R2: set_Pr5094982260447487303t_unit,A2: product_unit,B: product_unit] :
% 0.34/0.55 ( ( equiv_4189356295871601603t_unit @ A @ R2 )
% 0.34/0.55 => ( ( member7821138191923116944t_unit @ ( produc5692694829384537815t_unit @ A2 @ B ) @ R2 )
% 0.34/0.55 => ( ord_le3507040750410214029t_unit @ ( image_8431784415008483091t_unit @ R2 @ ( insert_Product_unit @ A2 @ bot_bo3957492148770167129t_unit ) ) @ ( image_8431784415008483091t_unit @ R2 @ ( insert_Product_unit @ B @ bot_bo3957492148770167129t_unit ) ) ) ) ) ).
% 0.34/0.55
% 0.34/0.55 % equiv_class_subset
% 0.34/0.55 thf(fact_1783_equiv__class__subset,axiom,
% 0.34/0.55 ! [A: set_o,R2: set_Product_prod_o_o,A2: $o,B: $o] :
% 0.34/0.55 ( ( equiv_equiv_o @ A @ R2 )
% 0.34/0.55 => ( ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ A2 @ B ) @ R2 )
% 0.34/0.55 => ( ord_less_eq_set_o @ ( image_o_o @ R2 @ ( insert_o @ A2 @ bot_bot_set_o ) ) @ ( image_o_o @ R2 @ ( insert_o @ B @ bot_bot_set_o ) ) ) ) ) ).
% 0.34/0.55
% 0.34/0.55 % equiv_class_subset
% 0.34/0.55 thf(fact_1784_equiv__class__subset,axiom,
% 0.34/0.55 ! [A: set_int,R2: set_Pr958786334691620121nt_int,A2: int,B: int] :
% 0.34/0.55 ( ( equiv_equiv_int @ A @ R2 )
% 0.34/0.55 => ( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ A2 @ B ) @ R2 )
% 0.34/0.55 => ( ord_less_eq_set_int @ ( image_int_int @ R2 @ ( insert_int @ A2 @ bot_bot_set_int ) ) @ ( image_int_int @ R2 @ ( insert_int @ B @ bot_bot_set_int ) ) ) ) ) ).
% 0.34/0.55
% 0.34/0.55 % equiv_class_subset
% 0.34/0.55 thf(fact_1785_equiv__class__subset,axiom,
% 0.34/0.55 ! [A: set_Pr4532377907799695533_nat_o,R2: set_Pr2161125870931222855_nat_o,A2: produc3658429121746597890et_nat > $o,B: produc3658429121746597890et_nat > $o] :
% 0.34/0.55 ( ( equiv_6726761681920839555_nat_o @ A @ R2 )
% 0.34/0.55 => ( ( member8781333585448626064_nat_o @ ( produc7368190662567826135_nat_o @ A2 @ B ) @ R2 )
% 0.34/0.55 => ( ord_le2965882846123202637_nat_o @ ( image_2232984160439939859_nat_o @ R2 @ ( insert5175938949040314269_nat_o @ A2 @ bot_bo7824918357723582233_nat_o ) ) @ ( image_2232984160439939859_nat_o @ R2 @ ( insert5175938949040314269_nat_o @ B @ bot_bo7824918357723582233_nat_o ) ) ) ) ) ).
% 0.34/0.55
% 0.34/0.55 % equiv_class_subset
% 0.34/0.55 thf(fact_1786_equiv__class__subset,axiom,
% 0.34/0.55 ! [A: set_nat,R2: set_Pr1261947904930325089at_nat,A2: nat,B: nat] :
% 0.34/0.55 ( ( equiv_equiv_nat @ A @ R2 )
% 0.34/0.55 => ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A2 @ B ) @ R2 )
% 0.34/0.55 => ( ord_less_eq_set_nat @ ( image_nat_nat @ R2 @ ( insert_nat @ A2 @ bot_bot_set_nat ) ) @ ( image_nat_nat @ R2 @ ( insert_nat @ B @ bot_bot_set_nat ) ) ) ) ) ).
% 0.34/0.55
% 0.34/0.55 % equiv_class_subset
% 0.34/0.55 thf(fact_1787_subset__equiv__class,axiom,
% 0.34/0.55 ! [A: set_Product_unit,R2: set_Pr5094982260447487303t_unit,B: product_unit,A2: product_unit] :
% 0.34/0.55 ( ( equiv_4189356295871601603t_unit @ A @ R2 )
% 0.34/0.55 => ( ( ord_le3507040750410214029t_unit @ ( image_8431784415008483091t_unit @ R2 @ ( insert_Product_unit @ B @ bot_bo3957492148770167129t_unit ) ) @ ( image_8431784415008483091t_unit @ R2 @ ( insert_Product_unit @ A2 @ bot_bo3957492148770167129t_unit ) ) )
% 0.34/0.55 => ( ( member_Product_unit @ B @ A )
% 0.34/0.55 => ( member7821138191923116944t_unit @ ( produc5692694829384537815t_unit @ A2 @ B ) @ R2 ) ) ) ) ).
% 0.34/0.55
% 0.34/0.55 % subset_equiv_class
% 0.34/0.55 thf(fact_1788_subset__equiv__class,axiom,
% 0.34/0.55 ! [A: set_o,R2: set_Product_prod_o_o,B: $o,A2: $o] :
% 0.34/0.55 ( ( equiv_equiv_o @ A @ R2 )
% 0.34/0.55 => ( ( ord_less_eq_set_o @ ( image_o_o @ R2 @ ( insert_o @ B @ bot_bot_set_o ) ) @ ( image_o_o @ R2 @ ( insert_o @ A2 @ bot_bot_set_o ) ) )
% 0.34/0.55 => ( ( member_o @ B @ A )
% 0.34/0.55 => ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ A2 @ B ) @ R2 ) ) ) ) ).
% 0.34/0.55
% 0.34/0.55 % subset_equiv_class
% 0.34/0.55 thf(fact_1789_subset__equiv__class,axiom,
% 0.34/0.55 ! [A: set_int,R2: set_Pr958786334691620121nt_int,B: int,A2: int] :
% 0.34/0.55 ( ( equiv_equiv_int @ A @ R2 )
% 0.34/0.55 => ( ( ord_less_eq_set_int @ ( image_int_int @ R2 @ ( insert_int @ B @ bot_bot_set_int ) ) @ ( image_int_int @ R2 @ ( insert_int @ A2 @ bot_bot_set_int ) ) )
% 0.34/0.55 => ( ( member_int @ B @ A )
% 0.34/0.55 => ( member5262025264175285858nt_int @ ( product_Pair_int_int @ A2 @ B ) @ R2 ) ) ) ) ).
% 0.34/0.55
% 0.34/0.55 % subset_equiv_class
% 0.34/0.55 thf(fact_1790_subset__equiv__class,axiom,
% 0.34/0.55 ! [A: set_Pr4532377907799695533_nat_o,R2: set_Pr2161125870931222855_nat_o,B: produc3658429121746597890et_nat > $o,A2: produc3658429121746597890et_nat > $o] :
% 0.34/0.55 ( ( equiv_6726761681920839555_nat_o @ A @ R2 )
% 0.34/0.55 => ( ( ord_le2965882846123202637_nat_o @ ( image_2232984160439939859_nat_o @ R2 @ ( insert5175938949040314269_nat_o @ B @ bot_bo7824918357723582233_nat_o ) ) @ ( image_2232984160439939859_nat_o @ R2 @ ( insert5175938949040314269_nat_o @ A2 @ bot_bo7824918357723582233_nat_o ) ) )
% 0.34/0.55 => ( ( member6576561426505652726_nat_o @ B @ A )
% 0.34/0.55 => ( member8781333585448626064_nat_o @ ( produc7368190662567826135_nat_o @ A2 @ B ) @ R2 ) ) ) ) ).
% 0.34/0.55
% 0.34/0.55 % subset_equiv_class
% 0.34/0.55 thf(fact_1791_subset__equiv__class,axiom,
% 0.34/0.55 ! [A: set_nat,R2: set_Pr1261947904930325089at_nat,B: nat,A2: nat] :
% 0.34/0.55 ( ( equiv_equiv_nat @ A @ R2 )
% 0.34/0.55 => ( ( ord_less_eq_set_nat @ ( image_nat_nat @ R2 @ ( insert_nat @ B @ bot_bot_set_nat ) ) @ ( image_nat_nat @ R2 @ ( insert_nat @ A2 @ bot_bot_set_nat ) ) )
% 0.34/0.55 => ( ( member_nat @ B @ A )
% 0.34/0.55 => ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A2 @ B ) @ R2 ) ) ) ) ).
% 0.34/0.55
% 0.34/0.55 % subset_equiv_class
% 0.34/0.55 thf(fact_1792_finite__finite__vimage__IntI,axiom,
% 0.34/0.55 ! [F2: set_Code_integer,H2: code_integer > code_integer,A: set_Code_integer] :
% 0.34/0.55 ( ( finite6017078050557962740nteger @ F2 )
% 0.34/0.55 => ( ! [Y2: code_integer] :
% 0.34/0.55 ( ( member_Code_integer @ Y2 @ F2 )
% 0.34/0.55 => ( finite6017078050557962740nteger @ ( inf_in1364745209274528805nteger @ ( vimage1318793397690208119nteger @ H2 @ ( insert_Code_integer @ Y2 @ bot_bo3990330152332043303nteger ) ) @ A ) ) )
% 0.34/0.55 => ( finite6017078050557962740nteger @ ( inf_in1364745209274528805nteger @ ( vimage1318793397690208119nteger @ H2 @ F2 ) @ A ) ) ) ) ).
% 0.34/0.55
% 0.34/0.55 % finite_finite_vimage_IntI
% 0.34/0.55 thf(fact_1793_finite__finite__vimage__IntI,axiom,
% 0.34/0.55 ! [F2: set_Code_integer,H2: nat > code_integer,A: set_nat] :
% 0.34/0.55 ( ( finite6017078050557962740nteger @ F2 )
% 0.34/0.55 => ( ! [Y2: code_integer] :
% 0.34/0.55 ( ( member_Code_integer @ Y2 @ F2 )
% 0.34/0.55 => ( finite_finite_nat @ ( inf_inf_set_nat @ ( vimage6104394758324456170nteger @ H2 @ ( insert_Code_integer @ Y2 @ bot_bo3990330152332043303nteger ) ) @ A ) ) )
% 0.34/0.55 => ( finite_finite_nat @ ( inf_inf_set_nat @ ( vimage6104394758324456170nteger @ H2 @ F2 ) @ A ) ) ) ) ).
% 0.34/0.55
% 0.34/0.55 % finite_finite_vimage_IntI
% 0.34/0.55 thf(fact_1794_finite__finite__vimage__IntI,axiom,
% 0.34/0.55 ! [F2: set_Product_unit,H2: code_integer > product_unit,A: set_Code_integer] :
% 0.34/0.55 ( ( finite4290736615968046902t_unit @ F2 )
% 0.34/0.55 => ( ! [Y2: product_unit] :
% 0.34/0.55 ( ( member_Product_unit @ Y2 @ F2 )
% 0.34/0.55 => ( finite6017078050557962740nteger @ ( inf_in1364745209274528805nteger @ ( vimage5651688891684837235t_unit @ H2 @ ( insert_Product_unit @ Y2 @ bot_bo3957492148770167129t_unit ) ) @ A ) ) )
% 0.34/0.55 => ( finite6017078050557962740nteger @ ( inf_in1364745209274528805nteger @ ( vimage5651688891684837235t_unit @ H2 @ F2 ) @ A ) ) ) ) ).
% 0.34/0.55
% 0.34/0.55 % finite_finite_vimage_IntI
% 0.34/0.55 thf(fact_1795_finite__finite__vimage__IntI,axiom,
% 0.34/0.55 ! [F2: set_Product_unit,H2: nat > product_unit,A: set_nat] :
% 0.34/0.55 ( ( finite4290736615968046902t_unit @ F2 )
% 0.34/0.55 => ( ! [Y2: product_unit] :
% 0.34/0.55 ( ( member_Product_unit @ Y2 @ F2 )
% 0.34/0.55 => ( finite_finite_nat @ ( inf_inf_set_nat @ ( vimage4884490618288580032t_unit @ H2 @ ( insert_Product_unit @ Y2 @ bot_bo3957492148770167129t_unit ) ) @ A ) ) )
% 0.34/0.55 => ( finite_finite_nat @ ( inf_inf_set_nat @ ( vimage4884490618288580032t_unit @ H2 @ F2 ) @ A ) ) ) ) ).
% 0.34/0.55
% 0.34/0.55 % finite_finite_vimage_IntI
% 0.34/0.55 thf(fact_1796_finite__finite__vimage__IntI,axiom,
% 0.34/0.55 ! [F2: set_o,H2: code_integer > $o,A: set_Code_integer] :
% 0.34/0.55 ( ( finite_finite_o @ F2 )
% 0.34/0.55 => ( ! [Y2: $o] :
% 0.34/0.55 ( ( member_o @ Y2 @ F2 )
% 0.34/0.55 => ( finite6017078050557962740nteger @ ( inf_in1364745209274528805nteger @ ( vimage3005800680117336574eger_o @ H2 @ ( insert_o @ Y2 @ bot_bot_set_o ) ) @ A ) ) )
% 0.34/0.55 => ( finite6017078050557962740nteger @ ( inf_in1364745209274528805nteger @ ( vimage3005800680117336574eger_o @ H2 @ F2 ) @ A ) ) ) ) ).
% 0.34/0.55
% 0.34/0.55 % finite_finite_vimage_IntI
% 0.34/0.55 thf(fact_1797_finite__finite__vimage__IntI,axiom,
% 0.34/0.55 ! [F2: set_o,H2: nat > $o,A: set_nat] :
% 0.34/0.55 ( ( finite_finite_o @ F2 )
% 0.34/0.55 => ( ! [Y2: $o] :
% 0.34/0.55 ( ( member_o @ Y2 @ F2 )
% 0.34/0.55 => ( finite_finite_nat @ ( inf_inf_set_nat @ ( vimage_nat_o @ H2 @ ( insert_o @ Y2 @ bot_bot_set_o ) ) @ A ) ) )
% 0.34/0.55 => ( finite_finite_nat @ ( inf_inf_set_nat @ ( vimage_nat_o @ H2 @ F2 ) @ A ) ) ) ) ).
% 0.34/0.55
% 0.34/0.55 % finite_finite_vimage_IntI
% 0.34/0.55 thf(fact_1798_finite__finite__vimage__IntI,axiom,
% 0.34/0.55 ! [F2: set_nat,H2: code_integer > nat,A: set_Code_integer] :
% 0.34/0.55 ( ( finite_finite_nat @ F2 )
% 0.34/0.55 => ( ! [Y2: nat] :
% 0.34/0.55 ( ( member_nat @ Y2 @ F2 )
% 0.34/0.55 => ( finite6017078050557962740nteger @ ( inf_in1364745209274528805nteger @ ( vimage5839839309545413354er_nat @ H2 @ ( insert_nat @ Y2 @ bot_bot_set_nat ) ) @ A ) ) )
% 0.34/0.55 => ( finite6017078050557962740nteger @ ( inf_in1364745209274528805nteger @ ( vimage5839839309545413354er_nat @ H2 @ F2 ) @ A ) ) ) ) ).
% 0.34/0.55
% 0.34/0.55 % finite_finite_vimage_IntI
% 0.34/0.55 thf(fact_1799_finite__finite__vimage__IntI,axiom,
% 0.34/0.55 ! [F2: set_nat,H2: nat > nat,A: set_nat] :
% 0.34/0.55 ( ( finite_finite_nat @ F2 )
% 0.34/0.55 => ( ! [Y2: nat] :
% 0.34/0.55 ( ( member_nat @ Y2 @ F2 )
% 0.34/0.55 => ( finite_finite_nat @ ( inf_inf_set_nat @ ( vimage_nat_nat @ H2 @ ( insert_nat @ Y2 @ bot_bot_set_nat ) ) @ A ) ) )
% 0.34/0.55 => ( finite_finite_nat @ ( inf_inf_set_nat @ ( vimage_nat_nat @ H2 @ F2 ) @ A ) ) ) ) ).
% 0.34/0.55
% 0.34/0.55 % finite_finite_vimage_IntI
% 0.34/0.55 thf(fact_1800_finite__finite__vimage__IntI,axiom,
% 0.34/0.55 ! [F2: set_int,H2: code_integer > int,A: set_Code_integer] :
% 0.34/0.55 ( ( finite_finite_int @ F2 )
% 0.34/0.55 => ( ! [Y2: int] :
% 0.34/0.55 ( ( member_int @ Y2 @ F2 )
% 0.34/0.55 => ( finite6017078050557962740nteger @ ( inf_in1364745209274528805nteger @ ( vimage5837348839036363078er_int @ H2 @ ( insert_int @ Y2 @ bot_bot_set_int ) ) @ A ) ) )
% 0.34/0.55 => ( finite6017078050557962740nteger @ ( inf_in1364745209274528805nteger @ ( vimage5837348839036363078er_int @ H2 @ F2 ) @ A ) ) ) ) ).
% 0.34/0.55
% 0.34/0.55 % finite_finite_vimage_IntI
% 0.34/0.55 thf(fact_1801_finite__finite__vimage__IntI,axiom,
% 0.34/0.55 ! [F2: set_int,H2: nat > int,A: set_nat] :
% 0.34/0.55 ( ( finite_finite_int @ F2 )
% 0.34/0.55 => ( ! [Y2: int] :
% 0.34/0.55 ( ( member_int @ Y2 @ F2 )
% 0.34/0.55 => ( finite_finite_nat @ ( inf_inf_set_nat @ ( vimage_nat_int @ H2 @ ( insert_int @ Y2 @ bot_bot_set_int ) ) @ A ) ) )
% 0.34/0.55 => ( finite_finite_nat @ ( inf_inf_set_nat @ ( vimage_nat_int @ H2 @ F2 ) @ A ) ) ) ) ).
% 0.34/0.55
% 0.34/0.55 % finite_finite_vimage_IntI
% 0.34/0.55 thf(fact_1802_finite__insert,axiom,
% 0.34/0.55 ! [A2: int,A: set_int] :
% 0.34/0.55 ( ( finite_finite_int @ ( insert_int @ A2 @ A ) )
% 0.34/0.55 = ( finite_finite_int @ A ) ) ).
% 0.34/0.55
% 0.34/0.55 % finite_insert
% 0.34/0.55 thf(fact_1803_finite__insert,axiom,
% 0.34/0.55 ! [A2: $o,A: set_o] :
% 0.34/0.55 ( ( finite_finite_o @ ( insert_o @ A2 @ A ) )
% 0.34/0.55 = ( finite_finite_o @ A ) ) ).
% 0.34/0.55
% 0.34/0.55 % finite_insert
% 0.34/0.55 thf(fact_1804_finite__insert,axiom,
% 0.34/0.55 ! [A2: product_unit,A: set_Product_unit] :
% 0.34/0.55 ( ( finite4290736615968046902t_unit @ ( insert_Product_unit @ A2 @ A ) )
% 0.34/0.55 = ( finite4290736615968046902t_unit @ A ) ) ).
% 0.34/0.55
% 0.34/0.55 % finite_insert
% 0.34/0.55 thf(fact_1805_finite__insert,axiom,
% 0.34/0.55 ! [A2: nat,A: set_nat] :
% 0.34/0.55 ( ( finite_finite_nat @ ( insert_nat @ A2 @ A ) )
% 0.34/0.55 = ( finite_finite_nat @ A ) ) ).
% 0.34/0.55
% 0.34/0.55 % finite_insert
% 0.34/0.55 thf(fact_1806_finite__insert,axiom,
% 0.34/0.55 ! [A2: code_integer,A: set_Code_integer] :
% 0.34/0.55 ( ( finite6017078050557962740nteger @ ( insert_Code_integer @ A2 @ A ) )
% 0.34/0.55 = ( finite6017078050557962740nteger @ A ) ) ).
% 0.34/0.55
% 0.34/0.55 % finite_insert
% 0.34/0.55 thf(fact_1807_finite__subset__induct_H,axiom,
% 0.34/0.55 ! [F2: set_Code_integer,A: set_Code_integer,P: set_Code_integer > $o] :
% 0.34/0.55 ( ( finite6017078050557962740nteger @ F2 )
% 0.34/0.55 => ( ( ord_le7084787975880047091nteger @ F2 @ A )
% 0.34/0.55 => ( ( P @ bot_bo3990330152332043303nteger )
% 0.34/0.55 => ( ! [A4: code_integer,F3: set_Code_integer] :
% 0.34/0.55 ( ( finite6017078050557962740nteger @ F3 )
% 0.34/0.55 => ( ( member_Code_integer @ A4 @ A )
% 0.34/0.55 => ( ( ord_le7084787975880047091nteger @ F3 @ A )
% 0.34/0.55 => ( ~ ( member_Code_integer @ A4 @ F3 )
% 0.34/0.55 => ( ( P @ F3 )
% 0.34/0.55 => ( P @ ( insert_Code_integer @ A4 @ F3 ) ) ) ) ) ) )
% 0.34/0.55 => ( P @ F2 ) ) ) ) ) ).
% 0.34/0.55
% 0.34/0.55 % finite_subset_induct'
% 0.34/0.55 thf(fact_1808_finite__subset__induct_H,axiom,
% 0.34/0.55 ! [F2: set_Product_unit,A: set_Product_unit,P: set_Product_unit > $o] :
% 0.34/0.55 ( ( finite4290736615968046902t_unit @ F2 )
% 0.34/0.55 => ( ( ord_le3507040750410214029t_unit @ F2 @ A )
% 0.34/0.55 => ( ( P @ bot_bo3957492148770167129t_unit )
% 0.34/0.55 => ( ! [A4: product_unit,F3: set_Product_unit] :
% 0.34/0.55 ( ( finite4290736615968046902t_unit @ F3 )
% 0.34/0.55 => ( ( member_Product_unit @ A4 @ A )
% 0.34/0.55 => ( ( ord_le3507040750410214029t_unit @ F3 @ A )
% 0.34/0.55 => ( ~ ( member_Product_unit @ A4 @ F3 )
% 0.34/0.55 => ( ( P @ F3 )
% 0.34/0.55 => ( P @ ( insert_Product_unit @ A4 @ F3 ) ) ) ) ) ) )
% 0.34/0.55 => ( P @ F2 ) ) ) ) ) ).
% 0.34/0.55
% 0.34/0.55 % finite_subset_induct'
% 0.34/0.55 thf(fact_1809_finite__subset__induct_H,axiom,
% 0.34/0.55 ! [F2: set_o,A: set_o,P: set_o > $o] :
% 0.34/0.55 ( ( finite_finite_o @ F2 )
% 0.34/0.55 => ( ( ord_less_eq_set_o @ F2 @ A )
% 0.34/0.55 => ( ( P @ bot_bot_set_o )
% 0.34/0.55 => ( ! [A4: $o,F3: set_o] :
% 0.34/0.55 ( ( finite_finite_o @ F3 )
% 0.34/0.55 => ( ( member_o @ A4 @ A )
% 0.34/0.55 => ( ( ord_less_eq_set_o @ F3 @ A )
% 0.34/0.55 => ( ~ ( member_o @ A4 @ F3 )
% 0.34/0.55 => ( ( P @ F3 )
% 0.34/0.55 => ( P @ ( insert_o @ A4 @ F3 ) ) ) ) ) ) )
% 0.34/0.55 => ( P @ F2 ) ) ) ) ) ).
% 0.34/0.55
% 0.34/0.55 % finite_subset_induct'
% 0.34/0.55 thf(fact_1810_finite__subset__induct_H,axiom,
% 0.34/0.55 ! [F2: set_int,A: set_int,P: set_int > $o] :
% 0.34/0.55 ( ( finite_finite_int @ F2 )
% 0.34/0.55 => ( ( ord_less_eq_set_int @ F2 @ A )
% 0.34/0.55 => ( ( P @ bot_bot_set_int )
% 0.34/0.55 => ( ! [A4: int,F3: set_int] :
% 0.34/0.55 ( ( finite_finite_int @ F3 )
% 0.34/0.55 => ( ( member_int @ A4 @ A )
% 0.34/0.55 => ( ( ord_less_eq_set_int @ F3 @ A )
% 0.34/0.55 => ( ~ ( member_int @ A4 @ F3 )
% 0.34/0.55 => ( ( P @ F3 )
% 0.34/0.55 => ( P @ ( insert_int @ A4 @ F3 ) ) ) ) ) ) )
% 0.34/0.55 => ( P @ F2 ) ) ) ) ) ).
% 0.34/0.55
% 0.34/0.55 % finite_subset_induct'
% 0.34/0.55 thf(fact_1811_finite__subset__induct_H,axiom,
% 0.34/0.55 ! [F2: set_Pr4532377907799695533_nat_o,A: set_Pr4532377907799695533_nat_o,P: set_Pr4532377907799695533_nat_o > $o] :
% 0.34/0.55 ( ( finite3252695134891459830_nat_o @ F2 )
% 0.34/0.55 => ( ( ord_le2965882846123202637_nat_o @ F2 @ A )
% 0.34/0.55 => ( ( P @ bot_bo7824918357723582233_nat_o )
% 0.34/0.55 => ( ! [A4: produc3658429121746597890et_nat > $o,F3: set_Pr4532377907799695533_nat_o] :
% 0.34/0.55 ( ( finite3252695134891459830_nat_o @ F3 )
% 0.34/0.55 => ( ( member6576561426505652726_nat_o @ A4 @ A )
% 0.34/0.55 => ( ( ord_le2965882846123202637_nat_o @ F3 @ A )
% 0.34/0.55 => ( ~ ( member6576561426505652726_nat_o @ A4 @ F3 )
% 0.34/0.55 => ( ( P @ F3 )
% 0.34/0.55 => ( P @ ( insert5175938949040314269_nat_o @ A4 @ F3 ) ) ) ) ) ) )
% 0.34/0.55 => ( P @ F2 ) ) ) ) ) ).
% 0.34/0.55
% 0.34/0.55 % finite_subset_induct'
% 0.34/0.55 thf(fact_1812_finite__subset__induct_H,axiom,
% 0.34/0.55 ! [F2: set_nat,A: set_nat,P: set_nat > $o] :
% 0.34/0.55 ( ( finite_finite_nat @ F2 )
% 0.34/0.55 => ( ( ord_less_eq_set_nat @ F2 @ A )
% 0.34/0.55 => ( ( P @ bot_bot_set_nat )
% 0.34/0.55 => ( ! [A4: nat,F3: set_nat] :
% 0.34/0.55 ( ( finite_finite_nat @ F3 )
% 0.34/0.55 => ( ( member_nat @ A4 @ A )
% 0.34/0.55 => ( ( ord_less_eq_set_nat @ F3 @ A )
% 0.34/0.55 => ( ~ ( member_nat @ A4 @ F3 )
% 0.34/0.55 => ( ( P @ F3 )
% 0.34/0.55 => ( P @ ( insert_nat @ A4 @ F3 ) ) ) ) ) ) )
% 0.34/0.55 => ( P @ F2 ) ) ) ) ) ).
% 0.34/0.55
% 0.34/0.55 % finite_subset_induct'
% 0.34/0.55 thf(fact_1813_finite__subset__induct,axiom,
% 0.34/0.55 ! [F2: set_Code_integer,A: set_Code_integer,P: set_Code_integer > $o] :
% 0.34/0.55 ( ( finite6017078050557962740nteger @ F2 )
% 0.34/0.55 => ( ( ord_le7084787975880047091nteger @ F2 @ A )
% 0.34/0.55 => ( ( P @ bot_bo3990330152332043303nteger )
% 0.34/0.55 => ( ! [A4: code_integer,F3: set_Code_integer] :
% 0.34/0.55 ( ( finite6017078050557962740nteger @ F3 )
% 0.34/0.55 => ( ( member_Code_integer @ A4 @ A )
% 0.34/0.55 => ( ~ ( member_Code_integer @ A4 @ F3 )
% 0.34/0.55 => ( ( P @ F3 )
% 0.34/0.55 => ( P @ ( insert_Code_integer @ A4 @ F3 ) ) ) ) ) )
% 0.34/0.55 => ( P @ F2 ) ) ) ) ) ).
% 0.34/0.55
% 0.34/0.55 % finite_subset_induct
% 0.34/0.55 thf(fact_1814_finite__subset__induct,axiom,
% 0.34/0.55 ! [F2: set_Product_unit,A: set_Product_unit,P: set_Product_unit > $o] :
% 0.34/0.55 ( ( finite4290736615968046902t_unit @ F2 )
% 0.34/0.55 => ( ( ord_le3507040750410214029t_unit @ F2 @ A )
% 0.34/0.55 => ( ( P @ bot_bo3957492148770167129t_unit )
% 0.34/0.55 => ( ! [A4: product_unit,F3: set_Product_unit] :
% 0.34/0.55 ( ( finite4290736615968046902t_unit @ F3 )
% 0.34/0.55 => ( ( member_Product_unit @ A4 @ A )
% 0.34/0.55 => ( ~ ( member_Product_unit @ A4 @ F3 )
% 0.34/0.55 => ( ( P @ F3 )
% 0.34/0.55 => ( P @ ( insert_Product_unit @ A4 @ F3 ) ) ) ) ) )
% 0.34/0.55 => ( P @ F2 ) ) ) ) ) ).
% 0.34/0.55
% 0.34/0.55 % finite_subset_induct
% 0.34/0.55 thf(fact_1815_finite__subset__induct,axiom,
% 0.34/0.55 ! [F2: set_o,A: set_o,P: set_o > $o] :
% 0.34/0.55 ( ( finite_finite_o @ F2 )
% 0.34/0.55 => ( ( ord_less_eq_set_o @ F2 @ A )
% 0.34/0.55 => ( ( P @ bot_bot_set_o )
% 0.34/0.55 => ( ! [A4: $o,F3: set_o] :
% 0.34/0.55 ( ( finite_finite_o @ F3 )
% 0.34/0.55 => ( ( member_o @ A4 @ A )
% 0.34/0.55 => ( ~ ( member_o @ A4 @ F3 )
% 0.34/0.55 => ( ( P @ F3 )
% 0.34/0.55 => ( P @ ( insert_o @ A4 @ F3 ) ) ) ) ) )
% 0.34/0.55 => ( P @ F2 ) ) ) ) ) ).
% 0.34/0.55
% 0.34/0.55 % finite_subset_induct
% 0.34/0.55 thf(fact_1816_finite__subset__induct,axiom,
% 0.34/0.55 ! [F2: set_int,A: set_int,P: set_int > $o] :
% 0.34/0.55 ( ( finite_finite_int @ F2 )
% 0.34/0.55 => ( ( ord_less_eq_set_int @ F2 @ A )
% 0.34/0.55 => ( ( P @ bot_bot_set_int )
% 0.34/0.55 => ( ! [A4: int,F3: set_int] :
% 0.34/0.55 ( ( finite_finite_int @ F3 )
% 0.34/0.55 => ( ( member_int @ A4 @ A )
% 0.34/0.55 => ( ~ ( member_int @ A4 @ F3 )
% 0.34/0.55 => ( ( P @ F3 )
% 0.34/0.55 => ( P @ ( insert_int @ A4 @ F3 ) ) ) ) ) )
% 0.34/0.55 => ( P @ F2 ) ) ) ) ) ).
% 0.34/0.55
% 0.34/0.55 % finite_subset_induct
% 0.34/0.55 thf(fact_1817_finite__subset__induct,axiom,
% 0.34/0.55 ! [F2: set_Pr4532377907799695533_nat_o,A: set_Pr4532377907799695533_nat_o,P: set_Pr4532377907799695533_nat_o > $o] :
% 0.34/0.55 ( ( finite3252695134891459830_nat_o @ F2 )
% 0.34/0.55 => ( ( ord_le2965882846123202637_nat_o @ F2 @ A )
% 0.34/0.55 => ( ( P @ bot_bo7824918357723582233_nat_o )
% 0.34/0.55 => ( ! [A4: produc3658429121746597890et_nat > $o,F3: set_Pr4532377907799695533_nat_o] :
% 0.34/0.55 ( ( finite3252695134891459830_nat_o @ F3 )
% 0.34/0.55 => ( ( member6576561426505652726_nat_o @ A4 @ A )
% 0.34/0.55 => ( ~ ( member6576561426505652726_nat_o @ A4 @ F3 )
% 0.34/0.55 => ( ( P @ F3 )
% 0.34/0.55 => ( P @ ( insert5175938949040314269_nat_o @ A4 @ F3 ) ) ) ) ) )
% 0.34/0.55 => ( P @ F2 ) ) ) ) ) ).
% 0.34/0.55
% 0.34/0.55 % finite_subset_induct
% 0.34/0.55 thf(fact_1818_finite__subset__induct,axiom,
% 0.34/0.55 ! [F2: set_nat,A: set_nat,P: set_nat > $o] :
% 0.34/0.55 ( ( finite_finite_nat @ F2 )
% 0.34/0.55 => ( ( ord_less_eq_set_nat @ F2 @ A )
% 0.34/0.55 => ( ( P @ bot_bot_set_nat )
% 0.34/0.55 => ( ! [A4: nat,F3: set_nat] :
% 0.34/0.55 ( ( finite_finite_nat @ F3 )
% 0.34/0.55 => ( ( member_nat @ A4 @ A )
% 0.34/0.55 => ( ~ ( member_nat @ A4 @ F3 )
% 0.34/0.55 => ( ( P @ F3 )
% 0.34/0.55 => ( P @ ( insert_nat @ A4 @ F3 ) ) ) ) ) )
% 0.34/0.55 => ( P @ F2 ) ) ) ) ) ).
% 0.34/0.55
% 0.34/0.55 % finite_subset_induct
% 0.34/0.55 thf(fact_1819_finite__ranking__induct,axiom,
% 0.34/0.55 ! [S: set_Code_integer,P: set_Code_integer > $o,F: code_integer > rat] :
% 0.34/0.55 ( ( finite6017078050557962740nteger @ S )
% 0.34/0.55 => ( ( P @ bot_bo3990330152332043303nteger )
% 0.34/0.55 => ( ! [X3: code_integer,S5: set_Code_integer] :
% 0.34/0.55 ( ( finite6017078050557962740nteger @ S5 )
% 0.34/0.55 => ( ! [Y5: code_integer] :
% 0.34/0.55 ( ( member_Code_integer @ Y5 @ S5 )
% 0.34/0.55 => ( ord_less_eq_rat @ ( F @ Y5 ) @ ( F @ X3 ) ) )
% 0.34/0.55 => ( ( P @ S5 )
% 0.34/0.55 => ( P @ ( insert_Code_integer @ X3 @ S5 ) ) ) ) )
% 0.34/0.55 => ( P @ S ) ) ) ) ).
% 0.34/0.55
% 0.34/0.55 % finite_ranking_induct
% 0.34/0.55 thf(fact_1820_finite__ranking__induct,axiom,
% 0.34/0.55 ! [S: set_Product_unit,P: set_Product_unit > $o,F: product_unit > rat] :
% 0.34/0.55 ( ( finite4290736615968046902t_unit @ S )
% 0.34/0.55 => ( ( P @ bot_bo3957492148770167129t_unit )
% 0.34/0.55 => ( ! [X3: product_unit,S5: set_Product_unit] :
% 0.34/0.55 ( ( finite4290736615968046902t_unit @ S5 )
% 0.34/0.55 => ( ! [Y5: product_unit] :
% 0.34/0.55 ( ( member_Product_unit @ Y5 @ S5 )
% 0.34/0.55 => ( ord_less_eq_rat @ ( F @ Y5 ) @ ( F @ X3 ) ) )
% 0.34/0.55 => ( ( P @ S5 )
% 0.34/0.55 => ( P @ ( insert_Product_unit @ X3 @ S5 ) ) ) ) )
% 0.34/0.55 => ( P @ S ) ) ) ) ).
% 0.34/0.55
% 0.34/0.55 % finite_ranking_induct
% 0.34/0.55 thf(fact_1821_finite__ranking__induct,axiom,
% 0.34/0.55 ! [S: set_o,P: set_o > $o,F: $o > rat] :
% 0.34/0.55 ( ( finite_finite_o @ S )
% 0.34/0.55 => ( ( P @ bot_bot_set_o )
% 0.34/0.55 => ( ! [X3: $o,S5: set_o] :
% 0.34/0.55 ( ( finite_finite_o @ S5 )
% 0.34/0.55 => ( ! [Y5: $o] :
% 0.34/0.55 ( ( member_o @ Y5 @ S5 )
% 0.34/0.55 => ( ord_less_eq_rat @ ( F @ Y5 ) @ ( F @ X3 ) ) )
% 0.34/0.55 => ( ( P @ S5 )
% 0.34/0.55 => ( P @ ( insert_o @ X3 @ S5 ) ) ) ) )
% 0.34/0.55 => ( P @ S ) ) ) ) ).
% 0.34/0.55
% 0.34/0.55 % finite_ranking_induct
% 0.34/0.55 thf(fact_1822_finite__ranking__induct,axiom,
% 0.34/0.55 ! [S: set_nat,P: set_nat > $o,F: nat > rat] :
% 0.34/0.55 ( ( finite_finite_nat @ S )
% 0.34/0.55 => ( ( P @ bot_bot_set_nat )
% 0.34/0.55 => ( ! [X3: nat,S5: set_nat] :
% 0.34/0.55 ( ( finite_finite_nat @ S5 )
% 0.34/0.55 => ( ! [Y5: nat] :
% 0.34/0.55 ( ( member_nat @ Y5 @ S5 )
% 0.34/0.55 => ( ord_less_eq_rat @ ( F @ Y5 ) @ ( F @ X3 ) ) )
% 0.34/0.55 => ( ( P @ S5 )
% 0.34/0.55 => ( P @ ( insert_nat @ X3 @ S5 ) ) ) ) )
% 0.34/0.55 => ( P @ S ) ) ) ) ).
% 0.34/0.55
% 0.34/0.55 % finite_ranking_induct
% 0.34/0.55 thf(fact_1823_finite__ranking__induct,axiom,
% 0.34/0.55 ! [S: set_int,P: set_int > $o,F: int > rat] :
% 0.34/0.55 ( ( finite_finite_int @ S )
% 0.34/0.55 => ( ( P @ bot_bot_set_int )
% 0.34/0.55 => ( ! [X3: int,S5: set_int] :
% 0.34/0.55 ( ( finite_finite_int @ S5 )
% 0.34/0.55 => ( ! [Y5: int] :
% 0.34/0.55 ( ( member_int @ Y5 @ S5 )
% 0.34/0.55 => ( ord_less_eq_rat @ ( F @ Y5 ) @ ( F @ X3 ) ) )
% 0.34/0.55 => ( ( P @ S5 )
% 0.34/0.55 => ( P @ ( insert_int @ X3 @ S5 ) ) ) ) )
% 0.34/0.55 => ( P @ S ) ) ) ) ).
% 0.34/0.55
% 0.34/0.55 % finite_ranking_induct
% 0.34/0.55 thf(fact_1824_finite__ranking__induct,axiom,
% 0.34/0.55 ! [S: set_Code_integer,P: set_Code_integer > $o,F: code_integer > nat] :
% 0.34/0.55 ( ( finite6017078050557962740nteger @ S )
% 0.34/0.55 => ( ( P @ bot_bo3990330152332043303nteger )
% 0.34/0.55 => ( ! [X3: code_integer,S5: set_Code_integer] :
% 0.34/0.55 ( ( finite6017078050557962740nteger @ S5 )
% 0.34/0.55 => ( ! [Y5: code_integer] :
% 0.34/0.55 ( ( member_Code_integer @ Y5 @ S5 )
% 0.34/0.55 => ( ord_less_eq_nat @ ( F @ Y5 ) @ ( F @ X3 ) ) )
% 0.34/0.55 => ( ( P @ S5 )
% 0.34/0.55 => ( P @ ( insert_Code_integer @ X3 @ S5 ) ) ) ) )
% 0.34/0.55 => ( P @ S ) ) ) ) ).
% 0.34/0.55
% 0.34/0.55 % finite_ranking_induct
% 0.34/0.55 thf(fact_1825_finite__ranking__induct,axiom,
% 0.34/0.55 ! [S: set_Product_unit,P: set_Product_unit > $o,F: product_unit > nat] :
% 0.34/0.55 ( ( finite4290736615968046902t_unit @ S )
% 0.34/0.55 => ( ( P @ bot_bo3957492148770167129t_unit )
% 0.34/0.55 => ( ! [X3: product_unit,S5: set_Product_unit] :
% 0.34/0.55 ( ( finite4290736615968046902t_unit @ S5 )
% 0.34/0.55 => ( ! [Y5: product_unit] :
% 0.34/0.55 ( ( member_Product_unit @ Y5 @ S5 )
% 0.34/0.55 => ( ord_less_eq_nat @ ( F @ Y5 ) @ ( F @ X3 ) ) )
% 0.34/0.55 => ( ( P @ S5 )
% 0.34/0.55 => ( P @ ( insert_Product_unit @ X3 @ S5 ) ) ) ) )
% 0.34/0.55 => ( P @ S ) ) ) ) ).
% 0.34/0.55
% 0.34/0.55 % finite_ranking_induct
% 0.34/0.55 thf(fact_1826_finite__ranking__induct,axiom,
% 0.34/0.55 ! [S: set_o,P: set_o > $o,F: $o > nat] :
% 0.34/0.55 ( ( finite_finite_o @ S )
% 0.34/0.55 => ( ( P @ bot_bot_set_o )
% 0.34/0.55 => ( ! [X3: $o,S5: set_o] :
% 0.34/0.55 ( ( finite_finite_o @ S5 )
% 0.34/0.55 => ( ! [Y5: $o] :
% 0.34/0.55 ( ( member_o @ Y5 @ S5 )
% 0.34/0.55 => ( ord_less_eq_nat @ ( F @ Y5 ) @ ( F @ X3 ) ) )
% 0.34/0.55 => ( ( P @ S5 )
% 0.34/0.55 => ( P @ ( insert_o @ X3 @ S5 ) ) ) ) )
% 0.34/0.55 => ( P @ S ) ) ) ) ).
% 0.34/0.55
% 0.34/0.55 % finite_ranking_induct
% 0.34/0.55 thf(fact_1827_finite__ranking__induct,axiom,
% 0.34/0.55 ! [S: set_nat,P: set_nat > $o,F: nat > nat] :
% 0.34/0.55 ( ( finite_finite_nat @ S )
% 0.34/0.55 => ( ( P @ bot_bot_set_nat )
% 0.34/0.55 => ( ! [X3: nat,S5: set_nat] :
% 0.34/0.55 ( ( finite_finite_nat @ S5 )
% 0.34/0.55 => ( ! [Y5: nat] :
% 0.34/0.55 ( ( member_nat @ Y5 @ S5 )
% 0.34/0.55 => ( ord_less_eq_nat @ ( F @ Y5 ) @ ( F @ X3 ) ) )
% 0.34/0.55 => ( ( P @ S5 )
% 0.34/0.55 => ( P @ ( insert_nat @ X3 @ S5 ) ) ) ) )
% 0.34/0.55 => ( P @ S ) ) ) ) ).
% 0.34/0.55
% 0.34/0.55 % finite_ranking_induct
% 0.34/0.55 thf(fact_1828_finite__ranking__induct,axiom,
% 0.34/0.55 ! [S: set_int,P: set_int > $o,F: int > nat] :
% 0.34/0.55 ( ( finite_finite_int @ S )
% 0.34/0.55 => ( ( P @ bot_bot_set_int )
% 0.34/0.55 => ( ! [X3: int,S5: set_int] :
% 0.34/0.55 ( ( finite_finite_int @ S5 )
% 0.34/0.55 => ( ! [Y5: int] :
% 0.34/0.55 ( ( member_int @ Y5 @ S5 )
% 0.34/0.55 => ( ord_less_eq_nat @ ( F @ Y5 ) @ ( F @ X3 ) ) )
% 0.34/0.55 => ( ( P @ S5 )
% 0.34/0.55 => ( P @ ( insert_int @ X3 @ S5 ) ) ) ) )
% 0.34/0.55 => ( P @ S ) ) ) ) ).
% 0.34/0.55
% 0.34/0.55 % finite_ranking_induct
% 0.34/0.55 thf(fact_1829_infinite__finite__induct,axiom,
% 0.34/0.55 ! [P: set_Code_integer > $o,A: set_Code_integer] :
% 0.34/0.55 ( ! [A9: set_Code_integer] :
% 0.34/0.55 ( ~ ( finite6017078050557962740nteger @ A9 )
% 0.34/0.55 => ( P @ A9 ) )
% 0.34/0.55 => ( ( P @ bot_bo3990330152332043303nteger )
% 0.34/0.55 => ( ! [X3: code_integer,F3: set_Code_integer] :
% 0.34/0.55 ( ( finite6017078050557962740nteger @ F3 )
% 0.34/0.55 => ( ~ ( member_Code_integer @ X3 @ F3 )
% 0.34/0.55 => ( ( P @ F3 )
% 0.34/0.55 => ( P @ ( insert_Code_integer @ X3 @ F3 ) ) ) ) )
% 0.34/0.55 => ( P @ A ) ) ) ) ).
% 0.34/0.55
% 0.34/0.55 % infinite_finite_induct
% 0.34/0.55 thf(fact_1830_infinite__finite__induct,axiom,
% 0.34/0.55 ! [P: set_Product_unit > $o,A: set_Product_unit] :
% 0.34/0.55 ( ! [A9: set_Product_unit] :
% 0.34/0.55 ( ~ ( finite4290736615968046902t_unit @ A9 )
% 0.34/0.55 => ( P @ A9 ) )
% 0.34/0.55 => ( ( P @ bot_bo3957492148770167129t_unit )
% 0.34/0.55 => ( ! [X3: product_unit,F3: set_Product_unit] :
% 0.34/0.55 ( ( finite4290736615968046902t_unit @ F3 )
% 0.34/0.55 => ( ~ ( member_Product_unit @ X3 @ F3 )
% 0.34/0.55 => ( ( P @ F3 )
% 0.34/0.55 => ( P @ ( insert_Product_unit @ X3 @ F3 ) ) ) ) )
% 0.34/0.55 => ( P @ A ) ) ) ) ).
% 0.34/0.55
% 0.34/0.55 % infinite_finite_induct
% 0.34/0.55 thf(fact_1831_infinite__finite__induct,axiom,
% 0.34/0.55 ! [P: set_o > $o,A: set_o] :
% 0.34/0.55 ( ! [A9: set_o] :
% 0.34/0.55 ( ~ ( finite_finite_o @ A9 )
% 0.34/0.55 => ( P @ A9 ) )
% 0.34/0.55 => ( ( P @ bot_bot_set_o )
% 0.34/0.55 => ( ! [X3: $o,F3: set_o] :
% 0.34/0.55 ( ( finite_finite_o @ F3 )
% 0.34/0.55 => ( ~ ( member_o @ X3 @ F3 )
% 0.34/0.55 => ( ( P @ F3 )
% 0.34/0.55 => ( P @ ( insert_o @ X3 @ F3 ) ) ) ) )
% 0.34/0.55 => ( P @ A ) ) ) ) ).
% 0.34/0.55
% 0.34/0.55 % infinite_finite_induct
% 0.34/0.55 thf(fact_1832_infinite__finite__induct,axiom,
% 0.34/0.55 ! [P: set_nat > $o,A: set_nat] :
% 0.34/0.55 ( ! [A9: set_nat] :
% 0.34/0.55 ( ~ ( finite_finite_nat @ A9 )
% 0.34/0.55 => ( P @ A9 ) )
% 0.34/0.55 => ( ( P @ bot_bot_set_nat )
% 0.34/0.55 => ( ! [X3: nat,F3: set_nat] :
% 0.34/0.55 ( ( finite_finite_nat @ F3 )
% 0.34/0.55 => ( ~ ( member_nat @ X3 @ F3 )
% 0.34/0.55 => ( ( P @ F3 )
% 0.34/0.55 => ( P @ ( insert_nat @ X3 @ F3 ) ) ) ) )
% 0.34/0.55 => ( P @ A ) ) ) ) ).
% 0.34/0.55
% 0.34/0.55 % infinite_finite_induct
% 0.34/0.55 thf(fact_1833_infinite__finite__induct,axiom,
% 0.34/0.55 ! [P: set_int > $o,A: set_int] :
% 0.34/0.55 ( ! [A9: set_int] :
% 0.34/0.55 ( ~ ( finite_finite_int @ A9 )
% 0.34/0.55 => ( P @ A9 ) )
% 0.34/0.55 => ( ( P @ bot_bot_set_int )
% 0.34/0.55 => ( ! [X3: int,F3: set_int] :
% 0.34/0.55 ( ( finite_finite_int @ F3 )
% 0.34/0.55 => ( ~ ( member_int @ X3 @ F3 )
% 0.34/0.55 => ( ( P @ F3 )
% 0.34/0.55 => ( P @ ( insert_int @ X3 @ F3 ) ) ) ) )
% 0.34/0.55 => ( P @ A ) ) ) ) ).
% 0.34/0.55
% 0.34/0.55 % infinite_finite_induct
% 0.34/0.55 thf(fact_1834_infinite__finite__induct,axiom,
% 0.34/0.55 ! [P: set_Pr4532377907799695533_nat_o > $o,A: set_Pr4532377907799695533_nat_o] :
% 0.34/0.55 ( ! [A9: set_Pr4532377907799695533_nat_o] :
% 0.34/0.55 ( ~ ( finite3252695134891459830_nat_o @ A9 )
% 0.34/0.55 => ( P @ A9 ) )
% 0.34/0.55 => ( ( P @ bot_bo7824918357723582233_nat_o )
% 0.34/0.55 => ( ! [X3: produc3658429121746597890et_nat > $o,F3: set_Pr4532377907799695533_nat_o] :
% 0.34/0.55 ( ( finite3252695134891459830_nat_o @ F3 )
% 0.34/0.55 => ( ~ ( member6576561426505652726_nat_o @ X3 @ F3 )
% 0.34/0.55 => ( ( P @ F3 )
% 0.34/0.55 => ( P @ ( insert5175938949040314269_nat_o @ X3 @ F3 ) ) ) ) )
% 0.34/0.55 => ( P @ A ) ) ) ) ).
% 0.34/0.55
% 0.34/0.55 % infinite_finite_induct
% 0.34/0.55 thf(fact_1835_finite__ne__induct,axiom,
% 0.34/0.55 ! [F2: set_Code_integer,P: set_Code_integer > $o] :
% 0.34/0.55 ( ( finite6017078050557962740nteger @ F2 )
% 0.34/0.55 => ( ( F2 != bot_bo3990330152332043303nteger )
% 0.34/0.55 => ( ! [X3: code_integer] : ( P @ ( insert_Code_integer @ X3 @ bot_bo3990330152332043303nteger ) )
% 0.34/0.55 => ( ! [X3: code_integer,F3: set_Code_integer] :
% 0.34/0.55 ( ( finite6017078050557962740nteger @ F3 )
% 0.34/0.55 => ( ( F3 != bot_bo3990330152332043303nteger )
% 0.34/0.55 => ( ~ ( member_Code_integer @ X3 @ F3 )
% 0.34/0.55 => ( ( P @ F3 )
% 0.34/0.55 => ( P @ ( insert_Code_integer @ X3 @ F3 ) ) ) ) ) )
% 0.34/0.55 => ( P @ F2 ) ) ) ) ) ).
% 0.34/0.55
% 0.34/0.55 % finite_ne_induct
% 0.34/0.55 thf(fact_1836_finite__ne__induct,axiom,
% 0.34/0.55 ! [F2: set_Product_unit,P: set_Product_unit > $o] :
% 0.34/0.55 ( ( finite4290736615968046902t_unit @ F2 )
% 0.34/0.55 => ( ( F2 != bot_bo3957492148770167129t_unit )
% 0.34/0.55 => ( ! [X3: product_unit] : ( P @ ( insert_Product_unit @ X3 @ bot_bo3957492148770167129t_unit ) )
% 0.34/0.55 => ( ! [X3: product_unit,F3: set_Product_unit] :
% 0.34/0.55 ( ( finite4290736615968046902t_unit @ F3 )
% 0.34/0.55 => ( ( F3 != bot_bo3957492148770167129t_unit )
% 0.34/0.55 => ( ~ ( member_Product_unit @ X3 @ F3 )
% 0.34/0.55 => ( ( P @ F3 )
% 0.34/0.55 => ( P @ ( insert_Product_unit @ X3 @ F3 ) ) ) ) ) )
% 0.34/0.55 => ( P @ F2 ) ) ) ) ) ).
% 0.34/0.55
% 0.34/0.55 % finite_ne_induct
% 0.34/0.55 thf(fact_1837_finite__ne__induct,axiom,
% 0.34/0.55 ! [F2: set_o,P: set_o > $o] :
% 0.34/0.55 ( ( finite_finite_o @ F2 )
% 0.34/0.55 => ( ( F2 != bot_bot_set_o )
% 0.34/0.55 => ( ! [X3: $o] : ( P @ ( insert_o @ X3 @ bot_bot_set_o ) )
% 0.34/0.55 => ( ! [X3: $o,F3: set_o] :
% 0.34/0.55 ( ( finite_finite_o @ F3 )
% 0.34/0.55 => ( ( F3 != bot_bot_set_o )
% 0.34/0.55 => ( ~ ( member_o @ X3 @ F3 )
% 0.34/0.55 => ( ( P @ F3 )
% 0.34/0.55 => ( P @ ( insert_o @ X3 @ F3 ) ) ) ) ) )
% 0.34/0.55 => ( P @ F2 ) ) ) ) ) ).
% 0.34/0.55
% 0.34/0.55 % finite_ne_induct
% 0.34/0.55 thf(fact_1838_finite__ne__induct,axiom,
% 0.34/0.55 ! [F2: set_nat,P: set_nat > $o] :
% 0.34/0.55 ( ( finite_finite_nat @ F2 )
% 0.34/0.55 => ( ( F2 != bot_bot_set_nat )
% 0.34/0.55 => ( ! [X3: nat] : ( P @ ( insert_nat @ X3 @ bot_bot_set_nat ) )
% 0.34/0.55 => ( ! [X3: nat,F3: set_nat] :
% 0.34/0.55 ( ( finite_finite_nat @ F3 )
% 0.34/0.55 => ( ( F3 != bot_bot_set_nat )
% 0.34/0.55 => ( ~ ( member_nat @ X3 @ F3 )
% 0.34/0.55 => ( ( P @ F3 )
% 0.34/0.55 => ( P @ ( insert_nat @ X3 @ F3 ) ) ) ) ) )
% 0.34/0.55 => ( P @ F2 ) ) ) ) ) ).
% 0.34/0.55
% 0.34/0.55 % finite_ne_induct
% 0.34/0.55 thf(fact_1839_finite__ne__induct,axiom,
% 0.34/0.55 ! [F2: set_int,P: set_int > $o] :
% 0.34/0.55 ( ( finite_finite_int @ F2 )
% 0.34/0.55 => ( ( F2 != bot_bot_set_int )
% 0.34/0.55 => ( ! [X3: int] : ( P @ ( insert_int @ X3 @ bot_bot_set_int ) )
% 0.34/0.55 => ( ! [X3: int,F3: set_int] :
% 0.34/0.55 ( ( finite_finite_int @ F3 )
% 0.34/0.55 => ( ( F3 != bot_bot_set_int )
% 0.34/0.55 => ( ~ ( member_int @ X3 @ F3 )
% 0.34/0.55 => ( ( P @ F3 )
% 0.34/0.55 => ( P @ ( insert_int @ X3 @ F3 ) ) ) ) ) )
% 0.34/0.55 => ( P @ F2 ) ) ) ) ) ).
% 0.34/0.55
% 0.34/0.55 % finite_ne_induct
% 0.34/0.55 thf(fact_1840_finite__ne__induct,axiom,
% 0.34/0.55 ! [F2: set_Pr4532377907799695533_nat_o,P: set_Pr4532377907799695533_nat_o > $o] :
% 0.34/0.55 ( ( finite3252695134891459830_nat_o @ F2 )
% 0.34/0.55 => ( ( F2 != bot_bo7824918357723582233_nat_o )
% 0.34/0.55 => ( ! [X3: produc3658429121746597890et_nat > $o] : ( P @ ( insert5175938949040314269_nat_o @ X3 @ bot_bo7824918357723582233_nat_o ) )
% 0.34/0.55 => ( ! [X3: produc3658429121746597890et_nat > $o,F3: set_Pr4532377907799695533_nat_o] :
% 0.34/0.55 ( ( finite3252695134891459830_nat_o @ F3 )
% 0.34/0.55 => ( ( F3 != bot_bo7824918357723582233_nat_o )
% 0.34/0.55 => ( ~ ( member6576561426505652726_nat_o @ X3 @ F3 )
% 0.34/0.55 => ( ( P @ F3 )
% 0.34/0.55 => ( P @ ( insert5175938949040314269_nat_o @ X3 @ F3 ) ) ) ) ) )
% 0.34/0.55 => ( P @ F2 ) ) ) ) ) ).
% 0.34/0.55
% 0.34/0.55 % finite_ne_induct
% 0.34/0.55 thf(fact_1841_finite__induct,axiom,
% 0.34/0.55 ! [F2: set_Code_integer,P: set_Code_integer > $o] :
% 0.34/0.55 ( ( finite6017078050557962740nteger @ F2 )
% 0.34/0.55 => ( ( P @ bot_bo3990330152332043303nteger )
% 0.34/0.55 => ( ! [X3: code_integer,F3: set_Code_integer] :
% 0.34/0.55 ( ( finite6017078050557962740nteger @ F3 )
% 0.34/0.55 => ( ~ ( member_Code_integer @ X3 @ F3 )
% 0.34/0.55 => ( ( P @ F3 )
% 0.34/0.55 => ( P @ ( insert_Code_integer @ X3 @ F3 ) ) ) ) )
% 0.34/0.55 => ( P @ F2 ) ) ) ) ).
% 0.34/0.55
% 0.34/0.55 % finite_induct
% 0.34/0.55 thf(fact_1842_finite__induct,axiom,
% 0.34/0.55 ! [F2: set_Product_unit,P: set_Product_unit > $o] :
% 0.34/0.55 ( ( finite4290736615968046902t_unit @ F2 )
% 0.34/0.55 => ( ( P @ bot_bo3957492148770167129t_unit )
% 0.34/0.55 => ( ! [X3: product_unit,F3: set_Product_unit] :
% 0.34/0.55 ( ( finite4290736615968046902t_unit @ F3 )
% 0.34/0.55 => ( ~ ( member_Product_unit @ X3 @ F3 )
% 0.34/0.55 => ( ( P @ F3 )
% 0.34/0.55 => ( P @ ( insert_Product_unit @ X3 @ F3 ) ) ) ) )
% 0.34/0.55 => ( P @ F2 ) ) ) ) ).
% 0.34/0.55
% 0.34/0.55 % finite_induct
% 0.34/0.55 thf(fact_1843_finite__induct,axiom,
% 0.34/0.55 ! [F2: set_o,P: set_o > $o] :
% 0.34/0.55 ( ( finite_finite_o @ F2 )
% 0.34/0.55 => ( ( P @ bot_bot_set_o )
% 0.34/0.55 => ( ! [X3: $o,F3: set_o] :
% 0.34/0.55 ( ( finite_finite_o @ F3 )
% 0.34/0.55 => ( ~ ( member_o @ X3 @ F3 )
% 0.34/0.55 => ( ( P @ F3 )
% 0.34/0.55 => ( P @ ( insert_o @ X3 @ F3 ) ) ) ) )
% 0.34/0.55 => ( P @ F2 ) ) ) ) ).
% 0.34/0.55
% 0.34/0.55 % finite_induct
% 0.34/0.55 thf(fact_1844_finite__induct,axiom,
% 0.34/0.55 ! [F2: set_nat,P: set_nat > $o] :
% 0.34/0.55 ( ( finite_finite_nat @ F2 )
% 0.34/0.55 => ( ( P @ bot_bot_set_nat )
% 0.34/0.55 => ( ! [X3: nat,F3: set_nat] :
% 0.34/0.55 ( ( finite_finite_nat @ F3 )
% 0.34/0.55 => ( ~ ( member_nat @ X3 @ F3 )
% 0.34/0.55 => ( ( P @ F3 )
% 0.34/0.55 => ( P @ ( insert_nat @ X3 @ F3 ) ) ) ) )
% 0.34/0.55 => ( P @ F2 ) ) ) ) ).
% 0.34/0.55
% 0.34/0.55 % finite_induct
% 0.34/0.55 thf(fact_1845_finite__induct,axiom,
% 0.34/0.55 ! [F2: set_int,P: set_int > $o] :
% 0.34/0.55 ( ( finite_finite_int @ F2 )
% 0.34/0.55 => ( ( P @ bot_bot_set_int )
% 0.34/0.55 => ( ! [X3: int,F3: set_int] :
% 0.34/0.55 ( ( finite_finite_int @ F3 )
% 0.34/0.55 => ( ~ ( member_int @ X3 @ F3 )
% 0.34/0.55 => ( ( P @ F3 )
% 0.34/0.55 => ( P @ ( insert_int @ X3 @ F3 ) ) ) ) )
% 0.34/0.55 => ( P @ F2 ) ) ) ) ).
% 0.34/0.55
% 0.34/0.55 % finite_induct
% 0.34/0.55 thf(fact_1846_finite__induct,axiom,
% 0.34/0.55 ! [F2: set_Pr4532377907799695533_nat_o,P: set_Pr4532377907799695533_nat_o > $o] :
% 0.34/0.55 ( ( finite3252695134891459830_nat_o @ F2 )
% 0.34/0.55 => ( ( P @ bot_bo7824918357723582233_nat_o )
% 0.34/0.55 => ( ! [X3: produc3658429121746597890et_nat > $o,F3: set_Pr4532377907799695533_nat_o] :
% 0.34/0.55 ( ( finite3252695134891459830_nat_o @ F3 )
% 0.34/0.55 => ( ~ ( member6576561426505652726_nat_o @ X3 @ F3 )
% 0.34/0.55 => ( ( P @ F3 )
% 0.34/0.55 => ( P @ ( insert5175938949040314269_nat_o @ X3 @ F3 ) ) ) ) )
% 0.34/0.55 => ( P @ F2 ) ) ) ) ).
% 0.34/0.55
% 0.34/0.55 % finite_induct
% 0.34/0.55 thf(fact_1847_finite_Osimps,axiom,
% 0.34/0.55 ( finite6017078050557962740nteger
% 0.34/0.55 = ( ^ [A6: set_Code_integer] :
% 0.34/0.55 ( ( A6 = bot_bo3990330152332043303nteger )
% 0.34/0.55 | ? [A3: set_Code_integer,B6: code_integer] :
% 0.34/0.55 ( ( A6
% 0.34/0.55 = ( insert_Code_integer @ B6 @ A3 ) )
% 0.34/0.55 & ( finite6017078050557962740nteger @ A3 ) ) ) ) ) ).
% 0.34/0.55
% 0.34/0.55 % finite.simps
% 0.34/0.55 thf(fact_1848_finite_Osimps,axiom,
% 0.34/0.55 ( finite4290736615968046902t_unit
% 0.34/0.55 = ( ^ [A6: set_Product_unit] :
% 0.34/0.55 ( ( A6 = bot_bo3957492148770167129t_unit )
% 0.34/0.55 | ? [A3: set_Product_unit,B6: product_unit] :
% 0.34/0.55 ( ( A6
% 0.34/0.55 = ( insert_Product_unit @ B6 @ A3 ) )
% 0.34/0.55 & ( finite4290736615968046902t_unit @ A3 ) ) ) ) ) ).
% 0.34/0.55
% 0.34/0.55 % finite.simps
% 0.34/0.55 thf(fact_1849_finite_Osimps,axiom,
% 0.34/0.55 ( finite_finite_o
% 0.34/0.55 = ( ^ [A6: set_o] :
% 0.34/0.55 ( ( A6 = bot_bot_set_o )
% 0.34/0.55 | ? [A3: set_o,B6: $o] :
% 0.34/0.55 ( ( A6
% 0.34/0.55 = ( insert_o @ B6 @ A3 ) )
% 0.34/0.55 & ( finite_finite_o @ A3 ) ) ) ) ) ).
% 0.34/0.55
% 0.34/0.55 % finite.simps
% 0.34/0.55 thf(fact_1850_finite_Osimps,axiom,
% 0.34/0.55 ( finite_finite_nat
% 0.34/0.55 = ( ^ [A6: set_nat] :
% 0.34/0.55 ( ( A6 = bot_bot_set_nat )
% 0.34/0.55 | ? [A3: set_nat,B6: nat] :
% 0.34/0.55 ( ( A6
% 0.34/0.55 = ( insert_nat @ B6 @ A3 ) )
% 0.34/0.55 & ( finite_finite_nat @ A3 ) ) ) ) ) ).
% 0.34/0.55
% 0.34/0.55 % finite.simps
% 0.34/0.55 thf(fact_1851_finite_Osimps,axiom,
% 0.34/0.55 ( finite_finite_int
% 0.34/0.55 = ( ^ [A6: set_int] :
% 0.34/0.55 ( ( A6 = bot_bot_set_int )
% 0.34/0.55 | ? [A3: set_int,B6: int] :
% 0.34/0.55 ( ( A6
% 0.34/0.55 = ( insert_int @ B6 @ A3 ) )
% 0.34/0.55 & ( finite_finite_int @ A3 ) ) ) ) ) ).
% 0.34/0.55
% 0.34/0.55 % finite.simps
% 0.34/0.55 thf(fact_1852_finite_Osimps,axiom,
% 0.34/0.55 ( finite3252695134891459830_nat_o
% 0.34/0.55 = ( ^ [A6: set_Pr4532377907799695533_nat_o] :
% 0.34/0.55 ( ( A6 = bot_bo7824918357723582233_nat_o )
% 0.34/0.55 | ? [A3: set_Pr4532377907799695533_nat_o,B6: produc3658429121746597890et_nat > $o] :
% 0.34/0.55 ( ( A6
% 0.34/0.55 = ( insert5175938949040314269_nat_o @ B6 @ A3 ) )
% 0.34/0.55 & ( finite3252695134891459830_nat_o @ A3 ) ) ) ) ) ).
% 0.34/0.55
% 0.34/0.55 % finite.simps
% 0.34/0.55 thf(fact_1853_infinite__imp__nonempty,axiom,
% 0.34/0.55 ! [S: set_Code_integer] :
% 0.34/0.55 ( ~ ( finite6017078050557962740nteger @ S )
% 0.34/0.55 => ( S != bot_bo3990330152332043303nteger ) ) ).
% 0.34/0.55
% 0.34/0.55 % infinite_imp_nonempty
% 0.34/0.55 thf(fact_1854_infinite__imp__nonempty,axiom,
% 0.34/0.55 ! [S: set_Product_unit] :
% 0.34/0.55 ( ~ ( finite4290736615968046902t_unit @ S )
% 0.34/0.55 => ( S != bot_bo3957492148770167129t_unit ) ) ).
% 0.34/0.55
% 0.34/0.55 % infinite_imp_nonempty
% 0.34/0.55 thf(fact_1855_infinite__imp__nonempty,axiom,
% 0.34/0.55 ! [S: set_o] :
% 0.34/0.55 ( ~ ( finite_finite_o @ S )
% 0.34/0.55 => ( S != bot_bot_set_o ) ) ).
% 0.34/0.55
% 0.34/0.55 % infinite_imp_nonempty
% 0.34/0.55 thf(fact_1856_infinite__imp__nonempty,axiom,
% 0.34/0.55 ! [S: set_nat] :
% 0.34/0.55 ( ~ ( finite_finite_nat @ S )
% 0.34/0.55 => ( S != bot_bot_set_nat ) ) ).
% 0.34/0.55
% 0.34/0.55 % infinite_imp_nonempty
% 0.34/0.55 thf(fact_1857_infinite__imp__nonempty,axiom,
% 0.34/0.55 ! [S: set_int] :
% 0.34/0.55 ( ~ ( finite_finite_int @ S )
% 0.34/0.55 => ( S != bot_bot_set_int ) ) ).
% 0.34/0.55
% 0.34/0.55 % infinite_imp_nonempty
% 0.34/0.55 thf(fact_1858_infinite__imp__nonempty,axiom,
% 0.34/0.55 ! [S: set_Pr4532377907799695533_nat_o] :
% 0.34/0.55 ( ~ ( finite3252695134891459830_nat_o @ S )
% 0.34/0.55 => ( S != bot_bo7824918357723582233_nat_o ) ) ).
% 0.34/0.55
% 0.34/0.55 % infinite_imp_nonempty
% 0.34/0.55 thf(fact_1859_finite_OemptyI,axiom,
% 0.34/0.55 finite6017078050557962740nteger @ bot_bo3990330152332043303nteger ).
% 0.34/0.55
% 0.34/0.55 % finite.emptyI
% 0.34/0.55 thf(fact_1860_finite_OemptyI,axiom,
% 0.34/0.55 finite4290736615968046902t_unit @ bot_bo3957492148770167129t_unit ).
% 0.34/0.55
% 0.34/0.55 % finite.emptyI
% 0.34/0.55 thf(fact_1861_finite_OemptyI,axiom,
% 0.34/0.55 finite_finite_o @ bot_bot_set_o ).
% 0.34/0.55
% 0.34/0.55 % finite.emptyI
% 0.34/0.55 thf(fact_1862_finite_OemptyI,axiom,
% 0.34/0.55 finite_finite_nat @ bot_bot_set_nat ).
% 0.34/0.55
% 0.34/0.55 % finite.emptyI
% 0.34/0.55 thf(fact_1863_finite_OemptyI,axiom,
% 0.34/0.55 finite_finite_int @ bot_bot_set_int ).
% 0.34/0.55
% 0.34/0.55 % finite.emptyI
% 0.34/0.55 thf(fact_1864_finite_OemptyI,axiom,
% 0.34/0.55 finite3252695134891459830_nat_o @ bot_bo7824918357723582233_nat_o ).
% 0.34/0.55
% 0.34/0.55 % finite.emptyI
% 0.34/0.55 thf(fact_1865_finite_OinsertI,axiom,
% 0.34/0.55 ! [A: set_int,A2: int] :
% 0.34/0.55 ( ( finite_finite_int @ A )
% 0.34/0.55 => ( finite_finite_int @ ( insert_int @ A2 @ A ) ) ) ).
% 0.34/0.55
% 0.34/0.55 % finite.insertI
% 0.34/0.55 thf(fact_1866_finite_OinsertI,axiom,
% 0.34/0.55 ! [A: set_o,A2: $o] :
% 0.34/0.55 ( ( finite_finite_o @ A )
% 0.34/0.55 => ( finite_finite_o @ ( insert_o @ A2 @ A ) ) ) ).
% 0.34/0.55
% 0.34/0.55 % finite.insertI
% 0.34/0.55 thf(fact_1867_finite_OinsertI,axiom,
% 0.34/0.55 ! [A: set_Product_unit,A2: product_unit] :
% 0.34/0.55 ( ( finite4290736615968046902t_unit @ A )
% 0.34/0.55 => ( finite4290736615968046902t_unit @ ( insert_Product_unit @ A2 @ A ) ) ) ).
% 0.34/0.55
% 0.34/0.55 % finite.insertI
% 0.34/0.55 thf(fact_1868_finite_OinsertI,axiom,
% 0.34/0.55 ! [A: set_nat,A2: nat] :
% 0.34/0.55 ( ( finite_finite_nat @ A )
% 0.34/0.55 => ( finite_finite_nat @ ( insert_nat @ A2 @ A ) ) ) ).
% 0.34/0.55
% 0.34/0.55 % finite.insertI
% 0.34/0.55 thf(fact_1869_finite_OinsertI,axiom,
% 0.34/0.55 ! [A: set_Code_integer,A2: code_integer] :
% 0.34/0.55 ( ( finite6017078050557962740nteger @ A )
% 0.34/0.55 => ( finite6017078050557962740nteger @ ( insert_Code_integer @ A2 @ A ) ) ) ).
% 0.34/0.55
% 0.34/0.55 % finite.insertI
% 0.34/0.55 thf(fact_1870_finite__has__minimal,axiom,
% 0.34/0.55 ! [A: set_Code_integer] :
% 0.34/0.55 ( ( finite6017078050557962740nteger @ A )
% 0.34/0.55 => ( ( A != bot_bo3990330152332043303nteger )
% 0.34/0.55 => ? [X3: code_integer] :
% 0.34/0.55 ( ( member_Code_integer @ X3 @ A )
% 0.34/0.55 & ! [Xa2: code_integer] :
% 0.34/0.55 ( ( member_Code_integer @ Xa2 @ A )
% 0.34/0.55 => ( ( ord_le3102999989581377725nteger @ Xa2 @ X3 )
% 0.34/0.55 => ( X3 = Xa2 ) ) ) ) ) ) ).
% 0.34/0.55
% 0.34/0.55 % finite_has_minimal
% 0.34/0.55 thf(fact_1871_finite__has__minimal,axiom,
% 0.34/0.55 ! [A: set_Product_unit] :
% 0.34/0.55 ( ( finite4290736615968046902t_unit @ A )
% 0.34/0.55 => ( ( A != bot_bo3957492148770167129t_unit )
% 0.34/0.55 => ? [X3: product_unit] :
% 0.34/0.55 ( ( member_Product_unit @ X3 @ A )
% 0.34/0.55 & ! [Xa2: product_unit] :
% 0.34/0.55 ( ( member_Product_unit @ Xa2 @ A )
% 0.34/0.55 => ( ( ord_le3221252021190050221t_unit @ Xa2 @ X3 )
% 0.34/0.55 => ( X3 = Xa2 ) ) ) ) ) ) ).
% 0.34/0.55
% 0.34/0.55 % finite_has_minimal
% 0.34/0.55 thf(fact_1872_finite__has__minimal,axiom,
% 0.34/0.55 ! [A: set_o] :
% 0.34/0.55 ( ( finite_finite_o @ A )
% 0.34/0.55 => ( ( A != bot_bot_set_o )
% 0.34/0.55 => ? [X3: $o] :
% 0.34/0.55 ( ( member_o @ X3 @ A )
% 0.34/0.55 & ! [Xa2: $o] :
% 0.34/0.55 ( ( member_o @ Xa2 @ A )
% 0.34/0.55 => ( ( ord_less_eq_o @ Xa2 @ X3 )
% 0.34/0.55 => ( X3 = Xa2 ) ) ) ) ) ) ).
% 0.34/0.55
% 0.34/0.55 % finite_has_minimal
% 0.34/0.55 thf(fact_1873_finite__has__minimal,axiom,
% 0.34/0.55 ! [A: set_Pr4532377907799695533_nat_o] :
% 0.34/0.55 ( ( finite3252695134891459830_nat_o @ A )
% 0.34/0.55 => ( ( A != bot_bo7824918357723582233_nat_o )
% 0.34/0.55 => ? [X3: produc3658429121746597890et_nat > $o] :
% 0.34/0.55 ( ( member6576561426505652726_nat_o @ X3 @ A )
% 0.34/0.55 & ! [Xa2: produc3658429121746597890et_nat > $o] :
% 0.34/0.55 ( ( member6576561426505652726_nat_o @ Xa2 @ A )
% 0.34/0.55 => ( ( ord_le729326519192465773_nat_o @ Xa2 @ X3 )
% 0.34/0.55 => ( X3 = Xa2 ) ) ) ) ) ) ).
% 0.34/0.55
% 0.34/0.55 % finite_has_minimal
% 0.34/0.55 thf(fact_1874_finite__has__minimal,axiom,
% 0.34/0.55 ! [A: set_filter_nat] :
% 0.34/0.55 ( ( finite2119507909894593271er_nat @ A )
% 0.34/0.55 => ( ( A != bot_bo498966703094740906er_nat )
% 0.34/0.55 => ? [X3: filter_nat] :
% 0.34/0.55 ( ( member_filter_nat @ X3 @ A )
% 0.34/0.55 & ! [Xa2: filter_nat] :
% 0.34/0.55 ( ( member_filter_nat @ Xa2 @ A )
% 0.34/0.55 => ( ( ord_le2510731241096832064er_nat @ Xa2 @ X3 )
% 0.34/0.55 => ( X3 = Xa2 ) ) ) ) ) ) ).
% 0.34/0.55
% 0.34/0.55 % finite_has_minimal
% 0.34/0.55 thf(fact_1875_finite__has__minimal,axiom,
% 0.34/0.55 ! [A: set_set_nat] :
% 0.34/0.55 ( ( finite1152437895449049373et_nat @ A )
% 0.34/0.55 => ( ( A != bot_bot_set_set_nat )
% 0.34/0.55 => ? [X3: set_nat] :
% 0.34/0.55 ( ( member_set_nat @ X3 @ A )
% 0.34/0.55 & ! [Xa2: set_nat] :
% 0.34/0.55 ( ( member_set_nat @ Xa2 @ A )
% 0.34/0.55 => ( ( ord_less_eq_set_nat @ Xa2 @ X3 )
% 0.34/0.55 => ( X3 = Xa2 ) ) ) ) ) ) ).
% 0.34/0.55
% 0.34/0.55 % finite_has_minimal
% 0.34/0.55 thf(fact_1876_finite__has__minimal,axiom,
% 0.34/0.55 ! [A: set_rat] :
% 0.34/0.55 ( ( finite_finite_rat @ A )
% 0.34/0.55 => ( ( A != bot_bot_set_rat )
% 0.34/0.55 => ? [X3: rat] :
% 0.34/0.55 ( ( member_rat @ X3 @ A )
% 0.34/0.55 & ! [Xa2: rat] :
% 0.34/0.55 ( ( member_rat @ Xa2 @ A )
% 0.34/0.55 => ( ( ord_less_eq_rat @ Xa2 @ X3 )
% 0.34/0.55 => ( X3 = Xa2 ) ) ) ) ) ) ).
% 0.34/0.55
% 0.34/0.55 % finite_has_minimal
% 0.34/0.55 thf(fact_1877_finite__has__minimal,axiom,
% 0.34/0.55 ! [A: set_nat] :
% 0.34/0.55 ( ( finite_finite_nat @ A )
% 0.34/0.55 => ( ( A != bot_bot_set_nat )
% 0.34/0.55 => ? [X3: nat] :
% 0.34/0.55 ( ( member_nat @ X3 @ A )
% 0.34/0.55 & ! [Xa2: nat] :
% 0.34/0.55 ( ( member_nat @ Xa2 @ A )
% 0.34/0.55 => ( ( ord_less_eq_nat @ Xa2 @ X3 )
% 0.34/0.55 => ( X3 = Xa2 ) ) ) ) ) ) ).
% 0.34/0.55
% 0.34/0.55 % finite_has_minimal
% 0.34/0.55 thf(fact_1878_finite__has__minimal,axiom,
% 0.34/0.55 ! [A: set_int] :
% 0.34/0.55 ( ( finite_finite_int @ A )
% 0.34/0.55 => ( ( A != bot_bot_set_int )
% 0.34/0.55 => ? [X3: int] :
% 0.34/0.55 ( ( member_int @ X3 @ A )
% 0.34/0.55 & ! [Xa2: int] :
% 0.34/0.55 ( ( member_int @ Xa2 @ A )
% 0.34/0.55 => ( ( ord_less_eq_int @ Xa2 @ X3 )
% 0.34/0.55 => ( X3 = Xa2 ) ) ) ) ) ) ).
% 0.34/0.55
% 0.34/0.55 % finite_has_minimal
% 0.34/0.55 thf(fact_1879_finite__has__maximal,axiom,
% 0.34/0.55 ! [A: set_Code_integer] :
% 0.34/0.55 ( ( finite6017078050557962740nteger @ A )
% 0.34/0.55 => ( ( A != bot_bo3990330152332043303nteger )
% 0.34/0.55 => ? [X3: code_integer] :
% 0.34/0.55 ( ( member_Code_integer @ X3 @ A )
% 0.34/0.55 & ! [Xa2: code_integer] :
% 0.34/0.55 ( ( member_Code_integer @ Xa2 @ A )
% 0.34/0.55 => ( ( ord_le3102999989581377725nteger @ X3 @ Xa2 )
% 0.34/0.55 => ( X3 = Xa2 ) ) ) ) ) ) ).
% 0.34/0.55
% 0.34/0.55 % finite_has_maximal
% 0.34/0.55 thf(fact_1880_finite__has__maximal,axiom,
% 0.34/0.55 ! [A: set_Product_unit] :
% 0.34/0.55 ( ( finite4290736615968046902t_unit @ A )
% 0.34/0.55 => ( ( A != bot_bo3957492148770167129t_unit )
% 0.34/0.55 => ? [X3: product_unit] :
% 0.34/0.55 ( ( member_Product_unit @ X3 @ A )
% 0.34/0.55 & ! [Xa2: product_unit] :
% 0.34/0.55 ( ( member_Product_unit @ Xa2 @ A )
% 0.34/0.55 => ( ( ord_le3221252021190050221t_unit @ X3 @ Xa2 )
% 0.34/0.55 => ( X3 = Xa2 ) ) ) ) ) ) ).
% 0.34/0.55
% 0.34/0.55 % finite_has_maximal
% 0.34/0.55 thf(fact_1881_finite__has__maximal,axiom,
% 0.34/0.55 ! [A: set_o] :
% 0.34/0.55 ( ( finite_finite_o @ A )
% 0.34/0.55 => ( ( A != bot_bot_set_o )
% 0.34/0.55 => ? [X3: $o] :
% 0.34/0.55 ( ( member_o @ X3 @ A )
% 0.34/0.55 & ! [Xa2: $o] :
% 0.34/0.55 ( ( member_o @ Xa2 @ A )
% 0.34/0.55 => ( ( ord_less_eq_o @ X3 @ Xa2 )
% 0.34/0.55 => ( X3 = Xa2 ) ) ) ) ) ) ).
% 0.34/0.55
% 0.34/0.55 % finite_has_maximal
% 0.34/0.55 thf(fact_1882_finite__has__maximal,axiom,
% 0.34/0.55 ! [A: set_Pr4532377907799695533_nat_o] :
% 0.34/0.55 ( ( finite3252695134891459830_nat_o @ A )
% 0.34/0.55 => ( ( A != bot_bo7824918357723582233_nat_o )
% 0.34/0.55 => ? [X3: produc3658429121746597890et_nat > $o] :
% 0.34/0.55 ( ( member6576561426505652726_nat_o @ X3 @ A )
% 0.34/0.55 & ! [Xa2: produc3658429121746597890et_nat > $o] :
% 0.34/0.55 ( ( member6576561426505652726_nat_o @ Xa2 @ A )
% 0.34/0.55 => ( ( ord_le729326519192465773_nat_o @ X3 @ Xa2 )
% 0.34/0.55 => ( X3 = Xa2 ) ) ) ) ) ) ).
% 0.34/0.55
% 0.34/0.55 % finite_has_maximal
% 0.34/0.55 thf(fact_1883_finite__has__maximal,axiom,
% 0.34/0.55 ! [A: set_filter_nat] :
% 0.34/0.55 ( ( finite2119507909894593271er_nat @ A )
% 0.34/0.55 => ( ( A != bot_bo498966703094740906er_nat )
% 0.34/0.55 => ? [X3: filter_nat] :
% 0.34/0.55 ( ( member_filter_nat @ X3 @ A )
% 0.34/0.55 & ! [Xa2: filter_nat] :
% 0.34/0.55 ( ( member_filter_nat @ Xa2 @ A )
% 0.34/0.55 => ( ( ord_le2510731241096832064er_nat @ X3 @ Xa2 )
% 0.34/0.55 => ( X3 = Xa2 ) ) ) ) ) ) ).
% 0.34/0.55
% 0.34/0.55 % finite_has_maximal
% 0.34/0.55 thf(fact_1884_finite__has__maximal,axiom,
% 0.34/0.55 ! [A: set_set_nat] :
% 0.34/0.55 ( ( finite1152437895449049373et_nat @ A )
% 0.34/0.55 => ( ( A != bot_bot_set_set_nat )
% 0.34/0.55 => ? [X3: set_nat] :
% 0.34/0.55 ( ( member_set_nat @ X3 @ A )
% 0.34/0.55 & ! [Xa2: set_nat] :
% 0.34/0.55 ( ( member_set_nat @ Xa2 @ A )
% 0.34/0.55 => ( ( ord_less_eq_set_nat @ X3 @ Xa2 )
% 0.34/0.55 => ( X3 = Xa2 ) ) ) ) ) ) ).
% 0.34/0.55
% 0.34/0.55 % finite_has_maximal
% 0.34/0.55 thf(fact_1885_finite__has__maximal,axiom,
% 0.34/0.55 ! [A: set_rat] :
% 0.34/0.55 ( ( finite_finite_rat @ A )
% 0.34/0.55 => ( ( A != bot_bot_set_rat )
% 0.34/0.55 => ? [X3: rat] :
% 0.34/0.55 ( ( member_rat @ X3 @ A )
% 0.34/0.55 & ! [Xa2: rat] :
% 0.34/0.55 ( ( member_rat @ Xa2 @ A )
% 0.34/0.55 => ( ( ord_less_eq_rat @ X3 @ Xa2 )
% 0.34/0.55 => ( X3 = Xa2 ) ) ) ) ) ) ).
% 0.34/0.55
% 0.34/0.55 % finite_has_maximal
% 0.34/0.55 thf(fact_1886_finite__has__maximal,axiom,
% 0.34/0.55 ! [A: set_nat] :
% 0.34/0.55 ( ( finite_finite_nat @ A )
% 0.34/0.55 => ( ( A != bot_bot_set_nat )
% 0.34/0.55 => ? [X3: nat] :
% 0.34/0.55 ( ( member_nat @ X3 @ A )
% 0.34/0.55 & ! [Xa2: nat] :
% 0.34/0.55 ( ( member_nat @ Xa2 @ A )
% 0.34/0.55 => ( ( ord_less_eq_nat @ X3 @ Xa2 )
% 0.34/0.55 => ( X3 = Xa2 ) ) ) ) ) ) ).
% 0.34/0.55
% 0.34/0.55 % finite_has_maximal
% 0.34/0.55 thf(fact_1887_finite__has__maximal,axiom,
% 0.34/0.55 ! [A: set_int] :
% 0.34/0.55 ( ( finite_finite_int @ A )
% 0.34/0.55 => ( ( A != bot_bot_set_int )
% 0.34/0.55 => ? [X3: int] :
% 0.34/0.55 ( ( member_int @ X3 @ A )
% 0.34/0.55 & ! [Xa2: int] :
% 0.34/0.55 ( ( member_int @ Xa2 @ A )
% 0.34/0.55 => ( ( ord_less_eq_int @ X3 @ Xa2 )
% 0.34/0.55 => ( X3 = Xa2 ) ) ) ) ) ) ).
% 0.34/0.55
% 0.34/0.55 % finite_has_maximal
% 0.34/0.55 thf(fact_1888_finite_Ocases,axiom,
% 0.34/0.55 ! [A2: set_Code_integer] :
% 0.34/0.55 ( ( finite6017078050557962740nteger @ A2 )
% 0.34/0.55 => ( ( A2 != bot_bo3990330152332043303nteger )
% 0.34/0.55 => ~ ! [A9: set_Code_integer] :
% 0.34/0.55 ( ? [A4: code_integer] :
% 0.34/0.55 ( A2
% 0.34/0.55 = ( insert_Code_integer @ A4 @ A9 ) )
% 0.34/0.55 => ~ ( finite6017078050557962740nteger @ A9 ) ) ) ) ).
% 0.34/0.55
% 0.34/0.55 % finite.cases
% 0.34/0.55 thf(fact_1889_finite_Ocases,axiom,
% 0.34/0.55 ! [A2: set_Product_unit] :
% 0.34/0.55 ( ( finite4290736615968046902t_unit @ A2 )
% 0.34/0.55 => ( ( A2 != bot_bo3957492148770167129t_unit )
% 0.34/0.55 => ~ ! [A9: set_Product_unit] :
% 0.34/0.55 ( ? [A4: product_unit] :
% 0.34/0.55 ( A2
% 0.34/0.55 = ( insert_Product_unit @ A4 @ A9 ) )
% 0.34/0.55 => ~ ( finite4290736615968046902t_unit @ A9 ) ) ) ) ).
% 0.34/0.55
% 0.34/0.55 % finite.cases
% 0.34/0.55 thf(fact_1890_finite_Ocases,axiom,
% 0.34/0.55 ! [A2: set_o] :
% 0.34/0.55 ( ( finite_finite_o @ A2 )
% 0.34/0.55 => ( ( A2 != bot_bot_set_o )
% 0.34/0.55 => ~ ! [A9: set_o] :
% 0.34/0.55 ( ? [A4: $o] :
% 0.34/0.55 ( A2
% 0.34/0.55 = ( insert_o @ A4 @ A9 ) )
% 0.34/0.55 => ~ ( finite_finite_o @ A9 ) ) ) ) ).
% 0.34/0.55
% 0.34/0.55 % finite.cases
% 0.34/0.55 thf(fact_1891_finite_Ocases,axiom,
% 0.34/0.55 ! [A2: set_nat] :
% 0.34/0.55 ( ( finite_finite_nat @ A2 )
% 0.34/0.55 => ( ( A2 != bot_bot_set_nat )
% 0.34/0.55 => ~ ! [A9: set_nat] :
% 0.34/0.55 ( ? [A4: nat] :
% 0.34/0.55 ( A2
% 0.34/0.55 = ( insert_nat @ A4 @ A9 ) )
% 0.34/0.55 => ~ ( finite_finite_nat @ A9 ) ) ) ) ).
% 0.34/0.55
% 0.34/0.55 % finite.cases
% 0.34/0.55 thf(fact_1892_finite_Ocases,axiom,
% 0.34/0.55 ! [A2: set_int] :
% 0.34/0.55 ( ( finite_finite_int @ A2 )
% 0.34/0.55 => ( ( A2 != bot_bot_set_int )
% 0.34/0.55 => ~ ! [A9: set_int] :
% 0.34/0.55 ( ? [A4: int] :
% 0.34/0.55 ( A2
% 0.34/0.55 = ( insert_int @ A4 @ A9 ) )
% 0.34/0.55 => ~ ( finite_finite_int @ A9 ) ) ) ) ).
% 0.34/0.55
% 0.34/0.55 % finite.cases
% 0.34/0.55 thf(fact_1893_finite_Ocases,axiom,
% 0.34/0.55 ! [A2: set_Pr4532377907799695533_nat_o] :
% 0.34/0.55 ( ( finite3252695134891459830_nat_o @ A2 )
% 0.34/0.55 => ( ( A2 != bot_bo7824918357723582233_nat_o )
% 0.34/0.55 => ~ ! [A9: set_Pr4532377907799695533_nat_o] :
% 0.34/0.55 ( ? [A4: produc3658429121746597890et_nat > $o] :
% 0.34/0.55 ( A2
% 0.34/0.55 = ( insert5175938949040314269_nat_o @ A4 @ A9 ) )
% 0.34/0.55 => ~ ( finite3252695134891459830_nat_o @ A9 ) ) ) ) ).
% 0.34/0.55
% 0.34/0.55 % finite.cases
% 0.34/0.55 thf(fact_1894_min__ext__compat,axiom,
% 0.34/0.55 ! [R3: set_Pr5094982260447487303t_unit,S: set_Pr5094982260447487303t_unit] :
% 0.34/0.55 ( ( ord_le930853871177014503t_unit @ ( relcom848357998638694028t_unit @ R3 @ S ) @ R3 )
% 0.34/0.55 => ( ord_le8072594200992201959t_unit @ ( relcom7108816970757792620t_unit @ ( min_ext_Product_unit @ R3 ) @ ( sup_su3573767163274163995t_unit @ ( min_ext_Product_unit @ S ) @ ( insert6793119152790066487t_unit @ ( produc7269801343787132631t_unit @ bot_bo3957492148770167129t_unit @ bot_bo3957492148770167129t_unit ) @ bot_bo5113747506523071411t_unit ) ) ) @ ( min_ext_Product_unit @ R3 ) ) ) ).
% 0.34/0.55
% 0.34/0.55 % min_ext_compat
% 0.34/0.55 thf(fact_1895_min__ext__compat,axiom,
% 0.34/0.55 ! [R3: set_Product_prod_o_o,S: set_Product_prod_o_o] :
% 0.34/0.55 ( ( ord_le192195778054999463od_o_o @ ( relcomp_o_o_o @ R3 @ S ) @ R3 )
% 0.34/0.55 => ( ord_le2885120122233735591_set_o @ ( relcom7534720040753365047_set_o @ ( min_ext_o @ R3 ) @ ( sup_su7766833992073560027_set_o @ ( min_ext_o @ S ) @ ( insert1494450971503307511_set_o @ ( produc5838405689764958487_set_o @ bot_bot_set_o @ bot_bot_set_o ) @ bot_bo8143257439550059635_set_o ) ) ) @ ( min_ext_o @ R3 ) ) ) ).
% 0.34/0.55
% 0.34/0.55 % min_ext_compat
% 0.34/0.55 thf(fact_1896_min__ext__compat,axiom,
% 0.34/0.55 ! [R3: set_Pr1261947904930325089at_nat,S: set_Pr1261947904930325089at_nat] :
% 0.34/0.55 ( ( ord_le3146513528884898305at_nat @ ( relcomp_nat_nat_nat @ R3 @ S ) @ R3 )
% 0.34/0.55 => ( ord_le4402255868550298733et_nat @ ( relcom236561585145426837et_nat @ ( min_ext_nat @ R3 ) @ ( sup_su4251893418135750817et_nat @ ( min_ext_nat @ S ) @ ( insert3810226134351308605et_nat @ ( produc4532415448927165861et_nat @ bot_bot_set_nat @ bot_bot_set_nat ) @ bot_bo4952753636535259449et_nat ) ) ) @ ( min_ext_nat @ R3 ) ) ) ).
% 0.34/0.55
% 0.34/0.55 % min_ext_compat
% 0.34/0.55 thf(fact_1897_min__ext__compat,axiom,
% 0.34/0.55 ! [R3: set_Pr958786334691620121nt_int,S: set_Pr958786334691620121nt_int] :
% 0.34/0.55 ( ( ord_le2843351958646193337nt_int @ ( relcomp_int_int_int @ R3 @ S ) @ R3 )
% 0.34/0.55 => ( ord_le1436784781161120549et_int @ ( relcom4034950705519254697et_int @ ( min_ext_int @ R3 ) @ ( sup_su1286422330746572633et_int @ ( min_ext_int @ S ) @ ( insert7328952640415156725et_int @ ( produc6363374080413544029et_int @ bot_bot_set_int @ bot_bot_set_int ) @ bot_bo1987282549146081265et_int ) ) ) @ ( min_ext_int @ R3 ) ) ) ).
% 0.34/0.55
% 0.34/0.55 % min_ext_compat
% 0.34/0.55 thf(fact_1898_min__ext__compat,axiom,
% 0.34/0.55 ! [R3: set_Pr2161125870931222855_nat_o,S: set_Pr2161125870931222855_nat_o] :
% 0.34/0.55 ( ( ord_le6659818918159358183_nat_o @ ( relcom6251035627269349452_nat_o @ R3 @ S ) @ R3 )
% 0.34/0.55 => ( ord_le8001614231985977575_nat_o @ ( relcom7236094467932586796_nat_o @ ( min_ex3730736537596016502_nat_o @ R3 ) @ ( sup_su4863159966000489243_nat_o @ ( min_ex3730736537596016502_nat_o @ S ) @ ( insert3244108804717745463_nat_o @ ( produc5149316574908490455_nat_o @ bot_bo7824918357723582233_nat_o @ bot_bo7824918357723582233_nat_o ) @ bot_bo7535334947429593011_nat_o ) ) ) @ ( min_ex3730736537596016502_nat_o @ R3 ) ) ) ).
% 0.34/0.55
% 0.34/0.55 % min_ext_compat
% 0.34/0.55 thf(fact_1899_arg__min__least,axiom,
% 0.34/0.55 ! [S: set_Code_integer,Y: code_integer,F: code_integer > rat] :
% 0.34/0.55 ( ( finite6017078050557962740nteger @ S )
% 0.34/0.55 => ( ( S != bot_bo3990330152332043303nteger )
% 0.34/0.55 => ( ( member_Code_integer @ Y @ S )
% 0.34/0.55 => ( ord_less_eq_rat @ ( F @ ( lattic5439806495466278992er_rat @ F @ S ) ) @ ( F @ Y ) ) ) ) ) ).
% 0.34/0.55
% 0.34/0.55 % arg_min_least
% 0.34/0.55 thf(fact_1900_arg__min__least,axiom,
% 0.34/0.55 ! [S: set_Product_unit,Y: product_unit,F: product_unit > rat] :
% 0.34/0.55 ( ( finite4290736615968046902t_unit @ S )
% 0.34/0.55 => ( ( S != bot_bo3957492148770167129t_unit )
% 0.34/0.55 => ( ( member_Product_unit @ Y @ S )
% 0.34/0.55 => ( ord_less_eq_rat @ ( F @ ( lattic725702332050428812it_rat @ F @ S ) ) @ ( F @ Y ) ) ) ) ) ).
% 0.34/0.55
% 0.34/0.55 % arg_min_least
% 0.34/0.55 thf(fact_1901_arg__min__least,axiom,
% 0.34/0.55 ! [S: set_o,Y: $o,F: $o > rat] :
% 0.34/0.55 ( ( finite_finite_o @ S )
% 0.34/0.55 => ( ( S != bot_bot_set_o )
% 0.34/0.55 => ( ( member_o @ Y @ S )
% 0.34/0.55 => ( ord_less_eq_rat @ ( F @ ( lattic2140725968369957399_o_rat @ F @ S ) ) @ ( F @ Y ) ) ) ) ) ).
% 0.34/0.55
% 0.34/0.55 % arg_min_least
% 0.34/0.55 thf(fact_1902_arg__min__least,axiom,
% 0.34/0.55 ! [S: set_nat,Y: nat,F: nat > rat] :
% 0.34/0.55 ( ( finite_finite_nat @ S )
% 0.34/0.55 => ( ( S != bot_bot_set_nat )
% 0.34/0.55 => ( ( member_nat @ Y @ S )
% 0.34/0.55 => ( ord_less_eq_rat @ ( F @ ( lattic6811802900495863747at_rat @ F @ S ) ) @ ( F @ Y ) ) ) ) ) ).
% 0.34/0.55
% 0.34/0.55 % arg_min_least
% 0.34/0.55 thf(fact_1903_arg__min__least,axiom,
% 0.34/0.55 ! [S: set_int,Y: int,F: int > rat] :
% 0.34/0.55 ( ( finite_finite_int @ S )
% 0.34/0.55 => ( ( S != bot_bot_set_int )
% 0.34/0.55 => ( ( member_int @ Y @ S )
% 0.34/0.55 => ( ord_less_eq_rat @ ( F @ ( lattic7811156612396918303nt_rat @ F @ S ) ) @ ( F @ Y ) ) ) ) ) ).
% 0.34/0.55
% 0.34/0.55 % arg_min_least
% 0.34/0.55 thf(fact_1904_arg__min__least,axiom,
% 0.34/0.55 ! [S: set_Code_integer,Y: code_integer,F: code_integer > nat] :
% 0.34/0.55 ( ( finite6017078050557962740nteger @ S )
% 0.34/0.55 => ( ( S != bot_bo3990330152332043303nteger )
% 0.34/0.55 => ( ( member_Code_integer @ Y @ S )
% 0.34/0.55 => ( ord_less_eq_nat @ ( F @ ( lattic6074936555552774728er_nat @ F @ S ) ) @ ( F @ Y ) ) ) ) ) ).
% 0.34/0.55
% 0.34/0.55 % arg_min_least
% 0.34/0.55 thf(fact_1905_arg__min__least,axiom,
% 0.34/0.55 ! [S: set_Product_unit,Y: product_unit,F: product_unit > nat] :
% 0.34/0.55 ( ( finite4290736615968046902t_unit @ S )
% 0.34/0.55 => ( ( S != bot_bo3957492148770167129t_unit )
% 0.34/0.55 => ( ( member_Product_unit @ Y @ S )
% 0.34/0.55 => ( ord_less_eq_nat @ ( F @ ( lattic1360832392136924548it_nat @ F @ S ) ) @ ( F @ Y ) ) ) ) ) ).
% 0.34/0.55
% 0.34/0.55 % arg_min_least
% 0.34/0.55 thf(fact_1906_arg__min__least,axiom,
% 0.34/0.55 ! [S: set_o,Y: $o,F: $o > nat] :
% 0.34/0.55 ( ( finite_finite_o @ S )
% 0.34/0.55 => ( ( S != bot_bot_set_o )
% 0.34/0.55 => ( ( member_o @ Y @ S )
% 0.34/0.55 => ( ord_less_eq_nat @ ( F @ ( lattic2775856028456453135_o_nat @ F @ S ) ) @ ( F @ Y ) ) ) ) ) ).
% 0.34/0.55
% 0.34/0.55 % arg_min_least
% 0.34/0.55 thf(fact_1907_arg__min__least,axiom,
% 0.34/0.55 ! [S: set_nat,Y: nat,F: nat > nat] :
% 0.34/0.55 ( ( finite_finite_nat @ S )
% 0.34/0.55 => ( ( S != bot_bot_set_nat )
% 0.34/0.55 => ( ( member_nat @ Y @ S )
% 0.34/0.55 => ( ord_less_eq_nat @ ( F @ ( lattic7446932960582359483at_nat @ F @ S ) ) @ ( F @ Y ) ) ) ) ) ).
% 0.34/0.55
% 0.34/0.55 % arg_min_least
% 0.34/0.55 thf(fact_1908_arg__min__least,axiom,
% 0.34/0.55 ! [S: set_int,Y: int,F: int > nat] :
% 0.34/0.55 ( ( finite_finite_int @ S )
% 0.34/0.55 => ( ( S != bot_bot_set_int )
% 0.34/0.55 => ( ( member_int @ Y @ S )
% 0.34/0.55 => ( ord_less_eq_nat @ ( F @ ( lattic8446286672483414039nt_nat @ F @ S ) ) @ ( F @ Y ) ) ) ) ) ).
% 0.34/0.55
% 0.34/0.55 % arg_min_least
% 0.34/0.55 thf(fact_1909_Total__subset__Id,axiom,
% 0.34/0.55 ! [R2: set_Pr1261947904930325089at_nat] :
% 0.34/0.55 ( ( total_on_nat @ ( field_nat @ R2 ) @ R2 )
% 0.34/0.55 => ( ( ord_le3146513528884898305at_nat @ R2 @ id_nat )
% 0.34/0.55 => ( ( R2 = bot_bo2099793752762293965at_nat )
% 0.34/0.55 | ? [A4: nat] :
% 0.34/0.55 ( R2
% 0.34/0.55 = ( insert8211810215607154385at_nat @ ( product_Pair_nat_nat @ A4 @ A4 ) @ bot_bo2099793752762293965at_nat ) ) ) ) ) ).
% 0.34/0.55
% 0.34/0.55 % Total_subset_Id
% 0.34/0.55 thf(fact_1910_Total__subset__Id,axiom,
% 0.34/0.55 ! [R2: set_Pr958786334691620121nt_int] :
% 0.34/0.55 ( ( total_on_int @ ( field_int @ R2 ) @ R2 )
% 0.34/0.55 => ( ( ord_le2843351958646193337nt_int @ R2 @ id_int )
% 0.34/0.55 => ( ( R2 = bot_bo1796632182523588997nt_int )
% 0.34/0.55 | ? [A4: int] :
% 0.34/0.55 ( R2
% 0.34/0.55 = ( insert5033312907999012233nt_int @ ( product_Pair_int_int @ A4 @ A4 ) @ bot_bo1796632182523588997nt_int ) ) ) ) ) ).
% 0.34/0.55
% 0.34/0.55 % Total_subset_Id
% 0.34/0.55 thf(fact_1911_finite__transitivity__chain,axiom,
% 0.34/0.55 ! [A: set_Code_integer,R3: code_integer > code_integer > $o] :
% 0.34/0.55 ( ( finite6017078050557962740nteger @ A )
% 0.34/0.55 => ( ! [X3: code_integer] :
% 0.34/0.55 ~ ( R3 @ X3 @ X3 )
% 0.34/0.55 => ( ! [X3: code_integer,Y2: code_integer,Z3: code_integer] :
% 0.34/0.55 ( ( R3 @ X3 @ Y2 )
% 0.34/0.55 => ( ( R3 @ Y2 @ Z3 )
% 0.34/0.55 => ( R3 @ X3 @ Z3 ) ) )
% 0.34/0.55 => ( ! [X3: code_integer] :
% 0.34/0.55 ( ( member_Code_integer @ X3 @ A )
% 0.34/0.55 => ? [Y5: code_integer] :
% 0.34/0.55 ( ( member_Code_integer @ Y5 @ A )
% 0.34/0.55 & ( R3 @ X3 @ Y5 ) ) )
% 0.34/0.55 => ( A = bot_bo3990330152332043303nteger ) ) ) ) ) ).
% 0.34/0.55
% 0.34/0.55 % finite_transitivity_chain
% 0.34/0.55 thf(fact_1912_finite__transitivity__chain,axiom,
% 0.34/0.55 ! [A: set_Product_unit,R3: product_unit > product_unit > $o] :
% 0.34/0.55 ( ( finite4290736615968046902t_unit @ A )
% 0.34/0.55 => ( ! [X3: product_unit] :
% 0.34/0.55 ~ ( R3 @ X3 @ X3 )
% 0.34/0.55 => ( ! [X3: product_unit,Y2: product_unit,Z3: product_unit] :
% 0.34/0.55 ( ( R3 @ X3 @ Y2 )
% 0.34/0.55 => ( ( R3 @ Y2 @ Z3 )
% 0.34/0.55 => ( R3 @ X3 @ Z3 ) ) )
% 0.34/0.55 => ( ! [X3: product_unit] :
% 0.34/0.55 ( ( member_Product_unit @ X3 @ A )
% 0.34/0.55 => ? [Y5: product_unit] :
% 0.34/0.55 ( ( member_Product_unit @ Y5 @ A )
% 0.34/0.55 & ( R3 @ X3 @ Y5 ) ) )
% 0.34/0.55 => ( A = bot_bo3957492148770167129t_unit ) ) ) ) ) ).
% 0.34/0.55
% 0.34/0.55 % finite_transitivity_chain
% 0.34/0.55 thf(fact_1913_finite__transitivity__chain,axiom,
% 0.34/0.55 ! [A: set_o,R3: $o > $o > $o] :
% 0.34/0.55 ( ( finite_finite_o @ A )
% 0.34/0.55 => ( ! [X3: $o] :
% 0.34/0.55 ~ ( R3 @ X3 @ X3 )
% 0.34/0.55 => ( ! [X3: $o,Y2: $o,Z3: $o] :
% 0.34/0.55 ( ( R3 @ X3 @ Y2 )
% 0.34/0.55 => ( ( R3 @ Y2 @ Z3 )
% 0.34/0.55 => ( R3 @ X3 @ Z3 ) ) )
% 0.34/0.55 => ( ! [X3: $o] :
% 0.34/0.55 ( ( member_o @ X3 @ A )
% 0.34/0.55 => ? [Y5: $o] :
% 0.34/0.55 ( ( member_o @ Y5 @ A )
% 0.34/0.55 & ( R3 @ X3 @ Y5 ) ) )
% 0.34/0.55 => ( A = bot_bot_set_o ) ) ) ) ) ).
% 0.34/0.55
% 0.34/0.55 % finite_transitivity_chain
% 0.34/0.55 thf(fact_1914_finite__transitivity__chain,axiom,
% 0.34/0.55 ! [A: set_nat,R3: nat > nat > $o] :
% 0.34/0.55 ( ( finite_finite_nat @ A )
% 0.34/0.55 => ( ! [X3: nat] :
% 0.34/0.55 ~ ( R3 @ X3 @ X3 )
% 0.34/0.55 => ( ! [X3: nat,Y2: nat,Z3: nat] :
% 0.34/0.55 ( ( R3 @ X3 @ Y2 )
% 0.34/0.55 => ( ( R3 @ Y2 @ Z3 )
% 0.34/0.55 => ( R3 @ X3 @ Z3 ) ) )
% 0.34/0.55 => ( ! [X3: nat] :
% 0.34/0.55 ( ( member_nat @ X3 @ A )
% 0.34/0.55 => ? [Y5: nat] :
% 0.34/0.55 ( ( member_nat @ Y5 @ A )
% 0.34/0.55 & ( R3 @ X3 @ Y5 ) ) )
% 0.34/0.55 => ( A = bot_bot_set_nat ) ) ) ) ) ).
% 0.34/0.55
% 0.34/0.55 % finite_transitivity_chain
% 0.34/0.55 thf(fact_1915_finite__transitivity__chain,axiom,
% 0.34/0.55 ! [A: set_int,R3: int > int > $o] :
% 0.34/0.55 ( ( finite_finite_int @ A )
% 0.34/0.55 => ( ! [X3: int] :
% 0.34/0.55 ~ ( R3 @ X3 @ X3 )
% 0.34/0.55 => ( ! [X3: int,Y2: int,Z3: int] :
% 0.34/0.55 ( ( R3 @ X3 @ Y2 )
% 0.34/0.55 => ( ( R3 @ Y2 @ Z3 )
% 0.34/0.55 => ( R3 @ X3 @ Z3 ) ) )
% 0.34/0.55 => ( ! [X3: int] :
% 0.34/0.55 ( ( member_int @ X3 @ A )
% 0.34/0.55 => ? [Y5: int] :
% 0.34/0.55 ( ( member_int @ Y5 @ A )
% 0.34/0.55 & ( R3 @ X3 @ Y5 ) ) )
% 0.34/0.55 => ( A = bot_bot_set_int ) ) ) ) ) ).
% 0.34/0.55
% 0.34/0.55 % finite_transitivity_chain
% 0.34/0.55 thf(fact_1916_finite__transitivity__chain,axiom,
% 0.34/0.55 ! [A: set_Pr4532377907799695533_nat_o,R3: ( produc3658429121746597890et_nat > $o ) > ( produc3658429121746597890et_nat > $o ) > $o] :
% 0.34/0.55 ( ( finite3252695134891459830_nat_o @ A )
% 0.34/0.55 => ( ! [X3: produc3658429121746597890et_nat > $o] :
% 0.34/0.55 ~ ( R3 @ X3 @ X3 )
% 0.34/0.55 => ( ! [X3: produc3658429121746597890et_nat > $o,Y2: produc3658429121746597890et_nat > $o,Z3: produc3658429121746597890et_nat > $o] :
% 0.34/0.55 ( ( R3 @ X3 @ Y2 )
% 0.34/0.55 => ( ( R3 @ Y2 @ Z3 )
% 0.34/0.55 => ( R3 @ X3 @ Z3 ) ) )
% 0.34/0.55 => ( ! [X3: produc3658429121746597890et_nat > $o] :
% 0.34/0.55 ( ( member6576561426505652726_nat_o @ X3 @ A )
% 0.34/0.55 => ? [Y5: produc3658429121746597890et_nat > $o] :
% 0.34/0.55 ( ( member6576561426505652726_nat_o @ Y5 @ A )
% 0.34/0.55 & ( R3 @ X3 @ Y5 ) ) )
% 0.34/0.55 => ( A = bot_bo7824918357723582233_nat_o ) ) ) ) ) ).
% 0.34/0.55
% 0.34/0.55 % finite_transitivity_chain
% 0.34/0.55 thf(fact_1917_disjnt__equiv__class,axiom,
% 0.34/0.55 ! [A: set_Product_unit,R2: set_Pr5094982260447487303t_unit,A2: product_unit,B: product_unit] :
% 0.34/0.55 ( ( equiv_4189356295871601603t_unit @ A @ R2 )
% 0.34/0.55 => ( ( disjnt_Product_unit @ ( image_8431784415008483091t_unit @ R2 @ ( insert_Product_unit @ A2 @ bot_bo3957492148770167129t_unit ) ) @ ( image_8431784415008483091t_unit @ R2 @ ( insert_Product_unit @ B @ bot_bo3957492148770167129t_unit ) ) )
% 0.34/0.55 = ( ~ ( member7821138191923116944t_unit @ ( produc5692694829384537815t_unit @ A2 @ B ) @ R2 ) ) ) ) ).
% 0.34/0.55
% 0.34/0.55 % disjnt_equiv_class
% 0.34/0.55 thf(fact_1918_disjnt__equiv__class,axiom,
% 0.34/0.55 ! [A: set_o,R2: set_Product_prod_o_o,A2: $o,B: $o] :
% 0.34/0.55 ( ( equiv_equiv_o @ A @ R2 )
% 0.34/0.55 => ( ( disjnt_o @ ( image_o_o @ R2 @ ( insert_o @ A2 @ bot_bot_set_o ) ) @ ( image_o_o @ R2 @ ( insert_o @ B @ bot_bot_set_o ) ) )
% 0.34/0.55 = ( ~ ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ A2 @ B ) @ R2 ) ) ) ) ).
% 0.34/0.55
% 0.34/0.55 % disjnt_equiv_class
% 0.34/0.55 thf(fact_1919_disjnt__equiv__class,axiom,
% 0.34/0.55 ! [A: set_nat,R2: set_Pr1261947904930325089at_nat,A2: nat,B: nat] :
% 0.34/0.55 ( ( equiv_equiv_nat @ A @ R2 )
% 0.34/0.55 => ( ( disjnt_nat @ ( image_nat_nat @ R2 @ ( insert_nat @ A2 @ bot_bot_set_nat ) ) @ ( image_nat_nat @ R2 @ ( insert_nat @ B @ bot_bot_set_nat ) ) )
% 0.34/0.55 = ( ~ ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A2 @ B ) @ R2 ) ) ) ) ).
% 0.34/0.55
% 0.34/0.55 % disjnt_equiv_class
% 0.34/0.55 thf(fact_1920_disjnt__equiv__class,axiom,
% 0.34/0.55 ! [A: set_int,R2: set_Pr958786334691620121nt_int,A2: int,B: int] :
% 0.34/0.55 ( ( equiv_equiv_int @ A @ R2 )
% 0.34/0.55 => ( ( disjnt_int @ ( image_int_int @ R2 @ ( insert_int @ A2 @ bot_bot_set_int ) ) @ ( image_int_int @ R2 @ ( insert_int @ B @ bot_bot_set_int ) ) )
% 0.34/0.55 = ( ~ ( member5262025264175285858nt_int @ ( product_Pair_int_int @ A2 @ B ) @ R2 ) ) ) ) ).
% 0.34/0.55
% 0.34/0.55 % disjnt_equiv_class
% 0.34/0.55 thf(fact_1921_disjnt__equiv__class,axiom,
% 0.34/0.55 ! [A: set_Pr4532377907799695533_nat_o,R2: set_Pr2161125870931222855_nat_o,A2: produc3658429121746597890et_nat > $o,B: produc3658429121746597890et_nat > $o] :
% 0.34/0.55 ( ( equiv_6726761681920839555_nat_o @ A @ R2 )
% 0.34/0.55 => ( ( disjnt7869448817988085652_nat_o @ ( image_2232984160439939859_nat_o @ R2 @ ( insert5175938949040314269_nat_o @ A2 @ bot_bo7824918357723582233_nat_o ) ) @ ( image_2232984160439939859_nat_o @ R2 @ ( insert5175938949040314269_nat_o @ B @ bot_bo7824918357723582233_nat_o ) ) )
% 0.34/0.55 = ( ~ ( member8781333585448626064_nat_o @ ( produc7368190662567826135_nat_o @ A2 @ B ) @ R2 ) ) ) ) ).
% 0.34/0.55
% 0.34/0.55 % disjnt_equiv_class
% 0.34/0.55 thf(fact_1922_Sup__fin_Oinsert,axiom,
% 0.34/0.55 ! [A: set_set_nat,X: set_nat] :
% 0.34/0.55 ( ( finite1152437895449049373et_nat @ A )
% 0.34/0.55 => ( ( A != bot_bot_set_set_nat )
% 0.34/0.55 => ( ( lattic3835124923745554447et_nat @ ( insert_set_nat @ X @ A ) )
% 0.34/0.55 = ( sup_sup_set_nat @ X @ ( lattic3835124923745554447et_nat @ A ) ) ) ) ) ).
% 0.34/0.55
% 0.34/0.55 % Sup_fin.insert
% 0.34/0.55 thf(fact_1923_Sup__fin_Oinsert,axiom,
% 0.34/0.55 ! [A: set_assn,X: assn] :
% 0.34/0.55 ( ( finite_finite_assn @ A )
% 0.34/0.55 => ( ( A != bot_bot_set_assn )
% 0.34/0.55 => ( ( lattic2150320897289308081n_assn @ ( insert_assn @ X @ A ) )
% 0.34/0.55 = ( sup_sup_assn @ X @ ( lattic2150320897289308081n_assn @ A ) ) ) ) ) ).
% 0.34/0.55
% 0.34/0.55 % Sup_fin.insert
% 0.34/0.55 thf(fact_1924_Sup__fin_Oinsert,axiom,
% 0.34/0.55 ! [A: set_Product_unit,X: product_unit] :
% 0.34/0.55 ( ( finite4290736615968046902t_unit @ A )
% 0.34/0.55 => ( ( A != bot_bo3957492148770167129t_unit )
% 0.34/0.55 => ( ( lattic5294303975357428420t_unit @ ( insert_Product_unit @ X @ A ) )
% 0.34/0.55 = ( sup_sup_Product_unit @ X @ ( lattic5294303975357428420t_unit @ A ) ) ) ) ) ).
% 0.34/0.55
% 0.34/0.55 % Sup_fin.insert
% 0.34/0.55 thf(fact_1925_Sup__fin_Oinsert,axiom,
% 0.34/0.55 ! [A: set_o,X: $o] :
% 0.34/0.55 ( ( finite_finite_o @ A )
% 0.34/0.55 => ( ( A != bot_bot_set_o )
% 0.34/0.55 => ( ( lattic1508158080041050831_fin_o @ ( insert_o @ X @ A ) )
% 0.34/0.55 = ( sup_sup_o @ X @ ( lattic1508158080041050831_fin_o @ A ) ) ) ) ) ).
% 0.34/0.55
% 0.34/0.55 % Sup_fin.insert
% 0.34/0.55 thf(fact_1926_Sup__fin_Oinsert,axiom,
% 0.34/0.55 ! [A: set_nat,X: nat] :
% 0.34/0.55 ( ( finite_finite_nat @ A )
% 0.34/0.55 => ( ( A != bot_bot_set_nat )
% 0.34/0.55 => ( ( lattic1093996805478795353in_nat @ ( insert_nat @ X @ A ) )
% 0.34/0.55 = ( sup_sup_nat @ X @ ( lattic1093996805478795353in_nat @ A ) ) ) ) ) ).
% 0.34/0.55
% 0.34/0.55 % Sup_fin.insert
% 0.34/0.55 thf(fact_1927_Sup__fin_Oinsert,axiom,
% 0.34/0.55 ! [A: set_int,X: int] :
% 0.34/0.55 ( ( finite_finite_int @ A )
% 0.34/0.55 => ( ( A != bot_bot_set_int )
% 0.34/0.55 => ( ( lattic1091506334969745077in_int @ ( insert_int @ X @ A ) )
% 0.34/0.55 = ( sup_sup_int @ X @ ( lattic1091506334969745077in_int @ A ) ) ) ) ) ).
% 0.34/0.55
% 0.34/0.55 % Sup_fin.insert
% 0.34/0.55 thf(fact_1928_Sup__fin_Oinsert,axiom,
% 0.34/0.55 ! [A: set_Pr4532377907799695533_nat_o,X: produc3658429121746597890et_nat > $o] :
% 0.34/0.55 ( ( finite3252695134891459830_nat_o @ A )
% 0.34/0.55 => ( ( A != bot_bo7824918357723582233_nat_o )
% 0.34/0.55 => ( ( lattic7320199455484906628_nat_o @ ( insert5175938949040314269_nat_o @ X @ A ) )
% 0.34/0.55 = ( sup_su5453871518329203617_nat_o @ X @ ( lattic7320199455484906628_nat_o @ A ) ) ) ) ) ).
% 0.34/0.55
% 0.34/0.55 % Sup_fin.insert
% 0.34/0.55 thf(fact_1929_Inf__fin_Oinsert,axiom,
% 0.34/0.55 ! [A: set_set_nat,X: set_nat] :
% 0.34/0.55 ( ( finite1152437895449049373et_nat @ A )
% 0.34/0.55 => ( ( A != bot_bot_set_set_nat )
% 0.34/0.55 => ( ( lattic3014633134055518761et_nat @ ( insert_set_nat @ X @ A ) )
% 0.34/0.55 = ( inf_inf_set_nat @ X @ ( lattic3014633134055518761et_nat @ A ) ) ) ) ) ).
% 0.34/0.55
% 0.34/0.55 % Inf_fin.insert
% 0.34/0.55 thf(fact_1930_Inf__fin_Oinsert,axiom,
% 0.34/0.55 ! [A: set_assn,X: assn] :
% 0.34/0.55 ( ( finite_finite_assn @ A )
% 0.34/0.55 => ( ( A != bot_bot_set_assn )
% 0.34/0.55 => ( ( lattic47131356835913163n_assn @ ( insert_assn @ X @ A ) )
% 0.34/0.55 = ( inf_inf_assn @ X @ ( lattic47131356835913163n_assn @ A ) ) ) ) ) ).
% 0.34/0.55
% 0.34/0.55 % Inf_fin.insert
% 0.34/0.55 thf(fact_1931_Inf__fin_Oinsert,axiom,
% 0.34/0.55 ! [A: set_Product_unit,X: product_unit] :
% 0.34/0.55 ( ( finite4290736615968046902t_unit @ A )
% 0.34/0.55 => ( ( A != bot_bo3957492148770167129t_unit )
% 0.34/0.55 => ( ( lattic1263872656861969706t_unit @ ( insert_Product_unit @ X @ A ) )
% 0.34/0.55 = ( inf_inf_Product_unit @ X @ ( lattic1263872656861969706t_unit @ A ) ) ) ) ) ).
% 0.34/0.55
% 0.34/0.55 % Inf_fin.insert
% 0.34/0.55 thf(fact_1932_Inf__fin_Oinsert,axiom,
% 0.34/0.55 ! [A: set_o,X: $o] :
% 0.34/0.55 ( ( finite_finite_o @ A )
% 0.34/0.55 => ( ( A != bot_bot_set_o )
% 0.34/0.55 => ( ( lattic4107685809792843317_fin_o @ ( insert_o @ X @ A ) )
% 0.34/0.55 = ( inf_inf_o @ X @ ( lattic4107685809792843317_fin_o @ A ) ) ) ) ) ).
% 0.34/0.55
% 0.34/0.55 % Inf_fin.insert
% 0.34/0.55 thf(fact_1933_Inf__fin_Oinsert,axiom,
% 0.34/0.55 ! [A: set_nat,X: nat] :
% 0.34/0.55 ( ( finite_finite_nat @ A )
% 0.34/0.55 => ( ( A != bot_bot_set_nat )
% 0.34/0.55 => ( ( lattic5238388535129920115in_nat @ ( insert_nat @ X @ A ) )
% 0.34/0.55 = ( inf_inf_nat @ X @ ( lattic5238388535129920115in_nat @ A ) ) ) ) ) ).
% 0.34/0.55
% 0.34/0.55 % Inf_fin.insert
% 0.34/0.55 thf(fact_1934_Inf__fin_Oinsert,axiom,
% 0.34/0.55 ! [A: set_int,X: int] :
% 0.34/0.55 ( ( finite_finite_int @ A )
% 0.34/0.55 => ( ( A != bot_bot_set_int )
% 0.34/0.55 => ( ( lattic5235898064620869839in_int @ ( insert_int @ X @ A ) )
% 0.34/0.55 = ( inf_inf_int @ X @ ( lattic5235898064620869839in_int @ A ) ) ) ) ) ).
% 0.34/0.55
% 0.34/0.55 % Inf_fin.insert
% 0.34/0.55 thf(fact_1935_Inf__fin_Oinsert,axiom,
% 0.34/0.55 ! [A: set_Pr4532377907799695533_nat_o,X: produc3658429121746597890et_nat > $o] :
% 0.34/0.55 ( ( finite3252695134891459830_nat_o @ A )
% 0.34/0.55 => ( ( A != bot_bo7824918357723582233_nat_o )
% 0.34/0.55 => ( ( lattic956194824204696298_nat_o @ ( insert5175938949040314269_nat_o @ X @ A ) )
% 0.34/0.55 = ( inf_in1318976480646536635_nat_o @ X @ ( lattic956194824204696298_nat_o @ A ) ) ) ) ) ).
% 0.34/0.55
% 0.34/0.55 % Inf_fin.insert
% 0.34/0.55 thf(fact_1936_in__quotient__imp__in__rel,axiom,
% 0.34/0.55 ! [A: set_Product_unit,R2: set_Pr5094982260447487303t_unit,X4: set_Product_unit,X: product_unit,Y: product_unit] :
% 0.34/0.55 ( ( equiv_4189356295871601603t_unit @ A @ R2 )
% 0.34/0.55 => ( ( member5877623283571906838t_unit @ X4 @ ( equiv_5172566880711946708t_unit @ A @ R2 ) )
% 0.34/0.55 => ( ( ord_le3507040750410214029t_unit @ ( insert_Product_unit @ X @ ( insert_Product_unit @ Y @ bot_bo3957492148770167129t_unit ) ) @ X4 )
% 0.34/0.55 => ( member7821138191923116944t_unit @ ( produc5692694829384537815t_unit @ X @ Y ) @ R2 ) ) ) ) ).
% 0.34/0.55
% 0.34/0.55 % in_quotient_imp_in_rel
% 0.34/0.55 thf(fact_1937_in__quotient__imp__in__rel,axiom,
% 0.34/0.55 ! [A: set_o,R2: set_Product_prod_o_o,X4: set_o,X: $o,Y: $o] :
% 0.34/0.55 ( ( equiv_equiv_o @ A @ R2 )
% 0.34/0.55 => ( ( member_set_o @ X4 @ ( equiv_quotient_o @ A @ R2 ) )
% 0.34/0.55 => ( ( ord_less_eq_set_o @ ( insert_o @ X @ ( insert_o @ Y @ bot_bot_set_o ) ) @ X4 )
% 0.34/0.55 => ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ X @ Y ) @ R2 ) ) ) ) ).
% 0.34/0.55
% 0.34/0.55 % in_quotient_imp_in_rel
% 0.34/0.55 thf(fact_1938_in__quotient__imp__in__rel,axiom,
% 0.34/0.55 ! [A: set_int,R2: set_Pr958786334691620121nt_int,X4: set_int,X: int,Y: int] :
% 0.34/0.55 ( ( equiv_equiv_int @ A @ R2 )
% 0.34/0.55 => ( ( member_set_int @ X4 @ ( equiv_quotient_int @ A @ R2 ) )
% 0.34/0.55 => ( ( ord_less_eq_set_int @ ( insert_int @ X @ ( insert_int @ Y @ bot_bot_set_int ) ) @ X4 )
% 0.34/0.55 => ( member5262025264175285858nt_int @ ( product_Pair_int_int @ X @ Y ) @ R2 ) ) ) ) ).
% 0.34/0.55
% 0.34/0.55 % in_quotient_imp_in_rel
% 0.34/0.55 thf(fact_1939_in__quotient__imp__in__rel,axiom,
% 0.34/0.55 ! [A: set_Pr4532377907799695533_nat_o,R2: set_Pr2161125870931222855_nat_o,X4: set_Pr4532377907799695533_nat_o,X: produc3658429121746597890et_nat > $o,Y: produc3658429121746597890et_nat > $o] :
% 0.34/0.55 ( ( equiv_6726761681920839555_nat_o @ A @ R2 )
% 0.34/0.55 => ( ( member1452989533377600726_nat_o @ X4 @ ( equiv_7899955228204526484_nat_o @ A @ R2 ) )
% 0.34/0.55 => ( ( ord_le2965882846123202637_nat_o @ ( insert5175938949040314269_nat_o @ X @ ( insert5175938949040314269_nat_o @ Y @ bot_bo7824918357723582233_nat_o ) ) @ X4 )
% 0.34/0.55 => ( member8781333585448626064_nat_o @ ( produc7368190662567826135_nat_o @ X @ Y ) @ R2 ) ) ) ) ).
% 0.34/0.55
% 0.34/0.55 % in_quotient_imp_in_rel
% 0.34/0.55 thf(fact_1940_in__quotient__imp__in__rel,axiom,
% 0.34/0.55 ! [A: set_nat,R2: set_Pr1261947904930325089at_nat,X4: set_nat,X: nat,Y: nat] :
% 0.34/0.55 ( ( equiv_equiv_nat @ A @ R2 )
% 0.34/0.55 => ( ( member_set_nat @ X4 @ ( equiv_quotient_nat @ A @ R2 ) )
% 0.34/0.55 => ( ( ord_less_eq_set_nat @ ( insert_nat @ X @ ( insert_nat @ Y @ bot_bot_set_nat ) ) @ X4 )
% 0.34/0.55 => ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ Y ) @ R2 ) ) ) ) ).
% 0.34/0.55
% 0.34/0.55 % in_quotient_imp_in_rel
% 0.34/0.55 thf(fact_1941_prod__set__simps_I1_J,axiom,
% 0.34/0.55 ! [X: heap_e7401611519738050253t_unit,Y: set_nat] :
% 0.34/0.55 ( ( basic_7833836874476830220et_nat @ ( produc7507926704131184380et_nat @ X @ Y ) )
% 0.34/0.55 = ( insert8563697753282147485t_unit @ X @ bot_bo3686245532977126937t_unit ) ) ).
% 0.34/0.55
% 0.34/0.55 % prod_set_simps(1)
% 0.34/0.55 thf(fact_1942_prod__set__simps_I1_J,axiom,
% 0.34/0.55 ! [X: nat,Y: nat] :
% 0.34/0.55 ( ( basic_fsts_nat_nat @ ( product_Pair_nat_nat @ X @ Y ) )
% 0.34/0.55 = ( insert_nat @ X @ bot_bot_set_nat ) ) ).
% 0.34/0.55
% 0.34/0.55 % prod_set_simps(1)
% 0.34/0.55 thf(fact_1943_prod__set__simps_I1_J,axiom,
% 0.34/0.55 ! [X: int,Y: int] :
% 0.34/0.55 ( ( basic_fsts_int_int @ ( product_Pair_int_int @ X @ Y ) )
% 0.34/0.55 = ( insert_int @ X @ bot_bot_set_int ) ) ).
% 0.34/0.55
% 0.34/0.55 % prod_set_simps(1)
% 0.34/0.55 thf(fact_1944_prod__set__simps_I1_J,axiom,
% 0.34/0.55 ! [X: produc3658429121746597890et_nat > $o,Y: produc3658429121746597890et_nat] :
% 0.34/0.55 ( ( basic_8654530971723826568et_nat @ ( produc5001842942810119800et_nat @ X @ Y ) )
% 0.34/0.55 = ( insert5175938949040314269_nat_o @ X @ bot_bo7824918357723582233_nat_o ) ) ).
% 0.34/0.55
% 0.34/0.55 % prod_set_simps(1)
% 0.34/0.55 thf(fact_1945_prod__set__simps_I1_J,axiom,
% 0.34/0.55 ! [X: produc3658429121746597890et_nat > $o,Y: produc3925858234332021118et_nat] :
% 0.34/0.55 ( ( basic_3966130907827183364et_nat @ ( produc2245416461498447860et_nat @ X @ Y ) )
% 0.34/0.56 = ( insert5175938949040314269_nat_o @ X @ bot_bo7824918357723582233_nat_o ) ) ).
% 0.34/0.56
% 0.34/0.56 % prod_set_simps(1)
% 0.34/0.56 thf(fact_1946_rel__restrict__tranclI,axiom,
% 0.34/0.56 ! [X: produc3658429121746597890et_nat > $o,Y: produc3658429121746597890et_nat > $o,E: set_Pr2161125870931222855_nat_o,R3: set_Pr4532377907799695533_nat_o] :
% 0.34/0.56 ( ( member8781333585448626064_nat_o @ ( produc7368190662567826135_nat_o @ X @ Y ) @ ( transi4009562729418357614_nat_o @ E ) )
% 0.34/0.56 => ( ~ ( member6576561426505652726_nat_o @ X @ R3 )
% 0.34/0.56 => ( ~ ( member6576561426505652726_nat_o @ Y @ R3 )
% 0.34/0.56 => ( ( ord_le2965882846123202637_nat_o @ ( image_2232984160439939859_nat_o @ E @ R3 ) @ R3 )
% 0.34/0.56 => ( member8781333585448626064_nat_o @ ( produc7368190662567826135_nat_o @ X @ Y ) @ ( transi4009562729418357614_nat_o @ ( rel_re3331361160397388719_nat_o @ E @ R3 ) ) ) ) ) ) ) ).
% 0.34/0.56
% 0.34/0.56 % rel_restrict_tranclI
% 0.34/0.56 thf(fact_1947_rel__restrict__tranclI,axiom,
% 0.34/0.56 ! [X: $o,Y: $o,E: set_Product_prod_o_o,R3: set_o] :
% 0.34/0.56 ( ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ X @ Y ) @ ( transitive_trancl_o @ E ) )
% 0.34/0.56 => ( ~ ( member_o @ X @ R3 )
% 0.34/0.56 => ( ~ ( member_o @ Y @ R3 )
% 0.34/0.56 => ( ( ord_less_eq_set_o @ ( image_o_o @ E @ R3 ) @ R3 )
% 0.34/0.56 => ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ X @ Y ) @ ( transitive_trancl_o @ ( rel_restrict_o @ E @ R3 ) ) ) ) ) ) ) ).
% 0.34/0.56
% 0.34/0.56 % rel_restrict_tranclI
% 0.34/0.56 thf(fact_1948_rel__restrict__tranclI,axiom,
% 0.34/0.56 ! [X: product_unit,Y: product_unit,E: set_Pr5094982260447487303t_unit,R3: set_Product_unit] :
% 0.34/0.56 ( ( member7821138191923116944t_unit @ ( produc5692694829384537815t_unit @ X @ Y ) @ ( transi430408012122777006t_unit @ E ) )
% 0.34/0.56 => ( ~ ( member_Product_unit @ X @ R3 )
% 0.34/0.56 => ( ~ ( member_Product_unit @ Y @ R3 )
% 0.34/0.56 => ( ( ord_le3507040750410214029t_unit @ ( image_8431784415008483091t_unit @ E @ R3 ) @ R3 )
% 0.34/0.56 => ( member7821138191923116944t_unit @ ( produc5692694829384537815t_unit @ X @ Y ) @ ( transi430408012122777006t_unit @ ( rel_re9207996470947987439t_unit @ E @ R3 ) ) ) ) ) ) ) ).
% 0.34/0.56
% 0.34/0.56 % rel_restrict_tranclI
% 0.34/0.56 thf(fact_1949_rel__restrict__tranclI,axiom,
% 0.34/0.56 ! [X: int,Y: int,E: set_Pr958786334691620121nt_int,R3: set_int] :
% 0.34/0.56 ( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ X @ Y ) @ ( transi6261509568448316235cl_int @ E ) )
% 0.34/0.56 => ( ~ ( member_int @ X @ R3 )
% 0.34/0.56 => ( ~ ( member_int @ Y @ R3 )
% 0.34/0.56 => ( ( ord_less_eq_set_int @ ( image_int_int @ E @ R3 ) @ R3 )
% 0.34/0.56 => ( member5262025264175285858nt_int @ ( product_Pair_int_int @ X @ Y ) @ ( transi6261509568448316235cl_int @ ( rel_restrict_int @ E @ R3 ) ) ) ) ) ) ) ).
% 0.34/0.56
% 0.34/0.56 % rel_restrict_tranclI
% 0.34/0.56 thf(fact_1950_rel__restrict__tranclI,axiom,
% 0.34/0.56 ! [X: nat,Y: nat,E: set_Pr1261947904930325089at_nat,R3: set_nat] :
% 0.34/0.56 ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ Y ) @ ( transi6264000038957366511cl_nat @ E ) )
% 0.34/0.56 => ( ~ ( member_nat @ X @ R3 )
% 0.34/0.56 => ( ~ ( member_nat @ Y @ R3 )
% 0.34/0.56 => ( ( ord_less_eq_set_nat @ ( image_nat_nat @ E @ R3 ) @ R3 )
% 0.34/0.56 => ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ Y ) @ ( transi6264000038957366511cl_nat @ ( rel_restrict_nat @ E @ R3 ) ) ) ) ) ) ) ).
% 0.34/0.56
% 0.34/0.56 % rel_restrict_tranclI
% 0.34/0.56 thf(fact_1951_pair__in__Id__conv,axiom,
% 0.34/0.56 ! [A2: nat,B: nat] :
% 0.34/0.56 ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A2 @ B ) @ id_nat )
% 0.34/0.56 = ( A2 = B ) ) ).
% 0.34/0.56
% 0.34/0.56 % pair_in_Id_conv
% 0.34/0.56 thf(fact_1952_pair__in__Id__conv,axiom,
% 0.34/0.56 ! [A2: int,B: int] :
% 0.34/0.56 ( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ A2 @ B ) @ id_int )
% 0.34/0.56 = ( A2 = B ) ) ).
% 0.34/0.56
% 0.34/0.56 % pair_in_Id_conv
% 0.34/0.56 thf(fact_1953_IdI,axiom,
% 0.34/0.56 ! [A2: nat] : ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A2 @ A2 ) @ id_nat ) ).
% 0.34/0.56
% 0.34/0.56 % IdI
% 0.34/0.56 thf(fact_1954_IdI,axiom,
% 0.34/0.56 ! [A2: int] : ( member5262025264175285858nt_int @ ( product_Pair_int_int @ A2 @ A2 ) @ id_int ) ).
% 0.34/0.56
% 0.34/0.56 % IdI
% 0.34/0.56 thf(fact_1955_quotient__empty,axiom,
% 0.34/0.56 ! [R2: set_Pr5094982260447487303t_unit] :
% 0.34/0.56 ( ( equiv_5172566880711946708t_unit @ bot_bo3957492148770167129t_unit @ R2 )
% 0.34/0.56 = bot_bo6284674557190688313t_unit ) ).
% 0.34/0.56
% 0.34/0.56 % quotient_empty
% 0.34/0.56 thf(fact_1956_quotient__empty,axiom,
% 0.34/0.56 ! [R2: set_Product_prod_o_o] :
% 0.34/0.56 ( ( equiv_quotient_o @ bot_bot_set_o @ R2 )
% 0.34/0.56 = bot_bot_set_set_o ) ).
% 0.34/0.56
% 0.34/0.56 % quotient_empty
% 0.34/0.56 thf(fact_1957_quotient__empty,axiom,
% 0.34/0.56 ! [R2: set_Pr1261947904930325089at_nat] :
% 0.34/0.56 ( ( equiv_quotient_nat @ bot_bot_set_nat @ R2 )
% 0.34/0.56 = bot_bot_set_set_nat ) ).
% 0.34/0.56
% 0.34/0.56 % quotient_empty
% 0.34/0.56 thf(fact_1958_quotient__empty,axiom,
% 0.34/0.56 ! [R2: set_Pr958786334691620121nt_int] :
% 0.34/0.56 ( ( equiv_quotient_int @ bot_bot_set_int @ R2 )
% 0.34/0.56 = bot_bot_set_set_int ) ).
% 0.34/0.56
% 0.34/0.56 % quotient_empty
% 0.34/0.56 thf(fact_1959_quotient__empty,axiom,
% 0.34/0.56 ! [R2: set_Pr2161125870931222855_nat_o] :
% 0.34/0.56 ( ( equiv_7899955228204526484_nat_o @ bot_bo7824918357723582233_nat_o @ R2 )
% 0.34/0.56 = bot_bo3736544561982865913_nat_o ) ).
% 0.34/0.56
% 0.34/0.56 % quotient_empty
% 0.34/0.56 thf(fact_1960_quotient__is__empty,axiom,
% 0.34/0.56 ! [A: set_Product_unit,R2: set_Pr5094982260447487303t_unit] :
% 0.34/0.56 ( ( ( equiv_5172566880711946708t_unit @ A @ R2 )
% 0.34/0.56 = bot_bo6284674557190688313t_unit )
% 0.34/0.56 = ( A = bot_bo3957492148770167129t_unit ) ) ).
% 0.34/0.56
% 0.34/0.56 % quotient_is_empty
% 0.34/0.56 thf(fact_1961_quotient__is__empty,axiom,
% 0.34/0.56 ! [A: set_o,R2: set_Product_prod_o_o] :
% 0.34/0.56 ( ( ( equiv_quotient_o @ A @ R2 )
% 0.34/0.56 = bot_bot_set_set_o )
% 0.34/0.56 = ( A = bot_bot_set_o ) ) ).
% 0.34/0.56
% 0.34/0.56 % quotient_is_empty
% 0.34/0.56 thf(fact_1962_quotient__is__empty,axiom,
% 0.34/0.56 ! [A: set_nat,R2: set_Pr1261947904930325089at_nat] :
% 0.34/0.56 ( ( ( equiv_quotient_nat @ A @ R2 )
% 0.34/0.56 = bot_bot_set_set_nat )
% 0.34/0.56 = ( A = bot_bot_set_nat ) ) ).
% 0.34/0.56
% 0.34/0.56 % quotient_is_empty
% 0.34/0.56 thf(fact_1963_quotient__is__empty,axiom,
% 0.34/0.56 ! [A: set_int,R2: set_Pr958786334691620121nt_int] :
% 0.34/0.56 ( ( ( equiv_quotient_int @ A @ R2 )
% 0.34/0.56 = bot_bot_set_set_int )
% 0.34/0.56 = ( A = bot_bot_set_int ) ) ).
% 0.34/0.56
% 0.34/0.56 % quotient_is_empty
% 0.34/0.56 thf(fact_1964_quotient__is__empty,axiom,
% 0.34/0.56 ! [A: set_Pr4532377907799695533_nat_o,R2: set_Pr2161125870931222855_nat_o] :
% 0.34/0.56 ( ( ( equiv_7899955228204526484_nat_o @ A @ R2 )
% 0.34/0.56 = bot_bo3736544561982865913_nat_o )
% 0.34/0.56 = ( A = bot_bo7824918357723582233_nat_o ) ) ).
% 0.34/0.56
% 0.34/0.56 % quotient_is_empty
% 0.34/0.56 thf(fact_1965_quotient__is__empty2,axiom,
% 0.34/0.56 ! [A: set_Product_unit,R2: set_Pr5094982260447487303t_unit] :
% 0.34/0.56 ( ( bot_bo6284674557190688313t_unit
% 0.34/0.56 = ( equiv_5172566880711946708t_unit @ A @ R2 ) )
% 0.34/0.56 = ( A = bot_bo3957492148770167129t_unit ) ) ).
% 0.34/0.56
% 0.34/0.56 % quotient_is_empty2
% 0.34/0.56 thf(fact_1966_quotient__is__empty2,axiom,
% 0.34/0.56 ! [A: set_o,R2: set_Product_prod_o_o] :
% 0.34/0.56 ( ( bot_bot_set_set_o
% 0.34/0.56 = ( equiv_quotient_o @ A @ R2 ) )
% 0.34/0.56 = ( A = bot_bot_set_o ) ) ).
% 0.34/0.56
% 0.34/0.56 % quotient_is_empty2
% 0.34/0.56 thf(fact_1967_quotient__is__empty2,axiom,
% 0.34/0.56 ! [A: set_nat,R2: set_Pr1261947904930325089at_nat] :
% 0.34/0.56 ( ( bot_bot_set_set_nat
% 0.34/0.56 = ( equiv_quotient_nat @ A @ R2 ) )
% 0.34/0.56 = ( A = bot_bot_set_nat ) ) ).
% 0.34/0.56
% 0.34/0.56 % quotient_is_empty2
% 0.34/0.56 thf(fact_1968_quotient__is__empty2,axiom,
% 0.34/0.56 ! [A: set_int,R2: set_Pr958786334691620121nt_int] :
% 0.34/0.56 ( ( bot_bot_set_set_int
% 0.34/0.56 = ( equiv_quotient_int @ A @ R2 ) )
% 0.34/0.56 = ( A = bot_bot_set_int ) ) ).
% 0.34/0.56
% 0.34/0.56 % quotient_is_empty2
% 0.34/0.56 thf(fact_1969_quotient__is__empty2,axiom,
% 0.34/0.56 ! [A: set_Pr4532377907799695533_nat_o,R2: set_Pr2161125870931222855_nat_o] :
% 0.34/0.56 ( ( bot_bo3736544561982865913_nat_o
% 0.34/0.56 = ( equiv_7899955228204526484_nat_o @ A @ R2 ) )
% 0.34/0.56 = ( A = bot_bo7824918357723582233_nat_o ) ) ).
% 0.34/0.56
% 0.34/0.56 % quotient_is_empty2
% 0.34/0.56 thf(fact_1970_disjnt__self__iff__empty,axiom,
% 0.34/0.56 ! [S: set_Product_unit] :
% 0.34/0.56 ( ( disjnt_Product_unit @ S @ S )
% 0.34/0.56 = ( S = bot_bo3957492148770167129t_unit ) ) ).
% 0.34/0.56
% 0.34/0.56 % disjnt_self_iff_empty
% 0.34/0.56 thf(fact_1971_disjnt__self__iff__empty,axiom,
% 0.34/0.56 ! [S: set_o] :
% 0.34/0.56 ( ( disjnt_o @ S @ S )
% 0.34/0.56 = ( S = bot_bot_set_o ) ) ).
% 0.34/0.56
% 0.34/0.56 % disjnt_self_iff_empty
% 0.34/0.56 thf(fact_1972_disjnt__self__iff__empty,axiom,
% 0.34/0.56 ! [S: set_nat] :
% 0.34/0.56 ( ( disjnt_nat @ S @ S )
% 0.34/0.56 = ( S = bot_bot_set_nat ) ) ).
% 0.34/0.56
% 0.34/0.56 % disjnt_self_iff_empty
% 0.34/0.56 thf(fact_1973_disjnt__self__iff__empty,axiom,
% 0.34/0.56 ! [S: set_int] :
% 0.34/0.56 ( ( disjnt_int @ S @ S )
% 0.34/0.56 = ( S = bot_bot_set_int ) ) ).
% 0.34/0.56
% 0.34/0.56 % disjnt_self_iff_empty
% 0.34/0.56 thf(fact_1974_disjnt__self__iff__empty,axiom,
% 0.34/0.56 ! [S: set_Pr4532377907799695533_nat_o] :
% 0.34/0.56 ( ( disjnt7869448817988085652_nat_o @ S @ S )
% 0.34/0.56 = ( S = bot_bo7824918357723582233_nat_o ) ) ).
% 0.34/0.56
% 0.34/0.56 % disjnt_self_iff_empty
% 0.34/0.56 thf(fact_1975_disjnt__insert1,axiom,
% 0.34/0.56 ! [A2: produc3658429121746597890et_nat > $o,X4: set_Pr4532377907799695533_nat_o,Y6: set_Pr4532377907799695533_nat_o] :
% 0.34/0.56 ( ( disjnt7869448817988085652_nat_o @ ( insert5175938949040314269_nat_o @ A2 @ X4 ) @ Y6 )
% 0.34/0.56 = ( ~ ( member6576561426505652726_nat_o @ A2 @ Y6 )
% 0.34/0.56 & ( disjnt7869448817988085652_nat_o @ X4 @ Y6 ) ) ) ).
% 0.34/0.56
% 0.34/0.56 % disjnt_insert1
% 0.34/0.56 thf(fact_1976_disjnt__insert1,axiom,
% 0.34/0.56 ! [A2: nat,X4: set_nat,Y6: set_nat] :
% 0.34/0.56 ( ( disjnt_nat @ ( insert_nat @ A2 @ X4 ) @ Y6 )
% 0.34/0.56 = ( ~ ( member_nat @ A2 @ Y6 )
% 0.34/0.56 & ( disjnt_nat @ X4 @ Y6 ) ) ) ).
% 0.34/0.56
% 0.34/0.56 % disjnt_insert1
% 0.34/0.56 thf(fact_1977_disjnt__insert1,axiom,
% 0.34/0.56 ! [A2: int,X4: set_int,Y6: set_int] :
% 0.34/0.56 ( ( disjnt_int @ ( insert_int @ A2 @ X4 ) @ Y6 )
% 0.34/0.56 = ( ~ ( member_int @ A2 @ Y6 )
% 0.34/0.56 & ( disjnt_int @ X4 @ Y6 ) ) ) ).
% 0.34/0.56
% 0.34/0.56 % disjnt_insert1
% 0.34/0.56 thf(fact_1978_disjnt__insert1,axiom,
% 0.34/0.56 ! [A2: $o,X4: set_o,Y6: set_o] :
% 0.34/0.56 ( ( disjnt_o @ ( insert_o @ A2 @ X4 ) @ Y6 )
% 0.34/0.56 = ( ~ ( member_o @ A2 @ Y6 )
% 0.34/0.56 & ( disjnt_o @ X4 @ Y6 ) ) ) ).
% 0.34/0.56
% 0.34/0.56 % disjnt_insert1
% 0.34/0.56 thf(fact_1979_disjnt__insert1,axiom,
% 0.34/0.56 ! [A2: product_unit,X4: set_Product_unit,Y6: set_Product_unit] :
% 0.34/0.56 ( ( disjnt_Product_unit @ ( insert_Product_unit @ A2 @ X4 ) @ Y6 )
% 0.34/0.56 = ( ~ ( member_Product_unit @ A2 @ Y6 )
% 0.34/0.56 & ( disjnt_Product_unit @ X4 @ Y6 ) ) ) ).
% 0.34/0.56
% 0.34/0.56 % disjnt_insert1
% 0.34/0.56 thf(fact_1980_disjnt__insert2,axiom,
% 0.34/0.56 ! [Y6: set_Pr4532377907799695533_nat_o,A2: produc3658429121746597890et_nat > $o,X4: set_Pr4532377907799695533_nat_o] :
% 0.34/0.56 ( ( disjnt7869448817988085652_nat_o @ Y6 @ ( insert5175938949040314269_nat_o @ A2 @ X4 ) )
% 0.34/0.56 = ( ~ ( member6576561426505652726_nat_o @ A2 @ Y6 )
% 0.34/0.56 & ( disjnt7869448817988085652_nat_o @ Y6 @ X4 ) ) ) ).
% 0.34/0.56
% 0.34/0.56 % disjnt_insert2
% 0.34/0.56 thf(fact_1981_disjnt__insert2,axiom,
% 0.34/0.56 ! [Y6: set_nat,A2: nat,X4: set_nat] :
% 0.34/0.56 ( ( disjnt_nat @ Y6 @ ( insert_nat @ A2 @ X4 ) )
% 0.34/0.56 = ( ~ ( member_nat @ A2 @ Y6 )
% 0.34/0.56 & ( disjnt_nat @ Y6 @ X4 ) ) ) ).
% 0.34/0.56
% 0.34/0.56 % disjnt_insert2
% 0.34/0.56 thf(fact_1982_disjnt__insert2,axiom,
% 0.34/0.56 ! [Y6: set_int,A2: int,X4: set_int] :
% 0.34/0.56 ( ( disjnt_int @ Y6 @ ( insert_int @ A2 @ X4 ) )
% 0.34/0.56 = ( ~ ( member_int @ A2 @ Y6 )
% 0.34/0.56 & ( disjnt_int @ Y6 @ X4 ) ) ) ).
% 0.34/0.56
% 0.34/0.56 % disjnt_insert2
% 0.34/0.56 thf(fact_1983_disjnt__insert2,axiom,
% 0.34/0.56 ! [Y6: set_o,A2: $o,X4: set_o] :
% 0.34/0.56 ( ( disjnt_o @ Y6 @ ( insert_o @ A2 @ X4 ) )
% 0.34/0.56 = ( ~ ( member_o @ A2 @ Y6 )
% 0.34/0.56 & ( disjnt_o @ Y6 @ X4 ) ) ) ).
% 0.34/0.56
% 0.34/0.56 % disjnt_insert2
% 0.34/0.56 thf(fact_1984_disjnt__insert2,axiom,
% 0.34/0.56 ! [Y6: set_Product_unit,A2: product_unit,X4: set_Product_unit] :
% 0.34/0.56 ( ( disjnt_Product_unit @ Y6 @ ( insert_Product_unit @ A2 @ X4 ) )
% 0.34/0.56 = ( ~ ( member_Product_unit @ A2 @ Y6 )
% 0.34/0.56 & ( disjnt_Product_unit @ Y6 @ X4 ) ) ) ).
% 0.34/0.56
% 0.34/0.56 % disjnt_insert2
% 0.34/0.56 thf(fact_1985_rel__restrict__empty,axiom,
% 0.34/0.56 ! [R3: set_Pr5094982260447487303t_unit] :
% 0.34/0.56 ( ( rel_re9207996470947987439t_unit @ R3 @ bot_bo3957492148770167129t_unit )
% 0.34/0.56 = R3 ) ).
% 0.34/0.56
% 0.34/0.56 % rel_restrict_empty
% 0.34/0.56 thf(fact_1986_rel__restrict__empty,axiom,
% 0.34/0.56 ! [R3: set_Product_prod_o_o] :
% 0.34/0.56 ( ( rel_restrict_o @ R3 @ bot_bot_set_o )
% 0.34/0.56 = R3 ) ).
% 0.34/0.56
% 0.34/0.56 % rel_restrict_empty
% 0.34/0.56 thf(fact_1987_rel__restrict__empty,axiom,
% 0.34/0.56 ! [R3: set_Pr1261947904930325089at_nat] :
% 0.34/0.56 ( ( rel_restrict_nat @ R3 @ bot_bot_set_nat )
% 0.34/0.56 = R3 ) ).
% 0.34/0.56
% 0.34/0.56 % rel_restrict_empty
% 0.34/0.56 thf(fact_1988_rel__restrict__empty,axiom,
% 0.34/0.56 ! [R3: set_Pr958786334691620121nt_int] :
% 0.34/0.56 ( ( rel_restrict_int @ R3 @ bot_bot_set_int )
% 0.34/0.56 = R3 ) ).
% 0.34/0.56
% 0.34/0.56 % rel_restrict_empty
% 0.34/0.56 thf(fact_1989_rel__restrict__empty,axiom,
% 0.34/0.56 ! [R3: set_Pr2161125870931222855_nat_o] :
% 0.34/0.56 ( ( rel_re3331361160397388719_nat_o @ R3 @ bot_bo7824918357723582233_nat_o )
% 0.34/0.56 = R3 ) ).
% 0.34/0.56
% 0.34/0.56 % rel_restrict_empty
% 0.34/0.56 thf(fact_1990_disjnt__Un1,axiom,
% 0.34/0.56 ! [A: set_nat,B4: set_nat,C5: set_nat] :
% 0.34/0.56 ( ( disjnt_nat @ ( sup_sup_set_nat @ A @ B4 ) @ C5 )
% 0.34/0.56 = ( ( disjnt_nat @ A @ C5 )
% 0.34/0.56 & ( disjnt_nat @ B4 @ C5 ) ) ) ).
% 0.34/0.56
% 0.34/0.56 % disjnt_Un1
% 0.34/0.56 thf(fact_1991_disjnt__Un2,axiom,
% 0.34/0.56 ! [C5: set_nat,A: set_nat,B4: set_nat] :
% 0.34/0.56 ( ( disjnt_nat @ C5 @ ( sup_sup_set_nat @ A @ B4 ) )
% 0.34/0.56 = ( ( disjnt_nat @ C5 @ A )
% 0.34/0.56 & ( disjnt_nat @ C5 @ B4 ) ) ) ).
% 0.34/0.56
% 0.34/0.56 % disjnt_Un2
% 0.34/0.56 thf(fact_1992_Sup__fin_Osingleton,axiom,
% 0.34/0.56 ! [X: product_unit] :
% 0.34/0.56 ( ( lattic5294303975357428420t_unit @ ( insert_Product_unit @ X @ bot_bo3957492148770167129t_unit ) )
% 0.34/0.56 = X ) ).
% 0.34/0.56
% 0.34/0.56 % Sup_fin.singleton
% 0.34/0.56 thf(fact_1993_Sup__fin_Osingleton,axiom,
% 0.34/0.56 ! [X: $o] :
% 0.34/0.56 ( ( lattic1508158080041050831_fin_o @ ( insert_o @ X @ bot_bot_set_o ) )
% 0.34/0.56 = X ) ).
% 0.34/0.56
% 0.34/0.56 % Sup_fin.singleton
% 0.34/0.56 thf(fact_1994_Sup__fin_Osingleton,axiom,
% 0.34/0.56 ! [X: nat] :
% 0.34/0.56 ( ( lattic1093996805478795353in_nat @ ( insert_nat @ X @ bot_bot_set_nat ) )
% 0.34/0.56 = X ) ).
% 0.34/0.56
% 0.34/0.56 % Sup_fin.singleton
% 0.34/0.56 thf(fact_1995_Sup__fin_Osingleton,axiom,
% 0.34/0.56 ! [X: int] :
% 0.34/0.56 ( ( lattic1091506334969745077in_int @ ( insert_int @ X @ bot_bot_set_int ) )
% 0.34/0.56 = X ) ).
% 0.34/0.56
% 0.34/0.56 % Sup_fin.singleton
% 0.34/0.56 thf(fact_1996_Sup__fin_Osingleton,axiom,
% 0.34/0.56 ! [X: produc3658429121746597890et_nat > $o] :
% 0.34/0.56 ( ( lattic7320199455484906628_nat_o @ ( insert5175938949040314269_nat_o @ X @ bot_bo7824918357723582233_nat_o ) )
% 0.34/0.56 = X ) ).
% 0.34/0.56
% 0.34/0.56 % Sup_fin.singleton
% 0.34/0.56 thf(fact_1997_Inf__fin_Osingleton,axiom,
% 0.34/0.56 ! [X: product_unit] :
% 0.34/0.56 ( ( lattic1263872656861969706t_unit @ ( insert_Product_unit @ X @ bot_bo3957492148770167129t_unit ) )
% 0.34/0.56 = X ) ).
% 0.34/0.56
% 0.34/0.56 % Inf_fin.singleton
% 0.34/0.56 thf(fact_1998_Inf__fin_Osingleton,axiom,
% 0.34/0.56 ! [X: $o] :
% 0.34/0.56 ( ( lattic4107685809792843317_fin_o @ ( insert_o @ X @ bot_bot_set_o ) )
% 0.34/0.56 = X ) ).
% 0.34/0.56
% 0.34/0.56 % Inf_fin.singleton
% 0.34/0.56 thf(fact_1999_Inf__fin_Osingleton,axiom,
% 0.34/0.56 ! [X: nat] :
% 0.34/0.56 ( ( lattic5238388535129920115in_nat @ ( insert_nat @ X @ bot_bot_set_nat ) )
% 0.34/0.56 = X ) ).
% 0.34/0.56
% 0.34/0.56 % Inf_fin.singleton
% 0.34/0.56 thf(fact_2000_Inf__fin_Osingleton,axiom,
% 0.34/0.56 ! [X: int] :
% 0.34/0.56 ( ( lattic5235898064620869839in_int @ ( insert_int @ X @ bot_bot_set_int ) )
% 0.34/0.56 = X ) ).
% 0.34/0.56
% 0.34/0.56 % Inf_fin.singleton
% 0.34/0.56 thf(fact_2001_Inf__fin_Osingleton,axiom,
% 0.34/0.56 ! [X: produc3658429121746597890et_nat > $o] :
% 0.34/0.56 ( ( lattic956194824204696298_nat_o @ ( insert5175938949040314269_nat_o @ X @ bot_bo7824918357723582233_nat_o ) )
% 0.34/0.56 = X ) ).
% 0.34/0.56
% 0.34/0.56 % Inf_fin.singleton
% 0.34/0.56 thf(fact_2002_disjnt__iff,axiom,
% 0.34/0.56 ( disjnt7869448817988085652_nat_o
% 0.34/0.56 = ( ^ [A3: set_Pr4532377907799695533_nat_o,B7: set_Pr4532377907799695533_nat_o] :
% 0.34/0.56 ! [X2: produc3658429121746597890et_nat > $o] :
% 0.34/0.56 ~ ( ( member6576561426505652726_nat_o @ X2 @ A3 )
% 0.34/0.56 & ( member6576561426505652726_nat_o @ X2 @ B7 ) ) ) ) ).
% 0.34/0.56
% 0.34/0.56 % disjnt_iff
% 0.34/0.56 thf(fact_2003_disjnt__iff,axiom,
% 0.34/0.56 ( disjnt_nat
% 0.34/0.56 = ( ^ [A3: set_nat,B7: set_nat] :
% 0.34/0.56 ! [X2: nat] :
% 0.34/0.56 ~ ( ( member_nat @ X2 @ A3 )
% 0.34/0.56 & ( member_nat @ X2 @ B7 ) ) ) ) ).
% 0.34/0.56
% 0.34/0.56 % disjnt_iff
% 0.34/0.56 thf(fact_2004_disjnt__iff,axiom,
% 0.34/0.56 ( disjnt_int
% 0.34/0.56 = ( ^ [A3: set_int,B7: set_int] :
% 0.34/0.56 ! [X2: int] :
% 0.34/0.56 ~ ( ( member_int @ X2 @ A3 )
% 0.34/0.56 & ( member_int @ X2 @ B7 ) ) ) ) ).
% 0.34/0.56
% 0.34/0.56 % disjnt_iff
% 0.34/0.56 thf(fact_2005_disjnt__iff,axiom,
% 0.34/0.56 ( disjnt_o
% 0.34/0.56 = ( ^ [A3: set_o,B7: set_o] :
% 0.34/0.56 ! [X2: $o] :
% 0.34/0.56 ~ ( ( member_o @ X2 @ A3 )
% 0.34/0.56 & ( member_o @ X2 @ B7 ) ) ) ) ).
% 0.34/0.56
% 0.34/0.56 % disjnt_iff
% 0.34/0.56 thf(fact_2006_disjnt__iff,axiom,
% 0.34/0.56 ( disjnt_Product_unit
% 0.34/0.56 = ( ^ [A3: set_Product_unit,B7: set_Product_unit] :
% 0.34/0.56 ! [X2: product_unit] :
% 0.34/0.56 ~ ( ( member_Product_unit @ X2 @ A3 )
% 0.34/0.56 & ( member_Product_unit @ X2 @ B7 ) ) ) ) ).
% 0.34/0.56
% 0.34/0.56 % disjnt_iff
% 0.34/0.56 thf(fact_2007_IdE,axiom,
% 0.34/0.56 ! [P3: product_prod_nat_nat] :
% 0.34/0.56 ( ( member8440522571783428010at_nat @ P3 @ id_nat )
% 0.34/0.56 => ~ ! [X3: nat] :
% 0.34/0.56 ( P3
% 0.34/0.56 != ( product_Pair_nat_nat @ X3 @ X3 ) ) ) ).
% 0.34/0.56
% 0.34/0.56 % IdE
% 0.34/0.56 thf(fact_2008_IdE,axiom,
% 0.34/0.56 ! [P3: product_prod_int_int] :
% 0.34/0.56 ( ( member5262025264175285858nt_int @ P3 @ id_int )
% 0.34/0.56 => ~ ! [X3: int] :
% 0.34/0.56 ( P3
% 0.34/0.56 != ( product_Pair_int_int @ X3 @ X3 ) ) ) ).
% 0.34/0.56
% 0.34/0.56 % IdE
% 0.34/0.56 thf(fact_2009_disjnt__empty1,axiom,
% 0.34/0.56 ! [A: set_Product_unit] : ( disjnt_Product_unit @ bot_bo3957492148770167129t_unit @ A ) ).
% 0.34/0.56
% 0.34/0.56 % disjnt_empty1
% 0.34/0.56 thf(fact_2010_disjnt__empty1,axiom,
% 0.34/0.56 ! [A: set_o] : ( disjnt_o @ bot_bot_set_o @ A ) ).
% 0.34/0.56
% 0.34/0.56 % disjnt_empty1
% 0.34/0.56 thf(fact_2011_disjnt__empty1,axiom,
% 0.34/0.56 ! [A: set_nat] : ( disjnt_nat @ bot_bot_set_nat @ A ) ).
% 0.34/0.56
% 0.34/0.56 % disjnt_empty1
% 0.34/0.56 thf(fact_2012_disjnt__empty1,axiom,
% 0.34/0.56 ! [A: set_int] : ( disjnt_int @ bot_bot_set_int @ A ) ).
% 0.34/0.56
% 0.34/0.56 % disjnt_empty1
% 0.34/0.56 thf(fact_2013_disjnt__empty1,axiom,
% 0.34/0.56 ! [A: set_Pr4532377907799695533_nat_o] : ( disjnt7869448817988085652_nat_o @ bot_bo7824918357723582233_nat_o @ A ) ).
% 0.34/0.56
% 0.34/0.56 % disjnt_empty1
% 0.34/0.56 thf(fact_2014_disjnt__empty2,axiom,
% 0.34/0.56 ! [A: set_Product_unit] : ( disjnt_Product_unit @ A @ bot_bo3957492148770167129t_unit ) ).
% 0.34/0.56
% 0.34/0.56 % disjnt_empty2
% 0.34/0.56 thf(fact_2015_disjnt__empty2,axiom,
% 0.34/0.56 ! [A: set_o] : ( disjnt_o @ A @ bot_bot_set_o ) ).
% 0.34/0.56
% 0.34/0.56 % disjnt_empty2
% 0.34/0.56 thf(fact_2016_disjnt__empty2,axiom,
% 0.34/0.56 ! [A: set_nat] : ( disjnt_nat @ A @ bot_bot_set_nat ) ).
% 0.34/0.56
% 0.34/0.56 % disjnt_empty2
% 0.34/0.56 thf(fact_2017_disjnt__empty2,axiom,
% 0.34/0.56 ! [A: set_int] : ( disjnt_int @ A @ bot_bot_set_int ) ).
% 0.34/0.56
% 0.34/0.56 % disjnt_empty2
% 0.34/0.56 thf(fact_2018_disjnt__empty2,axiom,
% 0.34/0.56 ! [A: set_Pr4532377907799695533_nat_o] : ( disjnt7869448817988085652_nat_o @ A @ bot_bo7824918357723582233_nat_o ) ).
% 0.34/0.56
% 0.34/0.56 % disjnt_empty2
% 0.34/0.56 thf(fact_2019_disjnt__subset1,axiom,
% 0.34/0.56 ! [X4: set_nat,Y6: set_nat,Z5: set_nat] :
% 0.34/0.56 ( ( disjnt_nat @ X4 @ Y6 )
% 0.34/0.56 => ( ( ord_less_eq_set_nat @ Z5 @ X4 )
% 0.34/0.56 => ( disjnt_nat @ Z5 @ Y6 ) ) ) ).
% 0.34/0.56
% 0.34/0.56 % disjnt_subset1
% 0.34/0.56 thf(fact_2020_disjnt__subset2,axiom,
% 0.34/0.56 ! [X4: set_nat,Y6: set_nat,Z5: set_nat] :
% 0.34/0.56 ( ( disjnt_nat @ X4 @ Y6 )
% 0.34/0.56 => ( ( ord_less_eq_set_nat @ Z5 @ Y6 )
% 0.34/0.56 => ( disjnt_nat @ X4 @ Z5 ) ) ) ).
% 0.34/0.56
% 0.34/0.56 % disjnt_subset2
% 0.34/0.56 thf(fact_2021_disjnt__insert,axiom,
% 0.34/0.56 ! [X: produc3658429121746597890et_nat > $o,N: set_Pr4532377907799695533_nat_o,M: set_Pr4532377907799695533_nat_o] :
% 0.34/0.56 ( ~ ( member6576561426505652726_nat_o @ X @ N )
% 0.34/0.56 => ( ( disjnt7869448817988085652_nat_o @ M @ N )
% 0.34/0.56 => ( disjnt7869448817988085652_nat_o @ ( insert5175938949040314269_nat_o @ X @ M ) @ N ) ) ) ).
% 0.34/0.56
% 0.34/0.56 % disjnt_insert
% 0.34/0.56 thf(fact_2022_disjnt__insert,axiom,
% 0.34/0.56 ! [X: nat,N: set_nat,M: set_nat] :
% 0.34/0.56 ( ~ ( member_nat @ X @ N )
% 0.34/0.56 => ( ( disjnt_nat @ M @ N )
% 0.34/0.56 => ( disjnt_nat @ ( insert_nat @ X @ M ) @ N ) ) ) ).
% 0.34/0.56
% 0.34/0.56 % disjnt_insert
% 0.34/0.56 thf(fact_2023_disjnt__insert,axiom,
% 0.34/0.56 ! [X: int,N: set_int,M: set_int] :
% 0.34/0.56 ( ~ ( member_int @ X @ N )
% 0.34/0.56 => ( ( disjnt_int @ M @ N )
% 0.34/0.56 => ( disjnt_int @ ( insert_int @ X @ M ) @ N ) ) ) ).
% 0.34/0.56
% 0.34/0.56 % disjnt_insert
% 0.34/0.56 thf(fact_2024_disjnt__insert,axiom,
% 0.34/0.56 ! [X: $o,N: set_o,M: set_o] :
% 0.34/0.56 ( ~ ( member_o @ X @ N )
% 0.34/0.56 => ( ( disjnt_o @ M @ N )
% 0.34/0.56 => ( disjnt_o @ ( insert_o @ X @ M ) @ N ) ) ) ).
% 0.34/0.56
% 0.34/0.56 % disjnt_insert
% 0.34/0.56 thf(fact_2025_disjnt__insert,axiom,
% 0.34/0.56 ! [X: product_unit,N: set_Product_unit,M: set_Product_unit] :
% 0.34/0.56 ( ~ ( member_Product_unit @ X @ N )
% 0.34/0.56 => ( ( disjnt_Product_unit @ M @ N )
% 0.34/0.56 => ( disjnt_Product_unit @ ( insert_Product_unit @ X @ M ) @ N ) ) ) ).
% 0.34/0.56
% 0.34/0.56 % disjnt_insert
% 0.34/0.56 thf(fact_2026_rel__restrict__notR_I2_J,axiom,
% 0.34/0.56 ! [X: produc3658429121746597890et_nat > $o,Y: produc3658429121746597890et_nat > $o,A: set_Pr2161125870931222855_nat_o,R3: set_Pr4532377907799695533_nat_o] :
% 0.34/0.56 ( ( member8781333585448626064_nat_o @ ( produc7368190662567826135_nat_o @ X @ Y ) @ ( rel_re3331361160397388719_nat_o @ A @ R3 ) )
% 0.34/0.56 => ~ ( member6576561426505652726_nat_o @ Y @ R3 ) ) ).
% 0.34/0.56
% 0.34/0.56 % rel_restrict_notR(2)
% 0.34/0.56 thf(fact_2027_rel__restrict__notR_I2_J,axiom,
% 0.34/0.56 ! [X: $o,Y: $o,A: set_Product_prod_o_o,R3: set_o] :
% 0.34/0.56 ( ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ X @ Y ) @ ( rel_restrict_o @ A @ R3 ) )
% 0.34/0.56 => ~ ( member_o @ Y @ R3 ) ) ).
% 0.34/0.56
% 0.34/0.56 % rel_restrict_notR(2)
% 0.34/0.56 thf(fact_2028_rel__restrict__notR_I2_J,axiom,
% 0.34/0.56 ! [X: product_unit,Y: product_unit,A: set_Pr5094982260447487303t_unit,R3: set_Product_unit] :
% 0.34/0.56 ( ( member7821138191923116944t_unit @ ( produc5692694829384537815t_unit @ X @ Y ) @ ( rel_re9207996470947987439t_unit @ A @ R3 ) )
% 0.34/0.56 => ~ ( member_Product_unit @ Y @ R3 ) ) ).
% 0.34/0.56
% 0.34/0.56 % rel_restrict_notR(2)
% 0.34/0.56 thf(fact_2029_rel__restrict__notR_I2_J,axiom,
% 0.34/0.56 ! [X: nat,Y: nat,A: set_Pr1261947904930325089at_nat,R3: set_nat] :
% 0.34/0.56 ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ Y ) @ ( rel_restrict_nat @ A @ R3 ) )
% 0.34/0.56 => ~ ( member_nat @ Y @ R3 ) ) ).
% 0.34/0.56
% 0.34/0.56 % rel_restrict_notR(2)
% 0.34/0.56 thf(fact_2030_rel__restrict__notR_I2_J,axiom,
% 0.34/0.56 ! [X: int,Y: int,A: set_Pr958786334691620121nt_int,R3: set_int] :
% 0.34/0.56 ( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ X @ Y ) @ ( rel_restrict_int @ A @ R3 ) )
% 0.34/0.56 => ~ ( member_int @ Y @ R3 ) ) ).
% 0.34/0.56
% 0.34/0.56 % rel_restrict_notR(2)
% 0.34/0.56 thf(fact_2031_rel__restrict__notR_I1_J,axiom,
% 0.34/0.56 ! [X: produc3658429121746597890et_nat > $o,Y: produc3658429121746597890et_nat > $o,A: set_Pr2161125870931222855_nat_o,R3: set_Pr4532377907799695533_nat_o] :
% 0.34/0.56 ( ( member8781333585448626064_nat_o @ ( produc7368190662567826135_nat_o @ X @ Y ) @ ( rel_re3331361160397388719_nat_o @ A @ R3 ) )
% 0.34/0.56 => ~ ( member6576561426505652726_nat_o @ X @ R3 ) ) ).
% 0.34/0.56
% 0.34/0.56 % rel_restrict_notR(1)
% 0.34/0.56 thf(fact_2032_rel__restrict__notR_I1_J,axiom,
% 0.34/0.56 ! [X: $o,Y: $o,A: set_Product_prod_o_o,R3: set_o] :
% 0.34/0.56 ( ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ X @ Y ) @ ( rel_restrict_o @ A @ R3 ) )
% 0.34/0.56 => ~ ( member_o @ X @ R3 ) ) ).
% 0.34/0.56
% 0.34/0.56 % rel_restrict_notR(1)
% 0.34/0.56 thf(fact_2033_rel__restrict__notR_I1_J,axiom,
% 0.34/0.56 ! [X: product_unit,Y: product_unit,A: set_Pr5094982260447487303t_unit,R3: set_Product_unit] :
% 0.34/0.56 ( ( member7821138191923116944t_unit @ ( produc5692694829384537815t_unit @ X @ Y ) @ ( rel_re9207996470947987439t_unit @ A @ R3 ) )
% 0.34/0.56 => ~ ( member_Product_unit @ X @ R3 ) ) ).
% 0.34/0.56
% 0.34/0.56 % rel_restrict_notR(1)
% 0.34/0.56 thf(fact_2034_rel__restrict__notR_I1_J,axiom,
% 0.34/0.56 ! [X: nat,Y: nat,A: set_Pr1261947904930325089at_nat,R3: set_nat] :
% 0.34/0.56 ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ Y ) @ ( rel_restrict_nat @ A @ R3 ) )
% 0.34/0.56 => ~ ( member_nat @ X @ R3 ) ) ).
% 0.34/0.56
% 0.34/0.56 % rel_restrict_notR(1)
% 0.34/0.56 thf(fact_2035_rel__restrict__notR_I1_J,axiom,
% 0.34/0.56 ! [X: int,Y: int,A: set_Pr958786334691620121nt_int,R3: set_int] :
% 0.34/0.56 ( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ X @ Y ) @ ( rel_restrict_int @ A @ R3 ) )
% 0.34/0.56 => ~ ( member_int @ X @ R3 ) ) ).
% 0.34/0.56
% 0.34/0.56 % rel_restrict_notR(1)
% 0.34/0.56 thf(fact_2036_rel__restrictI,axiom,
% 0.34/0.56 ! [X: produc3658429121746597890et_nat > $o,R3: set_Pr4532377907799695533_nat_o,Y: produc3658429121746597890et_nat > $o,E: set_Pr2161125870931222855_nat_o] :
% 0.34/0.56 ( ~ ( member6576561426505652726_nat_o @ X @ R3 )
% 0.34/0.56 => ( ~ ( member6576561426505652726_nat_o @ Y @ R3 )
% 0.34/0.56 => ( ( member8781333585448626064_nat_o @ ( produc7368190662567826135_nat_o @ X @ Y ) @ E )
% 0.34/0.56 => ( member8781333585448626064_nat_o @ ( produc7368190662567826135_nat_o @ X @ Y ) @ ( rel_re3331361160397388719_nat_o @ E @ R3 ) ) ) ) ) ).
% 0.34/0.56
% 0.34/0.56 % rel_restrictI
% 0.34/0.56 thf(fact_2037_rel__restrictI,axiom,
% 0.34/0.56 ! [X: $o,R3: set_o,Y: $o,E: set_Product_prod_o_o] :
% 0.34/0.56 ( ~ ( member_o @ X @ R3 )
% 0.34/0.56 => ( ~ ( member_o @ Y @ R3 )
% 0.34/0.56 => ( ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ X @ Y ) @ E )
% 0.34/0.56 => ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ X @ Y ) @ ( rel_restrict_o @ E @ R3 ) ) ) ) ) ).
% 0.34/0.56
% 0.34/0.56 % rel_restrictI
% 0.34/0.56 thf(fact_2038_rel__restrictI,axiom,
% 0.34/0.56 ! [X: product_unit,R3: set_Product_unit,Y: product_unit,E: set_Pr5094982260447487303t_unit] :
% 0.34/0.56 ( ~ ( member_Product_unit @ X @ R3 )
% 0.34/0.56 => ( ~ ( member_Product_unit @ Y @ R3 )
% 0.34/0.56 => ( ( member7821138191923116944t_unit @ ( produc5692694829384537815t_unit @ X @ Y ) @ E )
% 0.34/0.56 => ( member7821138191923116944t_unit @ ( produc5692694829384537815t_unit @ X @ Y ) @ ( rel_re9207996470947987439t_unit @ E @ R3 ) ) ) ) ) ).
% 0.34/0.56
% 0.34/0.56 % rel_restrictI
% 0.34/0.56 thf(fact_2039_rel__restrictI,axiom,
% 0.34/0.56 ! [X: nat,R3: set_nat,Y: nat,E: set_Pr1261947904930325089at_nat] :
% 0.34/0.56 ( ~ ( member_nat @ X @ R3 )
% 0.34/0.56 => ( ~ ( member_nat @ Y @ R3 )
% 0.34/0.56 => ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ Y ) @ E )
% 0.34/0.56 => ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ Y ) @ ( rel_restrict_nat @ E @ R3 ) ) ) ) ) ).
% 0.34/0.56
% 0.34/0.56 % rel_restrictI
% 0.34/0.56 thf(fact_2040_rel__restrictI,axiom,
% 0.34/0.56 ! [X: int,R3: set_int,Y: int,E: set_Pr958786334691620121nt_int] :
% 0.34/0.56 ( ~ ( member_int @ X @ R3 )
% 0.34/0.56 => ( ~ ( member_int @ Y @ R3 )
% 0.34/0.56 => ( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ X @ Y ) @ E )
% 0.34/0.56 => ( member5262025264175285858nt_int @ ( product_Pair_int_int @ X @ Y ) @ ( rel_restrict_int @ E @ R3 ) ) ) ) ) ).
% 0.34/0.56
% 0.34/0.56 % rel_restrictI
% 0.34/0.56 thf(fact_2041_rel__restrict__lift,axiom,
% 0.34/0.56 ! [X: nat,Y: nat,E: set_Pr1261947904930325089at_nat,R3: set_nat] :
% 0.34/0.56 ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ Y ) @ ( rel_restrict_nat @ E @ R3 ) )
% 0.34/0.56 => ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ Y ) @ E ) ) ).
% 0.34/0.56
% 0.34/0.56 % rel_restrict_lift
% 0.34/0.56 thf(fact_2042_rel__restrict__lift,axiom,
% 0.34/0.56 ! [X: int,Y: int,E: set_Pr958786334691620121nt_int,R3: set_int] :
% 0.34/0.56 ( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ X @ Y ) @ ( rel_restrict_int @ E @ R3 ) )
% 0.34/0.56 => ( member5262025264175285858nt_int @ ( product_Pair_int_int @ X @ Y ) @ E ) ) ).
% 0.34/0.56
% 0.34/0.56 % rel_restrict_lift
% 0.34/0.56 thf(fact_2043_rel__restrict__union,axiom,
% 0.34/0.56 ! [R3: set_Pr1261947904930325089at_nat,A: set_nat,B4: set_nat] :
% 0.34/0.56 ( ( rel_restrict_nat @ R3 @ ( sup_sup_set_nat @ A @ B4 ) )
% 0.34/0.56 = ( rel_restrict_nat @ ( rel_restrict_nat @ R3 @ A ) @ B4 ) ) ).
% 0.34/0.56
% 0.34/0.56 % rel_restrict_union
% 0.34/0.56 thf(fact_2044_Inf__fin__le__Sup__fin,axiom,
% 0.34/0.56 ! [A: set_Product_unit] :
% 0.34/0.56 ( ( finite4290736615968046902t_unit @ A )
% 0.34/0.56 => ( ( A != bot_bo3957492148770167129t_unit )
% 0.34/0.56 => ( ord_le3221252021190050221t_unit @ ( lattic1263872656861969706t_unit @ A ) @ ( lattic5294303975357428420t_unit @ A ) ) ) ) ).
% 0.34/0.56
% 0.34/0.56 % Inf_fin_le_Sup_fin
% 0.34/0.56 thf(fact_2045_Inf__fin__le__Sup__fin,axiom,
% 0.34/0.56 ! [A: set_o] :
% 0.34/0.56 ( ( finite_finite_o @ A )
% 0.34/0.56 => ( ( A != bot_bot_set_o )
% 0.34/0.56 => ( ord_less_eq_o @ ( lattic4107685809792843317_fin_o @ A ) @ ( lattic1508158080041050831_fin_o @ A ) ) ) ) ).
% 0.34/0.56
% 0.34/0.56 % Inf_fin_le_Sup_fin
% 0.34/0.56 thf(fact_2046_Inf__fin__le__Sup__fin,axiom,
% 0.34/0.56 ! [A: set_Pr4532377907799695533_nat_o] :
% 0.34/0.56 ( ( finite3252695134891459830_nat_o @ A )
% 0.34/0.56 => ( ( A != bot_bo7824918357723582233_nat_o )
% 0.34/0.56 => ( ord_le729326519192465773_nat_o @ ( lattic956194824204696298_nat_o @ A ) @ ( lattic7320199455484906628_nat_o @ A ) ) ) ) ).
% 0.34/0.56
% 0.34/0.56 % Inf_fin_le_Sup_fin
% 0.34/0.56 thf(fact_2047_Inf__fin__le__Sup__fin,axiom,
% 0.34/0.56 ! [A: set_filter_nat] :
% 0.34/0.56 ( ( finite2119507909894593271er_nat @ A )
% 0.34/0.56 => ( ( A != bot_bo498966703094740906er_nat )
% 0.34/0.56 => ( ord_le2510731241096832064er_nat @ ( lattic9128708996678779395er_nat @ A ) @ ( lattic5930898082463196905er_nat @ A ) ) ) ) ).
% 0.34/0.56
% 0.34/0.56 % Inf_fin_le_Sup_fin
% 0.34/0.56 thf(fact_2048_Inf__fin__le__Sup__fin,axiom,
% 0.34/0.56 ! [A: set_set_nat] :
% 0.34/0.56 ( ( finite1152437895449049373et_nat @ A )
% 0.34/0.56 => ( ( A != bot_bot_set_set_nat )
% 0.34/0.56 => ( ord_less_eq_set_nat @ ( lattic3014633134055518761et_nat @ A ) @ ( lattic3835124923745554447et_nat @ A ) ) ) ) ).
% 0.34/0.56
% 0.34/0.56 % Inf_fin_le_Sup_fin
% 0.34/0.56 thf(fact_2049_Inf__fin__le__Sup__fin,axiom,
% 0.34/0.56 ! [A: set_rat] :
% 0.34/0.56 ( ( finite_finite_rat @ A )
% 0.34/0.56 => ( ( A != bot_bot_set_rat )
% 0.34/0.56 => ( ord_less_eq_rat @ ( lattic4603258475043424379in_rat @ A ) @ ( lattic458866745392299617in_rat @ A ) ) ) ) ).
% 0.34/0.56
% 0.34/0.56 % Inf_fin_le_Sup_fin
% 0.34/0.56 thf(fact_2050_Inf__fin__le__Sup__fin,axiom,
% 0.34/0.56 ! [A: set_nat] :
% 0.34/0.56 ( ( finite_finite_nat @ A )
% 0.34/0.56 => ( ( A != bot_bot_set_nat )
% 0.34/0.56 => ( ord_less_eq_nat @ ( lattic5238388535129920115in_nat @ A ) @ ( lattic1093996805478795353in_nat @ A ) ) ) ) ).
% 0.34/0.56
% 0.34/0.56 % Inf_fin_le_Sup_fin
% 0.34/0.56 thf(fact_2051_Inf__fin__le__Sup__fin,axiom,
% 0.34/0.56 ! [A: set_int] :
% 0.34/0.56 ( ( finite_finite_int @ A )
% 0.34/0.56 => ( ( A != bot_bot_set_int )
% 0.34/0.56 => ( ord_less_eq_int @ ( lattic5235898064620869839in_int @ A ) @ ( lattic1091506334969745077in_int @ A ) ) ) ) ).
% 0.34/0.56
% 0.34/0.56 % Inf_fin_le_Sup_fin
% 0.34/0.56 thf(fact_2052_disjnt__Sigma__iff,axiom,
% 0.34/0.56 ! [A: set_nat,C5: nat > set_Product_unit,B4: set_nat] :
% 0.34/0.56 ( ( disjnt595863415355979947t_unit @ ( produc5506539147108004339t_unit @ A @ C5 ) @ ( produc5506539147108004339t_unit @ B4 @ C5 ) )
% 0.34/0.56 = ( ! [X2: nat] :
% 0.34/0.56 ( ( member_nat @ X2 @ ( inf_inf_set_nat @ A @ B4 ) )
% 0.34/0.56 => ( ( C5 @ X2 )
% 0.34/0.56 = bot_bo3957492148770167129t_unit ) )
% 0.34/0.56 | ( disjnt_nat @ A @ B4 ) ) ) ).
% 0.34/0.56
% 0.34/0.56 % disjnt_Sigma_iff
% 0.34/0.56 thf(fact_2053_disjnt__Sigma__iff,axiom,
% 0.34/0.56 ! [A: set_nat,C5: nat > set_o,B4: set_nat] :
% 0.34/0.56 ( ( disjnt5652777605541633334_nat_o @ ( product_Sigma_nat_o @ A @ C5 ) @ ( product_Sigma_nat_o @ B4 @ C5 ) )
% 0.34/0.56 = ( ! [X2: nat] :
% 0.34/0.56 ( ( member_nat @ X2 @ ( inf_inf_set_nat @ A @ B4 ) )
% 0.34/0.56 => ( ( C5 @ X2 )
% 0.34/0.56 = bot_bot_set_o ) )
% 0.34/0.56 | ( disjnt_nat @ A @ B4 ) ) ) ).
% 0.34/0.56
% 0.34/0.56 % disjnt_Sigma_iff
% 0.34/0.56 thf(fact_2054_disjnt__Sigma__iff,axiom,
% 0.34/0.56 ! [A: set_nat,C5: nat > set_nat,B4: set_nat] :
% 0.34/0.56 ( ( disjnt2713546992526078024at_nat @ ( produc457027306803732586at_nat @ A @ C5 ) @ ( produc457027306803732586at_nat @ B4 @ C5 ) )
% 0.34/0.56 = ( ! [X2: nat] :
% 0.34/0.56 ( ( member_nat @ X2 @ ( inf_inf_set_nat @ A @ B4 ) )
% 0.34/0.56 => ( ( C5 @ X2 )
% 0.34/0.56 = bot_bot_set_nat ) )
% 0.34/0.56 | ( disjnt_nat @ A @ B4 ) ) ) ).
% 0.34/0.56
% 0.34/0.56 % disjnt_Sigma_iff
% 0.34/0.56 thf(fact_2055_disjnt__Sigma__iff,axiom,
% 0.34/0.56 ! [A: set_nat,C5: nat > set_int,B4: set_nat] :
% 0.34/0.56 ( ( disjnt7759068009871657124at_int @ ( produc454536836294682310at_int @ A @ C5 ) @ ( produc454536836294682310at_int @ B4 @ C5 ) )
% 0.34/0.56 = ( ! [X2: nat] :
% 0.34/0.56 ( ( member_nat @ X2 @ ( inf_inf_set_nat @ A @ B4 ) )
% 0.34/0.56 => ( ( C5 @ X2 )
% 0.34/0.56 = bot_bot_set_int ) )
% 0.34/0.56 | ( disjnt_nat @ A @ B4 ) ) ) ).
% 0.34/0.56
% 0.34/0.56 % disjnt_Sigma_iff
% 0.34/0.56 thf(fact_2056_disjnt__Sigma__iff,axiom,
% 0.34/0.56 ! [A: set_nat,C5: nat > set_Pr4532377907799695533_nat_o,B4: set_nat] :
% 0.34/0.56 ( ( disjnt4476176636800211563_nat_o @ ( produc5650067098820117939_nat_o @ A @ C5 ) @ ( produc5650067098820117939_nat_o @ B4 @ C5 ) )
% 0.34/0.56 = ( ! [X2: nat] :
% 0.34/0.56 ( ( member_nat @ X2 @ ( inf_inf_set_nat @ A @ B4 ) )
% 0.34/0.56 => ( ( C5 @ X2 )
% 0.34/0.56 = bot_bo7824918357723582233_nat_o ) )
% 0.34/0.56 | ( disjnt_nat @ A @ B4 ) ) ) ).
% 0.34/0.56
% 0.34/0.56 % disjnt_Sigma_iff
% 0.34/0.56 thf(fact_2057_disjnt__def,axiom,
% 0.34/0.56 ( disjnt_Product_unit
% 0.34/0.56 = ( ^ [A3: set_Product_unit,B7: set_Product_unit] :
% 0.34/0.56 ( ( inf_in4660618365625256667t_unit @ A3 @ B7 )
% 0.34/0.56 = bot_bo3957492148770167129t_unit ) ) ) ).
% 0.34/0.56
% 0.34/0.56 % disjnt_def
% 0.34/0.56 thf(fact_2058_disjnt__def,axiom,
% 0.34/0.56 ( disjnt_o
% 0.34/0.56 = ( ^ [A3: set_o,B7: set_o] :
% 0.34/0.56 ( ( inf_inf_set_o @ A3 @ B7 )
% 0.34/0.56 = bot_bot_set_o ) ) ) ).
% 0.34/0.56
% 0.34/0.56 % disjnt_def
% 0.34/0.56 thf(fact_2059_disjnt__def,axiom,
% 0.34/0.56 ( disjnt_nat
% 0.34/0.56 = ( ^ [A3: set_nat,B7: set_nat] :
% 0.34/0.56 ( ( inf_inf_set_nat @ A3 @ B7 )
% 0.34/0.56 = bot_bot_set_nat ) ) ) ).
% 0.34/0.56
% 0.34/0.56 % disjnt_def
% 0.34/0.56 thf(fact_2060_disjnt__def,axiom,
% 0.34/0.56 ( disjnt_int
% 0.34/0.56 = ( ^ [A3: set_int,B7: set_int] :
% 0.34/0.56 ( ( inf_inf_set_int @ A3 @ B7 )
% 0.34/0.56 = bot_bot_set_int ) ) ) ).
% 0.34/0.56
% 0.34/0.56 % disjnt_def
% 0.34/0.56 thf(fact_2061_disjnt__def,axiom,
% 0.34/0.56 ( disjnt7869448817988085652_nat_o
% 0.34/0.56 = ( ^ [A3: set_Pr4532377907799695533_nat_o,B7: set_Pr4532377907799695533_nat_o] :
% 0.34/0.56 ( ( inf_in1906310914598751387_nat_o @ A3 @ B7 )
% 0.34/0.56 = bot_bo7824918357723582233_nat_o ) ) ) ).
% 0.34/0.56
% 0.34/0.56 % disjnt_def
% 0.34/0.56 thf(fact_2062_quotient__eqI,axiom,
% 0.34/0.56 ! [A: set_Pr4532377907799695533_nat_o,R2: set_Pr2161125870931222855_nat_o,X4: set_Pr4532377907799695533_nat_o,Y6: set_Pr4532377907799695533_nat_o,X: produc3658429121746597890et_nat > $o,Y: produc3658429121746597890et_nat > $o] :
% 0.34/0.56 ( ( equiv_6726761681920839555_nat_o @ A @ R2 )
% 0.34/0.56 => ( ( member1452989533377600726_nat_o @ X4 @ ( equiv_7899955228204526484_nat_o @ A @ R2 ) )
% 0.34/0.56 => ( ( member1452989533377600726_nat_o @ Y6 @ ( equiv_7899955228204526484_nat_o @ A @ R2 ) )
% 0.34/0.56 => ( ( member6576561426505652726_nat_o @ X @ X4 )
% 0.34/0.56 => ( ( member6576561426505652726_nat_o @ Y @ Y6 )
% 0.34/0.56 => ( ( member8781333585448626064_nat_o @ ( produc7368190662567826135_nat_o @ X @ Y ) @ R2 )
% 0.34/0.56 => ( X4 = Y6 ) ) ) ) ) ) ) ).
% 0.34/0.56
% 0.34/0.56 % quotient_eqI
% 0.34/0.56 thf(fact_2063_quotient__eqI,axiom,
% 0.34/0.56 ! [A: set_o,R2: set_Product_prod_o_o,X4: set_o,Y6: set_o,X: $o,Y: $o] :
% 0.34/0.56 ( ( equiv_equiv_o @ A @ R2 )
% 0.34/0.56 => ( ( member_set_o @ X4 @ ( equiv_quotient_o @ A @ R2 ) )
% 0.34/0.56 => ( ( member_set_o @ Y6 @ ( equiv_quotient_o @ A @ R2 ) )
% 0.34/0.56 => ( ( member_o @ X @ X4 )
% 0.34/0.56 => ( ( member_o @ Y @ Y6 )
% 0.34/0.56 => ( ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ X @ Y ) @ R2 )
% 0.34/0.56 => ( X4 = Y6 ) ) ) ) ) ) ) ).
% 0.34/0.56
% 0.34/0.56 % quotient_eqI
% 0.34/0.56 thf(fact_2064_quotient__eqI,axiom,
% 0.34/0.56 ! [A: set_Product_unit,R2: set_Pr5094982260447487303t_unit,X4: set_Product_unit,Y6: set_Product_unit,X: product_unit,Y: product_unit] :
% 0.34/0.56 ( ( equiv_4189356295871601603t_unit @ A @ R2 )
% 0.34/0.56 => ( ( member5877623283571906838t_unit @ X4 @ ( equiv_5172566880711946708t_unit @ A @ R2 ) )
% 0.34/0.56 => ( ( member5877623283571906838t_unit @ Y6 @ ( equiv_5172566880711946708t_unit @ A @ R2 ) )
% 0.34/0.56 => ( ( member_Product_unit @ X @ X4 )
% 0.34/0.56 => ( ( member_Product_unit @ Y @ Y6 )
% 0.34/0.56 => ( ( member7821138191923116944t_unit @ ( produc5692694829384537815t_unit @ X @ Y ) @ R2 )
% 0.34/0.56 => ( X4 = Y6 ) ) ) ) ) ) ) ).
% 0.34/0.56
% 0.34/0.56 % quotient_eqI
% 0.34/0.56 thf(fact_2065_quotient__eqI,axiom,
% 0.34/0.56 ! [A: set_nat,R2: set_Pr1261947904930325089at_nat,X4: set_nat,Y6: set_nat,X: nat,Y: nat] :
% 0.34/0.56 ( ( equiv_equiv_nat @ A @ R2 )
% 0.34/0.56 => ( ( member_set_nat @ X4 @ ( equiv_quotient_nat @ A @ R2 ) )
% 0.34/0.56 => ( ( member_set_nat @ Y6 @ ( equiv_quotient_nat @ A @ R2 ) )
% 0.34/0.56 => ( ( member_nat @ X @ X4 )
% 0.34/0.56 => ( ( member_nat @ Y @ Y6 )
% 0.34/0.56 => ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ Y ) @ R2 )
% 0.34/0.56 => ( X4 = Y6 ) ) ) ) ) ) ) ).
% 0.34/0.56
% 0.34/0.56 % quotient_eqI
% 0.34/0.56 thf(fact_2066_quotient__eqI,axiom,
% 0.34/0.56 ! [A: set_int,R2: set_Pr958786334691620121nt_int,X4: set_int,Y6: set_int,X: int,Y: int] :
% 0.34/0.56 ( ( equiv_equiv_int @ A @ R2 )
% 0.34/0.56 => ( ( member_set_int @ X4 @ ( equiv_quotient_int @ A @ R2 ) )
% 0.34/0.56 => ( ( member_set_int @ Y6 @ ( equiv_quotient_int @ A @ R2 ) )
% 0.34/0.56 => ( ( member_int @ X @ X4 )
% 0.34/0.56 => ( ( member_int @ Y @ Y6 )
% 0.34/0.56 => ( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ X @ Y ) @ R2 )
% 0.34/0.56 => ( X4 = Y6 ) ) ) ) ) ) ) ).
% 0.34/0.56
% 0.34/0.56 % quotient_eqI
% 0.34/0.56 thf(fact_2067_quotient__eq__iff,axiom,
% 0.34/0.56 ! [A: set_Pr4532377907799695533_nat_o,R2: set_Pr2161125870931222855_nat_o,X4: set_Pr4532377907799695533_nat_o,Y6: set_Pr4532377907799695533_nat_o,X: produc3658429121746597890et_nat > $o,Y: produc3658429121746597890et_nat > $o] :
% 0.34/0.56 ( ( equiv_6726761681920839555_nat_o @ A @ R2 )
% 0.34/0.56 => ( ( member1452989533377600726_nat_o @ X4 @ ( equiv_7899955228204526484_nat_o @ A @ R2 ) )
% 0.34/0.56 => ( ( member1452989533377600726_nat_o @ Y6 @ ( equiv_7899955228204526484_nat_o @ A @ R2 ) )
% 0.34/0.56 => ( ( member6576561426505652726_nat_o @ X @ X4 )
% 0.34/0.56 => ( ( member6576561426505652726_nat_o @ Y @ Y6 )
% 0.34/0.56 => ( ( X4 = Y6 )
% 0.34/0.56 = ( member8781333585448626064_nat_o @ ( produc7368190662567826135_nat_o @ X @ Y ) @ R2 ) ) ) ) ) ) ) ).
% 0.34/0.56
% 0.34/0.56 % quotient_eq_iff
% 0.34/0.56 thf(fact_2068_quotient__eq__iff,axiom,
% 0.34/0.56 ! [A: set_o,R2: set_Product_prod_o_o,X4: set_o,Y6: set_o,X: $o,Y: $o] :
% 0.34/0.56 ( ( equiv_equiv_o @ A @ R2 )
% 0.34/0.56 => ( ( member_set_o @ X4 @ ( equiv_quotient_o @ A @ R2 ) )
% 0.34/0.56 => ( ( member_set_o @ Y6 @ ( equiv_quotient_o @ A @ R2 ) )
% 0.34/0.56 => ( ( member_o @ X @ X4 )
% 0.34/0.56 => ( ( member_o @ Y @ Y6 )
% 0.34/0.56 => ( ( X4 = Y6 )
% 0.34/0.56 = ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ X @ Y ) @ R2 ) ) ) ) ) ) ) ).
% 0.34/0.56
% 0.34/0.56 % quotient_eq_iff
% 0.34/0.56 thf(fact_2069_quotient__eq__iff,axiom,
% 0.34/0.56 ! [A: set_Product_unit,R2: set_Pr5094982260447487303t_unit,X4: set_Product_unit,Y6: set_Product_unit,X: product_unit,Y: product_unit] :
% 0.34/0.56 ( ( equiv_4189356295871601603t_unit @ A @ R2 )
% 0.34/0.56 => ( ( member5877623283571906838t_unit @ X4 @ ( equiv_5172566880711946708t_unit @ A @ R2 ) )
% 0.34/0.56 => ( ( member5877623283571906838t_unit @ Y6 @ ( equiv_5172566880711946708t_unit @ A @ R2 ) )
% 0.34/0.56 => ( ( member_Product_unit @ X @ X4 )
% 0.34/0.56 => ( ( member_Product_unit @ Y @ Y6 )
% 0.34/0.56 => ( ( X4 = Y6 )
% 0.34/0.56 = ( member7821138191923116944t_unit @ ( produc5692694829384537815t_unit @ X @ Y ) @ R2 ) ) ) ) ) ) ) ).
% 0.34/0.56
% 0.34/0.56 % quotient_eq_iff
% 0.34/0.56 thf(fact_2070_quotient__eq__iff,axiom,
% 0.34/0.56 ! [A: set_nat,R2: set_Pr1261947904930325089at_nat,X4: set_nat,Y6: set_nat,X: nat,Y: nat] :
% 0.34/0.56 ( ( equiv_equiv_nat @ A @ R2 )
% 0.34/0.56 => ( ( member_set_nat @ X4 @ ( equiv_quotient_nat @ A @ R2 ) )
% 0.34/0.56 => ( ( member_set_nat @ Y6 @ ( equiv_quotient_nat @ A @ R2 ) )
% 0.34/0.56 => ( ( member_nat @ X @ X4 )
% 0.34/0.56 => ( ( member_nat @ Y @ Y6 )
% 0.34/0.56 => ( ( X4 = Y6 )
% 0.34/0.56 = ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ Y ) @ R2 ) ) ) ) ) ) ) ).
% 0.34/0.56
% 0.34/0.56 % quotient_eq_iff
% 0.34/0.56 thf(fact_2071_quotient__eq__iff,axiom,
% 0.34/0.56 ! [A: set_int,R2: set_Pr958786334691620121nt_int,X4: set_int,Y6: set_int,X: int,Y: int] :
% 0.34/0.56 ( ( equiv_equiv_int @ A @ R2 )
% 0.34/0.56 => ( ( member_set_int @ X4 @ ( equiv_quotient_int @ A @ R2 ) )
% 0.34/0.56 => ( ( member_set_int @ Y6 @ ( equiv_quotient_int @ A @ R2 ) )
% 0.34/0.56 => ( ( member_int @ X @ X4 )
% 0.34/0.56 => ( ( member_int @ Y @ Y6 )
% 0.34/0.56 => ( ( X4 = Y6 )
% 0.34/0.56 = ( member5262025264175285858nt_int @ ( product_Pair_int_int @ X @ Y ) @ R2 ) ) ) ) ) ) ) ).
% 0.34/0.56
% 0.34/0.56 % quotient_eq_iff
% 0.34/0.56 thf(fact_2072_in__quotient__imp__closed,axiom,
% 0.34/0.56 ! [A: set_Pr4532377907799695533_nat_o,R2: set_Pr2161125870931222855_nat_o,X4: set_Pr4532377907799695533_nat_o,X: produc3658429121746597890et_nat > $o,Y: produc3658429121746597890et_nat > $o] :
% 0.34/0.56 ( ( equiv_6726761681920839555_nat_o @ A @ R2 )
% 0.34/0.56 => ( ( member1452989533377600726_nat_o @ X4 @ ( equiv_7899955228204526484_nat_o @ A @ R2 ) )
% 0.34/0.56 => ( ( member6576561426505652726_nat_o @ X @ X4 )
% 0.34/0.56 => ( ( member8781333585448626064_nat_o @ ( produc7368190662567826135_nat_o @ X @ Y ) @ R2 )
% 0.34/0.56 => ( member6576561426505652726_nat_o @ Y @ X4 ) ) ) ) ) ).
% 0.34/0.56
% 0.34/0.56 % in_quotient_imp_closed
% 0.34/0.56 thf(fact_2073_in__quotient__imp__closed,axiom,
% 0.34/0.56 ! [A: set_o,R2: set_Product_prod_o_o,X4: set_o,X: $o,Y: $o] :
% 0.34/0.56 ( ( equiv_equiv_o @ A @ R2 )
% 0.34/0.56 => ( ( member_set_o @ X4 @ ( equiv_quotient_o @ A @ R2 ) )
% 0.34/0.56 => ( ( member_o @ X @ X4 )
% 0.34/0.56 => ( ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ X @ Y ) @ R2 )
% 0.34/0.56 => ( member_o @ Y @ X4 ) ) ) ) ) ).
% 0.34/0.56
% 0.34/0.56 % in_quotient_imp_closed
% 0.34/0.56 thf(fact_2074_in__quotient__imp__closed,axiom,
% 0.34/0.56 ! [A: set_Product_unit,R2: set_Pr5094982260447487303t_unit,X4: set_Product_unit,X: product_unit,Y: product_unit] :
% 0.34/0.56 ( ( equiv_4189356295871601603t_unit @ A @ R2 )
% 0.34/0.56 => ( ( member5877623283571906838t_unit @ X4 @ ( equiv_5172566880711946708t_unit @ A @ R2 ) )
% 0.34/0.56 => ( ( member_Product_unit @ X @ X4 )
% 0.34/0.56 => ( ( member7821138191923116944t_unit @ ( produc5692694829384537815t_unit @ X @ Y ) @ R2 )
% 0.34/0.56 => ( member_Product_unit @ Y @ X4 ) ) ) ) ) ).
% 0.34/0.56
% 0.34/0.56 % in_quotient_imp_closed
% 0.34/0.56 thf(fact_2075_in__quotient__imp__closed,axiom,
% 0.34/0.56 ! [A: set_nat,R2: set_Pr1261947904930325089at_nat,X4: set_nat,X: nat,Y: nat] :
% 0.34/0.56 ( ( equiv_equiv_nat @ A @ R2 )
% 0.34/0.56 => ( ( member_set_nat @ X4 @ ( equiv_quotient_nat @ A @ R2 ) )
% 0.34/0.56 => ( ( member_nat @ X @ X4 )
% 0.34/0.56 => ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ Y ) @ R2 )
% 0.34/0.56 => ( member_nat @ Y @ X4 ) ) ) ) ) ).
% 0.34/0.56
% 0.34/0.56 % in_quotient_imp_closed
% 0.34/0.56 thf(fact_2076_in__quotient__imp__closed,axiom,
% 0.34/0.56 ! [A: set_int,R2: set_Pr958786334691620121nt_int,X4: set_int,X: int,Y: int] :
% 0.34/0.56 ( ( equiv_equiv_int @ A @ R2 )
% 0.34/0.56 => ( ( member_set_int @ X4 @ ( equiv_quotient_int @ A @ R2 ) )
% 0.34/0.56 => ( ( member_int @ X @ X4 )
% 0.34/0.56 => ( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ X @ Y ) @ R2 )
% 0.34/0.56 => ( member_int @ Y @ X4 ) ) ) ) ) ).
% 0.34/0.56
% 0.34/0.56 % in_quotient_imp_closed
% 0.34/0.56 thf(fact_2077_in__quotient__imp__non__empty,axiom,
% 0.34/0.56 ! [A: set_Product_unit,R2: set_Pr5094982260447487303t_unit,X4: set_Product_unit] :
% 0.34/0.56 ( ( equiv_4189356295871601603t_unit @ A @ R2 )
% 0.34/0.56 => ( ( member5877623283571906838t_unit @ X4 @ ( equiv_5172566880711946708t_unit @ A @ R2 ) )
% 0.34/0.56 => ( X4 != bot_bo3957492148770167129t_unit ) ) ) ).
% 0.34/0.56
% 0.34/0.56 % in_quotient_imp_non_empty
% 0.34/0.56 thf(fact_2078_in__quotient__imp__non__empty,axiom,
% 0.34/0.56 ! [A: set_o,R2: set_Product_prod_o_o,X4: set_o] :
% 0.34/0.56 ( ( equiv_equiv_o @ A @ R2 )
% 0.34/0.56 => ( ( member_set_o @ X4 @ ( equiv_quotient_o @ A @ R2 ) )
% 0.34/0.56 => ( X4 != bot_bot_set_o ) ) ) ).
% 0.34/0.56
% 0.34/0.56 % in_quotient_imp_non_empty
% 0.34/0.56 thf(fact_2079_in__quotient__imp__non__empty,axiom,
% 0.34/0.56 ! [A: set_nat,R2: set_Pr1261947904930325089at_nat,X4: set_nat] :
% 0.34/0.56 ( ( equiv_equiv_nat @ A @ R2 )
% 0.34/0.56 => ( ( member_set_nat @ X4 @ ( equiv_quotient_nat @ A @ R2 ) )
% 0.34/0.56 => ( X4 != bot_bot_set_nat ) ) ) ).
% 0.34/0.56
% 0.34/0.56 % in_quotient_imp_non_empty
% 0.34/0.56 thf(fact_2080_in__quotient__imp__non__empty,axiom,
% 0.34/0.56 ! [A: set_int,R2: set_Pr958786334691620121nt_int,X4: set_int] :
% 0.34/0.56 ( ( equiv_equiv_int @ A @ R2 )
% 0.34/0.56 => ( ( member_set_int @ X4 @ ( equiv_quotient_int @ A @ R2 ) )
% 0.34/0.56 => ( X4 != bot_bot_set_int ) ) ) ).
% 0.34/0.56
% 0.34/0.56 % in_quotient_imp_non_empty
% 0.34/0.56 thf(fact_2081_in__quotient__imp__non__empty,axiom,
% 0.34/0.56 ! [A: set_Pr4532377907799695533_nat_o,R2: set_Pr2161125870931222855_nat_o,X4: set_Pr4532377907799695533_nat_o] :
% 0.34/0.56 ( ( equiv_6726761681920839555_nat_o @ A @ R2 )
% 0.34/0.56 => ( ( member1452989533377600726_nat_o @ X4 @ ( equiv_7899955228204526484_nat_o @ A @ R2 ) )
% 0.34/0.56 => ( X4 != bot_bo7824918357723582233_nat_o ) ) ) ).
% 0.34/0.56
% 0.34/0.56 % in_quotient_imp_non_empty
% 0.34/0.56 thf(fact_2082_rel__restrict__trancl__notR_I2_J,axiom,
% 0.34/0.56 ! [V: produc3658429121746597890et_nat > $o,W: produc3658429121746597890et_nat > $o,E: set_Pr2161125870931222855_nat_o,R3: set_Pr4532377907799695533_nat_o] :
% 0.34/0.56 ( ( member8781333585448626064_nat_o @ ( produc7368190662567826135_nat_o @ V @ W ) @ ( transi4009562729418357614_nat_o @ ( rel_re3331361160397388719_nat_o @ E @ R3 ) ) )
% 0.34/0.56 => ~ ( member6576561426505652726_nat_o @ W @ R3 ) ) ).
% 0.34/0.56
% 0.34/0.56 % rel_restrict_trancl_notR(2)
% 0.34/0.56 thf(fact_2083_rel__restrict__trancl__notR_I2_J,axiom,
% 0.34/0.56 ! [V: $o,W: $o,E: set_Product_prod_o_o,R3: set_o] :
% 0.34/0.56 ( ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ V @ W ) @ ( transitive_trancl_o @ ( rel_restrict_o @ E @ R3 ) ) )
% 0.34/0.56 => ~ ( member_o @ W @ R3 ) ) ).
% 0.34/0.56
% 0.34/0.56 % rel_restrict_trancl_notR(2)
% 0.34/0.56 thf(fact_2084_rel__restrict__trancl__notR_I2_J,axiom,
% 0.34/0.56 ! [V: product_unit,W: product_unit,E: set_Pr5094982260447487303t_unit,R3: set_Product_unit] :
% 0.34/0.56 ( ( member7821138191923116944t_unit @ ( produc5692694829384537815t_unit @ V @ W ) @ ( transi430408012122777006t_unit @ ( rel_re9207996470947987439t_unit @ E @ R3 ) ) )
% 0.34/0.56 => ~ ( member_Product_unit @ W @ R3 ) ) ).
% 0.34/0.56
% 0.34/0.56 % rel_restrict_trancl_notR(2)
% 0.34/0.56 thf(fact_2085_rel__restrict__trancl__notR_I2_J,axiom,
% 0.34/0.56 ! [V: nat,W: nat,E: set_Pr1261947904930325089at_nat,R3: set_nat] :
% 0.34/0.56 ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ V @ W ) @ ( transi6264000038957366511cl_nat @ ( rel_restrict_nat @ E @ R3 ) ) )
% 0.34/0.56 => ~ ( member_nat @ W @ R3 ) ) ).
% 0.34/0.56
% 0.34/0.56 % rel_restrict_trancl_notR(2)
% 0.34/0.56 thf(fact_2086_rel__restrict__trancl__notR_I2_J,axiom,
% 0.34/0.56 ! [V: int,W: int,E: set_Pr958786334691620121nt_int,R3: set_int] :
% 0.34/0.56 ( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ V @ W ) @ ( transi6261509568448316235cl_int @ ( rel_restrict_int @ E @ R3 ) ) )
% 0.34/0.56 => ~ ( member_int @ W @ R3 ) ) ).
% 0.34/0.56
% 0.34/0.56 % rel_restrict_trancl_notR(2)
% 0.34/0.56 thf(fact_2087_rel__restrict__trancl__notR_I1_J,axiom,
% 0.34/0.56 ! [V: produc3658429121746597890et_nat > $o,W: produc3658429121746597890et_nat > $o,E: set_Pr2161125870931222855_nat_o,R3: set_Pr4532377907799695533_nat_o] :
% 0.34/0.56 ( ( member8781333585448626064_nat_o @ ( produc7368190662567826135_nat_o @ V @ W ) @ ( transi4009562729418357614_nat_o @ ( rel_re3331361160397388719_nat_o @ E @ R3 ) ) )
% 0.34/0.56 => ~ ( member6576561426505652726_nat_o @ V @ R3 ) ) ).
% 0.34/0.56
% 0.34/0.56 % rel_restrict_trancl_notR(1)
% 0.34/0.56 thf(fact_2088_rel__restrict__trancl__notR_I1_J,axiom,
% 0.34/0.56 ! [V: $o,W: $o,E: set_Product_prod_o_o,R3: set_o] :
% 0.34/0.56 ( ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ V @ W ) @ ( transitive_trancl_o @ ( rel_restrict_o @ E @ R3 ) ) )
% 0.34/0.56 => ~ ( member_o @ V @ R3 ) ) ).
% 0.34/0.56
% 0.34/0.56 % rel_restrict_trancl_notR(1)
% 0.34/0.56 thf(fact_2089_rel__restrict__trancl__notR_I1_J,axiom,
% 0.34/0.56 ! [V: product_unit,W: product_unit,E: set_Pr5094982260447487303t_unit,R3: set_Product_unit] :
% 0.34/0.56 ( ( member7821138191923116944t_unit @ ( produc5692694829384537815t_unit @ V @ W ) @ ( transi430408012122777006t_unit @ ( rel_re9207996470947987439t_unit @ E @ R3 ) ) )
% 0.34/0.56 => ~ ( member_Product_unit @ V @ R3 ) ) ).
% 0.34/0.56
% 0.34/0.56 % rel_restrict_trancl_notR(1)
% 0.34/0.56 thf(fact_2090_rel__restrict__trancl__notR_I1_J,axiom,
% 0.34/0.56 ! [V: nat,W: nat,E: set_Pr1261947904930325089at_nat,R3: set_nat] :
% 0.34/0.56 ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ V @ W ) @ ( transi6264000038957366511cl_nat @ ( rel_restrict_nat @ E @ R3 ) ) )
% 0.34/0.56 => ~ ( member_nat @ V @ R3 ) ) ).
% 0.34/0.56
% 0.34/0.56 % rel_restrict_trancl_notR(1)
% 0.34/0.56 thf(fact_2091_rel__restrict__trancl__notR_I1_J,axiom,
% 0.34/0.56 ! [V: int,W: int,E: set_Pr958786334691620121nt_int,R3: set_int] :
% 0.34/0.56 ( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ V @ W ) @ ( transi6261509568448316235cl_int @ ( rel_restrict_int @ E @ R3 ) ) )
% 0.34/0.56 => ~ ( member_int @ V @ R3 ) ) ).
% 0.34/0.56
% 0.34/0.56 % rel_restrict_trancl_notR(1)
% 0.34/0.56 thf(fact_2092_rel__restrict__trancl__mem,axiom,
% 0.34/0.56 ! [A2: nat,B: nat,A: set_Pr1261947904930325089at_nat,R3: set_nat] :
% 0.34/0.56 ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A2 @ B ) @ ( transi6264000038957366511cl_nat @ ( rel_restrict_nat @ A @ R3 ) ) )
% 0.34/0.56 => ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A2 @ B ) @ ( rel_restrict_nat @ ( transi6264000038957366511cl_nat @ A ) @ R3 ) ) ) ).
% 0.34/0.56
% 0.34/0.56 % rel_restrict_trancl_mem
% 0.34/0.56 thf(fact_2093_rel__restrict__trancl__mem,axiom,
% 0.34/0.56 ! [A2: int,B: int,A: set_Pr958786334691620121nt_int,R3: set_int] :
% 0.34/0.56 ( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ A2 @ B ) @ ( transi6261509568448316235cl_int @ ( rel_restrict_int @ A @ R3 ) ) )
% 0.34/0.56 => ( member5262025264175285858nt_int @ ( product_Pair_int_int @ A2 @ B ) @ ( rel_restrict_int @ ( transi6261509568448316235cl_int @ A ) @ R3 ) ) ) ).
% 0.34/0.56
% 0.34/0.56 % rel_restrict_trancl_mem
% 0.34/0.56 thf(fact_2094_rel__restrict__mono2,axiom,
% 0.34/0.56 ! [R3: set_nat,S: set_nat,A: set_Pr1261947904930325089at_nat] :
% 0.34/0.56 ( ( ord_less_eq_set_nat @ R3 @ S )
% 0.34/0.56 => ( ord_le3146513528884898305at_nat @ ( rel_restrict_nat @ A @ S ) @ ( rel_restrict_nat @ A @ R3 ) ) ) ).
% 0.34/0.56
% 0.34/0.56 % rel_restrict_mono2
% 0.34/0.56 thf(fact_2095_singleton__quotient,axiom,
% 0.34/0.56 ! [X: product_unit,R2: set_Pr5094982260447487303t_unit] :
% 0.34/0.56 ( ( equiv_5172566880711946708t_unit @ ( insert_Product_unit @ X @ bot_bo3957492148770167129t_unit ) @ R2 )
% 0.34/0.56 = ( insert8986028188087627965t_unit @ ( image_8431784415008483091t_unit @ R2 @ ( insert_Product_unit @ X @ bot_bo3957492148770167129t_unit ) ) @ bot_bo6284674557190688313t_unit ) ) ).
% 0.34/0.56
% 0.34/0.56 % singleton_quotient
% 0.34/0.56 thf(fact_2096_singleton__quotient,axiom,
% 0.34/0.56 ! [X: $o,R2: set_Product_prod_o_o] :
% 0.34/0.56 ( ( equiv_quotient_o @ ( insert_o @ X @ bot_bot_set_o ) @ R2 )
% 0.34/0.56 = ( insert_set_o @ ( image_o_o @ R2 @ ( insert_o @ X @ bot_bot_set_o ) ) @ bot_bot_set_set_o ) ) ).
% 0.34/0.56
% 0.34/0.56 % singleton_quotient
% 0.34/0.56 thf(fact_2097_singleton__quotient,axiom,
% 0.34/0.56 ! [X: nat,R2: set_Pr1261947904930325089at_nat] :
% 0.34/0.56 ( ( equiv_quotient_nat @ ( insert_nat @ X @ bot_bot_set_nat ) @ R2 )
% 0.34/0.56 = ( insert_set_nat @ ( image_nat_nat @ R2 @ ( insert_nat @ X @ bot_bot_set_nat ) ) @ bot_bot_set_set_nat ) ) ).
% 0.34/0.56
% 0.34/0.56 % singleton_quotient
% 0.34/0.56 thf(fact_2098_singleton__quotient,axiom,
% 0.34/0.56 ! [X: int,R2: set_Pr958786334691620121nt_int] :
% 0.34/0.56 ( ( equiv_quotient_int @ ( insert_int @ X @ bot_bot_set_int ) @ R2 )
% 0.34/0.56 = ( insert_set_int @ ( image_int_int @ R2 @ ( insert_int @ X @ bot_bot_set_int ) ) @ bot_bot_set_set_int ) ) ).
% 0.34/0.56
% 0.34/0.56 % singleton_quotient
% 0.34/0.56 thf(fact_2099_singleton__quotient,axiom,
% 0.34/0.56 ! [X: produc3658429121746597890et_nat > $o,R2: set_Pr2161125870931222855_nat_o] :
% 0.34/0.56 ( ( equiv_7899955228204526484_nat_o @ ( insert5175938949040314269_nat_o @ X @ bot_bo7824918357723582233_nat_o ) @ R2 )
% 0.34/0.56 = ( insert2507855216428400253_nat_o @ ( image_2232984160439939859_nat_o @ R2 @ ( insert5175938949040314269_nat_o @ X @ bot_bo7824918357723582233_nat_o ) ) @ bot_bo3736544561982865913_nat_o ) ) ).
% 0.34/0.56
% 0.34/0.56 % singleton_quotient
% 0.34/0.56 thf(fact_2100_quotientI,axiom,
% 0.34/0.56 ! [X: product_unit,A: set_Product_unit,R2: set_Pr5094982260447487303t_unit] :
% 0.34/0.56 ( ( member_Product_unit @ X @ A )
% 0.34/0.56 => ( member5877623283571906838t_unit @ ( image_8431784415008483091t_unit @ R2 @ ( insert_Product_unit @ X @ bot_bo3957492148770167129t_unit ) ) @ ( equiv_5172566880711946708t_unit @ A @ R2 ) ) ) ).
% 0.34/0.56
% 0.34/0.56 % quotientI
% 0.34/0.56 thf(fact_2101_quotientI,axiom,
% 0.34/0.56 ! [X: $o,A: set_o,R2: set_Product_prod_o_o] :
% 0.34/0.56 ( ( member_o @ X @ A )
% 0.34/0.56 => ( member_set_o @ ( image_o_o @ R2 @ ( insert_o @ X @ bot_bot_set_o ) ) @ ( equiv_quotient_o @ A @ R2 ) ) ) ).
% 0.34/0.56
% 0.34/0.56 % quotientI
% 0.34/0.56 thf(fact_2102_quotientI,axiom,
% 0.34/0.56 ! [X: nat,A: set_nat,R2: set_Pr1261947904930325089at_nat] :
% 0.34/0.56 ( ( member_nat @ X @ A )
% 0.34/0.56 => ( member_set_nat @ ( image_nat_nat @ R2 @ ( insert_nat @ X @ bot_bot_set_nat ) ) @ ( equiv_quotient_nat @ A @ R2 ) ) ) ).
% 0.34/0.56
% 0.34/0.56 % quotientI
% 0.34/0.56 thf(fact_2103_quotientI,axiom,
% 0.34/0.56 ! [X: int,A: set_int,R2: set_Pr958786334691620121nt_int] :
% 0.34/0.56 ( ( member_int @ X @ A )
% 0.34/0.56 => ( member_set_int @ ( image_int_int @ R2 @ ( insert_int @ X @ bot_bot_set_int ) ) @ ( equiv_quotient_int @ A @ R2 ) ) ) ).
% 0.34/0.56
% 0.34/0.56 % quotientI
% 0.34/0.56 thf(fact_2104_quotientI,axiom,
% 0.34/0.56 ! [X: produc3658429121746597890et_nat > $o,A: set_Pr4532377907799695533_nat_o,R2: set_Pr2161125870931222855_nat_o] :
% 0.34/0.56 ( ( member6576561426505652726_nat_o @ X @ A )
% 0.34/0.56 => ( member1452989533377600726_nat_o @ ( image_2232984160439939859_nat_o @ R2 @ ( insert5175938949040314269_nat_o @ X @ bot_bo7824918357723582233_nat_o ) ) @ ( equiv_7899955228204526484_nat_o @ A @ R2 ) ) ) ).
% 0.34/0.56
% 0.34/0.56 % quotientI
% 0.34/0.56 thf(fact_2105_quotientE,axiom,
% 0.34/0.56 ! [X4: set_Product_unit,A: set_Product_unit,R2: set_Pr5094982260447487303t_unit] :
% 0.34/0.56 ( ( member5877623283571906838t_unit @ X4 @ ( equiv_5172566880711946708t_unit @ A @ R2 ) )
% 0.34/0.56 => ~ ! [X3: product_unit] :
% 0.34/0.56 ( ( X4
% 0.34/0.56 = ( image_8431784415008483091t_unit @ R2 @ ( insert_Product_unit @ X3 @ bot_bo3957492148770167129t_unit ) ) )
% 0.34/0.56 => ~ ( member_Product_unit @ X3 @ A ) ) ) ).
% 0.34/0.56
% 0.34/0.56 % quotientE
% 0.34/0.56 thf(fact_2106_quotientE,axiom,
% 0.34/0.56 ! [X4: set_o,A: set_o,R2: set_Product_prod_o_o] :
% 0.34/0.56 ( ( member_set_o @ X4 @ ( equiv_quotient_o @ A @ R2 ) )
% 0.34/0.56 => ~ ! [X3: $o] :
% 0.34/0.56 ( ( X4
% 0.34/0.56 = ( image_o_o @ R2 @ ( insert_o @ X3 @ bot_bot_set_o ) ) )
% 0.34/0.56 => ~ ( member_o @ X3 @ A ) ) ) ).
% 0.34/0.56
% 0.34/0.56 % quotientE
% 0.34/0.56 thf(fact_2107_quotientE,axiom,
% 0.34/0.56 ! [X4: set_nat,A: set_nat,R2: set_Pr1261947904930325089at_nat] :
% 0.34/0.56 ( ( member_set_nat @ X4 @ ( equiv_quotient_nat @ A @ R2 ) )
% 0.34/0.56 => ~ ! [X3: nat] :
% 0.34/0.56 ( ( X4
% 0.34/0.56 = ( image_nat_nat @ R2 @ ( insert_nat @ X3 @ bot_bot_set_nat ) ) )
% 0.34/0.56 => ~ ( member_nat @ X3 @ A ) ) ) ).
% 0.34/0.56
% 0.34/0.56 % quotientE
% 0.34/0.56 thf(fact_2108_quotientE,axiom,
% 0.34/0.56 ! [X4: set_int,A: set_int,R2: set_Pr958786334691620121nt_int] :
% 0.34/0.56 ( ( member_set_int @ X4 @ ( equiv_quotient_int @ A @ R2 ) )
% 0.34/0.56 => ~ ! [X3: int] :
% 0.34/0.56 ( ( X4
% 0.34/0.56 = ( image_int_int @ R2 @ ( insert_int @ X3 @ bot_bot_set_int ) ) )
% 0.34/0.56 => ~ ( member_int @ X3 @ A ) ) ) ).
% 0.34/0.56
% 0.34/0.56 % quotientE
% 0.34/0.56 thf(fact_2109_quotientE,axiom,
% 0.34/0.56 ! [X4: set_Pr4532377907799695533_nat_o,A: set_Pr4532377907799695533_nat_o,R2: set_Pr2161125870931222855_nat_o] :
% 0.34/0.56 ( ( member1452989533377600726_nat_o @ X4 @ ( equiv_7899955228204526484_nat_o @ A @ R2 ) )
% 0.34/0.56 => ~ ! [X3: produc3658429121746597890et_nat > $o] :
% 0.34/0.56 ( ( X4
% 0.34/0.56 = ( image_2232984160439939859_nat_o @ R2 @ ( insert5175938949040314269_nat_o @ X3 @ bot_bo7824918357723582233_nat_o ) ) )
% 0.34/0.56 => ~ ( member6576561426505652726_nat_o @ X3 @ A ) ) ) ).
% 0.34/0.56
% 0.34/0.56 % quotientE
% 0.34/0.56 thf(fact_2110_Inf__fin_OboundedE,axiom,
% 0.34/0.56 ! [A: set_Product_unit,X: product_unit] :
% 0.34/0.56 ( ( finite4290736615968046902t_unit @ A )
% 0.34/0.56 => ( ( A != bot_bo3957492148770167129t_unit )
% 0.34/0.56 => ( ( ord_le3221252021190050221t_unit @ X @ ( lattic1263872656861969706t_unit @ A ) )
% 0.34/0.56 => ! [A10: product_unit] :
% 0.34/0.56 ( ( member_Product_unit @ A10 @ A )
% 0.34/0.56 => ( ord_le3221252021190050221t_unit @ X @ A10 ) ) ) ) ) ).
% 0.34/0.56
% 0.34/0.56 % Inf_fin.boundedE
% 0.34/0.56 thf(fact_2111_Inf__fin_OboundedE,axiom,
% 0.34/0.56 ! [A: set_o,X: $o] :
% 0.34/0.56 ( ( finite_finite_o @ A )
% 0.34/0.56 => ( ( A != bot_bot_set_o )
% 0.34/0.56 => ( ( ord_less_eq_o @ X @ ( lattic4107685809792843317_fin_o @ A ) )
% 0.34/0.56 => ! [A10: $o] :
% 0.34/0.56 ( ( member_o @ A10 @ A )
% 0.34/0.56 => ( ord_less_eq_o @ X @ A10 ) ) ) ) ) ).
% 0.34/0.56
% 0.34/0.56 % Inf_fin.boundedE
% 0.34/0.56 thf(fact_2112_Inf__fin_OboundedE,axiom,
% 0.34/0.56 ! [A: set_Pr4532377907799695533_nat_o,X: produc3658429121746597890et_nat > $o] :
% 0.34/0.56 ( ( finite3252695134891459830_nat_o @ A )
% 0.34/0.56 => ( ( A != bot_bo7824918357723582233_nat_o )
% 0.34/0.56 => ( ( ord_le729326519192465773_nat_o @ X @ ( lattic956194824204696298_nat_o @ A ) )
% 0.34/0.56 => ! [A10: produc3658429121746597890et_nat > $o] :
% 0.34/0.56 ( ( member6576561426505652726_nat_o @ A10 @ A )
% 0.34/0.56 => ( ord_le729326519192465773_nat_o @ X @ A10 ) ) ) ) ) ).
% 0.34/0.56
% 0.34/0.56 % Inf_fin.boundedE
% 0.34/0.56 thf(fact_2113_Inf__fin_OboundedE,axiom,
% 0.34/0.56 ! [A: set_filter_nat,X: filter_nat] :
% 0.34/0.56 ( ( finite2119507909894593271er_nat @ A )
% 0.34/0.56 => ( ( A != bot_bo498966703094740906er_nat )
% 0.34/0.56 => ( ( ord_le2510731241096832064er_nat @ X @ ( lattic9128708996678779395er_nat @ A ) )
% 0.34/0.56 => ! [A10: filter_nat] :
% 0.34/0.56 ( ( member_filter_nat @ A10 @ A )
% 0.34/0.56 => ( ord_le2510731241096832064er_nat @ X @ A10 ) ) ) ) ) ).
% 0.34/0.56
% 0.34/0.56 % Inf_fin.boundedE
% 0.34/0.56 thf(fact_2114_Inf__fin_OboundedE,axiom,
% 0.34/0.56 ! [A: set_set_nat,X: set_nat] :
% 0.34/0.56 ( ( finite1152437895449049373et_nat @ A )
% 0.34/0.56 => ( ( A != bot_bot_set_set_nat )
% 0.34/0.56 => ( ( ord_less_eq_set_nat @ X @ ( lattic3014633134055518761et_nat @ A ) )
% 0.34/0.56 => ! [A10: set_nat] :
% 0.34/0.56 ( ( member_set_nat @ A10 @ A )
% 0.34/0.56 => ( ord_less_eq_set_nat @ X @ A10 ) ) ) ) ) ).
% 0.34/0.56
% 0.34/0.56 % Inf_fin.boundedE
% 0.34/0.56 thf(fact_2115_Inf__fin_OboundedE,axiom,
% 0.34/0.56 ! [A: set_rat,X: rat] :
% 0.34/0.56 ( ( finite_finite_rat @ A )
% 0.34/0.56 => ( ( A != bot_bot_set_rat )
% 0.34/0.56 => ( ( ord_less_eq_rat @ X @ ( lattic4603258475043424379in_rat @ A ) )
% 0.34/0.56 => ! [A10: rat] :
% 0.34/0.56 ( ( member_rat @ A10 @ A )
% 0.34/0.56 => ( ord_less_eq_rat @ X @ A10 ) ) ) ) ) ).
% 0.34/0.56
% 0.34/0.56 % Inf_fin.boundedE
% 0.34/0.56 thf(fact_2116_Inf__fin_OboundedE,axiom,
% 0.34/0.56 ! [A: set_nat,X: nat] :
% 0.34/0.56 ( ( finite_finite_nat @ A )
% 0.34/0.56 => ( ( A != bot_bot_set_nat )
% 0.34/0.56 => ( ( ord_less_eq_nat @ X @ ( lattic5238388535129920115in_nat @ A ) )
% 0.34/0.56 => ! [A10: nat] :
% 0.34/0.56 ( ( member_nat @ A10 @ A )
% 0.34/0.56 => ( ord_less_eq_nat @ X @ A10 ) ) ) ) ) ).
% 0.34/0.56
% 0.34/0.56 % Inf_fin.boundedE
% 0.34/0.56 thf(fact_2117_Inf__fin_OboundedE,axiom,
% 0.34/0.56 ! [A: set_int,X: int] :
% 0.34/0.56 ( ( finite_finite_int @ A )
% 0.34/0.56 => ( ( A != bot_bot_set_int )
% 0.34/0.56 => ( ( ord_less_eq_int @ X @ ( lattic5235898064620869839in_int @ A ) )
% 0.34/0.56 => ! [A10: int] :
% 0.34/0.56 ( ( member_int @ A10 @ A )
% 0.34/0.56 => ( ord_less_eq_int @ X @ A10 ) ) ) ) ) ).
% 0.34/0.56
% 0.34/0.56 % Inf_fin.boundedE
% 0.34/0.56 thf(fact_2118_Inf__fin_OboundedI,axiom,
% 0.34/0.56 ! [A: set_Product_unit,X: product_unit] :
% 0.34/0.56 ( ( finite4290736615968046902t_unit @ A )
% 0.34/0.56 => ( ( A != bot_bo3957492148770167129t_unit )
% 0.34/0.56 => ( ! [A4: product_unit] :
% 0.34/0.56 ( ( member_Product_unit @ A4 @ A )
% 0.34/0.56 => ( ord_le3221252021190050221t_unit @ X @ A4 ) )
% 0.34/0.56 => ( ord_le3221252021190050221t_unit @ X @ ( lattic1263872656861969706t_unit @ A ) ) ) ) ) ).
% 0.34/0.56
% 0.34/0.56 % Inf_fin.boundedI
% 0.34/0.56 thf(fact_2119_Inf__fin_OboundedI,axiom,
% 0.34/0.56 ! [A: set_o,X: $o] :
% 0.34/0.56 ( ( finite_finite_o @ A )
% 0.34/0.56 => ( ( A != bot_bot_set_o )
% 0.34/0.56 => ( ! [A4: $o] :
% 0.34/0.56 ( ( member_o @ A4 @ A )
% 0.34/0.56 => ( ord_less_eq_o @ X @ A4 ) )
% 0.34/0.56 => ( ord_less_eq_o @ X @ ( lattic4107685809792843317_fin_o @ A ) ) ) ) ) ).
% 0.34/0.56
% 0.34/0.56 % Inf_fin.boundedI
% 0.34/0.56 thf(fact_2120_Inf__fin_OboundedI,axiom,
% 0.34/0.56 ! [A: set_Pr4532377907799695533_nat_o,X: produc3658429121746597890et_nat > $o] :
% 0.34/0.56 ( ( finite3252695134891459830_nat_o @ A )
% 0.34/0.56 => ( ( A != bot_bo7824918357723582233_nat_o )
% 0.34/0.56 => ( ! [A4: produc3658429121746597890et_nat > $o] :
% 0.34/0.56 ( ( member6576561426505652726_nat_o @ A4 @ A )
% 0.34/0.56 => ( ord_le729326519192465773_nat_o @ X @ A4 ) )
% 0.34/0.56 => ( ord_le729326519192465773_nat_o @ X @ ( lattic956194824204696298_nat_o @ A ) ) ) ) ) ).
% 0.34/0.56
% 0.34/0.56 % Inf_fin.boundedI
% 0.34/0.56 thf(fact_2121_Inf__fin_OboundedI,axiom,
% 0.34/0.56 ! [A: set_filter_nat,X: filter_nat] :
% 0.34/0.56 ( ( finite2119507909894593271er_nat @ A )
% 0.34/0.56 => ( ( A != bot_bo498966703094740906er_nat )
% 0.34/0.56 => ( ! [A4: filter_nat] :
% 0.34/0.56 ( ( member_filter_nat @ A4 @ A )
% 0.34/0.56 => ( ord_le2510731241096832064er_nat @ X @ A4 ) )
% 0.34/0.56 => ( ord_le2510731241096832064er_nat @ X @ ( lattic9128708996678779395er_nat @ A ) ) ) ) ) ).
% 0.34/0.56
% 0.34/0.56 % Inf_fin.boundedI
% 0.34/0.56 thf(fact_2122_Inf__fin_OboundedI,axiom,
% 0.34/0.56 ! [A: set_set_nat,X: set_nat] :
% 0.34/0.56 ( ( finite1152437895449049373et_nat @ A )
% 0.34/0.56 => ( ( A != bot_bot_set_set_nat )
% 0.34/0.56 => ( ! [A4: set_nat] :
% 0.34/0.56 ( ( member_set_nat @ A4 @ A )
% 0.34/0.56 => ( ord_less_eq_set_nat @ X @ A4 ) )
% 0.34/0.56 => ( ord_less_eq_set_nat @ X @ ( lattic3014633134055518761et_nat @ A ) ) ) ) ) ).
% 0.34/0.56
% 0.34/0.56 % Inf_fin.boundedI
% 0.34/0.56 thf(fact_2123_Inf__fin_OboundedI,axiom,
% 0.34/0.56 ! [A: set_rat,X: rat] :
% 0.34/0.56 ( ( finite_finite_rat @ A )
% 0.34/0.56 => ( ( A != bot_bot_set_rat )
% 0.34/0.56 => ( ! [A4: rat] :
% 0.34/0.56 ( ( member_rat @ A4 @ A )
% 0.34/0.56 => ( ord_less_eq_rat @ X @ A4 ) )
% 0.34/0.56 => ( ord_less_eq_rat @ X @ ( lattic4603258475043424379in_rat @ A ) ) ) ) ) ).
% 0.34/0.56
% 0.34/0.56 % Inf_fin.boundedI
% 0.34/0.56 thf(fact_2124_Inf__fin_OboundedI,axiom,
% 0.34/0.56 ! [A: set_nat,X: nat] :
% 0.34/0.56 ( ( finite_finite_nat @ A )
% 0.34/0.56 => ( ( A != bot_bot_set_nat )
% 0.34/0.56 => ( ! [A4: nat] :
% 0.34/0.56 ( ( member_nat @ A4 @ A )
% 0.34/0.56 => ( ord_less_eq_nat @ X @ A4 ) )
% 0.34/0.56 => ( ord_less_eq_nat @ X @ ( lattic5238388535129920115in_nat @ A ) ) ) ) ) ).
% 0.34/0.56
% 0.34/0.56 % Inf_fin.boundedI
% 0.34/0.56 thf(fact_2125_Inf__fin_OboundedI,axiom,
% 0.34/0.56 ! [A: set_int,X: int] :
% 0.34/0.56 ( ( finite_finite_int @ A )
% 0.34/0.56 => ( ( A != bot_bot_set_int )
% 0.34/0.56 => ( ! [A4: int] :
% 0.34/0.56 ( ( member_int @ A4 @ A )
% 0.34/0.56 => ( ord_less_eq_int @ X @ A4 ) )
% 0.34/0.56 => ( ord_less_eq_int @ X @ ( lattic5235898064620869839in_int @ A ) ) ) ) ) ).
% 0.34/0.56
% 0.34/0.56 % Inf_fin.boundedI
% 0.34/0.56 thf(fact_2126_Sup__fin_OboundedE,axiom,
% 0.34/0.56 ! [A: set_Product_unit,X: product_unit] :
% 0.34/0.56 ( ( finite4290736615968046902t_unit @ A )
% 0.34/0.56 => ( ( A != bot_bo3957492148770167129t_unit )
% 0.34/0.56 => ( ( ord_le3221252021190050221t_unit @ ( lattic5294303975357428420t_unit @ A ) @ X )
% 0.34/0.56 => ! [A10: product_unit] :
% 0.34/0.56 ( ( member_Product_unit @ A10 @ A )
% 0.34/0.56 => ( ord_le3221252021190050221t_unit @ A10 @ X ) ) ) ) ) ).
% 0.34/0.56
% 0.34/0.56 % Sup_fin.boundedE
% 0.34/0.56 thf(fact_2127_Sup__fin_OboundedE,axiom,
% 0.34/0.56 ! [A: set_o,X: $o] :
% 0.34/0.56 ( ( finite_finite_o @ A )
% 0.34/0.56 => ( ( A != bot_bot_set_o )
% 0.34/0.56 => ( ( ord_less_eq_o @ ( lattic1508158080041050831_fin_o @ A ) @ X )
% 0.34/0.56 => ! [A10: $o] :
% 0.34/0.56 ( ( member_o @ A10 @ A )
% 0.34/0.56 => ( ord_less_eq_o @ A10 @ X ) ) ) ) ) ).
% 0.34/0.56
% 0.34/0.56 % Sup_fin.boundedE
% 0.34/0.56 thf(fact_2128_Sup__fin_OboundedE,axiom,
% 0.34/0.56 ! [A: set_Pr4532377907799695533_nat_o,X: produc3658429121746597890et_nat > $o] :
% 0.34/0.56 ( ( finite3252695134891459830_nat_o @ A )
% 0.34/0.56 => ( ( A != bot_bo7824918357723582233_nat_o )
% 0.34/0.56 => ( ( ord_le729326519192465773_nat_o @ ( lattic7320199455484906628_nat_o @ A ) @ X )
% 0.34/0.56 => ! [A10: produc3658429121746597890et_nat > $o] :
% 0.34/0.56 ( ( member6576561426505652726_nat_o @ A10 @ A )
% 0.34/0.56 => ( ord_le729326519192465773_nat_o @ A10 @ X ) ) ) ) ) ).
% 0.34/0.56
% 0.34/0.56 % Sup_fin.boundedE
% 0.34/0.56 thf(fact_2129_Sup__fin_OboundedE,axiom,
% 0.34/0.56 ! [A: set_filter_nat,X: filter_nat] :
% 0.34/0.56 ( ( finite2119507909894593271er_nat @ A )
% 0.34/0.56 => ( ( A != bot_bo498966703094740906er_nat )
% 0.34/0.56 => ( ( ord_le2510731241096832064er_nat @ ( lattic5930898082463196905er_nat @ A ) @ X )
% 0.34/0.56 => ! [A10: filter_nat] :
% 0.34/0.56 ( ( member_filter_nat @ A10 @ A )
% 0.34/0.56 => ( ord_le2510731241096832064er_nat @ A10 @ X ) ) ) ) ) ).
% 0.34/0.56
% 0.34/0.56 % Sup_fin.boundedE
% 0.34/0.56 thf(fact_2130_Sup__fin_OboundedE,axiom,
% 0.34/0.56 ! [A: set_set_nat,X: set_nat] :
% 0.34/0.56 ( ( finite1152437895449049373et_nat @ A )
% 0.34/0.56 => ( ( A != bot_bot_set_set_nat )
% 0.34/0.56 => ( ( ord_less_eq_set_nat @ ( lattic3835124923745554447et_nat @ A ) @ X )
% 0.34/0.56 => ! [A10: set_nat] :
% 0.34/0.56 ( ( member_set_nat @ A10 @ A )
% 0.34/0.56 => ( ord_less_eq_set_nat @ A10 @ X ) ) ) ) ) ).
% 0.34/0.56
% 0.34/0.56 % Sup_fin.boundedE
% 0.34/0.56 thf(fact_2131_Sup__fin_OboundedE,axiom,
% 0.34/0.56 ! [A: set_rat,X: rat] :
% 0.34/0.56 ( ( finite_finite_rat @ A )
% 0.34/0.56 => ( ( A != bot_bot_set_rat )
% 0.34/0.56 => ( ( ord_less_eq_rat @ ( lattic458866745392299617in_rat @ A ) @ X )
% 0.34/0.56 => ! [A10: rat] :
% 0.34/0.56 ( ( member_rat @ A10 @ A )
% 0.34/0.56 => ( ord_less_eq_rat @ A10 @ X ) ) ) ) ) ).
% 0.34/0.56
% 0.34/0.56 % Sup_fin.boundedE
% 0.34/0.56 thf(fact_2132_Sup__fin_OboundedE,axiom,
% 0.34/0.56 ! [A: set_nat,X: nat] :
% 0.34/0.56 ( ( finite_finite_nat @ A )
% 0.34/0.56 => ( ( A != bot_bot_set_nat )
% 0.34/0.56 => ( ( ord_less_eq_nat @ ( lattic1093996805478795353in_nat @ A ) @ X )
% 0.34/0.56 => ! [A10: nat] :
% 0.34/0.56 ( ( member_nat @ A10 @ A )
% 0.34/0.56 => ( ord_less_eq_nat @ A10 @ X ) ) ) ) ) ).
% 0.34/0.56
% 0.34/0.56 % Sup_fin.boundedE
% 0.34/0.56 thf(fact_2133_Sup__fin_OboundedE,axiom,
% 0.34/0.56 ! [A: set_int,X: int] :
% 0.34/0.56 ( ( finite_finite_int @ A )
% 0.34/0.56 => ( ( A != bot_bot_set_int )
% 0.34/0.56 => ( ( ord_less_eq_int @ ( lattic1091506334969745077in_int @ A ) @ X )
% 0.34/0.56 => ! [A10: int] :
% 0.34/0.56 ( ( member_int @ A10 @ A )
% 0.34/0.56 => ( ord_less_eq_int @ A10 @ X ) ) ) ) ) ).
% 0.34/0.56
% 0.34/0.56 % Sup_fin.boundedE
% 0.34/0.56 thf(fact_2134_Sup__fin_OboundedI,axiom,
% 0.34/0.56 ! [A: set_Product_unit,X: product_unit] :
% 0.34/0.56 ( ( finite4290736615968046902t_unit @ A )
% 0.34/0.56 => ( ( A != bot_bo3957492148770167129t_unit )
% 0.34/0.56 => ( ! [A4: product_unit] :
% 0.34/0.56 ( ( member_Product_unit @ A4 @ A )
% 0.34/0.56 => ( ord_le3221252021190050221t_unit @ A4 @ X ) )
% 0.34/0.56 => ( ord_le3221252021190050221t_unit @ ( lattic5294303975357428420t_unit @ A ) @ X ) ) ) ) ).
% 0.34/0.56
% 0.34/0.56 % Sup_fin.boundedI
% 0.34/0.56 thf(fact_2135_Sup__fin_OboundedI,axiom,
% 0.34/0.56 ! [A: set_o,X: $o] :
% 0.34/0.56 ( ( finite_finite_o @ A )
% 0.34/0.56 => ( ( A != bot_bot_set_o )
% 0.34/0.56 => ( ! [A4: $o] :
% 0.34/0.56 ( ( member_o @ A4 @ A )
% 0.34/0.56 => ( ord_less_eq_o @ A4 @ X ) )
% 0.34/0.56 => ( ord_less_eq_o @ ( lattic1508158080041050831_fin_o @ A ) @ X ) ) ) ) ).
% 0.34/0.56
% 0.34/0.56 % Sup_fin.boundedI
% 0.34/0.56 thf(fact_2136_Sup__fin_OboundedI,axiom,
% 0.34/0.56 ! [A: set_Pr4532377907799695533_nat_o,X: produc3658429121746597890et_nat > $o] :
% 0.34/0.56 ( ( finite3252695134891459830_nat_o @ A )
% 0.34/0.56 => ( ( A != bot_bo7824918357723582233_nat_o )
% 0.34/0.56 => ( ! [A4: produc3658429121746597890et_nat > $o] :
% 0.34/0.56 ( ( member6576561426505652726_nat_o @ A4 @ A )
% 0.34/0.56 => ( ord_le729326519192465773_nat_o @ A4 @ X ) )
% 0.34/0.56 => ( ord_le729326519192465773_nat_o @ ( lattic7320199455484906628_nat_o @ A ) @ X ) ) ) ) ).
% 0.34/0.56
% 0.34/0.56 % Sup_fin.boundedI
% 0.34/0.56 thf(fact_2137_Sup__fin_OboundedI,axiom,
% 0.34/0.56 ! [A: set_filter_nat,X: filter_nat] :
% 0.34/0.56 ( ( finite2119507909894593271er_nat @ A )
% 0.34/0.56 => ( ( A != bot_bo498966703094740906er_nat )
% 0.34/0.56 => ( ! [A4: filter_nat] :
% 0.34/0.56 ( ( member_filter_nat @ A4 @ A )
% 0.34/0.56 => ( ord_le2510731241096832064er_nat @ A4 @ X ) )
% 0.34/0.56 => ( ord_le2510731241096832064er_nat @ ( lattic5930898082463196905er_nat @ A ) @ X ) ) ) ) ).
% 0.34/0.56
% 0.34/0.56 % Sup_fin.boundedI
% 0.34/0.56 thf(fact_2138_Sup__fin_OboundedI,axiom,
% 0.34/0.56 ! [A: set_set_nat,X: set_nat] :
% 0.34/0.56 ( ( finite1152437895449049373et_nat @ A )
% 0.34/0.56 => ( ( A != bot_bot_set_set_nat )
% 0.34/0.56 => ( ! [A4: set_nat] :
% 0.34/0.56 ( ( member_set_nat @ A4 @ A )
% 0.34/0.56 => ( ord_less_eq_set_nat @ A4 @ X ) )
% 0.34/0.56 => ( ord_less_eq_set_nat @ ( lattic3835124923745554447et_nat @ A ) @ X ) ) ) ) ).
% 0.34/0.56
% 0.34/0.56 % Sup_fin.boundedI
% 0.34/0.56 thf(fact_2139_Sup__fin_OboundedI,axiom,
% 0.34/0.56 ! [A: set_rat,X: rat] :
% 0.34/0.56 ( ( finite_finite_rat @ A )
% 0.34/0.56 => ( ( A != bot_bot_set_rat )
% 0.34/0.56 => ( ! [A4: rat] :
% 0.34/0.56 ( ( member_rat @ A4 @ A )
% 0.34/0.56 => ( ord_less_eq_rat @ A4 @ X ) )
% 0.34/0.56 => ( ord_less_eq_rat @ ( lattic458866745392299617in_rat @ A ) @ X ) ) ) ) ).
% 0.34/0.56
% 0.34/0.56 % Sup_fin.boundedI
% 0.34/0.56 thf(fact_2140_Sup__fin_OboundedI,axiom,
% 0.34/0.56 ! [A: set_nat,X: nat] :
% 0.34/0.56 ( ( finite_finite_nat @ A )
% 0.34/0.56 => ( ( A != bot_bot_set_nat )
% 0.34/0.56 => ( ! [A4: nat] :
% 0.34/0.56 ( ( member_nat @ A4 @ A )
% 0.34/0.56 => ( ord_less_eq_nat @ A4 @ X ) )
% 0.34/0.56 => ( ord_less_eq_nat @ ( lattic1093996805478795353in_nat @ A ) @ X ) ) ) ) ).
% 0.34/0.56
% 0.34/0.56 % Sup_fin.boundedI
% 0.34/0.56 thf(fact_2141_Sup__fin_OboundedI,axiom,
% 0.34/0.56 ! [A: set_int,X: int] :
% 0.34/0.56 ( ( finite_finite_int @ A )
% 0.34/0.56 => ( ( A != bot_bot_set_int )
% 0.34/0.56 => ( ! [A4: int] :
% 0.34/0.56 ( ( member_int @ A4 @ A )
% 0.34/0.56 => ( ord_less_eq_int @ A4 @ X ) )
% 0.34/0.56 => ( ord_less_eq_int @ ( lattic1091506334969745077in_int @ A ) @ X ) ) ) ) ).
% 0.34/0.56
% 0.34/0.56 % Sup_fin.boundedI
% 0.34/0.56 thf(fact_2142_Inf__fin_Obounded__iff,axiom,
% 0.34/0.56 ! [A: set_Product_unit,X: product_unit] :
% 0.34/0.56 ( ( finite4290736615968046902t_unit @ A )
% 0.34/0.56 => ( ( A != bot_bo3957492148770167129t_unit )
% 0.34/0.56 => ( ( ord_le3221252021190050221t_unit @ X @ ( lattic1263872656861969706t_unit @ A ) )
% 0.34/0.56 = ( ! [X2: product_unit] :
% 0.34/0.56 ( ( member_Product_unit @ X2 @ A )
% 0.34/0.56 => ( ord_le3221252021190050221t_unit @ X @ X2 ) ) ) ) ) ) ).
% 0.34/0.56
% 0.34/0.56 % Inf_fin.bounded_iff
% 0.34/0.56 thf(fact_2143_Inf__fin_Obounded__iff,axiom,
% 0.34/0.56 ! [A: set_o,X: $o] :
% 0.34/0.56 ( ( finite_finite_o @ A )
% 0.34/0.56 => ( ( A != bot_bot_set_o )
% 0.34/0.56 => ( ( ord_less_eq_o @ X @ ( lattic4107685809792843317_fin_o @ A ) )
% 0.34/0.56 = ( ! [X2: $o] :
% 0.34/0.56 ( ( member_o @ X2 @ A )
% 0.34/0.56 => ( ord_less_eq_o @ X @ X2 ) ) ) ) ) ) ).
% 0.34/0.56
% 0.34/0.56 % Inf_fin.bounded_iff
% 0.34/0.56 thf(fact_2144_Inf__fin_Obounded__iff,axiom,
% 0.34/0.56 ! [A: set_Pr4532377907799695533_nat_o,X: produc3658429121746597890et_nat > $o] :
% 0.34/0.56 ( ( finite3252695134891459830_nat_o @ A )
% 0.34/0.56 => ( ( A != bot_bo7824918357723582233_nat_o )
% 0.34/0.56 => ( ( ord_le729326519192465773_nat_o @ X @ ( lattic956194824204696298_nat_o @ A ) )
% 0.34/0.56 = ( ! [X2: produc3658429121746597890et_nat > $o] :
% 0.34/0.56 ( ( member6576561426505652726_nat_o @ X2 @ A )
% 0.34/0.56 => ( ord_le729326519192465773_nat_o @ X @ X2 ) ) ) ) ) ) ).
% 0.34/0.56
% 0.34/0.56 % Inf_fin.bounded_iff
% 0.34/0.56 thf(fact_2145_Inf__fin_Obounded__iff,axiom,
% 0.34/0.56 ! [A: set_filter_nat,X: filter_nat] :
% 0.34/0.56 ( ( finite2119507909894593271er_nat @ A )
% 0.34/0.56 => ( ( A != bot_bo498966703094740906er_nat )
% 0.34/0.56 => ( ( ord_le2510731241096832064er_nat @ X @ ( lattic9128708996678779395er_nat @ A ) )
% 0.34/0.56 = ( ! [X2: filter_nat] :
% 0.34/0.56 ( ( member_filter_nat @ X2 @ A )
% 0.34/0.56 => ( ord_le2510731241096832064er_nat @ X @ X2 ) ) ) ) ) ) ).
% 0.34/0.56
% 0.34/0.56 % Inf_fin.bounded_iff
% 0.34/0.56 thf(fact_2146_Inf__fin_Obounded__iff,axiom,
% 0.34/0.56 ! [A: set_set_nat,X: set_nat] :
% 0.34/0.56 ( ( finite1152437895449049373et_nat @ A )
% 0.34/0.56 => ( ( A != bot_bot_set_set_nat )
% 0.34/0.56 => ( ( ord_less_eq_set_nat @ X @ ( lattic3014633134055518761et_nat @ A ) )
% 0.34/0.56 = ( ! [X2: set_nat] :
% 0.34/0.56 ( ( member_set_nat @ X2 @ A )
% 0.34/0.56 => ( ord_less_eq_set_nat @ X @ X2 ) ) ) ) ) ) ).
% 0.34/0.56
% 0.34/0.56 % Inf_fin.bounded_iff
% 0.34/0.56 thf(fact_2147_Inf__fin_Obounded__iff,axiom,
% 0.34/0.56 ! [A: set_rat,X: rat] :
% 0.34/0.56 ( ( finite_finite_rat @ A )
% 0.34/0.56 => ( ( A != bot_bot_set_rat )
% 0.34/0.56 => ( ( ord_less_eq_rat @ X @ ( lattic4603258475043424379in_rat @ A ) )
% 0.34/0.56 = ( ! [X2: rat] :
% 0.34/0.56 ( ( member_rat @ X2 @ A )
% 0.34/0.56 => ( ord_less_eq_rat @ X @ X2 ) ) ) ) ) ) ).
% 0.34/0.56
% 0.34/0.56 % Inf_fin.bounded_iff
% 0.34/0.56 thf(fact_2148_Inf__fin_Obounded__iff,axiom,
% 0.34/0.56 ! [A: set_nat,X: nat] :
% 0.34/0.56 ( ( finite_finite_nat @ A )
% 0.34/0.56 => ( ( A != bot_bot_set_nat )
% 0.34/0.56 => ( ( ord_less_eq_nat @ X @ ( lattic5238388535129920115in_nat @ A ) )
% 0.34/0.56 = ( ! [X2: nat] :
% 0.34/0.56 ( ( member_nat @ X2 @ A )
% 0.34/0.56 => ( ord_less_eq_nat @ X @ X2 ) ) ) ) ) ) ).
% 0.34/0.56
% 0.34/0.56 % Inf_fin.bounded_iff
% 0.34/0.56 thf(fact_2149_Inf__fin_Obounded__iff,axiom,
% 0.34/0.56 ! [A: set_int,X: int] :
% 0.34/0.56 ( ( finite_finite_int @ A )
% 0.34/0.56 => ( ( A != bot_bot_set_int )
% 0.34/0.56 => ( ( ord_less_eq_int @ X @ ( lattic5235898064620869839in_int @ A ) )
% 0.34/0.56 = ( ! [X2: int] :
% 0.34/0.56 ( ( member_int @ X2 @ A )
% 0.34/0.56 => ( ord_less_eq_int @ X @ X2 ) ) ) ) ) ) ).
% 0.34/0.56
% 0.34/0.56 % Inf_fin.bounded_iff
% 0.34/0.56 thf(fact_2150_Sup__fin_Obounded__iff,axiom,
% 0.34/0.56 ! [A: set_Product_unit,X: product_unit] :
% 0.34/0.56 ( ( finite4290736615968046902t_unit @ A )
% 0.34/0.56 => ( ( A != bot_bo3957492148770167129t_unit )
% 0.34/0.56 => ( ( ord_le3221252021190050221t_unit @ ( lattic5294303975357428420t_unit @ A ) @ X )
% 0.34/0.56 = ( ! [X2: product_unit] :
% 0.34/0.56 ( ( member_Product_unit @ X2 @ A )
% 0.34/0.56 => ( ord_le3221252021190050221t_unit @ X2 @ X ) ) ) ) ) ) ).
% 0.34/0.56
% 0.34/0.56 % Sup_fin.bounded_iff
% 0.34/0.56 thf(fact_2151_Sup__fin_Obounded__iff,axiom,
% 0.34/0.56 ! [A: set_o,X: $o] :
% 0.34/0.56 ( ( finite_finite_o @ A )
% 0.34/0.56 => ( ( A != bot_bot_set_o )
% 0.34/0.56 => ( ( ord_less_eq_o @ ( lattic1508158080041050831_fin_o @ A ) @ X )
% 0.34/0.56 = ( ! [X2: $o] :
% 0.34/0.56 ( ( member_o @ X2 @ A )
% 0.34/0.56 => ( ord_less_eq_o @ X2 @ X ) ) ) ) ) ) ).
% 0.34/0.56
% 0.34/0.56 % Sup_fin.bounded_iff
% 0.34/0.56 thf(fact_2152_Sup__fin_Obounded__iff,axiom,
% 0.34/0.56 ! [A: set_Pr4532377907799695533_nat_o,X: produc3658429121746597890et_nat > $o] :
% 0.34/0.56 ( ( finite3252695134891459830_nat_o @ A )
% 0.34/0.56 => ( ( A != bot_bo7824918357723582233_nat_o )
% 0.34/0.56 => ( ( ord_le729326519192465773_nat_o @ ( lattic7320199455484906628_nat_o @ A ) @ X )
% 0.34/0.56 = ( ! [X2: produc3658429121746597890et_nat > $o] :
% 0.34/0.56 ( ( member6576561426505652726_nat_o @ X2 @ A )
% 0.34/0.56 => ( ord_le729326519192465773_nat_o @ X2 @ X ) ) ) ) ) ) ).
% 0.34/0.56
% 0.34/0.56 % Sup_fin.bounded_iff
% 0.34/0.56 thf(fact_2153_Sup__fin_Obounded__iff,axiom,
% 0.34/0.56 ! [A: set_filter_nat,X: filter_nat] :
% 0.34/0.56 ( ( finite2119507909894593271er_nat @ A )
% 0.34/0.56 => ( ( A != bot_bo498966703094740906er_nat )
% 0.34/0.56 => ( ( ord_le2510731241096832064er_nat @ ( lattic5930898082463196905er_nat @ A ) @ X )
% 0.34/0.56 = ( ! [X2: filter_nat] :
% 0.34/0.56 ( ( member_filter_nat @ X2 @ A )
% 0.34/0.56 => ( ord_le2510731241096832064er_nat @ X2 @ X ) ) ) ) ) ) ).
% 0.34/0.56
% 0.34/0.56 % Sup_fin.bounded_iff
% 0.34/0.56 thf(fact_2154_Sup__fin_Obounded__iff,axiom,
% 0.34/0.56 ! [A: set_set_nat,X: set_nat] :
% 0.34/0.56 ( ( finite1152437895449049373et_nat @ A )
% 0.34/0.56 => ( ( A != bot_bot_set_set_nat )
% 0.34/0.56 => ( ( ord_less_eq_set_nat @ ( lattic3835124923745554447et_nat @ A ) @ X )
% 0.34/0.56 = ( ! [X2: set_nat] :
% 0.34/0.56 ( ( member_set_nat @ X2 @ A )
% 0.34/0.56 => ( ord_less_eq_set_nat @ X2 @ X ) ) ) ) ) ) ).
% 0.34/0.56
% 0.34/0.56 % Sup_fin.bounded_iff
% 0.34/0.56 thf(fact_2155_Sup__fin_Obounded__iff,axiom,
% 0.34/0.56 ! [A: set_rat,X: rat] :
% 0.34/0.56 ( ( finite_finite_rat @ A )
% 0.34/0.56 => ( ( A != bot_bot_set_rat )
% 0.34/0.56 => ( ( ord_less_eq_rat @ ( lattic458866745392299617in_rat @ A ) @ X )
% 0.34/0.56 = ( ! [X2: rat] :
% 0.34/0.56 ( ( member_rat @ X2 @ A )
% 0.34/0.56 => ( ord_less_eq_rat @ X2 @ X ) ) ) ) ) ) ).
% 0.34/0.56
% 0.34/0.56 % Sup_fin.bounded_iff
% 0.34/0.56 thf(fact_2156_Sup__fin_Obounded__iff,axiom,
% 0.34/0.56 ! [A: set_nat,X: nat] :
% 0.34/0.56 ( ( finite_finite_nat @ A )
% 0.34/0.56 => ( ( A != bot_bot_set_nat )
% 0.34/0.56 => ( ( ord_less_eq_nat @ ( lattic1093996805478795353in_nat @ A ) @ X )
% 0.34/0.56 = ( ! [X2: nat] :
% 0.34/0.56 ( ( member_nat @ X2 @ A )
% 0.34/0.56 => ( ord_less_eq_nat @ X2 @ X ) ) ) ) ) ) ).
% 0.34/0.56
% 0.34/0.56 % Sup_fin.bounded_iff
% 0.34/0.56 thf(fact_2157_Sup__fin_Obounded__iff,axiom,
% 0.34/0.56 ! [A: set_int,X: int] :
% 0.34/0.56 ( ( finite_finite_int @ A )
% 0.34/0.56 => ( ( A != bot_bot_set_int )
% 0.34/0.56 => ( ( ord_less_eq_int @ ( lattic1091506334969745077in_int @ A ) @ X )
% 0.34/0.56 = ( ! [X2: int] :
% 0.34/0.56 ( ( member_int @ X2 @ A )
% 0.34/0.56 => ( ord_less_eq_int @ X2 @ X ) ) ) ) ) ) ).
% 0.34/0.56
% 0.34/0.56 % Sup_fin.bounded_iff
% 0.34/0.56 thf(fact_2158_quotient__disj,axiom,
% 0.34/0.56 ! [A: set_Product_unit,R2: set_Pr5094982260447487303t_unit,X4: set_Product_unit,Y6: set_Product_unit] :
% 0.34/0.56 ( ( equiv_4189356295871601603t_unit @ A @ R2 )
% 0.34/0.56 => ( ( member5877623283571906838t_unit @ X4 @ ( equiv_5172566880711946708t_unit @ A @ R2 ) )
% 0.34/0.56 => ( ( member5877623283571906838t_unit @ Y6 @ ( equiv_5172566880711946708t_unit @ A @ R2 ) )
% 0.34/0.56 => ( ( X4 = Y6 )
% 0.34/0.56 | ( ( inf_in4660618365625256667t_unit @ X4 @ Y6 )
% 0.34/0.56 = bot_bo3957492148770167129t_unit ) ) ) ) ) ).
% 0.34/0.56
% 0.34/0.56 % quotient_disj
% 0.34/0.56 thf(fact_2159_quotient__disj,axiom,
% 0.34/0.56 ! [A: set_o,R2: set_Product_prod_o_o,X4: set_o,Y6: set_o] :
% 0.34/0.56 ( ( equiv_equiv_o @ A @ R2 )
% 0.34/0.56 => ( ( member_set_o @ X4 @ ( equiv_quotient_o @ A @ R2 ) )
% 0.34/0.56 => ( ( member_set_o @ Y6 @ ( equiv_quotient_o @ A @ R2 ) )
% 0.34/0.56 => ( ( X4 = Y6 )
% 0.34/0.56 | ( ( inf_inf_set_o @ X4 @ Y6 )
% 0.34/0.56 = bot_bot_set_o ) ) ) ) ) ).
% 0.34/0.56
% 0.34/0.56 % quotient_disj
% 0.34/0.56 thf(fact_2160_quotient__disj,axiom,
% 0.34/0.56 ! [A: set_nat,R2: set_Pr1261947904930325089at_nat,X4: set_nat,Y6: set_nat] :
% 0.34/0.56 ( ( equiv_equiv_nat @ A @ R2 )
% 0.34/0.56 => ( ( member_set_nat @ X4 @ ( equiv_quotient_nat @ A @ R2 ) )
% 0.34/0.56 => ( ( member_set_nat @ Y6 @ ( equiv_quotient_nat @ A @ R2 ) )
% 0.34/0.56 => ( ( X4 = Y6 )
% 0.34/0.56 | ( ( inf_inf_set_nat @ X4 @ Y6 )
% 0.34/0.56 = bot_bot_set_nat ) ) ) ) ) ).
% 0.34/0.56
% 0.34/0.56 % quotient_disj
% 0.34/0.56 thf(fact_2161_quotient__disj,axiom,
% 0.34/0.56 ! [A: set_int,R2: set_Pr958786334691620121nt_int,X4: set_int,Y6: set_int] :
% 0.34/0.56 ( ( equiv_equiv_int @ A @ R2 )
% 0.34/0.56 => ( ( member_set_int @ X4 @ ( equiv_quotient_int @ A @ R2 ) )
% 0.34/0.56 => ( ( member_set_int @ Y6 @ ( equiv_quotient_int @ A @ R2 ) )
% 0.34/0.56 => ( ( X4 = Y6 )
% 0.34/0.56 | ( ( inf_inf_set_int @ X4 @ Y6 )
% 0.34/0.56 = bot_bot_set_int ) ) ) ) ) ).
% 0.34/0.56
% 0.34/0.56 % quotient_disj
% 0.34/0.56 thf(fact_2162_quotient__disj,axiom,
% 0.34/0.56 ! [A: set_Pr4532377907799695533_nat_o,R2: set_Pr2161125870931222855_nat_o,X4: set_Pr4532377907799695533_nat_o,Y6: set_Pr4532377907799695533_nat_o] :
% 0.34/0.56 ( ( equiv_6726761681920839555_nat_o @ A @ R2 )
% 0.34/0.56 => ( ( member1452989533377600726_nat_o @ X4 @ ( equiv_7899955228204526484_nat_o @ A @ R2 ) )
% 0.34/0.56 => ( ( member1452989533377600726_nat_o @ Y6 @ ( equiv_7899955228204526484_nat_o @ A @ R2 ) )
% 0.34/0.56 => ( ( X4 = Y6 )
% 0.34/0.56 | ( ( inf_in1906310914598751387_nat_o @ X4 @ Y6 )
% 0.34/0.56 = bot_bo7824918357723582233_nat_o ) ) ) ) ) ).
% 0.34/0.56
% 0.34/0.56 % quotient_disj
% 0.34/0.56 thf(fact_2163_rel__restrict__Int__empty,axiom,
% 0.34/0.56 ! [A: set_Product_unit,R3: set_Pr5094982260447487303t_unit] :
% 0.34/0.56 ( ( ( inf_in4660618365625256667t_unit @ A @ ( field_Product_unit @ R3 ) )
% 0.34/0.56 = bot_bo3957492148770167129t_unit )
% 0.34/0.56 => ( ( rel_re9207996470947987439t_unit @ R3 @ A )
% 0.34/0.56 = R3 ) ) ).
% 0.34/0.56
% 0.34/0.56 % rel_restrict_Int_empty
% 0.34/0.56 thf(fact_2164_rel__restrict__Int__empty,axiom,
% 0.34/0.56 ! [A: set_o,R3: set_Product_prod_o_o] :
% 0.34/0.56 ( ( ( inf_inf_set_o @ A @ ( field_o @ R3 ) )
% 0.34/0.56 = bot_bot_set_o )
% 0.34/0.56 => ( ( rel_restrict_o @ R3 @ A )
% 0.34/0.56 = R3 ) ) ).
% 0.34/0.56
% 0.34/0.56 % rel_restrict_Int_empty
% 0.34/0.56 thf(fact_2165_rel__restrict__Int__empty,axiom,
% 0.34/0.56 ! [A: set_nat,R3: set_Pr1261947904930325089at_nat] :
% 0.34/0.56 ( ( ( inf_inf_set_nat @ A @ ( field_nat @ R3 ) )
% 0.34/0.56 = bot_bot_set_nat )
% 0.34/0.56 => ( ( rel_restrict_nat @ R3 @ A )
% 0.34/0.56 = R3 ) ) ).
% 0.34/0.56
% 0.34/0.56 % rel_restrict_Int_empty
% 0.34/0.56 thf(fact_2166_rel__restrict__Int__empty,axiom,
% 0.34/0.56 ! [A: set_int,R3: set_Pr958786334691620121nt_int] :
% 0.34/0.56 ( ( ( inf_inf_set_int @ A @ ( field_int @ R3 ) )
% 0.34/0.56 = bot_bot_set_int )
% 0.34/0.56 => ( ( rel_restrict_int @ R3 @ A )
% 0.34/0.56 = R3 ) ) ).
% 0.34/0.56
% 0.34/0.56 % rel_restrict_Int_empty
% 0.34/0.56 thf(fact_2167_rel__restrict__Int__empty,axiom,
% 0.34/0.56 ! [A: set_Pr4532377907799695533_nat_o,R3: set_Pr2161125870931222855_nat_o] :
% 0.34/0.56 ( ( ( inf_in1906310914598751387_nat_o @ A @ ( field_81821745747897882_nat_o @ R3 ) )
% 0.34/0.56 = bot_bo7824918357723582233_nat_o )
% 0.34/0.56 => ( ( rel_re3331361160397388719_nat_o @ R3 @ A )
% 0.34/0.56 = R3 ) ) ).
% 0.34/0.56
% 0.34/0.56 % rel_restrict_Int_empty
% 0.34/0.56 thf(fact_2168_eq__equiv__class__iff2,axiom,
% 0.34/0.56 ! [A: set_Product_unit,R2: set_Pr5094982260447487303t_unit,X: product_unit,Y: product_unit] :
% 0.34/0.56 ( ( equiv_4189356295871601603t_unit @ A @ R2 )
% 0.34/0.56 => ( ( member_Product_unit @ X @ A )
% 0.34/0.56 => ( ( member_Product_unit @ Y @ A )
% 0.34/0.56 => ( ( ( equiv_5172566880711946708t_unit @ ( insert_Product_unit @ X @ bot_bo3957492148770167129t_unit ) @ R2 )
% 0.34/0.56 = ( equiv_5172566880711946708t_unit @ ( insert_Product_unit @ Y @ bot_bo3957492148770167129t_unit ) @ R2 ) )
% 0.34/0.56 = ( member7821138191923116944t_unit @ ( produc5692694829384537815t_unit @ X @ Y ) @ R2 ) ) ) ) ) ).
% 0.34/0.56
% 0.34/0.56 % eq_equiv_class_iff2
% 0.34/0.56 thf(fact_2169_eq__equiv__class__iff2,axiom,
% 0.34/0.56 ! [A: set_o,R2: set_Product_prod_o_o,X: $o,Y: $o] :
% 0.34/0.56 ( ( equiv_equiv_o @ A @ R2 )
% 0.34/0.56 => ( ( member_o @ X @ A )
% 0.34/0.56 => ( ( member_o @ Y @ A )
% 0.34/0.56 => ( ( ( equiv_quotient_o @ ( insert_o @ X @ bot_bot_set_o ) @ R2 )
% 0.34/0.56 = ( equiv_quotient_o @ ( insert_o @ Y @ bot_bot_set_o ) @ R2 ) )
% 0.34/0.56 = ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ X @ Y ) @ R2 ) ) ) ) ) ).
% 0.34/0.56
% 0.34/0.56 % eq_equiv_class_iff2
% 0.34/0.56 thf(fact_2170_eq__equiv__class__iff2,axiom,
% 0.34/0.56 ! [A: set_nat,R2: set_Pr1261947904930325089at_nat,X: nat,Y: nat] :
% 0.34/0.56 ( ( equiv_equiv_nat @ A @ R2 )
% 0.34/0.56 => ( ( member_nat @ X @ A )
% 0.34/0.56 => ( ( member_nat @ Y @ A )
% 0.34/0.56 => ( ( ( equiv_quotient_nat @ ( insert_nat @ X @ bot_bot_set_nat ) @ R2 )
% 0.34/0.56 = ( equiv_quotient_nat @ ( insert_nat @ Y @ bot_bot_set_nat ) @ R2 ) )
% 0.34/0.56 = ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ Y ) @ R2 ) ) ) ) ) ).
% 0.34/0.56
% 0.34/0.56 % eq_equiv_class_iff2
% 0.34/0.56 thf(fact_2171_eq__equiv__class__iff2,axiom,
% 0.34/0.56 ! [A: set_int,R2: set_Pr958786334691620121nt_int,X: int,Y: int] :
% 0.34/0.56 ( ( equiv_equiv_int @ A @ R2 )
% 0.34/0.56 => ( ( member_int @ X @ A )
% 0.34/0.56 => ( ( member_int @ Y @ A )
% 0.34/0.56 => ( ( ( equiv_quotient_int @ ( insert_int @ X @ bot_bot_set_int ) @ R2 )
% 0.34/0.56 = ( equiv_quotient_int @ ( insert_int @ Y @ bot_bot_set_int ) @ R2 ) )
% 0.34/0.56 = ( member5262025264175285858nt_int @ ( product_Pair_int_int @ X @ Y ) @ R2 ) ) ) ) ) ).
% 0.34/0.56
% 0.34/0.56 % eq_equiv_class_iff2
% 0.34/0.56 thf(fact_2172_eq__equiv__class__iff2,axiom,
% 0.34/0.56 ! [A: set_Pr4532377907799695533_nat_o,R2: set_Pr2161125870931222855_nat_o,X: produc3658429121746597890et_nat > $o,Y: produc3658429121746597890et_nat > $o] :
% 0.34/0.56 ( ( equiv_6726761681920839555_nat_o @ A @ R2 )
% 0.34/0.56 => ( ( member6576561426505652726_nat_o @ X @ A )
% 0.34/0.56 => ( ( member6576561426505652726_nat_o @ Y @ A )
% 0.34/0.56 => ( ( ( equiv_7899955228204526484_nat_o @ ( insert5175938949040314269_nat_o @ X @ bot_bo7824918357723582233_nat_o ) @ R2 )
% 0.34/0.56 = ( equiv_7899955228204526484_nat_o @ ( insert5175938949040314269_nat_o @ Y @ bot_bo7824918357723582233_nat_o ) @ R2 ) )
% 0.34/0.56 = ( member8781333585448626064_nat_o @ ( produc7368190662567826135_nat_o @ X @ Y ) @ R2 ) ) ) ) ) ).
% 0.34/0.56
% 0.34/0.56 % eq_equiv_class_iff2
% 0.34/0.56 thf(fact_2173_Inf__fin_Osubset__imp,axiom,
% 0.34/0.56 ! [A: set_Product_unit,B4: set_Product_unit] :
% 0.34/0.56 ( ( ord_le3507040750410214029t_unit @ A @ B4 )
% 0.34/0.56 => ( ( A != bot_bo3957492148770167129t_unit )
% 0.34/0.56 => ( ( finite4290736615968046902t_unit @ B4 )
% 0.34/0.56 => ( ord_le3221252021190050221t_unit @ ( lattic1263872656861969706t_unit @ B4 ) @ ( lattic1263872656861969706t_unit @ A ) ) ) ) ) ).
% 0.34/0.56
% 0.34/0.56 % Inf_fin.subset_imp
% 0.34/0.56 thf(fact_2174_Inf__fin_Osubset__imp,axiom,
% 0.34/0.56 ! [A: set_o,B4: set_o] :
% 0.34/0.56 ( ( ord_less_eq_set_o @ A @ B4 )
% 0.34/0.56 => ( ( A != bot_bot_set_o )
% 0.34/0.56 => ( ( finite_finite_o @ B4 )
% 0.34/0.56 => ( ord_less_eq_o @ ( lattic4107685809792843317_fin_o @ B4 ) @ ( lattic4107685809792843317_fin_o @ A ) ) ) ) ) ).
% 0.34/0.56
% 0.34/0.56 % Inf_fin.subset_imp
% 0.34/0.56 thf(fact_2175_Inf__fin_Osubset__imp,axiom,
% 0.34/0.56 ! [A: set_Pr4532377907799695533_nat_o,B4: set_Pr4532377907799695533_nat_o] :
% 0.34/0.56 ( ( ord_le2965882846123202637_nat_o @ A @ B4 )
% 0.34/0.56 => ( ( A != bot_bo7824918357723582233_nat_o )
% 0.34/0.56 => ( ( finite3252695134891459830_nat_o @ B4 )
% 0.34/0.56 => ( ord_le729326519192465773_nat_o @ ( lattic956194824204696298_nat_o @ B4 ) @ ( lattic956194824204696298_nat_o @ A ) ) ) ) ) ).
% 0.34/0.56
% 0.34/0.56 % Inf_fin.subset_imp
% 0.34/0.56 thf(fact_2176_Inf__fin_Osubset__imp,axiom,
% 0.34/0.56 ! [A: set_filter_nat,B4: set_filter_nat] :
% 0.34/0.56 ( ( ord_le2426478655948331894er_nat @ A @ B4 )
% 0.34/0.56 => ( ( A != bot_bo498966703094740906er_nat )
% 0.34/0.56 => ( ( finite2119507909894593271er_nat @ B4 )
% 0.34/0.56 => ( ord_le2510731241096832064er_nat @ ( lattic9128708996678779395er_nat @ B4 ) @ ( lattic9128708996678779395er_nat @ A ) ) ) ) ) ).
% 0.34/0.56
% 0.34/0.56 % Inf_fin.subset_imp
% 0.34/0.56 thf(fact_2177_Inf__fin_Osubset__imp,axiom,
% 0.34/0.56 ! [A: set_set_nat,B4: set_set_nat] :
% 0.34/0.56 ( ( ord_le6893508408891458716et_nat @ A @ B4 )
% 0.34/0.56 => ( ( A != bot_bot_set_set_nat )
% 0.34/0.56 => ( ( finite1152437895449049373et_nat @ B4 )
% 0.34/0.56 => ( ord_less_eq_set_nat @ ( lattic3014633134055518761et_nat @ B4 ) @ ( lattic3014633134055518761et_nat @ A ) ) ) ) ) ).
% 0.34/0.56
% 0.34/0.56 % Inf_fin.subset_imp
% 0.34/0.56 thf(fact_2178_Inf__fin_Osubset__imp,axiom,
% 0.34/0.56 ! [A: set_rat,B4: set_rat] :
% 0.34/0.56 ( ( ord_less_eq_set_rat @ A @ B4 )
% 0.34/0.56 => ( ( A != bot_bot_set_rat )
% 0.34/0.56 => ( ( finite_finite_rat @ B4 )
% 0.34/0.56 => ( ord_less_eq_rat @ ( lattic4603258475043424379in_rat @ B4 ) @ ( lattic4603258475043424379in_rat @ A ) ) ) ) ) ).
% 0.34/0.56
% 0.34/0.56 % Inf_fin.subset_imp
% 0.34/0.56 thf(fact_2179_Inf__fin_Osubset__imp,axiom,
% 0.34/0.56 ! [A: set_int,B4: set_int] :
% 0.34/0.56 ( ( ord_less_eq_set_int @ A @ B4 )
% 0.34/0.56 => ( ( A != bot_bot_set_int )
% 0.34/0.56 => ( ( finite_finite_int @ B4 )
% 0.34/0.56 => ( ord_less_eq_int @ ( lattic5235898064620869839in_int @ B4 ) @ ( lattic5235898064620869839in_int @ A ) ) ) ) ) ).
% 0.34/0.56
% 0.34/0.56 % Inf_fin.subset_imp
% 0.34/0.56 thf(fact_2180_Inf__fin_Osubset__imp,axiom,
% 0.34/0.56 ! [A: set_nat,B4: set_nat] :
% 0.34/0.56 ( ( ord_less_eq_set_nat @ A @ B4 )
% 0.34/0.56 => ( ( A != bot_bot_set_nat )
% 0.34/0.56 => ( ( finite_finite_nat @ B4 )
% 0.34/0.56 => ( ord_less_eq_nat @ ( lattic5238388535129920115in_nat @ B4 ) @ ( lattic5238388535129920115in_nat @ A ) ) ) ) ) ).
% 0.34/0.56
% 0.34/0.56 % Inf_fin.subset_imp
% 0.34/0.56 thf(fact_2181_Sup__fin_Osubset__imp,axiom,
% 0.34/0.56 ! [A: set_Product_unit,B4: set_Product_unit] :
% 0.34/0.56 ( ( ord_le3507040750410214029t_unit @ A @ B4 )
% 0.34/0.56 => ( ( A != bot_bo3957492148770167129t_unit )
% 0.34/0.56 => ( ( finite4290736615968046902t_unit @ B4 )
% 0.34/0.56 => ( ord_le3221252021190050221t_unit @ ( lattic5294303975357428420t_unit @ A ) @ ( lattic5294303975357428420t_unit @ B4 ) ) ) ) ) ).
% 0.34/0.56
% 0.34/0.56 % Sup_fin.subset_imp
% 0.34/0.56 thf(fact_2182_Sup__fin_Osubset__imp,axiom,
% 0.34/0.56 ! [A: set_o,B4: set_o] :
% 0.34/0.56 ( ( ord_less_eq_set_o @ A @ B4 )
% 0.34/0.56 => ( ( A != bot_bot_set_o )
% 0.34/0.56 => ( ( finite_finite_o @ B4 )
% 0.34/0.56 => ( ord_less_eq_o @ ( lattic1508158080041050831_fin_o @ A ) @ ( lattic1508158080041050831_fin_o @ B4 ) ) ) ) ) ).
% 0.34/0.56
% 0.34/0.56 % Sup_fin.subset_imp
% 0.34/0.56 thf(fact_2183_Sup__fin_Osubset__imp,axiom,
% 0.34/0.56 ! [A: set_Pr4532377907799695533_nat_o,B4: set_Pr4532377907799695533_nat_o] :
% 0.34/0.56 ( ( ord_le2965882846123202637_nat_o @ A @ B4 )
% 0.34/0.56 => ( ( A != bot_bo7824918357723582233_nat_o )
% 0.34/0.56 => ( ( finite3252695134891459830_nat_o @ B4 )
% 0.34/0.56 => ( ord_le729326519192465773_nat_o @ ( lattic7320199455484906628_nat_o @ A ) @ ( lattic7320199455484906628_nat_o @ B4 ) ) ) ) ) ).
% 0.34/0.56
% 0.34/0.56 % Sup_fin.subset_imp
% 0.34/0.56 thf(fact_2184_Sup__fin_Osubset__imp,axiom,
% 0.34/0.56 ! [A: set_filter_nat,B4: set_filter_nat] :
% 0.34/0.56 ( ( ord_le2426478655948331894er_nat @ A @ B4 )
% 0.34/0.56 => ( ( A != bot_bo498966703094740906er_nat )
% 0.34/0.56 => ( ( finite2119507909894593271er_nat @ B4 )
% 0.34/0.56 => ( ord_le2510731241096832064er_nat @ ( lattic5930898082463196905er_nat @ A ) @ ( lattic5930898082463196905er_nat @ B4 ) ) ) ) ) ).
% 0.34/0.56
% 0.34/0.56 % Sup_fin.subset_imp
% 0.34/0.56 thf(fact_2185_Sup__fin_Osubset__imp,axiom,
% 0.34/0.56 ! [A: set_set_nat,B4: set_set_nat] :
% 0.34/0.56 ( ( ord_le6893508408891458716et_nat @ A @ B4 )
% 0.34/0.56 => ( ( A != bot_bot_set_set_nat )
% 0.34/0.56 => ( ( finite1152437895449049373et_nat @ B4 )
% 0.34/0.56 => ( ord_less_eq_set_nat @ ( lattic3835124923745554447et_nat @ A ) @ ( lattic3835124923745554447et_nat @ B4 ) ) ) ) ) ).
% 0.34/0.56
% 0.34/0.56 % Sup_fin.subset_imp
% 0.34/0.56 thf(fact_2186_Sup__fin_Osubset__imp,axiom,
% 0.34/0.56 ! [A: set_rat,B4: set_rat] :
% 0.34/0.56 ( ( ord_less_eq_set_rat @ A @ B4 )
% 0.34/0.56 => ( ( A != bot_bot_set_rat )
% 0.34/0.56 => ( ( finite_finite_rat @ B4 )
% 0.34/0.56 => ( ord_less_eq_rat @ ( lattic458866745392299617in_rat @ A ) @ ( lattic458866745392299617in_rat @ B4 ) ) ) ) ) ).
% 0.34/0.56
% 0.34/0.56 % Sup_fin.subset_imp
% 0.34/0.56 thf(fact_2187_Sup__fin_Osubset__imp,axiom,
% 0.34/0.56 ! [A: set_int,B4: set_int] :
% 0.34/0.56 ( ( ord_less_eq_set_int @ A @ B4 )
% 0.34/0.56 => ( ( A != bot_bot_set_int )
% 0.34/0.56 => ( ( finite_finite_int @ B4 )
% 0.34/0.56 => ( ord_less_eq_int @ ( lattic1091506334969745077in_int @ A ) @ ( lattic1091506334969745077in_int @ B4 ) ) ) ) ) ).
% 0.34/0.56
% 0.34/0.56 % Sup_fin.subset_imp
% 0.34/0.56 thf(fact_2188_Sup__fin_Osubset__imp,axiom,
% 0.34/0.56 ! [A: set_nat,B4: set_nat] :
% 0.34/0.56 ( ( ord_less_eq_set_nat @ A @ B4 )
% 0.34/0.56 => ( ( A != bot_bot_set_nat )
% 0.34/0.56 => ( ( finite_finite_nat @ B4 )
% 0.34/0.56 => ( ord_less_eq_nat @ ( lattic1093996805478795353in_nat @ A ) @ ( lattic1093996805478795353in_nat @ B4 ) ) ) ) ) ).
% 0.34/0.56
% 0.34/0.56 % Sup_fin.subset_imp
% 0.34/0.56 thf(fact_2189_Inf__fin_Osubset,axiom,
% 0.34/0.56 ! [A: set_set_nat,B4: set_set_nat] :
% 0.34/0.56 ( ( finite1152437895449049373et_nat @ A )
% 0.34/0.56 => ( ( B4 != bot_bot_set_set_nat )
% 0.34/0.56 => ( ( ord_le6893508408891458716et_nat @ B4 @ A )
% 0.34/0.56 => ( ( inf_inf_set_nat @ ( lattic3014633134055518761et_nat @ B4 ) @ ( lattic3014633134055518761et_nat @ A ) )
% 0.34/0.56 = ( lattic3014633134055518761et_nat @ A ) ) ) ) ) ).
% 0.34/0.56
% 0.34/0.56 % Inf_fin.subset
% 0.34/0.56 thf(fact_2190_Inf__fin_Osubset,axiom,
% 0.34/0.56 ! [A: set_assn,B4: set_assn] :
% 0.34/0.56 ( ( finite_finite_assn @ A )
% 0.34/0.56 => ( ( B4 != bot_bot_set_assn )
% 0.34/0.56 => ( ( ord_less_eq_set_assn @ B4 @ A )
% 0.34/0.56 => ( ( inf_inf_assn @ ( lattic47131356835913163n_assn @ B4 ) @ ( lattic47131356835913163n_assn @ A ) )
% 0.34/0.56 = ( lattic47131356835913163n_assn @ A ) ) ) ) ) ).
% 0.34/0.56
% 0.34/0.56 % Inf_fin.subset
% 0.34/0.56 thf(fact_2191_Inf__fin_Osubset,axiom,
% 0.34/0.56 ! [A: set_Product_unit,B4: set_Product_unit] :
% 0.34/0.56 ( ( finite4290736615968046902t_unit @ A )
% 0.34/0.56 => ( ( B4 != bot_bo3957492148770167129t_unit )
% 0.34/0.56 => ( ( ord_le3507040750410214029t_unit @ B4 @ A )
% 0.34/0.56 => ( ( inf_inf_Product_unit @ ( lattic1263872656861969706t_unit @ B4 ) @ ( lattic1263872656861969706t_unit @ A ) )
% 0.34/0.56 = ( lattic1263872656861969706t_unit @ A ) ) ) ) ) ).
% 0.34/0.56
% 0.34/0.56 % Inf_fin.subset
% 0.34/0.56 thf(fact_2192_Inf__fin_Osubset,axiom,
% 0.34/0.56 ! [A: set_o,B4: set_o] :
% 0.34/0.56 ( ( finite_finite_o @ A )
% 0.34/0.56 => ( ( B4 != bot_bot_set_o )
% 0.34/0.56 => ( ( ord_less_eq_set_o @ B4 @ A )
% 0.34/0.56 => ( ( inf_inf_o @ ( lattic4107685809792843317_fin_o @ B4 ) @ ( lattic4107685809792843317_fin_o @ A ) )
% 0.34/0.56 = ( lattic4107685809792843317_fin_o @ A ) ) ) ) ) ).
% 0.34/0.56
% 0.34/0.56 % Inf_fin.subset
% 0.34/0.56 thf(fact_2193_Inf__fin_Osubset,axiom,
% 0.34/0.56 ! [A: set_int,B4: set_int] :
% 0.34/0.56 ( ( finite_finite_int @ A )
% 0.34/0.56 => ( ( B4 != bot_bot_set_int )
% 0.34/0.56 => ( ( ord_less_eq_set_int @ B4 @ A )
% 0.34/0.56 => ( ( inf_inf_int @ ( lattic5235898064620869839in_int @ B4 ) @ ( lattic5235898064620869839in_int @ A ) )
% 0.34/0.56 = ( lattic5235898064620869839in_int @ A ) ) ) ) ) ).
% 0.34/0.56
% 0.34/0.56 % Inf_fin.subset
% 0.34/0.56 thf(fact_2194_Inf__fin_Osubset,axiom,
% 0.34/0.56 ! [A: set_Pr4532377907799695533_nat_o,B4: set_Pr4532377907799695533_nat_o] :
% 0.34/0.56 ( ( finite3252695134891459830_nat_o @ A )
% 0.34/0.56 => ( ( B4 != bot_bo7824918357723582233_nat_o )
% 0.34/0.56 => ( ( ord_le2965882846123202637_nat_o @ B4 @ A )
% 0.34/0.56 => ( ( inf_in1318976480646536635_nat_o @ ( lattic956194824204696298_nat_o @ B4 ) @ ( lattic956194824204696298_nat_o @ A ) )
% 0.34/0.56 = ( lattic956194824204696298_nat_o @ A ) ) ) ) ) ).
% 0.34/0.56
% 0.34/0.56 % Inf_fin.subset
% 0.34/0.56 thf(fact_2195_Inf__fin_Osubset,axiom,
% 0.34/0.56 ! [A: set_nat,B4: set_nat] :
% 0.34/0.56 ( ( finite_finite_nat @ A )
% 0.34/0.56 => ( ( B4 != bot_bot_set_nat )
% 0.34/0.56 => ( ( ord_less_eq_set_nat @ B4 @ A )
% 0.34/0.56 => ( ( inf_inf_nat @ ( lattic5238388535129920115in_nat @ B4 ) @ ( lattic5238388535129920115in_nat @ A ) )
% 0.34/0.56 = ( lattic5238388535129920115in_nat @ A ) ) ) ) ) ).
% 0.34/0.56
% 0.34/0.56 % Inf_fin.subset
% 0.34/0.56 thf(fact_2196_Sup__fin_Osubset,axiom,
% 0.34/0.56 ! [A: set_set_nat,B4: set_set_nat] :
% 0.34/0.56 ( ( finite1152437895449049373et_nat @ A )
% 0.34/0.56 => ( ( B4 != bot_bot_set_set_nat )
% 0.34/0.56 => ( ( ord_le6893508408891458716et_nat @ B4 @ A )
% 0.34/0.56 => ( ( sup_sup_set_nat @ ( lattic3835124923745554447et_nat @ B4 ) @ ( lattic3835124923745554447et_nat @ A ) )
% 0.34/0.56 = ( lattic3835124923745554447et_nat @ A ) ) ) ) ) ).
% 0.34/0.56
% 0.34/0.56 % Sup_fin.subset
% 0.34/0.56 thf(fact_2197_Sup__fin_Osubset,axiom,
% 0.34/0.56 ! [A: set_assn,B4: set_assn] :
% 0.34/0.56 ( ( finite_finite_assn @ A )
% 0.34/0.56 => ( ( B4 != bot_bot_set_assn )
% 0.34/0.56 => ( ( ord_less_eq_set_assn @ B4 @ A )
% 0.34/0.56 => ( ( sup_sup_assn @ ( lattic2150320897289308081n_assn @ B4 ) @ ( lattic2150320897289308081n_assn @ A ) )
% 0.34/0.56 = ( lattic2150320897289308081n_assn @ A ) ) ) ) ) ).
% 0.34/0.56
% 0.34/0.56 % Sup_fin.subset
% 0.34/0.56 thf(fact_2198_Sup__fin_Osubset,axiom,
% 0.34/0.56 ! [A: set_Product_unit,B4: set_Product_unit] :
% 0.34/0.56 ( ( finite4290736615968046902t_unit @ A )
% 0.34/0.56 => ( ( B4 != bot_bo3957492148770167129t_unit )
% 0.34/0.56 => ( ( ord_le3507040750410214029t_unit @ B4 @ A )
% 0.34/0.56 => ( ( sup_sup_Product_unit @ ( lattic5294303975357428420t_unit @ B4 ) @ ( lattic5294303975357428420t_unit @ A ) )
% 0.34/0.56 = ( lattic5294303975357428420t_unit @ A ) ) ) ) ) ).
% 0.34/0.56
% 0.34/0.56 % Sup_fin.subset
% 0.34/0.56 thf(fact_2199_Sup__fin_Osubset,axiom,
% 0.34/0.56 ! [A: set_o,B4: set_o] :
% 0.34/0.56 ( ( finite_finite_o @ A )
% 0.34/0.56 => ( ( B4 != bot_bot_set_o )
% 0.34/0.56 => ( ( ord_less_eq_set_o @ B4 @ A )
% 0.34/0.56 => ( ( sup_sup_o @ ( lattic1508158080041050831_fin_o @ B4 ) @ ( lattic1508158080041050831_fin_o @ A ) )
% 0.34/0.56 = ( lattic1508158080041050831_fin_o @ A ) ) ) ) ) ).
% 0.34/0.56
% 0.34/0.56 % Sup_fin.subset
% 0.34/0.56 thf(fact_2200_Sup__fin_Osubset,axiom,
% 0.34/0.56 ! [A: set_int,B4: set_int] :
% 0.34/0.56 ( ( finite_finite_int @ A )
% 0.34/0.56 => ( ( B4 != bot_bot_set_int )
% 0.34/0.56 => ( ( ord_less_eq_set_int @ B4 @ A )
% 0.34/0.56 => ( ( sup_sup_int @ ( lattic1091506334969745077in_int @ B4 ) @ ( lattic1091506334969745077in_int @ A ) )
% 0.34/0.56 = ( lattic1091506334969745077in_int @ A ) ) ) ) ) ).
% 0.34/0.56
% 0.34/0.56 % Sup_fin.subset
% 0.34/0.56 thf(fact_2201_Sup__fin_Osubset,axiom,
% 0.34/0.56 ! [A: set_Pr4532377907799695533_nat_o,B4: set_Pr4532377907799695533_nat_o] :
% 0.34/0.56 ( ( finite3252695134891459830_nat_o @ A )
% 0.34/0.56 => ( ( B4 != bot_bo7824918357723582233_nat_o )
% 0.34/0.56 => ( ( ord_le2965882846123202637_nat_o @ B4 @ A )
% 0.34/0.56 => ( ( sup_su5453871518329203617_nat_o @ ( lattic7320199455484906628_nat_o @ B4 ) @ ( lattic7320199455484906628_nat_o @ A ) )
% 0.34/0.56 = ( lattic7320199455484906628_nat_o @ A ) ) ) ) ) ).
% 0.34/0.56
% 0.34/0.56 % Sup_fin.subset
% 0.34/0.56 thf(fact_2202_Sup__fin_Osubset,axiom,
% 0.34/0.56 ! [A: set_nat,B4: set_nat] :
% 0.34/0.56 ( ( finite_finite_nat @ A )
% 0.34/0.56 => ( ( B4 != bot_bot_set_nat )
% 0.34/0.56 => ( ( ord_less_eq_set_nat @ B4 @ A )
% 0.34/0.56 => ( ( sup_sup_nat @ ( lattic1093996805478795353in_nat @ B4 ) @ ( lattic1093996805478795353in_nat @ A ) )
% 0.34/0.56 = ( lattic1093996805478795353in_nat @ A ) ) ) ) ) ).
% 0.34/0.56
% 0.34/0.56 % Sup_fin.subset
% 0.34/0.56 thf(fact_2203_Inf__fin_Oclosed,axiom,
% 0.34/0.56 ! [A: set_set_nat] :
% 0.34/0.56 ( ( finite1152437895449049373et_nat @ A )
% 0.34/0.56 => ( ( A != bot_bot_set_set_nat )
% 0.34/0.56 => ( ! [X3: set_nat,Y2: set_nat] : ( member_set_nat @ ( inf_inf_set_nat @ X3 @ Y2 ) @ ( insert_set_nat @ X3 @ ( insert_set_nat @ Y2 @ bot_bot_set_set_nat ) ) )
% 0.34/0.56 => ( member_set_nat @ ( lattic3014633134055518761et_nat @ A ) @ A ) ) ) ) ).
% 0.34/0.56
% 0.34/0.56 % Inf_fin.closed
% 0.34/0.56 thf(fact_2204_Inf__fin_Oclosed,axiom,
% 0.34/0.56 ! [A: set_assn] :
% 0.34/0.56 ( ( finite_finite_assn @ A )
% 0.34/0.56 => ( ( A != bot_bot_set_assn )
% 0.34/0.56 => ( ! [X3: assn,Y2: assn] : ( member_assn @ ( inf_inf_assn @ X3 @ Y2 ) @ ( insert_assn @ X3 @ ( insert_assn @ Y2 @ bot_bot_set_assn ) ) )
% 0.34/0.56 => ( member_assn @ ( lattic47131356835913163n_assn @ A ) @ A ) ) ) ) ).
% 0.34/0.56
% 0.34/0.56 % Inf_fin.closed
% 0.34/0.56 thf(fact_2205_Inf__fin_Oclosed,axiom,
% 0.34/0.56 ! [A: set_Product_unit] :
% 0.34/0.56 ( ( finite4290736615968046902t_unit @ A )
% 0.34/0.56 => ( ( A != bot_bo3957492148770167129t_unit )
% 0.34/0.56 => ( ! [X3: product_unit,Y2: product_unit] : ( member_Product_unit @ ( inf_inf_Product_unit @ X3 @ Y2 ) @ ( insert_Product_unit @ X3 @ ( insert_Product_unit @ Y2 @ bot_bo3957492148770167129t_unit ) ) )
% 0.34/0.56 => ( member_Product_unit @ ( lattic1263872656861969706t_unit @ A ) @ A ) ) ) ) ).
% 0.34/0.56
% 0.34/0.56 % Inf_fin.closed
% 0.34/0.56 thf(fact_2206_Inf__fin_Oclosed,axiom,
% 0.34/0.56 ! [A: set_o] :
% 0.34/0.56 ( ( finite_finite_o @ A )
% 0.34/0.56 => ( ( A != bot_bot_set_o )
% 0.34/0.56 => ( ! [X3: $o,Y2: $o] : ( member_o @ ( inf_inf_o @ X3 @ Y2 ) @ ( insert_o @ X3 @ ( insert_o @ Y2 @ bot_bot_set_o ) ) )
% 0.34/0.56 => ( member_o @ ( lattic4107685809792843317_fin_o @ A ) @ A ) ) ) ) ).
% 0.34/0.56
% 0.34/0.56 % Inf_fin.closed
% 0.34/0.56 thf(fact_2207_Inf__fin_Oclosed,axiom,
% 0.34/0.56 ! [A: set_nat] :
% 0.34/0.56 ( ( finite_finite_nat @ A )
% 0.34/0.56 => ( ( A != bot_bot_set_nat )
% 0.34/0.56 => ( ! [X3: nat,Y2: nat] : ( member_nat @ ( inf_inf_nat @ X3 @ Y2 ) @ ( insert_nat @ X3 @ ( insert_nat @ Y2 @ bot_bot_set_nat ) ) )
% 0.34/0.56 => ( member_nat @ ( lattic5238388535129920115in_nat @ A ) @ A ) ) ) ) ).
% 0.34/0.56
% 0.34/0.56 % Inf_fin.closed
% 0.34/0.56 thf(fact_2208_Inf__fin_Oclosed,axiom,
% 0.34/0.56 ! [A: set_int] :
% 0.34/0.56 ( ( finite_finite_int @ A )
% 0.34/0.56 => ( ( A != bot_bot_set_int )
% 0.34/0.56 => ( ! [X3: int,Y2: int] : ( member_int @ ( inf_inf_int @ X3 @ Y2 ) @ ( insert_int @ X3 @ ( insert_int @ Y2 @ bot_bot_set_int ) ) )
% 0.34/0.56 => ( member_int @ ( lattic5235898064620869839in_int @ A ) @ A ) ) ) ) ).
% 0.34/0.56
% 0.34/0.56 % Inf_fin.closed
% 0.34/0.56 thf(fact_2209_Inf__fin_Oclosed,axiom,
% 0.34/0.56 ! [A: set_Pr4532377907799695533_nat_o] :
% 0.34/0.56 ( ( finite3252695134891459830_nat_o @ A )
% 0.34/0.56 => ( ( A != bot_bo7824918357723582233_nat_o )
% 0.34/0.56 => ( ! [X3: produc3658429121746597890et_nat > $o,Y2: produc3658429121746597890et_nat > $o] : ( member6576561426505652726_nat_o @ ( inf_in1318976480646536635_nat_o @ X3 @ Y2 ) @ ( insert5175938949040314269_nat_o @ X3 @ ( insert5175938949040314269_nat_o @ Y2 @ bot_bo7824918357723582233_nat_o ) ) )
% 0.34/0.56 => ( member6576561426505652726_nat_o @ ( lattic956194824204696298_nat_o @ A ) @ A ) ) ) ) ).
% 0.34/0.56
% 0.34/0.56 % Inf_fin.closed
% 0.34/0.56 thf(fact_2210_Inf__fin_Oinsert__not__elem,axiom,
% 0.34/0.56 ! [A: set_set_nat,X: set_nat] :
% 0.34/0.56 ( ( finite1152437895449049373et_nat @ A )
% 0.34/0.56 => ( ~ ( member_set_nat @ X @ A )
% 0.34/0.56 => ( ( A != bot_bot_set_set_nat )
% 0.34/0.56 => ( ( lattic3014633134055518761et_nat @ ( insert_set_nat @ X @ A ) )
% 0.34/0.56 = ( inf_inf_set_nat @ X @ ( lattic3014633134055518761et_nat @ A ) ) ) ) ) ) ).
% 0.34/0.56
% 0.34/0.56 % Inf_fin.insert_not_elem
% 0.34/0.56 thf(fact_2211_Inf__fin_Oinsert__not__elem,axiom,
% 0.34/0.56 ! [A: set_assn,X: assn] :
% 0.34/0.56 ( ( finite_finite_assn @ A )
% 0.34/0.56 => ( ~ ( member_assn @ X @ A )
% 0.34/0.56 => ( ( A != bot_bot_set_assn )
% 0.34/0.56 => ( ( lattic47131356835913163n_assn @ ( insert_assn @ X @ A ) )
% 0.34/0.56 = ( inf_inf_assn @ X @ ( lattic47131356835913163n_assn @ A ) ) ) ) ) ) ).
% 0.34/0.56
% 0.34/0.56 % Inf_fin.insert_not_elem
% 0.34/0.56 thf(fact_2212_Inf__fin_Oinsert__not__elem,axiom,
% 0.34/0.56 ! [A: set_Product_unit,X: product_unit] :
% 0.34/0.56 ( ( finite4290736615968046902t_unit @ A )
% 0.34/0.56 => ( ~ ( member_Product_unit @ X @ A )
% 0.34/0.56 => ( ( A != bot_bo3957492148770167129t_unit )
% 0.34/0.56 => ( ( lattic1263872656861969706t_unit @ ( insert_Product_unit @ X @ A ) )
% 0.34/0.56 = ( inf_inf_Product_unit @ X @ ( lattic1263872656861969706t_unit @ A ) ) ) ) ) ) ).
% 0.34/0.56
% 0.34/0.56 % Inf_fin.insert_not_elem
% 0.34/0.56 thf(fact_2213_Inf__fin_Oinsert__not__elem,axiom,
% 0.34/0.56 ! [A: set_o,X: $o] :
% 0.34/0.56 ( ( finite_finite_o @ A )
% 0.34/0.56 => ( ~ ( member_o @ X @ A )
% 0.34/0.56 => ( ( A != bot_bot_set_o )
% 0.34/0.56 => ( ( lattic4107685809792843317_fin_o @ ( insert_o @ X @ A ) )
% 0.34/0.56 = ( inf_inf_o @ X @ ( lattic4107685809792843317_fin_o @ A ) ) ) ) ) ) ).
% 0.34/0.56
% 0.34/0.56 % Inf_fin.insert_not_elem
% 0.34/0.56 thf(fact_2214_Inf__fin_Oinsert__not__elem,axiom,
% 0.34/0.56 ! [A: set_nat,X: nat] :
% 0.34/0.56 ( ( finite_finite_nat @ A )
% 0.34/0.56 => ( ~ ( member_nat @ X @ A )
% 0.34/0.56 => ( ( A != bot_bot_set_nat )
% 0.34/0.56 => ( ( lattic5238388535129920115in_nat @ ( insert_nat @ X @ A ) )
% 0.34/0.56 = ( inf_inf_nat @ X @ ( lattic5238388535129920115in_nat @ A ) ) ) ) ) ) ).
% 0.34/0.56
% 0.34/0.56 % Inf_fin.insert_not_elem
% 0.34/0.56 thf(fact_2215_Inf__fin_Oinsert__not__elem,axiom,
% 0.34/0.56 ! [A: set_int,X: int] :
% 0.34/0.56 ( ( finite_finite_int @ A )
% 0.34/0.56 => ( ~ ( member_int @ X @ A )
% 0.34/0.56 => ( ( A != bot_bot_set_int )
% 0.34/0.56 => ( ( lattic5235898064620869839in_int @ ( insert_int @ X @ A ) )
% 0.34/0.56 = ( inf_inf_int @ X @ ( lattic5235898064620869839in_int @ A ) ) ) ) ) ) ).
% 0.34/0.56
% 0.34/0.56 % Inf_fin.insert_not_elem
% 0.34/0.56 thf(fact_2216_Inf__fin_Oinsert__not__elem,axiom,
% 0.34/0.56 ! [A: set_Pr4532377907799695533_nat_o,X: produc3658429121746597890et_nat > $o] :
% 0.34/0.56 ( ( finite3252695134891459830_nat_o @ A )
% 0.34/0.56 => ( ~ ( member6576561426505652726_nat_o @ X @ A )
% 0.34/0.56 => ( ( A != bot_bo7824918357723582233_nat_o )
% 0.34/0.56 => ( ( lattic956194824204696298_nat_o @ ( insert5175938949040314269_nat_o @ X @ A ) )
% 0.34/0.56 = ( inf_in1318976480646536635_nat_o @ X @ ( lattic956194824204696298_nat_o @ A ) ) ) ) ) ) ).
% 0.34/0.56
% 0.34/0.56 % Inf_fin.insert_not_elem
% 0.34/0.56 thf(fact_2217_Sup__fin_Oclosed,axiom,
% 0.34/0.56 ! [A: set_set_nat] :
% 0.34/0.56 ( ( finite1152437895449049373et_nat @ A )
% 0.34/0.56 => ( ( A != bot_bot_set_set_nat )
% 0.34/0.56 => ( ! [X3: set_nat,Y2: set_nat] : ( member_set_nat @ ( sup_sup_set_nat @ X3 @ Y2 ) @ ( insert_set_nat @ X3 @ ( insert_set_nat @ Y2 @ bot_bot_set_set_nat ) ) )
% 0.34/0.56 => ( member_set_nat @ ( lattic3835124923745554447et_nat @ A ) @ A ) ) ) ) ).
% 0.34/0.56
% 0.34/0.56 % Sup_fin.closed
% 0.34/0.56 thf(fact_2218_Sup__fin_Oclosed,axiom,
% 0.34/0.56 ! [A: set_assn] :
% 0.34/0.56 ( ( finite_finite_assn @ A )
% 0.34/0.56 => ( ( A != bot_bot_set_assn )
% 0.34/0.56 => ( ! [X3: assn,Y2: assn] : ( member_assn @ ( sup_sup_assn @ X3 @ Y2 ) @ ( insert_assn @ X3 @ ( insert_assn @ Y2 @ bot_bot_set_assn ) ) )
% 0.34/0.56 => ( member_assn @ ( lattic2150320897289308081n_assn @ A ) @ A ) ) ) ) ).
% 0.34/0.56
% 0.34/0.56 % Sup_fin.closed
% 0.34/0.56 thf(fact_2219_Sup__fin_Oclosed,axiom,
% 0.34/0.56 ! [A: set_Product_unit] :
% 0.34/0.56 ( ( finite4290736615968046902t_unit @ A )
% 0.34/0.56 => ( ( A != bot_bo3957492148770167129t_unit )
% 0.34/0.56 => ( ! [X3: product_unit,Y2: product_unit] : ( member_Product_unit @ ( sup_sup_Product_unit @ X3 @ Y2 ) @ ( insert_Product_unit @ X3 @ ( insert_Product_unit @ Y2 @ bot_bo3957492148770167129t_unit ) ) )
% 0.34/0.56 => ( member_Product_unit @ ( lattic5294303975357428420t_unit @ A ) @ A ) ) ) ) ).
% 0.34/0.56
% 0.34/0.56 % Sup_fin.closed
% 0.34/0.56 thf(fact_2220_Sup__fin_Oclosed,axiom,
% 0.34/0.56 ! [A: set_o] :
% 0.34/0.56 ( ( finite_finite_o @ A )
% 0.34/0.56 => ( ( A != bot_bot_set_o )
% 0.34/0.56 => ( ! [X3: $o,Y2: $o] : ( member_o @ ( sup_sup_o @ X3 @ Y2 ) @ ( insert_o @ X3 @ ( insert_o @ Y2 @ bot_bot_set_o ) ) )
% 0.34/0.56 => ( member_o @ ( lattic1508158080041050831_fin_o @ A ) @ A ) ) ) ) ).
% 0.34/0.56
% 0.34/0.56 % Sup_fin.closed
% 0.34/0.56 thf(fact_2221_Sup__fin_Oclosed,axiom,
% 0.34/0.56 ! [A: set_nat] :
% 0.34/0.56 ( ( finite_finite_nat @ A )
% 0.34/0.56 => ( ( A != bot_bot_set_nat )
% 0.34/0.56 => ( ! [X3: nat,Y2: nat] : ( member_nat @ ( sup_sup_nat @ X3 @ Y2 ) @ ( insert_nat @ X3 @ ( insert_nat @ Y2 @ bot_bot_set_nat ) ) )
% 0.34/0.56 => ( member_nat @ ( lattic1093996805478795353in_nat @ A ) @ A ) ) ) ) ).
% 0.34/0.56
% 0.34/0.56 % Sup_fin.closed
% 0.34/0.56 thf(fact_2222_Sup__fin_Oclosed,axiom,
% 0.34/0.56 ! [A: set_int] :
% 0.34/0.56 ( ( finite_finite_int @ A )
% 0.34/0.56 => ( ( A != bot_bot_set_int )
% 0.34/0.56 => ( ! [X3: int,Y2: int] : ( member_int @ ( sup_sup_int @ X3 @ Y2 ) @ ( insert_int @ X3 @ ( insert_int @ Y2 @ bot_bot_set_int ) ) )
% 0.34/0.56 => ( member_int @ ( lattic1091506334969745077in_int @ A ) @ A ) ) ) ) ).
% 0.34/0.56
% 0.34/0.56 % Sup_fin.closed
% 0.34/0.56 thf(fact_2223_Sup__fin_Oclosed,axiom,
% 0.34/0.56 ! [A: set_Pr4532377907799695533_nat_o] :
% 0.34/0.56 ( ( finite3252695134891459830_nat_o @ A )
% 0.34/0.56 => ( ( A != bot_bo7824918357723582233_nat_o )
% 0.34/0.56 => ( ! [X3: produc3658429121746597890et_nat > $o,Y2: produc3658429121746597890et_nat > $o] : ( member6576561426505652726_nat_o @ ( sup_su5453871518329203617_nat_o @ X3 @ Y2 ) @ ( insert5175938949040314269_nat_o @ X3 @ ( insert5175938949040314269_nat_o @ Y2 @ bot_bo7824918357723582233_nat_o ) ) )
% 0.34/0.56 => ( member6576561426505652726_nat_o @ ( lattic7320199455484906628_nat_o @ A ) @ A ) ) ) ) ).
% 0.34/0.56
% 0.34/0.56 % Sup_fin.closed
% 0.34/0.56 thf(fact_2224_Sup__fin_Oinsert__not__elem,axiom,
% 0.34/0.56 ! [A: set_set_nat,X: set_nat] :
% 0.34/0.56 ( ( finite1152437895449049373et_nat @ A )
% 0.34/0.56 => ( ~ ( member_set_nat @ X @ A )
% 0.34/0.56 => ( ( A != bot_bot_set_set_nat )
% 0.34/0.56 => ( ( lattic3835124923745554447et_nat @ ( insert_set_nat @ X @ A ) )
% 0.34/0.56 = ( sup_sup_set_nat @ X @ ( lattic3835124923745554447et_nat @ A ) ) ) ) ) ) ).
% 0.34/0.56
% 0.34/0.56 % Sup_fin.insert_not_elem
% 0.34/0.56 thf(fact_2225_Sup__fin_Oinsert__not__elem,axiom,
% 0.34/0.56 ! [A: set_assn,X: assn] :
% 0.34/0.56 ( ( finite_finite_assn @ A )
% 0.34/0.56 => ( ~ ( member_assn @ X @ A )
% 0.34/0.56 => ( ( A != bot_bot_set_assn )
% 0.34/0.56 => ( ( lattic2150320897289308081n_assn @ ( insert_assn @ X @ A ) )
% 0.34/0.56 = ( sup_sup_assn @ X @ ( lattic2150320897289308081n_assn @ A ) ) ) ) ) ) ).
% 0.34/0.56
% 0.34/0.56 % Sup_fin.insert_not_elem
% 0.34/0.56 thf(fact_2226_Sup__fin_Oinsert__not__elem,axiom,
% 0.34/0.56 ! [A: set_Product_unit,X: product_unit] :
% 0.34/0.56 ( ( finite4290736615968046902t_unit @ A )
% 0.34/0.56 => ( ~ ( member_Product_unit @ X @ A )
% 0.34/0.56 => ( ( A != bot_bo3957492148770167129t_unit )
% 0.34/0.56 => ( ( lattic5294303975357428420t_unit @ ( insert_Product_unit @ X @ A ) )
% 0.34/0.56 = ( sup_sup_Product_unit @ X @ ( lattic5294303975357428420t_unit @ A ) ) ) ) ) ) ).
% 0.34/0.56
% 0.34/0.56 % Sup_fin.insert_not_elem
% 0.34/0.56 thf(fact_2227_Sup__fin_Oinsert__not__elem,axiom,
% 0.34/0.56 ! [A: set_o,X: $o] :
% 0.34/0.56 ( ( finite_finite_o @ A )
% 0.34/0.56 => ( ~ ( member_o @ X @ A )
% 0.34/0.56 => ( ( A != bot_bot_set_o )
% 0.34/0.56 => ( ( lattic1508158080041050831_fin_o @ ( insert_o @ X @ A ) )
% 0.34/0.56 = ( sup_sup_o @ X @ ( lattic1508158080041050831_fin_o @ A ) ) ) ) ) ) ).
% 0.34/0.56
% 0.34/0.56 % Sup_fin.insert_not_elem
% 0.34/0.56 thf(fact_2228_Sup__fin_Oinsert__not__elem,axiom,
% 0.34/0.56 ! [A: set_nat,X: nat] :
% 0.34/0.56 ( ( finite_finite_nat @ A )
% 0.34/0.56 => ( ~ ( member_nat @ X @ A )
% 0.34/0.56 => ( ( A != bot_bot_set_nat )
% 0.34/0.56 => ( ( lattic1093996805478795353in_nat @ ( insert_nat @ X @ A ) )
% 0.34/0.56 = ( sup_sup_nat @ X @ ( lattic1093996805478795353in_nat @ A ) ) ) ) ) ) ).
% 0.34/0.56
% 0.34/0.56 % Sup_fin.insert_not_elem
% 0.34/0.56 thf(fact_2229_Sup__fin_Oinsert__not__elem,axiom,
% 0.34/0.56 ! [A: set_int,X: int] :
% 0.34/0.56 ( ( finite_finite_int @ A )
% 0.34/0.56 => ( ~ ( member_int @ X @ A )
% 0.34/0.56 => ( ( A != bot_bot_set_int )
% 0.34/0.56 => ( ( lattic1091506334969745077in_int @ ( insert_int @ X @ A ) )
% 0.34/0.56 = ( sup_sup_int @ X @ ( lattic1091506334969745077in_int @ A ) ) ) ) ) ) ).
% 0.34/0.56
% 0.34/0.56 % Sup_fin.insert_not_elem
% 0.34/0.56 thf(fact_2230_Sup__fin_Oinsert__not__elem,axiom,
% 0.34/0.56 ! [A: set_Pr4532377907799695533_nat_o,X: produc3658429121746597890et_nat > $o] :
% 0.34/0.56 ( ( finite3252695134891459830_nat_o @ A )
% 0.34/0.56 => ( ~ ( member6576561426505652726_nat_o @ X @ A )
% 0.34/0.56 => ( ( A != bot_bo7824918357723582233_nat_o )
% 0.34/0.56 => ( ( lattic7320199455484906628_nat_o @ ( insert5175938949040314269_nat_o @ X @ A ) )
% 0.34/0.56 = ( sup_su5453871518329203617_nat_o @ X @ ( lattic7320199455484906628_nat_o @ A ) ) ) ) ) ) ).
% 0.34/0.56
% 0.34/0.56 % Sup_fin.insert_not_elem
% 0.34/0.56 thf(fact_2231_Inf__fin_Ounion,axiom,
% 0.34/0.56 ! [A: set_set_nat,B4: set_set_nat] :
% 0.34/0.56 ( ( finite1152437895449049373et_nat @ A )
% 0.34/0.56 => ( ( A != bot_bot_set_set_nat )
% 0.34/0.56 => ( ( finite1152437895449049373et_nat @ B4 )
% 0.34/0.56 => ( ( B4 != bot_bot_set_set_nat )
% 0.34/0.56 => ( ( lattic3014633134055518761et_nat @ ( sup_sup_set_set_nat @ A @ B4 ) )
% 0.34/0.56 = ( inf_inf_set_nat @ ( lattic3014633134055518761et_nat @ A ) @ ( lattic3014633134055518761et_nat @ B4 ) ) ) ) ) ) ) ).
% 0.34/0.56
% 0.34/0.56 % Inf_fin.union
% 0.34/0.56 thf(fact_2232_Inf__fin_Ounion,axiom,
% 0.34/0.56 ! [A: set_assn,B4: set_assn] :
% 0.34/0.56 ( ( finite_finite_assn @ A )
% 0.34/0.56 => ( ( A != bot_bot_set_assn )
% 0.34/0.56 => ( ( finite_finite_assn @ B4 )
% 0.34/0.56 => ( ( B4 != bot_bot_set_assn )
% 0.34/0.56 => ( ( lattic47131356835913163n_assn @ ( sup_sup_set_assn @ A @ B4 ) )
% 0.34/0.56 = ( inf_inf_assn @ ( lattic47131356835913163n_assn @ A ) @ ( lattic47131356835913163n_assn @ B4 ) ) ) ) ) ) ) ).
% 0.34/0.56
% 0.34/0.56 % Inf_fin.union
% 0.34/0.56 thf(fact_2233_Inf__fin_Ounion,axiom,
% 0.34/0.56 ! [A: set_Product_unit,B4: set_Product_unit] :
% 0.34/0.56 ( ( finite4290736615968046902t_unit @ A )
% 0.34/0.56 => ( ( A != bot_bo3957492148770167129t_unit )
% 0.34/0.56 => ( ( finite4290736615968046902t_unit @ B4 )
% 0.34/0.56 => ( ( B4 != bot_bo3957492148770167129t_unit )
% 0.34/0.56 => ( ( lattic1263872656861969706t_unit @ ( sup_su793286257634532545t_unit @ A @ B4 ) )
% 0.34/0.56 = ( inf_inf_Product_unit @ ( lattic1263872656861969706t_unit @ A ) @ ( lattic1263872656861969706t_unit @ B4 ) ) ) ) ) ) ) ).
% 0.34/0.56
% 0.34/0.56 % Inf_fin.union
% 0.34/0.56 thf(fact_2234_Inf__fin_Ounion,axiom,
% 0.34/0.56 ! [A: set_o,B4: set_o] :
% 0.34/0.56 ( ( finite_finite_o @ A )
% 0.34/0.56 => ( ( A != bot_bot_set_o )
% 0.34/0.56 => ( ( finite_finite_o @ B4 )
% 0.34/0.56 => ( ( B4 != bot_bot_set_o )
% 0.34/0.56 => ( ( lattic4107685809792843317_fin_o @ ( sup_sup_set_o @ A @ B4 ) )
% 0.34/0.56 = ( inf_inf_o @ ( lattic4107685809792843317_fin_o @ A ) @ ( lattic4107685809792843317_fin_o @ B4 ) ) ) ) ) ) ) ).
% 0.34/0.56
% 0.34/0.56 % Inf_fin.union
% 0.34/0.56 thf(fact_2235_Inf__fin_Ounion,axiom,
% 0.34/0.56 ! [A: set_nat,B4: set_nat] :
% 0.34/0.56 ( ( finite_finite_nat @ A )
% 0.34/0.56 => ( ( A != bot_bot_set_nat )
% 0.34/0.56 => ( ( finite_finite_nat @ B4 )
% 0.34/0.56 => ( ( B4 != bot_bot_set_nat )
% 0.34/0.56 => ( ( lattic5238388535129920115in_nat @ ( sup_sup_set_nat @ A @ B4 ) )
% 0.34/0.56 = ( inf_inf_nat @ ( lattic5238388535129920115in_nat @ A ) @ ( lattic5238388535129920115in_nat @ B4 ) ) ) ) ) ) ) ).
% 0.34/0.56
% 0.34/0.56 % Inf_fin.union
% 0.34/0.56 thf(fact_2236_Inf__fin_Ounion,axiom,
% 0.34/0.56 ! [A: set_int,B4: set_int] :
% 0.34/0.56 ( ( finite_finite_int @ A )
% 0.34/0.56 => ( ( A != bot_bot_set_int )
% 0.34/0.56 => ( ( finite_finite_int @ B4 )
% 0.34/0.56 => ( ( B4 != bot_bot_set_int )
% 0.34/0.56 => ( ( lattic5235898064620869839in_int @ ( sup_sup_set_int @ A @ B4 ) )
% 0.34/0.56 = ( inf_inf_int @ ( lattic5235898064620869839in_int @ A ) @ ( lattic5235898064620869839in_int @ B4 ) ) ) ) ) ) ) ).
% 0.34/0.56
% 0.34/0.56 % Inf_fin.union
% 0.34/0.56 thf(fact_2237_Inf__fin_Ounion,axiom,
% 0.34/0.56 ! [A: set_Pr4532377907799695533_nat_o,B4: set_Pr4532377907799695533_nat_o] :
% 0.34/0.56 ( ( finite3252695134891459830_nat_o @ A )
% 0.34/0.56 => ( ( A != bot_bo7824918357723582233_nat_o )
% 0.34/0.56 => ( ( finite3252695134891459830_nat_o @ B4 )
% 0.34/0.56 => ( ( B4 != bot_bo7824918357723582233_nat_o )
% 0.34/0.56 => ( ( lattic956194824204696298_nat_o @ ( sup_su5209123915105501825_nat_o @ A @ B4 ) )
% 0.34/0.56 = ( inf_in1318976480646536635_nat_o @ ( lattic956194824204696298_nat_o @ A ) @ ( lattic956194824204696298_nat_o @ B4 ) ) ) ) ) ) ) ).
% 0.34/0.56
% 0.34/0.56 % Inf_fin.union
% 0.34/0.56 thf(fact_2238_Sup__fin_Ounion,axiom,
% 0.34/0.56 ! [A: set_set_nat,B4: set_set_nat] :
% 0.34/0.56 ( ( finite1152437895449049373et_nat @ A )
% 0.34/0.56 => ( ( A != bot_bot_set_set_nat )
% 0.34/0.56 => ( ( finite1152437895449049373et_nat @ B4 )
% 0.34/0.56 => ( ( B4 != bot_bot_set_set_nat )
% 0.34/0.56 => ( ( lattic3835124923745554447et_nat @ ( sup_sup_set_set_nat @ A @ B4 ) )
% 0.34/0.56 = ( sup_sup_set_nat @ ( lattic3835124923745554447et_nat @ A ) @ ( lattic3835124923745554447et_nat @ B4 ) ) ) ) ) ) ) ).
% 0.34/0.56
% 0.34/0.56 % Sup_fin.union
% 0.34/0.56 thf(fact_2239_Sup__fin_Ounion,axiom,
% 0.34/0.56 ! [A: set_assn,B4: set_assn] :
% 0.34/0.56 ( ( finite_finite_assn @ A )
% 0.34/0.56 => ( ( A != bot_bot_set_assn )
% 0.34/0.56 => ( ( finite_finite_assn @ B4 )
% 0.34/0.56 => ( ( B4 != bot_bot_set_assn )
% 0.34/0.56 => ( ( lattic2150320897289308081n_assn @ ( sup_sup_set_assn @ A @ B4 ) )
% 0.34/0.56 = ( sup_sup_assn @ ( lattic2150320897289308081n_assn @ A ) @ ( lattic2150320897289308081n_assn @ B4 ) ) ) ) ) ) ) ).
% 0.34/0.56
% 0.34/0.56 % Sup_fin.union
% 0.34/0.56 thf(fact_2240_Sup__fin_Ounion,axiom,
% 0.34/0.56 ! [A: set_Product_unit,B4: set_Product_unit] :
% 0.34/0.56 ( ( finite4290736615968046902t_unit @ A )
% 0.34/0.56 => ( ( A != bot_bo3957492148770167129t_unit )
% 0.34/0.56 => ( ( finite4290736615968046902t_unit @ B4 )
% 0.34/0.56 => ( ( B4 != bot_bo3957492148770167129t_unit )
% 0.34/0.56 => ( ( lattic5294303975357428420t_unit @ ( sup_su793286257634532545t_unit @ A @ B4 ) )
% 0.34/0.56 = ( sup_sup_Product_unit @ ( lattic5294303975357428420t_unit @ A ) @ ( lattic5294303975357428420t_unit @ B4 ) ) ) ) ) ) ) ).
% 0.34/0.56
% 0.34/0.56 % Sup_fin.union
% 0.34/0.56 thf(fact_2241_Sup__fin_Ounion,axiom,
% 0.34/0.56 ! [A: set_o,B4: set_o] :
% 0.34/0.56 ( ( finite_finite_o @ A )
% 0.34/0.56 => ( ( A != bot_bot_set_o )
% 0.34/0.56 => ( ( finite_finite_o @ B4 )
% 0.34/0.56 => ( ( B4 != bot_bot_set_o )
% 0.34/0.56 => ( ( lattic1508158080041050831_fin_o @ ( sup_sup_set_o @ A @ B4 ) )
% 0.34/0.56 = ( sup_sup_o @ ( lattic1508158080041050831_fin_o @ A ) @ ( lattic1508158080041050831_fin_o @ B4 ) ) ) ) ) ) ) ).
% 0.34/0.56
% 0.34/0.56 % Sup_fin.union
% 0.34/0.56 thf(fact_2242_Sup__fin_Ounion,axiom,
% 0.34/0.56 ! [A: set_nat,B4: set_nat] :
% 0.34/0.56 ( ( finite_finite_nat @ A )
% 0.34/0.56 => ( ( A != bot_bot_set_nat )
% 0.34/0.56 => ( ( finite_finite_nat @ B4 )
% 0.34/0.56 => ( ( B4 != bot_bot_set_nat )
% 0.34/0.56 => ( ( lattic1093996805478795353in_nat @ ( sup_sup_set_nat @ A @ B4 ) )
% 0.34/0.56 = ( sup_sup_nat @ ( lattic1093996805478795353in_nat @ A ) @ ( lattic1093996805478795353in_nat @ B4 ) ) ) ) ) ) ) ).
% 0.34/0.56
% 0.34/0.56 % Sup_fin.union
% 0.34/0.56 thf(fact_2243_Sup__fin_Ounion,axiom,
% 0.34/0.56 ! [A: set_int,B4: set_int] :
% 0.34/0.56 ( ( finite_finite_int @ A )
% 0.34/0.56 => ( ( A != bot_bot_set_int )
% 0.34/0.56 => ( ( finite_finite_int @ B4 )
% 0.34/0.56 => ( ( B4 != bot_bot_set_int )
% 0.34/0.56 => ( ( lattic1091506334969745077in_int @ ( sup_sup_set_int @ A @ B4 ) )
% 0.34/0.56 = ( sup_sup_int @ ( lattic1091506334969745077in_int @ A ) @ ( lattic1091506334969745077in_int @ B4 ) ) ) ) ) ) ) ).
% 0.34/0.56
% 0.34/0.56 % Sup_fin.union
% 0.34/0.56 thf(fact_2244_Sup__fin_Ounion,axiom,
% 0.34/0.56 ! [A: set_Pr4532377907799695533_nat_o,B4: set_Pr4532377907799695533_nat_o] :
% 0.34/0.56 ( ( finite3252695134891459830_nat_o @ A )
% 0.34/0.56 => ( ( A != bot_bo7824918357723582233_nat_o )
% 0.34/0.56 => ( ( finite3252695134891459830_nat_o @ B4 )
% 0.34/0.56 => ( ( B4 != bot_bo7824918357723582233_nat_o )
% 0.34/0.56 => ( ( lattic7320199455484906628_nat_o @ ( sup_su5209123915105501825_nat_o @ A @ B4 ) )
% 0.34/0.56 = ( sup_su5453871518329203617_nat_o @ ( lattic7320199455484906628_nat_o @ A ) @ ( lattic7320199455484906628_nat_o @ B4 ) ) ) ) ) ) ) ).
% 0.34/0.56
% 0.34/0.56 % Sup_fin.union
% 0.34/0.56 thf(fact_2245_refl__on__reflcl__Image,axiom,
% 0.34/0.56 ! [B4: set_nat,A: set_Pr1261947904930325089at_nat,C5: set_nat] :
% 0.34/0.56 ( ( refl_on_nat @ B4 @ A )
% 0.34/0.56 => ( ( ord_less_eq_set_nat @ C5 @ B4 )
% 0.34/0.56 => ( ( image_nat_nat @ ( sup_su6327502436637775413at_nat @ A @ id_nat ) @ C5 )
% 0.34/0.56 = ( image_nat_nat @ A @ C5 ) ) ) ) ).
% 0.34/0.56
% 0.34/0.56 % refl_on_reflcl_Image
% 0.34/0.56 thf(fact_2246_E__closed__restr__reach__cases,axiom,
% 0.34/0.56 ! [U: produc3658429121746597890et_nat > $o,V: produc3658429121746597890et_nat > $o,E: set_Pr2161125870931222855_nat_o,R3: set_Pr4532377907799695533_nat_o] :
% 0.34/0.56 ( ( member8781333585448626064_nat_o @ ( produc7368190662567826135_nat_o @ U @ V ) @ ( transi712980445759334096_nat_o @ E ) )
% 0.34/0.56 => ( ( ord_le2965882846123202637_nat_o @ ( image_2232984160439939859_nat_o @ E @ R3 ) @ R3 )
% 0.34/0.56 => ( ~ ( member6576561426505652726_nat_o @ V @ R3 )
% 0.34/0.56 => ~ ( ~ ( member6576561426505652726_nat_o @ U @ R3 )
% 0.34/0.56 => ~ ( member8781333585448626064_nat_o @ ( produc7368190662567826135_nat_o @ U @ V ) @ ( transi712980445759334096_nat_o @ ( rel_re3331361160397388719_nat_o @ E @ R3 ) ) ) ) ) ) ) ).
% 0.34/0.56
% 0.34/0.56 % E_closed_restr_reach_cases
% 0.34/0.56 thf(fact_2247_E__closed__restr__reach__cases,axiom,
% 0.34/0.56 ! [U: $o,V: $o,E: set_Product_prod_o_o,R3: set_o] :
% 0.34/0.56 ( ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ U @ V ) @ ( transitive_rtrancl_o @ E ) )
% 0.34/0.56 => ( ( ord_less_eq_set_o @ ( image_o_o @ E @ R3 ) @ R3 )
% 0.34/0.56 => ( ~ ( member_o @ V @ R3 )
% 0.34/0.56 => ~ ( ~ ( member_o @ U @ R3 )
% 0.34/0.56 => ~ ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ U @ V ) @ ( transitive_rtrancl_o @ ( rel_restrict_o @ E @ R3 ) ) ) ) ) ) ) ).
% 0.34/0.56
% 0.34/0.56 % E_closed_restr_reach_cases
% 0.34/0.56 thf(fact_2248_E__closed__restr__reach__cases,axiom,
% 0.34/0.56 ! [U: product_unit,V: product_unit,E: set_Pr5094982260447487303t_unit,R3: set_Product_unit] :
% 0.34/0.56 ( ( member7821138191923116944t_unit @ ( produc5692694829384537815t_unit @ U @ V ) @ ( transi6481887417741755664t_unit @ E ) )
% 0.34/0.56 => ( ( ord_le3507040750410214029t_unit @ ( image_8431784415008483091t_unit @ E @ R3 ) @ R3 )
% 0.34/0.56 => ( ~ ( member_Product_unit @ V @ R3 )
% 0.34/0.56 => ~ ( ~ ( member_Product_unit @ U @ R3 )
% 0.34/0.56 => ~ ( member7821138191923116944t_unit @ ( produc5692694829384537815t_unit @ U @ V ) @ ( transi6481887417741755664t_unit @ ( rel_re9207996470947987439t_unit @ E @ R3 ) ) ) ) ) ) ) ).
% 0.34/0.56
% 0.34/0.56 % E_closed_restr_reach_cases
% 0.34/0.56 thf(fact_2249_E__closed__restr__reach__cases,axiom,
% 0.34/0.56 ! [U: int,V: int,E: set_Pr958786334691620121nt_int,R3: set_int] :
% 0.34/0.56 ( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ U @ V ) @ ( transi2902850859426252137cl_int @ E ) )
% 0.34/0.56 => ( ( ord_less_eq_set_int @ ( image_int_int @ E @ R3 ) @ R3 )
% 0.34/0.56 => ( ~ ( member_int @ V @ R3 )
% 0.34/0.56 => ~ ( ~ ( member_int @ U @ R3 )
% 0.34/0.56 => ~ ( member5262025264175285858nt_int @ ( product_Pair_int_int @ U @ V ) @ ( transi2902850859426252137cl_int @ ( rel_restrict_int @ E @ R3 ) ) ) ) ) ) ) ).
% 0.34/0.56
% 0.34/0.56 % E_closed_restr_reach_cases
% 0.34/0.56 thf(fact_2250_E__closed__restr__reach__cases,axiom,
% 0.34/0.56 ! [U: nat,V: nat,E: set_Pr1261947904930325089at_nat,R3: set_nat] :
% 0.34/0.56 ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ U @ V ) @ ( transi2905341329935302413cl_nat @ E ) )
% 0.34/0.56 => ( ( ord_less_eq_set_nat @ ( image_nat_nat @ E @ R3 ) @ R3 )
% 0.34/0.56 => ( ~ ( member_nat @ V @ R3 )
% 0.34/0.56 => ~ ( ~ ( member_nat @ U @ R3 )
% 0.34/0.56 => ~ ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ U @ V ) @ ( transi2905341329935302413cl_nat @ ( rel_restrict_nat @ E @ R3 ) ) ) ) ) ) ) ).
% 0.34/0.56
% 0.34/0.56 % E_closed_restr_reach_cases
% 0.34/0.56 thf(fact_2251_max__ext__compat,axiom,
% 0.34/0.56 ! [R3: set_Pr5094982260447487303t_unit,S: set_Pr5094982260447487303t_unit] :
% 0.34/0.56 ( ( ord_le930853871177014503t_unit @ ( relcom848357998638694028t_unit @ R3 @ S ) @ R3 )
% 0.34/0.56 => ( ord_le8072594200992201959t_unit @ ( relcom7108816970757792620t_unit @ ( max_ext_Product_unit @ R3 ) @ ( sup_su3573767163274163995t_unit @ ( max_ext_Product_unit @ S ) @ ( insert6793119152790066487t_unit @ ( produc7269801343787132631t_unit @ bot_bo3957492148770167129t_unit @ bot_bo3957492148770167129t_unit ) @ bot_bo5113747506523071411t_unit ) ) ) @ ( max_ext_Product_unit @ R3 ) ) ) ).
% 0.34/0.56
% 0.34/0.56 % max_ext_compat
% 0.34/0.56 thf(fact_2252_max__ext__compat,axiom,
% 0.34/0.56 ! [R3: set_Product_prod_o_o,S: set_Product_prod_o_o] :
% 0.34/0.56 ( ( ord_le192195778054999463od_o_o @ ( relcomp_o_o_o @ R3 @ S ) @ R3 )
% 0.34/0.56 => ( ord_le2885120122233735591_set_o @ ( relcom7534720040753365047_set_o @ ( max_ext_o @ R3 ) @ ( sup_su7766833992073560027_set_o @ ( max_ext_o @ S ) @ ( insert1494450971503307511_set_o @ ( produc5838405689764958487_set_o @ bot_bot_set_o @ bot_bot_set_o ) @ bot_bo8143257439550059635_set_o ) ) ) @ ( max_ext_o @ R3 ) ) ) ).
% 0.34/0.56
% 0.34/0.56 % max_ext_compat
% 0.34/0.56 thf(fact_2253_max__ext__compat,axiom,
% 0.34/0.56 ! [R3: set_Pr1261947904930325089at_nat,S: set_Pr1261947904930325089at_nat] :
% 0.34/0.56 ( ( ord_le3146513528884898305at_nat @ ( relcomp_nat_nat_nat @ R3 @ S ) @ R3 )
% 0.34/0.56 => ( ord_le4402255868550298733et_nat @ ( relcom236561585145426837et_nat @ ( max_ext_nat @ R3 ) @ ( sup_su4251893418135750817et_nat @ ( max_ext_nat @ S ) @ ( insert3810226134351308605et_nat @ ( produc4532415448927165861et_nat @ bot_bot_set_nat @ bot_bot_set_nat ) @ bot_bo4952753636535259449et_nat ) ) ) @ ( max_ext_nat @ R3 ) ) ) ).
% 0.34/0.56
% 0.34/0.56 % max_ext_compat
% 0.34/0.56 thf(fact_2254_max__ext__compat,axiom,
% 0.34/0.56 ! [R3: set_Pr958786334691620121nt_int,S: set_Pr958786334691620121nt_int] :
% 0.34/0.56 ( ( ord_le2843351958646193337nt_int @ ( relcomp_int_int_int @ R3 @ S ) @ R3 )
% 0.34/0.56 => ( ord_le1436784781161120549et_int @ ( relcom4034950705519254697et_int @ ( max_ext_int @ R3 ) @ ( sup_su1286422330746572633et_int @ ( max_ext_int @ S ) @ ( insert7328952640415156725et_int @ ( produc6363374080413544029et_int @ bot_bot_set_int @ bot_bot_set_int ) @ bot_bo1987282549146081265et_int ) ) ) @ ( max_ext_int @ R3 ) ) ) ).
% 0.34/0.56
% 0.34/0.56 % max_ext_compat
% 0.34/0.56 thf(fact_2255_max__ext__compat,axiom,
% 0.34/0.56 ! [R3: set_Pr2161125870931222855_nat_o,S: set_Pr2161125870931222855_nat_o] :
% 0.34/0.56 ( ( ord_le6659818918159358183_nat_o @ ( relcom6251035627269349452_nat_o @ R3 @ S ) @ R3 )
% 0.34/0.56 => ( ord_le8001614231985977575_nat_o @ ( relcom7236094467932586796_nat_o @ ( max_ex5265246208886859912_nat_o @ R3 ) @ ( sup_su4863159966000489243_nat_o @ ( max_ex5265246208886859912_nat_o @ S ) @ ( insert3244108804717745463_nat_o @ ( produc5149316574908490455_nat_o @ bot_bo7824918357723582233_nat_o @ bot_bo7824918357723582233_nat_o ) @ bot_bo7535334947429593011_nat_o ) ) ) @ ( max_ex5265246208886859912_nat_o @ R3 ) ) ) ).
% 0.34/0.56
% 0.34/0.56 % max_ext_compat
% 0.34/0.56 thf(fact_2256_BNF__Greatest__Fixpoint_OIdD,axiom,
% 0.34/0.56 ! [A2: nat,B: nat] :
% 0.34/0.56 ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A2 @ B ) @ id_nat )
% 0.34/0.56 => ( A2 = B ) ) ).
% 0.34/0.56
% 0.34/0.56 % BNF_Greatest_Fixpoint.IdD
% 0.34/0.56 thf(fact_2257_BNF__Greatest__Fixpoint_OIdD,axiom,
% 0.34/0.56 ! [A2: int,B: int] :
% 0.34/0.56 ( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ A2 @ B ) @ id_int )
% 0.34/0.56 => ( A2 = B ) ) ).
% 0.34/0.56
% 0.34/0.56 % BNF_Greatest_Fixpoint.IdD
% 0.34/0.56 thf(fact_2258_Sup__fin_Oremove,axiom,
% 0.34/0.56 ! [A: set_set_nat,X: set_nat] :
% 0.34/0.56 ( ( finite1152437895449049373et_nat @ A )
% 0.34/0.56 => ( ( member_set_nat @ X @ A )
% 0.34/0.56 => ( ( ( ( minus_2163939370556025621et_nat @ A @ ( insert_set_nat @ X @ bot_bot_set_set_nat ) )
% 0.34/0.56 = bot_bot_set_set_nat )
% 0.34/0.56 => ( ( lattic3835124923745554447et_nat @ A )
% 0.34/0.56 = X ) )
% 0.34/0.56 & ( ( ( minus_2163939370556025621et_nat @ A @ ( insert_set_nat @ X @ bot_bot_set_set_nat ) )
% 0.34/0.56 != bot_bot_set_set_nat )
% 0.34/0.56 => ( ( lattic3835124923745554447et_nat @ A )
% 0.34/0.56 = ( sup_sup_set_nat @ X @ ( lattic3835124923745554447et_nat @ ( minus_2163939370556025621et_nat @ A @ ( insert_set_nat @ X @ bot_bot_set_set_nat ) ) ) ) ) ) ) ) ) ).
% 0.34/0.56
% 0.34/0.56 % Sup_fin.remove
% 0.34/0.56 thf(fact_2259_Sup__fin_Oremove,axiom,
% 0.34/0.56 ! [A: set_assn,X: assn] :
% 0.34/0.56 ( ( finite_finite_assn @ A )
% 0.34/0.56 => ( ( member_assn @ X @ A )
% 0.34/0.56 => ( ( ( ( minus_minus_set_assn @ A @ ( insert_assn @ X @ bot_bot_set_assn ) )
% 0.34/0.56 = bot_bot_set_assn )
% 0.34/0.56 => ( ( lattic2150320897289308081n_assn @ A )
% 0.34/0.56 = X ) )
% 0.34/0.56 & ( ( ( minus_minus_set_assn @ A @ ( insert_assn @ X @ bot_bot_set_assn ) )
% 0.34/0.56 != bot_bot_set_assn )
% 0.34/0.56 => ( ( lattic2150320897289308081n_assn @ A )
% 0.34/0.56 = ( sup_sup_assn @ X @ ( lattic2150320897289308081n_assn @ ( minus_minus_set_assn @ A @ ( insert_assn @ X @ bot_bot_set_assn ) ) ) ) ) ) ) ) ) ).
% 0.34/0.56
% 0.34/0.56 % Sup_fin.remove
% 0.34/0.56 thf(fact_2260_Sup__fin_Oremove,axiom,
% 0.34/0.56 ! [A: set_Product_unit,X: product_unit] :
% 0.34/0.56 ( ( finite4290736615968046902t_unit @ A )
% 0.34/0.56 => ( ( member_Product_unit @ X @ A )
% 0.34/0.56 => ( ( ( ( minus_6452836326544984404t_unit @ A @ ( insert_Product_unit @ X @ bot_bo3957492148770167129t_unit ) )
% 0.34/0.56 = bot_bo3957492148770167129t_unit )
% 0.34/0.56 => ( ( lattic5294303975357428420t_unit @ A )
% 0.34/0.56 = X ) )
% 0.34/0.56 & ( ( ( minus_6452836326544984404t_unit @ A @ ( insert_Product_unit @ X @ bot_bo3957492148770167129t_unit ) )
% 0.34/0.56 != bot_bo3957492148770167129t_unit )
% 0.34/0.56 => ( ( lattic5294303975357428420t_unit @ A )
% 0.34/0.56 = ( sup_sup_Product_unit @ X @ ( lattic5294303975357428420t_unit @ ( minus_6452836326544984404t_unit @ A @ ( insert_Product_unit @ X @ bot_bo3957492148770167129t_unit ) ) ) ) ) ) ) ) ) ).
% 0.34/0.56
% 0.34/0.56 % Sup_fin.remove
% 0.34/0.56 thf(fact_2261_Sup__fin_Oremove,axiom,
% 0.34/0.56 ! [A: set_o,X: $o] :
% 0.34/0.56 ( ( finite_finite_o @ A )
% 0.34/0.56 => ( ( member_o @ X @ A )
% 0.34/0.56 => ( ( lattic1508158080041050831_fin_o @ A )
% 0.34/0.56 = ( ( ( ( minus_minus_set_o @ A @ ( insert_o @ X @ bot_bot_set_o ) )
% 0.34/0.56 = bot_bot_set_o )
% 0.34/0.56 => X )
% 0.34/0.56 & ( ( ( minus_minus_set_o @ A @ ( insert_o @ X @ bot_bot_set_o ) )
% 0.34/0.56 != bot_bot_set_o )
% 0.34/0.56 => ( sup_sup_o @ X @ ( lattic1508158080041050831_fin_o @ ( minus_minus_set_o @ A @ ( insert_o @ X @ bot_bot_set_o ) ) ) ) ) ) ) ) ) ).
% 0.34/0.56
% 0.34/0.56 % Sup_fin.remove
% 0.34/0.56 thf(fact_2262_Sup__fin_Oremove,axiom,
% 0.34/0.56 ! [A: set_int,X: int] :
% 0.34/0.56 ( ( finite_finite_int @ A )
% 0.34/0.56 => ( ( member_int @ X @ A )
% 0.34/0.56 => ( ( ( ( minus_minus_set_int @ A @ ( insert_int @ X @ bot_bot_set_int ) )
% 0.34/0.56 = bot_bot_set_int )
% 0.34/0.56 => ( ( lattic1091506334969745077in_int @ A )
% 0.34/0.56 = X ) )
% 0.34/0.56 & ( ( ( minus_minus_set_int @ A @ ( insert_int @ X @ bot_bot_set_int ) )
% 0.34/0.56 != bot_bot_set_int )
% 0.34/0.56 => ( ( lattic1091506334969745077in_int @ A )
% 0.34/0.56 = ( sup_sup_int @ X @ ( lattic1091506334969745077in_int @ ( minus_minus_set_int @ A @ ( insert_int @ X @ bot_bot_set_int ) ) ) ) ) ) ) ) ) ).
% 0.34/0.56
% 0.34/0.56 % Sup_fin.remove
% 0.34/0.56 thf(fact_2263_Sup__fin_Oremove,axiom,
% 0.34/0.56 ! [A: set_Pr4532377907799695533_nat_o,X: produc3658429121746597890et_nat > $o] :
% 0.34/0.56 ( ( finite3252695134891459830_nat_o @ A )
% 0.34/0.56 => ( ( member6576561426505652726_nat_o @ X @ A )
% 0.34/0.56 => ( ( ( ( minus_1801376950450012436_nat_o @ A @ ( insert5175938949040314269_nat_o @ X @ bot_bo7824918357723582233_nat_o ) )
% 0.34/0.56 = bot_bo7824918357723582233_nat_o )
% 0.34/0.56 => ( ( lattic7320199455484906628_nat_o @ A )
% 0.34/0.56 = X ) )
% 0.34/0.56 & ( ( ( minus_1801376950450012436_nat_o @ A @ ( insert5175938949040314269_nat_o @ X @ bot_bo7824918357723582233_nat_o ) )
% 0.34/0.56 != bot_bo7824918357723582233_nat_o )
% 0.34/0.56 => ( ( lattic7320199455484906628_nat_o @ A )
% 0.34/0.56 = ( sup_su5453871518329203617_nat_o @ X @ ( lattic7320199455484906628_nat_o @ ( minus_1801376950450012436_nat_o @ A @ ( insert5175938949040314269_nat_o @ X @ bot_bo7824918357723582233_nat_o ) ) ) ) ) ) ) ) ) ).
% 0.34/0.56
% 0.34/0.56 % Sup_fin.remove
% 0.34/0.56 thf(fact_2264_Sup__fin_Oremove,axiom,
% 0.34/0.56 ! [A: set_nat,X: nat] :
% 0.34/0.56 ( ( finite_finite_nat @ A )
% 0.34/0.56 => ( ( member_nat @ X @ A )
% 0.34/0.56 => ( ( ( ( minus_minus_set_nat @ A @ ( insert_nat @ X @ bot_bot_set_nat ) )
% 0.34/0.56 = bot_bot_set_nat )
% 0.34/0.56 => ( ( lattic1093996805478795353in_nat @ A )
% 0.34/0.56 = X ) )
% 0.34/0.56 & ( ( ( minus_minus_set_nat @ A @ ( insert_nat @ X @ bot_bot_set_nat ) )
% 0.34/0.56 != bot_bot_set_nat )
% 0.34/0.56 => ( ( lattic1093996805478795353in_nat @ A )
% 0.34/0.56 = ( sup_sup_nat @ X @ ( lattic1093996805478795353in_nat @ ( minus_minus_set_nat @ A @ ( insert_nat @ X @ bot_bot_set_nat ) ) ) ) ) ) ) ) ) ).
% 0.34/0.56
% 0.34/0.56 % Sup_fin.remove
% 0.34/0.56 thf(fact_2265_Sup__fin_Oinsert__remove,axiom,
% 0.34/0.56 ! [A: set_set_nat,X: set_nat] :
% 0.34/0.56 ( ( finite1152437895449049373et_nat @ A )
% 0.34/0.56 => ( ( ( ( minus_2163939370556025621et_nat @ A @ ( insert_set_nat @ X @ bot_bot_set_set_nat ) )
% 0.34/0.56 = bot_bot_set_set_nat )
% 0.34/0.56 => ( ( lattic3835124923745554447et_nat @ ( insert_set_nat @ X @ A ) )
% 0.34/0.56 = X ) )
% 0.34/0.56 & ( ( ( minus_2163939370556025621et_nat @ A @ ( insert_set_nat @ X @ bot_bot_set_set_nat ) )
% 0.34/0.56 != bot_bot_set_set_nat )
% 0.34/0.56 => ( ( lattic3835124923745554447et_nat @ ( insert_set_nat @ X @ A ) )
% 0.34/0.56 = ( sup_sup_set_nat @ X @ ( lattic3835124923745554447et_nat @ ( minus_2163939370556025621et_nat @ A @ ( insert_set_nat @ X @ bot_bot_set_set_nat ) ) ) ) ) ) ) ) ).
% 0.34/0.56
% 0.34/0.56 % Sup_fin.insert_remove
% 0.34/0.56 thf(fact_2266_Sup__fin_Oinsert__remove,axiom,
% 0.34/0.56 ! [A: set_assn,X: assn] :
% 0.34/0.56 ( ( finite_finite_assn @ A )
% 0.34/0.56 => ( ( ( ( minus_minus_set_assn @ A @ ( insert_assn @ X @ bot_bot_set_assn ) )
% 0.34/0.56 = bot_bot_set_assn )
% 0.34/0.56 => ( ( lattic2150320897289308081n_assn @ ( insert_assn @ X @ A ) )
% 0.34/0.56 = X ) )
% 0.34/0.56 & ( ( ( minus_minus_set_assn @ A @ ( insert_assn @ X @ bot_bot_set_assn ) )
% 0.34/0.56 != bot_bot_set_assn )
% 0.34/0.56 => ( ( lattic2150320897289308081n_assn @ ( insert_assn @ X @ A ) )
% 0.34/0.56 = ( sup_sup_assn @ X @ ( lattic2150320897289308081n_assn @ ( minus_minus_set_assn @ A @ ( insert_assn @ X @ bot_bot_set_assn ) ) ) ) ) ) ) ) ).
% 0.34/0.56
% 0.34/0.56 % Sup_fin.insert_remove
% 0.34/0.56 thf(fact_2267_Sup__fin_Oinsert__remove,axiom,
% 0.34/0.56 ! [A: set_Product_unit,X: product_unit] :
% 0.34/0.56 ( ( finite4290736615968046902t_unit @ A )
% 0.34/0.56 => ( ( ( ( minus_6452836326544984404t_unit @ A @ ( insert_Product_unit @ X @ bot_bo3957492148770167129t_unit ) )
% 0.34/0.56 = bot_bo3957492148770167129t_unit )
% 0.34/0.56 => ( ( lattic5294303975357428420t_unit @ ( insert_Product_unit @ X @ A ) )
% 0.34/0.56 = X ) )
% 0.34/0.56 & ( ( ( minus_6452836326544984404t_unit @ A @ ( insert_Product_unit @ X @ bot_bo3957492148770167129t_unit ) )
% 0.34/0.56 != bot_bo3957492148770167129t_unit )
% 0.34/0.56 => ( ( lattic5294303975357428420t_unit @ ( insert_Product_unit @ X @ A ) )
% 0.34/0.56 = ( sup_sup_Product_unit @ X @ ( lattic5294303975357428420t_unit @ ( minus_6452836326544984404t_unit @ A @ ( insert_Product_unit @ X @ bot_bo3957492148770167129t_unit ) ) ) ) ) ) ) ) ).
% 0.34/0.56
% 0.34/0.56 % Sup_fin.insert_remove
% 0.34/0.56 thf(fact_2268_Sup__fin_Oinsert__remove,axiom,
% 0.34/0.56 ! [A: set_o,X: $o] :
% 0.34/0.56 ( ( finite_finite_o @ A )
% 0.34/0.56 => ( ( lattic1508158080041050831_fin_o @ ( insert_o @ X @ A ) )
% 0.34/0.56 = ( ( ( ( minus_minus_set_o @ A @ ( insert_o @ X @ bot_bot_set_o ) )
% 0.34/0.56 = bot_bot_set_o )
% 0.34/0.56 => X )
% 0.34/0.56 & ( ( ( minus_minus_set_o @ A @ ( insert_o @ X @ bot_bot_set_o ) )
% 0.34/0.56 != bot_bot_set_o )
% 0.34/0.56 => ( sup_sup_o @ X @ ( lattic1508158080041050831_fin_o @ ( minus_minus_set_o @ A @ ( insert_o @ X @ bot_bot_set_o ) ) ) ) ) ) ) ) ).
% 0.34/0.56
% 0.34/0.56 % Sup_fin.insert_remove
% 0.34/0.56 thf(fact_2269_Sup__fin_Oinsert__remove,axiom,
% 0.34/0.56 ! [A: set_int,X: int] :
% 0.34/0.56 ( ( finite_finite_int @ A )
% 0.34/0.56 => ( ( ( ( minus_minus_set_int @ A @ ( insert_int @ X @ bot_bot_set_int ) )
% 0.34/0.56 = bot_bot_set_int )
% 0.34/0.56 => ( ( lattic1091506334969745077in_int @ ( insert_int @ X @ A ) )
% 0.34/0.56 = X ) )
% 0.34/0.56 & ( ( ( minus_minus_set_int @ A @ ( insert_int @ X @ bot_bot_set_int ) )
% 0.34/0.56 != bot_bot_set_int )
% 0.34/0.56 => ( ( lattic1091506334969745077in_int @ ( insert_int @ X @ A ) )
% 0.34/0.56 = ( sup_sup_int @ X @ ( lattic1091506334969745077in_int @ ( minus_minus_set_int @ A @ ( insert_int @ X @ bot_bot_set_int ) ) ) ) ) ) ) ) ).
% 0.34/0.56
% 0.34/0.56 % Sup_fin.insert_remove
% 0.34/0.56 thf(fact_2270_Sup__fin_Oinsert__remove,axiom,
% 0.34/0.56 ! [A: set_Pr4532377907799695533_nat_o,X: produc3658429121746597890et_nat > $o] :
% 0.34/0.56 ( ( finite3252695134891459830_nat_o @ A )
% 0.34/0.56 => ( ( ( ( minus_1801376950450012436_nat_o @ A @ ( insert5175938949040314269_nat_o @ X @ bot_bo7824918357723582233_nat_o ) )
% 0.34/0.56 = bot_bo7824918357723582233_nat_o )
% 0.34/0.56 => ( ( lattic7320199455484906628_nat_o @ ( insert5175938949040314269_nat_o @ X @ A ) )
% 0.34/0.56 = X ) )
% 0.34/0.56 & ( ( ( minus_1801376950450012436_nat_o @ A @ ( insert5175938949040314269_nat_o @ X @ bot_bo7824918357723582233_nat_o ) )
% 0.34/0.56 != bot_bo7824918357723582233_nat_o )
% 0.34/0.56 => ( ( lattic7320199455484906628_nat_o @ ( insert5175938949040314269_nat_o @ X @ A ) )
% 0.34/0.56 = ( sup_su5453871518329203617_nat_o @ X @ ( lattic7320199455484906628_nat_o @ ( minus_1801376950450012436_nat_o @ A @ ( insert5175938949040314269_nat_o @ X @ bot_bo7824918357723582233_nat_o ) ) ) ) ) ) ) ) ).
% 0.34/0.56
% 0.34/0.56 % Sup_fin.insert_remove
% 0.34/0.56 thf(fact_2271_Sup__fin_Oinsert__remove,axiom,
% 0.34/0.56 ! [A: set_nat,X: nat] :
% 0.34/0.56 ( ( finite_finite_nat @ A )
% 0.34/0.56 => ( ( ( ( minus_minus_set_nat @ A @ ( insert_nat @ X @ bot_bot_set_nat ) )
% 0.34/0.56 = bot_bot_set_nat )
% 0.34/0.56 => ( ( lattic1093996805478795353in_nat @ ( insert_nat @ X @ A ) )
% 0.34/0.56 = X ) )
% 0.34/0.56 & ( ( ( minus_minus_set_nat @ A @ ( insert_nat @ X @ bot_bot_set_nat ) )
% 0.34/0.56 != bot_bot_set_nat )
% 0.34/0.56 => ( ( lattic1093996805478795353in_nat @ ( insert_nat @ X @ A ) )
% 0.34/0.56 = ( sup_sup_nat @ X @ ( lattic1093996805478795353in_nat @ ( minus_minus_set_nat @ A @ ( insert_nat @ X @ bot_bot_set_nat ) ) ) ) ) ) ) ) ).
% 0.34/0.56
% 0.34/0.56 % Sup_fin.insert_remove
% 0.34/0.56 thf(fact_2272_Inf__fin_Oremove,axiom,
% 0.34/0.56 ! [A: set_set_nat,X: set_nat] :
% 0.34/0.56 ( ( finite1152437895449049373et_nat @ A )
% 0.34/0.56 => ( ( member_set_nat @ X @ A )
% 0.34/0.56 => ( ( ( ( minus_2163939370556025621et_nat @ A @ ( insert_set_nat @ X @ bot_bot_set_set_nat ) )
% 0.34/0.56 = bot_bot_set_set_nat )
% 0.34/0.56 => ( ( lattic3014633134055518761et_nat @ A )
% 0.34/0.56 = X ) )
% 0.34/0.56 & ( ( ( minus_2163939370556025621et_nat @ A @ ( insert_set_nat @ X @ bot_bot_set_set_nat ) )
% 0.34/0.56 != bot_bot_set_set_nat )
% 0.34/0.56 => ( ( lattic3014633134055518761et_nat @ A )
% 0.34/0.56 = ( inf_inf_set_nat @ X @ ( lattic3014633134055518761et_nat @ ( minus_2163939370556025621et_nat @ A @ ( insert_set_nat @ X @ bot_bot_set_set_nat ) ) ) ) ) ) ) ) ) ).
% 0.34/0.56
% 0.34/0.56 % Inf_fin.remove
% 0.34/0.56 thf(fact_2273_Inf__fin_Oremove,axiom,
% 0.34/0.56 ! [A: set_assn,X: assn] :
% 0.34/0.56 ( ( finite_finite_assn @ A )
% 0.34/0.56 => ( ( member_assn @ X @ A )
% 0.34/0.56 => ( ( ( ( minus_minus_set_assn @ A @ ( insert_assn @ X @ bot_bot_set_assn ) )
% 0.34/0.56 = bot_bot_set_assn )
% 0.34/0.56 => ( ( lattic47131356835913163n_assn @ A )
% 0.34/0.56 = X ) )
% 0.34/0.56 & ( ( ( minus_minus_set_assn @ A @ ( insert_assn @ X @ bot_bot_set_assn ) )
% 0.34/0.56 != bot_bot_set_assn )
% 0.34/0.56 => ( ( lattic47131356835913163n_assn @ A )
% 0.34/0.56 = ( inf_inf_assn @ X @ ( lattic47131356835913163n_assn @ ( minus_minus_set_assn @ A @ ( insert_assn @ X @ bot_bot_set_assn ) ) ) ) ) ) ) ) ) ).
% 0.34/0.56
% 0.34/0.56 % Inf_fin.remove
% 0.34/0.56 thf(fact_2274_Inf__fin_Oremove,axiom,
% 0.34/0.56 ! [A: set_Product_unit,X: product_unit] :
% 0.34/0.56 ( ( finite4290736615968046902t_unit @ A )
% 0.34/0.56 => ( ( member_Product_unit @ X @ A )
% 0.34/0.56 => ( ( ( ( minus_6452836326544984404t_unit @ A @ ( insert_Product_unit @ X @ bot_bo3957492148770167129t_unit ) )
% 0.34/0.56 = bot_bo3957492148770167129t_unit )
% 0.34/0.56 => ( ( lattic1263872656861969706t_unit @ A )
% 0.34/0.56 = X ) )
% 0.34/0.56 & ( ( ( minus_6452836326544984404t_unit @ A @ ( insert_Product_unit @ X @ bot_bo3957492148770167129t_unit ) )
% 0.34/0.56 != bot_bo3957492148770167129t_unit )
% 0.34/0.56 => ( ( lattic1263872656861969706t_unit @ A )
% 0.34/0.56 = ( inf_inf_Product_unit @ X @ ( lattic1263872656861969706t_unit @ ( minus_6452836326544984404t_unit @ A @ ( insert_Product_unit @ X @ bot_bo3957492148770167129t_unit ) ) ) ) ) ) ) ) ) ).
% 0.34/0.56
% 0.34/0.56 % Inf_fin.remove
% 0.34/0.56 thf(fact_2275_Inf__fin_Oremove,axiom,
% 0.34/0.56 ! [A: set_o,X: $o] :
% 0.34/0.56 ( ( finite_finite_o @ A )
% 0.34/0.56 => ( ( member_o @ X @ A )
% 0.34/0.56 => ( ( lattic4107685809792843317_fin_o @ A )
% 0.34/0.56 = ( ( ( ( minus_minus_set_o @ A @ ( insert_o @ X @ bot_bot_set_o ) )
% 0.34/0.56 = bot_bot_set_o )
% 0.34/0.56 => X )
% 0.34/0.56 & ( ( ( minus_minus_set_o @ A @ ( insert_o @ X @ bot_bot_set_o ) )
% 0.34/0.56 != bot_bot_set_o )
% 0.34/0.56 => ( inf_inf_o @ X @ ( lattic4107685809792843317_fin_o @ ( minus_minus_set_o @ A @ ( insert_o @ X @ bot_bot_set_o ) ) ) ) ) ) ) ) ) ).
% 0.34/0.56
% 0.34/0.56 % Inf_fin.remove
% 0.34/0.56 thf(fact_2276_Inf__fin_Oremove,axiom,
% 0.34/0.56 ! [A: set_int,X: int] :
% 0.34/0.56 ( ( finite_finite_int @ A )
% 0.34/0.56 => ( ( member_int @ X @ A )
% 0.34/0.56 => ( ( ( ( minus_minus_set_int @ A @ ( insert_int @ X @ bot_bot_set_int ) )
% 0.34/0.56 = bot_bot_set_int )
% 0.34/0.56 => ( ( lattic5235898064620869839in_int @ A )
% 0.34/0.56 = X ) )
% 0.34/0.56 & ( ( ( minus_minus_set_int @ A @ ( insert_int @ X @ bot_bot_set_int ) )
% 0.34/0.56 != bot_bot_set_int )
% 0.34/0.56 => ( ( lattic5235898064620869839in_int @ A )
% 0.34/0.56 = ( inf_inf_int @ X @ ( lattic5235898064620869839in_int @ ( minus_minus_set_int @ A @ ( insert_int @ X @ bot_bot_set_int ) ) ) ) ) ) ) ) ) ).
% 0.34/0.56
% 0.34/0.56 % Inf_fin.remove
% 0.34/0.56 thf(fact_2277_Inf__fin_Oremove,axiom,
% 0.34/0.56 ! [A: set_Pr4532377907799695533_nat_o,X: produc3658429121746597890et_nat > $o] :
% 0.34/0.56 ( ( finite3252695134891459830_nat_o @ A )
% 0.34/0.56 => ( ( member6576561426505652726_nat_o @ X @ A )
% 0.34/0.56 => ( ( ( ( minus_1801376950450012436_nat_o @ A @ ( insert5175938949040314269_nat_o @ X @ bot_bo7824918357723582233_nat_o ) )
% 0.34/0.56 = bot_bo7824918357723582233_nat_o )
% 0.34/0.56 => ( ( lattic956194824204696298_nat_o @ A )
% 0.34/0.56 = X ) )
% 0.34/0.56 & ( ( ( minus_1801376950450012436_nat_o @ A @ ( insert5175938949040314269_nat_o @ X @ bot_bo7824918357723582233_nat_o ) )
% 0.34/0.56 != bot_bo7824918357723582233_nat_o )
% 0.34/0.56 => ( ( lattic956194824204696298_nat_o @ A )
% 0.34/0.56 = ( inf_in1318976480646536635_nat_o @ X @ ( lattic956194824204696298_nat_o @ ( minus_1801376950450012436_nat_o @ A @ ( insert5175938949040314269_nat_o @ X @ bot_bo7824918357723582233_nat_o ) ) ) ) ) ) ) ) ) ).
% 0.34/0.56
% 0.34/0.56 % Inf_fin.remove
% 0.34/0.56 thf(fact_2278_Inf__fin_Oremove,axiom,
% 0.34/0.56 ! [A: set_nat,X: nat] :
% 0.34/0.56 ( ( finite_finite_nat @ A )
% 0.34/0.56 => ( ( member_nat @ X @ A )
% 0.34/0.56 => ( ( ( ( minus_minus_set_nat @ A @ ( insert_nat @ X @ bot_bot_set_nat ) )
% 0.34/0.56 = bot_bot_set_nat )
% 0.34/0.56 => ( ( lattic5238388535129920115in_nat @ A )
% 0.34/0.56 = X ) )
% 0.34/0.56 & ( ( ( minus_minus_set_nat @ A @ ( insert_nat @ X @ bot_bot_set_nat ) )
% 0.34/0.56 != bot_bot_set_nat )
% 0.34/0.56 => ( ( lattic5238388535129920115in_nat @ A )
% 0.34/0.56 = ( inf_inf_nat @ X @ ( lattic5238388535129920115in_nat @ ( minus_minus_set_nat @ A @ ( insert_nat @ X @ bot_bot_set_nat ) ) ) ) ) ) ) ) ) ).
% 0.34/0.56
% 0.34/0.56 % Inf_fin.remove
% 0.34/0.56 thf(fact_2279_Inf__fin_Oinsert__remove,axiom,
% 0.34/0.56 ! [A: set_set_nat,X: set_nat] :
% 0.34/0.56 ( ( finite1152437895449049373et_nat @ A )
% 0.34/0.56 => ( ( ( ( minus_2163939370556025621et_nat @ A @ ( insert_set_nat @ X @ bot_bot_set_set_nat ) )
% 0.34/0.56 = bot_bot_set_set_nat )
% 0.34/0.56 => ( ( lattic3014633134055518761et_nat @ ( insert_set_nat @ X @ A ) )
% 0.34/0.56 = X ) )
% 0.34/0.56 & ( ( ( minus_2163939370556025621et_nat @ A @ ( insert_set_nat @ X @ bot_bot_set_set_nat ) )
% 0.34/0.56 != bot_bot_set_set_nat )
% 0.34/0.56 => ( ( lattic3014633134055518761et_nat @ ( insert_set_nat @ X @ A ) )
% 0.34/0.56 = ( inf_inf_set_nat @ X @ ( lattic3014633134055518761et_nat @ ( minus_2163939370556025621et_nat @ A @ ( insert_set_nat @ X @ bot_bot_set_set_nat ) ) ) ) ) ) ) ) ).
% 0.34/0.56
% 0.34/0.56 % Inf_fin.insert_remove
% 0.34/0.56 thf(fact_2280_Inf__fin_Oinsert__remove,axiom,
% 0.34/0.56 ! [A: set_assn,X: assn] :
% 0.34/0.56 ( ( finite_finite_assn @ A )
% 0.34/0.56 => ( ( ( ( minus_minus_set_assn @ A @ ( insert_assn @ X @ bot_bot_set_assn ) )
% 0.34/0.56 = bot_bot_set_assn )
% 0.34/0.56 => ( ( lattic47131356835913163n_assn @ ( insert_assn @ X @ A ) )
% 0.34/0.56 = X ) )
% 0.34/0.56 & ( ( ( minus_minus_set_assn @ A @ ( insert_assn @ X @ bot_bot_set_assn ) )
% 0.34/0.56 != bot_bot_set_assn )
% 0.34/0.56 => ( ( lattic47131356835913163n_assn @ ( insert_assn @ X @ A ) )
% 0.34/0.56 = ( inf_inf_assn @ X @ ( lattic47131356835913163n_assn @ ( minus_minus_set_assn @ A @ ( insert_assn @ X @ bot_bot_set_assn ) ) ) ) ) ) ) ) ).
% 0.34/0.56
% 0.34/0.56 % Inf_fin.insert_remove
% 0.34/0.56 thf(fact_2281_Inf__fin_Oinsert__remove,axiom,
% 0.34/0.56 ! [A: set_Product_unit,X: product_unit] :
% 0.34/0.56 ( ( finite4290736615968046902t_unit @ A )
% 0.34/0.56 => ( ( ( ( minus_6452836326544984404t_unit @ A @ ( insert_Product_unit @ X @ bot_bo3957492148770167129t_unit ) )
% 0.34/0.56 = bot_bo3957492148770167129t_unit )
% 0.34/0.56 => ( ( lattic1263872656861969706t_unit @ ( insert_Product_unit @ X @ A ) )
% 0.34/0.56 = X ) )
% 0.34/0.56 & ( ( ( minus_6452836326544984404t_unit @ A @ ( insert_Product_unit @ X @ bot_bo3957492148770167129t_unit ) )
% 0.34/0.56 != bot_bo3957492148770167129t_unit )
% 0.34/0.56 => ( ( lattic1263872656861969706t_unit @ ( insert_Product_unit @ X @ A ) )
% 0.34/0.56 = ( inf_inf_Product_unit @ X @ ( lattic1263872656861969706t_unit @ ( minus_6452836326544984404t_unit @ A @ ( insert_Product_unit @ X @ bot_bo3957492148770167129t_unit ) ) ) ) ) ) ) ) ).
% 0.34/0.56
% 0.34/0.56 % Inf_fin.insert_remove
% 0.34/0.56 thf(fact_2282_Inf__fin_Oinsert__remove,axiom,
% 0.34/0.56 ! [A: set_o,X: $o] :
% 0.34/0.56 ( ( finite_finite_o @ A )
% 0.34/0.56 => ( ( lattic4107685809792843317_fin_o @ ( insert_o @ X @ A ) )
% 0.34/0.56 = ( ( ( ( minus_minus_set_o @ A @ ( insert_o @ X @ bot_bot_set_o ) )
% 0.34/0.56 = bot_bot_set_o )
% 0.34/0.56 => X )
% 0.34/0.56 & ( ( ( minus_minus_set_o @ A @ ( insert_o @ X @ bot_bot_set_o ) )
% 0.34/0.56 != bot_bot_set_o )
% 0.34/0.56 => ( inf_inf_o @ X @ ( lattic4107685809792843317_fin_o @ ( minus_minus_set_o @ A @ ( insert_o @ X @ bot_bot_set_o ) ) ) ) ) ) ) ) ).
% 0.34/0.56
% 0.34/0.56 % Inf_fin.insert_remove
% 0.34/0.56 thf(fact_2283_Inf__fin_Oinsert__remove,axiom,
% 0.34/0.56 ! [A: set_int,X: int] :
% 0.34/0.56 ( ( finite_finite_int @ A )
% 0.34/0.56 => ( ( ( ( minus_minus_set_int @ A @ ( insert_int @ X @ bot_bot_set_int ) )
% 0.34/0.56 = bot_bot_set_int )
% 0.34/0.56 => ( ( lattic5235898064620869839in_int @ ( insert_int @ X @ A ) )
% 0.34/0.56 = X ) )
% 0.34/0.56 & ( ( ( minus_minus_set_int @ A @ ( insert_int @ X @ bot_bot_set_int ) )
% 0.34/0.56 != bot_bot_set_int )
% 0.34/0.56 => ( ( lattic5235898064620869839in_int @ ( insert_int @ X @ A ) )
% 0.34/0.56 = ( inf_inf_int @ X @ ( lattic5235898064620869839in_int @ ( minus_minus_set_int @ A @ ( insert_int @ X @ bot_bot_set_int ) ) ) ) ) ) ) ) ).
% 0.34/0.56
% 0.34/0.56 % Inf_fin.insert_remove
% 0.34/0.56 thf(fact_2284_Inf__fin_Oinsert__remove,axiom,
% 0.34/0.56 ! [A: set_Pr4532377907799695533_nat_o,X: produc3658429121746597890et_nat > $o] :
% 0.34/0.56 ( ( finite3252695134891459830_nat_o @ A )
% 0.34/0.56 => ( ( ( ( minus_1801376950450012436_nat_o @ A @ ( insert5175938949040314269_nat_o @ X @ bot_bo7824918357723582233_nat_o ) )
% 0.34/0.56 = bot_bo7824918357723582233_nat_o )
% 0.34/0.56 => ( ( lattic956194824204696298_nat_o @ ( insert5175938949040314269_nat_o @ X @ A ) )
% 0.34/0.56 = X ) )
% 0.34/0.56 & ( ( ( minus_1801376950450012436_nat_o @ A @ ( insert5175938949040314269_nat_o @ X @ bot_bo7824918357723582233_nat_o ) )
% 0.34/0.56 != bot_bo7824918357723582233_nat_o )
% 0.34/0.56 => ( ( lattic956194824204696298_nat_o @ ( insert5175938949040314269_nat_o @ X @ A ) )
% 0.34/0.56 = ( inf_in1318976480646536635_nat_o @ X @ ( lattic956194824204696298_nat_o @ ( minus_1801376950450012436_nat_o @ A @ ( insert5175938949040314269_nat_o @ X @ bot_bo7824918357723582233_nat_o ) ) ) ) ) ) ) ) ).
% 0.34/0.56
% 0.34/0.56 % Inf_fin.insert_remove
% 0.34/0.56 thf(fact_2285_Inf__fin_Oinsert__remove,axiom,
% 0.34/0.56 ! [A: set_nat,X: nat] :
% 0.34/0.56 ( ( finite_finite_nat @ A )
% 0.34/0.56 => ( ( ( ( minus_minus_set_nat @ A @ ( insert_nat @ X @ bot_bot_set_nat ) )
% 0.34/0.56 = bot_bot_set_nat )
% 0.34/0.56 => ( ( lattic5238388535129920115in_nat @ ( insert_nat @ X @ A ) )
% 0.34/0.56 = X ) )
% 0.34/0.56 & ( ( ( minus_minus_set_nat @ A @ ( insert_nat @ X @ bot_bot_set_nat ) )
% 0.34/0.56 != bot_bot_set_nat )
% 0.34/0.56 => ( ( lattic5238388535129920115in_nat @ ( insert_nat @ X @ A ) )
% 0.34/0.56 = ( inf_inf_nat @ X @ ( lattic5238388535129920115in_nat @ ( minus_minus_set_nat @ A @ ( insert_nat @ X @ bot_bot_set_nat ) ) ) ) ) ) ) ) ).
% 0.34/0.56
% 0.34/0.56 % Inf_fin.insert_remove
% 0.34/0.56 thf(fact_2286_acyclic__insert,axiom,
% 0.34/0.56 ! [Y: nat,X: nat,R2: set_Pr1261947904930325089at_nat] :
% 0.34/0.56 ( ( transi3139058337892597825ic_nat @ ( insert8211810215607154385at_nat @ ( product_Pair_nat_nat @ Y @ X ) @ R2 ) )
% 0.34/0.56 = ( ( transi3139058337892597825ic_nat @ R2 )
% 0.34/0.56 & ~ ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ Y ) @ ( transi2905341329935302413cl_nat @ R2 ) ) ) ) ).
% 0.34/0.56
% 0.34/0.56 % acyclic_insert
% 0.34/0.56 thf(fact_2287_acyclic__insert,axiom,
% 0.34/0.56 ! [Y: int,X: int,R2: set_Pr958786334691620121nt_int] :
% 0.34/0.56 ( ( transi3136567867383547549ic_int @ ( insert5033312907999012233nt_int @ ( product_Pair_int_int @ Y @ X ) @ R2 ) )
% 0.34/0.56 = ( ( transi3136567867383547549ic_int @ R2 )
% 0.34/0.56 & ~ ( member5262025264175285858nt_int @ ( product_Pair_int_int @ X @ Y ) @ ( transi2902850859426252137cl_int @ R2 ) ) ) ) ).
% 0.34/0.56
% 0.34/0.56 % acyclic_insert
% 0.34/0.56 thf(fact_2288_irrefl__tranclI,axiom,
% 0.34/0.56 ! [R2: set_Pr1261947904930325089at_nat,X: nat] :
% 0.34/0.56 ( ( ( inf_in2572325071724192079at_nat @ ( converse_nat_nat @ R2 ) @ ( transi2905341329935302413cl_nat @ R2 ) )
% 0.34/0.56 = bot_bo2099793752762293965at_nat )
% 0.34/0.56 => ~ ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ X ) @ ( transi6264000038957366511cl_nat @ R2 ) ) ) ).
% 0.34/0.56
% 0.34/0.56 % irrefl_tranclI
% 0.34/0.56 thf(fact_2289_irrefl__tranclI,axiom,
% 0.34/0.56 ! [R2: set_Pr958786334691620121nt_int,X: int] :
% 0.34/0.56 ( ( ( inf_in2269163501485487111nt_int @ ( converse_int_int @ R2 ) @ ( transi2902850859426252137cl_int @ R2 ) )
% 0.34/0.56 = bot_bo1796632182523588997nt_int )
% 0.34/0.56 => ~ ( member5262025264175285858nt_int @ ( product_Pair_int_int @ X @ X ) @ ( transi6261509568448316235cl_int @ R2 ) ) ) ).
% 0.34/0.56
% 0.34/0.56 % irrefl_tranclI
% 0.34/0.56 thf(fact_2290_inf__img__fin__dom_H,axiom,
% 0.34/0.56 ! [F: code_integer > code_integer,A: set_Code_integer] :
% 0.34/0.56 ( ( finite6017078050557962740nteger @ ( image_4470545334726330049nteger @ F @ A ) )
% 0.34/0.56 => ( ~ ( finite6017078050557962740nteger @ A )
% 0.34/0.56 => ? [X3: code_integer] :
% 0.34/0.56 ( ( member_Code_integer @ X3 @ ( image_4470545334726330049nteger @ F @ A ) )
% 0.34/0.56 & ~ ( finite6017078050557962740nteger @ ( inf_in1364745209274528805nteger @ ( vimage1318793397690208119nteger @ F @ ( insert_Code_integer @ X3 @ bot_bo3990330152332043303nteger ) ) @ A ) ) ) ) ) ).
% 0.34/0.56
% 0.34/0.56 % inf_img_fin_dom'
% 0.34/0.56 thf(fact_2291_inf__img__fin__dom_H,axiom,
% 0.34/0.56 ! [F: nat > code_integer,A: set_nat] :
% 0.34/0.56 ( ( finite6017078050557962740nteger @ ( image_1215581382706833972nteger @ F @ A ) )
% 0.34/0.56 => ( ~ ( finite_finite_nat @ A )
% 0.34/0.56 => ? [X3: code_integer] :
% 0.34/0.56 ( ( member_Code_integer @ X3 @ ( image_1215581382706833972nteger @ F @ A ) )
% 0.34/0.56 & ~ ( finite_finite_nat @ ( inf_inf_set_nat @ ( vimage6104394758324456170nteger @ F @ ( insert_Code_integer @ X3 @ bot_bo3990330152332043303nteger ) ) @ A ) ) ) ) ) ).
% 0.34/0.56
% 0.34/0.56 % inf_img_fin_dom'
% 0.34/0.56 thf(fact_2292_inf__img__fin__dom_H,axiom,
% 0.34/0.56 ! [F: code_integer > product_unit,A: set_Code_integer] :
% 0.34/0.56 ( ( finite4290736615968046902t_unit @ ( image_7182656606676592809t_unit @ F @ A ) )
% 0.34/0.56 => ( ~ ( finite6017078050557962740nteger @ A )
% 0.34/0.56 => ? [X3: product_unit] :
% 0.34/0.56 ( ( member_Product_unit @ X3 @ ( image_7182656606676592809t_unit @ F @ A ) )
% 0.34/0.56 & ~ ( finite6017078050557962740nteger @ ( inf_in1364745209274528805nteger @ ( vimage5651688891684837235t_unit @ F @ ( insert_Product_unit @ X3 @ bot_bo3957492148770167129t_unit ) ) @ A ) ) ) ) ) ).
% 0.34/0.56
% 0.34/0.56 % inf_img_fin_dom'
% 0.34/0.56 thf(fact_2293_inf__img__fin__dom_H,axiom,
% 0.34/0.56 ! [F: nat > product_unit,A: set_nat] :
% 0.34/0.56 ( ( finite4290736615968046902t_unit @ ( image_8730104196221521654t_unit @ F @ A ) )
% 0.34/0.56 => ( ~ ( finite_finite_nat @ A )
% 0.34/0.56 => ? [X3: product_unit] :
% 0.34/0.56 ( ( member_Product_unit @ X3 @ ( image_8730104196221521654t_unit @ F @ A ) )
% 0.34/0.56 & ~ ( finite_finite_nat @ ( inf_inf_set_nat @ ( vimage4884490618288580032t_unit @ F @ ( insert_Product_unit @ X3 @ bot_bo3957492148770167129t_unit ) ) @ A ) ) ) ) ) ).
% 0.34/0.56
% 0.34/0.56 % inf_img_fin_dom'
% 0.34/0.56 thf(fact_2294_inf__img__fin__dom_H,axiom,
% 0.34/0.56 ! [F: code_integer > $o,A: set_Code_integer] :
% 0.34/0.56 ( ( finite_finite_o @ ( image_Code_integer_o @ F @ A ) )
% 0.34/0.56 => ( ~ ( finite6017078050557962740nteger @ A )
% 0.34/0.56 => ? [X3: $o] :
% 0.34/0.56 ( ( member_o @ X3 @ ( image_Code_integer_o @ F @ A ) )
% 0.34/0.56 & ~ ( finite6017078050557962740nteger @ ( inf_in1364745209274528805nteger @ ( vimage3005800680117336574eger_o @ F @ ( insert_o @ X3 @ bot_bot_set_o ) ) @ A ) ) ) ) ) ).
% 0.34/0.56
% 0.34/0.56 % inf_img_fin_dom'
% 0.34/0.56 thf(fact_2295_inf__img__fin__dom_H,axiom,
% 0.34/0.56 ! [F: nat > $o,A: set_nat] :
% 0.34/0.56 ( ( finite_finite_o @ ( image_nat_o2 @ F @ A ) )
% 0.34/0.56 => ( ~ ( finite_finite_nat @ A )
% 0.34/0.56 => ? [X3: $o] :
% 0.34/0.56 ( ( member_o @ X3 @ ( image_nat_o2 @ F @ A ) )
% 0.34/0.56 & ~ ( finite_finite_nat @ ( inf_inf_set_nat @ ( vimage_nat_o @ F @ ( insert_o @ X3 @ bot_bot_set_o ) ) @ A ) ) ) ) ) ).
% 0.34/0.56
% 0.34/0.56 % inf_img_fin_dom'
% 0.34/0.56 thf(fact_2296_inf__img__fin__dom_H,axiom,
% 0.34/0.56 ! [F: code_integer > nat,A: set_Code_integer] :
% 0.34/0.56 ( ( finite_finite_nat @ ( image_951025933927791156er_nat @ F @ A ) )
% 0.34/0.56 => ( ~ ( finite6017078050557962740nteger @ A )
% 0.34/0.56 => ? [X3: nat] :
% 0.34/0.56 ( ( member_nat @ X3 @ ( image_951025933927791156er_nat @ F @ A ) )
% 0.34/0.56 & ~ ( finite6017078050557962740nteger @ ( inf_in1364745209274528805nteger @ ( vimage5839839309545413354er_nat @ F @ ( insert_nat @ X3 @ bot_bot_set_nat ) ) @ A ) ) ) ) ) ).
% 0.34/0.56
% 0.34/0.56 % inf_img_fin_dom'
% 0.34/0.56 thf(fact_2297_inf__img__fin__dom_H,axiom,
% 0.34/0.56 ! [F: nat > nat,A: set_nat] :
% 0.34/0.56 ( ( finite_finite_nat @ ( image_nat_nat2 @ F @ A ) )
% 0.34/0.56 => ( ~ ( finite_finite_nat @ A )
% 0.34/0.56 => ? [X3: nat] :
% 0.34/0.56 ( ( member_nat @ X3 @ ( image_nat_nat2 @ F @ A ) )
% 0.34/0.56 & ~ ( finite_finite_nat @ ( inf_inf_set_nat @ ( vimage_nat_nat @ F @ ( insert_nat @ X3 @ bot_bot_set_nat ) ) @ A ) ) ) ) ) ).
% 0.34/0.56
% 0.34/0.56 % inf_img_fin_dom'
% 0.34/0.56 thf(fact_2298_inf__img__fin__dom_H,axiom,
% 0.34/0.56 ! [F: code_integer > int,A: set_Code_integer] :
% 0.34/0.56 ( ( finite_finite_int @ ( image_948535463418740880er_int @ F @ A ) )
% 0.34/0.56 => ( ~ ( finite6017078050557962740nteger @ A )
% 0.34/0.56 => ? [X3: int] :
% 0.34/0.56 ( ( member_int @ X3 @ ( image_948535463418740880er_int @ F @ A ) )
% 0.34/0.56 & ~ ( finite6017078050557962740nteger @ ( inf_in1364745209274528805nteger @ ( vimage5837348839036363078er_int @ F @ ( insert_int @ X3 @ bot_bot_set_int ) ) @ A ) ) ) ) ) ).
% 0.34/0.56
% 0.34/0.56 % inf_img_fin_dom'
% 0.34/0.56 thf(fact_2299_inf__img__fin__dom_H,axiom,
% 0.34/0.56 ! [F: nat > int,A: set_nat] :
% 0.34/0.56 ( ( finite_finite_int @ ( image_nat_int2 @ F @ A ) )
% 0.34/0.56 => ( ~ ( finite_finite_nat @ A )
% 0.34/0.56 => ? [X3: int] :
% 0.34/0.56 ( ( member_int @ X3 @ ( image_nat_int2 @ F @ A ) )
% 0.34/0.56 & ~ ( finite_finite_nat @ ( inf_inf_set_nat @ ( vimage_nat_int @ F @ ( insert_int @ X3 @ bot_bot_set_int ) ) @ A ) ) ) ) ) ).
% 0.34/0.56
% 0.34/0.56 % inf_img_fin_dom'
% 0.34/0.56 thf(fact_2300_inf__img__fin__domE_H,axiom,
% 0.34/0.56 ! [F: code_integer > code_integer,A: set_Code_integer] :
% 0.34/0.56 ( ( finite6017078050557962740nteger @ ( image_4470545334726330049nteger @ F @ A ) )
% 0.34/0.56 => ( ~ ( finite6017078050557962740nteger @ A )
% 0.34/0.56 => ~ ! [Y2: code_integer] :
% 0.34/0.56 ( ( member_Code_integer @ Y2 @ ( image_4470545334726330049nteger @ F @ A ) )
% 0.34/0.56 => ( finite6017078050557962740nteger @ ( inf_in1364745209274528805nteger @ ( vimage1318793397690208119nteger @ F @ ( insert_Code_integer @ Y2 @ bot_bo3990330152332043303nteger ) ) @ A ) ) ) ) ) ).
% 0.34/0.56
% 0.34/0.56 % inf_img_fin_domE'
% 0.34/0.56 thf(fact_2301_inf__img__fin__domE_H,axiom,
% 0.34/0.56 ! [F: nat > code_integer,A: set_nat] :
% 0.34/0.56 ( ( finite6017078050557962740nteger @ ( image_1215581382706833972nteger @ F @ A ) )
% 0.34/0.56 => ( ~ ( finite_finite_nat @ A )
% 0.34/0.56 => ~ ! [Y2: code_integer] :
% 0.34/0.56 ( ( member_Code_integer @ Y2 @ ( image_1215581382706833972nteger @ F @ A ) )
% 0.34/0.56 => ( finite_finite_nat @ ( inf_inf_set_nat @ ( vimage6104394758324456170nteger @ F @ ( insert_Code_integer @ Y2 @ bot_bo3990330152332043303nteger ) ) @ A ) ) ) ) ) ).
% 0.34/0.56
% 0.34/0.56 % inf_img_fin_domE'
% 0.34/0.56 thf(fact_2302_inf__img__fin__domE_H,axiom,
% 0.34/0.56 ! [F: code_integer > product_unit,A: set_Code_integer] :
% 0.34/0.56 ( ( finite4290736615968046902t_unit @ ( image_7182656606676592809t_unit @ F @ A ) )
% 0.34/0.56 => ( ~ ( finite6017078050557962740nteger @ A )
% 0.34/0.56 => ~ ! [Y2: product_unit] :
% 0.34/0.56 ( ( member_Product_unit @ Y2 @ ( image_7182656606676592809t_unit @ F @ A ) )
% 0.34/0.56 => ( finite6017078050557962740nteger @ ( inf_in1364745209274528805nteger @ ( vimage5651688891684837235t_unit @ F @ ( insert_Product_unit @ Y2 @ bot_bo3957492148770167129t_unit ) ) @ A ) ) ) ) ) ).
% 0.34/0.56
% 0.34/0.56 % inf_img_fin_domE'
% 0.34/0.56 thf(fact_2303_inf__img__fin__domE_H,axiom,
% 0.34/0.56 ! [F: nat > product_unit,A: set_nat] :
% 0.34/0.56 ( ( finite4290736615968046902t_unit @ ( image_8730104196221521654t_unit @ F @ A ) )
% 0.34/0.56 => ( ~ ( finite_finite_nat @ A )
% 0.34/0.56 => ~ ! [Y2: product_unit] :
% 0.34/0.56 ( ( member_Product_unit @ Y2 @ ( image_8730104196221521654t_unit @ F @ A ) )
% 0.34/0.56 => ( finite_finite_nat @ ( inf_inf_set_nat @ ( vimage4884490618288580032t_unit @ F @ ( insert_Product_unit @ Y2 @ bot_bo3957492148770167129t_unit ) ) @ A ) ) ) ) ) ).
% 0.34/0.56
% 0.34/0.56 % inf_img_fin_domE'
% 0.34/0.56 thf(fact_2304_inf__img__fin__domE_H,axiom,
% 0.34/0.56 ! [F: code_integer > $o,A: set_Code_integer] :
% 0.34/0.56 ( ( finite_finite_o @ ( image_Code_integer_o @ F @ A ) )
% 0.34/0.56 => ( ~ ( finite6017078050557962740nteger @ A )
% 0.34/0.56 => ~ ! [Y2: $o] :
% 0.34/0.56 ( ( member_o @ Y2 @ ( image_Code_integer_o @ F @ A ) )
% 0.34/0.56 => ( finite6017078050557962740nteger @ ( inf_in1364745209274528805nteger @ ( vimage3005800680117336574eger_o @ F @ ( insert_o @ Y2 @ bot_bot_set_o ) ) @ A ) ) ) ) ) ).
% 0.34/0.56
% 0.34/0.56 % inf_img_fin_domE'
% 0.34/0.56 thf(fact_2305_inf__img__fin__domE_H,axiom,
% 0.34/0.56 ! [F: nat > $o,A: set_nat] :
% 0.34/0.56 ( ( finite_finite_o @ ( image_nat_o2 @ F @ A ) )
% 0.34/0.56 => ( ~ ( finite_finite_nat @ A )
% 0.34/0.56 => ~ ! [Y2: $o] :
% 0.34/0.56 ( ( member_o @ Y2 @ ( image_nat_o2 @ F @ A ) )
% 0.34/0.56 => ( finite_finite_nat @ ( inf_inf_set_nat @ ( vimage_nat_o @ F @ ( insert_o @ Y2 @ bot_bot_set_o ) ) @ A ) ) ) ) ) ).
% 0.34/0.56
% 0.34/0.56 % inf_img_fin_domE'
% 0.34/0.56 thf(fact_2306_inf__img__fin__domE_H,axiom,
% 0.34/0.56 ! [F: code_integer > nat,A: set_Code_integer] :
% 0.34/0.56 ( ( finite_finite_nat @ ( image_951025933927791156er_nat @ F @ A ) )
% 0.34/0.56 => ( ~ ( finite6017078050557962740nteger @ A )
% 0.34/0.56 => ~ ! [Y2: nat] :
% 0.34/0.56 ( ( member_nat @ Y2 @ ( image_951025933927791156er_nat @ F @ A ) )
% 0.34/0.56 => ( finite6017078050557962740nteger @ ( inf_in1364745209274528805nteger @ ( vimage5839839309545413354er_nat @ F @ ( insert_nat @ Y2 @ bot_bot_set_nat ) ) @ A ) ) ) ) ) ).
% 0.34/0.56
% 0.34/0.56 % inf_img_fin_domE'
% 0.34/0.56 thf(fact_2307_inf__img__fin__domE_H,axiom,
% 0.34/0.56 ! [F: nat > nat,A: set_nat] :
% 0.34/0.56 ( ( finite_finite_nat @ ( image_nat_nat2 @ F @ A ) )
% 0.34/0.56 => ( ~ ( finite_finite_nat @ A )
% 0.34/0.56 => ~ ! [Y2: nat] :
% 0.34/0.56 ( ( member_nat @ Y2 @ ( image_nat_nat2 @ F @ A ) )
% 0.34/0.56 => ( finite_finite_nat @ ( inf_inf_set_nat @ ( vimage_nat_nat @ F @ ( insert_nat @ Y2 @ bot_bot_set_nat ) ) @ A ) ) ) ) ) ).
% 0.34/0.56
% 0.34/0.56 % inf_img_fin_domE'
% 0.34/0.56 thf(fact_2308_inf__img__fin__domE_H,axiom,
% 0.34/0.56 ! [F: code_integer > int,A: set_Code_integer] :
% 0.34/0.56 ( ( finite_finite_int @ ( image_948535463418740880er_int @ F @ A ) )
% 0.34/0.56 => ( ~ ( finite6017078050557962740nteger @ A )
% 0.34/0.56 => ~ ! [Y2: int] :
% 0.34/0.56 ( ( member_int @ Y2 @ ( image_948535463418740880er_int @ F @ A ) )
% 0.34/0.56 => ( finite6017078050557962740nteger @ ( inf_in1364745209274528805nteger @ ( vimage5837348839036363078er_int @ F @ ( insert_int @ Y2 @ bot_bot_set_int ) ) @ A ) ) ) ) ) ).
% 0.34/0.56
% 0.34/0.56 % inf_img_fin_domE'
% 0.34/0.56 thf(fact_2309_inf__img__fin__domE_H,axiom,
% 0.34/0.56 ! [F: nat > int,A: set_nat] :
% 0.34/0.56 ( ( finite_finite_int @ ( image_nat_int2 @ F @ A ) )
% 0.34/0.56 => ( ~ ( finite_finite_nat @ A )
% 0.34/0.56 => ~ ! [Y2: int] :
% 0.34/0.56 ( ( member_int @ Y2 @ ( image_nat_int2 @ F @ A ) )
% 0.34/0.56 => ( finite_finite_nat @ ( inf_inf_set_nat @ ( vimage_nat_int @ F @ ( insert_int @ Y2 @ bot_bot_set_int ) ) @ A ) ) ) ) ) ).
% 0.34/0.56
% 0.34/0.56 % inf_img_fin_domE'
% 0.34/0.56 thf(fact_2310_image__eqI,axiom,
% 0.34/0.56 ! [B: code_integer,F: code_integer > code_integer,X: code_integer,A: set_Code_integer] :
% 0.34/0.56 ( ( B
% 0.34/0.56 = ( F @ X ) )
% 0.34/0.56 => ( ( member_Code_integer @ X @ A )
% 0.34/0.56 => ( member_Code_integer @ B @ ( image_4470545334726330049nteger @ F @ A ) ) ) ) ).
% 0.34/0.56
% 0.34/0.56 % image_eqI
% 0.34/0.56 thf(fact_2311_image__eqI,axiom,
% 0.34/0.56 ! [B: code_integer,F: nat > code_integer,X: nat,A: set_nat] :
% 0.34/0.56 ( ( B
% 0.34/0.56 = ( F @ X ) )
% 0.34/0.56 => ( ( member_nat @ X @ A )
% 0.34/0.56 => ( member_Code_integer @ B @ ( image_1215581382706833972nteger @ F @ A ) ) ) ) ).
% 0.34/0.56
% 0.34/0.56 % image_eqI
% 0.34/0.56 thf(fact_2312_image__eqI,axiom,
% 0.34/0.56 ! [B: nat,F: nat > nat,X: nat,A: set_nat] :
% 0.34/0.56 ( ( B
% 0.34/0.56 = ( F @ X ) )
% 0.34/0.56 => ( ( member_nat @ X @ A )
% 0.34/0.56 => ( member_nat @ B @ ( image_nat_nat2 @ F @ A ) ) ) ) ).
% 0.34/0.56
% 0.34/0.56 % image_eqI
% 0.34/0.56 thf(fact_2313_image__eqI,axiom,
% 0.34/0.56 ! [B: int,F: nat > int,X: nat,A: set_nat] :
% 0.34/0.56 ( ( B
% 0.34/0.56 = ( F @ X ) )
% 0.34/0.56 => ( ( member_nat @ X @ A )
% 0.34/0.56 => ( member_int @ B @ ( image_nat_int2 @ F @ A ) ) ) ) ).
% 0.34/0.56
% 0.34/0.56 % image_eqI
% 0.34/0.56 thf(fact_2314_image__eqI,axiom,
% 0.34/0.56 ! [B: $o,F: nat > $o,X: nat,A: set_nat] :
% 0.34/0.56 ( ( B
% 0.34/0.56 = ( F @ X ) )
% 0.34/0.56 => ( ( member_nat @ X @ A )
% 0.34/0.56 => ( member_o @ B @ ( image_nat_o2 @ F @ A ) ) ) ) ).
% 0.34/0.56
% 0.34/0.56 % image_eqI
% 0.34/0.56 thf(fact_2315_image__eqI,axiom,
% 0.34/0.56 ! [B: product_unit,F: nat > product_unit,X: nat,A: set_nat] :
% 0.34/0.56 ( ( B
% 0.34/0.56 = ( F @ X ) )
% 0.34/0.56 => ( ( member_nat @ X @ A )
% 0.34/0.56 => ( member_Product_unit @ B @ ( image_8730104196221521654t_unit @ F @ A ) ) ) ) ).
% 0.34/0.56
% 0.34/0.56 % image_eqI
% 0.34/0.56 thf(fact_2316_image__eqI,axiom,
% 0.34/0.56 ! [B: nat,F: int > nat,X: int,A: set_int] :
% 0.34/0.56 ( ( B
% 0.34/0.56 = ( F @ X ) )
% 0.34/0.56 => ( ( member_int @ X @ A )
% 0.34/0.56 => ( member_nat @ B @ ( image_int_nat2 @ F @ A ) ) ) ) ).
% 0.34/0.56
% 0.34/0.56 % image_eqI
% 0.34/0.56 thf(fact_2317_image__eqI,axiom,
% 0.34/0.56 ! [B: int,F: int > int,X: int,A: set_int] :
% 0.34/0.56 ( ( B
% 0.34/0.56 = ( F @ X ) )
% 0.34/0.56 => ( ( member_int @ X @ A )
% 0.34/0.56 => ( member_int @ B @ ( image_int_int2 @ F @ A ) ) ) ) ).
% 0.34/0.56
% 0.34/0.56 % image_eqI
% 0.34/0.56 thf(fact_2318_image__eqI,axiom,
% 0.34/0.57 ! [B: $o,F: int > $o,X: int,A: set_int] :
% 0.34/0.57 ( ( B
% 0.34/0.57 = ( F @ X ) )
% 0.34/0.57 => ( ( member_int @ X @ A )
% 0.34/0.57 => ( member_o @ B @ ( image_int_o2 @ F @ A ) ) ) ) ).
% 0.34/0.57
% 0.34/0.57 % image_eqI
% 0.34/0.57 thf(fact_2319_image__eqI,axiom,
% 0.34/0.57 ! [B: product_unit,F: int > product_unit,X: int,A: set_int] :
% 0.34/0.57 ( ( B
% 0.34/0.57 = ( F @ X ) )
% 0.34/0.57 => ( ( member_int @ X @ A )
% 0.34/0.57 => ( member_Product_unit @ B @ ( image_5846526143423701914t_unit @ F @ A ) ) ) ) ).
% 0.34/0.57
% 0.34/0.57 % image_eqI
% 0.34/0.57 thf(fact_2320_Diff__idemp,axiom,
% 0.34/0.57 ! [A: set_nat,B4: set_nat] :
% 0.34/0.57 ( ( minus_minus_set_nat @ ( minus_minus_set_nat @ A @ B4 ) @ B4 )
% 0.34/0.57 = ( minus_minus_set_nat @ A @ B4 ) ) ).
% 0.34/0.57
% 0.34/0.57 % Diff_idemp
% 0.34/0.57 thf(fact_2321_Diff__iff,axiom,
% 0.34/0.57 ! [C: produc3658429121746597890et_nat > $o,A: set_Pr4532377907799695533_nat_o,B4: set_Pr4532377907799695533_nat_o] :
% 0.34/0.57 ( ( member6576561426505652726_nat_o @ C @ ( minus_1801376950450012436_nat_o @ A @ B4 ) )
% 0.34/0.57 = ( ( member6576561426505652726_nat_o @ C @ A )
% 0.34/0.57 & ~ ( member6576561426505652726_nat_o @ C @ B4 ) ) ) ).
% 0.34/0.57
% 0.34/0.57 % Diff_iff
% 0.34/0.57 thf(fact_2322_Diff__iff,axiom,
% 0.34/0.57 ! [C: int,A: set_int,B4: set_int] :
% 0.34/0.57 ( ( member_int @ C @ ( minus_minus_set_int @ A @ B4 ) )
% 0.34/0.57 = ( ( member_int @ C @ A )
% 0.34/0.57 & ~ ( member_int @ C @ B4 ) ) ) ).
% 0.34/0.57
% 0.34/0.57 % Diff_iff
% 0.34/0.57 thf(fact_2323_Diff__iff,axiom,
% 0.34/0.57 ! [C: $o,A: set_o,B4: set_o] :
% 0.34/0.57 ( ( member_o @ C @ ( minus_minus_set_o @ A @ B4 ) )
% 0.34/0.57 = ( ( member_o @ C @ A )
% 0.34/0.57 & ~ ( member_o @ C @ B4 ) ) ) ).
% 0.34/0.57
% 0.34/0.57 % Diff_iff
% 0.34/0.57 thf(fact_2324_Diff__iff,axiom,
% 0.34/0.57 ! [C: product_unit,A: set_Product_unit,B4: set_Product_unit] :
% 0.34/0.57 ( ( member_Product_unit @ C @ ( minus_6452836326544984404t_unit @ A @ B4 ) )
% 0.34/0.57 = ( ( member_Product_unit @ C @ A )
% 0.34/0.57 & ~ ( member_Product_unit @ C @ B4 ) ) ) ).
% 0.34/0.57
% 0.34/0.57 % Diff_iff
% 0.34/0.57 thf(fact_2325_Diff__iff,axiom,
% 0.34/0.57 ! [C: nat,A: set_nat,B4: set_nat] :
% 0.34/0.57 ( ( member_nat @ C @ ( minus_minus_set_nat @ A @ B4 ) )
% 0.34/0.57 = ( ( member_nat @ C @ A )
% 0.34/0.57 & ~ ( member_nat @ C @ B4 ) ) ) ).
% 0.34/0.57
% 0.34/0.57 % Diff_iff
% 0.34/0.57 thf(fact_2326_DiffI,axiom,
% 0.34/0.57 ! [C: produc3658429121746597890et_nat > $o,A: set_Pr4532377907799695533_nat_o,B4: set_Pr4532377907799695533_nat_o] :
% 0.34/0.57 ( ( member6576561426505652726_nat_o @ C @ A )
% 0.34/0.57 => ( ~ ( member6576561426505652726_nat_o @ C @ B4 )
% 0.34/0.57 => ( member6576561426505652726_nat_o @ C @ ( minus_1801376950450012436_nat_o @ A @ B4 ) ) ) ) ).
% 0.34/0.57
% 0.34/0.57 % DiffI
% 0.34/0.57 thf(fact_2327_DiffI,axiom,
% 0.34/0.57 ! [C: int,A: set_int,B4: set_int] :
% 0.34/0.57 ( ( member_int @ C @ A )
% 0.34/0.57 => ( ~ ( member_int @ C @ B4 )
% 0.34/0.57 => ( member_int @ C @ ( minus_minus_set_int @ A @ B4 ) ) ) ) ).
% 0.34/0.57
% 0.34/0.57 % DiffI
% 0.34/0.57 thf(fact_2328_DiffI,axiom,
% 0.34/0.57 ! [C: $o,A: set_o,B4: set_o] :
% 0.34/0.57 ( ( member_o @ C @ A )
% 0.34/0.57 => ( ~ ( member_o @ C @ B4 )
% 0.34/0.57 => ( member_o @ C @ ( minus_minus_set_o @ A @ B4 ) ) ) ) ).
% 0.34/0.57
% 0.34/0.57 % DiffI
% 0.34/0.57 thf(fact_2329_DiffI,axiom,
% 0.34/0.57 ! [C: product_unit,A: set_Product_unit,B4: set_Product_unit] :
% 0.34/0.57 ( ( member_Product_unit @ C @ A )
% 0.34/0.57 => ( ~ ( member_Product_unit @ C @ B4 )
% 0.34/0.57 => ( member_Product_unit @ C @ ( minus_6452836326544984404t_unit @ A @ B4 ) ) ) ) ).
% 0.34/0.57
% 0.34/0.57 % DiffI
% 0.34/0.57 thf(fact_2330_DiffI,axiom,
% 0.34/0.57 ! [C: nat,A: set_nat,B4: set_nat] :
% 0.34/0.57 ( ( member_nat @ C @ A )
% 0.34/0.57 => ( ~ ( member_nat @ C @ B4 )
% 0.34/0.57 => ( member_nat @ C @ ( minus_minus_set_nat @ A @ B4 ) ) ) ) ).
% 0.34/0.57
% 0.34/0.57 % DiffI
% 0.34/0.57 thf(fact_2331_image__is__empty,axiom,
% 0.34/0.57 ! [F: code_integer > code_integer,A: set_Code_integer] :
% 0.34/0.57 ( ( ( image_4470545334726330049nteger @ F @ A )
% 0.34/0.57 = bot_bo3990330152332043303nteger )
% 0.34/0.57 = ( A = bot_bo3990330152332043303nteger ) ) ).
% 0.34/0.57
% 0.34/0.57 % image_is_empty
% 0.34/0.57 thf(fact_2332_image__is__empty,axiom,
% 0.34/0.57 ! [F: nat > code_integer,A: set_nat] :
% 0.34/0.57 ( ( ( image_1215581382706833972nteger @ F @ A )
% 0.34/0.57 = bot_bo3990330152332043303nteger )
% 0.34/0.57 = ( A = bot_bot_set_nat ) ) ).
% 0.34/0.57
% 0.34/0.57 % image_is_empty
% 0.34/0.57 thf(fact_2333_image__is__empty,axiom,
% 0.34/0.57 ! [F: product_unit > product_unit,A: set_Product_unit] :
% 0.34/0.57 ( ( ( image_405062704495631173t_unit @ F @ A )
% 0.34/0.57 = bot_bo3957492148770167129t_unit )
% 0.34/0.57 = ( A = bot_bo3957492148770167129t_unit ) ) ).
% 0.34/0.57
% 0.34/0.57 % image_is_empty
% 0.34/0.57 thf(fact_2334_image__is__empty,axiom,
% 0.34/0.57 ! [F: $o > product_unit,A: set_o] :
% 0.34/0.57 ( ( ( image_o_Product_unit2 @ F @ A )
% 0.34/0.57 = bot_bo3957492148770167129t_unit )
% 0.34/0.57 = ( A = bot_bot_set_o ) ) ).
% 0.34/0.57
% 0.34/0.57 % image_is_empty
% 0.34/0.57 thf(fact_2335_image__is__empty,axiom,
% 0.34/0.57 ! [F: nat > product_unit,A: set_nat] :
% 0.34/0.57 ( ( ( image_8730104196221521654t_unit @ F @ A )
% 0.34/0.57 = bot_bo3957492148770167129t_unit )
% 0.34/0.57 = ( A = bot_bot_set_nat ) ) ).
% 0.34/0.57
% 0.34/0.57 % image_is_empty
% 0.34/0.57 thf(fact_2336_image__is__empty,axiom,
% 0.34/0.57 ! [F: int > product_unit,A: set_int] :
% 0.34/0.57 ( ( ( image_5846526143423701914t_unit @ F @ A )
% 0.34/0.57 = bot_bo3957492148770167129t_unit )
% 0.34/0.57 = ( A = bot_bot_set_int ) ) ).
% 0.34/0.57
% 0.34/0.57 % image_is_empty
% 0.34/0.57 thf(fact_2337_image__is__empty,axiom,
% 0.34/0.57 ! [F: product_unit > $o,A: set_Product_unit] :
% 0.34/0.57 ( ( ( image_Product_unit_o2 @ F @ A )
% 0.34/0.57 = bot_bot_set_o )
% 0.34/0.57 = ( A = bot_bo3957492148770167129t_unit ) ) ).
% 0.34/0.57
% 0.34/0.57 % image_is_empty
% 0.34/0.57 thf(fact_2338_image__is__empty,axiom,
% 0.34/0.57 ! [F: $o > $o,A: set_o] :
% 0.34/0.57 ( ( ( image_o_o2 @ F @ A )
% 0.34/0.57 = bot_bot_set_o )
% 0.34/0.57 = ( A = bot_bot_set_o ) ) ).
% 0.34/0.57
% 0.34/0.57 % image_is_empty
% 0.34/0.57 thf(fact_2339_image__is__empty,axiom,
% 0.34/0.57 ! [F: nat > $o,A: set_nat] :
% 0.34/0.57 ( ( ( image_nat_o2 @ F @ A )
% 0.34/0.57 = bot_bot_set_o )
% 0.34/0.57 = ( A = bot_bot_set_nat ) ) ).
% 0.34/0.57
% 0.34/0.57 % image_is_empty
% 0.34/0.57 thf(fact_2340_image__is__empty,axiom,
% 0.34/0.57 ! [F: int > $o,A: set_int] :
% 0.34/0.57 ( ( ( image_int_o2 @ F @ A )
% 0.34/0.57 = bot_bot_set_o )
% 0.34/0.57 = ( A = bot_bot_set_int ) ) ).
% 0.34/0.57
% 0.34/0.57 % image_is_empty
% 0.34/0.57 thf(fact_2341_empty__is__image,axiom,
% 0.34/0.57 ! [F: code_integer > code_integer,A: set_Code_integer] :
% 0.34/0.57 ( ( bot_bo3990330152332043303nteger
% 0.34/0.57 = ( image_4470545334726330049nteger @ F @ A ) )
% 0.34/0.57 = ( A = bot_bo3990330152332043303nteger ) ) ).
% 0.34/0.57
% 0.34/0.57 % empty_is_image
% 0.34/0.57 thf(fact_2342_empty__is__image,axiom,
% 0.34/0.57 ! [F: nat > code_integer,A: set_nat] :
% 0.34/0.57 ( ( bot_bo3990330152332043303nteger
% 0.34/0.57 = ( image_1215581382706833972nteger @ F @ A ) )
% 0.34/0.57 = ( A = bot_bot_set_nat ) ) ).
% 0.34/0.57
% 0.34/0.57 % empty_is_image
% 0.34/0.57 thf(fact_2343_empty__is__image,axiom,
% 0.34/0.57 ! [F: product_unit > product_unit,A: set_Product_unit] :
% 0.34/0.57 ( ( bot_bo3957492148770167129t_unit
% 0.34/0.57 = ( image_405062704495631173t_unit @ F @ A ) )
% 0.34/0.57 = ( A = bot_bo3957492148770167129t_unit ) ) ).
% 0.34/0.57
% 0.34/0.57 % empty_is_image
% 0.34/0.57 thf(fact_2344_empty__is__image,axiom,
% 0.34/0.57 ! [F: $o > product_unit,A: set_o] :
% 0.34/0.57 ( ( bot_bo3957492148770167129t_unit
% 0.34/0.57 = ( image_o_Product_unit2 @ F @ A ) )
% 0.34/0.57 = ( A = bot_bot_set_o ) ) ).
% 0.34/0.57
% 0.34/0.57 % empty_is_image
% 0.34/0.57 thf(fact_2345_empty__is__image,axiom,
% 0.34/0.57 ! [F: nat > product_unit,A: set_nat] :
% 0.34/0.57 ( ( bot_bo3957492148770167129t_unit
% 0.34/0.57 = ( image_8730104196221521654t_unit @ F @ A ) )
% 0.34/0.57 = ( A = bot_bot_set_nat ) ) ).
% 0.34/0.57
% 0.34/0.57 % empty_is_image
% 0.34/0.57 thf(fact_2346_empty__is__image,axiom,
% 0.34/0.57 ! [F: int > product_unit,A: set_int] :
% 0.34/0.57 ( ( bot_bo3957492148770167129t_unit
% 0.34/0.57 = ( image_5846526143423701914t_unit @ F @ A ) )
% 0.34/0.57 = ( A = bot_bot_set_int ) ) ).
% 0.34/0.57
% 0.34/0.57 % empty_is_image
% 0.34/0.57 thf(fact_2347_empty__is__image,axiom,
% 0.34/0.57 ! [F: product_unit > $o,A: set_Product_unit] :
% 0.34/0.57 ( ( bot_bot_set_o
% 0.34/0.57 = ( image_Product_unit_o2 @ F @ A ) )
% 0.34/0.57 = ( A = bot_bo3957492148770167129t_unit ) ) ).
% 0.34/0.57
% 0.34/0.57 % empty_is_image
% 0.34/0.57 thf(fact_2348_empty__is__image,axiom,
% 0.34/0.57 ! [F: $o > $o,A: set_o] :
% 0.34/0.57 ( ( bot_bot_set_o
% 0.34/0.57 = ( image_o_o2 @ F @ A ) )
% 0.34/0.57 = ( A = bot_bot_set_o ) ) ).
% 0.34/0.57
% 0.34/0.57 % empty_is_image
% 0.34/0.57 thf(fact_2349_empty__is__image,axiom,
% 0.34/0.57 ! [F: nat > $o,A: set_nat] :
% 0.34/0.57 ( ( bot_bot_set_o
% 0.34/0.57 = ( image_nat_o2 @ F @ A ) )
% 0.34/0.57 = ( A = bot_bot_set_nat ) ) ).
% 0.34/0.57
% 0.34/0.57 % empty_is_image
% 0.34/0.57 thf(fact_2350_empty__is__image,axiom,
% 0.34/0.57 ! [F: int > $o,A: set_int] :
% 0.34/0.57 ( ( bot_bot_set_o
% 0.34/0.57 = ( image_int_o2 @ F @ A ) )
% 0.34/0.57 = ( A = bot_bot_set_int ) ) ).
% 0.34/0.57
% 0.34/0.57 % empty_is_image
% 0.34/0.57 thf(fact_2351_image__empty,axiom,
% 0.34/0.57 ! [F: code_integer > code_integer] :
% 0.34/0.57 ( ( image_4470545334726330049nteger @ F @ bot_bo3990330152332043303nteger )
% 0.34/0.57 = bot_bo3990330152332043303nteger ) ).
% 0.34/0.57
% 0.34/0.57 % image_empty
% 0.34/0.57 thf(fact_2352_image__empty,axiom,
% 0.34/0.57 ! [F: product_unit > product_unit] :
% 0.34/0.57 ( ( image_405062704495631173t_unit @ F @ bot_bo3957492148770167129t_unit )
% 0.34/0.57 = bot_bo3957492148770167129t_unit ) ).
% 0.34/0.57
% 0.34/0.57 % image_empty
% 0.34/0.57 thf(fact_2353_image__empty,axiom,
% 0.34/0.57 ! [F: product_unit > $o] :
% 0.34/0.57 ( ( image_Product_unit_o2 @ F @ bot_bo3957492148770167129t_unit )
% 0.34/0.57 = bot_bot_set_o ) ).
% 0.34/0.57
% 0.34/0.57 % image_empty
% 0.34/0.57 thf(fact_2354_image__empty,axiom,
% 0.34/0.57 ! [F: product_unit > nat] :
% 0.34/0.57 ( ( image_875570014554754200it_nat @ F @ bot_bo3957492148770167129t_unit )
% 0.34/0.57 = bot_bot_set_nat ) ).
% 0.34/0.57
% 0.34/0.57 % image_empty
% 0.34/0.57 thf(fact_2355_image__empty,axiom,
% 0.34/0.57 ! [F: product_unit > int] :
% 0.34/0.57 ( ( image_873079544045703924it_int @ F @ bot_bo3957492148770167129t_unit )
% 0.34/0.57 = bot_bot_set_int ) ).
% 0.34/0.57
% 0.34/0.57 % image_empty
% 0.34/0.57 thf(fact_2356_image__empty,axiom,
% 0.34/0.57 ! [F: $o > product_unit] :
% 0.34/0.57 ( ( image_o_Product_unit2 @ F @ bot_bot_set_o )
% 0.34/0.57 = bot_bo3957492148770167129t_unit ) ).
% 0.34/0.57
% 0.34/0.57 % image_empty
% 0.34/0.57 thf(fact_2357_image__empty,axiom,
% 0.34/0.57 ! [F: $o > $o] :
% 0.34/0.57 ( ( image_o_o2 @ F @ bot_bot_set_o )
% 0.34/0.57 = bot_bot_set_o ) ).
% 0.34/0.57
% 0.34/0.57 % image_empty
% 0.34/0.57 thf(fact_2358_image__empty,axiom,
% 0.34/0.57 ! [F: $o > nat] :
% 0.34/0.57 ( ( image_o_nat2 @ F @ bot_bot_set_o )
% 0.34/0.57 = bot_bot_set_nat ) ).
% 0.34/0.57
% 0.34/0.57 % image_empty
% 0.34/0.57 thf(fact_2359_image__empty,axiom,
% 0.34/0.57 ! [F: $o > int] :
% 0.34/0.57 ( ( image_o_int2 @ F @ bot_bot_set_o )
% 0.34/0.57 = bot_bot_set_int ) ).
% 0.34/0.57
% 0.34/0.57 % image_empty
% 0.34/0.57 thf(fact_2360_image__empty,axiom,
% 0.34/0.57 ! [F: nat > code_integer] :
% 0.34/0.57 ( ( image_1215581382706833972nteger @ F @ bot_bot_set_nat )
% 0.34/0.57 = bot_bo3990330152332043303nteger ) ).
% 0.34/0.57
% 0.34/0.57 % image_empty
% 0.34/0.57 thf(fact_2361_insert__image,axiom,
% 0.34/0.57 ! [X: code_integer,A: set_Code_integer,F: code_integer > code_integer] :
% 0.34/0.57 ( ( member_Code_integer @ X @ A )
% 0.34/0.57 => ( ( insert_Code_integer @ ( F @ X ) @ ( image_4470545334726330049nteger @ F @ A ) )
% 0.34/0.57 = ( image_4470545334726330049nteger @ F @ A ) ) ) ).
% 0.34/0.57
% 0.34/0.57 % insert_image
% 0.34/0.57 thf(fact_2362_insert__image,axiom,
% 0.34/0.57 ! [X: nat,A: set_nat,F: nat > code_integer] :
% 0.34/0.57 ( ( member_nat @ X @ A )
% 0.34/0.57 => ( ( insert_Code_integer @ ( F @ X ) @ ( image_1215581382706833972nteger @ F @ A ) )
% 0.34/0.57 = ( image_1215581382706833972nteger @ F @ A ) ) ) ).
% 0.34/0.57
% 0.34/0.57 % insert_image
% 0.34/0.57 thf(fact_2363_insert__image,axiom,
% 0.34/0.57 ! [X: nat,A: set_nat,F: nat > nat] :
% 0.34/0.57 ( ( member_nat @ X @ A )
% 0.34/0.57 => ( ( insert_nat @ ( F @ X ) @ ( image_nat_nat2 @ F @ A ) )
% 0.34/0.57 = ( image_nat_nat2 @ F @ A ) ) ) ).
% 0.34/0.57
% 0.34/0.57 % insert_image
% 0.34/0.57 thf(fact_2364_insert__image,axiom,
% 0.34/0.57 ! [X: nat,A: set_nat,F: nat > int] :
% 0.34/0.57 ( ( member_nat @ X @ A )
% 0.34/0.57 => ( ( insert_int @ ( F @ X ) @ ( image_nat_int2 @ F @ A ) )
% 0.34/0.57 = ( image_nat_int2 @ F @ A ) ) ) ).
% 0.34/0.57
% 0.34/0.57 % insert_image
% 0.34/0.57 thf(fact_2365_insert__image,axiom,
% 0.34/0.57 ! [X: nat,A: set_nat,F: nat > $o] :
% 0.34/0.57 ( ( member_nat @ X @ A )
% 0.34/0.57 => ( ( insert_o @ ( F @ X ) @ ( image_nat_o2 @ F @ A ) )
% 0.34/0.57 = ( image_nat_o2 @ F @ A ) ) ) ).
% 0.34/0.57
% 0.34/0.57 % insert_image
% 0.34/0.57 thf(fact_2366_insert__image,axiom,
% 0.34/0.57 ! [X: nat,A: set_nat,F: nat > product_unit] :
% 0.34/0.57 ( ( member_nat @ X @ A )
% 0.34/0.57 => ( ( insert_Product_unit @ ( F @ X ) @ ( image_8730104196221521654t_unit @ F @ A ) )
% 0.34/0.57 = ( image_8730104196221521654t_unit @ F @ A ) ) ) ).
% 0.34/0.57
% 0.34/0.57 % insert_image
% 0.34/0.57 thf(fact_2367_insert__image,axiom,
% 0.34/0.57 ! [X: int,A: set_int,F: int > nat] :
% 0.34/0.57 ( ( member_int @ X @ A )
% 0.34/0.57 => ( ( insert_nat @ ( F @ X ) @ ( image_int_nat2 @ F @ A ) )
% 0.34/0.57 = ( image_int_nat2 @ F @ A ) ) ) ).
% 0.34/0.57
% 0.34/0.57 % insert_image
% 0.34/0.57 thf(fact_2368_insert__image,axiom,
% 0.34/0.57 ! [X: int,A: set_int,F: int > int] :
% 0.34/0.57 ( ( member_int @ X @ A )
% 0.34/0.57 => ( ( insert_int @ ( F @ X ) @ ( image_int_int2 @ F @ A ) )
% 0.34/0.57 = ( image_int_int2 @ F @ A ) ) ) ).
% 0.34/0.57
% 0.34/0.57 % insert_image
% 0.34/0.57 thf(fact_2369_insert__image,axiom,
% 0.34/0.57 ! [X: int,A: set_int,F: int > $o] :
% 0.34/0.57 ( ( member_int @ X @ A )
% 0.34/0.57 => ( ( insert_o @ ( F @ X ) @ ( image_int_o2 @ F @ A ) )
% 0.34/0.57 = ( image_int_o2 @ F @ A ) ) ) ).
% 0.34/0.57
% 0.34/0.57 % insert_image
% 0.34/0.57 thf(fact_2370_insert__image,axiom,
% 0.34/0.57 ! [X: int,A: set_int,F: int > product_unit] :
% 0.34/0.57 ( ( member_int @ X @ A )
% 0.34/0.57 => ( ( insert_Product_unit @ ( F @ X ) @ ( image_5846526143423701914t_unit @ F @ A ) )
% 0.34/0.57 = ( image_5846526143423701914t_unit @ F @ A ) ) ) ).
% 0.34/0.57
% 0.34/0.57 % insert_image
% 0.34/0.57 thf(fact_2371_image__insert,axiom,
% 0.34/0.57 ! [F: code_integer > code_integer,A2: code_integer,B4: set_Code_integer] :
% 0.34/0.57 ( ( image_4470545334726330049nteger @ F @ ( insert_Code_integer @ A2 @ B4 ) )
% 0.34/0.57 = ( insert_Code_integer @ ( F @ A2 ) @ ( image_4470545334726330049nteger @ F @ B4 ) ) ) ).
% 0.34/0.57
% 0.34/0.57 % image_insert
% 0.34/0.57 thf(fact_2372_image__insert,axiom,
% 0.34/0.57 ! [F: nat > code_integer,A2: nat,B4: set_nat] :
% 0.34/0.57 ( ( image_1215581382706833972nteger @ F @ ( insert_nat @ A2 @ B4 ) )
% 0.34/0.57 = ( insert_Code_integer @ ( F @ A2 ) @ ( image_1215581382706833972nteger @ F @ B4 ) ) ) ).
% 0.34/0.57
% 0.34/0.57 % image_insert
% 0.34/0.57 thf(fact_2373_image__insert,axiom,
% 0.34/0.57 ! [F: nat > nat,A2: nat,B4: set_nat] :
% 0.34/0.57 ( ( image_nat_nat2 @ F @ ( insert_nat @ A2 @ B4 ) )
% 0.34/0.57 = ( insert_nat @ ( F @ A2 ) @ ( image_nat_nat2 @ F @ B4 ) ) ) ).
% 0.34/0.57
% 0.34/0.57 % image_insert
% 0.34/0.57 thf(fact_2374_image__insert,axiom,
% 0.34/0.57 ! [F: nat > int,A2: nat,B4: set_nat] :
% 0.34/0.57 ( ( image_nat_int2 @ F @ ( insert_nat @ A2 @ B4 ) )
% 0.34/0.57 = ( insert_int @ ( F @ A2 ) @ ( image_nat_int2 @ F @ B4 ) ) ) ).
% 0.34/0.57
% 0.34/0.57 % image_insert
% 0.34/0.57 thf(fact_2375_image__insert,axiom,
% 0.34/0.57 ! [F: nat > $o,A2: nat,B4: set_nat] :
% 0.34/0.57 ( ( image_nat_o2 @ F @ ( insert_nat @ A2 @ B4 ) )
% 0.34/0.57 = ( insert_o @ ( F @ A2 ) @ ( image_nat_o2 @ F @ B4 ) ) ) ).
% 0.34/0.57
% 0.34/0.57 % image_insert
% 0.34/0.57 thf(fact_2376_image__insert,axiom,
% 0.34/0.57 ! [F: nat > product_unit,A2: nat,B4: set_nat] :
% 0.34/0.57 ( ( image_8730104196221521654t_unit @ F @ ( insert_nat @ A2 @ B4 ) )
% 0.34/0.57 = ( insert_Product_unit @ ( F @ A2 ) @ ( image_8730104196221521654t_unit @ F @ B4 ) ) ) ).
% 0.34/0.57
% 0.34/0.57 % image_insert
% 0.34/0.57 thf(fact_2377_image__insert,axiom,
% 0.34/0.57 ! [F: int > nat,A2: int,B4: set_int] :
% 0.34/0.57 ( ( image_int_nat2 @ F @ ( insert_int @ A2 @ B4 ) )
% 0.34/0.57 = ( insert_nat @ ( F @ A2 ) @ ( image_int_nat2 @ F @ B4 ) ) ) ).
% 0.34/0.57
% 0.34/0.57 % image_insert
% 0.34/0.57 thf(fact_2378_image__insert,axiom,
% 0.34/0.57 ! [F: int > int,A2: int,B4: set_int] :
% 0.34/0.57 ( ( image_int_int2 @ F @ ( insert_int @ A2 @ B4 ) )
% 0.34/0.57 = ( insert_int @ ( F @ A2 ) @ ( image_int_int2 @ F @ B4 ) ) ) ).
% 0.34/0.57
% 0.34/0.57 % image_insert
% 0.34/0.57 thf(fact_2379_image__insert,axiom,
% 0.34/0.57 ! [F: int > $o,A2: int,B4: set_int] :
% 0.34/0.57 ( ( image_int_o2 @ F @ ( insert_int @ A2 @ B4 ) )
% 0.34/0.57 = ( insert_o @ ( F @ A2 ) @ ( image_int_o2 @ F @ B4 ) ) ) ).
% 0.34/0.57
% 0.34/0.57 % image_insert
% 0.34/0.57 thf(fact_2380_image__insert,axiom,
% 0.34/0.57 ! [F: int > product_unit,A2: int,B4: set_int] :
% 0.34/0.57 ( ( image_5846526143423701914t_unit @ F @ ( insert_int @ A2 @ B4 ) )
% 0.34/0.57 = ( insert_Product_unit @ ( F @ A2 ) @ ( image_5846526143423701914t_unit @ F @ B4 ) ) ) ).
% 0.34/0.57
% 0.34/0.57 % image_insert
% 0.34/0.57 thf(fact_2381_img__snd,axiom,
% 0.34/0.57 ! [A2: nat,B: nat,S: set_Pr1261947904930325089at_nat] :
% 0.34/0.57 ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A2 @ B ) @ S )
% 0.34/0.57 => ( member_nat @ B @ ( image_2486076414777270412at_nat @ product_snd_nat_nat @ S ) ) ) ).
% 0.34/0.57
% 0.34/0.57 % img_snd
% 0.34/0.57 thf(fact_2382_img__snd,axiom,
% 0.34/0.57 ! [A2: produc3658429121746597890et_nat > $o,B: produc3658429121746597890et_nat,S: set_Pr3286484037609594932et_nat] :
% 0.34/0.57 ( ( member1996754912294343701et_nat @ ( produc5001842942810119800et_nat @ A2 @ B ) @ S )
% 0.34/0.57 => ( member6260224972018164377et_nat @ B @ ( image_4390311411407041927et_nat @ produc4011572625026189258et_nat @ S ) ) ) ).
% 0.34/0.57
% 0.34/0.57 % img_snd
% 0.34/0.57 thf(fact_2383_img__snd,axiom,
% 0.34/0.57 ! [A2: produc3658429121746597890et_nat > $o,B: produc3925858234332021118et_nat,S: set_Pr8536935166611901872et_nat] :
% 0.34/0.57 ( ( member6124377750444531601et_nat @ ( produc2245416461498447860et_nat @ A2 @ B ) @ S )
% 0.34/0.57 => ( member1996754912294343701et_nat @ B @ ( image_4557310310409863807et_nat @ produc5374455773327741254et_nat @ S ) ) ) ).
% 0.34/0.57
% 0.34/0.57 % img_snd
% 0.34/0.57 thf(fact_2384_img__snd,axiom,
% 0.34/0.57 ! [A2: heap_e7401611519738050253t_unit,B: set_nat,S: set_Pr3948176798113811640et_nat] :
% 0.34/0.57 ( ( member6260224972018164377et_nat @ ( produc7507926704131184380et_nat @ A2 @ B ) @ S )
% 0.34/0.57 => ( member_set_nat @ B @ ( image_6088997261112373647et_nat @ produc8586169260539613262et_nat @ S ) ) ) ).
% 0.34/0.57
% 0.34/0.57 % img_snd
% 0.34/0.57 thf(fact_2385_img__snd,axiom,
% 0.34/0.57 ! [A2: int,B: int,S: set_Pr958786334691620121nt_int] :
% 0.34/0.57 ( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ A2 @ B ) @ S )
% 0.34/0.57 => ( member_int @ B @ ( image_5042161079198086560nt_int @ product_snd_int_int @ S ) ) ) ).
% 0.34/0.57
% 0.34/0.57 % img_snd
% 0.34/0.57 thf(fact_2386_Diff__cancel,axiom,
% 0.34/0.57 ! [A: set_Product_unit] :
% 0.34/0.57 ( ( minus_6452836326544984404t_unit @ A @ A )
% 0.34/0.57 = bot_bo3957492148770167129t_unit ) ).
% 0.34/0.57
% 0.34/0.57 % Diff_cancel
% 0.34/0.57 thf(fact_2387_Diff__cancel,axiom,
% 0.34/0.57 ! [A: set_o] :
% 0.34/0.57 ( ( minus_minus_set_o @ A @ A )
% 0.34/0.57 = bot_bot_set_o ) ).
% 0.34/0.57
% 0.34/0.57 % Diff_cancel
% 0.34/0.57 thf(fact_2388_Diff__cancel,axiom,
% 0.34/0.57 ! [A: set_int] :
% 0.34/0.57 ( ( minus_minus_set_int @ A @ A )
% 0.34/0.57 = bot_bot_set_int ) ).
% 0.34/0.57
% 0.34/0.57 % Diff_cancel
% 0.34/0.57 thf(fact_2389_Diff__cancel,axiom,
% 0.34/0.57 ! [A: set_Pr4532377907799695533_nat_o] :
% 0.34/0.57 ( ( minus_1801376950450012436_nat_o @ A @ A )
% 0.34/0.57 = bot_bo7824918357723582233_nat_o ) ).
% 0.34/0.57
% 0.34/0.57 % Diff_cancel
% 0.34/0.57 thf(fact_2390_Diff__cancel,axiom,
% 0.34/0.57 ! [A: set_nat] :
% 0.34/0.57 ( ( minus_minus_set_nat @ A @ A )
% 0.34/0.57 = bot_bot_set_nat ) ).
% 0.34/0.57
% 0.34/0.57 % Diff_cancel
% 0.34/0.57 thf(fact_2391_empty__Diff,axiom,
% 0.34/0.57 ! [A: set_Product_unit] :
% 0.34/0.57 ( ( minus_6452836326544984404t_unit @ bot_bo3957492148770167129t_unit @ A )
% 0.34/0.57 = bot_bo3957492148770167129t_unit ) ).
% 0.34/0.57
% 0.34/0.57 % empty_Diff
% 0.34/0.57 thf(fact_2392_empty__Diff,axiom,
% 0.34/0.57 ! [A: set_o] :
% 0.34/0.57 ( ( minus_minus_set_o @ bot_bot_set_o @ A )
% 0.34/0.57 = bot_bot_set_o ) ).
% 0.34/0.57
% 0.34/0.57 % empty_Diff
% 0.34/0.57 thf(fact_2393_empty__Diff,axiom,
% 0.34/0.57 ! [A: set_int] :
% 0.34/0.57 ( ( minus_minus_set_int @ bot_bot_set_int @ A )
% 0.34/0.57 = bot_bot_set_int ) ).
% 0.34/0.57
% 0.34/0.57 % empty_Diff
% 0.34/0.57 thf(fact_2394_empty__Diff,axiom,
% 0.34/0.57 ! [A: set_Pr4532377907799695533_nat_o] :
% 0.34/0.57 ( ( minus_1801376950450012436_nat_o @ bot_bo7824918357723582233_nat_o @ A )
% 0.34/0.57 = bot_bo7824918357723582233_nat_o ) ).
% 0.34/0.57
% 0.34/0.57 % empty_Diff
% 0.34/0.57 thf(fact_2395_empty__Diff,axiom,
% 0.34/0.57 ! [A: set_nat] :
% 0.34/0.57 ( ( minus_minus_set_nat @ bot_bot_set_nat @ A )
% 0.34/0.57 = bot_bot_set_nat ) ).
% 0.34/0.57
% 0.34/0.57 % empty_Diff
% 0.34/0.57 thf(fact_2396_Diff__empty,axiom,
% 0.34/0.57 ! [A: set_Product_unit] :
% 0.34/0.57 ( ( minus_6452836326544984404t_unit @ A @ bot_bo3957492148770167129t_unit )
% 0.34/0.57 = A ) ).
% 0.34/0.57
% 0.34/0.57 % Diff_empty
% 0.34/0.57 thf(fact_2397_Diff__empty,axiom,
% 0.34/0.57 ! [A: set_o] :
% 0.34/0.57 ( ( minus_minus_set_o @ A @ bot_bot_set_o )
% 0.34/0.57 = A ) ).
% 0.34/0.57
% 0.34/0.57 % Diff_empty
% 0.34/0.57 thf(fact_2398_Diff__empty,axiom,
% 0.34/0.57 ! [A: set_int] :
% 0.34/0.57 ( ( minus_minus_set_int @ A @ bot_bot_set_int )
% 0.34/0.57 = A ) ).
% 0.34/0.57
% 0.34/0.57 % Diff_empty
% 0.34/0.57 thf(fact_2399_Diff__empty,axiom,
% 0.34/0.57 ! [A: set_Pr4532377907799695533_nat_o] :
% 0.34/0.57 ( ( minus_1801376950450012436_nat_o @ A @ bot_bo7824918357723582233_nat_o )
% 0.34/0.57 = A ) ).
% 0.34/0.57
% 0.34/0.57 % Diff_empty
% 0.34/0.57 thf(fact_2400_Diff__empty,axiom,
% 0.34/0.57 ! [A: set_nat] :
% 0.34/0.57 ( ( minus_minus_set_nat @ A @ bot_bot_set_nat )
% 0.34/0.57 = A ) ).
% 0.34/0.57
% 0.34/0.57 % Diff_empty
% 0.34/0.57 thf(fact_2401_insert__Diff1,axiom,
% 0.34/0.57 ! [X: produc3658429121746597890et_nat > $o,B4: set_Pr4532377907799695533_nat_o,A: set_Pr4532377907799695533_nat_o] :
% 0.34/0.57 ( ( member6576561426505652726_nat_o @ X @ B4 )
% 0.34/0.57 => ( ( minus_1801376950450012436_nat_o @ ( insert5175938949040314269_nat_o @ X @ A ) @ B4 )
% 0.34/0.57 = ( minus_1801376950450012436_nat_o @ A @ B4 ) ) ) ).
% 0.34/0.57
% 0.34/0.57 % insert_Diff1
% 0.34/0.57 thf(fact_2402_insert__Diff1,axiom,
% 0.34/0.57 ! [X: int,B4: set_int,A: set_int] :
% 0.34/0.57 ( ( member_int @ X @ B4 )
% 0.34/0.57 => ( ( minus_minus_set_int @ ( insert_int @ X @ A ) @ B4 )
% 0.34/0.57 = ( minus_minus_set_int @ A @ B4 ) ) ) ).
% 0.34/0.57
% 0.34/0.57 % insert_Diff1
% 0.34/0.57 thf(fact_2403_insert__Diff1,axiom,
% 0.34/0.57 ! [X: $o,B4: set_o,A: set_o] :
% 0.34/0.57 ( ( member_o @ X @ B4 )
% 0.34/0.57 => ( ( minus_minus_set_o @ ( insert_o @ X @ A ) @ B4 )
% 0.34/0.57 = ( minus_minus_set_o @ A @ B4 ) ) ) ).
% 0.34/0.57
% 0.34/0.57 % insert_Diff1
% 0.34/0.57 thf(fact_2404_insert__Diff1,axiom,
% 0.34/0.57 ! [X: product_unit,B4: set_Product_unit,A: set_Product_unit] :
% 0.34/0.57 ( ( member_Product_unit @ X @ B4 )
% 0.34/0.57 => ( ( minus_6452836326544984404t_unit @ ( insert_Product_unit @ X @ A ) @ B4 )
% 0.34/0.57 = ( minus_6452836326544984404t_unit @ A @ B4 ) ) ) ).
% 0.34/0.57
% 0.34/0.57 % insert_Diff1
% 0.34/0.57 thf(fact_2405_insert__Diff1,axiom,
% 0.34/0.57 ! [X: nat,B4: set_nat,A: set_nat] :
% 0.34/0.57 ( ( member_nat @ X @ B4 )
% 0.34/0.57 => ( ( minus_minus_set_nat @ ( insert_nat @ X @ A ) @ B4 )
% 0.34/0.57 = ( minus_minus_set_nat @ A @ B4 ) ) ) ).
% 0.34/0.57
% 0.34/0.57 % insert_Diff1
% 0.34/0.57 thf(fact_2406_Diff__insert0,axiom,
% 0.34/0.57 ! [X: produc3658429121746597890et_nat > $o,A: set_Pr4532377907799695533_nat_o,B4: set_Pr4532377907799695533_nat_o] :
% 0.34/0.57 ( ~ ( member6576561426505652726_nat_o @ X @ A )
% 0.34/0.57 => ( ( minus_1801376950450012436_nat_o @ A @ ( insert5175938949040314269_nat_o @ X @ B4 ) )
% 0.34/0.57 = ( minus_1801376950450012436_nat_o @ A @ B4 ) ) ) ).
% 0.34/0.57
% 0.34/0.57 % Diff_insert0
% 0.34/0.57 thf(fact_2407_Diff__insert0,axiom,
% 0.34/0.57 ! [X: int,A: set_int,B4: set_int] :
% 0.34/0.57 ( ~ ( member_int @ X @ A )
% 0.34/0.57 => ( ( minus_minus_set_int @ A @ ( insert_int @ X @ B4 ) )
% 0.34/0.57 = ( minus_minus_set_int @ A @ B4 ) ) ) ).
% 0.34/0.57
% 0.34/0.57 % Diff_insert0
% 0.34/0.57 thf(fact_2408_Diff__insert0,axiom,
% 0.34/0.57 ! [X: $o,A: set_o,B4: set_o] :
% 0.34/0.57 ( ~ ( member_o @ X @ A )
% 0.34/0.57 => ( ( minus_minus_set_o @ A @ ( insert_o @ X @ B4 ) )
% 0.34/0.57 = ( minus_minus_set_o @ A @ B4 ) ) ) ).
% 0.34/0.57
% 0.34/0.57 % Diff_insert0
% 0.34/0.57 thf(fact_2409_Diff__insert0,axiom,
% 0.34/0.57 ! [X: product_unit,A: set_Product_unit,B4: set_Product_unit] :
% 0.34/0.57 ( ~ ( member_Product_unit @ X @ A )
% 0.34/0.57 => ( ( minus_6452836326544984404t_unit @ A @ ( insert_Product_unit @ X @ B4 ) )
% 0.34/0.57 = ( minus_6452836326544984404t_unit @ A @ B4 ) ) ) ).
% 0.34/0.57
% 0.34/0.57 % Diff_insert0
% 0.34/0.57 thf(fact_2410_Diff__insert0,axiom,
% 0.34/0.57 ! [X: nat,A: set_nat,B4: set_nat] :
% 0.34/0.57 ( ~ ( member_nat @ X @ A )
% 0.34/0.57 => ( ( minus_minus_set_nat @ A @ ( insert_nat @ X @ B4 ) )
% 0.34/0.57 = ( minus_minus_set_nat @ A @ B4 ) ) ) ).
% 0.34/0.57
% 0.34/0.57 % Diff_insert0
% 0.34/0.57 thf(fact_2411_Un__Diff__cancel2,axiom,
% 0.34/0.57 ! [B4: set_nat,A: set_nat] :
% 0.34/0.57 ( ( sup_sup_set_nat @ ( minus_minus_set_nat @ B4 @ A ) @ A )
% 0.34/0.57 = ( sup_sup_set_nat @ B4 @ A ) ) ).
% 0.34/0.57
% 0.34/0.57 % Un_Diff_cancel2
% 0.34/0.57 thf(fact_2412_Un__Diff__cancel,axiom,
% 0.34/0.57 ! [A: set_nat,B4: set_nat] :
% 0.34/0.57 ( ( sup_sup_set_nat @ A @ ( minus_minus_set_nat @ B4 @ A ) )
% 0.34/0.57 = ( sup_sup_set_nat @ A @ B4 ) ) ).
% 0.34/0.57
% 0.34/0.57 % Un_Diff_cancel
% 0.34/0.57 thf(fact_2413_converse__iff,axiom,
% 0.34/0.57 ! [A2: set_nat,B: heap_e7401611519738050253t_unit,R2: set_Pr3948176798113811640et_nat] :
% 0.34/0.57 ( ( member727419437372278867t_unit @ ( produc2471974422597015918t_unit @ A2 @ B ) @ ( conver7272282955447398104et_nat @ R2 ) )
% 0.34/0.57 = ( member6260224972018164377et_nat @ ( produc7507926704131184380et_nat @ B @ A2 ) @ R2 ) ) ).
% 0.34/0.57
% 0.34/0.57 % converse_iff
% 0.34/0.57 thf(fact_2414_converse__iff,axiom,
% 0.34/0.57 ! [A2: produc3658429121746597890et_nat,B: produc3658429121746597890et_nat > $o,R2: set_Pr3286484037609594932et_nat] :
% 0.34/0.57 ( ( member4978015717948814295_nat_o @ ( produc2829238748666232050_nat_o @ A2 @ B ) @ ( conver4003969491454187092et_nat @ R2 ) )
% 0.34/0.57 = ( member1996754912294343701et_nat @ ( produc5001842942810119800et_nat @ B @ A2 ) @ R2 ) ) ).
% 0.34/0.57
% 0.34/0.57 % converse_iff
% 0.34/0.57 thf(fact_2415_converse__iff,axiom,
% 0.34/0.57 ! [A2: produc3925858234332021118et_nat,B: produc3658429121746597890et_nat > $o,R2: set_Pr8536935166611901872et_nat] :
% 0.34/0.57 ( ( member8354763867512313179_nat_o @ ( produc3674526432113299062_nat_o @ A2 @ B ) @ ( conver3577387295554834384et_nat @ R2 ) )
% 0.34/0.57 = ( member6124377750444531601et_nat @ ( produc2245416461498447860et_nat @ B @ A2 ) @ R2 ) ) ).
% 0.34/0.57
% 0.34/0.57 % converse_iff
% 0.34/0.57 thf(fact_2416_converse__iff,axiom,
% 0.34/0.57 ! [A2: heap_e7401611519738050253t_unit,B: set_nat,R2: set_Pr4071100604818519410t_unit] :
% 0.34/0.57 ( ( member6260224972018164377et_nat @ ( produc7507926704131184380et_nat @ A2 @ B ) @ ( conver2236330673913229642t_unit @ R2 ) )
% 0.34/0.57 = ( member727419437372278867t_unit @ ( produc2471974422597015918t_unit @ B @ A2 ) @ R2 ) ) ).
% 0.34/0.57
% 0.34/0.57 % converse_iff
% 0.34/0.57 thf(fact_2417_converse__iff,axiom,
% 0.34/0.57 ! [A2: nat,B: nat,R2: set_Pr1261947904930325089at_nat] :
% 0.34/0.57 ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A2 @ B ) @ ( converse_nat_nat @ R2 ) )
% 0.34/0.57 = ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ B @ A2 ) @ R2 ) ) ).
% 0.34/0.57
% 0.34/0.57 % converse_iff
% 0.34/0.57 thf(fact_2418_converse__iff,axiom,
% 0.34/0.57 ! [A2: int,B: int,R2: set_Pr958786334691620121nt_int] :
% 0.34/0.57 ( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ A2 @ B ) @ ( converse_int_int @ R2 ) )
% 0.34/0.57 = ( member5262025264175285858nt_int @ ( product_Pair_int_int @ B @ A2 ) @ R2 ) ) ).
% 0.34/0.57
% 0.34/0.57 % converse_iff
% 0.34/0.57 thf(fact_2419_converse__iff,axiom,
% 0.34/0.57 ! [A2: produc3658429121746597890et_nat > $o,B: produc3658429121746597890et_nat,R2: set_Pr4208978187861016310_nat_o] :
% 0.34/0.57 ( ( member1996754912294343701et_nat @ ( produc5001842942810119800et_nat @ A2 @ B ) @ ( conver1831365297310299342_nat_o @ R2 ) )
% 0.34/0.57 = ( member4978015717948814295_nat_o @ ( produc2829238748666232050_nat_o @ B @ A2 ) @ R2 ) ) ).
% 0.34/0.57
% 0.34/0.57 % converse_iff
% 0.34/0.57 thf(fact_2420_converse__iff,axiom,
% 0.34/0.57 ! [A2: produc3658429121746597890et_nat > $o,B: produc3925858234332021118et_nat,R2: set_Pr7705059561495379578_nat_o] :
% 0.34/0.57 ( ( member6124377750444531601et_nat @ ( produc2245416461498447860et_nat @ A2 @ B ) @ ( conver5006497266169685586_nat_o @ R2 ) )
% 0.34/0.57 = ( member8354763867512313179_nat_o @ ( produc3674526432113299062_nat_o @ B @ A2 ) @ R2 ) ) ).
% 0.34/0.57
% 0.34/0.57 % converse_iff
% 0.34/0.57 thf(fact_2421_Diff__eq__empty__iff,axiom,
% 0.34/0.57 ! [A: set_Product_unit,B4: set_Product_unit] :
% 0.34/0.57 ( ( ( minus_6452836326544984404t_unit @ A @ B4 )
% 0.34/0.57 = bot_bo3957492148770167129t_unit )
% 0.34/0.57 = ( ord_le3507040750410214029t_unit @ A @ B4 ) ) ).
% 0.34/0.57
% 0.34/0.57 % Diff_eq_empty_iff
% 0.34/0.57 thf(fact_2422_Diff__eq__empty__iff,axiom,
% 0.34/0.57 ! [A: set_o,B4: set_o] :
% 0.34/0.57 ( ( ( minus_minus_set_o @ A @ B4 )
% 0.34/0.57 = bot_bot_set_o )
% 0.34/0.57 = ( ord_less_eq_set_o @ A @ B4 ) ) ).
% 0.34/0.57
% 0.34/0.57 % Diff_eq_empty_iff
% 0.34/0.57 thf(fact_2423_Diff__eq__empty__iff,axiom,
% 0.34/0.57 ! [A: set_int,B4: set_int] :
% 0.34/0.57 ( ( ( minus_minus_set_int @ A @ B4 )
% 0.34/0.57 = bot_bot_set_int )
% 0.34/0.57 = ( ord_less_eq_set_int @ A @ B4 ) ) ).
% 0.34/0.57
% 0.34/0.57 % Diff_eq_empty_iff
% 0.34/0.57 thf(fact_2424_Diff__eq__empty__iff,axiom,
% 0.34/0.57 ! [A: set_Pr4532377907799695533_nat_o,B4: set_Pr4532377907799695533_nat_o] :
% 0.34/0.57 ( ( ( minus_1801376950450012436_nat_o @ A @ B4 )
% 0.34/0.57 = bot_bo7824918357723582233_nat_o )
% 0.34/0.57 = ( ord_le2965882846123202637_nat_o @ A @ B4 ) ) ).
% 0.34/0.57
% 0.34/0.57 % Diff_eq_empty_iff
% 0.34/0.57 thf(fact_2425_Diff__eq__empty__iff,axiom,
% 0.34/0.57 ! [A: set_nat,B4: set_nat] :
% 0.34/0.57 ( ( ( minus_minus_set_nat @ A @ B4 )
% 0.34/0.57 = bot_bot_set_nat )
% 0.34/0.57 = ( ord_less_eq_set_nat @ A @ B4 ) ) ).
% 0.34/0.57
% 0.34/0.57 % Diff_eq_empty_iff
% 0.34/0.57 thf(fact_2426_insert__Diff__single,axiom,
% 0.34/0.57 ! [A2: product_unit,A: set_Product_unit] :
% 0.34/0.57 ( ( insert_Product_unit @ A2 @ ( minus_6452836326544984404t_unit @ A @ ( insert_Product_unit @ A2 @ bot_bo3957492148770167129t_unit ) ) )
% 0.34/0.57 = ( insert_Product_unit @ A2 @ A ) ) ).
% 0.34/0.57
% 0.34/0.57 % insert_Diff_single
% 0.34/0.57 thf(fact_2427_insert__Diff__single,axiom,
% 0.34/0.57 ! [A2: $o,A: set_o] :
% 0.34/0.57 ( ( insert_o @ A2 @ ( minus_minus_set_o @ A @ ( insert_o @ A2 @ bot_bot_set_o ) ) )
% 0.34/0.57 = ( insert_o @ A2 @ A ) ) ).
% 0.34/0.57
% 0.34/0.57 % insert_Diff_single
% 0.34/0.57 thf(fact_2428_insert__Diff__single,axiom,
% 0.34/0.57 ! [A2: int,A: set_int] :
% 0.34/0.57 ( ( insert_int @ A2 @ ( minus_minus_set_int @ A @ ( insert_int @ A2 @ bot_bot_set_int ) ) )
% 0.34/0.57 = ( insert_int @ A2 @ A ) ) ).
% 0.34/0.57
% 0.34/0.57 % insert_Diff_single
% 0.34/0.57 thf(fact_2429_insert__Diff__single,axiom,
% 0.34/0.57 ! [A2: produc3658429121746597890et_nat > $o,A: set_Pr4532377907799695533_nat_o] :
% 0.34/0.57 ( ( insert5175938949040314269_nat_o @ A2 @ ( minus_1801376950450012436_nat_o @ A @ ( insert5175938949040314269_nat_o @ A2 @ bot_bo7824918357723582233_nat_o ) ) )
% 0.34/0.57 = ( insert5175938949040314269_nat_o @ A2 @ A ) ) ).
% 0.34/0.57
% 0.34/0.57 % insert_Diff_single
% 0.34/0.57 thf(fact_2430_insert__Diff__single,axiom,
% 0.34/0.57 ! [A2: nat,A: set_nat] :
% 0.34/0.57 ( ( insert_nat @ A2 @ ( minus_minus_set_nat @ A @ ( insert_nat @ A2 @ bot_bot_set_nat ) ) )
% 0.34/0.57 = ( insert_nat @ A2 @ A ) ) ).
% 0.34/0.57
% 0.34/0.57 % insert_Diff_single
% 0.34/0.57 thf(fact_2431_finite__Diff__insert,axiom,
% 0.34/0.57 ! [A: set_int,A2: int,B4: set_int] :
% 0.34/0.57 ( ( finite_finite_int @ ( minus_minus_set_int @ A @ ( insert_int @ A2 @ B4 ) ) )
% 0.34/0.57 = ( finite_finite_int @ ( minus_minus_set_int @ A @ B4 ) ) ) ).
% 0.34/0.57
% 0.34/0.57 % finite_Diff_insert
% 0.34/0.57 thf(fact_2432_finite__Diff__insert,axiom,
% 0.34/0.57 ! [A: set_o,A2: $o,B4: set_o] :
% 0.34/0.57 ( ( finite_finite_o @ ( minus_minus_set_o @ A @ ( insert_o @ A2 @ B4 ) ) )
% 0.34/0.57 = ( finite_finite_o @ ( minus_minus_set_o @ A @ B4 ) ) ) ).
% 0.34/0.57
% 0.34/0.57 % finite_Diff_insert
% 0.34/0.57 thf(fact_2433_finite__Diff__insert,axiom,
% 0.34/0.57 ! [A: set_Product_unit,A2: product_unit,B4: set_Product_unit] :
% 0.34/0.57 ( ( finite4290736615968046902t_unit @ ( minus_6452836326544984404t_unit @ A @ ( insert_Product_unit @ A2 @ B4 ) ) )
% 0.34/0.57 = ( finite4290736615968046902t_unit @ ( minus_6452836326544984404t_unit @ A @ B4 ) ) ) ).
% 0.34/0.57
% 0.34/0.57 % finite_Diff_insert
% 0.34/0.57 thf(fact_2434_finite__Diff__insert,axiom,
% 0.34/0.57 ! [A: set_Code_integer,A2: code_integer,B4: set_Code_integer] :
% 0.34/0.57 ( ( finite6017078050557962740nteger @ ( minus_2355218937544613996nteger @ A @ ( insert_Code_integer @ A2 @ B4 ) ) )
% 0.34/0.57 = ( finite6017078050557962740nteger @ ( minus_2355218937544613996nteger @ A @ B4 ) ) ) ).
% 0.34/0.57
% 0.34/0.57 % finite_Diff_insert
% 0.34/0.57 thf(fact_2435_finite__Diff__insert,axiom,
% 0.34/0.57 ! [A: set_nat,A2: nat,B4: set_nat] :
% 0.34/0.57 ( ( finite_finite_nat @ ( minus_minus_set_nat @ A @ ( insert_nat @ A2 @ B4 ) ) )
% 0.34/0.57 = ( finite_finite_nat @ ( minus_minus_set_nat @ A @ B4 ) ) ) ).
% 0.34/0.57
% 0.34/0.57 % finite_Diff_insert
% 0.34/0.57 thf(fact_2436_Diff__disjoint,axiom,
% 0.34/0.57 ! [A: set_Product_unit,B4: set_Product_unit] :
% 0.34/0.57 ( ( inf_in4660618365625256667t_unit @ A @ ( minus_6452836326544984404t_unit @ B4 @ A ) )
% 0.34/0.57 = bot_bo3957492148770167129t_unit ) ).
% 0.34/0.57
% 0.34/0.57 % Diff_disjoint
% 0.34/0.57 thf(fact_2437_Diff__disjoint,axiom,
% 0.34/0.57 ! [A: set_o,B4: set_o] :
% 0.34/0.57 ( ( inf_inf_set_o @ A @ ( minus_minus_set_o @ B4 @ A ) )
% 0.34/0.57 = bot_bot_set_o ) ).
% 0.34/0.57
% 0.34/0.57 % Diff_disjoint
% 0.34/0.57 thf(fact_2438_Diff__disjoint,axiom,
% 0.34/0.57 ! [A: set_int,B4: set_int] :
% 0.34/0.57 ( ( inf_inf_set_int @ A @ ( minus_minus_set_int @ B4 @ A ) )
% 0.34/0.57 = bot_bot_set_int ) ).
% 0.34/0.57
% 0.34/0.57 % Diff_disjoint
% 0.34/0.57 thf(fact_2439_Diff__disjoint,axiom,
% 0.34/0.57 ! [A: set_Pr4532377907799695533_nat_o,B4: set_Pr4532377907799695533_nat_o] :
% 0.34/0.57 ( ( inf_in1906310914598751387_nat_o @ A @ ( minus_1801376950450012436_nat_o @ B4 @ A ) )
% 0.34/0.57 = bot_bo7824918357723582233_nat_o ) ).
% 0.34/0.57
% 0.34/0.57 % Diff_disjoint
% 0.34/0.57 thf(fact_2440_Diff__disjoint,axiom,
% 0.34/0.57 ! [A: set_nat,B4: set_nat] :
% 0.34/0.57 ( ( inf_inf_set_nat @ A @ ( minus_minus_set_nat @ B4 @ A ) )
% 0.34/0.57 = bot_bot_set_nat ) ).
% 0.34/0.57
% 0.34/0.57 % Diff_disjoint
% 0.34/0.57 thf(fact_2441_image__diff__subset,axiom,
% 0.34/0.57 ! [F: code_integer > code_integer,A: set_Code_integer,B4: set_Code_integer] : ( ord_le7084787975880047091nteger @ ( minus_2355218937544613996nteger @ ( image_4470545334726330049nteger @ F @ A ) @ ( image_4470545334726330049nteger @ F @ B4 ) ) @ ( image_4470545334726330049nteger @ F @ ( minus_2355218937544613996nteger @ A @ B4 ) ) ) ).
% 0.34/0.57
% 0.34/0.57 % image_diff_subset
% 0.34/0.57 thf(fact_2442_image__diff__subset,axiom,
% 0.34/0.57 ! [F: nat > set_nat,A: set_nat,B4: set_nat] : ( ord_le6893508408891458716et_nat @ ( minus_2163939370556025621et_nat @ ( image_nat_set_nat @ F @ A ) @ ( image_nat_set_nat @ F @ B4 ) ) @ ( image_nat_set_nat @ F @ ( minus_minus_set_nat @ A @ B4 ) ) ) ).
% 0.34/0.57
% 0.34/0.57 % image_diff_subset
% 0.34/0.57 thf(fact_2443_image__diff__subset,axiom,
% 0.34/0.57 ! [F: nat > code_integer,A: set_nat,B4: set_nat] : ( ord_le7084787975880047091nteger @ ( minus_2355218937544613996nteger @ ( image_1215581382706833972nteger @ F @ A ) @ ( image_1215581382706833972nteger @ F @ B4 ) ) @ ( image_1215581382706833972nteger @ F @ ( minus_minus_set_nat @ A @ B4 ) ) ) ).
% 0.34/0.57
% 0.34/0.57 % image_diff_subset
% 0.34/0.57 thf(fact_2444_image__diff__subset,axiom,
% 0.34/0.57 ! [F: nat > nat,A: set_nat,B4: set_nat] : ( ord_less_eq_set_nat @ ( minus_minus_set_nat @ ( image_nat_nat2 @ F @ A ) @ ( image_nat_nat2 @ F @ B4 ) ) @ ( image_nat_nat2 @ F @ ( minus_minus_set_nat @ A @ B4 ) ) ) ).
% 0.34/0.57
% 0.34/0.57 % image_diff_subset
% 0.34/0.57 thf(fact_2445_rev__image__eqI,axiom,
% 0.34/0.57 ! [X: code_integer,A: set_Code_integer,B: code_integer,F: code_integer > code_integer] :
% 0.34/0.57 ( ( member_Code_integer @ X @ A )
% 0.34/0.57 => ( ( B
% 0.34/0.57 = ( F @ X ) )
% 0.34/0.57 => ( member_Code_integer @ B @ ( image_4470545334726330049nteger @ F @ A ) ) ) ) ).
% 0.34/0.57
% 0.34/0.57 % rev_image_eqI
% 0.34/0.57 thf(fact_2446_rev__image__eqI,axiom,
% 0.34/0.57 ! [X: nat,A: set_nat,B: code_integer,F: nat > code_integer] :
% 0.34/0.57 ( ( member_nat @ X @ A )
% 0.34/0.57 => ( ( B
% 0.34/0.57 = ( F @ X ) )
% 0.34/0.57 => ( member_Code_integer @ B @ ( image_1215581382706833972nteger @ F @ A ) ) ) ) ).
% 0.34/0.57
% 0.34/0.57 % rev_image_eqI
% 0.34/0.57 thf(fact_2447_rev__image__eqI,axiom,
% 0.34/0.57 ! [X: nat,A: set_nat,B: nat,F: nat > nat] :
% 0.34/0.57 ( ( member_nat @ X @ A )
% 0.34/0.57 => ( ( B
% 0.34/0.57 = ( F @ X ) )
% 0.34/0.57 => ( member_nat @ B @ ( image_nat_nat2 @ F @ A ) ) ) ) ).
% 0.34/0.57
% 0.34/0.57 % rev_image_eqI
% 0.34/0.57 thf(fact_2448_rev__image__eqI,axiom,
% 0.34/0.57 ! [X: nat,A: set_nat,B: int,F: nat > int] :
% 0.34/0.57 ( ( member_nat @ X @ A )
% 0.34/0.57 => ( ( B
% 0.34/0.57 = ( F @ X ) )
% 0.34/0.57 => ( member_int @ B @ ( image_nat_int2 @ F @ A ) ) ) ) ).
% 0.34/0.57
% 0.34/0.57 % rev_image_eqI
% 0.34/0.57 thf(fact_2449_rev__image__eqI,axiom,
% 0.34/0.57 ! [X: nat,A: set_nat,B: $o,F: nat > $o] :
% 0.34/0.57 ( ( member_nat @ X @ A )
% 0.34/0.57 => ( ( B
% 0.34/0.57 = ( F @ X ) )
% 0.34/0.57 => ( member_o @ B @ ( image_nat_o2 @ F @ A ) ) ) ) ).
% 0.34/0.57
% 0.34/0.57 % rev_image_eqI
% 0.34/0.57 thf(fact_2450_rev__image__eqI,axiom,
% 0.34/0.57 ! [X: nat,A: set_nat,B: product_unit,F: nat > product_unit] :
% 0.34/0.57 ( ( member_nat @ X @ A )
% 0.34/0.57 => ( ( B
% 0.34/0.57 = ( F @ X ) )
% 0.34/0.57 => ( member_Product_unit @ B @ ( image_8730104196221521654t_unit @ F @ A ) ) ) ) ).
% 0.34/0.57
% 0.34/0.57 % rev_image_eqI
% 0.34/0.57 thf(fact_2451_rev__image__eqI,axiom,
% 0.34/0.57 ! [X: int,A: set_int,B: nat,F: int > nat] :
% 0.34/0.57 ( ( member_int @ X @ A )
% 0.34/0.57 => ( ( B
% 0.34/0.57 = ( F @ X ) )
% 0.34/0.57 => ( member_nat @ B @ ( image_int_nat2 @ F @ A ) ) ) ) ).
% 0.34/0.57
% 0.34/0.57 % rev_image_eqI
% 0.34/0.57 thf(fact_2452_rev__image__eqI,axiom,
% 0.34/0.57 ! [X: int,A: set_int,B: int,F: int > int] :
% 0.34/0.57 ( ( member_int @ X @ A )
% 0.34/0.57 => ( ( B
% 0.34/0.57 = ( F @ X ) )
% 0.34/0.57 => ( member_int @ B @ ( image_int_int2 @ F @ A ) ) ) ) ).
% 0.34/0.57
% 0.34/0.57 % rev_image_eqI
% 0.34/0.57 thf(fact_2453_rev__image__eqI,axiom,
% 0.34/0.57 ! [X: int,A: set_int,B: $o,F: int > $o] :
% 0.34/0.57 ( ( member_int @ X @ A )
% 0.34/0.57 => ( ( B
% 0.34/0.57 = ( F @ X ) )
% 0.34/0.57 => ( member_o @ B @ ( image_int_o2 @ F @ A ) ) ) ) ).
% 0.34/0.57
% 0.34/0.57 % rev_image_eqI
% 0.34/0.57 thf(fact_2454_rev__image__eqI,axiom,
% 0.34/0.57 ! [X: int,A: set_int,B: product_unit,F: int > product_unit] :
% 0.34/0.57 ( ( member_int @ X @ A )
% 0.34/0.57 => ( ( B
% 0.34/0.57 = ( F @ X ) )
% 0.34/0.57 => ( member_Product_unit @ B @ ( image_5846526143423701914t_unit @ F @ A ) ) ) ) ).
% 0.34/0.57
% 0.34/0.57 % rev_image_eqI
% 0.34/0.57 thf(fact_2455_ball__imageD,axiom,
% 0.34/0.57 ! [F: nat > set_nat,A: set_nat,P: set_nat > $o] :
% 0.34/0.57 ( ! [X3: set_nat] :
% 0.34/0.57 ( ( member_set_nat @ X3 @ ( image_nat_set_nat @ F @ A ) )
% 0.34/0.57 => ( P @ X3 ) )
% 0.34/0.57 => ! [X6: nat] :
% 0.34/0.57 ( ( member_nat @ X6 @ A )
% 0.34/0.57 => ( P @ ( F @ X6 ) ) ) ) ).
% 0.34/0.57
% 0.34/0.57 % ball_imageD
% 0.34/0.57 thf(fact_2456_ball__imageD,axiom,
% 0.34/0.57 ! [F: nat > nat,A: set_nat,P: nat > $o] :
% 0.34/0.57 ( ! [X3: nat] :
% 0.34/0.57 ( ( member_nat @ X3 @ ( image_nat_nat2 @ F @ A ) )
% 0.34/0.57 => ( P @ X3 ) )
% 0.34/0.57 => ! [X6: nat] :
% 0.34/0.57 ( ( member_nat @ X6 @ A )
% 0.34/0.57 => ( P @ ( F @ X6 ) ) ) ) ).
% 0.34/0.57
% 0.34/0.57 % ball_imageD
% 0.34/0.57 thf(fact_2457_ball__imageD,axiom,
% 0.34/0.57 ! [F: code_integer > code_integer,A: set_Code_integer,P: code_integer > $o] :
% 0.34/0.57 ( ! [X3: code_integer] :
% 0.34/0.57 ( ( member_Code_integer @ X3 @ ( image_4470545334726330049nteger @ F @ A ) )
% 0.34/0.57 => ( P @ X3 ) )
% 0.34/0.57 => ! [X6: code_integer] :
% 0.34/0.57 ( ( member_Code_integer @ X6 @ A )
% 0.34/0.57 => ( P @ ( F @ X6 ) ) ) ) ).
% 0.34/0.57
% 0.34/0.57 % ball_imageD
% 0.34/0.57 thf(fact_2458_ball__imageD,axiom,
% 0.34/0.57 ! [F: nat > code_integer,A: set_nat,P: code_integer > $o] :
% 0.34/0.57 ( ! [X3: code_integer] :
% 0.34/0.57 ( ( member_Code_integer @ X3 @ ( image_1215581382706833972nteger @ F @ A ) )
% 0.34/0.57 => ( P @ X3 ) )
% 0.34/0.57 => ! [X6: nat] :
% 0.34/0.57 ( ( member_nat @ X6 @ A )
% 0.34/0.57 => ( P @ ( F @ X6 ) ) ) ) ).
% 0.34/0.57
% 0.34/0.57 % ball_imageD
% 0.34/0.57 thf(fact_2459_image__cong,axiom,
% 0.34/0.57 ! [M: set_Code_integer,N: set_Code_integer,F: code_integer > code_integer,G: code_integer > code_integer] :
% 0.34/0.57 ( ( M = N )
% 0.34/0.57 => ( ! [X3: code_integer] :
% 0.34/0.57 ( ( member_Code_integer @ X3 @ N )
% 0.34/0.57 => ( ( F @ X3 )
% 0.34/0.57 = ( G @ X3 ) ) )
% 0.34/0.57 => ( ( image_4470545334726330049nteger @ F @ M )
% 0.34/0.57 = ( image_4470545334726330049nteger @ G @ N ) ) ) ) ).
% 0.34/0.57
% 0.34/0.57 % image_cong
% 0.34/0.57 thf(fact_2460_image__cong,axiom,
% 0.34/0.57 ! [M: set_nat,N: set_nat,F: nat > set_nat,G: nat > set_nat] :
% 0.34/0.57 ( ( M = N )
% 0.34/0.57 => ( ! [X3: nat] :
% 0.34/0.57 ( ( member_nat @ X3 @ N )
% 0.34/0.57 => ( ( F @ X3 )
% 0.34/0.57 = ( G @ X3 ) ) )
% 0.34/0.57 => ( ( image_nat_set_nat @ F @ M )
% 0.34/0.57 = ( image_nat_set_nat @ G @ N ) ) ) ) ).
% 0.34/0.57
% 0.34/0.57 % image_cong
% 0.34/0.57 thf(fact_2461_image__cong,axiom,
% 0.34/0.57 ! [M: set_nat,N: set_nat,F: nat > nat,G: nat > nat] :
% 0.34/0.57 ( ( M = N )
% 0.34/0.57 => ( ! [X3: nat] :
% 0.34/0.57 ( ( member_nat @ X3 @ N )
% 0.34/0.57 => ( ( F @ X3 )
% 0.34/0.57 = ( G @ X3 ) ) )
% 0.34/0.57 => ( ( image_nat_nat2 @ F @ M )
% 0.34/0.57 = ( image_nat_nat2 @ G @ N ) ) ) ) ).
% 0.34/0.57
% 0.34/0.57 % image_cong
% 0.34/0.57 thf(fact_2462_image__cong,axiom,
% 0.34/0.57 ! [M: set_nat,N: set_nat,F: nat > code_integer,G: nat > code_integer] :
% 0.34/0.57 ( ( M = N )
% 0.34/0.57 => ( ! [X3: nat] :
% 0.34/0.57 ( ( member_nat @ X3 @ N )
% 0.34/0.57 => ( ( F @ X3 )
% 0.34/0.57 = ( G @ X3 ) ) )
% 0.34/0.57 => ( ( image_1215581382706833972nteger @ F @ M )
% 0.34/0.57 = ( image_1215581382706833972nteger @ G @ N ) ) ) ) ).
% 0.34/0.57
% 0.34/0.57 % image_cong
% 0.34/0.57 thf(fact_2463_bex__imageD,axiom,
% 0.34/0.57 ! [F: nat > set_nat,A: set_nat,P: set_nat > $o] :
% 0.34/0.57 ( ? [X6: set_nat] :
% 0.34/0.57 ( ( member_set_nat @ X6 @ ( image_nat_set_nat @ F @ A ) )
% 0.34/0.57 & ( P @ X6 ) )
% 0.34/0.57 => ? [X3: nat] :
% 0.34/0.57 ( ( member_nat @ X3 @ A )
% 0.34/0.57 & ( P @ ( F @ X3 ) ) ) ) ).
% 0.34/0.57
% 0.34/0.57 % bex_imageD
% 0.34/0.57 thf(fact_2464_bex__imageD,axiom,
% 0.34/0.57 ! [F: nat > nat,A: set_nat,P: nat > $o] :
% 0.34/0.57 ( ? [X6: nat] :
% 0.34/0.57 ( ( member_nat @ X6 @ ( image_nat_nat2 @ F @ A ) )
% 0.34/0.57 & ( P @ X6 ) )
% 0.34/0.57 => ? [X3: nat] :
% 0.34/0.57 ( ( member_nat @ X3 @ A )
% 0.34/0.57 & ( P @ ( F @ X3 ) ) ) ) ).
% 0.34/0.57
% 0.34/0.57 % bex_imageD
% 0.34/0.57 thf(fact_2465_bex__imageD,axiom,
% 0.34/0.57 ! [F: code_integer > code_integer,A: set_Code_integer,P: code_integer > $o] :
% 0.34/0.57 ( ? [X6: code_integer] :
% 0.34/0.57 ( ( member_Code_integer @ X6 @ ( image_4470545334726330049nteger @ F @ A ) )
% 0.34/0.57 & ( P @ X6 ) )
% 0.34/0.57 => ? [X3: code_integer] :
% 0.34/0.57 ( ( member_Code_integer @ X3 @ A )
% 0.34/0.57 & ( P @ ( F @ X3 ) ) ) ) ).
% 0.34/0.57
% 0.34/0.57 % bex_imageD
% 0.34/0.57 thf(fact_2466_bex__imageD,axiom,
% 0.34/0.57 ! [F: nat > code_integer,A: set_nat,P: code_integer > $o] :
% 0.34/0.57 ( ? [X6: code_integer] :
% 0.34/0.57 ( ( member_Code_integer @ X6 @ ( image_1215581382706833972nteger @ F @ A ) )
% 0.34/0.57 & ( P @ X6 ) )
% 0.34/0.57 => ? [X3: nat] :
% 0.34/0.57 ( ( member_nat @ X3 @ A )
% 0.34/0.57 & ( P @ ( F @ X3 ) ) ) ) ).
% 0.34/0.57
% 0.34/0.57 % bex_imageD
% 0.34/0.57 thf(fact_2467_image__iff,axiom,
% 0.34/0.57 ! [Z: set_nat,F: nat > set_nat,A: set_nat] :
% 0.34/0.57 ( ( member_set_nat @ Z @ ( image_nat_set_nat @ F @ A ) )
% 0.34/0.57 = ( ? [X2: nat] :
% 0.34/0.57 ( ( member_nat @ X2 @ A )
% 0.34/0.57 & ( Z
% 0.34/0.57 = ( F @ X2 ) ) ) ) ) ).
% 0.34/0.57
% 0.34/0.57 % image_iff
% 0.34/0.57 thf(fact_2468_image__iff,axiom,
% 0.34/0.57 ! [Z: code_integer,F: code_integer > code_integer,A: set_Code_integer] :
% 0.34/0.57 ( ( member_Code_integer @ Z @ ( image_4470545334726330049nteger @ F @ A ) )
% 0.34/0.57 = ( ? [X2: code_integer] :
% 0.34/0.57 ( ( member_Code_integer @ X2 @ A )
% 0.34/0.57 & ( Z
% 0.34/0.57 = ( F @ X2 ) ) ) ) ) ).
% 0.34/0.57
% 0.34/0.57 % image_iff
% 0.34/0.57 thf(fact_2469_image__iff,axiom,
% 0.34/0.57 ! [Z: code_integer,F: nat > code_integer,A: set_nat] :
% 0.34/0.57 ( ( member_Code_integer @ Z @ ( image_1215581382706833972nteger @ F @ A ) )
% 0.34/0.57 = ( ? [X2: nat] :
% 0.34/0.57 ( ( member_nat @ X2 @ A )
% 0.34/0.57 & ( Z
% 0.34/0.57 = ( F @ X2 ) ) ) ) ) ).
% 0.34/0.57
% 0.34/0.57 % image_iff
% 0.34/0.57 thf(fact_2470_image__iff,axiom,
% 0.34/0.57 ! [Z: nat,F: nat > nat,A: set_nat] :
% 0.34/0.57 ( ( member_nat @ Z @ ( image_nat_nat2 @ F @ A ) )
% 0.34/0.57 = ( ? [X2: nat] :
% 0.34/0.57 ( ( member_nat @ X2 @ A )
% 0.34/0.57 & ( Z
% 0.34/0.57 = ( F @ X2 ) ) ) ) ) ).
% 0.34/0.57
% 0.34/0.57 % image_iff
% 0.34/0.57 thf(fact_2471_imageI,axiom,
% 0.34/0.57 ! [X: code_integer,A: set_Code_integer,F: code_integer > code_integer] :
% 0.34/0.57 ( ( member_Code_integer @ X @ A )
% 0.34/0.57 => ( member_Code_integer @ ( F @ X ) @ ( image_4470545334726330049nteger @ F @ A ) ) ) ).
% 0.34/0.57
% 0.34/0.57 % imageI
% 0.34/0.57 thf(fact_2472_imageI,axiom,
% 0.34/0.57 ! [X: nat,A: set_nat,F: nat > code_integer] :
% 0.34/0.57 ( ( member_nat @ X @ A )
% 0.34/0.57 => ( member_Code_integer @ ( F @ X ) @ ( image_1215581382706833972nteger @ F @ A ) ) ) ).
% 0.34/0.57
% 0.34/0.57 % imageI
% 0.34/0.57 thf(fact_2473_imageI,axiom,
% 0.34/0.57 ! [X: nat,A: set_nat,F: nat > nat] :
% 0.34/0.57 ( ( member_nat @ X @ A )
% 0.34/0.57 => ( member_nat @ ( F @ X ) @ ( image_nat_nat2 @ F @ A ) ) ) ).
% 0.34/0.57
% 0.34/0.57 % imageI
% 0.34/0.57 thf(fact_2474_imageI,axiom,
% 0.34/0.57 ! [X: nat,A: set_nat,F: nat > int] :
% 0.34/0.57 ( ( member_nat @ X @ A )
% 0.34/0.57 => ( member_int @ ( F @ X ) @ ( image_nat_int2 @ F @ A ) ) ) ).
% 0.34/0.57
% 0.34/0.57 % imageI
% 0.34/0.57 thf(fact_2475_imageI,axiom,
% 0.34/0.57 ! [X: nat,A: set_nat,F: nat > $o] :
% 0.34/0.57 ( ( member_nat @ X @ A )
% 0.34/0.57 => ( member_o @ ( F @ X ) @ ( image_nat_o2 @ F @ A ) ) ) ).
% 0.34/0.57
% 0.34/0.57 % imageI
% 0.34/0.57 thf(fact_2476_imageI,axiom,
% 0.34/0.57 ! [X: nat,A: set_nat,F: nat > product_unit] :
% 0.34/0.57 ( ( member_nat @ X @ A )
% 0.34/0.57 => ( member_Product_unit @ ( F @ X ) @ ( image_8730104196221521654t_unit @ F @ A ) ) ) ).
% 0.34/0.57
% 0.34/0.57 % imageI
% 0.34/0.57 thf(fact_2477_imageI,axiom,
% 0.34/0.57 ! [X: int,A: set_int,F: int > nat] :
% 0.34/0.57 ( ( member_int @ X @ A )
% 0.34/0.57 => ( member_nat @ ( F @ X ) @ ( image_int_nat2 @ F @ A ) ) ) ).
% 0.34/0.57
% 0.34/0.57 % imageI
% 0.34/0.57 thf(fact_2478_imageI,axiom,
% 0.34/0.57 ! [X: int,A: set_int,F: int > int] :
% 0.34/0.57 ( ( member_int @ X @ A )
% 0.34/0.57 => ( member_int @ ( F @ X ) @ ( image_int_int2 @ F @ A ) ) ) ).
% 0.34/0.57
% 0.34/0.57 % imageI
% 0.34/0.57 thf(fact_2479_imageI,axiom,
% 0.34/0.57 ! [X: int,A: set_int,F: int > $o] :
% 0.34/0.57 ( ( member_int @ X @ A )
% 0.34/0.57 => ( member_o @ ( F @ X ) @ ( image_int_o2 @ F @ A ) ) ) ).
% 0.34/0.57
% 0.34/0.57 % imageI
% 0.34/0.57 thf(fact_2480_imageI,axiom,
% 0.34/0.57 ! [X: int,A: set_int,F: int > product_unit] :
% 0.34/0.57 ( ( member_int @ X @ A )
% 0.34/0.57 => ( member_Product_unit @ ( F @ X ) @ ( image_5846526143423701914t_unit @ F @ A ) ) ) ).
% 0.34/0.57
% 0.34/0.57 % imageI
% 0.34/0.57 thf(fact_2481_DiffD2,axiom,
% 0.34/0.57 ! [C: produc3658429121746597890et_nat > $o,A: set_Pr4532377907799695533_nat_o,B4: set_Pr4532377907799695533_nat_o] :
% 0.34/0.57 ( ( member6576561426505652726_nat_o @ C @ ( minus_1801376950450012436_nat_o @ A @ B4 ) )
% 0.34/0.57 => ~ ( member6576561426505652726_nat_o @ C @ B4 ) ) ).
% 0.34/0.57
% 0.34/0.57 % DiffD2
% 0.34/0.57 thf(fact_2482_DiffD2,axiom,
% 0.34/0.57 ! [C: int,A: set_int,B4: set_int] :
% 0.34/0.57 ( ( member_int @ C @ ( minus_minus_set_int @ A @ B4 ) )
% 0.34/0.57 => ~ ( member_int @ C @ B4 ) ) ).
% 0.34/0.57
% 0.34/0.57 % DiffD2
% 0.34/0.57 thf(fact_2483_DiffD2,axiom,
% 0.34/0.57 ! [C: $o,A: set_o,B4: set_o] :
% 0.34/0.57 ( ( member_o @ C @ ( minus_minus_set_o @ A @ B4 ) )
% 0.34/0.57 => ~ ( member_o @ C @ B4 ) ) ).
% 0.34/0.57
% 0.34/0.57 % DiffD2
% 0.34/0.57 thf(fact_2484_DiffD2,axiom,
% 0.34/0.57 ! [C: product_unit,A: set_Product_unit,B4: set_Product_unit] :
% 0.34/0.57 ( ( member_Product_unit @ C @ ( minus_6452836326544984404t_unit @ A @ B4 ) )
% 0.34/0.57 => ~ ( member_Product_unit @ C @ B4 ) ) ).
% 0.34/0.57
% 0.34/0.57 % DiffD2
% 0.34/0.57 thf(fact_2485_DiffD2,axiom,
% 0.34/0.57 ! [C: nat,A: set_nat,B4: set_nat] :
% 0.34/0.57 ( ( member_nat @ C @ ( minus_minus_set_nat @ A @ B4 ) )
% 0.34/0.57 => ~ ( member_nat @ C @ B4 ) ) ).
% 0.34/0.57
% 0.34/0.57 % DiffD2
% 0.34/0.57 thf(fact_2486_DiffD1,axiom,
% 0.34/0.57 ! [C: produc3658429121746597890et_nat > $o,A: set_Pr4532377907799695533_nat_o,B4: set_Pr4532377907799695533_nat_o] :
% 0.34/0.57 ( ( member6576561426505652726_nat_o @ C @ ( minus_1801376950450012436_nat_o @ A @ B4 ) )
% 0.34/0.57 => ( member6576561426505652726_nat_o @ C @ A ) ) ).
% 0.34/0.57
% 0.34/0.57 % DiffD1
% 0.34/0.57 thf(fact_2487_DiffD1,axiom,
% 0.34/0.57 ! [C: int,A: set_int,B4: set_int] :
% 0.34/0.57 ( ( member_int @ C @ ( minus_minus_set_int @ A @ B4 ) )
% 0.34/0.57 => ( member_int @ C @ A ) ) ).
% 0.34/0.57
% 0.34/0.57 % DiffD1
% 0.34/0.57 thf(fact_2488_DiffD1,axiom,
% 0.34/0.57 ! [C: $o,A: set_o,B4: set_o] :
% 0.34/0.57 ( ( member_o @ C @ ( minus_minus_set_o @ A @ B4 ) )
% 0.34/0.57 => ( member_o @ C @ A ) ) ).
% 0.34/0.57
% 0.34/0.57 % DiffD1
% 0.34/0.57 thf(fact_2489_DiffD1,axiom,
% 0.34/0.57 ! [C: product_unit,A: set_Product_unit,B4: set_Product_unit] :
% 0.34/0.57 ( ( member_Product_unit @ C @ ( minus_6452836326544984404t_unit @ A @ B4 ) )
% 0.34/0.57 => ( member_Product_unit @ C @ A ) ) ).
% 0.34/0.57
% 0.34/0.57 % DiffD1
% 0.34/0.57 thf(fact_2490_DiffD1,axiom,
% 0.34/0.57 ! [C: nat,A: set_nat,B4: set_nat] :
% 0.34/0.57 ( ( member_nat @ C @ ( minus_minus_set_nat @ A @ B4 ) )
% 0.34/0.57 => ( member_nat @ C @ A ) ) ).
% 0.34/0.57
% 0.34/0.57 % DiffD1
% 0.34/0.57 thf(fact_2491_DiffE,axiom,
% 0.34/0.57 ! [C: produc3658429121746597890et_nat > $o,A: set_Pr4532377907799695533_nat_o,B4: set_Pr4532377907799695533_nat_o] :
% 0.34/0.57 ( ( member6576561426505652726_nat_o @ C @ ( minus_1801376950450012436_nat_o @ A @ B4 ) )
% 0.34/0.57 => ~ ( ( member6576561426505652726_nat_o @ C @ A )
% 0.34/0.57 => ( member6576561426505652726_nat_o @ C @ B4 ) ) ) ).
% 0.34/0.57
% 0.34/0.57 % DiffE
% 0.34/0.57 thf(fact_2492_DiffE,axiom,
% 0.34/0.57 ! [C: int,A: set_int,B4: set_int] :
% 0.34/0.57 ( ( member_int @ C @ ( minus_minus_set_int @ A @ B4 ) )
% 0.34/0.57 => ~ ( ( member_int @ C @ A )
% 0.34/0.57 => ( member_int @ C @ B4 ) ) ) ).
% 0.34/0.57
% 0.34/0.57 % DiffE
% 0.34/0.57 thf(fact_2493_DiffE,axiom,
% 0.34/0.57 ! [C: $o,A: set_o,B4: set_o] :
% 0.34/0.57 ( ( member_o @ C @ ( minus_minus_set_o @ A @ B4 ) )
% 0.34/0.57 => ~ ( ( member_o @ C @ A )
% 0.34/0.57 => ( member_o @ C @ B4 ) ) ) ).
% 0.34/0.57
% 0.34/0.57 % DiffE
% 0.34/0.57 thf(fact_2494_DiffE,axiom,
% 0.34/0.57 ! [C: product_unit,A: set_Product_unit,B4: set_Product_unit] :
% 0.34/0.57 ( ( member_Product_unit @ C @ ( minus_6452836326544984404t_unit @ A @ B4 ) )
% 0.34/0.57 => ~ ( ( member_Product_unit @ C @ A )
% 0.34/0.57 => ( member_Product_unit @ C @ B4 ) ) ) ).
% 0.34/0.57
% 0.34/0.57 % DiffE
% 0.34/0.57 thf(fact_2495_DiffE,axiom,
% 0.34/0.57 ! [C: nat,A: set_nat,B4: set_nat] :
% 0.34/0.57 ( ( member_nat @ C @ ( minus_minus_set_nat @ A @ B4 ) )
% 0.34/0.57 => ~ ( ( member_nat @ C @ A )
% 0.34/0.57 => ( member_nat @ C @ B4 ) ) ) ).
% 0.34/0.57
% 0.34/0.57 % DiffE
% 0.34/0.57 thf(fact_2496_cancel__ab__semigroup__add__class_Odiff__right__commute,axiom,
% 0.34/0.57 ! [A2: rat,C: rat,B: rat] :
% 0.34/0.57 ( ( minus_minus_rat @ ( minus_minus_rat @ A2 @ C ) @ B )
% 0.34/0.57 = ( minus_minus_rat @ ( minus_minus_rat @ A2 @ B ) @ C ) ) ).
% 0.34/0.57
% 0.34/0.57 % cancel_ab_semigroup_add_class.diff_right_commute
% 0.34/0.57 thf(fact_2497_cancel__ab__semigroup__add__class_Odiff__right__commute,axiom,
% 0.34/0.57 ! [A2: nat,C: nat,B: nat] :
% 0.34/0.57 ( ( minus_minus_nat @ ( minus_minus_nat @ A2 @ C ) @ B )
% 0.34/0.57 = ( minus_minus_nat @ ( minus_minus_nat @ A2 @ B ) @ C ) ) ).
% 0.34/0.57
% 0.34/0.57 % cancel_ab_semigroup_add_class.diff_right_commute
% 0.34/0.57 thf(fact_2498_cancel__ab__semigroup__add__class_Odiff__right__commute,axiom,
% 0.34/0.57 ! [A2: int,C: int,B: int] :
% 0.34/0.57 ( ( minus_minus_int @ ( minus_minus_int @ A2 @ C ) @ B )
% 0.34/0.57 = ( minus_minus_int @ ( minus_minus_int @ A2 @ B ) @ C ) ) ).
% 0.34/0.57
% 0.34/0.57 % cancel_ab_semigroup_add_class.diff_right_commute
% 0.34/0.57 thf(fact_2499_diff__eq__diff__eq,axiom,
% 0.34/0.57 ! [A2: rat,B: rat,C: rat,D2: rat] :
% 0.34/0.57 ( ( ( minus_minus_rat @ A2 @ B )
% 0.34/0.57 = ( minus_minus_rat @ C @ D2 ) )
% 0.34/0.57 => ( ( A2 = B )
% 0.34/0.57 = ( C = D2 ) ) ) ).
% 0.34/0.57
% 0.34/0.57 % diff_eq_diff_eq
% 0.34/0.57 thf(fact_2500_diff__eq__diff__eq,axiom,
% 0.34/0.57 ! [A2: int,B: int,C: int,D2: int] :
% 0.34/0.57 ( ( ( minus_minus_int @ A2 @ B )
% 0.34/0.57 = ( minus_minus_int @ C @ D2 ) )
% 0.34/0.57 => ( ( A2 = B )
% 0.34/0.57 = ( C = D2 ) ) ) ).
% 0.34/0.57
% 0.34/0.57 % diff_eq_diff_eq
% 0.34/0.57 thf(fact_2501_in__image__insert__iff,axiom,
% 0.34/0.57 ! [B4: set_set_Product_unit,X: product_unit,A: set_Product_unit] :
% 0.34/0.57 ( ! [C6: set_Product_unit] :
% 0.34/0.57 ( ( member5877623283571906838t_unit @ C6 @ B4 )
% 0.34/0.57 => ~ ( member_Product_unit @ X @ C6 ) )
% 0.34/0.57 => ( ( member5877623283571906838t_unit @ A @ ( image_4019945301284682565t_unit @ ( insert_Product_unit @ X ) @ B4 ) )
% 0.34/0.57 = ( ( member_Product_unit @ X @ A )
% 0.34/0.57 & ( member5877623283571906838t_unit @ ( minus_6452836326544984404t_unit @ A @ ( insert_Product_unit @ X @ bot_bo3957492148770167129t_unit ) ) @ B4 ) ) ) ) ).
% 0.34/0.57
% 0.34/0.57 % in_image_insert_iff
% 0.34/0.57 thf(fact_2502_in__image__insert__iff,axiom,
% 0.34/0.57 ! [B4: set_set_o,X: $o,A: set_o] :
% 0.34/0.57 ( ! [C6: set_o] :
% 0.34/0.57 ( ( member_set_o @ C6 @ B4 )
% 0.34/0.57 => ~ ( member_o @ X @ C6 ) )
% 0.34/0.57 => ( ( member_set_o @ A @ ( image_set_o_set_o @ ( insert_o @ X ) @ B4 ) )
% 0.34/0.57 = ( ( member_o @ X @ A )
% 0.34/0.57 & ( member_set_o @ ( minus_minus_set_o @ A @ ( insert_o @ X @ bot_bot_set_o ) ) @ B4 ) ) ) ) ).
% 0.34/0.57
% 0.34/0.57 % in_image_insert_iff
% 0.34/0.57 thf(fact_2503_in__image__insert__iff,axiom,
% 0.34/0.57 ! [B4: set_set_int,X: int,A: set_int] :
% 0.34/0.57 ( ! [C6: set_int] :
% 0.34/0.57 ( ( member_set_int @ C6 @ B4 )
% 0.34/0.57 => ~ ( member_int @ X @ C6 ) )
% 0.34/0.57 => ( ( member_set_int @ A @ ( image_524474410958335435et_int @ ( insert_int @ X ) @ B4 ) )
% 0.34/0.57 = ( ( member_int @ X @ A )
% 0.34/0.57 & ( member_set_int @ ( minus_minus_set_int @ A @ ( insert_int @ X @ bot_bot_set_int ) ) @ B4 ) ) ) ) ).
% 0.34/0.57
% 0.34/0.57 % in_image_insert_iff
% 0.34/0.57 thf(fact_2504_in__image__insert__iff,axiom,
% 0.34/0.57 ! [B4: set_se1353816476878259085_nat_o,X: produc3658429121746597890et_nat > $o,A: set_Pr4532377907799695533_nat_o] :
% 0.34/0.57 ( ! [C6: set_Pr4532377907799695533_nat_o] :
% 0.34/0.57 ( ( member1452989533377600726_nat_o @ C6 @ B4 )
% 0.34/0.57 => ~ ( member6576561426505652726_nat_o @ X @ C6 ) )
% 0.34/0.57 => ( ( member1452989533377600726_nat_o @ A @ ( image_8925030434418351941_nat_o @ ( insert5175938949040314269_nat_o @ X ) @ B4 ) )
% 0.34/0.57 = ( ( member6576561426505652726_nat_o @ X @ A )
% 0.34/0.57 & ( member1452989533377600726_nat_o @ ( minus_1801376950450012436_nat_o @ A @ ( insert5175938949040314269_nat_o @ X @ bot_bo7824918357723582233_nat_o ) ) @ B4 ) ) ) ) ).
% 0.34/0.57
% 0.34/0.57 % in_image_insert_iff
% 0.34/0.57 thf(fact_2505_in__image__insert__iff,axiom,
% 0.34/0.57 ! [B4: set_set_nat,X: nat,A: set_nat] :
% 0.34/0.57 ( ! [C6: set_nat] :
% 0.34/0.57 ( ( member_set_nat @ C6 @ B4 )
% 0.34/0.57 => ~ ( member_nat @ X @ C6 ) )
% 0.34/0.57 => ( ( member_set_nat @ A @ ( image_7916887816326733075et_nat @ ( insert_nat @ X ) @ B4 ) )
% 0.34/0.57 = ( ( member_nat @ X @ A )
% 0.34/0.57 & ( member_set_nat @ ( minus_minus_set_nat @ A @ ( insert_nat @ X @ bot_bot_set_nat ) ) @ B4 ) ) ) ) ).
% 0.34/0.57
% 0.34/0.57 % in_image_insert_iff
% 0.34/0.57 thf(fact_2506_converse_Ocases,axiom,
% 0.34/0.57 ! [A1: set_nat,A22: heap_e7401611519738050253t_unit,R2: set_Pr3948176798113811640et_nat] :
% 0.34/0.57 ( ( member727419437372278867t_unit @ ( produc2471974422597015918t_unit @ A1 @ A22 ) @ ( conver7272282955447398104et_nat @ R2 ) )
% 0.34/0.57 => ( member6260224972018164377et_nat @ ( produc7507926704131184380et_nat @ A22 @ A1 ) @ R2 ) ) ).
% 0.34/0.57
% 0.34/0.57 % converse.cases
% 0.34/0.57 thf(fact_2507_converse_Ocases,axiom,
% 0.34/0.57 ! [A1: produc3658429121746597890et_nat,A22: produc3658429121746597890et_nat > $o,R2: set_Pr3286484037609594932et_nat] :
% 0.34/0.57 ( ( member4978015717948814295_nat_o @ ( produc2829238748666232050_nat_o @ A1 @ A22 ) @ ( conver4003969491454187092et_nat @ R2 ) )
% 0.34/0.57 => ( member1996754912294343701et_nat @ ( produc5001842942810119800et_nat @ A22 @ A1 ) @ R2 ) ) ).
% 0.34/0.57
% 0.34/0.57 % converse.cases
% 0.34/0.57 thf(fact_2508_converse_Ocases,axiom,
% 0.34/0.57 ! [A1: produc3925858234332021118et_nat,A22: produc3658429121746597890et_nat > $o,R2: set_Pr8536935166611901872et_nat] :
% 0.34/0.57 ( ( member8354763867512313179_nat_o @ ( produc3674526432113299062_nat_o @ A1 @ A22 ) @ ( conver3577387295554834384et_nat @ R2 ) )
% 0.34/0.57 => ( member6124377750444531601et_nat @ ( produc2245416461498447860et_nat @ A22 @ A1 ) @ R2 ) ) ).
% 0.34/0.57
% 0.34/0.57 % converse.cases
% 0.34/0.57 thf(fact_2509_converse_Ocases,axiom,
% 0.34/0.57 ! [A1: heap_e7401611519738050253t_unit,A22: set_nat,R2: set_Pr4071100604818519410t_unit] :
% 0.34/0.57 ( ( member6260224972018164377et_nat @ ( produc7507926704131184380et_nat @ A1 @ A22 ) @ ( conver2236330673913229642t_unit @ R2 ) )
% 0.34/0.57 => ( member727419437372278867t_unit @ ( produc2471974422597015918t_unit @ A22 @ A1 ) @ R2 ) ) ).
% 0.34/0.57
% 0.34/0.57 % converse.cases
% 0.34/0.57 thf(fact_2510_converse_Ocases,axiom,
% 0.34/0.57 ! [A1: nat,A22: nat,R2: set_Pr1261947904930325089at_nat] :
% 0.34/0.57 ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A1 @ A22 ) @ ( converse_nat_nat @ R2 ) )
% 0.34/0.57 => ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A22 @ A1 ) @ R2 ) ) ).
% 0.34/0.57
% 0.34/0.57 % converse.cases
% 0.34/0.57 thf(fact_2511_converse_Ocases,axiom,
% 0.34/0.57 ! [A1: int,A22: int,R2: set_Pr958786334691620121nt_int] :
% 0.34/0.57 ( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ A1 @ A22 ) @ ( converse_int_int @ R2 ) )
% 0.34/0.57 => ( member5262025264175285858nt_int @ ( product_Pair_int_int @ A22 @ A1 ) @ R2 ) ) ).
% 0.34/0.57
% 0.34/0.57 % converse.cases
% 0.34/0.57 thf(fact_2512_converse_Ocases,axiom,
% 0.34/0.57 ! [A1: produc3658429121746597890et_nat > $o,A22: produc3658429121746597890et_nat,R2: set_Pr4208978187861016310_nat_o] :
% 0.34/0.57 ( ( member1996754912294343701et_nat @ ( produc5001842942810119800et_nat @ A1 @ A22 ) @ ( conver1831365297310299342_nat_o @ R2 ) )
% 0.34/0.57 => ( member4978015717948814295_nat_o @ ( produc2829238748666232050_nat_o @ A22 @ A1 ) @ R2 ) ) ).
% 0.34/0.57
% 0.34/0.57 % converse.cases
% 0.34/0.57 thf(fact_2513_converse_Ocases,axiom,
% 0.34/0.57 ! [A1: produc3658429121746597890et_nat > $o,A22: produc3925858234332021118et_nat,R2: set_Pr7705059561495379578_nat_o] :
% 0.34/0.57 ( ( member6124377750444531601et_nat @ ( produc2245416461498447860et_nat @ A1 @ A22 ) @ ( conver5006497266169685586_nat_o @ R2 ) )
% 0.34/0.57 => ( member8354763867512313179_nat_o @ ( produc3674526432113299062_nat_o @ A22 @ A1 ) @ R2 ) ) ).
% 0.34/0.57
% 0.34/0.57 % converse.cases
% 0.34/0.57 thf(fact_2514_converse_Osimps,axiom,
% 0.34/0.57 ! [A1: set_nat,A22: heap_e7401611519738050253t_unit,R2: set_Pr3948176798113811640et_nat] :
% 0.34/0.57 ( ( member727419437372278867t_unit @ ( produc2471974422597015918t_unit @ A1 @ A22 ) @ ( conver7272282955447398104et_nat @ R2 ) )
% 0.34/0.57 = ( ? [A6: heap_e7401611519738050253t_unit,B6: set_nat] :
% 0.34/0.57 ( ( A1 = B6 )
% 0.34/0.57 & ( A22 = A6 )
% 0.34/0.57 & ( member6260224972018164377et_nat @ ( produc7507926704131184380et_nat @ A6 @ B6 ) @ R2 ) ) ) ) ).
% 0.34/0.57
% 0.34/0.57 % converse.simps
% 0.34/0.57 thf(fact_2515_converse_Osimps,axiom,
% 0.34/0.57 ! [A1: produc3658429121746597890et_nat,A22: produc3658429121746597890et_nat > $o,R2: set_Pr3286484037609594932et_nat] :
% 0.34/0.57 ( ( member4978015717948814295_nat_o @ ( produc2829238748666232050_nat_o @ A1 @ A22 ) @ ( conver4003969491454187092et_nat @ R2 ) )
% 0.34/0.57 = ( ? [A6: produc3658429121746597890et_nat > $o,B6: produc3658429121746597890et_nat] :
% 0.34/0.57 ( ( A1 = B6 )
% 0.34/0.57 & ( A22 = A6 )
% 0.34/0.57 & ( member1996754912294343701et_nat @ ( produc5001842942810119800et_nat @ A6 @ B6 ) @ R2 ) ) ) ) ).
% 0.34/0.57
% 0.34/0.57 % converse.simps
% 0.34/0.57 thf(fact_2516_converse_Osimps,axiom,
% 0.34/0.57 ! [A1: produc3925858234332021118et_nat,A22: produc3658429121746597890et_nat > $o,R2: set_Pr8536935166611901872et_nat] :
% 0.34/0.57 ( ( member8354763867512313179_nat_o @ ( produc3674526432113299062_nat_o @ A1 @ A22 ) @ ( conver3577387295554834384et_nat @ R2 ) )
% 0.34/0.57 = ( ? [A6: produc3658429121746597890et_nat > $o,B6: produc3925858234332021118et_nat] :
% 0.34/0.57 ( ( A1 = B6 )
% 0.34/0.57 & ( A22 = A6 )
% 0.34/0.57 & ( member6124377750444531601et_nat @ ( produc2245416461498447860et_nat @ A6 @ B6 ) @ R2 ) ) ) ) ).
% 0.34/0.57
% 0.34/0.57 % converse.simps
% 0.34/0.57 thf(fact_2517_converse_Osimps,axiom,
% 0.34/0.57 ! [A1: heap_e7401611519738050253t_unit,A22: set_nat,R2: set_Pr4071100604818519410t_unit] :
% 0.34/0.57 ( ( member6260224972018164377et_nat @ ( produc7507926704131184380et_nat @ A1 @ A22 ) @ ( conver2236330673913229642t_unit @ R2 ) )
% 0.34/0.57 = ( ? [A6: set_nat,B6: heap_e7401611519738050253t_unit] :
% 0.34/0.57 ( ( A1 = B6 )
% 0.34/0.57 & ( A22 = A6 )
% 0.34/0.57 & ( member727419437372278867t_unit @ ( produc2471974422597015918t_unit @ A6 @ B6 ) @ R2 ) ) ) ) ).
% 0.34/0.57
% 0.34/0.57 % converse.simps
% 0.34/0.57 thf(fact_2518_converse_Osimps,axiom,
% 0.34/0.57 ! [A1: nat,A22: nat,R2: set_Pr1261947904930325089at_nat] :
% 0.34/0.57 ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A1 @ A22 ) @ ( converse_nat_nat @ R2 ) )
% 0.34/0.57 = ( ? [A6: nat,B6: nat] :
% 0.34/0.57 ( ( A1 = B6 )
% 0.34/0.57 & ( A22 = A6 )
% 0.34/0.57 & ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A6 @ B6 ) @ R2 ) ) ) ) ).
% 0.34/0.57
% 0.34/0.57 % converse.simps
% 0.34/0.57 thf(fact_2519_converse_Osimps,axiom,
% 0.34/0.57 ! [A1: int,A22: int,R2: set_Pr958786334691620121nt_int] :
% 0.34/0.57 ( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ A1 @ A22 ) @ ( converse_int_int @ R2 ) )
% 0.34/0.57 = ( ? [A6: int,B6: int] :
% 0.34/0.57 ( ( A1 = B6 )
% 0.34/0.57 & ( A22 = A6 )
% 0.34/0.57 & ( member5262025264175285858nt_int @ ( product_Pair_int_int @ A6 @ B6 ) @ R2 ) ) ) ) ).
% 0.34/0.57
% 0.34/0.57 % converse.simps
% 0.34/0.57 thf(fact_2520_converse_Osimps,axiom,
% 0.34/0.57 ! [A1: produc3658429121746597890et_nat > $o,A22: produc3658429121746597890et_nat,R2: set_Pr4208978187861016310_nat_o] :
% 0.34/0.57 ( ( member1996754912294343701et_nat @ ( produc5001842942810119800et_nat @ A1 @ A22 ) @ ( conver1831365297310299342_nat_o @ R2 ) )
% 0.34/0.57 = ( ? [A6: produc3658429121746597890et_nat,B6: produc3658429121746597890et_nat > $o] :
% 0.34/0.57 ( ( A1 = B6 )
% 0.34/0.57 & ( A22 = A6 )
% 0.34/0.57 & ( member4978015717948814295_nat_o @ ( produc2829238748666232050_nat_o @ A6 @ B6 ) @ R2 ) ) ) ) ).
% 0.34/0.57
% 0.34/0.57 % converse.simps
% 0.34/0.57 thf(fact_2521_converse_Osimps,axiom,
% 0.34/0.57 ! [A1: produc3658429121746597890et_nat > $o,A22: produc3925858234332021118et_nat,R2: set_Pr7705059561495379578_nat_o] :
% 0.34/0.57 ( ( member6124377750444531601et_nat @ ( produc2245416461498447860et_nat @ A1 @ A22 ) @ ( conver5006497266169685586_nat_o @ R2 ) )
% 0.34/0.57 = ( ? [A6: produc3925858234332021118et_nat,B6: produc3658429121746597890et_nat > $o] :
% 0.34/0.57 ( ( A1 = B6 )
% 0.34/0.57 & ( A22 = A6 )
% 0.34/0.57 & ( member8354763867512313179_nat_o @ ( produc3674526432113299062_nat_o @ A6 @ B6 ) @ R2 ) ) ) ) ).
% 0.34/0.57
% 0.34/0.57 % converse.simps
% 0.34/0.57 thf(fact_2522_converseD,axiom,
% 0.34/0.57 ! [A2: set_nat,B: heap_e7401611519738050253t_unit,R2: set_Pr3948176798113811640et_nat] :
% 0.34/0.57 ( ( member727419437372278867t_unit @ ( produc2471974422597015918t_unit @ A2 @ B ) @ ( conver7272282955447398104et_nat @ R2 ) )
% 0.34/0.57 => ( member6260224972018164377et_nat @ ( produc7507926704131184380et_nat @ B @ A2 ) @ R2 ) ) ).
% 0.34/0.57
% 0.34/0.57 % converseD
% 0.34/0.57 thf(fact_2523_converseD,axiom,
% 0.34/0.57 ! [A2: produc3658429121746597890et_nat,B: produc3658429121746597890et_nat > $o,R2: set_Pr3286484037609594932et_nat] :
% 0.34/0.57 ( ( member4978015717948814295_nat_o @ ( produc2829238748666232050_nat_o @ A2 @ B ) @ ( conver4003969491454187092et_nat @ R2 ) )
% 0.34/0.57 => ( member1996754912294343701et_nat @ ( produc5001842942810119800et_nat @ B @ A2 ) @ R2 ) ) ).
% 0.34/0.57
% 0.34/0.57 % converseD
% 0.34/0.57 thf(fact_2524_converseD,axiom,
% 0.34/0.57 ! [A2: produc3925858234332021118et_nat,B: produc3658429121746597890et_nat > $o,R2: set_Pr8536935166611901872et_nat] :
% 0.34/0.57 ( ( member8354763867512313179_nat_o @ ( produc3674526432113299062_nat_o @ A2 @ B ) @ ( conver3577387295554834384et_nat @ R2 ) )
% 0.34/0.57 => ( member6124377750444531601et_nat @ ( produc2245416461498447860et_nat @ B @ A2 ) @ R2 ) ) ).
% 0.34/0.57
% 0.34/0.57 % converseD
% 0.34/0.57 thf(fact_2525_converseD,axiom,
% 0.34/0.57 ! [A2: heap_e7401611519738050253t_unit,B: set_nat,R2: set_Pr4071100604818519410t_unit] :
% 0.34/0.57 ( ( member6260224972018164377et_nat @ ( produc7507926704131184380et_nat @ A2 @ B ) @ ( conver2236330673913229642t_unit @ R2 ) )
% 0.34/0.57 => ( member727419437372278867t_unit @ ( produc2471974422597015918t_unit @ B @ A2 ) @ R2 ) ) ).
% 0.34/0.57
% 0.34/0.57 % converseD
% 0.34/0.57 thf(fact_2526_converseD,axiom,
% 0.34/0.57 ! [A2: nat,B: nat,R2: set_Pr1261947904930325089at_nat] :
% 0.34/0.57 ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A2 @ B ) @ ( converse_nat_nat @ R2 ) )
% 0.34/0.57 => ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ B @ A2 ) @ R2 ) ) ).
% 0.34/0.57
% 0.34/0.57 % converseD
% 0.34/0.57 thf(fact_2527_converseD,axiom,
% 0.34/0.57 ! [A2: int,B: int,R2: set_Pr958786334691620121nt_int] :
% 0.34/0.57 ( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ A2 @ B ) @ ( converse_int_int @ R2 ) )
% 0.34/0.57 => ( member5262025264175285858nt_int @ ( product_Pair_int_int @ B @ A2 ) @ R2 ) ) ).
% 0.34/0.57
% 0.34/0.57 % converseD
% 0.34/0.57 thf(fact_2528_converseD,axiom,
% 0.34/0.57 ! [A2: produc3658429121746597890et_nat > $o,B: produc3658429121746597890et_nat,R2: set_Pr4208978187861016310_nat_o] :
% 0.34/0.57 ( ( member1996754912294343701et_nat @ ( produc5001842942810119800et_nat @ A2 @ B ) @ ( conver1831365297310299342_nat_o @ R2 ) )
% 0.34/0.57 => ( member4978015717948814295_nat_o @ ( produc2829238748666232050_nat_o @ B @ A2 ) @ R2 ) ) ).
% 0.34/0.57
% 0.34/0.57 % converseD
% 0.34/0.57 thf(fact_2529_converseD,axiom,
% 0.34/0.57 ! [A2: produc3658429121746597890et_nat > $o,B: produc3925858234332021118et_nat,R2: set_Pr7705059561495379578_nat_o] :
% 0.34/0.57 ( ( member6124377750444531601et_nat @ ( produc2245416461498447860et_nat @ A2 @ B ) @ ( conver5006497266169685586_nat_o @ R2 ) )
% 0.34/0.57 => ( member8354763867512313179_nat_o @ ( produc3674526432113299062_nat_o @ B @ A2 ) @ R2 ) ) ).
% 0.34/0.57
% 0.34/0.57 % converseD
% 0.34/0.57 thf(fact_2530_converseE,axiom,
% 0.34/0.57 ! [Yx: produc7348995623955488188t_unit,R2: set_Pr3948176798113811640et_nat] :
% 0.34/0.57 ( ( member727419437372278867t_unit @ Yx @ ( conver7272282955447398104et_nat @ R2 ) )
% 0.34/0.57 => ~ ! [X3: heap_e7401611519738050253t_unit,Y2: set_nat] :
% 0.34/0.57 ( ( Yx
% 0.34/0.57 = ( produc2471974422597015918t_unit @ Y2 @ X3 ) )
% 0.34/0.57 => ~ ( member6260224972018164377et_nat @ ( produc7507926704131184380et_nat @ X3 @ Y2 ) @ R2 ) ) ) ).
% 0.34/0.57
% 0.34/0.57 % converseE
% 0.34/0.57 thf(fact_2531_converseE,axiom,
% 0.34/0.57 ! [Yx: produc6907119039986491712_nat_o,R2: set_Pr3286484037609594932et_nat] :
% 0.34/0.57 ( ( member4978015717948814295_nat_o @ Yx @ ( conver4003969491454187092et_nat @ R2 ) )
% 0.34/0.57 => ~ ! [X3: produc3658429121746597890et_nat > $o,Y2: produc3658429121746597890et_nat] :
% 0.34/0.57 ( ( Yx
% 0.34/0.57 = ( produc2829238748666232050_nat_o @ Y2 @ X3 ) )
% 0.34/0.57 => ~ ( member1996754912294343701et_nat @ ( produc5001842942810119800et_nat @ X3 @ Y2 ) @ R2 ) ) ) ).
% 0.34/0.57
% 0.34/0.57 % converseE
% 0.34/0.57 thf(fact_2532_converseE,axiom,
% 0.34/0.57 ! [Yx: produc4962441903510821572_nat_o,R2: set_Pr8536935166611901872et_nat] :
% 0.34/0.57 ( ( member8354763867512313179_nat_o @ Yx @ ( conver3577387295554834384et_nat @ R2 ) )
% 0.34/0.57 => ~ ! [X3: produc3658429121746597890et_nat > $o,Y2: produc3925858234332021118et_nat] :
% 0.34/0.57 ( ( Yx
% 0.34/0.57 = ( produc3674526432113299062_nat_o @ Y2 @ X3 ) )
% 0.34/0.57 => ~ ( member6124377750444531601et_nat @ ( produc2245416461498447860et_nat @ X3 @ Y2 ) @ R2 ) ) ) ).
% 0.34/0.57
% 0.34/0.57 % converseE
% 0.34/0.57 thf(fact_2533_converseE,axiom,
% 0.34/0.57 ! [Yx: produc3658429121746597890et_nat,R2: set_Pr4071100604818519410t_unit] :
% 0.34/0.57 ( ( member6260224972018164377et_nat @ Yx @ ( conver2236330673913229642t_unit @ R2 ) )
% 0.34/0.57 => ~ ! [X3: set_nat,Y2: heap_e7401611519738050253t_unit] :
% 0.34/0.57 ( ( Yx
% 0.34/0.57 = ( produc7507926704131184380et_nat @ Y2 @ X3 ) )
% 0.34/0.57 => ~ ( member727419437372278867t_unit @ ( produc2471974422597015918t_unit @ X3 @ Y2 ) @ R2 ) ) ) ).
% 0.34/0.57
% 0.34/0.57 % converseE
% 0.34/0.57 thf(fact_2534_converseE,axiom,
% 0.34/0.57 ! [Yx: product_prod_nat_nat,R2: set_Pr1261947904930325089at_nat] :
% 0.34/0.57 ( ( member8440522571783428010at_nat @ Yx @ ( converse_nat_nat @ R2 ) )
% 0.34/0.57 => ~ ! [X3: nat,Y2: nat] :
% 0.34/0.57 ( ( Yx
% 0.34/0.57 = ( product_Pair_nat_nat @ Y2 @ X3 ) )
% 0.34/0.57 => ~ ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X3 @ Y2 ) @ R2 ) ) ) ).
% 0.34/0.57
% 0.34/0.57 % converseE
% 0.34/0.57 thf(fact_2535_converseE,axiom,
% 0.34/0.57 ! [Yx: product_prod_int_int,R2: set_Pr958786334691620121nt_int] :
% 0.34/0.57 ( ( member5262025264175285858nt_int @ Yx @ ( converse_int_int @ R2 ) )
% 0.34/0.57 => ~ ! [X3: int,Y2: int] :
% 0.34/0.57 ( ( Yx
% 0.34/0.57 = ( product_Pair_int_int @ Y2 @ X3 ) )
% 0.34/0.57 => ~ ( member5262025264175285858nt_int @ ( product_Pair_int_int @ X3 @ Y2 ) @ R2 ) ) ) ).
% 0.34/0.57
% 0.34/0.57 % converseE
% 0.34/0.57 thf(fact_2536_converseE,axiom,
% 0.34/0.57 ! [Yx: produc3925858234332021118et_nat,R2: set_Pr4208978187861016310_nat_o] :
% 0.34/0.57 ( ( member1996754912294343701et_nat @ Yx @ ( conver1831365297310299342_nat_o @ R2 ) )
% 0.34/0.57 => ~ ! [X3: produc3658429121746597890et_nat,Y2: produc3658429121746597890et_nat > $o] :
% 0.34/0.57 ( ( Yx
% 0.34/0.57 = ( produc5001842942810119800et_nat @ Y2 @ X3 ) )
% 0.34/0.57 => ~ ( member4978015717948814295_nat_o @ ( produc2829238748666232050_nat_o @ X3 @ Y2 ) @ R2 ) ) ) ).
% 0.34/0.57
% 0.34/0.57 % converseE
% 0.34/0.57 thf(fact_2537_converseE,axiom,
% 0.34/0.57 ! [Yx: produc2732055786443039994et_nat,R2: set_Pr7705059561495379578_nat_o] :
% 0.34/0.57 ( ( member6124377750444531601et_nat @ Yx @ ( conver5006497266169685586_nat_o @ R2 ) )
% 0.34/0.57 => ~ ! [X3: produc3925858234332021118et_nat,Y2: produc3658429121746597890et_nat > $o] :
% 0.34/0.57 ( ( Yx
% 0.34/0.57 = ( produc2245416461498447860et_nat @ Y2 @ X3 ) )
% 0.34/0.57 => ~ ( member8354763867512313179_nat_o @ ( produc3674526432113299062_nat_o @ X3 @ Y2 ) @ R2 ) ) ) ).
% 0.34/0.57
% 0.34/0.57 % converseE
% 0.34/0.57 thf(fact_2538_converseI,axiom,
% 0.34/0.57 ! [A2: set_nat,B: heap_e7401611519738050253t_unit,R2: set_Pr4071100604818519410t_unit] :
% 0.34/0.57 ( ( member727419437372278867t_unit @ ( produc2471974422597015918t_unit @ A2 @ B ) @ R2 )
% 0.34/0.57 => ( member6260224972018164377et_nat @ ( produc7507926704131184380et_nat @ B @ A2 ) @ ( conver2236330673913229642t_unit @ R2 ) ) ) ).
% 0.34/0.57
% 0.34/0.57 % converseI
% 0.34/0.57 thf(fact_2539_converseI,axiom,
% 0.34/0.57 ! [A2: produc3658429121746597890et_nat,B: produc3658429121746597890et_nat > $o,R2: set_Pr4208978187861016310_nat_o] :
% 0.34/0.57 ( ( member4978015717948814295_nat_o @ ( produc2829238748666232050_nat_o @ A2 @ B ) @ R2 )
% 0.34/0.57 => ( member1996754912294343701et_nat @ ( produc5001842942810119800et_nat @ B @ A2 ) @ ( conver1831365297310299342_nat_o @ R2 ) ) ) ).
% 0.34/0.57
% 0.34/0.57 % converseI
% 0.34/0.57 thf(fact_2540_converseI,axiom,
% 0.34/0.57 ! [A2: produc3925858234332021118et_nat,B: produc3658429121746597890et_nat > $o,R2: set_Pr7705059561495379578_nat_o] :
% 0.34/0.57 ( ( member8354763867512313179_nat_o @ ( produc3674526432113299062_nat_o @ A2 @ B ) @ R2 )
% 0.34/0.57 => ( member6124377750444531601et_nat @ ( produc2245416461498447860et_nat @ B @ A2 ) @ ( conver5006497266169685586_nat_o @ R2 ) ) ) ).
% 0.34/0.57
% 0.34/0.57 % converseI
% 0.34/0.57 thf(fact_2541_converseI,axiom,
% 0.34/0.57 ! [A2: heap_e7401611519738050253t_unit,B: set_nat,R2: set_Pr3948176798113811640et_nat] :
% 0.34/0.57 ( ( member6260224972018164377et_nat @ ( produc7507926704131184380et_nat @ A2 @ B ) @ R2 )
% 0.34/0.57 => ( member727419437372278867t_unit @ ( produc2471974422597015918t_unit @ B @ A2 ) @ ( conver7272282955447398104et_nat @ R2 ) ) ) ).
% 0.34/0.57
% 0.34/0.57 % converseI
% 0.34/0.57 thf(fact_2542_converseI,axiom,
% 0.34/0.57 ! [A2: nat,B: nat,R2: set_Pr1261947904930325089at_nat] :
% 0.34/0.57 ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A2 @ B ) @ R2 )
% 0.34/0.57 => ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ B @ A2 ) @ ( converse_nat_nat @ R2 ) ) ) ).
% 0.34/0.57
% 0.34/0.57 % converseI
% 0.34/0.57 thf(fact_2543_converseI,axiom,
% 0.34/0.57 ! [A2: int,B: int,R2: set_Pr958786334691620121nt_int] :
% 0.34/0.57 ( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ A2 @ B ) @ R2 )
% 0.34/0.57 => ( member5262025264175285858nt_int @ ( product_Pair_int_int @ B @ A2 ) @ ( converse_int_int @ R2 ) ) ) ).
% 0.34/0.57
% 0.34/0.57 % converseI
% 0.34/0.57 thf(fact_2544_converseI,axiom,
% 0.34/0.57 ! [A2: produc3658429121746597890et_nat > $o,B: produc3658429121746597890et_nat,R2: set_Pr3286484037609594932et_nat] :
% 0.34/0.57 ( ( member1996754912294343701et_nat @ ( produc5001842942810119800et_nat @ A2 @ B ) @ R2 )
% 0.34/0.57 => ( member4978015717948814295_nat_o @ ( produc2829238748666232050_nat_o @ B @ A2 ) @ ( conver4003969491454187092et_nat @ R2 ) ) ) ).
% 0.34/0.57
% 0.34/0.57 % converseI
% 0.34/0.57 thf(fact_2545_converseI,axiom,
% 0.34/0.57 ! [A2: produc3658429121746597890et_nat > $o,B: produc3925858234332021118et_nat,R2: set_Pr8536935166611901872et_nat] :
% 0.34/0.57 ( ( member6124377750444531601et_nat @ ( produc2245416461498447860et_nat @ A2 @ B ) @ R2 )
% 0.34/0.57 => ( member8354763867512313179_nat_o @ ( produc3674526432113299062_nat_o @ B @ A2 ) @ ( conver3577387295554834384et_nat @ R2 ) ) ) ).
% 0.34/0.57
% 0.34/0.57 % converseI
% 0.34/0.57 thf(fact_2546_diff__eq__diff__less__eq,axiom,
% 0.34/0.57 ! [A2: rat,B: rat,C: rat,D2: rat] :
% 0.34/0.57 ( ( ( minus_minus_rat @ A2 @ B )
% 0.34/0.57 = ( minus_minus_rat @ C @ D2 ) )
% 0.34/0.57 => ( ( ord_less_eq_rat @ A2 @ B )
% 0.34/0.57 = ( ord_less_eq_rat @ C @ D2 ) ) ) ).
% 0.34/0.57
% 0.34/0.57 % diff_eq_diff_less_eq
% 0.34/0.57 thf(fact_2547_diff__eq__diff__less__eq,axiom,
% 0.34/0.57 ! [A2: int,B: int,C: int,D2: int] :
% 0.34/0.57 ( ( ( minus_minus_int @ A2 @ B )
% 0.34/0.57 = ( minus_minus_int @ C @ D2 ) )
% 0.34/0.57 => ( ( ord_less_eq_int @ A2 @ B )
% 0.34/0.57 = ( ord_less_eq_int @ C @ D2 ) ) ) ).
% 0.34/0.57
% 0.34/0.57 % diff_eq_diff_less_eq
% 0.34/0.57 thf(fact_2548_diff__right__mono,axiom,
% 0.34/0.57 ! [A2: rat,B: rat,C: rat] :
% 0.34/0.57 ( ( ord_less_eq_rat @ A2 @ B )
% 0.34/0.57 => ( ord_less_eq_rat @ ( minus_minus_rat @ A2 @ C ) @ ( minus_minus_rat @ B @ C ) ) ) ).
% 0.34/0.57
% 0.34/0.57 % diff_right_mono
% 0.34/0.57 thf(fact_2549_diff__right__mono,axiom,
% 0.34/0.57 ! [A2: int,B: int,C: int] :
% 0.34/0.57 ( ( ord_less_eq_int @ A2 @ B )
% 0.34/0.57 => ( ord_less_eq_int @ ( minus_minus_int @ A2 @ C ) @ ( minus_minus_int @ B @ C ) ) ) ).
% 0.34/0.57
% 0.34/0.57 % diff_right_mono
% 0.34/0.57 thf(fact_2550_diff__left__mono,axiom,
% 0.34/0.57 ! [B: rat,A2: rat,C: rat] :
% 0.34/0.57 ( ( ord_less_eq_rat @ B @ A2 )
% 0.34/0.57 => ( ord_less_eq_rat @ ( minus_minus_rat @ C @ A2 ) @ ( minus_minus_rat @ C @ B ) ) ) ).
% 0.34/0.57
% 0.34/0.57 % diff_left_mono
% 0.34/0.57 thf(fact_2551_diff__left__mono,axiom,
% 0.34/0.57 ! [B: int,A2: int,C: int] :
% 0.34/0.57 ( ( ord_less_eq_int @ B @ A2 )
% 0.34/0.57 => ( ord_less_eq_int @ ( minus_minus_int @ C @ A2 ) @ ( minus_minus_int @ C @ B ) ) ) ).
% 0.34/0.57
% 0.34/0.57 % diff_left_mono
% 0.34/0.57 thf(fact_2552_diff__mono,axiom,
% 0.34/0.57 ! [A2: rat,B: rat,D2: rat,C: rat] :
% 0.34/0.57 ( ( ord_less_eq_rat @ A2 @ B )
% 0.34/0.57 => ( ( ord_less_eq_rat @ D2 @ C )
% 0.34/0.57 => ( ord_less_eq_rat @ ( minus_minus_rat @ A2 @ C ) @ ( minus_minus_rat @ B @ D2 ) ) ) ) ).
% 0.34/0.57
% 0.34/0.57 % diff_mono
% 0.34/0.57 thf(fact_2553_diff__mono,axiom,
% 0.34/0.57 ! [A2: int,B: int,D2: int,C: int] :
% 0.34/0.57 ( ( ord_less_eq_int @ A2 @ B )
% 0.34/0.57 => ( ( ord_less_eq_int @ D2 @ C )
% 0.34/0.57 => ( ord_less_eq_int @ ( minus_minus_int @ A2 @ C ) @ ( minus_minus_int @ B @ D2 ) ) ) ) ).
% 0.34/0.57
% 0.34/0.57 % diff_mono
% 0.34/0.57 thf(fact_2554_subset__image__iff,axiom,
% 0.34/0.57 ! [B4: set_Code_integer,F: code_integer > code_integer,A: set_Code_integer] :
% 0.34/0.57 ( ( ord_le7084787975880047091nteger @ B4 @ ( image_4470545334726330049nteger @ F @ A ) )
% 0.34/0.57 = ( ? [AA: set_Code_integer] :
% 0.34/0.57 ( ( ord_le7084787975880047091nteger @ AA @ A )
% 0.34/0.57 & ( B4
% 0.34/0.57 = ( image_4470545334726330049nteger @ F @ AA ) ) ) ) ) ).
% 0.34/0.57
% 0.34/0.57 % subset_image_iff
% 0.34/0.57 thf(fact_2555_subset__image__iff,axiom,
% 0.34/0.57 ! [B4: set_set_nat,F: nat > set_nat,A: set_nat] :
% 0.34/0.57 ( ( ord_le6893508408891458716et_nat @ B4 @ ( image_nat_set_nat @ F @ A ) )
% 0.34/0.57 = ( ? [AA: set_nat] :
% 0.34/0.57 ( ( ord_less_eq_set_nat @ AA @ A )
% 0.34/0.57 & ( B4
% 0.34/0.57 = ( image_nat_set_nat @ F @ AA ) ) ) ) ) ).
% 0.34/0.57
% 0.34/0.57 % subset_image_iff
% 0.34/0.57 thf(fact_2556_subset__image__iff,axiom,
% 0.34/0.57 ! [B4: set_Code_integer,F: nat > code_integer,A: set_nat] :
% 0.34/0.57 ( ( ord_le7084787975880047091nteger @ B4 @ ( image_1215581382706833972nteger @ F @ A ) )
% 0.34/0.57 = ( ? [AA: set_nat] :
% 0.34/0.57 ( ( ord_less_eq_set_nat @ AA @ A )
% 0.34/0.57 & ( B4
% 0.34/0.57 = ( image_1215581382706833972nteger @ F @ AA ) ) ) ) ) ).
% 0.34/0.57
% 0.34/0.57 % subset_image_iff
% 0.34/0.57 thf(fact_2557_subset__image__iff,axiom,
% 0.34/0.57 ! [B4: set_nat,F: nat > nat,A: set_nat] :
% 0.34/0.57 ( ( ord_less_eq_set_nat @ B4 @ ( image_nat_nat2 @ F @ A ) )
% 0.34/0.57 = ( ? [AA: set_nat] :
% 0.34/0.57 ( ( ord_less_eq_set_nat @ AA @ A )
% 0.34/0.57 & ( B4
% 0.34/0.57 = ( image_nat_nat2 @ F @ AA ) ) ) ) ) ).
% 0.34/0.57
% 0.34/0.57 % subset_image_iff
% 0.34/0.57 thf(fact_2558_image__subset__iff,axiom,
% 0.34/0.57 ! [F: nat > set_nat,A: set_nat,B4: set_set_nat] :
% 0.34/0.57 ( ( ord_le6893508408891458716et_nat @ ( image_nat_set_nat @ F @ A ) @ B4 )
% 0.34/0.57 = ( ! [X2: nat] :
% 0.34/0.57 ( ( member_nat @ X2 @ A )
% 0.34/0.57 => ( member_set_nat @ ( F @ X2 ) @ B4 ) ) ) ) ).
% 0.34/0.57
% 0.34/0.57 % image_subset_iff
% 0.34/0.57 thf(fact_2559_image__subset__iff,axiom,
% 0.34/0.57 ! [F: code_integer > code_integer,A: set_Code_integer,B4: set_Code_integer] :
% 0.34/0.57 ( ( ord_le7084787975880047091nteger @ ( image_4470545334726330049nteger @ F @ A ) @ B4 )
% 0.34/0.57 = ( ! [X2: code_integer] :
% 0.34/0.57 ( ( member_Code_integer @ X2 @ A )
% 0.34/0.57 => ( member_Code_integer @ ( F @ X2 ) @ B4 ) ) ) ) ).
% 0.34/0.57
% 0.34/0.57 % image_subset_iff
% 0.34/0.57 thf(fact_2560_image__subset__iff,axiom,
% 0.34/0.57 ! [F: nat > code_integer,A: set_nat,B4: set_Code_integer] :
% 0.34/0.57 ( ( ord_le7084787975880047091nteger @ ( image_1215581382706833972nteger @ F @ A ) @ B4 )
% 0.34/0.57 = ( ! [X2: nat] :
% 0.34/0.57 ( ( member_nat @ X2 @ A )
% 0.34/0.57 => ( member_Code_integer @ ( F @ X2 ) @ B4 ) ) ) ) ).
% 0.34/0.57
% 0.34/0.57 % image_subset_iff
% 0.34/0.57 thf(fact_2561_image__subset__iff,axiom,
% 0.34/0.57 ! [F: nat > nat,A: set_nat,B4: set_nat] :
% 0.34/0.57 ( ( ord_less_eq_set_nat @ ( image_nat_nat2 @ F @ A ) @ B4 )
% 0.34/0.57 = ( ! [X2: nat] :
% 0.34/0.57 ( ( member_nat @ X2 @ A )
% 0.34/0.57 => ( member_nat @ ( F @ X2 ) @ B4 ) ) ) ) ).
% 0.34/0.57
% 0.34/0.57 % image_subset_iff
% 0.34/0.57 thf(fact_2562_subset__imageE,axiom,
% 0.34/0.57 ! [B4: set_Code_integer,F: code_integer > code_integer,A: set_Code_integer] :
% 0.34/0.57 ( ( ord_le7084787975880047091nteger @ B4 @ ( image_4470545334726330049nteger @ F @ A ) )
% 0.34/0.57 => ~ ! [C6: set_Code_integer] :
% 0.34/0.57 ( ( ord_le7084787975880047091nteger @ C6 @ A )
% 0.34/0.57 => ( B4
% 0.34/0.57 != ( image_4470545334726330049nteger @ F @ C6 ) ) ) ) ).
% 0.34/0.57
% 0.34/0.57 % subset_imageE
% 0.34/0.57 thf(fact_2563_subset__imageE,axiom,
% 0.34/0.57 ! [B4: set_set_nat,F: nat > set_nat,A: set_nat] :
% 0.34/0.57 ( ( ord_le6893508408891458716et_nat @ B4 @ ( image_nat_set_nat @ F @ A ) )
% 0.34/0.57 => ~ ! [C6: set_nat] :
% 0.34/0.57 ( ( ord_less_eq_set_nat @ C6 @ A )
% 0.34/0.57 => ( B4
% 0.34/0.57 != ( image_nat_set_nat @ F @ C6 ) ) ) ) ).
% 0.34/0.57
% 0.34/0.57 % subset_imageE
% 0.34/0.57 thf(fact_2564_subset__imageE,axiom,
% 0.34/0.57 ! [B4: set_Code_integer,F: nat > code_integer,A: set_nat] :
% 0.34/0.57 ( ( ord_le7084787975880047091nteger @ B4 @ ( image_1215581382706833972nteger @ F @ A ) )
% 0.34/0.57 => ~ ! [C6: set_nat] :
% 0.34/0.57 ( ( ord_less_eq_set_nat @ C6 @ A )
% 0.34/0.57 => ( B4
% 0.34/0.57 != ( image_1215581382706833972nteger @ F @ C6 ) ) ) ) ).
% 0.34/0.57
% 0.34/0.57 % subset_imageE
% 0.34/0.57 thf(fact_2565_subset__imageE,axiom,
% 0.34/0.57 ! [B4: set_nat,F: nat > nat,A: set_nat] :
% 0.34/0.57 ( ( ord_less_eq_set_nat @ B4 @ ( image_nat_nat2 @ F @ A ) )
% 0.34/0.57 => ~ ! [C6: set_nat] :
% 0.34/0.57 ( ( ord_less_eq_set_nat @ C6 @ A )
% 0.34/0.57 => ( B4
% 0.34/0.57 != ( image_nat_nat2 @ F @ C6 ) ) ) ) ).
% 0.34/0.57
% 0.34/0.57 % subset_imageE
% 0.34/0.57 thf(fact_2566_image__subsetI,axiom,
% 0.34/0.57 ! [A: set_Code_integer,F: code_integer > code_integer,B4: set_Code_integer] :
% 0.34/0.57 ( ! [X3: code_integer] :
% 0.34/0.57 ( ( member_Code_integer @ X3 @ A )
% 0.34/0.57 => ( member_Code_integer @ ( F @ X3 ) @ B4 ) )
% 0.34/0.57 => ( ord_le7084787975880047091nteger @ ( image_4470545334726330049nteger @ F @ A ) @ B4 ) ) ).
% 0.34/0.57
% 0.34/0.57 % image_subsetI
% 0.34/0.57 thf(fact_2567_image__subsetI,axiom,
% 0.34/0.57 ! [A: set_nat,F: nat > code_integer,B4: set_Code_integer] :
% 0.34/0.57 ( ! [X3: nat] :
% 0.34/0.57 ( ( member_nat @ X3 @ A )
% 0.34/0.57 => ( member_Code_integer @ ( F @ X3 ) @ B4 ) )
% 0.34/0.57 => ( ord_le7084787975880047091nteger @ ( image_1215581382706833972nteger @ F @ A ) @ B4 ) ) ).
% 0.34/0.57
% 0.34/0.57 % image_subsetI
% 0.34/0.57 thf(fact_2568_image__subsetI,axiom,
% 0.34/0.57 ! [A: set_nat,F: nat > int,B4: set_int] :
% 0.34/0.57 ( ! [X3: nat] :
% 0.34/0.57 ( ( member_nat @ X3 @ A )
% 0.34/0.57 => ( member_int @ ( F @ X3 ) @ B4 ) )
% 0.34/0.57 => ( ord_less_eq_set_int @ ( image_nat_int2 @ F @ A ) @ B4 ) ) ).
% 0.34/0.57
% 0.34/0.57 % image_subsetI
% 0.34/0.57 thf(fact_2569_image__subsetI,axiom,
% 0.34/0.57 ! [A: set_nat,F: nat > $o,B4: set_o] :
% 0.34/0.57 ( ! [X3: nat] :
% 0.34/0.57 ( ( member_nat @ X3 @ A )
% 0.34/0.57 => ( member_o @ ( F @ X3 ) @ B4 ) )
% 0.34/0.57 => ( ord_less_eq_set_o @ ( image_nat_o2 @ F @ A ) @ B4 ) ) ).
% 0.34/0.57
% 0.34/0.57 % image_subsetI
% 0.34/0.57 thf(fact_2570_image__subsetI,axiom,
% 0.34/0.57 ! [A: set_nat,F: nat > product_unit,B4: set_Product_unit] :
% 0.34/0.57 ( ! [X3: nat] :
% 0.34/0.57 ( ( member_nat @ X3 @ A )
% 0.34/0.57 => ( member_Product_unit @ ( F @ X3 ) @ B4 ) )
% 0.34/0.57 => ( ord_le3507040750410214029t_unit @ ( image_8730104196221521654t_unit @ F @ A ) @ B4 ) ) ).
% 0.34/0.57
% 0.34/0.57 % image_subsetI
% 0.34/0.57 thf(fact_2571_image__subsetI,axiom,
% 0.34/0.57 ! [A: set_int,F: int > int,B4: set_int] :
% 0.34/0.57 ( ! [X3: int] :
% 0.34/0.57 ( ( member_int @ X3 @ A )
% 0.34/0.57 => ( member_int @ ( F @ X3 ) @ B4 ) )
% 0.34/0.57 => ( ord_less_eq_set_int @ ( image_int_int2 @ F @ A ) @ B4 ) ) ).
% 0.34/0.57
% 0.34/0.57 % image_subsetI
% 0.34/0.57 thf(fact_2572_image__subsetI,axiom,
% 0.34/0.57 ! [A: set_int,F: int > $o,B4: set_o] :
% 0.34/0.57 ( ! [X3: int] :
% 0.34/0.57 ( ( member_int @ X3 @ A )
% 0.34/0.57 => ( member_o @ ( F @ X3 ) @ B4 ) )
% 0.34/0.57 => ( ord_less_eq_set_o @ ( image_int_o2 @ F @ A ) @ B4 ) ) ).
% 0.34/0.57
% 0.34/0.57 % image_subsetI
% 0.34/0.57 thf(fact_2573_image__subsetI,axiom,
% 0.34/0.57 ! [A: set_int,F: int > product_unit,B4: set_Product_unit] :
% 0.34/0.57 ( ! [X3: int] :
% 0.34/0.57 ( ( member_int @ X3 @ A )
% 0.34/0.57 => ( member_Product_unit @ ( F @ X3 ) @ B4 ) )
% 0.34/0.57 => ( ord_le3507040750410214029t_unit @ ( image_5846526143423701914t_unit @ F @ A ) @ B4 ) ) ).
% 0.34/0.57
% 0.34/0.57 % image_subsetI
% 0.34/0.57 thf(fact_2574_image__subsetI,axiom,
% 0.34/0.57 ! [A: set_o,F: $o > int,B4: set_int] :
% 0.34/0.57 ( ! [X3: $o] :
% 0.34/0.57 ( ( member_o @ X3 @ A )
% 0.34/0.57 => ( member_int @ ( F @ X3 ) @ B4 ) )
% 0.34/0.57 => ( ord_less_eq_set_int @ ( image_o_int2 @ F @ A ) @ B4 ) ) ).
% 0.34/0.57
% 0.34/0.57 % image_subsetI
% 0.34/0.57 thf(fact_2575_image__subsetI,axiom,
% 0.34/0.57 ! [A: set_o,F: $o > $o,B4: set_o] :
% 0.34/0.57 ( ! [X3: $o] :
% 0.34/0.57 ( ( member_o @ X3 @ A )
% 0.34/0.57 => ( member_o @ ( F @ X3 ) @ B4 ) )
% 0.34/0.57 => ( ord_less_eq_set_o @ ( image_o_o2 @ F @ A ) @ B4 ) ) ).
% 0.34/0.57
% 0.34/0.57 % image_subsetI
% 0.34/0.57 thf(fact_2576_image__mono,axiom,
% 0.34/0.57 ! [A: set_Code_integer,B4: set_Code_integer,F: code_integer > code_integer] :
% 0.34/0.57 ( ( ord_le7084787975880047091nteger @ A @ B4 )
% 0.34/0.57 => ( ord_le7084787975880047091nteger @ ( image_4470545334726330049nteger @ F @ A ) @ ( image_4470545334726330049nteger @ F @ B4 ) ) ) ).
% 0.34/0.57
% 0.34/0.57 % image_mono
% 0.34/0.57 thf(fact_2577_image__mono,axiom,
% 0.34/0.57 ! [A: set_nat,B4: set_nat,F: nat > set_nat] :
% 0.34/0.57 ( ( ord_less_eq_set_nat @ A @ B4 )
% 0.34/0.57 => ( ord_le6893508408891458716et_nat @ ( image_nat_set_nat @ F @ A ) @ ( image_nat_set_nat @ F @ B4 ) ) ) ).
% 0.34/0.57
% 0.34/0.57 % image_mono
% 0.34/0.57 thf(fact_2578_image__mono,axiom,
% 0.34/0.57 ! [A: set_nat,B4: set_nat,F: nat > code_integer] :
% 0.34/0.57 ( ( ord_less_eq_set_nat @ A @ B4 )
% 0.34/0.57 => ( ord_le7084787975880047091nteger @ ( image_1215581382706833972nteger @ F @ A ) @ ( image_1215581382706833972nteger @ F @ B4 ) ) ) ).
% 0.34/0.57
% 0.34/0.57 % image_mono
% 0.34/0.57 thf(fact_2579_image__mono,axiom,
% 0.34/0.57 ! [A: set_nat,B4: set_nat,F: nat > nat] :
% 0.34/0.57 ( ( ord_less_eq_set_nat @ A @ B4 )
% 0.34/0.57 => ( ord_less_eq_set_nat @ ( image_nat_nat2 @ F @ A ) @ ( image_nat_nat2 @ F @ B4 ) ) ) ).
% 0.34/0.57
% 0.34/0.57 % image_mono
% 0.34/0.57 thf(fact_2580_image__Un,axiom,
% 0.34/0.57 ! [F: code_integer > code_integer,A: set_Code_integer,B4: set_Code_integer] :
% 0.34/0.57 ( ( image_4470545334726330049nteger @ F @ ( sup_su848401254843788991nteger @ A @ B4 ) )
% 0.34/0.57 = ( sup_su848401254843788991nteger @ ( image_4470545334726330049nteger @ F @ A ) @ ( image_4470545334726330049nteger @ F @ B4 ) ) ) ).
% 0.34/0.57
% 0.34/0.57 % image_Un
% 0.34/0.57 thf(fact_2581_image__Un,axiom,
% 0.34/0.57 ! [F: nat > set_nat,A: set_nat,B4: set_nat] :
% 0.34/0.57 ( ( image_nat_set_nat @ F @ ( sup_sup_set_nat @ A @ B4 ) )
% 0.34/0.57 = ( sup_sup_set_set_nat @ ( image_nat_set_nat @ F @ A ) @ ( image_nat_set_nat @ F @ B4 ) ) ) ).
% 0.34/0.57
% 0.34/0.57 % image_Un
% 0.34/0.57 thf(fact_2582_image__Un,axiom,
% 0.34/0.57 ! [F: nat > code_integer,A: set_nat,B4: set_nat] :
% 0.34/0.57 ( ( image_1215581382706833972nteger @ F @ ( sup_sup_set_nat @ A @ B4 ) )
% 0.34/0.57 = ( sup_su848401254843788991nteger @ ( image_1215581382706833972nteger @ F @ A ) @ ( image_1215581382706833972nteger @ F @ B4 ) ) ) ).
% 0.34/0.57
% 0.34/0.57 % image_Un
% 0.34/0.57 thf(fact_2583_image__Un,axiom,
% 0.34/0.57 ! [F: nat > nat,A: set_nat,B4: set_nat] :
% 0.34/0.57 ( ( image_nat_nat2 @ F @ ( sup_sup_set_nat @ A @ B4 ) )
% 0.34/0.57 = ( sup_sup_set_nat @ ( image_nat_nat2 @ F @ A ) @ ( image_nat_nat2 @ F @ B4 ) ) ) ).
% 0.34/0.57
% 0.34/0.57 % image_Un
% 0.34/0.57 thf(fact_2584_double__diff,axiom,
% 0.34/0.57 ! [A: set_nat,B4: set_nat,C5: set_nat] :
% 0.34/0.57 ( ( ord_less_eq_set_nat @ A @ B4 )
% 0.34/0.57 => ( ( ord_less_eq_set_nat @ B4 @ C5 )
% 0.34/0.57 => ( ( minus_minus_set_nat @ B4 @ ( minus_minus_set_nat @ C5 @ A ) )
% 0.34/0.57 = A ) ) ) ).
% 0.34/0.57
% 0.34/0.57 % double_diff
% 0.34/0.57 thf(fact_2585_Diff__subset,axiom,
% 0.34/0.57 ! [A: set_nat,B4: set_nat] : ( ord_less_eq_set_nat @ ( minus_minus_set_nat @ A @ B4 ) @ A ) ).
% 0.34/0.57
% 0.34/0.57 % Diff_subset
% 0.34/0.57 thf(fact_2586_Diff__mono,axiom,
% 0.34/0.57 ! [A: set_nat,C5: set_nat,D3: set_nat,B4: set_nat] :
% 0.34/0.57 ( ( ord_less_eq_set_nat @ A @ C5 )
% 0.34/0.57 => ( ( ord_less_eq_set_nat @ D3 @ B4 )
% 0.34/0.57 => ( ord_less_eq_set_nat @ ( minus_minus_set_nat @ A @ B4 ) @ ( minus_minus_set_nat @ C5 @ D3 ) ) ) ) ).
% 0.34/0.57
% 0.34/0.57 % Diff_mono
% 0.34/0.57 thf(fact_2587_insert__Diff__if,axiom,
% 0.34/0.57 ! [X: produc3658429121746597890et_nat > $o,B4: set_Pr4532377907799695533_nat_o,A: set_Pr4532377907799695533_nat_o] :
% 0.34/0.57 ( ( ( member6576561426505652726_nat_o @ X @ B4 )
% 0.34/0.57 => ( ( minus_1801376950450012436_nat_o @ ( insert5175938949040314269_nat_o @ X @ A ) @ B4 )
% 0.34/0.57 = ( minus_1801376950450012436_nat_o @ A @ B4 ) ) )
% 0.34/0.57 & ( ~ ( member6576561426505652726_nat_o @ X @ B4 )
% 0.34/0.57 => ( ( minus_1801376950450012436_nat_o @ ( insert5175938949040314269_nat_o @ X @ A ) @ B4 )
% 0.34/0.57 = ( insert5175938949040314269_nat_o @ X @ ( minus_1801376950450012436_nat_o @ A @ B4 ) ) ) ) ) ).
% 0.34/0.57
% 0.34/0.57 % insert_Diff_if
% 0.34/0.57 thf(fact_2588_insert__Diff__if,axiom,
% 0.34/0.57 ! [X: int,B4: set_int,A: set_int] :
% 0.34/0.57 ( ( ( member_int @ X @ B4 )
% 0.34/0.57 => ( ( minus_minus_set_int @ ( insert_int @ X @ A ) @ B4 )
% 0.34/0.57 = ( minus_minus_set_int @ A @ B4 ) ) )
% 0.34/0.57 & ( ~ ( member_int @ X @ B4 )
% 0.34/0.57 => ( ( minus_minus_set_int @ ( insert_int @ X @ A ) @ B4 )
% 0.34/0.57 = ( insert_int @ X @ ( minus_minus_set_int @ A @ B4 ) ) ) ) ) ).
% 0.34/0.57
% 0.34/0.57 % insert_Diff_if
% 0.34/0.57 thf(fact_2589_insert__Diff__if,axiom,
% 0.34/0.57 ! [X: $o,B4: set_o,A: set_o] :
% 0.34/0.57 ( ( ( member_o @ X @ B4 )
% 0.34/0.57 => ( ( minus_minus_set_o @ ( insert_o @ X @ A ) @ B4 )
% 0.34/0.57 = ( minus_minus_set_o @ A @ B4 ) ) )
% 0.34/0.57 & ( ~ ( member_o @ X @ B4 )
% 0.34/0.57 => ( ( minus_minus_set_o @ ( insert_o @ X @ A ) @ B4 )
% 0.34/0.57 = ( insert_o @ X @ ( minus_minus_set_o @ A @ B4 ) ) ) ) ) ).
% 0.34/0.57
% 0.34/0.57 % insert_Diff_if
% 0.34/0.57 thf(fact_2590_insert__Diff__if,axiom,
% 0.34/0.57 ! [X: product_unit,B4: set_Product_unit,A: set_Product_unit] :
% 0.34/0.57 ( ( ( member_Product_unit @ X @ B4 )
% 0.34/0.57 => ( ( minus_6452836326544984404t_unit @ ( insert_Product_unit @ X @ A ) @ B4 )
% 0.34/0.57 = ( minus_6452836326544984404t_unit @ A @ B4 ) ) )
% 0.34/0.57 & ( ~ ( member_Product_unit @ X @ B4 )
% 0.34/0.57 => ( ( minus_6452836326544984404t_unit @ ( insert_Product_unit @ X @ A ) @ B4 )
% 0.34/0.57 = ( insert_Product_unit @ X @ ( minus_6452836326544984404t_unit @ A @ B4 ) ) ) ) ) ).
% 0.34/0.57
% 0.34/0.57 % insert_Diff_if
% 0.34/0.57 thf(fact_2591_insert__Diff__if,axiom,
% 0.34/0.57 ! [X: nat,B4: set_nat,A: set_nat] :
% 0.34/0.57 ( ( ( member_nat @ X @ B4 )
% 0.34/0.57 => ( ( minus_minus_set_nat @ ( insert_nat @ X @ A ) @ B4 )
% 0.34/0.57 = ( minus_minus_set_nat @ A @ B4 ) ) )
% 0.34/0.57 & ( ~ ( member_nat @ X @ B4 )
% 0.34/0.57 => ( ( minus_minus_set_nat @ ( insert_nat @ X @ A ) @ B4 )
% 0.34/0.57 = ( insert_nat @ X @ ( minus_minus_set_nat @ A @ B4 ) ) ) ) ) ).
% 0.34/0.57
% 0.34/0.57 % insert_Diff_if
% 0.34/0.57 thf(fact_2592_Diff__Int__distrib2,axiom,
% 0.34/0.57 ! [A: set_nat,B4: set_nat,C5: set_nat] :
% 0.34/0.57 ( ( inf_inf_set_nat @ ( minus_minus_set_nat @ A @ B4 ) @ C5 )
% 0.34/0.57 = ( minus_minus_set_nat @ ( inf_inf_set_nat @ A @ C5 ) @ ( inf_inf_set_nat @ B4 @ C5 ) ) ) ).
% 0.34/0.57
% 0.34/0.57 % Diff_Int_distrib2
% 0.34/0.57 thf(fact_2593_Diff__Int__distrib,axiom,
% 0.34/0.57 ! [C5: set_nat,A: set_nat,B4: set_nat] :
% 0.34/0.57 ( ( inf_inf_set_nat @ C5 @ ( minus_minus_set_nat @ A @ B4 ) )
% 0.34/0.57 = ( minus_minus_set_nat @ ( inf_inf_set_nat @ C5 @ A ) @ ( inf_inf_set_nat @ C5 @ B4 ) ) ) ).
% 0.34/0.57
% 0.34/0.57 % Diff_Int_distrib
% 0.34/0.57 thf(fact_2594_Diff__Diff__Int,axiom,
% 0.34/0.57 ! [A: set_nat,B4: set_nat] :
% 0.34/0.57 ( ( minus_minus_set_nat @ A @ ( minus_minus_set_nat @ A @ B4 ) )
% 0.34/0.57 = ( inf_inf_set_nat @ A @ B4 ) ) ).
% 0.34/0.57
% 0.34/0.57 % Diff_Diff_Int
% 0.34/0.57 thf(fact_2595_Diff__Int2,axiom,
% 0.34/0.57 ! [A: set_nat,C5: set_nat,B4: set_nat] :
% 0.34/0.57 ( ( minus_minus_set_nat @ ( inf_inf_set_nat @ A @ C5 ) @ ( inf_inf_set_nat @ B4 @ C5 ) )
% 0.34/0.57 = ( minus_minus_set_nat @ ( inf_inf_set_nat @ A @ C5 ) @ B4 ) ) ).
% 0.34/0.57
% 0.34/0.57 % Diff_Int2
% 0.34/0.57 thf(fact_2596_Int__Diff,axiom,
% 0.34/0.57 ! [A: set_nat,B4: set_nat,C5: set_nat] :
% 0.34/0.57 ( ( minus_minus_set_nat @ ( inf_inf_set_nat @ A @ B4 ) @ C5 )
% 0.34/0.57 = ( inf_inf_set_nat @ A @ ( minus_minus_set_nat @ B4 @ C5 ) ) ) ).
% 0.34/0.57
% 0.34/0.57 % Int_Diff
% 0.34/0.57 thf(fact_2597_set__diff__diff__left,axiom,
% 0.34/0.57 ! [A: set_nat,B4: set_nat,C5: set_nat] :
% 0.34/0.57 ( ( minus_minus_set_nat @ ( minus_minus_set_nat @ A @ B4 ) @ C5 )
% 0.34/0.57 = ( minus_minus_set_nat @ A @ ( sup_sup_set_nat @ B4 @ C5 ) ) ) ).
% 0.34/0.57
% 0.34/0.57 % set_diff_diff_left
% 0.34/0.57 thf(fact_2598_Un__Diff,axiom,
% 0.34/0.57 ! [A: set_nat,B4: set_nat,C5: set_nat] :
% 0.34/0.57 ( ( minus_minus_set_nat @ ( sup_sup_set_nat @ A @ B4 ) @ C5 )
% 0.34/0.57 = ( sup_sup_set_nat @ ( minus_minus_set_nat @ A @ C5 ) @ ( minus_minus_set_nat @ B4 @ C5 ) ) ) ).
% 0.34/0.57
% 0.34/0.57 % Un_Diff
% 0.34/0.57 thf(fact_2599_vimage__Diff,axiom,
% 0.34/0.57 ! [F: nat > nat,A: set_nat,B4: set_nat] :
% 0.34/0.57 ( ( vimage_nat_nat @ F @ ( minus_minus_set_nat @ A @ B4 ) )
% 0.34/0.57 = ( minus_minus_set_nat @ ( vimage_nat_nat @ F @ A ) @ ( vimage_nat_nat @ F @ B4 ) ) ) ).
% 0.34/0.57
% 0.34/0.57 % vimage_Diff
% 0.34/0.57 thf(fact_2600_rtrancl__converseI,axiom,
% 0.34/0.57 ! [Y: nat,X: nat,R2: set_Pr1261947904930325089at_nat] :
% 0.34/0.57 ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ Y @ X ) @ ( transi2905341329935302413cl_nat @ R2 ) )
% 0.34/0.57 => ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ Y ) @ ( transi2905341329935302413cl_nat @ ( converse_nat_nat @ R2 ) ) ) ) ).
% 0.34/0.57
% 0.34/0.57 % rtrancl_converseI
% 0.34/0.57 thf(fact_2601_rtrancl__converseI,axiom,
% 0.34/0.57 ! [Y: int,X: int,R2: set_Pr958786334691620121nt_int] :
% 0.34/0.57 ( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ Y @ X ) @ ( transi2902850859426252137cl_int @ R2 ) )
% 0.34/0.57 => ( member5262025264175285858nt_int @ ( product_Pair_int_int @ X @ Y ) @ ( transi2902850859426252137cl_int @ ( converse_int_int @ R2 ) ) ) ) ).
% 0.34/0.57
% 0.34/0.57 % rtrancl_converseI
% 0.34/0.57 thf(fact_2602_rtrancl__converseD,axiom,
% 0.34/0.57 ! [X: nat,Y: nat,R2: set_Pr1261947904930325089at_nat] :
% 0.34/0.57 ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ Y ) @ ( transi2905341329935302413cl_nat @ ( converse_nat_nat @ R2 ) ) )
% 0.34/0.57 => ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ Y @ X ) @ ( transi2905341329935302413cl_nat @ R2 ) ) ) ).
% 0.34/0.57
% 0.34/0.57 % rtrancl_converseD
% 0.34/0.57 thf(fact_2603_rtrancl__converseD,axiom,
% 0.34/0.57 ! [X: int,Y: int,R2: set_Pr958786334691620121nt_int] :
% 0.34/0.57 ( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ X @ Y ) @ ( transi2902850859426252137cl_int @ ( converse_int_int @ R2 ) ) )
% 0.34/0.57 => ( member5262025264175285858nt_int @ ( product_Pair_int_int @ Y @ X ) @ ( transi2902850859426252137cl_int @ R2 ) ) ) ).
% 0.34/0.57
% 0.34/0.57 % rtrancl_converseD
% 0.34/0.57 thf(fact_2604_trancl__converseI,axiom,
% 0.34/0.57 ! [X: nat,Y: nat,R2: set_Pr1261947904930325089at_nat] :
% 0.34/0.57 ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ Y ) @ ( converse_nat_nat @ ( transi6264000038957366511cl_nat @ R2 ) ) )
% 0.34/0.57 => ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ Y ) @ ( transi6264000038957366511cl_nat @ ( converse_nat_nat @ R2 ) ) ) ) ).
% 0.34/0.57
% 0.34/0.57 % trancl_converseI
% 0.34/0.57 thf(fact_2605_trancl__converseI,axiom,
% 0.34/0.57 ! [X: int,Y: int,R2: set_Pr958786334691620121nt_int] :
% 0.34/0.57 ( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ X @ Y ) @ ( converse_int_int @ ( transi6261509568448316235cl_int @ R2 ) ) )
% 0.34/0.57 => ( member5262025264175285858nt_int @ ( product_Pair_int_int @ X @ Y ) @ ( transi6261509568448316235cl_int @ ( converse_int_int @ R2 ) ) ) ) ).
% 0.34/0.57
% 0.34/0.57 % trancl_converseI
% 0.34/0.57 thf(fact_2606_trancl__converseD,axiom,
% 0.34/0.57 ! [X: nat,Y: nat,R2: set_Pr1261947904930325089at_nat] :
% 0.34/0.57 ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ Y ) @ ( transi6264000038957366511cl_nat @ ( converse_nat_nat @ R2 ) ) )
% 0.34/0.57 => ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ Y ) @ ( converse_nat_nat @ ( transi6264000038957366511cl_nat @ R2 ) ) ) ) ).
% 0.34/0.57
% 0.34/0.57 % trancl_converseD
% 0.34/0.57 thf(fact_2607_trancl__converseD,axiom,
% 0.34/0.57 ! [X: int,Y: int,R2: set_Pr958786334691620121nt_int] :
% 0.34/0.57 ( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ X @ Y ) @ ( transi6261509568448316235cl_int @ ( converse_int_int @ R2 ) ) )
% 0.34/0.57 => ( member5262025264175285858nt_int @ ( product_Pair_int_int @ X @ Y ) @ ( converse_int_int @ ( transi6261509568448316235cl_int @ R2 ) ) ) ) ).
% 0.34/0.57
% 0.34/0.57 % trancl_converseD
% 0.34/0.57 thf(fact_2608_diff__shunt__var,axiom,
% 0.34/0.57 ! [X: set_Product_unit,Y: set_Product_unit] :
% 0.34/0.57 ( ( ( minus_6452836326544984404t_unit @ X @ Y )
% 0.34/0.57 = bot_bo3957492148770167129t_unit )
% 0.34/0.57 = ( ord_le3507040750410214029t_unit @ X @ Y ) ) ).
% 0.34/0.57
% 0.34/0.57 % diff_shunt_var
% 0.34/0.57 thf(fact_2609_diff__shunt__var,axiom,
% 0.34/0.57 ! [X: set_o,Y: set_o] :
% 0.34/0.57 ( ( ( minus_minus_set_o @ X @ Y )
% 0.34/0.57 = bot_bot_set_o )
% 0.34/0.57 = ( ord_less_eq_set_o @ X @ Y ) ) ).
% 0.34/0.57
% 0.34/0.57 % diff_shunt_var
% 0.34/0.57 thf(fact_2610_diff__shunt__var,axiom,
% 0.34/0.57 ! [X: set_int,Y: set_int] :
% 0.34/0.57 ( ( ( minus_minus_set_int @ X @ Y )
% 0.34/0.57 = bot_bot_set_int )
% 0.34/0.57 = ( ord_less_eq_set_int @ X @ Y ) ) ).
% 0.34/0.57
% 0.34/0.57 % diff_shunt_var
% 0.34/0.57 thf(fact_2611_diff__shunt__var,axiom,
% 0.34/0.57 ! [X: set_Pr4532377907799695533_nat_o,Y: set_Pr4532377907799695533_nat_o] :
% 0.34/0.57 ( ( ( minus_1801376950450012436_nat_o @ X @ Y )
% 0.34/0.57 = bot_bo7824918357723582233_nat_o )
% 0.34/0.57 = ( ord_le2965882846123202637_nat_o @ X @ Y ) ) ).
% 0.34/0.57
% 0.34/0.57 % diff_shunt_var
% 0.34/0.57 thf(fact_2612_diff__shunt__var,axiom,
% 0.34/0.57 ! [X: assn,Y: assn] :
% 0.34/0.57 ( ( ( minus_minus_assn @ X @ Y )
% 0.34/0.57 = bot_bot_assn )
% 0.34/0.57 = ( ord_less_eq_assn @ X @ Y ) ) ).
% 0.34/0.57
% 0.34/0.57 % diff_shunt_var
% 0.34/0.57 thf(fact_2613_diff__shunt__var,axiom,
% 0.34/0.57 ! [X: set_nat,Y: set_nat] :
% 0.34/0.57 ( ( ( minus_minus_set_nat @ X @ Y )
% 0.34/0.57 = bot_bot_set_nat )
% 0.34/0.57 = ( ord_less_eq_set_nat @ X @ Y ) ) ).
% 0.34/0.57
% 0.34/0.57 % diff_shunt_var
% 0.34/0.57 thf(fact_2614_in__snd__imageE,axiom,
% 0.34/0.57 ! [Y: nat,S: set_Pr1261947904930325089at_nat] :
% 0.34/0.57 ( ( member_nat @ Y @ ( image_2486076414777270412at_nat @ product_snd_nat_nat @ S ) )
% 0.34/0.57 => ~ ! [X3: nat] :
% 0.34/0.57 ~ ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X3 @ Y ) @ S ) ) ).
% 0.34/0.57
% 0.34/0.57 % in_snd_imageE
% 0.34/0.57 thf(fact_2615_in__snd__imageE,axiom,
% 0.34/0.57 ! [Y: produc3658429121746597890et_nat,S: set_Pr3286484037609594932et_nat] :
% 0.34/0.57 ( ( member6260224972018164377et_nat @ Y @ ( image_4390311411407041927et_nat @ produc4011572625026189258et_nat @ S ) )
% 0.34/0.57 => ~ ! [X3: produc3658429121746597890et_nat > $o] :
% 0.34/0.57 ~ ( member1996754912294343701et_nat @ ( produc5001842942810119800et_nat @ X3 @ Y ) @ S ) ) ).
% 0.34/0.57
% 0.34/0.57 % in_snd_imageE
% 0.34/0.57 thf(fact_2616_in__snd__imageE,axiom,
% 0.34/0.57 ! [Y: produc3925858234332021118et_nat,S: set_Pr8536935166611901872et_nat] :
% 0.34/0.57 ( ( member1996754912294343701et_nat @ Y @ ( image_4557310310409863807et_nat @ produc5374455773327741254et_nat @ S ) )
% 0.34/0.57 => ~ ! [X3: produc3658429121746597890et_nat > $o] :
% 0.34/0.57 ~ ( member6124377750444531601et_nat @ ( produc2245416461498447860et_nat @ X3 @ Y ) @ S ) ) ).
% 0.34/0.57
% 0.34/0.57 % in_snd_imageE
% 0.34/0.57 thf(fact_2617_in__snd__imageE,axiom,
% 0.34/0.57 ! [Y: set_nat,S: set_Pr3948176798113811640et_nat] :
% 0.34/0.57 ( ( member_set_nat @ Y @ ( image_6088997261112373647et_nat @ produc8586169260539613262et_nat @ S ) )
% 0.34/0.57 => ~ ! [X3: heap_e7401611519738050253t_unit] :
% 0.34/0.57 ~ ( member6260224972018164377et_nat @ ( produc7507926704131184380et_nat @ X3 @ Y ) @ S ) ) ).
% 0.34/0.57
% 0.34/0.57 % in_snd_imageE
% 0.34/0.57 thf(fact_2618_in__snd__imageE,axiom,
% 0.34/0.57 ! [Y: int,S: set_Pr958786334691620121nt_int] :
% 0.34/0.57 ( ( member_int @ Y @ ( image_5042161079198086560nt_int @ product_snd_int_int @ S ) )
% 0.34/0.57 => ~ ! [X3: int] :
% 0.34/0.57 ~ ( member5262025264175285858nt_int @ ( product_Pair_int_int @ X3 @ Y ) @ S ) ) ).
% 0.34/0.57
% 0.34/0.57 % in_snd_imageE
% 0.34/0.57 thf(fact_2619_image__Int__subset,axiom,
% 0.34/0.57 ! [F: code_integer > code_integer,A: set_Code_integer,B4: set_Code_integer] : ( ord_le7084787975880047091nteger @ ( image_4470545334726330049nteger @ F @ ( inf_in1364745209274528805nteger @ A @ B4 ) ) @ ( inf_in1364745209274528805nteger @ ( image_4470545334726330049nteger @ F @ A ) @ ( image_4470545334726330049nteger @ F @ B4 ) ) ) ).
% 0.34/0.57
% 0.34/0.57 % image_Int_subset
% 0.34/0.57 thf(fact_2620_image__Int__subset,axiom,
% 0.34/0.57 ! [F: nat > set_nat,A: set_nat,B4: set_nat] : ( ord_le6893508408891458716et_nat @ ( image_nat_set_nat @ F @ ( inf_inf_set_nat @ A @ B4 ) ) @ ( inf_inf_set_set_nat @ ( image_nat_set_nat @ F @ A ) @ ( image_nat_set_nat @ F @ B4 ) ) ) ).
% 0.34/0.57
% 0.34/0.57 % image_Int_subset
% 0.34/0.57 thf(fact_2621_image__Int__subset,axiom,
% 0.34/0.57 ! [F: nat > code_integer,A: set_nat,B4: set_nat] : ( ord_le7084787975880047091nteger @ ( image_1215581382706833972nteger @ F @ ( inf_inf_set_nat @ A @ B4 ) ) @ ( inf_in1364745209274528805nteger @ ( image_1215581382706833972nteger @ F @ A ) @ ( image_1215581382706833972nteger @ F @ B4 ) ) ) ).
% 0.34/0.57
% 0.34/0.57 % image_Int_subset
% 0.34/0.57 thf(fact_2622_image__Int__subset,axiom,
% 0.34/0.57 ! [F: nat > nat,A: set_nat,B4: set_nat] : ( ord_less_eq_set_nat @ ( image_nat_nat2 @ F @ ( inf_inf_set_nat @ A @ B4 ) ) @ ( inf_inf_set_nat @ ( image_nat_nat2 @ F @ A ) @ ( image_nat_nat2 @ F @ B4 ) ) ) ).
% 0.34/0.57
% 0.34/0.57 % image_Int_subset
% 0.34/0.57 thf(fact_2623_subset__minus__empty,axiom,
% 0.34/0.57 ! [A: set_Product_unit,B4: set_Product_unit] :
% 0.34/0.57 ( ( ord_le3507040750410214029t_unit @ A @ B4 )
% 0.34/0.57 => ( ( minus_6452836326544984404t_unit @ A @ B4 )
% 0.34/0.57 = bot_bo3957492148770167129t_unit ) ) ).
% 0.34/0.57
% 0.34/0.57 % subset_minus_empty
% 0.34/0.57 thf(fact_2624_subset__minus__empty,axiom,
% 0.34/0.57 ! [A: set_o,B4: set_o] :
% 0.34/0.57 ( ( ord_less_eq_set_o @ A @ B4 )
% 0.34/0.57 => ( ( minus_minus_set_o @ A @ B4 )
% 0.34/0.57 = bot_bot_set_o ) ) ).
% 0.34/0.57
% 0.34/0.57 % subset_minus_empty
% 0.34/0.57 thf(fact_2625_subset__minus__empty,axiom,
% 0.34/0.57 ! [A: set_int,B4: set_int] :
% 0.34/0.57 ( ( ord_less_eq_set_int @ A @ B4 )
% 0.34/0.57 => ( ( minus_minus_set_int @ A @ B4 )
% 0.34/0.57 = bot_bot_set_int ) ) ).
% 0.34/0.57
% 0.34/0.57 % subset_minus_empty
% 0.34/0.57 thf(fact_2626_subset__minus__empty,axiom,
% 0.34/0.57 ! [A: set_Pr4532377907799695533_nat_o,B4: set_Pr4532377907799695533_nat_o] :
% 0.34/0.57 ( ( ord_le2965882846123202637_nat_o @ A @ B4 )
% 0.34/0.57 => ( ( minus_1801376950450012436_nat_o @ A @ B4 )
% 0.34/0.57 = bot_bo7824918357723582233_nat_o ) ) ).
% 0.34/0.57
% 0.34/0.57 % subset_minus_empty
% 0.34/0.57 thf(fact_2627_subset__minus__empty,axiom,
% 0.34/0.57 ! [A: set_nat,B4: set_nat] :
% 0.34/0.57 ( ( ord_less_eq_set_nat @ A @ B4 )
% 0.34/0.57 => ( ( minus_minus_set_nat @ A @ B4 )
% 0.34/0.57 = bot_bot_set_nat ) ) ).
% 0.34/0.57
% 0.34/0.57 % subset_minus_empty
% 0.34/0.57 thf(fact_2628_image__subset__iff__subset__vimage,axiom,
% 0.34/0.57 ! [F: code_integer > code_integer,A: set_Code_integer,B4: set_Code_integer] :
% 0.34/0.57 ( ( ord_le7084787975880047091nteger @ ( image_4470545334726330049nteger @ F @ A ) @ B4 )
% 0.34/0.57 = ( ord_le7084787975880047091nteger @ A @ ( vimage1318793397690208119nteger @ F @ B4 ) ) ) ).
% 0.34/0.57
% 0.34/0.57 % image_subset_iff_subset_vimage
% 0.34/0.57 thf(fact_2629_image__subset__iff__subset__vimage,axiom,
% 0.34/0.57 ! [F: nat > set_nat,A: set_nat,B4: set_set_nat] :
% 0.34/0.57 ( ( ord_le6893508408891458716et_nat @ ( image_nat_set_nat @ F @ A ) @ B4 )
% 0.34/0.57 = ( ord_less_eq_set_nat @ A @ ( vimage_nat_set_nat @ F @ B4 ) ) ) ).
% 0.34/0.57
% 0.34/0.57 % image_subset_iff_subset_vimage
% 0.34/0.57 thf(fact_2630_image__subset__iff__subset__vimage,axiom,
% 0.34/0.57 ! [F: nat > code_integer,A: set_nat,B4: set_Code_integer] :
% 0.34/0.57 ( ( ord_le7084787975880047091nteger @ ( image_1215581382706833972nteger @ F @ A ) @ B4 )
% 0.34/0.57 = ( ord_less_eq_set_nat @ A @ ( vimage6104394758324456170nteger @ F @ B4 ) ) ) ).
% 0.34/0.57
% 0.34/0.57 % image_subset_iff_subset_vimage
% 0.34/0.57 thf(fact_2631_image__subset__iff__subset__vimage,axiom,
% 0.34/0.57 ! [F: nat > nat,A: set_nat,B4: set_nat] :
% 0.34/0.57 ( ( ord_less_eq_set_nat @ ( image_nat_nat2 @ F @ A ) @ B4 )
% 0.34/0.57 = ( ord_less_eq_set_nat @ A @ ( vimage_nat_nat @ F @ B4 ) ) ) ).
% 0.34/0.57
% 0.34/0.57 % image_subset_iff_subset_vimage
% 0.34/0.57 thf(fact_2632_image__vimage__subset,axiom,
% 0.34/0.57 ! [F: nat > set_nat,A: set_set_nat] : ( ord_le6893508408891458716et_nat @ ( image_nat_set_nat @ F @ ( vimage_nat_set_nat @ F @ A ) ) @ A ) ).
% 0.34/0.57
% 0.34/0.57 % image_vimage_subset
% 0.34/0.57 thf(fact_2633_image__vimage__subset,axiom,
% 0.34/0.57 ! [F: code_integer > code_integer,A: set_Code_integer] : ( ord_le7084787975880047091nteger @ ( image_4470545334726330049nteger @ F @ ( vimage1318793397690208119nteger @ F @ A ) ) @ A ) ).
% 0.34/0.57
% 0.34/0.57 % image_vimage_subset
% 0.34/0.57 thf(fact_2634_image__vimage__subset,axiom,
% 0.34/0.57 ! [F: nat > code_integer,A: set_Code_integer] : ( ord_le7084787975880047091nteger @ ( image_1215581382706833972nteger @ F @ ( vimage6104394758324456170nteger @ F @ A ) ) @ A ) ).
% 0.34/0.57
% 0.34/0.57 % image_vimage_subset
% 0.34/0.57 thf(fact_2635_image__vimage__subset,axiom,
% 0.34/0.57 ! [F: nat > nat,A: set_nat] : ( ord_less_eq_set_nat @ ( image_nat_nat2 @ F @ ( vimage_nat_nat @ F @ A ) ) @ A ) ).
% 0.34/0.57
% 0.34/0.57 % image_vimage_subset
% 0.34/0.57 thf(fact_2636_Diff__insert,axiom,
% 0.34/0.57 ! [A: set_Product_unit,A2: product_unit,B4: set_Product_unit] :
% 0.34/0.57 ( ( minus_6452836326544984404t_unit @ A @ ( insert_Product_unit @ A2 @ B4 ) )
% 0.34/0.57 = ( minus_6452836326544984404t_unit @ ( minus_6452836326544984404t_unit @ A @ B4 ) @ ( insert_Product_unit @ A2 @ bot_bo3957492148770167129t_unit ) ) ) ).
% 0.34/0.57
% 0.34/0.57 % Diff_insert
% 0.34/0.57 thf(fact_2637_Diff__insert,axiom,
% 0.34/0.57 ! [A: set_o,A2: $o,B4: set_o] :
% 0.34/0.57 ( ( minus_minus_set_o @ A @ ( insert_o @ A2 @ B4 ) )
% 0.34/0.57 = ( minus_minus_set_o @ ( minus_minus_set_o @ A @ B4 ) @ ( insert_o @ A2 @ bot_bot_set_o ) ) ) ).
% 0.34/0.57
% 0.34/0.57 % Diff_insert
% 0.34/0.57 thf(fact_2638_Diff__insert,axiom,
% 0.34/0.57 ! [A: set_int,A2: int,B4: set_int] :
% 0.34/0.57 ( ( minus_minus_set_int @ A @ ( insert_int @ A2 @ B4 ) )
% 0.34/0.57 = ( minus_minus_set_int @ ( minus_minus_set_int @ A @ B4 ) @ ( insert_int @ A2 @ bot_bot_set_int ) ) ) ).
% 0.34/0.57
% 0.34/0.57 % Diff_insert
% 0.34/0.57 thf(fact_2639_Diff__insert,axiom,
% 0.34/0.57 ! [A: set_Pr4532377907799695533_nat_o,A2: produc3658429121746597890et_nat > $o,B4: set_Pr4532377907799695533_nat_o] :
% 0.34/0.57 ( ( minus_1801376950450012436_nat_o @ A @ ( insert5175938949040314269_nat_o @ A2 @ B4 ) )
% 0.34/0.57 = ( minus_1801376950450012436_nat_o @ ( minus_1801376950450012436_nat_o @ A @ B4 ) @ ( insert5175938949040314269_nat_o @ A2 @ bot_bo7824918357723582233_nat_o ) ) ) ).
% 0.34/0.57
% 0.34/0.57 % Diff_insert
% 0.34/0.57 thf(fact_2640_Diff__insert,axiom,
% 0.34/0.57 ! [A: set_nat,A2: nat,B4: set_nat] :
% 0.34/0.57 ( ( minus_minus_set_nat @ A @ ( insert_nat @ A2 @ B4 ) )
% 0.34/0.57 = ( minus_minus_set_nat @ ( minus_minus_set_nat @ A @ B4 ) @ ( insert_nat @ A2 @ bot_bot_set_nat ) ) ) ).
% 0.34/0.57
% 0.34/0.57 % Diff_insert
% 0.34/0.57 thf(fact_2641_insert__Diff,axiom,
% 0.34/0.57 ! [A2: product_unit,A: set_Product_unit] :
% 0.34/0.57 ( ( member_Product_unit @ A2 @ A )
% 0.34/0.57 => ( ( insert_Product_unit @ A2 @ ( minus_6452836326544984404t_unit @ A @ ( insert_Product_unit @ A2 @ bot_bo3957492148770167129t_unit ) ) )
% 0.34/0.57 = A ) ) ).
% 0.34/0.57
% 0.34/0.57 % insert_Diff
% 0.34/0.57 thf(fact_2642_insert__Diff,axiom,
% 0.34/0.57 ! [A2: $o,A: set_o] :
% 0.34/0.57 ( ( member_o @ A2 @ A )
% 0.34/0.57 => ( ( insert_o @ A2 @ ( minus_minus_set_o @ A @ ( insert_o @ A2 @ bot_bot_set_o ) ) )
% 0.34/0.57 = A ) ) ).
% 0.34/0.57
% 0.34/0.57 % insert_Diff
% 0.34/0.57 thf(fact_2643_insert__Diff,axiom,
% 0.34/0.57 ! [A2: int,A: set_int] :
% 0.34/0.57 ( ( member_int @ A2 @ A )
% 0.34/0.57 => ( ( insert_int @ A2 @ ( minus_minus_set_int @ A @ ( insert_int @ A2 @ bot_bot_set_int ) ) )
% 0.34/0.57 = A ) ) ).
% 0.34/0.57
% 0.34/0.57 % insert_Diff
% 0.34/0.57 thf(fact_2644_insert__Diff,axiom,
% 0.34/0.57 ! [A2: produc3658429121746597890et_nat > $o,A: set_Pr4532377907799695533_nat_o] :
% 0.34/0.57 ( ( member6576561426505652726_nat_o @ A2 @ A )
% 0.34/0.57 => ( ( insert5175938949040314269_nat_o @ A2 @ ( minus_1801376950450012436_nat_o @ A @ ( insert5175938949040314269_nat_o @ A2 @ bot_bo7824918357723582233_nat_o ) ) )
% 0.34/0.57 = A ) ) ).
% 0.34/0.57
% 0.34/0.57 % insert_Diff
% 0.34/0.57 thf(fact_2645_insert__Diff,axiom,
% 0.34/0.57 ! [A2: nat,A: set_nat] :
% 0.34/0.57 ( ( member_nat @ A2 @ A )
% 0.34/0.57 => ( ( insert_nat @ A2 @ ( minus_minus_set_nat @ A @ ( insert_nat @ A2 @ bot_bot_set_nat ) ) )
% 0.34/0.57 = A ) ) ).
% 0.34/0.57
% 0.34/0.57 % insert_Diff
% 0.34/0.57 thf(fact_2646_Diff__insert2,axiom,
% 0.34/0.57 ! [A: set_Product_unit,A2: product_unit,B4: set_Product_unit] :
% 0.34/0.57 ( ( minus_6452836326544984404t_unit @ A @ ( insert_Product_unit @ A2 @ B4 ) )
% 0.34/0.57 = ( minus_6452836326544984404t_unit @ ( minus_6452836326544984404t_unit @ A @ ( insert_Product_unit @ A2 @ bot_bo3957492148770167129t_unit ) ) @ B4 ) ) ).
% 0.34/0.57
% 0.34/0.57 % Diff_insert2
% 0.34/0.57 thf(fact_2647_Diff__insert2,axiom,
% 0.34/0.57 ! [A: set_o,A2: $o,B4: set_o] :
% 0.34/0.57 ( ( minus_minus_set_o @ A @ ( insert_o @ A2 @ B4 ) )
% 0.34/0.57 = ( minus_minus_set_o @ ( minus_minus_set_o @ A @ ( insert_o @ A2 @ bot_bot_set_o ) ) @ B4 ) ) ).
% 0.34/0.57
% 0.34/0.57 % Diff_insert2
% 0.34/0.57 thf(fact_2648_Diff__insert2,axiom,
% 0.34/0.57 ! [A: set_int,A2: int,B4: set_int] :
% 0.34/0.57 ( ( minus_minus_set_int @ A @ ( insert_int @ A2 @ B4 ) )
% 0.34/0.57 = ( minus_minus_set_int @ ( minus_minus_set_int @ A @ ( insert_int @ A2 @ bot_bot_set_int ) ) @ B4 ) ) ).
% 0.34/0.57
% 0.34/0.57 % Diff_insert2
% 0.34/0.57 thf(fact_2649_Diff__insert2,axiom,
% 0.34/0.57 ! [A: set_Pr4532377907799695533_nat_o,A2: produc3658429121746597890et_nat > $o,B4: set_Pr4532377907799695533_nat_o] :
% 0.34/0.57 ( ( minus_1801376950450012436_nat_o @ A @ ( insert5175938949040314269_nat_o @ A2 @ B4 ) )
% 0.34/0.57 = ( minus_1801376950450012436_nat_o @ ( minus_1801376950450012436_nat_o @ A @ ( insert5175938949040314269_nat_o @ A2 @ bot_bo7824918357723582233_nat_o ) ) @ B4 ) ) ).
% 0.34/0.57
% 0.34/0.57 % Diff_insert2
% 0.34/0.57 thf(fact_2650_Diff__insert2,axiom,
% 0.34/0.57 ! [A: set_nat,A2: nat,B4: set_nat] :
% 0.34/0.57 ( ( minus_minus_set_nat @ A @ ( insert_nat @ A2 @ B4 ) )
% 0.34/0.57 = ( minus_minus_set_nat @ ( minus_minus_set_nat @ A @ ( insert_nat @ A2 @ bot_bot_set_nat ) ) @ B4 ) ) ).
% 0.34/0.57
% 0.34/0.57 % Diff_insert2
% 0.34/0.57 thf(fact_2651_insert__minus__eq,axiom,
% 0.34/0.57 ! [X: product_unit,Y: product_unit,A: set_Product_unit] :
% 0.34/0.57 ( ( X != Y )
% 0.34/0.57 => ( ( minus_6452836326544984404t_unit @ ( insert_Product_unit @ X @ A ) @ ( insert_Product_unit @ Y @ bot_bo3957492148770167129t_unit ) )
% 0.34/0.57 = ( insert_Product_unit @ X @ ( minus_6452836326544984404t_unit @ A @ ( insert_Product_unit @ Y @ bot_bo3957492148770167129t_unit ) ) ) ) ) ).
% 0.34/0.57
% 0.34/0.57 % insert_minus_eq
% 0.34/0.57 thf(fact_2652_insert__minus__eq,axiom,
% 0.34/0.57 ! [X: $o,Y: $o,A: set_o] :
% 0.34/0.57 ( ( X != Y )
% 0.34/0.57 => ( ( minus_minus_set_o @ ( insert_o @ X @ A ) @ ( insert_o @ Y @ bot_bot_set_o ) )
% 0.34/0.57 = ( insert_o @ X @ ( minus_minus_set_o @ A @ ( insert_o @ Y @ bot_bot_set_o ) ) ) ) ) ).
% 0.34/0.57
% 0.34/0.57 % insert_minus_eq
% 0.34/0.57 thf(fact_2653_insert__minus__eq,axiom,
% 0.34/0.57 ! [X: int,Y: int,A: set_int] :
% 0.34/0.57 ( ( X != Y )
% 0.34/0.57 => ( ( minus_minus_set_int @ ( insert_int @ X @ A ) @ ( insert_int @ Y @ bot_bot_set_int ) )
% 0.34/0.57 = ( insert_int @ X @ ( minus_minus_set_int @ A @ ( insert_int @ Y @ bot_bot_set_int ) ) ) ) ) ).
% 0.34/0.57
% 0.34/0.57 % insert_minus_eq
% 0.34/0.57 thf(fact_2654_insert__minus__eq,axiom,
% 0.34/0.57 ! [X: produc3658429121746597890et_nat > $o,Y: produc3658429121746597890et_nat > $o,A: set_Pr4532377907799695533_nat_o] :
% 0.34/0.57 ( ( X != Y )
% 0.34/0.57 => ( ( minus_1801376950450012436_nat_o @ ( insert5175938949040314269_nat_o @ X @ A ) @ ( insert5175938949040314269_nat_o @ Y @ bot_bo7824918357723582233_nat_o ) )
% 0.34/0.57 = ( insert5175938949040314269_nat_o @ X @ ( minus_1801376950450012436_nat_o @ A @ ( insert5175938949040314269_nat_o @ Y @ bot_bo7824918357723582233_nat_o ) ) ) ) ) ).
% 0.34/0.57
% 0.34/0.57 % insert_minus_eq
% 0.34/0.57 thf(fact_2655_insert__minus__eq,axiom,
% 0.34/0.57 ! [X: nat,Y: nat,A: set_nat] :
% 0.34/0.57 ( ( X != Y )
% 0.34/0.57 => ( ( minus_minus_set_nat @ ( insert_nat @ X @ A ) @ ( insert_nat @ Y @ bot_bot_set_nat ) )
% 0.34/0.57 = ( insert_nat @ X @ ( minus_minus_set_nat @ A @ ( insert_nat @ Y @ bot_bot_set_nat ) ) ) ) ) ).
% 0.34/0.57
% 0.34/0.57 % insert_minus_eq
% 0.34/0.57 thf(fact_2656_Diff__insert__absorb,axiom,
% 0.34/0.57 ! [X: product_unit,A: set_Product_unit] :
% 0.34/0.57 ( ~ ( member_Product_unit @ X @ A )
% 0.34/0.57 => ( ( minus_6452836326544984404t_unit @ ( insert_Product_unit @ X @ A ) @ ( insert_Product_unit @ X @ bot_bo3957492148770167129t_unit ) )
% 0.34/0.57 = A ) ) ).
% 0.34/0.57
% 0.34/0.57 % Diff_insert_absorb
% 0.34/0.57 thf(fact_2657_Diff__insert__absorb,axiom,
% 0.34/0.57 ! [X: $o,A: set_o] :
% 0.34/0.57 ( ~ ( member_o @ X @ A )
% 0.34/0.57 => ( ( minus_minus_set_o @ ( insert_o @ X @ A ) @ ( insert_o @ X @ bot_bot_set_o ) )
% 0.34/0.57 = A ) ) ).
% 0.34/0.57
% 0.34/0.57 % Diff_insert_absorb
% 0.34/0.57 thf(fact_2658_Diff__insert__absorb,axiom,
% 0.34/0.57 ! [X: int,A: set_int] :
% 0.34/0.57 ( ~ ( member_int @ X @ A )
% 0.34/0.57 => ( ( minus_minus_set_int @ ( insert_int @ X @ A ) @ ( insert_int @ X @ bot_bot_set_int ) )
% 0.34/0.57 = A ) ) ).
% 0.34/0.57
% 0.34/0.57 % Diff_insert_absorb
% 0.34/0.57 thf(fact_2659_Diff__insert__absorb,axiom,
% 0.34/0.57 ! [X: produc3658429121746597890et_nat > $o,A: set_Pr4532377907799695533_nat_o] :
% 0.34/0.57 ( ~ ( member6576561426505652726_nat_o @ X @ A )
% 0.34/0.57 => ( ( minus_1801376950450012436_nat_o @ ( insert5175938949040314269_nat_o @ X @ A ) @ ( insert5175938949040314269_nat_o @ X @ bot_bo7824918357723582233_nat_o ) )
% 0.34/0.57 = A ) ) ).
% 0.34/0.57
% 0.34/0.57 % Diff_insert_absorb
% 0.34/0.57 thf(fact_2660_Diff__insert__absorb,axiom,
% 0.34/0.57 ! [X: nat,A: set_nat] :
% 0.34/0.57 ( ~ ( member_nat @ X @ A )
% 0.34/0.57 => ( ( minus_minus_set_nat @ ( insert_nat @ X @ A ) @ ( insert_nat @ X @ bot_bot_set_nat ) )
% 0.34/0.57 = A ) ) ).
% 0.34/0.57
% 0.34/0.57 % Diff_insert_absorb
% 0.34/0.57 thf(fact_2661_set__minus__singleton__eq,axiom,
% 0.34/0.57 ! [X: product_unit,X4: set_Product_unit] :
% 0.34/0.57 ( ~ ( member_Product_unit @ X @ X4 )
% 0.34/0.57 => ( ( minus_6452836326544984404t_unit @ X4 @ ( insert_Product_unit @ X @ bot_bo3957492148770167129t_unit ) )
% 0.34/0.57 = X4 ) ) ).
% 0.34/0.57
% 0.34/0.57 % set_minus_singleton_eq
% 0.34/0.57 thf(fact_2662_set__minus__singleton__eq,axiom,
% 0.34/0.57 ! [X: $o,X4: set_o] :
% 0.34/0.57 ( ~ ( member_o @ X @ X4 )
% 0.34/0.57 => ( ( minus_minus_set_o @ X4 @ ( insert_o @ X @ bot_bot_set_o ) )
% 0.34/0.57 = X4 ) ) ).
% 0.34/0.57
% 0.34/0.57 % set_minus_singleton_eq
% 0.34/0.57 thf(fact_2663_set__minus__singleton__eq,axiom,
% 0.34/0.57 ! [X: int,X4: set_int] :
% 0.34/0.57 ( ~ ( member_int @ X @ X4 )
% 0.34/0.57 => ( ( minus_minus_set_int @ X4 @ ( insert_int @ X @ bot_bot_set_int ) )
% 0.34/0.57 = X4 ) ) ).
% 0.34/0.57
% 0.34/0.57 % set_minus_singleton_eq
% 0.34/0.57 thf(fact_2664_set__minus__singleton__eq,axiom,
% 0.34/0.57 ! [X: produc3658429121746597890et_nat > $o,X4: set_Pr4532377907799695533_nat_o] :
% 0.34/0.57 ( ~ ( member6576561426505652726_nat_o @ X @ X4 )
% 0.34/0.57 => ( ( minus_1801376950450012436_nat_o @ X4 @ ( insert5175938949040314269_nat_o @ X @ bot_bo7824918357723582233_nat_o ) )
% 0.34/0.57 = X4 ) ) ).
% 0.34/0.57
% 0.34/0.57 % set_minus_singleton_eq
% 0.34/0.57 thf(fact_2665_set__minus__singleton__eq,axiom,
% 0.34/0.57 ! [X: nat,X4: set_nat] :
% 0.34/0.57 ( ~ ( member_nat @ X @ X4 )
% 0.34/0.57 => ( ( minus_minus_set_nat @ X4 @ ( insert_nat @ X @ bot_bot_set_nat ) )
% 0.34/0.57 = X4 ) ) ).
% 0.34/0.57
% 0.34/0.57 % set_minus_singleton_eq
% 0.34/0.57 thf(fact_2666_subset__Diff__insert,axiom,
% 0.34/0.57 ! [A: set_Pr4532377907799695533_nat_o,B4: set_Pr4532377907799695533_nat_o,X: produc3658429121746597890et_nat > $o,C5: set_Pr4532377907799695533_nat_o] :
% 0.34/0.57 ( ( ord_le2965882846123202637_nat_o @ A @ ( minus_1801376950450012436_nat_o @ B4 @ ( insert5175938949040314269_nat_o @ X @ C5 ) ) )
% 0.34/0.57 = ( ( ord_le2965882846123202637_nat_o @ A @ ( minus_1801376950450012436_nat_o @ B4 @ C5 ) )
% 0.34/0.57 & ~ ( member6576561426505652726_nat_o @ X @ A ) ) ) ).
% 0.34/0.57
% 0.34/0.57 % subset_Diff_insert
% 0.34/0.57 thf(fact_2667_subset__Diff__insert,axiom,
% 0.34/0.57 ! [A: set_int,B4: set_int,X: int,C5: set_int] :
% 0.34/0.57 ( ( ord_less_eq_set_int @ A @ ( minus_minus_set_int @ B4 @ ( insert_int @ X @ C5 ) ) )
% 0.34/0.57 = ( ( ord_less_eq_set_int @ A @ ( minus_minus_set_int @ B4 @ C5 ) )
% 0.34/0.57 & ~ ( member_int @ X @ A ) ) ) ).
% 0.34/0.57
% 0.34/0.57 % subset_Diff_insert
% 0.34/0.57 thf(fact_2668_subset__Diff__insert,axiom,
% 0.34/0.57 ! [A: set_o,B4: set_o,X: $o,C5: set_o] :
% 0.34/0.57 ( ( ord_less_eq_set_o @ A @ ( minus_minus_set_o @ B4 @ ( insert_o @ X @ C5 ) ) )
% 0.34/0.57 = ( ( ord_less_eq_set_o @ A @ ( minus_minus_set_o @ B4 @ C5 ) )
% 0.34/0.57 & ~ ( member_o @ X @ A ) ) ) ).
% 0.34/0.57
% 0.34/0.57 % subset_Diff_insert
% 0.34/0.57 thf(fact_2669_subset__Diff__insert,axiom,
% 0.34/0.57 ! [A: set_Product_unit,B4: set_Product_unit,X: product_unit,C5: set_Product_unit] :
% 0.34/0.57 ( ( ord_le3507040750410214029t_unit @ A @ ( minus_6452836326544984404t_unit @ B4 @ ( insert_Product_unit @ X @ C5 ) ) )
% 0.34/0.57 = ( ( ord_le3507040750410214029t_unit @ A @ ( minus_6452836326544984404t_unit @ B4 @ C5 ) )
% 0.34/0.57 & ~ ( member_Product_unit @ X @ A ) ) ) ).
% 0.34/0.57
% 0.34/0.57 % subset_Diff_insert
% 0.34/0.57 thf(fact_2670_subset__Diff__insert,axiom,
% 0.34/0.57 ! [A: set_nat,B4: set_nat,X: nat,C5: set_nat] :
% 0.34/0.57 ( ( ord_less_eq_set_nat @ A @ ( minus_minus_set_nat @ B4 @ ( insert_nat @ X @ C5 ) ) )
% 0.34/0.57 = ( ( ord_less_eq_set_nat @ A @ ( minus_minus_set_nat @ B4 @ C5 ) )
% 0.34/0.57 & ~ ( member_nat @ X @ A ) ) ) ).
% 0.34/0.57
% 0.34/0.57 % subset_Diff_insert
% 0.34/0.57 thf(fact_2671_Diff__triv,axiom,
% 0.34/0.57 ! [A: set_Product_unit,B4: set_Product_unit] :
% 0.34/0.57 ( ( ( inf_in4660618365625256667t_unit @ A @ B4 )
% 0.34/0.57 = bot_bo3957492148770167129t_unit )
% 0.34/0.57 => ( ( minus_6452836326544984404t_unit @ A @ B4 )
% 0.34/0.57 = A ) ) ).
% 0.34/0.57
% 0.34/0.57 % Diff_triv
% 0.34/0.57 thf(fact_2672_Diff__triv,axiom,
% 0.34/0.57 ! [A: set_o,B4: set_o] :
% 0.34/0.57 ( ( ( inf_inf_set_o @ A @ B4 )
% 0.34/0.57 = bot_bot_set_o )
% 0.34/0.57 => ( ( minus_minus_set_o @ A @ B4 )
% 0.34/0.57 = A ) ) ).
% 0.34/0.57
% 0.34/0.57 % Diff_triv
% 0.34/0.57 thf(fact_2673_Diff__triv,axiom,
% 0.34/0.57 ! [A: set_int,B4: set_int] :
% 0.34/0.57 ( ( ( inf_inf_set_int @ A @ B4 )
% 0.34/0.57 = bot_bot_set_int )
% 0.34/0.57 => ( ( minus_minus_set_int @ A @ B4 )
% 0.34/0.57 = A ) ) ).
% 0.34/0.57
% 0.34/0.57 % Diff_triv
% 0.34/0.57 thf(fact_2674_Diff__triv,axiom,
% 0.34/0.57 ! [A: set_Pr4532377907799695533_nat_o,B4: set_Pr4532377907799695533_nat_o] :
% 0.34/0.57 ( ( ( inf_in1906310914598751387_nat_o @ A @ B4 )
% 0.34/0.57 = bot_bo7824918357723582233_nat_o )
% 0.34/0.57 => ( ( minus_1801376950450012436_nat_o @ A @ B4 )
% 0.34/0.57 = A ) ) ).
% 0.34/0.57
% 0.34/0.57 % Diff_triv
% 0.34/0.57 thf(fact_2675_Diff__triv,axiom,
% 0.34/0.57 ! [A: set_nat,B4: set_nat] :
% 0.34/0.57 ( ( ( inf_inf_set_nat @ A @ B4 )
% 0.34/0.57 = bot_bot_set_nat )
% 0.34/0.57 => ( ( minus_minus_set_nat @ A @ B4 )
% 0.34/0.57 = A ) ) ).
% 0.34/0.57
% 0.34/0.57 % Diff_triv
% 0.34/0.57 thf(fact_2676_Int__Diff__disjoint,axiom,
% 0.34/0.57 ! [A: set_Product_unit,B4: set_Product_unit] :
% 0.34/0.57 ( ( inf_in4660618365625256667t_unit @ ( inf_in4660618365625256667t_unit @ A @ B4 ) @ ( minus_6452836326544984404t_unit @ A @ B4 ) )
% 0.34/0.57 = bot_bo3957492148770167129t_unit ) ).
% 0.34/0.57
% 0.34/0.57 % Int_Diff_disjoint
% 0.34/0.57 thf(fact_2677_Int__Diff__disjoint,axiom,
% 0.34/0.57 ! [A: set_o,B4: set_o] :
% 0.34/0.57 ( ( inf_inf_set_o @ ( inf_inf_set_o @ A @ B4 ) @ ( minus_minus_set_o @ A @ B4 ) )
% 0.34/0.57 = bot_bot_set_o ) ).
% 0.34/0.57
% 0.34/0.57 % Int_Diff_disjoint
% 0.34/0.57 thf(fact_2678_Int__Diff__disjoint,axiom,
% 0.34/0.57 ! [A: set_int,B4: set_int] :
% 0.34/0.57 ( ( inf_inf_set_int @ ( inf_inf_set_int @ A @ B4 ) @ ( minus_minus_set_int @ A @ B4 ) )
% 0.34/0.57 = bot_bot_set_int ) ).
% 0.34/0.57
% 0.34/0.57 % Int_Diff_disjoint
% 0.34/0.57 thf(fact_2679_Int__Diff__disjoint,axiom,
% 0.34/0.57 ! [A: set_Pr4532377907799695533_nat_o,B4: set_Pr4532377907799695533_nat_o] :
% 0.34/0.57 ( ( inf_in1906310914598751387_nat_o @ ( inf_in1906310914598751387_nat_o @ A @ B4 ) @ ( minus_1801376950450012436_nat_o @ A @ B4 ) )
% 0.34/0.57 = bot_bo7824918357723582233_nat_o ) ).
% 0.34/0.57
% 0.34/0.57 % Int_Diff_disjoint
% 0.34/0.57 thf(fact_2680_Int__Diff__disjoint,axiom,
% 0.34/0.57 ! [A: set_nat,B4: set_nat] :
% 0.34/0.57 ( ( inf_inf_set_nat @ ( inf_inf_set_nat @ A @ B4 ) @ ( minus_minus_set_nat @ A @ B4 ) )
% 0.34/0.57 = bot_bot_set_nat ) ).
% 0.34/0.57
% 0.34/0.57 % Int_Diff_disjoint
% 0.34/0.57 thf(fact_2681_disjoint__alt__simp1,axiom,
% 0.34/0.57 ! [A: set_Product_unit,B4: set_Product_unit] :
% 0.34/0.57 ( ( ( minus_6452836326544984404t_unit @ A @ B4 )
% 0.34/0.57 = A )
% 0.34/0.57 = ( ( inf_in4660618365625256667t_unit @ A @ B4 )
% 0.34/0.57 = bot_bo3957492148770167129t_unit ) ) ).
% 0.34/0.57
% 0.34/0.57 % disjoint_alt_simp1
% 0.34/0.57 thf(fact_2682_disjoint__alt__simp1,axiom,
% 0.34/0.57 ! [A: set_o,B4: set_o] :
% 0.34/0.57 ( ( ( minus_minus_set_o @ A @ B4 )
% 0.34/0.57 = A )
% 0.34/0.57 = ( ( inf_inf_set_o @ A @ B4 )
% 0.34/0.57 = bot_bot_set_o ) ) ).
% 0.34/0.57
% 0.34/0.57 % disjoint_alt_simp1
% 0.34/0.57 thf(fact_2683_disjoint__alt__simp1,axiom,
% 0.34/0.57 ! [A: set_int,B4: set_int] :
% 0.34/0.57 ( ( ( minus_minus_set_int @ A @ B4 )
% 0.34/0.57 = A )
% 0.34/0.57 = ( ( inf_inf_set_int @ A @ B4 )
% 0.34/0.57 = bot_bot_set_int ) ) ).
% 0.34/0.57
% 0.34/0.57 % disjoint_alt_simp1
% 0.34/0.57 thf(fact_2684_disjoint__alt__simp1,axiom,
% 0.34/0.57 ! [A: set_Pr4532377907799695533_nat_o,B4: set_Pr4532377907799695533_nat_o] :
% 0.34/0.57 ( ( ( minus_1801376950450012436_nat_o @ A @ B4 )
% 0.34/0.57 = A )
% 0.34/0.57 = ( ( inf_in1906310914598751387_nat_o @ A @ B4 )
% 0.34/0.57 = bot_bo7824918357723582233_nat_o ) ) ).
% 0.34/0.57
% 0.34/0.57 % disjoint_alt_simp1
% 0.34/0.57 thf(fact_2685_disjoint__alt__simp1,axiom,
% 0.34/0.57 ! [A: set_nat,B4: set_nat] :
% 0.34/0.57 ( ( ( minus_minus_set_nat @ A @ B4 )
% 0.34/0.57 = A )
% 0.34/0.57 = ( ( inf_inf_set_nat @ A @ B4 )
% 0.34/0.57 = bot_bot_set_nat ) ) ).
% 0.34/0.57
% 0.34/0.57 % disjoint_alt_simp1
% 0.34/0.57 thf(fact_2686_disjoint__alt__simp2,axiom,
% 0.34/0.57 ! [A: set_Product_unit,B4: set_Product_unit] :
% 0.34/0.57 ( ( ( minus_6452836326544984404t_unit @ A @ B4 )
% 0.34/0.57 != A )
% 0.34/0.57 = ( ( inf_in4660618365625256667t_unit @ A @ B4 )
% 0.34/0.57 != bot_bo3957492148770167129t_unit ) ) ).
% 0.34/0.57
% 0.34/0.57 % disjoint_alt_simp2
% 0.34/0.57 thf(fact_2687_disjoint__alt__simp2,axiom,
% 0.34/0.57 ! [A: set_o,B4: set_o] :
% 0.34/0.57 ( ( ( minus_minus_set_o @ A @ B4 )
% 0.34/0.57 != A )
% 0.34/0.57 = ( ( inf_inf_set_o @ A @ B4 )
% 0.34/0.57 != bot_bot_set_o ) ) ).
% 0.34/0.57
% 0.34/0.57 % disjoint_alt_simp2
% 0.34/0.57 thf(fact_2688_disjoint__alt__simp2,axiom,
% 0.34/0.57 ! [A: set_int,B4: set_int] :
% 0.34/0.57 ( ( ( minus_minus_set_int @ A @ B4 )
% 0.34/0.57 != A )
% 0.34/0.57 = ( ( inf_inf_set_int @ A @ B4 )
% 0.34/0.57 != bot_bot_set_int ) ) ).
% 0.34/0.57
% 0.34/0.57 % disjoint_alt_simp2
% 0.34/0.57 thf(fact_2689_disjoint__alt__simp2,axiom,
% 0.34/0.57 ! [A: set_Pr4532377907799695533_nat_o,B4: set_Pr4532377907799695533_nat_o] :
% 0.34/0.57 ( ( ( minus_1801376950450012436_nat_o @ A @ B4 )
% 0.34/0.57 != A )
% 0.34/0.57 = ( ( inf_in1906310914598751387_nat_o @ A @ B4 )
% 0.34/0.57 != bot_bo7824918357723582233_nat_o ) ) ).
% 0.34/0.57
% 0.34/0.57 % disjoint_alt_simp2
% 0.34/0.57 thf(fact_2690_disjoint__alt__simp2,axiom,
% 0.34/0.57 ! [A: set_nat,B4: set_nat] :
% 0.34/0.57 ( ( ( minus_minus_set_nat @ A @ B4 )
% 0.34/0.57 != A )
% 0.34/0.57 = ( ( inf_inf_set_nat @ A @ B4 )
% 0.34/0.57 != bot_bot_set_nat ) ) ).
% 0.34/0.57
% 0.34/0.57 % disjoint_alt_simp2
% 0.34/0.57 thf(fact_2691_Diff__subset__conv,axiom,
% 0.34/0.57 ! [A: set_nat,B4: set_nat,C5: set_nat] :
% 0.34/0.57 ( ( ord_less_eq_set_nat @ ( minus_minus_set_nat @ A @ B4 ) @ C5 )
% 0.34/0.57 = ( ord_less_eq_set_nat @ A @ ( sup_sup_set_nat @ B4 @ C5 ) ) ) ).
% 0.34/0.57
% 0.34/0.57 % Diff_subset_conv
% 0.34/0.57 thf(fact_2692_Diff__partition,axiom,
% 0.34/0.57 ! [A: set_nat,B4: set_nat] :
% 0.34/0.57 ( ( ord_less_eq_set_nat @ A @ B4 )
% 0.34/0.57 => ( ( sup_sup_set_nat @ A @ ( minus_minus_set_nat @ B4 @ A ) )
% 0.34/0.57 = B4 ) ) ).
% 0.34/0.57
% 0.34/0.57 % Diff_partition
% 0.34/0.57 thf(fact_2693_Un__Diff__Int,axiom,
% 0.34/0.57 ! [A: set_nat,B4: set_nat] :
% 0.34/0.57 ( ( sup_sup_set_nat @ ( minus_minus_set_nat @ A @ B4 ) @ ( inf_inf_set_nat @ A @ B4 ) )
% 0.34/0.57 = A ) ).
% 0.34/0.57
% 0.34/0.57 % Un_Diff_Int
% 0.34/0.57 thf(fact_2694_Int__Diff__Un,axiom,
% 0.34/0.57 ! [A: set_nat,B4: set_nat] :
% 0.34/0.57 ( ( sup_sup_set_nat @ ( inf_inf_set_nat @ A @ B4 ) @ ( minus_minus_set_nat @ A @ B4 ) )
% 0.34/0.57 = A ) ).
% 0.34/0.57
% 0.34/0.57 % Int_Diff_Un
% 0.34/0.57 thf(fact_2695_Diff__Int,axiom,
% 0.34/0.57 ! [A: set_nat,B4: set_nat,C5: set_nat] :
% 0.34/0.57 ( ( minus_minus_set_nat @ A @ ( inf_inf_set_nat @ B4 @ C5 ) )
% 0.34/0.57 = ( sup_sup_set_nat @ ( minus_minus_set_nat @ A @ B4 ) @ ( minus_minus_set_nat @ A @ C5 ) ) ) ).
% 0.34/0.57
% 0.34/0.57 % Diff_Int
% 0.34/0.57 thf(fact_2696_Diff__Un,axiom,
% 0.34/0.57 ! [A: set_nat,B4: set_nat,C5: set_nat] :
% 0.34/0.57 ( ( minus_minus_set_nat @ A @ ( sup_sup_set_nat @ B4 @ C5 ) )
% 0.34/0.57 = ( inf_inf_set_nat @ ( minus_minus_set_nat @ A @ B4 ) @ ( minus_minus_set_nat @ A @ C5 ) ) ) ).
% 0.34/0.57
% 0.34/0.57 % Diff_Un
% 0.34/0.57 thf(fact_2697_Range__snd,axiom,
% 0.34/0.57 ( range_9132311542094983838et_nat
% 0.34/0.57 = ( image_6088997261112373647et_nat @ produc8586169260539613262et_nat ) ) ).
% 0.34/0.57
% 0.34/0.57 % Range_snd
% 0.34/0.57 thf(fact_2698_Range__snd,axiom,
% 0.34/0.57 ( range_int_int
% 0.34/0.57 = ( image_5042161079198086560nt_int @ product_snd_int_int ) ) ).
% 0.34/0.57
% 0.34/0.57 % Range_snd
% 0.34/0.57 thf(fact_2699_snd__eq__Range,axiom,
% 0.34/0.57 ! [R3: set_Pr3948176798113811640et_nat] :
% 0.34/0.57 ( ( image_6088997261112373647et_nat @ produc8586169260539613262et_nat @ R3 )
% 0.34/0.57 = ( range_9132311542094983838et_nat @ R3 ) ) ).
% 0.34/0.57
% 0.34/0.57 % snd_eq_Range
% 0.34/0.57 thf(fact_2700_snd__eq__Range,axiom,
% 0.34/0.57 ! [R3: set_Pr958786334691620121nt_int] :
% 0.34/0.57 ( ( image_5042161079198086560nt_int @ product_snd_int_int @ R3 )
% 0.34/0.57 = ( range_int_int @ R3 ) ) ).
% 0.34/0.57
% 0.34/0.57 % snd_eq_Range
% 0.34/0.57 thf(fact_2701_the__elem__image__unique,axiom,
% 0.34/0.57 ! [A: set_Code_integer,F: code_integer > code_integer,X: code_integer] :
% 0.34/0.57 ( ( A != bot_bo3990330152332043303nteger )
% 0.34/0.57 => ( ! [Y2: code_integer] :
% 0.34/0.57 ( ( member_Code_integer @ Y2 @ A )
% 0.34/0.57 => ( ( F @ Y2 )
% 0.34/0.57 = ( F @ X ) ) )
% 0.34/0.57 => ( ( the_el4656377885804969768nteger @ ( image_4470545334726330049nteger @ F @ A ) )
% 0.34/0.57 = ( F @ X ) ) ) ) ).
% 0.34/0.57
% 0.34/0.57 % the_elem_image_unique
% 0.34/0.57 thf(fact_2702_the__elem__image__unique,axiom,
% 0.34/0.57 ! [A: set_nat,F: nat > set_nat,X: nat] :
% 0.34/0.57 ( ( A != bot_bot_set_nat )
% 0.34/0.57 => ( ! [Y2: nat] :
% 0.34/0.57 ( ( member_nat @ Y2 @ A )
% 0.34/0.57 => ( ( F @ Y2 )
% 0.34/0.57 = ( F @ X ) ) )
% 0.34/0.57 => ( ( the_elem_set_nat @ ( image_nat_set_nat @ F @ A ) )
% 0.34/0.57 = ( F @ X ) ) ) ) ).
% 0.34/0.57
% 0.34/0.57 % the_elem_image_unique
% 0.34/0.57 thf(fact_2703_the__elem__image__unique,axiom,
% 0.34/0.57 ! [A: set_nat,F: nat > nat,X: nat] :
% 0.34/0.57 ( ( A != bot_bot_set_nat )
% 0.34/0.57 => ( ! [Y2: nat] :
% 0.34/0.57 ( ( member_nat @ Y2 @ A )
% 0.34/0.57 => ( ( F @ Y2 )
% 0.34/0.57 = ( F @ X ) ) )
% 0.34/0.57 => ( ( the_elem_nat @ ( image_nat_nat2 @ F @ A ) )
% 0.34/0.57 = ( F @ X ) ) ) ) ).
% 0.34/0.57
% 0.34/0.57 % the_elem_image_unique
% 0.34/0.57 thf(fact_2704_the__elem__image__unique,axiom,
% 0.34/0.57 ! [A: set_nat,F: nat > code_integer,X: nat] :
% 0.34/0.57 ( ( A != bot_bot_set_nat )
% 0.34/0.57 => ( ! [Y2: nat] :
% 0.34/0.57 ( ( member_nat @ Y2 @ A )
% 0.34/0.57 => ( ( F @ Y2 )
% 0.34/0.57 = ( F @ X ) ) )
% 0.34/0.57 => ( ( the_el4656377885804969768nteger @ ( image_1215581382706833972nteger @ F @ A ) )
% 0.34/0.57 = ( F @ X ) ) ) ) ).
% 0.34/0.57
% 0.34/0.57 % the_elem_image_unique
% 0.34/0.57 thf(fact_2705_cyclicE,axiom,
% 0.34/0.57 ! [G: set_Pr1261947904930325089at_nat] :
% 0.34/0.57 ( ~ ( transi3139058337892597825ic_nat @ G )
% 0.34/0.57 => ~ ! [X3: nat] :
% 0.34/0.57 ~ ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X3 @ X3 ) @ ( transi6264000038957366511cl_nat @ G ) ) ) ).
% 0.34/0.57
% 0.34/0.57 % cyclicE
% 0.34/0.57 thf(fact_2706_cyclicE,axiom,
% 0.34/0.57 ! [G: set_Pr958786334691620121nt_int] :
% 0.34/0.57 ( ~ ( transi3136567867383547549ic_int @ G )
% 0.34/0.57 => ~ ! [X3: int] :
% 0.34/0.57 ~ ( member5262025264175285858nt_int @ ( product_Pair_int_int @ X3 @ X3 ) @ ( transi6261509568448316235cl_int @ G ) ) ) ).
% 0.34/0.57
% 0.34/0.57 % cyclicE
% 0.34/0.57 thf(fact_2707_acyclic__def,axiom,
% 0.34/0.57 ( transi3139058337892597825ic_nat
% 0.34/0.57 = ( ^ [R4: set_Pr1261947904930325089at_nat] :
% 0.34/0.57 ! [X2: nat] :
% 0.34/0.57 ~ ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X2 @ X2 ) @ ( transi6264000038957366511cl_nat @ R4 ) ) ) ) ).
% 0.34/0.57
% 0.34/0.57 % acyclic_def
% 0.34/0.57 thf(fact_2708_acyclic__def,axiom,
% 0.34/0.57 ( transi3136567867383547549ic_int
% 0.34/0.57 = ( ^ [R4: set_Pr958786334691620121nt_int] :
% 0.34/0.57 ! [X2: int] :
% 0.34/0.57 ~ ( member5262025264175285858nt_int @ ( product_Pair_int_int @ X2 @ X2 ) @ ( transi6261509568448316235cl_int @ R4 ) ) ) ) ).
% 0.34/0.57
% 0.34/0.57 % acyclic_def
% 0.34/0.57 thf(fact_2709_acyclicI,axiom,
% 0.34/0.57 ! [R2: set_Pr1261947904930325089at_nat] :
% 0.34/0.57 ( ! [X3: nat] :
% 0.34/0.57 ~ ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X3 @ X3 ) @ ( transi6264000038957366511cl_nat @ R2 ) )
% 0.34/0.57 => ( transi3139058337892597825ic_nat @ R2 ) ) ).
% 0.34/0.57
% 0.34/0.57 % acyclicI
% 0.34/0.57 thf(fact_2710_acyclicI,axiom,
% 0.34/0.57 ! [R2: set_Pr958786334691620121nt_int] :
% 0.34/0.57 ( ! [X3: int] :
% 0.34/0.57 ~ ( member5262025264175285858nt_int @ ( product_Pair_int_int @ X3 @ X3 ) @ ( transi6261509568448316235cl_int @ R2 ) )
% 0.34/0.57 => ( transi3136567867383547549ic_int @ R2 ) ) ).
% 0.34/0.57
% 0.34/0.57 % acyclicI
% 0.34/0.57 thf(fact_2711_snd__image__mp,axiom,
% 0.34/0.57 ! [A: set_Pr3286484037609594932et_nat,B4: set_Pr3948176798113811640et_nat,X: produc3658429121746597890et_nat > $o,Y: produc3658429121746597890et_nat] :
% 0.34/0.57 ( ( ord_le8090042543978450200et_nat @ ( image_4390311411407041927et_nat @ produc4011572625026189258et_nat @ A ) @ B4 )
% 0.34/0.57 => ( ( member1996754912294343701et_nat @ ( produc5001842942810119800et_nat @ X @ Y ) @ A )
% 0.34/0.57 => ( member6260224972018164377et_nat @ Y @ B4 ) ) ) ).
% 0.34/0.57
% 0.34/0.57 % snd_image_mp
% 0.34/0.57 thf(fact_2712_snd__image__mp,axiom,
% 0.34/0.57 ! [A: set_Pr8536935166611901872et_nat,B4: set_Pr3286484037609594932et_nat,X: produc3658429121746597890et_nat > $o,Y: produc3925858234332021118et_nat] :
% 0.34/0.57 ( ( ord_le5966269811547037844et_nat @ ( image_4557310310409863807et_nat @ produc5374455773327741254et_nat @ A ) @ B4 )
% 0.34/0.57 => ( ( member6124377750444531601et_nat @ ( produc2245416461498447860et_nat @ X @ Y ) @ A )
% 0.34/0.57 => ( member1996754912294343701et_nat @ Y @ B4 ) ) ) ).
% 0.34/0.57
% 0.34/0.57 % snd_image_mp
% 0.34/0.57 thf(fact_2713_snd__image__mp,axiom,
% 0.34/0.57 ! [A: set_Pr3948176798113811640et_nat,B4: set_set_nat,X: heap_e7401611519738050253t_unit,Y: set_nat] :
% 0.34/0.57 ( ( ord_le6893508408891458716et_nat @ ( image_6088997261112373647et_nat @ produc8586169260539613262et_nat @ A ) @ B4 )
% 0.34/0.57 => ( ( member6260224972018164377et_nat @ ( produc7507926704131184380et_nat @ X @ Y ) @ A )
% 0.34/0.57 => ( member_set_nat @ Y @ B4 ) ) ) ).
% 0.34/0.57
% 0.34/0.57 % snd_image_mp
% 0.34/0.57 thf(fact_2714_snd__image__mp,axiom,
% 0.34/0.57 ! [A: set_Pr958786334691620121nt_int,B4: set_int,X: int,Y: int] :
% 0.34/0.57 ( ( ord_less_eq_set_int @ ( image_5042161079198086560nt_int @ product_snd_int_int @ A ) @ B4 )
% 0.34/0.57 => ( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ X @ Y ) @ A )
% 0.34/0.57 => ( member_int @ Y @ B4 ) ) ) ).
% 0.34/0.57
% 0.34/0.57 % snd_image_mp
% 0.34/0.57 thf(fact_2715_snd__image__mp,axiom,
% 0.34/0.57 ! [A: set_Pr1261947904930325089at_nat,B4: set_nat,X: nat,Y: nat] :
% 0.34/0.57 ( ( ord_less_eq_set_nat @ ( image_2486076414777270412at_nat @ product_snd_nat_nat @ A ) @ B4 )
% 0.34/0.57 => ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ Y ) @ A )
% 0.34/0.57 => ( member_nat @ Y @ B4 ) ) ) ).
% 0.34/0.57
% 0.34/0.57 % snd_image_mp
% 0.34/0.57 thf(fact_2716_snd__in__Field,axiom,
% 0.34/0.57 ! [R3: set_Pr958786334691620121nt_int] : ( ord_less_eq_set_int @ ( image_5042161079198086560nt_int @ product_snd_int_int @ R3 ) @ ( field_int @ R3 ) ) ).
% 0.34/0.57
% 0.34/0.57 % snd_in_Field
% 0.34/0.57 thf(fact_2717_snd__in__Field,axiom,
% 0.34/0.57 ! [R3: set_Pr1261947904930325089at_nat] : ( ord_less_eq_set_nat @ ( image_2486076414777270412at_nat @ product_snd_nat_nat @ R3 ) @ ( field_nat @ R3 ) ) ).
% 0.34/0.57
% 0.34/0.57 % snd_in_Field
% 0.34/0.57 thf(fact_2718_Image__subset__snd__image,axiom,
% 0.34/0.57 ! [A: set_Pr3948176798113811640et_nat,B4: set_he8644174446132254637t_unit] : ( ord_le6893508408891458716et_nat @ ( image_3942343239045813184et_nat @ A @ B4 ) @ ( image_6088997261112373647et_nat @ produc8586169260539613262et_nat @ A ) ) ).
% 0.34/0.57
% 0.34/0.57 % Image_subset_snd_image
% 0.34/0.57 thf(fact_2719_Image__subset__snd__image,axiom,
% 0.34/0.57 ! [A: set_Pr958786334691620121nt_int,B4: set_int] : ( ord_less_eq_set_int @ ( image_int_int @ A @ B4 ) @ ( image_5042161079198086560nt_int @ product_snd_int_int @ A ) ) ).
% 0.34/0.57
% 0.34/0.57 % Image_subset_snd_image
% 0.34/0.57 thf(fact_2720_infinite__remove,axiom,
% 0.34/0.57 ! [S: set_Code_integer,A2: code_integer] :
% 0.34/0.57 ( ~ ( finite6017078050557962740nteger @ S )
% 0.34/0.57 => ~ ( finite6017078050557962740nteger @ ( minus_2355218937544613996nteger @ S @ ( insert_Code_integer @ A2 @ bot_bo3990330152332043303nteger ) ) ) ) ).
% 0.34/0.57
% 0.34/0.57 % infinite_remove
% 0.34/0.57 thf(fact_2721_infinite__remove,axiom,
% 0.34/0.57 ! [S: set_Product_unit,A2: product_unit] :
% 0.34/0.57 ( ~ ( finite4290736615968046902t_unit @ S )
% 0.34/0.57 => ~ ( finite4290736615968046902t_unit @ ( minus_6452836326544984404t_unit @ S @ ( insert_Product_unit @ A2 @ bot_bo3957492148770167129t_unit ) ) ) ) ).
% 0.34/0.57
% 0.34/0.57 % infinite_remove
% 0.34/0.57 thf(fact_2722_infinite__remove,axiom,
% 0.34/0.57 ! [S: set_o,A2: $o] :
% 0.34/0.57 ( ~ ( finite_finite_o @ S )
% 0.34/0.57 => ~ ( finite_finite_o @ ( minus_minus_set_o @ S @ ( insert_o @ A2 @ bot_bot_set_o ) ) ) ) ).
% 0.34/0.57
% 0.34/0.57 % infinite_remove
% 0.34/0.57 thf(fact_2723_infinite__remove,axiom,
% 0.34/0.57 ! [S: set_int,A2: int] :
% 0.34/0.57 ( ~ ( finite_finite_int @ S )
% 0.34/0.57 => ~ ( finite_finite_int @ ( minus_minus_set_int @ S @ ( insert_int @ A2 @ bot_bot_set_int ) ) ) ) ).
% 0.34/0.57
% 0.34/0.57 % infinite_remove
% 0.34/0.57 thf(fact_2724_infinite__remove,axiom,
% 0.34/0.57 ! [S: set_Pr4532377907799695533_nat_o,A2: produc3658429121746597890et_nat > $o] :
% 0.34/0.57 ( ~ ( finite3252695134891459830_nat_o @ S )
% 0.34/0.57 => ~ ( finite3252695134891459830_nat_o @ ( minus_1801376950450012436_nat_o @ S @ ( insert5175938949040314269_nat_o @ A2 @ bot_bo7824918357723582233_nat_o ) ) ) ) ).
% 0.34/0.57
% 0.34/0.57 % infinite_remove
% 0.34/0.57 thf(fact_2725_infinite__remove,axiom,
% 0.34/0.57 ! [S: set_nat,A2: nat] :
% 0.34/0.57 ( ~ ( finite_finite_nat @ S )
% 0.34/0.57 => ~ ( finite_finite_nat @ ( minus_minus_set_nat @ S @ ( insert_nat @ A2 @ bot_bot_set_nat ) ) ) ) ).
% 0.34/0.57
% 0.34/0.57 % infinite_remove
% 0.34/0.57 thf(fact_2726_infinite__coinduct,axiom,
% 0.34/0.57 ! [X4: set_Code_integer > $o,A: set_Code_integer] :
% 0.34/0.57 ( ( X4 @ A )
% 0.34/0.57 => ( ! [A9: set_Code_integer] :
% 0.34/0.57 ( ( X4 @ A9 )
% 0.34/0.57 => ? [X6: code_integer] :
% 0.34/0.57 ( ( member_Code_integer @ X6 @ A9 )
% 0.34/0.57 & ( ( X4 @ ( minus_2355218937544613996nteger @ A9 @ ( insert_Code_integer @ X6 @ bot_bo3990330152332043303nteger ) ) )
% 0.34/0.57 | ~ ( finite6017078050557962740nteger @ ( minus_2355218937544613996nteger @ A9 @ ( insert_Code_integer @ X6 @ bot_bo3990330152332043303nteger ) ) ) ) ) )
% 0.34/0.57 => ~ ( finite6017078050557962740nteger @ A ) ) ) ).
% 0.34/0.57
% 0.34/0.57 % infinite_coinduct
% 0.34/0.57 thf(fact_2727_infinite__coinduct,axiom,
% 0.34/0.57 ! [X4: set_Product_unit > $o,A: set_Product_unit] :
% 0.34/0.57 ( ( X4 @ A )
% 0.34/0.57 => ( ! [A9: set_Product_unit] :
% 0.34/0.57 ( ( X4 @ A9 )
% 0.34/0.57 => ? [X6: product_unit] :
% 0.34/0.57 ( ( member_Product_unit @ X6 @ A9 )
% 0.34/0.57 & ( ( X4 @ ( minus_6452836326544984404t_unit @ A9 @ ( insert_Product_unit @ X6 @ bot_bo3957492148770167129t_unit ) ) )
% 0.34/0.57 | ~ ( finite4290736615968046902t_unit @ ( minus_6452836326544984404t_unit @ A9 @ ( insert_Product_unit @ X6 @ bot_bo3957492148770167129t_unit ) ) ) ) ) )
% 0.34/0.57 => ~ ( finite4290736615968046902t_unit @ A ) ) ) ).
% 0.34/0.57
% 0.34/0.57 % infinite_coinduct
% 0.34/0.57 thf(fact_2728_infinite__coinduct,axiom,
% 0.34/0.57 ! [X4: set_o > $o,A: set_o] :
% 0.34/0.57 ( ( X4 @ A )
% 0.34/0.57 => ( ! [A9: set_o] :
% 0.34/0.57 ( ( X4 @ A9 )
% 0.34/0.57 => ? [X6: $o] :
% 0.34/0.57 ( ( member_o @ X6 @ A9 )
% 0.34/0.57 & ( ( X4 @ ( minus_minus_set_o @ A9 @ ( insert_o @ X6 @ bot_bot_set_o ) ) )
% 0.34/0.57 | ~ ( finite_finite_o @ ( minus_minus_set_o @ A9 @ ( insert_o @ X6 @ bot_bot_set_o ) ) ) ) ) )
% 0.34/0.57 => ~ ( finite_finite_o @ A ) ) ) ).
% 0.34/0.57
% 0.34/0.57 % infinite_coinduct
% 0.34/0.57 thf(fact_2729_infinite__coinduct,axiom,
% 0.34/0.57 ! [X4: set_int > $o,A: set_int] :
% 0.34/0.57 ( ( X4 @ A )
% 0.34/0.57 => ( ! [A9: set_int] :
% 0.34/0.57 ( ( X4 @ A9 )
% 0.34/0.57 => ? [X6: int] :
% 0.34/0.57 ( ( member_int @ X6 @ A9 )
% 0.34/0.57 & ( ( X4 @ ( minus_minus_set_int @ A9 @ ( insert_int @ X6 @ bot_bot_set_int ) ) )
% 0.34/0.57 | ~ ( finite_finite_int @ ( minus_minus_set_int @ A9 @ ( insert_int @ X6 @ bot_bot_set_int ) ) ) ) ) )
% 0.34/0.57 => ~ ( finite_finite_int @ A ) ) ) ).
% 0.34/0.57
% 0.34/0.57 % infinite_coinduct
% 0.34/0.57 thf(fact_2730_infinite__coinduct,axiom,
% 0.34/0.57 ! [X4: set_Pr4532377907799695533_nat_o > $o,A: set_Pr4532377907799695533_nat_o] :
% 0.34/0.57 ( ( X4 @ A )
% 0.34/0.57 => ( ! [A9: set_Pr4532377907799695533_nat_o] :
% 0.34/0.57 ( ( X4 @ A9 )
% 0.34/0.57 => ? [X6: produc3658429121746597890et_nat > $o] :
% 0.34/0.57 ( ( member6576561426505652726_nat_o @ X6 @ A9 )
% 0.34/0.57 & ( ( X4 @ ( minus_1801376950450012436_nat_o @ A9 @ ( insert5175938949040314269_nat_o @ X6 @ bot_bo7824918357723582233_nat_o ) ) )
% 0.34/0.57 | ~ ( finite3252695134891459830_nat_o @ ( minus_1801376950450012436_nat_o @ A9 @ ( insert5175938949040314269_nat_o @ X6 @ bot_bo7824918357723582233_nat_o ) ) ) ) ) )
% 0.34/0.57 => ~ ( finite3252695134891459830_nat_o @ A ) ) ) ).
% 0.34/0.57
% 0.34/0.57 % infinite_coinduct
% 0.34/0.57 thf(fact_2731_infinite__coinduct,axiom,
% 0.34/0.57 ! [X4: set_nat > $o,A: set_nat] :
% 0.34/0.57 ( ( X4 @ A )
% 0.34/0.57 => ( ! [A9: set_nat] :
% 0.34/0.57 ( ( X4 @ A9 )
% 0.34/0.57 => ? [X6: nat] :
% 0.34/0.57 ( ( member_nat @ X6 @ A9 )
% 0.34/0.57 & ( ( X4 @ ( minus_minus_set_nat @ A9 @ ( insert_nat @ X6 @ bot_bot_set_nat ) ) )
% 0.34/0.57 | ~ ( finite_finite_nat @ ( minus_minus_set_nat @ A9 @ ( insert_nat @ X6 @ bot_bot_set_nat ) ) ) ) ) )
% 0.34/0.57 => ~ ( finite_finite_nat @ A ) ) ) ).
% 0.34/0.57
% 0.34/0.57 % infinite_coinduct
% 0.34/0.57 thf(fact_2732_finite__empty__induct,axiom,
% 0.34/0.57 ! [A: set_Code_integer,P: set_Code_integer > $o] :
% 0.34/0.57 ( ( finite6017078050557962740nteger @ A )
% 0.34/0.57 => ( ( P @ A )
% 0.34/0.57 => ( ! [A4: code_integer,A9: set_Code_integer] :
% 0.34/0.57 ( ( finite6017078050557962740nteger @ A9 )
% 0.34/0.57 => ( ( member_Code_integer @ A4 @ A9 )
% 0.34/0.57 => ( ( P @ A9 )
% 0.34/0.57 => ( P @ ( minus_2355218937544613996nteger @ A9 @ ( insert_Code_integer @ A4 @ bot_bo3990330152332043303nteger ) ) ) ) ) )
% 0.34/0.57 => ( P @ bot_bo3990330152332043303nteger ) ) ) ) ).
% 0.34/0.57
% 0.34/0.57 % finite_empty_induct
% 0.34/0.57 thf(fact_2733_finite__empty__induct,axiom,
% 0.34/0.57 ! [A: set_Product_unit,P: set_Product_unit > $o] :
% 0.34/0.57 ( ( finite4290736615968046902t_unit @ A )
% 0.34/0.57 => ( ( P @ A )
% 0.34/0.57 => ( ! [A4: product_unit,A9: set_Product_unit] :
% 0.34/0.57 ( ( finite4290736615968046902t_unit @ A9 )
% 0.34/0.57 => ( ( member_Product_unit @ A4 @ A9 )
% 0.34/0.57 => ( ( P @ A9 )
% 0.34/0.57 => ( P @ ( minus_6452836326544984404t_unit @ A9 @ ( insert_Product_unit @ A4 @ bot_bo3957492148770167129t_unit ) ) ) ) ) )
% 0.34/0.57 => ( P @ bot_bo3957492148770167129t_unit ) ) ) ) ).
% 0.34/0.57
% 0.34/0.57 % finite_empty_induct
% 0.34/0.57 thf(fact_2734_finite__empty__induct,axiom,
% 0.34/0.57 ! [A: set_o,P: set_o > $o] :
% 0.34/0.57 ( ( finite_finite_o @ A )
% 0.34/0.57 => ( ( P @ A )
% 0.34/0.57 => ( ! [A4: $o,A9: set_o] :
% 0.34/0.57 ( ( finite_finite_o @ A9 )
% 0.34/0.57 => ( ( member_o @ A4 @ A9 )
% 0.34/0.57 => ( ( P @ A9 )
% 0.34/0.57 => ( P @ ( minus_minus_set_o @ A9 @ ( insert_o @ A4 @ bot_bot_set_o ) ) ) ) ) )
% 0.34/0.57 => ( P @ bot_bot_set_o ) ) ) ) ).
% 0.34/0.57
% 0.34/0.57 % finite_empty_induct
% 0.34/0.57 thf(fact_2735_finite__empty__induct,axiom,
% 0.34/0.57 ! [A: set_int,P: set_int > $o] :
% 0.34/0.57 ( ( finite_finite_int @ A )
% 0.34/0.57 => ( ( P @ A )
% 0.34/0.57 => ( ! [A4: int,A9: set_int] :
% 0.34/0.57 ( ( finite_finite_int @ A9 )
% 0.34/0.57 => ( ( member_int @ A4 @ A9 )
% 0.34/0.57 => ( ( P @ A9 )
% 0.34/0.57 => ( P @ ( minus_minus_set_int @ A9 @ ( insert_int @ A4 @ bot_bot_set_int ) ) ) ) ) )
% 0.34/0.57 => ( P @ bot_bot_set_int ) ) ) ) ).
% 0.34/0.57
% 0.34/0.57 % finite_empty_induct
% 0.34/0.57 thf(fact_2736_finite__empty__induct,axiom,
% 0.34/0.57 ! [A: set_Pr4532377907799695533_nat_o,P: set_Pr4532377907799695533_nat_o > $o] :
% 0.34/0.57 ( ( finite3252695134891459830_nat_o @ A )
% 0.34/0.57 => ( ( P @ A )
% 0.34/0.57 => ( ! [A4: produc3658429121746597890et_nat > $o,A9: set_Pr4532377907799695533_nat_o] :
% 0.34/0.57 ( ( finite3252695134891459830_nat_o @ A9 )
% 0.34/0.57 => ( ( member6576561426505652726_nat_o @ A4 @ A9 )
% 0.34/0.57 => ( ( P @ A9 )
% 0.34/0.57 => ( P @ ( minus_1801376950450012436_nat_o @ A9 @ ( insert5175938949040314269_nat_o @ A4 @ bot_bo7824918357723582233_nat_o ) ) ) ) ) )
% 0.34/0.57 => ( P @ bot_bo7824918357723582233_nat_o ) ) ) ) ).
% 0.34/0.57
% 0.34/0.57 % finite_empty_induct
% 0.34/0.57 thf(fact_2737_finite__empty__induct,axiom,
% 0.34/0.57 ! [A: set_nat,P: set_nat > $o] :
% 0.34/0.57 ( ( finite_finite_nat @ A )
% 0.34/0.57 => ( ( P @ A )
% 0.34/0.57 => ( ! [A4: nat,A9: set_nat] :
% 0.34/0.57 ( ( finite_finite_nat @ A9 )
% 0.34/0.57 => ( ( member_nat @ A4 @ A9 )
% 0.34/0.57 => ( ( P @ A9 )
% 0.34/0.57 => ( P @ ( minus_minus_set_nat @ A9 @ ( insert_nat @ A4 @ bot_bot_set_nat ) ) ) ) ) )
% 0.34/0.57 => ( P @ bot_bot_set_nat ) ) ) ) ).
% 0.34/0.57
% 0.34/0.57 % finite_empty_induct
% 0.34/0.57 thf(fact_2738_subset__insert__iff,axiom,
% 0.34/0.57 ! [A: set_Product_unit,X: product_unit,B4: set_Product_unit] :
% 0.34/0.57 ( ( ord_le3507040750410214029t_unit @ A @ ( insert_Product_unit @ X @ B4 ) )
% 0.34/0.57 = ( ( ( member_Product_unit @ X @ A )
% 0.34/0.57 => ( ord_le3507040750410214029t_unit @ ( minus_6452836326544984404t_unit @ A @ ( insert_Product_unit @ X @ bot_bo3957492148770167129t_unit ) ) @ B4 ) )
% 0.34/0.57 & ( ~ ( member_Product_unit @ X @ A )
% 0.34/0.57 => ( ord_le3507040750410214029t_unit @ A @ B4 ) ) ) ) ).
% 0.34/0.57
% 0.34/0.57 % subset_insert_iff
% 0.34/0.57 thf(fact_2739_subset__insert__iff,axiom,
% 0.34/0.57 ! [A: set_o,X: $o,B4: set_o] :
% 0.34/0.57 ( ( ord_less_eq_set_o @ A @ ( insert_o @ X @ B4 ) )
% 0.34/0.57 = ( ( ( member_o @ X @ A )
% 0.34/0.57 => ( ord_less_eq_set_o @ ( minus_minus_set_o @ A @ ( insert_o @ X @ bot_bot_set_o ) ) @ B4 ) )
% 0.34/0.57 & ( ~ ( member_o @ X @ A )
% 0.34/0.57 => ( ord_less_eq_set_o @ A @ B4 ) ) ) ) ).
% 0.34/0.57
% 0.34/0.57 % subset_insert_iff
% 0.34/0.57 thf(fact_2740_subset__insert__iff,axiom,
% 0.34/0.57 ! [A: set_int,X: int,B4: set_int] :
% 0.34/0.57 ( ( ord_less_eq_set_int @ A @ ( insert_int @ X @ B4 ) )
% 0.34/0.57 = ( ( ( member_int @ X @ A )
% 0.34/0.57 => ( ord_less_eq_set_int @ ( minus_minus_set_int @ A @ ( insert_int @ X @ bot_bot_set_int ) ) @ B4 ) )
% 0.34/0.57 & ( ~ ( member_int @ X @ A )
% 0.34/0.57 => ( ord_less_eq_set_int @ A @ B4 ) ) ) ) ).
% 0.34/0.57
% 0.34/0.57 % subset_insert_iff
% 0.34/0.57 thf(fact_2741_subset__insert__iff,axiom,
% 0.34/0.57 ! [A: set_Pr4532377907799695533_nat_o,X: produc3658429121746597890et_nat > $o,B4: set_Pr4532377907799695533_nat_o] :
% 0.34/0.57 ( ( ord_le2965882846123202637_nat_o @ A @ ( insert5175938949040314269_nat_o @ X @ B4 ) )
% 0.34/0.57 = ( ( ( member6576561426505652726_nat_o @ X @ A )
% 0.34/0.57 => ( ord_le2965882846123202637_nat_o @ ( minus_1801376950450012436_nat_o @ A @ ( insert5175938949040314269_nat_o @ X @ bot_bo7824918357723582233_nat_o ) ) @ B4 ) )
% 0.34/0.57 & ( ~ ( member6576561426505652726_nat_o @ X @ A )
% 0.34/0.57 => ( ord_le2965882846123202637_nat_o @ A @ B4 ) ) ) ) ).
% 0.34/0.57
% 0.34/0.57 % subset_insert_iff
% 0.34/0.57 thf(fact_2742_subset__insert__iff,axiom,
% 0.34/0.57 ! [A: set_nat,X: nat,B4: set_nat] :
% 0.34/0.57 ( ( ord_less_eq_set_nat @ A @ ( insert_nat @ X @ B4 ) )
% 0.34/0.57 = ( ( ( member_nat @ X @ A )
% 0.34/0.57 => ( ord_less_eq_set_nat @ ( minus_minus_set_nat @ A @ ( insert_nat @ X @ bot_bot_set_nat ) ) @ B4 ) )
% 0.34/0.57 & ( ~ ( member_nat @ X @ A )
% 0.34/0.57 => ( ord_less_eq_set_nat @ A @ B4 ) ) ) ) ).
% 0.34/0.57
% 0.34/0.57 % subset_insert_iff
% 0.34/0.57 thf(fact_2743_Diff__single__insert,axiom,
% 0.34/0.57 ! [A: set_Product_unit,X: product_unit,B4: set_Product_unit] :
% 0.34/0.57 ( ( ord_le3507040750410214029t_unit @ ( minus_6452836326544984404t_unit @ A @ ( insert_Product_unit @ X @ bot_bo3957492148770167129t_unit ) ) @ B4 )
% 0.34/0.57 => ( ord_le3507040750410214029t_unit @ A @ ( insert_Product_unit @ X @ B4 ) ) ) ).
% 0.34/0.57
% 0.34/0.57 % Diff_single_insert
% 0.34/0.57 thf(fact_2744_Diff__single__insert,axiom,
% 0.34/0.57 ! [A: set_o,X: $o,B4: set_o] :
% 0.34/0.57 ( ( ord_less_eq_set_o @ ( minus_minus_set_o @ A @ ( insert_o @ X @ bot_bot_set_o ) ) @ B4 )
% 0.34/0.57 => ( ord_less_eq_set_o @ A @ ( insert_o @ X @ B4 ) ) ) ).
% 0.34/0.57
% 0.34/0.57 % Diff_single_insert
% 0.34/0.57 thf(fact_2745_Diff__single__insert,axiom,
% 0.34/0.57 ! [A: set_int,X: int,B4: set_int] :
% 0.34/0.57 ( ( ord_less_eq_set_int @ ( minus_minus_set_int @ A @ ( insert_int @ X @ bot_bot_set_int ) ) @ B4 )
% 0.34/0.57 => ( ord_less_eq_set_int @ A @ ( insert_int @ X @ B4 ) ) ) ).
% 0.34/0.57
% 0.34/0.57 % Diff_single_insert
% 0.34/0.57 thf(fact_2746_Diff__single__insert,axiom,
% 0.34/0.57 ! [A: set_Pr4532377907799695533_nat_o,X: produc3658429121746597890et_nat > $o,B4: set_Pr4532377907799695533_nat_o] :
% 0.34/0.57 ( ( ord_le2965882846123202637_nat_o @ ( minus_1801376950450012436_nat_o @ A @ ( insert5175938949040314269_nat_o @ X @ bot_bo7824918357723582233_nat_o ) ) @ B4 )
% 0.34/0.57 => ( ord_le2965882846123202637_nat_o @ A @ ( insert5175938949040314269_nat_o @ X @ B4 ) ) ) ).
% 0.34/0.57
% 0.34/0.57 % Diff_single_insert
% 0.34/0.57 thf(fact_2747_Diff__single__insert,axiom,
% 0.34/0.57 ! [A: set_nat,X: nat,B4: set_nat] :
% 0.34/0.57 ( ( ord_less_eq_set_nat @ ( minus_minus_set_nat @ A @ ( insert_nat @ X @ bot_bot_set_nat ) ) @ B4 )
% 0.34/0.57 => ( ord_less_eq_set_nat @ A @ ( insert_nat @ X @ B4 ) ) ) ).
% 0.34/0.57
% 0.34/0.57 % Diff_single_insert
% 0.34/0.57 thf(fact_2748_Field__rel__restrict,axiom,
% 0.34/0.57 ! [R3: set_Pr1261947904930325089at_nat,A: set_nat] : ( ord_less_eq_set_nat @ ( field_nat @ ( rel_restrict_nat @ R3 @ A ) ) @ ( minus_minus_set_nat @ ( field_nat @ R3 ) @ A ) ) ).
% 0.34/0.57
% 0.34/0.57 % Field_rel_restrict
% 0.34/0.57 thf(fact_2749_Domain__rel__restrict,axiom,
% 0.34/0.57 ! [R3: set_Pr1261947904930325089at_nat,A: set_nat] : ( ord_less_eq_set_nat @ ( domain_nat_nat @ ( rel_restrict_nat @ R3 @ A ) ) @ ( minus_minus_set_nat @ ( domain_nat_nat @ R3 ) @ A ) ) ).
% 0.34/0.57
% 0.34/0.57 % Domain_rel_restrict
% 0.34/0.57 thf(fact_2750_Range__rel__restrict,axiom,
% 0.34/0.57 ! [R3: set_Pr1261947904930325089at_nat,A: set_nat] : ( ord_less_eq_set_nat @ ( range_nat_nat @ ( rel_restrict_nat @ R3 @ A ) ) @ ( minus_minus_set_nat @ ( range_nat_nat @ R3 ) @ A ) ) ).
% 0.34/0.57
% 0.34/0.57 % Range_rel_restrict
% 0.34/0.57 thf(fact_2751_inf__img__fin__domE,axiom,
% 0.34/0.57 ! [F: nat > code_integer,A: set_nat] :
% 0.34/0.57 ( ( finite6017078050557962740nteger @ ( image_1215581382706833972nteger @ F @ A ) )
% 0.34/0.57 => ( ~ ( finite_finite_nat @ A )
% 0.34/0.57 => ~ ! [Y2: code_integer] :
% 0.34/0.57 ( ( member_Code_integer @ Y2 @ ( image_1215581382706833972nteger @ F @ A ) )
% 0.34/0.57 => ( finite_finite_nat @ ( vimage6104394758324456170nteger @ F @ ( insert_Code_integer @ Y2 @ bot_bo3990330152332043303nteger ) ) ) ) ) ) ).
% 0.34/0.57
% 0.34/0.57 % inf_img_fin_domE
% 0.34/0.57 thf(fact_2752_inf__img__fin__domE,axiom,
% 0.34/0.57 ! [F: code_integer > code_integer,A: set_Code_integer] :
% 0.34/0.57 ( ( finite6017078050557962740nteger @ ( image_4470545334726330049nteger @ F @ A ) )
% 0.34/0.57 => ( ~ ( finite6017078050557962740nteger @ A )
% 0.34/0.57 => ~ ! [Y2: code_integer] :
% 0.34/0.57 ( ( member_Code_integer @ Y2 @ ( image_4470545334726330049nteger @ F @ A ) )
% 0.34/0.57 => ( finite6017078050557962740nteger @ ( vimage1318793397690208119nteger @ F @ ( insert_Code_integer @ Y2 @ bot_bo3990330152332043303nteger ) ) ) ) ) ) ).
% 0.34/0.57
% 0.34/0.57 % inf_img_fin_domE
% 0.34/0.57 thf(fact_2753_inf__img__fin__domE,axiom,
% 0.34/0.57 ! [F: nat > product_unit,A: set_nat] :
% 0.34/0.57 ( ( finite4290736615968046902t_unit @ ( image_8730104196221521654t_unit @ F @ A ) )
% 0.34/0.57 => ( ~ ( finite_finite_nat @ A )
% 0.34/0.57 => ~ ! [Y2: product_unit] :
% 0.34/0.57 ( ( member_Product_unit @ Y2 @ ( image_8730104196221521654t_unit @ F @ A ) )
% 0.34/0.57 => ( finite_finite_nat @ ( vimage4884490618288580032t_unit @ F @ ( insert_Product_unit @ Y2 @ bot_bo3957492148770167129t_unit ) ) ) ) ) ) ).
% 0.34/0.57
% 0.34/0.57 % inf_img_fin_domE
% 0.34/0.57 thf(fact_2754_inf__img__fin__domE,axiom,
% 0.34/0.57 ! [F: code_integer > product_unit,A: set_Code_integer] :
% 0.34/0.57 ( ( finite4290736615968046902t_unit @ ( image_7182656606676592809t_unit @ F @ A ) )
% 0.34/0.57 => ( ~ ( finite6017078050557962740nteger @ A )
% 0.34/0.57 => ~ ! [Y2: product_unit] :
% 0.34/0.57 ( ( member_Product_unit @ Y2 @ ( image_7182656606676592809t_unit @ F @ A ) )
% 0.34/0.57 => ( finite6017078050557962740nteger @ ( vimage5651688891684837235t_unit @ F @ ( insert_Product_unit @ Y2 @ bot_bo3957492148770167129t_unit ) ) ) ) ) ) ).
% 0.34/0.57
% 0.34/0.57 % inf_img_fin_domE
% 0.34/0.57 thf(fact_2755_inf__img__fin__domE,axiom,
% 0.34/0.57 ! [F: nat > $o,A: set_nat] :
% 0.34/0.57 ( ( finite_finite_o @ ( image_nat_o2 @ F @ A ) )
% 0.34/0.57 => ( ~ ( finite_finite_nat @ A )
% 0.34/0.57 => ~ ! [Y2: $o] :
% 0.34/0.57 ( ( member_o @ Y2 @ ( image_nat_o2 @ F @ A ) )
% 0.34/0.57 => ( finite_finite_nat @ ( vimage_nat_o @ F @ ( insert_o @ Y2 @ bot_bot_set_o ) ) ) ) ) ) ).
% 0.34/0.57
% 0.34/0.57 % inf_img_fin_domE
% 0.34/0.57 thf(fact_2756_inf__img__fin__domE,axiom,
% 0.34/0.57 ! [F: code_integer > $o,A: set_Code_integer] :
% 0.34/0.57 ( ( finite_finite_o @ ( image_Code_integer_o @ F @ A ) )
% 0.34/0.57 => ( ~ ( finite6017078050557962740nteger @ A )
% 0.34/0.57 => ~ ! [Y2: $o] :
% 0.34/0.57 ( ( member_o @ Y2 @ ( image_Code_integer_o @ F @ A ) )
% 0.34/0.57 => ( finite6017078050557962740nteger @ ( vimage3005800680117336574eger_o @ F @ ( insert_o @ Y2 @ bot_bot_set_o ) ) ) ) ) ) ).
% 0.34/0.57
% 0.34/0.57 % inf_img_fin_domE
% 0.34/0.57 thf(fact_2757_inf__img__fin__domE,axiom,
% 0.34/0.57 ! [F: nat > nat,A: set_nat] :
% 0.34/0.57 ( ( finite_finite_nat @ ( image_nat_nat2 @ F @ A ) )
% 0.34/0.57 => ( ~ ( finite_finite_nat @ A )
% 0.34/0.57 => ~ ! [Y2: nat] :
% 0.34/0.57 ( ( member_nat @ Y2 @ ( image_nat_nat2 @ F @ A ) )
% 0.34/0.57 => ( finite_finite_nat @ ( vimage_nat_nat @ F @ ( insert_nat @ Y2 @ bot_bot_set_nat ) ) ) ) ) ) ).
% 0.34/0.57
% 0.34/0.57 % inf_img_fin_domE
% 0.34/0.57 thf(fact_2758_inf__img__fin__domE,axiom,
% 0.34/0.57 ! [F: code_integer > nat,A: set_Code_integer] :
% 0.34/0.57 ( ( finite_finite_nat @ ( image_951025933927791156er_nat @ F @ A ) )
% 0.34/0.57 => ( ~ ( finite6017078050557962740nteger @ A )
% 0.34/0.57 => ~ ! [Y2: nat] :
% 0.34/0.57 ( ( member_nat @ Y2 @ ( image_951025933927791156er_nat @ F @ A ) )
% 0.34/0.57 => ( finite6017078050557962740nteger @ ( vimage5839839309545413354er_nat @ F @ ( insert_nat @ Y2 @ bot_bot_set_nat ) ) ) ) ) ) ).
% 0.34/0.57
% 0.34/0.57 % inf_img_fin_domE
% 0.34/0.57 thf(fact_2759_inf__img__fin__domE,axiom,
% 0.34/0.57 ! [F: nat > int,A: set_nat] :
% 0.34/0.57 ( ( finite_finite_int @ ( image_nat_int2 @ F @ A ) )
% 0.34/0.57 => ( ~ ( finite_finite_nat @ A )
% 0.34/0.57 => ~ ! [Y2: int] :
% 0.34/0.57 ( ( member_int @ Y2 @ ( image_nat_int2 @ F @ A ) )
% 0.34/0.57 => ( finite_finite_nat @ ( vimage_nat_int @ F @ ( insert_int @ Y2 @ bot_bot_set_int ) ) ) ) ) ) ).
% 0.34/0.57
% 0.34/0.57 % inf_img_fin_domE
% 0.34/0.57 thf(fact_2760_inf__img__fin__domE,axiom,
% 0.34/0.57 ! [F: code_integer > int,A: set_Code_integer] :
% 0.34/0.57 ( ( finite_finite_int @ ( image_948535463418740880er_int @ F @ A ) )
% 0.34/0.57 => ( ~ ( finite6017078050557962740nteger @ A )
% 0.34/0.57 => ~ ! [Y2: int] :
% 0.34/0.57 ( ( member_int @ Y2 @ ( image_948535463418740880er_int @ F @ A ) )
% 0.34/0.57 => ( finite6017078050557962740nteger @ ( vimage5837348839036363078er_int @ F @ ( insert_int @ Y2 @ bot_bot_set_int ) ) ) ) ) ) ).
% 0.34/0.57
% 0.34/0.57 % inf_img_fin_domE
% 0.34/0.57 thf(fact_2761_inf__img__fin__dom,axiom,
% 0.34/0.57 ! [F: nat > code_integer,A: set_nat] :
% 0.34/0.57 ( ( finite6017078050557962740nteger @ ( image_1215581382706833972nteger @ F @ A ) )
% 0.34/0.57 => ( ~ ( finite_finite_nat @ A )
% 0.34/0.57 => ? [X3: code_integer] :
% 0.34/0.57 ( ( member_Code_integer @ X3 @ ( image_1215581382706833972nteger @ F @ A ) )
% 0.34/0.57 & ~ ( finite_finite_nat @ ( vimage6104394758324456170nteger @ F @ ( insert_Code_integer @ X3 @ bot_bo3990330152332043303nteger ) ) ) ) ) ) ).
% 0.34/0.57
% 0.34/0.57 % inf_img_fin_dom
% 0.34/0.57 thf(fact_2762_inf__img__fin__dom,axiom,
% 0.34/0.57 ! [F: code_integer > code_integer,A: set_Code_integer] :
% 0.34/0.57 ( ( finite6017078050557962740nteger @ ( image_4470545334726330049nteger @ F @ A ) )
% 0.34/0.57 => ( ~ ( finite6017078050557962740nteger @ A )
% 0.34/0.57 => ? [X3: code_integer] :
% 0.34/0.57 ( ( member_Code_integer @ X3 @ ( image_4470545334726330049nteger @ F @ A ) )
% 0.34/0.57 & ~ ( finite6017078050557962740nteger @ ( vimage1318793397690208119nteger @ F @ ( insert_Code_integer @ X3 @ bot_bo3990330152332043303nteger ) ) ) ) ) ) ).
% 0.34/0.57
% 0.34/0.57 % inf_img_fin_dom
% 0.34/0.57 thf(fact_2763_inf__img__fin__dom,axiom,
% 0.34/0.57 ! [F: nat > product_unit,A: set_nat] :
% 0.34/0.57 ( ( finite4290736615968046902t_unit @ ( image_8730104196221521654t_unit @ F @ A ) )
% 0.34/0.57 => ( ~ ( finite_finite_nat @ A )
% 0.34/0.57 => ? [X3: product_unit] :
% 0.34/0.57 ( ( member_Product_unit @ X3 @ ( image_8730104196221521654t_unit @ F @ A ) )
% 0.34/0.57 & ~ ( finite_finite_nat @ ( vimage4884490618288580032t_unit @ F @ ( insert_Product_unit @ X3 @ bot_bo3957492148770167129t_unit ) ) ) ) ) ) ).
% 0.34/0.57
% 0.34/0.57 % inf_img_fin_dom
% 0.34/0.57 thf(fact_2764_inf__img__fin__dom,axiom,
% 0.34/0.57 ! [F: code_integer > product_unit,A: set_Code_integer] :
% 0.34/0.57 ( ( finite4290736615968046902t_unit @ ( image_7182656606676592809t_unit @ F @ A ) )
% 0.34/0.57 => ( ~ ( finite6017078050557962740nteger @ A )
% 0.34/0.57 => ? [X3: product_unit] :
% 0.34/0.57 ( ( member_Product_unit @ X3 @ ( image_7182656606676592809t_unit @ F @ A ) )
% 0.34/0.57 & ~ ( finite6017078050557962740nteger @ ( vimage5651688891684837235t_unit @ F @ ( insert_Product_unit @ X3 @ bot_bo3957492148770167129t_unit ) ) ) ) ) ) ).
% 0.34/0.57
% 0.34/0.57 % inf_img_fin_dom
% 0.34/0.57 thf(fact_2765_inf__img__fin__dom,axiom,
% 0.34/0.57 ! [F: nat > $o,A: set_nat] :
% 0.34/0.57 ( ( finite_finite_o @ ( image_nat_o2 @ F @ A ) )
% 0.34/0.57 => ( ~ ( finite_finite_nat @ A )
% 0.34/0.57 => ? [X3: $o] :
% 0.34/0.57 ( ( member_o @ X3 @ ( image_nat_o2 @ F @ A ) )
% 0.34/0.57 & ~ ( finite_finite_nat @ ( vimage_nat_o @ F @ ( insert_o @ X3 @ bot_bot_set_o ) ) ) ) ) ) ).
% 0.34/0.57
% 0.34/0.57 % inf_img_fin_dom
% 0.34/0.57 thf(fact_2766_inf__img__fin__dom,axiom,
% 0.34/0.57 ! [F: code_integer > $o,A: set_Code_integer] :
% 0.34/0.57 ( ( finite_finite_o @ ( image_Code_integer_o @ F @ A ) )
% 0.34/0.57 => ( ~ ( finite6017078050557962740nteger @ A )
% 0.34/0.57 => ? [X3: $o] :
% 0.34/0.57 ( ( member_o @ X3 @ ( image_Code_integer_o @ F @ A ) )
% 0.34/0.57 & ~ ( finite6017078050557962740nteger @ ( vimage3005800680117336574eger_o @ F @ ( insert_o @ X3 @ bot_bot_set_o ) ) ) ) ) ) ).
% 0.34/0.57
% 0.34/0.57 % inf_img_fin_dom
% 0.34/0.57 thf(fact_2767_inf__img__fin__dom,axiom,
% 0.34/0.57 ! [F: nat > nat,A: set_nat] :
% 0.34/0.57 ( ( finite_finite_nat @ ( image_nat_nat2 @ F @ A ) )
% 0.34/0.57 => ( ~ ( finite_finite_nat @ A )
% 0.34/0.57 => ? [X3: nat] :
% 0.34/0.57 ( ( member_nat @ X3 @ ( image_nat_nat2 @ F @ A ) )
% 0.34/0.57 & ~ ( finite_finite_nat @ ( vimage_nat_nat @ F @ ( insert_nat @ X3 @ bot_bot_set_nat ) ) ) ) ) ) ).
% 0.34/0.57
% 0.34/0.57 % inf_img_fin_dom
% 0.34/0.57 thf(fact_2768_inf__img__fin__dom,axiom,
% 0.34/0.57 ! [F: code_integer > nat,A: set_Code_integer] :
% 0.34/0.57 ( ( finite_finite_nat @ ( image_951025933927791156er_nat @ F @ A ) )
% 0.34/0.57 => ( ~ ( finite6017078050557962740nteger @ A )
% 0.34/0.57 => ? [X3: nat] :
% 0.34/0.57 ( ( member_nat @ X3 @ ( image_951025933927791156er_nat @ F @ A ) )
% 0.34/0.57 & ~ ( finite6017078050557962740nteger @ ( vimage5839839309545413354er_nat @ F @ ( insert_nat @ X3 @ bot_bot_set_nat ) ) ) ) ) ) ).
% 0.34/0.57
% 0.34/0.57 % inf_img_fin_dom
% 0.34/0.57 thf(fact_2769_inf__img__fin__dom,axiom,
% 0.34/0.57 ! [F: nat > int,A: set_nat] :
% 0.34/0.57 ( ( finite_finite_int @ ( image_nat_int2 @ F @ A ) )
% 0.34/0.57 => ( ~ ( finite_finite_nat @ A )
% 0.34/0.57 => ? [X3: int] :
% 0.34/0.57 ( ( member_int @ X3 @ ( image_nat_int2 @ F @ A ) )
% 0.34/0.57 & ~ ( finite_finite_nat @ ( vimage_nat_int @ F @ ( insert_int @ X3 @ bot_bot_set_int ) ) ) ) ) ) ).
% 0.34/0.57
% 0.34/0.57 % inf_img_fin_dom
% 0.34/0.57 thf(fact_2770_inf__img__fin__dom,axiom,
% 0.34/0.57 ! [F: code_integer > int,A: set_Code_integer] :
% 0.34/0.57 ( ( finite_finite_int @ ( image_948535463418740880er_int @ F @ A ) )
% 0.34/0.57 => ( ~ ( finite6017078050557962740nteger @ A )
% 0.34/0.57 => ? [X3: int] :
% 0.34/0.57 ( ( member_int @ X3 @ ( image_948535463418740880er_int @ F @ A ) )
% 0.34/0.58 & ~ ( finite6017078050557962740nteger @ ( vimage5837348839036363078er_int @ F @ ( insert_int @ X3 @ bot_bot_set_int ) ) ) ) ) ) ).
% 0.34/0.58
% 0.34/0.58 % inf_img_fin_dom
% 0.34/0.58 thf(fact_2771_acyclic__insert__cyclic,axiom,
% 0.34/0.58 ! [G: set_Pr1261947904930325089at_nat,X: nat,Y: nat] :
% 0.34/0.58 ( ( transi3139058337892597825ic_nat @ G )
% 0.34/0.58 => ( ~ ( transi3139058337892597825ic_nat @ ( insert8211810215607154385at_nat @ ( product_Pair_nat_nat @ X @ Y ) @ G ) )
% 0.34/0.58 => ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ Y @ X ) @ ( transi2905341329935302413cl_nat @ G ) ) ) ) ).
% 0.34/0.58
% 0.34/0.58 % acyclic_insert_cyclic
% 0.34/0.58 thf(fact_2772_acyclic__insert__cyclic,axiom,
% 0.34/0.58 ! [G: set_Pr958786334691620121nt_int,X: int,Y: int] :
% 0.34/0.58 ( ( transi3136567867383547549ic_int @ G )
% 0.34/0.58 => ( ~ ( transi3136567867383547549ic_int @ ( insert5033312907999012233nt_int @ ( product_Pair_int_int @ X @ Y ) @ G ) )
% 0.34/0.58 => ( member5262025264175285858nt_int @ ( product_Pair_int_int @ Y @ X ) @ ( transi2902850859426252137cl_int @ G ) ) ) ) ).
% 0.34/0.58
% 0.34/0.58 % acyclic_insert_cyclic
% 0.34/0.58 thf(fact_2773_Inf__fin_Ohom__commute,axiom,
% 0.34/0.58 ! [H2: set_nat > set_nat,N: set_set_nat] :
% 0.34/0.58 ( ! [X3: set_nat,Y2: set_nat] :
% 0.34/0.58 ( ( H2 @ ( inf_inf_set_nat @ X3 @ Y2 ) )
% 0.34/0.58 = ( inf_inf_set_nat @ ( H2 @ X3 ) @ ( H2 @ Y2 ) ) )
% 0.34/0.58 => ( ( finite1152437895449049373et_nat @ N )
% 0.34/0.58 => ( ( N != bot_bot_set_set_nat )
% 0.34/0.58 => ( ( H2 @ ( lattic3014633134055518761et_nat @ N ) )
% 0.34/0.58 = ( lattic3014633134055518761et_nat @ ( image_7916887816326733075et_nat @ H2 @ N ) ) ) ) ) ) ).
% 0.34/0.58
% 0.34/0.58 % Inf_fin.hom_commute
% 0.34/0.58 thf(fact_2774_Inf__fin_Ohom__commute,axiom,
% 0.34/0.58 ! [H2: assn > assn,N: set_assn] :
% 0.34/0.58 ( ! [X3: assn,Y2: assn] :
% 0.34/0.58 ( ( H2 @ ( inf_inf_assn @ X3 @ Y2 ) )
% 0.34/0.58 = ( inf_inf_assn @ ( H2 @ X3 ) @ ( H2 @ Y2 ) ) )
% 0.34/0.58 => ( ( finite_finite_assn @ N )
% 0.34/0.58 => ( ( N != bot_bot_set_assn )
% 0.34/0.58 => ( ( H2 @ ( lattic47131356835913163n_assn @ N ) )
% 0.34/0.58 = ( lattic47131356835913163n_assn @ ( image_assn_assn @ H2 @ N ) ) ) ) ) ) ).
% 0.34/0.58
% 0.34/0.58 % Inf_fin.hom_commute
% 0.34/0.58 thf(fact_2775_Inf__fin_Ohom__commute,axiom,
% 0.34/0.58 ! [H2: product_unit > product_unit,N: set_Product_unit] :
% 0.34/0.58 ( ! [X3: product_unit,Y2: product_unit] :
% 0.34/0.58 ( ( H2 @ ( inf_inf_Product_unit @ X3 @ Y2 ) )
% 0.34/0.58 = ( inf_inf_Product_unit @ ( H2 @ X3 ) @ ( H2 @ Y2 ) ) )
% 0.34/0.58 => ( ( finite4290736615968046902t_unit @ N )
% 0.34/0.58 => ( ( N != bot_bo3957492148770167129t_unit )
% 0.34/0.58 => ( ( H2 @ ( lattic1263872656861969706t_unit @ N ) )
% 0.34/0.58 = ( lattic1263872656861969706t_unit @ ( image_405062704495631173t_unit @ H2 @ N ) ) ) ) ) ) ).
% 0.34/0.58
% 0.34/0.58 % Inf_fin.hom_commute
% 0.34/0.58 thf(fact_2776_Inf__fin_Ohom__commute,axiom,
% 0.34/0.58 ! [H2: $o > $o,N: set_o] :
% 0.34/0.58 ( ! [X3: $o,Y2: $o] :
% 0.34/0.58 ( ( H2 @ ( inf_inf_o @ X3 @ Y2 ) )
% 0.34/0.58 = ( inf_inf_o @ ( H2 @ X3 ) @ ( H2 @ Y2 ) ) )
% 0.34/0.58 => ( ( finite_finite_o @ N )
% 0.34/0.58 => ( ( N != bot_bot_set_o )
% 0.34/0.58 => ( ( H2 @ ( lattic4107685809792843317_fin_o @ N ) )
% 0.34/0.58 = ( lattic4107685809792843317_fin_o @ ( image_o_o2 @ H2 @ N ) ) ) ) ) ) ).
% 0.34/0.58
% 0.34/0.58 % Inf_fin.hom_commute
% 0.34/0.58 thf(fact_2777_Inf__fin_Ohom__commute,axiom,
% 0.34/0.58 ! [H2: nat > nat,N: set_nat] :
% 0.34/0.58 ( ! [X3: nat,Y2: nat] :
% 0.34/0.58 ( ( H2 @ ( inf_inf_nat @ X3 @ Y2 ) )
% 0.34/0.58 = ( inf_inf_nat @ ( H2 @ X3 ) @ ( H2 @ Y2 ) ) )
% 0.34/0.58 => ( ( finite_finite_nat @ N )
% 0.34/0.58 => ( ( N != bot_bot_set_nat )
% 0.34/0.58 => ( ( H2 @ ( lattic5238388535129920115in_nat @ N ) )
% 0.34/0.58 = ( lattic5238388535129920115in_nat @ ( image_nat_nat2 @ H2 @ N ) ) ) ) ) ) ).
% 0.34/0.58
% 0.34/0.58 % Inf_fin.hom_commute
% 0.34/0.58 thf(fact_2778_Inf__fin_Ohom__commute,axiom,
% 0.34/0.58 ! [H2: int > int,N: set_int] :
% 0.34/0.58 ( ! [X3: int,Y2: int] :
% 0.34/0.58 ( ( H2 @ ( inf_inf_int @ X3 @ Y2 ) )
% 0.34/0.58 = ( inf_inf_int @ ( H2 @ X3 ) @ ( H2 @ Y2 ) ) )
% 0.34/0.58 => ( ( finite_finite_int @ N )
% 0.34/0.58 => ( ( N != bot_bot_set_int )
% 0.34/0.58 => ( ( H2 @ ( lattic5235898064620869839in_int @ N ) )
% 0.34/0.58 = ( lattic5235898064620869839in_int @ ( image_int_int2 @ H2 @ N ) ) ) ) ) ) ).
% 0.34/0.58
% 0.34/0.58 % Inf_fin.hom_commute
% 0.34/0.58 thf(fact_2779_Inf__fin_Ohom__commute,axiom,
% 0.34/0.58 ! [H2: ( produc3658429121746597890et_nat > $o ) > produc3658429121746597890et_nat > $o,N: set_Pr4532377907799695533_nat_o] :
% 0.34/0.58 ( ! [X3: produc3658429121746597890et_nat > $o,Y2: produc3658429121746597890et_nat > $o] :
% 0.34/0.58 ( ( H2 @ ( inf_in1318976480646536635_nat_o @ X3 @ Y2 ) )
% 0.34/0.58 = ( inf_in1318976480646536635_nat_o @ ( H2 @ X3 ) @ ( H2 @ Y2 ) ) )
% 0.34/0.58 => ( ( finite3252695134891459830_nat_o @ N )
% 0.34/0.58 => ( ( N != bot_bo7824918357723582233_nat_o )
% 0.34/0.58 => ( ( H2 @ ( lattic956194824204696298_nat_o @ N ) )
% 0.34/0.58 = ( lattic956194824204696298_nat_o @ ( image_3399141053782081349_nat_o @ H2 @ N ) ) ) ) ) ) ).
% 0.34/0.58
% 0.34/0.58 % Inf_fin.hom_commute
% 0.34/0.58 thf(fact_2780_Sup__fin_Ohom__commute,axiom,
% 0.34/0.58 ! [H2: set_nat > set_nat,N: set_set_nat] :
% 0.34/0.58 ( ! [X3: set_nat,Y2: set_nat] :
% 0.34/0.58 ( ( H2 @ ( sup_sup_set_nat @ X3 @ Y2 ) )
% 0.34/0.58 = ( sup_sup_set_nat @ ( H2 @ X3 ) @ ( H2 @ Y2 ) ) )
% 0.34/0.58 => ( ( finite1152437895449049373et_nat @ N )
% 0.34/0.58 => ( ( N != bot_bot_set_set_nat )
% 0.34/0.58 => ( ( H2 @ ( lattic3835124923745554447et_nat @ N ) )
% 0.34/0.58 = ( lattic3835124923745554447et_nat @ ( image_7916887816326733075et_nat @ H2 @ N ) ) ) ) ) ) ).
% 0.34/0.58
% 0.34/0.58 % Sup_fin.hom_commute
% 0.34/0.58 thf(fact_2781_Sup__fin_Ohom__commute,axiom,
% 0.34/0.58 ! [H2: assn > assn,N: set_assn] :
% 0.34/0.58 ( ! [X3: assn,Y2: assn] :
% 0.34/0.58 ( ( H2 @ ( sup_sup_assn @ X3 @ Y2 ) )
% 0.34/0.58 = ( sup_sup_assn @ ( H2 @ X3 ) @ ( H2 @ Y2 ) ) )
% 0.34/0.58 => ( ( finite_finite_assn @ N )
% 0.34/0.58 => ( ( N != bot_bot_set_assn )
% 0.34/0.58 => ( ( H2 @ ( lattic2150320897289308081n_assn @ N ) )
% 0.34/0.58 = ( lattic2150320897289308081n_assn @ ( image_assn_assn @ H2 @ N ) ) ) ) ) ) ).
% 0.34/0.58
% 0.34/0.58 % Sup_fin.hom_commute
% 0.34/0.58 thf(fact_2782_Sup__fin_Ohom__commute,axiom,
% 0.34/0.58 ! [H2: product_unit > product_unit,N: set_Product_unit] :
% 0.34/0.58 ( ! [X3: product_unit,Y2: product_unit] :
% 0.34/0.58 ( ( H2 @ ( sup_sup_Product_unit @ X3 @ Y2 ) )
% 0.34/0.58 = ( sup_sup_Product_unit @ ( H2 @ X3 ) @ ( H2 @ Y2 ) ) )
% 0.34/0.58 => ( ( finite4290736615968046902t_unit @ N )
% 0.34/0.58 => ( ( N != bot_bo3957492148770167129t_unit )
% 0.34/0.58 => ( ( H2 @ ( lattic5294303975357428420t_unit @ N ) )
% 0.34/0.58 = ( lattic5294303975357428420t_unit @ ( image_405062704495631173t_unit @ H2 @ N ) ) ) ) ) ) ).
% 0.34/0.58
% 0.34/0.58 % Sup_fin.hom_commute
% 0.34/0.58 thf(fact_2783_Sup__fin_Ohom__commute,axiom,
% 0.34/0.58 ! [H2: $o > $o,N: set_o] :
% 0.34/0.58 ( ! [X3: $o,Y2: $o] :
% 0.34/0.58 ( ( H2 @ ( sup_sup_o @ X3 @ Y2 ) )
% 0.34/0.58 = ( sup_sup_o @ ( H2 @ X3 ) @ ( H2 @ Y2 ) ) )
% 0.34/0.58 => ( ( finite_finite_o @ N )
% 0.34/0.58 => ( ( N != bot_bot_set_o )
% 0.34/0.58 => ( ( H2 @ ( lattic1508158080041050831_fin_o @ N ) )
% 0.34/0.58 = ( lattic1508158080041050831_fin_o @ ( image_o_o2 @ H2 @ N ) ) ) ) ) ) ).
% 0.34/0.58
% 0.34/0.58 % Sup_fin.hom_commute
% 0.34/0.58 thf(fact_2784_Sup__fin_Ohom__commute,axiom,
% 0.34/0.58 ! [H2: nat > nat,N: set_nat] :
% 0.34/0.58 ( ! [X3: nat,Y2: nat] :
% 0.34/0.58 ( ( H2 @ ( sup_sup_nat @ X3 @ Y2 ) )
% 0.34/0.58 = ( sup_sup_nat @ ( H2 @ X3 ) @ ( H2 @ Y2 ) ) )
% 0.34/0.58 => ( ( finite_finite_nat @ N )
% 0.34/0.58 => ( ( N != bot_bot_set_nat )
% 0.34/0.58 => ( ( H2 @ ( lattic1093996805478795353in_nat @ N ) )
% 0.34/0.58 = ( lattic1093996805478795353in_nat @ ( image_nat_nat2 @ H2 @ N ) ) ) ) ) ) ).
% 0.34/0.58
% 0.34/0.58 % Sup_fin.hom_commute
% 0.34/0.58 thf(fact_2785_Sup__fin_Ohom__commute,axiom,
% 0.34/0.58 ! [H2: int > int,N: set_int] :
% 0.34/0.58 ( ! [X3: int,Y2: int] :
% 0.34/0.58 ( ( H2 @ ( sup_sup_int @ X3 @ Y2 ) )
% 0.34/0.58 = ( sup_sup_int @ ( H2 @ X3 ) @ ( H2 @ Y2 ) ) )
% 0.34/0.58 => ( ( finite_finite_int @ N )
% 0.34/0.58 => ( ( N != bot_bot_set_int )
% 0.34/0.58 => ( ( H2 @ ( lattic1091506334969745077in_int @ N ) )
% 0.34/0.58 = ( lattic1091506334969745077in_int @ ( image_int_int2 @ H2 @ N ) ) ) ) ) ) ).
% 0.34/0.58
% 0.34/0.58 % Sup_fin.hom_commute
% 0.34/0.58 thf(fact_2786_Sup__fin_Ohom__commute,axiom,
% 0.34/0.58 ! [H2: ( produc3658429121746597890et_nat > $o ) > produc3658429121746597890et_nat > $o,N: set_Pr4532377907799695533_nat_o] :
% 0.34/0.58 ( ! [X3: produc3658429121746597890et_nat > $o,Y2: produc3658429121746597890et_nat > $o] :
% 0.34/0.58 ( ( H2 @ ( sup_su5453871518329203617_nat_o @ X3 @ Y2 ) )
% 0.34/0.58 = ( sup_su5453871518329203617_nat_o @ ( H2 @ X3 ) @ ( H2 @ Y2 ) ) )
% 0.34/0.58 => ( ( finite3252695134891459830_nat_o @ N )
% 0.34/0.58 => ( ( N != bot_bo7824918357723582233_nat_o )
% 0.34/0.58 => ( ( H2 @ ( lattic7320199455484906628_nat_o @ N ) )
% 0.34/0.58 = ( lattic7320199455484906628_nat_o @ ( image_3399141053782081349_nat_o @ H2 @ N ) ) ) ) ) ) ).
% 0.34/0.58
% 0.34/0.58 % Sup_fin.hom_commute
% 0.34/0.58 thf(fact_2787_finite__remove__induct,axiom,
% 0.34/0.58 ! [B4: set_Code_integer,P: set_Code_integer > $o] :
% 0.34/0.58 ( ( finite6017078050557962740nteger @ B4 )
% 0.34/0.58 => ( ( P @ bot_bo3990330152332043303nteger )
% 0.34/0.58 => ( ! [A9: set_Code_integer] :
% 0.34/0.58 ( ( finite6017078050557962740nteger @ A9 )
% 0.34/0.58 => ( ( A9 != bot_bo3990330152332043303nteger )
% 0.34/0.58 => ( ( ord_le7084787975880047091nteger @ A9 @ B4 )
% 0.34/0.58 => ( ! [X6: code_integer] :
% 0.34/0.58 ( ( member_Code_integer @ X6 @ A9 )
% 0.34/0.58 => ( P @ ( minus_2355218937544613996nteger @ A9 @ ( insert_Code_integer @ X6 @ bot_bo3990330152332043303nteger ) ) ) )
% 0.34/0.58 => ( P @ A9 ) ) ) ) )
% 0.34/0.58 => ( P @ B4 ) ) ) ) ).
% 0.34/0.58
% 0.34/0.58 % finite_remove_induct
% 0.34/0.58 thf(fact_2788_finite__remove__induct,axiom,
% 0.34/0.58 ! [B4: set_Product_unit,P: set_Product_unit > $o] :
% 0.34/0.58 ( ( finite4290736615968046902t_unit @ B4 )
% 0.34/0.58 => ( ( P @ bot_bo3957492148770167129t_unit )
% 0.34/0.58 => ( ! [A9: set_Product_unit] :
% 0.34/0.58 ( ( finite4290736615968046902t_unit @ A9 )
% 0.34/0.58 => ( ( A9 != bot_bo3957492148770167129t_unit )
% 0.34/0.58 => ( ( ord_le3507040750410214029t_unit @ A9 @ B4 )
% 0.34/0.58 => ( ! [X6: product_unit] :
% 0.34/0.58 ( ( member_Product_unit @ X6 @ A9 )
% 0.34/0.58 => ( P @ ( minus_6452836326544984404t_unit @ A9 @ ( insert_Product_unit @ X6 @ bot_bo3957492148770167129t_unit ) ) ) )
% 0.34/0.58 => ( P @ A9 ) ) ) ) )
% 0.34/0.58 => ( P @ B4 ) ) ) ) ).
% 0.34/0.58
% 0.34/0.58 % finite_remove_induct
% 0.34/0.58 thf(fact_2789_finite__remove__induct,axiom,
% 0.34/0.58 ! [B4: set_o,P: set_o > $o] :
% 0.34/0.58 ( ( finite_finite_o @ B4 )
% 0.34/0.58 => ( ( P @ bot_bot_set_o )
% 0.34/0.58 => ( ! [A9: set_o] :
% 0.34/0.58 ( ( finite_finite_o @ A9 )
% 0.34/0.58 => ( ( A9 != bot_bot_set_o )
% 0.34/0.58 => ( ( ord_less_eq_set_o @ A9 @ B4 )
% 0.34/0.58 => ( ! [X6: $o] :
% 0.34/0.58 ( ( member_o @ X6 @ A9 )
% 0.34/0.58 => ( P @ ( minus_minus_set_o @ A9 @ ( insert_o @ X6 @ bot_bot_set_o ) ) ) )
% 0.34/0.58 => ( P @ A9 ) ) ) ) )
% 0.34/0.58 => ( P @ B4 ) ) ) ) ).
% 0.34/0.58
% 0.34/0.58 % finite_remove_induct
% 0.34/0.58 thf(fact_2790_finite__remove__induct,axiom,
% 0.34/0.58 ! [B4: set_int,P: set_int > $o] :
% 0.34/0.58 ( ( finite_finite_int @ B4 )
% 0.34/0.58 => ( ( P @ bot_bot_set_int )
% 0.34/0.58 => ( ! [A9: set_int] :
% 0.34/0.58 ( ( finite_finite_int @ A9 )
% 0.34/0.58 => ( ( A9 != bot_bot_set_int )
% 0.34/0.58 => ( ( ord_less_eq_set_int @ A9 @ B4 )
% 0.34/0.58 => ( ! [X6: int] :
% 0.34/0.58 ( ( member_int @ X6 @ A9 )
% 0.34/0.58 => ( P @ ( minus_minus_set_int @ A9 @ ( insert_int @ X6 @ bot_bot_set_int ) ) ) )
% 0.34/0.58 => ( P @ A9 ) ) ) ) )
% 0.34/0.58 => ( P @ B4 ) ) ) ) ).
% 0.34/0.58
% 0.34/0.58 % finite_remove_induct
% 0.34/0.58 thf(fact_2791_finite__remove__induct,axiom,
% 0.34/0.58 ! [B4: set_Pr4532377907799695533_nat_o,P: set_Pr4532377907799695533_nat_o > $o] :
% 0.34/0.58 ( ( finite3252695134891459830_nat_o @ B4 )
% 0.34/0.58 => ( ( P @ bot_bo7824918357723582233_nat_o )
% 0.34/0.58 => ( ! [A9: set_Pr4532377907799695533_nat_o] :
% 0.34/0.58 ( ( finite3252695134891459830_nat_o @ A9 )
% 0.34/0.58 => ( ( A9 != bot_bo7824918357723582233_nat_o )
% 0.34/0.58 => ( ( ord_le2965882846123202637_nat_o @ A9 @ B4 )
% 0.34/0.58 => ( ! [X6: produc3658429121746597890et_nat > $o] :
% 0.34/0.58 ( ( member6576561426505652726_nat_o @ X6 @ A9 )
% 0.34/0.58 => ( P @ ( minus_1801376950450012436_nat_o @ A9 @ ( insert5175938949040314269_nat_o @ X6 @ bot_bo7824918357723582233_nat_o ) ) ) )
% 0.34/0.58 => ( P @ A9 ) ) ) ) )
% 0.34/0.58 => ( P @ B4 ) ) ) ) ).
% 0.34/0.58
% 0.34/0.58 % finite_remove_induct
% 0.34/0.58 thf(fact_2792_finite__remove__induct,axiom,
% 0.34/0.58 ! [B4: set_nat,P: set_nat > $o] :
% 0.34/0.58 ( ( finite_finite_nat @ B4 )
% 0.34/0.58 => ( ( P @ bot_bot_set_nat )
% 0.34/0.58 => ( ! [A9: set_nat] :
% 0.34/0.58 ( ( finite_finite_nat @ A9 )
% 0.34/0.58 => ( ( A9 != bot_bot_set_nat )
% 0.34/0.58 => ( ( ord_less_eq_set_nat @ A9 @ B4 )
% 0.34/0.58 => ( ! [X6: nat] :
% 0.34/0.58 ( ( member_nat @ X6 @ A9 )
% 0.34/0.58 => ( P @ ( minus_minus_set_nat @ A9 @ ( insert_nat @ X6 @ bot_bot_set_nat ) ) ) )
% 0.34/0.58 => ( P @ A9 ) ) ) ) )
% 0.34/0.58 => ( P @ B4 ) ) ) ) ).
% 0.34/0.58
% 0.34/0.58 % finite_remove_induct
% 0.34/0.58 thf(fact_2793_remove__induct,axiom,
% 0.34/0.58 ! [P: set_Code_integer > $o,B4: set_Code_integer] :
% 0.34/0.58 ( ( P @ bot_bo3990330152332043303nteger )
% 0.34/0.58 => ( ( ~ ( finite6017078050557962740nteger @ B4 )
% 0.34/0.58 => ( P @ B4 ) )
% 0.34/0.58 => ( ! [A9: set_Code_integer] :
% 0.34/0.58 ( ( finite6017078050557962740nteger @ A9 )
% 0.34/0.58 => ( ( A9 != bot_bo3990330152332043303nteger )
% 0.34/0.58 => ( ( ord_le7084787975880047091nteger @ A9 @ B4 )
% 0.34/0.58 => ( ! [X6: code_integer] :
% 0.34/0.58 ( ( member_Code_integer @ X6 @ A9 )
% 0.34/0.58 => ( P @ ( minus_2355218937544613996nteger @ A9 @ ( insert_Code_integer @ X6 @ bot_bo3990330152332043303nteger ) ) ) )
% 0.34/0.58 => ( P @ A9 ) ) ) ) )
% 0.34/0.58 => ( P @ B4 ) ) ) ) ).
% 0.34/0.58
% 0.34/0.58 % remove_induct
% 0.34/0.58 thf(fact_2794_remove__induct,axiom,
% 0.34/0.58 ! [P: set_Product_unit > $o,B4: set_Product_unit] :
% 0.34/0.58 ( ( P @ bot_bo3957492148770167129t_unit )
% 0.34/0.58 => ( ( ~ ( finite4290736615968046902t_unit @ B4 )
% 0.34/0.58 => ( P @ B4 ) )
% 0.34/0.58 => ( ! [A9: set_Product_unit] :
% 0.34/0.58 ( ( finite4290736615968046902t_unit @ A9 )
% 0.34/0.58 => ( ( A9 != bot_bo3957492148770167129t_unit )
% 0.34/0.58 => ( ( ord_le3507040750410214029t_unit @ A9 @ B4 )
% 0.34/0.58 => ( ! [X6: product_unit] :
% 0.34/0.58 ( ( member_Product_unit @ X6 @ A9 )
% 0.34/0.58 => ( P @ ( minus_6452836326544984404t_unit @ A9 @ ( insert_Product_unit @ X6 @ bot_bo3957492148770167129t_unit ) ) ) )
% 0.34/0.58 => ( P @ A9 ) ) ) ) )
% 0.34/0.58 => ( P @ B4 ) ) ) ) ).
% 0.34/0.58
% 0.34/0.58 % remove_induct
% 0.34/0.58 thf(fact_2795_remove__induct,axiom,
% 0.34/0.58 ! [P: set_o > $o,B4: set_o] :
% 0.34/0.58 ( ( P @ bot_bot_set_o )
% 0.34/0.58 => ( ( ~ ( finite_finite_o @ B4 )
% 0.34/0.58 => ( P @ B4 ) )
% 0.34/0.58 => ( ! [A9: set_o] :
% 0.34/0.58 ( ( finite_finite_o @ A9 )
% 0.34/0.58 => ( ( A9 != bot_bot_set_o )
% 0.34/0.58 => ( ( ord_less_eq_set_o @ A9 @ B4 )
% 0.34/0.58 => ( ! [X6: $o] :
% 0.34/0.58 ( ( member_o @ X6 @ A9 )
% 0.34/0.58 => ( P @ ( minus_minus_set_o @ A9 @ ( insert_o @ X6 @ bot_bot_set_o ) ) ) )
% 0.34/0.58 => ( P @ A9 ) ) ) ) )
% 0.34/0.58 => ( P @ B4 ) ) ) ) ).
% 0.34/0.58
% 0.34/0.58 % remove_induct
% 0.34/0.58 thf(fact_2796_remove__induct,axiom,
% 0.34/0.58 ! [P: set_int > $o,B4: set_int] :
% 0.34/0.58 ( ( P @ bot_bot_set_int )
% 0.34/0.58 => ( ( ~ ( finite_finite_int @ B4 )
% 0.34/0.58 => ( P @ B4 ) )
% 0.34/0.58 => ( ! [A9: set_int] :
% 0.34/0.58 ( ( finite_finite_int @ A9 )
% 0.34/0.58 => ( ( A9 != bot_bot_set_int )
% 0.34/0.58 => ( ( ord_less_eq_set_int @ A9 @ B4 )
% 0.34/0.58 => ( ! [X6: int] :
% 0.34/0.58 ( ( member_int @ X6 @ A9 )
% 0.34/0.58 => ( P @ ( minus_minus_set_int @ A9 @ ( insert_int @ X6 @ bot_bot_set_int ) ) ) )
% 0.34/0.58 => ( P @ A9 ) ) ) ) )
% 0.34/0.58 => ( P @ B4 ) ) ) ) ).
% 0.34/0.58
% 0.34/0.58 % remove_induct
% 0.34/0.58 thf(fact_2797_remove__induct,axiom,
% 0.34/0.58 ! [P: set_Pr4532377907799695533_nat_o > $o,B4: set_Pr4532377907799695533_nat_o] :
% 0.34/0.58 ( ( P @ bot_bo7824918357723582233_nat_o )
% 0.34/0.58 => ( ( ~ ( finite3252695134891459830_nat_o @ B4 )
% 0.34/0.58 => ( P @ B4 ) )
% 0.34/0.58 => ( ! [A9: set_Pr4532377907799695533_nat_o] :
% 0.34/0.58 ( ( finite3252695134891459830_nat_o @ A9 )
% 0.34/0.58 => ( ( A9 != bot_bo7824918357723582233_nat_o )
% 0.34/0.58 => ( ( ord_le2965882846123202637_nat_o @ A9 @ B4 )
% 0.34/0.58 => ( ! [X6: produc3658429121746597890et_nat > $o] :
% 0.34/0.58 ( ( member6576561426505652726_nat_o @ X6 @ A9 )
% 0.34/0.58 => ( P @ ( minus_1801376950450012436_nat_o @ A9 @ ( insert5175938949040314269_nat_o @ X6 @ bot_bo7824918357723582233_nat_o ) ) ) )
% 0.34/0.58 => ( P @ A9 ) ) ) ) )
% 0.34/0.58 => ( P @ B4 ) ) ) ) ).
% 0.34/0.58
% 0.34/0.58 % remove_induct
% 0.34/0.58 thf(fact_2798_remove__induct,axiom,
% 0.34/0.58 ! [P: set_nat > $o,B4: set_nat] :
% 0.34/0.58 ( ( P @ bot_bot_set_nat )
% 0.34/0.58 => ( ( ~ ( finite_finite_nat @ B4 )
% 0.34/0.58 => ( P @ B4 ) )
% 0.34/0.58 => ( ! [A9: set_nat] :
% 0.34/0.58 ( ( finite_finite_nat @ A9 )
% 0.34/0.58 => ( ( A9 != bot_bot_set_nat )
% 0.34/0.58 => ( ( ord_less_eq_set_nat @ A9 @ B4 )
% 0.34/0.58 => ( ! [X6: nat] :
% 0.34/0.58 ( ( member_nat @ X6 @ A9 )
% 0.34/0.58 => ( P @ ( minus_minus_set_nat @ A9 @ ( insert_nat @ X6 @ bot_bot_set_nat ) ) ) )
% 0.34/0.58 => ( P @ A9 ) ) ) ) )
% 0.34/0.58 => ( P @ B4 ) ) ) ) ).
% 0.34/0.58
% 0.34/0.58 % remove_induct
% 0.34/0.58 thf(fact_2799_Linear__order__in__diff__Id,axiom,
% 0.34/0.58 ! [R2: set_Pr2161125870931222855_nat_o,A2: produc3658429121746597890et_nat > $o,B: produc3658429121746597890et_nat > $o] :
% 0.34/0.58 ( ( order_1145710701291701250_nat_o @ ( field_81821745747897882_nat_o @ R2 ) @ R2 )
% 0.34/0.58 => ( ( member6576561426505652726_nat_o @ A2 @ ( field_81821745747897882_nat_o @ R2 ) )
% 0.34/0.58 => ( ( member6576561426505652726_nat_o @ B @ ( field_81821745747897882_nat_o @ R2 ) )
% 0.34/0.58 => ( ( member8781333585448626064_nat_o @ ( produc7368190662567826135_nat_o @ A2 @ B ) @ R2 )
% 0.34/0.58 = ( ~ ( member8781333585448626064_nat_o @ ( produc7368190662567826135_nat_o @ B @ A2 ) @ ( minus_9189903262233110958_nat_o @ R2 @ id_Pro6775173067513470761_nat_o ) ) ) ) ) ) ) ).
% 0.34/0.58
% 0.34/0.58 % Linear_order_in_diff_Id
% 0.34/0.58 thf(fact_2800_Linear__order__in__diff__Id,axiom,
% 0.34/0.58 ! [R2: set_Product_prod_o_o,A2: $o,B: $o] :
% 0.34/0.58 ( ( order_6238756238976269133r_on_o @ ( field_o @ R2 ) @ R2 )
% 0.34/0.58 => ( ( member_o @ A2 @ ( field_o @ R2 ) )
% 0.34/0.58 => ( ( member_o @ B @ ( field_o @ R2 ) )
% 0.34/0.58 => ( ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ A2 @ B ) @ R2 )
% 0.34/0.58 = ( ~ ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ B @ A2 ) @ ( minus_7052513330764642158od_o_o @ R2 @ id_o2 ) ) ) ) ) ) ) ).
% 0.34/0.58
% 0.34/0.58 % Linear_order_in_diff_Id
% 0.34/0.58 thf(fact_2801_Linear__order__in__diff__Id,axiom,
% 0.34/0.58 ! [R2: set_Pr5094982260447487303t_unit,A2: product_unit,B: product_unit] :
% 0.34/0.58 ( ( order_888282469287196226t_unit @ ( field_Product_unit @ R2 ) @ R2 )
% 0.34/0.58 => ( ( member_Product_unit @ A2 @ ( field_Product_unit @ R2 ) )
% 0.34/0.58 => ( ( member_Product_unit @ B @ ( field_Product_unit @ R2 ) )
% 0.34/0.58 => ( ( member7821138191923116944t_unit @ ( produc5692694829384537815t_unit @ A2 @ B ) @ R2 )
% 0.34/0.58 = ( ~ ( member7821138191923116944t_unit @ ( produc5692694829384537815t_unit @ B @ A2 ) @ ( minus_6985556733573110190t_unit @ R2 @ id_Product_unit ) ) ) ) ) ) ) ).
% 0.34/0.58
% 0.34/0.58 % Linear_order_in_diff_Id
% 0.34/0.58 thf(fact_2802_Linear__order__in__diff__Id,axiom,
% 0.34/0.58 ! [R2: set_Pr1261947904930325089at_nat,A2: nat,B: nat] :
% 0.34/0.58 ( ( order_4473980167227706203on_nat @ ( field_nat @ R2 ) @ R2 )
% 0.34/0.58 => ( ( member_nat @ A2 @ ( field_nat @ R2 ) )
% 0.34/0.58 => ( ( member_nat @ B @ ( field_nat @ R2 ) )
% 0.34/0.58 => ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A2 @ B ) @ R2 )
% 0.34/0.58 = ( ~ ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ B @ A2 ) @ ( minus_1356011639430497352at_nat @ R2 @ id_nat ) ) ) ) ) ) ) ).
% 0.34/0.58
% 0.34/0.58 % Linear_order_in_diff_Id
% 0.34/0.58 thf(fact_2803_Linear__order__in__diff__Id,axiom,
% 0.34/0.58 ! [R2: set_Pr958786334691620121nt_int,A2: int,B: int] :
% 0.34/0.58 ( ( order_4471489696718655927on_int @ ( field_int @ R2 ) @ R2 )
% 0.34/0.58 => ( ( member_int @ A2 @ ( field_int @ R2 ) )
% 0.34/0.58 => ( ( member_int @ B @ ( field_int @ R2 ) )
% 0.34/0.58 => ( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ A2 @ B ) @ R2 )
% 0.34/0.58 = ( ~ ( member5262025264175285858nt_int @ ( product_Pair_int_int @ B @ A2 ) @ ( minus_1052850069191792384nt_int @ R2 @ id_int ) ) ) ) ) ) ) ).
% 0.34/0.58
% 0.34/0.58 % Linear_order_in_diff_Id
% 0.34/0.58 thf(fact_2804_max__ext_Omax__extI,axiom,
% 0.34/0.58 ! [X4: set_Code_integer,Y6: set_Code_integer,R3: set_Pr4811707699266497531nteger] :
% 0.34/0.58 ( ( finite6017078050557962740nteger @ X4 )
% 0.34/0.58 => ( ( finite6017078050557962740nteger @ Y6 )
% 0.34/0.58 => ( ( Y6 != bot_bo3990330152332043303nteger )
% 0.34/0.58 => ( ! [X3: code_integer] :
% 0.34/0.58 ( ( member_Code_integer @ X3 @ X4 )
% 0.34/0.58 => ? [Xa2: code_integer] :
% 0.34/0.58 ( ( member_Code_integer @ Xa2 @ Y6 )
% 0.34/0.58 & ( member157494554546826820nteger @ ( produc1086072967326762835nteger @ X3 @ Xa2 ) @ R3 ) ) )
% 0.34/0.58 => ( member4307123515891402160nteger @ ( produc7443773368509356479nteger @ X4 @ Y6 ) @ ( max_ext_Code_integer @ R3 ) ) ) ) ) ) ).
% 0.34/0.58
% 0.34/0.58 % max_ext.max_extI
% 0.34/0.58 thf(fact_2805_max__ext_Omax__extI,axiom,
% 0.34/0.58 ! [X4: set_Product_unit,Y6: set_Product_unit,R3: set_Pr5094982260447487303t_unit] :
% 0.34/0.58 ( ( finite4290736615968046902t_unit @ X4 )
% 0.34/0.58 => ( ( finite4290736615968046902t_unit @ Y6 )
% 0.34/0.58 => ( ( Y6 != bot_bo3957492148770167129t_unit )
% 0.34/0.58 => ( ! [X3: product_unit] :
% 0.34/0.58 ( ( member_Product_unit @ X3 @ X4 )
% 0.34/0.58 => ? [Xa2: product_unit] :
% 0.34/0.58 ( ( member_Product_unit @ Xa2 @ Y6 )
% 0.34/0.58 & ( member7821138191923116944t_unit @ ( produc5692694829384537815t_unit @ X3 @ Xa2 ) @ R3 ) ) )
% 0.34/0.58 => ( member4189953012975926160t_unit @ ( produc7269801343787132631t_unit @ X4 @ Y6 ) @ ( max_ext_Product_unit @ R3 ) ) ) ) ) ) ).
% 0.34/0.58
% 0.34/0.58 % max_ext.max_extI
% 0.34/0.58 thf(fact_2806_max__ext_Omax__extI,axiom,
% 0.34/0.58 ! [X4: set_o,Y6: set_o,R3: set_Product_prod_o_o] :
% 0.34/0.58 ( ( finite_finite_o @ X4 )
% 0.34/0.58 => ( ( finite_finite_o @ Y6 )
% 0.34/0.58 => ( ( Y6 != bot_bot_set_o )
% 0.34/0.58 => ( ! [X3: $o] :
% 0.34/0.58 ( ( member_o @ X3 @ X4 )
% 0.34/0.58 => ? [Xa2: $o] :
% 0.34/0.58 ( ( member_o @ Xa2 @ Y6 )
% 0.34/0.58 & ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ X3 @ Xa2 ) @ R3 ) ) )
% 0.34/0.58 => ( member9116954335612470352_set_o @ ( produc5838405689764958487_set_o @ X4 @ Y6 ) @ ( max_ext_o @ R3 ) ) ) ) ) ) ).
% 0.34/0.58
% 0.34/0.58 % max_ext.max_extI
% 0.34/0.58 thf(fact_2807_max__ext_Omax__extI,axiom,
% 0.34/0.58 ! [X4: set_nat,Y6: set_nat,R3: set_Pr1261947904930325089at_nat] :
% 0.34/0.58 ( ( finite_finite_nat @ X4 )
% 0.34/0.58 => ( ( finite_finite_nat @ Y6 )
% 0.34/0.58 => ( ( Y6 != bot_bot_set_nat )
% 0.34/0.58 => ( ! [X3: nat] :
% 0.34/0.58 ( ( member_nat @ X3 @ X4 )
% 0.34/0.58 => ? [Xa2: nat] :
% 0.34/0.58 ( ( member_nat @ Xa2 @ Y6 )
% 0.34/0.58 & ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X3 @ Xa2 ) @ R3 ) ) )
% 0.34/0.58 => ( member8277197624267554838et_nat @ ( produc4532415448927165861et_nat @ X4 @ Y6 ) @ ( max_ext_nat @ R3 ) ) ) ) ) ) ).
% 0.34/0.58
% 0.34/0.58 % max_ext.max_extI
% 0.34/0.58 thf(fact_2808_max__ext_Omax__extI,axiom,
% 0.34/0.58 ! [X4: set_int,Y6: set_int,R3: set_Pr958786334691620121nt_int] :
% 0.34/0.58 ( ( finite_finite_int @ X4 )
% 0.34/0.58 => ( ( finite_finite_int @ Y6 )
% 0.34/0.58 => ( ( Y6 != bot_bot_set_int )
% 0.34/0.58 => ( ! [X3: int] :
% 0.34/0.58 ( ( member_int @ X3 @ X4 )
% 0.34/0.58 => ? [Xa2: int] :
% 0.34/0.58 ( ( member_int @ Xa2 @ Y6 )
% 0.34/0.58 & ( member5262025264175285858nt_int @ ( product_Pair_int_int @ X3 @ Xa2 ) @ R3 ) ) )
% 0.34/0.58 => ( member2572552093476627150et_int @ ( produc6363374080413544029et_int @ X4 @ Y6 ) @ ( max_ext_int @ R3 ) ) ) ) ) ) ).
% 0.34/0.58
% 0.34/0.58 % max_ext.max_extI
% 0.34/0.58 thf(fact_2809_max__ext_Omax__extI,axiom,
% 0.34/0.58 ! [X4: set_Pr4532377907799695533_nat_o,Y6: set_Pr4532377907799695533_nat_o,R3: set_Pr2161125870931222855_nat_o] :
% 0.34/0.58 ( ( finite3252695134891459830_nat_o @ X4 )
% 0.34/0.58 => ( ( finite3252695134891459830_nat_o @ Y6 )
% 0.34/0.58 => ( ( Y6 != bot_bo7824918357723582233_nat_o )
% 0.34/0.58 => ( ! [X3: produc3658429121746597890et_nat > $o] :
% 0.34/0.58 ( ( member6576561426505652726_nat_o @ X3 @ X4 )
% 0.34/0.58 => ? [Xa2: produc3658429121746597890et_nat > $o] :
% 0.34/0.58 ( ( member6576561426505652726_nat_o @ Xa2 @ Y6 )
% 0.34/0.58 & ( member8781333585448626064_nat_o @ ( produc7368190662567826135_nat_o @ X3 @ Xa2 ) @ R3 ) ) )
% 0.34/0.58 => ( member6623172407746373520_nat_o @ ( produc5149316574908490455_nat_o @ X4 @ Y6 ) @ ( max_ex5265246208886859912_nat_o @ R3 ) ) ) ) ) ) ).
% 0.34/0.58
% 0.34/0.58 % max_ext.max_extI
% 0.34/0.58 thf(fact_2810_max__ext_Osimps,axiom,
% 0.34/0.58 ! [A1: set_Code_integer,A22: set_Code_integer,R3: set_Pr4811707699266497531nteger] :
% 0.34/0.58 ( ( member4307123515891402160nteger @ ( produc7443773368509356479nteger @ A1 @ A22 ) @ ( max_ext_Code_integer @ R3 ) )
% 0.34/0.58 = ( ( finite6017078050557962740nteger @ A1 )
% 0.34/0.58 & ( finite6017078050557962740nteger @ A22 )
% 0.34/0.58 & ( A22 != bot_bo3990330152332043303nteger )
% 0.34/0.58 & ! [X2: code_integer] :
% 0.34/0.58 ( ( member_Code_integer @ X2 @ A1 )
% 0.34/0.58 => ? [Y3: code_integer] :
% 0.34/0.58 ( ( member_Code_integer @ Y3 @ A22 )
% 0.34/0.58 & ( member157494554546826820nteger @ ( produc1086072967326762835nteger @ X2 @ Y3 ) @ R3 ) ) ) ) ) ).
% 0.34/0.58
% 0.34/0.58 % max_ext.simps
% 0.34/0.58 thf(fact_2811_max__ext_Osimps,axiom,
% 0.34/0.58 ! [A1: set_Product_unit,A22: set_Product_unit,R3: set_Pr5094982260447487303t_unit] :
% 0.34/0.58 ( ( member4189953012975926160t_unit @ ( produc7269801343787132631t_unit @ A1 @ A22 ) @ ( max_ext_Product_unit @ R3 ) )
% 0.34/0.58 = ( ( finite4290736615968046902t_unit @ A1 )
% 0.34/0.58 & ( finite4290736615968046902t_unit @ A22 )
% 0.34/0.58 & ( A22 != bot_bo3957492148770167129t_unit )
% 0.34/0.58 & ! [X2: product_unit] :
% 0.34/0.58 ( ( member_Product_unit @ X2 @ A1 )
% 0.34/0.58 => ? [Y3: product_unit] :
% 0.34/0.58 ( ( member_Product_unit @ Y3 @ A22 )
% 0.34/0.58 & ( member7821138191923116944t_unit @ ( produc5692694829384537815t_unit @ X2 @ Y3 ) @ R3 ) ) ) ) ) ).
% 0.34/0.58
% 0.34/0.58 % max_ext.simps
% 0.34/0.58 thf(fact_2812_max__ext_Osimps,axiom,
% 0.34/0.58 ! [A1: set_o,A22: set_o,R3: set_Product_prod_o_o] :
% 0.34/0.58 ( ( member9116954335612470352_set_o @ ( produc5838405689764958487_set_o @ A1 @ A22 ) @ ( max_ext_o @ R3 ) )
% 0.34/0.58 = ( ( finite_finite_o @ A1 )
% 0.34/0.58 & ( finite_finite_o @ A22 )
% 0.34/0.58 & ( A22 != bot_bot_set_o )
% 0.34/0.58 & ! [X2: $o] :
% 0.34/0.58 ( ( member_o @ X2 @ A1 )
% 0.34/0.58 => ? [Y3: $o] :
% 0.34/0.58 ( ( member_o @ Y3 @ A22 )
% 0.34/0.58 & ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ X2 @ Y3 ) @ R3 ) ) ) ) ) ).
% 0.34/0.58
% 0.34/0.58 % max_ext.simps
% 0.34/0.58 thf(fact_2813_max__ext_Osimps,axiom,
% 0.34/0.58 ! [A1: set_nat,A22: set_nat,R3: set_Pr1261947904930325089at_nat] :
% 0.34/0.58 ( ( member8277197624267554838et_nat @ ( produc4532415448927165861et_nat @ A1 @ A22 ) @ ( max_ext_nat @ R3 ) )
% 0.34/0.58 = ( ( finite_finite_nat @ A1 )
% 0.34/0.58 & ( finite_finite_nat @ A22 )
% 0.34/0.58 & ( A22 != bot_bot_set_nat )
% 0.34/0.58 & ! [X2: nat] :
% 0.34/0.58 ( ( member_nat @ X2 @ A1 )
% 0.34/0.58 => ? [Y3: nat] :
% 0.34/0.58 ( ( member_nat @ Y3 @ A22 )
% 0.34/0.58 & ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X2 @ Y3 ) @ R3 ) ) ) ) ) ).
% 0.34/0.58
% 0.34/0.58 % max_ext.simps
% 0.34/0.58 thf(fact_2814_max__ext_Osimps,axiom,
% 0.34/0.58 ! [A1: set_int,A22: set_int,R3: set_Pr958786334691620121nt_int] :
% 0.34/0.58 ( ( member2572552093476627150et_int @ ( produc6363374080413544029et_int @ A1 @ A22 ) @ ( max_ext_int @ R3 ) )
% 0.34/0.58 = ( ( finite_finite_int @ A1 )
% 0.34/0.58 & ( finite_finite_int @ A22 )
% 0.34/0.58 & ( A22 != bot_bot_set_int )
% 0.34/0.58 & ! [X2: int] :
% 0.34/0.58 ( ( member_int @ X2 @ A1 )
% 0.34/0.58 => ? [Y3: int] :
% 0.34/0.58 ( ( member_int @ Y3 @ A22 )
% 0.34/0.58 & ( member5262025264175285858nt_int @ ( product_Pair_int_int @ X2 @ Y3 ) @ R3 ) ) ) ) ) ).
% 0.34/0.58
% 0.34/0.58 % max_ext.simps
% 0.34/0.58 thf(fact_2815_max__ext_Osimps,axiom,
% 0.34/0.58 ! [A1: set_Pr4532377907799695533_nat_o,A22: set_Pr4532377907799695533_nat_o,R3: set_Pr2161125870931222855_nat_o] :
% 0.34/0.58 ( ( member6623172407746373520_nat_o @ ( produc5149316574908490455_nat_o @ A1 @ A22 ) @ ( max_ex5265246208886859912_nat_o @ R3 ) )
% 0.34/0.58 = ( ( finite3252695134891459830_nat_o @ A1 )
% 0.34/0.58 & ( finite3252695134891459830_nat_o @ A22 )
% 0.34/0.58 & ( A22 != bot_bo7824918357723582233_nat_o )
% 0.34/0.58 & ! [X2: produc3658429121746597890et_nat > $o] :
% 0.34/0.58 ( ( member6576561426505652726_nat_o @ X2 @ A1 )
% 0.34/0.58 => ? [Y3: produc3658429121746597890et_nat > $o] :
% 0.34/0.58 ( ( member6576561426505652726_nat_o @ Y3 @ A22 )
% 0.34/0.58 & ( member8781333585448626064_nat_o @ ( produc7368190662567826135_nat_o @ X2 @ Y3 ) @ R3 ) ) ) ) ) ).
% 0.34/0.58
% 0.34/0.58 % max_ext.simps
% 0.34/0.58 thf(fact_2816_max__ext_Ocases,axiom,
% 0.34/0.58 ! [A1: set_Code_integer,A22: set_Code_integer,R3: set_Pr4811707699266497531nteger] :
% 0.34/0.58 ( ( member4307123515891402160nteger @ ( produc7443773368509356479nteger @ A1 @ A22 ) @ ( max_ext_Code_integer @ R3 ) )
% 0.34/0.58 => ~ ( ( finite6017078050557962740nteger @ A1 )
% 0.34/0.58 => ( ( finite6017078050557962740nteger @ A22 )
% 0.34/0.58 => ( ( A22 != bot_bo3990330152332043303nteger )
% 0.34/0.58 => ~ ! [X6: code_integer] :
% 0.34/0.58 ( ( member_Code_integer @ X6 @ A1 )
% 0.34/0.58 => ? [Xa3: code_integer] :
% 0.34/0.58 ( ( member_Code_integer @ Xa3 @ A22 )
% 0.34/0.58 & ( member157494554546826820nteger @ ( produc1086072967326762835nteger @ X6 @ Xa3 ) @ R3 ) ) ) ) ) ) ) ).
% 0.34/0.58
% 0.34/0.58 % max_ext.cases
% 0.34/0.58 thf(fact_2817_max__ext_Ocases,axiom,
% 0.34/0.58 ! [A1: set_Product_unit,A22: set_Product_unit,R3: set_Pr5094982260447487303t_unit] :
% 0.34/0.58 ( ( member4189953012975926160t_unit @ ( produc7269801343787132631t_unit @ A1 @ A22 ) @ ( max_ext_Product_unit @ R3 ) )
% 0.34/0.58 => ~ ( ( finite4290736615968046902t_unit @ A1 )
% 0.34/0.58 => ( ( finite4290736615968046902t_unit @ A22 )
% 0.34/0.58 => ( ( A22 != bot_bo3957492148770167129t_unit )
% 0.34/0.58 => ~ ! [X6: product_unit] :
% 0.34/0.58 ( ( member_Product_unit @ X6 @ A1 )
% 0.34/0.58 => ? [Xa3: product_unit] :
% 0.34/0.58 ( ( member_Product_unit @ Xa3 @ A22 )
% 0.34/0.58 & ( member7821138191923116944t_unit @ ( produc5692694829384537815t_unit @ X6 @ Xa3 ) @ R3 ) ) ) ) ) ) ) ).
% 0.34/0.58
% 0.34/0.58 % max_ext.cases
% 0.34/0.58 thf(fact_2818_max__ext_Ocases,axiom,
% 0.34/0.58 ! [A1: set_o,A22: set_o,R3: set_Product_prod_o_o] :
% 0.34/0.58 ( ( member9116954335612470352_set_o @ ( produc5838405689764958487_set_o @ A1 @ A22 ) @ ( max_ext_o @ R3 ) )
% 0.34/0.58 => ~ ( ( finite_finite_o @ A1 )
% 0.34/0.58 => ( ( finite_finite_o @ A22 )
% 0.34/0.58 => ( ( A22 != bot_bot_set_o )
% 0.34/0.58 => ~ ! [X6: $o] :
% 0.34/0.58 ( ( member_o @ X6 @ A1 )
% 0.34/0.58 => ? [Xa3: $o] :
% 0.34/0.58 ( ( member_o @ Xa3 @ A22 )
% 0.34/0.58 & ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ X6 @ Xa3 ) @ R3 ) ) ) ) ) ) ) ).
% 0.34/0.58
% 0.34/0.58 % max_ext.cases
% 0.34/0.58 thf(fact_2819_max__ext_Ocases,axiom,
% 0.34/0.58 ! [A1: set_nat,A22: set_nat,R3: set_Pr1261947904930325089at_nat] :
% 0.34/0.58 ( ( member8277197624267554838et_nat @ ( produc4532415448927165861et_nat @ A1 @ A22 ) @ ( max_ext_nat @ R3 ) )
% 0.34/0.58 => ~ ( ( finite_finite_nat @ A1 )
% 0.34/0.58 => ( ( finite_finite_nat @ A22 )
% 0.34/0.58 => ( ( A22 != bot_bot_set_nat )
% 0.34/0.58 => ~ ! [X6: nat] :
% 0.34/0.58 ( ( member_nat @ X6 @ A1 )
% 0.34/0.58 => ? [Xa3: nat] :
% 0.34/0.58 ( ( member_nat @ Xa3 @ A22 )
% 0.34/0.58 & ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X6 @ Xa3 ) @ R3 ) ) ) ) ) ) ) ).
% 0.34/0.58
% 0.34/0.58 % max_ext.cases
% 0.34/0.58 thf(fact_2820_max__ext_Ocases,axiom,
% 0.34/0.58 ! [A1: set_int,A22: set_int,R3: set_Pr958786334691620121nt_int] :
% 0.34/0.58 ( ( member2572552093476627150et_int @ ( produc6363374080413544029et_int @ A1 @ A22 ) @ ( max_ext_int @ R3 ) )
% 0.34/0.58 => ~ ( ( finite_finite_int @ A1 )
% 0.34/0.58 => ( ( finite_finite_int @ A22 )
% 0.34/0.58 => ( ( A22 != bot_bot_set_int )
% 0.34/0.58 => ~ ! [X6: int] :
% 0.34/0.58 ( ( member_int @ X6 @ A1 )
% 0.34/0.58 => ? [Xa3: int] :
% 0.34/0.58 ( ( member_int @ Xa3 @ A22 )
% 0.34/0.58 & ( member5262025264175285858nt_int @ ( product_Pair_int_int @ X6 @ Xa3 ) @ R3 ) ) ) ) ) ) ) ).
% 0.34/0.58
% 0.34/0.58 % max_ext.cases
% 0.34/0.58 thf(fact_2821_max__ext_Ocases,axiom,
% 0.34/0.58 ! [A1: set_Pr4532377907799695533_nat_o,A22: set_Pr4532377907799695533_nat_o,R3: set_Pr2161125870931222855_nat_o] :
% 0.34/0.58 ( ( member6623172407746373520_nat_o @ ( produc5149316574908490455_nat_o @ A1 @ A22 ) @ ( max_ex5265246208886859912_nat_o @ R3 ) )
% 0.34/0.58 => ~ ( ( finite3252695134891459830_nat_o @ A1 )
% 0.34/0.58 => ( ( finite3252695134891459830_nat_o @ A22 )
% 0.34/0.58 => ( ( A22 != bot_bo7824918357723582233_nat_o )
% 0.34/0.58 => ~ ! [X6: produc3658429121746597890et_nat > $o] :
% 0.34/0.58 ( ( member6576561426505652726_nat_o @ X6 @ A1 )
% 0.34/0.58 => ? [Xa3: produc3658429121746597890et_nat > $o] :
% 0.34/0.58 ( ( member6576561426505652726_nat_o @ Xa3 @ A22 )
% 0.34/0.58 & ( member8781333585448626064_nat_o @ ( produc7368190662567826135_nat_o @ X6 @ Xa3 ) @ R3 ) ) ) ) ) ) ) ).
% 0.34/0.58
% 0.34/0.58 % max_ext.cases
% 0.34/0.58 thf(fact_2822_inf__period_I2_J,axiom,
% 0.34/0.58 ! [P: rat > $o,D3: rat,Q2: rat > $o] :
% 0.34/0.58 ( ! [X3: rat,K: rat] :
% 0.34/0.58 ( ( P @ X3 )
% 0.34/0.58 = ( P @ ( minus_minus_rat @ X3 @ ( times_times_rat @ K @ D3 ) ) ) )
% 0.34/0.58 => ( ! [X3: rat,K: rat] :
% 0.34/0.58 ( ( Q2 @ X3 )
% 0.34/0.58 = ( Q2 @ ( minus_minus_rat @ X3 @ ( times_times_rat @ K @ D3 ) ) ) )
% 0.34/0.58 => ! [X6: rat,K2: rat] :
% 0.34/0.58 ( ( ( P @ X6 )
% 0.34/0.58 | ( Q2 @ X6 ) )
% 0.34/0.58 = ( ( P @ ( minus_minus_rat @ X6 @ ( times_times_rat @ K2 @ D3 ) ) )
% 0.34/0.58 | ( Q2 @ ( minus_minus_rat @ X6 @ ( times_times_rat @ K2 @ D3 ) ) ) ) ) ) ) ).
% 0.34/0.58
% 0.34/0.58 % inf_period(2)
% 0.34/0.58 thf(fact_2823_inf__period_I2_J,axiom,
% 0.34/0.58 ! [P: int > $o,D3: int,Q2: int > $o] :
% 0.34/0.58 ( ! [X3: int,K: int] :
% 0.34/0.58 ( ( P @ X3 )
% 0.34/0.58 = ( P @ ( minus_minus_int @ X3 @ ( times_times_int @ K @ D3 ) ) ) )
% 0.34/0.58 => ( ! [X3: int,K: int] :
% 0.34/0.58 ( ( Q2 @ X3 )
% 0.34/0.58 = ( Q2 @ ( minus_minus_int @ X3 @ ( times_times_int @ K @ D3 ) ) ) )
% 0.34/0.58 => ! [X6: int,K2: int] :
% 0.34/0.58 ( ( ( P @ X6 )
% 0.34/0.58 | ( Q2 @ X6 ) )
% 0.34/0.58 = ( ( P @ ( minus_minus_int @ X6 @ ( times_times_int @ K2 @ D3 ) ) )
% 0.34/0.58 | ( Q2 @ ( minus_minus_int @ X6 @ ( times_times_int @ K2 @ D3 ) ) ) ) ) ) ) ).
% 0.34/0.58
% 0.34/0.58 % inf_period(2)
% 0.34/0.58 thf(fact_2824_inf__period_I1_J,axiom,
% 0.34/0.58 ! [P: rat > $o,D3: rat,Q2: rat > $o] :
% 0.34/0.58 ( ! [X3: rat,K: rat] :
% 0.34/0.58 ( ( P @ X3 )
% 0.34/0.58 = ( P @ ( minus_minus_rat @ X3 @ ( times_times_rat @ K @ D3 ) ) ) )
% 0.34/0.58 => ( ! [X3: rat,K: rat] :
% 0.34/0.58 ( ( Q2 @ X3 )
% 0.34/0.58 = ( Q2 @ ( minus_minus_rat @ X3 @ ( times_times_rat @ K @ D3 ) ) ) )
% 0.34/0.58 => ! [X6: rat,K2: rat] :
% 0.34/0.58 ( ( ( P @ X6 )
% 0.34/0.58 & ( Q2 @ X6 ) )
% 0.34/0.58 = ( ( P @ ( minus_minus_rat @ X6 @ ( times_times_rat @ K2 @ D3 ) ) )
% 0.34/0.58 & ( Q2 @ ( minus_minus_rat @ X6 @ ( times_times_rat @ K2 @ D3 ) ) ) ) ) ) ) ).
% 0.34/0.58
% 0.34/0.58 % inf_period(1)
% 0.34/0.58 thf(fact_2825_inf__period_I1_J,axiom,
% 0.34/0.58 ! [P: int > $o,D3: int,Q2: int > $o] :
% 0.34/0.58 ( ! [X3: int,K: int] :
% 0.34/0.58 ( ( P @ X3 )
% 0.34/0.58 = ( P @ ( minus_minus_int @ X3 @ ( times_times_int @ K @ D3 ) ) ) )
% 0.34/0.58 => ( ! [X3: int,K: int] :
% 0.34/0.58 ( ( Q2 @ X3 )
% 0.34/0.58 = ( Q2 @ ( minus_minus_int @ X3 @ ( times_times_int @ K @ D3 ) ) ) )
% 0.34/0.58 => ! [X6: int,K2: int] :
% 0.34/0.58 ( ( ( P @ X6 )
% 0.34/0.58 & ( Q2 @ X6 ) )
% 0.34/0.58 = ( ( P @ ( minus_minus_int @ X6 @ ( times_times_int @ K2 @ D3 ) ) )
% 0.34/0.58 & ( Q2 @ ( minus_minus_int @ X6 @ ( times_times_int @ K2 @ D3 ) ) ) ) ) ) ) ).
% 0.34/0.58
% 0.34/0.58 % inf_period(1)
% 0.34/0.58 thf(fact_2826_left__diff__distrib,axiom,
% 0.34/0.58 ! [A2: rat,B: rat,C: rat] :
% 0.34/0.58 ( ( times_times_rat @ ( minus_minus_rat @ A2 @ B ) @ C )
% 0.34/0.58 = ( minus_minus_rat @ ( times_times_rat @ A2 @ C ) @ ( times_times_rat @ B @ C ) ) ) ).
% 0.34/0.58
% 0.34/0.58 % left_diff_distrib
% 0.34/0.58 thf(fact_2827_left__diff__distrib,axiom,
% 0.34/0.58 ! [A2: int,B: int,C: int] :
% 0.34/0.58 ( ( times_times_int @ ( minus_minus_int @ A2 @ B ) @ C )
% 0.34/0.58 = ( minus_minus_int @ ( times_times_int @ A2 @ C ) @ ( times_times_int @ B @ C ) ) ) ).
% 0.34/0.58
% 0.34/0.58 % left_diff_distrib
% 0.34/0.58 thf(fact_2828_right__diff__distrib,axiom,
% 0.34/0.58 ! [A2: rat,B: rat,C: rat] :
% 0.34/0.58 ( ( times_times_rat @ A2 @ ( minus_minus_rat @ B @ C ) )
% 0.34/0.58 = ( minus_minus_rat @ ( times_times_rat @ A2 @ B ) @ ( times_times_rat @ A2 @ C ) ) ) ).
% 0.34/0.58
% 0.34/0.58 % right_diff_distrib
% 0.34/0.58 thf(fact_2829_right__diff__distrib,axiom,
% 0.34/0.58 ! [A2: int,B: int,C: int] :
% 0.34/0.58 ( ( times_times_int @ A2 @ ( minus_minus_int @ B @ C ) )
% 0.34/0.58 = ( minus_minus_int @ ( times_times_int @ A2 @ B ) @ ( times_times_int @ A2 @ C ) ) ) ).
% 0.34/0.58
% 0.34/0.58 % right_diff_distrib
% 0.34/0.58 thf(fact_2830_left__diff__distrib_H,axiom,
% 0.34/0.58 ! [B: rat,C: rat,A2: rat] :
% 0.34/0.58 ( ( times_times_rat @ ( minus_minus_rat @ B @ C ) @ A2 )
% 0.34/0.58 = ( minus_minus_rat @ ( times_times_rat @ B @ A2 ) @ ( times_times_rat @ C @ A2 ) ) ) ).
% 0.34/0.58
% 0.34/0.58 % left_diff_distrib'
% 0.34/0.58 thf(fact_2831_left__diff__distrib_H,axiom,
% 0.34/0.58 ! [B: nat,C: nat,A2: nat] :
% 0.34/0.58 ( ( times_times_nat @ ( minus_minus_nat @ B @ C ) @ A2 )
% 0.34/0.58 = ( minus_minus_nat @ ( times_times_nat @ B @ A2 ) @ ( times_times_nat @ C @ A2 ) ) ) ).
% 0.34/0.58
% 0.34/0.58 % left_diff_distrib'
% 0.34/0.58 thf(fact_2832_left__diff__distrib_H,axiom,
% 0.34/0.58 ! [B: int,C: int,A2: int] :
% 0.34/0.58 ( ( times_times_int @ ( minus_minus_int @ B @ C ) @ A2 )
% 0.34/0.58 = ( minus_minus_int @ ( times_times_int @ B @ A2 ) @ ( times_times_int @ C @ A2 ) ) ) ).
% 0.34/0.58
% 0.34/0.58 % left_diff_distrib'
% 0.34/0.58 thf(fact_2833_right__diff__distrib_H,axiom,
% 0.34/0.58 ! [A2: rat,B: rat,C: rat] :
% 0.34/0.58 ( ( times_times_rat @ A2 @ ( minus_minus_rat @ B @ C ) )
% 0.34/0.58 = ( minus_minus_rat @ ( times_times_rat @ A2 @ B ) @ ( times_times_rat @ A2 @ C ) ) ) ).
% 0.34/0.58
% 0.34/0.58 % right_diff_distrib'
% 0.34/0.58 thf(fact_2834_right__diff__distrib_H,axiom,
% 0.34/0.58 ! [A2: nat,B: nat,C: nat] :
% 0.34/0.58 ( ( times_times_nat @ A2 @ ( minus_minus_nat @ B @ C ) )
% 0.34/0.58 = ( minus_minus_nat @ ( times_times_nat @ A2 @ B ) @ ( times_times_nat @ A2 @ C ) ) ) ).
% 0.34/0.58
% 0.34/0.58 % right_diff_distrib'
% 0.34/0.58 thf(fact_2835_right__diff__distrib_H,axiom,
% 0.34/0.58 ! [A2: int,B: int,C: int] :
% 0.34/0.58 ( ( times_times_int @ A2 @ ( minus_minus_int @ B @ C ) )
% 0.34/0.58 = ( minus_minus_int @ ( times_times_int @ A2 @ B ) @ ( times_times_int @ A2 @ C ) ) ) ).
% 0.34/0.58
% 0.34/0.58 % right_diff_distrib'
% 0.34/0.58 thf(fact_2836_in__lex__prod,axiom,
% 0.34/0.58 ! [A2: nat,B: int,A5: nat,B3: int,R2: set_Pr1261947904930325089at_nat,S2: set_Pr958786334691620121nt_int] :
% 0.34/0.58 ( ( member4619661753630440592at_int @ ( produc886916002739626967at_int @ ( product_Pair_nat_int @ A2 @ B ) @ ( product_Pair_nat_int @ A5 @ B3 ) ) @ ( lex_prod_nat_int @ R2 @ S2 ) )
% 0.34/0.58 = ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A2 @ A5 ) @ R2 )
% 0.34/0.58 | ( ( A2 = A5 )
% 0.34/0.58 & ( member5262025264175285858nt_int @ ( product_Pair_int_int @ B @ B3 ) @ S2 ) ) ) ) ).
% 0.34/0.58
% 0.34/0.58 % in_lex_prod
% 0.34/0.58 thf(fact_2837_in__lex__prod,axiom,
% 0.34/0.58 ! [A2: int,B: nat,A5: int,B3: nat,R2: set_Pr958786334691620121nt_int,S2: set_Pr1261947904930325089at_nat] :
% 0.34/0.58 ( ( member2930414080797523088nt_nat @ ( produc8921240378546987991nt_nat @ ( product_Pair_int_nat @ A2 @ B ) @ ( product_Pair_int_nat @ A5 @ B3 ) ) @ ( lex_prod_int_nat @ R2 @ S2 ) )
% 0.34/0.58 = ( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ A2 @ A5 ) @ R2 )
% 0.34/0.58 | ( ( A2 = A5 )
% 0.34/0.58 & ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ B @ B3 ) @ S2 ) ) ) ) ).
% 0.34/0.58
% 0.34/0.58 % in_lex_prod
% 0.34/0.58 thf(fact_2838_in__lex__prod,axiom,
% 0.34/0.58 ! [A2: heap_e7401611519738050253t_unit,B: set_nat,A5: heap_e7401611519738050253t_unit,B3: set_nat,R2: set_Pr1776176089890104647t_unit,S2: set_Pr5488025237498180813et_nat] :
% 0.34/0.58 ( ( member6099555550032318734et_nat @ ( produc8199053930788261021et_nat @ ( produc7507926704131184380et_nat @ A2 @ B ) @ ( produc7507926704131184380et_nat @ A5 @ B3 ) ) @ ( lex_pr538634069021852900et_nat @ R2 @ S2 ) )
% 0.34/0.58 = ( ( member5046970461208427408t_unit @ ( produc8256076736985659095t_unit @ A2 @ A5 ) @ R2 )
% 0.34/0.58 | ( ( A2 = A5 )
% 0.34/0.58 & ( member8277197624267554838et_nat @ ( produc4532415448927165861et_nat @ B @ B3 ) @ S2 ) ) ) ) ).
% 0.34/0.58
% 0.34/0.58 % in_lex_prod
% 0.34/0.58 thf(fact_2839_in__lex__prod,axiom,
% 0.34/0.58 ! [A2: nat,B: nat,A5: nat,B3: nat,R2: set_Pr1261947904930325089at_nat,S2: set_Pr1261947904930325089at_nat] :
% 0.34/0.58 ( ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ ( product_Pair_nat_nat @ A2 @ B ) @ ( product_Pair_nat_nat @ A5 @ B3 ) ) @ ( lex_prod_nat_nat @ R2 @ S2 ) )
% 0.34/0.58 = ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A2 @ A5 ) @ R2 )
% 0.34/0.58 | ( ( A2 = A5 )
% 0.34/0.58 & ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ B @ B3 ) @ S2 ) ) ) ) ).
% 0.34/0.58
% 0.34/0.58 % in_lex_prod
% 0.34/0.58 thf(fact_2840_in__lex__prod,axiom,
% 0.34/0.58 ! [A2: int,B: int,A5: int,B3: int,R2: set_Pr958786334691620121nt_int,S2: set_Pr958786334691620121nt_int] :
% 0.34/0.58 ( ( member8566619992076573584nt_int @ ( produc3646306378393792727nt_int @ ( product_Pair_int_int @ A2 @ B ) @ ( product_Pair_int_int @ A5 @ B3 ) ) @ ( lex_prod_int_int @ R2 @ S2 ) )
% 0.34/0.58 = ( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ A2 @ A5 ) @ R2 )
% 0.34/0.58 | ( ( A2 = A5 )
% 0.34/0.58 & ( member5262025264175285858nt_int @ ( product_Pair_int_int @ B @ B3 ) @ S2 ) ) ) ) ).
% 0.34/0.58
% 0.34/0.58 % in_lex_prod
% 0.34/0.58 thf(fact_2841_in__lex__prod,axiom,
% 0.34/0.58 ! [A2: produc3658429121746597890et_nat > $o,B: produc3658429121746597890et_nat,A5: produc3658429121746597890et_nat > $o,B3: produc3658429121746597890et_nat,R2: set_Pr2161125870931222855_nat_o,S2: set_Pr719794911490849221et_nat] :
% 0.34/0.58 ( ( member4763271486408492550et_nat @ ( produc8599840265553166229et_nat @ ( produc5001842942810119800et_nat @ A2 @ B ) @ ( produc5001842942810119800et_nat @ A5 @ B3 ) ) @ ( lex_pr5340572901959109728et_nat @ R2 @ S2 ) )
% 0.34/0.58 = ( ( member8781333585448626064_nat_o @ ( produc7368190662567826135_nat_o @ A2 @ A5 ) @ R2 )
% 0.34/0.58 | ( ( A2 = A5 )
% 0.34/0.58 & ( member6099555550032318734et_nat @ ( produc8199053930788261021et_nat @ B @ B3 ) @ S2 ) ) ) ) ).
% 0.34/0.58
% 0.34/0.58 % in_lex_prod
% 0.34/0.58 thf(fact_2842_in__lex__prod,axiom,
% 0.34/0.58 ! [A2: produc3658429121746597890et_nat > $o,B: produc3925858234332021118et_nat,A5: produc3658429121746597890et_nat > $o,B3: produc3925858234332021118et_nat,R2: set_Pr2161125870931222855_nat_o,S2: set_Pr7928877670098842301et_nat] :
% 0.34/0.58 ( ( member6341495586645257982et_nat @ ( produc1940133919992309389et_nat @ ( produc2245416461498447860et_nat @ A2 @ B ) @ ( produc2245416461498447860et_nat @ A5 @ B3 ) ) @ ( lex_pr4722427456421979612et_nat @ R2 @ S2 ) )
% 0.34/0.58 = ( ( member8781333585448626064_nat_o @ ( produc7368190662567826135_nat_o @ A2 @ A5 ) @ R2 )
% 0.34/0.58 | ( ( A2 = A5 )
% 0.34/0.58 & ( member4763271486408492550et_nat @ ( produc8599840265553166229et_nat @ B @ B3 ) @ S2 ) ) ) ) ).
% 0.34/0.58
% 0.34/0.58 % in_lex_prod
% 0.34/0.58 thf(fact_2843_Linear__order__wf__diff__Id,axiom,
% 0.34/0.58 ! [R2: set_Pr5094982260447487303t_unit] :
% 0.34/0.58 ( ( order_888282469287196226t_unit @ ( field_Product_unit @ R2 ) @ R2 )
% 0.34/0.58 => ( ( wf_Product_unit @ ( minus_6985556733573110190t_unit @ R2 @ id_Product_unit ) )
% 0.34/0.58 = ( ! [A3: set_Product_unit] :
% 0.34/0.58 ( ( ord_le3507040750410214029t_unit @ A3 @ ( field_Product_unit @ R2 ) )
% 0.34/0.58 => ( ( A3 != bot_bo3957492148770167129t_unit )
% 0.34/0.58 => ? [X2: product_unit] :
% 0.34/0.58 ( ( member_Product_unit @ X2 @ A3 )
% 0.34/0.58 & ! [Y3: product_unit] :
% 0.34/0.58 ( ( member_Product_unit @ Y3 @ A3 )
% 0.34/0.58 => ( member7821138191923116944t_unit @ ( produc5692694829384537815t_unit @ X2 @ Y3 ) @ R2 ) ) ) ) ) ) ) ) ).
% 0.34/0.58
% 0.34/0.58 % Linear_order_wf_diff_Id
% 0.34/0.58 thf(fact_2844_Linear__order__wf__diff__Id,axiom,
% 0.34/0.58 ! [R2: set_Product_prod_o_o] :
% 0.34/0.58 ( ( order_6238756238976269133r_on_o @ ( field_o @ R2 ) @ R2 )
% 0.34/0.58 => ( ( wf_o @ ( minus_7052513330764642158od_o_o @ R2 @ id_o2 ) )
% 0.34/0.58 = ( ! [A3: set_o] :
% 0.34/0.58 ( ( ord_less_eq_set_o @ A3 @ ( field_o @ R2 ) )
% 0.34/0.58 => ( ( A3 != bot_bot_set_o )
% 0.34/0.58 => ? [X2: $o] :
% 0.34/0.58 ( ( member_o @ X2 @ A3 )
% 0.34/0.58 & ! [Y3: $o] :
% 0.34/0.58 ( ( member_o @ Y3 @ A3 )
% 0.34/0.58 => ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ X2 @ Y3 ) @ R2 ) ) ) ) ) ) ) ) ).
% 0.34/0.58
% 0.34/0.58 % Linear_order_wf_diff_Id
% 0.34/0.58 thf(fact_2845_Linear__order__wf__diff__Id,axiom,
% 0.34/0.58 ! [R2: set_Pr958786334691620121nt_int] :
% 0.34/0.58 ( ( order_4471489696718655927on_int @ ( field_int @ R2 ) @ R2 )
% 0.34/0.58 => ( ( wf_int @ ( minus_1052850069191792384nt_int @ R2 @ id_int ) )
% 0.34/0.58 = ( ! [A3: set_int] :
% 0.34/0.58 ( ( ord_less_eq_set_int @ A3 @ ( field_int @ R2 ) )
% 0.34/0.58 => ( ( A3 != bot_bot_set_int )
% 0.34/0.58 => ? [X2: int] :
% 0.34/0.58 ( ( member_int @ X2 @ A3 )
% 0.34/0.58 & ! [Y3: int] :
% 0.34/0.58 ( ( member_int @ Y3 @ A3 )
% 0.34/0.58 => ( member5262025264175285858nt_int @ ( product_Pair_int_int @ X2 @ Y3 ) @ R2 ) ) ) ) ) ) ) ) ).
% 0.34/0.58
% 0.34/0.58 % Linear_order_wf_diff_Id
% 0.34/0.58 thf(fact_2846_Linear__order__wf__diff__Id,axiom,
% 0.34/0.58 ! [R2: set_Pr2161125870931222855_nat_o] :
% 0.34/0.58 ( ( order_1145710701291701250_nat_o @ ( field_81821745747897882_nat_o @ R2 ) @ R2 )
% 0.34/0.58 => ( ( wf_Pro8787509627421085110_nat_o @ ( minus_9189903262233110958_nat_o @ R2 @ id_Pro6775173067513470761_nat_o ) )
% 0.34/0.58 = ( ! [A3: set_Pr4532377907799695533_nat_o] :
% 0.34/0.58 ( ( ord_le2965882846123202637_nat_o @ A3 @ ( field_81821745747897882_nat_o @ R2 ) )
% 0.34/0.58 => ( ( A3 != bot_bo7824918357723582233_nat_o )
% 0.34/0.58 => ? [X2: produc3658429121746597890et_nat > $o] :
% 0.34/0.58 ( ( member6576561426505652726_nat_o @ X2 @ A3 )
% 0.34/0.58 & ! [Y3: produc3658429121746597890et_nat > $o] :
% 0.34/0.58 ( ( member6576561426505652726_nat_o @ Y3 @ A3 )
% 0.34/0.58 => ( member8781333585448626064_nat_o @ ( produc7368190662567826135_nat_o @ X2 @ Y3 ) @ R2 ) ) ) ) ) ) ) ) ).
% 0.34/0.58
% 0.34/0.58 % Linear_order_wf_diff_Id
% 0.34/0.58 thf(fact_2847_Linear__order__wf__diff__Id,axiom,
% 0.34/0.58 ! [R2: set_Pr1261947904930325089at_nat] :
% 0.34/0.58 ( ( order_4473980167227706203on_nat @ ( field_nat @ R2 ) @ R2 )
% 0.34/0.58 => ( ( wf_nat @ ( minus_1356011639430497352at_nat @ R2 @ id_nat ) )
% 0.34/0.58 = ( ! [A3: set_nat] :
% 0.34/0.58 ( ( ord_less_eq_set_nat @ A3 @ ( field_nat @ R2 ) )
% 0.34/0.58 => ( ( A3 != bot_bot_set_nat )
% 0.34/0.58 => ? [X2: nat] :
% 0.34/0.58 ( ( member_nat @ X2 @ A3 )
% 0.34/0.58 & ! [Y3: nat] :
% 0.34/0.58 ( ( member_nat @ Y3 @ A3 )
% 0.34/0.58 => ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X2 @ Y3 ) @ R2 ) ) ) ) ) ) ) ) ).
% 0.34/0.58
% 0.34/0.58 % Linear_order_wf_diff_Id
% 0.34/0.58 thf(fact_2848_remove__def,axiom,
% 0.34/0.58 ( remove_Product_unit
% 0.34/0.58 = ( ^ [X2: product_unit,A3: set_Product_unit] : ( minus_6452836326544984404t_unit @ A3 @ ( insert_Product_unit @ X2 @ bot_bo3957492148770167129t_unit ) ) ) ) ).
% 0.34/0.58
% 0.34/0.58 % remove_def
% 0.34/0.58 thf(fact_2849_remove__def,axiom,
% 0.34/0.58 ( remove_o
% 0.34/0.58 = ( ^ [X2: $o,A3: set_o] : ( minus_minus_set_o @ A3 @ ( insert_o @ X2 @ bot_bot_set_o ) ) ) ) ).
% 0.34/0.58
% 0.34/0.58 % remove_def
% 0.34/0.58 thf(fact_2850_remove__def,axiom,
% 0.34/0.58 ( remove_int
% 0.34/0.58 = ( ^ [X2: int,A3: set_int] : ( minus_minus_set_int @ A3 @ ( insert_int @ X2 @ bot_bot_set_int ) ) ) ) ).
% 0.34/0.58
% 0.34/0.58 % remove_def
% 0.34/0.58 thf(fact_2851_remove__def,axiom,
% 0.34/0.58 ( remove4651630035290841522_nat_o
% 0.34/0.58 = ( ^ [X2: produc3658429121746597890et_nat > $o,A3: set_Pr4532377907799695533_nat_o] : ( minus_1801376950450012436_nat_o @ A3 @ ( insert5175938949040314269_nat_o @ X2 @ bot_bo7824918357723582233_nat_o ) ) ) ) ).
% 0.34/0.58
% 0.34/0.58 % remove_def
% 0.34/0.58 thf(fact_2852_remove__def,axiom,
% 0.34/0.58 ( remove_nat
% 0.34/0.58 = ( ^ [X2: nat,A3: set_nat] : ( minus_minus_set_nat @ A3 @ ( insert_nat @ X2 @ bot_bot_set_nat ) ) ) ) ).
% 0.34/0.58
% 0.34/0.58 % remove_def
% 0.34/0.58 thf(fact_2853_wo__rel_Ocases__Total3,axiom,
% 0.34/0.58 ! [R2: set_Pr5094982260447487303t_unit,A2: product_unit,B: product_unit,Phi: product_unit > product_unit > $o] :
% 0.34/0.58 ( ( bNF_We2091578341578594809t_unit @ R2 )
% 0.34/0.58 => ( ( ord_le3507040750410214029t_unit @ ( insert_Product_unit @ A2 @ ( insert_Product_unit @ B @ bot_bo3957492148770167129t_unit ) ) @ ( field_Product_unit @ R2 ) )
% 0.34/0.58 => ( ( ( ( member7821138191923116944t_unit @ ( produc5692694829384537815t_unit @ A2 @ B ) @ ( minus_6985556733573110190t_unit @ R2 @ id_Product_unit ) )
% 0.34/0.58 | ( member7821138191923116944t_unit @ ( produc5692694829384537815t_unit @ B @ A2 ) @ ( minus_6985556733573110190t_unit @ R2 @ id_Product_unit ) ) )
% 0.34/0.58 => ( Phi @ A2 @ B ) )
% 0.34/0.58 => ( ( ( A2 = B )
% 0.34/0.58 => ( Phi @ A2 @ B ) )
% 0.34/0.58 => ( Phi @ A2 @ B ) ) ) ) ) ).
% 0.34/0.58
% 0.34/0.58 % wo_rel.cases_Total3
% 0.34/0.58 thf(fact_2854_wo__rel_Ocases__Total3,axiom,
% 0.34/0.58 ! [R2: set_Product_prod_o_o,A2: $o,B: $o,Phi: $o > $o > $o] :
% 0.34/0.58 ( ( bNF_We6437023980930612612_rel_o @ R2 )
% 0.34/0.58 => ( ( ord_less_eq_set_o @ ( insert_o @ A2 @ ( insert_o @ B @ bot_bot_set_o ) ) @ ( field_o @ R2 ) )
% 0.34/0.58 => ( ( ( ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ A2 @ B ) @ ( minus_7052513330764642158od_o_o @ R2 @ id_o2 ) )
% 0.34/0.58 | ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ B @ A2 ) @ ( minus_7052513330764642158od_o_o @ R2 @ id_o2 ) ) )
% 0.34/0.58 => ( Phi @ A2 @ B ) )
% 0.34/0.58 => ( ( ( A2 = B )
% 0.34/0.58 => ( Phi @ A2 @ B ) )
% 0.34/0.58 => ( Phi @ A2 @ B ) ) ) ) ) ).
% 0.34/0.58
% 0.34/0.58 % wo_rel.cases_Total3
% 0.34/0.58 thf(fact_2855_wo__rel_Ocases__Total3,axiom,
% 0.34/0.58 ! [R2: set_Pr958786334691620121nt_int,A2: int,B: int,Phi: int > int > $o] :
% 0.34/0.58 ( ( bNF_We3815749466139970368el_int @ R2 )
% 0.34/0.58 => ( ( ord_less_eq_set_int @ ( insert_int @ A2 @ ( insert_int @ B @ bot_bot_set_int ) ) @ ( field_int @ R2 ) )
% 0.34/0.58 => ( ( ( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ A2 @ B ) @ ( minus_1052850069191792384nt_int @ R2 @ id_int ) )
% 0.34/0.58 | ( member5262025264175285858nt_int @ ( product_Pair_int_int @ B @ A2 ) @ ( minus_1052850069191792384nt_int @ R2 @ id_int ) ) )
% 0.34/0.58 => ( Phi @ A2 @ B ) )
% 0.34/0.58 => ( ( ( A2 = B )
% 0.34/0.58 => ( Phi @ A2 @ B ) )
% 0.34/0.58 => ( Phi @ A2 @ B ) ) ) ) ) ).
% 0.34/0.58
% 0.34/0.58 % wo_rel.cases_Total3
% 0.34/0.58 thf(fact_2856_wo__rel_Ocases__Total3,axiom,
% 0.34/0.58 ! [R2: set_Pr2161125870931222855_nat_o,A2: produc3658429121746597890et_nat > $o,B: produc3658429121746597890et_nat > $o,Phi: ( produc3658429121746597890et_nat > $o ) > ( produc3658429121746597890et_nat > $o ) > $o] :
% 0.34/0.58 ( ( bNF_We740377018556551609_nat_o @ R2 )
% 0.34/0.58 => ( ( ord_le2965882846123202637_nat_o @ ( insert5175938949040314269_nat_o @ A2 @ ( insert5175938949040314269_nat_o @ B @ bot_bo7824918357723582233_nat_o ) ) @ ( field_81821745747897882_nat_o @ R2 ) )
% 0.34/0.58 => ( ( ( ( member8781333585448626064_nat_o @ ( produc7368190662567826135_nat_o @ A2 @ B ) @ ( minus_9189903262233110958_nat_o @ R2 @ id_Pro6775173067513470761_nat_o ) )
% 0.34/0.58 | ( member8781333585448626064_nat_o @ ( produc7368190662567826135_nat_o @ B @ A2 ) @ ( minus_9189903262233110958_nat_o @ R2 @ id_Pro6775173067513470761_nat_o ) ) )
% 0.34/0.58 => ( Phi @ A2 @ B ) )
% 0.34/0.58 => ( ( ( A2 = B )
% 0.34/0.58 => ( Phi @ A2 @ B ) )
% 0.34/0.58 => ( Phi @ A2 @ B ) ) ) ) ) ).
% 0.34/0.58
% 0.34/0.58 % wo_rel.cases_Total3
% 0.34/0.58 thf(fact_2857_wo__rel_Ocases__Total3,axiom,
% 0.34/0.58 ! [R2: set_Pr1261947904930325089at_nat,A2: nat,B: nat,Phi: nat > nat > $o] :
% 0.34/0.58 ( ( bNF_We3818239936649020644el_nat @ R2 )
% 0.34/0.58 => ( ( ord_less_eq_set_nat @ ( insert_nat @ A2 @ ( insert_nat @ B @ bot_bot_set_nat ) ) @ ( field_nat @ R2 ) )
% 0.34/0.58 => ( ( ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A2 @ B ) @ ( minus_1356011639430497352at_nat @ R2 @ id_nat ) )
% 0.34/0.58 | ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ B @ A2 ) @ ( minus_1356011639430497352at_nat @ R2 @ id_nat ) ) )
% 0.34/0.58 => ( Phi @ A2 @ B ) )
% 0.34/0.58 => ( ( ( A2 = B )
% 0.34/0.58 => ( Phi @ A2 @ B ) )
% 0.34/0.58 => ( Phi @ A2 @ B ) ) ) ) ) ).
% 0.34/0.58
% 0.34/0.58 % wo_rel.cases_Total3
% 0.34/0.58 thf(fact_2858_wf__max,axiom,
% 0.34/0.58 ! [R3: set_Pr2161125870931222855_nat_o] :
% 0.34/0.58 ( ( wf_Pro8787509627421085110_nat_o @ ( conver2764731894203590139_nat_o @ R3 ) )
% 0.34/0.58 => ( ( R3 != bot_bo3657187498359500723_nat_o )
% 0.34/0.58 => ~ ! [M2: produc3658429121746597890et_nat > $o] :
% 0.34/0.58 ~ ( member6576561426505652726_nat_o @ M2 @ ( minus_1801376950450012436_nat_o @ ( range_5289318983786108341_nat_o @ R3 ) @ ( domain4881241903163524940_nat_o @ R3 ) ) ) ) ) ).
% 0.34/0.58
% 0.34/0.58 % wf_max
% 0.34/0.58 thf(fact_2859_wf__max,axiom,
% 0.34/0.58 ! [R3: set_Pr958786334691620121nt_int] :
% 0.34/0.58 ( ( wf_int @ ( converse_int_int @ R3 ) )
% 0.34/0.58 => ( ( R3 != bot_bo1796632182523588997nt_int )
% 0.34/0.58 => ~ ! [M2: int] :
% 0.34/0.58 ~ ( member_int @ M2 @ ( minus_minus_set_int @ ( range_int_int @ R3 ) @ ( domain_int_int @ R3 ) ) ) ) ) ).
% 0.34/0.58
% 0.34/0.58 % wf_max
% 0.34/0.58 thf(fact_2860_wf__max,axiom,
% 0.34/0.58 ! [R3: set_Product_prod_o_o] :
% 0.34/0.58 ( ( wf_o @ ( converse_o_o @ R3 ) )
% 0.34/0.58 => ( ( R3 != bot_bo7073875226086086771od_o_o )
% 0.34/0.58 => ~ ! [M2: $o] :
% 0.34/0.58 ~ ( member_o @ M2 @ ( minus_minus_set_o @ ( range_o_o @ R3 ) @ ( domain_o_o @ R3 ) ) ) ) ) ).
% 0.34/0.58
% 0.34/0.58 % wf_max
% 0.34/0.58 thf(fact_2861_wf__max,axiom,
% 0.34/0.58 ! [R3: set_Pr5094982260447487303t_unit] :
% 0.34/0.58 ( ( wf_Product_unit @ ( conver2495244158383847931t_unit @ R3 ) )
% 0.34/0.58 => ( ( R3 != bot_bo1654022762305692595t_unit )
% 0.34/0.58 => ~ ! [M2: product_unit] :
% 0.34/0.58 ~ ( member_Product_unit @ M2 @ ( minus_6452836326544984404t_unit @ ( range_6617724647562260917t_unit @ R3 ) @ ( domain4292024440186670924t_unit @ R3 ) ) ) ) ) ).
% 0.34/0.58
% 0.34/0.58 % wf_max
% 0.34/0.58 thf(fact_2862_wf__max,axiom,
% 0.34/0.58 ! [R3: set_Pr1261947904930325089at_nat] :
% 0.34/0.58 ( ( wf_nat @ ( converse_nat_nat @ R3 ) )
% 0.34/0.58 => ( ( R3 != bot_bo2099793752762293965at_nat )
% 0.34/0.58 => ~ ! [M2: nat] :
% 0.34/0.58 ~ ( member_nat @ M2 @ ( minus_minus_set_nat @ ( range_nat_nat @ R3 ) @ ( domain_nat_nat @ R3 ) ) ) ) ) ).
% 0.34/0.58
% 0.34/0.58 % wf_max
% 0.34/0.58 thf(fact_2863_wf__Un,axiom,
% 0.34/0.58 ! [R2: set_Pr5094982260447487303t_unit,S2: set_Pr5094982260447487303t_unit] :
% 0.34/0.58 ( ( wf_Product_unit @ R2 )
% 0.34/0.58 => ( ( wf_Product_unit @ S2 )
% 0.34/0.58 => ( ( ( inf_in4660618365625256667t_unit @ ( domain4292024440186670924t_unit @ R2 ) @ ( range_6617724647562260917t_unit @ S2 ) )
% 0.34/0.58 = bot_bo3957492148770167129t_unit )
% 0.34/0.58 => ( wf_Product_unit @ ( sup_su6481247232241170203t_unit @ R2 @ S2 ) ) ) ) ) ).
% 0.34/0.58
% 0.34/0.58 % wf_Un
% 0.34/0.58 thf(fact_2864_wf__Un,axiom,
% 0.34/0.58 ! [R2: set_Product_prod_o_o,S2: set_Product_prod_o_o] :
% 0.34/0.58 ( ( wf_o @ R2 )
% 0.34/0.58 => ( ( wf_o @ S2 )
% 0.34/0.58 => ( ( ( inf_inf_set_o @ ( domain_o_o @ R2 ) @ ( range_o_o @ S2 ) )
% 0.34/0.58 = bot_bot_set_o )
% 0.34/0.58 => ( wf_o @ ( sup_su5769328420594410459od_o_o @ R2 @ S2 ) ) ) ) ) ).
% 0.34/0.58
% 0.34/0.58 % wf_Un
% 0.34/0.58 thf(fact_2865_wf__Un,axiom,
% 0.34/0.58 ! [R2: set_Pr1261947904930325089at_nat,S2: set_Pr1261947904930325089at_nat] :
% 0.34/0.58 ( ( wf_nat @ R2 )
% 0.34/0.58 => ( ( wf_nat @ S2 )
% 0.34/0.58 => ( ( ( inf_inf_set_nat @ ( domain_nat_nat @ R2 ) @ ( range_nat_nat @ S2 ) )
% 0.34/0.58 = bot_bot_set_nat )
% 0.34/0.58 => ( wf_nat @ ( sup_su6327502436637775413at_nat @ R2 @ S2 ) ) ) ) ) ).
% 0.34/0.58
% 0.34/0.58 % wf_Un
% 0.34/0.58 thf(fact_2866_wf__Un,axiom,
% 0.34/0.58 ! [R2: set_Pr958786334691620121nt_int,S2: set_Pr958786334691620121nt_int] :
% 0.34/0.58 ( ( wf_int @ R2 )
% 0.34/0.58 => ( ( wf_int @ S2 )
% 0.34/0.58 => ( ( ( inf_inf_set_int @ ( domain_int_int @ R2 ) @ ( range_int_int @ S2 ) )
% 0.34/0.58 = bot_bot_set_int )
% 0.34/0.58 => ( wf_int @ ( sup_su6024340866399070445nt_int @ R2 @ S2 ) ) ) ) ) ).
% 0.34/0.58
% 0.34/0.58 % wf_Un
% 0.34/0.58 thf(fact_2867_wf__Un,axiom,
% 0.34/0.58 ! [R2: set_Pr2161125870931222855_nat_o,S2: set_Pr2161125870931222855_nat_o] :
% 0.34/0.58 ( ( wf_Pro8787509627421085110_nat_o @ R2 )
% 0.34/0.58 => ( ( wf_Pro8787509627421085110_nat_o @ S2 )
% 0.34/0.58 => ( ( ( inf_in1906310914598751387_nat_o @ ( domain4881241903163524940_nat_o @ R2 ) @ ( range_5289318983786108341_nat_o @ S2 ) )
% 0.34/0.58 = bot_bo7824918357723582233_nat_o )
% 0.34/0.58 => ( wf_Pro8787509627421085110_nat_o @ ( sup_su8328026650783933211_nat_o @ R2 @ S2 ) ) ) ) ) ).
% 0.34/0.58
% 0.34/0.58 % wf_Un
% 0.34/0.58 thf(fact_2868_Linear__order__Well__order__iff,axiom,
% 0.34/0.58 ! [R2: set_Pr5094982260447487303t_unit] :
% 0.34/0.58 ( ( order_888282469287196226t_unit @ ( field_Product_unit @ R2 ) @ R2 )
% 0.34/0.58 => ( ( order_3164716736300062379t_unit @ ( field_Product_unit @ R2 ) @ R2 )
% 0.34/0.58 = ( ! [A3: set_Product_unit] :
% 0.34/0.58 ( ( ord_le3507040750410214029t_unit @ A3 @ ( field_Product_unit @ R2 ) )
% 0.34/0.58 => ( ( A3 != bot_bo3957492148770167129t_unit )
% 0.34/0.58 => ? [X2: product_unit] :
% 0.34/0.58 ( ( member_Product_unit @ X2 @ A3 )
% 0.34/0.58 & ! [Y3: product_unit] :
% 0.34/0.58 ( ( member_Product_unit @ Y3 @ A3 )
% 0.34/0.58 => ( member7821138191923116944t_unit @ ( produc5692694829384537815t_unit @ X2 @ Y3 ) @ R2 ) ) ) ) ) ) ) ) ).
% 0.34/0.58
% 0.34/0.58 % Linear_order_Well_order_iff
% 0.34/0.58 thf(fact_2869_Linear__order__Well__order__iff,axiom,
% 0.34/0.58 ! [R2: set_Product_prod_o_o] :
% 0.34/0.58 ( ( order_6238756238976269133r_on_o @ ( field_o @ R2 ) @ R2 )
% 0.34/0.58 => ( ( order_2615775362358406454r_on_o @ ( field_o @ R2 ) @ R2 )
% 0.34/0.58 = ( ! [A3: set_o] :
% 0.34/0.58 ( ( ord_less_eq_set_o @ A3 @ ( field_o @ R2 ) )
% 0.34/0.58 => ( ( A3 != bot_bot_set_o )
% 0.34/0.58 => ? [X2: $o] :
% 0.34/0.58 ( ( member_o @ X2 @ A3 )
% 0.34/0.58 & ! [Y3: $o] :
% 0.34/0.58 ( ( member_o @ Y3 @ A3 )
% 0.34/0.58 => ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ X2 @ Y3 ) @ R2 ) ) ) ) ) ) ) ) ).
% 0.34/0.58
% 0.34/0.58 % Linear_order_Well_order_iff
% 0.34/0.58 thf(fact_2870_Linear__order__Well__order__iff,axiom,
% 0.34/0.58 ! [R2: set_Pr958786334691620121nt_int] :
% 0.34/0.58 ( ( order_4471489696718655927on_int @ ( field_int @ R2 ) @ R2 )
% 0.34/0.58 => ( ( order_2886507596567047182on_int @ ( field_int @ R2 ) @ R2 )
% 0.34/0.58 = ( ! [A3: set_int] :
% 0.34/0.58 ( ( ord_less_eq_set_int @ A3 @ ( field_int @ R2 ) )
% 0.34/0.58 => ( ( A3 != bot_bot_set_int )
% 0.34/0.58 => ? [X2: int] :
% 0.34/0.58 ( ( member_int @ X2 @ A3 )
% 0.34/0.58 & ! [Y3: int] :
% 0.34/0.58 ( ( member_int @ Y3 @ A3 )
% 0.34/0.58 => ( member5262025264175285858nt_int @ ( product_Pair_int_int @ X2 @ Y3 ) @ R2 ) ) ) ) ) ) ) ) ).
% 0.34/0.58
% 0.34/0.58 % Linear_order_Well_order_iff
% 0.34/0.58 thf(fact_2871_Linear__order__Well__order__iff,axiom,
% 0.34/0.58 ! [R2: set_Pr2161125870931222855_nat_o] :
% 0.34/0.58 ( ( order_1145710701291701250_nat_o @ ( field_81821745747897882_nat_o @ R2 ) @ R2 )
% 0.34/0.58 => ( ( order_2118566763991480939_nat_o @ ( field_81821745747897882_nat_o @ R2 ) @ R2 )
% 0.34/0.58 = ( ! [A3: set_Pr4532377907799695533_nat_o] :
% 0.34/0.58 ( ( ord_le2965882846123202637_nat_o @ A3 @ ( field_81821745747897882_nat_o @ R2 ) )
% 0.34/0.58 => ( ( A3 != bot_bo7824918357723582233_nat_o )
% 0.34/0.58 => ? [X2: produc3658429121746597890et_nat > $o] :
% 0.34/0.58 ( ( member6576561426505652726_nat_o @ X2 @ A3 )
% 0.34/0.58 & ! [Y3: produc3658429121746597890et_nat > $o] :
% 0.34/0.58 ( ( member6576561426505652726_nat_o @ Y3 @ A3 )
% 0.34/0.58 => ( member8781333585448626064_nat_o @ ( produc7368190662567826135_nat_o @ X2 @ Y3 ) @ R2 ) ) ) ) ) ) ) ) ).
% 0.34/0.58
% 0.34/0.58 % Linear_order_Well_order_iff
% 0.34/0.58 thf(fact_2872_Linear__order__Well__order__iff,axiom,
% 0.34/0.58 ! [R2: set_Pr1261947904930325089at_nat] :
% 0.34/0.58 ( ( order_4473980167227706203on_nat @ ( field_nat @ R2 ) @ R2 )
% 0.34/0.58 => ( ( order_2888998067076097458on_nat @ ( field_nat @ R2 ) @ R2 )
% 0.34/0.58 = ( ! [A3: set_nat] :
% 0.34/0.58 ( ( ord_less_eq_set_nat @ A3 @ ( field_nat @ R2 ) )
% 0.34/0.58 => ( ( A3 != bot_bot_set_nat )
% 0.34/0.58 => ? [X2: nat] :
% 0.34/0.58 ( ( member_nat @ X2 @ A3 )
% 0.34/0.58 & ! [Y3: nat] :
% 0.34/0.58 ( ( member_nat @ Y3 @ A3 )
% 0.34/0.58 => ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X2 @ Y3 ) @ R2 ) ) ) ) ) ) ) ) ).
% 0.34/0.58
% 0.34/0.58 % Linear_order_Well_order_iff
% 0.34/0.58 thf(fact_2873_underS__incr,axiom,
% 0.34/0.58 ! [R2: set_Pr958786334691620121nt_int,A2: int,B: int] :
% 0.34/0.58 ( ( trans_int @ R2 )
% 0.34/0.58 => ( ( antisym_int @ R2 )
% 0.34/0.58 => ( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ A2 @ B ) @ R2 )
% 0.34/0.58 => ( ord_less_eq_set_int @ ( order_underS_int @ R2 @ A2 ) @ ( order_underS_int @ R2 @ B ) ) ) ) ) ).
% 0.34/0.58
% 0.34/0.58 % underS_incr
% 0.34/0.58 thf(fact_2874_underS__incr,axiom,
% 0.34/0.58 ! [R2: set_Pr1261947904930325089at_nat,A2: nat,B: nat] :
% 0.34/0.58 ( ( trans_nat @ R2 )
% 0.34/0.58 => ( ( antisym_nat @ R2 )
% 0.34/0.58 => ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A2 @ B ) @ R2 )
% 0.34/0.58 => ( ord_less_eq_set_nat @ ( order_underS_nat @ R2 @ A2 ) @ ( order_underS_nat @ R2 @ B ) ) ) ) ) ).
% 0.34/0.58
% 0.34/0.58 % underS_incr
% 0.34/0.58 thf(fact_2875_member__remove,axiom,
% 0.34/0.58 ! [X: produc3658429121746597890et_nat > $o,Y: produc3658429121746597890et_nat > $o,A: set_Pr4532377907799695533_nat_o] :
% 0.34/0.58 ( ( member6576561426505652726_nat_o @ X @ ( remove4651630035290841522_nat_o @ Y @ A ) )
% 0.34/0.58 = ( ( member6576561426505652726_nat_o @ X @ A )
% 0.34/0.58 & ( X != Y ) ) ) ).
% 0.34/0.58
% 0.34/0.58 % member_remove
% 0.34/0.58 thf(fact_2876_member__remove,axiom,
% 0.34/0.58 ! [X: nat,Y: nat,A: set_nat] :
% 0.34/0.58 ( ( member_nat @ X @ ( remove_nat @ Y @ A ) )
% 0.34/0.58 = ( ( member_nat @ X @ A )
% 0.34/0.58 & ( X != Y ) ) ) ).
% 0.34/0.58
% 0.34/0.58 % member_remove
% 0.34/0.58 thf(fact_2877_member__remove,axiom,
% 0.34/0.58 ! [X: int,Y: int,A: set_int] :
% 0.34/0.58 ( ( member_int @ X @ ( remove_int @ Y @ A ) )
% 0.34/0.58 = ( ( member_int @ X @ A )
% 0.34/0.58 & ( X != Y ) ) ) ).
% 0.34/0.58
% 0.34/0.58 % member_remove
% 0.34/0.58 thf(fact_2878_member__remove,axiom,
% 0.34/0.58 ! [X: $o,Y: $o,A: set_o] :
% 0.34/0.58 ( ( member_o @ X @ ( remove_o @ Y @ A ) )
% 0.34/0.58 = ( ( member_o @ X @ A )
% 0.34/0.58 & ( X != Y ) ) ) ).
% 0.34/0.58
% 0.34/0.58 % member_remove
% 0.34/0.58 thf(fact_2879_member__remove,axiom,
% 0.34/0.58 ! [X: product_unit,Y: product_unit,A: set_Product_unit] :
% 0.34/0.58 ( ( member_Product_unit @ X @ ( remove_Product_unit @ Y @ A ) )
% 0.34/0.58 = ( ( member_Product_unit @ X @ A )
% 0.34/0.58 & ( X != Y ) ) ) ).
% 0.34/0.58
% 0.34/0.58 % member_remove
% 0.34/0.58 thf(fact_2880_wf__insert,axiom,
% 0.34/0.58 ! [Y: nat,X: nat,R2: set_Pr1261947904930325089at_nat] :
% 0.34/0.58 ( ( wf_nat @ ( insert8211810215607154385at_nat @ ( product_Pair_nat_nat @ Y @ X ) @ R2 ) )
% 0.34/0.58 = ( ( wf_nat @ R2 )
% 0.34/0.58 & ~ ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ Y ) @ ( transi2905341329935302413cl_nat @ R2 ) ) ) ) ).
% 0.34/0.58
% 0.34/0.58 % wf_insert
% 0.34/0.58 thf(fact_2881_wf__insert,axiom,
% 0.34/0.58 ! [Y: int,X: int,R2: set_Pr958786334691620121nt_int] :
% 0.34/0.58 ( ( wf_int @ ( insert5033312907999012233nt_int @ ( product_Pair_int_int @ Y @ X ) @ R2 ) )
% 0.34/0.58 = ( ( wf_int @ R2 )
% 0.34/0.58 & ~ ( member5262025264175285858nt_int @ ( product_Pair_int_int @ X @ Y ) @ ( transi2902850859426252137cl_int @ R2 ) ) ) ) ).
% 0.34/0.58
% 0.34/0.58 % wf_insert
% 0.34/0.58 thf(fact_2882_wo__rel_Owell__order__induct,axiom,
% 0.34/0.58 ! [R2: set_Pr1261947904930325089at_nat,P: nat > $o,A2: nat] :
% 0.34/0.58 ( ( bNF_We3818239936649020644el_nat @ R2 )
% 0.34/0.58 => ( ! [X3: nat] :
% 0.34/0.58 ( ! [Y5: nat] :
% 0.34/0.58 ( ( ( Y5 != X3 )
% 0.34/0.58 & ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ Y5 @ X3 ) @ R2 ) )
% 0.34/0.58 => ( P @ Y5 ) )
% 0.34/0.58 => ( P @ X3 ) )
% 0.34/0.58 => ( P @ A2 ) ) ) ).
% 0.34/0.58
% 0.34/0.58 % wo_rel.well_order_induct
% 0.34/0.58 thf(fact_2883_wo__rel_Owell__order__induct,axiom,
% 0.34/0.58 ! [R2: set_Pr958786334691620121nt_int,P: int > $o,A2: int] :
% 0.34/0.58 ( ( bNF_We3815749466139970368el_int @ R2 )
% 0.34/0.58 => ( ! [X3: int] :
% 0.34/0.58 ( ! [Y5: int] :
% 0.34/0.58 ( ( ( Y5 != X3 )
% 0.34/0.58 & ( member5262025264175285858nt_int @ ( product_Pair_int_int @ Y5 @ X3 ) @ R2 ) )
% 0.34/0.58 => ( P @ Y5 ) )
% 0.34/0.58 => ( P @ X3 ) )
% 0.34/0.58 => ( P @ A2 ) ) ) ).
% 0.34/0.58
% 0.34/0.58 % wo_rel.well_order_induct
% 0.34/0.58 thf(fact_2884_wf__induct__rule,axiom,
% 0.34/0.58 ! [R2: set_Pr1261947904930325089at_nat,P: nat > $o,A2: nat] :
% 0.34/0.58 ( ( wf_nat @ R2 )
% 0.34/0.58 => ( ! [X3: nat] :
% 0.34/0.58 ( ! [Y5: nat] :
% 0.34/0.58 ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ Y5 @ X3 ) @ R2 )
% 0.34/0.58 => ( P @ Y5 ) )
% 0.34/0.58 => ( P @ X3 ) )
% 0.34/0.58 => ( P @ A2 ) ) ) ).
% 0.34/0.58
% 0.34/0.58 % wf_induct_rule
% 0.34/0.58 thf(fact_2885_wf__induct__rule,axiom,
% 0.34/0.58 ! [R2: set_Pr958786334691620121nt_int,P: int > $o,A2: int] :
% 0.34/0.58 ( ( wf_int @ R2 )
% 0.34/0.58 => ( ! [X3: int] :
% 0.34/0.58 ( ! [Y5: int] :
% 0.34/0.58 ( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ Y5 @ X3 ) @ R2 )
% 0.34/0.58 => ( P @ Y5 ) )
% 0.34/0.58 => ( P @ X3 ) )
% 0.34/0.58 => ( P @ A2 ) ) ) ).
% 0.34/0.58
% 0.34/0.58 % wf_induct_rule
% 0.34/0.58 thf(fact_2886_wf__eq__minimal,axiom,
% 0.34/0.58 ( wf_Pro8787509627421085110_nat_o
% 0.34/0.58 = ( ^ [R4: set_Pr2161125870931222855_nat_o] :
% 0.34/0.58 ! [Q: set_Pr4532377907799695533_nat_o] :
% 0.34/0.58 ( ? [X2: produc3658429121746597890et_nat > $o] : ( member6576561426505652726_nat_o @ X2 @ Q )
% 0.34/0.58 => ? [X2: produc3658429121746597890et_nat > $o] :
% 0.34/0.58 ( ( member6576561426505652726_nat_o @ X2 @ Q )
% 0.34/0.58 & ! [Y3: produc3658429121746597890et_nat > $o] :
% 0.34/0.58 ( ( member8781333585448626064_nat_o @ ( produc7368190662567826135_nat_o @ Y3 @ X2 ) @ R4 )
% 0.34/0.58 => ~ ( member6576561426505652726_nat_o @ Y3 @ Q ) ) ) ) ) ) ).
% 0.34/0.58
% 0.34/0.58 % wf_eq_minimal
% 0.34/0.58 thf(fact_2887_wf__eq__minimal,axiom,
% 0.34/0.58 ( wf_o
% 0.34/0.58 = ( ^ [R4: set_Product_prod_o_o] :
% 0.34/0.58 ! [Q: set_o] :
% 0.34/0.58 ( ? [X2: $o] : ( member_o @ X2 @ Q )
% 0.34/0.58 => ? [X2: $o] :
% 0.34/0.58 ( ( member_o @ X2 @ Q )
% 0.34/0.58 & ! [Y3: $o] :
% 0.34/0.58 ( ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ Y3 @ X2 ) @ R4 )
% 0.34/0.58 => ~ ( member_o @ Y3 @ Q ) ) ) ) ) ) ).
% 0.34/0.58
% 0.34/0.58 % wf_eq_minimal
% 0.34/0.58 thf(fact_2888_wf__eq__minimal,axiom,
% 0.34/0.58 ( wf_Product_unit
% 0.34/0.58 = ( ^ [R4: set_Pr5094982260447487303t_unit] :
% 0.34/0.58 ! [Q: set_Product_unit] :
% 0.34/0.58 ( ? [X2: product_unit] : ( member_Product_unit @ X2 @ Q )
% 0.34/0.58 => ? [X2: product_unit] :
% 0.34/0.58 ( ( member_Product_unit @ X2 @ Q )
% 0.34/0.58 & ! [Y3: product_unit] :
% 0.34/0.58 ( ( member7821138191923116944t_unit @ ( produc5692694829384537815t_unit @ Y3 @ X2 ) @ R4 )
% 0.34/0.58 => ~ ( member_Product_unit @ Y3 @ Q ) ) ) ) ) ) ).
% 0.34/0.58
% 0.34/0.58 % wf_eq_minimal
% 0.34/0.58 thf(fact_2889_wf__eq__minimal,axiom,
% 0.34/0.58 ( wf_nat
% 0.34/0.58 = ( ^ [R4: set_Pr1261947904930325089at_nat] :
% 0.34/0.58 ! [Q: set_nat] :
% 0.34/0.58 ( ? [X2: nat] : ( member_nat @ X2 @ Q )
% 0.34/0.58 => ? [X2: nat] :
% 0.34/0.58 ( ( member_nat @ X2 @ Q )
% 0.34/0.58 & ! [Y3: nat] :
% 0.34/0.58 ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ Y3 @ X2 ) @ R4 )
% 0.34/0.58 => ~ ( member_nat @ Y3 @ Q ) ) ) ) ) ) ).
% 0.34/0.58
% 0.34/0.58 % wf_eq_minimal
% 0.34/0.58 thf(fact_2890_wf__eq__minimal,axiom,
% 0.34/0.58 ( wf_int
% 0.34/0.58 = ( ^ [R4: set_Pr958786334691620121nt_int] :
% 0.34/0.58 ! [Q: set_int] :
% 0.34/0.58 ( ? [X2: int] : ( member_int @ X2 @ Q )
% 0.34/0.58 => ? [X2: int] :
% 0.34/0.58 ( ( member_int @ X2 @ Q )
% 0.34/0.58 & ! [Y3: int] :
% 0.34/0.58 ( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ Y3 @ X2 ) @ R4 )
% 0.34/0.58 => ~ ( member_int @ Y3 @ Q ) ) ) ) ) ) ).
% 0.34/0.58
% 0.34/0.58 % wf_eq_minimal
% 0.34/0.58 thf(fact_2891_wf__not__refl,axiom,
% 0.34/0.58 ! [R2: set_Pr1261947904930325089at_nat,A2: nat] :
% 0.34/0.58 ( ( wf_nat @ R2 )
% 0.34/0.58 => ~ ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A2 @ A2 ) @ R2 ) ) ).
% 0.34/0.58
% 0.34/0.58 % wf_not_refl
% 0.34/0.58 thf(fact_2892_wf__not__refl,axiom,
% 0.34/0.58 ! [R2: set_Pr958786334691620121nt_int,A2: int] :
% 0.34/0.58 ( ( wf_int @ R2 )
% 0.34/0.58 => ~ ( member5262025264175285858nt_int @ ( product_Pair_int_int @ A2 @ A2 ) @ R2 ) ) ).
% 0.34/0.58
% 0.34/0.58 % wf_not_refl
% 0.34/0.58 thf(fact_2893_wf__not__sym,axiom,
% 0.34/0.58 ! [R2: set_Pr1261947904930325089at_nat,A2: nat,X: nat] :
% 0.34/0.58 ( ( wf_nat @ R2 )
% 0.34/0.58 => ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A2 @ X ) @ R2 )
% 0.34/0.58 => ~ ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ A2 ) @ R2 ) ) ) ).
% 0.34/0.58
% 0.34/0.58 % wf_not_sym
% 0.34/0.58 thf(fact_2894_wf__not__sym,axiom,
% 0.34/0.58 ! [R2: set_Pr958786334691620121nt_int,A2: int,X: int] :
% 0.34/0.58 ( ( wf_int @ R2 )
% 0.34/0.58 => ( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ A2 @ X ) @ R2 )
% 0.34/0.58 => ~ ( member5262025264175285858nt_int @ ( product_Pair_int_int @ X @ A2 ) @ R2 ) ) ) ).
% 0.34/0.58
% 0.34/0.58 % wf_not_sym
% 0.34/0.58 thf(fact_2895_wf__irrefl,axiom,
% 0.34/0.58 ! [R2: set_Pr1261947904930325089at_nat,A2: nat] :
% 0.34/0.58 ( ( wf_nat @ R2 )
% 0.34/0.58 => ~ ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A2 @ A2 ) @ R2 ) ) ).
% 0.34/0.58
% 0.34/0.58 % wf_irrefl
% 0.34/0.58 thf(fact_2896_wf__irrefl,axiom,
% 0.34/0.58 ! [R2: set_Pr958786334691620121nt_int,A2: int] :
% 0.34/0.58 ( ( wf_int @ R2 )
% 0.34/0.58 => ~ ( member5262025264175285858nt_int @ ( product_Pair_int_int @ A2 @ A2 ) @ R2 ) ) ).
% 0.34/0.58
% 0.34/0.58 % wf_irrefl
% 0.34/0.58 thf(fact_2897_wf__induct,axiom,
% 0.34/0.58 ! [R2: set_Pr1261947904930325089at_nat,P: nat > $o,A2: nat] :
% 0.34/0.58 ( ( wf_nat @ R2 )
% 0.34/0.58 => ( ! [X3: nat] :
% 0.34/0.58 ( ! [Y5: nat] :
% 0.34/0.58 ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ Y5 @ X3 ) @ R2 )
% 0.34/0.58 => ( P @ Y5 ) )
% 0.34/0.58 => ( P @ X3 ) )
% 0.34/0.58 => ( P @ A2 ) ) ) ).
% 0.34/0.58
% 0.34/0.58 % wf_induct
% 0.34/0.58 thf(fact_2898_wf__induct,axiom,
% 0.34/0.58 ! [R2: set_Pr958786334691620121nt_int,P: int > $o,A2: int] :
% 0.34/0.58 ( ( wf_int @ R2 )
% 0.34/0.58 => ( ! [X3: int] :
% 0.34/0.58 ( ! [Y5: int] :
% 0.34/0.58 ( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ Y5 @ X3 ) @ R2 )
% 0.34/0.58 => ( P @ Y5 ) )
% 0.34/0.58 => ( P @ X3 ) )
% 0.34/0.58 => ( P @ A2 ) ) ) ).
% 0.34/0.58
% 0.34/0.58 % wf_induct
% 0.34/0.58 thf(fact_2899_wf__asym,axiom,
% 0.34/0.58 ! [R2: set_Pr1261947904930325089at_nat,A2: nat,X: nat] :
% 0.34/0.58 ( ( wf_nat @ R2 )
% 0.34/0.58 => ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A2 @ X ) @ R2 )
% 0.34/0.58 => ~ ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ A2 ) @ R2 ) ) ) ).
% 0.34/0.58
% 0.34/0.58 % wf_asym
% 0.34/0.58 thf(fact_2900_wf__asym,axiom,
% 0.34/0.58 ! [R2: set_Pr958786334691620121nt_int,A2: int,X: int] :
% 0.34/0.58 ( ( wf_int @ R2 )
% 0.34/0.58 => ( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ A2 @ X ) @ R2 )
% 0.34/0.58 => ~ ( member5262025264175285858nt_int @ ( product_Pair_int_int @ X @ A2 ) @ R2 ) ) ) ).
% 0.34/0.58
% 0.34/0.58 % wf_asym
% 0.34/0.58 thf(fact_2901_wfUNIVI,axiom,
% 0.34/0.58 ! [R2: set_Pr1261947904930325089at_nat] :
% 0.34/0.58 ( ! [P4: nat > $o,X3: nat] :
% 0.34/0.58 ( ! [Xa2: nat] :
% 0.34/0.58 ( ! [Y2: nat] :
% 0.34/0.58 ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ Y2 @ Xa2 ) @ R2 )
% 0.34/0.58 => ( P4 @ Y2 ) )
% 0.34/0.58 => ( P4 @ Xa2 ) )
% 0.34/0.58 => ( P4 @ X3 ) )
% 0.34/0.58 => ( wf_nat @ R2 ) ) ).
% 0.34/0.58
% 0.34/0.58 % wfUNIVI
% 0.34/0.58 thf(fact_2902_wfUNIVI,axiom,
% 0.34/0.58 ! [R2: set_Pr958786334691620121nt_int] :
% 0.34/0.58 ( ! [P4: int > $o,X3: int] :
% 0.34/0.58 ( ! [Xa2: int] :
% 0.34/0.58 ( ! [Y2: int] :
% 0.34/0.58 ( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ Y2 @ Xa2 ) @ R2 )
% 0.34/0.58 => ( P4 @ Y2 ) )
% 0.34/0.58 => ( P4 @ Xa2 ) )
% 0.34/0.58 => ( P4 @ X3 ) )
% 0.34/0.58 => ( wf_int @ R2 ) ) ).
% 0.34/0.58
% 0.34/0.58 % wfUNIVI
% 0.34/0.58 thf(fact_2903_wfI__min,axiom,
% 0.34/0.58 ! [R3: set_Pr2161125870931222855_nat_o] :
% 0.34/0.58 ( ! [X3: produc3658429121746597890et_nat > $o,Q3: set_Pr4532377907799695533_nat_o] :
% 0.34/0.58 ( ( member6576561426505652726_nat_o @ X3 @ Q3 )
% 0.34/0.58 => ? [Xa2: produc3658429121746597890et_nat > $o] :
% 0.34/0.58 ( ( member6576561426505652726_nat_o @ Xa2 @ Q3 )
% 0.34/0.58 & ! [Y2: produc3658429121746597890et_nat > $o] :
% 0.34/0.58 ( ( member8781333585448626064_nat_o @ ( produc7368190662567826135_nat_o @ Y2 @ Xa2 ) @ R3 )
% 0.34/0.58 => ~ ( member6576561426505652726_nat_o @ Y2 @ Q3 ) ) ) )
% 0.34/0.58 => ( wf_Pro8787509627421085110_nat_o @ R3 ) ) ).
% 0.34/0.58
% 0.34/0.58 % wfI_min
% 0.34/0.58 thf(fact_2904_wfI__min,axiom,
% 0.34/0.58 ! [R3: set_Product_prod_o_o] :
% 0.34/0.58 ( ! [X3: $o,Q3: set_o] :
% 0.34/0.58 ( ( member_o @ X3 @ Q3 )
% 0.34/0.58 => ? [Xa2: $o] :
% 0.34/0.58 ( ( member_o @ Xa2 @ Q3 )
% 0.34/0.58 & ! [Y2: $o] :
% 0.34/0.58 ( ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ Y2 @ Xa2 ) @ R3 )
% 0.34/0.58 => ~ ( member_o @ Y2 @ Q3 ) ) ) )
% 0.34/0.58 => ( wf_o @ R3 ) ) ).
% 0.34/0.58
% 0.34/0.58 % wfI_min
% 0.34/0.58 thf(fact_2905_wfI__min,axiom,
% 0.34/0.58 ! [R3: set_Pr5094982260447487303t_unit] :
% 0.34/0.58 ( ! [X3: product_unit,Q3: set_Product_unit] :
% 0.34/0.58 ( ( member_Product_unit @ X3 @ Q3 )
% 0.34/0.58 => ? [Xa2: product_unit] :
% 0.34/0.58 ( ( member_Product_unit @ Xa2 @ Q3 )
% 0.34/0.58 & ! [Y2: product_unit] :
% 0.34/0.58 ( ( member7821138191923116944t_unit @ ( produc5692694829384537815t_unit @ Y2 @ Xa2 ) @ R3 )
% 0.34/0.58 => ~ ( member_Product_unit @ Y2 @ Q3 ) ) ) )
% 0.34/0.58 => ( wf_Product_unit @ R3 ) ) ).
% 0.34/0.58
% 0.34/0.58 % wfI_min
% 0.34/0.58 thf(fact_2906_wfI__min,axiom,
% 0.34/0.58 ! [R3: set_Pr1261947904930325089at_nat] :
% 0.34/0.58 ( ! [X3: nat,Q3: set_nat] :
% 0.34/0.58 ( ( member_nat @ X3 @ Q3 )
% 0.34/0.58 => ? [Xa2: nat] :
% 0.34/0.58 ( ( member_nat @ Xa2 @ Q3 )
% 0.34/0.58 & ! [Y2: nat] :
% 0.34/0.58 ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ Y2 @ Xa2 ) @ R3 )
% 0.34/0.58 => ~ ( member_nat @ Y2 @ Q3 ) ) ) )
% 0.34/0.58 => ( wf_nat @ R3 ) ) ).
% 0.34/0.58
% 0.34/0.58 % wfI_min
% 0.34/0.58 thf(fact_2907_wfI__min,axiom,
% 0.34/0.58 ! [R3: set_Pr958786334691620121nt_int] :
% 0.34/0.58 ( ! [X3: int,Q3: set_int] :
% 0.34/0.58 ( ( member_int @ X3 @ Q3 )
% 0.34/0.58 => ? [Xa2: int] :
% 0.34/0.58 ( ( member_int @ Xa2 @ Q3 )
% 0.34/0.58 & ! [Y2: int] :
% 0.34/0.58 ( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ Y2 @ Xa2 ) @ R3 )
% 0.34/0.58 => ~ ( member_int @ Y2 @ Q3 ) ) ) )
% 0.34/0.58 => ( wf_int @ R3 ) ) ).
% 0.34/0.58
% 0.34/0.58 % wfI_min
% 0.34/0.58 thf(fact_2908_wfE__min,axiom,
% 0.34/0.58 ! [R3: set_Pr2161125870931222855_nat_o,X: produc3658429121746597890et_nat > $o,Q2: set_Pr4532377907799695533_nat_o] :
% 0.34/0.58 ( ( wf_Pro8787509627421085110_nat_o @ R3 )
% 0.34/0.58 => ( ( member6576561426505652726_nat_o @ X @ Q2 )
% 0.34/0.58 => ~ ! [Z3: produc3658429121746597890et_nat > $o] :
% 0.34/0.58 ( ( member6576561426505652726_nat_o @ Z3 @ Q2 )
% 0.34/0.58 => ~ ! [Y5: produc3658429121746597890et_nat > $o] :
% 0.34/0.58 ( ( member8781333585448626064_nat_o @ ( produc7368190662567826135_nat_o @ Y5 @ Z3 ) @ R3 )
% 0.34/0.58 => ~ ( member6576561426505652726_nat_o @ Y5 @ Q2 ) ) ) ) ) ).
% 0.34/0.58
% 0.34/0.58 % wfE_min
% 0.34/0.58 thf(fact_2909_wfE__min,axiom,
% 0.34/0.58 ! [R3: set_Product_prod_o_o,X: $o,Q2: set_o] :
% 0.34/0.58 ( ( wf_o @ R3 )
% 0.34/0.58 => ( ( member_o @ X @ Q2 )
% 0.34/0.58 => ~ ! [Z3: $o] :
% 0.34/0.58 ( ( member_o @ Z3 @ Q2 )
% 0.34/0.58 => ~ ! [Y5: $o] :
% 0.34/0.58 ( ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ Y5 @ Z3 ) @ R3 )
% 0.34/0.58 => ~ ( member_o @ Y5 @ Q2 ) ) ) ) ) ).
% 0.34/0.58
% 0.34/0.58 % wfE_min
% 0.34/0.58 thf(fact_2910_wfE__min,axiom,
% 0.34/0.58 ! [R3: set_Pr5094982260447487303t_unit,X: product_unit,Q2: set_Product_unit] :
% 0.34/0.58 ( ( wf_Product_unit @ R3 )
% 0.34/0.58 => ( ( member_Product_unit @ X @ Q2 )
% 0.34/0.58 => ~ ! [Z3: product_unit] :
% 0.34/0.58 ( ( member_Product_unit @ Z3 @ Q2 )
% 0.34/0.58 => ~ ! [Y5: product_unit] :
% 0.34/0.58 ( ( member7821138191923116944t_unit @ ( produc5692694829384537815t_unit @ Y5 @ Z3 ) @ R3 )
% 0.34/0.58 => ~ ( member_Product_unit @ Y5 @ Q2 ) ) ) ) ) ).
% 0.34/0.58
% 0.34/0.58 % wfE_min
% 0.34/0.58 thf(fact_2911_wfE__min,axiom,
% 0.34/0.58 ! [R3: set_Pr1261947904930325089at_nat,X: nat,Q2: set_nat] :
% 0.34/0.58 ( ( wf_nat @ R3 )
% 0.34/0.58 => ( ( member_nat @ X @ Q2 )
% 0.34/0.58 => ~ ! [Z3: nat] :
% 0.34/0.58 ( ( member_nat @ Z3 @ Q2 )
% 0.34/0.58 => ~ ! [Y5: nat] :
% 0.34/0.58 ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ Y5 @ Z3 ) @ R3 )
% 0.34/0.58 => ~ ( member_nat @ Y5 @ Q2 ) ) ) ) ) ).
% 0.34/0.58
% 0.34/0.58 % wfE_min
% 0.34/0.58 thf(fact_2912_wfE__min,axiom,
% 0.34/0.58 ! [R3: set_Pr958786334691620121nt_int,X: int,Q2: set_int] :
% 0.34/0.58 ( ( wf_int @ R3 )
% 0.34/0.58 => ( ( member_int @ X @ Q2 )
% 0.34/0.58 => ~ ! [Z3: int] :
% 0.34/0.58 ( ( member_int @ Z3 @ Q2 )
% 0.34/0.58 => ~ ! [Y5: int] :
% 0.34/0.58 ( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ Y5 @ Z3 ) @ R3 )
% 0.34/0.58 => ~ ( member_int @ Y5 @ Q2 ) ) ) ) ) ).
% 0.34/0.58
% 0.34/0.58 % wfE_min
% 0.34/0.58 thf(fact_2913_wf__def,axiom,
% 0.34/0.58 ( wf_nat
% 0.34/0.58 = ( ^ [R4: set_Pr1261947904930325089at_nat] :
% 0.34/0.58 ! [P2: nat > $o] :
% 0.34/0.58 ( ! [X2: nat] :
% 0.34/0.58 ( ! [Y3: nat] :
% 0.34/0.58 ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ Y3 @ X2 ) @ R4 )
% 0.34/0.58 => ( P2 @ Y3 ) )
% 0.34/0.58 => ( P2 @ X2 ) )
% 0.34/0.58 => ! [X7: nat] : ( P2 @ X7 ) ) ) ) ).
% 0.34/0.58
% 0.34/0.58 % wf_def
% 0.34/0.58 thf(fact_2914_wf__def,axiom,
% 0.34/0.58 ( wf_int
% 0.34/0.58 = ( ^ [R4: set_Pr958786334691620121nt_int] :
% 0.34/0.58 ! [P2: int > $o] :
% 0.34/0.58 ( ! [X2: int] :
% 0.34/0.58 ( ! [Y3: int] :
% 0.34/0.58 ( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ Y3 @ X2 ) @ R4 )
% 0.34/0.58 => ( P2 @ Y3 ) )
% 0.34/0.58 => ( P2 @ X2 ) )
% 0.34/0.58 => ! [X7: int] : ( P2 @ X7 ) ) ) ) ).
% 0.34/0.58
% 0.34/0.58 % wf_def
% 0.34/0.58 thf(fact_2915_well__order__on__domain,axiom,
% 0.34/0.58 ! [A: set_Pr4532377907799695533_nat_o,R2: set_Pr2161125870931222855_nat_o,A2: produc3658429121746597890et_nat > $o,B: produc3658429121746597890et_nat > $o] :
% 0.34/0.58 ( ( order_2118566763991480939_nat_o @ A @ R2 )
% 0.34/0.58 => ( ( member8781333585448626064_nat_o @ ( produc7368190662567826135_nat_o @ A2 @ B ) @ R2 )
% 0.34/0.58 => ( ( member6576561426505652726_nat_o @ A2 @ A )
% 0.34/0.58 & ( member6576561426505652726_nat_o @ B @ A ) ) ) ) ).
% 0.34/0.58
% 0.34/0.58 % well_order_on_domain
% 0.34/0.58 thf(fact_2916_well__order__on__domain,axiom,
% 0.34/0.58 ! [A: set_o,R2: set_Product_prod_o_o,A2: $o,B: $o] :
% 0.34/0.58 ( ( order_2615775362358406454r_on_o @ A @ R2 )
% 0.34/0.58 => ( ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ A2 @ B ) @ R2 )
% 0.34/0.58 => ( ( member_o @ A2 @ A )
% 0.34/0.58 & ( member_o @ B @ A ) ) ) ) ).
% 0.34/0.58
% 0.34/0.58 % well_order_on_domain
% 0.34/0.58 thf(fact_2917_well__order__on__domain,axiom,
% 0.34/0.58 ! [A: set_Product_unit,R2: set_Pr5094982260447487303t_unit,A2: product_unit,B: product_unit] :
% 0.34/0.58 ( ( order_3164716736300062379t_unit @ A @ R2 )
% 0.34/0.58 => ( ( member7821138191923116944t_unit @ ( produc5692694829384537815t_unit @ A2 @ B ) @ R2 )
% 0.34/0.58 => ( ( member_Product_unit @ A2 @ A )
% 0.34/0.58 & ( member_Product_unit @ B @ A ) ) ) ) ).
% 0.34/0.58
% 0.34/0.58 % well_order_on_domain
% 0.34/0.58 thf(fact_2918_well__order__on__domain,axiom,
% 0.34/0.58 ! [A: set_nat,R2: set_Pr1261947904930325089at_nat,A2: nat,B: nat] :
% 0.34/0.58 ( ( order_2888998067076097458on_nat @ A @ R2 )
% 0.34/0.58 => ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A2 @ B ) @ R2 )
% 0.34/0.58 => ( ( member_nat @ A2 @ A )
% 0.34/0.58 & ( member_nat @ B @ A ) ) ) ) ).
% 0.34/0.58
% 0.34/0.58 % well_order_on_domain
% 0.34/0.58 thf(fact_2919_well__order__on__domain,axiom,
% 0.34/0.58 ! [A: set_int,R2: set_Pr958786334691620121nt_int,A2: int,B: int] :
% 0.34/0.58 ( ( order_2886507596567047182on_int @ A @ R2 )
% 0.34/0.58 => ( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ A2 @ B ) @ R2 )
% 0.34/0.58 => ( ( member_int @ A2 @ A )
% 0.34/0.58 & ( member_int @ B @ A ) ) ) ) ).
% 0.34/0.58
% 0.34/0.58 % well_order_on_domain
% 0.34/0.58 thf(fact_2920_underS__E,axiom,
% 0.34/0.58 ! [I2: produc3658429121746597890et_nat > $o,R3: set_Pr2161125870931222855_nat_o,J: produc3658429121746597890et_nat > $o] :
% 0.34/0.58 ( ( member6576561426505652726_nat_o @ I2 @ ( order_7262550146012401043_nat_o @ R3 @ J ) )
% 0.34/0.58 => ( ( I2 != J )
% 0.34/0.58 & ( member8781333585448626064_nat_o @ ( produc7368190662567826135_nat_o @ I2 @ J ) @ R3 ) ) ) ).
% 0.34/0.58
% 0.34/0.58 % underS_E
% 0.34/0.58 thf(fact_2921_underS__E,axiom,
% 0.34/0.58 ! [I2: $o,R3: set_Product_prod_o_o,J: $o] :
% 0.34/0.58 ( ( member_o @ I2 @ ( order_underS_o @ R3 @ J ) )
% 0.34/0.58 => ( ( I2 != J )
% 0.34/0.58 & ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ I2 @ J ) @ R3 ) ) ) ).
% 0.34/0.58
% 0.34/0.58 % underS_E
% 0.34/0.58 thf(fact_2922_underS__E,axiom,
% 0.34/0.58 ! [I2: product_unit,R3: set_Pr5094982260447487303t_unit,J: product_unit] :
% 0.34/0.58 ( ( member_Product_unit @ I2 @ ( order_5809128134054296019t_unit @ R3 @ J ) )
% 0.34/0.58 => ( ( I2 != J )
% 0.34/0.58 & ( member7821138191923116944t_unit @ ( produc5692694829384537815t_unit @ I2 @ J ) @ R3 ) ) ) ).
% 0.34/0.58
% 0.34/0.58 % underS_E
% 0.34/0.58 thf(fact_2923_underS__E,axiom,
% 0.34/0.58 ! [I2: nat,R3: set_Pr1261947904930325089at_nat,J: nat] :
% 0.34/0.58 ( ( member_nat @ I2 @ ( order_underS_nat @ R3 @ J ) )
% 0.34/0.58 => ( ( I2 != J )
% 0.34/0.58 & ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ I2 @ J ) @ R3 ) ) ) ).
% 0.34/0.58
% 0.34/0.58 % underS_E
% 0.34/0.58 thf(fact_2924_underS__E,axiom,
% 0.34/0.58 ! [I2: int,R3: set_Pr958786334691620121nt_int,J: int] :
% 0.34/0.58 ( ( member_int @ I2 @ ( order_underS_int @ R3 @ J ) )
% 0.34/0.58 => ( ( I2 != J )
% 0.34/0.58 & ( member5262025264175285858nt_int @ ( product_Pair_int_int @ I2 @ J ) @ R3 ) ) ) ).
% 0.34/0.58
% 0.34/0.58 % underS_E
% 0.34/0.58 thf(fact_2925_underS__I,axiom,
% 0.34/0.58 ! [I2: produc3658429121746597890et_nat > $o,J: produc3658429121746597890et_nat > $o,R3: set_Pr2161125870931222855_nat_o] :
% 0.34/0.58 ( ( I2 != J )
% 0.34/0.58 => ( ( member8781333585448626064_nat_o @ ( produc7368190662567826135_nat_o @ I2 @ J ) @ R3 )
% 0.34/0.58 => ( member6576561426505652726_nat_o @ I2 @ ( order_7262550146012401043_nat_o @ R3 @ J ) ) ) ) ).
% 0.34/0.58
% 0.34/0.58 % underS_I
% 0.34/0.58 thf(fact_2926_underS__I,axiom,
% 0.34/0.58 ! [I2: $o,J: $o,R3: set_Product_prod_o_o] :
% 0.34/0.58 ( ( I2 != J )
% 0.34/0.58 => ( ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ I2 @ J ) @ R3 )
% 0.34/0.58 => ( member_o @ I2 @ ( order_underS_o @ R3 @ J ) ) ) ) ).
% 0.34/0.58
% 0.34/0.58 % underS_I
% 0.34/0.58 thf(fact_2927_underS__I,axiom,
% 0.34/0.58 ! [I2: product_unit,J: product_unit,R3: set_Pr5094982260447487303t_unit] :
% 0.34/0.58 ( ( I2 != J )
% 0.34/0.58 => ( ( member7821138191923116944t_unit @ ( produc5692694829384537815t_unit @ I2 @ J ) @ R3 )
% 0.34/0.58 => ( member_Product_unit @ I2 @ ( order_5809128134054296019t_unit @ R3 @ J ) ) ) ) ).
% 0.34/0.58
% 0.34/0.58 % underS_I
% 0.34/0.58 thf(fact_2928_underS__I,axiom,
% 0.34/0.58 ! [I2: nat,J: nat,R3: set_Pr1261947904930325089at_nat] :
% 0.34/0.58 ( ( I2 != J )
% 0.34/0.58 => ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ I2 @ J ) @ R3 )
% 0.34/0.58 => ( member_nat @ I2 @ ( order_underS_nat @ R3 @ J ) ) ) ) ).
% 0.34/0.58
% 0.34/0.58 % underS_I
% 0.34/0.58 thf(fact_2929_underS__I,axiom,
% 0.34/0.58 ! [I2: int,J: int,R3: set_Pr958786334691620121nt_int] :
% 0.34/0.58 ( ( I2 != J )
% 0.34/0.58 => ( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ I2 @ J ) @ R3 )
% 0.34/0.58 => ( member_int @ I2 @ ( order_underS_int @ R3 @ J ) ) ) ) ).
% 0.34/0.58
% 0.34/0.58 % underS_I
% 0.34/0.58 thf(fact_2930_BNF__Least__Fixpoint_OunderS__Field,axiom,
% 0.34/0.58 ! [I2: produc3658429121746597890et_nat > $o,R3: set_Pr2161125870931222855_nat_o,J: produc3658429121746597890et_nat > $o] :
% 0.34/0.58 ( ( member6576561426505652726_nat_o @ I2 @ ( order_7262550146012401043_nat_o @ R3 @ J ) )
% 0.34/0.58 => ( member6576561426505652726_nat_o @ I2 @ ( field_81821745747897882_nat_o @ R3 ) ) ) ).
% 0.34/0.58
% 0.34/0.58 % BNF_Least_Fixpoint.underS_Field
% 0.34/0.58 thf(fact_2931_BNF__Least__Fixpoint_OunderS__Field,axiom,
% 0.34/0.58 ! [I2: nat,R3: set_Pr1261947904930325089at_nat,J: nat] :
% 0.34/0.58 ( ( member_nat @ I2 @ ( order_underS_nat @ R3 @ J ) )
% 0.34/0.58 => ( member_nat @ I2 @ ( field_nat @ R3 ) ) ) ).
% 0.34/0.58
% 0.34/0.58 % BNF_Least_Fixpoint.underS_Field
% 0.34/0.58 thf(fact_2932_BNF__Least__Fixpoint_OunderS__Field,axiom,
% 0.34/0.58 ! [I2: int,R3: set_Pr958786334691620121nt_int,J: int] :
% 0.34/0.58 ( ( member_int @ I2 @ ( order_underS_int @ R3 @ J ) )
% 0.34/0.58 => ( member_int @ I2 @ ( field_int @ R3 ) ) ) ).
% 0.34/0.58
% 0.34/0.58 % BNF_Least_Fixpoint.underS_Field
% 0.34/0.58 thf(fact_2933_BNF__Least__Fixpoint_OunderS__Field,axiom,
% 0.34/0.58 ! [I2: $o,R3: set_Product_prod_o_o,J: $o] :
% 0.34/0.58 ( ( member_o @ I2 @ ( order_underS_o @ R3 @ J ) )
% 0.34/0.58 => ( member_o @ I2 @ ( field_o @ R3 ) ) ) ).
% 0.34/0.58
% 0.34/0.58 % BNF_Least_Fixpoint.underS_Field
% 0.34/0.58 thf(fact_2934_BNF__Least__Fixpoint_OunderS__Field,axiom,
% 0.34/0.58 ! [I2: product_unit,R3: set_Pr5094982260447487303t_unit,J: product_unit] :
% 0.34/0.58 ( ( member_Product_unit @ I2 @ ( order_5809128134054296019t_unit @ R3 @ J ) )
% 0.34/0.58 => ( member_Product_unit @ I2 @ ( field_Product_unit @ R3 ) ) ) ).
% 0.34/0.58
% 0.34/0.58 % BNF_Least_Fixpoint.underS_Field
% 0.34/0.58 thf(fact_2935_wo__rel_OTOTALS,axiom,
% 0.34/0.58 ! [R2: set_Pr1261947904930325089at_nat] :
% 0.34/0.58 ( ( bNF_We3818239936649020644el_nat @ R2 )
% 0.34/0.58 => ! [X6: nat] :
% 0.34/0.58 ( ( member_nat @ X6 @ ( field_nat @ R2 ) )
% 0.34/0.58 => ! [Xa2: nat] :
% 0.34/0.58 ( ( member_nat @ Xa2 @ ( field_nat @ R2 ) )
% 0.34/0.58 => ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X6 @ Xa2 ) @ R2 )
% 0.34/0.58 | ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ Xa2 @ X6 ) @ R2 ) ) ) ) ) ).
% 0.34/0.58
% 0.34/0.58 % wo_rel.TOTALS
% 0.34/0.58 thf(fact_2936_wo__rel_OTOTALS,axiom,
% 0.34/0.58 ! [R2: set_Pr958786334691620121nt_int] :
% 0.34/0.58 ( ( bNF_We3815749466139970368el_int @ R2 )
% 0.34/0.58 => ! [X6: int] :
% 0.34/0.58 ( ( member_int @ X6 @ ( field_int @ R2 ) )
% 0.34/0.58 => ! [Xa2: int] :
% 0.34/0.58 ( ( member_int @ Xa2 @ ( field_int @ R2 ) )
% 0.34/0.58 => ( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ X6 @ Xa2 ) @ R2 )
% 0.34/0.58 | ( member5262025264175285858nt_int @ ( product_Pair_int_int @ Xa2 @ X6 ) @ R2 ) ) ) ) ) ).
% 0.34/0.58
% 0.34/0.58 % wo_rel.TOTALS
% 0.34/0.58 thf(fact_2937_wfE__min_H,axiom,
% 0.34/0.58 ! [R3: set_Pr5094982260447487303t_unit,Q2: set_Product_unit] :
% 0.34/0.58 ( ( wf_Product_unit @ R3 )
% 0.34/0.58 => ( ( Q2 != bot_bo3957492148770167129t_unit )
% 0.34/0.58 => ~ ! [Z3: product_unit] :
% 0.34/0.58 ( ( member_Product_unit @ Z3 @ Q2 )
% 0.34/0.58 => ~ ! [Y5: product_unit] :
% 0.34/0.58 ( ( member7821138191923116944t_unit @ ( produc5692694829384537815t_unit @ Y5 @ Z3 ) @ R3 )
% 0.34/0.58 => ~ ( member_Product_unit @ Y5 @ Q2 ) ) ) ) ) ).
% 0.34/0.58
% 0.34/0.58 % wfE_min'
% 0.34/0.58 thf(fact_2938_wfE__min_H,axiom,
% 0.34/0.58 ! [R3: set_Product_prod_o_o,Q2: set_o] :
% 0.34/0.58 ( ( wf_o @ R3 )
% 0.34/0.58 => ( ( Q2 != bot_bot_set_o )
% 0.34/0.58 => ~ ! [Z3: $o] :
% 0.34/0.58 ( ( member_o @ Z3 @ Q2 )
% 0.34/0.58 => ~ ! [Y5: $o] :
% 0.34/0.58 ( ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ Y5 @ Z3 ) @ R3 )
% 0.34/0.58 => ~ ( member_o @ Y5 @ Q2 ) ) ) ) ) ).
% 0.34/0.58
% 0.34/0.58 % wfE_min'
% 0.34/0.58 thf(fact_2939_wfE__min_H,axiom,
% 0.34/0.58 ! [R3: set_Pr1261947904930325089at_nat,Q2: set_nat] :
% 0.34/0.58 ( ( wf_nat @ R3 )
% 0.34/0.58 => ( ( Q2 != bot_bot_set_nat )
% 0.34/0.58 => ~ ! [Z3: nat] :
% 0.34/0.58 ( ( member_nat @ Z3 @ Q2 )
% 0.34/0.58 => ~ ! [Y5: nat] :
% 0.34/0.58 ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ Y5 @ Z3 ) @ R3 )
% 0.34/0.58 => ~ ( member_nat @ Y5 @ Q2 ) ) ) ) ) ).
% 0.34/0.58
% 0.34/0.58 % wfE_min'
% 0.34/0.58 thf(fact_2940_wfE__min_H,axiom,
% 0.34/0.58 ! [R3: set_Pr958786334691620121nt_int,Q2: set_int] :
% 0.34/0.58 ( ( wf_int @ R3 )
% 0.34/0.58 => ( ( Q2 != bot_bot_set_int )
% 0.34/0.58 => ~ ! [Z3: int] :
% 0.34/0.58 ( ( member_int @ Z3 @ Q2 )
% 0.34/0.58 => ~ ! [Y5: int] :
% 0.34/0.58 ( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ Y5 @ Z3 ) @ R3 )
% 0.34/0.58 => ~ ( member_int @ Y5 @ Q2 ) ) ) ) ) ).
% 0.34/0.58
% 0.34/0.58 % wfE_min'
% 0.34/0.58 thf(fact_2941_wfE__min_H,axiom,
% 0.34/0.58 ! [R3: set_Pr2161125870931222855_nat_o,Q2: set_Pr4532377907799695533_nat_o] :
% 0.34/0.58 ( ( wf_Pro8787509627421085110_nat_o @ R3 )
% 0.34/0.58 => ( ( Q2 != bot_bo7824918357723582233_nat_o )
% 0.34/0.58 => ~ ! [Z3: produc3658429121746597890et_nat > $o] :
% 0.34/0.58 ( ( member6576561426505652726_nat_o @ Z3 @ Q2 )
% 0.34/0.58 => ~ ! [Y5: produc3658429121746597890et_nat > $o] :
% 0.34/0.58 ( ( member8781333585448626064_nat_o @ ( produc7368190662567826135_nat_o @ Y5 @ Z3 ) @ R3 )
% 0.34/0.58 => ~ ( member6576561426505652726_nat_o @ Y5 @ Q2 ) ) ) ) ) ).
% 0.34/0.58
% 0.34/0.58 % wfE_min'
% 0.34/0.58 thf(fact_2942_well__order__on__empty,axiom,
% 0.34/0.58 order_3164716736300062379t_unit @ bot_bo3957492148770167129t_unit @ bot_bo1654022762305692595t_unit ).
% 0.34/0.58
% 0.34/0.58 % well_order_on_empty
% 0.34/0.58 thf(fact_2943_well__order__on__empty,axiom,
% 0.34/0.58 order_2615775362358406454r_on_o @ bot_bot_set_o @ bot_bo7073875226086086771od_o_o ).
% 0.34/0.58
% 0.34/0.58 % well_order_on_empty
% 0.34/0.58 thf(fact_2944_well__order__on__empty,axiom,
% 0.34/0.58 order_2888998067076097458on_nat @ bot_bot_set_nat @ bot_bo2099793752762293965at_nat ).
% 0.34/0.58
% 0.34/0.58 % well_order_on_empty
% 0.34/0.58 thf(fact_2945_well__order__on__empty,axiom,
% 0.34/0.58 order_2886507596567047182on_int @ bot_bot_set_int @ bot_bo1796632182523588997nt_int ).
% 0.34/0.58
% 0.34/0.58 % well_order_on_empty
% 0.34/0.58 thf(fact_2946_well__order__on__empty,axiom,
% 0.34/0.58 order_2118566763991480939_nat_o @ bot_bo7824918357723582233_nat_o @ bot_bo3657187498359500723_nat_o ).
% 0.34/0.58
% 0.34/0.58 % well_order_on_empty
% 0.34/0.58 thf(fact_2947_well__order__induct__imp,axiom,
% 0.34/0.58 ! [R2: set_Pr2161125870931222855_nat_o,P: ( produc3658429121746597890et_nat > $o ) > $o,A2: produc3658429121746597890et_nat > $o] :
% 0.34/0.58 ( ( bNF_We740377018556551609_nat_o @ R2 )
% 0.34/0.58 => ( ! [X3: produc3658429121746597890et_nat > $o] :
% 0.34/0.58 ( ! [Y5: produc3658429121746597890et_nat > $o] :
% 0.34/0.58 ( ( ( Y5 != X3 )
% 0.34/0.58 & ( member8781333585448626064_nat_o @ ( produc7368190662567826135_nat_o @ Y5 @ X3 ) @ R2 ) )
% 0.34/0.58 => ( ( member6576561426505652726_nat_o @ Y5 @ ( field_81821745747897882_nat_o @ R2 ) )
% 0.34/0.58 => ( P @ Y5 ) ) )
% 0.34/0.58 => ( ( member6576561426505652726_nat_o @ X3 @ ( field_81821745747897882_nat_o @ R2 ) )
% 0.34/0.58 => ( P @ X3 ) ) )
% 0.34/0.58 => ( ( member6576561426505652726_nat_o @ A2 @ ( field_81821745747897882_nat_o @ R2 ) )
% 0.34/0.58 => ( P @ A2 ) ) ) ) ).
% 0.34/0.58
% 0.34/0.58 % well_order_induct_imp
% 0.34/0.58 thf(fact_2948_well__order__induct__imp,axiom,
% 0.34/0.58 ! [R2: set_Product_prod_o_o,P: $o > $o,A2: $o] :
% 0.34/0.58 ( ( bNF_We6437023980930612612_rel_o @ R2 )
% 0.34/0.58 => ( ! [X3: $o] :
% 0.34/0.58 ( ! [Y5: $o] :
% 0.34/0.58 ( ( ( Y5 != X3 )
% 0.34/0.58 & ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ Y5 @ X3 ) @ R2 ) )
% 0.34/0.58 => ( ( member_o @ Y5 @ ( field_o @ R2 ) )
% 0.34/0.58 => ( P @ Y5 ) ) )
% 0.34/0.58 => ( ( member_o @ X3 @ ( field_o @ R2 ) )
% 0.34/0.58 => ( P @ X3 ) ) )
% 0.34/0.58 => ( ( member_o @ A2 @ ( field_o @ R2 ) )
% 0.34/0.58 => ( P @ A2 ) ) ) ) ).
% 0.34/0.58
% 0.34/0.58 % well_order_induct_imp
% 0.34/0.58 thf(fact_2949_well__order__induct__imp,axiom,
% 0.34/0.58 ! [R2: set_Pr5094982260447487303t_unit,P: product_unit > $o,A2: product_unit] :
% 0.34/0.58 ( ( bNF_We2091578341578594809t_unit @ R2 )
% 0.34/0.58 => ( ! [X3: product_unit] :
% 0.34/0.58 ( ! [Y5: product_unit] :
% 0.34/0.58 ( ( ( Y5 != X3 )
% 0.34/0.58 & ( member7821138191923116944t_unit @ ( produc5692694829384537815t_unit @ Y5 @ X3 ) @ R2 ) )
% 0.34/0.58 => ( ( member_Product_unit @ Y5 @ ( field_Product_unit @ R2 ) )
% 0.34/0.58 => ( P @ Y5 ) ) )
% 0.34/0.58 => ( ( member_Product_unit @ X3 @ ( field_Product_unit @ R2 ) )
% 0.34/0.58 => ( P @ X3 ) ) )
% 0.34/0.58 => ( ( member_Product_unit @ A2 @ ( field_Product_unit @ R2 ) )
% 0.34/0.58 => ( P @ A2 ) ) ) ) ).
% 0.34/0.58
% 0.34/0.58 % well_order_induct_imp
% 0.34/0.58 thf(fact_2950_well__order__induct__imp,axiom,
% 0.34/0.58 ! [R2: set_Pr1261947904930325089at_nat,P: nat > $o,A2: nat] :
% 0.34/0.58 ( ( bNF_We3818239936649020644el_nat @ R2 )
% 0.34/0.58 => ( ! [X3: nat] :
% 0.34/0.58 ( ! [Y5: nat] :
% 0.34/0.58 ( ( ( Y5 != X3 )
% 0.34/0.58 & ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ Y5 @ X3 ) @ R2 ) )
% 0.34/0.58 => ( ( member_nat @ Y5 @ ( field_nat @ R2 ) )
% 0.34/0.58 => ( P @ Y5 ) ) )
% 0.34/0.58 => ( ( member_nat @ X3 @ ( field_nat @ R2 ) )
% 0.34/0.58 => ( P @ X3 ) ) )
% 0.34/0.58 => ( ( member_nat @ A2 @ ( field_nat @ R2 ) )
% 0.34/0.58 => ( P @ A2 ) ) ) ) ).
% 0.34/0.58
% 0.34/0.58 % well_order_induct_imp
% 0.34/0.58 thf(fact_2951_well__order__induct__imp,axiom,
% 0.34/0.58 ! [R2: set_Pr958786334691620121nt_int,P: int > $o,A2: int] :
% 0.34/0.58 ( ( bNF_We3815749466139970368el_int @ R2 )
% 0.34/0.58 => ( ! [X3: int] :
% 0.34/0.58 ( ! [Y5: int] :
% 0.34/0.58 ( ( ( Y5 != X3 )
% 0.34/0.58 & ( member5262025264175285858nt_int @ ( product_Pair_int_int @ Y5 @ X3 ) @ R2 ) )
% 0.34/0.58 => ( ( member_int @ Y5 @ ( field_int @ R2 ) )
% 0.34/0.58 => ( P @ Y5 ) ) )
% 0.34/0.58 => ( ( member_int @ X3 @ ( field_int @ R2 ) )
% 0.34/0.58 => ( P @ X3 ) ) )
% 0.34/0.58 => ( ( member_int @ A2 @ ( field_int @ R2 ) )
% 0.34/0.58 => ( P @ A2 ) ) ) ) ).
% 0.34/0.58
% 0.34/0.58 % well_order_induct_imp
% 0.34/0.58 thf(fact_2952_underS__empty,axiom,
% 0.34/0.58 ! [A2: product_unit,R2: set_Pr5094982260447487303t_unit] :
% 0.34/0.58 ( ~ ( member_Product_unit @ A2 @ ( field_Product_unit @ R2 ) )
% 0.34/0.58 => ( ( order_5809128134054296019t_unit @ R2 @ A2 )
% 0.34/0.58 = bot_bo3957492148770167129t_unit ) ) ).
% 0.34/0.58
% 0.34/0.58 % underS_empty
% 0.34/0.58 thf(fact_2953_underS__empty,axiom,
% 0.34/0.58 ! [A2: $o,R2: set_Product_prod_o_o] :
% 0.34/0.58 ( ~ ( member_o @ A2 @ ( field_o @ R2 ) )
% 0.34/0.58 => ( ( order_underS_o @ R2 @ A2 )
% 0.34/0.58 = bot_bot_set_o ) ) ).
% 0.34/0.58
% 0.34/0.58 % underS_empty
% 0.34/0.58 thf(fact_2954_underS__empty,axiom,
% 0.34/0.58 ! [A2: nat,R2: set_Pr1261947904930325089at_nat] :
% 0.34/0.58 ( ~ ( member_nat @ A2 @ ( field_nat @ R2 ) )
% 0.34/0.58 => ( ( order_underS_nat @ R2 @ A2 )
% 0.34/0.58 = bot_bot_set_nat ) ) ).
% 0.34/0.58
% 0.34/0.58 % underS_empty
% 0.34/0.58 thf(fact_2955_underS__empty,axiom,
% 0.34/0.58 ! [A2: int,R2: set_Pr958786334691620121nt_int] :
% 0.34/0.58 ( ~ ( member_int @ A2 @ ( field_int @ R2 ) )
% 0.34/0.58 => ( ( order_underS_int @ R2 @ A2 )
% 0.34/0.58 = bot_bot_set_int ) ) ).
% 0.34/0.58
% 0.34/0.58 % underS_empty
% 0.34/0.58 thf(fact_2956_underS__empty,axiom,
% 0.34/0.58 ! [A2: produc3658429121746597890et_nat > $o,R2: set_Pr2161125870931222855_nat_o] :
% 0.34/0.58 ( ~ ( member6576561426505652726_nat_o @ A2 @ ( field_81821745747897882_nat_o @ R2 ) )
% 0.34/0.58 => ( ( order_7262550146012401043_nat_o @ R2 @ A2 )
% 0.34/0.58 = bot_bo7824918357723582233_nat_o ) ) ).
% 0.34/0.58
% 0.34/0.58 % underS_empty
% 0.34/0.58 thf(fact_2957_wfI__pf,axiom,
% 0.34/0.58 ! [R3: set_Pr5094982260447487303t_unit] :
% 0.34/0.58 ( ! [A9: set_Product_unit] :
% 0.34/0.58 ( ( ord_le3507040750410214029t_unit @ A9 @ ( image_8431784415008483091t_unit @ R3 @ A9 ) )
% 0.34/0.58 => ( A9 = bot_bo3957492148770167129t_unit ) )
% 0.34/0.58 => ( wf_Product_unit @ R3 ) ) ).
% 0.34/0.58
% 0.34/0.58 % wfI_pf
% 0.34/0.58 thf(fact_2958_wfI__pf,axiom,
% 0.34/0.58 ! [R3: set_Product_prod_o_o] :
% 0.34/0.58 ( ! [A9: set_o] :
% 0.34/0.58 ( ( ord_less_eq_set_o @ A9 @ ( image_o_o @ R3 @ A9 ) )
% 0.34/0.58 => ( A9 = bot_bot_set_o ) )
% 0.34/0.58 => ( wf_o @ R3 ) ) ).
% 0.34/0.58
% 0.34/0.58 % wfI_pf
% 0.34/0.58 thf(fact_2959_wfI__pf,axiom,
% 0.34/0.58 ! [R3: set_Pr958786334691620121nt_int] :
% 0.34/0.58 ( ! [A9: set_int] :
% 0.34/0.58 ( ( ord_less_eq_set_int @ A9 @ ( image_int_int @ R3 @ A9 ) )
% 0.34/0.58 => ( A9 = bot_bot_set_int ) )
% 0.34/0.58 => ( wf_int @ R3 ) ) ).
% 0.34/0.58
% 0.34/0.58 % wfI_pf
% 0.34/0.58 thf(fact_2960_wfI__pf,axiom,
% 0.34/0.58 ! [R3: set_Pr2161125870931222855_nat_o] :
% 0.34/0.58 ( ! [A9: set_Pr4532377907799695533_nat_o] :
% 0.34/0.58 ( ( ord_le2965882846123202637_nat_o @ A9 @ ( image_2232984160439939859_nat_o @ R3 @ A9 ) )
% 0.34/0.58 => ( A9 = bot_bo7824918357723582233_nat_o ) )
% 0.34/0.58 => ( wf_Pro8787509627421085110_nat_o @ R3 ) ) ).
% 0.34/0.58
% 0.34/0.58 % wfI_pf
% 0.34/0.58 thf(fact_2961_wfI__pf,axiom,
% 0.34/0.58 ! [R3: set_Pr1261947904930325089at_nat] :
% 0.34/0.58 ( ! [A9: set_nat] :
% 0.34/0.58 ( ( ord_less_eq_set_nat @ A9 @ ( image_nat_nat @ R3 @ A9 ) )
% 0.34/0.58 => ( A9 = bot_bot_set_nat ) )
% 0.34/0.58 => ( wf_nat @ R3 ) ) ).
% 0.34/0.58
% 0.34/0.58 % wfI_pf
% 0.34/0.58 thf(fact_2962_wfE__pf,axiom,
% 0.34/0.58 ! [R3: set_Pr5094982260447487303t_unit,A: set_Product_unit] :
% 0.34/0.58 ( ( wf_Product_unit @ R3 )
% 0.34/0.58 => ( ( ord_le3507040750410214029t_unit @ A @ ( image_8431784415008483091t_unit @ R3 @ A ) )
% 0.34/0.58 => ( A = bot_bo3957492148770167129t_unit ) ) ) ).
% 0.34/0.58
% 0.34/0.58 % wfE_pf
% 0.34/0.58 thf(fact_2963_wfE__pf,axiom,
% 0.34/0.58 ! [R3: set_Product_prod_o_o,A: set_o] :
% 0.34/0.58 ( ( wf_o @ R3 )
% 0.34/0.58 => ( ( ord_less_eq_set_o @ A @ ( image_o_o @ R3 @ A ) )
% 0.34/0.58 => ( A = bot_bot_set_o ) ) ) ).
% 0.34/0.58
% 0.34/0.58 % wfE_pf
% 0.34/0.58 thf(fact_2964_wfE__pf,axiom,
% 0.34/0.58 ! [R3: set_Pr958786334691620121nt_int,A: set_int] :
% 0.34/0.58 ( ( wf_int @ R3 )
% 0.34/0.58 => ( ( ord_less_eq_set_int @ A @ ( image_int_int @ R3 @ A ) )
% 0.34/0.58 => ( A = bot_bot_set_int ) ) ) ).
% 0.34/0.58
% 0.34/0.58 % wfE_pf
% 0.34/0.58 thf(fact_2965_wfE__pf,axiom,
% 0.34/0.58 ! [R3: set_Pr2161125870931222855_nat_o,A: set_Pr4532377907799695533_nat_o] :
% 0.34/0.58 ( ( wf_Pro8787509627421085110_nat_o @ R3 )
% 0.34/0.58 => ( ( ord_le2965882846123202637_nat_o @ A @ ( image_2232984160439939859_nat_o @ R3 @ A ) )
% 0.34/0.58 => ( A = bot_bo7824918357723582233_nat_o ) ) ) ).
% 0.34/0.58
% 0.34/0.58 % wfE_pf
% 0.34/0.58 thf(fact_2966_wfE__pf,axiom,
% 0.34/0.58 ! [R3: set_Pr1261947904930325089at_nat,A: set_nat] :
% 0.34/0.58 ( ( wf_nat @ R3 )
% 0.34/0.58 => ( ( ord_less_eq_set_nat @ A @ ( image_nat_nat @ R3 @ A ) )
% 0.34/0.58 => ( A = bot_bot_set_nat ) ) ) ).
% 0.34/0.58
% 0.34/0.58 % wfE_pf
% 0.34/0.58 thf(fact_2967_wf__eq__minimal2,axiom,
% 0.34/0.58 ( wf_Product_unit
% 0.34/0.58 = ( ^ [R4: set_Pr5094982260447487303t_unit] :
% 0.34/0.58 ! [A3: set_Product_unit] :
% 0.34/0.58 ( ( ( ord_le3507040750410214029t_unit @ A3 @ ( field_Product_unit @ R4 ) )
% 0.34/0.58 & ( A3 != bot_bo3957492148770167129t_unit ) )
% 0.34/0.58 => ? [X2: product_unit] :
% 0.34/0.58 ( ( member_Product_unit @ X2 @ A3 )
% 0.34/0.58 & ! [Y3: product_unit] :
% 0.34/0.58 ( ( member_Product_unit @ Y3 @ A3 )
% 0.34/0.58 => ~ ( member7821138191923116944t_unit @ ( produc5692694829384537815t_unit @ Y3 @ X2 ) @ R4 ) ) ) ) ) ) ).
% 0.34/0.58
% 0.34/0.58 % wf_eq_minimal2
% 0.34/0.58 thf(fact_2968_wf__eq__minimal2,axiom,
% 0.34/0.58 ( wf_o
% 0.34/0.58 = ( ^ [R4: set_Product_prod_o_o] :
% 0.34/0.58 ! [A3: set_o] :
% 0.34/0.58 ( ( ( ord_less_eq_set_o @ A3 @ ( field_o @ R4 ) )
% 0.34/0.58 & ( A3 != bot_bot_set_o ) )
% 0.34/0.58 => ? [X2: $o] :
% 0.34/0.58 ( ( member_o @ X2 @ A3 )
% 0.34/0.58 & ! [Y3: $o] :
% 0.34/0.58 ( ( member_o @ Y3 @ A3 )
% 0.34/0.58 => ~ ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ Y3 @ X2 ) @ R4 ) ) ) ) ) ) ).
% 0.34/0.58
% 0.34/0.58 % wf_eq_minimal2
% 0.34/0.58 thf(fact_2969_wf__eq__minimal2,axiom,
% 0.34/0.58 ( wf_int
% 0.34/0.58 = ( ^ [R4: set_Pr958786334691620121nt_int] :
% 0.34/0.58 ! [A3: set_int] :
% 0.34/0.58 ( ( ( ord_less_eq_set_int @ A3 @ ( field_int @ R4 ) )
% 0.34/0.58 & ( A3 != bot_bot_set_int ) )
% 0.34/0.58 => ? [X2: int] :
% 0.34/0.58 ( ( member_int @ X2 @ A3 )
% 0.34/0.58 & ! [Y3: int] :
% 0.34/0.58 ( ( member_int @ Y3 @ A3 )
% 0.34/0.58 => ~ ( member5262025264175285858nt_int @ ( product_Pair_int_int @ Y3 @ X2 ) @ R4 ) ) ) ) ) ) ).
% 0.34/0.58
% 0.34/0.58 % wf_eq_minimal2
% 0.34/0.58 thf(fact_2970_wf__eq__minimal2,axiom,
% 0.34/0.58 ( wf_Pro8787509627421085110_nat_o
% 0.34/0.58 = ( ^ [R4: set_Pr2161125870931222855_nat_o] :
% 0.34/0.58 ! [A3: set_Pr4532377907799695533_nat_o] :
% 0.34/0.58 ( ( ( ord_le2965882846123202637_nat_o @ A3 @ ( field_81821745747897882_nat_o @ R4 ) )
% 0.34/0.58 & ( A3 != bot_bo7824918357723582233_nat_o ) )
% 0.34/0.58 => ? [X2: produc3658429121746597890et_nat > $o] :
% 0.34/0.58 ( ( member6576561426505652726_nat_o @ X2 @ A3 )
% 0.34/0.58 & ! [Y3: produc3658429121746597890et_nat > $o] :
% 0.34/0.58 ( ( member6576561426505652726_nat_o @ Y3 @ A3 )
% 0.34/0.58 => ~ ( member8781333585448626064_nat_o @ ( produc7368190662567826135_nat_o @ Y3 @ X2 ) @ R4 ) ) ) ) ) ) ).
% 0.34/0.58
% 0.34/0.58 % wf_eq_minimal2
% 0.34/0.58 thf(fact_2971_wf__eq__minimal2,axiom,
% 0.34/0.58 ( wf_nat
% 0.34/0.58 = ( ^ [R4: set_Pr1261947904930325089at_nat] :
% 0.34/0.58 ! [A3: set_nat] :
% 0.34/0.58 ( ( ( ord_less_eq_set_nat @ A3 @ ( field_nat @ R4 ) )
% 0.34/0.58 & ( A3 != bot_bot_set_nat ) )
% 0.34/0.58 => ? [X2: nat] :
% 0.34/0.58 ( ( member_nat @ X2 @ A3 )
% 0.34/0.58 & ! [Y3: nat] :
% 0.34/0.58 ( ( member_nat @ Y3 @ A3 )
% 0.34/0.58 => ~ ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ Y3 @ X2 ) @ R4 ) ) ) ) ) ) ).
% 0.34/0.58
% 0.34/0.58 % wf_eq_minimal2
% 0.34/0.58 thf(fact_2972_wo__rel_Ocases__Total,axiom,
% 0.34/0.58 ! [R2: set_Pr5094982260447487303t_unit,A2: product_unit,B: product_unit,Phi: product_unit > product_unit > $o] :
% 0.34/0.58 ( ( bNF_We2091578341578594809t_unit @ R2 )
% 0.34/0.58 => ( ( ord_le3507040750410214029t_unit @ ( insert_Product_unit @ A2 @ ( insert_Product_unit @ B @ bot_bo3957492148770167129t_unit ) ) @ ( field_Product_unit @ R2 ) )
% 0.34/0.58 => ( ( ( member7821138191923116944t_unit @ ( produc5692694829384537815t_unit @ A2 @ B ) @ R2 )
% 0.34/0.58 => ( Phi @ A2 @ B ) )
% 0.34/0.58 => ( ( ( member7821138191923116944t_unit @ ( produc5692694829384537815t_unit @ B @ A2 ) @ R2 )
% 0.34/0.58 => ( Phi @ A2 @ B ) )
% 0.34/0.58 => ( Phi @ A2 @ B ) ) ) ) ) ).
% 0.34/0.58
% 0.34/0.58 % wo_rel.cases_Total
% 0.34/0.58 thf(fact_2973_wo__rel_Ocases__Total,axiom,
% 0.34/0.58 ! [R2: set_Product_prod_o_o,A2: $o,B: $o,Phi: $o > $o > $o] :
% 0.34/0.58 ( ( bNF_We6437023980930612612_rel_o @ R2 )
% 0.34/0.58 => ( ( ord_less_eq_set_o @ ( insert_o @ A2 @ ( insert_o @ B @ bot_bot_set_o ) ) @ ( field_o @ R2 ) )
% 0.34/0.58 => ( ( ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ A2 @ B ) @ R2 )
% 0.34/0.58 => ( Phi @ A2 @ B ) )
% 0.34/0.58 => ( ( ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ B @ A2 ) @ R2 )
% 0.34/0.58 => ( Phi @ A2 @ B ) )
% 0.34/0.58 => ( Phi @ A2 @ B ) ) ) ) ) ).
% 0.34/0.58
% 0.34/0.58 % wo_rel.cases_Total
% 0.34/0.58 thf(fact_2974_wo__rel_Ocases__Total,axiom,
% 0.34/0.58 ! [R2: set_Pr958786334691620121nt_int,A2: int,B: int,Phi: int > int > $o] :
% 0.34/0.58 ( ( bNF_We3815749466139970368el_int @ R2 )
% 0.34/0.58 => ( ( ord_less_eq_set_int @ ( insert_int @ A2 @ ( insert_int @ B @ bot_bot_set_int ) ) @ ( field_int @ R2 ) )
% 0.34/0.58 => ( ( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ A2 @ B ) @ R2 )
% 0.34/0.58 => ( Phi @ A2 @ B ) )
% 0.34/0.58 => ( ( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ B @ A2 ) @ R2 )
% 0.34/0.58 => ( Phi @ A2 @ B ) )
% 0.34/0.58 => ( Phi @ A2 @ B ) ) ) ) ) ).
% 0.34/0.58
% 0.34/0.58 % wo_rel.cases_Total
% 0.34/0.58 thf(fact_2975_wo__rel_Ocases__Total,axiom,
% 0.34/0.58 ! [R2: set_Pr2161125870931222855_nat_o,A2: produc3658429121746597890et_nat > $o,B: produc3658429121746597890et_nat > $o,Phi: ( produc3658429121746597890et_nat > $o ) > ( produc3658429121746597890et_nat > $o ) > $o] :
% 0.34/0.58 ( ( bNF_We740377018556551609_nat_o @ R2 )
% 0.34/0.58 => ( ( ord_le2965882846123202637_nat_o @ ( insert5175938949040314269_nat_o @ A2 @ ( insert5175938949040314269_nat_o @ B @ bot_bo7824918357723582233_nat_o ) ) @ ( field_81821745747897882_nat_o @ R2 ) )
% 0.34/0.58 => ( ( ( member8781333585448626064_nat_o @ ( produc7368190662567826135_nat_o @ A2 @ B ) @ R2 )
% 0.34/0.58 => ( Phi @ A2 @ B ) )
% 0.34/0.58 => ( ( ( member8781333585448626064_nat_o @ ( produc7368190662567826135_nat_o @ B @ A2 ) @ R2 )
% 0.34/0.58 => ( Phi @ A2 @ B ) )
% 0.34/0.58 => ( Phi @ A2 @ B ) ) ) ) ) ).
% 0.34/0.58
% 0.34/0.58 % wo_rel.cases_Total
% 0.34/0.58 thf(fact_2976_wo__rel_Ocases__Total,axiom,
% 0.34/0.58 ! [R2: set_Pr1261947904930325089at_nat,A2: nat,B: nat,Phi: nat > nat > $o] :
% 0.34/0.58 ( ( bNF_We3818239936649020644el_nat @ R2 )
% 0.34/0.58 => ( ( ord_less_eq_set_nat @ ( insert_nat @ A2 @ ( insert_nat @ B @ bot_bot_set_nat ) ) @ ( field_nat @ R2 ) )
% 0.34/0.58 => ( ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A2 @ B ) @ R2 )
% 0.34/0.58 => ( Phi @ A2 @ B ) )
% 0.34/0.58 => ( ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ B @ A2 ) @ R2 )
% 0.34/0.58 => ( Phi @ A2 @ B ) )
% 0.34/0.58 => ( Phi @ A2 @ B ) ) ) ) ) ).
% 0.34/0.58
% 0.34/0.58 % wo_rel.cases_Total
% 0.34/0.58 thf(fact_2977_wf__no__path,axiom,
% 0.34/0.58 ! [R3: set_Pr5094982260447487303t_unit] :
% 0.34/0.58 ( ( ( inf_in4660618365625256667t_unit @ ( domain4292024440186670924t_unit @ R3 ) @ ( range_6617724647562260917t_unit @ R3 ) )
% 0.34/0.58 = bot_bo3957492148770167129t_unit )
% 0.34/0.58 => ( wf_Product_unit @ R3 ) ) ).
% 0.34/0.58
% 0.34/0.58 % wf_no_path
% 0.34/0.58 thf(fact_2978_wf__no__path,axiom,
% 0.34/0.58 ! [R3: set_Product_prod_o_o] :
% 0.34/0.58 ( ( ( inf_inf_set_o @ ( domain_o_o @ R3 ) @ ( range_o_o @ R3 ) )
% 0.34/0.58 = bot_bot_set_o )
% 0.34/0.58 => ( wf_o @ R3 ) ) ).
% 0.34/0.58
% 0.34/0.58 % wf_no_path
% 0.34/0.58 thf(fact_2979_wf__no__path,axiom,
% 0.34/0.58 ! [R3: set_Pr1261947904930325089at_nat] :
% 0.34/0.58 ( ( ( inf_inf_set_nat @ ( domain_nat_nat @ R3 ) @ ( range_nat_nat @ R3 ) )
% 0.34/0.58 = bot_bot_set_nat )
% 0.34/0.58 => ( wf_nat @ R3 ) ) ).
% 0.34/0.58
% 0.34/0.58 % wf_no_path
% 0.34/0.58 thf(fact_2980_wf__no__path,axiom,
% 0.34/0.58 ! [R3: set_Pr958786334691620121nt_int] :
% 0.34/0.58 ( ( ( inf_inf_set_int @ ( domain_int_int @ R3 ) @ ( range_int_int @ R3 ) )
% 0.34/0.58 = bot_bot_set_int )
% 0.34/0.58 => ( wf_int @ R3 ) ) ).
% 0.34/0.58
% 0.34/0.58 % wf_no_path
% 0.34/0.58 thf(fact_2981_wf__no__path,axiom,
% 0.34/0.58 ! [R3: set_Pr2161125870931222855_nat_o] :
% 0.34/0.58 ( ( ( inf_in1906310914598751387_nat_o @ ( domain4881241903163524940_nat_o @ R3 ) @ ( range_5289318983786108341_nat_o @ R3 ) )
% 0.34/0.58 = bot_bo7824918357723582233_nat_o )
% 0.34/0.58 => ( wf_Pro8787509627421085110_nat_o @ R3 ) ) ).
% 0.34/0.58
% 0.34/0.58 % wf_no_path
% 0.34/0.58 thf(fact_2982_wf__min,axiom,
% 0.34/0.58 ! [R3: set_Pr2161125870931222855_nat_o] :
% 0.34/0.58 ( ( wf_Pro8787509627421085110_nat_o @ R3 )
% 0.34/0.58 => ( ( R3 != bot_bo3657187498359500723_nat_o )
% 0.34/0.58 => ~ ! [M2: produc3658429121746597890et_nat > $o] :
% 0.34/0.58 ~ ( member6576561426505652726_nat_o @ M2 @ ( minus_1801376950450012436_nat_o @ ( domain4881241903163524940_nat_o @ R3 ) @ ( range_5289318983786108341_nat_o @ R3 ) ) ) ) ) ).
% 0.34/0.58
% 0.34/0.58 % wf_min
% 0.34/0.58 thf(fact_2983_wf__min,axiom,
% 0.34/0.58 ! [R3: set_Pr958786334691620121nt_int] :
% 0.34/0.58 ( ( wf_int @ R3 )
% 0.34/0.58 => ( ( R3 != bot_bo1796632182523588997nt_int )
% 0.34/0.58 => ~ ! [M2: int] :
% 0.34/0.58 ~ ( member_int @ M2 @ ( minus_minus_set_int @ ( domain_int_int @ R3 ) @ ( range_int_int @ R3 ) ) ) ) ) ).
% 0.34/0.58
% 0.34/0.58 % wf_min
% 0.34/0.58 thf(fact_2984_wf__min,axiom,
% 0.34/0.58 ! [R3: set_Product_prod_o_o] :
% 0.34/0.58 ( ( wf_o @ R3 )
% 0.34/0.58 => ( ( R3 != bot_bo7073875226086086771od_o_o )
% 0.34/0.58 => ~ ! [M2: $o] :
% 0.34/0.58 ~ ( member_o @ M2 @ ( minus_minus_set_o @ ( domain_o_o @ R3 ) @ ( range_o_o @ R3 ) ) ) ) ) ).
% 0.34/0.58
% 0.34/0.58 % wf_min
% 0.34/0.58 thf(fact_2985_wf__min,axiom,
% 0.34/0.58 ! [R3: set_Pr5094982260447487303t_unit] :
% 0.34/0.58 ( ( wf_Product_unit @ R3 )
% 0.34/0.58 => ( ( R3 != bot_bo1654022762305692595t_unit )
% 0.34/0.58 => ~ ! [M2: product_unit] :
% 0.34/0.58 ~ ( member_Product_unit @ M2 @ ( minus_6452836326544984404t_unit @ ( domain4292024440186670924t_unit @ R3 ) @ ( range_6617724647562260917t_unit @ R3 ) ) ) ) ) ).
% 0.34/0.58
% 0.34/0.58 % wf_min
% 0.34/0.58 thf(fact_2986_wf__min,axiom,
% 0.34/0.58 ! [R3: set_Pr1261947904930325089at_nat] :
% 0.34/0.58 ( ( wf_nat @ R3 )
% 0.34/0.58 => ( ( R3 != bot_bo2099793752762293965at_nat )
% 0.34/0.58 => ~ ! [M2: nat] :
% 0.34/0.58 ~ ( member_nat @ M2 @ ( minus_minus_set_nat @ ( domain_nat_nat @ R3 ) @ ( range_nat_nat @ R3 ) ) ) ) ) ).
% 0.34/0.58
% 0.34/0.58 % wf_min
% 0.34/0.58 thf(fact_2987_underS__incl__iff,axiom,
% 0.34/0.58 ! [R2: set_Pr2161125870931222855_nat_o,A2: produc3658429121746597890et_nat > $o,B: produc3658429121746597890et_nat > $o] :
% 0.34/0.58 ( ( order_1145710701291701250_nat_o @ ( field_81821745747897882_nat_o @ R2 ) @ R2 )
% 0.34/0.58 => ( ( member6576561426505652726_nat_o @ A2 @ ( field_81821745747897882_nat_o @ R2 ) )
% 0.34/0.58 => ( ( member6576561426505652726_nat_o @ B @ ( field_81821745747897882_nat_o @ R2 ) )
% 0.34/0.58 => ( ( ord_le2965882846123202637_nat_o @ ( order_7262550146012401043_nat_o @ R2 @ A2 ) @ ( order_7262550146012401043_nat_o @ R2 @ B ) )
% 0.34/0.58 = ( member8781333585448626064_nat_o @ ( produc7368190662567826135_nat_o @ A2 @ B ) @ R2 ) ) ) ) ) ).
% 0.34/0.58
% 0.34/0.58 % underS_incl_iff
% 0.34/0.58 thf(fact_2988_underS__incl__iff,axiom,
% 0.34/0.58 ! [R2: set_Product_prod_o_o,A2: $o,B: $o] :
% 0.34/0.58 ( ( order_6238756238976269133r_on_o @ ( field_o @ R2 ) @ R2 )
% 0.34/0.58 => ( ( member_o @ A2 @ ( field_o @ R2 ) )
% 0.34/0.58 => ( ( member_o @ B @ ( field_o @ R2 ) )
% 0.34/0.58 => ( ( ord_less_eq_set_o @ ( order_underS_o @ R2 @ A2 ) @ ( order_underS_o @ R2 @ B ) )
% 0.34/0.58 = ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ A2 @ B ) @ R2 ) ) ) ) ) ).
% 0.34/0.58
% 0.34/0.58 % underS_incl_iff
% 0.34/0.58 thf(fact_2989_underS__incl__iff,axiom,
% 0.34/0.58 ! [R2: set_Pr5094982260447487303t_unit,A2: product_unit,B: product_unit] :
% 0.34/0.58 ( ( order_888282469287196226t_unit @ ( field_Product_unit @ R2 ) @ R2 )
% 0.34/0.58 => ( ( member_Product_unit @ A2 @ ( field_Product_unit @ R2 ) )
% 0.34/0.58 => ( ( member_Product_unit @ B @ ( field_Product_unit @ R2 ) )
% 0.34/0.58 => ( ( ord_le3507040750410214029t_unit @ ( order_5809128134054296019t_unit @ R2 @ A2 ) @ ( order_5809128134054296019t_unit @ R2 @ B ) )
% 0.34/0.58 = ( member7821138191923116944t_unit @ ( produc5692694829384537815t_unit @ A2 @ B ) @ R2 ) ) ) ) ) ).
% 0.34/0.58
% 0.34/0.58 % underS_incl_iff
% 0.34/0.58 thf(fact_2990_underS__incl__iff,axiom,
% 0.34/0.58 ! [R2: set_Pr958786334691620121nt_int,A2: int,B: int] :
% 0.34/0.58 ( ( order_4471489696718655927on_int @ ( field_int @ R2 ) @ R2 )
% 0.34/0.58 => ( ( member_int @ A2 @ ( field_int @ R2 ) )
% 0.34/0.58 => ( ( member_int @ B @ ( field_int @ R2 ) )
% 0.34/0.58 => ( ( ord_less_eq_set_int @ ( order_underS_int @ R2 @ A2 ) @ ( order_underS_int @ R2 @ B ) )
% 0.34/0.58 = ( member5262025264175285858nt_int @ ( product_Pair_int_int @ A2 @ B ) @ R2 ) ) ) ) ) ).
% 0.34/0.58
% 0.34/0.58 % underS_incl_iff
% 0.34/0.58 thf(fact_2991_underS__incl__iff,axiom,
% 0.34/0.58 ! [R2: set_Pr1261947904930325089at_nat,A2: nat,B: nat] :
% 0.34/0.58 ( ( order_4473980167227706203on_nat @ ( field_nat @ R2 ) @ R2 )
% 0.34/0.58 => ( ( member_nat @ A2 @ ( field_nat @ R2 ) )
% 0.34/0.58 => ( ( member_nat @ B @ ( field_nat @ R2 ) )
% 0.34/0.58 => ( ( ord_less_eq_set_nat @ ( order_underS_nat @ R2 @ A2 ) @ ( order_underS_nat @ R2 @ B ) )
% 0.34/0.58 = ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A2 @ B ) @ R2 ) ) ) ) ) ).
% 0.34/0.58
% 0.34/0.58 % underS_incl_iff
% 0.34/0.58 thf(fact_2992_wo__rel_OWell__order__isMinim__exists,axiom,
% 0.34/0.58 ! [R2: set_Pr5094982260447487303t_unit,B4: set_Product_unit] :
% 0.34/0.58 ( ( bNF_We2091578341578594809t_unit @ R2 )
% 0.34/0.58 => ( ( ord_le3507040750410214029t_unit @ B4 @ ( field_Product_unit @ R2 ) )
% 0.34/0.58 => ( ( B4 != bot_bo3957492148770167129t_unit )
% 0.34/0.58 => ? [X_1: product_unit] : ( bNF_We909621271159000003t_unit @ R2 @ B4 @ X_1 ) ) ) ) ).
% 0.34/0.58
% 0.34/0.58 % wo_rel.Well_order_isMinim_exists
% 0.34/0.58 thf(fact_2993_wo__rel_OWell__order__isMinim__exists,axiom,
% 0.34/0.58 ! [R2: set_Product_prod_o_o,B4: set_o] :
% 0.34/0.58 ( ( bNF_We6437023980930612612_rel_o @ R2 )
% 0.34/0.58 => ( ( ord_less_eq_set_o @ B4 @ ( field_o @ R2 ) )
% 0.34/0.58 => ( ( B4 != bot_bot_set_o )
% 0.34/0.58 => ? [X_1: $o] : ( bNF_We4647864369647134286inim_o @ R2 @ B4 @ X_1 ) ) ) ) ).
% 0.34/0.58
% 0.34/0.58 % wo_rel.Well_order_isMinim_exists
% 0.34/0.58 thf(fact_2994_wo__rel_OWell__order__isMinim__exists,axiom,
% 0.34/0.58 ! [R2: set_Pr958786334691620121nt_int,B4: set_int] :
% 0.34/0.58 ( ( bNF_We3815749466139970368el_int @ R2 )
% 0.34/0.58 => ( ( ord_less_eq_set_int @ B4 @ ( field_int @ R2 ) )
% 0.34/0.58 => ( ( B4 != bot_bot_set_int )
% 0.34/0.58 => ? [X_1: int] : ( bNF_We6511469253196982518im_int @ R2 @ B4 @ X_1 ) ) ) ) ).
% 0.34/0.58
% 0.34/0.58 % wo_rel.Well_order_isMinim_exists
% 0.34/0.58 thf(fact_2995_wo__rel_OWell__order__isMinim__exists,axiom,
% 0.34/0.58 ! [R2: set_Pr2161125870931222855_nat_o,B4: set_Pr4532377907799695533_nat_o] :
% 0.34/0.58 ( ( bNF_We740377018556551609_nat_o @ R2 )
% 0.34/0.58 => ( ( ord_le2965882846123202637_nat_o @ B4 @ ( field_81821745747897882_nat_o @ R2 ) )
% 0.34/0.58 => ( ( B4 != bot_bo7824918357723582233_nat_o )
% 0.34/0.58 => ? [X_1: produc3658429121746597890et_nat > $o] : ( bNF_We5608535359669600131_nat_o @ R2 @ B4 @ X_1 ) ) ) ) ).
% 0.34/0.58
% 0.34/0.58 % wo_rel.Well_order_isMinim_exists
% 0.34/0.58 thf(fact_2996_wo__rel_OWell__order__isMinim__exists,axiom,
% 0.34/0.58 ! [R2: set_Pr1261947904930325089at_nat,B4: set_nat] :
% 0.34/0.58 ( ( bNF_We3818239936649020644el_nat @ R2 )
% 0.34/0.58 => ( ( ord_less_eq_set_nat @ B4 @ ( field_nat @ R2 ) )
% 0.34/0.58 => ( ( B4 != bot_bot_set_nat )
% 0.34/0.58 => ? [X_1: nat] : ( bNF_We6513959723706032794im_nat @ R2 @ B4 @ X_1 ) ) ) ) ).
% 0.34/0.58
% 0.34/0.58 % wo_rel.Well_order_isMinim_exists
% 0.34/0.58 thf(fact_2997_Refl__under__underS,axiom,
% 0.34/0.58 ! [R2: set_Pr5094982260447487303t_unit,A2: product_unit] :
% 0.34/0.58 ( ( refl_on_Product_unit @ ( field_Product_unit @ R2 ) @ R2 )
% 0.34/0.58 => ( ( member_Product_unit @ A2 @ ( field_Product_unit @ R2 ) )
% 0.34/0.58 => ( ( order_6949322460238004850t_unit @ R2 @ A2 )
% 0.34/0.58 = ( sup_su793286257634532545t_unit @ ( order_5809128134054296019t_unit @ R2 @ A2 ) @ ( insert_Product_unit @ A2 @ bot_bo3957492148770167129t_unit ) ) ) ) ) ).
% 0.34/0.58
% 0.34/0.58 % Refl_under_underS
% 0.34/0.58 thf(fact_2998_Refl__under__underS,axiom,
% 0.34/0.58 ! [R2: set_Product_prod_o_o,A2: $o] :
% 0.34/0.58 ( ( refl_on_o @ ( field_o @ R2 ) @ R2 )
% 0.34/0.58 => ( ( member_o @ A2 @ ( field_o @ R2 ) )
% 0.34/0.58 => ( ( order_under_o @ R2 @ A2 )
% 0.34/0.58 = ( sup_sup_set_o @ ( order_underS_o @ R2 @ A2 ) @ ( insert_o @ A2 @ bot_bot_set_o ) ) ) ) ) ).
% 0.34/0.58
% 0.34/0.58 % Refl_under_underS
% 0.34/0.58 thf(fact_2999_Refl__under__underS,axiom,
% 0.34/0.58 ! [R2: set_Pr1261947904930325089at_nat,A2: nat] :
% 0.34/0.58 ( ( refl_on_nat @ ( field_nat @ R2 ) @ R2 )
% 0.34/0.58 => ( ( member_nat @ A2 @ ( field_nat @ R2 ) )
% 0.34/0.58 => ( ( order_under_nat @ R2 @ A2 )
% 0.34/0.58 = ( sup_sup_set_nat @ ( order_underS_nat @ R2 @ A2 ) @ ( insert_nat @ A2 @ bot_bot_set_nat ) ) ) ) ) ).
% 0.34/0.58
% 0.34/0.58 % Refl_under_underS
% 0.34/0.58 thf(fact_3000_Refl__under__underS,axiom,
% 0.34/0.58 ! [R2: set_Pr958786334691620121nt_int,A2: int] :
% 0.34/0.58 ( ( refl_on_int @ ( field_int @ R2 ) @ R2 )
% 0.34/0.58 => ( ( member_int @ A2 @ ( field_int @ R2 ) )
% 0.34/0.58 => ( ( order_under_int @ R2 @ A2 )
% 0.34/0.58 = ( sup_sup_set_int @ ( order_underS_int @ R2 @ A2 ) @ ( insert_int @ A2 @ bot_bot_set_int ) ) ) ) ) ).
% 0.34/0.58
% 0.34/0.58 % Refl_under_underS
% 0.34/0.58 thf(fact_3001_Refl__under__underS,axiom,
% 0.34/0.58 ! [R2: set_Pr2161125870931222855_nat_o,A2: produc3658429121746597890et_nat > $o] :
% 0.34/0.58 ( ( refl_o1075864025857821388_nat_o @ ( field_81821745747897882_nat_o @ R2 ) @ R2 )
% 0.34/0.58 => ( ( member6576561426505652726_nat_o @ A2 @ ( field_81821745747897882_nat_o @ R2 ) )
% 0.34/0.58 => ( ( order_2885205318754725938_nat_o @ R2 @ A2 )
% 0.34/0.58 = ( sup_su5209123915105501825_nat_o @ ( order_7262550146012401043_nat_o @ R2 @ A2 ) @ ( insert5175938949040314269_nat_o @ A2 @ bot_bo7824918357723582233_nat_o ) ) ) ) ) ).
% 0.34/0.58
% 0.34/0.58 % Refl_under_underS
% 0.34/0.58 thf(fact_3002_wo__rel_Ominim__inField,axiom,
% 0.34/0.58 ! [R2: set_Pr5094982260447487303t_unit,B4: set_Product_unit] :
% 0.34/0.58 ( ( bNF_We2091578341578594809t_unit @ R2 )
% 0.34/0.58 => ( ( ord_le3507040750410214029t_unit @ B4 @ ( field_Product_unit @ R2 ) )
% 0.34/0.58 => ( ( B4 != bot_bo3957492148770167129t_unit )
% 0.34/0.58 => ( member_Product_unit @ ( bNF_We7185707985517244377t_unit @ R2 @ B4 ) @ ( field_Product_unit @ R2 ) ) ) ) ) ).
% 0.34/0.58
% 0.34/0.58 % wo_rel.minim_inField
% 0.34/0.58 thf(fact_3003_wo__rel_Ominim__inField,axiom,
% 0.34/0.58 ! [R2: set_Product_prod_o_o,B4: set_o] :
% 0.34/0.58 ( ( bNF_We6437023980930612612_rel_o @ R2 )
% 0.34/0.58 => ( ( ord_less_eq_set_o @ B4 @ ( field_o @ R2 ) )
% 0.34/0.58 => ( ( B4 != bot_bot_set_o )
% 0.34/0.58 => ( member_o @ ( bNF_We537455116209922916inim_o @ R2 @ B4 ) @ ( field_o @ R2 ) ) ) ) ) ).
% 0.34/0.58
% 0.34/0.58 % wo_rel.minim_inField
% 0.34/0.58 thf(fact_3004_wo__rel_Ominim__inField,axiom,
% 0.34/0.58 ! [R2: set_Pr958786334691620121nt_int,B4: set_int] :
% 0.34/0.58 ( ( bNF_We3815749466139970368el_int @ R2 )
% 0.34/0.58 => ( ( ord_less_eq_set_int @ B4 @ ( field_int @ R2 ) )
% 0.34/0.58 => ( ( B4 != bot_bot_set_int )
% 0.34/0.58 => ( member_int @ ( bNF_We3272633801743244640im_int @ R2 @ B4 ) @ ( field_int @ R2 ) ) ) ) ) ).
% 0.34/0.58
% 0.34/0.58 % wo_rel.minim_inField
% 0.34/0.58 thf(fact_3005_wo__rel_Ominim__inField,axiom,
% 0.34/0.58 ! [R2: set_Pr2161125870931222855_nat_o,B4: set_Pr4532377907799695533_nat_o] :
% 0.34/0.58 ( ( bNF_We740377018556551609_nat_o @ R2 )
% 0.34/0.58 => ( ( ord_le2965882846123202637_nat_o @ B4 @ ( field_81821745747897882_nat_o @ R2 ) )
% 0.34/0.58 => ( ( B4 != bot_bo7824918357723582233_nat_o )
% 0.34/0.58 => ( member6576561426505652726_nat_o @ ( bNF_We8236795211600914329_nat_o @ R2 @ B4 ) @ ( field_81821745747897882_nat_o @ R2 ) ) ) ) ) ).
% 0.34/0.58
% 0.34/0.58 % wo_rel.minim_inField
% 0.34/0.58 thf(fact_3006_wo__rel_Ominim__inField,axiom,
% 0.34/0.58 ! [R2: set_Pr1261947904930325089at_nat,B4: set_nat] :
% 0.34/0.58 ( ( bNF_We3818239936649020644el_nat @ R2 )
% 0.34/0.58 => ( ( ord_less_eq_set_nat @ B4 @ ( field_nat @ R2 ) )
% 0.34/0.58 => ( ( B4 != bot_bot_set_nat )
% 0.34/0.58 => ( member_nat @ ( bNF_We3275124272252294916im_nat @ R2 @ B4 ) @ ( field_nat @ R2 ) ) ) ) ) ).
% 0.34/0.58
% 0.34/0.58 % wo_rel.minim_inField
% 0.34/0.58 thf(fact_3007_wo__rel_Ominim__in,axiom,
% 0.34/0.58 ! [R2: set_Pr5094982260447487303t_unit,B4: set_Product_unit] :
% 0.34/0.58 ( ( bNF_We2091578341578594809t_unit @ R2 )
% 0.34/0.58 => ( ( ord_le3507040750410214029t_unit @ B4 @ ( field_Product_unit @ R2 ) )
% 0.34/0.58 => ( ( B4 != bot_bo3957492148770167129t_unit )
% 0.34/0.58 => ( member_Product_unit @ ( bNF_We7185707985517244377t_unit @ R2 @ B4 ) @ B4 ) ) ) ) ).
% 0.34/0.58
% 0.34/0.58 % wo_rel.minim_in
% 0.34/0.58 thf(fact_3008_wo__rel_Ominim__in,axiom,
% 0.34/0.58 ! [R2: set_Product_prod_o_o,B4: set_o] :
% 0.34/0.58 ( ( bNF_We6437023980930612612_rel_o @ R2 )
% 0.34/0.58 => ( ( ord_less_eq_set_o @ B4 @ ( field_o @ R2 ) )
% 0.34/0.58 => ( ( B4 != bot_bot_set_o )
% 0.34/0.58 => ( member_o @ ( bNF_We537455116209922916inim_o @ R2 @ B4 ) @ B4 ) ) ) ) ).
% 0.34/0.58
% 0.34/0.58 % wo_rel.minim_in
% 0.34/0.58 thf(fact_3009_wo__rel_Ominim__in,axiom,
% 0.34/0.58 ! [R2: set_Pr958786334691620121nt_int,B4: set_int] :
% 0.34/0.58 ( ( bNF_We3815749466139970368el_int @ R2 )
% 0.34/0.58 => ( ( ord_less_eq_set_int @ B4 @ ( field_int @ R2 ) )
% 0.34/0.58 => ( ( B4 != bot_bot_set_int )
% 0.34/0.58 => ( member_int @ ( bNF_We3272633801743244640im_int @ R2 @ B4 ) @ B4 ) ) ) ) ).
% 0.34/0.58
% 0.34/0.58 % wo_rel.minim_in
% 0.34/0.58 thf(fact_3010_wo__rel_Ominim__in,axiom,
% 0.34/0.58 ! [R2: set_Pr2161125870931222855_nat_o,B4: set_Pr4532377907799695533_nat_o] :
% 0.34/0.58 ( ( bNF_We740377018556551609_nat_o @ R2 )
% 0.34/0.58 => ( ( ord_le2965882846123202637_nat_o @ B4 @ ( field_81821745747897882_nat_o @ R2 ) )
% 0.34/0.58 => ( ( B4 != bot_bo7824918357723582233_nat_o )
% 0.34/0.58 => ( member6576561426505652726_nat_o @ ( bNF_We8236795211600914329_nat_o @ R2 @ B4 ) @ B4 ) ) ) ) ).
% 0.34/0.58
% 0.34/0.58 % wo_rel.minim_in
% 0.34/0.58 thf(fact_3011_wo__rel_Ominim__in,axiom,
% 0.34/0.58 ! [R2: set_Pr1261947904930325089at_nat,B4: set_nat] :
% 0.34/0.58 ( ( bNF_We3818239936649020644el_nat @ R2 )
% 0.34/0.58 => ( ( ord_less_eq_set_nat @ B4 @ ( field_nat @ R2 ) )
% 0.34/0.58 => ( ( B4 != bot_bot_set_nat )
% 0.34/0.58 => ( member_nat @ ( bNF_We3275124272252294916im_nat @ R2 @ B4 ) @ B4 ) ) ) ) ).
% 0.34/0.58
% 0.34/0.58 % wo_rel.minim_in
% 0.34/0.58 thf(fact_3012_wo__rel_Oequals__minim,axiom,
% 0.34/0.58 ! [R2: set_Pr2161125870931222855_nat_o,B4: set_Pr4532377907799695533_nat_o,A2: produc3658429121746597890et_nat > $o] :
% 0.34/0.58 ( ( bNF_We740377018556551609_nat_o @ R2 )
% 0.34/0.58 => ( ( ord_le2965882846123202637_nat_o @ B4 @ ( field_81821745747897882_nat_o @ R2 ) )
% 0.34/0.58 => ( ( member6576561426505652726_nat_o @ A2 @ B4 )
% 0.34/0.58 => ( ! [B2: produc3658429121746597890et_nat > $o] :
% 0.34/0.58 ( ( member6576561426505652726_nat_o @ B2 @ B4 )
% 0.34/0.58 => ( member8781333585448626064_nat_o @ ( produc7368190662567826135_nat_o @ A2 @ B2 ) @ R2 ) )
% 0.34/0.58 => ( A2
% 0.34/0.58 = ( bNF_We8236795211600914329_nat_o @ R2 @ B4 ) ) ) ) ) ) ).
% 0.34/0.58
% 0.34/0.58 % wo_rel.equals_minim
% 0.34/0.58 thf(fact_3013_wo__rel_Oequals__minim,axiom,
% 0.34/0.58 ! [R2: set_Product_prod_o_o,B4: set_o,A2: $o] :
% 0.34/0.58 ( ( bNF_We6437023980930612612_rel_o @ R2 )
% 0.34/0.58 => ( ( ord_less_eq_set_o @ B4 @ ( field_o @ R2 ) )
% 0.34/0.58 => ( ( member_o @ A2 @ B4 )
% 0.34/0.58 => ( ! [B2: $o] :
% 0.34/0.58 ( ( member_o @ B2 @ B4 )
% 0.34/0.58 => ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ A2 @ B2 ) @ R2 ) )
% 0.34/0.58 => ( A2
% 0.34/0.58 = ( bNF_We537455116209922916inim_o @ R2 @ B4 ) ) ) ) ) ) ).
% 0.34/0.58
% 0.34/0.58 % wo_rel.equals_minim
% 0.34/0.58 thf(fact_3014_wo__rel_Oequals__minim,axiom,
% 0.34/0.58 ! [R2: set_Pr5094982260447487303t_unit,B4: set_Product_unit,A2: product_unit] :
% 0.34/0.58 ( ( bNF_We2091578341578594809t_unit @ R2 )
% 0.34/0.58 => ( ( ord_le3507040750410214029t_unit @ B4 @ ( field_Product_unit @ R2 ) )
% 0.34/0.58 => ( ( member_Product_unit @ A2 @ B4 )
% 0.34/0.58 => ( ! [B2: product_unit] :
% 0.34/0.58 ( ( member_Product_unit @ B2 @ B4 )
% 0.34/0.58 => ( member7821138191923116944t_unit @ ( produc5692694829384537815t_unit @ A2 @ B2 ) @ R2 ) )
% 0.34/0.58 => ( A2
% 0.34/0.58 = ( bNF_We7185707985517244377t_unit @ R2 @ B4 ) ) ) ) ) ) ).
% 0.34/0.58
% 0.34/0.58 % wo_rel.equals_minim
% 0.34/0.58 thf(fact_3015_wo__rel_Oequals__minim,axiom,
% 0.34/0.58 ! [R2: set_Pr958786334691620121nt_int,B4: set_int,A2: int] :
% 0.34/0.58 ( ( bNF_We3815749466139970368el_int @ R2 )
% 0.34/0.58 => ( ( ord_less_eq_set_int @ B4 @ ( field_int @ R2 ) )
% 0.34/0.58 => ( ( member_int @ A2 @ B4 )
% 0.34/0.58 => ( ! [B2: int] :
% 0.34/0.58 ( ( member_int @ B2 @ B4 )
% 0.34/0.58 => ( member5262025264175285858nt_int @ ( product_Pair_int_int @ A2 @ B2 ) @ R2 ) )
% 0.34/0.58 => ( A2
% 0.34/0.58 = ( bNF_We3272633801743244640im_int @ R2 @ B4 ) ) ) ) ) ) ).
% 0.34/0.58
% 0.34/0.58 % wo_rel.equals_minim
% 0.34/0.58 thf(fact_3016_wo__rel_Oequals__minim,axiom,
% 0.34/0.58 ! [R2: set_Pr1261947904930325089at_nat,B4: set_nat,A2: nat] :
% 0.34/0.58 ( ( bNF_We3818239936649020644el_nat @ R2 )
% 0.34/0.58 => ( ( ord_less_eq_set_nat @ B4 @ ( field_nat @ R2 ) )
% 0.34/0.58 => ( ( member_nat @ A2 @ B4 )
% 0.34/0.58 => ( ! [B2: nat] :
% 0.34/0.58 ( ( member_nat @ B2 @ B4 )
% 0.34/0.58 => ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A2 @ B2 ) @ R2 ) )
% 0.34/0.58 => ( A2
% 0.34/0.58 = ( bNF_We3275124272252294916im_nat @ R2 @ B4 ) ) ) ) ) ) ).
% 0.34/0.58
% 0.34/0.58 % wo_rel.equals_minim
% 0.34/0.58 thf(fact_3017_wo__rel_Ominim__least,axiom,
% 0.34/0.58 ! [R2: set_Pr2161125870931222855_nat_o,B4: set_Pr4532377907799695533_nat_o,B: produc3658429121746597890et_nat > $o] :
% 0.34/0.58 ( ( bNF_We740377018556551609_nat_o @ R2 )
% 0.34/0.58 => ( ( ord_le2965882846123202637_nat_o @ B4 @ ( field_81821745747897882_nat_o @ R2 ) )
% 0.34/0.58 => ( ( member6576561426505652726_nat_o @ B @ B4 )
% 0.34/0.58 => ( member8781333585448626064_nat_o @ ( produc7368190662567826135_nat_o @ ( bNF_We8236795211600914329_nat_o @ R2 @ B4 ) @ B ) @ R2 ) ) ) ) ).
% 0.34/0.58
% 0.34/0.58 % wo_rel.minim_least
% 0.34/0.58 thf(fact_3018_wo__rel_Ominim__least,axiom,
% 0.34/0.58 ! [R2: set_Product_prod_o_o,B4: set_o,B: $o] :
% 0.34/0.58 ( ( bNF_We6437023980930612612_rel_o @ R2 )
% 0.34/0.58 => ( ( ord_less_eq_set_o @ B4 @ ( field_o @ R2 ) )
% 0.34/0.58 => ( ( member_o @ B @ B4 )
% 0.34/0.58 => ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ ( bNF_We537455116209922916inim_o @ R2 @ B4 ) @ B ) @ R2 ) ) ) ) ).
% 0.34/0.58
% 0.34/0.58 % wo_rel.minim_least
% 0.34/0.58 thf(fact_3019_wo__rel_Ominim__least,axiom,
% 0.34/0.58 ! [R2: set_Pr5094982260447487303t_unit,B4: set_Product_unit,B: product_unit] :
% 0.34/0.58 ( ( bNF_We2091578341578594809t_unit @ R2 )
% 0.34/0.58 => ( ( ord_le3507040750410214029t_unit @ B4 @ ( field_Product_unit @ R2 ) )
% 0.34/0.58 => ( ( member_Product_unit @ B @ B4 )
% 0.34/0.58 => ( member7821138191923116944t_unit @ ( produc5692694829384537815t_unit @ ( bNF_We7185707985517244377t_unit @ R2 @ B4 ) @ B ) @ R2 ) ) ) ) ).
% 0.34/0.58
% 0.34/0.58 % wo_rel.minim_least
% 0.34/0.58 thf(fact_3020_wo__rel_Ominim__least,axiom,
% 0.34/0.58 ! [R2: set_Pr958786334691620121nt_int,B4: set_int,B: int] :
% 0.34/0.58 ( ( bNF_We3815749466139970368el_int @ R2 )
% 0.34/0.58 => ( ( ord_less_eq_set_int @ B4 @ ( field_int @ R2 ) )
% 0.34/0.58 => ( ( member_int @ B @ B4 )
% 0.34/0.58 => ( member5262025264175285858nt_int @ ( product_Pair_int_int @ ( bNF_We3272633801743244640im_int @ R2 @ B4 ) @ B ) @ R2 ) ) ) ) ).
% 0.34/0.58
% 0.34/0.58 % wo_rel.minim_least
% 0.34/0.58 thf(fact_3021_wo__rel_Ominim__least,axiom,
% 0.34/0.58 ! [R2: set_Pr1261947904930325089at_nat,B4: set_nat,B: nat] :
% 0.34/0.58 ( ( bNF_We3818239936649020644el_nat @ R2 )
% 0.34/0.58 => ( ( ord_less_eq_set_nat @ B4 @ ( field_nat @ R2 ) )
% 0.34/0.58 => ( ( member_nat @ B @ B4 )
% 0.34/0.58 => ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ ( bNF_We3275124272252294916im_nat @ R2 @ B4 ) @ B ) @ R2 ) ) ) ) ).
% 0.34/0.58
% 0.34/0.58 % wo_rel.minim_least
% 0.34/0.58 thf(fact_3022_wo__rel_Omax2__greater__among,axiom,
% 0.34/0.58 ! [R2: set_Pr5094982260447487303t_unit,A2: product_unit,B: product_unit] :
% 0.34/0.58 ( ( bNF_We2091578341578594809t_unit @ R2 )
% 0.34/0.58 => ( ( member_Product_unit @ A2 @ ( field_Product_unit @ R2 ) )
% 0.34/0.58 => ( ( member_Product_unit @ B @ ( field_Product_unit @ R2 ) )
% 0.34/0.58 => ( ( member7821138191923116944t_unit @ ( produc5692694829384537815t_unit @ A2 @ ( bNF_We7204996500647184731t_unit @ R2 @ A2 @ B ) ) @ R2 )
% 0.34/0.58 & ( member7821138191923116944t_unit @ ( produc5692694829384537815t_unit @ B @ ( bNF_We7204996500647184731t_unit @ R2 @ A2 @ B ) ) @ R2 )
% 0.34/0.58 & ( member_Product_unit @ ( bNF_We7204996500647184731t_unit @ R2 @ A2 @ B ) @ ( insert_Product_unit @ A2 @ ( insert_Product_unit @ B @ bot_bo3957492148770167129t_unit ) ) ) ) ) ) ) ).
% 0.34/0.58
% 0.34/0.58 % wo_rel.max2_greater_among
% 0.34/0.58 thf(fact_3023_wo__rel_Omax2__greater__among,axiom,
% 0.34/0.58 ! [R2: set_Product_prod_o_o,A2: $o,B: $o] :
% 0.34/0.58 ( ( bNF_We6437023980930612612_rel_o @ R2 )
% 0.34/0.58 => ( ( member_o @ A2 @ ( field_o @ R2 ) )
% 0.34/0.58 => ( ( member_o @ B @ ( field_o @ R2 ) )
% 0.34/0.58 => ( ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ A2 @ ( bNF_We3104086342435940326max2_o @ R2 @ A2 @ B ) ) @ R2 )
% 0.34/0.58 & ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ B @ ( bNF_We3104086342435940326max2_o @ R2 @ A2 @ B ) ) @ R2 )
% 0.34/0.58 & ( member_o @ ( bNF_We3104086342435940326max2_o @ R2 @ A2 @ B ) @ ( insert_o @ A2 @ ( insert_o @ B @ bot_bot_set_o ) ) ) ) ) ) ) ).
% 0.34/0.58
% 0.34/0.58 % wo_rel.max2_greater_among
% 0.34/0.58 thf(fact_3024_wo__rel_Omax2__greater__among,axiom,
% 0.34/0.58 ! [R2: set_Pr1261947904930325089at_nat,A2: nat,B: nat] :
% 0.34/0.58 ( ( bNF_We3818239936649020644el_nat @ R2 )
% 0.34/0.58 => ( ( member_nat @ A2 @ ( field_nat @ R2 ) )
% 0.34/0.58 => ( ( member_nat @ B @ ( field_nat @ R2 ) )
% 0.34/0.58 => ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A2 @ ( bNF_We2380939065524664066x2_nat @ R2 @ A2 @ B ) ) @ R2 )
% 0.34/0.58 & ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ B @ ( bNF_We2380939065524664066x2_nat @ R2 @ A2 @ B ) ) @ R2 )
% 0.34/0.58 & ( member_nat @ ( bNF_We2380939065524664066x2_nat @ R2 @ A2 @ B ) @ ( insert_nat @ A2 @ ( insert_nat @ B @ bot_bot_set_nat ) ) ) ) ) ) ) ).
% 0.34/0.58
% 0.34/0.58 % wo_rel.max2_greater_among
% 0.34/0.58 thf(fact_3025_wo__rel_Omax2__greater__among,axiom,
% 0.34/0.58 ! [R2: set_Pr958786334691620121nt_int,A2: int,B: int] :
% 0.34/0.58 ( ( bNF_We3815749466139970368el_int @ R2 )
% 0.34/0.58 => ( ( member_int @ A2 @ ( field_int @ R2 ) )
% 0.34/0.58 => ( ( member_int @ B @ ( field_int @ R2 ) )
% 0.34/0.58 => ( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ A2 @ ( bNF_We2378448595015613790x2_int @ R2 @ A2 @ B ) ) @ R2 )
% 0.34/0.58 & ( member5262025264175285858nt_int @ ( product_Pair_int_int @ B @ ( bNF_We2378448595015613790x2_int @ R2 @ A2 @ B ) ) @ R2 )
% 0.34/0.58 & ( member_int @ ( bNF_We2378448595015613790x2_int @ R2 @ A2 @ B ) @ ( insert_int @ A2 @ ( insert_int @ B @ bot_bot_set_int ) ) ) ) ) ) ) ).
% 0.34/0.58
% 0.34/0.58 % wo_rel.max2_greater_among
% 0.34/0.58 thf(fact_3026_wo__rel_Omax2__greater__among,axiom,
% 0.34/0.58 ! [R2: set_Pr2161125870931222855_nat_o,A2: produc3658429121746597890et_nat > $o,B: produc3658429121746597890et_nat > $o] :
% 0.34/0.58 ( ( bNF_We740377018556551609_nat_o @ R2 )
% 0.34/0.58 => ( ( member6576561426505652726_nat_o @ A2 @ ( field_81821745747897882_nat_o @ R2 ) )
% 0.34/0.58 => ( ( member6576561426505652726_nat_o @ B @ ( field_81821745747897882_nat_o @ R2 ) )
% 0.34/0.58 => ( ( member8781333585448626064_nat_o @ ( produc7368190662567826135_nat_o @ A2 @ ( bNF_We6400040844883348763_nat_o @ R2 @ A2 @ B ) ) @ R2 )
% 0.34/0.58 & ( member8781333585448626064_nat_o @ ( produc7368190662567826135_nat_o @ B @ ( bNF_We6400040844883348763_nat_o @ R2 @ A2 @ B ) ) @ R2 )
% 0.34/0.58 & ( member6576561426505652726_nat_o @ ( bNF_We6400040844883348763_nat_o @ R2 @ A2 @ B ) @ ( insert5175938949040314269_nat_o @ A2 @ ( insert5175938949040314269_nat_o @ B @ bot_bo7824918357723582233_nat_o ) ) ) ) ) ) ) ).
% 0.34/0.58
% 0.34/0.58 % wo_rel.max2_greater_among
% 0.34/0.58 thf(fact_3027_wo__rel_Omax2__def,axiom,
% 0.34/0.58 ! [R2: set_Pr1261947904930325089at_nat,A2: nat,B: nat] :
% 0.34/0.58 ( ( bNF_We3818239936649020644el_nat @ R2 )
% 0.34/0.58 => ( ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A2 @ B ) @ R2 )
% 0.34/0.58 => ( ( bNF_We2380939065524664066x2_nat @ R2 @ A2 @ B )
% 0.34/0.58 = B ) )
% 0.34/0.58 & ( ~ ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A2 @ B ) @ R2 )
% 0.34/0.58 => ( ( bNF_We2380939065524664066x2_nat @ R2 @ A2 @ B )
% 0.34/0.58 = A2 ) ) ) ) ).
% 0.34/0.58
% 0.34/0.58 % wo_rel.max2_def
% 0.34/0.58 thf(fact_3028_wo__rel_Omax2__def,axiom,
% 0.34/0.58 ! [R2: set_Pr958786334691620121nt_int,A2: int,B: int] :
% 0.34/0.58 ( ( bNF_We3815749466139970368el_int @ R2 )
% 0.34/0.58 => ( ( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ A2 @ B ) @ R2 )
% 0.34/0.58 => ( ( bNF_We2378448595015613790x2_int @ R2 @ A2 @ B )
% 0.34/0.58 = B ) )
% 0.34/0.58 & ( ~ ( member5262025264175285858nt_int @ ( product_Pair_int_int @ A2 @ B ) @ R2 )
% 0.34/0.58 => ( ( bNF_We2378448595015613790x2_int @ R2 @ A2 @ B )
% 0.34/0.58 = A2 ) ) ) ) ).
% 0.34/0.58
% 0.34/0.58 % wo_rel.max2_def
% 0.34/0.58 thf(fact_3029_under__incr,axiom,
% 0.34/0.58 ! [R2: set_Pr958786334691620121nt_int,A2: int,B: int] :
% 0.34/0.58 ( ( trans_int @ R2 )
% 0.34/0.58 => ( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ A2 @ B ) @ R2 )
% 0.34/0.58 => ( ord_less_eq_set_int @ ( order_under_int @ R2 @ A2 ) @ ( order_under_int @ R2 @ B ) ) ) ) ).
% 0.34/0.58
% 0.34/0.58 % under_incr
% 0.34/0.58 thf(fact_3030_under__incr,axiom,
% 0.34/0.58 ! [R2: set_Pr1261947904930325089at_nat,A2: nat,B: nat] :
% 0.34/0.58 ( ( trans_nat @ R2 )
% 0.34/0.58 => ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A2 @ B ) @ R2 )
% 0.34/0.58 => ( ord_less_eq_set_nat @ ( order_under_nat @ R2 @ A2 ) @ ( order_under_nat @ R2 @ B ) ) ) ) ).
% 0.34/0.58
% 0.34/0.58 % under_incr
% 0.34/0.58 thf(fact_3031_wo__rel_Ominim__isMinim,axiom,
% 0.34/0.58 ! [R2: set_Pr5094982260447487303t_unit,B4: set_Product_unit] :
% 0.34/0.58 ( ( bNF_We2091578341578594809t_unit @ R2 )
% 0.34/0.58 => ( ( ord_le3507040750410214029t_unit @ B4 @ ( field_Product_unit @ R2 ) )
% 0.34/0.58 => ( ( B4 != bot_bo3957492148770167129t_unit )
% 0.34/0.58 => ( bNF_We909621271159000003t_unit @ R2 @ B4 @ ( bNF_We7185707985517244377t_unit @ R2 @ B4 ) ) ) ) ) ).
% 0.34/0.58
% 0.34/0.58 % wo_rel.minim_isMinim
% 0.34/0.58 thf(fact_3032_wo__rel_Ominim__isMinim,axiom,
% 0.34/0.58 ! [R2: set_Product_prod_o_o,B4: set_o] :
% 0.34/0.58 ( ( bNF_We6437023980930612612_rel_o @ R2 )
% 0.34/0.58 => ( ( ord_less_eq_set_o @ B4 @ ( field_o @ R2 ) )
% 0.34/0.58 => ( ( B4 != bot_bot_set_o )
% 0.34/0.58 => ( bNF_We4647864369647134286inim_o @ R2 @ B4 @ ( bNF_We537455116209922916inim_o @ R2 @ B4 ) ) ) ) ) ).
% 0.34/0.58
% 0.34/0.58 % wo_rel.minim_isMinim
% 0.34/0.58 thf(fact_3033_wo__rel_Ominim__isMinim,axiom,
% 0.34/0.58 ! [R2: set_Pr958786334691620121nt_int,B4: set_int] :
% 0.34/0.58 ( ( bNF_We3815749466139970368el_int @ R2 )
% 0.34/0.58 => ( ( ord_less_eq_set_int @ B4 @ ( field_int @ R2 ) )
% 0.34/0.58 => ( ( B4 != bot_bot_set_int )
% 0.34/0.58 => ( bNF_We6511469253196982518im_int @ R2 @ B4 @ ( bNF_We3272633801743244640im_int @ R2 @ B4 ) ) ) ) ) ).
% 0.34/0.58
% 0.34/0.58 % wo_rel.minim_isMinim
% 0.34/0.58 thf(fact_3034_wo__rel_Ominim__isMinim,axiom,
% 0.34/0.58 ! [R2: set_Pr2161125870931222855_nat_o,B4: set_Pr4532377907799695533_nat_o] :
% 0.34/0.58 ( ( bNF_We740377018556551609_nat_o @ R2 )
% 0.34/0.58 => ( ( ord_le2965882846123202637_nat_o @ B4 @ ( field_81821745747897882_nat_o @ R2 ) )
% 0.34/0.58 => ( ( B4 != bot_bo7824918357723582233_nat_o )
% 0.34/0.58 => ( bNF_We5608535359669600131_nat_o @ R2 @ B4 @ ( bNF_We8236795211600914329_nat_o @ R2 @ B4 ) ) ) ) ) ).
% 0.34/0.58
% 0.34/0.58 % wo_rel.minim_isMinim
% 0.34/0.58 thf(fact_3035_wo__rel_Ominim__isMinim,axiom,
% 0.34/0.58 ! [R2: set_Pr1261947904930325089at_nat,B4: set_nat] :
% 0.34/0.58 ( ( bNF_We3818239936649020644el_nat @ R2 )
% 0.34/0.58 => ( ( ord_less_eq_set_nat @ B4 @ ( field_nat @ R2 ) )
% 0.34/0.58 => ( ( B4 != bot_bot_set_nat )
% 0.34/0.58 => ( bNF_We6513959723706032794im_nat @ R2 @ B4 @ ( bNF_We3275124272252294916im_nat @ R2 @ B4 ) ) ) ) ) ).
% 0.34/0.58
% 0.34/0.58 % wo_rel.minim_isMinim
% 0.34/0.58 thf(fact_3036_wo__rel_OisMinim__def,axiom,
% 0.34/0.58 ! [R2: set_Pr2161125870931222855_nat_o,A: set_Pr4532377907799695533_nat_o,B: produc3658429121746597890et_nat > $o] :
% 0.34/0.58 ( ( bNF_We740377018556551609_nat_o @ R2 )
% 0.34/0.58 => ( ( bNF_We5608535359669600131_nat_o @ R2 @ A @ B )
% 0.34/0.58 = ( ( member6576561426505652726_nat_o @ B @ A )
% 0.34/0.58 & ! [X2: produc3658429121746597890et_nat > $o] :
% 0.34/0.58 ( ( member6576561426505652726_nat_o @ X2 @ A )
% 0.34/0.58 => ( member8781333585448626064_nat_o @ ( produc7368190662567826135_nat_o @ B @ X2 ) @ R2 ) ) ) ) ) ).
% 0.34/0.58
% 0.34/0.58 % wo_rel.isMinim_def
% 0.34/0.58 thf(fact_3037_wo__rel_OisMinim__def,axiom,
% 0.34/0.58 ! [R2: set_Product_prod_o_o,A: set_o,B: $o] :
% 0.34/0.58 ( ( bNF_We6437023980930612612_rel_o @ R2 )
% 0.34/0.58 => ( ( bNF_We4647864369647134286inim_o @ R2 @ A @ B )
% 0.34/0.58 = ( ( member_o @ B @ A )
% 0.34/0.58 & ! [X2: $o] :
% 0.34/0.58 ( ( member_o @ X2 @ A )
% 0.34/0.58 => ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ B @ X2 ) @ R2 ) ) ) ) ) ).
% 0.34/0.58
% 0.34/0.58 % wo_rel.isMinim_def
% 0.34/0.58 thf(fact_3038_wo__rel_OisMinim__def,axiom,
% 0.34/0.58 ! [R2: set_Pr5094982260447487303t_unit,A: set_Product_unit,B: product_unit] :
% 0.34/0.58 ( ( bNF_We2091578341578594809t_unit @ R2 )
% 0.34/0.58 => ( ( bNF_We909621271159000003t_unit @ R2 @ A @ B )
% 0.34/0.58 = ( ( member_Product_unit @ B @ A )
% 0.34/0.58 & ! [X2: product_unit] :
% 0.34/0.58 ( ( member_Product_unit @ X2 @ A )
% 0.34/0.58 => ( member7821138191923116944t_unit @ ( produc5692694829384537815t_unit @ B @ X2 ) @ R2 ) ) ) ) ) ).
% 0.34/0.58
% 0.34/0.58 % wo_rel.isMinim_def
% 0.34/0.58 thf(fact_3039_wo__rel_OisMinim__def,axiom,
% 0.34/0.58 ! [R2: set_Pr1261947904930325089at_nat,A: set_nat,B: nat] :
% 0.34/0.58 ( ( bNF_We3818239936649020644el_nat @ R2 )
% 0.34/0.58 => ( ( bNF_We6513959723706032794im_nat @ R2 @ A @ B )
% 0.34/0.58 = ( ( member_nat @ B @ A )
% 0.34/0.58 & ! [X2: nat] :
% 0.34/0.58 ( ( member_nat @ X2 @ A )
% 0.34/0.58 => ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ B @ X2 ) @ R2 ) ) ) ) ) ).
% 0.34/0.58
% 0.34/0.58 % wo_rel.isMinim_def
% 0.34/0.58 thf(fact_3040_wo__rel_OisMinim__def,axiom,
% 0.34/0.58 ! [R2: set_Pr958786334691620121nt_int,A: set_int,B: int] :
% 0.34/0.58 ( ( bNF_We3815749466139970368el_int @ R2 )
% 0.34/0.58 => ( ( bNF_We6511469253196982518im_int @ R2 @ A @ B )
% 0.34/0.58 = ( ( member_int @ B @ A )
% 0.34/0.58 & ! [X2: int] :
% 0.34/0.58 ( ( member_int @ X2 @ A )
% 0.34/0.58 => ( member5262025264175285858nt_int @ ( product_Pair_int_int @ B @ X2 ) @ R2 ) ) ) ) ) ).
% 0.34/0.58
% 0.34/0.58 % wo_rel.isMinim_def
% 0.34/0.58 thf(fact_3041_wo__rel_Omax2__greater,axiom,
% 0.34/0.58 ! [R2: set_Pr2161125870931222855_nat_o,A2: produc3658429121746597890et_nat > $o,B: produc3658429121746597890et_nat > $o] :
% 0.34/0.58 ( ( bNF_We740377018556551609_nat_o @ R2 )
% 0.34/0.58 => ( ( member6576561426505652726_nat_o @ A2 @ ( field_81821745747897882_nat_o @ R2 ) )
% 0.34/0.58 => ( ( member6576561426505652726_nat_o @ B @ ( field_81821745747897882_nat_o @ R2 ) )
% 0.34/0.58 => ( ( member8781333585448626064_nat_o @ ( produc7368190662567826135_nat_o @ A2 @ ( bNF_We6400040844883348763_nat_o @ R2 @ A2 @ B ) ) @ R2 )
% 0.34/0.58 & ( member8781333585448626064_nat_o @ ( produc7368190662567826135_nat_o @ B @ ( bNF_We6400040844883348763_nat_o @ R2 @ A2 @ B ) ) @ R2 ) ) ) ) ) ).
% 0.34/0.58
% 0.34/0.58 % wo_rel.max2_greater
% 0.34/0.58 thf(fact_3042_wo__rel_Omax2__greater,axiom,
% 0.34/0.58 ! [R2: set_Product_prod_o_o,A2: $o,B: $o] :
% 0.34/0.58 ( ( bNF_We6437023980930612612_rel_o @ R2 )
% 0.34/0.58 => ( ( member_o @ A2 @ ( field_o @ R2 ) )
% 0.34/0.58 => ( ( member_o @ B @ ( field_o @ R2 ) )
% 0.34/0.58 => ( ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ A2 @ ( bNF_We3104086342435940326max2_o @ R2 @ A2 @ B ) ) @ R2 )
% 0.34/0.58 & ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ B @ ( bNF_We3104086342435940326max2_o @ R2 @ A2 @ B ) ) @ R2 ) ) ) ) ) ).
% 0.34/0.58
% 0.34/0.58 % wo_rel.max2_greater
% 0.34/0.58 thf(fact_3043_wo__rel_Omax2__greater,axiom,
% 0.34/0.58 ! [R2: set_Pr5094982260447487303t_unit,A2: product_unit,B: product_unit] :
% 0.34/0.58 ( ( bNF_We2091578341578594809t_unit @ R2 )
% 0.34/0.58 => ( ( member_Product_unit @ A2 @ ( field_Product_unit @ R2 ) )
% 0.34/0.58 => ( ( member_Product_unit @ B @ ( field_Product_unit @ R2 ) )
% 0.34/0.58 => ( ( member7821138191923116944t_unit @ ( produc5692694829384537815t_unit @ A2 @ ( bNF_We7204996500647184731t_unit @ R2 @ A2 @ B ) ) @ R2 )
% 0.34/0.58 & ( member7821138191923116944t_unit @ ( produc5692694829384537815t_unit @ B @ ( bNF_We7204996500647184731t_unit @ R2 @ A2 @ B ) ) @ R2 ) ) ) ) ) ).
% 0.34/0.58
% 0.34/0.58 % wo_rel.max2_greater
% 0.34/0.58 thf(fact_3044_wo__rel_Omax2__greater,axiom,
% 0.34/0.58 ! [R2: set_Pr1261947904930325089at_nat,A2: nat,B: nat] :
% 0.34/0.58 ( ( bNF_We3818239936649020644el_nat @ R2 )
% 0.34/0.58 => ( ( member_nat @ A2 @ ( field_nat @ R2 ) )
% 0.34/0.58 => ( ( member_nat @ B @ ( field_nat @ R2 ) )
% 0.34/0.58 => ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A2 @ ( bNF_We2380939065524664066x2_nat @ R2 @ A2 @ B ) ) @ R2 )
% 0.34/0.58 & ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ B @ ( bNF_We2380939065524664066x2_nat @ R2 @ A2 @ B ) ) @ R2 ) ) ) ) ) ).
% 0.34/0.58
% 0.34/0.58 % wo_rel.max2_greater
% 0.34/0.58 thf(fact_3045_wo__rel_Omax2__greater,axiom,
% 0.34/0.58 ! [R2: set_Pr958786334691620121nt_int,A2: int,B: int] :
% 0.34/0.58 ( ( bNF_We3815749466139970368el_int @ R2 )
% 0.34/0.58 => ( ( member_int @ A2 @ ( field_int @ R2 ) )
% 0.34/0.58 => ( ( member_int @ B @ ( field_int @ R2 ) )
% 0.34/0.58 => ( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ A2 @ ( bNF_We2378448595015613790x2_int @ R2 @ A2 @ B ) ) @ R2 )
% 0.34/0.58 & ( member5262025264175285858nt_int @ ( product_Pair_int_int @ B @ ( bNF_We2378448595015613790x2_int @ R2 @ A2 @ B ) ) @ R2 ) ) ) ) ) ).
% 0.34/0.58
% 0.34/0.58 % wo_rel.max2_greater
% 0.34/0.58 thf(fact_3046_wo__rel_Omax2__equals2,axiom,
% 0.34/0.58 ! [R2: set_Pr2161125870931222855_nat_o,A2: produc3658429121746597890et_nat > $o,B: produc3658429121746597890et_nat > $o] :
% 0.34/0.58 ( ( bNF_We740377018556551609_nat_o @ R2 )
% 0.34/0.58 => ( ( member6576561426505652726_nat_o @ A2 @ ( field_81821745747897882_nat_o @ R2 ) )
% 0.34/0.58 => ( ( member6576561426505652726_nat_o @ B @ ( field_81821745747897882_nat_o @ R2 ) )
% 0.34/0.58 => ( ( ( bNF_We6400040844883348763_nat_o @ R2 @ A2 @ B )
% 0.34/0.58 = B )
% 0.34/0.58 = ( member8781333585448626064_nat_o @ ( produc7368190662567826135_nat_o @ A2 @ B ) @ R2 ) ) ) ) ) ).
% 0.34/0.58
% 0.34/0.58 % wo_rel.max2_equals2
% 0.34/0.58 thf(fact_3047_wo__rel_Omax2__equals2,axiom,
% 0.34/0.58 ! [R2: set_Product_prod_o_o,A2: $o,B: $o] :
% 0.34/0.58 ( ( bNF_We6437023980930612612_rel_o @ R2 )
% 0.34/0.58 => ( ( member_o @ A2 @ ( field_o @ R2 ) )
% 0.34/0.58 => ( ( member_o @ B @ ( field_o @ R2 ) )
% 0.34/0.58 => ( ( ( bNF_We3104086342435940326max2_o @ R2 @ A2 @ B )
% 0.34/0.58 = B )
% 0.34/0.58 = ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ A2 @ B ) @ R2 ) ) ) ) ) ).
% 0.34/0.58
% 0.34/0.58 % wo_rel.max2_equals2
% 0.34/0.58 thf(fact_3048_wo__rel_Omax2__equals2,axiom,
% 0.34/0.58 ! [R2: set_Pr5094982260447487303t_unit,A2: product_unit,B: product_unit] :
% 0.34/0.58 ( ( bNF_We2091578341578594809t_unit @ R2 )
% 0.34/0.58 => ( ( member_Product_unit @ A2 @ ( field_Product_unit @ R2 ) )
% 0.34/0.58 => ( ( member_Product_unit @ B @ ( field_Product_unit @ R2 ) )
% 0.34/0.58 => ( ( ( bNF_We7204996500647184731t_unit @ R2 @ A2 @ B )
% 0.34/0.58 = B )
% 0.34/0.58 = ( member7821138191923116944t_unit @ ( produc5692694829384537815t_unit @ A2 @ B ) @ R2 ) ) ) ) ) ).
% 0.34/0.58
% 0.34/0.58 % wo_rel.max2_equals2
% 0.34/0.58 thf(fact_3049_wo__rel_Omax2__equals2,axiom,
% 0.34/0.58 ! [R2: set_Pr1261947904930325089at_nat,A2: nat,B: nat] :
% 0.34/0.58 ( ( bNF_We3818239936649020644el_nat @ R2 )
% 0.34/0.58 => ( ( member_nat @ A2 @ ( field_nat @ R2 ) )
% 0.34/0.58 => ( ( member_nat @ B @ ( field_nat @ R2 ) )
% 0.34/0.58 => ( ( ( bNF_We2380939065524664066x2_nat @ R2 @ A2 @ B )
% 0.34/0.58 = B )
% 0.34/0.58 = ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A2 @ B ) @ R2 ) ) ) ) ) ).
% 0.34/0.58
% 0.34/0.58 % wo_rel.max2_equals2
% 0.34/0.58 thf(fact_3050_wo__rel_Omax2__equals2,axiom,
% 0.34/0.58 ! [R2: set_Pr958786334691620121nt_int,A2: int,B: int] :
% 0.34/0.58 ( ( bNF_We3815749466139970368el_int @ R2 )
% 0.34/0.58 => ( ( member_int @ A2 @ ( field_int @ R2 ) )
% 0.34/0.58 => ( ( member_int @ B @ ( field_int @ R2 ) )
% 0.34/0.58 => ( ( ( bNF_We2378448595015613790x2_int @ R2 @ A2 @ B )
% 0.34/0.58 = B )
% 0.34/0.58 = ( member5262025264175285858nt_int @ ( product_Pair_int_int @ A2 @ B ) @ R2 ) ) ) ) ) ).
% 0.34/0.58
% 0.34/0.58 % wo_rel.max2_equals2
% 0.34/0.58 thf(fact_3051_wo__rel_Omax2__equals1,axiom,
% 0.34/0.58 ! [R2: set_Pr2161125870931222855_nat_o,A2: produc3658429121746597890et_nat > $o,B: produc3658429121746597890et_nat > $o] :
% 0.34/0.58 ( ( bNF_We740377018556551609_nat_o @ R2 )
% 0.34/0.58 => ( ( member6576561426505652726_nat_o @ A2 @ ( field_81821745747897882_nat_o @ R2 ) )
% 0.34/0.58 => ( ( member6576561426505652726_nat_o @ B @ ( field_81821745747897882_nat_o @ R2 ) )
% 0.34/0.58 => ( ( ( bNF_We6400040844883348763_nat_o @ R2 @ A2 @ B )
% 0.34/0.58 = A2 )
% 0.34/0.58 = ( member8781333585448626064_nat_o @ ( produc7368190662567826135_nat_o @ B @ A2 ) @ R2 ) ) ) ) ) ).
% 0.34/0.58
% 0.34/0.58 % wo_rel.max2_equals1
% 0.34/0.58 thf(fact_3052_wo__rel_Omax2__equals1,axiom,
% 0.34/0.58 ! [R2: set_Product_prod_o_o,A2: $o,B: $o] :
% 0.34/0.58 ( ( bNF_We6437023980930612612_rel_o @ R2 )
% 0.34/0.58 => ( ( member_o @ A2 @ ( field_o @ R2 ) )
% 0.34/0.58 => ( ( member_o @ B @ ( field_o @ R2 ) )
% 0.34/0.58 => ( ( ( bNF_We3104086342435940326max2_o @ R2 @ A2 @ B )
% 0.34/0.58 = A2 )
% 0.34/0.58 = ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ B @ A2 ) @ R2 ) ) ) ) ) ).
% 0.34/0.58
% 0.34/0.58 % wo_rel.max2_equals1
% 0.34/0.58 thf(fact_3053_wo__rel_Omax2__equals1,axiom,
% 0.34/0.58 ! [R2: set_Pr5094982260447487303t_unit,A2: product_unit,B: product_unit] :
% 0.34/0.58 ( ( bNF_We2091578341578594809t_unit @ R2 )
% 0.34/0.58 => ( ( member_Product_unit @ A2 @ ( field_Product_unit @ R2 ) )
% 0.34/0.58 => ( ( member_Product_unit @ B @ ( field_Product_unit @ R2 ) )
% 0.34/0.58 => ( ( ( bNF_We7204996500647184731t_unit @ R2 @ A2 @ B )
% 0.34/0.58 = A2 )
% 0.34/0.58 = ( member7821138191923116944t_unit @ ( produc5692694829384537815t_unit @ B @ A2 ) @ R2 ) ) ) ) ) ).
% 0.34/0.58
% 0.34/0.58 % wo_rel.max2_equals1
% 0.34/0.58 thf(fact_3054_wo__rel_Omax2__equals1,axiom,
% 0.34/0.58 ! [R2: set_Pr1261947904930325089at_nat,A2: nat,B: nat] :
% 0.34/0.58 ( ( bNF_We3818239936649020644el_nat @ R2 )
% 0.34/0.58 => ( ( member_nat @ A2 @ ( field_nat @ R2 ) )
% 0.34/0.58 => ( ( member_nat @ B @ ( field_nat @ R2 ) )
% 0.34/0.58 => ( ( ( bNF_We2380939065524664066x2_nat @ R2 @ A2 @ B )
% 0.34/0.58 = A2 )
% 0.34/0.58 = ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ B @ A2 ) @ R2 ) ) ) ) ) ).
% 0.34/0.58
% 0.34/0.58 % wo_rel.max2_equals1
% 0.34/0.58 thf(fact_3055_wo__rel_Omax2__equals1,axiom,
% 0.34/0.58 ! [R2: set_Pr958786334691620121nt_int,A2: int,B: int] :
% 0.34/0.58 ( ( bNF_We3815749466139970368el_int @ R2 )
% 0.34/0.58 => ( ( member_int @ A2 @ ( field_int @ R2 ) )
% 0.34/0.58 => ( ( member_int @ B @ ( field_int @ R2 ) )
% 0.34/0.58 => ( ( ( bNF_We2378448595015613790x2_int @ R2 @ A2 @ B )
% 0.34/0.58 = A2 )
% 0.34/0.58 = ( member5262025264175285858nt_int @ ( product_Pair_int_int @ B @ A2 ) @ R2 ) ) ) ) ) ).
% 0.34/0.58
% 0.34/0.58 % wo_rel.max2_equals1
% 0.34/0.58 thf(fact_3056_wo__rel_Omax2__among,axiom,
% 0.34/0.58 ! [R2: set_Pr5094982260447487303t_unit,A2: product_unit,B: product_unit] :
% 0.34/0.58 ( ( bNF_We2091578341578594809t_unit @ R2 )
% 0.34/0.58 => ( ( member_Product_unit @ A2 @ ( field_Product_unit @ R2 ) )
% 0.34/0.58 => ( ( member_Product_unit @ B @ ( field_Product_unit @ R2 ) )
% 0.34/0.58 => ( member_Product_unit @ ( bNF_We7204996500647184731t_unit @ R2 @ A2 @ B ) @ ( insert_Product_unit @ A2 @ ( insert_Product_unit @ B @ bot_bo3957492148770167129t_unit ) ) ) ) ) ) ).
% 0.34/0.58
% 0.34/0.58 % wo_rel.max2_among
% 0.34/0.58 thf(fact_3057_wo__rel_Omax2__among,axiom,
% 0.34/0.58 ! [R2: set_Product_prod_o_o,A2: $o,B: $o] :
% 0.34/0.58 ( ( bNF_We6437023980930612612_rel_o @ R2 )
% 0.34/0.58 => ( ( member_o @ A2 @ ( field_o @ R2 ) )
% 0.34/0.58 => ( ( member_o @ B @ ( field_o @ R2 ) )
% 0.34/0.58 => ( member_o @ ( bNF_We3104086342435940326max2_o @ R2 @ A2 @ B ) @ ( insert_o @ A2 @ ( insert_o @ B @ bot_bot_set_o ) ) ) ) ) ) ).
% 0.34/0.58
% 0.34/0.58 % wo_rel.max2_among
% 0.34/0.58 thf(fact_3058_wo__rel_Omax2__among,axiom,
% 0.34/0.58 ! [R2: set_Pr1261947904930325089at_nat,A2: nat,B: nat] :
% 0.34/0.58 ( ( bNF_We3818239936649020644el_nat @ R2 )
% 0.34/0.58 => ( ( member_nat @ A2 @ ( field_nat @ R2 ) )
% 0.34/0.58 => ( ( member_nat @ B @ ( field_nat @ R2 ) )
% 0.34/0.58 => ( member_nat @ ( bNF_We2380939065524664066x2_nat @ R2 @ A2 @ B ) @ ( insert_nat @ A2 @ ( insert_nat @ B @ bot_bot_set_nat ) ) ) ) ) ) ).
% 0.34/0.58
% 0.34/0.58 % wo_rel.max2_among
% 0.34/0.58 thf(fact_3059_wo__rel_Omax2__among,axiom,
% 0.34/0.58 ! [R2: set_Pr958786334691620121nt_int,A2: int,B: int] :
% 0.34/0.58 ( ( bNF_We3815749466139970368el_int @ R2 )
% 0.34/0.58 => ( ( member_int @ A2 @ ( field_int @ R2 ) )
% 0.34/0.58 => ( ( member_int @ B @ ( field_int @ R2 ) )
% 0.34/0.58 => ( member_int @ ( bNF_We2378448595015613790x2_int @ R2 @ A2 @ B ) @ ( insert_int @ A2 @ ( insert_int @ B @ bot_bot_set_int ) ) ) ) ) ) ).
% 0.34/0.58
% 0.34/0.58 % wo_rel.max2_among
% 0.34/0.58 thf(fact_3060_wo__rel_Omax2__among,axiom,
% 0.34/0.58 ! [R2: set_Pr2161125870931222855_nat_o,A2: produc3658429121746597890et_nat > $o,B: produc3658429121746597890et_nat > $o] :
% 0.34/0.58 ( ( bNF_We740377018556551609_nat_o @ R2 )
% 0.34/0.58 => ( ( member6576561426505652726_nat_o @ A2 @ ( field_81821745747897882_nat_o @ R2 ) )
% 0.34/0.58 => ( ( member6576561426505652726_nat_o @ B @ ( field_81821745747897882_nat_o @ R2 ) )
% 0.34/0.58 => ( member6576561426505652726_nat_o @ ( bNF_We6400040844883348763_nat_o @ R2 @ A2 @ B ) @ ( insert5175938949040314269_nat_o @ A2 @ ( insert5175938949040314269_nat_o @ B @ bot_bo7824918357723582233_nat_o ) ) ) ) ) ) ).
% 0.34/0.58
% 0.34/0.58 % wo_rel.max2_among
% 0.34/0.58 thf(fact_3061_chains__extend,axiom,
% 0.34/0.58 ! [C: set_set_nat,S: set_set_nat,Z: set_nat] :
% 0.34/0.58 ( ( member_set_set_nat @ C @ ( chains_nat2 @ S ) )
% 0.34/0.58 => ( ( member_set_nat @ Z @ S )
% 0.34/0.58 => ( ! [X3: set_nat] :
% 0.34/0.58 ( ( member_set_nat @ X3 @ C )
% 0.34/0.58 => ( ord_less_eq_set_nat @ X3 @ Z ) )
% 0.34/0.58 => ( member_set_set_nat @ ( sup_sup_set_set_nat @ ( insert_set_nat @ Z @ bot_bot_set_set_nat ) @ C ) @ ( chains_nat2 @ S ) ) ) ) ) ).
% 0.34/0.58
% 0.34/0.58 % chains_extend
% 0.34/0.58 thf(fact_3062_Zorns__po__lemma,axiom,
% 0.34/0.58 ! [R2: set_Pr1261947904930325089at_nat] :
% 0.34/0.58 ( ( order_5251275573222108571on_nat @ ( field_nat @ R2 ) @ R2 )
% 0.34/0.58 => ( ! [C6: set_nat] :
% 0.34/0.58 ( ( member_set_nat @ C6 @ ( chains_nat @ R2 ) )
% 0.34/0.58 => ? [X6: nat] :
% 0.34/0.58 ( ( member_nat @ X6 @ ( field_nat @ R2 ) )
% 0.34/0.58 & ! [Xa3: nat] :
% 0.34/0.58 ( ( member_nat @ Xa3 @ C6 )
% 0.34/0.58 => ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ Xa3 @ X6 ) @ R2 ) ) ) )
% 0.34/0.58 => ? [X3: nat] :
% 0.34/0.58 ( ( member_nat @ X3 @ ( field_nat @ R2 ) )
% 0.34/0.58 & ! [Xa2: nat] :
% 0.34/0.58 ( ( member_nat @ Xa2 @ ( field_nat @ R2 ) )
% 0.34/0.58 => ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X3 @ Xa2 ) @ R2 )
% 0.34/0.58 => ( Xa2 = X3 ) ) ) ) ) ) ).
% 0.34/0.58
% 0.34/0.58 % Zorns_po_lemma
% 0.34/0.58 thf(fact_3063_Zorns__po__lemma,axiom,
% 0.34/0.58 ! [R2: set_Pr958786334691620121nt_int] :
% 0.34/0.58 ( ( order_5248785102713058295on_int @ ( field_int @ R2 ) @ R2 )
% 0.34/0.58 => ( ! [C6: set_int] :
% 0.34/0.58 ( ( member_set_int @ C6 @ ( chains_int @ R2 ) )
% 0.34/0.58 => ? [X6: int] :
% 0.34/0.58 ( ( member_int @ X6 @ ( field_int @ R2 ) )
% 0.34/0.58 & ! [Xa3: int] :
% 0.34/0.58 ( ( member_int @ Xa3 @ C6 )
% 0.34/0.58 => ( member5262025264175285858nt_int @ ( product_Pair_int_int @ Xa3 @ X6 ) @ R2 ) ) ) )
% 0.34/0.58 => ? [X3: int] :
% 0.34/0.58 ( ( member_int @ X3 @ ( field_int @ R2 ) )
% 0.34/0.58 & ! [Xa2: int] :
% 0.34/0.58 ( ( member_int @ Xa2 @ ( field_int @ R2 ) )
% 0.34/0.58 => ( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ X3 @ Xa2 ) @ R2 )
% 0.34/0.58 => ( Xa2 = X3 ) ) ) ) ) ) ).
% 0.34/0.58
% 0.34/0.58 % Zorns_po_lemma
% 0.34/0.58 thf(fact_3064_bsqr__max2,axiom,
% 0.34/0.58 ! [R2: set_Pr1261947904930325089at_nat,A1: nat,A22: nat,B1: nat,B22: nat] :
% 0.34/0.58 ( ( order_2888998067076097458on_nat @ ( field_nat @ R2 ) @ R2 )
% 0.34/0.58 => ( ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ ( product_Pair_nat_nat @ A1 @ A22 ) @ ( product_Pair_nat_nat @ B1 @ B22 ) ) @ ( bNF_We1732890022510290291qr_nat @ R2 ) )
% 0.34/0.58 => ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ ( bNF_We2380939065524664066x2_nat @ R2 @ A1 @ A22 ) @ ( bNF_We2380939065524664066x2_nat @ R2 @ B1 @ B22 ) ) @ R2 ) ) ) ).
% 0.34/0.58
% 0.34/0.58 % bsqr_max2
% 0.34/0.58 thf(fact_3065_bsqr__max2,axiom,
% 0.34/0.58 ! [R2: set_Pr958786334691620121nt_int,A1: int,A22: int,B1: int,B22: int] :
% 0.34/0.58 ( ( order_2886507596567047182on_int @ ( field_int @ R2 ) @ R2 )
% 0.34/0.58 => ( ( member8566619992076573584nt_int @ ( produc3646306378393792727nt_int @ ( product_Pair_int_int @ A1 @ A22 ) @ ( product_Pair_int_int @ B1 @ B22 ) ) @ ( bNF_We1730399552001240015qr_int @ R2 ) )
% 0.34/0.58 => ( member5262025264175285858nt_int @ ( product_Pair_int_int @ ( bNF_We2378448595015613790x2_int @ R2 @ A1 @ A22 ) @ ( bNF_We2378448595015613790x2_int @ R2 @ B1 @ B22 ) ) @ R2 ) ) ) ).
% 0.34/0.58
% 0.34/0.58 % bsqr_max2
% 0.34/0.58 thf(fact_3066_wf__Union,axiom,
% 0.34/0.58 ! [R3: set_se8994758365886191399t_unit] :
% 0.34/0.58 ( ! [X3: set_Pr5094982260447487303t_unit] :
% 0.34/0.58 ( ( member1248447801810324848t_unit @ X3 @ R3 )
% 0.34/0.58 => ( wf_Product_unit @ X3 ) )
% 0.34/0.58 => ( ! [X3: set_Pr5094982260447487303t_unit] :
% 0.34/0.58 ( ( member1248447801810324848t_unit @ X3 @ R3 )
% 0.34/0.58 => ! [Xa3: set_Pr5094982260447487303t_unit] :
% 0.34/0.58 ( ( member1248447801810324848t_unit @ Xa3 @ R3 )
% 0.34/0.58 => ( ( X3 != Xa3 )
% 0.34/0.58 => ( ( inf_in4660618365625256667t_unit @ ( domain4292024440186670924t_unit @ X3 ) @ ( range_6617724647562260917t_unit @ Xa3 ) )
% 0.34/0.58 = bot_bo3957492148770167129t_unit ) ) ) )
% 0.34/0.58 => ( wf_Product_unit @ ( comple2798140142350441396t_unit @ R3 ) ) ) ) ).
% 0.34/0.58
% 0.34/0.58 % wf_Union
% 0.34/0.58 thf(fact_3067_wf__Union,axiom,
% 0.34/0.58 ! [R3: set_se491199571785101031od_o_o] :
% 0.34/0.58 ( ! [X3: set_Product_prod_o_o] :
% 0.34/0.58 ( ( member3157015407008874800od_o_o @ X3 @ R3 )
% 0.34/0.58 => ( wf_o @ X3 ) )
% 0.34/0.58 => ( ! [X3: set_Product_prod_o_o] :
% 0.34/0.58 ( ( member3157015407008874800od_o_o @ X3 @ R3 )
% 0.34/0.58 => ! [Xa3: set_Product_prod_o_o] :
% 0.34/0.58 ( ( member3157015407008874800od_o_o @ Xa3 @ R3 )
% 0.34/0.58 => ( ( X3 != Xa3 )
% 0.34/0.58 => ( ( inf_inf_set_o @ ( domain_o_o @ X3 ) @ ( range_o_o @ Xa3 ) )
% 0.34/0.58 = bot_bot_set_o ) ) ) )
% 0.34/0.58 => ( wf_o @ ( comple2567785453016644468od_o_o @ R3 ) ) ) ) ).
% 0.34/0.58
% 0.34/0.58 % wf_Union
% 0.34/0.58 thf(fact_3068_wf__Union,axiom,
% 0.34/0.58 ! [R3: set_se7855581050983116737at_nat] :
% 0.34/0.58 ( ! [X3: set_Pr1261947904930325089at_nat] :
% 0.34/0.58 ( ( member2643936169264416010at_nat @ X3 @ R3 )
% 0.34/0.58 => ( wf_nat @ X3 ) )
% 0.34/0.58 => ( ! [X3: set_Pr1261947904930325089at_nat] :
% 0.34/0.58 ( ( member2643936169264416010at_nat @ X3 @ R3 )
% 0.34/0.58 => ! [Xa3: set_Pr1261947904930325089at_nat] :
% 0.34/0.58 ( ( member2643936169264416010at_nat @ Xa3 @ R3 )
% 0.34/0.58 => ( ( X3 != Xa3 )
% 0.34/0.58 => ( ( inf_inf_set_nat @ ( domain_nat_nat @ X3 ) @ ( range_nat_nat @ Xa3 ) )
% 0.34/0.58 = bot_bot_set_nat ) ) ) )
% 0.34/0.58 => ( wf_nat @ ( comple5685304695842803022at_nat @ R3 ) ) ) ) ).
% 0.34/0.58
% 0.34/0.58 % wf_Union
% 0.34/0.58 thf(fact_3069_wf__Union,axiom,
% 0.34/0.58 ! [R3: set_se6260736226359567993nt_int] :
% 0.34/0.58 ( ! [X3: set_Pr958786334691620121nt_int] :
% 0.34/0.58 ( ( member2340774599025711042nt_int @ X3 @ R3 )
% 0.34/0.58 => ( wf_int @ X3 ) )
% 0.34/0.58 => ( ! [X3: set_Pr958786334691620121nt_int] :
% 0.34/0.58 ( ( member2340774599025711042nt_int @ X3 @ R3 )
% 0.34/0.58 => ! [Xa3: set_Pr958786334691620121nt_int] :
% 0.34/0.58 ( ( member2340774599025711042nt_int @ Xa3 @ R3 )
% 0.34/0.58 => ( ( X3 != Xa3 )
% 0.34/0.58 => ( ( inf_inf_set_int @ ( domain_int_int @ X3 ) @ ( range_int_int @ Xa3 ) )
% 0.34/0.58 = bot_bot_set_int ) ) ) )
% 0.34/0.58 => ( wf_int @ ( comple5382143125604098054nt_int @ R3 ) ) ) ) ).
% 0.34/0.58
% 0.34/0.58 % wf_Union
% 0.34/0.58 thf(fact_3070_wf__Union,axiom,
% 0.34/0.58 ! [R3: set_se7663487083892625191_nat_o] :
% 0.34/0.58 ( ! [X3: set_Pr2161125870931222855_nat_o] :
% 0.34/0.58 ( ( member5475208728337331568_nat_o @ X3 @ R3 )
% 0.34/0.58 => ( wf_Pro8787509627421085110_nat_o @ X3 ) )
% 0.34/0.58 => ( ! [X3: set_Pr2161125870931222855_nat_o] :
% 0.34/0.58 ( ( member5475208728337331568_nat_o @ X3 @ R3 )
% 0.34/0.58 => ! [Xa3: set_Pr2161125870931222855_nat_o] :
% 0.34/0.58 ( ( member5475208728337331568_nat_o @ Xa3 @ R3 )
% 0.34/0.58 => ( ( X3 != Xa3 )
% 0.34/0.58 => ( ( inf_in1906310914598751387_nat_o @ ( domain4881241903163524940_nat_o @ X3 ) @ ( range_5289318983786108341_nat_o @ Xa3 ) )
% 0.34/0.58 = bot_bo7824918357723582233_nat_o ) ) ) )
% 0.34/0.58 => ( wf_Pro8787509627421085110_nat_o @ ( comple8164645552537203636_nat_o @ R3 ) ) ) ) ).
% 0.34/0.58
% 0.34/0.58 % wf_Union
% 0.34/0.58 thf(fact_3071_Pow__singleton__iff,axiom,
% 0.34/0.58 ! [X4: set_Product_unit,Y6: set_Product_unit] :
% 0.34/0.58 ( ( ( pow_Product_unit @ X4 )
% 0.34/0.58 = ( insert8986028188087627965t_unit @ Y6 @ bot_bo6284674557190688313t_unit ) )
% 0.34/0.58 = ( ( X4 = bot_bo3957492148770167129t_unit )
% 0.34/0.58 & ( Y6 = bot_bo3957492148770167129t_unit ) ) ) ).
% 0.34/0.58
% 0.34/0.58 % Pow_singleton_iff
% 0.34/0.58 thf(fact_3072_Pow__singleton__iff,axiom,
% 0.34/0.58 ! [X4: set_o,Y6: set_o] :
% 0.34/0.58 ( ( ( pow_o @ X4 )
% 0.34/0.58 = ( insert_set_o @ Y6 @ bot_bot_set_set_o ) )
% 0.34/0.58 = ( ( X4 = bot_bot_set_o )
% 0.34/0.58 & ( Y6 = bot_bot_set_o ) ) ) ).
% 0.34/0.58
% 0.34/0.58 % Pow_singleton_iff
% 0.34/0.58 thf(fact_3073_Pow__singleton__iff,axiom,
% 0.34/0.58 ! [X4: set_nat,Y6: set_nat] :
% 0.34/0.58 ( ( ( pow_nat @ X4 )
% 0.34/0.58 = ( insert_set_nat @ Y6 @ bot_bot_set_set_nat ) )
% 0.34/0.58 = ( ( X4 = bot_bot_set_nat )
% 0.34/0.58 & ( Y6 = bot_bot_set_nat ) ) ) ).
% 0.34/0.58
% 0.34/0.58 % Pow_singleton_iff
% 0.34/0.58 thf(fact_3074_Pow__singleton__iff,axiom,
% 0.34/0.58 ! [X4: set_int,Y6: set_int] :
% 0.34/0.58 ( ( ( pow_int @ X4 )
% 0.34/0.58 = ( insert_set_int @ Y6 @ bot_bot_set_set_int ) )
% 0.34/0.58 = ( ( X4 = bot_bot_set_int )
% 0.34/0.58 & ( Y6 = bot_bot_set_int ) ) ) ).
% 0.34/0.58
% 0.34/0.58 % Pow_singleton_iff
% 0.34/0.58 thf(fact_3075_Pow__singleton__iff,axiom,
% 0.34/0.58 ! [X4: set_Pr4532377907799695533_nat_o,Y6: set_Pr4532377907799695533_nat_o] :
% 0.34/0.58 ( ( ( pow_Pr7829143134171777386_nat_o @ X4 )
% 0.34/0.58 = ( insert2507855216428400253_nat_o @ Y6 @ bot_bo3736544561982865913_nat_o ) )
% 0.34/0.58 = ( ( X4 = bot_bo7824918357723582233_nat_o )
% 0.34/0.58 & ( Y6 = bot_bo7824918357723582233_nat_o ) ) ) ).
% 0.34/0.58
% 0.34/0.58 % Pow_singleton_iff
% 0.34/0.58 thf(fact_3076_Pow__empty,axiom,
% 0.34/0.58 ( ( pow_Product_unit @ bot_bo3957492148770167129t_unit )
% 0.34/0.58 = ( insert8986028188087627965t_unit @ bot_bo3957492148770167129t_unit @ bot_bo6284674557190688313t_unit ) ) ).
% 0.34/0.58
% 0.34/0.58 % Pow_empty
% 0.34/0.58 thf(fact_3077_Pow__empty,axiom,
% 0.34/0.58 ( ( pow_o @ bot_bot_set_o )
% 0.34/0.58 = ( insert_set_o @ bot_bot_set_o @ bot_bot_set_set_o ) ) ).
% 0.34/0.58
% 0.34/0.58 % Pow_empty
% 0.34/0.58 thf(fact_3078_Pow__empty,axiom,
% 0.34/0.58 ( ( pow_nat @ bot_bot_set_nat )
% 0.34/0.58 = ( insert_set_nat @ bot_bot_set_nat @ bot_bot_set_set_nat ) ) ).
% 0.34/0.58
% 0.34/0.58 % Pow_empty
% 0.34/0.58 thf(fact_3079_Pow__empty,axiom,
% 0.34/0.58 ( ( pow_int @ bot_bot_set_int )
% 0.34/0.58 = ( insert_set_int @ bot_bot_set_int @ bot_bot_set_set_int ) ) ).
% 0.34/0.58
% 0.34/0.58 % Pow_empty
% 0.34/0.58 thf(fact_3080_Pow__empty,axiom,
% 0.34/0.58 ( ( pow_Pr7829143134171777386_nat_o @ bot_bo7824918357723582233_nat_o )
% 0.34/0.58 = ( insert2507855216428400253_nat_o @ bot_bo7824918357723582233_nat_o @ bot_bo3736544561982865913_nat_o ) ) ).
% 0.34/0.58
% 0.34/0.58 % Pow_empty
% 0.34/0.58 thf(fact_3081_rtrancl__mapI,axiom,
% 0.34/0.58 ! [A2: nat,B: nat,E: set_Pr1261947904930325089at_nat,F: nat > nat] :
% 0.34/0.58 ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A2 @ B ) @ ( transi2905341329935302413cl_nat @ E ) )
% 0.34/0.58 => ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ ( F @ A2 ) @ ( F @ B ) ) @ ( transi2905341329935302413cl_nat @ ( image_5168914502847457605at_nat @ ( pairself_nat_nat @ F ) @ E ) ) ) ) ).
% 0.34/0.58
% 0.34/0.58 % rtrancl_mapI
% 0.34/0.58 thf(fact_3082_rtrancl__mapI,axiom,
% 0.34/0.58 ! [A2: nat,B: nat,E: set_Pr1261947904930325089at_nat,F: nat > int] :
% 0.34/0.58 ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A2 @ B ) @ ( transi2905341329935302413cl_nat @ E ) )
% 0.34/0.58 => ( member5262025264175285858nt_int @ ( product_Pair_int_int @ ( F @ A2 ) @ ( F @ B ) ) @ ( transi2902850859426252137cl_int @ ( image_1990417195239315453nt_int @ ( pairself_nat_int @ F ) @ E ) ) ) ) ).
% 0.34/0.58
% 0.34/0.58 % rtrancl_mapI
% 0.34/0.58 thf(fact_3083_rtrancl__mapI,axiom,
% 0.34/0.58 ! [A2: int,B: int,E: set_Pr958786334691620121nt_int,F: int > nat] :
% 0.34/0.58 ( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ A2 @ B ) @ ( transi2902850859426252137cl_int @ E ) )
% 0.34/0.58 => ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ ( F @ A2 ) @ ( F @ B ) ) @ ( transi2905341329935302413cl_nat @ ( image_5831868185956570253at_nat @ ( pairself_int_nat @ F ) @ E ) ) ) ) ).
% 0.34/0.58
% 0.34/0.58 % rtrancl_mapI
% 0.34/0.58 thf(fact_3084_rtrancl__mapI,axiom,
% 0.34/0.58 ! [A2: int,B: int,E: set_Pr958786334691620121nt_int,F: int > int] :
% 0.34/0.58 ( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ A2 @ B ) @ ( transi2902850859426252137cl_int @ E ) )
% 0.34/0.58 => ( member5262025264175285858nt_int @ ( product_Pair_int_int @ ( F @ A2 ) @ ( F @ B ) ) @ ( transi2902850859426252137cl_int @ ( image_2653370878348428101nt_int @ ( pairself_int_int @ F ) @ E ) ) ) ) ).
% 0.34/0.58
% 0.34/0.58 % rtrancl_mapI
% 0.34/0.58 thf(fact_3085_disjoint__image__subset,axiom,
% 0.34/0.58 ! [A11: set_set_nat,F: set_nat > set_nat] :
% 0.34/0.58 ( ( pairwise_set_nat @ disjnt_nat @ A11 )
% 0.34/0.58 => ( ! [X8: set_nat] :
% 0.34/0.58 ( ( member_set_nat @ X8 @ A11 )
% 0.34/0.58 => ( ord_less_eq_set_nat @ ( F @ X8 ) @ X8 ) )
% 0.34/0.58 => ( pairwise_set_nat @ disjnt_nat @ ( image_7916887816326733075et_nat @ F @ A11 ) ) ) ) ).
% 0.34/0.58
% 0.34/0.58 % disjoint_image_subset
% 0.34/0.58 thf(fact_3086_pairself_Opelims,axiom,
% 0.34/0.58 ! [X: nat > nat,Xa: product_prod_nat_nat,Y: product_prod_nat_nat] :
% 0.34/0.58 ( ( ( pairself_nat_nat @ X @ Xa )
% 0.34/0.58 = Y )
% 0.34/0.58 => ( ( accp_P4147487641329543982at_nat @ pairself_rel_nat_nat @ ( produc4237282728380940073at_nat @ X @ Xa ) )
% 0.34/0.58 => ~ ! [A4: nat,B2: nat] :
% 0.34/0.58 ( ( Xa
% 0.34/0.58 = ( product_Pair_nat_nat @ A4 @ B2 ) )
% 0.34/0.58 => ( ( Y
% 0.34/0.58 = ( product_Pair_nat_nat @ ( X @ A4 ) @ ( X @ B2 ) ) )
% 0.34/0.58 => ~ ( accp_P4147487641329543982at_nat @ pairself_rel_nat_nat @ ( produc4237282728380940073at_nat @ X @ ( product_Pair_nat_nat @ A4 @ B2 ) ) ) ) ) ) ) ).
% 0.34/0.58
% 0.34/0.58 % pairself.pelims
% 0.34/0.58 thf(fact_3087_pairself_Opelims,axiom,
% 0.34/0.58 ! [X: nat > int,Xa: product_prod_nat_nat,Y: product_prod_int_int] :
% 0.34/0.58 ( ( ( pairself_nat_int @ X @ Xa )
% 0.34/0.58 = Y )
% 0.34/0.58 => ( ( accp_P3050404660685545426at_nat @ pairself_rel_nat_int @ ( produc3140199747736941517at_nat @ X @ Xa ) )
% 0.34/0.58 => ~ ! [A4: nat,B2: nat] :
% 0.34/0.58 ( ( Xa
% 0.34/0.58 = ( product_Pair_nat_nat @ A4 @ B2 ) )
% 0.34/0.58 => ( ( Y
% 0.34/0.58 = ( product_Pair_int_int @ ( X @ A4 ) @ ( X @ B2 ) ) )
% 0.34/0.58 => ~ ( accp_P3050404660685545426at_nat @ pairself_rel_nat_int @ ( produc3140199747736941517at_nat @ X @ ( product_Pair_nat_nat @ A4 @ B2 ) ) ) ) ) ) ) ).
% 0.34/0.58
% 0.34/0.58 % pairself.pelims
% 0.34/0.58 thf(fact_3088_pairself_Opelims,axiom,
% 0.34/0.58 ! [X: int > nat,Xa: product_prod_int_int,Y: product_prod_nat_nat] :
% 0.34/0.58 ( ( ( pairself_int_nat @ X @ Xa )
% 0.34/0.58 = Y )
% 0.34/0.58 => ( ( accp_P5604362734843950218nt_int @ pairself_rel_int_nat @ ( produc2818822084525909125nt_int @ X @ Xa ) )
% 0.34/0.58 => ~ ! [A4: int,B2: int] :
% 0.34/0.58 ( ( Xa
% 0.34/0.58 = ( product_Pair_int_int @ A4 @ B2 ) )
% 0.34/0.58 => ( ( Y
% 0.34/0.58 = ( product_Pair_nat_nat @ ( X @ A4 ) @ ( X @ B2 ) ) )
% 0.34/0.58 => ~ ( accp_P5604362734843950218nt_int @ pairself_rel_int_nat @ ( produc2818822084525909125nt_int @ X @ ( product_Pair_int_int @ A4 @ B2 ) ) ) ) ) ) ) ).
% 0.34/0.58
% 0.34/0.58 % pairself.pelims
% 0.34/0.58 thf(fact_3089_pairself_Opelims,axiom,
% 0.34/0.58 ! [X: int > int,Xa: product_prod_int_int,Y: product_prod_int_int] :
% 0.34/0.58 ( ( ( pairself_int_int @ X @ Xa )
% 0.34/0.58 = Y )
% 0.34/0.58 => ( ( accp_P4507279754199951662nt_int @ pairself_rel_int_int @ ( produc1721739103881910569nt_int @ X @ Xa ) )
% 0.34/0.58 => ~ ! [A4: int,B2: int] :
% 0.34/0.58 ( ( Xa
% 0.34/0.58 = ( product_Pair_int_int @ A4 @ B2 ) )
% 0.34/0.58 => ( ( Y
% 0.34/0.58 = ( product_Pair_int_int @ ( X @ A4 ) @ ( X @ B2 ) ) )
% 0.34/0.58 => ~ ( accp_P4507279754199951662nt_int @ pairself_rel_int_int @ ( produc1721739103881910569nt_int @ X @ ( product_Pair_int_int @ A4 @ B2 ) ) ) ) ) ) ) ).
% 0.34/0.58
% 0.34/0.58 % pairself.pelims
% 0.34/0.58 thf(fact_3090_Pow__iff,axiom,
% 0.34/0.58 ! [A: set_nat,B4: set_nat] :
% 0.34/0.58 ( ( member_set_nat @ A @ ( pow_nat @ B4 ) )
% 0.34/0.58 = ( ord_less_eq_set_nat @ A @ B4 ) ) ).
% 0.34/0.58
% 0.34/0.58 % Pow_iff
% 0.34/0.58 thf(fact_3091_PowI,axiom,
% 0.34/0.58 ! [A: set_nat,B4: set_nat] :
% 0.34/0.58 ( ( ord_less_eq_set_nat @ A @ B4 )
% 0.34/0.58 => ( member_set_nat @ A @ ( pow_nat @ B4 ) ) ) ).
% 0.34/0.58
% 0.34/0.58 % PowI
% 0.34/0.58 thf(fact_3092_Pow__Int__eq,axiom,
% 0.34/0.58 ! [A: set_nat,B4: set_nat] :
% 0.34/0.58 ( ( pow_nat @ ( inf_inf_set_nat @ A @ B4 ) )
% 0.34/0.58 = ( inf_inf_set_set_nat @ ( pow_nat @ A ) @ ( pow_nat @ B4 ) ) ) ).
% 0.34/0.58
% 0.34/0.58 % Pow_Int_eq
% 0.34/0.58 thf(fact_3093_pairwiseI,axiom,
% 0.34/0.58 ! [S: set_Pr4532377907799695533_nat_o,R3: ( produc3658429121746597890et_nat > $o ) > ( produc3658429121746597890et_nat > $o ) > $o] :
% 0.34/0.58 ( ! [X3: produc3658429121746597890et_nat > $o,Y2: produc3658429121746597890et_nat > $o] :
% 0.34/0.58 ( ( member6576561426505652726_nat_o @ X3 @ S )
% 0.34/0.58 => ( ( member6576561426505652726_nat_o @ Y2 @ S )
% 0.34/0.58 => ( ( X3 != Y2 )
% 0.34/0.58 => ( R3 @ X3 @ Y2 ) ) ) )
% 0.34/0.58 => ( pairwi2432143822833250616_nat_o @ R3 @ S ) ) ).
% 0.34/0.58
% 0.34/0.58 % pairwiseI
% 0.34/0.58 thf(fact_3094_pairwiseI,axiom,
% 0.34/0.58 ! [S: set_nat,R3: nat > nat > $o] :
% 0.34/0.58 ( ! [X3: nat,Y2: nat] :
% 0.34/0.58 ( ( member_nat @ X3 @ S )
% 0.34/0.58 => ( ( member_nat @ Y2 @ S )
% 0.34/0.58 => ( ( X3 != Y2 )
% 0.34/0.58 => ( R3 @ X3 @ Y2 ) ) ) )
% 0.34/0.58 => ( pairwise_nat @ R3 @ S ) ) ).
% 0.34/0.58
% 0.34/0.58 % pairwiseI
% 0.34/0.58 thf(fact_3095_pairwiseI,axiom,
% 0.34/0.58 ! [S: set_int,R3: int > int > $o] :
% 0.34/0.58 ( ! [X3: int,Y2: int] :
% 0.34/0.58 ( ( member_int @ X3 @ S )
% 0.34/0.58 => ( ( member_int @ Y2 @ S )
% 0.34/0.58 => ( ( X3 != Y2 )
% 0.34/0.58 => ( R3 @ X3 @ Y2 ) ) ) )
% 0.34/0.58 => ( pairwise_int @ R3 @ S ) ) ).
% 0.34/0.58
% 0.34/0.58 % pairwiseI
% 0.34/0.58 thf(fact_3096_pairwiseI,axiom,
% 0.34/0.58 ! [S: set_o,R3: $o > $o > $o] :
% 0.34/0.58 ( ! [X3: $o,Y2: $o] :
% 0.34/0.58 ( ( member_o @ X3 @ S )
% 0.34/0.58 => ( ( member_o @ Y2 @ S )
% 0.34/0.58 => ( ( X3 != Y2 )
% 0.34/0.58 => ( R3 @ X3 @ Y2 ) ) ) )
% 0.34/0.58 => ( pairwise_o @ R3 @ S ) ) ).
% 0.34/0.58
% 0.34/0.58 % pairwiseI
% 0.34/0.58 thf(fact_3097_pairwiseI,axiom,
% 0.34/0.58 ! [S: set_Product_unit,R3: product_unit > product_unit > $o] :
% 0.34/0.58 ( ! [X3: product_unit,Y2: product_unit] :
% 0.34/0.58 ( ( member_Product_unit @ X3 @ S )
% 0.34/0.58 => ( ( member_Product_unit @ Y2 @ S )
% 0.34/0.58 => ( ( X3 != Y2 )
% 0.34/0.58 => ( R3 @ X3 @ Y2 ) ) ) )
% 0.34/0.58 => ( pairwi7919127943033184120t_unit @ R3 @ S ) ) ).
% 0.34/0.58
% 0.34/0.58 % pairwiseI
% 0.34/0.58 thf(fact_3098_pairwiseD,axiom,
% 0.34/0.58 ! [R3: ( produc3658429121746597890et_nat > $o ) > ( produc3658429121746597890et_nat > $o ) > $o,S: set_Pr4532377907799695533_nat_o,X: produc3658429121746597890et_nat > $o,Y: produc3658429121746597890et_nat > $o] :
% 0.34/0.58 ( ( pairwi2432143822833250616_nat_o @ R3 @ S )
% 0.34/0.58 => ( ( member6576561426505652726_nat_o @ X @ S )
% 0.34/0.58 => ( ( member6576561426505652726_nat_o @ Y @ S )
% 0.34/0.58 => ( ( X != Y )
% 0.34/0.58 => ( R3 @ X @ Y ) ) ) ) ) ).
% 0.34/0.58
% 0.34/0.58 % pairwiseD
% 0.34/0.58 thf(fact_3099_pairwiseD,axiom,
% 0.34/0.58 ! [R3: nat > nat > $o,S: set_nat,X: nat,Y: nat] :
% 0.34/0.58 ( ( pairwise_nat @ R3 @ S )
% 0.34/0.58 => ( ( member_nat @ X @ S )
% 0.34/0.58 => ( ( member_nat @ Y @ S )
% 0.34/0.58 => ( ( X != Y )
% 0.34/0.58 => ( R3 @ X @ Y ) ) ) ) ) ).
% 0.34/0.58
% 0.34/0.58 % pairwiseD
% 0.34/0.58 thf(fact_3100_pairwiseD,axiom,
% 0.34/0.58 ! [R3: int > int > $o,S: set_int,X: int,Y: int] :
% 0.34/0.58 ( ( pairwise_int @ R3 @ S )
% 0.34/0.58 => ( ( member_int @ X @ S )
% 0.34/0.58 => ( ( member_int @ Y @ S )
% 0.34/0.58 => ( ( X != Y )
% 0.34/0.58 => ( R3 @ X @ Y ) ) ) ) ) ).
% 0.34/0.58
% 0.34/0.58 % pairwiseD
% 0.34/0.58 thf(fact_3101_pairwiseD,axiom,
% 0.34/0.58 ! [R3: $o > $o > $o,S: set_o,X: $o,Y: $o] :
% 0.34/0.58 ( ( pairwise_o @ R3 @ S )
% 0.34/0.58 => ( ( member_o @ X @ S )
% 0.34/0.58 => ( ( member_o @ Y @ S )
% 0.34/0.58 => ( ( X != Y )
% 0.34/0.58 => ( R3 @ X @ Y ) ) ) ) ) ).
% 0.34/0.58
% 0.34/0.58 % pairwiseD
% 0.34/0.58 thf(fact_3102_pairwiseD,axiom,
% 0.34/0.58 ! [R3: product_unit > product_unit > $o,S: set_Product_unit,X: product_unit,Y: product_unit] :
% 0.34/0.58 ( ( pairwi7919127943033184120t_unit @ R3 @ S )
% 0.34/0.58 => ( ( member_Product_unit @ X @ S )
% 0.34/0.58 => ( ( member_Product_unit @ Y @ S )
% 0.34/0.58 => ( ( X != Y )
% 0.34/0.58 => ( R3 @ X @ Y ) ) ) ) ) ).
% 0.34/0.58
% 0.34/0.58 % pairwiseD
% 0.34/0.58 thf(fact_3103_refl__on__UNION,axiom,
% 0.34/0.58 ! [S: set_nat,A: nat > set_nat,R2: nat > set_Pr1261947904930325089at_nat] :
% 0.34/0.58 ( ! [X3: nat] :
% 0.34/0.58 ( ( member_nat @ X3 @ S )
% 0.34/0.58 => ( refl_on_nat @ ( A @ X3 ) @ ( R2 @ X3 ) ) )
% 0.34/0.58 => ( refl_on_nat @ ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ A @ S ) ) @ ( comple5685304695842803022at_nat @ ( image_7178329752028323786at_nat @ R2 @ S ) ) ) ) ).
% 0.34/0.58
% 0.34/0.58 % refl_on_UNION
% 0.34/0.58 thf(fact_3104_Pow__bottom,axiom,
% 0.34/0.58 ! [B4: set_Product_unit] : ( member5877623283571906838t_unit @ bot_bo3957492148770167129t_unit @ ( pow_Product_unit @ B4 ) ) ).
% 0.34/0.58
% 0.34/0.58 % Pow_bottom
% 0.34/0.58 thf(fact_3105_Pow__bottom,axiom,
% 0.34/0.58 ! [B4: set_o] : ( member_set_o @ bot_bot_set_o @ ( pow_o @ B4 ) ) ).
% 0.34/0.58
% 0.34/0.58 % Pow_bottom
% 0.34/0.58 thf(fact_3106_Pow__bottom,axiom,
% 0.34/0.58 ! [B4: set_nat] : ( member_set_nat @ bot_bot_set_nat @ ( pow_nat @ B4 ) ) ).
% 0.34/0.58
% 0.34/0.58 % Pow_bottom
% 0.34/0.58 thf(fact_3107_Pow__bottom,axiom,
% 0.34/0.58 ! [B4: set_int] : ( member_set_int @ bot_bot_set_int @ ( pow_int @ B4 ) ) ).
% 0.34/0.58
% 0.34/0.58 % Pow_bottom
% 0.34/0.58 thf(fact_3108_Pow__bottom,axiom,
% 0.34/0.58 ! [B4: set_Pr4532377907799695533_nat_o] : ( member1452989533377600726_nat_o @ bot_bo7824918357723582233_nat_o @ ( pow_Pr7829143134171777386_nat_o @ B4 ) ) ).
% 0.34/0.58
% 0.34/0.58 % Pow_bottom
% 0.34/0.58 thf(fact_3109_Pow__mono,axiom,
% 0.34/0.58 ! [A: set_nat,B4: set_nat] :
% 0.34/0.58 ( ( ord_less_eq_set_nat @ A @ B4 )
% 0.34/0.58 => ( ord_le6893508408891458716et_nat @ ( pow_nat @ A ) @ ( pow_nat @ B4 ) ) ) ).
% 0.34/0.58
% 0.34/0.58 % Pow_mono
% 0.34/0.58 thf(fact_3110_PowD,axiom,
% 0.34/0.58 ! [A: set_nat,B4: set_nat] :
% 0.34/0.58 ( ( member_set_nat @ A @ ( pow_nat @ B4 ) )
% 0.34/0.58 => ( ord_less_eq_set_nat @ A @ B4 ) ) ).
% 0.34/0.58
% 0.34/0.58 % PowD
% 0.34/0.58 thf(fact_3111_Un__Pow__subset,axiom,
% 0.34/0.58 ! [A: set_nat,B4: set_nat] : ( ord_le6893508408891458716et_nat @ ( sup_sup_set_set_nat @ ( pow_nat @ A ) @ ( pow_nat @ B4 ) ) @ ( pow_nat @ ( sup_sup_set_nat @ A @ B4 ) ) ) ).
% 0.34/0.59
% 0.34/0.59 % Un_Pow_subset
% 0.34/0.59 thf(fact_3112_pairwise__imageI,axiom,
% 0.34/0.59 ! [A: set_Code_integer,F: code_integer > code_integer,P: code_integer > code_integer > $o] :
% 0.34/0.59 ( ! [X3: code_integer,Y2: code_integer] :
% 0.34/0.59 ( ( member_Code_integer @ X3 @ A )
% 0.34/0.59 => ( ( member_Code_integer @ Y2 @ A )
% 0.34/0.59 => ( ( X3 != Y2 )
% 0.34/0.59 => ( ( ( F @ X3 )
% 0.34/0.59 != ( F @ Y2 ) )
% 0.34/0.59 => ( P @ ( F @ X3 ) @ ( F @ Y2 ) ) ) ) ) )
% 0.34/0.59 => ( pairwi4741737654671186738nteger @ P @ ( image_4470545334726330049nteger @ F @ A ) ) ) ).
% 0.34/0.59
% 0.34/0.59 % pairwise_imageI
% 0.34/0.59 thf(fact_3113_pairwise__imageI,axiom,
% 0.34/0.59 ! [A: set_nat,F: nat > set_nat,P: set_nat > set_nat > $o] :
% 0.34/0.59 ( ! [X3: nat,Y2: nat] :
% 0.34/0.59 ( ( member_nat @ X3 @ A )
% 0.34/0.59 => ( ( member_nat @ Y2 @ A )
% 0.34/0.59 => ( ( X3 != Y2 )
% 0.34/0.59 => ( ( ( F @ X3 )
% 0.34/0.59 != ( F @ Y2 ) )
% 0.34/0.59 => ( P @ ( F @ X3 ) @ ( F @ Y2 ) ) ) ) ) )
% 0.34/0.59 => ( pairwise_set_nat @ P @ ( image_nat_set_nat @ F @ A ) ) ) ).
% 0.34/0.59
% 0.34/0.59 % pairwise_imageI
% 0.34/0.59 thf(fact_3114_pairwise__imageI,axiom,
% 0.34/0.59 ! [A: set_nat,F: nat > nat,P: nat > nat > $o] :
% 0.34/0.59 ( ! [X3: nat,Y2: nat] :
% 0.34/0.59 ( ( member_nat @ X3 @ A )
% 0.34/0.59 => ( ( member_nat @ Y2 @ A )
% 0.34/0.59 => ( ( X3 != Y2 )
% 0.34/0.59 => ( ( ( F @ X3 )
% 0.34/0.59 != ( F @ Y2 ) )
% 0.34/0.59 => ( P @ ( F @ X3 ) @ ( F @ Y2 ) ) ) ) ) )
% 0.34/0.59 => ( pairwise_nat @ P @ ( image_nat_nat2 @ F @ A ) ) ) ).
% 0.34/0.59
% 0.34/0.59 % pairwise_imageI
% 0.34/0.59 thf(fact_3115_pairwise__imageI,axiom,
% 0.34/0.59 ! [A: set_nat,F: nat > code_integer,P: code_integer > code_integer > $o] :
% 0.34/0.59 ( ! [X3: nat,Y2: nat] :
% 0.34/0.59 ( ( member_nat @ X3 @ A )
% 0.34/0.59 => ( ( member_nat @ Y2 @ A )
% 0.34/0.59 => ( ( X3 != Y2 )
% 0.34/0.59 => ( ( ( F @ X3 )
% 0.34/0.59 != ( F @ Y2 ) )
% 0.34/0.59 => ( P @ ( F @ X3 ) @ ( F @ Y2 ) ) ) ) ) )
% 0.34/0.59 => ( pairwi4741737654671186738nteger @ P @ ( image_1215581382706833972nteger @ F @ A ) ) ) ).
% 0.34/0.59
% 0.34/0.59 % pairwise_imageI
% 0.34/0.59 thf(fact_3116_pairwise__empty,axiom,
% 0.34/0.59 ! [P: product_unit > product_unit > $o] : ( pairwi7919127943033184120t_unit @ P @ bot_bo3957492148770167129t_unit ) ).
% 0.34/0.59
% 0.34/0.59 % pairwise_empty
% 0.34/0.59 thf(fact_3117_pairwise__empty,axiom,
% 0.34/0.59 ! [P: $o > $o > $o] : ( pairwise_o @ P @ bot_bot_set_o ) ).
% 0.34/0.59
% 0.34/0.59 % pairwise_empty
% 0.34/0.59 thf(fact_3118_pairwise__empty,axiom,
% 0.34/0.59 ! [P: nat > nat > $o] : ( pairwise_nat @ P @ bot_bot_set_nat ) ).
% 0.34/0.59
% 0.34/0.59 % pairwise_empty
% 0.34/0.59 thf(fact_3119_pairwise__empty,axiom,
% 0.34/0.59 ! [P: int > int > $o] : ( pairwise_int @ P @ bot_bot_set_int ) ).
% 0.34/0.59
% 0.34/0.59 % pairwise_empty
% 0.34/0.59 thf(fact_3120_pairwise__empty,axiom,
% 0.34/0.59 ! [P: ( produc3658429121746597890et_nat > $o ) > ( produc3658429121746597890et_nat > $o ) > $o] : ( pairwi2432143822833250616_nat_o @ P @ bot_bo7824918357723582233_nat_o ) ).
% 0.34/0.59
% 0.34/0.59 % pairwise_empty
% 0.34/0.59 thf(fact_3121_pairwise__subset,axiom,
% 0.34/0.59 ! [P: nat > nat > $o,S: set_nat,T2: set_nat] :
% 0.34/0.59 ( ( pairwise_nat @ P @ S )
% 0.34/0.59 => ( ( ord_less_eq_set_nat @ T2 @ S )
% 0.34/0.59 => ( pairwise_nat @ P @ T2 ) ) ) ).
% 0.34/0.59
% 0.34/0.59 % pairwise_subset
% 0.34/0.59 thf(fact_3122_pairwise__mono,axiom,
% 0.34/0.59 ! [P: nat > nat > $o,A: set_nat,Q2: nat > nat > $o,B4: set_nat] :
% 0.34/0.59 ( ( pairwise_nat @ P @ A )
% 0.34/0.59 => ( ! [X3: nat,Y2: nat] :
% 0.34/0.59 ( ( P @ X3 @ Y2 )
% 0.34/0.59 => ( Q2 @ X3 @ Y2 ) )
% 0.34/0.59 => ( ( ord_less_eq_set_nat @ B4 @ A )
% 0.34/0.59 => ( pairwise_nat @ Q2 @ B4 ) ) ) ) ).
% 0.34/0.59
% 0.34/0.59 % pairwise_mono
% 0.34/0.59 thf(fact_3123_pairwise__insert,axiom,
% 0.34/0.59 ! [R2: ( produc3658429121746597890et_nat > $o ) > ( produc3658429121746597890et_nat > $o ) > $o,X: produc3658429121746597890et_nat > $o,S2: set_Pr4532377907799695533_nat_o] :
% 0.34/0.59 ( ( pairwi2432143822833250616_nat_o @ R2 @ ( insert5175938949040314269_nat_o @ X @ S2 ) )
% 0.34/0.59 = ( ! [Y3: produc3658429121746597890et_nat > $o] :
% 0.34/0.59 ( ( ( member6576561426505652726_nat_o @ Y3 @ S2 )
% 0.34/0.59 & ( Y3 != X ) )
% 0.34/0.59 => ( ( R2 @ X @ Y3 )
% 0.34/0.59 & ( R2 @ Y3 @ X ) ) )
% 0.34/0.59 & ( pairwi2432143822833250616_nat_o @ R2 @ S2 ) ) ) ).
% 0.34/0.59
% 0.34/0.59 % pairwise_insert
% 0.34/0.59 thf(fact_3124_pairwise__insert,axiom,
% 0.34/0.59 ! [R2: nat > nat > $o,X: nat,S2: set_nat] :
% 0.34/0.59 ( ( pairwise_nat @ R2 @ ( insert_nat @ X @ S2 ) )
% 0.34/0.59 = ( ! [Y3: nat] :
% 0.34/0.59 ( ( ( member_nat @ Y3 @ S2 )
% 0.34/0.59 & ( Y3 != X ) )
% 0.34/0.59 => ( ( R2 @ X @ Y3 )
% 0.34/0.59 & ( R2 @ Y3 @ X ) ) )
% 0.34/0.59 & ( pairwise_nat @ R2 @ S2 ) ) ) ).
% 0.34/0.59
% 0.34/0.59 % pairwise_insert
% 0.34/0.59 thf(fact_3125_pairwise__insert,axiom,
% 0.34/0.59 ! [R2: int > int > $o,X: int,S2: set_int] :
% 0.34/0.59 ( ( pairwise_int @ R2 @ ( insert_int @ X @ S2 ) )
% 0.34/0.59 = ( ! [Y3: int] :
% 0.34/0.59 ( ( ( member_int @ Y3 @ S2 )
% 0.34/0.59 & ( Y3 != X ) )
% 0.34/0.59 => ( ( R2 @ X @ Y3 )
% 0.34/0.59 & ( R2 @ Y3 @ X ) ) )
% 0.34/0.59 & ( pairwise_int @ R2 @ S2 ) ) ) ).
% 0.34/0.59
% 0.34/0.59 % pairwise_insert
% 0.34/0.59 thf(fact_3126_pairwise__insert,axiom,
% 0.34/0.59 ! [R2: $o > $o > $o,X: $o,S2: set_o] :
% 0.34/0.59 ( ( pairwise_o @ R2 @ ( insert_o @ X @ S2 ) )
% 0.34/0.59 = ( ! [Y3: $o] :
% 0.34/0.59 ( ( ( member_o @ Y3 @ S2 )
% 0.34/0.59 & ( Y3 != X ) )
% 0.34/0.59 => ( ( R2 @ X @ Y3 )
% 0.34/0.59 & ( R2 @ Y3 @ X ) ) )
% 0.34/0.59 & ( pairwise_o @ R2 @ S2 ) ) ) ).
% 0.34/0.59
% 0.34/0.59 % pairwise_insert
% 0.34/0.59 thf(fact_3127_pairwise__insert,axiom,
% 0.34/0.59 ! [R2: product_unit > product_unit > $o,X: product_unit,S2: set_Product_unit] :
% 0.34/0.59 ( ( pairwi7919127943033184120t_unit @ R2 @ ( insert_Product_unit @ X @ S2 ) )
% 0.34/0.59 = ( ! [Y3: product_unit] :
% 0.34/0.59 ( ( ( member_Product_unit @ Y3 @ S2 )
% 0.34/0.59 & ( Y3 != X ) )
% 0.34/0.59 => ( ( R2 @ X @ Y3 )
% 0.34/0.59 & ( R2 @ Y3 @ X ) ) )
% 0.34/0.59 & ( pairwi7919127943033184120t_unit @ R2 @ S2 ) ) ) ).
% 0.34/0.59
% 0.34/0.59 % pairwise_insert
% 0.34/0.59 thf(fact_3128_finite__UNION__then__finite,axiom,
% 0.34/0.59 ! [B4: ( produc3658429121746597890et_nat > $o ) > set_nat,A: set_Pr4532377907799695533_nat_o,A2: produc3658429121746597890et_nat > $o] :
% 0.34/0.59 ( ( finite_finite_nat @ ( comple7399068483239264473et_nat @ ( image_6888641473154130958et_nat @ B4 @ A ) ) )
% 0.34/0.59 => ( ( member6576561426505652726_nat_o @ A2 @ A )
% 0.34/0.59 => ( finite_finite_nat @ ( B4 @ A2 ) ) ) ) ).
% 0.34/0.59
% 0.34/0.59 % finite_UNION_then_finite
% 0.34/0.59 thf(fact_3129_finite__UNION__then__finite,axiom,
% 0.34/0.59 ! [B4: nat > set_nat,A: set_nat,A2: nat] :
% 0.34/0.59 ( ( finite_finite_nat @ ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ B4 @ A ) ) )
% 0.34/0.59 => ( ( member_nat @ A2 @ A )
% 0.34/0.59 => ( finite_finite_nat @ ( B4 @ A2 ) ) ) ) ).
% 0.34/0.59
% 0.34/0.59 % finite_UNION_then_finite
% 0.34/0.59 thf(fact_3130_finite__UNION__then__finite,axiom,
% 0.34/0.59 ! [B4: int > set_nat,A: set_int,A2: int] :
% 0.34/0.59 ( ( finite_finite_nat @ ( comple7399068483239264473et_nat @ ( image_int_set_nat @ B4 @ A ) ) )
% 0.34/0.59 => ( ( member_int @ A2 @ A )
% 0.34/0.59 => ( finite_finite_nat @ ( B4 @ A2 ) ) ) ) ).
% 0.34/0.59
% 0.34/0.59 % finite_UNION_then_finite
% 0.34/0.59 thf(fact_3131_finite__UNION__then__finite,axiom,
% 0.34/0.59 ! [B4: $o > set_nat,A: set_o,A2: $o] :
% 0.34/0.59 ( ( finite_finite_nat @ ( comple7399068483239264473et_nat @ ( image_o_set_nat @ B4 @ A ) ) )
% 0.34/0.59 => ( ( member_o @ A2 @ A )
% 0.34/0.59 => ( finite_finite_nat @ ( B4 @ A2 ) ) ) ) ).
% 0.34/0.59
% 0.34/0.59 % finite_UNION_then_finite
% 0.34/0.59 thf(fact_3132_finite__UNION__then__finite,axiom,
% 0.34/0.59 ! [B4: product_unit > set_nat,A: set_Product_unit,A2: product_unit] :
% 0.34/0.59 ( ( finite_finite_nat @ ( comple7399068483239264473et_nat @ ( image_8173849991827986510et_nat @ B4 @ A ) ) )
% 0.34/0.59 => ( ( member_Product_unit @ A2 @ A )
% 0.34/0.59 => ( finite_finite_nat @ ( B4 @ A2 ) ) ) ) ).
% 0.34/0.59
% 0.34/0.59 % finite_UNION_then_finite
% 0.34/0.59 thf(fact_3133_finite__UNION__then__finite,axiom,
% 0.34/0.59 ! [B4: ( produc3658429121746597890et_nat > $o ) > set_Code_integer,A: set_Pr4532377907799695533_nat_o,A2: produc3658429121746597890et_nat > $o] :
% 0.34/0.59 ( ( finite6017078050557962740nteger @ ( comple739944243200306918nteger @ ( image_8657708916238040603nteger @ B4 @ A ) ) )
% 0.34/0.59 => ( ( member6576561426505652726_nat_o @ A2 @ A )
% 0.34/0.59 => ( finite6017078050557962740nteger @ ( B4 @ A2 ) ) ) ) ).
% 0.34/0.59
% 0.34/0.59 % finite_UNION_then_finite
% 0.34/0.59 thf(fact_3134_finite__UNION__then__finite,axiom,
% 0.34/0.59 ! [B4: nat > set_Code_integer,A: set_nat,A2: nat] :
% 0.34/0.59 ( ( finite6017078050557962740nteger @ ( comple739944243200306918nteger @ ( image_2385391725447763818nteger @ B4 @ A ) ) )
% 0.34/0.59 => ( ( member_nat @ A2 @ A )
% 0.34/0.59 => ( finite6017078050557962740nteger @ ( B4 @ A2 ) ) ) ) ).
% 0.34/0.59
% 0.34/0.59 % finite_UNION_then_finite
% 0.34/0.59 thf(fact_3135_finite__UNION__then__finite,axiom,
% 0.34/0.59 ! [B4: int > set_Code_integer,A: set_int,A2: int] :
% 0.34/0.59 ( ( finite6017078050557962740nteger @ ( comple739944243200306918nteger @ ( image_75573793354812870nteger @ B4 @ A ) ) )
% 0.34/0.59 => ( ( member_int @ A2 @ A )
% 0.34/0.59 => ( finite6017078050557962740nteger @ ( B4 @ A2 ) ) ) ) ).
% 0.34/0.59
% 0.34/0.59 % finite_UNION_then_finite
% 0.34/0.59 thf(fact_3136_finite__UNION__then__finite,axiom,
% 0.34/0.59 ! [B4: $o > set_Code_integer,A: set_o,A2: $o] :
% 0.34/0.59 ( ( finite6017078050557962740nteger @ ( comple739944243200306918nteger @ ( image_6491638074533506790nteger @ B4 @ A ) ) )
% 0.34/0.59 => ( ( member_o @ A2 @ A )
% 0.34/0.59 => ( finite6017078050557962740nteger @ ( B4 @ A2 ) ) ) ) ).
% 0.34/0.59
% 0.34/0.59 % finite_UNION_then_finite
% 0.34/0.59 thf(fact_3137_finite__UNION__then__finite,axiom,
% 0.34/0.59 ! [B4: product_unit > set_Code_integer,A: set_Product_unit,A2: product_unit] :
% 0.34/0.59 ( ( finite6017078050557962740nteger @ ( comple739944243200306918nteger @ ( image_4571654851662262875nteger @ B4 @ A ) ) )
% 0.34/0.59 => ( ( member_Product_unit @ A2 @ A )
% 0.34/0.59 => ( finite6017078050557962740nteger @ ( B4 @ A2 ) ) ) ) ).
% 0.34/0.59
% 0.34/0.59 % finite_UNION_then_finite
% 0.34/0.59 thf(fact_3138_image__Pow__surj,axiom,
% 0.34/0.59 ! [F: nat > set_nat,A: set_nat,B4: set_set_nat] :
% 0.34/0.59 ( ( ( image_nat_set_nat @ F @ A )
% 0.34/0.59 = B4 )
% 0.34/0.59 => ( ( image_6725021117256019401et_nat @ ( image_nat_set_nat @ F ) @ ( pow_nat @ A ) )
% 0.34/0.59 = ( pow_set_nat @ B4 ) ) ) ).
% 0.34/0.59
% 0.34/0.59 % image_Pow_surj
% 0.34/0.59 thf(fact_3139_image__Pow__surj,axiom,
% 0.34/0.59 ! [F: nat > nat,A: set_nat,B4: set_nat] :
% 0.34/0.59 ( ( ( image_nat_nat2 @ F @ A )
% 0.34/0.59 = B4 )
% 0.34/0.59 => ( ( image_7916887816326733075et_nat @ ( image_nat_nat2 @ F ) @ ( pow_nat @ A ) )
% 0.34/0.59 = ( pow_nat @ B4 ) ) ) ).
% 0.34/0.59
% 0.34/0.59 % image_Pow_surj
% 0.34/0.59 thf(fact_3140_image__Pow__surj,axiom,
% 0.34/0.59 ! [F: code_integer > code_integer,A: set_Code_integer,B4: set_Code_integer] :
% 0.34/0.59 ( ( ( image_4470545334726330049nteger @ F @ A )
% 0.34/0.59 = B4 )
% 0.34/0.59 => ( ( image_6270750266955655469nteger @ ( image_4470545334726330049nteger @ F ) @ ( pow_Code_integer @ A ) )
% 0.34/0.59 = ( pow_Code_integer @ B4 ) ) ) ).
% 0.34/0.59
% 0.34/0.59 % image_Pow_surj
% 0.34/0.59 thf(fact_3141_image__Pow__surj,axiom,
% 0.34/0.59 ! [F: nat > code_integer,A: set_nat,B4: set_Code_integer] :
% 0.34/0.59 ( ( ( image_1215581382706833972nteger @ F @ A )
% 0.34/0.59 = B4 )
% 0.34/0.59 => ( ( image_6916300684244607776nteger @ ( image_1215581382706833972nteger @ F ) @ ( pow_nat @ A ) )
% 0.34/0.59 = ( pow_Code_integer @ B4 ) ) ) ).
% 0.34/0.59
% 0.34/0.59 % image_Pow_surj
% 0.34/0.59 thf(fact_3142_Pow__insert,axiom,
% 0.34/0.59 ! [A2: nat,A: set_nat] :
% 0.34/0.59 ( ( pow_nat @ ( insert_nat @ A2 @ A ) )
% 0.34/0.59 = ( sup_sup_set_set_nat @ ( pow_nat @ A ) @ ( image_7916887816326733075et_nat @ ( insert_nat @ A2 ) @ ( pow_nat @ A ) ) ) ) ).
% 0.34/0.59
% 0.34/0.59 % Pow_insert
% 0.34/0.59 thf(fact_3143_Pow__insert,axiom,
% 0.34/0.59 ! [A2: int,A: set_int] :
% 0.34/0.59 ( ( pow_int @ ( insert_int @ A2 @ A ) )
% 0.34/0.59 = ( sup_sup_set_set_int @ ( pow_int @ A ) @ ( image_524474410958335435et_int @ ( insert_int @ A2 ) @ ( pow_int @ A ) ) ) ) ).
% 0.34/0.59
% 0.34/0.59 % Pow_insert
% 0.34/0.59 thf(fact_3144_Pow__insert,axiom,
% 0.34/0.59 ! [A2: $o,A: set_o] :
% 0.34/0.59 ( ( pow_o @ ( insert_o @ A2 @ A ) )
% 0.34/0.59 = ( sup_sup_set_set_o @ ( pow_o @ A ) @ ( image_set_o_set_o @ ( insert_o @ A2 ) @ ( pow_o @ A ) ) ) ) ).
% 0.34/0.59
% 0.34/0.59 % Pow_insert
% 0.34/0.59 thf(fact_3145_Pow__insert,axiom,
% 0.34/0.59 ! [A2: product_unit,A: set_Product_unit] :
% 0.34/0.59 ( ( pow_Product_unit @ ( insert_Product_unit @ A2 @ A ) )
% 0.34/0.59 = ( sup_su5975557051335651745t_unit @ ( pow_Product_unit @ A ) @ ( image_4019945301284682565t_unit @ ( insert_Product_unit @ A2 ) @ ( pow_Product_unit @ A ) ) ) ) ).
% 0.34/0.59
% 0.34/0.59 % Pow_insert
% 0.34/0.59 thf(fact_3146_pairself_Osimps,axiom,
% 0.34/0.59 ! [F: nat > nat,A2: nat,B: nat] :
% 0.34/0.59 ( ( pairself_nat_nat @ F @ ( product_Pair_nat_nat @ A2 @ B ) )
% 0.34/0.59 = ( product_Pair_nat_nat @ ( F @ A2 ) @ ( F @ B ) ) ) ).
% 0.34/0.59
% 0.34/0.59 % pairself.simps
% 0.34/0.59 thf(fact_3147_pairself_Osimps,axiom,
% 0.34/0.59 ! [F: nat > int,A2: nat,B: nat] :
% 0.34/0.59 ( ( pairself_nat_int @ F @ ( product_Pair_nat_nat @ A2 @ B ) )
% 0.34/0.59 = ( product_Pair_int_int @ ( F @ A2 ) @ ( F @ B ) ) ) ).
% 0.34/0.59
% 0.34/0.59 % pairself.simps
% 0.34/0.59 thf(fact_3148_pairself_Osimps,axiom,
% 0.34/0.59 ! [F: int > nat,A2: int,B: int] :
% 0.34/0.59 ( ( pairself_int_nat @ F @ ( product_Pair_int_int @ A2 @ B ) )
% 0.34/0.59 = ( product_Pair_nat_nat @ ( F @ A2 ) @ ( F @ B ) ) ) ).
% 0.34/0.59
% 0.34/0.59 % pairself.simps
% 0.34/0.59 thf(fact_3149_pairself_Osimps,axiom,
% 0.34/0.59 ! [F: int > int,A2: int,B: int] :
% 0.34/0.59 ( ( pairself_int_int @ F @ ( product_Pair_int_int @ A2 @ B ) )
% 0.34/0.59 = ( product_Pair_int_int @ ( F @ A2 ) @ ( F @ B ) ) ) ).
% 0.34/0.59
% 0.34/0.59 % pairself.simps
% 0.34/0.59 thf(fact_3150_pairself_Oelims,axiom,
% 0.34/0.59 ! [X: nat > nat,Xa: product_prod_nat_nat,Y: product_prod_nat_nat] :
% 0.34/0.59 ( ( ( pairself_nat_nat @ X @ Xa )
% 0.34/0.59 = Y )
% 0.34/0.59 => ~ ! [A4: nat,B2: nat] :
% 0.34/0.59 ( ( Xa
% 0.34/0.59 = ( product_Pair_nat_nat @ A4 @ B2 ) )
% 0.34/0.59 => ( Y
% 0.34/0.59 != ( product_Pair_nat_nat @ ( X @ A4 ) @ ( X @ B2 ) ) ) ) ) ).
% 0.34/0.59
% 0.34/0.59 % pairself.elims
% 0.34/0.59 thf(fact_3151_pairself_Oelims,axiom,
% 0.34/0.59 ! [X: nat > int,Xa: product_prod_nat_nat,Y: product_prod_int_int] :
% 0.34/0.59 ( ( ( pairself_nat_int @ X @ Xa )
% 0.34/0.59 = Y )
% 0.34/0.59 => ~ ! [A4: nat,B2: nat] :
% 0.34/0.59 ( ( Xa
% 0.34/0.59 = ( product_Pair_nat_nat @ A4 @ B2 ) )
% 0.34/0.59 => ( Y
% 0.34/0.59 != ( product_Pair_int_int @ ( X @ A4 ) @ ( X @ B2 ) ) ) ) ) ).
% 0.34/0.59
% 0.34/0.59 % pairself.elims
% 0.34/0.59 thf(fact_3152_pairself_Oelims,axiom,
% 0.34/0.59 ! [X: int > nat,Xa: product_prod_int_int,Y: product_prod_nat_nat] :
% 0.34/0.59 ( ( ( pairself_int_nat @ X @ Xa )
% 0.34/0.59 = Y )
% 0.34/0.59 => ~ ! [A4: int,B2: int] :
% 0.34/0.59 ( ( Xa
% 0.34/0.59 = ( product_Pair_int_int @ A4 @ B2 ) )
% 0.34/0.59 => ( Y
% 0.34/0.59 != ( product_Pair_nat_nat @ ( X @ A4 ) @ ( X @ B2 ) ) ) ) ) ).
% 0.34/0.59
% 0.34/0.59 % pairself.elims
% 0.34/0.59 thf(fact_3153_pairself_Oelims,axiom,
% 0.34/0.59 ! [X: int > int,Xa: product_prod_int_int,Y: product_prod_int_int] :
% 0.34/0.59 ( ( ( pairself_int_int @ X @ Xa )
% 0.34/0.59 = Y )
% 0.34/0.59 => ~ ! [A4: int,B2: int] :
% 0.34/0.59 ( ( Xa
% 0.34/0.59 = ( product_Pair_int_int @ A4 @ B2 ) )
% 0.34/0.59 => ( Y
% 0.34/0.59 != ( product_Pair_int_int @ ( X @ A4 ) @ ( X @ B2 ) ) ) ) ) ).
% 0.34/0.59
% 0.34/0.59 % pairself.elims
% 0.34/0.59 thf(fact_3154_pairwise__singleton,axiom,
% 0.34/0.59 ! [P: product_unit > product_unit > $o,A: product_unit] : ( pairwi7919127943033184120t_unit @ P @ ( insert_Product_unit @ A @ bot_bo3957492148770167129t_unit ) ) ).
% 0.34/0.59
% 0.34/0.59 % pairwise_singleton
% 0.34/0.59 thf(fact_3155_pairwise__singleton,axiom,
% 0.34/0.59 ! [P: $o > $o > $o,A: $o] : ( pairwise_o @ P @ ( insert_o @ A @ bot_bot_set_o ) ) ).
% 0.34/0.59
% 0.34/0.59 % pairwise_singleton
% 0.34/0.59 thf(fact_3156_pairwise__singleton,axiom,
% 0.34/0.59 ! [P: nat > nat > $o,A: nat] : ( pairwise_nat @ P @ ( insert_nat @ A @ bot_bot_set_nat ) ) ).
% 0.34/0.59
% 0.34/0.59 % pairwise_singleton
% 0.34/0.59 thf(fact_3157_pairwise__singleton,axiom,
% 0.34/0.59 ! [P: int > int > $o,A: int] : ( pairwise_int @ P @ ( insert_int @ A @ bot_bot_set_int ) ) ).
% 0.34/0.59
% 0.34/0.59 % pairwise_singleton
% 0.34/0.59 thf(fact_3158_pairwise__singleton,axiom,
% 0.34/0.59 ! [P: ( produc3658429121746597890et_nat > $o ) > ( produc3658429121746597890et_nat > $o ) > $o,A: produc3658429121746597890et_nat > $o] : ( pairwi2432143822833250616_nat_o @ P @ ( insert5175938949040314269_nat_o @ A @ bot_bo7824918357723582233_nat_o ) ) ).
% 0.34/0.59
% 0.34/0.59 % pairwise_singleton
% 0.34/0.59 thf(fact_3159_finite__Sup__in,axiom,
% 0.34/0.59 ! [A: set_set_nat] :
% 0.34/0.59 ( ( finite1152437895449049373et_nat @ A )
% 0.34/0.59 => ( ( A != bot_bot_set_set_nat )
% 0.34/0.59 => ( ! [X3: set_nat,Y2: set_nat] :
% 0.34/0.59 ( ( member_set_nat @ X3 @ A )
% 0.34/0.59 => ( ( member_set_nat @ Y2 @ A )
% 0.34/0.59 => ( member_set_nat @ ( sup_sup_set_nat @ X3 @ Y2 ) @ A ) ) )
% 0.34/0.59 => ( member_set_nat @ ( comple7399068483239264473et_nat @ A ) @ A ) ) ) ) ).
% 0.34/0.59
% 0.34/0.59 % finite_Sup_in
% 0.34/0.59 thf(fact_3160_finite__Sup__in,axiom,
% 0.34/0.59 ! [A: set_o] :
% 0.34/0.59 ( ( finite_finite_o @ A )
% 0.34/0.59 => ( ( A != bot_bot_set_o )
% 0.34/0.59 => ( ! [X3: $o,Y2: $o] :
% 0.34/0.59 ( ( member_o @ X3 @ A )
% 0.34/0.59 => ( ( member_o @ Y2 @ A )
% 0.34/0.59 => ( member_o @ ( sup_sup_o @ X3 @ Y2 ) @ A ) ) )
% 0.34/0.59 => ( member_o @ ( complete_Sup_Sup_o @ A ) @ A ) ) ) ) ).
% 0.34/0.59
% 0.34/0.59 % finite_Sup_in
% 0.34/0.59 thf(fact_3161_finite__Sup__in,axiom,
% 0.34/0.59 ! [A: set_Pr4532377907799695533_nat_o] :
% 0.34/0.59 ( ( finite3252695134891459830_nat_o @ A )
% 0.34/0.59 => ( ( A != bot_bo7824918357723582233_nat_o )
% 0.34/0.59 => ( ! [X3: produc3658429121746597890et_nat > $o,Y2: produc3658429121746597890et_nat > $o] :
% 0.34/0.59 ( ( member6576561426505652726_nat_o @ X3 @ A )
% 0.34/0.59 => ( ( member6576561426505652726_nat_o @ Y2 @ A )
% 0.34/0.59 => ( member6576561426505652726_nat_o @ ( sup_su5453871518329203617_nat_o @ X3 @ Y2 ) @ A ) ) )
% 0.34/0.59 => ( member6576561426505652726_nat_o @ ( comple4225350640907012154_nat_o @ A ) @ A ) ) ) ) ).
% 0.34/0.59
% 0.34/0.59 % finite_Sup_in
% 0.34/0.59 thf(fact_3162_finite__Sup__in,axiom,
% 0.34/0.59 ! [A: set_Product_unit] :
% 0.34/0.59 ( ( finite4290736615968046902t_unit @ A )
% 0.34/0.59 => ( ( A != bot_bo3957492148770167129t_unit )
% 0.34/0.59 => ( ! [X3: product_unit,Y2: product_unit] :
% 0.34/0.59 ( ( member_Product_unit @ X3 @ A )
% 0.34/0.59 => ( ( member_Product_unit @ Y2 @ A )
% 0.34/0.59 => ( member_Product_unit @ ( sup_sup_Product_unit @ X3 @ Y2 ) @ A ) ) )
% 0.34/0.59 => ( member_Product_unit @ ( comple4687483117567863418t_unit @ A ) @ A ) ) ) ) ).
% 0.34/0.59
% 0.34/0.59 % finite_Sup_in
% 0.34/0.59 thf(fact_3163_Sup__fin__Sup,axiom,
% 0.34/0.59 ! [A: set_o] :
% 0.34/0.59 ( ( finite_finite_o @ A )
% 0.34/0.59 => ( ( A != bot_bot_set_o )
% 0.34/0.59 => ( ( lattic1508158080041050831_fin_o @ A )
% 0.34/0.59 = ( complete_Sup_Sup_o @ A ) ) ) ) ).
% 0.34/0.59
% 0.34/0.59 % Sup_fin_Sup
% 0.34/0.59 thf(fact_3164_Sup__fin__Sup,axiom,
% 0.34/0.59 ! [A: set_Pr4532377907799695533_nat_o] :
% 0.34/0.59 ( ( finite3252695134891459830_nat_o @ A )
% 0.34/0.59 => ( ( A != bot_bo7824918357723582233_nat_o )
% 0.34/0.59 => ( ( lattic7320199455484906628_nat_o @ A )
% 0.34/0.59 = ( comple4225350640907012154_nat_o @ A ) ) ) ) ).
% 0.34/0.59
% 0.34/0.59 % Sup_fin_Sup
% 0.34/0.59 thf(fact_3165_Sup__fin__Sup,axiom,
% 0.34/0.59 ! [A: set_Product_unit] :
% 0.34/0.59 ( ( finite4290736615968046902t_unit @ A )
% 0.34/0.59 => ( ( A != bot_bo3957492148770167129t_unit )
% 0.34/0.59 => ( ( lattic5294303975357428420t_unit @ A )
% 0.34/0.59 = ( comple4687483117567863418t_unit @ A ) ) ) ) ).
% 0.34/0.59
% 0.34/0.59 % Sup_fin_Sup
% 0.34/0.59 thf(fact_3166_insert__partition,axiom,
% 0.34/0.59 ! [X: set_Product_unit,F2: set_set_Product_unit] :
% 0.34/0.59 ( ~ ( member5877623283571906838t_unit @ X @ F2 )
% 0.34/0.59 => ( ! [X3: set_Product_unit] :
% 0.34/0.59 ( ( member5877623283571906838t_unit @ X3 @ ( insert8986028188087627965t_unit @ X @ F2 ) )
% 0.34/0.59 => ! [Xa3: set_Product_unit] :
% 0.34/0.59 ( ( member5877623283571906838t_unit @ Xa3 @ ( insert8986028188087627965t_unit @ X @ F2 ) )
% 0.34/0.59 => ( ( X3 != Xa3 )
% 0.34/0.59 => ( ( inf_in4660618365625256667t_unit @ X3 @ Xa3 )
% 0.34/0.59 = bot_bo3957492148770167129t_unit ) ) ) )
% 0.34/0.59 => ( ( inf_in4660618365625256667t_unit @ X @ ( comple3349607820504768346t_unit @ F2 ) )
% 0.34/0.59 = bot_bo3957492148770167129t_unit ) ) ) ).
% 0.34/0.59
% 0.34/0.59 % insert_partition
% 0.34/0.59 thf(fact_3167_insert__partition,axiom,
% 0.34/0.59 ! [X: set_o,F2: set_set_o] :
% 0.34/0.59 ( ~ ( member_set_o @ X @ F2 )
% 0.34/0.59 => ( ! [X3: set_o] :
% 0.34/0.59 ( ( member_set_o @ X3 @ ( insert_set_o @ X @ F2 ) )
% 0.34/0.59 => ! [Xa3: set_o] :
% 0.34/0.59 ( ( member_set_o @ Xa3 @ ( insert_set_o @ X @ F2 ) )
% 0.34/0.59 => ( ( X3 != Xa3 )
% 0.34/0.59 => ( ( inf_inf_set_o @ X3 @ Xa3 )
% 0.34/0.59 = bot_bot_set_o ) ) ) )
% 0.34/0.59 => ( ( inf_inf_set_o @ X @ ( comple90263536869209701_set_o @ F2 ) )
% 0.34/0.59 = bot_bot_set_o ) ) ) ).
% 0.34/0.59
% 0.34/0.59 % insert_partition
% 0.34/0.59 thf(fact_3168_insert__partition,axiom,
% 0.34/0.59 ! [X: set_nat,F2: set_set_nat] :
% 0.34/0.59 ( ~ ( member_set_nat @ X @ F2 )
% 0.34/0.59 => ( ! [X3: set_nat] :
% 0.34/0.59 ( ( member_set_nat @ X3 @ ( insert_set_nat @ X @ F2 ) )
% 0.34/0.59 => ! [Xa3: set_nat] :
% 0.34/0.59 ( ( member_set_nat @ Xa3 @ ( insert_set_nat @ X @ F2 ) )
% 0.34/0.59 => ( ( X3 != Xa3 )
% 0.34/0.59 => ( ( inf_inf_set_nat @ X3 @ Xa3 )
% 0.34/0.59 = bot_bot_set_nat ) ) ) )
% 0.34/0.59 => ( ( inf_inf_set_nat @ X @ ( comple7399068483239264473et_nat @ F2 ) )
% 0.34/0.59 = bot_bot_set_nat ) ) ) ).
% 0.34/0.59
% 0.34/0.59 % insert_partition
% 0.34/0.59 thf(fact_3169_insert__partition,axiom,
% 0.34/0.59 ! [X: set_int,F2: set_set_int] :
% 0.34/0.59 ( ~ ( member_set_int @ X @ F2 )
% 0.34/0.59 => ( ! [X3: set_int] :
% 0.34/0.59 ( ( member_set_int @ X3 @ ( insert_set_int @ X @ F2 ) )
% 0.34/0.59 => ! [Xa3: set_int] :
% 0.34/0.59 ( ( member_set_int @ Xa3 @ ( insert_set_int @ X @ F2 ) )
% 0.34/0.59 => ( ( X3 != Xa3 )
% 0.34/0.59 => ( ( inf_inf_set_int @ X3 @ Xa3 )
% 0.34/0.59 = bot_bot_set_int ) ) ) )
% 0.34/0.59 => ( ( inf_inf_set_int @ X @ ( comple3221217463730067765et_int @ F2 ) )
% 0.34/0.59 = bot_bot_set_int ) ) ) ).
% 0.34/0.59
% 0.34/0.59 % insert_partition
% 0.34/0.59 thf(fact_3170_insert__partition,axiom,
% 0.34/0.59 ! [X: set_Pr4532377907799695533_nat_o,F2: set_se1353816476878259085_nat_o] :
% 0.34/0.59 ( ~ ( member1452989533377600726_nat_o @ X @ F2 )
% 0.34/0.59 => ( ! [X3: set_Pr4532377907799695533_nat_o] :
% 0.34/0.59 ( ( member1452989533377600726_nat_o @ X3 @ ( insert2507855216428400253_nat_o @ X @ F2 ) )
% 0.34/0.59 => ! [Xa3: set_Pr4532377907799695533_nat_o] :
% 0.34/0.59 ( ( member1452989533377600726_nat_o @ Xa3 @ ( insert2507855216428400253_nat_o @ X @ F2 ) )
% 0.34/0.59 => ( ( X3 != Xa3 )
% 0.34/0.59 => ( ( inf_in1906310914598751387_nat_o @ X3 @ Xa3 )
% 0.34/0.59 = bot_bo7824918357723582233_nat_o ) ) ) )
% 0.34/0.59 => ( ( inf_in1906310914598751387_nat_o @ X @ ( comple3382087578465172250_nat_o @ F2 ) )
% 0.34/0.59 = bot_bo7824918357723582233_nat_o ) ) ) ).
% 0.34/0.59
% 0.34/0.59 % insert_partition
% 0.34/0.59 thf(fact_3171_image__Pow__mono,axiom,
% 0.34/0.59 ! [F: nat > set_nat,A: set_nat,B4: set_set_nat] :
% 0.34/0.59 ( ( ord_le6893508408891458716et_nat @ ( image_nat_set_nat @ F @ A ) @ B4 )
% 0.34/0.59 => ( ord_le9131159989063066194et_nat @ ( image_6725021117256019401et_nat @ ( image_nat_set_nat @ F ) @ ( pow_nat @ A ) ) @ ( pow_set_nat @ B4 ) ) ) ).
% 0.34/0.59
% 0.34/0.59 % image_Pow_mono
% 0.34/0.59 thf(fact_3172_image__Pow__mono,axiom,
% 0.34/0.59 ! [F: code_integer > code_integer,A: set_Code_integer,B4: set_Code_integer] :
% 0.34/0.59 ( ( ord_le7084787975880047091nteger @ ( image_4470545334726330049nteger @ F @ A ) @ B4 )
% 0.34/0.59 => ( ord_le1914454125413604393nteger @ ( image_6270750266955655469nteger @ ( image_4470545334726330049nteger @ F ) @ ( pow_Code_integer @ A ) ) @ ( pow_Code_integer @ B4 ) ) ) ).
% 0.34/0.59
% 0.34/0.59 % image_Pow_mono
% 0.34/0.59 thf(fact_3173_image__Pow__mono,axiom,
% 0.34/0.59 ! [F: nat > code_integer,A: set_nat,B4: set_Code_integer] :
% 0.34/0.59 ( ( ord_le7084787975880047091nteger @ ( image_1215581382706833972nteger @ F @ A ) @ B4 )
% 0.34/0.59 => ( ord_le1914454125413604393nteger @ ( image_6916300684244607776nteger @ ( image_1215581382706833972nteger @ F ) @ ( pow_nat @ A ) ) @ ( pow_Code_integer @ B4 ) ) ) ).
% 0.34/0.59
% 0.34/0.59 % image_Pow_mono
% 0.34/0.59 thf(fact_3174_image__Pow__mono,axiom,
% 0.34/0.59 ! [F: nat > nat,A: set_nat,B4: set_nat] :
% 0.34/0.59 ( ( ord_less_eq_set_nat @ ( image_nat_nat2 @ F @ A ) @ B4 )
% 0.34/0.59 => ( ord_le6893508408891458716et_nat @ ( image_7916887816326733075et_nat @ ( image_nat_nat2 @ F ) @ ( pow_nat @ A ) ) @ ( pow_nat @ B4 ) ) ) ).
% 0.34/0.59
% 0.34/0.59 % image_Pow_mono
% 0.34/0.59 thf(fact_3175_pairwise__alt,axiom,
% 0.34/0.59 ( pairwi7919127943033184120t_unit
% 0.34/0.59 = ( ^ [R: product_unit > product_unit > $o,S6: set_Product_unit] :
% 0.34/0.59 ! [X2: product_unit] :
% 0.34/0.59 ( ( member_Product_unit @ X2 @ S6 )
% 0.34/0.59 => ! [Y3: product_unit] :
% 0.34/0.59 ( ( member_Product_unit @ Y3 @ ( minus_6452836326544984404t_unit @ S6 @ ( insert_Product_unit @ X2 @ bot_bo3957492148770167129t_unit ) ) )
% 0.34/0.59 => ( R @ X2 @ Y3 ) ) ) ) ) ).
% 0.34/0.59
% 0.34/0.59 % pairwise_alt
% 0.34/0.59 thf(fact_3176_pairwise__alt,axiom,
% 0.34/0.59 ( pairwise_o
% 0.34/0.59 = ( ^ [R: $o > $o > $o,S6: set_o] :
% 0.34/0.59 ! [X2: $o] :
% 0.34/0.59 ( ( member_o @ X2 @ S6 )
% 0.34/0.59 => ! [Y3: $o] :
% 0.34/0.59 ( ( member_o @ Y3 @ ( minus_minus_set_o @ S6 @ ( insert_o @ X2 @ bot_bot_set_o ) ) )
% 0.34/0.59 => ( R @ X2 @ Y3 ) ) ) ) ) ).
% 0.34/0.59
% 0.34/0.59 % pairwise_alt
% 0.34/0.59 thf(fact_3177_pairwise__alt,axiom,
% 0.34/0.59 ( pairwise_int
% 0.34/0.59 = ( ^ [R: int > int > $o,S6: set_int] :
% 0.34/0.59 ! [X2: int] :
% 0.34/0.59 ( ( member_int @ X2 @ S6 )
% 0.34/0.59 => ! [Y3: int] :
% 0.34/0.59 ( ( member_int @ Y3 @ ( minus_minus_set_int @ S6 @ ( insert_int @ X2 @ bot_bot_set_int ) ) )
% 0.34/0.59 => ( R @ X2 @ Y3 ) ) ) ) ) ).
% 0.34/0.59
% 0.34/0.59 % pairwise_alt
% 0.34/0.59 thf(fact_3178_pairwise__alt,axiom,
% 0.34/0.59 ( pairwi2432143822833250616_nat_o
% 0.34/0.59 = ( ^ [R: ( produc3658429121746597890et_nat > $o ) > ( produc3658429121746597890et_nat > $o ) > $o,S6: set_Pr4532377907799695533_nat_o] :
% 0.34/0.59 ! [X2: produc3658429121746597890et_nat > $o] :
% 0.34/0.59 ( ( member6576561426505652726_nat_o @ X2 @ S6 )
% 0.34/0.59 => ! [Y3: produc3658429121746597890et_nat > $o] :
% 0.34/0.59 ( ( member6576561426505652726_nat_o @ Y3 @ ( minus_1801376950450012436_nat_o @ S6 @ ( insert5175938949040314269_nat_o @ X2 @ bot_bo7824918357723582233_nat_o ) ) )
% 0.34/0.59 => ( R @ X2 @ Y3 ) ) ) ) ) ).
% 0.34/0.59
% 0.34/0.59 % pairwise_alt
% 0.34/0.59 thf(fact_3179_pairwise__alt,axiom,
% 0.34/0.59 ( pairwise_nat
% 0.34/0.59 = ( ^ [R: nat > nat > $o,S6: set_nat] :
% 0.34/0.59 ! [X2: nat] :
% 0.34/0.59 ( ( member_nat @ X2 @ S6 )
% 0.34/0.59 => ! [Y3: nat] :
% 0.34/0.59 ( ( member_nat @ Y3 @ ( minus_minus_set_nat @ S6 @ ( insert_nat @ X2 @ bot_bot_set_nat ) ) )
% 0.34/0.59 => ( R @ X2 @ Y3 ) ) ) ) ) ).
% 0.34/0.59
% 0.34/0.59 % pairwise_alt
% 0.34/0.59 thf(fact_3180_Sup__insert,axiom,
% 0.34/0.59 ! [A2: $o,A: set_o] :
% 0.34/0.59 ( ( complete_Sup_Sup_o @ ( insert_o @ A2 @ A ) )
% 0.34/0.59 = ( sup_sup_o @ A2 @ ( complete_Sup_Sup_o @ A ) ) ) ).
% 0.34/0.59
% 0.34/0.59 % Sup_insert
% 0.34/0.59 thf(fact_3181_Sup__insert,axiom,
% 0.34/0.59 ! [A2: set_nat,A: set_set_nat] :
% 0.34/0.59 ( ( comple7399068483239264473et_nat @ ( insert_set_nat @ A2 @ A ) )
% 0.34/0.59 = ( sup_sup_set_nat @ A2 @ ( comple7399068483239264473et_nat @ A ) ) ) ).
% 0.34/0.59
% 0.34/0.59 % Sup_insert
% 0.34/0.59 thf(fact_3182_Sup__insert,axiom,
% 0.34/0.59 ! [A2: product_unit,A: set_Product_unit] :
% 0.34/0.59 ( ( comple4687483117567863418t_unit @ ( insert_Product_unit @ A2 @ A ) )
% 0.34/0.59 = ( sup_sup_Product_unit @ A2 @ ( comple4687483117567863418t_unit @ A ) ) ) ).
% 0.34/0.59
% 0.34/0.59 % Sup_insert
% 0.34/0.59 thf(fact_3183_ccpo__Sup__singleton,axiom,
% 0.34/0.59 ! [X: $o] :
% 0.34/0.59 ( ( complete_Sup_Sup_o @ ( insert_o @ X @ bot_bot_set_o ) )
% 0.34/0.59 = X ) ).
% 0.34/0.59
% 0.34/0.59 % ccpo_Sup_singleton
% 0.34/0.59 thf(fact_3184_ccpo__Sup__singleton,axiom,
% 0.34/0.59 ! [X: produc3658429121746597890et_nat > $o] :
% 0.34/0.59 ( ( comple4225350640907012154_nat_o @ ( insert5175938949040314269_nat_o @ X @ bot_bo7824918357723582233_nat_o ) )
% 0.34/0.59 = X ) ).
% 0.34/0.59
% 0.34/0.59 % ccpo_Sup_singleton
% 0.34/0.59 thf(fact_3185_ccpo__Sup__singleton,axiom,
% 0.34/0.59 ! [X: product_unit] :
% 0.34/0.59 ( ( comple4687483117567863418t_unit @ ( insert_Product_unit @ X @ bot_bo3957492148770167129t_unit ) )
% 0.34/0.59 = X ) ).
% 0.34/0.59
% 0.34/0.59 % ccpo_Sup_singleton
% 0.34/0.59 thf(fact_3186_cSup__singleton,axiom,
% 0.34/0.59 ! [X: $o] :
% 0.34/0.59 ( ( complete_Sup_Sup_o @ ( insert_o @ X @ bot_bot_set_o ) )
% 0.34/0.59 = X ) ).
% 0.34/0.59
% 0.34/0.59 % cSup_singleton
% 0.34/0.59 thf(fact_3187_cSup__singleton,axiom,
% 0.34/0.59 ! [X: int] :
% 0.34/0.59 ( ( complete_Sup_Sup_int @ ( insert_int @ X @ bot_bot_set_int ) )
% 0.34/0.59 = X ) ).
% 0.34/0.59
% 0.34/0.59 % cSup_singleton
% 0.34/0.59 thf(fact_3188_cSup__singleton,axiom,
% 0.34/0.59 ! [X: produc3658429121746597890et_nat > $o] :
% 0.34/0.59 ( ( comple4225350640907012154_nat_o @ ( insert5175938949040314269_nat_o @ X @ bot_bo7824918357723582233_nat_o ) )
% 0.34/0.59 = X ) ).
% 0.34/0.59
% 0.34/0.59 % cSup_singleton
% 0.34/0.59 thf(fact_3189_cSup__singleton,axiom,
% 0.34/0.59 ! [X: nat] :
% 0.34/0.59 ( ( complete_Sup_Sup_nat @ ( insert_nat @ X @ bot_bot_set_nat ) )
% 0.34/0.59 = X ) ).
% 0.34/0.59
% 0.34/0.59 % cSup_singleton
% 0.34/0.59 thf(fact_3190_cSup__singleton,axiom,
% 0.34/0.59 ! [X: product_unit] :
% 0.34/0.59 ( ( comple4687483117567863418t_unit @ ( insert_Product_unit @ X @ bot_bo3957492148770167129t_unit ) )
% 0.34/0.59 = X ) ).
% 0.34/0.59
% 0.34/0.59 % cSup_singleton
% 0.34/0.59 thf(fact_3191_Sup__empty,axiom,
% 0.34/0.59 ( ( comple3349607820504768346t_unit @ bot_bo6284674557190688313t_unit )
% 0.34/0.59 = bot_bo3957492148770167129t_unit ) ).
% 0.34/0.59
% 0.34/0.59 % Sup_empty
% 0.34/0.59 thf(fact_3192_Sup__empty,axiom,
% 0.34/0.59 ( ( comple90263536869209701_set_o @ bot_bot_set_set_o )
% 0.34/0.59 = bot_bot_set_o ) ).
% 0.34/0.59
% 0.34/0.59 % Sup_empty
% 0.34/0.59 thf(fact_3193_Sup__empty,axiom,
% 0.34/0.59 ( ( comple7399068483239264473et_nat @ bot_bot_set_set_nat )
% 0.34/0.59 = bot_bot_set_nat ) ).
% 0.34/0.59
% 0.34/0.59 % Sup_empty
% 0.34/0.59 thf(fact_3194_Sup__empty,axiom,
% 0.34/0.59 ( ( comple3221217463730067765et_int @ bot_bot_set_set_int )
% 0.34/0.59 = bot_bot_set_int ) ).
% 0.34/0.59
% 0.34/0.59 % Sup_empty
% 0.34/0.59 thf(fact_3195_Sup__empty,axiom,
% 0.34/0.59 ( ( comple3382087578465172250_nat_o @ bot_bo3736544561982865913_nat_o )
% 0.34/0.59 = bot_bo7824918357723582233_nat_o ) ).
% 0.34/0.59
% 0.34/0.59 % Sup_empty
% 0.34/0.59 thf(fact_3196_Sup__empty,axiom,
% 0.34/0.59 ( ( complete_Sup_Sup_o @ bot_bot_set_o )
% 0.34/0.59 = bot_bot_o ) ).
% 0.34/0.59
% 0.34/0.59 % Sup_empty
% 0.34/0.59 thf(fact_3197_Sup__empty,axiom,
% 0.34/0.59 ( ( comple4225350640907012154_nat_o @ bot_bo7824918357723582233_nat_o )
% 0.34/0.59 = bot_bo6993851861578111033_nat_o ) ).
% 0.34/0.59
% 0.34/0.59 % Sup_empty
% 0.34/0.59 thf(fact_3198_Sup__empty,axiom,
% 0.34/0.59 ( ( comple4687483117567863418t_unit @ bot_bo3957492148770167129t_unit )
% 0.34/0.59 = bot_bot_Product_unit ) ).
% 0.34/0.59
% 0.34/0.59 % Sup_empty
% 0.34/0.59 thf(fact_3199_Sup__bot__conv_I1_J,axiom,
% 0.34/0.59 ! [A: set_set_Product_unit] :
% 0.34/0.59 ( ( ( comple3349607820504768346t_unit @ A )
% 0.34/0.59 = bot_bo3957492148770167129t_unit )
% 0.34/0.59 = ( ! [X2: set_Product_unit] :
% 0.34/0.59 ( ( member5877623283571906838t_unit @ X2 @ A )
% 0.34/0.59 => ( X2 = bot_bo3957492148770167129t_unit ) ) ) ) ).
% 0.34/0.59
% 0.34/0.59 % Sup_bot_conv(1)
% 0.34/0.59 thf(fact_3200_Sup__bot__conv_I1_J,axiom,
% 0.34/0.59 ! [A: set_set_o] :
% 0.34/0.59 ( ( ( comple90263536869209701_set_o @ A )
% 0.34/0.59 = bot_bot_set_o )
% 0.34/0.59 = ( ! [X2: set_o] :
% 0.34/0.59 ( ( member_set_o @ X2 @ A )
% 0.34/0.59 => ( X2 = bot_bot_set_o ) ) ) ) ).
% 0.34/0.59
% 0.34/0.59 % Sup_bot_conv(1)
% 0.34/0.59 thf(fact_3201_Sup__bot__conv_I1_J,axiom,
% 0.34/0.59 ! [A: set_set_nat] :
% 0.34/0.59 ( ( ( comple7399068483239264473et_nat @ A )
% 0.34/0.59 = bot_bot_set_nat )
% 0.34/0.59 = ( ! [X2: set_nat] :
% 0.34/0.59 ( ( member_set_nat @ X2 @ A )
% 0.34/0.59 => ( X2 = bot_bot_set_nat ) ) ) ) ).
% 0.34/0.59
% 0.34/0.59 % Sup_bot_conv(1)
% 0.34/0.59 thf(fact_3202_Sup__bot__conv_I1_J,axiom,
% 0.34/0.59 ! [A: set_set_int] :
% 0.34/0.59 ( ( ( comple3221217463730067765et_int @ A )
% 0.34/0.59 = bot_bot_set_int )
% 0.34/0.59 = ( ! [X2: set_int] :
% 0.34/0.59 ( ( member_set_int @ X2 @ A )
% 0.34/0.59 => ( X2 = bot_bot_set_int ) ) ) ) ).
% 0.34/0.59
% 0.34/0.59 % Sup_bot_conv(1)
% 0.34/0.59 thf(fact_3203_Sup__bot__conv_I1_J,axiom,
% 0.34/0.59 ! [A: set_se1353816476878259085_nat_o] :
% 0.34/0.59 ( ( ( comple3382087578465172250_nat_o @ A )
% 0.34/0.59 = bot_bo7824918357723582233_nat_o )
% 0.34/0.59 = ( ! [X2: set_Pr4532377907799695533_nat_o] :
% 0.34/0.59 ( ( member1452989533377600726_nat_o @ X2 @ A )
% 0.34/0.59 => ( X2 = bot_bo7824918357723582233_nat_o ) ) ) ) ).
% 0.34/0.59
% 0.34/0.59 % Sup_bot_conv(1)
% 0.34/0.59 thf(fact_3204_Sup__bot__conv_I1_J,axiom,
% 0.34/0.59 ! [A: set_Product_unit] :
% 0.34/0.59 ( ( ( comple4687483117567863418t_unit @ A )
% 0.34/0.59 = bot_bot_Product_unit )
% 0.34/0.59 = ( ! [X2: product_unit] :
% 0.34/0.59 ( ( member_Product_unit @ X2 @ A )
% 0.34/0.59 => ( X2 = bot_bot_Product_unit ) ) ) ) ).
% 0.34/0.59
% 0.34/0.59 % Sup_bot_conv(1)
% 0.34/0.59 thf(fact_3205_Sup__bot__conv_I2_J,axiom,
% 0.34/0.59 ! [A: set_set_Product_unit] :
% 0.34/0.59 ( ( bot_bo3957492148770167129t_unit
% 0.34/0.59 = ( comple3349607820504768346t_unit @ A ) )
% 0.34/0.59 = ( ! [X2: set_Product_unit] :
% 0.34/0.59 ( ( member5877623283571906838t_unit @ X2 @ A )
% 0.34/0.59 => ( X2 = bot_bo3957492148770167129t_unit ) ) ) ) ).
% 0.34/0.59
% 0.34/0.59 % Sup_bot_conv(2)
% 0.34/0.59 thf(fact_3206_Sup__bot__conv_I2_J,axiom,
% 0.34/0.59 ! [A: set_set_o] :
% 0.34/0.59 ( ( bot_bot_set_o
% 0.34/0.59 = ( comple90263536869209701_set_o @ A ) )
% 0.34/0.59 = ( ! [X2: set_o] :
% 0.34/0.59 ( ( member_set_o @ X2 @ A )
% 0.34/0.59 => ( X2 = bot_bot_set_o ) ) ) ) ).
% 0.34/0.59
% 0.34/0.59 % Sup_bot_conv(2)
% 0.34/0.59 thf(fact_3207_Sup__bot__conv_I2_J,axiom,
% 0.34/0.59 ! [A: set_set_nat] :
% 0.34/0.59 ( ( bot_bot_set_nat
% 0.34/0.59 = ( comple7399068483239264473et_nat @ A ) )
% 0.34/0.59 = ( ! [X2: set_nat] :
% 0.34/0.59 ( ( member_set_nat @ X2 @ A )
% 0.34/0.59 => ( X2 = bot_bot_set_nat ) ) ) ) ).
% 0.34/0.59
% 0.34/0.59 % Sup_bot_conv(2)
% 0.34/0.59 thf(fact_3208_Sup__bot__conv_I2_J,axiom,
% 0.34/0.59 ! [A: set_set_int] :
% 0.34/0.59 ( ( bot_bot_set_int
% 0.34/0.59 = ( comple3221217463730067765et_int @ A ) )
% 0.34/0.59 = ( ! [X2: set_int] :
% 0.34/0.59 ( ( member_set_int @ X2 @ A )
% 0.34/0.59 => ( X2 = bot_bot_set_int ) ) ) ) ).
% 0.34/0.59
% 0.34/0.59 % Sup_bot_conv(2)
% 0.34/0.59 thf(fact_3209_Sup__bot__conv_I2_J,axiom,
% 0.34/0.59 ! [A: set_se1353816476878259085_nat_o] :
% 0.34/0.59 ( ( bot_bo7824918357723582233_nat_o
% 0.34/0.59 = ( comple3382087578465172250_nat_o @ A ) )
% 0.34/0.59 = ( ! [X2: set_Pr4532377907799695533_nat_o] :
% 0.34/0.59 ( ( member1452989533377600726_nat_o @ X2 @ A )
% 0.34/0.59 => ( X2 = bot_bo7824918357723582233_nat_o ) ) ) ) ).
% 0.34/0.59
% 0.34/0.59 % Sup_bot_conv(2)
% 0.34/0.59 thf(fact_3210_Sup__bot__conv_I2_J,axiom,
% 0.34/0.59 ! [A: set_Product_unit] :
% 0.34/0.59 ( ( bot_bot_Product_unit
% 0.34/0.59 = ( comple4687483117567863418t_unit @ A ) )
% 0.34/0.59 = ( ! [X2: product_unit] :
% 0.34/0.59 ( ( member_Product_unit @ X2 @ A )
% 0.34/0.59 => ( X2 = bot_bot_Product_unit ) ) ) ) ).
% 0.34/0.59
% 0.34/0.59 % Sup_bot_conv(2)
% 0.34/0.59 thf(fact_3211_cSup__eq__Sup__fin,axiom,
% 0.34/0.59 ! [X4: set_o] :
% 0.34/0.59 ( ( finite_finite_o @ X4 )
% 0.34/0.59 => ( ( X4 != bot_bot_set_o )
% 0.34/0.59 => ( ( complete_Sup_Sup_o @ X4 )
% 0.34/0.59 = ( lattic1508158080041050831_fin_o @ X4 ) ) ) ) ).
% 0.34/0.59
% 0.34/0.59 % cSup_eq_Sup_fin
% 0.34/0.59 thf(fact_3212_cSup__eq__Sup__fin,axiom,
% 0.34/0.59 ! [X4: set_int] :
% 0.34/0.59 ( ( finite_finite_int @ X4 )
% 0.34/0.59 => ( ( X4 != bot_bot_set_int )
% 0.34/0.59 => ( ( complete_Sup_Sup_int @ X4 )
% 0.34/0.59 = ( lattic1091506334969745077in_int @ X4 ) ) ) ) ).
% 0.34/0.59
% 0.34/0.59 % cSup_eq_Sup_fin
% 0.34/0.59 thf(fact_3213_cSup__eq__Sup__fin,axiom,
% 0.34/0.59 ! [X4: set_Pr4532377907799695533_nat_o] :
% 0.34/0.59 ( ( finite3252695134891459830_nat_o @ X4 )
% 0.34/0.59 => ( ( X4 != bot_bo7824918357723582233_nat_o )
% 0.34/0.59 => ( ( comple4225350640907012154_nat_o @ X4 )
% 0.34/0.59 = ( lattic7320199455484906628_nat_o @ X4 ) ) ) ) ).
% 0.34/0.59
% 0.34/0.59 % cSup_eq_Sup_fin
% 0.34/0.59 thf(fact_3214_cSup__eq__Sup__fin,axiom,
% 0.34/0.59 ! [X4: set_nat] :
% 0.34/0.59 ( ( finite_finite_nat @ X4 )
% 0.34/0.59 => ( ( X4 != bot_bot_set_nat )
% 0.34/0.59 => ( ( complete_Sup_Sup_nat @ X4 )
% 0.34/0.59 = ( lattic1093996805478795353in_nat @ X4 ) ) ) ) ).
% 0.34/0.59
% 0.34/0.59 % cSup_eq_Sup_fin
% 0.34/0.59 thf(fact_3215_cSup__eq__Sup__fin,axiom,
% 0.34/0.59 ! [X4: set_Product_unit] :
% 0.34/0.59 ( ( finite4290736615968046902t_unit @ X4 )
% 0.34/0.59 => ( ( X4 != bot_bo3957492148770167129t_unit )
% 0.34/0.59 => ( ( comple4687483117567863418t_unit @ X4 )
% 0.34/0.59 = ( lattic5294303975357428420t_unit @ X4 ) ) ) ) ).
% 0.34/0.59
% 0.34/0.59 % cSup_eq_Sup_fin
% 0.34/0.59 thf(fact_3216_Union__image__insert,axiom,
% 0.34/0.59 ! [F: nat > set_nat,A2: nat,B4: set_nat] :
% 0.34/0.59 ( ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ F @ ( insert_nat @ A2 @ B4 ) ) )
% 0.34/0.59 = ( sup_sup_set_nat @ ( F @ A2 ) @ ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ F @ B4 ) ) ) ) ).
% 0.34/0.59
% 0.34/0.59 % Union_image_insert
% 0.34/0.59 thf(fact_3217_Union__image__insert,axiom,
% 0.34/0.59 ! [F: int > set_nat,A2: int,B4: set_int] :
% 0.34/0.59 ( ( comple7399068483239264473et_nat @ ( image_int_set_nat @ F @ ( insert_int @ A2 @ B4 ) ) )
% 0.34/0.59 = ( sup_sup_set_nat @ ( F @ A2 ) @ ( comple7399068483239264473et_nat @ ( image_int_set_nat @ F @ B4 ) ) ) ) ).
% 0.34/0.59
% 0.34/0.59 % Union_image_insert
% 0.34/0.59 thf(fact_3218_Union__image__insert,axiom,
% 0.34/0.59 ! [F: $o > set_nat,A2: $o,B4: set_o] :
% 0.34/0.59 ( ( comple7399068483239264473et_nat @ ( image_o_set_nat @ F @ ( insert_o @ A2 @ B4 ) ) )
% 0.34/0.59 = ( sup_sup_set_nat @ ( F @ A2 ) @ ( comple7399068483239264473et_nat @ ( image_o_set_nat @ F @ B4 ) ) ) ) ).
% 0.34/0.59
% 0.34/0.59 % Union_image_insert
% 0.34/0.59 thf(fact_3219_Union__image__insert,axiom,
% 0.34/0.59 ! [F: product_unit > set_nat,A2: product_unit,B4: set_Product_unit] :
% 0.34/0.59 ( ( comple7399068483239264473et_nat @ ( image_8173849991827986510et_nat @ F @ ( insert_Product_unit @ A2 @ B4 ) ) )
% 0.34/0.59 = ( sup_sup_set_nat @ ( F @ A2 ) @ ( comple7399068483239264473et_nat @ ( image_8173849991827986510et_nat @ F @ B4 ) ) ) ) ).
% 0.34/0.59
% 0.34/0.59 % Union_image_insert
% 0.34/0.59 thf(fact_3220_Union__image__empty,axiom,
% 0.34/0.59 ! [A: set_nat,F: product_unit > set_nat] :
% 0.34/0.59 ( ( sup_sup_set_nat @ A @ ( comple7399068483239264473et_nat @ ( image_8173849991827986510et_nat @ F @ bot_bo3957492148770167129t_unit ) ) )
% 0.34/0.59 = A ) ).
% 0.34/0.59
% 0.34/0.59 % Union_image_empty
% 0.34/0.59 thf(fact_3221_Union__image__empty,axiom,
% 0.34/0.59 ! [A: set_nat,F: $o > set_nat] :
% 0.34/0.59 ( ( sup_sup_set_nat @ A @ ( comple7399068483239264473et_nat @ ( image_o_set_nat @ F @ bot_bot_set_o ) ) )
% 0.34/0.59 = A ) ).
% 0.34/0.59
% 0.34/0.59 % Union_image_empty
% 0.34/0.59 thf(fact_3222_Union__image__empty,axiom,
% 0.34/0.59 ! [A: set_nat,F: nat > set_nat] :
% 0.34/0.59 ( ( sup_sup_set_nat @ A @ ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ F @ bot_bot_set_nat ) ) )
% 0.34/0.59 = A ) ).
% 0.34/0.59
% 0.34/0.59 % Union_image_empty
% 0.34/0.59 thf(fact_3223_Union__image__empty,axiom,
% 0.34/0.59 ! [A: set_nat,F: int > set_nat] :
% 0.34/0.59 ( ( sup_sup_set_nat @ A @ ( comple7399068483239264473et_nat @ ( image_int_set_nat @ F @ bot_bot_set_int ) ) )
% 0.34/0.59 = A ) ).
% 0.34/0.59
% 0.34/0.59 % Union_image_empty
% 0.34/0.59 thf(fact_3224_Union__image__empty,axiom,
% 0.34/0.59 ! [A: set_nat,F: ( produc3658429121746597890et_nat > $o ) > set_nat] :
% 0.34/0.59 ( ( sup_sup_set_nat @ A @ ( comple7399068483239264473et_nat @ ( image_6888641473154130958et_nat @ F @ bot_bo7824918357723582233_nat_o ) ) )
% 0.34/0.59 = A ) ).
% 0.34/0.59
% 0.34/0.59 % Union_image_empty
% 0.34/0.59 thf(fact_3225_Sup__SUP__eq,axiom,
% 0.34/0.59 ( comple3590721164184202819at_o_o
% 0.34/0.59 = ( ^ [S6: set_Pr6409761941060870310at_o_o,X2: produc3658429121746597890et_nat > $o] : ( member6576561426505652726_nat_o @ X2 @ ( comple3382087578465172250_nat_o @ ( image_8639947063239330806_nat_o @ collec939566748876313656_nat_o @ S6 ) ) ) ) ) ).
% 0.34/0.59
% 0.34/0.59 % Sup_SUP_eq
% 0.34/0.59 thf(fact_3226_Sup__SUP__eq,axiom,
% 0.34/0.59 ( comple8317665133742190828_nat_o
% 0.34/0.59 = ( ^ [S6: set_nat_o,X2: nat] : ( member_nat @ X2 @ ( comple7399068483239264473et_nat @ ( image_nat_o_set_nat @ collect_nat @ S6 ) ) ) ) ) ).
% 0.34/0.59
% 0.34/0.59 % Sup_SUP_eq
% 0.34/0.59 thf(fact_3227_Sup__SUP__eq,axiom,
% 0.34/0.59 ( comple6496622788309502864_int_o
% 0.34/0.59 = ( ^ [S6: set_int_o,X2: int] : ( member_int @ X2 @ ( comple3221217463730067765et_int @ ( image_int_o_set_int @ collect_int @ S6 ) ) ) ) ) ).
% 0.34/0.59
% 0.34/0.59 % Sup_SUP_eq
% 0.34/0.59 thf(fact_3228_Sup__SUP__eq,axiom,
% 0.34/0.59 ( complete_Sup_Sup_o_o
% 0.34/0.59 = ( ^ [S6: set_o_o,X2: $o] : ( member_o @ X2 @ ( comple90263536869209701_set_o @ ( image_o_o_set_o @ collect_o @ S6 ) ) ) ) ) ).
% 0.34/0.59
% 0.34/0.59 % Sup_SUP_eq
% 0.34/0.59 thf(fact_3229_Sup__SUP__eq,axiom,
% 0.34/0.59 ( comple4773641153059738115unit_o
% 0.34/0.59 = ( ^ [S6: set_Product_unit_o,X2: product_unit] : ( member_Product_unit @ X2 @ ( comple3349607820504768346t_unit @ ( image_5138372075374782070t_unit @ collect_Product_unit @ S6 ) ) ) ) ) ).
% 0.34/0.59
% 0.34/0.59 % Sup_SUP_eq
% 0.34/0.59 thf(fact_3230_empty__Union__conv,axiom,
% 0.34/0.59 ! [A: set_set_Product_unit] :
% 0.34/0.59 ( ( bot_bo3957492148770167129t_unit
% 0.34/0.59 = ( comple3349607820504768346t_unit @ A ) )
% 0.34/0.59 = ( ! [X2: set_Product_unit] :
% 0.34/0.59 ( ( member5877623283571906838t_unit @ X2 @ A )
% 0.34/0.59 => ( X2 = bot_bo3957492148770167129t_unit ) ) ) ) ).
% 0.34/0.59
% 0.34/0.59 % empty_Union_conv
% 0.34/0.59 thf(fact_3231_empty__Union__conv,axiom,
% 0.34/0.59 ! [A: set_set_o] :
% 0.34/0.59 ( ( bot_bot_set_o
% 0.34/0.59 = ( comple90263536869209701_set_o @ A ) )
% 0.34/0.59 = ( ! [X2: set_o] :
% 0.34/0.59 ( ( member_set_o @ X2 @ A )
% 0.34/0.59 => ( X2 = bot_bot_set_o ) ) ) ) ).
% 0.34/0.59
% 0.34/0.59 % empty_Union_conv
% 0.34/0.59 thf(fact_3232_empty__Union__conv,axiom,
% 0.34/0.59 ! [A: set_set_nat] :
% 0.34/0.59 ( ( bot_bot_set_nat
% 0.34/0.59 = ( comple7399068483239264473et_nat @ A ) )
% 0.34/0.59 = ( ! [X2: set_nat] :
% 0.34/0.59 ( ( member_set_nat @ X2 @ A )
% 0.34/0.59 => ( X2 = bot_bot_set_nat ) ) ) ) ).
% 0.34/0.59
% 0.34/0.59 % empty_Union_conv
% 0.34/0.59 thf(fact_3233_empty__Union__conv,axiom,
% 0.34/0.59 ! [A: set_set_int] :
% 0.34/0.59 ( ( bot_bot_set_int
% 0.34/0.59 = ( comple3221217463730067765et_int @ A ) )
% 0.34/0.59 = ( ! [X2: set_int] :
% 0.34/0.59 ( ( member_set_int @ X2 @ A )
% 0.34/0.59 => ( X2 = bot_bot_set_int ) ) ) ) ).
% 0.34/0.59
% 0.34/0.59 % empty_Union_conv
% 0.34/0.59 thf(fact_3234_empty__Union__conv,axiom,
% 0.34/0.59 ! [A: set_se1353816476878259085_nat_o] :
% 0.34/0.59 ( ( bot_bo7824918357723582233_nat_o
% 0.34/0.59 = ( comple3382087578465172250_nat_o @ A ) )
% 0.34/0.59 = ( ! [X2: set_Pr4532377907799695533_nat_o] :
% 0.34/0.59 ( ( member1452989533377600726_nat_o @ X2 @ A )
% 0.34/0.59 => ( X2 = bot_bo7824918357723582233_nat_o ) ) ) ) ).
% 0.34/0.59
% 0.34/0.59 % empty_Union_conv
% 0.34/0.59 thf(fact_3235_Union__empty__conv,axiom,
% 0.34/0.59 ! [A: set_set_Product_unit] :
% 0.34/0.59 ( ( ( comple3349607820504768346t_unit @ A )
% 0.34/0.59 = bot_bo3957492148770167129t_unit )
% 0.34/0.59 = ( ! [X2: set_Product_unit] :
% 0.34/0.59 ( ( member5877623283571906838t_unit @ X2 @ A )
% 0.34/0.59 => ( X2 = bot_bo3957492148770167129t_unit ) ) ) ) ).
% 0.34/0.59
% 0.34/0.59 % Union_empty_conv
% 0.34/0.59 thf(fact_3236_Union__empty__conv,axiom,
% 0.34/0.59 ! [A: set_set_o] :
% 0.34/0.59 ( ( ( comple90263536869209701_set_o @ A )
% 0.34/0.59 = bot_bot_set_o )
% 0.34/0.59 = ( ! [X2: set_o] :
% 0.34/0.59 ( ( member_set_o @ X2 @ A )
% 0.34/0.59 => ( X2 = bot_bot_set_o ) ) ) ) ).
% 0.34/0.59
% 0.34/0.59 % Union_empty_conv
% 0.34/0.59 thf(fact_3237_Union__empty__conv,axiom,
% 0.34/0.59 ! [A: set_set_nat] :
% 0.34/0.59 ( ( ( comple7399068483239264473et_nat @ A )
% 0.34/0.59 = bot_bot_set_nat )
% 0.34/0.59 = ( ! [X2: set_nat] :
% 0.34/0.59 ( ( member_set_nat @ X2 @ A )
% 0.34/0.59 => ( X2 = bot_bot_set_nat ) ) ) ) ).
% 0.34/0.59
% 0.34/0.59 % Union_empty_conv
% 0.34/0.59 thf(fact_3238_Union__empty__conv,axiom,
% 0.34/0.59 ! [A: set_set_int] :
% 0.34/0.59 ( ( ( comple3221217463730067765et_int @ A )
% 0.34/0.59 = bot_bot_set_int )
% 0.34/0.59 = ( ! [X2: set_int] :
% 0.34/0.59 ( ( member_set_int @ X2 @ A )
% 0.34/0.59 => ( X2 = bot_bot_set_int ) ) ) ) ).
% 0.34/0.59
% 0.34/0.59 % Union_empty_conv
% 0.34/0.59 thf(fact_3239_Union__empty__conv,axiom,
% 0.34/0.59 ! [A: set_se1353816476878259085_nat_o] :
% 0.34/0.59 ( ( ( comple3382087578465172250_nat_o @ A )
% 0.34/0.59 = bot_bo7824918357723582233_nat_o )
% 0.34/0.59 = ( ! [X2: set_Pr4532377907799695533_nat_o] :
% 0.34/0.59 ( ( member1452989533377600726_nat_o @ X2 @ A )
% 0.34/0.59 => ( X2 = bot_bo7824918357723582233_nat_o ) ) ) ) ).
% 0.34/0.59
% 0.34/0.59 % Union_empty_conv
% 0.34/0.59 thf(fact_3240_less__eq__Sup,axiom,
% 0.34/0.59 ! [A: set_o,U: $o] :
% 0.34/0.59 ( ! [V3: $o] :
% 0.34/0.59 ( ( member_o @ V3 @ A )
% 0.34/0.59 => ( ord_less_eq_o @ U @ V3 ) )
% 0.34/0.59 => ( ( A != bot_bot_set_o )
% 0.34/0.59 => ( ord_less_eq_o @ U @ ( complete_Sup_Sup_o @ A ) ) ) ) ).
% 0.34/0.59
% 0.34/0.59 % less_eq_Sup
% 0.34/0.59 thf(fact_3241_less__eq__Sup,axiom,
% 0.34/0.59 ! [A: set_Pr4532377907799695533_nat_o,U: produc3658429121746597890et_nat > $o] :
% 0.34/0.59 ( ! [V3: produc3658429121746597890et_nat > $o] :
% 0.34/0.59 ( ( member6576561426505652726_nat_o @ V3 @ A )
% 0.34/0.59 => ( ord_le729326519192465773_nat_o @ U @ V3 ) )
% 0.34/0.59 => ( ( A != bot_bo7824918357723582233_nat_o )
% 0.34/0.59 => ( ord_le729326519192465773_nat_o @ U @ ( comple4225350640907012154_nat_o @ A ) ) ) ) ).
% 0.34/0.59
% 0.34/0.59 % less_eq_Sup
% 0.34/0.59 thf(fact_3242_less__eq__Sup,axiom,
% 0.34/0.59 ! [A: set_filter_nat,U: filter_nat] :
% 0.34/0.59 ( ! [V3: filter_nat] :
% 0.34/0.59 ( ( member_filter_nat @ V3 @ A )
% 0.34/0.59 => ( ord_le2510731241096832064er_nat @ U @ V3 ) )
% 0.34/0.59 => ( ( A != bot_bo498966703094740906er_nat )
% 0.34/0.59 => ( ord_le2510731241096832064er_nat @ U @ ( comple2454491259644266675er_nat @ A ) ) ) ) ).
% 0.34/0.59
% 0.34/0.59 % less_eq_Sup
% 0.34/0.59 thf(fact_3243_less__eq__Sup,axiom,
% 0.34/0.59 ! [A: set_set_nat,U: set_nat] :
% 0.34/0.59 ( ! [V3: set_nat] :
% 0.34/0.59 ( ( member_set_nat @ V3 @ A )
% 0.34/0.59 => ( ord_less_eq_set_nat @ U @ V3 ) )
% 0.34/0.59 => ( ( A != bot_bot_set_set_nat )
% 0.34/0.59 => ( ord_less_eq_set_nat @ U @ ( comple7399068483239264473et_nat @ A ) ) ) ) ).
% 0.34/0.59
% 0.34/0.59 % less_eq_Sup
% 0.34/0.59 thf(fact_3244_less__eq__Sup,axiom,
% 0.34/0.59 ! [A: set_Product_unit,U: product_unit] :
% 0.34/0.59 ( ! [V3: product_unit] :
% 0.34/0.59 ( ( member_Product_unit @ V3 @ A )
% 0.34/0.59 => ( ord_le3221252021190050221t_unit @ U @ V3 ) )
% 0.34/0.59 => ( ( A != bot_bo3957492148770167129t_unit )
% 0.34/0.59 => ( ord_le3221252021190050221t_unit @ U @ ( comple4687483117567863418t_unit @ A ) ) ) ) ).
% 0.34/0.59
% 0.34/0.59 % less_eq_Sup
% 0.34/0.59 thf(fact_3245_cSup__least,axiom,
% 0.34/0.59 ! [X4: set_o,Z: $o] :
% 0.34/0.59 ( ( X4 != bot_bot_set_o )
% 0.34/0.59 => ( ! [X3: $o] :
% 0.34/0.59 ( ( member_o @ X3 @ X4 )
% 0.34/0.59 => ( ord_less_eq_o @ X3 @ Z ) )
% 0.34/0.59 => ( ord_less_eq_o @ ( complete_Sup_Sup_o @ X4 ) @ Z ) ) ) ).
% 0.34/0.59
% 0.34/0.59 % cSup_least
% 0.34/0.59 thf(fact_3246_cSup__least,axiom,
% 0.34/0.59 ! [X4: set_Pr4532377907799695533_nat_o,Z: produc3658429121746597890et_nat > $o] :
% 0.34/0.59 ( ( X4 != bot_bo7824918357723582233_nat_o )
% 0.34/0.59 => ( ! [X3: produc3658429121746597890et_nat > $o] :
% 0.34/0.59 ( ( member6576561426505652726_nat_o @ X3 @ X4 )
% 0.34/0.59 => ( ord_le729326519192465773_nat_o @ X3 @ Z ) )
% 0.34/0.59 => ( ord_le729326519192465773_nat_o @ ( comple4225350640907012154_nat_o @ X4 ) @ Z ) ) ) ).
% 0.34/0.59
% 0.34/0.59 % cSup_least
% 0.34/0.59 thf(fact_3247_cSup__least,axiom,
% 0.34/0.59 ! [X4: set_filter_nat,Z: filter_nat] :
% 0.34/0.59 ( ( X4 != bot_bo498966703094740906er_nat )
% 0.34/0.59 => ( ! [X3: filter_nat] :
% 0.34/0.59 ( ( member_filter_nat @ X3 @ X4 )
% 0.34/0.59 => ( ord_le2510731241096832064er_nat @ X3 @ Z ) )
% 0.34/0.59 => ( ord_le2510731241096832064er_nat @ ( comple2454491259644266675er_nat @ X4 ) @ Z ) ) ) ).
% 0.34/0.59
% 0.34/0.59 % cSup_least
% 0.34/0.59 thf(fact_3248_cSup__least,axiom,
% 0.34/0.59 ! [X4: set_set_nat,Z: set_nat] :
% 0.34/0.59 ( ( X4 != bot_bot_set_set_nat )
% 0.34/0.59 => ( ! [X3: set_nat] :
% 0.34/0.59 ( ( member_set_nat @ X3 @ X4 )
% 0.34/0.59 => ( ord_less_eq_set_nat @ X3 @ Z ) )
% 0.34/0.59 => ( ord_less_eq_set_nat @ ( comple7399068483239264473et_nat @ X4 ) @ Z ) ) ) ).
% 0.34/0.59
% 0.34/0.59 % cSup_least
% 0.34/0.59 thf(fact_3249_cSup__least,axiom,
% 0.34/0.59 ! [X4: set_int,Z: int] :
% 0.34/0.59 ( ( X4 != bot_bot_set_int )
% 0.34/0.59 => ( ! [X3: int] :
% 0.34/0.59 ( ( member_int @ X3 @ X4 )
% 0.34/0.59 => ( ord_less_eq_int @ X3 @ Z ) )
% 0.34/0.59 => ( ord_less_eq_int @ ( complete_Sup_Sup_int @ X4 ) @ Z ) ) ) ).
% 0.34/0.59
% 0.34/0.59 % cSup_least
% 0.34/0.59 thf(fact_3250_cSup__least,axiom,
% 0.34/0.59 ! [X4: set_nat,Z: nat] :
% 0.34/0.59 ( ( X4 != bot_bot_set_nat )
% 0.34/0.59 => ( ! [X3: nat] :
% 0.34/0.59 ( ( member_nat @ X3 @ X4 )
% 0.34/0.59 => ( ord_less_eq_nat @ X3 @ Z ) )
% 0.34/0.59 => ( ord_less_eq_nat @ ( complete_Sup_Sup_nat @ X4 ) @ Z ) ) ) ).
% 0.34/0.59
% 0.34/0.59 % cSup_least
% 0.34/0.59 thf(fact_3251_cSup__least,axiom,
% 0.34/0.59 ! [X4: set_Product_unit,Z: product_unit] :
% 0.34/0.59 ( ( X4 != bot_bo3957492148770167129t_unit )
% 0.34/0.59 => ( ! [X3: product_unit] :
% 0.34/0.59 ( ( member_Product_unit @ X3 @ X4 )
% 0.34/0.59 => ( ord_le3221252021190050221t_unit @ X3 @ Z ) )
% 0.34/0.59 => ( ord_le3221252021190050221t_unit @ ( comple4687483117567863418t_unit @ X4 ) @ Z ) ) ) ).
% 0.34/0.59
% 0.34/0.59 % cSup_least
% 0.34/0.59 thf(fact_3252_cSup__eq__non__empty,axiom,
% 0.34/0.59 ! [X4: set_o,A2: $o] :
% 0.34/0.59 ( ( X4 != bot_bot_set_o )
% 0.34/0.59 => ( ! [X3: $o] :
% 0.34/0.59 ( ( member_o @ X3 @ X4 )
% 0.34/0.59 => ( ord_less_eq_o @ X3 @ A2 ) )
% 0.34/0.59 => ( ! [Y2: $o] :
% 0.34/0.59 ( ! [X6: $o] :
% 0.34/0.59 ( ( member_o @ X6 @ X4 )
% 0.34/0.59 => ( ord_less_eq_o @ X6 @ Y2 ) )
% 0.34/0.59 => ( ord_less_eq_o @ A2 @ Y2 ) )
% 0.34/0.59 => ( ( complete_Sup_Sup_o @ X4 )
% 0.34/0.59 = A2 ) ) ) ) ).
% 0.34/0.59
% 0.34/0.59 % cSup_eq_non_empty
% 0.34/0.59 thf(fact_3253_cSup__eq__non__empty,axiom,
% 0.34/0.59 ! [X4: set_Pr4532377907799695533_nat_o,A2: produc3658429121746597890et_nat > $o] :
% 0.34/0.59 ( ( X4 != bot_bo7824918357723582233_nat_o )
% 0.34/0.59 => ( ! [X3: produc3658429121746597890et_nat > $o] :
% 0.34/0.59 ( ( member6576561426505652726_nat_o @ X3 @ X4 )
% 0.34/0.59 => ( ord_le729326519192465773_nat_o @ X3 @ A2 ) )
% 0.34/0.59 => ( ! [Y2: produc3658429121746597890et_nat > $o] :
% 0.34/0.59 ( ! [X6: produc3658429121746597890et_nat > $o] :
% 0.34/0.59 ( ( member6576561426505652726_nat_o @ X6 @ X4 )
% 0.34/0.59 => ( ord_le729326519192465773_nat_o @ X6 @ Y2 ) )
% 0.34/0.59 => ( ord_le729326519192465773_nat_o @ A2 @ Y2 ) )
% 0.34/0.59 => ( ( comple4225350640907012154_nat_o @ X4 )
% 0.34/0.59 = A2 ) ) ) ) ).
% 0.34/0.59
% 0.34/0.59 % cSup_eq_non_empty
% 0.34/0.59 thf(fact_3254_cSup__eq__non__empty,axiom,
% 0.34/0.59 ! [X4: set_filter_nat,A2: filter_nat] :
% 0.34/0.59 ( ( X4 != bot_bo498966703094740906er_nat )
% 0.34/0.59 => ( ! [X3: filter_nat] :
% 0.34/0.59 ( ( member_filter_nat @ X3 @ X4 )
% 0.34/0.59 => ( ord_le2510731241096832064er_nat @ X3 @ A2 ) )
% 0.34/0.59 => ( ! [Y2: filter_nat] :
% 0.34/0.59 ( ! [X6: filter_nat] :
% 0.34/0.59 ( ( member_filter_nat @ X6 @ X4 )
% 0.34/0.59 => ( ord_le2510731241096832064er_nat @ X6 @ Y2 ) )
% 0.34/0.59 => ( ord_le2510731241096832064er_nat @ A2 @ Y2 ) )
% 0.34/0.59 => ( ( comple2454491259644266675er_nat @ X4 )
% 0.34/0.59 = A2 ) ) ) ) ).
% 0.34/0.59
% 0.34/0.59 % cSup_eq_non_empty
% 0.34/0.59 thf(fact_3255_cSup__eq__non__empty,axiom,
% 0.34/0.59 ! [X4: set_set_nat,A2: set_nat] :
% 0.34/0.59 ( ( X4 != bot_bot_set_set_nat )
% 0.34/0.59 => ( ! [X3: set_nat] :
% 0.34/0.59 ( ( member_set_nat @ X3 @ X4 )
% 0.34/0.59 => ( ord_less_eq_set_nat @ X3 @ A2 ) )
% 0.34/0.59 => ( ! [Y2: set_nat] :
% 0.34/0.59 ( ! [X6: set_nat] :
% 0.34/0.59 ( ( member_set_nat @ X6 @ X4 )
% 0.34/0.59 => ( ord_less_eq_set_nat @ X6 @ Y2 ) )
% 0.34/0.59 => ( ord_less_eq_set_nat @ A2 @ Y2 ) )
% 0.34/0.59 => ( ( comple7399068483239264473et_nat @ X4 )
% 0.34/0.59 = A2 ) ) ) ) ).
% 0.34/0.59
% 0.34/0.59 % cSup_eq_non_empty
% 0.34/0.59 thf(fact_3256_cSup__eq__non__empty,axiom,
% 0.34/0.59 ! [X4: set_int,A2: int] :
% 0.34/0.59 ( ( X4 != bot_bot_set_int )
% 0.34/0.59 => ( ! [X3: int] :
% 0.34/0.59 ( ( member_int @ X3 @ X4 )
% 0.34/0.59 => ( ord_less_eq_int @ X3 @ A2 ) )
% 0.34/0.59 => ( ! [Y2: int] :
% 0.34/0.59 ( ! [X6: int] :
% 0.34/0.59 ( ( member_int @ X6 @ X4 )
% 0.34/0.59 => ( ord_less_eq_int @ X6 @ Y2 ) )
% 0.34/0.59 => ( ord_less_eq_int @ A2 @ Y2 ) )
% 0.34/0.59 => ( ( complete_Sup_Sup_int @ X4 )
% 0.34/0.59 = A2 ) ) ) ) ).
% 0.34/0.59
% 0.34/0.59 % cSup_eq_non_empty
% 0.34/0.59 thf(fact_3257_cSup__eq__non__empty,axiom,
% 0.34/0.59 ! [X4: set_nat,A2: nat] :
% 0.34/0.59 ( ( X4 != bot_bot_set_nat )
% 0.34/0.59 => ( ! [X3: nat] :
% 0.34/0.59 ( ( member_nat @ X3 @ X4 )
% 0.34/0.59 => ( ord_less_eq_nat @ X3 @ A2 ) )
% 0.34/0.59 => ( ! [Y2: nat] :
% 0.34/0.59 ( ! [X6: nat] :
% 0.34/0.59 ( ( member_nat @ X6 @ X4 )
% 0.34/0.59 => ( ord_less_eq_nat @ X6 @ Y2 ) )
% 0.34/0.59 => ( ord_less_eq_nat @ A2 @ Y2 ) )
% 0.34/0.59 => ( ( complete_Sup_Sup_nat @ X4 )
% 0.34/0.59 = A2 ) ) ) ) ).
% 0.34/0.59
% 0.34/0.59 % cSup_eq_non_empty
% 0.34/0.59 thf(fact_3258_cSup__eq__non__empty,axiom,
% 0.34/0.59 ! [X4: set_Product_unit,A2: product_unit] :
% 0.34/0.59 ( ( X4 != bot_bo3957492148770167129t_unit )
% 0.34/0.59 => ( ! [X3: product_unit] :
% 0.34/0.59 ( ( member_Product_unit @ X3 @ X4 )
% 0.34/0.59 => ( ord_le3221252021190050221t_unit @ X3 @ A2 ) )
% 0.34/0.59 => ( ! [Y2: product_unit] :
% 0.34/0.59 ( ! [X6: product_unit] :
% 0.34/0.59 ( ( member_Product_unit @ X6 @ X4 )
% 0.34/0.59 => ( ord_le3221252021190050221t_unit @ X6 @ Y2 ) )
% 0.34/0.59 => ( ord_le3221252021190050221t_unit @ A2 @ Y2 ) )
% 0.34/0.59 => ( ( comple4687483117567863418t_unit @ X4 )
% 0.34/0.59 = A2 ) ) ) ) ).
% 0.34/0.59
% 0.34/0.59 % cSup_eq_non_empty
% 0.34/0.59 thf(fact_3259_SUP__eq__const,axiom,
% 0.34/0.59 ! [I: set_nat,F: nat > set_nat,X: set_nat] :
% 0.34/0.59 ( ( I != bot_bot_set_nat )
% 0.34/0.59 => ( ! [I3: nat] :
% 0.34/0.59 ( ( member_nat @ I3 @ I )
% 0.34/0.59 => ( ( F @ I3 )
% 0.34/0.59 = X ) )
% 0.34/0.59 => ( ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ F @ I ) )
% 0.34/0.59 = X ) ) ) ).
% 0.34/0.59
% 0.34/0.59 % SUP_eq_const
% 0.34/0.59 thf(fact_3260_SUP__eq__const,axiom,
% 0.34/0.59 ! [I: set_Product_unit,F: product_unit > product_unit,X: product_unit] :
% 0.34/0.59 ( ( I != bot_bo3957492148770167129t_unit )
% 0.34/0.59 => ( ! [I3: product_unit] :
% 0.34/0.59 ( ( member_Product_unit @ I3 @ I )
% 0.34/0.59 => ( ( F @ I3 )
% 0.34/0.59 = X ) )
% 0.34/0.59 => ( ( comple4687483117567863418t_unit @ ( image_405062704495631173t_unit @ F @ I ) )
% 0.34/0.59 = X ) ) ) ).
% 0.34/0.59
% 0.34/0.59 % SUP_eq_const
% 0.34/0.59 thf(fact_3261_SUP__eq__const,axiom,
% 0.34/0.59 ! [I: set_o,F: $o > product_unit,X: product_unit] :
% 0.34/0.59 ( ( I != bot_bot_set_o )
% 0.34/0.59 => ( ! [I3: $o] :
% 0.34/0.59 ( ( member_o @ I3 @ I )
% 0.34/0.59 => ( ( F @ I3 )
% 0.34/0.59 = X ) )
% 0.34/0.59 => ( ( comple4687483117567863418t_unit @ ( image_o_Product_unit2 @ F @ I ) )
% 0.34/0.59 = X ) ) ) ).
% 0.34/0.59
% 0.34/0.59 % SUP_eq_const
% 0.34/0.59 thf(fact_3262_SUP__eq__const,axiom,
% 0.34/0.59 ! [I: set_nat,F: nat > product_unit,X: product_unit] :
% 0.34/0.59 ( ( I != bot_bot_set_nat )
% 0.34/0.59 => ( ! [I3: nat] :
% 0.34/0.59 ( ( member_nat @ I3 @ I )
% 0.34/0.59 => ( ( F @ I3 )
% 0.34/0.59 = X ) )
% 0.34/0.59 => ( ( comple4687483117567863418t_unit @ ( image_8730104196221521654t_unit @ F @ I ) )
% 0.34/0.59 = X ) ) ) ).
% 0.34/0.59
% 0.34/0.59 % SUP_eq_const
% 0.34/0.59 thf(fact_3263_SUP__eq__const,axiom,
% 0.34/0.59 ! [I: set_int,F: int > product_unit,X: product_unit] :
% 0.34/0.59 ( ( I != bot_bot_set_int )
% 0.34/0.59 => ( ! [I3: int] :
% 0.34/0.59 ( ( member_int @ I3 @ I )
% 0.34/0.59 => ( ( F @ I3 )
% 0.34/0.59 = X ) )
% 0.34/0.59 => ( ( comple4687483117567863418t_unit @ ( image_5846526143423701914t_unit @ F @ I ) )
% 0.34/0.59 = X ) ) ) ).
% 0.34/0.59
% 0.34/0.59 % SUP_eq_const
% 0.34/0.59 thf(fact_3264_SUP__eq__const,axiom,
% 0.34/0.59 ! [I: set_Pr4532377907799695533_nat_o,F: ( produc3658429121746597890et_nat > $o ) > product_unit,X: product_unit] :
% 0.34/0.59 ( ( I != bot_bo7824918357723582233_nat_o )
% 0.34/0.59 => ( ! [I3: produc3658429121746597890et_nat > $o] :
% 0.34/0.59 ( ( member6576561426505652726_nat_o @ I3 @ I )
% 0.34/0.59 => ( ( F @ I3 )
% 0.34/0.59 = X ) )
% 0.34/0.59 => ( ( comple4687483117567863418t_unit @ ( image_8451000719036923781t_unit @ F @ I ) )
% 0.34/0.59 = X ) ) ) ).
% 0.34/0.59
% 0.34/0.59 % SUP_eq_const
% 0.34/0.59 thf(fact_3265_Union__disjoint,axiom,
% 0.34/0.59 ! [C5: set_set_Product_unit,A: set_Product_unit] :
% 0.34/0.59 ( ( ( inf_in4660618365625256667t_unit @ ( comple3349607820504768346t_unit @ C5 ) @ A )
% 0.34/0.59 = bot_bo3957492148770167129t_unit )
% 0.34/0.59 = ( ! [X2: set_Product_unit] :
% 0.34/0.59 ( ( member5877623283571906838t_unit @ X2 @ C5 )
% 0.34/0.59 => ( ( inf_in4660618365625256667t_unit @ X2 @ A )
% 0.34/0.59 = bot_bo3957492148770167129t_unit ) ) ) ) ).
% 0.34/0.59
% 0.34/0.59 % Union_disjoint
% 0.34/0.59 thf(fact_3266_Union__disjoint,axiom,
% 0.34/0.59 ! [C5: set_set_o,A: set_o] :
% 0.34/0.59 ( ( ( inf_inf_set_o @ ( comple90263536869209701_set_o @ C5 ) @ A )
% 0.34/0.59 = bot_bot_set_o )
% 0.34/0.59 = ( ! [X2: set_o] :
% 0.34/0.59 ( ( member_set_o @ X2 @ C5 )
% 0.34/0.59 => ( ( inf_inf_set_o @ X2 @ A )
% 0.34/0.59 = bot_bot_set_o ) ) ) ) ).
% 0.34/0.59
% 0.34/0.59 % Union_disjoint
% 0.34/0.59 thf(fact_3267_Union__disjoint,axiom,
% 0.34/0.59 ! [C5: set_set_nat,A: set_nat] :
% 0.34/0.59 ( ( ( inf_inf_set_nat @ ( comple7399068483239264473et_nat @ C5 ) @ A )
% 0.34/0.59 = bot_bot_set_nat )
% 0.34/0.59 = ( ! [X2: set_nat] :
% 0.34/0.59 ( ( member_set_nat @ X2 @ C5 )
% 0.34/0.59 => ( ( inf_inf_set_nat @ X2 @ A )
% 0.34/0.59 = bot_bot_set_nat ) ) ) ) ).
% 0.34/0.59
% 0.34/0.59 % Union_disjoint
% 0.34/0.59 thf(fact_3268_Union__disjoint,axiom,
% 0.34/0.59 ! [C5: set_set_int,A: set_int] :
% 0.34/0.59 ( ( ( inf_inf_set_int @ ( comple3221217463730067765et_int @ C5 ) @ A )
% 0.34/0.59 = bot_bot_set_int )
% 0.34/0.59 = ( ! [X2: set_int] :
% 0.34/0.59 ( ( member_set_int @ X2 @ C5 )
% 0.34/0.59 => ( ( inf_inf_set_int @ X2 @ A )
% 0.34/0.59 = bot_bot_set_int ) ) ) ) ).
% 0.34/0.59
% 0.34/0.59 % Union_disjoint
% 0.34/0.59 thf(fact_3269_Union__disjoint,axiom,
% 0.34/0.59 ! [C5: set_se1353816476878259085_nat_o,A: set_Pr4532377907799695533_nat_o] :
% 0.34/0.59 ( ( ( inf_in1906310914598751387_nat_o @ ( comple3382087578465172250_nat_o @ C5 ) @ A )
% 0.34/0.59 = bot_bo7824918357723582233_nat_o )
% 0.34/0.59 = ( ! [X2: set_Pr4532377907799695533_nat_o] :
% 0.34/0.59 ( ( member1452989533377600726_nat_o @ X2 @ C5 )
% 0.34/0.59 => ( ( inf_in1906310914598751387_nat_o @ X2 @ A )
% 0.34/0.59 = bot_bo7824918357723582233_nat_o ) ) ) ) ).
% 0.34/0.59
% 0.34/0.59 % Union_disjoint
% 0.34/0.59 thf(fact_3270_Union__empty,axiom,
% 0.34/0.59 ( ( comple3349607820504768346t_unit @ bot_bo6284674557190688313t_unit )
% 0.34/0.59 = bot_bo3957492148770167129t_unit ) ).
% 0.34/0.59
% 0.34/0.59 % Union_empty
% 0.34/0.59 thf(fact_3271_Union__empty,axiom,
% 0.34/0.59 ( ( comple90263536869209701_set_o @ bot_bot_set_set_o )
% 0.34/0.59 = bot_bot_set_o ) ).
% 0.34/0.59
% 0.34/0.59 % Union_empty
% 0.34/0.59 thf(fact_3272_Union__empty,axiom,
% 0.34/0.59 ( ( comple7399068483239264473et_nat @ bot_bot_set_set_nat )
% 0.34/0.59 = bot_bot_set_nat ) ).
% 0.34/0.59
% 0.34/0.59 % Union_empty
% 0.34/0.59 thf(fact_3273_Union__empty,axiom,
% 0.34/0.59 ( ( comple3221217463730067765et_int @ bot_bot_set_set_int )
% 0.34/0.59 = bot_bot_set_int ) ).
% 0.34/0.59
% 0.34/0.59 % Union_empty
% 0.34/0.59 thf(fact_3274_Union__empty,axiom,
% 0.34/0.59 ( ( comple3382087578465172250_nat_o @ bot_bo3736544561982865913_nat_o )
% 0.34/0.59 = bot_bo7824918357723582233_nat_o ) ).
% 0.34/0.59
% 0.34/0.59 % Union_empty
% 0.34/0.59 thf(fact_3275_SUP__eq__iff,axiom,
% 0.34/0.59 ! [I: set_Product_unit,C: product_unit,F: product_unit > product_unit] :
% 0.34/0.59 ( ( I != bot_bo3957492148770167129t_unit )
% 0.34/0.59 => ( ! [I3: product_unit] :
% 0.34/0.59 ( ( member_Product_unit @ I3 @ I )
% 0.34/0.59 => ( ord_le3221252021190050221t_unit @ C @ ( F @ I3 ) ) )
% 0.34/0.59 => ( ( ( comple4687483117567863418t_unit @ ( image_405062704495631173t_unit @ F @ I ) )
% 0.34/0.59 = C )
% 0.34/0.59 = ( ! [X2: product_unit] :
% 0.34/0.59 ( ( member_Product_unit @ X2 @ I )
% 0.34/0.59 => ( ( F @ X2 )
% 0.34/0.59 = C ) ) ) ) ) ) ).
% 0.34/0.59
% 0.34/0.59 % SUP_eq_iff
% 0.34/0.59 thf(fact_3276_SUP__eq__iff,axiom,
% 0.34/0.59 ! [I: set_o,C: product_unit,F: $o > product_unit] :
% 0.34/0.59 ( ( I != bot_bot_set_o )
% 0.34/0.59 => ( ! [I3: $o] :
% 0.34/0.59 ( ( member_o @ I3 @ I )
% 0.34/0.59 => ( ord_le3221252021190050221t_unit @ C @ ( F @ I3 ) ) )
% 0.34/0.59 => ( ( ( comple4687483117567863418t_unit @ ( image_o_Product_unit2 @ F @ I ) )
% 0.34/0.59 = C )
% 0.34/0.59 = ( ! [X2: $o] :
% 0.34/0.59 ( ( member_o @ X2 @ I )
% 0.34/0.59 => ( ( F @ X2 )
% 0.34/0.59 = C ) ) ) ) ) ) ).
% 0.34/0.59
% 0.34/0.59 % SUP_eq_iff
% 0.34/0.59 thf(fact_3277_SUP__eq__iff,axiom,
% 0.34/0.59 ! [I: set_nat,C: product_unit,F: nat > product_unit] :
% 0.34/0.59 ( ( I != bot_bot_set_nat )
% 0.34/0.59 => ( ! [I3: nat] :
% 0.34/0.59 ( ( member_nat @ I3 @ I )
% 0.34/0.59 => ( ord_le3221252021190050221t_unit @ C @ ( F @ I3 ) ) )
% 0.34/0.59 => ( ( ( comple4687483117567863418t_unit @ ( image_8730104196221521654t_unit @ F @ I ) )
% 0.34/0.59 = C )
% 0.34/0.59 = ( ! [X2: nat] :
% 0.34/0.59 ( ( member_nat @ X2 @ I )
% 0.34/0.59 => ( ( F @ X2 )
% 0.34/0.59 = C ) ) ) ) ) ) ).
% 0.34/0.59
% 0.34/0.59 % SUP_eq_iff
% 0.34/0.59 thf(fact_3278_SUP__eq__iff,axiom,
% 0.34/0.59 ! [I: set_int,C: product_unit,F: int > product_unit] :
% 0.34/0.59 ( ( I != bot_bot_set_int )
% 0.34/0.59 => ( ! [I3: int] :
% 0.34/0.59 ( ( member_int @ I3 @ I )
% 0.34/0.59 => ( ord_le3221252021190050221t_unit @ C @ ( F @ I3 ) ) )
% 0.34/0.59 => ( ( ( comple4687483117567863418t_unit @ ( image_5846526143423701914t_unit @ F @ I ) )
% 0.34/0.59 = C )
% 0.34/0.59 = ( ! [X2: int] :
% 0.34/0.59 ( ( member_int @ X2 @ I )
% 0.34/0.59 => ( ( F @ X2 )
% 0.34/0.59 = C ) ) ) ) ) ) ).
% 0.34/0.59
% 0.34/0.59 % SUP_eq_iff
% 0.34/0.59 thf(fact_3279_SUP__eq__iff,axiom,
% 0.34/0.59 ! [I: set_Product_unit,C: filter_nat,F: product_unit > filter_nat] :
% 0.34/0.59 ( ( I != bot_bo3957492148770167129t_unit )
% 0.34/0.59 => ( ! [I3: product_unit] :
% 0.34/0.59 ( ( member_Product_unit @ I3 @ I )
% 0.34/0.59 => ( ord_le2510731241096832064er_nat @ C @ ( F @ I3 ) ) )
% 0.34/0.59 => ( ( ( comple2454491259644266675er_nat @ ( image_1853478764106424488er_nat @ F @ I ) )
% 0.34/0.59 = C )
% 0.34/0.59 = ( ! [X2: product_unit] :
% 0.34/0.59 ( ( member_Product_unit @ X2 @ I )
% 0.34/0.59 => ( ( F @ X2 )
% 0.34/0.59 = C ) ) ) ) ) ) ).
% 0.34/0.59
% 0.34/0.59 % SUP_eq_iff
% 0.34/0.59 thf(fact_3280_SUP__eq__iff,axiom,
% 0.34/0.59 ! [I: set_o,C: filter_nat,F: $o > filter_nat] :
% 0.34/0.59 ( ( I != bot_bot_set_o )
% 0.34/0.59 => ( ! [I3: $o] :
% 0.34/0.59 ( ( member_o @ I3 @ I )
% 0.34/0.59 => ( ord_le2510731241096832064er_nat @ C @ ( F @ I3 ) ) )
% 0.34/0.59 => ( ( ( comple2454491259644266675er_nat @ ( image_o_filter_nat @ F @ I ) )
% 0.34/0.59 = C )
% 0.34/0.59 = ( ! [X2: $o] :
% 0.34/0.59 ( ( member_o @ X2 @ I )
% 0.34/0.59 => ( ( F @ X2 )
% 0.34/0.59 = C ) ) ) ) ) ) ).
% 0.34/0.59
% 0.34/0.59 % SUP_eq_iff
% 0.34/0.59 thf(fact_3281_SUP__eq__iff,axiom,
% 0.34/0.59 ! [I: set_nat,C: filter_nat,F: nat > filter_nat] :
% 0.34/0.59 ( ( I != bot_bot_set_nat )
% 0.34/0.59 => ( ! [I3: nat] :
% 0.34/0.59 ( ( member_nat @ I3 @ I )
% 0.34/0.59 => ( ord_le2510731241096832064er_nat @ C @ ( F @ I3 ) ) )
% 0.34/0.59 => ( ( ( comple2454491259644266675er_nat @ ( image_nat_filter_nat @ F @ I ) )
% 0.34/0.59 = C )
% 0.34/0.59 = ( ! [X2: nat] :
% 0.34/0.59 ( ( member_nat @ X2 @ I )
% 0.34/0.59 => ( ( F @ X2 )
% 0.34/0.59 = C ) ) ) ) ) ) ).
% 0.34/0.59
% 0.34/0.59 % SUP_eq_iff
% 0.34/0.59 thf(fact_3282_SUP__eq__iff,axiom,
% 0.34/0.59 ! [I: set_int,C: filter_nat,F: int > filter_nat] :
% 0.34/0.59 ( ( I != bot_bot_set_int )
% 0.34/0.59 => ( ! [I3: int] :
% 0.34/0.59 ( ( member_int @ I3 @ I )
% 0.34/0.59 => ( ord_le2510731241096832064er_nat @ C @ ( F @ I3 ) ) )
% 0.34/0.59 => ( ( ( comple2454491259644266675er_nat @ ( image_int_filter_nat @ F @ I ) )
% 0.34/0.59 = C )
% 0.34/0.59 = ( ! [X2: int] :
% 0.34/0.59 ( ( member_int @ X2 @ I )
% 0.34/0.59 => ( ( F @ X2 )
% 0.34/0.59 = C ) ) ) ) ) ) ).
% 0.34/0.59
% 0.34/0.59 % SUP_eq_iff
% 0.34/0.59 thf(fact_3283_SUP__eq__iff,axiom,
% 0.34/0.59 ! [I: set_Product_unit,C: set_nat,F: product_unit > set_nat] :
% 0.34/0.59 ( ( I != bot_bo3957492148770167129t_unit )
% 0.34/0.59 => ( ! [I3: product_unit] :
% 0.34/0.59 ( ( member_Product_unit @ I3 @ I )
% 0.34/0.59 => ( ord_less_eq_set_nat @ C @ ( F @ I3 ) ) )
% 0.34/0.59 => ( ( ( comple7399068483239264473et_nat @ ( image_8173849991827986510et_nat @ F @ I ) )
% 0.34/0.59 = C )
% 0.34/0.59 = ( ! [X2: product_unit] :
% 0.34/0.59 ( ( member_Product_unit @ X2 @ I )
% 0.34/0.59 => ( ( F @ X2 )
% 0.34/0.59 = C ) ) ) ) ) ) ).
% 0.34/0.59
% 0.34/0.59 % SUP_eq_iff
% 0.34/0.59 thf(fact_3284_SUP__eq__iff,axiom,
% 0.34/0.59 ! [I: set_o,C: set_nat,F: $o > set_nat] :
% 0.34/0.59 ( ( I != bot_bot_set_o )
% 0.34/0.59 => ( ! [I3: $o] :
% 0.34/0.59 ( ( member_o @ I3 @ I )
% 0.34/0.59 => ( ord_less_eq_set_nat @ C @ ( F @ I3 ) ) )
% 0.34/0.59 => ( ( ( comple7399068483239264473et_nat @ ( image_o_set_nat @ F @ I ) )
% 0.34/0.59 = C )
% 0.34/0.59 = ( ! [X2: $o] :
% 0.34/0.59 ( ( member_o @ X2 @ I )
% 0.34/0.59 => ( ( F @ X2 )
% 0.34/0.59 = C ) ) ) ) ) ) ).
% 0.34/0.59
% 0.34/0.59 % SUP_eq_iff
% 0.34/0.59 thf(fact_3285_cSUP__least,axiom,
% 0.34/0.59 ! [A: set_Product_unit,F: product_unit > int,M: int] :
% 0.34/0.59 ( ( A != bot_bo3957492148770167129t_unit )
% 0.34/0.59 => ( ! [X3: product_unit] :
% 0.34/0.59 ( ( member_Product_unit @ X3 @ A )
% 0.34/0.59 => ( ord_less_eq_int @ ( F @ X3 ) @ M ) )
% 0.34/0.59 => ( ord_less_eq_int @ ( complete_Sup_Sup_int @ ( image_873079544045703924it_int @ F @ A ) ) @ M ) ) ) ).
% 0.34/0.59
% 0.34/0.59 % cSUP_least
% 0.34/0.59 thf(fact_3286_cSUP__least,axiom,
% 0.34/0.59 ! [A: set_o,F: $o > int,M: int] :
% 0.34/0.59 ( ( A != bot_bot_set_o )
% 0.34/0.59 => ( ! [X3: $o] :
% 0.34/0.59 ( ( member_o @ X3 @ A )
% 0.34/0.59 => ( ord_less_eq_int @ ( F @ X3 ) @ M ) )
% 0.34/0.59 => ( ord_less_eq_int @ ( complete_Sup_Sup_int @ ( image_o_int2 @ F @ A ) ) @ M ) ) ) ).
% 0.34/0.59
% 0.34/0.59 % cSUP_least
% 0.34/0.59 thf(fact_3287_cSUP__least,axiom,
% 0.34/0.59 ! [A: set_nat,F: nat > int,M: int] :
% 0.34/0.59 ( ( A != bot_bot_set_nat )
% 0.34/0.59 => ( ! [X3: nat] :
% 0.34/0.59 ( ( member_nat @ X3 @ A )
% 0.34/0.59 => ( ord_less_eq_int @ ( F @ X3 ) @ M ) )
% 0.34/0.59 => ( ord_less_eq_int @ ( complete_Sup_Sup_int @ ( image_nat_int2 @ F @ A ) ) @ M ) ) ) ).
% 0.34/0.59
% 0.34/0.59 % cSUP_least
% 0.34/0.59 thf(fact_3288_cSUP__least,axiom,
% 0.34/0.59 ! [A: set_int,F: int > int,M: int] :
% 0.34/0.59 ( ( A != bot_bot_set_int )
% 0.34/0.59 => ( ! [X3: int] :
% 0.34/0.59 ( ( member_int @ X3 @ A )
% 0.34/0.59 => ( ord_less_eq_int @ ( F @ X3 ) @ M ) )
% 0.34/0.59 => ( ord_less_eq_int @ ( complete_Sup_Sup_int @ ( image_int_int2 @ F @ A ) ) @ M ) ) ) ).
% 0.34/0.59
% 0.34/0.59 % cSUP_least
% 0.34/0.59 thf(fact_3289_cSUP__least,axiom,
% 0.34/0.59 ! [A: set_Product_unit,F: product_unit > nat,M: nat] :
% 0.34/0.59 ( ( A != bot_bo3957492148770167129t_unit )
% 0.34/0.59 => ( ! [X3: product_unit] :
% 0.34/0.59 ( ( member_Product_unit @ X3 @ A )
% 0.34/0.59 => ( ord_less_eq_nat @ ( F @ X3 ) @ M ) )
% 0.34/0.59 => ( ord_less_eq_nat @ ( complete_Sup_Sup_nat @ ( image_875570014554754200it_nat @ F @ A ) ) @ M ) ) ) ).
% 0.34/0.59
% 0.34/0.59 % cSUP_least
% 0.34/0.59 thf(fact_3290_cSUP__least,axiom,
% 0.34/0.59 ! [A: set_o,F: $o > nat,M: nat] :
% 0.34/0.59 ( ( A != bot_bot_set_o )
% 0.34/0.59 => ( ! [X3: $o] :
% 0.34/0.59 ( ( member_o @ X3 @ A )
% 0.34/0.59 => ( ord_less_eq_nat @ ( F @ X3 ) @ M ) )
% 0.34/0.59 => ( ord_less_eq_nat @ ( complete_Sup_Sup_nat @ ( image_o_nat2 @ F @ A ) ) @ M ) ) ) ).
% 0.34/0.59
% 0.34/0.59 % cSUP_least
% 0.34/0.59 thf(fact_3291_cSUP__least,axiom,
% 0.34/0.59 ! [A: set_nat,F: nat > nat,M: nat] :
% 0.34/0.59 ( ( A != bot_bot_set_nat )
% 0.34/0.59 => ( ! [X3: nat] :
% 0.34/0.59 ( ( member_nat @ X3 @ A )
% 0.34/0.59 => ( ord_less_eq_nat @ ( F @ X3 ) @ M ) )
% 0.34/0.59 => ( ord_less_eq_nat @ ( complete_Sup_Sup_nat @ ( image_nat_nat2 @ F @ A ) ) @ M ) ) ) ).
% 0.34/0.59
% 0.34/0.59 % cSUP_least
% 0.34/0.59 thf(fact_3292_cSUP__least,axiom,
% 0.34/0.59 ! [A: set_int,F: int > nat,M: nat] :
% 0.34/0.59 ( ( A != bot_bot_set_int )
% 0.34/0.59 => ( ! [X3: int] :
% 0.34/0.59 ( ( member_int @ X3 @ A )
% 0.34/0.59 => ( ord_less_eq_nat @ ( F @ X3 ) @ M ) )
% 0.34/0.59 => ( ord_less_eq_nat @ ( complete_Sup_Sup_nat @ ( image_int_nat2 @ F @ A ) ) @ M ) ) ) ).
% 0.34/0.59
% 0.34/0.59 % cSUP_least
% 0.34/0.59 thf(fact_3293_cSUP__least,axiom,
% 0.34/0.59 ! [A: set_Product_unit,F: product_unit > product_unit,M: product_unit] :
% 0.34/0.59 ( ( A != bot_bo3957492148770167129t_unit )
% 0.34/0.59 => ( ! [X3: product_unit] :
% 0.34/0.59 ( ( member_Product_unit @ X3 @ A )
% 0.34/0.59 => ( ord_le3221252021190050221t_unit @ ( F @ X3 ) @ M ) )
% 0.34/0.59 => ( ord_le3221252021190050221t_unit @ ( comple4687483117567863418t_unit @ ( image_405062704495631173t_unit @ F @ A ) ) @ M ) ) ) ).
% 0.34/0.59
% 0.34/0.59 % cSUP_least
% 0.34/0.59 thf(fact_3294_cSUP__least,axiom,
% 0.34/0.59 ! [A: set_o,F: $o > product_unit,M: product_unit] :
% 0.34/0.59 ( ( A != bot_bot_set_o )
% 0.34/0.59 => ( ! [X3: $o] :
% 0.34/0.59 ( ( member_o @ X3 @ A )
% 0.34/0.59 => ( ord_le3221252021190050221t_unit @ ( F @ X3 ) @ M ) )
% 0.34/0.59 => ( ord_le3221252021190050221t_unit @ ( comple4687483117567863418t_unit @ ( image_o_Product_unit2 @ F @ A ) ) @ M ) ) ) ).
% 0.34/0.59
% 0.34/0.59 % cSUP_least
% 0.34/0.59 thf(fact_3295_Sup__inf__eq__bot__iff,axiom,
% 0.34/0.59 ! [B4: set_set_Product_unit,A2: set_Product_unit] :
% 0.34/0.59 ( ( ( inf_in4660618365625256667t_unit @ ( comple3349607820504768346t_unit @ B4 ) @ A2 )
% 0.34/0.59 = bot_bo3957492148770167129t_unit )
% 0.34/0.59 = ( ! [X2: set_Product_unit] :
% 0.34/0.59 ( ( member5877623283571906838t_unit @ X2 @ B4 )
% 0.34/0.59 => ( ( inf_in4660618365625256667t_unit @ X2 @ A2 )
% 0.34/0.59 = bot_bo3957492148770167129t_unit ) ) ) ) ).
% 0.34/0.59
% 0.34/0.59 % Sup_inf_eq_bot_iff
% 0.34/0.59 thf(fact_3296_Sup__inf__eq__bot__iff,axiom,
% 0.34/0.59 ! [B4: set_set_o,A2: set_o] :
% 0.34/0.59 ( ( ( inf_inf_set_o @ ( comple90263536869209701_set_o @ B4 ) @ A2 )
% 0.34/0.59 = bot_bot_set_o )
% 0.34/0.59 = ( ! [X2: set_o] :
% 0.34/0.59 ( ( member_set_o @ X2 @ B4 )
% 0.34/0.59 => ( ( inf_inf_set_o @ X2 @ A2 )
% 0.34/0.59 = bot_bot_set_o ) ) ) ) ).
% 0.34/0.59
% 0.34/0.59 % Sup_inf_eq_bot_iff
% 0.34/0.59 thf(fact_3297_Sup__inf__eq__bot__iff,axiom,
% 0.34/0.59 ! [B4: set_set_nat,A2: set_nat] :
% 0.34/0.59 ( ( ( inf_inf_set_nat @ ( comple7399068483239264473et_nat @ B4 ) @ A2 )
% 0.34/0.59 = bot_bot_set_nat )
% 0.34/0.59 = ( ! [X2: set_nat] :
% 0.34/0.59 ( ( member_set_nat @ X2 @ B4 )
% 0.34/0.59 => ( ( inf_inf_set_nat @ X2 @ A2 )
% 0.34/0.59 = bot_bot_set_nat ) ) ) ) ).
% 0.34/0.59
% 0.34/0.59 % Sup_inf_eq_bot_iff
% 0.34/0.59 thf(fact_3298_Sup__inf__eq__bot__iff,axiom,
% 0.34/0.59 ! [B4: set_set_int,A2: set_int] :
% 0.34/0.59 ( ( ( inf_inf_set_int @ ( comple3221217463730067765et_int @ B4 ) @ A2 )
% 0.34/0.59 = bot_bot_set_int )
% 0.34/0.59 = ( ! [X2: set_int] :
% 0.34/0.59 ( ( member_set_int @ X2 @ B4 )
% 0.34/0.59 => ( ( inf_inf_set_int @ X2 @ A2 )
% 0.34/0.59 = bot_bot_set_int ) ) ) ) ).
% 0.34/0.59
% 0.34/0.59 % Sup_inf_eq_bot_iff
% 0.34/0.59 thf(fact_3299_Sup__inf__eq__bot__iff,axiom,
% 0.34/0.59 ! [B4: set_se1353816476878259085_nat_o,A2: set_Pr4532377907799695533_nat_o] :
% 0.34/0.59 ( ( ( inf_in1906310914598751387_nat_o @ ( comple3382087578465172250_nat_o @ B4 ) @ A2 )
% 0.34/0.59 = bot_bo7824918357723582233_nat_o )
% 0.34/0.59 = ( ! [X2: set_Pr4532377907799695533_nat_o] :
% 0.34/0.59 ( ( member1452989533377600726_nat_o @ X2 @ B4 )
% 0.34/0.59 => ( ( inf_in1906310914598751387_nat_o @ X2 @ A2 )
% 0.34/0.59 = bot_bo7824918357723582233_nat_o ) ) ) ) ).
% 0.34/0.59
% 0.34/0.59 % Sup_inf_eq_bot_iff
% 0.34/0.59 thf(fact_3300_Sup__inf__eq__bot__iff,axiom,
% 0.34/0.59 ! [B4: set_Product_unit,A2: product_unit] :
% 0.34/0.59 ( ( ( inf_inf_Product_unit @ ( comple4687483117567863418t_unit @ B4 ) @ A2 )
% 0.34/0.59 = bot_bot_Product_unit )
% 0.34/0.59 = ( ! [X2: product_unit] :
% 0.34/0.59 ( ( member_Product_unit @ X2 @ B4 )
% 0.34/0.59 => ( ( inf_inf_Product_unit @ X2 @ A2 )
% 0.34/0.59 = bot_bot_Product_unit ) ) ) ) ).
% 0.34/0.59
% 0.34/0.59 % Sup_inf_eq_bot_iff
% 0.34/0.59 thf(fact_3301_UNION__fun__upd,axiom,
% 0.34/0.59 ! [A: product_unit > set_Product_unit,I2: product_unit,B4: set_Product_unit,J2: set_Product_unit] :
% 0.34/0.59 ( ( comple3349607820504768346t_unit @ ( image_7778121751606633509t_unit @ ( fun_up488859312337516701t_unit @ A @ I2 @ B4 ) @ J2 ) )
% 0.34/0.59 = ( sup_su793286257634532545t_unit @ ( comple3349607820504768346t_unit @ ( image_7778121751606633509t_unit @ A @ ( minus_6452836326544984404t_unit @ J2 @ ( insert_Product_unit @ I2 @ bot_bo3957492148770167129t_unit ) ) ) ) @ ( if_set_Product_unit @ ( member_Product_unit @ I2 @ J2 ) @ B4 @ bot_bo3957492148770167129t_unit ) ) ) ).
% 0.34/0.59
% 0.34/0.59 % UNION_fun_upd
% 0.34/0.59 thf(fact_3302_UNION__fun__upd,axiom,
% 0.34/0.59 ! [A: product_unit > set_o,I2: product_unit,B4: set_o,J2: set_Product_unit] :
% 0.34/0.59 ( ( comple90263536869209701_set_o @ ( image_6930536745178679216_set_o @ ( fun_up1762901246852280360_set_o @ A @ I2 @ B4 ) @ J2 ) )
% 0.34/0.59 = ( sup_sup_set_o @ ( comple90263536869209701_set_o @ ( image_6930536745178679216_set_o @ A @ ( minus_6452836326544984404t_unit @ J2 @ ( insert_Product_unit @ I2 @ bot_bo3957492148770167129t_unit ) ) ) ) @ ( if_set_o @ ( member_Product_unit @ I2 @ J2 ) @ B4 @ bot_bot_set_o ) ) ) ).
% 0.34/0.59
% 0.34/0.59 % UNION_fun_upd
% 0.34/0.59 thf(fact_3303_UNION__fun__upd,axiom,
% 0.34/0.59 ! [A: product_unit > set_nat,I2: product_unit,B4: set_nat,J2: set_Product_unit] :
% 0.34/0.59 ( ( comple7399068483239264473et_nat @ ( image_8173849991827986510et_nat @ ( fun_up4569834200710200790et_nat @ A @ I2 @ B4 ) @ J2 ) )
% 0.34/0.59 = ( sup_sup_set_nat @ ( comple7399068483239264473et_nat @ ( image_8173849991827986510et_nat @ A @ ( minus_6452836326544984404t_unit @ J2 @ ( insert_Product_unit @ I2 @ bot_bo3957492148770167129t_unit ) ) ) ) @ ( if_set_nat @ ( member_Product_unit @ I2 @ J2 ) @ B4 @ bot_bot_set_nat ) ) ) ).
% 0.34/0.59
% 0.34/0.59 % UNION_fun_upd
% 0.34/0.59 thf(fact_3304_UNION__fun__upd,axiom,
% 0.34/0.59 ! [A: product_unit > set_int,I2: product_unit,B4: set_int,J2: set_Product_unit] :
% 0.34/0.59 ( ( comple3221217463730067765et_int @ ( image_3995998972318789802et_int @ ( fun_up391983181201004082et_int @ A @ I2 @ B4 ) @ J2 ) )
% 0.34/0.59 = ( sup_sup_set_int @ ( comple3221217463730067765et_int @ ( image_3995998972318789802et_int @ A @ ( minus_6452836326544984404t_unit @ J2 @ ( insert_Product_unit @ I2 @ bot_bo3957492148770167129t_unit ) ) ) ) @ ( if_set_int @ ( member_Product_unit @ I2 @ J2 ) @ B4 @ bot_bot_set_int ) ) ) ).
% 0.34/0.59
% 0.34/0.59 % UNION_fun_upd
% 0.34/0.59 thf(fact_3305_UNION__fun__upd,axiom,
% 0.34/0.59 ! [A: $o > set_Product_unit,I2: $o,B4: set_Product_unit,J2: set_o] :
% 0.34/0.59 ( ( comple3349607820504768346t_unit @ ( image_663195431845683034t_unit @ ( fun_up4718931970374059986t_unit @ A @ I2 @ B4 ) @ J2 ) )
% 0.34/0.59 = ( sup_su793286257634532545t_unit @ ( comple3349607820504768346t_unit @ ( image_663195431845683034t_unit @ A @ ( minus_minus_set_o @ J2 @ ( insert_o @ I2 @ bot_bot_set_o ) ) ) ) @ ( if_set_Product_unit @ ( member_o @ I2 @ J2 ) @ B4 @ bot_bo3957492148770167129t_unit ) ) ) ).
% 0.34/0.59
% 0.34/0.59 % UNION_fun_upd
% 0.34/0.59 thf(fact_3306_UNION__fun__upd,axiom,
% 0.34/0.59 ! [A: $o > set_o,I2: $o,B4: set_o,J2: set_o] :
% 0.34/0.59 ( ( comple90263536869209701_set_o @ ( image_o_set_o @ ( fun_upd_o_set_o @ A @ I2 @ B4 ) @ J2 ) )
% 0.34/0.59 = ( sup_sup_set_o @ ( comple90263536869209701_set_o @ ( image_o_set_o @ A @ ( minus_minus_set_o @ J2 @ ( insert_o @ I2 @ bot_bot_set_o ) ) ) ) @ ( if_set_o @ ( member_o @ I2 @ J2 ) @ B4 @ bot_bot_set_o ) ) ) ).
% 0.34/0.59
% 0.34/0.59 % UNION_fun_upd
% 0.34/0.59 thf(fact_3307_UNION__fun__upd,axiom,
% 0.34/0.59 ! [A: $o > set_nat,I2: $o,B4: set_nat,J2: set_o] :
% 0.34/0.59 ( ( comple7399068483239264473et_nat @ ( image_o_set_nat @ ( fun_upd_o_set_nat @ A @ I2 @ B4 ) @ J2 ) )
% 0.34/0.59 = ( sup_sup_set_nat @ ( comple7399068483239264473et_nat @ ( image_o_set_nat @ A @ ( minus_minus_set_o @ J2 @ ( insert_o @ I2 @ bot_bot_set_o ) ) ) ) @ ( if_set_nat @ ( member_o @ I2 @ J2 ) @ B4 @ bot_bot_set_nat ) ) ) ).
% 0.34/0.59
% 0.34/0.59 % UNION_fun_upd
% 0.34/0.59 thf(fact_3308_UNION__fun__upd,axiom,
% 0.34/0.59 ! [A: $o > set_int,I2: $o,B4: set_int,J2: set_o] :
% 0.34/0.59 ( ( comple3221217463730067765et_int @ ( image_o_set_int @ ( fun_upd_o_set_int @ A @ I2 @ B4 ) @ J2 ) )
% 0.34/0.59 = ( sup_sup_set_int @ ( comple3221217463730067765et_int @ ( image_o_set_int @ A @ ( minus_minus_set_o @ J2 @ ( insert_o @ I2 @ bot_bot_set_o ) ) ) ) @ ( if_set_int @ ( member_o @ I2 @ J2 ) @ B4 @ bot_bot_set_int ) ) ) ).
% 0.34/0.59
% 0.34/0.59 % UNION_fun_upd
% 0.34/0.59 thf(fact_3309_UNION__fun__upd,axiom,
% 0.34/0.59 ! [A: int > set_Product_unit,I2: int,B4: set_Product_unit,J2: set_int] :
% 0.34/0.59 ( ( comple3349607820504768346t_unit @ ( image_8658272795163236474t_unit @ ( fun_up5054257004045450754t_unit @ A @ I2 @ B4 ) @ J2 ) )
% 0.34/0.59 = ( sup_su793286257634532545t_unit @ ( comple3349607820504768346t_unit @ ( image_8658272795163236474t_unit @ A @ ( minus_minus_set_int @ J2 @ ( insert_int @ I2 @ bot_bot_set_int ) ) ) ) @ ( if_set_Product_unit @ ( member_int @ I2 @ J2 ) @ B4 @ bot_bo3957492148770167129t_unit ) ) ) ).
% 0.34/0.59
% 0.34/0.59 % UNION_fun_upd
% 0.34/0.59 thf(fact_3310_UNION__fun__upd,axiom,
% 0.34/0.59 ! [A: int > set_o,I2: int,B4: set_o,J2: set_int] :
% 0.34/0.59 ( ( comple90263536869209701_set_o @ ( image_int_set_o @ ( fun_upd_int_set_o @ A @ I2 @ B4 ) @ J2 ) )
% 0.34/0.59 = ( sup_sup_set_o @ ( comple90263536869209701_set_o @ ( image_int_set_o @ A @ ( minus_minus_set_int @ J2 @ ( insert_int @ I2 @ bot_bot_set_int ) ) ) ) @ ( if_set_o @ ( member_int @ I2 @ J2 ) @ B4 @ bot_bot_set_o ) ) ) ).
% 0.34/0.59
% 0.34/0.59 % UNION_fun_upd
% 0.34/0.59 thf(fact_3311_cSUP__union,axiom,
% 0.34/0.59 ! [A: set_Product_unit,F: product_unit > nat,B4: set_Product_unit] :
% 0.34/0.59 ( ( A != bot_bo3957492148770167129t_unit )
% 0.34/0.59 => ( ( condit2214826472909112428ve_nat @ ( image_875570014554754200it_nat @ F @ A ) )
% 0.34/0.59 => ( ( B4 != bot_bo3957492148770167129t_unit )
% 0.34/0.59 => ( ( condit2214826472909112428ve_nat @ ( image_875570014554754200it_nat @ F @ B4 ) )
% 0.34/0.59 => ( ( complete_Sup_Sup_nat @ ( image_875570014554754200it_nat @ F @ ( sup_su793286257634532545t_unit @ A @ B4 ) ) )
% 0.34/0.59 = ( sup_sup_nat @ ( complete_Sup_Sup_nat @ ( image_875570014554754200it_nat @ F @ A ) ) @ ( complete_Sup_Sup_nat @ ( image_875570014554754200it_nat @ F @ B4 ) ) ) ) ) ) ) ) ).
% 0.34/0.59
% 0.34/0.59 % cSUP_union
% 0.34/0.59 thf(fact_3312_cSUP__union,axiom,
% 0.34/0.59 ! [A: set_o,F: $o > nat,B4: set_o] :
% 0.34/0.59 ( ( A != bot_bot_set_o )
% 0.34/0.59 => ( ( condit2214826472909112428ve_nat @ ( image_o_nat2 @ F @ A ) )
% 0.34/0.59 => ( ( B4 != bot_bot_set_o )
% 0.34/0.59 => ( ( condit2214826472909112428ve_nat @ ( image_o_nat2 @ F @ B4 ) )
% 0.34/0.59 => ( ( complete_Sup_Sup_nat @ ( image_o_nat2 @ F @ ( sup_sup_set_o @ A @ B4 ) ) )
% 0.34/0.59 = ( sup_sup_nat @ ( complete_Sup_Sup_nat @ ( image_o_nat2 @ F @ A ) ) @ ( complete_Sup_Sup_nat @ ( image_o_nat2 @ F @ B4 ) ) ) ) ) ) ) ) ).
% 0.34/0.59
% 0.34/0.59 % cSUP_union
% 0.34/0.59 thf(fact_3313_cSUP__union,axiom,
% 0.34/0.59 ! [A: set_nat,F: nat > nat,B4: set_nat] :
% 0.34/0.59 ( ( A != bot_bot_set_nat )
% 0.34/0.59 => ( ( condit2214826472909112428ve_nat @ ( image_nat_nat2 @ F @ A ) )
% 0.34/0.59 => ( ( B4 != bot_bot_set_nat )
% 0.34/0.59 => ( ( condit2214826472909112428ve_nat @ ( image_nat_nat2 @ F @ B4 ) )
% 0.34/0.59 => ( ( complete_Sup_Sup_nat @ ( image_nat_nat2 @ F @ ( sup_sup_set_nat @ A @ B4 ) ) )
% 0.34/0.59 = ( sup_sup_nat @ ( complete_Sup_Sup_nat @ ( image_nat_nat2 @ F @ A ) ) @ ( complete_Sup_Sup_nat @ ( image_nat_nat2 @ F @ B4 ) ) ) ) ) ) ) ) ).
% 0.34/0.59
% 0.34/0.59 % cSUP_union
% 0.34/0.59 thf(fact_3314_cSUP__union,axiom,
% 0.34/0.59 ! [A: set_int,F: int > nat,B4: set_int] :
% 0.34/0.59 ( ( A != bot_bot_set_int )
% 0.34/0.59 => ( ( condit2214826472909112428ve_nat @ ( image_int_nat2 @ F @ A ) )
% 0.34/0.59 => ( ( B4 != bot_bot_set_int )
% 0.34/0.59 => ( ( condit2214826472909112428ve_nat @ ( image_int_nat2 @ F @ B4 ) )
% 0.34/0.59 => ( ( complete_Sup_Sup_nat @ ( image_int_nat2 @ F @ ( sup_sup_set_int @ A @ B4 ) ) )
% 0.34/0.59 = ( sup_sup_nat @ ( complete_Sup_Sup_nat @ ( image_int_nat2 @ F @ A ) ) @ ( complete_Sup_Sup_nat @ ( image_int_nat2 @ F @ B4 ) ) ) ) ) ) ) ) ).
% 0.34/0.59
% 0.34/0.59 % cSUP_union
% 0.34/0.59 thf(fact_3315_cSUP__union,axiom,
% 0.34/0.59 ! [A: set_Product_unit,F: product_unit > product_unit,B4: set_Product_unit] :
% 0.34/0.59 ( ( A != bot_bo3957492148770167129t_unit )
% 0.34/0.59 => ( ( condit2080475194861589617t_unit @ ( image_405062704495631173t_unit @ F @ A ) )
% 0.34/0.59 => ( ( B4 != bot_bo3957492148770167129t_unit )
% 0.34/0.59 => ( ( condit2080475194861589617t_unit @ ( image_405062704495631173t_unit @ F @ B4 ) )
% 0.34/0.59 => ( ( comple4687483117567863418t_unit @ ( image_405062704495631173t_unit @ F @ ( sup_su793286257634532545t_unit @ A @ B4 ) ) )
% 0.34/0.59 = ( sup_sup_Product_unit @ ( comple4687483117567863418t_unit @ ( image_405062704495631173t_unit @ F @ A ) ) @ ( comple4687483117567863418t_unit @ ( image_405062704495631173t_unit @ F @ B4 ) ) ) ) ) ) ) ) ).
% 0.34/0.59
% 0.34/0.59 % cSUP_union
% 0.34/0.59 thf(fact_3316_cSUP__union,axiom,
% 0.34/0.59 ! [A: set_o,F: $o > product_unit,B4: set_o] :
% 0.34/0.59 ( ( A != bot_bot_set_o )
% 0.34/0.59 => ( ( condit2080475194861589617t_unit @ ( image_o_Product_unit2 @ F @ A ) )
% 0.34/0.59 => ( ( B4 != bot_bot_set_o )
% 0.34/0.59 => ( ( condit2080475194861589617t_unit @ ( image_o_Product_unit2 @ F @ B4 ) )
% 0.34/0.59 => ( ( comple4687483117567863418t_unit @ ( image_o_Product_unit2 @ F @ ( sup_sup_set_o @ A @ B4 ) ) )
% 0.34/0.59 = ( sup_sup_Product_unit @ ( comple4687483117567863418t_unit @ ( image_o_Product_unit2 @ F @ A ) ) @ ( comple4687483117567863418t_unit @ ( image_o_Product_unit2 @ F @ B4 ) ) ) ) ) ) ) ) ).
% 0.34/0.59
% 0.34/0.59 % cSUP_union
% 0.34/0.59 thf(fact_3317_cSUP__union,axiom,
% 0.34/0.59 ! [A: set_nat,F: nat > product_unit,B4: set_nat] :
% 0.34/0.59 ( ( A != bot_bot_set_nat )
% 0.34/0.59 => ( ( condit2080475194861589617t_unit @ ( image_8730104196221521654t_unit @ F @ A ) )
% 0.34/0.59 => ( ( B4 != bot_bot_set_nat )
% 0.34/0.59 => ( ( condit2080475194861589617t_unit @ ( image_8730104196221521654t_unit @ F @ B4 ) )
% 0.34/0.59 => ( ( comple4687483117567863418t_unit @ ( image_8730104196221521654t_unit @ F @ ( sup_sup_set_nat @ A @ B4 ) ) )
% 0.34/0.59 = ( sup_sup_Product_unit @ ( comple4687483117567863418t_unit @ ( image_8730104196221521654t_unit @ F @ A ) ) @ ( comple4687483117567863418t_unit @ ( image_8730104196221521654t_unit @ F @ B4 ) ) ) ) ) ) ) ) ).
% 0.34/0.59
% 0.34/0.59 % cSUP_union
% 0.34/0.59 thf(fact_3318_cSUP__union,axiom,
% 0.34/0.59 ! [A: set_int,F: int > product_unit,B4: set_int] :
% 0.34/0.59 ( ( A != bot_bot_set_int )
% 0.34/0.59 => ( ( condit2080475194861589617t_unit @ ( image_5846526143423701914t_unit @ F @ A ) )
% 0.34/0.59 => ( ( B4 != bot_bot_set_int )
% 0.34/0.59 => ( ( condit2080475194861589617t_unit @ ( image_5846526143423701914t_unit @ F @ B4 ) )
% 0.34/0.59 => ( ( comple4687483117567863418t_unit @ ( image_5846526143423701914t_unit @ F @ ( sup_sup_set_int @ A @ B4 ) ) )
% 0.34/0.59 = ( sup_sup_Product_unit @ ( comple4687483117567863418t_unit @ ( image_5846526143423701914t_unit @ F @ A ) ) @ ( comple4687483117567863418t_unit @ ( image_5846526143423701914t_unit @ F @ B4 ) ) ) ) ) ) ) ) ).
% 0.34/0.59
% 0.34/0.59 % cSUP_union
% 0.34/0.59 thf(fact_3319_cSUP__union,axiom,
% 0.34/0.59 ! [A: set_Product_unit,F: product_unit > set_nat,B4: set_Product_unit] :
% 0.34/0.59 ( ( A != bot_bo3957492148770167129t_unit )
% 0.34/0.59 => ( ( condit5477540289124974626et_nat @ ( image_8173849991827986510et_nat @ F @ A ) )
% 0.34/0.59 => ( ( B4 != bot_bo3957492148770167129t_unit )
% 0.34/0.59 => ( ( condit5477540289124974626et_nat @ ( image_8173849991827986510et_nat @ F @ B4 ) )
% 0.34/0.59 => ( ( comple7399068483239264473et_nat @ ( image_8173849991827986510et_nat @ F @ ( sup_su793286257634532545t_unit @ A @ B4 ) ) )
% 0.34/0.59 = ( sup_sup_set_nat @ ( comple7399068483239264473et_nat @ ( image_8173849991827986510et_nat @ F @ A ) ) @ ( comple7399068483239264473et_nat @ ( image_8173849991827986510et_nat @ F @ B4 ) ) ) ) ) ) ) ) ).
% 0.34/0.59
% 0.34/0.59 % cSUP_union
% 0.34/0.59 thf(fact_3320_cSUP__union,axiom,
% 0.34/0.59 ! [A: set_o,F: $o > set_nat,B4: set_o] :
% 0.34/0.59 ( ( A != bot_bot_set_o )
% 0.34/0.59 => ( ( condit5477540289124974626et_nat @ ( image_o_set_nat @ F @ A ) )
% 0.34/0.59 => ( ( B4 != bot_bot_set_o )
% 0.34/0.59 => ( ( condit5477540289124974626et_nat @ ( image_o_set_nat @ F @ B4 ) )
% 0.34/0.59 => ( ( comple7399068483239264473et_nat @ ( image_o_set_nat @ F @ ( sup_sup_set_o @ A @ B4 ) ) )
% 0.34/0.59 = ( sup_sup_set_nat @ ( comple7399068483239264473et_nat @ ( image_o_set_nat @ F @ A ) ) @ ( comple7399068483239264473et_nat @ ( image_o_set_nat @ F @ B4 ) ) ) ) ) ) ) ) ).
% 0.34/0.59
% 0.34/0.59 % cSUP_union
% 0.34/0.59 thf(fact_3321_pair__in__swap__image,axiom,
% 0.34/0.59 ! [Y: set_nat,X: heap_e7401611519738050253t_unit,A: set_Pr3948176798113811640et_nat] :
% 0.34/0.59 ( ( member727419437372278867t_unit @ ( produc2471974422597015918t_unit @ Y @ X ) @ ( image_1893317993577548229t_unit @ produc6300697153377358172et_nat @ A ) )
% 0.34/0.59 = ( member6260224972018164377et_nat @ ( produc7507926704131184380et_nat @ X @ Y ) @ A ) ) ).
% 0.34/0.59
% 0.34/0.59 % pair_in_swap_image
% 0.34/0.59 thf(fact_3322_pair__in__swap__image,axiom,
% 0.34/0.59 ! [Y: produc3658429121746597890et_nat,X: produc3658429121746597890et_nat > $o,A: set_Pr3286484037609594932et_nat] :
% 0.34/0.59 ( ( member4978015717948814295_nat_o @ ( produc2829238748666232050_nat_o @ Y @ X ) @ ( image_6229134628169569989_nat_o @ produc3251101204053596376et_nat @ A ) )
% 0.34/0.59 = ( member1996754912294343701et_nat @ ( produc5001842942810119800et_nat @ X @ Y ) @ A ) ) ).
% 0.34/0.59
% 0.34/0.59 % pair_in_swap_image
% 0.34/0.59 thf(fact_3323_pair__in__swap__image,axiom,
% 0.34/0.59 ! [Y: produc3925858234332021118et_nat,X: produc3658429121746597890et_nat > $o,A: set_Pr8536935166611901872et_nat] :
% 0.34/0.59 ( ( member8354763867512313179_nat_o @ ( produc3674526432113299062_nat_o @ Y @ X ) @ ( image_4071802983491761093_nat_o @ produc1151471490888544852et_nat @ A ) )
% 0.34/0.59 = ( member6124377750444531601et_nat @ ( produc2245416461498447860et_nat @ X @ Y ) @ A ) ) ).
% 0.34/0.59
% 0.34/0.59 % pair_in_swap_image
% 0.34/0.59 thf(fact_3324_pair__in__swap__image,axiom,
% 0.34/0.59 ! [Y: heap_e7401611519738050253t_unit,X: set_nat,A: set_Pr4071100604818519410t_unit] :
% 0.34/0.59 ( ( member6260224972018164377et_nat @ ( produc7507926704131184380et_nat @ Y @ X ) @ ( image_5838825082199883205et_nat @ produc1264744871843189710t_unit @ A ) )
% 0.34/0.59 = ( member727419437372278867t_unit @ ( produc2471974422597015918t_unit @ X @ Y ) @ A ) ) ).
% 0.34/0.59
% 0.34/0.59 % pair_in_swap_image
% 0.34/0.59 thf(fact_3325_pair__in__swap__image,axiom,
% 0.34/0.59 ! [Y: nat,X: nat,A: set_Pr1261947904930325089at_nat] :
% 0.34/0.59 ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ Y @ X ) @ ( image_5168914502847457605at_nat @ product_swap_nat_nat @ A ) )
% 0.34/0.59 = ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ Y ) @ A ) ) ).
% 0.34/0.59
% 0.34/0.59 % pair_in_swap_image
% 0.34/0.59 thf(fact_3326_pair__in__swap__image,axiom,
% 0.34/0.59 ! [Y: int,X: int,A: set_Pr958786334691620121nt_int] :
% 0.34/0.59 ( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ Y @ X ) @ ( image_2653370878348428101nt_int @ product_swap_int_int @ A ) )
% 0.34/0.59 = ( member5262025264175285858nt_int @ ( product_Pair_int_int @ X @ Y ) @ A ) ) ).
% 0.34/0.59
% 0.34/0.59 % pair_in_swap_image
% 0.34/0.59 thf(fact_3327_pair__in__swap__image,axiom,
% 0.34/0.59 ! [Y: produc3658429121746597890et_nat > $o,X: produc3658429121746597890et_nat,A: set_Pr4208978187861016310_nat_o] :
% 0.34/0.59 ( ( member1996754912294343701et_nat @ ( produc5001842942810119800et_nat @ Y @ X ) @ ( image_4501728140019976901et_nat @ produc1078497009909708626_nat_o @ A ) )
% 0.34/0.59 = ( member4978015717948814295_nat_o @ ( produc2829238748666232050_nat_o @ X @ Y ) @ A ) ) ).
% 0.34/0.59
% 0.34/0.59 % pair_in_swap_image
% 0.34/0.59 thf(fact_3328_pair__in__swap__image,axiom,
% 0.34/0.59 ! [Y: produc3658429121746597890et_nat > $o,X: produc3925858234332021118et_nat,A: set_Pr7705059561495379578_nat_o] :
% 0.34/0.59 ( ( member6124377750444531601et_nat @ ( produc2245416461498447860et_nat @ Y @ X ) @ ( image_7255662321850427333et_nat @ produc2580581461503396054_nat_o @ A ) )
% 0.34/0.59 = ( member8354763867512313179_nat_o @ ( produc3674526432113299062_nat_o @ X @ Y ) @ A ) ) ).
% 0.34/0.59
% 0.34/0.59 % pair_in_swap_image
% 0.34/0.59 thf(fact_3329_cSUP__insert,axiom,
% 0.34/0.59 ! [A: set_Product_unit,F: product_unit > nat,A2: product_unit] :
% 0.34/0.59 ( ( A != bot_bo3957492148770167129t_unit )
% 0.34/0.59 => ( ( condit2214826472909112428ve_nat @ ( image_875570014554754200it_nat @ F @ A ) )
% 0.34/0.59 => ( ( complete_Sup_Sup_nat @ ( image_875570014554754200it_nat @ F @ ( insert_Product_unit @ A2 @ A ) ) )
% 0.34/0.59 = ( sup_sup_nat @ ( F @ A2 ) @ ( complete_Sup_Sup_nat @ ( image_875570014554754200it_nat @ F @ A ) ) ) ) ) ) ).
% 0.34/0.59
% 0.34/0.59 % cSUP_insert
% 0.34/0.59 thf(fact_3330_cSUP__insert,axiom,
% 0.34/0.59 ! [A: set_o,F: $o > nat,A2: $o] :
% 0.34/0.59 ( ( A != bot_bot_set_o )
% 0.34/0.59 => ( ( condit2214826472909112428ve_nat @ ( image_o_nat2 @ F @ A ) )
% 0.34/0.59 => ( ( complete_Sup_Sup_nat @ ( image_o_nat2 @ F @ ( insert_o @ A2 @ A ) ) )
% 0.34/0.59 = ( sup_sup_nat @ ( F @ A2 ) @ ( complete_Sup_Sup_nat @ ( image_o_nat2 @ F @ A ) ) ) ) ) ) ).
% 0.34/0.59
% 0.34/0.59 % cSUP_insert
% 0.34/0.59 thf(fact_3331_cSUP__insert,axiom,
% 0.34/0.59 ! [A: set_nat,F: nat > nat,A2: nat] :
% 0.34/0.59 ( ( A != bot_bot_set_nat )
% 0.34/0.59 => ( ( condit2214826472909112428ve_nat @ ( image_nat_nat2 @ F @ A ) )
% 0.34/0.59 => ( ( complete_Sup_Sup_nat @ ( image_nat_nat2 @ F @ ( insert_nat @ A2 @ A ) ) )
% 0.34/0.59 = ( sup_sup_nat @ ( F @ A2 ) @ ( complete_Sup_Sup_nat @ ( image_nat_nat2 @ F @ A ) ) ) ) ) ) ).
% 0.34/0.59
% 0.34/0.59 % cSUP_insert
% 0.34/0.59 thf(fact_3332_cSUP__insert,axiom,
% 0.34/0.59 ! [A: set_int,F: int > nat,A2: int] :
% 0.34/0.59 ( ( A != bot_bot_set_int )
% 0.34/0.59 => ( ( condit2214826472909112428ve_nat @ ( image_int_nat2 @ F @ A ) )
% 0.34/0.59 => ( ( complete_Sup_Sup_nat @ ( image_int_nat2 @ F @ ( insert_int @ A2 @ A ) ) )
% 0.34/0.59 = ( sup_sup_nat @ ( F @ A2 ) @ ( complete_Sup_Sup_nat @ ( image_int_nat2 @ F @ A ) ) ) ) ) ) ).
% 0.34/0.59
% 0.34/0.59 % cSUP_insert
% 0.34/0.59 thf(fact_3333_cSUP__insert,axiom,
% 0.34/0.59 ! [A: set_Product_unit,F: product_unit > product_unit,A2: product_unit] :
% 0.34/0.59 ( ( A != bot_bo3957492148770167129t_unit )
% 0.34/0.59 => ( ( condit2080475194861589617t_unit @ ( image_405062704495631173t_unit @ F @ A ) )
% 0.34/0.59 => ( ( comple4687483117567863418t_unit @ ( image_405062704495631173t_unit @ F @ ( insert_Product_unit @ A2 @ A ) ) )
% 0.34/0.59 = ( sup_sup_Product_unit @ ( F @ A2 ) @ ( comple4687483117567863418t_unit @ ( image_405062704495631173t_unit @ F @ A ) ) ) ) ) ) ).
% 0.34/0.59
% 0.34/0.59 % cSUP_insert
% 0.34/0.59 thf(fact_3334_cSUP__insert,axiom,
% 0.34/0.59 ! [A: set_o,F: $o > product_unit,A2: $o] :
% 0.34/0.59 ( ( A != bot_bot_set_o )
% 0.34/0.59 => ( ( condit2080475194861589617t_unit @ ( image_o_Product_unit2 @ F @ A ) )
% 0.34/0.59 => ( ( comple4687483117567863418t_unit @ ( image_o_Product_unit2 @ F @ ( insert_o @ A2 @ A ) ) )
% 0.34/0.59 = ( sup_sup_Product_unit @ ( F @ A2 ) @ ( comple4687483117567863418t_unit @ ( image_o_Product_unit2 @ F @ A ) ) ) ) ) ) ).
% 0.34/0.59
% 0.34/0.59 % cSUP_insert
% 0.34/0.59 thf(fact_3335_cSUP__insert,axiom,
% 0.34/0.59 ! [A: set_nat,F: nat > product_unit,A2: nat] :
% 0.34/0.59 ( ( A != bot_bot_set_nat )
% 0.34/0.59 => ( ( condit2080475194861589617t_unit @ ( image_8730104196221521654t_unit @ F @ A ) )
% 0.34/0.59 => ( ( comple4687483117567863418t_unit @ ( image_8730104196221521654t_unit @ F @ ( insert_nat @ A2 @ A ) ) )
% 0.34/0.59 = ( sup_sup_Product_unit @ ( F @ A2 ) @ ( comple4687483117567863418t_unit @ ( image_8730104196221521654t_unit @ F @ A ) ) ) ) ) ) ).
% 0.34/0.59
% 0.34/0.59 % cSUP_insert
% 0.34/0.59 thf(fact_3336_cSUP__insert,axiom,
% 0.34/0.59 ! [A: set_int,F: int > product_unit,A2: int] :
% 0.34/0.59 ( ( A != bot_bot_set_int )
% 0.34/0.59 => ( ( condit2080475194861589617t_unit @ ( image_5846526143423701914t_unit @ F @ A ) )
% 0.34/0.59 => ( ( comple4687483117567863418t_unit @ ( image_5846526143423701914t_unit @ F @ ( insert_int @ A2 @ A ) ) )
% 0.34/0.59 = ( sup_sup_Product_unit @ ( F @ A2 ) @ ( comple4687483117567863418t_unit @ ( image_5846526143423701914t_unit @ F @ A ) ) ) ) ) ) ).
% 0.34/0.59
% 0.34/0.59 % cSUP_insert
% 0.34/0.59 thf(fact_3337_cSUP__insert,axiom,
% 0.34/0.59 ! [A: set_Product_unit,F: product_unit > set_nat,A2: product_unit] :
% 0.34/0.59 ( ( A != bot_bo3957492148770167129t_unit )
% 0.34/0.59 => ( ( condit5477540289124974626et_nat @ ( image_8173849991827986510et_nat @ F @ A ) )
% 0.34/0.59 => ( ( comple7399068483239264473et_nat @ ( image_8173849991827986510et_nat @ F @ ( insert_Product_unit @ A2 @ A ) ) )
% 0.34/0.59 = ( sup_sup_set_nat @ ( F @ A2 ) @ ( comple7399068483239264473et_nat @ ( image_8173849991827986510et_nat @ F @ A ) ) ) ) ) ) ).
% 0.34/0.59
% 0.34/0.59 % cSUP_insert
% 0.34/0.59 thf(fact_3338_cSUP__insert,axiom,
% 0.34/0.59 ! [A: set_o,F: $o > set_nat,A2: $o] :
% 0.34/0.59 ( ( A != bot_bot_set_o )
% 0.34/0.59 => ( ( condit5477540289124974626et_nat @ ( image_o_set_nat @ F @ A ) )
% 0.34/0.59 => ( ( comple7399068483239264473et_nat @ ( image_o_set_nat @ F @ ( insert_o @ A2 @ A ) ) )
% 0.34/0.59 = ( sup_sup_set_nat @ ( F @ A2 ) @ ( comple7399068483239264473et_nat @ ( image_o_set_nat @ F @ A ) ) ) ) ) ) ).
% 0.34/0.59
% 0.34/0.59 % cSUP_insert
% 0.34/0.59 thf(fact_3339_cSup__inter__less__eq,axiom,
% 0.34/0.59 ! [A: set_o,B4: set_o] :
% 0.34/0.59 ( ( condit5488710616941104124bove_o @ A )
% 0.34/0.59 => ( ( condit5488710616941104124bove_o @ B4 )
% 0.34/0.59 => ( ( ( inf_inf_set_o @ A @ B4 )
% 0.34/0.59 != bot_bot_set_o )
% 0.34/0.59 => ( ord_less_eq_o @ ( complete_Sup_Sup_o @ ( inf_inf_set_o @ A @ B4 ) ) @ ( sup_sup_o @ ( complete_Sup_Sup_o @ A ) @ ( complete_Sup_Sup_o @ B4 ) ) ) ) ) ) ).
% 0.34/0.59
% 0.34/0.59 % cSup_inter_less_eq
% 0.34/0.59 thf(fact_3340_cSup__inter__less__eq,axiom,
% 0.34/0.59 ! [A: set_Pr4532377907799695533_nat_o,B4: set_Pr4532377907799695533_nat_o] :
% 0.34/0.59 ( ( condit4352657709669162033_nat_o @ A )
% 0.34/0.59 => ( ( condit4352657709669162033_nat_o @ B4 )
% 0.34/0.59 => ( ( ( inf_in1906310914598751387_nat_o @ A @ B4 )
% 0.34/0.59 != bot_bo7824918357723582233_nat_o )
% 0.34/0.59 => ( ord_le729326519192465773_nat_o @ ( comple4225350640907012154_nat_o @ ( inf_in1906310914598751387_nat_o @ A @ B4 ) ) @ ( sup_su5453871518329203617_nat_o @ ( comple4225350640907012154_nat_o @ A ) @ ( comple4225350640907012154_nat_o @ B4 ) ) ) ) ) ) ).
% 0.34/0.59
% 0.34/0.59 % cSup_inter_less_eq
% 0.34/0.59 thf(fact_3341_cSup__inter__less__eq,axiom,
% 0.34/0.59 ! [A: set_filter_nat,B4: set_filter_nat] :
% 0.34/0.59 ( ( condit4029664723990530684er_nat @ A )
% 0.34/0.59 => ( ( condit4029664723990530684er_nat @ B4 )
% 0.34/0.59 => ( ( ( inf_in7395780696554941096er_nat @ A @ B4 )
% 0.34/0.59 != bot_bo498966703094740906er_nat )
% 0.34/0.59 => ( ord_le2510731241096832064er_nat @ ( comple2454491259644266675er_nat @ ( inf_in7395780696554941096er_nat @ A @ B4 ) ) @ ( sup_sup_filter_nat @ ( comple2454491259644266675er_nat @ A ) @ ( comple2454491259644266675er_nat @ B4 ) ) ) ) ) ) ).
% 0.34/0.59
% 0.34/0.59 % cSup_inter_less_eq
% 0.34/0.59 thf(fact_3342_cSup__inter__less__eq,axiom,
% 0.34/0.59 ! [A: set_set_nat,B4: set_set_nat] :
% 0.34/0.59 ( ( condit5477540289124974626et_nat @ A )
% 0.34/0.59 => ( ( condit5477540289124974626et_nat @ B4 )
% 0.34/0.59 => ( ( ( inf_inf_set_set_nat @ A @ B4 )
% 0.34/0.59 != bot_bot_set_set_nat )
% 0.34/0.59 => ( ord_less_eq_set_nat @ ( comple7399068483239264473et_nat @ ( inf_inf_set_set_nat @ A @ B4 ) ) @ ( sup_sup_set_nat @ ( comple7399068483239264473et_nat @ A ) @ ( comple7399068483239264473et_nat @ B4 ) ) ) ) ) ) ).
% 0.34/0.59
% 0.34/0.59 % cSup_inter_less_eq
% 0.34/0.59 thf(fact_3343_cSup__inter__less__eq,axiom,
% 0.34/0.59 ! [A: set_int,B4: set_int] :
% 0.34/0.59 ( ( condit2212336002400062152ve_int @ A )
% 0.34/0.59 => ( ( condit2212336002400062152ve_int @ B4 )
% 0.34/0.59 => ( ( ( inf_inf_set_int @ A @ B4 )
% 0.34/0.59 != bot_bot_set_int )
% 0.34/0.59 => ( ord_less_eq_int @ ( complete_Sup_Sup_int @ ( inf_inf_set_int @ A @ B4 ) ) @ ( sup_sup_int @ ( complete_Sup_Sup_int @ A ) @ ( complete_Sup_Sup_int @ B4 ) ) ) ) ) ) ).
% 0.34/0.59
% 0.34/0.59 % cSup_inter_less_eq
% 0.34/0.59 thf(fact_3344_cSup__inter__less__eq,axiom,
% 0.34/0.59 ! [A: set_nat,B4: set_nat] :
% 0.34/0.59 ( ( condit2214826472909112428ve_nat @ A )
% 0.34/0.59 => ( ( condit2214826472909112428ve_nat @ B4 )
% 0.34/0.59 => ( ( ( inf_inf_set_nat @ A @ B4 )
% 0.34/0.59 != bot_bot_set_nat )
% 0.34/0.59 => ( ord_less_eq_nat @ ( complete_Sup_Sup_nat @ ( inf_inf_set_nat @ A @ B4 ) ) @ ( sup_sup_nat @ ( complete_Sup_Sup_nat @ A ) @ ( complete_Sup_Sup_nat @ B4 ) ) ) ) ) ) ).
% 0.34/0.59
% 0.34/0.59 % cSup_inter_less_eq
% 0.34/0.59 thf(fact_3345_cSup__inter__less__eq,axiom,
% 0.34/0.59 ! [A: set_Product_unit,B4: set_Product_unit] :
% 0.34/0.59 ( ( condit2080475194861589617t_unit @ A )
% 0.34/0.59 => ( ( condit2080475194861589617t_unit @ B4 )
% 0.34/0.59 => ( ( ( inf_in4660618365625256667t_unit @ A @ B4 )
% 0.34/0.59 != bot_bo3957492148770167129t_unit )
% 0.34/0.59 => ( ord_le3221252021190050221t_unit @ ( comple4687483117567863418t_unit @ ( inf_in4660618365625256667t_unit @ A @ B4 ) ) @ ( sup_sup_Product_unit @ ( comple4687483117567863418t_unit @ A ) @ ( comple4687483117567863418t_unit @ B4 ) ) ) ) ) ) ).
% 0.34/0.59
% 0.34/0.59 % cSup_inter_less_eq
% 0.34/0.59 thf(fact_3346_cSUP__subset__mono,axiom,
% 0.34/0.59 ! [A: set_Product_unit,G: product_unit > int,B4: set_Product_unit,F: product_unit > int] :
% 0.34/0.59 ( ( A != bot_bo3957492148770167129t_unit )
% 0.34/0.59 => ( ( condit2212336002400062152ve_int @ ( image_873079544045703924it_int @ G @ B4 ) )
% 0.34/0.59 => ( ( ord_le3507040750410214029t_unit @ A @ B4 )
% 0.34/0.59 => ( ! [X3: product_unit] :
% 0.34/0.59 ( ( member_Product_unit @ X3 @ A )
% 0.34/0.59 => ( ord_less_eq_int @ ( F @ X3 ) @ ( G @ X3 ) ) )
% 0.34/0.59 => ( ord_less_eq_int @ ( complete_Sup_Sup_int @ ( image_873079544045703924it_int @ F @ A ) ) @ ( complete_Sup_Sup_int @ ( image_873079544045703924it_int @ G @ B4 ) ) ) ) ) ) ) ).
% 0.34/0.59
% 0.34/0.59 % cSUP_subset_mono
% 0.34/0.59 thf(fact_3347_cSUP__subset__mono,axiom,
% 0.34/0.59 ! [A: set_o,G: $o > int,B4: set_o,F: $o > int] :
% 0.34/0.59 ( ( A != bot_bot_set_o )
% 0.34/0.59 => ( ( condit2212336002400062152ve_int @ ( image_o_int2 @ G @ B4 ) )
% 0.34/0.59 => ( ( ord_less_eq_set_o @ A @ B4 )
% 0.34/0.59 => ( ! [X3: $o] :
% 0.34/0.59 ( ( member_o @ X3 @ A )
% 0.34/0.59 => ( ord_less_eq_int @ ( F @ X3 ) @ ( G @ X3 ) ) )
% 0.34/0.59 => ( ord_less_eq_int @ ( complete_Sup_Sup_int @ ( image_o_int2 @ F @ A ) ) @ ( complete_Sup_Sup_int @ ( image_o_int2 @ G @ B4 ) ) ) ) ) ) ) ).
% 0.34/0.59
% 0.34/0.59 % cSUP_subset_mono
% 0.34/0.59 thf(fact_3348_cSUP__subset__mono,axiom,
% 0.34/0.59 ! [A: set_int,G: int > int,B4: set_int,F: int > int] :
% 0.34/0.59 ( ( A != bot_bot_set_int )
% 0.34/0.59 => ( ( condit2212336002400062152ve_int @ ( image_int_int2 @ G @ B4 ) )
% 0.34/0.59 => ( ( ord_less_eq_set_int @ A @ B4 )
% 0.34/0.59 => ( ! [X3: int] :
% 0.34/0.59 ( ( member_int @ X3 @ A )
% 0.34/0.59 => ( ord_less_eq_int @ ( F @ X3 ) @ ( G @ X3 ) ) )
% 0.34/0.59 => ( ord_less_eq_int @ ( complete_Sup_Sup_int @ ( image_int_int2 @ F @ A ) ) @ ( complete_Sup_Sup_int @ ( image_int_int2 @ G @ B4 ) ) ) ) ) ) ) ).
% 0.34/0.59
% 0.34/0.59 % cSUP_subset_mono
% 0.34/0.59 thf(fact_3349_cSUP__subset__mono,axiom,
% 0.34/0.59 ! [A: set_nat,G: nat > int,B4: set_nat,F: nat > int] :
% 0.34/0.59 ( ( A != bot_bot_set_nat )
% 0.34/0.59 => ( ( condit2212336002400062152ve_int @ ( image_nat_int2 @ G @ B4 ) )
% 0.34/0.59 => ( ( ord_less_eq_set_nat @ A @ B4 )
% 0.34/0.59 => ( ! [X3: nat] :
% 0.34/0.59 ( ( member_nat @ X3 @ A )
% 0.34/0.59 => ( ord_less_eq_int @ ( F @ X3 ) @ ( G @ X3 ) ) )
% 0.34/0.59 => ( ord_less_eq_int @ ( complete_Sup_Sup_int @ ( image_nat_int2 @ F @ A ) ) @ ( complete_Sup_Sup_int @ ( image_nat_int2 @ G @ B4 ) ) ) ) ) ) ) ).
% 0.34/0.59
% 0.34/0.59 % cSUP_subset_mono
% 0.34/0.59 thf(fact_3350_cSUP__subset__mono,axiom,
% 0.34/0.59 ! [A: set_Product_unit,G: product_unit > nat,B4: set_Product_unit,F: product_unit > nat] :
% 0.34/0.59 ( ( A != bot_bo3957492148770167129t_unit )
% 0.34/0.59 => ( ( condit2214826472909112428ve_nat @ ( image_875570014554754200it_nat @ G @ B4 ) )
% 0.34/0.59 => ( ( ord_le3507040750410214029t_unit @ A @ B4 )
% 0.34/0.59 => ( ! [X3: product_unit] :
% 0.34/0.59 ( ( member_Product_unit @ X3 @ A )
% 0.34/0.59 => ( ord_less_eq_nat @ ( F @ X3 ) @ ( G @ X3 ) ) )
% 0.34/0.59 => ( ord_less_eq_nat @ ( complete_Sup_Sup_nat @ ( image_875570014554754200it_nat @ F @ A ) ) @ ( complete_Sup_Sup_nat @ ( image_875570014554754200it_nat @ G @ B4 ) ) ) ) ) ) ) ).
% 0.34/0.59
% 0.34/0.59 % cSUP_subset_mono
% 0.34/0.59 thf(fact_3351_cSUP__subset__mono,axiom,
% 0.34/0.59 ! [A: set_o,G: $o > nat,B4: set_o,F: $o > nat] :
% 0.34/0.59 ( ( A != bot_bot_set_o )
% 0.34/0.59 => ( ( condit2214826472909112428ve_nat @ ( image_o_nat2 @ G @ B4 ) )
% 0.34/0.59 => ( ( ord_less_eq_set_o @ A @ B4 )
% 0.34/0.59 => ( ! [X3: $o] :
% 0.34/0.59 ( ( member_o @ X3 @ A )
% 0.34/0.59 => ( ord_less_eq_nat @ ( F @ X3 ) @ ( G @ X3 ) ) )
% 0.34/0.59 => ( ord_less_eq_nat @ ( complete_Sup_Sup_nat @ ( image_o_nat2 @ F @ A ) ) @ ( complete_Sup_Sup_nat @ ( image_o_nat2 @ G @ B4 ) ) ) ) ) ) ) ).
% 0.34/0.59
% 0.34/0.59 % cSUP_subset_mono
% 0.34/0.59 thf(fact_3352_cSUP__subset__mono,axiom,
% 0.34/0.59 ! [A: set_int,G: int > nat,B4: set_int,F: int > nat] :
% 0.34/0.59 ( ( A != bot_bot_set_int )
% 0.34/0.59 => ( ( condit2214826472909112428ve_nat @ ( image_int_nat2 @ G @ B4 ) )
% 0.34/0.59 => ( ( ord_less_eq_set_int @ A @ B4 )
% 0.34/0.59 => ( ! [X3: int] :
% 0.34/0.59 ( ( member_int @ X3 @ A )
% 0.34/0.59 => ( ord_less_eq_nat @ ( F @ X3 ) @ ( G @ X3 ) ) )
% 0.34/0.59 => ( ord_less_eq_nat @ ( complete_Sup_Sup_nat @ ( image_int_nat2 @ F @ A ) ) @ ( complete_Sup_Sup_nat @ ( image_int_nat2 @ G @ B4 ) ) ) ) ) ) ) ).
% 0.34/0.59
% 0.34/0.59 % cSUP_subset_mono
% 0.34/0.59 thf(fact_3353_cSUP__subset__mono,axiom,
% 0.34/0.59 ! [A: set_nat,G: nat > nat,B4: set_nat,F: nat > nat] :
% 0.34/0.59 ( ( A != bot_bot_set_nat )
% 0.34/0.59 => ( ( condit2214826472909112428ve_nat @ ( image_nat_nat2 @ G @ B4 ) )
% 0.34/0.59 => ( ( ord_less_eq_set_nat @ A @ B4 )
% 0.34/0.59 => ( ! [X3: nat] :
% 0.34/0.59 ( ( member_nat @ X3 @ A )
% 0.34/0.59 => ( ord_less_eq_nat @ ( F @ X3 ) @ ( G @ X3 ) ) )
% 0.34/0.59 => ( ord_less_eq_nat @ ( complete_Sup_Sup_nat @ ( image_nat_nat2 @ F @ A ) ) @ ( complete_Sup_Sup_nat @ ( image_nat_nat2 @ G @ B4 ) ) ) ) ) ) ) ).
% 0.34/0.59
% 0.34/0.59 % cSUP_subset_mono
% 0.34/0.59 thf(fact_3354_cSUP__subset__mono,axiom,
% 0.34/0.59 ! [A: set_Product_unit,G: product_unit > product_unit,B4: set_Product_unit,F: product_unit > product_unit] :
% 0.34/0.59 ( ( A != bot_bo3957492148770167129t_unit )
% 0.34/0.59 => ( ( condit2080475194861589617t_unit @ ( image_405062704495631173t_unit @ G @ B4 ) )
% 0.34/0.59 => ( ( ord_le3507040750410214029t_unit @ A @ B4 )
% 0.34/0.59 => ( ! [X3: product_unit] :
% 0.34/0.59 ( ( member_Product_unit @ X3 @ A )
% 0.34/0.59 => ( ord_le3221252021190050221t_unit @ ( F @ X3 ) @ ( G @ X3 ) ) )
% 0.34/0.59 => ( ord_le3221252021190050221t_unit @ ( comple4687483117567863418t_unit @ ( image_405062704495631173t_unit @ F @ A ) ) @ ( comple4687483117567863418t_unit @ ( image_405062704495631173t_unit @ G @ B4 ) ) ) ) ) ) ) ).
% 0.34/0.59
% 0.34/0.59 % cSUP_subset_mono
% 0.34/0.59 thf(fact_3355_cSUP__subset__mono,axiom,
% 0.34/0.59 ! [A: set_o,G: $o > product_unit,B4: set_o,F: $o > product_unit] :
% 0.34/0.59 ( ( A != bot_bot_set_o )
% 0.34/0.59 => ( ( condit2080475194861589617t_unit @ ( image_o_Product_unit2 @ G @ B4 ) )
% 0.34/0.59 => ( ( ord_less_eq_set_o @ A @ B4 )
% 0.34/0.59 => ( ! [X3: $o] :
% 0.34/0.59 ( ( member_o @ X3 @ A )
% 0.34/0.59 => ( ord_le3221252021190050221t_unit @ ( F @ X3 ) @ ( G @ X3 ) ) )
% 0.34/0.59 => ( ord_le3221252021190050221t_unit @ ( comple4687483117567863418t_unit @ ( image_o_Product_unit2 @ F @ A ) ) @ ( comple4687483117567863418t_unit @ ( image_o_Product_unit2 @ G @ B4 ) ) ) ) ) ) ) ).
% 0.34/0.59
% 0.34/0.59 % cSUP_subset_mono
% 0.34/0.59 thf(fact_3356_Image__Int__eq,axiom,
% 0.34/0.59 ! [R3: set_Pr1261947904930325089at_nat,A: set_nat,B4: set_nat] :
% 0.34/0.59 ( ( single5507084439971879973at_nat @ ( converse_nat_nat @ R3 ) )
% 0.34/0.59 => ( ( image_nat_nat @ R3 @ ( inf_inf_set_nat @ A @ B4 ) )
% 0.34/0.59 = ( inf_inf_set_nat @ ( image_nat_nat @ R3 @ A ) @ ( image_nat_nat @ R3 @ B4 ) ) ) ) ).
% 0.34/0.59
% 0.34/0.59 % Image_Int_eq
% 0.34/0.59 thf(fact_3357_image__update,axiom,
% 0.34/0.59 ! [X: code_integer,A: set_Code_integer,F: code_integer > code_integer,N2: code_integer] :
% 0.34/0.59 ( ~ ( member_Code_integer @ X @ A )
% 0.34/0.59 => ( ( image_4470545334726330049nteger @ ( fun_up8355007323143964473nteger @ F @ X @ N2 ) @ A )
% 0.34/0.59 = ( image_4470545334726330049nteger @ F @ A ) ) ) ).
% 0.34/0.59
% 0.34/0.59 % image_update
% 0.34/0.59 thf(fact_3358_image__update,axiom,
% 0.34/0.59 ! [X: nat,A: set_nat,F: nat > set_nat,N2: set_nat] :
% 0.34/0.59 ( ~ ( member_nat @ X @ A )
% 0.34/0.59 => ( ( image_nat_set_nat @ ( fun_upd_nat_set_nat @ F @ X @ N2 ) @ A )
% 0.34/0.59 = ( image_nat_set_nat @ F @ A ) ) ) ).
% 0.34/0.59
% 0.34/0.59 % image_update
% 0.34/0.59 thf(fact_3359_image__update,axiom,
% 0.34/0.59 ! [X: nat,A: set_nat,F: nat > nat,N2: nat] :
% 0.34/0.59 ( ~ ( member_nat @ X @ A )
% 0.34/0.59 => ( ( image_nat_nat2 @ ( fun_upd_nat_nat @ F @ X @ N2 ) @ A )
% 0.34/0.59 = ( image_nat_nat2 @ F @ A ) ) ) ).
% 0.34/0.59
% 0.34/0.59 % image_update
% 0.34/0.59 thf(fact_3360_image__update,axiom,
% 0.34/0.59 ! [X: nat,A: set_nat,F: nat > code_integer,N2: code_integer] :
% 0.34/0.59 ( ~ ( member_nat @ X @ A )
% 0.34/0.59 => ( ( image_1215581382706833972nteger @ ( fun_up3904115110581602988nteger @ F @ X @ N2 ) @ A )
% 0.34/0.59 = ( image_1215581382706833972nteger @ F @ A ) ) ) ).
% 0.34/0.59
% 0.34/0.59 % image_update
% 0.34/0.59 thf(fact_3361_bdd__above__empty,axiom,
% 0.34/0.59 condit2080475194861589617t_unit @ bot_bo3957492148770167129t_unit ).
% 0.34/0.59
% 0.34/0.59 % bdd_above_empty
% 0.34/0.59 thf(fact_3362_bdd__above__empty,axiom,
% 0.34/0.59 condit5488710616941104124bove_o @ bot_bot_set_o ).
% 0.34/0.59
% 0.34/0.59 % bdd_above_empty
% 0.34/0.59 thf(fact_3363_bdd__above__empty,axiom,
% 0.34/0.59 condit2214826472909112428ve_nat @ bot_bot_set_nat ).
% 0.34/0.59
% 0.34/0.59 % bdd_above_empty
% 0.34/0.59 thf(fact_3364_bdd__above__empty,axiom,
% 0.34/0.59 condit2212336002400062152ve_int @ bot_bot_set_int ).
% 0.34/0.59
% 0.34/0.59 % bdd_above_empty
% 0.34/0.59 thf(fact_3365_bdd__above__empty,axiom,
% 0.34/0.59 condit4352657709669162033_nat_o @ bot_bo7824918357723582233_nat_o ).
% 0.34/0.59
% 0.34/0.59 % bdd_above_empty
% 0.34/0.59 thf(fact_3366_bdd__above__insert,axiom,
% 0.34/0.59 ! [A2: nat,A: set_nat] :
% 0.34/0.59 ( ( condit2214826472909112428ve_nat @ ( insert_nat @ A2 @ A ) )
% 0.34/0.59 = ( condit2214826472909112428ve_nat @ A ) ) ).
% 0.34/0.59
% 0.34/0.59 % bdd_above_insert
% 0.34/0.59 thf(fact_3367_bdd__above__insert,axiom,
% 0.34/0.59 ! [A2: int,A: set_int] :
% 0.34/0.59 ( ( condit2212336002400062152ve_int @ ( insert_int @ A2 @ A ) )
% 0.34/0.59 = ( condit2212336002400062152ve_int @ A ) ) ).
% 0.34/0.59
% 0.34/0.59 % bdd_above_insert
% 0.34/0.59 thf(fact_3368_bdd__above__insert,axiom,
% 0.34/0.59 ! [A2: $o,A: set_o] :
% 0.34/0.59 ( ( condit5488710616941104124bove_o @ ( insert_o @ A2 @ A ) )
% 0.34/0.59 = ( condit5488710616941104124bove_o @ A ) ) ).
% 0.34/0.59
% 0.34/0.59 % bdd_above_insert
% 0.34/0.59 thf(fact_3369_bdd__above__insert,axiom,
% 0.34/0.59 ! [A2: product_unit,A: set_Product_unit] :
% 0.34/0.59 ( ( condit2080475194861589617t_unit @ ( insert_Product_unit @ A2 @ A ) )
% 0.34/0.59 = ( condit2080475194861589617t_unit @ A ) ) ).
% 0.34/0.59
% 0.34/0.59 % bdd_above_insert
% 0.34/0.59 thf(fact_3370_swap__simp,axiom,
% 0.34/0.59 ! [X: set_nat,Y: heap_e7401611519738050253t_unit] :
% 0.34/0.59 ( ( produc1264744871843189710t_unit @ ( produc2471974422597015918t_unit @ X @ Y ) )
% 0.34/0.59 = ( produc7507926704131184380et_nat @ Y @ X ) ) ).
% 0.34/0.59
% 0.34/0.59 % swap_simp
% 0.34/0.59 thf(fact_3371_swap__simp,axiom,
% 0.34/0.59 ! [X: produc3658429121746597890et_nat,Y: produc3658429121746597890et_nat > $o] :
% 0.34/0.59 ( ( produc1078497009909708626_nat_o @ ( produc2829238748666232050_nat_o @ X @ Y ) )
% 0.34/0.59 = ( produc5001842942810119800et_nat @ Y @ X ) ) ).
% 0.34/0.59
% 0.34/0.59 % swap_simp
% 0.34/0.59 thf(fact_3372_swap__simp,axiom,
% 0.34/0.59 ! [X: produc3925858234332021118et_nat,Y: produc3658429121746597890et_nat > $o] :
% 0.34/0.59 ( ( produc2580581461503396054_nat_o @ ( produc3674526432113299062_nat_o @ X @ Y ) )
% 0.34/0.59 = ( produc2245416461498447860et_nat @ Y @ X ) ) ).
% 0.34/0.59
% 0.34/0.59 % swap_simp
% 0.34/0.59 thf(fact_3373_swap__simp,axiom,
% 0.34/0.59 ! [X: heap_e7401611519738050253t_unit,Y: set_nat] :
% 0.34/0.59 ( ( produc6300697153377358172et_nat @ ( produc7507926704131184380et_nat @ X @ Y ) )
% 0.34/0.59 = ( produc2471974422597015918t_unit @ Y @ X ) ) ).
% 0.34/0.59
% 0.34/0.59 % swap_simp
% 0.34/0.59 thf(fact_3374_swap__simp,axiom,
% 0.34/0.59 ! [X: nat,Y: nat] :
% 0.34/0.59 ( ( product_swap_nat_nat @ ( product_Pair_nat_nat @ X @ Y ) )
% 0.34/0.59 = ( product_Pair_nat_nat @ Y @ X ) ) ).
% 0.34/0.59
% 0.34/0.59 % swap_simp
% 0.34/0.59 thf(fact_3375_swap__simp,axiom,
% 0.34/0.59 ! [X: int,Y: int] :
% 0.34/0.59 ( ( product_swap_int_int @ ( product_Pair_int_int @ X @ Y ) )
% 0.34/0.59 = ( product_Pair_int_int @ Y @ X ) ) ).
% 0.34/0.59
% 0.34/0.59 % swap_simp
% 0.34/0.59 thf(fact_3376_swap__simp,axiom,
% 0.34/0.59 ! [X: produc3658429121746597890et_nat > $o,Y: produc3658429121746597890et_nat] :
% 0.34/0.59 ( ( produc3251101204053596376et_nat @ ( produc5001842942810119800et_nat @ X @ Y ) )
% 0.34/0.59 = ( produc2829238748666232050_nat_o @ Y @ X ) ) ).
% 0.34/0.59
% 0.34/0.59 % swap_simp
% 0.34/0.59 thf(fact_3377_swap__simp,axiom,
% 0.34/0.59 ! [X: produc3658429121746597890et_nat > $o,Y: produc3925858234332021118et_nat] :
% 0.34/0.59 ( ( produc1151471490888544852et_nat @ ( produc2245416461498447860et_nat @ X @ Y ) )
% 0.34/0.59 = ( produc3674526432113299062_nat_o @ Y @ X ) ) ).
% 0.34/0.59
% 0.34/0.59 % swap_simp
% 0.34/0.59 thf(fact_3378_single__valuedD,axiom,
% 0.34/0.59 ! [R2: set_Pr3948176798113811640et_nat,X: heap_e7401611519738050253t_unit,Y: set_nat,Z: set_nat] :
% 0.34/0.59 ( ( single6620584844202787856et_nat @ R2 )
% 0.34/0.59 => ( ( member6260224972018164377et_nat @ ( produc7507926704131184380et_nat @ X @ Y ) @ R2 )
% 0.34/0.59 => ( ( member6260224972018164377et_nat @ ( produc7507926704131184380et_nat @ X @ Z ) @ R2 )
% 0.34/0.59 => ( Y = Z ) ) ) ) ).
% 0.34/0.59
% 0.34/0.59 % single_valuedD
% 0.34/0.59 thf(fact_3379_single__valuedD,axiom,
% 0.34/0.59 ! [R2: set_Pr1261947904930325089at_nat,X: nat,Y: nat,Z: nat] :
% 0.34/0.59 ( ( single5507084439971879973at_nat @ R2 )
% 0.34/0.59 => ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ Y ) @ R2 )
% 0.34/0.59 => ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ Z ) @ R2 )
% 0.34/0.59 => ( Y = Z ) ) ) ) ).
% 0.34/0.59
% 0.34/0.59 % single_valuedD
% 0.34/0.59 thf(fact_3380_single__valuedD,axiom,
% 0.34/0.59 ! [R2: set_Pr958786334691620121nt_int,X: int,Y: int,Z: int] :
% 0.34/0.59 ( ( single6503947681363884253nt_int @ R2 )
% 0.34/0.59 => ( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ X @ Y ) @ R2 )
% 0.34/0.59 => ( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ X @ Z ) @ R2 )
% 0.34/0.59 => ( Y = Z ) ) ) ) ).
% 0.34/0.59
% 0.34/0.59 % single_valuedD
% 0.34/0.59 thf(fact_3381_single__valuedD,axiom,
% 0.34/0.59 ! [R2: set_Pr3286484037609594932et_nat,X: produc3658429121746597890et_nat > $o,Y: produc3658429121746597890et_nat,Z: produc3658429121746597890et_nat] :
% 0.34/0.59 ( ( single7715245794981374860et_nat @ R2 )
% 0.34/0.59 => ( ( member1996754912294343701et_nat @ ( produc5001842942810119800et_nat @ X @ Y ) @ R2 )
% 0.34/0.59 => ( ( member1996754912294343701et_nat @ ( produc5001842942810119800et_nat @ X @ Z ) @ R2 )
% 0.34/0.59 => ( Y = Z ) ) ) ) ).
% 0.34/0.59
% 0.34/0.59 % single_valuedD
% 0.34/0.59 thf(fact_3382_single__valuedD,axiom,
% 0.34/0.59 ! [R2: set_Pr8536935166611901872et_nat,X: produc3658429121746597890et_nat > $o,Y: produc3925858234332021118et_nat,Z: produc3925858234332021118et_nat] :
% 0.34/0.59 ( ( single261872189793494280et_nat @ R2 )
% 0.34/0.59 => ( ( member6124377750444531601et_nat @ ( produc2245416461498447860et_nat @ X @ Y ) @ R2 )
% 0.34/0.59 => ( ( member6124377750444531601et_nat @ ( produc2245416461498447860et_nat @ X @ Z ) @ R2 )
% 0.34/0.59 => ( Y = Z ) ) ) ) ).
% 0.34/0.59
% 0.34/0.59 % single_valuedD
% 0.34/0.59 thf(fact_3383_single__valuedI,axiom,
% 0.34/0.59 ! [R2: set_Pr3948176798113811640et_nat] :
% 0.34/0.59 ( ! [X3: heap_e7401611519738050253t_unit,Y2: set_nat,Z3: set_nat] :
% 0.34/0.59 ( ( member6260224972018164377et_nat @ ( produc7507926704131184380et_nat @ X3 @ Y2 ) @ R2 )
% 0.34/0.59 => ( ( member6260224972018164377et_nat @ ( produc7507926704131184380et_nat @ X3 @ Z3 ) @ R2 )
% 0.34/0.59 => ( Y2 = Z3 ) ) )
% 0.34/0.59 => ( single6620584844202787856et_nat @ R2 ) ) ).
% 0.34/0.59
% 0.34/0.59 % single_valuedI
% 0.34/0.59 thf(fact_3384_single__valuedI,axiom,
% 0.34/0.59 ! [R2: set_Pr1261947904930325089at_nat] :
% 0.34/0.59 ( ! [X3: nat,Y2: nat,Z3: nat] :
% 0.34/0.59 ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X3 @ Y2 ) @ R2 )
% 0.34/0.59 => ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X3 @ Z3 ) @ R2 )
% 0.34/0.59 => ( Y2 = Z3 ) ) )
% 0.34/0.59 => ( single5507084439971879973at_nat @ R2 ) ) ).
% 0.34/0.59
% 0.34/0.59 % single_valuedI
% 0.34/0.59 thf(fact_3385_single__valuedI,axiom,
% 0.34/0.59 ! [R2: set_Pr958786334691620121nt_int] :
% 0.34/0.59 ( ! [X3: int,Y2: int,Z3: int] :
% 0.34/0.59 ( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ X3 @ Y2 ) @ R2 )
% 0.34/0.59 => ( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ X3 @ Z3 ) @ R2 )
% 0.34/0.59 => ( Y2 = Z3 ) ) )
% 0.34/0.59 => ( single6503947681363884253nt_int @ R2 ) ) ).
% 0.34/0.59
% 0.34/0.59 % single_valuedI
% 0.34/0.59 thf(fact_3386_single__valuedI,axiom,
% 0.34/0.59 ! [R2: set_Pr3286484037609594932et_nat] :
% 0.34/0.59 ( ! [X3: produc3658429121746597890et_nat > $o,Y2: produc3658429121746597890et_nat,Z3: produc3658429121746597890et_nat] :
% 0.34/0.59 ( ( member1996754912294343701et_nat @ ( produc5001842942810119800et_nat @ X3 @ Y2 ) @ R2 )
% 0.34/0.59 => ( ( member1996754912294343701et_nat @ ( produc5001842942810119800et_nat @ X3 @ Z3 ) @ R2 )
% 0.34/0.59 => ( Y2 = Z3 ) ) )
% 0.34/0.59 => ( single7715245794981374860et_nat @ R2 ) ) ).
% 0.34/0.59
% 0.34/0.59 % single_valuedI
% 0.34/0.59 thf(fact_3387_single__valuedI,axiom,
% 0.34/0.59 ! [R2: set_Pr8536935166611901872et_nat] :
% 0.34/0.59 ( ! [X3: produc3658429121746597890et_nat > $o,Y2: produc3925858234332021118et_nat,Z3: produc3925858234332021118et_nat] :
% 0.34/0.59 ( ( member6124377750444531601et_nat @ ( produc2245416461498447860et_nat @ X3 @ Y2 ) @ R2 )
% 0.34/0.59 => ( ( member6124377750444531601et_nat @ ( produc2245416461498447860et_nat @ X3 @ Z3 ) @ R2 )
% 0.34/0.59 => ( Y2 = Z3 ) ) )
% 0.34/0.59 => ( single261872189793494280et_nat @ R2 ) ) ).
% 0.34/0.59
% 0.34/0.59 % single_valuedI
% 0.34/0.59 thf(fact_3388_single__valued__def,axiom,
% 0.34/0.59 ( single6620584844202787856et_nat
% 0.34/0.59 = ( ^ [R4: set_Pr3948176798113811640et_nat] :
% 0.34/0.59 ! [X2: heap_e7401611519738050253t_unit,Y3: set_nat] :
% 0.34/0.59 ( ( member6260224972018164377et_nat @ ( produc7507926704131184380et_nat @ X2 @ Y3 ) @ R4 )
% 0.34/0.59 => ! [Z2: set_nat] :
% 0.34/0.59 ( ( member6260224972018164377et_nat @ ( produc7507926704131184380et_nat @ X2 @ Z2 ) @ R4 )
% 0.34/0.59 => ( Y3 = Z2 ) ) ) ) ) ).
% 0.34/0.59
% 0.34/0.59 % single_valued_def
% 0.34/0.59 thf(fact_3389_single__valued__def,axiom,
% 0.34/0.59 ( single5507084439971879973at_nat
% 0.34/0.59 = ( ^ [R4: set_Pr1261947904930325089at_nat] :
% 0.34/0.59 ! [X2: nat,Y3: nat] :
% 0.34/0.59 ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X2 @ Y3 ) @ R4 )
% 0.34/0.59 => ! [Z2: nat] :
% 0.34/0.59 ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X2 @ Z2 ) @ R4 )
% 0.34/0.59 => ( Y3 = Z2 ) ) ) ) ) ).
% 0.34/0.59
% 0.34/0.59 % single_valued_def
% 0.34/0.59 thf(fact_3390_single__valued__def,axiom,
% 0.34/0.59 ( single6503947681363884253nt_int
% 0.34/0.59 = ( ^ [R4: set_Pr958786334691620121nt_int] :
% 0.34/0.59 ! [X2: int,Y3: int] :
% 0.34/0.59 ( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ X2 @ Y3 ) @ R4 )
% 0.34/0.59 => ! [Z2: int] :
% 0.34/0.59 ( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ X2 @ Z2 ) @ R4 )
% 0.34/0.59 => ( Y3 = Z2 ) ) ) ) ) ).
% 0.34/0.59
% 0.34/0.59 % single_valued_def
% 0.34/0.59 thf(fact_3391_single__valued__def,axiom,
% 0.34/0.59 ( single7715245794981374860et_nat
% 0.34/0.59 = ( ^ [R4: set_Pr3286484037609594932et_nat] :
% 0.34/0.59 ! [X2: produc3658429121746597890et_nat > $o,Y3: produc3658429121746597890et_nat] :
% 0.34/0.59 ( ( member1996754912294343701et_nat @ ( produc5001842942810119800et_nat @ X2 @ Y3 ) @ R4 )
% 0.34/0.59 => ! [Z2: produc3658429121746597890et_nat] :
% 0.34/0.59 ( ( member1996754912294343701et_nat @ ( produc5001842942810119800et_nat @ X2 @ Z2 ) @ R4 )
% 0.34/0.59 => ( Y3 = Z2 ) ) ) ) ) ).
% 0.34/0.59
% 0.34/0.59 % single_valued_def
% 0.34/0.59 thf(fact_3392_single__valued__def,axiom,
% 0.34/0.59 ( single261872189793494280et_nat
% 0.34/0.59 = ( ^ [R4: set_Pr8536935166611901872et_nat] :
% 0.34/0.59 ! [X2: produc3658429121746597890et_nat > $o,Y3: produc3925858234332021118et_nat] :
% 0.34/0.59 ( ( member6124377750444531601et_nat @ ( produc2245416461498447860et_nat @ X2 @ Y3 ) @ R4 )
% 0.34/0.59 => ! [Z2: produc3925858234332021118et_nat] :
% 0.34/0.59 ( ( member6124377750444531601et_nat @ ( produc2245416461498447860et_nat @ X2 @ Z2 ) @ R4 )
% 0.34/0.59 => ( Y3 = Z2 ) ) ) ) ) ).
% 0.34/0.59
% 0.34/0.59 % single_valued_def
% 0.34/0.59 thf(fact_3393_single__valued__confluent,axiom,
% 0.34/0.59 ! [R2: set_Pr1261947904930325089at_nat,X: nat,Y: nat,Z: nat] :
% 0.34/0.59 ( ( single5507084439971879973at_nat @ R2 )
% 0.34/0.59 => ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ Y ) @ ( transi2905341329935302413cl_nat @ R2 ) )
% 0.34/0.59 => ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ Z ) @ ( transi2905341329935302413cl_nat @ R2 ) )
% 0.34/0.59 => ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ Y @ Z ) @ ( transi2905341329935302413cl_nat @ R2 ) )
% 0.34/0.59 | ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ Z @ Y ) @ ( transi2905341329935302413cl_nat @ R2 ) ) ) ) ) ) ).
% 0.34/0.59
% 0.34/0.59 % single_valued_confluent
% 0.34/0.59 thf(fact_3394_single__valued__confluent,axiom,
% 0.34/0.59 ! [R2: set_Pr958786334691620121nt_int,X: int,Y: int,Z: int] :
% 0.34/0.59 ( ( single6503947681363884253nt_int @ R2 )
% 0.34/0.59 => ( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ X @ Y ) @ ( transi2902850859426252137cl_int @ R2 ) )
% 0.34/0.59 => ( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ X @ Z ) @ ( transi2902850859426252137cl_int @ R2 ) )
% 0.34/0.59 => ( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ Y @ Z ) @ ( transi2902850859426252137cl_int @ R2 ) )
% 0.34/0.59 | ( member5262025264175285858nt_int @ ( product_Pair_int_int @ Z @ Y ) @ ( transi2902850859426252137cl_int @ R2 ) ) ) ) ) ) ).
% 0.34/0.59
% 0.34/0.59 % single_valued_confluent
% 0.34/0.59 thf(fact_3395_cSup__le__iff,axiom,
% 0.34/0.59 ! [S: set_o,A2: $o] :
% 0.34/0.59 ( ( S != bot_bot_set_o )
% 0.34/0.59 => ( ( condit5488710616941104124bove_o @ S )
% 0.34/0.59 => ( ( ord_less_eq_o @ ( complete_Sup_Sup_o @ S ) @ A2 )
% 0.34/0.59 = ( ! [X2: $o] :
% 0.34/0.59 ( ( member_o @ X2 @ S )
% 0.34/0.59 => ( ord_less_eq_o @ X2 @ A2 ) ) ) ) ) ) ).
% 0.34/0.59
% 0.34/0.59 % cSup_le_iff
% 0.34/0.59 thf(fact_3396_cSup__le__iff,axiom,
% 0.34/0.59 ! [S: set_Pr4532377907799695533_nat_o,A2: produc3658429121746597890et_nat > $o] :
% 0.34/0.59 ( ( S != bot_bo7824918357723582233_nat_o )
% 0.34/0.59 => ( ( condit4352657709669162033_nat_o @ S )
% 0.34/0.59 => ( ( ord_le729326519192465773_nat_o @ ( comple4225350640907012154_nat_o @ S ) @ A2 )
% 0.34/0.59 = ( ! [X2: produc3658429121746597890et_nat > $o] :
% 0.34/0.59 ( ( member6576561426505652726_nat_o @ X2 @ S )
% 0.34/0.59 => ( ord_le729326519192465773_nat_o @ X2 @ A2 ) ) ) ) ) ) ).
% 0.34/0.59
% 0.34/0.59 % cSup_le_iff
% 0.34/0.59 thf(fact_3397_cSup__le__iff,axiom,
% 0.34/0.59 ! [S: set_filter_nat,A2: filter_nat] :
% 0.34/0.59 ( ( S != bot_bo498966703094740906er_nat )
% 0.34/0.59 => ( ( condit4029664723990530684er_nat @ S )
% 0.34/0.59 => ( ( ord_le2510731241096832064er_nat @ ( comple2454491259644266675er_nat @ S ) @ A2 )
% 0.34/0.59 = ( ! [X2: filter_nat] :
% 0.34/0.59 ( ( member_filter_nat @ X2 @ S )
% 0.34/0.59 => ( ord_le2510731241096832064er_nat @ X2 @ A2 ) ) ) ) ) ) ).
% 0.34/0.59
% 0.34/0.59 % cSup_le_iff
% 0.34/0.59 thf(fact_3398_cSup__le__iff,axiom,
% 0.34/0.59 ! [S: set_set_nat,A2: set_nat] :
% 0.34/0.59 ( ( S != bot_bot_set_set_nat )
% 0.34/0.59 => ( ( condit5477540289124974626et_nat @ S )
% 0.34/0.59 => ( ( ord_less_eq_set_nat @ ( comple7399068483239264473et_nat @ S ) @ A2 )
% 0.34/0.59 = ( ! [X2: set_nat] :
% 0.34/0.59 ( ( member_set_nat @ X2 @ S )
% 0.34/0.59 => ( ord_less_eq_set_nat @ X2 @ A2 ) ) ) ) ) ) ).
% 0.34/0.59
% 0.34/0.59 % cSup_le_iff
% 0.34/0.59 thf(fact_3399_cSup__le__iff,axiom,
% 0.34/0.59 ! [S: set_int,A2: int] :
% 0.34/0.59 ( ( S != bot_bot_set_int )
% 0.34/0.59 => ( ( condit2212336002400062152ve_int @ S )
% 0.34/0.59 => ( ( ord_less_eq_int @ ( complete_Sup_Sup_int @ S ) @ A2 )
% 0.34/0.59 = ( ! [X2: int] :
% 0.34/0.59 ( ( member_int @ X2 @ S )
% 0.34/0.59 => ( ord_less_eq_int @ X2 @ A2 ) ) ) ) ) ) ).
% 0.34/0.59
% 0.34/0.59 % cSup_le_iff
% 0.34/0.59 thf(fact_3400_cSup__le__iff,axiom,
% 0.34/0.59 ! [S: set_nat,A2: nat] :
% 0.34/0.59 ( ( S != bot_bot_set_nat )
% 0.34/0.59 => ( ( condit2214826472909112428ve_nat @ S )
% 0.34/0.59 => ( ( ord_less_eq_nat @ ( complete_Sup_Sup_nat @ S ) @ A2 )
% 0.34/0.59 = ( ! [X2: nat] :
% 0.34/0.59 ( ( member_nat @ X2 @ S )
% 0.34/0.59 => ( ord_less_eq_nat @ X2 @ A2 ) ) ) ) ) ) ).
% 0.34/0.59
% 0.34/0.59 % cSup_le_iff
% 0.34/0.59 thf(fact_3401_cSup__le__iff,axiom,
% 0.34/0.59 ! [S: set_Product_unit,A2: product_unit] :
% 0.34/0.59 ( ( S != bot_bo3957492148770167129t_unit )
% 0.34/0.59 => ( ( condit2080475194861589617t_unit @ S )
% 0.34/0.59 => ( ( ord_le3221252021190050221t_unit @ ( comple4687483117567863418t_unit @ S ) @ A2 )
% 0.34/0.59 = ( ! [X2: product_unit] :
% 0.34/0.59 ( ( member_Product_unit @ X2 @ S )
% 0.34/0.59 => ( ord_le3221252021190050221t_unit @ X2 @ A2 ) ) ) ) ) ) ).
% 0.34/0.59
% 0.34/0.59 % cSup_le_iff
% 0.34/0.59 thf(fact_3402_cSup__mono,axiom,
% 0.34/0.59 ! [B4: set_o,A: set_o] :
% 0.34/0.59 ( ( B4 != bot_bot_set_o )
% 0.34/0.59 => ( ( condit5488710616941104124bove_o @ A )
% 0.34/0.59 => ( ! [B2: $o] :
% 0.34/0.59 ( ( member_o @ B2 @ B4 )
% 0.34/0.59 => ? [X6: $o] :
% 0.34/0.59 ( ( member_o @ X6 @ A )
% 0.34/0.59 & ( ord_less_eq_o @ B2 @ X6 ) ) )
% 0.34/0.59 => ( ord_less_eq_o @ ( complete_Sup_Sup_o @ B4 ) @ ( complete_Sup_Sup_o @ A ) ) ) ) ) ).
% 0.34/0.59
% 0.34/0.59 % cSup_mono
% 0.34/0.59 thf(fact_3403_cSup__mono,axiom,
% 0.34/0.59 ! [B4: set_Pr4532377907799695533_nat_o,A: set_Pr4532377907799695533_nat_o] :
% 0.34/0.59 ( ( B4 != bot_bo7824918357723582233_nat_o )
% 0.34/0.59 => ( ( condit4352657709669162033_nat_o @ A )
% 0.34/0.59 => ( ! [B2: produc3658429121746597890et_nat > $o] :
% 0.34/0.59 ( ( member6576561426505652726_nat_o @ B2 @ B4 )
% 0.34/0.59 => ? [X6: produc3658429121746597890et_nat > $o] :
% 0.34/0.59 ( ( member6576561426505652726_nat_o @ X6 @ A )
% 0.34/0.59 & ( ord_le729326519192465773_nat_o @ B2 @ X6 ) ) )
% 0.34/0.59 => ( ord_le729326519192465773_nat_o @ ( comple4225350640907012154_nat_o @ B4 ) @ ( comple4225350640907012154_nat_o @ A ) ) ) ) ) ).
% 0.34/0.59
% 0.34/0.59 % cSup_mono
% 0.34/0.59 thf(fact_3404_cSup__mono,axiom,
% 0.34/0.59 ! [B4: set_filter_nat,A: set_filter_nat] :
% 0.34/0.59 ( ( B4 != bot_bo498966703094740906er_nat )
% 0.34/0.59 => ( ( condit4029664723990530684er_nat @ A )
% 0.34/0.59 => ( ! [B2: filter_nat] :
% 0.34/0.59 ( ( member_filter_nat @ B2 @ B4 )
% 0.34/0.59 => ? [X6: filter_nat] :
% 0.34/0.59 ( ( member_filter_nat @ X6 @ A )
% 0.34/0.59 & ( ord_le2510731241096832064er_nat @ B2 @ X6 ) ) )
% 0.34/0.59 => ( ord_le2510731241096832064er_nat @ ( comple2454491259644266675er_nat @ B4 ) @ ( comple2454491259644266675er_nat @ A ) ) ) ) ) ).
% 0.34/0.59
% 0.34/0.59 % cSup_mono
% 0.34/0.59 thf(fact_3405_cSup__mono,axiom,
% 0.34/0.59 ! [B4: set_set_nat,A: set_set_nat] :
% 0.34/0.59 ( ( B4 != bot_bot_set_set_nat )
% 0.34/0.59 => ( ( condit5477540289124974626et_nat @ A )
% 0.34/0.59 => ( ! [B2: set_nat] :
% 0.34/0.59 ( ( member_set_nat @ B2 @ B4 )
% 0.34/0.59 => ? [X6: set_nat] :
% 0.34/0.59 ( ( member_set_nat @ X6 @ A )
% 0.34/0.59 & ( ord_less_eq_set_nat @ B2 @ X6 ) ) )
% 0.34/0.59 => ( ord_less_eq_set_nat @ ( comple7399068483239264473et_nat @ B4 ) @ ( comple7399068483239264473et_nat @ A ) ) ) ) ) ).
% 0.34/0.59
% 0.34/0.59 % cSup_mono
% 0.34/0.59 thf(fact_3406_cSup__mono,axiom,
% 0.34/0.59 ! [B4: set_int,A: set_int] :
% 0.34/0.59 ( ( B4 != bot_bot_set_int )
% 0.34/0.59 => ( ( condit2212336002400062152ve_int @ A )
% 0.34/0.59 => ( ! [B2: int] :
% 0.34/0.59 ( ( member_int @ B2 @ B4 )
% 0.34/0.59 => ? [X6: int] :
% 0.34/0.59 ( ( member_int @ X6 @ A )
% 0.34/0.59 & ( ord_less_eq_int @ B2 @ X6 ) ) )
% 0.34/0.59 => ( ord_less_eq_int @ ( complete_Sup_Sup_int @ B4 ) @ ( complete_Sup_Sup_int @ A ) ) ) ) ) ).
% 0.34/0.59
% 0.34/0.59 % cSup_mono
% 0.34/0.59 thf(fact_3407_cSup__mono,axiom,
% 0.34/0.59 ! [B4: set_nat,A: set_nat] :
% 0.34/0.59 ( ( B4 != bot_bot_set_nat )
% 0.34/0.59 => ( ( condit2214826472909112428ve_nat @ A )
% 0.34/0.59 => ( ! [B2: nat] :
% 0.34/0.59 ( ( member_nat @ B2 @ B4 )
% 0.34/0.59 => ? [X6: nat] :
% 0.34/0.59 ( ( member_nat @ X6 @ A )
% 0.34/0.59 & ( ord_less_eq_nat @ B2 @ X6 ) ) )
% 0.34/0.59 => ( ord_less_eq_nat @ ( complete_Sup_Sup_nat @ B4 ) @ ( complete_Sup_Sup_nat @ A ) ) ) ) ) ).
% 0.34/0.59
% 0.34/0.59 % cSup_mono
% 0.34/0.59 thf(fact_3408_cSup__mono,axiom,
% 0.34/0.59 ! [B4: set_Product_unit,A: set_Product_unit] :
% 0.34/0.59 ( ( B4 != bot_bo3957492148770167129t_unit )
% 0.34/0.59 => ( ( condit2080475194861589617t_unit @ A )
% 0.34/0.59 => ( ! [B2: product_unit] :
% 0.34/0.59 ( ( member_Product_unit @ B2 @ B4 )
% 0.34/0.59 => ? [X6: product_unit] :
% 0.34/0.59 ( ( member_Product_unit @ X6 @ A )
% 0.34/0.59 & ( ord_le3221252021190050221t_unit @ B2 @ X6 ) ) )
% 0.34/0.59 => ( ord_le3221252021190050221t_unit @ ( comple4687483117567863418t_unit @ B4 ) @ ( comple4687483117567863418t_unit @ A ) ) ) ) ) ).
% 0.34/0.59
% 0.34/0.59 % cSup_mono
% 0.34/0.59 thf(fact_3409_cSUP__le__iff,axiom,
% 0.34/0.59 ! [A: set_Product_unit,F: product_unit > int,U: int] :
% 0.34/0.59 ( ( A != bot_bo3957492148770167129t_unit )
% 0.34/0.59 => ( ( condit2212336002400062152ve_int @ ( image_873079544045703924it_int @ F @ A ) )
% 0.34/0.59 => ( ( ord_less_eq_int @ ( complete_Sup_Sup_int @ ( image_873079544045703924it_int @ F @ A ) ) @ U )
% 0.34/0.59 = ( ! [X2: product_unit] :
% 0.34/0.59 ( ( member_Product_unit @ X2 @ A )
% 0.34/0.59 => ( ord_less_eq_int @ ( F @ X2 ) @ U ) ) ) ) ) ) ).
% 0.34/0.59
% 0.34/0.59 % cSUP_le_iff
% 0.34/0.59 thf(fact_3410_cSUP__le__iff,axiom,
% 0.34/0.59 ! [A: set_o,F: $o > int,U: int] :
% 0.34/0.59 ( ( A != bot_bot_set_o )
% 0.34/0.59 => ( ( condit2212336002400062152ve_int @ ( image_o_int2 @ F @ A ) )
% 0.34/0.59 => ( ( ord_less_eq_int @ ( complete_Sup_Sup_int @ ( image_o_int2 @ F @ A ) ) @ U )
% 0.34/0.59 = ( ! [X2: $o] :
% 0.34/0.59 ( ( member_o @ X2 @ A )
% 0.34/0.59 => ( ord_less_eq_int @ ( F @ X2 ) @ U ) ) ) ) ) ) ).
% 0.34/0.59
% 0.34/0.59 % cSUP_le_iff
% 0.34/0.59 thf(fact_3411_cSUP__le__iff,axiom,
% 0.34/0.59 ! [A: set_nat,F: nat > int,U: int] :
% 0.34/0.59 ( ( A != bot_bot_set_nat )
% 0.34/0.59 => ( ( condit2212336002400062152ve_int @ ( image_nat_int2 @ F @ A ) )
% 0.34/0.59 => ( ( ord_less_eq_int @ ( complete_Sup_Sup_int @ ( image_nat_int2 @ F @ A ) ) @ U )
% 0.34/0.59 = ( ! [X2: nat] :
% 0.34/0.59 ( ( member_nat @ X2 @ A )
% 0.34/0.59 => ( ord_less_eq_int @ ( F @ X2 ) @ U ) ) ) ) ) ) ).
% 0.34/0.59
% 0.34/0.59 % cSUP_le_iff
% 0.34/0.59 thf(fact_3412_cSUP__le__iff,axiom,
% 0.34/0.59 ! [A: set_int,F: int > int,U: int] :
% 0.34/0.59 ( ( A != bot_bot_set_int )
% 0.34/0.59 => ( ( condit2212336002400062152ve_int @ ( image_int_int2 @ F @ A ) )
% 0.34/0.59 => ( ( ord_less_eq_int @ ( complete_Sup_Sup_int @ ( image_int_int2 @ F @ A ) ) @ U )
% 0.34/0.59 = ( ! [X2: int] :
% 0.34/0.59 ( ( member_int @ X2 @ A )
% 0.34/0.59 => ( ord_less_eq_int @ ( F @ X2 ) @ U ) ) ) ) ) ) ).
% 0.34/0.59
% 0.34/0.59 % cSUP_le_iff
% 0.34/0.59 thf(fact_3413_cSUP__le__iff,axiom,
% 0.34/0.59 ! [A: set_Product_unit,F: product_unit > nat,U: nat] :
% 0.34/0.59 ( ( A != bot_bo3957492148770167129t_unit )
% 0.34/0.59 => ( ( condit2214826472909112428ve_nat @ ( image_875570014554754200it_nat @ F @ A ) )
% 0.34/0.59 => ( ( ord_less_eq_nat @ ( complete_Sup_Sup_nat @ ( image_875570014554754200it_nat @ F @ A ) ) @ U )
% 0.34/0.59 = ( ! [X2: product_unit] :
% 0.34/0.59 ( ( member_Product_unit @ X2 @ A )
% 0.34/0.59 => ( ord_less_eq_nat @ ( F @ X2 ) @ U ) ) ) ) ) ) ).
% 0.34/0.59
% 0.34/0.59 % cSUP_le_iff
% 0.34/0.59 thf(fact_3414_cSUP__le__iff,axiom,
% 0.34/0.59 ! [A: set_o,F: $o > nat,U: nat] :
% 0.34/0.59 ( ( A != bot_bot_set_o )
% 0.34/0.59 => ( ( condit2214826472909112428ve_nat @ ( image_o_nat2 @ F @ A ) )
% 0.34/0.59 => ( ( ord_less_eq_nat @ ( complete_Sup_Sup_nat @ ( image_o_nat2 @ F @ A ) ) @ U )
% 0.34/0.59 = ( ! [X2: $o] :
% 0.34/0.59 ( ( member_o @ X2 @ A )
% 0.34/0.59 => ( ord_less_eq_nat @ ( F @ X2 ) @ U ) ) ) ) ) ) ).
% 0.34/0.59
% 0.34/0.59 % cSUP_le_iff
% 0.34/0.59 thf(fact_3415_cSUP__le__iff,axiom,
% 0.34/0.59 ! [A: set_nat,F: nat > nat,U: nat] :
% 0.34/0.59 ( ( A != bot_bot_set_nat )
% 0.34/0.59 => ( ( condit2214826472909112428ve_nat @ ( image_nat_nat2 @ F @ A ) )
% 0.34/0.59 => ( ( ord_less_eq_nat @ ( complete_Sup_Sup_nat @ ( image_nat_nat2 @ F @ A ) ) @ U )
% 0.34/0.59 = ( ! [X2: nat] :
% 0.34/0.59 ( ( member_nat @ X2 @ A )
% 0.34/0.59 => ( ord_less_eq_nat @ ( F @ X2 ) @ U ) ) ) ) ) ) ).
% 0.34/0.59
% 0.34/0.59 % cSUP_le_iff
% 0.34/0.59 thf(fact_3416_cSUP__le__iff,axiom,
% 0.34/0.59 ! [A: set_int,F: int > nat,U: nat] :
% 0.34/0.59 ( ( A != bot_bot_set_int )
% 0.34/0.59 => ( ( condit2214826472909112428ve_nat @ ( image_int_nat2 @ F @ A ) )
% 0.34/0.59 => ( ( ord_less_eq_nat @ ( complete_Sup_Sup_nat @ ( image_int_nat2 @ F @ A ) ) @ U )
% 0.34/0.59 = ( ! [X2: int] :
% 0.34/0.59 ( ( member_int @ X2 @ A )
% 0.34/0.59 => ( ord_less_eq_nat @ ( F @ X2 ) @ U ) ) ) ) ) ) ).
% 0.34/0.59
% 0.34/0.59 % cSUP_le_iff
% 0.34/0.59 thf(fact_3417_cSUP__le__iff,axiom,
% 0.34/0.59 ! [A: set_Product_unit,F: product_unit > product_unit,U: product_unit] :
% 0.34/0.59 ( ( A != bot_bo3957492148770167129t_unit )
% 0.34/0.59 => ( ( condit2080475194861589617t_unit @ ( image_405062704495631173t_unit @ F @ A ) )
% 0.34/0.59 => ( ( ord_le3221252021190050221t_unit @ ( comple4687483117567863418t_unit @ ( image_405062704495631173t_unit @ F @ A ) ) @ U )
% 0.34/0.59 = ( ! [X2: product_unit] :
% 0.34/0.59 ( ( member_Product_unit @ X2 @ A )
% 0.34/0.59 => ( ord_le3221252021190050221t_unit @ ( F @ X2 ) @ U ) ) ) ) ) ) ).
% 0.34/0.59
% 0.34/0.59 % cSUP_le_iff
% 0.34/0.59 thf(fact_3418_cSUP__le__iff,axiom,
% 0.34/0.59 ! [A: set_o,F: $o > product_unit,U: product_unit] :
% 0.34/0.59 ( ( A != bot_bot_set_o )
% 0.34/0.59 => ( ( condit2080475194861589617t_unit @ ( image_o_Product_unit2 @ F @ A ) )
% 0.34/0.59 => ( ( ord_le3221252021190050221t_unit @ ( comple4687483117567863418t_unit @ ( image_o_Product_unit2 @ F @ A ) ) @ U )
% 0.34/0.59 = ( ! [X2: $o] :
% 0.34/0.59 ( ( member_o @ X2 @ A )
% 0.34/0.59 => ( ord_le3221252021190050221t_unit @ ( F @ X2 ) @ U ) ) ) ) ) ) ).
% 0.34/0.59
% 0.34/0.59 % cSUP_le_iff
% 0.34/0.59 thf(fact_3419_cSUP__mono,axiom,
% 0.34/0.59 ! [A: set_Product_unit,G: nat > set_nat,B4: set_nat,F: product_unit > set_nat] :
% 0.34/0.59 ( ( A != bot_bo3957492148770167129t_unit )
% 0.34/0.59 => ( ( condit5477540289124974626et_nat @ ( image_nat_set_nat @ G @ B4 ) )
% 0.34/0.59 => ( ! [N3: product_unit] :
% 0.34/0.59 ( ( member_Product_unit @ N3 @ A )
% 0.34/0.59 => ? [X6: nat] :
% 0.34/0.59 ( ( member_nat @ X6 @ B4 )
% 0.34/0.59 & ( ord_less_eq_set_nat @ ( F @ N3 ) @ ( G @ X6 ) ) ) )
% 0.34/0.59 => ( ord_less_eq_set_nat @ ( comple7399068483239264473et_nat @ ( image_8173849991827986510et_nat @ F @ A ) ) @ ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ G @ B4 ) ) ) ) ) ) ).
% 0.34/0.59
% 0.34/0.59 % cSUP_mono
% 0.34/0.59 thf(fact_3420_cSUP__mono,axiom,
% 0.34/0.59 ! [A: set_o,G: nat > set_nat,B4: set_nat,F: $o > set_nat] :
% 0.34/0.59 ( ( A != bot_bot_set_o )
% 0.34/0.59 => ( ( condit5477540289124974626et_nat @ ( image_nat_set_nat @ G @ B4 ) )
% 0.34/0.59 => ( ! [N3: $o] :
% 0.34/0.59 ( ( member_o @ N3 @ A )
% 0.34/0.59 => ? [X6: nat] :
% 0.34/0.59 ( ( member_nat @ X6 @ B4 )
% 0.34/0.59 & ( ord_less_eq_set_nat @ ( F @ N3 ) @ ( G @ X6 ) ) ) )
% 0.34/0.59 => ( ord_less_eq_set_nat @ ( comple7399068483239264473et_nat @ ( image_o_set_nat @ F @ A ) ) @ ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ G @ B4 ) ) ) ) ) ) ).
% 0.34/0.59
% 0.34/0.59 % cSUP_mono
% 0.34/0.59 thf(fact_3421_cSUP__mono,axiom,
% 0.34/0.59 ! [A: set_nat,G: nat > set_nat,B4: set_nat,F: nat > set_nat] :
% 0.34/0.59 ( ( A != bot_bot_set_nat )
% 0.34/0.59 => ( ( condit5477540289124974626et_nat @ ( image_nat_set_nat @ G @ B4 ) )
% 0.34/0.59 => ( ! [N3: nat] :
% 0.34/0.59 ( ( member_nat @ N3 @ A )
% 0.34/0.59 => ? [X6: nat] :
% 0.34/0.59 ( ( member_nat @ X6 @ B4 )
% 0.34/0.59 & ( ord_less_eq_set_nat @ ( F @ N3 ) @ ( G @ X6 ) ) ) )
% 0.34/0.59 => ( ord_less_eq_set_nat @ ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ F @ A ) ) @ ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ G @ B4 ) ) ) ) ) ) ).
% 0.34/0.59
% 0.34/0.59 % cSUP_mono
% 0.34/0.59 thf(fact_3422_cSUP__mono,axiom,
% 0.34/0.59 ! [A: set_int,G: nat > set_nat,B4: set_nat,F: int > set_nat] :
% 0.34/0.59 ( ( A != bot_bot_set_int )
% 0.34/0.59 => ( ( condit5477540289124974626et_nat @ ( image_nat_set_nat @ G @ B4 ) )
% 0.34/0.59 => ( ! [N3: int] :
% 0.34/0.59 ( ( member_int @ N3 @ A )
% 0.34/0.59 => ? [X6: nat] :
% 0.34/0.59 ( ( member_nat @ X6 @ B4 )
% 0.34/0.59 & ( ord_less_eq_set_nat @ ( F @ N3 ) @ ( G @ X6 ) ) ) )
% 0.34/0.59 => ( ord_less_eq_set_nat @ ( comple7399068483239264473et_nat @ ( image_int_set_nat @ F @ A ) ) @ ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ G @ B4 ) ) ) ) ) ) ).
% 0.34/0.59
% 0.34/0.59 % cSUP_mono
% 0.34/0.59 thf(fact_3423_cSUP__mono,axiom,
% 0.34/0.59 ! [A: set_Pr4532377907799695533_nat_o,G: nat > set_nat,B4: set_nat,F: ( produc3658429121746597890et_nat > $o ) > set_nat] :
% 0.34/0.59 ( ( A != bot_bo7824918357723582233_nat_o )
% 0.34/0.59 => ( ( condit5477540289124974626et_nat @ ( image_nat_set_nat @ G @ B4 ) )
% 0.34/0.59 => ( ! [N3: produc3658429121746597890et_nat > $o] :
% 0.34/0.59 ( ( member6576561426505652726_nat_o @ N3 @ A )
% 0.34/0.59 => ? [X6: nat] :
% 0.34/0.59 ( ( member_nat @ X6 @ B4 )
% 0.34/0.59 & ( ord_less_eq_set_nat @ ( F @ N3 ) @ ( G @ X6 ) ) ) )
% 0.34/0.59 => ( ord_less_eq_set_nat @ ( comple7399068483239264473et_nat @ ( image_6888641473154130958et_nat @ F @ A ) ) @ ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ G @ B4 ) ) ) ) ) ) ).
% 0.34/0.59
% 0.34/0.59 % cSUP_mono
% 0.34/0.59 thf(fact_3424_cSUP__mono,axiom,
% 0.34/0.59 ! [A: set_Product_unit,G: nat > nat,B4: set_nat,F: product_unit > nat] :
% 0.34/0.59 ( ( A != bot_bo3957492148770167129t_unit )
% 0.34/0.59 => ( ( condit2214826472909112428ve_nat @ ( image_nat_nat2 @ G @ B4 ) )
% 0.34/0.59 => ( ! [N3: product_unit] :
% 0.34/0.59 ( ( member_Product_unit @ N3 @ A )
% 0.34/0.59 => ? [X6: nat] :
% 0.34/0.59 ( ( member_nat @ X6 @ B4 )
% 0.34/0.59 & ( ord_less_eq_nat @ ( F @ N3 ) @ ( G @ X6 ) ) ) )
% 0.34/0.59 => ( ord_less_eq_nat @ ( complete_Sup_Sup_nat @ ( image_875570014554754200it_nat @ F @ A ) ) @ ( complete_Sup_Sup_nat @ ( image_nat_nat2 @ G @ B4 ) ) ) ) ) ) ).
% 0.34/0.59
% 0.34/0.59 % cSUP_mono
% 0.34/0.59 thf(fact_3425_cSUP__mono,axiom,
% 0.34/0.59 ! [A: set_o,G: nat > nat,B4: set_nat,F: $o > nat] :
% 0.34/0.59 ( ( A != bot_bot_set_o )
% 0.34/0.59 => ( ( condit2214826472909112428ve_nat @ ( image_nat_nat2 @ G @ B4 ) )
% 0.34/0.59 => ( ! [N3: $o] :
% 0.34/0.59 ( ( member_o @ N3 @ A )
% 0.34/0.59 => ? [X6: nat] :
% 0.34/0.59 ( ( member_nat @ X6 @ B4 )
% 0.34/0.59 & ( ord_less_eq_nat @ ( F @ N3 ) @ ( G @ X6 ) ) ) )
% 0.34/0.59 => ( ord_less_eq_nat @ ( complete_Sup_Sup_nat @ ( image_o_nat2 @ F @ A ) ) @ ( complete_Sup_Sup_nat @ ( image_nat_nat2 @ G @ B4 ) ) ) ) ) ) ).
% 0.34/0.59
% 0.34/0.59 % cSUP_mono
% 0.34/0.59 thf(fact_3426_cSUP__mono,axiom,
% 0.34/0.59 ! [A: set_nat,G: nat > nat,B4: set_nat,F: nat > nat] :
% 0.34/0.59 ( ( A != bot_bot_set_nat )
% 0.34/0.59 => ( ( condit2214826472909112428ve_nat @ ( image_nat_nat2 @ G @ B4 ) )
% 0.34/0.59 => ( ! [N3: nat] :
% 0.34/0.59 ( ( member_nat @ N3 @ A )
% 0.34/0.59 => ? [X6: nat] :
% 0.34/0.59 ( ( member_nat @ X6 @ B4 )
% 0.34/0.59 & ( ord_less_eq_nat @ ( F @ N3 ) @ ( G @ X6 ) ) ) )
% 0.34/0.59 => ( ord_less_eq_nat @ ( complete_Sup_Sup_nat @ ( image_nat_nat2 @ F @ A ) ) @ ( complete_Sup_Sup_nat @ ( image_nat_nat2 @ G @ B4 ) ) ) ) ) ) ).
% 0.34/0.59
% 0.34/0.59 % cSUP_mono
% 0.34/0.59 thf(fact_3427_cSUP__mono,axiom,
% 0.34/0.59 ! [A: set_int,G: nat > nat,B4: set_nat,F: int > nat] :
% 0.34/0.59 ( ( A != bot_bot_set_int )
% 0.34/0.59 => ( ( condit2214826472909112428ve_nat @ ( image_nat_nat2 @ G @ B4 ) )
% 0.34/0.59 => ( ! [N3: int] :
% 0.34/0.59 ( ( member_int @ N3 @ A )
% 0.34/0.59 => ? [X6: nat] :
% 0.34/0.59 ( ( member_nat @ X6 @ B4 )
% 0.34/0.59 & ( ord_less_eq_nat @ ( F @ N3 ) @ ( G @ X6 ) ) ) )
% 0.34/0.59 => ( ord_less_eq_nat @ ( complete_Sup_Sup_nat @ ( image_int_nat2 @ F @ A ) ) @ ( complete_Sup_Sup_nat @ ( image_nat_nat2 @ G @ B4 ) ) ) ) ) ) ).
% 0.34/0.59
% 0.34/0.59 % cSUP_mono
% 0.34/0.59 thf(fact_3428_cSUP__mono,axiom,
% 0.34/0.59 ! [A: set_Pr4532377907799695533_nat_o,G: nat > nat,B4: set_nat,F: ( produc3658429121746597890et_nat > $o ) > nat] :
% 0.34/0.59 ( ( A != bot_bo7824918357723582233_nat_o )
% 0.34/0.59 => ( ( condit2214826472909112428ve_nat @ ( image_nat_nat2 @ G @ B4 ) )
% 0.34/0.59 => ( ! [N3: produc3658429121746597890et_nat > $o] :
% 0.34/0.59 ( ( member6576561426505652726_nat_o @ N3 @ A )
% 0.34/0.59 => ? [X6: nat] :
% 0.34/0.59 ( ( member_nat @ X6 @ B4 )
% 0.34/0.59 & ( ord_less_eq_nat @ ( F @ N3 ) @ ( G @ X6 ) ) ) )
% 0.34/0.59 => ( ord_less_eq_nat @ ( complete_Sup_Sup_nat @ ( image_2174381888560928856_o_nat @ F @ A ) ) @ ( complete_Sup_Sup_nat @ ( image_nat_nat2 @ G @ B4 ) ) ) ) ) ) ).
% 0.34/0.59
% 0.34/0.59 % cSUP_mono
% 0.34/0.59 thf(fact_3429_cSup__subset__mono,axiom,
% 0.34/0.59 ! [A: set_o,B4: set_o] :
% 0.34/0.59 ( ( A != bot_bot_set_o )
% 0.34/0.59 => ( ( condit5488710616941104124bove_o @ B4 )
% 0.34/0.59 => ( ( ord_less_eq_set_o @ A @ B4 )
% 0.34/0.59 => ( ord_less_eq_o @ ( complete_Sup_Sup_o @ A ) @ ( complete_Sup_Sup_o @ B4 ) ) ) ) ) ).
% 0.34/0.59
% 0.34/0.59 % cSup_subset_mono
% 0.34/0.59 thf(fact_3430_cSup__subset__mono,axiom,
% 0.34/0.59 ! [A: set_Pr4532377907799695533_nat_o,B4: set_Pr4532377907799695533_nat_o] :
% 0.34/0.59 ( ( A != bot_bo7824918357723582233_nat_o )
% 0.34/0.59 => ( ( condit4352657709669162033_nat_o @ B4 )
% 0.34/0.59 => ( ( ord_le2965882846123202637_nat_o @ A @ B4 )
% 0.34/0.59 => ( ord_le729326519192465773_nat_o @ ( comple4225350640907012154_nat_o @ A ) @ ( comple4225350640907012154_nat_o @ B4 ) ) ) ) ) ).
% 0.34/0.59
% 0.34/0.59 % cSup_subset_mono
% 0.34/0.59 thf(fact_3431_cSup__subset__mono,axiom,
% 0.34/0.59 ! [A: set_filter_nat,B4: set_filter_nat] :
% 0.34/0.59 ( ( A != bot_bo498966703094740906er_nat )
% 0.34/0.59 => ( ( condit4029664723990530684er_nat @ B4 )
% 0.34/0.59 => ( ( ord_le2426478655948331894er_nat @ A @ B4 )
% 0.34/0.59 => ( ord_le2510731241096832064er_nat @ ( comple2454491259644266675er_nat @ A ) @ ( comple2454491259644266675er_nat @ B4 ) ) ) ) ) ).
% 0.34/0.59
% 0.34/0.59 % cSup_subset_mono
% 0.34/0.59 thf(fact_3432_cSup__subset__mono,axiom,
% 0.34/0.59 ! [A: set_set_nat,B4: set_set_nat] :
% 0.34/0.59 ( ( A != bot_bot_set_set_nat )
% 0.34/0.59 => ( ( condit5477540289124974626et_nat @ B4 )
% 0.34/0.59 => ( ( ord_le6893508408891458716et_nat @ A @ B4 )
% 0.34/0.59 => ( ord_less_eq_set_nat @ ( comple7399068483239264473et_nat @ A ) @ ( comple7399068483239264473et_nat @ B4 ) ) ) ) ) ).
% 0.34/0.59
% 0.34/0.59 % cSup_subset_mono
% 0.34/0.59 thf(fact_3433_cSup__subset__mono,axiom,
% 0.34/0.59 ! [A: set_int,B4: set_int] :
% 0.34/0.59 ( ( A != bot_bot_set_int )
% 0.34/0.59 => ( ( condit2212336002400062152ve_int @ B4 )
% 0.34/0.59 => ( ( ord_less_eq_set_int @ A @ B4 )
% 0.34/0.59 => ( ord_less_eq_int @ ( complete_Sup_Sup_int @ A ) @ ( complete_Sup_Sup_int @ B4 ) ) ) ) ) ).
% 0.34/0.59
% 0.34/0.59 % cSup_subset_mono
% 0.34/0.59 thf(fact_3434_cSup__subset__mono,axiom,
% 0.34/0.59 ! [A: set_nat,B4: set_nat] :
% 0.34/0.59 ( ( A != bot_bot_set_nat )
% 0.34/0.59 => ( ( condit2214826472909112428ve_nat @ B4 )
% 0.34/0.59 => ( ( ord_less_eq_set_nat @ A @ B4 )
% 0.34/0.59 => ( ord_less_eq_nat @ ( complete_Sup_Sup_nat @ A ) @ ( complete_Sup_Sup_nat @ B4 ) ) ) ) ) ).
% 0.34/0.59
% 0.34/0.59 % cSup_subset_mono
% 0.34/0.59 thf(fact_3435_cSup__subset__mono,axiom,
% 0.34/0.59 ! [A: set_Product_unit,B4: set_Product_unit] :
% 0.34/0.59 ( ( A != bot_bo3957492148770167129t_unit )
% 0.34/0.59 => ( ( condit2080475194861589617t_unit @ B4 )
% 0.34/0.59 => ( ( ord_le3507040750410214029t_unit @ A @ B4 )
% 0.34/0.59 => ( ord_le3221252021190050221t_unit @ ( comple4687483117567863418t_unit @ A ) @ ( comple4687483117567863418t_unit @ B4 ) ) ) ) ) ).
% 0.34/0.59
% 0.34/0.59 % cSup_subset_mono
% 0.34/0.59 thf(fact_3436_cSup__insert__If,axiom,
% 0.34/0.59 ! [X4: set_set_nat,A2: set_nat] :
% 0.34/0.59 ( ( condit5477540289124974626et_nat @ X4 )
% 0.34/0.59 => ( ( ( X4 = bot_bot_set_set_nat )
% 0.34/0.59 => ( ( comple7399068483239264473et_nat @ ( insert_set_nat @ A2 @ X4 ) )
% 0.34/0.59 = A2 ) )
% 0.34/0.59 & ( ( X4 != bot_bot_set_set_nat )
% 0.34/0.59 => ( ( comple7399068483239264473et_nat @ ( insert_set_nat @ A2 @ X4 ) )
% 0.34/0.59 = ( sup_sup_set_nat @ A2 @ ( comple7399068483239264473et_nat @ X4 ) ) ) ) ) ) ).
% 0.34/0.59
% 0.34/0.59 % cSup_insert_If
% 0.34/0.59 thf(fact_3437_cSup__insert__If,axiom,
% 0.34/0.59 ! [X4: set_o,A2: $o] :
% 0.34/0.59 ( ( condit5488710616941104124bove_o @ X4 )
% 0.34/0.59 => ( ( complete_Sup_Sup_o @ ( insert_o @ A2 @ X4 ) )
% 0.34/0.59 = ( ( ( X4 = bot_bot_set_o )
% 0.34/0.59 => A2 )
% 0.34/0.59 & ( ( X4 != bot_bot_set_o )
% 0.34/0.59 => ( sup_sup_o @ A2 @ ( complete_Sup_Sup_o @ X4 ) ) ) ) ) ) ).
% 0.34/0.59
% 0.34/0.59 % cSup_insert_If
% 0.34/0.59 thf(fact_3438_cSup__insert__If,axiom,
% 0.34/0.59 ! [X4: set_int,A2: int] :
% 0.34/0.59 ( ( condit2212336002400062152ve_int @ X4 )
% 0.34/0.59 => ( ( ( X4 = bot_bot_set_int )
% 0.34/0.59 => ( ( complete_Sup_Sup_int @ ( insert_int @ A2 @ X4 ) )
% 0.34/0.59 = A2 ) )
% 0.34/0.59 & ( ( X4 != bot_bot_set_int )
% 0.34/0.59 => ( ( complete_Sup_Sup_int @ ( insert_int @ A2 @ X4 ) )
% 0.34/0.59 = ( sup_sup_int @ A2 @ ( complete_Sup_Sup_int @ X4 ) ) ) ) ) ) ).
% 0.34/0.59
% 0.34/0.59 % cSup_insert_If
% 0.34/0.59 thf(fact_3439_cSup__insert__If,axiom,
% 0.34/0.59 ! [X4: set_Pr4532377907799695533_nat_o,A2: produc3658429121746597890et_nat > $o] :
% 0.34/0.59 ( ( condit4352657709669162033_nat_o @ X4 )
% 0.34/0.59 => ( ( ( X4 = bot_bo7824918357723582233_nat_o )
% 0.34/0.59 => ( ( comple4225350640907012154_nat_o @ ( insert5175938949040314269_nat_o @ A2 @ X4 ) )
% 0.34/0.59 = A2 ) )
% 0.34/0.59 & ( ( X4 != bot_bo7824918357723582233_nat_o )
% 0.34/0.59 => ( ( comple4225350640907012154_nat_o @ ( insert5175938949040314269_nat_o @ A2 @ X4 ) )
% 0.34/0.59 = ( sup_su5453871518329203617_nat_o @ A2 @ ( comple4225350640907012154_nat_o @ X4 ) ) ) ) ) ) ).
% 0.34/0.59
% 0.34/0.59 % cSup_insert_If
% 0.34/0.59 thf(fact_3440_cSup__insert__If,axiom,
% 0.34/0.59 ! [X4: set_nat,A2: nat] :
% 0.34/0.59 ( ( condit2214826472909112428ve_nat @ X4 )
% 0.34/0.59 => ( ( ( X4 = bot_bot_set_nat )
% 0.34/0.59 => ( ( complete_Sup_Sup_nat @ ( insert_nat @ A2 @ X4 ) )
% 0.34/0.59 = A2 ) )
% 0.34/0.59 & ( ( X4 != bot_bot_set_nat )
% 0.34/0.59 => ( ( complete_Sup_Sup_nat @ ( insert_nat @ A2 @ X4 ) )
% 0.34/0.59 = ( sup_sup_nat @ A2 @ ( complete_Sup_Sup_nat @ X4 ) ) ) ) ) ) ).
% 0.34/0.59
% 0.34/0.59 % cSup_insert_If
% 0.34/0.59 thf(fact_3441_cSup__insert__If,axiom,
% 0.34/0.59 ! [X4: set_Product_unit,A2: product_unit] :
% 0.34/0.59 ( ( condit2080475194861589617t_unit @ X4 )
% 0.34/0.59 => ( ( ( X4 = bot_bo3957492148770167129t_unit )
% 0.34/0.59 => ( ( comple4687483117567863418t_unit @ ( insert_Product_unit @ A2 @ X4 ) )
% 0.34/0.59 = A2 ) )
% 0.34/0.59 & ( ( X4 != bot_bo3957492148770167129t_unit )
% 0.34/0.59 => ( ( comple4687483117567863418t_unit @ ( insert_Product_unit @ A2 @ X4 ) )
% 0.34/0.59 = ( sup_sup_Product_unit @ A2 @ ( comple4687483117567863418t_unit @ X4 ) ) ) ) ) ) ).
% 0.34/0.59
% 0.34/0.59 % cSup_insert_If
% 0.34/0.59 thf(fact_3442_cSup__insert,axiom,
% 0.34/0.59 ! [X4: set_set_nat,A2: set_nat] :
% 0.34/0.59 ( ( X4 != bot_bot_set_set_nat )
% 0.34/0.59 => ( ( condit5477540289124974626et_nat @ X4 )
% 0.34/0.59 => ( ( comple7399068483239264473et_nat @ ( insert_set_nat @ A2 @ X4 ) )
% 0.34/0.59 = ( sup_sup_set_nat @ A2 @ ( comple7399068483239264473et_nat @ X4 ) ) ) ) ) ).
% 0.34/0.59
% 0.34/0.59 % cSup_insert
% 0.34/0.59 thf(fact_3443_cSup__insert,axiom,
% 0.34/0.59 ! [X4: set_o,A2: $o] :
% 0.34/0.59 ( ( X4 != bot_bot_set_o )
% 0.34/0.59 => ( ( condit5488710616941104124bove_o @ X4 )
% 0.34/0.59 => ( ( complete_Sup_Sup_o @ ( insert_o @ A2 @ X4 ) )
% 0.34/0.59 = ( sup_sup_o @ A2 @ ( complete_Sup_Sup_o @ X4 ) ) ) ) ) ).
% 0.34/0.59
% 0.34/0.59 % cSup_insert
% 0.34/0.59 thf(fact_3444_cSup__insert,axiom,
% 0.34/0.59 ! [X4: set_int,A2: int] :
% 0.34/0.59 ( ( X4 != bot_bot_set_int )
% 0.34/0.59 => ( ( condit2212336002400062152ve_int @ X4 )
% 0.34/0.59 => ( ( complete_Sup_Sup_int @ ( insert_int @ A2 @ X4 ) )
% 0.34/0.59 = ( sup_sup_int @ A2 @ ( complete_Sup_Sup_int @ X4 ) ) ) ) ) ).
% 0.34/0.59
% 0.34/0.59 % cSup_insert
% 0.34/0.59 thf(fact_3445_cSup__insert,axiom,
% 0.34/0.59 ! [X4: set_Pr4532377907799695533_nat_o,A2: produc3658429121746597890et_nat > $o] :
% 0.34/0.59 ( ( X4 != bot_bo7824918357723582233_nat_o )
% 0.34/0.59 => ( ( condit4352657709669162033_nat_o @ X4 )
% 0.34/0.59 => ( ( comple4225350640907012154_nat_o @ ( insert5175938949040314269_nat_o @ A2 @ X4 ) )
% 0.34/0.59 = ( sup_su5453871518329203617_nat_o @ A2 @ ( comple4225350640907012154_nat_o @ X4 ) ) ) ) ) ).
% 0.34/0.59
% 0.34/0.59 % cSup_insert
% 0.34/0.59 thf(fact_3446_cSup__insert,axiom,
% 0.34/0.59 ! [X4: set_nat,A2: nat] :
% 0.34/0.59 ( ( X4 != bot_bot_set_nat )
% 0.34/0.59 => ( ( condit2214826472909112428ve_nat @ X4 )
% 0.34/0.59 => ( ( complete_Sup_Sup_nat @ ( insert_nat @ A2 @ X4 ) )
% 0.34/0.59 = ( sup_sup_nat @ A2 @ ( complete_Sup_Sup_nat @ X4 ) ) ) ) ) ).
% 0.34/0.59
% 0.34/0.59 % cSup_insert
% 0.34/0.59 thf(fact_3447_cSup__insert,axiom,
% 0.34/0.59 ! [X4: set_Product_unit,A2: product_unit] :
% 0.34/0.59 ( ( X4 != bot_bo3957492148770167129t_unit )
% 0.34/0.59 => ( ( condit2080475194861589617t_unit @ X4 )
% 0.34/0.59 => ( ( comple4687483117567863418t_unit @ ( insert_Product_unit @ A2 @ X4 ) )
% 0.34/0.59 = ( sup_sup_Product_unit @ A2 @ ( comple4687483117567863418t_unit @ X4 ) ) ) ) ) ).
% 0.34/0.59
% 0.34/0.59 % cSup_insert
% 0.34/0.59 thf(fact_3448_cSup__union__distrib,axiom,
% 0.34/0.59 ! [A: set_set_nat,B4: set_set_nat] :
% 0.34/0.59 ( ( A != bot_bot_set_set_nat )
% 0.34/0.59 => ( ( condit5477540289124974626et_nat @ A )
% 0.34/0.59 => ( ( B4 != bot_bot_set_set_nat )
% 0.34/0.59 => ( ( condit5477540289124974626et_nat @ B4 )
% 0.34/0.59 => ( ( comple7399068483239264473et_nat @ ( sup_sup_set_set_nat @ A @ B4 ) )
% 0.34/0.59 = ( sup_sup_set_nat @ ( comple7399068483239264473et_nat @ A ) @ ( comple7399068483239264473et_nat @ B4 ) ) ) ) ) ) ) ).
% 0.34/0.59
% 0.34/0.59 % cSup_union_distrib
% 0.34/0.59 thf(fact_3449_cSup__union__distrib,axiom,
% 0.34/0.59 ! [A: set_o,B4: set_o] :
% 0.34/0.59 ( ( A != bot_bot_set_o )
% 0.34/0.59 => ( ( condit5488710616941104124bove_o @ A )
% 0.34/0.59 => ( ( B4 != bot_bot_set_o )
% 0.34/0.59 => ( ( condit5488710616941104124bove_o @ B4 )
% 0.34/0.59 => ( ( complete_Sup_Sup_o @ ( sup_sup_set_o @ A @ B4 ) )
% 0.34/0.59 = ( sup_sup_o @ ( complete_Sup_Sup_o @ A ) @ ( complete_Sup_Sup_o @ B4 ) ) ) ) ) ) ) ).
% 0.34/0.59
% 0.34/0.59 % cSup_union_distrib
% 0.34/0.59 thf(fact_3450_cSup__union__distrib,axiom,
% 0.34/0.59 ! [A: set_int,B4: set_int] :
% 0.34/0.59 ( ( A != bot_bot_set_int )
% 0.34/0.59 => ( ( condit2212336002400062152ve_int @ A )
% 0.34/0.59 => ( ( B4 != bot_bot_set_int )
% 0.34/0.59 => ( ( condit2212336002400062152ve_int @ B4 )
% 0.34/0.59 => ( ( complete_Sup_Sup_int @ ( sup_sup_set_int @ A @ B4 ) )
% 0.34/0.59 = ( sup_sup_int @ ( complete_Sup_Sup_int @ A ) @ ( complete_Sup_Sup_int @ B4 ) ) ) ) ) ) ) ).
% 0.34/0.59
% 0.34/0.59 % cSup_union_distrib
% 0.34/0.59 thf(fact_3451_cSup__union__distrib,axiom,
% 0.34/0.59 ! [A: set_Pr4532377907799695533_nat_o,B4: set_Pr4532377907799695533_nat_o] :
% 0.34/0.59 ( ( A != bot_bo7824918357723582233_nat_o )
% 0.34/0.59 => ( ( condit4352657709669162033_nat_o @ A )
% 0.34/0.59 => ( ( B4 != bot_bo7824918357723582233_nat_o )
% 0.34/0.59 => ( ( condit4352657709669162033_nat_o @ B4 )
% 0.34/0.59 => ( ( comple4225350640907012154_nat_o @ ( sup_su5209123915105501825_nat_o @ A @ B4 ) )
% 0.34/0.59 = ( sup_su5453871518329203617_nat_o @ ( comple4225350640907012154_nat_o @ A ) @ ( comple4225350640907012154_nat_o @ B4 ) ) ) ) ) ) ) ).
% 0.34/0.59
% 0.34/0.59 % cSup_union_distrib
% 0.34/0.59 thf(fact_3452_cSup__union__distrib,axiom,
% 0.34/0.59 ! [A: set_nat,B4: set_nat] :
% 0.34/0.59 ( ( A != bot_bot_set_nat )
% 0.34/0.59 => ( ( condit2214826472909112428ve_nat @ A )
% 0.34/0.59 => ( ( B4 != bot_bot_set_nat )
% 0.34/0.59 => ( ( condit2214826472909112428ve_nat @ B4 )
% 0.34/0.59 => ( ( complete_Sup_Sup_nat @ ( sup_sup_set_nat @ A @ B4 ) )
% 0.34/0.59 = ( sup_sup_nat @ ( complete_Sup_Sup_nat @ A ) @ ( complete_Sup_Sup_nat @ B4 ) ) ) ) ) ) ) ).
% 0.34/0.59
% 0.34/0.59 % cSup_union_distrib
% 0.34/0.59 thf(fact_3453_cSup__union__distrib,axiom,
% 0.34/0.59 ! [A: set_Product_unit,B4: set_Product_unit] :
% 0.34/0.59 ( ( A != bot_bo3957492148770167129t_unit )
% 0.34/0.59 => ( ( condit2080475194861589617t_unit @ A )
% 0.34/0.59 => ( ( B4 != bot_bo3957492148770167129t_unit )
% 0.34/0.59 => ( ( condit2080475194861589617t_unit @ B4 )
% 0.34/0.59 => ( ( comple4687483117567863418t_unit @ ( sup_su793286257634532545t_unit @ A @ B4 ) )
% 0.34/0.59 = ( sup_sup_Product_unit @ ( comple4687483117567863418t_unit @ A ) @ ( comple4687483117567863418t_unit @ B4 ) ) ) ) ) ) ) ).
% 0.34/0.59
% 0.34/0.59 % cSup_union_distrib
% 0.34/0.59 thf(fact_3454_fun__upd__image,axiom,
% 0.34/0.59 ! [X: code_integer,A: set_Code_integer,F: code_integer > code_integer,Y: code_integer] :
% 0.34/0.59 ( ( ( member_Code_integer @ X @ A )
% 0.34/0.59 => ( ( image_4470545334726330049nteger @ ( fun_up8355007323143964473nteger @ F @ X @ Y ) @ A )
% 0.34/0.59 = ( insert_Code_integer @ Y @ ( image_4470545334726330049nteger @ F @ ( minus_2355218937544613996nteger @ A @ ( insert_Code_integer @ X @ bot_bo3990330152332043303nteger ) ) ) ) ) )
% 0.34/0.59 & ( ~ ( member_Code_integer @ X @ A )
% 0.34/0.59 => ( ( image_4470545334726330049nteger @ ( fun_up8355007323143964473nteger @ F @ X @ Y ) @ A )
% 0.34/0.59 = ( image_4470545334726330049nteger @ F @ A ) ) ) ) ).
% 0.34/0.59
% 0.34/0.59 % fun_upd_image
% 0.34/0.59 thf(fact_3455_fun__upd__image,axiom,
% 0.34/0.59 ! [X: product_unit,A: set_Product_unit,F: product_unit > nat,Y: nat] :
% 0.34/0.59 ( ( ( member_Product_unit @ X @ A )
% 0.34/0.59 => ( ( image_875570014554754200it_nat @ ( fun_up6473050886840373792it_nat @ F @ X @ Y ) @ A )
% 0.34/0.59 = ( insert_nat @ Y @ ( image_875570014554754200it_nat @ F @ ( minus_6452836326544984404t_unit @ A @ ( insert_Product_unit @ X @ bot_bo3957492148770167129t_unit ) ) ) ) ) )
% 0.34/0.59 & ( ~ ( member_Product_unit @ X @ A )
% 0.34/0.59 => ( ( image_875570014554754200it_nat @ ( fun_up6473050886840373792it_nat @ F @ X @ Y ) @ A )
% 0.34/0.59 = ( image_875570014554754200it_nat @ F @ A ) ) ) ) ).
% 0.34/0.59
% 0.34/0.59 % fun_upd_image
% 0.34/0.59 thf(fact_3456_fun__upd__image,axiom,
% 0.34/0.59 ! [X: product_unit,A: set_Product_unit,F: product_unit > int,Y: int] :
% 0.34/0.59 ( ( ( member_Product_unit @ X @ A )
% 0.34/0.59 => ( ( image_873079544045703924it_int @ ( fun_up6470560416331323516it_int @ F @ X @ Y ) @ A )
% 0.34/0.59 = ( insert_int @ Y @ ( image_873079544045703924it_int @ F @ ( minus_6452836326544984404t_unit @ A @ ( insert_Product_unit @ X @ bot_bo3957492148770167129t_unit ) ) ) ) ) )
% 0.34/0.59 & ( ~ ( member_Product_unit @ X @ A )
% 0.34/0.59 => ( ( image_873079544045703924it_int @ ( fun_up6470560416331323516it_int @ F @ X @ Y ) @ A )
% 0.34/0.59 = ( image_873079544045703924it_int @ F @ A ) ) ) ) ).
% 0.34/0.59
% 0.34/0.59 % fun_upd_image
% 0.34/0.59 thf(fact_3457_fun__upd__image,axiom,
% 0.34/0.59 ! [X: product_unit,A: set_Product_unit,F: product_unit > $o,Y: $o] :
% 0.34/0.59 ( ( ( member_Product_unit @ X @ A )
% 0.34/0.59 => ( ( image_Product_unit_o2 @ ( fun_up5432024876597553480unit_o @ F @ X @ Y ) @ A )
% 0.34/0.59 = ( insert_o @ Y @ ( image_Product_unit_o2 @ F @ ( minus_6452836326544984404t_unit @ A @ ( insert_Product_unit @ X @ bot_bo3957492148770167129t_unit ) ) ) ) ) )
% 0.34/0.59 & ( ~ ( member_Product_unit @ X @ A )
% 0.34/0.59 => ( ( image_Product_unit_o2 @ ( fun_up5432024876597553480unit_o @ F @ X @ Y ) @ A )
% 0.34/0.59 = ( image_Product_unit_o2 @ F @ A ) ) ) ) ).
% 0.34/0.59
% 0.34/0.59 % fun_upd_image
% 0.34/0.59 thf(fact_3458_fun__upd__image,axiom,
% 0.34/0.59 ! [X: product_unit,A: set_Product_unit,F: product_unit > product_unit,Y: product_unit] :
% 0.34/0.59 ( ( ( member_Product_unit @ X @ A )
% 0.34/0.59 => ( ( image_405062704495631173t_unit @ ( fun_up6875000483049295805t_unit @ F @ X @ Y ) @ A )
% 0.34/0.59 = ( insert_Product_unit @ Y @ ( image_405062704495631173t_unit @ F @ ( minus_6452836326544984404t_unit @ A @ ( insert_Product_unit @ X @ bot_bo3957492148770167129t_unit ) ) ) ) ) )
% 0.34/0.59 & ( ~ ( member_Product_unit @ X @ A )
% 0.34/0.59 => ( ( image_405062704495631173t_unit @ ( fun_up6875000483049295805t_unit @ F @ X @ Y ) @ A )
% 0.34/0.59 = ( image_405062704495631173t_unit @ F @ A ) ) ) ) ).
% 0.34/0.59
% 0.34/0.59 % fun_upd_image
% 0.34/0.59 thf(fact_3459_fun__upd__image,axiom,
% 0.34/0.59 ! [X: $o,A: set_o,F: $o > nat,Y: nat] :
% 0.34/0.59 ( ( ( member_o @ X @ A )
% 0.34/0.59 => ( ( image_o_nat2 @ ( fun_upd_o_nat @ F @ X @ Y ) @ A )
% 0.34/0.59 = ( insert_nat @ Y @ ( image_o_nat2 @ F @ ( minus_minus_set_o @ A @ ( insert_o @ X @ bot_bot_set_o ) ) ) ) ) )
% 0.34/0.59 & ( ~ ( member_o @ X @ A )
% 0.34/0.59 => ( ( image_o_nat2 @ ( fun_upd_o_nat @ F @ X @ Y ) @ A )
% 0.34/0.59 = ( image_o_nat2 @ F @ A ) ) ) ) ).
% 0.34/0.59
% 0.34/0.59 % fun_upd_image
% 0.34/0.59 thf(fact_3460_fun__upd__image,axiom,
% 0.34/0.59 ! [X: $o,A: set_o,F: $o > int,Y: int] :
% 0.34/0.59 ( ( ( member_o @ X @ A )
% 0.34/0.59 => ( ( image_o_int2 @ ( fun_upd_o_int @ F @ X @ Y ) @ A )
% 0.34/0.59 = ( insert_int @ Y @ ( image_o_int2 @ F @ ( minus_minus_set_o @ A @ ( insert_o @ X @ bot_bot_set_o ) ) ) ) ) )
% 0.34/0.59 & ( ~ ( member_o @ X @ A )
% 0.34/0.59 => ( ( image_o_int2 @ ( fun_upd_o_int @ F @ X @ Y ) @ A )
% 0.34/0.59 = ( image_o_int2 @ F @ A ) ) ) ) ).
% 0.34/0.59
% 0.34/0.59 % fun_upd_image
% 0.34/0.59 thf(fact_3461_fun__upd__image,axiom,
% 0.34/0.59 ! [X: $o,A: set_o,F: $o > $o,Y: $o] :
% 0.34/0.59 ( ( ( member_o @ X @ A )
% 0.34/0.59 => ( ( image_o_o2 @ ( fun_upd_o_o @ F @ X @ Y ) @ A )
% 0.34/0.59 = ( insert_o @ Y @ ( image_o_o2 @ F @ ( minus_minus_set_o @ A @ ( insert_o @ X @ bot_bot_set_o ) ) ) ) ) )
% 0.34/0.59 & ( ~ ( member_o @ X @ A )
% 0.34/0.59 => ( ( image_o_o2 @ ( fun_upd_o_o @ F @ X @ Y ) @ A )
% 0.34/0.59 = ( image_o_o2 @ F @ A ) ) ) ) ).
% 0.34/0.59
% 0.34/0.59 % fun_upd_image
% 0.34/0.59 thf(fact_3462_fun__upd__image,axiom,
% 0.34/0.59 ! [X: $o,A: set_o,F: $o > product_unit,Y: product_unit] :
% 0.34/0.59 ( ( ( member_o @ X @ A )
% 0.34/0.59 => ( ( image_o_Product_unit2 @ ( fun_up846054379331416818t_unit @ F @ X @ Y ) @ A )
% 0.34/0.59 = ( insert_Product_unit @ Y @ ( image_o_Product_unit2 @ F @ ( minus_minus_set_o @ A @ ( insert_o @ X @ bot_bot_set_o ) ) ) ) ) )
% 0.34/0.59 & ( ~ ( member_o @ X @ A )
% 0.34/0.59 => ( ( image_o_Product_unit2 @ ( fun_up846054379331416818t_unit @ F @ X @ Y ) @ A )
% 0.34/0.59 = ( image_o_Product_unit2 @ F @ A ) ) ) ) ).
% 0.34/0.59
% 0.34/0.59 % fun_upd_image
% 0.34/0.59 thf(fact_3463_fun__upd__image,axiom,
% 0.34/0.59 ! [X: int,A: set_int,F: int > nat,Y: nat] :
% 0.34/0.59 ( ( ( member_int @ X @ A )
% 0.34/0.59 => ( ( image_int_nat2 @ ( fun_upd_int_nat @ F @ X @ Y ) @ A )
% 0.34/0.59 = ( insert_nat @ Y @ ( image_int_nat2 @ F @ ( minus_minus_set_int @ A @ ( insert_int @ X @ bot_bot_set_int ) ) ) ) ) )
% 0.34/0.59 & ( ~ ( member_int @ X @ A )
% 0.34/0.59 => ( ( image_int_nat2 @ ( fun_upd_int_nat @ F @ X @ Y ) @ A )
% 0.34/0.59 = ( image_int_nat2 @ F @ A ) ) ) ) ).
% 0.34/0.59
% 0.34/0.59 % fun_upd_image
% 0.34/0.59 thf(fact_3464_Sup__fin_Oeq__fold,axiom,
% 0.34/0.59 ! [A: set_int,X: int] :
% 0.34/0.59 ( ( finite_finite_int @ A )
% 0.34/0.59 => ( ( lattic1091506334969745077in_int @ ( insert_int @ X @ A ) )
% 0.34/0.59 = ( finite_fold_int_int @ sup_sup_int @ X @ A ) ) ) ).
% 0.34/0.59
% 0.34/0.59 % Sup_fin.eq_fold
% 0.34/0.59 thf(fact_3465_Sup__fin_Oeq__fold,axiom,
% 0.34/0.59 ! [A: set_o,X: $o] :
% 0.34/0.59 ( ( finite_finite_o @ A )
% 0.34/0.59 => ( ( lattic1508158080041050831_fin_o @ ( insert_o @ X @ A ) )
% 0.34/0.59 = ( finite_fold_o_o @ sup_sup_o @ X @ A ) ) ) ).
% 0.34/0.59
% 0.34/0.59 % Sup_fin.eq_fold
% 0.34/0.59 thf(fact_3466_Sup__fin_Oeq__fold,axiom,
% 0.34/0.59 ! [A: set_nat,X: nat] :
% 0.34/0.59 ( ( finite_finite_nat @ A )
% 0.34/0.59 => ( ( lattic1093996805478795353in_nat @ ( insert_nat @ X @ A ) )
% 0.34/0.59 = ( finite_fold_nat_nat @ sup_sup_nat @ X @ A ) ) ) ).
% 0.34/0.59
% 0.34/0.59 % Sup_fin.eq_fold
% 0.34/0.59 thf(fact_3467_Sup__fin_Oeq__fold,axiom,
% 0.34/0.59 ! [A: set_set_nat,X: set_nat] :
% 0.34/0.59 ( ( finite1152437895449049373et_nat @ A )
% 0.34/0.59 => ( ( lattic3835124923745554447et_nat @ ( insert_set_nat @ X @ A ) )
% 0.34/0.59 = ( finite677925301803182934et_nat @ sup_sup_set_nat @ X @ A ) ) ) ).
% 0.34/0.59
% 0.34/0.59 % Sup_fin.eq_fold
% 0.34/0.59 thf(fact_3468_Sup__fin_Oeq__fold,axiom,
% 0.34/0.59 ! [A: set_assn,X: assn] :
% 0.34/0.59 ( ( finite_finite_assn @ A )
% 0.34/0.59 => ( ( lattic2150320897289308081n_assn @ ( insert_assn @ X @ A ) )
% 0.34/0.59 = ( finite1440822074540989594n_assn @ sup_sup_assn @ X @ A ) ) ) ).
% 0.34/0.59
% 0.34/0.59 % Sup_fin.eq_fold
% 0.34/0.59 thf(fact_3469_Sup__fin_Oeq__fold,axiom,
% 0.34/0.59 ! [A: set_Product_unit,X: product_unit] :
% 0.34/0.59 ( ( finite4290736615968046902t_unit @ A )
% 0.34/0.59 => ( ( lattic5294303975357428420t_unit @ ( insert_Product_unit @ X @ A ) )
% 0.34/0.59 = ( finite8834431295148555784t_unit @ sup_sup_Product_unit @ X @ A ) ) ) ).
% 0.34/0.59
% 0.34/0.59 % Sup_fin.eq_fold
% 0.34/0.59 thf(fact_3470_Inf__fin_Oeq__fold,axiom,
% 0.34/0.59 ! [A: set_int,X: int] :
% 0.34/0.59 ( ( finite_finite_int @ A )
% 0.34/0.59 => ( ( lattic5235898064620869839in_int @ ( insert_int @ X @ A ) )
% 0.34/0.59 = ( finite_fold_int_int @ inf_inf_int @ X @ A ) ) ) ).
% 0.34/0.59
% 0.34/0.59 % Inf_fin.eq_fold
% 0.34/0.59 thf(fact_3471_Inf__fin_Oeq__fold,axiom,
% 0.34/0.59 ! [A: set_o,X: $o] :
% 0.34/0.59 ( ( finite_finite_o @ A )
% 0.34/0.59 => ( ( lattic4107685809792843317_fin_o @ ( insert_o @ X @ A ) )
% 0.34/0.59 = ( finite_fold_o_o @ inf_inf_o @ X @ A ) ) ) ).
% 0.34/0.59
% 0.34/0.59 % Inf_fin.eq_fold
% 0.34/0.59 thf(fact_3472_Inf__fin_Oeq__fold,axiom,
% 0.34/0.59 ! [A: set_nat,X: nat] :
% 0.34/0.59 ( ( finite_finite_nat @ A )
% 0.34/0.59 => ( ( lattic5238388535129920115in_nat @ ( insert_nat @ X @ A ) )
% 0.34/0.59 = ( finite_fold_nat_nat @ inf_inf_nat @ X @ A ) ) ) ).
% 0.34/0.59
% 0.34/0.59 % Inf_fin.eq_fold
% 0.34/0.59 thf(fact_3473_Inf__fin_Oeq__fold,axiom,
% 0.34/0.59 ! [A: set_set_nat,X: set_nat] :
% 0.34/0.59 ( ( finite1152437895449049373et_nat @ A )
% 0.34/0.59 => ( ( lattic3014633134055518761et_nat @ ( insert_set_nat @ X @ A ) )
% 0.34/0.59 = ( finite677925301803182934et_nat @ inf_inf_set_nat @ X @ A ) ) ) ).
% 0.34/0.59
% 0.34/0.59 % Inf_fin.eq_fold
% 0.34/0.59 thf(fact_3474_Inf__fin_Oeq__fold,axiom,
% 0.34/0.59 ! [A: set_assn,X: assn] :
% 0.34/0.59 ( ( finite_finite_assn @ A )
% 0.34/0.59 => ( ( lattic47131356835913163n_assn @ ( insert_assn @ X @ A ) )
% 0.34/0.59 = ( finite1440822074540989594n_assn @ inf_inf_assn @ X @ A ) ) ) ).
% 0.34/0.59
% 0.34/0.59 % Inf_fin.eq_fold
% 0.34/0.59 thf(fact_3475_Inf__fin_Oeq__fold,axiom,
% 0.34/0.59 ! [A: set_Product_unit,X: product_unit] :
% 0.34/0.59 ( ( finite4290736615968046902t_unit @ A )
% 0.34/0.59 => ( ( lattic1263872656861969706t_unit @ ( insert_Product_unit @ X @ A ) )
% 0.34/0.59 = ( finite8834431295148555784t_unit @ inf_inf_Product_unit @ X @ A ) ) ) ).
% 0.34/0.59
% 0.34/0.59 % Inf_fin.eq_fold
% 0.34/0.59 thf(fact_3476_wf__bounded__set,axiom,
% 0.34/0.59 ! [R2: set_Pr1261947904930325089at_nat,Ub: nat > set_Code_integer,F: nat > set_Code_integer] :
% 0.34/0.59 ( ! [A4: nat,B2: nat] :
% 0.34/0.59 ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ B2 @ A4 ) @ R2 )
% 0.34/0.59 => ( ( finite6017078050557962740nteger @ ( Ub @ A4 ) )
% 0.34/0.59 & ( ord_le7084787975880047091nteger @ ( Ub @ B2 ) @ ( Ub @ A4 ) )
% 0.34/0.59 & ( ord_le7084787975880047091nteger @ ( F @ B2 ) @ ( Ub @ A4 ) )
% 0.34/0.59 & ( ord_le1307284697595431911nteger @ ( F @ A4 ) @ ( F @ B2 ) ) ) )
% 0.34/0.59 => ( wf_nat @ R2 ) ) ).
% 0.34/0.59
% 0.34/0.59 % wf_bounded_set
% 0.34/0.59 thf(fact_3477_wf__bounded__set,axiom,
% 0.34/0.59 ! [R2: set_Pr958786334691620121nt_int,Ub: int > set_Code_integer,F: int > set_Code_integer] :
% 0.34/0.59 ( ! [A4: int,B2: int] :
% 0.34/0.59 ( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ B2 @ A4 ) @ R2 )
% 0.34/0.59 => ( ( finite6017078050557962740nteger @ ( Ub @ A4 ) )
% 0.34/0.59 & ( ord_le7084787975880047091nteger @ ( Ub @ B2 ) @ ( Ub @ A4 ) )
% 0.34/0.59 & ( ord_le7084787975880047091nteger @ ( F @ B2 ) @ ( Ub @ A4 ) )
% 0.34/0.59 & ( ord_le1307284697595431911nteger @ ( F @ A4 ) @ ( F @ B2 ) ) ) )
% 0.34/0.59 => ( wf_int @ R2 ) ) ).
% 0.34/0.59
% 0.34/0.59 % wf_bounded_set
% 0.34/0.59 thf(fact_3478_wf__bounded__set,axiom,
% 0.34/0.59 ! [R2: set_Pr1261947904930325089at_nat,Ub: nat > set_nat,F: nat > set_nat] :
% 0.34/0.59 ( ! [A4: nat,B2: nat] :
% 0.34/0.59 ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ B2 @ A4 ) @ R2 )
% 0.34/0.59 => ( ( finite_finite_nat @ ( Ub @ A4 ) )
% 0.34/0.59 & ( ord_less_eq_set_nat @ ( Ub @ B2 ) @ ( Ub @ A4 ) )
% 0.34/0.59 & ( ord_less_eq_set_nat @ ( F @ B2 ) @ ( Ub @ A4 ) )
% 0.34/0.59 & ( ord_less_set_nat @ ( F @ A4 ) @ ( F @ B2 ) ) ) )
% 0.34/0.59 => ( wf_nat @ R2 ) ) ).
% 0.34/0.59
% 0.34/0.59 % wf_bounded_set
% 0.34/0.59 thf(fact_3479_wf__bounded__set,axiom,
% 0.34/0.59 ! [R2: set_Pr958786334691620121nt_int,Ub: int > set_nat,F: int > set_nat] :
% 0.34/0.59 ( ! [A4: int,B2: int] :
% 0.34/0.59 ( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ B2 @ A4 ) @ R2 )
% 0.34/0.59 => ( ( finite_finite_nat @ ( Ub @ A4 ) )
% 0.34/0.59 & ( ord_less_eq_set_nat @ ( Ub @ B2 ) @ ( Ub @ A4 ) )
% 0.34/0.59 & ( ord_less_eq_set_nat @ ( F @ B2 ) @ ( Ub @ A4 ) )
% 0.34/0.59 & ( ord_less_set_nat @ ( F @ A4 ) @ ( F @ B2 ) ) ) )
% 0.34/0.59 => ( wf_int @ R2 ) ) ).
% 0.34/0.59
% 0.34/0.59 % wf_bounded_set
% 0.34/0.59 thf(fact_3480_psubset__insert__iff,axiom,
% 0.34/0.59 ! [A: set_Product_unit,X: product_unit,B4: set_Product_unit] :
% 0.34/0.59 ( ( ord_le8056459307392131481t_unit @ A @ ( insert_Product_unit @ X @ B4 ) )
% 0.34/0.59 = ( ( ( member_Product_unit @ X @ B4 )
% 0.34/0.59 => ( ord_le8056459307392131481t_unit @ A @ B4 ) )
% 0.34/0.59 & ( ~ ( member_Product_unit @ X @ B4 )
% 0.34/0.59 => ( ( ( member_Product_unit @ X @ A )
% 0.34/0.59 => ( ord_le8056459307392131481t_unit @ ( minus_6452836326544984404t_unit @ A @ ( insert_Product_unit @ X @ bot_bo3957492148770167129t_unit ) ) @ B4 ) )
% 0.34/0.59 & ( ~ ( member_Product_unit @ X @ A )
% 0.34/0.59 => ( ord_le3507040750410214029t_unit @ A @ B4 ) ) ) ) ) ) ).
% 0.34/0.59
% 0.34/0.59 % psubset_insert_iff
% 0.34/0.59 thf(fact_3481_psubset__insert__iff,axiom,
% 0.34/0.59 ! [A: set_o,X: $o,B4: set_o] :
% 0.34/0.59 ( ( ord_less_set_o @ A @ ( insert_o @ X @ B4 ) )
% 0.34/0.59 = ( ( ( member_o @ X @ B4 )
% 0.34/0.59 => ( ord_less_set_o @ A @ B4 ) )
% 0.34/0.60 & ( ~ ( member_o @ X @ B4 )
% 0.34/0.60 => ( ( ( member_o @ X @ A )
% 0.34/0.60 => ( ord_less_set_o @ ( minus_minus_set_o @ A @ ( insert_o @ X @ bot_bot_set_o ) ) @ B4 ) )
% 0.34/0.60 & ( ~ ( member_o @ X @ A )
% 0.34/0.60 => ( ord_less_eq_set_o @ A @ B4 ) ) ) ) ) ) ).
% 0.34/0.60
% 0.34/0.60 % psubset_insert_iff
% 0.34/0.60 thf(fact_3482_psubset__insert__iff,axiom,
% 0.34/0.60 ! [A: set_int,X: int,B4: set_int] :
% 0.34/0.60 ( ( ord_less_set_int @ A @ ( insert_int @ X @ B4 ) )
% 0.34/0.60 = ( ( ( member_int @ X @ B4 )
% 0.34/0.60 => ( ord_less_set_int @ A @ B4 ) )
% 0.34/0.60 & ( ~ ( member_int @ X @ B4 )
% 0.34/0.60 => ( ( ( member_int @ X @ A )
% 0.34/0.60 => ( ord_less_set_int @ ( minus_minus_set_int @ A @ ( insert_int @ X @ bot_bot_set_int ) ) @ B4 ) )
% 0.34/0.60 & ( ~ ( member_int @ X @ A )
% 0.34/0.60 => ( ord_less_eq_set_int @ A @ B4 ) ) ) ) ) ) ).
% 0.34/0.60
% 0.34/0.60 % psubset_insert_iff
% 0.34/0.60 thf(fact_3483_psubset__insert__iff,axiom,
% 0.34/0.60 ! [A: set_Pr4532377907799695533_nat_o,X: produc3658429121746597890et_nat > $o,B4: set_Pr4532377907799695533_nat_o] :
% 0.34/0.60 ( ( ord_le2453136405763929_nat_o @ A @ ( insert5175938949040314269_nat_o @ X @ B4 ) )
% 0.34/0.60 = ( ( ( member6576561426505652726_nat_o @ X @ B4 )
% 0.34/0.60 => ( ord_le2453136405763929_nat_o @ A @ B4 ) )
% 0.34/0.60 & ( ~ ( member6576561426505652726_nat_o @ X @ B4 )
% 0.34/0.60 => ( ( ( member6576561426505652726_nat_o @ X @ A )
% 0.34/0.60 => ( ord_le2453136405763929_nat_o @ ( minus_1801376950450012436_nat_o @ A @ ( insert5175938949040314269_nat_o @ X @ bot_bo7824918357723582233_nat_o ) ) @ B4 ) )
% 0.34/0.60 & ( ~ ( member6576561426505652726_nat_o @ X @ A )
% 0.34/0.60 => ( ord_le2965882846123202637_nat_o @ A @ B4 ) ) ) ) ) ) ).
% 0.34/0.60
% 0.34/0.60 % psubset_insert_iff
% 0.34/0.60 thf(fact_3484_psubset__insert__iff,axiom,
% 0.34/0.60 ! [A: set_nat,X: nat,B4: set_nat] :
% 0.34/0.60 ( ( ord_less_set_nat @ A @ ( insert_nat @ X @ B4 ) )
% 0.34/0.60 = ( ( ( member_nat @ X @ B4 )
% 0.34/0.60 => ( ord_less_set_nat @ A @ B4 ) )
% 0.34/0.60 & ( ~ ( member_nat @ X @ B4 )
% 0.34/0.60 => ( ( ( member_nat @ X @ A )
% 0.34/0.60 => ( ord_less_set_nat @ ( minus_minus_set_nat @ A @ ( insert_nat @ X @ bot_bot_set_nat ) ) @ B4 ) )
% 0.34/0.60 & ( ~ ( member_nat @ X @ A )
% 0.34/0.60 => ( ord_less_eq_set_nat @ A @ B4 ) ) ) ) ) ) ).
% 0.34/0.60
% 0.34/0.60 % psubset_insert_iff
% 0.34/0.60 thf(fact_3485_finite__induct__select,axiom,
% 0.34/0.60 ! [S: set_Code_integer,P: set_Code_integer > $o] :
% 0.34/0.60 ( ( finite6017078050557962740nteger @ S )
% 0.34/0.60 => ( ( P @ bot_bo3990330152332043303nteger )
% 0.34/0.60 => ( ! [T3: set_Code_integer] :
% 0.34/0.60 ( ( ord_le1307284697595431911nteger @ T3 @ S )
% 0.34/0.60 => ( ( P @ T3 )
% 0.34/0.60 => ? [X6: code_integer] :
% 0.34/0.60 ( ( member_Code_integer @ X6 @ ( minus_2355218937544613996nteger @ S @ T3 ) )
% 0.34/0.60 & ( P @ ( insert_Code_integer @ X6 @ T3 ) ) ) ) )
% 0.34/0.60 => ( P @ S ) ) ) ) ).
% 0.34/0.60
% 0.34/0.60 % finite_induct_select
% 0.34/0.60 thf(fact_3486_finite__induct__select,axiom,
% 0.34/0.60 ! [S: set_Product_unit,P: set_Product_unit > $o] :
% 0.34/0.60 ( ( finite4290736615968046902t_unit @ S )
% 0.34/0.60 => ( ( P @ bot_bo3957492148770167129t_unit )
% 0.34/0.60 => ( ! [T3: set_Product_unit] :
% 0.34/0.60 ( ( ord_le8056459307392131481t_unit @ T3 @ S )
% 0.34/0.60 => ( ( P @ T3 )
% 0.34/0.60 => ? [X6: product_unit] :
% 0.34/0.60 ( ( member_Product_unit @ X6 @ ( minus_6452836326544984404t_unit @ S @ T3 ) )
% 0.34/0.60 & ( P @ ( insert_Product_unit @ X6 @ T3 ) ) ) ) )
% 0.34/0.60 => ( P @ S ) ) ) ) ).
% 0.34/0.60
% 0.34/0.60 % finite_induct_select
% 0.34/0.60 thf(fact_3487_finite__induct__select,axiom,
% 0.34/0.60 ! [S: set_o,P: set_o > $o] :
% 0.34/0.60 ( ( finite_finite_o @ S )
% 0.34/0.60 => ( ( P @ bot_bot_set_o )
% 0.34/0.60 => ( ! [T3: set_o] :
% 0.34/0.60 ( ( ord_less_set_o @ T3 @ S )
% 0.34/0.60 => ( ( P @ T3 )
% 0.34/0.60 => ? [X6: $o] :
% 0.34/0.60 ( ( member_o @ X6 @ ( minus_minus_set_o @ S @ T3 ) )
% 0.34/0.60 & ( P @ ( insert_o @ X6 @ T3 ) ) ) ) )
% 0.34/0.60 => ( P @ S ) ) ) ) ).
% 0.34/0.60
% 0.34/0.60 % finite_induct_select
% 0.34/0.60 thf(fact_3488_finite__induct__select,axiom,
% 0.34/0.60 ! [S: set_int,P: set_int > $o] :
% 0.34/0.60 ( ( finite_finite_int @ S )
% 0.34/0.60 => ( ( P @ bot_bot_set_int )
% 0.34/0.60 => ( ! [T3: set_int] :
% 0.34/0.60 ( ( ord_less_set_int @ T3 @ S )
% 0.34/0.60 => ( ( P @ T3 )
% 0.34/0.60 => ? [X6: int] :
% 0.34/0.60 ( ( member_int @ X6 @ ( minus_minus_set_int @ S @ T3 ) )
% 0.34/0.60 & ( P @ ( insert_int @ X6 @ T3 ) ) ) ) )
% 0.34/0.60 => ( P @ S ) ) ) ) ).
% 0.34/0.60
% 0.34/0.60 % finite_induct_select
% 0.34/0.60 thf(fact_3489_finite__induct__select,axiom,
% 0.34/0.60 ! [S: set_Pr4532377907799695533_nat_o,P: set_Pr4532377907799695533_nat_o > $o] :
% 0.34/0.60 ( ( finite3252695134891459830_nat_o @ S )
% 0.34/0.60 => ( ( P @ bot_bo7824918357723582233_nat_o )
% 0.34/0.60 => ( ! [T3: set_Pr4532377907799695533_nat_o] :
% 0.34/0.60 ( ( ord_le2453136405763929_nat_o @ T3 @ S )
% 0.34/0.60 => ( ( P @ T3 )
% 0.34/0.60 => ? [X6: produc3658429121746597890et_nat > $o] :
% 0.34/0.60 ( ( member6576561426505652726_nat_o @ X6 @ ( minus_1801376950450012436_nat_o @ S @ T3 ) )
% 0.34/0.60 & ( P @ ( insert5175938949040314269_nat_o @ X6 @ T3 ) ) ) ) )
% 0.34/0.60 => ( P @ S ) ) ) ) ).
% 0.34/0.60
% 0.34/0.60 % finite_induct_select
% 0.34/0.60 thf(fact_3490_finite__induct__select,axiom,
% 0.34/0.60 ! [S: set_nat,P: set_nat > $o] :
% 0.34/0.60 ( ( finite_finite_nat @ S )
% 0.34/0.60 => ( ( P @ bot_bot_set_nat )
% 0.34/0.60 => ( ! [T3: set_nat] :
% 0.34/0.60 ( ( ord_less_set_nat @ T3 @ S )
% 0.34/0.60 => ( ( P @ T3 )
% 0.34/0.60 => ? [X6: nat] :
% 0.34/0.60 ( ( member_nat @ X6 @ ( minus_minus_set_nat @ S @ T3 ) )
% 0.34/0.60 & ( P @ ( insert_nat @ X6 @ T3 ) ) ) ) )
% 0.34/0.60 => ( P @ S ) ) ) ) ).
% 0.34/0.60
% 0.34/0.60 % finite_induct_select
% 0.34/0.60 thf(fact_3491_Max_Osubset__imp,axiom,
% 0.34/0.60 ! [A: set_Code_integer,B4: set_Code_integer] :
% 0.34/0.60 ( ( ord_le7084787975880047091nteger @ A @ B4 )
% 0.34/0.60 => ( ( A != bot_bo3990330152332043303nteger )
% 0.34/0.60 => ( ( finite6017078050557962740nteger @ B4 )
% 0.34/0.60 => ( ord_le3102999989581377725nteger @ ( lattic4901227151466704046nteger @ A ) @ ( lattic4901227151466704046nteger @ B4 ) ) ) ) ) ).
% 0.34/0.60
% 0.34/0.60 % Max.subset_imp
% 0.34/0.60 thf(fact_3492_Max_Osubset__imp,axiom,
% 0.34/0.60 ! [A: set_Product_unit,B4: set_Product_unit] :
% 0.34/0.60 ( ( ord_le3507040750410214029t_unit @ A @ B4 )
% 0.34/0.60 => ( ( A != bot_bo3957492148770167129t_unit )
% 0.34/0.60 => ( ( finite4290736615968046902t_unit @ B4 )
% 0.34/0.60 => ( ord_le3221252021190050221t_unit @ ( lattic1210392890313158908t_unit @ A ) @ ( lattic1210392890313158908t_unit @ B4 ) ) ) ) ) ).
% 0.34/0.60
% 0.34/0.60 % Max.subset_imp
% 0.34/0.60 thf(fact_3493_Max_Osubset__imp,axiom,
% 0.34/0.60 ! [A: set_o,B4: set_o] :
% 0.34/0.60 ( ( ord_less_eq_set_o @ A @ B4 )
% 0.34/0.60 => ( ( A != bot_bot_set_o )
% 0.34/0.60 => ( ( finite_finite_o @ B4 )
% 0.34/0.60 => ( ord_less_eq_o @ ( lattic1921953407002678535_Max_o @ A ) @ ( lattic1921953407002678535_Max_o @ B4 ) ) ) ) ) ).
% 0.34/0.60
% 0.34/0.60 % Max.subset_imp
% 0.34/0.60 thf(fact_3494_Max_Osubset__imp,axiom,
% 0.34/0.60 ! [A: set_rat,B4: set_rat] :
% 0.34/0.60 ( ( ord_less_eq_set_rat @ A @ B4 )
% 0.34/0.60 => ( ( A != bot_bot_set_rat )
% 0.34/0.60 => ( ( finite_finite_rat @ B4 )
% 0.34/0.60 => ( ord_less_eq_rat @ ( lattic7630753665789217321ax_rat @ A ) @ ( lattic7630753665789217321ax_rat @ B4 ) ) ) ) ) ).
% 0.34/0.60
% 0.34/0.60 % Max.subset_imp
% 0.34/0.60 thf(fact_3495_Max_Osubset__imp,axiom,
% 0.34/0.60 ! [A: set_int,B4: set_int] :
% 0.34/0.60 ( ( ord_less_eq_set_int @ A @ B4 )
% 0.34/0.60 => ( ( A != bot_bot_set_int )
% 0.34/0.60 => ( ( finite_finite_int @ B4 )
% 0.34/0.60 => ( ord_less_eq_int @ ( lattic8263393255366662781ax_int @ A ) @ ( lattic8263393255366662781ax_int @ B4 ) ) ) ) ) ).
% 0.34/0.60
% 0.34/0.60 % Max.subset_imp
% 0.34/0.60 thf(fact_3496_Max_Osubset__imp,axiom,
% 0.34/0.60 ! [A: set_nat,B4: set_nat] :
% 0.34/0.60 ( ( ord_less_eq_set_nat @ A @ B4 )
% 0.34/0.60 => ( ( A != bot_bot_set_nat )
% 0.34/0.60 => ( ( finite_finite_nat @ B4 )
% 0.34/0.60 => ( ord_less_eq_nat @ ( lattic8265883725875713057ax_nat @ A ) @ ( lattic8265883725875713057ax_nat @ B4 ) ) ) ) ) ).
% 0.34/0.60
% 0.34/0.60 % Max.subset_imp
% 0.34/0.60 thf(fact_3497_Max__mono,axiom,
% 0.34/0.60 ! [M: set_Code_integer,N: set_Code_integer] :
% 0.34/0.60 ( ( ord_le7084787975880047091nteger @ M @ N )
% 0.34/0.60 => ( ( M != bot_bo3990330152332043303nteger )
% 0.34/0.60 => ( ( finite6017078050557962740nteger @ N )
% 0.34/0.60 => ( ord_le3102999989581377725nteger @ ( lattic4901227151466704046nteger @ M ) @ ( lattic4901227151466704046nteger @ N ) ) ) ) ) ).
% 0.34/0.60
% 0.34/0.60 % Max_mono
% 0.34/0.60 thf(fact_3498_Max__mono,axiom,
% 0.34/0.60 ! [M: set_Product_unit,N: set_Product_unit] :
% 0.34/0.60 ( ( ord_le3507040750410214029t_unit @ M @ N )
% 0.34/0.60 => ( ( M != bot_bo3957492148770167129t_unit )
% 0.34/0.60 => ( ( finite4290736615968046902t_unit @ N )
% 0.34/0.60 => ( ord_le3221252021190050221t_unit @ ( lattic1210392890313158908t_unit @ M ) @ ( lattic1210392890313158908t_unit @ N ) ) ) ) ) ).
% 0.34/0.60
% 0.34/0.60 % Max_mono
% 0.34/0.60 thf(fact_3499_Max__mono,axiom,
% 0.34/0.60 ! [M: set_o,N: set_o] :
% 0.34/0.60 ( ( ord_less_eq_set_o @ M @ N )
% 0.34/0.60 => ( ( M != bot_bot_set_o )
% 0.34/0.60 => ( ( finite_finite_o @ N )
% 0.34/0.60 => ( ord_less_eq_o @ ( lattic1921953407002678535_Max_o @ M ) @ ( lattic1921953407002678535_Max_o @ N ) ) ) ) ) ).
% 0.34/0.60
% 0.34/0.60 % Max_mono
% 0.34/0.60 thf(fact_3500_Max__mono,axiom,
% 0.34/0.60 ! [M: set_rat,N: set_rat] :
% 0.34/0.60 ( ( ord_less_eq_set_rat @ M @ N )
% 0.34/0.60 => ( ( M != bot_bot_set_rat )
% 0.34/0.60 => ( ( finite_finite_rat @ N )
% 0.34/0.60 => ( ord_less_eq_rat @ ( lattic7630753665789217321ax_rat @ M ) @ ( lattic7630753665789217321ax_rat @ N ) ) ) ) ) ).
% 0.34/0.60
% 0.34/0.60 % Max_mono
% 0.34/0.60 thf(fact_3501_Max__mono,axiom,
% 0.34/0.60 ! [M: set_int,N: set_int] :
% 0.34/0.60 ( ( ord_less_eq_set_int @ M @ N )
% 0.34/0.60 => ( ( M != bot_bot_set_int )
% 0.34/0.60 => ( ( finite_finite_int @ N )
% 0.34/0.60 => ( ord_less_eq_int @ ( lattic8263393255366662781ax_int @ M ) @ ( lattic8263393255366662781ax_int @ N ) ) ) ) ) ).
% 0.34/0.60
% 0.34/0.60 % Max_mono
% 0.34/0.60 thf(fact_3502_Max__mono,axiom,
% 0.34/0.60 ! [M: set_nat,N: set_nat] :
% 0.34/0.60 ( ( ord_less_eq_set_nat @ M @ N )
% 0.34/0.60 => ( ( M != bot_bot_set_nat )
% 0.34/0.60 => ( ( finite_finite_nat @ N )
% 0.34/0.60 => ( ord_less_eq_nat @ ( lattic8265883725875713057ax_nat @ M ) @ ( lattic8265883725875713057ax_nat @ N ) ) ) ) ) ).
% 0.34/0.60
% 0.34/0.60 % Max_mono
% 0.34/0.60 thf(fact_3503_image__vimage__eq,axiom,
% 0.34/0.60 ! [F: code_integer > code_integer,A: set_Code_integer] :
% 0.34/0.60 ( ( image_4470545334726330049nteger @ F @ ( vimage1318793397690208119nteger @ F @ A ) )
% 0.34/0.60 = ( inf_in1364745209274528805nteger @ A @ ( image_4470545334726330049nteger @ F @ top_to4645266643341252675nteger ) ) ) ).
% 0.34/0.60
% 0.34/0.60 % image_vimage_eq
% 0.34/0.60 thf(fact_3504_image__vimage__eq,axiom,
% 0.34/0.60 ! [F: product_unit > nat,A: set_nat] :
% 0.34/0.60 ( ( image_875570014554754200it_nat @ F @ ( vimage6253328473476588386it_nat @ F @ A ) )
% 0.34/0.60 = ( inf_inf_set_nat @ A @ ( image_875570014554754200it_nat @ F @ top_to1996260823553986621t_unit ) ) ) ).
% 0.34/0.60
% 0.34/0.60 % image_vimage_eq
% 0.34/0.60 thf(fact_3505_image__vimage__eq,axiom,
% 0.34/0.60 ! [F: $o > nat,A: set_nat] :
% 0.34/0.60 ( ( image_o_nat2 @ F @ ( vimage_o_nat @ F @ A ) )
% 0.34/0.60 = ( inf_inf_set_nat @ A @ ( image_o_nat2 @ F @ top_top_set_o ) ) ) ).
% 0.34/0.60
% 0.34/0.60 % image_vimage_eq
% 0.34/0.60 thf(fact_3506_image__vimage__eq,axiom,
% 0.34/0.60 ! [F: nat > set_nat,A: set_set_nat] :
% 0.34/0.60 ( ( image_nat_set_nat @ F @ ( vimage_nat_set_nat @ F @ A ) )
% 0.34/0.60 = ( inf_inf_set_set_nat @ A @ ( image_nat_set_nat @ F @ top_top_set_nat ) ) ) ).
% 0.34/0.60
% 0.34/0.60 % image_vimage_eq
% 0.34/0.60 thf(fact_3507_image__vimage__eq,axiom,
% 0.34/0.60 ! [F: nat > code_integer,A: set_Code_integer] :
% 0.34/0.60 ( ( image_1215581382706833972nteger @ F @ ( vimage6104394758324456170nteger @ F @ A ) )
% 0.34/0.60 = ( inf_in1364745209274528805nteger @ A @ ( image_1215581382706833972nteger @ F @ top_top_set_nat ) ) ) ).
% 0.34/0.60
% 0.34/0.60 % image_vimage_eq
% 0.34/0.60 thf(fact_3508_image__vimage__eq,axiom,
% 0.34/0.60 ! [F: nat > nat,A: set_nat] :
% 0.34/0.60 ( ( image_nat_nat2 @ F @ ( vimage_nat_nat @ F @ A ) )
% 0.34/0.60 = ( inf_inf_set_nat @ A @ ( image_nat_nat2 @ F @ top_top_set_nat ) ) ) ).
% 0.34/0.60
% 0.34/0.60 % image_vimage_eq
% 0.34/0.60 thf(fact_3509_image__vimage__eq,axiom,
% 0.34/0.60 ! [F: int > nat,A: set_nat] :
% 0.34/0.60 ( ( image_int_nat2 @ F @ ( vimage_int_nat @ F @ A ) )
% 0.34/0.60 = ( inf_inf_set_nat @ A @ ( image_int_nat2 @ F @ top_top_set_int ) ) ) ).
% 0.34/0.60
% 0.34/0.60 % image_vimage_eq
% 0.34/0.60 thf(fact_3510_UNIV__I,axiom,
% 0.34/0.60 ! [X: produc3658429121746597890et_nat > $o] : ( member6576561426505652726_nat_o @ X @ top_to4869887016220417533_nat_o ) ).
% 0.34/0.60
% 0.34/0.60 % UNIV_I
% 0.34/0.60 thf(fact_3511_UNIV__I,axiom,
% 0.34/0.60 ! [X: product_unit] : ( member_Product_unit @ X @ top_to1996260823553986621t_unit ) ).
% 0.34/0.60
% 0.34/0.60 % UNIV_I
% 0.34/0.60 thf(fact_3512_UNIV__I,axiom,
% 0.34/0.60 ! [X: $o] : ( member_o @ X @ top_top_set_o ) ).
% 0.34/0.60
% 0.34/0.60 % UNIV_I
% 0.34/0.60 thf(fact_3513_UNIV__I,axiom,
% 0.34/0.60 ! [X: nat] : ( member_nat @ X @ top_top_set_nat ) ).
% 0.34/0.60
% 0.34/0.60 % UNIV_I
% 0.34/0.60 thf(fact_3514_UNIV__I,axiom,
% 0.34/0.60 ! [X: int] : ( member_int @ X @ top_top_set_int ) ).
% 0.34/0.60
% 0.34/0.60 % UNIV_I
% 0.34/0.60 thf(fact_3515_Int__UNIV,axiom,
% 0.34/0.60 ! [A: set_Product_unit,B4: set_Product_unit] :
% 0.34/0.60 ( ( ( inf_in4660618365625256667t_unit @ A @ B4 )
% 0.34/0.60 = top_to1996260823553986621t_unit )
% 0.34/0.60 = ( ( A = top_to1996260823553986621t_unit )
% 0.34/0.60 & ( B4 = top_to1996260823553986621t_unit ) ) ) ).
% 0.34/0.60
% 0.34/0.60 % Int_UNIV
% 0.34/0.60 thf(fact_3516_Int__UNIV,axiom,
% 0.34/0.60 ! [A: set_o,B4: set_o] :
% 0.34/0.60 ( ( ( inf_inf_set_o @ A @ B4 )
% 0.34/0.60 = top_top_set_o )
% 0.34/0.60 = ( ( A = top_top_set_o )
% 0.34/0.60 & ( B4 = top_top_set_o ) ) ) ).
% 0.34/0.60
% 0.34/0.60 % Int_UNIV
% 0.34/0.60 thf(fact_3517_Int__UNIV,axiom,
% 0.34/0.60 ! [A: set_nat,B4: set_nat] :
% 0.34/0.60 ( ( ( inf_inf_set_nat @ A @ B4 )
% 0.34/0.60 = top_top_set_nat )
% 0.34/0.60 = ( ( A = top_top_set_nat )
% 0.34/0.60 & ( B4 = top_top_set_nat ) ) ) ).
% 0.34/0.60
% 0.34/0.60 % Int_UNIV
% 0.34/0.60 thf(fact_3518_Int__UNIV,axiom,
% 0.34/0.60 ! [A: set_int,B4: set_int] :
% 0.34/0.60 ( ( ( inf_inf_set_int @ A @ B4 )
% 0.34/0.60 = top_top_set_int )
% 0.34/0.60 = ( ( A = top_top_set_int )
% 0.34/0.60 & ( B4 = top_top_set_int ) ) ) ).
% 0.34/0.60
% 0.34/0.60 % Int_UNIV
% 0.34/0.60 thf(fact_3519_psubsetI,axiom,
% 0.34/0.60 ! [A: set_nat,B4: set_nat] :
% 0.34/0.60 ( ( ord_less_eq_set_nat @ A @ B4 )
% 0.34/0.60 => ( ( A != B4 )
% 0.34/0.60 => ( ord_less_set_nat @ A @ B4 ) ) ) ).
% 0.34/0.60
% 0.34/0.60 % psubsetI
% 0.34/0.60 thf(fact_3520_vimage__UNIV,axiom,
% 0.34/0.60 ! [F: product_unit > product_unit] :
% 0.34/0.60 ( ( vimage7995052115951654139t_unit @ F @ top_to1996260823553986621t_unit )
% 0.34/0.60 = top_to1996260823553986621t_unit ) ).
% 0.34/0.60
% 0.34/0.60 % vimage_UNIV
% 0.34/0.60 thf(fact_3521_vimage__UNIV,axiom,
% 0.34/0.60 ! [F: $o > product_unit] :
% 0.34/0.60 ( ( vimage4222377557387472176t_unit @ F @ top_to1996260823553986621t_unit )
% 0.34/0.60 = top_top_set_o ) ).
% 0.34/0.60
% 0.34/0.60 % vimage_UNIV
% 0.34/0.60 thf(fact_3522_vimage__UNIV,axiom,
% 0.34/0.60 ! [F: nat > product_unit] :
% 0.34/0.60 ( ( vimage4884490618288580032t_unit @ F @ top_to1996260823553986621t_unit )
% 0.34/0.60 = top_top_set_nat ) ).
% 0.34/0.60
% 0.34/0.60 % vimage_UNIV
% 0.34/0.60 thf(fact_3523_vimage__UNIV,axiom,
% 0.34/0.60 ! [F: int > product_unit] :
% 0.34/0.60 ( ( vimage2000912565490760292t_unit @ F @ top_to1996260823553986621t_unit )
% 0.34/0.60 = top_top_set_int ) ).
% 0.34/0.60
% 0.34/0.60 % vimage_UNIV
% 0.34/0.60 thf(fact_3524_vimage__UNIV,axiom,
% 0.34/0.60 ! [F: product_unit > $o] :
% 0.34/0.60 ( ( vimage8808348054653608838unit_o @ F @ top_top_set_o )
% 0.34/0.60 = top_to1996260823553986621t_unit ) ).
% 0.34/0.60
% 0.34/0.60 % vimage_UNIV
% 0.34/0.60 thf(fact_3525_vimage__UNIV,axiom,
% 0.34/0.60 ! [F: $o > $o] :
% 0.34/0.60 ( ( vimage_o_o @ F @ top_top_set_o )
% 0.34/0.60 = top_top_set_o ) ).
% 0.34/0.60
% 0.34/0.60 % vimage_UNIV
% 0.34/0.60 thf(fact_3526_vimage__UNIV,axiom,
% 0.34/0.60 ! [F: nat > $o] :
% 0.34/0.60 ( ( vimage_nat_o @ F @ top_top_set_o )
% 0.34/0.60 = top_top_set_nat ) ).
% 0.34/0.60
% 0.34/0.60 % vimage_UNIV
% 0.34/0.60 thf(fact_3527_vimage__UNIV,axiom,
% 0.34/0.60 ! [F: int > $o] :
% 0.34/0.60 ( ( vimage_int_o @ F @ top_top_set_o )
% 0.34/0.60 = top_top_set_int ) ).
% 0.34/0.60
% 0.34/0.60 % vimage_UNIV
% 0.34/0.60 thf(fact_3528_vimage__UNIV,axiom,
% 0.34/0.60 ! [F: product_unit > nat] :
% 0.34/0.60 ( ( vimage6253328473476588386it_nat @ F @ top_top_set_nat )
% 0.34/0.60 = top_to1996260823553986621t_unit ) ).
% 0.34/0.60
% 0.34/0.60 % vimage_UNIV
% 0.34/0.60 thf(fact_3529_vimage__UNIV,axiom,
% 0.34/0.60 ! [F: $o > nat] :
% 0.34/0.60 ( ( vimage_o_nat @ F @ top_top_set_nat )
% 0.34/0.60 = top_top_set_o ) ).
% 0.34/0.60
% 0.34/0.60 % vimage_UNIV
% 0.34/0.60 thf(fact_3530_Pow__UNIV,axiom,
% 0.34/0.60 ( ( pow_Product_unit @ top_to1996260823553986621t_unit )
% 0.34/0.60 = top_to1767297665138865437t_unit ) ).
% 0.34/0.60
% 0.34/0.60 % Pow_UNIV
% 0.34/0.60 thf(fact_3531_Pow__UNIV,axiom,
% 0.34/0.60 ( ( pow_o @ top_top_set_o )
% 0.34/0.60 = top_top_set_set_o ) ).
% 0.34/0.60
% 0.34/0.60 % Pow_UNIV
% 0.34/0.60 thf(fact_3532_Pow__UNIV,axiom,
% 0.34/0.60 ( ( pow_nat @ top_top_set_nat )
% 0.34/0.60 = top_top_set_set_nat ) ).
% 0.34/0.60
% 0.34/0.60 % Pow_UNIV
% 0.34/0.60 thf(fact_3533_Pow__UNIV,axiom,
% 0.34/0.60 ( ( pow_int @ top_top_set_int )
% 0.34/0.60 = top_top_set_set_int ) ).
% 0.34/0.60
% 0.34/0.60 % Pow_UNIV
% 0.34/0.60 thf(fact_3534_Diff__UNIV,axiom,
% 0.34/0.60 ! [A: set_Pr4532377907799695533_nat_o] :
% 0.34/0.60 ( ( minus_1801376950450012436_nat_o @ A @ top_to4869887016220417533_nat_o )
% 0.34/0.60 = bot_bo7824918357723582233_nat_o ) ).
% 0.34/0.60
% 0.34/0.60 % Diff_UNIV
% 0.34/0.60 thf(fact_3535_Diff__UNIV,axiom,
% 0.34/0.60 ! [A: set_Product_unit] :
% 0.34/0.60 ( ( minus_6452836326544984404t_unit @ A @ top_to1996260823553986621t_unit )
% 0.34/0.60 = bot_bo3957492148770167129t_unit ) ).
% 0.34/0.60
% 0.34/0.60 % Diff_UNIV
% 0.34/0.60 thf(fact_3536_Diff__UNIV,axiom,
% 0.34/0.60 ! [A: set_o] :
% 0.34/0.60 ( ( minus_minus_set_o @ A @ top_top_set_o )
% 0.34/0.60 = bot_bot_set_o ) ).
% 0.34/0.60
% 0.34/0.60 % Diff_UNIV
% 0.34/0.60 thf(fact_3537_Diff__UNIV,axiom,
% 0.34/0.60 ! [A: set_int] :
% 0.34/0.60 ( ( minus_minus_set_int @ A @ top_top_set_int )
% 0.34/0.60 = bot_bot_set_int ) ).
% 0.34/0.60
% 0.34/0.60 % Diff_UNIV
% 0.34/0.60 thf(fact_3538_Diff__UNIV,axiom,
% 0.34/0.60 ! [A: set_nat] :
% 0.34/0.60 ( ( minus_minus_set_nat @ A @ top_top_set_nat )
% 0.34/0.60 = bot_bot_set_nat ) ).
% 0.34/0.60
% 0.34/0.60 % Diff_UNIV
% 0.34/0.60 thf(fact_3539_Max__singleton,axiom,
% 0.34/0.60 ! [X: product_unit] :
% 0.34/0.60 ( ( lattic1210392890313158908t_unit @ ( insert_Product_unit @ X @ bot_bo3957492148770167129t_unit ) )
% 0.34/0.60 = X ) ).
% 0.34/0.60
% 0.34/0.60 % Max_singleton
% 0.34/0.60 thf(fact_3540_Max__singleton,axiom,
% 0.34/0.60 ! [X: $o] :
% 0.34/0.60 ( ( lattic1921953407002678535_Max_o @ ( insert_o @ X @ bot_bot_set_o ) )
% 0.34/0.60 = X ) ).
% 0.34/0.60
% 0.34/0.60 % Max_singleton
% 0.34/0.60 thf(fact_3541_Max__singleton,axiom,
% 0.34/0.60 ! [X: int] :
% 0.34/0.60 ( ( lattic8263393255366662781ax_int @ ( insert_int @ X @ bot_bot_set_int ) )
% 0.34/0.60 = X ) ).
% 0.34/0.60
% 0.34/0.60 % Max_singleton
% 0.34/0.60 thf(fact_3542_Max__singleton,axiom,
% 0.34/0.60 ! [X: nat] :
% 0.34/0.60 ( ( lattic8265883725875713057ax_nat @ ( insert_nat @ X @ bot_bot_set_nat ) )
% 0.34/0.60 = X ) ).
% 0.34/0.60
% 0.34/0.60 % Max_singleton
% 0.34/0.60 thf(fact_3543_range__snd,axiom,
% 0.34/0.60 ( ( image_6088997261112373647et_nat @ produc8586169260539613262et_nat @ top_to2524957550908328808et_nat )
% 0.34/0.60 = top_top_set_set_nat ) ).
% 0.34/0.60
% 0.34/0.60 % range_snd
% 0.34/0.60 thf(fact_3544_range__snd,axiom,
% 0.34/0.60 ( ( image_5042161079198086560nt_int @ product_snd_int_int @ top_to4366644338036079209nt_int )
% 0.34/0.60 = top_top_set_int ) ).
% 0.34/0.60
% 0.34/0.60 % range_snd
% 0.34/0.60 thf(fact_3545_Domain__Id,axiom,
% 0.34/0.60 ( ( domain4292024440186670924t_unit @ id_Product_unit )
% 0.34/0.60 = top_to1996260823553986621t_unit ) ).
% 0.34/0.60
% 0.34/0.60 % Domain_Id
% 0.34/0.60 thf(fact_3546_Domain__Id,axiom,
% 0.34/0.60 ( ( domain_o_o @ id_o2 )
% 0.34/0.60 = top_top_set_o ) ).
% 0.34/0.60
% 0.34/0.60 % Domain_Id
% 0.34/0.60 thf(fact_3547_Domain__Id,axiom,
% 0.34/0.60 ( ( domain_nat_nat @ id_nat )
% 0.34/0.60 = top_top_set_nat ) ).
% 0.34/0.60
% 0.34/0.60 % Domain_Id
% 0.34/0.60 thf(fact_3548_Domain__Id,axiom,
% 0.34/0.60 ( ( domain_int_int @ id_int )
% 0.34/0.60 = top_top_set_int ) ).
% 0.34/0.60
% 0.34/0.60 % Domain_Id
% 0.34/0.60 thf(fact_3549_Range__Id,axiom,
% 0.34/0.60 ( ( range_6617724647562260917t_unit @ id_Product_unit )
% 0.34/0.60 = top_to1996260823553986621t_unit ) ).
% 0.34/0.60
% 0.34/0.60 % Range_Id
% 0.34/0.60 thf(fact_3550_Range__Id,axiom,
% 0.34/0.60 ( ( range_o_o @ id_o2 )
% 0.34/0.60 = top_top_set_o ) ).
% 0.34/0.60
% 0.34/0.60 % Range_Id
% 0.34/0.60 thf(fact_3551_Range__Id,axiom,
% 0.34/0.60 ( ( range_nat_nat @ id_nat )
% 0.34/0.60 = top_top_set_nat ) ).
% 0.34/0.60
% 0.34/0.60 % Range_Id
% 0.34/0.60 thf(fact_3552_Range__Id,axiom,
% 0.34/0.60 ( ( range_int_int @ id_int )
% 0.34/0.60 = top_top_set_int ) ).
% 0.34/0.60
% 0.34/0.60 % Range_Id
% 0.34/0.60 thf(fact_3553_Max_Obounded__iff,axiom,
% 0.34/0.60 ! [A: set_Code_integer,X: code_integer] :
% 0.34/0.60 ( ( finite6017078050557962740nteger @ A )
% 0.34/0.60 => ( ( A != bot_bo3990330152332043303nteger )
% 0.34/0.60 => ( ( ord_le3102999989581377725nteger @ ( lattic4901227151466704046nteger @ A ) @ X )
% 0.34/0.60 = ( ! [X2: code_integer] :
% 0.34/0.60 ( ( member_Code_integer @ X2 @ A )
% 0.34/0.60 => ( ord_le3102999989581377725nteger @ X2 @ X ) ) ) ) ) ) ).
% 0.34/0.60
% 0.34/0.60 % Max.bounded_iff
% 0.34/0.60 thf(fact_3554_Max_Obounded__iff,axiom,
% 0.34/0.60 ! [A: set_Product_unit,X: product_unit] :
% 0.34/0.60 ( ( finite4290736615968046902t_unit @ A )
% 0.34/0.60 => ( ( A != bot_bo3957492148770167129t_unit )
% 0.34/0.60 => ( ( ord_le3221252021190050221t_unit @ ( lattic1210392890313158908t_unit @ A ) @ X )
% 0.34/0.60 = ( ! [X2: product_unit] :
% 0.34/0.60 ( ( member_Product_unit @ X2 @ A )
% 0.34/0.60 => ( ord_le3221252021190050221t_unit @ X2 @ X ) ) ) ) ) ) ).
% 0.34/0.60
% 0.34/0.60 % Max.bounded_iff
% 0.34/0.60 thf(fact_3555_Max_Obounded__iff,axiom,
% 0.34/0.60 ! [A: set_o,X: $o] :
% 0.34/0.60 ( ( finite_finite_o @ A )
% 0.34/0.60 => ( ( A != bot_bot_set_o )
% 0.34/0.60 => ( ( ord_less_eq_o @ ( lattic1921953407002678535_Max_o @ A ) @ X )
% 0.34/0.60 = ( ! [X2: $o] :
% 0.34/0.60 ( ( member_o @ X2 @ A )
% 0.34/0.60 => ( ord_less_eq_o @ X2 @ X ) ) ) ) ) ) ).
% 0.34/0.60
% 0.34/0.60 % Max.bounded_iff
% 0.34/0.60 thf(fact_3556_Max_Obounded__iff,axiom,
% 0.34/0.60 ! [A: set_rat,X: rat] :
% 0.34/0.60 ( ( finite_finite_rat @ A )
% 0.34/0.60 => ( ( A != bot_bot_set_rat )
% 0.34/0.60 => ( ( ord_less_eq_rat @ ( lattic7630753665789217321ax_rat @ A ) @ X )
% 0.34/0.60 = ( ! [X2: rat] :
% 0.34/0.60 ( ( member_rat @ X2 @ A )
% 0.34/0.60 => ( ord_less_eq_rat @ X2 @ X ) ) ) ) ) ) ).
% 0.34/0.60
% 0.34/0.60 % Max.bounded_iff
% 0.34/0.60 thf(fact_3557_Max_Obounded__iff,axiom,
% 0.34/0.60 ! [A: set_int,X: int] :
% 0.34/0.60 ( ( finite_finite_int @ A )
% 0.34/0.60 => ( ( A != bot_bot_set_int )
% 0.34/0.60 => ( ( ord_less_eq_int @ ( lattic8263393255366662781ax_int @ A ) @ X )
% 0.34/0.60 = ( ! [X2: int] :
% 0.34/0.60 ( ( member_int @ X2 @ A )
% 0.34/0.60 => ( ord_less_eq_int @ X2 @ X ) ) ) ) ) ) ).
% 0.34/0.60
% 0.34/0.60 % Max.bounded_iff
% 0.34/0.60 thf(fact_3558_Max_Obounded__iff,axiom,
% 0.34/0.60 ! [A: set_nat,X: nat] :
% 0.34/0.60 ( ( finite_finite_nat @ A )
% 0.34/0.60 => ( ( A != bot_bot_set_nat )
% 0.34/0.60 => ( ( ord_less_eq_nat @ ( lattic8265883725875713057ax_nat @ A ) @ X )
% 0.34/0.60 = ( ! [X2: nat] :
% 0.34/0.60 ( ( member_nat @ X2 @ A )
% 0.34/0.60 => ( ord_less_eq_nat @ X2 @ X ) ) ) ) ) ) ).
% 0.34/0.60
% 0.34/0.60 % Max.bounded_iff
% 0.34/0.60 thf(fact_3559_Max__less__iff,axiom,
% 0.34/0.60 ! [A: set_Product_unit,X: product_unit] :
% 0.34/0.60 ( ( finite4290736615968046902t_unit @ A )
% 0.34/0.60 => ( ( A != bot_bo3957492148770167129t_unit )
% 0.34/0.60 => ( ( ord_le361264281704409273t_unit @ ( lattic1210392890313158908t_unit @ A ) @ X )
% 0.34/0.60 = ( ! [X2: product_unit] :
% 0.34/0.60 ( ( member_Product_unit @ X2 @ A )
% 0.34/0.60 => ( ord_le361264281704409273t_unit @ X2 @ X ) ) ) ) ) ) ).
% 0.34/0.60
% 0.34/0.60 % Max_less_iff
% 0.34/0.60 thf(fact_3560_Max__less__iff,axiom,
% 0.34/0.60 ! [A: set_o,X: $o] :
% 0.34/0.60 ( ( finite_finite_o @ A )
% 0.34/0.60 => ( ( A != bot_bot_set_o )
% 0.34/0.60 => ( ( ord_less_o @ ( lattic1921953407002678535_Max_o @ A ) @ X )
% 0.34/0.60 = ( ! [X2: $o] :
% 0.34/0.60 ( ( member_o @ X2 @ A )
% 0.34/0.60 => ( ord_less_o @ X2 @ X ) ) ) ) ) ) ).
% 0.34/0.60
% 0.34/0.60 % Max_less_iff
% 0.34/0.60 thf(fact_3561_Max__less__iff,axiom,
% 0.34/0.60 ! [A: set_int,X: int] :
% 0.34/0.60 ( ( finite_finite_int @ A )
% 0.34/0.60 => ( ( A != bot_bot_set_int )
% 0.34/0.60 => ( ( ord_less_int @ ( lattic8263393255366662781ax_int @ A ) @ X )
% 0.34/0.60 = ( ! [X2: int] :
% 0.34/0.60 ( ( member_int @ X2 @ A )
% 0.34/0.60 => ( ord_less_int @ X2 @ X ) ) ) ) ) ) ).
% 0.34/0.60
% 0.34/0.60 % Max_less_iff
% 0.34/0.60 thf(fact_3562_Max__less__iff,axiom,
% 0.34/0.60 ! [A: set_Code_integer,X: code_integer] :
% 0.34/0.60 ( ( finite6017078050557962740nteger @ A )
% 0.34/0.60 => ( ( A != bot_bo3990330152332043303nteger )
% 0.34/0.60 => ( ( ord_le6747313008572928689nteger @ ( lattic4901227151466704046nteger @ A ) @ X )
% 0.34/0.60 = ( ! [X2: code_integer] :
% 0.34/0.60 ( ( member_Code_integer @ X2 @ A )
% 0.34/0.60 => ( ord_le6747313008572928689nteger @ X2 @ X ) ) ) ) ) ) ).
% 0.34/0.60
% 0.34/0.60 % Max_less_iff
% 0.34/0.60 thf(fact_3563_Max__less__iff,axiom,
% 0.34/0.60 ! [A: set_rat,X: rat] :
% 0.34/0.60 ( ( finite_finite_rat @ A )
% 0.34/0.60 => ( ( A != bot_bot_set_rat )
% 0.34/0.60 => ( ( ord_less_rat @ ( lattic7630753665789217321ax_rat @ A ) @ X )
% 0.34/0.60 = ( ! [X2: rat] :
% 0.34/0.60 ( ( member_rat @ X2 @ A )
% 0.34/0.60 => ( ord_less_rat @ X2 @ X ) ) ) ) ) ) ).
% 0.34/0.60
% 0.34/0.60 % Max_less_iff
% 0.34/0.60 thf(fact_3564_Max__less__iff,axiom,
% 0.34/0.60 ! [A: set_nat,X: nat] :
% 0.34/0.60 ( ( finite_finite_nat @ A )
% 0.34/0.60 => ( ( A != bot_bot_set_nat )
% 0.34/0.60 => ( ( ord_less_nat @ ( lattic8265883725875713057ax_nat @ A ) @ X )
% 0.34/0.60 = ( ! [X2: nat] :
% 0.34/0.60 ( ( member_nat @ X2 @ A )
% 0.34/0.60 => ( ord_less_nat @ X2 @ X ) ) ) ) ) ) ).
% 0.34/0.60
% 0.34/0.60 % Max_less_iff
% 0.34/0.60 thf(fact_3565_lt__ex,axiom,
% 0.34/0.60 ! [X: int] :
% 0.34/0.60 ? [Y2: int] : ( ord_less_int @ Y2 @ X ) ).
% 0.34/0.60
% 0.34/0.60 % lt_ex
% 0.34/0.60 thf(fact_3566_lt__ex,axiom,
% 0.34/0.60 ! [X: rat] :
% 0.34/0.60 ? [Y2: rat] : ( ord_less_rat @ Y2 @ X ) ).
% 0.34/0.60
% 0.34/0.60 % lt_ex
% 0.34/0.60 thf(fact_3567_gt__ex,axiom,
% 0.34/0.60 ! [X: nat] :
% 0.34/0.60 ? [X_1: nat] : ( ord_less_nat @ X @ X_1 ) ).
% 0.34/0.60
% 0.34/0.60 % gt_ex
% 0.34/0.60 thf(fact_3568_gt__ex,axiom,
% 0.34/0.60 ! [X: int] :
% 0.34/0.60 ? [X_1: int] : ( ord_less_int @ X @ X_1 ) ).
% 0.34/0.60
% 0.34/0.60 % gt_ex
% 0.34/0.60 thf(fact_3569_gt__ex,axiom,
% 0.34/0.60 ! [X: rat] :
% 0.34/0.60 ? [X_1: rat] : ( ord_less_rat @ X @ X_1 ) ).
% 0.34/0.60
% 0.34/0.60 % gt_ex
% 0.34/0.60 thf(fact_3570_dense,axiom,
% 0.34/0.60 ! [X: rat,Y: rat] :
% 0.34/0.60 ( ( ord_less_rat @ X @ Y )
% 0.34/0.60 => ? [Z3: rat] :
% 0.34/0.60 ( ( ord_less_rat @ X @ Z3 )
% 0.34/0.60 & ( ord_less_rat @ Z3 @ Y ) ) ) ).
% 0.34/0.60
% 0.34/0.60 % dense
% 0.34/0.60 thf(fact_3571_less__imp__neq,axiom,
% 0.34/0.60 ! [X: assn,Y: assn] :
% 0.34/0.60 ( ( ord_less_assn @ X @ Y )
% 0.34/0.60 => ( X != Y ) ) ).
% 0.34/0.60
% 0.34/0.60 % less_imp_neq
% 0.34/0.60 thf(fact_3572_less__imp__neq,axiom,
% 0.34/0.60 ! [X: nat,Y: nat] :
% 0.34/0.60 ( ( ord_less_nat @ X @ Y )
% 0.34/0.60 => ( X != Y ) ) ).
% 0.34/0.60
% 0.34/0.60 % less_imp_neq
% 0.34/0.60 thf(fact_3573_less__imp__neq,axiom,
% 0.34/0.60 ! [X: int,Y: int] :
% 0.34/0.60 ( ( ord_less_int @ X @ Y )
% 0.34/0.60 => ( X != Y ) ) ).
% 0.34/0.60
% 0.34/0.60 % less_imp_neq
% 0.34/0.60 thf(fact_3574_less__imp__neq,axiom,
% 0.34/0.60 ! [X: code_integer,Y: code_integer] :
% 0.34/0.60 ( ( ord_le6747313008572928689nteger @ X @ Y )
% 0.34/0.60 => ( X != Y ) ) ).
% 0.34/0.60
% 0.34/0.60 % less_imp_neq
% 0.34/0.60 thf(fact_3575_less__imp__neq,axiom,
% 0.34/0.60 ! [X: rat,Y: rat] :
% 0.34/0.60 ( ( ord_less_rat @ X @ Y )
% 0.34/0.60 => ( X != Y ) ) ).
% 0.34/0.60
% 0.34/0.60 % less_imp_neq
% 0.34/0.60 thf(fact_3576_order_Oasym,axiom,
% 0.34/0.60 ! [A2: assn,B: assn] :
% 0.34/0.60 ( ( ord_less_assn @ A2 @ B )
% 0.34/0.60 => ~ ( ord_less_assn @ B @ A2 ) ) ).
% 0.34/0.60
% 0.34/0.60 % order.asym
% 0.34/0.60 thf(fact_3577_order_Oasym,axiom,
% 0.34/0.60 ! [A2: nat,B: nat] :
% 0.34/0.60 ( ( ord_less_nat @ A2 @ B )
% 0.34/0.60 => ~ ( ord_less_nat @ B @ A2 ) ) ).
% 0.34/0.60
% 0.34/0.60 % order.asym
% 0.34/0.60 thf(fact_3578_order_Oasym,axiom,
% 0.34/0.60 ! [A2: int,B: int] :
% 0.34/0.60 ( ( ord_less_int @ A2 @ B )
% 0.34/0.60 => ~ ( ord_less_int @ B @ A2 ) ) ).
% 0.34/0.60
% 0.34/0.60 % order.asym
% 0.34/0.60 thf(fact_3579_order_Oasym,axiom,
% 0.34/0.60 ! [A2: code_integer,B: code_integer] :
% 0.34/0.60 ( ( ord_le6747313008572928689nteger @ A2 @ B )
% 0.34/0.60 => ~ ( ord_le6747313008572928689nteger @ B @ A2 ) ) ).
% 0.34/0.60
% 0.34/0.60 % order.asym
% 0.34/0.60 thf(fact_3580_order_Oasym,axiom,
% 0.34/0.60 ! [A2: rat,B: rat] :
% 0.34/0.60 ( ( ord_less_rat @ A2 @ B )
% 0.34/0.60 => ~ ( ord_less_rat @ B @ A2 ) ) ).
% 0.34/0.60
% 0.34/0.60 % order.asym
% 0.34/0.60 thf(fact_3581_ord__eq__less__trans,axiom,
% 0.34/0.60 ! [A2: assn,B: assn,C: assn] :
% 0.34/0.60 ( ( A2 = B )
% 0.34/0.60 => ( ( ord_less_assn @ B @ C )
% 0.34/0.60 => ( ord_less_assn @ A2 @ C ) ) ) ).
% 0.34/0.60
% 0.34/0.60 % ord_eq_less_trans
% 0.34/0.60 thf(fact_3582_ord__eq__less__trans,axiom,
% 0.34/0.60 ! [A2: nat,B: nat,C: nat] :
% 0.34/0.60 ( ( A2 = B )
% 0.34/0.60 => ( ( ord_less_nat @ B @ C )
% 0.34/0.60 => ( ord_less_nat @ A2 @ C ) ) ) ).
% 0.34/0.60
% 0.34/0.60 % ord_eq_less_trans
% 0.34/0.60 thf(fact_3583_ord__eq__less__trans,axiom,
% 0.34/0.60 ! [A2: int,B: int,C: int] :
% 0.34/0.60 ( ( A2 = B )
% 0.34/0.60 => ( ( ord_less_int @ B @ C )
% 0.34/0.60 => ( ord_less_int @ A2 @ C ) ) ) ).
% 0.34/0.60
% 0.34/0.60 % ord_eq_less_trans
% 0.34/0.60 thf(fact_3584_ord__eq__less__trans,axiom,
% 0.34/0.60 ! [A2: code_integer,B: code_integer,C: code_integer] :
% 0.34/0.60 ( ( A2 = B )
% 0.34/0.60 => ( ( ord_le6747313008572928689nteger @ B @ C )
% 0.34/0.60 => ( ord_le6747313008572928689nteger @ A2 @ C ) ) ) ).
% 0.34/0.60
% 0.34/0.60 % ord_eq_less_trans
% 0.34/0.60 thf(fact_3585_ord__eq__less__trans,axiom,
% 0.34/0.60 ! [A2: rat,B: rat,C: rat] :
% 0.34/0.60 ( ( A2 = B )
% 0.34/0.60 => ( ( ord_less_rat @ B @ C )
% 0.34/0.60 => ( ord_less_rat @ A2 @ C ) ) ) ).
% 0.34/0.60
% 0.34/0.60 % ord_eq_less_trans
% 0.34/0.60 thf(fact_3586_ord__less__eq__trans,axiom,
% 0.34/0.60 ! [A2: assn,B: assn,C: assn] :
% 0.34/0.60 ( ( ord_less_assn @ A2 @ B )
% 0.34/0.60 => ( ( B = C )
% 0.34/0.60 => ( ord_less_assn @ A2 @ C ) ) ) ).
% 0.34/0.60
% 0.34/0.60 % ord_less_eq_trans
% 0.34/0.60 thf(fact_3587_ord__less__eq__trans,axiom,
% 0.34/0.60 ! [A2: nat,B: nat,C: nat] :
% 0.34/0.60 ( ( ord_less_nat @ A2 @ B )
% 0.34/0.60 => ( ( B = C )
% 0.34/0.60 => ( ord_less_nat @ A2 @ C ) ) ) ).
% 0.34/0.60
% 0.34/0.60 % ord_less_eq_trans
% 0.34/0.60 thf(fact_3588_ord__less__eq__trans,axiom,
% 0.34/0.60 ! [A2: int,B: int,C: int] :
% 0.34/0.60 ( ( ord_less_int @ A2 @ B )
% 0.34/0.60 => ( ( B = C )
% 0.34/0.60 => ( ord_less_int @ A2 @ C ) ) ) ).
% 0.34/0.60
% 0.34/0.60 % ord_less_eq_trans
% 0.34/0.60 thf(fact_3589_ord__less__eq__trans,axiom,
% 0.34/0.60 ! [A2: code_integer,B: code_integer,C: code_integer] :
% 0.34/0.60 ( ( ord_le6747313008572928689nteger @ A2 @ B )
% 0.34/0.60 => ( ( B = C )
% 0.34/0.60 => ( ord_le6747313008572928689nteger @ A2 @ C ) ) ) ).
% 0.34/0.60
% 0.34/0.60 % ord_less_eq_trans
% 0.34/0.60 thf(fact_3590_ord__less__eq__trans,axiom,
% 0.34/0.60 ! [A2: rat,B: rat,C: rat] :
% 0.34/0.60 ( ( ord_less_rat @ A2 @ B )
% 0.34/0.60 => ( ( B = C )
% 0.34/0.60 => ( ord_less_rat @ A2 @ C ) ) ) ).
% 0.34/0.60
% 0.34/0.60 % ord_less_eq_trans
% 0.34/0.60 thf(fact_3591_less__induct,axiom,
% 0.34/0.60 ! [P: nat > $o,A2: nat] :
% 0.34/0.60 ( ! [X3: nat] :
% 0.34/0.60 ( ! [Y5: nat] :
% 0.34/0.60 ( ( ord_less_nat @ Y5 @ X3 )
% 0.34/0.60 => ( P @ Y5 ) )
% 0.34/0.60 => ( P @ X3 ) )
% 0.34/0.60 => ( P @ A2 ) ) ).
% 0.34/0.60
% 0.34/0.60 % less_induct
% 0.34/0.60 thf(fact_3592_antisym__conv3,axiom,
% 0.34/0.60 ! [Y: nat,X: nat] :
% 0.34/0.60 ( ~ ( ord_less_nat @ Y @ X )
% 0.34/0.60 => ( ( ~ ( ord_less_nat @ X @ Y ) )
% 0.34/0.60 = ( X = Y ) ) ) ).
% 0.34/0.60
% 0.34/0.60 % antisym_conv3
% 0.34/0.60 thf(fact_3593_antisym__conv3,axiom,
% 0.34/0.60 ! [Y: int,X: int] :
% 0.34/0.60 ( ~ ( ord_less_int @ Y @ X )
% 0.34/0.60 => ( ( ~ ( ord_less_int @ X @ Y ) )
% 0.34/0.60 = ( X = Y ) ) ) ).
% 0.34/0.60
% 0.34/0.60 % antisym_conv3
% 0.34/0.60 thf(fact_3594_antisym__conv3,axiom,
% 0.34/0.60 ! [Y: code_integer,X: code_integer] :
% 0.34/0.60 ( ~ ( ord_le6747313008572928689nteger @ Y @ X )
% 0.34/0.60 => ( ( ~ ( ord_le6747313008572928689nteger @ X @ Y ) )
% 0.34/0.60 = ( X = Y ) ) ) ).
% 0.34/0.60
% 0.34/0.60 % antisym_conv3
% 0.34/0.60 thf(fact_3595_antisym__conv3,axiom,
% 0.34/0.60 ! [Y: rat,X: rat] :
% 0.34/0.60 ( ~ ( ord_less_rat @ Y @ X )
% 0.34/0.60 => ( ( ~ ( ord_less_rat @ X @ Y ) )
% 0.34/0.60 = ( X = Y ) ) ) ).
% 0.34/0.60
% 0.34/0.60 % antisym_conv3
% 0.34/0.60 thf(fact_3596_linorder__cases,axiom,
% 0.34/0.60 ! [X: nat,Y: nat] :
% 0.34/0.60 ( ~ ( ord_less_nat @ X @ Y )
% 0.34/0.60 => ( ( X != Y )
% 0.34/0.60 => ( ord_less_nat @ Y @ X ) ) ) ).
% 0.34/0.60
% 0.34/0.60 % linorder_cases
% 0.34/0.60 thf(fact_3597_linorder__cases,axiom,
% 0.34/0.60 ! [X: int,Y: int] :
% 0.34/0.60 ( ~ ( ord_less_int @ X @ Y )
% 0.34/0.60 => ( ( X != Y )
% 0.34/0.60 => ( ord_less_int @ Y @ X ) ) ) ).
% 0.34/0.60
% 0.34/0.60 % linorder_cases
% 0.34/0.60 thf(fact_3598_linorder__cases,axiom,
% 0.34/0.60 ! [X: code_integer,Y: code_integer] :
% 0.34/0.60 ( ~ ( ord_le6747313008572928689nteger @ X @ Y )
% 0.34/0.60 => ( ( X != Y )
% 0.34/0.60 => ( ord_le6747313008572928689nteger @ Y @ X ) ) ) ).
% 0.34/0.60
% 0.34/0.60 % linorder_cases
% 0.34/0.60 thf(fact_3599_linorder__cases,axiom,
% 0.34/0.60 ! [X: rat,Y: rat] :
% 0.34/0.60 ( ~ ( ord_less_rat @ X @ Y )
% 0.34/0.60 => ( ( X != Y )
% 0.34/0.60 => ( ord_less_rat @ Y @ X ) ) ) ).
% 0.34/0.60
% 0.34/0.60 % linorder_cases
% 0.34/0.60 thf(fact_3600_dual__order_Oasym,axiom,
% 0.34/0.60 ! [B: assn,A2: assn] :
% 0.34/0.60 ( ( ord_less_assn @ B @ A2 )
% 0.34/0.60 => ~ ( ord_less_assn @ A2 @ B ) ) ).
% 0.34/0.60
% 0.34/0.60 % dual_order.asym
% 0.34/0.60 thf(fact_3601_dual__order_Oasym,axiom,
% 0.34/0.60 ! [B: nat,A2: nat] :
% 0.34/0.60 ( ( ord_less_nat @ B @ A2 )
% 0.34/0.60 => ~ ( ord_less_nat @ A2 @ B ) ) ).
% 0.34/0.60
% 0.34/0.60 % dual_order.asym
% 0.34/0.60 thf(fact_3602_dual__order_Oasym,axiom,
% 0.34/0.60 ! [B: int,A2: int] :
% 0.34/0.60 ( ( ord_less_int @ B @ A2 )
% 0.34/0.60 => ~ ( ord_less_int @ A2 @ B ) ) ).
% 0.34/0.60
% 0.34/0.60 % dual_order.asym
% 0.34/0.60 thf(fact_3603_dual__order_Oasym,axiom,
% 0.34/0.60 ! [B: code_integer,A2: code_integer] :
% 0.34/0.60 ( ( ord_le6747313008572928689nteger @ B @ A2 )
% 0.34/0.60 => ~ ( ord_le6747313008572928689nteger @ A2 @ B ) ) ).
% 0.34/0.60
% 0.34/0.60 % dual_order.asym
% 0.34/0.60 thf(fact_3604_dual__order_Oasym,axiom,
% 0.34/0.60 ! [B: rat,A2: rat] :
% 0.34/0.60 ( ( ord_less_rat @ B @ A2 )
% 0.34/0.60 => ~ ( ord_less_rat @ A2 @ B ) ) ).
% 0.34/0.60
% 0.34/0.60 % dual_order.asym
% 0.34/0.60 thf(fact_3605_dual__order_Oirrefl,axiom,
% 0.34/0.60 ! [A2: assn] :
% 0.34/0.60 ~ ( ord_less_assn @ A2 @ A2 ) ).
% 0.34/0.60
% 0.34/0.60 % dual_order.irrefl
% 0.34/0.60 thf(fact_3606_dual__order_Oirrefl,axiom,
% 0.34/0.60 ! [A2: nat] :
% 0.34/0.60 ~ ( ord_less_nat @ A2 @ A2 ) ).
% 0.34/0.60
% 0.34/0.60 % dual_order.irrefl
% 0.34/0.60 thf(fact_3607_dual__order_Oirrefl,axiom,
% 0.34/0.60 ! [A2: int] :
% 0.34/0.60 ~ ( ord_less_int @ A2 @ A2 ) ).
% 0.34/0.60
% 0.34/0.60 % dual_order.irrefl
% 0.34/0.60 thf(fact_3608_dual__order_Oirrefl,axiom,
% 0.34/0.60 ! [A2: code_integer] :
% 0.34/0.60 ~ ( ord_le6747313008572928689nteger @ A2 @ A2 ) ).
% 0.34/0.60
% 0.34/0.60 % dual_order.irrefl
% 0.34/0.60 thf(fact_3609_dual__order_Oirrefl,axiom,
% 0.34/0.60 ! [A2: rat] :
% 0.34/0.60 ~ ( ord_less_rat @ A2 @ A2 ) ).
% 0.34/0.60
% 0.34/0.60 % dual_order.irrefl
% 0.34/0.60 thf(fact_3610_exists__least__iff,axiom,
% 0.34/0.60 ( ( ^ [P6: nat > $o] :
% 0.34/0.60 ? [X9: nat] : ( P6 @ X9 ) )
% 0.34/0.60 = ( ^ [P2: nat > $o] :
% 0.34/0.60 ? [N4: nat] :
% 0.34/0.60 ( ( P2 @ N4 )
% 0.34/0.60 & ! [M3: nat] :
% 0.34/0.60 ( ( ord_less_nat @ M3 @ N4 )
% 0.34/0.60 => ~ ( P2 @ M3 ) ) ) ) ) ).
% 0.34/0.60
% 0.34/0.60 % exists_least_iff
% 0.34/0.60 thf(fact_3611_linorder__less__wlog,axiom,
% 0.34/0.60 ! [P: nat > nat > $o,A2: nat,B: nat] :
% 0.34/0.60 ( ! [A4: nat,B2: nat] :
% 0.34/0.60 ( ( ord_less_nat @ A4 @ B2 )
% 0.34/0.60 => ( P @ A4 @ B2 ) )
% 0.34/0.60 => ( ! [A4: nat] : ( P @ A4 @ A4 )
% 0.34/0.60 => ( ! [A4: nat,B2: nat] :
% 0.34/0.60 ( ( P @ B2 @ A4 )
% 0.34/0.60 => ( P @ A4 @ B2 ) )
% 0.34/0.60 => ( P @ A2 @ B ) ) ) ) ).
% 0.34/0.60
% 0.34/0.60 % linorder_less_wlog
% 0.34/0.60 thf(fact_3612_linorder__less__wlog,axiom,
% 0.34/0.60 ! [P: int > int > $o,A2: int,B: int] :
% 0.34/0.60 ( ! [A4: int,B2: int] :
% 0.34/0.60 ( ( ord_less_int @ A4 @ B2 )
% 0.34/0.60 => ( P @ A4 @ B2 ) )
% 0.34/0.60 => ( ! [A4: int] : ( P @ A4 @ A4 )
% 0.34/0.60 => ( ! [A4: int,B2: int] :
% 0.34/0.60 ( ( P @ B2 @ A4 )
% 0.34/0.60 => ( P @ A4 @ B2 ) )
% 0.34/0.60 => ( P @ A2 @ B ) ) ) ) ).
% 0.34/0.60
% 0.34/0.60 % linorder_less_wlog
% 0.34/0.60 thf(fact_3613_linorder__less__wlog,axiom,
% 0.34/0.60 ! [P: code_integer > code_integer > $o,A2: code_integer,B: code_integer] :
% 0.34/0.60 ( ! [A4: code_integer,B2: code_integer] :
% 0.34/0.60 ( ( ord_le6747313008572928689nteger @ A4 @ B2 )
% 0.34/0.60 => ( P @ A4 @ B2 ) )
% 0.34/0.60 => ( ! [A4: code_integer] : ( P @ A4 @ A4 )
% 0.34/0.60 => ( ! [A4: code_integer,B2: code_integer] :
% 0.34/0.60 ( ( P @ B2 @ A4 )
% 0.34/0.60 => ( P @ A4 @ B2 ) )
% 0.34/0.60 => ( P @ A2 @ B ) ) ) ) ).
% 0.34/0.60
% 0.34/0.60 % linorder_less_wlog
% 0.34/0.60 thf(fact_3614_linorder__less__wlog,axiom,
% 0.34/0.60 ! [P: rat > rat > $o,A2: rat,B: rat] :
% 0.34/0.60 ( ! [A4: rat,B2: rat] :
% 0.34/0.60 ( ( ord_less_rat @ A4 @ B2 )
% 0.34/0.60 => ( P @ A4 @ B2 ) )
% 0.34/0.60 => ( ! [A4: rat] : ( P @ A4 @ A4 )
% 0.34/0.60 => ( ! [A4: rat,B2: rat] :
% 0.34/0.60 ( ( P @ B2 @ A4 )
% 0.34/0.60 => ( P @ A4 @ B2 ) )
% 0.34/0.60 => ( P @ A2 @ B ) ) ) ) ).
% 0.34/0.60
% 0.34/0.60 % linorder_less_wlog
% 0.34/0.60 thf(fact_3615_order_Ostrict__trans,axiom,
% 0.34/0.60 ! [A2: assn,B: assn,C: assn] :
% 0.34/0.60 ( ( ord_less_assn @ A2 @ B )
% 0.34/0.60 => ( ( ord_less_assn @ B @ C )
% 0.34/0.60 => ( ord_less_assn @ A2 @ C ) ) ) ).
% 0.34/0.60
% 0.34/0.60 % order.strict_trans
% 0.34/0.60 thf(fact_3616_order_Ostrict__trans,axiom,
% 0.34/0.60 ! [A2: nat,B: nat,C: nat] :
% 0.34/0.60 ( ( ord_less_nat @ A2 @ B )
% 0.34/0.60 => ( ( ord_less_nat @ B @ C )
% 0.34/0.60 => ( ord_less_nat @ A2 @ C ) ) ) ).
% 0.34/0.60
% 0.34/0.60 % order.strict_trans
% 0.34/0.60 thf(fact_3617_order_Ostrict__trans,axiom,
% 0.34/0.60 ! [A2: int,B: int,C: int] :
% 0.34/0.60 ( ( ord_less_int @ A2 @ B )
% 0.34/0.60 => ( ( ord_less_int @ B @ C )
% 0.34/0.60 => ( ord_less_int @ A2 @ C ) ) ) ).
% 0.34/0.60
% 0.34/0.60 % order.strict_trans
% 0.34/0.60 thf(fact_3618_order_Ostrict__trans,axiom,
% 0.34/0.60 ! [A2: code_integer,B: code_integer,C: code_integer] :
% 0.34/0.60 ( ( ord_le6747313008572928689nteger @ A2 @ B )
% 0.34/0.60 => ( ( ord_le6747313008572928689nteger @ B @ C )
% 0.34/0.60 => ( ord_le6747313008572928689nteger @ A2 @ C ) ) ) ).
% 0.34/0.60
% 0.34/0.60 % order.strict_trans
% 0.34/0.60 thf(fact_3619_order_Ostrict__trans,axiom,
% 0.34/0.60 ! [A2: rat,B: rat,C: rat] :
% 0.34/0.60 ( ( ord_less_rat @ A2 @ B )
% 0.34/0.60 => ( ( ord_less_rat @ B @ C )
% 0.34/0.60 => ( ord_less_rat @ A2 @ C ) ) ) ).
% 0.34/0.60
% 0.34/0.60 % order.strict_trans
% 0.34/0.60 thf(fact_3620_top_Oextremum__strict,axiom,
% 0.34/0.60 ! [A2: set_Product_unit] :
% 0.34/0.60 ~ ( ord_le8056459307392131481t_unit @ top_to1996260823553986621t_unit @ A2 ) ).
% 0.34/0.60
% 0.34/0.60 % top.extremum_strict
% 0.34/0.60 thf(fact_3621_top_Oextremum__strict,axiom,
% 0.34/0.60 ! [A2: set_o] :
% 0.34/0.60 ~ ( ord_less_set_o @ top_top_set_o @ A2 ) ).
% 0.34/0.60
% 0.34/0.60 % top.extremum_strict
% 0.34/0.60 thf(fact_3622_top_Oextremum__strict,axiom,
% 0.34/0.60 ! [A2: set_nat] :
% 0.34/0.60 ~ ( ord_less_set_nat @ top_top_set_nat @ A2 ) ).
% 0.34/0.60
% 0.34/0.60 % top.extremum_strict
% 0.34/0.60 thf(fact_3623_top_Oextremum__strict,axiom,
% 0.34/0.60 ! [A2: set_int] :
% 0.34/0.60 ~ ( ord_less_set_int @ top_top_set_int @ A2 ) ).
% 0.34/0.60
% 0.34/0.60 % top.extremum_strict
% 0.34/0.60 thf(fact_3624_top_Oextremum__strict,axiom,
% 0.34/0.60 ! [A2: assn] :
% 0.34/0.60 ~ ( ord_less_assn @ top_top_assn @ A2 ) ).
% 0.34/0.60
% 0.34/0.60 % top.extremum_strict
% 0.34/0.60 thf(fact_3625_top_Onot__eq__extremum,axiom,
% 0.34/0.60 ! [A2: set_Product_unit] :
% 0.34/0.60 ( ( A2 != top_to1996260823553986621t_unit )
% 0.34/0.60 = ( ord_le8056459307392131481t_unit @ A2 @ top_to1996260823553986621t_unit ) ) ).
% 0.34/0.60
% 0.34/0.60 % top.not_eq_extremum
% 0.34/0.60 thf(fact_3626_top_Onot__eq__extremum,axiom,
% 0.34/0.60 ! [A2: set_o] :
% 0.34/0.60 ( ( A2 != top_top_set_o )
% 0.34/0.60 = ( ord_less_set_o @ A2 @ top_top_set_o ) ) ).
% 0.34/0.60
% 0.34/0.60 % top.not_eq_extremum
% 0.34/0.60 thf(fact_3627_top_Onot__eq__extremum,axiom,
% 0.34/0.60 ! [A2: set_nat] :
% 0.34/0.60 ( ( A2 != top_top_set_nat )
% 0.34/0.60 = ( ord_less_set_nat @ A2 @ top_top_set_nat ) ) ).
% 0.34/0.60
% 0.34/0.60 % top.not_eq_extremum
% 0.34/0.60 thf(fact_3628_top_Onot__eq__extremum,axiom,
% 0.34/0.60 ! [A2: set_int] :
% 0.34/0.60 ( ( A2 != top_top_set_int )
% 0.34/0.60 = ( ord_less_set_int @ A2 @ top_top_set_int ) ) ).
% 0.34/0.60
% 0.34/0.60 % top.not_eq_extremum
% 0.34/0.60 thf(fact_3629_top_Onot__eq__extremum,axiom,
% 0.34/0.60 ! [A2: assn] :
% 0.34/0.60 ( ( A2 != top_top_assn )
% 0.34/0.60 = ( ord_less_assn @ A2 @ top_top_assn ) ) ).
% 0.34/0.60
% 0.34/0.60 % top.not_eq_extremum
% 0.34/0.60 thf(fact_3630_not__less__iff__gr__or__eq,axiom,
% 0.34/0.60 ! [X: nat,Y: nat] :
% 0.34/0.60 ( ( ~ ( ord_less_nat @ X @ Y ) )
% 0.34/0.60 = ( ( ord_less_nat @ Y @ X )
% 0.34/0.60 | ( X = Y ) ) ) ).
% 0.34/0.60
% 0.34/0.60 % not_less_iff_gr_or_eq
% 0.34/0.60 thf(fact_3631_not__less__iff__gr__or__eq,axiom,
% 0.34/0.60 ! [X: int,Y: int] :
% 0.34/0.60 ( ( ~ ( ord_less_int @ X @ Y ) )
% 0.34/0.60 = ( ( ord_less_int @ Y @ X )
% 0.34/0.60 | ( X = Y ) ) ) ).
% 0.34/0.60
% 0.34/0.60 % not_less_iff_gr_or_eq
% 0.34/0.60 thf(fact_3632_not__less__iff__gr__or__eq,axiom,
% 0.34/0.60 ! [X: code_integer,Y: code_integer] :
% 0.34/0.60 ( ( ~ ( ord_le6747313008572928689nteger @ X @ Y ) )
% 0.34/0.60 = ( ( ord_le6747313008572928689nteger @ Y @ X )
% 0.34/0.60 | ( X = Y ) ) ) ).
% 0.34/0.60
% 0.34/0.60 % not_less_iff_gr_or_eq
% 0.34/0.60 thf(fact_3633_not__less__iff__gr__or__eq,axiom,
% 0.34/0.60 ! [X: rat,Y: rat] :
% 0.34/0.60 ( ( ~ ( ord_less_rat @ X @ Y ) )
% 0.34/0.60 = ( ( ord_less_rat @ Y @ X )
% 0.34/0.60 | ( X = Y ) ) ) ).
% 0.34/0.60
% 0.34/0.60 % not_less_iff_gr_or_eq
% 0.34/0.60 thf(fact_3634_dual__order_Ostrict__trans,axiom,
% 0.34/0.60 ! [B: assn,A2: assn,C: assn] :
% 0.34/0.60 ( ( ord_less_assn @ B @ A2 )
% 0.34/0.60 => ( ( ord_less_assn @ C @ B )
% 0.34/0.60 => ( ord_less_assn @ C @ A2 ) ) ) ).
% 0.34/0.60
% 0.34/0.60 % dual_order.strict_trans
% 0.34/0.60 thf(fact_3635_dual__order_Ostrict__trans,axiom,
% 0.34/0.60 ! [B: nat,A2: nat,C: nat] :
% 0.34/0.60 ( ( ord_less_nat @ B @ A2 )
% 0.34/0.60 => ( ( ord_less_nat @ C @ B )
% 0.34/0.60 => ( ord_less_nat @ C @ A2 ) ) ) ).
% 0.34/0.60
% 0.34/0.60 % dual_order.strict_trans
% 0.34/0.60 thf(fact_3636_dual__order_Ostrict__trans,axiom,
% 0.34/0.60 ! [B: int,A2: int,C: int] :
% 0.34/0.60 ( ( ord_less_int @ B @ A2 )
% 0.34/0.60 => ( ( ord_less_int @ C @ B )
% 0.34/0.60 => ( ord_less_int @ C @ A2 ) ) ) ).
% 0.34/0.60
% 0.34/0.60 % dual_order.strict_trans
% 0.34/0.60 thf(fact_3637_dual__order_Ostrict__trans,axiom,
% 0.34/0.60 ! [B: code_integer,A2: code_integer,C: code_integer] :
% 0.34/0.60 ( ( ord_le6747313008572928689nteger @ B @ A2 )
% 0.34/0.60 => ( ( ord_le6747313008572928689nteger @ C @ B )
% 0.34/0.60 => ( ord_le6747313008572928689nteger @ C @ A2 ) ) ) ).
% 0.34/0.60
% 0.34/0.60 % dual_order.strict_trans
% 0.34/0.60 thf(fact_3638_dual__order_Ostrict__trans,axiom,
% 0.34/0.60 ! [B: rat,A2: rat,C: rat] :
% 0.34/0.60 ( ( ord_less_rat @ B @ A2 )
% 0.34/0.60 => ( ( ord_less_rat @ C @ B )
% 0.34/0.60 => ( ord_less_rat @ C @ A2 ) ) ) ).
% 0.34/0.60
% 0.34/0.60 % dual_order.strict_trans
% 0.34/0.60 thf(fact_3639_order_Ostrict__implies__not__eq,axiom,
% 0.34/0.60 ! [A2: assn,B: assn] :
% 0.34/0.60 ( ( ord_less_assn @ A2 @ B )
% 0.34/0.60 => ( A2 != B ) ) ).
% 0.34/0.60
% 0.34/0.60 % order.strict_implies_not_eq
% 0.34/0.60 thf(fact_3640_order_Ostrict__implies__not__eq,axiom,
% 0.34/0.60 ! [A2: nat,B: nat] :
% 0.34/0.60 ( ( ord_less_nat @ A2 @ B )
% 0.34/0.60 => ( A2 != B ) ) ).
% 0.34/0.60
% 0.34/0.60 % order.strict_implies_not_eq
% 0.34/0.60 thf(fact_3641_order_Ostrict__implies__not__eq,axiom,
% 0.34/0.60 ! [A2: int,B: int] :
% 0.34/0.60 ( ( ord_less_int @ A2 @ B )
% 0.34/0.60 => ( A2 != B ) ) ).
% 0.34/0.60
% 0.34/0.60 % order.strict_implies_not_eq
% 0.34/0.60 thf(fact_3642_order_Ostrict__implies__not__eq,axiom,
% 0.34/0.60 ! [A2: code_integer,B: code_integer] :
% 0.34/0.60 ( ( ord_le6747313008572928689nteger @ A2 @ B )
% 0.34/0.60 => ( A2 != B ) ) ).
% 0.34/0.60
% 0.34/0.60 % order.strict_implies_not_eq
% 0.34/0.60 thf(fact_3643_order_Ostrict__implies__not__eq,axiom,
% 0.34/0.60 ! [A2: rat,B: rat] :
% 0.34/0.60 ( ( ord_less_rat @ A2 @ B )
% 0.34/0.60 => ( A2 != B ) ) ).
% 0.34/0.60
% 0.34/0.60 % order.strict_implies_not_eq
% 0.34/0.60 thf(fact_3644_dual__order_Ostrict__implies__not__eq,axiom,
% 0.34/0.60 ! [B: assn,A2: assn] :
% 0.34/0.60 ( ( ord_less_assn @ B @ A2 )
% 0.34/0.60 => ( A2 != B ) ) ).
% 0.34/0.60
% 0.34/0.60 % dual_order.strict_implies_not_eq
% 0.34/0.60 thf(fact_3645_dual__order_Ostrict__implies__not__eq,axiom,
% 0.34/0.60 ! [B: nat,A2: nat] :
% 0.34/0.60 ( ( ord_less_nat @ B @ A2 )
% 0.34/0.60 => ( A2 != B ) ) ).
% 0.34/0.60
% 0.34/0.60 % dual_order.strict_implies_not_eq
% 0.34/0.60 thf(fact_3646_dual__order_Ostrict__implies__not__eq,axiom,
% 0.34/0.60 ! [B: int,A2: int] :
% 0.34/0.60 ( ( ord_less_int @ B @ A2 )
% 0.34/0.60 => ( A2 != B ) ) ).
% 0.34/0.60
% 0.34/0.60 % dual_order.strict_implies_not_eq
% 0.34/0.60 thf(fact_3647_dual__order_Ostrict__implies__not__eq,axiom,
% 0.34/0.60 ! [B: code_integer,A2: code_integer] :
% 0.34/0.60 ( ( ord_le6747313008572928689nteger @ B @ A2 )
% 0.34/0.60 => ( A2 != B ) ) ).
% 0.34/0.60
% 0.34/0.60 % dual_order.strict_implies_not_eq
% 0.34/0.60 thf(fact_3648_dual__order_Ostrict__implies__not__eq,axiom,
% 0.34/0.60 ! [B: rat,A2: rat] :
% 0.34/0.60 ( ( ord_less_rat @ B @ A2 )
% 0.34/0.60 => ( A2 != B ) ) ).
% 0.34/0.60
% 0.34/0.60 % dual_order.strict_implies_not_eq
% 0.34/0.60 thf(fact_3649_psubsetD,axiom,
% 0.34/0.60 ! [A: set_Pr4532377907799695533_nat_o,B4: set_Pr4532377907799695533_nat_o,C: produc3658429121746597890et_nat > $o] :
% 0.34/0.60 ( ( ord_le2453136405763929_nat_o @ A @ B4 )
% 0.34/0.60 => ( ( member6576561426505652726_nat_o @ C @ A )
% 0.34/0.60 => ( member6576561426505652726_nat_o @ C @ B4 ) ) ) ).
% 0.34/0.60
% 0.34/0.60 % psubsetD
% 0.34/0.60 thf(fact_3650_psubsetD,axiom,
% 0.34/0.60 ! [A: set_nat,B4: set_nat,C: nat] :
% 0.34/0.60 ( ( ord_less_set_nat @ A @ B4 )
% 0.34/0.60 => ( ( member_nat @ C @ A )
% 0.34/0.60 => ( member_nat @ C @ B4 ) ) ) ).
% 0.34/0.60
% 0.34/0.60 % psubsetD
% 0.34/0.60 thf(fact_3651_psubsetD,axiom,
% 0.34/0.60 ! [A: set_int,B4: set_int,C: int] :
% 0.34/0.60 ( ( ord_less_set_int @ A @ B4 )
% 0.34/0.60 => ( ( member_int @ C @ A )
% 0.34/0.60 => ( member_int @ C @ B4 ) ) ) ).
% 0.34/0.60
% 0.34/0.60 % psubsetD
% 0.34/0.60 thf(fact_3652_psubsetD,axiom,
% 0.34/0.60 ! [A: set_o,B4: set_o,C: $o] :
% 0.34/0.60 ( ( ord_less_set_o @ A @ B4 )
% 0.34/0.60 => ( ( member_o @ C @ A )
% 0.34/0.60 => ( member_o @ C @ B4 ) ) ) ).
% 0.34/0.60
% 0.34/0.60 % psubsetD
% 0.34/0.60 thf(fact_3653_psubsetD,axiom,
% 0.34/0.60 ! [A: set_Product_unit,B4: set_Product_unit,C: product_unit] :
% 0.34/0.60 ( ( ord_le8056459307392131481t_unit @ A @ B4 )
% 0.34/0.60 => ( ( member_Product_unit @ C @ A )
% 0.34/0.60 => ( member_Product_unit @ C @ B4 ) ) ) ).
% 0.34/0.60
% 0.34/0.60 % psubsetD
% 0.34/0.60 thf(fact_3654_UNIV__eq__I,axiom,
% 0.34/0.60 ! [A: set_Pr4532377907799695533_nat_o] :
% 0.34/0.60 ( ! [X3: produc3658429121746597890et_nat > $o] : ( member6576561426505652726_nat_o @ X3 @ A )
% 0.34/0.60 => ( top_to4869887016220417533_nat_o = A ) ) ).
% 0.34/0.60
% 0.34/0.60 % UNIV_eq_I
% 0.34/0.60 thf(fact_3655_UNIV__eq__I,axiom,
% 0.34/0.60 ! [A: set_Product_unit] :
% 0.34/0.60 ( ! [X3: product_unit] : ( member_Product_unit @ X3 @ A )
% 0.34/0.60 => ( top_to1996260823553986621t_unit = A ) ) ).
% 0.34/0.60
% 0.34/0.60 % UNIV_eq_I
% 0.34/0.60 thf(fact_3656_UNIV__eq__I,axiom,
% 0.34/0.60 ! [A: set_o] :
% 0.34/0.60 ( ! [X3: $o] : ( member_o @ X3 @ A )
% 0.34/0.60 => ( top_top_set_o = A ) ) ).
% 0.34/0.60
% 0.34/0.60 % UNIV_eq_I
% 0.34/0.60 thf(fact_3657_UNIV__eq__I,axiom,
% 0.34/0.60 ! [A: set_nat] :
% 0.34/0.60 ( ! [X3: nat] : ( member_nat @ X3 @ A )
% 0.34/0.60 => ( top_top_set_nat = A ) ) ).
% 0.34/0.60
% 0.34/0.60 % UNIV_eq_I
% 0.34/0.60 thf(fact_3658_UNIV__eq__I,axiom,
% 0.34/0.60 ! [A: set_int] :
% 0.34/0.60 ( ! [X3: int] : ( member_int @ X3 @ A )
% 0.34/0.60 => ( top_top_set_int = A ) ) ).
% 0.34/0.60
% 0.34/0.60 % UNIV_eq_I
% 0.34/0.60 thf(fact_3659_UNIV__witness,axiom,
% 0.34/0.60 ? [X3: produc3658429121746597890et_nat > $o] : ( member6576561426505652726_nat_o @ X3 @ top_to4869887016220417533_nat_o ) ).
% 0.34/0.60
% 0.34/0.60 % UNIV_witness
% 0.34/0.60 thf(fact_3660_UNIV__witness,axiom,
% 0.34/0.60 ? [X3: product_unit] : ( member_Product_unit @ X3 @ top_to1996260823553986621t_unit ) ).
% 0.34/0.60
% 0.34/0.60 % UNIV_witness
% 0.34/0.60 thf(fact_3661_UNIV__witness,axiom,
% 0.34/0.60 ? [X3: $o] : ( member_o @ X3 @ top_top_set_o ) ).
% 0.34/0.60
% 0.34/0.60 % UNIV_witness
% 0.34/0.60 thf(fact_3662_UNIV__witness,axiom,
% 0.34/0.60 ? [X3: nat] : ( member_nat @ X3 @ top_top_set_nat ) ).
% 0.34/0.60
% 0.34/0.60 % UNIV_witness
% 0.34/0.60 thf(fact_3663_UNIV__witness,axiom,
% 0.34/0.60 ? [X3: int] : ( member_int @ X3 @ top_top_set_int ) ).
% 0.34/0.60
% 0.34/0.60 % UNIV_witness
% 0.34/0.60 thf(fact_3664_linorder__neqE,axiom,
% 0.34/0.60 ! [X: nat,Y: nat] :
% 0.34/0.60 ( ( X != Y )
% 0.34/0.60 => ( ~ ( ord_less_nat @ X @ Y )
% 0.34/0.60 => ( ord_less_nat @ Y @ X ) ) ) ).
% 0.34/0.60
% 0.34/0.60 % linorder_neqE
% 0.34/0.60 thf(fact_3665_linorder__neqE,axiom,
% 0.34/0.60 ! [X: int,Y: int] :
% 0.34/0.60 ( ( X != Y )
% 0.34/0.60 => ( ~ ( ord_less_int @ X @ Y )
% 0.34/0.60 => ( ord_less_int @ Y @ X ) ) ) ).
% 0.34/0.60
% 0.34/0.60 % linorder_neqE
% 0.34/0.60 thf(fact_3666_linorder__neqE,axiom,
% 0.34/0.60 ! [X: code_integer,Y: code_integer] :
% 0.34/0.60 ( ( X != Y )
% 0.34/0.60 => ( ~ ( ord_le6747313008572928689nteger @ X @ Y )
% 0.34/0.60 => ( ord_le6747313008572928689nteger @ Y @ X ) ) ) ).
% 0.34/0.60
% 0.34/0.60 % linorder_neqE
% 0.34/0.60 thf(fact_3667_linorder__neqE,axiom,
% 0.34/0.60 ! [X: rat,Y: rat] :
% 0.34/0.60 ( ( X != Y )
% 0.34/0.60 => ( ~ ( ord_less_rat @ X @ Y )
% 0.34/0.60 => ( ord_less_rat @ Y @ X ) ) ) ).
% 0.34/0.60
% 0.34/0.60 % linorder_neqE
% 0.34/0.60 thf(fact_3668_order__less__asym,axiom,
% 0.34/0.60 ! [X: assn,Y: assn] :
% 0.34/0.60 ( ( ord_less_assn @ X @ Y )
% 0.34/0.60 => ~ ( ord_less_assn @ Y @ X ) ) ).
% 0.34/0.60
% 0.34/0.60 % order_less_asym
% 0.34/0.60 thf(fact_3669_order__less__asym,axiom,
% 0.34/0.60 ! [X: nat,Y: nat] :
% 0.34/0.60 ( ( ord_less_nat @ X @ Y )
% 0.34/0.60 => ~ ( ord_less_nat @ Y @ X ) ) ).
% 0.34/0.60
% 0.34/0.60 % order_less_asym
% 0.34/0.60 thf(fact_3670_order__less__asym,axiom,
% 0.34/0.60 ! [X: int,Y: int] :
% 0.34/0.60 ( ( ord_less_int @ X @ Y )
% 0.34/0.60 => ~ ( ord_less_int @ Y @ X ) ) ).
% 0.34/0.60
% 0.34/0.60 % order_less_asym
% 0.34/0.60 thf(fact_3671_order__less__asym,axiom,
% 0.34/0.60 ! [X: code_integer,Y: code_integer] :
% 0.34/0.60 ( ( ord_le6747313008572928689nteger @ X @ Y )
% 0.34/0.60 => ~ ( ord_le6747313008572928689nteger @ Y @ X ) ) ).
% 0.34/0.60
% 0.34/0.60 % order_less_asym
% 0.34/0.60 thf(fact_3672_order__less__asym,axiom,
% 0.34/0.60 ! [X: rat,Y: rat] :
% 0.34/0.60 ( ( ord_less_rat @ X @ Y )
% 0.34/0.60 => ~ ( ord_less_rat @ Y @ X ) ) ).
% 0.34/0.60
% 0.34/0.60 % order_less_asym
% 0.34/0.60 thf(fact_3673_linorder__neq__iff,axiom,
% 0.34/0.60 ! [X: nat,Y: nat] :
% 0.34/0.60 ( ( X != Y )
% 0.34/0.60 = ( ( ord_less_nat @ X @ Y )
% 0.34/0.60 | ( ord_less_nat @ Y @ X ) ) ) ).
% 0.34/0.60
% 0.34/0.60 % linorder_neq_iff
% 0.34/0.60 thf(fact_3674_linorder__neq__iff,axiom,
% 0.34/0.60 ! [X: int,Y: int] :
% 0.34/0.60 ( ( X != Y )
% 0.34/0.60 = ( ( ord_less_int @ X @ Y )
% 0.34/0.60 | ( ord_less_int @ Y @ X ) ) ) ).
% 0.34/0.60
% 0.34/0.60 % linorder_neq_iff
% 0.34/0.60 thf(fact_3675_linorder__neq__iff,axiom,
% 0.34/0.60 ! [X: code_integer,Y: code_integer] :
% 0.34/0.60 ( ( X != Y )
% 0.34/0.60 = ( ( ord_le6747313008572928689nteger @ X @ Y )
% 0.34/0.60 | ( ord_le6747313008572928689nteger @ Y @ X ) ) ) ).
% 0.34/0.60
% 0.34/0.60 % linorder_neq_iff
% 0.34/0.60 thf(fact_3676_linorder__neq__iff,axiom,
% 0.34/0.60 ! [X: rat,Y: rat] :
% 0.34/0.60 ( ( X != Y )
% 0.34/0.60 = ( ( ord_less_rat @ X @ Y )
% 0.34/0.60 | ( ord_less_rat @ Y @ X ) ) ) ).
% 0.34/0.60
% 0.34/0.60 % linorder_neq_iff
% 0.34/0.60 thf(fact_3677_order__less__asym_H,axiom,
% 0.34/0.60 ! [A2: assn,B: assn] :
% 0.34/0.60 ( ( ord_less_assn @ A2 @ B )
% 0.34/0.60 => ~ ( ord_less_assn @ B @ A2 ) ) ).
% 0.34/0.60
% 0.34/0.60 % order_less_asym'
% 0.34/0.60 thf(fact_3678_order__less__asym_H,axiom,
% 0.34/0.60 ! [A2: nat,B: nat] :
% 0.34/0.60 ( ( ord_less_nat @ A2 @ B )
% 0.34/0.60 => ~ ( ord_less_nat @ B @ A2 ) ) ).
% 0.34/0.60
% 0.34/0.60 % order_less_asym'
% 0.34/0.60 thf(fact_3679_order__less__asym_H,axiom,
% 0.34/0.60 ! [A2: int,B: int] :
% 0.34/0.60 ( ( ord_less_int @ A2 @ B )
% 0.34/0.60 => ~ ( ord_less_int @ B @ A2 ) ) ).
% 0.34/0.60
% 0.34/0.60 % order_less_asym'
% 0.34/0.60 thf(fact_3680_order__less__asym_H,axiom,
% 0.34/0.60 ! [A2: code_integer,B: code_integer] :
% 0.34/0.60 ( ( ord_le6747313008572928689nteger @ A2 @ B )
% 0.34/0.60 => ~ ( ord_le6747313008572928689nteger @ B @ A2 ) ) ).
% 0.34/0.60
% 0.34/0.60 % order_less_asym'
% 0.34/0.60 thf(fact_3681_order__less__asym_H,axiom,
% 0.34/0.60 ! [A2: rat,B: rat] :
% 0.34/0.60 ( ( ord_less_rat @ A2 @ B )
% 0.34/0.60 => ~ ( ord_less_rat @ B @ A2 ) ) ).
% 0.34/0.60
% 0.34/0.60 % order_less_asym'
% 0.34/0.60 thf(fact_3682_order__less__trans,axiom,
% 0.34/0.60 ! [X: assn,Y: assn,Z: assn] :
% 0.34/0.60 ( ( ord_less_assn @ X @ Y )
% 0.34/0.60 => ( ( ord_less_assn @ Y @ Z )
% 0.34/0.60 => ( ord_less_assn @ X @ Z ) ) ) ).
% 0.34/0.60
% 0.34/0.60 % order_less_trans
% 0.34/0.60 thf(fact_3683_order__less__trans,axiom,
% 0.34/0.60 ! [X: nat,Y: nat,Z: nat] :
% 0.34/0.60 ( ( ord_less_nat @ X @ Y )
% 0.34/0.60 => ( ( ord_less_nat @ Y @ Z )
% 0.34/0.60 => ( ord_less_nat @ X @ Z ) ) ) ).
% 0.34/0.60
% 0.34/0.60 % order_less_trans
% 0.34/0.60 thf(fact_3684_order__less__trans,axiom,
% 0.34/0.60 ! [X: int,Y: int,Z: int] :
% 0.34/0.60 ( ( ord_less_int @ X @ Y )
% 0.34/0.60 => ( ( ord_less_int @ Y @ Z )
% 0.34/0.60 => ( ord_less_int @ X @ Z ) ) ) ).
% 0.34/0.60
% 0.34/0.60 % order_less_trans
% 0.34/0.60 thf(fact_3685_order__less__trans,axiom,
% 0.34/0.60 ! [X: code_integer,Y: code_integer,Z: code_integer] :
% 0.34/0.60 ( ( ord_le6747313008572928689nteger @ X @ Y )
% 0.34/0.60 => ( ( ord_le6747313008572928689nteger @ Y @ Z )
% 0.34/0.60 => ( ord_le6747313008572928689nteger @ X @ Z ) ) ) ).
% 0.34/0.60
% 0.34/0.60 % order_less_trans
% 0.34/0.60 thf(fact_3686_order__less__trans,axiom,
% 0.34/0.60 ! [X: rat,Y: rat,Z: rat] :
% 0.34/0.60 ( ( ord_less_rat @ X @ Y )
% 0.34/0.60 => ( ( ord_less_rat @ Y @ Z )
% 0.34/0.60 => ( ord_less_rat @ X @ Z ) ) ) ).
% 0.34/0.60
% 0.34/0.60 % order_less_trans
% 0.34/0.60 thf(fact_3687_ord__eq__less__subst,axiom,
% 0.34/0.60 ! [A2: assn,F: assn > assn,B: assn,C: assn] :
% 0.34/0.60 ( ( A2
% 0.34/0.60 = ( F @ B ) )
% 0.34/0.60 => ( ( ord_less_assn @ B @ C )
% 0.34/0.60 => ( ! [X3: assn,Y2: assn] :
% 0.34/0.60 ( ( ord_less_assn @ X3 @ Y2 )
% 0.34/0.60 => ( ord_less_assn @ ( F @ X3 ) @ ( F @ Y2 ) ) )
% 0.34/0.60 => ( ord_less_assn @ A2 @ ( F @ C ) ) ) ) ) ).
% 0.34/0.60
% 0.34/0.60 % ord_eq_less_subst
% 0.34/0.60 thf(fact_3688_ord__eq__less__subst,axiom,
% 0.34/0.60 ! [A2: nat,F: assn > nat,B: assn,C: assn] :
% 0.34/0.60 ( ( A2
% 0.34/0.60 = ( F @ B ) )
% 0.34/0.60 => ( ( ord_less_assn @ B @ C )
% 0.34/0.60 => ( ! [X3: assn,Y2: assn] :
% 0.34/0.60 ( ( ord_less_assn @ X3 @ Y2 )
% 0.34/0.60 => ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
% 0.34/0.60 => ( ord_less_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% 0.34/0.60
% 0.34/0.60 % ord_eq_less_subst
% 0.34/0.60 thf(fact_3689_ord__eq__less__subst,axiom,
% 0.34/0.60 ! [A2: int,F: assn > int,B: assn,C: assn] :
% 0.34/0.60 ( ( A2
% 0.34/0.60 = ( F @ B ) )
% 0.34/0.60 => ( ( ord_less_assn @ B @ C )
% 0.34/0.60 => ( ! [X3: assn,Y2: assn] :
% 0.34/0.60 ( ( ord_less_assn @ X3 @ Y2 )
% 0.34/0.60 => ( ord_less_int @ ( F @ X3 ) @ ( F @ Y2 ) ) )
% 0.34/0.60 => ( ord_less_int @ A2 @ ( F @ C ) ) ) ) ) ).
% 0.34/0.60
% 0.34/0.60 % ord_eq_less_subst
% 0.34/0.60 thf(fact_3690_ord__eq__less__subst,axiom,
% 0.34/0.60 ! [A2: code_integer,F: assn > code_integer,B: assn,C: assn] :
% 0.34/0.60 ( ( A2
% 0.34/0.60 = ( F @ B ) )
% 0.34/0.60 => ( ( ord_less_assn @ B @ C )
% 0.34/0.60 => ( ! [X3: assn,Y2: assn] :
% 0.34/0.60 ( ( ord_less_assn @ X3 @ Y2 )
% 0.34/0.60 => ( ord_le6747313008572928689nteger @ ( F @ X3 ) @ ( F @ Y2 ) ) )
% 0.34/0.60 => ( ord_le6747313008572928689nteger @ A2 @ ( F @ C ) ) ) ) ) ).
% 0.34/0.60
% 0.34/0.60 % ord_eq_less_subst
% 0.34/0.60 thf(fact_3691_ord__eq__less__subst,axiom,
% 0.34/0.60 ! [A2: rat,F: assn > rat,B: assn,C: assn] :
% 0.34/0.60 ( ( A2
% 0.34/0.60 = ( F @ B ) )
% 0.34/0.60 => ( ( ord_less_assn @ B @ C )
% 0.34/0.60 => ( ! [X3: assn,Y2: assn] :
% 0.34/0.60 ( ( ord_less_assn @ X3 @ Y2 )
% 0.34/0.60 => ( ord_less_rat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
% 0.34/0.60 => ( ord_less_rat @ A2 @ ( F @ C ) ) ) ) ) ).
% 0.34/0.60
% 0.34/0.60 % ord_eq_less_subst
% 0.34/0.60 thf(fact_3692_ord__eq__less__subst,axiom,
% 0.34/0.60 ! [A2: assn,F: nat > assn,B: nat,C: nat] :
% 0.34/0.60 ( ( A2
% 0.34/0.60 = ( F @ B ) )
% 0.34/0.60 => ( ( ord_less_nat @ B @ C )
% 0.34/0.60 => ( ! [X3: nat,Y2: nat] :
% 0.34/0.60 ( ( ord_less_nat @ X3 @ Y2 )
% 0.34/0.60 => ( ord_less_assn @ ( F @ X3 ) @ ( F @ Y2 ) ) )
% 0.34/0.60 => ( ord_less_assn @ A2 @ ( F @ C ) ) ) ) ) ).
% 0.34/0.60
% 0.34/0.60 % ord_eq_less_subst
% 0.34/0.60 thf(fact_3693_ord__eq__less__subst,axiom,
% 0.34/0.60 ! [A2: nat,F: nat > nat,B: nat,C: nat] :
% 0.34/0.60 ( ( A2
% 0.34/0.60 = ( F @ B ) )
% 0.34/0.60 => ( ( ord_less_nat @ B @ C )
% 0.34/0.60 => ( ! [X3: nat,Y2: nat] :
% 0.34/0.60 ( ( ord_less_nat @ X3 @ Y2 )
% 0.34/0.60 => ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
% 0.34/0.60 => ( ord_less_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% 0.34/0.60
% 0.34/0.60 % ord_eq_less_subst
% 0.34/0.60 thf(fact_3694_ord__eq__less__subst,axiom,
% 0.34/0.60 ! [A2: int,F: nat > int,B: nat,C: nat] :
% 0.34/0.60 ( ( A2
% 0.34/0.60 = ( F @ B ) )
% 0.34/0.60 => ( ( ord_less_nat @ B @ C )
% 0.34/0.60 => ( ! [X3: nat,Y2: nat] :
% 0.34/0.60 ( ( ord_less_nat @ X3 @ Y2 )
% 0.34/0.60 => ( ord_less_int @ ( F @ X3 ) @ ( F @ Y2 ) ) )
% 0.34/0.60 => ( ord_less_int @ A2 @ ( F @ C ) ) ) ) ) ).
% 0.34/0.60
% 0.34/0.60 % ord_eq_less_subst
% 0.34/0.60 thf(fact_3695_ord__eq__less__subst,axiom,
% 0.34/0.60 ! [A2: code_integer,F: nat > code_integer,B: nat,C: nat] :
% 0.34/0.60 ( ( A2
% 0.34/0.60 = ( F @ B ) )
% 0.34/0.60 => ( ( ord_less_nat @ B @ C )
% 0.34/0.60 => ( ! [X3: nat,Y2: nat] :
% 0.34/0.60 ( ( ord_less_nat @ X3 @ Y2 )
% 0.34/0.60 => ( ord_le6747313008572928689nteger @ ( F @ X3 ) @ ( F @ Y2 ) ) )
% 0.34/0.60 => ( ord_le6747313008572928689nteger @ A2 @ ( F @ C ) ) ) ) ) ).
% 0.34/0.60
% 0.34/0.60 % ord_eq_less_subst
% 0.34/0.60 thf(fact_3696_ord__eq__less__subst,axiom,
% 0.34/0.60 ! [A2: rat,F: nat > rat,B: nat,C: nat] :
% 0.34/0.60 ( ( A2
% 0.34/0.60 = ( F @ B ) )
% 0.34/0.60 => ( ( ord_less_nat @ B @ C )
% 0.34/0.60 => ( ! [X3: nat,Y2: nat] :
% 0.34/0.60 ( ( ord_less_nat @ X3 @ Y2 )
% 0.34/0.60 => ( ord_less_rat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
% 0.34/0.60 => ( ord_less_rat @ A2 @ ( F @ C ) ) ) ) ) ).
% 0.34/0.60
% 0.34/0.60 % ord_eq_less_subst
% 0.34/0.60 thf(fact_3697_ord__less__eq__subst,axiom,
% 0.34/0.60 ! [A2: assn,B: assn,F: assn > assn,C: assn] :
% 0.34/0.60 ( ( ord_less_assn @ A2 @ B )
% 0.34/0.60 => ( ( ( F @ B )
% 0.34/0.60 = C )
% 0.34/0.60 => ( ! [X3: assn,Y2: assn] :
% 0.34/0.60 ( ( ord_less_assn @ X3 @ Y2 )
% 0.34/0.60 => ( ord_less_assn @ ( F @ X3 ) @ ( F @ Y2 ) ) )
% 0.34/0.60 => ( ord_less_assn @ ( F @ A2 ) @ C ) ) ) ) ).
% 0.34/0.60
% 0.34/0.60 % ord_less_eq_subst
% 0.34/0.60 thf(fact_3698_ord__less__eq__subst,axiom,
% 0.34/0.60 ! [A2: assn,B: assn,F: assn > nat,C: nat] :
% 0.34/0.60 ( ( ord_less_assn @ A2 @ B )
% 0.34/0.60 => ( ( ( F @ B )
% 0.34/0.60 = C )
% 0.34/0.60 => ( ! [X3: assn,Y2: assn] :
% 0.34/0.60 ( ( ord_less_assn @ X3 @ Y2 )
% 0.34/0.60 => ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
% 0.34/0.60 => ( ord_less_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% 0.34/0.60
% 0.34/0.60 % ord_less_eq_subst
% 0.34/0.60 thf(fact_3699_ord__less__eq__subst,axiom,
% 0.34/0.60 ! [A2: assn,B: assn,F: assn > int,C: int] :
% 0.34/0.60 ( ( ord_less_assn @ A2 @ B )
% 0.34/0.60 => ( ( ( F @ B )
% 0.34/0.60 = C )
% 0.34/0.60 => ( ! [X3: assn,Y2: assn] :
% 0.34/0.60 ( ( ord_less_assn @ X3 @ Y2 )
% 0.34/0.60 => ( ord_less_int @ ( F @ X3 ) @ ( F @ Y2 ) ) )
% 0.34/0.60 => ( ord_less_int @ ( F @ A2 ) @ C ) ) ) ) ).
% 0.34/0.60
% 0.34/0.60 % ord_less_eq_subst
% 0.34/0.60 thf(fact_3700_ord__less__eq__subst,axiom,
% 0.34/0.60 ! [A2: assn,B: assn,F: assn > code_integer,C: code_integer] :
% 0.34/0.60 ( ( ord_less_assn @ A2 @ B )
% 0.34/0.60 => ( ( ( F @ B )
% 0.34/0.60 = C )
% 0.34/0.60 => ( ! [X3: assn,Y2: assn] :
% 0.34/0.60 ( ( ord_less_assn @ X3 @ Y2 )
% 0.34/0.60 => ( ord_le6747313008572928689nteger @ ( F @ X3 ) @ ( F @ Y2 ) ) )
% 0.34/0.60 => ( ord_le6747313008572928689nteger @ ( F @ A2 ) @ C ) ) ) ) ).
% 0.34/0.60
% 0.34/0.60 % ord_less_eq_subst
% 0.34/0.60 thf(fact_3701_ord__less__eq__subst,axiom,
% 0.34/0.60 ! [A2: assn,B: assn,F: assn > rat,C: rat] :
% 0.34/0.60 ( ( ord_less_assn @ A2 @ B )
% 0.34/0.60 => ( ( ( F @ B )
% 0.34/0.60 = C )
% 0.34/0.60 => ( ! [X3: assn,Y2: assn] :
% 0.34/0.60 ( ( ord_less_assn @ X3 @ Y2 )
% 0.34/0.60 => ( ord_less_rat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
% 0.34/0.60 => ( ord_less_rat @ ( F @ A2 ) @ C ) ) ) ) ).
% 0.34/0.60
% 0.34/0.60 % ord_less_eq_subst
% 0.34/0.60 thf(fact_3702_ord__less__eq__subst,axiom,
% 0.34/0.60 ! [A2: nat,B: nat,F: nat > assn,C: assn] :
% 0.34/0.60 ( ( ord_less_nat @ A2 @ B )
% 0.34/0.60 => ( ( ( F @ B )
% 0.34/0.60 = C )
% 0.34/0.60 => ( ! [X3: nat,Y2: nat] :
% 0.34/0.60 ( ( ord_less_nat @ X3 @ Y2 )
% 0.34/0.60 => ( ord_less_assn @ ( F @ X3 ) @ ( F @ Y2 ) ) )
% 0.34/0.60 => ( ord_less_assn @ ( F @ A2 ) @ C ) ) ) ) ).
% 0.34/0.60
% 0.34/0.60 % ord_less_eq_subst
% 0.34/0.60 thf(fact_3703_ord__less__eq__subst,axiom,
% 0.34/0.60 ! [A2: nat,B: nat,F: nat > nat,C: nat] :
% 0.34/0.60 ( ( ord_less_nat @ A2 @ B )
% 0.34/0.60 => ( ( ( F @ B )
% 0.34/0.60 = C )
% 0.34/0.60 => ( ! [X3: nat,Y2: nat] :
% 0.34/0.60 ( ( ord_less_nat @ X3 @ Y2 )
% 0.34/0.60 => ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
% 0.34/0.60 => ( ord_less_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% 0.34/0.60
% 0.34/0.60 % ord_less_eq_subst
% 0.34/0.60 thf(fact_3704_ord__less__eq__subst,axiom,
% 0.34/0.60 ! [A2: nat,B: nat,F: nat > int,C: int] :
% 0.34/0.60 ( ( ord_less_nat @ A2 @ B )
% 0.34/0.60 => ( ( ( F @ B )
% 0.34/0.60 = C )
% 0.34/0.60 => ( ! [X3: nat,Y2: nat] :
% 0.34/0.60 ( ( ord_less_nat @ X3 @ Y2 )
% 0.34/0.60 => ( ord_less_int @ ( F @ X3 ) @ ( F @ Y2 ) ) )
% 0.34/0.60 => ( ord_less_int @ ( F @ A2 ) @ C ) ) ) ) ).
% 0.34/0.60
% 0.34/0.60 % ord_less_eq_subst
% 0.34/0.60 thf(fact_3705_ord__less__eq__subst,axiom,
% 0.34/0.60 ! [A2: nat,B: nat,F: nat > code_integer,C: code_integer] :
% 0.34/0.60 ( ( ord_less_nat @ A2 @ B )
% 0.34/0.60 => ( ( ( F @ B )
% 0.34/0.60 = C )
% 0.34/0.60 => ( ! [X3: nat,Y2: nat] :
% 0.34/0.60 ( ( ord_less_nat @ X3 @ Y2 )
% 0.34/0.60 => ( ord_le6747313008572928689nteger @ ( F @ X3 ) @ ( F @ Y2 ) ) )
% 0.34/0.60 => ( ord_le6747313008572928689nteger @ ( F @ A2 ) @ C ) ) ) ) ).
% 0.34/0.60
% 0.34/0.60 % ord_less_eq_subst
% 0.34/0.60 thf(fact_3706_ord__less__eq__subst,axiom,
% 0.34/0.60 ! [A2: nat,B: nat,F: nat > rat,C: rat] :
% 0.34/0.60 ( ( ord_less_nat @ A2 @ B )
% 0.34/0.60 => ( ( ( F @ B )
% 0.34/0.60 = C )
% 0.34/0.60 => ( ! [X3: nat,Y2: nat] :
% 0.34/0.60 ( ( ord_less_nat @ X3 @ Y2 )
% 0.34/0.60 => ( ord_less_rat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
% 0.34/0.60 => ( ord_less_rat @ ( F @ A2 ) @ C ) ) ) ) ).
% 0.34/0.60
% 0.34/0.60 % ord_less_eq_subst
% 0.34/0.60 thf(fact_3707_order__less__irrefl,axiom,
% 0.34/0.60 ! [X: assn] :
% 0.34/0.60 ~ ( ord_less_assn @ X @ X ) ).
% 0.34/0.60
% 0.34/0.60 % order_less_irrefl
% 0.34/0.60 thf(fact_3708_order__less__irrefl,axiom,
% 0.34/0.60 ! [X: nat] :
% 0.34/0.60 ~ ( ord_less_nat @ X @ X ) ).
% 0.34/0.60
% 0.34/0.60 % order_less_irrefl
% 0.34/0.60 thf(fact_3709_order__less__irrefl,axiom,
% 0.34/0.60 ! [X: int] :
% 0.34/0.60 ~ ( ord_less_int @ X @ X ) ).
% 0.34/0.60
% 0.34/0.60 % order_less_irrefl
% 0.34/0.60 thf(fact_3710_order__less__irrefl,axiom,
% 0.34/0.60 ! [X: code_integer] :
% 0.34/0.60 ~ ( ord_le6747313008572928689nteger @ X @ X ) ).
% 0.34/0.60
% 0.34/0.60 % order_less_irrefl
% 0.34/0.60 thf(fact_3711_order__less__irrefl,axiom,
% 0.34/0.60 ! [X: rat] :
% 0.34/0.60 ~ ( ord_less_rat @ X @ X ) ).
% 0.34/0.60
% 0.34/0.60 % order_less_irrefl
% 0.34/0.60 thf(fact_3712_order__less__subst1,axiom,
% 0.34/0.60 ! [A2: assn,F: assn > assn,B: assn,C: assn] :
% 0.34/0.60 ( ( ord_less_assn @ A2 @ ( F @ B ) )
% 0.34/0.60 => ( ( ord_less_assn @ B @ C )
% 0.34/0.60 => ( ! [X3: assn,Y2: assn] :
% 0.34/0.60 ( ( ord_less_assn @ X3 @ Y2 )
% 0.34/0.60 => ( ord_less_assn @ ( F @ X3 ) @ ( F @ Y2 ) ) )
% 0.34/0.60 => ( ord_less_assn @ A2 @ ( F @ C ) ) ) ) ) ).
% 0.34/0.60
% 0.34/0.60 % order_less_subst1
% 0.34/0.60 thf(fact_3713_order__less__subst1,axiom,
% 0.34/0.60 ! [A2: assn,F: nat > assn,B: nat,C: nat] :
% 0.34/0.60 ( ( ord_less_assn @ A2 @ ( F @ B ) )
% 0.34/0.60 => ( ( ord_less_nat @ B @ C )
% 0.34/0.60 => ( ! [X3: nat,Y2: nat] :
% 0.34/0.60 ( ( ord_less_nat @ X3 @ Y2 )
% 0.34/0.60 => ( ord_less_assn @ ( F @ X3 ) @ ( F @ Y2 ) ) )
% 0.34/0.60 => ( ord_less_assn @ A2 @ ( F @ C ) ) ) ) ) ).
% 0.34/0.60
% 0.34/0.60 % order_less_subst1
% 0.34/0.60 thf(fact_3714_order__less__subst1,axiom,
% 0.34/0.60 ! [A2: assn,F: int > assn,B: int,C: int] :
% 0.34/0.60 ( ( ord_less_assn @ A2 @ ( F @ B ) )
% 0.34/0.60 => ( ( ord_less_int @ B @ C )
% 0.34/0.60 => ( ! [X3: int,Y2: int] :
% 0.34/0.60 ( ( ord_less_int @ X3 @ Y2 )
% 0.34/0.60 => ( ord_less_assn @ ( F @ X3 ) @ ( F @ Y2 ) ) )
% 0.34/0.60 => ( ord_less_assn @ A2 @ ( F @ C ) ) ) ) ) ).
% 0.34/0.60
% 0.34/0.60 % order_less_subst1
% 0.34/0.60 thf(fact_3715_order__less__subst1,axiom,
% 0.34/0.60 ! [A2: assn,F: code_integer > assn,B: code_integer,C: code_integer] :
% 0.34/0.60 ( ( ord_less_assn @ A2 @ ( F @ B ) )
% 0.34/0.60 => ( ( ord_le6747313008572928689nteger @ B @ C )
% 0.34/0.60 => ( ! [X3: code_integer,Y2: code_integer] :
% 0.34/0.60 ( ( ord_le6747313008572928689nteger @ X3 @ Y2 )
% 0.34/0.60 => ( ord_less_assn @ ( F @ X3 ) @ ( F @ Y2 ) ) )
% 0.34/0.60 => ( ord_less_assn @ A2 @ ( F @ C ) ) ) ) ) ).
% 0.34/0.60
% 0.34/0.60 % order_less_subst1
% 0.34/0.60 thf(fact_3716_order__less__subst1,axiom,
% 0.34/0.60 ! [A2: assn,F: rat > assn,B: rat,C: rat] :
% 0.34/0.60 ( ( ord_less_assn @ A2 @ ( F @ B ) )
% 0.34/0.60 => ( ( ord_less_rat @ B @ C )
% 0.34/0.60 => ( ! [X3: rat,Y2: rat] :
% 0.34/0.60 ( ( ord_less_rat @ X3 @ Y2 )
% 0.34/0.60 => ( ord_less_assn @ ( F @ X3 ) @ ( F @ Y2 ) ) )
% 0.34/0.60 => ( ord_less_assn @ A2 @ ( F @ C ) ) ) ) ) ).
% 0.34/0.60
% 0.34/0.60 % order_less_subst1
% 0.34/0.60 thf(fact_3717_order__less__subst1,axiom,
% 0.34/0.60 ! [A2: nat,F: assn > nat,B: assn,C: assn] :
% 0.34/0.60 ( ( ord_less_nat @ A2 @ ( F @ B ) )
% 0.34/0.60 => ( ( ord_less_assn @ B @ C )
% 0.34/0.60 => ( ! [X3: assn,Y2: assn] :
% 0.34/0.60 ( ( ord_less_assn @ X3 @ Y2 )
% 0.34/0.60 => ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
% 0.34/0.60 => ( ord_less_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% 0.34/0.60
% 0.34/0.60 % order_less_subst1
% 0.34/0.60 thf(fact_3718_order__less__subst1,axiom,
% 0.34/0.60 ! [A2: nat,F: nat > nat,B: nat,C: nat] :
% 0.34/0.60 ( ( ord_less_nat @ A2 @ ( F @ B ) )
% 0.34/0.60 => ( ( ord_less_nat @ B @ C )
% 0.34/0.60 => ( ! [X3: nat,Y2: nat] :
% 0.34/0.60 ( ( ord_less_nat @ X3 @ Y2 )
% 0.34/0.60 => ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
% 0.34/0.60 => ( ord_less_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% 0.34/0.60
% 0.34/0.60 % order_less_subst1
% 0.34/0.60 thf(fact_3719_order__less__subst1,axiom,
% 0.34/0.60 ! [A2: nat,F: int > nat,B: int,C: int] :
% 0.34/0.60 ( ( ord_less_nat @ A2 @ ( F @ B ) )
% 0.34/0.60 => ( ( ord_less_int @ B @ C )
% 0.34/0.60 => ( ! [X3: int,Y2: int] :
% 0.34/0.60 ( ( ord_less_int @ X3 @ Y2 )
% 0.34/0.60 => ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
% 0.34/0.60 => ( ord_less_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% 0.34/0.60
% 0.34/0.60 % order_less_subst1
% 0.34/0.60 thf(fact_3720_order__less__subst1,axiom,
% 0.34/0.60 ! [A2: nat,F: code_integer > nat,B: code_integer,C: code_integer] :
% 0.34/0.60 ( ( ord_less_nat @ A2 @ ( F @ B ) )
% 0.34/0.60 => ( ( ord_le6747313008572928689nteger @ B @ C )
% 0.34/0.60 => ( ! [X3: code_integer,Y2: code_integer] :
% 0.34/0.60 ( ( ord_le6747313008572928689nteger @ X3 @ Y2 )
% 0.34/0.60 => ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
% 0.34/0.60 => ( ord_less_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% 0.34/0.60
% 0.34/0.60 % order_less_subst1
% 0.34/0.60 thf(fact_3721_order__less__subst1,axiom,
% 0.34/0.60 ! [A2: nat,F: rat > nat,B: rat,C: rat] :
% 0.34/0.60 ( ( ord_less_nat @ A2 @ ( F @ B ) )
% 0.34/0.60 => ( ( ord_less_rat @ B @ C )
% 0.34/0.60 => ( ! [X3: rat,Y2: rat] :
% 0.34/0.60 ( ( ord_less_rat @ X3 @ Y2 )
% 0.34/0.60 => ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
% 0.34/0.60 => ( ord_less_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% 0.34/0.60
% 0.34/0.60 % order_less_subst1
% 0.34/0.60 thf(fact_3722_order__less__subst2,axiom,
% 0.34/0.60 ! [A2: assn,B: assn,F: assn > assn,C: assn] :
% 0.34/0.60 ( ( ord_less_assn @ A2 @ B )
% 0.34/0.60 => ( ( ord_less_assn @ ( F @ B ) @ C )
% 0.34/0.60 => ( ! [X3: assn,Y2: assn] :
% 0.34/0.60 ( ( ord_less_assn @ X3 @ Y2 )
% 0.34/0.60 => ( ord_less_assn @ ( F @ X3 ) @ ( F @ Y2 ) ) )
% 0.34/0.60 => ( ord_less_assn @ ( F @ A2 ) @ C ) ) ) ) ).
% 0.34/0.60
% 0.34/0.60 % order_less_subst2
% 0.34/0.60 thf(fact_3723_order__less__subst2,axiom,
% 0.34/0.60 ! [A2: assn,B: assn,F: assn > nat,C: nat] :
% 0.34/0.60 ( ( ord_less_assn @ A2 @ B )
% 0.34/0.60 => ( ( ord_less_nat @ ( F @ B ) @ C )
% 0.34/0.60 => ( ! [X3: assn,Y2: assn] :
% 0.34/0.60 ( ( ord_less_assn @ X3 @ Y2 )
% 0.34/0.60 => ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
% 0.34/0.60 => ( ord_less_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% 0.34/0.60
% 0.34/0.60 % order_less_subst2
% 0.34/0.60 thf(fact_3724_order__less__subst2,axiom,
% 0.34/0.60 ! [A2: assn,B: assn,F: assn > int,C: int] :
% 0.34/0.60 ( ( ord_less_assn @ A2 @ B )
% 0.34/0.60 => ( ( ord_less_int @ ( F @ B ) @ C )
% 0.34/0.60 => ( ! [X3: assn,Y2: assn] :
% 0.34/0.60 ( ( ord_less_assn @ X3 @ Y2 )
% 0.34/0.60 => ( ord_less_int @ ( F @ X3 ) @ ( F @ Y2 ) ) )
% 0.34/0.60 => ( ord_less_int @ ( F @ A2 ) @ C ) ) ) ) ).
% 0.34/0.60
% 0.34/0.60 % order_less_subst2
% 0.34/0.60 thf(fact_3725_order__less__subst2,axiom,
% 0.34/0.60 ! [A2: assn,B: assn,F: assn > code_integer,C: code_integer] :
% 0.34/0.60 ( ( ord_less_assn @ A2 @ B )
% 0.34/0.60 => ( ( ord_le6747313008572928689nteger @ ( F @ B ) @ C )
% 0.34/0.60 => ( ! [X3: assn,Y2: assn] :
% 0.34/0.60 ( ( ord_less_assn @ X3 @ Y2 )
% 0.34/0.60 => ( ord_le6747313008572928689nteger @ ( F @ X3 ) @ ( F @ Y2 ) ) )
% 0.34/0.60 => ( ord_le6747313008572928689nteger @ ( F @ A2 ) @ C ) ) ) ) ).
% 0.34/0.60
% 0.34/0.60 % order_less_subst2
% 0.34/0.60 thf(fact_3726_order__less__subst2,axiom,
% 0.34/0.60 ! [A2: assn,B: assn,F: assn > rat,C: rat] :
% 0.34/0.60 ( ( ord_less_assn @ A2 @ B )
% 0.34/0.60 => ( ( ord_less_rat @ ( F @ B ) @ C )
% 0.34/0.60 => ( ! [X3: assn,Y2: assn] :
% 0.34/0.60 ( ( ord_less_assn @ X3 @ Y2 )
% 0.34/0.60 => ( ord_less_rat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
% 0.34/0.60 => ( ord_less_rat @ ( F @ A2 ) @ C ) ) ) ) ).
% 0.34/0.60
% 0.34/0.60 % order_less_subst2
% 0.34/0.60 thf(fact_3727_order__less__subst2,axiom,
% 0.34/0.60 ! [A2: nat,B: nat,F: nat > assn,C: assn] :
% 0.34/0.60 ( ( ord_less_nat @ A2 @ B )
% 0.34/0.60 => ( ( ord_less_assn @ ( F @ B ) @ C )
% 0.34/0.60 => ( ! [X3: nat,Y2: nat] :
% 0.34/0.60 ( ( ord_less_nat @ X3 @ Y2 )
% 0.34/0.60 => ( ord_less_assn @ ( F @ X3 ) @ ( F @ Y2 ) ) )
% 0.34/0.60 => ( ord_less_assn @ ( F @ A2 ) @ C ) ) ) ) ).
% 0.34/0.60
% 0.34/0.60 % order_less_subst2
% 0.34/0.60 thf(fact_3728_order__less__subst2,axiom,
% 0.34/0.60 ! [A2: nat,B: nat,F: nat > nat,C: nat] :
% 0.34/0.60 ( ( ord_less_nat @ A2 @ B )
% 0.34/0.60 => ( ( ord_less_nat @ ( F @ B ) @ C )
% 0.34/0.60 => ( ! [X3: nat,Y2: nat] :
% 0.34/0.60 ( ( ord_less_nat @ X3 @ Y2 )
% 0.34/0.60 => ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
% 0.34/0.60 => ( ord_less_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% 0.34/0.60
% 0.34/0.60 % order_less_subst2
% 0.34/0.60 thf(fact_3729_order__less__subst2,axiom,
% 0.34/0.60 ! [A2: nat,B: nat,F: nat > int,C: int] :
% 0.34/0.60 ( ( ord_less_nat @ A2 @ B )
% 0.34/0.60 => ( ( ord_less_int @ ( F @ B ) @ C )
% 0.34/0.60 => ( ! [X3: nat,Y2: nat] :
% 0.34/0.60 ( ( ord_less_nat @ X3 @ Y2 )
% 0.34/0.60 => ( ord_less_int @ ( F @ X3 ) @ ( F @ Y2 ) ) )
% 0.34/0.60 => ( ord_less_int @ ( F @ A2 ) @ C ) ) ) ) ).
% 0.34/0.60
% 0.34/0.60 % order_less_subst2
% 0.34/0.60 thf(fact_3730_order__less__subst2,axiom,
% 0.34/0.60 ! [A2: nat,B: nat,F: nat > code_integer,C: code_integer] :
% 0.34/0.60 ( ( ord_less_nat @ A2 @ B )
% 0.34/0.60 => ( ( ord_le6747313008572928689nteger @ ( F @ B ) @ C )
% 0.34/0.60 => ( ! [X3: nat,Y2: nat] :
% 0.34/0.60 ( ( ord_less_nat @ X3 @ Y2 )
% 0.34/0.60 => ( ord_le6747313008572928689nteger @ ( F @ X3 ) @ ( F @ Y2 ) ) )
% 0.34/0.60 => ( ord_le6747313008572928689nteger @ ( F @ A2 ) @ C ) ) ) ) ).
% 0.34/0.60
% 0.34/0.60 % order_less_subst2
% 0.34/0.60 thf(fact_3731_order__less__subst2,axiom,
% 0.34/0.60 ! [A2: nat,B: nat,F: nat > rat,C: rat] :
% 0.34/0.60 ( ( ord_less_nat @ A2 @ B )
% 0.34/0.60 => ( ( ord_less_rat @ ( F @ B ) @ C )
% 0.34/0.60 => ( ! [X3: nat,Y2: nat] :
% 0.34/0.60 ( ( ord_less_nat @ X3 @ Y2 )
% 0.34/0.60 => ( ord_less_rat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
% 0.34/0.60 => ( ord_less_rat @ ( F @ A2 ) @ C ) ) ) ) ).
% 0.34/0.60
% 0.34/0.60 % order_less_subst2
% 0.34/0.60 thf(fact_3732_order__less__not__sym,axiom,
% 0.34/0.60 ! [X: assn,Y: assn] :
% 0.34/0.60 ( ( ord_less_assn @ X @ Y )
% 0.34/0.60 => ~ ( ord_less_assn @ Y @ X ) ) ).
% 0.34/0.60
% 0.34/0.60 % order_less_not_sym
% 0.34/0.60 thf(fact_3733_order__less__not__sym,axiom,
% 0.34/0.60 ! [X: nat,Y: nat] :
% 0.34/0.60 ( ( ord_less_nat @ X @ Y )
% 0.34/0.60 => ~ ( ord_less_nat @ Y @ X ) ) ).
% 0.34/0.60
% 0.34/0.60 % order_less_not_sym
% 0.34/0.60 thf(fact_3734_order__less__not__sym,axiom,
% 0.34/0.60 ! [X: int,Y: int] :
% 0.34/0.60 ( ( ord_less_int @ X @ Y )
% 0.34/0.60 => ~ ( ord_less_int @ Y @ X ) ) ).
% 0.34/0.60
% 0.34/0.60 % order_less_not_sym
% 0.34/0.60 thf(fact_3735_order__less__not__sym,axiom,
% 0.34/0.60 ! [X: code_integer,Y: code_integer] :
% 0.34/0.60 ( ( ord_le6747313008572928689nteger @ X @ Y )
% 0.34/0.60 => ~ ( ord_le6747313008572928689nteger @ Y @ X ) ) ).
% 0.34/0.60
% 0.34/0.60 % order_less_not_sym
% 0.34/0.60 thf(fact_3736_order__less__not__sym,axiom,
% 0.34/0.60 ! [X: rat,Y: rat] :
% 0.34/0.60 ( ( ord_less_rat @ X @ Y )
% 0.34/0.60 => ~ ( ord_less_rat @ Y @ X ) ) ).
% 0.34/0.60
% 0.34/0.60 % order_less_not_sym
% 0.34/0.60 thf(fact_3737_order__less__imp__triv,axiom,
% 0.34/0.60 ! [X: assn,Y: assn,P: $o] :
% 0.34/0.60 ( ( ord_less_assn @ X @ Y )
% 0.34/0.60 => ( ( ord_less_assn @ Y @ X )
% 0.34/0.60 => P ) ) ).
% 0.34/0.60
% 0.34/0.60 % order_less_imp_triv
% 0.34/0.60 thf(fact_3738_order__less__imp__triv,axiom,
% 0.34/0.60 ! [X: nat,Y: nat,P: $o] :
% 0.34/0.60 ( ( ord_less_nat @ X @ Y )
% 0.34/0.60 => ( ( ord_less_nat @ Y @ X )
% 0.34/0.60 => P ) ) ).
% 0.34/0.60
% 0.34/0.60 % order_less_imp_triv
% 0.34/0.60 thf(fact_3739_order__less__imp__triv,axiom,
% 0.34/0.60 ! [X: int,Y: int,P: $o] :
% 0.34/0.60 ( ( ord_less_int @ X @ Y )
% 0.34/0.60 => ( ( ord_less_int @ Y @ X )
% 0.34/0.60 => P ) ) ).
% 0.34/0.60
% 0.34/0.60 % order_less_imp_triv
% 0.34/0.60 thf(fact_3740_order__less__imp__triv,axiom,
% 0.34/0.60 ! [X: code_integer,Y: code_integer,P: $o] :
% 0.34/0.60 ( ( ord_le6747313008572928689nteger @ X @ Y )
% 0.34/0.60 => ( ( ord_le6747313008572928689nteger @ Y @ X )
% 0.34/0.60 => P ) ) ).
% 0.34/0.60
% 0.34/0.60 % order_less_imp_triv
% 0.34/0.60 thf(fact_3741_order__less__imp__triv,axiom,
% 0.34/0.60 ! [X: rat,Y: rat,P: $o] :
% 0.34/0.60 ( ( ord_less_rat @ X @ Y )
% 0.34/0.60 => ( ( ord_less_rat @ Y @ X )
% 0.34/0.60 => P ) ) ).
% 0.34/0.60
% 0.34/0.60 % order_less_imp_triv
% 0.34/0.60 thf(fact_3742_linorder__less__linear,axiom,
% 0.34/0.60 ! [X: nat,Y: nat] :
% 0.34/0.60 ( ( ord_less_nat @ X @ Y )
% 0.34/0.60 | ( X = Y )
% 0.34/0.60 | ( ord_less_nat @ Y @ X ) ) ).
% 0.34/0.60
% 0.34/0.60 % linorder_less_linear
% 0.34/0.60 thf(fact_3743_linorder__less__linear,axiom,
% 0.34/0.60 ! [X: int,Y: int] :
% 0.34/0.60 ( ( ord_less_int @ X @ Y )
% 0.34/0.60 | ( X = Y )
% 0.34/0.60 | ( ord_less_int @ Y @ X ) ) ).
% 0.34/0.60
% 0.34/0.60 % linorder_less_linear
% 0.34/0.60 thf(fact_3744_linorder__less__linear,axiom,
% 0.34/0.60 ! [X: code_integer,Y: code_integer] :
% 0.34/0.60 ( ( ord_le6747313008572928689nteger @ X @ Y )
% 0.34/0.60 | ( X = Y )
% 0.34/0.60 | ( ord_le6747313008572928689nteger @ Y @ X ) ) ).
% 0.34/0.60
% 0.34/0.60 % linorder_less_linear
% 0.34/0.60 thf(fact_3745_linorder__less__linear,axiom,
% 0.34/0.60 ! [X: rat,Y: rat] :
% 0.34/0.60 ( ( ord_less_rat @ X @ Y )
% 0.34/0.60 | ( X = Y )
% 0.34/0.60 | ( ord_less_rat @ Y @ X ) ) ).
% 0.34/0.60
% 0.34/0.60 % linorder_less_linear
% 0.34/0.60 thf(fact_3746_order__less__imp__not__eq,axiom,
% 0.34/0.60 ! [X: assn,Y: assn] :
% 0.34/0.60 ( ( ord_less_assn @ X @ Y )
% 0.34/0.60 => ( X != Y ) ) ).
% 0.34/0.60
% 0.34/0.60 % order_less_imp_not_eq
% 0.34/0.60 thf(fact_3747_order__less__imp__not__eq,axiom,
% 0.34/0.60 ! [X: nat,Y: nat] :
% 0.34/0.60 ( ( ord_less_nat @ X @ Y )
% 0.34/0.60 => ( X != Y ) ) ).
% 0.34/0.60
% 0.34/0.60 % order_less_imp_not_eq
% 0.34/0.60 thf(fact_3748_order__less__imp__not__eq,axiom,
% 0.34/0.60 ! [X: int,Y: int] :
% 0.34/0.60 ( ( ord_less_int @ X @ Y )
% 0.34/0.60 => ( X != Y ) ) ).
% 0.34/0.60
% 0.34/0.60 % order_less_imp_not_eq
% 0.34/0.60 thf(fact_3749_order__less__imp__not__eq,axiom,
% 0.34/0.60 ! [X: code_integer,Y: code_integer] :
% 0.34/0.60 ( ( ord_le6747313008572928689nteger @ X @ Y )
% 0.34/0.60 => ( X != Y ) ) ).
% 0.34/0.60
% 0.34/0.60 % order_less_imp_not_eq
% 0.34/0.60 thf(fact_3750_order__less__imp__not__eq,axiom,
% 0.34/0.60 ! [X: rat,Y: rat] :
% 0.34/0.60 ( ( ord_less_rat @ X @ Y )
% 0.34/0.60 => ( X != Y ) ) ).
% 0.34/0.60
% 0.34/0.60 % order_less_imp_not_eq
% 0.34/0.60 thf(fact_3751_order__less__imp__not__eq2,axiom,
% 0.34/0.60 ! [X: assn,Y: assn] :
% 0.34/0.60 ( ( ord_less_assn @ X @ Y )
% 0.34/0.60 => ( Y != X ) ) ).
% 0.34/0.60
% 0.34/0.60 % order_less_imp_not_eq2
% 0.34/0.60 thf(fact_3752_order__less__imp__not__eq2,axiom,
% 0.34/0.60 ! [X: nat,Y: nat] :
% 0.34/0.60 ( ( ord_less_nat @ X @ Y )
% 0.34/0.60 => ( Y != X ) ) ).
% 0.34/0.60
% 0.34/0.60 % order_less_imp_not_eq2
% 0.34/0.60 thf(fact_3753_order__less__imp__not__eq2,axiom,
% 0.34/0.60 ! [X: int,Y: int] :
% 0.34/0.60 ( ( ord_less_int @ X @ Y )
% 0.34/0.60 => ( Y != X ) ) ).
% 0.34/0.60
% 0.34/0.60 % order_less_imp_not_eq2
% 0.34/0.60 thf(fact_3754_order__less__imp__not__eq2,axiom,
% 0.34/0.60 ! [X: code_integer,Y: code_integer] :
% 0.34/0.60 ( ( ord_le6747313008572928689nteger @ X @ Y )
% 0.34/0.60 => ( Y != X ) ) ).
% 0.34/0.60
% 0.34/0.60 % order_less_imp_not_eq2
% 0.34/0.60 thf(fact_3755_order__less__imp__not__eq2,axiom,
% 0.34/0.60 ! [X: rat,Y: rat] :
% 0.34/0.60 ( ( ord_less_rat @ X @ Y )
% 0.34/0.60 => ( Y != X ) ) ).
% 0.34/0.60
% 0.34/0.60 % order_less_imp_not_eq2
% 0.34/0.60 thf(fact_3756_order__less__imp__not__less,axiom,
% 0.34/0.60 ! [X: assn,Y: assn] :
% 0.34/0.60 ( ( ord_less_assn @ X @ Y )
% 0.34/0.60 => ~ ( ord_less_assn @ Y @ X ) ) ).
% 0.34/0.60
% 0.34/0.60 % order_less_imp_not_less
% 0.34/0.60 thf(fact_3757_order__less__imp__not__less,axiom,
% 0.34/0.60 ! [X: nat,Y: nat] :
% 0.34/0.60 ( ( ord_less_nat @ X @ Y )
% 0.34/0.60 => ~ ( ord_less_nat @ Y @ X ) ) ).
% 0.34/0.60
% 0.34/0.60 % order_less_imp_not_less
% 0.34/0.60 thf(fact_3758_order__less__imp__not__less,axiom,
% 0.34/0.60 ! [X: int,Y: int] :
% 0.34/0.60 ( ( ord_less_int @ X @ Y )
% 0.34/0.60 => ~ ( ord_less_int @ Y @ X ) ) ).
% 0.34/0.60
% 0.34/0.60 % order_less_imp_not_less
% 0.34/0.60 thf(fact_3759_order__less__imp__not__less,axiom,
% 0.34/0.60 ! [X: code_integer,Y: code_integer] :
% 0.34/0.60 ( ( ord_le6747313008572928689nteger @ X @ Y )
% 0.34/0.60 => ~ ( ord_le6747313008572928689nteger @ Y @ X ) ) ).
% 0.34/0.60
% 0.34/0.60 % order_less_imp_not_less
% 0.34/0.60 thf(fact_3760_order__less__imp__not__less,axiom,
% 0.34/0.60 ! [X: rat,Y: rat] :
% 0.34/0.60 ( ( ord_less_rat @ X @ Y )
% 0.34/0.60 => ~ ( ord_less_rat @ Y @ X ) ) ).
% 0.34/0.60
% 0.34/0.60 % order_less_imp_not_less
% 0.34/0.60 thf(fact_3761_Max__gr__iff,axiom,
% 0.34/0.60 ! [A: set_Product_unit,X: product_unit] :
% 0.34/0.60 ( ( finite4290736615968046902t_unit @ A )
% 0.34/0.60 => ( ( A != bot_bo3957492148770167129t_unit )
% 0.34/0.60 => ( ( ord_le361264281704409273t_unit @ X @ ( lattic1210392890313158908t_unit @ A ) )
% 0.34/0.60 = ( ? [X2: product_unit] :
% 0.34/0.60 ( ( member_Product_unit @ X2 @ A )
% 0.34/0.60 & ( ord_le361264281704409273t_unit @ X @ X2 ) ) ) ) ) ) ).
% 0.34/0.60
% 0.34/0.60 % Max_gr_iff
% 0.34/0.60 thf(fact_3762_Max__gr__iff,axiom,
% 0.34/0.60 ! [A: set_o,X: $o] :
% 0.34/0.60 ( ( finite_finite_o @ A )
% 0.34/0.60 => ( ( A != bot_bot_set_o )
% 0.34/0.60 => ( ( ord_less_o @ X @ ( lattic1921953407002678535_Max_o @ A ) )
% 0.34/0.60 = ( ? [X2: $o] :
% 0.34/0.60 ( ( member_o @ X2 @ A )
% 0.34/0.60 & ( ord_less_o @ X @ X2 ) ) ) ) ) ) ).
% 0.34/0.60
% 0.34/0.60 % Max_gr_iff
% 0.34/0.60 thf(fact_3763_Max__gr__iff,axiom,
% 0.34/0.60 ! [A: set_int,X: int] :
% 0.34/0.60 ( ( finite_finite_int @ A )
% 0.34/0.60 => ( ( A != bot_bot_set_int )
% 0.34/0.60 => ( ( ord_less_int @ X @ ( lattic8263393255366662781ax_int @ A ) )
% 0.34/0.60 = ( ? [X2: int] :
% 0.34/0.60 ( ( member_int @ X2 @ A )
% 0.34/0.60 & ( ord_less_int @ X @ X2 ) ) ) ) ) ) ).
% 0.34/0.60
% 0.34/0.60 % Max_gr_iff
% 0.34/0.60 thf(fact_3764_Max__gr__iff,axiom,
% 0.34/0.60 ! [A: set_Code_integer,X: code_integer] :
% 0.34/0.60 ( ( finite6017078050557962740nteger @ A )
% 0.34/0.60 => ( ( A != bot_bo3990330152332043303nteger )
% 0.34/0.60 => ( ( ord_le6747313008572928689nteger @ X @ ( lattic4901227151466704046nteger @ A ) )
% 0.34/0.60 = ( ? [X2: code_integer] :
% 0.34/0.60 ( ( member_Code_integer @ X2 @ A )
% 0.34/0.60 & ( ord_le6747313008572928689nteger @ X @ X2 ) ) ) ) ) ) ).
% 0.34/0.60
% 0.34/0.60 % Max_gr_iff
% 0.34/0.60 thf(fact_3765_Max__gr__iff,axiom,
% 0.34/0.60 ! [A: set_rat,X: rat] :
% 0.34/0.60 ( ( finite_finite_rat @ A )
% 0.34/0.60 => ( ( A != bot_bot_set_rat )
% 0.34/0.60 => ( ( ord_less_rat @ X @ ( lattic7630753665789217321ax_rat @ A ) )
% 0.34/0.60 = ( ? [X2: rat] :
% 0.34/0.60 ( ( member_rat @ X2 @ A )
% 0.34/0.60 & ( ord_less_rat @ X @ X2 ) ) ) ) ) ) ).
% 0.34/0.60
% 0.34/0.60 % Max_gr_iff
% 0.34/0.60 thf(fact_3766_Max__gr__iff,axiom,
% 0.34/0.60 ! [A: set_nat,X: nat] :
% 0.34/0.60 ( ( finite_finite_nat @ A )
% 0.34/0.60 => ( ( A != bot_bot_set_nat )
% 0.34/0.60 => ( ( ord_less_nat @ X @ ( lattic8265883725875713057ax_nat @ A ) )
% 0.34/0.60 = ( ? [X2: nat] :
% 0.34/0.60 ( ( member_nat @ X2 @ A )
% 0.34/0.60 & ( ord_less_nat @ X @ X2 ) ) ) ) ) ) ).
% 0.34/0.60
% 0.34/0.60 % Max_gr_iff
% 0.34/0.60 thf(fact_3767_top__greatest,axiom,
% 0.34/0.60 ! [A2: assn] : ( ord_less_eq_assn @ A2 @ top_top_assn ) ).
% 0.34/0.60
% 0.34/0.60 % top_greatest
% 0.34/0.60 thf(fact_3768_top__greatest,axiom,
% 0.34/0.60 ! [A2: set_Product_unit] : ( ord_le3507040750410214029t_unit @ A2 @ top_to1996260823553986621t_unit ) ).
% 0.34/0.60
% 0.34/0.60 % top_greatest
% 0.34/0.60 thf(fact_3769_top__greatest,axiom,
% 0.34/0.60 ! [A2: set_o] : ( ord_less_eq_set_o @ A2 @ top_top_set_o ) ).
% 0.34/0.60
% 0.34/0.60 % top_greatest
% 0.34/0.60 thf(fact_3770_top__greatest,axiom,
% 0.34/0.60 ! [A2: set_int] : ( ord_less_eq_set_int @ A2 @ top_top_set_int ) ).
% 0.34/0.60
% 0.34/0.60 % top_greatest
% 0.34/0.60 thf(fact_3771_top__greatest,axiom,
% 0.34/0.60 ! [A2: filter_nat] : ( ord_le2510731241096832064er_nat @ A2 @ top_top_filter_nat ) ).
% 0.34/0.60
% 0.34/0.60 % top_greatest
% 0.34/0.60 thf(fact_3772_top__greatest,axiom,
% 0.34/0.60 ! [A2: set_nat] : ( ord_less_eq_set_nat @ A2 @ top_top_set_nat ) ).
% 0.34/0.60
% 0.34/0.60 % top_greatest
% 0.34/0.60 thf(fact_3773_top_Oextremum__unique,axiom,
% 0.34/0.60 ! [A2: assn] :
% 0.34/0.60 ( ( ord_less_eq_assn @ top_top_assn @ A2 )
% 0.34/0.60 = ( A2 = top_top_assn ) ) ).
% 0.34/0.60
% 0.34/0.60 % top.extremum_unique
% 0.34/0.60 thf(fact_3774_top_Oextremum__unique,axiom,
% 0.34/0.60 ! [A2: set_Product_unit] :
% 0.34/0.60 ( ( ord_le3507040750410214029t_unit @ top_to1996260823553986621t_unit @ A2 )
% 0.34/0.60 = ( A2 = top_to1996260823553986621t_unit ) ) ).
% 0.34/0.60
% 0.34/0.60 % top.extremum_unique
% 0.34/0.60 thf(fact_3775_top_Oextremum__unique,axiom,
% 0.34/0.60 ! [A2: set_o] :
% 0.34/0.60 ( ( ord_less_eq_set_o @ top_top_set_o @ A2 )
% 0.34/0.60 = ( A2 = top_top_set_o ) ) ).
% 0.34/0.60
% 0.34/0.60 % top.extremum_unique
% 0.34/0.60 thf(fact_3776_top_Oextremum__unique,axiom,
% 0.34/0.60 ! [A2: set_int] :
% 0.34/0.60 ( ( ord_less_eq_set_int @ top_top_set_int @ A2 )
% 0.34/0.60 = ( A2 = top_top_set_int ) ) ).
% 0.34/0.60
% 0.34/0.60 % top.extremum_unique
% 0.34/0.60 thf(fact_3777_top_Oextremum__unique,axiom,
% 0.34/0.60 ! [A2: filter_nat] :
% 0.34/0.60 ( ( ord_le2510731241096832064er_nat @ top_top_filter_nat @ A2 )
% 0.34/0.60 = ( A2 = top_top_filter_nat ) ) ).
% 0.34/0.60
% 0.34/0.60 % top.extremum_unique
% 0.34/0.60 thf(fact_3778_top_Oextremum__unique,axiom,
% 0.34/0.60 ! [A2: set_nat] :
% 0.34/0.60 ( ( ord_less_eq_set_nat @ top_top_set_nat @ A2 )
% 0.34/0.60 = ( A2 = top_top_set_nat ) ) ).
% 0.34/0.60
% 0.34/0.60 % top.extremum_unique
% 0.34/0.60 thf(fact_3779_top_Oextremum__uniqueI,axiom,
% 0.34/0.60 ! [A2: assn] :
% 0.34/0.60 ( ( ord_less_eq_assn @ top_top_assn @ A2 )
% 0.34/0.60 => ( A2 = top_top_assn ) ) ).
% 0.34/0.60
% 0.34/0.60 % top.extremum_uniqueI
% 0.34/0.60 thf(fact_3780_top_Oextremum__uniqueI,axiom,
% 0.34/0.60 ! [A2: set_Product_unit] :
% 0.34/0.60 ( ( ord_le3507040750410214029t_unit @ top_to1996260823553986621t_unit @ A2 )
% 0.34/0.60 => ( A2 = top_to1996260823553986621t_unit ) ) ).
% 0.34/0.60
% 0.34/0.60 % top.extremum_uniqueI
% 0.34/0.60 thf(fact_3781_top_Oextremum__uniqueI,axiom,
% 0.34/0.60 ! [A2: set_o] :
% 0.34/0.60 ( ( ord_less_eq_set_o @ top_top_set_o @ A2 )
% 0.34/0.60 => ( A2 = top_top_set_o ) ) ).
% 0.34/0.60
% 0.34/0.60 % top.extremum_uniqueI
% 0.34/0.60 thf(fact_3782_top_Oextremum__uniqueI,axiom,
% 0.34/0.60 ! [A2: set_int] :
% 0.34/0.60 ( ( ord_less_eq_set_int @ top_top_set_int @ A2 )
% 0.34/0.60 => ( A2 = top_top_set_int ) ) ).
% 0.34/0.60
% 0.34/0.60 % top.extremum_uniqueI
% 0.34/0.60 thf(fact_3783_top_Oextremum__uniqueI,axiom,
% 0.34/0.60 ! [A2: filter_nat] :
% 0.34/0.60 ( ( ord_le2510731241096832064er_nat @ top_top_filter_nat @ A2 )
% 0.34/0.60 => ( A2 = top_top_filter_nat ) ) ).
% 0.34/0.60
% 0.34/0.60 % top.extremum_uniqueI
% 0.34/0.60 thf(fact_3784_top_Oextremum__uniqueI,axiom,
% 0.34/0.60 ! [A2: set_nat] :
% 0.34/0.60 ( ( ord_less_eq_set_nat @ top_top_set_nat @ A2 )
% 0.34/0.60 => ( A2 = top_top_set_nat ) ) ).
% 0.34/0.60
% 0.34/0.60 % top.extremum_uniqueI
% 0.34/0.60 thf(fact_3785_leD,axiom,
% 0.34/0.60 ! [Y: assn,X: assn] :
% 0.34/0.60 ( ( ord_less_eq_assn @ Y @ X )
% 0.34/0.60 => ~ ( ord_less_assn @ X @ Y ) ) ).
% 0.34/0.60
% 0.34/0.60 % leD
% 0.34/0.60 thf(fact_3786_leD,axiom,
% 0.34/0.60 ! [Y: code_integer,X: code_integer] :
% 0.34/0.60 ( ( ord_le3102999989581377725nteger @ Y @ X )
% 0.34/0.60 => ~ ( ord_le6747313008572928689nteger @ X @ Y ) ) ).
% 0.34/0.60
% 0.34/0.60 % leD
% 0.34/0.60 thf(fact_3787_leD,axiom,
% 0.34/0.60 ! [Y: filter_nat,X: filter_nat] :
% 0.34/0.60 ( ( ord_le2510731241096832064er_nat @ Y @ X )
% 0.34/0.60 => ~ ( ord_less_filter_nat @ X @ Y ) ) ).
% 0.34/0.60
% 0.34/0.60 % leD
% 0.34/0.60 thf(fact_3788_leD,axiom,
% 0.34/0.60 ! [Y: set_nat,X: set_nat] :
% 0.34/0.60 ( ( ord_less_eq_set_nat @ Y @ X )
% 0.34/0.60 => ~ ( ord_less_set_nat @ X @ Y ) ) ).
% 0.34/0.60
% 0.34/0.60 % leD
% 0.34/0.60 thf(fact_3789_leD,axiom,
% 0.34/0.60 ! [Y: rat,X: rat] :
% 0.34/0.60 ( ( ord_less_eq_rat @ Y @ X )
% 0.34/0.60 => ~ ( ord_less_rat @ X @ Y ) ) ).
% 0.34/0.60
% 0.34/0.60 % leD
% 0.34/0.60 thf(fact_3790_leD,axiom,
% 0.34/0.60 ! [Y: nat,X: nat] :
% 0.34/0.60 ( ( ord_less_eq_nat @ Y @ X )
% 0.34/0.60 => ~ ( ord_less_nat @ X @ Y ) ) ).
% 0.34/0.60
% 0.34/0.60 % leD
% 0.34/0.60 thf(fact_3791_leD,axiom,
% 0.34/0.60 ! [Y: int,X: int] :
% 0.34/0.60 ( ( ord_less_eq_int @ Y @ X )
% 0.34/0.60 => ~ ( ord_less_int @ X @ Y ) ) ).
% 0.34/0.60
% 0.34/0.60 % leD
% 0.34/0.60 thf(fact_3792_leI,axiom,
% 0.34/0.60 ! [X: code_integer,Y: code_integer] :
% 0.34/0.60 ( ~ ( ord_le6747313008572928689nteger @ X @ Y )
% 0.34/0.60 => ( ord_le3102999989581377725nteger @ Y @ X ) ) ).
% 0.34/0.60
% 0.34/0.60 % leI
% 0.34/0.60 thf(fact_3793_leI,axiom,
% 0.34/0.60 ! [X: rat,Y: rat] :
% 0.34/0.60 ( ~ ( ord_less_rat @ X @ Y )
% 0.34/0.60 => ( ord_less_eq_rat @ Y @ X ) ) ).
% 0.34/0.60
% 0.34/0.60 % leI
% 0.34/0.60 thf(fact_3794_leI,axiom,
% 0.34/0.60 ! [X: nat,Y: nat] :
% 0.34/0.60 ( ~ ( ord_less_nat @ X @ Y )
% 0.34/0.60 => ( ord_less_eq_nat @ Y @ X ) ) ).
% 0.34/0.60
% 0.34/0.60 % leI
% 0.34/0.60 thf(fact_3795_leI,axiom,
% 0.34/0.60 ! [X: int,Y: int] :
% 0.34/0.60 ( ~ ( ord_less_int @ X @ Y )
% 0.34/0.60 => ( ord_less_eq_int @ Y @ X ) ) ).
% 0.34/0.60
% 0.34/0.60 % leI
% 0.34/0.60 thf(fact_3796_nless__le,axiom,
% 0.34/0.60 ! [A2: assn,B: assn] :
% 0.34/0.60 ( ( ~ ( ord_less_assn @ A2 @ B ) )
% 0.34/0.60 = ( ~ ( ord_less_eq_assn @ A2 @ B )
% 0.34/0.60 | ( A2 = B ) ) ) ).
% 0.34/0.60
% 0.34/0.60 % nless_le
% 0.34/0.60 thf(fact_3797_nless__le,axiom,
% 0.34/0.60 ! [A2: code_integer,B: code_integer] :
% 0.34/0.60 ( ( ~ ( ord_le6747313008572928689nteger @ A2 @ B ) )
% 0.34/0.60 = ( ~ ( ord_le3102999989581377725nteger @ A2 @ B )
% 0.34/0.60 | ( A2 = B ) ) ) ).
% 0.34/0.60
% 0.34/0.60 % nless_le
% 0.34/0.60 thf(fact_3798_nless__le,axiom,
% 0.34/0.60 ! [A2: filter_nat,B: filter_nat] :
% 0.34/0.60 ( ( ~ ( ord_less_filter_nat @ A2 @ B ) )
% 0.34/0.60 = ( ~ ( ord_le2510731241096832064er_nat @ A2 @ B )
% 0.34/0.60 | ( A2 = B ) ) ) ).
% 0.34/0.60
% 0.34/0.60 % nless_le
% 0.34/0.60 thf(fact_3799_nless__le,axiom,
% 0.34/0.60 ! [A2: set_nat,B: set_nat] :
% 0.34/0.60 ( ( ~ ( ord_less_set_nat @ A2 @ B ) )
% 0.34/0.60 = ( ~ ( ord_less_eq_set_nat @ A2 @ B )
% 0.34/0.60 | ( A2 = B ) ) ) ).
% 0.34/0.60
% 0.34/0.60 % nless_le
% 0.34/0.60 thf(fact_3800_nless__le,axiom,
% 0.34/0.60 ! [A2: rat,B: rat] :
% 0.34/0.60 ( ( ~ ( ord_less_rat @ A2 @ B ) )
% 0.34/0.60 = ( ~ ( ord_less_eq_rat @ A2 @ B )
% 0.34/0.60 | ( A2 = B ) ) ) ).
% 0.34/0.60
% 0.34/0.60 % nless_le
% 0.34/0.60 thf(fact_3801_nless__le,axiom,
% 0.34/0.60 ! [A2: nat,B: nat] :
% 0.34/0.60 ( ( ~ ( ord_less_nat @ A2 @ B ) )
% 0.34/0.60 = ( ~ ( ord_less_eq_nat @ A2 @ B )
% 0.34/0.60 | ( A2 = B ) ) ) ).
% 0.34/0.60
% 0.34/0.60 % nless_le
% 0.34/0.60 thf(fact_3802_nless__le,axiom,
% 0.34/0.60 ! [A2: int,B: int] :
% 0.34/0.60 ( ( ~ ( ord_less_int @ A2 @ B ) )
% 0.34/0.60 = ( ~ ( ord_less_eq_int @ A2 @ B )
% 0.34/0.60 | ( A2 = B ) ) ) ).
% 0.34/0.60
% 0.34/0.60 % nless_le
% 0.34/0.60 thf(fact_3803_antisym__conv1,axiom,
% 0.34/0.60 ! [X: assn,Y: assn] :
% 0.34/0.60 ( ~ ( ord_less_assn @ X @ Y )
% 0.34/0.60 => ( ( ord_less_eq_assn @ X @ Y )
% 0.34/0.60 = ( X = Y ) ) ) ).
% 0.34/0.60
% 0.34/0.60 % antisym_conv1
% 0.34/0.60 thf(fact_3804_antisym__conv1,axiom,
% 0.34/0.60 ! [X: code_integer,Y: code_integer] :
% 0.34/0.60 ( ~ ( ord_le6747313008572928689nteger @ X @ Y )
% 0.34/0.60 => ( ( ord_le3102999989581377725nteger @ X @ Y )
% 0.34/0.60 = ( X = Y ) ) ) ).
% 0.34/0.60
% 0.34/0.60 % antisym_conv1
% 0.34/0.60 thf(fact_3805_antisym__conv1,axiom,
% 0.34/0.60 ! [X: filter_nat,Y: filter_nat] :
% 0.34/0.60 ( ~ ( ord_less_filter_nat @ X @ Y )
% 0.34/0.60 => ( ( ord_le2510731241096832064er_nat @ X @ Y )
% 0.34/0.60 = ( X = Y ) ) ) ).
% 0.34/0.60
% 0.34/0.60 % antisym_conv1
% 0.34/0.60 thf(fact_3806_antisym__conv1,axiom,
% 0.34/0.60 ! [X: set_nat,Y: set_nat] :
% 0.34/0.60 ( ~ ( ord_less_set_nat @ X @ Y )
% 0.34/0.60 => ( ( ord_less_eq_set_nat @ X @ Y )
% 0.34/0.60 = ( X = Y ) ) ) ).
% 0.34/0.60
% 0.34/0.60 % antisym_conv1
% 0.34/0.60 thf(fact_3807_antisym__conv1,axiom,
% 0.34/0.60 ! [X: rat,Y: rat] :
% 0.34/0.60 ( ~ ( ord_less_rat @ X @ Y )
% 0.34/0.60 => ( ( ord_less_eq_rat @ X @ Y )
% 0.34/0.60 = ( X = Y ) ) ) ).
% 0.34/0.60
% 0.34/0.60 % antisym_conv1
% 0.34/0.60 thf(fact_3808_antisym__conv1,axiom,
% 0.34/0.60 ! [X: nat,Y: nat] :
% 0.34/0.60 ( ~ ( ord_less_nat @ X @ Y )
% 0.34/0.60 => ( ( ord_less_eq_nat @ X @ Y )
% 0.34/0.60 = ( X = Y ) ) ) ).
% 0.34/0.60
% 0.34/0.60 % antisym_conv1
% 0.34/0.60 thf(fact_3809_antisym__conv1,axiom,
% 0.34/0.60 ! [X: int,Y: int] :
% 0.34/0.60 ( ~ ( ord_less_int @ X @ Y )
% 0.34/0.60 => ( ( ord_less_eq_int @ X @ Y )
% 0.34/0.60 = ( X = Y ) ) ) ).
% 0.34/0.60
% 0.34/0.60 % antisym_conv1
% 0.34/0.60 thf(fact_3810_antisym__conv2,axiom,
% 0.34/0.60 ! [X: assn,Y: assn] :
% 0.34/0.60 ( ( ord_less_eq_assn @ X @ Y )
% 0.34/0.60 => ( ( ~ ( ord_less_assn @ X @ Y ) )
% 0.34/0.60 = ( X = Y ) ) ) ).
% 0.34/0.60
% 0.34/0.60 % antisym_conv2
% 0.34/0.60 thf(fact_3811_antisym__conv2,axiom,
% 0.34/0.60 ! [X: code_integer,Y: code_integer] :
% 0.34/0.60 ( ( ord_le3102999989581377725nteger @ X @ Y )
% 0.34/0.60 => ( ( ~ ( ord_le6747313008572928689nteger @ X @ Y ) )
% 0.34/0.60 = ( X = Y ) ) ) ).
% 0.34/0.60
% 0.34/0.60 % antisym_conv2
% 0.34/0.60 thf(fact_3812_antisym__conv2,axiom,
% 0.34/0.60 ! [X: filter_nat,Y: filter_nat] :
% 0.34/0.60 ( ( ord_le2510731241096832064er_nat @ X @ Y )
% 0.34/0.60 => ( ( ~ ( ord_less_filter_nat @ X @ Y ) )
% 0.34/0.60 = ( X = Y ) ) ) ).
% 0.34/0.60
% 0.34/0.60 % antisym_conv2
% 0.34/0.60 thf(fact_3813_antisym__conv2,axiom,
% 0.34/0.60 ! [X: set_nat,Y: set_nat] :
% 0.34/0.60 ( ( ord_less_eq_set_nat @ X @ Y )
% 0.34/0.60 => ( ( ~ ( ord_less_set_nat @ X @ Y ) )
% 0.34/0.60 = ( X = Y ) ) ) ).
% 0.34/0.60
% 0.34/0.60 % antisym_conv2
% 0.34/0.60 thf(fact_3814_antisym__conv2,axiom,
% 0.34/0.60 ! [X: rat,Y: rat] :
% 0.34/0.60 ( ( ord_less_eq_rat @ X @ Y )
% 0.34/0.60 => ( ( ~ ( ord_less_rat @ X @ Y ) )
% 0.34/0.60 = ( X = Y ) ) ) ).
% 0.34/0.60
% 0.34/0.60 % antisym_conv2
% 0.34/0.60 thf(fact_3815_antisym__conv2,axiom,
% 0.34/0.60 ! [X: nat,Y: nat] :
% 0.34/0.60 ( ( ord_less_eq_nat @ X @ Y )
% 0.34/0.60 => ( ( ~ ( ord_less_nat @ X @ Y ) )
% 0.34/0.60 = ( X = Y ) ) ) ).
% 0.34/0.60
% 0.34/0.60 % antisym_conv2
% 0.34/0.60 thf(fact_3816_antisym__conv2,axiom,
% 0.34/0.60 ! [X: int,Y: int] :
% 0.34/0.60 ( ( ord_less_eq_int @ X @ Y )
% 0.34/0.60 => ( ( ~ ( ord_less_int @ X @ Y ) )
% 0.34/0.60 = ( X = Y ) ) ) ).
% 0.34/0.60
% 0.34/0.60 % antisym_conv2
% 0.34/0.60 thf(fact_3817_dense__ge,axiom,
% 0.34/0.60 ! [Z: rat,Y: rat] :
% 0.34/0.60 ( ! [X3: rat] :
% 0.34/0.60 ( ( ord_less_rat @ Z @ X3 )
% 0.34/0.60 => ( ord_less_eq_rat @ Y @ X3 ) )
% 0.34/0.60 => ( ord_less_eq_rat @ Y @ Z ) ) ).
% 0.34/0.60
% 0.34/0.60 % dense_ge
% 0.34/0.60 thf(fact_3818_dense__le,axiom,
% 0.34/0.60 ! [Y: rat,Z: rat] :
% 0.34/0.60 ( ! [X3: rat] :
% 0.34/0.60 ( ( ord_less_rat @ X3 @ Y )
% 0.34/0.60 => ( ord_less_eq_rat @ X3 @ Z ) )
% 0.34/0.60 => ( ord_less_eq_rat @ Y @ Z ) ) ).
% 0.34/0.60
% 0.34/0.60 % dense_le
% 0.34/0.60 thf(fact_3819_less__le__not__le,axiom,
% 0.34/0.60 ( ord_less_assn
% 0.34/0.60 = ( ^ [X2: assn,Y3: assn] :
% 0.34/0.60 ( ( ord_less_eq_assn @ X2 @ Y3 )
% 0.34/0.60 & ~ ( ord_less_eq_assn @ Y3 @ X2 ) ) ) ) ).
% 0.34/0.60
% 0.34/0.60 % less_le_not_le
% 0.34/0.60 thf(fact_3820_less__le__not__le,axiom,
% 0.34/0.60 ( ord_le6747313008572928689nteger
% 0.34/0.60 = ( ^ [X2: code_integer,Y3: code_integer] :
% 0.34/0.60 ( ( ord_le3102999989581377725nteger @ X2 @ Y3 )
% 0.34/0.60 & ~ ( ord_le3102999989581377725nteger @ Y3 @ X2 ) ) ) ) ).
% 0.34/0.60
% 0.34/0.60 % less_le_not_le
% 0.34/0.60 thf(fact_3821_less__le__not__le,axiom,
% 0.34/0.60 ( ord_less_filter_nat
% 0.34/0.60 = ( ^ [X2: filter_nat,Y3: filter_nat] :
% 0.34/0.60 ( ( ord_le2510731241096832064er_nat @ X2 @ Y3 )
% 0.34/0.60 & ~ ( ord_le2510731241096832064er_nat @ Y3 @ X2 ) ) ) ) ).
% 0.34/0.60
% 0.34/0.60 % less_le_not_le
% 0.34/0.60 thf(fact_3822_less__le__not__le,axiom,
% 0.34/0.60 ( ord_less_set_nat
% 0.34/0.60 = ( ^ [X2: set_nat,Y3: set_nat] :
% 0.34/0.60 ( ( ord_less_eq_set_nat @ X2 @ Y3 )
% 0.34/0.60 & ~ ( ord_less_eq_set_nat @ Y3 @ X2 ) ) ) ) ).
% 0.34/0.60
% 0.34/0.60 % less_le_not_le
% 0.34/0.60 thf(fact_3823_less__le__not__le,axiom,
% 0.34/0.60 ( ord_less_rat
% 0.34/0.60 = ( ^ [X2: rat,Y3: rat] :
% 0.34/0.60 ( ( ord_less_eq_rat @ X2 @ Y3 )
% 0.34/0.60 & ~ ( ord_less_eq_rat @ Y3 @ X2 ) ) ) ) ).
% 0.34/0.60
% 0.34/0.60 % less_le_not_le
% 0.34/0.60 thf(fact_3824_less__le__not__le,axiom,
% 0.34/0.60 ( ord_less_nat
% 0.34/0.60 = ( ^ [X2: nat,Y3: nat] :
% 0.34/0.60 ( ( ord_less_eq_nat @ X2 @ Y3 )
% 0.34/0.60 & ~ ( ord_less_eq_nat @ Y3 @ X2 ) ) ) ) ).
% 0.34/0.60
% 0.34/0.60 % less_le_not_le
% 0.34/0.60 thf(fact_3825_less__le__not__le,axiom,
% 0.34/0.60 ( ord_less_int
% 0.34/0.60 = ( ^ [X2: int,Y3: int] :
% 0.34/0.60 ( ( ord_less_eq_int @ X2 @ Y3 )
% 0.34/0.60 & ~ ( ord_less_eq_int @ Y3 @ X2 ) ) ) ) ).
% 0.34/0.60
% 0.34/0.60 % less_le_not_le
% 0.34/0.60 thf(fact_3826_not__le__imp__less,axiom,
% 0.34/0.60 ! [Y: code_integer,X: code_integer] :
% 0.34/0.60 ( ~ ( ord_le3102999989581377725nteger @ Y @ X )
% 0.34/0.60 => ( ord_le6747313008572928689nteger @ X @ Y ) ) ).
% 0.34/0.60
% 0.34/0.60 % not_le_imp_less
% 0.34/0.60 thf(fact_3827_not__le__imp__less,axiom,
% 0.34/0.60 ! [Y: rat,X: rat] :
% 0.34/0.60 ( ~ ( ord_less_eq_rat @ Y @ X )
% 0.34/0.60 => ( ord_less_rat @ X @ Y ) ) ).
% 0.34/0.60
% 0.34/0.60 % not_le_imp_less
% 0.34/0.60 thf(fact_3828_not__le__imp__less,axiom,
% 0.34/0.60 ! [Y: nat,X: nat] :
% 0.34/0.60 ( ~ ( ord_less_eq_nat @ Y @ X )
% 0.34/0.60 => ( ord_less_nat @ X @ Y ) ) ).
% 0.34/0.60
% 0.34/0.60 % not_le_imp_less
% 0.34/0.60 thf(fact_3829_not__le__imp__less,axiom,
% 0.34/0.60 ! [Y: int,X: int] :
% 0.34/0.60 ( ~ ( ord_less_eq_int @ Y @ X )
% 0.34/0.60 => ( ord_less_int @ X @ Y ) ) ).
% 0.34/0.60
% 0.34/0.60 % not_le_imp_less
% 0.34/0.60 thf(fact_3830_order_Oorder__iff__strict,axiom,
% 0.34/0.60 ( ord_less_eq_assn
% 0.34/0.60 = ( ^ [A6: assn,B6: assn] :
% 0.34/0.60 ( ( ord_less_assn @ A6 @ B6 )
% 0.34/0.60 | ( A6 = B6 ) ) ) ) ).
% 0.34/0.60
% 0.34/0.60 % order.order_iff_strict
% 0.34/0.60 thf(fact_3831_order_Oorder__iff__strict,axiom,
% 0.34/0.60 ( ord_le3102999989581377725nteger
% 0.34/0.60 = ( ^ [A6: code_integer,B6: code_integer] :
% 0.34/0.60 ( ( ord_le6747313008572928689nteger @ A6 @ B6 )
% 0.34/0.60 | ( A6 = B6 ) ) ) ) ).
% 0.34/0.60
% 0.34/0.60 % order.order_iff_strict
% 0.34/0.60 thf(fact_3832_order_Oorder__iff__strict,axiom,
% 0.34/0.60 ( ord_le2510731241096832064er_nat
% 0.34/0.60 = ( ^ [A6: filter_nat,B6: filter_nat] :
% 0.34/0.60 ( ( ord_less_filter_nat @ A6 @ B6 )
% 0.34/0.60 | ( A6 = B6 ) ) ) ) ).
% 0.34/0.60
% 0.34/0.60 % order.order_iff_strict
% 0.34/0.60 thf(fact_3833_order_Oorder__iff__strict,axiom,
% 0.34/0.60 ( ord_less_eq_set_nat
% 0.34/0.60 = ( ^ [A6: set_nat,B6: set_nat] :
% 0.34/0.60 ( ( ord_less_set_nat @ A6 @ B6 )
% 0.34/0.60 | ( A6 = B6 ) ) ) ) ).
% 0.34/0.60
% 0.34/0.60 % order.order_iff_strict
% 0.34/0.60 thf(fact_3834_order_Oorder__iff__strict,axiom,
% 0.34/0.60 ( ord_less_eq_rat
% 0.34/0.60 = ( ^ [A6: rat,B6: rat] :
% 0.34/0.60 ( ( ord_less_rat @ A6 @ B6 )
% 0.34/0.60 | ( A6 = B6 ) ) ) ) ).
% 0.34/0.60
% 0.34/0.60 % order.order_iff_strict
% 0.34/0.60 thf(fact_3835_order_Oorder__iff__strict,axiom,
% 0.34/0.60 ( ord_less_eq_nat
% 0.34/0.60 = ( ^ [A6: nat,B6: nat] :
% 0.34/0.60 ( ( ord_less_nat @ A6 @ B6 )
% 0.34/0.60 | ( A6 = B6 ) ) ) ) ).
% 0.34/0.60
% 0.34/0.60 % order.order_iff_strict
% 0.34/0.60 thf(fact_3836_order_Oorder__iff__strict,axiom,
% 0.34/0.60 ( ord_less_eq_int
% 0.34/0.60 = ( ^ [A6: int,B6: int] :
% 0.34/0.60 ( ( ord_less_int @ A6 @ B6 )
% 0.34/0.60 | ( A6 = B6 ) ) ) ) ).
% 0.34/0.60
% 0.34/0.60 % order.order_iff_strict
% 0.34/0.60 thf(fact_3837_order_Ostrict__iff__order,axiom,
% 0.34/0.60 ( ord_less_assn
% 0.34/0.60 = ( ^ [A6: assn,B6: assn] :
% 0.34/0.60 ( ( ord_less_eq_assn @ A6 @ B6 )
% 0.34/0.60 & ( A6 != B6 ) ) ) ) ).
% 0.34/0.60
% 0.34/0.60 % order.strict_iff_order
% 0.34/0.60 thf(fact_3838_order_Ostrict__iff__order,axiom,
% 0.34/0.60 ( ord_le6747313008572928689nteger
% 0.34/0.60 = ( ^ [A6: code_integer,B6: code_integer] :
% 0.34/0.60 ( ( ord_le3102999989581377725nteger @ A6 @ B6 )
% 0.34/0.60 & ( A6 != B6 ) ) ) ) ).
% 0.34/0.60
% 0.34/0.60 % order.strict_iff_order
% 0.34/0.60 thf(fact_3839_order_Ostrict__iff__order,axiom,
% 0.34/0.60 ( ord_less_filter_nat
% 0.34/0.60 = ( ^ [A6: filter_nat,B6: filter_nat] :
% 0.34/0.60 ( ( ord_le2510731241096832064er_nat @ A6 @ B6 )
% 0.34/0.60 & ( A6 != B6 ) ) ) ) ).
% 0.34/0.60
% 0.34/0.60 % order.strict_iff_order
% 0.34/0.60 thf(fact_3840_order_Ostrict__iff__order,axiom,
% 0.34/0.60 ( ord_less_set_nat
% 0.34/0.60 = ( ^ [A6: set_nat,B6: set_nat] :
% 0.34/0.60 ( ( ord_less_eq_set_nat @ A6 @ B6 )
% 0.34/0.60 & ( A6 != B6 ) ) ) ) ).
% 0.34/0.60
% 0.34/0.60 % order.strict_iff_order
% 0.34/0.60 thf(fact_3841_order_Ostrict__iff__order,axiom,
% 0.34/0.60 ( ord_less_rat
% 0.34/0.60 = ( ^ [A6: rat,B6: rat] :
% 0.34/0.60 ( ( ord_less_eq_rat @ A6 @ B6 )
% 0.34/0.60 & ( A6 != B6 ) ) ) ) ).
% 0.34/0.60
% 0.34/0.60 % order.strict_iff_order
% 0.34/0.60 thf(fact_3842_order_Ostrict__iff__order,axiom,
% 0.34/0.60 ( ord_less_nat
% 0.34/0.60 = ( ^ [A6: nat,B6: nat] :
% 0.34/0.60 ( ( ord_less_eq_nat @ A6 @ B6 )
% 0.34/0.60 & ( A6 != B6 ) ) ) ) ).
% 0.34/0.60
% 0.34/0.60 % order.strict_iff_order
% 0.34/0.60 thf(fact_3843_order_Ostrict__iff__order,axiom,
% 0.34/0.60 ( ord_less_int
% 0.34/0.60 = ( ^ [A6: int,B6: int] :
% 0.34/0.60 ( ( ord_less_eq_int @ A6 @ B6 )
% 0.34/0.60 & ( A6 != B6 ) ) ) ) ).
% 0.34/0.60
% 0.34/0.60 % order.strict_iff_order
% 0.34/0.60 thf(fact_3844_order_Ostrict__trans1,axiom,
% 0.34/0.60 ! [A2: assn,B: assn,C: assn] :
% 0.34/0.60 ( ( ord_less_eq_assn @ A2 @ B )
% 0.34/0.60 => ( ( ord_less_assn @ B @ C )
% 0.34/0.60 => ( ord_less_assn @ A2 @ C ) ) ) ).
% 0.34/0.60
% 0.34/0.60 % order.strict_trans1
% 0.34/0.60 thf(fact_3845_order_Ostrict__trans1,axiom,
% 0.34/0.60 ! [A2: code_integer,B: code_integer,C: code_integer] :
% 0.34/0.60 ( ( ord_le3102999989581377725nteger @ A2 @ B )
% 0.34/0.60 => ( ( ord_le6747313008572928689nteger @ B @ C )
% 0.34/0.60 => ( ord_le6747313008572928689nteger @ A2 @ C ) ) ) ).
% 0.34/0.60
% 0.34/0.60 % order.strict_trans1
% 0.34/0.60 thf(fact_3846_order_Ostrict__trans1,axiom,
% 0.34/0.60 ! [A2: filter_nat,B: filter_nat,C: filter_nat] :
% 0.34/0.60 ( ( ord_le2510731241096832064er_nat @ A2 @ B )
% 0.34/0.60 => ( ( ord_less_filter_nat @ B @ C )
% 0.34/0.60 => ( ord_less_filter_nat @ A2 @ C ) ) ) ).
% 0.34/0.60
% 0.34/0.60 % order.strict_trans1
% 0.34/0.60 thf(fact_3847_order_Ostrict__trans1,axiom,
% 0.34/0.60 ! [A2: set_nat,B: set_nat,C: set_nat] :
% 0.34/0.60 ( ( ord_less_eq_set_nat @ A2 @ B )
% 0.34/0.60 => ( ( ord_less_set_nat @ B @ C )
% 0.34/0.60 => ( ord_less_set_nat @ A2 @ C ) ) ) ).
% 0.34/0.60
% 0.34/0.60 % order.strict_trans1
% 0.34/0.60 thf(fact_3848_order_Ostrict__trans1,axiom,
% 0.34/0.60 ! [A2: rat,B: rat,C: rat] :
% 0.34/0.60 ( ( ord_less_eq_rat @ A2 @ B )
% 0.34/0.60 => ( ( ord_less_rat @ B @ C )
% 0.34/0.60 => ( ord_less_rat @ A2 @ C ) ) ) ).
% 0.34/0.60
% 0.34/0.60 % order.strict_trans1
% 0.34/0.60 thf(fact_3849_order_Ostrict__trans1,axiom,
% 0.34/0.60 ! [A2: nat,B: nat,C: nat] :
% 0.34/0.60 ( ( ord_less_eq_nat @ A2 @ B )
% 0.34/0.60 => ( ( ord_less_nat @ B @ C )
% 0.34/0.60 => ( ord_less_nat @ A2 @ C ) ) ) ).
% 0.34/0.60
% 0.34/0.60 % order.strict_trans1
% 0.34/0.60 thf(fact_3850_order_Ostrict__trans1,axiom,
% 0.34/0.60 ! [A2: int,B: int,C: int] :
% 0.34/0.60 ( ( ord_less_eq_int @ A2 @ B )
% 0.34/0.60 => ( ( ord_less_int @ B @ C )
% 0.34/0.60 => ( ord_less_int @ A2 @ C ) ) ) ).
% 0.34/0.60
% 0.34/0.60 % order.strict_trans1
% 0.34/0.60 thf(fact_3851_order_Ostrict__trans2,axiom,
% 0.34/0.60 ! [A2: assn,B: assn,C: assn] :
% 0.34/0.60 ( ( ord_less_assn @ A2 @ B )
% 0.34/0.60 => ( ( ord_less_eq_assn @ B @ C )
% 0.34/0.60 => ( ord_less_assn @ A2 @ C ) ) ) ).
% 0.34/0.60
% 0.34/0.60 % order.strict_trans2
% 0.34/0.60 thf(fact_3852_order_Ostrict__trans2,axiom,
% 0.34/0.60 ! [A2: code_integer,B: code_integer,C: code_integer] :
% 0.34/0.60 ( ( ord_le6747313008572928689nteger @ A2 @ B )
% 0.34/0.60 => ( ( ord_le3102999989581377725nteger @ B @ C )
% 0.34/0.60 => ( ord_le6747313008572928689nteger @ A2 @ C ) ) ) ).
% 0.34/0.60
% 0.34/0.60 % order.strict_trans2
% 0.34/0.60 thf(fact_3853_order_Ostrict__trans2,axiom,
% 0.34/0.60 ! [A2: filter_nat,B: filter_nat,C: filter_nat] :
% 0.34/0.60 ( ( ord_less_filter_nat @ A2 @ B )
% 0.34/0.60 => ( ( ord_le2510731241096832064er_nat @ B @ C )
% 0.34/0.60 => ( ord_less_filter_nat @ A2 @ C ) ) ) ).
% 0.34/0.60
% 0.34/0.60 % order.strict_trans2
% 0.34/0.60 thf(fact_3854_order_Ostrict__trans2,axiom,
% 0.34/0.60 ! [A2: set_nat,B: set_nat,C: set_nat] :
% 0.34/0.60 ( ( ord_less_set_nat @ A2 @ B )
% 0.34/0.60 => ( ( ord_less_eq_set_nat @ B @ C )
% 0.34/0.60 => ( ord_less_set_nat @ A2 @ C ) ) ) ).
% 0.34/0.60
% 0.34/0.60 % order.strict_trans2
% 0.34/0.60 thf(fact_3855_order_Ostrict__trans2,axiom,
% 0.34/0.60 ! [A2: rat,B: rat,C: rat] :
% 0.34/0.60 ( ( ord_less_rat @ A2 @ B )
% 0.34/0.60 => ( ( ord_less_eq_rat @ B @ C )
% 0.34/0.60 => ( ord_less_rat @ A2 @ C ) ) ) ).
% 0.34/0.60
% 0.34/0.60 % order.strict_trans2
% 0.34/0.60 thf(fact_3856_order_Ostrict__trans2,axiom,
% 0.34/0.60 ! [A2: nat,B: nat,C: nat] :
% 0.34/0.60 ( ( ord_less_nat @ A2 @ B )
% 0.34/0.60 => ( ( ord_less_eq_nat @ B @ C )
% 0.34/0.60 => ( ord_less_nat @ A2 @ C ) ) ) ).
% 0.34/0.60
% 0.34/0.60 % order.strict_trans2
% 0.34/0.60 thf(fact_3857_order_Ostrict__trans2,axiom,
% 0.34/0.60 ! [A2: int,B: int,C: int] :
% 0.34/0.60 ( ( ord_less_int @ A2 @ B )
% 0.34/0.60 => ( ( ord_less_eq_int @ B @ C )
% 0.34/0.60 => ( ord_less_int @ A2 @ C ) ) ) ).
% 0.34/0.60
% 0.34/0.60 % order.strict_trans2
% 0.34/0.60 thf(fact_3858_order_Ostrict__iff__not,axiom,
% 0.34/0.60 ( ord_less_assn
% 0.34/0.60 = ( ^ [A6: assn,B6: assn] :
% 0.34/0.60 ( ( ord_less_eq_assn @ A6 @ B6 )
% 0.34/0.60 & ~ ( ord_less_eq_assn @ B6 @ A6 ) ) ) ) ).
% 0.34/0.60
% 0.34/0.60 % order.strict_iff_not
% 0.34/0.60 thf(fact_3859_order_Ostrict__iff__not,axiom,
% 0.34/0.60 ( ord_le6747313008572928689nteger
% 0.34/0.60 = ( ^ [A6: code_integer,B6: code_integer] :
% 0.34/0.60 ( ( ord_le3102999989581377725nteger @ A6 @ B6 )
% 0.34/0.60 & ~ ( ord_le3102999989581377725nteger @ B6 @ A6 ) ) ) ) ).
% 0.34/0.60
% 0.34/0.60 % order.strict_iff_not
% 0.34/0.60 thf(fact_3860_order_Ostrict__iff__not,axiom,
% 0.34/0.60 ( ord_less_filter_nat
% 0.34/0.60 = ( ^ [A6: filter_nat,B6: filter_nat] :
% 0.34/0.60 ( ( ord_le2510731241096832064er_nat @ A6 @ B6 )
% 0.34/0.60 & ~ ( ord_le2510731241096832064er_nat @ B6 @ A6 ) ) ) ) ).
% 0.34/0.60
% 0.34/0.60 % order.strict_iff_not
% 0.34/0.60 thf(fact_3861_order_Ostrict__iff__not,axiom,
% 0.34/0.60 ( ord_less_set_nat
% 0.34/0.60 = ( ^ [A6: set_nat,B6: set_nat] :
% 0.34/0.60 ( ( ord_less_eq_set_nat @ A6 @ B6 )
% 0.34/0.60 & ~ ( ord_less_eq_set_nat @ B6 @ A6 ) ) ) ) ).
% 0.34/0.60
% 0.34/0.60 % order.strict_iff_not
% 0.34/0.60 thf(fact_3862_order_Ostrict__iff__not,axiom,
% 0.34/0.60 ( ord_less_rat
% 0.34/0.60 = ( ^ [A6: rat,B6: rat] :
% 0.34/0.60 ( ( ord_less_eq_rat @ A6 @ B6 )
% 0.34/0.60 & ~ ( ord_less_eq_rat @ B6 @ A6 ) ) ) ) ).
% 0.34/0.60
% 0.34/0.60 % order.strict_iff_not
% 0.34/0.60 thf(fact_3863_order_Ostrict__iff__not,axiom,
% 0.34/0.60 ( ord_less_nat
% 0.34/0.60 = ( ^ [A6: nat,B6: nat] :
% 0.34/0.60 ( ( ord_less_eq_nat @ A6 @ B6 )
% 0.34/0.60 & ~ ( ord_less_eq_nat @ B6 @ A6 ) ) ) ) ).
% 0.34/0.60
% 0.34/0.60 % order.strict_iff_not
% 0.34/0.60 thf(fact_3864_order_Ostrict__iff__not,axiom,
% 0.34/0.60 ( ord_less_int
% 0.34/0.60 = ( ^ [A6: int,B6: int] :
% 0.34/0.60 ( ( ord_less_eq_int @ A6 @ B6 )
% 0.34/0.60 & ~ ( ord_less_eq_int @ B6 @ A6 ) ) ) ) ).
% 0.34/0.60
% 0.34/0.60 % order.strict_iff_not
% 0.34/0.60 thf(fact_3865_dense__ge__bounded,axiom,
% 0.34/0.60 ! [Z: rat,X: rat,Y: rat] :
% 0.34/0.60 ( ( ord_less_rat @ Z @ X )
% 0.34/0.60 => ( ! [W2: rat] :
% 0.34/0.60 ( ( ord_less_rat @ Z @ W2 )
% 0.34/0.60 => ( ( ord_less_rat @ W2 @ X )
% 0.34/0.60 => ( ord_less_eq_rat @ Y @ W2 ) ) )
% 0.34/0.60 => ( ord_less_eq_rat @ Y @ Z ) ) ) ).
% 0.34/0.60
% 0.34/0.60 % dense_ge_bounded
% 0.34/0.60 thf(fact_3866_dense__le__bounded,axiom,
% 0.34/0.60 ! [X: rat,Y: rat,Z: rat] :
% 0.34/0.60 ( ( ord_less_rat @ X @ Y )
% 0.34/0.60 => ( ! [W2: rat] :
% 0.34/0.60 ( ( ord_less_rat @ X @ W2 )
% 0.34/0.60 => ( ( ord_less_rat @ W2 @ Y )
% 0.34/0.60 => ( ord_less_eq_rat @ W2 @ Z ) ) )
% 0.34/0.60 => ( ord_less_eq_rat @ Y @ Z ) ) ) ).
% 0.34/0.60
% 0.34/0.60 % dense_le_bounded
% 0.34/0.60 thf(fact_3867_dual__order_Oorder__iff__strict,axiom,
% 0.34/0.60 ( ord_less_eq_assn
% 0.34/0.60 = ( ^ [B6: assn,A6: assn] :
% 0.34/0.60 ( ( ord_less_assn @ B6 @ A6 )
% 0.34/0.60 | ( A6 = B6 ) ) ) ) ).
% 0.34/0.60
% 0.34/0.60 % dual_order.order_iff_strict
% 0.34/0.60 thf(fact_3868_dual__order_Oorder__iff__strict,axiom,
% 0.34/0.60 ( ord_le3102999989581377725nteger
% 0.34/0.60 = ( ^ [B6: code_integer,A6: code_integer] :
% 0.34/0.60 ( ( ord_le6747313008572928689nteger @ B6 @ A6 )
% 0.34/0.60 | ( A6 = B6 ) ) ) ) ).
% 0.34/0.60
% 0.34/0.60 % dual_order.order_iff_strict
% 0.34/0.60 thf(fact_3869_dual__order_Oorder__iff__strict,axiom,
% 0.34/0.60 ( ord_le2510731241096832064er_nat
% 0.34/0.60 = ( ^ [B6: filter_nat,A6: filter_nat] :
% 0.34/0.60 ( ( ord_less_filter_nat @ B6 @ A6 )
% 0.34/0.60 | ( A6 = B6 ) ) ) ) ).
% 0.34/0.60
% 0.34/0.60 % dual_order.order_iff_strict
% 0.34/0.60 thf(fact_3870_dual__order_Oorder__iff__strict,axiom,
% 0.34/0.60 ( ord_less_eq_set_nat
% 0.34/0.60 = ( ^ [B6: set_nat,A6: set_nat] :
% 0.34/0.60 ( ( ord_less_set_nat @ B6 @ A6 )
% 0.34/0.60 | ( A6 = B6 ) ) ) ) ).
% 0.34/0.60
% 0.34/0.60 % dual_order.order_iff_strict
% 0.34/0.60 thf(fact_3871_dual__order_Oorder__iff__strict,axiom,
% 0.34/0.60 ( ord_less_eq_rat
% 0.34/0.60 = ( ^ [B6: rat,A6: rat] :
% 0.34/0.60 ( ( ord_less_rat @ B6 @ A6 )
% 0.34/0.60 | ( A6 = B6 ) ) ) ) ).
% 0.34/0.60
% 0.34/0.60 % dual_order.order_iff_strict
% 0.34/0.60 thf(fact_3872_dual__order_Oorder__iff__strict,axiom,
% 0.34/0.60 ( ord_less_eq_nat
% 0.34/0.60 = ( ^ [B6: nat,A6: nat] :
% 0.34/0.60 ( ( ord_less_nat @ B6 @ A6 )
% 0.34/0.60 | ( A6 = B6 ) ) ) ) ).
% 0.34/0.60
% 0.34/0.60 % dual_order.order_iff_strict
% 0.34/0.60 thf(fact_3873_dual__order_Oorder__iff__strict,axiom,
% 0.34/0.60 ( ord_less_eq_int
% 0.34/0.60 = ( ^ [B6: int,A6: int] :
% 0.34/0.60 ( ( ord_less_int @ B6 @ A6 )
% 0.34/0.60 | ( A6 = B6 ) ) ) ) ).
% 0.34/0.60
% 0.34/0.60 % dual_order.order_iff_strict
% 0.34/0.60 thf(fact_3874_dual__order_Ostrict__iff__order,axiom,
% 0.34/0.60 ( ord_less_assn
% 0.34/0.60 = ( ^ [B6: assn,A6: assn] :
% 0.34/0.60 ( ( ord_less_eq_assn @ B6 @ A6 )
% 0.34/0.60 & ( A6 != B6 ) ) ) ) ).
% 0.34/0.60
% 0.34/0.60 % dual_order.strict_iff_order
% 0.34/0.60 thf(fact_3875_dual__order_Ostrict__iff__order,axiom,
% 0.34/0.60 ( ord_le6747313008572928689nteger
% 0.34/0.60 = ( ^ [B6: code_integer,A6: code_integer] :
% 0.34/0.60 ( ( ord_le3102999989581377725nteger @ B6 @ A6 )
% 0.34/0.60 & ( A6 != B6 ) ) ) ) ).
% 0.34/0.60
% 0.34/0.60 % dual_order.strict_iff_order
% 0.34/0.60 thf(fact_3876_dual__order_Ostrict__iff__order,axiom,
% 0.34/0.60 ( ord_less_filter_nat
% 0.34/0.60 = ( ^ [B6: filter_nat,A6: filter_nat] :
% 0.34/0.60 ( ( ord_le2510731241096832064er_nat @ B6 @ A6 )
% 0.34/0.60 & ( A6 != B6 ) ) ) ) ).
% 0.34/0.60
% 0.34/0.60 % dual_order.strict_iff_order
% 0.34/0.60 thf(fact_3877_dual__order_Ostrict__iff__order,axiom,
% 0.34/0.60 ( ord_less_set_nat
% 0.34/0.60 = ( ^ [B6: set_nat,A6: set_nat] :
% 0.34/0.60 ( ( ord_less_eq_set_nat @ B6 @ A6 )
% 0.34/0.60 & ( A6 != B6 ) ) ) ) ).
% 0.34/0.60
% 0.34/0.60 % dual_order.strict_iff_order
% 0.34/0.60 thf(fact_3878_dual__order_Ostrict__iff__order,axiom,
% 0.34/0.60 ( ord_less_rat
% 0.34/0.60 = ( ^ [B6: rat,A6: rat] :
% 0.34/0.60 ( ( ord_less_eq_rat @ B6 @ A6 )
% 0.34/0.60 & ( A6 != B6 ) ) ) ) ).
% 0.34/0.60
% 0.34/0.60 % dual_order.strict_iff_order
% 0.34/0.60 thf(fact_3879_dual__order_Ostrict__iff__order,axiom,
% 0.34/0.60 ( ord_less_nat
% 0.34/0.60 = ( ^ [B6: nat,A6: nat] :
% 0.34/0.60 ( ( ord_less_eq_nat @ B6 @ A6 )
% 0.34/0.60 & ( A6 != B6 ) ) ) ) ).
% 0.34/0.60
% 0.34/0.60 % dual_order.strict_iff_order
% 0.34/0.60 thf(fact_3880_dual__order_Ostrict__iff__order,axiom,
% 0.34/0.60 ( ord_less_int
% 0.34/0.60 = ( ^ [B6: int,A6: int] :
% 0.34/0.60 ( ( ord_less_eq_int @ B6 @ A6 )
% 0.34/0.60 & ( A6 != B6 ) ) ) ) ).
% 0.34/0.60
% 0.34/0.60 % dual_order.strict_iff_order
% 0.34/0.60 thf(fact_3881_dual__order_Ostrict__trans1,axiom,
% 0.34/0.60 ! [B: assn,A2: assn,C: assn] :
% 0.34/0.60 ( ( ord_less_eq_assn @ B @ A2 )
% 0.34/0.60 => ( ( ord_less_assn @ C @ B )
% 0.34/0.60 => ( ord_less_assn @ C @ A2 ) ) ) ).
% 0.34/0.60
% 0.34/0.60 % dual_order.strict_trans1
% 0.34/0.60 thf(fact_3882_dual__order_Ostrict__trans1,axiom,
% 0.34/0.60 ! [B: code_integer,A2: code_integer,C: code_integer] :
% 0.34/0.60 ( ( ord_le3102999989581377725nteger @ B @ A2 )
% 0.34/0.60 => ( ( ord_le6747313008572928689nteger @ C @ B )
% 0.34/0.60 => ( ord_le6747313008572928689nteger @ C @ A2 ) ) ) ).
% 0.34/0.60
% 0.34/0.60 % dual_order.strict_trans1
% 0.34/0.60 thf(fact_3883_dual__order_Ostrict__trans1,axiom,
% 0.34/0.60 ! [B: filter_nat,A2: filter_nat,C: filter_nat] :
% 0.34/0.60 ( ( ord_le2510731241096832064er_nat @ B @ A2 )
% 0.34/0.60 => ( ( ord_less_filter_nat @ C @ B )
% 0.34/0.60 => ( ord_less_filter_nat @ C @ A2 ) ) ) ).
% 0.34/0.60
% 0.34/0.60 % dual_order.strict_trans1
% 0.34/0.60 thf(fact_3884_dual__order_Ostrict__trans1,axiom,
% 0.34/0.60 ! [B: set_nat,A2: set_nat,C: set_nat] :
% 0.34/0.60 ( ( ord_less_eq_set_nat @ B @ A2 )
% 0.34/0.60 => ( ( ord_less_set_nat @ C @ B )
% 0.34/0.60 => ( ord_less_set_nat @ C @ A2 ) ) ) ).
% 0.34/0.60
% 0.34/0.60 % dual_order.strict_trans1
% 0.34/0.60 thf(fact_3885_dual__order_Ostrict__trans1,axiom,
% 0.34/0.60 ! [B: rat,A2: rat,C: rat] :
% 0.34/0.60 ( ( ord_less_eq_rat @ B @ A2 )
% 0.34/0.60 => ( ( ord_less_rat @ C @ B )
% 0.34/0.60 => ( ord_less_rat @ C @ A2 ) ) ) ).
% 0.34/0.60
% 0.34/0.60 % dual_order.strict_trans1
% 0.34/0.60 thf(fact_3886_dual__order_Ostrict__trans1,axiom,
% 0.34/0.60 ! [B: nat,A2: nat,C: nat] :
% 0.34/0.60 ( ( ord_less_eq_nat @ B @ A2 )
% 0.34/0.60 => ( ( ord_less_nat @ C @ B )
% 0.34/0.60 => ( ord_less_nat @ C @ A2 ) ) ) ).
% 0.34/0.60
% 0.34/0.60 % dual_order.strict_trans1
% 0.34/0.60 thf(fact_3887_dual__order_Ostrict__trans1,axiom,
% 0.34/0.60 ! [B: int,A2: int,C: int] :
% 0.34/0.60 ( ( ord_less_eq_int @ B @ A2 )
% 0.34/0.60 => ( ( ord_less_int @ C @ B )
% 0.34/0.60 => ( ord_less_int @ C @ A2 ) ) ) ).
% 0.34/0.60
% 0.34/0.60 % dual_order.strict_trans1
% 0.34/0.60 thf(fact_3888_dual__order_Ostrict__trans2,axiom,
% 0.34/0.60 ! [B: assn,A2: assn,C: assn] :
% 0.34/0.60 ( ( ord_less_assn @ B @ A2 )
% 0.34/0.60 => ( ( ord_less_eq_assn @ C @ B )
% 0.34/0.60 => ( ord_less_assn @ C @ A2 ) ) ) ).
% 0.34/0.60
% 0.34/0.60 % dual_order.strict_trans2
% 0.34/0.60 thf(fact_3889_dual__order_Ostrict__trans2,axiom,
% 0.34/0.60 ! [B: code_integer,A2: code_integer,C: code_integer] :
% 0.34/0.60 ( ( ord_le6747313008572928689nteger @ B @ A2 )
% 0.34/0.60 => ( ( ord_le3102999989581377725nteger @ C @ B )
% 0.34/0.60 => ( ord_le6747313008572928689nteger @ C @ A2 ) ) ) ).
% 0.34/0.60
% 0.34/0.60 % dual_order.strict_trans2
% 0.34/0.60 thf(fact_3890_dual__order_Ostrict__trans2,axiom,
% 0.34/0.60 ! [B: filter_nat,A2: filter_nat,C: filter_nat] :
% 0.34/0.60 ( ( ord_less_filter_nat @ B @ A2 )
% 0.34/0.60 => ( ( ord_le2510731241096832064er_nat @ C @ B )
% 0.34/0.60 => ( ord_less_filter_nat @ C @ A2 ) ) ) ).
% 0.34/0.60
% 0.34/0.60 % dual_order.strict_trans2
% 0.34/0.60 thf(fact_3891_dual__order_Ostrict__trans2,axiom,
% 0.34/0.60 ! [B: set_nat,A2: set_nat,C: set_nat] :
% 0.34/0.60 ( ( ord_less_set_nat @ B @ A2 )
% 0.34/0.60 => ( ( ord_less_eq_set_nat @ C @ B )
% 0.34/0.60 => ( ord_less_set_nat @ C @ A2 ) ) ) ).
% 0.34/0.60
% 0.34/0.60 % dual_order.strict_trans2
% 0.34/0.60 thf(fact_3892_dual__order_Ostrict__trans2,axiom,
% 0.34/0.60 ! [B: rat,A2: rat,C: rat] :
% 0.34/0.60 ( ( ord_less_rat @ B @ A2 )
% 0.34/0.60 => ( ( ord_less_eq_rat @ C @ B )
% 0.34/0.60 => ( ord_less_rat @ C @ A2 ) ) ) ).
% 0.34/0.60
% 0.34/0.60 % dual_order.strict_trans2
% 0.34/0.60 thf(fact_3893_dual__order_Ostrict__trans2,axiom,
% 0.34/0.60 ! [B: nat,A2: nat,C: nat] :
% 0.34/0.60 ( ( ord_less_nat @ B @ A2 )
% 0.34/0.60 => ( ( ord_less_eq_nat @ C @ B )
% 0.34/0.60 => ( ord_less_nat @ C @ A2 ) ) ) ).
% 0.34/0.60
% 0.34/0.60 % dual_order.strict_trans2
% 0.34/0.60 thf(fact_3894_dual__order_Ostrict__trans2,axiom,
% 0.34/0.60 ! [B: int,A2: int,C: int] :
% 0.34/0.60 ( ( ord_less_int @ B @ A2 )
% 0.34/0.60 => ( ( ord_less_eq_int @ C @ B )
% 0.34/0.60 => ( ord_less_int @ C @ A2 ) ) ) ).
% 0.34/0.60
% 0.34/0.60 % dual_order.strict_trans2
% 0.34/0.60 thf(fact_3895_dual__order_Ostrict__iff__not,axiom,
% 0.34/0.60 ( ord_less_assn
% 0.34/0.60 = ( ^ [B6: assn,A6: assn] :
% 0.34/0.60 ( ( ord_less_eq_assn @ B6 @ A6 )
% 0.34/0.60 & ~ ( ord_less_eq_assn @ A6 @ B6 ) ) ) ) ).
% 0.34/0.60
% 0.34/0.60 % dual_order.strict_iff_not
% 0.34/0.60 thf(fact_3896_dual__order_Ostrict__iff__not,axiom,
% 0.34/0.60 ( ord_le6747313008572928689nteger
% 0.34/0.60 = ( ^ [B6: code_integer,A6: code_integer] :
% 0.34/0.60 ( ( ord_le3102999989581377725nteger @ B6 @ A6 )
% 0.34/0.60 & ~ ( ord_le3102999989581377725nteger @ A6 @ B6 ) ) ) ) ).
% 0.34/0.60
% 0.34/0.60 % dual_order.strict_iff_not
% 0.34/0.60 thf(fact_3897_dual__order_Ostrict__iff__not,axiom,
% 0.34/0.60 ( ord_less_filter_nat
% 0.34/0.60 = ( ^ [B6: filter_nat,A6: filter_nat] :
% 0.34/0.60 ( ( ord_le2510731241096832064er_nat @ B6 @ A6 )
% 0.34/0.60 & ~ ( ord_le2510731241096832064er_nat @ A6 @ B6 ) ) ) ) ).
% 0.34/0.60
% 0.34/0.60 % dual_order.strict_iff_not
% 0.34/0.60 thf(fact_3898_dual__order_Ostrict__iff__not,axiom,
% 0.34/0.60 ( ord_less_set_nat
% 0.34/0.60 = ( ^ [B6: set_nat,A6: set_nat] :
% 0.34/0.60 ( ( ord_less_eq_set_nat @ B6 @ A6 )
% 0.34/0.60 & ~ ( ord_less_eq_set_nat @ A6 @ B6 ) ) ) ) ).
% 0.34/0.60
% 0.34/0.60 % dual_order.strict_iff_not
% 0.34/0.60 thf(fact_3899_dual__order_Ostrict__iff__not,axiom,
% 0.34/0.60 ( ord_less_rat
% 0.34/0.60 = ( ^ [B6: rat,A6: rat] :
% 0.34/0.60 ( ( ord_less_eq_rat @ B6 @ A6 )
% 0.34/0.60 & ~ ( ord_less_eq_rat @ A6 @ B6 ) ) ) ) ).
% 0.34/0.60
% 0.34/0.60 % dual_order.strict_iff_not
% 0.34/0.60 thf(fact_3900_dual__order_Ostrict__iff__not,axiom,
% 0.34/0.60 ( ord_less_nat
% 0.34/0.60 = ( ^ [B6: nat,A6: nat] :
% 0.34/0.60 ( ( ord_less_eq_nat @ B6 @ A6 )
% 0.34/0.60 & ~ ( ord_less_eq_nat @ A6 @ B6 ) ) ) ) ).
% 0.34/0.60
% 0.34/0.60 % dual_order.strict_iff_not
% 0.34/0.60 thf(fact_3901_dual__order_Ostrict__iff__not,axiom,
% 0.34/0.60 ( ord_less_int
% 0.34/0.60 = ( ^ [B6: int,A6: int] :
% 0.34/0.60 ( ( ord_less_eq_int @ B6 @ A6 )
% 0.34/0.60 & ~ ( ord_less_eq_int @ A6 @ B6 ) ) ) ) ).
% 0.34/0.60
% 0.34/0.60 % dual_order.strict_iff_not
% 0.34/0.60 thf(fact_3902_order_Ostrict__implies__order,axiom,
% 0.34/0.60 ! [A2: assn,B: assn] :
% 0.34/0.60 ( ( ord_less_assn @ A2 @ B )
% 0.34/0.60 => ( ord_less_eq_assn @ A2 @ B ) ) ).
% 0.34/0.60
% 0.34/0.60 % order.strict_implies_order
% 0.34/0.60 thf(fact_3903_order_Ostrict__implies__order,axiom,
% 0.34/0.60 ! [A2: code_integer,B: code_integer] :
% 0.34/0.60 ( ( ord_le6747313008572928689nteger @ A2 @ B )
% 0.34/0.60 => ( ord_le3102999989581377725nteger @ A2 @ B ) ) ).
% 0.34/0.60
% 0.34/0.60 % order.strict_implies_order
% 0.34/0.60 thf(fact_3904_order_Ostrict__implies__order,axiom,
% 0.34/0.60 ! [A2: filter_nat,B: filter_nat] :
% 0.34/0.60 ( ( ord_less_filter_nat @ A2 @ B )
% 0.34/0.60 => ( ord_le2510731241096832064er_nat @ A2 @ B ) ) ).
% 0.34/0.60
% 0.34/0.60 % order.strict_implies_order
% 0.34/0.60 thf(fact_3905_order_Ostrict__implies__order,axiom,
% 0.34/0.60 ! [A2: set_nat,B: set_nat] :
% 0.34/0.60 ( ( ord_less_set_nat @ A2 @ B )
% 0.34/0.60 => ( ord_less_eq_set_nat @ A2 @ B ) ) ).
% 0.34/0.60
% 0.34/0.60 % order.strict_implies_order
% 0.34/0.60 thf(fact_3906_order_Ostrict__implies__order,axiom,
% 0.34/0.60 ! [A2: rat,B: rat] :
% 0.34/0.60 ( ( ord_less_rat @ A2 @ B )
% 0.34/0.60 => ( ord_less_eq_rat @ A2 @ B ) ) ).
% 0.34/0.60
% 0.34/0.60 % order.strict_implies_order
% 0.34/0.60 thf(fact_3907_order_Ostrict__implies__order,axiom,
% 0.34/0.60 ! [A2: nat,B: nat] :
% 0.34/0.60 ( ( ord_less_nat @ A2 @ B )
% 0.34/0.60 => ( ord_less_eq_nat @ A2 @ B ) ) ).
% 0.34/0.60
% 0.34/0.60 % order.strict_implies_order
% 0.34/0.60 thf(fact_3908_order_Ostrict__implies__order,axiom,
% 0.34/0.60 ! [A2: int,B: int] :
% 0.34/0.60 ( ( ord_less_int @ A2 @ B )
% 0.34/0.60 => ( ord_less_eq_int @ A2 @ B ) ) ).
% 0.34/0.60
% 0.34/0.60 % order.strict_implies_order
% 0.34/0.60 thf(fact_3909_dual__order_Ostrict__implies__order,axiom,
% 0.34/0.60 ! [B: assn,A2: assn] :
% 0.34/0.60 ( ( ord_less_assn @ B @ A2 )
% 0.34/0.60 => ( ord_less_eq_assn @ B @ A2 ) ) ).
% 0.34/0.60
% 0.34/0.60 % dual_order.strict_implies_order
% 0.34/0.60 thf(fact_3910_dual__order_Ostrict__implies__order,axiom,
% 0.34/0.60 ! [B: code_integer,A2: code_integer] :
% 0.34/0.60 ( ( ord_le6747313008572928689nteger @ B @ A2 )
% 0.34/0.60 => ( ord_le3102999989581377725nteger @ B @ A2 ) ) ).
% 0.34/0.60
% 0.34/0.60 % dual_order.strict_implies_order
% 0.34/0.60 thf(fact_3911_dual__order_Ostrict__implies__order,axiom,
% 0.34/0.60 ! [B: filter_nat,A2: filter_nat] :
% 0.34/0.60 ( ( ord_less_filter_nat @ B @ A2 )
% 0.34/0.60 => ( ord_le2510731241096832064er_nat @ B @ A2 ) ) ).
% 0.34/0.60
% 0.34/0.60 % dual_order.strict_implies_order
% 0.34/0.60 thf(fact_3912_dual__order_Ostrict__implies__order,axiom,
% 0.34/0.60 ! [B: set_nat,A2: set_nat] :
% 0.34/0.60 ( ( ord_less_set_nat @ B @ A2 )
% 0.34/0.60 => ( ord_less_eq_set_nat @ B @ A2 ) ) ).
% 0.34/0.60
% 0.34/0.60 % dual_order.strict_implies_order
% 0.34/0.60 thf(fact_3913_dual__order_Ostrict__implies__order,axiom,
% 0.34/0.60 ! [B: rat,A2: rat] :
% 0.34/0.60 ( ( ord_less_rat @ B @ A2 )
% 0.34/0.60 => ( ord_less_eq_rat @ B @ A2 ) ) ).
% 0.34/0.60
% 0.34/0.60 % dual_order.strict_implies_order
% 0.34/0.60 thf(fact_3914_dual__order_Ostrict__implies__order,axiom,
% 0.34/0.60 ! [B: nat,A2: nat] :
% 0.34/0.60 ( ( ord_less_nat @ B @ A2 )
% 0.34/0.60 => ( ord_less_eq_nat @ B @ A2 ) ) ).
% 0.34/0.60
% 0.34/0.60 % dual_order.strict_implies_order
% 0.34/0.60 thf(fact_3915_dual__order_Ostrict__implies__order,axiom,
% 0.34/0.60 ! [B: int,A2: int] :
% 0.34/0.60 ( ( ord_less_int @ B @ A2 )
% 0.34/0.60 => ( ord_less_eq_int @ B @ A2 ) ) ).
% 0.34/0.60
% 0.34/0.60 % dual_order.strict_implies_order
% 0.34/0.60 thf(fact_3916_order__le__less,axiom,
% 0.34/0.60 ( ord_less_eq_assn
% 0.34/0.60 = ( ^ [X2: assn,Y3: assn] :
% 0.34/0.60 ( ( ord_less_assn @ X2 @ Y3 )
% 0.34/0.60 | ( X2 = Y3 ) ) ) ) ).
% 0.34/0.60
% 0.34/0.60 % order_le_less
% 0.34/0.60 thf(fact_3917_order__le__less,axiom,
% 0.34/0.60 ( ord_le3102999989581377725nteger
% 0.34/0.60 = ( ^ [X2: code_integer,Y3: code_integer] :
% 0.34/0.60 ( ( ord_le6747313008572928689nteger @ X2 @ Y3 )
% 0.34/0.60 | ( X2 = Y3 ) ) ) ) ).
% 0.34/0.60
% 0.34/0.60 % order_le_less
% 0.34/0.60 thf(fact_3918_order__le__less,axiom,
% 0.34/0.60 ( ord_le2510731241096832064er_nat
% 0.34/0.60 = ( ^ [X2: filter_nat,Y3: filter_nat] :
% 0.34/0.60 ( ( ord_less_filter_nat @ X2 @ Y3 )
% 0.34/0.60 | ( X2 = Y3 ) ) ) ) ).
% 0.34/0.60
% 0.34/0.60 % order_le_less
% 0.34/0.60 thf(fact_3919_order__le__less,axiom,
% 0.34/0.60 ( ord_less_eq_set_nat
% 0.34/0.60 = ( ^ [X2: set_nat,Y3: set_nat] :
% 0.34/0.60 ( ( ord_less_set_nat @ X2 @ Y3 )
% 0.34/0.60 | ( X2 = Y3 ) ) ) ) ).
% 0.34/0.60
% 0.34/0.60 % order_le_less
% 0.34/0.60 thf(fact_3920_order__le__less,axiom,
% 0.34/0.60 ( ord_less_eq_rat
% 0.34/0.60 = ( ^ [X2: rat,Y3: rat] :
% 0.34/0.60 ( ( ord_less_rat @ X2 @ Y3 )
% 0.34/0.60 | ( X2 = Y3 ) ) ) ) ).
% 0.34/0.60
% 0.34/0.60 % order_le_less
% 0.34/0.60 thf(fact_3921_order__le__less,axiom,
% 0.34/0.60 ( ord_less_eq_nat
% 0.34/0.60 = ( ^ [X2: nat,Y3: nat] :
% 0.34/0.60 ( ( ord_less_nat @ X2 @ Y3 )
% 0.34/0.60 | ( X2 = Y3 ) ) ) ) ).
% 0.34/0.60
% 0.34/0.60 % order_le_less
% 0.34/0.60 thf(fact_3922_order__le__less,axiom,
% 0.34/0.60 ( ord_less_eq_int
% 0.34/0.60 = ( ^ [X2: int,Y3: int] :
% 0.34/0.60 ( ( ord_less_int @ X2 @ Y3 )
% 0.34/0.60 | ( X2 = Y3 ) ) ) ) ).
% 0.34/0.60
% 0.34/0.60 % order_le_less
% 0.34/0.60 thf(fact_3923_order__less__le,axiom,
% 0.34/0.60 ( ord_less_assn
% 0.34/0.60 = ( ^ [X2: assn,Y3: assn] :
% 0.34/0.60 ( ( ord_less_eq_assn @ X2 @ Y3 )
% 0.34/0.60 & ( X2 != Y3 ) ) ) ) ).
% 0.34/0.60
% 0.34/0.60 % order_less_le
% 0.34/0.60 thf(fact_3924_order__less__le,axiom,
% 0.34/0.60 ( ord_le6747313008572928689nteger
% 0.34/0.60 = ( ^ [X2: code_integer,Y3: code_integer] :
% 0.34/0.60 ( ( ord_le3102999989581377725nteger @ X2 @ Y3 )
% 0.34/0.60 & ( X2 != Y3 ) ) ) ) ).
% 0.34/0.60
% 0.34/0.60 % order_less_le
% 0.34/0.60 thf(fact_3925_order__less__le,axiom,
% 0.34/0.60 ( ord_less_filter_nat
% 0.34/0.60 = ( ^ [X2: filter_nat,Y3: filter_nat] :
% 0.34/0.60 ( ( ord_le2510731241096832064er_nat @ X2 @ Y3 )
% 0.34/0.60 & ( X2 != Y3 ) ) ) ) ).
% 0.34/0.60
% 0.34/0.60 % order_less_le
% 0.34/0.60 thf(fact_3926_order__less__le,axiom,
% 0.34/0.60 ( ord_less_set_nat
% 0.34/0.60 = ( ^ [X2: set_nat,Y3: set_nat] :
% 0.34/0.60 ( ( ord_less_eq_set_nat @ X2 @ Y3 )
% 0.34/0.60 & ( X2 != Y3 ) ) ) ) ).
% 0.34/0.60
% 0.34/0.60 % order_less_le
% 0.34/0.60 thf(fact_3927_order__less__le,axiom,
% 0.34/0.60 ( ord_less_rat
% 0.34/0.60 = ( ^ [X2: rat,Y3: rat] :
% 0.34/0.60 ( ( ord_less_eq_rat @ X2 @ Y3 )
% 0.34/0.60 & ( X2 != Y3 ) ) ) ) ).
% 0.34/0.60
% 0.34/0.60 % order_less_le
% 0.34/0.60 thf(fact_3928_order__less__le,axiom,
% 0.34/0.60 ( ord_less_nat
% 0.34/0.60 = ( ^ [X2: nat,Y3: nat] :
% 0.34/0.60 ( ( ord_less_eq_nat @ X2 @ Y3 )
% 0.34/0.60 & ( X2 != Y3 ) ) ) ) ).
% 0.34/0.60
% 0.34/0.60 % order_less_le
% 0.34/0.60 thf(fact_3929_order__less__le,axiom,
% 0.34/0.60 ( ord_less_int
% 0.34/0.60 = ( ^ [X2: int,Y3: int] :
% 0.34/0.60 ( ( ord_less_eq_int @ X2 @ Y3 )
% 0.34/0.60 & ( X2 != Y3 ) ) ) ) ).
% 0.34/0.60
% 0.34/0.60 % order_less_le
% 0.34/0.60 thf(fact_3930_linorder__not__le,axiom,
% 0.34/0.60 ! [X: code_integer,Y: code_integer] :
% 0.34/0.60 ( ( ~ ( ord_le3102999989581377725nteger @ X @ Y ) )
% 0.34/0.60 = ( ord_le6747313008572928689nteger @ Y @ X ) ) ).
% 0.34/0.60
% 0.34/0.60 % linorder_not_le
% 0.34/0.60 thf(fact_3931_linorder__not__le,axiom,
% 0.34/0.60 ! [X: rat,Y: rat] :
% 0.34/0.60 ( ( ~ ( ord_less_eq_rat @ X @ Y ) )
% 0.34/0.60 = ( ord_less_rat @ Y @ X ) ) ).
% 0.34/0.60
% 0.34/0.60 % linorder_not_le
% 0.34/0.60 thf(fact_3932_linorder__not__le,axiom,
% 0.34/0.60 ! [X: nat,Y: nat] :
% 0.34/0.60 ( ( ~ ( ord_less_eq_nat @ X @ Y ) )
% 0.34/0.60 = ( ord_less_nat @ Y @ X ) ) ).
% 0.34/0.60
% 0.34/0.60 % linorder_not_le
% 0.34/0.60 thf(fact_3933_linorder__not__le,axiom,
% 0.34/0.60 ! [X: int,Y: int] :
% 0.34/0.60 ( ( ~ ( ord_less_eq_int @ X @ Y ) )
% 0.34/0.60 = ( ord_less_int @ Y @ X ) ) ).
% 0.34/0.60
% 0.34/0.60 % linorder_not_le
% 0.34/0.60 thf(fact_3934_linorder__not__less,axiom,
% 0.34/0.60 ! [X: code_integer,Y: code_integer] :
% 0.34/0.60 ( ( ~ ( ord_le6747313008572928689nteger @ X @ Y ) )
% 0.34/0.60 = ( ord_le3102999989581377725nteger @ Y @ X ) ) ).
% 0.34/0.60
% 0.34/0.60 % linorder_not_less
% 0.34/0.60 thf(fact_3935_linorder__not__less,axiom,
% 0.34/0.60 ! [X: rat,Y: rat] :
% 0.34/0.60 ( ( ~ ( ord_less_rat @ X @ Y ) )
% 0.34/0.60 = ( ord_less_eq_rat @ Y @ X ) ) ).
% 0.34/0.60
% 0.34/0.60 % linorder_not_less
% 0.34/0.60 thf(fact_3936_linorder__not__less,axiom,
% 0.34/0.60 ! [X: nat,Y: nat] :
% 0.34/0.60 ( ( ~ ( ord_less_nat @ X @ Y ) )
% 0.34/0.60 = ( ord_less_eq_nat @ Y @ X ) ) ).
% 0.34/0.60
% 0.34/0.60 % linorder_not_less
% 0.34/0.60 thf(fact_3937_linorder__not__less,axiom,
% 0.34/0.60 ! [X: int,Y: int] :
% 0.34/0.60 ( ( ~ ( ord_less_int @ X @ Y ) )
% 0.34/0.60 = ( ord_less_eq_int @ Y @ X ) ) ).
% 0.34/0.60
% 0.34/0.60 % linorder_not_less
% 0.34/0.60 thf(fact_3938_order__less__imp__le,axiom,
% 0.34/0.60 ! [X: assn,Y: assn] :
% 0.34/0.60 ( ( ord_less_assn @ X @ Y )
% 0.34/0.60 => ( ord_less_eq_assn @ X @ Y ) ) ).
% 0.34/0.60
% 0.34/0.60 % order_less_imp_le
% 0.34/0.60 thf(fact_3939_order__less__imp__le,axiom,
% 0.34/0.60 ! [X: code_integer,Y: code_integer] :
% 0.34/0.60 ( ( ord_le6747313008572928689nteger @ X @ Y )
% 0.34/0.60 => ( ord_le3102999989581377725nteger @ X @ Y ) ) ).
% 0.34/0.60
% 0.34/0.60 % order_less_imp_le
% 0.34/0.60 thf(fact_3940_order__less__imp__le,axiom,
% 0.34/0.60 ! [X: filter_nat,Y: filter_nat] :
% 0.34/0.60 ( ( ord_less_filter_nat @ X @ Y )
% 0.34/0.60 => ( ord_le2510731241096832064er_nat @ X @ Y ) ) ).
% 0.34/0.60
% 0.34/0.60 % order_less_imp_le
% 0.34/0.60 thf(fact_3941_order__less__imp__le,axiom,
% 0.34/0.60 ! [X: set_nat,Y: set_nat] :
% 0.34/0.60 ( ( ord_less_set_nat @ X @ Y )
% 0.34/0.60 => ( ord_less_eq_set_nat @ X @ Y ) ) ).
% 0.34/0.60
% 0.34/0.60 % order_less_imp_le
% 0.34/0.60 thf(fact_3942_order__less__imp__le,axiom,
% 0.34/0.60 ! [X: rat,Y: rat] :
% 0.34/0.60 ( ( ord_less_rat @ X @ Y )
% 0.34/0.60 => ( ord_less_eq_rat @ X @ Y ) ) ).
% 0.34/0.60
% 0.34/0.60 % order_less_imp_le
% 0.34/0.60 thf(fact_3943_order__less__imp__le,axiom,
% 0.34/0.60 ! [X: nat,Y: nat] :
% 0.34/0.60 ( ( ord_less_nat @ X @ Y )
% 0.34/0.60 => ( ord_less_eq_nat @ X @ Y ) ) ).
% 0.34/0.60
% 0.34/0.60 % order_less_imp_le
% 0.34/0.60 thf(fact_3944_order__less__imp__le,axiom,
% 0.34/0.60 ! [X: int,Y: int] :
% 0.34/0.60 ( ( ord_less_int @ X @ Y )
% 0.34/0.60 => ( ord_less_eq_int @ X @ Y ) ) ).
% 0.34/0.60
% 0.34/0.60 % order_less_imp_le
% 0.34/0.60 thf(fact_3945_order__le__neq__trans,axiom,
% 0.34/0.60 ! [A2: assn,B: assn] :
% 0.34/0.60 ( ( ord_less_eq_assn @ A2 @ B )
% 0.34/0.60 => ( ( A2 != B )
% 0.34/0.60 => ( ord_less_assn @ A2 @ B ) ) ) ).
% 0.34/0.60
% 0.34/0.60 % order_le_neq_trans
% 0.34/0.60 thf(fact_3946_order__le__neq__trans,axiom,
% 0.34/0.60 ! [A2: code_integer,B: code_integer] :
% 0.34/0.60 ( ( ord_le3102999989581377725nteger @ A2 @ B )
% 0.34/0.60 => ( ( A2 != B )
% 0.34/0.60 => ( ord_le6747313008572928689nteger @ A2 @ B ) ) ) ).
% 0.34/0.60
% 0.34/0.60 % order_le_neq_trans
% 0.34/0.60 thf(fact_3947_order__le__neq__trans,axiom,
% 0.34/0.60 ! [A2: filter_nat,B: filter_nat] :
% 0.34/0.60 ( ( ord_le2510731241096832064er_nat @ A2 @ B )
% 0.34/0.60 => ( ( A2 != B )
% 0.34/0.60 => ( ord_less_filter_nat @ A2 @ B ) ) ) ).
% 0.34/0.60
% 0.34/0.60 % order_le_neq_trans
% 0.34/0.60 thf(fact_3948_order__le__neq__trans,axiom,
% 0.34/0.60 ! [A2: set_nat,B: set_nat] :
% 0.34/0.60 ( ( ord_less_eq_set_nat @ A2 @ B )
% 0.34/0.60 => ( ( A2 != B )
% 0.34/0.60 => ( ord_less_set_nat @ A2 @ B ) ) ) ).
% 0.34/0.60
% 0.34/0.60 % order_le_neq_trans
% 0.34/0.60 thf(fact_3949_order__le__neq__trans,axiom,
% 0.34/0.60 ! [A2: rat,B: rat] :
% 0.34/0.60 ( ( ord_less_eq_rat @ A2 @ B )
% 0.34/0.60 => ( ( A2 != B )
% 0.34/0.60 => ( ord_less_rat @ A2 @ B ) ) ) ).
% 0.34/0.60
% 0.34/0.60 % order_le_neq_trans
% 0.34/0.60 thf(fact_3950_order__le__neq__trans,axiom,
% 0.34/0.60 ! [A2: nat,B: nat] :
% 0.34/0.60 ( ( ord_less_eq_nat @ A2 @ B )
% 0.34/0.60 => ( ( A2 != B )
% 0.34/0.60 => ( ord_less_nat @ A2 @ B ) ) ) ).
% 0.34/0.60
% 0.34/0.60 % order_le_neq_trans
% 0.34/0.60 thf(fact_3951_order__le__neq__trans,axiom,
% 0.34/0.60 ! [A2: int,B: int] :
% 0.34/0.60 ( ( ord_less_eq_int @ A2 @ B )
% 0.34/0.60 => ( ( A2 != B )
% 0.34/0.60 => ( ord_less_int @ A2 @ B ) ) ) ).
% 0.34/0.60
% 0.34/0.60 % order_le_neq_trans
% 0.34/0.60 thf(fact_3952_order__neq__le__trans,axiom,
% 0.34/0.60 ! [A2: assn,B: assn] :
% 0.34/0.60 ( ( A2 != B )
% 0.34/0.60 => ( ( ord_less_eq_assn @ A2 @ B )
% 0.34/0.60 => ( ord_less_assn @ A2 @ B ) ) ) ).
% 0.34/0.60
% 0.34/0.60 % order_neq_le_trans
% 0.34/0.60 thf(fact_3953_order__neq__le__trans,axiom,
% 0.34/0.60 ! [A2: code_integer,B: code_integer] :
% 0.34/0.60 ( ( A2 != B )
% 0.34/0.60 => ( ( ord_le3102999989581377725nteger @ A2 @ B )
% 0.34/0.60 => ( ord_le6747313008572928689nteger @ A2 @ B ) ) ) ).
% 0.34/0.60
% 0.34/0.60 % order_neq_le_trans
% 0.34/0.60 thf(fact_3954_order__neq__le__trans,axiom,
% 0.34/0.60 ! [A2: filter_nat,B: filter_nat] :
% 0.34/0.60 ( ( A2 != B )
% 0.34/0.60 => ( ( ord_le2510731241096832064er_nat @ A2 @ B )
% 0.34/0.60 => ( ord_less_filter_nat @ A2 @ B ) ) ) ).
% 0.34/0.60
% 0.34/0.60 % order_neq_le_trans
% 0.34/0.60 thf(fact_3955_order__neq__le__trans,axiom,
% 0.34/0.60 ! [A2: set_nat,B: set_nat] :
% 0.34/0.60 ( ( A2 != B )
% 0.34/0.61 => ( ( ord_less_eq_set_nat @ A2 @ B )
% 0.34/0.61 => ( ord_less_set_nat @ A2 @ B ) ) ) ).
% 0.34/0.61
% 0.34/0.61 % order_neq_le_trans
% 0.34/0.61 thf(fact_3956_order__neq__le__trans,axiom,
% 0.34/0.61 ! [A2: rat,B: rat] :
% 0.34/0.61 ( ( A2 != B )
% 0.34/0.61 => ( ( ord_less_eq_rat @ A2 @ B )
% 0.34/0.61 => ( ord_less_rat @ A2 @ B ) ) ) ).
% 0.34/0.61
% 0.34/0.61 % order_neq_le_trans
% 0.34/0.61 thf(fact_3957_order__neq__le__trans,axiom,
% 0.34/0.61 ! [A2: nat,B: nat] :
% 0.34/0.61 ( ( A2 != B )
% 0.34/0.61 => ( ( ord_less_eq_nat @ A2 @ B )
% 0.34/0.61 => ( ord_less_nat @ A2 @ B ) ) ) ).
% 0.34/0.61
% 0.34/0.61 % order_neq_le_trans
% 0.34/0.61 thf(fact_3958_order__neq__le__trans,axiom,
% 0.34/0.61 ! [A2: int,B: int] :
% 0.34/0.61 ( ( A2 != B )
% 0.34/0.61 => ( ( ord_less_eq_int @ A2 @ B )
% 0.34/0.61 => ( ord_less_int @ A2 @ B ) ) ) ).
% 0.34/0.61
% 0.34/0.61 % order_neq_le_trans
% 0.34/0.61 thf(fact_3959_order__le__less__trans,axiom,
% 0.34/0.61 ! [X: assn,Y: assn,Z: assn] :
% 0.34/0.61 ( ( ord_less_eq_assn @ X @ Y )
% 0.34/0.61 => ( ( ord_less_assn @ Y @ Z )
% 0.34/0.61 => ( ord_less_assn @ X @ Z ) ) ) ).
% 0.34/0.61
% 0.34/0.61 % order_le_less_trans
% 0.34/0.61 thf(fact_3960_order__le__less__trans,axiom,
% 0.34/0.61 ! [X: code_integer,Y: code_integer,Z: code_integer] :
% 0.34/0.61 ( ( ord_le3102999989581377725nteger @ X @ Y )
% 0.34/0.61 => ( ( ord_le6747313008572928689nteger @ Y @ Z )
% 0.34/0.61 => ( ord_le6747313008572928689nteger @ X @ Z ) ) ) ).
% 0.34/0.61
% 0.34/0.61 % order_le_less_trans
% 0.34/0.61 thf(fact_3961_order__le__less__trans,axiom,
% 0.34/0.61 ! [X: filter_nat,Y: filter_nat,Z: filter_nat] :
% 0.34/0.61 ( ( ord_le2510731241096832064er_nat @ X @ Y )
% 0.34/0.61 => ( ( ord_less_filter_nat @ Y @ Z )
% 0.34/0.61 => ( ord_less_filter_nat @ X @ Z ) ) ) ).
% 0.34/0.61
% 0.34/0.61 % order_le_less_trans
% 0.34/0.61 thf(fact_3962_order__le__less__trans,axiom,
% 0.34/0.61 ! [X: set_nat,Y: set_nat,Z: set_nat] :
% 0.34/0.61 ( ( ord_less_eq_set_nat @ X @ Y )
% 0.34/0.61 => ( ( ord_less_set_nat @ Y @ Z )
% 0.34/0.61 => ( ord_less_set_nat @ X @ Z ) ) ) ).
% 0.34/0.61
% 0.34/0.61 % order_le_less_trans
% 0.34/0.61 thf(fact_3963_order__le__less__trans,axiom,
% 0.34/0.61 ! [X: rat,Y: rat,Z: rat] :
% 0.34/0.61 ( ( ord_less_eq_rat @ X @ Y )
% 0.34/0.61 => ( ( ord_less_rat @ Y @ Z )
% 0.34/0.61 => ( ord_less_rat @ X @ Z ) ) ) ).
% 0.34/0.61
% 0.34/0.61 % order_le_less_trans
% 0.34/0.61 thf(fact_3964_order__le__less__trans,axiom,
% 0.34/0.61 ! [X: nat,Y: nat,Z: nat] :
% 0.34/0.61 ( ( ord_less_eq_nat @ X @ Y )
% 0.34/0.61 => ( ( ord_less_nat @ Y @ Z )
% 0.34/0.61 => ( ord_less_nat @ X @ Z ) ) ) ).
% 0.34/0.61
% 0.34/0.61 % order_le_less_trans
% 0.34/0.61 thf(fact_3965_order__le__less__trans,axiom,
% 0.34/0.61 ! [X: int,Y: int,Z: int] :
% 0.34/0.61 ( ( ord_less_eq_int @ X @ Y )
% 0.34/0.61 => ( ( ord_less_int @ Y @ Z )
% 0.34/0.61 => ( ord_less_int @ X @ Z ) ) ) ).
% 0.34/0.61
% 0.34/0.61 % order_le_less_trans
% 0.34/0.61 thf(fact_3966_order__less__le__trans,axiom,
% 0.34/0.61 ! [X: assn,Y: assn,Z: assn] :
% 0.34/0.61 ( ( ord_less_assn @ X @ Y )
% 0.34/0.61 => ( ( ord_less_eq_assn @ Y @ Z )
% 0.34/0.61 => ( ord_less_assn @ X @ Z ) ) ) ).
% 0.34/0.61
% 0.34/0.61 % order_less_le_trans
% 0.34/0.61 thf(fact_3967_order__less__le__trans,axiom,
% 0.34/0.61 ! [X: code_integer,Y: code_integer,Z: code_integer] :
% 0.34/0.61 ( ( ord_le6747313008572928689nteger @ X @ Y )
% 0.34/0.61 => ( ( ord_le3102999989581377725nteger @ Y @ Z )
% 0.34/0.61 => ( ord_le6747313008572928689nteger @ X @ Z ) ) ) ).
% 0.34/0.61
% 0.34/0.61 % order_less_le_trans
% 0.34/0.61 thf(fact_3968_order__less__le__trans,axiom,
% 0.34/0.61 ! [X: filter_nat,Y: filter_nat,Z: filter_nat] :
% 0.34/0.61 ( ( ord_less_filter_nat @ X @ Y )
% 0.34/0.61 => ( ( ord_le2510731241096832064er_nat @ Y @ Z )
% 0.34/0.61 => ( ord_less_filter_nat @ X @ Z ) ) ) ).
% 0.34/0.61
% 0.34/0.61 % order_less_le_trans
% 0.34/0.61 thf(fact_3969_order__less__le__trans,axiom,
% 0.34/0.61 ! [X: set_nat,Y: set_nat,Z: set_nat] :
% 0.34/0.61 ( ( ord_less_set_nat @ X @ Y )
% 0.34/0.61 => ( ( ord_less_eq_set_nat @ Y @ Z )
% 0.34/0.61 => ( ord_less_set_nat @ X @ Z ) ) ) ).
% 0.34/0.61
% 0.34/0.61 % order_less_le_trans
% 0.34/0.61 thf(fact_3970_order__less__le__trans,axiom,
% 0.34/0.61 ! [X: rat,Y: rat,Z: rat] :
% 0.34/0.61 ( ( ord_less_rat @ X @ Y )
% 0.34/0.61 => ( ( ord_less_eq_rat @ Y @ Z )
% 0.34/0.61 => ( ord_less_rat @ X @ Z ) ) ) ).
% 0.34/0.61
% 0.34/0.61 % order_less_le_trans
% 0.34/0.61 thf(fact_3971_order__less__le__trans,axiom,
% 0.34/0.61 ! [X: nat,Y: nat,Z: nat] :
% 0.34/0.61 ( ( ord_less_nat @ X @ Y )
% 0.34/0.61 => ( ( ord_less_eq_nat @ Y @ Z )
% 0.34/0.61 => ( ord_less_nat @ X @ Z ) ) ) ).
% 0.34/0.61
% 0.34/0.61 % order_less_le_trans
% 0.34/0.61 thf(fact_3972_order__less__le__trans,axiom,
% 0.34/0.61 ! [X: int,Y: int,Z: int] :
% 0.34/0.61 ( ( ord_less_int @ X @ Y )
% 0.34/0.61 => ( ( ord_less_eq_int @ Y @ Z )
% 0.34/0.61 => ( ord_less_int @ X @ Z ) ) ) ).
% 0.34/0.61
% 0.34/0.61 % order_less_le_trans
% 0.34/0.61 thf(fact_3973_order__le__less__subst1,axiom,
% 0.34/0.61 ! [A2: assn,F: assn > assn,B: assn,C: assn] :
% 0.34/0.61 ( ( ord_less_eq_assn @ A2 @ ( F @ B ) )
% 0.34/0.61 => ( ( ord_less_assn @ B @ C )
% 0.34/0.61 => ( ! [X3: assn,Y2: assn] :
% 0.34/0.61 ( ( ord_less_assn @ X3 @ Y2 )
% 0.34/0.61 => ( ord_less_assn @ ( F @ X3 ) @ ( F @ Y2 ) ) )
% 0.34/0.61 => ( ord_less_assn @ A2 @ ( F @ C ) ) ) ) ) ).
% 0.34/0.61
% 0.34/0.61 % order_le_less_subst1
% 0.34/0.61 thf(fact_3974_order__le__less__subst1,axiom,
% 0.34/0.61 ! [A2: code_integer,F: assn > code_integer,B: assn,C: assn] :
% 0.34/0.61 ( ( ord_le3102999989581377725nteger @ A2 @ ( F @ B ) )
% 0.34/0.61 => ( ( ord_less_assn @ B @ C )
% 0.34/0.61 => ( ! [X3: assn,Y2: assn] :
% 0.34/0.61 ( ( ord_less_assn @ X3 @ Y2 )
% 0.34/0.61 => ( ord_le6747313008572928689nteger @ ( F @ X3 ) @ ( F @ Y2 ) ) )
% 0.34/0.61 => ( ord_le6747313008572928689nteger @ A2 @ ( F @ C ) ) ) ) ) ).
% 0.34/0.61
% 0.34/0.61 % order_le_less_subst1
% 0.34/0.61 thf(fact_3975_order__le__less__subst1,axiom,
% 0.34/0.61 ! [A2: assn,F: nat > assn,B: nat,C: nat] :
% 0.34/0.61 ( ( ord_less_eq_assn @ A2 @ ( F @ B ) )
% 0.34/0.61 => ( ( ord_less_nat @ B @ C )
% 0.34/0.61 => ( ! [X3: nat,Y2: nat] :
% 0.34/0.61 ( ( ord_less_nat @ X3 @ Y2 )
% 0.34/0.61 => ( ord_less_assn @ ( F @ X3 ) @ ( F @ Y2 ) ) )
% 0.34/0.61 => ( ord_less_assn @ A2 @ ( F @ C ) ) ) ) ) ).
% 0.34/0.61
% 0.34/0.61 % order_le_less_subst1
% 0.34/0.61 thf(fact_3976_order__le__less__subst1,axiom,
% 0.34/0.61 ! [A2: code_integer,F: nat > code_integer,B: nat,C: nat] :
% 0.34/0.61 ( ( ord_le3102999989581377725nteger @ A2 @ ( F @ B ) )
% 0.34/0.61 => ( ( ord_less_nat @ B @ C )
% 0.34/0.61 => ( ! [X3: nat,Y2: nat] :
% 0.34/0.61 ( ( ord_less_nat @ X3 @ Y2 )
% 0.34/0.61 => ( ord_le6747313008572928689nteger @ ( F @ X3 ) @ ( F @ Y2 ) ) )
% 0.34/0.61 => ( ord_le6747313008572928689nteger @ A2 @ ( F @ C ) ) ) ) ) ).
% 0.34/0.61
% 0.34/0.61 % order_le_less_subst1
% 0.34/0.61 thf(fact_3977_order__le__less__subst1,axiom,
% 0.34/0.61 ! [A2: assn,F: int > assn,B: int,C: int] :
% 0.34/0.61 ( ( ord_less_eq_assn @ A2 @ ( F @ B ) )
% 0.34/0.61 => ( ( ord_less_int @ B @ C )
% 0.34/0.61 => ( ! [X3: int,Y2: int] :
% 0.34/0.61 ( ( ord_less_int @ X3 @ Y2 )
% 0.34/0.61 => ( ord_less_assn @ ( F @ X3 ) @ ( F @ Y2 ) ) )
% 0.34/0.61 => ( ord_less_assn @ A2 @ ( F @ C ) ) ) ) ) ).
% 0.34/0.61
% 0.34/0.61 % order_le_less_subst1
% 0.34/0.61 thf(fact_3978_order__le__less__subst1,axiom,
% 0.34/0.61 ! [A2: code_integer,F: int > code_integer,B: int,C: int] :
% 0.34/0.61 ( ( ord_le3102999989581377725nteger @ A2 @ ( F @ B ) )
% 0.34/0.61 => ( ( ord_less_int @ B @ C )
% 0.34/0.61 => ( ! [X3: int,Y2: int] :
% 0.34/0.61 ( ( ord_less_int @ X3 @ Y2 )
% 0.34/0.61 => ( ord_le6747313008572928689nteger @ ( F @ X3 ) @ ( F @ Y2 ) ) )
% 0.34/0.61 => ( ord_le6747313008572928689nteger @ A2 @ ( F @ C ) ) ) ) ) ).
% 0.34/0.61
% 0.34/0.61 % order_le_less_subst1
% 0.34/0.61 thf(fact_3979_order__le__less__subst1,axiom,
% 0.34/0.61 ! [A2: assn,F: code_integer > assn,B: code_integer,C: code_integer] :
% 0.34/0.61 ( ( ord_less_eq_assn @ A2 @ ( F @ B ) )
% 0.34/0.61 => ( ( ord_le6747313008572928689nteger @ B @ C )
% 0.34/0.61 => ( ! [X3: code_integer,Y2: code_integer] :
% 0.34/0.61 ( ( ord_le6747313008572928689nteger @ X3 @ Y2 )
% 0.34/0.61 => ( ord_less_assn @ ( F @ X3 ) @ ( F @ Y2 ) ) )
% 0.34/0.61 => ( ord_less_assn @ A2 @ ( F @ C ) ) ) ) ) ).
% 0.34/0.61
% 0.34/0.61 % order_le_less_subst1
% 0.34/0.61 thf(fact_3980_order__le__less__subst1,axiom,
% 0.34/0.61 ! [A2: code_integer,F: code_integer > code_integer,B: code_integer,C: code_integer] :
% 0.34/0.61 ( ( ord_le3102999989581377725nteger @ A2 @ ( F @ B ) )
% 0.34/0.61 => ( ( ord_le6747313008572928689nteger @ B @ C )
% 0.34/0.61 => ( ! [X3: code_integer,Y2: code_integer] :
% 0.34/0.61 ( ( ord_le6747313008572928689nteger @ X3 @ Y2 )
% 0.34/0.61 => ( ord_le6747313008572928689nteger @ ( F @ X3 ) @ ( F @ Y2 ) ) )
% 0.34/0.61 => ( ord_le6747313008572928689nteger @ A2 @ ( F @ C ) ) ) ) ) ).
% 0.34/0.61
% 0.34/0.61 % order_le_less_subst1
% 0.34/0.61 thf(fact_3981_order__le__less__subst1,axiom,
% 0.34/0.61 ! [A2: assn,F: rat > assn,B: rat,C: rat] :
% 0.34/0.61 ( ( ord_less_eq_assn @ A2 @ ( F @ B ) )
% 0.34/0.61 => ( ( ord_less_rat @ B @ C )
% 0.34/0.61 => ( ! [X3: rat,Y2: rat] :
% 0.34/0.61 ( ( ord_less_rat @ X3 @ Y2 )
% 0.34/0.61 => ( ord_less_assn @ ( F @ X3 ) @ ( F @ Y2 ) ) )
% 0.34/0.61 => ( ord_less_assn @ A2 @ ( F @ C ) ) ) ) ) ).
% 0.34/0.61
% 0.34/0.61 % order_le_less_subst1
% 0.34/0.61 thf(fact_3982_order__le__less__subst1,axiom,
% 0.34/0.61 ! [A2: code_integer,F: rat > code_integer,B: rat,C: rat] :
% 0.34/0.61 ( ( ord_le3102999989581377725nteger @ A2 @ ( F @ B ) )
% 0.34/0.61 => ( ( ord_less_rat @ B @ C )
% 0.34/0.61 => ( ! [X3: rat,Y2: rat] :
% 0.34/0.61 ( ( ord_less_rat @ X3 @ Y2 )
% 0.34/0.61 => ( ord_le6747313008572928689nteger @ ( F @ X3 ) @ ( F @ Y2 ) ) )
% 0.34/0.61 => ( ord_le6747313008572928689nteger @ A2 @ ( F @ C ) ) ) ) ) ).
% 0.34/0.61
% 0.34/0.61 % order_le_less_subst1
% 0.34/0.61 thf(fact_3983_order__le__less__subst2,axiom,
% 0.34/0.61 ! [A2: rat,B: rat,F: rat > assn,C: assn] :
% 0.34/0.61 ( ( ord_less_eq_rat @ A2 @ B )
% 0.34/0.61 => ( ( ord_less_assn @ ( F @ B ) @ C )
% 0.34/0.61 => ( ! [X3: rat,Y2: rat] :
% 0.34/0.61 ( ( ord_less_eq_rat @ X3 @ Y2 )
% 0.34/0.61 => ( ord_less_eq_assn @ ( F @ X3 ) @ ( F @ Y2 ) ) )
% 0.34/0.61 => ( ord_less_assn @ ( F @ A2 ) @ C ) ) ) ) ).
% 0.34/0.61
% 0.34/0.61 % order_le_less_subst2
% 0.34/0.61 thf(fact_3984_order__le__less__subst2,axiom,
% 0.34/0.61 ! [A2: rat,B: rat,F: rat > code_integer,C: code_integer] :
% 0.34/0.61 ( ( ord_less_eq_rat @ A2 @ B )
% 0.34/0.61 => ( ( ord_le6747313008572928689nteger @ ( F @ B ) @ C )
% 0.34/0.61 => ( ! [X3: rat,Y2: rat] :
% 0.34/0.61 ( ( ord_less_eq_rat @ X3 @ Y2 )
% 0.34/0.61 => ( ord_le3102999989581377725nteger @ ( F @ X3 ) @ ( F @ Y2 ) ) )
% 0.34/0.61 => ( ord_le6747313008572928689nteger @ ( F @ A2 ) @ C ) ) ) ) ).
% 0.34/0.61
% 0.34/0.61 % order_le_less_subst2
% 0.34/0.61 thf(fact_3985_order__le__less__subst2,axiom,
% 0.34/0.61 ! [A2: rat,B: rat,F: rat > rat,C: rat] :
% 0.34/0.61 ( ( ord_less_eq_rat @ A2 @ B )
% 0.34/0.61 => ( ( ord_less_rat @ ( F @ B ) @ C )
% 0.34/0.61 => ( ! [X3: rat,Y2: rat] :
% 0.34/0.61 ( ( ord_less_eq_rat @ X3 @ Y2 )
% 0.34/0.61 => ( ord_less_eq_rat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
% 0.34/0.61 => ( ord_less_rat @ ( F @ A2 ) @ C ) ) ) ) ).
% 0.34/0.61
% 0.34/0.61 % order_le_less_subst2
% 0.34/0.61 thf(fact_3986_order__le__less__subst2,axiom,
% 0.34/0.61 ! [A2: rat,B: rat,F: rat > nat,C: nat] :
% 0.34/0.61 ( ( ord_less_eq_rat @ A2 @ B )
% 0.34/0.61 => ( ( ord_less_nat @ ( F @ B ) @ C )
% 0.34/0.61 => ( ! [X3: rat,Y2: rat] :
% 0.34/0.61 ( ( ord_less_eq_rat @ X3 @ Y2 )
% 0.34/0.61 => ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
% 0.34/0.61 => ( ord_less_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% 0.34/0.61
% 0.34/0.61 % order_le_less_subst2
% 0.34/0.61 thf(fact_3987_order__le__less__subst2,axiom,
% 0.34/0.61 ! [A2: rat,B: rat,F: rat > int,C: int] :
% 0.34/0.61 ( ( ord_less_eq_rat @ A2 @ B )
% 0.34/0.61 => ( ( ord_less_int @ ( F @ B ) @ C )
% 0.34/0.61 => ( ! [X3: rat,Y2: rat] :
% 0.34/0.61 ( ( ord_less_eq_rat @ X3 @ Y2 )
% 0.34/0.61 => ( ord_less_eq_int @ ( F @ X3 ) @ ( F @ Y2 ) ) )
% 0.34/0.61 => ( ord_less_int @ ( F @ A2 ) @ C ) ) ) ) ).
% 0.34/0.61
% 0.34/0.61 % order_le_less_subst2
% 0.34/0.61 thf(fact_3988_order__le__less__subst2,axiom,
% 0.34/0.61 ! [A2: nat,B: nat,F: nat > assn,C: assn] :
% 0.34/0.61 ( ( ord_less_eq_nat @ A2 @ B )
% 0.34/0.61 => ( ( ord_less_assn @ ( F @ B ) @ C )
% 0.34/0.61 => ( ! [X3: nat,Y2: nat] :
% 0.34/0.61 ( ( ord_less_eq_nat @ X3 @ Y2 )
% 0.34/0.61 => ( ord_less_eq_assn @ ( F @ X3 ) @ ( F @ Y2 ) ) )
% 0.34/0.61 => ( ord_less_assn @ ( F @ A2 ) @ C ) ) ) ) ).
% 0.34/0.61
% 0.34/0.61 % order_le_less_subst2
% 0.34/0.61 thf(fact_3989_order__le__less__subst2,axiom,
% 0.34/0.61 ! [A2: nat,B: nat,F: nat > code_integer,C: code_integer] :
% 0.34/0.61 ( ( ord_less_eq_nat @ A2 @ B )
% 0.34/0.61 => ( ( ord_le6747313008572928689nteger @ ( F @ B ) @ C )
% 0.34/0.61 => ( ! [X3: nat,Y2: nat] :
% 0.34/0.61 ( ( ord_less_eq_nat @ X3 @ Y2 )
% 0.34/0.61 => ( ord_le3102999989581377725nteger @ ( F @ X3 ) @ ( F @ Y2 ) ) )
% 0.34/0.61 => ( ord_le6747313008572928689nteger @ ( F @ A2 ) @ C ) ) ) ) ).
% 0.34/0.61
% 0.34/0.61 % order_le_less_subst2
% 0.34/0.61 thf(fact_3990_order__le__less__subst2,axiom,
% 0.34/0.61 ! [A2: nat,B: nat,F: nat > rat,C: rat] :
% 0.34/0.61 ( ( ord_less_eq_nat @ A2 @ B )
% 0.34/0.61 => ( ( ord_less_rat @ ( F @ B ) @ C )
% 0.34/0.61 => ( ! [X3: nat,Y2: nat] :
% 0.34/0.61 ( ( ord_less_eq_nat @ X3 @ Y2 )
% 0.34/0.61 => ( ord_less_eq_rat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
% 0.34/0.61 => ( ord_less_rat @ ( F @ A2 ) @ C ) ) ) ) ).
% 0.34/0.61
% 0.34/0.61 % order_le_less_subst2
% 0.34/0.61 thf(fact_3991_order__le__less__subst2,axiom,
% 0.34/0.61 ! [A2: nat,B: nat,F: nat > nat,C: nat] :
% 0.34/0.61 ( ( ord_less_eq_nat @ A2 @ B )
% 0.34/0.61 => ( ( ord_less_nat @ ( F @ B ) @ C )
% 0.34/0.61 => ( ! [X3: nat,Y2: nat] :
% 0.34/0.61 ( ( ord_less_eq_nat @ X3 @ Y2 )
% 0.34/0.61 => ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
% 0.34/0.61 => ( ord_less_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% 0.34/0.61
% 0.34/0.61 % order_le_less_subst2
% 0.34/0.61 thf(fact_3992_order__le__less__subst2,axiom,
% 0.34/0.61 ! [A2: nat,B: nat,F: nat > int,C: int] :
% 0.34/0.61 ( ( ord_less_eq_nat @ A2 @ B )
% 0.34/0.61 => ( ( ord_less_int @ ( F @ B ) @ C )
% 0.34/0.61 => ( ! [X3: nat,Y2: nat] :
% 0.34/0.61 ( ( ord_less_eq_nat @ X3 @ Y2 )
% 0.34/0.61 => ( ord_less_eq_int @ ( F @ X3 ) @ ( F @ Y2 ) ) )
% 0.34/0.61 => ( ord_less_int @ ( F @ A2 ) @ C ) ) ) ) ).
% 0.34/0.61
% 0.34/0.61 % order_le_less_subst2
% 0.34/0.61 thf(fact_3993_order__less__le__subst1,axiom,
% 0.34/0.61 ! [A2: assn,F: rat > assn,B: rat,C: rat] :
% 0.34/0.61 ( ( ord_less_assn @ A2 @ ( F @ B ) )
% 0.34/0.61 => ( ( ord_less_eq_rat @ B @ C )
% 0.34/0.61 => ( ! [X3: rat,Y2: rat] :
% 0.34/0.61 ( ( ord_less_eq_rat @ X3 @ Y2 )
% 0.34/0.61 => ( ord_less_eq_assn @ ( F @ X3 ) @ ( F @ Y2 ) ) )
% 0.34/0.61 => ( ord_less_assn @ A2 @ ( F @ C ) ) ) ) ) ).
% 0.34/0.61
% 0.34/0.61 % order_less_le_subst1
% 0.34/0.61 thf(fact_3994_order__less__le__subst1,axiom,
% 0.34/0.61 ! [A2: code_integer,F: rat > code_integer,B: rat,C: rat] :
% 0.34/0.61 ( ( ord_le6747313008572928689nteger @ A2 @ ( F @ B ) )
% 0.34/0.61 => ( ( ord_less_eq_rat @ B @ C )
% 0.34/0.61 => ( ! [X3: rat,Y2: rat] :
% 0.34/0.61 ( ( ord_less_eq_rat @ X3 @ Y2 )
% 0.34/0.61 => ( ord_le3102999989581377725nteger @ ( F @ X3 ) @ ( F @ Y2 ) ) )
% 0.34/0.61 => ( ord_le6747313008572928689nteger @ A2 @ ( F @ C ) ) ) ) ) ).
% 0.34/0.61
% 0.34/0.61 % order_less_le_subst1
% 0.34/0.61 thf(fact_3995_order__less__le__subst1,axiom,
% 0.34/0.61 ! [A2: rat,F: rat > rat,B: rat,C: rat] :
% 0.34/0.61 ( ( ord_less_rat @ A2 @ ( F @ B ) )
% 0.34/0.61 => ( ( ord_less_eq_rat @ B @ C )
% 0.34/0.61 => ( ! [X3: rat,Y2: rat] :
% 0.34/0.61 ( ( ord_less_eq_rat @ X3 @ Y2 )
% 0.34/0.61 => ( ord_less_eq_rat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
% 0.34/0.61 => ( ord_less_rat @ A2 @ ( F @ C ) ) ) ) ) ).
% 0.34/0.61
% 0.34/0.61 % order_less_le_subst1
% 0.34/0.61 thf(fact_3996_order__less__le__subst1,axiom,
% 0.34/0.61 ! [A2: nat,F: rat > nat,B: rat,C: rat] :
% 0.34/0.61 ( ( ord_less_nat @ A2 @ ( F @ B ) )
% 0.34/0.61 => ( ( ord_less_eq_rat @ B @ C )
% 0.34/0.61 => ( ! [X3: rat,Y2: rat] :
% 0.34/0.61 ( ( ord_less_eq_rat @ X3 @ Y2 )
% 0.34/0.61 => ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
% 0.34/0.61 => ( ord_less_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% 0.34/0.61
% 0.34/0.61 % order_less_le_subst1
% 0.34/0.61 thf(fact_3997_order__less__le__subst1,axiom,
% 0.34/0.61 ! [A2: int,F: rat > int,B: rat,C: rat] :
% 0.34/0.61 ( ( ord_less_int @ A2 @ ( F @ B ) )
% 0.34/0.61 => ( ( ord_less_eq_rat @ B @ C )
% 0.34/0.61 => ( ! [X3: rat,Y2: rat] :
% 0.34/0.61 ( ( ord_less_eq_rat @ X3 @ Y2 )
% 0.34/0.61 => ( ord_less_eq_int @ ( F @ X3 ) @ ( F @ Y2 ) ) )
% 0.34/0.61 => ( ord_less_int @ A2 @ ( F @ C ) ) ) ) ) ).
% 0.34/0.61
% 0.34/0.61 % order_less_le_subst1
% 0.34/0.61 thf(fact_3998_order__less__le__subst1,axiom,
% 0.34/0.61 ! [A2: assn,F: nat > assn,B: nat,C: nat] :
% 0.34/0.61 ( ( ord_less_assn @ A2 @ ( F @ B ) )
% 0.34/0.61 => ( ( ord_less_eq_nat @ B @ C )
% 0.34/0.61 => ( ! [X3: nat,Y2: nat] :
% 0.34/0.61 ( ( ord_less_eq_nat @ X3 @ Y2 )
% 0.34/0.61 => ( ord_less_eq_assn @ ( F @ X3 ) @ ( F @ Y2 ) ) )
% 0.34/0.61 => ( ord_less_assn @ A2 @ ( F @ C ) ) ) ) ) ).
% 0.34/0.61
% 0.34/0.61 % order_less_le_subst1
% 0.34/0.61 thf(fact_3999_order__less__le__subst1,axiom,
% 0.34/0.61 ! [A2: code_integer,F: nat > code_integer,B: nat,C: nat] :
% 0.34/0.61 ( ( ord_le6747313008572928689nteger @ A2 @ ( F @ B ) )
% 0.34/0.61 => ( ( ord_less_eq_nat @ B @ C )
% 0.34/0.61 => ( ! [X3: nat,Y2: nat] :
% 0.34/0.61 ( ( ord_less_eq_nat @ X3 @ Y2 )
% 0.34/0.61 => ( ord_le3102999989581377725nteger @ ( F @ X3 ) @ ( F @ Y2 ) ) )
% 0.34/0.61 => ( ord_le6747313008572928689nteger @ A2 @ ( F @ C ) ) ) ) ) ).
% 0.34/0.61
% 0.34/0.61 % order_less_le_subst1
% 0.34/0.61 thf(fact_4000_order__less__le__subst1,axiom,
% 0.34/0.61 ! [A2: rat,F: nat > rat,B: nat,C: nat] :
% 0.34/0.61 ( ( ord_less_rat @ A2 @ ( F @ B ) )
% 0.34/0.61 => ( ( ord_less_eq_nat @ B @ C )
% 0.34/0.61 => ( ! [X3: nat,Y2: nat] :
% 0.34/0.61 ( ( ord_less_eq_nat @ X3 @ Y2 )
% 0.34/0.61 => ( ord_less_eq_rat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
% 0.34/0.61 => ( ord_less_rat @ A2 @ ( F @ C ) ) ) ) ) ).
% 0.34/0.61
% 0.34/0.61 % order_less_le_subst1
% 0.34/0.61 thf(fact_4001_order__less__le__subst1,axiom,
% 0.34/0.61 ! [A2: nat,F: nat > nat,B: nat,C: nat] :
% 0.34/0.61 ( ( ord_less_nat @ A2 @ ( F @ B ) )
% 0.34/0.61 => ( ( ord_less_eq_nat @ B @ C )
% 0.34/0.61 => ( ! [X3: nat,Y2: nat] :
% 0.34/0.61 ( ( ord_less_eq_nat @ X3 @ Y2 )
% 0.34/0.61 => ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
% 0.34/0.61 => ( ord_less_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% 0.34/0.61
% 0.34/0.61 % order_less_le_subst1
% 0.34/0.61 thf(fact_4002_order__less__le__subst1,axiom,
% 0.34/0.61 ! [A2: int,F: nat > int,B: nat,C: nat] :
% 0.34/0.61 ( ( ord_less_int @ A2 @ ( F @ B ) )
% 0.34/0.61 => ( ( ord_less_eq_nat @ B @ C )
% 0.34/0.61 => ( ! [X3: nat,Y2: nat] :
% 0.34/0.61 ( ( ord_less_eq_nat @ X3 @ Y2 )
% 0.34/0.61 => ( ord_less_eq_int @ ( F @ X3 ) @ ( F @ Y2 ) ) )
% 0.34/0.61 => ( ord_less_int @ A2 @ ( F @ C ) ) ) ) ) ).
% 0.34/0.61
% 0.34/0.61 % order_less_le_subst1
% 0.34/0.61 thf(fact_4003_order__less__le__subst2,axiom,
% 0.34/0.61 ! [A2: assn,B: assn,F: assn > assn,C: assn] :
% 0.34/0.61 ( ( ord_less_assn @ A2 @ B )
% 0.34/0.61 => ( ( ord_less_eq_assn @ ( F @ B ) @ C )
% 0.34/0.61 => ( ! [X3: assn,Y2: assn] :
% 0.34/0.61 ( ( ord_less_assn @ X3 @ Y2 )
% 0.34/0.61 => ( ord_less_assn @ ( F @ X3 ) @ ( F @ Y2 ) ) )
% 0.34/0.61 => ( ord_less_assn @ ( F @ A2 ) @ C ) ) ) ) ).
% 0.34/0.61
% 0.34/0.61 % order_less_le_subst2
% 0.34/0.61 thf(fact_4004_order__less__le__subst2,axiom,
% 0.34/0.61 ! [A2: assn,B: assn,F: assn > code_integer,C: code_integer] :
% 0.34/0.61 ( ( ord_less_assn @ A2 @ B )
% 0.34/0.61 => ( ( ord_le3102999989581377725nteger @ ( F @ B ) @ C )
% 0.34/0.61 => ( ! [X3: assn,Y2: assn] :
% 0.34/0.61 ( ( ord_less_assn @ X3 @ Y2 )
% 0.34/0.61 => ( ord_le6747313008572928689nteger @ ( F @ X3 ) @ ( F @ Y2 ) ) )
% 0.34/0.61 => ( ord_le6747313008572928689nteger @ ( F @ A2 ) @ C ) ) ) ) ).
% 0.34/0.61
% 0.34/0.61 % order_less_le_subst2
% 0.34/0.61 thf(fact_4005_order__less__le__subst2,axiom,
% 0.34/0.61 ! [A2: nat,B: nat,F: nat > assn,C: assn] :
% 0.34/0.61 ( ( ord_less_nat @ A2 @ B )
% 0.34/0.61 => ( ( ord_less_eq_assn @ ( F @ B ) @ C )
% 0.34/0.61 => ( ! [X3: nat,Y2: nat] :
% 0.34/0.61 ( ( ord_less_nat @ X3 @ Y2 )
% 0.34/0.61 => ( ord_less_assn @ ( F @ X3 ) @ ( F @ Y2 ) ) )
% 0.34/0.61 => ( ord_less_assn @ ( F @ A2 ) @ C ) ) ) ) ).
% 0.34/0.61
% 0.34/0.61 % order_less_le_subst2
% 0.34/0.61 thf(fact_4006_order__less__le__subst2,axiom,
% 0.34/0.61 ! [A2: nat,B: nat,F: nat > code_integer,C: code_integer] :
% 0.34/0.61 ( ( ord_less_nat @ A2 @ B )
% 0.34/0.61 => ( ( ord_le3102999989581377725nteger @ ( F @ B ) @ C )
% 0.34/0.61 => ( ! [X3: nat,Y2: nat] :
% 0.34/0.61 ( ( ord_less_nat @ X3 @ Y2 )
% 0.34/0.61 => ( ord_le6747313008572928689nteger @ ( F @ X3 ) @ ( F @ Y2 ) ) )
% 0.34/0.61 => ( ord_le6747313008572928689nteger @ ( F @ A2 ) @ C ) ) ) ) ).
% 0.34/0.61
% 0.34/0.61 % order_less_le_subst2
% 0.34/0.61 thf(fact_4007_order__less__le__subst2,axiom,
% 0.34/0.61 ! [A2: int,B: int,F: int > assn,C: assn] :
% 0.34/0.61 ( ( ord_less_int @ A2 @ B )
% 0.34/0.61 => ( ( ord_less_eq_assn @ ( F @ B ) @ C )
% 0.34/0.61 => ( ! [X3: int,Y2: int] :
% 0.34/0.61 ( ( ord_less_int @ X3 @ Y2 )
% 0.34/0.61 => ( ord_less_assn @ ( F @ X3 ) @ ( F @ Y2 ) ) )
% 0.34/0.61 => ( ord_less_assn @ ( F @ A2 ) @ C ) ) ) ) ).
% 0.34/0.61
% 0.34/0.61 % order_less_le_subst2
% 0.34/0.61 thf(fact_4008_order__less__le__subst2,axiom,
% 0.34/0.61 ! [A2: int,B: int,F: int > code_integer,C: code_integer] :
% 0.34/0.61 ( ( ord_less_int @ A2 @ B )
% 0.34/0.61 => ( ( ord_le3102999989581377725nteger @ ( F @ B ) @ C )
% 0.34/0.61 => ( ! [X3: int,Y2: int] :
% 0.34/0.61 ( ( ord_less_int @ X3 @ Y2 )
% 0.34/0.61 => ( ord_le6747313008572928689nteger @ ( F @ X3 ) @ ( F @ Y2 ) ) )
% 0.34/0.61 => ( ord_le6747313008572928689nteger @ ( F @ A2 ) @ C ) ) ) ) ).
% 0.34/0.61
% 0.34/0.61 % order_less_le_subst2
% 0.34/0.61 thf(fact_4009_order__less__le__subst2,axiom,
% 0.34/0.61 ! [A2: code_integer,B: code_integer,F: code_integer > assn,C: assn] :
% 0.34/0.61 ( ( ord_le6747313008572928689nteger @ A2 @ B )
% 0.34/0.61 => ( ( ord_less_eq_assn @ ( F @ B ) @ C )
% 0.34/0.61 => ( ! [X3: code_integer,Y2: code_integer] :
% 0.34/0.61 ( ( ord_le6747313008572928689nteger @ X3 @ Y2 )
% 0.34/0.61 => ( ord_less_assn @ ( F @ X3 ) @ ( F @ Y2 ) ) )
% 0.34/0.61 => ( ord_less_assn @ ( F @ A2 ) @ C ) ) ) ) ).
% 0.34/0.61
% 0.34/0.61 % order_less_le_subst2
% 0.34/0.61 thf(fact_4010_order__less__le__subst2,axiom,
% 0.34/0.61 ! [A2: code_integer,B: code_integer,F: code_integer > code_integer,C: code_integer] :
% 0.34/0.61 ( ( ord_le6747313008572928689nteger @ A2 @ B )
% 0.34/0.61 => ( ( ord_le3102999989581377725nteger @ ( F @ B ) @ C )
% 0.34/0.61 => ( ! [X3: code_integer,Y2: code_integer] :
% 0.34/0.61 ( ( ord_le6747313008572928689nteger @ X3 @ Y2 )
% 0.34/0.61 => ( ord_le6747313008572928689nteger @ ( F @ X3 ) @ ( F @ Y2 ) ) )
% 0.34/0.61 => ( ord_le6747313008572928689nteger @ ( F @ A2 ) @ C ) ) ) ) ).
% 0.34/0.61
% 0.34/0.61 % order_less_le_subst2
% 0.34/0.61 thf(fact_4011_order__less__le__subst2,axiom,
% 0.34/0.61 ! [A2: rat,B: rat,F: rat > assn,C: assn] :
% 0.34/0.61 ( ( ord_less_rat @ A2 @ B )
% 0.34/0.61 => ( ( ord_less_eq_assn @ ( F @ B ) @ C )
% 0.34/0.61 => ( ! [X3: rat,Y2: rat] :
% 0.34/0.61 ( ( ord_less_rat @ X3 @ Y2 )
% 0.34/0.61 => ( ord_less_assn @ ( F @ X3 ) @ ( F @ Y2 ) ) )
% 0.34/0.61 => ( ord_less_assn @ ( F @ A2 ) @ C ) ) ) ) ).
% 0.34/0.61
% 0.34/0.61 % order_less_le_subst2
% 0.34/0.61 thf(fact_4012_order__less__le__subst2,axiom,
% 0.34/0.61 ! [A2: rat,B: rat,F: rat > code_integer,C: code_integer] :
% 0.34/0.61 ( ( ord_less_rat @ A2 @ B )
% 0.34/0.61 => ( ( ord_le3102999989581377725nteger @ ( F @ B ) @ C )
% 0.34/0.61 => ( ! [X3: rat,Y2: rat] :
% 0.34/0.61 ( ( ord_less_rat @ X3 @ Y2 )
% 0.34/0.61 => ( ord_le6747313008572928689nteger @ ( F @ X3 ) @ ( F @ Y2 ) ) )
% 0.34/0.61 => ( ord_le6747313008572928689nteger @ ( F @ A2 ) @ C ) ) ) ) ).
% 0.34/0.61
% 0.34/0.61 % order_less_le_subst2
% 0.34/0.61 thf(fact_4013_linorder__le__less__linear,axiom,
% 0.34/0.61 ! [X: code_integer,Y: code_integer] :
% 0.34/0.61 ( ( ord_le3102999989581377725nteger @ X @ Y )
% 0.34/0.61 | ( ord_le6747313008572928689nteger @ Y @ X ) ) ).
% 0.34/0.61
% 0.34/0.61 % linorder_le_less_linear
% 0.34/0.61 thf(fact_4014_linorder__le__less__linear,axiom,
% 0.34/0.61 ! [X: rat,Y: rat] :
% 0.34/0.61 ( ( ord_less_eq_rat @ X @ Y )
% 0.34/0.61 | ( ord_less_rat @ Y @ X ) ) ).
% 0.34/0.61
% 0.34/0.61 % linorder_le_less_linear
% 0.34/0.61 thf(fact_4015_linorder__le__less__linear,axiom,
% 0.34/0.61 ! [X: nat,Y: nat] :
% 0.34/0.61 ( ( ord_less_eq_nat @ X @ Y )
% 0.34/0.61 | ( ord_less_nat @ Y @ X ) ) ).
% 0.34/0.61
% 0.34/0.61 % linorder_le_less_linear
% 0.34/0.61 thf(fact_4016_linorder__le__less__linear,axiom,
% 0.34/0.61 ! [X: int,Y: int] :
% 0.34/0.61 ( ( ord_less_eq_int @ X @ Y )
% 0.34/0.61 | ( ord_less_int @ Y @ X ) ) ).
% 0.34/0.61
% 0.34/0.61 % linorder_le_less_linear
% 0.34/0.61 thf(fact_4017_order__le__imp__less__or__eq,axiom,
% 0.34/0.61 ! [X: assn,Y: assn] :
% 0.34/0.61 ( ( ord_less_eq_assn @ X @ Y )
% 0.34/0.61 => ( ( ord_less_assn @ X @ Y )
% 0.34/0.61 | ( X = Y ) ) ) ).
% 0.34/0.61
% 0.34/0.61 % order_le_imp_less_or_eq
% 0.34/0.61 thf(fact_4018_order__le__imp__less__or__eq,axiom,
% 0.34/0.61 ! [X: code_integer,Y: code_integer] :
% 0.34/0.61 ( ( ord_le3102999989581377725nteger @ X @ Y )
% 0.34/0.61 => ( ( ord_le6747313008572928689nteger @ X @ Y )
% 0.34/0.61 | ( X = Y ) ) ) ).
% 0.34/0.61
% 0.34/0.61 % order_le_imp_less_or_eq
% 0.34/0.61 thf(fact_4019_order__le__imp__less__or__eq,axiom,
% 0.34/0.61 ! [X: filter_nat,Y: filter_nat] :
% 0.34/0.61 ( ( ord_le2510731241096832064er_nat @ X @ Y )
% 0.34/0.61 => ( ( ord_less_filter_nat @ X @ Y )
% 0.34/0.61 | ( X = Y ) ) ) ).
% 0.34/0.61
% 0.34/0.61 % order_le_imp_less_or_eq
% 0.34/0.61 thf(fact_4020_order__le__imp__less__or__eq,axiom,
% 0.34/0.61 ! [X: set_nat,Y: set_nat] :
% 0.34/0.61 ( ( ord_less_eq_set_nat @ X @ Y )
% 0.34/0.61 => ( ( ord_less_set_nat @ X @ Y )
% 0.34/0.61 | ( X = Y ) ) ) ).
% 0.34/0.61
% 0.34/0.61 % order_le_imp_less_or_eq
% 0.34/0.61 thf(fact_4021_order__le__imp__less__or__eq,axiom,
% 0.34/0.61 ! [X: rat,Y: rat] :
% 0.34/0.61 ( ( ord_less_eq_rat @ X @ Y )
% 0.34/0.61 => ( ( ord_less_rat @ X @ Y )
% 0.34/0.61 | ( X = Y ) ) ) ).
% 0.34/0.61
% 0.34/0.61 % order_le_imp_less_or_eq
% 0.34/0.61 thf(fact_4022_order__le__imp__less__or__eq,axiom,
% 0.34/0.61 ! [X: nat,Y: nat] :
% 0.34/0.61 ( ( ord_less_eq_nat @ X @ Y )
% 0.34/0.61 => ( ( ord_less_nat @ X @ Y )
% 0.34/0.61 | ( X = Y ) ) ) ).
% 0.34/0.61
% 0.34/0.61 % order_le_imp_less_or_eq
% 0.34/0.61 thf(fact_4023_order__le__imp__less__or__eq,axiom,
% 0.34/0.61 ! [X: int,Y: int] :
% 0.34/0.61 ( ( ord_less_eq_int @ X @ Y )
% 0.34/0.61 => ( ( ord_less_int @ X @ Y )
% 0.34/0.61 | ( X = Y ) ) ) ).
% 0.34/0.61
% 0.34/0.61 % order_le_imp_less_or_eq
% 0.34/0.61 thf(fact_4024_less__numeral__extra_I4_J,axiom,
% 0.34/0.61 ~ ( ord_le5570908160329646204atural @ one_one_Code_natural @ one_one_Code_natural ) ).
% 0.34/0.61
% 0.34/0.61 % less_numeral_extra(4)
% 0.34/0.61 thf(fact_4025_less__numeral__extra_I4_J,axiom,
% 0.34/0.61 ~ ( ord_less_nat @ one_one_nat @ one_one_nat ) ).
% 0.34/0.61
% 0.34/0.61 % less_numeral_extra(4)
% 0.34/0.61 thf(fact_4026_less__numeral__extra_I4_J,axiom,
% 0.34/0.61 ~ ( ord_less_int @ one_one_int @ one_one_int ) ).
% 0.34/0.61
% 0.34/0.61 % less_numeral_extra(4)
% 0.34/0.61 thf(fact_4027_less__numeral__extra_I4_J,axiom,
% 0.34/0.61 ~ ( ord_le6747313008572928689nteger @ one_one_Code_integer @ one_one_Code_integer ) ).
% 0.34/0.61
% 0.34/0.61 % less_numeral_extra(4)
% 0.34/0.61 thf(fact_4028_less__numeral__extra_I4_J,axiom,
% 0.34/0.61 ~ ( ord_less_rat @ one_one_rat @ one_one_rat ) ).
% 0.34/0.61
% 0.34/0.61 % less_numeral_extra(4)
% 0.34/0.61 thf(fact_4029_diff__strict__mono,axiom,
% 0.34/0.61 ! [A2: int,B: int,D2: int,C: int] :
% 0.34/0.61 ( ( ord_less_int @ A2 @ B )
% 0.34/0.61 => ( ( ord_less_int @ D2 @ C )
% 0.34/0.61 => ( ord_less_int @ ( minus_minus_int @ A2 @ C ) @ ( minus_minus_int @ B @ D2 ) ) ) ) ).
% 0.34/0.61
% 0.34/0.61 % diff_strict_mono
% 0.34/0.61 thf(fact_4030_diff__strict__mono,axiom,
% 0.34/0.61 ! [A2: code_integer,B: code_integer,D2: code_integer,C: code_integer] :
% 0.34/0.61 ( ( ord_le6747313008572928689nteger @ A2 @ B )
% 0.34/0.61 => ( ( ord_le6747313008572928689nteger @ D2 @ C )
% 0.34/0.61 => ( ord_le6747313008572928689nteger @ ( minus_8373710615458151222nteger @ A2 @ C ) @ ( minus_8373710615458151222nteger @ B @ D2 ) ) ) ) ).
% 0.34/0.61
% 0.34/0.61 % diff_strict_mono
% 0.34/0.61 thf(fact_4031_diff__strict__mono,axiom,
% 0.34/0.61 ! [A2: rat,B: rat,D2: rat,C: rat] :
% 0.34/0.61 ( ( ord_less_rat @ A2 @ B )
% 0.34/0.61 => ( ( ord_less_rat @ D2 @ C )
% 0.34/0.61 => ( ord_less_rat @ ( minus_minus_rat @ A2 @ C ) @ ( minus_minus_rat @ B @ D2 ) ) ) ) ).
% 0.34/0.61
% 0.34/0.61 % diff_strict_mono
% 0.34/0.61 thf(fact_4032_diff__eq__diff__less,axiom,
% 0.34/0.61 ! [A2: int,B: int,C: int,D2: int] :
% 0.34/0.61 ( ( ( minus_minus_int @ A2 @ B )
% 0.34/0.61 = ( minus_minus_int @ C @ D2 ) )
% 0.34/0.61 => ( ( ord_less_int @ A2 @ B )
% 0.34/0.61 = ( ord_less_int @ C @ D2 ) ) ) ).
% 0.34/0.61
% 0.34/0.61 % diff_eq_diff_less
% 0.34/0.61 thf(fact_4033_diff__eq__diff__less,axiom,
% 0.34/0.61 ! [A2: code_integer,B: code_integer,C: code_integer,D2: code_integer] :
% 0.34/0.61 ( ( ( minus_8373710615458151222nteger @ A2 @ B )
% 0.34/0.61 = ( minus_8373710615458151222nteger @ C @ D2 ) )
% 0.34/0.61 => ( ( ord_le6747313008572928689nteger @ A2 @ B )
% 0.34/0.61 = ( ord_le6747313008572928689nteger @ C @ D2 ) ) ) ).
% 0.34/0.61
% 0.34/0.61 % diff_eq_diff_less
% 0.34/0.61 thf(fact_4034_diff__eq__diff__less,axiom,
% 0.34/0.61 ! [A2: rat,B: rat,C: rat,D2: rat] :
% 0.34/0.61 ( ( ( minus_minus_rat @ A2 @ B )
% 0.34/0.61 = ( minus_minus_rat @ C @ D2 ) )
% 0.34/0.61 => ( ( ord_less_rat @ A2 @ B )
% 0.34/0.61 = ( ord_less_rat @ C @ D2 ) ) ) ).
% 0.34/0.61
% 0.34/0.61 % diff_eq_diff_less
% 0.34/0.61 thf(fact_4035_diff__strict__left__mono,axiom,
% 0.34/0.61 ! [B: int,A2: int,C: int] :
% 0.34/0.61 ( ( ord_less_int @ B @ A2 )
% 0.34/0.61 => ( ord_less_int @ ( minus_minus_int @ C @ A2 ) @ ( minus_minus_int @ C @ B ) ) ) ).
% 0.34/0.61
% 0.34/0.61 % diff_strict_left_mono
% 0.34/0.61 thf(fact_4036_diff__strict__left__mono,axiom,
% 0.34/0.61 ! [B: code_integer,A2: code_integer,C: code_integer] :
% 0.34/0.61 ( ( ord_le6747313008572928689nteger @ B @ A2 )
% 0.34/0.61 => ( ord_le6747313008572928689nteger @ ( minus_8373710615458151222nteger @ C @ A2 ) @ ( minus_8373710615458151222nteger @ C @ B ) ) ) ).
% 0.34/0.61
% 0.34/0.61 % diff_strict_left_mono
% 0.34/0.61 thf(fact_4037_diff__strict__left__mono,axiom,
% 0.34/0.61 ! [B: rat,A2: rat,C: rat] :
% 0.34/0.61 ( ( ord_less_rat @ B @ A2 )
% 0.34/0.61 => ( ord_less_rat @ ( minus_minus_rat @ C @ A2 ) @ ( minus_minus_rat @ C @ B ) ) ) ).
% 0.34/0.61
% 0.34/0.61 % diff_strict_left_mono
% 0.34/0.61 thf(fact_4038_diff__strict__right__mono,axiom,
% 0.34/0.61 ! [A2: int,B: int,C: int] :
% 0.34/0.61 ( ( ord_less_int @ A2 @ B )
% 0.34/0.61 => ( ord_less_int @ ( minus_minus_int @ A2 @ C ) @ ( minus_minus_int @ B @ C ) ) ) ).
% 0.34/0.61
% 0.34/0.61 % diff_strict_right_mono
% 0.34/0.61 thf(fact_4039_diff__strict__right__mono,axiom,
% 0.34/0.61 ! [A2: code_integer,B: code_integer,C: code_integer] :
% 0.34/0.61 ( ( ord_le6747313008572928689nteger @ A2 @ B )
% 0.34/0.61 => ( ord_le6747313008572928689nteger @ ( minus_8373710615458151222nteger @ A2 @ C ) @ ( minus_8373710615458151222nteger @ B @ C ) ) ) ).
% 0.34/0.61
% 0.34/0.61 % diff_strict_right_mono
% 0.34/0.61 thf(fact_4040_diff__strict__right__mono,axiom,
% 0.34/0.61 ! [A2: rat,B: rat,C: rat] :
% 0.34/0.61 ( ( ord_less_rat @ A2 @ B )
% 0.34/0.61 => ( ord_less_rat @ ( minus_minus_rat @ A2 @ C ) @ ( minus_minus_rat @ B @ C ) ) ) ).
% 0.34/0.61
% 0.34/0.61 % diff_strict_right_mono
% 0.34/0.61 thf(fact_4041_bot_Oextremum__strict,axiom,
% 0.34/0.61 ! [A2: set_Product_unit] :
% 0.34/0.61 ~ ( ord_le8056459307392131481t_unit @ A2 @ bot_bo3957492148770167129t_unit ) ).
% 0.34/0.61
% 0.34/0.61 % bot.extremum_strict
% 0.34/0.61 thf(fact_4042_bot_Oextremum__strict,axiom,
% 0.34/0.61 ! [A2: set_o] :
% 0.34/0.61 ~ ( ord_less_set_o @ A2 @ bot_bot_set_o ) ).
% 0.34/0.61
% 0.34/0.61 % bot.extremum_strict
% 0.34/0.61 thf(fact_4043_bot_Oextremum__strict,axiom,
% 0.34/0.61 ! [A2: set_nat] :
% 0.34/0.61 ~ ( ord_less_set_nat @ A2 @ bot_bot_set_nat ) ).
% 0.34/0.61
% 0.34/0.61 % bot.extremum_strict
% 0.34/0.61 thf(fact_4044_bot_Oextremum__strict,axiom,
% 0.34/0.61 ! [A2: set_int] :
% 0.34/0.61 ~ ( ord_less_set_int @ A2 @ bot_bot_set_int ) ).
% 0.34/0.61
% 0.34/0.61 % bot.extremum_strict
% 0.34/0.61 thf(fact_4045_bot_Oextremum__strict,axiom,
% 0.34/0.61 ! [A2: set_Pr4532377907799695533_nat_o] :
% 0.34/0.61 ~ ( ord_le2453136405763929_nat_o @ A2 @ bot_bo7824918357723582233_nat_o ) ).
% 0.34/0.61
% 0.34/0.61 % bot.extremum_strict
% 0.34/0.61 thf(fact_4046_bot_Oextremum__strict,axiom,
% 0.34/0.61 ! [A2: assn] :
% 0.34/0.61 ~ ( ord_less_assn @ A2 @ bot_bot_assn ) ).
% 0.34/0.61
% 0.34/0.61 % bot.extremum_strict
% 0.34/0.61 thf(fact_4047_bot_Oextremum__strict,axiom,
% 0.34/0.61 ! [A2: nat] :
% 0.34/0.61 ~ ( ord_less_nat @ A2 @ bot_bot_nat ) ).
% 0.34/0.61
% 0.34/0.61 % bot.extremum_strict
% 0.34/0.61 thf(fact_4048_bot_Onot__eq__extremum,axiom,
% 0.34/0.61 ! [A2: set_Product_unit] :
% 0.34/0.61 ( ( A2 != bot_bo3957492148770167129t_unit )
% 0.34/0.61 = ( ord_le8056459307392131481t_unit @ bot_bo3957492148770167129t_unit @ A2 ) ) ).
% 0.34/0.61
% 0.34/0.61 % bot.not_eq_extremum
% 0.34/0.61 thf(fact_4049_bot_Onot__eq__extremum,axiom,
% 0.34/0.61 ! [A2: set_o] :
% 0.34/0.61 ( ( A2 != bot_bot_set_o )
% 0.34/0.61 = ( ord_less_set_o @ bot_bot_set_o @ A2 ) ) ).
% 0.34/0.61
% 0.34/0.61 % bot.not_eq_extremum
% 0.34/0.61 thf(fact_4050_bot_Onot__eq__extremum,axiom,
% 0.34/0.61 ! [A2: set_nat] :
% 0.34/0.61 ( ( A2 != bot_bot_set_nat )
% 0.34/0.61 = ( ord_less_set_nat @ bot_bot_set_nat @ A2 ) ) ).
% 0.34/0.61
% 0.34/0.61 % bot.not_eq_extremum
% 0.34/0.61 thf(fact_4051_bot_Onot__eq__extremum,axiom,
% 0.34/0.61 ! [A2: set_int] :
% 0.34/0.61 ( ( A2 != bot_bot_set_int )
% 0.34/0.61 = ( ord_less_set_int @ bot_bot_set_int @ A2 ) ) ).
% 0.34/0.61
% 0.34/0.61 % bot.not_eq_extremum
% 0.34/0.61 thf(fact_4052_bot_Onot__eq__extremum,axiom,
% 0.34/0.61 ! [A2: set_Pr4532377907799695533_nat_o] :
% 0.34/0.61 ( ( A2 != bot_bo7824918357723582233_nat_o )
% 0.34/0.61 = ( ord_le2453136405763929_nat_o @ bot_bo7824918357723582233_nat_o @ A2 ) ) ).
% 0.34/0.61
% 0.34/0.61 % bot.not_eq_extremum
% 0.34/0.61 thf(fact_4053_bot_Onot__eq__extremum,axiom,
% 0.34/0.61 ! [A2: assn] :
% 0.34/0.61 ( ( A2 != bot_bot_assn )
% 0.34/0.61 = ( ord_less_assn @ bot_bot_assn @ A2 ) ) ).
% 0.34/0.61
% 0.34/0.61 % bot.not_eq_extremum
% 0.34/0.61 thf(fact_4054_bot_Onot__eq__extremum,axiom,
% 0.34/0.61 ! [A2: nat] :
% 0.34/0.61 ( ( A2 != bot_bot_nat )
% 0.34/0.61 = ( ord_less_nat @ bot_bot_nat @ A2 ) ) ).
% 0.34/0.61
% 0.34/0.61 % bot.not_eq_extremum
% 0.34/0.61 thf(fact_4055_not__psubset__empty,axiom,
% 0.34/0.61 ! [A: set_Product_unit] :
% 0.34/0.61 ~ ( ord_le8056459307392131481t_unit @ A @ bot_bo3957492148770167129t_unit ) ).
% 0.34/0.61
% 0.34/0.61 % not_psubset_empty
% 0.34/0.61 thf(fact_4056_not__psubset__empty,axiom,
% 0.34/0.61 ! [A: set_o] :
% 0.34/0.61 ~ ( ord_less_set_o @ A @ bot_bot_set_o ) ).
% 0.34/0.61
% 0.34/0.61 % not_psubset_empty
% 0.34/0.61 thf(fact_4057_not__psubset__empty,axiom,
% 0.34/0.61 ! [A: set_nat] :
% 0.34/0.61 ~ ( ord_less_set_nat @ A @ bot_bot_set_nat ) ).
% 0.34/0.61
% 0.34/0.61 % not_psubset_empty
% 0.34/0.61 thf(fact_4058_not__psubset__empty,axiom,
% 0.34/0.61 ! [A: set_int] :
% 0.34/0.61 ~ ( ord_less_set_int @ A @ bot_bot_set_int ) ).
% 0.34/0.61
% 0.34/0.61 % not_psubset_empty
% 0.34/0.61 thf(fact_4059_not__psubset__empty,axiom,
% 0.34/0.61 ! [A: set_Pr4532377907799695533_nat_o] :
% 0.34/0.61 ~ ( ord_le2453136405763929_nat_o @ A @ bot_bo7824918357723582233_nat_o ) ).
% 0.34/0.61
% 0.34/0.61 % not_psubset_empty
% 0.34/0.61 thf(fact_4060_psubsetE,axiom,
% 0.34/0.61 ! [A: set_nat,B4: set_nat] :
% 0.34/0.61 ( ( ord_less_set_nat @ A @ B4 )
% 0.34/0.61 => ~ ( ( ord_less_eq_set_nat @ A @ B4 )
% 0.34/0.61 => ( ord_less_eq_set_nat @ B4 @ A ) ) ) ).
% 0.34/0.61
% 0.34/0.61 % psubsetE
% 0.34/0.61 thf(fact_4061_psubset__eq,axiom,
% 0.34/0.61 ( ord_less_set_nat
% 0.34/0.61 = ( ^ [A3: set_nat,B7: set_nat] :
% 0.34/0.61 ( ( ord_less_eq_set_nat @ A3 @ B7 )
% 0.34/0.61 & ( A3 != B7 ) ) ) ) ).
% 0.34/0.61
% 0.34/0.61 % psubset_eq
% 0.34/0.61 thf(fact_4062_psubset__imp__subset,axiom,
% 0.34/0.61 ! [A: set_nat,B4: set_nat] :
% 0.34/0.61 ( ( ord_less_set_nat @ A @ B4 )
% 0.34/0.61 => ( ord_less_eq_set_nat @ A @ B4 ) ) ).
% 0.34/0.61
% 0.34/0.61 % psubset_imp_subset
% 0.34/0.61 thf(fact_4063_psubset__subset__trans,axiom,
% 0.34/0.61 ! [A: set_nat,B4: set_nat,C5: set_nat] :
% 0.34/0.61 ( ( ord_less_set_nat @ A @ B4 )
% 0.34/0.61 => ( ( ord_less_eq_set_nat @ B4 @ C5 )
% 0.34/0.61 => ( ord_less_set_nat @ A @ C5 ) ) ) ).
% 0.34/0.61
% 0.34/0.61 % psubset_subset_trans
% 0.34/0.61 thf(fact_4064_subset__not__subset__eq,axiom,
% 0.34/0.61 ( ord_less_set_nat
% 0.34/0.61 = ( ^ [A3: set_nat,B7: set_nat] :
% 0.34/0.61 ( ( ord_less_eq_set_nat @ A3 @ B7 )
% 0.34/0.61 & ~ ( ord_less_eq_set_nat @ B7 @ A3 ) ) ) ) ).
% 0.34/0.61
% 0.34/0.61 % subset_not_subset_eq
% 0.34/0.61 thf(fact_4065_subset__psubset__trans,axiom,
% 0.34/0.61 ! [A: set_nat,B4: set_nat,C5: set_nat] :
% 0.34/0.61 ( ( ord_less_eq_set_nat @ A @ B4 )
% 0.34/0.61 => ( ( ord_less_set_nat @ B4 @ C5 )
% 0.34/0.61 => ( ord_less_set_nat @ A @ C5 ) ) ) ).
% 0.34/0.61
% 0.34/0.61 % subset_psubset_trans
% 0.34/0.61 thf(fact_4066_subset__iff__psubset__eq,axiom,
% 0.34/0.61 ( ord_less_eq_set_nat
% 0.34/0.61 = ( ^ [A3: set_nat,B7: set_nat] :
% 0.34/0.61 ( ( ord_less_set_nat @ A3 @ B7 )
% 0.34/0.61 | ( A3 = B7 ) ) ) ) ).
% 0.34/0.61
% 0.34/0.61 % subset_iff_psubset_eq
% 0.34/0.61 thf(fact_4067_rangeI,axiom,
% 0.34/0.61 ! [F: code_integer > code_integer,X: code_integer] : ( member_Code_integer @ ( F @ X ) @ ( image_4470545334726330049nteger @ F @ top_to4645266643341252675nteger ) ) ).
% 0.34/0.61
% 0.34/0.61 % rangeI
% 0.34/0.61 thf(fact_4068_rangeI,axiom,
% 0.34/0.61 ! [F: product_unit > nat,X: product_unit] : ( member_nat @ ( F @ X ) @ ( image_875570014554754200it_nat @ F @ top_to1996260823553986621t_unit ) ) ).
% 0.34/0.61
% 0.34/0.61 % rangeI
% 0.34/0.61 thf(fact_4069_rangeI,axiom,
% 0.34/0.61 ! [F: product_unit > int,X: product_unit] : ( member_int @ ( F @ X ) @ ( image_873079544045703924it_int @ F @ top_to1996260823553986621t_unit ) ) ).
% 0.34/0.61
% 0.34/0.61 % rangeI
% 0.34/0.61 thf(fact_4070_rangeI,axiom,
% 0.34/0.61 ! [F: product_unit > $o,X: product_unit] : ( member_o @ ( F @ X ) @ ( image_Product_unit_o2 @ F @ top_to1996260823553986621t_unit ) ) ).
% 0.34/0.61
% 0.34/0.61 % rangeI
% 0.34/0.61 thf(fact_4071_rangeI,axiom,
% 0.34/0.61 ! [F: product_unit > product_unit,X: product_unit] : ( member_Product_unit @ ( F @ X ) @ ( image_405062704495631173t_unit @ F @ top_to1996260823553986621t_unit ) ) ).
% 0.34/0.61
% 0.34/0.61 % rangeI
% 0.34/0.61 thf(fact_4072_rangeI,axiom,
% 0.34/0.61 ! [F: $o > nat,X: $o] : ( member_nat @ ( F @ X ) @ ( image_o_nat2 @ F @ top_top_set_o ) ) ).
% 0.34/0.61
% 0.34/0.61 % rangeI
% 0.34/0.61 thf(fact_4073_rangeI,axiom,
% 0.34/0.61 ! [F: $o > int,X: $o] : ( member_int @ ( F @ X ) @ ( image_o_int2 @ F @ top_top_set_o ) ) ).
% 0.34/0.61
% 0.34/0.61 % rangeI
% 0.34/0.61 thf(fact_4074_rangeI,axiom,
% 0.34/0.61 ! [F: $o > $o,X: $o] : ( member_o @ ( F @ X ) @ ( image_o_o2 @ F @ top_top_set_o ) ) ).
% 0.34/0.61
% 0.34/0.61 % rangeI
% 0.34/0.61 thf(fact_4075_rangeI,axiom,
% 0.34/0.61 ! [F: $o > product_unit,X: $o] : ( member_Product_unit @ ( F @ X ) @ ( image_o_Product_unit2 @ F @ top_top_set_o ) ) ).
% 0.34/0.61
% 0.34/0.61 % rangeI
% 0.34/0.61 thf(fact_4076_rangeI,axiom,
% 0.34/0.61 ! [F: nat > code_integer,X: nat] : ( member_Code_integer @ ( F @ X ) @ ( image_1215581382706833972nteger @ F @ top_top_set_nat ) ) ).
% 0.34/0.61
% 0.34/0.61 % rangeI
% 0.34/0.61 thf(fact_4077_range__eqI,axiom,
% 0.34/0.61 ! [B: code_integer,F: code_integer > code_integer,X: code_integer] :
% 0.34/0.61 ( ( B
% 0.34/0.61 = ( F @ X ) )
% 0.34/0.61 => ( member_Code_integer @ B @ ( image_4470545334726330049nteger @ F @ top_to4645266643341252675nteger ) ) ) ).
% 0.34/0.61
% 0.34/0.61 % range_eqI
% 0.34/0.61 thf(fact_4078_range__eqI,axiom,
% 0.34/0.61 ! [B: nat,F: product_unit > nat,X: product_unit] :
% 0.34/0.61 ( ( B
% 0.34/0.61 = ( F @ X ) )
% 0.34/0.61 => ( member_nat @ B @ ( image_875570014554754200it_nat @ F @ top_to1996260823553986621t_unit ) ) ) ).
% 0.34/0.61
% 0.34/0.61 % range_eqI
% 0.34/0.61 thf(fact_4079_range__eqI,axiom,
% 0.34/0.61 ! [B: int,F: product_unit > int,X: product_unit] :
% 0.34/0.61 ( ( B
% 0.34/0.61 = ( F @ X ) )
% 0.34/0.61 => ( member_int @ B @ ( image_873079544045703924it_int @ F @ top_to1996260823553986621t_unit ) ) ) ).
% 0.34/0.61
% 0.34/0.61 % range_eqI
% 0.34/0.61 thf(fact_4080_range__eqI,axiom,
% 0.34/0.61 ! [B: $o,F: product_unit > $o,X: product_unit] :
% 0.34/0.61 ( ( B
% 0.34/0.61 = ( F @ X ) )
% 0.34/0.61 => ( member_o @ B @ ( image_Product_unit_o2 @ F @ top_to1996260823553986621t_unit ) ) ) ).
% 0.34/0.61
% 0.34/0.61 % range_eqI
% 0.34/0.61 thf(fact_4081_range__eqI,axiom,
% 0.34/0.61 ! [B: product_unit,F: product_unit > product_unit,X: product_unit] :
% 0.34/0.61 ( ( B
% 0.34/0.61 = ( F @ X ) )
% 0.34/0.61 => ( member_Product_unit @ B @ ( image_405062704495631173t_unit @ F @ top_to1996260823553986621t_unit ) ) ) ).
% 0.34/0.61
% 0.34/0.61 % range_eqI
% 0.34/0.61 thf(fact_4082_range__eqI,axiom,
% 0.34/0.61 ! [B: nat,F: $o > nat,X: $o] :
% 0.34/0.61 ( ( B
% 0.34/0.61 = ( F @ X ) )
% 0.34/0.61 => ( member_nat @ B @ ( image_o_nat2 @ F @ top_top_set_o ) ) ) ).
% 0.34/0.61
% 0.34/0.61 % range_eqI
% 0.34/0.61 thf(fact_4083_range__eqI,axiom,
% 0.34/0.61 ! [B: int,F: $o > int,X: $o] :
% 0.34/0.61 ( ( B
% 0.34/0.61 = ( F @ X ) )
% 0.34/0.61 => ( member_int @ B @ ( image_o_int2 @ F @ top_top_set_o ) ) ) ).
% 0.34/0.61
% 0.34/0.61 % range_eqI
% 0.34/0.61 thf(fact_4084_range__eqI,axiom,
% 0.34/0.61 ! [B: $o,F: $o > $o,X: $o] :
% 0.34/0.61 ( ( B
% 0.34/0.61 = ( F @ X ) )
% 0.34/0.61 => ( member_o @ B @ ( image_o_o2 @ F @ top_top_set_o ) ) ) ).
% 0.34/0.61
% 0.34/0.61 % range_eqI
% 0.34/0.61 thf(fact_4085_range__eqI,axiom,
% 0.34/0.61 ! [B: product_unit,F: $o > product_unit,X: $o] :
% 0.34/0.61 ( ( B
% 0.34/0.61 = ( F @ X ) )
% 0.34/0.61 => ( member_Product_unit @ B @ ( image_o_Product_unit2 @ F @ top_top_set_o ) ) ) ).
% 0.34/0.61
% 0.34/0.61 % range_eqI
% 0.34/0.61 thf(fact_4086_range__eqI,axiom,
% 0.34/0.61 ! [B: code_integer,F: nat > code_integer,X: nat] :
% 0.34/0.61 ( ( B
% 0.34/0.61 = ( F @ X ) )
% 0.34/0.61 => ( member_Code_integer @ B @ ( image_1215581382706833972nteger @ F @ top_top_set_nat ) ) ) ).
% 0.34/0.61
% 0.34/0.61 % range_eqI
% 0.34/0.61 thf(fact_4087_empty__not__UNIV,axiom,
% 0.34/0.61 bot_bo7824918357723582233_nat_o != top_to4869887016220417533_nat_o ).
% 0.34/0.61
% 0.34/0.61 % empty_not_UNIV
% 0.34/0.61 thf(fact_4088_empty__not__UNIV,axiom,
% 0.34/0.61 bot_bo3957492148770167129t_unit != top_to1996260823553986621t_unit ).
% 0.34/0.61
% 0.34/0.61 % empty_not_UNIV
% 0.34/0.61 thf(fact_4089_empty__not__UNIV,axiom,
% 0.34/0.61 bot_bot_set_o != top_top_set_o ).
% 0.34/0.61
% 0.34/0.61 % empty_not_UNIV
% 0.34/0.61 thf(fact_4090_empty__not__UNIV,axiom,
% 0.34/0.61 bot_bot_set_nat != top_top_set_nat ).
% 0.34/0.61
% 0.34/0.61 % empty_not_UNIV
% 0.34/0.61 thf(fact_4091_empty__not__UNIV,axiom,
% 0.34/0.61 bot_bot_set_int != top_top_set_int ).
% 0.34/0.61
% 0.34/0.61 % empty_not_UNIV
% 0.34/0.61 thf(fact_4092_subset__UNIV,axiom,
% 0.34/0.61 ! [A: set_Product_unit] : ( ord_le3507040750410214029t_unit @ A @ top_to1996260823553986621t_unit ) ).
% 0.34/0.61
% 0.34/0.61 % subset_UNIV
% 0.34/0.61 thf(fact_4093_subset__UNIV,axiom,
% 0.34/0.61 ! [A: set_o] : ( ord_less_eq_set_o @ A @ top_top_set_o ) ).
% 0.34/0.61
% 0.34/0.61 % subset_UNIV
% 0.34/0.61 thf(fact_4094_subset__UNIV,axiom,
% 0.34/0.61 ! [A: set_int] : ( ord_less_eq_set_int @ A @ top_top_set_int ) ).
% 0.34/0.61
% 0.34/0.61 % subset_UNIV
% 0.34/0.61 thf(fact_4095_subset__UNIV,axiom,
% 0.34/0.61 ! [A: set_nat] : ( ord_less_eq_set_nat @ A @ top_top_set_nat ) ).
% 0.34/0.61
% 0.34/0.61 % subset_UNIV
% 0.34/0.61 thf(fact_4096_psubset__imp__ex__mem,axiom,
% 0.34/0.61 ! [A: set_Pr4532377907799695533_nat_o,B4: set_Pr4532377907799695533_nat_o] :
% 0.34/0.61 ( ( ord_le2453136405763929_nat_o @ A @ B4 )
% 0.34/0.61 => ? [B2: produc3658429121746597890et_nat > $o] : ( member6576561426505652726_nat_o @ B2 @ ( minus_1801376950450012436_nat_o @ B4 @ A ) ) ) ).
% 0.34/0.61
% 0.34/0.61 % psubset_imp_ex_mem
% 0.34/0.61 thf(fact_4097_psubset__imp__ex__mem,axiom,
% 0.34/0.61 ! [A: set_int,B4: set_int] :
% 0.34/0.61 ( ( ord_less_set_int @ A @ B4 )
% 0.34/0.61 => ? [B2: int] : ( member_int @ B2 @ ( minus_minus_set_int @ B4 @ A ) ) ) ).
% 0.34/0.61
% 0.34/0.61 % psubset_imp_ex_mem
% 0.34/0.61 thf(fact_4098_psubset__imp__ex__mem,axiom,
% 0.34/0.61 ! [A: set_o,B4: set_o] :
% 0.34/0.61 ( ( ord_less_set_o @ A @ B4 )
% 0.34/0.61 => ? [B2: $o] : ( member_o @ B2 @ ( minus_minus_set_o @ B4 @ A ) ) ) ).
% 0.34/0.61
% 0.34/0.61 % psubset_imp_ex_mem
% 0.34/0.61 thf(fact_4099_psubset__imp__ex__mem,axiom,
% 0.34/0.61 ! [A: set_Product_unit,B4: set_Product_unit] :
% 0.34/0.61 ( ( ord_le8056459307392131481t_unit @ A @ B4 )
% 0.34/0.61 => ? [B2: product_unit] : ( member_Product_unit @ B2 @ ( minus_6452836326544984404t_unit @ B4 @ A ) ) ) ).
% 0.34/0.61
% 0.34/0.61 % psubset_imp_ex_mem
% 0.34/0.61 thf(fact_4100_psubset__imp__ex__mem,axiom,
% 0.34/0.61 ! [A: set_nat,B4: set_nat] :
% 0.34/0.61 ( ( ord_less_set_nat @ A @ B4 )
% 0.34/0.61 => ? [B2: nat] : ( member_nat @ B2 @ ( minus_minus_set_nat @ B4 @ A ) ) ) ).
% 0.34/0.61
% 0.34/0.61 % psubset_imp_ex_mem
% 0.34/0.61 thf(fact_4101_insert__UNIV,axiom,
% 0.34/0.61 ! [X: product_unit] :
% 0.34/0.61 ( ( insert_Product_unit @ X @ top_to1996260823553986621t_unit )
% 0.34/0.61 = top_to1996260823553986621t_unit ) ).
% 0.34/0.61
% 0.34/0.61 % insert_UNIV
% 0.34/0.61 thf(fact_4102_insert__UNIV,axiom,
% 0.34/0.61 ! [X: $o] :
% 0.34/0.61 ( ( insert_o @ X @ top_top_set_o )
% 0.34/0.61 = top_top_set_o ) ).
% 0.34/0.61
% 0.34/0.61 % insert_UNIV
% 0.34/0.61 thf(fact_4103_insert__UNIV,axiom,
% 0.34/0.61 ! [X: nat] :
% 0.34/0.61 ( ( insert_nat @ X @ top_top_set_nat )
% 0.34/0.61 = top_top_set_nat ) ).
% 0.34/0.61
% 0.34/0.61 % insert_UNIV
% 0.34/0.61 thf(fact_4104_insert__UNIV,axiom,
% 0.34/0.61 ! [X: int] :
% 0.34/0.61 ( ( insert_int @ X @ top_top_set_int )
% 0.34/0.61 = top_top_set_int ) ).
% 0.34/0.61
% 0.34/0.61 % insert_UNIV
% 0.34/0.61 thf(fact_4105_Int__UNIV__left,axiom,
% 0.34/0.61 ! [B4: set_Product_unit] :
% 0.34/0.61 ( ( inf_in4660618365625256667t_unit @ top_to1996260823553986621t_unit @ B4 )
% 0.34/0.61 = B4 ) ).
% 0.34/0.61
% 0.34/0.61 % Int_UNIV_left
% 0.34/0.61 thf(fact_4106_Int__UNIV__left,axiom,
% 0.34/0.61 ! [B4: set_o] :
% 0.34/0.61 ( ( inf_inf_set_o @ top_top_set_o @ B4 )
% 0.34/0.61 = B4 ) ).
% 0.34/0.61
% 0.34/0.61 % Int_UNIV_left
% 0.34/0.61 thf(fact_4107_Int__UNIV__left,axiom,
% 0.34/0.61 ! [B4: set_nat] :
% 0.34/0.61 ( ( inf_inf_set_nat @ top_top_set_nat @ B4 )
% 0.34/0.61 = B4 ) ).
% 0.34/0.61
% 0.34/0.61 % Int_UNIV_left
% 0.34/0.61 thf(fact_4108_Int__UNIV__left,axiom,
% 0.34/0.61 ! [B4: set_int] :
% 0.34/0.61 ( ( inf_inf_set_int @ top_top_set_int @ B4 )
% 0.34/0.61 = B4 ) ).
% 0.34/0.61
% 0.34/0.61 % Int_UNIV_left
% 0.34/0.61 thf(fact_4109_Int__UNIV__right,axiom,
% 0.34/0.61 ! [A: set_Product_unit] :
% 0.34/0.61 ( ( inf_in4660618365625256667t_unit @ A @ top_to1996260823553986621t_unit )
% 0.34/0.61 = A ) ).
% 0.34/0.61
% 0.34/0.61 % Int_UNIV_right
% 0.34/0.61 thf(fact_4110_Int__UNIV__right,axiom,
% 0.34/0.61 ! [A: set_o] :
% 0.34/0.61 ( ( inf_inf_set_o @ A @ top_top_set_o )
% 0.34/0.61 = A ) ).
% 0.34/0.61
% 0.34/0.61 % Int_UNIV_right
% 0.34/0.61 thf(fact_4111_Int__UNIV__right,axiom,
% 0.34/0.61 ! [A: set_nat] :
% 0.34/0.61 ( ( inf_inf_set_nat @ A @ top_top_set_nat )
% 0.34/0.61 = A ) ).
% 0.34/0.61
% 0.34/0.61 % Int_UNIV_right
% 0.34/0.61 thf(fact_4112_Int__UNIV__right,axiom,
% 0.34/0.61 ! [A: set_int] :
% 0.34/0.61 ( ( inf_inf_set_int @ A @ top_top_set_int )
% 0.34/0.61 = A ) ).
% 0.34/0.61
% 0.34/0.61 % Int_UNIV_right
% 0.34/0.61 thf(fact_4113_Un__UNIV__left,axiom,
% 0.34/0.61 ! [B4: set_Product_unit] :
% 0.34/0.61 ( ( sup_su793286257634532545t_unit @ top_to1996260823553986621t_unit @ B4 )
% 0.34/0.61 = top_to1996260823553986621t_unit ) ).
% 0.34/0.61
% 0.34/0.61 % Un_UNIV_left
% 0.34/0.61 thf(fact_4114_Un__UNIV__left,axiom,
% 0.34/0.61 ! [B4: set_o] :
% 0.34/0.61 ( ( sup_sup_set_o @ top_top_set_o @ B4 )
% 0.34/0.61 = top_top_set_o ) ).
% 0.34/0.61
% 0.34/0.61 % Un_UNIV_left
% 0.34/0.61 thf(fact_4115_Un__UNIV__left,axiom,
% 0.34/0.61 ! [B4: set_nat] :
% 0.34/0.61 ( ( sup_sup_set_nat @ top_top_set_nat @ B4 )
% 0.34/0.61 = top_top_set_nat ) ).
% 0.34/0.61
% 0.34/0.61 % Un_UNIV_left
% 0.34/0.61 thf(fact_4116_Un__UNIV__left,axiom,
% 0.34/0.61 ! [B4: set_int] :
% 0.34/0.61 ( ( sup_sup_set_int @ top_top_set_int @ B4 )
% 0.34/0.61 = top_top_set_int ) ).
% 0.34/0.61
% 0.34/0.61 % Un_UNIV_left
% 0.34/0.61 thf(fact_4117_Un__UNIV__right,axiom,
% 0.34/0.61 ! [A: set_Product_unit] :
% 0.34/0.61 ( ( sup_su793286257634532545t_unit @ A @ top_to1996260823553986621t_unit )
% 0.34/0.61 = top_to1996260823553986621t_unit ) ).
% 0.34/0.61
% 0.34/0.61 % Un_UNIV_right
% 0.34/0.61 thf(fact_4118_Un__UNIV__right,axiom,
% 0.34/0.61 ! [A: set_o] :
% 0.34/0.61 ( ( sup_sup_set_o @ A @ top_top_set_o )
% 0.34/0.61 = top_top_set_o ) ).
% 0.34/0.61
% 0.34/0.61 % Un_UNIV_right
% 0.34/0.61 thf(fact_4119_Un__UNIV__right,axiom,
% 0.34/0.61 ! [A: set_nat] :
% 0.34/0.61 ( ( sup_sup_set_nat @ A @ top_top_set_nat )
% 0.34/0.61 = top_top_set_nat ) ).
% 0.34/0.61
% 0.34/0.61 % Un_UNIV_right
% 0.34/0.61 thf(fact_4120_Un__UNIV__right,axiom,
% 0.34/0.61 ! [A: set_int] :
% 0.34/0.61 ( ( sup_sup_set_int @ A @ top_top_set_int )
% 0.34/0.61 = top_top_set_int ) ).
% 0.34/0.61
% 0.34/0.61 % Un_UNIV_right
% 0.34/0.61 thf(fact_4121_type__copy__ex__RepI,axiom,
% 0.34/0.61 ! [Rep2: assn > produc3658429121746597890et_nat > $o,Abs2: ( produc3658429121746597890et_nat > $o ) > assn,F2: ( produc3658429121746597890et_nat > $o ) > $o] :
% 0.34/0.61 ( ( type_d3909072315231072503_nat_o @ Rep2 @ Abs2 @ top_to4869887016220417533_nat_o )
% 0.34/0.61 => ( ( ? [X7: produc3658429121746597890et_nat > $o] : ( F2 @ X7 ) )
% 0.34/0.61 = ( ? [B6: assn] : ( F2 @ ( Rep2 @ B6 ) ) ) ) ) ).
% 0.34/0.61
% 0.34/0.61 % type_copy_ex_RepI
% 0.34/0.61 thf(fact_4122_type__copy__ex__RepI,axiom,
% 0.34/0.61 ! [Rep2: code_integer > int,Abs2: int > code_integer,F2: int > $o] :
% 0.34/0.61 ( ( type_d8366093980585677751er_int @ Rep2 @ Abs2 @ top_top_set_int )
% 0.34/0.61 => ( ( ? [X7: int] : ( F2 @ X7 ) )
% 0.34/0.61 = ( ? [B6: code_integer] : ( F2 @ ( Rep2 @ B6 ) ) ) ) ) ).
% 0.34/0.61
% 0.34/0.61 % type_copy_ex_RepI
% 0.34/0.61 thf(fact_4123_type__copy__ex__RepI,axiom,
% 0.34/0.61 ! [Rep2: product_unit > $o,Abs2: $o > product_unit,F2: $o > $o] :
% 0.34/0.61 ( ( type_d6188575255521822967unit_o @ Rep2 @ Abs2 @ top_top_set_o )
% 0.34/0.61 => ( ( ? [X7: $o] : ( F2 @ X7 ) )
% 0.34/0.61 = ( ? [B6: product_unit] : ( F2 @ ( Rep2 @ B6 ) ) ) ) ) ).
% 0.34/0.61
% 0.34/0.61 % type_copy_ex_RepI
% 0.34/0.61 thf(fact_4124_type__copy__ex__RepI,axiom,
% 0.34/0.61 ! [Rep2: code_natural > nat,Abs2: nat > code_natural,F2: nat > $o] :
% 0.34/0.61 ( ( type_d4410041424927559462al_nat @ Rep2 @ Abs2 @ top_top_set_nat )
% 0.34/0.61 => ( ( ? [X7: nat] : ( F2 @ X7 ) )
% 0.34/0.61 = ( ? [B6: code_natural] : ( F2 @ ( Rep2 @ B6 ) ) ) ) ) ).
% 0.34/0.61
% 0.34/0.61 % type_copy_ex_RepI
% 0.34/0.61 thf(fact_4125_type__copy__obj__one__point__absE,axiom,
% 0.34/0.61 ! [Rep2: assn > produc3658429121746597890et_nat > $o,Abs2: ( produc3658429121746597890et_nat > $o ) > assn,S2: assn] :
% 0.34/0.61 ( ( type_d3909072315231072503_nat_o @ Rep2 @ Abs2 @ top_to4869887016220417533_nat_o )
% 0.34/0.61 => ~ ! [X3: produc3658429121746597890et_nat > $o] :
% 0.34/0.61 ( S2
% 0.34/0.61 != ( Abs2 @ X3 ) ) ) ).
% 0.34/0.61
% 0.34/0.61 % type_copy_obj_one_point_absE
% 0.34/0.61 thf(fact_4126_type__copy__obj__one__point__absE,axiom,
% 0.34/0.61 ! [Rep2: code_integer > int,Abs2: int > code_integer,S2: code_integer] :
% 0.34/0.61 ( ( type_d8366093980585677751er_int @ Rep2 @ Abs2 @ top_top_set_int )
% 0.34/0.61 => ~ ! [X3: int] :
% 0.34/0.61 ( S2
% 0.34/0.61 != ( Abs2 @ X3 ) ) ) ).
% 0.34/0.61
% 0.34/0.61 % type_copy_obj_one_point_absE
% 0.34/0.61 thf(fact_4127_type__copy__obj__one__point__absE,axiom,
% 0.34/0.61 ! [Rep2: product_unit > $o,Abs2: $o > product_unit,S2: product_unit] :
% 0.34/0.61 ( ( type_d6188575255521822967unit_o @ Rep2 @ Abs2 @ top_top_set_o )
% 0.34/0.61 => ~ ! [X3: $o] :
% 0.34/0.61 ( S2
% 0.34/0.61 != ( Abs2 @ X3 ) ) ) ).
% 0.34/0.61
% 0.34/0.61 % type_copy_obj_one_point_absE
% 0.34/0.61 thf(fact_4128_type__copy__obj__one__point__absE,axiom,
% 0.34/0.61 ! [Rep2: code_natural > nat,Abs2: nat > code_natural,S2: code_natural] :
% 0.34/0.61 ( ( type_d4410041424927559462al_nat @ Rep2 @ Abs2 @ top_top_set_nat )
% 0.34/0.61 => ~ ! [X3: nat] :
% 0.34/0.61 ( S2
% 0.34/0.61 != ( Abs2 @ X3 ) ) ) ).
% 0.34/0.61
% 0.34/0.61 % type_copy_obj_one_point_absE
% 0.34/0.61 thf(fact_4129_less__1__mult,axiom,
% 0.34/0.61 ! [M4: code_natural,N2: code_natural] :
% 0.34/0.61 ( ( ord_le5570908160329646204atural @ one_one_Code_natural @ M4 )
% 0.34/0.61 => ( ( ord_le5570908160329646204atural @ one_one_Code_natural @ N2 )
% 0.34/0.61 => ( ord_le5570908160329646204atural @ one_one_Code_natural @ ( times_2397367101498566445atural @ M4 @ N2 ) ) ) ) ).
% 0.34/0.61
% 0.34/0.61 % less_1_mult
% 0.34/0.61 thf(fact_4130_less__1__mult,axiom,
% 0.34/0.61 ! [M4: nat,N2: nat] :
% 0.34/0.61 ( ( ord_less_nat @ one_one_nat @ M4 )
% 0.34/0.61 => ( ( ord_less_nat @ one_one_nat @ N2 )
% 0.34/0.61 => ( ord_less_nat @ one_one_nat @ ( times_times_nat @ M4 @ N2 ) ) ) ) ).
% 0.34/0.61
% 0.34/0.61 % less_1_mult
% 0.34/0.61 thf(fact_4131_less__1__mult,axiom,
% 0.34/0.61 ! [M4: int,N2: int] :
% 0.34/0.61 ( ( ord_less_int @ one_one_int @ M4 )
% 0.34/0.61 => ( ( ord_less_int @ one_one_int @ N2 )
% 0.34/0.61 => ( ord_less_int @ one_one_int @ ( times_times_int @ M4 @ N2 ) ) ) ) ).
% 0.34/0.61
% 0.34/0.61 % less_1_mult
% 0.34/0.61 thf(fact_4132_less__1__mult,axiom,
% 0.34/0.61 ! [M4: code_integer,N2: code_integer] :
% 0.34/0.61 ( ( ord_le6747313008572928689nteger @ one_one_Code_integer @ M4 )
% 0.34/0.61 => ( ( ord_le6747313008572928689nteger @ one_one_Code_integer @ N2 )
% 0.34/0.61 => ( ord_le6747313008572928689nteger @ one_one_Code_integer @ ( times_3573771949741848930nteger @ M4 @ N2 ) ) ) ) ).
% 0.34/0.61
% 0.34/0.61 % less_1_mult
% 0.34/0.61 thf(fact_4133_less__1__mult,axiom,
% 0.34/0.61 ! [M4: rat,N2: rat] :
% 0.34/0.61 ( ( ord_less_rat @ one_one_rat @ M4 )
% 0.34/0.61 => ( ( ord_less_rat @ one_one_rat @ N2 )
% 0.34/0.61 => ( ord_less_rat @ one_one_rat @ ( times_times_rat @ M4 @ N2 ) ) ) ) ).
% 0.34/0.61
% 0.34/0.61 % less_1_mult
% 0.34/0.61 thf(fact_4134_infinite__growing,axiom,
% 0.34/0.61 ! [X4: set_Product_unit] :
% 0.34/0.61 ( ( X4 != bot_bo3957492148770167129t_unit )
% 0.34/0.61 => ( ! [X3: product_unit] :
% 0.34/0.61 ( ( member_Product_unit @ X3 @ X4 )
% 0.34/0.61 => ? [Xa2: product_unit] :
% 0.34/0.61 ( ( member_Product_unit @ Xa2 @ X4 )
% 0.34/0.61 & ( ord_le361264281704409273t_unit @ X3 @ Xa2 ) ) )
% 0.34/0.61 => ~ ( finite4290736615968046902t_unit @ X4 ) ) ) ).
% 0.34/0.61
% 0.34/0.61 % infinite_growing
% 0.34/0.61 thf(fact_4135_infinite__growing,axiom,
% 0.34/0.61 ! [X4: set_o] :
% 0.34/0.61 ( ( X4 != bot_bot_set_o )
% 0.34/0.61 => ( ! [X3: $o] :
% 0.34/0.61 ( ( member_o @ X3 @ X4 )
% 0.34/0.61 => ? [Xa2: $o] :
% 0.34/0.61 ( ( member_o @ Xa2 @ X4 )
% 0.34/0.61 & ( ord_less_o @ X3 @ Xa2 ) ) )
% 0.34/0.61 => ~ ( finite_finite_o @ X4 ) ) ) ).
% 0.34/0.61
% 0.34/0.61 % infinite_growing
% 0.34/0.61 thf(fact_4136_infinite__growing,axiom,
% 0.34/0.61 ! [X4: set_nat] :
% 0.34/0.61 ( ( X4 != bot_bot_set_nat )
% 0.34/0.61 => ( ! [X3: nat] :
% 0.34/0.61 ( ( member_nat @ X3 @ X4 )
% 0.34/0.61 => ? [Xa2: nat] :
% 0.34/0.61 ( ( member_nat @ Xa2 @ X4 )
% 0.34/0.61 & ( ord_less_nat @ X3 @ Xa2 ) ) )
% 0.34/0.61 => ~ ( finite_finite_nat @ X4 ) ) ) ).
% 0.34/0.61
% 0.34/0.61 % infinite_growing
% 0.34/0.61 thf(fact_4137_infinite__growing,axiom,
% 0.34/0.61 ! [X4: set_int] :
% 0.34/0.61 ( ( X4 != bot_bot_set_int )
% 0.34/0.61 => ( ! [X3: int] :
% 0.34/0.61 ( ( member_int @ X3 @ X4 )
% 0.34/0.61 => ? [Xa2: int] :
% 0.34/0.61 ( ( member_int @ Xa2 @ X4 )
% 0.34/0.61 & ( ord_less_int @ X3 @ Xa2 ) ) )
% 0.34/0.61 => ~ ( finite_finite_int @ X4 ) ) ) ).
% 0.34/0.61
% 0.34/0.61 % infinite_growing
% 0.34/0.61 thf(fact_4138_infinite__growing,axiom,
% 0.34/0.61 ! [X4: set_Code_integer] :
% 0.34/0.61 ( ( X4 != bot_bo3990330152332043303nteger )
% 0.34/0.61 => ( ! [X3: code_integer] :
% 0.34/0.61 ( ( member_Code_integer @ X3 @ X4 )
% 0.34/0.61 => ? [Xa2: code_integer] :
% 0.34/0.61 ( ( member_Code_integer @ Xa2 @ X4 )
% 0.34/0.61 & ( ord_le6747313008572928689nteger @ X3 @ Xa2 ) ) )
% 0.34/0.61 => ~ ( finite6017078050557962740nteger @ X4 ) ) ) ).
% 0.34/0.61
% 0.34/0.61 % infinite_growing
% 0.34/0.61 thf(fact_4139_infinite__growing,axiom,
% 0.34/0.61 ! [X4: set_rat] :
% 0.34/0.61 ( ( X4 != bot_bot_set_rat )
% 0.34/0.61 => ( ! [X3: rat] :
% 0.34/0.61 ( ( member_rat @ X3 @ X4 )
% 0.34/0.61 => ? [Xa2: rat] :
% 0.34/0.61 ( ( member_rat @ Xa2 @ X4 )
% 0.34/0.61 & ( ord_less_rat @ X3 @ Xa2 ) ) )
% 0.34/0.61 => ~ ( finite_finite_rat @ X4 ) ) ) ).
% 0.34/0.61
% 0.34/0.61 % infinite_growing
% 0.34/0.61 thf(fact_4140_ex__min__if__finite,axiom,
% 0.34/0.61 ! [S: set_Product_unit] :
% 0.34/0.61 ( ( finite4290736615968046902t_unit @ S )
% 0.34/0.61 => ( ( S != bot_bo3957492148770167129t_unit )
% 0.34/0.61 => ? [X3: product_unit] :
% 0.34/0.61 ( ( member_Product_unit @ X3 @ S )
% 0.34/0.61 & ~ ? [Xa2: product_unit] :
% 0.34/0.61 ( ( member_Product_unit @ Xa2 @ S )
% 0.34/0.61 & ( ord_le361264281704409273t_unit @ Xa2 @ X3 ) ) ) ) ) ).
% 0.34/0.61
% 0.34/0.61 % ex_min_if_finite
% 0.34/0.61 thf(fact_4141_ex__min__if__finite,axiom,
% 0.34/0.61 ! [S: set_o] :
% 0.34/0.61 ( ( finite_finite_o @ S )
% 0.34/0.61 => ( ( S != bot_bot_set_o )
% 0.34/0.61 => ? [X3: $o] :
% 0.34/0.61 ( ( member_o @ X3 @ S )
% 0.34/0.61 & ~ ? [Xa2: $o] :
% 0.34/0.61 ( ( member_o @ Xa2 @ S )
% 0.34/0.61 & ( ord_less_o @ Xa2 @ X3 ) ) ) ) ) ).
% 0.34/0.61
% 0.34/0.61 % ex_min_if_finite
% 0.34/0.61 thf(fact_4142_ex__min__if__finite,axiom,
% 0.34/0.61 ! [S: set_Pr4532377907799695533_nat_o] :
% 0.34/0.61 ( ( finite3252695134891459830_nat_o @ S )
% 0.34/0.61 => ( ( S != bot_bo7824918357723582233_nat_o )
% 0.34/0.61 => ? [X3: produc3658429121746597890et_nat > $o] :
% 0.34/0.61 ( ( member6576561426505652726_nat_o @ X3 @ S )
% 0.34/0.61 & ~ ? [Xa2: produc3658429121746597890et_nat > $o] :
% 0.34/0.61 ( ( member6576561426505652726_nat_o @ Xa2 @ S )
% 0.34/0.61 & ( ord_le8176553561544679545_nat_o @ Xa2 @ X3 ) ) ) ) ) ).
% 0.34/0.61
% 0.34/0.61 % ex_min_if_finite
% 0.34/0.61 thf(fact_4143_ex__min__if__finite,axiom,
% 0.34/0.61 ! [S: set_assn] :
% 0.34/0.61 ( ( finite_finite_assn @ S )
% 0.34/0.61 => ( ( S != bot_bot_set_assn )
% 0.34/0.61 => ? [X3: assn] :
% 0.34/0.61 ( ( member_assn @ X3 @ S )
% 0.34/0.61 & ~ ? [Xa2: assn] :
% 0.34/0.61 ( ( member_assn @ Xa2 @ S )
% 0.34/0.61 & ( ord_less_assn @ Xa2 @ X3 ) ) ) ) ) ).
% 0.34/0.61
% 0.34/0.61 % ex_min_if_finite
% 0.34/0.61 thf(fact_4144_ex__min__if__finite,axiom,
% 0.34/0.61 ! [S: set_nat] :
% 0.34/0.61 ( ( finite_finite_nat @ S )
% 0.34/0.61 => ( ( S != bot_bot_set_nat )
% 0.34/0.61 => ? [X3: nat] :
% 0.34/0.61 ( ( member_nat @ X3 @ S )
% 0.34/0.61 & ~ ? [Xa2: nat] :
% 0.34/0.61 ( ( member_nat @ Xa2 @ S )
% 0.34/0.61 & ( ord_less_nat @ Xa2 @ X3 ) ) ) ) ) ).
% 0.34/0.61
% 0.34/0.61 % ex_min_if_finite
% 0.34/0.61 thf(fact_4145_ex__min__if__finite,axiom,
% 0.34/0.61 ! [S: set_int] :
% 0.34/0.61 ( ( finite_finite_int @ S )
% 0.34/0.61 => ( ( S != bot_bot_set_int )
% 0.34/0.61 => ? [X3: int] :
% 0.34/0.61 ( ( member_int @ X3 @ S )
% 0.34/0.61 & ~ ? [Xa2: int] :
% 0.34/0.61 ( ( member_int @ Xa2 @ S )
% 0.34/0.61 & ( ord_less_int @ Xa2 @ X3 ) ) ) ) ) ).
% 0.34/0.61
% 0.34/0.61 % ex_min_if_finite
% 0.34/0.61 thf(fact_4146_ex__min__if__finite,axiom,
% 0.34/0.61 ! [S: set_Code_integer] :
% 0.34/0.61 ( ( finite6017078050557962740nteger @ S )
% 0.34/0.61 => ( ( S != bot_bo3990330152332043303nteger )
% 0.34/0.61 => ? [X3: code_integer] :
% 0.34/0.61 ( ( member_Code_integer @ X3 @ S )
% 0.34/0.61 & ~ ? [Xa2: code_integer] :
% 0.34/0.61 ( ( member_Code_integer @ Xa2 @ S )
% 0.34/0.61 & ( ord_le6747313008572928689nteger @ Xa2 @ X3 ) ) ) ) ) ).
% 0.34/0.61
% 0.34/0.61 % ex_min_if_finite
% 0.34/0.61 thf(fact_4147_ex__min__if__finite,axiom,
% 0.34/0.61 ! [S: set_rat] :
% 0.34/0.61 ( ( finite_finite_rat @ S )
% 0.34/0.61 => ( ( S != bot_bot_set_rat )
% 0.34/0.61 => ? [X3: rat] :
% 0.34/0.61 ( ( member_rat @ X3 @ S )
% 0.34/0.61 & ~ ? [Xa2: rat] :
% 0.34/0.61 ( ( member_rat @ Xa2 @ S )
% 0.34/0.61 & ( ord_less_rat @ Xa2 @ X3 ) ) ) ) ) ).
% 0.34/0.61
% 0.34/0.61 % ex_min_if_finite
% 0.34/0.61 thf(fact_4148_less__cSupD,axiom,
% 0.34/0.61 ! [X4: set_int,Z: int] :
% 0.34/0.61 ( ( X4 != bot_bot_set_int )
% 0.34/0.61 => ( ( ord_less_int @ Z @ ( complete_Sup_Sup_int @ X4 ) )
% 0.34/0.61 => ? [X3: int] :
% 0.34/0.61 ( ( member_int @ X3 @ X4 )
% 0.34/0.61 & ( ord_less_int @ Z @ X3 ) ) ) ) ).
% 0.34/0.61
% 0.34/0.61 % less_cSupD
% 0.34/0.61 thf(fact_4149_less__cSupD,axiom,
% 0.34/0.61 ! [X4: set_nat,Z: nat] :
% 0.34/0.61 ( ( X4 != bot_bot_set_nat )
% 0.34/0.61 => ( ( ord_less_nat @ Z @ ( complete_Sup_Sup_nat @ X4 ) )
% 0.34/0.61 => ? [X3: nat] :
% 0.34/0.61 ( ( member_nat @ X3 @ X4 )
% 0.34/0.61 & ( ord_less_nat @ Z @ X3 ) ) ) ) ).
% 0.34/0.61
% 0.34/0.61 % less_cSupD
% 0.34/0.61 thf(fact_4150_less__cSupD,axiom,
% 0.34/0.61 ! [X4: set_Product_unit,Z: product_unit] :
% 0.34/0.61 ( ( X4 != bot_bo3957492148770167129t_unit )
% 0.34/0.61 => ( ( ord_le361264281704409273t_unit @ Z @ ( comple4687483117567863418t_unit @ X4 ) )
% 0.34/0.61 => ? [X3: product_unit] :
% 0.34/0.61 ( ( member_Product_unit @ X3 @ X4 )
% 0.34/0.61 & ( ord_le361264281704409273t_unit @ Z @ X3 ) ) ) ) ).
% 0.34/0.61
% 0.34/0.61 % less_cSupD
% 0.34/0.61 thf(fact_4151_less__cSupE,axiom,
% 0.34/0.61 ! [Y: int,X4: set_int] :
% 0.34/0.61 ( ( ord_less_int @ Y @ ( complete_Sup_Sup_int @ X4 ) )
% 0.34/0.61 => ( ( X4 != bot_bot_set_int )
% 0.34/0.61 => ~ ! [X3: int] :
% 0.34/0.61 ( ( member_int @ X3 @ X4 )
% 0.34/0.61 => ~ ( ord_less_int @ Y @ X3 ) ) ) ) ).
% 0.34/0.61
% 0.34/0.61 % less_cSupE
% 0.34/0.61 thf(fact_4152_less__cSupE,axiom,
% 0.34/0.61 ! [Y: nat,X4: set_nat] :
% 0.34/0.61 ( ( ord_less_nat @ Y @ ( complete_Sup_Sup_nat @ X4 ) )
% 0.34/0.61 => ( ( X4 != bot_bot_set_nat )
% 0.34/0.61 => ~ ! [X3: nat] :
% 0.34/0.61 ( ( member_nat @ X3 @ X4 )
% 0.34/0.61 => ~ ( ord_less_nat @ Y @ X3 ) ) ) ) ).
% 0.34/0.61
% 0.34/0.61 % less_cSupE
% 0.34/0.61 thf(fact_4153_less__cSupE,axiom,
% 0.34/0.61 ! [Y: product_unit,X4: set_Product_unit] :
% 0.34/0.61 ( ( ord_le361264281704409273t_unit @ Y @ ( comple4687483117567863418t_unit @ X4 ) )
% 0.34/0.61 => ( ( X4 != bot_bo3957492148770167129t_unit )
% 0.34/0.61 => ~ ! [X3: product_unit] :
% 0.34/0.61 ( ( member_Product_unit @ X3 @ X4 )
% 0.34/0.61 => ~ ( ord_le361264281704409273t_unit @ Y @ X3 ) ) ) ) ).
% 0.34/0.61
% 0.34/0.61 % less_cSupE
% 0.34/0.61 thf(fact_4154_Max__in,axiom,
% 0.34/0.61 ! [A: set_Code_integer] :
% 0.34/0.61 ( ( finite6017078050557962740nteger @ A )
% 0.34/0.61 => ( ( A != bot_bo3990330152332043303nteger )
% 0.34/0.61 => ( member_Code_integer @ ( lattic4901227151466704046nteger @ A ) @ A ) ) ) ).
% 0.34/0.61
% 0.34/0.61 % Max_in
% 0.34/0.61 thf(fact_4155_Max__in,axiom,
% 0.34/0.61 ! [A: set_Product_unit] :
% 0.34/0.61 ( ( finite4290736615968046902t_unit @ A )
% 0.34/0.61 => ( ( A != bot_bo3957492148770167129t_unit )
% 0.34/0.61 => ( member_Product_unit @ ( lattic1210392890313158908t_unit @ A ) @ A ) ) ) ).
% 0.34/0.61
% 0.34/0.61 % Max_in
% 0.34/0.61 thf(fact_4156_Max__in,axiom,
% 0.34/0.61 ! [A: set_o] :
% 0.34/0.61 ( ( finite_finite_o @ A )
% 0.34/0.61 => ( ( A != bot_bot_set_o )
% 0.34/0.61 => ( member_o @ ( lattic1921953407002678535_Max_o @ A ) @ A ) ) ) ).
% 0.34/0.61
% 0.34/0.61 % Max_in
% 0.34/0.61 thf(fact_4157_Max__in,axiom,
% 0.34/0.61 ! [A: set_int] :
% 0.34/0.61 ( ( finite_finite_int @ A )
% 0.34/0.61 => ( ( A != bot_bot_set_int )
% 0.34/0.61 => ( member_int @ ( lattic8263393255366662781ax_int @ A ) @ A ) ) ) ).
% 0.34/0.61
% 0.34/0.61 % Max_in
% 0.34/0.61 thf(fact_4158_Max__in,axiom,
% 0.34/0.61 ! [A: set_nat] :
% 0.34/0.61 ( ( finite_finite_nat @ A )
% 0.34/0.61 => ( ( A != bot_bot_set_nat )
% 0.34/0.61 => ( member_nat @ ( lattic8265883725875713057ax_nat @ A ) @ A ) ) ) ).
% 0.34/0.61
% 0.34/0.61 % Max_in
% 0.34/0.61 thf(fact_4159_range__subsetD,axiom,
% 0.34/0.61 ! [F: code_integer > code_integer,B4: set_Code_integer,I2: code_integer] :
% 0.34/0.61 ( ( ord_le7084787975880047091nteger @ ( image_4470545334726330049nteger @ F @ top_to4645266643341252675nteger ) @ B4 )
% 0.34/0.61 => ( member_Code_integer @ ( F @ I2 ) @ B4 ) ) ).
% 0.34/0.61
% 0.34/0.61 % range_subsetD
% 0.34/0.61 thf(fact_4160_range__subsetD,axiom,
% 0.34/0.61 ! [F: product_unit > int,B4: set_int,I2: product_unit] :
% 0.34/0.61 ( ( ord_less_eq_set_int @ ( image_873079544045703924it_int @ F @ top_to1996260823553986621t_unit ) @ B4 )
% 0.34/0.61 => ( member_int @ ( F @ I2 ) @ B4 ) ) ).
% 0.34/0.61
% 0.34/0.61 % range_subsetD
% 0.34/0.61 thf(fact_4161_range__subsetD,axiom,
% 0.34/0.61 ! [F: product_unit > $o,B4: set_o,I2: product_unit] :
% 0.34/0.61 ( ( ord_less_eq_set_o @ ( image_Product_unit_o2 @ F @ top_to1996260823553986621t_unit ) @ B4 )
% 0.34/0.61 => ( member_o @ ( F @ I2 ) @ B4 ) ) ).
% 0.34/0.61
% 0.34/0.61 % range_subsetD
% 0.34/0.61 thf(fact_4162_range__subsetD,axiom,
% 0.34/0.61 ! [F: product_unit > product_unit,B4: set_Product_unit,I2: product_unit] :
% 0.34/0.61 ( ( ord_le3507040750410214029t_unit @ ( image_405062704495631173t_unit @ F @ top_to1996260823553986621t_unit ) @ B4 )
% 0.34/0.61 => ( member_Product_unit @ ( F @ I2 ) @ B4 ) ) ).
% 0.34/0.61
% 0.34/0.61 % range_subsetD
% 0.34/0.61 thf(fact_4163_range__subsetD,axiom,
% 0.34/0.61 ! [F: $o > int,B4: set_int,I2: $o] :
% 0.34/0.61 ( ( ord_less_eq_set_int @ ( image_o_int2 @ F @ top_top_set_o ) @ B4 )
% 0.34/0.61 => ( member_int @ ( F @ I2 ) @ B4 ) ) ).
% 0.34/0.61
% 0.34/0.61 % range_subsetD
% 0.34/0.61 thf(fact_4164_range__subsetD,axiom,
% 0.34/0.61 ! [F: $o > $o,B4: set_o,I2: $o] :
% 0.34/0.61 ( ( ord_less_eq_set_o @ ( image_o_o2 @ F @ top_top_set_o ) @ B4 )
% 0.34/0.61 => ( member_o @ ( F @ I2 ) @ B4 ) ) ).
% 0.34/0.61
% 0.34/0.61 % range_subsetD
% 0.34/0.61 thf(fact_4165_range__subsetD,axiom,
% 0.34/0.61 ! [F: $o > product_unit,B4: set_Product_unit,I2: $o] :
% 0.34/0.61 ( ( ord_le3507040750410214029t_unit @ ( image_o_Product_unit2 @ F @ top_top_set_o ) @ B4 )
% 0.34/0.61 => ( member_Product_unit @ ( F @ I2 ) @ B4 ) ) ).
% 0.34/0.61
% 0.34/0.61 % range_subsetD
% 0.34/0.61 thf(fact_4166_range__subsetD,axiom,
% 0.34/0.61 ! [F: nat > code_integer,B4: set_Code_integer,I2: nat] :
% 0.34/0.61 ( ( ord_le7084787975880047091nteger @ ( image_1215581382706833972nteger @ F @ top_top_set_nat ) @ B4 )
% 0.34/0.61 => ( member_Code_integer @ ( F @ I2 ) @ B4 ) ) ).
% 0.34/0.61
% 0.34/0.61 % range_subsetD
% 0.34/0.61 thf(fact_4167_range__subsetD,axiom,
% 0.34/0.61 ! [F: nat > int,B4: set_int,I2: nat] :
% 0.34/0.61 ( ( ord_less_eq_set_int @ ( image_nat_int2 @ F @ top_top_set_nat ) @ B4 )
% 0.34/0.61 => ( member_int @ ( F @ I2 ) @ B4 ) ) ).
% 0.34/0.61
% 0.34/0.61 % range_subsetD
% 0.34/0.61 thf(fact_4168_range__subsetD,axiom,
% 0.34/0.61 ! [F: nat > $o,B4: set_o,I2: nat] :
% 0.34/0.61 ( ( ord_less_eq_set_o @ ( image_nat_o2 @ F @ top_top_set_nat ) @ B4 )
% 0.34/0.61 => ( member_o @ ( F @ I2 ) @ B4 ) ) ).
% 0.34/0.61
% 0.34/0.61 % range_subsetD
% 0.34/0.61 thf(fact_4169_surj__vimage__empty,axiom,
% 0.34/0.61 ! [F: code_integer > code_integer,A: set_Code_integer] :
% 0.34/0.61 ( ( ( image_4470545334726330049nteger @ F @ top_to4645266643341252675nteger )
% 0.34/0.61 = top_to4645266643341252675nteger )
% 0.34/0.61 => ( ( ( vimage1318793397690208119nteger @ F @ A )
% 0.34/0.61 = bot_bo3990330152332043303nteger )
% 0.34/0.61 = ( A = bot_bo3990330152332043303nteger ) ) ) ).
% 0.34/0.61
% 0.34/0.61 % surj_vimage_empty
% 0.34/0.61 thf(fact_4170_surj__vimage__empty,axiom,
% 0.34/0.61 ! [F: product_unit > product_unit,A: set_Product_unit] :
% 0.34/0.61 ( ( ( image_405062704495631173t_unit @ F @ top_to1996260823553986621t_unit )
% 0.34/0.61 = top_to1996260823553986621t_unit )
% 0.34/0.61 => ( ( ( vimage7995052115951654139t_unit @ F @ A )
% 0.34/0.61 = bot_bo3957492148770167129t_unit )
% 0.34/0.61 = ( A = bot_bo3957492148770167129t_unit ) ) ) ).
% 0.34/0.61
% 0.34/0.61 % surj_vimage_empty
% 0.34/0.61 thf(fact_4171_surj__vimage__empty,axiom,
% 0.34/0.61 ! [F: product_unit > $o,A: set_o] :
% 0.34/0.61 ( ( ( image_Product_unit_o2 @ F @ top_to1996260823553986621t_unit )
% 0.34/0.61 = top_top_set_o )
% 0.34/0.61 => ( ( ( vimage8808348054653608838unit_o @ F @ A )
% 0.34/0.61 = bot_bo3957492148770167129t_unit )
% 0.34/0.61 = ( A = bot_bot_set_o ) ) ) ).
% 0.34/0.61
% 0.34/0.61 % surj_vimage_empty
% 0.34/0.61 thf(fact_4172_surj__vimage__empty,axiom,
% 0.34/0.61 ! [F: product_unit > nat,A: set_nat] :
% 0.34/0.61 ( ( ( image_875570014554754200it_nat @ F @ top_to1996260823553986621t_unit )
% 0.34/0.61 = top_top_set_nat )
% 0.34/0.61 => ( ( ( vimage6253328473476588386it_nat @ F @ A )
% 0.34/0.61 = bot_bo3957492148770167129t_unit )
% 0.34/0.61 = ( A = bot_bot_set_nat ) ) ) ).
% 0.34/0.61
% 0.34/0.61 % surj_vimage_empty
% 0.34/0.61 thf(fact_4173_surj__vimage__empty,axiom,
% 0.34/0.61 ! [F: product_unit > int,A: set_int] :
% 0.34/0.61 ( ( ( image_873079544045703924it_int @ F @ top_to1996260823553986621t_unit )
% 0.34/0.61 = top_top_set_int )
% 0.34/0.61 => ( ( ( vimage6250838002967538110it_int @ F @ A )
% 0.34/0.61 = bot_bo3957492148770167129t_unit )
% 0.34/0.61 = ( A = bot_bot_set_int ) ) ) ).
% 0.34/0.61
% 0.34/0.61 % surj_vimage_empty
% 0.34/0.61 thf(fact_4174_surj__vimage__empty,axiom,
% 0.34/0.61 ! [F: $o > product_unit,A: set_Product_unit] :
% 0.34/0.61 ( ( ( image_o_Product_unit2 @ F @ top_top_set_o )
% 0.34/0.61 = top_to1996260823553986621t_unit )
% 0.34/0.61 => ( ( ( vimage4222377557387472176t_unit @ F @ A )
% 0.34/0.61 = bot_bot_set_o )
% 0.34/0.61 = ( A = bot_bo3957492148770167129t_unit ) ) ) ).
% 0.34/0.61
% 0.34/0.61 % surj_vimage_empty
% 0.34/0.61 thf(fact_4175_surj__vimage__empty,axiom,
% 0.34/0.61 ! [F: $o > $o,A: set_o] :
% 0.34/0.61 ( ( ( image_o_o2 @ F @ top_top_set_o )
% 0.34/0.61 = top_top_set_o )
% 0.34/0.61 => ( ( ( vimage_o_o @ F @ A )
% 0.34/0.61 = bot_bot_set_o )
% 0.34/0.61 = ( A = bot_bot_set_o ) ) ) ).
% 0.34/0.61
% 0.34/0.61 % surj_vimage_empty
% 0.34/0.61 thf(fact_4176_surj__vimage__empty,axiom,
% 0.34/0.61 ! [F: $o > nat,A: set_nat] :
% 0.34/0.61 ( ( ( image_o_nat2 @ F @ top_top_set_o )
% 0.34/0.61 = top_top_set_nat )
% 0.34/0.61 => ( ( ( vimage_o_nat @ F @ A )
% 0.34/0.61 = bot_bot_set_o )
% 0.34/0.61 = ( A = bot_bot_set_nat ) ) ) ).
% 0.34/0.61
% 0.34/0.61 % surj_vimage_empty
% 0.34/0.61 thf(fact_4177_surj__vimage__empty,axiom,
% 0.34/0.61 ! [F: $o > int,A: set_int] :
% 0.34/0.61 ( ( ( image_o_int2 @ F @ top_top_set_o )
% 0.34/0.61 = top_top_set_int )
% 0.34/0.61 => ( ( ( vimage_o_int @ F @ A )
% 0.34/0.61 = bot_bot_set_o )
% 0.34/0.61 = ( A = bot_bot_set_int ) ) ) ).
% 0.34/0.61
% 0.34/0.61 % surj_vimage_empty
% 0.34/0.61 thf(fact_4178_surj__vimage__empty,axiom,
% 0.34/0.61 ! [F: nat > code_integer,A: set_Code_integer] :
% 0.34/0.61 ( ( ( image_1215581382706833972nteger @ F @ top_top_set_nat )
% 0.34/0.61 = top_to4645266643341252675nteger )
% 0.34/0.61 => ( ( ( vimage6104394758324456170nteger @ F @ A )
% 0.34/0.61 = bot_bot_set_nat )
% 0.34/0.61 = ( A = bot_bo3990330152332043303nteger ) ) ) ).
% 0.34/0.61
% 0.34/0.61 % surj_vimage_empty
% 0.34/0.61 thf(fact_4179_acyclicI__order,axiom,
% 0.34/0.61 ! [R2: set_Pr1261947904930325089at_nat,F: nat > assn] :
% 0.34/0.61 ( ! [A4: nat,B2: nat] :
% 0.34/0.61 ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A4 @ B2 ) @ R2 )
% 0.34/0.61 => ( ord_less_assn @ ( F @ B2 ) @ ( F @ A4 ) ) )
% 0.34/0.61 => ( transi3139058337892597825ic_nat @ R2 ) ) ).
% 0.34/0.61
% 0.34/0.61 % acyclicI_order
% 0.34/0.61 thf(fact_4180_acyclicI__order,axiom,
% 0.34/0.61 ! [R2: set_Pr958786334691620121nt_int,F: int > assn] :
% 0.34/0.61 ( ! [A4: int,B2: int] :
% 0.34/0.61 ( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ A4 @ B2 ) @ R2 )
% 0.34/0.61 => ( ord_less_assn @ ( F @ B2 ) @ ( F @ A4 ) ) )
% 0.34/0.61 => ( transi3136567867383547549ic_int @ R2 ) ) ).
% 0.34/0.61
% 0.34/0.61 % acyclicI_order
% 0.34/0.61 thf(fact_4181_acyclicI__order,axiom,
% 0.34/0.61 ! [R2: set_Pr1261947904930325089at_nat,F: nat > nat] :
% 0.34/0.61 ( ! [A4: nat,B2: nat] :
% 0.34/0.61 ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A4 @ B2 ) @ R2 )
% 0.34/0.61 => ( ord_less_nat @ ( F @ B2 ) @ ( F @ A4 ) ) )
% 0.34/0.61 => ( transi3139058337892597825ic_nat @ R2 ) ) ).
% 0.34/0.61
% 0.34/0.61 % acyclicI_order
% 0.34/0.61 thf(fact_4182_acyclicI__order,axiom,
% 0.34/0.61 ! [R2: set_Pr958786334691620121nt_int,F: int > nat] :
% 0.34/0.61 ( ! [A4: int,B2: int] :
% 0.34/0.61 ( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ A4 @ B2 ) @ R2 )
% 0.34/0.61 => ( ord_less_nat @ ( F @ B2 ) @ ( F @ A4 ) ) )
% 0.34/0.61 => ( transi3136567867383547549ic_int @ R2 ) ) ).
% 0.34/0.61
% 0.34/0.61 % acyclicI_order
% 0.34/0.61 thf(fact_4183_acyclicI__order,axiom,
% 0.34/0.61 ! [R2: set_Pr1261947904930325089at_nat,F: nat > int] :
% 0.34/0.61 ( ! [A4: nat,B2: nat] :
% 0.34/0.61 ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A4 @ B2 ) @ R2 )
% 0.34/0.61 => ( ord_less_int @ ( F @ B2 ) @ ( F @ A4 ) ) )
% 0.34/0.61 => ( transi3139058337892597825ic_nat @ R2 ) ) ).
% 0.34/0.61
% 0.34/0.61 % acyclicI_order
% 0.34/0.61 thf(fact_4184_acyclicI__order,axiom,
% 0.34/0.61 ! [R2: set_Pr958786334691620121nt_int,F: int > int] :
% 0.34/0.61 ( ! [A4: int,B2: int] :
% 0.34/0.61 ( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ A4 @ B2 ) @ R2 )
% 0.34/0.61 => ( ord_less_int @ ( F @ B2 ) @ ( F @ A4 ) ) )
% 0.34/0.61 => ( transi3136567867383547549ic_int @ R2 ) ) ).
% 0.34/0.61
% 0.34/0.61 % acyclicI_order
% 0.34/0.61 thf(fact_4185_acyclicI__order,axiom,
% 0.34/0.61 ! [R2: set_Pr1261947904930325089at_nat,F: nat > code_integer] :
% 0.34/0.61 ( ! [A4: nat,B2: nat] :
% 0.34/0.61 ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A4 @ B2 ) @ R2 )
% 0.34/0.61 => ( ord_le6747313008572928689nteger @ ( F @ B2 ) @ ( F @ A4 ) ) )
% 0.34/0.61 => ( transi3139058337892597825ic_nat @ R2 ) ) ).
% 0.34/0.61
% 0.34/0.61 % acyclicI_order
% 0.34/0.61 thf(fact_4186_acyclicI__order,axiom,
% 0.34/0.61 ! [R2: set_Pr958786334691620121nt_int,F: int > code_integer] :
% 0.34/0.61 ( ! [A4: int,B2: int] :
% 0.34/0.61 ( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ A4 @ B2 ) @ R2 )
% 0.34/0.61 => ( ord_le6747313008572928689nteger @ ( F @ B2 ) @ ( F @ A4 ) ) )
% 0.34/0.61 => ( transi3136567867383547549ic_int @ R2 ) ) ).
% 0.34/0.61
% 0.34/0.61 % acyclicI_order
% 0.34/0.61 thf(fact_4187_acyclicI__order,axiom,
% 0.34/0.61 ! [R2: set_Pr1261947904930325089at_nat,F: nat > rat] :
% 0.34/0.61 ( ! [A4: nat,B2: nat] :
% 0.34/0.61 ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A4 @ B2 ) @ R2 )
% 0.34/0.61 => ( ord_less_rat @ ( F @ B2 ) @ ( F @ A4 ) ) )
% 0.34/0.61 => ( transi3139058337892597825ic_nat @ R2 ) ) ).
% 0.34/0.61
% 0.34/0.61 % acyclicI_order
% 0.34/0.61 thf(fact_4188_acyclicI__order,axiom,
% 0.34/0.61 ! [R2: set_Pr958786334691620121nt_int,F: int > rat] :
% 0.34/0.61 ( ! [A4: int,B2: int] :
% 0.34/0.61 ( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ A4 @ B2 ) @ R2 )
% 0.34/0.61 => ( ord_less_rat @ ( F @ B2 ) @ ( F @ A4 ) ) )
% 0.34/0.61 => ( transi3136567867383547549ic_int @ R2 ) ) ).
% 0.34/0.61
% 0.34/0.61 % acyclicI_order
% 0.34/0.61 thf(fact_4189_type__definition_OAbs__image,axiom,
% 0.34/0.61 ! [Rep2: set_nat > nat,Abs2: nat > set_nat,A: set_nat] :
% 0.34/0.61 ( ( type_d202864840109488260at_nat @ Rep2 @ Abs2 @ A )
% 0.34/0.61 => ( ( image_nat_set_nat @ Abs2 @ A )
% 0.34/0.61 = top_top_set_set_nat ) ) ).
% 0.34/0.61
% 0.34/0.61 % type_definition.Abs_image
% 0.34/0.61 thf(fact_4190_type__definition_OAbs__image,axiom,
% 0.34/0.61 ! [Rep2: code_integer > code_integer,Abs2: code_integer > code_integer,A: set_Code_integer] :
% 0.34/0.61 ( ( type_d4796611474492816872nteger @ Rep2 @ Abs2 @ A )
% 0.34/0.61 => ( ( image_4470545334726330049nteger @ Abs2 @ A )
% 0.34/0.61 = top_to4645266643341252675nteger ) ) ).
% 0.34/0.61
% 0.34/0.61 % type_definition.Abs_image
% 0.34/0.61 thf(fact_4191_type__definition_OAbs__image,axiom,
% 0.34/0.61 ! [Rep2: code_integer > nat,Abs2: nat > code_integer,A: set_nat] :
% 0.34/0.61 ( ( type_d8368584451094728027er_nat @ Rep2 @ Abs2 @ A )
% 0.34/0.61 => ( ( image_1215581382706833972nteger @ Abs2 @ A )
% 0.34/0.61 = top_to4645266643341252675nteger ) ) ).
% 0.34/0.61
% 0.34/0.61 % type_definition.Abs_image
% 0.34/0.61 thf(fact_4192_type__definition_OAbs__image,axiom,
% 0.34/0.61 ! [Rep2: nat > nat,Abs2: nat > nat,A: set_nat] :
% 0.34/0.61 ( ( type_d6250493948777748686at_nat @ Rep2 @ Abs2 @ A )
% 0.34/0.61 => ( ( image_nat_nat2 @ Abs2 @ A )
% 0.34/0.61 = top_top_set_nat ) ) ).
% 0.34/0.61
% 0.34/0.61 % type_definition.Abs_image
% 0.34/0.61 thf(fact_4193_type__definition_OAbs__image,axiom,
% 0.34/0.61 ! [Rep2: assn > produc3658429121746597890et_nat > $o,Abs2: ( produc3658429121746597890et_nat > $o ) > assn,A: set_Pr4532377907799695533_nat_o] :
% 0.34/0.61 ( ( type_d3909072315231072503_nat_o @ Rep2 @ Abs2 @ A )
% 0.34/0.61 => ( ( image_6451031022632655280o_assn @ Abs2 @ A )
% 0.34/0.61 = top_top_set_assn ) ) ).
% 0.34/0.61
% 0.34/0.61 % type_definition.Abs_image
% 0.34/0.61 thf(fact_4194_type__definition_OAbs__image,axiom,
% 0.34/0.61 ! [Rep2: code_integer > int,Abs2: int > code_integer,A: set_int] :
% 0.34/0.61 ( ( type_d8366093980585677751er_int @ Rep2 @ Abs2 @ A )
% 0.34/0.61 => ( ( image_1587234942943678608nteger @ Abs2 @ A )
% 0.34/0.61 = top_to4645266643341252675nteger ) ) ).
% 0.34/0.61
% 0.34/0.61 % type_definition.Abs_image
% 0.34/0.61 thf(fact_4195_type__definition_OAbs__image,axiom,
% 0.34/0.61 ! [Rep2: product_unit > $o,Abs2: $o > product_unit,A: set_o] :
% 0.34/0.61 ( ( type_d6188575255521822967unit_o @ Rep2 @ Abs2 @ A )
% 0.34/0.61 => ( ( image_o_Product_unit2 @ Abs2 @ A )
% 0.34/0.61 = top_to1996260823553986621t_unit ) ) ).
% 0.34/0.61
% 0.34/0.61 % type_definition.Abs_image
% 0.34/0.61 thf(fact_4196_type__definition_OAbs__image,axiom,
% 0.34/0.61 ! [Rep2: code_natural > nat,Abs2: nat > code_natural,A: set_nat] :
% 0.34/0.61 ( ( type_d4410041424927559462al_nat @ Rep2 @ Abs2 @ A )
% 0.34/0.61 => ( ( image_39176534463551487atural @ Abs2 @ A )
% 0.34/0.61 = top_to9103716463852441742atural ) ) ).
% 0.34/0.61
% 0.34/0.61 % type_definition.Abs_image
% 0.34/0.61 thf(fact_4197_type__definition_ORep__range,axiom,
% 0.34/0.61 ! [Rep2: code_integer > code_integer,Abs2: code_integer > code_integer,A: set_Code_integer] :
% 0.34/0.61 ( ( type_d4796611474492816872nteger @ Rep2 @ Abs2 @ A )
% 0.34/0.61 => ( ( image_4470545334726330049nteger @ Rep2 @ top_to4645266643341252675nteger )
% 0.34/0.61 = A ) ) ).
% 0.34/0.61
% 0.34/0.61 % type_definition.Rep_range
% 0.34/0.61 thf(fact_4198_type__definition_ORep__range,axiom,
% 0.34/0.61 ! [Rep2: nat > set_nat,Abs2: set_nat > nat,A: set_set_nat] :
% 0.34/0.61 ( ( type_d3768499744764857476et_nat @ Rep2 @ Abs2 @ A )
% 0.34/0.61 => ( ( image_nat_set_nat @ Rep2 @ top_top_set_nat )
% 0.34/0.61 = A ) ) ).
% 0.34/0.61
% 0.34/0.61 % type_definition.Rep_range
% 0.34/0.61 thf(fact_4199_type__definition_ORep__range,axiom,
% 0.34/0.61 ! [Rep2: nat > nat,Abs2: nat > nat,A: set_nat] :
% 0.34/0.61 ( ( type_d6250493948777748686at_nat @ Rep2 @ Abs2 @ A )
% 0.34/0.61 => ( ( image_nat_nat2 @ Rep2 @ top_top_set_nat )
% 0.34/0.61 = A ) ) ).
% 0.34/0.61
% 0.34/0.61 % type_definition.Rep_range
% 0.34/0.61 thf(fact_4200_type__definition_ORep__range,axiom,
% 0.34/0.61 ! [Rep2: nat > code_integer,Abs2: code_integer > nat,A: set_Code_integer] :
% 0.34/0.61 ( ( type_d8633139899873770843nteger @ Rep2 @ Abs2 @ A )
% 0.34/0.61 => ( ( image_1215581382706833972nteger @ Rep2 @ top_top_set_nat )
% 0.34/0.61 = A ) ) ).
% 0.34/0.61
% 0.34/0.61 % type_definition.Rep_range
% 0.34/0.61 thf(fact_4201_type__definition_ORep__range,axiom,
% 0.34/0.61 ! [Rep2: assn > produc3658429121746597890et_nat > $o,Abs2: ( produc3658429121746597890et_nat > $o ) > assn,A: set_Pr4532377907799695533_nat_o] :
% 0.34/0.61 ( ( type_d3909072315231072503_nat_o @ Rep2 @ Abs2 @ A )
% 0.34/0.61 => ( ( image_5465535159827709278_nat_o @ Rep2 @ top_top_set_assn )
% 0.34/0.61 = A ) ) ).
% 0.34/0.61
% 0.34/0.61 % type_definition.Rep_range
% 0.34/0.61 thf(fact_4202_type__definition_ORep__range,axiom,
% 0.34/0.61 ! [Rep2: code_integer > int,Abs2: int > code_integer,A: set_int] :
% 0.34/0.61 ( ( type_d8366093980585677751er_int @ Rep2 @ Abs2 @ A )
% 0.34/0.61 => ( ( image_948535463418740880er_int @ Rep2 @ top_to4645266643341252675nteger )
% 0.34/0.61 = A ) ) ).
% 0.34/0.61
% 0.34/0.61 % type_definition.Rep_range
% 0.34/0.61 thf(fact_4203_type__definition_ORep__range,axiom,
% 0.34/0.61 ! [Rep2: product_unit > $o,Abs2: $o > product_unit,A: set_o] :
% 0.34/0.61 ( ( type_d6188575255521822967unit_o @ Rep2 @ Abs2 @ A )
% 0.34/0.61 => ( ( image_Product_unit_o2 @ Rep2 @ top_to1996260823553986621t_unit )
% 0.34/0.61 = A ) ) ).
% 0.34/0.61
% 0.34/0.61 % type_definition.Rep_range
% 0.34/0.61 thf(fact_4204_type__definition_ORep__range,axiom,
% 0.34/0.61 ! [Rep2: code_natural > nat,Abs2: nat > code_natural,A: set_nat] :
% 0.34/0.61 ( ( type_d4410041424927559462al_nat @ Rep2 @ Abs2 @ A )
% 0.34/0.61 => ( ( image_6215854944615398399al_nat @ Rep2 @ top_to9103716463852441742atural )
% 0.34/0.61 = A ) ) ).
% 0.34/0.61
% 0.34/0.61 % type_definition.Rep_range
% 0.34/0.61 thf(fact_4205_refl__Id,axiom,
% 0.34/0.61 refl_on_Product_unit @ top_to1996260823553986621t_unit @ id_Product_unit ).
% 0.34/0.61
% 0.34/0.61 % refl_Id
% 0.34/0.61 thf(fact_4206_refl__Id,axiom,
% 0.34/0.61 refl_on_o @ top_top_set_o @ id_o2 ).
% 0.34/0.61
% 0.34/0.61 % refl_Id
% 0.34/0.61 thf(fact_4207_refl__Id,axiom,
% 0.34/0.61 refl_on_nat @ top_top_set_nat @ id_nat ).
% 0.34/0.61
% 0.34/0.61 % refl_Id
% 0.34/0.61 thf(fact_4208_refl__Id,axiom,
% 0.34/0.61 refl_on_int @ top_top_set_int @ id_int ).
% 0.34/0.61
% 0.34/0.61 % refl_Id
% 0.34/0.61 thf(fact_4209_boolean__algebra_Ocomplement__unique,axiom,
% 0.34/0.61 ! [A2: product_unit,X: product_unit,Y: product_unit] :
% 0.34/0.61 ( ( ( inf_inf_Product_unit @ A2 @ X )
% 0.34/0.61 = bot_bot_Product_unit )
% 0.34/0.61 => ( ( ( sup_sup_Product_unit @ A2 @ X )
% 0.34/0.61 = top_top_Product_unit )
% 0.34/0.61 => ( ( ( inf_inf_Product_unit @ A2 @ Y )
% 0.34/0.61 = bot_bot_Product_unit )
% 0.34/0.61 => ( ( ( sup_sup_Product_unit @ A2 @ Y )
% 0.34/0.61 = top_top_Product_unit )
% 0.34/0.61 => ( X = Y ) ) ) ) ) ).
% 0.34/0.61
% 0.34/0.61 % boolean_algebra.complement_unique
% 0.34/0.61 thf(fact_4210_boolean__algebra_Ocomplement__unique,axiom,
% 0.34/0.61 ! [A2: set_Pr4532377907799695533_nat_o,X: set_Pr4532377907799695533_nat_o,Y: set_Pr4532377907799695533_nat_o] :
% 0.34/0.61 ( ( ( inf_in1906310914598751387_nat_o @ A2 @ X )
% 0.34/0.61 = bot_bo7824918357723582233_nat_o )
% 0.34/0.61 => ( ( ( sup_su5209123915105501825_nat_o @ A2 @ X )
% 0.34/0.61 = top_to4869887016220417533_nat_o )
% 0.34/0.61 => ( ( ( inf_in1906310914598751387_nat_o @ A2 @ Y )
% 0.34/0.61 = bot_bo7824918357723582233_nat_o )
% 0.34/0.61 => ( ( ( sup_su5209123915105501825_nat_o @ A2 @ Y )
% 0.34/0.61 = top_to4869887016220417533_nat_o )
% 0.34/0.61 => ( X = Y ) ) ) ) ) ).
% 0.34/0.61
% 0.34/0.61 % boolean_algebra.complement_unique
% 0.34/0.61 thf(fact_4211_boolean__algebra_Ocomplement__unique,axiom,
% 0.34/0.61 ! [A2: assn,X: assn,Y: assn] :
% 0.34/0.61 ( ( ( inf_inf_assn @ A2 @ X )
% 0.34/0.61 = bot_bot_assn )
% 0.34/0.61 => ( ( ( sup_sup_assn @ A2 @ X )
% 0.34/0.61 = top_top_assn )
% 0.34/0.61 => ( ( ( inf_inf_assn @ A2 @ Y )
% 0.34/0.61 = bot_bot_assn )
% 0.34/0.61 => ( ( ( sup_sup_assn @ A2 @ Y )
% 0.34/0.61 = top_top_assn )
% 0.34/0.61 => ( X = Y ) ) ) ) ) ).
% 0.34/0.61
% 0.34/0.61 % boolean_algebra.complement_unique
% 0.34/0.61 thf(fact_4212_boolean__algebra_Ocomplement__unique,axiom,
% 0.34/0.61 ! [A2: set_Product_unit,X: set_Product_unit,Y: set_Product_unit] :
% 0.34/0.61 ( ( ( inf_in4660618365625256667t_unit @ A2 @ X )
% 0.34/0.61 = bot_bo3957492148770167129t_unit )
% 0.34/0.61 => ( ( ( sup_su793286257634532545t_unit @ A2 @ X )
% 0.34/0.61 = top_to1996260823553986621t_unit )
% 0.34/0.61 => ( ( ( inf_in4660618365625256667t_unit @ A2 @ Y )
% 0.34/0.61 = bot_bo3957492148770167129t_unit )
% 0.34/0.61 => ( ( ( sup_su793286257634532545t_unit @ A2 @ Y )
% 0.34/0.61 = top_to1996260823553986621t_unit )
% 0.34/0.61 => ( X = Y ) ) ) ) ) ).
% 0.34/0.61
% 0.34/0.61 % boolean_algebra.complement_unique
% 0.34/0.61 thf(fact_4213_boolean__algebra_Ocomplement__unique,axiom,
% 0.34/0.61 ! [A2: set_o,X: set_o,Y: set_o] :
% 0.34/0.61 ( ( ( inf_inf_set_o @ A2 @ X )
% 0.34/0.61 = bot_bot_set_o )
% 0.34/0.61 => ( ( ( sup_sup_set_o @ A2 @ X )
% 0.34/0.61 = top_top_set_o )
% 0.34/0.61 => ( ( ( inf_inf_set_o @ A2 @ Y )
% 0.34/0.61 = bot_bot_set_o )
% 0.34/0.61 => ( ( ( sup_sup_set_o @ A2 @ Y )
% 0.34/0.61 = top_top_set_o )
% 0.34/0.61 => ( X = Y ) ) ) ) ) ).
% 0.34/0.61
% 0.34/0.61 % boolean_algebra.complement_unique
% 0.34/0.61 thf(fact_4214_boolean__algebra_Ocomplement__unique,axiom,
% 0.34/0.61 ! [A2: set_nat,X: set_nat,Y: set_nat] :
% 0.34/0.61 ( ( ( inf_inf_set_nat @ A2 @ X )
% 0.34/0.61 = bot_bot_set_nat )
% 0.34/0.61 => ( ( ( sup_sup_set_nat @ A2 @ X )
% 0.34/0.61 = top_top_set_nat )
% 0.34/0.61 => ( ( ( inf_inf_set_nat @ A2 @ Y )
% 0.34/0.61 = bot_bot_set_nat )
% 0.34/0.61 => ( ( ( sup_sup_set_nat @ A2 @ Y )
% 0.34/0.61 = top_top_set_nat )
% 0.34/0.61 => ( X = Y ) ) ) ) ) ).
% 0.34/0.61
% 0.34/0.61 % boolean_algebra.complement_unique
% 0.34/0.61 thf(fact_4215_boolean__algebra_Ocomplement__unique,axiom,
% 0.34/0.61 ! [A2: set_int,X: set_int,Y: set_int] :
% 0.34/0.61 ( ( ( inf_inf_set_int @ A2 @ X )
% 0.34/0.61 = bot_bot_set_int )
% 0.34/0.61 => ( ( ( sup_sup_set_int @ A2 @ X )
% 0.34/0.61 = top_top_set_int )
% 0.34/0.61 => ( ( ( inf_inf_set_int @ A2 @ Y )
% 0.34/0.61 = bot_bot_set_int )
% 0.34/0.61 => ( ( ( sup_sup_set_int @ A2 @ Y )
% 0.34/0.61 = top_top_set_int )
% 0.34/0.61 => ( X = Y ) ) ) ) ) ).
% 0.34/0.61
% 0.34/0.61 % boolean_algebra.complement_unique
% 0.34/0.61 thf(fact_4216_union__fold__insert,axiom,
% 0.34/0.61 ! [A: set_int,B4: set_int] :
% 0.34/0.61 ( ( finite_finite_int @ A )
% 0.34/0.61 => ( ( sup_sup_set_int @ A @ B4 )
% 0.34/0.61 = ( finite1723285575846220376et_int @ insert_int @ B4 @ A ) ) ) ).
% 0.34/0.61
% 0.34/0.61 % union_fold_insert
% 0.34/0.61 thf(fact_4217_union__fold__insert,axiom,
% 0.34/0.61 ! [A: set_o,B4: set_o] :
% 0.34/0.61 ( ( finite_finite_o @ A )
% 0.34/0.61 => ( ( sup_sup_set_o @ A @ B4 )
% 0.34/0.61 = ( finite_fold_o_set_o @ insert_o @ B4 @ A ) ) ) ).
% 0.34/0.61
% 0.34/0.61 % union_fold_insert
% 0.34/0.61 thf(fact_4218_union__fold__insert,axiom,
% 0.34/0.61 ! [A: set_Product_unit,B4: set_Product_unit] :
% 0.34/0.61 ( ( finite4290736615968046902t_unit @ A )
% 0.34/0.61 => ( ( sup_su793286257634532545t_unit @ A @ B4 )
% 0.34/0.61 = ( finite2740360166580943592t_unit @ insert_Product_unit @ B4 @ A ) ) ) ).
% 0.34/0.61
% 0.34/0.61 % union_fold_insert
% 0.34/0.61 thf(fact_4219_union__fold__insert,axiom,
% 0.34/0.61 ! [A: set_Code_integer,B4: set_Code_integer] :
% 0.34/0.61 ( ( finite6017078050557962740nteger @ A )
% 0.34/0.61 => ( ( sup_su848401254843788991nteger @ A @ B4 )
% 0.34/0.61 = ( finite7486036041278456378nteger @ insert_Code_integer @ B4 @ A ) ) ) ).
% 0.34/0.61
% 0.34/0.61 % union_fold_insert
% 0.34/0.61 thf(fact_4220_union__fold__insert,axiom,
% 0.34/0.61 ! [A: set_nat,B4: set_nat] :
% 0.34/0.61 ( ( finite_finite_nat @ A )
% 0.34/0.61 => ( ( sup_sup_set_nat @ A @ B4 )
% 0.34/0.61 = ( finite5529483035118572448et_nat @ insert_nat @ B4 @ A ) ) ) ).
% 0.34/0.61
% 0.34/0.61 % union_fold_insert
% 0.34/0.61 thf(fact_4221_finite__linorder__min__induct,axiom,
% 0.34/0.61 ! [A: set_Product_unit,P: set_Product_unit > $o] :
% 0.34/0.61 ( ( finite4290736615968046902t_unit @ A )
% 0.34/0.61 => ( ( P @ bot_bo3957492148770167129t_unit )
% 0.34/0.61 => ( ! [B2: product_unit,A9: set_Product_unit] :
% 0.34/0.61 ( ( finite4290736615968046902t_unit @ A9 )
% 0.34/0.61 => ( ! [X6: product_unit] :
% 0.34/0.61 ( ( member_Product_unit @ X6 @ A9 )
% 0.34/0.61 => ( ord_le361264281704409273t_unit @ B2 @ X6 ) )
% 0.34/0.61 => ( ( P @ A9 )
% 0.34/0.61 => ( P @ ( insert_Product_unit @ B2 @ A9 ) ) ) ) )
% 0.34/0.61 => ( P @ A ) ) ) ) ).
% 0.34/0.61
% 0.34/0.61 % finite_linorder_min_induct
% 0.34/0.61 thf(fact_4222_finite__linorder__min__induct,axiom,
% 0.34/0.61 ! [A: set_o,P: set_o > $o] :
% 0.34/0.61 ( ( finite_finite_o @ A )
% 0.34/0.61 => ( ( P @ bot_bot_set_o )
% 0.34/0.61 => ( ! [B2: $o,A9: set_o] :
% 0.34/0.61 ( ( finite_finite_o @ A9 )
% 0.34/0.61 => ( ! [X6: $o] :
% 0.34/0.61 ( ( member_o @ X6 @ A9 )
% 0.34/0.61 => ( ord_less_o @ B2 @ X6 ) )
% 0.34/0.61 => ( ( P @ A9 )
% 0.34/0.61 => ( P @ ( insert_o @ B2 @ A9 ) ) ) ) )
% 0.34/0.61 => ( P @ A ) ) ) ) ).
% 0.34/0.61
% 0.34/0.61 % finite_linorder_min_induct
% 0.34/0.61 thf(fact_4223_finite__linorder__min__induct,axiom,
% 0.34/0.61 ! [A: set_nat,P: set_nat > $o] :
% 0.34/0.61 ( ( finite_finite_nat @ A )
% 0.34/0.61 => ( ( P @ bot_bot_set_nat )
% 0.34/0.61 => ( ! [B2: nat,A9: set_nat] :
% 0.34/0.61 ( ( finite_finite_nat @ A9 )
% 0.34/0.61 => ( ! [X6: nat] :
% 0.34/0.61 ( ( member_nat @ X6 @ A9 )
% 0.34/0.61 => ( ord_less_nat @ B2 @ X6 ) )
% 0.34/0.61 => ( ( P @ A9 )
% 0.34/0.61 => ( P @ ( insert_nat @ B2 @ A9 ) ) ) ) )
% 0.34/0.61 => ( P @ A ) ) ) ) ).
% 0.34/0.61
% 0.34/0.61 % finite_linorder_min_induct
% 0.34/0.61 thf(fact_4224_finite__linorder__min__induct,axiom,
% 0.34/0.61 ! [A: set_int,P: set_int > $o] :
% 0.34/0.61 ( ( finite_finite_int @ A )
% 0.34/0.61 => ( ( P @ bot_bot_set_int )
% 0.34/0.61 => ( ! [B2: int,A9: set_int] :
% 0.34/0.61 ( ( finite_finite_int @ A9 )
% 0.34/0.61 => ( ! [X6: int] :
% 0.34/0.61 ( ( member_int @ X6 @ A9 )
% 0.34/0.61 => ( ord_less_int @ B2 @ X6 ) )
% 0.34/0.61 => ( ( P @ A9 )
% 0.34/0.61 => ( P @ ( insert_int @ B2 @ A9 ) ) ) ) )
% 0.34/0.61 => ( P @ A ) ) ) ) ).
% 0.34/0.61
% 0.34/0.61 % finite_linorder_min_induct
% 0.34/0.61 thf(fact_4225_finite__linorder__min__induct,axiom,
% 0.34/0.61 ! [A: set_Code_integer,P: set_Code_integer > $o] :
% 0.34/0.61 ( ( finite6017078050557962740nteger @ A )
% 0.34/0.61 => ( ( P @ bot_bo3990330152332043303nteger )
% 0.34/0.61 => ( ! [B2: code_integer,A9: set_Code_integer] :
% 0.34/0.61 ( ( finite6017078050557962740nteger @ A9 )
% 0.34/0.61 => ( ! [X6: code_integer] :
% 0.34/0.61 ( ( member_Code_integer @ X6 @ A9 )
% 0.34/0.61 => ( ord_le6747313008572928689nteger @ B2 @ X6 ) )
% 0.34/0.61 => ( ( P @ A9 )
% 0.34/0.61 => ( P @ ( insert_Code_integer @ B2 @ A9 ) ) ) ) )
% 0.34/0.61 => ( P @ A ) ) ) ) ).
% 0.34/0.61
% 0.34/0.61 % finite_linorder_min_induct
% 0.34/0.61 thf(fact_4226_finite__linorder__min__induct,axiom,
% 0.34/0.61 ! [A: set_rat,P: set_rat > $o] :
% 0.34/0.61 ( ( finite_finite_rat @ A )
% 0.34/0.61 => ( ( P @ bot_bot_set_rat )
% 0.34/0.61 => ( ! [B2: rat,A9: set_rat] :
% 0.34/0.61 ( ( finite_finite_rat @ A9 )
% 0.34/0.61 => ( ! [X6: rat] :
% 0.34/0.61 ( ( member_rat @ X6 @ A9 )
% 0.34/0.61 => ( ord_less_rat @ B2 @ X6 ) )
% 0.34/0.61 => ( ( P @ A9 )
% 0.34/0.61 => ( P @ ( insert_rat @ B2 @ A9 ) ) ) ) )
% 0.34/0.61 => ( P @ A ) ) ) ) ).
% 0.34/0.61
% 0.34/0.61 % finite_linorder_min_induct
% 0.34/0.61 thf(fact_4227_finite__linorder__max__induct,axiom,
% 0.34/0.61 ! [A: set_Product_unit,P: set_Product_unit > $o] :
% 0.34/0.61 ( ( finite4290736615968046902t_unit @ A )
% 0.34/0.61 => ( ( P @ bot_bo3957492148770167129t_unit )
% 0.34/0.61 => ( ! [B2: product_unit,A9: set_Product_unit] :
% 0.34/0.61 ( ( finite4290736615968046902t_unit @ A9 )
% 0.34/0.61 => ( ! [X6: product_unit] :
% 0.34/0.61 ( ( member_Product_unit @ X6 @ A9 )
% 0.34/0.61 => ( ord_le361264281704409273t_unit @ X6 @ B2 ) )
% 0.34/0.61 => ( ( P @ A9 )
% 0.34/0.61 => ( P @ ( insert_Product_unit @ B2 @ A9 ) ) ) ) )
% 0.34/0.61 => ( P @ A ) ) ) ) ).
% 0.34/0.61
% 0.34/0.61 % finite_linorder_max_induct
% 0.34/0.61 thf(fact_4228_finite__linorder__max__induct,axiom,
% 0.34/0.61 ! [A: set_o,P: set_o > $o] :
% 0.34/0.61 ( ( finite_finite_o @ A )
% 0.34/0.61 => ( ( P @ bot_bot_set_o )
% 0.34/0.61 => ( ! [B2: $o,A9: set_o] :
% 0.34/0.61 ( ( finite_finite_o @ A9 )
% 0.34/0.61 => ( ! [X6: $o] :
% 0.34/0.61 ( ( member_o @ X6 @ A9 )
% 0.34/0.61 => ( ord_less_o @ X6 @ B2 ) )
% 0.34/0.61 => ( ( P @ A9 )
% 0.34/0.61 => ( P @ ( insert_o @ B2 @ A9 ) ) ) ) )
% 0.34/0.61 => ( P @ A ) ) ) ) ).
% 0.34/0.61
% 0.34/0.61 % finite_linorder_max_induct
% 0.34/0.61 thf(fact_4229_finite__linorder__max__induct,axiom,
% 0.34/0.61 ! [A: set_nat,P: set_nat > $o] :
% 0.34/0.61 ( ( finite_finite_nat @ A )
% 0.34/0.61 => ( ( P @ bot_bot_set_nat )
% 0.34/0.61 => ( ! [B2: nat,A9: set_nat] :
% 0.34/0.61 ( ( finite_finite_nat @ A9 )
% 0.34/0.61 => ( ! [X6: nat] :
% 0.34/0.61 ( ( member_nat @ X6 @ A9 )
% 0.34/0.61 => ( ord_less_nat @ X6 @ B2 ) )
% 0.34/0.61 => ( ( P @ A9 )
% 0.34/0.61 => ( P @ ( insert_nat @ B2 @ A9 ) ) ) ) )
% 0.34/0.61 => ( P @ A ) ) ) ) ).
% 0.34/0.61
% 0.34/0.61 % finite_linorder_max_induct
% 0.34/0.61 thf(fact_4230_finite__linorder__max__induct,axiom,
% 0.34/0.61 ! [A: set_int,P: set_int > $o] :
% 0.34/0.61 ( ( finite_finite_int @ A )
% 0.34/0.61 => ( ( P @ bot_bot_set_int )
% 0.34/0.61 => ( ! [B2: int,A9: set_int] :
% 0.34/0.61 ( ( finite_finite_int @ A9 )
% 0.34/0.61 => ( ! [X6: int] :
% 0.34/0.61 ( ( member_int @ X6 @ A9 )
% 0.34/0.61 => ( ord_less_int @ X6 @ B2 ) )
% 0.34/0.61 => ( ( P @ A9 )
% 0.34/0.61 => ( P @ ( insert_int @ B2 @ A9 ) ) ) ) )
% 0.34/0.61 => ( P @ A ) ) ) ) ).
% 0.34/0.61
% 0.34/0.61 % finite_linorder_max_induct
% 0.34/0.61 thf(fact_4231_finite__linorder__max__induct,axiom,
% 0.34/0.61 ! [A: set_Code_integer,P: set_Code_integer > $o] :
% 0.34/0.61 ( ( finite6017078050557962740nteger @ A )
% 0.34/0.61 => ( ( P @ bot_bo3990330152332043303nteger )
% 0.34/0.61 => ( ! [B2: code_integer,A9: set_Code_integer] :
% 0.34/0.61 ( ( finite6017078050557962740nteger @ A9 )
% 0.34/0.61 => ( ! [X6: code_integer] :
% 0.34/0.61 ( ( member_Code_integer @ X6 @ A9 )
% 0.34/0.61 => ( ord_le6747313008572928689nteger @ X6 @ B2 ) )
% 0.34/0.61 => ( ( P @ A9 )
% 0.34/0.61 => ( P @ ( insert_Code_integer @ B2 @ A9 ) ) ) ) )
% 0.34/0.61 => ( P @ A ) ) ) ) ).
% 0.34/0.61
% 0.34/0.61 % finite_linorder_max_induct
% 0.34/0.61 thf(fact_4232_finite__linorder__max__induct,axiom,
% 0.34/0.61 ! [A: set_rat,P: set_rat > $o] :
% 0.34/0.61 ( ( finite_finite_rat @ A )
% 0.34/0.61 => ( ( P @ bot_bot_set_rat )
% 0.34/0.61 => ( ! [B2: rat,A9: set_rat] :
% 0.34/0.61 ( ( finite_finite_rat @ A9 )
% 0.34/0.61 => ( ! [X6: rat] :
% 0.34/0.61 ( ( member_rat @ X6 @ A9 )
% 0.34/0.61 => ( ord_less_rat @ X6 @ B2 ) )
% 0.34/0.61 => ( ( P @ A9 )
% 0.34/0.61 => ( P @ ( insert_rat @ B2 @ A9 ) ) ) ) )
% 0.34/0.61 => ( P @ A ) ) ) ) ).
% 0.34/0.61
% 0.34/0.61 % finite_linorder_max_induct
% 0.34/0.61 thf(fact_4233_finite__Sup__less__iff,axiom,
% 0.34/0.61 ! [X4: set_int,A2: int] :
% 0.34/0.61 ( ( finite_finite_int @ X4 )
% 0.34/0.61 => ( ( X4 != bot_bot_set_int )
% 0.34/0.61 => ( ( ord_less_int @ ( complete_Sup_Sup_int @ X4 ) @ A2 )
% 0.34/0.61 = ( ! [X2: int] :
% 0.34/0.61 ( ( member_int @ X2 @ X4 )
% 0.34/0.61 => ( ord_less_int @ X2 @ A2 ) ) ) ) ) ) ).
% 0.34/0.61
% 0.34/0.61 % finite_Sup_less_iff
% 0.34/0.61 thf(fact_4234_finite__Sup__less__iff,axiom,
% 0.34/0.61 ! [X4: set_nat,A2: nat] :
% 0.34/0.61 ( ( finite_finite_nat @ X4 )
% 0.34/0.61 => ( ( X4 != bot_bot_set_nat )
% 0.34/0.61 => ( ( ord_less_nat @ ( complete_Sup_Sup_nat @ X4 ) @ A2 )
% 0.34/0.61 = ( ! [X2: nat] :
% 0.34/0.61 ( ( member_nat @ X2 @ X4 )
% 0.34/0.61 => ( ord_less_nat @ X2 @ A2 ) ) ) ) ) ) ).
% 0.34/0.61
% 0.34/0.61 % finite_Sup_less_iff
% 0.34/0.61 thf(fact_4235_finite__Sup__less__iff,axiom,
% 0.34/0.61 ! [X4: set_Product_unit,A2: product_unit] :
% 0.34/0.61 ( ( finite4290736615968046902t_unit @ X4 )
% 0.34/0.61 => ( ( X4 != bot_bo3957492148770167129t_unit )
% 0.34/0.61 => ( ( ord_le361264281704409273t_unit @ ( comple4687483117567863418t_unit @ X4 ) @ A2 )
% 0.34/0.61 = ( ! [X2: product_unit] :
% 0.34/0.61 ( ( member_Product_unit @ X2 @ X4 )
% 0.34/0.61 => ( ord_le361264281704409273t_unit @ X2 @ A2 ) ) ) ) ) ) ).
% 0.34/0.61
% 0.34/0.61 % finite_Sup_less_iff
% 0.34/0.61 thf(fact_4236_Max__eq__iff,axiom,
% 0.34/0.61 ! [A: set_Code_integer,M4: code_integer] :
% 0.34/0.61 ( ( finite6017078050557962740nteger @ A )
% 0.34/0.61 => ( ( A != bot_bo3990330152332043303nteger )
% 0.34/0.61 => ( ( ( lattic4901227151466704046nteger @ A )
% 0.34/0.61 = M4 )
% 0.34/0.61 = ( ( member_Code_integer @ M4 @ A )
% 0.34/0.61 & ! [X2: code_integer] :
% 0.34/0.61 ( ( member_Code_integer @ X2 @ A )
% 0.34/0.61 => ( ord_le3102999989581377725nteger @ X2 @ M4 ) ) ) ) ) ) ).
% 0.34/0.61
% 0.34/0.61 % Max_eq_iff
% 0.34/0.61 thf(fact_4237_Max__eq__iff,axiom,
% 0.34/0.61 ! [A: set_Product_unit,M4: product_unit] :
% 0.34/0.61 ( ( finite4290736615968046902t_unit @ A )
% 0.34/0.61 => ( ( A != bot_bo3957492148770167129t_unit )
% 0.34/0.61 => ( ( ( lattic1210392890313158908t_unit @ A )
% 0.34/0.61 = M4 )
% 0.34/0.61 = ( ( member_Product_unit @ M4 @ A )
% 0.34/0.61 & ! [X2: product_unit] :
% 0.34/0.61 ( ( member_Product_unit @ X2 @ A )
% 0.34/0.61 => ( ord_le3221252021190050221t_unit @ X2 @ M4 ) ) ) ) ) ) ).
% 0.34/0.61
% 0.34/0.61 % Max_eq_iff
% 0.34/0.61 thf(fact_4238_Max__eq__iff,axiom,
% 0.34/0.61 ! [A: set_o,M4: $o] :
% 0.34/0.61 ( ( finite_finite_o @ A )
% 0.34/0.61 => ( ( A != bot_bot_set_o )
% 0.34/0.61 => ( ( ( lattic1921953407002678535_Max_o @ A )
% 0.34/0.61 = M4 )
% 0.34/0.61 = ( ( member_o @ M4 @ A )
% 0.34/0.61 & ! [X2: $o] :
% 0.34/0.61 ( ( member_o @ X2 @ A )
% 0.34/0.61 => ( ord_less_eq_o @ X2 @ M4 ) ) ) ) ) ) ).
% 0.34/0.61
% 0.34/0.61 % Max_eq_iff
% 0.34/0.61 thf(fact_4239_Max__eq__iff,axiom,
% 0.34/0.61 ! [A: set_rat,M4: rat] :
% 0.34/0.61 ( ( finite_finite_rat @ A )
% 0.34/0.61 => ( ( A != bot_bot_set_rat )
% 0.34/0.61 => ( ( ( lattic7630753665789217321ax_rat @ A )
% 0.34/0.61 = M4 )
% 0.34/0.61 = ( ( member_rat @ M4 @ A )
% 0.34/0.61 & ! [X2: rat] :
% 0.34/0.61 ( ( member_rat @ X2 @ A )
% 0.34/0.61 => ( ord_less_eq_rat @ X2 @ M4 ) ) ) ) ) ) ).
% 0.34/0.61
% 0.34/0.61 % Max_eq_iff
% 0.34/0.61 thf(fact_4240_Max__eq__iff,axiom,
% 0.34/0.61 ! [A: set_int,M4: int] :
% 0.34/0.61 ( ( finite_finite_int @ A )
% 0.34/0.61 => ( ( A != bot_bot_set_int )
% 0.34/0.61 => ( ( ( lattic8263393255366662781ax_int @ A )
% 0.34/0.61 = M4 )
% 0.34/0.61 = ( ( member_int @ M4 @ A )
% 0.34/0.61 & ! [X2: int] :
% 0.34/0.61 ( ( member_int @ X2 @ A )
% 0.34/0.61 => ( ord_less_eq_int @ X2 @ M4 ) ) ) ) ) ) ).
% 0.34/0.61
% 0.34/0.61 % Max_eq_iff
% 0.34/0.61 thf(fact_4241_Max__eq__iff,axiom,
% 0.34/0.61 ! [A: set_nat,M4: nat] :
% 0.34/0.61 ( ( finite_finite_nat @ A )
% 0.34/0.61 => ( ( A != bot_bot_set_nat )
% 0.34/0.61 => ( ( ( lattic8265883725875713057ax_nat @ A )
% 0.34/0.61 = M4 )
% 0.34/0.61 = ( ( member_nat @ M4 @ A )
% 0.34/0.61 & ! [X2: nat] :
% 0.34/0.61 ( ( member_nat @ X2 @ A )
% 0.34/0.61 => ( ord_less_eq_nat @ X2 @ M4 ) ) ) ) ) ) ).
% 0.34/0.61
% 0.34/0.61 % Max_eq_iff
% 0.34/0.61 thf(fact_4242_Max__ge__iff,axiom,
% 0.34/0.61 ! [A: set_Code_integer,X: code_integer] :
% 0.34/0.61 ( ( finite6017078050557962740nteger @ A )
% 0.34/0.61 => ( ( A != bot_bo3990330152332043303nteger )
% 0.34/0.61 => ( ( ord_le3102999989581377725nteger @ X @ ( lattic4901227151466704046nteger @ A ) )
% 0.34/0.61 = ( ? [X2: code_integer] :
% 0.34/0.61 ( ( member_Code_integer @ X2 @ A )
% 0.34/0.61 & ( ord_le3102999989581377725nteger @ X @ X2 ) ) ) ) ) ) ).
% 0.34/0.61
% 0.34/0.61 % Max_ge_iff
% 0.34/0.61 thf(fact_4243_Max__ge__iff,axiom,
% 0.34/0.61 ! [A: set_Product_unit,X: product_unit] :
% 0.34/0.61 ( ( finite4290736615968046902t_unit @ A )
% 0.34/0.61 => ( ( A != bot_bo3957492148770167129t_unit )
% 0.34/0.61 => ( ( ord_le3221252021190050221t_unit @ X @ ( lattic1210392890313158908t_unit @ A ) )
% 0.34/0.61 = ( ? [X2: product_unit] :
% 0.34/0.61 ( ( member_Product_unit @ X2 @ A )
% 0.34/0.61 & ( ord_le3221252021190050221t_unit @ X @ X2 ) ) ) ) ) ) ).
% 0.34/0.61
% 0.34/0.61 % Max_ge_iff
% 0.34/0.61 thf(fact_4244_Max__ge__iff,axiom,
% 0.34/0.61 ! [A: set_o,X: $o] :
% 0.34/0.61 ( ( finite_finite_o @ A )
% 0.34/0.61 => ( ( A != bot_bot_set_o )
% 0.34/0.61 => ( ( ord_less_eq_o @ X @ ( lattic1921953407002678535_Max_o @ A ) )
% 0.34/0.61 = ( ? [X2: $o] :
% 0.34/0.61 ( ( member_o @ X2 @ A )
% 0.34/0.61 & ( ord_less_eq_o @ X @ X2 ) ) ) ) ) ) ).
% 0.34/0.61
% 0.34/0.61 % Max_ge_iff
% 0.34/0.61 thf(fact_4245_Max__ge__iff,axiom,
% 0.34/0.61 ! [A: set_rat,X: rat] :
% 0.34/0.61 ( ( finite_finite_rat @ A )
% 0.34/0.61 => ( ( A != bot_bot_set_rat )
% 0.34/0.61 => ( ( ord_less_eq_rat @ X @ ( lattic7630753665789217321ax_rat @ A ) )
% 0.34/0.61 = ( ? [X2: rat] :
% 0.34/0.61 ( ( member_rat @ X2 @ A )
% 0.34/0.61 & ( ord_less_eq_rat @ X @ X2 ) ) ) ) ) ) ).
% 0.34/0.61
% 0.34/0.61 % Max_ge_iff
% 0.34/0.61 thf(fact_4246_Max__ge__iff,axiom,
% 0.34/0.61 ! [A: set_int,X: int] :
% 0.34/0.61 ( ( finite_finite_int @ A )
% 0.34/0.61 => ( ( A != bot_bot_set_int )
% 0.34/0.61 => ( ( ord_less_eq_int @ X @ ( lattic8263393255366662781ax_int @ A ) )
% 0.34/0.61 = ( ? [X2: int] :
% 0.34/0.61 ( ( member_int @ X2 @ A )
% 0.34/0.61 & ( ord_less_eq_int @ X @ X2 ) ) ) ) ) ) ).
% 0.34/0.61
% 0.34/0.61 % Max_ge_iff
% 0.34/0.61 thf(fact_4247_Max__ge__iff,axiom,
% 0.34/0.61 ! [A: set_nat,X: nat] :
% 0.34/0.61 ( ( finite_finite_nat @ A )
% 0.34/0.61 => ( ( A != bot_bot_set_nat )
% 0.34/0.61 => ( ( ord_less_eq_nat @ X @ ( lattic8265883725875713057ax_nat @ A ) )
% 0.34/0.61 = ( ? [X2: nat] :
% 0.34/0.61 ( ( member_nat @ X2 @ A )
% 0.34/0.61 & ( ord_less_eq_nat @ X @ X2 ) ) ) ) ) ) ).
% 0.34/0.61
% 0.34/0.61 % Max_ge_iff
% 0.34/0.61 thf(fact_4248_eq__Max__iff,axiom,
% 0.34/0.61 ! [A: set_Code_integer,M4: code_integer] :
% 0.34/0.61 ( ( finite6017078050557962740nteger @ A )
% 0.34/0.61 => ( ( A != bot_bo3990330152332043303nteger )
% 0.34/0.61 => ( ( M4
% 0.34/0.61 = ( lattic4901227151466704046nteger @ A ) )
% 0.34/0.61 = ( ( member_Code_integer @ M4 @ A )
% 0.34/0.61 & ! [X2: code_integer] :
% 0.34/0.61 ( ( member_Code_integer @ X2 @ A )
% 0.34/0.61 => ( ord_le3102999989581377725nteger @ X2 @ M4 ) ) ) ) ) ) ).
% 0.34/0.61
% 0.34/0.61 % eq_Max_iff
% 0.34/0.61 thf(fact_4249_eq__Max__iff,axiom,
% 0.34/0.61 ! [A: set_Product_unit,M4: product_unit] :
% 0.34/0.61 ( ( finite4290736615968046902t_unit @ A )
% 0.34/0.61 => ( ( A != bot_bo3957492148770167129t_unit )
% 0.34/0.61 => ( ( M4
% 0.34/0.61 = ( lattic1210392890313158908t_unit @ A ) )
% 0.34/0.61 = ( ( member_Product_unit @ M4 @ A )
% 0.34/0.61 & ! [X2: product_unit] :
% 0.34/0.61 ( ( member_Product_unit @ X2 @ A )
% 0.34/0.61 => ( ord_le3221252021190050221t_unit @ X2 @ M4 ) ) ) ) ) ) ).
% 0.34/0.61
% 0.34/0.61 % eq_Max_iff
% 0.34/0.61 thf(fact_4250_eq__Max__iff,axiom,
% 0.34/0.61 ! [A: set_o,M4: $o] :
% 0.34/0.61 ( ( finite_finite_o @ A )
% 0.34/0.61 => ( ( A != bot_bot_set_o )
% 0.34/0.61 => ( ( M4
% 0.34/0.61 = ( lattic1921953407002678535_Max_o @ A ) )
% 0.34/0.61 = ( ( member_o @ M4 @ A )
% 0.34/0.61 & ! [X2: $o] :
% 0.34/0.61 ( ( member_o @ X2 @ A )
% 0.34/0.61 => ( ord_less_eq_o @ X2 @ M4 ) ) ) ) ) ) ).
% 0.34/0.61
% 0.34/0.61 % eq_Max_iff
% 0.34/0.61 thf(fact_4251_eq__Max__iff,axiom,
% 0.34/0.61 ! [A: set_rat,M4: rat] :
% 0.34/0.61 ( ( finite_finite_rat @ A )
% 0.34/0.61 => ( ( A != bot_bot_set_rat )
% 0.34/0.61 => ( ( M4
% 0.34/0.61 = ( lattic7630753665789217321ax_rat @ A ) )
% 0.34/0.61 = ( ( member_rat @ M4 @ A )
% 0.34/0.61 & ! [X2: rat] :
% 0.34/0.61 ( ( member_rat @ X2 @ A )
% 0.34/0.61 => ( ord_less_eq_rat @ X2 @ M4 ) ) ) ) ) ) ).
% 0.34/0.61
% 0.34/0.61 % eq_Max_iff
% 0.34/0.61 thf(fact_4252_eq__Max__iff,axiom,
% 0.34/0.61 ! [A: set_int,M4: int] :
% 0.34/0.61 ( ( finite_finite_int @ A )
% 0.34/0.61 => ( ( A != bot_bot_set_int )
% 0.34/0.61 => ( ( M4
% 0.34/0.61 = ( lattic8263393255366662781ax_int @ A ) )
% 0.34/0.61 = ( ( member_int @ M4 @ A )
% 0.34/0.61 & ! [X2: int] :
% 0.34/0.61 ( ( member_int @ X2 @ A )
% 0.34/0.61 => ( ord_less_eq_int @ X2 @ M4 ) ) ) ) ) ) ).
% 0.34/0.61
% 0.34/0.61 % eq_Max_iff
% 0.34/0.61 thf(fact_4253_eq__Max__iff,axiom,
% 0.34/0.61 ! [A: set_nat,M4: nat] :
% 0.34/0.61 ( ( finite_finite_nat @ A )
% 0.34/0.61 => ( ( A != bot_bot_set_nat )
% 0.34/0.61 => ( ( M4
% 0.34/0.61 = ( lattic8265883725875713057ax_nat @ A ) )
% 0.34/0.61 = ( ( member_nat @ M4 @ A )
% 0.34/0.61 & ! [X2: nat] :
% 0.34/0.61 ( ( member_nat @ X2 @ A )
% 0.34/0.61 => ( ord_less_eq_nat @ X2 @ M4 ) ) ) ) ) ) ).
% 0.34/0.61
% 0.34/0.61 % eq_Max_iff
% 0.34/0.61 thf(fact_4254_Max_OboundedE,axiom,
% 0.34/0.61 ! [A: set_Code_integer,X: code_integer] :
% 0.34/0.61 ( ( finite6017078050557962740nteger @ A )
% 0.34/0.61 => ( ( A != bot_bo3990330152332043303nteger )
% 0.34/0.61 => ( ( ord_le3102999989581377725nteger @ ( lattic4901227151466704046nteger @ A ) @ X )
% 0.34/0.61 => ! [A10: code_integer] :
% 0.34/0.61 ( ( member_Code_integer @ A10 @ A )
% 0.34/0.61 => ( ord_le3102999989581377725nteger @ A10 @ X ) ) ) ) ) ).
% 0.34/0.61
% 0.34/0.61 % Max.boundedE
% 0.34/0.61 thf(fact_4255_Max_OboundedE,axiom,
% 0.34/0.61 ! [A: set_Product_unit,X: product_unit] :
% 0.34/0.61 ( ( finite4290736615968046902t_unit @ A )
% 0.34/0.61 => ( ( A != bot_bo3957492148770167129t_unit )
% 0.34/0.61 => ( ( ord_le3221252021190050221t_unit @ ( lattic1210392890313158908t_unit @ A ) @ X )
% 0.34/0.61 => ! [A10: product_unit] :
% 0.34/0.61 ( ( member_Product_unit @ A10 @ A )
% 0.34/0.61 => ( ord_le3221252021190050221t_unit @ A10 @ X ) ) ) ) ) ).
% 0.34/0.61
% 0.34/0.61 % Max.boundedE
% 0.34/0.61 thf(fact_4256_Max_OboundedE,axiom,
% 0.34/0.61 ! [A: set_o,X: $o] :
% 0.34/0.61 ( ( finite_finite_o @ A )
% 0.34/0.61 => ( ( A != bot_bot_set_o )
% 0.34/0.61 => ( ( ord_less_eq_o @ ( lattic1921953407002678535_Max_o @ A ) @ X )
% 0.34/0.61 => ! [A10: $o] :
% 0.34/0.61 ( ( member_o @ A10 @ A )
% 0.34/0.61 => ( ord_less_eq_o @ A10 @ X ) ) ) ) ) ).
% 0.34/0.61
% 0.34/0.61 % Max.boundedE
% 0.34/0.61 thf(fact_4257_Max_OboundedE,axiom,
% 0.34/0.61 ! [A: set_rat,X: rat] :
% 0.34/0.61 ( ( finite_finite_rat @ A )
% 0.34/0.61 => ( ( A != bot_bot_set_rat )
% 0.34/0.61 => ( ( ord_less_eq_rat @ ( lattic7630753665789217321ax_rat @ A ) @ X )
% 0.34/0.61 => ! [A10: rat] :
% 0.34/0.61 ( ( member_rat @ A10 @ A )
% 0.34/0.61 => ( ord_less_eq_rat @ A10 @ X ) ) ) ) ) ).
% 0.34/0.61
% 0.34/0.61 % Max.boundedE
% 0.34/0.61 thf(fact_4258_Max_OboundedE,axiom,
% 0.34/0.61 ! [A: set_int,X: int] :
% 0.34/0.61 ( ( finite_finite_int @ A )
% 0.34/0.61 => ( ( A != bot_bot_set_int )
% 0.34/0.61 => ( ( ord_less_eq_int @ ( lattic8263393255366662781ax_int @ A ) @ X )
% 0.34/0.61 => ! [A10: int] :
% 0.34/0.61 ( ( member_int @ A10 @ A )
% 0.34/0.61 => ( ord_less_eq_int @ A10 @ X ) ) ) ) ) ).
% 0.34/0.61
% 0.34/0.61 % Max.boundedE
% 0.34/0.61 thf(fact_4259_Max_OboundedE,axiom,
% 0.34/0.61 ! [A: set_nat,X: nat] :
% 0.34/0.61 ( ( finite_finite_nat @ A )
% 0.34/0.61 => ( ( A != bot_bot_set_nat )
% 0.34/0.61 => ( ( ord_less_eq_nat @ ( lattic8265883725875713057ax_nat @ A ) @ X )
% 0.34/0.61 => ! [A10: nat] :
% 0.34/0.61 ( ( member_nat @ A10 @ A )
% 0.34/0.61 => ( ord_less_eq_nat @ A10 @ X ) ) ) ) ) ).
% 0.34/0.61
% 0.34/0.61 % Max.boundedE
% 0.34/0.61 thf(fact_4260_Max_OboundedI,axiom,
% 0.34/0.61 ! [A: set_Code_integer,X: code_integer] :
% 0.34/0.61 ( ( finite6017078050557962740nteger @ A )
% 0.34/0.61 => ( ( A != bot_bo3990330152332043303nteger )
% 0.34/0.61 => ( ! [A4: code_integer] :
% 0.34/0.61 ( ( member_Code_integer @ A4 @ A )
% 0.34/0.61 => ( ord_le3102999989581377725nteger @ A4 @ X ) )
% 0.34/0.61 => ( ord_le3102999989581377725nteger @ ( lattic4901227151466704046nteger @ A ) @ X ) ) ) ) ).
% 0.34/0.61
% 0.34/0.61 % Max.boundedI
% 0.34/0.61 thf(fact_4261_Max_OboundedI,axiom,
% 0.34/0.61 ! [A: set_Product_unit,X: product_unit] :
% 0.34/0.61 ( ( finite4290736615968046902t_unit @ A )
% 0.34/0.61 => ( ( A != bot_bo3957492148770167129t_unit )
% 0.34/0.61 => ( ! [A4: product_unit] :
% 0.34/0.61 ( ( member_Product_unit @ A4 @ A )
% 0.34/0.61 => ( ord_le3221252021190050221t_unit @ A4 @ X ) )
% 0.34/0.61 => ( ord_le3221252021190050221t_unit @ ( lattic1210392890313158908t_unit @ A ) @ X ) ) ) ) ).
% 0.34/0.61
% 0.34/0.61 % Max.boundedI
% 0.34/0.61 thf(fact_4262_Max_OboundedI,axiom,
% 0.34/0.61 ! [A: set_o,X: $o] :
% 0.34/0.61 ( ( finite_finite_o @ A )
% 0.34/0.61 => ( ( A != bot_bot_set_o )
% 0.34/0.61 => ( ! [A4: $o] :
% 0.34/0.61 ( ( member_o @ A4 @ A )
% 0.34/0.61 => ( ord_less_eq_o @ A4 @ X ) )
% 0.34/0.61 => ( ord_less_eq_o @ ( lattic1921953407002678535_Max_o @ A ) @ X ) ) ) ) ).
% 0.34/0.61
% 0.34/0.61 % Max.boundedI
% 0.34/0.61 thf(fact_4263_Max_OboundedI,axiom,
% 0.34/0.61 ! [A: set_rat,X: rat] :
% 0.34/0.61 ( ( finite_finite_rat @ A )
% 0.34/0.61 => ( ( A != bot_bot_set_rat )
% 0.34/0.61 => ( ! [A4: rat] :
% 0.34/0.61 ( ( member_rat @ A4 @ A )
% 0.34/0.61 => ( ord_less_eq_rat @ A4 @ X ) )
% 0.34/0.61 => ( ord_less_eq_rat @ ( lattic7630753665789217321ax_rat @ A ) @ X ) ) ) ) ).
% 0.34/0.61
% 0.34/0.61 % Max.boundedI
% 0.34/0.61 thf(fact_4264_Max_OboundedI,axiom,
% 0.34/0.61 ! [A: set_int,X: int] :
% 0.34/0.61 ( ( finite_finite_int @ A )
% 0.34/0.61 => ( ( A != bot_bot_set_int )
% 0.34/0.61 => ( ! [A4: int] :
% 0.34/0.61 ( ( member_int @ A4 @ A )
% 0.34/0.61 => ( ord_less_eq_int @ A4 @ X ) )
% 0.34/0.61 => ( ord_less_eq_int @ ( lattic8263393255366662781ax_int @ A ) @ X ) ) ) ) ).
% 0.34/0.61
% 0.34/0.61 % Max.boundedI
% 0.34/0.61 thf(fact_4265_Max_OboundedI,axiom,
% 0.34/0.61 ! [A: set_nat,X: nat] :
% 0.34/0.61 ( ( finite_finite_nat @ A )
% 0.34/0.61 => ( ( A != bot_bot_set_nat )
% 0.34/0.61 => ( ! [A4: nat] :
% 0.34/0.61 ( ( member_nat @ A4 @ A )
% 0.34/0.61 => ( ord_less_eq_nat @ A4 @ X ) )
% 0.34/0.61 => ( ord_less_eq_nat @ ( lattic8265883725875713057ax_nat @ A ) @ X ) ) ) ) ).
% 0.34/0.61
% 0.34/0.61 % Max.boundedI
% 0.34/0.61 thf(fact_4266_Max__insert2,axiom,
% 0.34/0.61 ! [A: set_o,A2: $o] :
% 0.34/0.61 ( ( finite_finite_o @ A )
% 0.34/0.61 => ( ! [B2: $o] :
% 0.34/0.61 ( ( member_o @ B2 @ A )
% 0.34/0.61 => ( ord_less_eq_o @ B2 @ A2 ) )
% 0.34/0.61 => ( ( lattic1921953407002678535_Max_o @ ( insert_o @ A2 @ A ) )
% 0.34/0.61 = A2 ) ) ) ).
% 0.34/0.61
% 0.34/0.61 % Max_insert2
% 0.34/0.61 thf(fact_4267_Max__insert2,axiom,
% 0.34/0.61 ! [A: set_Product_unit,A2: product_unit] :
% 0.34/0.61 ( ( finite4290736615968046902t_unit @ A )
% 0.34/0.61 => ( ! [B2: product_unit] :
% 0.34/0.61 ( ( member_Product_unit @ B2 @ A )
% 0.34/0.61 => ( ord_le3221252021190050221t_unit @ B2 @ A2 ) )
% 0.34/0.61 => ( ( lattic1210392890313158908t_unit @ ( insert_Product_unit @ A2 @ A ) )
% 0.34/0.61 = A2 ) ) ) ).
% 0.34/0.61
% 0.34/0.61 % Max_insert2
% 0.34/0.61 thf(fact_4268_Max__insert2,axiom,
% 0.34/0.61 ! [A: set_Code_integer,A2: code_integer] :
% 0.34/0.61 ( ( finite6017078050557962740nteger @ A )
% 0.34/0.61 => ( ! [B2: code_integer] :
% 0.34/0.61 ( ( member_Code_integer @ B2 @ A )
% 0.34/0.61 => ( ord_le3102999989581377725nteger @ B2 @ A2 ) )
% 0.34/0.61 => ( ( lattic4901227151466704046nteger @ ( insert_Code_integer @ A2 @ A ) )
% 0.34/0.61 = A2 ) ) ) ).
% 0.34/0.61
% 0.34/0.61 % Max_insert2
% 0.34/0.61 thf(fact_4269_Max__insert2,axiom,
% 0.34/0.61 ! [A: set_rat,A2: rat] :
% 0.34/0.61 ( ( finite_finite_rat @ A )
% 0.34/0.61 => ( ! [B2: rat] :
% 0.34/0.61 ( ( member_rat @ B2 @ A )
% 0.34/0.61 => ( ord_less_eq_rat @ B2 @ A2 ) )
% 0.34/0.61 => ( ( lattic7630753665789217321ax_rat @ ( insert_rat @ A2 @ A ) )
% 0.34/0.61 = A2 ) ) ) ).
% 0.34/0.61
% 0.34/0.61 % Max_insert2
% 0.34/0.61 thf(fact_4270_Max__insert2,axiom,
% 0.34/0.61 ! [A: set_int,A2: int] :
% 0.34/0.61 ( ( finite_finite_int @ A )
% 0.34/0.61 => ( ! [B2: int] :
% 0.34/0.61 ( ( member_int @ B2 @ A )
% 0.34/0.61 => ( ord_less_eq_int @ B2 @ A2 ) )
% 0.34/0.61 => ( ( lattic8263393255366662781ax_int @ ( insert_int @ A2 @ A ) )
% 0.34/0.61 = A2 ) ) ) ).
% 0.34/0.61
% 0.34/0.61 % Max_insert2
% 0.34/0.61 thf(fact_4271_Max__insert2,axiom,
% 0.34/0.61 ! [A: set_nat,A2: nat] :
% 0.34/0.61 ( ( finite_finite_nat @ A )
% 0.34/0.61 => ( ! [B2: nat] :
% 0.34/0.61 ( ( member_nat @ B2 @ A )
% 0.34/0.61 => ( ord_less_eq_nat @ B2 @ A2 ) )
% 0.34/0.61 => ( ( lattic8265883725875713057ax_nat @ ( insert_nat @ A2 @ A ) )
% 0.34/0.61 = A2 ) ) ) ).
% 0.34/0.61
% 0.34/0.61 % Max_insert2
% 0.34/0.61 thf(fact_4272_remove__subset,axiom,
% 0.34/0.61 ! [X: product_unit,S: set_Product_unit] :
% 0.34/0.61 ( ( member_Product_unit @ X @ S )
% 0.34/0.61 => ( ord_le8056459307392131481t_unit @ ( minus_6452836326544984404t_unit @ S @ ( insert_Product_unit @ X @ bot_bo3957492148770167129t_unit ) ) @ S ) ) ).
% 0.34/0.61
% 0.34/0.61 % remove_subset
% 0.34/0.61 thf(fact_4273_remove__subset,axiom,
% 0.34/0.61 ! [X: $o,S: set_o] :
% 0.34/0.61 ( ( member_o @ X @ S )
% 0.34/0.61 => ( ord_less_set_o @ ( minus_minus_set_o @ S @ ( insert_o @ X @ bot_bot_set_o ) ) @ S ) ) ).
% 0.34/0.61
% 0.34/0.61 % remove_subset
% 0.34/0.61 thf(fact_4274_remove__subset,axiom,
% 0.34/0.61 ! [X: int,S: set_int] :
% 0.34/0.61 ( ( member_int @ X @ S )
% 0.34/0.61 => ( ord_less_set_int @ ( minus_minus_set_int @ S @ ( insert_int @ X @ bot_bot_set_int ) ) @ S ) ) ).
% 0.34/0.61
% 0.34/0.61 % remove_subset
% 0.34/0.61 thf(fact_4275_remove__subset,axiom,
% 0.34/0.61 ! [X: produc3658429121746597890et_nat > $o,S: set_Pr4532377907799695533_nat_o] :
% 0.34/0.61 ( ( member6576561426505652726_nat_o @ X @ S )
% 0.34/0.61 => ( ord_le2453136405763929_nat_o @ ( minus_1801376950450012436_nat_o @ S @ ( insert5175938949040314269_nat_o @ X @ bot_bo7824918357723582233_nat_o ) ) @ S ) ) ).
% 0.34/0.61
% 0.34/0.61 % remove_subset
% 0.34/0.61 thf(fact_4276_remove__subset,axiom,
% 0.34/0.61 ! [X: nat,S: set_nat] :
% 0.34/0.61 ( ( member_nat @ X @ S )
% 0.34/0.61 => ( ord_less_set_nat @ ( minus_minus_set_nat @ S @ ( insert_nat @ X @ bot_bot_set_nat ) ) @ S ) ) ).
% 0.34/0.61
% 0.34/0.61 % remove_subset
% 0.34/0.61 thf(fact_4277_cSup__eq__Max,axiom,
% 0.34/0.61 ! [X4: set_int] :
% 0.34/0.61 ( ( finite_finite_int @ X4 )
% 0.34/0.61 => ( ( X4 != bot_bot_set_int )
% 0.34/0.61 => ( ( complete_Sup_Sup_int @ X4 )
% 0.34/0.61 = ( lattic8263393255366662781ax_int @ X4 ) ) ) ) ).
% 0.34/0.61
% 0.34/0.61 % cSup_eq_Max
% 0.34/0.61 thf(fact_4278_cSup__eq__Max,axiom,
% 0.34/0.61 ! [X4: set_Product_unit] :
% 0.34/0.61 ( ( finite4290736615968046902t_unit @ X4 )
% 0.34/0.61 => ( ( X4 != bot_bo3957492148770167129t_unit )
% 0.34/0.61 => ( ( comple4687483117567863418t_unit @ X4 )
% 0.34/0.61 = ( lattic1210392890313158908t_unit @ X4 ) ) ) ) ).
% 0.34/0.61
% 0.34/0.61 % cSup_eq_Max
% 0.34/0.61 thf(fact_4279_cSup__eq__Max,axiom,
% 0.34/0.61 ! [X4: set_nat] :
% 0.34/0.61 ( ( finite_finite_nat @ X4 )
% 0.34/0.61 => ( ( X4 != bot_bot_set_nat )
% 0.34/0.61 => ( ( complete_Sup_Sup_nat @ X4 )
% 0.34/0.61 = ( lattic8265883725875713057ax_nat @ X4 ) ) ) ) ).
% 0.34/0.61
% 0.34/0.61 % cSup_eq_Max
% 0.34/0.61 thf(fact_4280_Max__Sup,axiom,
% 0.34/0.61 ! [A: set_Product_unit] :
% 0.34/0.61 ( ( finite4290736615968046902t_unit @ A )
% 0.34/0.61 => ( ( A != bot_bo3957492148770167129t_unit )
% 0.34/0.61 => ( ( lattic1210392890313158908t_unit @ A )
% 0.34/0.61 = ( comple4687483117567863418t_unit @ A ) ) ) ) ).
% 0.34/0.61
% 0.34/0.61 % Max_Sup
% 0.34/0.61 thf(fact_4281_disjoint__alt__simp3,axiom,
% 0.34/0.61 ! [A: set_Product_unit,B4: set_Product_unit] :
% 0.34/0.61 ( ( ord_le8056459307392131481t_unit @ ( minus_6452836326544984404t_unit @ A @ B4 ) @ A )
% 0.34/0.61 = ( ( inf_in4660618365625256667t_unit @ A @ B4 )
% 0.34/0.61 != bot_bo3957492148770167129t_unit ) ) ).
% 0.34/0.61
% 0.34/0.61 % disjoint_alt_simp3
% 0.34/0.61 thf(fact_4282_disjoint__alt__simp3,axiom,
% 0.34/0.61 ! [A: set_o,B4: set_o] :
% 0.34/0.61 ( ( ord_less_set_o @ ( minus_minus_set_o @ A @ B4 ) @ A )
% 0.34/0.61 = ( ( inf_inf_set_o @ A @ B4 )
% 0.34/0.61 != bot_bot_set_o ) ) ).
% 0.34/0.61
% 0.34/0.61 % disjoint_alt_simp3
% 0.34/0.61 thf(fact_4283_disjoint__alt__simp3,axiom,
% 0.34/0.61 ! [A: set_int,B4: set_int] :
% 0.34/0.61 ( ( ord_less_set_int @ ( minus_minus_set_int @ A @ B4 ) @ A )
% 0.34/0.61 = ( ( inf_inf_set_int @ A @ B4 )
% 0.34/0.61 != bot_bot_set_int ) ) ).
% 0.34/0.61
% 0.34/0.61 % disjoint_alt_simp3
% 0.34/0.61 thf(fact_4284_disjoint__alt__simp3,axiom,
% 0.34/0.61 ! [A: set_Pr4532377907799695533_nat_o,B4: set_Pr4532377907799695533_nat_o] :
% 0.34/0.61 ( ( ord_le2453136405763929_nat_o @ ( minus_1801376950450012436_nat_o @ A @ B4 ) @ A )
% 0.34/0.61 = ( ( inf_in1906310914598751387_nat_o @ A @ B4 )
% 0.34/0.61 != bot_bo7824918357723582233_nat_o ) ) ).
% 0.34/0.61
% 0.34/0.61 % disjoint_alt_simp3
% 0.34/0.61 thf(fact_4285_disjoint__alt__simp3,axiom,
% 0.34/0.61 ! [A: set_nat,B4: set_nat] :
% 0.34/0.61 ( ( ord_less_set_nat @ ( minus_minus_set_nat @ A @ B4 ) @ A )
% 0.34/0.61 = ( ( inf_inf_set_nat @ A @ B4 )
% 0.34/0.61 != bot_bot_set_nat ) ) ).
% 0.34/0.61
% 0.34/0.61 % disjoint_alt_simp3
% 0.34/0.61 thf(fact_4286_range__eq__singletonD,axiom,
% 0.34/0.61 ! [F: code_integer > code_integer,A2: code_integer,X: code_integer] :
% 0.34/0.61 ( ( ( image_4470545334726330049nteger @ F @ top_to4645266643341252675nteger )
% 0.34/0.61 = ( insert_Code_integer @ A2 @ bot_bo3990330152332043303nteger ) )
% 0.34/0.61 => ( ( F @ X )
% 0.34/0.61 = A2 ) ) ).
% 0.34/0.61
% 0.34/0.61 % range_eq_singletonD
% 0.34/0.61 thf(fact_4287_range__eq__singletonD,axiom,
% 0.34/0.61 ! [F: product_unit > product_unit,A2: product_unit,X: product_unit] :
% 0.34/0.61 ( ( ( image_405062704495631173t_unit @ F @ top_to1996260823553986621t_unit )
% 0.34/0.61 = ( insert_Product_unit @ A2 @ bot_bo3957492148770167129t_unit ) )
% 0.34/0.61 => ( ( F @ X )
% 0.34/0.61 = A2 ) ) ).
% 0.34/0.61
% 0.34/0.61 % range_eq_singletonD
% 0.34/0.61 thf(fact_4288_range__eq__singletonD,axiom,
% 0.34/0.61 ! [F: product_unit > $o,A2: $o,X: product_unit] :
% 0.34/0.61 ( ( ( image_Product_unit_o2 @ F @ top_to1996260823553986621t_unit )
% 0.34/0.61 = ( insert_o @ A2 @ bot_bot_set_o ) )
% 0.34/0.61 => ( ( F @ X )
% 0.34/0.61 = A2 ) ) ).
% 0.34/0.61
% 0.34/0.61 % range_eq_singletonD
% 0.34/0.61 thf(fact_4289_range__eq__singletonD,axiom,
% 0.34/0.61 ! [F: product_unit > nat,A2: nat,X: product_unit] :
% 0.34/0.61 ( ( ( image_875570014554754200it_nat @ F @ top_to1996260823553986621t_unit )
% 0.34/0.61 = ( insert_nat @ A2 @ bot_bot_set_nat ) )
% 0.34/0.61 => ( ( F @ X )
% 0.34/0.61 = A2 ) ) ).
% 0.34/0.61
% 0.34/0.61 % range_eq_singletonD
% 0.34/0.61 thf(fact_4290_range__eq__singletonD,axiom,
% 0.34/0.61 ! [F: product_unit > int,A2: int,X: product_unit] :
% 0.34/0.61 ( ( ( image_873079544045703924it_int @ F @ top_to1996260823553986621t_unit )
% 0.34/0.61 = ( insert_int @ A2 @ bot_bot_set_int ) )
% 0.34/0.61 => ( ( F @ X )
% 0.34/0.61 = A2 ) ) ).
% 0.34/0.61
% 0.34/0.61 % range_eq_singletonD
% 0.34/0.61 thf(fact_4291_range__eq__singletonD,axiom,
% 0.34/0.61 ! [F: $o > product_unit,A2: product_unit,X: $o] :
% 0.34/0.61 ( ( ( image_o_Product_unit2 @ F @ top_top_set_o )
% 0.34/0.61 = ( insert_Product_unit @ A2 @ bot_bo3957492148770167129t_unit ) )
% 0.34/0.61 => ( ( F @ X )
% 0.34/0.61 = A2 ) ) ).
% 0.34/0.61
% 0.34/0.61 % range_eq_singletonD
% 0.34/0.61 thf(fact_4292_range__eq__singletonD,axiom,
% 0.34/0.61 ! [F: $o > $o,A2: $o,X: $o] :
% 0.34/0.61 ( ( ( image_o_o2 @ F @ top_top_set_o )
% 0.34/0.61 = ( insert_o @ A2 @ bot_bot_set_o ) )
% 0.34/0.61 => ( ( F @ X )
% 0.34/0.61 = A2 ) ) ).
% 0.34/0.61
% 0.34/0.61 % range_eq_singletonD
% 0.34/0.61 thf(fact_4293_range__eq__singletonD,axiom,
% 0.34/0.61 ! [F: $o > nat,A2: nat,X: $o] :
% 0.34/0.61 ( ( ( image_o_nat2 @ F @ top_top_set_o )
% 0.34/0.61 = ( insert_nat @ A2 @ bot_bot_set_nat ) )
% 0.34/0.61 => ( ( F @ X )
% 0.34/0.61 = A2 ) ) ).
% 0.34/0.61
% 0.34/0.61 % range_eq_singletonD
% 0.34/0.61 thf(fact_4294_range__eq__singletonD,axiom,
% 0.34/0.61 ! [F: $o > int,A2: int,X: $o] :
% 0.34/0.61 ( ( ( image_o_int2 @ F @ top_top_set_o )
% 0.34/0.61 = ( insert_int @ A2 @ bot_bot_set_int ) )
% 0.34/0.61 => ( ( F @ X )
% 0.34/0.61 = A2 ) ) ).
% 0.34/0.61
% 0.34/0.61 % range_eq_singletonD
% 0.34/0.61 thf(fact_4295_range__eq__singletonD,axiom,
% 0.34/0.61 ! [F: nat > code_integer,A2: code_integer,X: nat] :
% 0.34/0.61 ( ( ( image_1215581382706833972nteger @ F @ top_top_set_nat )
% 0.34/0.61 = ( insert_Code_integer @ A2 @ bot_bo3990330152332043303nteger ) )
% 0.34/0.61 => ( ( F @ X )
% 0.34/0.61 = A2 ) ) ).
% 0.34/0.61
% 0.34/0.61 % range_eq_singletonD
% 0.34/0.61 thf(fact_4296_less__cSup__iff,axiom,
% 0.34/0.61 ! [X4: set_int,Y: int] :
% 0.34/0.61 ( ( X4 != bot_bot_set_int )
% 0.34/0.61 => ( ( condit2212336002400062152ve_int @ X4 )
% 0.34/0.61 => ( ( ord_less_int @ Y @ ( complete_Sup_Sup_int @ X4 ) )
% 0.34/0.61 = ( ? [X2: int] :
% 0.34/0.61 ( ( member_int @ X2 @ X4 )
% 0.34/0.61 & ( ord_less_int @ Y @ X2 ) ) ) ) ) ) ).
% 0.34/0.61
% 0.34/0.61 % less_cSup_iff
% 0.34/0.61 thf(fact_4297_less__cSup__iff,axiom,
% 0.34/0.61 ! [X4: set_nat,Y: nat] :
% 0.34/0.61 ( ( X4 != bot_bot_set_nat )
% 0.34/0.61 => ( ( condit2214826472909112428ve_nat @ X4 )
% 0.34/0.61 => ( ( ord_less_nat @ Y @ ( complete_Sup_Sup_nat @ X4 ) )
% 0.34/0.61 = ( ? [X2: nat] :
% 0.34/0.61 ( ( member_nat @ X2 @ X4 )
% 0.34/0.61 & ( ord_less_nat @ Y @ X2 ) ) ) ) ) ) ).
% 0.34/0.61
% 0.34/0.61 % less_cSup_iff
% 0.34/0.61 thf(fact_4298_less__cSup__iff,axiom,
% 0.34/0.61 ! [X4: set_Product_unit,Y: product_unit] :
% 0.34/0.61 ( ( X4 != bot_bo3957492148770167129t_unit )
% 0.34/0.61 => ( ( condit2080475194861589617t_unit @ X4 )
% 0.34/0.61 => ( ( ord_le361264281704409273t_unit @ Y @ ( comple4687483117567863418t_unit @ X4 ) )
% 0.34/0.61 = ( ? [X2: product_unit] :
% 0.34/0.61 ( ( member_Product_unit @ X2 @ X4 )
% 0.34/0.61 & ( ord_le361264281704409273t_unit @ Y @ X2 ) ) ) ) ) ) ).
% 0.34/0.61
% 0.34/0.61 % less_cSup_iff
% 0.34/0.61 thf(fact_4299_underS__Field3,axiom,
% 0.34/0.61 ! [R2: set_Pr5094982260447487303t_unit,A2: product_unit] :
% 0.34/0.61 ( ( ( field_Product_unit @ R2 )
% 0.34/0.61 != bot_bo3957492148770167129t_unit )
% 0.34/0.61 => ( ord_le8056459307392131481t_unit @ ( order_5809128134054296019t_unit @ R2 @ A2 ) @ ( field_Product_unit @ R2 ) ) ) ).
% 0.34/0.61
% 0.34/0.61 % underS_Field3
% 0.34/0.61 thf(fact_4300_underS__Field3,axiom,
% 0.34/0.61 ! [R2: set_Product_prod_o_o,A2: $o] :
% 0.34/0.61 ( ( ( field_o @ R2 )
% 0.34/0.61 != bot_bot_set_o )
% 0.34/0.61 => ( ord_less_set_o @ ( order_underS_o @ R2 @ A2 ) @ ( field_o @ R2 ) ) ) ).
% 0.34/0.61
% 0.34/0.61 % underS_Field3
% 0.34/0.61 thf(fact_4301_underS__Field3,axiom,
% 0.34/0.61 ! [R2: set_Pr1261947904930325089at_nat,A2: nat] :
% 0.34/0.61 ( ( ( field_nat @ R2 )
% 0.34/0.61 != bot_bot_set_nat )
% 0.34/0.61 => ( ord_less_set_nat @ ( order_underS_nat @ R2 @ A2 ) @ ( field_nat @ R2 ) ) ) ).
% 0.34/0.61
% 0.34/0.61 % underS_Field3
% 0.34/0.61 thf(fact_4302_underS__Field3,axiom,
% 0.34/0.61 ! [R2: set_Pr958786334691620121nt_int,A2: int] :
% 0.34/0.61 ( ( ( field_int @ R2 )
% 0.34/0.61 != bot_bot_set_int )
% 0.34/0.61 => ( ord_less_set_int @ ( order_underS_int @ R2 @ A2 ) @ ( field_int @ R2 ) ) ) ).
% 0.34/0.61
% 0.34/0.61 % underS_Field3
% 0.34/0.61 thf(fact_4303_underS__Field3,axiom,
% 0.34/0.61 ! [R2: set_Pr2161125870931222855_nat_o,A2: produc3658429121746597890et_nat > $o] :
% 0.34/0.61 ( ( ( field_81821745747897882_nat_o @ R2 )
% 0.34/0.61 != bot_bo7824918357723582233_nat_o )
% 0.34/0.61 => ( ord_le2453136405763929_nat_o @ ( order_7262550146012401043_nat_o @ R2 @ A2 ) @ ( field_81821745747897882_nat_o @ R2 ) ) ) ).
% 0.34/0.61
% 0.34/0.61 % underS_Field3
% 0.34/0.61 thf(fact_4304_arg__min__if__finite_I2_J,axiom,
% 0.34/0.61 ! [S: set_Code_integer,F: code_integer > assn] :
% 0.34/0.61 ( ( finite6017078050557962740nteger @ S )
% 0.34/0.61 => ( ( S != bot_bo3990330152332043303nteger )
% 0.34/0.61 => ~ ? [X6: code_integer] :
% 0.34/0.61 ( ( member_Code_integer @ X6 @ S )
% 0.34/0.61 & ( ord_less_assn @ ( F @ X6 ) @ ( F @ ( lattic4682378690128378784r_assn @ F @ S ) ) ) ) ) ) ).
% 0.34/0.61
% 0.34/0.61 % arg_min_if_finite(2)
% 0.34/0.61 thf(fact_4305_arg__min__if__finite_I2_J,axiom,
% 0.34/0.61 ! [S: set_Product_unit,F: product_unit > assn] :
% 0.34/0.61 ( ( finite4290736615968046902t_unit @ S )
% 0.34/0.61 => ( ( S != bot_bo3957492148770167129t_unit )
% 0.34/0.61 => ~ ? [X6: product_unit] :
% 0.34/0.61 ( ( member_Product_unit @ X6 @ S )
% 0.34/0.61 & ( ord_less_assn @ ( F @ X6 ) @ ( F @ ( lattic4376901495776413404t_assn @ F @ S ) ) ) ) ) ) ).
% 0.34/0.61
% 0.34/0.61 % arg_min_if_finite(2)
% 0.34/0.61 thf(fact_4306_arg__min__if__finite_I2_J,axiom,
% 0.34/0.61 ! [S: set_o,F: $o > assn] :
% 0.34/0.61 ( ( finite_finite_o @ S )
% 0.34/0.61 => ( ( S != bot_bot_set_o )
% 0.34/0.61 => ~ ? [X6: $o] :
% 0.34/0.61 ( ( member_o @ X6 @ S )
% 0.34/0.61 & ( ord_less_assn @ ( F @ X6 ) @ ( F @ ( lattic7708394068118247271o_assn @ F @ S ) ) ) ) ) ) ).
% 0.34/0.61
% 0.34/0.61 % arg_min_if_finite(2)
% 0.34/0.61 thf(fact_4307_arg__min__if__finite_I2_J,axiom,
% 0.34/0.61 ! [S: set_nat,F: nat > assn] :
% 0.34/0.61 ( ( finite_finite_nat @ S )
% 0.34/0.61 => ( ( S != bot_bot_set_nat )
% 0.34/0.61 => ~ ? [X6: nat] :
% 0.34/0.61 ( ( member_nat @ X6 @ S )
% 0.34/0.61 & ( ord_less_assn @ ( F @ X6 ) @ ( F @ ( lattic2187264146484958483t_assn @ F @ S ) ) ) ) ) ) ).
% 0.34/0.61
% 0.34/0.61 % arg_min_if_finite(2)
% 0.34/0.61 thf(fact_4308_arg__min__if__finite_I2_J,axiom,
% 0.34/0.61 ! [S: set_int,F: int > assn] :
% 0.34/0.61 ( ( finite_finite_int @ S )
% 0.34/0.61 => ( ( S != bot_bot_set_int )
% 0.34/0.61 => ~ ? [X6: int] :
% 0.34/0.61 ( ( member_int @ X6 @ S )
% 0.34/0.61 & ( ord_less_assn @ ( F @ X6 ) @ ( F @ ( lattic3162799611691135855t_assn @ F @ S ) ) ) ) ) ) ).
% 0.34/0.61
% 0.34/0.61 % arg_min_if_finite(2)
% 0.34/0.61 thf(fact_4309_arg__min__if__finite_I2_J,axiom,
% 0.34/0.61 ! [S: set_Code_integer,F: code_integer > nat] :
% 0.34/0.61 ( ( finite6017078050557962740nteger @ S )
% 0.34/0.61 => ( ( S != bot_bo3990330152332043303nteger )
% 0.34/0.61 => ~ ? [X6: code_integer] :
% 0.34/0.61 ( ( member_Code_integer @ X6 @ S )
% 0.34/0.61 & ( ord_less_nat @ ( F @ X6 ) @ ( F @ ( lattic6074936555552774728er_nat @ F @ S ) ) ) ) ) ) ).
% 0.34/0.61
% 0.34/0.61 % arg_min_if_finite(2)
% 0.34/0.61 thf(fact_4310_arg__min__if__finite_I2_J,axiom,
% 0.34/0.61 ! [S: set_Product_unit,F: product_unit > nat] :
% 0.34/0.61 ( ( finite4290736615968046902t_unit @ S )
% 0.34/0.61 => ( ( S != bot_bo3957492148770167129t_unit )
% 0.34/0.61 => ~ ? [X6: product_unit] :
% 0.34/0.61 ( ( member_Product_unit @ X6 @ S )
% 0.34/0.61 & ( ord_less_nat @ ( F @ X6 ) @ ( F @ ( lattic1360832392136924548it_nat @ F @ S ) ) ) ) ) ) ).
% 0.34/0.61
% 0.34/0.61 % arg_min_if_finite(2)
% 0.34/0.61 thf(fact_4311_arg__min__if__finite_I2_J,axiom,
% 0.34/0.61 ! [S: set_o,F: $o > nat] :
% 0.34/0.61 ( ( finite_finite_o @ S )
% 0.34/0.61 => ( ( S != bot_bot_set_o )
% 0.34/0.61 => ~ ? [X6: $o] :
% 0.34/0.61 ( ( member_o @ X6 @ S )
% 0.34/0.61 & ( ord_less_nat @ ( F @ X6 ) @ ( F @ ( lattic2775856028456453135_o_nat @ F @ S ) ) ) ) ) ) ).
% 0.34/0.61
% 0.34/0.61 % arg_min_if_finite(2)
% 0.34/0.61 thf(fact_4312_arg__min__if__finite_I2_J,axiom,
% 0.34/0.61 ! [S: set_nat,F: nat > nat] :
% 0.34/0.61 ( ( finite_finite_nat @ S )
% 0.34/0.61 => ( ( S != bot_bot_set_nat )
% 0.34/0.61 => ~ ? [X6: nat] :
% 0.34/0.61 ( ( member_nat @ X6 @ S )
% 0.34/0.61 & ( ord_less_nat @ ( F @ X6 ) @ ( F @ ( lattic7446932960582359483at_nat @ F @ S ) ) ) ) ) ) ).
% 0.34/0.61
% 0.34/0.61 % arg_min_if_finite(2)
% 0.34/0.61 thf(fact_4313_arg__min__if__finite_I2_J,axiom,
% 0.34/0.61 ! [S: set_int,F: int > nat] :
% 0.34/0.61 ( ( finite_finite_int @ S )
% 0.34/0.61 => ( ( S != bot_bot_set_int )
% 0.34/0.61 => ~ ? [X6: int] :
% 0.34/0.61 ( ( member_int @ X6 @ S )
% 0.34/0.61 & ( ord_less_nat @ ( F @ X6 ) @ ( F @ ( lattic8446286672483414039nt_nat @ F @ S ) ) ) ) ) ) ).
% 0.34/0.61
% 0.34/0.61 % arg_min_if_finite(2)
% 0.34/0.61 thf(fact_4314_Sup__fold__sup,axiom,
% 0.34/0.61 ! [A: set_set_Product_unit] :
% 0.34/0.61 ( ( finite1772178364199683094t_unit @ A )
% 0.34/0.61 => ( ( comple3349607820504768346t_unit @ A )
% 0.34/0.61 = ( finite4243028863286625800t_unit @ sup_su793286257634532545t_unit @ bot_bo3957492148770167129t_unit @ A ) ) ) ).
% 0.34/0.61
% 0.34/0.61 % Sup_fold_sup
% 0.34/0.61 thf(fact_4315_Sup__fold__sup,axiom,
% 0.34/0.61 ! [A: set_set_o] :
% 0.34/0.61 ( ( finite_finite_set_o @ A )
% 0.34/0.61 => ( ( comple90263536869209701_set_o @ A )
% 0.34/0.61 = ( finite4337638375924247368_set_o @ sup_sup_set_o @ bot_bot_set_o @ A ) ) ) ).
% 0.34/0.61
% 0.34/0.61 % Sup_fold_sup
% 0.34/0.61 thf(fact_4316_Sup__fold__sup,axiom,
% 0.34/0.61 ! [A: set_set_nat] :
% 0.34/0.61 ( ( finite1152437895449049373et_nat @ A )
% 0.34/0.61 => ( ( comple7399068483239264473et_nat @ A )
% 0.34/0.61 = ( finite677925301803182934et_nat @ sup_sup_set_nat @ bot_bot_set_nat @ A ) ) ) ).
% 0.34/0.61
% 0.34/0.61 % Sup_fold_sup
% 0.34/0.61 thf(fact_4317_Sup__fold__sup,axiom,
% 0.34/0.61 ! [A: set_set_int] :
% 0.34/0.61 ( ( finite6197958912794628473et_int @ A )
% 0.34/0.61 => ( ( comple3221217463730067765et_int @ A )
% 0.34/0.61 = ( finite2508883933289561102et_int @ sup_sup_set_int @ bot_bot_set_int @ A ) ) ) ).
% 0.34/0.61
% 0.34/0.61 % Sup_fold_sup
% 0.34/0.61 thf(fact_4318_Sup__fold__sup,axiom,
% 0.34/0.61 ! [A: set_se1353816476878259085_nat_o] :
% 0.34/0.61 ( ( finite6069017635265841110_nat_o @ A )
% 0.34/0.61 => ( ( comple3382087578465172250_nat_o @ A )
% 0.34/0.61 = ( finite8456856254070787592_nat_o @ sup_su5209123915105501825_nat_o @ bot_bo7824918357723582233_nat_o @ A ) ) ) ).
% 0.34/0.61
% 0.34/0.61 % Sup_fold_sup
% 0.34/0.61 thf(fact_4319_Sup__fold__sup,axiom,
% 0.34/0.61 ! [A: set_Product_unit] :
% 0.34/0.61 ( ( finite4290736615968046902t_unit @ A )
% 0.34/0.61 => ( ( comple4687483117567863418t_unit @ A )
% 0.34/0.61 = ( finite8834431295148555784t_unit @ sup_sup_Product_unit @ bot_bot_Product_unit @ A ) ) ) ).
% 0.34/0.61
% 0.34/0.61 % Sup_fold_sup
% 0.34/0.61 thf(fact_4320_SUP__fold__sup,axiom,
% 0.34/0.61 ! [A: set_nat,F: nat > product_unit] :
% 0.34/0.61 ( ( finite_finite_nat @ A )
% 0.34/0.61 => ( ( comple4687483117567863418t_unit @ ( image_8730104196221521654t_unit @ F @ A ) )
% 0.34/0.61 = ( finite6602901031542678131t_unit @ ( comp_P6204500334276302049it_nat @ sup_sup_Product_unit @ F ) @ bot_bot_Product_unit @ A ) ) ) ).
% 0.34/0.61
% 0.34/0.61 % SUP_fold_sup
% 0.34/0.61 thf(fact_4321_SUP__fold__sup,axiom,
% 0.34/0.61 ! [A: set_Code_integer,F: code_integer > product_unit] :
% 0.34/0.61 ( ( finite6017078050557962740nteger @ A )
% 0.34/0.61 => ( ( comple4687483117567863418t_unit @ ( image_7182656606676592809t_unit @ F @ A ) )
% 0.34/0.61 = ( finite2281032298429728806t_unit @ ( comp_P4558236313141964142nteger @ sup_sup_Product_unit @ F ) @ bot_bot_Product_unit @ A ) ) ) ).
% 0.34/0.61
% 0.34/0.61 % SUP_fold_sup
% 0.34/0.61 thf(fact_4322_SUP__fold__sup,axiom,
% 0.34/0.61 ! [A: set_nat,F: nat > set_Product_unit] :
% 0.34/0.61 ( ( finite_finite_nat @ A )
% 0.34/0.61 => ( ( comple3349607820504768346t_unit @ ( image_4824033694279209942t_unit @ F @ A ) )
% 0.34/0.61 = ( finite9145781422887121747t_unit @ ( comp_s6943286182646368705it_nat @ sup_su793286257634532545t_unit @ F ) @ bot_bo3957492148770167129t_unit @ A ) ) ) ).
% 0.34/0.61
% 0.34/0.61 % SUP_fold_sup
% 0.34/0.61 thf(fact_4323_SUP__fold__sup,axiom,
% 0.34/0.61 ! [A: set_Code_integer,F: code_integer > set_Product_unit] :
% 0.34/0.61 ( ( finite6017078050557962740nteger @ A )
% 0.34/0.61 => ( ( comple3349607820504768346t_unit @ ( image_5870443399019143561t_unit @ F @ A ) )
% 0.34/0.61 = ( finite3687622995246107910t_unit @ ( comp_s3199726385246003790nteger @ sup_su793286257634532545t_unit @ F ) @ bot_bo3957492148770167129t_unit @ A ) ) ) ).
% 0.34/0.61
% 0.34/0.61 % SUP_fold_sup
% 0.34/0.61 thf(fact_4324_SUP__fold__sup,axiom,
% 0.34/0.61 ! [A: set_nat,F: nat > set_o] :
% 0.34/0.61 ( ( finite_finite_nat @ A )
% 0.34/0.61 => ( ( comple90263536869209701_set_o @ ( image_nat_set_o @ F @ A ) )
% 0.34/0.61 = ( finite3217087857726763998_set_o @ ( comp_s4424444925706540684_o_nat @ sup_sup_set_o @ F ) @ bot_bot_set_o @ A ) ) ) ).
% 0.34/0.61
% 0.34/0.61 % SUP_fold_sup
% 0.34/0.61 thf(fact_4325_SUP__fold__sup,axiom,
% 0.34/0.61 ! [A: set_Code_integer,F: code_integer > set_o] :
% 0.34/0.61 ( ( finite6017078050557962740nteger @ A )
% 0.34/0.61 => ( ( comple90263536869209701_set_o @ ( image_1867029960091478804_set_o @ F @ A ) )
% 0.34/0.61 = ( finite1928688952044918289_set_o @ ( comp_s90868338361139993nteger @ sup_sup_set_o @ F ) @ bot_bot_set_o @ A ) ) ) ).
% 0.34/0.61
% 0.34/0.61 % SUP_fold_sup
% 0.34/0.61 thf(fact_4326_SUP__fold__sup,axiom,
% 0.34/0.61 ! [A: set_nat,F: nat > set_nat] :
% 0.34/0.61 ( ( finite_finite_nat @ A )
% 0.34/0.61 => ( ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ F @ A ) )
% 0.34/0.61 = ( finite5529483035118572448et_nat @ ( comp_s2351873599094224870at_nat @ sup_sup_set_nat @ F ) @ bot_bot_set_nat @ A ) ) ) ).
% 0.34/0.61
% 0.34/0.61 % SUP_fold_sup
% 0.34/0.61 thf(fact_4327_SUP__fold__sup,axiom,
% 0.34/0.61 ! [A: set_Code_integer,F: code_integer > set_nat] :
% 0.34/0.61 ( ( finite6017078050557962740nteger @ A )
% 0.34/0.61 => ( ( comple7399068483239264473et_nat @ ( image_8829277216472192490et_nat @ F @ A ) )
% 0.34/0.61 = ( finite1590314701948642349et_nat @ ( comp_s3169385134670675059nteger @ sup_sup_set_nat @ F ) @ bot_bot_set_nat @ A ) ) ) ).
% 0.34/0.61
% 0.34/0.61 % SUP_fold_sup
% 0.34/0.61 thf(fact_4328_SUP__fold__sup,axiom,
% 0.34/0.61 ! [A: set_nat,F: nat > set_int] :
% 0.34/0.61 ( ( finite_finite_nat @ A )
% 0.34/0.61 => ( ( comple3221217463730067765et_int @ ( image_nat_set_int @ F @ A ) )
% 0.34/0.61 = ( finite1351632015609375740et_int @ ( comp_s6689493721604790522nt_nat @ sup_sup_set_int @ F ) @ bot_bot_set_int @ A ) ) ) ).
% 0.34/0.61
% 0.34/0.61 % SUP_fold_sup
% 0.34/0.61 thf(fact_4329_SUP__fold__sup,axiom,
% 0.34/0.61 ! [A: set_Code_integer,F: code_integer > set_int] :
% 0.34/0.61 ( ( finite6017078050557962740nteger @ A )
% 0.34/0.61 => ( ( comple3221217463730067765et_int @ ( image_4651426196962995782et_int @ F @ A ) )
% 0.34/0.61 = ( finite6635835719294221449et_int @ ( comp_s3634918272309710215nteger @ sup_sup_set_int @ F ) @ bot_bot_set_int @ A ) ) ) ).
% 0.34/0.61
% 0.34/0.61 % SUP_fold_sup
% 0.34/0.61 thf(fact_4330_finite__subset__Union__chain,axiom,
% 0.34/0.61 ! [A: set_Code_integer,B9: set_set_Code_integer,A11: set_set_Code_integer] :
% 0.34/0.61 ( ( finite6017078050557962740nteger @ A )
% 0.34/0.61 => ( ( ord_le7084787975880047091nteger @ A @ ( comple739944243200306918nteger @ B9 ) )
% 0.34/0.61 => ( ( B9 != bot_bo9204850693367297629nteger )
% 0.34/0.61 => ( ( pred_c5891848916688490280nteger @ A11 @ ord_le1307284697595431911nteger @ B9 )
% 0.34/0.61 => ~ ! [B5: set_Code_integer] :
% 0.34/0.61 ( ( member1834298987299537450nteger @ B5 @ B9 )
% 0.34/0.61 => ~ ( ord_le7084787975880047091nteger @ A @ B5 ) ) ) ) ) ) ).
% 0.34/0.61
% 0.34/0.61 % finite_subset_Union_chain
% 0.34/0.61 thf(fact_4331_finite__subset__Union__chain,axiom,
% 0.34/0.61 ! [A: set_nat,B9: set_set_nat,A11: set_set_nat] :
% 0.34/0.61 ( ( finite_finite_nat @ A )
% 0.34/0.61 => ( ( ord_less_eq_set_nat @ A @ ( comple7399068483239264473et_nat @ B9 ) )
% 0.34/0.61 => ( ( B9 != bot_bot_set_set_nat )
% 0.34/0.61 => ( ( pred_chain_set_nat @ A11 @ ord_less_set_nat @ B9 )
% 0.34/0.61 => ~ ! [B5: set_nat] :
% 0.34/0.61 ( ( member_set_nat @ B5 @ B9 )
% 0.34/0.61 => ~ ( ord_less_eq_set_nat @ A @ B5 ) ) ) ) ) ) ).
% 0.34/0.61
% 0.34/0.61 % finite_subset_Union_chain
% 0.34/0.61 thf(fact_4332_merge__true__star,axiom,
% 0.34/0.61 ( ( times_times_assn @ top_top_assn @ top_top_assn )
% 0.34/0.61 = top_top_assn ) ).
% 0.34/0.61
% 0.34/0.61 % merge_true_star
% 0.34/0.61 thf(fact_4333_assn__basic__inequalities_I5_J,axiom,
% 0.34/0.61 top_top_assn != bot_bot_assn ).
% 0.34/0.61
% 0.34/0.61 % assn_basic_inequalities(5)
% 0.34/0.61 thf(fact_4334_assn__basic__inequalities_I1_J,axiom,
% 0.34/0.61 top_top_assn != one_one_assn ).
% 0.34/0.61
% 0.34/0.61 % assn_basic_inequalities(1)
% 0.34/0.61 thf(fact_4335_mod__true,axiom,
% 0.34/0.61 ! [H2: produc3658429121746597890et_nat] :
% 0.34/0.61 ( ( rep_assn @ top_top_assn @ H2 )
% 0.34/0.61 = ( in_range @ H2 ) ) ).
% 0.34/0.61
% 0.34/0.61 % mod_true
% 0.34/0.61 thf(fact_4336_fun__comp__eq__conv,axiom,
% 0.34/0.61 ! [F: code_integer > produc8923325533196201883nteger > produc8923325533196201883nteger,G: code_integer > code_integer,Fg: code_integer > produc8923325533196201883nteger > produc8923325533196201883nteger] :
% 0.34/0.61 ( ( ( comp_C1593894019821074884nteger @ F @ G )
% 0.34/0.61 = Fg )
% 0.34/0.61 = ( ! [X2: code_integer] :
% 0.34/0.61 ( ( F @ ( G @ X2 ) )
% 0.34/0.61 = ( Fg @ X2 ) ) ) ) ).
% 0.34/0.61
% 0.34/0.61 % fun_comp_eq_conv
% 0.34/0.61 thf(fact_4337_fun__comp__eq__conv,axiom,
% 0.34/0.61 ! [F: ( code_integer > code_integer ) > produc8923325533196201883nteger > produc8923325533196201883nteger,G: code_integer > code_integer > code_integer,Fg: code_integer > produc8923325533196201883nteger > produc8923325533196201883nteger] :
% 0.34/0.61 ( ( ( comp_C8797469213163452608nteger @ F @ G )
% 0.34/0.61 = Fg )
% 0.34/0.61 = ( ! [X2: code_integer] :
% 0.34/0.61 ( ( F @ ( G @ X2 ) )
% 0.34/0.61 = ( Fg @ X2 ) ) ) ) ).
% 0.34/0.61
% 0.34/0.61 % fun_comp_eq_conv
% 0.34/0.61 thf(fact_4338_comp__cong__right,axiom,
% 0.34/0.61 ! [X: code_integer > code_integer,Y: code_integer > code_integer,F: code_integer > produc8923325533196201883nteger > produc8923325533196201883nteger] :
% 0.34/0.61 ( ( X = Y )
% 0.34/0.61 => ( ( comp_C1593894019821074884nteger @ F @ X )
% 0.34/0.61 = ( comp_C1593894019821074884nteger @ F @ Y ) ) ) ).
% 0.34/0.61
% 0.34/0.61 % comp_cong_right
% 0.34/0.61 thf(fact_4339_comp__cong__right,axiom,
% 0.34/0.61 ! [X: code_integer > code_integer > code_integer,Y: code_integer > code_integer > code_integer,F: ( code_integer > code_integer ) > produc8923325533196201883nteger > produc8923325533196201883nteger] :
% 0.34/0.61 ( ( X = Y )
% 0.34/0.61 => ( ( comp_C8797469213163452608nteger @ F @ X )
% 0.34/0.61 = ( comp_C8797469213163452608nteger @ F @ Y ) ) ) ).
% 0.34/0.61
% 0.34/0.61 % comp_cong_right
% 0.34/0.61 thf(fact_4340_comp__cong__left,axiom,
% 0.34/0.61 ! [X: code_integer > produc8923325533196201883nteger > produc8923325533196201883nteger,Y: code_integer > produc8923325533196201883nteger > produc8923325533196201883nteger,F: code_integer > code_integer] :
% 0.34/0.61 ( ( X = Y )
% 0.34/0.61 => ( ( comp_C1593894019821074884nteger @ X @ F )
% 0.34/0.61 = ( comp_C1593894019821074884nteger @ Y @ F ) ) ) ).
% 0.34/0.61
% 0.34/0.61 % comp_cong_left
% 0.34/0.61 thf(fact_4341_comp__cong__left,axiom,
% 0.34/0.61 ! [X: ( code_integer > code_integer ) > produc8923325533196201883nteger > produc8923325533196201883nteger,Y: ( code_integer > code_integer ) > produc8923325533196201883nteger > produc8923325533196201883nteger,F: code_integer > code_integer > code_integer] :
% 0.34/0.61 ( ( X = Y )
% 0.34/0.61 => ( ( comp_C8797469213163452608nteger @ X @ F )
% 0.34/0.61 = ( comp_C8797469213163452608nteger @ Y @ F ) ) ) ).
% 0.34/0.61
% 0.34/0.61 % comp_cong_left
% 0.34/0.61 thf(fact_4342_comp__apply__eq,axiom,
% 0.34/0.61 ! [F: code_integer > produc8923325533196201883nteger > produc8923325533196201883nteger,G: code_integer > code_integer,X: code_integer,H2: code_integer > produc8923325533196201883nteger > produc8923325533196201883nteger,K3: code_integer > code_integer] :
% 0.34/0.61 ( ( ( F @ ( G @ X ) )
% 0.34/0.61 = ( H2 @ ( K3 @ X ) ) )
% 0.34/0.61 => ( ( comp_C1593894019821074884nteger @ F @ G @ X )
% 0.34/0.61 = ( comp_C1593894019821074884nteger @ H2 @ K3 @ X ) ) ) ).
% 0.34/0.61
% 0.34/0.61 % comp_apply_eq
% 0.34/0.61 thf(fact_4343_comp__apply__eq,axiom,
% 0.34/0.61 ! [F: code_integer > produc8923325533196201883nteger > produc8923325533196201883nteger,G: code_integer > code_integer,X: code_integer,H2: ( code_integer > code_integer ) > produc8923325533196201883nteger > produc8923325533196201883nteger,K3: code_integer > code_integer > code_integer] :
% 0.34/0.61 ( ( ( F @ ( G @ X ) )
% 0.34/0.61 = ( H2 @ ( K3 @ X ) ) )
% 0.34/0.61 => ( ( comp_C1593894019821074884nteger @ F @ G @ X )
% 0.34/0.61 = ( comp_C8797469213163452608nteger @ H2 @ K3 @ X ) ) ) ).
% 0.34/0.61
% 0.34/0.61 % comp_apply_eq
% 0.34/0.61 thf(fact_4344_comp__apply__eq,axiom,
% 0.34/0.61 ! [F: ( code_integer > code_integer ) > produc8923325533196201883nteger > produc8923325533196201883nteger,G: code_integer > code_integer > code_integer,X: code_integer,H2: code_integer > produc8923325533196201883nteger > produc8923325533196201883nteger,K3: code_integer > code_integer] :
% 0.34/0.61 ( ( ( F @ ( G @ X ) )
% 0.34/0.61 = ( H2 @ ( K3 @ X ) ) )
% 0.34/0.61 => ( ( comp_C8797469213163452608nteger @ F @ G @ X )
% 0.34/0.61 = ( comp_C1593894019821074884nteger @ H2 @ K3 @ X ) ) ) ).
% 0.34/0.61
% 0.34/0.61 % comp_apply_eq
% 0.34/0.61 thf(fact_4345_comp__apply__eq,axiom,
% 0.34/0.61 ! [F: ( code_integer > code_integer ) > produc8923325533196201883nteger > produc8923325533196201883nteger,G: code_integer > code_integer > code_integer,X: code_integer,H2: ( code_integer > code_integer ) > produc8923325533196201883nteger > produc8923325533196201883nteger,K3: code_integer > code_integer > code_integer] :
% 0.34/0.61 ( ( ( F @ ( G @ X ) )
% 0.34/0.61 = ( H2 @ ( K3 @ X ) ) )
% 0.34/0.61 => ( ( comp_C8797469213163452608nteger @ F @ G @ X )
% 0.34/0.61 = ( comp_C8797469213163452608nteger @ H2 @ K3 @ X ) ) ) ).
% 0.34/0.61
% 0.34/0.61 % comp_apply_eq
% 0.34/0.61 thf(fact_4346_type__copy__set__map0,axiom,
% 0.34/0.61 ! [Rep2: ( produc8923325533196201883nteger > produc8923325533196201883nteger ) > code_integer,Abs2: code_integer > produc8923325533196201883nteger > produc8923325533196201883nteger,S: code_integer > set_nat,M: code_integer > code_integer,F: nat > nat,S4: code_integer > set_nat,G: code_integer > code_integer] :
% 0.34/0.61 ( ( type_d6544512618915850416nteger @ Rep2 @ Abs2 @ top_to4645266643341252675nteger )
% 0.34/0.61 => ( ( ( comp_C6537567860429791781nteger @ S @ M )
% 0.34/0.61 = ( comp_s4605850120322998606nteger @ ( image_nat_nat2 @ F ) @ S4 ) )
% 0.34/0.61 => ( ( comp_P8262219321096090349nteger @ ( comp_C3184339336801759213nteger @ S @ Rep2 ) @ ( comp_C1593894019821074884nteger @ ( comp_C1593894019821074884nteger @ Abs2 @ M ) @ G ) )
% 0.34/0.61 = ( comp_s4605850120322998606nteger @ ( image_nat_nat2 @ F ) @ ( comp_C6537567860429791781nteger @ S4 @ G ) ) ) ) ) ).
% 0.34/0.61
% 0.34/0.61 % type_copy_set_map0
% 0.34/0.61 thf(fact_4347_type__copy__set__map0,axiom,
% 0.34/0.61 ! [Rep2: ( produc8923325533196201883nteger > produc8923325533196201883nteger ) > code_integer,Abs2: code_integer > produc8923325533196201883nteger > produc8923325533196201883nteger,S: code_integer > set_Code_integer,M: code_integer > code_integer,F: code_integer > code_integer,S4: code_integer > set_Code_integer,G: code_integer > code_integer] :
% 0.34/0.61 ( ( type_d6544512618915850416nteger @ Rep2 @ Abs2 @ top_to4645266643341252675nteger )
% 0.34/0.61 => ( ( ( comp_C8186931911283765298nteger @ S @ M )
% 0.34/0.61 = ( comp_s956550612328592744nteger @ ( image_4470545334726330049nteger @ F ) @ S4 ) )
% 0.34/0.61 => ( ( comp_P4885729792186999034nteger @ ( comp_C6344129587964912378nteger @ S @ Rep2 ) @ ( comp_C1593894019821074884nteger @ ( comp_C1593894019821074884nteger @ Abs2 @ M ) @ G ) )
% 0.34/0.61 = ( comp_s956550612328592744nteger @ ( image_4470545334726330049nteger @ F ) @ ( comp_C8186931911283765298nteger @ S4 @ G ) ) ) ) ) ).
% 0.34/0.61
% 0.34/0.61 % type_copy_set_map0
% 0.34/0.61 thf(fact_4348_type__copy__set__map0,axiom,
% 0.34/0.61 ! [Rep2: ( produc8923325533196201883nteger > produc8923325533196201883nteger ) > code_integer,Abs2: code_integer > produc8923325533196201883nteger > produc8923325533196201883nteger,S: code_integer > set_Code_integer,M: code_integer > code_integer,F: nat > code_integer,S4: code_integer > set_nat,G: code_integer > code_integer] :
% 0.34/0.61 ( ( type_d6544512618915850416nteger @ Rep2 @ Abs2 @ top_to4645266643341252675nteger )
% 0.34/0.61 => ( ( ( comp_C8186931911283765298nteger @ S @ M )
% 0.34/0.61 = ( comp_s8292494440407567195nteger @ ( image_1215581382706833972nteger @ F ) @ S4 ) )
% 0.34/0.61 => ( ( comp_P4885729792186999034nteger @ ( comp_C6344129587964912378nteger @ S @ Rep2 ) @ ( comp_C1593894019821074884nteger @ ( comp_C1593894019821074884nteger @ Abs2 @ M ) @ G ) )
% 0.34/0.61 = ( comp_s8292494440407567195nteger @ ( image_1215581382706833972nteger @ F ) @ ( comp_C6537567860429791781nteger @ S4 @ G ) ) ) ) ) ).
% 0.34/0.61
% 0.34/0.61 % type_copy_set_map0
% 0.34/0.61 thf(fact_4349_type__copy__set__map0,axiom,
% 0.34/0.61 ! [Rep2: ( produc8923325533196201883nteger > produc8923325533196201883nteger ) > product_unit,Abs2: product_unit > produc8923325533196201883nteger > produc8923325533196201883nteger,S: product_unit > set_nat,M: code_integer > product_unit,F: nat > nat,S4: code_integer > set_nat,G: code_integer > code_integer] :
% 0.34/0.61 ( ( type_d6996789741066924922t_unit @ Rep2 @ Abs2 @ top_to1996260823553986621t_unit )
% 0.34/0.61 => ( ( ( comp_P5878433733084393551nteger @ S @ M )
% 0.34/0.61 = ( comp_s4605850120322998606nteger @ ( image_nat_nat2 @ F ) @ S4 ) )
% 0.34/0.61 => ( ( comp_P8262219321096090349nteger @ ( comp_P6101996899264665623nteger @ S @ Rep2 ) @ ( comp_C1593894019821074884nteger @ ( comp_P4511551582283981294nteger @ Abs2 @ M ) @ G ) )
% 0.34/0.61 = ( comp_s4605850120322998606nteger @ ( image_nat_nat2 @ F ) @ ( comp_C6537567860429791781nteger @ S4 @ G ) ) ) ) ) ).
% 0.34/0.61
% 0.34/0.61 % type_copy_set_map0
% 0.34/0.61 thf(fact_4350_type__copy__set__map0,axiom,
% 0.34/0.61 ! [Rep2: ( produc8923325533196201883nteger > produc8923325533196201883nteger ) > product_unit,Abs2: product_unit > produc8923325533196201883nteger > produc8923325533196201883nteger,S: product_unit > set_Code_integer,M: code_integer > product_unit,F: code_integer > code_integer,S4: code_integer > set_Code_integer,G: code_integer > code_integer] :
% 0.34/0.61 ( ( type_d6996789741066924922t_unit @ Rep2 @ Abs2 @ top_to1996260823553986621t_unit )
% 0.34/0.61 => ( ( ( comp_P4767411901762164316nteger @ S @ M )
% 0.34/0.61 = ( comp_s956550612328592744nteger @ ( image_4470545334726330049nteger @ F ) @ S4 ) )
% 0.34/0.61 => ( ( comp_P4885729792186999034nteger @ ( comp_P4635212033913086756nteger @ S @ Rep2 ) @ ( comp_C1593894019821074884nteger @ ( comp_P4511551582283981294nteger @ Abs2 @ M ) @ G ) )
% 0.34/0.61 = ( comp_s956550612328592744nteger @ ( image_4470545334726330049nteger @ F ) @ ( comp_C8186931911283765298nteger @ S4 @ G ) ) ) ) ) ).
% 0.34/0.61
% 0.34/0.61 % type_copy_set_map0
% 0.34/0.61 thf(fact_4351_type__copy__set__map0,axiom,
% 0.34/0.61 ! [Rep2: ( produc8923325533196201883nteger > produc8923325533196201883nteger ) > product_unit,Abs2: product_unit > produc8923325533196201883nteger > produc8923325533196201883nteger,S: product_unit > set_Code_integer,M: code_integer > product_unit,F: nat > code_integer,S4: code_integer > set_nat,G: code_integer > code_integer] :
% 0.34/0.61 ( ( type_d6996789741066924922t_unit @ Rep2 @ Abs2 @ top_to1996260823553986621t_unit )
% 0.34/0.61 => ( ( ( comp_P4767411901762164316nteger @ S @ M )
% 0.34/0.61 = ( comp_s8292494440407567195nteger @ ( image_1215581382706833972nteger @ F ) @ S4 ) )
% 0.34/0.61 => ( ( comp_P4885729792186999034nteger @ ( comp_P4635212033913086756nteger @ S @ Rep2 ) @ ( comp_C1593894019821074884nteger @ ( comp_P4511551582283981294nteger @ Abs2 @ M ) @ G ) )
% 0.34/0.61 = ( comp_s8292494440407567195nteger @ ( image_1215581382706833972nteger @ F ) @ ( comp_C6537567860429791781nteger @ S4 @ G ) ) ) ) ) ).
% 0.34/0.61
% 0.34/0.61 % type_copy_set_map0
% 0.34/0.61 thf(fact_4352_type__copy__set__map0,axiom,
% 0.34/0.61 ! [Rep2: ( produc8923325533196201883nteger > produc8923325533196201883nteger ) > $o,Abs2: $o > produc8923325533196201883nteger > produc8923325533196201883nteger,S: $o > set_nat,M: code_integer > $o,F: nat > nat,S4: code_integer > set_nat,G: code_integer > code_integer] :
% 0.34/0.61 ( ( type_d5661033022651575429eger_o @ Rep2 @ Abs2 @ top_top_set_o )
% 0.34/0.61 => ( ( ( comp_o172754640513974618nteger @ S @ M )
% 0.34/0.61 = ( comp_s4605850120322998606nteger @ ( image_nat_nat2 @ F ) @ S4 ) )
% 0.34/0.61 => ( ( comp_P8262219321096090349nteger @ ( comp_o1289984117834210338nteger @ S @ Rep2 ) @ ( comp_C1593894019821074884nteger @ ( comp_o8922910837708301817nteger @ Abs2 @ M ) @ G ) )
% 0.34/0.61 = ( comp_s4605850120322998606nteger @ ( image_nat_nat2 @ F ) @ ( comp_C6537567860429791781nteger @ S4 @ G ) ) ) ) ) ).
% 0.34/0.61
% 0.34/0.61 % type_copy_set_map0
% 0.34/0.61 thf(fact_4353_type__copy__set__map0,axiom,
% 0.34/0.61 ! [Rep2: ( produc8923325533196201883nteger > produc8923325533196201883nteger ) > $o,Abs2: $o > produc8923325533196201883nteger > produc8923325533196201883nteger,S: $o > set_Code_integer,M: code_integer > $o,F: code_integer > code_integer,S4: code_integer > set_Code_integer,G: code_integer > code_integer] :
% 0.34/0.61 ( ( type_d5661033022651575429eger_o @ Rep2 @ Abs2 @ top_top_set_o )
% 0.34/0.61 => ( ( ( comp_o4291842030903409511nteger @ S @ M )
% 0.34/0.61 = ( comp_s956550612328592744nteger @ ( image_4470545334726330049nteger @ F ) @ S4 ) )
% 0.34/0.61 => ( ( comp_P4885729792186999034nteger @ ( comp_o7929101295636871983nteger @ S @ Rep2 ) @ ( comp_C1593894019821074884nteger @ ( comp_o8922910837708301817nteger @ Abs2 @ M ) @ G ) )
% 0.34/0.61 = ( comp_s956550612328592744nteger @ ( image_4470545334726330049nteger @ F ) @ ( comp_C8186931911283765298nteger @ S4 @ G ) ) ) ) ) ).
% 0.34/0.61
% 0.34/0.61 % type_copy_set_map0
% 0.34/0.61 thf(fact_4354_type__copy__set__map0,axiom,
% 0.34/0.61 ! [Rep2: ( produc8923325533196201883nteger > produc8923325533196201883nteger ) > $o,Abs2: $o > produc8923325533196201883nteger > produc8923325533196201883nteger,S: $o > set_Code_integer,M: code_integer > $o,F: nat > code_integer,S4: code_integer > set_nat,G: code_integer > code_integer] :
% 0.34/0.61 ( ( type_d5661033022651575429eger_o @ Rep2 @ Abs2 @ top_top_set_o )
% 0.34/0.61 => ( ( ( comp_o4291842030903409511nteger @ S @ M )
% 0.34/0.61 = ( comp_s8292494440407567195nteger @ ( image_1215581382706833972nteger @ F ) @ S4 ) )
% 0.34/0.61 => ( ( comp_P4885729792186999034nteger @ ( comp_o7929101295636871983nteger @ S @ Rep2 ) @ ( comp_C1593894019821074884nteger @ ( comp_o8922910837708301817nteger @ Abs2 @ M ) @ G ) )
% 0.34/0.61 = ( comp_s8292494440407567195nteger @ ( image_1215581382706833972nteger @ F ) @ ( comp_C6537567860429791781nteger @ S4 @ G ) ) ) ) ) ).
% 0.34/0.61
% 0.34/0.61 % type_copy_set_map0
% 0.34/0.61 thf(fact_4355_type__copy__set__map0,axiom,
% 0.34/0.61 ! [Rep2: ( produc8923325533196201883nteger > produc8923325533196201883nteger ) > nat,Abs2: nat > produc8923325533196201883nteger > produc8923325533196201883nteger,S: nat > set_nat,M: code_integer > nat,F: nat > nat,S4: code_integer > set_nat,G: code_integer > code_integer] :
% 0.34/0.61 ( ( type_d6278479724763174947er_nat @ Rep2 @ Abs2 @ top_top_set_nat )
% 0.34/0.61 => ( ( ( comp_n5963683855531483544nteger @ S @ M )
% 0.34/0.61 = ( comp_s4605850120322998606nteger @ ( image_nat_nat2 @ F ) @ S4 ) )
% 0.34/0.61 => ( ( comp_P8262219321096090349nteger @ ( comp_n3623798160786426464nteger @ S @ Rep2 ) @ ( comp_C1593894019821074884nteger @ ( comp_n2033352843805742135nteger @ Abs2 @ M ) @ G ) )
% 0.34/0.61 = ( comp_s4605850120322998606nteger @ ( image_nat_nat2 @ F ) @ ( comp_C6537567860429791781nteger @ S4 @ G ) ) ) ) ) ).
% 0.34/0.61
% 0.34/0.61 % type_copy_set_map0
% 0.34/0.61 thf(fact_4356_top__empty__eq,axiom,
% 0.34/0.61 ( top_to2462227307520033056at_o_o
% 0.34/0.61 = ( ^ [X2: produc3658429121746597890et_nat > $o] : ( member6576561426505652726_nat_o @ X2 @ top_to4869887016220417533_nat_o ) ) ) ).
% 0.34/0.61
% 0.34/0.61 % top_empty_eq
% 0.34/0.61 thf(fact_4357_top__empty__eq,axiom,
% 0.34/0.61 ( top_to2465898995584390880unit_o
% 0.34/0.61 = ( ^ [X2: product_unit] : ( member_Product_unit @ X2 @ top_to1996260823553986621t_unit ) ) ) ).
% 0.34/0.61
% 0.34/0.61 % top_empty_eq
% 0.34/0.61 thf(fact_4358_top__empty__eq,axiom,
% 0.34/0.61 ( top_top_o_o
% 0.34/0.61 = ( ^ [X2: $o] : ( member_o @ X2 @ top_top_set_o ) ) ) ).
% 0.34/0.61
% 0.34/0.61 % top_empty_eq
% 0.34/0.61 thf(fact_4359_top__empty__eq,axiom,
% 0.34/0.61 ( top_top_nat_o
% 0.34/0.61 = ( ^ [X2: nat] : ( member_nat @ X2 @ top_top_set_nat ) ) ) ).
% 0.34/0.61
% 0.34/0.61 % top_empty_eq
% 0.34/0.61 thf(fact_4360_top__empty__eq,axiom,
% 0.34/0.61 ( top_top_int_o
% 0.34/0.61 = ( ^ [X2: int] : ( member_int @ X2 @ top_top_set_int ) ) ) ).
% 0.34/0.61
% 0.34/0.61 % top_empty_eq
% 0.34/0.61 thf(fact_4361_top__set__def,axiom,
% 0.34/0.61 ( top_to4869887016220417533_nat_o
% 0.34/0.61 = ( collec939566748876313656_nat_o @ top_to2462227307520033056at_o_o ) ) ).
% 0.34/0.61
% 0.34/0.61 % top_set_def
% 0.34/0.61 thf(fact_4362_top__set__def,axiom,
% 0.34/0.61 ( top_to1996260823553986621t_unit
% 0.34/0.61 = ( collect_Product_unit @ top_to2465898995584390880unit_o ) ) ).
% 0.34/0.61
% 0.34/0.61 % top_set_def
% 0.34/0.61 thf(fact_4363_top__set__def,axiom,
% 0.34/0.61 ( top_top_set_o
% 0.34/0.61 = ( collect_o @ top_top_o_o ) ) ).
% 0.34/0.61
% 0.34/0.61 % top_set_def
% 0.34/0.61 thf(fact_4364_top__set__def,axiom,
% 0.34/0.61 ( top_top_set_nat
% 0.34/0.61 = ( collect_nat @ top_top_nat_o ) ) ).
% 0.34/0.61
% 0.34/0.61 % top_set_def
% 0.34/0.61 thf(fact_4365_top__set__def,axiom,
% 0.34/0.61 ( top_top_set_int
% 0.34/0.61 = ( collect_int @ top_top_int_o ) ) ).
% 0.34/0.61
% 0.34/0.61 % top_set_def
% 0.34/0.61 thf(fact_4366_pred__on_Ochain__empty,axiom,
% 0.34/0.61 ! [A: set_Product_unit,P: product_unit > product_unit > $o] : ( pred_c6915491785556283576t_unit @ A @ P @ bot_bo3957492148770167129t_unit ) ).
% 0.34/0.61
% 0.34/0.61 % pred_on.chain_empty
% 0.34/0.61 thf(fact_4367_pred__on_Ochain__empty,axiom,
% 0.34/0.61 ! [A: set_o,P: $o > $o > $o] : ( pred_chain_o @ A @ P @ bot_bot_set_o ) ).
% 0.34/0.61
% 0.34/0.61 % pred_on.chain_empty
% 0.34/0.61 thf(fact_4368_pred__on_Ochain__empty,axiom,
% 0.34/0.61 ! [A: set_nat,P: nat > nat > $o] : ( pred_chain_nat @ A @ P @ bot_bot_set_nat ) ).
% 0.34/0.61
% 0.34/0.61 % pred_on.chain_empty
% 0.34/0.61 thf(fact_4369_pred__on_Ochain__empty,axiom,
% 0.34/0.61 ! [A: set_int,P: int > int > $o] : ( pred_chain_int @ A @ P @ bot_bot_set_int ) ).
% 0.34/0.61
% 0.34/0.61 % pred_on.chain_empty
% 0.34/0.61 thf(fact_4370_pred__on_Ochain__empty,axiom,
% 0.34/0.61 ! [A: set_Pr4532377907799695533_nat_o,P: ( produc3658429121746597890et_nat > $o ) > ( produc3658429121746597890et_nat > $o ) > $o] : ( pred_c177362287423414904_nat_o @ A @ P @ bot_bo7824918357723582233_nat_o ) ).
% 0.34/0.61
% 0.34/0.61 % pred_on.chain_empty
% 0.34/0.61 thf(fact_4371_surj__fun__eq,axiom,
% 0.34/0.61 ! [F: code_integer > code_integer > code_integer,X4: set_Code_integer,G1: ( code_integer > code_integer ) > produc8923325533196201883nteger > produc8923325533196201883nteger,G2: ( code_integer > code_integer ) > produc8923325533196201883nteger > produc8923325533196201883nteger] :
% 0.34/0.61 ( ( ( image_2177946750938214461nteger @ F @ X4 )
% 0.34/0.61 = top_to3804862627464599871nteger )
% 0.34/0.61 => ( ! [X3: code_integer] :
% 0.34/0.61 ( ( member_Code_integer @ X3 @ X4 )
% 0.34/0.61 => ( ( comp_C8797469213163452608nteger @ G1 @ F @ X3 )
% 0.34/0.61 = ( comp_C8797469213163452608nteger @ G2 @ F @ X3 ) ) )
% 0.34/0.61 => ( G1 = G2 ) ) ) ).
% 0.34/0.61
% 0.34/0.61 % surj_fun_eq
% 0.34/0.61 thf(fact_4372_surj__fun__eq,axiom,
% 0.34/0.61 ! [F: code_integer > code_integer,X4: set_Code_integer,G1: code_integer > produc8923325533196201883nteger > produc8923325533196201883nteger,G2: code_integer > produc8923325533196201883nteger > produc8923325533196201883nteger] :
% 0.34/0.61 ( ( ( image_4470545334726330049nteger @ F @ X4 )
% 0.34/0.61 = top_to4645266643341252675nteger )
% 0.34/0.61 => ( ! [X3: code_integer] :
% 0.34/0.61 ( ( member_Code_integer @ X3 @ X4 )
% 0.34/0.61 => ( ( comp_C1593894019821074884nteger @ G1 @ F @ X3 )
% 0.34/0.61 = ( comp_C1593894019821074884nteger @ G2 @ F @ X3 ) ) )
% 0.34/0.61 => ( G1 = G2 ) ) ) ).
% 0.34/0.61
% 0.34/0.61 % surj_fun_eq
% 0.34/0.61 thf(fact_4373_type__copy__map__comp0,axiom,
% 0.34/0.61 ! [Rep2: code_integer > code_integer,Abs2: code_integer > code_integer,M: code_integer > code_integer,M1: code_integer > code_integer,M22: code_integer > code_integer,F: code_integer > produc8923325533196201883nteger > produc8923325533196201883nteger,G: code_integer > code_integer] :
% 0.34/0.61 ( ( type_d4796611474492816872nteger @ Rep2 @ Abs2 @ top_to4645266643341252675nteger )
% 0.34/0.61 => ( ( M
% 0.34/0.61 = ( comp_C7449957260575251196nteger @ M1 @ M22 ) )
% 0.34/0.61 => ( ( comp_C1593894019821074884nteger @ ( comp_C1593894019821074884nteger @ F @ M ) @ G )
% 0.34/0.61 = ( comp_C1593894019821074884nteger @ ( comp_C1593894019821074884nteger @ ( comp_C1593894019821074884nteger @ F @ M1 ) @ Rep2 ) @ ( comp_C7449957260575251196nteger @ ( comp_C7449957260575251196nteger @ Abs2 @ M22 ) @ G ) ) ) ) ) ).
% 0.34/0.61
% 0.34/0.61 % type_copy_map_comp0
% 0.34/0.61 thf(fact_4374_type__copy__map__comp0,axiom,
% 0.34/0.61 ! [Rep2: code_integer > product_unit,Abs2: product_unit > code_integer,M: code_integer > code_integer,M1: product_unit > code_integer,M22: code_integer > product_unit,F: code_integer > produc8923325533196201883nteger > produc8923325533196201883nteger,G: code_integer > code_integer] :
% 0.34/0.61 ( ( type_d9006111508436661314t_unit @ Rep2 @ Abs2 @ top_to1996260823553986621t_unit )
% 0.34/0.61 => ( ( M
% 0.34/0.61 = ( comp_P6790823133229852966nteger @ M1 @ M22 ) )
% 0.34/0.61 => ( ( comp_C1593894019821074884nteger @ ( comp_C1593894019821074884nteger @ F @ M ) @ G )
% 0.34/0.61 = ( comp_C1593894019821074884nteger @ ( comp_P4511551582283981294nteger @ ( comp_C5854136495726678886t_unit @ F @ M1 ) @ Rep2 ) @ ( comp_C7449957260575251196nteger @ ( comp_P6790823133229852966nteger @ Abs2 @ M22 ) @ G ) ) ) ) ) ).
% 0.34/0.61
% 0.34/0.61 % type_copy_map_comp0
% 0.34/0.61 thf(fact_4375_type__copy__map__comp0,axiom,
% 0.34/0.61 ! [Rep2: code_integer > $o,Abs2: $o > code_integer,M: code_integer > code_integer,M1: $o > code_integer,M22: code_integer > $o,F: code_integer > produc8923325533196201883nteger > produc8923325533196201883nteger,G: code_integer > code_integer] :
% 0.34/0.61 ( ( type_d6812736842639505741eger_o @ Rep2 @ Abs2 @ top_top_set_o )
% 0.34/0.61 => ( ( M
% 0.34/0.61 = ( comp_o1085144040659434033nteger @ M1 @ M22 ) )
% 0.34/0.61 => ( ( comp_C1593894019821074884nteger @ ( comp_C1593894019821074884nteger @ F @ M ) @ G )
% 0.34/0.61 = ( comp_C1593894019821074884nteger @ ( comp_o8922910837708301817nteger @ ( comp_C5733264393235242609eger_o @ F @ M1 ) @ Rep2 ) @ ( comp_C7449957260575251196nteger @ ( comp_o1085144040659434033nteger @ Abs2 @ M22 ) @ G ) ) ) ) ) ).
% 0.34/0.61
% 0.34/0.61 % type_copy_map_comp0
% 0.34/0.61 thf(fact_4376_type__copy__map__comp0,axiom,
% 0.34/0.61 ! [Rep2: code_integer > nat,Abs2: nat > code_integer,M: code_integer > code_integer,M1: nat > code_integer,M22: code_integer > nat,F: code_integer > produc8923325533196201883nteger > produc8923325533196201883nteger,G: code_integer > code_integer] :
% 0.34/0.62 ( ( type_d8368584451094728027er_nat @ Rep2 @ Abs2 @ top_top_set_nat )
% 0.34/0.62 => ( ( M
% 0.34/0.62 = ( comp_n6876073255676942959nteger @ M1 @ M22 ) )
% 0.34/0.62 => ( ( comp_C1593894019821074884nteger @ ( comp_C1593894019821074884nteger @ F @ M ) @ G )
% 0.34/0.62 = ( comp_C1593894019821074884nteger @ ( comp_n2033352843805742135nteger @ ( comp_C836733381939181879er_nat @ F @ M1 ) @ Rep2 ) @ ( comp_C7449957260575251196nteger @ ( comp_n6876073255676942959nteger @ Abs2 @ M22 ) @ G ) ) ) ) ) ).
% 0.34/0.62
% 0.34/0.62 % type_copy_map_comp0
% 0.34/0.62 thf(fact_4377_type__copy__map__comp0,axiom,
% 0.34/0.62 ! [Rep2: code_integer > int,Abs2: int > code_integer,M: code_integer > code_integer,M1: int > code_integer,M22: code_integer > int,F: code_integer > produc8923325533196201883nteger > produc8923325533196201883nteger,G: code_integer > code_integer] :
% 0.34/0.62 ( ( type_d8366093980585677751er_int @ Rep2 @ Abs2 @ top_top_set_int )
% 0.34/0.62 => ( ( M
% 0.34/0.62 = ( comp_i6014793192326455499nteger @ M1 @ M22 ) )
% 0.34/0.62 => ( ( comp_C1593894019821074884nteger @ ( comp_C1593894019821074884nteger @ F @ M ) @ G )
% 0.34/0.62 = ( comp_C1593894019821074884nteger @ ( comp_i2874689474467862163nteger @ ( comp_C834242911430131603er_int @ F @ M1 ) @ Rep2 ) @ ( comp_C7449957260575251196nteger @ ( comp_i6014793192326455499nteger @ Abs2 @ M22 ) @ G ) ) ) ) ) ).
% 0.34/0.62
% 0.34/0.62 % type_copy_map_comp0
% 0.34/0.62 thf(fact_4378_type__copy__map__comp0,axiom,
% 0.34/0.62 ! [Rep2: product_unit > $o,Abs2: $o > product_unit,M: code_integer > code_integer,M1: $o > code_integer,M22: code_integer > $o,F: code_integer > produc8923325533196201883nteger > produc8923325533196201883nteger,G: code_integer > code_integer] :
% 0.34/0.62 ( ( type_d6188575255521822967unit_o @ Rep2 @ Abs2 @ top_top_set_o )
% 0.34/0.62 => ( ( M
% 0.34/0.62 = ( comp_o1085144040659434033nteger @ M1 @ M22 ) )
% 0.34/0.62 => ( ( comp_C1593894019821074884nteger @ ( comp_C1593894019821074884nteger @ F @ M ) @ G )
% 0.34/0.62 = ( comp_P4511551582283981294nteger @ ( comp_o6961622296474427249t_unit @ ( comp_C5733264393235242609eger_o @ F @ M1 ) @ Rep2 ) @ ( comp_C8146212659816229034nteger @ ( comp_o6635994240230588853nteger @ Abs2 @ M22 ) @ G ) ) ) ) ) ).
% 0.34/0.62
% 0.34/0.62 % type_copy_map_comp0
% 0.34/0.62 thf(fact_4379_type__copy__map__comp0,axiom,
% 0.34/0.62 ! [Rep2: code_natural > nat,Abs2: nat > code_natural,M: code_integer > code_integer,M1: nat > code_integer,M22: code_integer > nat,F: code_integer > produc8923325533196201883nteger > produc8923325533196201883nteger,G: code_integer > code_integer] :
% 0.34/0.62 ( ( type_d4410041424927559462al_nat @ Rep2 @ Abs2 @ top_top_set_nat )
% 0.34/0.62 => ( ( M
% 0.34/0.62 = ( comp_n6876073255676942959nteger @ M1 @ M22 ) )
% 0.34/0.62 => ( ( comp_C1593894019821074884nteger @ ( comp_C1593894019821074884nteger @ F @ M ) @ G )
% 0.34/0.62 = ( comp_C456341964420721295nteger @ ( comp_n856947995562459650atural @ ( comp_C836733381939181879er_nat @ F @ M1 ) @ Rep2 ) @ ( comp_C1891057325546811591nteger @ ( comp_n1317173320648503354nteger @ Abs2 @ M22 ) @ G ) ) ) ) ) ).
% 0.34/0.62
% 0.34/0.62 % type_copy_map_comp0
% 0.34/0.62 thf(fact_4380_type__copy__map__comp0,axiom,
% 0.34/0.62 ! [Rep2: code_integer > code_integer > code_integer,Abs2: ( code_integer > code_integer ) > code_integer,M: code_integer > code_integer,M1: ( code_integer > code_integer ) > code_integer,M22: code_integer > code_integer > code_integer,F: code_integer > produc8923325533196201883nteger > produc8923325533196201883nteger,G: code_integer > code_integer] :
% 0.34/0.62 ( ( type_d669523880323286244nteger @ Rep2 @ Abs2 @ top_to3804862627464599871nteger )
% 0.34/0.62 => ( ( M
% 0.34/0.62 = ( comp_C5076291335850163704nteger @ M1 @ M22 ) )
% 0.34/0.62 => ( ( comp_C1593894019821074884nteger @ ( comp_C1593894019821074884nteger @ F @ M ) @ G )
% 0.34/0.62 = ( comp_C1593894019821074884nteger @ ( comp_C8797469213163452608nteger @ ( comp_C95226695241045696nteger @ F @ M1 ) @ Rep2 ) @ ( comp_C7449957260575251196nteger @ ( comp_C5076291335850163704nteger @ Abs2 @ M22 ) @ G ) ) ) ) ) ).
% 0.34/0.62
% 0.34/0.62 % type_copy_map_comp0
% 0.34/0.62 thf(fact_4381_type__copy__map__comp0,axiom,
% 0.34/0.62 ! [Rep2: ( code_integer > code_integer ) > code_integer,Abs2: code_integer > code_integer > code_integer,M: code_integer > code_integer,M1: code_integer > code_integer,M22: code_integer > code_integer,F: code_integer > produc8923325533196201883nteger > produc8923325533196201883nteger,G: code_integer > code_integer] :
% 0.34/0.62 ( ( type_d5515310227294319076nteger @ Rep2 @ Abs2 @ top_to4645266643341252675nteger )
% 0.34/0.62 => ( ( M
% 0.34/0.62 = ( comp_C7449957260575251196nteger @ M1 @ M22 ) )
% 0.34/0.62 => ( ( comp_C1593894019821074884nteger @ ( comp_C1593894019821074884nteger @ F @ M ) @ G )
% 0.34/0.62 = ( comp_C8797469213163452608nteger @ ( comp_C95226695241045696nteger @ ( comp_C1593894019821074884nteger @ F @ M1 ) @ Rep2 ) @ ( comp_C2931745410781319416nteger @ ( comp_C2931745410781319416nteger @ Abs2 @ M22 ) @ G ) ) ) ) ) ).
% 0.34/0.62
% 0.34/0.62 % type_copy_map_comp0
% 0.34/0.62 thf(fact_4382_type__copy__map__comp0,axiom,
% 0.34/0.62 ! [Rep2: code_integer > code_integer,Abs2: code_integer > code_integer,M: code_integer > code_integer > code_integer,M1: code_integer > code_integer > code_integer,M22: code_integer > code_integer,F: ( code_integer > code_integer ) > produc8923325533196201883nteger > produc8923325533196201883nteger,G: code_integer > code_integer] :
% 0.34/0.62 ( ( type_d4796611474492816872nteger @ Rep2 @ Abs2 @ top_to4645266643341252675nteger )
% 0.34/0.62 => ( ( M
% 0.34/0.62 = ( comp_C2931745410781319416nteger @ M1 @ M22 ) )
% 0.34/0.62 => ( ( comp_C1593894019821074884nteger @ ( comp_C8797469213163452608nteger @ F @ M ) @ G )
% 0.34/0.62 = ( comp_C1593894019821074884nteger @ ( comp_C1593894019821074884nteger @ ( comp_C8797469213163452608nteger @ F @ M1 ) @ Rep2 ) @ ( comp_C7449957260575251196nteger @ ( comp_C7449957260575251196nteger @ Abs2 @ M22 ) @ G ) ) ) ) ) ).
% 0.34/0.62
% 0.34/0.62 % type_copy_map_comp0
% 0.34/0.62 thf(fact_4383_type__copy__map__comp0__undo,axiom,
% 0.34/0.62 ! [Rep2: code_integer > int,Abs2: int > code_integer,Rep3: code_integer > int,Abs3: int > code_integer,Rep4: code_integer > int,Abs4: int > code_integer,M: int > int,M1: int > int,M22: int > int] :
% 0.34/0.62 ( ( type_d8366093980585677751er_int @ Rep2 @ Abs2 @ top_top_set_int )
% 0.34/0.62 => ( ( type_d8366093980585677751er_int @ Rep3 @ Abs3 @ top_top_set_int )
% 0.34/0.62 => ( ( type_d8366093980585677751er_int @ Rep4 @ Abs4 @ top_top_set_int )
% 0.34/0.62 => ( ( ( comp_i6014793192326455499nteger @ ( comp_i1585864551200866970er_int @ Abs3 @ M ) @ Rep4 )
% 0.34/0.62 = ( comp_C7449957260575251196nteger @ ( comp_i6014793192326455499nteger @ ( comp_i1585864551200866970er_int @ Abs3 @ M1 ) @ Rep2 ) @ ( comp_i6014793192326455499nteger @ ( comp_i1585864551200866970er_int @ Abs2 @ M22 ) @ Rep4 ) ) )
% 0.34/0.62 => ( ( comp_int_int_int @ M1 @ M22 )
% 0.34/0.62 = M ) ) ) ) ) ).
% 0.34/0.62
% 0.34/0.62 % type_copy_map_comp0_undo
% 0.34/0.62 thf(fact_4384_type__copy__map__comp0__undo,axiom,
% 0.34/0.62 ! [Rep2: code_integer > int,Abs2: int > code_integer,Rep3: code_integer > int,Abs3: int > code_integer,Rep4: product_unit > $o,Abs4: $o > product_unit,M: $o > int,M1: int > int,M22: $o > int] :
% 0.34/0.62 ( ( type_d8366093980585677751er_int @ Rep2 @ Abs2 @ top_top_set_int )
% 0.34/0.62 => ( ( type_d8366093980585677751er_int @ Rep3 @ Abs3 @ top_top_set_int )
% 0.34/0.62 => ( ( type_d6188575255521822967unit_o @ Rep4 @ Abs4 @ top_top_set_o )
% 0.34/0.62 => ( ( ( comp_o780160699970281785t_unit @ ( comp_i5286552343147518890eger_o @ Abs3 @ M ) @ Rep4 )
% 0.34/0.62 = ( comp_C2290379119555921966t_unit @ ( comp_i6014793192326455499nteger @ ( comp_i1585864551200866970er_int @ Abs3 @ M1 ) @ Rep2 ) @ ( comp_o780160699970281785t_unit @ ( comp_i5286552343147518890eger_o @ Abs2 @ M22 ) @ Rep4 ) ) )
% 0.34/0.62 => ( ( comp_int_int_o @ M1 @ M22 )
% 0.34/0.62 = M ) ) ) ) ) ).
% 0.34/0.62
% 0.34/0.62 % type_copy_map_comp0_undo
% 0.34/0.62 thf(fact_4385_type__copy__map__comp0__undo,axiom,
% 0.34/0.62 ! [Rep2: code_integer > int,Abs2: int > code_integer,Rep3: code_integer > int,Abs3: int > code_integer,Rep4: code_natural > nat,Abs4: nat > code_natural,M: nat > int,M1: int > int,M22: nat > int] :
% 0.34/0.62 ( ( type_d8366093980585677751er_int @ Rep2 @ Abs2 @ top_top_set_int )
% 0.34/0.62 => ( ( type_d8366093980585677751er_int @ Rep3 @ Abs3 @ top_top_set_int )
% 0.34/0.62 => ( ( type_d4410041424927559462al_nat @ Rep4 @ Abs4 @ top_top_set_nat )
% 0.34/0.62 => ( ( ( comp_n5699668407433660474atural @ ( comp_i1588355021709917246er_nat @ Abs3 @ M ) @ Rep4 )
% 0.34/0.62 = ( comp_C6273552412331968711atural @ ( comp_i6014793192326455499nteger @ ( comp_i1585864551200866970er_int @ Abs3 @ M1 ) @ Rep2 ) @ ( comp_n5699668407433660474atural @ ( comp_i1588355021709917246er_nat @ Abs2 @ M22 ) @ Rep4 ) ) )
% 0.34/0.62 => ( ( comp_int_int_nat @ M1 @ M22 )
% 0.34/0.62 = M ) ) ) ) ) ).
% 0.34/0.62
% 0.34/0.62 % type_copy_map_comp0_undo
% 0.34/0.62 thf(fact_4386_type__copy__map__comp0__undo,axiom,
% 0.34/0.62 ! [Rep2: code_integer > int,Abs2: int > code_integer,Rep3: product_unit > $o,Abs3: $o > product_unit,Rep4: code_integer > int,Abs4: int > code_integer,M: int > $o,M1: int > $o,M22: int > int] :
% 0.34/0.62 ( ( type_d8366093980585677751er_int @ Rep2 @ Abs2 @ top_top_set_int )
% 0.34/0.62 => ( ( type_d6188575255521822967unit_o @ Rep3 @ Abs3 @ top_top_set_o )
% 0.34/0.62 => ( ( type_d8366093980585677751er_int @ Rep4 @ Abs4 @ top_top_set_int )
% 0.34/0.62 => ( ( ( comp_i8439899303502915483nteger @ ( comp_o7049022088938261380it_int @ Abs3 @ M ) @ Rep4 )
% 0.34/0.62 = ( comp_C8146212659816229034nteger @ ( comp_i8439899303502915483nteger @ ( comp_o7049022088938261380it_int @ Abs3 @ M1 ) @ Rep2 ) @ ( comp_i6014793192326455499nteger @ ( comp_i1585864551200866970er_int @ Abs2 @ M22 ) @ Rep4 ) ) )
% 0.34/0.62 => ( ( comp_int_o_int @ M1 @ M22 )
% 0.34/0.62 = M ) ) ) ) ) ).
% 0.34/0.62
% 0.34/0.62 % type_copy_map_comp0_undo
% 0.34/0.62 thf(fact_4387_type__copy__map__comp0__undo,axiom,
% 0.34/0.62 ! [Rep2: code_integer > int,Abs2: int > code_integer,Rep3: product_unit > $o,Abs3: $o > product_unit,Rep4: product_unit > $o,Abs4: $o > product_unit,M: $o > $o,M1: int > $o,M22: $o > int] :
% 0.34/0.62 ( ( type_d8366093980585677751er_int @ Rep2 @ Abs2 @ top_top_set_int )
% 0.34/0.62 => ( ( type_d6188575255521822967unit_o @ Rep3 @ Abs3 @ top_top_set_o )
% 0.34/0.62 => ( ( type_d6188575255521822967unit_o @ Rep4 @ Abs4 @ top_top_set_o )
% 0.34/0.62 => ( ( ( comp_o7724735231917782709t_unit @ ( comp_o3916225744453444672unit_o @ Abs3 @ M ) @ Rep4 )
% 0.34/0.62 = ( comp_C2408090164605745152t_unit @ ( comp_i8439899303502915483nteger @ ( comp_o7049022088938261380it_int @ Abs3 @ M1 ) @ Rep2 ) @ ( comp_o780160699970281785t_unit @ ( comp_i5286552343147518890eger_o @ Abs2 @ M22 ) @ Rep4 ) ) )
% 0.34/0.62 => ( ( comp_int_o_o @ M1 @ M22 )
% 0.34/0.62 = M ) ) ) ) ) ).
% 0.34/0.62
% 0.34/0.62 % type_copy_map_comp0_undo
% 0.34/0.62 thf(fact_4388_type__copy__map__comp0__undo,axiom,
% 0.34/0.62 ! [Rep2: code_integer > int,Abs2: int > code_integer,Rep3: product_unit > $o,Abs3: $o > product_unit,Rep4: code_natural > nat,Abs4: nat > code_natural,M: nat > $o,M1: int > $o,M22: nat > int] :
% 0.34/0.62 ( ( type_d8366093980585677751er_int @ Rep2 @ Abs2 @ top_top_set_int )
% 0.34/0.62 => ( ( type_d6188575255521822967unit_o @ Rep3 @ Abs3 @ top_top_set_o )
% 0.34/0.62 => ( ( type_d4410041424927559462al_nat @ Rep4 @ Abs4 @ top_top_set_nat )
% 0.34/0.62 => ( ( ( comp_n8360577435903631554atural @ ( comp_o7051512559447311656it_nat @ Abs3 @ M ) @ Rep4 )
% 0.34/0.62 = ( comp_C6969807811572946549atural @ ( comp_i8439899303502915483nteger @ ( comp_o7049022088938261380it_int @ Abs3 @ M1 ) @ Rep2 ) @ ( comp_n5699668407433660474atural @ ( comp_i1588355021709917246er_nat @ Abs2 @ M22 ) @ Rep4 ) ) )
% 0.34/0.62 => ( ( comp_int_o_nat @ M1 @ M22 )
% 0.34/0.62 = M ) ) ) ) ) ).
% 0.34/0.62
% 0.34/0.62 % type_copy_map_comp0_undo
% 0.34/0.62 thf(fact_4389_type__copy__map__comp0__undo,axiom,
% 0.34/0.62 ! [Rep2: code_integer > int,Abs2: int > code_integer,Rep3: code_natural > nat,Abs3: nat > code_natural,Rep4: code_integer > int,Abs4: int > code_integer,M: int > nat,M1: int > nat,M22: int > int] :
% 0.34/0.62 ( ( type_d8366093980585677751er_int @ Rep2 @ Abs2 @ top_top_set_int )
% 0.34/0.62 => ( ( type_d4410041424927559462al_nat @ Rep3 @ Abs3 @ top_top_set_nat )
% 0.34/0.62 => ( ( type_d8366093980585677751er_int @ Rep4 @ Abs4 @ top_top_set_int )
% 0.34/0.62 => ( ( ( comp_i455893257298015894nteger @ ( comp_n9160511493981425161al_int @ Abs3 @ M ) @ Rep4 )
% 0.34/0.62 = ( comp_C1891057325546811591nteger @ ( comp_i455893257298015894nteger @ ( comp_n9160511493981425161al_int @ Abs3 @ M1 ) @ Rep2 ) @ ( comp_i6014793192326455499nteger @ ( comp_i1585864551200866970er_int @ Abs2 @ M22 ) @ Rep4 ) ) )
% 0.34/0.62 => ( ( comp_int_nat_int @ M1 @ M22 )
% 0.34/0.62 = M ) ) ) ) ) ).
% 0.34/0.62
% 0.34/0.62 % type_copy_map_comp0_undo
% 0.34/0.62 thf(fact_4390_type__copy__map__comp0__undo,axiom,
% 0.34/0.62 ! [Rep2: code_integer > int,Abs2: int > code_integer,Rep3: code_natural > nat,Abs3: nat > code_natural,Rep4: product_unit > $o,Abs4: $o > product_unit,M: $o > nat,M1: int > nat,M22: $o > int] :
% 0.34/0.62 ( ( type_d8366093980585677751er_int @ Rep2 @ Abs2 @ top_top_set_int )
% 0.34/0.62 => ( ( type_d4410041424927559462al_nat @ Rep3 @ Abs3 @ top_top_set_nat )
% 0.34/0.62 => ( ( type_d6188575255521822967unit_o @ Rep4 @ Abs4 @ top_top_set_o )
% 0.34/0.62 => ( ( ( comp_o2643362028263920430t_unit @ ( comp_n2857367870785791611ural_o @ Abs3 @ M ) @ Rep4 )
% 0.34/0.62 = ( comp_C4153580447849560611t_unit @ ( comp_i455893257298015894nteger @ ( comp_n9160511493981425161al_int @ Abs3 @ M1 ) @ Rep2 ) @ ( comp_o780160699970281785t_unit @ ( comp_i5286552343147518890eger_o @ Abs2 @ M22 ) @ Rep4 ) ) )
% 0.34/0.62 => ( ( comp_int_nat_o @ M1 @ M22 )
% 0.34/0.62 = M ) ) ) ) ) ).
% 0.34/0.62
% 0.34/0.62 % type_copy_map_comp0_undo
% 0.34/0.62 thf(fact_4391_type__copy__map__comp0__undo,axiom,
% 0.34/0.62 ! [Rep2: code_integer > int,Abs2: int > code_integer,Rep3: code_natural > nat,Abs3: nat > code_natural,Rep4: code_natural > nat,Abs4: nat > code_natural,M: nat > nat,M1: int > nat,M22: nat > int] :
% 0.34/0.62 ( ( type_d8366093980585677751er_int @ Rep2 @ Abs2 @ top_top_set_int )
% 0.34/0.62 => ( ( type_d4410041424927559462al_nat @ Rep3 @ Abs3 @ top_top_set_nat )
% 0.34/0.62 => ( ( type_d4410041424927559462al_nat @ Rep4 @ Abs4 @ top_top_set_nat )
% 0.34/0.62 => ( ( ( comp_n140768472405220869atural @ ( comp_n9163001964490475437al_nat @ Abs3 @ M ) @ Rep4 )
% 0.34/0.62 = ( comp_C714652477303529106atural @ ( comp_i455893257298015894nteger @ ( comp_n9160511493981425161al_int @ Abs3 @ M1 ) @ Rep2 ) @ ( comp_n5699668407433660474atural @ ( comp_i1588355021709917246er_nat @ Abs2 @ M22 ) @ Rep4 ) ) )
% 0.34/0.62 => ( ( comp_int_nat_nat @ M1 @ M22 )
% 0.34/0.62 = M ) ) ) ) ) ).
% 0.34/0.62
% 0.34/0.62 % type_copy_map_comp0_undo
% 0.34/0.62 thf(fact_4392_type__copy__map__comp0__undo,axiom,
% 0.34/0.62 ! [Rep2: product_unit > $o,Abs2: $o > product_unit,Rep3: code_integer > int,Abs3: int > code_integer,Rep4: code_integer > int,Abs4: int > code_integer,M: int > int,M1: $o > int,M22: int > $o] :
% 0.34/0.62 ( ( type_d6188575255521822967unit_o @ Rep2 @ Abs2 @ top_top_set_o )
% 0.34/0.62 => ( ( type_d8366093980585677751er_int @ Rep3 @ Abs3 @ top_top_set_int )
% 0.34/0.62 => ( ( type_d8366093980585677751er_int @ Rep4 @ Abs4 @ top_top_set_int )
% 0.34/0.62 => ( ( ( comp_i6014793192326455499nteger @ ( comp_i1585864551200866970er_int @ Abs3 @ M ) @ Rep4 )
% 0.34/0.62 = ( comp_P6790823133229852966nteger @ ( comp_o780160699970281785t_unit @ ( comp_i5286552343147518890eger_o @ Abs3 @ M1 ) @ Rep2 ) @ ( comp_i8439899303502915483nteger @ ( comp_o7049022088938261380it_int @ Abs2 @ M22 ) @ Rep4 ) ) )
% 0.34/0.62 => ( ( comp_o_int_int @ M1 @ M22 )
% 0.34/0.62 = M ) ) ) ) ) ).
% 0.34/0.62
% 0.34/0.62 % type_copy_map_comp0_undo
% 0.34/0.62 thf(fact_4393_wf__bounded__measure,axiom,
% 0.34/0.62 ! [R2: set_Pr1261947904930325089at_nat,Ub: nat > nat,F: nat > nat] :
% 0.34/0.62 ( ! [A4: nat,B2: nat] :
% 0.34/0.62 ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ B2 @ A4 ) @ R2 )
% 0.34/0.62 => ( ( ord_less_eq_nat @ ( Ub @ B2 ) @ ( Ub @ A4 ) )
% 0.34/0.62 & ( ord_less_eq_nat @ ( F @ B2 ) @ ( Ub @ A4 ) )
% 0.34/0.62 & ( ord_less_nat @ ( F @ A4 ) @ ( F @ B2 ) ) ) )
% 0.34/0.62 => ( wf_nat @ R2 ) ) ).
% 0.34/0.62
% 0.34/0.62 % wf_bounded_measure
% 0.34/0.62 thf(fact_4394_wf__bounded__measure,axiom,
% 0.34/0.62 ! [R2: set_Pr958786334691620121nt_int,Ub: int > nat,F: int > nat] :
% 0.34/0.62 ( ! [A4: int,B2: int] :
% 0.34/0.62 ( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ B2 @ A4 ) @ R2 )
% 0.34/0.62 => ( ( ord_less_eq_nat @ ( Ub @ B2 ) @ ( Ub @ A4 ) )
% 0.34/0.62 & ( ord_less_eq_nat @ ( F @ B2 ) @ ( Ub @ A4 ) )
% 0.34/0.62 & ( ord_less_nat @ ( F @ A4 ) @ ( F @ B2 ) ) ) )
% 0.34/0.62 => ( wf_int @ R2 ) ) ).
% 0.34/0.62
% 0.34/0.62 % wf_bounded_measure
% 0.34/0.62 thf(fact_4395_less__assn__def,axiom,
% 0.34/0.62 ( ord_less_assn
% 0.34/0.62 = ( ^ [A6: assn,B6: assn] :
% 0.34/0.62 ( ( ord_less_eq_assn @ A6 @ B6 )
% 0.34/0.62 & ( A6 != B6 ) ) ) ) ).
% 0.34/0.62
% 0.34/0.62 % less_assn_def
% 0.34/0.62 thf(fact_4396_top__assn__def,axiom,
% 0.34/0.62 ( top_top_assn
% 0.34/0.62 = ( abs_assn @ in_range ) ) ).
% 0.34/0.62
% 0.34/0.62 % top_assn_def
% 0.34/0.62 thf(fact_4397_pred__on_Ochain__extend,axiom,
% 0.34/0.62 ! [A: set_Product_unit,P: product_unit > product_unit > $o,C5: set_Product_unit,Z: product_unit] :
% 0.34/0.62 ( ( pred_c6915491785556283576t_unit @ A @ P @ C5 )
% 0.34/0.62 => ( ( member_Product_unit @ Z @ A )
% 0.34/0.62 => ( ! [X3: product_unit] :
% 0.34/0.62 ( ( member_Product_unit @ X3 @ C5 )
% 0.34/0.62 => ( sup_su3411595859437804162unit_o @ P
% 0.34/0.62 @ ^ [Y4: product_unit,Z4: product_unit] : ( Y4 = Z4 )
% 0.34/0.62 @ X3
% 0.34/0.62 @ Z ) )
% 0.34/0.62 => ( pred_c6915491785556283576t_unit @ A @ P @ ( sup_su793286257634532545t_unit @ ( insert_Product_unit @ Z @ bot_bo3957492148770167129t_unit ) @ C5 ) ) ) ) ) ).
% 0.34/0.62
% 0.34/0.62 % pred_on.chain_extend
% 0.34/0.62 thf(fact_4398_pred__on_Ochain__extend,axiom,
% 0.34/0.62 ! [A: set_o,P: $o > $o > $o,C5: set_o,Z: $o] :
% 0.34/0.62 ( ( pred_chain_o @ A @ P @ C5 )
% 0.34/0.62 => ( ( member_o @ Z @ A )
% 0.34/0.62 => ( ! [X3: $o] :
% 0.34/0.62 ( ( member_o @ X3 @ C5 )
% 0.34/0.62 => ( sup_sup_o_o_o @ P
% 0.34/0.62 @ ^ [Y4: $o,Z4: $o] : ( Y4 = Z4 )
% 0.34/0.62 @ X3
% 0.34/0.62 @ Z ) )
% 0.34/0.62 => ( pred_chain_o @ A @ P @ ( sup_sup_set_o @ ( insert_o @ Z @ bot_bot_set_o ) @ C5 ) ) ) ) ) ).
% 0.34/0.62
% 0.34/0.62 % pred_on.chain_extend
% 0.34/0.62 thf(fact_4399_pred__on_Ochain__extend,axiom,
% 0.34/0.62 ! [A: set_nat,P: nat > nat > $o,C5: set_nat,Z: nat] :
% 0.34/0.62 ( ( pred_chain_nat @ A @ P @ C5 )
% 0.34/0.62 => ( ( member_nat @ Z @ A )
% 0.34/0.62 => ( ! [X3: nat] :
% 0.34/0.62 ( ( member_nat @ X3 @ C5 )
% 0.34/0.62 => ( sup_sup_nat_nat_o @ P
% 0.34/0.62 @ ^ [Y4: nat,Z4: nat] : ( Y4 = Z4 )
% 0.34/0.62 @ X3
% 0.34/0.62 @ Z ) )
% 0.34/0.62 => ( pred_chain_nat @ A @ P @ ( sup_sup_set_nat @ ( insert_nat @ Z @ bot_bot_set_nat ) @ C5 ) ) ) ) ) ).
% 0.34/0.62
% 0.34/0.62 % pred_on.chain_extend
% 0.34/0.62 thf(fact_4400_pred__on_Ochain__extend,axiom,
% 0.34/0.62 ! [A: set_int,P: int > int > $o,C5: set_int,Z: int] :
% 0.34/0.62 ( ( pred_chain_int @ A @ P @ C5 )
% 0.34/0.62 => ( ( member_int @ Z @ A )
% 0.34/0.62 => ( ! [X3: int] :
% 0.34/0.62 ( ( member_int @ X3 @ C5 )
% 0.34/0.62 => ( sup_sup_int_int_o @ P
% 0.34/0.62 @ ^ [Y4: int,Z4: int] : ( Y4 = Z4 )
% 0.34/0.62 @ X3
% 0.34/0.62 @ Z ) )
% 0.34/0.62 => ( pred_chain_int @ A @ P @ ( sup_sup_set_int @ ( insert_int @ Z @ bot_bot_set_int ) @ C5 ) ) ) ) ) ).
% 0.34/0.62
% 0.34/0.62 % pred_on.chain_extend
% 0.34/0.62 thf(fact_4401_pred__on_Ochain__extend,axiom,
% 0.34/0.62 ! [A: set_Pr4532377907799695533_nat_o,P: ( produc3658429121746597890et_nat > $o ) > ( produc3658429121746597890et_nat > $o ) > $o,C5: set_Pr4532377907799695533_nat_o,Z: produc3658429121746597890et_nat > $o] :
% 0.34/0.62 ( ( pred_c177362287423414904_nat_o @ A @ P @ C5 )
% 0.34/0.62 => ( ( member6576561426505652726_nat_o @ Z @ A )
% 0.34/0.62 => ( ! [X3: produc3658429121746597890et_nat > $o] :
% 0.34/0.62 ( ( member6576561426505652726_nat_o @ X3 @ C5 )
% 0.34/0.62 => ( sup_su1942872429929737346at_o_o @ P
% 0.34/0.62 @ ^ [Y4: produc3658429121746597890et_nat > $o,Z4: produc3658429121746597890et_nat > $o] : ( Y4 = Z4 )
% 0.34/0.62 @ X3
% 0.34/0.62 @ Z ) )
% 0.34/0.62 => ( pred_c177362287423414904_nat_o @ A @ P @ ( sup_su5209123915105501825_nat_o @ ( insert5175938949040314269_nat_o @ Z @ bot_bo7824918357723582233_nat_o ) @ C5 ) ) ) ) ) ).
% 0.34/0.62
% 0.34/0.62 % pred_on.chain_extend
% 0.34/0.62 thf(fact_4402_subset__Zorn__nonempty,axiom,
% 0.34/0.62 ! [A11: set_set_nat] :
% 0.34/0.62 ( ( A11 != bot_bot_set_set_nat )
% 0.34/0.62 => ( ! [C7: set_set_nat] :
% 0.34/0.62 ( ( C7 != bot_bot_set_set_nat )
% 0.34/0.62 => ( ( pred_chain_set_nat @ A11 @ ord_less_set_nat @ C7 )
% 0.34/0.62 => ( member_set_nat @ ( comple7399068483239264473et_nat @ C7 ) @ A11 ) ) )
% 0.34/0.62 => ? [X3: set_nat] :
% 0.34/0.62 ( ( member_set_nat @ X3 @ A11 )
% 0.34/0.62 & ! [Xa2: set_nat] :
% 0.34/0.62 ( ( member_set_nat @ Xa2 @ A11 )
% 0.34/0.62 => ( ( ord_less_eq_set_nat @ X3 @ Xa2 )
% 0.34/0.62 => ( Xa2 = X3 ) ) ) ) ) ) ).
% 0.34/0.62
% 0.34/0.62 % subset_Zorn_nonempty
% 0.34/0.62 thf(fact_4403_in__measure,axiom,
% 0.34/0.62 ! [X: nat,Y: nat,F: nat > nat] :
% 0.34/0.62 ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ Y ) @ ( measure_nat @ F ) )
% 0.34/0.62 = ( ord_less_nat @ ( F @ X ) @ ( F @ Y ) ) ) ).
% 0.34/0.62
% 0.34/0.62 % in_measure
% 0.34/0.62 thf(fact_4404_in__measure,axiom,
% 0.34/0.62 ! [X: int,Y: int,F: int > nat] :
% 0.34/0.62 ( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ X @ Y ) @ ( measure_int @ F ) )
% 0.34/0.62 = ( ord_less_nat @ ( F @ X ) @ ( F @ Y ) ) ) ).
% 0.34/0.62
% 0.34/0.62 % in_measure
% 0.34/0.62 thf(fact_4405_top_Oordering__top__axioms,axiom,
% 0.34/0.62 orderi2935705153647960990t_unit @ ord_le3507040750410214029t_unit @ ord_le8056459307392131481t_unit @ top_to1996260823553986621t_unit ).
% 0.34/0.62
% 0.34/0.62 % top.ordering_top_axioms
% 0.34/0.62 thf(fact_4406_top_Oordering__top__axioms,axiom,
% 0.34/0.62 ordering_top_set_o @ ord_less_eq_set_o @ ord_less_set_o @ top_top_set_o ).
% 0.34/0.62
% 0.34/0.62 % top.ordering_top_axioms
% 0.34/0.62 thf(fact_4407_top_Oordering__top__axioms,axiom,
% 0.34/0.62 ordering_top_set_int @ ord_less_eq_set_int @ ord_less_set_int @ top_top_set_int ).
% 0.34/0.62
% 0.34/0.62 % top.ordering_top_axioms
% 0.34/0.62 thf(fact_4408_top_Oordering__top__axioms,axiom,
% 0.34/0.62 ordering_top_assn @ ord_less_eq_assn @ ord_less_assn @ top_top_assn ).
% 0.34/0.62
% 0.34/0.62 % top.ordering_top_axioms
% 0.34/0.62 thf(fact_4409_top_Oordering__top__axioms,axiom,
% 0.34/0.62 orderi2807725203697798511er_nat @ ord_le2510731241096832064er_nat @ ord_less_filter_nat @ top_top_filter_nat ).
% 0.34/0.62
% 0.34/0.62 % top.ordering_top_axioms
% 0.34/0.62 thf(fact_4410_top_Oordering__top__axioms,axiom,
% 0.34/0.62 ordering_top_set_nat @ ord_less_eq_set_nat @ ord_less_set_nat @ top_top_set_nat ).
% 0.34/0.62
% 0.34/0.62 % top.ordering_top_axioms
% 0.34/0.62 thf(fact_4411_Max_Oinsert__remove,axiom,
% 0.34/0.62 ! [A: set_Code_integer,X: code_integer] :
% 0.34/0.62 ( ( finite6017078050557962740nteger @ A )
% 0.34/0.62 => ( ( ( ( minus_2355218937544613996nteger @ A @ ( insert_Code_integer @ X @ bot_bo3990330152332043303nteger ) )
% 0.34/0.62 = bot_bo3990330152332043303nteger )
% 0.34/0.62 => ( ( lattic4901227151466704046nteger @ ( insert_Code_integer @ X @ A ) )
% 0.34/0.62 = X ) )
% 0.34/0.62 & ( ( ( minus_2355218937544613996nteger @ A @ ( insert_Code_integer @ X @ bot_bo3990330152332043303nteger ) )
% 0.34/0.62 != bot_bo3990330152332043303nteger )
% 0.34/0.62 => ( ( lattic4901227151466704046nteger @ ( insert_Code_integer @ X @ A ) )
% 0.34/0.62 = ( ord_max_Code_integer @ X @ ( lattic4901227151466704046nteger @ ( minus_2355218937544613996nteger @ A @ ( insert_Code_integer @ X @ bot_bo3990330152332043303nteger ) ) ) ) ) ) ) ) ).
% 0.34/0.62
% 0.34/0.62 % Max.insert_remove
% 0.34/0.62 thf(fact_4412_Max_Oinsert__remove,axiom,
% 0.34/0.62 ! [A: set_Product_unit,X: product_unit] :
% 0.34/0.62 ( ( finite4290736615968046902t_unit @ A )
% 0.34/0.62 => ( ( ( ( minus_6452836326544984404t_unit @ A @ ( insert_Product_unit @ X @ bot_bo3957492148770167129t_unit ) )
% 0.34/0.62 = bot_bo3957492148770167129t_unit )
% 0.34/0.62 => ( ( lattic1210392890313158908t_unit @ ( insert_Product_unit @ X @ A ) )
% 0.34/0.62 = X ) )
% 0.34/0.62 & ( ( ( minus_6452836326544984404t_unit @ A @ ( insert_Product_unit @ X @ bot_bo3957492148770167129t_unit ) )
% 0.34/0.62 != bot_bo3957492148770167129t_unit )
% 0.34/0.62 => ( ( lattic1210392890313158908t_unit @ ( insert_Product_unit @ X @ A ) )
% 0.34/0.62 = ( ord_max_Product_unit @ X @ ( lattic1210392890313158908t_unit @ ( minus_6452836326544984404t_unit @ A @ ( insert_Product_unit @ X @ bot_bo3957492148770167129t_unit ) ) ) ) ) ) ) ) ).
% 0.34/0.62
% 0.34/0.62 % Max.insert_remove
% 0.34/0.62 thf(fact_4413_Max_Oinsert__remove,axiom,
% 0.34/0.62 ! [A: set_o,X: $o] :
% 0.34/0.62 ( ( finite_finite_o @ A )
% 0.34/0.62 => ( ( lattic1921953407002678535_Max_o @ ( insert_o @ X @ A ) )
% 0.34/0.62 = ( ( ( ( minus_minus_set_o @ A @ ( insert_o @ X @ bot_bot_set_o ) )
% 0.34/0.62 = bot_bot_set_o )
% 0.34/0.62 => X )
% 0.34/0.62 & ( ( ( minus_minus_set_o @ A @ ( insert_o @ X @ bot_bot_set_o ) )
% 0.34/0.62 != bot_bot_set_o )
% 0.34/0.62 => ( ord_max_o @ X @ ( lattic1921953407002678535_Max_o @ ( minus_minus_set_o @ A @ ( insert_o @ X @ bot_bot_set_o ) ) ) ) ) ) ) ) ).
% 0.34/0.62
% 0.34/0.62 % Max.insert_remove
% 0.34/0.62 thf(fact_4414_Max_Oinsert__remove,axiom,
% 0.34/0.62 ! [A: set_int,X: int] :
% 0.34/0.62 ( ( finite_finite_int @ A )
% 0.34/0.62 => ( ( ( ( minus_minus_set_int @ A @ ( insert_int @ X @ bot_bot_set_int ) )
% 0.34/0.62 = bot_bot_set_int )
% 0.34/0.62 => ( ( lattic8263393255366662781ax_int @ ( insert_int @ X @ A ) )
% 0.34/0.62 = X ) )
% 0.34/0.62 & ( ( ( minus_minus_set_int @ A @ ( insert_int @ X @ bot_bot_set_int ) )
% 0.34/0.62 != bot_bot_set_int )
% 0.34/0.62 => ( ( lattic8263393255366662781ax_int @ ( insert_int @ X @ A ) )
% 0.34/0.62 = ( ord_max_int @ X @ ( lattic8263393255366662781ax_int @ ( minus_minus_set_int @ A @ ( insert_int @ X @ bot_bot_set_int ) ) ) ) ) ) ) ) ).
% 0.34/0.62
% 0.34/0.62 % Max.insert_remove
% 0.34/0.62 thf(fact_4415_Max_Oinsert__remove,axiom,
% 0.34/0.62 ! [A: set_nat,X: nat] :
% 0.34/0.62 ( ( finite_finite_nat @ A )
% 0.34/0.62 => ( ( ( ( minus_minus_set_nat @ A @ ( insert_nat @ X @ bot_bot_set_nat ) )
% 0.34/0.62 = bot_bot_set_nat )
% 0.34/0.62 => ( ( lattic8265883725875713057ax_nat @ ( insert_nat @ X @ A ) )
% 0.34/0.62 = X ) )
% 0.34/0.62 & ( ( ( minus_minus_set_nat @ A @ ( insert_nat @ X @ bot_bot_set_nat ) )
% 0.34/0.62 != bot_bot_set_nat )
% 0.34/0.62 => ( ( lattic8265883725875713057ax_nat @ ( insert_nat @ X @ A ) )
% 0.34/0.62 = ( ord_max_nat @ X @ ( lattic8265883725875713057ax_nat @ ( minus_minus_set_nat @ A @ ( insert_nat @ X @ bot_bot_set_nat ) ) ) ) ) ) ) ) ).
% 0.34/0.62
% 0.34/0.62 % Max.insert_remove
% 0.34/0.62 thf(fact_4416_Max_Oremove,axiom,
% 0.34/0.62 ! [A: set_Code_integer,X: code_integer] :
% 0.34/0.62 ( ( finite6017078050557962740nteger @ A )
% 0.34/0.62 => ( ( member_Code_integer @ X @ A )
% 0.34/0.62 => ( ( ( ( minus_2355218937544613996nteger @ A @ ( insert_Code_integer @ X @ bot_bo3990330152332043303nteger ) )
% 0.34/0.62 = bot_bo3990330152332043303nteger )
% 0.34/0.62 => ( ( lattic4901227151466704046nteger @ A )
% 0.34/0.62 = X ) )
% 0.34/0.62 & ( ( ( minus_2355218937544613996nteger @ A @ ( insert_Code_integer @ X @ bot_bo3990330152332043303nteger ) )
% 0.34/0.62 != bot_bo3990330152332043303nteger )
% 0.34/0.62 => ( ( lattic4901227151466704046nteger @ A )
% 0.34/0.62 = ( ord_max_Code_integer @ X @ ( lattic4901227151466704046nteger @ ( minus_2355218937544613996nteger @ A @ ( insert_Code_integer @ X @ bot_bo3990330152332043303nteger ) ) ) ) ) ) ) ) ) ).
% 0.34/0.62
% 0.34/0.62 % Max.remove
% 0.34/0.62 thf(fact_4417_Max_Oremove,axiom,
% 0.34/0.62 ! [A: set_Product_unit,X: product_unit] :
% 0.34/0.62 ( ( finite4290736615968046902t_unit @ A )
% 0.34/0.62 => ( ( member_Product_unit @ X @ A )
% 0.34/0.62 => ( ( ( ( minus_6452836326544984404t_unit @ A @ ( insert_Product_unit @ X @ bot_bo3957492148770167129t_unit ) )
% 0.34/0.62 = bot_bo3957492148770167129t_unit )
% 0.34/0.62 => ( ( lattic1210392890313158908t_unit @ A )
% 0.34/0.62 = X ) )
% 0.34/0.62 & ( ( ( minus_6452836326544984404t_unit @ A @ ( insert_Product_unit @ X @ bot_bo3957492148770167129t_unit ) )
% 0.34/0.62 != bot_bo3957492148770167129t_unit )
% 0.34/0.62 => ( ( lattic1210392890313158908t_unit @ A )
% 0.34/0.62 = ( ord_max_Product_unit @ X @ ( lattic1210392890313158908t_unit @ ( minus_6452836326544984404t_unit @ A @ ( insert_Product_unit @ X @ bot_bo3957492148770167129t_unit ) ) ) ) ) ) ) ) ) ).
% 0.34/0.62
% 0.34/0.62 % Max.remove
% 0.34/0.62 thf(fact_4418_Max_Oremove,axiom,
% 0.34/0.62 ! [A: set_o,X: $o] :
% 0.34/0.62 ( ( finite_finite_o @ A )
% 0.34/0.62 => ( ( member_o @ X @ A )
% 0.34/0.62 => ( ( lattic1921953407002678535_Max_o @ A )
% 0.34/0.62 = ( ( ( ( minus_minus_set_o @ A @ ( insert_o @ X @ bot_bot_set_o ) )
% 0.34/0.62 = bot_bot_set_o )
% 0.34/0.62 => X )
% 0.34/0.62 & ( ( ( minus_minus_set_o @ A @ ( insert_o @ X @ bot_bot_set_o ) )
% 0.34/0.62 != bot_bot_set_o )
% 0.34/0.62 => ( ord_max_o @ X @ ( lattic1921953407002678535_Max_o @ ( minus_minus_set_o @ A @ ( insert_o @ X @ bot_bot_set_o ) ) ) ) ) ) ) ) ) ).
% 0.34/0.62
% 0.34/0.62 % Max.remove
% 0.34/0.62 thf(fact_4419_Max_Oremove,axiom,
% 0.34/0.62 ! [A: set_int,X: int] :
% 0.34/0.62 ( ( finite_finite_int @ A )
% 0.34/0.62 => ( ( member_int @ X @ A )
% 0.34/0.62 => ( ( ( ( minus_minus_set_int @ A @ ( insert_int @ X @ bot_bot_set_int ) )
% 0.34/0.62 = bot_bot_set_int )
% 0.34/0.62 => ( ( lattic8263393255366662781ax_int @ A )
% 0.34/0.62 = X ) )
% 0.34/0.62 & ( ( ( minus_minus_set_int @ A @ ( insert_int @ X @ bot_bot_set_int ) )
% 0.34/0.62 != bot_bot_set_int )
% 0.34/0.62 => ( ( lattic8263393255366662781ax_int @ A )
% 0.34/0.62 = ( ord_max_int @ X @ ( lattic8263393255366662781ax_int @ ( minus_minus_set_int @ A @ ( insert_int @ X @ bot_bot_set_int ) ) ) ) ) ) ) ) ) ).
% 0.34/0.62
% 0.34/0.62 % Max.remove
% 0.34/0.62 thf(fact_4420_Max_Oremove,axiom,
% 0.34/0.62 ! [A: set_nat,X: nat] :
% 0.34/0.62 ( ( finite_finite_nat @ A )
% 0.34/0.62 => ( ( member_nat @ X @ A )
% 0.34/0.62 => ( ( ( ( minus_minus_set_nat @ A @ ( insert_nat @ X @ bot_bot_set_nat ) )
% 0.34/0.62 = bot_bot_set_nat )
% 0.34/0.62 => ( ( lattic8265883725875713057ax_nat @ A )
% 0.34/0.62 = X ) )
% 0.34/0.62 & ( ( ( minus_minus_set_nat @ A @ ( insert_nat @ X @ bot_bot_set_nat ) )
% 0.34/0.62 != bot_bot_set_nat )
% 0.34/0.62 => ( ( lattic8265883725875713057ax_nat @ A )
% 0.34/0.62 = ( ord_max_nat @ X @ ( lattic8265883725875713057ax_nat @ ( minus_minus_set_nat @ A @ ( insert_nat @ X @ bot_bot_set_nat ) ) ) ) ) ) ) ) ) ).
% 0.34/0.62
% 0.34/0.62 % Max.remove
% 0.34/0.62 thf(fact_4421_inj__on__Un,axiom,
% 0.34/0.62 ! [F: code_integer > code_integer,A: set_Code_integer,B4: set_Code_integer] :
% 0.34/0.62 ( ( inj_on1158064722259180589nteger @ F @ ( sup_su848401254843788991nteger @ A @ B4 ) )
% 0.34/0.62 = ( ( inj_on1158064722259180589nteger @ F @ A )
% 0.34/0.62 & ( inj_on1158064722259180589nteger @ F @ B4 )
% 0.34/0.62 & ( ( inf_in1364745209274528805nteger @ ( image_4470545334726330049nteger @ F @ ( minus_2355218937544613996nteger @ A @ B4 ) ) @ ( image_4470545334726330049nteger @ F @ ( minus_2355218937544613996nteger @ B4 @ A ) ) )
% 0.34/0.62 = bot_bo3990330152332043303nteger ) ) ) ).
% 0.34/0.62
% 0.34/0.62 % inj_on_Un
% 0.34/0.62 thf(fact_4422_inj__on__Un,axiom,
% 0.34/0.62 ! [F: nat > set_nat,A: set_nat,B4: set_nat] :
% 0.34/0.62 ( ( inj_on_nat_set_nat @ F @ ( sup_sup_set_nat @ A @ B4 ) )
% 0.34/0.62 = ( ( inj_on_nat_set_nat @ F @ A )
% 0.34/0.62 & ( inj_on_nat_set_nat @ F @ B4 )
% 0.34/0.62 & ( ( inf_inf_set_set_nat @ ( image_nat_set_nat @ F @ ( minus_minus_set_nat @ A @ B4 ) ) @ ( image_nat_set_nat @ F @ ( minus_minus_set_nat @ B4 @ A ) ) )
% 0.34/0.62 = bot_bot_set_set_nat ) ) ) ).
% 0.34/0.62
% 0.34/0.62 % inj_on_Un
% 0.34/0.62 thf(fact_4423_inj__on__Un,axiom,
% 0.34/0.62 ! [F: nat > code_integer,A: set_nat,B4: set_nat] :
% 0.34/0.62 ( ( inj_on2245834603434889120nteger @ F @ ( sup_sup_set_nat @ A @ B4 ) )
% 0.34/0.62 = ( ( inj_on2245834603434889120nteger @ F @ A )
% 0.34/0.62 & ( inj_on2245834603434889120nteger @ F @ B4 )
% 0.34/0.62 & ( ( inf_in1364745209274528805nteger @ ( image_1215581382706833972nteger @ F @ ( minus_minus_set_nat @ A @ B4 ) ) @ ( image_1215581382706833972nteger @ F @ ( minus_minus_set_nat @ B4 @ A ) ) )
% 0.34/0.62 = bot_bo3990330152332043303nteger ) ) ) ).
% 0.34/0.62
% 0.34/0.62 % inj_on_Un
% 0.34/0.62 thf(fact_4424_inj__on__Un,axiom,
% 0.34/0.62 ! [F: nat > product_unit,A: set_nat,B4: set_nat] :
% 0.34/0.62 ( ( inj_on7061601236592826506t_unit @ F @ ( sup_sup_set_nat @ A @ B4 ) )
% 0.34/0.62 = ( ( inj_on7061601236592826506t_unit @ F @ A )
% 0.34/0.62 & ( inj_on7061601236592826506t_unit @ F @ B4 )
% 0.34/0.62 & ( ( inf_in4660618365625256667t_unit @ ( image_8730104196221521654t_unit @ F @ ( minus_minus_set_nat @ A @ B4 ) ) @ ( image_8730104196221521654t_unit @ F @ ( minus_minus_set_nat @ B4 @ A ) ) )
% 0.34/0.62 = bot_bo3957492148770167129t_unit ) ) ) ).
% 0.34/0.62
% 0.34/0.62 % inj_on_Un
% 0.34/0.62 thf(fact_4425_inj__on__Un,axiom,
% 0.34/0.62 ! [F: nat > $o,A: set_nat,B4: set_nat] :
% 0.34/0.62 ( ( inj_on_nat_o @ F @ ( sup_sup_set_nat @ A @ B4 ) )
% 0.34/0.62 = ( ( inj_on_nat_o @ F @ A )
% 0.34/0.62 & ( inj_on_nat_o @ F @ B4 )
% 0.34/0.62 & ( ( inf_inf_set_o @ ( image_nat_o2 @ F @ ( minus_minus_set_nat @ A @ B4 ) ) @ ( image_nat_o2 @ F @ ( minus_minus_set_nat @ B4 @ A ) ) )
% 0.34/0.62 = bot_bot_set_o ) ) ) ).
% 0.34/0.62
% 0.34/0.62 % inj_on_Un
% 0.34/0.62 thf(fact_4426_inj__on__Un,axiom,
% 0.34/0.62 ! [F: nat > nat,A: set_nat,B4: set_nat] :
% 0.34/0.62 ( ( inj_on_nat_nat @ F @ ( sup_sup_set_nat @ A @ B4 ) )
% 0.34/0.62 = ( ( inj_on_nat_nat @ F @ A )
% 0.34/0.62 & ( inj_on_nat_nat @ F @ B4 )
% 0.34/0.62 & ( ( inf_inf_set_nat @ ( image_nat_nat2 @ F @ ( minus_minus_set_nat @ A @ B4 ) ) @ ( image_nat_nat2 @ F @ ( minus_minus_set_nat @ B4 @ A ) ) )
% 0.34/0.62 = bot_bot_set_nat ) ) ) ).
% 0.34/0.62
% 0.34/0.62 % inj_on_Un
% 0.34/0.62 thf(fact_4427_inj__on__Un,axiom,
% 0.34/0.62 ! [F: nat > int,A: set_nat,B4: set_nat] :
% 0.34/0.62 ( ( inj_on_nat_int @ F @ ( sup_sup_set_nat @ A @ B4 ) )
% 0.34/0.62 = ( ( inj_on_nat_int @ F @ A )
% 0.34/0.62 & ( inj_on_nat_int @ F @ B4 )
% 0.34/0.62 & ( ( inf_inf_set_int @ ( image_nat_int2 @ F @ ( minus_minus_set_nat @ A @ B4 ) ) @ ( image_nat_int2 @ F @ ( minus_minus_set_nat @ B4 @ A ) ) )
% 0.34/0.62 = bot_bot_set_int ) ) ) ).
% 0.34/0.62
% 0.34/0.62 % inj_on_Un
% 0.34/0.62 thf(fact_4428_inj__on__Un,axiom,
% 0.34/0.62 ! [F: nat > produc3658429121746597890et_nat > $o,A: set_nat,B4: set_nat] :
% 0.34/0.62 ( ( inj_on7779444506608719434_nat_o @ F @ ( sup_sup_set_nat @ A @ B4 ) )
% 0.34/0.62 = ( ( inj_on7779444506608719434_nat_o @ F @ A )
% 0.34/0.62 & ( inj_on7779444506608719434_nat_o @ F @ B4 )
% 0.34/0.62 & ( ( inf_in1906310914598751387_nat_o @ ( image_4682239675366380726_nat_o @ F @ ( minus_minus_set_nat @ A @ B4 ) ) @ ( image_4682239675366380726_nat_o @ F @ ( minus_minus_set_nat @ B4 @ A ) ) )
% 0.34/0.62 = bot_bo7824918357723582233_nat_o ) ) ) ).
% 0.34/0.62
% 0.34/0.62 % inj_on_Un
% 0.34/0.62 thf(fact_4429_Max__insert,axiom,
% 0.34/0.62 ! [A: set_Code_integer,X: code_integer] :
% 0.34/0.62 ( ( finite6017078050557962740nteger @ A )
% 0.34/0.62 => ( ( A != bot_bo3990330152332043303nteger )
% 0.34/0.62 => ( ( lattic4901227151466704046nteger @ ( insert_Code_integer @ X @ A ) )
% 0.34/0.62 = ( ord_max_Code_integer @ X @ ( lattic4901227151466704046nteger @ A ) ) ) ) ) ).
% 0.34/0.62
% 0.34/0.62 % Max_insert
% 0.34/0.62 thf(fact_4430_Max__insert,axiom,
% 0.34/0.62 ! [A: set_Product_unit,X: product_unit] :
% 0.34/0.62 ( ( finite4290736615968046902t_unit @ A )
% 0.34/0.62 => ( ( A != bot_bo3957492148770167129t_unit )
% 0.34/0.62 => ( ( lattic1210392890313158908t_unit @ ( insert_Product_unit @ X @ A ) )
% 0.34/0.62 = ( ord_max_Product_unit @ X @ ( lattic1210392890313158908t_unit @ A ) ) ) ) ) ).
% 0.34/0.62
% 0.34/0.62 % Max_insert
% 0.34/0.62 thf(fact_4431_Max__insert,axiom,
% 0.34/0.62 ! [A: set_o,X: $o] :
% 0.34/0.62 ( ( finite_finite_o @ A )
% 0.34/0.62 => ( ( A != bot_bot_set_o )
% 0.34/0.62 => ( ( lattic1921953407002678535_Max_o @ ( insert_o @ X @ A ) )
% 0.34/0.62 = ( ord_max_o @ X @ ( lattic1921953407002678535_Max_o @ A ) ) ) ) ) ).
% 0.34/0.62
% 0.34/0.62 % Max_insert
% 0.34/0.62 thf(fact_4432_Max__insert,axiom,
% 0.34/0.62 ! [A: set_int,X: int] :
% 0.34/0.62 ( ( finite_finite_int @ A )
% 0.34/0.62 => ( ( A != bot_bot_set_int )
% 0.34/0.62 => ( ( lattic8263393255366662781ax_int @ ( insert_int @ X @ A ) )
% 0.34/0.62 = ( ord_max_int @ X @ ( lattic8263393255366662781ax_int @ A ) ) ) ) ) ).
% 0.34/0.62
% 0.34/0.62 % Max_insert
% 0.34/0.62 thf(fact_4433_Max__insert,axiom,
% 0.34/0.62 ! [A: set_nat,X: nat] :
% 0.34/0.62 ( ( finite_finite_nat @ A )
% 0.34/0.62 => ( ( A != bot_bot_set_nat )
% 0.34/0.62 => ( ( lattic8265883725875713057ax_nat @ ( insert_nat @ X @ A ) )
% 0.34/0.62 = ( ord_max_nat @ X @ ( lattic8265883725875713057ax_nat @ A ) ) ) ) ) ).
% 0.34/0.62
% 0.34/0.62 % Max_insert
% 0.34/0.62 thf(fact_4434_map__prod__surj,axiom,
% 0.34/0.62 ! [F: code_integer > code_integer,G: code_integer > code_integer] :
% 0.34/0.62 ( ( ( image_4470545334726330049nteger @ F @ top_to4645266643341252675nteger )
% 0.34/0.62 = top_to4645266643341252675nteger )
% 0.34/0.62 => ( ( ( image_4470545334726330049nteger @ G @ top_to4645266643341252675nteger )
% 0.34/0.62 = top_to4645266643341252675nteger )
% 0.34/0.62 => ( ( image_7082141833288935621nteger @ ( produc1802105962441033582nteger @ F @ G ) @ top_to2363841527343591755nteger )
% 0.34/0.62 = top_to2363841527343591755nteger ) ) ) ).
% 0.34/0.62
% 0.34/0.62 % map_prod_surj
% 0.34/0.62 thf(fact_4435_map__prod__surj,axiom,
% 0.34/0.62 ! [F: code_integer > code_integer,G: product_unit > product_unit] :
% 0.34/0.62 ( ( ( image_4470545334726330049nteger @ F @ top_to4645266643341252675nteger )
% 0.34/0.62 = top_to4645266643341252675nteger )
% 0.34/0.62 => ( ( ( image_405062704495631173t_unit @ G @ top_to1996260823553986621t_unit )
% 0.34/0.62 = top_to1996260823553986621t_unit )
% 0.34/0.62 => ( ( image_3456600171786721233t_unit @ ( produc3346700480963112818t_unit @ F @ G ) @ top_to3244471725241573579t_unit )
% 0.34/0.62 = top_to3244471725241573579t_unit ) ) ) ).
% 0.34/0.62
% 0.34/0.62 % map_prod_surj
% 0.34/0.62 thf(fact_4436_map__prod__surj,axiom,
% 0.34/0.62 ! [F: code_integer > code_integer,G: product_unit > $o] :
% 0.34/0.62 ( ( ( image_4470545334726330049nteger @ F @ top_to4645266643341252675nteger )
% 0.34/0.62 = top_to4645266643341252675nteger )
% 0.34/0.62 => ( ( ( image_Product_unit_o2 @ G @ top_to1996260823553986621t_unit )
% 0.34/0.62 = top_top_set_o )
% 0.34/0.62 => ( ( image_7060974388358999388eger_o @ ( produc8116347370398923901unit_o @ F @ G ) @ top_to3244471725241573579t_unit )
% 0.34/0.62 = top_to2658285827425157974eger_o ) ) ) ).
% 0.34/0.62
% 0.34/0.62 % map_prod_surj
% 0.34/0.62 thf(fact_4437_map__prod__surj,axiom,
% 0.34/0.62 ! [F: code_integer > code_integer,G: product_unit > nat] :
% 0.34/0.62 ( ( ( image_4470545334726330049nteger @ F @ top_to4645266643341252675nteger )
% 0.34/0.62 = top_to4645266643341252675nteger )
% 0.34/0.62 => ( ( ( image_875570014554754200it_nat @ G @ top_to1996260823553986621t_unit )
% 0.34/0.62 = top_top_set_nat )
% 0.34/0.62 => ( ( image_8236373689398998434er_nat @ ( produc4022400886476617259it_nat @ F @ G ) @ top_to3244471725241573579t_unit )
% 0.34/0.62 = top_to8760411888770933182er_nat ) ) ) ).
% 0.34/0.62
% 0.34/0.62 % map_prod_surj
% 0.34/0.62 thf(fact_4438_map__prod__surj,axiom,
% 0.34/0.62 ! [F: code_integer > code_integer,G: product_unit > int] :
% 0.34/0.62 ( ( ( image_4470545334726330049nteger @ F @ top_to4645266643341252675nteger )
% 0.34/0.62 = top_to4645266643341252675nteger )
% 0.34/0.62 => ( ( ( image_873079544045703924it_int @ G @ top_to1996260823553986621t_unit )
% 0.34/0.62 = top_top_set_int )
% 0.34/0.62 => ( ( image_4058522669889801726er_int @ ( produc4019910415967566983it_int @ F @ G ) @ top_to3244471725241573579t_unit )
% 0.34/0.62 = top_to6270328743839206426er_int ) ) ) ).
% 0.34/0.62
% 0.34/0.62 % map_prod_surj
% 0.34/0.62 thf(fact_4439_map__prod__surj,axiom,
% 0.34/0.62 ! [F: code_integer > code_integer,G: $o > product_unit] :
% 0.34/0.62 ( ( ( image_4470545334726330049nteger @ F @ top_to4645266643341252675nteger )
% 0.34/0.62 = top_to4645266643341252675nteger )
% 0.34/0.62 => ( ( ( image_o_Product_unit2 @ G @ top_top_set_o )
% 0.34/0.62 = top_to1996260823553986621t_unit )
% 0.34/0.62 => ( ( image_2425804817718344284t_unit @ ( produc3530376873132787239t_unit @ F @ G ) @ top_to2658285827425157974eger_o )
% 0.34/0.62 = top_to3244471725241573579t_unit ) ) ) ).
% 0.34/0.62
% 0.34/0.62 % map_prod_surj
% 0.34/0.62 thf(fact_4440_map__prod__surj,axiom,
% 0.34/0.62 ! [F: code_integer > code_integer,G: $o > $o] :
% 0.34/0.62 ( ( ( image_4470545334726330049nteger @ F @ top_to4645266643341252675nteger )
% 0.34/0.62 = top_to4645266643341252675nteger )
% 0.34/0.62 => ( ( ( image_o_o2 @ G @ top_top_set_o )
% 0.34/0.62 = top_top_set_o )
% 0.34/0.62 => ( ( image_227735840630799463eger_o @ ( produc1084426483498010290er_o_o @ F @ G ) @ top_to2658285827425157974eger_o )
% 0.34/0.62 = top_to2658285827425157974eger_o ) ) ) ).
% 0.34/0.62
% 0.34/0.62 % map_prod_surj
% 0.34/0.62 thf(fact_4441_map__prod__surj,axiom,
% 0.34/0.62 ! [F: code_integer > code_integer,G: $o > nat] :
% 0.34/0.62 ( ( ( image_4470545334726330049nteger @ F @ top_to4645266643341252675nteger )
% 0.34/0.62 = top_to4645266643341252675nteger )
% 0.34/0.62 => ( ( ( image_o_nat2 @ G @ top_top_set_o )
% 0.34/0.62 = top_top_set_nat )
% 0.34/0.62 => ( ( image_1252617143230709591er_nat @ ( produc7550393335539917622_o_nat @ F @ G ) @ top_to2658285827425157974eger_o )
% 0.34/0.62 = top_to8760411888770933182er_nat ) ) ) ).
% 0.34/0.62
% 0.34/0.62 % map_prod_surj
% 0.34/0.62 thf(fact_4442_map__prod__surj,axiom,
% 0.34/0.62 ! [F: code_integer > code_integer,G: $o > int] :
% 0.34/0.62 ( ( ( image_4470545334726330049nteger @ F @ top_to4645266643341252675nteger )
% 0.34/0.62 = top_to4645266643341252675nteger )
% 0.34/0.62 => ( ( ( image_o_int2 @ G @ top_top_set_o )
% 0.34/0.62 = top_top_set_int )
% 0.34/0.62 => ( ( image_6298138160576288691er_int @ ( produc7547902865030867346_o_int @ F @ G ) @ top_to2658285827425157974eger_o )
% 0.34/0.62 = top_to6270328743839206426er_int ) ) ) ).
% 0.34/0.62
% 0.34/0.62 % map_prod_surj
% 0.34/0.62 thf(fact_4443_map__prod__surj,axiom,
% 0.34/0.62 ! [F: code_integer > code_integer,G: nat > code_integer] :
% 0.34/0.62 ( ( ( image_4470545334726330049nteger @ F @ top_to4645266643341252675nteger )
% 0.34/0.62 = top_to4645266643341252675nteger )
% 0.34/0.62 => ( ( ( image_1215581382706833972nteger @ G @ top_top_set_nat )
% 0.34/0.62 = top_to4645266643341252675nteger )
% 0.34/0.62 => ( ( image_4921214744877245202nteger @ ( produc1622059938646302433nteger @ F @ G ) @ top_to8760411888770933182er_nat )
% 0.34/0.62 = top_to2363841527343591755nteger ) ) ) ).
% 0.34/0.62
% 0.34/0.62 % map_prod_surj
% 0.34/0.62 thf(fact_4444_inj__on__insert,axiom,
% 0.34/0.62 ! [F: code_integer > code_integer,A2: code_integer,A: set_Code_integer] :
% 0.34/0.62 ( ( inj_on1158064722259180589nteger @ F @ ( insert_Code_integer @ A2 @ A ) )
% 0.34/0.62 = ( ( inj_on1158064722259180589nteger @ F @ A )
% 0.34/0.62 & ~ ( member_Code_integer @ ( F @ A2 ) @ ( image_4470545334726330049nteger @ F @ ( minus_2355218937544613996nteger @ A @ ( insert_Code_integer @ A2 @ bot_bo3990330152332043303nteger ) ) ) ) ) ) ).
% 0.34/0.62
% 0.34/0.62 % inj_on_insert
% 0.34/0.62 thf(fact_4445_inj__on__insert,axiom,
% 0.34/0.62 ! [F: product_unit > nat,A2: product_unit,A: set_Product_unit] :
% 0.34/0.62 ( ( inj_on8430439091780834860it_nat @ F @ ( insert_Product_unit @ A2 @ A ) )
% 0.34/0.62 = ( ( inj_on8430439091780834860it_nat @ F @ A )
% 0.34/0.62 & ~ ( member_nat @ ( F @ A2 ) @ ( image_875570014554754200it_nat @ F @ ( minus_6452836326544984404t_unit @ A @ ( insert_Product_unit @ A2 @ bot_bo3957492148770167129t_unit ) ) ) ) ) ) ).
% 0.34/0.62
% 0.34/0.62 % inj_on_insert
% 0.34/0.62 thf(fact_4446_inj__on__insert,axiom,
% 0.34/0.62 ! [F: product_unit > int,A2: product_unit,A: set_Product_unit] :
% 0.34/0.62 ( ( inj_on8427948621271784584it_int @ F @ ( insert_Product_unit @ A2 @ A ) )
% 0.34/0.62 = ( ( inj_on8427948621271784584it_int @ F @ A )
% 0.34/0.62 & ~ ( member_int @ ( F @ A2 ) @ ( image_873079544045703924it_int @ F @ ( minus_6452836326544984404t_unit @ A @ ( insert_Product_unit @ A2 @ bot_bo3957492148770167129t_unit ) ) ) ) ) ) ).
% 0.34/0.62
% 0.34/0.62 % inj_on_insert
% 0.34/0.62 thf(fact_4447_inj__on__insert,axiom,
% 0.34/0.62 ! [F: product_unit > $o,A2: product_unit,A: set_Product_unit] :
% 0.34/0.62 ( ( inj_on8854722734121536060unit_o @ F @ ( insert_Product_unit @ A2 @ A ) )
% 0.34/0.62 = ( ( inj_on8854722734121536060unit_o @ F @ A )
% 0.34/0.62 & ~ ( member_o @ ( F @ A2 ) @ ( image_Product_unit_o2 @ F @ ( minus_6452836326544984404t_unit @ A @ ( insert_Product_unit @ A2 @ bot_bo3957492148770167129t_unit ) ) ) ) ) ) ).
% 0.34/0.62
% 0.34/0.62 % inj_on_insert
% 0.34/0.62 thf(fact_4448_inj__on__insert,axiom,
% 0.34/0.62 ! [F: product_unit > product_unit,A2: product_unit,A: set_Product_unit] :
% 0.34/0.62 ( ( inj_on8151373323710067377t_unit @ F @ ( insert_Product_unit @ A2 @ A ) )
% 0.34/0.62 = ( ( inj_on8151373323710067377t_unit @ F @ A )
% 0.34/0.62 & ~ ( member_Product_unit @ ( F @ A2 ) @ ( image_405062704495631173t_unit @ F @ ( minus_6452836326544984404t_unit @ A @ ( insert_Product_unit @ A2 @ bot_bo3957492148770167129t_unit ) ) ) ) ) ) ).
% 0.34/0.62
% 0.34/0.62 % inj_on_insert
% 0.34/0.62 thf(fact_4449_inj__on__insert,axiom,
% 0.34/0.62 ! [F: $o > nat,A2: $o,A: set_o] :
% 0.34/0.62 ( ( inj_on_o_nat @ F @ ( insert_o @ A2 @ A ) )
% 0.34/0.62 = ( ( inj_on_o_nat @ F @ A )
% 0.34/0.62 & ~ ( member_nat @ ( F @ A2 ) @ ( image_o_nat2 @ F @ ( minus_minus_set_o @ A @ ( insert_o @ A2 @ bot_bot_set_o ) ) ) ) ) ) ).
% 0.34/0.62
% 0.34/0.62 % inj_on_insert
% 0.34/0.62 thf(fact_4450_inj__on__insert,axiom,
% 0.34/0.62 ! [F: $o > int,A2: $o,A: set_o] :
% 0.34/0.62 ( ( inj_on_o_int @ F @ ( insert_o @ A2 @ A ) )
% 0.34/0.62 = ( ( inj_on_o_int @ F @ A )
% 0.34/0.62 & ~ ( member_int @ ( F @ A2 ) @ ( image_o_int2 @ F @ ( minus_minus_set_o @ A @ ( insert_o @ A2 @ bot_bot_set_o ) ) ) ) ) ) ).
% 0.34/0.62
% 0.34/0.62 % inj_on_insert
% 0.34/0.62 thf(fact_4451_inj__on__insert,axiom,
% 0.34/0.62 ! [F: $o > $o,A2: $o,A: set_o] :
% 0.34/0.62 ( ( inj_on_o_o @ F @ ( insert_o @ A2 @ A ) )
% 0.34/0.62 = ( ( inj_on_o_o @ F @ A )
% 0.34/0.62 & ~ ( member_o @ ( F @ A2 ) @ ( image_o_o2 @ F @ ( minus_minus_set_o @ A @ ( insert_o @ A2 @ bot_bot_set_o ) ) ) ) ) ) ).
% 0.34/0.62
% 0.34/0.62 % inj_on_insert
% 0.34/0.62 thf(fact_4452_inj__on__insert,axiom,
% 0.34/0.62 ! [F: $o > product_unit,A2: $o,A: set_o] :
% 0.34/0.62 ( ( inj_on4268752236855399398t_unit @ F @ ( insert_o @ A2 @ A ) )
% 0.34/0.62 = ( ( inj_on4268752236855399398t_unit @ F @ A )
% 0.34/0.62 & ~ ( member_Product_unit @ ( F @ A2 ) @ ( image_o_Product_unit2 @ F @ ( minus_minus_set_o @ A @ ( insert_o @ A2 @ bot_bot_set_o ) ) ) ) ) ) ).
% 0.34/0.62
% 0.34/0.62 % inj_on_insert
% 0.34/0.62 thf(fact_4453_inj__on__insert,axiom,
% 0.34/0.62 ! [F: int > nat,A2: int,A: set_int] :
% 0.34/0.62 ( ( inj_on_int_nat @ F @ ( insert_int @ A2 @ A ) )
% 0.34/0.62 = ( ( inj_on_int_nat @ F @ A )
% 0.34/0.62 & ~ ( member_nat @ ( F @ A2 ) @ ( image_int_nat2 @ F @ ( minus_minus_set_int @ A @ ( insert_int @ A2 @ bot_bot_set_int ) ) ) ) ) ) ).
% 0.34/0.62
% 0.34/0.62 % inj_on_insert
% 0.34/0.62 thf(fact_4454_max__bot,axiom,
% 0.34/0.62 ! [X: set_Product_unit] :
% 0.34/0.62 ( ( ord_ma508538259134630082t_unit @ bot_bo3957492148770167129t_unit @ X )
% 0.34/0.62 = X ) ).
% 0.34/0.62
% 0.34/0.62 % max_bot
% 0.34/0.62 thf(fact_4455_max__bot,axiom,
% 0.34/0.62 ! [X: set_o] :
% 0.34/0.62 ( ( ord_max_set_o @ bot_bot_set_o @ X )
% 0.34/0.62 = X ) ).
% 0.34/0.62
% 0.34/0.62 % max_bot
% 0.34/0.62 thf(fact_4456_max__bot,axiom,
% 0.34/0.62 ! [X: set_nat] :
% 0.34/0.62 ( ( ord_max_set_nat @ bot_bot_set_nat @ X )
% 0.34/0.62 = X ) ).
% 0.34/0.62
% 0.34/0.62 % max_bot
% 0.34/0.62 thf(fact_4457_max__bot,axiom,
% 0.34/0.62 ! [X: set_int] :
% 0.34/0.62 ( ( ord_max_set_int @ bot_bot_set_int @ X )
% 0.34/0.62 = X ) ).
% 0.34/0.62
% 0.34/0.62 % max_bot
% 0.34/0.62 thf(fact_4458_max__bot,axiom,
% 0.34/0.62 ! [X: set_Pr4532377907799695533_nat_o] :
% 0.34/0.62 ( ( ord_ma8206644353667601538_nat_o @ bot_bo7824918357723582233_nat_o @ X )
% 0.34/0.62 = X ) ).
% 0.34/0.62
% 0.34/0.62 % max_bot
% 0.34/0.62 thf(fact_4459_max__bot,axiom,
% 0.34/0.62 ! [X: nat] :
% 0.34/0.62 ( ( ord_max_nat @ bot_bot_nat @ X )
% 0.34/0.62 = X ) ).
% 0.34/0.62
% 0.34/0.62 % max_bot
% 0.34/0.62 thf(fact_4460_max__bot2,axiom,
% 0.34/0.62 ! [X: set_Product_unit] :
% 0.34/0.62 ( ( ord_ma508538259134630082t_unit @ X @ bot_bo3957492148770167129t_unit )
% 0.34/0.62 = X ) ).
% 0.34/0.62
% 0.34/0.62 % max_bot2
% 0.34/0.62 thf(fact_4461_max__bot2,axiom,
% 0.34/0.62 ! [X: set_o] :
% 0.34/0.62 ( ( ord_max_set_o @ X @ bot_bot_set_o )
% 0.34/0.62 = X ) ).
% 0.34/0.62
% 0.34/0.62 % max_bot2
% 0.34/0.62 thf(fact_4462_max__bot2,axiom,
% 0.34/0.62 ! [X: set_nat] :
% 0.34/0.62 ( ( ord_max_set_nat @ X @ bot_bot_set_nat )
% 0.34/0.62 = X ) ).
% 0.34/0.62
% 0.34/0.62 % max_bot2
% 0.34/0.62 thf(fact_4463_max__bot2,axiom,
% 0.34/0.62 ! [X: set_int] :
% 0.34/0.62 ( ( ord_max_set_int @ X @ bot_bot_set_int )
% 0.34/0.62 = X ) ).
% 0.34/0.62
% 0.34/0.62 % max_bot2
% 0.34/0.62 thf(fact_4464_max__bot2,axiom,
% 0.34/0.62 ! [X: set_Pr4532377907799695533_nat_o] :
% 0.34/0.62 ( ( ord_ma8206644353667601538_nat_o @ X @ bot_bo7824918357723582233_nat_o )
% 0.34/0.62 = X ) ).
% 0.34/0.62
% 0.34/0.62 % max_bot2
% 0.34/0.62 thf(fact_4465_max__bot2,axiom,
% 0.34/0.62 ! [X: nat] :
% 0.34/0.62 ( ( ord_max_nat @ X @ bot_bot_nat )
% 0.34/0.62 = X ) ).
% 0.34/0.62
% 0.34/0.62 % max_bot2
% 0.34/0.62 thf(fact_4466_max__top,axiom,
% 0.34/0.62 ! [X: assn] :
% 0.34/0.62 ( ( ord_max_assn @ top_top_assn @ X )
% 0.34/0.62 = top_top_assn ) ).
% 0.34/0.62
% 0.34/0.62 % max_top
% 0.34/0.62 thf(fact_4467_max__top,axiom,
% 0.34/0.62 ! [X: set_Product_unit] :
% 0.34/0.62 ( ( ord_ma508538259134630082t_unit @ top_to1996260823553986621t_unit @ X )
% 0.34/0.62 = top_to1996260823553986621t_unit ) ).
% 0.34/0.62
% 0.34/0.62 % max_top
% 0.34/0.62 thf(fact_4468_max__top,axiom,
% 0.34/0.62 ! [X: set_o] :
% 0.34/0.62 ( ( ord_max_set_o @ top_top_set_o @ X )
% 0.34/0.62 = top_top_set_o ) ).
% 0.34/0.62
% 0.34/0.62 % max_top
% 0.34/0.62 thf(fact_4469_max__top,axiom,
% 0.34/0.62 ! [X: set_nat] :
% 0.34/0.62 ( ( ord_max_set_nat @ top_top_set_nat @ X )
% 0.34/0.62 = top_top_set_nat ) ).
% 0.34/0.62
% 0.34/0.62 % max_top
% 0.34/0.62 thf(fact_4470_max__top,axiom,
% 0.34/0.62 ! [X: set_int] :
% 0.34/0.62 ( ( ord_max_set_int @ top_top_set_int @ X )
% 0.34/0.62 = top_top_set_int ) ).
% 0.34/0.62
% 0.34/0.62 % max_top
% 0.34/0.62 thf(fact_4471_max__top2,axiom,
% 0.34/0.62 ! [X: assn] :
% 0.34/0.62 ( ( ord_max_assn @ X @ top_top_assn )
% 0.34/0.62 = top_top_assn ) ).
% 0.34/0.62
% 0.34/0.62 % max_top2
% 0.34/0.62 thf(fact_4472_max__top2,axiom,
% 0.34/0.62 ! [X: set_Product_unit] :
% 0.34/0.62 ( ( ord_ma508538259134630082t_unit @ X @ top_to1996260823553986621t_unit )
% 0.34/0.62 = top_to1996260823553986621t_unit ) ).
% 0.34/0.62
% 0.34/0.62 % max_top2
% 0.34/0.62 thf(fact_4473_max__top2,axiom,
% 0.34/0.62 ! [X: set_o] :
% 0.34/0.62 ( ( ord_max_set_o @ X @ top_top_set_o )
% 0.34/0.62 = top_top_set_o ) ).
% 0.34/0.62
% 0.34/0.62 % max_top2
% 0.34/0.62 thf(fact_4474_max__top2,axiom,
% 0.34/0.62 ! [X: set_nat] :
% 0.34/0.62 ( ( ord_max_set_nat @ X @ top_top_set_nat )
% 0.34/0.62 = top_top_set_nat ) ).
% 0.34/0.62
% 0.34/0.62 % max_top2
% 0.34/0.62 thf(fact_4475_max__top2,axiom,
% 0.34/0.62 ! [X: set_int] :
% 0.34/0.62 ( ( ord_max_set_int @ X @ top_top_set_int )
% 0.34/0.62 = top_top_set_int ) ).
% 0.34/0.62
% 0.34/0.62 % max_top2
% 0.34/0.62 thf(fact_4476_map__prod__simp,axiom,
% 0.34/0.62 ! [F: nat > nat,G: nat > nat,A2: nat,B: nat] :
% 0.34/0.62 ( ( produc6977886695330630970at_nat @ F @ G @ ( product_Pair_nat_nat @ A2 @ B ) )
% 0.34/0.62 = ( product_Pair_nat_nat @ ( F @ A2 ) @ ( G @ B ) ) ) ).
% 0.34/0.62
% 0.34/0.62 % map_prod_simp
% 0.34/0.62 thf(fact_4477_map__prod__simp,axiom,
% 0.34/0.62 ! [F: nat > int,G: nat > int,A2: nat,B: nat] :
% 0.34/0.62 ( ( produc7347049785058425330at_int @ F @ G @ ( product_Pair_nat_nat @ A2 @ B ) )
% 0.34/0.62 = ( product_Pair_int_int @ ( F @ A2 ) @ ( G @ B ) ) ) ).
% 0.34/0.62
% 0.34/0.62 % map_prod_simp
% 0.34/0.62 thf(fact_4478_map__prod__simp,axiom,
% 0.34/0.62 ! [F: int > nat,G: int > nat,A2: int,B: int] :
% 0.34/0.62 ( ( produc5667422475138734578nt_nat @ F @ G @ ( product_Pair_int_int @ A2 @ B ) )
% 0.34/0.62 = ( product_Pair_nat_nat @ ( F @ A2 ) @ ( G @ B ) ) ) ).
% 0.34/0.62
% 0.34/0.62 % map_prod_simp
% 0.34/0.62 thf(fact_4479_map__prod__simp,axiom,
% 0.34/0.62 ! [F: int > int,G: int > int,A2: int,B: int] :
% 0.34/0.62 ( ( produc6036585564866528938nt_int @ F @ G @ ( product_Pair_int_int @ A2 @ B ) )
% 0.34/0.62 = ( product_Pair_int_int @ ( F @ A2 ) @ ( G @ B ) ) ) ).
% 0.34/0.62
% 0.34/0.62 % map_prod_simp
% 0.34/0.62 thf(fact_4480_map__prod__simp,axiom,
% 0.34/0.62 ! [F: heap_e7401611519738050253t_unit > nat,G: set_nat > nat,A2: heap_e7401611519738050253t_unit,B: set_nat] :
% 0.34/0.62 ( ( produc8285742784527409791at_nat @ F @ G @ ( produc7507926704131184380et_nat @ A2 @ B ) )
% 0.34/0.62 = ( product_Pair_nat_nat @ ( F @ A2 ) @ ( G @ B ) ) ) ).
% 0.34/0.62
% 0.34/0.62 % map_prod_simp
% 0.34/0.62 thf(fact_4481_map__prod__simp,axiom,
% 0.34/0.62 ! [F: heap_e7401611519738050253t_unit > int,G: set_nat > int,A2: heap_e7401611519738050253t_unit,B: set_nat] :
% 0.34/0.62 ( ( produc5973434381925408567at_int @ F @ G @ ( produc7507926704131184380et_nat @ A2 @ B ) )
% 0.34/0.62 = ( product_Pair_int_int @ ( F @ A2 ) @ ( G @ B ) ) ) ).
% 0.34/0.62
% 0.34/0.62 % map_prod_simp
% 0.34/0.62 thf(fact_4482_map__prod__simp,axiom,
% 0.34/0.62 ! [F: nat > heap_e7401611519738050253t_unit,G: nat > set_nat,A2: nat,B: nat] :
% 0.34/0.62 ( ( produc2064153442785501et_nat @ F @ G @ ( product_Pair_nat_nat @ A2 @ B ) )
% 0.34/0.62 = ( produc7507926704131184380et_nat @ ( F @ A2 ) @ ( G @ B ) ) ) ).
% 0.34/0.62
% 0.34/0.62 % map_prod_simp
% 0.34/0.62 thf(fact_4483_map__prod__simp,axiom,
% 0.34/0.62 ! [F: int > heap_e7401611519738050253t_unit,G: int > set_nat,A2: int,B: int] :
% 0.34/0.62 ( ( produc3152090231653401565et_nat @ F @ G @ ( product_Pair_int_int @ A2 @ B ) )
% 0.34/0.62 = ( produc7507926704131184380et_nat @ ( F @ A2 ) @ ( G @ B ) ) ) ).
% 0.34/0.62
% 0.34/0.62 % map_prod_simp
% 0.34/0.62 thf(fact_4484_map__prod__simp,axiom,
% 0.34/0.62 ! [F: heap_e7401611519738050253t_unit > heap_e7401611519738050253t_unit,G: set_nat > set_nat,A2: heap_e7401611519738050253t_unit,B: set_nat] :
% 0.34/0.62 ( ( produc8234977478398262212et_nat @ F @ G @ ( produc7507926704131184380et_nat @ A2 @ B ) )
% 0.34/0.62 = ( produc7507926704131184380et_nat @ ( F @ A2 ) @ ( G @ B ) ) ) ).
% 0.34/0.62
% 0.34/0.62 % map_prod_simp
% 0.34/0.62 thf(fact_4485_map__prod__simp,axiom,
% 0.34/0.62 ! [F: nat > produc3658429121746597890et_nat > $o,G: nat > produc3658429121746597890et_nat,A2: nat,B: nat] :
% 0.34/0.62 ( ( produc79747477514316377et_nat @ F @ G @ ( product_Pair_nat_nat @ A2 @ B ) )
% 0.34/0.62 = ( produc5001842942810119800et_nat @ ( F @ A2 ) @ ( G @ B ) ) ) ).
% 0.34/0.62
% 0.34/0.62 % map_prod_simp
% 0.34/0.62 thf(fact_4486_snd__comp__map__prod,axiom,
% 0.34/0.62 ! [F: heap_e7401611519738050253t_unit > heap_e7401611519738050253t_unit,G: set_nat > set_nat] :
% 0.34/0.62 ( ( comp_P2930521774885306735et_nat @ produc8586169260539613262et_nat @ ( produc8234977478398262212et_nat @ F @ G ) )
% 0.34/0.62 = ( comp_s5696514601250091763et_nat @ G @ produc8586169260539613262et_nat ) ) ).
% 0.34/0.62
% 0.34/0.62 % snd_comp_map_prod
% 0.34/0.62 thf(fact_4487_snd__comp__map__prod,axiom,
% 0.34/0.62 ! [F: int > heap_e7401611519738050253t_unit,G: int > set_nat] :
% 0.34/0.62 ( ( comp_P8606889026643445772nt_int @ produc8586169260539613262et_nat @ ( produc3152090231653401565et_nat @ F @ G ) )
% 0.34/0.62 = ( comp_i6200299785623974754nt_int @ G @ product_snd_int_int ) ) ).
% 0.34/0.62
% 0.34/0.62 % snd_comp_map_prod
% 0.34/0.62 thf(fact_4488_snd__comp__map__prod,axiom,
% 0.34/0.62 ! [F: heap_e7401611519738050253t_unit > int,G: set_nat > int] :
% 0.34/0.62 ( ( comp_P8308861049717174982et_nat @ product_snd_int_int @ ( produc5973434381925408567at_int @ F @ G ) )
% 0.34/0.62 = ( comp_s1601710177542341017et_nat @ G @ produc8586169260539613262et_nat ) ) ).
% 0.34/0.62
% 0.34/0.62 % snd_comp_map_prod
% 0.34/0.62 thf(fact_4489_snd__comp__map__prod,axiom,
% 0.34/0.62 ! [F: int > int,G: int > int] :
% 0.34/0.62 ( ( comp_P5356232565778667509nt_int @ product_snd_int_int @ ( produc6036585564866528938nt_int @ F @ G ) )
% 0.34/0.62 = ( comp_i4330622015127193916nt_int @ G @ product_snd_int_int ) ) ).
% 0.34/0.62
% 0.34/0.62 % snd_comp_map_prod
% 0.34/0.62 thf(fact_4490_snd__map__prod,axiom,
% 0.34/0.62 ! [F: heap_e7401611519738050253t_unit > heap_e7401611519738050253t_unit,G: set_nat > set_nat,X: produc3658429121746597890et_nat] :
% 0.34/0.62 ( ( produc8586169260539613262et_nat @ ( produc8234977478398262212et_nat @ F @ G @ X ) )
% 0.34/0.62 = ( G @ ( produc8586169260539613262et_nat @ X ) ) ) ).
% 0.34/0.62
% 0.34/0.62 % snd_map_prod
% 0.34/0.62 thf(fact_4491_snd__map__prod,axiom,
% 0.34/0.62 ! [F: int > heap_e7401611519738050253t_unit,G: int > set_nat,X: product_prod_int_int] :
% 0.34/0.62 ( ( produc8586169260539613262et_nat @ ( produc3152090231653401565et_nat @ F @ G @ X ) )
% 0.34/0.62 = ( G @ ( product_snd_int_int @ X ) ) ) ).
% 0.34/0.62
% 0.34/0.62 % snd_map_prod
% 0.34/0.62 thf(fact_4492_snd__map__prod,axiom,
% 0.34/0.62 ! [F: heap_e7401611519738050253t_unit > int,G: set_nat > int,X: produc3658429121746597890et_nat] :
% 0.34/0.62 ( ( product_snd_int_int @ ( produc5973434381925408567at_int @ F @ G @ X ) )
% 0.34/0.62 = ( G @ ( produc8586169260539613262et_nat @ X ) ) ) ).
% 0.34/0.62
% 0.34/0.62 % snd_map_prod
% 0.34/0.62 thf(fact_4493_snd__map__prod,axiom,
% 0.34/0.62 ! [F: int > int,G: int > int,X: product_prod_int_int] :
% 0.34/0.62 ( ( product_snd_int_int @ ( produc6036585564866528938nt_int @ F @ G @ X ) )
% 0.34/0.62 = ( G @ ( product_snd_int_int @ X ) ) ) ).
% 0.34/0.62
% 0.34/0.62 % snd_map_prod
% 0.34/0.62 thf(fact_4494_map__prod__imageI,axiom,
% 0.34/0.62 ! [A2: nat,B: nat,R3: set_Pr1261947904930325089at_nat,F: nat > nat,G: nat > nat] :
% 0.34/0.62 ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A2 @ B ) @ R3 )
% 0.34/0.62 => ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ ( F @ A2 ) @ ( G @ B ) ) @ ( image_5168914502847457605at_nat @ ( produc6977886695330630970at_nat @ F @ G ) @ R3 ) ) ) ).
% 0.34/0.62
% 0.34/0.62 % map_prod_imageI
% 0.34/0.62 thf(fact_4495_map__prod__imageI,axiom,
% 0.34/0.62 ! [A2: nat,B: nat,R3: set_Pr1261947904930325089at_nat,F: nat > int,G: nat > int] :
% 0.34/0.62 ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A2 @ B ) @ R3 )
% 0.34/0.62 => ( member5262025264175285858nt_int @ ( product_Pair_int_int @ ( F @ A2 ) @ ( G @ B ) ) @ ( image_1990417195239315453nt_int @ ( produc7347049785058425330at_int @ F @ G ) @ R3 ) ) ) ).
% 0.34/0.62
% 0.34/0.62 % map_prod_imageI
% 0.34/0.62 thf(fact_4496_map__prod__imageI,axiom,
% 0.34/0.62 ! [A2: int,B: int,R3: set_Pr958786334691620121nt_int,F: int > nat,G: int > nat] :
% 0.34/0.62 ( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ A2 @ B ) @ R3 )
% 0.34/0.62 => ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ ( F @ A2 ) @ ( G @ B ) ) @ ( image_5831868185956570253at_nat @ ( produc5667422475138734578nt_nat @ F @ G ) @ R3 ) ) ) ).
% 0.34/0.62
% 0.34/0.62 % map_prod_imageI
% 0.34/0.62 thf(fact_4497_map__prod__imageI,axiom,
% 0.34/0.62 ! [A2: int,B: int,R3: set_Pr958786334691620121nt_int,F: int > int,G: int > int] :
% 0.34/0.62 ( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ A2 @ B ) @ R3 )
% 0.34/0.62 => ( member5262025264175285858nt_int @ ( product_Pair_int_int @ ( F @ A2 ) @ ( G @ B ) ) @ ( image_2653370878348428101nt_int @ ( produc6036585564866528938nt_int @ F @ G ) @ R3 ) ) ) ).
% 0.34/0.62
% 0.34/0.62 % map_prod_imageI
% 0.34/0.62 thf(fact_4498_map__prod__imageI,axiom,
% 0.34/0.62 ! [A2: heap_e7401611519738050253t_unit,B: set_nat,R3: set_Pr3948176798113811640et_nat,F: heap_e7401611519738050253t_unit > nat,G: set_nat > nat] :
% 0.34/0.62 ( ( member6260224972018164377et_nat @ ( produc7507926704131184380et_nat @ A2 @ B ) @ R3 )
% 0.34/0.62 => ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ ( F @ A2 ) @ ( G @ B ) ) @ ( image_7757775281999735736at_nat @ ( produc8285742784527409791at_nat @ F @ G ) @ R3 ) ) ) ).
% 0.34/0.62
% 0.34/0.62 % map_prod_imageI
% 0.34/0.62 thf(fact_4499_map__prod__imageI,axiom,
% 0.34/0.62 ! [A2: heap_e7401611519738050253t_unit,B: set_nat,R3: set_Pr3948176798113811640et_nat,F: heap_e7401611519738050253t_unit > int,G: set_nat > int] :
% 0.34/0.62 ( ( member6260224972018164377et_nat @ ( produc7507926704131184380et_nat @ A2 @ B ) @ R3 )
% 0.34/0.62 => ( member5262025264175285858nt_int @ ( product_Pair_int_int @ ( F @ A2 ) @ ( G @ B ) ) @ ( image_4579277974391593584nt_int @ ( produc5973434381925408567at_int @ F @ G ) @ R3 ) ) ) ).
% 0.34/0.62
% 0.34/0.62 % map_prod_imageI
% 0.34/0.62 thf(fact_4500_map__prod__imageI,axiom,
% 0.34/0.62 ! [A2: nat,B: nat,R3: set_Pr1261947904930325089at_nat,F: nat > heap_e7401611519738050253t_unit,G: nat > set_nat] :
% 0.34/0.62 ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A2 @ B ) @ R3 )
% 0.34/0.62 => ( member6260224972018164377et_nat @ ( produc7507926704131184380et_nat @ ( F @ A2 ) @ ( G @ B ) ) @ ( image_9068448769188195518et_nat @ ( produc2064153442785501et_nat @ F @ G ) @ R3 ) ) ) ).
% 0.34/0.62
% 0.34/0.62 % map_prod_imageI
% 0.34/0.62 thf(fact_4501_map__prod__imageI,axiom,
% 0.34/0.62 ! [A2: int,B: int,R3: set_Pr958786334691620121nt_int,F: int > heap_e7401611519738050253t_unit,G: int > set_nat] :
% 0.34/0.62 ( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ A2 @ B ) @ R3 )
% 0.34/0.62 => ( member6260224972018164377et_nat @ ( produc7507926704131184380et_nat @ ( F @ A2 ) @ ( G @ B ) ) @ ( image_2127501072796355446et_nat @ ( produc3152090231653401565et_nat @ F @ G ) @ R3 ) ) ) ).
% 0.34/0.62
% 0.34/0.62 % map_prod_imageI
% 0.34/0.62 thf(fact_4502_map__prod__imageI,axiom,
% 0.34/0.62 ! [A2: heap_e7401611519738050253t_unit,B: set_nat,R3: set_Pr3948176798113811640et_nat,F: heap_e7401611519738050253t_unit > heap_e7401611519738050253t_unit,G: set_nat > set_nat] :
% 0.34/0.62 ( ( member6260224972018164377et_nat @ ( produc7507926704131184380et_nat @ A2 @ B ) @ R3 )
% 0.34/0.62 => ( member6260224972018164377et_nat @ ( produc7507926704131184380et_nat @ ( F @ A2 ) @ ( G @ B ) ) @ ( image_7426123528223433739et_nat @ ( produc8234977478398262212et_nat @ F @ G ) @ R3 ) ) ) ).
% 0.34/0.62
% 0.34/0.62 % map_prod_imageI
% 0.34/0.62 thf(fact_4503_map__prod__imageI,axiom,
% 0.34/0.62 ! [A2: nat,B: nat,R3: set_Pr1261947904930325089at_nat,F: nat > produc3658429121746597890et_nat > $o,G: nat > produc3658429121746597890et_nat] :
% 0.34/0.62 ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A2 @ B ) @ R3 )
% 0.34/0.62 => ( member1996754912294343701et_nat @ ( produc5001842942810119800et_nat @ ( F @ A2 ) @ ( G @ B ) ) @ ( image_5744643625575355962et_nat @ ( produc79747477514316377et_nat @ F @ G ) @ R3 ) ) ) ).
% 0.34/0.62
% 0.34/0.62 % map_prod_imageI
% 0.34/0.62 thf(fact_4504_exists__leI,axiom,
% 0.34/0.62 ! [N2: nat,P: nat > $o] :
% 0.34/0.62 ( ( ! [N5: nat] :
% 0.34/0.62 ( ( ord_less_nat @ N5 @ N2 )
% 0.34/0.62 => ~ ( P @ N5 ) )
% 0.34/0.62 => ( P @ N2 ) )
% 0.34/0.62 => ? [N6: nat] :
% 0.34/0.62 ( ( ord_less_eq_nat @ N6 @ N2 )
% 0.34/0.62 & ( P @ N6 ) ) ) ).
% 0.34/0.62
% 0.34/0.62 % exists_leI
% 0.34/0.62 thf(fact_4505_max__def,axiom,
% 0.34/0.62 ( ord_max_filter_nat
% 0.34/0.62 = ( ^ [A6: filter_nat,B6: filter_nat] : ( if_filter_nat @ ( ord_le2510731241096832064er_nat @ A6 @ B6 ) @ B6 @ A6 ) ) ) ).
% 0.34/0.62
% 0.34/0.62 % max_def
% 0.34/0.62 thf(fact_4506_max__def,axiom,
% 0.34/0.62 ( ord_max_set_nat
% 0.34/0.62 = ( ^ [A6: set_nat,B6: set_nat] : ( if_set_nat @ ( ord_less_eq_set_nat @ A6 @ B6 ) @ B6 @ A6 ) ) ) ).
% 0.34/0.62
% 0.34/0.62 % max_def
% 0.34/0.62 thf(fact_4507_max__def,axiom,
% 0.34/0.62 ( ord_max_rat
% 0.34/0.62 = ( ^ [A6: rat,B6: rat] : ( if_rat @ ( ord_less_eq_rat @ A6 @ B6 ) @ B6 @ A6 ) ) ) ).
% 0.34/0.62
% 0.34/0.62 % max_def
% 0.34/0.62 thf(fact_4508_max__def,axiom,
% 0.34/0.62 ( ord_max_nat
% 0.34/0.62 = ( ^ [A6: nat,B6: nat] : ( if_nat @ ( ord_less_eq_nat @ A6 @ B6 ) @ B6 @ A6 ) ) ) ).
% 0.34/0.62
% 0.34/0.62 % max_def
% 0.34/0.62 thf(fact_4509_max__def,axiom,
% 0.34/0.62 ( ord_max_int
% 0.34/0.62 = ( ^ [A6: int,B6: int] : ( if_int @ ( ord_less_eq_int @ A6 @ B6 ) @ B6 @ A6 ) ) ) ).
% 0.34/0.62
% 0.34/0.62 % max_def
% 0.34/0.62 thf(fact_4510_max__absorb1,axiom,
% 0.34/0.62 ! [Y: filter_nat,X: filter_nat] :
% 0.34/0.62 ( ( ord_le2510731241096832064er_nat @ Y @ X )
% 0.34/0.62 => ( ( ord_max_filter_nat @ X @ Y )
% 0.34/0.62 = X ) ) ).
% 0.34/0.62
% 0.34/0.62 % max_absorb1
% 0.34/0.62 thf(fact_4511_max__absorb1,axiom,
% 0.34/0.62 ! [Y: set_nat,X: set_nat] :
% 0.34/0.62 ( ( ord_less_eq_set_nat @ Y @ X )
% 0.34/0.62 => ( ( ord_max_set_nat @ X @ Y )
% 0.34/0.62 = X ) ) ).
% 0.34/0.62
% 0.34/0.62 % max_absorb1
% 0.34/0.62 thf(fact_4512_max__absorb1,axiom,
% 0.34/0.62 ! [Y: rat,X: rat] :
% 0.34/0.62 ( ( ord_less_eq_rat @ Y @ X )
% 0.34/0.62 => ( ( ord_max_rat @ X @ Y )
% 0.34/0.62 = X ) ) ).
% 0.34/0.62
% 0.34/0.62 % max_absorb1
% 0.34/0.62 thf(fact_4513_max__absorb1,axiom,
% 0.34/0.62 ! [Y: nat,X: nat] :
% 0.34/0.62 ( ( ord_less_eq_nat @ Y @ X )
% 0.34/0.62 => ( ( ord_max_nat @ X @ Y )
% 0.34/0.62 = X ) ) ).
% 0.34/0.62
% 0.34/0.62 % max_absorb1
% 0.34/0.62 thf(fact_4514_max__absorb1,axiom,
% 0.34/0.62 ! [Y: int,X: int] :
% 0.34/0.62 ( ( ord_less_eq_int @ Y @ X )
% 0.34/0.62 => ( ( ord_max_int @ X @ Y )
% 0.34/0.62 = X ) ) ).
% 0.34/0.62
% 0.34/0.62 % max_absorb1
% 0.34/0.62 thf(fact_4515_max__absorb2,axiom,
% 0.34/0.62 ! [X: filter_nat,Y: filter_nat] :
% 0.34/0.62 ( ( ord_le2510731241096832064er_nat @ X @ Y )
% 0.34/0.62 => ( ( ord_max_filter_nat @ X @ Y )
% 0.34/0.62 = Y ) ) ).
% 0.34/0.62
% 0.34/0.62 % max_absorb2
% 0.34/0.62 thf(fact_4516_max__absorb2,axiom,
% 0.34/0.62 ! [X: set_nat,Y: set_nat] :
% 0.34/0.62 ( ( ord_less_eq_set_nat @ X @ Y )
% 0.34/0.62 => ( ( ord_max_set_nat @ X @ Y )
% 0.34/0.62 = Y ) ) ).
% 0.34/0.62
% 0.34/0.62 % max_absorb2
% 0.34/0.62 thf(fact_4517_max__absorb2,axiom,
% 0.34/0.62 ! [X: rat,Y: rat] :
% 0.34/0.62 ( ( ord_less_eq_rat @ X @ Y )
% 0.34/0.62 => ( ( ord_max_rat @ X @ Y )
% 0.34/0.62 = Y ) ) ).
% 0.34/0.62
% 0.34/0.62 % max_absorb2
% 0.34/0.62 thf(fact_4518_max__absorb2,axiom,
% 0.34/0.62 ! [X: nat,Y: nat] :
% 0.34/0.62 ( ( ord_less_eq_nat @ X @ Y )
% 0.34/0.62 => ( ( ord_max_nat @ X @ Y )
% 0.34/0.62 = Y ) ) ).
% 0.34/0.62
% 0.34/0.62 % max_absorb2
% 0.34/0.62 thf(fact_4519_max__absorb2,axiom,
% 0.34/0.62 ! [X: int,Y: int] :
% 0.34/0.62 ( ( ord_less_eq_int @ X @ Y )
% 0.34/0.62 => ( ( ord_max_int @ X @ Y )
% 0.34/0.62 = Y ) ) ).
% 0.34/0.62
% 0.34/0.62 % max_absorb2
% 0.34/0.62 thf(fact_4520_max__diff__distrib__left,axiom,
% 0.34/0.62 ! [X: rat,Y: rat,Z: rat] :
% 0.34/0.62 ( ( minus_minus_rat @ ( ord_max_rat @ X @ Y ) @ Z )
% 0.34/0.62 = ( ord_max_rat @ ( minus_minus_rat @ X @ Z ) @ ( minus_minus_rat @ Y @ Z ) ) ) ).
% 0.34/0.62
% 0.34/0.62 % max_diff_distrib_left
% 0.34/0.62 thf(fact_4521_max__diff__distrib__left,axiom,
% 0.34/0.62 ! [X: int,Y: int,Z: int] :
% 0.34/0.62 ( ( minus_minus_int @ ( ord_max_int @ X @ Y ) @ Z )
% 0.34/0.62 = ( ord_max_int @ ( minus_minus_int @ X @ Z ) @ ( minus_minus_int @ Y @ Z ) ) ) ).
% 0.34/0.62
% 0.34/0.62 % max_diff_distrib_left
% 0.34/0.62 thf(fact_4522_map__prod_Ocomp,axiom,
% 0.34/0.62 ! [F: code_integer > produc8923325533196201883nteger > produc8923325533196201883nteger,G: code_integer > produc8923325533196201883nteger > produc8923325533196201883nteger,H2: code_integer > code_integer,I2: code_integer > code_integer] :
% 0.34/0.62 ( ( comp_P234064951390806312nteger @ ( produc5081366107664765182nteger @ F @ G ) @ ( produc1802105962441033582nteger @ H2 @ I2 ) )
% 0.34/0.62 = ( produc5081366107664765182nteger @ ( comp_C1593894019821074884nteger @ F @ H2 ) @ ( comp_C1593894019821074884nteger @ G @ I2 ) ) ) ).
% 0.34/0.62
% 0.34/0.62 % map_prod.comp
% 0.34/0.62 thf(fact_4523_map__prod_Ocomp,axiom,
% 0.34/0.62 ! [F: code_integer > produc8923325533196201883nteger > produc8923325533196201883nteger,G: ( code_integer > code_integer ) > produc8923325533196201883nteger > produc8923325533196201883nteger,H2: code_integer > code_integer,I2: code_integer > code_integer > code_integer] :
% 0.34/0.62 ( ( comp_P6312383820422015140nteger @ ( produc3090791198996863994nteger @ F @ G ) @ ( produc6023007152014423402nteger @ H2 @ I2 ) )
% 0.34/0.62 = ( produc5081366107664765182nteger @ ( comp_C1593894019821074884nteger @ F @ H2 ) @ ( comp_C8797469213163452608nteger @ G @ I2 ) ) ) ).
% 0.34/0.62
% 0.34/0.62 % map_prod.comp
% 0.34/0.62 thf(fact_4524_map__prod_Ocomp,axiom,
% 0.34/0.62 ! [F: ( code_integer > code_integer ) > produc8923325533196201883nteger > produc8923325533196201883nteger,G: code_integer > produc8923325533196201883nteger > produc8923325533196201883nteger,H2: code_integer > code_integer > code_integer,I2: code_integer > code_integer] :
% 0.34/0.62 ( ( comp_P1414880308642389412nteger @ ( produc6158254626097298938nteger @ F @ G ) @ ( produc3789967387199524714nteger @ H2 @ I2 ) )
% 0.34/0.62 = ( produc5081366107664765182nteger @ ( comp_C8797469213163452608nteger @ F @ H2 ) @ ( comp_C1593894019821074884nteger @ G @ I2 ) ) ) ).
% 0.34/0.62
% 0.34/0.62 % map_prod.comp
% 0.34/0.62 thf(fact_4525_map__prod_Ocomp,axiom,
% 0.34/0.62 ! [F: ( code_integer > code_integer ) > produc8923325533196201883nteger > produc8923325533196201883nteger,G: ( code_integer > code_integer ) > produc8923325533196201883nteger > produc8923325533196201883nteger,H2: code_integer > code_integer > code_integer,I2: code_integer > code_integer > code_integer] :
% 0.34/0.62 ( ( comp_P5544053077415136032nteger @ ( produc4724772261320961782nteger @ F @ G ) @ ( produc7818937854452556646nteger @ H2 @ I2 ) )
% 0.34/0.62 = ( produc5081366107664765182nteger @ ( comp_C8797469213163452608nteger @ F @ H2 ) @ ( comp_C8797469213163452608nteger @ G @ I2 ) ) ) ).
% 0.34/0.62
% 0.34/0.62 % map_prod.comp
% 0.34/0.62 thf(fact_4526_map__prod_Ocompositionality,axiom,
% 0.34/0.62 ! [F: code_integer > produc8923325533196201883nteger > produc8923325533196201883nteger,G: code_integer > produc8923325533196201883nteger > produc8923325533196201883nteger,H2: code_integer > code_integer,I2: code_integer > code_integer,Prod: produc8923325533196201883nteger] :
% 0.34/0.62 ( ( produc5081366107664765182nteger @ F @ G @ ( produc1802105962441033582nteger @ H2 @ I2 @ Prod ) )
% 0.34/0.62 = ( produc5081366107664765182nteger @ ( comp_C1593894019821074884nteger @ F @ H2 ) @ ( comp_C1593894019821074884nteger @ G @ I2 ) @ Prod ) ) ).
% 0.34/0.62
% 0.34/0.62 % map_prod.compositionality
% 0.34/0.62 thf(fact_4527_map__prod_Ocompositionality,axiom,
% 0.34/0.62 ! [F: code_integer > produc8923325533196201883nteger > produc8923325533196201883nteger,G: ( code_integer > code_integer ) > produc8923325533196201883nteger > produc8923325533196201883nteger,H2: code_integer > code_integer,I2: code_integer > code_integer > code_integer,Prod: produc8923325533196201883nteger] :
% 0.34/0.62 ( ( produc3090791198996863994nteger @ F @ G @ ( produc6023007152014423402nteger @ H2 @ I2 @ Prod ) )
% 0.34/0.62 = ( produc5081366107664765182nteger @ ( comp_C1593894019821074884nteger @ F @ H2 ) @ ( comp_C8797469213163452608nteger @ G @ I2 ) @ Prod ) ) ).
% 0.34/0.62
% 0.34/0.62 % map_prod.compositionality
% 0.34/0.62 thf(fact_4528_map__prod_Ocompositionality,axiom,
% 0.34/0.62 ! [F: ( code_integer > code_integer ) > produc8923325533196201883nteger > produc8923325533196201883nteger,G: code_integer > produc8923325533196201883nteger > produc8923325533196201883nteger,H2: code_integer > code_integer > code_integer,I2: code_integer > code_integer,Prod: produc8923325533196201883nteger] :
% 0.34/0.62 ( ( produc6158254626097298938nteger @ F @ G @ ( produc3789967387199524714nteger @ H2 @ I2 @ Prod ) )
% 0.34/0.62 = ( produc5081366107664765182nteger @ ( comp_C8797469213163452608nteger @ F @ H2 ) @ ( comp_C1593894019821074884nteger @ G @ I2 ) @ Prod ) ) ).
% 0.34/0.62
% 0.34/0.62 % map_prod.compositionality
% 0.34/0.62 thf(fact_4529_map__prod_Ocompositionality,axiom,
% 0.34/0.62 ! [F: ( code_integer > code_integer ) > produc8923325533196201883nteger > produc8923325533196201883nteger,G: ( code_integer > code_integer ) > produc8923325533196201883nteger > produc8923325533196201883nteger,H2: code_integer > code_integer > code_integer,I2: code_integer > code_integer > code_integer,Prod: produc8923325533196201883nteger] :
% 0.34/0.62 ( ( produc4724772261320961782nteger @ F @ G @ ( produc7818937854452556646nteger @ H2 @ I2 @ Prod ) )
% 0.34/0.62 = ( produc5081366107664765182nteger @ ( comp_C8797469213163452608nteger @ F @ H2 ) @ ( comp_C8797469213163452608nteger @ G @ I2 ) @ Prod ) ) ).
% 0.34/0.62
% 0.34/0.62 % map_prod.compositionality
% 0.34/0.62 thf(fact_4530_map__prod__compose,axiom,
% 0.34/0.62 ! [F1: code_integer > produc8923325533196201883nteger > produc8923325533196201883nteger,F22: code_integer > code_integer,G1: code_integer > produc8923325533196201883nteger > produc8923325533196201883nteger,G2: code_integer > code_integer] :
% 0.34/0.62 ( ( produc5081366107664765182nteger @ ( comp_C1593894019821074884nteger @ F1 @ F22 ) @ ( comp_C1593894019821074884nteger @ G1 @ G2 ) )
% 0.34/0.62 = ( comp_P234064951390806312nteger @ ( produc5081366107664765182nteger @ F1 @ G1 ) @ ( produc1802105962441033582nteger @ F22 @ G2 ) ) ) ).
% 0.34/0.62
% 0.34/0.62 % map_prod_compose
% 0.34/0.62 thf(fact_4531_map__prod__compose,axiom,
% 0.34/0.62 ! [F1: code_integer > produc8923325533196201883nteger > produc8923325533196201883nteger,F22: code_integer > code_integer,G1: ( code_integer > code_integer ) > produc8923325533196201883nteger > produc8923325533196201883nteger,G2: code_integer > code_integer > code_integer] :
% 0.34/0.62 ( ( produc5081366107664765182nteger @ ( comp_C1593894019821074884nteger @ F1 @ F22 ) @ ( comp_C8797469213163452608nteger @ G1 @ G2 ) )
% 0.34/0.62 = ( comp_P6312383820422015140nteger @ ( produc3090791198996863994nteger @ F1 @ G1 ) @ ( produc6023007152014423402nteger @ F22 @ G2 ) ) ) ).
% 0.34/0.62
% 0.34/0.62 % map_prod_compose
% 0.34/0.62 thf(fact_4532_map__prod__compose,axiom,
% 0.34/0.62 ! [F1: ( code_integer > code_integer ) > produc8923325533196201883nteger > produc8923325533196201883nteger,F22: code_integer > code_integer > code_integer,G1: code_integer > produc8923325533196201883nteger > produc8923325533196201883nteger,G2: code_integer > code_integer] :
% 0.34/0.62 ( ( produc5081366107664765182nteger @ ( comp_C8797469213163452608nteger @ F1 @ F22 ) @ ( comp_C1593894019821074884nteger @ G1 @ G2 ) )
% 0.34/0.62 = ( comp_P1414880308642389412nteger @ ( produc6158254626097298938nteger @ F1 @ G1 ) @ ( produc3789967387199524714nteger @ F22 @ G2 ) ) ) ).
% 0.34/0.62
% 0.34/0.62 % map_prod_compose
% 0.34/0.62 thf(fact_4533_map__prod__compose,axiom,
% 0.34/0.62 ! [F1: ( code_integer > code_integer ) > produc8923325533196201883nteger > produc8923325533196201883nteger,F22: code_integer > code_integer > code_integer,G1: ( code_integer > code_integer ) > produc8923325533196201883nteger > produc8923325533196201883nteger,G2: code_integer > code_integer > code_integer] :
% 0.34/0.62 ( ( produc5081366107664765182nteger @ ( comp_C8797469213163452608nteger @ F1 @ F22 ) @ ( comp_C8797469213163452608nteger @ G1 @ G2 ) )
% 0.34/0.62 = ( comp_P5544053077415136032nteger @ ( produc4724772261320961782nteger @ F1 @ G1 ) @ ( produc7818937854452556646nteger @ F22 @ G2 ) ) ) ).
% 0.34/0.62
% 0.34/0.62 % map_prod_compose
% 0.34/0.62 thf(fact_4534_inj__img__insertE,axiom,
% 0.34/0.62 ! [F: code_integer > code_integer,A: set_Code_integer,X: code_integer,B4: set_Code_integer] :
% 0.34/0.62 ( ( inj_on1158064722259180589nteger @ F @ A )
% 0.34/0.62 => ( ~ ( member_Code_integer @ X @ B4 )
% 0.34/0.62 => ( ( ( insert_Code_integer @ X @ B4 )
% 0.34/0.62 = ( image_4470545334726330049nteger @ F @ A ) )
% 0.34/0.62 => ~ ! [X10: code_integer,A7: set_Code_integer] :
% 0.34/0.62 ( ~ ( member_Code_integer @ X10 @ A7 )
% 0.34/0.62 => ( ( A
% 0.34/0.62 = ( insert_Code_integer @ X10 @ A7 ) )
% 0.34/0.62 => ( ( X
% 0.34/0.62 = ( F @ X10 ) )
% 0.34/0.62 => ( B4
% 0.34/0.62 != ( image_4470545334726330049nteger @ F @ A7 ) ) ) ) ) ) ) ) ).
% 0.34/0.62
% 0.34/0.62 % inj_img_insertE
% 0.34/0.62 thf(fact_4535_inj__img__insertE,axiom,
% 0.34/0.62 ! [F: nat > code_integer,A: set_nat,X: code_integer,B4: set_Code_integer] :
% 0.34/0.62 ( ( inj_on2245834603434889120nteger @ F @ A )
% 0.34/0.62 => ( ~ ( member_Code_integer @ X @ B4 )
% 0.34/0.62 => ( ( ( insert_Code_integer @ X @ B4 )
% 0.34/0.62 = ( image_1215581382706833972nteger @ F @ A ) )
% 0.34/0.62 => ~ ! [X10: nat,A7: set_nat] :
% 0.34/0.62 ( ~ ( member_nat @ X10 @ A7 )
% 0.34/0.62 => ( ( A
% 0.34/0.62 = ( insert_nat @ X10 @ A7 ) )
% 0.34/0.62 => ( ( X
% 0.34/0.62 = ( F @ X10 ) )
% 0.34/0.62 => ( B4
% 0.34/0.62 != ( image_1215581382706833972nteger @ F @ A7 ) ) ) ) ) ) ) ) ).
% 0.34/0.62
% 0.34/0.62 % inj_img_insertE
% 0.34/0.62 thf(fact_4536_inj__img__insertE,axiom,
% 0.34/0.62 ! [F: nat > nat,A: set_nat,X: nat,B4: set_nat] :
% 0.34/0.62 ( ( inj_on_nat_nat @ F @ A )
% 0.34/0.62 => ( ~ ( member_nat @ X @ B4 )
% 0.34/0.62 => ( ( ( insert_nat @ X @ B4 )
% 0.34/0.62 = ( image_nat_nat2 @ F @ A ) )
% 0.34/0.62 => ~ ! [X10: nat,A7: set_nat] :
% 0.34/0.62 ( ~ ( member_nat @ X10 @ A7 )
% 0.34/0.62 => ( ( A
% 0.34/0.62 = ( insert_nat @ X10 @ A7 ) )
% 0.34/0.62 => ( ( X
% 0.34/0.62 = ( F @ X10 ) )
% 0.34/0.62 => ( B4
% 0.34/0.62 != ( image_nat_nat2 @ F @ A7 ) ) ) ) ) ) ) ) ).
% 0.34/0.62
% 0.34/0.62 % inj_img_insertE
% 0.34/0.62 thf(fact_4537_inj__img__insertE,axiom,
% 0.34/0.62 ! [F: int > nat,A: set_int,X: nat,B4: set_nat] :
% 0.34/0.62 ( ( inj_on_int_nat @ F @ A )
% 0.34/0.62 => ( ~ ( member_nat @ X @ B4 )
% 0.34/0.62 => ( ( ( insert_nat @ X @ B4 )
% 0.34/0.62 = ( image_int_nat2 @ F @ A ) )
% 0.34/0.62 => ~ ! [X10: int,A7: set_int] :
% 0.34/0.62 ( ~ ( member_int @ X10 @ A7 )
% 0.34/0.62 => ( ( A
% 0.34/0.62 = ( insert_int @ X10 @ A7 ) )
% 0.34/0.62 => ( ( X
% 0.34/0.62 = ( F @ X10 ) )
% 0.34/0.62 => ( B4
% 0.34/0.62 != ( image_int_nat2 @ F @ A7 ) ) ) ) ) ) ) ) ).
% 0.34/0.62
% 0.34/0.62 % inj_img_insertE
% 0.34/0.62 thf(fact_4538_inj__img__insertE,axiom,
% 0.34/0.62 ! [F: $o > nat,A: set_o,X: nat,B4: set_nat] :
% 0.34/0.62 ( ( inj_on_o_nat @ F @ A )
% 0.34/0.62 => ( ~ ( member_nat @ X @ B4 )
% 0.34/0.62 => ( ( ( insert_nat @ X @ B4 )
% 0.34/0.62 = ( image_o_nat2 @ F @ A ) )
% 0.34/0.62 => ~ ! [X10: $o,A7: set_o] :
% 0.34/0.62 ( ~ ( member_o @ X10 @ A7 )
% 0.34/0.62 => ( ( A
% 0.34/0.62 = ( insert_o @ X10 @ A7 ) )
% 0.34/0.62 => ( ( X
% 0.34/0.62 = ( F @ X10 ) )
% 0.34/0.62 => ( B4
% 0.34/0.62 != ( image_o_nat2 @ F @ A7 ) ) ) ) ) ) ) ) ).
% 0.34/0.62
% 0.34/0.62 % inj_img_insertE
% 0.34/0.62 thf(fact_4539_inj__img__insertE,axiom,
% 0.34/0.62 ! [F: product_unit > nat,A: set_Product_unit,X: nat,B4: set_nat] :
% 0.34/0.62 ( ( inj_on8430439091780834860it_nat @ F @ A )
% 0.34/0.62 => ( ~ ( member_nat @ X @ B4 )
% 0.34/0.62 => ( ( ( insert_nat @ X @ B4 )
% 0.34/0.62 = ( image_875570014554754200it_nat @ F @ A ) )
% 0.34/0.62 => ~ ! [X10: product_unit,A7: set_Product_unit] :
% 0.34/0.62 ( ~ ( member_Product_unit @ X10 @ A7 )
% 0.34/0.62 => ( ( A
% 0.34/0.62 = ( insert_Product_unit @ X10 @ A7 ) )
% 0.34/0.62 => ( ( X
% 0.34/0.62 = ( F @ X10 ) )
% 0.34/0.62 => ( B4
% 0.34/0.62 != ( image_875570014554754200it_nat @ F @ A7 ) ) ) ) ) ) ) ) ).
% 0.34/0.62
% 0.34/0.62 % inj_img_insertE
% 0.34/0.62 thf(fact_4540_inj__img__insertE,axiom,
% 0.34/0.62 ! [F: nat > int,A: set_nat,X: int,B4: set_int] :
% 0.34/0.62 ( ( inj_on_nat_int @ F @ A )
% 0.34/0.62 => ( ~ ( member_int @ X @ B4 )
% 0.34/0.62 => ( ( ( insert_int @ X @ B4 )
% 0.34/0.62 = ( image_nat_int2 @ F @ A ) )
% 0.34/0.62 => ~ ! [X10: nat,A7: set_nat] :
% 0.34/0.62 ( ~ ( member_nat @ X10 @ A7 )
% 0.34/0.62 => ( ( A
% 0.34/0.62 = ( insert_nat @ X10 @ A7 ) )
% 0.34/0.62 => ( ( X
% 0.34/0.62 = ( F @ X10 ) )
% 0.34/0.62 => ( B4
% 0.34/0.62 != ( image_nat_int2 @ F @ A7 ) ) ) ) ) ) ) ) ).
% 0.34/0.62
% 0.34/0.62 % inj_img_insertE
% 0.34/0.62 thf(fact_4541_inj__img__insertE,axiom,
% 0.34/0.62 ! [F: int > int,A: set_int,X: int,B4: set_int] :
% 0.34/0.62 ( ( inj_on_int_int @ F @ A )
% 0.34/0.62 => ( ~ ( member_int @ X @ B4 )
% 0.34/0.62 => ( ( ( insert_int @ X @ B4 )
% 0.34/0.62 = ( image_int_int2 @ F @ A ) )
% 0.34/0.62 => ~ ! [X10: int,A7: set_int] :
% 0.34/0.62 ( ~ ( member_int @ X10 @ A7 )
% 0.34/0.62 => ( ( A
% 0.34/0.62 = ( insert_int @ X10 @ A7 ) )
% 0.34/0.62 => ( ( X
% 0.34/0.62 = ( F @ X10 ) )
% 0.34/0.62 => ( B4
% 0.34/0.62 != ( image_int_int2 @ F @ A7 ) ) ) ) ) ) ) ) ).
% 0.34/0.62
% 0.34/0.62 % inj_img_insertE
% 0.34/0.62 thf(fact_4542_inj__img__insertE,axiom,
% 0.34/0.62 ! [F: $o > int,A: set_o,X: int,B4: set_int] :
% 0.34/0.62 ( ( inj_on_o_int @ F @ A )
% 0.34/0.62 => ( ~ ( member_int @ X @ B4 )
% 0.34/0.62 => ( ( ( insert_int @ X @ B4 )
% 0.34/0.62 = ( image_o_int2 @ F @ A ) )
% 0.34/0.62 => ~ ! [X10: $o,A7: set_o] :
% 0.34/0.62 ( ~ ( member_o @ X10 @ A7 )
% 0.34/0.62 => ( ( A
% 0.34/0.62 = ( insert_o @ X10 @ A7 ) )
% 0.34/0.62 => ( ( X
% 0.34/0.62 = ( F @ X10 ) )
% 0.34/0.62 => ( B4
% 0.34/0.62 != ( image_o_int2 @ F @ A7 ) ) ) ) ) ) ) ) ).
% 0.34/0.62
% 0.34/0.62 % inj_img_insertE
% 0.34/0.62 thf(fact_4543_inj__img__insertE,axiom,
% 0.34/0.62 ! [F: product_unit > int,A: set_Product_unit,X: int,B4: set_int] :
% 0.34/0.62 ( ( inj_on8427948621271784584it_int @ F @ A )
% 0.34/0.62 => ( ~ ( member_int @ X @ B4 )
% 0.34/0.62 => ( ( ( insert_int @ X @ B4 )
% 0.34/0.62 = ( image_873079544045703924it_int @ F @ A ) )
% 0.34/0.62 => ~ ! [X10: product_unit,A7: set_Product_unit] :
% 0.34/0.62 ( ~ ( member_Product_unit @ X10 @ A7 )
% 0.34/0.62 => ( ( A
% 0.34/0.62 = ( insert_Product_unit @ X10 @ A7 ) )
% 0.34/0.62 => ( ( X
% 0.34/0.62 = ( F @ X10 ) )
% 0.34/0.62 => ( B4
% 0.34/0.62 != ( image_873079544045703924it_int @ F @ A7 ) ) ) ) ) ) ) ) ).
% 0.34/0.62
% 0.34/0.62 % inj_img_insertE
% 0.34/0.62 thf(fact_4544_type__copy__wit,axiom,
% 0.34/0.62 ! [Rep2: code_integer > int,Abs2: int > code_integer,X: nat,S: int > set_nat,Y: int] :
% 0.34/0.62 ( ( type_d8366093980585677751er_int @ Rep2 @ Abs2 @ top_top_set_int )
% 0.34/0.62 => ( ( member_nat @ X @ ( comp_i5102403792180996084nteger @ S @ Rep2 @ ( Abs2 @ Y ) ) )
% 0.34/0.62 => ( member_nat @ X @ ( S @ Y ) ) ) ) ).
% 0.34/0.62
% 0.34/0.62 % type_copy_wit
% 0.34/0.62 thf(fact_4545_type__copy__wit,axiom,
% 0.34/0.62 ! [Rep2: code_integer > int,Abs2: int > code_integer,X: int,S: int > set_int,Y: int] :
% 0.34/0.62 ( ( type_d8366093980585677751er_int @ Rep2 @ Abs2 @ top_top_set_int )
% 0.34/0.62 => ( ( member_int @ X @ ( comp_i1887841406321795152nteger @ S @ Rep2 @ ( Abs2 @ Y ) ) )
% 0.34/0.62 => ( member_int @ X @ ( S @ Y ) ) ) ) ).
% 0.34/0.62
% 0.34/0.62 % type_copy_wit
% 0.34/0.62 thf(fact_4546_type__copy__wit,axiom,
% 0.34/0.62 ! [Rep2: code_integer > int,Abs2: int > code_integer,X: $o,S: int > set_o,Y: int] :
% 0.34/0.62 ( ( type_d8366093980585677751er_int @ Rep2 @ Abs2 @ top_top_set_int )
% 0.34/0.62 => ( ( member_o @ X @ ( comp_i5479505385629679622nteger @ S @ Rep2 @ ( Abs2 @ Y ) ) )
% 0.34/0.62 => ( member_o @ X @ ( S @ Y ) ) ) ) ).
% 0.34/0.62
% 0.34/0.62 % type_copy_wit
% 0.34/0.62 thf(fact_4547_type__copy__wit,axiom,
% 0.34/0.62 ! [Rep2: code_integer > int,Abs2: int > code_integer,X: product_unit,S: int > set_Product_unit,Y: int] :
% 0.34/0.62 ( ( type_d8366093980585677751er_int @ Rep2 @ Abs2 @ top_top_set_int )
% 0.34/0.62 => ( ( member_Product_unit @ X @ ( comp_i3319814986458681467nteger @ S @ Rep2 @ ( Abs2 @ Y ) ) )
% 0.34/0.62 => ( member_Product_unit @ X @ ( S @ Y ) ) ) ) ).
% 0.34/0.62
% 0.34/0.62 % type_copy_wit
% 0.34/0.62 thf(fact_4548_type__copy__wit,axiom,
% 0.34/0.62 ! [Rep2: product_unit > $o,Abs2: $o > product_unit,X: nat,S: $o > set_nat,Y: $o] :
% 0.34/0.62 ( ( type_d6188575255521822967unit_o @ Rep2 @ Abs2 @ top_top_set_o )
% 0.34/0.62 => ( ( member_nat @ X @ ( comp_o9190160103024335696t_unit @ S @ Rep2 @ ( Abs2 @ Y ) ) )
% 0.34/0.62 => ( member_nat @ X @ ( S @ Y ) ) ) ) ).
% 0.34/0.62
% 0.34/0.62 % type_copy_wit
% 0.34/0.62 thf(fact_4549_type__copy__wit,axiom,
% 0.34/0.62 ! [Rep2: product_unit > $o,Abs2: $o > product_unit,X: int,S: $o > set_int,Y: $o] :
% 0.34/0.62 ( ( type_d6188575255521822967unit_o @ Rep2 @ Abs2 @ top_top_set_o )
% 0.34/0.62 => ( ( member_int @ X @ ( comp_o1521574261646630388t_unit @ S @ Rep2 @ ( Abs2 @ Y ) ) )
% 0.34/0.62 => ( member_int @ X @ ( S @ Y ) ) ) ) ).
% 0.34/0.62
% 0.34/0.62 % type_copy_wit
% 0.34/0.62 thf(fact_4550_type__copy__wit,axiom,
% 0.34/0.62 ! [Rep2: product_unit > $o,Abs2: $o > product_unit,X: $o,S: $o > set_o,Y: $o] :
% 0.34/0.62 ( ( type_d6188575255521822967unit_o @ Rep2 @ Abs2 @ top_top_set_o )
% 0.34/0.62 => ( ( member_o @ X @ ( comp_o6895741221744486154t_unit @ S @ Rep2 @ ( Abs2 @ Y ) ) )
% 0.34/0.62 => ( member_o @ X @ ( S @ Y ) ) ) ) ).
% 0.34/0.62
% 0.34/0.62 % type_copy_wit
% 0.34/0.62 thf(fact_4551_type__copy__wit,axiom,
% 0.34/0.62 ! [Rep2: product_unit > $o,Abs2: $o > product_unit,X: product_unit,S: $o > set_Product_unit,Y: $o] :
% 0.34/0.62 ( ( type_d6188575255521822967unit_o @ Rep2 @ Abs2 @ top_top_set_o )
% 0.34/0.62 => ( ( member_Product_unit @ X @ ( comp_o2086126948152544725t_unit @ S @ Rep2 @ ( Abs2 @ Y ) ) )
% 0.34/0.62 => ( member_Product_unit @ X @ ( S @ Y ) ) ) ) ).
% 0.34/0.62
% 0.34/0.62 % type_copy_wit
% 0.34/0.62 thf(fact_4552_type__copy__wit,axiom,
% 0.34/0.62 ! [Rep2: code_natural > nat,Abs2: nat > code_natural,X: nat,S: nat > set_nat,Y: nat] :
% 0.34/0.62 ( ( type_d4410041424927559462al_nat @ Rep2 @ Abs2 @ top_top_set_nat )
% 0.34/0.62 => ( ( member_nat @ X @ ( comp_n4787279007288201059atural @ S @ Rep2 @ ( Abs2 @ Y ) ) )
% 0.34/0.62 => ( member_nat @ X @ ( S @ Y ) ) ) ) ).
% 0.34/0.62
% 0.34/0.62 % type_copy_wit
% 0.34/0.62 thf(fact_4553_type__copy__wit,axiom,
% 0.34/0.62 ! [Rep2: code_natural > nat,Abs2: nat > code_natural,X: int,S: nat > set_int,Y: nat] :
% 0.34/0.62 ( ( type_d4410041424927559462al_nat @ Rep2 @ Abs2 @ top_top_set_nat )
% 0.34/0.62 => ( ( member_int @ X @ ( comp_n1572716621429000127atural @ S @ Rep2 @ ( Abs2 @ Y ) ) )
% 0.34/0.62 => ( member_int @ X @ ( S @ Y ) ) ) ) ).
% 0.34/0.62
% 0.34/0.62 % type_copy_wit
% 0.34/0.62 thf(fact_4554_prod__fun__imageE,axiom,
% 0.34/0.62 ! [C: product_prod_nat_nat,F: nat > nat,G: nat > nat,R3: set_Pr1261947904930325089at_nat] :
% 0.34/0.62 ( ( member8440522571783428010at_nat @ C @ ( image_5168914502847457605at_nat @ ( produc6977886695330630970at_nat @ F @ G ) @ R3 ) )
% 0.34/0.62 => ~ ! [X3: nat,Y2: nat] :
% 0.34/0.62 ( ( C
% 0.34/0.62 = ( product_Pair_nat_nat @ ( F @ X3 ) @ ( G @ Y2 ) ) )
% 0.34/0.62 => ~ ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X3 @ Y2 ) @ R3 ) ) ) ).
% 0.34/0.62
% 0.34/0.62 % prod_fun_imageE
% 0.34/0.62 thf(fact_4555_prod__fun__imageE,axiom,
% 0.34/0.62 ! [C: product_prod_nat_nat,F: int > nat,G: int > nat,R3: set_Pr958786334691620121nt_int] :
% 0.34/0.62 ( ( member8440522571783428010at_nat @ C @ ( image_5831868185956570253at_nat @ ( produc5667422475138734578nt_nat @ F @ G ) @ R3 ) )
% 0.34/0.62 => ~ ! [X3: int,Y2: int] :
% 0.34/0.62 ( ( C
% 0.34/0.62 = ( product_Pair_nat_nat @ ( F @ X3 ) @ ( G @ Y2 ) ) )
% 0.34/0.62 => ~ ( member5262025264175285858nt_int @ ( product_Pair_int_int @ X3 @ Y2 ) @ R3 ) ) ) ).
% 0.34/0.62
% 0.34/0.62 % prod_fun_imageE
% 0.34/0.62 thf(fact_4556_prod__fun__imageE,axiom,
% 0.34/0.62 ! [C: product_prod_int_int,F: nat > int,G: nat > int,R3: set_Pr1261947904930325089at_nat] :
% 0.34/0.62 ( ( member5262025264175285858nt_int @ C @ ( image_1990417195239315453nt_int @ ( produc7347049785058425330at_int @ F @ G ) @ R3 ) )
% 0.34/0.62 => ~ ! [X3: nat,Y2: nat] :
% 0.34/0.62 ( ( C
% 0.34/0.62 = ( product_Pair_int_int @ ( F @ X3 ) @ ( G @ Y2 ) ) )
% 0.34/0.62 => ~ ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X3 @ Y2 ) @ R3 ) ) ) ).
% 0.34/0.62
% 0.34/0.62 % prod_fun_imageE
% 0.34/0.62 thf(fact_4557_prod__fun__imageE,axiom,
% 0.34/0.62 ! [C: product_prod_int_int,F: int > int,G: int > int,R3: set_Pr958786334691620121nt_int] :
% 0.34/0.62 ( ( member5262025264175285858nt_int @ C @ ( image_2653370878348428101nt_int @ ( produc6036585564866528938nt_int @ F @ G ) @ R3 ) )
% 0.34/0.62 => ~ ! [X3: int,Y2: int] :
% 0.34/0.62 ( ( C
% 0.34/0.62 = ( product_Pair_int_int @ ( F @ X3 ) @ ( G @ Y2 ) ) )
% 0.34/0.62 => ~ ( member5262025264175285858nt_int @ ( product_Pair_int_int @ X3 @ Y2 ) @ R3 ) ) ) ).
% 0.34/0.62
% 0.34/0.62 % prod_fun_imageE
% 0.34/0.62 thf(fact_4558_prod__fun__imageE,axiom,
% 0.34/0.62 ! [C: produc3658429121746597890et_nat,F: nat > heap_e7401611519738050253t_unit,G: nat > set_nat,R3: set_Pr1261947904930325089at_nat] :
% 0.34/0.62 ( ( member6260224972018164377et_nat @ C @ ( image_9068448769188195518et_nat @ ( produc2064153442785501et_nat @ F @ G ) @ R3 ) )
% 0.34/0.62 => ~ ! [X3: nat,Y2: nat] :
% 0.34/0.62 ( ( C
% 0.34/0.62 = ( produc7507926704131184380et_nat @ ( F @ X3 ) @ ( G @ Y2 ) ) )
% 0.34/0.62 => ~ ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X3 @ Y2 ) @ R3 ) ) ) ).
% 0.34/0.62
% 0.34/0.62 % prod_fun_imageE
% 0.34/0.62 thf(fact_4559_prod__fun__imageE,axiom,
% 0.34/0.62 ! [C: produc3658429121746597890et_nat,F: int > heap_e7401611519738050253t_unit,G: int > set_nat,R3: set_Pr958786334691620121nt_int] :
% 0.34/0.62 ( ( member6260224972018164377et_nat @ C @ ( image_2127501072796355446et_nat @ ( produc3152090231653401565et_nat @ F @ G ) @ R3 ) )
% 0.34/0.62 => ~ ! [X3: int,Y2: int] :
% 0.34/0.62 ( ( C
% 0.34/0.62 = ( produc7507926704131184380et_nat @ ( F @ X3 ) @ ( G @ Y2 ) ) )
% 0.34/0.62 => ~ ( member5262025264175285858nt_int @ ( product_Pair_int_int @ X3 @ Y2 ) @ R3 ) ) ) ).
% 0.34/0.62
% 0.34/0.62 % prod_fun_imageE
% 0.34/0.62 thf(fact_4560_prod__fun__imageE,axiom,
% 0.34/0.62 ! [C: product_prod_nat_nat,F: heap_e7401611519738050253t_unit > nat,G: set_nat > nat,R3: set_Pr3948176798113811640et_nat] :
% 0.34/0.62 ( ( member8440522571783428010at_nat @ C @ ( image_7757775281999735736at_nat @ ( produc8285742784527409791at_nat @ F @ G ) @ R3 ) )
% 0.34/0.62 => ~ ! [X3: heap_e7401611519738050253t_unit,Y2: set_nat] :
% 0.34/0.62 ( ( C
% 0.34/0.62 = ( product_Pair_nat_nat @ ( F @ X3 ) @ ( G @ Y2 ) ) )
% 0.34/0.62 => ~ ( member6260224972018164377et_nat @ ( produc7507926704131184380et_nat @ X3 @ Y2 ) @ R3 ) ) ) ).
% 0.34/0.62
% 0.34/0.62 % prod_fun_imageE
% 0.34/0.62 thf(fact_4561_prod__fun__imageE,axiom,
% 0.34/0.62 ! [C: product_prod_int_int,F: heap_e7401611519738050253t_unit > int,G: set_nat > int,R3: set_Pr3948176798113811640et_nat] :
% 0.34/0.62 ( ( member5262025264175285858nt_int @ C @ ( image_4579277974391593584nt_int @ ( produc5973434381925408567at_int @ F @ G ) @ R3 ) )
% 0.34/0.62 => ~ ! [X3: heap_e7401611519738050253t_unit,Y2: set_nat] :
% 0.34/0.62 ( ( C
% 0.34/0.62 = ( product_Pair_int_int @ ( F @ X3 ) @ ( G @ Y2 ) ) )
% 0.34/0.62 => ~ ( member6260224972018164377et_nat @ ( produc7507926704131184380et_nat @ X3 @ Y2 ) @ R3 ) ) ) ).
% 0.34/0.62
% 0.34/0.62 % prod_fun_imageE
% 0.34/0.62 thf(fact_4562_prod__fun__imageE,axiom,
% 0.34/0.62 ! [C: produc3658429121746597890et_nat,F: heap_e7401611519738050253t_unit > heap_e7401611519738050253t_unit,G: set_nat > set_nat,R3: set_Pr3948176798113811640et_nat] :
% 0.34/0.62 ( ( member6260224972018164377et_nat @ C @ ( image_7426123528223433739et_nat @ ( produc8234977478398262212et_nat @ F @ G ) @ R3 ) )
% 0.34/0.62 => ~ ! [X3: heap_e7401611519738050253t_unit,Y2: set_nat] :
% 0.34/0.62 ( ( C
% 0.34/0.62 = ( produc7507926704131184380et_nat @ ( F @ X3 ) @ ( G @ Y2 ) ) )
% 0.34/0.62 => ~ ( member6260224972018164377et_nat @ ( produc7507926704131184380et_nat @ X3 @ Y2 ) @ R3 ) ) ) ).
% 0.34/0.62
% 0.34/0.62 % prod_fun_imageE
% 0.34/0.62 thf(fact_4563_prod__fun__imageE,axiom,
% 0.34/0.62 ! [C: product_prod_nat_nat,F: ( produc3658429121746597890et_nat > $o ) > nat,G: produc3658429121746597890et_nat > nat,R3: set_Pr3286484037609594932et_nat] :
% 0.34/0.62 ( ( member8440522571783428010at_nat @ C @ ( image_7073998564157519676at_nat @ ( produc2784369760062397435at_nat @ F @ G ) @ R3 ) )
% 0.34/0.62 => ~ ! [X3: produc3658429121746597890et_nat > $o,Y2: produc3658429121746597890et_nat] :
% 0.34/0.62 ( ( C
% 0.34/0.62 = ( product_Pair_nat_nat @ ( F @ X3 ) @ ( G @ Y2 ) ) )
% 0.34/0.62 => ~ ( member1996754912294343701et_nat @ ( produc5001842942810119800et_nat @ X3 @ Y2 ) @ R3 ) ) ) ).
% 0.34/0.62
% 0.34/0.62 % prod_fun_imageE
% 0.34/0.62 thf(fact_4564_inj__on__iff__surj,axiom,
% 0.34/0.62 ! [A: set_Code_integer,A8: set_Code_integer] :
% 0.34/0.62 ( ( A != bot_bo3990330152332043303nteger )
% 0.34/0.62 => ( ( ? [F4: code_integer > code_integer] :
% 0.34/0.62 ( ( inj_on1158064722259180589nteger @ F4 @ A )
% 0.34/0.62 & ( ord_le7084787975880047091nteger @ ( image_4470545334726330049nteger @ F4 @ A ) @ A8 ) ) )
% 0.34/0.62 = ( ? [G3: code_integer > code_integer] :
% 0.34/0.62 ( ( image_4470545334726330049nteger @ G3 @ A8 )
% 0.34/0.62 = A ) ) ) ) ).
% 0.34/0.62
% 0.34/0.62 % inj_on_iff_surj
% 0.34/0.62 thf(fact_4565_inj__on__iff__surj,axiom,
% 0.34/0.62 ! [A: set_nat,A8: set_set_nat] :
% 0.34/0.62 ( ( A != bot_bot_set_nat )
% 0.34/0.62 => ( ( ? [F4: nat > set_nat] :
% 0.34/0.62 ( ( inj_on_nat_set_nat @ F4 @ A )
% 0.34/0.62 & ( ord_le6893508408891458716et_nat @ ( image_nat_set_nat @ F4 @ A ) @ A8 ) ) )
% 0.34/0.62 = ( ? [G3: set_nat > nat] :
% 0.34/0.62 ( ( image_set_nat_nat @ G3 @ A8 )
% 0.34/0.62 = A ) ) ) ) ).
% 0.34/0.62
% 0.34/0.62 % inj_on_iff_surj
% 0.34/0.62 thf(fact_4566_inj__on__iff__surj,axiom,
% 0.34/0.62 ! [A: set_nat,A8: set_Code_integer] :
% 0.34/0.62 ( ( A != bot_bot_set_nat )
% 0.34/0.62 => ( ( ? [F4: nat > code_integer] :
% 0.34/0.62 ( ( inj_on2245834603434889120nteger @ F4 @ A )
% 0.34/0.62 & ( ord_le7084787975880047091nteger @ ( image_1215581382706833972nteger @ F4 @ A ) @ A8 ) ) )
% 0.34/0.62 = ( ? [G3: code_integer > nat] :
% 0.34/0.62 ( ( image_951025933927791156er_nat @ G3 @ A8 )
% 0.34/0.62 = A ) ) ) ) ).
% 0.34/0.62
% 0.34/0.62 % inj_on_iff_surj
% 0.34/0.62 thf(fact_4567_inj__on__iff__surj,axiom,
% 0.34/0.62 ! [A: set_set_nat,A8: set_nat] :
% 0.34/0.62 ( ( A != bot_bot_set_set_nat )
% 0.34/0.62 => ( ( ? [F4: set_nat > nat] :
% 0.34/0.62 ( ( inj_on_set_nat_nat @ F4 @ A )
% 0.34/0.62 & ( ord_less_eq_set_nat @ ( image_set_nat_nat @ F4 @ A ) @ A8 ) ) )
% 0.34/0.62 = ( ? [G3: nat > set_nat] :
% 0.34/0.62 ( ( image_nat_set_nat @ G3 @ A8 )
% 0.34/0.62 = A ) ) ) ) ).
% 0.34/0.62
% 0.34/0.62 % inj_on_iff_surj
% 0.34/0.62 thf(fact_4568_inj__on__iff__surj,axiom,
% 0.34/0.62 ! [A: set_Code_integer,A8: set_nat] :
% 0.34/0.62 ( ( A != bot_bo3990330152332043303nteger )
% 0.34/0.62 => ( ( ? [F4: code_integer > nat] :
% 0.34/0.62 ( ( inj_on1981279154655846304er_nat @ F4 @ A )
% 0.34/0.62 & ( ord_less_eq_set_nat @ ( image_951025933927791156er_nat @ F4 @ A ) @ A8 ) ) )
% 0.34/0.62 = ( ? [G3: nat > code_integer] :
% 0.34/0.62 ( ( image_1215581382706833972nteger @ G3 @ A8 )
% 0.34/0.62 = A ) ) ) ) ).
% 0.34/0.62
% 0.34/0.62 % inj_on_iff_surj
% 0.34/0.62 thf(fact_4569_inj__on__iff__surj,axiom,
% 0.34/0.62 ! [A: set_Product_unit,A8: set_nat] :
% 0.34/0.62 ( ( A != bot_bo3957492148770167129t_unit )
% 0.34/0.62 => ( ( ? [F4: product_unit > nat] :
% 0.34/0.62 ( ( inj_on8430439091780834860it_nat @ F4 @ A )
% 0.34/0.62 & ( ord_less_eq_set_nat @ ( image_875570014554754200it_nat @ F4 @ A ) @ A8 ) ) )
% 0.34/0.62 = ( ? [G3: nat > product_unit] :
% 0.34/0.62 ( ( image_8730104196221521654t_unit @ G3 @ A8 )
% 0.34/0.62 = A ) ) ) ) ).
% 0.34/0.62
% 0.34/0.62 % inj_on_iff_surj
% 0.34/0.62 thf(fact_4570_inj__on__iff__surj,axiom,
% 0.34/0.62 ! [A: set_o,A8: set_nat] :
% 0.34/0.62 ( ( A != bot_bot_set_o )
% 0.34/0.62 => ( ( ? [F4: $o > nat] :
% 0.34/0.62 ( ( inj_on_o_nat @ F4 @ A )
% 0.34/0.62 & ( ord_less_eq_set_nat @ ( image_o_nat2 @ F4 @ A ) @ A8 ) ) )
% 0.34/0.62 = ( ? [G3: nat > $o] :
% 0.34/0.62 ( ( image_nat_o2 @ G3 @ A8 )
% 0.34/0.62 = A ) ) ) ) ).
% 0.34/0.62
% 0.34/0.62 % inj_on_iff_surj
% 0.34/0.62 thf(fact_4571_inj__on__iff__surj,axiom,
% 0.34/0.62 ! [A: set_nat,A8: set_nat] :
% 0.34/0.62 ( ( A != bot_bot_set_nat )
% 0.34/0.62 => ( ( ? [F4: nat > nat] :
% 0.34/0.62 ( ( inj_on_nat_nat @ F4 @ A )
% 0.34/0.62 & ( ord_less_eq_set_nat @ ( image_nat_nat2 @ F4 @ A ) @ A8 ) ) )
% 0.34/0.62 = ( ? [G3: nat > nat] :
% 0.34/0.62 ( ( image_nat_nat2 @ G3 @ A8 )
% 0.34/0.62 = A ) ) ) ) ).
% 0.34/0.62
% 0.34/0.62 % inj_on_iff_surj
% 0.34/0.62 thf(fact_4572_inj__on__iff__surj,axiom,
% 0.34/0.62 ! [A: set_int,A8: set_nat] :
% 0.34/0.62 ( ( A != bot_bot_set_int )
% 0.34/0.62 => ( ( ? [F4: int > nat] :
% 0.34/0.62 ( ( inj_on_int_nat @ F4 @ A )
% 0.34/0.62 & ( ord_less_eq_set_nat @ ( image_int_nat2 @ F4 @ A ) @ A8 ) ) )
% 0.34/0.62 = ( ? [G3: nat > int] :
% 0.34/0.62 ( ( image_nat_int2 @ G3 @ A8 )
% 0.34/0.62 = A ) ) ) ) ).
% 0.34/0.62
% 0.34/0.62 % inj_on_iff_surj
% 0.34/0.62 thf(fact_4573_inj__on__iff__surj,axiom,
% 0.34/0.62 ! [A: set_Pr4532377907799695533_nat_o,A8: set_nat] :
% 0.34/0.62 ( ( A != bot_bo7824918357723582233_nat_o )
% 0.34/0.62 => ( ( ? [F4: ( produc3658429121746597890et_nat > $o ) > nat] :
% 0.34/0.62 ( ( inj_on5271586719803267564_o_nat @ F4 @ A )
% 0.34/0.62 & ( ord_less_eq_set_nat @ ( image_2174381888560928856_o_nat @ F4 @ A ) @ A8 ) ) )
% 0.34/0.62 = ( ? [G3: nat > produc3658429121746597890et_nat > $o] :
% 0.34/0.62 ( ( image_4682239675366380726_nat_o @ G3 @ A8 )
% 0.34/0.62 = A ) ) ) ) ).
% 0.34/0.62
% 0.34/0.62 % inj_on_iff_surj
% 0.34/0.62 thf(fact_4574_Sup__insert__finite,axiom,
% 0.34/0.62 ! [S: set_int,X: int] :
% 0.34/0.62 ( ( finite_finite_int @ S )
% 0.34/0.62 => ( ( ( S = bot_bot_set_int )
% 0.34/0.62 => ( ( complete_Sup_Sup_int @ ( insert_int @ X @ S ) )
% 0.34/0.62 = X ) )
% 0.34/0.62 & ( ( S != bot_bot_set_int )
% 0.34/0.62 => ( ( complete_Sup_Sup_int @ ( insert_int @ X @ S ) )
% 0.34/0.62 = ( ord_max_int @ X @ ( complete_Sup_Sup_int @ S ) ) ) ) ) ) ).
% 0.34/0.62
% 0.34/0.62 % Sup_insert_finite
% 0.34/0.62 thf(fact_4575_Sup__insert__finite,axiom,
% 0.34/0.62 ! [S: set_nat,X: nat] :
% 0.34/0.62 ( ( finite_finite_nat @ S )
% 0.34/0.62 => ( ( ( S = bot_bot_set_nat )
% 0.34/0.62 => ( ( complete_Sup_Sup_nat @ ( insert_nat @ X @ S ) )
% 0.34/0.62 = X ) )
% 0.34/0.62 & ( ( S != bot_bot_set_nat )
% 0.34/0.62 => ( ( complete_Sup_Sup_nat @ ( insert_nat @ X @ S ) )
% 0.34/0.62 = ( ord_max_nat @ X @ ( complete_Sup_Sup_nat @ S ) ) ) ) ) ) ).
% 0.34/0.62
% 0.34/0.62 % Sup_insert_finite
% 0.34/0.62 thf(fact_4576_Sup__insert__finite,axiom,
% 0.34/0.62 ! [S: set_Product_unit,X: product_unit] :
% 0.34/0.62 ( ( finite4290736615968046902t_unit @ S )
% 0.34/0.62 => ( ( ( S = bot_bo3957492148770167129t_unit )
% 0.34/0.62 => ( ( comple4687483117567863418t_unit @ ( insert_Product_unit @ X @ S ) )
% 0.34/0.62 = X ) )
% 0.34/0.62 & ( ( S != bot_bo3957492148770167129t_unit )
% 0.34/0.62 => ( ( comple4687483117567863418t_unit @ ( insert_Product_unit @ X @ S ) )
% 0.34/0.62 = ( ord_max_Product_unit @ X @ ( comple4687483117567863418t_unit @ S ) ) ) ) ) ) ).
% 0.34/0.62
% 0.34/0.62 % Sup_insert_finite
% 0.34/0.62 thf(fact_4577_hom__Max__commute,axiom,
% 0.34/0.62 ! [H2: code_integer > code_integer,N: set_Code_integer] :
% 0.34/0.62 ( ! [X3: code_integer,Y2: code_integer] :
% 0.34/0.62 ( ( H2 @ ( ord_max_Code_integer @ X3 @ Y2 ) )
% 0.34/0.62 = ( ord_max_Code_integer @ ( H2 @ X3 ) @ ( H2 @ Y2 ) ) )
% 0.34/0.62 => ( ( finite6017078050557962740nteger @ N )
% 0.34/0.62 => ( ( N != bot_bo3990330152332043303nteger )
% 0.34/0.62 => ( ( H2 @ ( lattic4901227151466704046nteger @ N ) )
% 0.34/0.62 = ( lattic4901227151466704046nteger @ ( image_4470545334726330049nteger @ H2 @ N ) ) ) ) ) ) ).
% 0.34/0.62
% 0.34/0.62 % hom_Max_commute
% 0.34/0.62 thf(fact_4578_hom__Max__commute,axiom,
% 0.34/0.62 ! [H2: product_unit > product_unit,N: set_Product_unit] :
% 0.34/0.62 ( ! [X3: product_unit,Y2: product_unit] :
% 0.34/0.62 ( ( H2 @ ( ord_max_Product_unit @ X3 @ Y2 ) )
% 0.34/0.62 = ( ord_max_Product_unit @ ( H2 @ X3 ) @ ( H2 @ Y2 ) ) )
% 0.34/0.62 => ( ( finite4290736615968046902t_unit @ N )
% 0.34/0.62 => ( ( N != bot_bo3957492148770167129t_unit )
% 0.34/0.62 => ( ( H2 @ ( lattic1210392890313158908t_unit @ N ) )
% 0.34/0.62 = ( lattic1210392890313158908t_unit @ ( image_405062704495631173t_unit @ H2 @ N ) ) ) ) ) ) ).
% 0.34/0.62
% 0.34/0.62 % hom_Max_commute
% 0.34/0.62 thf(fact_4579_hom__Max__commute,axiom,
% 0.34/0.62 ! [H2: $o > $o,N: set_o] :
% 0.34/0.62 ( ! [X3: $o,Y2: $o] :
% 0.34/0.62 ( ( H2 @ ( ord_max_o @ X3 @ Y2 ) )
% 0.34/0.62 = ( ord_max_o @ ( H2 @ X3 ) @ ( H2 @ Y2 ) ) )
% 0.34/0.62 => ( ( finite_finite_o @ N )
% 0.34/0.62 => ( ( N != bot_bot_set_o )
% 0.34/0.62 => ( ( H2 @ ( lattic1921953407002678535_Max_o @ N ) )
% 0.34/0.62 = ( lattic1921953407002678535_Max_o @ ( image_o_o2 @ H2 @ N ) ) ) ) ) ) ).
% 0.34/0.62
% 0.34/0.62 % hom_Max_commute
% 0.34/0.62 thf(fact_4580_hom__Max__commute,axiom,
% 0.34/0.62 ! [H2: int > int,N: set_int] :
% 0.34/0.62 ( ! [X3: int,Y2: int] :
% 0.34/0.62 ( ( H2 @ ( ord_max_int @ X3 @ Y2 ) )
% 0.34/0.62 = ( ord_max_int @ ( H2 @ X3 ) @ ( H2 @ Y2 ) ) )
% 0.34/0.62 => ( ( finite_finite_int @ N )
% 0.34/0.62 => ( ( N != bot_bot_set_int )
% 0.34/0.62 => ( ( H2 @ ( lattic8263393255366662781ax_int @ N ) )
% 0.34/0.62 = ( lattic8263393255366662781ax_int @ ( image_int_int2 @ H2 @ N ) ) ) ) ) ) ).
% 0.34/0.62
% 0.34/0.62 % hom_Max_commute
% 0.34/0.62 thf(fact_4581_hom__Max__commute,axiom,
% 0.34/0.62 ! [H2: nat > nat,N: set_nat] :
% 0.34/0.62 ( ! [X3: nat,Y2: nat] :
% 0.34/0.62 ( ( H2 @ ( ord_max_nat @ X3 @ Y2 ) )
% 0.34/0.62 = ( ord_max_nat @ ( H2 @ X3 ) @ ( H2 @ Y2 ) ) )
% 0.34/0.62 => ( ( finite_finite_nat @ N )
% 0.34/0.62 => ( ( N != bot_bot_set_nat )
% 0.34/0.62 => ( ( H2 @ ( lattic8265883725875713057ax_nat @ N ) )
% 0.34/0.62 = ( lattic8265883725875713057ax_nat @ ( image_nat_nat2 @ H2 @ N ) ) ) ) ) ) ).
% 0.34/0.62
% 0.34/0.62 % hom_Max_commute
% 0.34/0.62 thf(fact_4582_Max_Osubset,axiom,
% 0.34/0.62 ! [A: set_Code_integer,B4: set_Code_integer] :
% 0.34/0.62 ( ( finite6017078050557962740nteger @ A )
% 0.34/0.62 => ( ( B4 != bot_bo3990330152332043303nteger )
% 0.34/0.62 => ( ( ord_le7084787975880047091nteger @ B4 @ A )
% 0.34/0.62 => ( ( ord_max_Code_integer @ ( lattic4901227151466704046nteger @ B4 ) @ ( lattic4901227151466704046nteger @ A ) )
% 0.34/0.62 = ( lattic4901227151466704046nteger @ A ) ) ) ) ) ).
% 0.34/0.62
% 0.34/0.62 % Max.subset
% 0.34/0.62 thf(fact_4583_Max_Osubset,axiom,
% 0.34/0.62 ! [A: set_Product_unit,B4: set_Product_unit] :
% 0.34/0.62 ( ( finite4290736615968046902t_unit @ A )
% 0.34/0.62 => ( ( B4 != bot_bo3957492148770167129t_unit )
% 0.34/0.62 => ( ( ord_le3507040750410214029t_unit @ B4 @ A )
% 0.34/0.62 => ( ( ord_max_Product_unit @ ( lattic1210392890313158908t_unit @ B4 ) @ ( lattic1210392890313158908t_unit @ A ) )
% 0.34/0.62 = ( lattic1210392890313158908t_unit @ A ) ) ) ) ) ).
% 0.34/0.62
% 0.34/0.62 % Max.subset
% 0.34/0.62 thf(fact_4584_Max_Osubset,axiom,
% 0.34/0.62 ! [A: set_o,B4: set_o] :
% 0.34/0.62 ( ( finite_finite_o @ A )
% 0.34/0.62 => ( ( B4 != bot_bot_set_o )
% 0.34/0.62 => ( ( ord_less_eq_set_o @ B4 @ A )
% 0.34/0.62 => ( ( ord_max_o @ ( lattic1921953407002678535_Max_o @ B4 ) @ ( lattic1921953407002678535_Max_o @ A ) )
% 0.34/0.62 = ( lattic1921953407002678535_Max_o @ A ) ) ) ) ) ).
% 0.34/0.62
% 0.34/0.62 % Max.subset
% 0.34/0.62 thf(fact_4585_Max_Osubset,axiom,
% 0.34/0.62 ! [A: set_int,B4: set_int] :
% 0.34/0.62 ( ( finite_finite_int @ A )
% 0.34/0.62 => ( ( B4 != bot_bot_set_int )
% 0.34/0.62 => ( ( ord_less_eq_set_int @ B4 @ A )
% 0.34/0.62 => ( ( ord_max_int @ ( lattic8263393255366662781ax_int @ B4 ) @ ( lattic8263393255366662781ax_int @ A ) )
% 0.34/0.62 = ( lattic8263393255366662781ax_int @ A ) ) ) ) ) ).
% 0.34/0.62
% 0.34/0.62 % Max.subset
% 0.34/0.62 thf(fact_4586_Max_Osubset,axiom,
% 0.34/0.62 ! [A: set_nat,B4: set_nat] :
% 0.34/0.62 ( ( finite_finite_nat @ A )
% 0.34/0.62 => ( ( B4 != bot_bot_set_nat )
% 0.34/0.62 => ( ( ord_less_eq_set_nat @ B4 @ A )
% 0.34/0.62 => ( ( ord_max_nat @ ( lattic8265883725875713057ax_nat @ B4 ) @ ( lattic8265883725875713057ax_nat @ A ) )
% 0.34/0.62 = ( lattic8265883725875713057ax_nat @ A ) ) ) ) ) ).
% 0.34/0.62
% 0.34/0.62 % Max.subset
% 0.34/0.62 thf(fact_4587_Max_Oinsert__not__elem,axiom,
% 0.34/0.62 ! [A: set_Code_integer,X: code_integer] :
% 0.34/0.62 ( ( finite6017078050557962740nteger @ A )
% 0.34/0.62 => ( ~ ( member_Code_integer @ X @ A )
% 0.34/0.62 => ( ( A != bot_bo3990330152332043303nteger )
% 0.34/0.62 => ( ( lattic4901227151466704046nteger @ ( insert_Code_integer @ X @ A ) )
% 0.34/0.62 = ( ord_max_Code_integer @ X @ ( lattic4901227151466704046nteger @ A ) ) ) ) ) ) ).
% 0.34/0.62
% 0.34/0.62 % Max.insert_not_elem
% 0.34/0.62 thf(fact_4588_Max_Oinsert__not__elem,axiom,
% 0.34/0.62 ! [A: set_Product_unit,X: product_unit] :
% 0.34/0.62 ( ( finite4290736615968046902t_unit @ A )
% 0.34/0.62 => ( ~ ( member_Product_unit @ X @ A )
% 0.34/0.62 => ( ( A != bot_bo3957492148770167129t_unit )
% 0.34/0.62 => ( ( lattic1210392890313158908t_unit @ ( insert_Product_unit @ X @ A ) )
% 0.34/0.62 = ( ord_max_Product_unit @ X @ ( lattic1210392890313158908t_unit @ A ) ) ) ) ) ) ).
% 0.34/0.62
% 0.34/0.62 % Max.insert_not_elem
% 0.34/0.62 thf(fact_4589_Max_Oinsert__not__elem,axiom,
% 0.34/0.62 ! [A: set_o,X: $o] :
% 0.34/0.62 ( ( finite_finite_o @ A )
% 0.34/0.62 => ( ~ ( member_o @ X @ A )
% 0.34/0.62 => ( ( A != bot_bot_set_o )
% 0.34/0.62 => ( ( lattic1921953407002678535_Max_o @ ( insert_o @ X @ A ) )
% 0.34/0.62 = ( ord_max_o @ X @ ( lattic1921953407002678535_Max_o @ A ) ) ) ) ) ) ).
% 0.34/0.62
% 0.34/0.62 % Max.insert_not_elem
% 0.34/0.62 thf(fact_4590_Max_Oinsert__not__elem,axiom,
% 0.34/0.62 ! [A: set_int,X: int] :
% 0.34/0.62 ( ( finite_finite_int @ A )
% 0.34/0.62 => ( ~ ( member_int @ X @ A )
% 0.34/0.62 => ( ( A != bot_bot_set_int )
% 0.34/0.62 => ( ( lattic8263393255366662781ax_int @ ( insert_int @ X @ A ) )
% 0.34/0.62 = ( ord_max_int @ X @ ( lattic8263393255366662781ax_int @ A ) ) ) ) ) ) ).
% 0.34/0.62
% 0.34/0.62 % Max.insert_not_elem
% 0.34/0.62 thf(fact_4591_Max_Oinsert__not__elem,axiom,
% 0.34/0.62 ! [A: set_nat,X: nat] :
% 0.34/0.62 ( ( finite_finite_nat @ A )
% 0.34/0.62 => ( ~ ( member_nat @ X @ A )
% 0.34/0.62 => ( ( A != bot_bot_set_nat )
% 0.34/0.62 => ( ( lattic8265883725875713057ax_nat @ ( insert_nat @ X @ A ) )
% 0.34/0.62 = ( ord_max_nat @ X @ ( lattic8265883725875713057ax_nat @ A ) ) ) ) ) ) ).
% 0.34/0.62
% 0.34/0.62 % Max.insert_not_elem
% 0.34/0.62 thf(fact_4592_Max_Oclosed,axiom,
% 0.34/0.62 ! [A: set_Code_integer] :
% 0.34/0.62 ( ( finite6017078050557962740nteger @ A )
% 0.34/0.62 => ( ( A != bot_bo3990330152332043303nteger )
% 0.34/0.62 => ( ! [X3: code_integer,Y2: code_integer] : ( member_Code_integer @ ( ord_max_Code_integer @ X3 @ Y2 ) @ ( insert_Code_integer @ X3 @ ( insert_Code_integer @ Y2 @ bot_bo3990330152332043303nteger ) ) )
% 0.34/0.62 => ( member_Code_integer @ ( lattic4901227151466704046nteger @ A ) @ A ) ) ) ) ).
% 0.34/0.62
% 0.34/0.62 % Max.closed
% 0.34/0.62 thf(fact_4593_Max_Oclosed,axiom,
% 0.34/0.62 ! [A: set_Product_unit] :
% 0.34/0.62 ( ( finite4290736615968046902t_unit @ A )
% 0.34/0.62 => ( ( A != bot_bo3957492148770167129t_unit )
% 0.34/0.62 => ( ! [X3: product_unit,Y2: product_unit] : ( member_Product_unit @ ( ord_max_Product_unit @ X3 @ Y2 ) @ ( insert_Product_unit @ X3 @ ( insert_Product_unit @ Y2 @ bot_bo3957492148770167129t_unit ) ) )
% 0.34/0.62 => ( member_Product_unit @ ( lattic1210392890313158908t_unit @ A ) @ A ) ) ) ) ).
% 0.34/0.62
% 0.34/0.62 % Max.closed
% 0.34/0.62 thf(fact_4594_Max_Oclosed,axiom,
% 0.34/0.62 ! [A: set_o] :
% 0.34/0.62 ( ( finite_finite_o @ A )
% 0.34/0.62 => ( ( A != bot_bot_set_o )
% 0.34/0.62 => ( ! [X3: $o,Y2: $o] : ( member_o @ ( ord_max_o @ X3 @ Y2 ) @ ( insert_o @ X3 @ ( insert_o @ Y2 @ bot_bot_set_o ) ) )
% 0.34/0.62 => ( member_o @ ( lattic1921953407002678535_Max_o @ A ) @ A ) ) ) ) ).
% 0.34/0.62
% 0.34/0.62 % Max.closed
% 0.34/0.62 thf(fact_4595_Max_Oclosed,axiom,
% 0.34/0.62 ! [A: set_int] :
% 0.34/0.62 ( ( finite_finite_int @ A )
% 0.34/0.62 => ( ( A != bot_bot_set_int )
% 0.34/0.62 => ( ! [X3: int,Y2: int] : ( member_int @ ( ord_max_int @ X3 @ Y2 ) @ ( insert_int @ X3 @ ( insert_int @ Y2 @ bot_bot_set_int ) ) )
% 0.34/0.62 => ( member_int @ ( lattic8263393255366662781ax_int @ A ) @ A ) ) ) ) ).
% 0.34/0.62
% 0.34/0.62 % Max.closed
% 0.34/0.62 thf(fact_4596_Max_Oclosed,axiom,
% 0.34/0.62 ! [A: set_nat] :
% 0.34/0.62 ( ( finite_finite_nat @ A )
% 0.34/0.62 => ( ( A != bot_bot_set_nat )
% 0.34/0.62 => ( ! [X3: nat,Y2: nat] : ( member_nat @ ( ord_max_nat @ X3 @ Y2 ) @ ( insert_nat @ X3 @ ( insert_nat @ Y2 @ bot_bot_set_nat ) ) )
% 0.34/0.62 => ( member_nat @ ( lattic8265883725875713057ax_nat @ A ) @ A ) ) ) ) ).
% 0.34/0.62
% 0.34/0.62 % Max.closed
% 0.34/0.62 thf(fact_4597_Max_Ounion,axiom,
% 0.34/0.62 ! [A: set_Code_integer,B4: set_Code_integer] :
% 0.34/0.62 ( ( finite6017078050557962740nteger @ A )
% 0.34/0.62 => ( ( A != bot_bo3990330152332043303nteger )
% 0.34/0.62 => ( ( finite6017078050557962740nteger @ B4 )
% 0.34/0.62 => ( ( B4 != bot_bo3990330152332043303nteger )
% 0.34/0.62 => ( ( lattic4901227151466704046nteger @ ( sup_su848401254843788991nteger @ A @ B4 ) )
% 0.34/0.62 = ( ord_max_Code_integer @ ( lattic4901227151466704046nteger @ A ) @ ( lattic4901227151466704046nteger @ B4 ) ) ) ) ) ) ) ).
% 0.34/0.62
% 0.34/0.62 % Max.union
% 0.34/0.62 thf(fact_4598_Max_Ounion,axiom,
% 0.34/0.62 ! [A: set_Product_unit,B4: set_Product_unit] :
% 0.34/0.62 ( ( finite4290736615968046902t_unit @ A )
% 0.34/0.62 => ( ( A != bot_bo3957492148770167129t_unit )
% 0.34/0.62 => ( ( finite4290736615968046902t_unit @ B4 )
% 0.34/0.62 => ( ( B4 != bot_bo3957492148770167129t_unit )
% 0.34/0.62 => ( ( lattic1210392890313158908t_unit @ ( sup_su793286257634532545t_unit @ A @ B4 ) )
% 0.34/0.62 = ( ord_max_Product_unit @ ( lattic1210392890313158908t_unit @ A ) @ ( lattic1210392890313158908t_unit @ B4 ) ) ) ) ) ) ) ).
% 0.34/0.62
% 0.34/0.62 % Max.union
% 0.34/0.62 thf(fact_4599_Max_Ounion,axiom,
% 0.34/0.62 ! [A: set_o,B4: set_o] :
% 0.34/0.62 ( ( finite_finite_o @ A )
% 0.34/0.62 => ( ( A != bot_bot_set_o )
% 0.34/0.62 => ( ( finite_finite_o @ B4 )
% 0.34/0.62 => ( ( B4 != bot_bot_set_o )
% 0.34/0.62 => ( ( lattic1921953407002678535_Max_o @ ( sup_sup_set_o @ A @ B4 ) )
% 0.34/0.62 = ( ord_max_o @ ( lattic1921953407002678535_Max_o @ A ) @ ( lattic1921953407002678535_Max_o @ B4 ) ) ) ) ) ) ) ).
% 0.34/0.62
% 0.34/0.62 % Max.union
% 0.34/0.62 thf(fact_4600_Max_Ounion,axiom,
% 0.34/0.62 ! [A: set_int,B4: set_int] :
% 0.34/0.62 ( ( finite_finite_int @ A )
% 0.34/0.62 => ( ( A != bot_bot_set_int )
% 0.34/0.62 => ( ( finite_finite_int @ B4 )
% 0.34/0.62 => ( ( B4 != bot_bot_set_int )
% 0.34/0.62 => ( ( lattic8263393255366662781ax_int @ ( sup_sup_set_int @ A @ B4 ) )
% 0.34/0.62 = ( ord_max_int @ ( lattic8263393255366662781ax_int @ A ) @ ( lattic8263393255366662781ax_int @ B4 ) ) ) ) ) ) ) ).
% 0.34/0.62
% 0.34/0.62 % Max.union
% 0.34/0.62 thf(fact_4601_Max_Ounion,axiom,
% 0.34/0.62 ! [A: set_nat,B4: set_nat] :
% 0.34/0.62 ( ( finite_finite_nat @ A )
% 0.34/0.62 => ( ( A != bot_bot_set_nat )
% 0.34/0.62 => ( ( finite_finite_nat @ B4 )
% 0.34/0.62 => ( ( B4 != bot_bot_set_nat )
% 0.34/0.62 => ( ( lattic8265883725875713057ax_nat @ ( sup_sup_set_nat @ A @ B4 ) )
% 0.34/0.62 = ( ord_max_nat @ ( lattic8265883725875713057ax_nat @ A ) @ ( lattic8265883725875713057ax_nat @ B4 ) ) ) ) ) ) ) ).
% 0.34/0.62
% 0.34/0.62 % Max.union
% 0.34/0.62 thf(fact_4602_Max_Oeq__fold,axiom,
% 0.34/0.62 ! [A: set_o,X: $o] :
% 0.34/0.62 ( ( finite_finite_o @ A )
% 0.34/0.62 => ( ( lattic1921953407002678535_Max_o @ ( insert_o @ X @ A ) )
% 0.34/0.62 = ( finite_fold_o_o @ ord_max_o @ X @ A ) ) ) ).
% 0.34/0.62
% 0.34/0.62 % Max.eq_fold
% 0.34/0.62 thf(fact_4603_Max_Oeq__fold,axiom,
% 0.34/0.62 ! [A: set_Product_unit,X: product_unit] :
% 0.34/0.62 ( ( finite4290736615968046902t_unit @ A )
% 0.34/0.62 => ( ( lattic1210392890313158908t_unit @ ( insert_Product_unit @ X @ A ) )
% 0.34/0.62 = ( finite8834431295148555784t_unit @ ord_max_Product_unit @ X @ A ) ) ) ).
% 0.34/0.62
% 0.34/0.62 % Max.eq_fold
% 0.34/0.62 thf(fact_4604_Max_Oeq__fold,axiom,
% 0.34/0.62 ! [A: set_Code_integer,X: code_integer] :
% 0.34/0.62 ( ( finite6017078050557962740nteger @ A )
% 0.34/0.62 => ( ( lattic4901227151466704046nteger @ ( insert_Code_integer @ X @ A ) )
% 0.34/0.62 = ( finite6454954857057555716nteger @ ord_max_Code_integer @ X @ A ) ) ) ).
% 0.34/0.62
% 0.34/0.62 % Max.eq_fold
% 0.34/0.62 thf(fact_4605_Max_Oeq__fold,axiom,
% 0.34/0.62 ! [A: set_int,X: int] :
% 0.34/0.62 ( ( finite_finite_int @ A )
% 0.34/0.62 => ( ( lattic8263393255366662781ax_int @ ( insert_int @ X @ A ) )
% 0.34/0.62 = ( finite_fold_int_int @ ord_max_int @ X @ A ) ) ) ).
% 0.34/0.62
% 0.34/0.62 % Max.eq_fold
% 0.34/0.62 thf(fact_4606_Max_Oeq__fold,axiom,
% 0.34/0.62 ! [A: set_nat,X: nat] :
% 0.34/0.62 ( ( finite_finite_nat @ A )
% 0.34/0.62 => ( ( lattic8265883725875713057ax_nat @ ( insert_nat @ X @ A ) )
% 0.34/0.62 = ( finite_fold_nat_nat @ ord_max_nat @ X @ A ) ) ) ).
% 0.34/0.62
% 0.34/0.62 % Max.eq_fold
% 0.34/0.62 thf(fact_4607_finite__range__prod,axiom,
% 0.34/0.62 ! [F: code_integer > produc2591615631378489102nteger] :
% 0.34/0.62 ( ( finite_finite_nat @ ( image_951025933927791156er_nat @ ( comp_P5725896914063526938nteger @ produc8897218493789310642nteger @ F ) @ top_to4645266643341252675nteger ) )
% 0.34/0.62 => ( ( finite6017078050557962740nteger @ ( image_4470545334726330049nteger @ ( comp_P6122662217367900327nteger @ produc1781891006948753140nteger @ F ) @ top_to4645266643341252675nteger ) )
% 0.34/0.62 => ( finite1154042800025562295nteger @ ( image_3408592110290467754nteger @ F @ top_to4645266643341252675nteger ) ) ) ) ).
% 0.34/0.62
% 0.34/0.62 % finite_range_prod
% 0.34/0.62 thf(fact_4608_finite__range__prod,axiom,
% 0.34/0.62 ! [F: code_integer > produc6433947425880249742er_nat] :
% 0.34/0.62 ( ( finite6017078050557962740nteger @ ( image_4470545334726330049nteger @ ( comp_P7101464685127059239nteger @ produc8632663045010267826er_nat @ F ) @ top_to4645266643341252675nteger ) )
% 0.34/0.62 => ( ( finite_finite_nat @ ( image_951025933927791156er_nat @ ( comp_P5964733648194011802nteger @ produc1517335558169710324er_nat @ F ) @ top_to4645266643341252675nteger ) )
% 0.34/0.62 => ( finite4996374594527322935er_nat @ ( image_7250923904792228394er_nat @ F @ top_to4645266643341252675nteger ) ) ) ) ).
% 0.34/0.62
% 0.34/0.62 % finite_range_prod
% 0.34/0.62 thf(fact_4609_finite__range__prod,axiom,
% 0.34/0.62 ! [F: code_integer > produc8923325533196201883nteger] :
% 0.34/0.62 ( ( finite6017078050557962740nteger @ ( image_4470545334726330049nteger @ ( comp_P7111796090210262324nteger @ produc8508995932063986495nteger @ F ) @ top_to4645266643341252675nteger ) )
% 0.34/0.62 => ( ( finite6017078050557962740nteger @ ( image_4470545334726330049nteger @ ( comp_P7111796090210262324nteger @ produc6174133586879617921nteger @ F ) @ top_to4645266643341252675nteger ) )
% 0.34/0.62 => ( finite2154694888287295812nteger @ ( image_8049062792287394871nteger @ F @ top_to4645266643341252675nteger ) ) ) ) ).
% 0.34/0.62
% 0.34/0.62 % finite_range_prod
% 0.34/0.62 thf(fact_4610_finite__range__prod,axiom,
% 0.34/0.62 ! [F: product_unit > product_prod_nat_nat] :
% 0.34/0.62 ( ( finite_finite_nat @ ( image_875570014554754200it_nat @ ( comp_P2980510830910040157t_unit @ product_fst_nat_nat @ F ) @ top_to1996260823553986621t_unit ) )
% 0.34/0.62 => ( ( finite_finite_nat @ ( image_875570014554754200it_nat @ ( comp_P2980510830910040157t_unit @ product_snd_nat_nat @ F ) @ top_to1996260823553986621t_unit ) )
% 0.34/0.62 => ( finite6177210948735845034at_nat @ ( image_7281959405765017145at_nat @ F @ top_to1996260823553986621t_unit ) ) ) ) ).
% 0.34/0.62
% 0.34/0.62 % finite_range_prod
% 0.34/0.62 thf(fact_4611_finite__range__prod,axiom,
% 0.34/0.62 ! [F: product_unit > produc2591615631378489102nteger] :
% 0.34/0.62 ( ( finite_finite_nat @ ( image_875570014554754200it_nat @ ( comp_P7621350044865347728t_unit @ produc8897218493789310642nteger @ F ) @ top_to1996260823553986621t_unit ) )
% 0.34/0.62 => ( ( finite6017078050557962740nteger @ ( image_3815118110082124069nteger @ ( comp_P7320554561093743171t_unit @ produc1781891006948753140nteger @ F ) @ top_to1996260823553986621t_unit ) )
% 0.34/0.62 => ( finite1154042800025562295nteger @ ( image_4946983642536848454nteger @ F @ top_to1996260823553986621t_unit ) ) ) ) ).
% 0.34/0.62
% 0.34/0.62 % finite_range_prod
% 0.34/0.62 thf(fact_4612_finite__range__prod,axiom,
% 0.34/0.62 ! [F: product_unit > produc6433947425880249742er_nat] :
% 0.34/0.62 ( ( finite6017078050557962740nteger @ ( image_3815118110082124069nteger @ ( comp_P7581484221039074243t_unit @ produc8632663045010267826er_nat @ F ) @ top_to1996260823553986621t_unit ) )
% 0.34/0.62 => ( ( finite_finite_nat @ ( image_875570014554754200it_nat @ ( comp_P6821003776282389008t_unit @ produc1517335558169710324er_nat @ F ) @ top_to1996260823553986621t_unit ) )
% 0.34/0.62 => ( finite4996374594527322935er_nat @ ( image_8789315437038609094er_nat @ F @ top_to1996260823553986621t_unit ) ) ) ) ).
% 0.34/0.62
% 0.34/0.62 % finite_range_prod
% 0.34/0.62 thf(fact_4613_finite__range__prod,axiom,
% 0.34/0.62 ! [F: product_unit > produc8923325533196201883nteger] :
% 0.34/0.62 ( ( finite6017078050557962740nteger @ ( image_3815118110082124069nteger @ ( comp_P659088277599099894t_unit @ produc8508995932063986495nteger @ F ) @ top_to1996260823553986621t_unit ) )
% 0.34/0.62 => ( ( finite6017078050557962740nteger @ ( image_3815118110082124069nteger @ ( comp_P659088277599099894t_unit @ produc6174133586879617921nteger @ F ) @ top_to1996260823553986621t_unit ) )
% 0.34/0.62 => ( finite2154694888287295812nteger @ ( image_8089323762101378259nteger @ F @ top_to1996260823553986621t_unit ) ) ) ) ).
% 0.34/0.62
% 0.34/0.62 % finite_range_prod
% 0.34/0.62 thf(fact_4614_finite__range__prod,axiom,
% 0.34/0.62 ! [F: product_unit > product_prod_int_int] :
% 0.34/0.62 ( ( finite_finite_int @ ( image_873079544045703924it_int @ ( comp_P6796073005956588617t_unit @ product_fst_int_int @ F ) @ top_to1996260823553986621t_unit ) )
% 0.34/0.62 => ( ( finite_finite_int @ ( image_873079544045703924it_int @ ( comp_P6796073005956588617t_unit @ product_snd_int_int @ F ) @ top_to1996260823553986621t_unit ) )
% 0.34/0.62 => ( finite2998713641127702882nt_int @ ( image_4103462098156874993nt_int @ F @ top_to1996260823553986621t_unit ) ) ) ) ).
% 0.34/0.62
% 0.34/0.62 % finite_range_prod
% 0.34/0.62 thf(fact_4615_finite__range__prod,axiom,
% 0.34/0.62 ! [F: $o > product_prod_nat_nat] :
% 0.34/0.62 ( ( finite_finite_nat @ ( image_o_nat2 @ ( comp_P4394204175247225320_nat_o @ product_fst_nat_nat @ F ) @ top_top_set_o ) )
% 0.34/0.62 => ( ( finite_finite_nat @ ( image_o_nat2 @ ( comp_P4394204175247225320_nat_o @ product_snd_nat_nat @ F ) @ top_top_set_o ) )
% 0.34/0.62 => ( finite6177210948735845034at_nat @ ( image_3855930084881510382at_nat @ F @ top_top_set_o ) ) ) ) ).
% 0.34/0.62
% 0.34/0.62 % finite_range_prod
% 0.34/0.62 thf(fact_4616_finite__range__prod,axiom,
% 0.34/0.62 ! [F: $o > produc2591615631378489102nteger] :
% 0.34/0.62 ( ( finite_finite_nat @ ( image_o_nat2 @ ( comp_P7179991235626930843_nat_o @ produc8897218493789310642nteger @ F ) @ top_top_set_o ) )
% 0.34/0.62 => ( ( finite6017078050557962740nteger @ ( image_o_Code_integer @ ( comp_P7177471617286032590eger_o @ produc1781891006948753140nteger @ F ) @ top_top_set_o ) )
% 0.34/0.62 => ( finite1154042800025562295nteger @ ( image_8407096251609565691nteger @ F @ top_top_set_o ) ) ) ) ).
% 0.34/0.62
% 0.34/0.62 % finite_range_prod
% 0.34/0.62 thf(fact_4617_inj__apsnd,axiom,
% 0.34/0.62 ! [F: code_integer > code_integer] :
% 0.34/0.62 ( ( inj_on20428373856169009nteger @ ( produc6499014454317279255nteger @ F ) @ top_to2363841527343591755nteger )
% 0.34/0.62 = ( inj_on1158064722259180589nteger @ F @ top_to4645266643341252675nteger ) ) ).
% 0.34/0.62
% 0.34/0.62 % inj_apsnd
% 0.34/0.62 thf(fact_4618_total__inv__image,axiom,
% 0.34/0.62 ! [F: product_unit > product_unit,R2: set_Pr5094982260447487303t_unit] :
% 0.34/0.62 ( ( inj_on8151373323710067377t_unit @ F @ top_to1996260823553986621t_unit )
% 0.34/0.62 => ( ( total_5665700600833547665t_unit @ top_to1996260823553986621t_unit @ R2 )
% 0.34/0.62 => ( total_5665700600833547665t_unit @ top_to1996260823553986621t_unit @ ( inv_im8770536620514187154t_unit @ R2 @ F ) ) ) ) ).
% 0.34/0.62
% 0.34/0.62 % total_inv_image
% 0.34/0.62 thf(fact_4619_total__inv__image,axiom,
% 0.34/0.62 ! [F: product_unit > $o,R2: set_Product_prod_o_o] :
% 0.34/0.62 ( ( inj_on8854722734121536060unit_o @ F @ top_to1996260823553986621t_unit )
% 0.34/0.62 => ( ( total_on_o @ top_top_set_o @ R2 )
% 0.34/0.62 => ( total_5665700600833547665t_unit @ top_to1996260823553986621t_unit @ ( inv_im4602722609573551687t_unit @ R2 @ F ) ) ) ) ).
% 0.34/0.62
% 0.34/0.62 % total_inv_image
% 0.34/0.62 thf(fact_4620_total__inv__image,axiom,
% 0.34/0.62 ! [F: product_unit > nat,R2: set_Pr1261947904930325089at_nat] :
% 0.34/0.62 ( ( inj_on8430439091780834860it_nat @ F @ top_to1996260823553986621t_unit )
% 0.34/0.62 => ( ( total_on_nat @ top_top_set_nat @ R2 )
% 0.34/0.62 => ( total_5665700600833547665t_unit @ top_to1996260823553986621t_unit @ ( inv_im7624054837341743209t_unit @ R2 @ F ) ) ) ) ).
% 0.34/0.62
% 0.34/0.62 % total_inv_image
% 0.34/0.62 thf(fact_4621_total__inv__image,axiom,
% 0.34/0.62 ! [F: product_unit > int,R2: set_Pr958786334691620121nt_int] :
% 0.34/0.62 ( ( inj_on8427948621271784584it_int @ F @ top_to1996260823553986621t_unit )
% 0.34/0.62 => ( ( total_on_int @ top_top_set_int @ R2 )
% 0.34/0.62 => ( total_5665700600833547665t_unit @ top_to1996260823553986621t_unit @ ( inv_im4740476784543923469t_unit @ R2 @ F ) ) ) ) ).
% 0.34/0.62
% 0.34/0.62 % total_inv_image
% 0.34/0.62 thf(fact_4622_total__inv__image,axiom,
% 0.34/0.62 ! [F: $o > product_unit,R2: set_Pr5094982260447487303t_unit] :
% 0.34/0.62 ( ( inj_on4268752236855399398t_unit @ F @ top_top_set_o )
% 0.34/0.62 => ( ( total_5665700600833547665t_unit @ top_to1996260823553986621t_unit @ R2 )
% 0.34/0.62 => ( total_on_o @ top_top_set_o @ ( inv_im9188693106839688349unit_o @ R2 @ F ) ) ) ) ).
% 0.34/0.62
% 0.34/0.62 % total_inv_image
% 0.34/0.62 thf(fact_4623_total__inv__image,axiom,
% 0.34/0.62 ! [F: $o > $o,R2: set_Product_prod_o_o] :
% 0.34/0.62 ( ( inj_on_o_o @ F @ top_top_set_o )
% 0.34/0.62 => ( ( total_on_o @ top_top_set_o @ R2 )
% 0.34/0.62 => ( total_on_o @ top_top_set_o @ ( inv_image_o_o @ R2 @ F ) ) ) ) ).
% 0.34/0.62
% 0.34/0.62 % total_inv_image
% 0.34/0.62 thf(fact_4624_total__inv__image,axiom,
% 0.34/0.62 ! [F: $o > nat,R2: set_Pr1261947904930325089at_nat] :
% 0.34/0.62 ( ( inj_on_o_nat @ F @ top_top_set_o )
% 0.34/0.62 => ( ( total_on_nat @ top_top_set_nat @ R2 )
% 0.34/0.62 => ( total_on_o @ top_top_set_o @ ( inv_image_nat_o @ R2 @ F ) ) ) ) ).
% 0.34/0.62
% 0.34/0.62 % total_inv_image
% 0.34/0.62 thf(fact_4625_total__inv__image,axiom,
% 0.34/0.62 ! [F: $o > int,R2: set_Pr958786334691620121nt_int] :
% 0.34/0.62 ( ( inj_on_o_int @ F @ top_top_set_o )
% 0.34/0.62 => ( ( total_on_int @ top_top_set_int @ R2 )
% 0.34/0.62 => ( total_on_o @ top_top_set_o @ ( inv_image_int_o @ R2 @ F ) ) ) ) ).
% 0.34/0.62
% 0.34/0.62 % total_inv_image
% 0.34/0.62 thf(fact_4626_total__inv__image,axiom,
% 0.34/0.62 ! [F: nat > product_unit,R2: set_Pr5094982260447487303t_unit] :
% 0.34/0.62 ( ( inj_on7061601236592826506t_unit @ F @ top_top_set_nat )
% 0.34/0.62 => ( ( total_5665700600833547665t_unit @ top_to1996260823553986621t_unit @ R2 )
% 0.34/0.62 => ( total_on_nat @ top_top_set_nat @ ( inv_im8992892692529751563it_nat @ R2 @ F ) ) ) ) ).
% 0.34/0.62
% 0.34/0.62 % total_inv_image
% 0.34/0.62 thf(fact_4627_total__inv__image,axiom,
% 0.34/0.62 ! [F: nat > $o,R2: set_Product_prod_o_o] :
% 0.34/0.62 ( ( inj_on_nat_o @ F @ top_top_set_nat )
% 0.34/0.62 => ( ( total_on_o @ top_top_set_o @ R2 )
% 0.34/0.62 => ( total_on_nat @ top_top_set_nat @ ( inv_image_o_nat @ R2 @ F ) ) ) ) ).
% 0.34/0.62
% 0.34/0.62 % total_inv_image
% 0.34/0.62 thf(fact_4628_card__Diff1__less,axiom,
% 0.34/0.62 ! [A: set_Code_integer,X: code_integer] :
% 0.34/0.62 ( ( finite6017078050557962740nteger @ A )
% 0.34/0.62 => ( ( member_Code_integer @ X @ A )
% 0.34/0.62 => ( ord_less_nat @ ( finite4902975817058060853nteger @ ( minus_2355218937544613996nteger @ A @ ( insert_Code_integer @ X @ bot_bo3990330152332043303nteger ) ) ) @ ( finite4902975817058060853nteger @ A ) ) ) ) ).
% 0.34/0.62
% 0.34/0.62 % card_Diff1_less
% 0.34/0.62 thf(fact_4629_card__Diff1__less,axiom,
% 0.34/0.62 ! [A: set_Product_unit,X: product_unit] :
% 0.34/0.62 ( ( finite4290736615968046902t_unit @ A )
% 0.34/0.62 => ( ( member_Product_unit @ X @ A )
% 0.34/0.62 => ( ord_less_nat @ ( finite410649719033368117t_unit @ ( minus_6452836326544984404t_unit @ A @ ( insert_Product_unit @ X @ bot_bo3957492148770167129t_unit ) ) ) @ ( finite410649719033368117t_unit @ A ) ) ) ) ).
% 0.34/0.62
% 0.34/0.62 % card_Diff1_less
% 0.34/0.62 thf(fact_4630_card__Diff1__less,axiom,
% 0.34/0.62 ! [A: set_o,X: $o] :
% 0.34/0.62 ( ( finite_finite_o @ A )
% 0.34/0.62 => ( ( member_o @ X @ A )
% 0.34/0.62 => ( ord_less_nat @ ( finite_card_o @ ( minus_minus_set_o @ A @ ( insert_o @ X @ bot_bot_set_o ) ) ) @ ( finite_card_o @ A ) ) ) ) ).
% 0.34/0.62
% 0.34/0.62 % card_Diff1_less
% 0.34/0.62 thf(fact_4631_card__Diff1__less,axiom,
% 0.34/0.62 ! [A: set_int,X: int] :
% 0.34/0.62 ( ( finite_finite_int @ A )
% 0.34/0.62 => ( ( member_int @ X @ A )
% 0.34/0.62 => ( ord_less_nat @ ( finite_card_int @ ( minus_minus_set_int @ A @ ( insert_int @ X @ bot_bot_set_int ) ) ) @ ( finite_card_int @ A ) ) ) ) ).
% 0.34/0.62
% 0.34/0.62 % card_Diff1_less
% 0.34/0.62 thf(fact_4632_card__Diff1__less,axiom,
% 0.34/0.62 ! [A: set_Pr4532377907799695533_nat_o,X: produc3658429121746597890et_nat > $o] :
% 0.34/0.62 ( ( finite3252695134891459830_nat_o @ A )
% 0.34/0.62 => ( ( member6576561426505652726_nat_o @ X @ A )
% 0.34/0.62 => ( ord_less_nat @ ( finite1363419556375932405_nat_o @ ( minus_1801376950450012436_nat_o @ A @ ( insert5175938949040314269_nat_o @ X @ bot_bo7824918357723582233_nat_o ) ) ) @ ( finite1363419556375932405_nat_o @ A ) ) ) ) ).
% 0.34/0.62
% 0.34/0.62 % card_Diff1_less
% 0.34/0.62 thf(fact_4633_card__Diff1__less,axiom,
% 0.34/0.62 ! [A: set_nat,X: nat] :
% 0.34/0.62 ( ( finite_finite_nat @ A )
% 0.34/0.62 => ( ( member_nat @ X @ A )
% 0.34/0.62 => ( ord_less_nat @ ( finite_card_nat @ ( minus_minus_set_nat @ A @ ( insert_nat @ X @ bot_bot_set_nat ) ) ) @ ( finite_card_nat @ A ) ) ) ) ).
% 0.34/0.62
% 0.34/0.62 % card_Diff1_less
% 0.34/0.62 thf(fact_4634_card__Diff2__less,axiom,
% 0.34/0.62 ! [A: set_Code_integer,X: code_integer,Y: code_integer] :
% 0.34/0.62 ( ( finite6017078050557962740nteger @ A )
% 0.34/0.62 => ( ( member_Code_integer @ X @ A )
% 0.34/0.62 => ( ( member_Code_integer @ Y @ A )
% 0.34/0.62 => ( ord_less_nat @ ( finite4902975817058060853nteger @ ( minus_2355218937544613996nteger @ ( minus_2355218937544613996nteger @ A @ ( insert_Code_integer @ X @ bot_bo3990330152332043303nteger ) ) @ ( insert_Code_integer @ Y @ bot_bo3990330152332043303nteger ) ) ) @ ( finite4902975817058060853nteger @ A ) ) ) ) ) ).
% 0.34/0.62
% 0.34/0.62 % card_Diff2_less
% 0.34/0.62 thf(fact_4635_card__Diff2__less,axiom,
% 0.34/0.62 ! [A: set_Product_unit,X: product_unit,Y: product_unit] :
% 0.34/0.62 ( ( finite4290736615968046902t_unit @ A )
% 0.34/0.62 => ( ( member_Product_unit @ X @ A )
% 0.34/0.62 => ( ( member_Product_unit @ Y @ A )
% 0.34/0.62 => ( ord_less_nat @ ( finite410649719033368117t_unit @ ( minus_6452836326544984404t_unit @ ( minus_6452836326544984404t_unit @ A @ ( insert_Product_unit @ X @ bot_bo3957492148770167129t_unit ) ) @ ( insert_Product_unit @ Y @ bot_bo3957492148770167129t_unit ) ) ) @ ( finite410649719033368117t_unit @ A ) ) ) ) ) ).
% 0.34/0.62
% 0.34/0.62 % card_Diff2_less
% 0.34/0.62 thf(fact_4636_card__Diff2__less,axiom,
% 0.34/0.62 ! [A: set_o,X: $o,Y: $o] :
% 0.34/0.62 ( ( finite_finite_o @ A )
% 0.34/0.62 => ( ( member_o @ X @ A )
% 0.34/0.62 => ( ( member_o @ Y @ A )
% 0.34/0.62 => ( ord_less_nat @ ( finite_card_o @ ( minus_minus_set_o @ ( minus_minus_set_o @ A @ ( insert_o @ X @ bot_bot_set_o ) ) @ ( insert_o @ Y @ bot_bot_set_o ) ) ) @ ( finite_card_o @ A ) ) ) ) ) ).
% 0.34/0.62
% 0.34/0.62 % card_Diff2_less
% 0.34/0.62 thf(fact_4637_card__Diff2__less,axiom,
% 0.34/0.62 ! [A: set_int,X: int,Y: int] :
% 0.34/0.62 ( ( finite_finite_int @ A )
% 0.34/0.62 => ( ( member_int @ X @ A )
% 0.34/0.62 => ( ( member_int @ Y @ A )
% 0.34/0.62 => ( ord_less_nat @ ( finite_card_int @ ( minus_minus_set_int @ ( minus_minus_set_int @ A @ ( insert_int @ X @ bot_bot_set_int ) ) @ ( insert_int @ Y @ bot_bot_set_int ) ) ) @ ( finite_card_int @ A ) ) ) ) ) ).
% 0.34/0.62
% 0.34/0.62 % card_Diff2_less
% 0.34/0.62 thf(fact_4638_card__Diff2__less,axiom,
% 0.34/0.62 ! [A: set_Pr4532377907799695533_nat_o,X: produc3658429121746597890et_nat > $o,Y: produc3658429121746597890et_nat > $o] :
% 0.34/0.62 ( ( finite3252695134891459830_nat_o @ A )
% 0.34/0.62 => ( ( member6576561426505652726_nat_o @ X @ A )
% 0.34/0.62 => ( ( member6576561426505652726_nat_o @ Y @ A )
% 0.34/0.62 => ( ord_less_nat @ ( finite1363419556375932405_nat_o @ ( minus_1801376950450012436_nat_o @ ( minus_1801376950450012436_nat_o @ A @ ( insert5175938949040314269_nat_o @ X @ bot_bo7824918357723582233_nat_o ) ) @ ( insert5175938949040314269_nat_o @ Y @ bot_bo7824918357723582233_nat_o ) ) ) @ ( finite1363419556375932405_nat_o @ A ) ) ) ) ) ).
% 0.34/0.62
% 0.34/0.62 % card_Diff2_less
% 0.34/0.62 thf(fact_4639_card__Diff2__less,axiom,
% 0.34/0.62 ! [A: set_nat,X: nat,Y: nat] :
% 0.34/0.62 ( ( finite_finite_nat @ A )
% 0.34/0.62 => ( ( member_nat @ X @ A )
% 0.34/0.62 => ( ( member_nat @ Y @ A )
% 0.34/0.62 => ( ord_less_nat @ ( finite_card_nat @ ( minus_minus_set_nat @ ( minus_minus_set_nat @ A @ ( insert_nat @ X @ bot_bot_set_nat ) ) @ ( insert_nat @ Y @ bot_bot_set_nat ) ) ) @ ( finite_card_nat @ A ) ) ) ) ) ).
% 0.34/0.62
% 0.34/0.62 % card_Diff2_less
% 0.34/0.62 thf(fact_4640_card__Diff1__less__iff,axiom,
% 0.34/0.62 ! [A: set_Code_integer,X: code_integer] :
% 0.34/0.62 ( ( ord_less_nat @ ( finite4902975817058060853nteger @ ( minus_2355218937544613996nteger @ A @ ( insert_Code_integer @ X @ bot_bo3990330152332043303nteger ) ) ) @ ( finite4902975817058060853nteger @ A ) )
% 0.34/0.62 = ( ( finite6017078050557962740nteger @ A )
% 0.34/0.62 & ( member_Code_integer @ X @ A ) ) ) ).
% 0.34/0.62
% 0.34/0.62 % card_Diff1_less_iff
% 0.34/0.62 thf(fact_4641_card__Diff1__less__iff,axiom,
% 0.34/0.62 ! [A: set_Product_unit,X: product_unit] :
% 0.34/0.62 ( ( ord_less_nat @ ( finite410649719033368117t_unit @ ( minus_6452836326544984404t_unit @ A @ ( insert_Product_unit @ X @ bot_bo3957492148770167129t_unit ) ) ) @ ( finite410649719033368117t_unit @ A ) )
% 0.34/0.62 = ( ( finite4290736615968046902t_unit @ A )
% 0.34/0.62 & ( member_Product_unit @ X @ A ) ) ) ).
% 0.34/0.62
% 0.34/0.62 % card_Diff1_less_iff
% 0.34/0.62 thf(fact_4642_card__Diff1__less__iff,axiom,
% 0.34/0.62 ! [A: set_o,X: $o] :
% 0.34/0.62 ( ( ord_less_nat @ ( finite_card_o @ ( minus_minus_set_o @ A @ ( insert_o @ X @ bot_bot_set_o ) ) ) @ ( finite_card_o @ A ) )
% 0.34/0.62 = ( ( finite_finite_o @ A )
% 0.34/0.62 & ( member_o @ X @ A ) ) ) ).
% 0.34/0.62
% 0.34/0.62 % card_Diff1_less_iff
% 0.34/0.62 thf(fact_4643_card__Diff1__less__iff,axiom,
% 0.34/0.62 ! [A: set_int,X: int] :
% 0.34/0.62 ( ( ord_less_nat @ ( finite_card_int @ ( minus_minus_set_int @ A @ ( insert_int @ X @ bot_bot_set_int ) ) ) @ ( finite_card_int @ A ) )
% 0.34/0.62 = ( ( finite_finite_int @ A )
% 0.34/0.62 & ( member_int @ X @ A ) ) ) ).
% 0.34/0.62
% 0.34/0.62 % card_Diff1_less_iff
% 0.34/0.62 thf(fact_4644_card__Diff1__less__iff,axiom,
% 0.34/0.62 ! [A: set_Pr4532377907799695533_nat_o,X: produc3658429121746597890et_nat > $o] :
% 0.34/0.62 ( ( ord_less_nat @ ( finite1363419556375932405_nat_o @ ( minus_1801376950450012436_nat_o @ A @ ( insert5175938949040314269_nat_o @ X @ bot_bo7824918357723582233_nat_o ) ) ) @ ( finite1363419556375932405_nat_o @ A ) )
% 0.34/0.62 = ( ( finite3252695134891459830_nat_o @ A )
% 0.34/0.62 & ( member6576561426505652726_nat_o @ X @ A ) ) ) ).
% 0.34/0.62
% 0.34/0.62 % card_Diff1_less_iff
% 0.34/0.62 thf(fact_4645_card__Diff1__less__iff,axiom,
% 0.34/0.62 ! [A: set_nat,X: nat] :
% 0.34/0.62 ( ( ord_less_nat @ ( finite_card_nat @ ( minus_minus_set_nat @ A @ ( insert_nat @ X @ bot_bot_set_nat ) ) ) @ ( finite_card_nat @ A ) )
% 0.34/0.62 = ( ( finite_finite_nat @ A )
% 0.34/0.62 & ( member_nat @ X @ A ) ) ) ).
% 0.34/0.62
% 0.34/0.62 % card_Diff1_less_iff
% 0.34/0.62 thf(fact_4646_Inter__in__chain,axiom,
% 0.34/0.62 ! [B9: set_set_nat,A11: set_set_nat] :
% 0.34/0.62 ( ( finite1152437895449049373et_nat @ B9 )
% 0.34/0.62 => ( ( B9 != bot_bot_set_set_nat )
% 0.34/0.62 => ( ( pred_chain_set_nat @ A11 @ ord_less_set_nat @ B9 )
% 0.34/0.62 => ( member_set_nat @ ( comple7806235888213564991et_nat @ B9 ) @ B9 ) ) ) ) ).
% 0.34/0.62
% 0.34/0.62 % Inter_in_chain
% 0.34/0.62 thf(fact_4647_card__eq__UNIV2,axiom,
% 0.34/0.62 ! [S: set_Product_unit] :
% 0.34/0.62 ( ( ( finite410649719033368117t_unit @ top_to1996260823553986621t_unit )
% 0.34/0.62 = ( finite410649719033368117t_unit @ S ) )
% 0.34/0.62 = ( S = top_to1996260823553986621t_unit ) ) ).
% 0.34/0.62
% 0.34/0.62 % card_eq_UNIV2
% 0.34/0.62 thf(fact_4648_card__eq__UNIV2,axiom,
% 0.34/0.62 ! [S: set_o] :
% 0.34/0.62 ( ( ( finite_card_o @ top_top_set_o )
% 0.34/0.62 = ( finite_card_o @ S ) )
% 0.34/0.62 = ( S = top_top_set_o ) ) ).
% 0.34/0.62
% 0.34/0.62 % card_eq_UNIV2
% 0.34/0.62 thf(fact_4649_card__eq__UNIV,axiom,
% 0.34/0.62 ! [S: set_Product_unit] :
% 0.34/0.62 ( ( ( finite410649719033368117t_unit @ S )
% 0.34/0.62 = ( finite410649719033368117t_unit @ top_to1996260823553986621t_unit ) )
% 0.34/0.62 = ( S = top_to1996260823553986621t_unit ) ) ).
% 0.34/0.62
% 0.34/0.62 % card_eq_UNIV
% 0.34/0.62 thf(fact_4650_card__eq__UNIV,axiom,
% 0.34/0.62 ! [S: set_o] :
% 0.34/0.62 ( ( ( finite_card_o @ S )
% 0.34/0.62 = ( finite_card_o @ top_top_set_o ) )
% 0.34/0.62 = ( S = top_top_set_o ) ) ).
% 0.34/0.62
% 0.34/0.62 % card_eq_UNIV
% 0.34/0.62 thf(fact_4651_fst__map__prod,axiom,
% 0.34/0.62 ! [F: int > int,G: int > int,X: product_prod_int_int] :
% 0.34/0.62 ( ( product_fst_int_int @ ( produc6036585564866528938nt_int @ F @ G @ X ) )
% 0.34/0.62 = ( F @ ( product_fst_int_int @ X ) ) ) ).
% 0.34/0.62
% 0.34/0.62 % fst_map_prod
% 0.34/0.62 thf(fact_4652_apfst__conv,axiom,
% 0.34/0.62 ! [F: heap_e7401611519738050253t_unit > heap_e7401611519738050253t_unit,X: heap_e7401611519738050253t_unit,Y: set_nat] :
% 0.34/0.62 ( ( produc4172003873565521026et_nat @ F @ ( produc7507926704131184380et_nat @ X @ Y ) )
% 0.34/0.62 = ( produc7507926704131184380et_nat @ ( F @ X ) @ Y ) ) ).
% 0.34/0.62
% 0.34/0.62 % apfst_conv
% 0.34/0.62 thf(fact_4653_apfst__conv,axiom,
% 0.34/0.62 ! [F: nat > nat,X: nat,Y: nat] :
% 0.34/0.62 ( ( produc986720760941809198at_nat @ F @ ( product_Pair_nat_nat @ X @ Y ) )
% 0.34/0.62 = ( product_Pair_nat_nat @ ( F @ X ) @ Y ) ) ).
% 0.34/0.62
% 0.34/0.62 % apfst_conv
% 0.34/0.62 thf(fact_4654_apfst__conv,axiom,
% 0.34/0.62 ! [F: int > int,X: int,Y: int] :
% 0.34/0.62 ( ( produc2355237562570658114nt_int @ F @ ( product_Pair_int_int @ X @ Y ) )
% 0.34/0.62 = ( product_Pair_int_int @ ( F @ X ) @ Y ) ) ).
% 0.34/0.62
% 0.34/0.62 % apfst_conv
% 0.34/0.62 thf(fact_4655_apfst__conv,axiom,
% 0.34/0.62 ! [F: ( produc3658429121746597890et_nat > $o ) > produc3658429121746597890et_nat > $o,X: produc3658429121746597890et_nat > $o,Y: produc3658429121746597890et_nat] :
% 0.34/0.62 ( ( produc3198230245617193214et_nat @ F @ ( produc5001842942810119800et_nat @ X @ Y ) )
% 0.34/0.62 = ( produc5001842942810119800et_nat @ ( F @ X ) @ Y ) ) ).
% 0.34/0.62
% 0.34/0.62 % apfst_conv
% 0.34/0.62 thf(fact_4656_apfst__conv,axiom,
% 0.34/0.62 ! [F: ( produc3658429121746597890et_nat > $o ) > produc3658429121746597890et_nat > $o,X: produc3658429121746597890et_nat > $o,Y: produc3925858234332021118et_nat] :
% 0.34/0.62 ( ( produc1442628905706551930et_nat @ F @ ( produc2245416461498447860et_nat @ X @ Y ) )
% 0.34/0.62 = ( produc2245416461498447860et_nat @ ( F @ X ) @ Y ) ) ).
% 0.34/0.62
% 0.34/0.62 % apfst_conv
% 0.34/0.62 thf(fact_4657_apsnd__conv,axiom,
% 0.34/0.62 ! [F: set_nat > set_nat,X: heap_e7401611519738050253t_unit,Y: set_nat] :
% 0.34/0.62 ( ( produc3286488329990770625t_unit @ F @ ( produc7507926704131184380et_nat @ X @ Y ) )
% 0.34/0.62 = ( produc7507926704131184380et_nat @ X @ ( F @ Y ) ) ) ).
% 0.34/0.62
% 0.34/0.62 % apsnd_conv
% 0.34/0.62 thf(fact_4658_apsnd__conv,axiom,
% 0.34/0.62 ! [F: nat > nat,X: nat,Y: nat] :
% 0.34/0.62 ( ( produc3094765310956027504at_nat @ F @ ( product_Pair_nat_nat @ X @ Y ) )
% 0.34/0.62 = ( product_Pair_nat_nat @ X @ ( F @ Y ) ) ) ).
% 0.34/0.62
% 0.34/0.62 % apsnd_conv
% 0.34/0.62 thf(fact_4659_apsnd__conv,axiom,
% 0.34/0.62 ! [F: int > int,X: int,Y: int] :
% 0.34/0.62 ( ( produc4463282112584876420nt_int @ F @ ( product_Pair_int_int @ X @ Y ) )
% 0.34/0.62 = ( product_Pair_int_int @ X @ ( F @ Y ) ) ) ).
% 0.34/0.62
% 0.34/0.62 % apsnd_conv
% 0.34/0.62 thf(fact_4660_apsnd__conv,axiom,
% 0.34/0.62 ! [F: produc3658429121746597890et_nat > produc3658429121746597890et_nat,X: produc3658429121746597890et_nat > $o,Y: produc3658429121746597890et_nat] :
% 0.34/0.62 ( ( produc8750854537940449737_nat_o @ F @ ( produc5001842942810119800et_nat @ X @ Y ) )
% 0.34/0.62 = ( produc5001842942810119800et_nat @ X @ ( F @ Y ) ) ) ).
% 0.34/0.62
% 0.34/0.62 % apsnd_conv
% 0.34/0.62 thf(fact_4661_apsnd__conv,axiom,
% 0.34/0.62 ! [F: produc3658429121746597890et_nat > produc3925858234332021118et_nat,X: produc3658429121746597890et_nat > $o,Y: produc3658429121746597890et_nat] :
% 0.34/0.62 ( ( produc7714581247149323085_nat_o @ F @ ( produc5001842942810119800et_nat @ X @ Y ) )
% 0.34/0.62 = ( produc2245416461498447860et_nat @ X @ ( F @ Y ) ) ) ).
% 0.34/0.62
% 0.34/0.62 % apsnd_conv
% 0.34/0.62 thf(fact_4662_apsnd__conv,axiom,
% 0.34/0.62 ! [F: produc3925858234332021118et_nat > produc3658429121746597890et_nat,X: produc3658429121746597890et_nat > $o,Y: produc3925858234332021118et_nat] :
% 0.34/0.62 ( ( produc1515462096303866701_nat_o @ F @ ( produc2245416461498447860et_nat @ X @ Y ) )
% 0.34/0.62 = ( produc5001842942810119800et_nat @ X @ ( F @ Y ) ) ) ).
% 0.34/0.62
% 0.34/0.62 % apsnd_conv
% 0.34/0.62 thf(fact_4663_apsnd__conv,axiom,
% 0.34/0.62 ! [F: produc3925858234332021118et_nat > produc3925858234332021118et_nat,X: produc3658429121746597890et_nat > $o,Y: produc3925858234332021118et_nat] :
% 0.34/0.62 ( ( produc969530845752564945_nat_o @ F @ ( produc2245416461498447860et_nat @ X @ Y ) )
% 0.34/0.62 = ( produc2245416461498447860et_nat @ X @ ( F @ Y ) ) ) ).
% 0.34/0.62
% 0.34/0.62 % apsnd_conv
% 0.34/0.62 thf(fact_4664_apsnd__conv,axiom,
% 0.34/0.62 ! [F: code_integer > code_integer,X: code_integer,Y: code_integer] :
% 0.34/0.62 ( ( produc6499014454317279255nteger @ F @ ( produc1086072967326762835nteger @ X @ Y ) )
% 0.34/0.62 = ( produc1086072967326762835nteger @ X @ ( F @ Y ) ) ) ).
% 0.34/0.62
% 0.34/0.62 % apsnd_conv
% 0.34/0.62 thf(fact_4665_fst__apfst,axiom,
% 0.34/0.62 ! [F: int > int,X: product_prod_int_int] :
% 0.34/0.62 ( ( product_fst_int_int @ ( produc2355237562570658114nt_int @ F @ X ) )
% 0.34/0.62 = ( F @ ( product_fst_int_int @ X ) ) ) ).
% 0.34/0.62
% 0.34/0.62 % fst_apfst
% 0.34/0.62 thf(fact_4666_snd__apfst,axiom,
% 0.34/0.62 ! [F: heap_e7401611519738050253t_unit > heap_e7401611519738050253t_unit,X: produc3658429121746597890et_nat] :
% 0.34/0.62 ( ( produc8586169260539613262et_nat @ ( produc4172003873565521026et_nat @ F @ X ) )
% 0.34/0.62 = ( produc8586169260539613262et_nat @ X ) ) ).
% 0.34/0.62
% 0.34/0.62 % snd_apfst
% 0.34/0.62 thf(fact_4667_snd__apfst,axiom,
% 0.34/0.62 ! [F: int > int,X: product_prod_int_int] :
% 0.34/0.62 ( ( product_snd_int_int @ ( produc2355237562570658114nt_int @ F @ X ) )
% 0.34/0.62 = ( product_snd_int_int @ X ) ) ).
% 0.34/0.62
% 0.34/0.62 % snd_apfst
% 0.34/0.62 thf(fact_4668_in__inv__image,axiom,
% 0.34/0.62 ! [X: nat,Y: nat,R2: set_Pr1261947904930325089at_nat,F: nat > nat] :
% 0.34/0.62 ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ Y ) @ ( inv_image_nat_nat @ R2 @ F ) )
% 0.34/0.62 = ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ ( F @ X ) @ ( F @ Y ) ) @ R2 ) ) ).
% 0.34/0.62
% 0.34/0.62 % in_inv_image
% 0.34/0.62 thf(fact_4669_in__inv__image,axiom,
% 0.34/0.62 ! [X: nat,Y: nat,R2: set_Pr958786334691620121nt_int,F: nat > int] :
% 0.34/0.62 ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ Y ) @ ( inv_image_int_nat @ R2 @ F ) )
% 0.34/0.62 = ( member5262025264175285858nt_int @ ( product_Pair_int_int @ ( F @ X ) @ ( F @ Y ) ) @ R2 ) ) ).
% 0.34/0.62
% 0.34/0.62 % in_inv_image
% 0.34/0.62 thf(fact_4670_in__inv__image,axiom,
% 0.34/0.62 ! [X: int,Y: int,R2: set_Pr1261947904930325089at_nat,F: int > nat] :
% 0.34/0.62 ( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ X @ Y ) @ ( inv_image_nat_int @ R2 @ F ) )
% 0.34/0.62 = ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ ( F @ X ) @ ( F @ Y ) ) @ R2 ) ) ).
% 0.34/0.62
% 0.34/0.62 % in_inv_image
% 0.34/0.62 thf(fact_4671_in__inv__image,axiom,
% 0.34/0.62 ! [X: int,Y: int,R2: set_Pr958786334691620121nt_int,F: int > int] :
% 0.34/0.62 ( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ X @ Y ) @ ( inv_image_int_int @ R2 @ F ) )
% 0.34/0.62 = ( member5262025264175285858nt_int @ ( product_Pair_int_int @ ( F @ X ) @ ( F @ Y ) ) @ R2 ) ) ).
% 0.34/0.62
% 0.34/0.62 % in_inv_image
% 0.34/0.62 thf(fact_4672_fst__apsnd,axiom,
% 0.34/0.62 ! [F: code_integer > code_integer,X: produc8923325533196201883nteger] :
% 0.34/0.62 ( ( produc8508995932063986495nteger @ ( produc6499014454317279255nteger @ F @ X ) )
% 0.34/0.62 = ( produc8508995932063986495nteger @ X ) ) ).
% 0.34/0.62
% 0.34/0.62 % fst_apsnd
% 0.34/0.62 thf(fact_4673_fst__apsnd,axiom,
% 0.34/0.62 ! [F: int > int,X: product_prod_int_int] :
% 0.34/0.62 ( ( product_fst_int_int @ ( produc4463282112584876420nt_int @ F @ X ) )
% 0.34/0.62 = ( product_fst_int_int @ X ) ) ).
% 0.34/0.62
% 0.34/0.62 % fst_apsnd
% 0.34/0.62 thf(fact_4674_apsnd__eq__conv,axiom,
% 0.34/0.62 ! [F: code_integer > code_integer,X: produc8923325533196201883nteger,G: code_integer > code_integer] :
% 0.34/0.62 ( ( ( produc6499014454317279255nteger @ F @ X )
% 0.34/0.62 = ( produc6499014454317279255nteger @ G @ X ) )
% 0.34/0.62 = ( ( F @ ( produc6174133586879617921nteger @ X ) )
% 0.34/0.62 = ( G @ ( produc6174133586879617921nteger @ X ) ) ) ) ).
% 0.34/0.62
% 0.34/0.62 % apsnd_eq_conv
% 0.34/0.62 thf(fact_4675_snd__apsnd,axiom,
% 0.34/0.62 ! [F: code_integer > code_integer,X: produc8923325533196201883nteger] :
% 0.34/0.62 ( ( produc6174133586879617921nteger @ ( produc6499014454317279255nteger @ F @ X ) )
% 0.34/0.62 = ( F @ ( produc6174133586879617921nteger @ X ) ) ) ).
% 0.34/0.62
% 0.34/0.62 % snd_apsnd
% 0.34/0.62 thf(fact_4676_snd__apsnd,axiom,
% 0.34/0.62 ! [F: set_nat > set_nat,X: produc3658429121746597890et_nat] :
% 0.34/0.62 ( ( produc8586169260539613262et_nat @ ( produc3286488329990770625t_unit @ F @ X ) )
% 0.34/0.62 = ( F @ ( produc8586169260539613262et_nat @ X ) ) ) ).
% 0.34/0.62
% 0.34/0.62 % snd_apsnd
% 0.34/0.62 thf(fact_4677_snd__apsnd,axiom,
% 0.34/0.62 ! [F: int > int,X: product_prod_int_int] :
% 0.34/0.62 ( ( product_snd_int_int @ ( produc4463282112584876420nt_int @ F @ X ) )
% 0.34/0.62 = ( F @ ( product_snd_int_int @ X ) ) ) ).
% 0.34/0.62
% 0.34/0.62 % snd_apsnd
% 0.34/0.62 thf(fact_4678_Inf__eq__bot__iff,axiom,
% 0.34/0.62 ! [A: set_Product_unit] :
% 0.34/0.62 ( ( ( comple2584293577114468500t_unit @ A )
% 0.34/0.62 = bot_bot_Product_unit )
% 0.34/0.62 = ( ! [X2: product_unit] :
% 0.34/0.62 ( ( ord_le361264281704409273t_unit @ bot_bot_Product_unit @ X2 )
% 0.34/0.62 => ? [Y3: product_unit] :
% 0.34/0.62 ( ( member_Product_unit @ Y3 @ A )
% 0.34/0.62 & ( ord_le361264281704409273t_unit @ Y3 @ X2 ) ) ) ) ) ).
% 0.34/0.62
% 0.34/0.62 % Inf_eq_bot_iff
% 0.34/0.62 thf(fact_4679_Inf__empty,axiom,
% 0.34/0.62 ( ( complete_Inf_Inf_o @ bot_bot_set_o )
% 0.34/0.62 = top_top_o ) ).
% 0.34/0.62
% 0.34/0.62 % Inf_empty
% 0.34/0.62 thf(fact_4680_Inf__empty,axiom,
% 0.34/0.62 ( ( comple90455603224345172_nat_o @ bot_bo7824918357723582233_nat_o )
% 0.34/0.62 = top_to5213382387105117981_nat_o ) ).
% 0.34/0.62
% 0.34/0.62 % Inf_empty
% 0.34/0.62 thf(fact_4681_Inf__empty,axiom,
% 0.34/0.62 ( ( comple7216939928495492468t_unit @ bot_bo6284674557190688313t_unit )
% 0.34/0.62 = top_to1996260823553986621t_unit ) ).
% 0.34/0.62
% 0.34/0.62 % Inf_empty
% 0.34/0.62 thf(fact_4682_Inf__empty,axiom,
% 0.34/0.62 ( ( comple3063163877087187839_set_o @ bot_bot_set_set_o )
% 0.34/0.62 = top_top_set_o ) ).
% 0.34/0.62
% 0.34/0.62 % Inf_empty
% 0.34/0.62 thf(fact_4683_Inf__empty,axiom,
% 0.34/0.62 ( ( comple3628384868704368283et_int @ bot_bot_set_set_int )
% 0.34/0.62 = top_top_set_int ) ).
% 0.34/0.62
% 0.34/0.62 % Inf_empty
% 0.34/0.62 thf(fact_4684_Inf__empty,axiom,
% 0.34/0.62 ( ( comple7806235888213564991et_nat @ bot_bot_set_set_nat )
% 0.34/0.62 = top_top_set_nat ) ).
% 0.34/0.62
% 0.34/0.62 % Inf_empty
% 0.34/0.62 thf(fact_4685_Inf__empty,axiom,
% 0.34/0.62 ( ( comple2584293577114468500t_unit @ bot_bo3957492148770167129t_unit )
% 0.34/0.62 = top_top_Product_unit ) ).
% 0.34/0.62
% 0.34/0.62 % Inf_empty
% 0.34/0.62 thf(fact_4686_Inf__UNIV,axiom,
% 0.34/0.62 ( ( comple7216939928495492468t_unit @ top_to1767297665138865437t_unit )
% 0.34/0.62 = bot_bo3957492148770167129t_unit ) ).
% 0.34/0.62
% 0.34/0.62 % Inf_UNIV
% 0.34/0.62 thf(fact_4687_Inf__UNIV,axiom,
% 0.34/0.62 ( ( comple3063163877087187839_set_o @ top_top_set_set_o )
% 0.34/0.62 = bot_bot_set_o ) ).
% 0.34/0.62
% 0.34/0.62 % Inf_UNIV
% 0.34/0.62 thf(fact_4688_Inf__UNIV,axiom,
% 0.34/0.62 ( ( comple3628384868704368283et_int @ top_top_set_set_int )
% 0.34/0.62 = bot_bot_set_int ) ).
% 0.34/0.62
% 0.34/0.62 % Inf_UNIV
% 0.34/0.62 thf(fact_4689_Inf__UNIV,axiom,
% 0.34/0.62 ( ( comple79274577958421812_nat_o @ top_to8515903933492314333_nat_o )
% 0.34/0.62 = bot_bo7824918357723582233_nat_o ) ).
% 0.34/0.62
% 0.34/0.62 % Inf_UNIV
% 0.34/0.62 thf(fact_4690_Inf__UNIV,axiom,
% 0.34/0.62 ( ( complete_Inf_Inf_o @ top_top_set_o )
% 0.34/0.62 = bot_bot_o ) ).
% 0.34/0.62
% 0.34/0.62 % Inf_UNIV
% 0.34/0.62 thf(fact_4691_Inf__UNIV,axiom,
% 0.34/0.62 ( ( comple7806235888213564991et_nat @ top_top_set_set_nat )
% 0.34/0.62 = bot_bot_set_nat ) ).
% 0.34/0.62
% 0.34/0.62 % Inf_UNIV
% 0.34/0.62 thf(fact_4692_Inf__UNIV,axiom,
% 0.34/0.62 ( ( comple2584293577114468500t_unit @ top_to1996260823553986621t_unit )
% 0.34/0.62 = bot_bot_Product_unit ) ).
% 0.34/0.62
% 0.34/0.62 % Inf_UNIV
% 0.34/0.62 thf(fact_4693_cInf__singleton,axiom,
% 0.34/0.62 ! [X: $o] :
% 0.34/0.62 ( ( complete_Inf_Inf_o @ ( insert_o @ X @ bot_bot_set_o ) )
% 0.34/0.62 = X ) ).
% 0.34/0.62
% 0.34/0.62 % cInf_singleton
% 0.34/0.62 thf(fact_4694_cInf__singleton,axiom,
% 0.34/0.62 ! [X: int] :
% 0.34/0.62 ( ( complete_Inf_Inf_int @ ( insert_int @ X @ bot_bot_set_int ) )
% 0.34/0.62 = X ) ).
% 0.34/0.62
% 0.34/0.62 % cInf_singleton
% 0.34/0.62 thf(fact_4695_cInf__singleton,axiom,
% 0.34/0.62 ! [X: produc3658429121746597890et_nat > $o] :
% 0.34/0.62 ( ( comple90455603224345172_nat_o @ ( insert5175938949040314269_nat_o @ X @ bot_bo7824918357723582233_nat_o ) )
% 0.34/0.62 = X ) ).
% 0.34/0.62
% 0.34/0.62 % cInf_singleton
% 0.34/0.62 thf(fact_4696_cInf__singleton,axiom,
% 0.34/0.62 ! [X: nat] :
% 0.34/0.62 ( ( complete_Inf_Inf_nat @ ( insert_nat @ X @ bot_bot_set_nat ) )
% 0.34/0.62 = X ) ).
% 0.34/0.62
% 0.34/0.62 % cInf_singleton
% 0.34/0.62 thf(fact_4697_cInf__singleton,axiom,
% 0.34/0.62 ! [X: set_nat] :
% 0.34/0.62 ( ( comple7806235888213564991et_nat @ ( insert_set_nat @ X @ bot_bot_set_set_nat ) )
% 0.34/0.62 = X ) ).
% 0.34/0.62
% 0.34/0.62 % cInf_singleton
% 0.34/0.62 thf(fact_4698_cInf__singleton,axiom,
% 0.34/0.62 ! [X: product_unit] :
% 0.34/0.62 ( ( comple2584293577114468500t_unit @ ( insert_Product_unit @ X @ bot_bo3957492148770167129t_unit ) )
% 0.34/0.62 = X ) ).
% 0.34/0.62
% 0.34/0.62 % cInf_singleton
% 0.34/0.62 thf(fact_4699_Inf__insert,axiom,
% 0.34/0.62 ! [A2: $o,A: set_o] :
% 0.34/0.62 ( ( complete_Inf_Inf_o @ ( insert_o @ A2 @ A ) )
% 0.34/0.62 = ( inf_inf_o @ A2 @ ( complete_Inf_Inf_o @ A ) ) ) ).
% 0.34/0.62
% 0.34/0.62 % Inf_insert
% 0.34/0.62 thf(fact_4700_Inf__insert,axiom,
% 0.34/0.62 ! [A2: set_nat,A: set_set_nat] :
% 0.34/0.62 ( ( comple7806235888213564991et_nat @ ( insert_set_nat @ A2 @ A ) )
% 0.34/0.62 = ( inf_inf_set_nat @ A2 @ ( comple7806235888213564991et_nat @ A ) ) ) ).
% 0.34/0.62
% 0.34/0.62 % Inf_insert
% 0.34/0.62 thf(fact_4701_Inf__insert,axiom,
% 0.34/0.62 ! [A2: product_unit,A: set_Product_unit] :
% 0.34/0.62 ( ( comple2584293577114468500t_unit @ ( insert_Product_unit @ A2 @ A ) )
% 0.34/0.62 = ( inf_inf_Product_unit @ A2 @ ( comple2584293577114468500t_unit @ A ) ) ) ).
% 0.34/0.62
% 0.34/0.62 % Inf_insert
% 0.34/0.62 thf(fact_4702_prod_Ocollapse,axiom,
% 0.34/0.62 ! [Prod: product_prod_nat_nat] :
% 0.34/0.62 ( ( product_Pair_nat_nat @ ( product_fst_nat_nat @ Prod ) @ ( product_snd_nat_nat @ Prod ) )
% 0.34/0.62 = Prod ) ).
% 0.34/0.62
% 0.34/0.62 % prod.collapse
% 0.34/0.62 thf(fact_4703_prod_Ocollapse,axiom,
% 0.34/0.62 ! [Prod: produc3925858234332021118et_nat] :
% 0.34/0.62 ( ( produc5001842942810119800et_nat @ ( produc995936583742144908et_nat @ Prod ) @ ( produc4011572625026189258et_nat @ Prod ) )
% 0.34/0.62 = Prod ) ).
% 0.34/0.62
% 0.34/0.62 % prod.collapse
% 0.34/0.62 thf(fact_4704_prod_Ocollapse,axiom,
% 0.34/0.62 ! [Prod: produc2732055786443039994et_nat] :
% 0.34/0.62 ( ( produc2245416461498447860et_nat @ ( produc180342877477747464et_nat @ Prod ) @ ( produc5374455773327741254et_nat @ Prod ) )
% 0.34/0.62 = Prod ) ).
% 0.34/0.62
% 0.34/0.62 % prod.collapse
% 0.34/0.62 thf(fact_4705_prod_Ocollapse,axiom,
% 0.34/0.62 ! [Prod: produc3658429121746597890et_nat] :
% 0.34/0.62 ( ( produc7507926704131184380et_nat @ ( produc1824681642469235216et_nat @ Prod ) @ ( produc8586169260539613262et_nat @ Prod ) )
% 0.34/0.62 = Prod ) ).
% 0.34/0.62
% 0.34/0.62 % prod.collapse
% 0.34/0.62 thf(fact_4706_prod_Ocollapse,axiom,
% 0.34/0.62 ! [Prod: product_prod_int_int] :
% 0.34/0.62 ( ( product_Pair_int_int @ ( product_fst_int_int @ Prod ) @ ( product_snd_int_int @ Prod ) )
% 0.34/0.62 = Prod ) ).
% 0.34/0.62
% 0.34/0.62 % prod.collapse
% 0.34/0.62 thf(fact_4707_card__ge__UNIV,axiom,
% 0.34/0.62 ! [S: set_Product_unit] :
% 0.34/0.62 ( ( ord_less_eq_nat @ ( finite410649719033368117t_unit @ top_to1996260823553986621t_unit ) @ ( finite410649719033368117t_unit @ S ) )
% 0.34/0.62 = ( S = top_to1996260823553986621t_unit ) ) ).
% 0.34/0.62
% 0.34/0.62 % card_ge_UNIV
% 0.34/0.62 thf(fact_4708_card__ge__UNIV,axiom,
% 0.34/0.62 ! [S: set_o] :
% 0.34/0.62 ( ( ord_less_eq_nat @ ( finite_card_o @ top_top_set_o ) @ ( finite_card_o @ S ) )
% 0.34/0.62 = ( S = top_top_set_o ) ) ).
% 0.34/0.62
% 0.34/0.62 % card_ge_UNIV
% 0.34/0.62 thf(fact_4709_card__Diff__insert,axiom,
% 0.34/0.62 ! [A2: produc3658429121746597890et_nat > $o,A: set_Pr4532377907799695533_nat_o,B4: set_Pr4532377907799695533_nat_o] :
% 0.34/0.62 ( ( member6576561426505652726_nat_o @ A2 @ A )
% 0.34/0.62 => ( ~ ( member6576561426505652726_nat_o @ A2 @ B4 )
% 0.34/0.62 => ( ( finite1363419556375932405_nat_o @ ( minus_1801376950450012436_nat_o @ A @ ( insert5175938949040314269_nat_o @ A2 @ B4 ) ) )
% 0.34/0.62 = ( minus_minus_nat @ ( finite1363419556375932405_nat_o @ ( minus_1801376950450012436_nat_o @ A @ B4 ) ) @ one_one_nat ) ) ) ) ).
% 0.34/0.62
% 0.34/0.62 % card_Diff_insert
% 0.34/0.62 thf(fact_4710_card__Diff__insert,axiom,
% 0.34/0.62 ! [A2: int,A: set_int,B4: set_int] :
% 0.34/0.62 ( ( member_int @ A2 @ A )
% 0.34/0.62 => ( ~ ( member_int @ A2 @ B4 )
% 0.34/0.62 => ( ( finite_card_int @ ( minus_minus_set_int @ A @ ( insert_int @ A2 @ B4 ) ) )
% 0.34/0.62 = ( minus_minus_nat @ ( finite_card_int @ ( minus_minus_set_int @ A @ B4 ) ) @ one_one_nat ) ) ) ) ).
% 0.34/0.62
% 0.34/0.62 % card_Diff_insert
% 0.34/0.62 thf(fact_4711_card__Diff__insert,axiom,
% 0.34/0.62 ! [A2: $o,A: set_o,B4: set_o] :
% 0.34/0.62 ( ( member_o @ A2 @ A )
% 0.34/0.62 => ( ~ ( member_o @ A2 @ B4 )
% 0.34/0.62 => ( ( finite_card_o @ ( minus_minus_set_o @ A @ ( insert_o @ A2 @ B4 ) ) )
% 0.34/0.62 = ( minus_minus_nat @ ( finite_card_o @ ( minus_minus_set_o @ A @ B4 ) ) @ one_one_nat ) ) ) ) ).
% 0.34/0.62
% 0.34/0.62 % card_Diff_insert
% 0.34/0.62 thf(fact_4712_card__Diff__insert,axiom,
% 0.34/0.62 ! [A2: product_unit,A: set_Product_unit,B4: set_Product_unit] :
% 0.34/0.62 ( ( member_Product_unit @ A2 @ A )
% 0.34/0.62 => ( ~ ( member_Product_unit @ A2 @ B4 )
% 0.34/0.62 => ( ( finite410649719033368117t_unit @ ( minus_6452836326544984404t_unit @ A @ ( insert_Product_unit @ A2 @ B4 ) ) )
% 0.34/0.62 = ( minus_minus_nat @ ( finite410649719033368117t_unit @ ( minus_6452836326544984404t_unit @ A @ B4 ) ) @ one_one_nat ) ) ) ) ).
% 0.34/0.62
% 0.34/0.62 % card_Diff_insert
% 0.34/0.62 thf(fact_4713_card__Diff__insert,axiom,
% 0.34/0.62 ! [A2: nat,A: set_nat,B4: set_nat] :
% 0.34/0.62 ( ( member_nat @ A2 @ A )
% 0.34/0.62 => ( ~ ( member_nat @ A2 @ B4 )
% 0.34/0.62 => ( ( finite_card_nat @ ( minus_minus_set_nat @ A @ ( insert_nat @ A2 @ B4 ) ) )
% 0.34/0.62 = ( minus_minus_nat @ ( finite_card_nat @ ( minus_minus_set_nat @ A @ B4 ) ) @ one_one_nat ) ) ) ) ).
% 0.34/0.62
% 0.34/0.62 % card_Diff_insert
% 0.34/0.62 thf(fact_4714_img__fst,axiom,
% 0.34/0.62 ! [A2: heap_e7401611519738050253t_unit,B: set_nat,S: set_Pr3948176798113811640et_nat] :
% 0.34/0.62 ( ( member6260224972018164377et_nat @ ( produc7507926704131184380et_nat @ A2 @ B ) @ S )
% 0.34/0.62 => ( member242389114213480694t_unit @ A2 @ ( image_6824039242672219652t_unit @ produc1824681642469235216et_nat @ S ) ) ) ).
% 0.34/0.62
% 0.34/0.62 % img_fst
% 0.34/0.62 thf(fact_4715_img__fst,axiom,
% 0.34/0.62 ! [A2: nat,B: nat,S: set_Pr1261947904930325089at_nat] :
% 0.34/0.62 ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A2 @ B ) @ S )
% 0.34/0.62 => ( member_nat @ A2 @ ( image_2486076414777270412at_nat @ product_fst_nat_nat @ S ) ) ) ).
% 0.34/0.62
% 0.34/0.62 % img_fst
% 0.34/0.62 thf(fact_4716_img__fst,axiom,
% 0.34/0.62 ! [A2: produc3658429121746597890et_nat > $o,B: produc3658429121746597890et_nat,S: set_Pr3286484037609594932et_nat] :
% 0.34/0.62 ( ( member1996754912294343701et_nat @ ( produc5001842942810119800et_nat @ A2 @ B ) @ S )
% 0.34/0.62 => ( member6576561426505652726_nat_o @ A2 @ ( image_6802380912191114376_nat_o @ produc995936583742144908et_nat @ S ) ) ) ).
% 0.34/0.62
% 0.34/0.62 % img_fst
% 0.34/0.62 thf(fact_4717_img__fst,axiom,
% 0.34/0.62 ! [A2: produc3658429121746597890et_nat > $o,B: produc3925858234332021118et_nat,S: set_Pr8536935166611901872et_nat] :
% 0.34/0.62 ( ( member6124377750444531601et_nat @ ( produc2245416461498447860et_nat @ A2 @ B ) @ S )
% 0.34/0.62 => ( member6576561426505652726_nat_o @ A2 @ ( image_2517677226577294348_nat_o @ produc180342877477747464et_nat @ S ) ) ) ).
% 0.34/0.62
% 0.34/0.62 % img_fst
% 0.34/0.62 thf(fact_4718_img__fst,axiom,
% 0.34/0.62 ! [A2: int,B: int,S: set_Pr958786334691620121nt_int] :
% 0.34/0.62 ( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ A2 @ B ) @ S )
% 0.34/0.62 => ( member_int @ A2 @ ( image_5042161079198086560nt_int @ product_fst_int_int @ S ) ) ) ).
% 0.34/0.62
% 0.34/0.62 % img_fst
% 0.34/0.62 thf(fact_4719_fst__swap,axiom,
% 0.34/0.62 ! [X: produc3658429121746597890et_nat] :
% 0.34/0.62 ( ( produc6012101397789842562t_unit @ ( produc6300697153377358172et_nat @ X ) )
% 0.34/0.62 = ( produc8586169260539613262et_nat @ X ) ) ).
% 0.34/0.62
% 0.34/0.62 % fst_swap
% 0.34/0.62 thf(fact_4720_fst__swap,axiom,
% 0.34/0.62 ! [X: product_prod_int_int] :
% 0.34/0.62 ( ( product_fst_int_int @ ( product_swap_int_int @ X ) )
% 0.34/0.62 = ( product_snd_int_int @ X ) ) ).
% 0.34/0.62
% 0.34/0.62 % fst_swap
% 0.34/0.62 thf(fact_4721_snd__swap,axiom,
% 0.34/0.62 ! [X: produc7348995623955488188t_unit] :
% 0.34/0.62 ( ( produc8586169260539613262et_nat @ ( produc1264744871843189710t_unit @ X ) )
% 0.34/0.62 = ( produc6012101397789842562t_unit @ X ) ) ).
% 0.34/0.62
% 0.34/0.62 % snd_swap
% 0.34/0.62 thf(fact_4722_snd__swap,axiom,
% 0.34/0.62 ! [X: product_prod_int_int] :
% 0.34/0.62 ( ( product_snd_int_int @ ( product_swap_int_int @ X ) )
% 0.34/0.62 = ( product_fst_int_int @ X ) ) ).
% 0.34/0.62
% 0.34/0.62 % snd_swap
% 0.34/0.62 thf(fact_4723_snd__comp__apfst,axiom,
% 0.34/0.62 ! [F: heap_e7401611519738050253t_unit > heap_e7401611519738050253t_unit] :
% 0.34/0.62 ( ( comp_P2930521774885306735et_nat @ produc8586169260539613262et_nat @ ( produc4172003873565521026et_nat @ F ) )
% 0.34/0.62 = produc8586169260539613262et_nat ) ).
% 0.34/0.62
% 0.34/0.62 % snd_comp_apfst
% 0.34/0.62 thf(fact_4724_snd__comp__apfst,axiom,
% 0.34/0.62 ! [F: int > int] :
% 0.34/0.62 ( ( comp_P5356232565778667509nt_int @ product_snd_int_int @ ( produc2355237562570658114nt_int @ F ) )
% 0.34/0.62 = product_snd_int_int ) ).
% 0.34/0.62
% 0.34/0.62 % snd_comp_apfst
% 0.34/0.62 thf(fact_4725_fst__comp__apsnd,axiom,
% 0.34/0.62 ! [F: code_integer > code_integer] :
% 0.34/0.62 ( ( comp_P3428181540658402820nteger @ produc8508995932063986495nteger @ ( produc6499014454317279255nteger @ F ) )
% 0.34/0.62 = produc8508995932063986495nteger ) ).
% 0.34/0.62
% 0.34/0.62 % fst_comp_apsnd
% 0.34/0.62 thf(fact_4726_fst__comp__apsnd,axiom,
% 0.34/0.62 ! [F: int > int] :
% 0.34/0.62 ( ( comp_P5356232565778667509nt_int @ product_fst_int_int @ ( produc4463282112584876420nt_int @ F ) )
% 0.34/0.62 = product_fst_int_int ) ).
% 0.34/0.62
% 0.34/0.62 % fst_comp_apsnd
% 0.34/0.62 thf(fact_4727_range__fst,axiom,
% 0.34/0.62 ( ( image_5042161079198086560nt_int @ product_fst_int_int @ top_to4366644338036079209nt_int )
% 0.34/0.62 = top_top_set_int ) ).
% 0.34/0.62
% 0.34/0.62 % range_fst
% 0.34/0.62 thf(fact_4728_fst__comp__map__prod,axiom,
% 0.34/0.62 ! [F: int > int,G: int > int] :
% 0.34/0.62 ( ( comp_P5356232565778667509nt_int @ product_fst_int_int @ ( produc6036585564866528938nt_int @ F @ G ) )
% 0.34/0.62 = ( comp_i4330622015127193916nt_int @ F @ product_fst_int_int ) ) ).
% 0.34/0.62
% 0.34/0.62 % fst_comp_map_prod
% 0.34/0.62 thf(fact_4729_fst__comp__apfst,axiom,
% 0.34/0.62 ! [F: int > int] :
% 0.34/0.62 ( ( comp_P5356232565778667509nt_int @ product_fst_int_int @ ( produc2355237562570658114nt_int @ F ) )
% 0.34/0.62 = ( comp_i4330622015127193916nt_int @ F @ product_fst_int_int ) ) ).
% 0.34/0.62
% 0.34/0.62 % fst_comp_apfst
% 0.34/0.62 thf(fact_4730_snd__comp__apsnd,axiom,
% 0.34/0.62 ! [F: code_integer > code_integer] :
% 0.34/0.62 ( ( comp_P3428181540658402820nteger @ produc6174133586879617921nteger @ ( produc6499014454317279255nteger @ F ) )
% 0.34/0.62 = ( comp_C1043215000170864188nteger @ F @ produc6174133586879617921nteger ) ) ).
% 0.34/0.62
% 0.34/0.62 % snd_comp_apsnd
% 0.34/0.62 thf(fact_4731_snd__comp__apsnd,axiom,
% 0.34/0.62 ! [F: set_nat > set_nat] :
% 0.34/0.62 ( ( comp_P2930521774885306735et_nat @ produc8586169260539613262et_nat @ ( produc3286488329990770625t_unit @ F ) )
% 0.34/0.62 = ( comp_s5696514601250091763et_nat @ F @ produc8586169260539613262et_nat ) ) ).
% 0.34/0.62
% 0.34/0.62 % snd_comp_apsnd
% 0.34/0.62 thf(fact_4732_snd__comp__apsnd,axiom,
% 0.34/0.62 ! [F: int > int] :
% 0.34/0.62 ( ( comp_P5356232565778667509nt_int @ product_snd_int_int @ ( produc4463282112584876420nt_int @ F ) )
% 0.34/0.62 = ( comp_i4330622015127193916nt_int @ F @ product_snd_int_int ) ) ).
% 0.34/0.62
% 0.34/0.62 % snd_comp_apsnd
% 0.34/0.62 thf(fact_4733_apfst__apsnd,axiom,
% 0.34/0.62 ! [F: heap_e7401611519738050253t_unit > heap_e7401611519738050253t_unit,G: set_nat > set_nat,X: produc3658429121746597890et_nat] :
% 0.34/0.62 ( ( produc4172003873565521026et_nat @ F @ ( produc3286488329990770625t_unit @ G @ X ) )
% 0.34/0.62 = ( produc7507926704131184380et_nat @ ( F @ ( produc1824681642469235216et_nat @ X ) ) @ ( G @ ( produc8586169260539613262et_nat @ X ) ) ) ) ).
% 0.34/0.62
% 0.34/0.62 % apfst_apsnd
% 0.34/0.62 thf(fact_4734_apfst__apsnd,axiom,
% 0.34/0.62 ! [F: heap_e7401611519738050253t_unit > nat,G: set_nat > nat,X: produc3658429121746597890et_nat] :
% 0.34/0.62 ( ( produc8233931525780029747at_nat @ F @ ( produc8442695800884449271t_unit @ G @ X ) )
% 0.34/0.62 = ( product_Pair_nat_nat @ ( F @ ( produc1824681642469235216et_nat @ X ) ) @ ( G @ ( produc8586169260539613262et_nat @ X ) ) ) ) ).
% 0.34/0.62
% 0.34/0.62 % apfst_apsnd
% 0.34/0.62 thf(fact_4735_apfst__apsnd,axiom,
% 0.34/0.62 ! [F: heap_e7401611519738050253t_unit > int,G: set_nat > int,X: produc3658429121746597890et_nat] :
% 0.34/0.62 ( ( produc7422730317258219nt_int @ F @ ( produc7345612820240450715t_unit @ G @ X ) )
% 0.34/0.62 = ( product_Pair_int_int @ ( F @ ( produc1824681642469235216et_nat @ X ) ) @ ( G @ ( produc8586169260539613262et_nat @ X ) ) ) ) ).
% 0.34/0.62
% 0.34/0.62 % apfst_apsnd
% 0.34/0.62 thf(fact_4736_apfst__apsnd,axiom,
% 0.34/0.62 ! [F: heap_e7401611519738050253t_unit > produc3658429121746597890et_nat > $o,G: set_nat > produc3658429121746597890et_nat,X: produc3658429121746597890et_nat] :
% 0.34/0.62 ( ( produc6095824665904122878et_nat @ F @ ( produc7441988914112407621t_unit @ G @ X ) )
% 0.34/0.62 = ( produc5001842942810119800et_nat @ ( F @ ( produc1824681642469235216et_nat @ X ) ) @ ( G @ ( produc8586169260539613262et_nat @ X ) ) ) ) ).
% 0.34/0.62
% 0.34/0.62 % apfst_apsnd
% 0.34/0.62 thf(fact_4737_apfst__apsnd,axiom,
% 0.34/0.62 ! [F: heap_e7401611519738050253t_unit > produc3658429121746597890et_nat > $o,G: set_nat > produc3925858234332021118et_nat,X: produc3658429121746597890et_nat] :
% 0.34/0.62 ( ( produc4273547285574235514et_nat @ F @ ( produc7176435795764513225t_unit @ G @ X ) )
% 0.34/0.62 = ( produc2245416461498447860et_nat @ ( F @ ( produc1824681642469235216et_nat @ X ) ) @ ( G @ ( produc8586169260539613262et_nat @ X ) ) ) ) ).
% 0.34/0.62
% 0.34/0.62 % apfst_apsnd
% 0.34/0.62 thf(fact_4738_apfst__apsnd,axiom,
% 0.34/0.62 ! [F: int > heap_e7401611519738050253t_unit,G: int > set_nat,X: product_prod_int_int] :
% 0.34/0.62 ( ( produc7086134074927288299et_nat @ F @ ( produc1253519502038700766at_int @ G @ X ) )
% 0.34/0.62 = ( produc7507926704131184380et_nat @ ( F @ ( product_fst_int_int @ X ) ) @ ( G @ ( product_snd_int_int @ X ) ) ) ) ).
% 0.34/0.62
% 0.34/0.62 % apfst_apsnd
% 0.34/0.62 thf(fact_4739_apfst__apsnd,axiom,
% 0.34/0.62 ! [F: int > nat,G: int > nat,X: product_prod_int_int] :
% 0.34/0.62 ( ( produc1358374321178653834at_nat @ F @ ( produc3463928400683821864at_int @ G @ X ) )
% 0.34/0.62 = ( product_Pair_nat_nat @ ( F @ ( product_fst_int_int @ X ) ) @ ( G @ ( product_snd_int_int @ X ) ) ) ) ).
% 0.34/0.62
% 0.34/0.62 % apfst_apsnd
% 0.34/0.62 thf(fact_4740_apfst__apsnd,axiom,
% 0.34/0.62 ! [F: int > int,G: int > int,X: product_prod_int_int] :
% 0.34/0.62 ( ( produc2355237562570658114nt_int @ F @ ( produc4463282112584876420nt_int @ G @ X ) )
% 0.34/0.62 = ( product_Pair_int_int @ ( F @ ( product_fst_int_int @ X ) ) @ ( G @ ( product_snd_int_int @ X ) ) ) ) ).
% 0.34/0.62
% 0.34/0.62 % apfst_apsnd
% 0.34/0.62 thf(fact_4741_apfst__apsnd,axiom,
% 0.34/0.62 ! [F: int > produc3658429121746597890et_nat > $o,G: int > produc3658429121746597890et_nat,X: product_prod_int_int] :
% 0.34/0.62 ( ( produc5259063908904861543et_nat @ F @ ( produc9121118195053341018at_int @ G @ X ) )
% 0.34/0.62 = ( produc5001842942810119800et_nat @ ( F @ ( product_fst_int_int @ X ) ) @ ( G @ ( product_snd_int_int @ X ) ) ) ) ).
% 0.34/0.62
% 0.34/0.62 % apfst_apsnd
% 0.34/0.62 thf(fact_4742_apfst__apsnd,axiom,
% 0.34/0.62 ! [F: int > produc3658429121746597890et_nat > $o,G: int > produc3925858234332021118et_nat,X: product_prod_int_int] :
% 0.34/0.62 ( ( produc3602029435013415139et_nat @ F @ ( produc2878475440847162070at_int @ G @ X ) )
% 0.34/0.62 = ( produc2245416461498447860et_nat @ ( F @ ( product_fst_int_int @ X ) ) @ ( G @ ( product_snd_int_int @ X ) ) ) ) ).
% 0.34/0.62
% 0.34/0.62 % apfst_apsnd
% 0.34/0.62 thf(fact_4743_apsnd__apfst,axiom,
% 0.34/0.62 ! [F: set_nat > set_nat,G: heap_e7401611519738050253t_unit > heap_e7401611519738050253t_unit,X: produc3658429121746597890et_nat] :
% 0.34/0.62 ( ( produc3286488329990770625t_unit @ F @ ( produc4172003873565521026et_nat @ G @ X ) )
% 0.34/0.62 = ( produc7507926704131184380et_nat @ ( G @ ( produc1824681642469235216et_nat @ X ) ) @ ( F @ ( produc8586169260539613262et_nat @ X ) ) ) ) ).
% 0.34/0.62
% 0.34/0.62 % apsnd_apfst
% 0.34/0.62 thf(fact_4744_apsnd__apfst,axiom,
% 0.34/0.62 ! [F: set_nat > nat,G: heap_e7401611519738050253t_unit > nat,X: produc3658429121746597890et_nat] :
% 0.34/0.62 ( ( produc1747169720485888550at_nat @ F @ ( produc2974404638222180585et_nat @ G @ X ) )
% 0.34/0.62 = ( product_Pair_nat_nat @ ( G @ ( produc1824681642469235216et_nat @ X ) ) @ ( F @ ( produc8586169260539613262et_nat @ X ) ) ) ) ).
% 0.34/0.62
% 0.34/0.62 % apsnd_apfst
% 0.34/0.62 thf(fact_4745_apsnd__apfst,axiom,
% 0.34/0.62 ! [F: set_nat > int,G: heap_e7401611519738050253t_unit > int,X: produc3658429121746597890et_nat] :
% 0.34/0.62 ( ( produc2744032961877892830nt_int @ F @ ( produc3346058198459025221et_nat @ G @ X ) )
% 0.34/0.62 = ( product_Pair_int_int @ ( G @ ( produc1824681642469235216et_nat @ X ) ) @ ( F @ ( produc8586169260539613262et_nat @ X ) ) ) ) ).
% 0.34/0.62
% 0.34/0.62 % apsnd_apfst
% 0.34/0.62 thf(fact_4746_apsnd__apfst,axiom,
% 0.34/0.62 ! [F: set_nat > produc3658429121746597890et_nat,G: heap_e7401611519738050253t_unit > produc3658429121746597890et_nat > $o,X: produc3658429121746597890et_nat] :
% 0.34/0.62 ( ( produc1718959742329256773_nat_o @ F @ ( produc962520821680634754et_nat @ G @ X ) )
% 0.34/0.62 = ( produc5001842942810119800et_nat @ ( G @ ( produc1824681642469235216et_nat @ X ) ) @ ( F @ ( produc8586169260539613262et_nat @ X ) ) ) ) ).
% 0.34/0.62
% 0.34/0.62 % apsnd_apfst
% 0.34/0.62 thf(fact_4747_apsnd__apfst,axiom,
% 0.34/0.62 ! [F: set_nat > produc3925858234332021118et_nat,G: heap_e7401611519738050253t_unit > produc3658429121746597890et_nat > $o,X: produc3658429121746597890et_nat] :
% 0.34/0.62 ( ( produc3009152965630989513_nat_o @ F @ ( produc962520821680634754et_nat @ G @ X ) )
% 0.34/0.62 = ( produc2245416461498447860et_nat @ ( G @ ( produc1824681642469235216et_nat @ X ) ) @ ( F @ ( produc8586169260539613262et_nat @ X ) ) ) ) ).
% 0.34/0.62
% 0.34/0.62 % apsnd_apfst
% 0.34/0.62 thf(fact_4748_apsnd__apfst,axiom,
% 0.34/0.62 ! [F: int > set_nat,G: int > heap_e7401611519738050253t_unit,X: product_prod_int_int] :
% 0.34/0.62 ( ( produc2122153897363312795t_unit @ F @ ( produc1158806734383251089it_int @ G @ X ) )
% 0.34/0.62 = ( produc7507926704131184380et_nat @ ( G @ ( product_fst_int_int @ X ) ) @ ( F @ ( product_snd_int_int @ X ) ) ) ) ).
% 0.34/0.62
% 0.34/0.62 % apsnd_apfst
% 0.34/0.62 thf(fact_4749_apsnd__apfst,axiom,
% 0.34/0.62 ! [F: int > nat,G: int > nat,X: product_prod_int_int] :
% 0.34/0.62 ( ( produc3466418871192872140at_nat @ F @ ( produc1355883850669603558at_int @ G @ X ) )
% 0.34/0.62 = ( product_Pair_nat_nat @ ( G @ ( product_fst_int_int @ X ) ) @ ( F @ ( product_snd_int_int @ X ) ) ) ) ).
% 0.34/0.62
% 0.34/0.62 % apsnd_apfst
% 0.34/0.62 thf(fact_4750_apsnd__apfst,axiom,
% 0.34/0.62 ! [F: int > int,G: int > int,X: product_prod_int_int] :
% 0.34/0.62 ( ( produc4463282112584876420nt_int @ F @ ( produc2355237562570658114nt_int @ G @ X ) )
% 0.34/0.62 = ( product_Pair_int_int @ ( G @ ( product_fst_int_int @ X ) ) @ ( F @ ( product_snd_int_int @ X ) ) ) ) ).
% 0.34/0.62
% 0.34/0.62 % apsnd_apfst
% 0.34/0.62 thf(fact_4751_apsnd__apfst,axiom,
% 0.34/0.62 ! [F: int > produc3658429121746597890et_nat,G: int > produc3658429121746597890et_nat > $o,X: product_prod_int_int] :
% 0.34/0.62 ( ( produc3447581998352395295_nat_o @ F @ ( produc1933289243015553937_o_int @ G @ X ) )
% 0.34/0.62 = ( produc5001842942810119800et_nat @ ( G @ ( product_fst_int_int @ X ) ) @ ( F @ ( product_snd_int_int @ X ) ) ) ) ).
% 0.34/0.62
% 0.34/0.62 % apsnd_apfst
% 0.34/0.62 thf(fact_4752_apsnd__apfst,axiom,
% 0.34/0.62 ! [F: int > produc3925858234332021118et_nat,G: int > produc3658429121746597890et_nat > $o,X: product_prod_int_int] :
% 0.34/0.62 ( ( produc1783401272518604195_nat_o @ F @ ( produc1933289243015553937_o_int @ G @ X ) )
% 0.34/0.62 = ( produc2245416461498447860et_nat @ ( G @ ( product_fst_int_int @ X ) ) @ ( F @ ( product_snd_int_int @ X ) ) ) ) ).
% 0.34/0.62
% 0.34/0.62 % apsnd_apfst
% 0.34/0.62 thf(fact_4753_apsnd__apfst__commute,axiom,
% 0.34/0.62 ! [F: code_integer > code_integer,G: code_integer > code_integer,P3: produc8923325533196201883nteger] :
% 0.34/0.62 ( ( produc6499014454317279255nteger @ F @ ( produc3195571438000033749nteger @ G @ P3 ) )
% 0.34/0.62 = ( produc3195571438000033749nteger @ G @ ( produc6499014454317279255nteger @ F @ P3 ) ) ) ).
% 0.34/0.62
% 0.34/0.62 % apsnd_apfst_commute
% 0.34/0.62 thf(fact_4754_fstI,axiom,
% 0.34/0.62 ! [X: produc3658429121746597890et_nat,Y: heap_e7401611519738050253t_unit,Z: set_nat] :
% 0.34/0.62 ( ( X
% 0.34/0.62 = ( produc7507926704131184380et_nat @ Y @ Z ) )
% 0.34/0.62 => ( ( produc1824681642469235216et_nat @ X )
% 0.34/0.62 = Y ) ) ).
% 0.34/0.62
% 0.34/0.62 % fstI
% 0.34/0.62 thf(fact_4755_fstI,axiom,
% 0.34/0.62 ! [X: product_prod_nat_nat,Y: nat,Z: nat] :
% 0.34/0.62 ( ( X
% 0.34/0.62 = ( product_Pair_nat_nat @ Y @ Z ) )
% 0.34/0.62 => ( ( product_fst_nat_nat @ X )
% 0.34/0.62 = Y ) ) ).
% 0.34/0.62
% 0.34/0.62 % fstI
% 0.34/0.62 thf(fact_4756_fstI,axiom,
% 0.34/0.62 ! [X: produc3925858234332021118et_nat,Y: produc3658429121746597890et_nat > $o,Z: produc3658429121746597890et_nat] :
% 0.34/0.62 ( ( X
% 0.34/0.62 = ( produc5001842942810119800et_nat @ Y @ Z ) )
% 0.34/0.62 => ( ( produc995936583742144908et_nat @ X )
% 0.34/0.62 = Y ) ) ).
% 0.34/0.62
% 0.34/0.62 % fstI
% 0.34/0.62 thf(fact_4757_fstI,axiom,
% 0.34/0.62 ! [X: produc2732055786443039994et_nat,Y: produc3658429121746597890et_nat > $o,Z: produc3925858234332021118et_nat] :
% 0.34/0.62 ( ( X
% 0.34/0.62 = ( produc2245416461498447860et_nat @ Y @ Z ) )
% 0.34/0.62 => ( ( produc180342877477747464et_nat @ X )
% 0.34/0.62 = Y ) ) ).
% 0.34/0.62
% 0.34/0.62 % fstI
% 0.34/0.62 thf(fact_4758_fstI,axiom,
% 0.34/0.62 ! [X: product_prod_int_int,Y: int,Z: int] :
% 0.34/0.62 ( ( X
% 0.34/0.62 = ( product_Pair_int_int @ Y @ Z ) )
% 0.34/0.62 => ( ( product_fst_int_int @ X )
% 0.34/0.62 = Y ) ) ).
% 0.34/0.62
% 0.34/0.62 % fstI
% 0.34/0.62 thf(fact_4759_fst__eqD,axiom,
% 0.34/0.62 ! [X: heap_e7401611519738050253t_unit,Y: set_nat,A2: heap_e7401611519738050253t_unit] :
% 0.34/0.62 ( ( ( produc1824681642469235216et_nat @ ( produc7507926704131184380et_nat @ X @ Y ) )
% 0.34/0.62 = A2 )
% 0.34/0.62 => ( X = A2 ) ) ).
% 0.34/0.62
% 0.34/0.62 % fst_eqD
% 0.34/0.62 thf(fact_4760_fst__eqD,axiom,
% 0.34/0.62 ! [X: nat,Y: nat,A2: nat] :
% 0.34/0.62 ( ( ( product_fst_nat_nat @ ( product_Pair_nat_nat @ X @ Y ) )
% 0.34/0.62 = A2 )
% 0.34/0.62 => ( X = A2 ) ) ).
% 0.34/0.62
% 0.34/0.62 % fst_eqD
% 0.34/0.62 thf(fact_4761_fst__eqD,axiom,
% 0.34/0.62 ! [X: produc3658429121746597890et_nat > $o,Y: produc3658429121746597890et_nat,A2: produc3658429121746597890et_nat > $o] :
% 0.34/0.62 ( ( ( produc995936583742144908et_nat @ ( produc5001842942810119800et_nat @ X @ Y ) )
% 0.34/0.62 = A2 )
% 0.34/0.62 => ( X = A2 ) ) ).
% 0.34/0.62
% 0.34/0.62 % fst_eqD
% 0.34/0.62 thf(fact_4762_fst__eqD,axiom,
% 0.34/0.62 ! [X: produc3658429121746597890et_nat > $o,Y: produc3925858234332021118et_nat,A2: produc3658429121746597890et_nat > $o] :
% 0.34/0.62 ( ( ( produc180342877477747464et_nat @ ( produc2245416461498447860et_nat @ X @ Y ) )
% 0.34/0.62 = A2 )
% 0.34/0.62 => ( X = A2 ) ) ).
% 0.34/0.62
% 0.34/0.62 % fst_eqD
% 0.34/0.62 thf(fact_4763_fst__eqD,axiom,
% 0.34/0.62 ! [X: int,Y: int,A2: int] :
% 0.34/0.62 ( ( ( product_fst_int_int @ ( product_Pair_int_int @ X @ Y ) )
% 0.34/0.62 = A2 )
% 0.34/0.62 => ( X = A2 ) ) ).
% 0.34/0.62
% 0.34/0.62 % fst_eqD
% 0.34/0.62 thf(fact_4764_fst__conv,axiom,
% 0.34/0.62 ! [X1: heap_e7401611519738050253t_unit,X22: set_nat] :
% 0.34/0.62 ( ( produc1824681642469235216et_nat @ ( produc7507926704131184380et_nat @ X1 @ X22 ) )
% 0.34/0.62 = X1 ) ).
% 0.34/0.62
% 0.34/0.62 % fst_conv
% 0.34/0.62 thf(fact_4765_fst__conv,axiom,
% 0.34/0.62 ! [X1: nat,X22: nat] :
% 0.34/0.62 ( ( product_fst_nat_nat @ ( product_Pair_nat_nat @ X1 @ X22 ) )
% 0.34/0.62 = X1 ) ).
% 0.34/0.62
% 0.34/0.62 % fst_conv
% 0.34/0.62 thf(fact_4766_fst__conv,axiom,
% 0.34/0.62 ! [X1: produc3658429121746597890et_nat > $o,X22: produc3658429121746597890et_nat] :
% 0.34/0.62 ( ( produc995936583742144908et_nat @ ( produc5001842942810119800et_nat @ X1 @ X22 ) )
% 0.34/0.62 = X1 ) ).
% 0.34/0.62
% 0.34/0.62 % fst_conv
% 0.34/0.62 thf(fact_4767_fst__conv,axiom,
% 0.34/0.62 ! [X1: produc3658429121746597890et_nat > $o,X22: produc3925858234332021118et_nat] :
% 0.34/0.62 ( ( produc180342877477747464et_nat @ ( produc2245416461498447860et_nat @ X1 @ X22 ) )
% 0.34/0.62 = X1 ) ).
% 0.34/0.62
% 0.34/0.62 % fst_conv
% 0.34/0.62 thf(fact_4768_fst__conv,axiom,
% 0.34/0.62 ! [X1: int,X22: int] :
% 0.34/0.62 ( ( product_fst_int_int @ ( product_Pair_int_int @ X1 @ X22 ) )
% 0.34/0.62 = X1 ) ).
% 0.34/0.62
% 0.34/0.62 % fst_conv
% 0.34/0.62 thf(fact_4769_eq__fst__iff,axiom,
% 0.34/0.62 ! [A2: heap_e7401611519738050253t_unit,P3: produc3658429121746597890et_nat] :
% 0.34/0.62 ( ( A2
% 0.34/0.62 = ( produc1824681642469235216et_nat @ P3 ) )
% 0.34/0.62 = ( ? [B6: set_nat] :
% 0.34/0.62 ( P3
% 0.34/0.62 = ( produc7507926704131184380et_nat @ A2 @ B6 ) ) ) ) ).
% 0.34/0.62
% 0.34/0.62 % eq_fst_iff
% 0.34/0.62 thf(fact_4770_eq__fst__iff,axiom,
% 0.34/0.62 ! [A2: nat,P3: product_prod_nat_nat] :
% 0.34/0.62 ( ( A2
% 0.34/0.62 = ( product_fst_nat_nat @ P3 ) )
% 0.34/0.62 = ( ? [B6: nat] :
% 0.34/0.62 ( P3
% 0.34/0.62 = ( product_Pair_nat_nat @ A2 @ B6 ) ) ) ) ).
% 0.34/0.62
% 0.34/0.62 % eq_fst_iff
% 0.34/0.62 thf(fact_4771_eq__fst__iff,axiom,
% 0.34/0.62 ! [A2: produc3658429121746597890et_nat > $o,P3: produc3925858234332021118et_nat] :
% 0.34/0.62 ( ( A2
% 0.34/0.62 = ( produc995936583742144908et_nat @ P3 ) )
% 0.34/0.62 = ( ? [B6: produc3658429121746597890et_nat] :
% 0.34/0.62 ( P3
% 0.34/0.62 = ( produc5001842942810119800et_nat @ A2 @ B6 ) ) ) ) ).
% 0.34/0.62
% 0.34/0.62 % eq_fst_iff
% 0.34/0.62 thf(fact_4772_eq__fst__iff,axiom,
% 0.34/0.62 ! [A2: produc3658429121746597890et_nat > $o,P3: produc2732055786443039994et_nat] :
% 0.34/0.62 ( ( A2
% 0.34/0.62 = ( produc180342877477747464et_nat @ P3 ) )
% 0.34/0.62 = ( ? [B6: produc3925858234332021118et_nat] :
% 0.34/0.62 ( P3
% 0.34/0.62 = ( produc2245416461498447860et_nat @ A2 @ B6 ) ) ) ) ).
% 0.34/0.62
% 0.34/0.62 % eq_fst_iff
% 0.34/0.62 thf(fact_4773_eq__fst__iff,axiom,
% 0.34/0.62 ! [A2: int,P3: product_prod_int_int] :
% 0.34/0.62 ( ( A2
% 0.34/0.62 = ( product_fst_int_int @ P3 ) )
% 0.34/0.62 = ( ? [B6: int] :
% 0.34/0.62 ( P3
% 0.34/0.62 = ( product_Pair_int_int @ A2 @ B6 ) ) ) ) ).
% 0.34/0.62
% 0.34/0.62 % eq_fst_iff
% 0.34/0.62 thf(fact_4774_fstE,axiom,
% 0.34/0.62 ! [X: produc3658429121746597890et_nat,A2: heap_e7401611519738050253t_unit,B: set_nat,P: heap_e7401611519738050253t_unit > $o] :
% 0.34/0.62 ( ( X
% 0.34/0.62 = ( produc7507926704131184380et_nat @ A2 @ B ) )
% 0.34/0.62 => ( ( P @ ( produc1824681642469235216et_nat @ X ) )
% 0.34/0.62 => ( P @ A2 ) ) ) ).
% 0.34/0.62
% 0.34/0.62 % fstE
% 0.34/0.62 thf(fact_4775_fstE,axiom,
% 0.34/0.62 ! [X: product_prod_nat_nat,A2: nat,B: nat,P: nat > $o] :
% 0.34/0.62 ( ( X
% 0.34/0.62 = ( product_Pair_nat_nat @ A2 @ B ) )
% 0.34/0.62 => ( ( P @ ( product_fst_nat_nat @ X ) )
% 0.34/0.62 => ( P @ A2 ) ) ) ).
% 0.34/0.62
% 0.34/0.62 % fstE
% 0.34/0.62 thf(fact_4776_fstE,axiom,
% 0.34/0.62 ! [X: produc3925858234332021118et_nat,A2: produc3658429121746597890et_nat > $o,B: produc3658429121746597890et_nat,P: ( produc3658429121746597890et_nat > $o ) > $o] :
% 0.34/0.62 ( ( X
% 0.34/0.62 = ( produc5001842942810119800et_nat @ A2 @ B ) )
% 0.34/0.62 => ( ( P @ ( produc995936583742144908et_nat @ X ) )
% 0.34/0.62 => ( P @ A2 ) ) ) ).
% 0.34/0.62
% 0.34/0.62 % fstE
% 0.34/0.62 thf(fact_4777_fstE,axiom,
% 0.34/0.62 ! [X: produc2732055786443039994et_nat,A2: produc3658429121746597890et_nat > $o,B: produc3925858234332021118et_nat,P: ( produc3658429121746597890et_nat > $o ) > $o] :
% 0.34/0.62 ( ( X
% 0.34/0.62 = ( produc2245416461498447860et_nat @ A2 @ B ) )
% 0.34/0.62 => ( ( P @ ( produc180342877477747464et_nat @ X ) )
% 0.34/0.62 => ( P @ A2 ) ) ) ).
% 0.34/0.62
% 0.34/0.62 % fstE
% 0.34/0.62 thf(fact_4778_fstE,axiom,
% 0.34/0.62 ! [X: product_prod_int_int,A2: int,B: int,P: int > $o] :
% 0.34/0.62 ( ( X
% 0.34/0.62 = ( product_Pair_int_int @ A2 @ B ) )
% 0.34/0.62 => ( ( P @ ( product_fst_int_int @ X ) )
% 0.34/0.62 => ( P @ A2 ) ) ) ).
% 0.34/0.62
% 0.34/0.62 % fstE
% 0.34/0.62 thf(fact_4779_prod__eq__iff,axiom,
% 0.34/0.62 ( ( ^ [Y4: produc3658429121746597890et_nat,Z4: produc3658429121746597890et_nat] : ( Y4 = Z4 ) )
% 0.34/0.62 = ( ^ [S7: produc3658429121746597890et_nat,T: produc3658429121746597890et_nat] :
% 0.34/0.62 ( ( ( produc1824681642469235216et_nat @ S7 )
% 0.34/0.62 = ( produc1824681642469235216et_nat @ T ) )
% 0.34/0.62 & ( ( produc8586169260539613262et_nat @ S7 )
% 0.34/0.62 = ( produc8586169260539613262et_nat @ T ) ) ) ) ) ).
% 0.34/0.62
% 0.34/0.62 % prod_eq_iff
% 0.34/0.62 thf(fact_4780_prod__eq__iff,axiom,
% 0.34/0.62 ( ( ^ [Y4: product_prod_int_int,Z4: product_prod_int_int] : ( Y4 = Z4 ) )
% 0.34/0.62 = ( ^ [S7: product_prod_int_int,T: product_prod_int_int] :
% 0.34/0.62 ( ( ( product_fst_int_int @ S7 )
% 0.34/0.62 = ( product_fst_int_int @ T ) )
% 0.34/0.62 & ( ( product_snd_int_int @ S7 )
% 0.34/0.62 = ( product_snd_int_int @ T ) ) ) ) ) ).
% 0.34/0.62
% 0.34/0.62 % prod_eq_iff
% 0.34/0.62 thf(fact_4781_prod__eqI,axiom,
% 0.34/0.62 ! [P3: produc3658429121746597890et_nat,Q4: produc3658429121746597890et_nat] :
% 0.34/0.62 ( ( ( produc1824681642469235216et_nat @ P3 )
% 0.34/0.62 = ( produc1824681642469235216et_nat @ Q4 ) )
% 0.34/0.62 => ( ( ( produc8586169260539613262et_nat @ P3 )
% 0.34/0.62 = ( produc8586169260539613262et_nat @ Q4 ) )
% 0.34/0.62 => ( P3 = Q4 ) ) ) ).
% 0.34/0.62
% 0.34/0.62 % prod_eqI
% 0.34/0.62 thf(fact_4782_prod__eqI,axiom,
% 0.34/0.62 ! [P3: product_prod_int_int,Q4: product_prod_int_int] :
% 0.34/0.62 ( ( ( product_fst_int_int @ P3 )
% 0.34/0.62 = ( product_fst_int_int @ Q4 ) )
% 0.34/0.62 => ( ( ( product_snd_int_int @ P3 )
% 0.34/0.62 = ( product_snd_int_int @ Q4 ) )
% 0.34/0.62 => ( P3 = Q4 ) ) ) ).
% 0.34/0.62
% 0.34/0.62 % prod_eqI
% 0.34/0.62 thf(fact_4783_prod_Oexpand,axiom,
% 0.34/0.62 ! [Prod: produc3658429121746597890et_nat,Prod2: produc3658429121746597890et_nat] :
% 0.34/0.62 ( ( ( ( produc1824681642469235216et_nat @ Prod )
% 0.34/0.62 = ( produc1824681642469235216et_nat @ Prod2 ) )
% 0.34/0.62 & ( ( produc8586169260539613262et_nat @ Prod )
% 0.34/0.62 = ( produc8586169260539613262et_nat @ Prod2 ) ) )
% 0.34/0.62 => ( Prod = Prod2 ) ) ).
% 0.34/0.62
% 0.34/0.62 % prod.expand
% 0.34/0.62 thf(fact_4784_prod_Oexpand,axiom,
% 0.34/0.62 ! [Prod: product_prod_int_int,Prod2: product_prod_int_int] :
% 0.34/0.62 ( ( ( ( product_fst_int_int @ Prod )
% 0.34/0.62 = ( product_fst_int_int @ Prod2 ) )
% 0.34/0.62 & ( ( product_snd_int_int @ Prod )
% 0.34/0.62 = ( product_snd_int_int @ Prod2 ) ) )
% 0.34/0.62 => ( Prod = Prod2 ) ) ).
% 0.34/0.62
% 0.34/0.62 % prod.expand
% 0.34/0.62 thf(fact_4785_All__prod__contract,axiom,
% 0.34/0.62 ! [P: heap_e7401611519738050253t_unit > set_nat > $o] :
% 0.34/0.62 ( ( ! [A6: heap_e7401611519738050253t_unit,X7: set_nat] : ( P @ A6 @ X7 ) )
% 0.34/0.62 = ( ! [Z2: produc3658429121746597890et_nat] : ( P @ ( produc1824681642469235216et_nat @ Z2 ) @ ( produc8586169260539613262et_nat @ Z2 ) ) ) ) ).
% 0.34/0.62
% 0.34/0.62 % All_prod_contract
% 0.34/0.62 thf(fact_4786_All__prod__contract,axiom,
% 0.34/0.62 ! [P: int > int > $o] :
% 0.34/0.62 ( ( ! [A6: int,X7: int] : ( P @ A6 @ X7 ) )
% 0.34/0.62 = ( ! [Z2: product_prod_int_int] : ( P @ ( product_fst_int_int @ Z2 ) @ ( product_snd_int_int @ Z2 ) ) ) ) ).
% 0.34/0.62
% 0.34/0.62 % All_prod_contract
% 0.34/0.62 thf(fact_4787_Ex__prod__contract,axiom,
% 0.34/0.62 ! [P: heap_e7401611519738050253t_unit > set_nat > $o] :
% 0.34/0.62 ( ( ? [A6: heap_e7401611519738050253t_unit,X7: set_nat] : ( P @ A6 @ X7 ) )
% 0.34/0.62 = ( ? [Z2: produc3658429121746597890et_nat] : ( P @ ( produc1824681642469235216et_nat @ Z2 ) @ ( produc8586169260539613262et_nat @ Z2 ) ) ) ) ).
% 0.34/0.62
% 0.34/0.62 % Ex_prod_contract
% 0.34/0.62 thf(fact_4788_Ex__prod__contract,axiom,
% 0.34/0.62 ! [P: int > int > $o] :
% 0.34/0.62 ( ( ? [A6: int,X7: int] : ( P @ A6 @ X7 ) )
% 0.34/0.62 = ( ? [Z2: product_prod_int_int] : ( P @ ( product_fst_int_int @ Z2 ) @ ( product_snd_int_int @ Z2 ) ) ) ) ).
% 0.34/0.62
% 0.34/0.62 % Ex_prod_contract
% 0.34/0.62 thf(fact_4789_apsnd__compose,axiom,
% 0.34/0.62 ! [F: code_integer > produc8923325533196201883nteger > produc8923325533196201883nteger,G: code_integer > code_integer,X: produc8923325533196201883nteger] :
% 0.34/0.62 ( ( produc9020447175693601247nteger @ F @ ( produc6499014454317279255nteger @ G @ X ) )
% 0.34/0.62 = ( produc9020447175693601247nteger @ ( comp_C1593894019821074884nteger @ F @ G ) @ X ) ) ).
% 0.34/0.62
% 0.34/0.62 % apsnd_compose
% 0.34/0.62 thf(fact_4790_apsnd__compose,axiom,
% 0.34/0.62 ! [F: code_integer > code_integer,G: code_integer > code_integer,X: produc8923325533196201883nteger] :
% 0.34/0.62 ( ( produc6499014454317279255nteger @ F @ ( produc6499014454317279255nteger @ G @ X ) )
% 0.34/0.62 = ( produc6499014454317279255nteger @ ( comp_C7449957260575251196nteger @ F @ G ) @ X ) ) ).
% 0.34/0.62
% 0.34/0.62 % apsnd_compose
% 0.34/0.62 thf(fact_4791_refl__on__INTER,axiom,
% 0.34/0.62 ! [S: set_nat,A: nat > set_nat,R2: nat > set_Pr1261947904930325089at_nat] :
% 0.34/0.62 ( ! [X3: nat] :
% 0.34/0.62 ( ( member_nat @ X3 @ S )
% 0.34/0.62 => ( refl_on_nat @ ( A @ X3 ) @ ( R2 @ X3 ) ) )
% 0.34/0.62 => ( refl_on_nat @ ( comple7806235888213564991et_nat @ ( image_nat_set_nat @ A @ S ) ) @ ( comple1930127330929219688at_nat @ ( image_7178329752028323786at_nat @ R2 @ S ) ) ) ) ).
% 0.34/0.62
% 0.34/0.62 % refl_on_INTER
% 0.34/0.62 thf(fact_4792_cInf__eq__non__empty,axiom,
% 0.34/0.62 ! [X4: set_o,A2: $o] :
% 0.34/0.62 ( ( X4 != bot_bot_set_o )
% 0.34/0.62 => ( ! [X3: $o] :
% 0.34/0.62 ( ( member_o @ X3 @ X4 )
% 0.34/0.62 => ( ord_less_eq_o @ A2 @ X3 ) )
% 0.34/0.62 => ( ! [Y2: $o] :
% 0.34/0.62 ( ! [X6: $o] :
% 0.34/0.62 ( ( member_o @ X6 @ X4 )
% 0.34/0.62 => ( ord_less_eq_o @ Y2 @ X6 ) )
% 0.34/0.62 => ( ord_less_eq_o @ Y2 @ A2 ) )
% 0.34/0.62 => ( ( complete_Inf_Inf_o @ X4 )
% 0.34/0.62 = A2 ) ) ) ) ).
% 0.34/0.62
% 0.34/0.62 % cInf_eq_non_empty
% 0.34/0.62 thf(fact_4793_cInf__eq__non__empty,axiom,
% 0.34/0.62 ! [X4: set_Pr4532377907799695533_nat_o,A2: produc3658429121746597890et_nat > $o] :
% 0.34/0.62 ( ( X4 != bot_bo7824918357723582233_nat_o )
% 0.34/0.62 => ( ! [X3: produc3658429121746597890et_nat > $o] :
% 0.34/0.62 ( ( member6576561426505652726_nat_o @ X3 @ X4 )
% 0.34/0.62 => ( ord_le729326519192465773_nat_o @ A2 @ X3 ) )
% 0.34/0.62 => ( ! [Y2: produc3658429121746597890et_nat > $o] :
% 0.34/0.62 ( ! [X6: produc3658429121746597890et_nat > $o] :
% 0.34/0.62 ( ( member6576561426505652726_nat_o @ X6 @ X4 )
% 0.34/0.62 => ( ord_le729326519192465773_nat_o @ Y2 @ X6 ) )
% 0.34/0.62 => ( ord_le729326519192465773_nat_o @ Y2 @ A2 ) )
% 0.34/0.62 => ( ( comple90455603224345172_nat_o @ X4 )
% 0.34/0.62 = A2 ) ) ) ) ).
% 0.34/0.62
% 0.34/0.62 % cInf_eq_non_empty
% 0.34/0.62 thf(fact_4794_cInf__eq__non__empty,axiom,
% 0.34/0.62 ! [X4: set_filter_nat,A2: filter_nat] :
% 0.34/0.62 ( ( X4 != bot_bo498966703094740906er_nat )
% 0.34/0.62 => ( ! [X3: filter_nat] :
% 0.34/0.62 ( ( member_filter_nat @ X3 @ X4 )
% 0.34/0.62 => ( ord_le2510731241096832064er_nat @ A2 @ X3 ) )
% 0.34/0.62 => ( ! [Y2: filter_nat] :
% 0.34/0.62 ( ! [X6: filter_nat] :
% 0.34/0.62 ( ( member_filter_nat @ X6 @ X4 )
% 0.34/0.62 => ( ord_le2510731241096832064er_nat @ Y2 @ X6 ) )
% 0.34/0.62 => ( ord_le2510731241096832064er_nat @ Y2 @ A2 ) )
% 0.34/0.62 => ( ( comple6612786467464916249er_nat @ X4 )
% 0.34/0.62 = A2 ) ) ) ) ).
% 0.34/0.62
% 0.34/0.62 % cInf_eq_non_empty
% 0.34/0.62 thf(fact_4795_cInf__eq__non__empty,axiom,
% 0.34/0.62 ! [X4: set_int,A2: int] :
% 0.34/0.62 ( ( X4 != bot_bot_set_int )
% 0.34/0.62 => ( ! [X3: int] :
% 0.34/0.62 ( ( member_int @ X3 @ X4 )
% 0.34/0.62 => ( ord_less_eq_int @ A2 @ X3 ) )
% 0.34/0.62 => ( ! [Y2: int] :
% 0.34/0.62 ( ! [X6: int] :
% 0.34/0.62 ( ( member_int @ X6 @ X4 )
% 0.34/0.62 => ( ord_less_eq_int @ Y2 @ X6 ) )
% 0.34/0.62 => ( ord_less_eq_int @ Y2 @ A2 ) )
% 0.34/0.62 => ( ( complete_Inf_Inf_int @ X4 )
% 0.34/0.62 = A2 ) ) ) ) ).
% 0.34/0.62
% 0.34/0.62 % cInf_eq_non_empty
% 0.34/0.62 thf(fact_4796_cInf__eq__non__empty,axiom,
% 0.34/0.62 ! [X4: set_nat,A2: nat] :
% 0.34/0.62 ( ( X4 != bot_bot_set_nat )
% 0.34/0.62 => ( ! [X3: nat] :
% 0.34/0.62 ( ( member_nat @ X3 @ X4 )
% 0.34/0.62 => ( ord_less_eq_nat @ A2 @ X3 ) )
% 0.34/0.62 => ( ! [Y2: nat] :
% 0.34/0.62 ( ! [X6: nat] :
% 0.34/0.62 ( ( member_nat @ X6 @ X4 )
% 0.34/0.62 => ( ord_less_eq_nat @ Y2 @ X6 ) )
% 0.34/0.62 => ( ord_less_eq_nat @ Y2 @ A2 ) )
% 0.34/0.62 => ( ( complete_Inf_Inf_nat @ X4 )
% 0.34/0.62 = A2 ) ) ) ) ).
% 0.34/0.62
% 0.34/0.62 % cInf_eq_non_empty
% 0.34/0.62 thf(fact_4797_cInf__eq__non__empty,axiom,
% 0.34/0.62 ! [X4: set_set_nat,A2: set_nat] :
% 0.34/0.62 ( ( X4 != bot_bot_set_set_nat )
% 0.34/0.62 => ( ! [X3: set_nat] :
% 0.34/0.62 ( ( member_set_nat @ X3 @ X4 )
% 0.34/0.62 => ( ord_less_eq_set_nat @ A2 @ X3 ) )
% 0.34/0.62 => ( ! [Y2: set_nat] :
% 0.34/0.62 ( ! [X6: set_nat] :
% 0.34/0.62 ( ( member_set_nat @ X6 @ X4 )
% 0.34/0.62 => ( ord_less_eq_set_nat @ Y2 @ X6 ) )
% 0.34/0.62 => ( ord_less_eq_set_nat @ Y2 @ A2 ) )
% 0.34/0.62 => ( ( comple7806235888213564991et_nat @ X4 )
% 0.34/0.62 = A2 ) ) ) ) ).
% 0.34/0.62
% 0.34/0.62 % cInf_eq_non_empty
% 0.34/0.62 thf(fact_4798_cInf__eq__non__empty,axiom,
% 0.34/0.62 ! [X4: set_Product_unit,A2: product_unit] :
% 0.34/0.62 ( ( X4 != bot_bo3957492148770167129t_unit )
% 0.34/0.62 => ( ! [X3: product_unit] :
% 0.34/0.62 ( ( member_Product_unit @ X3 @ X4 )
% 0.34/0.62 => ( ord_le3221252021190050221t_unit @ A2 @ X3 ) )
% 0.34/0.62 => ( ! [Y2: product_unit] :
% 0.34/0.62 ( ! [X6: product_unit] :
% 0.34/0.62 ( ( member_Product_unit @ X6 @ X4 )
% 0.34/0.62 => ( ord_le3221252021190050221t_unit @ Y2 @ X6 ) )
% 0.34/0.62 => ( ord_le3221252021190050221t_unit @ Y2 @ A2 ) )
% 0.34/0.62 => ( ( comple2584293577114468500t_unit @ X4 )
% 0.34/0.62 = A2 ) ) ) ) ).
% 0.34/0.62
% 0.34/0.62 % cInf_eq_non_empty
% 0.34/0.62 thf(fact_4799_cInf__greatest,axiom,
% 0.34/0.62 ! [X4: set_o,Z: $o] :
% 0.34/0.62 ( ( X4 != bot_bot_set_o )
% 0.34/0.62 => ( ! [X3: $o] :
% 0.34/0.62 ( ( member_o @ X3 @ X4 )
% 0.34/0.62 => ( ord_less_eq_o @ Z @ X3 ) )
% 0.34/0.62 => ( ord_less_eq_o @ Z @ ( complete_Inf_Inf_o @ X4 ) ) ) ) ).
% 0.34/0.62
% 0.34/0.62 % cInf_greatest
% 0.34/0.62 thf(fact_4800_cInf__greatest,axiom,
% 0.34/0.62 ! [X4: set_Pr4532377907799695533_nat_o,Z: produc3658429121746597890et_nat > $o] :
% 0.34/0.62 ( ( X4 != bot_bo7824918357723582233_nat_o )
% 0.34/0.62 => ( ! [X3: produc3658429121746597890et_nat > $o] :
% 0.34/0.62 ( ( member6576561426505652726_nat_o @ X3 @ X4 )
% 0.34/0.62 => ( ord_le729326519192465773_nat_o @ Z @ X3 ) )
% 0.34/0.63 => ( ord_le729326519192465773_nat_o @ Z @ ( comple90455603224345172_nat_o @ X4 ) ) ) ) ).
% 0.34/0.63
% 0.34/0.63 % cInf_greatest
% 0.34/0.63 thf(fact_4801_cInf__greatest,axiom,
% 0.34/0.63 ! [X4: set_filter_nat,Z: filter_nat] :
% 0.34/0.63 ( ( X4 != bot_bo498966703094740906er_nat )
% 0.34/0.63 => ( ! [X3: filter_nat] :
% 0.34/0.63 ( ( member_filter_nat @ X3 @ X4 )
% 0.34/0.63 => ( ord_le2510731241096832064er_nat @ Z @ X3 ) )
% 0.34/0.63 => ( ord_le2510731241096832064er_nat @ Z @ ( comple6612786467464916249er_nat @ X4 ) ) ) ) ).
% 0.34/0.63
% 0.34/0.63 % cInf_greatest
% 0.34/0.63 thf(fact_4802_cInf__greatest,axiom,
% 0.34/0.63 ! [X4: set_int,Z: int] :
% 0.34/0.63 ( ( X4 != bot_bot_set_int )
% 0.34/0.63 => ( ! [X3: int] :
% 0.34/0.63 ( ( member_int @ X3 @ X4 )
% 0.34/0.63 => ( ord_less_eq_int @ Z @ X3 ) )
% 0.34/0.63 => ( ord_less_eq_int @ Z @ ( complete_Inf_Inf_int @ X4 ) ) ) ) ).
% 0.34/0.63
% 0.34/0.63 % cInf_greatest
% 0.34/0.63 thf(fact_4803_cInf__greatest,axiom,
% 0.34/0.63 ! [X4: set_nat,Z: nat] :
% 0.34/0.63 ( ( X4 != bot_bot_set_nat )
% 0.34/0.63 => ( ! [X3: nat] :
% 0.34/0.63 ( ( member_nat @ X3 @ X4 )
% 0.34/0.63 => ( ord_less_eq_nat @ Z @ X3 ) )
% 0.34/0.63 => ( ord_less_eq_nat @ Z @ ( complete_Inf_Inf_nat @ X4 ) ) ) ) ).
% 0.34/0.63
% 0.34/0.63 % cInf_greatest
% 0.34/0.63 thf(fact_4804_cInf__greatest,axiom,
% 0.34/0.63 ! [X4: set_set_nat,Z: set_nat] :
% 0.34/0.63 ( ( X4 != bot_bot_set_set_nat )
% 0.34/0.63 => ( ! [X3: set_nat] :
% 0.34/0.63 ( ( member_set_nat @ X3 @ X4 )
% 0.34/0.63 => ( ord_less_eq_set_nat @ Z @ X3 ) )
% 0.34/0.63 => ( ord_less_eq_set_nat @ Z @ ( comple7806235888213564991et_nat @ X4 ) ) ) ) ).
% 0.34/0.63
% 0.34/0.63 % cInf_greatest
% 0.34/0.63 thf(fact_4805_cInf__greatest,axiom,
% 0.34/0.63 ! [X4: set_Product_unit,Z: product_unit] :
% 0.34/0.63 ( ( X4 != bot_bo3957492148770167129t_unit )
% 0.34/0.63 => ( ! [X3: product_unit] :
% 0.34/0.63 ( ( member_Product_unit @ X3 @ X4 )
% 0.34/0.63 => ( ord_le3221252021190050221t_unit @ Z @ X3 ) )
% 0.34/0.63 => ( ord_le3221252021190050221t_unit @ Z @ ( comple2584293577114468500t_unit @ X4 ) ) ) ) ).
% 0.34/0.63
% 0.34/0.63 % cInf_greatest
% 0.34/0.63 thf(fact_4806_Inf__less__eq,axiom,
% 0.34/0.63 ! [A: set_o,U: $o] :
% 0.34/0.63 ( ! [V3: $o] :
% 0.34/0.63 ( ( member_o @ V3 @ A )
% 0.34/0.63 => ( ord_less_eq_o @ V3 @ U ) )
% 0.34/0.63 => ( ( A != bot_bot_set_o )
% 0.34/0.63 => ( ord_less_eq_o @ ( complete_Inf_Inf_o @ A ) @ U ) ) ) ).
% 0.34/0.63
% 0.34/0.63 % Inf_less_eq
% 0.34/0.63 thf(fact_4807_Inf__less__eq,axiom,
% 0.34/0.63 ! [A: set_Pr4532377907799695533_nat_o,U: produc3658429121746597890et_nat > $o] :
% 0.34/0.63 ( ! [V3: produc3658429121746597890et_nat > $o] :
% 0.34/0.63 ( ( member6576561426505652726_nat_o @ V3 @ A )
% 0.34/0.63 => ( ord_le729326519192465773_nat_o @ V3 @ U ) )
% 0.34/0.63 => ( ( A != bot_bo7824918357723582233_nat_o )
% 0.34/0.63 => ( ord_le729326519192465773_nat_o @ ( comple90455603224345172_nat_o @ A ) @ U ) ) ) ).
% 0.34/0.63
% 0.34/0.63 % Inf_less_eq
% 0.34/0.63 thf(fact_4808_Inf__less__eq,axiom,
% 0.34/0.63 ! [A: set_filter_nat,U: filter_nat] :
% 0.34/0.63 ( ! [V3: filter_nat] :
% 0.34/0.63 ( ( member_filter_nat @ V3 @ A )
% 0.34/0.63 => ( ord_le2510731241096832064er_nat @ V3 @ U ) )
% 0.34/0.63 => ( ( A != bot_bo498966703094740906er_nat )
% 0.34/0.63 => ( ord_le2510731241096832064er_nat @ ( comple6612786467464916249er_nat @ A ) @ U ) ) ) ).
% 0.34/0.63
% 0.34/0.63 % Inf_less_eq
% 0.34/0.63 thf(fact_4809_Inf__less__eq,axiom,
% 0.34/0.63 ! [A: set_set_nat,U: set_nat] :
% 0.34/0.63 ( ! [V3: set_nat] :
% 0.34/0.63 ( ( member_set_nat @ V3 @ A )
% 0.34/0.63 => ( ord_less_eq_set_nat @ V3 @ U ) )
% 0.34/0.63 => ( ( A != bot_bot_set_set_nat )
% 0.34/0.63 => ( ord_less_eq_set_nat @ ( comple7806235888213564991et_nat @ A ) @ U ) ) ) ).
% 0.34/0.63
% 0.34/0.63 % Inf_less_eq
% 0.34/0.63 thf(fact_4810_Inf__less__eq,axiom,
% 0.34/0.63 ! [A: set_Product_unit,U: product_unit] :
% 0.34/0.63 ( ! [V3: product_unit] :
% 0.34/0.63 ( ( member_Product_unit @ V3 @ A )
% 0.34/0.63 => ( ord_le3221252021190050221t_unit @ V3 @ U ) )
% 0.34/0.63 => ( ( A != bot_bo3957492148770167129t_unit )
% 0.34/0.63 => ( ord_le3221252021190050221t_unit @ ( comple2584293577114468500t_unit @ A ) @ U ) ) ) ).
% 0.34/0.63
% 0.34/0.63 % Inf_less_eq
% 0.34/0.63 thf(fact_4811_cInf__lessD,axiom,
% 0.34/0.63 ! [X4: set_int,Z: int] :
% 0.34/0.63 ( ( X4 != bot_bot_set_int )
% 0.34/0.63 => ( ( ord_less_int @ ( complete_Inf_Inf_int @ X4 ) @ Z )
% 0.34/0.63 => ? [X3: int] :
% 0.34/0.63 ( ( member_int @ X3 @ X4 )
% 0.34/0.63 & ( ord_less_int @ X3 @ Z ) ) ) ) ).
% 0.34/0.63
% 0.34/0.63 % cInf_lessD
% 0.34/0.63 thf(fact_4812_cInf__lessD,axiom,
% 0.34/0.63 ! [X4: set_nat,Z: nat] :
% 0.34/0.63 ( ( X4 != bot_bot_set_nat )
% 0.34/0.63 => ( ( ord_less_nat @ ( complete_Inf_Inf_nat @ X4 ) @ Z )
% 0.34/0.63 => ? [X3: nat] :
% 0.34/0.63 ( ( member_nat @ X3 @ X4 )
% 0.34/0.63 & ( ord_less_nat @ X3 @ Z ) ) ) ) ).
% 0.34/0.63
% 0.34/0.63 % cInf_lessD
% 0.34/0.63 thf(fact_4813_cInf__lessD,axiom,
% 0.34/0.63 ! [X4: set_Product_unit,Z: product_unit] :
% 0.34/0.63 ( ( X4 != bot_bo3957492148770167129t_unit )
% 0.34/0.63 => ( ( ord_le361264281704409273t_unit @ ( comple2584293577114468500t_unit @ X4 ) @ Z )
% 0.34/0.63 => ? [X3: product_unit] :
% 0.34/0.63 ( ( member_Product_unit @ X3 @ X4 )
% 0.34/0.63 & ( ord_le361264281704409273t_unit @ X3 @ Z ) ) ) ) ).
% 0.34/0.63
% 0.34/0.63 % cInf_lessD
% 0.34/0.63 thf(fact_4814_INF__eq__const,axiom,
% 0.34/0.63 ! [I: set_Product_unit,F: product_unit > set_nat,X: set_nat] :
% 0.34/0.63 ( ( I != bot_bo3957492148770167129t_unit )
% 0.34/0.63 => ( ! [I3: product_unit] :
% 0.34/0.63 ( ( member_Product_unit @ I3 @ I )
% 0.34/0.63 => ( ( F @ I3 )
% 0.34/0.63 = X ) )
% 0.34/0.63 => ( ( comple7806235888213564991et_nat @ ( image_8173849991827986510et_nat @ F @ I ) )
% 0.34/0.63 = X ) ) ) ).
% 0.34/0.63
% 0.34/0.63 % INF_eq_const
% 0.34/0.63 thf(fact_4815_INF__eq__const,axiom,
% 0.34/0.63 ! [I: set_o,F: $o > set_nat,X: set_nat] :
% 0.34/0.63 ( ( I != bot_bot_set_o )
% 0.34/0.63 => ( ! [I3: $o] :
% 0.34/0.63 ( ( member_o @ I3 @ I )
% 0.34/0.63 => ( ( F @ I3 )
% 0.34/0.63 = X ) )
% 0.34/0.63 => ( ( comple7806235888213564991et_nat @ ( image_o_set_nat @ F @ I ) )
% 0.34/0.63 = X ) ) ) ).
% 0.34/0.63
% 0.34/0.63 % INF_eq_const
% 0.34/0.63 thf(fact_4816_INF__eq__const,axiom,
% 0.34/0.63 ! [I: set_nat,F: nat > set_nat,X: set_nat] :
% 0.34/0.63 ( ( I != bot_bot_set_nat )
% 0.34/0.63 => ( ! [I3: nat] :
% 0.34/0.63 ( ( member_nat @ I3 @ I )
% 0.34/0.63 => ( ( F @ I3 )
% 0.34/0.63 = X ) )
% 0.34/0.63 => ( ( comple7806235888213564991et_nat @ ( image_nat_set_nat @ F @ I ) )
% 0.34/0.63 = X ) ) ) ).
% 0.34/0.63
% 0.34/0.63 % INF_eq_const
% 0.34/0.63 thf(fact_4817_INF__eq__const,axiom,
% 0.34/0.63 ! [I: set_int,F: int > set_nat,X: set_nat] :
% 0.34/0.63 ( ( I != bot_bot_set_int )
% 0.34/0.63 => ( ! [I3: int] :
% 0.34/0.63 ( ( member_int @ I3 @ I )
% 0.34/0.63 => ( ( F @ I3 )
% 0.34/0.63 = X ) )
% 0.34/0.63 => ( ( comple7806235888213564991et_nat @ ( image_int_set_nat @ F @ I ) )
% 0.34/0.63 = X ) ) ) ).
% 0.34/0.63
% 0.34/0.63 % INF_eq_const
% 0.34/0.63 thf(fact_4818_INF__eq__const,axiom,
% 0.34/0.63 ! [I: set_Pr4532377907799695533_nat_o,F: ( produc3658429121746597890et_nat > $o ) > set_nat,X: set_nat] :
% 0.34/0.63 ( ( I != bot_bo7824918357723582233_nat_o )
% 0.34/0.63 => ( ! [I3: produc3658429121746597890et_nat > $o] :
% 0.34/0.63 ( ( member6576561426505652726_nat_o @ I3 @ I )
% 0.34/0.63 => ( ( F @ I3 )
% 0.34/0.63 = X ) )
% 0.34/0.63 => ( ( comple7806235888213564991et_nat @ ( image_6888641473154130958et_nat @ F @ I ) )
% 0.34/0.63 = X ) ) ) ).
% 0.34/0.63
% 0.34/0.63 % INF_eq_const
% 0.34/0.63 thf(fact_4819_INF__eq__const,axiom,
% 0.34/0.63 ! [I: set_Product_unit,F: product_unit > product_unit,X: product_unit] :
% 0.34/0.63 ( ( I != bot_bo3957492148770167129t_unit )
% 0.34/0.63 => ( ! [I3: product_unit] :
% 0.34/0.63 ( ( member_Product_unit @ I3 @ I )
% 0.34/0.63 => ( ( F @ I3 )
% 0.34/0.63 = X ) )
% 0.34/0.63 => ( ( comple2584293577114468500t_unit @ ( image_405062704495631173t_unit @ F @ I ) )
% 0.34/0.63 = X ) ) ) ).
% 0.34/0.63
% 0.34/0.63 % INF_eq_const
% 0.34/0.63 thf(fact_4820_INF__eq__const,axiom,
% 0.34/0.63 ! [I: set_o,F: $o > product_unit,X: product_unit] :
% 0.34/0.63 ( ( I != bot_bot_set_o )
% 0.34/0.63 => ( ! [I3: $o] :
% 0.34/0.63 ( ( member_o @ I3 @ I )
% 0.34/0.63 => ( ( F @ I3 )
% 0.34/0.63 = X ) )
% 0.34/0.63 => ( ( comple2584293577114468500t_unit @ ( image_o_Product_unit2 @ F @ I ) )
% 0.34/0.63 = X ) ) ) ).
% 0.34/0.63
% 0.34/0.63 % INF_eq_const
% 0.34/0.63 thf(fact_4821_INF__eq__const,axiom,
% 0.34/0.63 ! [I: set_nat,F: nat > product_unit,X: product_unit] :
% 0.34/0.63 ( ( I != bot_bot_set_nat )
% 0.34/0.63 => ( ! [I3: nat] :
% 0.34/0.63 ( ( member_nat @ I3 @ I )
% 0.34/0.63 => ( ( F @ I3 )
% 0.34/0.63 = X ) )
% 0.34/0.63 => ( ( comple2584293577114468500t_unit @ ( image_8730104196221521654t_unit @ F @ I ) )
% 0.34/0.63 = X ) ) ) ).
% 0.34/0.63
% 0.34/0.63 % INF_eq_const
% 0.34/0.63 thf(fact_4822_INF__eq__const,axiom,
% 0.34/0.63 ! [I: set_int,F: int > product_unit,X: product_unit] :
% 0.34/0.63 ( ( I != bot_bot_set_int )
% 0.34/0.63 => ( ! [I3: int] :
% 0.34/0.63 ( ( member_int @ I3 @ I )
% 0.34/0.63 => ( ( F @ I3 )
% 0.34/0.63 = X ) )
% 0.34/0.63 => ( ( comple2584293577114468500t_unit @ ( image_5846526143423701914t_unit @ F @ I ) )
% 0.34/0.63 = X ) ) ) ).
% 0.34/0.63
% 0.34/0.63 % INF_eq_const
% 0.34/0.63 thf(fact_4823_INF__eq__const,axiom,
% 0.34/0.63 ! [I: set_Pr4532377907799695533_nat_o,F: ( produc3658429121746597890et_nat > $o ) > product_unit,X: product_unit] :
% 0.34/0.63 ( ( I != bot_bo7824918357723582233_nat_o )
% 0.34/0.63 => ( ! [I3: produc3658429121746597890et_nat > $o] :
% 0.34/0.63 ( ( member6576561426505652726_nat_o @ I3 @ I )
% 0.34/0.63 => ( ( F @ I3 )
% 0.34/0.63 = X ) )
% 0.34/0.63 => ( ( comple2584293577114468500t_unit @ ( image_8451000719036923781t_unit @ F @ I ) )
% 0.34/0.63 = X ) ) ) ).
% 0.34/0.63
% 0.34/0.63 % INF_eq_const
% 0.34/0.63 thf(fact_4824_card__1__singletonE,axiom,
% 0.34/0.63 ! [A: set_Product_unit] :
% 0.34/0.63 ( ( ( finite410649719033368117t_unit @ A )
% 0.34/0.63 = one_one_nat )
% 0.34/0.63 => ~ ! [X3: product_unit] :
% 0.34/0.63 ( A
% 0.34/0.63 != ( insert_Product_unit @ X3 @ bot_bo3957492148770167129t_unit ) ) ) ).
% 0.34/0.63
% 0.34/0.63 % card_1_singletonE
% 0.34/0.63 thf(fact_4825_card__1__singletonE,axiom,
% 0.34/0.63 ! [A: set_o] :
% 0.34/0.63 ( ( ( finite_card_o @ A )
% 0.34/0.63 = one_one_nat )
% 0.34/0.63 => ~ ! [X3: $o] :
% 0.34/0.63 ( A
% 0.34/0.63 != ( insert_o @ X3 @ bot_bot_set_o ) ) ) ).
% 0.34/0.63
% 0.34/0.63 % card_1_singletonE
% 0.34/0.63 thf(fact_4826_card__1__singletonE,axiom,
% 0.34/0.63 ! [A: set_nat] :
% 0.34/0.63 ( ( ( finite_card_nat @ A )
% 0.34/0.63 = one_one_nat )
% 0.34/0.63 => ~ ! [X3: nat] :
% 0.34/0.63 ( A
% 0.34/0.63 != ( insert_nat @ X3 @ bot_bot_set_nat ) ) ) ).
% 0.34/0.63
% 0.34/0.63 % card_1_singletonE
% 0.34/0.63 thf(fact_4827_card__1__singletonE,axiom,
% 0.34/0.63 ! [A: set_int] :
% 0.34/0.63 ( ( ( finite_card_int @ A )
% 0.34/0.63 = one_one_nat )
% 0.34/0.63 => ~ ! [X3: int] :
% 0.34/0.63 ( A
% 0.34/0.63 != ( insert_int @ X3 @ bot_bot_set_int ) ) ) ).
% 0.34/0.63
% 0.34/0.63 % card_1_singletonE
% 0.34/0.63 thf(fact_4828_card__1__singletonE,axiom,
% 0.34/0.63 ! [A: set_Pr4532377907799695533_nat_o] :
% 0.34/0.63 ( ( ( finite1363419556375932405_nat_o @ A )
% 0.34/0.63 = one_one_nat )
% 0.34/0.63 => ~ ! [X3: produc3658429121746597890et_nat > $o] :
% 0.34/0.63 ( A
% 0.34/0.63 != ( insert5175938949040314269_nat_o @ X3 @ bot_bo7824918357723582233_nat_o ) ) ) ).
% 0.34/0.63
% 0.34/0.63 % card_1_singletonE
% 0.34/0.63 thf(fact_4829_BNF__Greatest__Fixpoint_Osubst__Pair,axiom,
% 0.34/0.63 ! [P: nat > nat > $o,X: nat,Y: nat,A2: product_prod_nat_nat] :
% 0.34/0.63 ( ( P @ X @ Y )
% 0.34/0.63 => ( ( A2
% 0.34/0.63 = ( product_Pair_nat_nat @ X @ Y ) )
% 0.34/0.63 => ( P @ ( product_fst_nat_nat @ A2 ) @ ( product_snd_nat_nat @ A2 ) ) ) ) ).
% 0.34/0.63
% 0.34/0.63 % BNF_Greatest_Fixpoint.subst_Pair
% 0.34/0.63 thf(fact_4830_BNF__Greatest__Fixpoint_Osubst__Pair,axiom,
% 0.34/0.63 ! [P: ( produc3658429121746597890et_nat > $o ) > produc3658429121746597890et_nat > $o,X: produc3658429121746597890et_nat > $o,Y: produc3658429121746597890et_nat,A2: produc3925858234332021118et_nat] :
% 0.34/0.63 ( ( P @ X @ Y )
% 0.34/0.63 => ( ( A2
% 0.34/0.63 = ( produc5001842942810119800et_nat @ X @ Y ) )
% 0.34/0.63 => ( P @ ( produc995936583742144908et_nat @ A2 ) @ ( produc4011572625026189258et_nat @ A2 ) ) ) ) ).
% 0.34/0.63
% 0.34/0.63 % BNF_Greatest_Fixpoint.subst_Pair
% 0.34/0.63 thf(fact_4831_BNF__Greatest__Fixpoint_Osubst__Pair,axiom,
% 0.34/0.63 ! [P: ( produc3658429121746597890et_nat > $o ) > produc3925858234332021118et_nat > $o,X: produc3658429121746597890et_nat > $o,Y: produc3925858234332021118et_nat,A2: produc2732055786443039994et_nat] :
% 0.34/0.63 ( ( P @ X @ Y )
% 0.34/0.63 => ( ( A2
% 0.34/0.63 = ( produc2245416461498447860et_nat @ X @ Y ) )
% 0.34/0.63 => ( P @ ( produc180342877477747464et_nat @ A2 ) @ ( produc5374455773327741254et_nat @ A2 ) ) ) ) ).
% 0.34/0.63
% 0.34/0.63 % BNF_Greatest_Fixpoint.subst_Pair
% 0.34/0.63 thf(fact_4832_BNF__Greatest__Fixpoint_Osubst__Pair,axiom,
% 0.34/0.63 ! [P: heap_e7401611519738050253t_unit > set_nat > $o,X: heap_e7401611519738050253t_unit,Y: set_nat,A2: produc3658429121746597890et_nat] :
% 0.34/0.63 ( ( P @ X @ Y )
% 0.34/0.63 => ( ( A2
% 0.34/0.63 = ( produc7507926704131184380et_nat @ X @ Y ) )
% 0.34/0.63 => ( P @ ( produc1824681642469235216et_nat @ A2 ) @ ( produc8586169260539613262et_nat @ A2 ) ) ) ) ).
% 0.34/0.63
% 0.34/0.63 % BNF_Greatest_Fixpoint.subst_Pair
% 0.34/0.63 thf(fact_4833_BNF__Greatest__Fixpoint_Osubst__Pair,axiom,
% 0.34/0.63 ! [P: int > int > $o,X: int,Y: int,A2: product_prod_int_int] :
% 0.34/0.63 ( ( P @ X @ Y )
% 0.34/0.63 => ( ( A2
% 0.34/0.63 = ( product_Pair_int_int @ X @ Y ) )
% 0.34/0.63 => ( P @ ( product_fst_int_int @ A2 ) @ ( product_snd_int_int @ A2 ) ) ) ) ).
% 0.34/0.63
% 0.34/0.63 % BNF_Greatest_Fixpoint.subst_Pair
% 0.34/0.63 thf(fact_4834_prod_Oexhaust__sel,axiom,
% 0.34/0.63 ! [Prod: product_prod_nat_nat] :
% 0.34/0.63 ( Prod
% 0.34/0.63 = ( product_Pair_nat_nat @ ( product_fst_nat_nat @ Prod ) @ ( product_snd_nat_nat @ Prod ) ) ) ).
% 0.34/0.63
% 0.34/0.63 % prod.exhaust_sel
% 0.34/0.63 thf(fact_4835_prod_Oexhaust__sel,axiom,
% 0.34/0.63 ! [Prod: produc3925858234332021118et_nat] :
% 0.34/0.63 ( Prod
% 0.34/0.63 = ( produc5001842942810119800et_nat @ ( produc995936583742144908et_nat @ Prod ) @ ( produc4011572625026189258et_nat @ Prod ) ) ) ).
% 0.34/0.63
% 0.34/0.63 % prod.exhaust_sel
% 0.34/0.63 thf(fact_4836_prod_Oexhaust__sel,axiom,
% 0.34/0.63 ! [Prod: produc2732055786443039994et_nat] :
% 0.34/0.63 ( Prod
% 0.34/0.63 = ( produc2245416461498447860et_nat @ ( produc180342877477747464et_nat @ Prod ) @ ( produc5374455773327741254et_nat @ Prod ) ) ) ).
% 0.34/0.63
% 0.34/0.63 % prod.exhaust_sel
% 0.34/0.63 thf(fact_4837_prod_Oexhaust__sel,axiom,
% 0.34/0.63 ! [Prod: produc3658429121746597890et_nat] :
% 0.34/0.63 ( Prod
% 0.34/0.63 = ( produc7507926704131184380et_nat @ ( produc1824681642469235216et_nat @ Prod ) @ ( produc8586169260539613262et_nat @ Prod ) ) ) ).
% 0.34/0.63
% 0.34/0.63 % prod.exhaust_sel
% 0.34/0.63 thf(fact_4838_prod_Oexhaust__sel,axiom,
% 0.34/0.63 ! [Prod: product_prod_int_int] :
% 0.34/0.63 ( Prod
% 0.34/0.63 = ( product_Pair_int_int @ ( product_fst_int_int @ Prod ) @ ( product_snd_int_int @ Prod ) ) ) ).
% 0.34/0.63
% 0.34/0.63 % prod.exhaust_sel
% 0.34/0.63 thf(fact_4839_surjective__pairing,axiom,
% 0.34/0.63 ! [T4: product_prod_nat_nat] :
% 0.34/0.63 ( T4
% 0.34/0.63 = ( product_Pair_nat_nat @ ( product_fst_nat_nat @ T4 ) @ ( product_snd_nat_nat @ T4 ) ) ) ).
% 0.34/0.63
% 0.34/0.63 % surjective_pairing
% 0.34/0.63 thf(fact_4840_surjective__pairing,axiom,
% 0.34/0.63 ! [T4: produc3925858234332021118et_nat] :
% 0.34/0.63 ( T4
% 0.34/0.63 = ( produc5001842942810119800et_nat @ ( produc995936583742144908et_nat @ T4 ) @ ( produc4011572625026189258et_nat @ T4 ) ) ) ).
% 0.34/0.63
% 0.34/0.63 % surjective_pairing
% 0.34/0.63 thf(fact_4841_surjective__pairing,axiom,
% 0.34/0.63 ! [T4: produc2732055786443039994et_nat] :
% 0.34/0.63 ( T4
% 0.34/0.63 = ( produc2245416461498447860et_nat @ ( produc180342877477747464et_nat @ T4 ) @ ( produc5374455773327741254et_nat @ T4 ) ) ) ).
% 0.34/0.63
% 0.34/0.63 % surjective_pairing
% 0.34/0.63 thf(fact_4842_surjective__pairing,axiom,
% 0.34/0.63 ! [T4: produc3658429121746597890et_nat] :
% 0.34/0.63 ( T4
% 0.34/0.63 = ( produc7507926704131184380et_nat @ ( produc1824681642469235216et_nat @ T4 ) @ ( produc8586169260539613262et_nat @ T4 ) ) ) ).
% 0.34/0.63
% 0.34/0.63 % surjective_pairing
% 0.34/0.63 thf(fact_4843_surjective__pairing,axiom,
% 0.34/0.63 ! [T4: product_prod_int_int] :
% 0.34/0.63 ( T4
% 0.34/0.63 = ( product_Pair_int_int @ ( product_fst_int_int @ T4 ) @ ( product_snd_int_int @ T4 ) ) ) ).
% 0.34/0.63
% 0.34/0.63 % surjective_pairing
% 0.34/0.63 thf(fact_4844_card__insert__le,axiom,
% 0.34/0.63 ! [A: set_nat,X: nat] : ( ord_less_eq_nat @ ( finite_card_nat @ A ) @ ( finite_card_nat @ ( insert_nat @ X @ A ) ) ) ).
% 0.34/0.63
% 0.34/0.63 % card_insert_le
% 0.34/0.63 thf(fact_4845_card__insert__le,axiom,
% 0.34/0.63 ! [A: set_int,X: int] : ( ord_less_eq_nat @ ( finite_card_int @ A ) @ ( finite_card_int @ ( insert_int @ X @ A ) ) ) ).
% 0.34/0.63
% 0.34/0.63 % card_insert_le
% 0.34/0.63 thf(fact_4846_card__insert__le,axiom,
% 0.34/0.63 ! [A: set_o,X: $o] : ( ord_less_eq_nat @ ( finite_card_o @ A ) @ ( finite_card_o @ ( insert_o @ X @ A ) ) ) ).
% 0.34/0.63
% 0.34/0.63 % card_insert_le
% 0.34/0.63 thf(fact_4847_card__insert__le,axiom,
% 0.34/0.63 ! [A: set_Product_unit,X: product_unit] : ( ord_less_eq_nat @ ( finite410649719033368117t_unit @ A ) @ ( finite410649719033368117t_unit @ ( insert_Product_unit @ X @ A ) ) ) ).
% 0.34/0.63
% 0.34/0.63 % card_insert_le
% 0.34/0.63 thf(fact_4848_in__fst__imageE,axiom,
% 0.34/0.63 ! [X: heap_e7401611519738050253t_unit,S: set_Pr3948176798113811640et_nat] :
% 0.34/0.63 ( ( member242389114213480694t_unit @ X @ ( image_6824039242672219652t_unit @ produc1824681642469235216et_nat @ S ) )
% 0.34/0.63 => ~ ! [Y2: set_nat] :
% 0.34/0.63 ~ ( member6260224972018164377et_nat @ ( produc7507926704131184380et_nat @ X @ Y2 ) @ S ) ) ).
% 0.34/0.63
% 0.34/0.63 % in_fst_imageE
% 0.34/0.63 thf(fact_4849_in__fst__imageE,axiom,
% 0.34/0.63 ! [X: nat,S: set_Pr1261947904930325089at_nat] :
% 0.34/0.63 ( ( member_nat @ X @ ( image_2486076414777270412at_nat @ product_fst_nat_nat @ S ) )
% 0.34/0.63 => ~ ! [Y2: nat] :
% 0.34/0.63 ~ ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ Y2 ) @ S ) ) ).
% 0.34/0.63
% 0.34/0.63 % in_fst_imageE
% 0.34/0.63 thf(fact_4850_in__fst__imageE,axiom,
% 0.34/0.63 ! [X: produc3658429121746597890et_nat > $o,S: set_Pr3286484037609594932et_nat] :
% 0.34/0.63 ( ( member6576561426505652726_nat_o @ X @ ( image_6802380912191114376_nat_o @ produc995936583742144908et_nat @ S ) )
% 0.34/0.63 => ~ ! [Y2: produc3658429121746597890et_nat] :
% 0.34/0.63 ~ ( member1996754912294343701et_nat @ ( produc5001842942810119800et_nat @ X @ Y2 ) @ S ) ) ).
% 0.34/0.63
% 0.34/0.63 % in_fst_imageE
% 0.34/0.63 thf(fact_4851_in__fst__imageE,axiom,
% 0.34/0.63 ! [X: produc3658429121746597890et_nat > $o,S: set_Pr8536935166611901872et_nat] :
% 0.34/0.63 ( ( member6576561426505652726_nat_o @ X @ ( image_2517677226577294348_nat_o @ produc180342877477747464et_nat @ S ) )
% 0.34/0.63 => ~ ! [Y2: produc3925858234332021118et_nat] :
% 0.34/0.63 ~ ( member6124377750444531601et_nat @ ( produc2245416461498447860et_nat @ X @ Y2 ) @ S ) ) ).
% 0.34/0.63
% 0.34/0.63 % in_fst_imageE
% 0.34/0.63 thf(fact_4852_in__fst__imageE,axiom,
% 0.34/0.63 ! [X: int,S: set_Pr958786334691620121nt_int] :
% 0.34/0.63 ( ( member_int @ X @ ( image_5042161079198086560nt_int @ product_fst_int_int @ S ) )
% 0.34/0.63 => ~ ! [Y2: int] :
% 0.34/0.63 ~ ( member5262025264175285858nt_int @ ( product_Pair_int_int @ X @ Y2 ) @ S ) ) ).
% 0.34/0.63
% 0.34/0.63 % in_fst_imageE
% 0.34/0.63 thf(fact_4853_Inter__subset,axiom,
% 0.34/0.63 ! [A: set_set_nat,B4: set_nat] :
% 0.34/0.63 ( ! [X8: set_nat] :
% 0.34/0.63 ( ( member_set_nat @ X8 @ A )
% 0.34/0.63 => ( ord_less_eq_set_nat @ X8 @ B4 ) )
% 0.34/0.63 => ( ( A != bot_bot_set_set_nat )
% 0.34/0.63 => ( ord_less_eq_set_nat @ ( comple7806235888213564991et_nat @ A ) @ B4 ) ) ) ).
% 0.34/0.63
% 0.34/0.63 % Inter_subset
% 0.34/0.63 thf(fact_4854_Inter__empty,axiom,
% 0.34/0.63 ( ( comple7216939928495492468t_unit @ bot_bo6284674557190688313t_unit )
% 0.34/0.63 = top_to1996260823553986621t_unit ) ).
% 0.34/0.63
% 0.34/0.63 % Inter_empty
% 0.34/0.63 thf(fact_4855_Inter__empty,axiom,
% 0.34/0.63 ( ( comple3063163877087187839_set_o @ bot_bot_set_set_o )
% 0.34/0.63 = top_top_set_o ) ).
% 0.34/0.63
% 0.34/0.63 % Inter_empty
% 0.34/0.63 thf(fact_4856_Inter__empty,axiom,
% 0.34/0.63 ( ( comple3628384868704368283et_int @ bot_bot_set_set_int )
% 0.34/0.63 = top_top_set_int ) ).
% 0.34/0.63
% 0.34/0.63 % Inter_empty
% 0.34/0.63 thf(fact_4857_Inter__empty,axiom,
% 0.34/0.63 ( ( comple7806235888213564991et_nat @ bot_bot_set_set_nat )
% 0.34/0.63 = top_top_set_nat ) ).
% 0.34/0.63
% 0.34/0.63 % Inter_empty
% 0.34/0.63 thf(fact_4858_fst__eq__Domain,axiom,
% 0.34/0.63 ! [R3: set_Pr958786334691620121nt_int] :
% 0.34/0.63 ( ( image_5042161079198086560nt_int @ product_fst_int_int @ R3 )
% 0.34/0.63 = ( domain_int_int @ R3 ) ) ).
% 0.34/0.63
% 0.34/0.63 % fst_eq_Domain
% 0.34/0.63 thf(fact_4859_Domain__fst,axiom,
% 0.34/0.63 ( domain_int_int
% 0.34/0.63 = ( image_5042161079198086560nt_int @ product_fst_int_int ) ) ).
% 0.34/0.63
% 0.34/0.63 % Domain_fst
% 0.34/0.63 thf(fact_4860_card__partition,axiom,
% 0.34/0.63 ! [C5: set_set_Code_integer,K3: nat] :
% 0.34/0.63 ( ( finite6931041176100689706nteger @ C5 )
% 0.34/0.63 => ( ( finite6017078050557962740nteger @ ( comple739944243200306918nteger @ C5 ) )
% 0.34/0.63 => ( ! [C2: set_Code_integer] :
% 0.34/0.63 ( ( member1834298987299537450nteger @ C2 @ C5 )
% 0.34/0.63 => ( ( finite4902975817058060853nteger @ C2 )
% 0.34/0.63 = K3 ) )
% 0.34/0.63 => ( ! [C1: set_Code_integer,C22: set_Code_integer] :
% 0.34/0.63 ( ( member1834298987299537450nteger @ C1 @ C5 )
% 0.34/0.63 => ( ( member1834298987299537450nteger @ C22 @ C5 )
% 0.34/0.63 => ( ( C1 != C22 )
% 0.34/0.63 => ( ( inf_in1364745209274528805nteger @ C1 @ C22 )
% 0.34/0.63 = bot_bo3990330152332043303nteger ) ) ) )
% 0.34/0.63 => ( ( times_times_nat @ K3 @ ( finite1340570857867686763nteger @ C5 ) )
% 0.34/0.63 = ( finite4902975817058060853nteger @ ( comple739944243200306918nteger @ C5 ) ) ) ) ) ) ) ).
% 0.34/0.63
% 0.34/0.63 % card_partition
% 0.34/0.63 thf(fact_4861_card__partition,axiom,
% 0.34/0.63 ! [C5: set_set_Product_unit,K3: nat] :
% 0.34/0.63 ( ( finite1772178364199683094t_unit @ C5 )
% 0.34/0.63 => ( ( finite4290736615968046902t_unit @ ( comple3349607820504768346t_unit @ C5 ) )
% 0.34/0.63 => ( ! [C2: set_Product_unit] :
% 0.34/0.63 ( ( member5877623283571906838t_unit @ C2 @ C5 )
% 0.34/0.63 => ( ( finite410649719033368117t_unit @ C2 )
% 0.34/0.63 = K3 ) )
% 0.34/0.63 => ( ! [C1: set_Product_unit,C22: set_Product_unit] :
% 0.34/0.63 ( ( member5877623283571906838t_unit @ C1 @ C5 )
% 0.34/0.63 => ( ( member5877623283571906838t_unit @ C22 @ C5 )
% 0.34/0.63 => ( ( C1 != C22 )
% 0.34/0.63 => ( ( inf_in4660618365625256667t_unit @ C1 @ C22 )
% 0.34/0.63 = bot_bo3957492148770167129t_unit ) ) ) )
% 0.34/0.63 => ( ( times_times_nat @ K3 @ ( finite22775419188893461t_unit @ C5 ) )
% 0.34/0.63 = ( finite410649719033368117t_unit @ ( comple3349607820504768346t_unit @ C5 ) ) ) ) ) ) ) ).
% 0.34/0.63
% 0.34/0.63 % card_partition
% 0.34/0.63 thf(fact_4862_card__partition,axiom,
% 0.34/0.63 ! [C5: set_set_o,K3: nat] :
% 0.34/0.63 ( ( finite_finite_set_o @ C5 )
% 0.34/0.63 => ( ( finite_finite_o @ ( comple90263536869209701_set_o @ C5 ) )
% 0.34/0.63 => ( ! [C2: set_o] :
% 0.34/0.63 ( ( member_set_o @ C2 @ C5 )
% 0.34/0.63 => ( ( finite_card_o @ C2 )
% 0.34/0.63 = K3 ) )
% 0.34/0.63 => ( ! [C1: set_o,C22: set_o] :
% 0.34/0.63 ( ( member_set_o @ C1 @ C5 )
% 0.34/0.63 => ( ( member_set_o @ C22 @ C5 )
% 0.34/0.63 => ( ( C1 != C22 )
% 0.34/0.63 => ( ( inf_inf_set_o @ C1 @ C22 )
% 0.34/0.63 = bot_bot_set_o ) ) ) )
% 0.34/0.63 => ( ( times_times_nat @ K3 @ ( finite_card_set_o @ C5 ) )
% 0.34/0.63 = ( finite_card_o @ ( comple90263536869209701_set_o @ C5 ) ) ) ) ) ) ) ).
% 0.34/0.63
% 0.34/0.63 % card_partition
% 0.34/0.63 thf(fact_4863_card__partition,axiom,
% 0.34/0.63 ! [C5: set_set_nat,K3: nat] :
% 0.34/0.63 ( ( finite1152437895449049373et_nat @ C5 )
% 0.34/0.63 => ( ( finite_finite_nat @ ( comple7399068483239264473et_nat @ C5 ) )
% 0.34/0.63 => ( ! [C2: set_nat] :
% 0.34/0.63 ( ( member_set_nat @ C2 @ C5 )
% 0.34/0.63 => ( ( finite_card_nat @ C2 )
% 0.34/0.63 = K3 ) )
% 0.34/0.63 => ( ! [C1: set_nat,C22: set_nat] :
% 0.34/0.63 ( ( member_set_nat @ C1 @ C5 )
% 0.34/0.63 => ( ( member_set_nat @ C22 @ C5 )
% 0.34/0.63 => ( ( C1 != C22 )
% 0.34/0.63 => ( ( inf_inf_set_nat @ C1 @ C22 )
% 0.34/0.63 = bot_bot_set_nat ) ) ) )
% 0.34/0.63 => ( ( times_times_nat @ K3 @ ( finite_card_set_nat @ C5 ) )
% 0.34/0.63 = ( finite_card_nat @ ( comple7399068483239264473et_nat @ C5 ) ) ) ) ) ) ) ).
% 0.34/0.63
% 0.34/0.63 % card_partition
% 0.34/0.63 thf(fact_4864_card__partition,axiom,
% 0.34/0.63 ! [C5: set_set_int,K3: nat] :
% 0.34/0.63 ( ( finite6197958912794628473et_int @ C5 )
% 0.34/0.63 => ( ( finite_finite_int @ ( comple3221217463730067765et_int @ C5 ) )
% 0.34/0.63 => ( ! [C2: set_int] :
% 0.34/0.63 ( ( member_set_int @ C2 @ C5 )
% 0.34/0.63 => ( ( finite_card_int @ C2 )
% 0.34/0.63 = K3 ) )
% 0.34/0.63 => ( ! [C1: set_int,C22: set_int] :
% 0.34/0.63 ( ( member_set_int @ C1 @ C5 )
% 0.34/0.63 => ( ( member_set_int @ C22 @ C5 )
% 0.34/0.63 => ( ( C1 != C22 )
% 0.34/0.63 => ( ( inf_inf_set_int @ C1 @ C22 )
% 0.34/0.63 = bot_bot_set_int ) ) ) )
% 0.34/0.63 => ( ( times_times_nat @ K3 @ ( finite_card_set_int @ C5 ) )
% 0.34/0.63 = ( finite_card_int @ ( comple3221217463730067765et_int @ C5 ) ) ) ) ) ) ) ).
% 0.34/0.63
% 0.34/0.63 % card_partition
% 0.34/0.63 thf(fact_4865_card__partition,axiom,
% 0.34/0.63 ! [C5: set_se1353816476878259085_nat_o,K3: nat] :
% 0.34/0.63 ( ( finite6069017635265841110_nat_o @ C5 )
% 0.34/0.63 => ( ( finite3252695134891459830_nat_o @ ( comple3382087578465172250_nat_o @ C5 ) )
% 0.34/0.63 => ( ! [C2: set_Pr4532377907799695533_nat_o] :
% 0.34/0.63 ( ( member1452989533377600726_nat_o @ C2 @ C5 )
% 0.34/0.63 => ( ( finite1363419556375932405_nat_o @ C2 )
% 0.34/0.63 = K3 ) )
% 0.34/0.63 => ( ! [C1: set_Pr4532377907799695533_nat_o,C22: set_Pr4532377907799695533_nat_o] :
% 0.34/0.63 ( ( member1452989533377600726_nat_o @ C1 @ C5 )
% 0.34/0.63 => ( ( member1452989533377600726_nat_o @ C22 @ C5 )
% 0.34/0.63 => ( ( C1 != C22 )
% 0.34/0.63 => ( ( inf_in1906310914598751387_nat_o @ C1 @ C22 )
% 0.34/0.63 = bot_bo7824918357723582233_nat_o ) ) ) )
% 0.34/0.63 => ( ( times_times_nat @ K3 @ ( finite4731484169140532437_nat_o @ C5 ) )
% 0.34/0.63 = ( finite1363419556375932405_nat_o @ ( comple3382087578465172250_nat_o @ C5 ) ) ) ) ) ) ) ).
% 0.34/0.63
% 0.34/0.63 % card_partition
% 0.34/0.63 thf(fact_4866_Inter__UNIV,axiom,
% 0.34/0.63 ( ( comple7216939928495492468t_unit @ top_to1767297665138865437t_unit )
% 0.34/0.63 = bot_bo3957492148770167129t_unit ) ).
% 0.34/0.63
% 0.34/0.63 % Inter_UNIV
% 0.34/0.63 thf(fact_4867_Inter__UNIV,axiom,
% 0.34/0.63 ( ( comple3063163877087187839_set_o @ top_top_set_set_o )
% 0.34/0.63 = bot_bot_set_o ) ).
% 0.34/0.63
% 0.34/0.63 % Inter_UNIV
% 0.34/0.63 thf(fact_4868_Inter__UNIV,axiom,
% 0.34/0.63 ( ( comple3628384868704368283et_int @ top_top_set_set_int )
% 0.34/0.63 = bot_bot_set_int ) ).
% 0.34/0.63
% 0.34/0.63 % Inter_UNIV
% 0.34/0.63 thf(fact_4869_Inter__UNIV,axiom,
% 0.34/0.63 ( ( comple79274577958421812_nat_o @ top_to8515903933492314333_nat_o )
% 0.34/0.63 = bot_bo7824918357723582233_nat_o ) ).
% 0.34/0.63
% 0.34/0.63 % Inter_UNIV
% 0.34/0.63 thf(fact_4870_Inter__UNIV,axiom,
% 0.34/0.63 ( ( comple7806235888213564991et_nat @ top_top_set_set_nat )
% 0.34/0.63 = bot_bot_set_nat ) ).
% 0.34/0.63
% 0.34/0.63 % Inter_UNIV
% 0.34/0.63 thf(fact_4871_inv__on__f__range,axiom,
% 0.34/0.63 ! [Y: code_integer,F: code_integer > code_integer,A: set_Code_integer] :
% 0.34/0.63 ( ( member_Code_integer @ Y @ ( image_4470545334726330049nteger @ F @ A ) )
% 0.34/0.63 => ( member_Code_integer @ ( inv_on4869000451854154612nteger @ F @ A @ Y ) @ A ) ) ).
% 0.34/0.63
% 0.34/0.63 % inv_on_f_range
% 0.34/0.63 thf(fact_4872_inv__on__f__range,axiom,
% 0.34/0.63 ! [Y: code_integer,F: nat > code_integer,A: set_nat] :
% 0.34/0.63 ( ( member_Code_integer @ Y @ ( image_1215581382706833972nteger @ F @ A ) )
% 0.34/0.63 => ( member_nat @ ( inv_on1369624593536199399nteger @ F @ A @ Y ) @ A ) ) ).
% 0.34/0.63
% 0.34/0.63 % inv_on_f_range
% 0.34/0.63 thf(fact_4873_inv__on__f__range,axiom,
% 0.34/0.63 ! [Y: nat,F: nat > nat,A: set_nat] :
% 0.34/0.63 ( ( member_nat @ Y @ ( image_nat_nat2 @ F @ A ) )
% 0.34/0.63 => ( member_nat @ ( inv_on_nat_nat @ F @ A @ Y ) @ A ) ) ).
% 0.34/0.63
% 0.34/0.63 % inv_on_f_range
% 0.34/0.63 thf(fact_4874_inv__on__f__range,axiom,
% 0.34/0.63 ! [Y: nat,F: int > nat,A: set_int] :
% 0.34/0.63 ( ( member_nat @ Y @ ( image_int_nat2 @ F @ A ) )
% 0.34/0.63 => ( member_int @ ( inv_on_int_nat @ F @ A @ Y ) @ A ) ) ).
% 0.34/0.63
% 0.34/0.63 % inv_on_f_range
% 0.34/0.63 thf(fact_4875_inv__on__f__range,axiom,
% 0.34/0.63 ! [Y: nat,F: $o > nat,A: set_o] :
% 0.34/0.63 ( ( member_nat @ Y @ ( image_o_nat2 @ F @ A ) )
% 0.34/0.63 => ( member_o @ ( inv_on_o_nat @ F @ A @ Y ) @ A ) ) ).
% 0.34/0.63
% 0.34/0.63 % inv_on_f_range
% 0.34/0.63 thf(fact_4876_inv__on__f__range,axiom,
% 0.34/0.63 ! [Y: nat,F: product_unit > nat,A: set_Product_unit] :
% 0.34/0.63 ( ( member_nat @ Y @ ( image_875570014554754200it_nat @ F @ A ) )
% 0.34/0.63 => ( member_Product_unit @ ( inv_on7727615724625388965it_nat @ F @ A @ Y ) @ A ) ) ).
% 0.34/0.63
% 0.34/0.63 % inv_on_f_range
% 0.34/0.63 thf(fact_4877_inv__on__f__range,axiom,
% 0.34/0.63 ! [Y: int,F: nat > int,A: set_nat] :
% 0.34/0.63 ( ( member_int @ Y @ ( image_nat_int2 @ F @ A ) )
% 0.34/0.63 => ( member_nat @ ( inv_on_nat_int @ F @ A @ Y ) @ A ) ) ).
% 0.34/0.63
% 0.34/0.63 % inv_on_f_range
% 0.34/0.63 thf(fact_4878_inv__on__f__range,axiom,
% 0.34/0.63 ! [Y: int,F: int > int,A: set_int] :
% 0.34/0.63 ( ( member_int @ Y @ ( image_int_int2 @ F @ A ) )
% 0.34/0.63 => ( member_int @ ( inv_on_int_int @ F @ A @ Y ) @ A ) ) ).
% 0.34/0.63
% 0.34/0.63 % inv_on_f_range
% 0.34/0.63 thf(fact_4879_inv__on__f__range,axiom,
% 0.34/0.63 ! [Y: int,F: $o > int,A: set_o] :
% 0.34/0.63 ( ( member_int @ Y @ ( image_o_int2 @ F @ A ) )
% 0.34/0.63 => ( member_o @ ( inv_on_o_int @ F @ A @ Y ) @ A ) ) ).
% 0.34/0.63
% 0.34/0.63 % inv_on_f_range
% 0.34/0.63 thf(fact_4880_inv__on__f__range,axiom,
% 0.34/0.63 ! [Y: int,F: product_unit > int,A: set_Product_unit] :
% 0.34/0.63 ( ( member_int @ Y @ ( image_873079544045703924it_int @ F @ A ) )
% 0.34/0.63 => ( member_Product_unit @ ( inv_on7725125254116338689it_int @ F @ A @ Y ) @ A ) ) ).
% 0.34/0.63
% 0.34/0.63 % inv_on_f_range
% 0.34/0.63 thf(fact_4881_f__inv__on__f,axiom,
% 0.34/0.63 ! [Y: set_nat,F: nat > set_nat,A: set_nat] :
% 0.34/0.63 ( ( member_set_nat @ Y @ ( image_nat_set_nat @ F @ A ) )
% 0.34/0.63 => ( ( F @ ( inv_on_nat_set_nat @ F @ A @ Y ) )
% 0.34/0.63 = Y ) ) ).
% 0.34/0.63
% 0.34/0.63 % f_inv_on_f
% 0.34/0.63 thf(fact_4882_f__inv__on__f,axiom,
% 0.34/0.63 ! [Y: code_integer,F: code_integer > code_integer,A: set_Code_integer] :
% 0.34/0.63 ( ( member_Code_integer @ Y @ ( image_4470545334726330049nteger @ F @ A ) )
% 0.34/0.63 => ( ( F @ ( inv_on4869000451854154612nteger @ F @ A @ Y ) )
% 0.34/0.63 = Y ) ) ).
% 0.34/0.63
% 0.34/0.63 % f_inv_on_f
% 0.34/0.63 thf(fact_4883_f__inv__on__f,axiom,
% 0.34/0.63 ! [Y: code_integer,F: nat > code_integer,A: set_nat] :
% 0.34/0.63 ( ( member_Code_integer @ Y @ ( image_1215581382706833972nteger @ F @ A ) )
% 0.34/0.63 => ( ( F @ ( inv_on1369624593536199399nteger @ F @ A @ Y ) )
% 0.34/0.63 = Y ) ) ).
% 0.34/0.63
% 0.34/0.63 % f_inv_on_f
% 0.34/0.63 thf(fact_4884_f__inv__on__f,axiom,
% 0.34/0.63 ! [Y: nat,F: nat > nat,A: set_nat] :
% 0.34/0.63 ( ( member_nat @ Y @ ( image_nat_nat2 @ F @ A ) )
% 0.34/0.63 => ( ( F @ ( inv_on_nat_nat @ F @ A @ Y ) )
% 0.34/0.63 = Y ) ) ).
% 0.34/0.63
% 0.34/0.63 % f_inv_on_f
% 0.34/0.63 thf(fact_4885_cINF__greatest,axiom,
% 0.34/0.63 ! [A: set_Product_unit,M4: int,F: product_unit > int] :
% 0.34/0.63 ( ( A != bot_bo3957492148770167129t_unit )
% 0.34/0.63 => ( ! [X3: product_unit] :
% 0.34/0.63 ( ( member_Product_unit @ X3 @ A )
% 0.34/0.63 => ( ord_less_eq_int @ M4 @ ( F @ X3 ) ) )
% 0.34/0.63 => ( ord_less_eq_int @ M4 @ ( complete_Inf_Inf_int @ ( image_873079544045703924it_int @ F @ A ) ) ) ) ) ).
% 0.34/0.63
% 0.34/0.63 % cINF_greatest
% 0.34/0.63 thf(fact_4886_cINF__greatest,axiom,
% 0.34/0.63 ! [A: set_o,M4: int,F: $o > int] :
% 0.34/0.63 ( ( A != bot_bot_set_o )
% 0.34/0.63 => ( ! [X3: $o] :
% 0.34/0.63 ( ( member_o @ X3 @ A )
% 0.34/0.63 => ( ord_less_eq_int @ M4 @ ( F @ X3 ) ) )
% 0.34/0.63 => ( ord_less_eq_int @ M4 @ ( complete_Inf_Inf_int @ ( image_o_int2 @ F @ A ) ) ) ) ) ).
% 0.34/0.63
% 0.34/0.63 % cINF_greatest
% 0.34/0.63 thf(fact_4887_cINF__greatest,axiom,
% 0.34/0.63 ! [A: set_nat,M4: int,F: nat > int] :
% 0.34/0.63 ( ( A != bot_bot_set_nat )
% 0.34/0.63 => ( ! [X3: nat] :
% 0.34/0.63 ( ( member_nat @ X3 @ A )
% 0.34/0.63 => ( ord_less_eq_int @ M4 @ ( F @ X3 ) ) )
% 0.34/0.63 => ( ord_less_eq_int @ M4 @ ( complete_Inf_Inf_int @ ( image_nat_int2 @ F @ A ) ) ) ) ) ).
% 0.34/0.63
% 0.34/0.63 % cINF_greatest
% 0.34/0.63 thf(fact_4888_cINF__greatest,axiom,
% 0.34/0.63 ! [A: set_int,M4: int,F: int > int] :
% 0.34/0.63 ( ( A != bot_bot_set_int )
% 0.34/0.63 => ( ! [X3: int] :
% 0.34/0.63 ( ( member_int @ X3 @ A )
% 0.34/0.63 => ( ord_less_eq_int @ M4 @ ( F @ X3 ) ) )
% 0.34/0.63 => ( ord_less_eq_int @ M4 @ ( complete_Inf_Inf_int @ ( image_int_int2 @ F @ A ) ) ) ) ) ).
% 0.34/0.63
% 0.34/0.63 % cINF_greatest
% 0.34/0.63 thf(fact_4889_cINF__greatest,axiom,
% 0.34/0.63 ! [A: set_Product_unit,M4: nat,F: product_unit > nat] :
% 0.34/0.63 ( ( A != bot_bo3957492148770167129t_unit )
% 0.34/0.63 => ( ! [X3: product_unit] :
% 0.34/0.63 ( ( member_Product_unit @ X3 @ A )
% 0.34/0.63 => ( ord_less_eq_nat @ M4 @ ( F @ X3 ) ) )
% 0.34/0.63 => ( ord_less_eq_nat @ M4 @ ( complete_Inf_Inf_nat @ ( image_875570014554754200it_nat @ F @ A ) ) ) ) ) ).
% 0.34/0.63
% 0.34/0.63 % cINF_greatest
% 0.34/0.63 thf(fact_4890_cINF__greatest,axiom,
% 0.34/0.63 ! [A: set_o,M4: nat,F: $o > nat] :
% 0.34/0.63 ( ( A != bot_bot_set_o )
% 0.34/0.63 => ( ! [X3: $o] :
% 0.34/0.63 ( ( member_o @ X3 @ A )
% 0.34/0.63 => ( ord_less_eq_nat @ M4 @ ( F @ X3 ) ) )
% 0.34/0.63 => ( ord_less_eq_nat @ M4 @ ( complete_Inf_Inf_nat @ ( image_o_nat2 @ F @ A ) ) ) ) ) ).
% 0.34/0.63
% 0.34/0.63 % cINF_greatest
% 0.34/0.63 thf(fact_4891_cINF__greatest,axiom,
% 0.34/0.63 ! [A: set_nat,M4: nat,F: nat > nat] :
% 0.34/0.63 ( ( A != bot_bot_set_nat )
% 0.34/0.63 => ( ! [X3: nat] :
% 0.34/0.63 ( ( member_nat @ X3 @ A )
% 0.34/0.63 => ( ord_less_eq_nat @ M4 @ ( F @ X3 ) ) )
% 0.34/0.63 => ( ord_less_eq_nat @ M4 @ ( complete_Inf_Inf_nat @ ( image_nat_nat2 @ F @ A ) ) ) ) ) ).
% 0.34/0.63
% 0.34/0.63 % cINF_greatest
% 0.34/0.63 thf(fact_4892_cINF__greatest,axiom,
% 0.34/0.63 ! [A: set_int,M4: nat,F: int > nat] :
% 0.34/0.63 ( ( A != bot_bot_set_int )
% 0.34/0.63 => ( ! [X3: int] :
% 0.34/0.63 ( ( member_int @ X3 @ A )
% 0.34/0.63 => ( ord_less_eq_nat @ M4 @ ( F @ X3 ) ) )
% 0.34/0.63 => ( ord_less_eq_nat @ M4 @ ( complete_Inf_Inf_nat @ ( image_int_nat2 @ F @ A ) ) ) ) ) ).
% 0.34/0.63
% 0.34/0.63 % cINF_greatest
% 0.34/0.63 thf(fact_4893_cINF__greatest,axiom,
% 0.34/0.63 ! [A: set_Product_unit,M4: product_unit,F: product_unit > product_unit] :
% 0.34/0.63 ( ( A != bot_bo3957492148770167129t_unit )
% 0.34/0.63 => ( ! [X3: product_unit] :
% 0.34/0.63 ( ( member_Product_unit @ X3 @ A )
% 0.34/0.63 => ( ord_le3221252021190050221t_unit @ M4 @ ( F @ X3 ) ) )
% 0.34/0.63 => ( ord_le3221252021190050221t_unit @ M4 @ ( comple2584293577114468500t_unit @ ( image_405062704495631173t_unit @ F @ A ) ) ) ) ) ).
% 0.34/0.63
% 0.34/0.63 % cINF_greatest
% 0.34/0.63 thf(fact_4894_cINF__greatest,axiom,
% 0.34/0.63 ! [A: set_o,M4: product_unit,F: $o > product_unit] :
% 0.34/0.63 ( ( A != bot_bot_set_o )
% 0.34/0.63 => ( ! [X3: $o] :
% 0.34/0.63 ( ( member_o @ X3 @ A )
% 0.34/0.63 => ( ord_le3221252021190050221t_unit @ M4 @ ( F @ X3 ) ) )
% 0.34/0.63 => ( ord_le3221252021190050221t_unit @ M4 @ ( comple2584293577114468500t_unit @ ( image_o_Product_unit2 @ F @ A ) ) ) ) ) ).
% 0.34/0.63
% 0.34/0.63 % cINF_greatest
% 0.34/0.63 thf(fact_4895_INF__eq__iff,axiom,
% 0.34/0.63 ! [I: set_Product_unit,F: product_unit > product_unit,C: product_unit] :
% 0.34/0.63 ( ( I != bot_bo3957492148770167129t_unit )
% 0.34/0.63 => ( ! [I3: product_unit] :
% 0.34/0.63 ( ( member_Product_unit @ I3 @ I )
% 0.34/0.63 => ( ord_le3221252021190050221t_unit @ ( F @ I3 ) @ C ) )
% 0.34/0.63 => ( ( ( comple2584293577114468500t_unit @ ( image_405062704495631173t_unit @ F @ I ) )
% 0.34/0.63 = C )
% 0.34/0.63 = ( ! [X2: product_unit] :
% 0.34/0.63 ( ( member_Product_unit @ X2 @ I )
% 0.34/0.63 => ( ( F @ X2 )
% 0.34/0.63 = C ) ) ) ) ) ) ).
% 0.34/0.63
% 0.34/0.63 % INF_eq_iff
% 0.34/0.63 thf(fact_4896_INF__eq__iff,axiom,
% 0.34/0.63 ! [I: set_o,F: $o > product_unit,C: product_unit] :
% 0.34/0.63 ( ( I != bot_bot_set_o )
% 0.34/0.63 => ( ! [I3: $o] :
% 0.34/0.63 ( ( member_o @ I3 @ I )
% 0.34/0.63 => ( ord_le3221252021190050221t_unit @ ( F @ I3 ) @ C ) )
% 0.34/0.63 => ( ( ( comple2584293577114468500t_unit @ ( image_o_Product_unit2 @ F @ I ) )
% 0.34/0.63 = C )
% 0.34/0.63 = ( ! [X2: $o] :
% 0.34/0.63 ( ( member_o @ X2 @ I )
% 0.34/0.63 => ( ( F @ X2 )
% 0.34/0.63 = C ) ) ) ) ) ) ).
% 0.34/0.63
% 0.34/0.63 % INF_eq_iff
% 0.34/0.63 thf(fact_4897_INF__eq__iff,axiom,
% 0.34/0.63 ! [I: set_nat,F: nat > product_unit,C: product_unit] :
% 0.34/0.63 ( ( I != bot_bot_set_nat )
% 0.34/0.63 => ( ! [I3: nat] :
% 0.34/0.63 ( ( member_nat @ I3 @ I )
% 0.34/0.63 => ( ord_le3221252021190050221t_unit @ ( F @ I3 ) @ C ) )
% 0.34/0.63 => ( ( ( comple2584293577114468500t_unit @ ( image_8730104196221521654t_unit @ F @ I ) )
% 0.34/0.63 = C )
% 0.34/0.63 = ( ! [X2: nat] :
% 0.34/0.63 ( ( member_nat @ X2 @ I )
% 0.34/0.63 => ( ( F @ X2 )
% 0.34/0.63 = C ) ) ) ) ) ) ).
% 0.34/0.63
% 0.34/0.63 % INF_eq_iff
% 0.34/0.63 thf(fact_4898_INF__eq__iff,axiom,
% 0.34/0.63 ! [I: set_int,F: int > product_unit,C: product_unit] :
% 0.34/0.63 ( ( I != bot_bot_set_int )
% 0.34/0.63 => ( ! [I3: int] :
% 0.34/0.63 ( ( member_int @ I3 @ I )
% 0.34/0.63 => ( ord_le3221252021190050221t_unit @ ( F @ I3 ) @ C ) )
% 0.34/0.63 => ( ( ( comple2584293577114468500t_unit @ ( image_5846526143423701914t_unit @ F @ I ) )
% 0.34/0.63 = C )
% 0.34/0.63 = ( ! [X2: int] :
% 0.34/0.63 ( ( member_int @ X2 @ I )
% 0.34/0.63 => ( ( F @ X2 )
% 0.34/0.63 = C ) ) ) ) ) ) ).
% 0.34/0.63
% 0.34/0.63 % INF_eq_iff
% 0.34/0.63 thf(fact_4899_INF__eq__iff,axiom,
% 0.34/0.63 ! [I: set_Product_unit,F: product_unit > filter_nat,C: filter_nat] :
% 0.34/0.63 ( ( I != bot_bo3957492148770167129t_unit )
% 0.34/0.63 => ( ! [I3: product_unit] :
% 0.34/0.63 ( ( member_Product_unit @ I3 @ I )
% 0.34/0.63 => ( ord_le2510731241096832064er_nat @ ( F @ I3 ) @ C ) )
% 0.34/0.63 => ( ( ( comple6612786467464916249er_nat @ ( image_1853478764106424488er_nat @ F @ I ) )
% 0.34/0.63 = C )
% 0.34/0.63 = ( ! [X2: product_unit] :
% 0.34/0.63 ( ( member_Product_unit @ X2 @ I )
% 0.34/0.63 => ( ( F @ X2 )
% 0.34/0.63 = C ) ) ) ) ) ) ).
% 0.34/0.63
% 0.34/0.63 % INF_eq_iff
% 0.34/0.63 thf(fact_4900_INF__eq__iff,axiom,
% 0.34/0.63 ! [I: set_o,F: $o > filter_nat,C: filter_nat] :
% 0.34/0.63 ( ( I != bot_bot_set_o )
% 0.34/0.63 => ( ! [I3: $o] :
% 0.34/0.63 ( ( member_o @ I3 @ I )
% 0.34/0.63 => ( ord_le2510731241096832064er_nat @ ( F @ I3 ) @ C ) )
% 0.34/0.63 => ( ( ( comple6612786467464916249er_nat @ ( image_o_filter_nat @ F @ I ) )
% 0.34/0.63 = C )
% 0.34/0.63 = ( ! [X2: $o] :
% 0.34/0.63 ( ( member_o @ X2 @ I )
% 0.34/0.63 => ( ( F @ X2 )
% 0.34/0.63 = C ) ) ) ) ) ) ).
% 0.34/0.63
% 0.34/0.63 % INF_eq_iff
% 0.34/0.63 thf(fact_4901_INF__eq__iff,axiom,
% 0.34/0.63 ! [I: set_nat,F: nat > filter_nat,C: filter_nat] :
% 0.34/0.63 ( ( I != bot_bot_set_nat )
% 0.34/0.63 => ( ! [I3: nat] :
% 0.34/0.63 ( ( member_nat @ I3 @ I )
% 0.34/0.63 => ( ord_le2510731241096832064er_nat @ ( F @ I3 ) @ C ) )
% 0.34/0.63 => ( ( ( comple6612786467464916249er_nat @ ( image_nat_filter_nat @ F @ I ) )
% 0.34/0.63 = C )
% 0.34/0.63 = ( ! [X2: nat] :
% 0.34/0.63 ( ( member_nat @ X2 @ I )
% 0.34/0.63 => ( ( F @ X2 )
% 0.34/0.63 = C ) ) ) ) ) ) ).
% 0.34/0.63
% 0.34/0.63 % INF_eq_iff
% 0.34/0.63 thf(fact_4902_INF__eq__iff,axiom,
% 0.34/0.63 ! [I: set_int,F: int > filter_nat,C: filter_nat] :
% 0.34/0.63 ( ( I != bot_bot_set_int )
% 0.34/0.63 => ( ! [I3: int] :
% 0.34/0.63 ( ( member_int @ I3 @ I )
% 0.34/0.63 => ( ord_le2510731241096832064er_nat @ ( F @ I3 ) @ C ) )
% 0.34/0.63 => ( ( ( comple6612786467464916249er_nat @ ( image_int_filter_nat @ F @ I ) )
% 0.34/0.63 = C )
% 0.34/0.63 = ( ! [X2: int] :
% 0.34/0.63 ( ( member_int @ X2 @ I )
% 0.34/0.63 => ( ( F @ X2 )
% 0.34/0.63 = C ) ) ) ) ) ) ).
% 0.34/0.63
% 0.34/0.63 % INF_eq_iff
% 0.34/0.63 thf(fact_4903_INF__eq__iff,axiom,
% 0.34/0.63 ! [I: set_Product_unit,F: product_unit > set_nat,C: set_nat] :
% 0.34/0.63 ( ( I != bot_bo3957492148770167129t_unit )
% 0.34/0.63 => ( ! [I3: product_unit] :
% 0.34/0.63 ( ( member_Product_unit @ I3 @ I )
% 0.34/0.63 => ( ord_less_eq_set_nat @ ( F @ I3 ) @ C ) )
% 0.34/0.63 => ( ( ( comple7806235888213564991et_nat @ ( image_8173849991827986510et_nat @ F @ I ) )
% 0.34/0.63 = C )
% 0.34/0.63 = ( ! [X2: product_unit] :
% 0.34/0.63 ( ( member_Product_unit @ X2 @ I )
% 0.34/0.63 => ( ( F @ X2 )
% 0.34/0.63 = C ) ) ) ) ) ) ).
% 0.34/0.63
% 0.34/0.63 % INF_eq_iff
% 0.34/0.63 thf(fact_4904_INF__eq__iff,axiom,
% 0.34/0.63 ! [I: set_o,F: $o > set_nat,C: set_nat] :
% 0.34/0.63 ( ( I != bot_bot_set_o )
% 0.34/0.63 => ( ! [I3: $o] :
% 0.34/0.63 ( ( member_o @ I3 @ I )
% 0.34/0.63 => ( ord_less_eq_set_nat @ ( F @ I3 ) @ C ) )
% 0.34/0.63 => ( ( ( comple7806235888213564991et_nat @ ( image_o_set_nat @ F @ I ) )
% 0.34/0.63 = C )
% 0.34/0.63 = ( ! [X2: $o] :
% 0.34/0.63 ( ( member_o @ X2 @ I )
% 0.34/0.63 => ( ( F @ X2 )
% 0.34/0.63 = C ) ) ) ) ) ) ).
% 0.34/0.63
% 0.34/0.63 % INF_eq_iff
% 0.34/0.63 thf(fact_4905_finite__less__Inf__iff,axiom,
% 0.34/0.63 ! [X4: set_int,A2: int] :
% 0.34/0.63 ( ( finite_finite_int @ X4 )
% 0.34/0.63 => ( ( X4 != bot_bot_set_int )
% 0.34/0.63 => ( ( ord_less_int @ A2 @ ( complete_Inf_Inf_int @ X4 ) )
% 0.34/0.63 = ( ! [X2: int] :
% 0.34/0.63 ( ( member_int @ X2 @ X4 )
% 0.34/0.63 => ( ord_less_int @ A2 @ X2 ) ) ) ) ) ) ).
% 0.34/0.63
% 0.34/0.63 % finite_less_Inf_iff
% 0.34/0.63 thf(fact_4906_finite__less__Inf__iff,axiom,
% 0.34/0.63 ! [X4: set_nat,A2: nat] :
% 0.34/0.63 ( ( finite_finite_nat @ X4 )
% 0.34/0.63 => ( ( X4 != bot_bot_set_nat )
% 0.34/0.63 => ( ( ord_less_nat @ A2 @ ( complete_Inf_Inf_nat @ X4 ) )
% 0.34/0.63 = ( ! [X2: nat] :
% 0.34/0.63 ( ( member_nat @ X2 @ X4 )
% 0.34/0.63 => ( ord_less_nat @ A2 @ X2 ) ) ) ) ) ) ).
% 0.34/0.63
% 0.34/0.63 % finite_less_Inf_iff
% 0.34/0.63 thf(fact_4907_finite__less__Inf__iff,axiom,
% 0.34/0.63 ! [X4: set_Product_unit,A2: product_unit] :
% 0.34/0.63 ( ( finite4290736615968046902t_unit @ X4 )
% 0.34/0.63 => ( ( X4 != bot_bo3957492148770167129t_unit )
% 0.34/0.63 => ( ( ord_le361264281704409273t_unit @ A2 @ ( comple2584293577114468500t_unit @ X4 ) )
% 0.34/0.63 = ( ! [X2: product_unit] :
% 0.34/0.63 ( ( member_Product_unit @ X2 @ X4 )
% 0.34/0.63 => ( ord_le361264281704409273t_unit @ A2 @ X2 ) ) ) ) ) ) ).
% 0.34/0.63
% 0.34/0.63 % finite_less_Inf_iff
% 0.34/0.63 thf(fact_4908_Inf__le__Sup,axiom,
% 0.34/0.63 ! [A: set_o] :
% 0.34/0.63 ( ( A != bot_bot_set_o )
% 0.34/0.63 => ( ord_less_eq_o @ ( complete_Inf_Inf_o @ A ) @ ( complete_Sup_Sup_o @ A ) ) ) ).
% 0.34/0.63
% 0.34/0.63 % Inf_le_Sup
% 0.34/0.63 thf(fact_4909_Inf__le__Sup,axiom,
% 0.34/0.63 ! [A: set_Pr4532377907799695533_nat_o] :
% 0.34/0.63 ( ( A != bot_bo7824918357723582233_nat_o )
% 0.34/0.63 => ( ord_le729326519192465773_nat_o @ ( comple90455603224345172_nat_o @ A ) @ ( comple4225350640907012154_nat_o @ A ) ) ) ).
% 0.34/0.63
% 0.34/0.63 % Inf_le_Sup
% 0.34/0.63 thf(fact_4910_Inf__le__Sup,axiom,
% 0.34/0.63 ! [A: set_filter_nat] :
% 0.34/0.63 ( ( A != bot_bo498966703094740906er_nat )
% 0.34/0.63 => ( ord_le2510731241096832064er_nat @ ( comple6612786467464916249er_nat @ A ) @ ( comple2454491259644266675er_nat @ A ) ) ) ).
% 0.34/0.63
% 0.34/0.63 % Inf_le_Sup
% 0.34/0.63 thf(fact_4911_Inf__le__Sup,axiom,
% 0.34/0.63 ! [A: set_set_nat] :
% 0.34/0.63 ( ( A != bot_bot_set_set_nat )
% 0.34/0.63 => ( ord_less_eq_set_nat @ ( comple7806235888213564991et_nat @ A ) @ ( comple7399068483239264473et_nat @ A ) ) ) ).
% 0.34/0.63
% 0.34/0.63 % Inf_le_Sup
% 0.34/0.63 thf(fact_4912_Inf__le__Sup,axiom,
% 0.34/0.63 ! [A: set_Product_unit] :
% 0.34/0.63 ( ( A != bot_bo3957492148770167129t_unit )
% 0.34/0.63 => ( ord_le3221252021190050221t_unit @ ( comple2584293577114468500t_unit @ A ) @ ( comple4687483117567863418t_unit @ A ) ) ) ).
% 0.34/0.63
% 0.34/0.63 % Inf_le_Sup
% 0.34/0.63 thf(fact_4913_finite__Inf__in,axiom,
% 0.34/0.63 ! [A: set_o] :
% 0.34/0.63 ( ( finite_finite_o @ A )
% 0.34/0.63 => ( ( A != bot_bot_set_o )
% 0.34/0.63 => ( ! [X3: $o,Y2: $o] :
% 0.34/0.63 ( ( member_o @ X3 @ A )
% 0.34/0.63 => ( ( member_o @ Y2 @ A )
% 0.34/0.63 => ( member_o @ ( inf_inf_o @ X3 @ Y2 ) @ A ) ) )
% 0.34/0.63 => ( member_o @ ( complete_Inf_Inf_o @ A ) @ A ) ) ) ) ).
% 0.34/0.63
% 0.34/0.63 % finite_Inf_in
% 0.34/0.63 thf(fact_4914_finite__Inf__in,axiom,
% 0.34/0.63 ! [A: set_Pr4532377907799695533_nat_o] :
% 0.34/0.63 ( ( finite3252695134891459830_nat_o @ A )
% 0.34/0.63 => ( ( A != bot_bo7824918357723582233_nat_o )
% 0.34/0.63 => ( ! [X3: produc3658429121746597890et_nat > $o,Y2: produc3658429121746597890et_nat > $o] :
% 0.34/0.63 ( ( member6576561426505652726_nat_o @ X3 @ A )
% 0.34/0.63 => ( ( member6576561426505652726_nat_o @ Y2 @ A )
% 0.34/0.63 => ( member6576561426505652726_nat_o @ ( inf_in1318976480646536635_nat_o @ X3 @ Y2 ) @ A ) ) )
% 0.34/0.63 => ( member6576561426505652726_nat_o @ ( comple90455603224345172_nat_o @ A ) @ A ) ) ) ) ).
% 0.34/0.63
% 0.34/0.63 % finite_Inf_in
% 0.34/0.63 thf(fact_4915_finite__Inf__in,axiom,
% 0.34/0.63 ! [A: set_set_nat] :
% 0.34/0.63 ( ( finite1152437895449049373et_nat @ A )
% 0.34/0.63 => ( ( A != bot_bot_set_set_nat )
% 0.34/0.63 => ( ! [X3: set_nat,Y2: set_nat] :
% 0.34/0.63 ( ( member_set_nat @ X3 @ A )
% 0.34/0.63 => ( ( member_set_nat @ Y2 @ A )
% 0.34/0.63 => ( member_set_nat @ ( inf_inf_set_nat @ X3 @ Y2 ) @ A ) ) )
% 0.34/0.63 => ( member_set_nat @ ( comple7806235888213564991et_nat @ A ) @ A ) ) ) ) ).
% 0.34/0.63
% 0.34/0.63 % finite_Inf_in
% 0.34/0.63 thf(fact_4916_finite__Inf__in,axiom,
% 0.34/0.63 ! [A: set_Product_unit] :
% 0.34/0.63 ( ( finite4290736615968046902t_unit @ A )
% 0.34/0.63 => ( ( A != bot_bo3957492148770167129t_unit )
% 0.34/0.63 => ( ! [X3: product_unit,Y2: product_unit] :
% 0.34/0.63 ( ( member_Product_unit @ X3 @ A )
% 0.34/0.63 => ( ( member_Product_unit @ Y2 @ A )
% 0.34/0.63 => ( member_Product_unit @ ( inf_inf_Product_unit @ X3 @ Y2 ) @ A ) ) )
% 0.34/0.63 => ( member_Product_unit @ ( comple2584293577114468500t_unit @ A ) @ A ) ) ) ) ).
% 0.34/0.63
% 0.34/0.63 % finite_Inf_in
% 0.34/0.63 thf(fact_4917_card__1__singletonI,axiom,
% 0.34/0.63 ! [S: set_Code_integer,X: code_integer] :
% 0.34/0.63 ( ( finite6017078050557962740nteger @ S )
% 0.34/0.63 => ( ( ( finite4902975817058060853nteger @ S )
% 0.34/0.63 = one_one_nat )
% 0.34/0.63 => ( ( member_Code_integer @ X @ S )
% 0.34/0.63 => ( S
% 0.34/0.63 = ( insert_Code_integer @ X @ bot_bo3990330152332043303nteger ) ) ) ) ) ).
% 0.34/0.63
% 0.34/0.63 % card_1_singletonI
% 0.34/0.63 thf(fact_4918_card__1__singletonI,axiom,
% 0.34/0.63 ! [S: set_Product_unit,X: product_unit] :
% 0.34/0.63 ( ( finite4290736615968046902t_unit @ S )
% 0.34/0.63 => ( ( ( finite410649719033368117t_unit @ S )
% 0.34/0.63 = one_one_nat )
% 0.34/0.63 => ( ( member_Product_unit @ X @ S )
% 0.34/0.63 => ( S
% 0.34/0.63 = ( insert_Product_unit @ X @ bot_bo3957492148770167129t_unit ) ) ) ) ) ).
% 0.34/0.63
% 0.34/0.63 % card_1_singletonI
% 0.34/0.63 thf(fact_4919_card__1__singletonI,axiom,
% 0.34/0.63 ! [S: set_o,X: $o] :
% 0.34/0.63 ( ( finite_finite_o @ S )
% 0.34/0.63 => ( ( ( finite_card_o @ S )
% 0.34/0.63 = one_one_nat )
% 0.34/0.63 => ( ( member_o @ X @ S )
% 0.34/0.63 => ( S
% 0.34/0.63 = ( insert_o @ X @ bot_bot_set_o ) ) ) ) ) ).
% 0.34/0.63
% 0.34/0.63 % card_1_singletonI
% 0.34/0.63 thf(fact_4920_card__1__singletonI,axiom,
% 0.34/0.63 ! [S: set_nat,X: nat] :
% 0.34/0.63 ( ( finite_finite_nat @ S )
% 0.34/0.63 => ( ( ( finite_card_nat @ S )
% 0.34/0.63 = one_one_nat )
% 0.34/0.63 => ( ( member_nat @ X @ S )
% 0.34/0.63 => ( S
% 0.34/0.63 = ( insert_nat @ X @ bot_bot_set_nat ) ) ) ) ) ).
% 0.34/0.63
% 0.34/0.63 % card_1_singletonI
% 0.34/0.63 thf(fact_4921_card__1__singletonI,axiom,
% 0.34/0.63 ! [S: set_int,X: int] :
% 0.34/0.63 ( ( finite_finite_int @ S )
% 0.34/0.63 => ( ( ( finite_card_int @ S )
% 0.34/0.63 = one_one_nat )
% 0.34/0.63 => ( ( member_int @ X @ S )
% 0.34/0.63 => ( S
% 0.34/0.63 = ( insert_int @ X @ bot_bot_set_int ) ) ) ) ) ).
% 0.34/0.63
% 0.34/0.63 % card_1_singletonI
% 0.34/0.63 thf(fact_4922_card__1__singletonI,axiom,
% 0.34/0.63 ! [S: set_Pr4532377907799695533_nat_o,X: produc3658429121746597890et_nat > $o] :
% 0.34/0.63 ( ( finite3252695134891459830_nat_o @ S )
% 0.34/0.63 => ( ( ( finite1363419556375932405_nat_o @ S )
% 0.34/0.63 = one_one_nat )
% 0.34/0.63 => ( ( member6576561426505652726_nat_o @ X @ S )
% 0.34/0.63 => ( S
% 0.34/0.63 = ( insert5175938949040314269_nat_o @ X @ bot_bo7824918357723582233_nat_o ) ) ) ) ) ).
% 0.34/0.63
% 0.34/0.63 % card_1_singletonI
% 0.34/0.63 thf(fact_4923_card__Diff__singleton__if,axiom,
% 0.34/0.63 ! [X: product_unit,A: set_Product_unit] :
% 0.34/0.63 ( ( ( member_Product_unit @ X @ A )
% 0.34/0.63 => ( ( finite410649719033368117t_unit @ ( minus_6452836326544984404t_unit @ A @ ( insert_Product_unit @ X @ bot_bo3957492148770167129t_unit ) ) )
% 0.34/0.63 = ( minus_minus_nat @ ( finite410649719033368117t_unit @ A ) @ one_one_nat ) ) )
% 0.34/0.63 & ( ~ ( member_Product_unit @ X @ A )
% 0.34/0.63 => ( ( finite410649719033368117t_unit @ ( minus_6452836326544984404t_unit @ A @ ( insert_Product_unit @ X @ bot_bo3957492148770167129t_unit ) ) )
% 0.34/0.63 = ( finite410649719033368117t_unit @ A ) ) ) ) ).
% 0.34/0.63
% 0.34/0.63 % card_Diff_singleton_if
% 0.34/0.63 thf(fact_4924_card__Diff__singleton__if,axiom,
% 0.34/0.63 ! [X: $o,A: set_o] :
% 0.34/0.63 ( ( ( member_o @ X @ A )
% 0.34/0.63 => ( ( finite_card_o @ ( minus_minus_set_o @ A @ ( insert_o @ X @ bot_bot_set_o ) ) )
% 0.34/0.63 = ( minus_minus_nat @ ( finite_card_o @ A ) @ one_one_nat ) ) )
% 0.34/0.63 & ( ~ ( member_o @ X @ A )
% 0.34/0.63 => ( ( finite_card_o @ ( minus_minus_set_o @ A @ ( insert_o @ X @ bot_bot_set_o ) ) )
% 0.34/0.63 = ( finite_card_o @ A ) ) ) ) ).
% 0.34/0.63
% 0.34/0.63 % card_Diff_singleton_if
% 0.34/0.63 thf(fact_4925_card__Diff__singleton__if,axiom,
% 0.34/0.63 ! [X: int,A: set_int] :
% 0.34/0.63 ( ( ( member_int @ X @ A )
% 0.34/0.63 => ( ( finite_card_int @ ( minus_minus_set_int @ A @ ( insert_int @ X @ bot_bot_set_int ) ) )
% 0.34/0.63 = ( minus_minus_nat @ ( finite_card_int @ A ) @ one_one_nat ) ) )
% 0.34/0.63 & ( ~ ( member_int @ X @ A )
% 0.34/0.63 => ( ( finite_card_int @ ( minus_minus_set_int @ A @ ( insert_int @ X @ bot_bot_set_int ) ) )
% 0.34/0.63 = ( finite_card_int @ A ) ) ) ) ).
% 0.34/0.63
% 0.34/0.63 % card_Diff_singleton_if
% 0.34/0.63 thf(fact_4926_card__Diff__singleton__if,axiom,
% 0.34/0.63 ! [X: produc3658429121746597890et_nat > $o,A: set_Pr4532377907799695533_nat_o] :
% 0.34/0.63 ( ( ( member6576561426505652726_nat_o @ X @ A )
% 0.34/0.63 => ( ( finite1363419556375932405_nat_o @ ( minus_1801376950450012436_nat_o @ A @ ( insert5175938949040314269_nat_o @ X @ bot_bo7824918357723582233_nat_o ) ) )
% 0.34/0.63 = ( minus_minus_nat @ ( finite1363419556375932405_nat_o @ A ) @ one_one_nat ) ) )
% 0.34/0.63 & ( ~ ( member6576561426505652726_nat_o @ X @ A )
% 0.34/0.63 => ( ( finite1363419556375932405_nat_o @ ( minus_1801376950450012436_nat_o @ A @ ( insert5175938949040314269_nat_o @ X @ bot_bo7824918357723582233_nat_o ) ) )
% 0.34/0.63 = ( finite1363419556375932405_nat_o @ A ) ) ) ) ).
% 0.34/0.63
% 0.34/0.63 % card_Diff_singleton_if
% 0.34/0.63 thf(fact_4927_card__Diff__singleton__if,axiom,
% 0.34/0.63 ! [X: nat,A: set_nat] :
% 0.34/0.63 ( ( ( member_nat @ X @ A )
% 0.34/0.63 => ( ( finite_card_nat @ ( minus_minus_set_nat @ A @ ( insert_nat @ X @ bot_bot_set_nat ) ) )
% 0.34/0.63 = ( minus_minus_nat @ ( finite_card_nat @ A ) @ one_one_nat ) ) )
% 0.34/0.63 & ( ~ ( member_nat @ X @ A )
% 0.34/0.63 => ( ( finite_card_nat @ ( minus_minus_set_nat @ A @ ( insert_nat @ X @ bot_bot_set_nat ) ) )
% 0.34/0.63 = ( finite_card_nat @ A ) ) ) ) ).
% 0.34/0.63
% 0.34/0.63 % card_Diff_singleton_if
% 0.34/0.63 thf(fact_4928_card__Diff__singleton,axiom,
% 0.34/0.63 ! [X: product_unit,A: set_Product_unit] :
% 0.34/0.63 ( ( member_Product_unit @ X @ A )
% 0.34/0.63 => ( ( finite410649719033368117t_unit @ ( minus_6452836326544984404t_unit @ A @ ( insert_Product_unit @ X @ bot_bo3957492148770167129t_unit ) ) )
% 0.34/0.63 = ( minus_minus_nat @ ( finite410649719033368117t_unit @ A ) @ one_one_nat ) ) ) ).
% 0.34/0.63
% 0.34/0.63 % card_Diff_singleton
% 0.34/0.63 thf(fact_4929_card__Diff__singleton,axiom,
% 0.34/0.63 ! [X: $o,A: set_o] :
% 0.34/0.63 ( ( member_o @ X @ A )
% 0.34/0.63 => ( ( finite_card_o @ ( minus_minus_set_o @ A @ ( insert_o @ X @ bot_bot_set_o ) ) )
% 0.34/0.63 = ( minus_minus_nat @ ( finite_card_o @ A ) @ one_one_nat ) ) ) ).
% 0.34/0.63
% 0.34/0.63 % card_Diff_singleton
% 0.34/0.63 thf(fact_4930_card__Diff__singleton,axiom,
% 0.34/0.63 ! [X: int,A: set_int] :
% 0.34/0.63 ( ( member_int @ X @ A )
% 0.34/0.63 => ( ( finite_card_int @ ( minus_minus_set_int @ A @ ( insert_int @ X @ bot_bot_set_int ) ) )
% 0.34/0.63 = ( minus_minus_nat @ ( finite_card_int @ A ) @ one_one_nat ) ) ) ).
% 0.34/0.63
% 0.34/0.63 % card_Diff_singleton
% 0.34/0.63 thf(fact_4931_card__Diff__singleton,axiom,
% 0.34/0.63 ! [X: produc3658429121746597890et_nat > $o,A: set_Pr4532377907799695533_nat_o] :
% 0.34/0.63 ( ( member6576561426505652726_nat_o @ X @ A )
% 0.34/0.63 => ( ( finite1363419556375932405_nat_o @ ( minus_1801376950450012436_nat_o @ A @ ( insert5175938949040314269_nat_o @ X @ bot_bo7824918357723582233_nat_o ) ) )
% 0.34/0.63 = ( minus_minus_nat @ ( finite1363419556375932405_nat_o @ A ) @ one_one_nat ) ) ) ).
% 0.34/0.63
% 0.34/0.63 % card_Diff_singleton
% 0.34/0.63 thf(fact_4932_card__Diff__singleton,axiom,
% 0.34/0.63 ! [X: nat,A: set_nat] :
% 0.34/0.63 ( ( member_nat @ X @ A )
% 0.34/0.63 => ( ( finite_card_nat @ ( minus_minus_set_nat @ A @ ( insert_nat @ X @ bot_bot_set_nat ) ) )
% 0.34/0.63 = ( minus_minus_nat @ ( finite_card_nat @ A ) @ one_one_nat ) ) ) ).
% 0.34/0.63
% 0.34/0.63 % card_Diff_singleton
% 0.34/0.63 thf(fact_4933_fst__image__mp,axiom,
% 0.34/0.63 ! [A: set_Pr3948176798113811640et_nat,B4: set_he8644174446132254637t_unit,X: heap_e7401611519738050253t_unit,Y: set_nat] :
% 0.34/0.63 ( ( ord_le4903435234951335757t_unit @ ( image_6824039242672219652t_unit @ produc1824681642469235216et_nat @ A ) @ B4 )
% 0.34/0.63 => ( ( member6260224972018164377et_nat @ ( produc7507926704131184380et_nat @ X @ Y ) @ A )
% 0.34/0.63 => ( member242389114213480694t_unit @ X @ B4 ) ) ) ).
% 0.34/0.63
% 0.34/0.63 % fst_image_mp
% 0.34/0.63 thf(fact_4934_fst__image__mp,axiom,
% 0.34/0.63 ! [A: set_Pr3286484037609594932et_nat,B4: set_Pr4532377907799695533_nat_o,X: produc3658429121746597890et_nat > $o,Y: produc3658429121746597890et_nat] :
% 0.34/0.63 ( ( ord_le2965882846123202637_nat_o @ ( image_6802380912191114376_nat_o @ produc995936583742144908et_nat @ A ) @ B4 )
% 0.34/0.63 => ( ( member1996754912294343701et_nat @ ( produc5001842942810119800et_nat @ X @ Y ) @ A )
% 0.34/0.63 => ( member6576561426505652726_nat_o @ X @ B4 ) ) ) ).
% 0.34/0.63
% 0.34/0.63 % fst_image_mp
% 0.34/0.63 thf(fact_4935_fst__image__mp,axiom,
% 0.34/0.63 ! [A: set_Pr8536935166611901872et_nat,B4: set_Pr4532377907799695533_nat_o,X: produc3658429121746597890et_nat > $o,Y: produc3925858234332021118et_nat] :
% 0.34/0.63 ( ( ord_le2965882846123202637_nat_o @ ( image_2517677226577294348_nat_o @ produc180342877477747464et_nat @ A ) @ B4 )
% 0.34/0.63 => ( ( member6124377750444531601et_nat @ ( produc2245416461498447860et_nat @ X @ Y ) @ A )
% 0.34/0.63 => ( member6576561426505652726_nat_o @ X @ B4 ) ) ) ).
% 0.34/0.63
% 0.34/0.63 % fst_image_mp
% 0.34/0.63 thf(fact_4936_fst__image__mp,axiom,
% 0.34/0.63 ! [A: set_Pr958786334691620121nt_int,B4: set_int,X: int,Y: int] :
% 0.34/0.63 ( ( ord_less_eq_set_int @ ( image_5042161079198086560nt_int @ product_fst_int_int @ A ) @ B4 )
% 0.34/0.63 => ( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ X @ Y ) @ A )
% 0.34/0.63 => ( member_int @ X @ B4 ) ) ) ).
% 0.34/0.63
% 0.34/0.63 % fst_image_mp
% 0.34/0.63 thf(fact_4937_fst__image__mp,axiom,
% 0.34/0.63 ! [A: set_Pr1261947904930325089at_nat,B4: set_nat,X: nat,Y: nat] :
% 0.34/0.63 ( ( ord_less_eq_set_nat @ ( image_2486076414777270412at_nat @ product_fst_nat_nat @ A ) @ B4 )
% 0.34/0.63 => ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ Y ) @ A )
% 0.34/0.63 => ( member_nat @ X @ B4 ) ) ) ).
% 0.34/0.63
% 0.34/0.63 % fst_image_mp
% 0.34/0.63 thf(fact_4938_fst__in__Field,axiom,
% 0.34/0.63 ! [R3: set_Pr958786334691620121nt_int] : ( ord_less_eq_set_int @ ( image_5042161079198086560nt_int @ product_fst_int_int @ R3 ) @ ( field_int @ R3 ) ) ).
% 0.34/0.63
% 0.34/0.63 % fst_in_Field
% 0.34/0.63 thf(fact_4939_fst__in__Field,axiom,
% 0.34/0.63 ! [R3: set_Pr1261947904930325089at_nat] : ( ord_less_eq_set_nat @ ( image_2486076414777270412at_nat @ product_fst_nat_nat @ R3 ) @ ( field_nat @ R3 ) ) ).
% 0.34/0.63
% 0.34/0.63 % fst_in_Field
% 0.34/0.63 thf(fact_4940_Inf__fin__Inf,axiom,
% 0.34/0.63 ! [A: set_o] :
% 0.34/0.63 ( ( finite_finite_o @ A )
% 0.34/0.63 => ( ( A != bot_bot_set_o )
% 0.34/0.63 => ( ( lattic4107685809792843317_fin_o @ A )
% 0.34/0.63 = ( complete_Inf_Inf_o @ A ) ) ) ) ).
% 0.34/0.63
% 0.34/0.63 % Inf_fin_Inf
% 0.34/0.63 thf(fact_4941_Inf__fin__Inf,axiom,
% 0.34/0.63 ! [A: set_Pr4532377907799695533_nat_o] :
% 0.34/0.63 ( ( finite3252695134891459830_nat_o @ A )
% 0.34/0.63 => ( ( A != bot_bo7824918357723582233_nat_o )
% 0.34/0.63 => ( ( lattic956194824204696298_nat_o @ A )
% 0.34/0.63 = ( comple90455603224345172_nat_o @ A ) ) ) ) ).
% 0.34/0.63
% 0.34/0.63 % Inf_fin_Inf
% 0.34/0.63 thf(fact_4942_Inf__fin__Inf,axiom,
% 0.34/0.63 ! [A: set_set_nat] :
% 0.34/0.63 ( ( finite1152437895449049373et_nat @ A )
% 0.34/0.63 => ( ( A != bot_bot_set_set_nat )
% 0.34/0.63 => ( ( lattic3014633134055518761et_nat @ A )
% 0.34/0.63 = ( comple7806235888213564991et_nat @ A ) ) ) ) ).
% 0.34/0.63
% 0.34/0.63 % Inf_fin_Inf
% 0.34/0.63 thf(fact_4943_Inf__fin__Inf,axiom,
% 0.34/0.63 ! [A: set_Product_unit] :
% 0.34/0.63 ( ( finite4290736615968046902t_unit @ A )
% 0.34/0.63 => ( ( A != bot_bo3957492148770167129t_unit )
% 0.34/0.63 => ( ( lattic1263872656861969706t_unit @ A )
% 0.34/0.63 = ( comple2584293577114468500t_unit @ A ) ) ) ) ).
% 0.34/0.63
% 0.34/0.63 % Inf_fin_Inf
% 0.34/0.63 thf(fact_4944_cInf__eq__Inf__fin,axiom,
% 0.34/0.63 ! [X4: set_o] :
% 0.34/0.63 ( ( finite_finite_o @ X4 )
% 0.34/0.63 => ( ( X4 != bot_bot_set_o )
% 0.34/0.63 => ( ( complete_Inf_Inf_o @ X4 )
% 0.34/0.63 = ( lattic4107685809792843317_fin_o @ X4 ) ) ) ) ).
% 0.34/0.63
% 0.34/0.63 % cInf_eq_Inf_fin
% 0.34/0.63 thf(fact_4945_cInf__eq__Inf__fin,axiom,
% 0.34/0.63 ! [X4: set_int] :
% 0.34/0.63 ( ( finite_finite_int @ X4 )
% 0.34/0.63 => ( ( X4 != bot_bot_set_int )
% 0.34/0.63 => ( ( complete_Inf_Inf_int @ X4 )
% 0.34/0.63 = ( lattic5235898064620869839in_int @ X4 ) ) ) ) ).
% 0.34/0.63
% 0.34/0.63 % cInf_eq_Inf_fin
% 0.34/0.63 thf(fact_4946_cInf__eq__Inf__fin,axiom,
% 0.34/0.63 ! [X4: set_Pr4532377907799695533_nat_o] :
% 0.34/0.63 ( ( finite3252695134891459830_nat_o @ X4 )
% 0.34/0.63 => ( ( X4 != bot_bo7824918357723582233_nat_o )
% 0.34/0.63 => ( ( comple90455603224345172_nat_o @ X4 )
% 0.34/0.63 = ( lattic956194824204696298_nat_o @ X4 ) ) ) ) ).
% 0.34/0.63
% 0.34/0.63 % cInf_eq_Inf_fin
% 0.34/0.63 thf(fact_4947_cInf__eq__Inf__fin,axiom,
% 0.34/0.63 ! [X4: set_nat] :
% 0.34/0.63 ( ( finite_finite_nat @ X4 )
% 0.34/0.63 => ( ( X4 != bot_bot_set_nat )
% 0.34/0.63 => ( ( complete_Inf_Inf_nat @ X4 )
% 0.34/0.63 = ( lattic5238388535129920115in_nat @ X4 ) ) ) ) ).
% 0.34/0.63
% 0.34/0.63 % cInf_eq_Inf_fin
% 0.34/0.63 thf(fact_4948_cInf__eq__Inf__fin,axiom,
% 0.34/0.63 ! [X4: set_set_nat] :
% 0.34/0.63 ( ( finite1152437895449049373et_nat @ X4 )
% 0.34/0.63 => ( ( X4 != bot_bot_set_set_nat )
% 0.34/0.63 => ( ( comple7806235888213564991et_nat @ X4 )
% 0.34/0.63 = ( lattic3014633134055518761et_nat @ X4 ) ) ) ) ).
% 0.34/0.63
% 0.34/0.63 % cInf_eq_Inf_fin
% 0.34/0.63 thf(fact_4949_cInf__eq__Inf__fin,axiom,
% 0.34/0.63 ! [X4: set_Product_unit] :
% 0.34/0.63 ( ( finite4290736615968046902t_unit @ X4 )
% 0.34/0.63 => ( ( X4 != bot_bo3957492148770167129t_unit )
% 0.34/0.63 => ( ( comple2584293577114468500t_unit @ X4 )
% 0.34/0.63 = ( lattic1263872656861969706t_unit @ X4 ) ) ) ) ).
% 0.34/0.63
% 0.34/0.63 % cInf_eq_Inf_fin
% 0.34/0.63 thf(fact_4950_prod__set__defs_I1_J,axiom,
% 0.34/0.63 ( basic_fsts_int_int
% 0.34/0.63 = ( ^ [P5: product_prod_int_int] : ( insert_int @ ( product_fst_int_int @ P5 ) @ bot_bot_set_int ) ) ) ).
% 0.34/0.63
% 0.34/0.63 % prod_set_defs(1)
% 0.34/0.63 thf(fact_4951_prod_Oswap__def,axiom,
% 0.34/0.63 ( produc1264744871843189710t_unit
% 0.34/0.63 = ( ^ [P5: produc7348995623955488188t_unit] : ( produc7507926704131184380et_nat @ ( produc3550216979005444800t_unit @ P5 ) @ ( produc6012101397789842562t_unit @ P5 ) ) ) ) ).
% 0.34/0.63
% 0.34/0.63 % prod.swap_def
% 0.34/0.63 thf(fact_4952_prod_Oswap__def,axiom,
% 0.34/0.63 ( product_swap_nat_nat
% 0.34/0.63 = ( ^ [P5: product_prod_nat_nat] : ( product_Pair_nat_nat @ ( product_snd_nat_nat @ P5 ) @ ( product_fst_nat_nat @ P5 ) ) ) ) ).
% 0.34/0.63
% 0.34/0.63 % prod.swap_def
% 0.34/0.63 thf(fact_4953_prod_Oswap__def,axiom,
% 0.34/0.63 ( produc1078497009909708626_nat_o
% 0.34/0.63 = ( ^ [P5: produc6907119039986491712_nat_o] : ( produc5001842942810119800et_nat @ ( produc1838968430882301508_nat_o @ P5 ) @ ( produc8046704426453032966_nat_o @ P5 ) ) ) ) ).
% 0.34/0.63
% 0.34/0.63 % prod.swap_def
% 0.34/0.63 thf(fact_4954_prod_Oswap__def,axiom,
% 0.34/0.63 ( produc2580581461503396054_nat_o
% 0.34/0.63 = ( ^ [P5: produc4962441903510821572_nat_o] : ( produc2245416461498447860et_nat @ ( produc6803565743942592456_nat_o @ P5 ) @ ( produc1609452848092598666_nat_o @ P5 ) ) ) ) ).
% 0.34/0.63
% 0.34/0.63 % prod.swap_def
% 0.34/0.63 thf(fact_4955_prod_Oswap__def,axiom,
% 0.34/0.63 ( produc6300697153377358172et_nat
% 0.34/0.63 = ( ^ [P5: produc3658429121746597890et_nat] : ( produc2471974422597015918t_unit @ ( produc8586169260539613262et_nat @ P5 ) @ ( produc1824681642469235216et_nat @ P5 ) ) ) ) ).
% 0.34/0.63
% 0.34/0.63 % prod.swap_def
% 0.34/0.63 thf(fact_4956_prod_Oswap__def,axiom,
% 0.34/0.63 ( product_swap_int_int
% 0.34/0.63 = ( ^ [P5: product_prod_int_int] : ( product_Pair_int_int @ ( product_snd_int_int @ P5 ) @ ( product_fst_int_int @ P5 ) ) ) ) ).
% 0.34/0.63
% 0.34/0.63 % prod.swap_def
% 0.34/0.63 thf(fact_4957_equiv__proj,axiom,
% 0.34/0.63 ! [A: set_int,R3: set_Pr958786334691620121nt_int,Z: product_prod_int_int] :
% 0.34/0.63 ( ( equiv_equiv_int @ A @ R3 )
% 0.34/0.63 => ( ( member5262025264175285858nt_int @ Z @ R3 )
% 0.34/0.63 => ( ( comp_i5103216804979976198nt_int @ ( equiv_proj_int_int @ R3 ) @ product_fst_int_int @ Z )
% 0.34/0.63 = ( comp_i5103216804979976198nt_int @ ( equiv_proj_int_int @ R3 ) @ product_snd_int_int @ Z ) ) ) ) ).
% 0.34/0.63
% 0.34/0.63 % equiv_proj
% 0.34/0.63 thf(fact_4958_INF__le__SUP,axiom,
% 0.34/0.63 ! [A: set_Product_unit,F: product_unit > product_unit] :
% 0.34/0.63 ( ( A != bot_bo3957492148770167129t_unit )
% 0.34/0.63 => ( ord_le3221252021190050221t_unit @ ( comple2584293577114468500t_unit @ ( image_405062704495631173t_unit @ F @ A ) ) @ ( comple4687483117567863418t_unit @ ( image_405062704495631173t_unit @ F @ A ) ) ) ) ).
% 0.34/0.63
% 0.34/0.63 % INF_le_SUP
% 0.34/0.63 thf(fact_4959_INF__le__SUP,axiom,
% 0.34/0.63 ! [A: set_o,F: $o > product_unit] :
% 0.34/0.63 ( ( A != bot_bot_set_o )
% 0.34/0.63 => ( ord_le3221252021190050221t_unit @ ( comple2584293577114468500t_unit @ ( image_o_Product_unit2 @ F @ A ) ) @ ( comple4687483117567863418t_unit @ ( image_o_Product_unit2 @ F @ A ) ) ) ) ).
% 0.34/0.63
% 0.34/0.63 % INF_le_SUP
% 0.34/0.63 thf(fact_4960_INF__le__SUP,axiom,
% 0.34/0.63 ! [A: set_nat,F: nat > product_unit] :
% 0.34/0.63 ( ( A != bot_bot_set_nat )
% 0.34/0.63 => ( ord_le3221252021190050221t_unit @ ( comple2584293577114468500t_unit @ ( image_8730104196221521654t_unit @ F @ A ) ) @ ( comple4687483117567863418t_unit @ ( image_8730104196221521654t_unit @ F @ A ) ) ) ) ).
% 0.34/0.63
% 0.34/0.63 % INF_le_SUP
% 0.34/0.63 thf(fact_4961_INF__le__SUP,axiom,
% 0.34/0.63 ! [A: set_int,F: int > product_unit] :
% 0.34/0.63 ( ( A != bot_bot_set_int )
% 0.34/0.63 => ( ord_le3221252021190050221t_unit @ ( comple2584293577114468500t_unit @ ( image_5846526143423701914t_unit @ F @ A ) ) @ ( comple4687483117567863418t_unit @ ( image_5846526143423701914t_unit @ F @ A ) ) ) ) ).
% 0.34/0.63
% 0.34/0.63 % INF_le_SUP
% 0.34/0.63 thf(fact_4962_INF__le__SUP,axiom,
% 0.34/0.63 ! [A: set_Product_unit,F: product_unit > filter_nat] :
% 0.34/0.63 ( ( A != bot_bo3957492148770167129t_unit )
% 0.34/0.63 => ( ord_le2510731241096832064er_nat @ ( comple6612786467464916249er_nat @ ( image_1853478764106424488er_nat @ F @ A ) ) @ ( comple2454491259644266675er_nat @ ( image_1853478764106424488er_nat @ F @ A ) ) ) ) ).
% 0.34/0.63
% 0.34/0.63 % INF_le_SUP
% 0.34/0.63 thf(fact_4963_INF__le__SUP,axiom,
% 0.34/0.63 ! [A: set_o,F: $o > filter_nat] :
% 0.34/0.63 ( ( A != bot_bot_set_o )
% 0.34/0.63 => ( ord_le2510731241096832064er_nat @ ( comple6612786467464916249er_nat @ ( image_o_filter_nat @ F @ A ) ) @ ( comple2454491259644266675er_nat @ ( image_o_filter_nat @ F @ A ) ) ) ) ).
% 0.34/0.63
% 0.34/0.63 % INF_le_SUP
% 0.34/0.63 thf(fact_4964_INF__le__SUP,axiom,
% 0.34/0.63 ! [A: set_nat,F: nat > filter_nat] :
% 0.34/0.63 ( ( A != bot_bot_set_nat )
% 0.34/0.63 => ( ord_le2510731241096832064er_nat @ ( comple6612786467464916249er_nat @ ( image_nat_filter_nat @ F @ A ) ) @ ( comple2454491259644266675er_nat @ ( image_nat_filter_nat @ F @ A ) ) ) ) ).
% 0.34/0.63
% 0.34/0.63 % INF_le_SUP
% 0.34/0.63 thf(fact_4965_INF__le__SUP,axiom,
% 0.34/0.63 ! [A: set_int,F: int > filter_nat] :
% 0.34/0.63 ( ( A != bot_bot_set_int )
% 0.34/0.63 => ( ord_le2510731241096832064er_nat @ ( comple6612786467464916249er_nat @ ( image_int_filter_nat @ F @ A ) ) @ ( comple2454491259644266675er_nat @ ( image_int_filter_nat @ F @ A ) ) ) ) ).
% 0.34/0.63
% 0.34/0.63 % INF_le_SUP
% 0.34/0.63 thf(fact_4966_INF__le__SUP,axiom,
% 0.34/0.63 ! [A: set_Product_unit,F: product_unit > set_nat] :
% 0.34/0.63 ( ( A != bot_bo3957492148770167129t_unit )
% 0.34/0.63 => ( ord_less_eq_set_nat @ ( comple7806235888213564991et_nat @ ( image_8173849991827986510et_nat @ F @ A ) ) @ ( comple7399068483239264473et_nat @ ( image_8173849991827986510et_nat @ F @ A ) ) ) ) ).
% 0.34/0.63
% 0.34/0.63 % INF_le_SUP
% 0.34/0.63 thf(fact_4967_INF__le__SUP,axiom,
% 0.34/0.63 ! [A: set_o,F: $o > set_nat] :
% 0.34/0.63 ( ( A != bot_bot_set_o )
% 0.34/0.63 => ( ord_less_eq_set_nat @ ( comple7806235888213564991et_nat @ ( image_o_set_nat @ F @ A ) ) @ ( comple7399068483239264473et_nat @ ( image_o_set_nat @ F @ A ) ) ) ) ).
% 0.34/0.63
% 0.34/0.63 % INF_le_SUP
% 0.34/0.63 thf(fact_4968_card__Diff1__le,axiom,
% 0.34/0.63 ! [A: set_Product_unit,X: product_unit] : ( ord_less_eq_nat @ ( finite410649719033368117t_unit @ ( minus_6452836326544984404t_unit @ A @ ( insert_Product_unit @ X @ bot_bo3957492148770167129t_unit ) ) ) @ ( finite410649719033368117t_unit @ A ) ) ).
% 0.34/0.63
% 0.34/0.63 % card_Diff1_le
% 0.34/0.63 thf(fact_4969_card__Diff1__le,axiom,
% 0.34/0.63 ! [A: set_o,X: $o] : ( ord_less_eq_nat @ ( finite_card_o @ ( minus_minus_set_o @ A @ ( insert_o @ X @ bot_bot_set_o ) ) ) @ ( finite_card_o @ A ) ) ).
% 0.34/0.63
% 0.34/0.63 % card_Diff1_le
% 0.34/0.63 thf(fact_4970_card__Diff1__le,axiom,
% 0.34/0.63 ! [A: set_int,X: int] : ( ord_less_eq_nat @ ( finite_card_int @ ( minus_minus_set_int @ A @ ( insert_int @ X @ bot_bot_set_int ) ) ) @ ( finite_card_int @ A ) ) ).
% 0.34/0.63
% 0.34/0.63 % card_Diff1_le
% 0.34/0.63 thf(fact_4971_card__Diff1__le,axiom,
% 0.34/0.63 ! [A: set_Pr4532377907799695533_nat_o,X: produc3658429121746597890et_nat > $o] : ( ord_less_eq_nat @ ( finite1363419556375932405_nat_o @ ( minus_1801376950450012436_nat_o @ A @ ( insert5175938949040314269_nat_o @ X @ bot_bo7824918357723582233_nat_o ) ) ) @ ( finite1363419556375932405_nat_o @ A ) ) ).
% 0.34/0.63
% 0.34/0.63 % card_Diff1_le
% 0.34/0.63 thf(fact_4972_card__Diff1__le,axiom,
% 0.34/0.63 ! [A: set_nat,X: nat] : ( ord_less_eq_nat @ ( finite_card_nat @ ( minus_minus_set_nat @ A @ ( insert_nat @ X @ bot_bot_set_nat ) ) ) @ ( finite_card_nat @ A ) ) ).
% 0.34/0.63
% 0.34/0.63 % card_Diff1_le
% 0.34/0.63 thf(fact_4973_exI__realizer,axiom,
% 0.34/0.63 ! [P: nat > nat > $o,Y: nat,X: nat] :
% 0.34/0.63 ( ( P @ Y @ X )
% 0.34/0.63 => ( P @ ( product_snd_nat_nat @ ( product_Pair_nat_nat @ X @ Y ) ) @ ( product_fst_nat_nat @ ( product_Pair_nat_nat @ X @ Y ) ) ) ) ).
% 0.34/0.63
% 0.34/0.63 % exI_realizer
% 0.34/0.63 thf(fact_4974_exI__realizer,axiom,
% 0.34/0.63 ! [P: produc3658429121746597890et_nat > ( produc3658429121746597890et_nat > $o ) > $o,Y: produc3658429121746597890et_nat,X: produc3658429121746597890et_nat > $o] :
% 0.34/0.63 ( ( P @ Y @ X )
% 0.34/0.63 => ( P @ ( produc4011572625026189258et_nat @ ( produc5001842942810119800et_nat @ X @ Y ) ) @ ( produc995936583742144908et_nat @ ( produc5001842942810119800et_nat @ X @ Y ) ) ) ) ).
% 0.34/0.63
% 0.34/0.63 % exI_realizer
% 0.34/0.63 thf(fact_4975_exI__realizer,axiom,
% 0.34/0.63 ! [P: produc3925858234332021118et_nat > ( produc3658429121746597890et_nat > $o ) > $o,Y: produc3925858234332021118et_nat,X: produc3658429121746597890et_nat > $o] :
% 0.34/0.63 ( ( P @ Y @ X )
% 0.34/0.63 => ( P @ ( produc5374455773327741254et_nat @ ( produc2245416461498447860et_nat @ X @ Y ) ) @ ( produc180342877477747464et_nat @ ( produc2245416461498447860et_nat @ X @ Y ) ) ) ) ).
% 0.34/0.63
% 0.34/0.63 % exI_realizer
% 0.34/0.63 thf(fact_4976_exI__realizer,axiom,
% 0.34/0.63 ! [P: set_nat > heap_e7401611519738050253t_unit > $o,Y: set_nat,X: heap_e7401611519738050253t_unit] :
% 0.34/0.63 ( ( P @ Y @ X )
% 0.34/0.63 => ( P @ ( produc8586169260539613262et_nat @ ( produc7507926704131184380et_nat @ X @ Y ) ) @ ( produc1824681642469235216et_nat @ ( produc7507926704131184380et_nat @ X @ Y ) ) ) ) ).
% 0.34/0.63
% 0.34/0.63 % exI_realizer
% 0.34/0.63 thf(fact_4977_exI__realizer,axiom,
% 0.34/0.63 ! [P: int > int > $o,Y: int,X: int] :
% 0.34/0.63 ( ( P @ Y @ X )
% 0.34/0.63 => ( P @ ( product_snd_int_int @ ( product_Pair_int_int @ X @ Y ) ) @ ( product_fst_int_int @ ( product_Pair_int_int @ X @ Y ) ) ) ) ).
% 0.34/0.63
% 0.34/0.63 % exI_realizer
% 0.34/0.63 thf(fact_4978_conjI__realizer,axiom,
% 0.34/0.63 ! [P: nat > $o,P3: nat,Q2: nat > $o,Q4: nat] :
% 0.34/0.63 ( ( P @ P3 )
% 0.34/0.63 => ( ( Q2 @ Q4 )
% 0.34/0.63 => ( ( P @ ( product_fst_nat_nat @ ( product_Pair_nat_nat @ P3 @ Q4 ) ) )
% 0.34/0.63 & ( Q2 @ ( product_snd_nat_nat @ ( product_Pair_nat_nat @ P3 @ Q4 ) ) ) ) ) ) ).
% 0.34/0.63
% 0.34/0.63 % conjI_realizer
% 0.34/0.63 thf(fact_4979_conjI__realizer,axiom,
% 0.34/0.63 ! [P: ( produc3658429121746597890et_nat > $o ) > $o,P3: produc3658429121746597890et_nat > $o,Q2: produc3658429121746597890et_nat > $o,Q4: produc3658429121746597890et_nat] :
% 0.34/0.63 ( ( P @ P3 )
% 0.34/0.63 => ( ( Q2 @ Q4 )
% 0.34/0.63 => ( ( P @ ( produc995936583742144908et_nat @ ( produc5001842942810119800et_nat @ P3 @ Q4 ) ) )
% 0.34/0.63 & ( Q2 @ ( produc4011572625026189258et_nat @ ( produc5001842942810119800et_nat @ P3 @ Q4 ) ) ) ) ) ) ).
% 0.34/0.63
% 0.34/0.63 % conjI_realizer
% 0.34/0.63 thf(fact_4980_conjI__realizer,axiom,
% 0.34/0.63 ! [P: ( produc3658429121746597890et_nat > $o ) > $o,P3: produc3658429121746597890et_nat > $o,Q2: produc3925858234332021118et_nat > $o,Q4: produc3925858234332021118et_nat] :
% 0.34/0.63 ( ( P @ P3 )
% 0.34/0.63 => ( ( Q2 @ Q4 )
% 0.34/0.63 => ( ( P @ ( produc180342877477747464et_nat @ ( produc2245416461498447860et_nat @ P3 @ Q4 ) ) )
% 0.34/0.63 & ( Q2 @ ( produc5374455773327741254et_nat @ ( produc2245416461498447860et_nat @ P3 @ Q4 ) ) ) ) ) ) ).
% 0.34/0.63
% 0.34/0.63 % conjI_realizer
% 0.34/0.63 thf(fact_4981_conjI__realizer,axiom,
% 0.34/0.63 ! [P: heap_e7401611519738050253t_unit > $o,P3: heap_e7401611519738050253t_unit,Q2: set_nat > $o,Q4: set_nat] :
% 0.34/0.63 ( ( P @ P3 )
% 0.34/0.63 => ( ( Q2 @ Q4 )
% 0.34/0.63 => ( ( P @ ( produc1824681642469235216et_nat @ ( produc7507926704131184380et_nat @ P3 @ Q4 ) ) )
% 0.34/0.63 & ( Q2 @ ( produc8586169260539613262et_nat @ ( produc7507926704131184380et_nat @ P3 @ Q4 ) ) ) ) ) ) ).
% 0.34/0.63
% 0.34/0.63 % conjI_realizer
% 0.34/0.63 thf(fact_4982_conjI__realizer,axiom,
% 0.34/0.63 ! [P: int > $o,P3: int,Q2: int > $o,Q4: int] :
% 0.34/0.63 ( ( P @ P3 )
% 0.34/0.63 => ( ( Q2 @ Q4 )
% 0.34/0.63 => ( ( P @ ( product_fst_int_int @ ( product_Pair_int_int @ P3 @ Q4 ) ) )
% 0.34/0.63 & ( Q2 @ ( product_snd_int_int @ ( product_Pair_int_int @ P3 @ Q4 ) ) ) ) ) ) ).
% 0.34/0.63
% 0.34/0.63 % conjI_realizer
% 0.34/0.63 thf(fact_4983_apfst__convE,axiom,
% 0.34/0.63 ! [Q4: produc3658429121746597890et_nat,F: heap_e7401611519738050253t_unit > heap_e7401611519738050253t_unit,P3: produc3658429121746597890et_nat] :
% 0.34/0.63 ( ( Q4
% 0.34/0.63 = ( produc4172003873565521026et_nat @ F @ P3 ) )
% 0.34/0.63 => ~ ! [X3: heap_e7401611519738050253t_unit,Y2: set_nat] :
% 0.34/0.63 ( ( P3
% 0.34/0.63 = ( produc7507926704131184380et_nat @ X3 @ Y2 ) )
% 0.34/0.63 => ( Q4
% 0.34/0.63 != ( produc7507926704131184380et_nat @ ( F @ X3 ) @ Y2 ) ) ) ) ).
% 0.34/0.63
% 0.34/0.63 % apfst_convE
% 0.34/0.63 thf(fact_4984_apfst__convE,axiom,
% 0.34/0.63 ! [Q4: product_prod_nat_nat,F: nat > nat,P3: product_prod_nat_nat] :
% 0.34/0.63 ( ( Q4
% 0.34/0.63 = ( produc986720760941809198at_nat @ F @ P3 ) )
% 0.34/0.63 => ~ ! [X3: nat,Y2: nat] :
% 0.34/0.63 ( ( P3
% 0.34/0.63 = ( product_Pair_nat_nat @ X3 @ Y2 ) )
% 0.34/0.63 => ( Q4
% 0.34/0.63 != ( product_Pair_nat_nat @ ( F @ X3 ) @ Y2 ) ) ) ) ).
% 0.34/0.63
% 0.34/0.63 % apfst_convE
% 0.34/0.63 thf(fact_4985_apfst__convE,axiom,
% 0.34/0.63 ! [Q4: product_prod_int_int,F: int > int,P3: product_prod_int_int] :
% 0.34/0.63 ( ( Q4
% 0.34/0.63 = ( produc2355237562570658114nt_int @ F @ P3 ) )
% 0.34/0.63 => ~ ! [X3: int,Y2: int] :
% 0.34/0.63 ( ( P3
% 0.34/0.63 = ( product_Pair_int_int @ X3 @ Y2 ) )
% 0.34/0.63 => ( Q4
% 0.34/0.63 != ( product_Pair_int_int @ ( F @ X3 ) @ Y2 ) ) ) ) ).
% 0.34/0.63
% 0.34/0.63 % apfst_convE
% 0.34/0.63 thf(fact_4986_apfst__convE,axiom,
% 0.34/0.63 ! [Q4: produc3925858234332021118et_nat,F: ( produc3658429121746597890et_nat > $o ) > produc3658429121746597890et_nat > $o,P3: produc3925858234332021118et_nat] :
% 0.34/0.63 ( ( Q4
% 0.34/0.63 = ( produc3198230245617193214et_nat @ F @ P3 ) )
% 0.34/0.63 => ~ ! [X3: produc3658429121746597890et_nat > $o,Y2: produc3658429121746597890et_nat] :
% 0.34/0.63 ( ( P3
% 0.34/0.63 = ( produc5001842942810119800et_nat @ X3 @ Y2 ) )
% 0.34/0.63 => ( Q4
% 0.34/0.63 != ( produc5001842942810119800et_nat @ ( F @ X3 ) @ Y2 ) ) ) ) ).
% 0.34/0.63
% 0.34/0.63 % apfst_convE
% 0.34/0.63 thf(fact_4987_apfst__convE,axiom,
% 0.34/0.63 ! [Q4: produc2732055786443039994et_nat,F: ( produc3658429121746597890et_nat > $o ) > produc3658429121746597890et_nat > $o,P3: produc2732055786443039994et_nat] :
% 0.34/0.63 ( ( Q4
% 0.34/0.63 = ( produc1442628905706551930et_nat @ F @ P3 ) )
% 0.34/0.63 => ~ ! [X3: produc3658429121746597890et_nat > $o,Y2: produc3925858234332021118et_nat] :
% 0.34/0.63 ( ( P3
% 0.34/0.63 = ( produc2245416461498447860et_nat @ X3 @ Y2 ) )
% 0.34/0.63 => ( Q4
% 0.34/0.63 != ( produc2245416461498447860et_nat @ ( F @ X3 ) @ Y2 ) ) ) ) ).
% 0.34/0.63
% 0.34/0.63 % apfst_convE
% 0.34/0.63 thf(fact_4988_exE__realizer_H,axiom,
% 0.34/0.63 ! [P: set_nat > heap_e7401611519738050253t_unit > $o,P3: produc3658429121746597890et_nat] :
% 0.34/0.63 ( ( P @ ( produc8586169260539613262et_nat @ P3 ) @ ( produc1824681642469235216et_nat @ P3 ) )
% 0.34/0.63 => ~ ! [X3: heap_e7401611519738050253t_unit,Y2: set_nat] :
% 0.34/0.63 ~ ( P @ Y2 @ X3 ) ) ).
% 0.34/0.63
% 0.34/0.63 % exE_realizer'
% 0.34/0.63 thf(fact_4989_exE__realizer_H,axiom,
% 0.34/0.63 ! [P: int > int > $o,P3: product_prod_int_int] :
% 0.34/0.63 ( ( P @ ( product_snd_int_int @ P3 ) @ ( product_fst_int_int @ P3 ) )
% 0.34/0.63 => ~ ! [X3: int,Y2: int] :
% 0.34/0.63 ~ ( P @ Y2 @ X3 ) ) ).
% 0.34/0.63
% 0.34/0.63 % exE_realizer'
% 0.34/0.63 thf(fact_4990_snd__sndOp,axiom,
% 0.34/0.63 ! [P: heap_e7401611519738050253t_unit > heap_e7401611519738050253t_unit > $o,Q2: heap_e7401611519738050253t_unit > set_nat > $o] :
% 0.34/0.63 ( produc8586169260539613262et_nat
% 0.34/0.63 = ( comp_P2930521774885306735et_nat @ produc8586169260539613262et_nat @ ( bNF_sn4039337771197282234et_nat @ P @ Q2 ) ) ) ).
% 0.34/0.63
% 0.34/0.63 % snd_sndOp
% 0.34/0.63 thf(fact_4991_snd__sndOp,axiom,
% 0.34/0.63 ! [P: int > int > $o,Q2: int > int > $o] :
% 0.34/0.63 ( product_snd_int_int
% 0.34/0.63 = ( comp_P5356232565778667509nt_int @ product_snd_int_int @ ( bNF_sn1062102010912252026nt_int @ P @ Q2 ) ) ) ).
% 0.34/0.63
% 0.34/0.63 % snd_sndOp
% 0.34/0.63 thf(fact_4992_fst__fstOp,axiom,
% 0.34/0.63 ! [P: int > int > $o,Q2: int > int > $o] :
% 0.34/0.63 ( product_fst_int_int
% 0.34/0.63 = ( comp_P5356232565778667509nt_int @ product_fst_int_int @ ( bNF_fs8167890477030535480nt_int @ P @ Q2 ) ) ) ).
% 0.34/0.63
% 0.34/0.63 % fst_fstOp
% 0.34/0.63 thf(fact_4993_card__Un__disjoint,axiom,
% 0.34/0.63 ! [A: set_Code_integer,B4: set_Code_integer] :
% 0.34/0.63 ( ( finite6017078050557962740nteger @ A )
% 0.34/0.63 => ( ( finite6017078050557962740nteger @ B4 )
% 0.34/0.63 => ( ( ( inf_in1364745209274528805nteger @ A @ B4 )
% 0.34/0.63 = bot_bo3990330152332043303nteger )
% 0.34/0.63 => ( ( finite4902975817058060853nteger @ ( sup_su848401254843788991nteger @ A @ B4 ) )
% 0.34/0.63 = ( plus_plus_nat @ ( finite4902975817058060853nteger @ A ) @ ( finite4902975817058060853nteger @ B4 ) ) ) ) ) ) ).
% 0.34/0.63
% 0.34/0.63 % card_Un_disjoint
% 0.34/0.63 thf(fact_4994_card__Un__disjoint,axiom,
% 0.34/0.63 ! [A: set_Product_unit,B4: set_Product_unit] :
% 0.34/0.63 ( ( finite4290736615968046902t_unit @ A )
% 0.34/0.63 => ( ( finite4290736615968046902t_unit @ B4 )
% 0.34/0.63 => ( ( ( inf_in4660618365625256667t_unit @ A @ B4 )
% 0.34/0.63 = bot_bo3957492148770167129t_unit )
% 0.34/0.63 => ( ( finite410649719033368117t_unit @ ( sup_su793286257634532545t_unit @ A @ B4 ) )
% 0.34/0.63 = ( plus_plus_nat @ ( finite410649719033368117t_unit @ A ) @ ( finite410649719033368117t_unit @ B4 ) ) ) ) ) ) ).
% 0.34/0.63
% 0.34/0.63 % card_Un_disjoint
% 0.34/0.63 thf(fact_4995_card__Un__disjoint,axiom,
% 0.34/0.63 ! [A: set_o,B4: set_o] :
% 0.34/0.63 ( ( finite_finite_o @ A )
% 0.34/0.63 => ( ( finite_finite_o @ B4 )
% 0.34/0.63 => ( ( ( inf_inf_set_o @ A @ B4 )
% 0.34/0.63 = bot_bot_set_o )
% 0.34/0.63 => ( ( finite_card_o @ ( sup_sup_set_o @ A @ B4 ) )
% 0.34/0.63 = ( plus_plus_nat @ ( finite_card_o @ A ) @ ( finite_card_o @ B4 ) ) ) ) ) ) ).
% 0.34/0.63
% 0.34/0.63 % card_Un_disjoint
% 0.34/0.63 thf(fact_4996_card__Un__disjoint,axiom,
% 0.34/0.63 ! [A: set_nat,B4: set_nat] :
% 0.34/0.63 ( ( finite_finite_nat @ A )
% 0.34/0.63 => ( ( finite_finite_nat @ B4 )
% 0.34/0.63 => ( ( ( inf_inf_set_nat @ A @ B4 )
% 0.34/0.63 = bot_bot_set_nat )
% 0.34/0.63 => ( ( finite_card_nat @ ( sup_sup_set_nat @ A @ B4 ) )
% 0.34/0.63 = ( plus_plus_nat @ ( finite_card_nat @ A ) @ ( finite_card_nat @ B4 ) ) ) ) ) ) ).
% 0.34/0.63
% 0.34/0.63 % card_Un_disjoint
% 0.34/0.63 thf(fact_4997_card__Un__disjoint,axiom,
% 0.34/0.63 ! [A: set_int,B4: set_int] :
% 0.34/0.63 ( ( finite_finite_int @ A )
% 0.34/0.63 => ( ( finite_finite_int @ B4 )
% 0.34/0.63 => ( ( ( inf_inf_set_int @ A @ B4 )
% 0.34/0.63 = bot_bot_set_int )
% 0.34/0.63 => ( ( finite_card_int @ ( sup_sup_set_int @ A @ B4 ) )
% 0.34/0.63 = ( plus_plus_nat @ ( finite_card_int @ A ) @ ( finite_card_int @ B4 ) ) ) ) ) ) ).
% 0.34/0.63
% 0.34/0.63 % card_Un_disjoint
% 0.34/0.63 thf(fact_4998_card__Un__disjoint,axiom,
% 0.34/0.63 ! [A: set_Pr4532377907799695533_nat_o,B4: set_Pr4532377907799695533_nat_o] :
% 0.34/0.63 ( ( finite3252695134891459830_nat_o @ A )
% 0.34/0.63 => ( ( finite3252695134891459830_nat_o @ B4 )
% 0.34/0.63 => ( ( ( inf_in1906310914598751387_nat_o @ A @ B4 )
% 0.34/0.63 = bot_bo7824918357723582233_nat_o )
% 0.34/0.63 => ( ( finite1363419556375932405_nat_o @ ( sup_su5209123915105501825_nat_o @ A @ B4 ) )
% 0.34/0.63 = ( plus_plus_nat @ ( finite1363419556375932405_nat_o @ A ) @ ( finite1363419556375932405_nat_o @ B4 ) ) ) ) ) ) ).
% 0.34/0.63
% 0.34/0.63 % card_Un_disjoint
% 0.34/0.63 thf(fact_4999_card__Suc__Diff1,axiom,
% 0.34/0.63 ! [A: set_Code_integer,X: code_integer] :
% 0.34/0.63 ( ( finite6017078050557962740nteger @ A )
% 0.34/0.63 => ( ( member_Code_integer @ X @ A )
% 0.34/0.63 => ( ( suc @ ( finite4902975817058060853nteger @ ( minus_2355218937544613996nteger @ A @ ( insert_Code_integer @ X @ bot_bo3990330152332043303nteger ) ) ) )
% 0.34/0.63 = ( finite4902975817058060853nteger @ A ) ) ) ) ).
% 0.34/0.63
% 0.34/0.63 % card_Suc_Diff1
% 0.34/0.63 thf(fact_5000_card__Suc__Diff1,axiom,
% 0.34/0.63 ! [A: set_Product_unit,X: product_unit] :
% 0.34/0.63 ( ( finite4290736615968046902t_unit @ A )
% 0.34/0.63 => ( ( member_Product_unit @ X @ A )
% 0.34/0.63 => ( ( suc @ ( finite410649719033368117t_unit @ ( minus_6452836326544984404t_unit @ A @ ( insert_Product_unit @ X @ bot_bo3957492148770167129t_unit ) ) ) )
% 0.34/0.63 = ( finite410649719033368117t_unit @ A ) ) ) ) ).
% 0.34/0.63
% 0.34/0.63 % card_Suc_Diff1
% 0.34/0.63 thf(fact_5001_card__Suc__Diff1,axiom,
% 0.34/0.63 ! [A: set_o,X: $o] :
% 0.34/0.63 ( ( finite_finite_o @ A )
% 0.34/0.63 => ( ( member_o @ X @ A )
% 0.34/0.63 => ( ( suc @ ( finite_card_o @ ( minus_minus_set_o @ A @ ( insert_o @ X @ bot_bot_set_o ) ) ) )
% 0.34/0.63 = ( finite_card_o @ A ) ) ) ) ).
% 0.34/0.63
% 0.34/0.63 % card_Suc_Diff1
% 0.34/0.63 thf(fact_5002_card__Suc__Diff1,axiom,
% 0.34/0.63 ! [A: set_int,X: int] :
% 0.34/0.63 ( ( finite_finite_int @ A )
% 0.34/0.63 => ( ( member_int @ X @ A )
% 0.34/0.63 => ( ( suc @ ( finite_card_int @ ( minus_minus_set_int @ A @ ( insert_int @ X @ bot_bot_set_int ) ) ) )
% 0.34/0.63 = ( finite_card_int @ A ) ) ) ) ).
% 0.34/0.63
% 0.34/0.63 % card_Suc_Diff1
% 0.34/0.63 thf(fact_5003_card__Suc__Diff1,axiom,
% 0.34/0.63 ! [A: set_Pr4532377907799695533_nat_o,X: produc3658429121746597890et_nat > $o] :
% 0.34/0.63 ( ( finite3252695134891459830_nat_o @ A )
% 0.34/0.63 => ( ( member6576561426505652726_nat_o @ X @ A )
% 0.34/0.63 => ( ( suc @ ( finite1363419556375932405_nat_o @ ( minus_1801376950450012436_nat_o @ A @ ( insert5175938949040314269_nat_o @ X @ bot_bo7824918357723582233_nat_o ) ) ) )
% 0.34/0.63 = ( finite1363419556375932405_nat_o @ A ) ) ) ) ).
% 0.34/0.63
% 0.34/0.63 % card_Suc_Diff1
% 0.34/0.63 thf(fact_5004_card__Suc__Diff1,axiom,
% 0.34/0.63 ! [A: set_nat,X: nat] :
% 0.34/0.63 ( ( finite_finite_nat @ A )
% 0.34/0.63 => ( ( member_nat @ X @ A )
% 0.34/0.63 => ( ( suc @ ( finite_card_nat @ ( minus_minus_set_nat @ A @ ( insert_nat @ X @ bot_bot_set_nat ) ) ) )
% 0.34/0.63 = ( finite_card_nat @ A ) ) ) ) ).
% 0.34/0.63
% 0.34/0.63 % card_Suc_Diff1
% 0.34/0.63 thf(fact_5005_add__left__cancel,axiom,
% 0.34/0.63 ! [A2: code_integer,B: code_integer,C: code_integer] :
% 0.34/0.63 ( ( ( plus_p5714425477246183910nteger @ A2 @ B )
% 0.34/0.63 = ( plus_p5714425477246183910nteger @ A2 @ C ) )
% 0.34/0.63 = ( B = C ) ) ).
% 0.34/0.63
% 0.34/0.63 % add_left_cancel
% 0.34/0.63 thf(fact_5006_add__left__cancel,axiom,
% 0.34/0.63 ! [A2: rat,B: rat,C: rat] :
% 0.34/0.63 ( ( ( plus_plus_rat @ A2 @ B )
% 0.34/0.63 = ( plus_plus_rat @ A2 @ C ) )
% 0.34/0.63 = ( B = C ) ) ).
% 0.34/0.63
% 0.34/0.63 % add_left_cancel
% 0.34/0.63 thf(fact_5007_add__left__cancel,axiom,
% 0.34/0.63 ! [A2: nat,B: nat,C: nat] :
% 0.34/0.63 ( ( ( plus_plus_nat @ A2 @ B )
% 0.34/0.63 = ( plus_plus_nat @ A2 @ C ) )
% 0.34/0.63 = ( B = C ) ) ).
% 0.34/0.63
% 0.34/0.63 % add_left_cancel
% 0.34/0.63 thf(fact_5008_add__left__cancel,axiom,
% 0.34/0.63 ! [A2: int,B: int,C: int] :
% 0.34/0.63 ( ( ( plus_plus_int @ A2 @ B )
% 0.34/0.63 = ( plus_plus_int @ A2 @ C ) )
% 0.34/0.63 = ( B = C ) ) ).
% 0.34/0.63
% 0.34/0.63 % add_left_cancel
% 0.34/0.63 thf(fact_5009_add__right__cancel,axiom,
% 0.34/0.63 ! [B: code_integer,A2: code_integer,C: code_integer] :
% 0.34/0.63 ( ( ( plus_p5714425477246183910nteger @ B @ A2 )
% 0.34/0.63 = ( plus_p5714425477246183910nteger @ C @ A2 ) )
% 0.34/0.63 = ( B = C ) ) ).
% 0.34/0.63
% 0.34/0.63 % add_right_cancel
% 0.34/0.63 thf(fact_5010_add__right__cancel,axiom,
% 0.34/0.63 ! [B: rat,A2: rat,C: rat] :
% 0.34/0.63 ( ( ( plus_plus_rat @ B @ A2 )
% 0.34/0.63 = ( plus_plus_rat @ C @ A2 ) )
% 0.34/0.63 = ( B = C ) ) ).
% 0.34/0.63
% 0.34/0.63 % add_right_cancel
% 0.34/0.63 thf(fact_5011_add__right__cancel,axiom,
% 0.34/0.63 ! [B: nat,A2: nat,C: nat] :
% 0.34/0.63 ( ( ( plus_plus_nat @ B @ A2 )
% 0.34/0.63 = ( plus_plus_nat @ C @ A2 ) )
% 0.34/0.63 = ( B = C ) ) ).
% 0.34/0.63
% 0.34/0.63 % add_right_cancel
% 0.34/0.63 thf(fact_5012_add__right__cancel,axiom,
% 0.34/0.63 ! [B: int,A2: int,C: int] :
% 0.34/0.63 ( ( ( plus_plus_int @ B @ A2 )
% 0.34/0.63 = ( plus_plus_int @ C @ A2 ) )
% 0.34/0.63 = ( B = C ) ) ).
% 0.34/0.63
% 0.34/0.63 % add_right_cancel
% 0.34/0.63 thf(fact_5013_add__le__cancel__left,axiom,
% 0.34/0.63 ! [C: code_integer,A2: code_integer,B: code_integer] :
% 0.34/0.63 ( ( ord_le3102999989581377725nteger @ ( plus_p5714425477246183910nteger @ C @ A2 ) @ ( plus_p5714425477246183910nteger @ C @ B ) )
% 0.34/0.63 = ( ord_le3102999989581377725nteger @ A2 @ B ) ) ).
% 0.34/0.63
% 0.34/0.63 % add_le_cancel_left
% 0.34/0.63 thf(fact_5014_add__le__cancel__left,axiom,
% 0.34/0.63 ! [C: rat,A2: rat,B: rat] :
% 0.34/0.63 ( ( ord_less_eq_rat @ ( plus_plus_rat @ C @ A2 ) @ ( plus_plus_rat @ C @ B ) )
% 0.34/0.63 = ( ord_less_eq_rat @ A2 @ B ) ) ).
% 0.34/0.63
% 0.34/0.63 % add_le_cancel_left
% 0.34/0.63 thf(fact_5015_add__le__cancel__left,axiom,
% 0.34/0.63 ! [C: nat,A2: nat,B: nat] :
% 0.34/0.63 ( ( ord_less_eq_nat @ ( plus_plus_nat @ C @ A2 ) @ ( plus_plus_nat @ C @ B ) )
% 0.34/0.63 = ( ord_less_eq_nat @ A2 @ B ) ) ).
% 0.34/0.63
% 0.34/0.63 % add_le_cancel_left
% 0.34/0.63 thf(fact_5016_add__le__cancel__left,axiom,
% 0.34/0.63 ! [C: int,A2: int,B: int] :
% 0.34/0.63 ( ( ord_less_eq_int @ ( plus_plus_int @ C @ A2 ) @ ( plus_plus_int @ C @ B ) )
% 0.34/0.63 = ( ord_less_eq_int @ A2 @ B ) ) ).
% 0.34/0.63
% 0.34/0.63 % add_le_cancel_left
% 0.34/0.63 thf(fact_5017_add__le__cancel__right,axiom,
% 0.34/0.63 ! [A2: code_integer,C: code_integer,B: code_integer] :
% 0.34/0.63 ( ( ord_le3102999989581377725nteger @ ( plus_p5714425477246183910nteger @ A2 @ C ) @ ( plus_p5714425477246183910nteger @ B @ C ) )
% 0.34/0.63 = ( ord_le3102999989581377725nteger @ A2 @ B ) ) ).
% 0.34/0.63
% 0.34/0.63 % add_le_cancel_right
% 0.34/0.63 thf(fact_5018_add__le__cancel__right,axiom,
% 0.34/0.63 ! [A2: rat,C: rat,B: rat] :
% 0.34/0.63 ( ( ord_less_eq_rat @ ( plus_plus_rat @ A2 @ C ) @ ( plus_plus_rat @ B @ C ) )
% 0.34/0.63 = ( ord_less_eq_rat @ A2 @ B ) ) ).
% 0.34/0.63
% 0.34/0.63 % add_le_cancel_right
% 0.34/0.63 thf(fact_5019_add__le__cancel__right,axiom,
% 0.34/0.63 ! [A2: nat,C: nat,B: nat] :
% 0.34/0.63 ( ( ord_less_eq_nat @ ( plus_plus_nat @ A2 @ C ) @ ( plus_plus_nat @ B @ C ) )
% 0.34/0.63 = ( ord_less_eq_nat @ A2 @ B ) ) ).
% 0.34/0.63
% 0.34/0.63 % add_le_cancel_right
% 0.34/0.63 thf(fact_5020_add__le__cancel__right,axiom,
% 0.34/0.63 ! [A2: int,C: int,B: int] :
% 0.34/0.63 ( ( ord_less_eq_int @ ( plus_plus_int @ A2 @ C ) @ ( plus_plus_int @ B @ C ) )
% 0.34/0.63 = ( ord_less_eq_int @ A2 @ B ) ) ).
% 0.34/0.63
% 0.34/0.63 % add_le_cancel_right
% 0.34/0.63 thf(fact_5021_add__less__cancel__left,axiom,
% 0.34/0.63 ! [C: nat,A2: nat,B: nat] :
% 0.34/0.63 ( ( ord_less_nat @ ( plus_plus_nat @ C @ A2 ) @ ( plus_plus_nat @ C @ B ) )
% 0.34/0.63 = ( ord_less_nat @ A2 @ B ) ) ).
% 0.34/0.63
% 0.34/0.63 % add_less_cancel_left
% 0.34/0.63 thf(fact_5022_add__less__cancel__left,axiom,
% 0.34/0.63 ! [C: int,A2: int,B: int] :
% 0.34/0.63 ( ( ord_less_int @ ( plus_plus_int @ C @ A2 ) @ ( plus_plus_int @ C @ B ) )
% 0.34/0.63 = ( ord_less_int @ A2 @ B ) ) ).
% 0.34/0.63
% 0.34/0.63 % add_less_cancel_left
% 0.34/0.63 thf(fact_5023_add__less__cancel__left,axiom,
% 0.34/0.63 ! [C: code_integer,A2: code_integer,B: code_integer] :
% 0.34/0.63 ( ( ord_le6747313008572928689nteger @ ( plus_p5714425477246183910nteger @ C @ A2 ) @ ( plus_p5714425477246183910nteger @ C @ B ) )
% 0.34/0.63 = ( ord_le6747313008572928689nteger @ A2 @ B ) ) ).
% 0.34/0.63
% 0.34/0.63 % add_less_cancel_left
% 0.34/0.63 thf(fact_5024_add__less__cancel__left,axiom,
% 0.34/0.63 ! [C: rat,A2: rat,B: rat] :
% 0.34/0.63 ( ( ord_less_rat @ ( plus_plus_rat @ C @ A2 ) @ ( plus_plus_rat @ C @ B ) )
% 0.34/0.63 = ( ord_less_rat @ A2 @ B ) ) ).
% 0.34/0.63
% 0.34/0.63 % add_less_cancel_left
% 0.34/0.63 thf(fact_5025_add__less__cancel__right,axiom,
% 0.34/0.63 ! [A2: nat,C: nat,B: nat] :
% 0.34/0.63 ( ( ord_less_nat @ ( plus_plus_nat @ A2 @ C ) @ ( plus_plus_nat @ B @ C ) )
% 0.34/0.63 = ( ord_less_nat @ A2 @ B ) ) ).
% 0.34/0.63
% 0.34/0.63 % add_less_cancel_right
% 0.34/0.63 thf(fact_5026_add__less__cancel__right,axiom,
% 0.34/0.63 ! [A2: int,C: int,B: int] :
% 0.34/0.63 ( ( ord_less_int @ ( plus_plus_int @ A2 @ C ) @ ( plus_plus_int @ B @ C ) )
% 0.34/0.63 = ( ord_less_int @ A2 @ B ) ) ).
% 0.34/0.63
% 0.34/0.63 % add_less_cancel_right
% 0.34/0.63 thf(fact_5027_add__less__cancel__right,axiom,
% 0.34/0.63 ! [A2: code_integer,C: code_integer,B: code_integer] :
% 0.34/0.63 ( ( ord_le6747313008572928689nteger @ ( plus_p5714425477246183910nteger @ A2 @ C ) @ ( plus_p5714425477246183910nteger @ B @ C ) )
% 0.34/0.63 = ( ord_le6747313008572928689nteger @ A2 @ B ) ) ).
% 0.34/0.63
% 0.34/0.63 % add_less_cancel_right
% 0.34/0.63 thf(fact_5028_add__less__cancel__right,axiom,
% 0.34/0.63 ! [A2: rat,C: rat,B: rat] :
% 0.34/0.63 ( ( ord_less_rat @ ( plus_plus_rat @ A2 @ C ) @ ( plus_plus_rat @ B @ C ) )
% 0.34/0.63 = ( ord_less_rat @ A2 @ B ) ) ).
% 0.34/0.63
% 0.34/0.63 % add_less_cancel_right
% 0.34/0.63 thf(fact_5029_add__diff__cancel,axiom,
% 0.34/0.63 ! [A2: code_integer,B: code_integer] :
% 0.34/0.63 ( ( minus_8373710615458151222nteger @ ( plus_p5714425477246183910nteger @ A2 @ B ) @ B )
% 0.34/0.63 = A2 ) ).
% 0.34/0.63
% 0.34/0.63 % add_diff_cancel
% 0.34/0.63 thf(fact_5030_add__diff__cancel,axiom,
% 0.34/0.63 ! [A2: rat,B: rat] :
% 0.34/0.63 ( ( minus_minus_rat @ ( plus_plus_rat @ A2 @ B ) @ B )
% 0.34/0.63 = A2 ) ).
% 0.34/0.63
% 0.34/0.63 % add_diff_cancel
% 0.34/0.63 thf(fact_5031_add__diff__cancel,axiom,
% 0.34/0.63 ! [A2: int,B: int] :
% 0.34/0.63 ( ( minus_minus_int @ ( plus_plus_int @ A2 @ B ) @ B )
% 0.34/0.63 = A2 ) ).
% 0.34/0.63
% 0.34/0.63 % add_diff_cancel
% 0.34/0.63 thf(fact_5032_diff__add__cancel,axiom,
% 0.34/0.63 ! [A2: code_integer,B: code_integer] :
% 0.34/0.63 ( ( plus_p5714425477246183910nteger @ ( minus_8373710615458151222nteger @ A2 @ B ) @ B )
% 0.34/0.63 = A2 ) ).
% 0.34/0.63
% 0.34/0.63 % diff_add_cancel
% 0.34/0.63 thf(fact_5033_diff__add__cancel,axiom,
% 0.34/0.63 ! [A2: rat,B: rat] :
% 0.34/0.63 ( ( plus_plus_rat @ ( minus_minus_rat @ A2 @ B ) @ B )
% 0.34/0.63 = A2 ) ).
% 0.34/0.63
% 0.34/0.63 % diff_add_cancel
% 0.34/0.63 thf(fact_5034_diff__add__cancel,axiom,
% 0.34/0.63 ! [A2: int,B: int] :
% 0.34/0.63 ( ( plus_plus_int @ ( minus_minus_int @ A2 @ B ) @ B )
% 0.34/0.63 = A2 ) ).
% 0.34/0.63
% 0.34/0.63 % diff_add_cancel
% 0.34/0.63 thf(fact_5035_add__diff__cancel__left,axiom,
% 0.34/0.63 ! [C: code_integer,A2: code_integer,B: code_integer] :
% 0.34/0.63 ( ( minus_8373710615458151222nteger @ ( plus_p5714425477246183910nteger @ C @ A2 ) @ ( plus_p5714425477246183910nteger @ C @ B ) )
% 0.34/0.63 = ( minus_8373710615458151222nteger @ A2 @ B ) ) ).
% 0.34/0.63
% 0.34/0.63 % add_diff_cancel_left
% 0.34/0.63 thf(fact_5036_add__diff__cancel__left,axiom,
% 0.34/0.63 ! [C: rat,A2: rat,B: rat] :
% 0.34/0.63 ( ( minus_minus_rat @ ( plus_plus_rat @ C @ A2 ) @ ( plus_plus_rat @ C @ B ) )
% 0.34/0.63 = ( minus_minus_rat @ A2 @ B ) ) ).
% 0.34/0.63
% 0.34/0.63 % add_diff_cancel_left
% 0.34/0.63 thf(fact_5037_add__diff__cancel__left,axiom,
% 0.34/0.63 ! [C: nat,A2: nat,B: nat] :
% 0.34/0.63 ( ( minus_minus_nat @ ( plus_plus_nat @ C @ A2 ) @ ( plus_plus_nat @ C @ B ) )
% 0.34/0.63 = ( minus_minus_nat @ A2 @ B ) ) ).
% 0.34/0.63
% 0.34/0.63 % add_diff_cancel_left
% 0.34/0.63 thf(fact_5038_add__diff__cancel__left,axiom,
% 0.34/0.63 ! [C: int,A2: int,B: int] :
% 0.34/0.63 ( ( minus_minus_int @ ( plus_plus_int @ C @ A2 ) @ ( plus_plus_int @ C @ B ) )
% 0.34/0.63 = ( minus_minus_int @ A2 @ B ) ) ).
% 0.34/0.63
% 0.34/0.63 % add_diff_cancel_left
% 0.34/0.63 thf(fact_5039_add__diff__cancel__left_H,axiom,
% 0.34/0.63 ! [A2: code_integer,B: code_integer] :
% 0.34/0.63 ( ( minus_8373710615458151222nteger @ ( plus_p5714425477246183910nteger @ A2 @ B ) @ A2 )
% 0.34/0.63 = B ) ).
% 0.34/0.63
% 0.34/0.63 % add_diff_cancel_left'
% 0.34/0.63 thf(fact_5040_add__diff__cancel__left_H,axiom,
% 0.34/0.63 ! [A2: rat,B: rat] :
% 0.34/0.63 ( ( minus_minus_rat @ ( plus_plus_rat @ A2 @ B ) @ A2 )
% 0.34/0.63 = B ) ).
% 0.34/0.63
% 0.34/0.63 % add_diff_cancel_left'
% 0.34/0.63 thf(fact_5041_add__diff__cancel__left_H,axiom,
% 0.34/0.63 ! [A2: nat,B: nat] :
% 0.34/0.63 ( ( minus_minus_nat @ ( plus_plus_nat @ A2 @ B ) @ A2 )
% 0.34/0.63 = B ) ).
% 0.34/0.63
% 0.34/0.63 % add_diff_cancel_left'
% 0.34/0.63 thf(fact_5042_add__diff__cancel__left_H,axiom,
% 0.34/0.63 ! [A2: int,B: int] :
% 0.34/0.63 ( ( minus_minus_int @ ( plus_plus_int @ A2 @ B ) @ A2 )
% 0.34/0.63 = B ) ).
% 0.34/0.63
% 0.34/0.63 % add_diff_cancel_left'
% 0.34/0.63 thf(fact_5043_add__diff__cancel__right,axiom,
% 0.34/0.63 ! [A2: code_integer,C: code_integer,B: code_integer] :
% 0.34/0.63 ( ( minus_8373710615458151222nteger @ ( plus_p5714425477246183910nteger @ A2 @ C ) @ ( plus_p5714425477246183910nteger @ B @ C ) )
% 0.34/0.63 = ( minus_8373710615458151222nteger @ A2 @ B ) ) ).
% 0.34/0.63
% 0.34/0.63 % add_diff_cancel_right
% 0.34/0.63 thf(fact_5044_add__diff__cancel__right,axiom,
% 0.34/0.63 ! [A2: rat,C: rat,B: rat] :
% 0.34/0.63 ( ( minus_minus_rat @ ( plus_plus_rat @ A2 @ C ) @ ( plus_plus_rat @ B @ C ) )
% 0.34/0.63 = ( minus_minus_rat @ A2 @ B ) ) ).
% 0.34/0.63
% 0.34/0.63 % add_diff_cancel_right
% 0.34/0.63 thf(fact_5045_add__diff__cancel__right,axiom,
% 0.34/0.63 ! [A2: nat,C: nat,B: nat] :
% 0.34/0.63 ( ( minus_minus_nat @ ( plus_plus_nat @ A2 @ C ) @ ( plus_plus_nat @ B @ C ) )
% 0.34/0.63 = ( minus_minus_nat @ A2 @ B ) ) ).
% 0.34/0.63
% 0.34/0.63 % add_diff_cancel_right
% 0.34/0.63 thf(fact_5046_add__diff__cancel__right,axiom,
% 0.34/0.63 ! [A2: int,C: int,B: int] :
% 0.34/0.63 ( ( minus_minus_int @ ( plus_plus_int @ A2 @ C ) @ ( plus_plus_int @ B @ C ) )
% 0.34/0.63 = ( minus_minus_int @ A2 @ B ) ) ).
% 0.34/0.63
% 0.34/0.63 % add_diff_cancel_right
% 0.34/0.63 thf(fact_5047_add__diff__cancel__right_H,axiom,
% 0.34/0.63 ! [A2: code_integer,B: code_integer] :
% 0.34/0.63 ( ( minus_8373710615458151222nteger @ ( plus_p5714425477246183910nteger @ A2 @ B ) @ B )
% 0.34/0.63 = A2 ) ).
% 0.34/0.63
% 0.34/0.63 % add_diff_cancel_right'
% 0.34/0.63 thf(fact_5048_add__diff__cancel__right_H,axiom,
% 0.34/0.63 ! [A2: rat,B: rat] :
% 0.34/0.63 ( ( minus_minus_rat @ ( plus_plus_rat @ A2 @ B ) @ B )
% 0.34/0.63 = A2 ) ).
% 0.34/0.63
% 0.34/0.63 % add_diff_cancel_right'
% 0.34/0.63 thf(fact_5049_add__diff__cancel__right_H,axiom,
% 0.34/0.63 ! [A2: nat,B: nat] :
% 0.34/0.63 ( ( minus_minus_nat @ ( plus_plus_nat @ A2 @ B ) @ B )
% 0.34/0.63 = A2 ) ).
% 0.34/0.63
% 0.34/0.63 % add_diff_cancel_right'
% 0.34/0.63 thf(fact_5050_add__diff__cancel__right_H,axiom,
% 0.34/0.63 ! [A2: int,B: int] :
% 0.34/0.63 ( ( minus_minus_int @ ( plus_plus_int @ A2 @ B ) @ B )
% 0.34/0.63 = A2 ) ).
% 0.34/0.63
% 0.34/0.63 % add_diff_cancel_right'
% 0.34/0.63 thf(fact_5051_card__insert__disjoint,axiom,
% 0.34/0.63 ! [A: set_Pr4532377907799695533_nat_o,X: produc3658429121746597890et_nat > $o] :
% 0.34/0.63 ( ( finite3252695134891459830_nat_o @ A )
% 0.34/0.63 => ( ~ ( member6576561426505652726_nat_o @ X @ A )
% 0.34/0.63 => ( ( finite1363419556375932405_nat_o @ ( insert5175938949040314269_nat_o @ X @ A ) )
% 0.34/0.63 = ( suc @ ( finite1363419556375932405_nat_o @ A ) ) ) ) ) ).
% 0.34/0.63
% 0.34/0.63 % card_insert_disjoint
% 0.34/0.63 thf(fact_5052_card__insert__disjoint,axiom,
% 0.34/0.63 ! [A: set_int,X: int] :
% 0.34/0.63 ( ( finite_finite_int @ A )
% 0.34/0.63 => ( ~ ( member_int @ X @ A )
% 0.34/0.63 => ( ( finite_card_int @ ( insert_int @ X @ A ) )
% 0.34/0.63 = ( suc @ ( finite_card_int @ A ) ) ) ) ) ).
% 0.34/0.63
% 0.34/0.63 % card_insert_disjoint
% 0.34/0.63 thf(fact_5053_card__insert__disjoint,axiom,
% 0.34/0.63 ! [A: set_o,X: $o] :
% 0.34/0.63 ( ( finite_finite_o @ A )
% 0.34/0.63 => ( ~ ( member_o @ X @ A )
% 0.34/0.63 => ( ( finite_card_o @ ( insert_o @ X @ A ) )
% 0.34/0.63 = ( suc @ ( finite_card_o @ A ) ) ) ) ) ).
% 0.34/0.63
% 0.34/0.63 % card_insert_disjoint
% 0.34/0.63 thf(fact_5054_card__insert__disjoint,axiom,
% 0.34/0.63 ! [A: set_Product_unit,X: product_unit] :
% 0.34/0.63 ( ( finite4290736615968046902t_unit @ A )
% 0.34/0.63 => ( ~ ( member_Product_unit @ X @ A )
% 0.34/0.63 => ( ( finite410649719033368117t_unit @ ( insert_Product_unit @ X @ A ) )
% 0.34/0.63 = ( suc @ ( finite410649719033368117t_unit @ A ) ) ) ) ) ).
% 0.34/0.63
% 0.34/0.63 % card_insert_disjoint
% 0.34/0.63 thf(fact_5055_card__insert__disjoint,axiom,
% 0.34/0.63 ! [A: set_nat,X: nat] :
% 0.34/0.63 ( ( finite_finite_nat @ A )
% 0.34/0.63 => ( ~ ( member_nat @ X @ A )
% 0.34/0.63 => ( ( finite_card_nat @ ( insert_nat @ X @ A ) )
% 0.34/0.63 = ( suc @ ( finite_card_nat @ A ) ) ) ) ) ).
% 0.34/0.63
% 0.34/0.63 % card_insert_disjoint
% 0.34/0.63 thf(fact_5056_card__insert__disjoint,axiom,
% 0.34/0.63 ! [A: set_Code_integer,X: code_integer] :
% 0.34/0.63 ( ( finite6017078050557962740nteger @ A )
% 0.34/0.63 => ( ~ ( member_Code_integer @ X @ A )
% 0.34/0.63 => ( ( finite4902975817058060853nteger @ ( insert_Code_integer @ X @ A ) )
% 0.34/0.63 = ( suc @ ( finite4902975817058060853nteger @ A ) ) ) ) ) ).
% 0.34/0.63
% 0.34/0.63 % card_insert_disjoint
% 0.34/0.63 thf(fact_5057_Suc__diff,axiom,
% 0.34/0.63 ! [M4: nat,N2: nat] :
% 0.34/0.63 ( ( ord_less_eq_nat @ M4 @ N2 )
% 0.34/0.63 => ( ( ord_less_eq_nat @ one_one_nat @ M4 )
% 0.34/0.63 => ( ( suc @ ( minus_minus_nat @ N2 @ M4 ) )
% 0.34/0.63 = ( minus_minus_nat @ N2 @ ( minus_minus_nat @ M4 @ one_one_nat ) ) ) ) ) ).
% 0.34/0.63
% 0.34/0.63 % Suc_diff
% 0.34/0.63 thf(fact_5058_add__mono__thms__linordered__semiring_I4_J,axiom,
% 0.34/0.63 ! [I2: code_integer,J: code_integer,K3: code_integer,L: code_integer] :
% 0.34/0.63 ( ( ( I2 = J )
% 0.34/0.63 & ( K3 = L ) )
% 0.34/0.63 => ( ( plus_p5714425477246183910nteger @ I2 @ K3 )
% 0.34/0.63 = ( plus_p5714425477246183910nteger @ J @ L ) ) ) ).
% 0.34/0.63
% 0.34/0.63 % add_mono_thms_linordered_semiring(4)
% 0.34/0.63 thf(fact_5059_add__mono__thms__linordered__semiring_I4_J,axiom,
% 0.34/0.63 ! [I2: rat,J: rat,K3: rat,L: rat] :
% 0.34/0.63 ( ( ( I2 = J )
% 0.34/0.63 & ( K3 = L ) )
% 0.34/0.63 => ( ( plus_plus_rat @ I2 @ K3 )
% 0.34/0.63 = ( plus_plus_rat @ J @ L ) ) ) ).
% 0.34/0.63
% 0.34/0.63 % add_mono_thms_linordered_semiring(4)
% 0.34/0.63 thf(fact_5060_add__mono__thms__linordered__semiring_I4_J,axiom,
% 0.34/0.63 ! [I2: nat,J: nat,K3: nat,L: nat] :
% 0.34/0.63 ( ( ( I2 = J )
% 0.34/0.63 & ( K3 = L ) )
% 0.34/0.63 => ( ( plus_plus_nat @ I2 @ K3 )
% 0.34/0.63 = ( plus_plus_nat @ J @ L ) ) ) ).
% 0.34/0.63
% 0.34/0.63 % add_mono_thms_linordered_semiring(4)
% 0.34/0.63 thf(fact_5061_add__mono__thms__linordered__semiring_I4_J,axiom,
% 0.34/0.63 ! [I2: int,J: int,K3: int,L: int] :
% 0.34/0.63 ( ( ( I2 = J )
% 0.34/0.63 & ( K3 = L ) )
% 0.34/0.63 => ( ( plus_plus_int @ I2 @ K3 )
% 0.34/0.63 = ( plus_plus_int @ J @ L ) ) ) ).
% 0.34/0.63
% 0.34/0.63 % add_mono_thms_linordered_semiring(4)
% 0.34/0.63 thf(fact_5062_group__cancel_Oadd1,axiom,
% 0.34/0.63 ! [A: code_integer,K3: code_integer,A2: code_integer,B: code_integer] :
% 0.34/0.63 ( ( A
% 0.34/0.63 = ( plus_p5714425477246183910nteger @ K3 @ A2 ) )
% 0.34/0.63 => ( ( plus_p5714425477246183910nteger @ A @ B )
% 0.34/0.63 = ( plus_p5714425477246183910nteger @ K3 @ ( plus_p5714425477246183910nteger @ A2 @ B ) ) ) ) ).
% 0.34/0.63
% 0.34/0.63 % group_cancel.add1
% 0.34/0.63 thf(fact_5063_group__cancel_Oadd1,axiom,
% 0.34/0.63 ! [A: rat,K3: rat,A2: rat,B: rat] :
% 0.34/0.63 ( ( A
% 0.34/0.63 = ( plus_plus_rat @ K3 @ A2 ) )
% 0.34/0.63 => ( ( plus_plus_rat @ A @ B )
% 0.34/0.63 = ( plus_plus_rat @ K3 @ ( plus_plus_rat @ A2 @ B ) ) ) ) ).
% 0.34/0.63
% 0.34/0.63 % group_cancel.add1
% 0.34/0.63 thf(fact_5064_group__cancel_Oadd1,axiom,
% 0.34/0.63 ! [A: nat,K3: nat,A2: nat,B: nat] :
% 0.34/0.63 ( ( A
% 0.34/0.63 = ( plus_plus_nat @ K3 @ A2 ) )
% 0.34/0.63 => ( ( plus_plus_nat @ A @ B )
% 0.34/0.63 = ( plus_plus_nat @ K3 @ ( plus_plus_nat @ A2 @ B ) ) ) ) ).
% 0.34/0.63
% 0.34/0.63 % group_cancel.add1
% 0.34/0.63 thf(fact_5065_group__cancel_Oadd1,axiom,
% 0.34/0.63 ! [A: int,K3: int,A2: int,B: int] :
% 0.34/0.63 ( ( A
% 0.34/0.63 = ( plus_plus_int @ K3 @ A2 ) )
% 0.34/0.63 => ( ( plus_plus_int @ A @ B )
% 0.34/0.63 = ( plus_plus_int @ K3 @ ( plus_plus_int @ A2 @ B ) ) ) ) ).
% 0.34/0.63
% 0.34/0.63 % group_cancel.add1
% 0.34/0.63 thf(fact_5066_group__cancel_Oadd2,axiom,
% 0.34/0.63 ! [B4: code_integer,K3: code_integer,B: code_integer,A2: code_integer] :
% 0.34/0.63 ( ( B4
% 0.34/0.63 = ( plus_p5714425477246183910nteger @ K3 @ B ) )
% 0.34/0.63 => ( ( plus_p5714425477246183910nteger @ A2 @ B4 )
% 0.34/0.63 = ( plus_p5714425477246183910nteger @ K3 @ ( plus_p5714425477246183910nteger @ A2 @ B ) ) ) ) ).
% 0.34/0.63
% 0.34/0.63 % group_cancel.add2
% 0.34/0.63 thf(fact_5067_group__cancel_Oadd2,axiom,
% 0.34/0.63 ! [B4: rat,K3: rat,B: rat,A2: rat] :
% 0.34/0.63 ( ( B4
% 0.34/0.63 = ( plus_plus_rat @ K3 @ B ) )
% 0.34/0.63 => ( ( plus_plus_rat @ A2 @ B4 )
% 0.34/0.63 = ( plus_plus_rat @ K3 @ ( plus_plus_rat @ A2 @ B ) ) ) ) ).
% 0.34/0.63
% 0.34/0.63 % group_cancel.add2
% 0.34/0.63 thf(fact_5068_group__cancel_Oadd2,axiom,
% 0.34/0.63 ! [B4: nat,K3: nat,B: nat,A2: nat] :
% 0.34/0.63 ( ( B4
% 0.34/0.63 = ( plus_plus_nat @ K3 @ B ) )
% 0.34/0.63 => ( ( plus_plus_nat @ A2 @ B4 )
% 0.34/0.63 = ( plus_plus_nat @ K3 @ ( plus_plus_nat @ A2 @ B ) ) ) ) ).
% 0.34/0.63
% 0.34/0.63 % group_cancel.add2
% 0.34/0.63 thf(fact_5069_group__cancel_Oadd2,axiom,
% 0.34/0.63 ! [B4: int,K3: int,B: int,A2: int] :
% 0.34/0.63 ( ( B4
% 0.34/0.63 = ( plus_plus_int @ K3 @ B ) )
% 0.34/0.63 => ( ( plus_plus_int @ A2 @ B4 )
% 0.34/0.63 = ( plus_plus_int @ K3 @ ( plus_plus_int @ A2 @ B ) ) ) ) ).
% 0.34/0.63
% 0.34/0.63 % group_cancel.add2
% 0.34/0.63 thf(fact_5070_add_Oassoc,axiom,
% 0.34/0.63 ! [A2: code_integer,B: code_integer,C: code_integer] :
% 0.34/0.63 ( ( plus_p5714425477246183910nteger @ ( plus_p5714425477246183910nteger @ A2 @ B ) @ C )
% 0.34/0.63 = ( plus_p5714425477246183910nteger @ A2 @ ( plus_p5714425477246183910nteger @ B @ C ) ) ) ).
% 0.34/0.63
% 0.34/0.63 % add.assoc
% 0.34/0.63 thf(fact_5071_add_Oassoc,axiom,
% 0.34/0.63 ! [A2: rat,B: rat,C: rat] :
% 0.34/0.63 ( ( plus_plus_rat @ ( plus_plus_rat @ A2 @ B ) @ C )
% 0.34/0.63 = ( plus_plus_rat @ A2 @ ( plus_plus_rat @ B @ C ) ) ) ).
% 0.34/0.63
% 0.34/0.63 % add.assoc
% 0.34/0.63 thf(fact_5072_add_Oassoc,axiom,
% 0.34/0.63 ! [A2: nat,B: nat,C: nat] :
% 0.34/0.63 ( ( plus_plus_nat @ ( plus_plus_nat @ A2 @ B ) @ C )
% 0.34/0.63 = ( plus_plus_nat @ A2 @ ( plus_plus_nat @ B @ C ) ) ) ).
% 0.34/0.63
% 0.34/0.63 % add.assoc
% 0.34/0.63 thf(fact_5073_add_Oassoc,axiom,
% 0.34/0.63 ! [A2: int,B: int,C: int] :
% 0.34/0.63 ( ( plus_plus_int @ ( plus_plus_int @ A2 @ B ) @ C )
% 0.34/0.63 = ( plus_plus_int @ A2 @ ( plus_plus_int @ B @ C ) ) ) ).
% 0.34/0.63
% 0.34/0.63 % add.assoc
% 0.34/0.63 thf(fact_5074_add_Oleft__cancel,axiom,
% 0.34/0.63 ! [A2: code_integer,B: code_integer,C: code_integer] :
% 0.34/0.63 ( ( ( plus_p5714425477246183910nteger @ A2 @ B )
% 0.34/0.63 = ( plus_p5714425477246183910nteger @ A2 @ C ) )
% 0.34/0.63 = ( B = C ) ) ).
% 0.34/0.63
% 0.34/0.63 % add.left_cancel
% 0.34/0.63 thf(fact_5075_add_Oleft__cancel,axiom,
% 0.34/0.63 ! [A2: rat,B: rat,C: rat] :
% 0.34/0.63 ( ( ( plus_plus_rat @ A2 @ B )
% 0.34/0.63 = ( plus_plus_rat @ A2 @ C ) )
% 0.34/0.63 = ( B = C ) ) ).
% 0.34/0.63
% 0.34/0.63 % add.left_cancel
% 0.34/0.63 thf(fact_5076_add_Oleft__cancel,axiom,
% 0.34/0.63 ! [A2: int,B: int,C: int] :
% 0.34/0.63 ( ( ( plus_plus_int @ A2 @ B )
% 0.34/0.63 = ( plus_plus_int @ A2 @ C ) )
% 0.34/0.63 = ( B = C ) ) ).
% 0.34/0.63
% 0.34/0.63 % add.left_cancel
% 0.34/0.63 thf(fact_5077_add_Oright__cancel,axiom,
% 0.34/0.63 ! [B: code_integer,A2: code_integer,C: code_integer] :
% 0.34/0.63 ( ( ( plus_p5714425477246183910nteger @ B @ A2 )
% 0.34/0.63 = ( plus_p5714425477246183910nteger @ C @ A2 ) )
% 0.34/0.63 = ( B = C ) ) ).
% 0.34/0.63
% 0.34/0.63 % add.right_cancel
% 0.34/0.63 thf(fact_5078_add_Oright__cancel,axiom,
% 0.34/0.63 ! [B: rat,A2: rat,C: rat] :
% 0.34/0.63 ( ( ( plus_plus_rat @ B @ A2 )
% 0.34/0.63 = ( plus_plus_rat @ C @ A2 ) )
% 0.34/0.63 = ( B = C ) ) ).
% 0.34/0.63
% 0.34/0.63 % add.right_cancel
% 0.34/0.63 thf(fact_5079_add_Oright__cancel,axiom,
% 0.34/0.63 ! [B: int,A2: int,C: int] :
% 0.34/0.63 ( ( ( plus_plus_int @ B @ A2 )
% 0.34/0.63 = ( plus_plus_int @ C @ A2 ) )
% 0.34/0.63 = ( B = C ) ) ).
% 0.34/0.63
% 0.34/0.63 % add.right_cancel
% 0.34/0.63 thf(fact_5080_ab__semigroup__add__class_Oadd_Ocommute,axiom,
% 0.34/0.63 ( plus_p5714425477246183910nteger
% 0.34/0.63 = ( ^ [A6: code_integer,B6: code_integer] : ( plus_p5714425477246183910nteger @ B6 @ A6 ) ) ) ).
% 0.34/0.63
% 0.34/0.63 % ab_semigroup_add_class.add.commute
% 0.34/0.63 thf(fact_5081_ab__semigroup__add__class_Oadd_Ocommute,axiom,
% 0.34/0.63 ( plus_plus_rat
% 0.34/0.63 = ( ^ [A6: rat,B6: rat] : ( plus_plus_rat @ B6 @ A6 ) ) ) ).
% 0.34/0.63
% 0.34/0.63 % ab_semigroup_add_class.add.commute
% 0.34/0.63 thf(fact_5082_ab__semigroup__add__class_Oadd_Ocommute,axiom,
% 0.34/0.63 ( plus_plus_nat
% 0.34/0.63 = ( ^ [A6: nat,B6: nat] : ( plus_plus_nat @ B6 @ A6 ) ) ) ).
% 0.34/0.63
% 0.34/0.63 % ab_semigroup_add_class.add.commute
% 0.34/0.63 thf(fact_5083_ab__semigroup__add__class_Oadd_Ocommute,axiom,
% 0.34/0.63 ( plus_plus_int
% 0.34/0.63 = ( ^ [A6: int,B6: int] : ( plus_plus_int @ B6 @ A6 ) ) ) ).
% 0.34/0.63
% 0.34/0.63 % ab_semigroup_add_class.add.commute
% 0.34/0.63 thf(fact_5084_ab__semigroup__add__class_Oadd_Oleft__commute,axiom,
% 0.34/0.63 ! [B: code_integer,A2: code_integer,C: code_integer] :
% 0.34/0.63 ( ( plus_p5714425477246183910nteger @ B @ ( plus_p5714425477246183910nteger @ A2 @ C ) )
% 0.34/0.63 = ( plus_p5714425477246183910nteger @ A2 @ ( plus_p5714425477246183910nteger @ B @ C ) ) ) ).
% 0.34/0.63
% 0.34/0.63 % ab_semigroup_add_class.add.left_commute
% 0.34/0.63 thf(fact_5085_ab__semigroup__add__class_Oadd_Oleft__commute,axiom,
% 0.34/0.63 ! [B: rat,A2: rat,C: rat] :
% 0.34/0.63 ( ( plus_plus_rat @ B @ ( plus_plus_rat @ A2 @ C ) )
% 0.34/0.63 = ( plus_plus_rat @ A2 @ ( plus_plus_rat @ B @ C ) ) ) ).
% 0.34/0.63
% 0.34/0.63 % ab_semigroup_add_class.add.left_commute
% 0.34/0.63 thf(fact_5086_ab__semigroup__add__class_Oadd_Oleft__commute,axiom,
% 0.34/0.63 ! [B: nat,A2: nat,C: nat] :
% 0.34/0.63 ( ( plus_plus_nat @ B @ ( plus_plus_nat @ A2 @ C ) )
% 0.34/0.63 = ( plus_plus_nat @ A2 @ ( plus_plus_nat @ B @ C ) ) ) ).
% 0.34/0.63
% 0.34/0.63 % ab_semigroup_add_class.add.left_commute
% 0.34/0.63 thf(fact_5087_ab__semigroup__add__class_Oadd_Oleft__commute,axiom,
% 0.34/0.63 ! [B: int,A2: int,C: int] :
% 0.34/0.63 ( ( plus_plus_int @ B @ ( plus_plus_int @ A2 @ C ) )
% 0.34/0.63 = ( plus_plus_int @ A2 @ ( plus_plus_int @ B @ C ) ) ) ).
% 0.34/0.63
% 0.34/0.63 % ab_semigroup_add_class.add.left_commute
% 0.34/0.63 thf(fact_5088_add__left__imp__eq,axiom,
% 0.34/0.63 ! [A2: code_integer,B: code_integer,C: code_integer] :
% 0.34/0.63 ( ( ( plus_p5714425477246183910nteger @ A2 @ B )
% 0.34/0.63 = ( plus_p5714425477246183910nteger @ A2 @ C ) )
% 0.34/0.63 => ( B = C ) ) ).
% 0.34/0.63
% 0.34/0.63 % add_left_imp_eq
% 0.34/0.63 thf(fact_5089_add__left__imp__eq,axiom,
% 0.34/0.63 ! [A2: rat,B: rat,C: rat] :
% 0.34/0.63 ( ( ( plus_plus_rat @ A2 @ B )
% 0.34/0.63 = ( plus_plus_rat @ A2 @ C ) )
% 0.34/0.63 => ( B = C ) ) ).
% 0.34/0.63
% 0.34/0.63 % add_left_imp_eq
% 0.34/0.63 thf(fact_5090_add__left__imp__eq,axiom,
% 0.34/0.63 ! [A2: nat,B: nat,C: nat] :
% 0.34/0.63 ( ( ( plus_plus_nat @ A2 @ B )
% 0.34/0.63 = ( plus_plus_nat @ A2 @ C ) )
% 0.34/0.63 => ( B = C ) ) ).
% 0.34/0.63
% 0.34/0.63 % add_left_imp_eq
% 0.34/0.63 thf(fact_5091_add__left__imp__eq,axiom,
% 0.34/0.63 ! [A2: int,B: int,C: int] :
% 0.34/0.63 ( ( ( plus_plus_int @ A2 @ B )
% 0.34/0.63 = ( plus_plus_int @ A2 @ C ) )
% 0.34/0.63 => ( B = C ) ) ).
% 0.34/0.63
% 0.34/0.63 % add_left_imp_eq
% 0.34/0.63 thf(fact_5092_add__right__imp__eq,axiom,
% 0.34/0.63 ! [B: code_integer,A2: code_integer,C: code_integer] :
% 0.34/0.63 ( ( ( plus_p5714425477246183910nteger @ B @ A2 )
% 0.34/0.63 = ( plus_p5714425477246183910nteger @ C @ A2 ) )
% 0.34/0.63 => ( B = C ) ) ).
% 0.34/0.63
% 0.34/0.63 % add_right_imp_eq
% 0.34/0.63 thf(fact_5093_add__right__imp__eq,axiom,
% 0.34/0.63 ! [B: rat,A2: rat,C: rat] :
% 0.34/0.63 ( ( ( plus_plus_rat @ B @ A2 )
% 0.34/0.63 = ( plus_plus_rat @ C @ A2 ) )
% 0.34/0.63 => ( B = C ) ) ).
% 0.34/0.63
% 0.34/0.63 % add_right_imp_eq
% 0.34/0.63 thf(fact_5094_add__right__imp__eq,axiom,
% 0.34/0.63 ! [B: nat,A2: nat,C: nat] :
% 0.34/0.63 ( ( ( plus_plus_nat @ B @ A2 )
% 0.34/0.63 = ( plus_plus_nat @ C @ A2 ) )
% 0.34/0.63 => ( B = C ) ) ).
% 0.34/0.63
% 0.34/0.63 % add_right_imp_eq
% 0.34/0.63 thf(fact_5095_add__right__imp__eq,axiom,
% 0.34/0.63 ! [B: int,A2: int,C: int] :
% 0.34/0.63 ( ( ( plus_plus_int @ B @ A2 )
% 0.34/0.63 = ( plus_plus_int @ C @ A2 ) )
% 0.34/0.63 => ( B = C ) ) ).
% 0.34/0.63
% 0.34/0.63 % add_right_imp_eq
% 0.34/0.63 thf(fact_5096_add__mono__thms__linordered__semiring_I3_J,axiom,
% 0.34/0.63 ! [I2: code_integer,J: code_integer,K3: code_integer,L: code_integer] :
% 0.34/0.63 ( ( ( ord_le3102999989581377725nteger @ I2 @ J )
% 0.34/0.63 & ( K3 = L ) )
% 0.34/0.63 => ( ord_le3102999989581377725nteger @ ( plus_p5714425477246183910nteger @ I2 @ K3 ) @ ( plus_p5714425477246183910nteger @ J @ L ) ) ) ).
% 0.34/0.63
% 0.34/0.63 % add_mono_thms_linordered_semiring(3)
% 0.34/0.63 thf(fact_5097_add__mono__thms__linordered__semiring_I3_J,axiom,
% 0.34/0.63 ! [I2: rat,J: rat,K3: rat,L: rat] :
% 0.34/0.63 ( ( ( ord_less_eq_rat @ I2 @ J )
% 0.34/0.63 & ( K3 = L ) )
% 0.34/0.63 => ( ord_less_eq_rat @ ( plus_plus_rat @ I2 @ K3 ) @ ( plus_plus_rat @ J @ L ) ) ) ).
% 0.34/0.63
% 0.34/0.63 % add_mono_thms_linordered_semiring(3)
% 0.34/0.63 thf(fact_5098_add__mono__thms__linordered__semiring_I3_J,axiom,
% 0.34/0.63 ! [I2: nat,J: nat,K3: nat,L: nat] :
% 0.34/0.63 ( ( ( ord_less_eq_nat @ I2 @ J )
% 0.34/0.63 & ( K3 = L ) )
% 0.34/0.63 => ( ord_less_eq_nat @ ( plus_plus_nat @ I2 @ K3 ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% 0.34/0.63
% 0.34/0.63 % add_mono_thms_linordered_semiring(3)
% 0.34/0.63 thf(fact_5099_add__mono__thms__linordered__semiring_I3_J,axiom,
% 0.34/0.63 ! [I2: int,J: int,K3: int,L: int] :
% 0.34/0.63 ( ( ( ord_less_eq_int @ I2 @ J )
% 0.34/0.63 & ( K3 = L ) )
% 0.34/0.63 => ( ord_less_eq_int @ ( plus_plus_int @ I2 @ K3 ) @ ( plus_plus_int @ J @ L ) ) ) ).
% 0.34/0.63
% 0.34/0.63 % add_mono_thms_linordered_semiring(3)
% 0.34/0.63 thf(fact_5100_add__mono__thms__linordered__semiring_I2_J,axiom,
% 0.34/0.63 ! [I2: code_integer,J: code_integer,K3: code_integer,L: code_integer] :
% 0.34/0.63 ( ( ( I2 = J )
% 0.34/0.63 & ( ord_le3102999989581377725nteger @ K3 @ L ) )
% 0.34/0.63 => ( ord_le3102999989581377725nteger @ ( plus_p5714425477246183910nteger @ I2 @ K3 ) @ ( plus_p5714425477246183910nteger @ J @ L ) ) ) ).
% 0.34/0.63
% 0.34/0.63 % add_mono_thms_linordered_semiring(2)
% 0.34/0.63 thf(fact_5101_add__mono__thms__linordered__semiring_I2_J,axiom,
% 0.34/0.63 ! [I2: rat,J: rat,K3: rat,L: rat] :
% 0.34/0.63 ( ( ( I2 = J )
% 0.34/0.63 & ( ord_less_eq_rat @ K3 @ L ) )
% 0.34/0.63 => ( ord_less_eq_rat @ ( plus_plus_rat @ I2 @ K3 ) @ ( plus_plus_rat @ J @ L ) ) ) ).
% 0.34/0.63
% 0.34/0.63 % add_mono_thms_linordered_semiring(2)
% 0.34/0.63 thf(fact_5102_add__mono__thms__linordered__semiring_I2_J,axiom,
% 0.34/0.63 ! [I2: nat,J: nat,K3: nat,L: nat] :
% 0.34/0.63 ( ( ( I2 = J )
% 0.34/0.63 & ( ord_less_eq_nat @ K3 @ L ) )
% 0.34/0.63 => ( ord_less_eq_nat @ ( plus_plus_nat @ I2 @ K3 ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% 0.34/0.63
% 0.34/0.63 % add_mono_thms_linordered_semiring(2)
% 0.34/0.63 thf(fact_5103_add__mono__thms__linordered__semiring_I2_J,axiom,
% 0.34/0.63 ! [I2: int,J: int,K3: int,L: int] :
% 0.34/0.63 ( ( ( I2 = J )
% 0.34/0.63 & ( ord_less_eq_int @ K3 @ L ) )
% 0.34/0.63 => ( ord_less_eq_int @ ( plus_plus_int @ I2 @ K3 ) @ ( plus_plus_int @ J @ L ) ) ) ).
% 0.34/0.63
% 0.34/0.63 % add_mono_thms_linordered_semiring(2)
% 0.34/0.63 thf(fact_5104_add__mono__thms__linordered__semiring_I1_J,axiom,
% 0.34/0.63 ! [I2: code_integer,J: code_integer,K3: code_integer,L: code_integer] :
% 0.34/0.63 ( ( ( ord_le3102999989581377725nteger @ I2 @ J )
% 0.34/0.63 & ( ord_le3102999989581377725nteger @ K3 @ L ) )
% 0.34/0.63 => ( ord_le3102999989581377725nteger @ ( plus_p5714425477246183910nteger @ I2 @ K3 ) @ ( plus_p5714425477246183910nteger @ J @ L ) ) ) ).
% 0.34/0.63
% 0.34/0.63 % add_mono_thms_linordered_semiring(1)
% 0.34/0.63 thf(fact_5105_add__mono__thms__linordered__semiring_I1_J,axiom,
% 0.34/0.63 ! [I2: rat,J: rat,K3: rat,L: rat] :
% 0.34/0.63 ( ( ( ord_less_eq_rat @ I2 @ J )
% 0.34/0.63 & ( ord_less_eq_rat @ K3 @ L ) )
% 0.34/0.63 => ( ord_less_eq_rat @ ( plus_plus_rat @ I2 @ K3 ) @ ( plus_plus_rat @ J @ L ) ) ) ).
% 0.34/0.63
% 0.34/0.63 % add_mono_thms_linordered_semiring(1)
% 0.34/0.63 thf(fact_5106_add__mono__thms__linordered__semiring_I1_J,axiom,
% 0.34/0.63 ! [I2: nat,J: nat,K3: nat,L: nat] :
% 0.34/0.63 ( ( ( ord_less_eq_nat @ I2 @ J )
% 0.34/0.63 & ( ord_less_eq_nat @ K3 @ L ) )
% 0.34/0.63 => ( ord_less_eq_nat @ ( plus_plus_nat @ I2 @ K3 ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% 0.34/0.63
% 0.34/0.63 % add_mono_thms_linordered_semiring(1)
% 0.34/0.63 thf(fact_5107_add__mono__thms__linordered__semiring_I1_J,axiom,
% 0.34/0.63 ! [I2: int,J: int,K3: int,L: int] :
% 0.34/0.63 ( ( ( ord_less_eq_int @ I2 @ J )
% 0.34/0.63 & ( ord_less_eq_int @ K3 @ L ) )
% 0.34/0.63 => ( ord_less_eq_int @ ( plus_plus_int @ I2 @ K3 ) @ ( plus_plus_int @ J @ L ) ) ) ).
% 0.34/0.63
% 0.34/0.63 % add_mono_thms_linordered_semiring(1)
% 0.34/0.63 thf(fact_5108_add__mono,axiom,
% 0.34/0.63 ! [A2: code_integer,B: code_integer,C: code_integer,D2: code_integer] :
% 0.34/0.63 ( ( ord_le3102999989581377725nteger @ A2 @ B )
% 0.34/0.63 => ( ( ord_le3102999989581377725nteger @ C @ D2 )
% 0.34/0.63 => ( ord_le3102999989581377725nteger @ ( plus_p5714425477246183910nteger @ A2 @ C ) @ ( plus_p5714425477246183910nteger @ B @ D2 ) ) ) ) ).
% 0.34/0.63
% 0.34/0.63 % add_mono
% 0.34/0.63 thf(fact_5109_add__mono,axiom,
% 0.34/0.63 ! [A2: rat,B: rat,C: rat,D2: rat] :
% 0.34/0.63 ( ( ord_less_eq_rat @ A2 @ B )
% 0.34/0.63 => ( ( ord_less_eq_rat @ C @ D2 )
% 0.34/0.63 => ( ord_less_eq_rat @ ( plus_plus_rat @ A2 @ C ) @ ( plus_plus_rat @ B @ D2 ) ) ) ) ).
% 0.34/0.63
% 0.34/0.63 % add_mono
% 0.34/0.63 thf(fact_5110_add__mono,axiom,
% 0.34/0.63 ! [A2: nat,B: nat,C: nat,D2: nat] :
% 0.34/0.63 ( ( ord_less_eq_nat @ A2 @ B )
% 0.34/0.63 => ( ( ord_less_eq_nat @ C @ D2 )
% 0.34/0.63 => ( ord_less_eq_nat @ ( plus_plus_nat @ A2 @ C ) @ ( plus_plus_nat @ B @ D2 ) ) ) ) ).
% 0.34/0.63
% 0.34/0.63 % add_mono
% 0.34/0.63 thf(fact_5111_add__mono,axiom,
% 0.34/0.63 ! [A2: int,B: int,C: int,D2: int] :
% 0.34/0.63 ( ( ord_less_eq_int @ A2 @ B )
% 0.34/0.63 => ( ( ord_less_eq_int @ C @ D2 )
% 0.34/0.63 => ( ord_less_eq_int @ ( plus_plus_int @ A2 @ C ) @ ( plus_plus_int @ B @ D2 ) ) ) ) ).
% 0.34/0.63
% 0.34/0.63 % add_mono
% 0.34/0.63 thf(fact_5112_add__left__mono,axiom,
% 0.34/0.63 ! [A2: code_integer,B: code_integer,C: code_integer] :
% 0.34/0.63 ( ( ord_le3102999989581377725nteger @ A2 @ B )
% 0.34/0.63 => ( ord_le3102999989581377725nteger @ ( plus_p5714425477246183910nteger @ C @ A2 ) @ ( plus_p5714425477246183910nteger @ C @ B ) ) ) ).
% 0.34/0.63
% 0.34/0.63 % add_left_mono
% 0.34/0.63 thf(fact_5113_add__left__mono,axiom,
% 0.34/0.63 ! [A2: rat,B: rat,C: rat] :
% 0.34/0.63 ( ( ord_less_eq_rat @ A2 @ B )
% 0.34/0.63 => ( ord_less_eq_rat @ ( plus_plus_rat @ C @ A2 ) @ ( plus_plus_rat @ C @ B ) ) ) ).
% 0.34/0.63
% 0.34/0.63 % add_left_mono
% 0.34/0.63 thf(fact_5114_add__left__mono,axiom,
% 0.34/0.63 ! [A2: nat,B: nat,C: nat] :
% 0.34/0.63 ( ( ord_less_eq_nat @ A2 @ B )
% 0.34/0.63 => ( ord_less_eq_nat @ ( plus_plus_nat @ C @ A2 ) @ ( plus_plus_nat @ C @ B ) ) ) ).
% 0.34/0.63
% 0.34/0.63 % add_left_mono
% 0.34/0.63 thf(fact_5115_add__left__mono,axiom,
% 0.34/0.63 ! [A2: int,B: int,C: int] :
% 0.34/0.63 ( ( ord_less_eq_int @ A2 @ B )
% 0.34/0.63 => ( ord_less_eq_int @ ( plus_plus_int @ C @ A2 ) @ ( plus_plus_int @ C @ B ) ) ) ).
% 0.34/0.63
% 0.34/0.63 % add_left_mono
% 0.34/0.63 thf(fact_5116_less__eqE,axiom,
% 0.34/0.63 ! [A2: nat,B: nat] :
% 0.34/0.63 ( ( ord_less_eq_nat @ A2 @ B )
% 0.34/0.63 => ~ ! [C2: nat] :
% 0.34/0.63 ( B
% 0.34/0.63 != ( plus_plus_nat @ A2 @ C2 ) ) ) ).
% 0.34/0.63
% 0.34/0.63 % less_eqE
% 0.34/0.63 thf(fact_5117_add__right__mono,axiom,
% 0.34/0.63 ! [A2: code_integer,B: code_integer,C: code_integer] :
% 0.34/0.63 ( ( ord_le3102999989581377725nteger @ A2 @ B )
% 0.34/0.63 => ( ord_le3102999989581377725nteger @ ( plus_p5714425477246183910nteger @ A2 @ C ) @ ( plus_p5714425477246183910nteger @ B @ C ) ) ) ).
% 0.34/0.63
% 0.34/0.63 % add_right_mono
% 0.34/0.63 thf(fact_5118_add__right__mono,axiom,
% 0.34/0.63 ! [A2: rat,B: rat,C: rat] :
% 0.34/0.63 ( ( ord_less_eq_rat @ A2 @ B )
% 0.34/0.63 => ( ord_less_eq_rat @ ( plus_plus_rat @ A2 @ C ) @ ( plus_plus_rat @ B @ C ) ) ) ).
% 0.34/0.63
% 0.34/0.63 % add_right_mono
% 0.34/0.63 thf(fact_5119_add__right__mono,axiom,
% 0.34/0.63 ! [A2: nat,B: nat,C: nat] :
% 0.34/0.63 ( ( ord_less_eq_nat @ A2 @ B )
% 0.34/0.63 => ( ord_less_eq_nat @ ( plus_plus_nat @ A2 @ C ) @ ( plus_plus_nat @ B @ C ) ) ) ).
% 0.34/0.63
% 0.34/0.63 % add_right_mono
% 0.34/0.63 thf(fact_5120_add__right__mono,axiom,
% 0.34/0.63 ! [A2: int,B: int,C: int] :
% 0.34/0.63 ( ( ord_less_eq_int @ A2 @ B )
% 0.34/0.63 => ( ord_less_eq_int @ ( plus_plus_int @ A2 @ C ) @ ( plus_plus_int @ B @ C ) ) ) ).
% 0.34/0.63
% 0.34/0.63 % add_right_mono
% 0.34/0.63 thf(fact_5121_le__iff__add,axiom,
% 0.34/0.63 ( ord_less_eq_nat
% 0.34/0.63 = ( ^ [A6: nat,B6: nat] :
% 0.34/0.63 ? [C4: nat] :
% 0.34/0.63 ( B6
% 0.34/0.63 = ( plus_plus_nat @ A6 @ C4 ) ) ) ) ).
% 0.34/0.63
% 0.34/0.63 % le_iff_add
% 0.34/0.63 thf(fact_5122_add__le__imp__le__left,axiom,
% 0.34/0.63 ! [C: code_integer,A2: code_integer,B: code_integer] :
% 0.34/0.63 ( ( ord_le3102999989581377725nteger @ ( plus_p5714425477246183910nteger @ C @ A2 ) @ ( plus_p5714425477246183910nteger @ C @ B ) )
% 0.34/0.63 => ( ord_le3102999989581377725nteger @ A2 @ B ) ) ).
% 0.34/0.63
% 0.34/0.63 % add_le_imp_le_left
% 0.34/0.63 thf(fact_5123_add__le__imp__le__left,axiom,
% 0.34/0.63 ! [C: rat,A2: rat,B: rat] :
% 0.34/0.63 ( ( ord_less_eq_rat @ ( plus_plus_rat @ C @ A2 ) @ ( plus_plus_rat @ C @ B ) )
% 0.34/0.63 => ( ord_less_eq_rat @ A2 @ B ) ) ).
% 0.34/0.63
% 0.34/0.63 % add_le_imp_le_left
% 0.34/0.63 thf(fact_5124_add__le__imp__le__left,axiom,
% 0.34/0.63 ! [C: nat,A2: nat,B: nat] :
% 0.34/0.63 ( ( ord_less_eq_nat @ ( plus_plus_nat @ C @ A2 ) @ ( plus_plus_nat @ C @ B ) )
% 0.34/0.63 => ( ord_less_eq_nat @ A2 @ B ) ) ).
% 0.34/0.63
% 0.34/0.63 % add_le_imp_le_left
% 0.34/0.63 thf(fact_5125_add__le__imp__le__left,axiom,
% 0.34/0.63 ! [C: int,A2: int,B: int] :
% 0.34/0.63 ( ( ord_less_eq_int @ ( plus_plus_int @ C @ A2 ) @ ( plus_plus_int @ C @ B ) )
% 0.34/0.63 => ( ord_less_eq_int @ A2 @ B ) ) ).
% 0.34/0.63
% 0.34/0.63 % add_le_imp_le_left
% 0.34/0.63 thf(fact_5126_add__le__imp__le__right,axiom,
% 0.34/0.63 ! [A2: code_integer,C: code_integer,B: code_integer] :
% 0.34/0.63 ( ( ord_le3102999989581377725nteger @ ( plus_p5714425477246183910nteger @ A2 @ C ) @ ( plus_p5714425477246183910nteger @ B @ C ) )
% 0.34/0.63 => ( ord_le3102999989581377725nteger @ A2 @ B ) ) ).
% 0.34/0.63
% 0.34/0.63 % add_le_imp_le_right
% 0.34/0.63 thf(fact_5127_add__le__imp__le__right,axiom,
% 0.34/0.63 ! [A2: rat,C: rat,B: rat] :
% 0.34/0.63 ( ( ord_less_eq_rat @ ( plus_plus_rat @ A2 @ C ) @ ( plus_plus_rat @ B @ C ) )
% 0.34/0.63 => ( ord_less_eq_rat @ A2 @ B ) ) ).
% 0.34/0.63
% 0.34/0.63 % add_le_imp_le_right
% 0.34/0.63 thf(fact_5128_add__le__imp__le__right,axiom,
% 0.34/0.63 ! [A2: nat,C: nat,B: nat] :
% 0.34/0.63 ( ( ord_less_eq_nat @ ( plus_plus_nat @ A2 @ C ) @ ( plus_plus_nat @ B @ C ) )
% 0.34/0.63 => ( ord_less_eq_nat @ A2 @ B ) ) ).
% 0.34/0.63
% 0.34/0.63 % add_le_imp_le_right
% 0.34/0.63 thf(fact_5129_add__le__imp__le__right,axiom,
% 0.34/0.63 ! [A2: int,C: int,B: int] :
% 0.34/0.63 ( ( ord_less_eq_int @ ( plus_plus_int @ A2 @ C ) @ ( plus_plus_int @ B @ C ) )
% 0.34/0.63 => ( ord_less_eq_int @ A2 @ B ) ) ).
% 0.34/0.63
% 0.34/0.63 % add_le_imp_le_right
% 0.34/0.63 thf(fact_5130_add__mono__thms__linordered__field_I5_J,axiom,
% 0.34/0.63 ! [I2: nat,J: nat,K3: nat,L: nat] :
% 0.34/0.63 ( ( ( ord_less_nat @ I2 @ J )
% 0.34/0.63 & ( ord_less_nat @ K3 @ L ) )
% 0.34/0.63 => ( ord_less_nat @ ( plus_plus_nat @ I2 @ K3 ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% 0.34/0.63
% 0.34/0.63 % add_mono_thms_linordered_field(5)
% 0.34/0.63 thf(fact_5131_add__mono__thms__linordered__field_I5_J,axiom,
% 0.34/0.63 ! [I2: int,J: int,K3: int,L: int] :
% 0.34/0.63 ( ( ( ord_less_int @ I2 @ J )
% 0.34/0.63 & ( ord_less_int @ K3 @ L ) )
% 0.34/0.63 => ( ord_less_int @ ( plus_plus_int @ I2 @ K3 ) @ ( plus_plus_int @ J @ L ) ) ) ).
% 0.34/0.63
% 0.34/0.63 % add_mono_thms_linordered_field(5)
% 0.34/0.63 thf(fact_5132_add__mono__thms__linordered__field_I5_J,axiom,
% 0.34/0.63 ! [I2: code_integer,J: code_integer,K3: code_integer,L: code_integer] :
% 0.34/0.63 ( ( ( ord_le6747313008572928689nteger @ I2 @ J )
% 0.34/0.63 & ( ord_le6747313008572928689nteger @ K3 @ L ) )
% 0.34/0.63 => ( ord_le6747313008572928689nteger @ ( plus_p5714425477246183910nteger @ I2 @ K3 ) @ ( plus_p5714425477246183910nteger @ J @ L ) ) ) ).
% 0.34/0.63
% 0.34/0.63 % add_mono_thms_linordered_field(5)
% 0.34/0.63 thf(fact_5133_add__mono__thms__linordered__field_I5_J,axiom,
% 0.34/0.63 ! [I2: rat,J: rat,K3: rat,L: rat] :
% 0.34/0.63 ( ( ( ord_less_rat @ I2 @ J )
% 0.34/0.63 & ( ord_less_rat @ K3 @ L ) )
% 0.34/0.63 => ( ord_less_rat @ ( plus_plus_rat @ I2 @ K3 ) @ ( plus_plus_rat @ J @ L ) ) ) ).
% 0.34/0.63
% 0.34/0.63 % add_mono_thms_linordered_field(5)
% 0.34/0.63 thf(fact_5134_add__mono__thms__linordered__field_I2_J,axiom,
% 0.34/0.63 ! [I2: nat,J: nat,K3: nat,L: nat] :
% 0.34/0.63 ( ( ( I2 = J )
% 0.34/0.63 & ( ord_less_nat @ K3 @ L ) )
% 0.34/0.63 => ( ord_less_nat @ ( plus_plus_nat @ I2 @ K3 ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% 0.34/0.63
% 0.34/0.63 % add_mono_thms_linordered_field(2)
% 0.34/0.63 thf(fact_5135_add__mono__thms__linordered__field_I2_J,axiom,
% 0.34/0.63 ! [I2: int,J: int,K3: int,L: int] :
% 0.34/0.63 ( ( ( I2 = J )
% 0.34/0.63 & ( ord_less_int @ K3 @ L ) )
% 0.34/0.63 => ( ord_less_int @ ( plus_plus_int @ I2 @ K3 ) @ ( plus_plus_int @ J @ L ) ) ) ).
% 0.34/0.63
% 0.34/0.63 % add_mono_thms_linordered_field(2)
% 0.34/0.63 thf(fact_5136_add__mono__thms__linordered__field_I2_J,axiom,
% 0.34/0.63 ! [I2: code_integer,J: code_integer,K3: code_integer,L: code_integer] :
% 0.34/0.63 ( ( ( I2 = J )
% 0.34/0.63 & ( ord_le6747313008572928689nteger @ K3 @ L ) )
% 0.34/0.63 => ( ord_le6747313008572928689nteger @ ( plus_p5714425477246183910nteger @ I2 @ K3 ) @ ( plus_p5714425477246183910nteger @ J @ L ) ) ) ).
% 0.34/0.63
% 0.34/0.63 % add_mono_thms_linordered_field(2)
% 0.34/0.63 thf(fact_5137_add__mono__thms__linordered__field_I2_J,axiom,
% 0.34/0.63 ! [I2: rat,J: rat,K3: rat,L: rat] :
% 0.34/0.63 ( ( ( I2 = J )
% 0.34/0.63 & ( ord_less_rat @ K3 @ L ) )
% 0.34/0.63 => ( ord_less_rat @ ( plus_plus_rat @ I2 @ K3 ) @ ( plus_plus_rat @ J @ L ) ) ) ).
% 0.34/0.63
% 0.34/0.63 % add_mono_thms_linordered_field(2)
% 0.34/0.63 thf(fact_5138_add__mono__thms__linordered__field_I1_J,axiom,
% 0.34/0.63 ! [I2: nat,J: nat,K3: nat,L: nat] :
% 0.34/0.63 ( ( ( ord_less_nat @ I2 @ J )
% 0.34/0.63 & ( K3 = L ) )
% 0.34/0.63 => ( ord_less_nat @ ( plus_plus_nat @ I2 @ K3 ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% 0.34/0.63
% 0.34/0.63 % add_mono_thms_linordered_field(1)
% 0.34/0.63 thf(fact_5139_add__mono__thms__linordered__field_I1_J,axiom,
% 0.34/0.63 ! [I2: int,J: int,K3: int,L: int] :
% 0.34/0.63 ( ( ( ord_less_int @ I2 @ J )
% 0.34/0.63 & ( K3 = L ) )
% 0.34/0.63 => ( ord_less_int @ ( plus_plus_int @ I2 @ K3 ) @ ( plus_plus_int @ J @ L ) ) ) ).
% 0.34/0.63
% 0.34/0.63 % add_mono_thms_linordered_field(1)
% 0.34/0.63 thf(fact_5140_add__mono__thms__linordered__field_I1_J,axiom,
% 0.34/0.63 ! [I2: code_integer,J: code_integer,K3: code_integer,L: code_integer] :
% 0.34/0.63 ( ( ( ord_le6747313008572928689nteger @ I2 @ J )
% 0.34/0.63 & ( K3 = L ) )
% 0.34/0.63 => ( ord_le6747313008572928689nteger @ ( plus_p5714425477246183910nteger @ I2 @ K3 ) @ ( plus_p5714425477246183910nteger @ J @ L ) ) ) ).
% 0.34/0.63
% 0.34/0.63 % add_mono_thms_linordered_field(1)
% 0.34/0.63 thf(fact_5141_add__mono__thms__linordered__field_I1_J,axiom,
% 0.34/0.63 ! [I2: rat,J: rat,K3: rat,L: rat] :
% 0.34/0.63 ( ( ( ord_less_rat @ I2 @ J )
% 0.34/0.63 & ( K3 = L ) )
% 0.34/0.63 => ( ord_less_rat @ ( plus_plus_rat @ I2 @ K3 ) @ ( plus_plus_rat @ J @ L ) ) ) ).
% 0.34/0.63
% 0.34/0.63 % add_mono_thms_linordered_field(1)
% 0.34/0.63 thf(fact_5142_add__strict__mono,axiom,
% 0.34/0.63 ! [A2: nat,B: nat,C: nat,D2: nat] :
% 0.34/0.63 ( ( ord_less_nat @ A2 @ B )
% 0.34/0.63 => ( ( ord_less_nat @ C @ D2 )
% 0.34/0.63 => ( ord_less_nat @ ( plus_plus_nat @ A2 @ C ) @ ( plus_plus_nat @ B @ D2 ) ) ) ) ).
% 0.34/0.63
% 0.34/0.63 % add_strict_mono
% 0.34/0.63 thf(fact_5143_add__strict__mono,axiom,
% 0.34/0.63 ! [A2: int,B: int,C: int,D2: int] :
% 0.34/0.63 ( ( ord_less_int @ A2 @ B )
% 0.34/0.63 => ( ( ord_less_int @ C @ D2 )
% 0.34/0.63 => ( ord_less_int @ ( plus_plus_int @ A2 @ C ) @ ( plus_plus_int @ B @ D2 ) ) ) ) ).
% 0.34/0.63
% 0.34/0.63 % add_strict_mono
% 0.34/0.63 thf(fact_5144_add__strict__mono,axiom,
% 0.34/0.63 ! [A2: code_integer,B: code_integer,C: code_integer,D2: code_integer] :
% 0.34/0.63 ( ( ord_le6747313008572928689nteger @ A2 @ B )
% 0.34/0.63 => ( ( ord_le6747313008572928689nteger @ C @ D2 )
% 0.34/0.63 => ( ord_le6747313008572928689nteger @ ( plus_p5714425477246183910nteger @ A2 @ C ) @ ( plus_p5714425477246183910nteger @ B @ D2 ) ) ) ) ).
% 0.34/0.63
% 0.34/0.63 % add_strict_mono
% 0.34/0.63 thf(fact_5145_add__strict__mono,axiom,
% 0.34/0.63 ! [A2: rat,B: rat,C: rat,D2: rat] :
% 0.34/0.63 ( ( ord_less_rat @ A2 @ B )
% 0.34/0.63 => ( ( ord_less_rat @ C @ D2 )
% 0.34/0.63 => ( ord_less_rat @ ( plus_plus_rat @ A2 @ C ) @ ( plus_plus_rat @ B @ D2 ) ) ) ) ).
% 0.34/0.63
% 0.34/0.63 % add_strict_mono
% 0.34/0.63 thf(fact_5146_add__strict__left__mono,axiom,
% 0.34/0.63 ! [A2: nat,B: nat,C: nat] :
% 0.34/0.63 ( ( ord_less_nat @ A2 @ B )
% 0.34/0.63 => ( ord_less_nat @ ( plus_plus_nat @ C @ A2 ) @ ( plus_plus_nat @ C @ B ) ) ) ).
% 0.34/0.63
% 0.34/0.63 % add_strict_left_mono
% 0.34/0.63 thf(fact_5147_add__strict__left__mono,axiom,
% 0.34/0.63 ! [A2: int,B: int,C: int] :
% 0.34/0.63 ( ( ord_less_int @ A2 @ B )
% 0.34/0.63 => ( ord_less_int @ ( plus_plus_int @ C @ A2 ) @ ( plus_plus_int @ C @ B ) ) ) ).
% 0.34/0.63
% 0.34/0.63 % add_strict_left_mono
% 0.34/0.63 thf(fact_5148_add__strict__left__mono,axiom,
% 0.34/0.63 ! [A2: code_integer,B: code_integer,C: code_integer] :
% 0.34/0.63 ( ( ord_le6747313008572928689nteger @ A2 @ B )
% 0.34/0.63 => ( ord_le6747313008572928689nteger @ ( plus_p5714425477246183910nteger @ C @ A2 ) @ ( plus_p5714425477246183910nteger @ C @ B ) ) ) ).
% 0.34/0.63
% 0.34/0.63 % add_strict_left_mono
% 0.34/0.63 thf(fact_5149_add__strict__left__mono,axiom,
% 0.34/0.63 ! [A2: rat,B: rat,C: rat] :
% 0.34/0.63 ( ( ord_less_rat @ A2 @ B )
% 0.34/0.63 => ( ord_less_rat @ ( plus_plus_rat @ C @ A2 ) @ ( plus_plus_rat @ C @ B ) ) ) ).
% 0.34/0.63
% 0.34/0.63 % add_strict_left_mono
% 0.34/0.63 thf(fact_5150_add__strict__right__mono,axiom,
% 0.34/0.63 ! [A2: nat,B: nat,C: nat] :
% 0.34/0.63 ( ( ord_less_nat @ A2 @ B )
% 0.34/0.63 => ( ord_less_nat @ ( plus_plus_nat @ A2 @ C ) @ ( plus_plus_nat @ B @ C ) ) ) ).
% 0.34/0.63
% 0.34/0.63 % add_strict_right_mono
% 0.34/0.63 thf(fact_5151_add__strict__right__mono,axiom,
% 0.34/0.63 ! [A2: int,B: int,C: int] :
% 0.34/0.63 ( ( ord_less_int @ A2 @ B )
% 0.34/0.63 => ( ord_less_int @ ( plus_plus_int @ A2 @ C ) @ ( plus_plus_int @ B @ C ) ) ) ).
% 0.34/0.63
% 0.34/0.63 % add_strict_right_mono
% 0.34/0.63 thf(fact_5152_add__strict__right__mono,axiom,
% 0.34/0.63 ! [A2: code_integer,B: code_integer,C: code_integer] :
% 0.34/0.63 ( ( ord_le6747313008572928689nteger @ A2 @ B )
% 0.34/0.63 => ( ord_le6747313008572928689nteger @ ( plus_p5714425477246183910nteger @ A2 @ C ) @ ( plus_p5714425477246183910nteger @ B @ C ) ) ) ).
% 0.34/0.63
% 0.34/0.63 % add_strict_right_mono
% 0.34/0.63 thf(fact_5153_add__strict__right__mono,axiom,
% 0.34/0.63 ! [A2: rat,B: rat,C: rat] :
% 0.34/0.63 ( ( ord_less_rat @ A2 @ B )
% 0.34/0.63 => ( ord_less_rat @ ( plus_plus_rat @ A2 @ C ) @ ( plus_plus_rat @ B @ C ) ) ) ).
% 0.34/0.63
% 0.34/0.63 % add_strict_right_mono
% 0.34/0.63 thf(fact_5154_add__less__imp__less__left,axiom,
% 0.34/0.63 ! [C: nat,A2: nat,B: nat] :
% 0.34/0.63 ( ( ord_less_nat @ ( plus_plus_nat @ C @ A2 ) @ ( plus_plus_nat @ C @ B ) )
% 0.34/0.63 => ( ord_less_nat @ A2 @ B ) ) ).
% 0.34/0.63
% 0.34/0.63 % add_less_imp_less_left
% 0.34/0.63 thf(fact_5155_add__less__imp__less__left,axiom,
% 0.34/0.63 ! [C: int,A2: int,B: int] :
% 0.34/0.63 ( ( ord_less_int @ ( plus_plus_int @ C @ A2 ) @ ( plus_plus_int @ C @ B ) )
% 0.34/0.63 => ( ord_less_int @ A2 @ B ) ) ).
% 0.34/0.63
% 0.34/0.63 % add_less_imp_less_left
% 0.34/0.63 thf(fact_5156_add__less__imp__less__left,axiom,
% 0.34/0.63 ! [C: code_integer,A2: code_integer,B: code_integer] :
% 0.34/0.63 ( ( ord_le6747313008572928689nteger @ ( plus_p5714425477246183910nteger @ C @ A2 ) @ ( plus_p5714425477246183910nteger @ C @ B ) )
% 0.34/0.63 => ( ord_le6747313008572928689nteger @ A2 @ B ) ) ).
% 0.34/0.63
% 0.34/0.63 % add_less_imp_less_left
% 0.34/0.63 thf(fact_5157_add__less__imp__less__left,axiom,
% 0.34/0.63 ! [C: rat,A2: rat,B: rat] :
% 0.34/0.63 ( ( ord_less_rat @ ( plus_plus_rat @ C @ A2 ) @ ( plus_plus_rat @ C @ B ) )
% 0.34/0.63 => ( ord_less_rat @ A2 @ B ) ) ).
% 0.34/0.63
% 0.34/0.63 % add_less_imp_less_left
% 0.34/0.63 thf(fact_5158_add__less__imp__less__right,axiom,
% 0.34/0.63 ! [A2: nat,C: nat,B: nat] :
% 0.34/0.63 ( ( ord_less_nat @ ( plus_plus_nat @ A2 @ C ) @ ( plus_plus_nat @ B @ C ) )
% 0.34/0.63 => ( ord_less_nat @ A2 @ B ) ) ).
% 0.34/0.63
% 0.34/0.63 % add_less_imp_less_right
% 0.34/0.63 thf(fact_5159_add__less__imp__less__right,axiom,
% 0.34/0.63 ! [A2: int,C: int,B: int] :
% 0.34/0.63 ( ( ord_less_int @ ( plus_plus_int @ A2 @ C ) @ ( plus_plus_int @ B @ C ) )
% 0.34/0.63 => ( ord_less_int @ A2 @ B ) ) ).
% 0.34/0.63
% 0.34/0.63 % add_less_imp_less_right
% 0.34/0.63 thf(fact_5160_add__less__imp__less__right,axiom,
% 0.34/0.63 ! [A2: code_integer,C: code_integer,B: code_integer] :
% 0.34/0.63 ( ( ord_le6747313008572928689nteger @ ( plus_p5714425477246183910nteger @ A2 @ C ) @ ( plus_p5714425477246183910nteger @ B @ C ) )
% 0.34/0.63 => ( ord_le6747313008572928689nteger @ A2 @ B ) ) ).
% 0.34/0.63
% 0.34/0.63 % add_less_imp_less_right
% 0.34/0.63 thf(fact_5161_add__less__imp__less__right,axiom,
% 0.34/0.63 ! [A2: rat,C: rat,B: rat] :
% 0.34/0.63 ( ( ord_less_rat @ ( plus_plus_rat @ A2 @ C ) @ ( plus_plus_rat @ B @ C ) )
% 0.34/0.63 => ( ord_less_rat @ A2 @ B ) ) ).
% 0.34/0.63
% 0.34/0.63 % add_less_imp_less_right
% 0.34/0.63 thf(fact_5162_combine__common__factor,axiom,
% 0.34/0.63 ! [A2: code_integer,E2: code_integer,B: code_integer,C: code_integer] :
% 0.34/0.63 ( ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ A2 @ E2 ) @ ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ B @ E2 ) @ C ) )
% 0.34/0.63 = ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ ( plus_p5714425477246183910nteger @ A2 @ B ) @ E2 ) @ C ) ) ).
% 0.34/0.63
% 0.34/0.63 % combine_common_factor
% 0.34/0.63 thf(fact_5163_combine__common__factor,axiom,
% 0.34/0.63 ! [A2: rat,E2: rat,B: rat,C: rat] :
% 0.34/0.63 ( ( plus_plus_rat @ ( times_times_rat @ A2 @ E2 ) @ ( plus_plus_rat @ ( times_times_rat @ B @ E2 ) @ C ) )
% 0.34/0.63 = ( plus_plus_rat @ ( times_times_rat @ ( plus_plus_rat @ A2 @ B ) @ E2 ) @ C ) ) ).
% 0.34/0.63
% 0.34/0.63 % combine_common_factor
% 0.34/0.63 thf(fact_5164_combine__common__factor,axiom,
% 0.34/0.63 ! [A2: nat,E2: nat,B: nat,C: nat] :
% 0.34/0.63 ( ( plus_plus_nat @ ( times_times_nat @ A2 @ E2 ) @ ( plus_plus_nat @ ( times_times_nat @ B @ E2 ) @ C ) )
% 0.34/0.63 = ( plus_plus_nat @ ( times_times_nat @ ( plus_plus_nat @ A2 @ B ) @ E2 ) @ C ) ) ).
% 0.34/0.63
% 0.34/0.63 % combine_common_factor
% 0.34/0.63 thf(fact_5165_combine__common__factor,axiom,
% 0.34/0.63 ! [A2: int,E2: int,B: int,C: int] :
% 0.34/0.63 ( ( plus_plus_int @ ( times_times_int @ A2 @ E2 ) @ ( plus_plus_int @ ( times_times_int @ B @ E2 ) @ C ) )
% 0.34/0.63 = ( plus_plus_int @ ( times_times_int @ ( plus_plus_int @ A2 @ B ) @ E2 ) @ C ) ) ).
% 0.34/0.63
% 0.34/0.63 % combine_common_factor
% 0.34/0.63 thf(fact_5166_distrib__right,axiom,
% 0.34/0.63 ! [A2: code_integer,B: code_integer,C: code_integer] :
% 0.34/0.63 ( ( times_3573771949741848930nteger @ ( plus_p5714425477246183910nteger @ A2 @ B ) @ C )
% 0.34/0.63 = ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ A2 @ C ) @ ( times_3573771949741848930nteger @ B @ C ) ) ) ).
% 0.34/0.63
% 0.34/0.63 % distrib_right
% 0.34/0.63 thf(fact_5167_distrib__right,axiom,
% 0.34/0.63 ! [A2: rat,B: rat,C: rat] :
% 0.34/0.63 ( ( times_times_rat @ ( plus_plus_rat @ A2 @ B ) @ C )
% 0.34/0.63 = ( plus_plus_rat @ ( times_times_rat @ A2 @ C ) @ ( times_times_rat @ B @ C ) ) ) ).
% 0.34/0.63
% 0.34/0.63 % distrib_right
% 0.34/0.63 thf(fact_5168_distrib__right,axiom,
% 0.34/0.63 ! [A2: nat,B: nat,C: nat] :
% 0.34/0.63 ( ( times_times_nat @ ( plus_plus_nat @ A2 @ B ) @ C )
% 0.34/0.63 = ( plus_plus_nat @ ( times_times_nat @ A2 @ C ) @ ( times_times_nat @ B @ C ) ) ) ).
% 0.34/0.63
% 0.34/0.63 % distrib_right
% 0.34/0.63 thf(fact_5169_distrib__right,axiom,
% 0.34/0.63 ! [A2: int,B: int,C: int] :
% 0.34/0.63 ( ( times_times_int @ ( plus_plus_int @ A2 @ B ) @ C )
% 0.34/0.63 = ( plus_plus_int @ ( times_times_int @ A2 @ C ) @ ( times_times_int @ B @ C ) ) ) ).
% 0.34/0.63
% 0.34/0.63 % distrib_right
% 0.34/0.63 thf(fact_5170_distrib__left,axiom,
% 0.34/0.63 ! [A2: code_integer,B: code_integer,C: code_integer] :
% 0.34/0.63 ( ( times_3573771949741848930nteger @ A2 @ ( plus_p5714425477246183910nteger @ B @ C ) )
% 0.34/0.63 = ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ A2 @ B ) @ ( times_3573771949741848930nteger @ A2 @ C ) ) ) ).
% 0.34/0.63
% 0.34/0.63 % distrib_left
% 0.34/0.63 thf(fact_5171_distrib__left,axiom,
% 0.34/0.63 ! [A2: rat,B: rat,C: rat] :
% 0.34/0.63 ( ( times_times_rat @ A2 @ ( plus_plus_rat @ B @ C ) )
% 0.34/0.63 = ( plus_plus_rat @ ( times_times_rat @ A2 @ B ) @ ( times_times_rat @ A2 @ C ) ) ) ).
% 0.34/0.63
% 0.34/0.63 % distrib_left
% 0.34/0.63 thf(fact_5172_distrib__left,axiom,
% 0.34/0.63 ! [A2: nat,B: nat,C: nat] :
% 0.34/0.63 ( ( times_times_nat @ A2 @ ( plus_plus_nat @ B @ C ) )
% 0.34/0.63 = ( plus_plus_nat @ ( times_times_nat @ A2 @ B ) @ ( times_times_nat @ A2 @ C ) ) ) ).
% 0.34/0.63
% 0.34/0.63 % distrib_left
% 0.34/0.63 thf(fact_5173_distrib__left,axiom,
% 0.34/0.63 ! [A2: int,B: int,C: int] :
% 0.34/0.63 ( ( times_times_int @ A2 @ ( plus_plus_int @ B @ C ) )
% 0.34/0.63 = ( plus_plus_int @ ( times_times_int @ A2 @ B ) @ ( times_times_int @ A2 @ C ) ) ) ).
% 0.34/0.63
% 0.34/0.63 % distrib_left
% 0.34/0.63 thf(fact_5174_comm__semiring__class_Odistrib,axiom,
% 0.34/0.63 ! [A2: code_integer,B: code_integer,C: code_integer] :
% 0.34/0.63 ( ( times_3573771949741848930nteger @ ( plus_p5714425477246183910nteger @ A2 @ B ) @ C )
% 0.34/0.63 = ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ A2 @ C ) @ ( times_3573771949741848930nteger @ B @ C ) ) ) ).
% 0.34/0.63
% 0.34/0.63 % comm_semiring_class.distrib
% 0.34/0.63 thf(fact_5175_comm__semiring__class_Odistrib,axiom,
% 0.34/0.63 ! [A2: rat,B: rat,C: rat] :
% 0.34/0.63 ( ( times_times_rat @ ( plus_plus_rat @ A2 @ B ) @ C )
% 0.34/0.63 = ( plus_plus_rat @ ( times_times_rat @ A2 @ C ) @ ( times_times_rat @ B @ C ) ) ) ).
% 0.34/0.63
% 0.34/0.63 % comm_semiring_class.distrib
% 0.34/0.63 thf(fact_5176_comm__semiring__class_Odistrib,axiom,
% 0.34/0.63 ! [A2: nat,B: nat,C: nat] :
% 0.34/0.63 ( ( times_times_nat @ ( plus_plus_nat @ A2 @ B ) @ C )
% 0.34/0.63 = ( plus_plus_nat @ ( times_times_nat @ A2 @ C ) @ ( times_times_nat @ B @ C ) ) ) ).
% 0.34/0.63
% 0.34/0.63 % comm_semiring_class.distrib
% 0.34/0.63 thf(fact_5177_comm__semiring__class_Odistrib,axiom,
% 0.34/0.63 ! [A2: int,B: int,C: int] :
% 0.34/0.63 ( ( times_times_int @ ( plus_plus_int @ A2 @ B ) @ C )
% 0.34/0.63 = ( plus_plus_int @ ( times_times_int @ A2 @ C ) @ ( times_times_int @ B @ C ) ) ) ).
% 0.34/0.63
% 0.34/0.63 % comm_semiring_class.distrib
% 0.34/0.63 thf(fact_5178_ring__class_Oring__distribs_I1_J,axiom,
% 0.34/0.63 ! [A2: code_integer,B: code_integer,C: code_integer] :
% 0.34/0.63 ( ( times_3573771949741848930nteger @ A2 @ ( plus_p5714425477246183910nteger @ B @ C ) )
% 0.34/0.63 = ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ A2 @ B ) @ ( times_3573771949741848930nteger @ A2 @ C ) ) ) ).
% 0.34/0.63
% 0.34/0.63 % ring_class.ring_distribs(1)
% 0.34/0.63 thf(fact_5179_ring__class_Oring__distribs_I1_J,axiom,
% 0.34/0.63 ! [A2: rat,B: rat,C: rat] :
% 0.34/0.63 ( ( times_times_rat @ A2 @ ( plus_plus_rat @ B @ C ) )
% 0.34/0.63 = ( plus_plus_rat @ ( times_times_rat @ A2 @ B ) @ ( times_times_rat @ A2 @ C ) ) ) ).
% 0.34/0.63
% 0.34/0.63 % ring_class.ring_distribs(1)
% 0.34/0.63 thf(fact_5180_ring__class_Oring__distribs_I1_J,axiom,
% 0.34/0.63 ! [A2: int,B: int,C: int] :
% 0.34/0.63 ( ( times_times_int @ A2 @ ( plus_plus_int @ B @ C ) )
% 0.34/0.63 = ( plus_plus_int @ ( times_times_int @ A2 @ B ) @ ( times_times_int @ A2 @ C ) ) ) ).
% 0.34/0.63
% 0.34/0.63 % ring_class.ring_distribs(1)
% 0.34/0.63 thf(fact_5181_ring__class_Oring__distribs_I2_J,axiom,
% 0.34/0.63 ! [A2: code_integer,B: code_integer,C: code_integer] :
% 0.34/0.63 ( ( times_3573771949741848930nteger @ ( plus_p5714425477246183910nteger @ A2 @ B ) @ C )
% 0.34/0.63 = ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ A2 @ C ) @ ( times_3573771949741848930nteger @ B @ C ) ) ) ).
% 0.34/0.63
% 0.34/0.63 % ring_class.ring_distribs(2)
% 0.34/0.63 thf(fact_5182_ring__class_Oring__distribs_I2_J,axiom,
% 0.34/0.63 ! [A2: rat,B: rat,C: rat] :
% 0.34/0.63 ( ( times_times_rat @ ( plus_plus_rat @ A2 @ B ) @ C )
% 0.34/0.63 = ( plus_plus_rat @ ( times_times_rat @ A2 @ C ) @ ( times_times_rat @ B @ C ) ) ) ).
% 0.34/0.63
% 0.34/0.63 % ring_class.ring_distribs(2)
% 0.34/0.63 thf(fact_5183_ring__class_Oring__distribs_I2_J,axiom,
% 0.34/0.63 ! [A2: int,B: int,C: int] :
% 0.34/0.63 ( ( times_times_int @ ( plus_plus_int @ A2 @ B ) @ C )
% 0.34/0.63 = ( plus_plus_int @ ( times_times_int @ A2 @ C ) @ ( times_times_int @ B @ C ) ) ) ).
% 0.34/0.63
% 0.34/0.63 % ring_class.ring_distribs(2)
% 0.34/0.63 thf(fact_5184_group__cancel_Osub1,axiom,
% 0.34/0.63 ! [A: code_integer,K3: code_integer,A2: code_integer,B: code_integer] :
% 0.34/0.63 ( ( A
% 0.34/0.63 = ( plus_p5714425477246183910nteger @ K3 @ A2 ) )
% 0.34/0.63 => ( ( minus_8373710615458151222nteger @ A @ B )
% 0.34/0.63 = ( plus_p5714425477246183910nteger @ K3 @ ( minus_8373710615458151222nteger @ A2 @ B ) ) ) ) ).
% 0.34/0.63
% 0.34/0.63 % group_cancel.sub1
% 0.34/0.63 thf(fact_5185_group__cancel_Osub1,axiom,
% 0.34/0.63 ! [A: rat,K3: rat,A2: rat,B: rat] :
% 0.34/0.63 ( ( A
% 0.34/0.63 = ( plus_plus_rat @ K3 @ A2 ) )
% 0.34/0.63 => ( ( minus_minus_rat @ A @ B )
% 0.34/0.63 = ( plus_plus_rat @ K3 @ ( minus_minus_rat @ A2 @ B ) ) ) ) ).
% 0.34/0.63
% 0.34/0.63 % group_cancel.sub1
% 0.34/0.63 thf(fact_5186_group__cancel_Osub1,axiom,
% 0.34/0.63 ! [A: int,K3: int,A2: int,B: int] :
% 0.34/0.63 ( ( A
% 0.34/0.63 = ( plus_plus_int @ K3 @ A2 ) )
% 0.34/0.63 => ( ( minus_minus_int @ A @ B )
% 0.34/0.63 = ( plus_plus_int @ K3 @ ( minus_minus_int @ A2 @ B ) ) ) ) ).
% 0.34/0.63
% 0.34/0.63 % group_cancel.sub1
% 0.34/0.63 thf(fact_5187_diff__eq__eq,axiom,
% 0.34/0.63 ! [A2: code_integer,B: code_integer,C: code_integer] :
% 0.34/0.63 ( ( ( minus_8373710615458151222nteger @ A2 @ B )
% 0.34/0.63 = C )
% 0.34/0.63 = ( A2
% 0.34/0.63 = ( plus_p5714425477246183910nteger @ C @ B ) ) ) ).
% 0.34/0.63
% 0.34/0.63 % diff_eq_eq
% 0.34/0.63 thf(fact_5188_diff__eq__eq,axiom,
% 0.34/0.63 ! [A2: rat,B: rat,C: rat] :
% 0.34/0.63 ( ( ( minus_minus_rat @ A2 @ B )
% 0.34/0.63 = C )
% 0.34/0.63 = ( A2
% 0.34/0.63 = ( plus_plus_rat @ C @ B ) ) ) ).
% 0.34/0.63
% 0.34/0.63 % diff_eq_eq
% 0.34/0.63 thf(fact_5189_diff__eq__eq,axiom,
% 0.34/0.63 ! [A2: int,B: int,C: int] :
% 0.34/0.63 ( ( ( minus_minus_int @ A2 @ B )
% 0.34/0.63 = C )
% 0.34/0.63 = ( A2
% 0.34/0.63 = ( plus_plus_int @ C @ B ) ) ) ).
% 0.34/0.63
% 0.34/0.63 % diff_eq_eq
% 0.34/0.63 thf(fact_5190_eq__diff__eq,axiom,
% 0.34/0.63 ! [A2: code_integer,C: code_integer,B: code_integer] :
% 0.34/0.63 ( ( A2
% 0.34/0.63 = ( minus_8373710615458151222nteger @ C @ B ) )
% 0.34/0.63 = ( ( plus_p5714425477246183910nteger @ A2 @ B )
% 0.34/0.63 = C ) ) ).
% 0.34/0.63
% 0.34/0.63 % eq_diff_eq
% 0.34/0.63 thf(fact_5191_eq__diff__eq,axiom,
% 0.34/0.63 ! [A2: rat,C: rat,B: rat] :
% 0.34/0.63 ( ( A2
% 0.34/0.63 = ( minus_minus_rat @ C @ B ) )
% 0.34/0.63 = ( ( plus_plus_rat @ A2 @ B )
% 0.34/0.63 = C ) ) ).
% 0.34/0.63
% 0.34/0.63 % eq_diff_eq
% 0.34/0.63 thf(fact_5192_eq__diff__eq,axiom,
% 0.34/0.63 ! [A2: int,C: int,B: int] :
% 0.34/0.63 ( ( A2
% 0.34/0.63 = ( minus_minus_int @ C @ B ) )
% 0.34/0.63 = ( ( plus_plus_int @ A2 @ B )
% 0.34/0.63 = C ) ) ).
% 0.34/0.63
% 0.34/0.63 % eq_diff_eq
% 0.34/0.63 thf(fact_5193_add__diff__eq,axiom,
% 0.34/0.63 ! [A2: code_integer,B: code_integer,C: code_integer] :
% 0.34/0.63 ( ( plus_p5714425477246183910nteger @ A2 @ ( minus_8373710615458151222nteger @ B @ C ) )
% 0.34/0.63 = ( minus_8373710615458151222nteger @ ( plus_p5714425477246183910nteger @ A2 @ B ) @ C ) ) ).
% 0.34/0.63
% 0.34/0.63 % add_diff_eq
% 0.34/0.63 thf(fact_5194_add__diff__eq,axiom,
% 0.34/0.63 ! [A2: rat,B: rat,C: rat] :
% 0.34/0.63 ( ( plus_plus_rat @ A2 @ ( minus_minus_rat @ B @ C ) )
% 0.34/0.63 = ( minus_minus_rat @ ( plus_plus_rat @ A2 @ B ) @ C ) ) ).
% 0.34/0.63
% 0.34/0.63 % add_diff_eq
% 0.34/0.63 thf(fact_5195_add__diff__eq,axiom,
% 0.34/0.63 ! [A2: int,B: int,C: int] :
% 0.34/0.63 ( ( plus_plus_int @ A2 @ ( minus_minus_int @ B @ C ) )
% 0.34/0.63 = ( minus_minus_int @ ( plus_plus_int @ A2 @ B ) @ C ) ) ).
% 0.34/0.63
% 0.34/0.63 % add_diff_eq
% 0.34/0.63 thf(fact_5196_diff__diff__eq2,axiom,
% 0.34/0.63 ! [A2: code_integer,B: code_integer,C: code_integer] :
% 0.34/0.63 ( ( minus_8373710615458151222nteger @ A2 @ ( minus_8373710615458151222nteger @ B @ C ) )
% 0.34/0.63 = ( minus_8373710615458151222nteger @ ( plus_p5714425477246183910nteger @ A2 @ C ) @ B ) ) ).
% 0.34/0.63
% 0.34/0.63 % diff_diff_eq2
% 0.34/0.63 thf(fact_5197_diff__diff__eq2,axiom,
% 0.34/0.63 ! [A2: rat,B: rat,C: rat] :
% 0.34/0.63 ( ( minus_minus_rat @ A2 @ ( minus_minus_rat @ B @ C ) )
% 0.34/0.63 = ( minus_minus_rat @ ( plus_plus_rat @ A2 @ C ) @ B ) ) ).
% 0.34/0.63
% 0.34/0.63 % diff_diff_eq2
% 0.34/0.63 thf(fact_5198_diff__diff__eq2,axiom,
% 0.34/0.63 ! [A2: int,B: int,C: int] :
% 0.34/0.63 ( ( minus_minus_int @ A2 @ ( minus_minus_int @ B @ C ) )
% 0.34/0.63 = ( minus_minus_int @ ( plus_plus_int @ A2 @ C ) @ B ) ) ).
% 0.34/0.63
% 0.34/0.63 % diff_diff_eq2
% 0.34/0.63 thf(fact_5199_diff__add__eq,axiom,
% 0.34/0.63 ! [A2: code_integer,B: code_integer,C: code_integer] :
% 0.34/0.63 ( ( plus_p5714425477246183910nteger @ ( minus_8373710615458151222nteger @ A2 @ B ) @ C )
% 0.34/0.63 = ( minus_8373710615458151222nteger @ ( plus_p5714425477246183910nteger @ A2 @ C ) @ B ) ) ).
% 0.34/0.63
% 0.34/0.63 % diff_add_eq
% 0.34/0.63 thf(fact_5200_diff__add__eq,axiom,
% 0.34/0.63 ! [A2: rat,B: rat,C: rat] :
% 0.34/0.63 ( ( plus_plus_rat @ ( minus_minus_rat @ A2 @ B ) @ C )
% 0.34/0.63 = ( minus_minus_rat @ ( plus_plus_rat @ A2 @ C ) @ B ) ) ).
% 0.34/0.63
% 0.34/0.63 % diff_add_eq
% 0.34/0.63 thf(fact_5201_diff__add__eq,axiom,
% 0.34/0.63 ! [A2: int,B: int,C: int] :
% 0.34/0.63 ( ( plus_plus_int @ ( minus_minus_int @ A2 @ B ) @ C )
% 0.34/0.63 = ( minus_minus_int @ ( plus_plus_int @ A2 @ C ) @ B ) ) ).
% 0.34/0.63
% 0.34/0.63 % diff_add_eq
% 0.34/0.63 thf(fact_5202_diff__add__eq__diff__diff__swap,axiom,
% 0.34/0.63 ! [A2: code_integer,B: code_integer,C: code_integer] :
% 0.34/0.63 ( ( minus_8373710615458151222nteger @ A2 @ ( plus_p5714425477246183910nteger @ B @ C ) )
% 0.34/0.63 = ( minus_8373710615458151222nteger @ ( minus_8373710615458151222nteger @ A2 @ C ) @ B ) ) ).
% 0.34/0.63
% 0.34/0.63 % diff_add_eq_diff_diff_swap
% 0.34/0.63 thf(fact_5203_diff__add__eq__diff__diff__swap,axiom,
% 0.34/0.63 ! [A2: rat,B: rat,C: rat] :
% 0.34/0.63 ( ( minus_minus_rat @ A2 @ ( plus_plus_rat @ B @ C ) )
% 0.34/0.63 = ( minus_minus_rat @ ( minus_minus_rat @ A2 @ C ) @ B ) ) ).
% 0.34/0.63
% 0.34/0.63 % diff_add_eq_diff_diff_swap
% 0.34/0.63 thf(fact_5204_diff__add__eq__diff__diff__swap,axiom,
% 0.34/0.63 ! [A2: int,B: int,C: int] :
% 0.34/0.63 ( ( minus_minus_int @ A2 @ ( plus_plus_int @ B @ C ) )
% 0.34/0.63 = ( minus_minus_int @ ( minus_minus_int @ A2 @ C ) @ B ) ) ).
% 0.34/0.63
% 0.34/0.63 % diff_add_eq_diff_diff_swap
% 0.34/0.63 thf(fact_5205_add__implies__diff,axiom,
% 0.34/0.63 ! [C: code_integer,B: code_integer,A2: code_integer] :
% 0.34/0.63 ( ( ( plus_p5714425477246183910nteger @ C @ B )
% 0.34/0.63 = A2 )
% 0.34/0.63 => ( C
% 0.34/0.63 = ( minus_8373710615458151222nteger @ A2 @ B ) ) ) ).
% 0.34/0.63
% 0.34/0.63 % add_implies_diff
% 0.34/0.63 thf(fact_5206_add__implies__diff,axiom,
% 0.34/0.63 ! [C: rat,B: rat,A2: rat] :
% 0.34/0.63 ( ( ( plus_plus_rat @ C @ B )
% 0.34/0.63 = A2 )
% 0.34/0.63 => ( C
% 0.34/0.63 = ( minus_minus_rat @ A2 @ B ) ) ) ).
% 0.34/0.63
% 0.34/0.63 % add_implies_diff
% 0.34/0.63 thf(fact_5207_add__implies__diff,axiom,
% 0.34/0.63 ! [C: nat,B: nat,A2: nat] :
% 0.34/0.63 ( ( ( plus_plus_nat @ C @ B )
% 0.34/0.63 = A2 )
% 0.34/0.63 => ( C
% 0.34/0.63 = ( minus_minus_nat @ A2 @ B ) ) ) ).
% 0.34/0.63
% 0.34/0.63 % add_implies_diff
% 0.34/0.63 thf(fact_5208_add__implies__diff,axiom,
% 0.34/0.63 ! [C: int,B: int,A2: int] :
% 0.34/0.63 ( ( ( plus_plus_int @ C @ B )
% 0.34/0.63 = A2 )
% 0.34/0.63 => ( C
% 0.34/0.63 = ( minus_minus_int @ A2 @ B ) ) ) ).
% 0.34/0.63
% 0.34/0.63 % add_implies_diff
% 0.34/0.63 thf(fact_5209_diff__diff__eq,axiom,
% 0.34/0.63 ! [A2: code_integer,B: code_integer,C: code_integer] :
% 0.34/0.63 ( ( minus_8373710615458151222nteger @ ( minus_8373710615458151222nteger @ A2 @ B ) @ C )
% 0.34/0.63 = ( minus_8373710615458151222nteger @ A2 @ ( plus_p5714425477246183910nteger @ B @ C ) ) ) ).
% 0.34/0.63
% 0.34/0.63 % diff_diff_eq
% 0.34/0.63 thf(fact_5210_diff__diff__eq,axiom,
% 0.34/0.63 ! [A2: rat,B: rat,C: rat] :
% 0.34/0.63 ( ( minus_minus_rat @ ( minus_minus_rat @ A2 @ B ) @ C )
% 0.34/0.63 = ( minus_minus_rat @ A2 @ ( plus_plus_rat @ B @ C ) ) ) ).
% 0.34/0.63
% 0.34/0.63 % diff_diff_eq
% 0.34/0.63 thf(fact_5211_diff__diff__eq,axiom,
% 0.34/0.63 ! [A2: nat,B: nat,C: nat] :
% 0.34/0.63 ( ( minus_minus_nat @ ( minus_minus_nat @ A2 @ B ) @ C )
% 0.34/0.63 = ( minus_minus_nat @ A2 @ ( plus_plus_nat @ B @ C ) ) ) ).
% 0.34/0.63
% 0.34/0.63 % diff_diff_eq
% 0.34/0.63 thf(fact_5212_diff__diff__eq,axiom,
% 0.34/0.63 ! [A2: int,B: int,C: int] :
% 0.34/0.63 ( ( minus_minus_int @ ( minus_minus_int @ A2 @ B ) @ C )
% 0.34/0.63 = ( minus_minus_int @ A2 @ ( plus_plus_int @ B @ C ) ) ) ).
% 0.34/0.63
% 0.34/0.63 % diff_diff_eq
% 0.34/0.63 thf(fact_5213_mlex__bound,axiom,
% 0.34/0.63 ! [A2: nat,A: nat,B: nat,N: nat] :
% 0.34/0.63 ( ( ord_less_nat @ A2 @ A )
% 0.34/0.63 => ( ( ord_less_nat @ B @ N )
% 0.34/0.63 => ( ord_less_nat @ ( plus_plus_nat @ ( times_times_nat @ A2 @ N ) @ B ) @ ( times_times_nat @ A @ N ) ) ) ) ).
% 0.34/0.63
% 0.34/0.63 % mlex_bound
% 0.34/0.63 thf(fact_5214_mlex__fst__decrI,axiom,
% 0.34/0.63 ! [A2: nat,A5: nat,B: nat,N: nat,B3: nat] :
% 0.34/0.63 ( ( ord_less_nat @ A2 @ A5 )
% 0.34/0.63 => ( ( ord_less_nat @ B @ N )
% 0.34/0.63 => ( ( ord_less_nat @ B3 @ N )
% 0.34/0.63 => ( ord_less_nat @ ( plus_plus_nat @ ( times_times_nat @ A2 @ N ) @ B ) @ ( plus_plus_nat @ ( times_times_nat @ A5 @ N ) @ B3 ) ) ) ) ) ).
% 0.34/0.63
% 0.34/0.63 % mlex_fst_decrI
% 0.34/0.63 thf(fact_5215_mlex__snd__decrI,axiom,
% 0.34/0.63 ! [A2: nat,A5: nat,B: nat,B3: nat,N: nat] :
% 0.34/0.63 ( ( A2 = A5 )
% 0.34/0.63 => ( ( ord_less_nat @ B @ B3 )
% 0.34/0.63 => ( ord_less_nat @ ( plus_plus_nat @ ( times_times_nat @ A2 @ N ) @ B ) @ ( plus_plus_nat @ ( times_times_nat @ A5 @ N ) @ B3 ) ) ) ) ).
% 0.34/0.63
% 0.34/0.63 % mlex_snd_decrI
% 0.34/0.63 thf(fact_5216_mlex__leI,axiom,
% 0.34/0.63 ! [A2: nat,A5: nat,B: nat,B3: nat,N: nat] :
% 0.34/0.63 ( ( ord_less_eq_nat @ A2 @ A5 )
% 0.34/0.63 => ( ( ord_less_eq_nat @ B @ B3 )
% 0.34/0.63 => ( ord_less_eq_nat @ ( plus_plus_nat @ ( times_times_nat @ A2 @ N ) @ B ) @ ( plus_plus_nat @ ( times_times_nat @ A5 @ N ) @ B3 ) ) ) ) ).
% 0.34/0.63
% 0.34/0.63 % mlex_leI
% 0.34/0.63 thf(fact_5217_Inf__nat__def1,axiom,
% 0.34/0.63 ! [K4: set_nat] :
% 0.34/0.63 ( ( K4 != bot_bot_set_nat )
% 0.34/0.63 => ( member_nat @ ( complete_Inf_Inf_nat @ K4 ) @ K4 ) ) ).
% 0.34/0.63
% 0.34/0.63 % Inf_nat_def1
% 0.34/0.63 thf(fact_5218_max__add__distrib__left,axiom,
% 0.34/0.63 ! [X: code_integer,Y: code_integer,Z: code_integer] :
% 0.34/0.63 ( ( plus_p5714425477246183910nteger @ ( ord_max_Code_integer @ X @ Y ) @ Z )
% 0.34/0.63 = ( ord_max_Code_integer @ ( plus_p5714425477246183910nteger @ X @ Z ) @ ( plus_p5714425477246183910nteger @ Y @ Z ) ) ) ).
% 0.34/0.63
% 0.34/0.63 % max_add_distrib_left
% 0.34/0.63 thf(fact_5219_max__add__distrib__left,axiom,
% 0.34/0.63 ! [X: rat,Y: rat,Z: rat] :
% 0.34/0.63 ( ( plus_plus_rat @ ( ord_max_rat @ X @ Y ) @ Z )
% 0.34/0.63 = ( ord_max_rat @ ( plus_plus_rat @ X @ Z ) @ ( plus_plus_rat @ Y @ Z ) ) ) ).
% 0.34/0.63
% 0.34/0.63 % max_add_distrib_left
% 0.34/0.63 thf(fact_5220_max__add__distrib__left,axiom,
% 0.34/0.63 ! [X: nat,Y: nat,Z: nat] :
% 0.34/0.63 ( ( plus_plus_nat @ ( ord_max_nat @ X @ Y ) @ Z )
% 0.34/0.63 = ( ord_max_nat @ ( plus_plus_nat @ X @ Z ) @ ( plus_plus_nat @ Y @ Z ) ) ) ).
% 0.34/0.63
% 0.34/0.63 % max_add_distrib_left
% 0.34/0.63 thf(fact_5221_max__add__distrib__left,axiom,
% 0.34/0.63 ! [X: int,Y: int,Z: int] :
% 0.34/0.63 ( ( plus_plus_int @ ( ord_max_int @ X @ Y ) @ Z )
% 0.34/0.63 = ( ord_max_int @ ( plus_plus_int @ X @ Z ) @ ( plus_plus_int @ Y @ Z ) ) ) ).
% 0.34/0.63
% 0.34/0.63 % max_add_distrib_left
% 0.34/0.63 thf(fact_5222_max__add__distrib__right,axiom,
% 0.34/0.63 ! [X: code_integer,Y: code_integer,Z: code_integer] :
% 0.34/0.63 ( ( plus_p5714425477246183910nteger @ X @ ( ord_max_Code_integer @ Y @ Z ) )
% 0.34/0.63 = ( ord_max_Code_integer @ ( plus_p5714425477246183910nteger @ X @ Y ) @ ( plus_p5714425477246183910nteger @ X @ Z ) ) ) ).
% 0.34/0.63
% 0.34/0.63 % max_add_distrib_right
% 0.34/0.63 thf(fact_5223_max__add__distrib__right,axiom,
% 0.34/0.63 ! [X: rat,Y: rat,Z: rat] :
% 0.34/0.63 ( ( plus_plus_rat @ X @ ( ord_max_rat @ Y @ Z ) )
% 0.34/0.63 = ( ord_max_rat @ ( plus_plus_rat @ X @ Y ) @ ( plus_plus_rat @ X @ Z ) ) ) ).
% 0.34/0.63
% 0.34/0.63 % max_add_distrib_right
% 0.34/0.63 thf(fact_5224_max__add__distrib__right,axiom,
% 0.34/0.63 ! [X: nat,Y: nat,Z: nat] :
% 0.34/0.63 ( ( plus_plus_nat @ X @ ( ord_max_nat @ Y @ Z ) )
% 0.34/0.63 = ( ord_max_nat @ ( plus_plus_nat @ X @ Y ) @ ( plus_plus_nat @ X @ Z ) ) ) ).
% 0.34/0.63
% 0.34/0.63 % max_add_distrib_right
% 0.34/0.63 thf(fact_5225_max__add__distrib__right,axiom,
% 0.34/0.63 ! [X: int,Y: int,Z: int] :
% 0.34/0.63 ( ( plus_plus_int @ X @ ( ord_max_int @ Y @ Z ) )
% 0.34/0.63 = ( ord_max_int @ ( plus_plus_int @ X @ Y ) @ ( plus_plus_int @ X @ Z ) ) ) ).
% 0.34/0.63
% 0.34/0.63 % max_add_distrib_right
% 0.34/0.63 thf(fact_5226_add__mono__thms__linordered__field_I4_J,axiom,
% 0.34/0.63 ! [I2: code_integer,J: code_integer,K3: code_integer,L: code_integer] :
% 0.34/0.63 ( ( ( ord_le3102999989581377725nteger @ I2 @ J )
% 0.34/0.63 & ( ord_le6747313008572928689nteger @ K3 @ L ) )
% 0.34/0.63 => ( ord_le6747313008572928689nteger @ ( plus_p5714425477246183910nteger @ I2 @ K3 ) @ ( plus_p5714425477246183910nteger @ J @ L ) ) ) ).
% 0.34/0.63
% 0.34/0.63 % add_mono_thms_linordered_field(4)
% 0.34/0.63 thf(fact_5227_add__mono__thms__linordered__field_I4_J,axiom,
% 0.34/0.63 ! [I2: rat,J: rat,K3: rat,L: rat] :
% 0.34/0.63 ( ( ( ord_less_eq_rat @ I2 @ J )
% 0.34/0.63 & ( ord_less_rat @ K3 @ L ) )
% 0.34/0.63 => ( ord_less_rat @ ( plus_plus_rat @ I2 @ K3 ) @ ( plus_plus_rat @ J @ L ) ) ) ).
% 0.34/0.63
% 0.34/0.63 % add_mono_thms_linordered_field(4)
% 0.34/0.63 thf(fact_5228_add__mono__thms__linordered__field_I4_J,axiom,
% 0.34/0.63 ! [I2: nat,J: nat,K3: nat,L: nat] :
% 0.34/0.63 ( ( ( ord_less_eq_nat @ I2 @ J )
% 0.34/0.63 & ( ord_less_nat @ K3 @ L ) )
% 0.34/0.63 => ( ord_less_nat @ ( plus_plus_nat @ I2 @ K3 ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% 0.34/0.63
% 0.34/0.63 % add_mono_thms_linordered_field(4)
% 0.34/0.63 thf(fact_5229_add__mono__thms__linordered__field_I4_J,axiom,
% 0.34/0.63 ! [I2: int,J: int,K3: int,L: int] :
% 0.34/0.63 ( ( ( ord_less_eq_int @ I2 @ J )
% 0.34/0.63 & ( ord_less_int @ K3 @ L ) )
% 0.34/0.63 => ( ord_less_int @ ( plus_plus_int @ I2 @ K3 ) @ ( plus_plus_int @ J @ L ) ) ) ).
% 0.34/0.63
% 0.34/0.63 % add_mono_thms_linordered_field(4)
% 0.34/0.63 thf(fact_5230_add__mono__thms__linordered__field_I3_J,axiom,
% 0.34/0.63 ! [I2: code_integer,J: code_integer,K3: code_integer,L: code_integer] :
% 0.34/0.63 ( ( ( ord_le6747313008572928689nteger @ I2 @ J )
% 0.34/0.63 & ( ord_le3102999989581377725nteger @ K3 @ L ) )
% 0.34/0.63 => ( ord_le6747313008572928689nteger @ ( plus_p5714425477246183910nteger @ I2 @ K3 ) @ ( plus_p5714425477246183910nteger @ J @ L ) ) ) ).
% 0.34/0.63
% 0.34/0.63 % add_mono_thms_linordered_field(3)
% 0.34/0.63 thf(fact_5231_add__mono__thms__linordered__field_I3_J,axiom,
% 0.34/0.63 ! [I2: rat,J: rat,K3: rat,L: rat] :
% 0.34/0.63 ( ( ( ord_less_rat @ I2 @ J )
% 0.34/0.63 & ( ord_less_eq_rat @ K3 @ L ) )
% 0.34/0.63 => ( ord_less_rat @ ( plus_plus_rat @ I2 @ K3 ) @ ( plus_plus_rat @ J @ L ) ) ) ).
% 0.34/0.63
% 0.34/0.63 % add_mono_thms_linordered_field(3)
% 0.34/0.63 thf(fact_5232_add__mono__thms__linordered__field_I3_J,axiom,
% 0.34/0.63 ! [I2: nat,J: nat,K3: nat,L: nat] :
% 0.34/0.63 ( ( ( ord_less_nat @ I2 @ J )
% 0.34/0.63 & ( ord_less_eq_nat @ K3 @ L ) )
% 0.34/0.63 => ( ord_less_nat @ ( plus_plus_nat @ I2 @ K3 ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% 0.34/0.63
% 0.34/0.63 % add_mono_thms_linordered_field(3)
% 0.34/0.63 thf(fact_5233_add__mono__thms__linordered__field_I3_J,axiom,
% 0.34/0.63 ! [I2: int,J: int,K3: int,L: int] :
% 0.34/0.63 ( ( ( ord_less_int @ I2 @ J )
% 0.34/0.63 & ( ord_less_eq_int @ K3 @ L ) )
% 0.34/0.63 => ( ord_less_int @ ( plus_plus_int @ I2 @ K3 ) @ ( plus_plus_int @ J @ L ) ) ) ).
% 0.34/0.63
% 0.34/0.63 % add_mono_thms_linordered_field(3)
% 0.34/0.63 thf(fact_5234_add__le__less__mono,axiom,
% 0.34/0.63 ! [A2: code_integer,B: code_integer,C: code_integer,D2: code_integer] :
% 0.34/0.63 ( ( ord_le3102999989581377725nteger @ A2 @ B )
% 0.34/0.63 => ( ( ord_le6747313008572928689nteger @ C @ D2 )
% 0.34/0.63 => ( ord_le6747313008572928689nteger @ ( plus_p5714425477246183910nteger @ A2 @ C ) @ ( plus_p5714425477246183910nteger @ B @ D2 ) ) ) ) ).
% 0.34/0.63
% 0.34/0.63 % add_le_less_mono
% 0.34/0.63 thf(fact_5235_add__le__less__mono,axiom,
% 0.34/0.63 ! [A2: rat,B: rat,C: rat,D2: rat] :
% 0.34/0.63 ( ( ord_less_eq_rat @ A2 @ B )
% 0.34/0.63 => ( ( ord_less_rat @ C @ D2 )
% 0.34/0.63 => ( ord_less_rat @ ( plus_plus_rat @ A2 @ C ) @ ( plus_plus_rat @ B @ D2 ) ) ) ) ).
% 0.34/0.63
% 0.34/0.63 % add_le_less_mono
% 0.34/0.63 thf(fact_5236_add__le__less__mono,axiom,
% 0.34/0.63 ! [A2: nat,B: nat,C: nat,D2: nat] :
% 0.34/0.63 ( ( ord_less_eq_nat @ A2 @ B )
% 0.34/0.63 => ( ( ord_less_nat @ C @ D2 )
% 0.34/0.63 => ( ord_less_nat @ ( plus_plus_nat @ A2 @ C ) @ ( plus_plus_nat @ B @ D2 ) ) ) ) ).
% 0.34/0.63
% 0.34/0.63 % add_le_less_mono
% 0.34/0.63 thf(fact_5237_add__le__less__mono,axiom,
% 0.34/0.63 ! [A2: int,B: int,C: int,D2: int] :
% 0.34/0.63 ( ( ord_less_eq_int @ A2 @ B )
% 0.34/0.63 => ( ( ord_less_int @ C @ D2 )
% 0.34/0.63 => ( ord_less_int @ ( plus_plus_int @ A2 @ C ) @ ( plus_plus_int @ B @ D2 ) ) ) ) ).
% 0.34/0.63
% 0.34/0.63 % add_le_less_mono
% 0.34/0.63 thf(fact_5238_add__less__le__mono,axiom,
% 0.34/0.63 ! [A2: code_integer,B: code_integer,C: code_integer,D2: code_integer] :
% 0.34/0.63 ( ( ord_le6747313008572928689nteger @ A2 @ B )
% 0.34/0.63 => ( ( ord_le3102999989581377725nteger @ C @ D2 )
% 0.34/0.63 => ( ord_le6747313008572928689nteger @ ( plus_p5714425477246183910nteger @ A2 @ C ) @ ( plus_p5714425477246183910nteger @ B @ D2 ) ) ) ) ).
% 0.34/0.64
% 0.34/0.64 % add_less_le_mono
% 0.34/0.64 thf(fact_5239_add__less__le__mono,axiom,
% 0.34/0.64 ! [A2: rat,B: rat,C: rat,D2: rat] :
% 0.34/0.64 ( ( ord_less_rat @ A2 @ B )
% 0.34/0.64 => ( ( ord_less_eq_rat @ C @ D2 )
% 0.34/0.64 => ( ord_less_rat @ ( plus_plus_rat @ A2 @ C ) @ ( plus_plus_rat @ B @ D2 ) ) ) ) ).
% 0.34/0.64
% 0.34/0.64 % add_less_le_mono
% 0.34/0.64 thf(fact_5240_add__less__le__mono,axiom,
% 0.34/0.64 ! [A2: nat,B: nat,C: nat,D2: nat] :
% 0.34/0.64 ( ( ord_less_nat @ A2 @ B )
% 0.34/0.64 => ( ( ord_less_eq_nat @ C @ D2 )
% 0.34/0.64 => ( ord_less_nat @ ( plus_plus_nat @ A2 @ C ) @ ( plus_plus_nat @ B @ D2 ) ) ) ) ).
% 0.34/0.64
% 0.34/0.64 % add_less_le_mono
% 0.34/0.64 thf(fact_5241_add__less__le__mono,axiom,
% 0.34/0.64 ! [A2: int,B: int,C: int,D2: int] :
% 0.34/0.64 ( ( ord_less_int @ A2 @ B )
% 0.34/0.64 => ( ( ord_less_eq_int @ C @ D2 )
% 0.34/0.64 => ( ord_less_int @ ( plus_plus_int @ A2 @ C ) @ ( plus_plus_int @ B @ D2 ) ) ) ) ).
% 0.34/0.64
% 0.34/0.64 % add_less_le_mono
% 0.34/0.64 thf(fact_5242_ordered__cancel__comm__monoid__diff__class_Ole__imp__diff__is__add,axiom,
% 0.34/0.64 ! [A2: nat,B: nat,C: nat] :
% 0.34/0.64 ( ( ord_less_eq_nat @ A2 @ B )
% 0.34/0.64 => ( ( ord_less_eq_nat @ A2 @ B )
% 0.34/0.64 => ( ( ( minus_minus_nat @ B @ A2 )
% 0.34/0.64 = C )
% 0.34/0.64 = ( B
% 0.34/0.64 = ( plus_plus_nat @ C @ A2 ) ) ) ) ) ).
% 0.34/0.64
% 0.34/0.64 % ordered_cancel_comm_monoid_diff_class.le_imp_diff_is_add
% 0.34/0.64 thf(fact_5243_ordered__cancel__comm__monoid__diff__class_Oadd__diff__inverse,axiom,
% 0.34/0.64 ! [A2: nat,B: nat] :
% 0.34/0.64 ( ( ord_less_eq_nat @ A2 @ B )
% 0.34/0.64 => ( ( plus_plus_nat @ A2 @ ( minus_minus_nat @ B @ A2 ) )
% 0.34/0.64 = B ) ) ).
% 0.34/0.64
% 0.34/0.64 % ordered_cancel_comm_monoid_diff_class.add_diff_inverse
% 0.34/0.64 thf(fact_5244_ordered__cancel__comm__monoid__diff__class_Odiff__diff__right,axiom,
% 0.34/0.64 ! [A2: nat,B: nat,C: nat] :
% 0.34/0.64 ( ( ord_less_eq_nat @ A2 @ B )
% 0.34/0.64 => ( ( minus_minus_nat @ C @ ( minus_minus_nat @ B @ A2 ) )
% 0.34/0.64 = ( minus_minus_nat @ ( plus_plus_nat @ C @ A2 ) @ B ) ) ) ).
% 0.34/0.64
% 0.34/0.64 % ordered_cancel_comm_monoid_diff_class.diff_diff_right
% 0.34/0.64 thf(fact_5245_ordered__cancel__comm__monoid__diff__class_Odiff__add__assoc2,axiom,
% 0.34/0.64 ! [A2: nat,B: nat,C: nat] :
% 0.34/0.64 ( ( ord_less_eq_nat @ A2 @ B )
% 0.34/0.64 => ( ( minus_minus_nat @ ( plus_plus_nat @ B @ C ) @ A2 )
% 0.34/0.64 = ( plus_plus_nat @ ( minus_minus_nat @ B @ A2 ) @ C ) ) ) ).
% 0.34/0.64
% 0.34/0.64 % ordered_cancel_comm_monoid_diff_class.diff_add_assoc2
% 0.34/0.64 thf(fact_5246_ordered__cancel__comm__monoid__diff__class_Oadd__diff__assoc2,axiom,
% 0.34/0.64 ! [A2: nat,B: nat,C: nat] :
% 0.34/0.64 ( ( ord_less_eq_nat @ A2 @ B )
% 0.34/0.64 => ( ( plus_plus_nat @ ( minus_minus_nat @ B @ A2 ) @ C )
% 0.34/0.64 = ( minus_minus_nat @ ( plus_plus_nat @ B @ C ) @ A2 ) ) ) ).
% 0.34/0.64
% 0.34/0.64 % ordered_cancel_comm_monoid_diff_class.add_diff_assoc2
% 0.34/0.64 thf(fact_5247_ordered__cancel__comm__monoid__diff__class_Odiff__add__assoc,axiom,
% 0.34/0.64 ! [A2: nat,B: nat,C: nat] :
% 0.34/0.64 ( ( ord_less_eq_nat @ A2 @ B )
% 0.34/0.64 => ( ( minus_minus_nat @ ( plus_plus_nat @ C @ B ) @ A2 )
% 0.34/0.64 = ( plus_plus_nat @ C @ ( minus_minus_nat @ B @ A2 ) ) ) ) ).
% 0.34/0.64
% 0.34/0.64 % ordered_cancel_comm_monoid_diff_class.diff_add_assoc
% 0.34/0.64 thf(fact_5248_ordered__cancel__comm__monoid__diff__class_Oadd__diff__assoc,axiom,
% 0.34/0.64 ! [A2: nat,B: nat,C: nat] :
% 0.34/0.64 ( ( ord_less_eq_nat @ A2 @ B )
% 0.34/0.64 => ( ( plus_plus_nat @ C @ ( minus_minus_nat @ B @ A2 ) )
% 0.34/0.64 = ( minus_minus_nat @ ( plus_plus_nat @ C @ B ) @ A2 ) ) ) ).
% 0.34/0.64
% 0.34/0.64 % ordered_cancel_comm_monoid_diff_class.add_diff_assoc
% 0.34/0.64 thf(fact_5249_ordered__cancel__comm__monoid__diff__class_Ole__diff__conv2,axiom,
% 0.34/0.64 ! [A2: nat,B: nat,C: nat] :
% 0.34/0.64 ( ( ord_less_eq_nat @ A2 @ B )
% 0.34/0.64 => ( ( ord_less_eq_nat @ C @ ( minus_minus_nat @ B @ A2 ) )
% 0.34/0.64 = ( ord_less_eq_nat @ ( plus_plus_nat @ C @ A2 ) @ B ) ) ) ).
% 0.34/0.64
% 0.34/0.64 % ordered_cancel_comm_monoid_diff_class.le_diff_conv2
% 0.34/0.64 thf(fact_5250_le__add__diff,axiom,
% 0.34/0.64 ! [A2: nat,B: nat,C: nat] :
% 0.34/0.64 ( ( ord_less_eq_nat @ A2 @ B )
% 0.34/0.64 => ( ord_less_eq_nat @ C @ ( minus_minus_nat @ ( plus_plus_nat @ B @ C ) @ A2 ) ) ) ).
% 0.34/0.64
% 0.34/0.64 % le_add_diff
% 0.34/0.64 thf(fact_5251_ordered__cancel__comm__monoid__diff__class_Odiff__add,axiom,
% 0.34/0.64 ! [A2: nat,B: nat] :
% 0.34/0.64 ( ( ord_less_eq_nat @ A2 @ B )
% 0.34/0.64 => ( ( plus_plus_nat @ ( minus_minus_nat @ B @ A2 ) @ A2 )
% 0.34/0.64 = B ) ) ).
% 0.34/0.64
% 0.34/0.64 % ordered_cancel_comm_monoid_diff_class.diff_add
% 0.34/0.64 thf(fact_5252_le__diff__eq,axiom,
% 0.34/0.64 ! [A2: code_integer,C: code_integer,B: code_integer] :
% 0.34/0.64 ( ( ord_le3102999989581377725nteger @ A2 @ ( minus_8373710615458151222nteger @ C @ B ) )
% 0.34/0.64 = ( ord_le3102999989581377725nteger @ ( plus_p5714425477246183910nteger @ A2 @ B ) @ C ) ) ).
% 0.34/0.64
% 0.34/0.64 % le_diff_eq
% 0.34/0.64 thf(fact_5253_le__diff__eq,axiom,
% 0.34/0.64 ! [A2: rat,C: rat,B: rat] :
% 0.34/0.64 ( ( ord_less_eq_rat @ A2 @ ( minus_minus_rat @ C @ B ) )
% 0.34/0.64 = ( ord_less_eq_rat @ ( plus_plus_rat @ A2 @ B ) @ C ) ) ).
% 0.34/0.64
% 0.34/0.64 % le_diff_eq
% 0.34/0.64 thf(fact_5254_le__diff__eq,axiom,
% 0.34/0.64 ! [A2: int,C: int,B: int] :
% 0.34/0.64 ( ( ord_less_eq_int @ A2 @ ( minus_minus_int @ C @ B ) )
% 0.34/0.64 = ( ord_less_eq_int @ ( plus_plus_int @ A2 @ B ) @ C ) ) ).
% 0.34/0.64
% 0.34/0.64 % le_diff_eq
% 0.34/0.64 thf(fact_5255_diff__le__eq,axiom,
% 0.34/0.64 ! [A2: code_integer,B: code_integer,C: code_integer] :
% 0.34/0.64 ( ( ord_le3102999989581377725nteger @ ( minus_8373710615458151222nteger @ A2 @ B ) @ C )
% 0.34/0.64 = ( ord_le3102999989581377725nteger @ A2 @ ( plus_p5714425477246183910nteger @ C @ B ) ) ) ).
% 0.34/0.64
% 0.34/0.64 % diff_le_eq
% 0.34/0.64 thf(fact_5256_diff__le__eq,axiom,
% 0.34/0.64 ! [A2: rat,B: rat,C: rat] :
% 0.34/0.64 ( ( ord_less_eq_rat @ ( minus_minus_rat @ A2 @ B ) @ C )
% 0.34/0.64 = ( ord_less_eq_rat @ A2 @ ( plus_plus_rat @ C @ B ) ) ) ).
% 0.34/0.64
% 0.34/0.64 % diff_le_eq
% 0.34/0.64 thf(fact_5257_diff__le__eq,axiom,
% 0.34/0.64 ! [A2: int,B: int,C: int] :
% 0.34/0.64 ( ( ord_less_eq_int @ ( minus_minus_int @ A2 @ B ) @ C )
% 0.34/0.64 = ( ord_less_eq_int @ A2 @ ( plus_plus_int @ C @ B ) ) ) ).
% 0.34/0.64
% 0.34/0.64 % diff_le_eq
% 0.34/0.64 thf(fact_5258_less__add__one,axiom,
% 0.34/0.64 ! [A2: code_natural] : ( ord_le5570908160329646204atural @ A2 @ ( plus_p4538020629002901425atural @ A2 @ one_one_Code_natural ) ) ).
% 0.34/0.64
% 0.34/0.64 % less_add_one
% 0.34/0.64 thf(fact_5259_less__add__one,axiom,
% 0.34/0.64 ! [A2: nat] : ( ord_less_nat @ A2 @ ( plus_plus_nat @ A2 @ one_one_nat ) ) ).
% 0.34/0.64
% 0.34/0.64 % less_add_one
% 0.34/0.64 thf(fact_5260_less__add__one,axiom,
% 0.34/0.64 ! [A2: int] : ( ord_less_int @ A2 @ ( plus_plus_int @ A2 @ one_one_int ) ) ).
% 0.34/0.64
% 0.34/0.64 % less_add_one
% 0.34/0.64 thf(fact_5261_less__add__one,axiom,
% 0.34/0.64 ! [A2: code_integer] : ( ord_le6747313008572928689nteger @ A2 @ ( plus_p5714425477246183910nteger @ A2 @ one_one_Code_integer ) ) ).
% 0.34/0.64
% 0.34/0.64 % less_add_one
% 0.34/0.64 thf(fact_5262_less__add__one,axiom,
% 0.34/0.64 ! [A2: rat] : ( ord_less_rat @ A2 @ ( plus_plus_rat @ A2 @ one_one_rat ) ) ).
% 0.34/0.64
% 0.34/0.64 % less_add_one
% 0.34/0.64 thf(fact_5263_add__mono1,axiom,
% 0.34/0.64 ! [A2: code_natural,B: code_natural] :
% 0.34/0.64 ( ( ord_le5570908160329646204atural @ A2 @ B )
% 0.34/0.64 => ( ord_le5570908160329646204atural @ ( plus_p4538020629002901425atural @ A2 @ one_one_Code_natural ) @ ( plus_p4538020629002901425atural @ B @ one_one_Code_natural ) ) ) ).
% 0.34/0.64
% 0.34/0.64 % add_mono1
% 0.34/0.64 thf(fact_5264_add__mono1,axiom,
% 0.34/0.64 ! [A2: nat,B: nat] :
% 0.34/0.64 ( ( ord_less_nat @ A2 @ B )
% 0.34/0.64 => ( ord_less_nat @ ( plus_plus_nat @ A2 @ one_one_nat ) @ ( plus_plus_nat @ B @ one_one_nat ) ) ) ).
% 0.34/0.64
% 0.34/0.64 % add_mono1
% 0.34/0.64 thf(fact_5265_add__mono1,axiom,
% 0.34/0.64 ! [A2: int,B: int] :
% 0.34/0.64 ( ( ord_less_int @ A2 @ B )
% 0.34/0.64 => ( ord_less_int @ ( plus_plus_int @ A2 @ one_one_int ) @ ( plus_plus_int @ B @ one_one_int ) ) ) ).
% 0.34/0.64
% 0.34/0.64 % add_mono1
% 0.34/0.64 thf(fact_5266_add__mono1,axiom,
% 0.34/0.64 ! [A2: code_integer,B: code_integer] :
% 0.34/0.64 ( ( ord_le6747313008572928689nteger @ A2 @ B )
% 0.34/0.64 => ( ord_le6747313008572928689nteger @ ( plus_p5714425477246183910nteger @ A2 @ one_one_Code_integer ) @ ( plus_p5714425477246183910nteger @ B @ one_one_Code_integer ) ) ) ).
% 0.34/0.64
% 0.34/0.64 % add_mono1
% 0.34/0.64 thf(fact_5267_add__mono1,axiom,
% 0.34/0.64 ! [A2: rat,B: rat] :
% 0.34/0.64 ( ( ord_less_rat @ A2 @ B )
% 0.34/0.64 => ( ord_less_rat @ ( plus_plus_rat @ A2 @ one_one_rat ) @ ( plus_plus_rat @ B @ one_one_rat ) ) ) ).
% 0.34/0.64
% 0.34/0.64 % add_mono1
% 0.34/0.64 thf(fact_5268_diff__less__eq,axiom,
% 0.34/0.64 ! [A2: int,B: int,C: int] :
% 0.34/0.64 ( ( ord_less_int @ ( minus_minus_int @ A2 @ B ) @ C )
% 0.34/0.64 = ( ord_less_int @ A2 @ ( plus_plus_int @ C @ B ) ) ) ).
% 0.34/0.64
% 0.34/0.64 % diff_less_eq
% 0.34/0.64 thf(fact_5269_diff__less__eq,axiom,
% 0.34/0.64 ! [A2: code_integer,B: code_integer,C: code_integer] :
% 0.34/0.64 ( ( ord_le6747313008572928689nteger @ ( minus_8373710615458151222nteger @ A2 @ B ) @ C )
% 0.34/0.64 = ( ord_le6747313008572928689nteger @ A2 @ ( plus_p5714425477246183910nteger @ C @ B ) ) ) ).
% 0.34/0.64
% 0.34/0.64 % diff_less_eq
% 0.34/0.64 thf(fact_5270_diff__less__eq,axiom,
% 0.34/0.64 ! [A2: rat,B: rat,C: rat] :
% 0.34/0.64 ( ( ord_less_rat @ ( minus_minus_rat @ A2 @ B ) @ C )
% 0.34/0.64 = ( ord_less_rat @ A2 @ ( plus_plus_rat @ C @ B ) ) ) ).
% 0.34/0.64
% 0.34/0.64 % diff_less_eq
% 0.34/0.64 thf(fact_5271_less__diff__eq,axiom,
% 0.34/0.64 ! [A2: int,C: int,B: int] :
% 0.34/0.64 ( ( ord_less_int @ A2 @ ( minus_minus_int @ C @ B ) )
% 0.34/0.64 = ( ord_less_int @ ( plus_plus_int @ A2 @ B ) @ C ) ) ).
% 0.34/0.64
% 0.34/0.64 % less_diff_eq
% 0.34/0.64 thf(fact_5272_less__diff__eq,axiom,
% 0.34/0.64 ! [A2: code_integer,C: code_integer,B: code_integer] :
% 0.34/0.64 ( ( ord_le6747313008572928689nteger @ A2 @ ( minus_8373710615458151222nteger @ C @ B ) )
% 0.34/0.64 = ( ord_le6747313008572928689nteger @ ( plus_p5714425477246183910nteger @ A2 @ B ) @ C ) ) ).
% 0.34/0.64
% 0.34/0.64 % less_diff_eq
% 0.34/0.64 thf(fact_5273_less__diff__eq,axiom,
% 0.34/0.64 ! [A2: rat,C: rat,B: rat] :
% 0.34/0.64 ( ( ord_less_rat @ A2 @ ( minus_minus_rat @ C @ B ) )
% 0.34/0.64 = ( ord_less_rat @ ( plus_plus_rat @ A2 @ B ) @ C ) ) ).
% 0.34/0.64
% 0.34/0.64 % less_diff_eq
% 0.34/0.64 thf(fact_5274_square__diff__square__factored,axiom,
% 0.34/0.64 ! [X: code_integer,Y: code_integer] :
% 0.34/0.64 ( ( minus_8373710615458151222nteger @ ( times_3573771949741848930nteger @ X @ X ) @ ( times_3573771949741848930nteger @ Y @ Y ) )
% 0.34/0.64 = ( times_3573771949741848930nteger @ ( plus_p5714425477246183910nteger @ X @ Y ) @ ( minus_8373710615458151222nteger @ X @ Y ) ) ) ).
% 0.34/0.64
% 0.34/0.64 % square_diff_square_factored
% 0.34/0.64 thf(fact_5275_square__diff__square__factored,axiom,
% 0.34/0.64 ! [X: rat,Y: rat] :
% 0.34/0.64 ( ( minus_minus_rat @ ( times_times_rat @ X @ X ) @ ( times_times_rat @ Y @ Y ) )
% 0.34/0.64 = ( times_times_rat @ ( plus_plus_rat @ X @ Y ) @ ( minus_minus_rat @ X @ Y ) ) ) ).
% 0.34/0.64
% 0.34/0.64 % square_diff_square_factored
% 0.34/0.64 thf(fact_5276_square__diff__square__factored,axiom,
% 0.34/0.64 ! [X: int,Y: int] :
% 0.34/0.64 ( ( minus_minus_int @ ( times_times_int @ X @ X ) @ ( times_times_int @ Y @ Y ) )
% 0.34/0.64 = ( times_times_int @ ( plus_plus_int @ X @ Y ) @ ( minus_minus_int @ X @ Y ) ) ) ).
% 0.34/0.64
% 0.34/0.64 % square_diff_square_factored
% 0.34/0.64 thf(fact_5277_eq__add__iff2,axiom,
% 0.34/0.64 ! [A2: code_integer,E2: code_integer,C: code_integer,B: code_integer,D2: code_integer] :
% 0.34/0.64 ( ( ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ A2 @ E2 ) @ C )
% 0.34/0.64 = ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ B @ E2 ) @ D2 ) )
% 0.34/0.64 = ( C
% 0.34/0.64 = ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ ( minus_8373710615458151222nteger @ B @ A2 ) @ E2 ) @ D2 ) ) ) ).
% 0.34/0.64
% 0.34/0.64 % eq_add_iff2
% 0.34/0.64 thf(fact_5278_eq__add__iff2,axiom,
% 0.34/0.64 ! [A2: rat,E2: rat,C: rat,B: rat,D2: rat] :
% 0.34/0.64 ( ( ( plus_plus_rat @ ( times_times_rat @ A2 @ E2 ) @ C )
% 0.34/0.64 = ( plus_plus_rat @ ( times_times_rat @ B @ E2 ) @ D2 ) )
% 0.34/0.64 = ( C
% 0.34/0.64 = ( plus_plus_rat @ ( times_times_rat @ ( minus_minus_rat @ B @ A2 ) @ E2 ) @ D2 ) ) ) ).
% 0.34/0.64
% 0.34/0.64 % eq_add_iff2
% 0.34/0.64 thf(fact_5279_eq__add__iff2,axiom,
% 0.34/0.64 ! [A2: int,E2: int,C: int,B: int,D2: int] :
% 0.34/0.64 ( ( ( plus_plus_int @ ( times_times_int @ A2 @ E2 ) @ C )
% 0.34/0.64 = ( plus_plus_int @ ( times_times_int @ B @ E2 ) @ D2 ) )
% 0.34/0.64 = ( C
% 0.34/0.64 = ( plus_plus_int @ ( times_times_int @ ( minus_minus_int @ B @ A2 ) @ E2 ) @ D2 ) ) ) ).
% 0.34/0.64
% 0.34/0.64 % eq_add_iff2
% 0.34/0.64 thf(fact_5280_eq__add__iff1,axiom,
% 0.34/0.64 ! [A2: code_integer,E2: code_integer,C: code_integer,B: code_integer,D2: code_integer] :
% 0.34/0.64 ( ( ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ A2 @ E2 ) @ C )
% 0.34/0.64 = ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ B @ E2 ) @ D2 ) )
% 0.34/0.64 = ( ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ ( minus_8373710615458151222nteger @ A2 @ B ) @ E2 ) @ C )
% 0.34/0.64 = D2 ) ) ).
% 0.34/0.64
% 0.34/0.64 % eq_add_iff1
% 0.34/0.64 thf(fact_5281_eq__add__iff1,axiom,
% 0.34/0.64 ! [A2: rat,E2: rat,C: rat,B: rat,D2: rat] :
% 0.34/0.64 ( ( ( plus_plus_rat @ ( times_times_rat @ A2 @ E2 ) @ C )
% 0.34/0.64 = ( plus_plus_rat @ ( times_times_rat @ B @ E2 ) @ D2 ) )
% 0.34/0.64 = ( ( plus_plus_rat @ ( times_times_rat @ ( minus_minus_rat @ A2 @ B ) @ E2 ) @ C )
% 0.34/0.64 = D2 ) ) ).
% 0.34/0.64
% 0.34/0.64 % eq_add_iff1
% 0.34/0.64 thf(fact_5282_eq__add__iff1,axiom,
% 0.34/0.64 ! [A2: int,E2: int,C: int,B: int,D2: int] :
% 0.34/0.64 ( ( ( plus_plus_int @ ( times_times_int @ A2 @ E2 ) @ C )
% 0.34/0.64 = ( plus_plus_int @ ( times_times_int @ B @ E2 ) @ D2 ) )
% 0.34/0.64 = ( ( plus_plus_int @ ( times_times_int @ ( minus_minus_int @ A2 @ B ) @ E2 ) @ C )
% 0.34/0.64 = D2 ) ) ).
% 0.34/0.64
% 0.34/0.64 % eq_add_iff1
% 0.34/0.64 thf(fact_5283_Suc__n__minus__m__eq,axiom,
% 0.34/0.64 ! [M4: nat,N2: nat] :
% 0.34/0.64 ( ( ord_less_eq_nat @ M4 @ N2 )
% 0.34/0.64 => ( ( ord_less_nat @ one_one_nat @ M4 )
% 0.34/0.64 => ( ( suc @ ( minus_minus_nat @ N2 @ M4 ) )
% 0.34/0.64 = ( minus_minus_nat @ N2 @ ( minus_minus_nat @ M4 @ one_one_nat ) ) ) ) ) ).
% 0.34/0.64
% 0.34/0.64 % Suc_n_minus_m_eq
% 0.34/0.64 thf(fact_5284_nat__in__between__eq_I1_J,axiom,
% 0.34/0.64 ! [A2: nat,B: nat] :
% 0.34/0.64 ( ( ( ord_less_nat @ A2 @ B )
% 0.34/0.64 & ( ord_less_eq_nat @ B @ ( suc @ A2 ) ) )
% 0.34/0.64 = ( B
% 0.34/0.64 = ( suc @ A2 ) ) ) ).
% 0.34/0.64
% 0.34/0.64 % nat_in_between_eq(1)
% 0.34/0.64 thf(fact_5285_nat__in__between__eq_I2_J,axiom,
% 0.34/0.64 ! [A2: nat,B: nat] :
% 0.34/0.64 ( ( ( ord_less_eq_nat @ A2 @ B )
% 0.34/0.64 & ( ord_less_nat @ B @ ( suc @ A2 ) ) )
% 0.34/0.64 = ( B = A2 ) ) ).
% 0.34/0.64
% 0.34/0.64 % nat_in_between_eq(2)
% 0.34/0.64 thf(fact_5286_wf__no__infinite__down__chainE,axiom,
% 0.34/0.64 ! [R2: set_Pr1261947904930325089at_nat,F: nat > nat] :
% 0.34/0.64 ( ( wf_nat @ R2 )
% 0.34/0.64 => ~ ! [K: nat] : ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ ( F @ ( suc @ K ) ) @ ( F @ K ) ) @ R2 ) ) ).
% 0.34/0.64
% 0.34/0.64 % wf_no_infinite_down_chainE
% 0.34/0.64 thf(fact_5287_wf__no__infinite__down__chainE,axiom,
% 0.34/0.64 ! [R2: set_Pr958786334691620121nt_int,F: nat > int] :
% 0.34/0.64 ( ( wf_int @ R2 )
% 0.34/0.64 => ~ ! [K: nat] : ( member5262025264175285858nt_int @ ( product_Pair_int_int @ ( F @ ( suc @ K ) ) @ ( F @ K ) ) @ R2 ) ) ).
% 0.34/0.64
% 0.34/0.64 % wf_no_infinite_down_chainE
% 0.34/0.64 thf(fact_5288_wf__iff__no__infinite__down__chain,axiom,
% 0.34/0.64 ( wf_nat
% 0.34/0.64 = ( ^ [R4: set_Pr1261947904930325089at_nat] :
% 0.34/0.64 ~ ? [F4: nat > nat] :
% 0.34/0.64 ! [I4: nat] : ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ ( F4 @ ( suc @ I4 ) ) @ ( F4 @ I4 ) ) @ R4 ) ) ) ).
% 0.34/0.64
% 0.34/0.64 % wf_iff_no_infinite_down_chain
% 0.34/0.64 thf(fact_5289_wf__iff__no__infinite__down__chain,axiom,
% 0.34/0.64 ( wf_int
% 0.34/0.64 = ( ^ [R4: set_Pr958786334691620121nt_int] :
% 0.34/0.64 ~ ? [F4: nat > int] :
% 0.34/0.64 ! [I4: nat] : ( member5262025264175285858nt_int @ ( product_Pair_int_int @ ( F4 @ ( suc @ I4 ) ) @ ( F4 @ I4 ) ) @ R4 ) ) ) ).
% 0.34/0.64
% 0.34/0.64 % wf_iff_no_infinite_down_chain
% 0.34/0.64 thf(fact_5290_Inf__INT__eq,axiom,
% 0.34/0.64 ( comple4605987646318907561at_o_o
% 0.34/0.64 = ( ^ [S6: set_Pr6409761941060870310at_o_o,X2: produc3658429121746597890et_nat > $o] : ( member6576561426505652726_nat_o @ X2 @ ( comple79274577958421812_nat_o @ ( image_8639947063239330806_nat_o @ collec939566748876313656_nat_o @ S6 ) ) ) ) ) ).
% 0.34/0.64
% 0.34/0.64 % Inf_INT_eq
% 0.34/0.64 thf(fact_5291_Inf__INT__eq,axiom,
% 0.34/0.64 ( comple4393433247856107946_int_o
% 0.34/0.64 = ( ^ [S6: set_int_o,X2: int] : ( member_int @ X2 @ ( comple3628384868704368283et_int @ ( image_int_o_set_int @ collect_int @ S6 ) ) ) ) ) ).
% 0.34/0.64
% 0.34/0.64 % Inf_INT_eq
% 0.34/0.64 thf(fact_5292_Inf__INT__eq,axiom,
% 0.34/0.64 ( complete_Inf_Inf_o_o
% 0.34/0.64 = ( ^ [S6: set_o_o,X2: $o] : ( member_o @ X2 @ ( comple3063163877087187839_set_o @ ( image_o_o_set_o @ collect_o @ S6 ) ) ) ) ) ).
% 0.34/0.64
% 0.34/0.64 % Inf_INT_eq
% 0.34/0.64 thf(fact_5293_Inf__INT__eq,axiom,
% 0.34/0.64 ( comple2092403528524206185unit_o
% 0.34/0.64 = ( ^ [S6: set_Product_unit_o,X2: product_unit] : ( member_Product_unit @ X2 @ ( comple7216939928495492468t_unit @ ( image_5138372075374782070t_unit @ collect_Product_unit @ S6 ) ) ) ) ) ).
% 0.34/0.64
% 0.34/0.64 % Inf_INT_eq
% 0.34/0.64 thf(fact_5294_Inf__INT__eq,axiom,
% 0.34/0.64 ( comple6214475593288795910_nat_o
% 0.34/0.64 = ( ^ [S6: set_nat_o,X2: nat] : ( member_nat @ X2 @ ( comple7806235888213564991et_nat @ ( image_nat_o_set_nat @ collect_nat @ S6 ) ) ) ) ) ).
% 0.34/0.64
% 0.34/0.64 % Inf_INT_eq
% 0.34/0.64 thf(fact_5295_le__add__iff2,axiom,
% 0.34/0.64 ! [A2: code_integer,E2: code_integer,C: code_integer,B: code_integer,D2: code_integer] :
% 0.34/0.64 ( ( ord_le3102999989581377725nteger @ ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ A2 @ E2 ) @ C ) @ ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ B @ E2 ) @ D2 ) )
% 0.34/0.64 = ( ord_le3102999989581377725nteger @ C @ ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ ( minus_8373710615458151222nteger @ B @ A2 ) @ E2 ) @ D2 ) ) ) ).
% 0.34/0.64
% 0.34/0.64 % le_add_iff2
% 0.34/0.64 thf(fact_5296_le__add__iff2,axiom,
% 0.34/0.64 ! [A2: rat,E2: rat,C: rat,B: rat,D2: rat] :
% 0.34/0.64 ( ( ord_less_eq_rat @ ( plus_plus_rat @ ( times_times_rat @ A2 @ E2 ) @ C ) @ ( plus_plus_rat @ ( times_times_rat @ B @ E2 ) @ D2 ) )
% 0.34/0.64 = ( ord_less_eq_rat @ C @ ( plus_plus_rat @ ( times_times_rat @ ( minus_minus_rat @ B @ A2 ) @ E2 ) @ D2 ) ) ) ).
% 0.34/0.64
% 0.34/0.64 % le_add_iff2
% 0.34/0.64 thf(fact_5297_le__add__iff2,axiom,
% 0.34/0.64 ! [A2: int,E2: int,C: int,B: int,D2: int] :
% 0.34/0.64 ( ( ord_less_eq_int @ ( plus_plus_int @ ( times_times_int @ A2 @ E2 ) @ C ) @ ( plus_plus_int @ ( times_times_int @ B @ E2 ) @ D2 ) )
% 0.34/0.64 = ( ord_less_eq_int @ C @ ( plus_plus_int @ ( times_times_int @ ( minus_minus_int @ B @ A2 ) @ E2 ) @ D2 ) ) ) ).
% 0.34/0.64
% 0.34/0.64 % le_add_iff2
% 0.34/0.64 thf(fact_5298_le__add__iff1,axiom,
% 0.34/0.64 ! [A2: code_integer,E2: code_integer,C: code_integer,B: code_integer,D2: code_integer] :
% 0.34/0.64 ( ( ord_le3102999989581377725nteger @ ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ A2 @ E2 ) @ C ) @ ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ B @ E2 ) @ D2 ) )
% 0.34/0.64 = ( ord_le3102999989581377725nteger @ ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ ( minus_8373710615458151222nteger @ A2 @ B ) @ E2 ) @ C ) @ D2 ) ) ).
% 0.34/0.64
% 0.34/0.64 % le_add_iff1
% 0.34/0.64 thf(fact_5299_le__add__iff1,axiom,
% 0.34/0.64 ! [A2: rat,E2: rat,C: rat,B: rat,D2: rat] :
% 0.34/0.64 ( ( ord_less_eq_rat @ ( plus_plus_rat @ ( times_times_rat @ A2 @ E2 ) @ C ) @ ( plus_plus_rat @ ( times_times_rat @ B @ E2 ) @ D2 ) )
% 0.34/0.64 = ( ord_less_eq_rat @ ( plus_plus_rat @ ( times_times_rat @ ( minus_minus_rat @ A2 @ B ) @ E2 ) @ C ) @ D2 ) ) ).
% 0.34/0.64
% 0.34/0.64 % le_add_iff1
% 0.34/0.64 thf(fact_5300_le__add__iff1,axiom,
% 0.34/0.64 ! [A2: int,E2: int,C: int,B: int,D2: int] :
% 0.34/0.64 ( ( ord_less_eq_int @ ( plus_plus_int @ ( times_times_int @ A2 @ E2 ) @ C ) @ ( plus_plus_int @ ( times_times_int @ B @ E2 ) @ D2 ) )
% 0.34/0.64 = ( ord_less_eq_int @ ( plus_plus_int @ ( times_times_int @ ( minus_minus_int @ A2 @ B ) @ E2 ) @ C ) @ D2 ) ) ).
% 0.34/0.64
% 0.34/0.64 % le_add_iff1
% 0.34/0.64 thf(fact_5301_less__add__iff2,axiom,
% 0.34/0.64 ! [A2: int,E2: int,C: int,B: int,D2: int] :
% 0.34/0.64 ( ( ord_less_int @ ( plus_plus_int @ ( times_times_int @ A2 @ E2 ) @ C ) @ ( plus_plus_int @ ( times_times_int @ B @ E2 ) @ D2 ) )
% 0.34/0.64 = ( ord_less_int @ C @ ( plus_plus_int @ ( times_times_int @ ( minus_minus_int @ B @ A2 ) @ E2 ) @ D2 ) ) ) ).
% 0.34/0.64
% 0.34/0.64 % less_add_iff2
% 0.34/0.64 thf(fact_5302_less__add__iff2,axiom,
% 0.34/0.64 ! [A2: code_integer,E2: code_integer,C: code_integer,B: code_integer,D2: code_integer] :
% 0.34/0.64 ( ( ord_le6747313008572928689nteger @ ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ A2 @ E2 ) @ C ) @ ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ B @ E2 ) @ D2 ) )
% 0.34/0.64 = ( ord_le6747313008572928689nteger @ C @ ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ ( minus_8373710615458151222nteger @ B @ A2 ) @ E2 ) @ D2 ) ) ) ).
% 0.34/0.64
% 0.34/0.64 % less_add_iff2
% 0.34/0.64 thf(fact_5303_less__add__iff2,axiom,
% 0.34/0.64 ! [A2: rat,E2: rat,C: rat,B: rat,D2: rat] :
% 0.34/0.64 ( ( ord_less_rat @ ( plus_plus_rat @ ( times_times_rat @ A2 @ E2 ) @ C ) @ ( plus_plus_rat @ ( times_times_rat @ B @ E2 ) @ D2 ) )
% 0.34/0.64 = ( ord_less_rat @ C @ ( plus_plus_rat @ ( times_times_rat @ ( minus_minus_rat @ B @ A2 ) @ E2 ) @ D2 ) ) ) ).
% 0.34/0.64
% 0.34/0.64 % less_add_iff2
% 0.34/0.64 thf(fact_5304_less__add__iff1,axiom,
% 0.34/0.64 ! [A2: int,E2: int,C: int,B: int,D2: int] :
% 0.34/0.64 ( ( ord_less_int @ ( plus_plus_int @ ( times_times_int @ A2 @ E2 ) @ C ) @ ( plus_plus_int @ ( times_times_int @ B @ E2 ) @ D2 ) )
% 0.34/0.64 = ( ord_less_int @ ( plus_plus_int @ ( times_times_int @ ( minus_minus_int @ A2 @ B ) @ E2 ) @ C ) @ D2 ) ) ).
% 0.34/0.64
% 0.34/0.64 % less_add_iff1
% 0.34/0.64 thf(fact_5305_less__add__iff1,axiom,
% 0.34/0.64 ! [A2: code_integer,E2: code_integer,C: code_integer,B: code_integer,D2: code_integer] :
% 0.34/0.64 ( ( ord_le6747313008572928689nteger @ ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ A2 @ E2 ) @ C ) @ ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ B @ E2 ) @ D2 ) )
% 0.34/0.64 = ( ord_le6747313008572928689nteger @ ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ ( minus_8373710615458151222nteger @ A2 @ B ) @ E2 ) @ C ) @ D2 ) ) ).
% 0.34/0.64
% 0.34/0.64 % less_add_iff1
% 0.34/0.64 thf(fact_5306_less__add__iff1,axiom,
% 0.34/0.64 ! [A2: rat,E2: rat,C: rat,B: rat,D2: rat] :
% 0.34/0.64 ( ( ord_less_rat @ ( plus_plus_rat @ ( times_times_rat @ A2 @ E2 ) @ C ) @ ( plus_plus_rat @ ( times_times_rat @ B @ E2 ) @ D2 ) )
% 0.34/0.64 = ( ord_less_rat @ ( plus_plus_rat @ ( times_times_rat @ ( minus_minus_rat @ A2 @ B ) @ E2 ) @ C ) @ D2 ) ) ).
% 0.34/0.64
% 0.34/0.64 % less_add_iff1
% 0.34/0.64 thf(fact_5307_square__diff__one__factored,axiom,
% 0.34/0.64 ! [X: code_integer] :
% 0.34/0.64 ( ( minus_8373710615458151222nteger @ ( times_3573771949741848930nteger @ X @ X ) @ one_one_Code_integer )
% 0.34/0.64 = ( times_3573771949741848930nteger @ ( plus_p5714425477246183910nteger @ X @ one_one_Code_integer ) @ ( minus_8373710615458151222nteger @ X @ one_one_Code_integer ) ) ) ).
% 0.34/0.64
% 0.34/0.64 % square_diff_one_factored
% 0.34/0.64 thf(fact_5308_square__diff__one__factored,axiom,
% 0.34/0.64 ! [X: rat] :
% 0.34/0.64 ( ( minus_minus_rat @ ( times_times_rat @ X @ X ) @ one_one_rat )
% 0.34/0.64 = ( times_times_rat @ ( plus_plus_rat @ X @ one_one_rat ) @ ( minus_minus_rat @ X @ one_one_rat ) ) ) ).
% 0.34/0.64
% 0.34/0.64 % square_diff_one_factored
% 0.34/0.64 thf(fact_5309_square__diff__one__factored,axiom,
% 0.34/0.64 ! [X: int] :
% 0.34/0.64 ( ( minus_minus_int @ ( times_times_int @ X @ X ) @ one_one_int )
% 0.34/0.64 = ( times_times_int @ ( plus_plus_int @ X @ one_one_int ) @ ( minus_minus_int @ X @ one_one_int ) ) ) ).
% 0.34/0.64
% 0.34/0.64 % square_diff_one_factored
% 0.34/0.64 thf(fact_5310_card__Suc__eq__finite,axiom,
% 0.34/0.64 ! [A: set_Pr4532377907799695533_nat_o,K3: nat] :
% 0.34/0.64 ( ( ( finite1363419556375932405_nat_o @ A )
% 0.34/0.64 = ( suc @ K3 ) )
% 0.34/0.64 = ( ? [B6: produc3658429121746597890et_nat > $o,B7: set_Pr4532377907799695533_nat_o] :
% 0.34/0.64 ( ( A
% 0.34/0.64 = ( insert5175938949040314269_nat_o @ B6 @ B7 ) )
% 0.34/0.64 & ~ ( member6576561426505652726_nat_o @ B6 @ B7 )
% 0.34/0.64 & ( ( finite1363419556375932405_nat_o @ B7 )
% 0.34/0.64 = K3 )
% 0.34/0.64 & ( finite3252695134891459830_nat_o @ B7 ) ) ) ) ).
% 0.34/0.64
% 0.34/0.64 % card_Suc_eq_finite
% 0.34/0.64 thf(fact_5311_card__Suc__eq__finite,axiom,
% 0.34/0.64 ! [A: set_int,K3: nat] :
% 0.34/0.64 ( ( ( finite_card_int @ A )
% 0.34/0.64 = ( suc @ K3 ) )
% 0.34/0.64 = ( ? [B6: int,B7: set_int] :
% 0.34/0.64 ( ( A
% 0.34/0.64 = ( insert_int @ B6 @ B7 ) )
% 0.34/0.64 & ~ ( member_int @ B6 @ B7 )
% 0.34/0.64 & ( ( finite_card_int @ B7 )
% 0.34/0.64 = K3 )
% 0.34/0.64 & ( finite_finite_int @ B7 ) ) ) ) ).
% 0.34/0.64
% 0.34/0.64 % card_Suc_eq_finite
% 0.34/0.64 thf(fact_5312_card__Suc__eq__finite,axiom,
% 0.34/0.64 ! [A: set_o,K3: nat] :
% 0.34/0.64 ( ( ( finite_card_o @ A )
% 0.34/0.64 = ( suc @ K3 ) )
% 0.34/0.64 = ( ? [B6: $o,B7: set_o] :
% 0.34/0.64 ( ( A
% 0.34/0.64 = ( insert_o @ B6 @ B7 ) )
% 0.34/0.64 & ~ ( member_o @ B6 @ B7 )
% 0.34/0.64 & ( ( finite_card_o @ B7 )
% 0.34/0.64 = K3 )
% 0.34/0.64 & ( finite_finite_o @ B7 ) ) ) ) ).
% 0.34/0.64
% 0.34/0.64 % card_Suc_eq_finite
% 0.34/0.64 thf(fact_5313_card__Suc__eq__finite,axiom,
% 0.34/0.64 ! [A: set_Product_unit,K3: nat] :
% 0.34/0.64 ( ( ( finite410649719033368117t_unit @ A )
% 0.34/0.64 = ( suc @ K3 ) )
% 0.34/0.64 = ( ? [B6: product_unit,B7: set_Product_unit] :
% 0.34/0.64 ( ( A
% 0.34/0.64 = ( insert_Product_unit @ B6 @ B7 ) )
% 0.34/0.64 & ~ ( member_Product_unit @ B6 @ B7 )
% 0.34/0.64 & ( ( finite410649719033368117t_unit @ B7 )
% 0.34/0.64 = K3 )
% 0.34/0.64 & ( finite4290736615968046902t_unit @ B7 ) ) ) ) ).
% 0.34/0.64
% 0.34/0.64 % card_Suc_eq_finite
% 0.34/0.64 thf(fact_5314_card__Suc__eq__finite,axiom,
% 0.34/0.64 ! [A: set_nat,K3: nat] :
% 0.34/0.64 ( ( ( finite_card_nat @ A )
% 0.34/0.64 = ( suc @ K3 ) )
% 0.34/0.64 = ( ? [B6: nat,B7: set_nat] :
% 0.34/0.64 ( ( A
% 0.34/0.64 = ( insert_nat @ B6 @ B7 ) )
% 0.34/0.64 & ~ ( member_nat @ B6 @ B7 )
% 0.34/0.64 & ( ( finite_card_nat @ B7 )
% 0.34/0.64 = K3 )
% 0.34/0.64 & ( finite_finite_nat @ B7 ) ) ) ) ).
% 0.34/0.64
% 0.34/0.64 % card_Suc_eq_finite
% 0.34/0.64 thf(fact_5315_card__Suc__eq__finite,axiom,
% 0.34/0.64 ! [A: set_Code_integer,K3: nat] :
% 0.34/0.64 ( ( ( finite4902975817058060853nteger @ A )
% 0.34/0.64 = ( suc @ K3 ) )
% 0.34/0.64 = ( ? [B6: code_integer,B7: set_Code_integer] :
% 0.34/0.64 ( ( A
% 0.34/0.64 = ( insert_Code_integer @ B6 @ B7 ) )
% 0.34/0.64 & ~ ( member_Code_integer @ B6 @ B7 )
% 0.34/0.64 & ( ( finite4902975817058060853nteger @ B7 )
% 0.34/0.64 = K3 )
% 0.34/0.64 & ( finite6017078050557962740nteger @ B7 ) ) ) ) ).
% 0.34/0.64
% 0.34/0.64 % card_Suc_eq_finite
% 0.34/0.64 thf(fact_5316_card__insert__if,axiom,
% 0.34/0.64 ! [A: set_Pr4532377907799695533_nat_o,X: produc3658429121746597890et_nat > $o] :
% 0.34/0.64 ( ( finite3252695134891459830_nat_o @ A )
% 0.34/0.64 => ( ( ( member6576561426505652726_nat_o @ X @ A )
% 0.34/0.64 => ( ( finite1363419556375932405_nat_o @ ( insert5175938949040314269_nat_o @ X @ A ) )
% 0.34/0.64 = ( finite1363419556375932405_nat_o @ A ) ) )
% 0.34/0.64 & ( ~ ( member6576561426505652726_nat_o @ X @ A )
% 0.34/0.64 => ( ( finite1363419556375932405_nat_o @ ( insert5175938949040314269_nat_o @ X @ A ) )
% 0.34/0.64 = ( suc @ ( finite1363419556375932405_nat_o @ A ) ) ) ) ) ) ).
% 0.34/0.64
% 0.34/0.64 % card_insert_if
% 0.34/0.64 thf(fact_5317_card__insert__if,axiom,
% 0.34/0.64 ! [A: set_int,X: int] :
% 0.34/0.64 ( ( finite_finite_int @ A )
% 0.34/0.64 => ( ( ( member_int @ X @ A )
% 0.34/0.64 => ( ( finite_card_int @ ( insert_int @ X @ A ) )
% 0.34/0.64 = ( finite_card_int @ A ) ) )
% 0.34/0.64 & ( ~ ( member_int @ X @ A )
% 0.34/0.64 => ( ( finite_card_int @ ( insert_int @ X @ A ) )
% 0.34/0.64 = ( suc @ ( finite_card_int @ A ) ) ) ) ) ) ).
% 0.34/0.64
% 0.34/0.64 % card_insert_if
% 0.34/0.64 thf(fact_5318_card__insert__if,axiom,
% 0.34/0.64 ! [A: set_o,X: $o] :
% 0.34/0.64 ( ( finite_finite_o @ A )
% 0.34/0.64 => ( ( ( member_o @ X @ A )
% 0.34/0.64 => ( ( finite_card_o @ ( insert_o @ X @ A ) )
% 0.34/0.64 = ( finite_card_o @ A ) ) )
% 0.34/0.64 & ( ~ ( member_o @ X @ A )
% 0.34/0.64 => ( ( finite_card_o @ ( insert_o @ X @ A ) )
% 0.34/0.64 = ( suc @ ( finite_card_o @ A ) ) ) ) ) ) ).
% 0.34/0.64
% 0.34/0.64 % card_insert_if
% 0.34/0.64 thf(fact_5319_card__insert__if,axiom,
% 0.34/0.64 ! [A: set_Product_unit,X: product_unit] :
% 0.34/0.64 ( ( finite4290736615968046902t_unit @ A )
% 0.34/0.64 => ( ( ( member_Product_unit @ X @ A )
% 0.34/0.64 => ( ( finite410649719033368117t_unit @ ( insert_Product_unit @ X @ A ) )
% 0.34/0.64 = ( finite410649719033368117t_unit @ A ) ) )
% 0.34/0.64 & ( ~ ( member_Product_unit @ X @ A )
% 0.34/0.64 => ( ( finite410649719033368117t_unit @ ( insert_Product_unit @ X @ A ) )
% 0.34/0.64 = ( suc @ ( finite410649719033368117t_unit @ A ) ) ) ) ) ) ).
% 0.34/0.64
% 0.34/0.64 % card_insert_if
% 0.34/0.64 thf(fact_5320_card__insert__if,axiom,
% 0.34/0.64 ! [A: set_nat,X: nat] :
% 0.34/0.64 ( ( finite_finite_nat @ A )
% 0.34/0.64 => ( ( ( member_nat @ X @ A )
% 0.34/0.64 => ( ( finite_card_nat @ ( insert_nat @ X @ A ) )
% 0.34/0.64 = ( finite_card_nat @ A ) ) )
% 0.34/0.64 & ( ~ ( member_nat @ X @ A )
% 0.34/0.64 => ( ( finite_card_nat @ ( insert_nat @ X @ A ) )
% 0.34/0.64 = ( suc @ ( finite_card_nat @ A ) ) ) ) ) ) ).
% 0.34/0.64
% 0.34/0.64 % card_insert_if
% 0.34/0.64 thf(fact_5321_card__insert__if,axiom,
% 0.34/0.64 ! [A: set_Code_integer,X: code_integer] :
% 0.34/0.64 ( ( finite6017078050557962740nteger @ A )
% 0.34/0.64 => ( ( ( member_Code_integer @ X @ A )
% 0.34/0.64 => ( ( finite4902975817058060853nteger @ ( insert_Code_integer @ X @ A ) )
% 0.34/0.64 = ( finite4902975817058060853nteger @ A ) ) )
% 0.34/0.64 & ( ~ ( member_Code_integer @ X @ A )
% 0.34/0.64 => ( ( finite4902975817058060853nteger @ ( insert_Code_integer @ X @ A ) )
% 0.34/0.64 = ( suc @ ( finite4902975817058060853nteger @ A ) ) ) ) ) ) ).
% 0.34/0.64
% 0.34/0.64 % card_insert_if
% 0.34/0.64 thf(fact_5322_card__le__Suc__iff,axiom,
% 0.34/0.64 ! [N2: nat,A: set_Pr4532377907799695533_nat_o] :
% 0.34/0.64 ( ( ord_less_eq_nat @ ( suc @ N2 ) @ ( finite1363419556375932405_nat_o @ A ) )
% 0.34/0.64 = ( ? [A6: produc3658429121746597890et_nat > $o,B7: set_Pr4532377907799695533_nat_o] :
% 0.34/0.64 ( ( A
% 0.34/0.64 = ( insert5175938949040314269_nat_o @ A6 @ B7 ) )
% 0.34/0.64 & ~ ( member6576561426505652726_nat_o @ A6 @ B7 )
% 0.34/0.64 & ( ord_less_eq_nat @ N2 @ ( finite1363419556375932405_nat_o @ B7 ) )
% 0.34/0.64 & ( finite3252695134891459830_nat_o @ B7 ) ) ) ) ).
% 0.34/0.64
% 0.34/0.64 % card_le_Suc_iff
% 0.34/0.64 thf(fact_5323_card__le__Suc__iff,axiom,
% 0.34/0.64 ! [N2: nat,A: set_int] :
% 0.34/0.64 ( ( ord_less_eq_nat @ ( suc @ N2 ) @ ( finite_card_int @ A ) )
% 0.34/0.64 = ( ? [A6: int,B7: set_int] :
% 0.34/0.64 ( ( A
% 0.34/0.64 = ( insert_int @ A6 @ B7 ) )
% 0.34/0.64 & ~ ( member_int @ A6 @ B7 )
% 0.34/0.64 & ( ord_less_eq_nat @ N2 @ ( finite_card_int @ B7 ) )
% 0.34/0.64 & ( finite_finite_int @ B7 ) ) ) ) ).
% 0.34/0.64
% 0.34/0.64 % card_le_Suc_iff
% 0.34/0.64 thf(fact_5324_card__le__Suc__iff,axiom,
% 0.34/0.64 ! [N2: nat,A: set_o] :
% 0.34/0.64 ( ( ord_less_eq_nat @ ( suc @ N2 ) @ ( finite_card_o @ A ) )
% 0.34/0.64 = ( ? [A6: $o,B7: set_o] :
% 0.34/0.64 ( ( A
% 0.34/0.64 = ( insert_o @ A6 @ B7 ) )
% 0.34/0.64 & ~ ( member_o @ A6 @ B7 )
% 0.34/0.64 & ( ord_less_eq_nat @ N2 @ ( finite_card_o @ B7 ) )
% 0.34/0.64 & ( finite_finite_o @ B7 ) ) ) ) ).
% 0.34/0.64
% 0.34/0.64 % card_le_Suc_iff
% 0.34/0.64 thf(fact_5325_card__le__Suc__iff,axiom,
% 0.34/0.64 ! [N2: nat,A: set_Product_unit] :
% 0.34/0.64 ( ( ord_less_eq_nat @ ( suc @ N2 ) @ ( finite410649719033368117t_unit @ A ) )
% 0.34/0.64 = ( ? [A6: product_unit,B7: set_Product_unit] :
% 0.34/0.64 ( ( A
% 0.34/0.64 = ( insert_Product_unit @ A6 @ B7 ) )
% 0.34/0.64 & ~ ( member_Product_unit @ A6 @ B7 )
% 0.34/0.64 & ( ord_less_eq_nat @ N2 @ ( finite410649719033368117t_unit @ B7 ) )
% 0.34/0.64 & ( finite4290736615968046902t_unit @ B7 ) ) ) ) ).
% 0.34/0.64
% 0.34/0.64 % card_le_Suc_iff
% 0.34/0.64 thf(fact_5326_card__le__Suc__iff,axiom,
% 0.34/0.64 ! [N2: nat,A: set_nat] :
% 0.34/0.64 ( ( ord_less_eq_nat @ ( suc @ N2 ) @ ( finite_card_nat @ A ) )
% 0.34/0.64 = ( ? [A6: nat,B7: set_nat] :
% 0.34/0.64 ( ( A
% 0.34/0.64 = ( insert_nat @ A6 @ B7 ) )
% 0.34/0.64 & ~ ( member_nat @ A6 @ B7 )
% 0.34/0.64 & ( ord_less_eq_nat @ N2 @ ( finite_card_nat @ B7 ) )
% 0.34/0.64 & ( finite_finite_nat @ B7 ) ) ) ) ).
% 0.34/0.64
% 0.34/0.64 % card_le_Suc_iff
% 0.34/0.64 thf(fact_5327_card__le__Suc__iff,axiom,
% 0.34/0.64 ! [N2: nat,A: set_Code_integer] :
% 0.34/0.64 ( ( ord_less_eq_nat @ ( suc @ N2 ) @ ( finite4902975817058060853nteger @ A ) )
% 0.34/0.64 = ( ? [A6: code_integer,B7: set_Code_integer] :
% 0.34/0.64 ( ( A
% 0.34/0.64 = ( insert_Code_integer @ A6 @ B7 ) )
% 0.34/0.64 & ~ ( member_Code_integer @ A6 @ B7 )
% 0.34/0.64 & ( ord_less_eq_nat @ N2 @ ( finite4902975817058060853nteger @ B7 ) )
% 0.34/0.64 & ( finite6017078050557962740nteger @ B7 ) ) ) ) ).
% 0.34/0.64
% 0.34/0.64 % card_le_Suc_iff
% 0.34/0.64 thf(fact_5328_card_Oremove,axiom,
% 0.34/0.64 ! [A: set_Code_integer,X: code_integer] :
% 0.34/0.64 ( ( finite6017078050557962740nteger @ A )
% 0.34/0.64 => ( ( member_Code_integer @ X @ A )
% 0.34/0.64 => ( ( finite4902975817058060853nteger @ A )
% 0.34/0.64 = ( suc @ ( finite4902975817058060853nteger @ ( minus_2355218937544613996nteger @ A @ ( insert_Code_integer @ X @ bot_bo3990330152332043303nteger ) ) ) ) ) ) ) ).
% 0.34/0.64
% 0.34/0.64 % card.remove
% 0.34/0.64 thf(fact_5329_card_Oremove,axiom,
% 0.34/0.64 ! [A: set_Product_unit,X: product_unit] :
% 0.34/0.64 ( ( finite4290736615968046902t_unit @ A )
% 0.34/0.64 => ( ( member_Product_unit @ X @ A )
% 0.34/0.64 => ( ( finite410649719033368117t_unit @ A )
% 0.34/0.64 = ( suc @ ( finite410649719033368117t_unit @ ( minus_6452836326544984404t_unit @ A @ ( insert_Product_unit @ X @ bot_bo3957492148770167129t_unit ) ) ) ) ) ) ) ).
% 0.34/0.64
% 0.34/0.64 % card.remove
% 0.34/0.64 thf(fact_5330_card_Oremove,axiom,
% 0.34/0.64 ! [A: set_o,X: $o] :
% 0.34/0.64 ( ( finite_finite_o @ A )
% 0.34/0.64 => ( ( member_o @ X @ A )
% 0.34/0.64 => ( ( finite_card_o @ A )
% 0.34/0.64 = ( suc @ ( finite_card_o @ ( minus_minus_set_o @ A @ ( insert_o @ X @ bot_bot_set_o ) ) ) ) ) ) ) ).
% 0.34/0.64
% 0.34/0.64 % card.remove
% 0.34/0.64 thf(fact_5331_card_Oremove,axiom,
% 0.34/0.64 ! [A: set_int,X: int] :
% 0.34/0.64 ( ( finite_finite_int @ A )
% 0.34/0.64 => ( ( member_int @ X @ A )
% 0.34/0.64 => ( ( finite_card_int @ A )
% 0.34/0.64 = ( suc @ ( finite_card_int @ ( minus_minus_set_int @ A @ ( insert_int @ X @ bot_bot_set_int ) ) ) ) ) ) ) ).
% 0.34/0.64
% 0.34/0.64 % card.remove
% 0.34/0.64 thf(fact_5332_card_Oremove,axiom,
% 0.34/0.64 ! [A: set_Pr4532377907799695533_nat_o,X: produc3658429121746597890et_nat > $o] :
% 0.34/0.64 ( ( finite3252695134891459830_nat_o @ A )
% 0.34/0.64 => ( ( member6576561426505652726_nat_o @ X @ A )
% 0.34/0.64 => ( ( finite1363419556375932405_nat_o @ A )
% 0.34/0.64 = ( suc @ ( finite1363419556375932405_nat_o @ ( minus_1801376950450012436_nat_o @ A @ ( insert5175938949040314269_nat_o @ X @ bot_bo7824918357723582233_nat_o ) ) ) ) ) ) ) ).
% 0.34/0.64
% 0.34/0.64 % card.remove
% 0.34/0.64 thf(fact_5333_card_Oremove,axiom,
% 0.34/0.64 ! [A: set_nat,X: nat] :
% 0.34/0.64 ( ( finite_finite_nat @ A )
% 0.34/0.64 => ( ( member_nat @ X @ A )
% 0.34/0.64 => ( ( finite_card_nat @ A )
% 0.34/0.64 = ( suc @ ( finite_card_nat @ ( minus_minus_set_nat @ A @ ( insert_nat @ X @ bot_bot_set_nat ) ) ) ) ) ) ) ).
% 0.34/0.64
% 0.34/0.64 % card.remove
% 0.34/0.64 thf(fact_5334_card_Oinsert__remove,axiom,
% 0.34/0.64 ! [A: set_Code_integer,X: code_integer] :
% 0.34/0.64 ( ( finite6017078050557962740nteger @ A )
% 0.34/0.64 => ( ( finite4902975817058060853nteger @ ( insert_Code_integer @ X @ A ) )
% 0.34/0.64 = ( suc @ ( finite4902975817058060853nteger @ ( minus_2355218937544613996nteger @ A @ ( insert_Code_integer @ X @ bot_bo3990330152332043303nteger ) ) ) ) ) ) ).
% 0.34/0.64
% 0.34/0.64 % card.insert_remove
% 0.34/0.64 thf(fact_5335_card_Oinsert__remove,axiom,
% 0.34/0.64 ! [A: set_Product_unit,X: product_unit] :
% 0.34/0.64 ( ( finite4290736615968046902t_unit @ A )
% 0.34/0.64 => ( ( finite410649719033368117t_unit @ ( insert_Product_unit @ X @ A ) )
% 0.34/0.64 = ( suc @ ( finite410649719033368117t_unit @ ( minus_6452836326544984404t_unit @ A @ ( insert_Product_unit @ X @ bot_bo3957492148770167129t_unit ) ) ) ) ) ) ).
% 0.34/0.64
% 0.34/0.64 % card.insert_remove
% 0.34/0.64 thf(fact_5336_card_Oinsert__remove,axiom,
% 0.34/0.64 ! [A: set_o,X: $o] :
% 0.34/0.64 ( ( finite_finite_o @ A )
% 0.34/0.64 => ( ( finite_card_o @ ( insert_o @ X @ A ) )
% 0.34/0.64 = ( suc @ ( finite_card_o @ ( minus_minus_set_o @ A @ ( insert_o @ X @ bot_bot_set_o ) ) ) ) ) ) ).
% 0.34/0.64
% 0.34/0.64 % card.insert_remove
% 0.34/0.64 thf(fact_5337_card_Oinsert__remove,axiom,
% 0.34/0.64 ! [A: set_int,X: int] :
% 0.34/0.64 ( ( finite_finite_int @ A )
% 0.34/0.64 => ( ( finite_card_int @ ( insert_int @ X @ A ) )
% 0.34/0.64 = ( suc @ ( finite_card_int @ ( minus_minus_set_int @ A @ ( insert_int @ X @ bot_bot_set_int ) ) ) ) ) ) ).
% 0.34/0.64
% 0.34/0.64 % card.insert_remove
% 0.34/0.64 thf(fact_5338_card_Oinsert__remove,axiom,
% 0.34/0.64 ! [A: set_Pr4532377907799695533_nat_o,X: produc3658429121746597890et_nat > $o] :
% 0.34/0.64 ( ( finite3252695134891459830_nat_o @ A )
% 0.34/0.64 => ( ( finite1363419556375932405_nat_o @ ( insert5175938949040314269_nat_o @ X @ A ) )
% 0.34/0.64 = ( suc @ ( finite1363419556375932405_nat_o @ ( minus_1801376950450012436_nat_o @ A @ ( insert5175938949040314269_nat_o @ X @ bot_bo7824918357723582233_nat_o ) ) ) ) ) ) ).
% 0.34/0.64
% 0.34/0.64 % card.insert_remove
% 0.34/0.64 thf(fact_5339_card_Oinsert__remove,axiom,
% 0.34/0.64 ! [A: set_nat,X: nat] :
% 0.34/0.64 ( ( finite_finite_nat @ A )
% 0.34/0.64 => ( ( finite_card_nat @ ( insert_nat @ X @ A ) )
% 0.34/0.64 = ( suc @ ( finite_card_nat @ ( minus_minus_set_nat @ A @ ( insert_nat @ X @ bot_bot_set_nat ) ) ) ) ) ) ).
% 0.34/0.64
% 0.34/0.64 % card.insert_remove
% 0.34/0.64 thf(fact_5340_mult__Suc__right,axiom,
% 0.34/0.64 ! [M4: nat,N2: nat] :
% 0.34/0.64 ( ( times_times_nat @ M4 @ ( suc @ N2 ) )
% 0.34/0.64 = ( plus_plus_nat @ M4 @ ( times_times_nat @ M4 @ N2 ) ) ) ).
% 0.34/0.64
% 0.34/0.64 % mult_Suc_right
% 0.34/0.64 thf(fact_5341_nat__mult__eq__1__iff,axiom,
% 0.34/0.64 ! [M4: nat,N2: nat] :
% 0.34/0.64 ( ( ( times_times_nat @ M4 @ N2 )
% 0.34/0.64 = one_one_nat )
% 0.34/0.64 = ( ( M4 = one_one_nat )
% 0.34/0.64 & ( N2 = one_one_nat ) ) ) ).
% 0.34/0.64
% 0.34/0.64 % nat_mult_eq_1_iff
% 0.34/0.64 thf(fact_5342_nat__1__eq__mult__iff,axiom,
% 0.34/0.64 ! [M4: nat,N2: nat] :
% 0.34/0.64 ( ( one_one_nat
% 0.34/0.64 = ( times_times_nat @ M4 @ N2 ) )
% 0.34/0.64 = ( ( M4 = one_one_nat )
% 0.34/0.64 & ( N2 = one_one_nat ) ) ) ).
% 0.34/0.64
% 0.34/0.64 % nat_1_eq_mult_iff
% 0.34/0.64 thf(fact_5343_nat__less__add__iff1,axiom,
% 0.34/0.64 ! [J: nat,I2: nat,U: nat,M4: nat,N2: nat] :
% 0.34/0.64 ( ( ord_less_eq_nat @ J @ I2 )
% 0.34/0.64 => ( ( ord_less_nat @ ( plus_plus_nat @ ( times_times_nat @ I2 @ U ) @ M4 ) @ ( plus_plus_nat @ ( times_times_nat @ J @ U ) @ N2 ) )
% 0.34/0.64 = ( ord_less_nat @ ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ I2 @ J ) @ U ) @ M4 ) @ N2 ) ) ) ).
% 0.34/0.64
% 0.34/0.64 % nat_less_add_iff1
% 0.34/0.64 thf(fact_5344_nat__less__add__iff2,axiom,
% 0.34/0.64 ! [I2: nat,J: nat,U: nat,M4: nat,N2: nat] :
% 0.34/0.64 ( ( ord_less_eq_nat @ I2 @ J )
% 0.34/0.64 => ( ( ord_less_nat @ ( plus_plus_nat @ ( times_times_nat @ I2 @ U ) @ M4 ) @ ( plus_plus_nat @ ( times_times_nat @ J @ U ) @ N2 ) )
% 0.34/0.64 = ( ord_less_nat @ M4 @ ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ J @ I2 ) @ U ) @ N2 ) ) ) ) ).
% 0.34/0.64
% 0.34/0.64 % nat_less_add_iff2
% 0.34/0.64 thf(fact_5345_nat__diff__add__eq2,axiom,
% 0.34/0.64 ! [I2: nat,J: nat,U: nat,M4: nat,N2: nat] :
% 0.34/0.64 ( ( ord_less_eq_nat @ I2 @ J )
% 0.34/0.64 => ( ( minus_minus_nat @ ( plus_plus_nat @ ( times_times_nat @ I2 @ U ) @ M4 ) @ ( plus_plus_nat @ ( times_times_nat @ J @ U ) @ N2 ) )
% 0.34/0.64 = ( minus_minus_nat @ M4 @ ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ J @ I2 ) @ U ) @ N2 ) ) ) ) ).
% 0.34/0.64
% 0.34/0.64 % nat_diff_add_eq2
% 0.34/0.64 thf(fact_5346_nat__diff__add__eq1,axiom,
% 0.34/0.64 ! [J: nat,I2: nat,U: nat,M4: nat,N2: nat] :
% 0.34/0.64 ( ( ord_less_eq_nat @ J @ I2 )
% 0.34/0.64 => ( ( minus_minus_nat @ ( plus_plus_nat @ ( times_times_nat @ I2 @ U ) @ M4 ) @ ( plus_plus_nat @ ( times_times_nat @ J @ U ) @ N2 ) )
% 0.34/0.64 = ( minus_minus_nat @ ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ I2 @ J ) @ U ) @ M4 ) @ N2 ) ) ) ).
% 0.34/0.64
% 0.34/0.64 % nat_diff_add_eq1
% 0.34/0.64 thf(fact_5347_nat__le__add__iff2,axiom,
% 0.34/0.64 ! [I2: nat,J: nat,U: nat,M4: nat,N2: nat] :
% 0.34/0.64 ( ( ord_less_eq_nat @ I2 @ J )
% 0.34/0.64 => ( ( ord_less_eq_nat @ ( plus_plus_nat @ ( times_times_nat @ I2 @ U ) @ M4 ) @ ( plus_plus_nat @ ( times_times_nat @ J @ U ) @ N2 ) )
% 0.34/0.64 = ( ord_less_eq_nat @ M4 @ ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ J @ I2 ) @ U ) @ N2 ) ) ) ) ).
% 0.34/0.64
% 0.34/0.64 % nat_le_add_iff2
% 0.34/0.64 thf(fact_5348_nat__mult__max__right,axiom,
% 0.34/0.64 ! [M4: nat,N2: nat,Q4: nat] :
% 0.34/0.64 ( ( times_times_nat @ M4 @ ( ord_max_nat @ N2 @ Q4 ) )
% 0.34/0.64 = ( ord_max_nat @ ( times_times_nat @ M4 @ N2 ) @ ( times_times_nat @ M4 @ Q4 ) ) ) ).
% 0.34/0.64
% 0.34/0.64 % nat_mult_max_right
% 0.34/0.64 thf(fact_5349_nat__mult__max__left,axiom,
% 0.34/0.64 ! [M4: nat,N2: nat,Q4: nat] :
% 0.34/0.64 ( ( times_times_nat @ ( ord_max_nat @ M4 @ N2 ) @ Q4 )
% 0.34/0.64 = ( ord_max_nat @ ( times_times_nat @ M4 @ Q4 ) @ ( times_times_nat @ N2 @ Q4 ) ) ) ).
% 0.34/0.64
% 0.34/0.64 % nat_mult_max_left
% 0.34/0.64 thf(fact_5350_Suc__mult__cancel1,axiom,
% 0.34/0.64 ! [K3: nat,M4: nat,N2: nat] :
% 0.34/0.64 ( ( ( times_times_nat @ ( suc @ K3 ) @ M4 )
% 0.34/0.64 = ( times_times_nat @ ( suc @ K3 ) @ N2 ) )
% 0.34/0.64 = ( M4 = N2 ) ) ).
% 0.34/0.64
% 0.34/0.64 % Suc_mult_cancel1
% 0.34/0.64 thf(fact_5351_mult__le__mono2,axiom,
% 0.34/0.64 ! [I2: nat,J: nat,K3: nat] :
% 0.34/0.64 ( ( ord_less_eq_nat @ I2 @ J )
% 0.34/0.64 => ( ord_less_eq_nat @ ( times_times_nat @ K3 @ I2 ) @ ( times_times_nat @ K3 @ J ) ) ) ).
% 0.34/0.64
% 0.34/0.64 % mult_le_mono2
% 0.34/0.64 thf(fact_5352_mult__le__mono1,axiom,
% 0.34/0.64 ! [I2: nat,J: nat,K3: nat] :
% 0.34/0.64 ( ( ord_less_eq_nat @ I2 @ J )
% 0.34/0.64 => ( ord_less_eq_nat @ ( times_times_nat @ I2 @ K3 ) @ ( times_times_nat @ J @ K3 ) ) ) ).
% 0.34/0.64
% 0.34/0.64 % mult_le_mono1
% 0.34/0.64 thf(fact_5353_mult__le__mono,axiom,
% 0.34/0.64 ! [I2: nat,J: nat,K3: nat,L: nat] :
% 0.34/0.64 ( ( ord_less_eq_nat @ I2 @ J )
% 0.34/0.64 => ( ( ord_less_eq_nat @ K3 @ L )
% 0.34/0.64 => ( ord_less_eq_nat @ ( times_times_nat @ I2 @ K3 ) @ ( times_times_nat @ J @ L ) ) ) ) ).
% 0.34/0.64
% 0.34/0.64 % mult_le_mono
% 0.34/0.64 thf(fact_5354_le__square,axiom,
% 0.34/0.64 ! [M4: nat] : ( ord_less_eq_nat @ M4 @ ( times_times_nat @ M4 @ M4 ) ) ).
% 0.34/0.64
% 0.34/0.64 % le_square
% 0.34/0.64 thf(fact_5355_le__cube,axiom,
% 0.34/0.64 ! [M4: nat] : ( ord_less_eq_nat @ M4 @ ( times_times_nat @ M4 @ ( times_times_nat @ M4 @ M4 ) ) ) ).
% 0.34/0.64
% 0.34/0.64 % le_cube
% 0.34/0.64 thf(fact_5356_add__mult__distrib,axiom,
% 0.34/0.64 ! [M4: nat,N2: nat,K3: nat] :
% 0.34/0.64 ( ( times_times_nat @ ( plus_plus_nat @ M4 @ N2 ) @ K3 )
% 0.34/0.64 = ( plus_plus_nat @ ( times_times_nat @ M4 @ K3 ) @ ( times_times_nat @ N2 @ K3 ) ) ) ).
% 0.34/0.64
% 0.34/0.64 % add_mult_distrib
% 0.34/0.64 thf(fact_5357_add__mult__distrib2,axiom,
% 0.34/0.64 ! [K3: nat,M4: nat,N2: nat] :
% 0.34/0.64 ( ( times_times_nat @ K3 @ ( plus_plus_nat @ M4 @ N2 ) )
% 0.34/0.64 = ( plus_plus_nat @ ( times_times_nat @ K3 @ M4 ) @ ( times_times_nat @ K3 @ N2 ) ) ) ).
% 0.34/0.64
% 0.34/0.64 % add_mult_distrib2
% 0.34/0.64 thf(fact_5358_left__add__mult__distrib,axiom,
% 0.34/0.64 ! [I2: nat,U: nat,J: nat,K3: nat] :
% 0.34/0.64 ( ( plus_plus_nat @ ( times_times_nat @ I2 @ U ) @ ( plus_plus_nat @ ( times_times_nat @ J @ U ) @ K3 ) )
% 0.34/0.64 = ( plus_plus_nat @ ( times_times_nat @ ( plus_plus_nat @ I2 @ J ) @ U ) @ K3 ) ) ).
% 0.34/0.64
% 0.34/0.64 % left_add_mult_distrib
% 0.34/0.64 thf(fact_5359_diff__mult__distrib,axiom,
% 0.34/0.64 ! [M4: nat,N2: nat,K3: nat] :
% 0.34/0.64 ( ( times_times_nat @ ( minus_minus_nat @ M4 @ N2 ) @ K3 )
% 0.34/0.64 = ( minus_minus_nat @ ( times_times_nat @ M4 @ K3 ) @ ( times_times_nat @ N2 @ K3 ) ) ) ).
% 0.34/0.64
% 0.34/0.64 % diff_mult_distrib
% 0.34/0.64 thf(fact_5360_diff__mult__distrib2,axiom,
% 0.34/0.64 ! [K3: nat,M4: nat,N2: nat] :
% 0.34/0.64 ( ( times_times_nat @ K3 @ ( minus_minus_nat @ M4 @ N2 ) )
% 0.34/0.64 = ( minus_minus_nat @ ( times_times_nat @ K3 @ M4 ) @ ( times_times_nat @ K3 @ N2 ) ) ) ).
% 0.34/0.64
% 0.34/0.64 % diff_mult_distrib2
% 0.34/0.64 thf(fact_5361_nat__mult__1,axiom,
% 0.34/0.64 ! [N2: nat] :
% 0.34/0.64 ( ( times_times_nat @ one_one_nat @ N2 )
% 0.34/0.64 = N2 ) ).
% 0.34/0.64
% 0.34/0.64 % nat_mult_1
% 0.34/0.64 thf(fact_5362_nat__mult__1__right,axiom,
% 0.34/0.64 ! [N2: nat] :
% 0.34/0.64 ( ( times_times_nat @ N2 @ one_one_nat )
% 0.34/0.64 = N2 ) ).
% 0.34/0.64
% 0.34/0.64 % nat_mult_1_right
% 0.34/0.64 thf(fact_5363_Suc__mult__less__cancel1,axiom,
% 0.34/0.64 ! [K3: nat,M4: nat,N2: nat] :
% 0.34/0.64 ( ( ord_less_nat @ ( times_times_nat @ ( suc @ K3 ) @ M4 ) @ ( times_times_nat @ ( suc @ K3 ) @ N2 ) )
% 0.34/0.64 = ( ord_less_nat @ M4 @ N2 ) ) ).
% 0.34/0.64
% 0.34/0.64 % Suc_mult_less_cancel1
% 0.34/0.64 thf(fact_5364_Suc__mult__le__cancel1,axiom,
% 0.34/0.64 ! [K3: nat,M4: nat,N2: nat] :
% 0.34/0.64 ( ( ord_less_eq_nat @ ( times_times_nat @ ( suc @ K3 ) @ M4 ) @ ( times_times_nat @ ( suc @ K3 ) @ N2 ) )
% 0.34/0.64 = ( ord_less_eq_nat @ M4 @ N2 ) ) ).
% 0.34/0.64
% 0.34/0.64 % Suc_mult_le_cancel1
% 0.34/0.64 thf(fact_5365_mult__Suc,axiom,
% 0.34/0.64 ! [M4: nat,N2: nat] :
% 0.34/0.64 ( ( times_times_nat @ ( suc @ M4 ) @ N2 )
% 0.34/0.64 = ( plus_plus_nat @ N2 @ ( times_times_nat @ M4 @ N2 ) ) ) ).
% 0.34/0.64
% 0.34/0.64 % mult_Suc
% 0.34/0.64 thf(fact_5366_nat__eq__add__iff1,axiom,
% 0.34/0.64 ! [J: nat,I2: nat,U: nat,M4: nat,N2: nat] :
% 0.34/0.64 ( ( ord_less_eq_nat @ J @ I2 )
% 0.34/0.64 => ( ( ( plus_plus_nat @ ( times_times_nat @ I2 @ U ) @ M4 )
% 0.34/0.64 = ( plus_plus_nat @ ( times_times_nat @ J @ U ) @ N2 ) )
% 0.34/0.64 = ( ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ I2 @ J ) @ U ) @ M4 )
% 0.34/0.64 = N2 ) ) ) ).
% 0.34/0.64
% 0.34/0.64 % nat_eq_add_iff1
% 0.34/0.64 thf(fact_5367_nat__eq__add__iff2,axiom,
% 0.34/0.64 ! [I2: nat,J: nat,U: nat,M4: nat,N2: nat] :
% 0.34/0.64 ( ( ord_less_eq_nat @ I2 @ J )
% 0.34/0.64 => ( ( ( plus_plus_nat @ ( times_times_nat @ I2 @ U ) @ M4 )
% 0.34/0.64 = ( plus_plus_nat @ ( times_times_nat @ J @ U ) @ N2 ) )
% 0.34/0.64 = ( M4
% 0.34/0.64 = ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ J @ I2 ) @ U ) @ N2 ) ) ) ) ).
% 0.34/0.64
% 0.34/0.64 % nat_eq_add_iff2
% 0.34/0.64 thf(fact_5368_nat__le__add__iff1,axiom,
% 0.34/0.64 ! [J: nat,I2: nat,U: nat,M4: nat,N2: nat] :
% 0.34/0.64 ( ( ord_less_eq_nat @ J @ I2 )
% 0.34/0.64 => ( ( ord_less_eq_nat @ ( plus_plus_nat @ ( times_times_nat @ I2 @ U ) @ M4 ) @ ( plus_plus_nat @ ( times_times_nat @ J @ U ) @ N2 ) )
% 0.34/0.64 = ( ord_less_eq_nat @ ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ I2 @ J ) @ U ) @ M4 ) @ N2 ) ) ) ).
% 0.34/0.64
% 0.34/0.64 % nat_le_add_iff1
% 0.34/0.64 thf(fact_5369_discrete,axiom,
% 0.34/0.64 ( ord_le6747313008572928689nteger
% 0.34/0.64 = ( ^ [A6: code_integer] : ( ord_le3102999989581377725nteger @ ( plus_p5714425477246183910nteger @ A6 @ one_one_Code_integer ) ) ) ) ).
% 0.34/0.64
% 0.34/0.64 % discrete
% 0.34/0.64 thf(fact_5370_discrete,axiom,
% 0.34/0.64 ( ord_less_nat
% 0.34/0.64 = ( ^ [A6: nat] : ( ord_less_eq_nat @ ( plus_plus_nat @ A6 @ one_one_nat ) ) ) ) ).
% 0.34/0.64
% 0.34/0.64 % discrete
% 0.34/0.64 thf(fact_5371_discrete,axiom,
% 0.34/0.64 ( ord_less_int
% 0.34/0.64 = ( ^ [A6: int] : ( ord_less_eq_int @ ( plus_plus_int @ A6 @ one_one_int ) ) ) ) ).
% 0.34/0.64
% 0.34/0.64 % discrete
% 0.34/0.64 thf(fact_5372_fstOp__def,axiom,
% 0.34/0.64 ( bNF_fs3871268833404974121et_nat
% 0.34/0.64 = ( ^ [P2: heap_e7401611519738050253t_unit > set_nat > $o,Q: set_nat > set_nat > $o,Ac: produc3658429121746597890et_nat] : ( produc7507926704131184380et_nat @ ( produc1824681642469235216et_nat @ Ac ) @ ( bNF_pi4042512755101277815et_nat @ P2 @ Q @ ( produc1824681642469235216et_nat @ Ac ) @ ( produc8586169260539613262et_nat @ Ac ) ) ) ) ) ).
% 0.34/0.64
% 0.34/0.64 % fstOp_def
% 0.34/0.64 thf(fact_5373_fstOp__def,axiom,
% 0.34/0.64 ( bNF_fs8167890477030535480nt_int
% 0.34/0.64 = ( ^ [P2: int > int > $o,Q: int > int > $o,Ac: product_prod_int_int] : ( product_Pair_int_int @ ( product_fst_int_int @ Ac ) @ ( bNF_pi8853482479757052266nt_int @ P2 @ Q @ ( product_fst_int_int @ Ac ) @ ( product_snd_int_int @ Ac ) ) ) ) ) ).
% 0.34/0.64
% 0.34/0.64 % fstOp_def
% 0.34/0.64 thf(fact_5374_sndOp__def,axiom,
% 0.34/0.64 ( bNF_sn4039337771197282234et_nat
% 0.34/0.64 = ( ^ [P2: heap_e7401611519738050253t_unit > heap_e7401611519738050253t_unit > $o,Q: heap_e7401611519738050253t_unit > set_nat > $o,Ac: produc3658429121746597890et_nat] : ( produc7507926704131184380et_nat @ ( bNF_pi3450177011487049898et_nat @ P2 @ Q @ ( produc1824681642469235216et_nat @ Ac ) @ ( produc8586169260539613262et_nat @ Ac ) ) @ ( produc8586169260539613262et_nat @ Ac ) ) ) ) ).
% 0.34/0.64
% 0.34/0.64 % sndOp_def
% 0.34/0.64 thf(fact_5375_sndOp__def,axiom,
% 0.34/0.64 ( bNF_sn1062102010912252026nt_int
% 0.34/0.64 = ( ^ [P2: int > int > $o,Q: int > int > $o,Ac: product_prod_int_int] : ( product_Pair_int_int @ ( bNF_pi8853482479757052266nt_int @ P2 @ Q @ ( product_fst_int_int @ Ac ) @ ( product_snd_int_int @ Ac ) ) @ ( product_snd_int_int @ Ac ) ) ) ) ).
% 0.34/0.64
% 0.34/0.64 % sndOp_def
% 0.34/0.64 thf(fact_5376_card__insert__le__m1,axiom,
% 0.34/0.64 ! [N2: nat,Y: set_nat,X: nat] :
% 0.34/0.64 ( ( ord_less_nat @ zero_zero_nat @ N2 )
% 0.34/0.64 => ( ( ord_less_eq_nat @ ( finite_card_nat @ Y ) @ ( minus_minus_nat @ N2 @ one_one_nat ) )
% 0.34/0.64 => ( ord_less_eq_nat @ ( finite_card_nat @ ( insert_nat @ X @ Y ) ) @ N2 ) ) ) ).
% 0.34/0.64
% 0.34/0.64 % card_insert_le_m1
% 0.34/0.64 thf(fact_5377_card__insert__le__m1,axiom,
% 0.34/0.64 ! [N2: nat,Y: set_int,X: int] :
% 0.34/0.64 ( ( ord_less_nat @ zero_zero_nat @ N2 )
% 0.34/0.64 => ( ( ord_less_eq_nat @ ( finite_card_int @ Y ) @ ( minus_minus_nat @ N2 @ one_one_nat ) )
% 0.34/0.64 => ( ord_less_eq_nat @ ( finite_card_int @ ( insert_int @ X @ Y ) ) @ N2 ) ) ) ).
% 0.34/0.64
% 0.34/0.64 % card_insert_le_m1
% 0.34/0.64 thf(fact_5378_card__insert__le__m1,axiom,
% 0.34/0.64 ! [N2: nat,Y: set_o,X: $o] :
% 0.34/0.64 ( ( ord_less_nat @ zero_zero_nat @ N2 )
% 0.34/0.64 => ( ( ord_less_eq_nat @ ( finite_card_o @ Y ) @ ( minus_minus_nat @ N2 @ one_one_nat ) )
% 0.34/0.64 => ( ord_less_eq_nat @ ( finite_card_o @ ( insert_o @ X @ Y ) ) @ N2 ) ) ) ).
% 0.34/0.64
% 0.34/0.64 % card_insert_le_m1
% 0.34/0.64 thf(fact_5379_card__insert__le__m1,axiom,
% 0.34/0.64 ! [N2: nat,Y: set_Product_unit,X: product_unit] :
% 0.34/0.64 ( ( ord_less_nat @ zero_zero_nat @ N2 )
% 0.34/0.64 => ( ( ord_less_eq_nat @ ( finite410649719033368117t_unit @ Y ) @ ( minus_minus_nat @ N2 @ one_one_nat ) )
% 0.34/0.64 => ( ord_less_eq_nat @ ( finite410649719033368117t_unit @ ( insert_Product_unit @ X @ Y ) ) @ N2 ) ) ) ).
% 0.34/0.64
% 0.34/0.64 % card_insert_le_m1
% 0.34/0.64 thf(fact_5380_crossproduct__noteq,axiom,
% 0.34/0.64 ! [A2: code_integer,B: code_integer,C: code_integer,D2: code_integer] :
% 0.34/0.64 ( ( ( A2 != B )
% 0.34/0.64 & ( C != D2 ) )
% 0.34/0.64 = ( ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ A2 @ C ) @ ( times_3573771949741848930nteger @ B @ D2 ) )
% 0.34/0.64 != ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ A2 @ D2 ) @ ( times_3573771949741848930nteger @ B @ C ) ) ) ) ).
% 0.34/0.64
% 0.34/0.64 % crossproduct_noteq
% 0.34/0.64 thf(fact_5381_crossproduct__noteq,axiom,
% 0.34/0.64 ! [A2: rat,B: rat,C: rat,D2: rat] :
% 0.34/0.64 ( ( ( A2 != B )
% 0.34/0.64 & ( C != D2 ) )
% 0.34/0.64 = ( ( plus_plus_rat @ ( times_times_rat @ A2 @ C ) @ ( times_times_rat @ B @ D2 ) )
% 0.34/0.64 != ( plus_plus_rat @ ( times_times_rat @ A2 @ D2 ) @ ( times_times_rat @ B @ C ) ) ) ) ).
% 0.34/0.64
% 0.34/0.64 % crossproduct_noteq
% 0.34/0.64 thf(fact_5382_crossproduct__noteq,axiom,
% 0.34/0.64 ! [A2: nat,B: nat,C: nat,D2: nat] :
% 0.34/0.64 ( ( ( A2 != B )
% 0.34/0.64 & ( C != D2 ) )
% 0.34/0.64 = ( ( plus_plus_nat @ ( times_times_nat @ A2 @ C ) @ ( times_times_nat @ B @ D2 ) )
% 0.34/0.64 != ( plus_plus_nat @ ( times_times_nat @ A2 @ D2 ) @ ( times_times_nat @ B @ C ) ) ) ) ).
% 0.34/0.64
% 0.34/0.64 % crossproduct_noteq
% 0.34/0.64 thf(fact_5383_crossproduct__noteq,axiom,
% 0.34/0.64 ! [A2: int,B: int,C: int,D2: int] :
% 0.34/0.64 ( ( ( A2 != B )
% 0.34/0.64 & ( C != D2 ) )
% 0.34/0.64 = ( ( plus_plus_int @ ( times_times_int @ A2 @ C ) @ ( times_times_int @ B @ D2 ) )
% 0.34/0.64 != ( plus_plus_int @ ( times_times_int @ A2 @ D2 ) @ ( times_times_int @ B @ C ) ) ) ) ).
% 0.34/0.64
% 0.34/0.64 % crossproduct_noteq
% 0.34/0.64 thf(fact_5384_crossproduct__eq,axiom,
% 0.34/0.64 ! [W: code_integer,Y: code_integer,X: code_integer,Z: code_integer] :
% 0.34/0.64 ( ( ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ W @ Y ) @ ( times_3573771949741848930nteger @ X @ Z ) )
% 0.34/0.64 = ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ W @ Z ) @ ( times_3573771949741848930nteger @ X @ Y ) ) )
% 0.34/0.64 = ( ( W = X )
% 0.34/0.64 | ( Y = Z ) ) ) ).
% 0.34/0.64
% 0.34/0.64 % crossproduct_eq
% 0.34/0.64 thf(fact_5385_crossproduct__eq,axiom,
% 0.34/0.64 ! [W: rat,Y: rat,X: rat,Z: rat] :
% 0.34/0.64 ( ( ( plus_plus_rat @ ( times_times_rat @ W @ Y ) @ ( times_times_rat @ X @ Z ) )
% 0.34/0.64 = ( plus_plus_rat @ ( times_times_rat @ W @ Z ) @ ( times_times_rat @ X @ Y ) ) )
% 0.34/0.64 = ( ( W = X )
% 0.34/0.64 | ( Y = Z ) ) ) ).
% 0.34/0.64
% 0.34/0.64 % crossproduct_eq
% 0.34/0.64 thf(fact_5386_crossproduct__eq,axiom,
% 0.34/0.64 ! [W: nat,Y: nat,X: nat,Z: nat] :
% 0.34/0.64 ( ( ( plus_plus_nat @ ( times_times_nat @ W @ Y ) @ ( times_times_nat @ X @ Z ) )
% 0.34/0.64 = ( plus_plus_nat @ ( times_times_nat @ W @ Z ) @ ( times_times_nat @ X @ Y ) ) )
% 0.34/0.64 = ( ( W = X )
% 0.34/0.64 | ( Y = Z ) ) ) ).
% 0.34/0.64
% 0.34/0.64 % crossproduct_eq
% 0.34/0.64 thf(fact_5387_crossproduct__eq,axiom,
% 0.34/0.64 ! [W: int,Y: int,X: int,Z: int] :
% 0.34/0.64 ( ( ( plus_plus_int @ ( times_times_int @ W @ Y ) @ ( times_times_int @ X @ Z ) )
% 0.34/0.64 = ( plus_plus_int @ ( times_times_int @ W @ Z ) @ ( times_times_int @ X @ Y ) ) )
% 0.34/0.64 = ( ( W = X )
% 0.34/0.64 | ( Y = Z ) ) ) ).
% 0.34/0.64
% 0.34/0.64 % crossproduct_eq
% 0.34/0.64 thf(fact_5388_dbl__dec__def,axiom,
% 0.34/0.64 ( neg_nu7757733837767384882nteger
% 0.34/0.64 = ( ^ [X2: code_integer] : ( minus_8373710615458151222nteger @ ( plus_p5714425477246183910nteger @ X2 @ X2 ) @ one_one_Code_integer ) ) ) ).
% 0.34/0.64
% 0.34/0.64 % dbl_dec_def
% 0.34/0.64 thf(fact_5389_dbl__dec__def,axiom,
% 0.34/0.64 ( neg_nu3179335615603231917ec_rat
% 0.34/0.64 = ( ^ [X2: rat] : ( minus_minus_rat @ ( plus_plus_rat @ X2 @ X2 ) @ one_one_rat ) ) ) ).
% 0.34/0.64
% 0.34/0.64 % dbl_dec_def
% 0.34/0.64 thf(fact_5390_dbl__dec__def,axiom,
% 0.34/0.64 ( neg_nu3811975205180677377ec_int
% 0.34/0.64 = ( ^ [X2: int] : ( minus_minus_int @ ( plus_plus_int @ X2 @ X2 ) @ one_one_int ) ) ) ).
% 0.34/0.64
% 0.34/0.64 % dbl_dec_def
% 0.34/0.64 thf(fact_5391_card__insert__disjoint_H,axiom,
% 0.34/0.64 ! [A: set_Pr4532377907799695533_nat_o,X: produc3658429121746597890et_nat > $o] :
% 0.34/0.64 ( ( finite3252695134891459830_nat_o @ A )
% 0.34/0.64 => ( ~ ( member6576561426505652726_nat_o @ X @ A )
% 0.34/0.64 => ( ( minus_minus_nat @ ( finite1363419556375932405_nat_o @ ( insert5175938949040314269_nat_o @ X @ A ) ) @ ( suc @ zero_zero_nat ) )
% 0.34/0.64 = ( finite1363419556375932405_nat_o @ A ) ) ) ) ).
% 0.34/0.64
% 0.34/0.64 % card_insert_disjoint'
% 0.34/0.64 thf(fact_5392_card__insert__disjoint_H,axiom,
% 0.34/0.64 ! [A: set_int,X: int] :
% 0.34/0.64 ( ( finite_finite_int @ A )
% 0.34/0.64 => ( ~ ( member_int @ X @ A )
% 0.34/0.64 => ( ( minus_minus_nat @ ( finite_card_int @ ( insert_int @ X @ A ) ) @ ( suc @ zero_zero_nat ) )
% 0.34/0.64 = ( finite_card_int @ A ) ) ) ) ).
% 0.34/0.64
% 0.34/0.64 % card_insert_disjoint'
% 0.34/0.64 thf(fact_5393_card__insert__disjoint_H,axiom,
% 0.34/0.64 ! [A: set_o,X: $o] :
% 0.34/0.64 ( ( finite_finite_o @ A )
% 0.34/0.64 => ( ~ ( member_o @ X @ A )
% 0.34/0.64 => ( ( minus_minus_nat @ ( finite_card_o @ ( insert_o @ X @ A ) ) @ ( suc @ zero_zero_nat ) )
% 0.34/0.64 = ( finite_card_o @ A ) ) ) ) ).
% 0.34/0.64
% 0.34/0.64 % card_insert_disjoint'
% 0.34/0.64 thf(fact_5394_card__insert__disjoint_H,axiom,
% 0.34/0.64 ! [A: set_Product_unit,X: product_unit] :
% 0.34/0.64 ( ( finite4290736615968046902t_unit @ A )
% 0.34/0.64 => ( ~ ( member_Product_unit @ X @ A )
% 0.34/0.64 => ( ( minus_minus_nat @ ( finite410649719033368117t_unit @ ( insert_Product_unit @ X @ A ) ) @ ( suc @ zero_zero_nat ) )
% 0.34/0.64 = ( finite410649719033368117t_unit @ A ) ) ) ) ).
% 0.34/0.64
% 0.34/0.64 % card_insert_disjoint'
% 0.34/0.64 thf(fact_5395_card__insert__disjoint_H,axiom,
% 0.34/0.64 ! [A: set_nat,X: nat] :
% 0.34/0.64 ( ( finite_finite_nat @ A )
% 0.34/0.64 => ( ~ ( member_nat @ X @ A )
% 0.34/0.64 => ( ( minus_minus_nat @ ( finite_card_nat @ ( insert_nat @ X @ A ) ) @ ( suc @ zero_zero_nat ) )
% 0.34/0.64 = ( finite_card_nat @ A ) ) ) ) ).
% 0.34/0.64
% 0.34/0.64 % card_insert_disjoint'
% 0.34/0.64 thf(fact_5396_card__insert__disjoint_H,axiom,
% 0.34/0.64 ! [A: set_Code_integer,X: code_integer] :
% 0.34/0.64 ( ( finite6017078050557962740nteger @ A )
% 0.34/0.64 => ( ~ ( member_Code_integer @ X @ A )
% 0.34/0.64 => ( ( minus_minus_nat @ ( finite4902975817058060853nteger @ ( insert_Code_integer @ X @ A ) ) @ ( suc @ zero_zero_nat ) )
% 0.34/0.64 = ( finite4902975817058060853nteger @ A ) ) ) ) ).
% 0.34/0.64
% 0.34/0.64 % card_insert_disjoint'
% 0.34/0.64 thf(fact_5397_le__zero__eq,axiom,
% 0.34/0.64 ! [N2: nat] :
% 0.34/0.64 ( ( ord_less_eq_nat @ N2 @ zero_zero_nat )
% 0.34/0.64 = ( N2 = zero_zero_nat ) ) ).
% 0.34/0.64
% 0.34/0.64 % le_zero_eq
% 0.34/0.64 thf(fact_5398_not__gr__zero,axiom,
% 0.34/0.64 ! [N2: nat] :
% 0.34/0.64 ( ( ~ ( ord_less_nat @ zero_zero_nat @ N2 ) )
% 0.34/0.64 = ( N2 = zero_zero_nat ) ) ).
% 0.34/0.64
% 0.34/0.64 % not_gr_zero
% 0.34/0.64 thf(fact_5399_add_Oright__neutral,axiom,
% 0.34/0.64 ! [A2: rat] :
% 0.34/0.64 ( ( plus_plus_rat @ A2 @ zero_zero_rat )
% 0.34/0.64 = A2 ) ).
% 0.34/0.64
% 0.34/0.64 % add.right_neutral
% 0.34/0.64 thf(fact_5400_add_Oright__neutral,axiom,
% 0.34/0.64 ! [A2: nat] :
% 0.34/0.64 ( ( plus_plus_nat @ A2 @ zero_zero_nat )
% 0.34/0.64 = A2 ) ).
% 0.34/0.64
% 0.34/0.64 % add.right_neutral
% 0.34/0.64 thf(fact_5401_add_Oright__neutral,axiom,
% 0.34/0.64 ! [A2: int] :
% 0.34/0.64 ( ( plus_plus_int @ A2 @ zero_zero_int )
% 0.34/0.64 = A2 ) ).
% 0.34/0.64
% 0.34/0.64 % add.right_neutral
% 0.34/0.64 thf(fact_5402_add_Oright__neutral,axiom,
% 0.34/0.64 ! [A2: code_integer] :
% 0.34/0.64 ( ( plus_p5714425477246183910nteger @ A2 @ zero_z3403309356797280102nteger )
% 0.34/0.64 = A2 ) ).
% 0.34/0.64
% 0.34/0.64 % add.right_neutral
% 0.34/0.64 thf(fact_5403_double__zero__sym,axiom,
% 0.34/0.64 ! [A2: rat] :
% 0.34/0.64 ( ( zero_zero_rat
% 0.34/0.64 = ( plus_plus_rat @ A2 @ A2 ) )
% 0.34/0.64 = ( A2 = zero_zero_rat ) ) ).
% 0.34/0.64
% 0.34/0.64 % double_zero_sym
% 0.34/0.64 thf(fact_5404_double__zero__sym,axiom,
% 0.34/0.64 ! [A2: int] :
% 0.34/0.64 ( ( zero_zero_int
% 0.34/0.64 = ( plus_plus_int @ A2 @ A2 ) )
% 0.34/0.64 = ( A2 = zero_zero_int ) ) ).
% 0.34/0.64
% 0.34/0.64 % double_zero_sym
% 0.34/0.64 thf(fact_5405_double__zero__sym,axiom,
% 0.34/0.64 ! [A2: code_integer] :
% 0.34/0.64 ( ( zero_z3403309356797280102nteger
% 0.34/0.64 = ( plus_p5714425477246183910nteger @ A2 @ A2 ) )
% 0.34/0.64 = ( A2 = zero_z3403309356797280102nteger ) ) ).
% 0.34/0.64
% 0.34/0.64 % double_zero_sym
% 0.34/0.64 thf(fact_5406_add__cancel__left__left,axiom,
% 0.34/0.64 ! [B: rat,A2: rat] :
% 0.34/0.64 ( ( ( plus_plus_rat @ B @ A2 )
% 0.34/0.64 = A2 )
% 0.34/0.64 = ( B = zero_zero_rat ) ) ).
% 0.34/0.64
% 0.34/0.64 % add_cancel_left_left
% 0.34/0.64 thf(fact_5407_add__cancel__left__left,axiom,
% 0.34/0.64 ! [B: nat,A2: nat] :
% 0.34/0.64 ( ( ( plus_plus_nat @ B @ A2 )
% 0.34/0.64 = A2 )
% 0.34/0.64 = ( B = zero_zero_nat ) ) ).
% 0.34/0.64
% 0.34/0.64 % add_cancel_left_left
% 0.34/0.64 thf(fact_5408_add__cancel__left__left,axiom,
% 0.34/0.64 ! [B: int,A2: int] :
% 0.34/0.64 ( ( ( plus_plus_int @ B @ A2 )
% 0.34/0.64 = A2 )
% 0.34/0.64 = ( B = zero_zero_int ) ) ).
% 0.34/0.64
% 0.34/0.64 % add_cancel_left_left
% 0.34/0.64 thf(fact_5409_add__cancel__left__left,axiom,
% 0.34/0.64 ! [B: code_integer,A2: code_integer] :
% 0.34/0.64 ( ( ( plus_p5714425477246183910nteger @ B @ A2 )
% 0.34/0.64 = A2 )
% 0.34/0.64 = ( B = zero_z3403309356797280102nteger ) ) ).
% 0.34/0.64
% 0.34/0.64 % add_cancel_left_left
% 0.34/0.64 thf(fact_5410_add__cancel__left__right,axiom,
% 0.34/0.64 ! [A2: rat,B: rat] :
% 0.34/0.64 ( ( ( plus_plus_rat @ A2 @ B )
% 0.34/0.64 = A2 )
% 0.34/0.64 = ( B = zero_zero_rat ) ) ).
% 0.34/0.64
% 0.34/0.64 % add_cancel_left_right
% 0.34/0.64 thf(fact_5411_add__cancel__left__right,axiom,
% 0.34/0.64 ! [A2: nat,B: nat] :
% 0.34/0.64 ( ( ( plus_plus_nat @ A2 @ B )
% 0.34/0.64 = A2 )
% 0.34/0.64 = ( B = zero_zero_nat ) ) ).
% 0.34/0.64
% 0.34/0.64 % add_cancel_left_right
% 0.34/0.64 thf(fact_5412_add__cancel__left__right,axiom,
% 0.34/0.64 ! [A2: int,B: int] :
% 0.34/0.64 ( ( ( plus_plus_int @ A2 @ B )
% 0.34/0.64 = A2 )
% 0.34/0.64 = ( B = zero_zero_int ) ) ).
% 0.34/0.64
% 0.34/0.64 % add_cancel_left_right
% 0.34/0.64 thf(fact_5413_add__cancel__left__right,axiom,
% 0.34/0.64 ! [A2: code_integer,B: code_integer] :
% 0.34/0.64 ( ( ( plus_p5714425477246183910nteger @ A2 @ B )
% 0.34/0.64 = A2 )
% 0.34/0.64 = ( B = zero_z3403309356797280102nteger ) ) ).
% 0.34/0.64
% 0.34/0.64 % add_cancel_left_right
% 0.34/0.64 thf(fact_5414_add__cancel__right__left,axiom,
% 0.34/0.64 ! [A2: rat,B: rat] :
% 0.34/0.64 ( ( A2
% 0.34/0.64 = ( plus_plus_rat @ B @ A2 ) )
% 0.34/0.64 = ( B = zero_zero_rat ) ) ).
% 0.34/0.64
% 0.34/0.64 % add_cancel_right_left
% 0.34/0.64 thf(fact_5415_add__cancel__right__left,axiom,
% 0.34/0.64 ! [A2: nat,B: nat] :
% 0.34/0.64 ( ( A2
% 0.34/0.64 = ( plus_plus_nat @ B @ A2 ) )
% 0.34/0.64 = ( B = zero_zero_nat ) ) ).
% 0.34/0.64
% 0.34/0.64 % add_cancel_right_left
% 0.34/0.64 thf(fact_5416_add__cancel__right__left,axiom,
% 0.34/0.64 ! [A2: int,B: int] :
% 0.34/0.64 ( ( A2
% 0.34/0.64 = ( plus_plus_int @ B @ A2 ) )
% 0.34/0.64 = ( B = zero_zero_int ) ) ).
% 0.34/0.64
% 0.34/0.64 % add_cancel_right_left
% 0.34/0.64 thf(fact_5417_add__cancel__right__left,axiom,
% 0.34/0.64 ! [A2: code_integer,B: code_integer] :
% 0.34/0.64 ( ( A2
% 0.34/0.64 = ( plus_p5714425477246183910nteger @ B @ A2 ) )
% 0.34/0.64 = ( B = zero_z3403309356797280102nteger ) ) ).
% 0.34/0.64
% 0.34/0.64 % add_cancel_right_left
% 0.34/0.64 thf(fact_5418_add__cancel__right__right,axiom,
% 0.34/0.64 ! [A2: rat,B: rat] :
% 0.34/0.64 ( ( A2
% 0.34/0.64 = ( plus_plus_rat @ A2 @ B ) )
% 0.34/0.64 = ( B = zero_zero_rat ) ) ).
% 0.34/0.64
% 0.34/0.64 % add_cancel_right_right
% 0.34/0.64 thf(fact_5419_add__cancel__right__right,axiom,
% 0.34/0.64 ! [A2: nat,B: nat] :
% 0.34/0.64 ( ( A2
% 0.34/0.64 = ( plus_plus_nat @ A2 @ B ) )
% 0.34/0.64 = ( B = zero_zero_nat ) ) ).
% 0.34/0.64
% 0.34/0.64 % add_cancel_right_right
% 0.34/0.64 thf(fact_5420_add__cancel__right__right,axiom,
% 0.34/0.64 ! [A2: int,B: int] :
% 0.34/0.64 ( ( A2
% 0.34/0.64 = ( plus_plus_int @ A2 @ B ) )
% 0.34/0.64 = ( B = zero_zero_int ) ) ).
% 0.34/0.64
% 0.34/0.64 % add_cancel_right_right
% 0.34/0.64 thf(fact_5421_add__cancel__right__right,axiom,
% 0.34/0.64 ! [A2: code_integer,B: code_integer] :
% 0.34/0.64 ( ( A2
% 0.34/0.64 = ( plus_p5714425477246183910nteger @ A2 @ B ) )
% 0.34/0.64 = ( B = zero_z3403309356797280102nteger ) ) ).
% 0.34/0.64
% 0.34/0.64 % add_cancel_right_right
% 0.34/0.64 thf(fact_5422_add__eq__0__iff__both__eq__0,axiom,
% 0.34/0.64 ! [X: nat,Y: nat] :
% 0.34/0.64 ( ( ( plus_plus_nat @ X @ Y )
% 0.34/0.64 = zero_zero_nat )
% 0.34/0.64 = ( ( X = zero_zero_nat )
% 0.34/0.64 & ( Y = zero_zero_nat ) ) ) ).
% 0.34/0.64
% 0.34/0.64 % add_eq_0_iff_both_eq_0
% 0.34/0.64 thf(fact_5423_zero__eq__add__iff__both__eq__0,axiom,
% 0.34/0.64 ! [X: nat,Y: nat] :
% 0.34/0.64 ( ( zero_zero_nat
% 0.34/0.64 = ( plus_plus_nat @ X @ Y ) )
% 0.34/0.64 = ( ( X = zero_zero_nat )
% 0.34/0.64 & ( Y = zero_zero_nat ) ) ) ).
% 0.34/0.64
% 0.34/0.64 % zero_eq_add_iff_both_eq_0
% 0.34/0.64 thf(fact_5424_add__0,axiom,
% 0.34/0.64 ! [A2: rat] :
% 0.34/0.64 ( ( plus_plus_rat @ zero_zero_rat @ A2 )
% 0.34/0.64 = A2 ) ).
% 0.34/0.64
% 0.34/0.64 % add_0
% 0.34/0.64 thf(fact_5425_add__0,axiom,
% 0.34/0.64 ! [A2: nat] :
% 0.34/0.64 ( ( plus_plus_nat @ zero_zero_nat @ A2 )
% 0.34/0.64 = A2 ) ).
% 0.34/0.64
% 0.34/0.64 % add_0
% 0.34/0.64 thf(fact_5426_add__0,axiom,
% 0.34/0.64 ! [A2: int] :
% 0.34/0.64 ( ( plus_plus_int @ zero_zero_int @ A2 )
% 0.34/0.64 = A2 ) ).
% 0.34/0.64
% 0.34/0.64 % add_0
% 0.34/0.64 thf(fact_5427_add__0,axiom,
% 0.34/0.64 ! [A2: code_integer] :
% 0.34/0.64 ( ( plus_p5714425477246183910nteger @ zero_z3403309356797280102nteger @ A2 )
% 0.34/0.64 = A2 ) ).
% 0.34/0.64
% 0.34/0.64 % add_0
% 0.34/0.64 thf(fact_5428_mult__cancel__right,axiom,
% 0.34/0.64 ! [A2: code_integer,C: code_integer,B: code_integer] :
% 0.34/0.64 ( ( ( times_3573771949741848930nteger @ A2 @ C )
% 0.34/0.64 = ( times_3573771949741848930nteger @ B @ C ) )
% 0.34/0.64 = ( ( C = zero_z3403309356797280102nteger )
% 0.34/0.64 | ( A2 = B ) ) ) ).
% 0.34/0.64
% 0.34/0.64 % mult_cancel_right
% 0.34/0.64 thf(fact_5429_mult__cancel__right,axiom,
% 0.34/0.64 ! [A2: rat,C: rat,B: rat] :
% 0.34/0.64 ( ( ( times_times_rat @ A2 @ C )
% 0.34/0.64 = ( times_times_rat @ B @ C ) )
% 0.34/0.64 = ( ( C = zero_zero_rat )
% 0.34/0.64 | ( A2 = B ) ) ) ).
% 0.34/0.64
% 0.34/0.64 % mult_cancel_right
% 0.34/0.64 thf(fact_5430_mult__cancel__right,axiom,
% 0.34/0.64 ! [A2: nat,C: nat,B: nat] :
% 0.34/0.64 ( ( ( times_times_nat @ A2 @ C )
% 0.34/0.64 = ( times_times_nat @ B @ C ) )
% 0.34/0.64 = ( ( C = zero_zero_nat )
% 0.34/0.64 | ( A2 = B ) ) ) ).
% 0.34/0.64
% 0.34/0.64 % mult_cancel_right
% 0.34/0.64 thf(fact_5431_mult__cancel__right,axiom,
% 0.34/0.64 ! [A2: int,C: int,B: int] :
% 0.34/0.64 ( ( ( times_times_int @ A2 @ C )
% 0.34/0.64 = ( times_times_int @ B @ C ) )
% 0.34/0.64 = ( ( C = zero_zero_int )
% 0.34/0.64 | ( A2 = B ) ) ) ).
% 0.34/0.64
% 0.34/0.64 % mult_cancel_right
% 0.34/0.64 thf(fact_5432_mult__cancel__left,axiom,
% 0.34/0.64 ! [C: code_integer,A2: code_integer,B: code_integer] :
% 0.34/0.64 ( ( ( times_3573771949741848930nteger @ C @ A2 )
% 0.34/0.64 = ( times_3573771949741848930nteger @ C @ B ) )
% 0.34/0.64 = ( ( C = zero_z3403309356797280102nteger )
% 0.34/0.64 | ( A2 = B ) ) ) ).
% 0.34/0.64
% 0.34/0.64 % mult_cancel_left
% 0.34/0.64 thf(fact_5433_mult__cancel__left,axiom,
% 0.34/0.64 ! [C: rat,A2: rat,B: rat] :
% 0.34/0.64 ( ( ( times_times_rat @ C @ A2 )
% 0.34/0.64 = ( times_times_rat @ C @ B ) )
% 0.34/0.64 = ( ( C = zero_zero_rat )
% 0.34/0.64 | ( A2 = B ) ) ) ).
% 0.34/0.64
% 0.34/0.64 % mult_cancel_left
% 0.34/0.64 thf(fact_5434_mult__cancel__left,axiom,
% 0.34/0.64 ! [C: nat,A2: nat,B: nat] :
% 0.34/0.64 ( ( ( times_times_nat @ C @ A2 )
% 0.34/0.64 = ( times_times_nat @ C @ B ) )
% 0.34/0.64 = ( ( C = zero_zero_nat )
% 0.34/0.64 | ( A2 = B ) ) ) ).
% 0.34/0.64
% 0.34/0.64 % mult_cancel_left
% 0.34/0.64 thf(fact_5435_mult__cancel__left,axiom,
% 0.34/0.64 ! [C: int,A2: int,B: int] :
% 0.34/0.64 ( ( ( times_times_int @ C @ A2 )
% 0.34/0.64 = ( times_times_int @ C @ B ) )
% 0.34/0.64 = ( ( C = zero_zero_int )
% 0.34/0.64 | ( A2 = B ) ) ) ).
% 0.34/0.64
% 0.34/0.64 % mult_cancel_left
% 0.34/0.64 thf(fact_5436_mult__eq__0__iff,axiom,
% 0.34/0.64 ! [A2: code_integer,B: code_integer] :
% 0.34/0.64 ( ( ( times_3573771949741848930nteger @ A2 @ B )
% 0.34/0.64 = zero_z3403309356797280102nteger )
% 0.34/0.64 = ( ( A2 = zero_z3403309356797280102nteger )
% 0.34/0.64 | ( B = zero_z3403309356797280102nteger ) ) ) ).
% 0.34/0.64
% 0.34/0.64 % mult_eq_0_iff
% 0.34/0.64 thf(fact_5437_mult__eq__0__iff,axiom,
% 0.34/0.64 ! [A2: rat,B: rat] :
% 0.34/0.64 ( ( ( times_times_rat @ A2 @ B )
% 0.34/0.64 = zero_zero_rat )
% 0.34/0.64 = ( ( A2 = zero_zero_rat )
% 0.34/0.64 | ( B = zero_zero_rat ) ) ) ).
% 0.34/0.64
% 0.34/0.64 % mult_eq_0_iff
% 0.34/0.64 thf(fact_5438_mult__eq__0__iff,axiom,
% 0.34/0.64 ! [A2: nat,B: nat] :
% 0.34/0.64 ( ( ( times_times_nat @ A2 @ B )
% 0.34/0.64 = zero_zero_nat )
% 0.34/0.64 = ( ( A2 = zero_zero_nat )
% 0.34/0.64 | ( B = zero_zero_nat ) ) ) ).
% 0.34/0.64
% 0.34/0.64 % mult_eq_0_iff
% 0.34/0.64 thf(fact_5439_mult__eq__0__iff,axiom,
% 0.34/0.64 ! [A2: int,B: int] :
% 0.34/0.64 ( ( ( times_times_int @ A2 @ B )
% 0.34/0.64 = zero_zero_int )
% 0.34/0.64 = ( ( A2 = zero_zero_int )
% 0.34/0.64 | ( B = zero_zero_int ) ) ) ).
% 0.34/0.64
% 0.34/0.64 % mult_eq_0_iff
% 0.34/0.64 thf(fact_5440_mult__zero__right,axiom,
% 0.34/0.64 ! [A2: code_integer] :
% 0.34/0.64 ( ( times_3573771949741848930nteger @ A2 @ zero_z3403309356797280102nteger )
% 0.34/0.64 = zero_z3403309356797280102nteger ) ).
% 0.34/0.64
% 0.34/0.64 % mult_zero_right
% 0.34/0.64 thf(fact_5441_mult__zero__right,axiom,
% 0.34/0.64 ! [A2: rat] :
% 0.34/0.64 ( ( times_times_rat @ A2 @ zero_zero_rat )
% 0.34/0.64 = zero_zero_rat ) ).
% 0.34/0.64
% 0.34/0.64 % mult_zero_right
% 0.34/0.64 thf(fact_5442_mult__zero__right,axiom,
% 0.34/0.64 ! [A2: nat] :
% 0.34/0.64 ( ( times_times_nat @ A2 @ zero_zero_nat )
% 0.34/0.64 = zero_zero_nat ) ).
% 0.34/0.64
% 0.34/0.64 % mult_zero_right
% 0.34/0.64 thf(fact_5443_mult__zero__right,axiom,
% 0.34/0.64 ! [A2: int] :
% 0.34/0.64 ( ( times_times_int @ A2 @ zero_zero_int )
% 0.34/0.64 = zero_zero_int ) ).
% 0.34/0.64
% 0.34/0.64 % mult_zero_right
% 0.34/0.64 thf(fact_5444_mult__zero__left,axiom,
% 0.34/0.64 ! [A2: code_integer] :
% 0.34/0.64 ( ( times_3573771949741848930nteger @ zero_z3403309356797280102nteger @ A2 )
% 0.34/0.64 = zero_z3403309356797280102nteger ) ).
% 0.34/0.64
% 0.34/0.64 % mult_zero_left
% 0.34/0.64 thf(fact_5445_mult__zero__left,axiom,
% 0.34/0.64 ! [A2: rat] :
% 0.34/0.64 ( ( times_times_rat @ zero_zero_rat @ A2 )
% 0.34/0.64 = zero_zero_rat ) ).
% 0.34/0.64
% 0.34/0.64 % mult_zero_left
% 0.34/0.64 thf(fact_5446_mult__zero__left,axiom,
% 0.34/0.64 ! [A2: nat] :
% 0.34/0.64 ( ( times_times_nat @ zero_zero_nat @ A2 )
% 0.34/0.64 = zero_zero_nat ) ).
% 0.34/0.64
% 0.34/0.64 % mult_zero_left
% 0.34/0.64 thf(fact_5447_mult__zero__left,axiom,
% 0.34/0.64 ! [A2: int] :
% 0.34/0.64 ( ( times_times_int @ zero_zero_int @ A2 )
% 0.34/0.64 = zero_zero_int ) ).
% 0.34/0.64
% 0.34/0.64 % mult_zero_left
% 0.34/0.64 thf(fact_5448_diff__self,axiom,
% 0.34/0.64 ! [A2: code_integer] :
% 0.34/0.64 ( ( minus_8373710615458151222nteger @ A2 @ A2 )
% 0.34/0.64 = zero_z3403309356797280102nteger ) ).
% 0.34/0.64
% 0.34/0.64 % diff_self
% 0.34/0.64 thf(fact_5449_diff__self,axiom,
% 0.34/0.64 ! [A2: rat] :
% 0.34/0.64 ( ( minus_minus_rat @ A2 @ A2 )
% 0.34/0.64 = zero_zero_rat ) ).
% 0.34/0.64
% 0.34/0.64 % diff_self
% 0.34/0.64 thf(fact_5450_diff__self,axiom,
% 0.34/0.64 ! [A2: int] :
% 0.34/0.64 ( ( minus_minus_int @ A2 @ A2 )
% 0.34/0.64 = zero_zero_int ) ).
% 0.34/0.64
% 0.34/0.64 % diff_self
% 0.34/0.64 thf(fact_5451_diff__0__right,axiom,
% 0.34/0.64 ! [A2: code_integer] :
% 0.34/0.64 ( ( minus_8373710615458151222nteger @ A2 @ zero_z3403309356797280102nteger )
% 0.34/0.64 = A2 ) ).
% 0.34/0.64
% 0.34/0.64 % diff_0_right
% 0.34/0.64 thf(fact_5452_diff__0__right,axiom,
% 0.34/0.64 ! [A2: rat] :
% 0.34/0.64 ( ( minus_minus_rat @ A2 @ zero_zero_rat )
% 0.34/0.64 = A2 ) ).
% 0.34/0.64
% 0.34/0.64 % diff_0_right
% 0.34/0.64 thf(fact_5453_diff__0__right,axiom,
% 0.34/0.64 ! [A2: int] :
% 0.34/0.64 ( ( minus_minus_int @ A2 @ zero_zero_int )
% 0.34/0.64 = A2 ) ).
% 0.34/0.64
% 0.34/0.64 % diff_0_right
% 0.34/0.64 thf(fact_5454_zero__diff,axiom,
% 0.34/0.64 ! [A2: nat] :
% 0.34/0.64 ( ( minus_minus_nat @ zero_zero_nat @ A2 )
% 0.34/0.64 = zero_zero_nat ) ).
% 0.34/0.64
% 0.34/0.64 % zero_diff
% 0.34/0.64 thf(fact_5455_diff__zero,axiom,
% 0.34/0.64 ! [A2: code_integer] :
% 0.34/0.64 ( ( minus_8373710615458151222nteger @ A2 @ zero_z3403309356797280102nteger )
% 0.34/0.64 = A2 ) ).
% 0.34/0.64
% 0.34/0.64 % diff_zero
% 0.34/0.64 thf(fact_5456_diff__zero,axiom,
% 0.34/0.64 ! [A2: rat] :
% 0.34/0.64 ( ( minus_minus_rat @ A2 @ zero_zero_rat )
% 0.34/0.64 = A2 ) ).
% 0.34/0.64
% 0.34/0.64 % diff_zero
% 0.34/0.64 thf(fact_5457_diff__zero,axiom,
% 0.34/0.64 ! [A2: nat] :
% 0.34/0.64 ( ( minus_minus_nat @ A2 @ zero_zero_nat )
% 0.34/0.64 = A2 ) ).
% 0.34/0.64
% 0.34/0.64 % diff_zero
% 0.34/0.64 thf(fact_5458_diff__zero,axiom,
% 0.34/0.64 ! [A2: int] :
% 0.34/0.64 ( ( minus_minus_int @ A2 @ zero_zero_int )
% 0.34/0.64 = A2 ) ).
% 0.34/0.64
% 0.34/0.64 % diff_zero
% 0.34/0.64 thf(fact_5459_cancel__comm__monoid__add__class_Odiff__cancel,axiom,
% 0.34/0.64 ! [A2: code_integer] :
% 0.34/0.64 ( ( minus_8373710615458151222nteger @ A2 @ A2 )
% 0.34/0.64 = zero_z3403309356797280102nteger ) ).
% 0.34/0.64
% 0.34/0.64 % cancel_comm_monoid_add_class.diff_cancel
% 0.34/0.64 thf(fact_5460_cancel__comm__monoid__add__class_Odiff__cancel,axiom,
% 0.34/0.64 ! [A2: rat] :
% 0.34/0.64 ( ( minus_minus_rat @ A2 @ A2 )
% 0.34/0.64 = zero_zero_rat ) ).
% 0.34/0.64
% 0.34/0.64 % cancel_comm_monoid_add_class.diff_cancel
% 0.34/0.64 thf(fact_5461_cancel__comm__monoid__add__class_Odiff__cancel,axiom,
% 0.34/0.64 ! [A2: nat] :
% 0.34/0.64 ( ( minus_minus_nat @ A2 @ A2 )
% 0.34/0.64 = zero_zero_nat ) ).
% 0.34/0.64
% 0.34/0.64 % cancel_comm_monoid_add_class.diff_cancel
% 0.34/0.64 thf(fact_5462_cancel__comm__monoid__add__class_Odiff__cancel,axiom,
% 0.34/0.64 ! [A2: int] :
% 0.34/0.64 ( ( minus_minus_int @ A2 @ A2 )
% 0.34/0.64 = zero_zero_int ) ).
% 0.34/0.64
% 0.34/0.64 % cancel_comm_monoid_add_class.diff_cancel
% 0.34/0.64 thf(fact_5463_mult__cancel2,axiom,
% 0.34/0.64 ! [M4: nat,K3: nat,N2: nat] :
% 0.34/0.64 ( ( ( times_times_nat @ M4 @ K3 )
% 0.34/0.64 = ( times_times_nat @ N2 @ K3 ) )
% 0.34/0.64 = ( ( M4 = N2 )
% 0.34/0.64 | ( K3 = zero_zero_nat ) ) ) ).
% 0.34/0.64
% 0.34/0.64 % mult_cancel2
% 0.34/0.64 thf(fact_5464_mult__cancel1,axiom,
% 0.34/0.64 ! [K3: nat,M4: nat,N2: nat] :
% 0.34/0.64 ( ( ( times_times_nat @ K3 @ M4 )
% 0.34/0.64 = ( times_times_nat @ K3 @ N2 ) )
% 0.34/0.64 = ( ( M4 = N2 )
% 0.34/0.64 | ( K3 = zero_zero_nat ) ) ) ).
% 0.34/0.64
% 0.34/0.64 % mult_cancel1
% 0.34/0.64 thf(fact_5465_mult__0__right,axiom,
% 0.34/0.64 ! [M4: nat] :
% 0.34/0.64 ( ( times_times_nat @ M4 @ zero_zero_nat )
% 0.34/0.64 = zero_zero_nat ) ).
% 0.34/0.64
% 0.34/0.64 % mult_0_right
% 0.34/0.64 thf(fact_5466_mult__is__0,axiom,
% 0.34/0.64 ! [M4: nat,N2: nat] :
% 0.34/0.64 ( ( ( times_times_nat @ M4 @ N2 )
% 0.34/0.64 = zero_zero_nat )
% 0.34/0.64 = ( ( M4 = zero_zero_nat )
% 0.34/0.64 | ( N2 = zero_zero_nat ) ) ) ).
% 0.34/0.64
% 0.34/0.64 % mult_is_0
% 0.34/0.64 thf(fact_5467_Sup__nat__empty,axiom,
% 0.34/0.64 ( ( complete_Sup_Sup_nat @ bot_bot_set_nat )
% 0.34/0.64 = zero_zero_nat ) ).
% 0.34/0.64
% 0.34/0.64 % Sup_nat_empty
% 0.34/0.64 thf(fact_5468_dbl__dec__simps_I3_J,axiom,
% 0.34/0.64 ( ( neg_nu3811975205180677377ec_int @ one_one_int )
% 0.34/0.64 = one_one_int ) ).
% 0.34/0.64
% 0.34/0.64 % dbl_dec_simps(3)
% 0.34/0.64 thf(fact_5469_dbl__dec__simps_I3_J,axiom,
% 0.34/0.64 ( ( neg_nu7757733837767384882nteger @ one_one_Code_integer )
% 0.34/0.64 = one_one_Code_integer ) ).
% 0.34/0.64
% 0.34/0.64 % dbl_dec_simps(3)
% 0.34/0.64 thf(fact_5470_zero__le__double__add__iff__zero__le__single__add,axiom,
% 0.34/0.64 ! [A2: code_integer] :
% 0.34/0.64 ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ ( plus_p5714425477246183910nteger @ A2 @ A2 ) )
% 0.34/0.64 = ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ A2 ) ) ).
% 0.34/0.64
% 0.34/0.64 % zero_le_double_add_iff_zero_le_single_add
% 0.34/0.64 thf(fact_5471_zero__le__double__add__iff__zero__le__single__add,axiom,
% 0.34/0.64 ! [A2: rat] :
% 0.34/0.64 ( ( ord_less_eq_rat @ zero_zero_rat @ ( plus_plus_rat @ A2 @ A2 ) )
% 0.34/0.64 = ( ord_less_eq_rat @ zero_zero_rat @ A2 ) ) ).
% 0.34/0.64
% 0.34/0.64 % zero_le_double_add_iff_zero_le_single_add
% 0.34/0.64 thf(fact_5472_zero__le__double__add__iff__zero__le__single__add,axiom,
% 0.34/0.64 ! [A2: int] :
% 0.34/0.64 ( ( ord_less_eq_int @ zero_zero_int @ ( plus_plus_int @ A2 @ A2 ) )
% 0.34/0.64 = ( ord_less_eq_int @ zero_zero_int @ A2 ) ) ).
% 0.34/0.64
% 0.34/0.64 % zero_le_double_add_iff_zero_le_single_add
% 0.34/0.64 thf(fact_5473_double__add__le__zero__iff__single__add__le__zero,axiom,
% 0.34/0.64 ! [A2: code_integer] :
% 0.34/0.64 ( ( ord_le3102999989581377725nteger @ ( plus_p5714425477246183910nteger @ A2 @ A2 ) @ zero_z3403309356797280102nteger )
% 0.34/0.64 = ( ord_le3102999989581377725nteger @ A2 @ zero_z3403309356797280102nteger ) ) ).
% 0.34/0.64
% 0.34/0.64 % double_add_le_zero_iff_single_add_le_zero
% 0.34/0.64 thf(fact_5474_double__add__le__zero__iff__single__add__le__zero,axiom,
% 0.34/0.64 ! [A2: rat] :
% 0.34/0.64 ( ( ord_less_eq_rat @ ( plus_plus_rat @ A2 @ A2 ) @ zero_zero_rat )
% 0.34/0.64 = ( ord_less_eq_rat @ A2 @ zero_zero_rat ) ) ).
% 0.34/0.64
% 0.34/0.64 % double_add_le_zero_iff_single_add_le_zero
% 0.34/0.64 thf(fact_5475_double__add__le__zero__iff__single__add__le__zero,axiom,
% 0.34/0.64 ! [A2: int] :
% 0.34/0.64 ( ( ord_less_eq_int @ ( plus_plus_int @ A2 @ A2 ) @ zero_zero_int )
% 0.34/0.64 = ( ord_less_eq_int @ A2 @ zero_zero_int ) ) ).
% 0.34/0.64
% 0.34/0.64 % double_add_le_zero_iff_single_add_le_zero
% 0.34/0.64 thf(fact_5476_le__add__same__cancel2,axiom,
% 0.34/0.64 ! [A2: code_integer,B: code_integer] :
% 0.34/0.64 ( ( ord_le3102999989581377725nteger @ A2 @ ( plus_p5714425477246183910nteger @ B @ A2 ) )
% 0.34/0.64 = ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ B ) ) ).
% 0.34/0.64
% 0.34/0.64 % le_add_same_cancel2
% 0.34/0.64 thf(fact_5477_le__add__same__cancel2,axiom,
% 0.34/0.64 ! [A2: rat,B: rat] :
% 0.34/0.64 ( ( ord_less_eq_rat @ A2 @ ( plus_plus_rat @ B @ A2 ) )
% 0.34/0.64 = ( ord_less_eq_rat @ zero_zero_rat @ B ) ) ).
% 0.34/0.64
% 0.34/0.64 % le_add_same_cancel2
% 0.34/0.64 thf(fact_5478_le__add__same__cancel2,axiom,
% 0.34/0.64 ! [A2: nat,B: nat] :
% 0.34/0.64 ( ( ord_less_eq_nat @ A2 @ ( plus_plus_nat @ B @ A2 ) )
% 0.34/0.64 = ( ord_less_eq_nat @ zero_zero_nat @ B ) ) ).
% 0.34/0.64
% 0.34/0.64 % le_add_same_cancel2
% 0.34/0.64 thf(fact_5479_le__add__same__cancel2,axiom,
% 0.34/0.64 ! [A2: int,B: int] :
% 0.34/0.64 ( ( ord_less_eq_int @ A2 @ ( plus_plus_int @ B @ A2 ) )
% 0.34/0.64 = ( ord_less_eq_int @ zero_zero_int @ B ) ) ).
% 0.34/0.64
% 0.34/0.64 % le_add_same_cancel2
% 0.34/0.64 thf(fact_5480_le__add__same__cancel1,axiom,
% 0.34/0.64 ! [A2: code_integer,B: code_integer] :
% 0.34/0.64 ( ( ord_le3102999989581377725nteger @ A2 @ ( plus_p5714425477246183910nteger @ A2 @ B ) )
% 0.34/0.64 = ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ B ) ) ).
% 0.34/0.64
% 0.34/0.64 % le_add_same_cancel1
% 0.34/0.64 thf(fact_5481_le__add__same__cancel1,axiom,
% 0.34/0.64 ! [A2: rat,B: rat] :
% 0.34/0.64 ( ( ord_less_eq_rat @ A2 @ ( plus_plus_rat @ A2 @ B ) )
% 0.34/0.64 = ( ord_less_eq_rat @ zero_zero_rat @ B ) ) ).
% 0.34/0.64
% 0.34/0.64 % le_add_same_cancel1
% 0.34/0.64 thf(fact_5482_le__add__same__cancel1,axiom,
% 0.34/0.64 ! [A2: nat,B: nat] :
% 0.34/0.64 ( ( ord_less_eq_nat @ A2 @ ( plus_plus_nat @ A2 @ B ) )
% 0.34/0.64 = ( ord_less_eq_nat @ zero_zero_nat @ B ) ) ).
% 0.34/0.64
% 0.34/0.64 % le_add_same_cancel1
% 0.34/0.64 thf(fact_5483_le__add__same__cancel1,axiom,
% 0.34/0.64 ! [A2: int,B: int] :
% 0.34/0.64 ( ( ord_less_eq_int @ A2 @ ( plus_plus_int @ A2 @ B ) )
% 0.34/0.64 = ( ord_less_eq_int @ zero_zero_int @ B ) ) ).
% 0.34/0.64
% 0.34/0.64 % le_add_same_cancel1
% 0.34/0.64 thf(fact_5484_add__le__same__cancel2,axiom,
% 0.34/0.64 ! [A2: code_integer,B: code_integer] :
% 0.34/0.64 ( ( ord_le3102999989581377725nteger @ ( plus_p5714425477246183910nteger @ A2 @ B ) @ B )
% 0.34/0.64 = ( ord_le3102999989581377725nteger @ A2 @ zero_z3403309356797280102nteger ) ) ).
% 0.34/0.64
% 0.34/0.64 % add_le_same_cancel2
% 0.34/0.64 thf(fact_5485_add__le__same__cancel2,axiom,
% 0.34/0.64 ! [A2: rat,B: rat] :
% 0.34/0.64 ( ( ord_less_eq_rat @ ( plus_plus_rat @ A2 @ B ) @ B )
% 0.34/0.64 = ( ord_less_eq_rat @ A2 @ zero_zero_rat ) ) ).
% 0.34/0.64
% 0.34/0.64 % add_le_same_cancel2
% 0.34/0.64 thf(fact_5486_add__le__same__cancel2,axiom,
% 0.34/0.64 ! [A2: nat,B: nat] :
% 0.34/0.64 ( ( ord_less_eq_nat @ ( plus_plus_nat @ A2 @ B ) @ B )
% 0.34/0.64 = ( ord_less_eq_nat @ A2 @ zero_zero_nat ) ) ).
% 0.34/0.64
% 0.34/0.64 % add_le_same_cancel2
% 0.34/0.64 thf(fact_5487_add__le__same__cancel2,axiom,
% 0.34/0.64 ! [A2: int,B: int] :
% 0.34/0.64 ( ( ord_less_eq_int @ ( plus_plus_int @ A2 @ B ) @ B )
% 0.34/0.64 = ( ord_less_eq_int @ A2 @ zero_zero_int ) ) ).
% 0.34/0.64
% 0.34/0.64 % add_le_same_cancel2
% 0.34/0.64 thf(fact_5488_add__le__same__cancel1,axiom,
% 0.34/0.64 ! [B: code_integer,A2: code_integer] :
% 0.34/0.64 ( ( ord_le3102999989581377725nteger @ ( plus_p5714425477246183910nteger @ B @ A2 ) @ B )
% 0.34/0.64 = ( ord_le3102999989581377725nteger @ A2 @ zero_z3403309356797280102nteger ) ) ).
% 0.34/0.64
% 0.34/0.64 % add_le_same_cancel1
% 0.34/0.64 thf(fact_5489_add__le__same__cancel1,axiom,
% 0.34/0.64 ! [B: rat,A2: rat] :
% 0.34/0.64 ( ( ord_less_eq_rat @ ( plus_plus_rat @ B @ A2 ) @ B )
% 0.34/0.64 = ( ord_less_eq_rat @ A2 @ zero_zero_rat ) ) ).
% 0.34/0.64
% 0.34/0.64 % add_le_same_cancel1
% 0.34/0.64 thf(fact_5490_add__le__same__cancel1,axiom,
% 0.34/0.64 ! [B: nat,A2: nat] :
% 0.34/0.64 ( ( ord_less_eq_nat @ ( plus_plus_nat @ B @ A2 ) @ B )
% 0.34/0.64 = ( ord_less_eq_nat @ A2 @ zero_zero_nat ) ) ).
% 0.34/0.64
% 0.34/0.64 % add_le_same_cancel1
% 0.34/0.64 thf(fact_5491_add__le__same__cancel1,axiom,
% 0.34/0.64 ! [B: int,A2: int] :
% 0.34/0.64 ( ( ord_less_eq_int @ ( plus_plus_int @ B @ A2 ) @ B )
% 0.34/0.64 = ( ord_less_eq_int @ A2 @ zero_zero_int ) ) ).
% 0.34/0.64
% 0.34/0.64 % add_le_same_cancel1
% 0.34/0.64 thf(fact_5492_diff__ge__0__iff__ge,axiom,
% 0.34/0.64 ! [A2: code_integer,B: code_integer] :
% 0.34/0.64 ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ ( minus_8373710615458151222nteger @ A2 @ B ) )
% 0.34/0.64 = ( ord_le3102999989581377725nteger @ B @ A2 ) ) ).
% 0.34/0.64
% 0.34/0.64 % diff_ge_0_iff_ge
% 0.34/0.64 thf(fact_5493_diff__ge__0__iff__ge,axiom,
% 0.34/0.64 ! [A2: rat,B: rat] :
% 0.34/0.64 ( ( ord_less_eq_rat @ zero_zero_rat @ ( minus_minus_rat @ A2 @ B ) )
% 0.34/0.64 = ( ord_less_eq_rat @ B @ A2 ) ) ).
% 0.34/0.64
% 0.34/0.64 % diff_ge_0_iff_ge
% 0.34/0.64 thf(fact_5494_diff__ge__0__iff__ge,axiom,
% 0.34/0.64 ! [A2: int,B: int] :
% 0.34/0.64 ( ( ord_less_eq_int @ zero_zero_int @ ( minus_minus_int @ A2 @ B ) )
% 0.34/0.64 = ( ord_less_eq_int @ B @ A2 ) ) ).
% 0.34/0.64
% 0.34/0.64 % diff_ge_0_iff_ge
% 0.34/0.64 thf(fact_5495_zero__comp__diff__simps_I1_J,axiom,
% 0.34/0.64 ! [A2: code_integer,B: code_integer] :
% 0.34/0.64 ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ ( minus_8373710615458151222nteger @ A2 @ B ) )
% 0.34/0.64 = ( ord_le3102999989581377725nteger @ B @ A2 ) ) ).
% 0.34/0.64
% 0.34/0.64 % zero_comp_diff_simps(1)
% 0.34/0.64 thf(fact_5496_zero__comp__diff__simps_I1_J,axiom,
% 0.34/0.64 ! [A2: rat,B: rat] :
% 0.34/0.64 ( ( ord_less_eq_rat @ zero_zero_rat @ ( minus_minus_rat @ A2 @ B ) )
% 0.34/0.64 = ( ord_less_eq_rat @ B @ A2 ) ) ).
% 0.34/0.64
% 0.34/0.64 % zero_comp_diff_simps(1)
% 0.34/0.64 thf(fact_5497_zero__comp__diff__simps_I1_J,axiom,
% 0.34/0.64 ! [A2: int,B: int] :
% 0.34/0.64 ( ( ord_less_eq_int @ zero_zero_int @ ( minus_minus_int @ A2 @ B ) )
% 0.34/0.64 = ( ord_less_eq_int @ B @ A2 ) ) ).
% 0.34/0.64
% 0.34/0.64 % zero_comp_diff_simps(1)
% 0.34/0.64 thf(fact_5498_zero__less__double__add__iff__zero__less__single__add,axiom,
% 0.34/0.64 ! [A2: int] :
% 0.34/0.64 ( ( ord_less_int @ zero_zero_int @ ( plus_plus_int @ A2 @ A2 ) )
% 0.34/0.64 = ( ord_less_int @ zero_zero_int @ A2 ) ) ).
% 0.34/0.64
% 0.34/0.64 % zero_less_double_add_iff_zero_less_single_add
% 0.34/0.64 thf(fact_5499_zero__less__double__add__iff__zero__less__single__add,axiom,
% 0.34/0.64 ! [A2: code_integer] :
% 0.34/0.64 ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ ( plus_p5714425477246183910nteger @ A2 @ A2 ) )
% 0.34/0.64 = ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ A2 ) ) ).
% 0.34/0.64
% 0.34/0.64 % zero_less_double_add_iff_zero_less_single_add
% 0.34/0.64 thf(fact_5500_zero__less__double__add__iff__zero__less__single__add,axiom,
% 0.34/0.64 ! [A2: rat] :
% 0.34/0.64 ( ( ord_less_rat @ zero_zero_rat @ ( plus_plus_rat @ A2 @ A2 ) )
% 0.34/0.64 = ( ord_less_rat @ zero_zero_rat @ A2 ) ) ).
% 0.34/0.64
% 0.34/0.64 % zero_less_double_add_iff_zero_less_single_add
% 0.34/0.64 thf(fact_5501_double__add__less__zero__iff__single__add__less__zero,axiom,
% 0.34/0.64 ! [A2: int] :
% 0.34/0.64 ( ( ord_less_int @ ( plus_plus_int @ A2 @ A2 ) @ zero_zero_int )
% 0.34/0.64 = ( ord_less_int @ A2 @ zero_zero_int ) ) ).
% 0.34/0.64
% 0.34/0.64 % double_add_less_zero_iff_single_add_less_zero
% 0.34/0.64 thf(fact_5502_double__add__less__zero__iff__single__add__less__zero,axiom,
% 0.34/0.64 ! [A2: code_integer] :
% 0.34/0.64 ( ( ord_le6747313008572928689nteger @ ( plus_p5714425477246183910nteger @ A2 @ A2 ) @ zero_z3403309356797280102nteger )
% 0.34/0.64 = ( ord_le6747313008572928689nteger @ A2 @ zero_z3403309356797280102nteger ) ) ).
% 0.34/0.64
% 0.34/0.64 % double_add_less_zero_iff_single_add_less_zero
% 0.34/0.64 thf(fact_5503_double__add__less__zero__iff__single__add__less__zero,axiom,
% 0.34/0.64 ! [A2: rat] :
% 0.34/0.64 ( ( ord_less_rat @ ( plus_plus_rat @ A2 @ A2 ) @ zero_zero_rat )
% 0.34/0.64 = ( ord_less_rat @ A2 @ zero_zero_rat ) ) ).
% 0.34/0.64
% 0.34/0.64 % double_add_less_zero_iff_single_add_less_zero
% 0.34/0.64 thf(fact_5504_less__add__same__cancel2,axiom,
% 0.34/0.64 ! [A2: nat,B: nat] :
% 0.34/0.64 ( ( ord_less_nat @ A2 @ ( plus_plus_nat @ B @ A2 ) )
% 0.34/0.64 = ( ord_less_nat @ zero_zero_nat @ B ) ) ).
% 0.34/0.64
% 0.34/0.64 % less_add_same_cancel2
% 0.34/0.64 thf(fact_5505_less__add__same__cancel2,axiom,
% 0.34/0.64 ! [A2: int,B: int] :
% 0.34/0.64 ( ( ord_less_int @ A2 @ ( plus_plus_int @ B @ A2 ) )
% 0.34/0.64 = ( ord_less_int @ zero_zero_int @ B ) ) ).
% 0.34/0.64
% 0.34/0.64 % less_add_same_cancel2
% 0.34/0.64 thf(fact_5506_less__add__same__cancel2,axiom,
% 0.34/0.64 ! [A2: code_integer,B: code_integer] :
% 0.34/0.64 ( ( ord_le6747313008572928689nteger @ A2 @ ( plus_p5714425477246183910nteger @ B @ A2 ) )
% 0.34/0.64 = ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ B ) ) ).
% 0.34/0.64
% 0.34/0.64 % less_add_same_cancel2
% 0.34/0.64 thf(fact_5507_less__add__same__cancel2,axiom,
% 0.34/0.64 ! [A2: rat,B: rat] :
% 0.34/0.64 ( ( ord_less_rat @ A2 @ ( plus_plus_rat @ B @ A2 ) )
% 0.34/0.64 = ( ord_less_rat @ zero_zero_rat @ B ) ) ).
% 0.34/0.64
% 0.34/0.64 % less_add_same_cancel2
% 0.34/0.64 thf(fact_5508_less__add__same__cancel1,axiom,
% 0.34/0.64 ! [A2: nat,B: nat] :
% 0.34/0.64 ( ( ord_less_nat @ A2 @ ( plus_plus_nat @ A2 @ B ) )
% 0.34/0.64 = ( ord_less_nat @ zero_zero_nat @ B ) ) ).
% 0.34/0.64
% 0.34/0.64 % less_add_same_cancel1
% 0.34/0.64 thf(fact_5509_less__add__same__cancel1,axiom,
% 0.34/0.64 ! [A2: int,B: int] :
% 0.34/0.64 ( ( ord_less_int @ A2 @ ( plus_plus_int @ A2 @ B ) )
% 0.34/0.64 = ( ord_less_int @ zero_zero_int @ B ) ) ).
% 0.34/0.64
% 0.34/0.64 % less_add_same_cancel1
% 0.34/0.64 thf(fact_5510_less__add__same__cancel1,axiom,
% 0.34/0.64 ! [A2: code_integer,B: code_integer] :
% 0.34/0.64 ( ( ord_le6747313008572928689nteger @ A2 @ ( plus_p5714425477246183910nteger @ A2 @ B ) )
% 0.34/0.64 = ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ B ) ) ).
% 0.34/0.64
% 0.34/0.64 % less_add_same_cancel1
% 0.34/0.64 thf(fact_5511_less__add__same__cancel1,axiom,
% 0.34/0.64 ! [A2: rat,B: rat] :
% 0.34/0.64 ( ( ord_less_rat @ A2 @ ( plus_plus_rat @ A2 @ B ) )
% 0.34/0.64 = ( ord_less_rat @ zero_zero_rat @ B ) ) ).
% 0.34/0.64
% 0.34/0.64 % less_add_same_cancel1
% 0.34/0.64 thf(fact_5512_add__less__same__cancel2,axiom,
% 0.34/0.64 ! [A2: nat,B: nat] :
% 0.34/0.64 ( ( ord_less_nat @ ( plus_plus_nat @ A2 @ B ) @ B )
% 0.34/0.64 = ( ord_less_nat @ A2 @ zero_zero_nat ) ) ).
% 0.34/0.64
% 0.34/0.64 % add_less_same_cancel2
% 0.34/0.64 thf(fact_5513_add__less__same__cancel2,axiom,
% 0.34/0.64 ! [A2: int,B: int] :
% 0.34/0.64 ( ( ord_less_int @ ( plus_plus_int @ A2 @ B ) @ B )
% 0.34/0.64 = ( ord_less_int @ A2 @ zero_zero_int ) ) ).
% 0.34/0.64
% 0.34/0.64 % add_less_same_cancel2
% 0.34/0.64 thf(fact_5514_add__less__same__cancel2,axiom,
% 0.34/0.64 ! [A2: code_integer,B: code_integer] :
% 0.34/0.64 ( ( ord_le6747313008572928689nteger @ ( plus_p5714425477246183910nteger @ A2 @ B ) @ B )
% 0.34/0.64 = ( ord_le6747313008572928689nteger @ A2 @ zero_z3403309356797280102nteger ) ) ).
% 0.34/0.64
% 0.34/0.64 % add_less_same_cancel2
% 0.34/0.64 thf(fact_5515_add__less__same__cancel2,axiom,
% 0.34/0.64 ! [A2: rat,B: rat] :
% 0.34/0.64 ( ( ord_less_rat @ ( plus_plus_rat @ A2 @ B ) @ B )
% 0.34/0.64 = ( ord_less_rat @ A2 @ zero_zero_rat ) ) ).
% 0.34/0.64
% 0.34/0.64 % add_less_same_cancel2
% 0.34/0.64 thf(fact_5516_add__less__same__cancel1,axiom,
% 0.34/0.64 ! [B: nat,A2: nat] :
% 0.34/0.64 ( ( ord_less_nat @ ( plus_plus_nat @ B @ A2 ) @ B )
% 0.34/0.64 = ( ord_less_nat @ A2 @ zero_zero_nat ) ) ).
% 0.34/0.64
% 0.34/0.64 % add_less_same_cancel1
% 0.34/0.64 thf(fact_5517_add__less__same__cancel1,axiom,
% 0.34/0.64 ! [B: int,A2: int] :
% 0.34/0.64 ( ( ord_less_int @ ( plus_plus_int @ B @ A2 ) @ B )
% 0.34/0.64 = ( ord_less_int @ A2 @ zero_zero_int ) ) ).
% 0.34/0.64
% 0.34/0.64 % add_less_same_cancel1
% 0.34/0.64 thf(fact_5518_add__less__same__cancel1,axiom,
% 0.34/0.64 ! [B: code_integer,A2: code_integer] :
% 0.34/0.64 ( ( ord_le6747313008572928689nteger @ ( plus_p5714425477246183910nteger @ B @ A2 ) @ B )
% 0.34/0.64 = ( ord_le6747313008572928689nteger @ A2 @ zero_z3403309356797280102nteger ) ) ).
% 0.34/0.64
% 0.34/0.64 % add_less_same_cancel1
% 0.34/0.64 thf(fact_5519_add__less__same__cancel1,axiom,
% 0.34/0.64 ! [B: rat,A2: rat] :
% 0.34/0.64 ( ( ord_less_rat @ ( plus_plus_rat @ B @ A2 ) @ B )
% 0.34/0.64 = ( ord_less_rat @ A2 @ zero_zero_rat ) ) ).
% 0.34/0.64
% 0.34/0.64 % add_less_same_cancel1
% 0.34/0.64 thf(fact_5520_zero__comp__diff__simps_I2_J,axiom,
% 0.34/0.64 ! [A2: int,B: int] :
% 0.34/0.64 ( ( ord_less_int @ zero_zero_int @ ( minus_minus_int @ A2 @ B ) )
% 0.34/0.64 = ( ord_less_int @ B @ A2 ) ) ).
% 0.34/0.64
% 0.34/0.64 % zero_comp_diff_simps(2)
% 0.34/0.64 thf(fact_5521_zero__comp__diff__simps_I2_J,axiom,
% 0.34/0.64 ! [A2: code_integer,B: code_integer] :
% 0.34/0.64 ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ ( minus_8373710615458151222nteger @ A2 @ B ) )
% 0.34/0.64 = ( ord_le6747313008572928689nteger @ B @ A2 ) ) ).
% 0.34/0.64
% 0.34/0.64 % zero_comp_diff_simps(2)
% 0.34/0.64 thf(fact_5522_zero__comp__diff__simps_I2_J,axiom,
% 0.34/0.64 ! [A2: rat,B: rat] :
% 0.34/0.64 ( ( ord_less_rat @ zero_zero_rat @ ( minus_minus_rat @ A2 @ B ) )
% 0.34/0.64 = ( ord_less_rat @ B @ A2 ) ) ).
% 0.34/0.64
% 0.34/0.64 % zero_comp_diff_simps(2)
% 0.34/0.64 thf(fact_5523_diff__gt__0__iff__gt,axiom,
% 0.34/0.64 ! [A2: int,B: int] :
% 0.34/0.64 ( ( ord_less_int @ zero_zero_int @ ( minus_minus_int @ A2 @ B ) )
% 0.34/0.64 = ( ord_less_int @ B @ A2 ) ) ).
% 0.34/0.64
% 0.34/0.64 % diff_gt_0_iff_gt
% 0.34/0.64 thf(fact_5524_diff__gt__0__iff__gt,axiom,
% 0.34/0.64 ! [A2: code_integer,B: code_integer] :
% 0.34/0.64 ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ ( minus_8373710615458151222nteger @ A2 @ B ) )
% 0.34/0.64 = ( ord_le6747313008572928689nteger @ B @ A2 ) ) ).
% 0.34/0.64
% 0.34/0.64 % diff_gt_0_iff_gt
% 0.34/0.64 thf(fact_5525_diff__gt__0__iff__gt,axiom,
% 0.34/0.64 ! [A2: rat,B: rat] :
% 0.34/0.64 ( ( ord_less_rat @ zero_zero_rat @ ( minus_minus_rat @ A2 @ B ) )
% 0.34/0.64 = ( ord_less_rat @ B @ A2 ) ) ).
% 0.34/0.64
% 0.34/0.64 % diff_gt_0_iff_gt
% 0.34/0.64 thf(fact_5526_diff__add__zero,axiom,
% 0.34/0.64 ! [A2: nat,B: nat] :
% 0.34/0.64 ( ( minus_minus_nat @ A2 @ ( plus_plus_nat @ A2 @ B ) )
% 0.34/0.64 = zero_zero_nat ) ).
% 0.34/0.64
% 0.34/0.64 % diff_add_zero
% 0.34/0.64 thf(fact_5527_mult__cancel__right2,axiom,
% 0.34/0.64 ! [A2: code_integer,C: code_integer] :
% 0.34/0.64 ( ( ( times_3573771949741848930nteger @ A2 @ C )
% 0.34/0.64 = C )
% 0.34/0.64 = ( ( C = zero_z3403309356797280102nteger )
% 0.34/0.64 | ( A2 = one_one_Code_integer ) ) ) ).
% 0.34/0.64
% 0.34/0.64 % mult_cancel_right2
% 0.34/0.64 thf(fact_5528_mult__cancel__right2,axiom,
% 0.34/0.64 ! [A2: rat,C: rat] :
% 0.34/0.64 ( ( ( times_times_rat @ A2 @ C )
% 0.34/0.64 = C )
% 0.34/0.64 = ( ( C = zero_zero_rat )
% 0.34/0.64 | ( A2 = one_one_rat ) ) ) ).
% 0.34/0.64
% 0.34/0.64 % mult_cancel_right2
% 0.34/0.64 thf(fact_5529_mult__cancel__right2,axiom,
% 0.34/0.64 ! [A2: int,C: int] :
% 0.34/0.64 ( ( ( times_times_int @ A2 @ C )
% 0.34/0.64 = C )
% 0.34/0.64 = ( ( C = zero_zero_int )
% 0.34/0.64 | ( A2 = one_one_int ) ) ) ).
% 0.34/0.64
% 0.34/0.64 % mult_cancel_right2
% 0.34/0.64 thf(fact_5530_mult__cancel__right1,axiom,
% 0.34/0.64 ! [C: code_integer,B: code_integer] :
% 0.34/0.64 ( ( C
% 0.34/0.64 = ( times_3573771949741848930nteger @ B @ C ) )
% 0.34/0.64 = ( ( C = zero_z3403309356797280102nteger )
% 0.34/0.64 | ( B = one_one_Code_integer ) ) ) ).
% 0.34/0.64
% 0.34/0.64 % mult_cancel_right1
% 0.34/0.64 thf(fact_5531_mult__cancel__right1,axiom,
% 0.34/0.64 ! [C: rat,B: rat] :
% 0.34/0.64 ( ( C
% 0.34/0.64 = ( times_times_rat @ B @ C ) )
% 0.34/0.64 = ( ( C = zero_zero_rat )
% 0.34/0.64 | ( B = one_one_rat ) ) ) ).
% 0.34/0.64
% 0.34/0.64 % mult_cancel_right1
% 0.34/0.64 thf(fact_5532_mult__cancel__right1,axiom,
% 0.34/0.64 ! [C: int,B: int] :
% 0.34/0.64 ( ( C
% 0.34/0.64 = ( times_times_int @ B @ C ) )
% 0.34/0.64 = ( ( C = zero_zero_int )
% 0.34/0.64 | ( B = one_one_int ) ) ) ).
% 0.34/0.64
% 0.34/0.64 % mult_cancel_right1
% 0.34/0.64 thf(fact_5533_mult__cancel__left2,axiom,
% 0.34/0.64 ! [C: code_integer,A2: code_integer] :
% 0.34/0.64 ( ( ( times_3573771949741848930nteger @ C @ A2 )
% 0.34/0.64 = C )
% 0.34/0.64 = ( ( C = zero_z3403309356797280102nteger )
% 0.34/0.64 | ( A2 = one_one_Code_integer ) ) ) ).
% 0.34/0.64
% 0.34/0.64 % mult_cancel_left2
% 0.34/0.64 thf(fact_5534_mult__cancel__left2,axiom,
% 0.34/0.64 ! [C: rat,A2: rat] :
% 0.34/0.64 ( ( ( times_times_rat @ C @ A2 )
% 0.34/0.64 = C )
% 0.34/0.64 = ( ( C = zero_zero_rat )
% 0.34/0.64 | ( A2 = one_one_rat ) ) ) ).
% 0.34/0.64
% 0.34/0.64 % mult_cancel_left2
% 0.34/0.64 thf(fact_5535_mult__cancel__left2,axiom,
% 0.34/0.64 ! [C: int,A2: int] :
% 0.34/0.64 ( ( ( times_times_int @ C @ A2 )
% 0.34/0.64 = C )
% 0.34/0.64 = ( ( C = zero_zero_int )
% 0.34/0.64 | ( A2 = one_one_int ) ) ) ).
% 0.34/0.64
% 0.34/0.64 % mult_cancel_left2
% 0.34/0.64 thf(fact_5536_mult__cancel__left1,axiom,
% 0.34/0.64 ! [C: code_integer,B: code_integer] :
% 0.34/0.64 ( ( C
% 0.34/0.64 = ( times_3573771949741848930nteger @ C @ B ) )
% 0.34/0.64 = ( ( C = zero_z3403309356797280102nteger )
% 0.34/0.64 | ( B = one_one_Code_integer ) ) ) ).
% 0.34/0.64
% 0.34/0.64 % mult_cancel_left1
% 0.34/0.64 thf(fact_5537_mult__cancel__left1,axiom,
% 0.34/0.64 ! [C: rat,B: rat] :
% 0.34/0.64 ( ( C
% 0.34/0.64 = ( times_times_rat @ C @ B ) )
% 0.34/0.64 = ( ( C = zero_zero_rat )
% 0.34/0.64 | ( B = one_one_rat ) ) ) ).
% 0.34/0.64
% 0.34/0.64 % mult_cancel_left1
% 0.34/0.64 thf(fact_5538_mult__cancel__left1,axiom,
% 0.34/0.64 ! [C: int,B: int] :
% 0.34/0.64 ( ( C
% 0.34/0.64 = ( times_times_int @ C @ B ) )
% 0.34/0.64 = ( ( C = zero_zero_int )
% 0.34/0.64 | ( B = one_one_int ) ) ) ).
% 0.34/0.64
% 0.34/0.64 % mult_cancel_left1
% 0.34/0.64 thf(fact_5539_image__add__0,axiom,
% 0.34/0.64 ! [S: set_rat] :
% 0.34/0.64 ( ( image_rat_rat @ ( plus_plus_rat @ zero_zero_rat ) @ S )
% 0.34/0.64 = S ) ).
% 0.34/0.64
% 0.34/0.64 % image_add_0
% 0.34/0.64 thf(fact_5540_image__add__0,axiom,
% 0.34/0.64 ! [S: set_nat] :
% 0.34/0.64 ( ( image_nat_nat2 @ ( plus_plus_nat @ zero_zero_nat ) @ S )
% 0.34/0.64 = S ) ).
% 0.34/0.64
% 0.34/0.64 % image_add_0
% 0.34/0.64 thf(fact_5541_image__add__0,axiom,
% 0.34/0.64 ! [S: set_int] :
% 0.34/0.64 ( ( image_int_int2 @ ( plus_plus_int @ zero_zero_int ) @ S )
% 0.34/0.64 = S ) ).
% 0.34/0.64
% 0.34/0.64 % image_add_0
% 0.34/0.64 thf(fact_5542_image__add__0,axiom,
% 0.34/0.64 ! [S: set_Code_integer] :
% 0.34/0.64 ( ( image_4470545334726330049nteger @ ( plus_p5714425477246183910nteger @ zero_z3403309356797280102nteger ) @ S )
% 0.34/0.64 = S ) ).
% 0.34/0.64
% 0.34/0.64 % image_add_0
% 0.34/0.64 thf(fact_5543_diff__numeral__special_I9_J,axiom,
% 0.34/0.64 ( ( minus_8373710615458151222nteger @ one_one_Code_integer @ one_one_Code_integer )
% 0.34/0.64 = zero_z3403309356797280102nteger ) ).
% 0.34/0.64
% 0.34/0.64 % diff_numeral_special(9)
% 0.34/0.64 thf(fact_5544_diff__numeral__special_I9_J,axiom,
% 0.34/0.64 ( ( minus_minus_rat @ one_one_rat @ one_one_rat )
% 0.34/0.64 = zero_zero_rat ) ).
% 0.34/0.64
% 0.34/0.64 % diff_numeral_special(9)
% 0.34/0.64 thf(fact_5545_diff__numeral__special_I9_J,axiom,
% 0.34/0.64 ( ( minus_minus_int @ one_one_int @ one_one_int )
% 0.34/0.64 = zero_zero_int ) ).
% 0.34/0.64
% 0.34/0.64 % diff_numeral_special(9)
% 0.34/0.64 thf(fact_5546_max__0__1_I2_J,axiom,
% 0.34/0.64 ( ( ord_max_Code_natural @ one_one_Code_natural @ zero_z2226904508553997617atural )
% 0.34/0.64 = one_one_Code_natural ) ).
% 0.34/0.64
% 0.34/0.64 % max_0_1(2)
% 0.34/0.64 thf(fact_5547_max__0__1_I2_J,axiom,
% 0.34/0.64 ( ( ord_max_Code_integer @ one_one_Code_integer @ zero_z3403309356797280102nteger )
% 0.34/0.64 = one_one_Code_integer ) ).
% 0.34/0.64
% 0.34/0.64 % max_0_1(2)
% 0.34/0.64 thf(fact_5548_max__0__1_I2_J,axiom,
% 0.34/0.64 ( ( ord_max_nat @ one_one_nat @ zero_zero_nat )
% 0.34/0.64 = one_one_nat ) ).
% 0.34/0.64
% 0.34/0.64 % max_0_1(2)
% 0.34/0.64 thf(fact_5549_max__0__1_I2_J,axiom,
% 0.34/0.64 ( ( ord_max_int @ one_one_int @ zero_zero_int )
% 0.34/0.64 = one_one_int ) ).
% 0.34/0.64
% 0.34/0.64 % max_0_1(2)
% 0.34/0.64 thf(fact_5550_max__0__1_I1_J,axiom,
% 0.34/0.64 ( ( ord_max_Code_natural @ zero_z2226904508553997617atural @ one_one_Code_natural )
% 0.34/0.64 = one_one_Code_natural ) ).
% 0.34/0.64
% 0.34/0.64 % max_0_1(1)
% 0.34/0.64 thf(fact_5551_max__0__1_I1_J,axiom,
% 0.34/0.64 ( ( ord_max_Code_integer @ zero_z3403309356797280102nteger @ one_one_Code_integer )
% 0.34/0.64 = one_one_Code_integer ) ).
% 0.34/0.64
% 0.34/0.64 % max_0_1(1)
% 0.34/0.64 thf(fact_5552_max__0__1_I1_J,axiom,
% 0.34/0.64 ( ( ord_max_nat @ zero_zero_nat @ one_one_nat )
% 0.34/0.64 = one_one_nat ) ).
% 0.34/0.64
% 0.34/0.64 % max_0_1(1)
% 0.34/0.64 thf(fact_5553_max__0__1_I1_J,axiom,
% 0.34/0.64 ( ( ord_max_int @ zero_zero_int @ one_one_int )
% 0.34/0.64 = one_one_int ) ).
% 0.34/0.64
% 0.34/0.64 % max_0_1(1)
% 0.34/0.64 thf(fact_5554_card_Oempty,axiom,
% 0.34/0.64 ( ( finite410649719033368117t_unit @ bot_bo3957492148770167129t_unit )
% 0.34/0.64 = zero_zero_nat ) ).
% 0.34/0.64
% 0.34/0.64 % card.empty
% 0.34/0.64 thf(fact_5555_card_Oempty,axiom,
% 0.34/0.64 ( ( finite_card_o @ bot_bot_set_o )
% 0.34/0.64 = zero_zero_nat ) ).
% 0.34/0.64
% 0.34/0.64 % card.empty
% 0.34/0.64 thf(fact_5556_card_Oempty,axiom,
% 0.34/0.64 ( ( finite_card_nat @ bot_bot_set_nat )
% 0.34/0.64 = zero_zero_nat ) ).
% 0.34/0.64
% 0.34/0.64 % card.empty
% 0.34/0.64 thf(fact_5557_card_Oempty,axiom,
% 0.34/0.64 ( ( finite_card_int @ bot_bot_set_int )
% 0.34/0.64 = zero_zero_nat ) ).
% 0.34/0.64
% 0.34/0.64 % card.empty
% 0.34/0.64 thf(fact_5558_card_Oempty,axiom,
% 0.34/0.64 ( ( finite1363419556375932405_nat_o @ bot_bo7824918357723582233_nat_o )
% 0.34/0.64 = zero_zero_nat ) ).
% 0.34/0.64
% 0.34/0.64 % card.empty
% 0.34/0.64 thf(fact_5559_one__eq__mult__iff,axiom,
% 0.34/0.64 ! [M4: nat,N2: nat] :
% 0.34/0.64 ( ( ( suc @ zero_zero_nat )
% 0.34/0.64 = ( times_times_nat @ M4 @ N2 ) )
% 0.34/0.64 = ( ( M4
% 0.34/0.64 = ( suc @ zero_zero_nat ) )
% 0.34/0.64 & ( N2
% 0.34/0.64 = ( suc @ zero_zero_nat ) ) ) ) ).
% 0.34/0.64
% 0.34/0.64 % one_eq_mult_iff
% 0.34/0.64 thf(fact_5560_mult__eq__1__iff,axiom,
% 0.34/0.64 ! [M4: nat,N2: nat] :
% 0.34/0.64 ( ( ( times_times_nat @ M4 @ N2 )
% 0.34/0.64 = ( suc @ zero_zero_nat ) )
% 0.34/0.64 = ( ( M4
% 0.34/0.64 = ( suc @ zero_zero_nat ) )
% 0.34/0.64 & ( N2
% 0.34/0.64 = ( suc @ zero_zero_nat ) ) ) ) ).
% 0.34/0.64
% 0.34/0.64 % mult_eq_1_iff
% 0.34/0.64 thf(fact_5561_nat__mult__less__cancel__disj,axiom,
% 0.34/0.64 ! [K3: nat,M4: nat,N2: nat] :
% 0.34/0.64 ( ( ord_less_nat @ ( times_times_nat @ K3 @ M4 ) @ ( times_times_nat @ K3 @ N2 ) )
% 0.34/0.64 = ( ( ord_less_nat @ zero_zero_nat @ K3 )
% 0.34/0.64 & ( ord_less_nat @ M4 @ N2 ) ) ) ).
% 0.34/0.64
% 0.34/0.64 % nat_mult_less_cancel_disj
% 0.34/0.64 thf(fact_5562_nat__0__less__mult__iff,axiom,
% 0.34/0.64 ! [M4: nat,N2: nat] :
% 0.34/0.64 ( ( ord_less_nat @ zero_zero_nat @ ( times_times_nat @ M4 @ N2 ) )
% 0.34/0.64 = ( ( ord_less_nat @ zero_zero_nat @ M4 )
% 0.34/0.64 & ( ord_less_nat @ zero_zero_nat @ N2 ) ) ) ).
% 0.34/0.64
% 0.34/0.64 % nat_0_less_mult_iff
% 0.34/0.64 thf(fact_5563_mult__less__cancel2,axiom,
% 0.34/0.64 ! [M4: nat,K3: nat,N2: nat] :
% 0.34/0.64 ( ( ord_less_nat @ ( times_times_nat @ M4 @ K3 ) @ ( times_times_nat @ N2 @ K3 ) )
% 0.34/0.64 = ( ( ord_less_nat @ zero_zero_nat @ K3 )
% 0.34/0.64 & ( ord_less_nat @ M4 @ N2 ) ) ) ).
% 0.34/0.64
% 0.34/0.64 % mult_less_cancel2
% 0.34/0.64 thf(fact_5564_card__0__eq,axiom,
% 0.34/0.64 ! [A: set_Code_integer] :
% 0.34/0.64 ( ( finite6017078050557962740nteger @ A )
% 0.34/0.64 => ( ( ( finite4902975817058060853nteger @ A )
% 0.34/0.64 = zero_zero_nat )
% 0.34/0.64 = ( A = bot_bo3990330152332043303nteger ) ) ) ).
% 0.34/0.64
% 0.34/0.64 % card_0_eq
% 0.34/0.64 thf(fact_5565_card__0__eq,axiom,
% 0.34/0.64 ! [A: set_Product_unit] :
% 0.34/0.64 ( ( finite4290736615968046902t_unit @ A )
% 0.34/0.64 => ( ( ( finite410649719033368117t_unit @ A )
% 0.34/0.64 = zero_zero_nat )
% 0.34/0.64 = ( A = bot_bo3957492148770167129t_unit ) ) ) ).
% 0.34/0.64
% 0.34/0.64 % card_0_eq
% 0.34/0.64 thf(fact_5566_card__0__eq,axiom,
% 0.34/0.64 ! [A: set_o] :
% 0.34/0.64 ( ( finite_finite_o @ A )
% 0.34/0.64 => ( ( ( finite_card_o @ A )
% 0.34/0.64 = zero_zero_nat )
% 0.34/0.64 = ( A = bot_bot_set_o ) ) ) ).
% 0.34/0.64
% 0.34/0.64 % card_0_eq
% 0.34/0.64 thf(fact_5567_card__0__eq,axiom,
% 0.34/0.64 ! [A: set_nat] :
% 0.34/0.64 ( ( finite_finite_nat @ A )
% 0.34/0.64 => ( ( ( finite_card_nat @ A )
% 0.34/0.64 = zero_zero_nat )
% 0.34/0.64 = ( A = bot_bot_set_nat ) ) ) ).
% 0.34/0.64
% 0.34/0.64 % card_0_eq
% 0.34/0.64 thf(fact_5568_card__0__eq,axiom,
% 0.34/0.64 ! [A: set_int] :
% 0.34/0.64 ( ( finite_finite_int @ A )
% 0.34/0.64 => ( ( ( finite_card_int @ A )
% 0.34/0.64 = zero_zero_nat )
% 0.34/0.64 = ( A = bot_bot_set_int ) ) ) ).
% 0.34/0.64
% 0.34/0.64 % card_0_eq
% 0.34/0.64 thf(fact_5569_card__0__eq,axiom,
% 0.34/0.64 ! [A: set_Pr4532377907799695533_nat_o] :
% 0.34/0.64 ( ( finite3252695134891459830_nat_o @ A )
% 0.34/0.64 => ( ( ( finite1363419556375932405_nat_o @ A )
% 0.34/0.64 = zero_zero_nat )
% 0.34/0.64 = ( A = bot_bo7824918357723582233_nat_o ) ) ) ).
% 0.34/0.64
% 0.34/0.64 % card_0_eq
% 0.34/0.64 thf(fact_5570_one__le__mult__iff,axiom,
% 0.34/0.64 ! [M4: nat,N2: nat] :
% 0.34/0.64 ( ( ord_less_eq_nat @ ( suc @ zero_zero_nat ) @ ( times_times_nat @ M4 @ N2 ) )
% 0.34/0.64 = ( ( ord_less_eq_nat @ ( suc @ zero_zero_nat ) @ M4 )
% 0.34/0.64 & ( ord_less_eq_nat @ ( suc @ zero_zero_nat ) @ N2 ) ) ) ).
% 0.34/0.64
% 0.34/0.64 % one_le_mult_iff
% 0.34/0.64 thf(fact_5571_mult__le__cancel2,axiom,
% 0.34/0.64 ! [M4: nat,K3: nat,N2: nat] :
% 0.34/0.64 ( ( ord_less_eq_nat @ ( times_times_nat @ M4 @ K3 ) @ ( times_times_nat @ N2 @ K3 ) )
% 0.34/0.64 = ( ( ord_less_nat @ zero_zero_nat @ K3 )
% 0.34/0.64 => ( ord_less_eq_nat @ M4 @ N2 ) ) ) ).
% 0.34/0.64
% 0.34/0.64 % mult_le_cancel2
% 0.34/0.64 thf(fact_5572_nat__mult__le__cancel__disj,axiom,
% 0.34/0.64 ! [K3: nat,M4: nat,N2: nat] :
% 0.34/0.64 ( ( ord_less_eq_nat @ ( times_times_nat @ K3 @ M4 ) @ ( times_times_nat @ K3 @ N2 ) )
% 0.34/0.64 = ( ( ord_less_nat @ zero_zero_nat @ K3 )
% 0.34/0.64 => ( ord_less_eq_nat @ M4 @ N2 ) ) ) ).
% 0.34/0.64
% 0.34/0.64 % nat_mult_le_cancel_disj
% 0.34/0.64 thf(fact_5573_bot__nat__def,axiom,
% 0.34/0.64 bot_bot_nat = zero_zero_nat ).
% 0.34/0.64
% 0.34/0.64 % bot_nat_def
% 0.34/0.64 thf(fact_5574_zero__reorient,axiom,
% 0.34/0.64 ! [X: nat] :
% 0.34/0.64 ( ( zero_zero_nat = X )
% 0.34/0.64 = ( X = zero_zero_nat ) ) ).
% 0.34/0.64
% 0.34/0.64 % zero_reorient
% 0.34/0.64 thf(fact_5575_zero__reorient,axiom,
% 0.34/0.64 ! [X: int] :
% 0.34/0.64 ( ( zero_zero_int = X )
% 0.34/0.64 = ( X = zero_zero_int ) ) ).
% 0.34/0.64
% 0.34/0.64 % zero_reorient
% 0.34/0.64 thf(fact_5576_zero__reorient,axiom,
% 0.34/0.64 ! [X: code_integer] :
% 0.34/0.64 ( ( zero_z3403309356797280102nteger = X )
% 0.34/0.64 = ( X = zero_z3403309356797280102nteger ) ) ).
% 0.34/0.64
% 0.34/0.64 % zero_reorient
% 0.34/0.64 thf(fact_5577_zero__le,axiom,
% 0.34/0.64 ! [X: nat] : ( ord_less_eq_nat @ zero_zero_nat @ X ) ).
% 0.34/0.64
% 0.34/0.64 % zero_le
% 0.34/0.64 thf(fact_5578_gr__zeroI,axiom,
% 0.34/0.64 ! [N2: nat] :
% 0.34/0.64 ( ( N2 != zero_zero_nat )
% 0.34/0.64 => ( ord_less_nat @ zero_zero_nat @ N2 ) ) ).
% 0.34/0.64
% 0.34/0.64 % gr_zeroI
% 0.34/0.64 thf(fact_5579_not__less__zero,axiom,
% 0.34/0.64 ! [N2: nat] :
% 0.34/0.64 ~ ( ord_less_nat @ N2 @ zero_zero_nat ) ).
% 0.34/0.64
% 0.34/0.64 % not_less_zero
% 0.34/0.64 thf(fact_5580_gr__implies__not__zero,axiom,
% 0.34/0.64 ! [M4: nat,N2: nat] :
% 0.34/0.64 ( ( ord_less_nat @ M4 @ N2 )
% 0.34/0.64 => ( N2 != zero_zero_nat ) ) ).
% 0.34/0.64
% 0.34/0.64 % gr_implies_not_zero
% 0.34/0.64 thf(fact_5581_zero__less__iff__neq__zero,axiom,
% 0.34/0.64 ! [N2: nat] :
% 0.34/0.64 ( ( ord_less_nat @ zero_zero_nat @ N2 )
% 0.34/0.64 = ( N2 != zero_zero_nat ) ) ).
% 0.34/0.64
% 0.34/0.64 % zero_less_iff_neq_zero
% 0.34/0.64 thf(fact_5582_comm__monoid__add__class_Oadd__0,axiom,
% 0.34/0.64 ! [A2: rat] :
% 0.34/0.64 ( ( plus_plus_rat @ zero_zero_rat @ A2 )
% 0.34/0.64 = A2 ) ).
% 0.34/0.64
% 0.34/0.64 % comm_monoid_add_class.add_0
% 0.34/0.64 thf(fact_5583_comm__monoid__add__class_Oadd__0,axiom,
% 0.34/0.64 ! [A2: nat] :
% 0.34/0.64 ( ( plus_plus_nat @ zero_zero_nat @ A2 )
% 0.34/0.64 = A2 ) ).
% 0.34/0.64
% 0.34/0.64 % comm_monoid_add_class.add_0
% 0.34/0.64 thf(fact_5584_comm__monoid__add__class_Oadd__0,axiom,
% 0.34/0.64 ! [A2: int] :
% 0.34/0.64 ( ( plus_plus_int @ zero_zero_int @ A2 )
% 0.34/0.64 = A2 ) ).
% 0.34/0.64
% 0.34/0.64 % comm_monoid_add_class.add_0
% 0.34/0.64 thf(fact_5585_comm__monoid__add__class_Oadd__0,axiom,
% 0.34/0.64 ! [A2: code_integer] :
% 0.34/0.64 ( ( plus_p5714425477246183910nteger @ zero_z3403309356797280102nteger @ A2 )
% 0.34/0.64 = A2 ) ).
% 0.34/0.64
% 0.34/0.64 % comm_monoid_add_class.add_0
% 0.34/0.64 thf(fact_5586_add_Ocomm__neutral,axiom,
% 0.34/0.64 ! [A2: rat] :
% 0.34/0.64 ( ( plus_plus_rat @ A2 @ zero_zero_rat )
% 0.34/0.64 = A2 ) ).
% 0.34/0.64
% 0.34/0.64 % add.comm_neutral
% 0.34/0.64 thf(fact_5587_add_Ocomm__neutral,axiom,
% 0.34/0.64 ! [A2: nat] :
% 0.34/0.64 ( ( plus_plus_nat @ A2 @ zero_zero_nat )
% 0.34/0.64 = A2 ) ).
% 0.34/0.64
% 0.34/0.64 % add.comm_neutral
% 0.34/0.64 thf(fact_5588_add_Ocomm__neutral,axiom,
% 0.34/0.64 ! [A2: int] :
% 0.34/0.64 ( ( plus_plus_int @ A2 @ zero_zero_int )
% 0.34/0.64 = A2 ) ).
% 0.34/0.64
% 0.34/0.64 % add.comm_neutral
% 0.34/0.64 thf(fact_5589_add_Ocomm__neutral,axiom,
% 0.34/0.64 ! [A2: code_integer] :
% 0.34/0.64 ( ( plus_p5714425477246183910nteger @ A2 @ zero_z3403309356797280102nteger )
% 0.34/0.64 = A2 ) ).
% 0.34/0.64
% 0.34/0.64 % add.comm_neutral
% 0.34/0.64 thf(fact_5590_add_Ogroup__left__neutral,axiom,
% 0.34/0.64 ! [A2: rat] :
% 0.34/0.64 ( ( plus_plus_rat @ zero_zero_rat @ A2 )
% 0.34/0.64 = A2 ) ).
% 0.34/0.64
% 0.34/0.64 % add.group_left_neutral
% 0.34/0.64 thf(fact_5591_add_Ogroup__left__neutral,axiom,
% 0.34/0.64 ! [A2: int] :
% 0.34/0.64 ( ( plus_plus_int @ zero_zero_int @ A2 )
% 0.34/0.64 = A2 ) ).
% 0.34/0.64
% 0.34/0.64 % add.group_left_neutral
% 0.34/0.64 thf(fact_5592_add_Ogroup__left__neutral,axiom,
% 0.34/0.64 ! [A2: code_integer] :
% 0.34/0.64 ( ( plus_p5714425477246183910nteger @ zero_z3403309356797280102nteger @ A2 )
% 0.34/0.64 = A2 ) ).
% 0.34/0.64
% 0.34/0.64 % add.group_left_neutral
% 0.34/0.64 thf(fact_5593_mult__right__cancel,axiom,
% 0.34/0.64 ! [C: code_integer,A2: code_integer,B: code_integer] :
% 0.34/0.64 ( ( C != zero_z3403309356797280102nteger )
% 0.34/0.64 => ( ( ( times_3573771949741848930nteger @ A2 @ C )
% 0.34/0.64 = ( times_3573771949741848930nteger @ B @ C ) )
% 0.34/0.64 = ( A2 = B ) ) ) ).
% 0.34/0.64
% 0.34/0.64 % mult_right_cancel
% 0.34/0.64 thf(fact_5594_mult__right__cancel,axiom,
% 0.34/0.64 ! [C: rat,A2: rat,B: rat] :
% 0.34/0.64 ( ( C != zero_zero_rat )
% 0.34/0.64 => ( ( ( times_times_rat @ A2 @ C )
% 0.34/0.64 = ( times_times_rat @ B @ C ) )
% 0.34/0.64 = ( A2 = B ) ) ) ).
% 0.34/0.64
% 0.34/0.64 % mult_right_cancel
% 0.34/0.64 thf(fact_5595_mult__right__cancel,axiom,
% 0.34/0.64 ! [C: nat,A2: nat,B: nat] :
% 0.34/0.64 ( ( C != zero_zero_nat )
% 0.34/0.64 => ( ( ( times_times_nat @ A2 @ C )
% 0.34/0.64 = ( times_times_nat @ B @ C ) )
% 0.34/0.64 = ( A2 = B ) ) ) ).
% 0.34/0.64
% 0.34/0.64 % mult_right_cancel
% 0.34/0.64 thf(fact_5596_mult__right__cancel,axiom,
% 0.34/0.64 ! [C: int,A2: int,B: int] :
% 0.34/0.64 ( ( C != zero_zero_int )
% 0.34/0.64 => ( ( ( times_times_int @ A2 @ C )
% 0.34/0.64 = ( times_times_int @ B @ C ) )
% 0.34/0.64 = ( A2 = B ) ) ) ).
% 0.34/0.64
% 0.34/0.64 % mult_right_cancel
% 0.34/0.64 thf(fact_5597_mult__left__cancel,axiom,
% 0.34/0.64 ! [C: code_integer,A2: code_integer,B: code_integer] :
% 0.34/0.64 ( ( C != zero_z3403309356797280102nteger )
% 0.34/0.64 => ( ( ( times_3573771949741848930nteger @ C @ A2 )
% 0.34/0.64 = ( times_3573771949741848930nteger @ C @ B ) )
% 0.34/0.64 = ( A2 = B ) ) ) ).
% 0.34/0.64
% 0.34/0.64 % mult_left_cancel
% 0.34/0.64 thf(fact_5598_mult__left__cancel,axiom,
% 0.34/0.64 ! [C: rat,A2: rat,B: rat] :
% 0.34/0.64 ( ( C != zero_zero_rat )
% 0.34/0.64 => ( ( ( times_times_rat @ C @ A2 )
% 0.34/0.64 = ( times_times_rat @ C @ B ) )
% 0.34/0.64 = ( A2 = B ) ) ) ).
% 0.34/0.64
% 0.34/0.64 % mult_left_cancel
% 0.34/0.64 thf(fact_5599_mult__left__cancel,axiom,
% 0.34/0.64 ! [C: nat,A2: nat,B: nat] :
% 0.34/0.64 ( ( C != zero_zero_nat )
% 0.34/0.64 => ( ( ( times_times_nat @ C @ A2 )
% 0.34/0.64 = ( times_times_nat @ C @ B ) )
% 0.34/0.64 = ( A2 = B ) ) ) ).
% 0.34/0.64
% 0.34/0.64 % mult_left_cancel
% 0.34/0.64 thf(fact_5600_mult__left__cancel,axiom,
% 0.34/0.64 ! [C: int,A2: int,B: int] :
% 0.34/0.64 ( ( C != zero_zero_int )
% 0.34/0.64 => ( ( ( times_times_int @ C @ A2 )
% 0.34/0.64 = ( times_times_int @ C @ B ) )
% 0.34/0.64 = ( A2 = B ) ) ) ).
% 0.34/0.64
% 0.34/0.64 % mult_left_cancel
% 0.34/0.64 thf(fact_5601_no__zero__divisors,axiom,
% 0.34/0.64 ! [A2: code_integer,B: code_integer] :
% 0.34/0.64 ( ( A2 != zero_z3403309356797280102nteger )
% 0.34/0.64 => ( ( B != zero_z3403309356797280102nteger )
% 0.34/0.64 => ( ( times_3573771949741848930nteger @ A2 @ B )
% 0.34/0.64 != zero_z3403309356797280102nteger ) ) ) ).
% 0.34/0.64
% 0.34/0.64 % no_zero_divisors
% 0.34/0.64 thf(fact_5602_no__zero__divisors,axiom,
% 0.34/0.64 ! [A2: rat,B: rat] :
% 0.34/0.64 ( ( A2 != zero_zero_rat )
% 0.34/0.64 => ( ( B != zero_zero_rat )
% 0.34/0.64 => ( ( times_times_rat @ A2 @ B )
% 0.34/0.64 != zero_zero_rat ) ) ) ).
% 0.34/0.64
% 0.34/0.64 % no_zero_divisors
% 0.34/0.64 thf(fact_5603_no__zero__divisors,axiom,
% 0.34/0.64 ! [A2: nat,B: nat] :
% 0.34/0.64 ( ( A2 != zero_zero_nat )
% 0.34/0.64 => ( ( B != zero_zero_nat )
% 0.34/0.64 => ( ( times_times_nat @ A2 @ B )
% 0.34/0.64 != zero_zero_nat ) ) ) ).
% 0.34/0.64
% 0.34/0.64 % no_zero_divisors
% 0.34/0.64 thf(fact_5604_no__zero__divisors,axiom,
% 0.34/0.64 ! [A2: int,B: int] :
% 0.34/0.64 ( ( A2 != zero_zero_int )
% 0.34/0.64 => ( ( B != zero_zero_int )
% 0.34/0.64 => ( ( times_times_int @ A2 @ B )
% 0.34/0.64 != zero_zero_int ) ) ) ).
% 0.34/0.64
% 0.34/0.64 % no_zero_divisors
% 0.34/0.64 thf(fact_5605_divisors__zero,axiom,
% 0.34/0.64 ! [A2: code_integer,B: code_integer] :
% 0.34/0.64 ( ( ( times_3573771949741848930nteger @ A2 @ B )
% 0.34/0.64 = zero_z3403309356797280102nteger )
% 0.34/0.64 => ( ( A2 = zero_z3403309356797280102nteger )
% 0.34/0.64 | ( B = zero_z3403309356797280102nteger ) ) ) ).
% 0.34/0.64
% 0.34/0.64 % divisors_zero
% 0.34/0.64 thf(fact_5606_divisors__zero,axiom,
% 0.34/0.64 ! [A2: rat,B: rat] :
% 0.34/0.64 ( ( ( times_times_rat @ A2 @ B )
% 0.34/0.64 = zero_zero_rat )
% 0.34/0.64 => ( ( A2 = zero_zero_rat )
% 0.34/0.64 | ( B = zero_zero_rat ) ) ) ).
% 0.34/0.64
% 0.34/0.64 % divisors_zero
% 0.34/0.64 thf(fact_5607_divisors__zero,axiom,
% 0.34/0.64 ! [A2: nat,B: nat] :
% 0.34/0.64 ( ( ( times_times_nat @ A2 @ B )
% 0.34/0.64 = zero_zero_nat )
% 0.34/0.64 => ( ( A2 = zero_zero_nat )
% 0.34/0.64 | ( B = zero_zero_nat ) ) ) ).
% 0.34/0.64
% 0.34/0.64 % divisors_zero
% 0.34/0.64 thf(fact_5608_divisors__zero,axiom,
% 0.34/0.64 ! [A2: int,B: int] :
% 0.34/0.64 ( ( ( times_times_int @ A2 @ B )
% 0.34/0.64 = zero_zero_int )
% 0.34/0.64 => ( ( A2 = zero_zero_int )
% 0.34/0.64 | ( B = zero_zero_int ) ) ) ).
% 0.34/0.64
% 0.34/0.64 % divisors_zero
% 0.34/0.64 thf(fact_5609_mult__not__zero,axiom,
% 0.34/0.64 ! [A2: code_integer,B: code_integer] :
% 0.34/0.64 ( ( ( times_3573771949741848930nteger @ A2 @ B )
% 0.34/0.64 != zero_z3403309356797280102nteger )
% 0.34/0.64 => ( ( A2 != zero_z3403309356797280102nteger )
% 0.34/0.64 & ( B != zero_z3403309356797280102nteger ) ) ) ).
% 0.34/0.64
% 0.34/0.64 % mult_not_zero
% 0.34/0.64 thf(fact_5610_mult__not__zero,axiom,
% 0.34/0.64 ! [A2: rat,B: rat] :
% 0.34/0.64 ( ( ( times_times_rat @ A2 @ B )
% 0.34/0.64 != zero_zero_rat )
% 0.34/0.64 => ( ( A2 != zero_zero_rat )
% 0.34/0.64 & ( B != zero_zero_rat ) ) ) ).
% 0.34/0.64
% 0.34/0.64 % mult_not_zero
% 0.34/0.64 thf(fact_5611_mult__not__zero,axiom,
% 0.34/0.64 ! [A2: nat,B: nat] :
% 0.34/0.64 ( ( ( times_times_nat @ A2 @ B )
% 0.34/0.64 != zero_zero_nat )
% 0.34/0.64 => ( ( A2 != zero_zero_nat )
% 0.34/0.64 & ( B != zero_zero_nat ) ) ) ).
% 0.34/0.64
% 0.34/0.64 % mult_not_zero
% 0.34/0.64 thf(fact_5612_mult__not__zero,axiom,
% 0.34/0.64 ! [A2: int,B: int] :
% 0.34/0.64 ( ( ( times_times_int @ A2 @ B )
% 0.34/0.64 != zero_zero_int )
% 0.34/0.64 => ( ( A2 != zero_zero_int )
% 0.34/0.64 & ( B != zero_zero_int ) ) ) ).
% 0.34/0.64
% 0.34/0.64 % mult_not_zero
% 0.34/0.64 thf(fact_5613_zero__neq__one,axiom,
% 0.34/0.64 zero_z2226904508553997617atural != one_one_Code_natural ).
% 0.34/0.64
% 0.34/0.64 % zero_neq_one
% 0.34/0.64 thf(fact_5614_zero__neq__one,axiom,
% 0.34/0.64 zero_zero_nat != one_one_nat ).
% 0.34/0.64
% 0.34/0.64 % zero_neq_one
% 0.34/0.64 thf(fact_5615_zero__neq__one,axiom,
% 0.34/0.64 zero_zero_int != one_one_int ).
% 0.34/0.64
% 0.34/0.64 % zero_neq_one
% 0.34/0.64 thf(fact_5616_zero__neq__one,axiom,
% 0.34/0.64 zero_z3403309356797280102nteger != one_one_Code_integer ).
% 0.34/0.64
% 0.34/0.64 % zero_neq_one
% 0.34/0.64 thf(fact_5617_eq__iff__diff__eq__0,axiom,
% 0.34/0.64 ( ( ^ [Y4: code_integer,Z4: code_integer] : ( Y4 = Z4 ) )
% 0.34/0.64 = ( ^ [A6: code_integer,B6: code_integer] :
% 0.34/0.64 ( ( minus_8373710615458151222nteger @ A6 @ B6 )
% 0.34/0.64 = zero_z3403309356797280102nteger ) ) ) ).
% 0.34/0.64
% 0.34/0.64 % eq_iff_diff_eq_0
% 0.34/0.64 thf(fact_5618_eq__iff__diff__eq__0,axiom,
% 0.34/0.64 ( ( ^ [Y4: rat,Z4: rat] : ( Y4 = Z4 ) )
% 0.34/0.64 = ( ^ [A6: rat,B6: rat] :
% 0.34/0.64 ( ( minus_minus_rat @ A6 @ B6 )
% 0.34/0.64 = zero_zero_rat ) ) ) ).
% 0.34/0.64
% 0.34/0.64 % eq_iff_diff_eq_0
% 0.34/0.64 thf(fact_5619_eq__iff__diff__eq__0,axiom,
% 0.34/0.64 ( ( ^ [Y4: int,Z4: int] : ( Y4 = Z4 ) )
% 0.34/0.64 = ( ^ [A6: int,B6: int] :
% 0.34/0.64 ( ( minus_minus_int @ A6 @ B6 )
% 0.34/0.64 = zero_zero_int ) ) ) ).
% 0.34/0.64
% 0.34/0.64 % eq_iff_diff_eq_0
% 0.34/0.64 thf(fact_5620_mult__0,axiom,
% 0.34/0.64 ! [N2: nat] :
% 0.34/0.64 ( ( times_times_nat @ zero_zero_nat @ N2 )
% 0.34/0.64 = zero_zero_nat ) ).
% 0.34/0.64
% 0.34/0.64 % mult_0
% 0.34/0.64 thf(fact_5621_nat__mult__eq__cancel__disj,axiom,
% 0.34/0.64 ! [K3: nat,M4: nat,N2: nat] :
% 0.34/0.64 ( ( ( times_times_nat @ K3 @ M4 )
% 0.34/0.64 = ( times_times_nat @ K3 @ N2 ) )
% 0.34/0.64 = ( ( K3 = zero_zero_nat )
% 0.34/0.64 | ( M4 = N2 ) ) ) ).
% 0.34/0.64
% 0.34/0.64 % nat_mult_eq_cancel_disj
% 0.34/0.64 thf(fact_5622_add__scale__eq__noteq,axiom,
% 0.34/0.64 ! [R2: code_integer,A2: code_integer,B: code_integer,C: code_integer,D2: code_integer] :
% 0.34/0.64 ( ( R2 != zero_z3403309356797280102nteger )
% 0.34/0.64 => ( ( ( A2 = B )
% 0.34/0.64 & ( C != D2 ) )
% 0.34/0.64 => ( ( plus_p5714425477246183910nteger @ A2 @ ( times_3573771949741848930nteger @ R2 @ C ) )
% 0.34/0.64 != ( plus_p5714425477246183910nteger @ B @ ( times_3573771949741848930nteger @ R2 @ D2 ) ) ) ) ) ).
% 0.34/0.64
% 0.34/0.64 % add_scale_eq_noteq
% 0.34/0.64 thf(fact_5623_add__scale__eq__noteq,axiom,
% 0.34/0.64 ! [R2: rat,A2: rat,B: rat,C: rat,D2: rat] :
% 0.34/0.64 ( ( R2 != zero_zero_rat )
% 0.34/0.64 => ( ( ( A2 = B )
% 0.34/0.64 & ( C != D2 ) )
% 0.34/0.64 => ( ( plus_plus_rat @ A2 @ ( times_times_rat @ R2 @ C ) )
% 0.34/0.64 != ( plus_plus_rat @ B @ ( times_times_rat @ R2 @ D2 ) ) ) ) ) ).
% 0.34/0.64
% 0.34/0.64 % add_scale_eq_noteq
% 0.34/0.64 thf(fact_5624_add__scale__eq__noteq,axiom,
% 0.34/0.64 ! [R2: nat,A2: nat,B: nat,C: nat,D2: nat] :
% 0.34/0.64 ( ( R2 != zero_zero_nat )
% 0.34/0.64 => ( ( ( A2 = B )
% 0.34/0.64 & ( C != D2 ) )
% 0.34/0.64 => ( ( plus_plus_nat @ A2 @ ( times_times_nat @ R2 @ C ) )
% 0.34/0.64 != ( plus_plus_nat @ B @ ( times_times_nat @ R2 @ D2 ) ) ) ) ) ).
% 0.34/0.64
% 0.34/0.64 % add_scale_eq_noteq
% 0.34/0.64 thf(fact_5625_add__scale__eq__noteq,axiom,
% 0.34/0.64 ! [R2: int,A2: int,B: int,C: int,D2: int] :
% 0.34/0.64 ( ( R2 != zero_zero_int )
% 0.34/0.64 => ( ( ( A2 = B )
% 0.34/0.64 & ( C != D2 ) )
% 0.34/0.64 => ( ( plus_plus_int @ A2 @ ( times_times_int @ R2 @ C ) )
% 0.34/0.64 != ( plus_plus_int @ B @ ( times_times_int @ R2 @ D2 ) ) ) ) ) ).
% 0.34/0.64
% 0.34/0.64 % add_scale_eq_noteq
% 0.34/0.64 thf(fact_5626_Sup__nat__def,axiom,
% 0.34/0.64 ( complete_Sup_Sup_nat
% 0.34/0.64 = ( ^ [X7: set_nat] : ( if_nat @ ( X7 = bot_bot_set_nat ) @ zero_zero_nat @ ( lattic8265883725875713057ax_nat @ X7 ) ) ) ) ).
% 0.34/0.64
% 0.34/0.64 % Sup_nat_def
% 0.34/0.64 thf(fact_5627_add__decreasing,axiom,
% 0.34/0.64 ! [A2: code_integer,C: code_integer,B: code_integer] :
% 0.34/0.64 ( ( ord_le3102999989581377725nteger @ A2 @ zero_z3403309356797280102nteger )
% 0.34/0.64 => ( ( ord_le3102999989581377725nteger @ C @ B )
% 0.34/0.64 => ( ord_le3102999989581377725nteger @ ( plus_p5714425477246183910nteger @ A2 @ C ) @ B ) ) ) ).
% 0.34/0.64
% 0.34/0.64 % add_decreasing
% 0.34/0.64 thf(fact_5628_add__decreasing,axiom,
% 0.34/0.64 ! [A2: rat,C: rat,B: rat] :
% 0.34/0.64 ( ( ord_less_eq_rat @ A2 @ zero_zero_rat )
% 0.34/0.64 => ( ( ord_less_eq_rat @ C @ B )
% 0.34/0.64 => ( ord_less_eq_rat @ ( plus_plus_rat @ A2 @ C ) @ B ) ) ) ).
% 0.34/0.64
% 0.34/0.64 % add_decreasing
% 0.34/0.64 thf(fact_5629_add__decreasing,axiom,
% 0.34/0.64 ! [A2: nat,C: nat,B: nat] :
% 0.34/0.64 ( ( ord_less_eq_nat @ A2 @ zero_zero_nat )
% 0.34/0.64 => ( ( ord_less_eq_nat @ C @ B )
% 0.34/0.64 => ( ord_less_eq_nat @ ( plus_plus_nat @ A2 @ C ) @ B ) ) ) ).
% 0.34/0.64
% 0.34/0.64 % add_decreasing
% 0.34/0.64 thf(fact_5630_add__decreasing,axiom,
% 0.34/0.64 ! [A2: int,C: int,B: int] :
% 0.34/0.64 ( ( ord_less_eq_int @ A2 @ zero_zero_int )
% 0.34/0.64 => ( ( ord_less_eq_int @ C @ B )
% 0.34/0.64 => ( ord_less_eq_int @ ( plus_plus_int @ A2 @ C ) @ B ) ) ) ).
% 0.34/0.64
% 0.34/0.64 % add_decreasing
% 0.34/0.64 thf(fact_5631_add__increasing,axiom,
% 0.34/0.64 ! [A2: code_integer,B: code_integer,C: code_integer] :
% 0.34/0.64 ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ A2 )
% 0.34/0.64 => ( ( ord_le3102999989581377725nteger @ B @ C )
% 0.34/0.64 => ( ord_le3102999989581377725nteger @ B @ ( plus_p5714425477246183910nteger @ A2 @ C ) ) ) ) ).
% 0.34/0.64
% 0.34/0.64 % add_increasing
% 0.34/0.64 thf(fact_5632_add__increasing,axiom,
% 0.34/0.64 ! [A2: rat,B: rat,C: rat] :
% 0.34/0.64 ( ( ord_less_eq_rat @ zero_zero_rat @ A2 )
% 0.34/0.64 => ( ( ord_less_eq_rat @ B @ C )
% 0.34/0.64 => ( ord_less_eq_rat @ B @ ( plus_plus_rat @ A2 @ C ) ) ) ) ).
% 0.34/0.64
% 0.34/0.64 % add_increasing
% 0.34/0.64 thf(fact_5633_add__increasing,axiom,
% 0.34/0.64 ! [A2: nat,B: nat,C: nat] :
% 0.34/0.64 ( ( ord_less_eq_nat @ zero_zero_nat @ A2 )
% 0.34/0.64 => ( ( ord_less_eq_nat @ B @ C )
% 0.34/0.64 => ( ord_less_eq_nat @ B @ ( plus_plus_nat @ A2 @ C ) ) ) ) ).
% 0.34/0.64
% 0.34/0.64 % add_increasing
% 0.34/0.64 thf(fact_5634_add__increasing,axiom,
% 0.34/0.64 ! [A2: int,B: int,C: int] :
% 0.34/0.64 ( ( ord_less_eq_int @ zero_zero_int @ A2 )
% 0.34/0.64 => ( ( ord_less_eq_int @ B @ C )
% 0.34/0.64 => ( ord_less_eq_int @ B @ ( plus_plus_int @ A2 @ C ) ) ) ) ).
% 0.34/0.64
% 0.34/0.64 % add_increasing
% 0.34/0.64 thf(fact_5635_add__decreasing2,axiom,
% 0.34/0.64 ! [C: code_integer,A2: code_integer,B: code_integer] :
% 0.34/0.64 ( ( ord_le3102999989581377725nteger @ C @ zero_z3403309356797280102nteger )
% 0.34/0.64 => ( ( ord_le3102999989581377725nteger @ A2 @ B )
% 0.34/0.64 => ( ord_le3102999989581377725nteger @ ( plus_p5714425477246183910nteger @ A2 @ C ) @ B ) ) ) ).
% 0.34/0.64
% 0.34/0.64 % add_decreasing2
% 0.34/0.64 thf(fact_5636_add__decreasing2,axiom,
% 0.34/0.64 ! [C: rat,A2: rat,B: rat] :
% 0.34/0.64 ( ( ord_less_eq_rat @ C @ zero_zero_rat )
% 0.34/0.64 => ( ( ord_less_eq_rat @ A2 @ B )
% 0.34/0.64 => ( ord_less_eq_rat @ ( plus_plus_rat @ A2 @ C ) @ B ) ) ) ).
% 0.34/0.64
% 0.34/0.64 % add_decreasing2
% 0.34/0.64 thf(fact_5637_add__decreasing2,axiom,
% 0.34/0.64 ! [C: nat,A2: nat,B: nat] :
% 0.34/0.64 ( ( ord_less_eq_nat @ C @ zero_zero_nat )
% 0.34/0.64 => ( ( ord_less_eq_nat @ A2 @ B )
% 0.34/0.64 => ( ord_less_eq_nat @ ( plus_plus_nat @ A2 @ C ) @ B ) ) ) ).
% 0.34/0.64
% 0.34/0.64 % add_decreasing2
% 0.34/0.64 thf(fact_5638_add__decreasing2,axiom,
% 0.34/0.64 ! [C: int,A2: int,B: int] :
% 0.34/0.64 ( ( ord_less_eq_int @ C @ zero_zero_int )
% 0.34/0.64 => ( ( ord_less_eq_int @ A2 @ B )
% 0.34/0.64 => ( ord_less_eq_int @ ( plus_plus_int @ A2 @ C ) @ B ) ) ) ).
% 0.34/0.64
% 0.34/0.64 % add_decreasing2
% 0.34/0.64 thf(fact_5639_add__increasing2,axiom,
% 0.34/0.64 ! [C: code_integer,B: code_integer,A2: code_integer] :
% 0.34/0.64 ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ C )
% 0.34/0.64 => ( ( ord_le3102999989581377725nteger @ B @ A2 )
% 0.34/0.64 => ( ord_le3102999989581377725nteger @ B @ ( plus_p5714425477246183910nteger @ A2 @ C ) ) ) ) ).
% 0.34/0.64
% 0.34/0.64 % add_increasing2
% 0.34/0.64 thf(fact_5640_add__increasing2,axiom,
% 0.34/0.64 ! [C: rat,B: rat,A2: rat] :
% 0.34/0.64 ( ( ord_less_eq_rat @ zero_zero_rat @ C )
% 0.34/0.64 => ( ( ord_less_eq_rat @ B @ A2 )
% 0.34/0.64 => ( ord_less_eq_rat @ B @ ( plus_plus_rat @ A2 @ C ) ) ) ) ).
% 0.34/0.64
% 0.34/0.64 % add_increasing2
% 0.34/0.64 thf(fact_5641_add__increasing2,axiom,
% 0.34/0.64 ! [C: nat,B: nat,A2: nat] :
% 0.34/0.64 ( ( ord_less_eq_nat @ zero_zero_nat @ C )
% 0.34/0.64 => ( ( ord_less_eq_nat @ B @ A2 )
% 0.34/0.64 => ( ord_less_eq_nat @ B @ ( plus_plus_nat @ A2 @ C ) ) ) ) ).
% 0.34/0.64
% 0.34/0.64 % add_increasing2
% 0.34/0.64 thf(fact_5642_add__increasing2,axiom,
% 0.34/0.64 ! [C: int,B: int,A2: int] :
% 0.34/0.64 ( ( ord_less_eq_int @ zero_zero_int @ C )
% 0.34/0.64 => ( ( ord_less_eq_int @ B @ A2 )
% 0.34/0.64 => ( ord_less_eq_int @ B @ ( plus_plus_int @ A2 @ C ) ) ) ) ).
% 0.34/0.64
% 0.34/0.64 % add_increasing2
% 0.34/0.64 thf(fact_5643_add__nonneg__nonneg,axiom,
% 0.34/0.64 ! [A2: code_integer,B: code_integer] :
% 0.34/0.64 ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ A2 )
% 0.34/0.64 => ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ B )
% 0.34/0.64 => ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ ( plus_p5714425477246183910nteger @ A2 @ B ) ) ) ) ).
% 0.34/0.64
% 0.34/0.64 % add_nonneg_nonneg
% 0.34/0.64 thf(fact_5644_add__nonneg__nonneg,axiom,
% 0.34/0.64 ! [A2: rat,B: rat] :
% 0.34/0.64 ( ( ord_less_eq_rat @ zero_zero_rat @ A2 )
% 0.34/0.64 => ( ( ord_less_eq_rat @ zero_zero_rat @ B )
% 0.34/0.64 => ( ord_less_eq_rat @ zero_zero_rat @ ( plus_plus_rat @ A2 @ B ) ) ) ) ).
% 0.34/0.64
% 0.34/0.64 % add_nonneg_nonneg
% 0.34/0.64 thf(fact_5645_add__nonneg__nonneg,axiom,
% 0.34/0.64 ! [A2: nat,B: nat] :
% 0.34/0.64 ( ( ord_less_eq_nat @ zero_zero_nat @ A2 )
% 0.34/0.64 => ( ( ord_less_eq_nat @ zero_zero_nat @ B )
% 0.34/0.64 => ( ord_less_eq_nat @ zero_zero_nat @ ( plus_plus_nat @ A2 @ B ) ) ) ) ).
% 0.34/0.64
% 0.34/0.64 % add_nonneg_nonneg
% 0.34/0.64 thf(fact_5646_add__nonneg__nonneg,axiom,
% 0.34/0.64 ! [A2: int,B: int] :
% 0.34/0.64 ( ( ord_less_eq_int @ zero_zero_int @ A2 )
% 0.34/0.64 => ( ( ord_less_eq_int @ zero_zero_int @ B )
% 0.34/0.64 => ( ord_less_eq_int @ zero_zero_int @ ( plus_plus_int @ A2 @ B ) ) ) ) ).
% 0.34/0.64
% 0.34/0.64 % add_nonneg_nonneg
% 0.34/0.64 thf(fact_5647_add__nonpos__nonpos,axiom,
% 0.34/0.64 ! [A2: code_integer,B: code_integer] :
% 0.34/0.64 ( ( ord_le3102999989581377725nteger @ A2 @ zero_z3403309356797280102nteger )
% 0.34/0.64 => ( ( ord_le3102999989581377725nteger @ B @ zero_z3403309356797280102nteger )
% 0.34/0.64 => ( ord_le3102999989581377725nteger @ ( plus_p5714425477246183910nteger @ A2 @ B ) @ zero_z3403309356797280102nteger ) ) ) ).
% 0.34/0.64
% 0.34/0.64 % add_nonpos_nonpos
% 0.34/0.64 thf(fact_5648_add__nonpos__nonpos,axiom,
% 0.34/0.64 ! [A2: rat,B: rat] :
% 0.34/0.64 ( ( ord_less_eq_rat @ A2 @ zero_zero_rat )
% 0.34/0.64 => ( ( ord_less_eq_rat @ B @ zero_zero_rat )
% 0.34/0.64 => ( ord_less_eq_rat @ ( plus_plus_rat @ A2 @ B ) @ zero_zero_rat ) ) ) ).
% 0.34/0.64
% 0.34/0.64 % add_nonpos_nonpos
% 0.34/0.64 thf(fact_5649_add__nonpos__nonpos,axiom,
% 0.34/0.64 ! [A2: nat,B: nat] :
% 0.34/0.64 ( ( ord_less_eq_nat @ A2 @ zero_zero_nat )
% 0.34/0.64 => ( ( ord_less_eq_nat @ B @ zero_zero_nat )
% 0.34/0.64 => ( ord_less_eq_nat @ ( plus_plus_nat @ A2 @ B ) @ zero_zero_nat ) ) ) ).
% 0.34/0.64
% 0.34/0.64 % add_nonpos_nonpos
% 0.34/0.64 thf(fact_5650_add__nonpos__nonpos,axiom,
% 0.34/0.64 ! [A2: int,B: int] :
% 0.34/0.64 ( ( ord_less_eq_int @ A2 @ zero_zero_int )
% 0.34/0.64 => ( ( ord_less_eq_int @ B @ zero_zero_int )
% 0.34/0.64 => ( ord_less_eq_int @ ( plus_plus_int @ A2 @ B ) @ zero_zero_int ) ) ) ).
% 0.34/0.64
% 0.34/0.64 % add_nonpos_nonpos
% 0.34/0.64 thf(fact_5651_add__nonneg__eq__0__iff,axiom,
% 0.34/0.64 ! [X: code_integer,Y: code_integer] :
% 0.34/0.64 ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ X )
% 0.34/0.64 => ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ Y )
% 0.34/0.64 => ( ( ( plus_p5714425477246183910nteger @ X @ Y )
% 0.34/0.64 = zero_z3403309356797280102nteger )
% 0.34/0.64 = ( ( X = zero_z3403309356797280102nteger )
% 0.34/0.64 & ( Y = zero_z3403309356797280102nteger ) ) ) ) ) ).
% 0.34/0.64
% 0.34/0.64 % add_nonneg_eq_0_iff
% 0.34/0.64 thf(fact_5652_add__nonneg__eq__0__iff,axiom,
% 0.34/0.64 ! [X: rat,Y: rat] :
% 0.34/0.64 ( ( ord_less_eq_rat @ zero_zero_rat @ X )
% 0.34/0.64 => ( ( ord_less_eq_rat @ zero_zero_rat @ Y )
% 0.34/0.64 => ( ( ( plus_plus_rat @ X @ Y )
% 0.34/0.64 = zero_zero_rat )
% 0.34/0.64 = ( ( X = zero_zero_rat )
% 0.34/0.64 & ( Y = zero_zero_rat ) ) ) ) ) ).
% 0.34/0.64
% 0.34/0.64 % add_nonneg_eq_0_iff
% 0.34/0.64 thf(fact_5653_add__nonneg__eq__0__iff,axiom,
% 0.34/0.64 ! [X: nat,Y: nat] :
% 0.34/0.64 ( ( ord_less_eq_nat @ zero_zero_nat @ X )
% 0.34/0.64 => ( ( ord_less_eq_nat @ zero_zero_nat @ Y )
% 0.34/0.64 => ( ( ( plus_plus_nat @ X @ Y )
% 0.34/0.64 = zero_zero_nat )
% 0.34/0.64 = ( ( X = zero_zero_nat )
% 0.34/0.64 & ( Y = zero_zero_nat ) ) ) ) ) ).
% 0.34/0.64
% 0.34/0.64 % add_nonneg_eq_0_iff
% 0.34/0.64 thf(fact_5654_add__nonneg__eq__0__iff,axiom,
% 0.34/0.64 ! [X: int,Y: int] :
% 0.34/0.64 ( ( ord_less_eq_int @ zero_zero_int @ X )
% 0.34/0.64 => ( ( ord_less_eq_int @ zero_zero_int @ Y )
% 0.34/0.64 => ( ( ( plus_plus_int @ X @ Y )
% 0.34/0.64 = zero_zero_int )
% 0.34/0.64 = ( ( X = zero_zero_int )
% 0.34/0.64 & ( Y = zero_zero_int ) ) ) ) ) ).
% 0.34/0.64
% 0.34/0.64 % add_nonneg_eq_0_iff
% 0.34/0.64 thf(fact_5655_add__nonpos__eq__0__iff,axiom,
% 0.34/0.64 ! [X: code_integer,Y: code_integer] :
% 0.34/0.64 ( ( ord_le3102999989581377725nteger @ X @ zero_z3403309356797280102nteger )
% 0.34/0.64 => ( ( ord_le3102999989581377725nteger @ Y @ zero_z3403309356797280102nteger )
% 0.34/0.64 => ( ( ( plus_p5714425477246183910nteger @ X @ Y )
% 0.34/0.64 = zero_z3403309356797280102nteger )
% 0.34/0.64 = ( ( X = zero_z3403309356797280102nteger )
% 0.34/0.64 & ( Y = zero_z3403309356797280102nteger ) ) ) ) ) ).
% 0.34/0.64
% 0.34/0.64 % add_nonpos_eq_0_iff
% 0.34/0.64 thf(fact_5656_add__nonpos__eq__0__iff,axiom,
% 0.34/0.64 ! [X: rat,Y: rat] :
% 0.34/0.64 ( ( ord_less_eq_rat @ X @ zero_zero_rat )
% 0.34/0.64 => ( ( ord_less_eq_rat @ Y @ zero_zero_rat )
% 0.34/0.64 => ( ( ( plus_plus_rat @ X @ Y )
% 0.34/0.64 = zero_zero_rat )
% 0.34/0.64 = ( ( X = zero_zero_rat )
% 0.34/0.64 & ( Y = zero_zero_rat ) ) ) ) ) ).
% 0.34/0.64
% 0.34/0.64 % add_nonpos_eq_0_iff
% 0.34/0.64 thf(fact_5657_add__nonpos__eq__0__iff,axiom,
% 0.34/0.64 ! [X: nat,Y: nat] :
% 0.34/0.64 ( ( ord_less_eq_nat @ X @ zero_zero_nat )
% 0.34/0.64 => ( ( ord_less_eq_nat @ Y @ zero_zero_nat )
% 0.34/0.64 => ( ( ( plus_plus_nat @ X @ Y )
% 0.34/0.64 = zero_zero_nat )
% 0.34/0.64 = ( ( X = zero_zero_nat )
% 0.34/0.64 & ( Y = zero_zero_nat ) ) ) ) ) ).
% 0.34/0.64
% 0.34/0.64 % add_nonpos_eq_0_iff
% 0.34/0.64 thf(fact_5658_add__nonpos__eq__0__iff,axiom,
% 0.34/0.64 ! [X: int,Y: int] :
% 0.34/0.64 ( ( ord_less_eq_int @ X @ zero_zero_int )
% 0.34/0.64 => ( ( ord_less_eq_int @ Y @ zero_zero_int )
% 0.34/0.64 => ( ( ( plus_plus_int @ X @ Y )
% 0.34/0.64 = zero_zero_int )
% 0.34/0.64 = ( ( X = zero_zero_int )
% 0.34/0.64 & ( Y = zero_zero_int ) ) ) ) ) ).
% 0.34/0.64
% 0.34/0.64 % add_nonpos_eq_0_iff
% 0.34/0.64 thf(fact_5659_ordered__comm__semiring__class_Ocomm__mult__left__mono,axiom,
% 0.34/0.64 ! [A2: code_integer,B: code_integer,C: code_integer] :
% 0.34/0.64 ( ( ord_le3102999989581377725nteger @ A2 @ B )
% 0.34/0.64 => ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ C )
% 0.34/0.64 => ( ord_le3102999989581377725nteger @ ( times_3573771949741848930nteger @ C @ A2 ) @ ( times_3573771949741848930nteger @ C @ B ) ) ) ) ).
% 0.34/0.64
% 0.34/0.64 % ordered_comm_semiring_class.comm_mult_left_mono
% 0.34/0.64 thf(fact_5660_ordered__comm__semiring__class_Ocomm__mult__left__mono,axiom,
% 0.34/0.64 ! [A2: rat,B: rat,C: rat] :
% 0.34/0.64 ( ( ord_less_eq_rat @ A2 @ B )
% 0.34/0.64 => ( ( ord_less_eq_rat @ zero_zero_rat @ C )
% 0.34/0.64 => ( ord_less_eq_rat @ ( times_times_rat @ C @ A2 ) @ ( times_times_rat @ C @ B ) ) ) ) ).
% 0.34/0.64
% 0.34/0.64 % ordered_comm_semiring_class.comm_mult_left_mono
% 0.34/0.64 thf(fact_5661_ordered__comm__semiring__class_Ocomm__mult__left__mono,axiom,
% 0.34/0.64 ! [A2: nat,B: nat,C: nat] :
% 0.34/0.64 ( ( ord_less_eq_nat @ A2 @ B )
% 0.34/0.64 => ( ( ord_less_eq_nat @ zero_zero_nat @ C )
% 0.34/0.64 => ( ord_less_eq_nat @ ( times_times_nat @ C @ A2 ) @ ( times_times_nat @ C @ B ) ) ) ) ).
% 0.34/0.64
% 0.34/0.64 % ordered_comm_semiring_class.comm_mult_left_mono
% 0.34/0.64 thf(fact_5662_ordered__comm__semiring__class_Ocomm__mult__left__mono,axiom,
% 0.34/0.64 ! [A2: int,B: int,C: int] :
% 0.34/0.64 ( ( ord_less_eq_int @ A2 @ B )
% 0.34/0.64 => ( ( ord_less_eq_int @ zero_zero_int @ C )
% 0.34/0.64 => ( ord_less_eq_int @ ( times_times_int @ C @ A2 ) @ ( times_times_int @ C @ B ) ) ) ) ).
% 0.34/0.64
% 0.34/0.64 % ordered_comm_semiring_class.comm_mult_left_mono
% 0.34/0.64 thf(fact_5663_zero__le__mult__iff,axiom,
% 0.34/0.64 ! [A2: code_integer,B: code_integer] :
% 0.34/0.64 ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ ( times_3573771949741848930nteger @ A2 @ B ) )
% 0.34/0.64 = ( ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ A2 )
% 0.34/0.64 & ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ B ) )
% 0.34/0.64 | ( ( ord_le3102999989581377725nteger @ A2 @ zero_z3403309356797280102nteger )
% 0.34/0.64 & ( ord_le3102999989581377725nteger @ B @ zero_z3403309356797280102nteger ) ) ) ) ).
% 0.34/0.64
% 0.34/0.64 % zero_le_mult_iff
% 0.34/0.64 thf(fact_5664_zero__le__mult__iff,axiom,
% 0.34/0.64 ! [A2: rat,B: rat] :
% 0.34/0.64 ( ( ord_less_eq_rat @ zero_zero_rat @ ( times_times_rat @ A2 @ B ) )
% 0.34/0.64 = ( ( ( ord_less_eq_rat @ zero_zero_rat @ A2 )
% 0.34/0.64 & ( ord_less_eq_rat @ zero_zero_rat @ B ) )
% 0.34/0.64 | ( ( ord_less_eq_rat @ A2 @ zero_zero_rat )
% 0.34/0.64 & ( ord_less_eq_rat @ B @ zero_zero_rat ) ) ) ) ).
% 0.34/0.64
% 0.34/0.64 % zero_le_mult_iff
% 0.34/0.64 thf(fact_5665_zero__le__mult__iff,axiom,
% 0.34/0.64 ! [A2: int,B: int] :
% 0.34/0.64 ( ( ord_less_eq_int @ zero_zero_int @ ( times_times_int @ A2 @ B ) )
% 0.34/0.64 = ( ( ( ord_less_eq_int @ zero_zero_int @ A2 )
% 0.34/0.64 & ( ord_less_eq_int @ zero_zero_int @ B ) )
% 0.34/0.64 | ( ( ord_less_eq_int @ A2 @ zero_zero_int )
% 0.34/0.64 & ( ord_less_eq_int @ B @ zero_zero_int ) ) ) ) ).
% 0.34/0.64
% 0.34/0.64 % zero_le_mult_iff
% 0.34/0.64 thf(fact_5666_mult__nonneg__nonpos2,axiom,
% 0.34/0.64 ! [A2: code_integer,B: code_integer] :
% 0.34/0.64 ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ A2 )
% 0.34/0.64 => ( ( ord_le3102999989581377725nteger @ B @ zero_z3403309356797280102nteger )
% 0.34/0.64 => ( ord_le3102999989581377725nteger @ ( times_3573771949741848930nteger @ B @ A2 ) @ zero_z3403309356797280102nteger ) ) ) ).
% 0.34/0.64
% 0.34/0.64 % mult_nonneg_nonpos2
% 0.34/0.64 thf(fact_5667_mult__nonneg__nonpos2,axiom,
% 0.34/0.64 ! [A2: rat,B: rat] :
% 0.34/0.64 ( ( ord_less_eq_rat @ zero_zero_rat @ A2 )
% 0.34/0.64 => ( ( ord_less_eq_rat @ B @ zero_zero_rat )
% 0.34/0.64 => ( ord_less_eq_rat @ ( times_times_rat @ B @ A2 ) @ zero_zero_rat ) ) ) ).
% 0.34/0.64
% 0.34/0.64 % mult_nonneg_nonpos2
% 0.34/0.64 thf(fact_5668_mult__nonneg__nonpos2,axiom,
% 0.34/0.64 ! [A2: nat,B: nat] :
% 0.34/0.64 ( ( ord_less_eq_nat @ zero_zero_nat @ A2 )
% 0.34/0.64 => ( ( ord_less_eq_nat @ B @ zero_zero_nat )
% 0.34/0.64 => ( ord_less_eq_nat @ ( times_times_nat @ B @ A2 ) @ zero_zero_nat ) ) ) ).
% 0.34/0.64
% 0.34/0.64 % mult_nonneg_nonpos2
% 0.34/0.64 thf(fact_5669_mult__nonneg__nonpos2,axiom,
% 0.34/0.64 ! [A2: int,B: int] :
% 0.34/0.64 ( ( ord_less_eq_int @ zero_zero_int @ A2 )
% 0.34/0.64 => ( ( ord_less_eq_int @ B @ zero_zero_int )
% 0.34/0.64 => ( ord_less_eq_int @ ( times_times_int @ B @ A2 ) @ zero_zero_int ) ) ) ).
% 0.34/0.64
% 0.34/0.64 % mult_nonneg_nonpos2
% 0.34/0.64 thf(fact_5670_mult__nonpos__nonneg,axiom,
% 0.34/0.64 ! [A2: code_integer,B: code_integer] :
% 0.34/0.64 ( ( ord_le3102999989581377725nteger @ A2 @ zero_z3403309356797280102nteger )
% 0.34/0.64 => ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ B )
% 0.34/0.64 => ( ord_le3102999989581377725nteger @ ( times_3573771949741848930nteger @ A2 @ B ) @ zero_z3403309356797280102nteger ) ) ) ).
% 0.34/0.64
% 0.34/0.64 % mult_nonpos_nonneg
% 0.34/0.64 thf(fact_5671_mult__nonpos__nonneg,axiom,
% 0.34/0.64 ! [A2: rat,B: rat] :
% 0.34/0.64 ( ( ord_less_eq_rat @ A2 @ zero_zero_rat )
% 0.34/0.64 => ( ( ord_less_eq_rat @ zero_zero_rat @ B )
% 0.34/0.64 => ( ord_less_eq_rat @ ( times_times_rat @ A2 @ B ) @ zero_zero_rat ) ) ) ).
% 0.34/0.64
% 0.34/0.64 % mult_nonpos_nonneg
% 0.34/0.64 thf(fact_5672_mult__nonpos__nonneg,axiom,
% 0.34/0.64 ! [A2: nat,B: nat] :
% 0.34/0.64 ( ( ord_less_eq_nat @ A2 @ zero_zero_nat )
% 0.34/0.64 => ( ( ord_less_eq_nat @ zero_zero_nat @ B )
% 0.34/0.64 => ( ord_less_eq_nat @ ( times_times_nat @ A2 @ B ) @ zero_zero_nat ) ) ) ).
% 0.34/0.64
% 0.34/0.64 % mult_nonpos_nonneg
% 0.34/0.64 thf(fact_5673_mult__nonpos__nonneg,axiom,
% 0.34/0.64 ! [A2: int,B: int] :
% 0.34/0.64 ( ( ord_less_eq_int @ A2 @ zero_zero_int )
% 0.34/0.64 => ( ( ord_less_eq_int @ zero_zero_int @ B )
% 0.34/0.64 => ( ord_less_eq_int @ ( times_times_int @ A2 @ B ) @ zero_zero_int ) ) ) ).
% 0.34/0.64
% 0.34/0.64 % mult_nonpos_nonneg
% 0.34/0.64 thf(fact_5674_mult__nonneg__nonpos,axiom,
% 0.34/0.64 ! [A2: code_integer,B: code_integer] :
% 0.34/0.64 ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ A2 )
% 0.34/0.64 => ( ( ord_le3102999989581377725nteger @ B @ zero_z3403309356797280102nteger )
% 0.34/0.64 => ( ord_le3102999989581377725nteger @ ( times_3573771949741848930nteger @ A2 @ B ) @ zero_z3403309356797280102nteger ) ) ) ).
% 0.34/0.64
% 0.34/0.64 % mult_nonneg_nonpos
% 0.34/0.64 thf(fact_5675_mult__nonneg__nonpos,axiom,
% 0.34/0.64 ! [A2: rat,B: rat] :
% 0.34/0.64 ( ( ord_less_eq_rat @ zero_zero_rat @ A2 )
% 0.34/0.64 => ( ( ord_less_eq_rat @ B @ zero_zero_rat )
% 0.34/0.64 => ( ord_less_eq_rat @ ( times_times_rat @ A2 @ B ) @ zero_zero_rat ) ) ) ).
% 0.34/0.64
% 0.34/0.64 % mult_nonneg_nonpos
% 0.34/0.64 thf(fact_5676_mult__nonneg__nonpos,axiom,
% 0.34/0.64 ! [A2: nat,B: nat] :
% 0.34/0.64 ( ( ord_less_eq_nat @ zero_zero_nat @ A2 )
% 0.34/0.64 => ( ( ord_less_eq_nat @ B @ zero_zero_nat )
% 0.34/0.64 => ( ord_less_eq_nat @ ( times_times_nat @ A2 @ B ) @ zero_zero_nat ) ) ) ).
% 0.34/0.64
% 0.34/0.64 % mult_nonneg_nonpos
% 0.34/0.64 thf(fact_5677_mult__nonneg__nonpos,axiom,
% 0.34/0.64 ! [A2: int,B: int] :
% 0.34/0.64 ( ( ord_less_eq_int @ zero_zero_int @ A2 )
% 0.34/0.64 => ( ( ord_less_eq_int @ B @ zero_zero_int )
% 0.34/0.64 => ( ord_less_eq_int @ ( times_times_int @ A2 @ B ) @ zero_zero_int ) ) ) ).
% 0.34/0.64
% 0.34/0.64 % mult_nonneg_nonpos
% 0.34/0.64 thf(fact_5678_mult__nonneg__nonneg,axiom,
% 0.34/0.64 ! [A2: code_integer,B: code_integer] :
% 0.34/0.64 ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ A2 )
% 0.34/0.64 => ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ B )
% 0.34/0.64 => ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ ( times_3573771949741848930nteger @ A2 @ B ) ) ) ) ).
% 0.34/0.64
% 0.34/0.64 % mult_nonneg_nonneg
% 0.34/0.64 thf(fact_5679_mult__nonneg__nonneg,axiom,
% 0.34/0.64 ! [A2: rat,B: rat] :
% 0.34/0.64 ( ( ord_less_eq_rat @ zero_zero_rat @ A2 )
% 0.34/0.64 => ( ( ord_less_eq_rat @ zero_zero_rat @ B )
% 0.34/0.64 => ( ord_less_eq_rat @ zero_zero_rat @ ( times_times_rat @ A2 @ B ) ) ) ) ).
% 0.34/0.64
% 0.34/0.64 % mult_nonneg_nonneg
% 0.34/0.64 thf(fact_5680_mult__nonneg__nonneg,axiom,
% 0.34/0.64 ! [A2: nat,B: nat] :
% 0.34/0.64 ( ( ord_less_eq_nat @ zero_zero_nat @ A2 )
% 0.34/0.64 => ( ( ord_less_eq_nat @ zero_zero_nat @ B )
% 0.34/0.64 => ( ord_less_eq_nat @ zero_zero_nat @ ( times_times_nat @ A2 @ B ) ) ) ) ).
% 0.34/0.64
% 0.34/0.64 % mult_nonneg_nonneg
% 0.34/0.64 thf(fact_5681_mult__nonneg__nonneg,axiom,
% 0.34/0.64 ! [A2: int,B: int] :
% 0.34/0.64 ( ( ord_less_eq_int @ zero_zero_int @ A2 )
% 0.34/0.64 => ( ( ord_less_eq_int @ zero_zero_int @ B )
% 0.34/0.64 => ( ord_less_eq_int @ zero_zero_int @ ( times_times_int @ A2 @ B ) ) ) ) ).
% 0.34/0.64
% 0.34/0.64 % mult_nonneg_nonneg
% 0.34/0.64 thf(fact_5682_split__mult__neg__le,axiom,
% 0.34/0.64 ! [A2: code_integer,B: code_integer] :
% 0.34/0.64 ( ( ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ A2 )
% 0.34/0.64 & ( ord_le3102999989581377725nteger @ B @ zero_z3403309356797280102nteger ) )
% 0.34/0.64 | ( ( ord_le3102999989581377725nteger @ A2 @ zero_z3403309356797280102nteger )
% 0.34/0.64 & ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ B ) ) )
% 0.34/0.64 => ( ord_le3102999989581377725nteger @ ( times_3573771949741848930nteger @ A2 @ B ) @ zero_z3403309356797280102nteger ) ) ).
% 0.34/0.64
% 0.34/0.64 % split_mult_neg_le
% 0.34/0.64 thf(fact_5683_split__mult__neg__le,axiom,
% 0.34/0.64 ! [A2: rat,B: rat] :
% 0.34/0.64 ( ( ( ( ord_less_eq_rat @ zero_zero_rat @ A2 )
% 0.34/0.64 & ( ord_less_eq_rat @ B @ zero_zero_rat ) )
% 0.34/0.64 | ( ( ord_less_eq_rat @ A2 @ zero_zero_rat )
% 0.34/0.64 & ( ord_less_eq_rat @ zero_zero_rat @ B ) ) )
% 0.34/0.64 => ( ord_less_eq_rat @ ( times_times_rat @ A2 @ B ) @ zero_zero_rat ) ) ).
% 0.34/0.64
% 0.34/0.64 % split_mult_neg_le
% 0.34/0.64 thf(fact_5684_split__mult__neg__le,axiom,
% 0.34/0.64 ! [A2: nat,B: nat] :
% 0.34/0.64 ( ( ( ( ord_less_eq_nat @ zero_zero_nat @ A2 )
% 0.34/0.64 & ( ord_less_eq_nat @ B @ zero_zero_nat ) )
% 0.34/0.64 | ( ( ord_less_eq_nat @ A2 @ zero_zero_nat )
% 0.34/0.64 & ( ord_less_eq_nat @ zero_zero_nat @ B ) ) )
% 0.34/0.64 => ( ord_less_eq_nat @ ( times_times_nat @ A2 @ B ) @ zero_zero_nat ) ) ).
% 0.34/0.64
% 0.34/0.64 % split_mult_neg_le
% 0.34/0.64 thf(fact_5685_split__mult__neg__le,axiom,
% 0.34/0.64 ! [A2: int,B: int] :
% 0.34/0.64 ( ( ( ( ord_less_eq_int @ zero_zero_int @ A2 )
% 0.34/0.64 & ( ord_less_eq_int @ B @ zero_zero_int ) )
% 0.34/0.64 | ( ( ord_less_eq_int @ A2 @ zero_zero_int )
% 0.34/0.64 & ( ord_less_eq_int @ zero_zero_int @ B ) ) )
% 0.34/0.64 => ( ord_less_eq_int @ ( times_times_int @ A2 @ B ) @ zero_zero_int ) ) ).
% 0.34/0.64
% 0.34/0.64 % split_mult_neg_le
% 0.34/0.64 thf(fact_5686_mult__le__0__iff,axiom,
% 0.34/0.64 ! [A2: code_integer,B: code_integer] :
% 0.34/0.64 ( ( ord_le3102999989581377725nteger @ ( times_3573771949741848930nteger @ A2 @ B ) @ zero_z3403309356797280102nteger )
% 0.34/0.64 = ( ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ A2 )
% 0.34/0.64 & ( ord_le3102999989581377725nteger @ B @ zero_z3403309356797280102nteger ) )
% 0.34/0.64 | ( ( ord_le3102999989581377725nteger @ A2 @ zero_z3403309356797280102nteger )
% 0.34/0.64 & ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ B ) ) ) ) ).
% 0.34/0.64
% 0.34/0.64 % mult_le_0_iff
% 0.34/0.64 thf(fact_5687_mult__le__0__iff,axiom,
% 0.34/0.64 ! [A2: rat,B: rat] :
% 0.34/0.64 ( ( ord_less_eq_rat @ ( times_times_rat @ A2 @ B ) @ zero_zero_rat )
% 0.34/0.64 = ( ( ( ord_less_eq_rat @ zero_zero_rat @ A2 )
% 0.34/0.64 & ( ord_less_eq_rat @ B @ zero_zero_rat ) )
% 0.34/0.64 | ( ( ord_less_eq_rat @ A2 @ zero_zero_rat )
% 0.34/0.64 & ( ord_less_eq_rat @ zero_zero_rat @ B ) ) ) ) ).
% 0.34/0.64
% 0.34/0.64 % mult_le_0_iff
% 0.34/0.64 thf(fact_5688_mult__le__0__iff,axiom,
% 0.34/0.64 ! [A2: int,B: int] :
% 0.34/0.64 ( ( ord_less_eq_int @ ( times_times_int @ A2 @ B ) @ zero_zero_int )
% 0.34/0.64 = ( ( ( ord_less_eq_int @ zero_zero_int @ A2 )
% 0.34/0.64 & ( ord_less_eq_int @ B @ zero_zero_int ) )
% 0.34/0.64 | ( ( ord_less_eq_int @ A2 @ zero_zero_int )
% 0.34/0.64 & ( ord_less_eq_int @ zero_zero_int @ B ) ) ) ) ).
% 0.34/0.64
% 0.34/0.64 % mult_le_0_iff
% 0.34/0.64 thf(fact_5689_mult__right__mono,axiom,
% 0.34/0.64 ! [A2: code_integer,B: code_integer,C: code_integer] :
% 0.34/0.64 ( ( ord_le3102999989581377725nteger @ A2 @ B )
% 0.34/0.64 => ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ C )
% 0.34/0.64 => ( ord_le3102999989581377725nteger @ ( times_3573771949741848930nteger @ A2 @ C ) @ ( times_3573771949741848930nteger @ B @ C ) ) ) ) ).
% 0.34/0.64
% 0.34/0.64 % mult_right_mono
% 0.34/0.64 thf(fact_5690_mult__right__mono,axiom,
% 0.34/0.64 ! [A2: rat,B: rat,C: rat] :
% 0.34/0.64 ( ( ord_less_eq_rat @ A2 @ B )
% 0.34/0.64 => ( ( ord_less_eq_rat @ zero_zero_rat @ C )
% 0.34/0.64 => ( ord_less_eq_rat @ ( times_times_rat @ A2 @ C ) @ ( times_times_rat @ B @ C ) ) ) ) ).
% 0.34/0.64
% 0.34/0.64 % mult_right_mono
% 0.34/0.64 thf(fact_5691_mult__right__mono,axiom,
% 0.34/0.64 ! [A2: nat,B: nat,C: nat] :
% 0.34/0.64 ( ( ord_less_eq_nat @ A2 @ B )
% 0.34/0.64 => ( ( ord_less_eq_nat @ zero_zero_nat @ C )
% 0.34/0.64 => ( ord_less_eq_nat @ ( times_times_nat @ A2 @ C ) @ ( times_times_nat @ B @ C ) ) ) ) ).
% 0.34/0.64
% 0.34/0.64 % mult_right_mono
% 0.34/0.64 thf(fact_5692_mult__right__mono,axiom,
% 0.34/0.64 ! [A2: int,B: int,C: int] :
% 0.34/0.64 ( ( ord_less_eq_int @ A2 @ B )
% 0.34/0.64 => ( ( ord_less_eq_int @ zero_zero_int @ C )
% 0.34/0.64 => ( ord_less_eq_int @ ( times_times_int @ A2 @ C ) @ ( times_times_int @ B @ C ) ) ) ) ).
% 0.34/0.64
% 0.34/0.64 % mult_right_mono
% 0.34/0.64 thf(fact_5693_mult__right__mono__neg,axiom,
% 0.34/0.64 ! [B: code_integer,A2: code_integer,C: code_integer] :
% 0.34/0.64 ( ( ord_le3102999989581377725nteger @ B @ A2 )
% 0.34/0.64 => ( ( ord_le3102999989581377725nteger @ C @ zero_z3403309356797280102nteger )
% 0.34/0.64 => ( ord_le3102999989581377725nteger @ ( times_3573771949741848930nteger @ A2 @ C ) @ ( times_3573771949741848930nteger @ B @ C ) ) ) ) ).
% 0.34/0.64
% 0.34/0.64 % mult_right_mono_neg
% 0.34/0.64 thf(fact_5694_mult__right__mono__neg,axiom,
% 0.34/0.64 ! [B: rat,A2: rat,C: rat] :
% 0.34/0.64 ( ( ord_less_eq_rat @ B @ A2 )
% 0.34/0.64 => ( ( ord_less_eq_rat @ C @ zero_zero_rat )
% 0.34/0.64 => ( ord_less_eq_rat @ ( times_times_rat @ A2 @ C ) @ ( times_times_rat @ B @ C ) ) ) ) ).
% 0.34/0.64
% 0.34/0.64 % mult_right_mono_neg
% 0.34/0.64 thf(fact_5695_mult__right__mono__neg,axiom,
% 0.34/0.64 ! [B: int,A2: int,C: int] :
% 0.34/0.64 ( ( ord_less_eq_int @ B @ A2 )
% 0.34/0.64 => ( ( ord_less_eq_int @ C @ zero_zero_int )
% 0.34/0.64 => ( ord_less_eq_int @ ( times_times_int @ A2 @ C ) @ ( times_times_int @ B @ C ) ) ) ) ).
% 0.34/0.64
% 0.34/0.64 % mult_right_mono_neg
% 0.34/0.64 thf(fact_5696_mult__left__mono,axiom,
% 0.34/0.64 ! [A2: code_integer,B: code_integer,C: code_integer] :
% 0.34/0.64 ( ( ord_le3102999989581377725nteger @ A2 @ B )
% 0.34/0.64 => ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ C )
% 0.34/0.64 => ( ord_le3102999989581377725nteger @ ( times_3573771949741848930nteger @ C @ A2 ) @ ( times_3573771949741848930nteger @ C @ B ) ) ) ) ).
% 0.34/0.64
% 0.34/0.64 % mult_left_mono
% 0.34/0.64 thf(fact_5697_mult__left__mono,axiom,
% 0.34/0.64 ! [A2: rat,B: rat,C: rat] :
% 0.34/0.64 ( ( ord_less_eq_rat @ A2 @ B )
% 0.34/0.64 => ( ( ord_less_eq_rat @ zero_zero_rat @ C )
% 0.34/0.64 => ( ord_less_eq_rat @ ( times_times_rat @ C @ A2 ) @ ( times_times_rat @ C @ B ) ) ) ) ).
% 0.34/0.64
% 0.34/0.64 % mult_left_mono
% 0.34/0.64 thf(fact_5698_mult__left__mono,axiom,
% 0.34/0.64 ! [A2: nat,B: nat,C: nat] :
% 0.34/0.64 ( ( ord_less_eq_nat @ A2 @ B )
% 0.34/0.64 => ( ( ord_less_eq_nat @ zero_zero_nat @ C )
% 0.34/0.64 => ( ord_less_eq_nat @ ( times_times_nat @ C @ A2 ) @ ( times_times_nat @ C @ B ) ) ) ) ).
% 0.34/0.64
% 0.34/0.64 % mult_left_mono
% 0.34/0.64 thf(fact_5699_mult__left__mono,axiom,
% 0.34/0.64 ! [A2: int,B: int,C: int] :
% 0.34/0.64 ( ( ord_less_eq_int @ A2 @ B )
% 0.34/0.64 => ( ( ord_less_eq_int @ zero_zero_int @ C )
% 0.34/0.64 => ( ord_less_eq_int @ ( times_times_int @ C @ A2 ) @ ( times_times_int @ C @ B ) ) ) ) ).
% 0.34/0.64
% 0.34/0.64 % mult_left_mono
% 0.34/0.64 thf(fact_5700_mult__nonpos__nonpos,axiom,
% 0.34/0.64 ! [A2: code_integer,B: code_integer] :
% 0.34/0.64 ( ( ord_le3102999989581377725nteger @ A2 @ zero_z3403309356797280102nteger )
% 0.34/0.64 => ( ( ord_le3102999989581377725nteger @ B @ zero_z3403309356797280102nteger )
% 0.34/0.64 => ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ ( times_3573771949741848930nteger @ A2 @ B ) ) ) ) ).
% 0.34/0.64
% 0.34/0.64 % mult_nonpos_nonpos
% 0.34/0.64 thf(fact_5701_mult__nonpos__nonpos,axiom,
% 0.34/0.64 ! [A2: rat,B: rat] :
% 0.34/0.64 ( ( ord_less_eq_rat @ A2 @ zero_zero_rat )
% 0.34/0.64 => ( ( ord_less_eq_rat @ B @ zero_zero_rat )
% 0.34/0.64 => ( ord_less_eq_rat @ zero_zero_rat @ ( times_times_rat @ A2 @ B ) ) ) ) ).
% 0.34/0.64
% 0.34/0.64 % mult_nonpos_nonpos
% 0.34/0.64 thf(fact_5702_mult__nonpos__nonpos,axiom,
% 0.34/0.64 ! [A2: int,B: int] :
% 0.34/0.64 ( ( ord_less_eq_int @ A2 @ zero_zero_int )
% 0.34/0.64 => ( ( ord_less_eq_int @ B @ zero_zero_int )
% 0.34/0.64 => ( ord_less_eq_int @ zero_zero_int @ ( times_times_int @ A2 @ B ) ) ) ) ).
% 0.34/0.64
% 0.34/0.64 % mult_nonpos_nonpos
% 0.34/0.64 thf(fact_5703_mult__left__mono__neg,axiom,
% 0.34/0.64 ! [B: code_integer,A2: code_integer,C: code_integer] :
% 0.34/0.64 ( ( ord_le3102999989581377725nteger @ B @ A2 )
% 0.34/0.64 => ( ( ord_le3102999989581377725nteger @ C @ zero_z3403309356797280102nteger )
% 0.34/0.64 => ( ord_le3102999989581377725nteger @ ( times_3573771949741848930nteger @ C @ A2 ) @ ( times_3573771949741848930nteger @ C @ B ) ) ) ) ).
% 0.34/0.64
% 0.34/0.64 % mult_left_mono_neg
% 0.34/0.64 thf(fact_5704_mult__left__mono__neg,axiom,
% 0.34/0.64 ! [B: rat,A2: rat,C: rat] :
% 0.34/0.64 ( ( ord_less_eq_rat @ B @ A2 )
% 0.34/0.64 => ( ( ord_less_eq_rat @ C @ zero_zero_rat )
% 0.34/0.64 => ( ord_less_eq_rat @ ( times_times_rat @ C @ A2 ) @ ( times_times_rat @ C @ B ) ) ) ) ).
% 0.34/0.64
% 0.34/0.64 % mult_left_mono_neg
% 0.34/0.64 thf(fact_5705_mult__left__mono__neg,axiom,
% 0.34/0.64 ! [B: int,A2: int,C: int] :
% 0.34/0.64 ( ( ord_less_eq_int @ B @ A2 )
% 0.34/0.64 => ( ( ord_less_eq_int @ C @ zero_zero_int )
% 0.34/0.64 => ( ord_less_eq_int @ ( times_times_int @ C @ A2 ) @ ( times_times_int @ C @ B ) ) ) ) ).
% 0.34/0.64
% 0.34/0.64 % mult_left_mono_neg
% 0.34/0.64 thf(fact_5706_split__mult__pos__le,axiom,
% 0.34/0.64 ! [A2: code_integer,B: code_integer] :
% 0.34/0.64 ( ( ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ A2 )
% 0.34/0.64 & ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ B ) )
% 0.34/0.64 | ( ( ord_le3102999989581377725nteger @ A2 @ zero_z3403309356797280102nteger )
% 0.34/0.64 & ( ord_le3102999989581377725nteger @ B @ zero_z3403309356797280102nteger ) ) )
% 0.34/0.64 => ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ ( times_3573771949741848930nteger @ A2 @ B ) ) ) ).
% 0.34/0.64
% 0.34/0.64 % split_mult_pos_le
% 0.34/0.64 thf(fact_5707_split__mult__pos__le,axiom,
% 0.34/0.64 ! [A2: rat,B: rat] :
% 0.34/0.64 ( ( ( ( ord_less_eq_rat @ zero_zero_rat @ A2 )
% 0.34/0.64 & ( ord_less_eq_rat @ zero_zero_rat @ B ) )
% 0.34/0.64 | ( ( ord_less_eq_rat @ A2 @ zero_zero_rat )
% 0.34/0.64 & ( ord_less_eq_rat @ B @ zero_zero_rat ) ) )
% 0.34/0.64 => ( ord_less_eq_rat @ zero_zero_rat @ ( times_times_rat @ A2 @ B ) ) ) ).
% 0.34/0.64
% 0.34/0.64 % split_mult_pos_le
% 0.34/0.64 thf(fact_5708_split__mult__pos__le,axiom,
% 0.34/0.64 ! [A2: int,B: int] :
% 0.34/0.64 ( ( ( ( ord_less_eq_int @ zero_zero_int @ A2 )
% 0.34/0.64 & ( ord_less_eq_int @ zero_zero_int @ B ) )
% 0.34/0.64 | ( ( ord_less_eq_int @ A2 @ zero_zero_int )
% 0.34/0.64 & ( ord_less_eq_int @ B @ zero_zero_int ) ) )
% 0.34/0.64 => ( ord_less_eq_int @ zero_zero_int @ ( times_times_int @ A2 @ B ) ) ) ).
% 0.34/0.64
% 0.34/0.64 % split_mult_pos_le
% 0.34/0.64 thf(fact_5709_zero__le__square,axiom,
% 0.34/0.64 ! [A2: code_integer] : ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ ( times_3573771949741848930nteger @ A2 @ A2 ) ) ).
% 0.34/0.64
% 0.34/0.64 % zero_le_square
% 0.34/0.64 thf(fact_5710_zero__le__square,axiom,
% 0.34/0.64 ! [A2: rat] : ( ord_less_eq_rat @ zero_zero_rat @ ( times_times_rat @ A2 @ A2 ) ) ).
% 0.34/0.64
% 0.34/0.64 % zero_le_square
% 0.34/0.64 thf(fact_5711_zero__le__square,axiom,
% 0.34/0.64 ! [A2: int] : ( ord_less_eq_int @ zero_zero_int @ ( times_times_int @ A2 @ A2 ) ) ).
% 0.34/0.64
% 0.34/0.64 % zero_le_square
% 0.34/0.64 thf(fact_5712_mult__mono_H,axiom,
% 0.34/0.64 ! [A2: code_integer,B: code_integer,C: code_integer,D2: code_integer] :
% 0.34/0.64 ( ( ord_le3102999989581377725nteger @ A2 @ B )
% 0.34/0.64 => ( ( ord_le3102999989581377725nteger @ C @ D2 )
% 0.34/0.64 => ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ A2 )
% 0.34/0.64 => ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ C )
% 0.34/0.64 => ( ord_le3102999989581377725nteger @ ( times_3573771949741848930nteger @ A2 @ C ) @ ( times_3573771949741848930nteger @ B @ D2 ) ) ) ) ) ) ).
% 0.34/0.64
% 0.34/0.64 % mult_mono'
% 0.34/0.64 thf(fact_5713_mult__mono_H,axiom,
% 0.34/0.64 ! [A2: rat,B: rat,C: rat,D2: rat] :
% 0.34/0.64 ( ( ord_less_eq_rat @ A2 @ B )
% 0.34/0.64 => ( ( ord_less_eq_rat @ C @ D2 )
% 0.34/0.64 => ( ( ord_less_eq_rat @ zero_zero_rat @ A2 )
% 0.34/0.64 => ( ( ord_less_eq_rat @ zero_zero_rat @ C )
% 0.34/0.64 => ( ord_less_eq_rat @ ( times_times_rat @ A2 @ C ) @ ( times_times_rat @ B @ D2 ) ) ) ) ) ) ).
% 0.34/0.64
% 0.34/0.64 % mult_mono'
% 0.34/0.64 thf(fact_5714_mult__mono_H,axiom,
% 0.34/0.64 ! [A2: nat,B: nat,C: nat,D2: nat] :
% 0.34/0.64 ( ( ord_less_eq_nat @ A2 @ B )
% 0.34/0.64 => ( ( ord_less_eq_nat @ C @ D2 )
% 0.34/0.64 => ( ( ord_less_eq_nat @ zero_zero_nat @ A2 )
% 0.34/0.64 => ( ( ord_less_eq_nat @ zero_zero_nat @ C )
% 0.34/0.64 => ( ord_less_eq_nat @ ( times_times_nat @ A2 @ C ) @ ( times_times_nat @ B @ D2 ) ) ) ) ) ) ).
% 0.34/0.64
% 0.34/0.64 % mult_mono'
% 0.34/0.64 thf(fact_5715_mult__mono_H,axiom,
% 0.34/0.64 ! [A2: int,B: int,C: int,D2: int] :
% 0.34/0.64 ( ( ord_less_eq_int @ A2 @ B )
% 0.34/0.64 => ( ( ord_less_eq_int @ C @ D2 )
% 0.34/0.64 => ( ( ord_less_eq_int @ zero_zero_int @ A2 )
% 0.34/0.64 => ( ( ord_less_eq_int @ zero_zero_int @ C )
% 0.34/0.64 => ( ord_less_eq_int @ ( times_times_int @ A2 @ C ) @ ( times_times_int @ B @ D2 ) ) ) ) ) ) ).
% 0.34/0.65
% 0.34/0.65 % mult_mono'
% 0.34/0.65 thf(fact_5716_mult__mono,axiom,
% 0.34/0.65 ! [A2: code_integer,B: code_integer,C: code_integer,D2: code_integer] :
% 0.34/0.65 ( ( ord_le3102999989581377725nteger @ A2 @ B )
% 0.34/0.65 => ( ( ord_le3102999989581377725nteger @ C @ D2 )
% 0.34/0.65 => ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ B )
% 0.34/0.65 => ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ C )
% 0.34/0.65 => ( ord_le3102999989581377725nteger @ ( times_3573771949741848930nteger @ A2 @ C ) @ ( times_3573771949741848930nteger @ B @ D2 ) ) ) ) ) ) ).
% 0.34/0.65
% 0.34/0.65 % mult_mono
% 0.34/0.65 thf(fact_5717_mult__mono,axiom,
% 0.34/0.65 ! [A2: rat,B: rat,C: rat,D2: rat] :
% 0.34/0.65 ( ( ord_less_eq_rat @ A2 @ B )
% 0.34/0.65 => ( ( ord_less_eq_rat @ C @ D2 )
% 0.34/0.65 => ( ( ord_less_eq_rat @ zero_zero_rat @ B )
% 0.34/0.65 => ( ( ord_less_eq_rat @ zero_zero_rat @ C )
% 0.34/0.65 => ( ord_less_eq_rat @ ( times_times_rat @ A2 @ C ) @ ( times_times_rat @ B @ D2 ) ) ) ) ) ) ).
% 0.34/0.65
% 0.34/0.65 % mult_mono
% 0.34/0.65 thf(fact_5718_mult__mono,axiom,
% 0.34/0.65 ! [A2: nat,B: nat,C: nat,D2: nat] :
% 0.34/0.65 ( ( ord_less_eq_nat @ A2 @ B )
% 0.34/0.65 => ( ( ord_less_eq_nat @ C @ D2 )
% 0.34/0.65 => ( ( ord_less_eq_nat @ zero_zero_nat @ B )
% 0.34/0.65 => ( ( ord_less_eq_nat @ zero_zero_nat @ C )
% 0.34/0.65 => ( ord_less_eq_nat @ ( times_times_nat @ A2 @ C ) @ ( times_times_nat @ B @ D2 ) ) ) ) ) ) ).
% 0.34/0.65
% 0.34/0.65 % mult_mono
% 0.34/0.65 thf(fact_5719_mult__mono,axiom,
% 0.34/0.65 ! [A2: int,B: int,C: int,D2: int] :
% 0.34/0.65 ( ( ord_less_eq_int @ A2 @ B )
% 0.34/0.65 => ( ( ord_less_eq_int @ C @ D2 )
% 0.34/0.65 => ( ( ord_less_eq_int @ zero_zero_int @ B )
% 0.34/0.65 => ( ( ord_less_eq_int @ zero_zero_int @ C )
% 0.34/0.65 => ( ord_less_eq_int @ ( times_times_int @ A2 @ C ) @ ( times_times_int @ B @ D2 ) ) ) ) ) ) ).
% 0.34/0.65
% 0.34/0.65 % mult_mono
% 0.34/0.65 thf(fact_5720_not__one__le__zero,axiom,
% 0.34/0.65 ~ ( ord_le1926595141338095240atural @ one_one_Code_natural @ zero_z2226904508553997617atural ) ).
% 0.34/0.65
% 0.34/0.65 % not_one_le_zero
% 0.34/0.65 thf(fact_5721_not__one__le__zero,axiom,
% 0.34/0.65 ~ ( ord_le3102999989581377725nteger @ one_one_Code_integer @ zero_z3403309356797280102nteger ) ).
% 0.34/0.65
% 0.34/0.65 % not_one_le_zero
% 0.34/0.65 thf(fact_5722_not__one__le__zero,axiom,
% 0.34/0.65 ~ ( ord_less_eq_rat @ one_one_rat @ zero_zero_rat ) ).
% 0.34/0.65
% 0.34/0.65 % not_one_le_zero
% 0.34/0.65 thf(fact_5723_not__one__le__zero,axiom,
% 0.34/0.65 ~ ( ord_less_eq_nat @ one_one_nat @ zero_zero_nat ) ).
% 0.34/0.65
% 0.34/0.65 % not_one_le_zero
% 0.34/0.65 thf(fact_5724_not__one__le__zero,axiom,
% 0.34/0.65 ~ ( ord_less_eq_int @ one_one_int @ zero_zero_int ) ).
% 0.34/0.65
% 0.34/0.65 % not_one_le_zero
% 0.34/0.65 thf(fact_5725_linordered__nonzero__semiring__class_Ozero__le__one,axiom,
% 0.34/0.65 ord_le1926595141338095240atural @ zero_z2226904508553997617atural @ one_one_Code_natural ).
% 0.34/0.65
% 0.34/0.65 % linordered_nonzero_semiring_class.zero_le_one
% 0.34/0.65 thf(fact_5726_linordered__nonzero__semiring__class_Ozero__le__one,axiom,
% 0.34/0.65 ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ one_one_Code_integer ).
% 0.34/0.65
% 0.34/0.65 % linordered_nonzero_semiring_class.zero_le_one
% 0.34/0.65 thf(fact_5727_linordered__nonzero__semiring__class_Ozero__le__one,axiom,
% 0.34/0.65 ord_less_eq_rat @ zero_zero_rat @ one_one_rat ).
% 0.34/0.65
% 0.34/0.65 % linordered_nonzero_semiring_class.zero_le_one
% 0.34/0.65 thf(fact_5728_linordered__nonzero__semiring__class_Ozero__le__one,axiom,
% 0.34/0.65 ord_less_eq_nat @ zero_zero_nat @ one_one_nat ).
% 0.34/0.65
% 0.34/0.65 % linordered_nonzero_semiring_class.zero_le_one
% 0.34/0.65 thf(fact_5729_linordered__nonzero__semiring__class_Ozero__le__one,axiom,
% 0.34/0.65 ord_less_eq_int @ zero_zero_int @ one_one_int ).
% 0.34/0.65
% 0.34/0.65 % linordered_nonzero_semiring_class.zero_le_one
% 0.34/0.65 thf(fact_5730_zero__less__one__class_Ozero__le__one,axiom,
% 0.34/0.65 ord_le1926595141338095240atural @ zero_z2226904508553997617atural @ one_one_Code_natural ).
% 0.34/0.65
% 0.34/0.65 % zero_less_one_class.zero_le_one
% 0.34/0.65 thf(fact_5731_zero__less__one__class_Ozero__le__one,axiom,
% 0.34/0.65 ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ one_one_Code_integer ).
% 0.34/0.65
% 0.34/0.65 % zero_less_one_class.zero_le_one
% 0.34/0.65 thf(fact_5732_zero__less__one__class_Ozero__le__one,axiom,
% 0.34/0.65 ord_less_eq_rat @ zero_zero_rat @ one_one_rat ).
% 0.34/0.65
% 0.34/0.65 % zero_less_one_class.zero_le_one
% 0.34/0.65 thf(fact_5733_zero__less__one__class_Ozero__le__one,axiom,
% 0.34/0.65 ord_less_eq_nat @ zero_zero_nat @ one_one_nat ).
% 0.34/0.65
% 0.34/0.65 % zero_less_one_class.zero_le_one
% 0.34/0.65 thf(fact_5734_zero__less__one__class_Ozero__le__one,axiom,
% 0.34/0.65 ord_less_eq_int @ zero_zero_int @ one_one_int ).
% 0.34/0.65
% 0.34/0.65 % zero_less_one_class.zero_le_one
% 0.34/0.65 thf(fact_5735_le__iff__diff__le__0,axiom,
% 0.34/0.65 ( ord_le3102999989581377725nteger
% 0.34/0.65 = ( ^ [A6: code_integer,B6: code_integer] : ( ord_le3102999989581377725nteger @ ( minus_8373710615458151222nteger @ A6 @ B6 ) @ zero_z3403309356797280102nteger ) ) ) ).
% 0.34/0.65
% 0.34/0.65 % le_iff_diff_le_0
% 0.34/0.65 thf(fact_5736_le__iff__diff__le__0,axiom,
% 0.34/0.65 ( ord_less_eq_rat
% 0.34/0.65 = ( ^ [A6: rat,B6: rat] : ( ord_less_eq_rat @ ( minus_minus_rat @ A6 @ B6 ) @ zero_zero_rat ) ) ) ).
% 0.34/0.65
% 0.34/0.65 % le_iff_diff_le_0
% 0.34/0.65 thf(fact_5737_le__iff__diff__le__0,axiom,
% 0.34/0.65 ( ord_less_eq_int
% 0.34/0.65 = ( ^ [A6: int,B6: int] : ( ord_less_eq_int @ ( minus_minus_int @ A6 @ B6 ) @ zero_zero_int ) ) ) ).
% 0.34/0.65
% 0.34/0.65 % le_iff_diff_le_0
% 0.34/0.65 thf(fact_5738_pos__add__strict,axiom,
% 0.34/0.65 ! [A2: nat,B: nat,C: nat] :
% 0.34/0.65 ( ( ord_less_nat @ zero_zero_nat @ A2 )
% 0.34/0.65 => ( ( ord_less_nat @ B @ C )
% 0.34/0.65 => ( ord_less_nat @ B @ ( plus_plus_nat @ A2 @ C ) ) ) ) ).
% 0.34/0.65
% 0.34/0.65 % pos_add_strict
% 0.34/0.65 thf(fact_5739_pos__add__strict,axiom,
% 0.34/0.65 ! [A2: int,B: int,C: int] :
% 0.34/0.65 ( ( ord_less_int @ zero_zero_int @ A2 )
% 0.34/0.65 => ( ( ord_less_int @ B @ C )
% 0.34/0.65 => ( ord_less_int @ B @ ( plus_plus_int @ A2 @ C ) ) ) ) ).
% 0.34/0.65
% 0.34/0.65 % pos_add_strict
% 0.34/0.65 thf(fact_5740_pos__add__strict,axiom,
% 0.34/0.65 ! [A2: code_integer,B: code_integer,C: code_integer] :
% 0.34/0.65 ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ A2 )
% 0.34/0.65 => ( ( ord_le6747313008572928689nteger @ B @ C )
% 0.34/0.65 => ( ord_le6747313008572928689nteger @ B @ ( plus_p5714425477246183910nteger @ A2 @ C ) ) ) ) ).
% 0.34/0.65
% 0.34/0.65 % pos_add_strict
% 0.34/0.65 thf(fact_5741_pos__add__strict,axiom,
% 0.34/0.65 ! [A2: rat,B: rat,C: rat] :
% 0.34/0.65 ( ( ord_less_rat @ zero_zero_rat @ A2 )
% 0.34/0.65 => ( ( ord_less_rat @ B @ C )
% 0.34/0.65 => ( ord_less_rat @ B @ ( plus_plus_rat @ A2 @ C ) ) ) ) ).
% 0.34/0.65
% 0.34/0.65 % pos_add_strict
% 0.34/0.65 thf(fact_5742_canonically__ordered__monoid__add__class_OlessE,axiom,
% 0.34/0.65 ! [A2: nat,B: nat] :
% 0.34/0.65 ( ( ord_less_nat @ A2 @ B )
% 0.34/0.65 => ~ ! [C2: nat] :
% 0.34/0.65 ( ( B
% 0.34/0.65 = ( plus_plus_nat @ A2 @ C2 ) )
% 0.34/0.65 => ( C2 = zero_zero_nat ) ) ) ).
% 0.34/0.65
% 0.34/0.65 % canonically_ordered_monoid_add_class.lessE
% 0.34/0.65 thf(fact_5743_add__pos__pos,axiom,
% 0.34/0.65 ! [A2: nat,B: nat] :
% 0.34/0.65 ( ( ord_less_nat @ zero_zero_nat @ A2 )
% 0.34/0.65 => ( ( ord_less_nat @ zero_zero_nat @ B )
% 0.34/0.65 => ( ord_less_nat @ zero_zero_nat @ ( plus_plus_nat @ A2 @ B ) ) ) ) ).
% 0.34/0.65
% 0.34/0.65 % add_pos_pos
% 0.34/0.65 thf(fact_5744_add__pos__pos,axiom,
% 0.34/0.65 ! [A2: int,B: int] :
% 0.34/0.65 ( ( ord_less_int @ zero_zero_int @ A2 )
% 0.34/0.65 => ( ( ord_less_int @ zero_zero_int @ B )
% 0.34/0.65 => ( ord_less_int @ zero_zero_int @ ( plus_plus_int @ A2 @ B ) ) ) ) ).
% 0.34/0.65
% 0.34/0.65 % add_pos_pos
% 0.34/0.65 thf(fact_5745_add__pos__pos,axiom,
% 0.34/0.65 ! [A2: code_integer,B: code_integer] :
% 0.34/0.65 ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ A2 )
% 0.34/0.65 => ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ B )
% 0.34/0.65 => ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ ( plus_p5714425477246183910nteger @ A2 @ B ) ) ) ) ).
% 0.34/0.65
% 0.34/0.65 % add_pos_pos
% 0.34/0.65 thf(fact_5746_add__pos__pos,axiom,
% 0.34/0.65 ! [A2: rat,B: rat] :
% 0.34/0.65 ( ( ord_less_rat @ zero_zero_rat @ A2 )
% 0.34/0.65 => ( ( ord_less_rat @ zero_zero_rat @ B )
% 0.34/0.65 => ( ord_less_rat @ zero_zero_rat @ ( plus_plus_rat @ A2 @ B ) ) ) ) ).
% 0.34/0.65
% 0.34/0.65 % add_pos_pos
% 0.34/0.65 thf(fact_5747_add__neg__neg,axiom,
% 0.34/0.65 ! [A2: nat,B: nat] :
% 0.34/0.65 ( ( ord_less_nat @ A2 @ zero_zero_nat )
% 0.34/0.65 => ( ( ord_less_nat @ B @ zero_zero_nat )
% 0.34/0.65 => ( ord_less_nat @ ( plus_plus_nat @ A2 @ B ) @ zero_zero_nat ) ) ) ).
% 0.34/0.65
% 0.34/0.65 % add_neg_neg
% 0.34/0.65 thf(fact_5748_add__neg__neg,axiom,
% 0.34/0.65 ! [A2: int,B: int] :
% 0.34/0.65 ( ( ord_less_int @ A2 @ zero_zero_int )
% 0.34/0.65 => ( ( ord_less_int @ B @ zero_zero_int )
% 0.34/0.65 => ( ord_less_int @ ( plus_plus_int @ A2 @ B ) @ zero_zero_int ) ) ) ).
% 0.34/0.65
% 0.34/0.65 % add_neg_neg
% 0.34/0.65 thf(fact_5749_add__neg__neg,axiom,
% 0.34/0.65 ! [A2: code_integer,B: code_integer] :
% 0.34/0.65 ( ( ord_le6747313008572928689nteger @ A2 @ zero_z3403309356797280102nteger )
% 0.34/0.65 => ( ( ord_le6747313008572928689nteger @ B @ zero_z3403309356797280102nteger )
% 0.34/0.65 => ( ord_le6747313008572928689nteger @ ( plus_p5714425477246183910nteger @ A2 @ B ) @ zero_z3403309356797280102nteger ) ) ) ).
% 0.34/0.65
% 0.34/0.65 % add_neg_neg
% 0.34/0.65 thf(fact_5750_add__neg__neg,axiom,
% 0.34/0.65 ! [A2: rat,B: rat] :
% 0.34/0.65 ( ( ord_less_rat @ A2 @ zero_zero_rat )
% 0.34/0.65 => ( ( ord_less_rat @ B @ zero_zero_rat )
% 0.34/0.65 => ( ord_less_rat @ ( plus_plus_rat @ A2 @ B ) @ zero_zero_rat ) ) ) ).
% 0.34/0.65
% 0.34/0.65 % add_neg_neg
% 0.34/0.65 thf(fact_5751_mult__neg__neg,axiom,
% 0.34/0.65 ! [A2: int,B: int] :
% 0.34/0.65 ( ( ord_less_int @ A2 @ zero_zero_int )
% 0.34/0.65 => ( ( ord_less_int @ B @ zero_zero_int )
% 0.34/0.65 => ( ord_less_int @ zero_zero_int @ ( times_times_int @ A2 @ B ) ) ) ) ).
% 0.34/0.65
% 0.34/0.65 % mult_neg_neg
% 0.34/0.65 thf(fact_5752_mult__neg__neg,axiom,
% 0.34/0.65 ! [A2: code_integer,B: code_integer] :
% 0.34/0.65 ( ( ord_le6747313008572928689nteger @ A2 @ zero_z3403309356797280102nteger )
% 0.34/0.65 => ( ( ord_le6747313008572928689nteger @ B @ zero_z3403309356797280102nteger )
% 0.34/0.65 => ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ ( times_3573771949741848930nteger @ A2 @ B ) ) ) ) ).
% 0.34/0.65
% 0.34/0.65 % mult_neg_neg
% 0.34/0.65 thf(fact_5753_mult__neg__neg,axiom,
% 0.34/0.65 ! [A2: rat,B: rat] :
% 0.34/0.65 ( ( ord_less_rat @ A2 @ zero_zero_rat )
% 0.34/0.65 => ( ( ord_less_rat @ B @ zero_zero_rat )
% 0.34/0.65 => ( ord_less_rat @ zero_zero_rat @ ( times_times_rat @ A2 @ B ) ) ) ) ).
% 0.34/0.65
% 0.34/0.65 % mult_neg_neg
% 0.34/0.65 thf(fact_5754_not__square__less__zero,axiom,
% 0.34/0.65 ! [A2: int] :
% 0.34/0.65 ~ ( ord_less_int @ ( times_times_int @ A2 @ A2 ) @ zero_zero_int ) ).
% 0.34/0.65
% 0.34/0.65 % not_square_less_zero
% 0.34/0.65 thf(fact_5755_not__square__less__zero,axiom,
% 0.34/0.65 ! [A2: code_integer] :
% 0.34/0.65 ~ ( ord_le6747313008572928689nteger @ ( times_3573771949741848930nteger @ A2 @ A2 ) @ zero_z3403309356797280102nteger ) ).
% 0.34/0.65
% 0.34/0.65 % not_square_less_zero
% 0.34/0.65 thf(fact_5756_not__square__less__zero,axiom,
% 0.34/0.65 ! [A2: rat] :
% 0.34/0.65 ~ ( ord_less_rat @ ( times_times_rat @ A2 @ A2 ) @ zero_zero_rat ) ).
% 0.34/0.65
% 0.34/0.65 % not_square_less_zero
% 0.34/0.65 thf(fact_5757_mult__less__0__iff,axiom,
% 0.34/0.65 ! [A2: int,B: int] :
% 0.34/0.65 ( ( ord_less_int @ ( times_times_int @ A2 @ B ) @ zero_zero_int )
% 0.34/0.65 = ( ( ( ord_less_int @ zero_zero_int @ A2 )
% 0.34/0.65 & ( ord_less_int @ B @ zero_zero_int ) )
% 0.34/0.65 | ( ( ord_less_int @ A2 @ zero_zero_int )
% 0.34/0.65 & ( ord_less_int @ zero_zero_int @ B ) ) ) ) ).
% 0.34/0.65
% 0.34/0.65 % mult_less_0_iff
% 0.34/0.65 thf(fact_5758_mult__less__0__iff,axiom,
% 0.34/0.65 ! [A2: code_integer,B: code_integer] :
% 0.34/0.65 ( ( ord_le6747313008572928689nteger @ ( times_3573771949741848930nteger @ A2 @ B ) @ zero_z3403309356797280102nteger )
% 0.34/0.65 = ( ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ A2 )
% 0.34/0.65 & ( ord_le6747313008572928689nteger @ B @ zero_z3403309356797280102nteger ) )
% 0.34/0.65 | ( ( ord_le6747313008572928689nteger @ A2 @ zero_z3403309356797280102nteger )
% 0.34/0.65 & ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ B ) ) ) ) ).
% 0.34/0.65
% 0.34/0.65 % mult_less_0_iff
% 0.34/0.65 thf(fact_5759_mult__less__0__iff,axiom,
% 0.34/0.65 ! [A2: rat,B: rat] :
% 0.34/0.65 ( ( ord_less_rat @ ( times_times_rat @ A2 @ B ) @ zero_zero_rat )
% 0.34/0.65 = ( ( ( ord_less_rat @ zero_zero_rat @ A2 )
% 0.34/0.65 & ( ord_less_rat @ B @ zero_zero_rat ) )
% 0.34/0.65 | ( ( ord_less_rat @ A2 @ zero_zero_rat )
% 0.34/0.65 & ( ord_less_rat @ zero_zero_rat @ B ) ) ) ) ).
% 0.34/0.65
% 0.34/0.65 % mult_less_0_iff
% 0.34/0.65 thf(fact_5760_mult__neg__pos,axiom,
% 0.34/0.65 ! [A2: nat,B: nat] :
% 0.34/0.65 ( ( ord_less_nat @ A2 @ zero_zero_nat )
% 0.34/0.65 => ( ( ord_less_nat @ zero_zero_nat @ B )
% 0.34/0.65 => ( ord_less_nat @ ( times_times_nat @ A2 @ B ) @ zero_zero_nat ) ) ) ).
% 0.34/0.65
% 0.34/0.65 % mult_neg_pos
% 0.34/0.65 thf(fact_5761_mult__neg__pos,axiom,
% 0.34/0.65 ! [A2: int,B: int] :
% 0.34/0.65 ( ( ord_less_int @ A2 @ zero_zero_int )
% 0.34/0.65 => ( ( ord_less_int @ zero_zero_int @ B )
% 0.34/0.65 => ( ord_less_int @ ( times_times_int @ A2 @ B ) @ zero_zero_int ) ) ) ).
% 0.34/0.65
% 0.34/0.65 % mult_neg_pos
% 0.34/0.65 thf(fact_5762_mult__neg__pos,axiom,
% 0.34/0.65 ! [A2: code_integer,B: code_integer] :
% 0.34/0.65 ( ( ord_le6747313008572928689nteger @ A2 @ zero_z3403309356797280102nteger )
% 0.34/0.65 => ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ B )
% 0.34/0.65 => ( ord_le6747313008572928689nteger @ ( times_3573771949741848930nteger @ A2 @ B ) @ zero_z3403309356797280102nteger ) ) ) ).
% 0.34/0.65
% 0.34/0.65 % mult_neg_pos
% 0.34/0.65 thf(fact_5763_mult__neg__pos,axiom,
% 0.34/0.65 ! [A2: rat,B: rat] :
% 0.34/0.65 ( ( ord_less_rat @ A2 @ zero_zero_rat )
% 0.34/0.65 => ( ( ord_less_rat @ zero_zero_rat @ B )
% 0.34/0.65 => ( ord_less_rat @ ( times_times_rat @ A2 @ B ) @ zero_zero_rat ) ) ) ).
% 0.34/0.65
% 0.34/0.65 % mult_neg_pos
% 0.34/0.65 thf(fact_5764_mult__pos__neg,axiom,
% 0.34/0.65 ! [A2: nat,B: nat] :
% 0.34/0.65 ( ( ord_less_nat @ zero_zero_nat @ A2 )
% 0.34/0.65 => ( ( ord_less_nat @ B @ zero_zero_nat )
% 0.34/0.65 => ( ord_less_nat @ ( times_times_nat @ A2 @ B ) @ zero_zero_nat ) ) ) ).
% 0.34/0.65
% 0.34/0.65 % mult_pos_neg
% 0.34/0.65 thf(fact_5765_mult__pos__neg,axiom,
% 0.34/0.65 ! [A2: int,B: int] :
% 0.34/0.65 ( ( ord_less_int @ zero_zero_int @ A2 )
% 0.34/0.65 => ( ( ord_less_int @ B @ zero_zero_int )
% 0.34/0.65 => ( ord_less_int @ ( times_times_int @ A2 @ B ) @ zero_zero_int ) ) ) ).
% 0.34/0.65
% 0.34/0.65 % mult_pos_neg
% 0.34/0.65 thf(fact_5766_mult__pos__neg,axiom,
% 0.34/0.65 ! [A2: code_integer,B: code_integer] :
% 0.34/0.65 ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ A2 )
% 0.34/0.65 => ( ( ord_le6747313008572928689nteger @ B @ zero_z3403309356797280102nteger )
% 0.34/0.65 => ( ord_le6747313008572928689nteger @ ( times_3573771949741848930nteger @ A2 @ B ) @ zero_z3403309356797280102nteger ) ) ) ).
% 0.34/0.65
% 0.34/0.65 % mult_pos_neg
% 0.34/0.65 thf(fact_5767_mult__pos__neg,axiom,
% 0.34/0.65 ! [A2: rat,B: rat] :
% 0.34/0.65 ( ( ord_less_rat @ zero_zero_rat @ A2 )
% 0.34/0.65 => ( ( ord_less_rat @ B @ zero_zero_rat )
% 0.34/0.65 => ( ord_less_rat @ ( times_times_rat @ A2 @ B ) @ zero_zero_rat ) ) ) ).
% 0.34/0.65
% 0.34/0.65 % mult_pos_neg
% 0.34/0.65 thf(fact_5768_mult__pos__pos,axiom,
% 0.34/0.65 ! [A2: nat,B: nat] :
% 0.34/0.65 ( ( ord_less_nat @ zero_zero_nat @ A2 )
% 0.34/0.65 => ( ( ord_less_nat @ zero_zero_nat @ B )
% 0.34/0.65 => ( ord_less_nat @ zero_zero_nat @ ( times_times_nat @ A2 @ B ) ) ) ) ).
% 0.34/0.65
% 0.34/0.65 % mult_pos_pos
% 0.34/0.65 thf(fact_5769_mult__pos__pos,axiom,
% 0.34/0.65 ! [A2: int,B: int] :
% 0.34/0.65 ( ( ord_less_int @ zero_zero_int @ A2 )
% 0.34/0.65 => ( ( ord_less_int @ zero_zero_int @ B )
% 0.34/0.65 => ( ord_less_int @ zero_zero_int @ ( times_times_int @ A2 @ B ) ) ) ) ).
% 0.34/0.65
% 0.34/0.65 % mult_pos_pos
% 0.34/0.65 thf(fact_5770_mult__pos__pos,axiom,
% 0.34/0.65 ! [A2: code_integer,B: code_integer] :
% 0.34/0.65 ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ A2 )
% 0.34/0.65 => ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ B )
% 0.34/0.65 => ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ ( times_3573771949741848930nteger @ A2 @ B ) ) ) ) ).
% 0.34/0.65
% 0.34/0.65 % mult_pos_pos
% 0.34/0.65 thf(fact_5771_mult__pos__pos,axiom,
% 0.34/0.65 ! [A2: rat,B: rat] :
% 0.34/0.65 ( ( ord_less_rat @ zero_zero_rat @ A2 )
% 0.34/0.65 => ( ( ord_less_rat @ zero_zero_rat @ B )
% 0.34/0.65 => ( ord_less_rat @ zero_zero_rat @ ( times_times_rat @ A2 @ B ) ) ) ) ).
% 0.34/0.65
% 0.34/0.65 % mult_pos_pos
% 0.34/0.65 thf(fact_5772_mult__pos__neg2,axiom,
% 0.34/0.65 ! [A2: nat,B: nat] :
% 0.34/0.65 ( ( ord_less_nat @ zero_zero_nat @ A2 )
% 0.34/0.65 => ( ( ord_less_nat @ B @ zero_zero_nat )
% 0.34/0.65 => ( ord_less_nat @ ( times_times_nat @ B @ A2 ) @ zero_zero_nat ) ) ) ).
% 0.34/0.65
% 0.34/0.65 % mult_pos_neg2
% 0.34/0.65 thf(fact_5773_mult__pos__neg2,axiom,
% 0.34/0.65 ! [A2: int,B: int] :
% 0.34/0.65 ( ( ord_less_int @ zero_zero_int @ A2 )
% 0.34/0.65 => ( ( ord_less_int @ B @ zero_zero_int )
% 0.34/0.65 => ( ord_less_int @ ( times_times_int @ B @ A2 ) @ zero_zero_int ) ) ) ).
% 0.34/0.65
% 0.34/0.65 % mult_pos_neg2
% 0.34/0.65 thf(fact_5774_mult__pos__neg2,axiom,
% 0.34/0.65 ! [A2: code_integer,B: code_integer] :
% 0.34/0.65 ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ A2 )
% 0.34/0.65 => ( ( ord_le6747313008572928689nteger @ B @ zero_z3403309356797280102nteger )
% 0.34/0.65 => ( ord_le6747313008572928689nteger @ ( times_3573771949741848930nteger @ B @ A2 ) @ zero_z3403309356797280102nteger ) ) ) ).
% 0.34/0.65
% 0.34/0.65 % mult_pos_neg2
% 0.34/0.65 thf(fact_5775_mult__pos__neg2,axiom,
% 0.34/0.65 ! [A2: rat,B: rat] :
% 0.34/0.65 ( ( ord_less_rat @ zero_zero_rat @ A2 )
% 0.34/0.65 => ( ( ord_less_rat @ B @ zero_zero_rat )
% 0.34/0.65 => ( ord_less_rat @ ( times_times_rat @ B @ A2 ) @ zero_zero_rat ) ) ) ).
% 0.34/0.65
% 0.34/0.65 % mult_pos_neg2
% 0.34/0.65 thf(fact_5776_zero__less__mult__iff,axiom,
% 0.34/0.65 ! [A2: int,B: int] :
% 0.34/0.65 ( ( ord_less_int @ zero_zero_int @ ( times_times_int @ A2 @ B ) )
% 0.34/0.65 = ( ( ( ord_less_int @ zero_zero_int @ A2 )
% 0.34/0.65 & ( ord_less_int @ zero_zero_int @ B ) )
% 0.34/0.65 | ( ( ord_less_int @ A2 @ zero_zero_int )
% 0.34/0.65 & ( ord_less_int @ B @ zero_zero_int ) ) ) ) ).
% 0.34/0.65
% 0.34/0.65 % zero_less_mult_iff
% 0.34/0.65 thf(fact_5777_zero__less__mult__iff,axiom,
% 0.34/0.65 ! [A2: code_integer,B: code_integer] :
% 0.34/0.65 ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ ( times_3573771949741848930nteger @ A2 @ B ) )
% 0.34/0.65 = ( ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ A2 )
% 0.34/0.65 & ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ B ) )
% 0.34/0.65 | ( ( ord_le6747313008572928689nteger @ A2 @ zero_z3403309356797280102nteger )
% 0.34/0.65 & ( ord_le6747313008572928689nteger @ B @ zero_z3403309356797280102nteger ) ) ) ) ).
% 0.34/0.65
% 0.34/0.65 % zero_less_mult_iff
% 0.34/0.65 thf(fact_5778_zero__less__mult__iff,axiom,
% 0.34/0.65 ! [A2: rat,B: rat] :
% 0.34/0.65 ( ( ord_less_rat @ zero_zero_rat @ ( times_times_rat @ A2 @ B ) )
% 0.34/0.65 = ( ( ( ord_less_rat @ zero_zero_rat @ A2 )
% 0.34/0.65 & ( ord_less_rat @ zero_zero_rat @ B ) )
% 0.34/0.65 | ( ( ord_less_rat @ A2 @ zero_zero_rat )
% 0.34/0.65 & ( ord_less_rat @ B @ zero_zero_rat ) ) ) ) ).
% 0.34/0.65
% 0.34/0.65 % zero_less_mult_iff
% 0.34/0.65 thf(fact_5779_zero__less__mult__pos,axiom,
% 0.34/0.65 ! [A2: nat,B: nat] :
% 0.34/0.65 ( ( ord_less_nat @ zero_zero_nat @ ( times_times_nat @ A2 @ B ) )
% 0.34/0.65 => ( ( ord_less_nat @ zero_zero_nat @ A2 )
% 0.34/0.65 => ( ord_less_nat @ zero_zero_nat @ B ) ) ) ).
% 0.34/0.65
% 0.34/0.65 % zero_less_mult_pos
% 0.34/0.65 thf(fact_5780_zero__less__mult__pos,axiom,
% 0.34/0.65 ! [A2: int,B: int] :
% 0.34/0.65 ( ( ord_less_int @ zero_zero_int @ ( times_times_int @ A2 @ B ) )
% 0.34/0.65 => ( ( ord_less_int @ zero_zero_int @ A2 )
% 0.34/0.65 => ( ord_less_int @ zero_zero_int @ B ) ) ) ).
% 0.34/0.65
% 0.34/0.65 % zero_less_mult_pos
% 0.34/0.65 thf(fact_5781_zero__less__mult__pos,axiom,
% 0.34/0.65 ! [A2: code_integer,B: code_integer] :
% 0.34/0.65 ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ ( times_3573771949741848930nteger @ A2 @ B ) )
% 0.34/0.65 => ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ A2 )
% 0.34/0.65 => ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ B ) ) ) ).
% 0.34/0.65
% 0.34/0.65 % zero_less_mult_pos
% 0.34/0.65 thf(fact_5782_zero__less__mult__pos,axiom,
% 0.34/0.65 ! [A2: rat,B: rat] :
% 0.34/0.65 ( ( ord_less_rat @ zero_zero_rat @ ( times_times_rat @ A2 @ B ) )
% 0.34/0.65 => ( ( ord_less_rat @ zero_zero_rat @ A2 )
% 0.34/0.65 => ( ord_less_rat @ zero_zero_rat @ B ) ) ) ).
% 0.34/0.65
% 0.34/0.65 % zero_less_mult_pos
% 0.34/0.65 thf(fact_5783_zero__less__mult__pos2,axiom,
% 0.34/0.65 ! [B: nat,A2: nat] :
% 0.34/0.65 ( ( ord_less_nat @ zero_zero_nat @ ( times_times_nat @ B @ A2 ) )
% 0.34/0.65 => ( ( ord_less_nat @ zero_zero_nat @ A2 )
% 0.34/0.65 => ( ord_less_nat @ zero_zero_nat @ B ) ) ) ).
% 0.34/0.65
% 0.34/0.65 % zero_less_mult_pos2
% 0.34/0.65 thf(fact_5784_zero__less__mult__pos2,axiom,
% 0.34/0.65 ! [B: int,A2: int] :
% 0.34/0.65 ( ( ord_less_int @ zero_zero_int @ ( times_times_int @ B @ A2 ) )
% 0.34/0.65 => ( ( ord_less_int @ zero_zero_int @ A2 )
% 0.34/0.65 => ( ord_less_int @ zero_zero_int @ B ) ) ) ).
% 0.34/0.65
% 0.34/0.65 % zero_less_mult_pos2
% 0.34/0.65 thf(fact_5785_zero__less__mult__pos2,axiom,
% 0.34/0.65 ! [B: code_integer,A2: code_integer] :
% 0.34/0.65 ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ ( times_3573771949741848930nteger @ B @ A2 ) )
% 0.34/0.65 => ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ A2 )
% 0.34/0.65 => ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ B ) ) ) ).
% 0.34/0.65
% 0.34/0.65 % zero_less_mult_pos2
% 0.34/0.65 thf(fact_5786_zero__less__mult__pos2,axiom,
% 0.34/0.65 ! [B: rat,A2: rat] :
% 0.34/0.65 ( ( ord_less_rat @ zero_zero_rat @ ( times_times_rat @ B @ A2 ) )
% 0.34/0.65 => ( ( ord_less_rat @ zero_zero_rat @ A2 )
% 0.34/0.65 => ( ord_less_rat @ zero_zero_rat @ B ) ) ) ).
% 0.34/0.65
% 0.34/0.65 % zero_less_mult_pos2
% 0.34/0.65 thf(fact_5787_mult__less__cancel__left__neg,axiom,
% 0.34/0.65 ! [C: int,A2: int,B: int] :
% 0.34/0.65 ( ( ord_less_int @ C @ zero_zero_int )
% 0.34/0.65 => ( ( ord_less_int @ ( times_times_int @ C @ A2 ) @ ( times_times_int @ C @ B ) )
% 0.34/0.65 = ( ord_less_int @ B @ A2 ) ) ) ).
% 0.34/0.65
% 0.34/0.65 % mult_less_cancel_left_neg
% 0.34/0.65 thf(fact_5788_mult__less__cancel__left__neg,axiom,
% 0.34/0.65 ! [C: code_integer,A2: code_integer,B: code_integer] :
% 0.34/0.65 ( ( ord_le6747313008572928689nteger @ C @ zero_z3403309356797280102nteger )
% 0.34/0.65 => ( ( ord_le6747313008572928689nteger @ ( times_3573771949741848930nteger @ C @ A2 ) @ ( times_3573771949741848930nteger @ C @ B ) )
% 0.34/0.65 = ( ord_le6747313008572928689nteger @ B @ A2 ) ) ) ).
% 0.34/0.65
% 0.34/0.65 % mult_less_cancel_left_neg
% 0.34/0.65 thf(fact_5789_mult__less__cancel__left__neg,axiom,
% 0.34/0.65 ! [C: rat,A2: rat,B: rat] :
% 0.34/0.65 ( ( ord_less_rat @ C @ zero_zero_rat )
% 0.34/0.65 => ( ( ord_less_rat @ ( times_times_rat @ C @ A2 ) @ ( times_times_rat @ C @ B ) )
% 0.34/0.65 = ( ord_less_rat @ B @ A2 ) ) ) ).
% 0.34/0.65
% 0.34/0.65 % mult_less_cancel_left_neg
% 0.34/0.65 thf(fact_5790_mult__less__cancel__left__pos,axiom,
% 0.34/0.65 ! [C: int,A2: int,B: int] :
% 0.34/0.65 ( ( ord_less_int @ zero_zero_int @ C )
% 0.34/0.65 => ( ( ord_less_int @ ( times_times_int @ C @ A2 ) @ ( times_times_int @ C @ B ) )
% 0.34/0.65 = ( ord_less_int @ A2 @ B ) ) ) ).
% 0.34/0.65
% 0.34/0.65 % mult_less_cancel_left_pos
% 0.34/0.65 thf(fact_5791_mult__less__cancel__left__pos,axiom,
% 0.34/0.65 ! [C: code_integer,A2: code_integer,B: code_integer] :
% 0.34/0.65 ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ C )
% 0.34/0.65 => ( ( ord_le6747313008572928689nteger @ ( times_3573771949741848930nteger @ C @ A2 ) @ ( times_3573771949741848930nteger @ C @ B ) )
% 0.34/0.65 = ( ord_le6747313008572928689nteger @ A2 @ B ) ) ) ).
% 0.34/0.65
% 0.34/0.65 % mult_less_cancel_left_pos
% 0.34/0.65 thf(fact_5792_mult__less__cancel__left__pos,axiom,
% 0.34/0.65 ! [C: rat,A2: rat,B: rat] :
% 0.34/0.65 ( ( ord_less_rat @ zero_zero_rat @ C )
% 0.34/0.65 => ( ( ord_less_rat @ ( times_times_rat @ C @ A2 ) @ ( times_times_rat @ C @ B ) )
% 0.34/0.65 = ( ord_less_rat @ A2 @ B ) ) ) ).
% 0.34/0.65
% 0.34/0.65 % mult_less_cancel_left_pos
% 0.34/0.65 thf(fact_5793_mult__strict__left__mono__neg,axiom,
% 0.34/0.65 ! [B: int,A2: int,C: int] :
% 0.34/0.65 ( ( ord_less_int @ B @ A2 )
% 0.34/0.65 => ( ( ord_less_int @ C @ zero_zero_int )
% 0.34/0.65 => ( ord_less_int @ ( times_times_int @ C @ A2 ) @ ( times_times_int @ C @ B ) ) ) ) ).
% 0.34/0.65
% 0.34/0.65 % mult_strict_left_mono_neg
% 0.34/0.65 thf(fact_5794_mult__strict__left__mono__neg,axiom,
% 0.34/0.65 ! [B: code_integer,A2: code_integer,C: code_integer] :
% 0.34/0.65 ( ( ord_le6747313008572928689nteger @ B @ A2 )
% 0.34/0.65 => ( ( ord_le6747313008572928689nteger @ C @ zero_z3403309356797280102nteger )
% 0.34/0.65 => ( ord_le6747313008572928689nteger @ ( times_3573771949741848930nteger @ C @ A2 ) @ ( times_3573771949741848930nteger @ C @ B ) ) ) ) ).
% 0.34/0.65
% 0.34/0.65 % mult_strict_left_mono_neg
% 0.34/0.65 thf(fact_5795_mult__strict__left__mono__neg,axiom,
% 0.34/0.65 ! [B: rat,A2: rat,C: rat] :
% 0.34/0.65 ( ( ord_less_rat @ B @ A2 )
% 0.34/0.65 => ( ( ord_less_rat @ C @ zero_zero_rat )
% 0.34/0.65 => ( ord_less_rat @ ( times_times_rat @ C @ A2 ) @ ( times_times_rat @ C @ B ) ) ) ) ).
% 0.34/0.65
% 0.34/0.65 % mult_strict_left_mono_neg
% 0.34/0.65 thf(fact_5796_mult__strict__left__mono,axiom,
% 0.34/0.65 ! [A2: nat,B: nat,C: nat] :
% 0.34/0.65 ( ( ord_less_nat @ A2 @ B )
% 0.34/0.65 => ( ( ord_less_nat @ zero_zero_nat @ C )
% 0.34/0.65 => ( ord_less_nat @ ( times_times_nat @ C @ A2 ) @ ( times_times_nat @ C @ B ) ) ) ) ).
% 0.34/0.65
% 0.34/0.65 % mult_strict_left_mono
% 0.34/0.65 thf(fact_5797_mult__strict__left__mono,axiom,
% 0.34/0.65 ! [A2: int,B: int,C: int] :
% 0.34/0.65 ( ( ord_less_int @ A2 @ B )
% 0.34/0.65 => ( ( ord_less_int @ zero_zero_int @ C )
% 0.34/0.65 => ( ord_less_int @ ( times_times_int @ C @ A2 ) @ ( times_times_int @ C @ B ) ) ) ) ).
% 0.34/0.65
% 0.34/0.65 % mult_strict_left_mono
% 0.34/0.65 thf(fact_5798_mult__strict__left__mono,axiom,
% 0.34/0.65 ! [A2: code_integer,B: code_integer,C: code_integer] :
% 0.34/0.65 ( ( ord_le6747313008572928689nteger @ A2 @ B )
% 0.34/0.65 => ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ C )
% 0.34/0.65 => ( ord_le6747313008572928689nteger @ ( times_3573771949741848930nteger @ C @ A2 ) @ ( times_3573771949741848930nteger @ C @ B ) ) ) ) ).
% 0.34/0.65
% 0.34/0.65 % mult_strict_left_mono
% 0.34/0.65 thf(fact_5799_mult__strict__left__mono,axiom,
% 0.34/0.65 ! [A2: rat,B: rat,C: rat] :
% 0.34/0.65 ( ( ord_less_rat @ A2 @ B )
% 0.34/0.65 => ( ( ord_less_rat @ zero_zero_rat @ C )
% 0.34/0.65 => ( ord_less_rat @ ( times_times_rat @ C @ A2 ) @ ( times_times_rat @ C @ B ) ) ) ) ).
% 0.34/0.65
% 0.34/0.65 % mult_strict_left_mono
% 0.34/0.65 thf(fact_5800_mult__less__cancel__left__disj,axiom,
% 0.34/0.65 ! [C: int,A2: int,B: int] :
% 0.34/0.65 ( ( ord_less_int @ ( times_times_int @ C @ A2 ) @ ( times_times_int @ C @ B ) )
% 0.34/0.65 = ( ( ( ord_less_int @ zero_zero_int @ C )
% 0.34/0.65 & ( ord_less_int @ A2 @ B ) )
% 0.34/0.65 | ( ( ord_less_int @ C @ zero_zero_int )
% 0.34/0.65 & ( ord_less_int @ B @ A2 ) ) ) ) ).
% 0.34/0.65
% 0.34/0.65 % mult_less_cancel_left_disj
% 0.34/0.65 thf(fact_5801_mult__less__cancel__left__disj,axiom,
% 0.34/0.65 ! [C: code_integer,A2: code_integer,B: code_integer] :
% 0.34/0.65 ( ( ord_le6747313008572928689nteger @ ( times_3573771949741848930nteger @ C @ A2 ) @ ( times_3573771949741848930nteger @ C @ B ) )
% 0.34/0.65 = ( ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ C )
% 0.34/0.65 & ( ord_le6747313008572928689nteger @ A2 @ B ) )
% 0.34/0.65 | ( ( ord_le6747313008572928689nteger @ C @ zero_z3403309356797280102nteger )
% 0.34/0.65 & ( ord_le6747313008572928689nteger @ B @ A2 ) ) ) ) ).
% 0.34/0.65
% 0.34/0.65 % mult_less_cancel_left_disj
% 0.34/0.65 thf(fact_5802_mult__less__cancel__left__disj,axiom,
% 0.34/0.65 ! [C: rat,A2: rat,B: rat] :
% 0.34/0.65 ( ( ord_less_rat @ ( times_times_rat @ C @ A2 ) @ ( times_times_rat @ C @ B ) )
% 0.34/0.65 = ( ( ( ord_less_rat @ zero_zero_rat @ C )
% 0.34/0.65 & ( ord_less_rat @ A2 @ B ) )
% 0.34/0.65 | ( ( ord_less_rat @ C @ zero_zero_rat )
% 0.34/0.65 & ( ord_less_rat @ B @ A2 ) ) ) ) ).
% 0.34/0.65
% 0.34/0.65 % mult_less_cancel_left_disj
% 0.34/0.65 thf(fact_5803_mult__strict__right__mono__neg,axiom,
% 0.34/0.65 ! [B: int,A2: int,C: int] :
% 0.34/0.65 ( ( ord_less_int @ B @ A2 )
% 0.34/0.65 => ( ( ord_less_int @ C @ zero_zero_int )
% 0.34/0.65 => ( ord_less_int @ ( times_times_int @ A2 @ C ) @ ( times_times_int @ B @ C ) ) ) ) ).
% 0.34/0.65
% 0.34/0.65 % mult_strict_right_mono_neg
% 0.34/0.65 thf(fact_5804_mult__strict__right__mono__neg,axiom,
% 0.34/0.65 ! [B: code_integer,A2: code_integer,C: code_integer] :
% 0.34/0.65 ( ( ord_le6747313008572928689nteger @ B @ A2 )
% 0.34/0.65 => ( ( ord_le6747313008572928689nteger @ C @ zero_z3403309356797280102nteger )
% 0.34/0.65 => ( ord_le6747313008572928689nteger @ ( times_3573771949741848930nteger @ A2 @ C ) @ ( times_3573771949741848930nteger @ B @ C ) ) ) ) ).
% 0.34/0.65
% 0.34/0.65 % mult_strict_right_mono_neg
% 0.34/0.65 thf(fact_5805_mult__strict__right__mono__neg,axiom,
% 0.34/0.65 ! [B: rat,A2: rat,C: rat] :
% 0.34/0.65 ( ( ord_less_rat @ B @ A2 )
% 0.34/0.65 => ( ( ord_less_rat @ C @ zero_zero_rat )
% 0.34/0.65 => ( ord_less_rat @ ( times_times_rat @ A2 @ C ) @ ( times_times_rat @ B @ C ) ) ) ) ).
% 0.34/0.65
% 0.34/0.65 % mult_strict_right_mono_neg
% 0.34/0.65 thf(fact_5806_mult__strict__right__mono,axiom,
% 0.34/0.65 ! [A2: nat,B: nat,C: nat] :
% 0.34/0.65 ( ( ord_less_nat @ A2 @ B )
% 0.34/0.65 => ( ( ord_less_nat @ zero_zero_nat @ C )
% 0.34/0.65 => ( ord_less_nat @ ( times_times_nat @ A2 @ C ) @ ( times_times_nat @ B @ C ) ) ) ) ).
% 0.34/0.65
% 0.34/0.65 % mult_strict_right_mono
% 0.34/0.65 thf(fact_5807_mult__strict__right__mono,axiom,
% 0.34/0.65 ! [A2: int,B: int,C: int] :
% 0.34/0.65 ( ( ord_less_int @ A2 @ B )
% 0.34/0.65 => ( ( ord_less_int @ zero_zero_int @ C )
% 0.34/0.65 => ( ord_less_int @ ( times_times_int @ A2 @ C ) @ ( times_times_int @ B @ C ) ) ) ) ).
% 0.34/0.65
% 0.34/0.65 % mult_strict_right_mono
% 0.34/0.65 thf(fact_5808_mult__strict__right__mono,axiom,
% 0.34/0.65 ! [A2: code_integer,B: code_integer,C: code_integer] :
% 0.34/0.65 ( ( ord_le6747313008572928689nteger @ A2 @ B )
% 0.34/0.65 => ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ C )
% 0.34/0.65 => ( ord_le6747313008572928689nteger @ ( times_3573771949741848930nteger @ A2 @ C ) @ ( times_3573771949741848930nteger @ B @ C ) ) ) ) ).
% 0.34/0.65
% 0.34/0.65 % mult_strict_right_mono
% 0.34/0.65 thf(fact_5809_mult__strict__right__mono,axiom,
% 0.34/0.65 ! [A2: rat,B: rat,C: rat] :
% 0.34/0.65 ( ( ord_less_rat @ A2 @ B )
% 0.34/0.65 => ( ( ord_less_rat @ zero_zero_rat @ C )
% 0.34/0.65 => ( ord_less_rat @ ( times_times_rat @ A2 @ C ) @ ( times_times_rat @ B @ C ) ) ) ) ).
% 0.34/0.65
% 0.34/0.65 % mult_strict_right_mono
% 0.34/0.65 thf(fact_5810_mult__less__cancel__right__disj,axiom,
% 0.34/0.65 ! [A2: int,C: int,B: int] :
% 0.34/0.65 ( ( ord_less_int @ ( times_times_int @ A2 @ C ) @ ( times_times_int @ B @ C ) )
% 0.34/0.65 = ( ( ( ord_less_int @ zero_zero_int @ C )
% 0.34/0.65 & ( ord_less_int @ A2 @ B ) )
% 0.34/0.65 | ( ( ord_less_int @ C @ zero_zero_int )
% 0.34/0.65 & ( ord_less_int @ B @ A2 ) ) ) ) ).
% 0.34/0.65
% 0.34/0.65 % mult_less_cancel_right_disj
% 0.34/0.65 thf(fact_5811_mult__less__cancel__right__disj,axiom,
% 0.34/0.65 ! [A2: code_integer,C: code_integer,B: code_integer] :
% 0.34/0.65 ( ( ord_le6747313008572928689nteger @ ( times_3573771949741848930nteger @ A2 @ C ) @ ( times_3573771949741848930nteger @ B @ C ) )
% 0.34/0.65 = ( ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ C )
% 0.34/0.65 & ( ord_le6747313008572928689nteger @ A2 @ B ) )
% 0.34/0.65 | ( ( ord_le6747313008572928689nteger @ C @ zero_z3403309356797280102nteger )
% 0.34/0.65 & ( ord_le6747313008572928689nteger @ B @ A2 ) ) ) ) ).
% 0.34/0.65
% 0.34/0.65 % mult_less_cancel_right_disj
% 0.34/0.65 thf(fact_5812_mult__less__cancel__right__disj,axiom,
% 0.34/0.65 ! [A2: rat,C: rat,B: rat] :
% 0.34/0.65 ( ( ord_less_rat @ ( times_times_rat @ A2 @ C ) @ ( times_times_rat @ B @ C ) )
% 0.34/0.65 = ( ( ( ord_less_rat @ zero_zero_rat @ C )
% 0.34/0.65 & ( ord_less_rat @ A2 @ B ) )
% 0.34/0.65 | ( ( ord_less_rat @ C @ zero_zero_rat )
% 0.34/0.65 & ( ord_less_rat @ B @ A2 ) ) ) ) ).
% 0.34/0.65
% 0.34/0.65 % mult_less_cancel_right_disj
% 0.34/0.65 thf(fact_5813_linordered__comm__semiring__strict__class_Ocomm__mult__strict__left__mono,axiom,
% 0.34/0.65 ! [A2: nat,B: nat,C: nat] :
% 0.34/0.65 ( ( ord_less_nat @ A2 @ B )
% 0.34/0.65 => ( ( ord_less_nat @ zero_zero_nat @ C )
% 0.34/0.65 => ( ord_less_nat @ ( times_times_nat @ C @ A2 ) @ ( times_times_nat @ C @ B ) ) ) ) ).
% 0.34/0.65
% 0.34/0.65 % linordered_comm_semiring_strict_class.comm_mult_strict_left_mono
% 0.34/0.65 thf(fact_5814_linordered__comm__semiring__strict__class_Ocomm__mult__strict__left__mono,axiom,
% 0.34/0.65 ! [A2: int,B: int,C: int] :
% 0.34/0.65 ( ( ord_less_int @ A2 @ B )
% 0.34/0.65 => ( ( ord_less_int @ zero_zero_int @ C )
% 0.34/0.65 => ( ord_less_int @ ( times_times_int @ C @ A2 ) @ ( times_times_int @ C @ B ) ) ) ) ).
% 0.34/0.65
% 0.34/0.65 % linordered_comm_semiring_strict_class.comm_mult_strict_left_mono
% 0.34/0.65 thf(fact_5815_linordered__comm__semiring__strict__class_Ocomm__mult__strict__left__mono,axiom,
% 0.34/0.65 ! [A2: code_integer,B: code_integer,C: code_integer] :
% 0.34/0.65 ( ( ord_le6747313008572928689nteger @ A2 @ B )
% 0.34/0.65 => ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ C )
% 0.34/0.65 => ( ord_le6747313008572928689nteger @ ( times_3573771949741848930nteger @ C @ A2 ) @ ( times_3573771949741848930nteger @ C @ B ) ) ) ) ).
% 0.34/0.65
% 0.34/0.65 % linordered_comm_semiring_strict_class.comm_mult_strict_left_mono
% 0.34/0.65 thf(fact_5816_linordered__comm__semiring__strict__class_Ocomm__mult__strict__left__mono,axiom,
% 0.34/0.65 ! [A2: rat,B: rat,C: rat] :
% 0.34/0.65 ( ( ord_less_rat @ A2 @ B )
% 0.34/0.65 => ( ( ord_less_rat @ zero_zero_rat @ C )
% 0.34/0.65 => ( ord_less_rat @ ( times_times_rat @ C @ A2 ) @ ( times_times_rat @ C @ B ) ) ) ) ).
% 0.34/0.65
% 0.34/0.65 % linordered_comm_semiring_strict_class.comm_mult_strict_left_mono
% 0.34/0.65 thf(fact_5817_less__numeral__extra_I1_J,axiom,
% 0.34/0.65 ord_le5570908160329646204atural @ zero_z2226904508553997617atural @ one_one_Code_natural ).
% 0.34/0.65
% 0.34/0.65 % less_numeral_extra(1)
% 0.34/0.65 thf(fact_5818_less__numeral__extra_I1_J,axiom,
% 0.34/0.65 ord_less_nat @ zero_zero_nat @ one_one_nat ).
% 0.34/0.65
% 0.34/0.65 % less_numeral_extra(1)
% 0.34/0.65 thf(fact_5819_less__numeral__extra_I1_J,axiom,
% 0.34/0.65 ord_less_int @ zero_zero_int @ one_one_int ).
% 0.34/0.65
% 0.34/0.65 % less_numeral_extra(1)
% 0.34/0.65 thf(fact_5820_less__numeral__extra_I1_J,axiom,
% 0.34/0.65 ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ one_one_Code_integer ).
% 0.34/0.65
% 0.34/0.65 % less_numeral_extra(1)
% 0.34/0.65 thf(fact_5821_less__numeral__extra_I1_J,axiom,
% 0.34/0.65 ord_less_rat @ zero_zero_rat @ one_one_rat ).
% 0.34/0.65
% 0.34/0.65 % less_numeral_extra(1)
% 0.34/0.65 thf(fact_5822_zero__less__one,axiom,
% 0.34/0.65 ord_le5570908160329646204atural @ zero_z2226904508553997617atural @ one_one_Code_natural ).
% 0.34/0.65
% 0.34/0.65 % zero_less_one
% 0.34/0.65 thf(fact_5823_zero__less__one,axiom,
% 0.34/0.65 ord_less_nat @ zero_zero_nat @ one_one_nat ).
% 0.34/0.65
% 0.34/0.65 % zero_less_one
% 0.34/0.65 thf(fact_5824_zero__less__one,axiom,
% 0.34/0.65 ord_less_int @ zero_zero_int @ one_one_int ).
% 0.34/0.65
% 0.34/0.65 % zero_less_one
% 0.34/0.65 thf(fact_5825_zero__less__one,axiom,
% 0.34/0.65 ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ one_one_Code_integer ).
% 0.34/0.65
% 0.34/0.65 % zero_less_one
% 0.34/0.65 thf(fact_5826_zero__less__one,axiom,
% 0.34/0.65 ord_less_rat @ zero_zero_rat @ one_one_rat ).
% 0.34/0.65
% 0.34/0.65 % zero_less_one
% 0.34/0.65 thf(fact_5827_not__one__less__zero,axiom,
% 0.34/0.65 ~ ( ord_le5570908160329646204atural @ one_one_Code_natural @ zero_z2226904508553997617atural ) ).
% 0.34/0.65
% 0.34/0.65 % not_one_less_zero
% 0.34/0.65 thf(fact_5828_not__one__less__zero,axiom,
% 0.34/0.65 ~ ( ord_less_nat @ one_one_nat @ zero_zero_nat ) ).
% 0.34/0.65
% 0.34/0.65 % not_one_less_zero
% 0.34/0.65 thf(fact_5829_not__one__less__zero,axiom,
% 0.34/0.65 ~ ( ord_less_int @ one_one_int @ zero_zero_int ) ).
% 0.34/0.65
% 0.34/0.65 % not_one_less_zero
% 0.34/0.65 thf(fact_5830_not__one__less__zero,axiom,
% 0.34/0.65 ~ ( ord_le6747313008572928689nteger @ one_one_Code_integer @ zero_z3403309356797280102nteger ) ).
% 0.34/0.65
% 0.34/0.65 % not_one_less_zero
% 0.34/0.65 thf(fact_5831_not__one__less__zero,axiom,
% 0.34/0.65 ~ ( ord_less_rat @ one_one_rat @ zero_zero_rat ) ).
% 0.34/0.65
% 0.34/0.65 % not_one_less_zero
% 0.34/0.65 thf(fact_5832_less__iff__diff__less__0,axiom,
% 0.34/0.65 ( ord_less_int
% 0.34/0.65 = ( ^ [A6: int,B6: int] : ( ord_less_int @ ( minus_minus_int @ A6 @ B6 ) @ zero_zero_int ) ) ) ).
% 0.34/0.65
% 0.34/0.65 % less_iff_diff_less_0
% 0.34/0.65 thf(fact_5833_less__iff__diff__less__0,axiom,
% 0.34/0.65 ( ord_le6747313008572928689nteger
% 0.34/0.65 = ( ^ [A6: code_integer,B6: code_integer] : ( ord_le6747313008572928689nteger @ ( minus_8373710615458151222nteger @ A6 @ B6 ) @ zero_z3403309356797280102nteger ) ) ) ).
% 0.34/0.65
% 0.34/0.65 % less_iff_diff_less_0
% 0.34/0.65 thf(fact_5834_less__iff__diff__less__0,axiom,
% 0.34/0.65 ( ord_less_rat
% 0.34/0.65 = ( ^ [A6: rat,B6: rat] : ( ord_less_rat @ ( minus_minus_rat @ A6 @ B6 ) @ zero_zero_rat ) ) ) ).
% 0.34/0.65
% 0.34/0.65 % less_iff_diff_less_0
% 0.34/0.65 thf(fact_5835_inj__on__mult,axiom,
% 0.34/0.65 ! [A2: code_integer,A: set_Code_integer] :
% 0.34/0.65 ( ( A2 != zero_z3403309356797280102nteger )
% 0.34/0.65 => ( inj_on1158064722259180589nteger @ ( times_3573771949741848930nteger @ A2 ) @ A ) ) ).
% 0.34/0.65
% 0.34/0.65 % inj_on_mult
% 0.34/0.65 thf(fact_5836_inj__on__mult,axiom,
% 0.34/0.65 ! [A2: rat,A: set_rat] :
% 0.34/0.65 ( ( A2 != zero_zero_rat )
% 0.34/0.65 => ( inj_on_rat_rat @ ( times_times_rat @ A2 ) @ A ) ) ).
% 0.34/0.65
% 0.34/0.65 % inj_on_mult
% 0.34/0.65 thf(fact_5837_inj__on__mult,axiom,
% 0.34/0.65 ! [A2: nat,A: set_nat] :
% 0.34/0.65 ( ( A2 != zero_zero_nat )
% 0.34/0.65 => ( inj_on_nat_nat @ ( times_times_nat @ A2 ) @ A ) ) ).
% 0.34/0.65
% 0.34/0.65 % inj_on_mult
% 0.34/0.65 thf(fact_5838_inj__on__mult,axiom,
% 0.34/0.65 ! [A2: int,A: set_int] :
% 0.34/0.65 ( ( A2 != zero_zero_int )
% 0.34/0.65 => ( inj_on_int_int @ ( times_times_int @ A2 ) @ A ) ) ).
% 0.34/0.65
% 0.34/0.65 % inj_on_mult
% 0.34/0.65 thf(fact_5839_nat__compl__induct,axiom,
% 0.34/0.65 ! [P: nat > $o,N2: nat] :
% 0.34/0.65 ( ( P @ zero_zero_nat )
% 0.34/0.65 => ( ! [N3: nat] :
% 0.34/0.65 ( ! [Nn: nat] :
% 0.34/0.65 ( ( ord_less_eq_nat @ Nn @ N3 )
% 0.34/0.65 => ( P @ Nn ) )
% 0.34/0.65 => ( P @ ( suc @ N3 ) ) )
% 0.34/0.65 => ( P @ N2 ) ) ) ).
% 0.34/0.65
% 0.34/0.65 % nat_compl_induct
% 0.34/0.65 thf(fact_5840_nat__compl__induct_H,axiom,
% 0.34/0.65 ! [P: nat > $o,N2: nat] :
% 0.34/0.65 ( ( P @ zero_zero_nat )
% 0.34/0.65 => ( ! [N3: nat] :
% 0.34/0.65 ( ! [Nn: nat] :
% 0.34/0.65 ( ( ord_less_eq_nat @ Nn @ N3 )
% 0.34/0.65 => ( P @ Nn ) )
% 0.34/0.65 => ( P @ ( suc @ N3 ) ) )
% 0.34/0.65 => ( P @ N2 ) ) ) ).
% 0.34/0.65
% 0.34/0.65 % nat_compl_induct'
% 0.34/0.65 thf(fact_5841_Suc__to__right,axiom,
% 0.34/0.65 ! [N2: nat,M4: nat] :
% 0.34/0.65 ( ( ( suc @ N2 )
% 0.34/0.65 = M4 )
% 0.34/0.65 => ( N2
% 0.34/0.65 = ( minus_minus_nat @ M4 @ ( suc @ zero_zero_nat ) ) ) ) ).
% 0.34/0.65
% 0.34/0.65 % Suc_to_right
% 0.34/0.65 thf(fact_5842_mult__less__mono1,axiom,
% 0.34/0.65 ! [I2: nat,J: nat,K3: nat] :
% 0.34/0.65 ( ( ord_less_nat @ I2 @ J )
% 0.34/0.65 => ( ( ord_less_nat @ zero_zero_nat @ K3 )
% 0.34/0.65 => ( ord_less_nat @ ( times_times_nat @ I2 @ K3 ) @ ( times_times_nat @ J @ K3 ) ) ) ) ).
% 0.34/0.65
% 0.34/0.65 % mult_less_mono1
% 0.34/0.65 thf(fact_5843_mult__less__mono2,axiom,
% 0.34/0.65 ! [I2: nat,J: nat,K3: nat] :
% 0.34/0.65 ( ( ord_less_nat @ I2 @ J )
% 0.34/0.65 => ( ( ord_less_nat @ zero_zero_nat @ K3 )
% 0.34/0.65 => ( ord_less_nat @ ( times_times_nat @ K3 @ I2 ) @ ( times_times_nat @ K3 @ J ) ) ) ) ).
% 0.34/0.65
% 0.34/0.65 % mult_less_mono2
% 0.34/0.65 thf(fact_5844_nat__mult__eq__cancel1,axiom,
% 0.34/0.65 ! [K3: nat,M4: nat,N2: nat] :
% 0.34/0.65 ( ( ord_less_nat @ zero_zero_nat @ K3 )
% 0.34/0.65 => ( ( ( times_times_nat @ K3 @ M4 )
% 0.34/0.65 = ( times_times_nat @ K3 @ N2 ) )
% 0.34/0.65 = ( M4 = N2 ) ) ) ).
% 0.34/0.65
% 0.34/0.65 % nat_mult_eq_cancel1
% 0.34/0.65 thf(fact_5845_nat__mult__less__cancel1,axiom,
% 0.34/0.65 ! [K3: nat,M4: nat,N2: nat] :
% 0.34/0.65 ( ( ord_less_nat @ zero_zero_nat @ K3 )
% 0.34/0.65 => ( ( ord_less_nat @ ( times_times_nat @ K3 @ M4 ) @ ( times_times_nat @ K3 @ N2 ) )
% 0.34/0.65 = ( ord_less_nat @ M4 @ N2 ) ) ) ).
% 0.34/0.65
% 0.34/0.65 % nat_mult_less_cancel1
% 0.34/0.65 thf(fact_5846_nat__geq__1__eq__neqz,axiom,
% 0.34/0.65 ! [X: nat] :
% 0.34/0.65 ( ( ord_less_eq_nat @ one_one_nat @ X )
% 0.34/0.65 = ( X != zero_zero_nat ) ) ).
% 0.34/0.65
% 0.34/0.65 % nat_geq_1_eq_neqz
% 0.34/0.65 thf(fact_5847_mult__eq__self__implies__10,axiom,
% 0.34/0.65 ! [M4: nat,N2: nat] :
% 0.34/0.65 ( ( M4
% 0.34/0.65 = ( times_times_nat @ M4 @ N2 ) )
% 0.34/0.65 => ( ( N2 = one_one_nat )
% 0.34/0.65 | ( M4 = zero_zero_nat ) ) ) ).
% 0.34/0.65
% 0.34/0.65 % mult_eq_self_implies_10
% 0.34/0.65 thf(fact_5848_add__strict__increasing2,axiom,
% 0.34/0.65 ! [A2: code_integer,B: code_integer,C: code_integer] :
% 0.34/0.65 ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ A2 )
% 0.34/0.65 => ( ( ord_le6747313008572928689nteger @ B @ C )
% 0.34/0.65 => ( ord_le6747313008572928689nteger @ B @ ( plus_p5714425477246183910nteger @ A2 @ C ) ) ) ) ).
% 0.34/0.65
% 0.34/0.65 % add_strict_increasing2
% 0.34/0.65 thf(fact_5849_add__strict__increasing2,axiom,
% 0.34/0.65 ! [A2: rat,B: rat,C: rat] :
% 0.34/0.65 ( ( ord_less_eq_rat @ zero_zero_rat @ A2 )
% 0.34/0.65 => ( ( ord_less_rat @ B @ C )
% 0.34/0.65 => ( ord_less_rat @ B @ ( plus_plus_rat @ A2 @ C ) ) ) ) ).
% 0.34/0.65
% 0.34/0.65 % add_strict_increasing2
% 0.34/0.65 thf(fact_5850_add__strict__increasing2,axiom,
% 0.34/0.65 ! [A2: nat,B: nat,C: nat] :
% 0.34/0.65 ( ( ord_less_eq_nat @ zero_zero_nat @ A2 )
% 0.34/0.65 => ( ( ord_less_nat @ B @ C )
% 0.34/0.65 => ( ord_less_nat @ B @ ( plus_plus_nat @ A2 @ C ) ) ) ) ).
% 0.34/0.65
% 0.34/0.65 % add_strict_increasing2
% 0.34/0.65 thf(fact_5851_add__strict__increasing2,axiom,
% 0.34/0.65 ! [A2: int,B: int,C: int] :
% 0.34/0.65 ( ( ord_less_eq_int @ zero_zero_int @ A2 )
% 0.34/0.65 => ( ( ord_less_int @ B @ C )
% 0.34/0.65 => ( ord_less_int @ B @ ( plus_plus_int @ A2 @ C ) ) ) ) ).
% 0.34/0.65
% 0.34/0.65 % add_strict_increasing2
% 0.34/0.65 thf(fact_5852_add__strict__increasing,axiom,
% 0.34/0.65 ! [A2: code_integer,B: code_integer,C: code_integer] :
% 0.34/0.65 ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ A2 )
% 0.34/0.65 => ( ( ord_le3102999989581377725nteger @ B @ C )
% 0.34/0.65 => ( ord_le6747313008572928689nteger @ B @ ( plus_p5714425477246183910nteger @ A2 @ C ) ) ) ) ).
% 0.34/0.65
% 0.34/0.65 % add_strict_increasing
% 0.34/0.65 thf(fact_5853_add__strict__increasing,axiom,
% 0.34/0.65 ! [A2: rat,B: rat,C: rat] :
% 0.34/0.65 ( ( ord_less_rat @ zero_zero_rat @ A2 )
% 0.34/0.65 => ( ( ord_less_eq_rat @ B @ C )
% 0.34/0.65 => ( ord_less_rat @ B @ ( plus_plus_rat @ A2 @ C ) ) ) ) ).
% 0.34/0.65
% 0.34/0.65 % add_strict_increasing
% 0.34/0.65 thf(fact_5854_add__strict__increasing,axiom,
% 0.34/0.65 ! [A2: nat,B: nat,C: nat] :
% 0.34/0.65 ( ( ord_less_nat @ zero_zero_nat @ A2 )
% 0.34/0.65 => ( ( ord_less_eq_nat @ B @ C )
% 0.34/0.65 => ( ord_less_nat @ B @ ( plus_plus_nat @ A2 @ C ) ) ) ) ).
% 0.34/0.65
% 0.34/0.65 % add_strict_increasing
% 0.34/0.65 thf(fact_5855_add__strict__increasing,axiom,
% 0.34/0.65 ! [A2: int,B: int,C: int] :
% 0.34/0.65 ( ( ord_less_int @ zero_zero_int @ A2 )
% 0.34/0.65 => ( ( ord_less_eq_int @ B @ C )
% 0.34/0.65 => ( ord_less_int @ B @ ( plus_plus_int @ A2 @ C ) ) ) ) ).
% 0.34/0.65
% 0.34/0.65 % add_strict_increasing
% 0.34/0.65 thf(fact_5856_add__pos__nonneg,axiom,
% 0.34/0.65 ! [A2: code_integer,B: code_integer] :
% 0.34/0.65 ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ A2 )
% 0.34/0.65 => ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ B )
% 0.34/0.65 => ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ ( plus_p5714425477246183910nteger @ A2 @ B ) ) ) ) ).
% 0.34/0.65
% 0.34/0.65 % add_pos_nonneg
% 0.34/0.65 thf(fact_5857_add__pos__nonneg,axiom,
% 0.34/0.65 ! [A2: rat,B: rat] :
% 0.34/0.65 ( ( ord_less_rat @ zero_zero_rat @ A2 )
% 0.34/0.65 => ( ( ord_less_eq_rat @ zero_zero_rat @ B )
% 0.34/0.65 => ( ord_less_rat @ zero_zero_rat @ ( plus_plus_rat @ A2 @ B ) ) ) ) ).
% 0.34/0.65
% 0.34/0.65 % add_pos_nonneg
% 0.34/0.65 thf(fact_5858_add__pos__nonneg,axiom,
% 0.34/0.65 ! [A2: nat,B: nat] :
% 0.34/0.65 ( ( ord_less_nat @ zero_zero_nat @ A2 )
% 0.34/0.65 => ( ( ord_less_eq_nat @ zero_zero_nat @ B )
% 0.34/0.65 => ( ord_less_nat @ zero_zero_nat @ ( plus_plus_nat @ A2 @ B ) ) ) ) ).
% 0.34/0.65
% 0.34/0.65 % add_pos_nonneg
% 0.34/0.65 thf(fact_5859_add__pos__nonneg,axiom,
% 0.34/0.65 ! [A2: int,B: int] :
% 0.34/0.65 ( ( ord_less_int @ zero_zero_int @ A2 )
% 0.34/0.65 => ( ( ord_less_eq_int @ zero_zero_int @ B )
% 0.34/0.65 => ( ord_less_int @ zero_zero_int @ ( plus_plus_int @ A2 @ B ) ) ) ) ).
% 0.34/0.65
% 0.34/0.65 % add_pos_nonneg
% 0.34/0.65 thf(fact_5860_add__nonpos__neg,axiom,
% 0.34/0.65 ! [A2: code_integer,B: code_integer] :
% 0.34/0.65 ( ( ord_le3102999989581377725nteger @ A2 @ zero_z3403309356797280102nteger )
% 0.34/0.65 => ( ( ord_le6747313008572928689nteger @ B @ zero_z3403309356797280102nteger )
% 0.34/0.65 => ( ord_le6747313008572928689nteger @ ( plus_p5714425477246183910nteger @ A2 @ B ) @ zero_z3403309356797280102nteger ) ) ) ).
% 0.34/0.65
% 0.34/0.65 % add_nonpos_neg
% 0.34/0.65 thf(fact_5861_add__nonpos__neg,axiom,
% 0.34/0.65 ! [A2: rat,B: rat] :
% 0.34/0.65 ( ( ord_less_eq_rat @ A2 @ zero_zero_rat )
% 0.34/0.65 => ( ( ord_less_rat @ B @ zero_zero_rat )
% 0.34/0.65 => ( ord_less_rat @ ( plus_plus_rat @ A2 @ B ) @ zero_zero_rat ) ) ) ).
% 0.34/0.65
% 0.34/0.65 % add_nonpos_neg
% 0.34/0.65 thf(fact_5862_add__nonpos__neg,axiom,
% 0.34/0.65 ! [A2: nat,B: nat] :
% 0.34/0.65 ( ( ord_less_eq_nat @ A2 @ zero_zero_nat )
% 0.34/0.65 => ( ( ord_less_nat @ B @ zero_zero_nat )
% 0.34/0.65 => ( ord_less_nat @ ( plus_plus_nat @ A2 @ B ) @ zero_zero_nat ) ) ) ).
% 0.34/0.65
% 0.34/0.65 % add_nonpos_neg
% 0.34/0.65 thf(fact_5863_add__nonpos__neg,axiom,
% 0.34/0.65 ! [A2: int,B: int] :
% 0.34/0.65 ( ( ord_less_eq_int @ A2 @ zero_zero_int )
% 0.34/0.65 => ( ( ord_less_int @ B @ zero_zero_int )
% 0.34/0.65 => ( ord_less_int @ ( plus_plus_int @ A2 @ B ) @ zero_zero_int ) ) ) ).
% 0.34/0.65
% 0.34/0.65 % add_nonpos_neg
% 0.34/0.65 thf(fact_5864_add__nonneg__pos,axiom,
% 0.34/0.65 ! [A2: code_integer,B: code_integer] :
% 0.34/0.65 ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ A2 )
% 0.34/0.65 => ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ B )
% 0.34/0.65 => ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ ( plus_p5714425477246183910nteger @ A2 @ B ) ) ) ) ).
% 0.34/0.65
% 0.34/0.65 % add_nonneg_pos
% 0.34/0.65 thf(fact_5865_add__nonneg__pos,axiom,
% 0.34/0.65 ! [A2: rat,B: rat] :
% 0.34/0.65 ( ( ord_less_eq_rat @ zero_zero_rat @ A2 )
% 0.34/0.65 => ( ( ord_less_rat @ zero_zero_rat @ B )
% 0.34/0.65 => ( ord_less_rat @ zero_zero_rat @ ( plus_plus_rat @ A2 @ B ) ) ) ) ).
% 0.34/0.65
% 0.34/0.65 % add_nonneg_pos
% 0.34/0.65 thf(fact_5866_add__nonneg__pos,axiom,
% 0.34/0.65 ! [A2: nat,B: nat] :
% 0.34/0.65 ( ( ord_less_eq_nat @ zero_zero_nat @ A2 )
% 0.34/0.65 => ( ( ord_less_nat @ zero_zero_nat @ B )
% 0.34/0.65 => ( ord_less_nat @ zero_zero_nat @ ( plus_plus_nat @ A2 @ B ) ) ) ) ).
% 0.34/0.65
% 0.34/0.65 % add_nonneg_pos
% 0.34/0.65 thf(fact_5867_add__nonneg__pos,axiom,
% 0.34/0.65 ! [A2: int,B: int] :
% 0.34/0.65 ( ( ord_less_eq_int @ zero_zero_int @ A2 )
% 0.34/0.65 => ( ( ord_less_int @ zero_zero_int @ B )
% 0.34/0.65 => ( ord_less_int @ zero_zero_int @ ( plus_plus_int @ A2 @ B ) ) ) ) ).
% 0.34/0.65
% 0.34/0.65 % add_nonneg_pos
% 0.34/0.65 thf(fact_5868_add__neg__nonpos,axiom,
% 0.34/0.65 ! [A2: code_integer,B: code_integer] :
% 0.34/0.65 ( ( ord_le6747313008572928689nteger @ A2 @ zero_z3403309356797280102nteger )
% 0.34/0.65 => ( ( ord_le3102999989581377725nteger @ B @ zero_z3403309356797280102nteger )
% 0.34/0.65 => ( ord_le6747313008572928689nteger @ ( plus_p5714425477246183910nteger @ A2 @ B ) @ zero_z3403309356797280102nteger ) ) ) ).
% 0.34/0.65
% 0.34/0.65 % add_neg_nonpos
% 0.34/0.65 thf(fact_5869_add__neg__nonpos,axiom,
% 0.34/0.65 ! [A2: rat,B: rat] :
% 0.34/0.65 ( ( ord_less_rat @ A2 @ zero_zero_rat )
% 0.34/0.65 => ( ( ord_less_eq_rat @ B @ zero_zero_rat )
% 0.34/0.65 => ( ord_less_rat @ ( plus_plus_rat @ A2 @ B ) @ zero_zero_rat ) ) ) ).
% 0.34/0.65
% 0.34/0.65 % add_neg_nonpos
% 0.34/0.65 thf(fact_5870_add__neg__nonpos,axiom,
% 0.34/0.65 ! [A2: nat,B: nat] :
% 0.34/0.65 ( ( ord_less_nat @ A2 @ zero_zero_nat )
% 0.34/0.65 => ( ( ord_less_eq_nat @ B @ zero_zero_nat )
% 0.34/0.65 => ( ord_less_nat @ ( plus_plus_nat @ A2 @ B ) @ zero_zero_nat ) ) ) ).
% 0.34/0.65
% 0.34/0.65 % add_neg_nonpos
% 0.34/0.65 thf(fact_5871_add__neg__nonpos,axiom,
% 0.34/0.65 ! [A2: int,B: int] :
% 0.34/0.65 ( ( ord_less_int @ A2 @ zero_zero_int )
% 0.34/0.65 => ( ( ord_less_eq_int @ B @ zero_zero_int )
% 0.34/0.65 => ( ord_less_int @ ( plus_plus_int @ A2 @ B ) @ zero_zero_int ) ) ) ).
% 0.34/0.65
% 0.34/0.65 % add_neg_nonpos
% 0.34/0.65 thf(fact_5872_mult__less__le__imp__less,axiom,
% 0.34/0.65 ! [A2: code_integer,B: code_integer,C: code_integer,D2: code_integer] :
% 0.34/0.65 ( ( ord_le6747313008572928689nteger @ A2 @ B )
% 0.34/0.65 => ( ( ord_le3102999989581377725nteger @ C @ D2 )
% 0.34/0.65 => ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ A2 )
% 0.34/0.65 => ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ C )
% 0.34/0.65 => ( ord_le6747313008572928689nteger @ ( times_3573771949741848930nteger @ A2 @ C ) @ ( times_3573771949741848930nteger @ B @ D2 ) ) ) ) ) ) ).
% 0.34/0.65
% 0.34/0.65 % mult_less_le_imp_less
% 0.34/0.65 thf(fact_5873_mult__less__le__imp__less,axiom,
% 0.34/0.65 ! [A2: rat,B: rat,C: rat,D2: rat] :
% 0.34/0.65 ( ( ord_less_rat @ A2 @ B )
% 0.34/0.65 => ( ( ord_less_eq_rat @ C @ D2 )
% 0.34/0.65 => ( ( ord_less_eq_rat @ zero_zero_rat @ A2 )
% 0.34/0.65 => ( ( ord_less_rat @ zero_zero_rat @ C )
% 0.34/0.65 => ( ord_less_rat @ ( times_times_rat @ A2 @ C ) @ ( times_times_rat @ B @ D2 ) ) ) ) ) ) ).
% 0.34/0.65
% 0.34/0.65 % mult_less_le_imp_less
% 0.34/0.65 thf(fact_5874_mult__less__le__imp__less,axiom,
% 0.34/0.65 ! [A2: nat,B: nat,C: nat,D2: nat] :
% 0.34/0.65 ( ( ord_less_nat @ A2 @ B )
% 0.34/0.65 => ( ( ord_less_eq_nat @ C @ D2 )
% 0.34/0.65 => ( ( ord_less_eq_nat @ zero_zero_nat @ A2 )
% 0.34/0.65 => ( ( ord_less_nat @ zero_zero_nat @ C )
% 0.34/0.65 => ( ord_less_nat @ ( times_times_nat @ A2 @ C ) @ ( times_times_nat @ B @ D2 ) ) ) ) ) ) ).
% 0.34/0.65
% 0.34/0.65 % mult_less_le_imp_less
% 0.34/0.65 thf(fact_5875_mult__less__le__imp__less,axiom,
% 0.34/0.65 ! [A2: int,B: int,C: int,D2: int] :
% 0.34/0.65 ( ( ord_less_int @ A2 @ B )
% 0.34/0.65 => ( ( ord_less_eq_int @ C @ D2 )
% 0.34/0.65 => ( ( ord_less_eq_int @ zero_zero_int @ A2 )
% 0.34/0.65 => ( ( ord_less_int @ zero_zero_int @ C )
% 0.34/0.65 => ( ord_less_int @ ( times_times_int @ A2 @ C ) @ ( times_times_int @ B @ D2 ) ) ) ) ) ) ).
% 0.34/0.65
% 0.34/0.65 % mult_less_le_imp_less
% 0.34/0.65 thf(fact_5876_mult__le__less__imp__less,axiom,
% 0.34/0.65 ! [A2: code_integer,B: code_integer,C: code_integer,D2: code_integer] :
% 0.34/0.65 ( ( ord_le3102999989581377725nteger @ A2 @ B )
% 0.34/0.65 => ( ( ord_le6747313008572928689nteger @ C @ D2 )
% 0.34/0.65 => ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ A2 )
% 0.34/0.65 => ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ C )
% 0.34/0.65 => ( ord_le6747313008572928689nteger @ ( times_3573771949741848930nteger @ A2 @ C ) @ ( times_3573771949741848930nteger @ B @ D2 ) ) ) ) ) ) ).
% 0.34/0.65
% 0.34/0.65 % mult_le_less_imp_less
% 0.34/0.65 thf(fact_5877_mult__le__less__imp__less,axiom,
% 0.34/0.65 ! [A2: rat,B: rat,C: rat,D2: rat] :
% 0.34/0.65 ( ( ord_less_eq_rat @ A2 @ B )
% 0.34/0.65 => ( ( ord_less_rat @ C @ D2 )
% 0.34/0.65 => ( ( ord_less_rat @ zero_zero_rat @ A2 )
% 0.34/0.65 => ( ( ord_less_eq_rat @ zero_zero_rat @ C )
% 0.34/0.65 => ( ord_less_rat @ ( times_times_rat @ A2 @ C ) @ ( times_times_rat @ B @ D2 ) ) ) ) ) ) ).
% 0.34/0.65
% 0.34/0.65 % mult_le_less_imp_less
% 0.34/0.65 thf(fact_5878_mult__le__less__imp__less,axiom,
% 0.34/0.65 ! [A2: nat,B: nat,C: nat,D2: nat] :
% 0.34/0.65 ( ( ord_less_eq_nat @ A2 @ B )
% 0.34/0.65 => ( ( ord_less_nat @ C @ D2 )
% 0.34/0.65 => ( ( ord_less_nat @ zero_zero_nat @ A2 )
% 0.34/0.65 => ( ( ord_less_eq_nat @ zero_zero_nat @ C )
% 0.34/0.65 => ( ord_less_nat @ ( times_times_nat @ A2 @ C ) @ ( times_times_nat @ B @ D2 ) ) ) ) ) ) ).
% 0.34/0.65
% 0.34/0.65 % mult_le_less_imp_less
% 0.34/0.65 thf(fact_5879_mult__le__less__imp__less,axiom,
% 0.34/0.65 ! [A2: int,B: int,C: int,D2: int] :
% 0.34/0.65 ( ( ord_less_eq_int @ A2 @ B )
% 0.34/0.65 => ( ( ord_less_int @ C @ D2 )
% 0.34/0.65 => ( ( ord_less_int @ zero_zero_int @ A2 )
% 0.34/0.65 => ( ( ord_less_eq_int @ zero_zero_int @ C )
% 0.34/0.65 => ( ord_less_int @ ( times_times_int @ A2 @ C ) @ ( times_times_int @ B @ D2 ) ) ) ) ) ) ).
% 0.34/0.65
% 0.34/0.65 % mult_le_less_imp_less
% 0.34/0.65 thf(fact_5880_mult__right__le__imp__le,axiom,
% 0.34/0.65 ! [A2: code_integer,C: code_integer,B: code_integer] :
% 0.34/0.65 ( ( ord_le3102999989581377725nteger @ ( times_3573771949741848930nteger @ A2 @ C ) @ ( times_3573771949741848930nteger @ B @ C ) )
% 0.34/0.65 => ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ C )
% 0.34/0.65 => ( ord_le3102999989581377725nteger @ A2 @ B ) ) ) ).
% 0.34/0.65
% 0.34/0.65 % mult_right_le_imp_le
% 0.34/0.65 thf(fact_5881_mult__right__le__imp__le,axiom,
% 0.34/0.65 ! [A2: rat,C: rat,B: rat] :
% 0.34/0.65 ( ( ord_less_eq_rat @ ( times_times_rat @ A2 @ C ) @ ( times_times_rat @ B @ C ) )
% 0.34/0.65 => ( ( ord_less_rat @ zero_zero_rat @ C )
% 0.34/0.65 => ( ord_less_eq_rat @ A2 @ B ) ) ) ).
% 0.34/0.65
% 0.34/0.65 % mult_right_le_imp_le
% 0.34/0.65 thf(fact_5882_mult__right__le__imp__le,axiom,
% 0.34/0.65 ! [A2: nat,C: nat,B: nat] :
% 0.34/0.65 ( ( ord_less_eq_nat @ ( times_times_nat @ A2 @ C ) @ ( times_times_nat @ B @ C ) )
% 0.34/0.65 => ( ( ord_less_nat @ zero_zero_nat @ C )
% 0.34/0.65 => ( ord_less_eq_nat @ A2 @ B ) ) ) ).
% 0.34/0.65
% 0.34/0.65 % mult_right_le_imp_le
% 0.34/0.65 thf(fact_5883_mult__right__le__imp__le,axiom,
% 0.34/0.65 ! [A2: int,C: int,B: int] :
% 0.34/0.65 ( ( ord_less_eq_int @ ( times_times_int @ A2 @ C ) @ ( times_times_int @ B @ C ) )
% 0.34/0.65 => ( ( ord_less_int @ zero_zero_int @ C )
% 0.34/0.65 => ( ord_less_eq_int @ A2 @ B ) ) ) ).
% 0.34/0.65
% 0.34/0.65 % mult_right_le_imp_le
% 0.34/0.65 thf(fact_5884_mult__left__le__imp__le,axiom,
% 0.34/0.65 ! [C: code_integer,A2: code_integer,B: code_integer] :
% 0.34/0.65 ( ( ord_le3102999989581377725nteger @ ( times_3573771949741848930nteger @ C @ A2 ) @ ( times_3573771949741848930nteger @ C @ B ) )
% 0.34/0.65 => ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ C )
% 0.34/0.65 => ( ord_le3102999989581377725nteger @ A2 @ B ) ) ) ).
% 0.34/0.65
% 0.34/0.65 % mult_left_le_imp_le
% 0.34/0.65 thf(fact_5885_mult__left__le__imp__le,axiom,
% 0.34/0.65 ! [C: rat,A2: rat,B: rat] :
% 0.34/0.65 ( ( ord_less_eq_rat @ ( times_times_rat @ C @ A2 ) @ ( times_times_rat @ C @ B ) )
% 0.34/0.65 => ( ( ord_less_rat @ zero_zero_rat @ C )
% 0.34/0.65 => ( ord_less_eq_rat @ A2 @ B ) ) ) ).
% 0.34/0.65
% 0.34/0.65 % mult_left_le_imp_le
% 0.34/0.65 thf(fact_5886_mult__left__le__imp__le,axiom,
% 0.34/0.65 ! [C: nat,A2: nat,B: nat] :
% 0.34/0.65 ( ( ord_less_eq_nat @ ( times_times_nat @ C @ A2 ) @ ( times_times_nat @ C @ B ) )
% 0.34/0.65 => ( ( ord_less_nat @ zero_zero_nat @ C )
% 0.34/0.65 => ( ord_less_eq_nat @ A2 @ B ) ) ) ).
% 0.34/0.65
% 0.34/0.65 % mult_left_le_imp_le
% 0.34/0.65 thf(fact_5887_mult__left__le__imp__le,axiom,
% 0.34/0.65 ! [C: int,A2: int,B: int] :
% 0.34/0.65 ( ( ord_less_eq_int @ ( times_times_int @ C @ A2 ) @ ( times_times_int @ C @ B ) )
% 0.34/0.65 => ( ( ord_less_int @ zero_zero_int @ C )
% 0.34/0.65 => ( ord_less_eq_int @ A2 @ B ) ) ) ).
% 0.34/0.65
% 0.34/0.65 % mult_left_le_imp_le
% 0.34/0.65 thf(fact_5888_mult__le__cancel__left__pos,axiom,
% 0.34/0.65 ! [C: code_integer,A2: code_integer,B: code_integer] :
% 0.34/0.65 ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ C )
% 0.34/0.65 => ( ( ord_le3102999989581377725nteger @ ( times_3573771949741848930nteger @ C @ A2 ) @ ( times_3573771949741848930nteger @ C @ B ) )
% 0.34/0.65 = ( ord_le3102999989581377725nteger @ A2 @ B ) ) ) ).
% 0.34/0.65
% 0.34/0.65 % mult_le_cancel_left_pos
% 0.34/0.65 thf(fact_5889_mult__le__cancel__left__pos,axiom,
% 0.34/0.65 ! [C: rat,A2: rat,B: rat] :
% 0.34/0.65 ( ( ord_less_rat @ zero_zero_rat @ C )
% 0.34/0.65 => ( ( ord_less_eq_rat @ ( times_times_rat @ C @ A2 ) @ ( times_times_rat @ C @ B ) )
% 0.34/0.65 = ( ord_less_eq_rat @ A2 @ B ) ) ) ).
% 0.34/0.65
% 0.34/0.65 % mult_le_cancel_left_pos
% 0.34/0.65 thf(fact_5890_mult__le__cancel__left__pos,axiom,
% 0.34/0.65 ! [C: int,A2: int,B: int] :
% 0.34/0.65 ( ( ord_less_int @ zero_zero_int @ C )
% 0.34/0.65 => ( ( ord_less_eq_int @ ( times_times_int @ C @ A2 ) @ ( times_times_int @ C @ B ) )
% 0.34/0.65 = ( ord_less_eq_int @ A2 @ B ) ) ) ).
% 0.34/0.65
% 0.34/0.65 % mult_le_cancel_left_pos
% 0.34/0.65 thf(fact_5891_mult__le__cancel__left__neg,axiom,
% 0.34/0.65 ! [C: code_integer,A2: code_integer,B: code_integer] :
% 0.34/0.65 ( ( ord_le6747313008572928689nteger @ C @ zero_z3403309356797280102nteger )
% 0.34/0.65 => ( ( ord_le3102999989581377725nteger @ ( times_3573771949741848930nteger @ C @ A2 ) @ ( times_3573771949741848930nteger @ C @ B ) )
% 0.34/0.65 = ( ord_le3102999989581377725nteger @ B @ A2 ) ) ) ).
% 0.34/0.65
% 0.34/0.65 % mult_le_cancel_left_neg
% 0.34/0.65 thf(fact_5892_mult__le__cancel__left__neg,axiom,
% 0.34/0.65 ! [C: rat,A2: rat,B: rat] :
% 0.34/0.65 ( ( ord_less_rat @ C @ zero_zero_rat )
% 0.34/0.65 => ( ( ord_less_eq_rat @ ( times_times_rat @ C @ A2 ) @ ( times_times_rat @ C @ B ) )
% 0.34/0.65 = ( ord_less_eq_rat @ B @ A2 ) ) ) ).
% 0.34/0.65
% 0.34/0.65 % mult_le_cancel_left_neg
% 0.34/0.65 thf(fact_5893_mult__le__cancel__left__neg,axiom,
% 0.34/0.65 ! [C: int,A2: int,B: int] :
% 0.34/0.65 ( ( ord_less_int @ C @ zero_zero_int )
% 0.34/0.65 => ( ( ord_less_eq_int @ ( times_times_int @ C @ A2 ) @ ( times_times_int @ C @ B ) )
% 0.34/0.65 = ( ord_less_eq_int @ B @ A2 ) ) ) ).
% 0.34/0.65
% 0.34/0.65 % mult_le_cancel_left_neg
% 0.34/0.65 thf(fact_5894_mult__less__cancel__right,axiom,
% 0.34/0.65 ! [A2: code_integer,C: code_integer,B: code_integer] :
% 0.34/0.65 ( ( ord_le6747313008572928689nteger @ ( times_3573771949741848930nteger @ A2 @ C ) @ ( times_3573771949741848930nteger @ B @ C ) )
% 0.34/0.65 = ( ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ C )
% 0.34/0.65 => ( ord_le6747313008572928689nteger @ A2 @ B ) )
% 0.34/0.65 & ( ( ord_le3102999989581377725nteger @ C @ zero_z3403309356797280102nteger )
% 0.34/0.65 => ( ord_le6747313008572928689nteger @ B @ A2 ) ) ) ) ).
% 0.34/0.65
% 0.34/0.65 % mult_less_cancel_right
% 0.34/0.65 thf(fact_5895_mult__less__cancel__right,axiom,
% 0.34/0.65 ! [A2: rat,C: rat,B: rat] :
% 0.34/0.65 ( ( ord_less_rat @ ( times_times_rat @ A2 @ C ) @ ( times_times_rat @ B @ C ) )
% 0.34/0.65 = ( ( ( ord_less_eq_rat @ zero_zero_rat @ C )
% 0.34/0.65 => ( ord_less_rat @ A2 @ B ) )
% 0.34/0.65 & ( ( ord_less_eq_rat @ C @ zero_zero_rat )
% 0.34/0.65 => ( ord_less_rat @ B @ A2 ) ) ) ) ).
% 0.34/0.65
% 0.34/0.65 % mult_less_cancel_right
% 0.34/0.65 thf(fact_5896_mult__less__cancel__right,axiom,
% 0.34/0.65 ! [A2: int,C: int,B: int] :
% 0.34/0.65 ( ( ord_less_int @ ( times_times_int @ A2 @ C ) @ ( times_times_int @ B @ C ) )
% 0.34/0.65 = ( ( ( ord_less_eq_int @ zero_zero_int @ C )
% 0.34/0.65 => ( ord_less_int @ A2 @ B ) )
% 0.34/0.65 & ( ( ord_less_eq_int @ C @ zero_zero_int )
% 0.34/0.65 => ( ord_less_int @ B @ A2 ) ) ) ) ).
% 0.34/0.65
% 0.34/0.65 % mult_less_cancel_right
% 0.34/0.65 thf(fact_5897_mult__strict__mono_H,axiom,
% 0.34/0.65 ! [A2: code_integer,B: code_integer,C: code_integer,D2: code_integer] :
% 0.34/0.65 ( ( ord_le6747313008572928689nteger @ A2 @ B )
% 0.34/0.65 => ( ( ord_le6747313008572928689nteger @ C @ D2 )
% 0.34/0.65 => ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ A2 )
% 0.34/0.65 => ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ C )
% 0.34/0.65 => ( ord_le6747313008572928689nteger @ ( times_3573771949741848930nteger @ A2 @ C ) @ ( times_3573771949741848930nteger @ B @ D2 ) ) ) ) ) ) ).
% 0.34/0.65
% 0.34/0.65 % mult_strict_mono'
% 0.34/0.65 thf(fact_5898_mult__strict__mono_H,axiom,
% 0.34/0.65 ! [A2: rat,B: rat,C: rat,D2: rat] :
% 0.34/0.65 ( ( ord_less_rat @ A2 @ B )
% 0.34/0.65 => ( ( ord_less_rat @ C @ D2 )
% 0.34/0.65 => ( ( ord_less_eq_rat @ zero_zero_rat @ A2 )
% 0.34/0.65 => ( ( ord_less_eq_rat @ zero_zero_rat @ C )
% 0.34/0.65 => ( ord_less_rat @ ( times_times_rat @ A2 @ C ) @ ( times_times_rat @ B @ D2 ) ) ) ) ) ) ).
% 0.34/0.65
% 0.34/0.65 % mult_strict_mono'
% 0.34/0.65 thf(fact_5899_mult__strict__mono_H,axiom,
% 0.34/0.65 ! [A2: nat,B: nat,C: nat,D2: nat] :
% 0.34/0.65 ( ( ord_less_nat @ A2 @ B )
% 0.34/0.65 => ( ( ord_less_nat @ C @ D2 )
% 0.34/0.65 => ( ( ord_less_eq_nat @ zero_zero_nat @ A2 )
% 0.34/0.65 => ( ( ord_less_eq_nat @ zero_zero_nat @ C )
% 0.34/0.65 => ( ord_less_nat @ ( times_times_nat @ A2 @ C ) @ ( times_times_nat @ B @ D2 ) ) ) ) ) ) ).
% 0.34/0.65
% 0.34/0.65 % mult_strict_mono'
% 0.34/0.65 thf(fact_5900_mult__strict__mono_H,axiom,
% 0.34/0.65 ! [A2: int,B: int,C: int,D2: int] :
% 0.34/0.65 ( ( ord_less_int @ A2 @ B )
% 0.34/0.65 => ( ( ord_less_int @ C @ D2 )
% 0.34/0.65 => ( ( ord_less_eq_int @ zero_zero_int @ A2 )
% 0.34/0.65 => ( ( ord_less_eq_int @ zero_zero_int @ C )
% 0.34/0.65 => ( ord_less_int @ ( times_times_int @ A2 @ C ) @ ( times_times_int @ B @ D2 ) ) ) ) ) ) ).
% 0.34/0.65
% 0.34/0.65 % mult_strict_mono'
% 0.34/0.65 thf(fact_5901_mult__right__less__imp__less,axiom,
% 0.34/0.65 ! [A2: code_integer,C: code_integer,B: code_integer] :
% 0.34/0.65 ( ( ord_le6747313008572928689nteger @ ( times_3573771949741848930nteger @ A2 @ C ) @ ( times_3573771949741848930nteger @ B @ C ) )
% 0.34/0.65 => ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ C )
% 0.34/0.65 => ( ord_le6747313008572928689nteger @ A2 @ B ) ) ) ).
% 0.34/0.65
% 0.34/0.65 % mult_right_less_imp_less
% 0.34/0.65 thf(fact_5902_mult__right__less__imp__less,axiom,
% 0.34/0.65 ! [A2: rat,C: rat,B: rat] :
% 0.34/0.65 ( ( ord_less_rat @ ( times_times_rat @ A2 @ C ) @ ( times_times_rat @ B @ C ) )
% 0.34/0.65 => ( ( ord_less_eq_rat @ zero_zero_rat @ C )
% 0.34/0.65 => ( ord_less_rat @ A2 @ B ) ) ) ).
% 0.34/0.65
% 0.34/0.65 % mult_right_less_imp_less
% 0.34/0.65 thf(fact_5903_mult__right__less__imp__less,axiom,
% 0.34/0.65 ! [A2: nat,C: nat,B: nat] :
% 0.34/0.65 ( ( ord_less_nat @ ( times_times_nat @ A2 @ C ) @ ( times_times_nat @ B @ C ) )
% 0.34/0.65 => ( ( ord_less_eq_nat @ zero_zero_nat @ C )
% 0.34/0.65 => ( ord_less_nat @ A2 @ B ) ) ) ).
% 0.34/0.65
% 0.34/0.65 % mult_right_less_imp_less
% 0.34/0.65 thf(fact_5904_mult__right__less__imp__less,axiom,
% 0.34/0.65 ! [A2: int,C: int,B: int] :
% 0.34/0.65 ( ( ord_less_int @ ( times_times_int @ A2 @ C ) @ ( times_times_int @ B @ C ) )
% 0.34/0.65 => ( ( ord_less_eq_int @ zero_zero_int @ C )
% 0.34/0.65 => ( ord_less_int @ A2 @ B ) ) ) ).
% 0.34/0.65
% 0.34/0.65 % mult_right_less_imp_less
% 0.34/0.65 thf(fact_5905_mult__less__cancel__left,axiom,
% 0.34/0.65 ! [C: code_integer,A2: code_integer,B: code_integer] :
% 0.34/0.65 ( ( ord_le6747313008572928689nteger @ ( times_3573771949741848930nteger @ C @ A2 ) @ ( times_3573771949741848930nteger @ C @ B ) )
% 0.34/0.65 = ( ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ C )
% 0.34/0.65 => ( ord_le6747313008572928689nteger @ A2 @ B ) )
% 0.34/0.65 & ( ( ord_le3102999989581377725nteger @ C @ zero_z3403309356797280102nteger )
% 0.34/0.65 => ( ord_le6747313008572928689nteger @ B @ A2 ) ) ) ) ).
% 0.34/0.65
% 0.34/0.65 % mult_less_cancel_left
% 0.34/0.65 thf(fact_5906_mult__less__cancel__left,axiom,
% 0.34/0.65 ! [C: rat,A2: rat,B: rat] :
% 0.34/0.65 ( ( ord_less_rat @ ( times_times_rat @ C @ A2 ) @ ( times_times_rat @ C @ B ) )
% 0.34/0.65 = ( ( ( ord_less_eq_rat @ zero_zero_rat @ C )
% 0.34/0.65 => ( ord_less_rat @ A2 @ B ) )
% 0.34/0.65 & ( ( ord_less_eq_rat @ C @ zero_zero_rat )
% 0.34/0.65 => ( ord_less_rat @ B @ A2 ) ) ) ) ).
% 0.34/0.65
% 0.34/0.65 % mult_less_cancel_left
% 0.34/0.65 thf(fact_5907_mult__less__cancel__left,axiom,
% 0.34/0.65 ! [C: int,A2: int,B: int] :
% 0.34/0.65 ( ( ord_less_int @ ( times_times_int @ C @ A2 ) @ ( times_times_int @ C @ B ) )
% 0.34/0.65 = ( ( ( ord_less_eq_int @ zero_zero_int @ C )
% 0.34/0.65 => ( ord_less_int @ A2 @ B ) )
% 0.34/0.65 & ( ( ord_less_eq_int @ C @ zero_zero_int )
% 0.34/0.65 => ( ord_less_int @ B @ A2 ) ) ) ) ).
% 0.34/0.65
% 0.34/0.65 % mult_less_cancel_left
% 0.34/0.65 thf(fact_5908_mult__strict__mono,axiom,
% 0.34/0.65 ! [A2: code_integer,B: code_integer,C: code_integer,D2: code_integer] :
% 0.34/0.65 ( ( ord_le6747313008572928689nteger @ A2 @ B )
% 0.34/0.65 => ( ( ord_le6747313008572928689nteger @ C @ D2 )
% 0.34/0.65 => ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ B )
% 0.34/0.65 => ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ C )
% 0.34/0.65 => ( ord_le6747313008572928689nteger @ ( times_3573771949741848930nteger @ A2 @ C ) @ ( times_3573771949741848930nteger @ B @ D2 ) ) ) ) ) ) ).
% 0.34/0.65
% 0.34/0.65 % mult_strict_mono
% 0.34/0.65 thf(fact_5909_mult__strict__mono,axiom,
% 0.34/0.65 ! [A2: rat,B: rat,C: rat,D2: rat] :
% 0.34/0.65 ( ( ord_less_rat @ A2 @ B )
% 0.34/0.65 => ( ( ord_less_rat @ C @ D2 )
% 0.34/0.65 => ( ( ord_less_rat @ zero_zero_rat @ B )
% 0.34/0.65 => ( ( ord_less_eq_rat @ zero_zero_rat @ C )
% 0.34/0.65 => ( ord_less_rat @ ( times_times_rat @ A2 @ C ) @ ( times_times_rat @ B @ D2 ) ) ) ) ) ) ).
% 0.34/0.65
% 0.34/0.65 % mult_strict_mono
% 0.34/0.65 thf(fact_5910_mult__strict__mono,axiom,
% 0.34/0.65 ! [A2: nat,B: nat,C: nat,D2: nat] :
% 0.34/0.65 ( ( ord_less_nat @ A2 @ B )
% 0.34/0.65 => ( ( ord_less_nat @ C @ D2 )
% 0.34/0.65 => ( ( ord_less_nat @ zero_zero_nat @ B )
% 0.34/0.65 => ( ( ord_less_eq_nat @ zero_zero_nat @ C )
% 0.34/0.65 => ( ord_less_nat @ ( times_times_nat @ A2 @ C ) @ ( times_times_nat @ B @ D2 ) ) ) ) ) ) ).
% 0.34/0.65
% 0.34/0.65 % mult_strict_mono
% 0.34/0.65 thf(fact_5911_mult__strict__mono,axiom,
% 0.34/0.65 ! [A2: int,B: int,C: int,D2: int] :
% 0.34/0.65 ( ( ord_less_int @ A2 @ B )
% 0.34/0.65 => ( ( ord_less_int @ C @ D2 )
% 0.34/0.65 => ( ( ord_less_int @ zero_zero_int @ B )
% 0.34/0.65 => ( ( ord_less_eq_int @ zero_zero_int @ C )
% 0.34/0.65 => ( ord_less_int @ ( times_times_int @ A2 @ C ) @ ( times_times_int @ B @ D2 ) ) ) ) ) ) ).
% 0.34/0.65
% 0.34/0.65 % mult_strict_mono
% 0.34/0.65 thf(fact_5912_mult__left__less__imp__less,axiom,
% 0.34/0.65 ! [C: code_integer,A2: code_integer,B: code_integer] :
% 0.34/0.65 ( ( ord_le6747313008572928689nteger @ ( times_3573771949741848930nteger @ C @ A2 ) @ ( times_3573771949741848930nteger @ C @ B ) )
% 0.34/0.65 => ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ C )
% 0.34/0.65 => ( ord_le6747313008572928689nteger @ A2 @ B ) ) ) ).
% 0.34/0.65
% 0.34/0.65 % mult_left_less_imp_less
% 0.34/0.65 thf(fact_5913_mult__left__less__imp__less,axiom,
% 0.34/0.65 ! [C: rat,A2: rat,B: rat] :
% 0.34/0.65 ( ( ord_less_rat @ ( times_times_rat @ C @ A2 ) @ ( times_times_rat @ C @ B ) )
% 0.34/0.65 => ( ( ord_less_eq_rat @ zero_zero_rat @ C )
% 0.34/0.65 => ( ord_less_rat @ A2 @ B ) ) ) ).
% 0.34/0.65
% 0.34/0.65 % mult_left_less_imp_less
% 0.34/0.65 thf(fact_5914_mult__left__less__imp__less,axiom,
% 0.34/0.65 ! [C: nat,A2: nat,B: nat] :
% 0.34/0.65 ( ( ord_less_nat @ ( times_times_nat @ C @ A2 ) @ ( times_times_nat @ C @ B ) )
% 0.34/0.65 => ( ( ord_less_eq_nat @ zero_zero_nat @ C )
% 0.34/0.65 => ( ord_less_nat @ A2 @ B ) ) ) ).
% 0.34/0.65
% 0.34/0.65 % mult_left_less_imp_less
% 0.34/0.65 thf(fact_5915_mult__left__less__imp__less,axiom,
% 0.34/0.65 ! [C: int,A2: int,B: int] :
% 0.34/0.65 ( ( ord_less_int @ ( times_times_int @ C @ A2 ) @ ( times_times_int @ C @ B ) )
% 0.34/0.65 => ( ( ord_less_eq_int @ zero_zero_int @ C )
% 0.34/0.65 => ( ord_less_int @ A2 @ B ) ) ) ).
% 0.34/0.65
% 0.34/0.65 % mult_left_less_imp_less
% 0.34/0.65 thf(fact_5916_mult__le__cancel__right,axiom,
% 0.34/0.65 ! [A2: code_integer,C: code_integer,B: code_integer] :
% 0.34/0.65 ( ( ord_le3102999989581377725nteger @ ( times_3573771949741848930nteger @ A2 @ C ) @ ( times_3573771949741848930nteger @ B @ C ) )
% 0.34/0.65 = ( ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ C )
% 0.34/0.65 => ( ord_le3102999989581377725nteger @ A2 @ B ) )
% 0.34/0.65 & ( ( ord_le6747313008572928689nteger @ C @ zero_z3403309356797280102nteger )
% 0.34/0.65 => ( ord_le3102999989581377725nteger @ B @ A2 ) ) ) ) ).
% 0.34/0.65
% 0.34/0.65 % mult_le_cancel_right
% 0.34/0.65 thf(fact_5917_mult__le__cancel__right,axiom,
% 0.34/0.65 ! [A2: rat,C: rat,B: rat] :
% 0.34/0.65 ( ( ord_less_eq_rat @ ( times_times_rat @ A2 @ C ) @ ( times_times_rat @ B @ C ) )
% 0.34/0.65 = ( ( ( ord_less_rat @ zero_zero_rat @ C )
% 0.34/0.65 => ( ord_less_eq_rat @ A2 @ B ) )
% 0.34/0.65 & ( ( ord_less_rat @ C @ zero_zero_rat )
% 0.34/0.65 => ( ord_less_eq_rat @ B @ A2 ) ) ) ) ).
% 0.34/0.65
% 0.34/0.65 % mult_le_cancel_right
% 0.34/0.65 thf(fact_5918_mult__le__cancel__right,axiom,
% 0.34/0.65 ! [A2: int,C: int,B: int] :
% 0.34/0.65 ( ( ord_less_eq_int @ ( times_times_int @ A2 @ C ) @ ( times_times_int @ B @ C ) )
% 0.34/0.65 = ( ( ( ord_less_int @ zero_zero_int @ C )
% 0.34/0.65 => ( ord_less_eq_int @ A2 @ B ) )
% 0.34/0.65 & ( ( ord_less_int @ C @ zero_zero_int )
% 0.34/0.65 => ( ord_less_eq_int @ B @ A2 ) ) ) ) ).
% 0.34/0.65
% 0.34/0.65 % mult_le_cancel_right
% 0.34/0.65 thf(fact_5919_mult__le__cancel__left,axiom,
% 0.34/0.65 ! [C: code_integer,A2: code_integer,B: code_integer] :
% 0.34/0.65 ( ( ord_le3102999989581377725nteger @ ( times_3573771949741848930nteger @ C @ A2 ) @ ( times_3573771949741848930nteger @ C @ B ) )
% 0.34/0.65 = ( ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ C )
% 0.34/0.65 => ( ord_le3102999989581377725nteger @ A2 @ B ) )
% 0.34/0.65 & ( ( ord_le6747313008572928689nteger @ C @ zero_z3403309356797280102nteger )
% 0.34/0.65 => ( ord_le3102999989581377725nteger @ B @ A2 ) ) ) ) ).
% 0.34/0.65
% 0.34/0.65 % mult_le_cancel_left
% 0.34/0.65 thf(fact_5920_mult__le__cancel__left,axiom,
% 0.34/0.65 ! [C: rat,A2: rat,B: rat] :
% 0.34/0.65 ( ( ord_less_eq_rat @ ( times_times_rat @ C @ A2 ) @ ( times_times_rat @ C @ B ) )
% 0.34/0.65 = ( ( ( ord_less_rat @ zero_zero_rat @ C )
% 0.34/0.65 => ( ord_less_eq_rat @ A2 @ B ) )
% 0.34/0.65 & ( ( ord_less_rat @ C @ zero_zero_rat )
% 0.34/0.65 => ( ord_less_eq_rat @ B @ A2 ) ) ) ) ).
% 0.34/0.65
% 0.34/0.65 % mult_le_cancel_left
% 0.34/0.65 thf(fact_5921_mult__le__cancel__left,axiom,
% 0.34/0.65 ! [C: int,A2: int,B: int] :
% 0.34/0.65 ( ( ord_less_eq_int @ ( times_times_int @ C @ A2 ) @ ( times_times_int @ C @ B ) )
% 0.34/0.65 = ( ( ( ord_less_int @ zero_zero_int @ C )
% 0.34/0.65 => ( ord_less_eq_int @ A2 @ B ) )
% 0.34/0.65 & ( ( ord_less_int @ C @ zero_zero_int )
% 0.34/0.65 => ( ord_less_eq_int @ B @ A2 ) ) ) ) ).
% 0.34/0.65
% 0.34/0.65 % mult_le_cancel_left
% 0.34/0.65 thf(fact_5922_sum__squares__ge__zero,axiom,
% 0.34/0.65 ! [X: code_integer,Y: code_integer] : ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ X @ X ) @ ( times_3573771949741848930nteger @ Y @ Y ) ) ) ).
% 0.34/0.65
% 0.34/0.65 % sum_squares_ge_zero
% 0.34/0.65 thf(fact_5923_sum__squares__ge__zero,axiom,
% 0.34/0.65 ! [X: rat,Y: rat] : ( ord_less_eq_rat @ zero_zero_rat @ ( plus_plus_rat @ ( times_times_rat @ X @ X ) @ ( times_times_rat @ Y @ Y ) ) ) ).
% 0.34/0.65
% 0.34/0.65 % sum_squares_ge_zero
% 0.34/0.65 thf(fact_5924_sum__squares__ge__zero,axiom,
% 0.34/0.65 ! [X: int,Y: int] : ( ord_less_eq_int @ zero_zero_int @ ( plus_plus_int @ ( times_times_int @ X @ X ) @ ( times_times_int @ Y @ Y ) ) ) ).
% 0.34/0.65
% 0.34/0.65 % sum_squares_ge_zero
% 0.34/0.65 thf(fact_5925_mult__left__le,axiom,
% 0.34/0.65 ! [C: code_natural,A2: code_natural] :
% 0.34/0.65 ( ( ord_le1926595141338095240atural @ C @ one_one_Code_natural )
% 0.34/0.65 => ( ( ord_le1926595141338095240atural @ zero_z2226904508553997617atural @ A2 )
% 0.34/0.65 => ( ord_le1926595141338095240atural @ ( times_2397367101498566445atural @ A2 @ C ) @ A2 ) ) ) ).
% 0.34/0.65
% 0.34/0.65 % mult_left_le
% 0.34/0.65 thf(fact_5926_mult__left__le,axiom,
% 0.34/0.65 ! [C: code_integer,A2: code_integer] :
% 0.34/0.65 ( ( ord_le3102999989581377725nteger @ C @ one_one_Code_integer )
% 0.34/0.65 => ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ A2 )
% 0.34/0.65 => ( ord_le3102999989581377725nteger @ ( times_3573771949741848930nteger @ A2 @ C ) @ A2 ) ) ) ).
% 0.34/0.65
% 0.34/0.65 % mult_left_le
% 0.34/0.65 thf(fact_5927_mult__left__le,axiom,
% 0.34/0.65 ! [C: rat,A2: rat] :
% 0.34/0.65 ( ( ord_less_eq_rat @ C @ one_one_rat )
% 0.34/0.65 => ( ( ord_less_eq_rat @ zero_zero_rat @ A2 )
% 0.34/0.65 => ( ord_less_eq_rat @ ( times_times_rat @ A2 @ C ) @ A2 ) ) ) ).
% 0.34/0.65
% 0.34/0.65 % mult_left_le
% 0.34/0.65 thf(fact_5928_mult__left__le,axiom,
% 0.34/0.65 ! [C: nat,A2: nat] :
% 0.34/0.65 ( ( ord_less_eq_nat @ C @ one_one_nat )
% 0.34/0.65 => ( ( ord_less_eq_nat @ zero_zero_nat @ A2 )
% 0.34/0.65 => ( ord_less_eq_nat @ ( times_times_nat @ A2 @ C ) @ A2 ) ) ) ).
% 0.34/0.65
% 0.34/0.65 % mult_left_le
% 0.34/0.65 thf(fact_5929_mult__left__le,axiom,
% 0.34/0.65 ! [C: int,A2: int] :
% 0.34/0.65 ( ( ord_less_eq_int @ C @ one_one_int )
% 0.34/0.65 => ( ( ord_less_eq_int @ zero_zero_int @ A2 )
% 0.34/0.65 => ( ord_less_eq_int @ ( times_times_int @ A2 @ C ) @ A2 ) ) ) ).
% 0.34/0.65
% 0.34/0.65 % mult_left_le
% 0.34/0.65 thf(fact_5930_mult__le__one,axiom,
% 0.34/0.65 ! [A2: code_natural,B: code_natural] :
% 0.34/0.65 ( ( ord_le1926595141338095240atural @ A2 @ one_one_Code_natural )
% 0.34/0.65 => ( ( ord_le1926595141338095240atural @ zero_z2226904508553997617atural @ B )
% 0.34/0.65 => ( ( ord_le1926595141338095240atural @ B @ one_one_Code_natural )
% 0.34/0.65 => ( ord_le1926595141338095240atural @ ( times_2397367101498566445atural @ A2 @ B ) @ one_one_Code_natural ) ) ) ) ).
% 0.34/0.65
% 0.34/0.65 % mult_le_one
% 0.34/0.65 thf(fact_5931_mult__le__one,axiom,
% 0.34/0.65 ! [A2: code_integer,B: code_integer] :
% 0.34/0.65 ( ( ord_le3102999989581377725nteger @ A2 @ one_one_Code_integer )
% 0.34/0.65 => ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ B )
% 0.34/0.65 => ( ( ord_le3102999989581377725nteger @ B @ one_one_Code_integer )
% 0.34/0.65 => ( ord_le3102999989581377725nteger @ ( times_3573771949741848930nteger @ A2 @ B ) @ one_one_Code_integer ) ) ) ) ).
% 0.34/0.65
% 0.34/0.65 % mult_le_one
% 0.34/0.65 thf(fact_5932_mult__le__one,axiom,
% 0.34/0.65 ! [A2: rat,B: rat] :
% 0.34/0.65 ( ( ord_less_eq_rat @ A2 @ one_one_rat )
% 0.34/0.65 => ( ( ord_less_eq_rat @ zero_zero_rat @ B )
% 0.34/0.65 => ( ( ord_less_eq_rat @ B @ one_one_rat )
% 0.34/0.65 => ( ord_less_eq_rat @ ( times_times_rat @ A2 @ B ) @ one_one_rat ) ) ) ) ).
% 0.34/0.65
% 0.34/0.65 % mult_le_one
% 0.34/0.65 thf(fact_5933_mult__le__one,axiom,
% 0.34/0.65 ! [A2: nat,B: nat] :
% 0.34/0.65 ( ( ord_less_eq_nat @ A2 @ one_one_nat )
% 0.34/0.65 => ( ( ord_less_eq_nat @ zero_zero_nat @ B )
% 0.34/0.65 => ( ( ord_less_eq_nat @ B @ one_one_nat )
% 0.34/0.65 => ( ord_less_eq_nat @ ( times_times_nat @ A2 @ B ) @ one_one_nat ) ) ) ) ).
% 0.34/0.65
% 0.34/0.65 % mult_le_one
% 0.34/0.65 thf(fact_5934_mult__le__one,axiom,
% 0.34/0.65 ! [A2: int,B: int] :
% 0.34/0.65 ( ( ord_less_eq_int @ A2 @ one_one_int )
% 0.34/0.65 => ( ( ord_less_eq_int @ zero_zero_int @ B )
% 0.34/0.65 => ( ( ord_less_eq_int @ B @ one_one_int )
% 0.34/0.65 => ( ord_less_eq_int @ ( times_times_int @ A2 @ B ) @ one_one_int ) ) ) ) ).
% 0.34/0.65
% 0.34/0.65 % mult_le_one
% 0.34/0.65 thf(fact_5935_mult__right__le__one__le,axiom,
% 0.34/0.65 ! [X: code_integer,Y: code_integer] :
% 0.34/0.65 ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ X )
% 0.34/0.65 => ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ Y )
% 0.34/0.65 => ( ( ord_le3102999989581377725nteger @ Y @ one_one_Code_integer )
% 0.34/0.65 => ( ord_le3102999989581377725nteger @ ( times_3573771949741848930nteger @ X @ Y ) @ X ) ) ) ) ).
% 0.34/0.65
% 0.34/0.65 % mult_right_le_one_le
% 0.34/0.65 thf(fact_5936_mult__right__le__one__le,axiom,
% 0.34/0.65 ! [X: rat,Y: rat] :
% 0.34/0.65 ( ( ord_less_eq_rat @ zero_zero_rat @ X )
% 0.34/0.65 => ( ( ord_less_eq_rat @ zero_zero_rat @ Y )
% 0.34/0.65 => ( ( ord_less_eq_rat @ Y @ one_one_rat )
% 0.34/0.65 => ( ord_less_eq_rat @ ( times_times_rat @ X @ Y ) @ X ) ) ) ) ).
% 0.34/0.65
% 0.34/0.65 % mult_right_le_one_le
% 0.34/0.65 thf(fact_5937_mult__right__le__one__le,axiom,
% 0.34/0.65 ! [X: int,Y: int] :
% 0.34/0.65 ( ( ord_less_eq_int @ zero_zero_int @ X )
% 0.34/0.65 => ( ( ord_less_eq_int @ zero_zero_int @ Y )
% 0.34/0.65 => ( ( ord_less_eq_int @ Y @ one_one_int )
% 0.34/0.65 => ( ord_less_eq_int @ ( times_times_int @ X @ Y ) @ X ) ) ) ) ).
% 0.34/0.65
% 0.34/0.65 % mult_right_le_one_le
% 0.34/0.65 thf(fact_5938_mult__left__le__one__le,axiom,
% 0.34/0.65 ! [X: code_integer,Y: code_integer] :
% 0.34/0.65 ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ X )
% 0.34/0.65 => ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ Y )
% 0.34/0.65 => ( ( ord_le3102999989581377725nteger @ Y @ one_one_Code_integer )
% 0.34/0.65 => ( ord_le3102999989581377725nteger @ ( times_3573771949741848930nteger @ Y @ X ) @ X ) ) ) ) ).
% 0.34/0.65
% 0.34/0.65 % mult_left_le_one_le
% 0.34/0.65 thf(fact_5939_mult__left__le__one__le,axiom,
% 0.34/0.65 ! [X: rat,Y: rat] :
% 0.34/0.65 ( ( ord_less_eq_rat @ zero_zero_rat @ X )
% 0.34/0.65 => ( ( ord_less_eq_rat @ zero_zero_rat @ Y )
% 0.34/0.65 => ( ( ord_less_eq_rat @ Y @ one_one_rat )
% 0.34/0.65 => ( ord_less_eq_rat @ ( times_times_rat @ Y @ X ) @ X ) ) ) ) ).
% 0.34/0.65
% 0.34/0.65 % mult_left_le_one_le
% 0.34/0.65 thf(fact_5940_mult__left__le__one__le,axiom,
% 0.34/0.65 ! [X: int,Y: int] :
% 0.34/0.65 ( ( ord_less_eq_int @ zero_zero_int @ X )
% 0.34/0.65 => ( ( ord_less_eq_int @ zero_zero_int @ Y )
% 0.34/0.65 => ( ( ord_less_eq_int @ Y @ one_one_int )
% 0.34/0.65 => ( ord_less_eq_int @ ( times_times_int @ Y @ X ) @ X ) ) ) ) ).
% 0.34/0.65
% 0.34/0.65 % mult_left_le_one_le
% 0.34/0.65 thf(fact_5941_not__sum__squares__lt__zero,axiom,
% 0.34/0.65 ! [X: int,Y: int] :
% 0.34/0.65 ~ ( ord_less_int @ ( plus_plus_int @ ( times_times_int @ X @ X ) @ ( times_times_int @ Y @ Y ) ) @ zero_zero_int ) ).
% 0.34/0.65
% 0.34/0.65 % not_sum_squares_lt_zero
% 0.34/0.65 thf(fact_5942_not__sum__squares__lt__zero,axiom,
% 0.34/0.65 ! [X: code_integer,Y: code_integer] :
% 0.34/0.65 ~ ( ord_le6747313008572928689nteger @ ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ X @ X ) @ ( times_3573771949741848930nteger @ Y @ Y ) ) @ zero_z3403309356797280102nteger ) ).
% 0.34/0.65
% 0.34/0.65 % not_sum_squares_lt_zero
% 0.34/0.65 thf(fact_5943_not__sum__squares__lt__zero,axiom,
% 0.34/0.65 ! [X: rat,Y: rat] :
% 0.34/0.65 ~ ( ord_less_rat @ ( plus_plus_rat @ ( times_times_rat @ X @ X ) @ ( times_times_rat @ Y @ Y ) ) @ zero_zero_rat ) ).
% 0.34/0.65
% 0.34/0.65 % not_sum_squares_lt_zero
% 0.34/0.65 thf(fact_5944_zero__less__two,axiom,
% 0.34/0.65 ord_le5570908160329646204atural @ zero_z2226904508553997617atural @ ( plus_p4538020629002901425atural @ one_one_Code_natural @ one_one_Code_natural ) ).
% 0.34/0.65
% 0.34/0.65 % zero_less_two
% 0.34/0.65 thf(fact_5945_zero__less__two,axiom,
% 0.34/0.65 ord_less_nat @ zero_zero_nat @ ( plus_plus_nat @ one_one_nat @ one_one_nat ) ).
% 0.34/0.65
% 0.34/0.65 % zero_less_two
% 0.34/0.65 thf(fact_5946_zero__less__two,axiom,
% 0.34/0.65 ord_less_int @ zero_zero_int @ ( plus_plus_int @ one_one_int @ one_one_int ) ).
% 0.34/0.65
% 0.34/0.65 % zero_less_two
% 0.34/0.65 thf(fact_5947_zero__less__two,axiom,
% 0.34/0.65 ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ ( plus_p5714425477246183910nteger @ one_one_Code_integer @ one_one_Code_integer ) ).
% 0.34/0.65
% 0.34/0.65 % zero_less_two
% 0.34/0.65 thf(fact_5948_zero__less__two,axiom,
% 0.34/0.65 ord_less_rat @ zero_zero_rat @ ( plus_plus_rat @ one_one_rat @ one_one_rat ) ).
% 0.34/0.65
% 0.34/0.65 % zero_less_two
% 0.34/0.65 thf(fact_5949_card__eq__0__iff,axiom,
% 0.34/0.65 ! [A: set_Code_integer] :
% 0.34/0.65 ( ( ( finite4902975817058060853nteger @ A )
% 0.34/0.65 = zero_zero_nat )
% 0.34/0.65 = ( ( A = bot_bo3990330152332043303nteger )
% 0.34/0.65 | ~ ( finite6017078050557962740nteger @ A ) ) ) ).
% 0.34/0.65
% 0.34/0.65 % card_eq_0_iff
% 0.34/0.65 thf(fact_5950_card__eq__0__iff,axiom,
% 0.34/0.65 ! [A: set_Product_unit] :
% 0.34/0.65 ( ( ( finite410649719033368117t_unit @ A )
% 0.34/0.65 = zero_zero_nat )
% 0.34/0.65 = ( ( A = bot_bo3957492148770167129t_unit )
% 0.34/0.65 | ~ ( finite4290736615968046902t_unit @ A ) ) ) ).
% 0.34/0.65
% 0.34/0.65 % card_eq_0_iff
% 0.34/0.65 thf(fact_5951_card__eq__0__iff,axiom,
% 0.34/0.65 ! [A: set_o] :
% 0.34/0.65 ( ( ( finite_card_o @ A )
% 0.34/0.65 = zero_zero_nat )
% 0.34/0.65 = ( ( A = bot_bot_set_o )
% 0.34/0.65 | ~ ( finite_finite_o @ A ) ) ) ).
% 0.34/0.65
% 0.34/0.65 % card_eq_0_iff
% 0.34/0.65 thf(fact_5952_card__eq__0__iff,axiom,
% 0.34/0.65 ! [A: set_nat] :
% 0.34/0.65 ( ( ( finite_card_nat @ A )
% 0.34/0.65 = zero_zero_nat )
% 0.34/0.65 = ( ( A = bot_bot_set_nat )
% 0.34/0.65 | ~ ( finite_finite_nat @ A ) ) ) ).
% 0.34/0.65
% 0.34/0.65 % card_eq_0_iff
% 0.34/0.65 thf(fact_5953_card__eq__0__iff,axiom,
% 0.34/0.65 ! [A: set_int] :
% 0.34/0.65 ( ( ( finite_card_int @ A )
% 0.34/0.65 = zero_zero_nat )
% 0.34/0.65 = ( ( A = bot_bot_set_int )
% 0.34/0.65 | ~ ( finite_finite_int @ A ) ) ) ).
% 0.34/0.65
% 0.34/0.65 % card_eq_0_iff
% 0.34/0.65 thf(fact_5954_card__eq__0__iff,axiom,
% 0.34/0.65 ! [A: set_Pr4532377907799695533_nat_o] :
% 0.34/0.65 ( ( ( finite1363419556375932405_nat_o @ A )
% 0.34/0.65 = zero_zero_nat )
% 0.34/0.65 = ( ( A = bot_bo7824918357723582233_nat_o )
% 0.34/0.65 | ~ ( finite3252695134891459830_nat_o @ A ) ) ) ).
% 0.34/0.65
% 0.34/0.65 % card_eq_0_iff
% 0.34/0.65 thf(fact_5955_one__less__mult,axiom,
% 0.34/0.65 ! [N2: nat,M4: nat] :
% 0.34/0.65 ( ( ord_less_nat @ ( suc @ zero_zero_nat ) @ N2 )
% 0.34/0.65 => ( ( ord_less_nat @ ( suc @ zero_zero_nat ) @ M4 )
% 0.34/0.65 => ( ord_less_nat @ ( suc @ zero_zero_nat ) @ ( times_times_nat @ M4 @ N2 ) ) ) ) ).
% 0.34/0.65
% 0.34/0.65 % one_less_mult
% 0.34/0.65 thf(fact_5956_n__less__m__mult__n,axiom,
% 0.34/0.65 ! [N2: nat,M4: nat] :
% 0.34/0.65 ( ( ord_less_nat @ zero_zero_nat @ N2 )
% 0.34/0.65 => ( ( ord_less_nat @ ( suc @ zero_zero_nat ) @ M4 )
% 0.34/0.65 => ( ord_less_nat @ N2 @ ( times_times_nat @ M4 @ N2 ) ) ) ) ).
% 0.34/0.65
% 0.34/0.65 % n_less_m_mult_n
% 0.34/0.65 thf(fact_5957_n__less__n__mult__m,axiom,
% 0.34/0.65 ! [N2: nat,M4: nat] :
% 0.34/0.65 ( ( ord_less_nat @ zero_zero_nat @ N2 )
% 0.34/0.65 => ( ( ord_less_nat @ ( suc @ zero_zero_nat ) @ M4 )
% 0.34/0.65 => ( ord_less_nat @ N2 @ ( times_times_nat @ N2 @ M4 ) ) ) ) ).
% 0.34/0.65
% 0.34/0.65 % n_less_n_mult_m
% 0.34/0.65 thf(fact_5958_nat__mult__le__cancel1,axiom,
% 0.34/0.65 ! [K3: nat,M4: nat,N2: nat] :
% 0.34/0.65 ( ( ord_less_nat @ zero_zero_nat @ K3 )
% 0.34/0.65 => ( ( ord_less_eq_nat @ ( times_times_nat @ K3 @ M4 ) @ ( times_times_nat @ K3 @ N2 ) )
% 0.34/0.65 = ( ord_less_eq_nat @ M4 @ N2 ) ) ) ).
% 0.34/0.65
% 0.34/0.65 % nat_mult_le_cancel1
% 0.34/0.65 thf(fact_5959_mult__le__cancel__left1,axiom,
% 0.34/0.65 ! [C: code_integer,B: code_integer] :
% 0.34/0.65 ( ( ord_le3102999989581377725nteger @ C @ ( times_3573771949741848930nteger @ C @ B ) )
% 0.34/0.65 = ( ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ C )
% 0.34/0.65 => ( ord_le3102999989581377725nteger @ one_one_Code_integer @ B ) )
% 0.34/0.65 & ( ( ord_le6747313008572928689nteger @ C @ zero_z3403309356797280102nteger )
% 0.34/0.65 => ( ord_le3102999989581377725nteger @ B @ one_one_Code_integer ) ) ) ) ).
% 0.34/0.65
% 0.34/0.65 % mult_le_cancel_left1
% 0.34/0.65 thf(fact_5960_mult__le__cancel__left1,axiom,
% 0.34/0.65 ! [C: rat,B: rat] :
% 0.34/0.65 ( ( ord_less_eq_rat @ C @ ( times_times_rat @ C @ B ) )
% 0.34/0.65 = ( ( ( ord_less_rat @ zero_zero_rat @ C )
% 0.34/0.65 => ( ord_less_eq_rat @ one_one_rat @ B ) )
% 0.34/0.65 & ( ( ord_less_rat @ C @ zero_zero_rat )
% 0.34/0.65 => ( ord_less_eq_rat @ B @ one_one_rat ) ) ) ) ).
% 0.34/0.65
% 0.34/0.65 % mult_le_cancel_left1
% 0.34/0.65 thf(fact_5961_mult__le__cancel__left1,axiom,
% 0.34/0.65 ! [C: int,B: int] :
% 0.34/0.65 ( ( ord_less_eq_int @ C @ ( times_times_int @ C @ B ) )
% 0.34/0.65 = ( ( ( ord_less_int @ zero_zero_int @ C )
% 0.34/0.65 => ( ord_less_eq_int @ one_one_int @ B ) )
% 0.34/0.65 & ( ( ord_less_int @ C @ zero_zero_int )
% 0.34/0.65 => ( ord_less_eq_int @ B @ one_one_int ) ) ) ) ).
% 0.34/0.65
% 0.34/0.65 % mult_le_cancel_left1
% 0.34/0.65 thf(fact_5962_mult__le__cancel__left2,axiom,
% 0.34/0.65 ! [C: code_integer,A2: code_integer] :
% 0.34/0.65 ( ( ord_le3102999989581377725nteger @ ( times_3573771949741848930nteger @ C @ A2 ) @ C )
% 0.34/0.65 = ( ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ C )
% 0.34/0.65 => ( ord_le3102999989581377725nteger @ A2 @ one_one_Code_integer ) )
% 0.34/0.65 & ( ( ord_le6747313008572928689nteger @ C @ zero_z3403309356797280102nteger )
% 0.34/0.65 => ( ord_le3102999989581377725nteger @ one_one_Code_integer @ A2 ) ) ) ) ).
% 0.34/0.65
% 0.34/0.65 % mult_le_cancel_left2
% 0.34/0.65 thf(fact_5963_mult__le__cancel__left2,axiom,
% 0.34/0.65 ! [C: rat,A2: rat] :
% 0.34/0.65 ( ( ord_less_eq_rat @ ( times_times_rat @ C @ A2 ) @ C )
% 0.34/0.65 = ( ( ( ord_less_rat @ zero_zero_rat @ C )
% 0.34/0.65 => ( ord_less_eq_rat @ A2 @ one_one_rat ) )
% 0.34/0.65 & ( ( ord_less_rat @ C @ zero_zero_rat )
% 0.34/0.65 => ( ord_less_eq_rat @ one_one_rat @ A2 ) ) ) ) ).
% 0.34/0.65
% 0.34/0.65 % mult_le_cancel_left2
% 0.34/0.65 thf(fact_5964_mult__le__cancel__left2,axiom,
% 0.34/0.65 ! [C: int,A2: int] :
% 0.34/0.65 ( ( ord_less_eq_int @ ( times_times_int @ C @ A2 ) @ C )
% 0.34/0.65 = ( ( ( ord_less_int @ zero_zero_int @ C )
% 0.34/0.65 => ( ord_less_eq_int @ A2 @ one_one_int ) )
% 0.34/0.65 & ( ( ord_less_int @ C @ zero_zero_int )
% 0.34/0.65 => ( ord_less_eq_int @ one_one_int @ A2 ) ) ) ) ).
% 0.34/0.65
% 0.34/0.65 % mult_le_cancel_left2
% 0.34/0.65 thf(fact_5965_mult__le__cancel__right1,axiom,
% 0.34/0.65 ! [C: code_integer,B: code_integer] :
% 0.34/0.65 ( ( ord_le3102999989581377725nteger @ C @ ( times_3573771949741848930nteger @ B @ C ) )
% 0.34/0.65 = ( ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ C )
% 0.34/0.65 => ( ord_le3102999989581377725nteger @ one_one_Code_integer @ B ) )
% 0.34/0.65 & ( ( ord_le6747313008572928689nteger @ C @ zero_z3403309356797280102nteger )
% 0.34/0.65 => ( ord_le3102999989581377725nteger @ B @ one_one_Code_integer ) ) ) ) ).
% 0.34/0.65
% 0.34/0.65 % mult_le_cancel_right1
% 0.34/0.65 thf(fact_5966_mult__le__cancel__right1,axiom,
% 0.34/0.65 ! [C: rat,B: rat] :
% 0.34/0.65 ( ( ord_less_eq_rat @ C @ ( times_times_rat @ B @ C ) )
% 0.34/0.65 = ( ( ( ord_less_rat @ zero_zero_rat @ C )
% 0.34/0.65 => ( ord_less_eq_rat @ one_one_rat @ B ) )
% 0.34/0.65 & ( ( ord_less_rat @ C @ zero_zero_rat )
% 0.34/0.65 => ( ord_less_eq_rat @ B @ one_one_rat ) ) ) ) ).
% 0.34/0.65
% 0.34/0.65 % mult_le_cancel_right1
% 0.34/0.65 thf(fact_5967_mult__le__cancel__right1,axiom,
% 0.34/0.65 ! [C: int,B: int] :
% 0.34/0.65 ( ( ord_less_eq_int @ C @ ( times_times_int @ B @ C ) )
% 0.34/0.65 = ( ( ( ord_less_int @ zero_zero_int @ C )
% 0.34/0.65 => ( ord_less_eq_int @ one_one_int @ B ) )
% 0.34/0.65 & ( ( ord_less_int @ C @ zero_zero_int )
% 0.34/0.65 => ( ord_less_eq_int @ B @ one_one_int ) ) ) ) ).
% 0.34/0.65
% 0.34/0.65 % mult_le_cancel_right1
% 0.34/0.65 thf(fact_5968_mult__le__cancel__right2,axiom,
% 0.34/0.65 ! [A2: code_integer,C: code_integer] :
% 0.34/0.65 ( ( ord_le3102999989581377725nteger @ ( times_3573771949741848930nteger @ A2 @ C ) @ C )
% 0.34/0.65 = ( ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ C )
% 0.34/0.65 => ( ord_le3102999989581377725nteger @ A2 @ one_one_Code_integer ) )
% 0.34/0.65 & ( ( ord_le6747313008572928689nteger @ C @ zero_z3403309356797280102nteger )
% 0.34/0.65 => ( ord_le3102999989581377725nteger @ one_one_Code_integer @ A2 ) ) ) ) ).
% 0.34/0.65
% 0.34/0.65 % mult_le_cancel_right2
% 0.34/0.65 thf(fact_5969_mult__le__cancel__right2,axiom,
% 0.34/0.65 ! [A2: rat,C: rat] :
% 0.34/0.65 ( ( ord_less_eq_rat @ ( times_times_rat @ A2 @ C ) @ C )
% 0.34/0.65 = ( ( ( ord_less_rat @ zero_zero_rat @ C )
% 0.34/0.65 => ( ord_less_eq_rat @ A2 @ one_one_rat ) )
% 0.34/0.65 & ( ( ord_less_rat @ C @ zero_zero_rat )
% 0.34/0.65 => ( ord_less_eq_rat @ one_one_rat @ A2 ) ) ) ) ).
% 0.34/0.65
% 0.34/0.65 % mult_le_cancel_right2
% 0.34/0.65 thf(fact_5970_mult__le__cancel__right2,axiom,
% 0.34/0.65 ! [A2: int,C: int] :
% 0.34/0.65 ( ( ord_less_eq_int @ ( times_times_int @ A2 @ C ) @ C )
% 0.34/0.65 = ( ( ( ord_less_int @ zero_zero_int @ C )
% 0.34/0.65 => ( ord_less_eq_int @ A2 @ one_one_int ) )
% 0.34/0.65 & ( ( ord_less_int @ C @ zero_zero_int )
% 0.34/0.65 => ( ord_less_eq_int @ one_one_int @ A2 ) ) ) ) ).
% 0.34/0.65
% 0.34/0.65 % mult_le_cancel_right2
% 0.34/0.65 thf(fact_5971_mult__less__cancel__left1,axiom,
% 0.34/0.65 ! [C: code_integer,B: code_integer] :
% 0.34/0.65 ( ( ord_le6747313008572928689nteger @ C @ ( times_3573771949741848930nteger @ C @ B ) )
% 0.34/0.65 = ( ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ C )
% 0.34/0.65 => ( ord_le6747313008572928689nteger @ one_one_Code_integer @ B ) )
% 0.34/0.65 & ( ( ord_le3102999989581377725nteger @ C @ zero_z3403309356797280102nteger )
% 0.34/0.65 => ( ord_le6747313008572928689nteger @ B @ one_one_Code_integer ) ) ) ) ).
% 0.34/0.65
% 0.34/0.65 % mult_less_cancel_left1
% 0.34/0.65 thf(fact_5972_mult__less__cancel__left1,axiom,
% 0.34/0.65 ! [C: rat,B: rat] :
% 0.34/0.65 ( ( ord_less_rat @ C @ ( times_times_rat @ C @ B ) )
% 0.34/0.65 = ( ( ( ord_less_eq_rat @ zero_zero_rat @ C )
% 0.34/0.65 => ( ord_less_rat @ one_one_rat @ B ) )
% 0.34/0.65 & ( ( ord_less_eq_rat @ C @ zero_zero_rat )
% 0.34/0.65 => ( ord_less_rat @ B @ one_one_rat ) ) ) ) ).
% 0.34/0.65
% 0.34/0.65 % mult_less_cancel_left1
% 0.34/0.65 thf(fact_5973_mult__less__cancel__left1,axiom,
% 0.34/0.65 ! [C: int,B: int] :
% 0.34/0.65 ( ( ord_less_int @ C @ ( times_times_int @ C @ B ) )
% 0.34/0.65 = ( ( ( ord_less_eq_int @ zero_zero_int @ C )
% 0.34/0.65 => ( ord_less_int @ one_one_int @ B ) )
% 0.34/0.65 & ( ( ord_less_eq_int @ C @ zero_zero_int )
% 0.34/0.65 => ( ord_less_int @ B @ one_one_int ) ) ) ) ).
% 0.34/0.65
% 0.34/0.65 % mult_less_cancel_left1
% 0.34/0.65 thf(fact_5974_mult__less__cancel__left2,axiom,
% 0.34/0.65 ! [C: code_integer,A2: code_integer] :
% 0.34/0.65 ( ( ord_le6747313008572928689nteger @ ( times_3573771949741848930nteger @ C @ A2 ) @ C )
% 0.34/0.65 = ( ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ C )
% 0.34/0.65 => ( ord_le6747313008572928689nteger @ A2 @ one_one_Code_integer ) )
% 0.34/0.65 & ( ( ord_le3102999989581377725nteger @ C @ zero_z3403309356797280102nteger )
% 0.34/0.65 => ( ord_le6747313008572928689nteger @ one_one_Code_integer @ A2 ) ) ) ) ).
% 0.34/0.65
% 0.34/0.65 % mult_less_cancel_left2
% 0.34/0.65 thf(fact_5975_mult__less__cancel__left2,axiom,
% 0.34/0.65 ! [C: rat,A2: rat] :
% 0.34/0.65 ( ( ord_less_rat @ ( times_times_rat @ C @ A2 ) @ C )
% 0.34/0.65 = ( ( ( ord_less_eq_rat @ zero_zero_rat @ C )
% 0.34/0.65 => ( ord_less_rat @ A2 @ one_one_rat ) )
% 0.34/0.65 & ( ( ord_less_eq_rat @ C @ zero_zero_rat )
% 0.34/0.65 => ( ord_less_rat @ one_one_rat @ A2 ) ) ) ) ).
% 0.34/0.65
% 0.34/0.65 % mult_less_cancel_left2
% 0.34/0.65 thf(fact_5976_mult__less__cancel__left2,axiom,
% 0.34/0.65 ! [C: int,A2: int] :
% 0.34/0.65 ( ( ord_less_int @ ( times_times_int @ C @ A2 ) @ C )
% 0.34/0.65 = ( ( ( ord_less_eq_int @ zero_zero_int @ C )
% 0.34/0.65 => ( ord_less_int @ A2 @ one_one_int ) )
% 0.34/0.65 & ( ( ord_less_eq_int @ C @ zero_zero_int )
% 0.34/0.65 => ( ord_less_int @ one_one_int @ A2 ) ) ) ) ).
% 0.34/0.65
% 0.34/0.65 % mult_less_cancel_left2
% 0.34/0.65 thf(fact_5977_mult__less__cancel__right1,axiom,
% 0.34/0.65 ! [C: code_integer,B: code_integer] :
% 0.34/0.65 ( ( ord_le6747313008572928689nteger @ C @ ( times_3573771949741848930nteger @ B @ C ) )
% 0.34/0.65 = ( ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ C )
% 0.34/0.65 => ( ord_le6747313008572928689nteger @ one_one_Code_integer @ B ) )
% 0.34/0.65 & ( ( ord_le3102999989581377725nteger @ C @ zero_z3403309356797280102nteger )
% 0.34/0.65 => ( ord_le6747313008572928689nteger @ B @ one_one_Code_integer ) ) ) ) ).
% 0.34/0.65
% 0.34/0.65 % mult_less_cancel_right1
% 0.34/0.65 thf(fact_5978_mult__less__cancel__right1,axiom,
% 0.34/0.65 ! [C: rat,B: rat] :
% 0.34/0.65 ( ( ord_less_rat @ C @ ( times_times_rat @ B @ C ) )
% 0.34/0.65 = ( ( ( ord_less_eq_rat @ zero_zero_rat @ C )
% 0.34/0.65 => ( ord_less_rat @ one_one_rat @ B ) )
% 0.34/0.65 & ( ( ord_less_eq_rat @ C @ zero_zero_rat )
% 0.34/0.65 => ( ord_less_rat @ B @ one_one_rat ) ) ) ) ).
% 0.34/0.65
% 0.34/0.65 % mult_less_cancel_right1
% 0.34/0.65 thf(fact_5979_mult__less__cancel__right1,axiom,
% 0.34/0.65 ! [C: int,B: int] :
% 0.34/0.65 ( ( ord_less_int @ C @ ( times_times_int @ B @ C ) )
% 0.34/0.65 = ( ( ( ord_less_eq_int @ zero_zero_int @ C )
% 0.34/0.65 => ( ord_less_int @ one_one_int @ B ) )
% 0.34/0.65 & ( ( ord_less_eq_int @ C @ zero_zero_int )
% 0.34/0.65 => ( ord_less_int @ B @ one_one_int ) ) ) ) ).
% 0.34/0.65
% 0.34/0.65 % mult_less_cancel_right1
% 0.34/0.65 thf(fact_5980_mult__less__cancel__right2,axiom,
% 0.34/0.65 ! [A2: code_integer,C: code_integer] :
% 0.34/0.65 ( ( ord_le6747313008572928689nteger @ ( times_3573771949741848930nteger @ A2 @ C ) @ C )
% 0.34/0.65 = ( ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ C )
% 0.34/0.65 => ( ord_le6747313008572928689nteger @ A2 @ one_one_Code_integer ) )
% 0.34/0.65 & ( ( ord_le3102999989581377725nteger @ C @ zero_z3403309356797280102nteger )
% 0.34/0.65 => ( ord_le6747313008572928689nteger @ one_one_Code_integer @ A2 ) ) ) ) ).
% 0.34/0.65
% 0.34/0.65 % mult_less_cancel_right2
% 0.34/0.65 thf(fact_5981_mult__less__cancel__right2,axiom,
% 0.34/0.65 ! [A2: rat,C: rat] :
% 0.34/0.65 ( ( ord_less_rat @ ( times_times_rat @ A2 @ C ) @ C )
% 0.34/0.65 = ( ( ( ord_less_eq_rat @ zero_zero_rat @ C )
% 0.34/0.65 => ( ord_less_rat @ A2 @ one_one_rat ) )
% 0.34/0.65 & ( ( ord_less_eq_rat @ C @ zero_zero_rat )
% 0.34/0.65 => ( ord_less_rat @ one_one_rat @ A2 ) ) ) ) ).
% 0.34/0.65
% 0.34/0.65 % mult_less_cancel_right2
% 0.34/0.65 thf(fact_5982_mult__less__cancel__right2,axiom,
% 0.34/0.65 ! [A2: int,C: int] :
% 0.34/0.65 ( ( ord_less_int @ ( times_times_int @ A2 @ C ) @ C )
% 0.34/0.65 = ( ( ( ord_less_eq_int @ zero_zero_int @ C )
% 0.34/0.65 => ( ord_less_int @ A2 @ one_one_int ) )
% 0.34/0.65 & ( ( ord_less_eq_int @ C @ zero_zero_int )
% 0.34/0.65 => ( ord_less_int @ one_one_int @ A2 ) ) ) ) ).
% 0.34/0.65
% 0.34/0.65 % mult_less_cancel_right2
% 0.34/0.65 thf(fact_5983_convex__bound__le,axiom,
% 0.34/0.65 ! [X: code_integer,A2: code_integer,Y: code_integer,U: code_integer,V: code_integer] :
% 0.34/0.65 ( ( ord_le3102999989581377725nteger @ X @ A2 )
% 0.34/0.65 => ( ( ord_le3102999989581377725nteger @ Y @ A2 )
% 0.34/0.65 => ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ U )
% 0.34/0.65 => ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ V )
% 0.34/0.65 => ( ( ( plus_p5714425477246183910nteger @ U @ V )
% 0.34/0.65 = one_one_Code_integer )
% 0.34/0.65 => ( ord_le3102999989581377725nteger @ ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ U @ X ) @ ( times_3573771949741848930nteger @ V @ Y ) ) @ A2 ) ) ) ) ) ) ).
% 0.34/0.65
% 0.34/0.65 % convex_bound_le
% 0.34/0.65 thf(fact_5984_convex__bound__le,axiom,
% 0.34/0.65 ! [X: rat,A2: rat,Y: rat,U: rat,V: rat] :
% 0.34/0.65 ( ( ord_less_eq_rat @ X @ A2 )
% 0.34/0.65 => ( ( ord_less_eq_rat @ Y @ A2 )
% 0.34/0.65 => ( ( ord_less_eq_rat @ zero_zero_rat @ U )
% 0.34/0.65 => ( ( ord_less_eq_rat @ zero_zero_rat @ V )
% 0.34/0.65 => ( ( ( plus_plus_rat @ U @ V )
% 0.34/0.65 = one_one_rat )
% 0.34/0.65 => ( ord_less_eq_rat @ ( plus_plus_rat @ ( times_times_rat @ U @ X ) @ ( times_times_rat @ V @ Y ) ) @ A2 ) ) ) ) ) ) ).
% 0.34/0.65
% 0.34/0.65 % convex_bound_le
% 0.34/0.65 thf(fact_5985_convex__bound__le,axiom,
% 0.34/0.65 ! [X: int,A2: int,Y: int,U: int,V: int] :
% 0.34/0.65 ( ( ord_less_eq_int @ X @ A2 )
% 0.34/0.65 => ( ( ord_less_eq_int @ Y @ A2 )
% 0.34/0.65 => ( ( ord_less_eq_int @ zero_zero_int @ U )
% 0.34/0.65 => ( ( ord_less_eq_int @ zero_zero_int @ V )
% 0.34/0.65 => ( ( ( plus_plus_int @ U @ V )
% 0.34/0.65 = one_one_int )
% 0.34/0.65 => ( ord_less_eq_int @ ( plus_plus_int @ ( times_times_int @ U @ X ) @ ( times_times_int @ V @ Y ) ) @ A2 ) ) ) ) ) ) ).
% 0.34/0.65
% 0.34/0.65 % convex_bound_le
% 0.34/0.65 thf(fact_5986_card__gt__0__iff,axiom,
% 0.34/0.65 ! [A: set_Code_integer] :
% 0.34/0.65 ( ( ord_less_nat @ zero_zero_nat @ ( finite4902975817058060853nteger @ A ) )
% 0.34/0.65 = ( ( A != bot_bo3990330152332043303nteger )
% 0.34/0.65 & ( finite6017078050557962740nteger @ A ) ) ) ).
% 0.34/0.65
% 0.34/0.65 % card_gt_0_iff
% 0.34/0.65 thf(fact_5987_card__gt__0__iff,axiom,
% 0.34/0.65 ! [A: set_Product_unit] :
% 0.34/0.65 ( ( ord_less_nat @ zero_zero_nat @ ( finite410649719033368117t_unit @ A ) )
% 0.34/0.65 = ( ( A != bot_bo3957492148770167129t_unit )
% 0.34/0.65 & ( finite4290736615968046902t_unit @ A ) ) ) ).
% 0.34/0.65
% 0.34/0.65 % card_gt_0_iff
% 0.34/0.65 thf(fact_5988_card__gt__0__iff,axiom,
% 0.34/0.65 ! [A: set_o] :
% 0.34/0.65 ( ( ord_less_nat @ zero_zero_nat @ ( finite_card_o @ A ) )
% 0.34/0.65 = ( ( A != bot_bot_set_o )
% 0.34/0.65 & ( finite_finite_o @ A ) ) ) ).
% 0.34/0.65
% 0.34/0.65 % card_gt_0_iff
% 0.34/0.65 thf(fact_5989_card__gt__0__iff,axiom,
% 0.34/0.65 ! [A: set_nat] :
% 0.34/0.65 ( ( ord_less_nat @ zero_zero_nat @ ( finite_card_nat @ A ) )
% 0.34/0.65 = ( ( A != bot_bot_set_nat )
% 0.34/0.65 & ( finite_finite_nat @ A ) ) ) ).
% 0.34/0.65
% 0.34/0.65 % card_gt_0_iff
% 0.34/0.65 thf(fact_5990_card__gt__0__iff,axiom,
% 0.34/0.65 ! [A: set_int] :
% 0.34/0.65 ( ( ord_less_nat @ zero_zero_nat @ ( finite_card_int @ A ) )
% 0.34/0.65 = ( ( A != bot_bot_set_int )
% 0.34/0.65 & ( finite_finite_int @ A ) ) ) ).
% 0.34/0.65
% 0.34/0.65 % card_gt_0_iff
% 0.34/0.65 thf(fact_5991_card__gt__0__iff,axiom,
% 0.34/0.65 ! [A: set_Pr4532377907799695533_nat_o] :
% 0.34/0.65 ( ( ord_less_nat @ zero_zero_nat @ ( finite1363419556375932405_nat_o @ A ) )
% 0.34/0.65 = ( ( A != bot_bo7824918357723582233_nat_o )
% 0.34/0.65 & ( finite3252695134891459830_nat_o @ A ) ) ) ).
% 0.34/0.65
% 0.34/0.65 % card_gt_0_iff
% 0.34/0.65 thf(fact_5992_card__1__singleton__iff,axiom,
% 0.34/0.65 ! [A: set_Product_unit] :
% 0.34/0.65 ( ( ( finite410649719033368117t_unit @ A )
% 0.34/0.65 = ( suc @ zero_zero_nat ) )
% 0.34/0.65 = ( ? [X2: product_unit] :
% 0.34/0.65 ( A
% 0.34/0.65 = ( insert_Product_unit @ X2 @ bot_bo3957492148770167129t_unit ) ) ) ) ).
% 0.34/0.65
% 0.34/0.65 % card_1_singleton_iff
% 0.34/0.65 thf(fact_5993_card__1__singleton__iff,axiom,
% 0.34/0.65 ! [A: set_o] :
% 0.34/0.65 ( ( ( finite_card_o @ A )
% 0.34/0.65 = ( suc @ zero_zero_nat ) )
% 0.34/0.65 = ( ? [X2: $o] :
% 0.34/0.65 ( A
% 0.34/0.65 = ( insert_o @ X2 @ bot_bot_set_o ) ) ) ) ).
% 0.34/0.65
% 0.34/0.65 % card_1_singleton_iff
% 0.34/0.65 thf(fact_5994_card__1__singleton__iff,axiom,
% 0.34/0.65 ! [A: set_nat] :
% 0.34/0.65 ( ( ( finite_card_nat @ A )
% 0.34/0.65 = ( suc @ zero_zero_nat ) )
% 0.34/0.65 = ( ? [X2: nat] :
% 0.34/0.65 ( A
% 0.34/0.65 = ( insert_nat @ X2 @ bot_bot_set_nat ) ) ) ) ).
% 0.34/0.65
% 0.34/0.65 % card_1_singleton_iff
% 0.34/0.65 thf(fact_5995_card__1__singleton__iff,axiom,
% 0.34/0.65 ! [A: set_int] :
% 0.34/0.65 ( ( ( finite_card_int @ A )
% 0.34/0.65 = ( suc @ zero_zero_nat ) )
% 0.34/0.65 = ( ? [X2: int] :
% 0.34/0.65 ( A
% 0.34/0.65 = ( insert_int @ X2 @ bot_bot_set_int ) ) ) ) ).
% 0.34/0.65
% 0.34/0.65 % card_1_singleton_iff
% 0.34/0.65 thf(fact_5996_card__1__singleton__iff,axiom,
% 0.34/0.65 ! [A: set_Pr4532377907799695533_nat_o] :
% 0.34/0.65 ( ( ( finite1363419556375932405_nat_o @ A )
% 0.34/0.65 = ( suc @ zero_zero_nat ) )
% 0.34/0.65 = ( ? [X2: produc3658429121746597890et_nat > $o] :
% 0.34/0.65 ( A
% 0.34/0.65 = ( insert5175938949040314269_nat_o @ X2 @ bot_bo7824918357723582233_nat_o ) ) ) ) ).
% 0.34/0.65
% 0.34/0.65 % card_1_singleton_iff
% 0.34/0.65 thf(fact_5997_card__eq__SucD,axiom,
% 0.34/0.65 ! [A: set_Product_unit,K3: nat] :
% 0.34/0.65 ( ( ( finite410649719033368117t_unit @ A )
% 0.34/0.65 = ( suc @ K3 ) )
% 0.34/0.65 => ? [B2: product_unit,B5: set_Product_unit] :
% 0.34/0.65 ( ( A
% 0.34/0.65 = ( insert_Product_unit @ B2 @ B5 ) )
% 0.34/0.65 & ~ ( member_Product_unit @ B2 @ B5 )
% 0.34/0.65 & ( ( finite410649719033368117t_unit @ B5 )
% 0.34/0.65 = K3 )
% 0.34/0.65 & ( ( K3 = zero_zero_nat )
% 0.34/0.65 => ( B5 = bot_bo3957492148770167129t_unit ) ) ) ) ).
% 0.34/0.65
% 0.34/0.65 % card_eq_SucD
% 0.34/0.65 thf(fact_5998_card__eq__SucD,axiom,
% 0.34/0.65 ! [A: set_o,K3: nat] :
% 0.34/0.65 ( ( ( finite_card_o @ A )
% 0.34/0.65 = ( suc @ K3 ) )
% 0.34/0.65 => ? [B2: $o,B5: set_o] :
% 0.34/0.65 ( ( A
% 0.34/0.65 = ( insert_o @ B2 @ B5 ) )
% 0.34/0.65 & ~ ( member_o @ B2 @ B5 )
% 0.34/0.65 & ( ( finite_card_o @ B5 )
% 0.34/0.65 = K3 )
% 0.34/0.65 & ( ( K3 = zero_zero_nat )
% 0.34/0.65 => ( B5 = bot_bot_set_o ) ) ) ) ).
% 0.34/0.65
% 0.34/0.65 % card_eq_SucD
% 0.34/0.65 thf(fact_5999_card__eq__SucD,axiom,
% 0.34/0.65 ! [A: set_nat,K3: nat] :
% 0.34/0.65 ( ( ( finite_card_nat @ A )
% 0.34/0.65 = ( suc @ K3 ) )
% 0.34/0.65 => ? [B2: nat,B5: set_nat] :
% 0.34/0.65 ( ( A
% 0.34/0.65 = ( insert_nat @ B2 @ B5 ) )
% 0.34/0.65 & ~ ( member_nat @ B2 @ B5 )
% 0.34/0.65 & ( ( finite_card_nat @ B5 )
% 0.34/0.65 = K3 )
% 0.34/0.65 & ( ( K3 = zero_zero_nat )
% 0.34/0.65 => ( B5 = bot_bot_set_nat ) ) ) ) ).
% 0.34/0.65
% 0.34/0.65 % card_eq_SucD
% 0.34/0.65 thf(fact_6000_card__eq__SucD,axiom,
% 0.34/0.65 ! [A: set_int,K3: nat] :
% 0.34/0.65 ( ( ( finite_card_int @ A )
% 0.34/0.65 = ( suc @ K3 ) )
% 0.34/0.65 => ? [B2: int,B5: set_int] :
% 0.34/0.65 ( ( A
% 0.34/0.65 = ( insert_int @ B2 @ B5 ) )
% 0.34/0.65 & ~ ( member_int @ B2 @ B5 )
% 0.34/0.65 & ( ( finite_card_int @ B5 )
% 0.34/0.65 = K3 )
% 0.34/0.65 & ( ( K3 = zero_zero_nat )
% 0.34/0.65 => ( B5 = bot_bot_set_int ) ) ) ) ).
% 0.34/0.65
% 0.34/0.65 % card_eq_SucD
% 0.34/0.65 thf(fact_6001_card__eq__SucD,axiom,
% 0.34/0.65 ! [A: set_Pr4532377907799695533_nat_o,K3: nat] :
% 0.34/0.65 ( ( ( finite1363419556375932405_nat_o @ A )
% 0.34/0.65 = ( suc @ K3 ) )
% 0.34/0.65 => ? [B2: produc3658429121746597890et_nat > $o,B5: set_Pr4532377907799695533_nat_o] :
% 0.34/0.65 ( ( A
% 0.34/0.65 = ( insert5175938949040314269_nat_o @ B2 @ B5 ) )
% 0.34/0.65 & ~ ( member6576561426505652726_nat_o @ B2 @ B5 )
% 0.34/0.65 & ( ( finite1363419556375932405_nat_o @ B5 )
% 0.34/0.65 = K3 )
% 0.34/0.65 & ( ( K3 = zero_zero_nat )
% 0.34/0.65 => ( B5 = bot_bo7824918357723582233_nat_o ) ) ) ) ).
% 0.34/0.65
% 0.34/0.65 % card_eq_SucD
% 0.34/0.65 thf(fact_6002_card__Suc__eq,axiom,
% 0.34/0.65 ! [A: set_Product_unit,K3: nat] :
% 0.34/0.65 ( ( ( finite410649719033368117t_unit @ A )
% 0.34/0.65 = ( suc @ K3 ) )
% 0.34/0.65 = ( ? [B6: product_unit,B7: set_Product_unit] :
% 0.34/0.65 ( ( A
% 0.34/0.65 = ( insert_Product_unit @ B6 @ B7 ) )
% 0.34/0.65 & ~ ( member_Product_unit @ B6 @ B7 )
% 0.34/0.65 & ( ( finite410649719033368117t_unit @ B7 )
% 0.34/0.65 = K3 )
% 0.34/0.65 & ( ( K3 = zero_zero_nat )
% 0.34/0.65 => ( B7 = bot_bo3957492148770167129t_unit ) ) ) ) ) ).
% 0.34/0.65
% 0.34/0.65 % card_Suc_eq
% 0.34/0.65 thf(fact_6003_card__Suc__eq,axiom,
% 0.34/0.65 ! [A: set_o,K3: nat] :
% 0.34/0.65 ( ( ( finite_card_o @ A )
% 0.34/0.65 = ( suc @ K3 ) )
% 0.34/0.65 = ( ? [B6: $o,B7: set_o] :
% 0.34/0.65 ( ( A
% 0.34/0.65 = ( insert_o @ B6 @ B7 ) )
% 0.34/0.65 & ~ ( member_o @ B6 @ B7 )
% 0.34/0.65 & ( ( finite_card_o @ B7 )
% 0.34/0.65 = K3 )
% 0.34/0.65 & ( ( K3 = zero_zero_nat )
% 0.34/0.65 => ( B7 = bot_bot_set_o ) ) ) ) ) ).
% 0.34/0.65
% 0.34/0.65 % card_Suc_eq
% 0.34/0.65 thf(fact_6004_card__Suc__eq,axiom,
% 0.34/0.65 ! [A: set_nat,K3: nat] :
% 0.34/0.65 ( ( ( finite_card_nat @ A )
% 0.34/0.65 = ( suc @ K3 ) )
% 0.34/0.65 = ( ? [B6: nat,B7: set_nat] :
% 0.34/0.65 ( ( A
% 0.34/0.65 = ( insert_nat @ B6 @ B7 ) )
% 0.34/0.65 & ~ ( member_nat @ B6 @ B7 )
% 0.34/0.65 & ( ( finite_card_nat @ B7 )
% 0.34/0.65 = K3 )
% 0.34/0.65 & ( ( K3 = zero_zero_nat )
% 0.34/0.65 => ( B7 = bot_bot_set_nat ) ) ) ) ) ).
% 0.34/0.65
% 0.34/0.65 % card_Suc_eq
% 0.34/0.65 thf(fact_6005_card__Suc__eq,axiom,
% 0.34/0.65 ! [A: set_int,K3: nat] :
% 0.34/0.65 ( ( ( finite_card_int @ A )
% 0.34/0.65 = ( suc @ K3 ) )
% 0.34/0.65 = ( ? [B6: int,B7: set_int] :
% 0.34/0.65 ( ( A
% 0.34/0.65 = ( insert_int @ B6 @ B7 ) )
% 0.34/0.65 & ~ ( member_int @ B6 @ B7 )
% 0.34/0.65 & ( ( finite_card_int @ B7 )
% 0.34/0.65 = K3 )
% 0.34/0.65 & ( ( K3 = zero_zero_nat )
% 0.34/0.65 => ( B7 = bot_bot_set_int ) ) ) ) ) ).
% 0.34/0.65
% 0.34/0.65 % card_Suc_eq
% 0.34/0.65 thf(fact_6006_card__Suc__eq,axiom,
% 0.34/0.65 ! [A: set_Pr4532377907799695533_nat_o,K3: nat] :
% 0.34/0.65 ( ( ( finite1363419556375932405_nat_o @ A )
% 0.34/0.65 = ( suc @ K3 ) )
% 0.34/0.65 = ( ? [B6: produc3658429121746597890et_nat > $o,B7: set_Pr4532377907799695533_nat_o] :
% 0.34/0.65 ( ( A
% 0.34/0.65 = ( insert5175938949040314269_nat_o @ B6 @ B7 ) )
% 0.34/0.65 & ~ ( member6576561426505652726_nat_o @ B6 @ B7 )
% 0.34/0.65 & ( ( finite1363419556375932405_nat_o @ B7 )
% 0.34/0.65 = K3 )
% 0.34/0.65 & ( ( K3 = zero_zero_nat )
% 0.34/0.65 => ( B7 = bot_bo7824918357723582233_nat_o ) ) ) ) ) ).
% 0.34/0.65
% 0.34/0.65 % card_Suc_eq
% 0.34/0.65 thf(fact_6007_nz__le__conv__less,axiom,
% 0.34/0.65 ! [K3: nat,M4: nat] :
% 0.34/0.65 ( ( ord_less_nat @ zero_zero_nat @ K3 )
% 0.34/0.65 => ( ( ord_less_eq_nat @ K3 @ M4 )
% 0.34/0.65 => ( ord_less_nat @ ( minus_minus_nat @ K3 @ ( suc @ zero_zero_nat ) ) @ M4 ) ) ) ).
% 0.34/0.65
% 0.34/0.65 % nz_le_conv_less
% 0.34/0.65 thf(fact_6008_mult__eq__if,axiom,
% 0.34/0.65 ( times_times_nat
% 0.34/0.65 = ( ^ [M3: nat,N4: nat] : ( if_nat @ ( M3 = zero_zero_nat ) @ zero_zero_nat @ ( plus_plus_nat @ N4 @ ( times_times_nat @ ( minus_minus_nat @ M3 @ one_one_nat ) @ N4 ) ) ) ) ) ).
% 0.34/0.65
% 0.34/0.65 % mult_eq_if
% 0.34/0.65 thf(fact_6009_convex__bound__lt,axiom,
% 0.34/0.65 ! [X: code_integer,A2: code_integer,Y: code_integer,U: code_integer,V: code_integer] :
% 0.34/0.65 ( ( ord_le6747313008572928689nteger @ X @ A2 )
% 0.34/0.65 => ( ( ord_le6747313008572928689nteger @ Y @ A2 )
% 0.34/0.65 => ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ U )
% 0.34/0.65 => ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ V )
% 0.34/0.65 => ( ( ( plus_p5714425477246183910nteger @ U @ V )
% 0.34/0.65 = one_one_Code_integer )
% 0.34/0.65 => ( ord_le6747313008572928689nteger @ ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ U @ X ) @ ( times_3573771949741848930nteger @ V @ Y ) ) @ A2 ) ) ) ) ) ) ).
% 0.34/0.65
% 0.34/0.65 % convex_bound_lt
% 0.34/0.65 thf(fact_6010_convex__bound__lt,axiom,
% 0.34/0.65 ! [X: rat,A2: rat,Y: rat,U: rat,V: rat] :
% 0.34/0.65 ( ( ord_less_rat @ X @ A2 )
% 0.34/0.65 => ( ( ord_less_rat @ Y @ A2 )
% 0.34/0.65 => ( ( ord_less_eq_rat @ zero_zero_rat @ U )
% 0.34/0.65 => ( ( ord_less_eq_rat @ zero_zero_rat @ V )
% 0.34/0.65 => ( ( ( plus_plus_rat @ U @ V )
% 0.34/0.65 = one_one_rat )
% 0.34/0.65 => ( ord_less_rat @ ( plus_plus_rat @ ( times_times_rat @ U @ X ) @ ( times_times_rat @ V @ Y ) ) @ A2 ) ) ) ) ) ) ).
% 0.34/0.65
% 0.34/0.65 % convex_bound_lt
% 0.34/0.65 thf(fact_6011_convex__bound__lt,axiom,
% 0.34/0.65 ! [X: int,A2: int,Y: int,U: int,V: int] :
% 0.34/0.65 ( ( ord_less_int @ X @ A2 )
% 0.34/0.65 => ( ( ord_less_int @ Y @ A2 )
% 0.34/0.65 => ( ( ord_less_eq_int @ zero_zero_int @ U )
% 0.34/0.65 => ( ( ord_less_eq_int @ zero_zero_int @ V )
% 0.34/0.65 => ( ( ( plus_plus_int @ U @ V )
% 0.34/0.65 = one_one_int )
% 0.34/0.65 => ( ord_less_int @ ( plus_plus_int @ ( times_times_int @ U @ X ) @ ( times_times_int @ V @ Y ) ) @ A2 ) ) ) ) ) ) ).
% 0.34/0.65
% 0.34/0.65 % convex_bound_lt
% 0.34/0.65 thf(fact_6012_inj__mult__left,axiom,
% 0.34/0.65 ! [A2: code_integer] :
% 0.34/0.65 ( ( inj_on1158064722259180589nteger @ ( times_3573771949741848930nteger @ A2 ) @ top_to4645266643341252675nteger )
% 0.34/0.65 = ( A2 != zero_z3403309356797280102nteger ) ) ).
% 0.34/0.65
% 0.34/0.65 % inj_mult_left
% 0.34/0.65 thf(fact_6013_inj__mult__left,axiom,
% 0.34/0.65 ! [A2: rat] :
% 0.34/0.65 ( ( inj_on_rat_rat @ ( times_times_rat @ A2 ) @ top_top_set_rat )
% 0.34/0.65 = ( A2 != zero_zero_rat ) ) ).
% 0.34/0.65
% 0.34/0.65 % inj_mult_left
% 0.34/0.65 thf(fact_6014_inj__mult__left,axiom,
% 0.34/0.65 ! [A2: int] :
% 0.34/0.65 ( ( inj_on_int_int @ ( times_times_int @ A2 ) @ top_top_set_int )
% 0.34/0.65 = ( A2 != zero_zero_int ) ) ).
% 0.34/0.65
% 0.34/0.65 % inj_mult_left
% 0.34/0.65 thf(fact_6015_field__le__mult__one__interval,axiom,
% 0.34/0.65 ! [X: rat,Y: rat] :
% 0.34/0.65 ( ! [Z3: rat] :
% 0.34/0.65 ( ( ord_less_rat @ zero_zero_rat @ Z3 )
% 0.34/0.65 => ( ( ord_less_rat @ Z3 @ one_one_rat )
% 0.34/0.65 => ( ord_less_eq_rat @ ( times_times_rat @ Z3 @ X ) @ Y ) ) )
% 0.34/0.65 => ( ord_less_eq_rat @ X @ Y ) ) ).
% 0.34/0.65
% 0.34/0.65 % field_le_mult_one_interval
% 0.34/0.65 thf(fact_6016_size__prod__simp,axiom,
% 0.34/0.65 ( basic_5567952622663070111et_nat
% 0.34/0.65 = ( ^ [F4: heap_e7401611519738050253t_unit > nat,G3: set_nat > nat,P5: produc3658429121746597890et_nat] : ( plus_plus_nat @ ( plus_plus_nat @ ( F4 @ ( produc1824681642469235216et_nat @ P5 ) ) @ ( G3 @ ( produc8586169260539613262et_nat @ P5 ) ) ) @ ( suc @ zero_zero_nat ) ) ) ) ).
% 0.34/0.65
% 0.34/0.65 % size_prod_simp
% 0.34/0.65 thf(fact_6017_size__prod__simp,axiom,
% 0.34/0.65 ( basic_1872990034501187214nt_int
% 0.34/0.65 = ( ^ [F4: int > nat,G3: int > nat,P5: product_prod_int_int] : ( plus_plus_nat @ ( plus_plus_nat @ ( F4 @ ( product_fst_int_int @ P5 ) ) @ ( G3 @ ( product_snd_int_int @ P5 ) ) ) @ ( suc @ zero_zero_nat ) ) ) ) ).
% 0.34/0.65
% 0.34/0.65 % size_prod_simp
% 0.34/0.65 thf(fact_6018_sum__squares__gt__zero__iff,axiom,
% 0.34/0.65 ! [X: int,Y: int] :
% 0.34/0.65 ( ( ord_less_int @ zero_zero_int @ ( plus_plus_int @ ( times_times_int @ X @ X ) @ ( times_times_int @ Y @ Y ) ) )
% 0.34/0.65 = ( ( X != zero_zero_int )
% 0.34/0.65 | ( Y != zero_zero_int ) ) ) ).
% 0.34/0.65
% 0.34/0.65 % sum_squares_gt_zero_iff
% 0.34/0.65 thf(fact_6019_sum__squares__gt__zero__iff,axiom,
% 0.34/0.65 ! [X: code_integer,Y: code_integer] :
% 0.34/0.65 ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ X @ X ) @ ( times_3573771949741848930nteger @ Y @ Y ) ) )
% 0.34/0.65 = ( ( X != zero_z3403309356797280102nteger )
% 0.34/0.65 | ( Y != zero_z3403309356797280102nteger ) ) ) ).
% 0.34/0.65
% 0.34/0.65 % sum_squares_gt_zero_iff
% 0.34/0.65 thf(fact_6020_sum__squares__gt__zero__iff,axiom,
% 0.34/0.65 ! [X: rat,Y: rat] :
% 0.34/0.65 ( ( ord_less_rat @ zero_zero_rat @ ( plus_plus_rat @ ( times_times_rat @ X @ X ) @ ( times_times_rat @ Y @ Y ) ) )
% 0.34/0.65 = ( ( X != zero_zero_rat )
% 0.34/0.65 | ( Y != zero_zero_rat ) ) ) ).
% 0.34/0.65
% 0.34/0.65 % sum_squares_gt_zero_iff
% 0.34/0.65 thf(fact_6021_sum__squares__le__zero__iff,axiom,
% 0.34/0.65 ! [X: code_integer,Y: code_integer] :
% 0.34/0.65 ( ( ord_le3102999989581377725nteger @ ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ X @ X ) @ ( times_3573771949741848930nteger @ Y @ Y ) ) @ zero_z3403309356797280102nteger )
% 0.34/0.65 = ( ( X = zero_z3403309356797280102nteger )
% 0.34/0.65 & ( Y = zero_z3403309356797280102nteger ) ) ) ).
% 0.34/0.65
% 0.34/0.65 % sum_squares_le_zero_iff
% 0.34/0.65 thf(fact_6022_sum__squares__le__zero__iff,axiom,
% 0.34/0.65 ! [X: rat,Y: rat] :
% 0.34/0.65 ( ( ord_less_eq_rat @ ( plus_plus_rat @ ( times_times_rat @ X @ X ) @ ( times_times_rat @ Y @ Y ) ) @ zero_zero_rat )
% 0.34/0.65 = ( ( X = zero_zero_rat )
% 0.34/0.65 & ( Y = zero_zero_rat ) ) ) ).
% 0.34/0.65
% 0.34/0.65 % sum_squares_le_zero_iff
% 0.34/0.65 thf(fact_6023_sum__squares__le__zero__iff,axiom,
% 0.34/0.65 ! [X: int,Y: int] :
% 0.34/0.65 ( ( ord_less_eq_int @ ( plus_plus_int @ ( times_times_int @ X @ X ) @ ( times_times_int @ Y @ Y ) ) @ zero_zero_int )
% 0.34/0.65 = ( ( X = zero_zero_int )
% 0.34/0.65 & ( Y = zero_zero_int ) ) ) ).
% 0.34/0.65
% 0.34/0.65 % sum_squares_le_zero_iff
% 0.34/0.65 thf(fact_6024_mult__le__cancel__iff2,axiom,
% 0.34/0.65 ! [Z: code_integer,X: code_integer,Y: code_integer] :
% 0.34/0.65 ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ Z )
% 0.34/0.65 => ( ( ord_le3102999989581377725nteger @ ( times_3573771949741848930nteger @ Z @ X ) @ ( times_3573771949741848930nteger @ Z @ Y ) )
% 0.34/0.65 = ( ord_le3102999989581377725nteger @ X @ Y ) ) ) ).
% 0.34/0.65
% 0.34/0.65 % mult_le_cancel_iff2
% 0.34/0.65 thf(fact_6025_mult__le__cancel__iff2,axiom,
% 0.34/0.65 ! [Z: rat,X: rat,Y: rat] :
% 0.34/0.65 ( ( ord_less_rat @ zero_zero_rat @ Z )
% 0.34/0.65 => ( ( ord_less_eq_rat @ ( times_times_rat @ Z @ X ) @ ( times_times_rat @ Z @ Y ) )
% 0.34/0.65 = ( ord_less_eq_rat @ X @ Y ) ) ) ).
% 0.34/0.65
% 0.34/0.65 % mult_le_cancel_iff2
% 0.34/0.65 thf(fact_6026_mult__le__cancel__iff2,axiom,
% 0.34/0.65 ! [Z: int,X: int,Y: int] :
% 0.34/0.65 ( ( ord_less_int @ zero_zero_int @ Z )
% 0.34/0.65 => ( ( ord_less_eq_int @ ( times_times_int @ Z @ X ) @ ( times_times_int @ Z @ Y ) )
% 0.34/0.65 = ( ord_less_eq_int @ X @ Y ) ) ) ).
% 0.34/0.65
% 0.34/0.65 % mult_le_cancel_iff2
% 0.34/0.65 thf(fact_6027_mult__le__cancel__iff1,axiom,
% 0.34/0.65 ! [Z: code_integer,X: code_integer,Y: code_integer] :
% 0.34/0.65 ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ Z )
% 0.34/0.65 => ( ( ord_le3102999989581377725nteger @ ( times_3573771949741848930nteger @ X @ Z ) @ ( times_3573771949741848930nteger @ Y @ Z ) )
% 0.34/0.65 = ( ord_le3102999989581377725nteger @ X @ Y ) ) ) ).
% 0.34/0.65
% 0.34/0.65 % mult_le_cancel_iff1
% 0.34/0.65 thf(fact_6028_mult__le__cancel__iff1,axiom,
% 0.34/0.65 ! [Z: rat,X: rat,Y: rat] :
% 0.34/0.65 ( ( ord_less_rat @ zero_zero_rat @ Z )
% 0.34/0.65 => ( ( ord_less_eq_rat @ ( times_times_rat @ X @ Z ) @ ( times_times_rat @ Y @ Z ) )
% 0.34/0.65 = ( ord_less_eq_rat @ X @ Y ) ) ) ).
% 0.34/0.65
% 0.34/0.65 % mult_le_cancel_iff1
% 0.34/0.65 thf(fact_6029_mult__le__cancel__iff1,axiom,
% 0.34/0.65 ! [Z: int,X: int,Y: int] :
% 0.34/0.65 ( ( ord_less_int @ zero_zero_int @ Z )
% 0.34/0.65 => ( ( ord_less_eq_int @ ( times_times_int @ X @ Z ) @ ( times_times_int @ Y @ Z ) )
% 0.34/0.65 = ( ord_less_eq_int @ X @ Y ) ) ) ).
% 0.34/0.65
% 0.34/0.65 % mult_le_cancel_iff1
% 0.34/0.65 thf(fact_6030_divides__aux__eq,axiom,
% 0.34/0.65 ! [Q4: nat,R2: nat] :
% 0.34/0.65 ( ( unique6322359934112328802ux_nat @ ( product_Pair_nat_nat @ Q4 @ R2 ) )
% 0.34/0.65 = ( R2 = zero_zero_nat ) ) ).
% 0.34/0.65
% 0.34/0.65 % divides_aux_eq
% 0.34/0.65 thf(fact_6031_divides__aux__eq,axiom,
% 0.34/0.65 ! [Q4: int,R2: int] :
% 0.34/0.65 ( ( unique6319869463603278526ux_int @ ( product_Pair_int_int @ Q4 @ R2 ) )
% 0.34/0.65 = ( R2 = zero_zero_int ) ) ).
% 0.34/0.65
% 0.34/0.65 % divides_aux_eq
% 0.34/0.65 thf(fact_6032_divides__aux__eq,axiom,
% 0.34/0.65 ! [Q4: code_integer,R2: code_integer] :
% 0.34/0.65 ( ( unique5706413561485394159nteger @ ( produc1086072967326762835nteger @ Q4 @ R2 ) )
% 0.34/0.65 = ( R2 = zero_z3403309356797280102nteger ) ) ).
% 0.34/0.65
% 0.34/0.65 % divides_aux_eq
% 0.34/0.65 thf(fact_6033_divides__aux__def,axiom,
% 0.34/0.65 ( unique6322359934112328802ux_nat
% 0.34/0.65 = ( ^ [Qr: product_prod_nat_nat] :
% 0.34/0.65 ( ( product_snd_nat_nat @ Qr )
% 0.34/0.65 = zero_zero_nat ) ) ) ).
% 0.34/0.65
% 0.34/0.65 % divides_aux_def
% 0.34/0.65 thf(fact_6034_divides__aux__def,axiom,
% 0.34/0.65 ( unique6319869463603278526ux_int
% 0.34/0.65 = ( ^ [Qr: product_prod_int_int] :
% 0.34/0.65 ( ( product_snd_int_int @ Qr )
% 0.34/0.65 = zero_zero_int ) ) ) ).
% 0.34/0.65
% 0.34/0.65 % divides_aux_def
% 0.34/0.65 thf(fact_6035_divides__aux__def,axiom,
% 0.34/0.65 ( unique5706413561485394159nteger
% 0.34/0.65 = ( ^ [Qr: produc8923325533196201883nteger] :
% 0.34/0.65 ( ( produc6174133586879617921nteger @ Qr )
% 0.34/0.65 = zero_z3403309356797280102nteger ) ) ) ).
% 0.34/0.65
% 0.34/0.65 % divides_aux_def
% 0.34/0.65 thf(fact_6036_mult__less__iff1,axiom,
% 0.34/0.65 ! [Z: int,X: int,Y: int] :
% 0.34/0.65 ( ( ord_less_int @ zero_zero_int @ Z )
% 0.34/0.65 => ( ( ord_less_int @ ( times_times_int @ X @ Z ) @ ( times_times_int @ Y @ Z ) )
% 0.34/0.65 = ( ord_less_int @ X @ Y ) ) ) ).
% 0.34/0.65
% 0.34/0.65 % mult_less_iff1
% 0.34/0.65 thf(fact_6037_mult__less__iff1,axiom,
% 0.34/0.65 ! [Z: code_integer,X: code_integer,Y: code_integer] :
% 0.34/0.65 ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ Z )
% 0.34/0.65 => ( ( ord_le6747313008572928689nteger @ ( times_3573771949741848930nteger @ X @ Z ) @ ( times_3573771949741848930nteger @ Y @ Z ) )
% 0.34/0.65 = ( ord_le6747313008572928689nteger @ X @ Y ) ) ) ).
% 0.34/0.65
% 0.34/0.65 % mult_less_iff1
% 0.34/0.65 thf(fact_6038_mult__less__iff1,axiom,
% 0.34/0.65 ! [Z: rat,X: rat,Y: rat] :
% 0.34/0.65 ( ( ord_less_rat @ zero_zero_rat @ Z )
% 0.34/0.65 => ( ( ord_less_rat @ ( times_times_rat @ X @ Z ) @ ( times_times_rat @ Y @ Z ) )
% 0.34/0.65 = ( ord_less_rat @ X @ Y ) ) ) ).
% 0.34/0.65
% 0.34/0.65 % mult_less_iff1
% 0.34/0.65 thf(fact_6039_sum__squares__eq__zero__iff,axiom,
% 0.34/0.65 ! [X: code_integer,Y: code_integer] :
% 0.34/0.65 ( ( ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ X @ X ) @ ( times_3573771949741848930nteger @ Y @ Y ) )
% 0.34/0.65 = zero_z3403309356797280102nteger )
% 0.34/0.65 = ( ( X = zero_z3403309356797280102nteger )
% 0.34/0.65 & ( Y = zero_z3403309356797280102nteger ) ) ) ).
% 0.34/0.65
% 0.34/0.65 % sum_squares_eq_zero_iff
% 0.34/0.65 thf(fact_6040_sum__squares__eq__zero__iff,axiom,
% 0.34/0.65 ! [X: rat,Y: rat] :
% 0.34/0.65 ( ( ( plus_plus_rat @ ( times_times_rat @ X @ X ) @ ( times_times_rat @ Y @ Y ) )
% 0.34/0.65 = zero_zero_rat )
% 0.34/0.65 = ( ( X = zero_zero_rat )
% 0.34/0.65 & ( Y = zero_zero_rat ) ) ) ).
% 0.34/0.65
% 0.34/0.65 % sum_squares_eq_zero_iff
% 0.34/0.65 thf(fact_6041_sum__squares__eq__zero__iff,axiom,
% 0.34/0.65 ! [X: int,Y: int] :
% 0.34/0.65 ( ( ( plus_plus_int @ ( times_times_int @ X @ X ) @ ( times_times_int @ Y @ Y ) )
% 0.34/0.65 = zero_zero_int )
% 0.34/0.65 = ( ( X = zero_zero_int )
% 0.34/0.65 & ( Y = zero_zero_int ) ) ) ).
% 0.34/0.65
% 0.34/0.65 % sum_squares_eq_zero_iff
% 0.34/0.65 thf(fact_6042_dbl__inc__simps_I2_J,axiom,
% 0.34/0.65 ( ( neg_nu5851722552734809277nc_int @ zero_zero_int )
% 0.34/0.65 = one_one_int ) ).
% 0.34/0.65
% 0.34/0.65 % dbl_inc_simps(2)
% 0.34/0.65 thf(fact_6043_dbl__inc__simps_I2_J,axiom,
% 0.34/0.65 ( ( neg_nu5831290666863070958nteger @ zero_z3403309356797280102nteger )
% 0.34/0.65 = one_one_Code_integer ) ).
% 0.34/0.65
% 0.34/0.65 % dbl_inc_simps(2)
% 0.34/0.65 thf(fact_6044_enumerate__Suc_H,axiom,
% 0.34/0.65 ! [S: set_Product_unit,N2: nat] :
% 0.34/0.65 ( ( infini7930543730640340914t_unit @ S @ ( suc @ N2 ) )
% 0.34/0.65 = ( infini7930543730640340914t_unit @ ( minus_6452836326544984404t_unit @ S @ ( insert_Product_unit @ ( infini7930543730640340914t_unit @ S @ zero_zero_nat ) @ bot_bo3957492148770167129t_unit ) ) @ N2 ) ) ).
% 0.34/0.65
% 0.34/0.65 % enumerate_Suc'
% 0.34/0.65 thf(fact_6045_enumerate__Suc_H,axiom,
% 0.34/0.65 ! [S: set_nat,N2: nat] :
% 0.34/0.65 ( ( infini8530281810654367211te_nat @ S @ ( suc @ N2 ) )
% 0.34/0.65 = ( infini8530281810654367211te_nat @ ( minus_minus_set_nat @ S @ ( insert_nat @ ( infini8530281810654367211te_nat @ S @ zero_zero_nat ) @ bot_bot_set_nat ) ) @ N2 ) ) ).
% 0.34/0.65
% 0.34/0.65 % enumerate_Suc'
% 0.34/0.65 thf(fact_6046_power__decreasing__iff,axiom,
% 0.34/0.65 ! [B: code_natural,M4: nat,N2: nat] :
% 0.34/0.65 ( ( ord_le5570908160329646204atural @ zero_z2226904508553997617atural @ B )
% 0.34/0.65 => ( ( ord_le5570908160329646204atural @ B @ one_one_Code_natural )
% 0.34/0.65 => ( ( ord_le1926595141338095240atural @ ( power_7079662738309270450atural @ B @ M4 ) @ ( power_7079662738309270450atural @ B @ N2 ) )
% 0.34/0.65 = ( ord_less_eq_nat @ N2 @ M4 ) ) ) ) ).
% 0.34/0.65
% 0.34/0.65 % power_decreasing_iff
% 0.34/0.65 thf(fact_6047_power__decreasing__iff,axiom,
% 0.34/0.65 ! [B: code_integer,M4: nat,N2: nat] :
% 0.34/0.65 ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ B )
% 0.34/0.65 => ( ( ord_le6747313008572928689nteger @ B @ one_one_Code_integer )
% 0.34/0.65 => ( ( ord_le3102999989581377725nteger @ ( power_8256067586552552935nteger @ B @ M4 ) @ ( power_8256067586552552935nteger @ B @ N2 ) )
% 0.34/0.65 = ( ord_less_eq_nat @ N2 @ M4 ) ) ) ) ).
% 0.34/0.65
% 0.34/0.65 % power_decreasing_iff
% 0.34/0.65 thf(fact_6048_power__decreasing__iff,axiom,
% 0.34/0.65 ! [B: rat,M4: nat,N2: nat] :
% 0.34/0.65 ( ( ord_less_rat @ zero_zero_rat @ B )
% 0.34/0.65 => ( ( ord_less_rat @ B @ one_one_rat )
% 0.34/0.65 => ( ( ord_less_eq_rat @ ( power_power_rat @ B @ M4 ) @ ( power_power_rat @ B @ N2 ) )
% 0.34/0.65 = ( ord_less_eq_nat @ N2 @ M4 ) ) ) ) ).
% 0.34/0.65
% 0.34/0.65 % power_decreasing_iff
% 0.34/0.65 thf(fact_6049_power__decreasing__iff,axiom,
% 0.34/0.65 ! [B: nat,M4: nat,N2: nat] :
% 0.34/0.65 ( ( ord_less_nat @ zero_zero_nat @ B )
% 0.34/0.65 => ( ( ord_less_nat @ B @ one_one_nat )
% 0.34/0.65 => ( ( ord_less_eq_nat @ ( power_power_nat @ B @ M4 ) @ ( power_power_nat @ B @ N2 ) )
% 0.34/0.65 = ( ord_less_eq_nat @ N2 @ M4 ) ) ) ) ).
% 0.34/0.65
% 0.34/0.65 % power_decreasing_iff
% 0.34/0.65 thf(fact_6050_power__decreasing__iff,axiom,
% 0.34/0.65 ! [B: int,M4: nat,N2: nat] :
% 0.34/0.65 ( ( ord_less_int @ zero_zero_int @ B )
% 0.34/0.65 => ( ( ord_less_int @ B @ one_one_int )
% 0.34/0.65 => ( ( ord_less_eq_int @ ( power_power_int @ B @ M4 ) @ ( power_power_int @ B @ N2 ) )
% 0.34/0.65 = ( ord_less_eq_nat @ N2 @ M4 ) ) ) ) ).
% 0.34/0.65
% 0.34/0.65 % power_decreasing_iff
% 0.34/0.65 thf(fact_6051_dvd__partition,axiom,
% 0.34/0.65 ! [C5: set_set_Code_integer,K3: nat] :
% 0.34/0.65 ( ( finite6017078050557962740nteger @ ( comple739944243200306918nteger @ C5 ) )
% 0.34/0.65 => ( ! [X3: set_Code_integer] :
% 0.34/0.65 ( ( member1834298987299537450nteger @ X3 @ C5 )
% 0.34/0.65 => ( dvd_dvd_nat @ K3 @ ( finite4902975817058060853nteger @ X3 ) ) )
% 0.34/0.65 => ( ! [X3: set_Code_integer] :
% 0.34/0.65 ( ( member1834298987299537450nteger @ X3 @ C5 )
% 0.34/0.65 => ! [Xa3: set_Code_integer] :
% 0.34/0.65 ( ( member1834298987299537450nteger @ Xa3 @ C5 )
% 0.34/0.65 => ( ( X3 != Xa3 )
% 0.34/0.65 => ( ( inf_in1364745209274528805nteger @ X3 @ Xa3 )
% 0.34/0.65 = bot_bo3990330152332043303nteger ) ) ) )
% 0.34/0.65 => ( dvd_dvd_nat @ K3 @ ( finite4902975817058060853nteger @ ( comple739944243200306918nteger @ C5 ) ) ) ) ) ) ).
% 0.34/0.65
% 0.34/0.65 % dvd_partition
% 0.34/0.65 thf(fact_6052_dvd__partition,axiom,
% 0.34/0.65 ! [C5: set_set_Product_unit,K3: nat] :
% 0.34/0.65 ( ( finite4290736615968046902t_unit @ ( comple3349607820504768346t_unit @ C5 ) )
% 0.34/0.65 => ( ! [X3: set_Product_unit] :
% 0.34/0.65 ( ( member5877623283571906838t_unit @ X3 @ C5 )
% 0.34/0.65 => ( dvd_dvd_nat @ K3 @ ( finite410649719033368117t_unit @ X3 ) ) )
% 0.34/0.65 => ( ! [X3: set_Product_unit] :
% 0.34/0.65 ( ( member5877623283571906838t_unit @ X3 @ C5 )
% 0.34/0.65 => ! [Xa3: set_Product_unit] :
% 0.34/0.65 ( ( member5877623283571906838t_unit @ Xa3 @ C5 )
% 0.34/0.65 => ( ( X3 != Xa3 )
% 0.34/0.65 => ( ( inf_in4660618365625256667t_unit @ X3 @ Xa3 )
% 0.34/0.65 = bot_bo3957492148770167129t_unit ) ) ) )
% 0.34/0.65 => ( dvd_dvd_nat @ K3 @ ( finite410649719033368117t_unit @ ( comple3349607820504768346t_unit @ C5 ) ) ) ) ) ) ).
% 0.34/0.65
% 0.34/0.65 % dvd_partition
% 0.34/0.65 thf(fact_6053_dvd__partition,axiom,
% 0.34/0.65 ! [C5: set_set_o,K3: nat] :
% 0.34/0.65 ( ( finite_finite_o @ ( comple90263536869209701_set_o @ C5 ) )
% 0.34/0.65 => ( ! [X3: set_o] :
% 0.34/0.65 ( ( member_set_o @ X3 @ C5 )
% 0.34/0.65 => ( dvd_dvd_nat @ K3 @ ( finite_card_o @ X3 ) ) )
% 0.34/0.65 => ( ! [X3: set_o] :
% 0.34/0.65 ( ( member_set_o @ X3 @ C5 )
% 0.34/0.65 => ! [Xa3: set_o] :
% 0.34/0.65 ( ( member_set_o @ Xa3 @ C5 )
% 0.34/0.65 => ( ( X3 != Xa3 )
% 0.34/0.65 => ( ( inf_inf_set_o @ X3 @ Xa3 )
% 0.34/0.65 = bot_bot_set_o ) ) ) )
% 0.34/0.65 => ( dvd_dvd_nat @ K3 @ ( finite_card_o @ ( comple90263536869209701_set_o @ C5 ) ) ) ) ) ) ).
% 0.34/0.65
% 0.34/0.65 % dvd_partition
% 0.34/0.65 thf(fact_6054_dvd__partition,axiom,
% 0.34/0.65 ! [C5: set_set_nat,K3: nat] :
% 0.34/0.65 ( ( finite_finite_nat @ ( comple7399068483239264473et_nat @ C5 ) )
% 0.34/0.65 => ( ! [X3: set_nat] :
% 0.34/0.65 ( ( member_set_nat @ X3 @ C5 )
% 0.34/0.65 => ( dvd_dvd_nat @ K3 @ ( finite_card_nat @ X3 ) ) )
% 0.34/0.65 => ( ! [X3: set_nat] :
% 0.34/0.65 ( ( member_set_nat @ X3 @ C5 )
% 0.34/0.65 => ! [Xa3: set_nat] :
% 0.34/0.65 ( ( member_set_nat @ Xa3 @ C5 )
% 0.34/0.65 => ( ( X3 != Xa3 )
% 0.34/0.65 => ( ( inf_inf_set_nat @ X3 @ Xa3 )
% 0.34/0.65 = bot_bot_set_nat ) ) ) )
% 0.34/0.65 => ( dvd_dvd_nat @ K3 @ ( finite_card_nat @ ( comple7399068483239264473et_nat @ C5 ) ) ) ) ) ) ).
% 0.34/0.65
% 0.34/0.65 % dvd_partition
% 0.34/0.65 thf(fact_6055_dvd__partition,axiom,
% 0.34/0.65 ! [C5: set_set_int,K3: nat] :
% 0.34/0.65 ( ( finite_finite_int @ ( comple3221217463730067765et_int @ C5 ) )
% 0.34/0.65 => ( ! [X3: set_int] :
% 0.34/0.65 ( ( member_set_int @ X3 @ C5 )
% 0.34/0.65 => ( dvd_dvd_nat @ K3 @ ( finite_card_int @ X3 ) ) )
% 0.34/0.65 => ( ! [X3: set_int] :
% 0.34/0.65 ( ( member_set_int @ X3 @ C5 )
% 0.34/0.65 => ! [Xa3: set_int] :
% 0.34/0.65 ( ( member_set_int @ Xa3 @ C5 )
% 0.34/0.65 => ( ( X3 != Xa3 )
% 0.34/0.65 => ( ( inf_inf_set_int @ X3 @ Xa3 )
% 0.34/0.65 = bot_bot_set_int ) ) ) )
% 0.34/0.65 => ( dvd_dvd_nat @ K3 @ ( finite_card_int @ ( comple3221217463730067765et_int @ C5 ) ) ) ) ) ) ).
% 0.34/0.65
% 0.34/0.65 % dvd_partition
% 0.34/0.65 thf(fact_6056_dvd__partition,axiom,
% 0.34/0.65 ! [C5: set_se1353816476878259085_nat_o,K3: nat] :
% 0.34/0.65 ( ( finite3252695134891459830_nat_o @ ( comple3382087578465172250_nat_o @ C5 ) )
% 0.34/0.65 => ( ! [X3: set_Pr4532377907799695533_nat_o] :
% 0.34/0.65 ( ( member1452989533377600726_nat_o @ X3 @ C5 )
% 0.34/0.65 => ( dvd_dvd_nat @ K3 @ ( finite1363419556375932405_nat_o @ X3 ) ) )
% 0.34/0.65 => ( ! [X3: set_Pr4532377907799695533_nat_o] :
% 0.34/0.65 ( ( member1452989533377600726_nat_o @ X3 @ C5 )
% 0.34/0.65 => ! [Xa3: set_Pr4532377907799695533_nat_o] :
% 0.34/0.65 ( ( member1452989533377600726_nat_o @ Xa3 @ C5 )
% 0.34/0.65 => ( ( X3 != Xa3 )
% 0.34/0.65 => ( ( inf_in1906310914598751387_nat_o @ X3 @ Xa3 )
% 0.34/0.65 = bot_bo7824918357723582233_nat_o ) ) ) )
% 0.34/0.65 => ( dvd_dvd_nat @ K3 @ ( finite1363419556375932405_nat_o @ ( comple3382087578465172250_nat_o @ C5 ) ) ) ) ) ) ).
% 0.34/0.65
% 0.34/0.65 % dvd_partition
% 0.34/0.65 thf(fact_6057_cInf__le__cSup,axiom,
% 0.34/0.65 ! [A: set_o] :
% 0.34/0.65 ( ( A != bot_bot_set_o )
% 0.34/0.65 => ( ( condit5488710616941104124bove_o @ A )
% 0.34/0.65 => ( ( condit5413489452508810728elow_o @ A )
% 0.34/0.65 => ( ord_less_eq_o @ ( complete_Inf_Inf_o @ A ) @ ( complete_Sup_Sup_o @ A ) ) ) ) ) ).
% 0.34/0.65
% 0.34/0.65 % cInf_le_cSup
% 0.34/0.65 thf(fact_6058_cInf__le__cSup,axiom,
% 0.34/0.65 ! [A: set_Pr4532377907799695533_nat_o] :
% 0.34/0.65 ( ( A != bot_bo7824918357723582233_nat_o )
% 0.34/0.65 => ( ( condit4352657709669162033_nat_o @ A )
% 0.34/0.65 => ( ( condit8399017556247434269_nat_o @ A )
% 0.34/0.65 => ( ord_le729326519192465773_nat_o @ ( comple90455603224345172_nat_o @ A ) @ ( comple4225350640907012154_nat_o @ A ) ) ) ) ) ).
% 0.34/0.65
% 0.34/0.65 % cInf_le_cSup
% 0.34/0.65 thf(fact_6059_cInf__le__cSup,axiom,
% 0.34/0.65 ! [A: set_filter_nat] :
% 0.34/0.65 ( ( A != bot_bo498966703094740906er_nat )
% 0.34/0.65 => ( ( condit4029664723990530684er_nat @ A )
% 0.34/0.65 => ( ( condit3133331395780078992er_nat @ A )
% 0.34/0.65 => ( ord_le2510731241096832064er_nat @ ( comple6612786467464916249er_nat @ A ) @ ( comple2454491259644266675er_nat @ A ) ) ) ) ) ).
% 0.34/0.65
% 0.34/0.65 % cInf_le_cSup
% 0.34/0.65 thf(fact_6060_cInf__le__cSup,axiom,
% 0.34/0.65 ! [A: set_int] :
% 0.34/0.65 ( ( A != bot_bot_set_int )
% 0.34/0.65 => ( ( condit2212336002400062152ve_int @ A )
% 0.34/0.65 => ( ( condit1735850657277959132ow_int @ A )
% 0.34/0.65 => ( ord_less_eq_int @ ( complete_Inf_Inf_int @ A ) @ ( complete_Sup_Sup_int @ A ) ) ) ) ) ).
% 0.34/0.65
% 0.34/0.65 % cInf_le_cSup
% 0.34/0.65 thf(fact_6061_cInf__le__cSup,axiom,
% 0.34/0.65 ! [A: set_set_nat] :
% 0.34/0.65 ( ( A != bot_bot_set_set_nat )
% 0.34/0.65 => ( ( condit5477540289124974626et_nat @ A )
% 0.34/0.65 => ( ( condit68592940725977398et_nat @ A )
% 0.34/0.65 => ( ord_less_eq_set_nat @ ( comple7806235888213564991et_nat @ A ) @ ( comple7399068483239264473et_nat @ A ) ) ) ) ) ).
% 0.34/0.65
% 0.34/0.65 % cInf_le_cSup
% 0.34/0.65 thf(fact_6062_cInf__le__cSup,axiom,
% 0.34/0.65 ! [A: set_nat] :
% 0.34/0.65 ( ( A != bot_bot_set_nat )
% 0.34/0.65 => ( ( condit2214826472909112428ve_nat @ A )
% 0.34/0.65 => ( ( condit1738341127787009408ow_nat @ A )
% 0.34/0.65 => ( ord_less_eq_nat @ ( complete_Inf_Inf_nat @ A ) @ ( complete_Sup_Sup_nat @ A ) ) ) ) ) ).
% 0.34/0.65
% 0.34/0.65 % cInf_le_cSup
% 0.34/0.65 thf(fact_6063_cInf__le__cSup,axiom,
% 0.34/0.65 ! [A: set_Product_unit] :
% 0.34/0.65 ( ( A != bot_bo3957492148770167129t_unit )
% 0.34/0.65 => ( ( condit2080475194861589617t_unit @ A )
% 0.34/0.65 => ( ( condit9161856309968323677t_unit @ A )
% 0.34/0.65 => ( ord_le3221252021190050221t_unit @ ( comple2584293577114468500t_unit @ A ) @ ( comple4687483117567863418t_unit @ A ) ) ) ) ) ).
% 0.34/0.65
% 0.34/0.65 % cInf_le_cSup
% 0.34/0.65 thf(fact_6064_sum__le__card__Max,axiom,
% 0.34/0.65 ! [A: set_Code_integer,F: code_integer > nat] :
% 0.34/0.65 ( ( finite6017078050557962740nteger @ A )
% 0.34/0.65 => ( ord_less_eq_nat @ ( groups7237345082560585321er_nat @ F @ A ) @ ( times_times_nat @ ( finite4902975817058060853nteger @ A ) @ ( lattic8265883725875713057ax_nat @ ( image_951025933927791156er_nat @ F @ A ) ) ) ) ) ).
% 0.34/0.65
% 0.34/0.65 % sum_le_card_Max
% 0.34/0.65 thf(fact_6065_sum__le__card__Max,axiom,
% 0.34/0.65 ! [A: set_nat,F: nat > nat] :
% 0.34/0.65 ( ( finite_finite_nat @ A )
% 0.34/0.65 => ( ord_less_eq_nat @ ( groups3542108847815614940at_nat @ F @ A ) @ ( times_times_nat @ ( finite_card_nat @ A ) @ ( lattic8265883725875713057ax_nat @ ( image_nat_nat2 @ F @ A ) ) ) ) ) ).
% 0.34/0.65
% 0.34/0.65 % sum_le_card_Max
% 0.34/0.65 thf(fact_6066_gbinomial__reduce__nat,axiom,
% 0.34/0.65 ! [K3: nat,A2: rat] :
% 0.34/0.65 ( ( ord_less_nat @ zero_zero_nat @ K3 )
% 0.34/0.65 => ( ( gbinomial_rat @ A2 @ K3 )
% 0.34/0.65 = ( plus_plus_rat @ ( gbinomial_rat @ ( minus_minus_rat @ A2 @ one_one_rat ) @ ( minus_minus_nat @ K3 @ one_one_nat ) ) @ ( gbinomial_rat @ ( minus_minus_rat @ A2 @ one_one_rat ) @ K3 ) ) ) ) ).
% 0.34/0.65
% 0.34/0.65 % gbinomial_reduce_nat
% 0.34/0.65 thf(fact_6067_power__one,axiom,
% 0.34/0.65 ! [N2: nat] :
% 0.34/0.65 ( ( power_power_assn @ one_one_assn @ N2 )
% 0.34/0.65 = one_one_assn ) ).
% 0.34/0.65
% 0.34/0.65 % power_one
% 0.34/0.65 thf(fact_6068_power__one,axiom,
% 0.34/0.65 ! [N2: nat] :
% 0.34/0.65 ( ( power_8256067586552552935nteger @ one_one_Code_integer @ N2 )
% 0.34/0.65 = one_one_Code_integer ) ).
% 0.34/0.65
% 0.34/0.65 % power_one
% 0.34/0.65 thf(fact_6069_power__one,axiom,
% 0.34/0.65 ! [N2: nat] :
% 0.34/0.65 ( ( power_7079662738309270450atural @ one_one_Code_natural @ N2 )
% 0.34/0.65 = one_one_Code_natural ) ).
% 0.34/0.65
% 0.34/0.65 % power_one
% 0.34/0.65 thf(fact_6070_power__one,axiom,
% 0.34/0.65 ! [N2: nat] :
% 0.34/0.65 ( ( power_power_int @ one_one_int @ N2 )
% 0.34/0.65 = one_one_int ) ).
% 0.34/0.65
% 0.34/0.65 % power_one
% 0.34/0.65 thf(fact_6071_power__one,axiom,
% 0.34/0.65 ! [N2: nat] :
% 0.34/0.65 ( ( power_power_nat @ one_one_nat @ N2 )
% 0.34/0.65 = one_one_nat ) ).
% 0.34/0.65
% 0.34/0.65 % power_one
% 0.34/0.65 thf(fact_6072_bdd__below__empty,axiom,
% 0.34/0.65 condit9161856309968323677t_unit @ bot_bo3957492148770167129t_unit ).
% 0.34/0.65
% 0.34/0.65 % bdd_below_empty
% 0.34/0.65 thf(fact_6073_bdd__below__empty,axiom,
% 0.34/0.65 condit5413489452508810728elow_o @ bot_bot_set_o ).
% 0.34/0.65
% 0.34/0.65 % bdd_below_empty
% 0.34/0.65 thf(fact_6074_bdd__below__empty,axiom,
% 0.34/0.65 condit1738341127787009408ow_nat @ bot_bot_set_nat ).
% 0.34/0.65
% 0.34/0.65 % bdd_below_empty
% 0.34/0.65 thf(fact_6075_bdd__below__empty,axiom,
% 0.34/0.65 condit1735850657277959132ow_int @ bot_bot_set_int ).
% 0.34/0.65
% 0.34/0.65 % bdd_below_empty
% 0.34/0.65 thf(fact_6076_bdd__below__empty,axiom,
% 0.34/0.65 condit8399017556247434269_nat_o @ bot_bo7824918357723582233_nat_o ).
% 0.34/0.65
% 0.34/0.65 % bdd_below_empty
% 0.34/0.65 thf(fact_6077_bdd__below__insert,axiom,
% 0.34/0.65 ! [A2: nat,A: set_nat] :
% 0.34/0.65 ( ( condit1738341127787009408ow_nat @ ( insert_nat @ A2 @ A ) )
% 0.34/0.65 = ( condit1738341127787009408ow_nat @ A ) ) ).
% 0.34/0.65
% 0.34/0.65 % bdd_below_insert
% 0.34/0.65 thf(fact_6078_bdd__below__insert,axiom,
% 0.34/0.65 ! [A2: int,A: set_int] :
% 0.34/0.65 ( ( condit1735850657277959132ow_int @ ( insert_int @ A2 @ A ) )
% 0.34/0.65 = ( condit1735850657277959132ow_int @ A ) ) ).
% 0.34/0.65
% 0.34/0.65 % bdd_below_insert
% 0.34/0.65 thf(fact_6079_bdd__below__insert,axiom,
% 0.34/0.65 ! [A2: $o,A: set_o] :
% 0.34/0.65 ( ( condit5413489452508810728elow_o @ ( insert_o @ A2 @ A ) )
% 0.34/0.65 = ( condit5413489452508810728elow_o @ A ) ) ).
% 0.34/0.65
% 0.34/0.65 % bdd_below_insert
% 0.34/0.65 thf(fact_6080_bdd__below__insert,axiom,
% 0.34/0.65 ! [A2: product_unit,A: set_Product_unit] :
% 0.34/0.65 ( ( condit9161856309968323677t_unit @ ( insert_Product_unit @ A2 @ A ) )
% 0.34/0.65 = ( condit9161856309968323677t_unit @ A ) ) ).
% 0.34/0.65
% 0.34/0.65 % bdd_below_insert
% 0.34/0.65 thf(fact_6081_dvd__times__right__cancel__iff,axiom,
% 0.34/0.65 ! [A2: code_integer,B: code_integer,C: code_integer] :
% 0.34/0.65 ( ( A2 != zero_z3403309356797280102nteger )
% 0.34/0.65 => ( ( dvd_dvd_Code_integer @ ( times_3573771949741848930nteger @ B @ A2 ) @ ( times_3573771949741848930nteger @ C @ A2 ) )
% 0.34/0.65 = ( dvd_dvd_Code_integer @ B @ C ) ) ) ).
% 0.34/0.65
% 0.34/0.65 % dvd_times_right_cancel_iff
% 0.34/0.65 thf(fact_6082_dvd__times__right__cancel__iff,axiom,
% 0.34/0.65 ! [A2: nat,B: nat,C: nat] :
% 0.34/0.65 ( ( A2 != zero_zero_nat )
% 0.34/0.65 => ( ( dvd_dvd_nat @ ( times_times_nat @ B @ A2 ) @ ( times_times_nat @ C @ A2 ) )
% 0.34/0.65 = ( dvd_dvd_nat @ B @ C ) ) ) ).
% 0.34/0.65
% 0.34/0.65 % dvd_times_right_cancel_iff
% 0.34/0.65 thf(fact_6083_dvd__times__right__cancel__iff,axiom,
% 0.34/0.65 ! [A2: int,B: int,C: int] :
% 0.34/0.65 ( ( A2 != zero_zero_int )
% 0.34/0.65 => ( ( dvd_dvd_int @ ( times_times_int @ B @ A2 ) @ ( times_times_int @ C @ A2 ) )
% 0.34/0.65 = ( dvd_dvd_int @ B @ C ) ) ) ).
% 0.34/0.65
% 0.34/0.65 % dvd_times_right_cancel_iff
% 0.34/0.65 thf(fact_6084_dvd__times__left__cancel__iff,axiom,
% 0.34/0.65 ! [A2: code_integer,B: code_integer,C: code_integer] :
% 0.34/0.65 ( ( A2 != zero_z3403309356797280102nteger )
% 0.34/0.65 => ( ( dvd_dvd_Code_integer @ ( times_3573771949741848930nteger @ A2 @ B ) @ ( times_3573771949741848930nteger @ A2 @ C ) )
% 0.34/0.65 = ( dvd_dvd_Code_integer @ B @ C ) ) ) ).
% 0.34/0.65
% 0.34/0.65 % dvd_times_left_cancel_iff
% 0.34/0.65 thf(fact_6085_dvd__times__left__cancel__iff,axiom,
% 0.34/0.65 ! [A2: nat,B: nat,C: nat] :
% 0.34/0.65 ( ( A2 != zero_zero_nat )
% 0.34/0.65 => ( ( dvd_dvd_nat @ ( times_times_nat @ A2 @ B ) @ ( times_times_nat @ A2 @ C ) )
% 0.34/0.65 = ( dvd_dvd_nat @ B @ C ) ) ) ).
% 0.34/0.65
% 0.34/0.65 % dvd_times_left_cancel_iff
% 0.34/0.65 thf(fact_6086_dvd__times__left__cancel__iff,axiom,
% 0.34/0.65 ! [A2: int,B: int,C: int] :
% 0.34/0.65 ( ( A2 != zero_zero_int )
% 0.34/0.65 => ( ( dvd_dvd_int @ ( times_times_int @ A2 @ B ) @ ( times_times_int @ A2 @ C ) )
% 0.34/0.65 = ( dvd_dvd_int @ B @ C ) ) ) ).
% 0.34/0.65
% 0.34/0.65 % dvd_times_left_cancel_iff
% 0.34/0.65 thf(fact_6087_dvd__mult__cancel__right,axiom,
% 0.34/0.65 ! [A2: code_integer,C: code_integer,B: code_integer] :
% 0.34/0.65 ( ( dvd_dvd_Code_integer @ ( times_3573771949741848930nteger @ A2 @ C ) @ ( times_3573771949741848930nteger @ B @ C ) )
% 0.34/0.65 = ( ( C = zero_z3403309356797280102nteger )
% 0.34/0.65 | ( dvd_dvd_Code_integer @ A2 @ B ) ) ) ).
% 0.34/0.65
% 0.34/0.65 % dvd_mult_cancel_right
% 0.34/0.65 thf(fact_6088_dvd__mult__cancel__right,axiom,
% 0.34/0.65 ! [A2: rat,C: rat,B: rat] :
% 0.34/0.65 ( ( dvd_dvd_rat @ ( times_times_rat @ A2 @ C ) @ ( times_times_rat @ B @ C ) )
% 0.34/0.65 = ( ( C = zero_zero_rat )
% 0.34/0.65 | ( dvd_dvd_rat @ A2 @ B ) ) ) ).
% 0.34/0.65
% 0.34/0.65 % dvd_mult_cancel_right
% 0.34/0.65 thf(fact_6089_dvd__mult__cancel__right,axiom,
% 0.34/0.65 ! [A2: int,C: int,B: int] :
% 0.34/0.65 ( ( dvd_dvd_int @ ( times_times_int @ A2 @ C ) @ ( times_times_int @ B @ C ) )
% 0.34/0.65 = ( ( C = zero_zero_int )
% 0.34/0.65 | ( dvd_dvd_int @ A2 @ B ) ) ) ).
% 0.34/0.65
% 0.34/0.65 % dvd_mult_cancel_right
% 0.34/0.65 thf(fact_6090_dvd__mult__cancel__left,axiom,
% 0.34/0.65 ! [C: code_integer,A2: code_integer,B: code_integer] :
% 0.34/0.65 ( ( dvd_dvd_Code_integer @ ( times_3573771949741848930nteger @ C @ A2 ) @ ( times_3573771949741848930nteger @ C @ B ) )
% 0.34/0.65 = ( ( C = zero_z3403309356797280102nteger )
% 0.34/0.65 | ( dvd_dvd_Code_integer @ A2 @ B ) ) ) ).
% 0.34/0.65
% 0.34/0.65 % dvd_mult_cancel_left
% 0.34/0.65 thf(fact_6091_dvd__mult__cancel__left,axiom,
% 0.34/0.65 ! [C: rat,A2: rat,B: rat] :
% 0.34/0.65 ( ( dvd_dvd_rat @ ( times_times_rat @ C @ A2 ) @ ( times_times_rat @ C @ B ) )
% 0.34/0.65 = ( ( C = zero_zero_rat )
% 0.34/0.65 | ( dvd_dvd_rat @ A2 @ B ) ) ) ).
% 0.34/0.65
% 0.34/0.65 % dvd_mult_cancel_left
% 0.34/0.65 thf(fact_6092_dvd__mult__cancel__left,axiom,
% 0.34/0.65 ! [C: int,A2: int,B: int] :
% 0.34/0.65 ( ( dvd_dvd_int @ ( times_times_int @ C @ A2 ) @ ( times_times_int @ C @ B ) )
% 0.34/0.65 = ( ( C = zero_zero_int )
% 0.34/0.65 | ( dvd_dvd_int @ A2 @ B ) ) ) ).
% 0.34/0.65
% 0.34/0.65 % dvd_mult_cancel_left
% 0.34/0.65 thf(fact_6093_dvd__add__times__triv__left__iff,axiom,
% 0.34/0.65 ! [A2: code_integer,C: code_integer,B: code_integer] :
% 0.34/0.65 ( ( dvd_dvd_Code_integer @ A2 @ ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ C @ A2 ) @ B ) )
% 0.34/0.65 = ( dvd_dvd_Code_integer @ A2 @ B ) ) ).
% 0.34/0.65
% 0.34/0.65 % dvd_add_times_triv_left_iff
% 0.34/0.65 thf(fact_6094_dvd__add__times__triv__left__iff,axiom,
% 0.34/0.65 ! [A2: rat,C: rat,B: rat] :
% 0.34/0.65 ( ( dvd_dvd_rat @ A2 @ ( plus_plus_rat @ ( times_times_rat @ C @ A2 ) @ B ) )
% 0.34/0.65 = ( dvd_dvd_rat @ A2 @ B ) ) ).
% 0.34/0.65
% 0.34/0.65 % dvd_add_times_triv_left_iff
% 0.34/0.65 thf(fact_6095_dvd__add__times__triv__left__iff,axiom,
% 0.34/0.65 ! [A2: nat,C: nat,B: nat] :
% 0.34/0.65 ( ( dvd_dvd_nat @ A2 @ ( plus_plus_nat @ ( times_times_nat @ C @ A2 ) @ B ) )
% 0.34/0.65 = ( dvd_dvd_nat @ A2 @ B ) ) ).
% 0.34/0.65
% 0.34/0.65 % dvd_add_times_triv_left_iff
% 0.34/0.65 thf(fact_6096_dvd__add__times__triv__left__iff,axiom,
% 0.34/0.65 ! [A2: int,C: int,B: int] :
% 0.34/0.65 ( ( dvd_dvd_int @ A2 @ ( plus_plus_int @ ( times_times_int @ C @ A2 ) @ B ) )
% 0.34/0.65 = ( dvd_dvd_int @ A2 @ B ) ) ).
% 0.34/0.65
% 0.34/0.65 % dvd_add_times_triv_left_iff
% 0.34/0.65 thf(fact_6097_dvd__add__times__triv__right__iff,axiom,
% 0.34/0.65 ! [A2: code_integer,B: code_integer,C: code_integer] :
% 0.34/0.65 ( ( dvd_dvd_Code_integer @ A2 @ ( plus_p5714425477246183910nteger @ B @ ( times_3573771949741848930nteger @ C @ A2 ) ) )
% 0.34/0.65 = ( dvd_dvd_Code_integer @ A2 @ B ) ) ).
% 0.34/0.65
% 0.34/0.65 % dvd_add_times_triv_right_iff
% 0.34/0.65 thf(fact_6098_dvd__add__times__triv__right__iff,axiom,
% 0.34/0.65 ! [A2: rat,B: rat,C: rat] :
% 0.34/0.65 ( ( dvd_dvd_rat @ A2 @ ( plus_plus_rat @ B @ ( times_times_rat @ C @ A2 ) ) )
% 0.34/0.65 = ( dvd_dvd_rat @ A2 @ B ) ) ).
% 0.34/0.65
% 0.34/0.65 % dvd_add_times_triv_right_iff
% 0.34/0.65 thf(fact_6099_dvd__add__times__triv__right__iff,axiom,
% 0.34/0.65 ! [A2: nat,B: nat,C: nat] :
% 0.34/0.65 ( ( dvd_dvd_nat @ A2 @ ( plus_plus_nat @ B @ ( times_times_nat @ C @ A2 ) ) )
% 0.34/0.65 = ( dvd_dvd_nat @ A2 @ B ) ) ).
% 0.34/0.65
% 0.34/0.65 % dvd_add_times_triv_right_iff
% 0.34/0.65 thf(fact_6100_dvd__add__times__triv__right__iff,axiom,
% 0.34/0.65 ! [A2: int,B: int,C: int] :
% 0.34/0.65 ( ( dvd_dvd_int @ A2 @ ( plus_plus_int @ B @ ( times_times_int @ C @ A2 ) ) )
% 0.34/0.65 = ( dvd_dvd_int @ A2 @ B ) ) ).
% 0.34/0.65
% 0.34/0.65 % dvd_add_times_triv_right_iff
% 0.34/0.65 thf(fact_6101_unit__prod,axiom,
% 0.34/0.65 ! [A2: code_integer,B: code_integer] :
% 0.34/0.65 ( ( dvd_dvd_Code_integer @ A2 @ one_one_Code_integer )
% 0.34/0.65 => ( ( dvd_dvd_Code_integer @ B @ one_one_Code_integer )
% 0.34/0.65 => ( dvd_dvd_Code_integer @ ( times_3573771949741848930nteger @ A2 @ B ) @ one_one_Code_integer ) ) ) ).
% 0.34/0.65
% 0.34/0.65 % unit_prod
% 0.34/0.65 thf(fact_6102_unit__prod,axiom,
% 0.34/0.65 ! [A2: code_natural,B: code_natural] :
% 0.34/0.65 ( ( dvd_dvd_Code_natural @ A2 @ one_one_Code_natural )
% 0.34/0.65 => ( ( dvd_dvd_Code_natural @ B @ one_one_Code_natural )
% 0.34/0.65 => ( dvd_dvd_Code_natural @ ( times_2397367101498566445atural @ A2 @ B ) @ one_one_Code_natural ) ) ) ).
% 0.34/0.65
% 0.34/0.65 % unit_prod
% 0.34/0.65 thf(fact_6103_unit__prod,axiom,
% 0.34/0.65 ! [A2: nat,B: nat] :
% 0.34/0.65 ( ( dvd_dvd_nat @ A2 @ one_one_nat )
% 0.34/0.65 => ( ( dvd_dvd_nat @ B @ one_one_nat )
% 0.34/0.65 => ( dvd_dvd_nat @ ( times_times_nat @ A2 @ B ) @ one_one_nat ) ) ) ).
% 0.34/0.65
% 0.34/0.65 % unit_prod
% 0.34/0.65 thf(fact_6104_unit__prod,axiom,
% 0.34/0.65 ! [A2: int,B: int] :
% 0.34/0.65 ( ( dvd_dvd_int @ A2 @ one_one_int )
% 0.34/0.65 => ( ( dvd_dvd_int @ B @ one_one_int )
% 0.34/0.65 => ( dvd_dvd_int @ ( times_times_int @ A2 @ B ) @ one_one_int ) ) ) ).
% 0.34/0.65
% 0.34/0.65 % unit_prod
% 0.34/0.65 thf(fact_6105_power__inject__exp,axiom,
% 0.34/0.65 ! [A2: code_natural,M4: nat,N2: nat] :
% 0.34/0.65 ( ( ord_le5570908160329646204atural @ one_one_Code_natural @ A2 )
% 0.34/0.65 => ( ( ( power_7079662738309270450atural @ A2 @ M4 )
% 0.34/0.65 = ( power_7079662738309270450atural @ A2 @ N2 ) )
% 0.34/0.65 = ( M4 = N2 ) ) ) ).
% 0.34/0.65
% 0.34/0.65 % power_inject_exp
% 0.34/0.65 thf(fact_6106_power__inject__exp,axiom,
% 0.34/0.65 ! [A2: nat,M4: nat,N2: nat] :
% 0.34/0.65 ( ( ord_less_nat @ one_one_nat @ A2 )
% 0.34/0.65 => ( ( ( power_power_nat @ A2 @ M4 )
% 0.34/0.65 = ( power_power_nat @ A2 @ N2 ) )
% 0.34/0.65 = ( M4 = N2 ) ) ) ).
% 0.34/0.65
% 0.34/0.65 % power_inject_exp
% 0.34/0.65 thf(fact_6107_power__inject__exp,axiom,
% 0.34/0.65 ! [A2: int,M4: nat,N2: nat] :
% 0.34/0.65 ( ( ord_less_int @ one_one_int @ A2 )
% 0.34/0.65 => ( ( ( power_power_int @ A2 @ M4 )
% 0.34/0.65 = ( power_power_int @ A2 @ N2 ) )
% 0.34/0.65 = ( M4 = N2 ) ) ) ).
% 0.34/0.65
% 0.34/0.65 % power_inject_exp
% 0.34/0.65 thf(fact_6108_power__inject__exp,axiom,
% 0.34/0.65 ! [A2: code_integer,M4: nat,N2: nat] :
% 0.34/0.65 ( ( ord_le6747313008572928689nteger @ one_one_Code_integer @ A2 )
% 0.34/0.65 => ( ( ( power_8256067586552552935nteger @ A2 @ M4 )
% 0.34/0.65 = ( power_8256067586552552935nteger @ A2 @ N2 ) )
% 0.34/0.65 = ( M4 = N2 ) ) ) ).
% 0.34/0.65
% 0.34/0.65 % power_inject_exp
% 0.34/0.65 thf(fact_6109_power__inject__exp,axiom,
% 0.34/0.65 ! [A2: rat,M4: nat,N2: nat] :
% 0.34/0.65 ( ( ord_less_rat @ one_one_rat @ A2 )
% 0.34/0.65 => ( ( ( power_power_rat @ A2 @ M4 )
% 0.34/0.65 = ( power_power_rat @ A2 @ N2 ) )
% 0.34/0.65 = ( M4 = N2 ) ) ) ).
% 0.34/0.65
% 0.34/0.65 % power_inject_exp
% 0.34/0.65 thf(fact_6110_nat__mult__dvd__cancel__disj,axiom,
% 0.34/0.65 ! [K3: nat,M4: nat,N2: nat] :
% 0.34/0.65 ( ( dvd_dvd_nat @ ( times_times_nat @ K3 @ M4 ) @ ( times_times_nat @ K3 @ N2 ) )
% 0.34/0.65 = ( ( K3 = zero_zero_nat )
% 0.34/0.65 | ( dvd_dvd_nat @ M4 @ N2 ) ) ) ).
% 0.34/0.65
% 0.34/0.65 % nat_mult_dvd_cancel_disj
% 0.34/0.65 thf(fact_6111_gbinomial__0_I1_J,axiom,
% 0.34/0.65 ! [A2: nat] :
% 0.34/0.65 ( ( gbinomial_nat @ A2 @ zero_zero_nat )
% 0.34/0.65 = one_one_nat ) ).
% 0.34/0.65
% 0.34/0.65 % gbinomial_0(1)
% 0.34/0.65 thf(fact_6112_gbinomial__0_I1_J,axiom,
% 0.34/0.65 ! [A2: int] :
% 0.34/0.65 ( ( gbinomial_int @ A2 @ zero_zero_nat )
% 0.34/0.65 = one_one_int ) ).
% 0.34/0.65
% 0.34/0.65 % gbinomial_0(1)
% 0.34/0.65 thf(fact_6113_gbinomial__0_I1_J,axiom,
% 0.34/0.65 ! [A2: code_integer] :
% 0.34/0.65 ( ( gbinom8545251970709558553nteger @ A2 @ zero_zero_nat )
% 0.34/0.65 = one_one_Code_integer ) ).
% 0.34/0.65
% 0.34/0.65 % gbinomial_0(1)
% 0.34/0.65 thf(fact_6114_gbinomial__0_I1_J,axiom,
% 0.34/0.65 ! [A2: code_natural] :
% 0.34/0.65 ( ( gbinom7368847122466276068atural @ A2 @ zero_zero_nat )
% 0.34/0.65 = one_one_Code_natural ) ).
% 0.34/0.65
% 0.34/0.65 % gbinomial_0(1)
% 0.34/0.65 thf(fact_6115_power__strict__increasing__iff,axiom,
% 0.34/0.65 ! [B: code_natural,X: nat,Y: nat] :
% 0.34/0.65 ( ( ord_le5570908160329646204atural @ one_one_Code_natural @ B )
% 0.34/0.65 => ( ( ord_le5570908160329646204atural @ ( power_7079662738309270450atural @ B @ X ) @ ( power_7079662738309270450atural @ B @ Y ) )
% 0.34/0.65 = ( ord_less_nat @ X @ Y ) ) ) ).
% 0.34/0.65
% 0.34/0.65 % power_strict_increasing_iff
% 0.34/0.65 thf(fact_6116_power__strict__increasing__iff,axiom,
% 0.34/0.65 ! [B: nat,X: nat,Y: nat] :
% 0.34/0.65 ( ( ord_less_nat @ one_one_nat @ B )
% 0.34/0.65 => ( ( ord_less_nat @ ( power_power_nat @ B @ X ) @ ( power_power_nat @ B @ Y ) )
% 0.34/0.65 = ( ord_less_nat @ X @ Y ) ) ) ).
% 0.34/0.65
% 0.34/0.65 % power_strict_increasing_iff
% 0.34/0.65 thf(fact_6117_power__strict__increasing__iff,axiom,
% 0.34/0.65 ! [B: int,X: nat,Y: nat] :
% 0.34/0.65 ( ( ord_less_int @ one_one_int @ B )
% 0.34/0.65 => ( ( ord_less_int @ ( power_power_int @ B @ X ) @ ( power_power_int @ B @ Y ) )
% 0.34/0.65 = ( ord_less_nat @ X @ Y ) ) ) ).
% 0.34/0.65
% 0.34/0.65 % power_strict_increasing_iff
% 0.34/0.65 thf(fact_6118_power__strict__increasing__iff,axiom,
% 0.34/0.65 ! [B: code_integer,X: nat,Y: nat] :
% 0.34/0.65 ( ( ord_le6747313008572928689nteger @ one_one_Code_integer @ B )
% 0.34/0.65 => ( ( ord_le6747313008572928689nteger @ ( power_8256067586552552935nteger @ B @ X ) @ ( power_8256067586552552935nteger @ B @ Y ) )
% 0.34/0.65 = ( ord_less_nat @ X @ Y ) ) ) ).
% 0.34/0.65
% 0.34/0.65 % power_strict_increasing_iff
% 0.34/0.65 thf(fact_6119_power__strict__increasing__iff,axiom,
% 0.34/0.65 ! [B: rat,X: nat,Y: nat] :
% 0.34/0.65 ( ( ord_less_rat @ one_one_rat @ B )
% 0.34/0.65 => ( ( ord_less_rat @ ( power_power_rat @ B @ X ) @ ( power_power_rat @ B @ Y ) )
% 0.34/0.65 = ( ord_less_nat @ X @ Y ) ) ) ).
% 0.34/0.65
% 0.34/0.65 % power_strict_increasing_iff
% 0.34/0.65 thf(fact_6120_power__strict__decreasing__iff,axiom,
% 0.34/0.65 ! [B: code_natural,M4: nat,N2: nat] :
% 0.34/0.65 ( ( ord_le5570908160329646204atural @ zero_z2226904508553997617atural @ B )
% 0.34/0.65 => ( ( ord_le5570908160329646204atural @ B @ one_one_Code_natural )
% 0.34/0.65 => ( ( ord_le5570908160329646204atural @ ( power_7079662738309270450atural @ B @ M4 ) @ ( power_7079662738309270450atural @ B @ N2 ) )
% 0.34/0.65 = ( ord_less_nat @ N2 @ M4 ) ) ) ) ).
% 0.34/0.65
% 0.34/0.65 % power_strict_decreasing_iff
% 0.34/0.65 thf(fact_6121_power__strict__decreasing__iff,axiom,
% 0.34/0.65 ! [B: nat,M4: nat,N2: nat] :
% 0.34/0.65 ( ( ord_less_nat @ zero_zero_nat @ B )
% 0.34/0.65 => ( ( ord_less_nat @ B @ one_one_nat )
% 0.34/0.65 => ( ( ord_less_nat @ ( power_power_nat @ B @ M4 ) @ ( power_power_nat @ B @ N2 ) )
% 0.34/0.65 = ( ord_less_nat @ N2 @ M4 ) ) ) ) ).
% 0.34/0.65
% 0.34/0.65 % power_strict_decreasing_iff
% 0.34/0.65 thf(fact_6122_power__strict__decreasing__iff,axiom,
% 0.34/0.65 ! [B: int,M4: nat,N2: nat] :
% 0.34/0.65 ( ( ord_less_int @ zero_zero_int @ B )
% 0.34/0.65 => ( ( ord_less_int @ B @ one_one_int )
% 0.34/0.65 => ( ( ord_less_int @ ( power_power_int @ B @ M4 ) @ ( power_power_int @ B @ N2 ) )
% 0.34/0.65 = ( ord_less_nat @ N2 @ M4 ) ) ) ) ).
% 0.34/0.65
% 0.34/0.65 % power_strict_decreasing_iff
% 0.34/0.65 thf(fact_6123_power__strict__decreasing__iff,axiom,
% 0.34/0.65 ! [B: code_integer,M4: nat,N2: nat] :
% 0.34/0.65 ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ B )
% 0.34/0.65 => ( ( ord_le6747313008572928689nteger @ B @ one_one_Code_integer )
% 0.34/0.65 => ( ( ord_le6747313008572928689nteger @ ( power_8256067586552552935nteger @ B @ M4 ) @ ( power_8256067586552552935nteger @ B @ N2 ) )
% 0.34/0.65 = ( ord_less_nat @ N2 @ M4 ) ) ) ) ).
% 0.34/0.65
% 0.34/0.65 % power_strict_decreasing_iff
% 0.34/0.65 thf(fact_6124_power__strict__decreasing__iff,axiom,
% 0.34/0.65 ! [B: rat,M4: nat,N2: nat] :
% 0.34/0.65 ( ( ord_less_rat @ zero_zero_rat @ B )
% 0.34/0.65 => ( ( ord_less_rat @ B @ one_one_rat )
% 0.34/0.65 => ( ( ord_less_rat @ ( power_power_rat @ B @ M4 ) @ ( power_power_rat @ B @ N2 ) )
% 0.34/0.65 = ( ord_less_nat @ N2 @ M4 ) ) ) ) ).
% 0.34/0.65
% 0.34/0.65 % power_strict_decreasing_iff
% 0.34/0.65 thf(fact_6125_power__increasing__iff,axiom,
% 0.34/0.65 ! [B: code_natural,X: nat,Y: nat] :
% 0.34/0.65 ( ( ord_le5570908160329646204atural @ one_one_Code_natural @ B )
% 0.34/0.65 => ( ( ord_le1926595141338095240atural @ ( power_7079662738309270450atural @ B @ X ) @ ( power_7079662738309270450atural @ B @ Y ) )
% 0.34/0.65 = ( ord_less_eq_nat @ X @ Y ) ) ) ).
% 0.34/0.65
% 0.34/0.65 % power_increasing_iff
% 0.34/0.65 thf(fact_6126_power__increasing__iff,axiom,
% 0.34/0.65 ! [B: code_integer,X: nat,Y: nat] :
% 0.34/0.65 ( ( ord_le6747313008572928689nteger @ one_one_Code_integer @ B )
% 0.34/0.65 => ( ( ord_le3102999989581377725nteger @ ( power_8256067586552552935nteger @ B @ X ) @ ( power_8256067586552552935nteger @ B @ Y ) )
% 0.34/0.65 = ( ord_less_eq_nat @ X @ Y ) ) ) ).
% 0.34/0.65
% 0.34/0.65 % power_increasing_iff
% 0.34/0.65 thf(fact_6127_power__increasing__iff,axiom,
% 0.34/0.65 ! [B: rat,X: nat,Y: nat] :
% 0.34/0.65 ( ( ord_less_rat @ one_one_rat @ B )
% 0.34/0.65 => ( ( ord_less_eq_rat @ ( power_power_rat @ B @ X ) @ ( power_power_rat @ B @ Y ) )
% 0.34/0.65 = ( ord_less_eq_nat @ X @ Y ) ) ) ).
% 0.34/0.65
% 0.34/0.65 % power_increasing_iff
% 0.34/0.65 thf(fact_6128_power__increasing__iff,axiom,
% 0.34/0.65 ! [B: nat,X: nat,Y: nat] :
% 0.34/0.65 ( ( ord_less_nat @ one_one_nat @ B )
% 0.34/0.65 => ( ( ord_less_eq_nat @ ( power_power_nat @ B @ X ) @ ( power_power_nat @ B @ Y ) )
% 0.34/0.65 = ( ord_less_eq_nat @ X @ Y ) ) ) ).
% 0.34/0.65
% 0.34/0.65 % power_increasing_iff
% 0.34/0.65 thf(fact_6129_power__increasing__iff,axiom,
% 0.34/0.65 ! [B: int,X: nat,Y: nat] :
% 0.34/0.65 ( ( ord_less_int @ one_one_int @ B )
% 0.34/0.65 => ( ( ord_less_eq_int @ ( power_power_int @ B @ X ) @ ( power_power_int @ B @ Y ) )
% 0.34/0.65 = ( ord_less_eq_nat @ X @ Y ) ) ) ).
% 0.34/0.65
% 0.34/0.65 % power_increasing_iff
% 0.34/0.65 thf(fact_6130_is__unit__power__iff,axiom,
% 0.34/0.65 ! [A2: code_integer,N2: nat] :
% 0.34/0.65 ( ( dvd_dvd_Code_integer @ ( power_8256067586552552935nteger @ A2 @ N2 ) @ one_one_Code_integer )
% 0.34/0.65 = ( ( dvd_dvd_Code_integer @ A2 @ one_one_Code_integer )
% 0.34/0.65 | ( N2 = zero_zero_nat ) ) ) ).
% 0.34/0.65
% 0.34/0.65 % is_unit_power_iff
% 0.34/0.65 thf(fact_6131_is__unit__power__iff,axiom,
% 0.34/0.65 ! [A2: code_natural,N2: nat] :
% 0.34/0.65 ( ( dvd_dvd_Code_natural @ ( power_7079662738309270450atural @ A2 @ N2 ) @ one_one_Code_natural )
% 0.34/0.65 = ( ( dvd_dvd_Code_natural @ A2 @ one_one_Code_natural )
% 0.34/0.65 | ( N2 = zero_zero_nat ) ) ) ).
% 0.34/0.65
% 0.34/0.65 % is_unit_power_iff
% 0.34/0.65 thf(fact_6132_is__unit__power__iff,axiom,
% 0.34/0.65 ! [A2: int,N2: nat] :
% 0.34/0.65 ( ( dvd_dvd_int @ ( power_power_int @ A2 @ N2 ) @ one_one_int )
% 0.34/0.65 = ( ( dvd_dvd_int @ A2 @ one_one_int )
% 0.34/0.65 | ( N2 = zero_zero_nat ) ) ) ).
% 0.34/0.65
% 0.34/0.65 % is_unit_power_iff
% 0.34/0.65 thf(fact_6133_is__unit__power__iff,axiom,
% 0.34/0.65 ! [A2: nat,N2: nat] :
% 0.34/0.65 ( ( dvd_dvd_nat @ ( power_power_nat @ A2 @ N2 ) @ one_one_nat )
% 0.34/0.65 = ( ( dvd_dvd_nat @ A2 @ one_one_nat )
% 0.34/0.65 | ( N2 = zero_zero_nat ) ) ) ).
% 0.34/0.65
% 0.34/0.65 % is_unit_power_iff
% 0.34/0.65 thf(fact_6134_power__commuting__commutes,axiom,
% 0.34/0.65 ! [X: assn,Y: assn,N2: nat] :
% 0.34/0.65 ( ( ( times_times_assn @ X @ Y )
% 0.34/0.65 = ( times_times_assn @ Y @ X ) )
% 0.34/0.65 => ( ( times_times_assn @ ( power_power_assn @ X @ N2 ) @ Y )
% 0.34/0.65 = ( times_times_assn @ Y @ ( power_power_assn @ X @ N2 ) ) ) ) ).
% 0.34/0.65
% 0.34/0.65 % power_commuting_commutes
% 0.34/0.65 thf(fact_6135_power__commuting__commutes,axiom,
% 0.34/0.65 ! [X: rat,Y: rat,N2: nat] :
% 0.34/0.65 ( ( ( times_times_rat @ X @ Y )
% 0.34/0.65 = ( times_times_rat @ Y @ X ) )
% 0.34/0.65 => ( ( times_times_rat @ ( power_power_rat @ X @ N2 ) @ Y )
% 0.34/0.65 = ( times_times_rat @ Y @ ( power_power_rat @ X @ N2 ) ) ) ) ).
% 0.34/0.65
% 0.34/0.65 % power_commuting_commutes
% 0.34/0.65 thf(fact_6136_power__commuting__commutes,axiom,
% 0.34/0.65 ! [X: nat,Y: nat,N2: nat] :
% 0.34/0.65 ( ( ( times_times_nat @ X @ Y )
% 0.34/0.65 = ( times_times_nat @ Y @ X ) )
% 0.34/0.65 => ( ( times_times_nat @ ( power_power_nat @ X @ N2 ) @ Y )
% 0.34/0.65 = ( times_times_nat @ Y @ ( power_power_nat @ X @ N2 ) ) ) ) ).
% 0.34/0.65
% 0.34/0.65 % power_commuting_commutes
% 0.34/0.65 thf(fact_6137_power__commuting__commutes,axiom,
% 0.34/0.65 ! [X: int,Y: int,N2: nat] :
% 0.34/0.65 ( ( ( times_times_int @ X @ Y )
% 0.34/0.65 = ( times_times_int @ Y @ X ) )
% 0.34/0.65 => ( ( times_times_int @ ( power_power_int @ X @ N2 ) @ Y )
% 0.34/0.65 = ( times_times_int @ Y @ ( power_power_int @ X @ N2 ) ) ) ) ).
% 0.34/0.65
% 0.34/0.65 % power_commuting_commutes
% 0.34/0.65 thf(fact_6138_power__mult__distrib,axiom,
% 0.34/0.65 ! [A2: assn,B: assn,N2: nat] :
% 0.34/0.65 ( ( power_power_assn @ ( times_times_assn @ A2 @ B ) @ N2 )
% 0.34/0.65 = ( times_times_assn @ ( power_power_assn @ A2 @ N2 ) @ ( power_power_assn @ B @ N2 ) ) ) ).
% 0.34/0.65
% 0.34/0.65 % power_mult_distrib
% 0.34/0.65 thf(fact_6139_power__mult__distrib,axiom,
% 0.34/0.65 ! [A2: rat,B: rat,N2: nat] :
% 0.34/0.65 ( ( power_power_rat @ ( times_times_rat @ A2 @ B ) @ N2 )
% 0.34/0.65 = ( times_times_rat @ ( power_power_rat @ A2 @ N2 ) @ ( power_power_rat @ B @ N2 ) ) ) ).
% 0.34/0.65
% 0.34/0.65 % power_mult_distrib
% 0.34/0.65 thf(fact_6140_power__mult__distrib,axiom,
% 0.34/0.65 ! [A2: nat,B: nat,N2: nat] :
% 0.34/0.65 ( ( power_power_nat @ ( times_times_nat @ A2 @ B ) @ N2 )
% 0.34/0.65 = ( times_times_nat @ ( power_power_nat @ A2 @ N2 ) @ ( power_power_nat @ B @ N2 ) ) ) ).
% 0.34/0.65
% 0.34/0.65 % power_mult_distrib
% 0.34/0.65 thf(fact_6141_power__mult__distrib,axiom,
% 0.34/0.65 ! [A2: int,B: int,N2: nat] :
% 0.34/0.65 ( ( power_power_int @ ( times_times_int @ A2 @ B ) @ N2 )
% 0.34/0.65 = ( times_times_int @ ( power_power_int @ A2 @ N2 ) @ ( power_power_int @ B @ N2 ) ) ) ).
% 0.34/0.65
% 0.34/0.65 % power_mult_distrib
% 0.34/0.65 thf(fact_6142_power__commutes,axiom,
% 0.34/0.65 ! [A2: assn,N2: nat] :
% 0.34/0.65 ( ( times_times_assn @ ( power_power_assn @ A2 @ N2 ) @ A2 )
% 0.34/0.65 = ( times_times_assn @ A2 @ ( power_power_assn @ A2 @ N2 ) ) ) ).
% 0.34/0.65
% 0.34/0.65 % power_commutes
% 0.34/0.65 thf(fact_6143_power__commutes,axiom,
% 0.34/0.65 ! [A2: rat,N2: nat] :
% 0.34/0.65 ( ( times_times_rat @ ( power_power_rat @ A2 @ N2 ) @ A2 )
% 0.34/0.65 = ( times_times_rat @ A2 @ ( power_power_rat @ A2 @ N2 ) ) ) ).
% 0.34/0.65
% 0.34/0.65 % power_commutes
% 0.34/0.65 thf(fact_6144_power__commutes,axiom,
% 0.34/0.65 ! [A2: nat,N2: nat] :
% 0.34/0.65 ( ( times_times_nat @ ( power_power_nat @ A2 @ N2 ) @ A2 )
% 0.34/0.65 = ( times_times_nat @ A2 @ ( power_power_nat @ A2 @ N2 ) ) ) ).
% 0.34/0.65
% 0.34/0.65 % power_commutes
% 0.34/0.65 thf(fact_6145_power__commutes,axiom,
% 0.34/0.65 ! [A2: int,N2: nat] :
% 0.34/0.65 ( ( times_times_int @ ( power_power_int @ A2 @ N2 ) @ A2 )
% 0.34/0.65 = ( times_times_int @ A2 @ ( power_power_int @ A2 @ N2 ) ) ) ).
% 0.34/0.65
% 0.34/0.65 % power_commutes
% 0.34/0.65 thf(fact_6146_dvd__triv__right,axiom,
% 0.34/0.65 ! [A2: assn,B: assn] : ( dvd_dvd_assn @ A2 @ ( times_times_assn @ B @ A2 ) ) ).
% 0.34/0.65
% 0.34/0.65 % dvd_triv_right
% 0.34/0.65 thf(fact_6147_dvd__triv__right,axiom,
% 0.34/0.65 ! [A2: rat,B: rat] : ( dvd_dvd_rat @ A2 @ ( times_times_rat @ B @ A2 ) ) ).
% 0.34/0.65
% 0.34/0.65 % dvd_triv_right
% 0.34/0.65 thf(fact_6148_dvd__triv__right,axiom,
% 0.34/0.65 ! [A2: nat,B: nat] : ( dvd_dvd_nat @ A2 @ ( times_times_nat @ B @ A2 ) ) ).
% 0.34/0.65
% 0.34/0.65 % dvd_triv_right
% 0.34/0.65 thf(fact_6149_dvd__triv__right,axiom,
% 0.34/0.65 ! [A2: int,B: int] : ( dvd_dvd_int @ A2 @ ( times_times_int @ B @ A2 ) ) ).
% 0.34/0.65
% 0.34/0.65 % dvd_triv_right
% 0.34/0.65 thf(fact_6150_dvd__mult__right,axiom,
% 0.34/0.65 ! [A2: assn,B: assn,C: assn] :
% 0.34/0.65 ( ( dvd_dvd_assn @ ( times_times_assn @ A2 @ B ) @ C )
% 0.34/0.65 => ( dvd_dvd_assn @ B @ C ) ) ).
% 0.34/0.65
% 0.34/0.65 % dvd_mult_right
% 0.34/0.65 thf(fact_6151_dvd__mult__right,axiom,
% 0.34/0.65 ! [A2: rat,B: rat,C: rat] :
% 0.34/0.65 ( ( dvd_dvd_rat @ ( times_times_rat @ A2 @ B ) @ C )
% 0.34/0.65 => ( dvd_dvd_rat @ B @ C ) ) ).
% 0.34/0.65
% 0.34/0.65 % dvd_mult_right
% 0.34/0.65 thf(fact_6152_dvd__mult__right,axiom,
% 0.34/0.65 ! [A2: nat,B: nat,C: nat] :
% 0.34/0.65 ( ( dvd_dvd_nat @ ( times_times_nat @ A2 @ B ) @ C )
% 0.34/0.65 => ( dvd_dvd_nat @ B @ C ) ) ).
% 0.34/0.65
% 0.34/0.65 % dvd_mult_right
% 0.34/0.65 thf(fact_6153_dvd__mult__right,axiom,
% 0.34/0.65 ! [A2: int,B: int,C: int] :
% 0.34/0.65 ( ( dvd_dvd_int @ ( times_times_int @ A2 @ B ) @ C )
% 0.34/0.65 => ( dvd_dvd_int @ B @ C ) ) ).
% 0.34/0.65
% 0.34/0.65 % dvd_mult_right
% 0.34/0.65 thf(fact_6154_mult__dvd__mono,axiom,
% 0.34/0.65 ! [A2: assn,B: assn,C: assn,D2: assn] :
% 0.34/0.65 ( ( dvd_dvd_assn @ A2 @ B )
% 0.34/0.65 => ( ( dvd_dvd_assn @ C @ D2 )
% 0.34/0.65 => ( dvd_dvd_assn @ ( times_times_assn @ A2 @ C ) @ ( times_times_assn @ B @ D2 ) ) ) ) ).
% 0.34/0.65
% 0.34/0.65 % mult_dvd_mono
% 0.34/0.65 thf(fact_6155_mult__dvd__mono,axiom,
% 0.34/0.65 ! [A2: rat,B: rat,C: rat,D2: rat] :
% 0.34/0.65 ( ( dvd_dvd_rat @ A2 @ B )
% 0.34/0.65 => ( ( dvd_dvd_rat @ C @ D2 )
% 0.34/0.65 => ( dvd_dvd_rat @ ( times_times_rat @ A2 @ C ) @ ( times_times_rat @ B @ D2 ) ) ) ) ).
% 0.34/0.65
% 0.34/0.65 % mult_dvd_mono
% 0.34/0.65 thf(fact_6156_mult__dvd__mono,axiom,
% 0.34/0.65 ! [A2: nat,B: nat,C: nat,D2: nat] :
% 0.34/0.65 ( ( dvd_dvd_nat @ A2 @ B )
% 0.34/0.65 => ( ( dvd_dvd_nat @ C @ D2 )
% 0.34/0.65 => ( dvd_dvd_nat @ ( times_times_nat @ A2 @ C ) @ ( times_times_nat @ B @ D2 ) ) ) ) ).
% 0.34/0.65
% 0.34/0.65 % mult_dvd_mono
% 0.34/0.65 thf(fact_6157_mult__dvd__mono,axiom,
% 0.34/0.65 ! [A2: int,B: int,C: int,D2: int] :
% 0.34/0.65 ( ( dvd_dvd_int @ A2 @ B )
% 0.34/0.65 => ( ( dvd_dvd_int @ C @ D2 )
% 0.34/0.65 => ( dvd_dvd_int @ ( times_times_int @ A2 @ C ) @ ( times_times_int @ B @ D2 ) ) ) ) ).
% 0.34/0.65
% 0.34/0.65 % mult_dvd_mono
% 0.34/0.65 thf(fact_6158_dvd__triv__left,axiom,
% 0.34/0.65 ! [A2: assn,B: assn] : ( dvd_dvd_assn @ A2 @ ( times_times_assn @ A2 @ B ) ) ).
% 0.34/0.65
% 0.34/0.65 % dvd_triv_left
% 0.34/0.65 thf(fact_6159_dvd__triv__left,axiom,
% 0.34/0.65 ! [A2: rat,B: rat] : ( dvd_dvd_rat @ A2 @ ( times_times_rat @ A2 @ B ) ) ).
% 0.34/0.65
% 0.34/0.65 % dvd_triv_left
% 0.34/0.65 thf(fact_6160_dvd__triv__left,axiom,
% 0.34/0.65 ! [A2: nat,B: nat] : ( dvd_dvd_nat @ A2 @ ( times_times_nat @ A2 @ B ) ) ).
% 0.34/0.65
% 0.34/0.65 % dvd_triv_left
% 0.34/0.65 thf(fact_6161_dvd__triv__left,axiom,
% 0.34/0.65 ! [A2: int,B: int] : ( dvd_dvd_int @ A2 @ ( times_times_int @ A2 @ B ) ) ).
% 0.34/0.65
% 0.34/0.65 % dvd_triv_left
% 0.34/0.65 thf(fact_6162_dvd__mult__left,axiom,
% 0.34/0.65 ! [A2: assn,B: assn,C: assn] :
% 0.34/0.65 ( ( dvd_dvd_assn @ ( times_times_assn @ A2 @ B ) @ C )
% 0.34/0.65 => ( dvd_dvd_assn @ A2 @ C ) ) ).
% 0.34/0.65
% 0.34/0.65 % dvd_mult_left
% 0.34/0.65 thf(fact_6163_dvd__mult__left,axiom,
% 0.34/0.65 ! [A2: rat,B: rat,C: rat] :
% 0.34/0.65 ( ( dvd_dvd_rat @ ( times_times_rat @ A2 @ B ) @ C )
% 0.34/0.65 => ( dvd_dvd_rat @ A2 @ C ) ) ).
% 0.34/0.65
% 0.34/0.65 % dvd_mult_left
% 0.34/0.65 thf(fact_6164_dvd__mult__left,axiom,
% 0.34/0.65 ! [A2: nat,B: nat,C: nat] :
% 0.34/0.65 ( ( dvd_dvd_nat @ ( times_times_nat @ A2 @ B ) @ C )
% 0.34/0.65 => ( dvd_dvd_nat @ A2 @ C ) ) ).
% 0.34/0.65
% 0.34/0.65 % dvd_mult_left
% 0.34/0.65 thf(fact_6165_dvd__mult__left,axiom,
% 0.34/0.65 ! [A2: int,B: int,C: int] :
% 0.34/0.65 ( ( dvd_dvd_int @ ( times_times_int @ A2 @ B ) @ C )
% 0.34/0.65 => ( dvd_dvd_int @ A2 @ C ) ) ).
% 0.34/0.65
% 0.34/0.65 % dvd_mult_left
% 0.34/0.65 thf(fact_6166_dvd__mult2,axiom,
% 0.34/0.65 ! [A2: assn,B: assn,C: assn] :
% 0.34/0.65 ( ( dvd_dvd_assn @ A2 @ B )
% 0.34/0.65 => ( dvd_dvd_assn @ A2 @ ( times_times_assn @ B @ C ) ) ) ).
% 0.34/0.65
% 0.34/0.65 % dvd_mult2
% 0.34/0.65 thf(fact_6167_dvd__mult2,axiom,
% 0.34/0.65 ! [A2: rat,B: rat,C: rat] :
% 0.34/0.65 ( ( dvd_dvd_rat @ A2 @ B )
% 0.34/0.65 => ( dvd_dvd_rat @ A2 @ ( times_times_rat @ B @ C ) ) ) ).
% 0.34/0.65
% 0.34/0.65 % dvd_mult2
% 0.34/0.65 thf(fact_6168_dvd__mult2,axiom,
% 0.34/0.65 ! [A2: nat,B: nat,C: nat] :
% 0.34/0.65 ( ( dvd_dvd_nat @ A2 @ B )
% 0.34/0.65 => ( dvd_dvd_nat @ A2 @ ( times_times_nat @ B @ C ) ) ) ).
% 0.34/0.65
% 0.34/0.65 % dvd_mult2
% 0.34/0.65 thf(fact_6169_dvd__mult2,axiom,
% 0.34/0.65 ! [A2: int,B: int,C: int] :
% 0.34/0.65 ( ( dvd_dvd_int @ A2 @ B )
% 0.34/0.65 => ( dvd_dvd_int @ A2 @ ( times_times_int @ B @ C ) ) ) ).
% 0.34/0.65
% 0.34/0.65 % dvd_mult2
% 0.34/0.65 thf(fact_6170_dvd__mult,axiom,
% 0.34/0.65 ! [A2: assn,C: assn,B: assn] :
% 0.34/0.65 ( ( dvd_dvd_assn @ A2 @ C )
% 0.34/0.65 => ( dvd_dvd_assn @ A2 @ ( times_times_assn @ B @ C ) ) ) ).
% 0.34/0.65
% 0.34/0.65 % dvd_mult
% 0.34/0.65 thf(fact_6171_dvd__mult,axiom,
% 0.34/0.65 ! [A2: rat,C: rat,B: rat] :
% 0.34/0.65 ( ( dvd_dvd_rat @ A2 @ C )
% 0.34/0.65 => ( dvd_dvd_rat @ A2 @ ( times_times_rat @ B @ C ) ) ) ).
% 0.34/0.65
% 0.34/0.65 % dvd_mult
% 0.34/0.65 thf(fact_6172_dvd__mult,axiom,
% 0.34/0.65 ! [A2: nat,C: nat,B: nat] :
% 0.34/0.65 ( ( dvd_dvd_nat @ A2 @ C )
% 0.34/0.65 => ( dvd_dvd_nat @ A2 @ ( times_times_nat @ B @ C ) ) ) ).
% 0.34/0.65
% 0.34/0.65 % dvd_mult
% 0.34/0.65 thf(fact_6173_dvd__mult,axiom,
% 0.34/0.65 ! [A2: int,C: int,B: int] :
% 0.34/0.65 ( ( dvd_dvd_int @ A2 @ C )
% 0.34/0.65 => ( dvd_dvd_int @ A2 @ ( times_times_int @ B @ C ) ) ) ).
% 0.34/0.65
% 0.34/0.65 % dvd_mult
% 0.34/0.65 thf(fact_6174_dvd__def,axiom,
% 0.34/0.65 ( dvd_dvd_assn
% 0.34/0.65 = ( ^ [B6: assn,A6: assn] :
% 0.34/0.65 ? [K5: assn] :
% 0.34/0.65 ( A6
% 0.34/0.65 = ( times_times_assn @ B6 @ K5 ) ) ) ) ).
% 0.34/0.65
% 0.34/0.65 % dvd_def
% 0.34/0.65 thf(fact_6175_dvd__def,axiom,
% 0.34/0.65 ( dvd_dvd_rat
% 0.34/0.65 = ( ^ [B6: rat,A6: rat] :
% 0.34/0.65 ? [K5: rat] :
% 0.34/0.65 ( A6
% 0.34/0.65 = ( times_times_rat @ B6 @ K5 ) ) ) ) ).
% 0.34/0.65
% 0.34/0.65 % dvd_def
% 0.34/0.65 thf(fact_6176_dvd__def,axiom,
% 0.34/0.65 ( dvd_dvd_nat
% 0.34/0.65 = ( ^ [B6: nat,A6: nat] :
% 0.34/0.65 ? [K5: nat] :
% 0.34/0.65 ( A6
% 0.34/0.65 = ( times_times_nat @ B6 @ K5 ) ) ) ) ).
% 0.34/0.65
% 0.34/0.65 % dvd_def
% 0.34/0.65 thf(fact_6177_dvd__def,axiom,
% 0.34/0.65 ( dvd_dvd_int
% 0.34/0.65 = ( ^ [B6: int,A6: int] :
% 0.34/0.65 ? [K5: int] :
% 0.34/0.65 ( A6
% 0.34/0.65 = ( times_times_int @ B6 @ K5 ) ) ) ) ).
% 0.34/0.65
% 0.34/0.65 % dvd_def
% 0.34/0.65 thf(fact_6178_dvdI,axiom,
% 0.34/0.65 ! [A2: assn,B: assn,K3: assn] :
% 0.34/0.65 ( ( A2
% 0.34/0.65 = ( times_times_assn @ B @ K3 ) )
% 0.34/0.65 => ( dvd_dvd_assn @ B @ A2 ) ) ).
% 0.34/0.65
% 0.34/0.65 % dvdI
% 0.34/0.65 thf(fact_6179_dvdI,axiom,
% 0.34/0.65 ! [A2: rat,B: rat,K3: rat] :
% 0.34/0.65 ( ( A2
% 0.34/0.65 = ( times_times_rat @ B @ K3 ) )
% 0.34/0.66 => ( dvd_dvd_rat @ B @ A2 ) ) ).
% 0.34/0.66
% 0.34/0.66 % dvdI
% 0.34/0.66 thf(fact_6180_dvdI,axiom,
% 0.34/0.66 ! [A2: nat,B: nat,K3: nat] :
% 0.34/0.66 ( ( A2
% 0.34/0.66 = ( times_times_nat @ B @ K3 ) )
% 0.34/0.66 => ( dvd_dvd_nat @ B @ A2 ) ) ).
% 0.34/0.66
% 0.34/0.66 % dvdI
% 0.34/0.66 thf(fact_6181_dvdI,axiom,
% 0.34/0.66 ! [A2: int,B: int,K3: int] :
% 0.34/0.66 ( ( A2
% 0.34/0.66 = ( times_times_int @ B @ K3 ) )
% 0.34/0.66 => ( dvd_dvd_int @ B @ A2 ) ) ).
% 0.34/0.66
% 0.34/0.66 % dvdI
% 0.34/0.66 thf(fact_6182_dvdE,axiom,
% 0.34/0.66 ! [B: assn,A2: assn] :
% 0.34/0.66 ( ( dvd_dvd_assn @ B @ A2 )
% 0.34/0.66 => ~ ! [K: assn] :
% 0.34/0.66 ( A2
% 0.34/0.66 != ( times_times_assn @ B @ K ) ) ) ).
% 0.34/0.66
% 0.34/0.66 % dvdE
% 0.34/0.66 thf(fact_6183_dvdE,axiom,
% 0.34/0.66 ! [B: rat,A2: rat] :
% 0.34/0.66 ( ( dvd_dvd_rat @ B @ A2 )
% 0.34/0.66 => ~ ! [K: rat] :
% 0.34/0.66 ( A2
% 0.34/0.66 != ( times_times_rat @ B @ K ) ) ) ).
% 0.34/0.66
% 0.34/0.66 % dvdE
% 0.34/0.66 thf(fact_6184_dvdE,axiom,
% 0.34/0.66 ! [B: nat,A2: nat] :
% 0.34/0.66 ( ( dvd_dvd_nat @ B @ A2 )
% 0.34/0.66 => ~ ! [K: nat] :
% 0.34/0.66 ( A2
% 0.34/0.66 != ( times_times_nat @ B @ K ) ) ) ).
% 0.34/0.66
% 0.34/0.66 % dvdE
% 0.34/0.66 thf(fact_6185_dvdE,axiom,
% 0.34/0.66 ! [B: int,A2: int] :
% 0.34/0.66 ( ( dvd_dvd_int @ B @ A2 )
% 0.34/0.66 => ~ ! [K: int] :
% 0.34/0.66 ( A2
% 0.34/0.66 != ( times_times_int @ B @ K ) ) ) ).
% 0.34/0.66
% 0.34/0.66 % dvdE
% 0.34/0.66 thf(fact_6186_dvd__unit__imp__unit,axiom,
% 0.34/0.66 ! [A2: nat,B: nat] :
% 0.34/0.66 ( ( dvd_dvd_nat @ A2 @ B )
% 0.34/0.66 => ( ( dvd_dvd_nat @ B @ one_one_nat )
% 0.34/0.66 => ( dvd_dvd_nat @ A2 @ one_one_nat ) ) ) ).
% 0.34/0.66
% 0.34/0.66 % dvd_unit_imp_unit
% 0.34/0.66 thf(fact_6187_dvd__unit__imp__unit,axiom,
% 0.34/0.66 ! [A2: int,B: int] :
% 0.34/0.66 ( ( dvd_dvd_int @ A2 @ B )
% 0.34/0.66 => ( ( dvd_dvd_int @ B @ one_one_int )
% 0.34/0.66 => ( dvd_dvd_int @ A2 @ one_one_int ) ) ) ).
% 0.34/0.66
% 0.34/0.66 % dvd_unit_imp_unit
% 0.34/0.66 thf(fact_6188_dvd__unit__imp__unit,axiom,
% 0.34/0.66 ! [A2: code_integer,B: code_integer] :
% 0.34/0.66 ( ( dvd_dvd_Code_integer @ A2 @ B )
% 0.34/0.66 => ( ( dvd_dvd_Code_integer @ B @ one_one_Code_integer )
% 0.34/0.66 => ( dvd_dvd_Code_integer @ A2 @ one_one_Code_integer ) ) ) ).
% 0.34/0.66
% 0.34/0.66 % dvd_unit_imp_unit
% 0.34/0.66 thf(fact_6189_dvd__unit__imp__unit,axiom,
% 0.34/0.66 ! [A2: code_natural,B: code_natural] :
% 0.34/0.66 ( ( dvd_dvd_Code_natural @ A2 @ B )
% 0.34/0.66 => ( ( dvd_dvd_Code_natural @ B @ one_one_Code_natural )
% 0.34/0.66 => ( dvd_dvd_Code_natural @ A2 @ one_one_Code_natural ) ) ) ).
% 0.34/0.66
% 0.34/0.66 % dvd_unit_imp_unit
% 0.34/0.66 thf(fact_6190_unit__imp__dvd,axiom,
% 0.34/0.66 ! [B: nat,A2: nat] :
% 0.34/0.66 ( ( dvd_dvd_nat @ B @ one_one_nat )
% 0.34/0.66 => ( dvd_dvd_nat @ B @ A2 ) ) ).
% 0.34/0.66
% 0.34/0.66 % unit_imp_dvd
% 0.34/0.66 thf(fact_6191_unit__imp__dvd,axiom,
% 0.34/0.66 ! [B: int,A2: int] :
% 0.34/0.66 ( ( dvd_dvd_int @ B @ one_one_int )
% 0.34/0.66 => ( dvd_dvd_int @ B @ A2 ) ) ).
% 0.34/0.66
% 0.34/0.66 % unit_imp_dvd
% 0.34/0.66 thf(fact_6192_unit__imp__dvd,axiom,
% 0.34/0.66 ! [B: code_integer,A2: code_integer] :
% 0.34/0.66 ( ( dvd_dvd_Code_integer @ B @ one_one_Code_integer )
% 0.34/0.66 => ( dvd_dvd_Code_integer @ B @ A2 ) ) ).
% 0.34/0.66
% 0.34/0.66 % unit_imp_dvd
% 0.34/0.66 thf(fact_6193_unit__imp__dvd,axiom,
% 0.34/0.66 ! [B: code_natural,A2: code_natural] :
% 0.34/0.66 ( ( dvd_dvd_Code_natural @ B @ one_one_Code_natural )
% 0.34/0.66 => ( dvd_dvd_Code_natural @ B @ A2 ) ) ).
% 0.34/0.66
% 0.34/0.66 % unit_imp_dvd
% 0.34/0.66 thf(fact_6194_one__dvd,axiom,
% 0.34/0.66 ! [A2: assn] : ( dvd_dvd_assn @ one_one_assn @ A2 ) ).
% 0.34/0.66
% 0.34/0.66 % one_dvd
% 0.34/0.66 thf(fact_6195_one__dvd,axiom,
% 0.34/0.66 ! [A2: nat] : ( dvd_dvd_nat @ one_one_nat @ A2 ) ).
% 0.34/0.66
% 0.34/0.66 % one_dvd
% 0.34/0.66 thf(fact_6196_one__dvd,axiom,
% 0.34/0.66 ! [A2: int] : ( dvd_dvd_int @ one_one_int @ A2 ) ).
% 0.34/0.66
% 0.34/0.66 % one_dvd
% 0.34/0.66 thf(fact_6197_one__dvd,axiom,
% 0.34/0.66 ! [A2: code_integer] : ( dvd_dvd_Code_integer @ one_one_Code_integer @ A2 ) ).
% 0.34/0.66
% 0.34/0.66 % one_dvd
% 0.34/0.66 thf(fact_6198_one__dvd,axiom,
% 0.34/0.66 ! [A2: code_natural] : ( dvd_dvd_Code_natural @ one_one_Code_natural @ A2 ) ).
% 0.34/0.66
% 0.34/0.66 % one_dvd
% 0.34/0.66 thf(fact_6199_power__mult,axiom,
% 0.34/0.66 ! [A2: int,M4: nat,N2: nat] :
% 0.34/0.66 ( ( power_power_int @ A2 @ ( times_times_nat @ M4 @ N2 ) )
% 0.34/0.66 = ( power_power_int @ ( power_power_int @ A2 @ M4 ) @ N2 ) ) ).
% 0.34/0.66
% 0.34/0.66 % power_mult
% 0.34/0.66 thf(fact_6200_power__mult,axiom,
% 0.34/0.66 ! [A2: nat,M4: nat,N2: nat] :
% 0.34/0.66 ( ( power_power_nat @ A2 @ ( times_times_nat @ M4 @ N2 ) )
% 0.34/0.66 = ( power_power_nat @ ( power_power_nat @ A2 @ M4 ) @ N2 ) ) ).
% 0.34/0.66
% 0.34/0.66 % power_mult
% 0.34/0.66 thf(fact_6201_dvd__power__iff,axiom,
% 0.34/0.66 ! [X: code_natural,M4: nat,N2: nat] :
% 0.34/0.66 ( ( X != zero_z2226904508553997617atural )
% 0.34/0.66 => ( ( dvd_dvd_Code_natural @ ( power_7079662738309270450atural @ X @ M4 ) @ ( power_7079662738309270450atural @ X @ N2 ) )
% 0.34/0.66 = ( ( dvd_dvd_Code_natural @ X @ one_one_Code_natural )
% 0.34/0.66 | ( ord_less_eq_nat @ M4 @ N2 ) ) ) ) ).
% 0.34/0.66
% 0.34/0.66 % dvd_power_iff
% 0.34/0.66 thf(fact_6202_dvd__power__iff,axiom,
% 0.34/0.66 ! [X: code_integer,M4: nat,N2: nat] :
% 0.34/0.66 ( ( X != zero_z3403309356797280102nteger )
% 0.34/0.66 => ( ( dvd_dvd_Code_integer @ ( power_8256067586552552935nteger @ X @ M4 ) @ ( power_8256067586552552935nteger @ X @ N2 ) )
% 0.34/0.66 = ( ( dvd_dvd_Code_integer @ X @ one_one_Code_integer )
% 0.34/0.66 | ( ord_less_eq_nat @ M4 @ N2 ) ) ) ) ).
% 0.34/0.66
% 0.34/0.66 % dvd_power_iff
% 0.34/0.66 thf(fact_6203_dvd__power__iff,axiom,
% 0.34/0.66 ! [X: int,M4: nat,N2: nat] :
% 0.34/0.66 ( ( X != zero_zero_int )
% 0.34/0.66 => ( ( dvd_dvd_int @ ( power_power_int @ X @ M4 ) @ ( power_power_int @ X @ N2 ) )
% 0.34/0.66 = ( ( dvd_dvd_int @ X @ one_one_int )
% 0.34/0.66 | ( ord_less_eq_nat @ M4 @ N2 ) ) ) ) ).
% 0.34/0.66
% 0.34/0.66 % dvd_power_iff
% 0.34/0.66 thf(fact_6204_dvd__power__iff,axiom,
% 0.34/0.66 ! [X: nat,M4: nat,N2: nat] :
% 0.34/0.66 ( ( X != zero_zero_nat )
% 0.34/0.66 => ( ( dvd_dvd_nat @ ( power_power_nat @ X @ M4 ) @ ( power_power_nat @ X @ N2 ) )
% 0.34/0.66 = ( ( dvd_dvd_nat @ X @ one_one_nat )
% 0.34/0.66 | ( ord_less_eq_nat @ M4 @ N2 ) ) ) ) ).
% 0.34/0.66
% 0.34/0.66 % dvd_power_iff
% 0.34/0.66 thf(fact_6205_dvd__power,axiom,
% 0.34/0.66 ! [N2: nat,X: code_integer] :
% 0.34/0.66 ( ( ( ord_less_nat @ zero_zero_nat @ N2 )
% 0.34/0.66 | ( X = one_one_Code_integer ) )
% 0.34/0.66 => ( dvd_dvd_Code_integer @ X @ ( power_8256067586552552935nteger @ X @ N2 ) ) ) ).
% 0.34/0.66
% 0.34/0.66 % dvd_power
% 0.34/0.66 thf(fact_6206_dvd__power,axiom,
% 0.34/0.66 ! [N2: nat,X: code_natural] :
% 0.34/0.66 ( ( ( ord_less_nat @ zero_zero_nat @ N2 )
% 0.34/0.66 | ( X = one_one_Code_natural ) )
% 0.34/0.66 => ( dvd_dvd_Code_natural @ X @ ( power_7079662738309270450atural @ X @ N2 ) ) ) ).
% 0.34/0.66
% 0.34/0.66 % dvd_power
% 0.34/0.66 thf(fact_6207_dvd__power,axiom,
% 0.34/0.66 ! [N2: nat,X: int] :
% 0.34/0.66 ( ( ( ord_less_nat @ zero_zero_nat @ N2 )
% 0.34/0.66 | ( X = one_one_int ) )
% 0.34/0.66 => ( dvd_dvd_int @ X @ ( power_power_int @ X @ N2 ) ) ) ).
% 0.34/0.66
% 0.34/0.66 % dvd_power
% 0.34/0.66 thf(fact_6208_dvd__power,axiom,
% 0.34/0.66 ! [N2: nat,X: nat] :
% 0.34/0.66 ( ( ( ord_less_nat @ zero_zero_nat @ N2 )
% 0.34/0.66 | ( X = one_one_nat ) )
% 0.34/0.66 => ( dvd_dvd_nat @ X @ ( power_power_nat @ X @ N2 ) ) ) ).
% 0.34/0.66
% 0.34/0.66 % dvd_power
% 0.34/0.66 thf(fact_6209_gbinomial__Suc__Suc,axiom,
% 0.34/0.66 ! [A2: rat,K3: nat] :
% 0.34/0.66 ( ( gbinomial_rat @ ( plus_plus_rat @ A2 @ one_one_rat ) @ ( suc @ K3 ) )
% 0.34/0.66 = ( plus_plus_rat @ ( gbinomial_rat @ A2 @ K3 ) @ ( gbinomial_rat @ A2 @ ( suc @ K3 ) ) ) ) ).
% 0.34/0.66
% 0.34/0.66 % gbinomial_Suc_Suc
% 0.34/0.66 thf(fact_6210_one__le__power,axiom,
% 0.34/0.66 ! [A2: code_integer,N2: nat] :
% 0.34/0.66 ( ( ord_le3102999989581377725nteger @ one_one_Code_integer @ A2 )
% 0.34/0.66 => ( ord_le3102999989581377725nteger @ one_one_Code_integer @ ( power_8256067586552552935nteger @ A2 @ N2 ) ) ) ).
% 0.34/0.66
% 0.34/0.66 % one_le_power
% 0.34/0.66 thf(fact_6211_one__le__power,axiom,
% 0.34/0.66 ! [A2: code_natural,N2: nat] :
% 0.34/0.66 ( ( ord_le1926595141338095240atural @ one_one_Code_natural @ A2 )
% 0.34/0.66 => ( ord_le1926595141338095240atural @ one_one_Code_natural @ ( power_7079662738309270450atural @ A2 @ N2 ) ) ) ).
% 0.34/0.66
% 0.34/0.66 % one_le_power
% 0.34/0.66 thf(fact_6212_one__le__power,axiom,
% 0.34/0.66 ! [A2: rat,N2: nat] :
% 0.34/0.66 ( ( ord_less_eq_rat @ one_one_rat @ A2 )
% 0.34/0.66 => ( ord_less_eq_rat @ one_one_rat @ ( power_power_rat @ A2 @ N2 ) ) ) ).
% 0.34/0.66
% 0.34/0.66 % one_le_power
% 0.34/0.66 thf(fact_6213_one__le__power,axiom,
% 0.34/0.66 ! [A2: nat,N2: nat] :
% 0.34/0.66 ( ( ord_less_eq_nat @ one_one_nat @ A2 )
% 0.34/0.66 => ( ord_less_eq_nat @ one_one_nat @ ( power_power_nat @ A2 @ N2 ) ) ) ).
% 0.34/0.66
% 0.34/0.66 % one_le_power
% 0.34/0.66 thf(fact_6214_one__le__power,axiom,
% 0.34/0.66 ! [A2: int,N2: nat] :
% 0.34/0.66 ( ( ord_less_eq_int @ one_one_int @ A2 )
% 0.34/0.66 => ( ord_less_eq_int @ one_one_int @ ( power_power_int @ A2 @ N2 ) ) ) ).
% 0.34/0.66
% 0.34/0.66 % one_le_power
% 0.34/0.66 thf(fact_6215_not__is__unit__0,axiom,
% 0.34/0.66 ~ ( dvd_dvd_Code_natural @ zero_z2226904508553997617atural @ one_one_Code_natural ) ).
% 0.34/0.66
% 0.34/0.66 % not_is_unit_0
% 0.34/0.66 thf(fact_6216_not__is__unit__0,axiom,
% 0.34/0.66 ~ ( dvd_dvd_nat @ zero_zero_nat @ one_one_nat ) ).
% 0.34/0.66
% 0.34/0.66 % not_is_unit_0
% 0.34/0.66 thf(fact_6217_not__is__unit__0,axiom,
% 0.34/0.66 ~ ( dvd_dvd_int @ zero_zero_int @ one_one_int ) ).
% 0.34/0.66
% 0.34/0.66 % not_is_unit_0
% 0.34/0.66 thf(fact_6218_not__is__unit__0,axiom,
% 0.34/0.66 ~ ( dvd_dvd_Code_integer @ zero_z3403309356797280102nteger @ one_one_Code_integer ) ).
% 0.34/0.66
% 0.34/0.66 % not_is_unit_0
% 0.34/0.66 thf(fact_6219_left__right__inverse__power,axiom,
% 0.34/0.66 ! [X: code_integer,Y: code_integer,N2: nat] :
% 0.34/0.66 ( ( ( times_3573771949741848930nteger @ X @ Y )
% 0.34/0.66 = one_one_Code_integer )
% 0.34/0.66 => ( ( times_3573771949741848930nteger @ ( power_8256067586552552935nteger @ X @ N2 ) @ ( power_8256067586552552935nteger @ Y @ N2 ) )
% 0.34/0.66 = one_one_Code_integer ) ) ).
% 0.34/0.66
% 0.34/0.66 % left_right_inverse_power
% 0.34/0.66 thf(fact_6220_left__right__inverse__power,axiom,
% 0.34/0.66 ! [X: code_natural,Y: code_natural,N2: nat] :
% 0.34/0.66 ( ( ( times_2397367101498566445atural @ X @ Y )
% 0.34/0.66 = one_one_Code_natural )
% 0.34/0.66 => ( ( times_2397367101498566445atural @ ( power_7079662738309270450atural @ X @ N2 ) @ ( power_7079662738309270450atural @ Y @ N2 ) )
% 0.34/0.66 = one_one_Code_natural ) ) ).
% 0.34/0.66
% 0.34/0.66 % left_right_inverse_power
% 0.34/0.66 thf(fact_6221_left__right__inverse__power,axiom,
% 0.34/0.66 ! [X: assn,Y: assn,N2: nat] :
% 0.34/0.66 ( ( ( times_times_assn @ X @ Y )
% 0.34/0.66 = one_one_assn )
% 0.34/0.66 => ( ( times_times_assn @ ( power_power_assn @ X @ N2 ) @ ( power_power_assn @ Y @ N2 ) )
% 0.34/0.66 = one_one_assn ) ) ).
% 0.34/0.66
% 0.34/0.66 % left_right_inverse_power
% 0.34/0.66 thf(fact_6222_left__right__inverse__power,axiom,
% 0.34/0.66 ! [X: rat,Y: rat,N2: nat] :
% 0.34/0.66 ( ( ( times_times_rat @ X @ Y )
% 0.34/0.66 = one_one_rat )
% 0.34/0.66 => ( ( times_times_rat @ ( power_power_rat @ X @ N2 ) @ ( power_power_rat @ Y @ N2 ) )
% 0.34/0.66 = one_one_rat ) ) ).
% 0.34/0.66
% 0.34/0.66 % left_right_inverse_power
% 0.34/0.66 thf(fact_6223_left__right__inverse__power,axiom,
% 0.34/0.66 ! [X: nat,Y: nat,N2: nat] :
% 0.34/0.66 ( ( ( times_times_nat @ X @ Y )
% 0.34/0.66 = one_one_nat )
% 0.34/0.66 => ( ( times_times_nat @ ( power_power_nat @ X @ N2 ) @ ( power_power_nat @ Y @ N2 ) )
% 0.34/0.66 = one_one_nat ) ) ).
% 0.34/0.66
% 0.34/0.66 % left_right_inverse_power
% 0.34/0.66 thf(fact_6224_left__right__inverse__power,axiom,
% 0.34/0.66 ! [X: int,Y: int,N2: nat] :
% 0.34/0.66 ( ( ( times_times_int @ X @ Y )
% 0.34/0.66 = one_one_int )
% 0.34/0.66 => ( ( times_times_int @ ( power_power_int @ X @ N2 ) @ ( power_power_int @ Y @ N2 ) )
% 0.34/0.66 = one_one_int ) ) ).
% 0.34/0.66
% 0.34/0.66 % left_right_inverse_power
% 0.34/0.66 thf(fact_6225_power__Suc2,axiom,
% 0.34/0.66 ! [A2: assn,N2: nat] :
% 0.34/0.66 ( ( power_power_assn @ A2 @ ( suc @ N2 ) )
% 0.34/0.66 = ( times_times_assn @ ( power_power_assn @ A2 @ N2 ) @ A2 ) ) ).
% 0.34/0.66
% 0.34/0.66 % power_Suc2
% 0.34/0.66 thf(fact_6226_power__Suc2,axiom,
% 0.34/0.66 ! [A2: rat,N2: nat] :
% 0.34/0.66 ( ( power_power_rat @ A2 @ ( suc @ N2 ) )
% 0.34/0.66 = ( times_times_rat @ ( power_power_rat @ A2 @ N2 ) @ A2 ) ) ).
% 0.34/0.66
% 0.34/0.66 % power_Suc2
% 0.34/0.66 thf(fact_6227_power__Suc2,axiom,
% 0.34/0.66 ! [A2: nat,N2: nat] :
% 0.34/0.66 ( ( power_power_nat @ A2 @ ( suc @ N2 ) )
% 0.34/0.66 = ( times_times_nat @ ( power_power_nat @ A2 @ N2 ) @ A2 ) ) ).
% 0.34/0.66
% 0.34/0.66 % power_Suc2
% 0.34/0.66 thf(fact_6228_power__Suc2,axiom,
% 0.34/0.66 ! [A2: int,N2: nat] :
% 0.34/0.66 ( ( power_power_int @ A2 @ ( suc @ N2 ) )
% 0.34/0.66 = ( times_times_int @ ( power_power_int @ A2 @ N2 ) @ A2 ) ) ).
% 0.34/0.66
% 0.34/0.66 % power_Suc2
% 0.34/0.66 thf(fact_6229_power__Suc,axiom,
% 0.34/0.66 ! [A2: assn,N2: nat] :
% 0.34/0.66 ( ( power_power_assn @ A2 @ ( suc @ N2 ) )
% 0.34/0.66 = ( times_times_assn @ A2 @ ( power_power_assn @ A2 @ N2 ) ) ) ).
% 0.34/0.66
% 0.34/0.66 % power_Suc
% 0.34/0.66 thf(fact_6230_power__Suc,axiom,
% 0.34/0.66 ! [A2: rat,N2: nat] :
% 0.34/0.66 ( ( power_power_rat @ A2 @ ( suc @ N2 ) )
% 0.34/0.66 = ( times_times_rat @ A2 @ ( power_power_rat @ A2 @ N2 ) ) ) ).
% 0.34/0.66
% 0.34/0.66 % power_Suc
% 0.34/0.66 thf(fact_6231_power__Suc,axiom,
% 0.34/0.66 ! [A2: nat,N2: nat] :
% 0.34/0.66 ( ( power_power_nat @ A2 @ ( suc @ N2 ) )
% 0.34/0.66 = ( times_times_nat @ A2 @ ( power_power_nat @ A2 @ N2 ) ) ) ).
% 0.34/0.66
% 0.34/0.66 % power_Suc
% 0.34/0.66 thf(fact_6232_power__Suc,axiom,
% 0.34/0.66 ! [A2: int,N2: nat] :
% 0.34/0.66 ( ( power_power_int @ A2 @ ( suc @ N2 ) )
% 0.34/0.66 = ( times_times_int @ A2 @ ( power_power_int @ A2 @ N2 ) ) ) ).
% 0.34/0.66
% 0.34/0.66 % power_Suc
% 0.34/0.66 thf(fact_6233_unit__mult__right__cancel,axiom,
% 0.34/0.66 ! [A2: code_integer,B: code_integer,C: code_integer] :
% 0.34/0.66 ( ( dvd_dvd_Code_integer @ A2 @ one_one_Code_integer )
% 0.34/0.66 => ( ( ( times_3573771949741848930nteger @ B @ A2 )
% 0.34/0.66 = ( times_3573771949741848930nteger @ C @ A2 ) )
% 0.34/0.66 = ( B = C ) ) ) ).
% 0.34/0.66
% 0.34/0.66 % unit_mult_right_cancel
% 0.34/0.66 thf(fact_6234_unit__mult__right__cancel,axiom,
% 0.34/0.66 ! [A2: code_natural,B: code_natural,C: code_natural] :
% 0.34/0.66 ( ( dvd_dvd_Code_natural @ A2 @ one_one_Code_natural )
% 0.34/0.66 => ( ( ( times_2397367101498566445atural @ B @ A2 )
% 0.34/0.66 = ( times_2397367101498566445atural @ C @ A2 ) )
% 0.34/0.66 = ( B = C ) ) ) ).
% 0.34/0.66
% 0.34/0.66 % unit_mult_right_cancel
% 0.34/0.66 thf(fact_6235_unit__mult__right__cancel,axiom,
% 0.34/0.66 ! [A2: nat,B: nat,C: nat] :
% 0.34/0.66 ( ( dvd_dvd_nat @ A2 @ one_one_nat )
% 0.34/0.66 => ( ( ( times_times_nat @ B @ A2 )
% 0.34/0.66 = ( times_times_nat @ C @ A2 ) )
% 0.34/0.66 = ( B = C ) ) ) ).
% 0.34/0.66
% 0.34/0.66 % unit_mult_right_cancel
% 0.34/0.66 thf(fact_6236_unit__mult__right__cancel,axiom,
% 0.34/0.66 ! [A2: int,B: int,C: int] :
% 0.34/0.66 ( ( dvd_dvd_int @ A2 @ one_one_int )
% 0.34/0.66 => ( ( ( times_times_int @ B @ A2 )
% 0.34/0.66 = ( times_times_int @ C @ A2 ) )
% 0.34/0.66 = ( B = C ) ) ) ).
% 0.34/0.66
% 0.34/0.66 % unit_mult_right_cancel
% 0.34/0.66 thf(fact_6237_unit__mult__left__cancel,axiom,
% 0.34/0.66 ! [A2: code_integer,B: code_integer,C: code_integer] :
% 0.34/0.66 ( ( dvd_dvd_Code_integer @ A2 @ one_one_Code_integer )
% 0.34/0.66 => ( ( ( times_3573771949741848930nteger @ A2 @ B )
% 0.34/0.66 = ( times_3573771949741848930nteger @ A2 @ C ) )
% 0.34/0.66 = ( B = C ) ) ) ).
% 0.34/0.66
% 0.34/0.66 % unit_mult_left_cancel
% 0.34/0.66 thf(fact_6238_unit__mult__left__cancel,axiom,
% 0.34/0.66 ! [A2: code_natural,B: code_natural,C: code_natural] :
% 0.34/0.66 ( ( dvd_dvd_Code_natural @ A2 @ one_one_Code_natural )
% 0.34/0.66 => ( ( ( times_2397367101498566445atural @ A2 @ B )
% 0.34/0.66 = ( times_2397367101498566445atural @ A2 @ C ) )
% 0.34/0.66 = ( B = C ) ) ) ).
% 0.34/0.66
% 0.34/0.66 % unit_mult_left_cancel
% 0.34/0.66 thf(fact_6239_unit__mult__left__cancel,axiom,
% 0.34/0.66 ! [A2: nat,B: nat,C: nat] :
% 0.34/0.66 ( ( dvd_dvd_nat @ A2 @ one_one_nat )
% 0.34/0.66 => ( ( ( times_times_nat @ A2 @ B )
% 0.34/0.66 = ( times_times_nat @ A2 @ C ) )
% 0.34/0.66 = ( B = C ) ) ) ).
% 0.34/0.66
% 0.34/0.66 % unit_mult_left_cancel
% 0.34/0.66 thf(fact_6240_unit__mult__left__cancel,axiom,
% 0.34/0.66 ! [A2: int,B: int,C: int] :
% 0.34/0.66 ( ( dvd_dvd_int @ A2 @ one_one_int )
% 0.34/0.66 => ( ( ( times_times_int @ A2 @ B )
% 0.34/0.66 = ( times_times_int @ A2 @ C ) )
% 0.34/0.66 = ( B = C ) ) ) ).
% 0.34/0.66
% 0.34/0.66 % unit_mult_left_cancel
% 0.34/0.66 thf(fact_6241_mult__unit__dvd__iff_H,axiom,
% 0.34/0.66 ! [A2: code_integer,B: code_integer,C: code_integer] :
% 0.34/0.66 ( ( dvd_dvd_Code_integer @ A2 @ one_one_Code_integer )
% 0.34/0.66 => ( ( dvd_dvd_Code_integer @ ( times_3573771949741848930nteger @ A2 @ B ) @ C )
% 0.34/0.66 = ( dvd_dvd_Code_integer @ B @ C ) ) ) ).
% 0.34/0.66
% 0.34/0.66 % mult_unit_dvd_iff'
% 0.34/0.66 thf(fact_6242_mult__unit__dvd__iff_H,axiom,
% 0.34/0.66 ! [A2: code_natural,B: code_natural,C: code_natural] :
% 0.34/0.66 ( ( dvd_dvd_Code_natural @ A2 @ one_one_Code_natural )
% 0.34/0.66 => ( ( dvd_dvd_Code_natural @ ( times_2397367101498566445atural @ A2 @ B ) @ C )
% 0.34/0.66 = ( dvd_dvd_Code_natural @ B @ C ) ) ) ).
% 0.34/0.66
% 0.34/0.66 % mult_unit_dvd_iff'
% 0.34/0.66 thf(fact_6243_mult__unit__dvd__iff_H,axiom,
% 0.34/0.66 ! [A2: nat,B: nat,C: nat] :
% 0.34/0.66 ( ( dvd_dvd_nat @ A2 @ one_one_nat )
% 0.34/0.66 => ( ( dvd_dvd_nat @ ( times_times_nat @ A2 @ B ) @ C )
% 0.34/0.66 = ( dvd_dvd_nat @ B @ C ) ) ) ).
% 0.34/0.66
% 0.34/0.66 % mult_unit_dvd_iff'
% 0.34/0.66 thf(fact_6244_mult__unit__dvd__iff_H,axiom,
% 0.34/0.66 ! [A2: int,B: int,C: int] :
% 0.34/0.66 ( ( dvd_dvd_int @ A2 @ one_one_int )
% 0.34/0.66 => ( ( dvd_dvd_int @ ( times_times_int @ A2 @ B ) @ C )
% 0.34/0.66 = ( dvd_dvd_int @ B @ C ) ) ) ).
% 0.34/0.66
% 0.34/0.66 % mult_unit_dvd_iff'
% 0.34/0.66 thf(fact_6245_dvd__mult__unit__iff_H,axiom,
% 0.34/0.66 ! [B: code_integer,A2: code_integer,C: code_integer] :
% 0.34/0.66 ( ( dvd_dvd_Code_integer @ B @ one_one_Code_integer )
% 0.34/0.66 => ( ( dvd_dvd_Code_integer @ A2 @ ( times_3573771949741848930nteger @ B @ C ) )
% 0.34/0.66 = ( dvd_dvd_Code_integer @ A2 @ C ) ) ) ).
% 0.34/0.66
% 0.34/0.66 % dvd_mult_unit_iff'
% 0.34/0.66 thf(fact_6246_dvd__mult__unit__iff_H,axiom,
% 0.34/0.66 ! [B: code_natural,A2: code_natural,C: code_natural] :
% 0.34/0.66 ( ( dvd_dvd_Code_natural @ B @ one_one_Code_natural )
% 0.34/0.66 => ( ( dvd_dvd_Code_natural @ A2 @ ( times_2397367101498566445atural @ B @ C ) )
% 0.34/0.66 = ( dvd_dvd_Code_natural @ A2 @ C ) ) ) ).
% 0.34/0.66
% 0.34/0.66 % dvd_mult_unit_iff'
% 0.34/0.66 thf(fact_6247_dvd__mult__unit__iff_H,axiom,
% 0.34/0.66 ! [B: nat,A2: nat,C: nat] :
% 0.34/0.66 ( ( dvd_dvd_nat @ B @ one_one_nat )
% 0.34/0.66 => ( ( dvd_dvd_nat @ A2 @ ( times_times_nat @ B @ C ) )
% 0.34/0.66 = ( dvd_dvd_nat @ A2 @ C ) ) ) ).
% 0.34/0.66
% 0.34/0.66 % dvd_mult_unit_iff'
% 0.34/0.66 thf(fact_6248_dvd__mult__unit__iff_H,axiom,
% 0.34/0.66 ! [B: int,A2: int,C: int] :
% 0.34/0.66 ( ( dvd_dvd_int @ B @ one_one_int )
% 0.34/0.66 => ( ( dvd_dvd_int @ A2 @ ( times_times_int @ B @ C ) )
% 0.34/0.66 = ( dvd_dvd_int @ A2 @ C ) ) ) ).
% 0.34/0.66
% 0.34/0.66 % dvd_mult_unit_iff'
% 0.34/0.66 thf(fact_6249_mult__unit__dvd__iff,axiom,
% 0.34/0.66 ! [B: code_integer,A2: code_integer,C: code_integer] :
% 0.34/0.66 ( ( dvd_dvd_Code_integer @ B @ one_one_Code_integer )
% 0.34/0.66 => ( ( dvd_dvd_Code_integer @ ( times_3573771949741848930nteger @ A2 @ B ) @ C )
% 0.34/0.66 = ( dvd_dvd_Code_integer @ A2 @ C ) ) ) ).
% 0.34/0.66
% 0.34/0.66 % mult_unit_dvd_iff
% 0.34/0.66 thf(fact_6250_mult__unit__dvd__iff,axiom,
% 0.34/0.66 ! [B: code_natural,A2: code_natural,C: code_natural] :
% 0.34/0.66 ( ( dvd_dvd_Code_natural @ B @ one_one_Code_natural )
% 0.34/0.66 => ( ( dvd_dvd_Code_natural @ ( times_2397367101498566445atural @ A2 @ B ) @ C )
% 0.34/0.66 = ( dvd_dvd_Code_natural @ A2 @ C ) ) ) ).
% 0.34/0.66
% 0.34/0.66 % mult_unit_dvd_iff
% 0.34/0.66 thf(fact_6251_mult__unit__dvd__iff,axiom,
% 0.34/0.66 ! [B: nat,A2: nat,C: nat] :
% 0.34/0.66 ( ( dvd_dvd_nat @ B @ one_one_nat )
% 0.34/0.66 => ( ( dvd_dvd_nat @ ( times_times_nat @ A2 @ B ) @ C )
% 0.34/0.66 = ( dvd_dvd_nat @ A2 @ C ) ) ) ).
% 0.34/0.66
% 0.34/0.66 % mult_unit_dvd_iff
% 0.34/0.66 thf(fact_6252_mult__unit__dvd__iff,axiom,
% 0.34/0.66 ! [B: int,A2: int,C: int] :
% 0.34/0.66 ( ( dvd_dvd_int @ B @ one_one_int )
% 0.34/0.66 => ( ( dvd_dvd_int @ ( times_times_int @ A2 @ B ) @ C )
% 0.34/0.66 = ( dvd_dvd_int @ A2 @ C ) ) ) ).
% 0.34/0.66
% 0.34/0.66 % mult_unit_dvd_iff
% 0.34/0.66 thf(fact_6253_dvd__mult__unit__iff,axiom,
% 0.34/0.66 ! [B: code_integer,A2: code_integer,C: code_integer] :
% 0.34/0.66 ( ( dvd_dvd_Code_integer @ B @ one_one_Code_integer )
% 0.34/0.66 => ( ( dvd_dvd_Code_integer @ A2 @ ( times_3573771949741848930nteger @ C @ B ) )
% 0.34/0.66 = ( dvd_dvd_Code_integer @ A2 @ C ) ) ) ).
% 0.34/0.66
% 0.34/0.66 % dvd_mult_unit_iff
% 0.34/0.66 thf(fact_6254_dvd__mult__unit__iff,axiom,
% 0.34/0.66 ! [B: code_natural,A2: code_natural,C: code_natural] :
% 0.34/0.66 ( ( dvd_dvd_Code_natural @ B @ one_one_Code_natural )
% 0.34/0.66 => ( ( dvd_dvd_Code_natural @ A2 @ ( times_2397367101498566445atural @ C @ B ) )
% 0.34/0.66 = ( dvd_dvd_Code_natural @ A2 @ C ) ) ) ).
% 0.34/0.66
% 0.34/0.66 % dvd_mult_unit_iff
% 0.34/0.66 thf(fact_6255_dvd__mult__unit__iff,axiom,
% 0.34/0.66 ! [B: nat,A2: nat,C: nat] :
% 0.34/0.66 ( ( dvd_dvd_nat @ B @ one_one_nat )
% 0.34/0.66 => ( ( dvd_dvd_nat @ A2 @ ( times_times_nat @ C @ B ) )
% 0.34/0.66 = ( dvd_dvd_nat @ A2 @ C ) ) ) ).
% 0.34/0.66
% 0.34/0.66 % dvd_mult_unit_iff
% 0.34/0.66 thf(fact_6256_dvd__mult__unit__iff,axiom,
% 0.34/0.66 ! [B: int,A2: int,C: int] :
% 0.34/0.66 ( ( dvd_dvd_int @ B @ one_one_int )
% 0.34/0.66 => ( ( dvd_dvd_int @ A2 @ ( times_times_int @ C @ B ) )
% 0.34/0.66 = ( dvd_dvd_int @ A2 @ C ) ) ) ).
% 0.34/0.66
% 0.34/0.66 % dvd_mult_unit_iff
% 0.34/0.66 thf(fact_6257_is__unit__mult__iff,axiom,
% 0.34/0.66 ! [A2: code_integer,B: code_integer] :
% 0.34/0.66 ( ( dvd_dvd_Code_integer @ ( times_3573771949741848930nteger @ A2 @ B ) @ one_one_Code_integer )
% 0.34/0.66 = ( ( dvd_dvd_Code_integer @ A2 @ one_one_Code_integer )
% 0.34/0.66 & ( dvd_dvd_Code_integer @ B @ one_one_Code_integer ) ) ) ).
% 0.34/0.66
% 0.34/0.66 % is_unit_mult_iff
% 0.34/0.66 thf(fact_6258_is__unit__mult__iff,axiom,
% 0.34/0.66 ! [A2: code_natural,B: code_natural] :
% 0.34/0.66 ( ( dvd_dvd_Code_natural @ ( times_2397367101498566445atural @ A2 @ B ) @ one_one_Code_natural )
% 0.34/0.66 = ( ( dvd_dvd_Code_natural @ A2 @ one_one_Code_natural )
% 0.34/0.66 & ( dvd_dvd_Code_natural @ B @ one_one_Code_natural ) ) ) ).
% 0.34/0.66
% 0.34/0.66 % is_unit_mult_iff
% 0.34/0.66 thf(fact_6259_is__unit__mult__iff,axiom,
% 0.34/0.66 ! [A2: nat,B: nat] :
% 0.34/0.66 ( ( dvd_dvd_nat @ ( times_times_nat @ A2 @ B ) @ one_one_nat )
% 0.34/0.66 = ( ( dvd_dvd_nat @ A2 @ one_one_nat )
% 0.34/0.66 & ( dvd_dvd_nat @ B @ one_one_nat ) ) ) ).
% 0.34/0.66
% 0.34/0.66 % is_unit_mult_iff
% 0.34/0.66 thf(fact_6260_is__unit__mult__iff,axiom,
% 0.34/0.66 ! [A2: int,B: int] :
% 0.34/0.66 ( ( dvd_dvd_int @ ( times_times_int @ A2 @ B ) @ one_one_int )
% 0.34/0.66 = ( ( dvd_dvd_int @ A2 @ one_one_int )
% 0.34/0.66 & ( dvd_dvd_int @ B @ one_one_int ) ) ) ).
% 0.34/0.66
% 0.34/0.66 % is_unit_mult_iff
% 0.34/0.66 thf(fact_6261_power__0,axiom,
% 0.34/0.66 ! [A2: assn] :
% 0.34/0.66 ( ( power_power_assn @ A2 @ zero_zero_nat )
% 0.34/0.66 = one_one_assn ) ).
% 0.34/0.66
% 0.34/0.66 % power_0
% 0.34/0.66 thf(fact_6262_power__0,axiom,
% 0.34/0.66 ! [A2: code_integer] :
% 0.34/0.66 ( ( power_8256067586552552935nteger @ A2 @ zero_zero_nat )
% 0.34/0.66 = one_one_Code_integer ) ).
% 0.34/0.66
% 0.34/0.66 % power_0
% 0.34/0.66 thf(fact_6263_power__0,axiom,
% 0.34/0.66 ! [A2: code_natural] :
% 0.34/0.66 ( ( power_7079662738309270450atural @ A2 @ zero_zero_nat )
% 0.34/0.66 = one_one_Code_natural ) ).
% 0.34/0.66
% 0.34/0.66 % power_0
% 0.34/0.66 thf(fact_6264_power__0,axiom,
% 0.34/0.66 ! [A2: int] :
% 0.34/0.66 ( ( power_power_int @ A2 @ zero_zero_nat )
% 0.34/0.66 = one_one_int ) ).
% 0.34/0.66
% 0.34/0.66 % power_0
% 0.34/0.66 thf(fact_6265_power__0,axiom,
% 0.34/0.66 ! [A2: nat] :
% 0.34/0.66 ( ( power_power_nat @ A2 @ zero_zero_nat )
% 0.34/0.66 = one_one_nat ) ).
% 0.34/0.66
% 0.34/0.66 % power_0
% 0.34/0.66 thf(fact_6266_power__add,axiom,
% 0.34/0.66 ! [A2: assn,M4: nat,N2: nat] :
% 0.34/0.66 ( ( power_power_assn @ A2 @ ( plus_plus_nat @ M4 @ N2 ) )
% 0.34/0.66 = ( times_times_assn @ ( power_power_assn @ A2 @ M4 ) @ ( power_power_assn @ A2 @ N2 ) ) ) ).
% 0.34/0.66
% 0.34/0.66 % power_add
% 0.34/0.66 thf(fact_6267_power__add,axiom,
% 0.34/0.66 ! [A2: rat,M4: nat,N2: nat] :
% 0.34/0.66 ( ( power_power_rat @ A2 @ ( plus_plus_nat @ M4 @ N2 ) )
% 0.34/0.66 = ( times_times_rat @ ( power_power_rat @ A2 @ M4 ) @ ( power_power_rat @ A2 @ N2 ) ) ) ).
% 0.34/0.66
% 0.34/0.66 % power_add
% 0.34/0.66 thf(fact_6268_power__add,axiom,
% 0.34/0.66 ! [A2: nat,M4: nat,N2: nat] :
% 0.34/0.66 ( ( power_power_nat @ A2 @ ( plus_plus_nat @ M4 @ N2 ) )
% 0.34/0.66 = ( times_times_nat @ ( power_power_nat @ A2 @ M4 ) @ ( power_power_nat @ A2 @ N2 ) ) ) ).
% 0.34/0.66
% 0.34/0.66 % power_add
% 0.34/0.66 thf(fact_6269_power__add,axiom,
% 0.34/0.66 ! [A2: int,M4: nat,N2: nat] :
% 0.34/0.66 ( ( power_power_int @ A2 @ ( plus_plus_nat @ M4 @ N2 ) )
% 0.34/0.66 = ( times_times_int @ ( power_power_int @ A2 @ M4 ) @ ( power_power_int @ A2 @ N2 ) ) ) ).
% 0.34/0.66
% 0.34/0.66 % power_add
% 0.34/0.66 thf(fact_6270_gbinomial__addition__formula,axiom,
% 0.34/0.66 ! [A2: rat,K3: nat] :
% 0.34/0.66 ( ( gbinomial_rat @ A2 @ ( suc @ K3 ) )
% 0.34/0.66 = ( plus_plus_rat @ ( gbinomial_rat @ ( minus_minus_rat @ A2 @ one_one_rat ) @ ( suc @ K3 ) ) @ ( gbinomial_rat @ ( minus_minus_rat @ A2 @ one_one_rat ) @ K3 ) ) ) ).
% 0.34/0.66
% 0.34/0.66 % gbinomial_addition_formula
% 0.34/0.66 thf(fact_6271_power__le__one,axiom,
% 0.34/0.66 ! [A2: code_natural,N2: nat] :
% 0.34/0.66 ( ( ord_le1926595141338095240atural @ zero_z2226904508553997617atural @ A2 )
% 0.34/0.66 => ( ( ord_le1926595141338095240atural @ A2 @ one_one_Code_natural )
% 0.34/0.66 => ( ord_le1926595141338095240atural @ ( power_7079662738309270450atural @ A2 @ N2 ) @ one_one_Code_natural ) ) ) ).
% 0.34/0.66
% 0.34/0.66 % power_le_one
% 0.34/0.66 thf(fact_6272_power__le__one,axiom,
% 0.34/0.66 ! [A2: code_integer,N2: nat] :
% 0.34/0.66 ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ A2 )
% 0.34/0.66 => ( ( ord_le3102999989581377725nteger @ A2 @ one_one_Code_integer )
% 0.34/0.66 => ( ord_le3102999989581377725nteger @ ( power_8256067586552552935nteger @ A2 @ N2 ) @ one_one_Code_integer ) ) ) ).
% 0.34/0.66
% 0.34/0.66 % power_le_one
% 0.34/0.66 thf(fact_6273_power__le__one,axiom,
% 0.34/0.66 ! [A2: rat,N2: nat] :
% 0.34/0.66 ( ( ord_less_eq_rat @ zero_zero_rat @ A2 )
% 0.34/0.66 => ( ( ord_less_eq_rat @ A2 @ one_one_rat )
% 0.34/0.66 => ( ord_less_eq_rat @ ( power_power_rat @ A2 @ N2 ) @ one_one_rat ) ) ) ).
% 0.34/0.66
% 0.34/0.66 % power_le_one
% 0.34/0.66 thf(fact_6274_power__le__one,axiom,
% 0.34/0.66 ! [A2: nat,N2: nat] :
% 0.34/0.66 ( ( ord_less_eq_nat @ zero_zero_nat @ A2 )
% 0.34/0.66 => ( ( ord_less_eq_nat @ A2 @ one_one_nat )
% 0.34/0.66 => ( ord_less_eq_nat @ ( power_power_nat @ A2 @ N2 ) @ one_one_nat ) ) ) ).
% 0.34/0.66
% 0.34/0.66 % power_le_one
% 0.34/0.66 thf(fact_6275_power__le__one,axiom,
% 0.34/0.66 ! [A2: int,N2: nat] :
% 0.34/0.66 ( ( ord_less_eq_int @ zero_zero_int @ A2 )
% 0.34/0.66 => ( ( ord_less_eq_int @ A2 @ one_one_int )
% 0.34/0.66 => ( ord_less_eq_int @ ( power_power_int @ A2 @ N2 ) @ one_one_int ) ) ) ).
% 0.34/0.66
% 0.34/0.66 % power_le_one
% 0.34/0.66 thf(fact_6276_power__less__power__Suc,axiom,
% 0.34/0.66 ! [A2: code_natural,N2: nat] :
% 0.34/0.66 ( ( ord_le5570908160329646204atural @ one_one_Code_natural @ A2 )
% 0.34/0.66 => ( ord_le5570908160329646204atural @ ( power_7079662738309270450atural @ A2 @ N2 ) @ ( times_2397367101498566445atural @ A2 @ ( power_7079662738309270450atural @ A2 @ N2 ) ) ) ) ).
% 0.34/0.66
% 0.34/0.66 % power_less_power_Suc
% 0.34/0.66 thf(fact_6277_power__less__power__Suc,axiom,
% 0.34/0.66 ! [A2: nat,N2: nat] :
% 0.34/0.66 ( ( ord_less_nat @ one_one_nat @ A2 )
% 0.34/0.66 => ( ord_less_nat @ ( power_power_nat @ A2 @ N2 ) @ ( times_times_nat @ A2 @ ( power_power_nat @ A2 @ N2 ) ) ) ) ).
% 0.34/0.66
% 0.34/0.66 % power_less_power_Suc
% 0.34/0.66 thf(fact_6278_power__less__power__Suc,axiom,
% 0.34/0.66 ! [A2: int,N2: nat] :
% 0.34/0.66 ( ( ord_less_int @ one_one_int @ A2 )
% 0.34/0.66 => ( ord_less_int @ ( power_power_int @ A2 @ N2 ) @ ( times_times_int @ A2 @ ( power_power_int @ A2 @ N2 ) ) ) ) ).
% 0.34/0.66
% 0.34/0.66 % power_less_power_Suc
% 0.34/0.66 thf(fact_6279_power__less__power__Suc,axiom,
% 0.34/0.66 ! [A2: code_integer,N2: nat] :
% 0.34/0.66 ( ( ord_le6747313008572928689nteger @ one_one_Code_integer @ A2 )
% 0.34/0.66 => ( ord_le6747313008572928689nteger @ ( power_8256067586552552935nteger @ A2 @ N2 ) @ ( times_3573771949741848930nteger @ A2 @ ( power_8256067586552552935nteger @ A2 @ N2 ) ) ) ) ).
% 0.34/0.66
% 0.34/0.66 % power_less_power_Suc
% 0.34/0.66 thf(fact_6280_power__less__power__Suc,axiom,
% 0.34/0.66 ! [A2: rat,N2: nat] :
% 0.34/0.66 ( ( ord_less_rat @ one_one_rat @ A2 )
% 0.34/0.66 => ( ord_less_rat @ ( power_power_rat @ A2 @ N2 ) @ ( times_times_rat @ A2 @ ( power_power_rat @ A2 @ N2 ) ) ) ) ).
% 0.34/0.66
% 0.34/0.66 % power_less_power_Suc
% 0.34/0.66 thf(fact_6281_power__gt1__lemma,axiom,
% 0.34/0.66 ! [A2: code_natural,N2: nat] :
% 0.34/0.66 ( ( ord_le5570908160329646204atural @ one_one_Code_natural @ A2 )
% 0.34/0.66 => ( ord_le5570908160329646204atural @ one_one_Code_natural @ ( times_2397367101498566445atural @ A2 @ ( power_7079662738309270450atural @ A2 @ N2 ) ) ) ) ).
% 0.34/0.66
% 0.34/0.66 % power_gt1_lemma
% 0.34/0.66 thf(fact_6282_power__gt1__lemma,axiom,
% 0.34/0.66 ! [A2: nat,N2: nat] :
% 0.34/0.66 ( ( ord_less_nat @ one_one_nat @ A2 )
% 0.34/0.66 => ( ord_less_nat @ one_one_nat @ ( times_times_nat @ A2 @ ( power_power_nat @ A2 @ N2 ) ) ) ) ).
% 0.34/0.66
% 0.34/0.66 % power_gt1_lemma
% 0.34/0.66 thf(fact_6283_power__gt1__lemma,axiom,
% 0.34/0.66 ! [A2: int,N2: nat] :
% 0.34/0.66 ( ( ord_less_int @ one_one_int @ A2 )
% 0.34/0.66 => ( ord_less_int @ one_one_int @ ( times_times_int @ A2 @ ( power_power_int @ A2 @ N2 ) ) ) ) ).
% 0.34/0.66
% 0.34/0.66 % power_gt1_lemma
% 0.34/0.66 thf(fact_6284_power__gt1__lemma,axiom,
% 0.34/0.66 ! [A2: code_integer,N2: nat] :
% 0.34/0.66 ( ( ord_le6747313008572928689nteger @ one_one_Code_integer @ A2 )
% 0.34/0.66 => ( ord_le6747313008572928689nteger @ one_one_Code_integer @ ( times_3573771949741848930nteger @ A2 @ ( power_8256067586552552935nteger @ A2 @ N2 ) ) ) ) ).
% 0.34/0.66
% 0.34/0.66 % power_gt1_lemma
% 0.34/0.66 thf(fact_6285_power__gt1__lemma,axiom,
% 0.34/0.66 ! [A2: rat,N2: nat] :
% 0.34/0.66 ( ( ord_less_rat @ one_one_rat @ A2 )
% 0.34/0.66 => ( ord_less_rat @ one_one_rat @ ( times_times_rat @ A2 @ ( power_power_rat @ A2 @ N2 ) ) ) ) ).
% 0.34/0.66
% 0.34/0.66 % power_gt1_lemma
% 0.34/0.66 thf(fact_6286_unity__coeff__ex,axiom,
% 0.34/0.66 ! [P: code_integer > $o,L: code_integer] :
% 0.34/0.66 ( ( ? [X2: code_integer] : ( P @ ( times_3573771949741848930nteger @ L @ X2 ) ) )
% 0.34/0.66 = ( ? [X2: code_integer] :
% 0.34/0.66 ( ( dvd_dvd_Code_integer @ L @ ( plus_p5714425477246183910nteger @ X2 @ zero_z3403309356797280102nteger ) )
% 0.34/0.66 & ( P @ X2 ) ) ) ) ).
% 0.34/0.66
% 0.34/0.66 % unity_coeff_ex
% 0.34/0.66 thf(fact_6287_unity__coeff__ex,axiom,
% 0.34/0.66 ! [P: rat > $o,L: rat] :
% 0.34/0.66 ( ( ? [X2: rat] : ( P @ ( times_times_rat @ L @ X2 ) ) )
% 0.34/0.66 = ( ? [X2: rat] :
% 0.34/0.66 ( ( dvd_dvd_rat @ L @ ( plus_plus_rat @ X2 @ zero_zero_rat ) )
% 0.34/0.66 & ( P @ X2 ) ) ) ) ).
% 0.34/0.66
% 0.34/0.66 % unity_coeff_ex
% 0.34/0.66 thf(fact_6288_unity__coeff__ex,axiom,
% 0.34/0.66 ! [P: nat > $o,L: nat] :
% 0.34/0.66 ( ( ? [X2: nat] : ( P @ ( times_times_nat @ L @ X2 ) ) )
% 0.34/0.66 = ( ? [X2: nat] :
% 0.34/0.66 ( ( dvd_dvd_nat @ L @ ( plus_plus_nat @ X2 @ zero_zero_nat ) )
% 0.34/0.66 & ( P @ X2 ) ) ) ) ).
% 0.34/0.66
% 0.34/0.66 % unity_coeff_ex
% 0.34/0.66 thf(fact_6289_unity__coeff__ex,axiom,
% 0.34/0.66 ! [P: int > $o,L: int] :
% 0.34/0.66 ( ( ? [X2: int] : ( P @ ( times_times_int @ L @ X2 ) ) )
% 0.34/0.66 = ( ? [X2: int] :
% 0.34/0.66 ( ( dvd_dvd_int @ L @ ( plus_plus_int @ X2 @ zero_zero_int ) )
% 0.34/0.66 & ( P @ X2 ) ) ) ) ).
% 0.34/0.66
% 0.34/0.66 % unity_coeff_ex
% 0.34/0.66 thf(fact_6290_power__0__left,axiom,
% 0.34/0.66 ! [N2: nat] :
% 0.34/0.66 ( ( ( N2 = zero_zero_nat )
% 0.34/0.66 => ( ( power_7079662738309270450atural @ zero_z2226904508553997617atural @ N2 )
% 0.34/0.66 = one_one_Code_natural ) )
% 0.34/0.66 & ( ( N2 != zero_zero_nat )
% 0.34/0.66 => ( ( power_7079662738309270450atural @ zero_z2226904508553997617atural @ N2 )
% 0.34/0.66 = zero_z2226904508553997617atural ) ) ) ).
% 0.34/0.66
% 0.34/0.66 % power_0_left
% 0.34/0.66 thf(fact_6291_power__0__left,axiom,
% 0.34/0.66 ! [N2: nat] :
% 0.34/0.66 ( ( ( N2 = zero_zero_nat )
% 0.34/0.66 => ( ( power_8256067586552552935nteger @ zero_z3403309356797280102nteger @ N2 )
% 0.34/0.66 = one_one_Code_integer ) )
% 0.34/0.66 & ( ( N2 != zero_zero_nat )
% 0.34/0.66 => ( ( power_8256067586552552935nteger @ zero_z3403309356797280102nteger @ N2 )
% 0.34/0.66 = zero_z3403309356797280102nteger ) ) ) ).
% 0.34/0.66
% 0.34/0.66 % power_0_left
% 0.34/0.66 thf(fact_6292_power__0__left,axiom,
% 0.34/0.66 ! [N2: nat] :
% 0.34/0.66 ( ( ( N2 = zero_zero_nat )
% 0.34/0.66 => ( ( power_power_int @ zero_zero_int @ N2 )
% 0.34/0.66 = one_one_int ) )
% 0.34/0.66 & ( ( N2 != zero_zero_nat )
% 0.34/0.66 => ( ( power_power_int @ zero_zero_int @ N2 )
% 0.34/0.66 = zero_zero_int ) ) ) ).
% 0.34/0.66
% 0.34/0.66 % power_0_left
% 0.34/0.66 thf(fact_6293_power__0__left,axiom,
% 0.34/0.66 ! [N2: nat] :
% 0.34/0.66 ( ( ( N2 = zero_zero_nat )
% 0.34/0.66 => ( ( power_power_nat @ zero_zero_nat @ N2 )
% 0.34/0.66 = one_one_nat ) )
% 0.34/0.66 & ( ( N2 != zero_zero_nat )
% 0.34/0.66 => ( ( power_power_nat @ zero_zero_nat @ N2 )
% 0.34/0.66 = zero_zero_nat ) ) ) ).
% 0.34/0.66
% 0.34/0.66 % power_0_left
% 0.34/0.66 thf(fact_6294_unit__dvdE,axiom,
% 0.34/0.66 ! [A2: code_natural,B: code_natural] :
% 0.34/0.66 ( ( dvd_dvd_Code_natural @ A2 @ one_one_Code_natural )
% 0.34/0.66 => ~ ( ( A2 != zero_z2226904508553997617atural )
% 0.34/0.66 => ! [C2: code_natural] :
% 0.34/0.66 ( B
% 0.34/0.66 != ( times_2397367101498566445atural @ A2 @ C2 ) ) ) ) ).
% 0.34/0.66
% 0.34/0.66 % unit_dvdE
% 0.34/0.66 thf(fact_6295_unit__dvdE,axiom,
% 0.34/0.66 ! [A2: code_integer,B: code_integer] :
% 0.34/0.66 ( ( dvd_dvd_Code_integer @ A2 @ one_one_Code_integer )
% 0.34/0.66 => ~ ( ( A2 != zero_z3403309356797280102nteger )
% 0.34/0.66 => ! [C2: code_integer] :
% 0.34/0.66 ( B
% 0.34/0.66 != ( times_3573771949741848930nteger @ A2 @ C2 ) ) ) ) ).
% 0.34/0.66
% 0.34/0.66 % unit_dvdE
% 0.34/0.66 thf(fact_6296_unit__dvdE,axiom,
% 0.34/0.66 ! [A2: nat,B: nat] :
% 0.34/0.66 ( ( dvd_dvd_nat @ A2 @ one_one_nat )
% 0.34/0.66 => ~ ( ( A2 != zero_zero_nat )
% 0.34/0.66 => ! [C2: nat] :
% 0.34/0.66 ( B
% 0.34/0.66 != ( times_times_nat @ A2 @ C2 ) ) ) ) ).
% 0.34/0.66
% 0.34/0.66 % unit_dvdE
% 0.34/0.66 thf(fact_6297_unit__dvdE,axiom,
% 0.34/0.66 ! [A2: int,B: int] :
% 0.34/0.66 ( ( dvd_dvd_int @ A2 @ one_one_int )
% 0.34/0.66 => ~ ( ( A2 != zero_zero_int )
% 0.34/0.66 => ! [C2: int] :
% 0.34/0.66 ( B
% 0.34/0.66 != ( times_times_int @ A2 @ C2 ) ) ) ) ).
% 0.34/0.66
% 0.34/0.66 % unit_dvdE
% 0.34/0.66 thf(fact_6298_power__gt1,axiom,
% 0.34/0.66 ! [A2: code_natural,N2: nat] :
% 0.34/0.66 ( ( ord_le5570908160329646204atural @ one_one_Code_natural @ A2 )
% 0.34/0.66 => ( ord_le5570908160329646204atural @ one_one_Code_natural @ ( power_7079662738309270450atural @ A2 @ ( suc @ N2 ) ) ) ) ).
% 0.34/0.66
% 0.34/0.66 % power_gt1
% 0.34/0.66 thf(fact_6299_power__gt1,axiom,
% 0.34/0.66 ! [A2: nat,N2: nat] :
% 0.34/0.66 ( ( ord_less_nat @ one_one_nat @ A2 )
% 0.34/0.66 => ( ord_less_nat @ one_one_nat @ ( power_power_nat @ A2 @ ( suc @ N2 ) ) ) ) ).
% 0.34/0.66
% 0.34/0.66 % power_gt1
% 0.34/0.66 thf(fact_6300_power__gt1,axiom,
% 0.34/0.66 ! [A2: int,N2: nat] :
% 0.34/0.66 ( ( ord_less_int @ one_one_int @ A2 )
% 0.34/0.66 => ( ord_less_int @ one_one_int @ ( power_power_int @ A2 @ ( suc @ N2 ) ) ) ) ).
% 0.34/0.66
% 0.34/0.66 % power_gt1
% 0.34/0.66 thf(fact_6301_power__gt1,axiom,
% 0.34/0.66 ! [A2: code_integer,N2: nat] :
% 0.34/0.66 ( ( ord_le6747313008572928689nteger @ one_one_Code_integer @ A2 )
% 0.34/0.66 => ( ord_le6747313008572928689nteger @ one_one_Code_integer @ ( power_8256067586552552935nteger @ A2 @ ( suc @ N2 ) ) ) ) ).
% 0.34/0.66
% 0.34/0.66 % power_gt1
% 0.34/0.66 thf(fact_6302_power__gt1,axiom,
% 0.34/0.66 ! [A2: rat,N2: nat] :
% 0.34/0.66 ( ( ord_less_rat @ one_one_rat @ A2 )
% 0.34/0.66 => ( ord_less_rat @ one_one_rat @ ( power_power_rat @ A2 @ ( suc @ N2 ) ) ) ) ).
% 0.34/0.66
% 0.34/0.66 % power_gt1
% 0.34/0.66 thf(fact_6303_power__strict__increasing,axiom,
% 0.34/0.66 ! [N2: nat,N: nat,A2: code_natural] :
% 0.34/0.66 ( ( ord_less_nat @ N2 @ N )
% 0.34/0.66 => ( ( ord_le5570908160329646204atural @ one_one_Code_natural @ A2 )
% 0.34/0.66 => ( ord_le5570908160329646204atural @ ( power_7079662738309270450atural @ A2 @ N2 ) @ ( power_7079662738309270450atural @ A2 @ N ) ) ) ) ).
% 0.34/0.66
% 0.34/0.66 % power_strict_increasing
% 0.34/0.66 thf(fact_6304_power__strict__increasing,axiom,
% 0.34/0.66 ! [N2: nat,N: nat,A2: nat] :
% 0.34/0.66 ( ( ord_less_nat @ N2 @ N )
% 0.34/0.66 => ( ( ord_less_nat @ one_one_nat @ A2 )
% 0.34/0.66 => ( ord_less_nat @ ( power_power_nat @ A2 @ N2 ) @ ( power_power_nat @ A2 @ N ) ) ) ) ).
% 0.34/0.66
% 0.34/0.66 % power_strict_increasing
% 0.34/0.66 thf(fact_6305_power__strict__increasing,axiom,
% 0.34/0.66 ! [N2: nat,N: nat,A2: int] :
% 0.34/0.66 ( ( ord_less_nat @ N2 @ N )
% 0.34/0.66 => ( ( ord_less_int @ one_one_int @ A2 )
% 0.34/0.66 => ( ord_less_int @ ( power_power_int @ A2 @ N2 ) @ ( power_power_int @ A2 @ N ) ) ) ) ).
% 0.34/0.66
% 0.34/0.66 % power_strict_increasing
% 0.34/0.66 thf(fact_6306_power__strict__increasing,axiom,
% 0.34/0.66 ! [N2: nat,N: nat,A2: code_integer] :
% 0.34/0.66 ( ( ord_less_nat @ N2 @ N )
% 0.34/0.66 => ( ( ord_le6747313008572928689nteger @ one_one_Code_integer @ A2 )
% 0.34/0.66 => ( ord_le6747313008572928689nteger @ ( power_8256067586552552935nteger @ A2 @ N2 ) @ ( power_8256067586552552935nteger @ A2 @ N ) ) ) ) ).
% 0.34/0.66
% 0.34/0.66 % power_strict_increasing
% 0.34/0.66 thf(fact_6307_power__strict__increasing,axiom,
% 0.34/0.66 ! [N2: nat,N: nat,A2: rat] :
% 0.34/0.66 ( ( ord_less_nat @ N2 @ N )
% 0.34/0.66 => ( ( ord_less_rat @ one_one_rat @ A2 )
% 0.34/0.66 => ( ord_less_rat @ ( power_power_rat @ A2 @ N2 ) @ ( power_power_rat @ A2 @ N ) ) ) ) ).
% 0.34/0.66
% 0.34/0.66 % power_strict_increasing
% 0.34/0.66 thf(fact_6308_power__less__imp__less__exp,axiom,
% 0.34/0.66 ! [A2: code_natural,M4: nat,N2: nat] :
% 0.34/0.66 ( ( ord_le5570908160329646204atural @ one_one_Code_natural @ A2 )
% 0.34/0.66 => ( ( ord_le5570908160329646204atural @ ( power_7079662738309270450atural @ A2 @ M4 ) @ ( power_7079662738309270450atural @ A2 @ N2 ) )
% 0.34/0.66 => ( ord_less_nat @ M4 @ N2 ) ) ) ).
% 0.34/0.66
% 0.34/0.66 % power_less_imp_less_exp
% 0.34/0.66 thf(fact_6309_power__less__imp__less__exp,axiom,
% 0.34/0.66 ! [A2: nat,M4: nat,N2: nat] :
% 0.34/0.66 ( ( ord_less_nat @ one_one_nat @ A2 )
% 0.34/0.66 => ( ( ord_less_nat @ ( power_power_nat @ A2 @ M4 ) @ ( power_power_nat @ A2 @ N2 ) )
% 0.34/0.66 => ( ord_less_nat @ M4 @ N2 ) ) ) ).
% 0.34/0.66
% 0.34/0.66 % power_less_imp_less_exp
% 0.34/0.66 thf(fact_6310_power__less__imp__less__exp,axiom,
% 0.34/0.66 ! [A2: int,M4: nat,N2: nat] :
% 0.34/0.66 ( ( ord_less_int @ one_one_int @ A2 )
% 0.34/0.66 => ( ( ord_less_int @ ( power_power_int @ A2 @ M4 ) @ ( power_power_int @ A2 @ N2 ) )
% 0.34/0.66 => ( ord_less_nat @ M4 @ N2 ) ) ) ).
% 0.34/0.66
% 0.34/0.66 % power_less_imp_less_exp
% 0.34/0.66 thf(fact_6311_power__less__imp__less__exp,axiom,
% 0.34/0.66 ! [A2: code_integer,M4: nat,N2: nat] :
% 0.34/0.66 ( ( ord_le6747313008572928689nteger @ one_one_Code_integer @ A2 )
% 0.34/0.66 => ( ( ord_le6747313008572928689nteger @ ( power_8256067586552552935nteger @ A2 @ M4 ) @ ( power_8256067586552552935nteger @ A2 @ N2 ) )
% 0.34/0.66 => ( ord_less_nat @ M4 @ N2 ) ) ) ).
% 0.34/0.66
% 0.34/0.66 % power_less_imp_less_exp
% 0.34/0.66 thf(fact_6312_power__less__imp__less__exp,axiom,
% 0.34/0.66 ! [A2: rat,M4: nat,N2: nat] :
% 0.34/0.66 ( ( ord_less_rat @ one_one_rat @ A2 )
% 0.34/0.66 => ( ( ord_less_rat @ ( power_power_rat @ A2 @ M4 ) @ ( power_power_rat @ A2 @ N2 ) )
% 0.34/0.66 => ( ord_less_nat @ M4 @ N2 ) ) ) ).
% 0.34/0.66
% 0.34/0.66 % power_less_imp_less_exp
% 0.34/0.66 thf(fact_6313_power__increasing,axiom,
% 0.34/0.66 ! [N2: nat,N: nat,A2: code_integer] :
% 0.34/0.66 ( ( ord_less_eq_nat @ N2 @ N )
% 0.34/0.66 => ( ( ord_le3102999989581377725nteger @ one_one_Code_integer @ A2 )
% 0.34/0.66 => ( ord_le3102999989581377725nteger @ ( power_8256067586552552935nteger @ A2 @ N2 ) @ ( power_8256067586552552935nteger @ A2 @ N ) ) ) ) ).
% 0.34/0.66
% 0.34/0.66 % power_increasing
% 0.34/0.66 thf(fact_6314_power__increasing,axiom,
% 0.34/0.66 ! [N2: nat,N: nat,A2: code_natural] :
% 0.34/0.66 ( ( ord_less_eq_nat @ N2 @ N )
% 0.34/0.66 => ( ( ord_le1926595141338095240atural @ one_one_Code_natural @ A2 )
% 0.34/0.66 => ( ord_le1926595141338095240atural @ ( power_7079662738309270450atural @ A2 @ N2 ) @ ( power_7079662738309270450atural @ A2 @ N ) ) ) ) ).
% 0.34/0.66
% 0.34/0.66 % power_increasing
% 0.34/0.66 thf(fact_6315_power__increasing,axiom,
% 0.34/0.66 ! [N2: nat,N: nat,A2: rat] :
% 0.34/0.66 ( ( ord_less_eq_nat @ N2 @ N )
% 0.34/0.66 => ( ( ord_less_eq_rat @ one_one_rat @ A2 )
% 0.34/0.66 => ( ord_less_eq_rat @ ( power_power_rat @ A2 @ N2 ) @ ( power_power_rat @ A2 @ N ) ) ) ) ).
% 0.34/0.66
% 0.34/0.66 % power_increasing
% 0.34/0.66 thf(fact_6316_power__increasing,axiom,
% 0.34/0.66 ! [N2: nat,N: nat,A2: nat] :
% 0.34/0.66 ( ( ord_less_eq_nat @ N2 @ N )
% 0.34/0.66 => ( ( ord_less_eq_nat @ one_one_nat @ A2 )
% 0.34/0.66 => ( ord_less_eq_nat @ ( power_power_nat @ A2 @ N2 ) @ ( power_power_nat @ A2 @ N ) ) ) ) ).
% 0.34/0.66
% 0.34/0.66 % power_increasing
% 0.34/0.66 thf(fact_6317_power__increasing,axiom,
% 0.34/0.66 ! [N2: nat,N: nat,A2: int] :
% 0.34/0.66 ( ( ord_less_eq_nat @ N2 @ N )
% 0.34/0.66 => ( ( ord_less_eq_int @ one_one_int @ A2 )
% 0.34/0.66 => ( ord_less_eq_int @ ( power_power_int @ A2 @ N2 ) @ ( power_power_int @ A2 @ N ) ) ) ) ).
% 0.34/0.66
% 0.34/0.66 % power_increasing
% 0.34/0.66 thf(fact_6318_inf__period_I3_J,axiom,
% 0.34/0.66 ! [D2: code_integer,D3: code_integer,T4: code_integer] :
% 0.34/0.66 ( ( dvd_dvd_Code_integer @ D2 @ D3 )
% 0.34/0.66 => ! [X6: code_integer,K2: code_integer] :
% 0.34/0.66 ( ( dvd_dvd_Code_integer @ D2 @ ( plus_p5714425477246183910nteger @ X6 @ T4 ) )
% 0.34/0.66 = ( dvd_dvd_Code_integer @ D2 @ ( plus_p5714425477246183910nteger @ ( minus_8373710615458151222nteger @ X6 @ ( times_3573771949741848930nteger @ K2 @ D3 ) ) @ T4 ) ) ) ) ).
% 0.34/0.66
% 0.34/0.66 % inf_period(3)
% 0.34/0.66 thf(fact_6319_inf__period_I3_J,axiom,
% 0.34/0.66 ! [D2: rat,D3: rat,T4: rat] :
% 0.34/0.66 ( ( dvd_dvd_rat @ D2 @ D3 )
% 0.34/0.66 => ! [X6: rat,K2: rat] :
% 0.34/0.66 ( ( dvd_dvd_rat @ D2 @ ( plus_plus_rat @ X6 @ T4 ) )
% 0.34/0.66 = ( dvd_dvd_rat @ D2 @ ( plus_plus_rat @ ( minus_minus_rat @ X6 @ ( times_times_rat @ K2 @ D3 ) ) @ T4 ) ) ) ) ).
% 0.34/0.66
% 0.34/0.66 % inf_period(3)
% 0.34/0.66 thf(fact_6320_inf__period_I3_J,axiom,
% 0.34/0.66 ! [D2: int,D3: int,T4: int] :
% 0.34/0.66 ( ( dvd_dvd_int @ D2 @ D3 )
% 0.34/0.66 => ! [X6: int,K2: int] :
% 0.34/0.66 ( ( dvd_dvd_int @ D2 @ ( plus_plus_int @ X6 @ T4 ) )
% 0.34/0.66 = ( dvd_dvd_int @ D2 @ ( plus_plus_int @ ( minus_minus_int @ X6 @ ( times_times_int @ K2 @ D3 ) ) @ T4 ) ) ) ) ).
% 0.34/0.66
% 0.34/0.66 % inf_period(3)
% 0.34/0.66 thf(fact_6321_inf__period_I4_J,axiom,
% 0.34/0.66 ! [D2: code_integer,D3: code_integer,T4: code_integer] :
% 0.34/0.66 ( ( dvd_dvd_Code_integer @ D2 @ D3 )
% 0.34/0.66 => ! [X6: code_integer,K2: code_integer] :
% 0.34/0.66 ( ( ~ ( dvd_dvd_Code_integer @ D2 @ ( plus_p5714425477246183910nteger @ X6 @ T4 ) ) )
% 0.34/0.66 = ( ~ ( dvd_dvd_Code_integer @ D2 @ ( plus_p5714425477246183910nteger @ ( minus_8373710615458151222nteger @ X6 @ ( times_3573771949741848930nteger @ K2 @ D3 ) ) @ T4 ) ) ) ) ) ).
% 0.34/0.66
% 0.34/0.66 % inf_period(4)
% 0.34/0.66 thf(fact_6322_inf__period_I4_J,axiom,
% 0.34/0.66 ! [D2: rat,D3: rat,T4: rat] :
% 0.34/0.66 ( ( dvd_dvd_rat @ D2 @ D3 )
% 0.34/0.66 => ! [X6: rat,K2: rat] :
% 0.34/0.66 ( ( ~ ( dvd_dvd_rat @ D2 @ ( plus_plus_rat @ X6 @ T4 ) ) )
% 0.34/0.66 = ( ~ ( dvd_dvd_rat @ D2 @ ( plus_plus_rat @ ( minus_minus_rat @ X6 @ ( times_times_rat @ K2 @ D3 ) ) @ T4 ) ) ) ) ) ).
% 0.34/0.66
% 0.34/0.66 % inf_period(4)
% 0.34/0.66 thf(fact_6323_inf__period_I4_J,axiom,
% 0.34/0.66 ! [D2: int,D3: int,T4: int] :
% 0.34/0.66 ( ( dvd_dvd_int @ D2 @ D3 )
% 0.34/0.66 => ! [X6: int,K2: int] :
% 0.34/0.66 ( ( ~ ( dvd_dvd_int @ D2 @ ( plus_plus_int @ X6 @ T4 ) ) )
% 0.34/0.66 = ( ~ ( dvd_dvd_int @ D2 @ ( plus_plus_int @ ( minus_minus_int @ X6 @ ( times_times_int @ K2 @ D3 ) ) @ T4 ) ) ) ) ) ).
% 0.34/0.66
% 0.34/0.66 % inf_period(4)
% 0.34/0.66 thf(fact_6324_cInf__mono,axiom,
% 0.34/0.66 ! [B4: set_o,A: set_o] :
% 0.34/0.66 ( ( B4 != bot_bot_set_o )
% 0.34/0.66 => ( ( condit5413489452508810728elow_o @ A )
% 0.34/0.66 => ( ! [B2: $o] :
% 0.34/0.66 ( ( member_o @ B2 @ B4 )
% 0.34/0.66 => ? [X6: $o] :
% 0.34/0.66 ( ( member_o @ X6 @ A )
% 0.34/0.66 & ( ord_less_eq_o @ X6 @ B2 ) ) )
% 0.34/0.66 => ( ord_less_eq_o @ ( complete_Inf_Inf_o @ A ) @ ( complete_Inf_Inf_o @ B4 ) ) ) ) ) ).
% 0.34/0.66
% 0.34/0.66 % cInf_mono
% 0.34/0.66 thf(fact_6325_cInf__mono,axiom,
% 0.34/0.66 ! [B4: set_Pr4532377907799695533_nat_o,A: set_Pr4532377907799695533_nat_o] :
% 0.34/0.66 ( ( B4 != bot_bo7824918357723582233_nat_o )
% 0.34/0.66 => ( ( condit8399017556247434269_nat_o @ A )
% 0.34/0.66 => ( ! [B2: produc3658429121746597890et_nat > $o] :
% 0.34/0.66 ( ( member6576561426505652726_nat_o @ B2 @ B4 )
% 0.34/0.66 => ? [X6: produc3658429121746597890et_nat > $o] :
% 0.34/0.66 ( ( member6576561426505652726_nat_o @ X6 @ A )
% 0.34/0.66 & ( ord_le729326519192465773_nat_o @ X6 @ B2 ) ) )
% 0.34/0.66 => ( ord_le729326519192465773_nat_o @ ( comple90455603224345172_nat_o @ A ) @ ( comple90455603224345172_nat_o @ B4 ) ) ) ) ) ).
% 0.34/0.66
% 0.34/0.66 % cInf_mono
% 0.34/0.66 thf(fact_6326_cInf__mono,axiom,
% 0.34/0.66 ! [B4: set_filter_nat,A: set_filter_nat] :
% 0.34/0.66 ( ( B4 != bot_bo498966703094740906er_nat )
% 0.34/0.66 => ( ( condit3133331395780078992er_nat @ A )
% 0.34/0.66 => ( ! [B2: filter_nat] :
% 0.34/0.66 ( ( member_filter_nat @ B2 @ B4 )
% 0.34/0.66 => ? [X6: filter_nat] :
% 0.34/0.66 ( ( member_filter_nat @ X6 @ A )
% 0.34/0.66 & ( ord_le2510731241096832064er_nat @ X6 @ B2 ) ) )
% 0.34/0.66 => ( ord_le2510731241096832064er_nat @ ( comple6612786467464916249er_nat @ A ) @ ( comple6612786467464916249er_nat @ B4 ) ) ) ) ) ).
% 0.34/0.66
% 0.34/0.66 % cInf_mono
% 0.34/0.66 thf(fact_6327_cInf__mono,axiom,
% 0.34/0.66 ! [B4: set_int,A: set_int] :
% 0.34/0.66 ( ( B4 != bot_bot_set_int )
% 0.34/0.66 => ( ( condit1735850657277959132ow_int @ A )
% 0.34/0.66 => ( ! [B2: int] :
% 0.34/0.66 ( ( member_int @ B2 @ B4 )
% 0.34/0.66 => ? [X6: int] :
% 0.34/0.66 ( ( member_int @ X6 @ A )
% 0.34/0.66 & ( ord_less_eq_int @ X6 @ B2 ) ) )
% 0.34/0.66 => ( ord_less_eq_int @ ( complete_Inf_Inf_int @ A ) @ ( complete_Inf_Inf_int @ B4 ) ) ) ) ) ).
% 0.34/0.66
% 0.34/0.66 % cInf_mono
% 0.34/0.66 thf(fact_6328_cInf__mono,axiom,
% 0.34/0.66 ! [B4: set_nat,A: set_nat] :
% 0.34/0.66 ( ( B4 != bot_bot_set_nat )
% 0.34/0.66 => ( ( condit1738341127787009408ow_nat @ A )
% 0.34/0.66 => ( ! [B2: nat] :
% 0.34/0.66 ( ( member_nat @ B2 @ B4 )
% 0.34/0.66 => ? [X6: nat] :
% 0.34/0.66 ( ( member_nat @ X6 @ A )
% 0.34/0.66 & ( ord_less_eq_nat @ X6 @ B2 ) ) )
% 0.34/0.66 => ( ord_less_eq_nat @ ( complete_Inf_Inf_nat @ A ) @ ( complete_Inf_Inf_nat @ B4 ) ) ) ) ) ).
% 0.34/0.66
% 0.34/0.66 % cInf_mono
% 0.34/0.66 thf(fact_6329_cInf__mono,axiom,
% 0.34/0.66 ! [B4: set_set_nat,A: set_set_nat] :
% 0.34/0.66 ( ( B4 != bot_bot_set_set_nat )
% 0.34/0.66 => ( ( condit68592940725977398et_nat @ A )
% 0.34/0.66 => ( ! [B2: set_nat] :
% 0.34/0.66 ( ( member_set_nat @ B2 @ B4 )
% 0.34/0.66 => ? [X6: set_nat] :
% 0.34/0.66 ( ( member_set_nat @ X6 @ A )
% 0.34/0.66 & ( ord_less_eq_set_nat @ X6 @ B2 ) ) )
% 0.34/0.66 => ( ord_less_eq_set_nat @ ( comple7806235888213564991et_nat @ A ) @ ( comple7806235888213564991et_nat @ B4 ) ) ) ) ) ).
% 0.34/0.66
% 0.34/0.66 % cInf_mono
% 0.34/0.66 thf(fact_6330_cInf__mono,axiom,
% 0.34/0.66 ! [B4: set_Product_unit,A: set_Product_unit] :
% 0.34/0.66 ( ( B4 != bot_bo3957492148770167129t_unit )
% 0.34/0.66 => ( ( condit9161856309968323677t_unit @ A )
% 0.34/0.66 => ( ! [B2: product_unit] :
% 0.34/0.66 ( ( member_Product_unit @ B2 @ B4 )
% 0.34/0.66 => ? [X6: product_unit] :
% 0.34/0.66 ( ( member_Product_unit @ X6 @ A )
% 0.34/0.66 & ( ord_le3221252021190050221t_unit @ X6 @ B2 ) ) )
% 0.34/0.66 => ( ord_le3221252021190050221t_unit @ ( comple2584293577114468500t_unit @ A ) @ ( comple2584293577114468500t_unit @ B4 ) ) ) ) ) ).
% 0.34/0.66
% 0.34/0.66 % cInf_mono
% 0.34/0.66 thf(fact_6331_le__cInf__iff,axiom,
% 0.34/0.66 ! [S: set_o,A2: $o] :
% 0.34/0.66 ( ( S != bot_bot_set_o )
% 0.34/0.66 => ( ( condit5413489452508810728elow_o @ S )
% 0.34/0.66 => ( ( ord_less_eq_o @ A2 @ ( complete_Inf_Inf_o @ S ) )
% 0.34/0.66 = ( ! [X2: $o] :
% 0.34/0.66 ( ( member_o @ X2 @ S )
% 0.34/0.66 => ( ord_less_eq_o @ A2 @ X2 ) ) ) ) ) ) ).
% 0.34/0.66
% 0.34/0.66 % le_cInf_iff
% 0.34/0.66 thf(fact_6332_le__cInf__iff,axiom,
% 0.34/0.66 ! [S: set_Pr4532377907799695533_nat_o,A2: produc3658429121746597890et_nat > $o] :
% 0.34/0.66 ( ( S != bot_bo7824918357723582233_nat_o )
% 0.34/0.66 => ( ( condit8399017556247434269_nat_o @ S )
% 0.34/0.66 => ( ( ord_le729326519192465773_nat_o @ A2 @ ( comple90455603224345172_nat_o @ S ) )
% 0.34/0.66 = ( ! [X2: produc3658429121746597890et_nat > $o] :
% 0.34/0.66 ( ( member6576561426505652726_nat_o @ X2 @ S )
% 0.34/0.66 => ( ord_le729326519192465773_nat_o @ A2 @ X2 ) ) ) ) ) ) ).
% 0.34/0.66
% 0.34/0.66 % le_cInf_iff
% 0.34/0.66 thf(fact_6333_le__cInf__iff,axiom,
% 0.34/0.66 ! [S: set_filter_nat,A2: filter_nat] :
% 0.34/0.66 ( ( S != bot_bo498966703094740906er_nat )
% 0.34/0.66 => ( ( condit3133331395780078992er_nat @ S )
% 0.34/0.66 => ( ( ord_le2510731241096832064er_nat @ A2 @ ( comple6612786467464916249er_nat @ S ) )
% 0.34/0.66 = ( ! [X2: filter_nat] :
% 0.34/0.66 ( ( member_filter_nat @ X2 @ S )
% 0.34/0.66 => ( ord_le2510731241096832064er_nat @ A2 @ X2 ) ) ) ) ) ) ).
% 0.34/0.66
% 0.34/0.66 % le_cInf_iff
% 0.34/0.66 thf(fact_6334_le__cInf__iff,axiom,
% 0.34/0.66 ! [S: set_int,A2: int] :
% 0.34/0.66 ( ( S != bot_bot_set_int )
% 0.34/0.66 => ( ( condit1735850657277959132ow_int @ S )
% 0.34/0.66 => ( ( ord_less_eq_int @ A2 @ ( complete_Inf_Inf_int @ S ) )
% 0.34/0.66 = ( ! [X2: int] :
% 0.34/0.66 ( ( member_int @ X2 @ S )
% 0.34/0.66 => ( ord_less_eq_int @ A2 @ X2 ) ) ) ) ) ) ).
% 0.34/0.66
% 0.34/0.66 % le_cInf_iff
% 0.34/0.66 thf(fact_6335_le__cInf__iff,axiom,
% 0.34/0.66 ! [S: set_nat,A2: nat] :
% 0.34/0.66 ( ( S != bot_bot_set_nat )
% 0.34/0.66 => ( ( condit1738341127787009408ow_nat @ S )
% 0.34/0.66 => ( ( ord_less_eq_nat @ A2 @ ( complete_Inf_Inf_nat @ S ) )
% 0.34/0.66 = ( ! [X2: nat] :
% 0.34/0.66 ( ( member_nat @ X2 @ S )
% 0.34/0.66 => ( ord_less_eq_nat @ A2 @ X2 ) ) ) ) ) ) ).
% 0.34/0.66
% 0.34/0.66 % le_cInf_iff
% 0.34/0.66 thf(fact_6336_le__cInf__iff,axiom,
% 0.34/0.66 ! [S: set_set_nat,A2: set_nat] :
% 0.34/0.66 ( ( S != bot_bot_set_set_nat )
% 0.34/0.66 => ( ( condit68592940725977398et_nat @ S )
% 0.34/0.66 => ( ( ord_less_eq_set_nat @ A2 @ ( comple7806235888213564991et_nat @ S ) )
% 0.34/0.66 = ( ! [X2: set_nat] :
% 0.34/0.66 ( ( member_set_nat @ X2 @ S )
% 0.34/0.66 => ( ord_less_eq_set_nat @ A2 @ X2 ) ) ) ) ) ) ).
% 0.34/0.66
% 0.34/0.66 % le_cInf_iff
% 0.34/0.66 thf(fact_6337_le__cInf__iff,axiom,
% 0.34/0.66 ! [S: set_Product_unit,A2: product_unit] :
% 0.34/0.66 ( ( S != bot_bo3957492148770167129t_unit )
% 0.34/0.66 => ( ( condit9161856309968323677t_unit @ S )
% 0.34/0.66 => ( ( ord_le3221252021190050221t_unit @ A2 @ ( comple2584293577114468500t_unit @ S ) )
% 0.34/0.66 = ( ! [X2: product_unit] :
% 0.34/0.66 ( ( member_Product_unit @ X2 @ S )
% 0.34/0.66 => ( ord_le3221252021190050221t_unit @ A2 @ X2 ) ) ) ) ) ) ).
% 0.34/0.66
% 0.34/0.66 % le_cInf_iff
% 0.34/0.66 thf(fact_6338_cInf__less__iff,axiom,
% 0.34/0.66 ! [X4: set_int,Y: int] :
% 0.34/0.66 ( ( X4 != bot_bot_set_int )
% 0.34/0.66 => ( ( condit1735850657277959132ow_int @ X4 )
% 0.34/0.66 => ( ( ord_less_int @ ( complete_Inf_Inf_int @ X4 ) @ Y )
% 0.34/0.66 = ( ? [X2: int] :
% 0.34/0.66 ( ( member_int @ X2 @ X4 )
% 0.34/0.66 & ( ord_less_int @ X2 @ Y ) ) ) ) ) ) ).
% 0.34/0.66
% 0.34/0.66 % cInf_less_iff
% 0.34/0.66 thf(fact_6339_cInf__less__iff,axiom,
% 0.34/0.66 ! [X4: set_nat,Y: nat] :
% 0.34/0.66 ( ( X4 != bot_bot_set_nat )
% 0.34/0.66 => ( ( condit1738341127787009408ow_nat @ X4 )
% 0.34/0.66 => ( ( ord_less_nat @ ( complete_Inf_Inf_nat @ X4 ) @ Y )
% 0.34/0.66 = ( ? [X2: nat] :
% 0.34/0.66 ( ( member_nat @ X2 @ X4 )
% 0.34/0.66 & ( ord_less_nat @ X2 @ Y ) ) ) ) ) ) ).
% 0.34/0.66
% 0.34/0.66 % cInf_less_iff
% 0.34/0.66 thf(fact_6340_cInf__less__iff,axiom,
% 0.34/0.66 ! [X4: set_Product_unit,Y: product_unit] :
% 0.34/0.66 ( ( X4 != bot_bo3957492148770167129t_unit )
% 0.34/0.66 => ( ( condit9161856309968323677t_unit @ X4 )
% 0.34/0.66 => ( ( ord_le361264281704409273t_unit @ ( comple2584293577114468500t_unit @ X4 ) @ Y )
% 0.34/0.66 = ( ? [X2: product_unit] :
% 0.34/0.66 ( ( member_Product_unit @ X2 @ X4 )
% 0.34/0.66 & ( ord_le361264281704409273t_unit @ X2 @ Y ) ) ) ) ) ) ).
% 0.34/0.66
% 0.34/0.66 % cInf_less_iff
% 0.34/0.66 thf(fact_6341_dvd__mult__cancel,axiom,
% 0.34/0.66 ! [K3: nat,M4: nat,N2: nat] :
% 0.34/0.66 ( ( dvd_dvd_nat @ ( times_times_nat @ K3 @ M4 ) @ ( times_times_nat @ K3 @ N2 ) )
% 0.34/0.66 => ( ( ord_less_nat @ zero_zero_nat @ K3 )
% 0.34/0.66 => ( dvd_dvd_nat @ M4 @ N2 ) ) ) ).
% 0.34/0.66
% 0.34/0.66 % dvd_mult_cancel
% 0.34/0.66 thf(fact_6342_nat__mult__dvd__cancel1,axiom,
% 0.34/0.66 ! [K3: nat,M4: nat,N2: nat] :
% 0.34/0.66 ( ( ord_less_nat @ zero_zero_nat @ K3 )
% 0.34/0.66 => ( ( dvd_dvd_nat @ ( times_times_nat @ K3 @ M4 ) @ ( times_times_nat @ K3 @ N2 ) )
% 0.34/0.66 = ( dvd_dvd_nat @ M4 @ N2 ) ) ) ).
% 0.34/0.66
% 0.34/0.66 % nat_mult_dvd_cancel1
% 0.34/0.66 thf(fact_6343_power__Suc__less,axiom,
% 0.34/0.66 ! [A2: code_natural,N2: nat] :
% 0.34/0.66 ( ( ord_le5570908160329646204atural @ zero_z2226904508553997617atural @ A2 )
% 0.34/0.66 => ( ( ord_le5570908160329646204atural @ A2 @ one_one_Code_natural )
% 0.34/0.66 => ( ord_le5570908160329646204atural @ ( times_2397367101498566445atural @ A2 @ ( power_7079662738309270450atural @ A2 @ N2 ) ) @ ( power_7079662738309270450atural @ A2 @ N2 ) ) ) ) ).
% 0.34/0.66
% 0.34/0.66 % power_Suc_less
% 0.34/0.66 thf(fact_6344_power__Suc__less,axiom,
% 0.34/0.66 ! [A2: nat,N2: nat] :
% 0.34/0.66 ( ( ord_less_nat @ zero_zero_nat @ A2 )
% 0.34/0.66 => ( ( ord_less_nat @ A2 @ one_one_nat )
% 0.34/0.66 => ( ord_less_nat @ ( times_times_nat @ A2 @ ( power_power_nat @ A2 @ N2 ) ) @ ( power_power_nat @ A2 @ N2 ) ) ) ) ).
% 0.34/0.66
% 0.34/0.66 % power_Suc_less
% 0.34/0.66 thf(fact_6345_power__Suc__less,axiom,
% 0.34/0.66 ! [A2: int,N2: nat] :
% 0.34/0.66 ( ( ord_less_int @ zero_zero_int @ A2 )
% 0.34/0.66 => ( ( ord_less_int @ A2 @ one_one_int )
% 0.34/0.66 => ( ord_less_int @ ( times_times_int @ A2 @ ( power_power_int @ A2 @ N2 ) ) @ ( power_power_int @ A2 @ N2 ) ) ) ) ).
% 0.34/0.66
% 0.34/0.66 % power_Suc_less
% 0.34/0.66 thf(fact_6346_power__Suc__less,axiom,
% 0.34/0.66 ! [A2: code_integer,N2: nat] :
% 0.34/0.66 ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ A2 )
% 0.34/0.66 => ( ( ord_le6747313008572928689nteger @ A2 @ one_one_Code_integer )
% 0.34/0.66 => ( ord_le6747313008572928689nteger @ ( times_3573771949741848930nteger @ A2 @ ( power_8256067586552552935nteger @ A2 @ N2 ) ) @ ( power_8256067586552552935nteger @ A2 @ N2 ) ) ) ) ).
% 0.34/0.66
% 0.34/0.66 % power_Suc_less
% 0.34/0.66 thf(fact_6347_power__Suc__less,axiom,
% 0.34/0.66 ! [A2: rat,N2: nat] :
% 0.34/0.66 ( ( ord_less_rat @ zero_zero_rat @ A2 )
% 0.34/0.66 => ( ( ord_less_rat @ A2 @ one_one_rat )
% 0.34/0.66 => ( ord_less_rat @ ( times_times_rat @ A2 @ ( power_power_rat @ A2 @ N2 ) ) @ ( power_power_rat @ A2 @ N2 ) ) ) ) ).
% 0.34/0.66
% 0.34/0.66 % power_Suc_less
% 0.34/0.66 thf(fact_6348_power__Suc__le__self,axiom,
% 0.34/0.66 ! [A2: code_natural,N2: nat] :
% 0.34/0.66 ( ( ord_le1926595141338095240atural @ zero_z2226904508553997617atural @ A2 )
% 0.34/0.66 => ( ( ord_le1926595141338095240atural @ A2 @ one_one_Code_natural )
% 0.34/0.66 => ( ord_le1926595141338095240atural @ ( power_7079662738309270450atural @ A2 @ ( suc @ N2 ) ) @ A2 ) ) ) ).
% 0.34/0.66
% 0.34/0.66 % power_Suc_le_self
% 0.34/0.66 thf(fact_6349_power__Suc__le__self,axiom,
% 0.34/0.66 ! [A2: code_integer,N2: nat] :
% 0.34/0.66 ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ A2 )
% 0.34/0.66 => ( ( ord_le3102999989581377725nteger @ A2 @ one_one_Code_integer )
% 0.34/0.66 => ( ord_le3102999989581377725nteger @ ( power_8256067586552552935nteger @ A2 @ ( suc @ N2 ) ) @ A2 ) ) ) ).
% 0.34/0.66
% 0.34/0.66 % power_Suc_le_self
% 0.34/0.66 thf(fact_6350_power__Suc__le__self,axiom,
% 0.34/0.66 ! [A2: rat,N2: nat] :
% 0.34/0.66 ( ( ord_less_eq_rat @ zero_zero_rat @ A2 )
% 0.34/0.66 => ( ( ord_less_eq_rat @ A2 @ one_one_rat )
% 0.34/0.66 => ( ord_less_eq_rat @ ( power_power_rat @ A2 @ ( suc @ N2 ) ) @ A2 ) ) ) ).
% 0.34/0.66
% 0.34/0.66 % power_Suc_le_self
% 0.34/0.66 thf(fact_6351_power__Suc__le__self,axiom,
% 0.34/0.66 ! [A2: nat,N2: nat] :
% 0.34/0.66 ( ( ord_less_eq_nat @ zero_zero_nat @ A2 )
% 0.34/0.66 => ( ( ord_less_eq_nat @ A2 @ one_one_nat )
% 0.34/0.66 => ( ord_less_eq_nat @ ( power_power_nat @ A2 @ ( suc @ N2 ) ) @ A2 ) ) ) ).
% 0.34/0.66
% 0.34/0.66 % power_Suc_le_self
% 0.34/0.66 thf(fact_6352_power__Suc__le__self,axiom,
% 0.34/0.66 ! [A2: int,N2: nat] :
% 0.34/0.66 ( ( ord_less_eq_int @ zero_zero_int @ A2 )
% 0.34/0.66 => ( ( ord_less_eq_int @ A2 @ one_one_int )
% 0.34/0.66 => ( ord_less_eq_int @ ( power_power_int @ A2 @ ( suc @ N2 ) ) @ A2 ) ) ) ).
% 0.34/0.66
% 0.34/0.66 % power_Suc_le_self
% 0.34/0.66 thf(fact_6353_power__Suc__less__one,axiom,
% 0.34/0.66 ! [A2: code_natural,N2: nat] :
% 0.34/0.66 ( ( ord_le5570908160329646204atural @ zero_z2226904508553997617atural @ A2 )
% 0.34/0.66 => ( ( ord_le5570908160329646204atural @ A2 @ one_one_Code_natural )
% 0.34/0.66 => ( ord_le5570908160329646204atural @ ( power_7079662738309270450atural @ A2 @ ( suc @ N2 ) ) @ one_one_Code_natural ) ) ) ).
% 0.34/0.66
% 0.34/0.66 % power_Suc_less_one
% 0.34/0.66 thf(fact_6354_power__Suc__less__one,axiom,
% 0.34/0.66 ! [A2: nat,N2: nat] :
% 0.34/0.66 ( ( ord_less_nat @ zero_zero_nat @ A2 )
% 0.34/0.66 => ( ( ord_less_nat @ A2 @ one_one_nat )
% 0.34/0.66 => ( ord_less_nat @ ( power_power_nat @ A2 @ ( suc @ N2 ) ) @ one_one_nat ) ) ) ).
% 0.34/0.66
% 0.34/0.66 % power_Suc_less_one
% 0.34/0.66 thf(fact_6355_power__Suc__less__one,axiom,
% 0.34/0.66 ! [A2: int,N2: nat] :
% 0.34/0.66 ( ( ord_less_int @ zero_zero_int @ A2 )
% 0.34/0.66 => ( ( ord_less_int @ A2 @ one_one_int )
% 0.34/0.66 => ( ord_less_int @ ( power_power_int @ A2 @ ( suc @ N2 ) ) @ one_one_int ) ) ) ).
% 0.34/0.66
% 0.34/0.66 % power_Suc_less_one
% 0.34/0.66 thf(fact_6356_power__Suc__less__one,axiom,
% 0.34/0.66 ! [A2: code_integer,N2: nat] :
% 0.34/0.66 ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ A2 )
% 0.34/0.66 => ( ( ord_le6747313008572928689nteger @ A2 @ one_one_Code_integer )
% 0.34/0.66 => ( ord_le6747313008572928689nteger @ ( power_8256067586552552935nteger @ A2 @ ( suc @ N2 ) ) @ one_one_Code_integer ) ) ) ).
% 0.34/0.66
% 0.34/0.66 % power_Suc_less_one
% 0.34/0.66 thf(fact_6357_power__Suc__less__one,axiom,
% 0.34/0.66 ! [A2: rat,N2: nat] :
% 0.34/0.66 ( ( ord_less_rat @ zero_zero_rat @ A2 )
% 0.34/0.66 => ( ( ord_less_rat @ A2 @ one_one_rat )
% 0.34/0.66 => ( ord_less_rat @ ( power_power_rat @ A2 @ ( suc @ N2 ) ) @ one_one_rat ) ) ) ).
% 0.34/0.66
% 0.34/0.66 % power_Suc_less_one
% 0.34/0.66 thf(fact_6358_power__strict__decreasing,axiom,
% 0.34/0.66 ! [N2: nat,N: nat,A2: code_natural] :
% 0.34/0.66 ( ( ord_less_nat @ N2 @ N )
% 0.34/0.66 => ( ( ord_le5570908160329646204atural @ zero_z2226904508553997617atural @ A2 )
% 0.34/0.66 => ( ( ord_le5570908160329646204atural @ A2 @ one_one_Code_natural )
% 0.34/0.66 => ( ord_le5570908160329646204atural @ ( power_7079662738309270450atural @ A2 @ N ) @ ( power_7079662738309270450atural @ A2 @ N2 ) ) ) ) ) ).
% 0.34/0.66
% 0.34/0.66 % power_strict_decreasing
% 0.34/0.66 thf(fact_6359_power__strict__decreasing,axiom,
% 0.34/0.66 ! [N2: nat,N: nat,A2: nat] :
% 0.34/0.66 ( ( ord_less_nat @ N2 @ N )
% 0.34/0.66 => ( ( ord_less_nat @ zero_zero_nat @ A2 )
% 0.34/0.66 => ( ( ord_less_nat @ A2 @ one_one_nat )
% 0.34/0.66 => ( ord_less_nat @ ( power_power_nat @ A2 @ N ) @ ( power_power_nat @ A2 @ N2 ) ) ) ) ) ).
% 0.34/0.66
% 0.34/0.66 % power_strict_decreasing
% 0.34/0.66 thf(fact_6360_power__strict__decreasing,axiom,
% 0.34/0.66 ! [N2: nat,N: nat,A2: int] :
% 0.34/0.66 ( ( ord_less_nat @ N2 @ N )
% 0.34/0.66 => ( ( ord_less_int @ zero_zero_int @ A2 )
% 0.34/0.66 => ( ( ord_less_int @ A2 @ one_one_int )
% 0.34/0.66 => ( ord_less_int @ ( power_power_int @ A2 @ N ) @ ( power_power_int @ A2 @ N2 ) ) ) ) ) ).
% 0.34/0.66
% 0.34/0.66 % power_strict_decreasing
% 0.34/0.66 thf(fact_6361_power__strict__decreasing,axiom,
% 0.34/0.66 ! [N2: nat,N: nat,A2: code_integer] :
% 0.34/0.66 ( ( ord_less_nat @ N2 @ N )
% 0.34/0.66 => ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ A2 )
% 0.34/0.66 => ( ( ord_le6747313008572928689nteger @ A2 @ one_one_Code_integer )
% 0.34/0.66 => ( ord_le6747313008572928689nteger @ ( power_8256067586552552935nteger @ A2 @ N ) @ ( power_8256067586552552935nteger @ A2 @ N2 ) ) ) ) ) ).
% 0.34/0.66
% 0.34/0.66 % power_strict_decreasing
% 0.34/0.66 thf(fact_6362_power__strict__decreasing,axiom,
% 0.34/0.66 ! [N2: nat,N: nat,A2: rat] :
% 0.34/0.66 ( ( ord_less_nat @ N2 @ N )
% 0.34/0.66 => ( ( ord_less_rat @ zero_zero_rat @ A2 )
% 0.34/0.66 => ( ( ord_less_rat @ A2 @ one_one_rat )
% 0.34/0.66 => ( ord_less_rat @ ( power_power_rat @ A2 @ N ) @ ( power_power_rat @ A2 @ N2 ) ) ) ) ) ).
% 0.34/0.66
% 0.34/0.66 % power_strict_decreasing
% 0.34/0.66 thf(fact_6363_power__decreasing,axiom,
% 0.34/0.66 ! [N2: nat,N: nat,A2: code_natural] :
% 0.34/0.66 ( ( ord_less_eq_nat @ N2 @ N )
% 0.34/0.66 => ( ( ord_le1926595141338095240atural @ zero_z2226904508553997617atural @ A2 )
% 0.34/0.66 => ( ( ord_le1926595141338095240atural @ A2 @ one_one_Code_natural )
% 0.34/0.66 => ( ord_le1926595141338095240atural @ ( power_7079662738309270450atural @ A2 @ N ) @ ( power_7079662738309270450atural @ A2 @ N2 ) ) ) ) ) ).
% 0.34/0.66
% 0.34/0.66 % power_decreasing
% 0.34/0.66 thf(fact_6364_power__decreasing,axiom,
% 0.34/0.66 ! [N2: nat,N: nat,A2: code_integer] :
% 0.34/0.66 ( ( ord_less_eq_nat @ N2 @ N )
% 0.34/0.66 => ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ A2 )
% 0.34/0.66 => ( ( ord_le3102999989581377725nteger @ A2 @ one_one_Code_integer )
% 0.34/0.66 => ( ord_le3102999989581377725nteger @ ( power_8256067586552552935nteger @ A2 @ N ) @ ( power_8256067586552552935nteger @ A2 @ N2 ) ) ) ) ) ).
% 0.34/0.66
% 0.34/0.66 % power_decreasing
% 0.34/0.66 thf(fact_6365_power__decreasing,axiom,
% 0.34/0.66 ! [N2: nat,N: nat,A2: rat] :
% 0.34/0.66 ( ( ord_less_eq_nat @ N2 @ N )
% 0.34/0.66 => ( ( ord_less_eq_rat @ zero_zero_rat @ A2 )
% 0.34/0.66 => ( ( ord_less_eq_rat @ A2 @ one_one_rat )
% 0.34/0.66 => ( ord_less_eq_rat @ ( power_power_rat @ A2 @ N ) @ ( power_power_rat @ A2 @ N2 ) ) ) ) ) ).
% 0.34/0.66
% 0.34/0.66 % power_decreasing
% 0.34/0.66 thf(fact_6366_power__decreasing,axiom,
% 0.34/0.66 ! [N2: nat,N: nat,A2: nat] :
% 0.34/0.66 ( ( ord_less_eq_nat @ N2 @ N )
% 0.34/0.66 => ( ( ord_less_eq_nat @ zero_zero_nat @ A2 )
% 0.34/0.66 => ( ( ord_less_eq_nat @ A2 @ one_one_nat )
% 0.34/0.66 => ( ord_less_eq_nat @ ( power_power_nat @ A2 @ N ) @ ( power_power_nat @ A2 @ N2 ) ) ) ) ) ).
% 0.34/0.66
% 0.34/0.66 % power_decreasing
% 0.34/0.66 thf(fact_6367_power__decreasing,axiom,
% 0.34/0.66 ! [N2: nat,N: nat,A2: int] :
% 0.34/0.66 ( ( ord_less_eq_nat @ N2 @ N )
% 0.34/0.66 => ( ( ord_less_eq_int @ zero_zero_int @ A2 )
% 0.34/0.66 => ( ( ord_less_eq_int @ A2 @ one_one_int )
% 0.34/0.66 => ( ord_less_eq_int @ ( power_power_int @ A2 @ N ) @ ( power_power_int @ A2 @ N2 ) ) ) ) ) ).
% 0.34/0.66
% 0.34/0.66 % power_decreasing
% 0.34/0.66 thf(fact_6368_power__le__imp__le__exp,axiom,
% 0.34/0.66 ! [A2: code_natural,M4: nat,N2: nat] :
% 0.34/0.66 ( ( ord_le5570908160329646204atural @ one_one_Code_natural @ A2 )
% 0.34/0.66 => ( ( ord_le1926595141338095240atural @ ( power_7079662738309270450atural @ A2 @ M4 ) @ ( power_7079662738309270450atural @ A2 @ N2 ) )
% 0.34/0.66 => ( ord_less_eq_nat @ M4 @ N2 ) ) ) ).
% 0.34/0.66
% 0.34/0.66 % power_le_imp_le_exp
% 0.34/0.66 thf(fact_6369_power__le__imp__le__exp,axiom,
% 0.34/0.66 ! [A2: code_integer,M4: nat,N2: nat] :
% 0.34/0.66 ( ( ord_le6747313008572928689nteger @ one_one_Code_integer @ A2 )
% 0.34/0.66 => ( ( ord_le3102999989581377725nteger @ ( power_8256067586552552935nteger @ A2 @ M4 ) @ ( power_8256067586552552935nteger @ A2 @ N2 ) )
% 0.34/0.66 => ( ord_less_eq_nat @ M4 @ N2 ) ) ) ).
% 0.34/0.66
% 0.34/0.66 % power_le_imp_le_exp
% 0.34/0.66 thf(fact_6370_power__le__imp__le__exp,axiom,
% 0.34/0.66 ! [A2: rat,M4: nat,N2: nat] :
% 0.34/0.66 ( ( ord_less_rat @ one_one_rat @ A2 )
% 0.34/0.66 => ( ( ord_less_eq_rat @ ( power_power_rat @ A2 @ M4 ) @ ( power_power_rat @ A2 @ N2 ) )
% 0.34/0.66 => ( ord_less_eq_nat @ M4 @ N2 ) ) ) ).
% 0.34/0.66
% 0.34/0.66 % power_le_imp_le_exp
% 0.34/0.66 thf(fact_6371_power__le__imp__le__exp,axiom,
% 0.34/0.66 ! [A2: nat,M4: nat,N2: nat] :
% 0.34/0.66 ( ( ord_less_nat @ one_one_nat @ A2 )
% 0.34/0.66 => ( ( ord_less_eq_nat @ ( power_power_nat @ A2 @ M4 ) @ ( power_power_nat @ A2 @ N2 ) )
% 0.34/0.66 => ( ord_less_eq_nat @ M4 @ N2 ) ) ) ).
% 0.34/0.66
% 0.34/0.66 % power_le_imp_le_exp
% 0.34/0.66 thf(fact_6372_power__le__imp__le__exp,axiom,
% 0.34/0.66 ! [A2: int,M4: nat,N2: nat] :
% 0.34/0.66 ( ( ord_less_int @ one_one_int @ A2 )
% 0.34/0.66 => ( ( ord_less_eq_int @ ( power_power_int @ A2 @ M4 ) @ ( power_power_int @ A2 @ N2 ) )
% 0.34/0.66 => ( ord_less_eq_nat @ M4 @ N2 ) ) ) ).
% 0.34/0.66
% 0.34/0.66 % power_le_imp_le_exp
% 0.34/0.66 thf(fact_6373_self__le__power,axiom,
% 0.34/0.66 ! [A2: code_integer,N2: nat] :
% 0.34/0.66 ( ( ord_le3102999989581377725nteger @ one_one_Code_integer @ A2 )
% 0.34/0.66 => ( ( ord_less_nat @ zero_zero_nat @ N2 )
% 0.34/0.66 => ( ord_le3102999989581377725nteger @ A2 @ ( power_8256067586552552935nteger @ A2 @ N2 ) ) ) ) ).
% 0.34/0.66
% 0.34/0.66 % self_le_power
% 0.34/0.66 thf(fact_6374_self__le__power,axiom,
% 0.34/0.66 ! [A2: code_natural,N2: nat] :
% 0.34/0.66 ( ( ord_le1926595141338095240atural @ one_one_Code_natural @ A2 )
% 0.34/0.66 => ( ( ord_less_nat @ zero_zero_nat @ N2 )
% 0.34/0.66 => ( ord_le1926595141338095240atural @ A2 @ ( power_7079662738309270450atural @ A2 @ N2 ) ) ) ) ).
% 0.34/0.66
% 0.34/0.66 % self_le_power
% 0.34/0.66 thf(fact_6375_self__le__power,axiom,
% 0.34/0.66 ! [A2: rat,N2: nat] :
% 0.34/0.66 ( ( ord_less_eq_rat @ one_one_rat @ A2 )
% 0.34/0.66 => ( ( ord_less_nat @ zero_zero_nat @ N2 )
% 0.34/0.66 => ( ord_less_eq_rat @ A2 @ ( power_power_rat @ A2 @ N2 ) ) ) ) ).
% 0.34/0.66
% 0.34/0.66 % self_le_power
% 0.34/0.66 thf(fact_6376_self__le__power,axiom,
% 0.34/0.66 ! [A2: nat,N2: nat] :
% 0.34/0.66 ( ( ord_less_eq_nat @ one_one_nat @ A2 )
% 0.34/0.66 => ( ( ord_less_nat @ zero_zero_nat @ N2 )
% 0.34/0.66 => ( ord_less_eq_nat @ A2 @ ( power_power_nat @ A2 @ N2 ) ) ) ) ).
% 0.34/0.66
% 0.34/0.66 % self_le_power
% 0.34/0.66 thf(fact_6377_self__le__power,axiom,
% 0.34/0.66 ! [A2: int,N2: nat] :
% 0.34/0.66 ( ( ord_less_eq_int @ one_one_int @ A2 )
% 0.34/0.66 => ( ( ord_less_nat @ zero_zero_nat @ N2 )
% 0.34/0.66 => ( ord_less_eq_int @ A2 @ ( power_power_int @ A2 @ N2 ) ) ) ) ).
% 0.34/0.66
% 0.34/0.66 % self_le_power
% 0.34/0.66 thf(fact_6378_one__less__power,axiom,
% 0.34/0.66 ! [A2: code_natural,N2: nat] :
% 0.34/0.66 ( ( ord_le5570908160329646204atural @ one_one_Code_natural @ A2 )
% 0.34/0.66 => ( ( ord_less_nat @ zero_zero_nat @ N2 )
% 0.34/0.66 => ( ord_le5570908160329646204atural @ one_one_Code_natural @ ( power_7079662738309270450atural @ A2 @ N2 ) ) ) ) ).
% 0.34/0.66
% 0.34/0.66 % one_less_power
% 0.34/0.66 thf(fact_6379_one__less__power,axiom,
% 0.34/0.66 ! [A2: nat,N2: nat] :
% 0.34/0.66 ( ( ord_less_nat @ one_one_nat @ A2 )
% 0.34/0.66 => ( ( ord_less_nat @ zero_zero_nat @ N2 )
% 0.34/0.66 => ( ord_less_nat @ one_one_nat @ ( power_power_nat @ A2 @ N2 ) ) ) ) ).
% 0.34/0.66
% 0.34/0.66 % one_less_power
% 0.34/0.66 thf(fact_6380_one__less__power,axiom,
% 0.34/0.66 ! [A2: int,N2: nat] :
% 0.34/0.66 ( ( ord_less_int @ one_one_int @ A2 )
% 0.34/0.66 => ( ( ord_less_nat @ zero_zero_nat @ N2 )
% 0.34/0.66 => ( ord_less_int @ one_one_int @ ( power_power_int @ A2 @ N2 ) ) ) ) ).
% 0.34/0.66
% 0.34/0.66 % one_less_power
% 0.34/0.66 thf(fact_6381_one__less__power,axiom,
% 0.34/0.66 ! [A2: code_integer,N2: nat] :
% 0.34/0.66 ( ( ord_le6747313008572928689nteger @ one_one_Code_integer @ A2 )
% 0.34/0.66 => ( ( ord_less_nat @ zero_zero_nat @ N2 )
% 0.34/0.66 => ( ord_le6747313008572928689nteger @ one_one_Code_integer @ ( power_8256067586552552935nteger @ A2 @ N2 ) ) ) ) ).
% 0.34/0.66
% 0.34/0.66 % one_less_power
% 0.34/0.66 thf(fact_6382_one__less__power,axiom,
% 0.34/0.66 ! [A2: rat,N2: nat] :
% 0.34/0.66 ( ( ord_less_rat @ one_one_rat @ A2 )
% 0.34/0.66 => ( ( ord_less_nat @ zero_zero_nat @ N2 )
% 0.34/0.66 => ( ord_less_rat @ one_one_rat @ ( power_power_rat @ A2 @ N2 ) ) ) ) ).
% 0.34/0.66
% 0.34/0.66 % one_less_power
% 0.34/0.66 thf(fact_6383_cINF__mono,axiom,
% 0.34/0.66 ! [B4: set_Product_unit,F: nat > nat,A: set_nat,G: product_unit > nat] :
% 0.34/0.66 ( ( B4 != bot_bo3957492148770167129t_unit )
% 0.34/0.66 => ( ( condit1738341127787009408ow_nat @ ( image_nat_nat2 @ F @ A ) )
% 0.34/0.66 => ( ! [M2: product_unit] :
% 0.34/0.66 ( ( member_Product_unit @ M2 @ B4 )
% 0.34/0.66 => ? [X6: nat] :
% 0.34/0.66 ( ( member_nat @ X6 @ A )
% 0.34/0.66 & ( ord_less_eq_nat @ ( F @ X6 ) @ ( G @ M2 ) ) ) )
% 0.34/0.66 => ( ord_less_eq_nat @ ( complete_Inf_Inf_nat @ ( image_nat_nat2 @ F @ A ) ) @ ( complete_Inf_Inf_nat @ ( image_875570014554754200it_nat @ G @ B4 ) ) ) ) ) ) ).
% 0.34/0.66
% 0.34/0.66 % cINF_mono
% 0.34/0.66 thf(fact_6384_cINF__mono,axiom,
% 0.34/0.66 ! [B4: set_o,F: nat > nat,A: set_nat,G: $o > nat] :
% 0.34/0.66 ( ( B4 != bot_bot_set_o )
% 0.34/0.66 => ( ( condit1738341127787009408ow_nat @ ( image_nat_nat2 @ F @ A ) )
% 0.34/0.66 => ( ! [M2: $o] :
% 0.34/0.66 ( ( member_o @ M2 @ B4 )
% 0.34/0.66 => ? [X6: nat] :
% 0.34/0.66 ( ( member_nat @ X6 @ A )
% 0.34/0.66 & ( ord_less_eq_nat @ ( F @ X6 ) @ ( G @ M2 ) ) ) )
% 0.34/0.66 => ( ord_less_eq_nat @ ( complete_Inf_Inf_nat @ ( image_nat_nat2 @ F @ A ) ) @ ( complete_Inf_Inf_nat @ ( image_o_nat2 @ G @ B4 ) ) ) ) ) ) ).
% 0.34/0.66
% 0.34/0.66 % cINF_mono
% 0.34/0.66 thf(fact_6385_cINF__mono,axiom,
% 0.34/0.66 ! [B4: set_nat,F: nat > nat,A: set_nat,G: nat > nat] :
% 0.34/0.66 ( ( B4 != bot_bot_set_nat )
% 0.34/0.66 => ( ( condit1738341127787009408ow_nat @ ( image_nat_nat2 @ F @ A ) )
% 0.34/0.66 => ( ! [M2: nat] :
% 0.34/0.66 ( ( member_nat @ M2 @ B4 )
% 0.34/0.66 => ? [X6: nat] :
% 0.34/0.66 ( ( member_nat @ X6 @ A )
% 0.34/0.66 & ( ord_less_eq_nat @ ( F @ X6 ) @ ( G @ M2 ) ) ) )
% 0.34/0.66 => ( ord_less_eq_nat @ ( complete_Inf_Inf_nat @ ( image_nat_nat2 @ F @ A ) ) @ ( complete_Inf_Inf_nat @ ( image_nat_nat2 @ G @ B4 ) ) ) ) ) ) ).
% 0.34/0.66
% 0.34/0.66 % cINF_mono
% 0.34/0.66 thf(fact_6386_cINF__mono,axiom,
% 0.34/0.66 ! [B4: set_int,F: nat > nat,A: set_nat,G: int > nat] :
% 0.34/0.66 ( ( B4 != bot_bot_set_int )
% 0.34/0.66 => ( ( condit1738341127787009408ow_nat @ ( image_nat_nat2 @ F @ A ) )
% 0.34/0.66 => ( ! [M2: int] :
% 0.34/0.66 ( ( member_int @ M2 @ B4 )
% 0.34/0.66 => ? [X6: nat] :
% 0.34/0.66 ( ( member_nat @ X6 @ A )
% 0.34/0.66 & ( ord_less_eq_nat @ ( F @ X6 ) @ ( G @ M2 ) ) ) )
% 0.34/0.66 => ( ord_less_eq_nat @ ( complete_Inf_Inf_nat @ ( image_nat_nat2 @ F @ A ) ) @ ( complete_Inf_Inf_nat @ ( image_int_nat2 @ G @ B4 ) ) ) ) ) ) ).
% 0.34/0.66
% 0.34/0.66 % cINF_mono
% 0.34/0.66 thf(fact_6387_cINF__mono,axiom,
% 0.34/0.66 ! [B4: set_Pr4532377907799695533_nat_o,F: nat > nat,A: set_nat,G: ( produc3658429121746597890et_nat > $o ) > nat] :
% 0.34/0.66 ( ( B4 != bot_bo7824918357723582233_nat_o )
% 0.34/0.66 => ( ( condit1738341127787009408ow_nat @ ( image_nat_nat2 @ F @ A ) )
% 0.34/0.66 => ( ! [M2: produc3658429121746597890et_nat > $o] :
% 0.34/0.66 ( ( member6576561426505652726_nat_o @ M2 @ B4 )
% 0.34/0.66 => ? [X6: nat] :
% 0.34/0.66 ( ( member_nat @ X6 @ A )
% 0.34/0.66 & ( ord_less_eq_nat @ ( F @ X6 ) @ ( G @ M2 ) ) ) )
% 0.34/0.66 => ( ord_less_eq_nat @ ( complete_Inf_Inf_nat @ ( image_nat_nat2 @ F @ A ) ) @ ( complete_Inf_Inf_nat @ ( image_2174381888560928856_o_nat @ G @ B4 ) ) ) ) ) ) ).
% 0.34/0.66
% 0.34/0.66 % cINF_mono
% 0.34/0.66 thf(fact_6388_cINF__mono,axiom,
% 0.34/0.66 ! [B4: set_Product_unit,F: nat > set_nat,A: set_nat,G: product_unit > set_nat] :
% 0.34/0.66 ( ( B4 != bot_bo3957492148770167129t_unit )
% 0.34/0.66 => ( ( condit68592940725977398et_nat @ ( image_nat_set_nat @ F @ A ) )
% 0.34/0.66 => ( ! [M2: product_unit] :
% 0.34/0.66 ( ( member_Product_unit @ M2 @ B4 )
% 0.34/0.66 => ? [X6: nat] :
% 0.34/0.66 ( ( member_nat @ X6 @ A )
% 0.34/0.66 & ( ord_less_eq_set_nat @ ( F @ X6 ) @ ( G @ M2 ) ) ) )
% 0.34/0.66 => ( ord_less_eq_set_nat @ ( comple7806235888213564991et_nat @ ( image_nat_set_nat @ F @ A ) ) @ ( comple7806235888213564991et_nat @ ( image_8173849991827986510et_nat @ G @ B4 ) ) ) ) ) ) ).
% 0.34/0.66
% 0.34/0.66 % cINF_mono
% 0.34/0.66 thf(fact_6389_cINF__mono,axiom,
% 0.34/0.66 ! [B4: set_o,F: nat > set_nat,A: set_nat,G: $o > set_nat] :
% 0.34/0.66 ( ( B4 != bot_bot_set_o )
% 0.34/0.66 => ( ( condit68592940725977398et_nat @ ( image_nat_set_nat @ F @ A ) )
% 0.34/0.66 => ( ! [M2: $o] :
% 0.34/0.66 ( ( member_o @ M2 @ B4 )
% 0.34/0.66 => ? [X6: nat] :
% 0.34/0.66 ( ( member_nat @ X6 @ A )
% 0.34/0.66 & ( ord_less_eq_set_nat @ ( F @ X6 ) @ ( G @ M2 ) ) ) )
% 0.34/0.66 => ( ord_less_eq_set_nat @ ( comple7806235888213564991et_nat @ ( image_nat_set_nat @ F @ A ) ) @ ( comple7806235888213564991et_nat @ ( image_o_set_nat @ G @ B4 ) ) ) ) ) ) ).
% 0.34/0.66
% 0.34/0.66 % cINF_mono
% 0.34/0.66 thf(fact_6390_cINF__mono,axiom,
% 0.34/0.66 ! [B4: set_nat,F: nat > set_nat,A: set_nat,G: nat > set_nat] :
% 0.34/0.66 ( ( B4 != bot_bot_set_nat )
% 0.34/0.66 => ( ( condit68592940725977398et_nat @ ( image_nat_set_nat @ F @ A ) )
% 0.34/0.66 => ( ! [M2: nat] :
% 0.34/0.66 ( ( member_nat @ M2 @ B4 )
% 0.34/0.66 => ? [X6: nat] :
% 0.34/0.66 ( ( member_nat @ X6 @ A )
% 0.34/0.66 & ( ord_less_eq_set_nat @ ( F @ X6 ) @ ( G @ M2 ) ) ) )
% 0.34/0.66 => ( ord_less_eq_set_nat @ ( comple7806235888213564991et_nat @ ( image_nat_set_nat @ F @ A ) ) @ ( comple7806235888213564991et_nat @ ( image_nat_set_nat @ G @ B4 ) ) ) ) ) ) ).
% 0.34/0.66
% 0.34/0.66 % cINF_mono
% 0.34/0.66 thf(fact_6391_cINF__mono,axiom,
% 0.34/0.66 ! [B4: set_int,F: nat > set_nat,A: set_nat,G: int > set_nat] :
% 0.34/0.66 ( ( B4 != bot_bot_set_int )
% 0.34/0.66 => ( ( condit68592940725977398et_nat @ ( image_nat_set_nat @ F @ A ) )
% 0.34/0.66 => ( ! [M2: int] :
% 0.34/0.66 ( ( member_int @ M2 @ B4 )
% 0.34/0.66 => ? [X6: nat] :
% 0.34/0.66 ( ( member_nat @ X6 @ A )
% 0.34/0.66 & ( ord_less_eq_set_nat @ ( F @ X6 ) @ ( G @ M2 ) ) ) )
% 0.34/0.66 => ( ord_less_eq_set_nat @ ( comple7806235888213564991et_nat @ ( image_nat_set_nat @ F @ A ) ) @ ( comple7806235888213564991et_nat @ ( image_int_set_nat @ G @ B4 ) ) ) ) ) ) ).
% 0.34/0.66
% 0.34/0.66 % cINF_mono
% 0.34/0.66 thf(fact_6392_cINF__mono,axiom,
% 0.34/0.66 ! [B4: set_Pr4532377907799695533_nat_o,F: nat > set_nat,A: set_nat,G: ( produc3658429121746597890et_nat > $o ) > set_nat] :
% 0.34/0.66 ( ( B4 != bot_bo7824918357723582233_nat_o )
% 0.34/0.66 => ( ( condit68592940725977398et_nat @ ( image_nat_set_nat @ F @ A ) )
% 0.34/0.66 => ( ! [M2: produc3658429121746597890et_nat > $o] :
% 0.34/0.66 ( ( member6576561426505652726_nat_o @ M2 @ B4 )
% 0.34/0.66 => ? [X6: nat] :
% 0.34/0.66 ( ( member_nat @ X6 @ A )
% 0.34/0.66 & ( ord_less_eq_set_nat @ ( F @ X6 ) @ ( G @ M2 ) ) ) )
% 0.34/0.66 => ( ord_less_eq_set_nat @ ( comple7806235888213564991et_nat @ ( image_nat_set_nat @ F @ A ) ) @ ( comple7806235888213564991et_nat @ ( image_6888641473154130958et_nat @ G @ B4 ) ) ) ) ) ) ).
% 0.34/0.66
% 0.34/0.66 % cINF_mono
% 0.34/0.66 thf(fact_6393_le__cINF__iff,axiom,
% 0.34/0.66 ! [A: set_Product_unit,F: product_unit > int,U: int] :
% 0.34/0.66 ( ( A != bot_bo3957492148770167129t_unit )
% 0.34/0.66 => ( ( condit1735850657277959132ow_int @ ( image_873079544045703924it_int @ F @ A ) )
% 0.34/0.66 => ( ( ord_less_eq_int @ U @ ( complete_Inf_Inf_int @ ( image_873079544045703924it_int @ F @ A ) ) )
% 0.34/0.66 = ( ! [X2: product_unit] :
% 0.34/0.66 ( ( member_Product_unit @ X2 @ A )
% 0.34/0.66 => ( ord_less_eq_int @ U @ ( F @ X2 ) ) ) ) ) ) ) ).
% 0.34/0.66
% 0.34/0.66 % le_cINF_iff
% 0.34/0.66 thf(fact_6394_le__cINF__iff,axiom,
% 0.34/0.66 ! [A: set_o,F: $o > int,U: int] :
% 0.34/0.66 ( ( A != bot_bot_set_o )
% 0.34/0.66 => ( ( condit1735850657277959132ow_int @ ( image_o_int2 @ F @ A ) )
% 0.34/0.66 => ( ( ord_less_eq_int @ U @ ( complete_Inf_Inf_int @ ( image_o_int2 @ F @ A ) ) )
% 0.34/0.66 = ( ! [X2: $o] :
% 0.34/0.66 ( ( member_o @ X2 @ A )
% 0.34/0.66 => ( ord_less_eq_int @ U @ ( F @ X2 ) ) ) ) ) ) ) ).
% 0.34/0.66
% 0.34/0.66 % le_cINF_iff
% 0.34/0.66 thf(fact_6395_le__cINF__iff,axiom,
% 0.34/0.66 ! [A: set_nat,F: nat > int,U: int] :
% 0.34/0.66 ( ( A != bot_bot_set_nat )
% 0.34/0.66 => ( ( condit1735850657277959132ow_int @ ( image_nat_int2 @ F @ A ) )
% 0.34/0.66 => ( ( ord_less_eq_int @ U @ ( complete_Inf_Inf_int @ ( image_nat_int2 @ F @ A ) ) )
% 0.34/0.66 = ( ! [X2: nat] :
% 0.34/0.66 ( ( member_nat @ X2 @ A )
% 0.34/0.66 => ( ord_less_eq_int @ U @ ( F @ X2 ) ) ) ) ) ) ) ).
% 0.34/0.66
% 0.34/0.66 % le_cINF_iff
% 0.34/0.66 thf(fact_6396_le__cINF__iff,axiom,
% 0.34/0.66 ! [A: set_int,F: int > int,U: int] :
% 0.34/0.66 ( ( A != bot_bot_set_int )
% 0.34/0.66 => ( ( condit1735850657277959132ow_int @ ( image_int_int2 @ F @ A ) )
% 0.34/0.66 => ( ( ord_less_eq_int @ U @ ( complete_Inf_Inf_int @ ( image_int_int2 @ F @ A ) ) )
% 0.34/0.66 = ( ! [X2: int] :
% 0.34/0.66 ( ( member_int @ X2 @ A )
% 0.34/0.66 => ( ord_less_eq_int @ U @ ( F @ X2 ) ) ) ) ) ) ) ).
% 0.34/0.66
% 0.34/0.66 % le_cINF_iff
% 0.34/0.66 thf(fact_6397_le__cINF__iff,axiom,
% 0.34/0.66 ! [A: set_Product_unit,F: product_unit > nat,U: nat] :
% 0.34/0.66 ( ( A != bot_bo3957492148770167129t_unit )
% 0.34/0.66 => ( ( condit1738341127787009408ow_nat @ ( image_875570014554754200it_nat @ F @ A ) )
% 0.34/0.66 => ( ( ord_less_eq_nat @ U @ ( complete_Inf_Inf_nat @ ( image_875570014554754200it_nat @ F @ A ) ) )
% 0.34/0.66 = ( ! [X2: product_unit] :
% 0.34/0.66 ( ( member_Product_unit @ X2 @ A )
% 0.34/0.66 => ( ord_less_eq_nat @ U @ ( F @ X2 ) ) ) ) ) ) ) ).
% 0.34/0.66
% 0.34/0.66 % le_cINF_iff
% 0.34/0.66 thf(fact_6398_le__cINF__iff,axiom,
% 0.34/0.66 ! [A: set_o,F: $o > nat,U: nat] :
% 0.34/0.66 ( ( A != bot_bot_set_o )
% 0.34/0.66 => ( ( condit1738341127787009408ow_nat @ ( image_o_nat2 @ F @ A ) )
% 0.34/0.66 => ( ( ord_less_eq_nat @ U @ ( complete_Inf_Inf_nat @ ( image_o_nat2 @ F @ A ) ) )
% 0.34/0.66 = ( ! [X2: $o] :
% 0.34/0.66 ( ( member_o @ X2 @ A )
% 0.34/0.66 => ( ord_less_eq_nat @ U @ ( F @ X2 ) ) ) ) ) ) ) ).
% 0.34/0.66
% 0.34/0.66 % le_cINF_iff
% 0.34/0.66 thf(fact_6399_le__cINF__iff,axiom,
% 0.34/0.66 ! [A: set_nat,F: nat > nat,U: nat] :
% 0.34/0.66 ( ( A != bot_bot_set_nat )
% 0.34/0.66 => ( ( condit1738341127787009408ow_nat @ ( image_nat_nat2 @ F @ A ) )
% 0.34/0.66 => ( ( ord_less_eq_nat @ U @ ( complete_Inf_Inf_nat @ ( image_nat_nat2 @ F @ A ) ) )
% 0.34/0.66 = ( ! [X2: nat] :
% 0.34/0.66 ( ( member_nat @ X2 @ A )
% 0.34/0.66 => ( ord_less_eq_nat @ U @ ( F @ X2 ) ) ) ) ) ) ) ).
% 0.34/0.66
% 0.34/0.66 % le_cINF_iff
% 0.34/0.66 thf(fact_6400_le__cINF__iff,axiom,
% 0.34/0.66 ! [A: set_int,F: int > nat,U: nat] :
% 0.34/0.66 ( ( A != bot_bot_set_int )
% 0.34/0.66 => ( ( condit1738341127787009408ow_nat @ ( image_int_nat2 @ F @ A ) )
% 0.34/0.66 => ( ( ord_less_eq_nat @ U @ ( complete_Inf_Inf_nat @ ( image_int_nat2 @ F @ A ) ) )
% 0.34/0.66 = ( ! [X2: int] :
% 0.34/0.66 ( ( member_int @ X2 @ A )
% 0.34/0.66 => ( ord_less_eq_nat @ U @ ( F @ X2 ) ) ) ) ) ) ) ).
% 0.34/0.66
% 0.34/0.66 % le_cINF_iff
% 0.34/0.66 thf(fact_6401_le__cINF__iff,axiom,
% 0.34/0.66 ! [A: set_Product_unit,F: product_unit > product_unit,U: product_unit] :
% 0.34/0.66 ( ( A != bot_bo3957492148770167129t_unit )
% 0.34/0.66 => ( ( condit9161856309968323677t_unit @ ( image_405062704495631173t_unit @ F @ A ) )
% 0.34/0.66 => ( ( ord_le3221252021190050221t_unit @ U @ ( comple2584293577114468500t_unit @ ( image_405062704495631173t_unit @ F @ A ) ) )
% 0.34/0.66 = ( ! [X2: product_unit] :
% 0.34/0.66 ( ( member_Product_unit @ X2 @ A )
% 0.34/0.66 => ( ord_le3221252021190050221t_unit @ U @ ( F @ X2 ) ) ) ) ) ) ) ).
% 0.34/0.66
% 0.34/0.66 % le_cINF_iff
% 0.34/0.66 thf(fact_6402_le__cINF__iff,axiom,
% 0.34/0.66 ! [A: set_o,F: $o > product_unit,U: product_unit] :
% 0.34/0.66 ( ( A != bot_bot_set_o )
% 0.34/0.66 => ( ( condit9161856309968323677t_unit @ ( image_o_Product_unit2 @ F @ A ) )
% 0.34/0.66 => ( ( ord_le3221252021190050221t_unit @ U @ ( comple2584293577114468500t_unit @ ( image_o_Product_unit2 @ F @ A ) ) )
% 0.34/0.66 = ( ! [X2: $o] :
% 0.34/0.66 ( ( member_o @ X2 @ A )
% 0.34/0.66 => ( ord_le3221252021190050221t_unit @ U @ ( F @ X2 ) ) ) ) ) ) ) ).
% 0.34/0.66
% 0.34/0.66 % le_cINF_iff
% 0.34/0.66 thf(fact_6403_cInf__superset__mono,axiom,
% 0.34/0.66 ! [A: set_o,B4: set_o] :
% 0.34/0.66 ( ( A != bot_bot_set_o )
% 0.34/0.66 => ( ( condit5413489452508810728elow_o @ B4 )
% 0.34/0.66 => ( ( ord_less_eq_set_o @ A @ B4 )
% 0.34/0.66 => ( ord_less_eq_o @ ( complete_Inf_Inf_o @ B4 ) @ ( complete_Inf_Inf_o @ A ) ) ) ) ) ).
% 0.34/0.66
% 0.34/0.66 % cInf_superset_mono
% 0.34/0.66 thf(fact_6404_cInf__superset__mono,axiom,
% 0.34/0.66 ! [A: set_Pr4532377907799695533_nat_o,B4: set_Pr4532377907799695533_nat_o] :
% 0.34/0.66 ( ( A != bot_bo7824918357723582233_nat_o )
% 0.34/0.66 => ( ( condit8399017556247434269_nat_o @ B4 )
% 0.34/0.66 => ( ( ord_le2965882846123202637_nat_o @ A @ B4 )
% 0.34/0.66 => ( ord_le729326519192465773_nat_o @ ( comple90455603224345172_nat_o @ B4 ) @ ( comple90455603224345172_nat_o @ A ) ) ) ) ) ).
% 0.34/0.66
% 0.34/0.66 % cInf_superset_mono
% 0.34/0.66 thf(fact_6405_cInf__superset__mono,axiom,
% 0.34/0.66 ! [A: set_filter_nat,B4: set_filter_nat] :
% 0.34/0.66 ( ( A != bot_bo498966703094740906er_nat )
% 0.34/0.66 => ( ( condit3133331395780078992er_nat @ B4 )
% 0.34/0.66 => ( ( ord_le2426478655948331894er_nat @ A @ B4 )
% 0.34/0.66 => ( ord_le2510731241096832064er_nat @ ( comple6612786467464916249er_nat @ B4 ) @ ( comple6612786467464916249er_nat @ A ) ) ) ) ) ).
% 0.34/0.66
% 0.34/0.66 % cInf_superset_mono
% 0.34/0.66 thf(fact_6406_cInf__superset__mono,axiom,
% 0.34/0.66 ! [A: set_int,B4: set_int] :
% 0.34/0.66 ( ( A != bot_bot_set_int )
% 0.34/0.66 => ( ( condit1735850657277959132ow_int @ B4 )
% 0.34/0.66 => ( ( ord_less_eq_set_int @ A @ B4 )
% 0.34/0.66 => ( ord_less_eq_int @ ( complete_Inf_Inf_int @ B4 ) @ ( complete_Inf_Inf_int @ A ) ) ) ) ) ).
% 0.34/0.66
% 0.34/0.66 % cInf_superset_mono
% 0.34/0.66 thf(fact_6407_cInf__superset__mono,axiom,
% 0.34/0.66 ! [A: set_nat,B4: set_nat] :
% 0.34/0.66 ( ( A != bot_bot_set_nat )
% 0.34/0.66 => ( ( condit1738341127787009408ow_nat @ B4 )
% 0.34/0.66 => ( ( ord_less_eq_set_nat @ A @ B4 )
% 0.34/0.66 => ( ord_less_eq_nat @ ( complete_Inf_Inf_nat @ B4 ) @ ( complete_Inf_Inf_nat @ A ) ) ) ) ) ).
% 0.34/0.66
% 0.34/0.66 % cInf_superset_mono
% 0.34/0.66 thf(fact_6408_cInf__superset__mono,axiom,
% 0.34/0.66 ! [A: set_set_nat,B4: set_set_nat] :
% 0.34/0.66 ( ( A != bot_bot_set_set_nat )
% 0.34/0.66 => ( ( condit68592940725977398et_nat @ B4 )
% 0.34/0.66 => ( ( ord_le6893508408891458716et_nat @ A @ B4 )
% 0.34/0.66 => ( ord_less_eq_set_nat @ ( comple7806235888213564991et_nat @ B4 ) @ ( comple7806235888213564991et_nat @ A ) ) ) ) ) ).
% 0.34/0.66
% 0.34/0.66 % cInf_superset_mono
% 0.34/0.66 thf(fact_6409_cInf__superset__mono,axiom,
% 0.34/0.66 ! [A: set_Product_unit,B4: set_Product_unit] :
% 0.34/0.66 ( ( A != bot_bo3957492148770167129t_unit )
% 0.34/0.66 => ( ( condit9161856309968323677t_unit @ B4 )
% 0.34/0.66 => ( ( ord_le3507040750410214029t_unit @ A @ B4 )
% 0.34/0.66 => ( ord_le3221252021190050221t_unit @ ( comple2584293577114468500t_unit @ B4 ) @ ( comple2584293577114468500t_unit @ A ) ) ) ) ) ).
% 0.34/0.66
% 0.34/0.66 % cInf_superset_mono
% 0.34/0.66 thf(fact_6410_cInf__insert,axiom,
% 0.34/0.66 ! [X4: set_o,A2: $o] :
% 0.34/0.66 ( ( X4 != bot_bot_set_o )
% 0.34/0.66 => ( ( condit5413489452508810728elow_o @ X4 )
% 0.34/0.66 => ( ( complete_Inf_Inf_o @ ( insert_o @ A2 @ X4 ) )
% 0.34/0.66 = ( inf_inf_o @ A2 @ ( complete_Inf_Inf_o @ X4 ) ) ) ) ) ).
% 0.34/0.66
% 0.34/0.66 % cInf_insert
% 0.34/0.66 thf(fact_6411_cInf__insert,axiom,
% 0.34/0.66 ! [X4: set_int,A2: int] :
% 0.34/0.66 ( ( X4 != bot_bot_set_int )
% 0.34/0.66 => ( ( condit1735850657277959132ow_int @ X4 )
% 0.34/0.66 => ( ( complete_Inf_Inf_int @ ( insert_int @ A2 @ X4 ) )
% 0.34/0.66 = ( inf_inf_int @ A2 @ ( complete_Inf_Inf_int @ X4 ) ) ) ) ) ).
% 0.34/0.66
% 0.34/0.66 % cInf_insert
% 0.34/0.66 thf(fact_6412_cInf__insert,axiom,
% 0.34/0.66 ! [X4: set_Pr4532377907799695533_nat_o,A2: produc3658429121746597890et_nat > $o] :
% 0.34/0.66 ( ( X4 != bot_bo7824918357723582233_nat_o )
% 0.34/0.66 => ( ( condit8399017556247434269_nat_o @ X4 )
% 0.34/0.66 => ( ( comple90455603224345172_nat_o @ ( insert5175938949040314269_nat_o @ A2 @ X4 ) )
% 0.34/0.66 = ( inf_in1318976480646536635_nat_o @ A2 @ ( comple90455603224345172_nat_o @ X4 ) ) ) ) ) ).
% 0.34/0.66
% 0.34/0.66 % cInf_insert
% 0.34/0.66 thf(fact_6413_cInf__insert,axiom,
% 0.34/0.66 ! [X4: set_nat,A2: nat] :
% 0.34/0.66 ( ( X4 != bot_bot_set_nat )
% 0.34/0.66 => ( ( condit1738341127787009408ow_nat @ X4 )
% 0.34/0.66 => ( ( complete_Inf_Inf_nat @ ( insert_nat @ A2 @ X4 ) )
% 0.34/0.66 = ( inf_inf_nat @ A2 @ ( complete_Inf_Inf_nat @ X4 ) ) ) ) ) ).
% 0.34/0.66
% 0.34/0.66 % cInf_insert
% 0.34/0.66 thf(fact_6414_cInf__insert,axiom,
% 0.34/0.66 ! [X4: set_set_nat,A2: set_nat] :
% 0.34/0.66 ( ( X4 != bot_bot_set_set_nat )
% 0.34/0.66 => ( ( condit68592940725977398et_nat @ X4 )
% 0.34/0.66 => ( ( comple7806235888213564991et_nat @ ( insert_set_nat @ A2 @ X4 ) )
% 0.34/0.66 = ( inf_inf_set_nat @ A2 @ ( comple7806235888213564991et_nat @ X4 ) ) ) ) ) ).
% 0.34/0.66
% 0.34/0.66 % cInf_insert
% 0.34/0.66 thf(fact_6415_cInf__insert,axiom,
% 0.34/0.66 ! [X4: set_Product_unit,A2: product_unit] :
% 0.34/0.66 ( ( X4 != bot_bo3957492148770167129t_unit )
% 0.34/0.66 => ( ( condit9161856309968323677t_unit @ X4 )
% 0.34/0.66 => ( ( comple2584293577114468500t_unit @ ( insert_Product_unit @ A2 @ X4 ) )
% 0.34/0.66 = ( inf_inf_Product_unit @ A2 @ ( comple2584293577114468500t_unit @ X4 ) ) ) ) ) ).
% 0.34/0.66
% 0.34/0.66 % cInf_insert
% 0.34/0.66 thf(fact_6416_cInf__insert__If,axiom,
% 0.34/0.66 ! [X4: set_o,A2: $o] :
% 0.34/0.66 ( ( condit5413489452508810728elow_o @ X4 )
% 0.34/0.66 => ( ( complete_Inf_Inf_o @ ( insert_o @ A2 @ X4 ) )
% 0.34/0.66 = ( ( ( X4 = bot_bot_set_o )
% 0.34/0.66 => A2 )
% 0.34/0.66 & ( ( X4 != bot_bot_set_o )
% 0.34/0.66 => ( inf_inf_o @ A2 @ ( complete_Inf_Inf_o @ X4 ) ) ) ) ) ) ).
% 0.34/0.66
% 0.34/0.66 % cInf_insert_If
% 0.34/0.66 thf(fact_6417_cInf__insert__If,axiom,
% 0.34/0.66 ! [X4: set_int,A2: int] :
% 0.34/0.66 ( ( condit1735850657277959132ow_int @ X4 )
% 0.34/0.66 => ( ( ( X4 = bot_bot_set_int )
% 0.34/0.66 => ( ( complete_Inf_Inf_int @ ( insert_int @ A2 @ X4 ) )
% 0.34/0.66 = A2 ) )
% 0.34/0.66 & ( ( X4 != bot_bot_set_int )
% 0.34/0.66 => ( ( complete_Inf_Inf_int @ ( insert_int @ A2 @ X4 ) )
% 0.34/0.66 = ( inf_inf_int @ A2 @ ( complete_Inf_Inf_int @ X4 ) ) ) ) ) ) ).
% 0.34/0.66
% 0.34/0.66 % cInf_insert_If
% 0.34/0.66 thf(fact_6418_cInf__insert__If,axiom,
% 0.34/0.66 ! [X4: set_Pr4532377907799695533_nat_o,A2: produc3658429121746597890et_nat > $o] :
% 0.34/0.66 ( ( condit8399017556247434269_nat_o @ X4 )
% 0.34/0.66 => ( ( ( X4 = bot_bo7824918357723582233_nat_o )
% 0.34/0.66 => ( ( comple90455603224345172_nat_o @ ( insert5175938949040314269_nat_o @ A2 @ X4 ) )
% 0.34/0.66 = A2 ) )
% 0.34/0.66 & ( ( X4 != bot_bo7824918357723582233_nat_o )
% 0.34/0.66 => ( ( comple90455603224345172_nat_o @ ( insert5175938949040314269_nat_o @ A2 @ X4 ) )
% 0.34/0.66 = ( inf_in1318976480646536635_nat_o @ A2 @ ( comple90455603224345172_nat_o @ X4 ) ) ) ) ) ) ).
% 0.34/0.66
% 0.34/0.66 % cInf_insert_If
% 0.34/0.66 thf(fact_6419_cInf__insert__If,axiom,
% 0.34/0.66 ! [X4: set_nat,A2: nat] :
% 0.34/0.66 ( ( condit1738341127787009408ow_nat @ X4 )
% 0.34/0.66 => ( ( ( X4 = bot_bot_set_nat )
% 0.34/0.66 => ( ( complete_Inf_Inf_nat @ ( insert_nat @ A2 @ X4 ) )
% 0.34/0.66 = A2 ) )
% 0.34/0.66 & ( ( X4 != bot_bot_set_nat )
% 0.34/0.66 => ( ( complete_Inf_Inf_nat @ ( insert_nat @ A2 @ X4 ) )
% 0.34/0.66 = ( inf_inf_nat @ A2 @ ( complete_Inf_Inf_nat @ X4 ) ) ) ) ) ) ).
% 0.34/0.66
% 0.34/0.66 % cInf_insert_If
% 0.34/0.66 thf(fact_6420_cInf__insert__If,axiom,
% 0.34/0.66 ! [X4: set_set_nat,A2: set_nat] :
% 0.34/0.66 ( ( condit68592940725977398et_nat @ X4 )
% 0.34/0.66 => ( ( ( X4 = bot_bot_set_set_nat )
% 0.34/0.66 => ( ( comple7806235888213564991et_nat @ ( insert_set_nat @ A2 @ X4 ) )
% 0.34/0.66 = A2 ) )
% 0.34/0.66 & ( ( X4 != bot_bot_set_set_nat )
% 0.34/0.66 => ( ( comple7806235888213564991et_nat @ ( insert_set_nat @ A2 @ X4 ) )
% 0.34/0.66 = ( inf_inf_set_nat @ A2 @ ( comple7806235888213564991et_nat @ X4 ) ) ) ) ) ) ).
% 0.34/0.66
% 0.34/0.66 % cInf_insert_If
% 0.34/0.66 thf(fact_6421_cInf__insert__If,axiom,
% 0.34/0.66 ! [X4: set_Product_unit,A2: product_unit] :
% 0.34/0.66 ( ( condit9161856309968323677t_unit @ X4 )
% 0.34/0.66 => ( ( ( X4 = bot_bo3957492148770167129t_unit )
% 0.34/0.66 => ( ( comple2584293577114468500t_unit @ ( insert_Product_unit @ A2 @ X4 ) )
% 0.34/0.66 = A2 ) )
% 0.34/0.66 & ( ( X4 != bot_bo3957492148770167129t_unit )
% 0.34/0.66 => ( ( comple2584293577114468500t_unit @ ( insert_Product_unit @ A2 @ X4 ) )
% 0.34/0.66 = ( inf_inf_Product_unit @ A2 @ ( comple2584293577114468500t_unit @ X4 ) ) ) ) ) ) ).
% 0.34/0.66
% 0.34/0.66 % cInf_insert_If
% 0.34/0.66 thf(fact_6422_cInf__union__distrib,axiom,
% 0.34/0.66 ! [A: set_o,B4: set_o] :
% 0.34/0.66 ( ( A != bot_bot_set_o )
% 0.34/0.66 => ( ( condit5413489452508810728elow_o @ A )
% 0.34/0.66 => ( ( B4 != bot_bot_set_o )
% 0.34/0.66 => ( ( condit5413489452508810728elow_o @ B4 )
% 0.34/0.66 => ( ( complete_Inf_Inf_o @ ( sup_sup_set_o @ A @ B4 ) )
% 0.34/0.66 = ( inf_inf_o @ ( complete_Inf_Inf_o @ A ) @ ( complete_Inf_Inf_o @ B4 ) ) ) ) ) ) ) ).
% 0.34/0.66
% 0.34/0.66 % cInf_union_distrib
% 0.34/0.66 thf(fact_6423_cInf__union__distrib,axiom,
% 0.34/0.66 ! [A: set_int,B4: set_int] :
% 0.34/0.66 ( ( A != bot_bot_set_int )
% 0.34/0.66 => ( ( condit1735850657277959132ow_int @ A )
% 0.34/0.66 => ( ( B4 != bot_bot_set_int )
% 0.34/0.66 => ( ( condit1735850657277959132ow_int @ B4 )
% 0.34/0.66 => ( ( complete_Inf_Inf_int @ ( sup_sup_set_int @ A @ B4 ) )
% 0.34/0.66 = ( inf_inf_int @ ( complete_Inf_Inf_int @ A ) @ ( complete_Inf_Inf_int @ B4 ) ) ) ) ) ) ) ).
% 0.34/0.66
% 0.34/0.66 % cInf_union_distrib
% 0.34/0.66 thf(fact_6424_cInf__union__distrib,axiom,
% 0.34/0.66 ! [A: set_Pr4532377907799695533_nat_o,B4: set_Pr4532377907799695533_nat_o] :
% 0.34/0.66 ( ( A != bot_bo7824918357723582233_nat_o )
% 0.34/0.66 => ( ( condit8399017556247434269_nat_o @ A )
% 0.34/0.66 => ( ( B4 != bot_bo7824918357723582233_nat_o )
% 0.34/0.66 => ( ( condit8399017556247434269_nat_o @ B4 )
% 0.34/0.66 => ( ( comple90455603224345172_nat_o @ ( sup_su5209123915105501825_nat_o @ A @ B4 ) )
% 0.34/0.66 = ( inf_in1318976480646536635_nat_o @ ( comple90455603224345172_nat_o @ A ) @ ( comple90455603224345172_nat_o @ B4 ) ) ) ) ) ) ) ).
% 0.34/0.66
% 0.34/0.66 % cInf_union_distrib
% 0.34/0.66 thf(fact_6425_cInf__union__distrib,axiom,
% 0.34/0.66 ! [A: set_nat,B4: set_nat] :
% 0.34/0.66 ( ( A != bot_bot_set_nat )
% 0.34/0.66 => ( ( condit1738341127787009408ow_nat @ A )
% 0.34/0.66 => ( ( B4 != bot_bot_set_nat )
% 0.34/0.66 => ( ( condit1738341127787009408ow_nat @ B4 )
% 0.34/0.66 => ( ( complete_Inf_Inf_nat @ ( sup_sup_set_nat @ A @ B4 ) )
% 0.34/0.66 = ( inf_inf_nat @ ( complete_Inf_Inf_nat @ A ) @ ( complete_Inf_Inf_nat @ B4 ) ) ) ) ) ) ) ).
% 0.34/0.66
% 0.34/0.66 % cInf_union_distrib
% 0.34/0.66 thf(fact_6426_cInf__union__distrib,axiom,
% 0.34/0.66 ! [A: set_set_nat,B4: set_set_nat] :
% 0.34/0.66 ( ( A != bot_bot_set_set_nat )
% 0.34/0.66 => ( ( condit68592940725977398et_nat @ A )
% 0.34/0.66 => ( ( B4 != bot_bot_set_set_nat )
% 0.34/0.66 => ( ( condit68592940725977398et_nat @ B4 )
% 0.34/0.66 => ( ( comple7806235888213564991et_nat @ ( sup_sup_set_set_nat @ A @ B4 ) )
% 0.34/0.66 = ( inf_inf_set_nat @ ( comple7806235888213564991et_nat @ A ) @ ( comple7806235888213564991et_nat @ B4 ) ) ) ) ) ) ) ).
% 0.34/0.66
% 0.34/0.66 % cInf_union_distrib
% 0.34/0.66 thf(fact_6427_cInf__union__distrib,axiom,
% 0.34/0.66 ! [A: set_Product_unit,B4: set_Product_unit] :
% 0.34/0.66 ( ( A != bot_bo3957492148770167129t_unit )
% 0.34/0.66 => ( ( condit9161856309968323677t_unit @ A )
% 0.34/0.66 => ( ( B4 != bot_bo3957492148770167129t_unit )
% 0.34/0.66 => ( ( condit9161856309968323677t_unit @ B4 )
% 0.34/0.66 => ( ( comple2584293577114468500t_unit @ ( sup_su793286257634532545t_unit @ A @ B4 ) )
% 0.34/0.66 = ( inf_inf_Product_unit @ ( comple2584293577114468500t_unit @ A ) @ ( comple2584293577114468500t_unit @ B4 ) ) ) ) ) ) ) ).
% 0.34/0.66
% 0.34/0.66 % cInf_union_distrib
% 0.34/0.66 thf(fact_6428_dbl__inc__def,axiom,
% 0.34/0.66 ( neg_nu5831290666863070958nteger
% 0.34/0.66 = ( ^ [X2: code_integer] : ( plus_p5714425477246183910nteger @ ( plus_p5714425477246183910nteger @ X2 @ X2 ) @ one_one_Code_integer ) ) ) ).
% 0.34/0.66
% 0.34/0.66 % dbl_inc_def
% 0.34/0.66 thf(fact_6429_dbl__inc__def,axiom,
% 0.34/0.66 ( neg_nu5219082963157363817nc_rat
% 0.34/0.66 = ( ^ [X2: rat] : ( plus_plus_rat @ ( plus_plus_rat @ X2 @ X2 ) @ one_one_rat ) ) ) ).
% 0.34/0.66
% 0.34/0.66 % dbl_inc_def
% 0.34/0.66 thf(fact_6430_dbl__inc__def,axiom,
% 0.34/0.66 ( neg_nu5851722552734809277nc_int
% 0.34/0.66 = ( ^ [X2: int] : ( plus_plus_int @ ( plus_plus_int @ X2 @ X2 ) @ one_one_int ) ) ) ).
% 0.34/0.66
% 0.34/0.66 % dbl_inc_def
% 0.34/0.66 thf(fact_6431_dvd__mult__cancel1,axiom,
% 0.34/0.66 ! [M4: nat,N2: nat] :
% 0.34/0.66 ( ( ord_less_nat @ zero_zero_nat @ M4 )
% 0.34/0.66 => ( ( dvd_dvd_nat @ ( times_times_nat @ M4 @ N2 ) @ M4 )
% 0.34/0.66 = ( N2 = one_one_nat ) ) ) ).
% 0.34/0.66
% 0.34/0.66 % dvd_mult_cancel1
% 0.34/0.66 thf(fact_6432_dvd__mult__cancel2,axiom,
% 0.34/0.66 ! [M4: nat,N2: nat] :
% 0.34/0.66 ( ( ord_less_nat @ zero_zero_nat @ M4 )
% 0.34/0.66 => ( ( dvd_dvd_nat @ ( times_times_nat @ N2 @ M4 ) @ M4 )
% 0.34/0.66 = ( N2 = one_one_nat ) ) ) ).
% 0.34/0.66
% 0.34/0.66 % dvd_mult_cancel2
% 0.34/0.66 thf(fact_6433_dvd__minus__add,axiom,
% 0.34/0.66 ! [Q4: nat,N2: nat,R2: nat,M4: nat] :
% 0.34/0.66 ( ( ord_less_eq_nat @ Q4 @ N2 )
% 0.34/0.66 => ( ( ord_less_eq_nat @ Q4 @ ( times_times_nat @ R2 @ M4 ) )
% 0.34/0.66 => ( ( dvd_dvd_nat @ M4 @ ( minus_minus_nat @ N2 @ Q4 ) )
% 0.34/0.66 = ( dvd_dvd_nat @ M4 @ ( plus_plus_nat @ N2 @ ( minus_minus_nat @ ( times_times_nat @ R2 @ M4 ) @ Q4 ) ) ) ) ) ) ).
% 0.34/0.66
% 0.34/0.66 % dvd_minus_add
% 0.34/0.66 thf(fact_6434_cINF__superset__mono,axiom,
% 0.34/0.66 ! [A: set_Product_unit,G: product_unit > int,B4: set_Product_unit,F: product_unit > int] :
% 0.34/0.66 ( ( A != bot_bo3957492148770167129t_unit )
% 0.34/0.66 => ( ( condit1735850657277959132ow_int @ ( image_873079544045703924it_int @ G @ B4 ) )
% 0.34/0.66 => ( ( ord_le3507040750410214029t_unit @ A @ B4 )
% 0.34/0.66 => ( ! [X3: product_unit] :
% 0.34/0.66 ( ( member_Product_unit @ X3 @ B4 )
% 0.34/0.66 => ( ord_less_eq_int @ ( G @ X3 ) @ ( F @ X3 ) ) )
% 0.34/0.66 => ( ord_less_eq_int @ ( complete_Inf_Inf_int @ ( image_873079544045703924it_int @ G @ B4 ) ) @ ( complete_Inf_Inf_int @ ( image_873079544045703924it_int @ F @ A ) ) ) ) ) ) ) ).
% 0.34/0.66
% 0.34/0.66 % cINF_superset_mono
% 0.34/0.66 thf(fact_6435_cINF__superset__mono,axiom,
% 0.34/0.66 ! [A: set_o,G: $o > int,B4: set_o,F: $o > int] :
% 0.34/0.66 ( ( A != bot_bot_set_o )
% 0.34/0.66 => ( ( condit1735850657277959132ow_int @ ( image_o_int2 @ G @ B4 ) )
% 0.34/0.66 => ( ( ord_less_eq_set_o @ A @ B4 )
% 0.34/0.66 => ( ! [X3: $o] :
% 0.34/0.66 ( ( member_o @ X3 @ B4 )
% 0.34/0.66 => ( ord_less_eq_int @ ( G @ X3 ) @ ( F @ X3 ) ) )
% 0.34/0.66 => ( ord_less_eq_int @ ( complete_Inf_Inf_int @ ( image_o_int2 @ G @ B4 ) ) @ ( complete_Inf_Inf_int @ ( image_o_int2 @ F @ A ) ) ) ) ) ) ) ).
% 0.34/0.66
% 0.34/0.66 % cINF_superset_mono
% 0.34/0.66 thf(fact_6436_cINF__superset__mono,axiom,
% 0.34/0.66 ! [A: set_int,G: int > int,B4: set_int,F: int > int] :
% 0.34/0.66 ( ( A != bot_bot_set_int )
% 0.34/0.66 => ( ( condit1735850657277959132ow_int @ ( image_int_int2 @ G @ B4 ) )
% 0.34/0.66 => ( ( ord_less_eq_set_int @ A @ B4 )
% 0.34/0.66 => ( ! [X3: int] :
% 0.34/0.66 ( ( member_int @ X3 @ B4 )
% 0.34/0.66 => ( ord_less_eq_int @ ( G @ X3 ) @ ( F @ X3 ) ) )
% 0.34/0.66 => ( ord_less_eq_int @ ( complete_Inf_Inf_int @ ( image_int_int2 @ G @ B4 ) ) @ ( complete_Inf_Inf_int @ ( image_int_int2 @ F @ A ) ) ) ) ) ) ) ).
% 0.34/0.66
% 0.34/0.66 % cINF_superset_mono
% 0.34/0.66 thf(fact_6437_cINF__superset__mono,axiom,
% 0.34/0.66 ! [A: set_nat,G: nat > int,B4: set_nat,F: nat > int] :
% 0.34/0.66 ( ( A != bot_bot_set_nat )
% 0.34/0.66 => ( ( condit1735850657277959132ow_int @ ( image_nat_int2 @ G @ B4 ) )
% 0.34/0.66 => ( ( ord_less_eq_set_nat @ A @ B4 )
% 0.34/0.66 => ( ! [X3: nat] :
% 0.34/0.66 ( ( member_nat @ X3 @ B4 )
% 0.34/0.66 => ( ord_less_eq_int @ ( G @ X3 ) @ ( F @ X3 ) ) )
% 0.34/0.66 => ( ord_less_eq_int @ ( complete_Inf_Inf_int @ ( image_nat_int2 @ G @ B4 ) ) @ ( complete_Inf_Inf_int @ ( image_nat_int2 @ F @ A ) ) ) ) ) ) ) ).
% 0.34/0.66
% 0.34/0.66 % cINF_superset_mono
% 0.34/0.66 thf(fact_6438_cINF__superset__mono,axiom,
% 0.34/0.66 ! [A: set_Product_unit,G: product_unit > nat,B4: set_Product_unit,F: product_unit > nat] :
% 0.34/0.66 ( ( A != bot_bo3957492148770167129t_unit )
% 0.34/0.66 => ( ( condit1738341127787009408ow_nat @ ( image_875570014554754200it_nat @ G @ B4 ) )
% 0.34/0.66 => ( ( ord_le3507040750410214029t_unit @ A @ B4 )
% 0.34/0.66 => ( ! [X3: product_unit] :
% 0.34/0.66 ( ( member_Product_unit @ X3 @ B4 )
% 0.34/0.66 => ( ord_less_eq_nat @ ( G @ X3 ) @ ( F @ X3 ) ) )
% 0.34/0.66 => ( ord_less_eq_nat @ ( complete_Inf_Inf_nat @ ( image_875570014554754200it_nat @ G @ B4 ) ) @ ( complete_Inf_Inf_nat @ ( image_875570014554754200it_nat @ F @ A ) ) ) ) ) ) ) ).
% 0.34/0.66
% 0.34/0.66 % cINF_superset_mono
% 0.34/0.66 thf(fact_6439_cINF__superset__mono,axiom,
% 0.34/0.66 ! [A: set_o,G: $o > nat,B4: set_o,F: $o > nat] :
% 0.34/0.66 ( ( A != bot_bot_set_o )
% 0.34/0.66 => ( ( condit1738341127787009408ow_nat @ ( image_o_nat2 @ G @ B4 ) )
% 0.34/0.66 => ( ( ord_less_eq_set_o @ A @ B4 )
% 0.34/0.66 => ( ! [X3: $o] :
% 0.34/0.66 ( ( member_o @ X3 @ B4 )
% 0.34/0.66 => ( ord_less_eq_nat @ ( G @ X3 ) @ ( F @ X3 ) ) )
% 0.34/0.66 => ( ord_less_eq_nat @ ( complete_Inf_Inf_nat @ ( image_o_nat2 @ G @ B4 ) ) @ ( complete_Inf_Inf_nat @ ( image_o_nat2 @ F @ A ) ) ) ) ) ) ) ).
% 0.34/0.66
% 0.34/0.66 % cINF_superset_mono
% 0.34/0.66 thf(fact_6440_cINF__superset__mono,axiom,
% 0.34/0.66 ! [A: set_int,G: int > nat,B4: set_int,F: int > nat] :
% 0.34/0.66 ( ( A != bot_bot_set_int )
% 0.34/0.66 => ( ( condit1738341127787009408ow_nat @ ( image_int_nat2 @ G @ B4 ) )
% 0.34/0.66 => ( ( ord_less_eq_set_int @ A @ B4 )
% 0.34/0.66 => ( ! [X3: int] :
% 0.34/0.66 ( ( member_int @ X3 @ B4 )
% 0.34/0.66 => ( ord_less_eq_nat @ ( G @ X3 ) @ ( F @ X3 ) ) )
% 0.34/0.66 => ( ord_less_eq_nat @ ( complete_Inf_Inf_nat @ ( image_int_nat2 @ G @ B4 ) ) @ ( complete_Inf_Inf_nat @ ( image_int_nat2 @ F @ A ) ) ) ) ) ) ) ).
% 0.34/0.66
% 0.34/0.66 % cINF_superset_mono
% 0.34/0.66 thf(fact_6441_cINF__superset__mono,axiom,
% 0.34/0.66 ! [A: set_nat,G: nat > nat,B4: set_nat,F: nat > nat] :
% 0.34/0.66 ( ( A != bot_bot_set_nat )
% 0.34/0.66 => ( ( condit1738341127787009408ow_nat @ ( image_nat_nat2 @ G @ B4 ) )
% 0.34/0.66 => ( ( ord_less_eq_set_nat @ A @ B4 )
% 0.34/0.66 => ( ! [X3: nat] :
% 0.34/0.66 ( ( member_nat @ X3 @ B4 )
% 0.34/0.66 => ( ord_less_eq_nat @ ( G @ X3 ) @ ( F @ X3 ) ) )
% 0.34/0.66 => ( ord_less_eq_nat @ ( complete_Inf_Inf_nat @ ( image_nat_nat2 @ G @ B4 ) ) @ ( complete_Inf_Inf_nat @ ( image_nat_nat2 @ F @ A ) ) ) ) ) ) ) ).
% 0.34/0.66
% 0.34/0.66 % cINF_superset_mono
% 0.34/0.66 thf(fact_6442_cINF__superset__mono,axiom,
% 0.34/0.66 ! [A: set_Product_unit,G: product_unit > product_unit,B4: set_Product_unit,F: product_unit > product_unit] :
% 0.34/0.66 ( ( A != bot_bo3957492148770167129t_unit )
% 0.34/0.66 => ( ( condit9161856309968323677t_unit @ ( image_405062704495631173t_unit @ G @ B4 ) )
% 0.34/0.66 => ( ( ord_le3507040750410214029t_unit @ A @ B4 )
% 0.34/0.66 => ( ! [X3: product_unit] :
% 0.34/0.66 ( ( member_Product_unit @ X3 @ B4 )
% 0.34/0.66 => ( ord_le3221252021190050221t_unit @ ( G @ X3 ) @ ( F @ X3 ) ) )
% 0.34/0.66 => ( ord_le3221252021190050221t_unit @ ( comple2584293577114468500t_unit @ ( image_405062704495631173t_unit @ G @ B4 ) ) @ ( comple2584293577114468500t_unit @ ( image_405062704495631173t_unit @ F @ A ) ) ) ) ) ) ) ).
% 0.34/0.66
% 0.34/0.66 % cINF_superset_mono
% 0.34/0.66 thf(fact_6443_cINF__superset__mono,axiom,
% 0.34/0.66 ! [A: set_o,G: $o > product_unit,B4: set_o,F: $o > product_unit] :
% 0.34/0.66 ( ( A != bot_bot_set_o )
% 0.34/0.66 => ( ( condit9161856309968323677t_unit @ ( image_o_Product_unit2 @ G @ B4 ) )
% 0.34/0.66 => ( ( ord_less_eq_set_o @ A @ B4 )
% 0.34/0.66 => ( ! [X3: $o] :
% 0.34/0.66 ( ( member_o @ X3 @ B4 )
% 0.34/0.66 => ( ord_le3221252021190050221t_unit @ ( G @ X3 ) @ ( F @ X3 ) ) )
% 0.34/0.66 => ( ord_le3221252021190050221t_unit @ ( comple2584293577114468500t_unit @ ( image_o_Product_unit2 @ G @ B4 ) ) @ ( comple2584293577114468500t_unit @ ( image_o_Product_unit2 @ F @ A ) ) ) ) ) ) ) ).
% 0.34/0.66
% 0.34/0.66 % cINF_superset_mono
% 0.34/0.66 thf(fact_6444_less__eq__cInf__inter,axiom,
% 0.34/0.66 ! [A: set_o,B4: set_o] :
% 0.34/0.66 ( ( condit5413489452508810728elow_o @ A )
% 0.34/0.66 => ( ( condit5413489452508810728elow_o @ B4 )
% 0.34/0.66 => ( ( ( inf_inf_set_o @ A @ B4 )
% 0.34/0.66 != bot_bot_set_o )
% 0.34/0.66 => ( ord_less_eq_o @ ( inf_inf_o @ ( complete_Inf_Inf_o @ A ) @ ( complete_Inf_Inf_o @ B4 ) ) @ ( complete_Inf_Inf_o @ ( inf_inf_set_o @ A @ B4 ) ) ) ) ) ) ).
% 0.34/0.66
% 0.34/0.66 % less_eq_cInf_inter
% 0.34/0.66 thf(fact_6445_less__eq__cInf__inter,axiom,
% 0.34/0.66 ! [A: set_Pr4532377907799695533_nat_o,B4: set_Pr4532377907799695533_nat_o] :
% 0.34/0.66 ( ( condit8399017556247434269_nat_o @ A )
% 0.34/0.66 => ( ( condit8399017556247434269_nat_o @ B4 )
% 0.34/0.66 => ( ( ( inf_in1906310914598751387_nat_o @ A @ B4 )
% 0.34/0.66 != bot_bo7824918357723582233_nat_o )
% 0.34/0.66 => ( ord_le729326519192465773_nat_o @ ( inf_in1318976480646536635_nat_o @ ( comple90455603224345172_nat_o @ A ) @ ( comple90455603224345172_nat_o @ B4 ) ) @ ( comple90455603224345172_nat_o @ ( inf_in1906310914598751387_nat_o @ A @ B4 ) ) ) ) ) ) ).
% 0.34/0.66
% 0.34/0.66 % less_eq_cInf_inter
% 0.34/0.66 thf(fact_6446_less__eq__cInf__inter,axiom,
% 0.34/0.66 ! [A: set_filter_nat,B4: set_filter_nat] :
% 0.34/0.66 ( ( condit3133331395780078992er_nat @ A )
% 0.34/0.66 => ( ( condit3133331395780078992er_nat @ B4 )
% 0.34/0.66 => ( ( ( inf_in7395780696554941096er_nat @ A @ B4 )
% 0.34/0.66 != bot_bo498966703094740906er_nat )
% 0.34/0.66 => ( ord_le2510731241096832064er_nat @ ( inf_inf_filter_nat @ ( comple6612786467464916249er_nat @ A ) @ ( comple6612786467464916249er_nat @ B4 ) ) @ ( comple6612786467464916249er_nat @ ( inf_in7395780696554941096er_nat @ A @ B4 ) ) ) ) ) ) ).
% 0.34/0.66
% 0.34/0.66 % less_eq_cInf_inter
% 0.34/0.66 thf(fact_6447_less__eq__cInf__inter,axiom,
% 0.34/0.66 ! [A: set_int,B4: set_int] :
% 0.34/0.66 ( ( condit1735850657277959132ow_int @ A )
% 0.34/0.66 => ( ( condit1735850657277959132ow_int @ B4 )
% 0.34/0.66 => ( ( ( inf_inf_set_int @ A @ B4 )
% 0.34/0.66 != bot_bot_set_int )
% 0.34/0.66 => ( ord_less_eq_int @ ( inf_inf_int @ ( complete_Inf_Inf_int @ A ) @ ( complete_Inf_Inf_int @ B4 ) ) @ ( complete_Inf_Inf_int @ ( inf_inf_set_int @ A @ B4 ) ) ) ) ) ) ).
% 0.34/0.66
% 0.34/0.66 % less_eq_cInf_inter
% 0.34/0.66 thf(fact_6448_less__eq__cInf__inter,axiom,
% 0.34/0.66 ! [A: set_nat,B4: set_nat] :
% 0.34/0.66 ( ( condit1738341127787009408ow_nat @ A )
% 0.34/0.66 => ( ( condit1738341127787009408ow_nat @ B4 )
% 0.34/0.66 => ( ( ( inf_inf_set_nat @ A @ B4 )
% 0.34/0.66 != bot_bot_set_nat )
% 0.34/0.66 => ( ord_less_eq_nat @ ( inf_inf_nat @ ( complete_Inf_Inf_nat @ A ) @ ( complete_Inf_Inf_nat @ B4 ) ) @ ( complete_Inf_Inf_nat @ ( inf_inf_set_nat @ A @ B4 ) ) ) ) ) ) ).
% 0.34/0.66
% 0.34/0.66 % less_eq_cInf_inter
% 0.34/0.66 thf(fact_6449_less__eq__cInf__inter,axiom,
% 0.34/0.66 ! [A: set_set_nat,B4: set_set_nat] :
% 0.34/0.66 ( ( condit68592940725977398et_nat @ A )
% 0.34/0.66 => ( ( condit68592940725977398et_nat @ B4 )
% 0.34/0.66 => ( ( ( inf_inf_set_set_nat @ A @ B4 )
% 0.34/0.66 != bot_bot_set_set_nat )
% 0.34/0.66 => ( ord_less_eq_set_nat @ ( inf_inf_set_nat @ ( comple7806235888213564991et_nat @ A ) @ ( comple7806235888213564991et_nat @ B4 ) ) @ ( comple7806235888213564991et_nat @ ( inf_inf_set_set_nat @ A @ B4 ) ) ) ) ) ) ).
% 0.34/0.66
% 0.34/0.66 % less_eq_cInf_inter
% 0.34/0.66 thf(fact_6450_less__eq__cInf__inter,axiom,
% 0.34/0.66 ! [A: set_Product_unit,B4: set_Product_unit] :
% 0.34/0.66 ( ( condit9161856309968323677t_unit @ A )
% 0.34/0.66 => ( ( condit9161856309968323677t_unit @ B4 )
% 0.34/0.66 => ( ( ( inf_in4660618365625256667t_unit @ A @ B4 )
% 0.34/0.66 != bot_bo3957492148770167129t_unit )
% 0.34/0.66 => ( ord_le3221252021190050221t_unit @ ( inf_inf_Product_unit @ ( comple2584293577114468500t_unit @ A ) @ ( comple2584293577114468500t_unit @ B4 ) ) @ ( comple2584293577114468500t_unit @ ( inf_in4660618365625256667t_unit @ A @ B4 ) ) ) ) ) ) ).
% 0.34/0.66
% 0.34/0.66 % less_eq_cInf_inter
% 0.34/0.66 thf(fact_6451_cINF__insert,axiom,
% 0.34/0.66 ! [A: set_Product_unit,F: product_unit > nat,A2: product_unit] :
% 0.34/0.66 ( ( A != bot_bo3957492148770167129t_unit )
% 0.34/0.66 => ( ( condit1738341127787009408ow_nat @ ( image_875570014554754200it_nat @ F @ A ) )
% 0.34/0.66 => ( ( complete_Inf_Inf_nat @ ( image_875570014554754200it_nat @ F @ ( insert_Product_unit @ A2 @ A ) ) )
% 0.34/0.66 = ( inf_inf_nat @ ( F @ A2 ) @ ( complete_Inf_Inf_nat @ ( image_875570014554754200it_nat @ F @ A ) ) ) ) ) ) ).
% 0.34/0.66
% 0.34/0.66 % cINF_insert
% 0.34/0.66 thf(fact_6452_cINF__insert,axiom,
% 0.34/0.66 ! [A: set_o,F: $o > nat,A2: $o] :
% 0.34/0.66 ( ( A != bot_bot_set_o )
% 0.34/0.66 => ( ( condit1738341127787009408ow_nat @ ( image_o_nat2 @ F @ A ) )
% 0.34/0.66 => ( ( complete_Inf_Inf_nat @ ( image_o_nat2 @ F @ ( insert_o @ A2 @ A ) ) )
% 0.34/0.66 = ( inf_inf_nat @ ( F @ A2 ) @ ( complete_Inf_Inf_nat @ ( image_o_nat2 @ F @ A ) ) ) ) ) ) ).
% 0.34/0.66
% 0.34/0.66 % cINF_insert
% 0.34/0.66 thf(fact_6453_cINF__insert,axiom,
% 0.34/0.66 ! [A: set_nat,F: nat > nat,A2: nat] :
% 0.34/0.66 ( ( A != bot_bot_set_nat )
% 0.34/0.66 => ( ( condit1738341127787009408ow_nat @ ( image_nat_nat2 @ F @ A ) )
% 0.34/0.66 => ( ( complete_Inf_Inf_nat @ ( image_nat_nat2 @ F @ ( insert_nat @ A2 @ A ) ) )
% 0.34/0.66 = ( inf_inf_nat @ ( F @ A2 ) @ ( complete_Inf_Inf_nat @ ( image_nat_nat2 @ F @ A ) ) ) ) ) ) ).
% 0.34/0.66
% 0.34/0.66 % cINF_insert
% 0.34/0.66 thf(fact_6454_cINF__insert,axiom,
% 0.34/0.66 ! [A: set_int,F: int > nat,A2: int] :
% 0.34/0.66 ( ( A != bot_bot_set_int )
% 0.34/0.66 => ( ( condit1738341127787009408ow_nat @ ( image_int_nat2 @ F @ A ) )
% 0.34/0.66 => ( ( complete_Inf_Inf_nat @ ( image_int_nat2 @ F @ ( insert_int @ A2 @ A ) ) )
% 0.34/0.66 = ( inf_inf_nat @ ( F @ A2 ) @ ( complete_Inf_Inf_nat @ ( image_int_nat2 @ F @ A ) ) ) ) ) ) ).
% 0.34/0.66
% 0.34/0.66 % cINF_insert
% 0.34/0.66 thf(fact_6455_cINF__insert,axiom,
% 0.34/0.66 ! [A: set_Product_unit,F: product_unit > product_unit,A2: product_unit] :
% 0.34/0.66 ( ( A != bot_bo3957492148770167129t_unit )
% 0.34/0.66 => ( ( condit9161856309968323677t_unit @ ( image_405062704495631173t_unit @ F @ A ) )
% 0.34/0.66 => ( ( comple2584293577114468500t_unit @ ( image_405062704495631173t_unit @ F @ ( insert_Product_unit @ A2 @ A ) ) )
% 0.34/0.66 = ( inf_inf_Product_unit @ ( F @ A2 ) @ ( comple2584293577114468500t_unit @ ( image_405062704495631173t_unit @ F @ A ) ) ) ) ) ) ).
% 0.34/0.66
% 0.34/0.66 % cINF_insert
% 0.34/0.66 thf(fact_6456_cINF__insert,axiom,
% 0.34/0.66 ! [A: set_o,F: $o > product_unit,A2: $o] :
% 0.34/0.66 ( ( A != bot_bot_set_o )
% 0.34/0.66 => ( ( condit9161856309968323677t_unit @ ( image_o_Product_unit2 @ F @ A ) )
% 0.34/0.66 => ( ( comple2584293577114468500t_unit @ ( image_o_Product_unit2 @ F @ ( insert_o @ A2 @ A ) ) )
% 0.34/0.66 = ( inf_inf_Product_unit @ ( F @ A2 ) @ ( comple2584293577114468500t_unit @ ( image_o_Product_unit2 @ F @ A ) ) ) ) ) ) ).
% 0.34/0.66
% 0.34/0.66 % cINF_insert
% 0.34/0.66 thf(fact_6457_cINF__insert,axiom,
% 0.34/0.66 ! [A: set_nat,F: nat > product_unit,A2: nat] :
% 0.34/0.66 ( ( A != bot_bot_set_nat )
% 0.34/0.66 => ( ( condit9161856309968323677t_unit @ ( image_8730104196221521654t_unit @ F @ A ) )
% 0.34/0.66 => ( ( comple2584293577114468500t_unit @ ( image_8730104196221521654t_unit @ F @ ( insert_nat @ A2 @ A ) ) )
% 0.34/0.66 = ( inf_inf_Product_unit @ ( F @ A2 ) @ ( comple2584293577114468500t_unit @ ( image_8730104196221521654t_unit @ F @ A ) ) ) ) ) ) ).
% 0.34/0.66
% 0.34/0.66 % cINF_insert
% 0.34/0.66 thf(fact_6458_cINF__insert,axiom,
% 0.34/0.66 ! [A: set_int,F: int > product_unit,A2: int] :
% 0.34/0.66 ( ( A != bot_bot_set_int )
% 0.34/0.66 => ( ( condit9161856309968323677t_unit @ ( image_5846526143423701914t_unit @ F @ A ) )
% 0.34/0.66 => ( ( comple2584293577114468500t_unit @ ( image_5846526143423701914t_unit @ F @ ( insert_int @ A2 @ A ) ) )
% 0.34/0.66 = ( inf_inf_Product_unit @ ( F @ A2 ) @ ( comple2584293577114468500t_unit @ ( image_5846526143423701914t_unit @ F @ A ) ) ) ) ) ) ).
% 0.34/0.66
% 0.34/0.66 % cINF_insert
% 0.34/0.66 thf(fact_6459_cINF__insert,axiom,
% 0.34/0.66 ! [A: set_Product_unit,F: product_unit > set_nat,A2: product_unit] :
% 0.34/0.66 ( ( A != bot_bo3957492148770167129t_unit )
% 0.34/0.66 => ( ( condit68592940725977398et_nat @ ( image_8173849991827986510et_nat @ F @ A ) )
% 0.34/0.66 => ( ( comple7806235888213564991et_nat @ ( image_8173849991827986510et_nat @ F @ ( insert_Product_unit @ A2 @ A ) ) )
% 0.34/0.66 = ( inf_inf_set_nat @ ( F @ A2 ) @ ( comple7806235888213564991et_nat @ ( image_8173849991827986510et_nat @ F @ A ) ) ) ) ) ) ).
% 0.34/0.66
% 0.34/0.66 % cINF_insert
% 0.34/0.66 thf(fact_6460_cINF__insert,axiom,
% 0.34/0.66 ! [A: set_o,F: $o > set_nat,A2: $o] :
% 0.34/0.66 ( ( A != bot_bot_set_o )
% 0.34/0.66 => ( ( condit68592940725977398et_nat @ ( image_o_set_nat @ F @ A ) )
% 0.34/0.66 => ( ( comple7806235888213564991et_nat @ ( image_o_set_nat @ F @ ( insert_o @ A2 @ A ) ) )
% 0.34/0.66 = ( inf_inf_set_nat @ ( F @ A2 ) @ ( comple7806235888213564991et_nat @ ( image_o_set_nat @ F @ A ) ) ) ) ) ) ).
% 0.34/0.66
% 0.34/0.66 % cINF_insert
% 0.34/0.66 thf(fact_6461_power__eq__if,axiom,
% 0.34/0.66 ( power_8256067586552552935nteger
% 0.34/0.66 = ( ^ [P5: code_integer,M3: nat] : ( if_Code_integer @ ( M3 = zero_zero_nat ) @ one_one_Code_integer @ ( times_3573771949741848930nteger @ P5 @ ( power_8256067586552552935nteger @ P5 @ ( minus_minus_nat @ M3 @ one_one_nat ) ) ) ) ) ) ).
% 0.34/0.66
% 0.34/0.66 % power_eq_if
% 0.34/0.66 thf(fact_6462_power__eq__if,axiom,
% 0.34/0.66 ( power_7079662738309270450atural
% 0.34/0.66 = ( ^ [P5: code_natural,M3: nat] : ( if_Code_natural @ ( M3 = zero_zero_nat ) @ one_one_Code_natural @ ( times_2397367101498566445atural @ P5 @ ( power_7079662738309270450atural @ P5 @ ( minus_minus_nat @ M3 @ one_one_nat ) ) ) ) ) ) ).
% 0.34/0.66
% 0.34/0.66 % power_eq_if
% 0.34/0.66 thf(fact_6463_power__eq__if,axiom,
% 0.34/0.66 ( power_power_assn
% 0.34/0.66 = ( ^ [P5: assn,M3: nat] : ( if_assn @ ( M3 = zero_zero_nat ) @ one_one_assn @ ( times_times_assn @ P5 @ ( power_power_assn @ P5 @ ( minus_minus_nat @ M3 @ one_one_nat ) ) ) ) ) ) ).
% 0.34/0.66
% 0.34/0.66 % power_eq_if
% 0.34/0.66 thf(fact_6464_power__eq__if,axiom,
% 0.34/0.66 ( power_power_rat
% 0.34/0.66 = ( ^ [P5: rat,M3: nat] : ( if_rat @ ( M3 = zero_zero_nat ) @ one_one_rat @ ( times_times_rat @ P5 @ ( power_power_rat @ P5 @ ( minus_minus_nat @ M3 @ one_one_nat ) ) ) ) ) ) ).
% 0.34/0.66
% 0.34/0.66 % power_eq_if
% 0.34/0.66 thf(fact_6465_power__eq__if,axiom,
% 0.34/0.66 ( power_power_nat
% 0.34/0.66 = ( ^ [P5: nat,M3: nat] : ( if_nat @ ( M3 = zero_zero_nat ) @ one_one_nat @ ( times_times_nat @ P5 @ ( power_power_nat @ P5 @ ( minus_minus_nat @ M3 @ one_one_nat ) ) ) ) ) ) ).
% 0.34/0.66
% 0.34/0.66 % power_eq_if
% 0.34/0.66 thf(fact_6466_power__eq__if,axiom,
% 0.34/0.66 ( power_power_int
% 0.34/0.66 = ( ^ [P5: int,M3: nat] : ( if_int @ ( M3 = zero_zero_nat ) @ one_one_int @ ( times_times_int @ P5 @ ( power_power_int @ P5 @ ( minus_minus_nat @ M3 @ one_one_nat ) ) ) ) ) ) ).
% 0.34/0.66
% 0.34/0.66 % power_eq_if
% 0.34/0.66 thf(fact_6467_power__minus__mult,axiom,
% 0.34/0.66 ! [N2: nat,A2: assn] :
% 0.34/0.66 ( ( ord_less_nat @ zero_zero_nat @ N2 )
% 0.34/0.66 => ( ( times_times_assn @ ( power_power_assn @ A2 @ ( minus_minus_nat @ N2 @ one_one_nat ) ) @ A2 )
% 0.34/0.66 = ( power_power_assn @ A2 @ N2 ) ) ) ).
% 0.34/0.66
% 0.34/0.66 % power_minus_mult
% 0.34/0.66 thf(fact_6468_power__minus__mult,axiom,
% 0.34/0.66 ! [N2: nat,A2: rat] :
% 0.34/0.66 ( ( ord_less_nat @ zero_zero_nat @ N2 )
% 0.34/0.66 => ( ( times_times_rat @ ( power_power_rat @ A2 @ ( minus_minus_nat @ N2 @ one_one_nat ) ) @ A2 )
% 0.34/0.66 = ( power_power_rat @ A2 @ N2 ) ) ) ).
% 0.34/0.66
% 0.34/0.66 % power_minus_mult
% 0.34/0.66 thf(fact_6469_power__minus__mult,axiom,
% 0.34/0.66 ! [N2: nat,A2: nat] :
% 0.34/0.66 ( ( ord_less_nat @ zero_zero_nat @ N2 )
% 0.34/0.66 => ( ( times_times_nat @ ( power_power_nat @ A2 @ ( minus_minus_nat @ N2 @ one_one_nat ) ) @ A2 )
% 0.34/0.66 = ( power_power_nat @ A2 @ N2 ) ) ) ).
% 0.34/0.66
% 0.34/0.66 % power_minus_mult
% 0.34/0.66 thf(fact_6470_power__minus__mult,axiom,
% 0.34/0.66 ! [N2: nat,A2: int] :
% 0.34/0.66 ( ( ord_less_nat @ zero_zero_nat @ N2 )
% 0.34/0.66 => ( ( times_times_int @ ( power_power_int @ A2 @ ( minus_minus_nat @ N2 @ one_one_nat ) ) @ A2 )
% 0.34/0.66 = ( power_power_int @ A2 @ N2 ) ) ) ).
% 0.34/0.66
% 0.34/0.66 % power_minus_mult
% 0.34/0.66 thf(fact_6471_cINF__union,axiom,
% 0.34/0.66 ! [A: set_Product_unit,F: product_unit > nat,B4: set_Product_unit] :
% 0.34/0.66 ( ( A != bot_bo3957492148770167129t_unit )
% 0.34/0.66 => ( ( condit1738341127787009408ow_nat @ ( image_875570014554754200it_nat @ F @ A ) )
% 0.34/0.66 => ( ( B4 != bot_bo3957492148770167129t_unit )
% 0.34/0.66 => ( ( condit1738341127787009408ow_nat @ ( image_875570014554754200it_nat @ F @ B4 ) )
% 0.34/0.66 => ( ( complete_Inf_Inf_nat @ ( image_875570014554754200it_nat @ F @ ( sup_su793286257634532545t_unit @ A @ B4 ) ) )
% 0.34/0.66 = ( inf_inf_nat @ ( complete_Inf_Inf_nat @ ( image_875570014554754200it_nat @ F @ A ) ) @ ( complete_Inf_Inf_nat @ ( image_875570014554754200it_nat @ F @ B4 ) ) ) ) ) ) ) ) ).
% 0.34/0.66
% 0.34/0.66 % cINF_union
% 0.34/0.66 thf(fact_6472_cINF__union,axiom,
% 0.34/0.66 ! [A: set_o,F: $o > nat,B4: set_o] :
% 0.34/0.66 ( ( A != bot_bot_set_o )
% 0.34/0.66 => ( ( condit1738341127787009408ow_nat @ ( image_o_nat2 @ F @ A ) )
% 0.34/0.66 => ( ( B4 != bot_bot_set_o )
% 0.34/0.66 => ( ( condit1738341127787009408ow_nat @ ( image_o_nat2 @ F @ B4 ) )
% 0.34/0.66 => ( ( complete_Inf_Inf_nat @ ( image_o_nat2 @ F @ ( sup_sup_set_o @ A @ B4 ) ) )
% 0.34/0.66 = ( inf_inf_nat @ ( complete_Inf_Inf_nat @ ( image_o_nat2 @ F @ A ) ) @ ( complete_Inf_Inf_nat @ ( image_o_nat2 @ F @ B4 ) ) ) ) ) ) ) ) ).
% 0.34/0.66
% 0.34/0.66 % cINF_union
% 0.34/0.66 thf(fact_6473_cINF__union,axiom,
% 0.34/0.66 ! [A: set_nat,F: nat > nat,B4: set_nat] :
% 0.34/0.66 ( ( A != bot_bot_set_nat )
% 0.34/0.66 => ( ( condit1738341127787009408ow_nat @ ( image_nat_nat2 @ F @ A ) )
% 0.34/0.66 => ( ( B4 != bot_bot_set_nat )
% 0.34/0.66 => ( ( condit1738341127787009408ow_nat @ ( image_nat_nat2 @ F @ B4 ) )
% 0.34/0.66 => ( ( complete_Inf_Inf_nat @ ( image_nat_nat2 @ F @ ( sup_sup_set_nat @ A @ B4 ) ) )
% 0.34/0.66 = ( inf_inf_nat @ ( complete_Inf_Inf_nat @ ( image_nat_nat2 @ F @ A ) ) @ ( complete_Inf_Inf_nat @ ( image_nat_nat2 @ F @ B4 ) ) ) ) ) ) ) ) ).
% 0.34/0.66
% 0.34/0.66 % cINF_union
% 0.34/0.66 thf(fact_6474_cINF__union,axiom,
% 0.34/0.66 ! [A: set_int,F: int > nat,B4: set_int] :
% 0.34/0.66 ( ( A != bot_bot_set_int )
% 0.34/0.66 => ( ( condit1738341127787009408ow_nat @ ( image_int_nat2 @ F @ A ) )
% 0.34/0.66 => ( ( B4 != bot_bot_set_int )
% 0.34/0.66 => ( ( condit1738341127787009408ow_nat @ ( image_int_nat2 @ F @ B4 ) )
% 0.34/0.66 => ( ( complete_Inf_Inf_nat @ ( image_int_nat2 @ F @ ( sup_sup_set_int @ A @ B4 ) ) )
% 0.34/0.66 = ( inf_inf_nat @ ( complete_Inf_Inf_nat @ ( image_int_nat2 @ F @ A ) ) @ ( complete_Inf_Inf_nat @ ( image_int_nat2 @ F @ B4 ) ) ) ) ) ) ) ) ).
% 0.34/0.66
% 0.34/0.66 % cINF_union
% 0.34/0.66 thf(fact_6475_cINF__union,axiom,
% 0.34/0.66 ! [A: set_Product_unit,F: product_unit > product_unit,B4: set_Product_unit] :
% 0.34/0.66 ( ( A != bot_bo3957492148770167129t_unit )
% 0.34/0.66 => ( ( condit9161856309968323677t_unit @ ( image_405062704495631173t_unit @ F @ A ) )
% 0.34/0.66 => ( ( B4 != bot_bo3957492148770167129t_unit )
% 0.34/0.66 => ( ( condit9161856309968323677t_unit @ ( image_405062704495631173t_unit @ F @ B4 ) )
% 0.34/0.66 => ( ( comple2584293577114468500t_unit @ ( image_405062704495631173t_unit @ F @ ( sup_su793286257634532545t_unit @ A @ B4 ) ) )
% 0.34/0.66 = ( inf_inf_Product_unit @ ( comple2584293577114468500t_unit @ ( image_405062704495631173t_unit @ F @ A ) ) @ ( comple2584293577114468500t_unit @ ( image_405062704495631173t_unit @ F @ B4 ) ) ) ) ) ) ) ) ).
% 0.34/0.66
% 0.34/0.66 % cINF_union
% 0.34/0.66 thf(fact_6476_cINF__union,axiom,
% 0.34/0.66 ! [A: set_o,F: $o > product_unit,B4: set_o] :
% 0.34/0.66 ( ( A != bot_bot_set_o )
% 0.34/0.66 => ( ( condit9161856309968323677t_unit @ ( image_o_Product_unit2 @ F @ A ) )
% 0.34/0.66 => ( ( B4 != bot_bot_set_o )
% 0.34/0.66 => ( ( condit9161856309968323677t_unit @ ( image_o_Product_unit2 @ F @ B4 ) )
% 0.34/0.66 => ( ( comple2584293577114468500t_unit @ ( image_o_Product_unit2 @ F @ ( sup_sup_set_o @ A @ B4 ) ) )
% 0.34/0.66 = ( inf_inf_Product_unit @ ( comple2584293577114468500t_unit @ ( image_o_Product_unit2 @ F @ A ) ) @ ( comple2584293577114468500t_unit @ ( image_o_Product_unit2 @ F @ B4 ) ) ) ) ) ) ) ) ).
% 0.34/0.66
% 0.34/0.66 % cINF_union
% 0.34/0.66 thf(fact_6477_cINF__union,axiom,
% 0.34/0.66 ! [A: set_nat,F: nat > product_unit,B4: set_nat] :
% 0.34/0.66 ( ( A != bot_bot_set_nat )
% 0.34/0.66 => ( ( condit9161856309968323677t_unit @ ( image_8730104196221521654t_unit @ F @ A ) )
% 0.34/0.66 => ( ( B4 != bot_bot_set_nat )
% 0.34/0.66 => ( ( condit9161856309968323677t_unit @ ( image_8730104196221521654t_unit @ F @ B4 ) )
% 0.34/0.66 => ( ( comple2584293577114468500t_unit @ ( image_8730104196221521654t_unit @ F @ ( sup_sup_set_nat @ A @ B4 ) ) )
% 0.34/0.66 = ( inf_inf_Product_unit @ ( comple2584293577114468500t_unit @ ( image_8730104196221521654t_unit @ F @ A ) ) @ ( comple2584293577114468500t_unit @ ( image_8730104196221521654t_unit @ F @ B4 ) ) ) ) ) ) ) ) ).
% 0.34/0.66
% 0.34/0.66 % cINF_union
% 0.34/0.66 thf(fact_6478_cINF__union,axiom,
% 0.34/0.66 ! [A: set_int,F: int > product_unit,B4: set_int] :
% 0.34/0.66 ( ( A != bot_bot_set_int )
% 0.34/0.66 => ( ( condit9161856309968323677t_unit @ ( image_5846526143423701914t_unit @ F @ A ) )
% 0.34/0.66 => ( ( B4 != bot_bot_set_int )
% 0.34/0.66 => ( ( condit9161856309968323677t_unit @ ( image_5846526143423701914t_unit @ F @ B4 ) )
% 0.34/0.66 => ( ( comple2584293577114468500t_unit @ ( image_5846526143423701914t_unit @ F @ ( sup_sup_set_int @ A @ B4 ) ) )
% 0.34/0.66 = ( inf_inf_Product_unit @ ( comple2584293577114468500t_unit @ ( image_5846526143423701914t_unit @ F @ A ) ) @ ( comple2584293577114468500t_unit @ ( image_5846526143423701914t_unit @ F @ B4 ) ) ) ) ) ) ) ) ).
% 0.34/0.66
% 0.34/0.66 % cINF_union
% 0.34/0.66 thf(fact_6479_cINF__union,axiom,
% 0.34/0.66 ! [A: set_Product_unit,F: product_unit > set_nat,B4: set_Product_unit] :
% 0.34/0.66 ( ( A != bot_bo3957492148770167129t_unit )
% 0.34/0.66 => ( ( condit68592940725977398et_nat @ ( image_8173849991827986510et_nat @ F @ A ) )
% 0.34/0.66 => ( ( B4 != bot_bo3957492148770167129t_unit )
% 0.34/0.66 => ( ( condit68592940725977398et_nat @ ( image_8173849991827986510et_nat @ F @ B4 ) )
% 0.34/0.66 => ( ( comple7806235888213564991et_nat @ ( image_8173849991827986510et_nat @ F @ ( sup_su793286257634532545t_unit @ A @ B4 ) ) )
% 0.34/0.66 = ( inf_inf_set_nat @ ( comple7806235888213564991et_nat @ ( image_8173849991827986510et_nat @ F @ A ) ) @ ( comple7806235888213564991et_nat @ ( image_8173849991827986510et_nat @ F @ B4 ) ) ) ) ) ) ) ) ).
% 0.34/0.66
% 0.34/0.66 % cINF_union
% 0.34/0.66 thf(fact_6480_cINF__union,axiom,
% 0.34/0.66 ! [A: set_o,F: $o > set_nat,B4: set_o] :
% 0.34/0.66 ( ( A != bot_bot_set_o )
% 0.34/0.66 => ( ( condit68592940725977398et_nat @ ( image_o_set_nat @ F @ A ) )
% 0.34/0.66 => ( ( B4 != bot_bot_set_o )
% 0.34/0.66 => ( ( condit68592940725977398et_nat @ ( image_o_set_nat @ F @ B4 ) )
% 0.34/0.66 => ( ( comple7806235888213564991et_nat @ ( image_o_set_nat @ F @ ( sup_sup_set_o @ A @ B4 ) ) )
% 0.34/0.66 = ( inf_inf_set_nat @ ( comple7806235888213564991et_nat @ ( image_o_set_nat @ F @ A ) ) @ ( comple7806235888213564991et_nat @ ( image_o_set_nat @ F @ B4 ) ) ) ) ) ) ) ) ).
% 0.34/0.66
% 0.34/0.66 % cINF_union
% 0.34/0.66 thf(fact_6481_sum_Oinsert,axiom,
% 0.34/0.66 ! [A: set_int,X: int,G: int > code_integer] :
% 0.34/0.66 ( ( finite_finite_int @ A )
% 0.34/0.66 => ( ~ ( member_int @ X @ A )
% 0.34/0.66 => ( ( groups7873554091576472773nteger @ G @ ( insert_int @ X @ A ) )
% 0.34/0.66 = ( plus_p5714425477246183910nteger @ ( G @ X ) @ ( groups7873554091576472773nteger @ G @ A ) ) ) ) ) ).
% 0.34/0.66
% 0.34/0.66 % sum.insert
% 0.34/0.66 thf(fact_6482_sum_Oinsert,axiom,
% 0.34/0.66 ! [A: set_o,X: $o,G: $o > code_integer] :
% 0.34/0.66 ( ( finite_finite_o @ A )
% 0.34/0.66 => ( ~ ( member_o @ X @ A )
% 0.34/0.66 => ( ( groups4406642042086082107nteger @ G @ ( insert_o @ X @ A ) )
% 0.34/0.66 = ( plus_p5714425477246183910nteger @ ( G @ X ) @ ( groups4406642042086082107nteger @ G @ A ) ) ) ) ) ).
% 0.34/0.66
% 0.34/0.66 % sum.insert
% 0.34/0.66 thf(fact_6483_sum_Oinsert,axiom,
% 0.34/0.66 ! [A: set_Product_unit,X: product_unit,G: product_unit > code_integer] :
% 0.34/0.66 ( ( finite4290736615968046902t_unit @ A )
% 0.34/0.66 => ( ~ ( member_Product_unit @ X @ A )
% 0.34/0.66 => ( ( groups5393439102872367408nteger @ G @ ( insert_Product_unit @ X @ A ) )
% 0.34/0.66 = ( plus_p5714425477246183910nteger @ ( G @ X ) @ ( groups5393439102872367408nteger @ G @ A ) ) ) ) ) ).
% 0.34/0.66
% 0.34/0.66 % sum.insert
% 0.34/0.66 thf(fact_6484_sum_Oinsert,axiom,
% 0.34/0.66 ! [A: set_nat,X: nat,G: nat > code_integer] :
% 0.34/0.66 ( ( finite_finite_nat @ A )
% 0.34/0.66 => ( ~ ( member_nat @ X @ A )
% 0.34/0.66 => ( ( groups7501900531339628137nteger @ G @ ( insert_nat @ X @ A ) )
% 0.34/0.66 = ( plus_p5714425477246183910nteger @ ( G @ X ) @ ( groups7501900531339628137nteger @ G @ A ) ) ) ) ) ).
% 0.34/0.66
% 0.34/0.66 % sum.insert
% 0.34/0.66 thf(fact_6485_sum_Oinsert,axiom,
% 0.34/0.66 ! [A: set_Code_integer,X: code_integer,G: code_integer > code_integer] :
% 0.34/0.66 ( ( finite6017078050557962740nteger @ A )
% 0.34/0.66 => ( ~ ( member_Code_integer @ X @ A )
% 0.34/0.66 => ( ( groups879477027807139574nteger @ G @ ( insert_Code_integer @ X @ A ) )
% 0.34/0.66 = ( plus_p5714425477246183910nteger @ ( G @ X ) @ ( groups879477027807139574nteger @ G @ A ) ) ) ) ) ).
% 0.34/0.66
% 0.34/0.66 % sum.insert
% 0.34/0.66 thf(fact_6486_sum_Oinsert,axiom,
% 0.34/0.66 ! [A: set_int,X: int,G: int > rat] :
% 0.34/0.66 ( ( finite_finite_int @ A )
% 0.34/0.66 => ( ~ ( member_int @ X @ A )
% 0.34/0.66 => ( ( groups3906332499630173760nt_rat @ G @ ( insert_int @ X @ A ) )
% 0.34/0.66 = ( plus_plus_rat @ ( G @ X ) @ ( groups3906332499630173760nt_rat @ G @ A ) ) ) ) ) ).
% 0.34/0.66
% 0.34/0.66 % sum.insert
% 0.34/0.66 thf(fact_6487_sum_Oinsert,axiom,
% 0.34/0.66 ! [A: set_o,X: $o,G: $o > rat] :
% 0.34/0.66 ( ( finite_finite_o @ A )
% 0.34/0.66 => ( ~ ( member_o @ X @ A )
% 0.34/0.66 => ( ( groups7872700643590313910_o_rat @ G @ ( insert_o @ X @ A ) )
% 0.34/0.66 = ( plus_plus_rat @ ( G @ X ) @ ( groups7872700643590313910_o_rat @ G @ A ) ) ) ) ) ).
% 0.34/0.66
% 0.34/0.66 % sum.insert
% 0.34/0.66 thf(fact_6488_sum_Oinsert,axiom,
% 0.34/0.66 ! [A: set_Product_unit,X: product_unit,G: product_unit > rat] :
% 0.34/0.66 ( ( finite4290736615968046902t_unit @ A )
% 0.34/0.66 => ( ~ ( member_Product_unit @ X @ A )
% 0.34/0.66 => ( ( groups8131667117488461483it_rat @ G @ ( insert_Product_unit @ X @ A ) )
% 0.34/0.66 = ( plus_plus_rat @ ( G @ X ) @ ( groups8131667117488461483it_rat @ G @ A ) ) ) ) ) ).
% 0.34/0.66
% 0.34/0.66 % sum.insert
% 0.34/0.66 thf(fact_6489_sum_Oinsert,axiom,
% 0.34/0.66 ! [A: set_nat,X: nat,G: nat > rat] :
% 0.34/0.66 ( ( finite_finite_nat @ A )
% 0.34/0.66 => ( ~ ( member_nat @ X @ A )
% 0.34/0.66 => ( ( groups2906978787729119204at_rat @ G @ ( insert_nat @ X @ A ) )
% 0.34/0.66 = ( plus_plus_rat @ ( G @ X ) @ ( groups2906978787729119204at_rat @ G @ A ) ) ) ) ) ).
% 0.34/0.66
% 0.34/0.66 % sum.insert
% 0.34/0.66 thf(fact_6490_sum_Oinsert,axiom,
% 0.34/0.66 ! [A: set_Code_integer,X: code_integer,G: code_integer > rat] :
% 0.34/0.66 ( ( finite6017078050557962740nteger @ A )
% 0.34/0.66 => ( ~ ( member_Code_integer @ X @ A )
% 0.34/0.66 => ( ( groups6602215022474089585er_rat @ G @ ( insert_Code_integer @ X @ A ) )
% 0.34/0.66 = ( plus_plus_rat @ ( G @ X ) @ ( groups6602215022474089585er_rat @ G @ A ) ) ) ) ) ).
% 0.34/0.66
% 0.34/0.66 % sum.insert
% 0.34/0.66 thf(fact_6491_sum_Oempty,axiom,
% 0.34/0.66 ! [G: product_unit > nat] :
% 0.34/0.66 ( ( groups8766797177574957219it_nat @ G @ bot_bo3957492148770167129t_unit )
% 0.34/0.66 = zero_zero_nat ) ).
% 0.34/0.66
% 0.34/0.66 % sum.empty
% 0.34/0.66 thf(fact_6492_sum_Oempty,axiom,
% 0.34/0.66 ! [G: product_unit > int] :
% 0.34/0.66 ( ( groups8764306707065906943it_int @ G @ bot_bo3957492148770167129t_unit )
% 0.34/0.66 = zero_zero_int ) ).
% 0.34/0.66
% 0.34/0.66 % sum.empty
% 0.34/0.66 thf(fact_6493_sum_Oempty,axiom,
% 0.34/0.66 ! [G: product_unit > code_integer] :
% 0.34/0.66 ( ( groups5393439102872367408nteger @ G @ bot_bo3957492148770167129t_unit )
% 0.34/0.66 = zero_z3403309356797280102nteger ) ).
% 0.34/0.66
% 0.34/0.66 % sum.empty
% 0.34/0.66 thf(fact_6494_sum_Oempty,axiom,
% 0.34/0.66 ! [G: $o > nat] :
% 0.34/0.66 ( ( groups8507830703676809646_o_nat @ G @ bot_bot_set_o )
% 0.34/0.66 = zero_zero_nat ) ).
% 0.34/0.66
% 0.34/0.66 % sum.empty
% 0.34/0.66 thf(fact_6495_sum_Oempty,axiom,
% 0.34/0.66 ! [G: $o > int] :
% 0.34/0.66 ( ( groups8505340233167759370_o_int @ G @ bot_bot_set_o )
% 0.34/0.66 = zero_zero_int ) ).
% 0.34/0.66
% 0.34/0.66 % sum.empty
% 0.34/0.66 thf(fact_6496_sum_Oempty,axiom,
% 0.34/0.66 ! [G: $o > code_integer] :
% 0.34/0.66 ( ( groups4406642042086082107nteger @ G @ bot_bot_set_o )
% 0.34/0.66 = zero_z3403309356797280102nteger ) ).
% 0.34/0.66
% 0.34/0.66 % sum.empty
% 0.34/0.66 thf(fact_6497_sum_Oempty,axiom,
% 0.34/0.66 ! [G: nat > int] :
% 0.34/0.66 ( ( groups3539618377306564664at_int @ G @ bot_bot_set_nat )
% 0.34/0.66 = zero_zero_int ) ).
% 0.34/0.66
% 0.34/0.66 % sum.empty
% 0.34/0.66 thf(fact_6498_sum_Oempty,axiom,
% 0.34/0.66 ! [G: nat > code_integer] :
% 0.34/0.66 ( ( groups7501900531339628137nteger @ G @ bot_bot_set_nat )
% 0.34/0.66 = zero_z3403309356797280102nteger ) ).
% 0.34/0.66
% 0.34/0.66 % sum.empty
% 0.34/0.66 thf(fact_6499_sum_Oempty,axiom,
% 0.34/0.66 ! [G: int > nat] :
% 0.34/0.66 ( ( groups4541462559716669496nt_nat @ G @ bot_bot_set_int )
% 0.34/0.66 = zero_zero_nat ) ).
% 0.34/0.66
% 0.34/0.66 % sum.empty
% 0.34/0.66 thf(fact_6500_sum_Oempty,axiom,
% 0.34/0.66 ! [G: int > code_integer] :
% 0.34/0.66 ( ( groups7873554091576472773nteger @ G @ bot_bot_set_int )
% 0.34/0.66 = zero_z3403309356797280102nteger ) ).
% 0.34/0.66
% 0.34/0.66 % sum.empty
% 0.34/0.66 thf(fact_6501_member__le__sum,axiom,
% 0.34/0.66 ! [I2: code_integer,A: set_Code_integer,F: code_integer > code_integer] :
% 0.34/0.66 ( ( member_Code_integer @ I2 @ A )
% 0.34/0.66 => ( ! [X3: code_integer] :
% 0.34/0.66 ( ( member_Code_integer @ X3 @ ( minus_2355218937544613996nteger @ A @ ( insert_Code_integer @ I2 @ bot_bo3990330152332043303nteger ) ) )
% 0.34/0.66 => ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ ( F @ X3 ) ) )
% 0.34/0.66 => ( ( finite6017078050557962740nteger @ A )
% 0.34/0.66 => ( ord_le3102999989581377725nteger @ ( F @ I2 ) @ ( groups879477027807139574nteger @ F @ A ) ) ) ) ) ).
% 0.34/0.66
% 0.34/0.66 % member_le_sum
% 0.34/0.66 thf(fact_6502_member__le__sum,axiom,
% 0.34/0.66 ! [I2: product_unit,A: set_Product_unit,F: product_unit > code_integer] :
% 0.34/0.66 ( ( member_Product_unit @ I2 @ A )
% 0.34/0.66 => ( ! [X3: product_unit] :
% 0.34/0.66 ( ( member_Product_unit @ X3 @ ( minus_6452836326544984404t_unit @ A @ ( insert_Product_unit @ I2 @ bot_bo3957492148770167129t_unit ) ) )
% 0.34/0.66 => ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ ( F @ X3 ) ) )
% 0.34/0.66 => ( ( finite4290736615968046902t_unit @ A )
% 0.34/0.66 => ( ord_le3102999989581377725nteger @ ( F @ I2 ) @ ( groups5393439102872367408nteger @ F @ A ) ) ) ) ) ).
% 0.34/0.66
% 0.34/0.66 % member_le_sum
% 0.34/0.66 thf(fact_6503_member__le__sum,axiom,
% 0.34/0.66 ! [I2: $o,A: set_o,F: $o > code_integer] :
% 0.34/0.66 ( ( member_o @ I2 @ A )
% 0.34/0.66 => ( ! [X3: $o] :
% 0.34/0.66 ( ( member_o @ X3 @ ( minus_minus_set_o @ A @ ( insert_o @ I2 @ bot_bot_set_o ) ) )
% 0.34/0.66 => ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ ( F @ X3 ) ) )
% 0.34/0.66 => ( ( finite_finite_o @ A )
% 0.34/0.66 => ( ord_le3102999989581377725nteger @ ( F @ I2 ) @ ( groups4406642042086082107nteger @ F @ A ) ) ) ) ) ).
% 0.34/0.66
% 0.34/0.66 % member_le_sum
% 0.34/0.66 thf(fact_6504_member__le__sum,axiom,
% 0.34/0.66 ! [I2: int,A: set_int,F: int > code_integer] :
% 0.34/0.66 ( ( member_int @ I2 @ A )
% 0.34/0.66 => ( ! [X3: int] :
% 0.34/0.66 ( ( member_int @ X3 @ ( minus_minus_set_int @ A @ ( insert_int @ I2 @ bot_bot_set_int ) ) )
% 0.34/0.66 => ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ ( F @ X3 ) ) )
% 0.34/0.66 => ( ( finite_finite_int @ A )
% 0.34/0.66 => ( ord_le3102999989581377725nteger @ ( F @ I2 ) @ ( groups7873554091576472773nteger @ F @ A ) ) ) ) ) ).
% 0.34/0.66
% 0.34/0.66 % member_le_sum
% 0.34/0.66 thf(fact_6505_member__le__sum,axiom,
% 0.34/0.66 ! [I2: nat,A: set_nat,F: nat > code_integer] :
% 0.34/0.66 ( ( member_nat @ I2 @ A )
% 0.34/0.66 => ( ! [X3: nat] :
% 0.34/0.66 ( ( member_nat @ X3 @ ( minus_minus_set_nat @ A @ ( insert_nat @ I2 @ bot_bot_set_nat ) ) )
% 0.34/0.66 => ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ ( F @ X3 ) ) )
% 0.34/0.66 => ( ( finite_finite_nat @ A )
% 0.34/0.66 => ( ord_le3102999989581377725nteger @ ( F @ I2 ) @ ( groups7501900531339628137nteger @ F @ A ) ) ) ) ) ).
% 0.34/0.66
% 0.34/0.66 % member_le_sum
% 0.34/0.66 thf(fact_6506_member__le__sum,axiom,
% 0.34/0.66 ! [I2: code_integer,A: set_Code_integer,F: code_integer > rat] :
% 0.34/0.66 ( ( member_Code_integer @ I2 @ A )
% 0.34/0.66 => ( ! [X3: code_integer] :
% 0.34/0.66 ( ( member_Code_integer @ X3 @ ( minus_2355218937544613996nteger @ A @ ( insert_Code_integer @ I2 @ bot_bo3990330152332043303nteger ) ) )
% 0.34/0.66 => ( ord_less_eq_rat @ zero_zero_rat @ ( F @ X3 ) ) )
% 0.34/0.66 => ( ( finite6017078050557962740nteger @ A )
% 0.34/0.66 => ( ord_less_eq_rat @ ( F @ I2 ) @ ( groups6602215022474089585er_rat @ F @ A ) ) ) ) ) ).
% 0.34/0.66
% 0.34/0.66 % member_le_sum
% 0.34/0.66 thf(fact_6507_member__le__sum,axiom,
% 0.34/0.66 ! [I2: product_unit,A: set_Product_unit,F: product_unit > rat] :
% 0.34/0.66 ( ( member_Product_unit @ I2 @ A )
% 0.34/0.66 => ( ! [X3: product_unit] :
% 0.34/0.66 ( ( member_Product_unit @ X3 @ ( minus_6452836326544984404t_unit @ A @ ( insert_Product_unit @ I2 @ bot_bo3957492148770167129t_unit ) ) )
% 0.34/0.66 => ( ord_less_eq_rat @ zero_zero_rat @ ( F @ X3 ) ) )
% 0.34/0.66 => ( ( finite4290736615968046902t_unit @ A )
% 0.34/0.66 => ( ord_less_eq_rat @ ( F @ I2 ) @ ( groups8131667117488461483it_rat @ F @ A ) ) ) ) ) ).
% 0.34/0.66
% 0.34/0.66 % member_le_sum
% 0.34/0.66 thf(fact_6508_member__le__sum,axiom,
% 0.34/0.66 ! [I2: $o,A: set_o,F: $o > rat] :
% 0.34/0.66 ( ( member_o @ I2 @ A )
% 0.34/0.66 => ( ! [X3: $o] :
% 0.34/0.66 ( ( member_o @ X3 @ ( minus_minus_set_o @ A @ ( insert_o @ I2 @ bot_bot_set_o ) ) )
% 0.34/0.66 => ( ord_less_eq_rat @ zero_zero_rat @ ( F @ X3 ) ) )
% 0.34/0.66 => ( ( finite_finite_o @ A )
% 0.34/0.66 => ( ord_less_eq_rat @ ( F @ I2 ) @ ( groups7872700643590313910_o_rat @ F @ A ) ) ) ) ) ).
% 0.34/0.66
% 0.34/0.66 % member_le_sum
% 0.34/0.66 thf(fact_6509_member__le__sum,axiom,
% 0.34/0.66 ! [I2: int,A: set_int,F: int > rat] :
% 0.34/0.66 ( ( member_int @ I2 @ A )
% 0.34/0.66 => ( ! [X3: int] :
% 0.34/0.66 ( ( member_int @ X3 @ ( minus_minus_set_int @ A @ ( insert_int @ I2 @ bot_bot_set_int ) ) )
% 0.34/0.66 => ( ord_less_eq_rat @ zero_zero_rat @ ( F @ X3 ) ) )
% 0.34/0.66 => ( ( finite_finite_int @ A )
% 0.34/0.66 => ( ord_less_eq_rat @ ( F @ I2 ) @ ( groups3906332499630173760nt_rat @ F @ A ) ) ) ) ) ).
% 0.34/0.66
% 0.34/0.66 % member_le_sum
% 0.34/0.66 thf(fact_6510_member__le__sum,axiom,
% 0.34/0.66 ! [I2: nat,A: set_nat,F: nat > rat] :
% 0.34/0.66 ( ( member_nat @ I2 @ A )
% 0.34/0.66 => ( ! [X3: nat] :
% 0.34/0.66 ( ( member_nat @ X3 @ ( minus_minus_set_nat @ A @ ( insert_nat @ I2 @ bot_bot_set_nat ) ) )
% 0.34/0.66 => ( ord_less_eq_rat @ zero_zero_rat @ ( F @ X3 ) ) )
% 0.34/0.66 => ( ( finite_finite_nat @ A )
% 0.34/0.66 => ( ord_less_eq_rat @ ( F @ I2 ) @ ( groups2906978787729119204at_rat @ F @ A ) ) ) ) ) ).
% 0.34/0.66
% 0.34/0.66 % member_le_sum
% 0.34/0.66 thf(fact_6511_sum_Ounion__disjoint,axiom,
% 0.34/0.66 ! [A: set_Code_integer,B4: set_Code_integer,G: code_integer > code_integer] :
% 0.34/0.66 ( ( finite6017078050557962740nteger @ A )
% 0.34/0.66 => ( ( finite6017078050557962740nteger @ B4 )
% 0.34/0.66 => ( ( ( inf_in1364745209274528805nteger @ A @ B4 )
% 0.34/0.66 = bot_bo3990330152332043303nteger )
% 0.34/0.66 => ( ( groups879477027807139574nteger @ G @ ( sup_su848401254843788991nteger @ A @ B4 ) )
% 0.34/0.66 = ( plus_p5714425477246183910nteger @ ( groups879477027807139574nteger @ G @ A ) @ ( groups879477027807139574nteger @ G @ B4 ) ) ) ) ) ) ).
% 0.34/0.66
% 0.34/0.66 % sum.union_disjoint
% 0.34/0.66 thf(fact_6512_sum_Ounion__disjoint,axiom,
% 0.34/0.66 ! [A: set_Code_integer,B4: set_Code_integer,G: code_integer > rat] :
% 0.34/0.66 ( ( finite6017078050557962740nteger @ A )
% 0.34/0.66 => ( ( finite6017078050557962740nteger @ B4 )
% 0.34/0.66 => ( ( ( inf_in1364745209274528805nteger @ A @ B4 )
% 0.34/0.66 = bot_bo3990330152332043303nteger )
% 0.34/0.66 => ( ( groups6602215022474089585er_rat @ G @ ( sup_su848401254843788991nteger @ A @ B4 ) )
% 0.34/0.66 = ( plus_plus_rat @ ( groups6602215022474089585er_rat @ G @ A ) @ ( groups6602215022474089585er_rat @ G @ B4 ) ) ) ) ) ) ).
% 0.34/0.66
% 0.34/0.66 % sum.union_disjoint
% 0.34/0.66 thf(fact_6513_sum_Ounion__disjoint,axiom,
% 0.34/0.66 ! [A: set_Code_integer,B4: set_Code_integer,G: code_integer > nat] :
% 0.34/0.66 ( ( finite6017078050557962740nteger @ A )
% 0.34/0.66 => ( ( finite6017078050557962740nteger @ B4 )
% 0.34/0.66 => ( ( ( inf_in1364745209274528805nteger @ A @ B4 )
% 0.34/0.66 = bot_bo3990330152332043303nteger )
% 0.34/0.66 => ( ( groups7237345082560585321er_nat @ G @ ( sup_su848401254843788991nteger @ A @ B4 ) )
% 0.34/0.66 = ( plus_plus_nat @ ( groups7237345082560585321er_nat @ G @ A ) @ ( groups7237345082560585321er_nat @ G @ B4 ) ) ) ) ) ) ).
% 0.34/0.66
% 0.34/0.66 % sum.union_disjoint
% 0.34/0.66 thf(fact_6514_sum_Ounion__disjoint,axiom,
% 0.34/0.66 ! [A: set_Code_integer,B4: set_Code_integer,G: code_integer > int] :
% 0.34/0.66 ( ( finite6017078050557962740nteger @ A )
% 0.34/0.66 => ( ( finite6017078050557962740nteger @ B4 )
% 0.34/0.66 => ( ( ( inf_in1364745209274528805nteger @ A @ B4 )
% 0.34/0.66 = bot_bo3990330152332043303nteger )
% 0.34/0.66 => ( ( groups7234854612051535045er_int @ G @ ( sup_su848401254843788991nteger @ A @ B4 ) )
% 0.34/0.66 = ( plus_plus_int @ ( groups7234854612051535045er_int @ G @ A ) @ ( groups7234854612051535045er_int @ G @ B4 ) ) ) ) ) ) ).
% 0.34/0.66
% 0.34/0.66 % sum.union_disjoint
% 0.34/0.66 thf(fact_6515_sum_Ounion__disjoint,axiom,
% 0.34/0.66 ! [A: set_Product_unit,B4: set_Product_unit,G: product_unit > code_integer] :
% 0.34/0.66 ( ( finite4290736615968046902t_unit @ A )
% 0.34/0.66 => ( ( finite4290736615968046902t_unit @ B4 )
% 0.34/0.66 => ( ( ( inf_in4660618365625256667t_unit @ A @ B4 )
% 0.34/0.66 = bot_bo3957492148770167129t_unit )
% 0.34/0.66 => ( ( groups5393439102872367408nteger @ G @ ( sup_su793286257634532545t_unit @ A @ B4 ) )
% 0.34/0.66 = ( plus_p5714425477246183910nteger @ ( groups5393439102872367408nteger @ G @ A ) @ ( groups5393439102872367408nteger @ G @ B4 ) ) ) ) ) ) ).
% 0.34/0.66
% 0.34/0.66 % sum.union_disjoint
% 0.34/0.66 thf(fact_6516_sum_Ounion__disjoint,axiom,
% 0.34/0.66 ! [A: set_Product_unit,B4: set_Product_unit,G: product_unit > rat] :
% 0.34/0.66 ( ( finite4290736615968046902t_unit @ A )
% 0.34/0.66 => ( ( finite4290736615968046902t_unit @ B4 )
% 0.34/0.66 => ( ( ( inf_in4660618365625256667t_unit @ A @ B4 )
% 0.34/0.66 = bot_bo3957492148770167129t_unit )
% 0.34/0.66 => ( ( groups8131667117488461483it_rat @ G @ ( sup_su793286257634532545t_unit @ A @ B4 ) )
% 0.34/0.66 = ( plus_plus_rat @ ( groups8131667117488461483it_rat @ G @ A ) @ ( groups8131667117488461483it_rat @ G @ B4 ) ) ) ) ) ) ).
% 0.34/0.66
% 0.34/0.66 % sum.union_disjoint
% 0.34/0.66 thf(fact_6517_sum_Ounion__disjoint,axiom,
% 0.34/0.66 ! [A: set_Product_unit,B4: set_Product_unit,G: product_unit > nat] :
% 0.34/0.66 ( ( finite4290736615968046902t_unit @ A )
% 0.34/0.66 => ( ( finite4290736615968046902t_unit @ B4 )
% 0.34/0.66 => ( ( ( inf_in4660618365625256667t_unit @ A @ B4 )
% 0.34/0.66 = bot_bo3957492148770167129t_unit )
% 0.34/0.66 => ( ( groups8766797177574957219it_nat @ G @ ( sup_su793286257634532545t_unit @ A @ B4 ) )
% 0.34/0.66 = ( plus_plus_nat @ ( groups8766797177574957219it_nat @ G @ A ) @ ( groups8766797177574957219it_nat @ G @ B4 ) ) ) ) ) ) ).
% 0.34/0.66
% 0.34/0.66 % sum.union_disjoint
% 0.34/0.66 thf(fact_6518_sum_Ounion__disjoint,axiom,
% 0.34/0.66 ! [A: set_Product_unit,B4: set_Product_unit,G: product_unit > int] :
% 0.34/0.66 ( ( finite4290736615968046902t_unit @ A )
% 0.34/0.66 => ( ( finite4290736615968046902t_unit @ B4 )
% 0.34/0.66 => ( ( ( inf_in4660618365625256667t_unit @ A @ B4 )
% 0.34/0.66 = bot_bo3957492148770167129t_unit )
% 0.34/0.66 => ( ( groups8764306707065906943it_int @ G @ ( sup_su793286257634532545t_unit @ A @ B4 ) )
% 0.34/0.66 = ( plus_plus_int @ ( groups8764306707065906943it_int @ G @ A ) @ ( groups8764306707065906943it_int @ G @ B4 ) ) ) ) ) ) ).
% 0.34/0.66
% 0.34/0.66 % sum.union_disjoint
% 0.34/0.66 thf(fact_6519_sum_Ounion__disjoint,axiom,
% 0.34/0.66 ! [A: set_o,B4: set_o,G: $o > code_integer] :
% 0.34/0.66 ( ( finite_finite_o @ A )
% 0.34/0.66 => ( ( finite_finite_o @ B4 )
% 0.34/0.66 => ( ( ( inf_inf_set_o @ A @ B4 )
% 0.34/0.66 = bot_bot_set_o )
% 0.34/0.66 => ( ( groups4406642042086082107nteger @ G @ ( sup_sup_set_o @ A @ B4 ) )
% 0.34/0.66 = ( plus_p5714425477246183910nteger @ ( groups4406642042086082107nteger @ G @ A ) @ ( groups4406642042086082107nteger @ G @ B4 ) ) ) ) ) ) ).
% 0.34/0.66
% 0.34/0.66 % sum.union_disjoint
% 0.34/0.66 thf(fact_6520_sum_Ounion__disjoint,axiom,
% 0.34/0.66 ! [A: set_o,B4: set_o,G: $o > rat] :
% 0.34/0.66 ( ( finite_finite_o @ A )
% 0.34/0.66 => ( ( finite_finite_o @ B4 )
% 0.34/0.66 => ( ( ( inf_inf_set_o @ A @ B4 )
% 0.34/0.66 = bot_bot_set_o )
% 0.34/0.66 => ( ( groups7872700643590313910_o_rat @ G @ ( sup_sup_set_o @ A @ B4 ) )
% 0.34/0.66 = ( plus_plus_rat @ ( groups7872700643590313910_o_rat @ G @ A ) @ ( groups7872700643590313910_o_rat @ G @ B4 ) ) ) ) ) ) ).
% 0.34/0.66
% 0.34/0.66 % sum.union_disjoint
% 0.34/0.66 thf(fact_6521_sum__diff1,axiom,
% 0.34/0.66 ! [A: set_Code_integer,A2: code_integer,F: code_integer > rat] :
% 0.34/0.66 ( ( finite6017078050557962740nteger @ A )
% 0.34/0.66 => ( ( ( member_Code_integer @ A2 @ A )
% 0.34/0.66 => ( ( groups6602215022474089585er_rat @ F @ ( minus_2355218937544613996nteger @ A @ ( insert_Code_integer @ A2 @ bot_bo3990330152332043303nteger ) ) )
% 0.34/0.66 = ( minus_minus_rat @ ( groups6602215022474089585er_rat @ F @ A ) @ ( F @ A2 ) ) ) )
% 0.34/0.66 & ( ~ ( member_Code_integer @ A2 @ A )
% 0.34/0.66 => ( ( groups6602215022474089585er_rat @ F @ ( minus_2355218937544613996nteger @ A @ ( insert_Code_integer @ A2 @ bot_bo3990330152332043303nteger ) ) )
% 0.34/0.66 = ( groups6602215022474089585er_rat @ F @ A ) ) ) ) ) ).
% 0.34/0.66
% 0.34/0.66 % sum_diff1
% 0.34/0.66 thf(fact_6522_sum__diff1,axiom,
% 0.34/0.66 ! [A: set_Product_unit,A2: product_unit,F: product_unit > rat] :
% 0.34/0.66 ( ( finite4290736615968046902t_unit @ A )
% 0.34/0.66 => ( ( ( member_Product_unit @ A2 @ A )
% 0.34/0.66 => ( ( groups8131667117488461483it_rat @ F @ ( minus_6452836326544984404t_unit @ A @ ( insert_Product_unit @ A2 @ bot_bo3957492148770167129t_unit ) ) )
% 0.34/0.66 = ( minus_minus_rat @ ( groups8131667117488461483it_rat @ F @ A ) @ ( F @ A2 ) ) ) )
% 0.34/0.66 & ( ~ ( member_Product_unit @ A2 @ A )
% 0.34/0.66 => ( ( groups8131667117488461483it_rat @ F @ ( minus_6452836326544984404t_unit @ A @ ( insert_Product_unit @ A2 @ bot_bo3957492148770167129t_unit ) ) )
% 0.34/0.66 = ( groups8131667117488461483it_rat @ F @ A ) ) ) ) ) ).
% 0.34/0.66
% 0.34/0.66 % sum_diff1
% 0.34/0.66 thf(fact_6523_sum__diff1,axiom,
% 0.34/0.66 ! [A: set_o,A2: $o,F: $o > rat] :
% 0.34/0.66 ( ( finite_finite_o @ A )
% 0.34/0.66 => ( ( ( member_o @ A2 @ A )
% 0.34/0.66 => ( ( groups7872700643590313910_o_rat @ F @ ( minus_minus_set_o @ A @ ( insert_o @ A2 @ bot_bot_set_o ) ) )
% 0.34/0.66 = ( minus_minus_rat @ ( groups7872700643590313910_o_rat @ F @ A ) @ ( F @ A2 ) ) ) )
% 0.34/0.66 & ( ~ ( member_o @ A2 @ A )
% 0.34/0.66 => ( ( groups7872700643590313910_o_rat @ F @ ( minus_minus_set_o @ A @ ( insert_o @ A2 @ bot_bot_set_o ) ) )
% 0.34/0.66 = ( groups7872700643590313910_o_rat @ F @ A ) ) ) ) ) ).
% 0.34/0.66
% 0.34/0.66 % sum_diff1
% 0.34/0.66 thf(fact_6524_sum__diff1,axiom,
% 0.34/0.66 ! [A: set_int,A2: int,F: int > rat] :
% 0.34/0.66 ( ( finite_finite_int @ A )
% 0.34/0.66 => ( ( ( member_int @ A2 @ A )
% 0.34/0.66 => ( ( groups3906332499630173760nt_rat @ F @ ( minus_minus_set_int @ A @ ( insert_int @ A2 @ bot_bot_set_int ) ) )
% 0.34/0.66 = ( minus_minus_rat @ ( groups3906332499630173760nt_rat @ F @ A ) @ ( F @ A2 ) ) ) )
% 0.34/0.66 & ( ~ ( member_int @ A2 @ A )
% 0.34/0.66 => ( ( groups3906332499630173760nt_rat @ F @ ( minus_minus_set_int @ A @ ( insert_int @ A2 @ bot_bot_set_int ) ) )
% 0.34/0.66 = ( groups3906332499630173760nt_rat @ F @ A ) ) ) ) ) ).
% 0.34/0.66
% 0.34/0.66 % sum_diff1
% 0.34/0.66 thf(fact_6525_sum__diff1,axiom,
% 0.34/0.66 ! [A: set_Code_integer,A2: code_integer,F: code_integer > int] :
% 0.34/0.66 ( ( finite6017078050557962740nteger @ A )
% 0.34/0.66 => ( ( ( member_Code_integer @ A2 @ A )
% 0.34/0.66 => ( ( groups7234854612051535045er_int @ F @ ( minus_2355218937544613996nteger @ A @ ( insert_Code_integer @ A2 @ bot_bo3990330152332043303nteger ) ) )
% 0.34/0.66 = ( minus_minus_int @ ( groups7234854612051535045er_int @ F @ A ) @ ( F @ A2 ) ) ) )
% 0.34/0.66 & ( ~ ( member_Code_integer @ A2 @ A )
% 0.34/0.66 => ( ( groups7234854612051535045er_int @ F @ ( minus_2355218937544613996nteger @ A @ ( insert_Code_integer @ A2 @ bot_bo3990330152332043303nteger ) ) )
% 0.34/0.66 = ( groups7234854612051535045er_int @ F @ A ) ) ) ) ) ).
% 0.34/0.66
% 0.34/0.66 % sum_diff1
% 0.34/0.66 thf(fact_6526_sum__diff1,axiom,
% 0.34/0.66 ! [A: set_Product_unit,A2: product_unit,F: product_unit > int] :
% 0.34/0.66 ( ( finite4290736615968046902t_unit @ A )
% 0.34/0.66 => ( ( ( member_Product_unit @ A2 @ A )
% 0.34/0.66 => ( ( groups8764306707065906943it_int @ F @ ( minus_6452836326544984404t_unit @ A @ ( insert_Product_unit @ A2 @ bot_bo3957492148770167129t_unit ) ) )
% 0.34/0.66 = ( minus_minus_int @ ( groups8764306707065906943it_int @ F @ A ) @ ( F @ A2 ) ) ) )
% 0.34/0.66 & ( ~ ( member_Product_unit @ A2 @ A )
% 0.34/0.66 => ( ( groups8764306707065906943it_int @ F @ ( minus_6452836326544984404t_unit @ A @ ( insert_Product_unit @ A2 @ bot_bo3957492148770167129t_unit ) ) )
% 0.34/0.66 = ( groups8764306707065906943it_int @ F @ A ) ) ) ) ) ).
% 0.34/0.66
% 0.34/0.66 % sum_diff1
% 0.34/0.66 thf(fact_6527_sum__diff1,axiom,
% 0.34/0.66 ! [A: set_o,A2: $o,F: $o > int] :
% 0.34/0.66 ( ( finite_finite_o @ A )
% 0.34/0.66 => ( ( ( member_o @ A2 @ A )
% 0.34/0.66 => ( ( groups8505340233167759370_o_int @ F @ ( minus_minus_set_o @ A @ ( insert_o @ A2 @ bot_bot_set_o ) ) )
% 0.34/0.66 = ( minus_minus_int @ ( groups8505340233167759370_o_int @ F @ A ) @ ( F @ A2 ) ) ) )
% 0.34/0.66 & ( ~ ( member_o @ A2 @ A )
% 0.34/0.66 => ( ( groups8505340233167759370_o_int @ F @ ( minus_minus_set_o @ A @ ( insert_o @ A2 @ bot_bot_set_o ) ) )
% 0.34/0.66 = ( groups8505340233167759370_o_int @ F @ A ) ) ) ) ) ).
% 0.34/0.66
% 0.34/0.66 % sum_diff1
% 0.34/0.66 thf(fact_6528_sum__diff1,axiom,
% 0.34/0.66 ! [A: set_nat,A2: nat,F: nat > rat] :
% 0.34/0.66 ( ( finite_finite_nat @ A )
% 0.34/0.66 => ( ( ( member_nat @ A2 @ A )
% 0.34/0.66 => ( ( groups2906978787729119204at_rat @ F @ ( minus_minus_set_nat @ A @ ( insert_nat @ A2 @ bot_bot_set_nat ) ) )
% 0.34/0.66 = ( minus_minus_rat @ ( groups2906978787729119204at_rat @ F @ A ) @ ( F @ A2 ) ) ) )
% 0.34/0.66 & ( ~ ( member_nat @ A2 @ A )
% 0.34/0.66 => ( ( groups2906978787729119204at_rat @ F @ ( minus_minus_set_nat @ A @ ( insert_nat @ A2 @ bot_bot_set_nat ) ) )
% 0.34/0.66 = ( groups2906978787729119204at_rat @ F @ A ) ) ) ) ) ).
% 0.34/0.66
% 0.34/0.66 % sum_diff1
% 0.34/0.66 thf(fact_6529_sum__diff1,axiom,
% 0.34/0.66 ! [A: set_nat,A2: nat,F: nat > int] :
% 0.34/0.66 ( ( finite_finite_nat @ A )
% 0.34/0.66 => ( ( ( member_nat @ A2 @ A )
% 0.34/0.66 => ( ( groups3539618377306564664at_int @ F @ ( minus_minus_set_nat @ A @ ( insert_nat @ A2 @ bot_bot_set_nat ) ) )
% 0.34/0.66 = ( minus_minus_int @ ( groups3539618377306564664at_int @ F @ A ) @ ( F @ A2 ) ) ) )
% 0.34/0.66 & ( ~ ( member_nat @ A2 @ A )
% 0.34/0.66 => ( ( groups3539618377306564664at_int @ F @ ( minus_minus_set_nat @ A @ ( insert_nat @ A2 @ bot_bot_set_nat ) ) )
% 0.34/0.66 = ( groups3539618377306564664at_int @ F @ A ) ) ) ) ) ).
% 0.34/0.66
% 0.34/0.66 % sum_diff1
% 0.34/0.66 thf(fact_6530_sum__diff1,axiom,
% 0.34/0.66 ! [A: set_int,A2: int,F: int > int] :
% 0.34/0.66 ( ( finite_finite_int @ A )
% 0.34/0.66 => ( ( ( member_int @ A2 @ A )
% 0.34/0.66 => ( ( groups4538972089207619220nt_int @ F @ ( minus_minus_set_int @ A @ ( insert_int @ A2 @ bot_bot_set_int ) ) )
% 0.34/0.66 = ( minus_minus_int @ ( groups4538972089207619220nt_int @ F @ A ) @ ( F @ A2 ) ) ) )
% 0.34/0.66 & ( ~ ( member_int @ A2 @ A )
% 0.34/0.66 => ( ( groups4538972089207619220nt_int @ F @ ( minus_minus_set_int @ A @ ( insert_int @ A2 @ bot_bot_set_int ) ) )
% 0.34/0.66 = ( groups4538972089207619220nt_int @ F @ A ) ) ) ) ) ).
% 0.34/0.66
% 0.34/0.66 % sum_diff1
% 0.34/0.66 thf(fact_6531_sum_Oremove,axiom,
% 0.34/0.66 ! [A: set_Code_integer,X: code_integer,G: code_integer > code_integer] :
% 0.34/0.66 ( ( finite6017078050557962740nteger @ A )
% 0.34/0.66 => ( ( member_Code_integer @ X @ A )
% 0.34/0.66 => ( ( groups879477027807139574nteger @ G @ A )
% 0.34/0.66 = ( plus_p5714425477246183910nteger @ ( G @ X ) @ ( groups879477027807139574nteger @ G @ ( minus_2355218937544613996nteger @ A @ ( insert_Code_integer @ X @ bot_bo3990330152332043303nteger ) ) ) ) ) ) ) ).
% 0.34/0.66
% 0.34/0.66 % sum.remove
% 0.34/0.66 thf(fact_6532_sum_Oremove,axiom,
% 0.34/0.66 ! [A: set_Code_integer,X: code_integer,G: code_integer > rat] :
% 0.34/0.66 ( ( finite6017078050557962740nteger @ A )
% 0.34/0.66 => ( ( member_Code_integer @ X @ A )
% 0.34/0.66 => ( ( groups6602215022474089585er_rat @ G @ A )
% 0.34/0.66 = ( plus_plus_rat @ ( G @ X ) @ ( groups6602215022474089585er_rat @ G @ ( minus_2355218937544613996nteger @ A @ ( insert_Code_integer @ X @ bot_bo3990330152332043303nteger ) ) ) ) ) ) ) ).
% 0.34/0.66
% 0.34/0.66 % sum.remove
% 0.34/0.66 thf(fact_6533_sum_Oremove,axiom,
% 0.34/0.66 ! [A: set_Code_integer,X: code_integer,G: code_integer > nat] :
% 0.34/0.66 ( ( finite6017078050557962740nteger @ A )
% 0.34/0.66 => ( ( member_Code_integer @ X @ A )
% 0.34/0.66 => ( ( groups7237345082560585321er_nat @ G @ A )
% 0.34/0.66 = ( plus_plus_nat @ ( G @ X ) @ ( groups7237345082560585321er_nat @ G @ ( minus_2355218937544613996nteger @ A @ ( insert_Code_integer @ X @ bot_bo3990330152332043303nteger ) ) ) ) ) ) ) ).
% 0.34/0.66
% 0.34/0.66 % sum.remove
% 0.34/0.66 thf(fact_6534_sum_Oremove,axiom,
% 0.34/0.66 ! [A: set_Code_integer,X: code_integer,G: code_integer > int] :
% 0.34/0.66 ( ( finite6017078050557962740nteger @ A )
% 0.34/0.66 => ( ( member_Code_integer @ X @ A )
% 0.34/0.66 => ( ( groups7234854612051535045er_int @ G @ A )
% 0.34/0.66 = ( plus_plus_int @ ( G @ X ) @ ( groups7234854612051535045er_int @ G @ ( minus_2355218937544613996nteger @ A @ ( insert_Code_integer @ X @ bot_bo3990330152332043303nteger ) ) ) ) ) ) ) ).
% 0.34/0.66
% 0.34/0.66 % sum.remove
% 0.34/0.66 thf(fact_6535_sum_Oremove,axiom,
% 0.34/0.66 ! [A: set_Product_unit,X: product_unit,G: product_unit > code_integer] :
% 0.34/0.66 ( ( finite4290736615968046902t_unit @ A )
% 0.34/0.66 => ( ( member_Product_unit @ X @ A )
% 0.34/0.66 => ( ( groups5393439102872367408nteger @ G @ A )
% 0.34/0.66 = ( plus_p5714425477246183910nteger @ ( G @ X ) @ ( groups5393439102872367408nteger @ G @ ( minus_6452836326544984404t_unit @ A @ ( insert_Product_unit @ X @ bot_bo3957492148770167129t_unit ) ) ) ) ) ) ) ).
% 0.34/0.66
% 0.34/0.66 % sum.remove
% 0.34/0.66 thf(fact_6536_sum_Oremove,axiom,
% 0.34/0.66 ! [A: set_Product_unit,X: product_unit,G: product_unit > rat] :
% 0.34/0.66 ( ( finite4290736615968046902t_unit @ A )
% 0.34/0.66 => ( ( member_Product_unit @ X @ A )
% 0.34/0.66 => ( ( groups8131667117488461483it_rat @ G @ A )
% 0.34/0.66 = ( plus_plus_rat @ ( G @ X ) @ ( groups8131667117488461483it_rat @ G @ ( minus_6452836326544984404t_unit @ A @ ( insert_Product_unit @ X @ bot_bo3957492148770167129t_unit ) ) ) ) ) ) ) ).
% 0.34/0.66
% 0.34/0.66 % sum.remove
% 0.34/0.66 thf(fact_6537_sum_Oremove,axiom,
% 0.34/0.66 ! [A: set_Product_unit,X: product_unit,G: product_unit > nat] :
% 0.34/0.66 ( ( finite4290736615968046902t_unit @ A )
% 0.34/0.66 => ( ( member_Product_unit @ X @ A )
% 0.34/0.66 => ( ( groups8766797177574957219it_nat @ G @ A )
% 0.34/0.66 = ( plus_plus_nat @ ( G @ X ) @ ( groups8766797177574957219it_nat @ G @ ( minus_6452836326544984404t_unit @ A @ ( insert_Product_unit @ X @ bot_bo3957492148770167129t_unit ) ) ) ) ) ) ) ).
% 0.34/0.66
% 0.34/0.66 % sum.remove
% 0.34/0.66 thf(fact_6538_sum_Oremove,axiom,
% 0.34/0.66 ! [A: set_Product_unit,X: product_unit,G: product_unit > int] :
% 0.34/0.66 ( ( finite4290736615968046902t_unit @ A )
% 0.34/0.66 => ( ( member_Product_unit @ X @ A )
% 0.34/0.66 => ( ( groups8764306707065906943it_int @ G @ A )
% 0.34/0.66 = ( plus_plus_int @ ( G @ X ) @ ( groups8764306707065906943it_int @ G @ ( minus_6452836326544984404t_unit @ A @ ( insert_Product_unit @ X @ bot_bo3957492148770167129t_unit ) ) ) ) ) ) ) ).
% 0.34/0.66
% 0.34/0.66 % sum.remove
% 0.34/0.66 thf(fact_6539_sum_Oremove,axiom,
% 0.34/0.66 ! [A: set_o,X: $o,G: $o > code_integer] :
% 0.34/0.66 ( ( finite_finite_o @ A )
% 0.34/0.66 => ( ( member_o @ X @ A )
% 0.34/0.66 => ( ( groups4406642042086082107nteger @ G @ A )
% 0.34/0.66 = ( plus_p5714425477246183910nteger @ ( G @ X ) @ ( groups4406642042086082107nteger @ G @ ( minus_minus_set_o @ A @ ( insert_o @ X @ bot_bot_set_o ) ) ) ) ) ) ) ).
% 0.34/0.66
% 0.34/0.66 % sum.remove
% 0.34/0.66 thf(fact_6540_sum_Oremove,axiom,
% 0.34/0.66 ! [A: set_o,X: $o,G: $o > rat] :
% 0.34/0.66 ( ( finite_finite_o @ A )
% 0.34/0.66 => ( ( member_o @ X @ A )
% 0.34/0.66 => ( ( groups7872700643590313910_o_rat @ G @ A )
% 0.34/0.66 = ( plus_plus_rat @ ( G @ X ) @ ( groups7872700643590313910_o_rat @ G @ ( minus_minus_set_o @ A @ ( insert_o @ X @ bot_bot_set_o ) ) ) ) ) ) ) ).
% 0.34/0.66
% 0.34/0.66 % sum.remove
% 0.34/0.66 thf(fact_6541_sum_Oinsert__remove,axiom,
% 0.34/0.66 ! [A: set_Code_integer,G: code_integer > code_integer,X: code_integer] :
% 0.34/0.66 ( ( finite6017078050557962740nteger @ A )
% 0.34/0.66 => ( ( groups879477027807139574nteger @ G @ ( insert_Code_integer @ X @ A ) )
% 0.34/0.66 = ( plus_p5714425477246183910nteger @ ( G @ X ) @ ( groups879477027807139574nteger @ G @ ( minus_2355218937544613996nteger @ A @ ( insert_Code_integer @ X @ bot_bo3990330152332043303nteger ) ) ) ) ) ) ).
% 0.34/0.66
% 0.34/0.66 % sum.insert_remove
% 0.34/0.66 thf(fact_6542_sum_Oinsert__remove,axiom,
% 0.34/0.66 ! [A: set_Code_integer,G: code_integer > rat,X: code_integer] :
% 0.34/0.66 ( ( finite6017078050557962740nteger @ A )
% 0.34/0.66 => ( ( groups6602215022474089585er_rat @ G @ ( insert_Code_integer @ X @ A ) )
% 0.34/0.66 = ( plus_plus_rat @ ( G @ X ) @ ( groups6602215022474089585er_rat @ G @ ( minus_2355218937544613996nteger @ A @ ( insert_Code_integer @ X @ bot_bo3990330152332043303nteger ) ) ) ) ) ) ).
% 0.34/0.66
% 0.34/0.66 % sum.insert_remove
% 0.34/0.66 thf(fact_6543_sum_Oinsert__remove,axiom,
% 0.34/0.66 ! [A: set_Code_integer,G: code_integer > nat,X: code_integer] :
% 0.34/0.66 ( ( finite6017078050557962740nteger @ A )
% 0.34/0.66 => ( ( groups7237345082560585321er_nat @ G @ ( insert_Code_integer @ X @ A ) )
% 0.34/0.66 = ( plus_plus_nat @ ( G @ X ) @ ( groups7237345082560585321er_nat @ G @ ( minus_2355218937544613996nteger @ A @ ( insert_Code_integer @ X @ bot_bo3990330152332043303nteger ) ) ) ) ) ) ).
% 0.34/0.66
% 0.34/0.66 % sum.insert_remove
% 0.34/0.66 thf(fact_6544_sum_Oinsert__remove,axiom,
% 0.34/0.66 ! [A: set_Code_integer,G: code_integer > int,X: code_integer] :
% 0.34/0.66 ( ( finite6017078050557962740nteger @ A )
% 0.34/0.66 => ( ( groups7234854612051535045er_int @ G @ ( insert_Code_integer @ X @ A ) )
% 0.34/0.66 = ( plus_plus_int @ ( G @ X ) @ ( groups7234854612051535045er_int @ G @ ( minus_2355218937544613996nteger @ A @ ( insert_Code_integer @ X @ bot_bo3990330152332043303nteger ) ) ) ) ) ) ).
% 0.34/0.66
% 0.34/0.66 % sum.insert_remove
% 0.34/0.66 thf(fact_6545_sum_Oinsert__remove,axiom,
% 0.34/0.66 ! [A: set_Product_unit,G: product_unit > code_integer,X: product_unit] :
% 0.34/0.66 ( ( finite4290736615968046902t_unit @ A )
% 0.34/0.66 => ( ( groups5393439102872367408nteger @ G @ ( insert_Product_unit @ X @ A ) )
% 0.34/0.66 = ( plus_p5714425477246183910nteger @ ( G @ X ) @ ( groups5393439102872367408nteger @ G @ ( minus_6452836326544984404t_unit @ A @ ( insert_Product_unit @ X @ bot_bo3957492148770167129t_unit ) ) ) ) ) ) ).
% 0.34/0.66
% 0.34/0.66 % sum.insert_remove
% 0.34/0.66 thf(fact_6546_sum_Oinsert__remove,axiom,
% 0.34/0.66 ! [A: set_Product_unit,G: product_unit > rat,X: product_unit] :
% 0.34/0.66 ( ( finite4290736615968046902t_unit @ A )
% 0.34/0.66 => ( ( groups8131667117488461483it_rat @ G @ ( insert_Product_unit @ X @ A ) )
% 0.34/0.66 = ( plus_plus_rat @ ( G @ X ) @ ( groups8131667117488461483it_rat @ G @ ( minus_6452836326544984404t_unit @ A @ ( insert_Product_unit @ X @ bot_bo3957492148770167129t_unit ) ) ) ) ) ) ).
% 0.34/0.66
% 0.34/0.66 % sum.insert_remove
% 0.34/0.66 thf(fact_6547_sum_Oinsert__remove,axiom,
% 0.34/0.66 ! [A: set_Product_unit,G: product_unit > nat,X: product_unit] :
% 0.34/0.66 ( ( finite4290736615968046902t_unit @ A )
% 0.34/0.66 => ( ( groups8766797177574957219it_nat @ G @ ( insert_Product_unit @ X @ A ) )
% 0.34/0.66 = ( plus_plus_nat @ ( G @ X ) @ ( groups8766797177574957219it_nat @ G @ ( minus_6452836326544984404t_unit @ A @ ( insert_Product_unit @ X @ bot_bo3957492148770167129t_unit ) ) ) ) ) ) ).
% 0.34/0.66
% 0.34/0.66 % sum.insert_remove
% 0.34/0.66 thf(fact_6548_sum_Oinsert__remove,axiom,
% 0.34/0.66 ! [A: set_Product_unit,G: product_unit > int,X: product_unit] :
% 0.34/0.66 ( ( finite4290736615968046902t_unit @ A )
% 0.34/0.66 => ( ( groups8764306707065906943it_int @ G @ ( insert_Product_unit @ X @ A ) )
% 0.34/0.66 = ( plus_plus_int @ ( G @ X ) @ ( groups8764306707065906943it_int @ G @ ( minus_6452836326544984404t_unit @ A @ ( insert_Product_unit @ X @ bot_bo3957492148770167129t_unit ) ) ) ) ) ) ).
% 0.34/0.66
% 0.34/0.66 % sum.insert_remove
% 0.34/0.66 thf(fact_6549_sum_Oinsert__remove,axiom,
% 0.34/0.66 ! [A: set_o,G: $o > code_integer,X: $o] :
% 0.34/0.66 ( ( finite_finite_o @ A )
% 0.34/0.66 => ( ( groups4406642042086082107nteger @ G @ ( insert_o @ X @ A ) )
% 0.34/0.66 = ( plus_p5714425477246183910nteger @ ( G @ X ) @ ( groups4406642042086082107nteger @ G @ ( minus_minus_set_o @ A @ ( insert_o @ X @ bot_bot_set_o ) ) ) ) ) ) ).
% 0.34/0.66
% 0.34/0.66 % sum.insert_remove
% 0.34/0.66 thf(fact_6550_sum_Oinsert__remove,axiom,
% 0.34/0.66 ! [A: set_o,G: $o > rat,X: $o] :
% 0.34/0.66 ( ( finite_finite_o @ A )
% 0.34/0.66 => ( ( groups7872700643590313910_o_rat @ G @ ( insert_o @ X @ A ) )
% 0.34/0.66 = ( plus_plus_rat @ ( G @ X ) @ ( groups7872700643590313910_o_rat @ G @ ( minus_minus_set_o @ A @ ( insert_o @ X @ bot_bot_set_o ) ) ) ) ) ) ).
% 0.34/0.66
% 0.34/0.66 % sum.insert_remove
% 0.34/0.66 thf(fact_6551_sum__strict__mono,axiom,
% 0.34/0.66 ! [A: set_Code_integer,F: code_integer > nat,G: code_integer > nat] :
% 0.34/0.66 ( ( finite6017078050557962740nteger @ A )
% 0.34/0.66 => ( ( A != bot_bo3990330152332043303nteger )
% 0.34/0.66 => ( ! [X3: code_integer] :
% 0.34/0.66 ( ( member_Code_integer @ X3 @ A )
% 0.34/0.66 => ( ord_less_nat @ ( F @ X3 ) @ ( G @ X3 ) ) )
% 0.34/0.66 => ( ord_less_nat @ ( groups7237345082560585321er_nat @ F @ A ) @ ( groups7237345082560585321er_nat @ G @ A ) ) ) ) ) ).
% 0.34/0.66
% 0.34/0.66 % sum_strict_mono
% 0.34/0.66 thf(fact_6552_sum__strict__mono,axiom,
% 0.34/0.66 ! [A: set_Product_unit,F: product_unit > nat,G: product_unit > nat] :
% 0.34/0.66 ( ( finite4290736615968046902t_unit @ A )
% 0.34/0.66 => ( ( A != bot_bo3957492148770167129t_unit )
% 0.34/0.66 => ( ! [X3: product_unit] :
% 0.34/0.66 ( ( member_Product_unit @ X3 @ A )
% 0.34/0.66 => ( ord_less_nat @ ( F @ X3 ) @ ( G @ X3 ) ) )
% 0.34/0.66 => ( ord_less_nat @ ( groups8766797177574957219it_nat @ F @ A ) @ ( groups8766797177574957219it_nat @ G @ A ) ) ) ) ) ).
% 0.34/0.66
% 0.34/0.66 % sum_strict_mono
% 0.34/0.66 thf(fact_6553_sum__strict__mono,axiom,
% 0.34/0.66 ! [A: set_o,F: $o > nat,G: $o > nat] :
% 0.34/0.66 ( ( finite_finite_o @ A )
% 0.34/0.66 => ( ( A != bot_bot_set_o )
% 0.34/0.66 => ( ! [X3: $o] :
% 0.34/0.66 ( ( member_o @ X3 @ A )
% 0.34/0.66 => ( ord_less_nat @ ( F @ X3 ) @ ( G @ X3 ) ) )
% 0.34/0.66 => ( ord_less_nat @ ( groups8507830703676809646_o_nat @ F @ A ) @ ( groups8507830703676809646_o_nat @ G @ A ) ) ) ) ) ).
% 0.34/0.66
% 0.34/0.66 % sum_strict_mono
% 0.34/0.66 thf(fact_6554_sum__strict__mono,axiom,
% 0.34/0.66 ! [A: set_int,F: int > nat,G: int > nat] :
% 0.34/0.66 ( ( finite_finite_int @ A )
% 0.34/0.66 => ( ( A != bot_bot_set_int )
% 0.34/0.66 => ( ! [X3: int] :
% 0.34/0.66 ( ( member_int @ X3 @ A )
% 0.34/0.66 => ( ord_less_nat @ ( F @ X3 ) @ ( G @ X3 ) ) )
% 0.34/0.66 => ( ord_less_nat @ ( groups4541462559716669496nt_nat @ F @ A ) @ ( groups4541462559716669496nt_nat @ G @ A ) ) ) ) ) ).
% 0.34/0.66
% 0.34/0.66 % sum_strict_mono
% 0.34/0.66 thf(fact_6555_sum__strict__mono,axiom,
% 0.34/0.66 ! [A: set_Code_integer,F: code_integer > int,G: code_integer > int] :
% 0.34/0.66 ( ( finite6017078050557962740nteger @ A )
% 0.34/0.66 => ( ( A != bot_bo3990330152332043303nteger )
% 0.34/0.66 => ( ! [X3: code_integer] :
% 0.34/0.66 ( ( member_Code_integer @ X3 @ A )
% 0.34/0.66 => ( ord_less_int @ ( F @ X3 ) @ ( G @ X3 ) ) )
% 0.34/0.66 => ( ord_less_int @ ( groups7234854612051535045er_int @ F @ A ) @ ( groups7234854612051535045er_int @ G @ A ) ) ) ) ) ).
% 0.34/0.66
% 0.34/0.66 % sum_strict_mono
% 0.34/0.66 thf(fact_6556_sum__strict__mono,axiom,
% 0.34/0.66 ! [A: set_Product_unit,F: product_unit > int,G: product_unit > int] :
% 0.34/0.66 ( ( finite4290736615968046902t_unit @ A )
% 0.34/0.66 => ( ( A != bot_bo3957492148770167129t_unit )
% 0.34/0.66 => ( ! [X3: product_unit] :
% 0.34/0.66 ( ( member_Product_unit @ X3 @ A )
% 0.34/0.66 => ( ord_less_int @ ( F @ X3 ) @ ( G @ X3 ) ) )
% 0.34/0.66 => ( ord_less_int @ ( groups8764306707065906943it_int @ F @ A ) @ ( groups8764306707065906943it_int @ G @ A ) ) ) ) ) ).
% 0.34/0.66
% 0.34/0.66 % sum_strict_mono
% 0.34/0.66 thf(fact_6557_sum__strict__mono,axiom,
% 0.34/0.66 ! [A: set_o,F: $o > int,G: $o > int] :
% 0.34/0.66 ( ( finite_finite_o @ A )
% 0.34/0.66 => ( ( A != bot_bot_set_o )
% 0.34/0.66 => ( ! [X3: $o] :
% 0.34/0.66 ( ( member_o @ X3 @ A )
% 0.34/0.66 => ( ord_less_int @ ( F @ X3 ) @ ( G @ X3 ) ) )
% 0.34/0.66 => ( ord_less_int @ ( groups8505340233167759370_o_int @ F @ A ) @ ( groups8505340233167759370_o_int @ G @ A ) ) ) ) ) ).
% 0.34/0.66
% 0.34/0.66 % sum_strict_mono
% 0.34/0.66 thf(fact_6558_sum__strict__mono,axiom,
% 0.34/0.66 ! [A: set_nat,F: nat > int,G: nat > int] :
% 0.34/0.66 ( ( finite_finite_nat @ A )
% 0.34/0.66 => ( ( A != bot_bot_set_nat )
% 0.34/0.66 => ( ! [X3: nat] :
% 0.34/0.66 ( ( member_nat @ X3 @ A )
% 0.34/0.66 => ( ord_less_int @ ( F @ X3 ) @ ( G @ X3 ) ) )
% 0.34/0.66 => ( ord_less_int @ ( groups3539618377306564664at_int @ F @ A ) @ ( groups3539618377306564664at_int @ G @ A ) ) ) ) ) ).
% 0.34/0.66
% 0.34/0.66 % sum_strict_mono
% 0.34/0.66 thf(fact_6559_sum__strict__mono,axiom,
% 0.34/0.66 ! [A: set_Code_integer,F: code_integer > code_integer,G: code_integer > code_integer] :
% 0.34/0.66 ( ( finite6017078050557962740nteger @ A )
% 0.34/0.66 => ( ( A != bot_bo3990330152332043303nteger )
% 0.34/0.66 => ( ! [X3: code_integer] :
% 0.34/0.66 ( ( member_Code_integer @ X3 @ A )
% 0.34/0.66 => ( ord_le6747313008572928689nteger @ ( F @ X3 ) @ ( G @ X3 ) ) )
% 0.34/0.66 => ( ord_le6747313008572928689nteger @ ( groups879477027807139574nteger @ F @ A ) @ ( groups879477027807139574nteger @ G @ A ) ) ) ) ) ).
% 0.34/0.66
% 0.34/0.66 % sum_strict_mono
% 0.34/0.66 thf(fact_6560_sum__strict__mono,axiom,
% 0.34/0.66 ! [A: set_Product_unit,F: product_unit > code_integer,G: product_unit > code_integer] :
% 0.34/0.66 ( ( finite4290736615968046902t_unit @ A )
% 0.34/0.66 => ( ( A != bot_bo3957492148770167129t_unit )
% 0.34/0.66 => ( ! [X3: product_unit] :
% 0.34/0.66 ( ( member_Product_unit @ X3 @ A )
% 0.34/0.66 => ( ord_le6747313008572928689nteger @ ( F @ X3 ) @ ( G @ X3 ) ) )
% 0.34/0.66 => ( ord_le6747313008572928689nteger @ ( groups5393439102872367408nteger @ F @ A ) @ ( groups5393439102872367408nteger @ G @ A ) ) ) ) ) ).
% 0.34/0.66
% 0.34/0.66 % sum_strict_mono
% 0.34/0.66 thf(fact_6561_sum_Oinsert__if,axiom,
% 0.34/0.66 ! [A: set_int,X: int,G: int > code_integer] :
% 0.34/0.66 ( ( finite_finite_int @ A )
% 0.34/0.66 => ( ( ( member_int @ X @ A )
% 0.34/0.66 => ( ( groups7873554091576472773nteger @ G @ ( insert_int @ X @ A ) )
% 0.34/0.66 = ( groups7873554091576472773nteger @ G @ A ) ) )
% 0.34/0.66 & ( ~ ( member_int @ X @ A )
% 0.34/0.66 => ( ( groups7873554091576472773nteger @ G @ ( insert_int @ X @ A ) )
% 0.34/0.66 = ( plus_p5714425477246183910nteger @ ( G @ X ) @ ( groups7873554091576472773nteger @ G @ A ) ) ) ) ) ) ).
% 0.34/0.66
% 0.34/0.66 % sum.insert_if
% 0.34/0.66 thf(fact_6562_sum_Oinsert__if,axiom,
% 0.34/0.66 ! [A: set_o,X: $o,G: $o > code_integer] :
% 0.34/0.66 ( ( finite_finite_o @ A )
% 0.34/0.66 => ( ( ( member_o @ X @ A )
% 0.34/0.66 => ( ( groups4406642042086082107nteger @ G @ ( insert_o @ X @ A ) )
% 0.34/0.66 = ( groups4406642042086082107nteger @ G @ A ) ) )
% 0.34/0.66 & ( ~ ( member_o @ X @ A )
% 0.34/0.66 => ( ( groups4406642042086082107nteger @ G @ ( insert_o @ X @ A ) )
% 0.34/0.66 = ( plus_p5714425477246183910nteger @ ( G @ X ) @ ( groups4406642042086082107nteger @ G @ A ) ) ) ) ) ) ).
% 0.34/0.66
% 0.34/0.66 % sum.insert_if
% 0.34/0.66 thf(fact_6563_sum_Oinsert__if,axiom,
% 0.34/0.66 ! [A: set_Product_unit,X: product_unit,G: product_unit > code_integer] :
% 0.34/0.66 ( ( finite4290736615968046902t_unit @ A )
% 0.34/0.66 => ( ( ( member_Product_unit @ X @ A )
% 0.34/0.66 => ( ( groups5393439102872367408nteger @ G @ ( insert_Product_unit @ X @ A ) )
% 0.34/0.66 = ( groups5393439102872367408nteger @ G @ A ) ) )
% 0.34/0.66 & ( ~ ( member_Product_unit @ X @ A )
% 0.34/0.66 => ( ( groups5393439102872367408nteger @ G @ ( insert_Product_unit @ X @ A ) )
% 0.34/0.66 = ( plus_p5714425477246183910nteger @ ( G @ X ) @ ( groups5393439102872367408nteger @ G @ A ) ) ) ) ) ) ).
% 0.34/0.66
% 0.34/0.66 % sum.insert_if
% 0.34/0.66 thf(fact_6564_sum_Oinsert__if,axiom,
% 0.34/0.66 ! [A: set_nat,X: nat,G: nat > code_integer] :
% 0.34/0.66 ( ( finite_finite_nat @ A )
% 0.34/0.66 => ( ( ( member_nat @ X @ A )
% 0.34/0.66 => ( ( groups7501900531339628137nteger @ G @ ( insert_nat @ X @ A ) )
% 0.34/0.66 = ( groups7501900531339628137nteger @ G @ A ) ) )
% 0.34/0.66 & ( ~ ( member_nat @ X @ A )
% 0.34/0.66 => ( ( groups7501900531339628137nteger @ G @ ( insert_nat @ X @ A ) )
% 0.34/0.66 = ( plus_p5714425477246183910nteger @ ( G @ X ) @ ( groups7501900531339628137nteger @ G @ A ) ) ) ) ) ) ).
% 0.34/0.66
% 0.34/0.66 % sum.insert_if
% 0.34/0.66 thf(fact_6565_sum_Oinsert__if,axiom,
% 0.34/0.66 ! [A: set_Code_integer,X: code_integer,G: code_integer > code_integer] :
% 0.34/0.66 ( ( finite6017078050557962740nteger @ A )
% 0.34/0.66 => ( ( ( member_Code_integer @ X @ A )
% 0.34/0.66 => ( ( groups879477027807139574nteger @ G @ ( insert_Code_integer @ X @ A ) )
% 0.34/0.66 = ( groups879477027807139574nteger @ G @ A ) ) )
% 0.34/0.66 & ( ~ ( member_Code_integer @ X @ A )
% 0.34/0.66 => ( ( groups879477027807139574nteger @ G @ ( insert_Code_integer @ X @ A ) )
% 0.34/0.66 = ( plus_p5714425477246183910nteger @ ( G @ X ) @ ( groups879477027807139574nteger @ G @ A ) ) ) ) ) ) ).
% 0.34/0.66
% 0.34/0.66 % sum.insert_if
% 0.34/0.66 thf(fact_6566_sum_Oinsert__if,axiom,
% 0.34/0.66 ! [A: set_int,X: int,G: int > rat] :
% 0.34/0.66 ( ( finite_finite_int @ A )
% 0.34/0.66 => ( ( ( member_int @ X @ A )
% 0.34/0.66 => ( ( groups3906332499630173760nt_rat @ G @ ( insert_int @ X @ A ) )
% 0.34/0.66 = ( groups3906332499630173760nt_rat @ G @ A ) ) )
% 0.34/0.66 & ( ~ ( member_int @ X @ A )
% 0.34/0.66 => ( ( groups3906332499630173760nt_rat @ G @ ( insert_int @ X @ A ) )
% 0.34/0.66 = ( plus_plus_rat @ ( G @ X ) @ ( groups3906332499630173760nt_rat @ G @ A ) ) ) ) ) ) ).
% 0.34/0.66
% 0.34/0.66 % sum.insert_if
% 0.34/0.66 thf(fact_6567_sum_Oinsert__if,axiom,
% 0.34/0.66 ! [A: set_o,X: $o,G: $o > rat] :
% 0.34/0.66 ( ( finite_finite_o @ A )
% 0.34/0.66 => ( ( ( member_o @ X @ A )
% 0.34/0.66 => ( ( groups7872700643590313910_o_rat @ G @ ( insert_o @ X @ A ) )
% 0.34/0.66 = ( groups7872700643590313910_o_rat @ G @ A ) ) )
% 0.34/0.66 & ( ~ ( member_o @ X @ A )
% 0.34/0.66 => ( ( groups7872700643590313910_o_rat @ G @ ( insert_o @ X @ A ) )
% 0.34/0.66 = ( plus_plus_rat @ ( G @ X ) @ ( groups7872700643590313910_o_rat @ G @ A ) ) ) ) ) ) ).
% 0.34/0.66
% 0.34/0.66 % sum.insert_if
% 0.34/0.66 thf(fact_6568_sum_Oinsert__if,axiom,
% 0.34/0.66 ! [A: set_Product_unit,X: product_unit,G: product_unit > rat] :
% 0.34/0.66 ( ( finite4290736615968046902t_unit @ A )
% 0.34/0.66 => ( ( ( member_Product_unit @ X @ A )
% 0.34/0.66 => ( ( groups8131667117488461483it_rat @ G @ ( insert_Product_unit @ X @ A ) )
% 0.34/0.66 = ( groups8131667117488461483it_rat @ G @ A ) ) )
% 0.34/0.66 & ( ~ ( member_Product_unit @ X @ A )
% 0.34/0.66 => ( ( groups8131667117488461483it_rat @ G @ ( insert_Product_unit @ X @ A ) )
% 0.34/0.66 = ( plus_plus_rat @ ( G @ X ) @ ( groups8131667117488461483it_rat @ G @ A ) ) ) ) ) ) ).
% 0.34/0.66
% 0.34/0.66 % sum.insert_if
% 0.34/0.66 thf(fact_6569_sum_Oinsert__if,axiom,
% 0.34/0.66 ! [A: set_nat,X: nat,G: nat > rat] :
% 0.34/0.66 ( ( finite_finite_nat @ A )
% 0.34/0.66 => ( ( ( member_nat @ X @ A )
% 0.34/0.66 => ( ( groups2906978787729119204at_rat @ G @ ( insert_nat @ X @ A ) )
% 0.34/0.66 = ( groups2906978787729119204at_rat @ G @ A ) ) )
% 0.34/0.66 & ( ~ ( member_nat @ X @ A )
% 0.34/0.66 => ( ( groups2906978787729119204at_rat @ G @ ( insert_nat @ X @ A ) )
% 0.34/0.66 = ( plus_plus_rat @ ( G @ X ) @ ( groups2906978787729119204at_rat @ G @ A ) ) ) ) ) ) ).
% 0.34/0.66
% 0.34/0.66 % sum.insert_if
% 0.34/0.66 thf(fact_6570_sum_Oinsert__if,axiom,
% 0.34/0.66 ! [A: set_Code_integer,X: code_integer,G: code_integer > rat] :
% 0.34/0.66 ( ( finite6017078050557962740nteger @ A )
% 0.34/0.66 => ( ( ( member_Code_integer @ X @ A )
% 0.34/0.66 => ( ( groups6602215022474089585er_rat @ G @ ( insert_Code_integer @ X @ A ) )
% 0.34/0.66 = ( groups6602215022474089585er_rat @ G @ A ) ) )
% 0.34/0.66 & ( ~ ( member_Code_integer @ X @ A )
% 0.34/0.66 => ( ( groups6602215022474089585er_rat @ G @ ( insert_Code_integer @ X @ A ) )
% 0.34/0.66 = ( plus_plus_rat @ ( G @ X ) @ ( groups6602215022474089585er_rat @ G @ A ) ) ) ) ) ) ).
% 0.34/0.66
% 0.34/0.66 % sum.insert_if
% 0.34/0.66 thf(fact_6571_sum__pos,axiom,
% 0.34/0.66 ! [I: set_Code_integer,F: code_integer > nat] :
% 0.34/0.66 ( ( finite6017078050557962740nteger @ I )
% 0.34/0.66 => ( ( I != bot_bo3990330152332043303nteger )
% 0.34/0.66 => ( ! [I3: code_integer] :
% 0.34/0.66 ( ( member_Code_integer @ I3 @ I )
% 0.34/0.66 => ( ord_less_nat @ zero_zero_nat @ ( F @ I3 ) ) )
% 0.34/0.66 => ( ord_less_nat @ zero_zero_nat @ ( groups7237345082560585321er_nat @ F @ I ) ) ) ) ) ).
% 0.34/0.66
% 0.34/0.66 % sum_pos
% 0.34/0.66 thf(fact_6572_sum__pos,axiom,
% 0.34/0.66 ! [I: set_Product_unit,F: product_unit > nat] :
% 0.34/0.66 ( ( finite4290736615968046902t_unit @ I )
% 0.34/0.66 => ( ( I != bot_bo3957492148770167129t_unit )
% 0.34/0.66 => ( ! [I3: product_unit] :
% 0.34/0.66 ( ( member_Product_unit @ I3 @ I )
% 0.34/0.66 => ( ord_less_nat @ zero_zero_nat @ ( F @ I3 ) ) )
% 0.34/0.66 => ( ord_less_nat @ zero_zero_nat @ ( groups8766797177574957219it_nat @ F @ I ) ) ) ) ) ).
% 0.34/0.66
% 0.34/0.66 % sum_pos
% 0.34/0.66 thf(fact_6573_sum__pos,axiom,
% 0.34/0.66 ! [I: set_o,F: $o > nat] :
% 0.34/0.66 ( ( finite_finite_o @ I )
% 0.34/0.66 => ( ( I != bot_bot_set_o )
% 0.34/0.66 => ( ! [I3: $o] :
% 0.34/0.66 ( ( member_o @ I3 @ I )
% 0.34/0.66 => ( ord_less_nat @ zero_zero_nat @ ( F @ I3 ) ) )
% 0.34/0.66 => ( ord_less_nat @ zero_zero_nat @ ( groups8507830703676809646_o_nat @ F @ I ) ) ) ) ) ).
% 0.34/0.66
% 0.34/0.66 % sum_pos
% 0.34/0.66 thf(fact_6574_sum__pos,axiom,
% 0.34/0.66 ! [I: set_int,F: int > nat] :
% 0.34/0.66 ( ( finite_finite_int @ I )
% 0.34/0.66 => ( ( I != bot_bot_set_int )
% 0.34/0.66 => ( ! [I3: int] :
% 0.34/0.66 ( ( member_int @ I3 @ I )
% 0.34/0.66 => ( ord_less_nat @ zero_zero_nat @ ( F @ I3 ) ) )
% 0.34/0.66 => ( ord_less_nat @ zero_zero_nat @ ( groups4541462559716669496nt_nat @ F @ I ) ) ) ) ) ).
% 0.34/0.66
% 0.34/0.66 % sum_pos
% 0.34/0.66 thf(fact_6575_sum__pos,axiom,
% 0.34/0.66 ! [I: set_Code_integer,F: code_integer > int] :
% 0.34/0.66 ( ( finite6017078050557962740nteger @ I )
% 0.34/0.66 => ( ( I != bot_bo3990330152332043303nteger )
% 0.34/0.66 => ( ! [I3: code_integer] :
% 0.34/0.66 ( ( member_Code_integer @ I3 @ I )
% 0.34/0.66 => ( ord_less_int @ zero_zero_int @ ( F @ I3 ) ) )
% 0.34/0.66 => ( ord_less_int @ zero_zero_int @ ( groups7234854612051535045er_int @ F @ I ) ) ) ) ) ).
% 0.34/0.66
% 0.34/0.66 % sum_pos
% 0.34/0.66 thf(fact_6576_sum__pos,axiom,
% 0.34/0.66 ! [I: set_Product_unit,F: product_unit > int] :
% 0.34/0.66 ( ( finite4290736615968046902t_unit @ I )
% 0.34/0.66 => ( ( I != bot_bo3957492148770167129t_unit )
% 0.34/0.66 => ( ! [I3: product_unit] :
% 0.34/0.66 ( ( member_Product_unit @ I3 @ I )
% 0.34/0.66 => ( ord_less_int @ zero_zero_int @ ( F @ I3 ) ) )
% 0.34/0.66 => ( ord_less_int @ zero_zero_int @ ( groups8764306707065906943it_int @ F @ I ) ) ) ) ) ).
% 0.34/0.66
% 0.34/0.66 % sum_pos
% 0.34/0.66 thf(fact_6577_sum__pos,axiom,
% 0.34/0.66 ! [I: set_o,F: $o > int] :
% 0.34/0.66 ( ( finite_finite_o @ I )
% 0.34/0.66 => ( ( I != bot_bot_set_o )
% 0.34/0.66 => ( ! [I3: $o] :
% 0.34/0.66 ( ( member_o @ I3 @ I )
% 0.34/0.66 => ( ord_less_int @ zero_zero_int @ ( F @ I3 ) ) )
% 0.34/0.66 => ( ord_less_int @ zero_zero_int @ ( groups8505340233167759370_o_int @ F @ I ) ) ) ) ) ).
% 0.34/0.66
% 0.34/0.66 % sum_pos
% 0.34/0.66 thf(fact_6578_sum__pos,axiom,
% 0.34/0.66 ! [I: set_nat,F: nat > int] :
% 0.34/0.66 ( ( finite_finite_nat @ I )
% 0.34/0.66 => ( ( I != bot_bot_set_nat )
% 0.34/0.66 => ( ! [I3: nat] :
% 0.34/0.66 ( ( member_nat @ I3 @ I )
% 0.34/0.66 => ( ord_less_int @ zero_zero_int @ ( F @ I3 ) ) )
% 0.34/0.66 => ( ord_less_int @ zero_zero_int @ ( groups3539618377306564664at_int @ F @ I ) ) ) ) ) ).
% 0.34/0.66
% 0.34/0.66 % sum_pos
% 0.34/0.66 thf(fact_6579_sum__pos,axiom,
% 0.34/0.66 ! [I: set_Code_integer,F: code_integer > code_integer] :
% 0.34/0.66 ( ( finite6017078050557962740nteger @ I )
% 0.34/0.66 => ( ( I != bot_bo3990330152332043303nteger )
% 0.34/0.66 => ( ! [I3: code_integer] :
% 0.34/0.66 ( ( member_Code_integer @ I3 @ I )
% 0.34/0.66 => ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ ( F @ I3 ) ) )
% 0.34/0.66 => ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ ( groups879477027807139574nteger @ F @ I ) ) ) ) ) ).
% 0.34/0.67
% 0.34/0.67 % sum_pos
% 0.34/0.67 thf(fact_6580_sum__pos,axiom,
% 0.34/0.67 ! [I: set_Product_unit,F: product_unit > code_integer] :
% 0.34/0.67 ( ( finite4290736615968046902t_unit @ I )
% 0.34/0.67 => ( ( I != bot_bo3957492148770167129t_unit )
% 0.34/0.67 => ( ! [I3: product_unit] :
% 0.34/0.67 ( ( member_Product_unit @ I3 @ I )
% 0.34/0.67 => ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ ( F @ I3 ) ) )
% 0.34/0.67 => ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ ( groups5393439102872367408nteger @ F @ I ) ) ) ) ) ).
% 0.34/0.67
% 0.34/0.67 % sum_pos
% 0.34/0.67 thf(fact_6581_sum_OUnion__disjoint,axiom,
% 0.34/0.67 ! [C5: set_set_nat,G: nat > nat] :
% 0.34/0.67 ( ! [X3: set_nat] :
% 0.34/0.67 ( ( member_set_nat @ X3 @ C5 )
% 0.34/0.67 => ( finite_finite_nat @ X3 ) )
% 0.34/0.67 => ( ! [X3: set_nat] :
% 0.34/0.67 ( ( member_set_nat @ X3 @ C5 )
% 0.34/0.67 => ! [Xa3: set_nat] :
% 0.34/0.67 ( ( member_set_nat @ Xa3 @ C5 )
% 0.34/0.67 => ( ( X3 != Xa3 )
% 0.34/0.67 => ( ( inf_inf_set_nat @ X3 @ Xa3 )
% 0.34/0.67 = bot_bot_set_nat ) ) ) )
% 0.34/0.67 => ( ( groups3542108847815614940at_nat @ G @ ( comple7399068483239264473et_nat @ C5 ) )
% 0.34/0.67 = ( comp_s5456809700045850692at_nat @ groups8294997508430121362at_nat @ groups3542108847815614940at_nat @ G @ C5 ) ) ) ) ).
% 0.34/0.67
% 0.34/0.67 % sum.Union_disjoint
% 0.34/0.67 thf(fact_6582_sum_OUnion__disjoint,axiom,
% 0.34/0.67 ! [C5: set_set_int,G: int > int] :
% 0.34/0.67 ( ! [X3: set_int] :
% 0.34/0.67 ( ( member_set_int @ X3 @ C5 )
% 0.34/0.67 => ( finite_finite_int @ X3 ) )
% 0.34/0.67 => ( ! [X3: set_int] :
% 0.34/0.67 ( ( member_set_int @ X3 @ C5 )
% 0.34/0.67 => ! [Xa3: set_int] :
% 0.34/0.67 ( ( member_set_int @ Xa3 @ C5 )
% 0.34/0.67 => ( ( X3 != Xa3 )
% 0.34/0.67 => ( ( inf_inf_set_int @ X3 @ Xa3 )
% 0.34/0.67 = bot_bot_set_int ) ) ) )
% 0.34/0.67 => ( ( groups4538972089207619220nt_int @ G @ ( comple3221217463730067765et_int @ C5 ) )
% 0.34/0.67 = ( comp_s9098231955081793132nt_int @ groups1256056575759317066nt_int @ groups4538972089207619220nt_int @ G @ C5 ) ) ) ) ).
% 0.34/0.67
% 0.34/0.67 % sum.Union_disjoint
% 0.34/0.67 thf(fact_6583_sum__diff1__nat,axiom,
% 0.34/0.67 ! [A2: product_unit,A: set_Product_unit,F: product_unit > nat] :
% 0.34/0.67 ( ( ( member_Product_unit @ A2 @ A )
% 0.34/0.67 => ( ( groups8766797177574957219it_nat @ F @ ( minus_6452836326544984404t_unit @ A @ ( insert_Product_unit @ A2 @ bot_bo3957492148770167129t_unit ) ) )
% 0.34/0.67 = ( minus_minus_nat @ ( groups8766797177574957219it_nat @ F @ A ) @ ( F @ A2 ) ) ) )
% 0.34/0.67 & ( ~ ( member_Product_unit @ A2 @ A )
% 0.34/0.67 => ( ( groups8766797177574957219it_nat @ F @ ( minus_6452836326544984404t_unit @ A @ ( insert_Product_unit @ A2 @ bot_bo3957492148770167129t_unit ) ) )
% 0.34/0.67 = ( groups8766797177574957219it_nat @ F @ A ) ) ) ) ).
% 0.34/0.67
% 0.34/0.67 % sum_diff1_nat
% 0.34/0.67 thf(fact_6584_sum__diff1__nat,axiom,
% 0.34/0.67 ! [A2: $o,A: set_o,F: $o > nat] :
% 0.34/0.67 ( ( ( member_o @ A2 @ A )
% 0.34/0.67 => ( ( groups8507830703676809646_o_nat @ F @ ( minus_minus_set_o @ A @ ( insert_o @ A2 @ bot_bot_set_o ) ) )
% 0.34/0.67 = ( minus_minus_nat @ ( groups8507830703676809646_o_nat @ F @ A ) @ ( F @ A2 ) ) ) )
% 0.34/0.67 & ( ~ ( member_o @ A2 @ A )
% 0.34/0.67 => ( ( groups8507830703676809646_o_nat @ F @ ( minus_minus_set_o @ A @ ( insert_o @ A2 @ bot_bot_set_o ) ) )
% 0.34/0.67 = ( groups8507830703676809646_o_nat @ F @ A ) ) ) ) ).
% 0.34/0.67
% 0.34/0.67 % sum_diff1_nat
% 0.34/0.67 thf(fact_6585_sum__diff1__nat,axiom,
% 0.34/0.67 ! [A2: int,A: set_int,F: int > nat] :
% 0.34/0.67 ( ( ( member_int @ A2 @ A )
% 0.34/0.67 => ( ( groups4541462559716669496nt_nat @ F @ ( minus_minus_set_int @ A @ ( insert_int @ A2 @ bot_bot_set_int ) ) )
% 0.34/0.67 = ( minus_minus_nat @ ( groups4541462559716669496nt_nat @ F @ A ) @ ( F @ A2 ) ) ) )
% 0.34/0.67 & ( ~ ( member_int @ A2 @ A )
% 0.34/0.67 => ( ( groups4541462559716669496nt_nat @ F @ ( minus_minus_set_int @ A @ ( insert_int @ A2 @ bot_bot_set_int ) ) )
% 0.34/0.67 = ( groups4541462559716669496nt_nat @ F @ A ) ) ) ) ).
% 0.34/0.67
% 0.34/0.67 % sum_diff1_nat
% 0.34/0.67 thf(fact_6586_sum__diff1__nat,axiom,
% 0.34/0.67 ! [A2: produc3658429121746597890et_nat > $o,A: set_Pr4532377907799695533_nat_o,F: ( produc3658429121746597890et_nat > $o ) > nat] :
% 0.34/0.67 ( ( ( member6576561426505652726_nat_o @ A2 @ A )
% 0.34/0.67 => ( ( groups6742167275697360483_o_nat @ F @ ( minus_1801376950450012436_nat_o @ A @ ( insert5175938949040314269_nat_o @ A2 @ bot_bo7824918357723582233_nat_o ) ) )
% 0.34/0.67 = ( minus_minus_nat @ ( groups6742167275697360483_o_nat @ F @ A ) @ ( F @ A2 ) ) ) )
% 0.34/0.67 & ( ~ ( member6576561426505652726_nat_o @ A2 @ A )
% 0.34/0.67 => ( ( groups6742167275697360483_o_nat @ F @ ( minus_1801376950450012436_nat_o @ A @ ( insert5175938949040314269_nat_o @ A2 @ bot_bo7824918357723582233_nat_o ) ) )
% 0.34/0.67 = ( groups6742167275697360483_o_nat @ F @ A ) ) ) ) ).
% 0.34/0.67
% 0.34/0.67 % sum_diff1_nat
% 0.34/0.67 thf(fact_6587_sum__diff1__nat,axiom,
% 0.34/0.67 ! [A2: nat,A: set_nat,F: nat > nat] :
% 0.34/0.67 ( ( ( member_nat @ A2 @ A )
% 0.34/0.67 => ( ( groups3542108847815614940at_nat @ F @ ( minus_minus_set_nat @ A @ ( insert_nat @ A2 @ bot_bot_set_nat ) ) )
% 0.34/0.67 = ( minus_minus_nat @ ( groups3542108847815614940at_nat @ F @ A ) @ ( F @ A2 ) ) ) )
% 0.34/0.67 & ( ~ ( member_nat @ A2 @ A )
% 0.34/0.67 => ( ( groups3542108847815614940at_nat @ F @ ( minus_minus_set_nat @ A @ ( insert_nat @ A2 @ bot_bot_set_nat ) ) )
% 0.34/0.67 = ( groups3542108847815614940at_nat @ F @ A ) ) ) ) ).
% 0.34/0.67
% 0.34/0.67 % sum_diff1_nat
% 0.34/0.67 thf(fact_6588_bezout__add__strong__nat,axiom,
% 0.34/0.67 ! [A2: nat,B: nat] :
% 0.34/0.67 ( ( A2 != zero_zero_nat )
% 0.34/0.67 => ? [D: nat,X3: nat,Y2: nat] :
% 0.34/0.67 ( ( dvd_dvd_nat @ D @ A2 )
% 0.34/0.67 & ( dvd_dvd_nat @ D @ B )
% 0.34/0.67 & ( ( times_times_nat @ A2 @ X3 )
% 0.34/0.67 = ( plus_plus_nat @ ( times_times_nat @ B @ Y2 ) @ D ) ) ) ) ).
% 0.34/0.67
% 0.34/0.67 % bezout_add_strong_nat
% 0.34/0.67 thf(fact_6589_bezout1__nat,axiom,
% 0.34/0.67 ! [A2: nat,B: nat] :
% 0.34/0.67 ? [D: nat,X3: nat,Y2: nat] :
% 0.34/0.67 ( ( dvd_dvd_nat @ D @ A2 )
% 0.34/0.67 & ( dvd_dvd_nat @ D @ B )
% 0.34/0.67 & ( ( ( minus_minus_nat @ ( times_times_nat @ A2 @ X3 ) @ ( times_times_nat @ B @ Y2 ) )
% 0.34/0.67 = D )
% 0.34/0.67 | ( ( minus_minus_nat @ ( times_times_nat @ B @ X3 ) @ ( times_times_nat @ A2 @ Y2 ) )
% 0.34/0.67 = D ) ) ) ).
% 0.34/0.67
% 0.34/0.67 % bezout1_nat
% 0.34/0.67 thf(fact_6590_bezout__add__nat,axiom,
% 0.34/0.67 ! [A2: nat,B: nat] :
% 0.34/0.67 ? [D: nat,X3: nat,Y2: nat] :
% 0.34/0.67 ( ( dvd_dvd_nat @ D @ A2 )
% 0.34/0.67 & ( dvd_dvd_nat @ D @ B )
% 0.34/0.67 & ( ( ( times_times_nat @ A2 @ X3 )
% 0.34/0.67 = ( plus_plus_nat @ ( times_times_nat @ B @ Y2 ) @ D ) )
% 0.34/0.67 | ( ( times_times_nat @ B @ X3 )
% 0.34/0.67 = ( plus_plus_nat @ ( times_times_nat @ A2 @ Y2 ) @ D ) ) ) ) ).
% 0.34/0.67
% 0.34/0.67 % bezout_add_nat
% 0.34/0.67 thf(fact_6591_bezout__lemma__nat,axiom,
% 0.34/0.67 ! [D2: nat,A2: nat,B: nat,X: nat,Y: nat] :
% 0.34/0.67 ( ( dvd_dvd_nat @ D2 @ A2 )
% 0.34/0.67 => ( ( dvd_dvd_nat @ D2 @ B )
% 0.34/0.67 => ( ( ( ( times_times_nat @ A2 @ X )
% 0.34/0.67 = ( plus_plus_nat @ ( times_times_nat @ B @ Y ) @ D2 ) )
% 0.34/0.67 | ( ( times_times_nat @ B @ X )
% 0.34/0.67 = ( plus_plus_nat @ ( times_times_nat @ A2 @ Y ) @ D2 ) ) )
% 0.34/0.67 => ? [X3: nat,Y2: nat] :
% 0.34/0.67 ( ( dvd_dvd_nat @ D2 @ A2 )
% 0.34/0.67 & ( dvd_dvd_nat @ D2 @ ( plus_plus_nat @ A2 @ B ) )
% 0.34/0.67 & ( ( ( times_times_nat @ A2 @ X3 )
% 0.34/0.67 = ( plus_plus_nat @ ( times_times_nat @ ( plus_plus_nat @ A2 @ B ) @ Y2 ) @ D2 ) )
% 0.34/0.67 | ( ( times_times_nat @ ( plus_plus_nat @ A2 @ B ) @ X3 )
% 0.34/0.67 = ( plus_plus_nat @ ( times_times_nat @ A2 @ Y2 ) @ D2 ) ) ) ) ) ) ) ).
% 0.34/0.67
% 0.34/0.67 % bezout_lemma_nat
% 0.34/0.67 thf(fact_6592_dvd__productE,axiom,
% 0.34/0.67 ! [P3: nat,A2: nat,B: nat] :
% 0.34/0.67 ( ( dvd_dvd_nat @ P3 @ ( times_times_nat @ A2 @ B ) )
% 0.34/0.67 => ~ ! [X3: nat,Y2: nat] :
% 0.34/0.67 ( ( P3
% 0.34/0.67 = ( times_times_nat @ X3 @ Y2 ) )
% 0.34/0.67 => ( ( dvd_dvd_nat @ X3 @ A2 )
% 0.34/0.67 => ~ ( dvd_dvd_nat @ Y2 @ B ) ) ) ) ).
% 0.34/0.67
% 0.34/0.67 % dvd_productE
% 0.34/0.67 thf(fact_6593_dvd__productE,axiom,
% 0.34/0.67 ! [P3: int,A2: int,B: int] :
% 0.34/0.67 ( ( dvd_dvd_int @ P3 @ ( times_times_int @ A2 @ B ) )
% 0.34/0.67 => ~ ! [X3: int,Y2: int] :
% 0.34/0.67 ( ( P3
% 0.34/0.67 = ( times_times_int @ X3 @ Y2 ) )
% 0.34/0.67 => ( ( dvd_dvd_int @ X3 @ A2 )
% 0.34/0.67 => ~ ( dvd_dvd_int @ Y2 @ B ) ) ) ) ).
% 0.34/0.67
% 0.34/0.67 % dvd_productE
% 0.34/0.67 thf(fact_6594_division__decomp,axiom,
% 0.34/0.67 ! [A2: nat,B: nat,C: nat] :
% 0.34/0.67 ( ( dvd_dvd_nat @ A2 @ ( times_times_nat @ B @ C ) )
% 0.34/0.67 => ? [B10: nat,C8: nat] :
% 0.34/0.67 ( ( A2
% 0.34/0.67 = ( times_times_nat @ B10 @ C8 ) )
% 0.34/0.67 & ( dvd_dvd_nat @ B10 @ B )
% 0.34/0.67 & ( dvd_dvd_nat @ C8 @ C ) ) ) ).
% 0.34/0.67
% 0.34/0.67 % division_decomp
% 0.34/0.67 thf(fact_6595_division__decomp,axiom,
% 0.34/0.67 ! [A2: int,B: int,C: int] :
% 0.34/0.67 ( ( dvd_dvd_int @ A2 @ ( times_times_int @ B @ C ) )
% 0.34/0.67 => ? [B10: int,C8: int] :
% 0.34/0.67 ( ( A2
% 0.34/0.67 = ( times_times_int @ B10 @ C8 ) )
% 0.34/0.67 & ( dvd_dvd_int @ B10 @ B )
% 0.34/0.67 & ( dvd_dvd_int @ C8 @ C ) ) ) ).
% 0.34/0.67
% 0.34/0.67 % division_decomp
% 0.34/0.67 thf(fact_6596_card__Min__le__sum,axiom,
% 0.34/0.67 ! [A: set_Code_integer,F: code_integer > nat] :
% 0.34/0.67 ( ( finite6017078050557962740nteger @ A )
% 0.34/0.67 => ( ord_less_eq_nat @ ( times_times_nat @ ( finite4902975817058060853nteger @ A ) @ ( lattic8721135487736765967in_nat @ ( image_951025933927791156er_nat @ F @ A ) ) ) @ ( groups7237345082560585321er_nat @ F @ A ) ) ) ).
% 0.34/0.67
% 0.34/0.67 % card_Min_le_sum
% 0.34/0.67 thf(fact_6597_card__Min__le__sum,axiom,
% 0.34/0.67 ! [A: set_nat,F: nat > nat] :
% 0.34/0.67 ( ( finite_finite_nat @ A )
% 0.34/0.67 => ( ord_less_eq_nat @ ( times_times_nat @ ( finite_card_nat @ A ) @ ( lattic8721135487736765967in_nat @ ( image_nat_nat2 @ F @ A ) ) ) @ ( groups3542108847815614940at_nat @ F @ A ) ) ) ).
% 0.34/0.67
% 0.34/0.67 % card_Min_le_sum
% 0.34/0.67 thf(fact_6598_Min__singleton,axiom,
% 0.34/0.67 ! [X: product_unit] :
% 0.34/0.67 ( ( lattic8554435388794137742t_unit @ ( insert_Product_unit @ X @ bot_bo3957492148770167129t_unit ) )
% 0.34/0.67 = X ) ).
% 0.34/0.67
% 0.34/0.67 % Min_singleton
% 0.34/0.67 thf(fact_6599_Min__singleton,axiom,
% 0.34/0.67 ! [X: $o] :
% 0.34/0.67 ( ( lattic1973801136483472281_Min_o @ ( insert_o @ X @ bot_bot_set_o ) )
% 0.34/0.67 = X ) ).
% 0.34/0.67
% 0.34/0.67 % Min_singleton
% 0.34/0.67 thf(fact_6600_Min__singleton,axiom,
% 0.34/0.67 ! [X: nat] :
% 0.34/0.67 ( ( lattic8721135487736765967in_nat @ ( insert_nat @ X @ bot_bot_set_nat ) )
% 0.34/0.67 = X ) ).
% 0.34/0.67
% 0.34/0.67 % Min_singleton
% 0.34/0.67 thf(fact_6601_Min__singleton,axiom,
% 0.34/0.67 ! [X: int] :
% 0.34/0.67 ( ( lattic8718645017227715691in_int @ ( insert_int @ X @ bot_bot_set_int ) )
% 0.34/0.67 = X ) ).
% 0.34/0.67
% 0.34/0.67 % Min_singleton
% 0.34/0.67 thf(fact_6602_Min_Obounded__iff,axiom,
% 0.34/0.67 ! [A: set_Code_integer,X: code_integer] :
% 0.34/0.67 ( ( finite6017078050557962740nteger @ A )
% 0.34/0.67 => ( ( A != bot_bo3990330152332043303nteger )
% 0.34/0.67 => ( ( ord_le3102999989581377725nteger @ X @ ( lattic1063845414844153500nteger @ A ) )
% 0.34/0.67 = ( ! [X2: code_integer] :
% 0.34/0.67 ( ( member_Code_integer @ X2 @ A )
% 0.34/0.67 => ( ord_le3102999989581377725nteger @ X @ X2 ) ) ) ) ) ) ).
% 0.34/0.67
% 0.34/0.67 % Min.bounded_iff
% 0.34/0.67 thf(fact_6603_Min_Obounded__iff,axiom,
% 0.34/0.67 ! [A: set_Product_unit,X: product_unit] :
% 0.34/0.67 ( ( finite4290736615968046902t_unit @ A )
% 0.34/0.67 => ( ( A != bot_bo3957492148770167129t_unit )
% 0.34/0.67 => ( ( ord_le3221252021190050221t_unit @ X @ ( lattic8554435388794137742t_unit @ A ) )
% 0.34/0.67 = ( ! [X2: product_unit] :
% 0.34/0.67 ( ( member_Product_unit @ X2 @ A )
% 0.34/0.67 => ( ord_le3221252021190050221t_unit @ X @ X2 ) ) ) ) ) ) ).
% 0.34/0.67
% 0.34/0.67 % Min.bounded_iff
% 0.34/0.67 thf(fact_6604_Min_Obounded__iff,axiom,
% 0.34/0.67 ! [A: set_o,X: $o] :
% 0.34/0.67 ( ( finite_finite_o @ A )
% 0.34/0.67 => ( ( A != bot_bot_set_o )
% 0.34/0.67 => ( ( ord_less_eq_o @ X @ ( lattic1973801136483472281_Min_o @ A ) )
% 0.34/0.67 = ( ! [X2: $o] :
% 0.34/0.67 ( ( member_o @ X2 @ A )
% 0.34/0.67 => ( ord_less_eq_o @ X @ X2 ) ) ) ) ) ) ).
% 0.34/0.67
% 0.34/0.67 % Min.bounded_iff
% 0.34/0.67 thf(fact_6605_Min_Obounded__iff,axiom,
% 0.34/0.67 ! [A: set_rat,X: rat] :
% 0.34/0.67 ( ( finite_finite_rat @ A )
% 0.34/0.67 => ( ( A != bot_bot_set_rat )
% 0.34/0.67 => ( ( ord_less_eq_rat @ X @ ( lattic8086005427650270231in_rat @ A ) )
% 0.34/0.67 = ( ! [X2: rat] :
% 0.34/0.67 ( ( member_rat @ X2 @ A )
% 0.34/0.67 => ( ord_less_eq_rat @ X @ X2 ) ) ) ) ) ) ).
% 0.34/0.67
% 0.34/0.67 % Min.bounded_iff
% 0.34/0.67 thf(fact_6606_Min_Obounded__iff,axiom,
% 0.34/0.67 ! [A: set_nat,X: nat] :
% 0.34/0.67 ( ( finite_finite_nat @ A )
% 0.34/0.67 => ( ( A != bot_bot_set_nat )
% 0.34/0.67 => ( ( ord_less_eq_nat @ X @ ( lattic8721135487736765967in_nat @ A ) )
% 0.34/0.67 = ( ! [X2: nat] :
% 0.34/0.67 ( ( member_nat @ X2 @ A )
% 0.34/0.67 => ( ord_less_eq_nat @ X @ X2 ) ) ) ) ) ) ).
% 0.34/0.67
% 0.34/0.67 % Min.bounded_iff
% 0.34/0.67 thf(fact_6607_Min_Obounded__iff,axiom,
% 0.34/0.67 ! [A: set_int,X: int] :
% 0.34/0.67 ( ( finite_finite_int @ A )
% 0.34/0.67 => ( ( A != bot_bot_set_int )
% 0.34/0.67 => ( ( ord_less_eq_int @ X @ ( lattic8718645017227715691in_int @ A ) )
% 0.34/0.67 = ( ! [X2: int] :
% 0.34/0.67 ( ( member_int @ X2 @ A )
% 0.34/0.67 => ( ord_less_eq_int @ X @ X2 ) ) ) ) ) ) ).
% 0.34/0.67
% 0.34/0.67 % Min.bounded_iff
% 0.34/0.67 thf(fact_6608_Min__gr__iff,axiom,
% 0.34/0.67 ! [A: set_Product_unit,X: product_unit] :
% 0.34/0.67 ( ( finite4290736615968046902t_unit @ A )
% 0.34/0.67 => ( ( A != bot_bo3957492148770167129t_unit )
% 0.34/0.67 => ( ( ord_le361264281704409273t_unit @ X @ ( lattic8554435388794137742t_unit @ A ) )
% 0.34/0.67 = ( ! [X2: product_unit] :
% 0.34/0.67 ( ( member_Product_unit @ X2 @ A )
% 0.34/0.67 => ( ord_le361264281704409273t_unit @ X @ X2 ) ) ) ) ) ) ).
% 0.34/0.67
% 0.34/0.67 % Min_gr_iff
% 0.34/0.67 thf(fact_6609_Min__gr__iff,axiom,
% 0.34/0.67 ! [A: set_o,X: $o] :
% 0.34/0.67 ( ( finite_finite_o @ A )
% 0.34/0.67 => ( ( A != bot_bot_set_o )
% 0.34/0.67 => ( ( ord_less_o @ X @ ( lattic1973801136483472281_Min_o @ A ) )
% 0.34/0.67 = ( ! [X2: $o] :
% 0.34/0.67 ( ( member_o @ X2 @ A )
% 0.34/0.67 => ( ord_less_o @ X @ X2 ) ) ) ) ) ) ).
% 0.34/0.67
% 0.34/0.67 % Min_gr_iff
% 0.34/0.67 thf(fact_6610_Min__gr__iff,axiom,
% 0.34/0.67 ! [A: set_nat,X: nat] :
% 0.34/0.67 ( ( finite_finite_nat @ A )
% 0.34/0.67 => ( ( A != bot_bot_set_nat )
% 0.34/0.67 => ( ( ord_less_nat @ X @ ( lattic8721135487736765967in_nat @ A ) )
% 0.34/0.67 = ( ! [X2: nat] :
% 0.34/0.67 ( ( member_nat @ X2 @ A )
% 0.34/0.67 => ( ord_less_nat @ X @ X2 ) ) ) ) ) ) ).
% 0.34/0.67
% 0.34/0.67 % Min_gr_iff
% 0.34/0.67 thf(fact_6611_Min__gr__iff,axiom,
% 0.34/0.67 ! [A: set_int,X: int] :
% 0.34/0.67 ( ( finite_finite_int @ A )
% 0.34/0.67 => ( ( A != bot_bot_set_int )
% 0.34/0.67 => ( ( ord_less_int @ X @ ( lattic8718645017227715691in_int @ A ) )
% 0.34/0.67 = ( ! [X2: int] :
% 0.34/0.67 ( ( member_int @ X2 @ A )
% 0.34/0.67 => ( ord_less_int @ X @ X2 ) ) ) ) ) ) ).
% 0.34/0.67
% 0.34/0.67 % Min_gr_iff
% 0.34/0.67 thf(fact_6612_Min__gr__iff,axiom,
% 0.34/0.67 ! [A: set_Code_integer,X: code_integer] :
% 0.34/0.67 ( ( finite6017078050557962740nteger @ A )
% 0.34/0.67 => ( ( A != bot_bo3990330152332043303nteger )
% 0.34/0.67 => ( ( ord_le6747313008572928689nteger @ X @ ( lattic1063845414844153500nteger @ A ) )
% 0.34/0.67 = ( ! [X2: code_integer] :
% 0.34/0.67 ( ( member_Code_integer @ X2 @ A )
% 0.34/0.67 => ( ord_le6747313008572928689nteger @ X @ X2 ) ) ) ) ) ) ).
% 0.34/0.67
% 0.34/0.67 % Min_gr_iff
% 0.34/0.67 thf(fact_6613_Min__gr__iff,axiom,
% 0.34/0.67 ! [A: set_rat,X: rat] :
% 0.34/0.67 ( ( finite_finite_rat @ A )
% 0.34/0.67 => ( ( A != bot_bot_set_rat )
% 0.34/0.67 => ( ( ord_less_rat @ X @ ( lattic8086005427650270231in_rat @ A ) )
% 0.34/0.67 = ( ! [X2: rat] :
% 0.34/0.67 ( ( member_rat @ X2 @ A )
% 0.34/0.67 => ( ord_less_rat @ X @ X2 ) ) ) ) ) ) ).
% 0.34/0.67
% 0.34/0.67 % Min_gr_iff
% 0.34/0.67 thf(fact_6614_Min__in,axiom,
% 0.34/0.67 ! [A: set_Code_integer] :
% 0.34/0.67 ( ( finite6017078050557962740nteger @ A )
% 0.34/0.67 => ( ( A != bot_bo3990330152332043303nteger )
% 0.34/0.67 => ( member_Code_integer @ ( lattic1063845414844153500nteger @ A ) @ A ) ) ) ).
% 0.34/0.67
% 0.34/0.67 % Min_in
% 0.34/0.67 thf(fact_6615_Min__in,axiom,
% 0.34/0.67 ! [A: set_Product_unit] :
% 0.34/0.67 ( ( finite4290736615968046902t_unit @ A )
% 0.34/0.67 => ( ( A != bot_bo3957492148770167129t_unit )
% 0.34/0.67 => ( member_Product_unit @ ( lattic8554435388794137742t_unit @ A ) @ A ) ) ) ).
% 0.34/0.67
% 0.34/0.67 % Min_in
% 0.34/0.67 thf(fact_6616_Min__in,axiom,
% 0.34/0.67 ! [A: set_o] :
% 0.34/0.67 ( ( finite_finite_o @ A )
% 0.34/0.67 => ( ( A != bot_bot_set_o )
% 0.34/0.67 => ( member_o @ ( lattic1973801136483472281_Min_o @ A ) @ A ) ) ) ).
% 0.34/0.67
% 0.34/0.67 % Min_in
% 0.34/0.67 thf(fact_6617_Min__in,axiom,
% 0.34/0.67 ! [A: set_nat] :
% 0.34/0.67 ( ( finite_finite_nat @ A )
% 0.34/0.67 => ( ( A != bot_bot_set_nat )
% 0.34/0.67 => ( member_nat @ ( lattic8721135487736765967in_nat @ A ) @ A ) ) ) ).
% 0.34/0.67
% 0.34/0.67 % Min_in
% 0.34/0.67 thf(fact_6618_Min__in,axiom,
% 0.34/0.67 ! [A: set_int] :
% 0.34/0.67 ( ( finite_finite_int @ A )
% 0.34/0.67 => ( ( A != bot_bot_set_int )
% 0.34/0.67 => ( member_int @ ( lattic8718645017227715691in_int @ A ) @ A ) ) ) ).
% 0.34/0.67
% 0.34/0.67 % Min_in
% 0.34/0.67 thf(fact_6619_Min__eq__iff,axiom,
% 0.34/0.67 ! [A: set_Code_integer,M4: code_integer] :
% 0.34/0.67 ( ( finite6017078050557962740nteger @ A )
% 0.34/0.67 => ( ( A != bot_bo3990330152332043303nteger )
% 0.34/0.67 => ( ( ( lattic1063845414844153500nteger @ A )
% 0.34/0.67 = M4 )
% 0.34/0.67 = ( ( member_Code_integer @ M4 @ A )
% 0.34/0.67 & ! [X2: code_integer] :
% 0.34/0.67 ( ( member_Code_integer @ X2 @ A )
% 0.34/0.67 => ( ord_le3102999989581377725nteger @ M4 @ X2 ) ) ) ) ) ) ).
% 0.34/0.67
% 0.34/0.67 % Min_eq_iff
% 0.34/0.67 thf(fact_6620_Min__eq__iff,axiom,
% 0.34/0.67 ! [A: set_Product_unit,M4: product_unit] :
% 0.34/0.67 ( ( finite4290736615968046902t_unit @ A )
% 0.34/0.67 => ( ( A != bot_bo3957492148770167129t_unit )
% 0.34/0.67 => ( ( ( lattic8554435388794137742t_unit @ A )
% 0.34/0.67 = M4 )
% 0.34/0.67 = ( ( member_Product_unit @ M4 @ A )
% 0.34/0.67 & ! [X2: product_unit] :
% 0.34/0.67 ( ( member_Product_unit @ X2 @ A )
% 0.34/0.67 => ( ord_le3221252021190050221t_unit @ M4 @ X2 ) ) ) ) ) ) ).
% 0.34/0.67
% 0.34/0.67 % Min_eq_iff
% 0.34/0.67 thf(fact_6621_Min__eq__iff,axiom,
% 0.34/0.67 ! [A: set_o,M4: $o] :
% 0.34/0.67 ( ( finite_finite_o @ A )
% 0.34/0.67 => ( ( A != bot_bot_set_o )
% 0.34/0.67 => ( ( ( lattic1973801136483472281_Min_o @ A )
% 0.34/0.67 = M4 )
% 0.34/0.67 = ( ( member_o @ M4 @ A )
% 0.34/0.67 & ! [X2: $o] :
% 0.34/0.67 ( ( member_o @ X2 @ A )
% 0.34/0.67 => ( ord_less_eq_o @ M4 @ X2 ) ) ) ) ) ) ).
% 0.34/0.67
% 0.34/0.67 % Min_eq_iff
% 0.34/0.67 thf(fact_6622_Min__eq__iff,axiom,
% 0.34/0.67 ! [A: set_rat,M4: rat] :
% 0.34/0.67 ( ( finite_finite_rat @ A )
% 0.34/0.67 => ( ( A != bot_bot_set_rat )
% 0.34/0.67 => ( ( ( lattic8086005427650270231in_rat @ A )
% 0.34/0.67 = M4 )
% 0.34/0.67 = ( ( member_rat @ M4 @ A )
% 0.34/0.67 & ! [X2: rat] :
% 0.34/0.67 ( ( member_rat @ X2 @ A )
% 0.34/0.67 => ( ord_less_eq_rat @ M4 @ X2 ) ) ) ) ) ) ).
% 0.34/0.67
% 0.34/0.67 % Min_eq_iff
% 0.34/0.67 thf(fact_6623_Min__eq__iff,axiom,
% 0.34/0.67 ! [A: set_nat,M4: nat] :
% 0.34/0.67 ( ( finite_finite_nat @ A )
% 0.34/0.67 => ( ( A != bot_bot_set_nat )
% 0.34/0.67 => ( ( ( lattic8721135487736765967in_nat @ A )
% 0.34/0.67 = M4 )
% 0.34/0.67 = ( ( member_nat @ M4 @ A )
% 0.34/0.67 & ! [X2: nat] :
% 0.34/0.67 ( ( member_nat @ X2 @ A )
% 0.34/0.67 => ( ord_less_eq_nat @ M4 @ X2 ) ) ) ) ) ) ).
% 0.34/0.67
% 0.34/0.67 % Min_eq_iff
% 0.34/0.67 thf(fact_6624_Min__eq__iff,axiom,
% 0.34/0.67 ! [A: set_int,M4: int] :
% 0.34/0.67 ( ( finite_finite_int @ A )
% 0.34/0.67 => ( ( A != bot_bot_set_int )
% 0.34/0.67 => ( ( ( lattic8718645017227715691in_int @ A )
% 0.34/0.67 = M4 )
% 0.34/0.67 = ( ( member_int @ M4 @ A )
% 0.34/0.67 & ! [X2: int] :
% 0.34/0.67 ( ( member_int @ X2 @ A )
% 0.34/0.67 => ( ord_less_eq_int @ M4 @ X2 ) ) ) ) ) ) ).
% 0.34/0.67
% 0.34/0.67 % Min_eq_iff
% 0.34/0.67 thf(fact_6625_Min__le__iff,axiom,
% 0.34/0.67 ! [A: set_Code_integer,X: code_integer] :
% 0.34/0.67 ( ( finite6017078050557962740nteger @ A )
% 0.34/0.67 => ( ( A != bot_bo3990330152332043303nteger )
% 0.34/0.67 => ( ( ord_le3102999989581377725nteger @ ( lattic1063845414844153500nteger @ A ) @ X )
% 0.34/0.67 = ( ? [X2: code_integer] :
% 0.34/0.67 ( ( member_Code_integer @ X2 @ A )
% 0.34/0.67 & ( ord_le3102999989581377725nteger @ X2 @ X ) ) ) ) ) ) ).
% 0.34/0.67
% 0.34/0.67 % Min_le_iff
% 0.34/0.67 thf(fact_6626_Min__le__iff,axiom,
% 0.34/0.67 ! [A: set_Product_unit,X: product_unit] :
% 0.34/0.67 ( ( finite4290736615968046902t_unit @ A )
% 0.34/0.67 => ( ( A != bot_bo3957492148770167129t_unit )
% 0.34/0.67 => ( ( ord_le3221252021190050221t_unit @ ( lattic8554435388794137742t_unit @ A ) @ X )
% 0.34/0.67 = ( ? [X2: product_unit] :
% 0.34/0.67 ( ( member_Product_unit @ X2 @ A )
% 0.34/0.67 & ( ord_le3221252021190050221t_unit @ X2 @ X ) ) ) ) ) ) ).
% 0.34/0.67
% 0.34/0.67 % Min_le_iff
% 0.34/0.67 thf(fact_6627_Min__le__iff,axiom,
% 0.34/0.67 ! [A: set_o,X: $o] :
% 0.34/0.67 ( ( finite_finite_o @ A )
% 0.34/0.67 => ( ( A != bot_bot_set_o )
% 0.34/0.67 => ( ( ord_less_eq_o @ ( lattic1973801136483472281_Min_o @ A ) @ X )
% 0.34/0.67 = ( ? [X2: $o] :
% 0.34/0.67 ( ( member_o @ X2 @ A )
% 0.34/0.67 & ( ord_less_eq_o @ X2 @ X ) ) ) ) ) ) ).
% 0.34/0.67
% 0.34/0.67 % Min_le_iff
% 0.34/0.67 thf(fact_6628_Min__le__iff,axiom,
% 0.34/0.67 ! [A: set_rat,X: rat] :
% 0.34/0.67 ( ( finite_finite_rat @ A )
% 0.34/0.67 => ( ( A != bot_bot_set_rat )
% 0.34/0.67 => ( ( ord_less_eq_rat @ ( lattic8086005427650270231in_rat @ A ) @ X )
% 0.34/0.67 = ( ? [X2: rat] :
% 0.34/0.67 ( ( member_rat @ X2 @ A )
% 0.34/0.67 & ( ord_less_eq_rat @ X2 @ X ) ) ) ) ) ) ).
% 0.34/0.67
% 0.34/0.67 % Min_le_iff
% 0.34/0.67 thf(fact_6629_Min__le__iff,axiom,
% 0.34/0.67 ! [A: set_nat,X: nat] :
% 0.34/0.67 ( ( finite_finite_nat @ A )
% 0.34/0.67 => ( ( A != bot_bot_set_nat )
% 0.34/0.67 => ( ( ord_less_eq_nat @ ( lattic8721135487736765967in_nat @ A ) @ X )
% 0.34/0.67 = ( ? [X2: nat] :
% 0.34/0.67 ( ( member_nat @ X2 @ A )
% 0.34/0.67 & ( ord_less_eq_nat @ X2 @ X ) ) ) ) ) ) ).
% 0.34/0.67
% 0.34/0.67 % Min_le_iff
% 0.34/0.67 thf(fact_6630_Min__le__iff,axiom,
% 0.34/0.67 ! [A: set_int,X: int] :
% 0.34/0.67 ( ( finite_finite_int @ A )
% 0.34/0.67 => ( ( A != bot_bot_set_int )
% 0.34/0.67 => ( ( ord_less_eq_int @ ( lattic8718645017227715691in_int @ A ) @ X )
% 0.34/0.67 = ( ? [X2: int] :
% 0.34/0.67 ( ( member_int @ X2 @ A )
% 0.34/0.67 & ( ord_less_eq_int @ X2 @ X ) ) ) ) ) ) ).
% 0.34/0.67
% 0.34/0.67 % Min_le_iff
% 0.34/0.67 thf(fact_6631_eq__Min__iff,axiom,
% 0.34/0.67 ! [A: set_Code_integer,M4: code_integer] :
% 0.34/0.67 ( ( finite6017078050557962740nteger @ A )
% 0.34/0.67 => ( ( A != bot_bo3990330152332043303nteger )
% 0.34/0.67 => ( ( M4
% 0.34/0.67 = ( lattic1063845414844153500nteger @ A ) )
% 0.34/0.67 = ( ( member_Code_integer @ M4 @ A )
% 0.34/0.67 & ! [X2: code_integer] :
% 0.34/0.67 ( ( member_Code_integer @ X2 @ A )
% 0.34/0.67 => ( ord_le3102999989581377725nteger @ M4 @ X2 ) ) ) ) ) ) ).
% 0.34/0.67
% 0.34/0.67 % eq_Min_iff
% 0.34/0.67 thf(fact_6632_eq__Min__iff,axiom,
% 0.34/0.67 ! [A: set_Product_unit,M4: product_unit] :
% 0.34/0.67 ( ( finite4290736615968046902t_unit @ A )
% 0.34/0.67 => ( ( A != bot_bo3957492148770167129t_unit )
% 0.34/0.67 => ( ( M4
% 0.34/0.67 = ( lattic8554435388794137742t_unit @ A ) )
% 0.34/0.67 = ( ( member_Product_unit @ M4 @ A )
% 0.34/0.67 & ! [X2: product_unit] :
% 0.34/0.67 ( ( member_Product_unit @ X2 @ A )
% 0.34/0.67 => ( ord_le3221252021190050221t_unit @ M4 @ X2 ) ) ) ) ) ) ).
% 0.34/0.67
% 0.34/0.67 % eq_Min_iff
% 0.34/0.67 thf(fact_6633_eq__Min__iff,axiom,
% 0.34/0.67 ! [A: set_o,M4: $o] :
% 0.34/0.67 ( ( finite_finite_o @ A )
% 0.34/0.67 => ( ( A != bot_bot_set_o )
% 0.34/0.67 => ( ( M4
% 0.34/0.67 = ( lattic1973801136483472281_Min_o @ A ) )
% 0.34/0.67 = ( ( member_o @ M4 @ A )
% 0.34/0.67 & ! [X2: $o] :
% 0.34/0.67 ( ( member_o @ X2 @ A )
% 0.34/0.67 => ( ord_less_eq_o @ M4 @ X2 ) ) ) ) ) ) ).
% 0.34/0.67
% 0.34/0.67 % eq_Min_iff
% 0.34/0.67 thf(fact_6634_eq__Min__iff,axiom,
% 0.34/0.67 ! [A: set_rat,M4: rat] :
% 0.34/0.67 ( ( finite_finite_rat @ A )
% 0.34/0.67 => ( ( A != bot_bot_set_rat )
% 0.34/0.67 => ( ( M4
% 0.34/0.67 = ( lattic8086005427650270231in_rat @ A ) )
% 0.34/0.67 = ( ( member_rat @ M4 @ A )
% 0.34/0.67 & ! [X2: rat] :
% 0.34/0.67 ( ( member_rat @ X2 @ A )
% 0.34/0.67 => ( ord_less_eq_rat @ M4 @ X2 ) ) ) ) ) ) ).
% 0.34/0.67
% 0.34/0.67 % eq_Min_iff
% 0.34/0.67 thf(fact_6635_eq__Min__iff,axiom,
% 0.34/0.67 ! [A: set_nat,M4: nat] :
% 0.34/0.67 ( ( finite_finite_nat @ A )
% 0.34/0.67 => ( ( A != bot_bot_set_nat )
% 0.34/0.67 => ( ( M4
% 0.34/0.67 = ( lattic8721135487736765967in_nat @ A ) )
% 0.34/0.67 = ( ( member_nat @ M4 @ A )
% 0.34/0.67 & ! [X2: nat] :
% 0.34/0.67 ( ( member_nat @ X2 @ A )
% 0.34/0.67 => ( ord_less_eq_nat @ M4 @ X2 ) ) ) ) ) ) ).
% 0.34/0.67
% 0.34/0.67 % eq_Min_iff
% 0.34/0.67 thf(fact_6636_eq__Min__iff,axiom,
% 0.34/0.67 ! [A: set_int,M4: int] :
% 0.34/0.67 ( ( finite_finite_int @ A )
% 0.34/0.67 => ( ( A != bot_bot_set_int )
% 0.34/0.67 => ( ( M4
% 0.34/0.67 = ( lattic8718645017227715691in_int @ A ) )
% 0.34/0.67 = ( ( member_int @ M4 @ A )
% 0.34/0.67 & ! [X2: int] :
% 0.34/0.67 ( ( member_int @ X2 @ A )
% 0.34/0.67 => ( ord_less_eq_int @ M4 @ X2 ) ) ) ) ) ) ).
% 0.34/0.67
% 0.34/0.67 % eq_Min_iff
% 0.34/0.67 thf(fact_6637_Min_OboundedE,axiom,
% 0.34/0.67 ! [A: set_Code_integer,X: code_integer] :
% 0.34/0.67 ( ( finite6017078050557962740nteger @ A )
% 0.34/0.67 => ( ( A != bot_bo3990330152332043303nteger )
% 0.34/0.67 => ( ( ord_le3102999989581377725nteger @ X @ ( lattic1063845414844153500nteger @ A ) )
% 0.34/0.67 => ! [A10: code_integer] :
% 0.34/0.67 ( ( member_Code_integer @ A10 @ A )
% 0.34/0.67 => ( ord_le3102999989581377725nteger @ X @ A10 ) ) ) ) ) ).
% 0.34/0.67
% 0.34/0.67 % Min.boundedE
% 0.34/0.67 thf(fact_6638_Min_OboundedE,axiom,
% 0.34/0.67 ! [A: set_Product_unit,X: product_unit] :
% 0.34/0.67 ( ( finite4290736615968046902t_unit @ A )
% 0.34/0.67 => ( ( A != bot_bo3957492148770167129t_unit )
% 0.34/0.67 => ( ( ord_le3221252021190050221t_unit @ X @ ( lattic8554435388794137742t_unit @ A ) )
% 0.34/0.67 => ! [A10: product_unit] :
% 0.34/0.67 ( ( member_Product_unit @ A10 @ A )
% 0.34/0.67 => ( ord_le3221252021190050221t_unit @ X @ A10 ) ) ) ) ) ).
% 0.34/0.67
% 0.34/0.67 % Min.boundedE
% 0.34/0.67 thf(fact_6639_Min_OboundedE,axiom,
% 0.34/0.67 ! [A: set_o,X: $o] :
% 0.34/0.67 ( ( finite_finite_o @ A )
% 0.34/0.67 => ( ( A != bot_bot_set_o )
% 0.34/0.67 => ( ( ord_less_eq_o @ X @ ( lattic1973801136483472281_Min_o @ A ) )
% 0.34/0.67 => ! [A10: $o] :
% 0.34/0.67 ( ( member_o @ A10 @ A )
% 0.34/0.67 => ( ord_less_eq_o @ X @ A10 ) ) ) ) ) ).
% 0.34/0.67
% 0.34/0.67 % Min.boundedE
% 0.34/0.67 thf(fact_6640_Min_OboundedE,axiom,
% 0.34/0.67 ! [A: set_rat,X: rat] :
% 0.34/0.67 ( ( finite_finite_rat @ A )
% 0.34/0.67 => ( ( A != bot_bot_set_rat )
% 0.34/0.67 => ( ( ord_less_eq_rat @ X @ ( lattic8086005427650270231in_rat @ A ) )
% 0.34/0.67 => ! [A10: rat] :
% 0.34/0.67 ( ( member_rat @ A10 @ A )
% 0.34/0.67 => ( ord_less_eq_rat @ X @ A10 ) ) ) ) ) ).
% 0.34/0.67
% 0.34/0.67 % Min.boundedE
% 0.34/0.67 thf(fact_6641_Min_OboundedE,axiom,
% 0.34/0.67 ! [A: set_nat,X: nat] :
% 0.34/0.67 ( ( finite_finite_nat @ A )
% 0.34/0.67 => ( ( A != bot_bot_set_nat )
% 0.34/0.67 => ( ( ord_less_eq_nat @ X @ ( lattic8721135487736765967in_nat @ A ) )
% 0.34/0.67 => ! [A10: nat] :
% 0.34/0.67 ( ( member_nat @ A10 @ A )
% 0.34/0.67 => ( ord_less_eq_nat @ X @ A10 ) ) ) ) ) ).
% 0.34/0.67
% 0.34/0.67 % Min.boundedE
% 0.34/0.67 thf(fact_6642_Min_OboundedE,axiom,
% 0.34/0.67 ! [A: set_int,X: int] :
% 0.34/0.67 ( ( finite_finite_int @ A )
% 0.34/0.67 => ( ( A != bot_bot_set_int )
% 0.34/0.67 => ( ( ord_less_eq_int @ X @ ( lattic8718645017227715691in_int @ A ) )
% 0.34/0.67 => ! [A10: int] :
% 0.34/0.67 ( ( member_int @ A10 @ A )
% 0.34/0.67 => ( ord_less_eq_int @ X @ A10 ) ) ) ) ) ).
% 0.34/0.67
% 0.34/0.67 % Min.boundedE
% 0.34/0.67 thf(fact_6643_Min_OboundedI,axiom,
% 0.34/0.67 ! [A: set_Code_integer,X: code_integer] :
% 0.34/0.67 ( ( finite6017078050557962740nteger @ A )
% 0.34/0.67 => ( ( A != bot_bo3990330152332043303nteger )
% 0.34/0.67 => ( ! [A4: code_integer] :
% 0.34/0.67 ( ( member_Code_integer @ A4 @ A )
% 0.34/0.67 => ( ord_le3102999989581377725nteger @ X @ A4 ) )
% 0.34/0.67 => ( ord_le3102999989581377725nteger @ X @ ( lattic1063845414844153500nteger @ A ) ) ) ) ) ).
% 0.34/0.67
% 0.34/0.67 % Min.boundedI
% 0.34/0.67 thf(fact_6644_Min_OboundedI,axiom,
% 0.34/0.67 ! [A: set_Product_unit,X: product_unit] :
% 0.34/0.67 ( ( finite4290736615968046902t_unit @ A )
% 0.34/0.67 => ( ( A != bot_bo3957492148770167129t_unit )
% 0.34/0.67 => ( ! [A4: product_unit] :
% 0.34/0.67 ( ( member_Product_unit @ A4 @ A )
% 0.34/0.67 => ( ord_le3221252021190050221t_unit @ X @ A4 ) )
% 0.34/0.67 => ( ord_le3221252021190050221t_unit @ X @ ( lattic8554435388794137742t_unit @ A ) ) ) ) ) ).
% 0.34/0.67
% 0.34/0.67 % Min.boundedI
% 0.34/0.67 thf(fact_6645_Min_OboundedI,axiom,
% 0.34/0.67 ! [A: set_o,X: $o] :
% 0.34/0.67 ( ( finite_finite_o @ A )
% 0.34/0.67 => ( ( A != bot_bot_set_o )
% 0.34/0.67 => ( ! [A4: $o] :
% 0.34/0.67 ( ( member_o @ A4 @ A )
% 0.34/0.67 => ( ord_less_eq_o @ X @ A4 ) )
% 0.34/0.67 => ( ord_less_eq_o @ X @ ( lattic1973801136483472281_Min_o @ A ) ) ) ) ) ).
% 0.34/0.67
% 0.34/0.67 % Min.boundedI
% 0.34/0.67 thf(fact_6646_Min_OboundedI,axiom,
% 0.34/0.67 ! [A: set_rat,X: rat] :
% 0.34/0.67 ( ( finite_finite_rat @ A )
% 0.34/0.67 => ( ( A != bot_bot_set_rat )
% 0.34/0.67 => ( ! [A4: rat] :
% 0.34/0.67 ( ( member_rat @ A4 @ A )
% 0.34/0.67 => ( ord_less_eq_rat @ X @ A4 ) )
% 0.34/0.67 => ( ord_less_eq_rat @ X @ ( lattic8086005427650270231in_rat @ A ) ) ) ) ) ).
% 0.34/0.67
% 0.34/0.67 % Min.boundedI
% 0.34/0.67 thf(fact_6647_Min_OboundedI,axiom,
% 0.34/0.67 ! [A: set_nat,X: nat] :
% 0.34/0.67 ( ( finite_finite_nat @ A )
% 0.34/0.67 => ( ( A != bot_bot_set_nat )
% 0.34/0.67 => ( ! [A4: nat] :
% 0.34/0.67 ( ( member_nat @ A4 @ A )
% 0.34/0.67 => ( ord_less_eq_nat @ X @ A4 ) )
% 0.34/0.67 => ( ord_less_eq_nat @ X @ ( lattic8721135487736765967in_nat @ A ) ) ) ) ) ).
% 0.34/0.67
% 0.34/0.67 % Min.boundedI
% 0.34/0.67 thf(fact_6648_Min_OboundedI,axiom,
% 0.34/0.67 ! [A: set_int,X: int] :
% 0.34/0.67 ( ( finite_finite_int @ A )
% 0.34/0.67 => ( ( A != bot_bot_set_int )
% 0.34/0.67 => ( ! [A4: int] :
% 0.34/0.67 ( ( member_int @ A4 @ A )
% 0.34/0.67 => ( ord_less_eq_int @ X @ A4 ) )
% 0.34/0.67 => ( ord_less_eq_int @ X @ ( lattic8718645017227715691in_int @ A ) ) ) ) ) ).
% 0.34/0.67
% 0.34/0.67 % Min.boundedI
% 0.34/0.67 thf(fact_6649_Min__less__iff,axiom,
% 0.34/0.67 ! [A: set_Product_unit,X: product_unit] :
% 0.34/0.67 ( ( finite4290736615968046902t_unit @ A )
% 0.34/0.67 => ( ( A != bot_bo3957492148770167129t_unit )
% 0.34/0.67 => ( ( ord_le361264281704409273t_unit @ ( lattic8554435388794137742t_unit @ A ) @ X )
% 0.34/0.67 = ( ? [X2: product_unit] :
% 0.34/0.67 ( ( member_Product_unit @ X2 @ A )
% 0.34/0.67 & ( ord_le361264281704409273t_unit @ X2 @ X ) ) ) ) ) ) ).
% 0.34/0.67
% 0.34/0.67 % Min_less_iff
% 0.34/0.67 thf(fact_6650_Min__less__iff,axiom,
% 0.34/0.67 ! [A: set_o,X: $o] :
% 0.34/0.67 ( ( finite_finite_o @ A )
% 0.34/0.67 => ( ( A != bot_bot_set_o )
% 0.34/0.67 => ( ( ord_less_o @ ( lattic1973801136483472281_Min_o @ A ) @ X )
% 0.34/0.67 = ( ? [X2: $o] :
% 0.34/0.67 ( ( member_o @ X2 @ A )
% 0.34/0.67 & ( ord_less_o @ X2 @ X ) ) ) ) ) ) ).
% 0.34/0.67
% 0.34/0.67 % Min_less_iff
% 0.34/0.67 thf(fact_6651_Min__less__iff,axiom,
% 0.34/0.67 ! [A: set_nat,X: nat] :
% 0.34/0.67 ( ( finite_finite_nat @ A )
% 0.34/0.67 => ( ( A != bot_bot_set_nat )
% 0.34/0.67 => ( ( ord_less_nat @ ( lattic8721135487736765967in_nat @ A ) @ X )
% 0.34/0.67 = ( ? [X2: nat] :
% 0.34/0.67 ( ( member_nat @ X2 @ A )
% 0.34/0.67 & ( ord_less_nat @ X2 @ X ) ) ) ) ) ) ).
% 0.34/0.67
% 0.34/0.67 % Min_less_iff
% 0.34/0.67 thf(fact_6652_Min__less__iff,axiom,
% 0.34/0.67 ! [A: set_int,X: int] :
% 0.34/0.67 ( ( finite_finite_int @ A )
% 0.34/0.67 => ( ( A != bot_bot_set_int )
% 0.34/0.67 => ( ( ord_less_int @ ( lattic8718645017227715691in_int @ A ) @ X )
% 0.34/0.67 = ( ? [X2: int] :
% 0.34/0.67 ( ( member_int @ X2 @ A )
% 0.34/0.67 & ( ord_less_int @ X2 @ X ) ) ) ) ) ) ).
% 0.34/0.67
% 0.34/0.67 % Min_less_iff
% 0.34/0.67 thf(fact_6653_Min__less__iff,axiom,
% 0.34/0.67 ! [A: set_Code_integer,X: code_integer] :
% 0.34/0.67 ( ( finite6017078050557962740nteger @ A )
% 0.34/0.67 => ( ( A != bot_bo3990330152332043303nteger )
% 0.34/0.67 => ( ( ord_le6747313008572928689nteger @ ( lattic1063845414844153500nteger @ A ) @ X )
% 0.34/0.67 = ( ? [X2: code_integer] :
% 0.34/0.67 ( ( member_Code_integer @ X2 @ A )
% 0.34/0.67 & ( ord_le6747313008572928689nteger @ X2 @ X ) ) ) ) ) ) ).
% 0.34/0.67
% 0.34/0.67 % Min_less_iff
% 0.34/0.67 thf(fact_6654_Min__less__iff,axiom,
% 0.34/0.67 ! [A: set_rat,X: rat] :
% 0.34/0.67 ( ( finite_finite_rat @ A )
% 0.34/0.67 => ( ( A != bot_bot_set_rat )
% 0.34/0.67 => ( ( ord_less_rat @ ( lattic8086005427650270231in_rat @ A ) @ X )
% 0.34/0.67 = ( ? [X2: rat] :
% 0.34/0.67 ( ( member_rat @ X2 @ A )
% 0.34/0.67 & ( ord_less_rat @ X2 @ X ) ) ) ) ) ) ).
% 0.34/0.67
% 0.34/0.67 % Min_less_iff
% 0.34/0.67 thf(fact_6655_Min__insert2,axiom,
% 0.34/0.67 ! [A: set_o,A2: $o] :
% 0.34/0.67 ( ( finite_finite_o @ A )
% 0.34/0.67 => ( ! [B2: $o] :
% 0.34/0.67 ( ( member_o @ B2 @ A )
% 0.34/0.67 => ( ord_less_eq_o @ A2 @ B2 ) )
% 0.34/0.67 => ( ( lattic1973801136483472281_Min_o @ ( insert_o @ A2 @ A ) )
% 0.34/0.67 = A2 ) ) ) ).
% 0.34/0.67
% 0.34/0.67 % Min_insert2
% 0.34/0.67 thf(fact_6656_Min__insert2,axiom,
% 0.34/0.67 ! [A: set_Product_unit,A2: product_unit] :
% 0.34/0.67 ( ( finite4290736615968046902t_unit @ A )
% 0.34/0.67 => ( ! [B2: product_unit] :
% 0.34/0.67 ( ( member_Product_unit @ B2 @ A )
% 0.34/0.67 => ( ord_le3221252021190050221t_unit @ A2 @ B2 ) )
% 0.34/0.67 => ( ( lattic8554435388794137742t_unit @ ( insert_Product_unit @ A2 @ A ) )
% 0.34/0.67 = A2 ) ) ) ).
% 0.34/0.67
% 0.34/0.67 % Min_insert2
% 0.34/0.67 thf(fact_6657_Min__insert2,axiom,
% 0.34/0.67 ! [A: set_Code_integer,A2: code_integer] :
% 0.34/0.67 ( ( finite6017078050557962740nteger @ A )
% 0.34/0.67 => ( ! [B2: code_integer] :
% 0.34/0.67 ( ( member_Code_integer @ B2 @ A )
% 0.34/0.67 => ( ord_le3102999989581377725nteger @ A2 @ B2 ) )
% 0.34/0.67 => ( ( lattic1063845414844153500nteger @ ( insert_Code_integer @ A2 @ A ) )
% 0.34/0.67 = A2 ) ) ) ).
% 0.34/0.67
% 0.34/0.67 % Min_insert2
% 0.34/0.67 thf(fact_6658_Min__insert2,axiom,
% 0.34/0.67 ! [A: set_rat,A2: rat] :
% 0.34/0.67 ( ( finite_finite_rat @ A )
% 0.34/0.67 => ( ! [B2: rat] :
% 0.34/0.67 ( ( member_rat @ B2 @ A )
% 0.34/0.67 => ( ord_less_eq_rat @ A2 @ B2 ) )
% 0.34/0.67 => ( ( lattic8086005427650270231in_rat @ ( insert_rat @ A2 @ A ) )
% 0.34/0.67 = A2 ) ) ) ).
% 0.34/0.67
% 0.34/0.67 % Min_insert2
% 0.34/0.67 thf(fact_6659_Min__insert2,axiom,
% 0.34/0.67 ! [A: set_nat,A2: nat] :
% 0.34/0.67 ( ( finite_finite_nat @ A )
% 0.34/0.67 => ( ! [B2: nat] :
% 0.34/0.67 ( ( member_nat @ B2 @ A )
% 0.34/0.67 => ( ord_less_eq_nat @ A2 @ B2 ) )
% 0.34/0.67 => ( ( lattic8721135487736765967in_nat @ ( insert_nat @ A2 @ A ) )
% 0.34/0.67 = A2 ) ) ) ).
% 0.34/0.67
% 0.34/0.67 % Min_insert2
% 0.34/0.67 thf(fact_6660_Min__insert2,axiom,
% 0.34/0.67 ! [A: set_int,A2: int] :
% 0.34/0.67 ( ( finite_finite_int @ A )
% 0.34/0.67 => ( ! [B2: int] :
% 0.34/0.67 ( ( member_int @ B2 @ A )
% 0.34/0.67 => ( ord_less_eq_int @ A2 @ B2 ) )
% 0.34/0.67 => ( ( lattic8718645017227715691in_int @ ( insert_int @ A2 @ A ) )
% 0.34/0.67 = A2 ) ) ) ).
% 0.34/0.67
% 0.34/0.67 % Min_insert2
% 0.34/0.67 thf(fact_6661_cInf__eq__Min,axiom,
% 0.34/0.67 ! [X4: set_int] :
% 0.34/0.67 ( ( finite_finite_int @ X4 )
% 0.34/0.67 => ( ( X4 != bot_bot_set_int )
% 0.34/0.67 => ( ( complete_Inf_Inf_int @ X4 )
% 0.34/0.67 = ( lattic8718645017227715691in_int @ X4 ) ) ) ) ).
% 0.34/0.67
% 0.34/0.67 % cInf_eq_Min
% 0.34/0.67 thf(fact_6662_cInf__eq__Min,axiom,
% 0.34/0.67 ! [X4: set_nat] :
% 0.34/0.67 ( ( finite_finite_nat @ X4 )
% 0.34/0.67 => ( ( X4 != bot_bot_set_nat )
% 0.34/0.67 => ( ( complete_Inf_Inf_nat @ X4 )
% 0.34/0.67 = ( lattic8721135487736765967in_nat @ X4 ) ) ) ) ).
% 0.34/0.67
% 0.34/0.67 % cInf_eq_Min
% 0.34/0.67 thf(fact_6663_cInf__eq__Min,axiom,
% 0.34/0.67 ! [X4: set_Product_unit] :
% 0.34/0.67 ( ( finite4290736615968046902t_unit @ X4 )
% 0.34/0.67 => ( ( X4 != bot_bo3957492148770167129t_unit )
% 0.34/0.67 => ( ( comple2584293577114468500t_unit @ X4 )
% 0.34/0.67 = ( lattic8554435388794137742t_unit @ X4 ) ) ) ) ).
% 0.34/0.67
% 0.34/0.67 % cInf_eq_Min
% 0.34/0.67 thf(fact_6664_Min__Inf,axiom,
% 0.34/0.67 ! [A: set_Product_unit] :
% 0.34/0.67 ( ( finite4290736615968046902t_unit @ A )
% 0.34/0.67 => ( ( A != bot_bo3957492148770167129t_unit )
% 0.34/0.67 => ( ( lattic8554435388794137742t_unit @ A )
% 0.34/0.67 = ( comple2584293577114468500t_unit @ A ) ) ) ) ).
% 0.34/0.67
% 0.34/0.67 % Min_Inf
% 0.34/0.67 thf(fact_6665_Min__antimono,axiom,
% 0.34/0.67 ! [M: set_Code_integer,N: set_Code_integer] :
% 0.34/0.67 ( ( ord_le7084787975880047091nteger @ M @ N )
% 0.34/0.67 => ( ( M != bot_bo3990330152332043303nteger )
% 0.34/0.67 => ( ( finite6017078050557962740nteger @ N )
% 0.34/0.67 => ( ord_le3102999989581377725nteger @ ( lattic1063845414844153500nteger @ N ) @ ( lattic1063845414844153500nteger @ M ) ) ) ) ) ).
% 0.34/0.67
% 0.34/0.67 % Min_antimono
% 0.34/0.67 thf(fact_6666_Min__antimono,axiom,
% 0.34/0.67 ! [M: set_Product_unit,N: set_Product_unit] :
% 0.34/0.67 ( ( ord_le3507040750410214029t_unit @ M @ N )
% 0.34/0.67 => ( ( M != bot_bo3957492148770167129t_unit )
% 0.34/0.67 => ( ( finite4290736615968046902t_unit @ N )
% 0.34/0.67 => ( ord_le3221252021190050221t_unit @ ( lattic8554435388794137742t_unit @ N ) @ ( lattic8554435388794137742t_unit @ M ) ) ) ) ) ).
% 0.34/0.67
% 0.34/0.67 % Min_antimono
% 0.34/0.67 thf(fact_6667_Min__antimono,axiom,
% 0.34/0.67 ! [M: set_o,N: set_o] :
% 0.34/0.67 ( ( ord_less_eq_set_o @ M @ N )
% 0.34/0.67 => ( ( M != bot_bot_set_o )
% 0.34/0.67 => ( ( finite_finite_o @ N )
% 0.34/0.67 => ( ord_less_eq_o @ ( lattic1973801136483472281_Min_o @ N ) @ ( lattic1973801136483472281_Min_o @ M ) ) ) ) ) ).
% 0.34/0.67
% 0.34/0.67 % Min_antimono
% 0.34/0.67 thf(fact_6668_Min__antimono,axiom,
% 0.34/0.67 ! [M: set_rat,N: set_rat] :
% 0.34/0.67 ( ( ord_less_eq_set_rat @ M @ N )
% 0.34/0.67 => ( ( M != bot_bot_set_rat )
% 0.34/0.67 => ( ( finite_finite_rat @ N )
% 0.34/0.67 => ( ord_less_eq_rat @ ( lattic8086005427650270231in_rat @ N ) @ ( lattic8086005427650270231in_rat @ M ) ) ) ) ) ).
% 0.34/0.67
% 0.34/0.67 % Min_antimono
% 0.34/0.67 thf(fact_6669_Min__antimono,axiom,
% 0.34/0.67 ! [M: set_int,N: set_int] :
% 0.34/0.67 ( ( ord_less_eq_set_int @ M @ N )
% 0.34/0.67 => ( ( M != bot_bot_set_int )
% 0.34/0.67 => ( ( finite_finite_int @ N )
% 0.34/0.67 => ( ord_less_eq_int @ ( lattic8718645017227715691in_int @ N ) @ ( lattic8718645017227715691in_int @ M ) ) ) ) ) ).
% 0.34/0.67
% 0.34/0.67 % Min_antimono
% 0.34/0.67 thf(fact_6670_Min__antimono,axiom,
% 0.34/0.67 ! [M: set_nat,N: set_nat] :
% 0.34/0.67 ( ( ord_less_eq_set_nat @ M @ N )
% 0.34/0.67 => ( ( M != bot_bot_set_nat )
% 0.34/0.67 => ( ( finite_finite_nat @ N )
% 0.34/0.67 => ( ord_less_eq_nat @ ( lattic8721135487736765967in_nat @ N ) @ ( lattic8721135487736765967in_nat @ M ) ) ) ) ) ).
% 0.34/0.67
% 0.34/0.67 % Min_antimono
% 0.34/0.67 thf(fact_6671_Min_Osubset__imp,axiom,
% 0.34/0.67 ! [A: set_Code_integer,B4: set_Code_integer] :
% 0.34/0.67 ( ( ord_le7084787975880047091nteger @ A @ B4 )
% 0.34/0.67 => ( ( A != bot_bo3990330152332043303nteger )
% 0.34/0.67 => ( ( finite6017078050557962740nteger @ B4 )
% 0.34/0.67 => ( ord_le3102999989581377725nteger @ ( lattic1063845414844153500nteger @ B4 ) @ ( lattic1063845414844153500nteger @ A ) ) ) ) ) ).
% 0.34/0.67
% 0.34/0.67 % Min.subset_imp
% 0.34/0.67 thf(fact_6672_Min_Osubset__imp,axiom,
% 0.34/0.67 ! [A: set_Product_unit,B4: set_Product_unit] :
% 0.34/0.67 ( ( ord_le3507040750410214029t_unit @ A @ B4 )
% 0.34/0.67 => ( ( A != bot_bo3957492148770167129t_unit )
% 0.34/0.67 => ( ( finite4290736615968046902t_unit @ B4 )
% 0.34/0.67 => ( ord_le3221252021190050221t_unit @ ( lattic8554435388794137742t_unit @ B4 ) @ ( lattic8554435388794137742t_unit @ A ) ) ) ) ) ).
% 0.34/0.67
% 0.34/0.67 % Min.subset_imp
% 0.34/0.67 thf(fact_6673_Min_Osubset__imp,axiom,
% 0.34/0.67 ! [A: set_o,B4: set_o] :
% 0.34/0.67 ( ( ord_less_eq_set_o @ A @ B4 )
% 0.34/0.67 => ( ( A != bot_bot_set_o )
% 0.34/0.67 => ( ( finite_finite_o @ B4 )
% 0.34/0.67 => ( ord_less_eq_o @ ( lattic1973801136483472281_Min_o @ B4 ) @ ( lattic1973801136483472281_Min_o @ A ) ) ) ) ) ).
% 0.34/0.67
% 0.34/0.67 % Min.subset_imp
% 0.34/0.67 thf(fact_6674_Min_Osubset__imp,axiom,
% 0.34/0.67 ! [A: set_rat,B4: set_rat] :
% 0.34/0.67 ( ( ord_less_eq_set_rat @ A @ B4 )
% 0.34/0.67 => ( ( A != bot_bot_set_rat )
% 0.34/0.67 => ( ( finite_finite_rat @ B4 )
% 0.34/0.67 => ( ord_less_eq_rat @ ( lattic8086005427650270231in_rat @ B4 ) @ ( lattic8086005427650270231in_rat @ A ) ) ) ) ) ).
% 0.34/0.67
% 0.34/0.67 % Min.subset_imp
% 0.34/0.67 thf(fact_6675_Min_Osubset__imp,axiom,
% 0.34/0.67 ! [A: set_int,B4: set_int] :
% 0.34/0.67 ( ( ord_less_eq_set_int @ A @ B4 )
% 0.34/0.67 => ( ( A != bot_bot_set_int )
% 0.34/0.67 => ( ( finite_finite_int @ B4 )
% 0.34/0.67 => ( ord_less_eq_int @ ( lattic8718645017227715691in_int @ B4 ) @ ( lattic8718645017227715691in_int @ A ) ) ) ) ) ).
% 0.34/0.67
% 0.34/0.67 % Min.subset_imp
% 0.34/0.67 thf(fact_6676_Min_Osubset__imp,axiom,
% 0.34/0.67 ! [A: set_nat,B4: set_nat] :
% 0.34/0.67 ( ( ord_less_eq_set_nat @ A @ B4 )
% 0.34/0.67 => ( ( A != bot_bot_set_nat )
% 0.34/0.67 => ( ( finite_finite_nat @ B4 )
% 0.34/0.67 => ( ord_less_eq_nat @ ( lattic8721135487736765967in_nat @ B4 ) @ ( lattic8721135487736765967in_nat @ A ) ) ) ) ) ).
% 0.34/0.67
% 0.34/0.67 % Min.subset_imp
% 0.34/0.67 thf(fact_6677_minus__Min__eq__Max,axiom,
% 0.34/0.67 ! [S: set_int] :
% 0.34/0.67 ( ( finite_finite_int @ S )
% 0.34/0.67 => ( ( S != bot_bot_set_int )
% 0.34/0.67 => ( ( uminus_uminus_int @ ( lattic8718645017227715691in_int @ S ) )
% 0.34/0.67 = ( lattic8263393255366662781ax_int @ ( image_int_int2 @ uminus_uminus_int @ S ) ) ) ) ) ).
% 0.34/0.67
% 0.34/0.67 % minus_Min_eq_Max
% 0.34/0.67 thf(fact_6678_minus__Min__eq__Max,axiom,
% 0.34/0.67 ! [S: set_Code_integer] :
% 0.34/0.67 ( ( finite6017078050557962740nteger @ S )
% 0.34/0.67 => ( ( S != bot_bo3990330152332043303nteger )
% 0.34/0.67 => ( ( uminus1351360451143612070nteger @ ( lattic1063845414844153500nteger @ S ) )
% 0.34/0.67 = ( lattic4901227151466704046nteger @ ( image_4470545334726330049nteger @ uminus1351360451143612070nteger @ S ) ) ) ) ) ).
% 0.34/0.67
% 0.34/0.67 % minus_Min_eq_Max
% 0.34/0.67 thf(fact_6679_minus__Max__eq__Min,axiom,
% 0.34/0.67 ! [S: set_int] :
% 0.34/0.67 ( ( finite_finite_int @ S )
% 0.34/0.67 => ( ( S != bot_bot_set_int )
% 0.34/0.67 => ( ( uminus_uminus_int @ ( lattic8263393255366662781ax_int @ S ) )
% 0.34/0.67 = ( lattic8718645017227715691in_int @ ( image_int_int2 @ uminus_uminus_int @ S ) ) ) ) ) ).
% 0.34/0.67
% 0.34/0.67 % minus_Max_eq_Min
% 0.34/0.67 thf(fact_6680_minus__Max__eq__Min,axiom,
% 0.34/0.67 ! [S: set_Code_integer] :
% 0.34/0.67 ( ( finite6017078050557962740nteger @ S )
% 0.34/0.67 => ( ( S != bot_bo3990330152332043303nteger )
% 0.34/0.67 => ( ( uminus1351360451143612070nteger @ ( lattic4901227151466704046nteger @ S ) )
% 0.34/0.67 = ( lattic1063845414844153500nteger @ ( image_4470545334726330049nteger @ uminus1351360451143612070nteger @ S ) ) ) ) ) ).
% 0.34/0.67
% 0.34/0.67 % minus_Max_eq_Min
% 0.34/0.67 thf(fact_6681_Gcd__0__iff,axiom,
% 0.34/0.67 ! [A: set_int] :
% 0.34/0.67 ( ( ( gcd_Gcd_int @ A )
% 0.34/0.67 = zero_zero_int )
% 0.34/0.67 = ( ord_less_eq_set_int @ A @ ( insert_int @ zero_zero_int @ bot_bot_set_int ) ) ) ).
% 0.34/0.67
% 0.34/0.67 % Gcd_0_iff
% 0.34/0.67 thf(fact_6682_Gcd__0__iff,axiom,
% 0.34/0.67 ! [A: set_nat] :
% 0.34/0.67 ( ( ( gcd_Gcd_nat @ A )
% 0.34/0.67 = zero_zero_nat )
% 0.34/0.67 = ( ord_less_eq_set_nat @ A @ ( insert_nat @ zero_zero_nat @ bot_bot_set_nat ) ) ) ).
% 0.34/0.67
% 0.34/0.67 % Gcd_0_iff
% 0.34/0.67 thf(fact_6683_Gcd__fin__0__iff,axiom,
% 0.34/0.67 ! [A: set_int] :
% 0.34/0.67 ( ( ( semiri4256215615220890538in_int @ A )
% 0.34/0.67 = zero_zero_int )
% 0.34/0.67 = ( ( ord_less_eq_set_int @ A @ ( insert_int @ zero_zero_int @ bot_bot_set_int ) )
% 0.34/0.67 & ( finite_finite_int @ A ) ) ) ).
% 0.34/0.67
% 0.34/0.67 % Gcd_fin_0_iff
% 0.34/0.67 thf(fact_6684_Gcd__fin__0__iff,axiom,
% 0.34/0.67 ! [A: set_nat] :
% 0.34/0.67 ( ( ( semiri4258706085729940814in_nat @ A )
% 0.34/0.67 = zero_zero_nat )
% 0.34/0.67 = ( ( ord_less_eq_set_nat @ A @ ( insert_nat @ zero_zero_nat @ bot_bot_set_nat ) )
% 0.34/0.67 & ( finite_finite_nat @ A ) ) ) ).
% 0.34/0.67
% 0.34/0.67 % Gcd_fin_0_iff
% 0.34/0.67 thf(fact_6685_sum__bounded__above__strict,axiom,
% 0.34/0.67 ! [A: set_nat,F: nat > rat,K4: rat] :
% 0.34/0.67 ( ! [I3: nat] :
% 0.34/0.67 ( ( member_nat @ I3 @ A )
% 0.34/0.67 => ( ord_less_rat @ ( F @ I3 ) @ K4 ) )
% 0.34/0.67 => ( ( ord_less_nat @ zero_zero_nat @ ( finite_card_nat @ A ) )
% 0.34/0.67 => ( ord_less_rat @ ( groups2906978787729119204at_rat @ F @ A ) @ ( times_times_rat @ ( semiri681578069525770553at_rat @ ( finite_card_nat @ A ) ) @ K4 ) ) ) ) ).
% 0.34/0.67
% 0.34/0.67 % sum_bounded_above_strict
% 0.34/0.67 thf(fact_6686_sum__bounded__above__strict,axiom,
% 0.34/0.67 ! [A: set_int,F: int > rat,K4: rat] :
% 0.34/0.67 ( ! [I3: int] :
% 0.34/0.67 ( ( member_int @ I3 @ A )
% 0.34/0.67 => ( ord_less_rat @ ( F @ I3 ) @ K4 ) )
% 0.34/0.67 => ( ( ord_less_nat @ zero_zero_nat @ ( finite_card_int @ A ) )
% 0.34/0.67 => ( ord_less_rat @ ( groups3906332499630173760nt_rat @ F @ A ) @ ( times_times_rat @ ( semiri681578069525770553at_rat @ ( finite_card_int @ A ) ) @ K4 ) ) ) ) ).
% 0.34/0.67
% 0.34/0.67 % sum_bounded_above_strict
% 0.34/0.67 thf(fact_6687_sum__bounded__above__strict,axiom,
% 0.34/0.67 ! [A: set_o,F: $o > rat,K4: rat] :
% 0.34/0.67 ( ! [I3: $o] :
% 0.34/0.67 ( ( member_o @ I3 @ A )
% 0.34/0.67 => ( ord_less_rat @ ( F @ I3 ) @ K4 ) )
% 0.34/0.67 => ( ( ord_less_nat @ zero_zero_nat @ ( finite_card_o @ A ) )
% 0.34/0.67 => ( ord_less_rat @ ( groups7872700643590313910_o_rat @ F @ A ) @ ( times_times_rat @ ( semiri681578069525770553at_rat @ ( finite_card_o @ A ) ) @ K4 ) ) ) ) ).
% 0.34/0.67
% 0.34/0.67 % sum_bounded_above_strict
% 0.34/0.67 thf(fact_6688_sum__bounded__above__strict,axiom,
% 0.34/0.67 ! [A: set_Product_unit,F: product_unit > rat,K4: rat] :
% 0.34/0.67 ( ! [I3: product_unit] :
% 0.34/0.67 ( ( member_Product_unit @ I3 @ A )
% 0.34/0.67 => ( ord_less_rat @ ( F @ I3 ) @ K4 ) )
% 0.34/0.67 => ( ( ord_less_nat @ zero_zero_nat @ ( finite410649719033368117t_unit @ A ) )
% 0.34/0.67 => ( ord_less_rat @ ( groups8131667117488461483it_rat @ F @ A ) @ ( times_times_rat @ ( semiri681578069525770553at_rat @ ( finite410649719033368117t_unit @ A ) ) @ K4 ) ) ) ) ).
% 0.34/0.67
% 0.34/0.67 % sum_bounded_above_strict
% 0.34/0.67 thf(fact_6689_sum__bounded__above__strict,axiom,
% 0.34/0.67 ! [A: set_nat,F: nat > int,K4: int] :
% 0.34/0.67 ( ! [I3: nat] :
% 0.34/0.67 ( ( member_nat @ I3 @ A )
% 0.34/0.67 => ( ord_less_int @ ( F @ I3 ) @ K4 ) )
% 0.34/0.67 => ( ( ord_less_nat @ zero_zero_nat @ ( finite_card_nat @ A ) )
% 0.34/0.67 => ( ord_less_int @ ( groups3539618377306564664at_int @ F @ A ) @ ( times_times_int @ ( semiri1314217659103216013at_int @ ( finite_card_nat @ A ) ) @ K4 ) ) ) ) ).
% 0.34/0.67
% 0.34/0.67 % sum_bounded_above_strict
% 0.34/0.67 thf(fact_6690_sum__bounded__above__strict,axiom,
% 0.34/0.67 ! [A: set_o,F: $o > int,K4: int] :
% 0.34/0.67 ( ! [I3: $o] :
% 0.34/0.67 ( ( member_o @ I3 @ A )
% 0.34/0.67 => ( ord_less_int @ ( F @ I3 ) @ K4 ) )
% 0.34/0.67 => ( ( ord_less_nat @ zero_zero_nat @ ( finite_card_o @ A ) )
% 0.34/0.67 => ( ord_less_int @ ( groups8505340233167759370_o_int @ F @ A ) @ ( times_times_int @ ( semiri1314217659103216013at_int @ ( finite_card_o @ A ) ) @ K4 ) ) ) ) ).
% 0.34/0.67
% 0.34/0.67 % sum_bounded_above_strict
% 0.34/0.67 thf(fact_6691_sum__bounded__above__strict,axiom,
% 0.34/0.67 ! [A: set_Product_unit,F: product_unit > int,K4: int] :
% 0.34/0.67 ( ! [I3: product_unit] :
% 0.34/0.67 ( ( member_Product_unit @ I3 @ A )
% 0.34/0.67 => ( ord_less_int @ ( F @ I3 ) @ K4 ) )
% 0.34/0.67 => ( ( ord_less_nat @ zero_zero_nat @ ( finite410649719033368117t_unit @ A ) )
% 0.34/0.67 => ( ord_less_int @ ( groups8764306707065906943it_int @ F @ A ) @ ( times_times_int @ ( semiri1314217659103216013at_int @ ( finite410649719033368117t_unit @ A ) ) @ K4 ) ) ) ) ).
% 0.34/0.67
% 0.34/0.67 % sum_bounded_above_strict
% 0.34/0.67 thf(fact_6692_sum__bounded__above__strict,axiom,
% 0.34/0.67 ! [A: set_int,F: int > nat,K4: nat] :
% 0.34/0.67 ( ! [I3: int] :
% 0.34/0.67 ( ( member_int @ I3 @ A )
% 0.34/0.67 => ( ord_less_nat @ ( F @ I3 ) @ K4 ) )
% 0.34/0.67 => ( ( ord_less_nat @ zero_zero_nat @ ( finite_card_int @ A ) )
% 0.34/0.67 => ( ord_less_nat @ ( groups4541462559716669496nt_nat @ F @ A ) @ ( times_times_nat @ ( semiri1316708129612266289at_nat @ ( finite_card_int @ A ) ) @ K4 ) ) ) ) ).
% 0.34/0.67
% 0.34/0.67 % sum_bounded_above_strict
% 0.34/0.67 thf(fact_6693_sum__bounded__above__strict,axiom,
% 0.34/0.67 ! [A: set_o,F: $o > nat,K4: nat] :
% 0.34/0.67 ( ! [I3: $o] :
% 0.34/0.67 ( ( member_o @ I3 @ A )
% 0.34/0.67 => ( ord_less_nat @ ( F @ I3 ) @ K4 ) )
% 0.34/0.67 => ( ( ord_less_nat @ zero_zero_nat @ ( finite_card_o @ A ) )
% 0.34/0.67 => ( ord_less_nat @ ( groups8507830703676809646_o_nat @ F @ A ) @ ( times_times_nat @ ( semiri1316708129612266289at_nat @ ( finite_card_o @ A ) ) @ K4 ) ) ) ) ).
% 0.34/0.67
% 0.34/0.67 % sum_bounded_above_strict
% 0.34/0.67 thf(fact_6694_sum__bounded__above__strict,axiom,
% 0.34/0.67 ! [A: set_Product_unit,F: product_unit > nat,K4: nat] :
% 0.34/0.67 ( ! [I3: product_unit] :
% 0.34/0.67 ( ( member_Product_unit @ I3 @ A )
% 0.34/0.67 => ( ord_less_nat @ ( F @ I3 ) @ K4 ) )
% 0.34/0.67 => ( ( ord_less_nat @ zero_zero_nat @ ( finite410649719033368117t_unit @ A ) )
% 0.34/0.67 => ( ord_less_nat @ ( groups8766797177574957219it_nat @ F @ A ) @ ( times_times_nat @ ( semiri1316708129612266289at_nat @ ( finite410649719033368117t_unit @ A ) ) @ K4 ) ) ) ) ).
% 0.34/0.67
% 0.34/0.67 % sum_bounded_above_strict
% 0.34/0.67 thf(fact_6695_prod__le__power,axiom,
% 0.34/0.67 ! [A: set_nat,F: nat > code_natural,N2: code_natural,K3: nat] :
% 0.34/0.67 ( ! [I3: nat] :
% 0.34/0.67 ( ( member_nat @ I3 @ A )
% 0.34/0.67 => ( ( ord_le1926595141338095240atural @ zero_z2226904508553997617atural @ ( F @ I3 ) )
% 0.34/0.67 & ( ord_le1926595141338095240atural @ ( F @ I3 ) @ N2 ) ) )
% 0.34/0.67 => ( ( ord_less_eq_nat @ ( finite_card_nat @ A ) @ K3 )
% 0.34/0.67 => ( ( ord_le1926595141338095240atural @ one_one_Code_natural @ N2 )
% 0.34/0.67 => ( ord_le1926595141338095240atural @ ( groups2279045934846249631atural @ F @ A ) @ ( power_7079662738309270450atural @ N2 @ K3 ) ) ) ) ) ).
% 0.34/0.67
% 0.34/0.67 % prod_le_power
% 0.34/0.67 thf(fact_6696_prod__le__power,axiom,
% 0.34/0.67 ! [A: set_int,F: int > code_natural,N2: code_natural,K3: nat] :
% 0.34/0.67 ( ! [I3: int] :
% 0.34/0.67 ( ( member_int @ I3 @ A )
% 0.34/0.67 => ( ( ord_le1926595141338095240atural @ zero_z2226904508553997617atural @ ( F @ I3 ) )
% 0.34/0.67 & ( ord_le1926595141338095240atural @ ( F @ I3 ) @ N2 ) ) )
% 0.34/0.67 => ( ( ord_less_eq_nat @ ( finite_card_int @ A ) @ K3 )
% 0.34/0.67 => ( ( ord_le1926595141338095240atural @ one_one_Code_natural @ N2 )
% 0.34/0.67 => ( ord_le1926595141338095240atural @ ( groups2650699495083094267atural @ F @ A ) @ ( power_7079662738309270450atural @ N2 @ K3 ) ) ) ) ) ).
% 0.34/0.67
% 0.34/0.67 % prod_le_power
% 0.34/0.67 thf(fact_6697_prod__le__power,axiom,
% 0.34/0.67 ! [A: set_o,F: $o > code_natural,N2: code_natural,K3: nat] :
% 0.34/0.67 ( ! [I3: $o] :
% 0.34/0.67 ( ( member_o @ I3 @ A )
% 0.34/0.67 => ( ( ord_le1926595141338095240atural @ zero_z2226904508553997617atural @ ( F @ I3 ) )
% 0.34/0.67 & ( ord_le1926595141338095240atural @ ( F @ I3 ) @ N2 ) ) )
% 0.34/0.67 => ( ( ord_less_eq_nat @ ( finite_card_o @ A ) @ K3 )
% 0.34/0.67 => ( ( ord_le1926595141338095240atural @ one_one_Code_natural @ N2 )
% 0.34/0.67 => ( ord_le1926595141338095240atural @ ( groups6518289543945209051atural @ F @ A ) @ ( power_7079662738309270450atural @ N2 @ K3 ) ) ) ) ) ).
% 0.34/0.67
% 0.34/0.67 % prod_le_power
% 0.34/0.67 thf(fact_6698_prod__le__power,axiom,
% 0.34/0.67 ! [A: set_Product_unit,F: product_unit > code_natural,N2: code_natural,K3: nat] :
% 0.34/0.67 ( ! [I3: product_unit] :
% 0.34/0.67 ( ( member_Product_unit @ I3 @ A )
% 0.34/0.67 => ( ( ord_le1926595141338095240atural @ zero_z2226904508553997617atural @ ( F @ I3 ) )
% 0.34/0.67 & ( ord_le1926595141338095240atural @ ( F @ I3 ) @ N2 ) ) )
% 0.34/0.67 => ( ( ord_less_eq_nat @ ( finite410649719033368117t_unit @ A ) @ K3 )
% 0.34/0.67 => ( ( ord_le1926595141338095240atural @ one_one_Code_natural @ N2 )
% 0.34/0.67 => ( ord_le1926595141338095240atural @ ( groups7008954963700796496atural @ F @ A ) @ ( power_7079662738309270450atural @ N2 @ K3 ) ) ) ) ) ).
% 0.34/0.67
% 0.34/0.67 % prod_le_power
% 0.34/0.67 thf(fact_6699_prod__le__power,axiom,
% 0.34/0.67 ! [A: set_nat,F: nat > code_integer,N2: code_integer,K3: nat] :
% 0.34/0.67 ( ! [I3: nat] :
% 0.34/0.67 ( ( member_nat @ I3 @ A )
% 0.34/0.67 => ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ ( F @ I3 ) )
% 0.34/0.67 & ( ord_le3102999989581377725nteger @ ( F @ I3 ) @ N2 ) ) )
% 0.34/0.67 => ( ( ord_less_eq_nat @ ( finite_card_nat @ A ) @ K3 )
% 0.34/0.67 => ( ( ord_le3102999989581377725nteger @ one_one_Code_integer @ N2 )
% 0.34/0.67 => ( ord_le3102999989581377725nteger @ ( groups3455450783089532116nteger @ F @ A ) @ ( power_8256067586552552935nteger @ N2 @ K3 ) ) ) ) ) ).
% 0.34/0.67
% 0.34/0.67 % prod_le_power
% 0.34/0.67 thf(fact_6700_prod__le__power,axiom,
% 0.34/0.67 ! [A: set_int,F: int > code_integer,N2: code_integer,K3: nat] :
% 0.34/0.67 ( ! [I3: int] :
% 0.34/0.67 ( ( member_int @ I3 @ A )
% 0.34/0.67 => ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ ( F @ I3 ) )
% 0.34/0.67 & ( ord_le3102999989581377725nteger @ ( F @ I3 ) @ N2 ) ) )
% 0.34/0.67 => ( ( ord_less_eq_nat @ ( finite_card_int @ A ) @ K3 )
% 0.34/0.67 => ( ( ord_le3102999989581377725nteger @ one_one_Code_integer @ N2 )
% 0.34/0.67 => ( ord_le3102999989581377725nteger @ ( groups3827104343326376752nteger @ F @ A ) @ ( power_8256067586552552935nteger @ N2 @ K3 ) ) ) ) ) ).
% 0.34/0.67
% 0.34/0.67 % prod_le_power
% 0.34/0.67 thf(fact_6701_prod__le__power,axiom,
% 0.34/0.67 ! [A: set_o,F: $o > code_integer,N2: code_integer,K3: nat] :
% 0.34/0.67 ( ! [I3: $o] :
% 0.34/0.67 ( ( member_o @ I3 @ A )
% 0.34/0.67 => ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ ( F @ I3 ) )
% 0.34/0.67 & ( ord_le3102999989581377725nteger @ ( F @ I3 ) @ N2 ) ) )
% 0.34/0.67 => ( ( ord_less_eq_nat @ ( finite_card_o @ A ) @ K3 )
% 0.34/0.67 => ( ( ord_le3102999989581377725nteger @ one_one_Code_integer @ N2 )
% 0.34/0.67 => ( ord_le3102999989581377725nteger @ ( groups7694694392188491536nteger @ F @ A ) @ ( power_8256067586552552935nteger @ N2 @ K3 ) ) ) ) ) ).
% 0.34/0.67
% 0.34/0.67 % prod_le_power
% 0.34/0.67 thf(fact_6702_prod__le__power,axiom,
% 0.34/0.67 ! [A: set_Product_unit,F: product_unit > code_integer,N2: code_integer,K3: nat] :
% 0.34/0.67 ( ! [I3: product_unit] :
% 0.34/0.67 ( ( member_Product_unit @ I3 @ A )
% 0.34/0.67 => ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ ( F @ I3 ) )
% 0.34/0.67 & ( ord_le3102999989581377725nteger @ ( F @ I3 ) @ N2 ) ) )
% 0.34/0.67 => ( ( ord_less_eq_nat @ ( finite410649719033368117t_unit @ A ) @ K3 )
% 0.34/0.67 => ( ( ord_le3102999989581377725nteger @ one_one_Code_integer @ N2 )
% 0.34/0.67 => ( ord_le3102999989581377725nteger @ ( groups8185359811944078981nteger @ F @ A ) @ ( power_8256067586552552935nteger @ N2 @ K3 ) ) ) ) ) ).
% 0.34/0.67
% 0.34/0.67 % prod_le_power
% 0.34/0.67 thf(fact_6703_prod__le__power,axiom,
% 0.34/0.67 ! [A: set_nat,F: nat > rat,N2: rat,K3: nat] :
% 0.34/0.67 ( ! [I3: nat] :
% 0.34/0.67 ( ( member_nat @ I3 @ A )
% 0.34/0.67 => ( ( ord_less_eq_rat @ zero_zero_rat @ ( F @ I3 ) )
% 0.34/0.67 & ( ord_less_eq_rat @ ( F @ I3 ) @ N2 ) ) )
% 0.34/0.67 => ( ( ord_less_eq_nat @ ( finite_card_nat @ A ) @ K3 )
% 0.34/0.67 => ( ( ord_less_eq_rat @ one_one_rat @ N2 )
% 0.34/0.67 => ( ord_less_eq_rat @ ( groups73079841787564623at_rat @ F @ A ) @ ( power_power_rat @ N2 @ K3 ) ) ) ) ) ).
% 0.34/0.67
% 0.34/0.67 % prod_le_power
% 0.34/0.67 thf(fact_6704_prod__le__power,axiom,
% 0.34/0.67 ! [A: set_int,F: int > rat,N2: rat,K3: nat] :
% 0.34/0.67 ( ! [I3: int] :
% 0.34/0.67 ( ( member_int @ I3 @ A )
% 0.34/0.67 => ( ( ord_less_eq_rat @ zero_zero_rat @ ( F @ I3 ) )
% 0.34/0.67 & ( ord_less_eq_rat @ ( F @ I3 ) @ N2 ) ) )
% 0.34/0.67 => ( ( ord_less_eq_nat @ ( finite_card_int @ A ) @ K3 )
% 0.34/0.67 => ( ( ord_less_eq_rat @ one_one_rat @ N2 )
% 0.34/0.67 => ( ord_less_eq_rat @ ( groups1072433553688619179nt_rat @ F @ A ) @ ( power_power_rat @ N2 @ K3 ) ) ) ) ) ).
% 0.34/0.67
% 0.34/0.67 % prod_le_power
% 0.34/0.67 thf(fact_6705_prod__mono2,axiom,
% 0.34/0.67 ! [B4: set_int,A: set_int,F: int > code_integer] :
% 0.34/0.67 ( ( finite_finite_int @ B4 )
% 0.34/0.67 => ( ( ord_less_eq_set_int @ A @ B4 )
% 0.34/0.67 => ( ! [B2: int] :
% 0.34/0.67 ( ( member_int @ B2 @ ( minus_minus_set_int @ B4 @ A ) )
% 0.34/0.67 => ( ord_le3102999989581377725nteger @ one_one_Code_integer @ ( F @ B2 ) ) )
% 0.34/0.67 => ( ! [A4: int] :
% 0.34/0.67 ( ( member_int @ A4 @ A )
% 0.34/0.67 => ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ ( F @ A4 ) ) )
% 0.34/0.67 => ( ord_le3102999989581377725nteger @ ( groups3827104343326376752nteger @ F @ A ) @ ( groups3827104343326376752nteger @ F @ B4 ) ) ) ) ) ) ).
% 0.34/0.67
% 0.34/0.67 % prod_mono2
% 0.34/0.67 thf(fact_6706_prod__mono2,axiom,
% 0.34/0.67 ! [B4: set_o,A: set_o,F: $o > code_integer] :
% 0.34/0.67 ( ( finite_finite_o @ B4 )
% 0.34/0.67 => ( ( ord_less_eq_set_o @ A @ B4 )
% 0.34/0.67 => ( ! [B2: $o] :
% 0.34/0.67 ( ( member_o @ B2 @ ( minus_minus_set_o @ B4 @ A ) )
% 0.34/0.67 => ( ord_le3102999989581377725nteger @ one_one_Code_integer @ ( F @ B2 ) ) )
% 0.34/0.67 => ( ! [A4: $o] :
% 0.34/0.67 ( ( member_o @ A4 @ A )
% 0.34/0.67 => ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ ( F @ A4 ) ) )
% 0.34/0.67 => ( ord_le3102999989581377725nteger @ ( groups7694694392188491536nteger @ F @ A ) @ ( groups7694694392188491536nteger @ F @ B4 ) ) ) ) ) ) ).
% 0.34/0.67
% 0.34/0.67 % prod_mono2
% 0.34/0.67 thf(fact_6707_prod__mono2,axiom,
% 0.34/0.67 ! [B4: set_Product_unit,A: set_Product_unit,F: product_unit > code_integer] :
% 0.34/0.67 ( ( finite4290736615968046902t_unit @ B4 )
% 0.34/0.67 => ( ( ord_le3507040750410214029t_unit @ A @ B4 )
% 0.34/0.67 => ( ! [B2: product_unit] :
% 0.34/0.67 ( ( member_Product_unit @ B2 @ ( minus_6452836326544984404t_unit @ B4 @ A ) )
% 0.34/0.67 => ( ord_le3102999989581377725nteger @ one_one_Code_integer @ ( F @ B2 ) ) )
% 0.34/0.67 => ( ! [A4: product_unit] :
% 0.34/0.67 ( ( member_Product_unit @ A4 @ A )
% 0.34/0.67 => ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ ( F @ A4 ) ) )
% 0.34/0.67 => ( ord_le3102999989581377725nteger @ ( groups8185359811944078981nteger @ F @ A ) @ ( groups8185359811944078981nteger @ F @ B4 ) ) ) ) ) ) ).
% 0.34/0.67
% 0.34/0.67 % prod_mono2
% 0.34/0.67 thf(fact_6708_prod__mono2,axiom,
% 0.34/0.67 ! [B4: set_Code_integer,A: set_Code_integer,F: code_integer > code_integer] :
% 0.34/0.67 ( ( finite6017078050557962740nteger @ B4 )
% 0.34/0.67 => ( ( ord_le7084787975880047091nteger @ A @ B4 )
% 0.34/0.67 => ( ! [B2: code_integer] :
% 0.34/0.67 ( ( member_Code_integer @ B2 @ ( minus_2355218937544613996nteger @ B4 @ A ) )
% 0.34/0.67 => ( ord_le3102999989581377725nteger @ one_one_Code_integer @ ( F @ B2 ) ) )
% 0.34/0.67 => ( ! [A4: code_integer] :
% 0.34/0.67 ( ( member_Code_integer @ A4 @ A )
% 0.34/0.67 => ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ ( F @ A4 ) ) )
% 0.34/0.67 => ( ord_le3102999989581377725nteger @ ( groups3674199335183972705nteger @ F @ A ) @ ( groups3674199335183972705nteger @ F @ B4 ) ) ) ) ) ) ).
% 0.34/0.67
% 0.34/0.67 % prod_mono2
% 0.34/0.67 thf(fact_6709_prod__mono2,axiom,
% 0.34/0.67 ! [B4: set_int,A: set_int,F: int > rat] :
% 0.34/0.67 ( ( finite_finite_int @ B4 )
% 0.34/0.67 => ( ( ord_less_eq_set_int @ A @ B4 )
% 0.34/0.67 => ( ! [B2: int] :
% 0.34/0.67 ( ( member_int @ B2 @ ( minus_minus_set_int @ B4 @ A ) )
% 0.34/0.67 => ( ord_less_eq_rat @ one_one_rat @ ( F @ B2 ) ) )
% 0.34/0.67 => ( ! [A4: int] :
% 0.34/0.67 ( ( member_int @ A4 @ A )
% 0.34/0.67 => ( ord_less_eq_rat @ zero_zero_rat @ ( F @ A4 ) ) )
% 0.34/0.67 => ( ord_less_eq_rat @ ( groups1072433553688619179nt_rat @ F @ A ) @ ( groups1072433553688619179nt_rat @ F @ B4 ) ) ) ) ) ) ).
% 0.34/0.67
% 0.34/0.67 % prod_mono2
% 0.34/0.67 thf(fact_6710_prod__mono2,axiom,
% 0.34/0.67 ! [B4: set_o,A: set_o,F: $o > rat] :
% 0.34/0.67 ( ( finite_finite_o @ B4 )
% 0.34/0.67 => ( ( ord_less_eq_set_o @ A @ B4 )
% 0.34/0.67 => ( ! [B2: $o] :
% 0.34/0.67 ( ( member_o @ B2 @ ( minus_minus_set_o @ B4 @ A ) )
% 0.34/0.67 => ( ord_less_eq_rat @ one_one_rat @ ( F @ B2 ) ) )
% 0.34/0.67 => ( ! [A4: $o] :
% 0.34/0.67 ( ( member_o @ A4 @ A )
% 0.34/0.67 => ( ord_less_eq_rat @ zero_zero_rat @ ( F @ A4 ) ) )
% 0.34/0.67 => ( ord_less_eq_rat @ ( groups2869687844427037835_o_rat @ F @ A ) @ ( groups2869687844427037835_o_rat @ F @ B4 ) ) ) ) ) ) ).
% 0.34/0.67
% 0.34/0.67 % prod_mono2
% 0.34/0.67 thf(fact_6711_prod__mono2,axiom,
% 0.34/0.67 ! [B4: set_Product_unit,A: set_Product_unit,F: product_unit > rat] :
% 0.34/0.67 ( ( finite4290736615968046902t_unit @ B4 )
% 0.34/0.67 => ( ( ord_le3507040750410214029t_unit @ A @ B4 )
% 0.34/0.67 => ( ! [B2: product_unit] :
% 0.34/0.67 ( ( member_Product_unit @ B2 @ ( minus_6452836326544984404t_unit @ B4 @ A ) )
% 0.34/0.67 => ( ord_less_eq_rat @ one_one_rat @ ( F @ B2 ) ) )
% 0.34/0.67 => ( ! [A4: product_unit] :
% 0.34/0.67 ( ( member_Product_unit @ A4 @ A )
% 0.34/0.67 => ( ord_less_eq_rat @ zero_zero_rat @ ( F @ A4 ) ) )
% 0.34/0.67 => ( ord_less_eq_rat @ ( groups8984663225794451456it_rat @ F @ A ) @ ( groups8984663225794451456it_rat @ F @ B4 ) ) ) ) ) ) ).
% 0.34/0.67
% 0.34/0.67 % prod_mono2
% 0.34/0.67 thf(fact_6712_prod__mono2,axiom,
% 0.34/0.67 ! [B4: set_Code_integer,A: set_Code_integer,F: code_integer > rat] :
% 0.34/0.67 ( ( finite6017078050557962740nteger @ B4 )
% 0.34/0.67 => ( ( ord_le7084787975880047091nteger @ A @ B4 )
% 0.34/0.67 => ( ! [B2: code_integer] :
% 0.34/0.67 ( ( member_Code_integer @ B2 @ ( minus_2355218937544613996nteger @ B4 @ A ) )
% 0.34/0.67 => ( ord_less_eq_rat @ one_one_rat @ ( F @ B2 ) ) )
% 0.34/0.67 => ( ! [A4: code_integer] :
% 0.34/0.67 ( ( member_Code_integer @ A4 @ A )
% 0.34/0.67 => ( ord_less_eq_rat @ zero_zero_rat @ ( F @ A4 ) ) )
% 0.34/0.67 => ( ord_less_eq_rat @ ( groups2555765274223993564er_rat @ F @ A ) @ ( groups2555765274223993564er_rat @ F @ B4 ) ) ) ) ) ) ).
% 0.34/0.67
% 0.34/0.67 % prod_mono2
% 0.34/0.67 thf(fact_6713_prod__mono2,axiom,
% 0.34/0.67 ! [B4: set_int,A: set_int,F: int > int] :
% 0.34/0.67 ( ( finite_finite_int @ B4 )
% 0.34/0.67 => ( ( ord_less_eq_set_int @ A @ B4 )
% 0.34/0.67 => ( ! [B2: int] :
% 0.34/0.67 ( ( member_int @ B2 @ ( minus_minus_set_int @ B4 @ A ) )
% 0.34/0.67 => ( ord_less_eq_int @ one_one_int @ ( F @ B2 ) ) )
% 0.34/0.67 => ( ! [A4: int] :
% 0.34/0.67 ( ( member_int @ A4 @ A )
% 0.34/0.67 => ( ord_less_eq_int @ zero_zero_int @ ( F @ A4 ) ) )
% 0.34/0.67 => ( ord_less_eq_int @ ( groups1705073143266064639nt_int @ F @ A ) @ ( groups1705073143266064639nt_int @ F @ B4 ) ) ) ) ) ) ).
% 0.34/0.67
% 0.34/0.67 % prod_mono2
% 0.34/0.67 thf(fact_6714_prod__mono2,axiom,
% 0.34/0.67 ! [B4: set_o,A: set_o,F: $o > int] :
% 0.34/0.67 ( ( finite_finite_o @ B4 )
% 0.34/0.67 => ( ( ord_less_eq_set_o @ A @ B4 )
% 0.34/0.67 => ( ! [B2: $o] :
% 0.34/0.67 ( ( member_o @ B2 @ ( minus_minus_set_o @ B4 @ A ) )
% 0.34/0.67 => ( ord_less_eq_int @ one_one_int @ ( F @ B2 ) ) )
% 0.34/0.67 => ( ! [A4: $o] :
% 0.34/0.67 ( ( member_o @ A4 @ A )
% 0.34/0.67 => ( ord_less_eq_int @ zero_zero_int @ ( F @ A4 ) ) )
% 0.34/0.67 => ( ord_less_eq_int @ ( groups3502327434004483295_o_int @ F @ A ) @ ( groups3502327434004483295_o_int @ F @ B4 ) ) ) ) ) ) ).
% 0.34/0.67
% 0.34/0.67 % prod_mono2
% 0.34/0.67 thf(fact_6715_neg__equal__iff__equal,axiom,
% 0.34/0.67 ! [A2: int,B: int] :
% 0.34/0.67 ( ( ( uminus_uminus_int @ A2 )
% 0.34/0.67 = ( uminus_uminus_int @ B ) )
% 0.34/0.67 = ( A2 = B ) ) ).
% 0.34/0.67
% 0.34/0.67 % neg_equal_iff_equal
% 0.34/0.67 thf(fact_6716_neg__equal__iff__equal,axiom,
% 0.34/0.67 ! [A2: code_integer,B: code_integer] :
% 0.34/0.67 ( ( ( uminus1351360451143612070nteger @ A2 )
% 0.34/0.67 = ( uminus1351360451143612070nteger @ B ) )
% 0.34/0.67 = ( A2 = B ) ) ).
% 0.34/0.67
% 0.34/0.67 % neg_equal_iff_equal
% 0.34/0.67 thf(fact_6717_add_Oinverse__inverse,axiom,
% 0.34/0.67 ! [A2: int] :
% 0.34/0.67 ( ( uminus_uminus_int @ ( uminus_uminus_int @ A2 ) )
% 0.34/0.67 = A2 ) ).
% 0.34/0.67
% 0.34/0.67 % add.inverse_inverse
% 0.34/0.67 thf(fact_6718_add_Oinverse__inverse,axiom,
% 0.34/0.67 ! [A2: code_integer] :
% 0.34/0.67 ( ( uminus1351360451143612070nteger @ ( uminus1351360451143612070nteger @ A2 ) )
% 0.34/0.67 = A2 ) ).
% 0.34/0.67
% 0.34/0.67 % add.inverse_inverse
% 0.34/0.67 thf(fact_6719_Compl__subset__Compl__iff,axiom,
% 0.34/0.67 ! [A: set_nat,B4: set_nat] :
% 0.34/0.67 ( ( ord_less_eq_set_nat @ ( uminus5710092332889474511et_nat @ A ) @ ( uminus5710092332889474511et_nat @ B4 ) )
% 0.34/0.67 = ( ord_less_eq_set_nat @ B4 @ A ) ) ).
% 0.34/0.67
% 0.34/0.67 % Compl_subset_Compl_iff
% 0.34/0.67 thf(fact_6720_Compl__anti__mono,axiom,
% 0.34/0.67 ! [A: set_nat,B4: set_nat] :
% 0.34/0.67 ( ( ord_less_eq_set_nat @ A @ B4 )
% 0.34/0.67 => ( ord_less_eq_set_nat @ ( uminus5710092332889474511et_nat @ B4 ) @ ( uminus5710092332889474511et_nat @ A ) ) ) ).
% 0.34/0.67
% 0.34/0.67 % Compl_anti_mono
% 0.34/0.67 thf(fact_6721_neg__le__iff__le,axiom,
% 0.34/0.67 ! [B: code_integer,A2: code_integer] :
% 0.34/0.67 ( ( ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ B ) @ ( uminus1351360451143612070nteger @ A2 ) )
% 0.34/0.67 = ( ord_le3102999989581377725nteger @ A2 @ B ) ) ).
% 0.34/0.67
% 0.34/0.67 % neg_le_iff_le
% 0.34/0.67 thf(fact_6722_neg__le__iff__le,axiom,
% 0.34/0.67 ! [B: rat,A2: rat] :
% 0.34/0.67 ( ( ord_less_eq_rat @ ( uminus_uminus_rat @ B ) @ ( uminus_uminus_rat @ A2 ) )
% 0.34/0.67 = ( ord_less_eq_rat @ A2 @ B ) ) ).
% 0.34/0.67
% 0.34/0.67 % neg_le_iff_le
% 0.34/0.67 thf(fact_6723_neg__le__iff__le,axiom,
% 0.34/0.67 ! [B: int,A2: int] :
% 0.34/0.67 ( ( ord_less_eq_int @ ( uminus_uminus_int @ B ) @ ( uminus_uminus_int @ A2 ) )
% 0.34/0.67 = ( ord_less_eq_int @ A2 @ B ) ) ).
% 0.34/0.67
% 0.34/0.67 % neg_le_iff_le
% 0.34/0.67 thf(fact_6724_neg__equal__zero,axiom,
% 0.34/0.67 ! [A2: int] :
% 0.34/0.67 ( ( ( uminus_uminus_int @ A2 )
% 0.34/0.67 = A2 )
% 0.34/0.67 = ( A2 = zero_zero_int ) ) ).
% 0.34/0.67
% 0.34/0.67 % neg_equal_zero
% 0.34/0.67 thf(fact_6725_neg__equal__zero,axiom,
% 0.34/0.67 ! [A2: code_integer] :
% 0.34/0.67 ( ( ( uminus1351360451143612070nteger @ A2 )
% 0.34/0.67 = A2 )
% 0.34/0.67 = ( A2 = zero_z3403309356797280102nteger ) ) ).
% 0.34/0.67
% 0.34/0.67 % neg_equal_zero
% 0.34/0.67 thf(fact_6726_equal__neg__zero,axiom,
% 0.34/0.67 ! [A2: int] :
% 0.34/0.67 ( ( A2
% 0.34/0.67 = ( uminus_uminus_int @ A2 ) )
% 0.34/0.67 = ( A2 = zero_zero_int ) ) ).
% 0.34/0.67
% 0.34/0.67 % equal_neg_zero
% 0.34/0.67 thf(fact_6727_equal__neg__zero,axiom,
% 0.34/0.67 ! [A2: code_integer] :
% 0.34/0.67 ( ( A2
% 0.34/0.67 = ( uminus1351360451143612070nteger @ A2 ) )
% 0.34/0.67 = ( A2 = zero_z3403309356797280102nteger ) ) ).
% 0.34/0.67
% 0.34/0.67 % equal_neg_zero
% 0.34/0.67 thf(fact_6728_neg__equal__0__iff__equal,axiom,
% 0.34/0.67 ! [A2: int] :
% 0.34/0.67 ( ( ( uminus_uminus_int @ A2 )
% 0.34/0.67 = zero_zero_int )
% 0.34/0.67 = ( A2 = zero_zero_int ) ) ).
% 0.34/0.67
% 0.34/0.67 % neg_equal_0_iff_equal
% 0.34/0.67 thf(fact_6729_neg__equal__0__iff__equal,axiom,
% 0.34/0.67 ! [A2: code_integer] :
% 0.34/0.67 ( ( ( uminus1351360451143612070nteger @ A2 )
% 0.34/0.67 = zero_z3403309356797280102nteger )
% 0.34/0.67 = ( A2 = zero_z3403309356797280102nteger ) ) ).
% 0.34/0.67
% 0.34/0.67 % neg_equal_0_iff_equal
% 0.34/0.67 thf(fact_6730_neg__0__equal__iff__equal,axiom,
% 0.34/0.67 ! [A2: int] :
% 0.34/0.67 ( ( zero_zero_int
% 0.34/0.67 = ( uminus_uminus_int @ A2 ) )
% 0.34/0.67 = ( zero_zero_int = A2 ) ) ).
% 0.34/0.67
% 0.34/0.67 % neg_0_equal_iff_equal
% 0.34/0.67 thf(fact_6731_neg__0__equal__iff__equal,axiom,
% 0.34/0.67 ! [A2: code_integer] :
% 0.34/0.67 ( ( zero_z3403309356797280102nteger
% 0.34/0.67 = ( uminus1351360451143612070nteger @ A2 ) )
% 0.34/0.67 = ( zero_z3403309356797280102nteger = A2 ) ) ).
% 0.34/0.67
% 0.34/0.67 % neg_0_equal_iff_equal
% 0.34/0.67 thf(fact_6732_add_Oinverse__neutral,axiom,
% 0.34/0.67 ( ( uminus_uminus_int @ zero_zero_int )
% 0.34/0.67 = zero_zero_int ) ).
% 0.34/0.67
% 0.34/0.67 % add.inverse_neutral
% 0.34/0.67 thf(fact_6733_add_Oinverse__neutral,axiom,
% 0.34/0.67 ( ( uminus1351360451143612070nteger @ zero_z3403309356797280102nteger )
% 0.34/0.67 = zero_z3403309356797280102nteger ) ).
% 0.34/0.67
% 0.34/0.67 % add.inverse_neutral
% 0.34/0.67 thf(fact_6734_neg__less__iff__less,axiom,
% 0.34/0.67 ! [B: rat,A2: rat] :
% 0.34/0.67 ( ( ord_less_rat @ ( uminus_uminus_rat @ B ) @ ( uminus_uminus_rat @ A2 ) )
% 0.34/0.67 = ( ord_less_rat @ A2 @ B ) ) ).
% 0.34/0.67
% 0.34/0.67 % neg_less_iff_less
% 0.34/0.67 thf(fact_6735_neg__less__iff__less,axiom,
% 0.34/0.67 ! [B: int,A2: int] :
% 0.34/0.67 ( ( ord_less_int @ ( uminus_uminus_int @ B ) @ ( uminus_uminus_int @ A2 ) )
% 0.34/0.67 = ( ord_less_int @ A2 @ B ) ) ).
% 0.34/0.67
% 0.34/0.67 % neg_less_iff_less
% 0.34/0.67 thf(fact_6736_neg__less__iff__less,axiom,
% 0.34/0.67 ! [B: code_integer,A2: code_integer] :
% 0.34/0.67 ( ( ord_le6747313008572928689nteger @ ( uminus1351360451143612070nteger @ B ) @ ( uminus1351360451143612070nteger @ A2 ) )
% 0.34/0.67 = ( ord_le6747313008572928689nteger @ A2 @ B ) ) ).
% 0.34/0.67
% 0.34/0.67 % neg_less_iff_less
% 0.34/0.67 thf(fact_6737_minus__add__distrib,axiom,
% 0.34/0.67 ! [A2: rat,B: rat] :
% 0.34/0.67 ( ( uminus_uminus_rat @ ( plus_plus_rat @ A2 @ B ) )
% 0.34/0.67 = ( plus_plus_rat @ ( uminus_uminus_rat @ A2 ) @ ( uminus_uminus_rat @ B ) ) ) ).
% 0.34/0.67
% 0.34/0.67 % minus_add_distrib
% 0.34/0.67 thf(fact_6738_minus__add__distrib,axiom,
% 0.34/0.67 ! [A2: int,B: int] :
% 0.34/0.67 ( ( uminus_uminus_int @ ( plus_plus_int @ A2 @ B ) )
% 0.34/0.67 = ( plus_plus_int @ ( uminus_uminus_int @ A2 ) @ ( uminus_uminus_int @ B ) ) ) ).
% 0.34/0.67
% 0.34/0.67 % minus_add_distrib
% 0.34/0.67 thf(fact_6739_minus__add__distrib,axiom,
% 0.34/0.67 ! [A2: code_integer,B: code_integer] :
% 0.34/0.67 ( ( uminus1351360451143612070nteger @ ( plus_p5714425477246183910nteger @ A2 @ B ) )
% 0.34/0.67 = ( plus_p5714425477246183910nteger @ ( uminus1351360451143612070nteger @ A2 ) @ ( uminus1351360451143612070nteger @ B ) ) ) ).
% 0.34/0.67
% 0.34/0.67 % minus_add_distrib
% 0.34/0.67 thf(fact_6740_minus__add__cancel,axiom,
% 0.34/0.67 ! [A2: rat,B: rat] :
% 0.34/0.67 ( ( plus_plus_rat @ ( uminus_uminus_rat @ A2 ) @ ( plus_plus_rat @ A2 @ B ) )
% 0.34/0.67 = B ) ).
% 0.34/0.67
% 0.34/0.67 % minus_add_cancel
% 0.34/0.67 thf(fact_6741_minus__add__cancel,axiom,
% 0.34/0.67 ! [A2: int,B: int] :
% 0.34/0.67 ( ( plus_plus_int @ ( uminus_uminus_int @ A2 ) @ ( plus_plus_int @ A2 @ B ) )
% 0.34/0.67 = B ) ).
% 0.34/0.67
% 0.34/0.67 % minus_add_cancel
% 0.34/0.67 thf(fact_6742_minus__add__cancel,axiom,
% 0.34/0.67 ! [A2: code_integer,B: code_integer] :
% 0.34/0.67 ( ( plus_p5714425477246183910nteger @ ( uminus1351360451143612070nteger @ A2 ) @ ( plus_p5714425477246183910nteger @ A2 @ B ) )
% 0.34/0.67 = B ) ).
% 0.34/0.67
% 0.34/0.67 % minus_add_cancel
% 0.34/0.67 thf(fact_6743_add__minus__cancel,axiom,
% 0.34/0.67 ! [A2: rat,B: rat] :
% 0.34/0.67 ( ( plus_plus_rat @ A2 @ ( plus_plus_rat @ ( uminus_uminus_rat @ A2 ) @ B ) )
% 0.34/0.67 = B ) ).
% 0.34/0.67
% 0.34/0.67 % add_minus_cancel
% 0.34/0.67 thf(fact_6744_add__minus__cancel,axiom,
% 0.34/0.67 ! [A2: int,B: int] :
% 0.34/0.67 ( ( plus_plus_int @ A2 @ ( plus_plus_int @ ( uminus_uminus_int @ A2 ) @ B ) )
% 0.34/0.67 = B ) ).
% 0.34/0.67
% 0.34/0.67 % add_minus_cancel
% 0.34/0.67 thf(fact_6745_add__minus__cancel,axiom,
% 0.34/0.67 ! [A2: code_integer,B: code_integer] :
% 0.34/0.67 ( ( plus_p5714425477246183910nteger @ A2 @ ( plus_p5714425477246183910nteger @ ( uminus1351360451143612070nteger @ A2 ) @ B ) )
% 0.34/0.67 = B ) ).
% 0.34/0.67
% 0.34/0.67 % add_minus_cancel
% 0.34/0.67 thf(fact_6746_mult__minus__left,axiom,
% 0.34/0.67 ! [A2: rat,B: rat] :
% 0.34/0.67 ( ( times_times_rat @ ( uminus_uminus_rat @ A2 ) @ B )
% 0.34/0.67 = ( uminus_uminus_rat @ ( times_times_rat @ A2 @ B ) ) ) ).
% 0.34/0.67
% 0.34/0.67 % mult_minus_left
% 0.34/0.67 thf(fact_6747_mult__minus__left,axiom,
% 0.34/0.67 ! [A2: int,B: int] :
% 0.34/0.67 ( ( times_times_int @ ( uminus_uminus_int @ A2 ) @ B )
% 0.34/0.67 = ( uminus_uminus_int @ ( times_times_int @ A2 @ B ) ) ) ).
% 0.34/0.67
% 0.34/0.67 % mult_minus_left
% 0.34/0.67 thf(fact_6748_mult__minus__left,axiom,
% 0.34/0.67 ! [A2: code_integer,B: code_integer] :
% 0.34/0.67 ( ( times_3573771949741848930nteger @ ( uminus1351360451143612070nteger @ A2 ) @ B )
% 0.34/0.67 = ( uminus1351360451143612070nteger @ ( times_3573771949741848930nteger @ A2 @ B ) ) ) ).
% 0.34/0.67
% 0.34/0.67 % mult_minus_left
% 0.34/0.67 thf(fact_6749_minus__mult__minus,axiom,
% 0.34/0.67 ! [A2: rat,B: rat] :
% 0.34/0.67 ( ( times_times_rat @ ( uminus_uminus_rat @ A2 ) @ ( uminus_uminus_rat @ B ) )
% 0.34/0.67 = ( times_times_rat @ A2 @ B ) ) ).
% 0.34/0.67
% 0.34/0.67 % minus_mult_minus
% 0.34/0.67 thf(fact_6750_minus__mult__minus,axiom,
% 0.34/0.67 ! [A2: int,B: int] :
% 0.34/0.67 ( ( times_times_int @ ( uminus_uminus_int @ A2 ) @ ( uminus_uminus_int @ B ) )
% 0.34/0.67 = ( times_times_int @ A2 @ B ) ) ).
% 0.34/0.67
% 0.34/0.67 % minus_mult_minus
% 0.34/0.67 thf(fact_6751_minus__mult__minus,axiom,
% 0.34/0.67 ! [A2: code_integer,B: code_integer] :
% 0.34/0.67 ( ( times_3573771949741848930nteger @ ( uminus1351360451143612070nteger @ A2 ) @ ( uminus1351360451143612070nteger @ B ) )
% 0.34/0.67 = ( times_3573771949741848930nteger @ A2 @ B ) ) ).
% 0.34/0.67
% 0.34/0.67 % minus_mult_minus
% 0.34/0.67 thf(fact_6752_mult__minus__right,axiom,
% 0.34/0.67 ! [A2: rat,B: rat] :
% 0.34/0.67 ( ( times_times_rat @ A2 @ ( uminus_uminus_rat @ B ) )
% 0.34/0.67 = ( uminus_uminus_rat @ ( times_times_rat @ A2 @ B ) ) ) ).
% 0.34/0.67
% 0.34/0.67 % mult_minus_right
% 0.34/0.67 thf(fact_6753_mult__minus__right,axiom,
% 0.34/0.67 ! [A2: int,B: int] :
% 0.34/0.67 ( ( times_times_int @ A2 @ ( uminus_uminus_int @ B ) )
% 0.34/0.67 = ( uminus_uminus_int @ ( times_times_int @ A2 @ B ) ) ) ).
% 0.34/0.67
% 0.34/0.67 % mult_minus_right
% 0.34/0.67 thf(fact_6754_mult__minus__right,axiom,
% 0.34/0.67 ! [A2: code_integer,B: code_integer] :
% 0.34/0.67 ( ( times_3573771949741848930nteger @ A2 @ ( uminus1351360451143612070nteger @ B ) )
% 0.34/0.67 = ( uminus1351360451143612070nteger @ ( times_3573771949741848930nteger @ A2 @ B ) ) ) ).
% 0.34/0.67
% 0.34/0.67 % mult_minus_right
% 0.34/0.67 thf(fact_6755_minus__diff__eq,axiom,
% 0.34/0.67 ! [A2: rat,B: rat] :
% 0.34/0.67 ( ( uminus_uminus_rat @ ( minus_minus_rat @ A2 @ B ) )
% 0.34/0.67 = ( minus_minus_rat @ B @ A2 ) ) ).
% 0.34/0.67
% 0.34/0.67 % minus_diff_eq
% 0.34/0.67 thf(fact_6756_minus__diff__eq,axiom,
% 0.34/0.67 ! [A2: int,B: int] :
% 0.34/0.67 ( ( uminus_uminus_int @ ( minus_minus_int @ A2 @ B ) )
% 0.34/0.67 = ( minus_minus_int @ B @ A2 ) ) ).
% 0.34/0.67
% 0.34/0.67 % minus_diff_eq
% 0.34/0.67 thf(fact_6757_minus__diff__eq,axiom,
% 0.34/0.67 ! [A2: code_integer,B: code_integer] :
% 0.34/0.67 ( ( uminus1351360451143612070nteger @ ( minus_8373710615458151222nteger @ A2 @ B ) )
% 0.34/0.67 = ( minus_8373710615458151222nteger @ B @ A2 ) ) ).
% 0.34/0.67
% 0.34/0.67 % minus_diff_eq
% 0.34/0.67 thf(fact_6758_Compl__disjoint,axiom,
% 0.34/0.67 ! [A: set_Product_unit] :
% 0.34/0.67 ( ( inf_in4660618365625256667t_unit @ A @ ( uminus5944136376168626660t_unit @ A ) )
% 0.34/0.67 = bot_bo3957492148770167129t_unit ) ).
% 0.34/0.67
% 0.34/0.67 % Compl_disjoint
% 0.34/0.67 thf(fact_6759_Compl__disjoint,axiom,
% 0.34/0.67 ! [A: set_o] :
% 0.34/0.67 ( ( inf_inf_set_o @ A @ ( uminus_uminus_set_o @ A ) )
% 0.34/0.67 = bot_bot_set_o ) ).
% 0.34/0.67
% 0.34/0.67 % Compl_disjoint
% 0.34/0.67 thf(fact_6760_Compl__disjoint,axiom,
% 0.34/0.67 ! [A: set_nat] :
% 0.34/0.67 ( ( inf_inf_set_nat @ A @ ( uminus5710092332889474511et_nat @ A ) )
% 0.34/0.67 = bot_bot_set_nat ) ).
% 0.34/0.67
% 0.34/0.67 % Compl_disjoint
% 0.34/0.67 thf(fact_6761_Compl__disjoint,axiom,
% 0.34/0.67 ! [A: set_int] :
% 0.34/0.67 ( ( inf_inf_set_int @ A @ ( uminus1532241313380277803et_int @ A ) )
% 0.34/0.67 = bot_bot_set_int ) ).
% 0.34/0.67
% 0.34/0.67 % Compl_disjoint
% 0.34/0.67 thf(fact_6762_Compl__disjoint,axiom,
% 0.34/0.67 ! [A: set_Pr4532377907799695533_nat_o] :
% 0.34/0.67 ( ( inf_in1906310914598751387_nat_o @ A @ ( uminus5254974436814262692_nat_o @ A ) )
% 0.34/0.67 = bot_bo7824918357723582233_nat_o ) ).
% 0.34/0.67
% 0.34/0.67 % Compl_disjoint
% 0.34/0.67 thf(fact_6763_Compl__disjoint2,axiom,
% 0.34/0.67 ! [A: set_Product_unit] :
% 0.34/0.67 ( ( inf_in4660618365625256667t_unit @ ( uminus5944136376168626660t_unit @ A ) @ A )
% 0.34/0.67 = bot_bo3957492148770167129t_unit ) ).
% 0.34/0.67
% 0.34/0.67 % Compl_disjoint2
% 0.34/0.67 thf(fact_6764_Compl__disjoint2,axiom,
% 0.34/0.67 ! [A: set_o] :
% 0.34/0.67 ( ( inf_inf_set_o @ ( uminus_uminus_set_o @ A ) @ A )
% 0.34/0.67 = bot_bot_set_o ) ).
% 0.34/0.67
% 0.34/0.67 % Compl_disjoint2
% 0.34/0.67 thf(fact_6765_Compl__disjoint2,axiom,
% 0.34/0.67 ! [A: set_nat] :
% 0.34/0.67 ( ( inf_inf_set_nat @ ( uminus5710092332889474511et_nat @ A ) @ A )
% 0.34/0.67 = bot_bot_set_nat ) ).
% 0.34/0.67
% 0.34/0.67 % Compl_disjoint2
% 0.34/0.67 thf(fact_6766_Compl__disjoint2,axiom,
% 0.34/0.67 ! [A: set_int] :
% 0.34/0.67 ( ( inf_inf_set_int @ ( uminus1532241313380277803et_int @ A ) @ A )
% 0.34/0.67 = bot_bot_set_int ) ).
% 0.34/0.67
% 0.34/0.67 % Compl_disjoint2
% 0.34/0.67 thf(fact_6767_Compl__disjoint2,axiom,
% 0.34/0.67 ! [A: set_Pr4532377907799695533_nat_o] :
% 0.34/0.67 ( ( inf_in1906310914598751387_nat_o @ ( uminus5254974436814262692_nat_o @ A ) @ A )
% 0.34/0.67 = bot_bo7824918357723582233_nat_o ) ).
% 0.34/0.67
% 0.34/0.67 % Compl_disjoint2
% 0.34/0.67 thf(fact_6768_inter__compl__diff__conv,axiom,
% 0.34/0.67 ! [A: set_nat,B4: set_nat] :
% 0.34/0.67 ( ( inf_inf_set_nat @ A @ ( uminus5710092332889474511et_nat @ B4 ) )
% 0.34/0.67 = ( minus_minus_set_nat @ A @ B4 ) ) ).
% 0.34/0.67
% 0.34/0.67 % inter_compl_diff_conv
% 0.34/0.67 thf(fact_6769_Diff__Compl,axiom,
% 0.34/0.67 ! [A: set_nat,B4: set_nat] :
% 0.34/0.67 ( ( minus_minus_set_nat @ A @ ( uminus5710092332889474511et_nat @ B4 ) )
% 0.34/0.67 = ( inf_inf_set_nat @ A @ B4 ) ) ).
% 0.34/0.67
% 0.34/0.67 % Diff_Compl
% 0.34/0.67 thf(fact_6770_Compl__Diff__eq,axiom,
% 0.34/0.67 ! [A: set_nat,B4: set_nat] :
% 0.34/0.67 ( ( uminus5710092332889474511et_nat @ ( minus_minus_set_nat @ A @ B4 ) )
% 0.34/0.67 = ( sup_sup_set_nat @ ( uminus5710092332889474511et_nat @ A ) @ B4 ) ) ).
% 0.34/0.67
% 0.34/0.67 % Compl_Diff_eq
% 0.34/0.67 thf(fact_6771_neg__0__le__iff__le,axiom,
% 0.34/0.67 ! [A2: code_integer] :
% 0.34/0.67 ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ ( uminus1351360451143612070nteger @ A2 ) )
% 0.34/0.67 = ( ord_le3102999989581377725nteger @ A2 @ zero_z3403309356797280102nteger ) ) ).
% 0.34/0.67
% 0.34/0.67 % neg_0_le_iff_le
% 0.34/0.67 thf(fact_6772_neg__0__le__iff__le,axiom,
% 0.34/0.67 ! [A2: rat] :
% 0.34/0.67 ( ( ord_less_eq_rat @ zero_zero_rat @ ( uminus_uminus_rat @ A2 ) )
% 0.34/0.67 = ( ord_less_eq_rat @ A2 @ zero_zero_rat ) ) ).
% 0.34/0.67
% 0.34/0.67 % neg_0_le_iff_le
% 0.34/0.67 thf(fact_6773_neg__0__le__iff__le,axiom,
% 0.34/0.67 ! [A2: int] :
% 0.34/0.67 ( ( ord_less_eq_int @ zero_zero_int @ ( uminus_uminus_int @ A2 ) )
% 0.34/0.67 = ( ord_less_eq_int @ A2 @ zero_zero_int ) ) ).
% 0.34/0.67
% 0.34/0.67 % neg_0_le_iff_le
% 0.34/0.67 thf(fact_6774_neg__le__0__iff__le,axiom,
% 0.34/0.67 ! [A2: code_integer] :
% 0.34/0.67 ( ( ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ A2 ) @ zero_z3403309356797280102nteger )
% 0.34/0.67 = ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ A2 ) ) ).
% 0.34/0.67
% 0.34/0.67 % neg_le_0_iff_le
% 0.34/0.67 thf(fact_6775_neg__le__0__iff__le,axiom,
% 0.34/0.67 ! [A2: rat] :
% 0.34/0.67 ( ( ord_less_eq_rat @ ( uminus_uminus_rat @ A2 ) @ zero_zero_rat )
% 0.34/0.67 = ( ord_less_eq_rat @ zero_zero_rat @ A2 ) ) ).
% 0.34/0.67
% 0.34/0.67 % neg_le_0_iff_le
% 0.34/0.67 thf(fact_6776_neg__le__0__iff__le,axiom,
% 0.34/0.67 ! [A2: int] :
% 0.34/0.67 ( ( ord_less_eq_int @ ( uminus_uminus_int @ A2 ) @ zero_zero_int )
% 0.34/0.67 = ( ord_less_eq_int @ zero_zero_int @ A2 ) ) ).
% 0.34/0.67
% 0.34/0.67 % neg_le_0_iff_le
% 0.34/0.67 thf(fact_6777_less__eq__neg__nonpos,axiom,
% 0.34/0.67 ! [A2: code_integer] :
% 0.34/0.67 ( ( ord_le3102999989581377725nteger @ A2 @ ( uminus1351360451143612070nteger @ A2 ) )
% 0.34/0.67 = ( ord_le3102999989581377725nteger @ A2 @ zero_z3403309356797280102nteger ) ) ).
% 0.34/0.67
% 0.34/0.67 % less_eq_neg_nonpos
% 0.34/0.67 thf(fact_6778_less__eq__neg__nonpos,axiom,
% 0.34/0.67 ! [A2: rat] :
% 0.34/0.67 ( ( ord_less_eq_rat @ A2 @ ( uminus_uminus_rat @ A2 ) )
% 0.34/0.67 = ( ord_less_eq_rat @ A2 @ zero_zero_rat ) ) ).
% 0.34/0.67
% 0.34/0.67 % less_eq_neg_nonpos
% 0.34/0.67 thf(fact_6779_less__eq__neg__nonpos,axiom,
% 0.34/0.67 ! [A2: int] :
% 0.34/0.67 ( ( ord_less_eq_int @ A2 @ ( uminus_uminus_int @ A2 ) )
% 0.34/0.67 = ( ord_less_eq_int @ A2 @ zero_zero_int ) ) ).
% 0.34/0.67
% 0.34/0.67 % less_eq_neg_nonpos
% 0.34/0.67 thf(fact_6780_neg__less__eq__nonneg,axiom,
% 0.34/0.67 ! [A2: code_integer] :
% 0.34/0.67 ( ( ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ A2 ) @ A2 )
% 0.34/0.67 = ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ A2 ) ) ).
% 0.34/0.67
% 0.34/0.67 % neg_less_eq_nonneg
% 0.34/0.67 thf(fact_6781_neg__less__eq__nonneg,axiom,
% 0.34/0.67 ! [A2: rat] :
% 0.34/0.67 ( ( ord_less_eq_rat @ ( uminus_uminus_rat @ A2 ) @ A2 )
% 0.34/0.67 = ( ord_less_eq_rat @ zero_zero_rat @ A2 ) ) ).
% 0.34/0.67
% 0.34/0.67 % neg_less_eq_nonneg
% 0.34/0.67 thf(fact_6782_neg__less__eq__nonneg,axiom,
% 0.34/0.67 ! [A2: int] :
% 0.34/0.67 ( ( ord_less_eq_int @ ( uminus_uminus_int @ A2 ) @ A2 )
% 0.34/0.67 = ( ord_less_eq_int @ zero_zero_int @ A2 ) ) ).
% 0.34/0.67
% 0.34/0.67 % neg_less_eq_nonneg
% 0.34/0.67 thf(fact_6783_less__neg__neg,axiom,
% 0.34/0.67 ! [A2: rat] :
% 0.34/0.67 ( ( ord_less_rat @ A2 @ ( uminus_uminus_rat @ A2 ) )
% 0.34/0.67 = ( ord_less_rat @ A2 @ zero_zero_rat ) ) ).
% 0.34/0.67
% 0.34/0.67 % less_neg_neg
% 0.34/0.67 thf(fact_6784_less__neg__neg,axiom,
% 0.34/0.67 ! [A2: int] :
% 0.34/0.67 ( ( ord_less_int @ A2 @ ( uminus_uminus_int @ A2 ) )
% 0.34/0.67 = ( ord_less_int @ A2 @ zero_zero_int ) ) ).
% 0.34/0.67
% 0.34/0.67 % less_neg_neg
% 0.34/0.67 thf(fact_6785_less__neg__neg,axiom,
% 0.34/0.67 ! [A2: code_integer] :
% 0.34/0.67 ( ( ord_le6747313008572928689nteger @ A2 @ ( uminus1351360451143612070nteger @ A2 ) )
% 0.34/0.67 = ( ord_le6747313008572928689nteger @ A2 @ zero_z3403309356797280102nteger ) ) ).
% 0.34/0.67
% 0.34/0.67 % less_neg_neg
% 0.34/0.67 thf(fact_6786_neg__less__pos,axiom,
% 0.34/0.67 ! [A2: rat] :
% 0.34/0.67 ( ( ord_less_rat @ ( uminus_uminus_rat @ A2 ) @ A2 )
% 0.34/0.67 = ( ord_less_rat @ zero_zero_rat @ A2 ) ) ).
% 0.34/0.67
% 0.34/0.67 % neg_less_pos
% 0.34/0.67 thf(fact_6787_neg__less__pos,axiom,
% 0.34/0.67 ! [A2: int] :
% 0.34/0.67 ( ( ord_less_int @ ( uminus_uminus_int @ A2 ) @ A2 )
% 0.34/0.67 = ( ord_less_int @ zero_zero_int @ A2 ) ) ).
% 0.34/0.67
% 0.34/0.67 % neg_less_pos
% 0.34/0.67 thf(fact_6788_neg__less__pos,axiom,
% 0.34/0.67 ! [A2: code_integer] :
% 0.34/0.67 ( ( ord_le6747313008572928689nteger @ ( uminus1351360451143612070nteger @ A2 ) @ A2 )
% 0.34/0.67 = ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ A2 ) ) ).
% 0.34/0.67
% 0.34/0.67 % neg_less_pos
% 0.34/0.67 thf(fact_6789_neg__0__less__iff__less,axiom,
% 0.34/0.67 ! [A2: rat] :
% 0.34/0.67 ( ( ord_less_rat @ zero_zero_rat @ ( uminus_uminus_rat @ A2 ) )
% 0.34/0.67 = ( ord_less_rat @ A2 @ zero_zero_rat ) ) ).
% 0.34/0.67
% 0.34/0.67 % neg_0_less_iff_less
% 0.34/0.67 thf(fact_6790_neg__0__less__iff__less,axiom,
% 0.34/0.67 ! [A2: int] :
% 0.34/0.67 ( ( ord_less_int @ zero_zero_int @ ( uminus_uminus_int @ A2 ) )
% 0.34/0.67 = ( ord_less_int @ A2 @ zero_zero_int ) ) ).
% 0.34/0.67
% 0.34/0.67 % neg_0_less_iff_less
% 0.34/0.67 thf(fact_6791_neg__0__less__iff__less,axiom,
% 0.34/0.67 ! [A2: code_integer] :
% 0.34/0.67 ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ ( uminus1351360451143612070nteger @ A2 ) )
% 0.34/0.67 = ( ord_le6747313008572928689nteger @ A2 @ zero_z3403309356797280102nteger ) ) ).
% 0.34/0.67
% 0.34/0.67 % neg_0_less_iff_less
% 0.34/0.67 thf(fact_6792_neg__less__0__iff__less,axiom,
% 0.34/0.67 ! [A2: rat] :
% 0.34/0.67 ( ( ord_less_rat @ ( uminus_uminus_rat @ A2 ) @ zero_zero_rat )
% 0.34/0.67 = ( ord_less_rat @ zero_zero_rat @ A2 ) ) ).
% 0.34/0.67
% 0.34/0.67 % neg_less_0_iff_less
% 0.34/0.67 thf(fact_6793_neg__less__0__iff__less,axiom,
% 0.34/0.67 ! [A2: int] :
% 0.34/0.67 ( ( ord_less_int @ ( uminus_uminus_int @ A2 ) @ zero_zero_int )
% 0.34/0.67 = ( ord_less_int @ zero_zero_int @ A2 ) ) ).
% 0.34/0.67
% 0.34/0.67 % neg_less_0_iff_less
% 0.34/0.67 thf(fact_6794_neg__less__0__iff__less,axiom,
% 0.34/0.67 ! [A2: code_integer] :
% 0.34/0.67 ( ( ord_le6747313008572928689nteger @ ( uminus1351360451143612070nteger @ A2 ) @ zero_z3403309356797280102nteger )
% 0.34/0.67 = ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ A2 ) ) ).
% 0.34/0.67
% 0.34/0.67 % neg_less_0_iff_less
% 0.34/0.67 thf(fact_6795_ab__left__minus,axiom,
% 0.34/0.67 ! [A2: rat] :
% 0.34/0.67 ( ( plus_plus_rat @ ( uminus_uminus_rat @ A2 ) @ A2 )
% 0.34/0.67 = zero_zero_rat ) ).
% 0.34/0.67
% 0.34/0.67 % ab_left_minus
% 0.34/0.67 thf(fact_6796_ab__left__minus,axiom,
% 0.34/0.67 ! [A2: int] :
% 0.34/0.67 ( ( plus_plus_int @ ( uminus_uminus_int @ A2 ) @ A2 )
% 0.34/0.67 = zero_zero_int ) ).
% 0.34/0.67
% 0.34/0.67 % ab_left_minus
% 0.34/0.67 thf(fact_6797_ab__left__minus,axiom,
% 0.34/0.67 ! [A2: code_integer] :
% 0.34/0.67 ( ( plus_p5714425477246183910nteger @ ( uminus1351360451143612070nteger @ A2 ) @ A2 )
% 0.34/0.67 = zero_z3403309356797280102nteger ) ).
% 0.34/0.67
% 0.34/0.67 % ab_left_minus
% 0.34/0.67 thf(fact_6798_add_Oright__inverse,axiom,
% 0.34/0.67 ! [A2: rat] :
% 0.34/0.67 ( ( plus_plus_rat @ A2 @ ( uminus_uminus_rat @ A2 ) )
% 0.34/0.67 = zero_zero_rat ) ).
% 0.34/0.67
% 0.34/0.67 % add.right_inverse
% 0.34/0.67 thf(fact_6799_add_Oright__inverse,axiom,
% 0.34/0.67 ! [A2: int] :
% 0.34/0.67 ( ( plus_plus_int @ A2 @ ( uminus_uminus_int @ A2 ) )
% 0.34/0.67 = zero_zero_int ) ).
% 0.34/0.67
% 0.34/0.67 % add.right_inverse
% 0.34/0.67 thf(fact_6800_add_Oright__inverse,axiom,
% 0.34/0.67 ! [A2: code_integer] :
% 0.34/0.67 ( ( plus_p5714425477246183910nteger @ A2 @ ( uminus1351360451143612070nteger @ A2 ) )
% 0.34/0.67 = zero_z3403309356797280102nteger ) ).
% 0.34/0.67
% 0.34/0.67 % add.right_inverse
% 0.34/0.67 thf(fact_6801_diff__0,axiom,
% 0.34/0.67 ! [A2: rat] :
% 0.34/0.67 ( ( minus_minus_rat @ zero_zero_rat @ A2 )
% 0.34/0.67 = ( uminus_uminus_rat @ A2 ) ) ).
% 0.34/0.67
% 0.34/0.67 % diff_0
% 0.34/0.67 thf(fact_6802_diff__0,axiom,
% 0.34/0.67 ! [A2: int] :
% 0.34/0.67 ( ( minus_minus_int @ zero_zero_int @ A2 )
% 0.34/0.67 = ( uminus_uminus_int @ A2 ) ) ).
% 0.34/0.67
% 0.34/0.67 % diff_0
% 0.34/0.67 thf(fact_6803_diff__0,axiom,
% 0.34/0.67 ! [A2: code_integer] :
% 0.34/0.67 ( ( minus_8373710615458151222nteger @ zero_z3403309356797280102nteger @ A2 )
% 0.34/0.67 = ( uminus1351360451143612070nteger @ A2 ) ) ).
% 0.34/0.67
% 0.34/0.67 % diff_0
% 0.34/0.67 thf(fact_6804_diff__minus__eq__add,axiom,
% 0.34/0.67 ! [A2: rat,B: rat] :
% 0.34/0.67 ( ( minus_minus_rat @ A2 @ ( uminus_uminus_rat @ B ) )
% 0.34/0.67 = ( plus_plus_rat @ A2 @ B ) ) ).
% 0.34/0.67
% 0.34/0.67 % diff_minus_eq_add
% 0.34/0.67 thf(fact_6805_diff__minus__eq__add,axiom,
% 0.34/0.67 ! [A2: int,B: int] :
% 0.34/0.67 ( ( minus_minus_int @ A2 @ ( uminus_uminus_int @ B ) )
% 0.34/0.67 = ( plus_plus_int @ A2 @ B ) ) ).
% 0.34/0.67
% 0.34/0.67 % diff_minus_eq_add
% 0.34/0.67 thf(fact_6806_diff__minus__eq__add,axiom,
% 0.34/0.67 ! [A2: code_integer,B: code_integer] :
% 0.34/0.67 ( ( minus_8373710615458151222nteger @ A2 @ ( uminus1351360451143612070nteger @ B ) )
% 0.34/0.67 = ( plus_p5714425477246183910nteger @ A2 @ B ) ) ).
% 0.34/0.67
% 0.34/0.67 % diff_minus_eq_add
% 0.34/0.67 thf(fact_6807_uminus__add__conv__diff,axiom,
% 0.34/0.67 ! [A2: rat,B: rat] :
% 0.34/0.67 ( ( plus_plus_rat @ ( uminus_uminus_rat @ A2 ) @ B )
% 0.34/0.67 = ( minus_minus_rat @ B @ A2 ) ) ).
% 0.34/0.67
% 0.34/0.67 % uminus_add_conv_diff
% 0.34/0.67 thf(fact_6808_uminus__add__conv__diff,axiom,
% 0.34/0.67 ! [A2: int,B: int] :
% 0.34/0.67 ( ( plus_plus_int @ ( uminus_uminus_int @ A2 ) @ B )
% 0.34/0.67 = ( minus_minus_int @ B @ A2 ) ) ).
% 0.34/0.67
% 0.34/0.67 % uminus_add_conv_diff
% 0.34/0.67 thf(fact_6809_uminus__add__conv__diff,axiom,
% 0.34/0.67 ! [A2: code_integer,B: code_integer] :
% 0.34/0.67 ( ( plus_p5714425477246183910nteger @ ( uminus1351360451143612070nteger @ A2 ) @ B )
% 0.34/0.67 = ( minus_8373710615458151222nteger @ B @ A2 ) ) ).
% 0.34/0.67
% 0.34/0.67 % uminus_add_conv_diff
% 0.34/0.67 thf(fact_6810_mult__minus1,axiom,
% 0.34/0.67 ! [Z: rat] :
% 0.34/0.67 ( ( times_times_rat @ ( uminus_uminus_rat @ one_one_rat ) @ Z )
% 0.34/0.67 = ( uminus_uminus_rat @ Z ) ) ).
% 0.34/0.67
% 0.34/0.67 % mult_minus1
% 0.34/0.67 thf(fact_6811_mult__minus1,axiom,
% 0.34/0.67 ! [Z: int] :
% 0.34/0.67 ( ( times_times_int @ ( uminus_uminus_int @ one_one_int ) @ Z )
% 0.34/0.67 = ( uminus_uminus_int @ Z ) ) ).
% 0.34/0.67
% 0.34/0.67 % mult_minus1
% 0.34/0.67 thf(fact_6812_mult__minus1,axiom,
% 0.34/0.67 ! [Z: code_integer] :
% 0.34/0.67 ( ( times_3573771949741848930nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ Z )
% 0.34/0.67 = ( uminus1351360451143612070nteger @ Z ) ) ).
% 0.34/0.67
% 0.34/0.67 % mult_minus1
% 0.34/0.67 thf(fact_6813_mult__minus1__right,axiom,
% 0.34/0.67 ! [Z: rat] :
% 0.34/0.67 ( ( times_times_rat @ Z @ ( uminus_uminus_rat @ one_one_rat ) )
% 0.34/0.67 = ( uminus_uminus_rat @ Z ) ) ).
% 0.34/0.67
% 0.34/0.67 % mult_minus1_right
% 0.34/0.67 thf(fact_6814_mult__minus1__right,axiom,
% 0.34/0.67 ! [Z: int] :
% 0.34/0.67 ( ( times_times_int @ Z @ ( uminus_uminus_int @ one_one_int ) )
% 0.34/0.67 = ( uminus_uminus_int @ Z ) ) ).
% 0.34/0.67
% 0.34/0.67 % mult_minus1_right
% 0.34/0.67 thf(fact_6815_mult__minus1__right,axiom,
% 0.34/0.67 ! [Z: code_integer] :
% 0.34/0.67 ( ( times_3573771949741848930nteger @ Z @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) )
% 0.34/0.67 = ( uminus1351360451143612070nteger @ Z ) ) ).
% 0.34/0.67
% 0.34/0.67 % mult_minus1_right
% 0.34/0.67 thf(fact_6816_boolean__algebra_Ocompl__one,axiom,
% 0.34/0.67 ( ( uminus5254974436814262692_nat_o @ top_to4869887016220417533_nat_o )
% 0.34/0.67 = bot_bo7824918357723582233_nat_o ) ).
% 0.34/0.67
% 0.34/0.67 % boolean_algebra.compl_one
% 0.34/0.67 thf(fact_6817_boolean__algebra_Ocompl__one,axiom,
% 0.34/0.67 ( ( uminus5944136376168626660t_unit @ top_to1996260823553986621t_unit )
% 0.34/0.67 = bot_bo3957492148770167129t_unit ) ).
% 0.34/0.67
% 0.34/0.67 % boolean_algebra.compl_one
% 0.34/0.67 thf(fact_6818_boolean__algebra_Ocompl__one,axiom,
% 0.34/0.67 ( ( uminus_uminus_set_o @ top_top_set_o )
% 0.34/0.67 = bot_bot_set_o ) ).
% 0.34/0.67
% 0.34/0.67 % boolean_algebra.compl_one
% 0.34/0.67 thf(fact_6819_boolean__algebra_Ocompl__one,axiom,
% 0.34/0.67 ( ( uminus5710092332889474511et_nat @ top_top_set_nat )
% 0.34/0.67 = bot_bot_set_nat ) ).
% 0.34/0.67
% 0.34/0.67 % boolean_algebra.compl_one
% 0.34/0.67 thf(fact_6820_boolean__algebra_Ocompl__one,axiom,
% 0.34/0.67 ( ( uminus1532241313380277803et_int @ top_top_set_int )
% 0.34/0.67 = bot_bot_set_int ) ).
% 0.34/0.67
% 0.34/0.67 % boolean_algebra.compl_one
% 0.34/0.67 thf(fact_6821_boolean__algebra_Ocompl__one,axiom,
% 0.34/0.67 ( ( uminus_uminus_assn @ top_top_assn )
% 0.34/0.67 = bot_bot_assn ) ).
% 0.34/0.67
% 0.34/0.67 % boolean_algebra.compl_one
% 0.34/0.67 thf(fact_6822_boolean__algebra_Ocompl__one,axiom,
% 0.34/0.67 ( ( uminus2952777764628376836t_unit @ top_top_Product_unit )
% 0.34/0.67 = bot_bot_Product_unit ) ).
% 0.34/0.67
% 0.34/0.67 % boolean_algebra.compl_one
% 0.34/0.67 thf(fact_6823_boolean__algebra_Ocompl__zero,axiom,
% 0.34/0.67 ( ( uminus5254974436814262692_nat_o @ bot_bo7824918357723582233_nat_o )
% 0.34/0.67 = top_to4869887016220417533_nat_o ) ).
% 0.34/0.67
% 0.34/0.67 % boolean_algebra.compl_zero
% 0.34/0.67 thf(fact_6824_boolean__algebra_Ocompl__zero,axiom,
% 0.34/0.67 ( ( uminus5944136376168626660t_unit @ bot_bo3957492148770167129t_unit )
% 0.34/0.67 = top_to1996260823553986621t_unit ) ).
% 0.34/0.67
% 0.34/0.67 % boolean_algebra.compl_zero
% 0.34/0.67 thf(fact_6825_boolean__algebra_Ocompl__zero,axiom,
% 0.34/0.67 ( ( uminus_uminus_set_o @ bot_bot_set_o )
% 0.34/0.67 = top_top_set_o ) ).
% 0.34/0.67
% 0.34/0.67 % boolean_algebra.compl_zero
% 0.34/0.67 thf(fact_6826_boolean__algebra_Ocompl__zero,axiom,
% 0.34/0.67 ( ( uminus5710092332889474511et_nat @ bot_bot_set_nat )
% 0.34/0.67 = top_top_set_nat ) ).
% 0.34/0.67
% 0.34/0.67 % boolean_algebra.compl_zero
% 0.34/0.67 thf(fact_6827_boolean__algebra_Ocompl__zero,axiom,
% 0.34/0.67 ( ( uminus1532241313380277803et_int @ bot_bot_set_int )
% 0.34/0.67 = top_top_set_int ) ).
% 0.34/0.67
% 0.34/0.67 % boolean_algebra.compl_zero
% 0.34/0.67 thf(fact_6828_boolean__algebra_Ocompl__zero,axiom,
% 0.34/0.67 ( ( uminus_uminus_assn @ bot_bot_assn )
% 0.34/0.67 = top_top_assn ) ).
% 0.34/0.67
% 0.34/0.67 % boolean_algebra.compl_zero
% 0.34/0.67 thf(fact_6829_boolean__algebra_Ocompl__zero,axiom,
% 0.34/0.67 ( ( uminus2952777764628376836t_unit @ bot_bot_Product_unit )
% 0.34/0.67 = top_top_Product_unit ) ).
% 0.34/0.67
% 0.34/0.67 % boolean_algebra.compl_zero
% 0.34/0.67 thf(fact_6830_boolean__algebra_Oconj__cancel__right,axiom,
% 0.34/0.67 ! [X: set_Product_unit] :
% 0.34/0.67 ( ( inf_in4660618365625256667t_unit @ X @ ( uminus5944136376168626660t_unit @ X ) )
% 0.34/0.67 = bot_bo3957492148770167129t_unit ) ).
% 0.34/0.67
% 0.34/0.67 % boolean_algebra.conj_cancel_right
% 0.34/0.67 thf(fact_6831_boolean__algebra_Oconj__cancel__right,axiom,
% 0.34/0.67 ! [X: set_o] :
% 0.34/0.67 ( ( inf_inf_set_o @ X @ ( uminus_uminus_set_o @ X ) )
% 0.34/0.67 = bot_bot_set_o ) ).
% 0.34/0.67
% 0.34/0.67 % boolean_algebra.conj_cancel_right
% 0.34/0.67 thf(fact_6832_boolean__algebra_Oconj__cancel__right,axiom,
% 0.34/0.67 ! [X: set_nat] :
% 0.34/0.67 ( ( inf_inf_set_nat @ X @ ( uminus5710092332889474511et_nat @ X ) )
% 0.34/0.67 = bot_bot_set_nat ) ).
% 0.34/0.67
% 0.34/0.67 % boolean_algebra.conj_cancel_right
% 0.34/0.67 thf(fact_6833_boolean__algebra_Oconj__cancel__right,axiom,
% 0.34/0.67 ! [X: set_int] :
% 0.34/0.67 ( ( inf_inf_set_int @ X @ ( uminus1532241313380277803et_int @ X ) )
% 0.34/0.67 = bot_bot_set_int ) ).
% 0.34/0.67
% 0.34/0.67 % boolean_algebra.conj_cancel_right
% 0.34/0.67 thf(fact_6834_boolean__algebra_Oconj__cancel__right,axiom,
% 0.34/0.67 ! [X: set_Pr4532377907799695533_nat_o] :
% 0.34/0.67 ( ( inf_in1906310914598751387_nat_o @ X @ ( uminus5254974436814262692_nat_o @ X ) )
% 0.34/0.67 = bot_bo7824918357723582233_nat_o ) ).
% 0.34/0.67
% 0.34/0.67 % boolean_algebra.conj_cancel_right
% 0.34/0.67 thf(fact_6835_boolean__algebra_Oconj__cancel__right,axiom,
% 0.34/0.67 ! [X: assn] :
% 0.34/0.67 ( ( inf_inf_assn @ X @ ( uminus_uminus_assn @ X ) )
% 0.34/0.67 = bot_bot_assn ) ).
% 0.34/0.67
% 0.34/0.67 % boolean_algebra.conj_cancel_right
% 0.34/0.67 thf(fact_6836_boolean__algebra_Oconj__cancel__right,axiom,
% 0.34/0.67 ! [X: product_unit] :
% 0.34/0.67 ( ( inf_inf_Product_unit @ X @ ( uminus2952777764628376836t_unit @ X ) )
% 0.34/0.67 = bot_bot_Product_unit ) ).
% 0.34/0.67
% 0.34/0.67 % boolean_algebra.conj_cancel_right
% 0.34/0.67 thf(fact_6837_boolean__algebra_Oconj__cancel__left,axiom,
% 0.34/0.67 ! [X: set_Product_unit] :
% 0.34/0.67 ( ( inf_in4660618365625256667t_unit @ ( uminus5944136376168626660t_unit @ X ) @ X )
% 0.34/0.67 = bot_bo3957492148770167129t_unit ) ).
% 0.34/0.67
% 0.34/0.67 % boolean_algebra.conj_cancel_left
% 0.34/0.67 thf(fact_6838_boolean__algebra_Oconj__cancel__left,axiom,
% 0.34/0.67 ! [X: set_o] :
% 0.34/0.67 ( ( inf_inf_set_o @ ( uminus_uminus_set_o @ X ) @ X )
% 0.34/0.67 = bot_bot_set_o ) ).
% 0.34/0.67
% 0.34/0.67 % boolean_algebra.conj_cancel_left
% 0.34/0.67 thf(fact_6839_boolean__algebra_Oconj__cancel__left,axiom,
% 0.34/0.67 ! [X: set_nat] :
% 0.34/0.67 ( ( inf_inf_set_nat @ ( uminus5710092332889474511et_nat @ X ) @ X )
% 0.34/0.67 = bot_bot_set_nat ) ).
% 0.34/0.67
% 0.34/0.67 % boolean_algebra.conj_cancel_left
% 0.34/0.67 thf(fact_6840_boolean__algebra_Oconj__cancel__left,axiom,
% 0.34/0.67 ! [X: set_int] :
% 0.34/0.67 ( ( inf_inf_set_int @ ( uminus1532241313380277803et_int @ X ) @ X )
% 0.34/0.67 = bot_bot_set_int ) ).
% 0.34/0.67
% 0.34/0.67 % boolean_algebra.conj_cancel_left
% 0.34/0.67 thf(fact_6841_boolean__algebra_Oconj__cancel__left,axiom,
% 0.34/0.67 ! [X: set_Pr4532377907799695533_nat_o] :
% 0.34/0.67 ( ( inf_in1906310914598751387_nat_o @ ( uminus5254974436814262692_nat_o @ X ) @ X )
% 0.34/0.67 = bot_bo7824918357723582233_nat_o ) ).
% 0.34/0.67
% 0.34/0.67 % boolean_algebra.conj_cancel_left
% 0.34/0.67 thf(fact_6842_boolean__algebra_Oconj__cancel__left,axiom,
% 0.34/0.67 ! [X: assn] :
% 0.34/0.67 ( ( inf_inf_assn @ ( uminus_uminus_assn @ X ) @ X )
% 0.34/0.67 = bot_bot_assn ) ).
% 0.34/0.67
% 0.34/0.67 % boolean_algebra.conj_cancel_left
% 0.34/0.67 thf(fact_6843_boolean__algebra_Oconj__cancel__left,axiom,
% 0.34/0.67 ! [X: product_unit] :
% 0.34/0.67 ( ( inf_inf_Product_unit @ ( uminus2952777764628376836t_unit @ X ) @ X )
% 0.34/0.67 = bot_bot_Product_unit ) ).
% 0.34/0.67
% 0.34/0.67 % boolean_algebra.conj_cancel_left
% 0.34/0.67 thf(fact_6844_inf__compl__bot__right,axiom,
% 0.34/0.67 ! [X: set_Product_unit,Y: set_Product_unit] :
% 0.34/0.67 ( ( inf_in4660618365625256667t_unit @ X @ ( inf_in4660618365625256667t_unit @ Y @ ( uminus5944136376168626660t_unit @ X ) ) )
% 0.34/0.67 = bot_bo3957492148770167129t_unit ) ).
% 0.34/0.67
% 0.34/0.67 % inf_compl_bot_right
% 0.34/0.67 thf(fact_6845_inf__compl__bot__right,axiom,
% 0.34/0.67 ! [X: set_o,Y: set_o] :
% 0.34/0.67 ( ( inf_inf_set_o @ X @ ( inf_inf_set_o @ Y @ ( uminus_uminus_set_o @ X ) ) )
% 0.34/0.67 = bot_bot_set_o ) ).
% 0.34/0.67
% 0.34/0.67 % inf_compl_bot_right
% 0.34/0.67 thf(fact_6846_inf__compl__bot__right,axiom,
% 0.34/0.67 ! [X: set_nat,Y: set_nat] :
% 0.34/0.67 ( ( inf_inf_set_nat @ X @ ( inf_inf_set_nat @ Y @ ( uminus5710092332889474511et_nat @ X ) ) )
% 0.34/0.67 = bot_bot_set_nat ) ).
% 0.34/0.67
% 0.34/0.67 % inf_compl_bot_right
% 0.34/0.67 thf(fact_6847_inf__compl__bot__right,axiom,
% 0.34/0.67 ! [X: set_int,Y: set_int] :
% 0.34/0.67 ( ( inf_inf_set_int @ X @ ( inf_inf_set_int @ Y @ ( uminus1532241313380277803et_int @ X ) ) )
% 0.34/0.67 = bot_bot_set_int ) ).
% 0.34/0.67
% 0.34/0.67 % inf_compl_bot_right
% 0.34/0.67 thf(fact_6848_inf__compl__bot__right,axiom,
% 0.34/0.67 ! [X: set_Pr4532377907799695533_nat_o,Y: set_Pr4532377907799695533_nat_o] :
% 0.34/0.67 ( ( inf_in1906310914598751387_nat_o @ X @ ( inf_in1906310914598751387_nat_o @ Y @ ( uminus5254974436814262692_nat_o @ X ) ) )
% 0.34/0.67 = bot_bo7824918357723582233_nat_o ) ).
% 0.34/0.67
% 0.34/0.67 % inf_compl_bot_right
% 0.34/0.67 thf(fact_6849_inf__compl__bot__right,axiom,
% 0.34/0.67 ! [X: assn,Y: assn] :
% 0.34/0.67 ( ( inf_inf_assn @ X @ ( inf_inf_assn @ Y @ ( uminus_uminus_assn @ X ) ) )
% 0.34/0.67 = bot_bot_assn ) ).
% 0.34/0.67
% 0.34/0.67 % inf_compl_bot_right
% 0.34/0.67 thf(fact_6850_inf__compl__bot__right,axiom,
% 0.34/0.67 ! [X: product_unit,Y: product_unit] :
% 0.34/0.67 ( ( inf_inf_Product_unit @ X @ ( inf_inf_Product_unit @ Y @ ( uminus2952777764628376836t_unit @ X ) ) )
% 0.34/0.67 = bot_bot_Product_unit ) ).
% 0.34/0.67
% 0.34/0.67 % inf_compl_bot_right
% 0.34/0.67 thf(fact_6851_inf__compl__bot__left2,axiom,
% 0.34/0.67 ! [X: set_Product_unit,Y: set_Product_unit] :
% 0.34/0.67 ( ( inf_in4660618365625256667t_unit @ X @ ( inf_in4660618365625256667t_unit @ ( uminus5944136376168626660t_unit @ X ) @ Y ) )
% 0.34/0.67 = bot_bo3957492148770167129t_unit ) ).
% 0.34/0.67
% 0.34/0.67 % inf_compl_bot_left2
% 0.34/0.67 thf(fact_6852_inf__compl__bot__left2,axiom,
% 0.34/0.67 ! [X: set_o,Y: set_o] :
% 0.34/0.67 ( ( inf_inf_set_o @ X @ ( inf_inf_set_o @ ( uminus_uminus_set_o @ X ) @ Y ) )
% 0.34/0.67 = bot_bot_set_o ) ).
% 0.34/0.67
% 0.34/0.67 % inf_compl_bot_left2
% 0.34/0.67 thf(fact_6853_inf__compl__bot__left2,axiom,
% 0.34/0.67 ! [X: set_nat,Y: set_nat] :
% 0.34/0.67 ( ( inf_inf_set_nat @ X @ ( inf_inf_set_nat @ ( uminus5710092332889474511et_nat @ X ) @ Y ) )
% 0.34/0.67 = bot_bot_set_nat ) ).
% 0.34/0.67
% 0.34/0.67 % inf_compl_bot_left2
% 0.34/0.67 thf(fact_6854_inf__compl__bot__left2,axiom,
% 0.34/0.67 ! [X: set_int,Y: set_int] :
% 0.34/0.67 ( ( inf_inf_set_int @ X @ ( inf_inf_set_int @ ( uminus1532241313380277803et_int @ X ) @ Y ) )
% 0.34/0.67 = bot_bot_set_int ) ).
% 0.34/0.67
% 0.34/0.67 % inf_compl_bot_left2
% 0.34/0.67 thf(fact_6855_inf__compl__bot__left2,axiom,
% 0.34/0.67 ! [X: set_Pr4532377907799695533_nat_o,Y: set_Pr4532377907799695533_nat_o] :
% 0.34/0.67 ( ( inf_in1906310914598751387_nat_o @ X @ ( inf_in1906310914598751387_nat_o @ ( uminus5254974436814262692_nat_o @ X ) @ Y ) )
% 0.34/0.67 = bot_bo7824918357723582233_nat_o ) ).
% 0.34/0.67
% 0.34/0.67 % inf_compl_bot_left2
% 0.34/0.67 thf(fact_6856_inf__compl__bot__left2,axiom,
% 0.34/0.67 ! [X: assn,Y: assn] :
% 0.34/0.67 ( ( inf_inf_assn @ X @ ( inf_inf_assn @ ( uminus_uminus_assn @ X ) @ Y ) )
% 0.34/0.67 = bot_bot_assn ) ).
% 0.34/0.67
% 0.34/0.67 % inf_compl_bot_left2
% 0.34/0.67 thf(fact_6857_inf__compl__bot__left2,axiom,
% 0.34/0.67 ! [X: product_unit,Y: product_unit] :
% 0.34/0.67 ( ( inf_inf_Product_unit @ X @ ( inf_inf_Product_unit @ ( uminus2952777764628376836t_unit @ X ) @ Y ) )
% 0.34/0.67 = bot_bot_Product_unit ) ).
% 0.34/0.67
% 0.34/0.67 % inf_compl_bot_left2
% 0.34/0.67 thf(fact_6858_inf__compl__bot__left1,axiom,
% 0.34/0.67 ! [X: set_Product_unit,Y: set_Product_unit] :
% 0.34/0.67 ( ( inf_in4660618365625256667t_unit @ ( uminus5944136376168626660t_unit @ X ) @ ( inf_in4660618365625256667t_unit @ X @ Y ) )
% 0.34/0.67 = bot_bo3957492148770167129t_unit ) ).
% 0.34/0.67
% 0.34/0.67 % inf_compl_bot_left1
% 0.34/0.67 thf(fact_6859_inf__compl__bot__left1,axiom,
% 0.34/0.67 ! [X: set_o,Y: set_o] :
% 0.34/0.67 ( ( inf_inf_set_o @ ( uminus_uminus_set_o @ X ) @ ( inf_inf_set_o @ X @ Y ) )
% 0.34/0.67 = bot_bot_set_o ) ).
% 0.34/0.67
% 0.34/0.67 % inf_compl_bot_left1
% 0.34/0.67 thf(fact_6860_inf__compl__bot__left1,axiom,
% 0.34/0.67 ! [X: set_nat,Y: set_nat] :
% 0.34/0.67 ( ( inf_inf_set_nat @ ( uminus5710092332889474511et_nat @ X ) @ ( inf_inf_set_nat @ X @ Y ) )
% 0.34/0.67 = bot_bot_set_nat ) ).
% 0.34/0.67
% 0.34/0.67 % inf_compl_bot_left1
% 0.34/0.67 thf(fact_6861_inf__compl__bot__left1,axiom,
% 0.34/0.67 ! [X: set_int,Y: set_int] :
% 0.34/0.67 ( ( inf_inf_set_int @ ( uminus1532241313380277803et_int @ X ) @ ( inf_inf_set_int @ X @ Y ) )
% 0.34/0.67 = bot_bot_set_int ) ).
% 0.34/0.67
% 0.34/0.67 % inf_compl_bot_left1
% 0.34/0.67 thf(fact_6862_inf__compl__bot__left1,axiom,
% 0.34/0.67 ! [X: set_Pr4532377907799695533_nat_o,Y: set_Pr4532377907799695533_nat_o] :
% 0.34/0.67 ( ( inf_in1906310914598751387_nat_o @ ( uminus5254974436814262692_nat_o @ X ) @ ( inf_in1906310914598751387_nat_o @ X @ Y ) )
% 0.34/0.67 = bot_bo7824918357723582233_nat_o ) ).
% 0.34/0.67
% 0.34/0.67 % inf_compl_bot_left1
% 0.34/0.67 thf(fact_6863_inf__compl__bot__left1,axiom,
% 0.34/0.67 ! [X: assn,Y: assn] :
% 0.34/0.67 ( ( inf_inf_assn @ ( uminus_uminus_assn @ X ) @ ( inf_inf_assn @ X @ Y ) )
% 0.34/0.67 = bot_bot_assn ) ).
% 0.34/0.67
% 0.34/0.67 % inf_compl_bot_left1
% 0.34/0.67 thf(fact_6864_inf__compl__bot__left1,axiom,
% 0.34/0.67 ! [X: product_unit,Y: product_unit] :
% 0.34/0.67 ( ( inf_inf_Product_unit @ ( uminus2952777764628376836t_unit @ X ) @ ( inf_inf_Product_unit @ X @ Y ) )
% 0.34/0.67 = bot_bot_Product_unit ) ).
% 0.34/0.67
% 0.34/0.67 % inf_compl_bot_left1
% 0.34/0.67 thf(fact_6865_subset__Compl__singleton,axiom,
% 0.34/0.67 ! [A: set_Product_unit,B: product_unit] :
% 0.34/0.67 ( ( ord_le3507040750410214029t_unit @ A @ ( uminus5944136376168626660t_unit @ ( insert_Product_unit @ B @ bot_bo3957492148770167129t_unit ) ) )
% 0.34/0.67 = ( ~ ( member_Product_unit @ B @ A ) ) ) ).
% 0.34/0.67
% 0.34/0.67 % subset_Compl_singleton
% 0.34/0.67 thf(fact_6866_subset__Compl__singleton,axiom,
% 0.34/0.67 ! [A: set_o,B: $o] :
% 0.34/0.67 ( ( ord_less_eq_set_o @ A @ ( uminus_uminus_set_o @ ( insert_o @ B @ bot_bot_set_o ) ) )
% 0.34/0.67 = ( ~ ( member_o @ B @ A ) ) ) ).
% 0.34/0.67
% 0.34/0.67 % subset_Compl_singleton
% 0.34/0.67 thf(fact_6867_subset__Compl__singleton,axiom,
% 0.34/0.67 ! [A: set_int,B: int] :
% 0.34/0.67 ( ( ord_less_eq_set_int @ A @ ( uminus1532241313380277803et_int @ ( insert_int @ B @ bot_bot_set_int ) ) )
% 0.34/0.67 = ( ~ ( member_int @ B @ A ) ) ) ).
% 0.34/0.67
% 0.34/0.67 % subset_Compl_singleton
% 0.34/0.67 thf(fact_6868_subset__Compl__singleton,axiom,
% 0.34/0.67 ! [A: set_Pr4532377907799695533_nat_o,B: produc3658429121746597890et_nat > $o] :
% 0.34/0.67 ( ( ord_le2965882846123202637_nat_o @ A @ ( uminus5254974436814262692_nat_o @ ( insert5175938949040314269_nat_o @ B @ bot_bo7824918357723582233_nat_o ) ) )
% 0.34/0.67 = ( ~ ( member6576561426505652726_nat_o @ B @ A ) ) ) ).
% 0.34/0.67
% 0.34/0.67 % subset_Compl_singleton
% 0.34/0.67 thf(fact_6869_subset__Compl__singleton,axiom,
% 0.34/0.67 ! [A: set_nat,B: nat] :
% 0.34/0.67 ( ( ord_less_eq_set_nat @ A @ ( uminus5710092332889474511et_nat @ ( insert_nat @ B @ bot_bot_set_nat ) ) )
% 0.34/0.67 = ( ~ ( member_nat @ B @ A ) ) ) ).
% 0.34/0.67
% 0.34/0.67 % subset_Compl_singleton
% 0.34/0.67 thf(fact_6870_prod_Oempty,axiom,
% 0.34/0.67 ! [G: product_unit > assn] :
% 0.34/0.67 ( ( groups5360233303959380816t_assn @ G @ bot_bo3957492148770167129t_unit )
% 0.34/0.67 = one_one_assn ) ).
% 0.34/0.67
% 0.34/0.67 % prod.empty
% 0.34/0.67 thf(fact_6871_prod_Oempty,axiom,
% 0.34/0.67 ! [G: product_unit > nat] :
% 0.34/0.67 ( ( groups396421249026171384it_nat @ G @ bot_bo3957492148770167129t_unit )
% 0.34/0.67 = one_one_nat ) ).
% 0.34/0.67
% 0.34/0.67 % prod.empty
% 0.34/0.67 thf(fact_6872_prod_Oempty,axiom,
% 0.34/0.67 ! [G: product_unit > int] :
% 0.34/0.67 ( ( groups393930778517121108it_int @ G @ bot_bo3957492148770167129t_unit )
% 0.34/0.67 = one_one_int ) ).
% 0.34/0.67
% 0.34/0.67 % prod.empty
% 0.34/0.67 thf(fact_6873_prod_Oempty,axiom,
% 0.34/0.67 ! [G: product_unit > code_integer] :
% 0.34/0.67 ( ( groups8185359811944078981nteger @ G @ bot_bo3957492148770167129t_unit )
% 0.34/0.67 = one_one_Code_integer ) ).
% 0.34/0.67
% 0.34/0.67 % prod.empty
% 0.34/0.67 thf(fact_6874_prod_Oempty,axiom,
% 0.34/0.67 ! [G: product_unit > code_natural] :
% 0.34/0.67 ( ( groups7008954963700796496atural @ G @ bot_bo3957492148770167129t_unit )
% 0.34/0.67 = one_one_Code_natural ) ).
% 0.34/0.67
% 0.34/0.67 % prod.empty
% 0.34/0.67 thf(fact_6875_prod_Oempty,axiom,
% 0.34/0.67 ! [G: $o > assn] :
% 0.34/0.67 ( ( groups5301882518646026715o_assn @ G @ bot_bot_set_o )
% 0.34/0.67 = one_one_assn ) ).
% 0.34/0.67
% 0.34/0.67 % prod.empty
% 0.34/0.67 thf(fact_6876_prod_Oempty,axiom,
% 0.34/0.67 ! [G: $o > nat] :
% 0.34/0.67 ( ( groups3504817904513533571_o_nat @ G @ bot_bot_set_o )
% 0.34/0.67 = one_one_nat ) ).
% 0.34/0.67
% 0.34/0.67 % prod.empty
% 0.34/0.67 thf(fact_6877_prod_Oempty,axiom,
% 0.34/0.67 ! [G: $o > int] :
% 0.34/0.67 ( ( groups3502327434004483295_o_int @ G @ bot_bot_set_o )
% 0.34/0.67 = one_one_int ) ).
% 0.34/0.67
% 0.34/0.67 % prod.empty
% 0.34/0.67 thf(fact_6878_prod_Oempty,axiom,
% 0.34/0.67 ! [G: $o > code_integer] :
% 0.34/0.67 ( ( groups7694694392188491536nteger @ G @ bot_bot_set_o )
% 0.34/0.67 = one_one_Code_integer ) ).
% 0.34/0.67
% 0.34/0.67 % prod.empty
% 0.34/0.67 thf(fact_6879_prod_Oempty,axiom,
% 0.34/0.67 ! [G: $o > code_natural] :
% 0.34/0.67 ( ( groups6518289543945209051atural @ G @ bot_bot_set_o )
% 0.34/0.67 = one_one_Code_natural ) ).
% 0.34/0.67
% 0.34/0.67 % prod.empty
% 0.34/0.67 thf(fact_6880_prod_Oinfinite,axiom,
% 0.34/0.67 ! [A: set_nat,G: nat > assn] :
% 0.34/0.67 ( ~ ( finite_finite_nat @ A )
% 0.34/0.67 => ( ( groups6906906614972039071t_assn @ G @ A )
% 0.34/0.67 = one_one_assn ) ) ).
% 0.34/0.67
% 0.34/0.67 % prod.infinite
% 0.34/0.67 thf(fact_6881_prod_Oinfinite,axiom,
% 0.34/0.67 ! [A: set_Code_integer,G: code_integer > assn] :
% 0.34/0.67 ( ~ ( finite6017078050557962740nteger @ A )
% 0.34/0.67 => ( ( groups1304777262505850412r_assn @ G @ A )
% 0.34/0.67 = one_one_assn ) ) ).
% 0.34/0.67
% 0.34/0.67 % prod.infinite
% 0.34/0.67 thf(fact_6882_prod_Oinfinite,axiom,
% 0.34/0.67 ! [A: set_Code_integer,G: code_integer > nat] :
% 0.34/0.67 ( ~ ( finite6017078050557962740nteger @ A )
% 0.34/0.67 => ( ( groups3190895334310489300er_nat @ G @ A )
% 0.34/0.67 = one_one_nat ) ) ).
% 0.34/0.67
% 0.34/0.67 % prod.infinite
% 0.34/0.67 thf(fact_6883_prod_Oinfinite,axiom,
% 0.34/0.67 ! [A: set_nat,G: nat > int] :
% 0.34/0.67 ( ~ ( finite_finite_nat @ A )
% 0.34/0.67 => ( ( groups705719431365010083at_int @ G @ A )
% 0.34/0.67 = one_one_int ) ) ).
% 0.34/0.67
% 0.34/0.67 % prod.infinite
% 0.34/0.67 thf(fact_6884_prod_Oinfinite,axiom,
% 0.34/0.67 ! [A: set_Code_integer,G: code_integer > int] :
% 0.34/0.67 ( ~ ( finite6017078050557962740nteger @ A )
% 0.34/0.67 => ( ( groups3188404863801439024er_int @ G @ A )
% 0.34/0.67 = one_one_int ) ) ).
% 0.34/0.67
% 0.34/0.67 % prod.infinite
% 0.34/0.67 thf(fact_6885_prod_Oinfinite,axiom,
% 0.34/0.67 ! [A: set_nat,G: nat > code_integer] :
% 0.34/0.67 ( ~ ( finite_finite_nat @ A )
% 0.34/0.67 => ( ( groups3455450783089532116nteger @ G @ A )
% 0.34/0.67 = one_one_Code_integer ) ) ).
% 0.34/0.67
% 0.34/0.67 % prod.infinite
% 0.34/0.67 thf(fact_6886_prod_Oinfinite,axiom,
% 0.34/0.67 ! [A: set_Code_integer,G: code_integer > code_integer] :
% 0.34/0.67 ( ~ ( finite6017078050557962740nteger @ A )
% 0.34/0.67 => ( ( groups3674199335183972705nteger @ G @ A )
% 0.34/0.67 = one_one_Code_integer ) ) ).
% 0.34/0.67
% 0.34/0.67 % prod.infinite
% 0.34/0.67 thf(fact_6887_prod_Oinfinite,axiom,
% 0.34/0.67 ! [A: set_nat,G: nat > code_natural] :
% 0.34/0.67 ( ~ ( finite_finite_nat @ A )
% 0.34/0.67 => ( ( groups2279045934846249631atural @ G @ A )
% 0.34/0.67 = one_one_Code_natural ) ) ).
% 0.34/0.67
% 0.34/0.67 % prod.infinite
% 0.34/0.67 thf(fact_6888_prod_Oinfinite,axiom,
% 0.34/0.67 ! [A: set_Code_integer,G: code_integer > code_natural] :
% 0.34/0.67 ( ~ ( finite6017078050557962740nteger @ A )
% 0.34/0.67 => ( ( groups2497794486940690220atural @ G @ A )
% 0.34/0.67 = one_one_Code_natural ) ) ).
% 0.34/0.67
% 0.34/0.67 % prod.infinite
% 0.34/0.67 thf(fact_6889_prod_Oinfinite,axiom,
% 0.34/0.67 ! [A: set_nat,G: nat > nat] :
% 0.34/0.67 ( ~ ( finite_finite_nat @ A )
% 0.34/0.67 => ( ( groups708209901874060359at_nat @ G @ A )
% 0.34/0.67 = one_one_nat ) ) ).
% 0.34/0.67
% 0.34/0.67 % prod.infinite
% 0.34/0.67 thf(fact_6890_of__nat__mult,axiom,
% 0.34/0.67 ! [M4: nat,N2: nat] :
% 0.34/0.67 ( ( semiri681578069525770553at_rat @ ( times_times_nat @ M4 @ N2 ) )
% 0.34/0.67 = ( times_times_rat @ ( semiri681578069525770553at_rat @ M4 ) @ ( semiri681578069525770553at_rat @ N2 ) ) ) ).
% 0.34/0.67
% 0.34/0.67 % of_nat_mult
% 0.34/0.67 thf(fact_6891_of__nat__mult,axiom,
% 0.34/0.67 ! [M4: nat,N2: nat] :
% 0.34/0.67 ( ( semiri1314217659103216013at_int @ ( times_times_nat @ M4 @ N2 ) )
% 0.34/0.67 = ( times_times_int @ ( semiri1314217659103216013at_int @ M4 ) @ ( semiri1314217659103216013at_int @ N2 ) ) ) ).
% 0.34/0.67
% 0.34/0.67 % of_nat_mult
% 0.34/0.67 thf(fact_6892_of__nat__mult,axiom,
% 0.34/0.67 ! [M4: nat,N2: nat] :
% 0.34/0.67 ( ( semiri1316708129612266289at_nat @ ( times_times_nat @ M4 @ N2 ) )
% 0.34/0.67 = ( times_times_nat @ ( semiri1316708129612266289at_nat @ M4 ) @ ( semiri1316708129612266289at_nat @ N2 ) ) ) ).
% 0.34/0.67
% 0.34/0.67 % of_nat_mult
% 0.34/0.67 thf(fact_6893_of__nat__mult,axiom,
% 0.34/0.67 ! [M4: nat,N2: nat] :
% 0.34/0.67 ( ( semiri4939895301339042750nteger @ ( times_times_nat @ M4 @ N2 ) )
% 0.34/0.67 = ( times_3573771949741848930nteger @ ( semiri4939895301339042750nteger @ M4 ) @ ( semiri4939895301339042750nteger @ N2 ) ) ) ).
% 0.34/0.67
% 0.34/0.67 % of_nat_mult
% 0.34/0.67 thf(fact_6894_of__nat__1,axiom,
% 0.34/0.67 ( ( semiri3763490453095760265atural @ one_one_nat )
% 0.34/0.67 = one_one_Code_natural ) ).
% 0.34/0.67
% 0.34/0.67 % of_nat_1
% 0.34/0.67 thf(fact_6895_of__nat__1,axiom,
% 0.34/0.67 ( ( semiri1314217659103216013at_int @ one_one_nat )
% 0.34/0.67 = one_one_int ) ).
% 0.34/0.67
% 0.34/0.67 % of_nat_1
% 0.34/0.67 thf(fact_6896_of__nat__1,axiom,
% 0.34/0.67 ( ( semiri1316708129612266289at_nat @ one_one_nat )
% 0.34/0.67 = one_one_nat ) ).
% 0.34/0.67
% 0.34/0.67 % of_nat_1
% 0.34/0.67 thf(fact_6897_of__nat__1,axiom,
% 0.34/0.67 ( ( semiri4939895301339042750nteger @ one_one_nat )
% 0.34/0.67 = one_one_Code_integer ) ).
% 0.34/0.67
% 0.34/0.67 % of_nat_1
% 0.34/0.67 thf(fact_6898_of__nat__1__eq__iff,axiom,
% 0.34/0.67 ! [N2: nat] :
% 0.34/0.67 ( ( one_one_Code_natural
% 0.34/0.67 = ( semiri3763490453095760265atural @ N2 ) )
% 0.34/0.67 = ( N2 = one_one_nat ) ) ).
% 0.34/0.67
% 0.34/0.67 % of_nat_1_eq_iff
% 0.34/0.67 thf(fact_6899_of__nat__1__eq__iff,axiom,
% 0.34/0.67 ! [N2: nat] :
% 0.34/0.67 ( ( one_one_int
% 0.34/0.67 = ( semiri1314217659103216013at_int @ N2 ) )
% 0.34/0.67 = ( N2 = one_one_nat ) ) ).
% 0.34/0.67
% 0.34/0.67 % of_nat_1_eq_iff
% 0.34/0.67 thf(fact_6900_of__nat__1__eq__iff,axiom,
% 0.34/0.67 ! [N2: nat] :
% 0.34/0.67 ( ( one_one_nat
% 0.34/0.67 = ( semiri1316708129612266289at_nat @ N2 ) )
% 0.34/0.67 = ( N2 = one_one_nat ) ) ).
% 0.34/0.67
% 0.34/0.67 % of_nat_1_eq_iff
% 0.34/0.67 thf(fact_6901_of__nat__1__eq__iff,axiom,
% 0.34/0.67 ! [N2: nat] :
% 0.34/0.67 ( ( one_one_Code_integer
% 0.34/0.67 = ( semiri4939895301339042750nteger @ N2 ) )
% 0.34/0.67 = ( N2 = one_one_nat ) ) ).
% 0.34/0.67
% 0.34/0.67 % of_nat_1_eq_iff
% 0.34/0.67 thf(fact_6902_of__nat__eq__1__iff,axiom,
% 0.34/0.67 ! [N2: nat] :
% 0.34/0.67 ( ( ( semiri3763490453095760265atural @ N2 )
% 0.34/0.67 = one_one_Code_natural )
% 0.34/0.67 = ( N2 = one_one_nat ) ) ).
% 0.34/0.67
% 0.34/0.67 % of_nat_eq_1_iff
% 0.34/0.67 thf(fact_6903_of__nat__eq__1__iff,axiom,
% 0.34/0.67 ! [N2: nat] :
% 0.34/0.67 ( ( ( semiri1314217659103216013at_int @ N2 )
% 0.34/0.67 = one_one_int )
% 0.34/0.67 = ( N2 = one_one_nat ) ) ).
% 0.34/0.67
% 0.34/0.67 % of_nat_eq_1_iff
% 0.34/0.67 thf(fact_6904_of__nat__eq__1__iff,axiom,
% 0.34/0.67 ! [N2: nat] :
% 0.34/0.67 ( ( ( semiri1316708129612266289at_nat @ N2 )
% 0.34/0.67 = one_one_nat )
% 0.34/0.67 = ( N2 = one_one_nat ) ) ).
% 0.34/0.67
% 0.34/0.67 % of_nat_eq_1_iff
% 0.34/0.67 thf(fact_6905_of__nat__eq__1__iff,axiom,
% 0.34/0.67 ! [N2: nat] :
% 0.34/0.67 ( ( ( semiri4939895301339042750nteger @ N2 )
% 0.34/0.67 = one_one_Code_integer )
% 0.34/0.67 = ( N2 = one_one_nat ) ) ).
% 0.34/0.67
% 0.34/0.67 % of_nat_eq_1_iff
% 0.34/0.67 thf(fact_6906_Gcd__empty,axiom,
% 0.34/0.67 ( ( gcd_Gcd_nat @ bot_bot_set_nat )
% 0.34/0.67 = zero_zero_nat ) ).
% 0.34/0.67
% 0.34/0.67 % Gcd_empty
% 0.34/0.67 thf(fact_6907_Gcd__empty,axiom,
% 0.34/0.67 ( ( gcd_Gcd_int @ bot_bot_set_int )
% 0.34/0.67 = zero_zero_int ) ).
% 0.34/0.67
% 0.34/0.67 % Gcd_empty
% 0.34/0.67 thf(fact_6908_Gcd__UNIV,axiom,
% 0.34/0.67 ( ( gcd_Gcd_nat @ top_top_set_nat )
% 0.34/0.67 = one_one_nat ) ).
% 0.34/0.67
% 0.34/0.67 % Gcd_UNIV
% 0.34/0.67 thf(fact_6909_Gcd__UNIV,axiom,
% 0.34/0.67 ( ( gcd_Gcd_int @ top_top_set_int )
% 0.34/0.67 = one_one_int ) ).
% 0.34/0.67
% 0.34/0.67 % Gcd_UNIV
% 0.34/0.67 thf(fact_6910_Gcd__fin_Oempty,axiom,
% 0.34/0.67 ( ( semiri4258706085729940814in_nat @ bot_bot_set_nat )
% 0.34/0.67 = zero_zero_nat ) ).
% 0.34/0.67
% 0.34/0.67 % Gcd_fin.empty
% 0.34/0.67 thf(fact_6911_Gcd__fin_Oempty,axiom,
% 0.34/0.67 ( ( semiri4256215615220890538in_int @ bot_bot_set_int )
% 0.34/0.67 = zero_zero_int ) ).
% 0.34/0.67
% 0.34/0.67 % Gcd_fin.empty
% 0.34/0.67 thf(fact_6912_Gcd__fin_Oinfinite,axiom,
% 0.34/0.67 ! [A: set_nat] :
% 0.34/0.67 ( ~ ( finite_finite_nat @ A )
% 0.34/0.67 => ( ( semiri4258706085729940814in_nat @ A )
% 0.34/0.67 = one_one_nat ) ) ).
% 0.34/0.67
% 0.34/0.67 % Gcd_fin.infinite
% 0.34/0.67 thf(fact_6913_Gcd__fin_Oinfinite,axiom,
% 0.34/0.67 ! [A: set_int] :
% 0.34/0.67 ( ~ ( finite_finite_int @ A )
% 0.34/0.67 => ( ( semiri4256215615220890538in_int @ A )
% 0.34/0.67 = one_one_int ) ) ).
% 0.34/0.67
% 0.34/0.67 % Gcd_fin.infinite
% 0.34/0.67 thf(fact_6914_is__unit__Gcd__fin__iff,axiom,
% 0.34/0.67 ! [A: set_nat] :
% 0.34/0.67 ( ( dvd_dvd_nat @ ( semiri4258706085729940814in_nat @ A ) @ one_one_nat )
% 0.34/0.67 = ( ( semiri4258706085729940814in_nat @ A )
% 0.34/0.67 = one_one_nat ) ) ).
% 0.34/0.67
% 0.34/0.67 % is_unit_Gcd_fin_iff
% 0.34/0.67 thf(fact_6915_is__unit__Gcd__fin__iff,axiom,
% 0.34/0.67 ! [A: set_int] :
% 0.34/0.67 ( ( dvd_dvd_int @ ( semiri4256215615220890538in_int @ A ) @ one_one_int )
% 0.34/0.67 = ( ( semiri4256215615220890538in_int @ A )
% 0.34/0.67 = one_one_int ) ) ).
% 0.34/0.67
% 0.34/0.67 % is_unit_Gcd_fin_iff
% 0.34/0.67 thf(fact_6916_dbl__inc__simps_I4_J,axiom,
% 0.34/0.67 ( ( neg_nu5851722552734809277nc_int @ ( uminus_uminus_int @ one_one_int ) )
% 0.34/0.67 = ( uminus_uminus_int @ one_one_int ) ) ).
% 0.34/0.67
% 0.34/0.67 % dbl_inc_simps(4)
% 0.34/0.67 thf(fact_6917_dbl__inc__simps_I4_J,axiom,
% 0.34/0.67 ( ( neg_nu5831290666863070958nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) )
% 0.34/0.67 = ( uminus1351360451143612070nteger @ one_one_Code_integer ) ) ).
% 0.34/0.67
% 0.34/0.67 % dbl_inc_simps(4)
% 0.34/0.67 thf(fact_6918_add__neg__numeral__special_I7_J,axiom,
% 0.34/0.67 ( ( plus_plus_rat @ one_one_rat @ ( uminus_uminus_rat @ one_one_rat ) )
% 0.34/0.67 = zero_zero_rat ) ).
% 0.34/0.67
% 0.34/0.67 % add_neg_numeral_special(7)
% 0.34/0.67 thf(fact_6919_add__neg__numeral__special_I7_J,axiom,
% 0.34/0.67 ( ( plus_plus_int @ one_one_int @ ( uminus_uminus_int @ one_one_int ) )
% 0.34/0.67 = zero_zero_int ) ).
% 0.34/0.67
% 0.34/0.67 % add_neg_numeral_special(7)
% 0.34/0.67 thf(fact_6920_add__neg__numeral__special_I7_J,axiom,
% 0.34/0.67 ( ( plus_p5714425477246183910nteger @ one_one_Code_integer @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) )
% 0.34/0.67 = zero_z3403309356797280102nteger ) ).
% 0.34/0.67
% 0.34/0.67 % add_neg_numeral_special(7)
% 0.34/0.67 thf(fact_6921_add__neg__numeral__special_I8_J,axiom,
% 0.34/0.67 ( ( plus_plus_rat @ ( uminus_uminus_rat @ one_one_rat ) @ one_one_rat )
% 0.34/0.67 = zero_zero_rat ) ).
% 0.34/0.67
% 0.34/0.67 % add_neg_numeral_special(8)
% 0.34/0.67 thf(fact_6922_add__neg__numeral__special_I8_J,axiom,
% 0.34/0.67 ( ( plus_plus_int @ ( uminus_uminus_int @ one_one_int ) @ one_one_int )
% 0.34/0.67 = zero_zero_int ) ).
% 0.34/0.67
% 0.34/0.67 % add_neg_numeral_special(8)
% 0.34/0.67 thf(fact_6923_add__neg__numeral__special_I8_J,axiom,
% 0.34/0.67 ( ( plus_p5714425477246183910nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ one_one_Code_integer )
% 0.34/0.67 = zero_z3403309356797280102nteger ) ).
% 0.34/0.67
% 0.34/0.67 % add_neg_numeral_special(8)
% 0.34/0.67 thf(fact_6924_diff__numeral__special_I12_J,axiom,
% 0.34/0.67 ( ( minus_minus_rat @ ( uminus_uminus_rat @ one_one_rat ) @ ( uminus_uminus_rat @ one_one_rat ) )
% 0.34/0.67 = zero_zero_rat ) ).
% 0.34/0.67
% 0.34/0.67 % diff_numeral_special(12)
% 0.34/0.67 thf(fact_6925_diff__numeral__special_I12_J,axiom,
% 0.34/0.67 ( ( minus_minus_int @ ( uminus_uminus_int @ one_one_int ) @ ( uminus_uminus_int @ one_one_int ) )
% 0.34/0.67 = zero_zero_int ) ).
% 0.34/0.67
% 0.34/0.67 % diff_numeral_special(12)
% 0.34/0.67 thf(fact_6926_diff__numeral__special_I12_J,axiom,
% 0.34/0.67 ( ( minus_8373710615458151222nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) )
% 0.34/0.67 = zero_z3403309356797280102nteger ) ).
% 0.34/0.67
% 0.34/0.67 % diff_numeral_special(12)
% 0.34/0.67 thf(fact_6927_minus__one__mult__self,axiom,
% 0.34/0.67 ! [N2: nat] :
% 0.34/0.67 ( ( times_times_rat @ ( power_power_rat @ ( uminus_uminus_rat @ one_one_rat ) @ N2 ) @ ( power_power_rat @ ( uminus_uminus_rat @ one_one_rat ) @ N2 ) )
% 0.34/0.67 = one_one_rat ) ).
% 0.34/0.67
% 0.34/0.67 % minus_one_mult_self
% 0.34/0.67 thf(fact_6928_minus__one__mult__self,axiom,
% 0.34/0.67 ! [N2: nat] :
% 0.34/0.67 ( ( times_times_int @ ( power_power_int @ ( uminus_uminus_int @ one_one_int ) @ N2 ) @ ( power_power_int @ ( uminus_uminus_int @ one_one_int ) @ N2 ) )
% 0.34/0.67 = one_one_int ) ).
% 0.34/0.67
% 0.34/0.67 % minus_one_mult_self
% 0.34/0.67 thf(fact_6929_minus__one__mult__self,axiom,
% 0.34/0.67 ! [N2: nat] :
% 0.34/0.67 ( ( times_3573771949741848930nteger @ ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ N2 ) @ ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ N2 ) )
% 0.34/0.67 = one_one_Code_integer ) ).
% 0.34/0.67
% 0.34/0.67 % minus_one_mult_self
% 0.34/0.67 thf(fact_6930_left__minus__one__mult__self,axiom,
% 0.34/0.67 ! [N2: nat,A2: rat] :
% 0.34/0.67 ( ( times_times_rat @ ( power_power_rat @ ( uminus_uminus_rat @ one_one_rat ) @ N2 ) @ ( times_times_rat @ ( power_power_rat @ ( uminus_uminus_rat @ one_one_rat ) @ N2 ) @ A2 ) )
% 0.34/0.67 = A2 ) ).
% 0.34/0.67
% 0.34/0.67 % left_minus_one_mult_self
% 0.34/0.67 thf(fact_6931_left__minus__one__mult__self,axiom,
% 0.34/0.67 ! [N2: nat,A2: int] :
% 0.34/0.67 ( ( times_times_int @ ( power_power_int @ ( uminus_uminus_int @ one_one_int ) @ N2 ) @ ( times_times_int @ ( power_power_int @ ( uminus_uminus_int @ one_one_int ) @ N2 ) @ A2 ) )
% 0.34/0.67 = A2 ) ).
% 0.34/0.67
% 0.34/0.67 % left_minus_one_mult_self
% 0.34/0.67 thf(fact_6932_left__minus__one__mult__self,axiom,
% 0.34/0.67 ! [N2: nat,A2: code_integer] :
% 0.34/0.67 ( ( times_3573771949741848930nteger @ ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ N2 ) @ ( times_3573771949741848930nteger @ ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ N2 ) @ A2 ) )
% 0.34/0.67 = A2 ) ).
% 0.34/0.67
% 0.34/0.67 % left_minus_one_mult_self
% 0.34/0.67 thf(fact_6933_of__nat__Suc,axiom,
% 0.34/0.67 ! [M4: nat] :
% 0.34/0.67 ( ( semiri3763490453095760265atural @ ( suc @ M4 ) )
% 0.34/0.67 = ( plus_p4538020629002901425atural @ one_one_Code_natural @ ( semiri3763490453095760265atural @ M4 ) ) ) ).
% 0.34/0.67
% 0.34/0.67 % of_nat_Suc
% 0.34/0.67 thf(fact_6934_of__nat__Suc,axiom,
% 0.34/0.67 ! [M4: nat] :
% 0.34/0.67 ( ( semiri681578069525770553at_rat @ ( suc @ M4 ) )
% 0.34/0.67 = ( plus_plus_rat @ one_one_rat @ ( semiri681578069525770553at_rat @ M4 ) ) ) ).
% 0.34/0.67
% 0.34/0.67 % of_nat_Suc
% 0.34/0.67 thf(fact_6935_of__nat__Suc,axiom,
% 0.34/0.67 ! [M4: nat] :
% 0.34/0.67 ( ( semiri1314217659103216013at_int @ ( suc @ M4 ) )
% 0.34/0.67 = ( plus_plus_int @ one_one_int @ ( semiri1314217659103216013at_int @ M4 ) ) ) ).
% 0.34/0.67
% 0.34/0.67 % of_nat_Suc
% 0.34/0.67 thf(fact_6936_of__nat__Suc,axiom,
% 0.34/0.67 ! [M4: nat] :
% 0.34/0.67 ( ( semiri1316708129612266289at_nat @ ( suc @ M4 ) )
% 0.34/0.67 = ( plus_plus_nat @ one_one_nat @ ( semiri1316708129612266289at_nat @ M4 ) ) ) ).
% 0.34/0.67
% 0.34/0.67 % of_nat_Suc
% 0.34/0.67 thf(fact_6937_of__nat__Suc,axiom,
% 0.34/0.67 ! [M4: nat] :
% 0.34/0.67 ( ( semiri4939895301339042750nteger @ ( suc @ M4 ) )
% 0.34/0.67 = ( plus_p5714425477246183910nteger @ one_one_Code_integer @ ( semiri4939895301339042750nteger @ M4 ) ) ) ).
% 0.34/0.67
% 0.34/0.67 % of_nat_Suc
% 0.34/0.67 thf(fact_6938_prod_Oinsert,axiom,
% 0.34/0.67 ! [A: set_int,X: int,G: int > assn] :
% 0.34/0.67 ( ( finite_finite_int @ A )
% 0.34/0.67 => ( ~ ( member_int @ X @ A )
% 0.34/0.67 => ( ( groups7882442080178216443t_assn @ G @ ( insert_int @ X @ A ) )
% 0.34/0.67 = ( times_times_assn @ ( G @ X ) @ ( groups7882442080178216443t_assn @ G @ A ) ) ) ) ) ).
% 0.34/0.67
% 0.34/0.67 % prod.insert
% 0.34/0.67 thf(fact_6939_prod_Oinsert,axiom,
% 0.34/0.67 ! [A: set_o,X: $o,G: $o > assn] :
% 0.34/0.67 ( ( finite_finite_o @ A )
% 0.34/0.67 => ( ~ ( member_o @ X @ A )
% 0.34/0.67 => ( ( groups5301882518646026715o_assn @ G @ ( insert_o @ X @ A ) )
% 0.34/0.67 = ( times_times_assn @ ( G @ X ) @ ( groups5301882518646026715o_assn @ G @ A ) ) ) ) ) ).
% 0.34/0.67
% 0.34/0.67 % prod.insert
% 0.34/0.67 thf(fact_6940_prod_Oinsert,axiom,
% 0.34/0.67 ! [A: set_Product_unit,X: product_unit,G: product_unit > assn] :
% 0.34/0.67 ( ( finite4290736615968046902t_unit @ A )
% 0.34/0.67 => ( ~ ( member_Product_unit @ X @ A )
% 0.34/0.67 => ( ( groups5360233303959380816t_assn @ G @ ( insert_Product_unit @ X @ A ) )
% 0.34/0.67 = ( times_times_assn @ ( G @ X ) @ ( groups5360233303959380816t_assn @ G @ A ) ) ) ) ) ).
% 0.34/0.67
% 0.34/0.67 % prod.insert
% 0.34/0.67 thf(fact_6941_prod_Oinsert,axiom,
% 0.34/0.67 ! [A: set_nat,X: nat,G: nat > assn] :
% 0.34/0.67 ( ( finite_finite_nat @ A )
% 0.34/0.67 => ( ~ ( member_nat @ X @ A )
% 0.34/0.67 => ( ( groups6906906614972039071t_assn @ G @ ( insert_nat @ X @ A ) )
% 0.34/0.67 = ( times_times_assn @ ( G @ X ) @ ( groups6906906614972039071t_assn @ G @ A ) ) ) ) ) ).
% 0.34/0.67
% 0.34/0.67 % prod.insert
% 0.34/0.67 thf(fact_6942_prod_Oinsert,axiom,
% 0.34/0.67 ! [A: set_Code_integer,X: code_integer,G: code_integer > assn] :
% 0.34/0.67 ( ( finite6017078050557962740nteger @ A )
% 0.34/0.67 => ( ~ ( member_Code_integer @ X @ A )
% 0.34/0.67 => ( ( groups1304777262505850412r_assn @ G @ ( insert_Code_integer @ X @ A ) )
% 0.34/0.67 = ( times_times_assn @ ( G @ X ) @ ( groups1304777262505850412r_assn @ G @ A ) ) ) ) ) ).
% 0.34/0.67
% 0.34/0.67 % prod.insert
% 0.34/0.67 thf(fact_6943_prod_Oinsert,axiom,
% 0.34/0.67 ! [A: set_int,X: int,G: int > rat] :
% 0.34/0.67 ( ( finite_finite_int @ A )
% 0.34/0.67 => ( ~ ( member_int @ X @ A )
% 0.34/0.67 => ( ( groups1072433553688619179nt_rat @ G @ ( insert_int @ X @ A ) )
% 0.34/0.67 = ( times_times_rat @ ( G @ X ) @ ( groups1072433553688619179nt_rat @ G @ A ) ) ) ) ) ).
% 0.34/0.67
% 0.34/0.67 % prod.insert
% 0.34/0.67 thf(fact_6944_prod_Oinsert,axiom,
% 0.34/0.67 ! [A: set_o,X: $o,G: $o > rat] :
% 0.34/0.67 ( ( finite_finite_o @ A )
% 0.34/0.67 => ( ~ ( member_o @ X @ A )
% 0.34/0.67 => ( ( groups2869687844427037835_o_rat @ G @ ( insert_o @ X @ A ) )
% 0.34/0.67 = ( times_times_rat @ ( G @ X ) @ ( groups2869687844427037835_o_rat @ G @ A ) ) ) ) ) ).
% 0.34/0.67
% 0.34/0.67 % prod.insert
% 0.34/0.67 thf(fact_6945_prod_Oinsert,axiom,
% 0.34/0.67 ! [A: set_Product_unit,X: product_unit,G: product_unit > rat] :
% 0.34/0.67 ( ( finite4290736615968046902t_unit @ A )
% 0.34/0.67 => ( ~ ( member_Product_unit @ X @ A )
% 0.34/0.67 => ( ( groups8984663225794451456it_rat @ G @ ( insert_Product_unit @ X @ A ) )
% 0.34/0.67 = ( times_times_rat @ ( G @ X ) @ ( groups8984663225794451456it_rat @ G @ A ) ) ) ) ) ).
% 0.34/0.67
% 0.34/0.67 % prod.insert
% 0.34/0.67 thf(fact_6946_prod_Oinsert,axiom,
% 0.34/0.67 ! [A: set_nat,X: nat,G: nat > rat] :
% 0.34/0.67 ( ( finite_finite_nat @ A )
% 0.34/0.67 => ( ~ ( member_nat @ X @ A )
% 0.34/0.67 => ( ( groups73079841787564623at_rat @ G @ ( insert_nat @ X @ A ) )
% 0.34/0.67 = ( times_times_rat @ ( G @ X ) @ ( groups73079841787564623at_rat @ G @ A ) ) ) ) ) ).
% 0.34/0.67
% 0.34/0.67 % prod.insert
% 0.34/0.67 thf(fact_6947_prod_Oinsert,axiom,
% 0.34/0.67 ! [A: set_Code_integer,X: code_integer,G: code_integer > rat] :
% 0.34/0.67 ( ( finite6017078050557962740nteger @ A )
% 0.34/0.67 => ( ~ ( member_Code_integer @ X @ A )
% 0.34/0.67 => ( ( groups2555765274223993564er_rat @ G @ ( insert_Code_integer @ X @ A ) )
% 0.34/0.67 = ( times_times_rat @ ( G @ X ) @ ( groups2555765274223993564er_rat @ G @ A ) ) ) ) ) ).
% 0.34/0.67
% 0.34/0.67 % prod.insert
% 0.34/0.67 thf(fact_6948_dbl__dec__simps_I2_J,axiom,
% 0.34/0.67 ( ( neg_nu3811975205180677377ec_int @ zero_zero_int )
% 0.34/0.67 = ( uminus_uminus_int @ one_one_int ) ) ).
% 0.34/0.67
% 0.34/0.67 % dbl_dec_simps(2)
% 0.34/0.67 thf(fact_6949_dbl__dec__simps_I2_J,axiom,
% 0.34/0.67 ( ( neg_nu7757733837767384882nteger @ zero_z3403309356797280102nteger )
% 0.34/0.67 = ( uminus1351360451143612070nteger @ one_one_Code_integer ) ) ).
% 0.34/0.67
% 0.34/0.67 % dbl_dec_simps(2)
% 0.34/0.67 thf(fact_6950_minus__equation__iff,axiom,
% 0.34/0.67 ! [A2: int,B: int] :
% 0.34/0.67 ( ( ( uminus_uminus_int @ A2 )
% 0.34/0.67 = B )
% 0.34/0.67 = ( ( uminus_uminus_int @ B )
% 0.34/0.67 = A2 ) ) ).
% 0.34/0.67
% 0.34/0.67 % minus_equation_iff
% 0.34/0.67 thf(fact_6951_minus__equation__iff,axiom,
% 0.34/0.67 ! [A2: code_integer,B: code_integer] :
% 0.34/0.67 ( ( ( uminus1351360451143612070nteger @ A2 )
% 0.34/0.67 = B )
% 0.34/0.67 = ( ( uminus1351360451143612070nteger @ B )
% 0.34/0.67 = A2 ) ) ).
% 0.34/0.67
% 0.34/0.67 % minus_equation_iff
% 0.34/0.67 thf(fact_6952_equation__minus__iff,axiom,
% 0.34/0.67 ! [A2: int,B: int] :
% 0.34/0.67 ( ( A2
% 0.34/0.67 = ( uminus_uminus_int @ B ) )
% 0.34/0.67 = ( B
% 0.34/0.67 = ( uminus_uminus_int @ A2 ) ) ) ).
% 0.34/0.67
% 0.34/0.67 % equation_minus_iff
% 0.34/0.67 thf(fact_6953_equation__minus__iff,axiom,
% 0.34/0.67 ! [A2: code_integer,B: code_integer] :
% 0.34/0.67 ( ( A2
% 0.34/0.67 = ( uminus1351360451143612070nteger @ B ) )
% 0.34/0.67 = ( B
% 0.34/0.67 = ( uminus1351360451143612070nteger @ A2 ) ) ) ).
% 0.34/0.67
% 0.34/0.67 % equation_minus_iff
% 0.34/0.67 thf(fact_6954_prod_Oneutral,axiom,
% 0.34/0.67 ! [A: set_nat,G: nat > nat] :
% 0.34/0.67 ( ! [X3: nat] :
% 0.34/0.67 ( ( member_nat @ X3 @ A )
% 0.34/0.67 => ( ( G @ X3 )
% 0.34/0.67 = one_one_nat ) )
% 0.34/0.67 => ( ( groups708209901874060359at_nat @ G @ A )
% 0.34/0.67 = one_one_nat ) ) ).
% 0.34/0.67
% 0.34/0.67 % prod.neutral
% 0.34/0.67 thf(fact_6955_prod_Onot__neutral__contains__not__neutral,axiom,
% 0.34/0.67 ! [G: nat > assn,A: set_nat] :
% 0.34/0.67 ( ( ( groups6906906614972039071t_assn @ G @ A )
% 0.34/0.67 != one_one_assn )
% 0.34/0.67 => ~ ! [A4: nat] :
% 0.34/0.67 ( ( member_nat @ A4 @ A )
% 0.34/0.67 => ( ( G @ A4 )
% 0.34/0.67 = one_one_assn ) ) ) ).
% 0.34/0.67
% 0.34/0.67 % prod.not_neutral_contains_not_neutral
% 0.34/0.67 thf(fact_6956_prod_Onot__neutral__contains__not__neutral,axiom,
% 0.34/0.67 ! [G: int > assn,A: set_int] :
% 0.34/0.67 ( ( ( groups7882442080178216443t_assn @ G @ A )
% 0.34/0.67 != one_one_assn )
% 0.34/0.67 => ~ ! [A4: int] :
% 0.34/0.67 ( ( member_int @ A4 @ A )
% 0.34/0.67 => ( ( G @ A4 )
% 0.34/0.67 = one_one_assn ) ) ) ).
% 0.34/0.67
% 0.34/0.67 % prod.not_neutral_contains_not_neutral
% 0.34/0.67 thf(fact_6957_prod_Onot__neutral__contains__not__neutral,axiom,
% 0.34/0.67 ! [G: $o > assn,A: set_o] :
% 0.34/0.67 ( ( ( groups5301882518646026715o_assn @ G @ A )
% 0.34/0.67 != one_one_assn )
% 0.34/0.67 => ~ ! [A4: $o] :
% 0.34/0.67 ( ( member_o @ A4 @ A )
% 0.34/0.67 => ( ( G @ A4 )
% 0.34/0.67 = one_one_assn ) ) ) ).
% 0.34/0.67
% 0.34/0.67 % prod.not_neutral_contains_not_neutral
% 0.34/0.67 thf(fact_6958_prod_Onot__neutral__contains__not__neutral,axiom,
% 0.34/0.67 ! [G: product_unit > assn,A: set_Product_unit] :
% 0.34/0.67 ( ( ( groups5360233303959380816t_assn @ G @ A )
% 0.34/0.67 != one_one_assn )
% 0.34/0.67 => ~ ! [A4: product_unit] :
% 0.34/0.67 ( ( member_Product_unit @ A4 @ A )
% 0.34/0.67 => ( ( G @ A4 )
% 0.34/0.67 = one_one_assn ) ) ) ).
% 0.34/0.67
% 0.34/0.67 % prod.not_neutral_contains_not_neutral
% 0.34/0.67 thf(fact_6959_prod_Onot__neutral__contains__not__neutral,axiom,
% 0.34/0.67 ! [G: int > nat,A: set_int] :
% 0.34/0.67 ( ( ( groups1707563613775114915nt_nat @ G @ A )
% 0.34/0.67 != one_one_nat )
% 0.34/0.67 => ~ ! [A4: int] :
% 0.34/0.67 ( ( member_int @ A4 @ A )
% 0.34/0.67 => ( ( G @ A4 )
% 0.34/0.67 = one_one_nat ) ) ) ).
% 0.34/0.67
% 0.34/0.67 % prod.not_neutral_contains_not_neutral
% 0.34/0.67 thf(fact_6960_prod_Onot__neutral__contains__not__neutral,axiom,
% 0.34/0.67 ! [G: $o > nat,A: set_o] :
% 0.34/0.67 ( ( ( groups3504817904513533571_o_nat @ G @ A )
% 0.34/0.67 != one_one_nat )
% 0.34/0.67 => ~ ! [A4: $o] :
% 0.34/0.67 ( ( member_o @ A4 @ A )
% 0.34/0.67 => ( ( G @ A4 )
% 0.34/0.67 = one_one_nat ) ) ) ).
% 0.34/0.67
% 0.34/0.67 % prod.not_neutral_contains_not_neutral
% 0.34/0.67 thf(fact_6961_prod_Onot__neutral__contains__not__neutral,axiom,
% 0.34/0.67 ! [G: product_unit > nat,A: set_Product_unit] :
% 0.34/0.67 ( ( ( groups396421249026171384it_nat @ G @ A )
% 0.34/0.67 != one_one_nat )
% 0.34/0.67 => ~ ! [A4: product_unit] :
% 0.34/0.67 ( ( member_Product_unit @ A4 @ A )
% 0.34/0.67 => ( ( G @ A4 )
% 0.34/0.67 = one_one_nat ) ) ) ).
% 0.34/0.67
% 0.34/0.67 % prod.not_neutral_contains_not_neutral
% 0.34/0.67 thf(fact_6962_prod_Onot__neutral__contains__not__neutral,axiom,
% 0.34/0.67 ! [G: nat > int,A: set_nat] :
% 0.34/0.67 ( ( ( groups705719431365010083at_int @ G @ A )
% 0.34/0.67 != one_one_int )
% 0.34/0.67 => ~ ! [A4: nat] :
% 0.34/0.67 ( ( member_nat @ A4 @ A )
% 0.34/0.67 => ( ( G @ A4 )
% 0.34/0.67 = one_one_int ) ) ) ).
% 0.34/0.67
% 0.34/0.67 % prod.not_neutral_contains_not_neutral
% 0.34/0.67 thf(fact_6963_prod_Onot__neutral__contains__not__neutral,axiom,
% 0.34/0.67 ! [G: int > int,A: set_int] :
% 0.34/0.67 ( ( ( groups1705073143266064639nt_int @ G @ A )
% 0.34/0.67 != one_one_int )
% 0.34/0.67 => ~ ! [A4: int] :
% 0.34/0.67 ( ( member_int @ A4 @ A )
% 0.34/0.67 => ( ( G @ A4 )
% 0.34/0.67 = one_one_int ) ) ) ).
% 0.34/0.67
% 0.34/0.67 % prod.not_neutral_contains_not_neutral
% 0.34/0.67 thf(fact_6964_prod_Onot__neutral__contains__not__neutral,axiom,
% 0.34/0.67 ! [G: $o > int,A: set_o] :
% 0.34/0.67 ( ( ( groups3502327434004483295_o_int @ G @ A )
% 0.34/0.67 != one_one_int )
% 0.34/0.67 => ~ ! [A4: $o] :
% 0.34/0.67 ( ( member_o @ A4 @ A )
% 0.34/0.67 => ( ( G @ A4 )
% 0.34/0.67 = one_one_int ) ) ) ).
% 0.34/0.67
% 0.34/0.67 % prod.not_neutral_contains_not_neutral
% 0.34/0.67 thf(fact_6965_mult__of__nat__commute,axiom,
% 0.34/0.67 ! [X: nat,Y: rat] :
% 0.34/0.67 ( ( times_times_rat @ ( semiri681578069525770553at_rat @ X ) @ Y )
% 0.34/0.67 = ( times_times_rat @ Y @ ( semiri681578069525770553at_rat @ X ) ) ) ).
% 0.34/0.67
% 0.34/0.67 % mult_of_nat_commute
% 0.34/0.67 thf(fact_6966_mult__of__nat__commute,axiom,
% 0.34/0.67 ! [X: nat,Y: int] :
% 0.34/0.67 ( ( times_times_int @ ( semiri1314217659103216013at_int @ X ) @ Y )
% 0.34/0.67 = ( times_times_int @ Y @ ( semiri1314217659103216013at_int @ X ) ) ) ).
% 0.34/0.67
% 0.34/0.67 % mult_of_nat_commute
% 0.34/0.67 thf(fact_6967_mult__of__nat__commute,axiom,
% 0.34/0.67 ! [X: nat,Y: nat] :
% 0.34/0.67 ( ( times_times_nat @ ( semiri1316708129612266289at_nat @ X ) @ Y )
% 0.34/0.67 = ( times_times_nat @ Y @ ( semiri1316708129612266289at_nat @ X ) ) ) ).
% 0.34/0.67
% 0.34/0.67 % mult_of_nat_commute
% 0.34/0.67 thf(fact_6968_mult__of__nat__commute,axiom,
% 0.34/0.67 ! [X: nat,Y: code_integer] :
% 0.34/0.67 ( ( times_3573771949741848930nteger @ ( semiri4939895301339042750nteger @ X ) @ Y )
% 0.34/0.67 = ( times_3573771949741848930nteger @ Y @ ( semiri4939895301339042750nteger @ X ) ) ) ).
% 0.34/0.67
% 0.34/0.67 % mult_of_nat_commute
% 0.34/0.67 thf(fact_6969_le__imp__neg__le,axiom,
% 0.34/0.67 ! [A2: code_integer,B: code_integer] :
% 0.34/0.67 ( ( ord_le3102999989581377725nteger @ A2 @ B )
% 0.34/0.67 => ( ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ B ) @ ( uminus1351360451143612070nteger @ A2 ) ) ) ).
% 0.34/0.67
% 0.34/0.67 % le_imp_neg_le
% 0.34/0.67 thf(fact_6970_le__imp__neg__le,axiom,
% 0.34/0.67 ! [A2: rat,B: rat] :
% 0.34/0.67 ( ( ord_less_eq_rat @ A2 @ B )
% 0.34/0.67 => ( ord_less_eq_rat @ ( uminus_uminus_rat @ B ) @ ( uminus_uminus_rat @ A2 ) ) ) ).
% 0.34/0.67
% 0.34/0.67 % le_imp_neg_le
% 0.34/0.67 thf(fact_6971_le__imp__neg__le,axiom,
% 0.34/0.67 ! [A2: int,B: int] :
% 0.34/0.67 ( ( ord_less_eq_int @ A2 @ B )
% 0.34/0.67 => ( ord_less_eq_int @ ( uminus_uminus_int @ B ) @ ( uminus_uminus_int @ A2 ) ) ) ).
% 0.34/0.67
% 0.34/0.67 % le_imp_neg_le
% 0.34/0.67 thf(fact_6972_minus__le__iff,axiom,
% 0.34/0.67 ! [A2: code_integer,B: code_integer] :
% 0.34/0.67 ( ( ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ A2 ) @ B )
% 0.34/0.67 = ( ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ B ) @ A2 ) ) ).
% 0.34/0.67
% 0.34/0.67 % minus_le_iff
% 0.34/0.67 thf(fact_6973_minus__le__iff,axiom,
% 0.34/0.67 ! [A2: rat,B: rat] :
% 0.34/0.67 ( ( ord_less_eq_rat @ ( uminus_uminus_rat @ A2 ) @ B )
% 0.34/0.67 = ( ord_less_eq_rat @ ( uminus_uminus_rat @ B ) @ A2 ) ) ).
% 0.34/0.67
% 0.34/0.67 % minus_le_iff
% 0.34/0.67 thf(fact_6974_minus__le__iff,axiom,
% 0.34/0.67 ! [A2: int,B: int] :
% 0.34/0.67 ( ( ord_less_eq_int @ ( uminus_uminus_int @ A2 ) @ B )
% 0.34/0.67 = ( ord_less_eq_int @ ( uminus_uminus_int @ B ) @ A2 ) ) ).
% 0.34/0.67
% 0.34/0.67 % minus_le_iff
% 0.34/0.67 thf(fact_6975_le__minus__iff,axiom,
% 0.34/0.67 ! [A2: code_integer,B: code_integer] :
% 0.34/0.67 ( ( ord_le3102999989581377725nteger @ A2 @ ( uminus1351360451143612070nteger @ B ) )
% 0.34/0.67 = ( ord_le3102999989581377725nteger @ B @ ( uminus1351360451143612070nteger @ A2 ) ) ) ).
% 0.34/0.67
% 0.34/0.67 % le_minus_iff
% 0.34/0.67 thf(fact_6976_le__minus__iff,axiom,
% 0.34/0.67 ! [A2: rat,B: rat] :
% 0.34/0.67 ( ( ord_less_eq_rat @ A2 @ ( uminus_uminus_rat @ B ) )
% 0.34/0.67 = ( ord_less_eq_rat @ B @ ( uminus_uminus_rat @ A2 ) ) ) ).
% 0.34/0.67
% 0.34/0.67 % le_minus_iff
% 0.34/0.67 thf(fact_6977_le__minus__iff,axiom,
% 0.34/0.67 ! [A2: int,B: int] :
% 0.34/0.67 ( ( ord_less_eq_int @ A2 @ ( uminus_uminus_int @ B ) )
% 0.34/0.67 = ( ord_less_eq_int @ B @ ( uminus_uminus_int @ A2 ) ) ) ).
% 0.34/0.67
% 0.34/0.67 % le_minus_iff
% 0.34/0.67 thf(fact_6978_minus__less__iff,axiom,
% 0.34/0.67 ! [A2: rat,B: rat] :
% 0.34/0.67 ( ( ord_less_rat @ ( uminus_uminus_rat @ A2 ) @ B )
% 0.34/0.67 = ( ord_less_rat @ ( uminus_uminus_rat @ B ) @ A2 ) ) ).
% 0.34/0.67
% 0.34/0.67 % minus_less_iff
% 0.34/0.67 thf(fact_6979_minus__less__iff,axiom,
% 0.34/0.67 ! [A2: int,B: int] :
% 0.34/0.67 ( ( ord_less_int @ ( uminus_uminus_int @ A2 ) @ B )
% 0.34/0.67 = ( ord_less_int @ ( uminus_uminus_int @ B ) @ A2 ) ) ).
% 0.34/0.67
% 0.34/0.67 % minus_less_iff
% 0.34/0.67 thf(fact_6980_minus__less__iff,axiom,
% 0.34/0.67 ! [A2: code_integer,B: code_integer] :
% 0.34/0.67 ( ( ord_le6747313008572928689nteger @ ( uminus1351360451143612070nteger @ A2 ) @ B )
% 0.34/0.67 = ( ord_le6747313008572928689nteger @ ( uminus1351360451143612070nteger @ B ) @ A2 ) ) ).
% 0.34/0.67
% 0.34/0.67 % minus_less_iff
% 0.34/0.67 thf(fact_6981_less__minus__iff,axiom,
% 0.34/0.67 ! [A2: rat,B: rat] :
% 0.34/0.67 ( ( ord_less_rat @ A2 @ ( uminus_uminus_rat @ B ) )
% 0.34/0.67 = ( ord_less_rat @ B @ ( uminus_uminus_rat @ A2 ) ) ) ).
% 0.34/0.67
% 0.34/0.67 % less_minus_iff
% 0.34/0.67 thf(fact_6982_less__minus__iff,axiom,
% 0.34/0.67 ! [A2: int,B: int] :
% 0.34/0.67 ( ( ord_less_int @ A2 @ ( uminus_uminus_int @ B ) )
% 0.34/0.67 = ( ord_less_int @ B @ ( uminus_uminus_int @ A2 ) ) ) ).
% 0.34/0.67
% 0.34/0.67 % less_minus_iff
% 0.34/0.67 thf(fact_6983_less__minus__iff,axiom,
% 0.34/0.67 ! [A2: code_integer,B: code_integer] :
% 0.34/0.67 ( ( ord_le6747313008572928689nteger @ A2 @ ( uminus1351360451143612070nteger @ B ) )
% 0.34/0.67 = ( ord_le6747313008572928689nteger @ B @ ( uminus1351360451143612070nteger @ A2 ) ) ) ).
% 0.34/0.67
% 0.34/0.67 % less_minus_iff
% 0.34/0.67 thf(fact_6984_add_Oinverse__distrib__swap,axiom,
% 0.34/0.67 ! [A2: rat,B: rat] :
% 0.34/0.67 ( ( uminus_uminus_rat @ ( plus_plus_rat @ A2 @ B ) )
% 0.34/0.67 = ( plus_plus_rat @ ( uminus_uminus_rat @ B ) @ ( uminus_uminus_rat @ A2 ) ) ) ).
% 0.34/0.67
% 0.34/0.67 % add.inverse_distrib_swap
% 0.34/0.67 thf(fact_6985_add_Oinverse__distrib__swap,axiom,
% 0.34/0.67 ! [A2: int,B: int] :
% 0.34/0.67 ( ( uminus_uminus_int @ ( plus_plus_int @ A2 @ B ) )
% 0.34/0.67 = ( plus_plus_int @ ( uminus_uminus_int @ B ) @ ( uminus_uminus_int @ A2 ) ) ) ).
% 0.34/0.67
% 0.34/0.67 % add.inverse_distrib_swap
% 0.34/0.67 thf(fact_6986_add_Oinverse__distrib__swap,axiom,
% 0.34/0.67 ! [A2: code_integer,B: code_integer] :
% 0.34/0.67 ( ( uminus1351360451143612070nteger @ ( plus_p5714425477246183910nteger @ A2 @ B ) )
% 0.34/0.67 = ( plus_p5714425477246183910nteger @ ( uminus1351360451143612070nteger @ B ) @ ( uminus1351360451143612070nteger @ A2 ) ) ) ).
% 0.34/0.67
% 0.34/0.67 % add.inverse_distrib_swap
% 0.34/0.67 thf(fact_6987_group__cancel_Oneg1,axiom,
% 0.34/0.67 ! [A: rat,K3: rat,A2: rat] :
% 0.34/0.67 ( ( A
% 0.34/0.67 = ( plus_plus_rat @ K3 @ A2 ) )
% 0.34/0.67 => ( ( uminus_uminus_rat @ A )
% 0.34/0.67 = ( plus_plus_rat @ ( uminus_uminus_rat @ K3 ) @ ( uminus_uminus_rat @ A2 ) ) ) ) ).
% 0.34/0.67
% 0.34/0.67 % group_cancel.neg1
% 0.34/0.67 thf(fact_6988_group__cancel_Oneg1,axiom,
% 0.34/0.67 ! [A: int,K3: int,A2: int] :
% 0.34/0.67 ( ( A
% 0.34/0.67 = ( plus_plus_int @ K3 @ A2 ) )
% 0.34/0.67 => ( ( uminus_uminus_int @ A )
% 0.34/0.67 = ( plus_plus_int @ ( uminus_uminus_int @ K3 ) @ ( uminus_uminus_int @ A2 ) ) ) ) ).
% 0.34/0.67
% 0.34/0.67 % group_cancel.neg1
% 0.34/0.67 thf(fact_6989_group__cancel_Oneg1,axiom,
% 0.34/0.67 ! [A: code_integer,K3: code_integer,A2: code_integer] :
% 0.34/0.67 ( ( A
% 0.34/0.67 = ( plus_p5714425477246183910nteger @ K3 @ A2 ) )
% 0.34/0.67 => ( ( uminus1351360451143612070nteger @ A )
% 0.34/0.67 = ( plus_p5714425477246183910nteger @ ( uminus1351360451143612070nteger @ K3 ) @ ( uminus1351360451143612070nteger @ A2 ) ) ) ) ).
% 0.34/0.67
% 0.34/0.67 % group_cancel.neg1
% 0.34/0.67 thf(fact_6990_square__eq__iff,axiom,
% 0.34/0.67 ! [A2: rat,B: rat] :
% 0.34/0.67 ( ( ( times_times_rat @ A2 @ A2 )
% 0.34/0.67 = ( times_times_rat @ B @ B ) )
% 0.34/0.67 = ( ( A2 = B )
% 0.34/0.67 | ( A2
% 0.34/0.67 = ( uminus_uminus_rat @ B ) ) ) ) ).
% 0.34/0.67
% 0.34/0.67 % square_eq_iff
% 0.34/0.67 thf(fact_6991_square__eq__iff,axiom,
% 0.34/0.67 ! [A2: int,B: int] :
% 0.34/0.67 ( ( ( times_times_int @ A2 @ A2 )
% 0.34/0.67 = ( times_times_int @ B @ B ) )
% 0.34/0.67 = ( ( A2 = B )
% 0.34/0.67 | ( A2
% 0.34/0.67 = ( uminus_uminus_int @ B ) ) ) ) ).
% 0.34/0.67
% 0.34/0.67 % square_eq_iff
% 0.34/0.67 thf(fact_6992_square__eq__iff,axiom,
% 0.34/0.67 ! [A2: code_integer,B: code_integer] :
% 0.34/0.67 ( ( ( times_3573771949741848930nteger @ A2 @ A2 )
% 0.34/0.67 = ( times_3573771949741848930nteger @ B @ B ) )
% 0.34/0.67 = ( ( A2 = B )
% 0.34/0.67 | ( A2
% 0.34/0.67 = ( uminus1351360451143612070nteger @ B ) ) ) ) ).
% 0.34/0.67
% 0.34/0.67 % square_eq_iff
% 0.34/0.67 thf(fact_6993_minus__mult__commute,axiom,
% 0.34/0.67 ! [A2: rat,B: rat] :
% 0.34/0.67 ( ( times_times_rat @ ( uminus_uminus_rat @ A2 ) @ B )
% 0.34/0.67 = ( times_times_rat @ A2 @ ( uminus_uminus_rat @ B ) ) ) ).
% 0.34/0.67
% 0.34/0.67 % minus_mult_commute
% 0.34/0.67 thf(fact_6994_minus__mult__commute,axiom,
% 0.34/0.67 ! [A2: int,B: int] :
% 0.34/0.67 ( ( times_times_int @ ( uminus_uminus_int @ A2 ) @ B )
% 0.34/0.67 = ( times_times_int @ A2 @ ( uminus_uminus_int @ B ) ) ) ).
% 0.34/0.67
% 0.34/0.67 % minus_mult_commute
% 0.34/0.67 thf(fact_6995_minus__mult__commute,axiom,
% 0.34/0.67 ! [A2: code_integer,B: code_integer] :
% 0.34/0.67 ( ( times_3573771949741848930nteger @ ( uminus1351360451143612070nteger @ A2 ) @ B )
% 0.34/0.67 = ( times_3573771949741848930nteger @ A2 @ ( uminus1351360451143612070nteger @ B ) ) ) ).
% 0.34/0.67
% 0.34/0.67 % minus_mult_commute
% 0.34/0.67 thf(fact_6996_one__neq__neg__one,axiom,
% 0.34/0.67 ( one_one_int
% 0.34/0.67 != ( uminus_uminus_int @ one_one_int ) ) ).
% 0.34/0.67
% 0.34/0.67 % one_neq_neg_one
% 0.34/0.67 thf(fact_6997_one__neq__neg__one,axiom,
% 0.34/0.67 ( one_one_Code_integer
% 0.34/0.67 != ( uminus1351360451143612070nteger @ one_one_Code_integer ) ) ).
% 0.34/0.67
% 0.34/0.67 % one_neq_neg_one
% 0.34/0.67 thf(fact_6998_minus__diff__commute,axiom,
% 0.34/0.67 ! [B: rat,A2: rat] :
% 0.34/0.67 ( ( minus_minus_rat @ ( uminus_uminus_rat @ B ) @ A2 )
% 0.34/0.67 = ( minus_minus_rat @ ( uminus_uminus_rat @ A2 ) @ B ) ) ).
% 0.34/0.67
% 0.34/0.67 % minus_diff_commute
% 0.34/0.67 thf(fact_6999_minus__diff__commute,axiom,
% 0.34/0.67 ! [B: int,A2: int] :
% 0.34/0.67 ( ( minus_minus_int @ ( uminus_uminus_int @ B ) @ A2 )
% 0.34/0.67 = ( minus_minus_int @ ( uminus_uminus_int @ A2 ) @ B ) ) ).
% 0.34/0.67
% 0.34/0.67 % minus_diff_commute
% 0.34/0.67 thf(fact_7000_minus__diff__commute,axiom,
% 0.34/0.67 ! [B: code_integer,A2: code_integer] :
% 0.34/0.67 ( ( minus_8373710615458151222nteger @ ( uminus1351360451143612070nteger @ B ) @ A2 )
% 0.34/0.67 = ( minus_8373710615458151222nteger @ ( uminus1351360451143612070nteger @ A2 ) @ B ) ) ).
% 0.34/0.67
% 0.34/0.67 % minus_diff_commute
% 0.34/0.67 thf(fact_7001_Gcd__1,axiom,
% 0.34/0.67 ! [A: set_nat] :
% 0.34/0.67 ( ( member_nat @ one_one_nat @ A )
% 0.34/0.67 => ( ( gcd_Gcd_nat @ A )
% 0.34/0.67 = one_one_nat ) ) ).
% 0.34/0.67
% 0.34/0.67 % Gcd_1
% 0.34/0.67 thf(fact_7002_Gcd__1,axiom,
% 0.34/0.67 ! [A: set_int] :
% 0.34/0.67 ( ( member_int @ one_one_int @ A )
% 0.34/0.67 => ( ( gcd_Gcd_int @ A )
% 0.34/0.67 = one_one_int ) ) ).
% 0.34/0.67
% 0.34/0.67 % Gcd_1
% 0.34/0.67 thf(fact_7003_minus__assn__def,axiom,
% 0.34/0.67 ( minus_minus_assn
% 0.34/0.67 = ( ^ [A6: assn,B6: assn] : ( inf_inf_assn @ A6 @ ( uminus_uminus_assn @ B6 ) ) ) ) ).
% 0.34/0.67
% 0.34/0.67 % minus_assn_def
% 0.34/0.67 thf(fact_7004_prod__ge__1,axiom,
% 0.34/0.67 ! [A: set_nat,F: nat > code_integer] :
% 0.34/0.67 ( ! [X3: nat] :
% 0.34/0.67 ( ( member_nat @ X3 @ A )
% 0.34/0.67 => ( ord_le3102999989581377725nteger @ one_one_Code_integer @ ( F @ X3 ) ) )
% 0.34/0.67 => ( ord_le3102999989581377725nteger @ one_one_Code_integer @ ( groups3455450783089532116nteger @ F @ A ) ) ) ).
% 0.34/0.67
% 0.34/0.67 % prod_ge_1
% 0.34/0.67 thf(fact_7005_prod__ge__1,axiom,
% 0.34/0.67 ! [A: set_int,F: int > code_integer] :
% 0.34/0.67 ( ! [X3: int] :
% 0.34/0.67 ( ( member_int @ X3 @ A )
% 0.34/0.67 => ( ord_le3102999989581377725nteger @ one_one_Code_integer @ ( F @ X3 ) ) )
% 0.34/0.67 => ( ord_le3102999989581377725nteger @ one_one_Code_integer @ ( groups3827104343326376752nteger @ F @ A ) ) ) ).
% 0.34/0.67
% 0.34/0.67 % prod_ge_1
% 0.34/0.67 thf(fact_7006_prod__ge__1,axiom,
% 0.34/0.67 ! [A: set_o,F: $o > code_integer] :
% 0.34/0.67 ( ! [X3: $o] :
% 0.34/0.67 ( ( member_o @ X3 @ A )
% 0.34/0.67 => ( ord_le3102999989581377725nteger @ one_one_Code_integer @ ( F @ X3 ) ) )
% 0.34/0.67 => ( ord_le3102999989581377725nteger @ one_one_Code_integer @ ( groups7694694392188491536nteger @ F @ A ) ) ) ).
% 0.34/0.67
% 0.34/0.67 % prod_ge_1
% 0.34/0.67 thf(fact_7007_prod__ge__1,axiom,
% 0.34/0.67 ! [A: set_Product_unit,F: product_unit > code_integer] :
% 0.34/0.67 ( ! [X3: product_unit] :
% 0.34/0.67 ( ( member_Product_unit @ X3 @ A )
% 0.34/0.67 => ( ord_le3102999989581377725nteger @ one_one_Code_integer @ ( F @ X3 ) ) )
% 0.34/0.67 => ( ord_le3102999989581377725nteger @ one_one_Code_integer @ ( groups8185359811944078981nteger @ F @ A ) ) ) ).
% 0.34/0.67
% 0.34/0.67 % prod_ge_1
% 0.34/0.67 thf(fact_7008_prod__ge__1,axiom,
% 0.34/0.67 ! [A: set_nat,F: nat > code_natural] :
% 0.34/0.67 ( ! [X3: nat] :
% 0.34/0.67 ( ( member_nat @ X3 @ A )
% 0.34/0.67 => ( ord_le1926595141338095240atural @ one_one_Code_natural @ ( F @ X3 ) ) )
% 0.34/0.67 => ( ord_le1926595141338095240atural @ one_one_Code_natural @ ( groups2279045934846249631atural @ F @ A ) ) ) ).
% 0.34/0.67
% 0.34/0.67 % prod_ge_1
% 0.34/0.67 thf(fact_7009_prod__ge__1,axiom,
% 0.34/0.67 ! [A: set_int,F: int > code_natural] :
% 0.34/0.67 ( ! [X3: int] :
% 0.34/0.67 ( ( member_int @ X3 @ A )
% 0.34/0.67 => ( ord_le1926595141338095240atural @ one_one_Code_natural @ ( F @ X3 ) ) )
% 0.34/0.67 => ( ord_le1926595141338095240atural @ one_one_Code_natural @ ( groups2650699495083094267atural @ F @ A ) ) ) ).
% 0.34/0.67
% 0.34/0.67 % prod_ge_1
% 0.34/0.67 thf(fact_7010_prod__ge__1,axiom,
% 0.34/0.67 ! [A: set_o,F: $o > code_natural] :
% 0.34/0.67 ( ! [X3: $o] :
% 0.34/0.67 ( ( member_o @ X3 @ A )
% 0.34/0.67 => ( ord_le1926595141338095240atural @ one_one_Code_natural @ ( F @ X3 ) ) )
% 0.34/0.67 => ( ord_le1926595141338095240atural @ one_one_Code_natural @ ( groups6518289543945209051atural @ F @ A ) ) ) ).
% 0.34/0.67
% 0.34/0.67 % prod_ge_1
% 0.34/0.67 thf(fact_7011_prod__ge__1,axiom,
% 0.34/0.67 ! [A: set_Product_unit,F: product_unit > code_natural] :
% 0.34/0.67 ( ! [X3: product_unit] :
% 0.34/0.67 ( ( member_Product_unit @ X3 @ A )
% 0.34/0.67 => ( ord_le1926595141338095240atural @ one_one_Code_natural @ ( F @ X3 ) ) )
% 0.34/0.67 => ( ord_le1926595141338095240atural @ one_one_Code_natural @ ( groups7008954963700796496atural @ F @ A ) ) ) ).
% 0.34/0.67
% 0.34/0.67 % prod_ge_1
% 0.34/0.67 thf(fact_7012_prod__ge__1,axiom,
% 0.34/0.67 ! [A: set_nat,F: nat > rat] :
% 0.34/0.67 ( ! [X3: nat] :
% 0.34/0.67 ( ( member_nat @ X3 @ A )
% 0.34/0.67 => ( ord_less_eq_rat @ one_one_rat @ ( F @ X3 ) ) )
% 0.34/0.67 => ( ord_less_eq_rat @ one_one_rat @ ( groups73079841787564623at_rat @ F @ A ) ) ) ).
% 0.34/0.67
% 0.34/0.67 % prod_ge_1
% 0.34/0.67 thf(fact_7013_prod__ge__1,axiom,
% 0.34/0.67 ! [A: set_int,F: int > rat] :
% 0.34/0.67 ( ! [X3: int] :
% 0.34/0.67 ( ( member_int @ X3 @ A )
% 0.34/0.67 => ( ord_less_eq_rat @ one_one_rat @ ( F @ X3 ) ) )
% 0.34/0.67 => ( ord_less_eq_rat @ one_one_rat @ ( groups1072433553688619179nt_rat @ F @ A ) ) ) ).
% 0.34/0.67
% 0.34/0.67 % prod_ge_1
% 0.34/0.67 thf(fact_7014_prod_OUnion__comp,axiom,
% 0.34/0.67 ! [B4: set_set_int,G: int > assn] :
% 0.34/0.67 ( ! [X3: set_int] :
% 0.34/0.67 ( ( member_set_int @ X3 @ B4 )
% 0.34/0.67 => ( finite_finite_int @ X3 ) )
% 0.34/0.67 => ( ! [A12: set_int] :
% 0.34/0.67 ( ( member_set_int @ A12 @ B4 )
% 0.34/0.67 => ! [A23: set_int] :
% 0.34/0.67 ( ( member_set_int @ A23 @ B4 )
% 0.34/0.67 => ( ( A12 != A23 )
% 0.34/0.67 => ! [X3: int] :
% 0.34/0.67 ( ( member_int @ X3 @ A12 )
% 0.34/0.67 => ( ( member_int @ X3 @ A23 )
% 0.34/0.67 => ( ( G @ X3 )
% 0.34/0.67 = one_one_assn ) ) ) ) ) )
% 0.34/0.67 => ( ( groups7882442080178216443t_assn @ G @ ( comple3221217463730067765et_int @ B4 ) )
% 0.34/0.67 = ( comp_s4595925057662634336t_assn @ groups5950967880310095793t_assn @ groups7882442080178216443t_assn @ G @ B4 ) ) ) ) ).
% 0.34/0.67
% 0.34/0.67 % prod.Union_comp
% 0.34/0.67 thf(fact_7015_prod_OUnion__comp,axiom,
% 0.34/0.67 ! [B4: set_set_o,G: $o > assn] :
% 0.34/0.67 ( ! [X3: set_o] :
% 0.34/0.67 ( ( member_set_o @ X3 @ B4 )
% 0.34/0.67 => ( finite_finite_o @ X3 ) )
% 0.34/0.67 => ( ! [A12: set_o] :
% 0.34/0.67 ( ( member_set_o @ A12 @ B4 )
% 0.34/0.67 => ! [A23: set_o] :
% 0.34/0.67 ( ( member_set_o @ A23 @ B4 )
% 0.34/0.67 => ( ( A12 != A23 )
% 0.34/0.67 => ! [X3: $o] :
% 0.34/0.67 ( ( member_o @ X3 @ A12 )
% 0.34/0.67 => ( ( member_o @ X3 @ A23 )
% 0.34/0.67 => ( ( G @ X3 )
% 0.34/0.67 = one_one_assn ) ) ) ) ) )
% 0.34/0.67 => ( ( groups5301882518646026715o_assn @ G @ ( comple90263536869209701_set_o @ B4 ) )
% 0.34/0.67 = ( comp_s518380907012905326o_assn @ groups6963733262650904763o_assn @ groups5301882518646026715o_assn @ G @ B4 ) ) ) ) ).
% 0.34/0.67
% 0.34/0.67 % prod.Union_comp
% 0.34/0.67 thf(fact_7016_prod_OUnion__comp,axiom,
% 0.34/0.67 ! [B4: set_set_Product_unit,G: product_unit > assn] :
% 0.34/0.67 ( ! [X3: set_Product_unit] :
% 0.34/0.67 ( ( member5877623283571906838t_unit @ X3 @ B4 )
% 0.34/0.67 => ( finite4290736615968046902t_unit @ X3 ) )
% 0.34/0.67 => ( ! [A12: set_Product_unit] :
% 0.34/0.67 ( ( member5877623283571906838t_unit @ A12 @ B4 )
% 0.34/0.67 => ! [A23: set_Product_unit] :
% 0.34/0.67 ( ( member5877623283571906838t_unit @ A23 @ B4 )
% 0.34/0.67 => ( ( A12 != A23 )
% 0.34/0.68 => ! [X3: product_unit] :
% 0.34/0.68 ( ( member_Product_unit @ X3 @ A12 )
% 0.34/0.68 => ( ( member_Product_unit @ X3 @ A23 )
% 0.34/0.68 => ( ( G @ X3 )
% 0.34/0.68 = one_one_assn ) ) ) ) ) )
% 0.34/0.68 => ( ( groups5360233303959380816t_assn @ G @ ( comple3349607820504768346t_unit @ B4 ) )
% 0.34/0.68 = ( comp_s8247530669214396195t_assn @ groups2392624107559085104t_assn @ groups5360233303959380816t_assn @ G @ B4 ) ) ) ) ).
% 0.34/0.68
% 0.34/0.68 % prod.Union_comp
% 0.34/0.68 thf(fact_7017_prod_OUnion__comp,axiom,
% 0.34/0.68 ! [B4: set_set_nat,G: nat > assn] :
% 0.34/0.68 ( ! [X3: set_nat] :
% 0.34/0.68 ( ( member_set_nat @ X3 @ B4 )
% 0.34/0.68 => ( finite_finite_nat @ X3 ) )
% 0.34/0.68 => ( ! [A12: set_nat] :
% 0.34/0.68 ( ( member_set_nat @ A12 @ B4 )
% 0.34/0.68 => ! [A23: set_nat] :
% 0.34/0.68 ( ( member_set_nat @ A23 @ B4 )
% 0.34/0.68 => ( ( A12 != A23 )
% 0.34/0.68 => ! [X3: nat] :
% 0.34/0.68 ( ( member_nat @ X3 @ A12 )
% 0.34/0.68 => ( ( member_nat @ X3 @ A23 )
% 0.34/0.68 => ( ( G @ X3 )
% 0.34/0.68 = one_one_assn ) ) ) ) ) )
% 0.34/0.68 => ( ( groups6906906614972039071t_assn @ G @ ( comple7399068483239264473et_nat @ B4 ) )
% 0.34/0.68 = ( comp_s2451231738238893644t_assn @ groups1535173586962943829t_assn @ groups6906906614972039071t_assn @ G @ B4 ) ) ) ) ).
% 0.34/0.68
% 0.34/0.68 % prod.Union_comp
% 0.34/0.68 thf(fact_7018_prod_OUnion__comp,axiom,
% 0.34/0.68 ! [B4: set_set_Code_integer,G: code_integer > assn] :
% 0.34/0.68 ( ! [X3: set_Code_integer] :
% 0.34/0.68 ( ( member1834298987299537450nteger @ X3 @ B4 )
% 0.34/0.68 => ( finite6017078050557962740nteger @ X3 ) )
% 0.34/0.68 => ( ! [A12: set_Code_integer] :
% 0.34/0.68 ( ( member1834298987299537450nteger @ A12 @ B4 )
% 0.34/0.68 => ! [A23: set_Code_integer] :
% 0.34/0.68 ( ( member1834298987299537450nteger @ A23 @ B4 )
% 0.34/0.68 => ( ( A12 != A23 )
% 0.34/0.68 => ! [X3: code_integer] :
% 0.34/0.68 ( ( member_Code_integer @ X3 @ A12 )
% 0.34/0.68 => ( ( member_Code_integer @ X3 @ A23 )
% 0.34/0.68 => ( ( G @ X3 )
% 0.34/0.68 = one_one_assn ) ) ) ) ) )
% 0.34/0.68 => ( ( groups1304777262505850412r_assn @ G @ ( comple739944243200306918nteger @ B4 ) )
% 0.34/0.68 = ( comp_s3531641609786562035r_assn @ groups4245288774754739042r_assn @ groups1304777262505850412r_assn @ G @ B4 ) ) ) ) ).
% 0.34/0.68
% 0.34/0.68 % prod.Union_comp
% 0.34/0.68 thf(fact_7019_prod_OUnion__comp,axiom,
% 0.34/0.68 ! [B4: set_set_int,G: int > nat] :
% 0.34/0.68 ( ! [X3: set_int] :
% 0.34/0.68 ( ( member_set_int @ X3 @ B4 )
% 0.34/0.68 => ( finite_finite_int @ X3 ) )
% 0.34/0.68 => ( ! [A12: set_int] :
% 0.34/0.68 ( ( member_set_int @ A12 @ B4 )
% 0.34/0.68 => ! [A23: set_int] :
% 0.34/0.68 ( ( member_set_int @ A23 @ B4 )
% 0.34/0.68 => ( ( A12 != A23 )
% 0.34/0.68 => ! [X3: int] :
% 0.34/0.68 ( ( member_int @ X3 @ A12 )
% 0.34/0.68 => ( ( member_int @ X3 @ A23 )
% 0.34/0.68 => ( ( G @ X3 )
% 0.34/0.68 = one_one_nat ) ) ) ) ) )
% 0.34/0.68 => ( ( groups1707563613775114915nt_nat @ G @ ( comple3221217463730067765et_int @ B4 ) )
% 0.34/0.68 = ( comp_s3237642481825588056nt_nat @ groups6435469334873047129nt_nat @ groups1707563613775114915nt_nat @ G @ B4 ) ) ) ) ).
% 0.34/0.68
% 0.34/0.68 % prod.Union_comp
% 0.34/0.68 thf(fact_7020_prod_OUnion__comp,axiom,
% 0.34/0.68 ! [B4: set_set_o,G: $o > nat] :
% 0.34/0.68 ( ! [X3: set_o] :
% 0.34/0.68 ( ( member_set_o @ X3 @ B4 )
% 0.34/0.68 => ( finite_finite_o @ X3 ) )
% 0.34/0.68 => ( ! [A12: set_o] :
% 0.34/0.68 ( ( member_set_o @ A12 @ B4 )
% 0.34/0.68 => ! [A23: set_o] :
% 0.34/0.68 ( ( member_set_o @ A23 @ B4 )
% 0.34/0.68 => ( ( A12 != A23 )
% 0.34/0.68 => ! [X3: $o] :
% 0.34/0.68 ( ( member_o @ X3 @ A12 )
% 0.34/0.68 => ( ( member_o @ X3 @ A23 )
% 0.34/0.68 => ( ( G @ X3 )
% 0.34/0.68 = one_one_nat ) ) ) ) ) )
% 0.34/0.68 => ( ( groups3504817904513533571_o_nat @ G @ ( comple90263536869209701_set_o @ B4 ) )
% 0.34/0.68 = ( comp_s6084864194472624150_o_nat @ groups4868773677410035043_o_nat @ groups3504817904513533571_o_nat @ G @ B4 ) ) ) ) ).
% 0.34/0.68
% 0.34/0.68 % prod.Union_comp
% 0.34/0.68 thf(fact_7021_prod_OUnion__comp,axiom,
% 0.34/0.68 ! [B4: set_set_Product_unit,G: product_unit > nat] :
% 0.34/0.68 ( ! [X3: set_Product_unit] :
% 0.34/0.68 ( ( member5877623283571906838t_unit @ X3 @ B4 )
% 0.34/0.68 => ( finite4290736615968046902t_unit @ X3 ) )
% 0.34/0.68 => ( ! [A12: set_Product_unit] :
% 0.34/0.68 ( ( member5877623283571906838t_unit @ A12 @ B4 )
% 0.34/0.68 => ! [A23: set_Product_unit] :
% 0.34/0.68 ( ( member5877623283571906838t_unit @ A23 @ B4 )
% 0.34/0.68 => ( ( A12 != A23 )
% 0.34/0.68 => ! [X3: product_unit] :
% 0.34/0.68 ( ( member_Product_unit @ X3 @ A12 )
% 0.34/0.68 => ( ( member_Product_unit @ X3 @ A23 )
% 0.34/0.68 => ( ( G @ X3 )
% 0.34/0.68 = one_one_nat ) ) ) ) ) )
% 0.34/0.68 => ( ( groups396421249026171384it_nat @ G @ ( comple3349607820504768346t_unit @ B4 ) )
% 0.34/0.68 = ( comp_s8928436226432208331it_nat @ groups8454329047868613336it_nat @ groups396421249026171384it_nat @ G @ B4 ) ) ) ) ).
% 0.34/0.68
% 0.34/0.68 % prod.Union_comp
% 0.34/0.68 thf(fact_7022_prod_OUnion__comp,axiom,
% 0.34/0.68 ! [B4: set_set_Code_integer,G: code_integer > nat] :
% 0.34/0.68 ( ! [X3: set_Code_integer] :
% 0.34/0.68 ( ( member1834298987299537450nteger @ X3 @ B4 )
% 0.34/0.68 => ( finite6017078050557962740nteger @ X3 ) )
% 0.34/0.68 => ( ! [A12: set_Code_integer] :
% 0.34/0.68 ( ( member1834298987299537450nteger @ A12 @ B4 )
% 0.34/0.68 => ! [A23: set_Code_integer] :
% 0.34/0.68 ( ( member1834298987299537450nteger @ A23 @ B4 )
% 0.34/0.68 => ( ( A12 != A23 )
% 0.34/0.68 => ! [X3: code_integer] :
% 0.34/0.68 ( ( member_Code_integer @ X3 @ A12 )
% 0.34/0.68 => ( ( member_Code_integer @ X3 @ A23 )
% 0.34/0.68 => ( ( G @ X3 )
% 0.34/0.68 = one_one_nat ) ) ) ) ) )
% 0.34/0.68 => ( ( groups3190895334310489300er_nat @ G @ ( comple739944243200306918nteger @ B4 ) )
% 0.34/0.68 = ( comp_s8759123648413916139er_nat @ groups6044384099095135754er_nat @ groups3190895334310489300er_nat @ G @ B4 ) ) ) ) ).
% 0.34/0.68
% 0.34/0.68 % prod.Union_comp
% 0.34/0.68 thf(fact_7023_prod_OUnion__comp,axiom,
% 0.34/0.68 ! [B4: set_set_int,G: int > int] :
% 0.34/0.68 ( ! [X3: set_int] :
% 0.34/0.68 ( ( member_set_int @ X3 @ B4 )
% 0.34/0.68 => ( finite_finite_int @ X3 ) )
% 0.34/0.68 => ( ! [A12: set_int] :
% 0.34/0.68 ( ( member_set_int @ A12 @ B4 )
% 0.34/0.68 => ! [A23: set_int] :
% 0.34/0.68 ( ( member_set_int @ A23 @ B4 )
% 0.34/0.68 => ( ( A12 != A23 )
% 0.34/0.68 => ! [X3: int] :
% 0.34/0.68 ( ( member_int @ X3 @ A12 )
% 0.34/0.68 => ( ( member_int @ X3 @ A23 )
% 0.34/0.68 => ( ( G @ X3 )
% 0.34/0.68 = one_one_int ) ) ) ) ) )
% 0.34/0.68 => ( ( groups1705073143266064639nt_int @ G @ ( comple3221217463730067765et_int @ B4 ) )
% 0.34/0.68 = ( comp_s9098231955081793132nt_int @ groups6432978864363996853nt_int @ groups1705073143266064639nt_int @ G @ B4 ) ) ) ) ).
% 0.34/0.68
% 0.34/0.68 % prod.Union_comp
% 0.34/0.68 thf(fact_7024_neg__eq__iff__add__eq__0,axiom,
% 0.34/0.68 ! [A2: rat,B: rat] :
% 0.34/0.68 ( ( ( uminus_uminus_rat @ A2 )
% 0.34/0.68 = B )
% 0.34/0.68 = ( ( plus_plus_rat @ A2 @ B )
% 0.34/0.68 = zero_zero_rat ) ) ).
% 0.34/0.68
% 0.34/0.68 % neg_eq_iff_add_eq_0
% 0.34/0.68 thf(fact_7025_neg__eq__iff__add__eq__0,axiom,
% 0.34/0.68 ! [A2: int,B: int] :
% 0.34/0.68 ( ( ( uminus_uminus_int @ A2 )
% 0.34/0.68 = B )
% 0.34/0.68 = ( ( plus_plus_int @ A2 @ B )
% 0.34/0.68 = zero_zero_int ) ) ).
% 0.34/0.68
% 0.34/0.68 % neg_eq_iff_add_eq_0
% 0.34/0.68 thf(fact_7026_neg__eq__iff__add__eq__0,axiom,
% 0.34/0.68 ! [A2: code_integer,B: code_integer] :
% 0.34/0.68 ( ( ( uminus1351360451143612070nteger @ A2 )
% 0.34/0.68 = B )
% 0.34/0.68 = ( ( plus_p5714425477246183910nteger @ A2 @ B )
% 0.34/0.68 = zero_z3403309356797280102nteger ) ) ).
% 0.34/0.68
% 0.34/0.68 % neg_eq_iff_add_eq_0
% 0.34/0.68 thf(fact_7027_eq__neg__iff__add__eq__0,axiom,
% 0.34/0.68 ! [A2: rat,B: rat] :
% 0.34/0.68 ( ( A2
% 0.34/0.68 = ( uminus_uminus_rat @ B ) )
% 0.34/0.68 = ( ( plus_plus_rat @ A2 @ B )
% 0.34/0.68 = zero_zero_rat ) ) ).
% 0.34/0.68
% 0.34/0.68 % eq_neg_iff_add_eq_0
% 0.34/0.68 thf(fact_7028_eq__neg__iff__add__eq__0,axiom,
% 0.34/0.68 ! [A2: int,B: int] :
% 0.34/0.68 ( ( A2
% 0.34/0.68 = ( uminus_uminus_int @ B ) )
% 0.34/0.68 = ( ( plus_plus_int @ A2 @ B )
% 0.34/0.68 = zero_zero_int ) ) ).
% 0.34/0.68
% 0.34/0.68 % eq_neg_iff_add_eq_0
% 0.34/0.68 thf(fact_7029_eq__neg__iff__add__eq__0,axiom,
% 0.34/0.68 ! [A2: code_integer,B: code_integer] :
% 0.34/0.68 ( ( A2
% 0.34/0.68 = ( uminus1351360451143612070nteger @ B ) )
% 0.34/0.68 = ( ( plus_p5714425477246183910nteger @ A2 @ B )
% 0.34/0.68 = zero_z3403309356797280102nteger ) ) ).
% 0.34/0.68
% 0.34/0.68 % eq_neg_iff_add_eq_0
% 0.34/0.68 thf(fact_7030_add_Oinverse__unique,axiom,
% 0.34/0.68 ! [A2: rat,B: rat] :
% 0.34/0.68 ( ( ( plus_plus_rat @ A2 @ B )
% 0.34/0.68 = zero_zero_rat )
% 0.34/0.68 => ( ( uminus_uminus_rat @ A2 )
% 0.34/0.68 = B ) ) ).
% 0.34/0.68
% 0.34/0.68 % add.inverse_unique
% 0.34/0.68 thf(fact_7031_add_Oinverse__unique,axiom,
% 0.34/0.68 ! [A2: int,B: int] :
% 0.34/0.68 ( ( ( plus_plus_int @ A2 @ B )
% 0.34/0.68 = zero_zero_int )
% 0.34/0.68 => ( ( uminus_uminus_int @ A2 )
% 0.34/0.68 = B ) ) ).
% 0.34/0.68
% 0.34/0.68 % add.inverse_unique
% 0.34/0.68 thf(fact_7032_add_Oinverse__unique,axiom,
% 0.34/0.68 ! [A2: code_integer,B: code_integer] :
% 0.34/0.68 ( ( ( plus_p5714425477246183910nteger @ A2 @ B )
% 0.34/0.68 = zero_z3403309356797280102nteger )
% 0.34/0.68 => ( ( uminus1351360451143612070nteger @ A2 )
% 0.34/0.68 = B ) ) ).
% 0.34/0.68
% 0.34/0.68 % add.inverse_unique
% 0.34/0.68 thf(fact_7033_ab__group__add__class_Oab__left__minus,axiom,
% 0.34/0.68 ! [A2: rat] :
% 0.34/0.68 ( ( plus_plus_rat @ ( uminus_uminus_rat @ A2 ) @ A2 )
% 0.34/0.68 = zero_zero_rat ) ).
% 0.34/0.68
% 0.34/0.68 % ab_group_add_class.ab_left_minus
% 0.34/0.68 thf(fact_7034_ab__group__add__class_Oab__left__minus,axiom,
% 0.34/0.68 ! [A2: int] :
% 0.34/0.68 ( ( plus_plus_int @ ( uminus_uminus_int @ A2 ) @ A2 )
% 0.34/0.68 = zero_zero_int ) ).
% 0.34/0.68
% 0.34/0.68 % ab_group_add_class.ab_left_minus
% 0.34/0.68 thf(fact_7035_ab__group__add__class_Oab__left__minus,axiom,
% 0.34/0.68 ! [A2: code_integer] :
% 0.34/0.68 ( ( plus_p5714425477246183910nteger @ ( uminus1351360451143612070nteger @ A2 ) @ A2 )
% 0.34/0.68 = zero_z3403309356797280102nteger ) ).
% 0.34/0.68
% 0.34/0.68 % ab_group_add_class.ab_left_minus
% 0.34/0.68 thf(fact_7036_add__eq__0__iff,axiom,
% 0.34/0.68 ! [A2: rat,B: rat] :
% 0.34/0.68 ( ( ( plus_plus_rat @ A2 @ B )
% 0.34/0.68 = zero_zero_rat )
% 0.34/0.68 = ( B
% 0.34/0.68 = ( uminus_uminus_rat @ A2 ) ) ) ).
% 0.34/0.68
% 0.34/0.68 % add_eq_0_iff
% 0.34/0.68 thf(fact_7037_add__eq__0__iff,axiom,
% 0.34/0.68 ! [A2: int,B: int] :
% 0.34/0.68 ( ( ( plus_plus_int @ A2 @ B )
% 0.34/0.68 = zero_zero_int )
% 0.34/0.68 = ( B
% 0.34/0.68 = ( uminus_uminus_int @ A2 ) ) ) ).
% 0.34/0.68
% 0.34/0.68 % add_eq_0_iff
% 0.34/0.68 thf(fact_7038_add__eq__0__iff,axiom,
% 0.34/0.68 ! [A2: code_integer,B: code_integer] :
% 0.34/0.68 ( ( ( plus_p5714425477246183910nteger @ A2 @ B )
% 0.34/0.68 = zero_z3403309356797280102nteger )
% 0.34/0.68 = ( B
% 0.34/0.68 = ( uminus1351360451143612070nteger @ A2 ) ) ) ).
% 0.34/0.68
% 0.34/0.68 % add_eq_0_iff
% 0.34/0.68 thf(fact_7039_le__minus__one__simps_I4_J,axiom,
% 0.34/0.68 ~ ( ord_le3102999989581377725nteger @ one_one_Code_integer @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) ) ).
% 0.34/0.68
% 0.34/0.68 % le_minus_one_simps(4)
% 0.34/0.68 thf(fact_7040_le__minus__one__simps_I4_J,axiom,
% 0.34/0.68 ~ ( ord_less_eq_rat @ one_one_rat @ ( uminus_uminus_rat @ one_one_rat ) ) ).
% 0.34/0.68
% 0.34/0.68 % le_minus_one_simps(4)
% 0.34/0.68 thf(fact_7041_le__minus__one__simps_I4_J,axiom,
% 0.34/0.68 ~ ( ord_less_eq_int @ one_one_int @ ( uminus_uminus_int @ one_one_int ) ) ).
% 0.34/0.68
% 0.34/0.68 % le_minus_one_simps(4)
% 0.34/0.68 thf(fact_7042_le__minus__one__simps_I2_J,axiom,
% 0.34/0.68 ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ one_one_Code_integer ).
% 0.34/0.68
% 0.34/0.68 % le_minus_one_simps(2)
% 0.34/0.68 thf(fact_7043_le__minus__one__simps_I2_J,axiom,
% 0.34/0.68 ord_less_eq_rat @ ( uminus_uminus_rat @ one_one_rat ) @ one_one_rat ).
% 0.34/0.68
% 0.34/0.68 % le_minus_one_simps(2)
% 0.34/0.68 thf(fact_7044_le__minus__one__simps_I2_J,axiom,
% 0.34/0.68 ord_less_eq_int @ ( uminus_uminus_int @ one_one_int ) @ one_one_int ).
% 0.34/0.68
% 0.34/0.68 % le_minus_one_simps(2)
% 0.34/0.68 thf(fact_7045_zero__neq__neg__one,axiom,
% 0.34/0.68 ( zero_zero_int
% 0.34/0.68 != ( uminus_uminus_int @ one_one_int ) ) ).
% 0.34/0.68
% 0.34/0.68 % zero_neq_neg_one
% 0.34/0.68 thf(fact_7046_zero__neq__neg__one,axiom,
% 0.34/0.68 ( zero_z3403309356797280102nteger
% 0.34/0.68 != ( uminus1351360451143612070nteger @ one_one_Code_integer ) ) ).
% 0.34/0.68
% 0.34/0.68 % zero_neq_neg_one
% 0.34/0.68 thf(fact_7047_less__minus__one__simps_I2_J,axiom,
% 0.34/0.68 ord_less_rat @ ( uminus_uminus_rat @ one_one_rat ) @ one_one_rat ).
% 0.34/0.68
% 0.34/0.68 % less_minus_one_simps(2)
% 0.34/0.68 thf(fact_7048_less__minus__one__simps_I2_J,axiom,
% 0.34/0.68 ord_less_int @ ( uminus_uminus_int @ one_one_int ) @ one_one_int ).
% 0.34/0.68
% 0.34/0.68 % less_minus_one_simps(2)
% 0.34/0.68 thf(fact_7049_less__minus__one__simps_I2_J,axiom,
% 0.34/0.68 ord_le6747313008572928689nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ one_one_Code_integer ).
% 0.34/0.68
% 0.34/0.68 % less_minus_one_simps(2)
% 0.34/0.68 thf(fact_7050_less__minus__one__simps_I4_J,axiom,
% 0.34/0.68 ~ ( ord_less_rat @ one_one_rat @ ( uminus_uminus_rat @ one_one_rat ) ) ).
% 0.34/0.68
% 0.34/0.68 % less_minus_one_simps(4)
% 0.34/0.68 thf(fact_7051_less__minus__one__simps_I4_J,axiom,
% 0.34/0.68 ~ ( ord_less_int @ one_one_int @ ( uminus_uminus_int @ one_one_int ) ) ).
% 0.34/0.68
% 0.34/0.68 % less_minus_one_simps(4)
% 0.34/0.68 thf(fact_7052_less__minus__one__simps_I4_J,axiom,
% 0.34/0.68 ~ ( ord_le6747313008572928689nteger @ one_one_Code_integer @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) ) ).
% 0.34/0.68
% 0.34/0.68 % less_minus_one_simps(4)
% 0.34/0.68 thf(fact_7053_ab__group__add__class_Oab__diff__conv__add__uminus,axiom,
% 0.34/0.68 ( minus_minus_rat
% 0.34/0.68 = ( ^ [A6: rat,B6: rat] : ( plus_plus_rat @ A6 @ ( uminus_uminus_rat @ B6 ) ) ) ) ).
% 0.34/0.68
% 0.34/0.68 % ab_group_add_class.ab_diff_conv_add_uminus
% 0.34/0.68 thf(fact_7054_ab__group__add__class_Oab__diff__conv__add__uminus,axiom,
% 0.34/0.68 ( minus_minus_int
% 0.34/0.68 = ( ^ [A6: int,B6: int] : ( plus_plus_int @ A6 @ ( uminus_uminus_int @ B6 ) ) ) ) ).
% 0.34/0.68
% 0.34/0.68 % ab_group_add_class.ab_diff_conv_add_uminus
% 0.34/0.68 thf(fact_7055_ab__group__add__class_Oab__diff__conv__add__uminus,axiom,
% 0.34/0.68 ( minus_8373710615458151222nteger
% 0.34/0.68 = ( ^ [A6: code_integer,B6: code_integer] : ( plus_p5714425477246183910nteger @ A6 @ ( uminus1351360451143612070nteger @ B6 ) ) ) ) ).
% 0.34/0.68
% 0.34/0.68 % ab_group_add_class.ab_diff_conv_add_uminus
% 0.34/0.68 thf(fact_7056_diff__conv__add__uminus,axiom,
% 0.34/0.68 ( minus_minus_rat
% 0.34/0.68 = ( ^ [A6: rat,B6: rat] : ( plus_plus_rat @ A6 @ ( uminus_uminus_rat @ B6 ) ) ) ) ).
% 0.34/0.68
% 0.34/0.68 % diff_conv_add_uminus
% 0.34/0.68 thf(fact_7057_diff__conv__add__uminus,axiom,
% 0.34/0.68 ( minus_minus_int
% 0.34/0.68 = ( ^ [A6: int,B6: int] : ( plus_plus_int @ A6 @ ( uminus_uminus_int @ B6 ) ) ) ) ).
% 0.34/0.68
% 0.34/0.68 % diff_conv_add_uminus
% 0.34/0.68 thf(fact_7058_diff__conv__add__uminus,axiom,
% 0.34/0.68 ( minus_8373710615458151222nteger
% 0.34/0.68 = ( ^ [A6: code_integer,B6: code_integer] : ( plus_p5714425477246183910nteger @ A6 @ ( uminus1351360451143612070nteger @ B6 ) ) ) ) ).
% 0.34/0.68
% 0.34/0.68 % diff_conv_add_uminus
% 0.34/0.68 thf(fact_7059_group__cancel_Osub2,axiom,
% 0.34/0.68 ! [B4: rat,K3: rat,B: rat,A2: rat] :
% 0.34/0.68 ( ( B4
% 0.34/0.68 = ( plus_plus_rat @ K3 @ B ) )
% 0.34/0.68 => ( ( minus_minus_rat @ A2 @ B4 )
% 0.34/0.68 = ( plus_plus_rat @ ( uminus_uminus_rat @ K3 ) @ ( minus_minus_rat @ A2 @ B ) ) ) ) ).
% 0.34/0.68
% 0.34/0.68 % group_cancel.sub2
% 0.34/0.68 thf(fact_7060_group__cancel_Osub2,axiom,
% 0.34/0.68 ! [B4: int,K3: int,B: int,A2: int] :
% 0.34/0.68 ( ( B4
% 0.34/0.68 = ( plus_plus_int @ K3 @ B ) )
% 0.34/0.68 => ( ( minus_minus_int @ A2 @ B4 )
% 0.34/0.68 = ( plus_plus_int @ ( uminus_uminus_int @ K3 ) @ ( minus_minus_int @ A2 @ B ) ) ) ) ).
% 0.34/0.68
% 0.34/0.68 % group_cancel.sub2
% 0.34/0.68 thf(fact_7061_group__cancel_Osub2,axiom,
% 0.34/0.68 ! [B4: code_integer,K3: code_integer,B: code_integer,A2: code_integer] :
% 0.34/0.68 ( ( B4
% 0.34/0.68 = ( plus_p5714425477246183910nteger @ K3 @ B ) )
% 0.34/0.68 => ( ( minus_8373710615458151222nteger @ A2 @ B4 )
% 0.34/0.68 = ( plus_p5714425477246183910nteger @ ( uminus1351360451143612070nteger @ K3 ) @ ( minus_8373710615458151222nteger @ A2 @ B ) ) ) ) ).
% 0.34/0.68
% 0.34/0.68 % group_cancel.sub2
% 0.34/0.68 thf(fact_7062_square__eq__1__iff,axiom,
% 0.34/0.68 ! [X: rat] :
% 0.34/0.68 ( ( ( times_times_rat @ X @ X )
% 0.34/0.68 = one_one_rat )
% 0.34/0.68 = ( ( X = one_one_rat )
% 0.34/0.68 | ( X
% 0.34/0.68 = ( uminus_uminus_rat @ one_one_rat ) ) ) ) ).
% 0.34/0.68
% 0.34/0.68 % square_eq_1_iff
% 0.34/0.68 thf(fact_7063_square__eq__1__iff,axiom,
% 0.34/0.68 ! [X: int] :
% 0.34/0.68 ( ( ( times_times_int @ X @ X )
% 0.34/0.68 = one_one_int )
% 0.34/0.68 = ( ( X = one_one_int )
% 0.34/0.68 | ( X
% 0.34/0.68 = ( uminus_uminus_int @ one_one_int ) ) ) ) ).
% 0.34/0.68
% 0.34/0.68 % square_eq_1_iff
% 0.34/0.68 thf(fact_7064_square__eq__1__iff,axiom,
% 0.34/0.68 ! [X: code_integer] :
% 0.34/0.68 ( ( ( times_3573771949741848930nteger @ X @ X )
% 0.34/0.68 = one_one_Code_integer )
% 0.34/0.68 = ( ( X = one_one_Code_integer )
% 0.34/0.68 | ( X
% 0.34/0.68 = ( uminus1351360451143612070nteger @ one_one_Code_integer ) ) ) ) ).
% 0.34/0.68
% 0.34/0.68 % square_eq_1_iff
% 0.34/0.68 thf(fact_7065_inf__cancel__left2,axiom,
% 0.34/0.68 ! [X: set_Product_unit,A2: set_Product_unit,B: set_Product_unit] :
% 0.34/0.68 ( ( inf_in4660618365625256667t_unit @ ( inf_in4660618365625256667t_unit @ ( uminus5944136376168626660t_unit @ X ) @ A2 ) @ ( inf_in4660618365625256667t_unit @ X @ B ) )
% 0.34/0.68 = bot_bo3957492148770167129t_unit ) ).
% 0.34/0.68
% 0.34/0.68 % inf_cancel_left2
% 0.34/0.68 thf(fact_7066_inf__cancel__left2,axiom,
% 0.34/0.68 ! [X: set_o,A2: set_o,B: set_o] :
% 0.34/0.68 ( ( inf_inf_set_o @ ( inf_inf_set_o @ ( uminus_uminus_set_o @ X ) @ A2 ) @ ( inf_inf_set_o @ X @ B ) )
% 0.34/0.68 = bot_bot_set_o ) ).
% 0.34/0.68
% 0.34/0.68 % inf_cancel_left2
% 0.34/0.68 thf(fact_7067_inf__cancel__left2,axiom,
% 0.34/0.68 ! [X: set_nat,A2: set_nat,B: set_nat] :
% 0.34/0.68 ( ( inf_inf_set_nat @ ( inf_inf_set_nat @ ( uminus5710092332889474511et_nat @ X ) @ A2 ) @ ( inf_inf_set_nat @ X @ B ) )
% 0.34/0.68 = bot_bot_set_nat ) ).
% 0.34/0.68
% 0.34/0.68 % inf_cancel_left2
% 0.34/0.68 thf(fact_7068_inf__cancel__left2,axiom,
% 0.34/0.68 ! [X: set_int,A2: set_int,B: set_int] :
% 0.34/0.68 ( ( inf_inf_set_int @ ( inf_inf_set_int @ ( uminus1532241313380277803et_int @ X ) @ A2 ) @ ( inf_inf_set_int @ X @ B ) )
% 0.34/0.68 = bot_bot_set_int ) ).
% 0.34/0.68
% 0.34/0.68 % inf_cancel_left2
% 0.34/0.68 thf(fact_7069_inf__cancel__left2,axiom,
% 0.34/0.68 ! [X: set_Pr4532377907799695533_nat_o,A2: set_Pr4532377907799695533_nat_o,B: set_Pr4532377907799695533_nat_o] :
% 0.34/0.68 ( ( inf_in1906310914598751387_nat_o @ ( inf_in1906310914598751387_nat_o @ ( uminus5254974436814262692_nat_o @ X ) @ A2 ) @ ( inf_in1906310914598751387_nat_o @ X @ B ) )
% 0.34/0.68 = bot_bo7824918357723582233_nat_o ) ).
% 0.34/0.68
% 0.34/0.68 % inf_cancel_left2
% 0.34/0.68 thf(fact_7070_inf__cancel__left2,axiom,
% 0.34/0.68 ! [X: assn,A2: assn,B: assn] :
% 0.34/0.68 ( ( inf_inf_assn @ ( inf_inf_assn @ ( uminus_uminus_assn @ X ) @ A2 ) @ ( inf_inf_assn @ X @ B ) )
% 0.34/0.68 = bot_bot_assn ) ).
% 0.34/0.68
% 0.34/0.68 % inf_cancel_left2
% 0.34/0.68 thf(fact_7071_inf__cancel__left2,axiom,
% 0.34/0.68 ! [X: product_unit,A2: product_unit,B: product_unit] :
% 0.34/0.68 ( ( inf_inf_Product_unit @ ( inf_inf_Product_unit @ ( uminus2952777764628376836t_unit @ X ) @ A2 ) @ ( inf_inf_Product_unit @ X @ B ) )
% 0.34/0.68 = bot_bot_Product_unit ) ).
% 0.34/0.68
% 0.34/0.68 % inf_cancel_left2
% 0.34/0.68 thf(fact_7072_inf__cancel__left1,axiom,
% 0.34/0.68 ! [X: set_Product_unit,A2: set_Product_unit,B: set_Product_unit] :
% 0.34/0.68 ( ( inf_in4660618365625256667t_unit @ ( inf_in4660618365625256667t_unit @ X @ A2 ) @ ( inf_in4660618365625256667t_unit @ ( uminus5944136376168626660t_unit @ X ) @ B ) )
% 0.34/0.68 = bot_bo3957492148770167129t_unit ) ).
% 0.34/0.68
% 0.34/0.68 % inf_cancel_left1
% 0.34/0.68 thf(fact_7073_inf__cancel__left1,axiom,
% 0.34/0.68 ! [X: set_o,A2: set_o,B: set_o] :
% 0.34/0.68 ( ( inf_inf_set_o @ ( inf_inf_set_o @ X @ A2 ) @ ( inf_inf_set_o @ ( uminus_uminus_set_o @ X ) @ B ) )
% 0.34/0.68 = bot_bot_set_o ) ).
% 0.34/0.68
% 0.34/0.68 % inf_cancel_left1
% 0.34/0.68 thf(fact_7074_inf__cancel__left1,axiom,
% 0.34/0.68 ! [X: set_nat,A2: set_nat,B: set_nat] :
% 0.34/0.68 ( ( inf_inf_set_nat @ ( inf_inf_set_nat @ X @ A2 ) @ ( inf_inf_set_nat @ ( uminus5710092332889474511et_nat @ X ) @ B ) )
% 0.34/0.68 = bot_bot_set_nat ) ).
% 0.34/0.68
% 0.34/0.68 % inf_cancel_left1
% 0.34/0.68 thf(fact_7075_inf__cancel__left1,axiom,
% 0.34/0.68 ! [X: set_int,A2: set_int,B: set_int] :
% 0.34/0.68 ( ( inf_inf_set_int @ ( inf_inf_set_int @ X @ A2 ) @ ( inf_inf_set_int @ ( uminus1532241313380277803et_int @ X ) @ B ) )
% 0.34/0.68 = bot_bot_set_int ) ).
% 0.34/0.68
% 0.34/0.68 % inf_cancel_left1
% 0.34/0.68 thf(fact_7076_inf__cancel__left1,axiom,
% 0.34/0.68 ! [X: set_Pr4532377907799695533_nat_o,A2: set_Pr4532377907799695533_nat_o,B: set_Pr4532377907799695533_nat_o] :
% 0.34/0.68 ( ( inf_in1906310914598751387_nat_o @ ( inf_in1906310914598751387_nat_o @ X @ A2 ) @ ( inf_in1906310914598751387_nat_o @ ( uminus5254974436814262692_nat_o @ X ) @ B ) )
% 0.34/0.68 = bot_bo7824918357723582233_nat_o ) ).
% 0.34/0.68
% 0.34/0.68 % inf_cancel_left1
% 0.34/0.68 thf(fact_7077_inf__cancel__left1,axiom,
% 0.34/0.68 ! [X: assn,A2: assn,B: assn] :
% 0.34/0.68 ( ( inf_inf_assn @ ( inf_inf_assn @ X @ A2 ) @ ( inf_inf_assn @ ( uminus_uminus_assn @ X ) @ B ) )
% 0.34/0.68 = bot_bot_assn ) ).
% 0.34/0.68
% 0.34/0.68 % inf_cancel_left1
% 0.34/0.68 thf(fact_7078_inf__cancel__left1,axiom,
% 0.34/0.68 ! [X: product_unit,A2: product_unit,B: product_unit] :
% 0.34/0.68 ( ( inf_inf_Product_unit @ ( inf_inf_Product_unit @ X @ A2 ) @ ( inf_inf_Product_unit @ ( uminus2952777764628376836t_unit @ X ) @ B ) )
% 0.34/0.68 = bot_bot_Product_unit ) ).
% 0.34/0.68
% 0.34/0.68 % inf_cancel_left1
% 0.34/0.68 thf(fact_7079_subset__Compl__self__eq,axiom,
% 0.34/0.68 ! [A: set_Product_unit] :
% 0.34/0.68 ( ( ord_le3507040750410214029t_unit @ A @ ( uminus5944136376168626660t_unit @ A ) )
% 0.34/0.68 = ( A = bot_bo3957492148770167129t_unit ) ) ).
% 0.34/0.68
% 0.34/0.68 % subset_Compl_self_eq
% 0.34/0.68 thf(fact_7080_subset__Compl__self__eq,axiom,
% 0.34/0.68 ! [A: set_o] :
% 0.34/0.68 ( ( ord_less_eq_set_o @ A @ ( uminus_uminus_set_o @ A ) )
% 0.34/0.68 = ( A = bot_bot_set_o ) ) ).
% 0.34/0.68
% 0.34/0.68 % subset_Compl_self_eq
% 0.34/0.68 thf(fact_7081_subset__Compl__self__eq,axiom,
% 0.34/0.68 ! [A: set_int] :
% 0.34/0.68 ( ( ord_less_eq_set_int @ A @ ( uminus1532241313380277803et_int @ A ) )
% 0.34/0.68 = ( A = bot_bot_set_int ) ) ).
% 0.34/0.68
% 0.34/0.68 % subset_Compl_self_eq
% 0.34/0.68 thf(fact_7082_subset__Compl__self__eq,axiom,
% 0.34/0.68 ! [A: set_Pr4532377907799695533_nat_o] :
% 0.34/0.68 ( ( ord_le2965882846123202637_nat_o @ A @ ( uminus5254974436814262692_nat_o @ A ) )
% 0.34/0.68 = ( A = bot_bo7824918357723582233_nat_o ) ) ).
% 0.34/0.68
% 0.34/0.68 % subset_Compl_self_eq
% 0.34/0.68 thf(fact_7083_subset__Compl__self__eq,axiom,
% 0.34/0.68 ! [A: set_nat] :
% 0.34/0.68 ( ( ord_less_eq_set_nat @ A @ ( uminus5710092332889474511et_nat @ A ) )
% 0.34/0.68 = ( A = bot_bot_set_nat ) ) ).
% 0.34/0.68
% 0.34/0.68 % subset_Compl_self_eq
% 0.34/0.68 thf(fact_7084_Compl__UNIV__eq,axiom,
% 0.34/0.68 ( ( uminus5254974436814262692_nat_o @ top_to4869887016220417533_nat_o )
% 0.34/0.68 = bot_bo7824918357723582233_nat_o ) ).
% 0.34/0.68
% 0.34/0.68 % Compl_UNIV_eq
% 0.34/0.68 thf(fact_7085_Compl__UNIV__eq,axiom,
% 0.34/0.68 ( ( uminus5944136376168626660t_unit @ top_to1996260823553986621t_unit )
% 0.34/0.68 = bot_bo3957492148770167129t_unit ) ).
% 0.34/0.68
% 0.34/0.68 % Compl_UNIV_eq
% 0.34/0.68 thf(fact_7086_Compl__UNIV__eq,axiom,
% 0.34/0.68 ( ( uminus_uminus_set_o @ top_top_set_o )
% 0.34/0.68 = bot_bot_set_o ) ).
% 0.34/0.68
% 0.34/0.68 % Compl_UNIV_eq
% 0.34/0.68 thf(fact_7087_Compl__UNIV__eq,axiom,
% 0.34/0.68 ( ( uminus5710092332889474511et_nat @ top_top_set_nat )
% 0.34/0.68 = bot_bot_set_nat ) ).
% 0.34/0.68
% 0.34/0.68 % Compl_UNIV_eq
% 0.34/0.68 thf(fact_7088_Compl__UNIV__eq,axiom,
% 0.34/0.68 ( ( uminus1532241313380277803et_int @ top_top_set_int )
% 0.34/0.68 = bot_bot_set_int ) ).
% 0.34/0.68
% 0.34/0.68 % Compl_UNIV_eq
% 0.34/0.68 thf(fact_7089_Compl__empty__eq,axiom,
% 0.34/0.68 ( ( uminus5254974436814262692_nat_o @ bot_bo7824918357723582233_nat_o )
% 0.34/0.68 = top_to4869887016220417533_nat_o ) ).
% 0.34/0.68
% 0.34/0.68 % Compl_empty_eq
% 0.34/0.68 thf(fact_7090_Compl__empty__eq,axiom,
% 0.34/0.68 ( ( uminus5944136376168626660t_unit @ bot_bo3957492148770167129t_unit )
% 0.34/0.68 = top_to1996260823553986621t_unit ) ).
% 0.34/0.68
% 0.34/0.68 % Compl_empty_eq
% 0.34/0.68 thf(fact_7091_Compl__empty__eq,axiom,
% 0.34/0.68 ( ( uminus_uminus_set_o @ bot_bot_set_o )
% 0.34/0.68 = top_top_set_o ) ).
% 0.34/0.68
% 0.34/0.68 % Compl_empty_eq
% 0.34/0.68 thf(fact_7092_Compl__empty__eq,axiom,
% 0.34/0.68 ( ( uminus5710092332889474511et_nat @ bot_bot_set_nat )
% 0.34/0.68 = top_top_set_nat ) ).
% 0.34/0.68
% 0.34/0.68 % Compl_empty_eq
% 0.34/0.68 thf(fact_7093_Compl__empty__eq,axiom,
% 0.34/0.68 ( ( uminus1532241313380277803et_int @ bot_bot_set_int )
% 0.34/0.68 = top_top_set_int ) ).
% 0.34/0.68
% 0.34/0.68 % Compl_empty_eq
% 0.34/0.68 thf(fact_7094_Compl__partition2,axiom,
% 0.34/0.68 ! [A: set_Product_unit] :
% 0.34/0.68 ( ( sup_su793286257634532545t_unit @ ( uminus5944136376168626660t_unit @ A ) @ A )
% 0.34/0.68 = top_to1996260823553986621t_unit ) ).
% 0.34/0.68
% 0.34/0.68 % Compl_partition2
% 0.34/0.68 thf(fact_7095_Compl__partition2,axiom,
% 0.34/0.68 ! [A: set_o] :
% 0.34/0.68 ( ( sup_sup_set_o @ ( uminus_uminus_set_o @ A ) @ A )
% 0.34/0.68 = top_top_set_o ) ).
% 0.34/0.68
% 0.34/0.68 % Compl_partition2
% 0.34/0.68 thf(fact_7096_Compl__partition2,axiom,
% 0.34/0.68 ! [A: set_nat] :
% 0.34/0.68 ( ( sup_sup_set_nat @ ( uminus5710092332889474511et_nat @ A ) @ A )
% 0.34/0.68 = top_top_set_nat ) ).
% 0.34/0.68
% 0.34/0.68 % Compl_partition2
% 0.34/0.68 thf(fact_7097_Compl__partition2,axiom,
% 0.34/0.68 ! [A: set_int] :
% 0.34/0.68 ( ( sup_sup_set_int @ ( uminus1532241313380277803et_int @ A ) @ A )
% 0.34/0.68 = top_top_set_int ) ).
% 0.34/0.68
% 0.34/0.68 % Compl_partition2
% 0.34/0.68 thf(fact_7098_Compl__partition,axiom,
% 0.34/0.68 ! [A: set_Product_unit] :
% 0.34/0.68 ( ( sup_su793286257634532545t_unit @ A @ ( uminus5944136376168626660t_unit @ A ) )
% 0.34/0.68 = top_to1996260823553986621t_unit ) ).
% 0.34/0.68
% 0.34/0.68 % Compl_partition
% 0.34/0.68 thf(fact_7099_Compl__partition,axiom,
% 0.34/0.68 ! [A: set_o] :
% 0.34/0.68 ( ( sup_sup_set_o @ A @ ( uminus_uminus_set_o @ A ) )
% 0.34/0.68 = top_top_set_o ) ).
% 0.34/0.68
% 0.34/0.68 % Compl_partition
% 0.34/0.68 thf(fact_7100_Compl__partition,axiom,
% 0.34/0.68 ! [A: set_nat] :
% 0.34/0.68 ( ( sup_sup_set_nat @ A @ ( uminus5710092332889474511et_nat @ A ) )
% 0.34/0.68 = top_top_set_nat ) ).
% 0.34/0.68
% 0.34/0.68 % Compl_partition
% 0.34/0.68 thf(fact_7101_Compl__partition,axiom,
% 0.34/0.68 ! [A: set_int] :
% 0.34/0.68 ( ( sup_sup_set_int @ A @ ( uminus1532241313380277803et_int @ A ) )
% 0.34/0.68 = top_top_set_int ) ).
% 0.34/0.68
% 0.34/0.68 % Compl_partition
% 0.34/0.68 thf(fact_7102_Compl__eq__Diff__UNIV,axiom,
% 0.34/0.68 ( uminus5944136376168626660t_unit
% 0.34/0.68 = ( minus_6452836326544984404t_unit @ top_to1996260823553986621t_unit ) ) ).
% 0.34/0.68
% 0.34/0.68 % Compl_eq_Diff_UNIV
% 0.34/0.68 thf(fact_7103_Compl__eq__Diff__UNIV,axiom,
% 0.34/0.68 ( uminus_uminus_set_o
% 0.34/0.68 = ( minus_minus_set_o @ top_top_set_o ) ) ).
% 0.34/0.68
% 0.34/0.68 % Compl_eq_Diff_UNIV
% 0.34/0.68 thf(fact_7104_Compl__eq__Diff__UNIV,axiom,
% 0.34/0.68 ( uminus1532241313380277803et_int
% 0.34/0.68 = ( minus_minus_set_int @ top_top_set_int ) ) ).
% 0.34/0.68
% 0.34/0.68 % Compl_eq_Diff_UNIV
% 0.34/0.68 thf(fact_7105_Compl__eq__Diff__UNIV,axiom,
% 0.34/0.68 ( uminus5710092332889474511et_nat
% 0.34/0.68 = ( minus_minus_set_nat @ top_top_set_nat ) ) ).
% 0.34/0.68
% 0.34/0.68 % Compl_eq_Diff_UNIV
% 0.34/0.68 thf(fact_7106_Compl__Int,axiom,
% 0.34/0.68 ! [A: set_nat,B4: set_nat] :
% 0.34/0.68 ( ( uminus5710092332889474511et_nat @ ( inf_inf_set_nat @ A @ B4 ) )
% 0.34/0.68 = ( sup_sup_set_nat @ ( uminus5710092332889474511et_nat @ A ) @ ( uminus5710092332889474511et_nat @ B4 ) ) ) ).
% 0.34/0.68
% 0.34/0.68 % Compl_Int
% 0.34/0.68 thf(fact_7107_Compl__Un,axiom,
% 0.34/0.68 ! [A: set_nat,B4: set_nat] :
% 0.34/0.68 ( ( uminus5710092332889474511et_nat @ ( sup_sup_set_nat @ A @ B4 ) )
% 0.34/0.68 = ( inf_inf_set_nat @ ( uminus5710092332889474511et_nat @ A ) @ ( uminus5710092332889474511et_nat @ B4 ) ) ) ).
% 0.34/0.68
% 0.34/0.68 % Compl_Un
% 0.34/0.68 thf(fact_7108_gbinomial__negated__upper,axiom,
% 0.34/0.68 ( gbinomial_rat
% 0.34/0.68 = ( ^ [A6: rat,K5: nat] : ( times_times_rat @ ( power_power_rat @ ( uminus_uminus_rat @ one_one_rat ) @ K5 ) @ ( gbinomial_rat @ ( minus_minus_rat @ ( minus_minus_rat @ ( semiri681578069525770553at_rat @ K5 ) @ A6 ) @ one_one_rat ) @ K5 ) ) ) ) ).
% 0.34/0.68
% 0.34/0.68 % gbinomial_negated_upper
% 0.34/0.68 thf(fact_7109_gbinomial__index__swap,axiom,
% 0.34/0.68 ! [K3: nat,N2: nat] :
% 0.34/0.68 ( ( times_times_rat @ ( power_power_rat @ ( uminus_uminus_rat @ one_one_rat ) @ K3 ) @ ( gbinomial_rat @ ( minus_minus_rat @ ( uminus_uminus_rat @ ( semiri681578069525770553at_rat @ N2 ) ) @ one_one_rat ) @ K3 ) )
% 0.34/0.68 = ( times_times_rat @ ( power_power_rat @ ( uminus_uminus_rat @ one_one_rat ) @ N2 ) @ ( gbinomial_rat @ ( minus_minus_rat @ ( uminus_uminus_rat @ ( semiri681578069525770553at_rat @ K3 ) ) @ one_one_rat ) @ N2 ) ) ) ).
% 0.34/0.68
% 0.34/0.68 % gbinomial_index_swap
% 0.34/0.68 thf(fact_7110_prod_OUnion__disjoint,axiom,
% 0.34/0.68 ! [C5: set_set_nat,G: nat > nat] :
% 0.34/0.68 ( ! [X3: set_nat] :
% 0.34/0.68 ( ( member_set_nat @ X3 @ C5 )
% 0.34/0.68 => ( finite_finite_nat @ X3 ) )
% 0.34/0.68 => ( ! [X3: set_nat] :
% 0.34/0.68 ( ( member_set_nat @ X3 @ C5 )
% 0.34/0.68 => ! [Xa3: set_nat] :
% 0.34/0.68 ( ( member_set_nat @ Xa3 @ C5 )
% 0.34/0.68 => ( ( X3 != Xa3 )
% 0.34/0.68 => ( ( inf_inf_set_nat @ X3 @ Xa3 )
% 0.34/0.68 = bot_bot_set_nat ) ) ) )
% 0.34/0.68 => ( ( groups708209901874060359at_nat @ G @ ( comple7399068483239264473et_nat @ C5 ) )
% 0.34/0.68 = ( comp_s5456809700045850692at_nat @ groups4248547760180025341at_nat @ groups708209901874060359at_nat @ G @ C5 ) ) ) ) ).
% 0.34/0.68
% 0.34/0.68 % prod.Union_disjoint
% 0.34/0.68 thf(fact_7111_Gcd__remove0__nat,axiom,
% 0.34/0.68 ! [M: set_nat] :
% 0.34/0.68 ( ( finite_finite_nat @ M )
% 0.34/0.68 => ( ( gcd_Gcd_nat @ M )
% 0.34/0.68 = ( gcd_Gcd_nat @ ( minus_minus_set_nat @ M @ ( insert_nat @ zero_zero_nat @ bot_bot_set_nat ) ) ) ) ) ).
% 0.34/0.68
% 0.34/0.68 % Gcd_remove0_nat
% 0.34/0.68 thf(fact_7112_Gcd__eq__1__I,axiom,
% 0.34/0.68 ! [A2: nat,A: set_nat] :
% 0.34/0.68 ( ( dvd_dvd_nat @ A2 @ one_one_nat )
% 0.34/0.68 => ( ( member_nat @ A2 @ A )
% 0.34/0.68 => ( ( gcd_Gcd_nat @ A )
% 0.34/0.68 = one_one_nat ) ) ) ).
% 0.34/0.68
% 0.34/0.68 % Gcd_eq_1_I
% 0.34/0.68 thf(fact_7113_Gcd__eq__1__I,axiom,
% 0.34/0.68 ! [A2: int,A: set_int] :
% 0.34/0.68 ( ( dvd_dvd_int @ A2 @ one_one_int )
% 0.34/0.68 => ( ( member_int @ A2 @ A )
% 0.34/0.68 => ( ( gcd_Gcd_int @ A )
% 0.34/0.68 = one_one_int ) ) ) ).
% 0.34/0.68
% 0.34/0.68 % Gcd_eq_1_I
% 0.34/0.68 thf(fact_7114_prod__le__1,axiom,
% 0.34/0.68 ! [A: set_nat,F: nat > code_natural] :
% 0.34/0.68 ( ! [X3: nat] :
% 0.34/0.68 ( ( member_nat @ X3 @ A )
% 0.34/0.68 => ( ( ord_le1926595141338095240atural @ zero_z2226904508553997617atural @ ( F @ X3 ) )
% 0.34/0.68 & ( ord_le1926595141338095240atural @ ( F @ X3 ) @ one_one_Code_natural ) ) )
% 0.34/0.68 => ( ord_le1926595141338095240atural @ ( groups2279045934846249631atural @ F @ A ) @ one_one_Code_natural ) ) ).
% 0.34/0.68
% 0.34/0.68 % prod_le_1
% 0.34/0.68 thf(fact_7115_prod__le__1,axiom,
% 0.34/0.68 ! [A: set_int,F: int > code_natural] :
% 0.34/0.68 ( ! [X3: int] :
% 0.34/0.68 ( ( member_int @ X3 @ A )
% 0.34/0.68 => ( ( ord_le1926595141338095240atural @ zero_z2226904508553997617atural @ ( F @ X3 ) )
% 0.34/0.68 & ( ord_le1926595141338095240atural @ ( F @ X3 ) @ one_one_Code_natural ) ) )
% 0.34/0.68 => ( ord_le1926595141338095240atural @ ( groups2650699495083094267atural @ F @ A ) @ one_one_Code_natural ) ) ).
% 0.34/0.68
% 0.34/0.68 % prod_le_1
% 0.34/0.68 thf(fact_7116_prod__le__1,axiom,
% 0.34/0.68 ! [A: set_o,F: $o > code_natural] :
% 0.34/0.68 ( ! [X3: $o] :
% 0.34/0.68 ( ( member_o @ X3 @ A )
% 0.34/0.68 => ( ( ord_le1926595141338095240atural @ zero_z2226904508553997617atural @ ( F @ X3 ) )
% 0.34/0.68 & ( ord_le1926595141338095240atural @ ( F @ X3 ) @ one_one_Code_natural ) ) )
% 0.34/0.68 => ( ord_le1926595141338095240atural @ ( groups6518289543945209051atural @ F @ A ) @ one_one_Code_natural ) ) ).
% 0.34/0.68
% 0.34/0.68 % prod_le_1
% 0.34/0.68 thf(fact_7117_prod__le__1,axiom,
% 0.34/0.68 ! [A: set_Product_unit,F: product_unit > code_natural] :
% 0.34/0.68 ( ! [X3: product_unit] :
% 0.34/0.68 ( ( member_Product_unit @ X3 @ A )
% 0.34/0.68 => ( ( ord_le1926595141338095240atural @ zero_z2226904508553997617atural @ ( F @ X3 ) )
% 0.34/0.68 & ( ord_le1926595141338095240atural @ ( F @ X3 ) @ one_one_Code_natural ) ) )
% 0.34/0.68 => ( ord_le1926595141338095240atural @ ( groups7008954963700796496atural @ F @ A ) @ one_one_Code_natural ) ) ).
% 0.34/0.68
% 0.34/0.68 % prod_le_1
% 0.34/0.68 thf(fact_7118_prod__le__1,axiom,
% 0.34/0.68 ! [A: set_nat,F: nat > code_integer] :
% 0.34/0.68 ( ! [X3: nat] :
% 0.34/0.68 ( ( member_nat @ X3 @ A )
% 0.34/0.68 => ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ ( F @ X3 ) )
% 0.34/0.68 & ( ord_le3102999989581377725nteger @ ( F @ X3 ) @ one_one_Code_integer ) ) )
% 0.34/0.68 => ( ord_le3102999989581377725nteger @ ( groups3455450783089532116nteger @ F @ A ) @ one_one_Code_integer ) ) ).
% 0.34/0.68
% 0.34/0.68 % prod_le_1
% 0.34/0.68 thf(fact_7119_prod__le__1,axiom,
% 0.34/0.68 ! [A: set_int,F: int > code_integer] :
% 0.34/0.68 ( ! [X3: int] :
% 0.34/0.68 ( ( member_int @ X3 @ A )
% 0.34/0.68 => ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ ( F @ X3 ) )
% 0.34/0.68 & ( ord_le3102999989581377725nteger @ ( F @ X3 ) @ one_one_Code_integer ) ) )
% 0.34/0.68 => ( ord_le3102999989581377725nteger @ ( groups3827104343326376752nteger @ F @ A ) @ one_one_Code_integer ) ) ).
% 0.34/0.68
% 0.34/0.68 % prod_le_1
% 0.34/0.68 thf(fact_7120_prod__le__1,axiom,
% 0.34/0.68 ! [A: set_o,F: $o > code_integer] :
% 0.34/0.68 ( ! [X3: $o] :
% 0.34/0.68 ( ( member_o @ X3 @ A )
% 0.34/0.68 => ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ ( F @ X3 ) )
% 0.34/0.68 & ( ord_le3102999989581377725nteger @ ( F @ X3 ) @ one_one_Code_integer ) ) )
% 0.34/0.68 => ( ord_le3102999989581377725nteger @ ( groups7694694392188491536nteger @ F @ A ) @ one_one_Code_integer ) ) ).
% 0.34/0.68
% 0.34/0.68 % prod_le_1
% 0.34/0.68 thf(fact_7121_prod__le__1,axiom,
% 0.34/0.68 ! [A: set_Product_unit,F: product_unit > code_integer] :
% 0.34/0.68 ( ! [X3: product_unit] :
% 0.34/0.68 ( ( member_Product_unit @ X3 @ A )
% 0.34/0.68 => ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ ( F @ X3 ) )
% 0.34/0.68 & ( ord_le3102999989581377725nteger @ ( F @ X3 ) @ one_one_Code_integer ) ) )
% 0.34/0.68 => ( ord_le3102999989581377725nteger @ ( groups8185359811944078981nteger @ F @ A ) @ one_one_Code_integer ) ) ).
% 0.34/0.68
% 0.34/0.68 % prod_le_1
% 0.34/0.68 thf(fact_7122_prod__le__1,axiom,
% 0.34/0.68 ! [A: set_nat,F: nat > rat] :
% 0.34/0.68 ( ! [X3: nat] :
% 0.34/0.68 ( ( member_nat @ X3 @ A )
% 0.34/0.68 => ( ( ord_less_eq_rat @ zero_zero_rat @ ( F @ X3 ) )
% 0.34/0.68 & ( ord_less_eq_rat @ ( F @ X3 ) @ one_one_rat ) ) )
% 0.34/0.68 => ( ord_less_eq_rat @ ( groups73079841787564623at_rat @ F @ A ) @ one_one_rat ) ) ).
% 0.34/0.68
% 0.34/0.68 % prod_le_1
% 0.34/0.68 thf(fact_7123_prod__le__1,axiom,
% 0.34/0.68 ! [A: set_int,F: int > rat] :
% 0.34/0.68 ( ! [X3: int] :
% 0.34/0.68 ( ( member_int @ X3 @ A )
% 0.34/0.68 => ( ( ord_less_eq_rat @ zero_zero_rat @ ( F @ X3 ) )
% 0.34/0.68 & ( ord_less_eq_rat @ ( F @ X3 ) @ one_one_rat ) ) )
% 0.34/0.68 => ( ord_less_eq_rat @ ( groups1072433553688619179nt_rat @ F @ A ) @ one_one_rat ) ) ).
% 0.34/0.68
% 0.34/0.68 % prod_le_1
% 0.34/0.68 thf(fact_7124_prod_Orelated,axiom,
% 0.34/0.68 ! [R3: code_integer > code_integer > $o,S: set_nat,H2: nat > code_integer,G: nat > code_integer] :
% 0.34/0.68 ( ( R3 @ one_one_Code_integer @ one_one_Code_integer )
% 0.34/0.68 => ( ! [X12: code_integer,Y12: code_integer,X23: code_integer,Y23: code_integer] :
% 0.34/0.68 ( ( ( R3 @ X12 @ X23 )
% 0.34/0.68 & ( R3 @ Y12 @ Y23 ) )
% 0.34/0.68 => ( R3 @ ( times_3573771949741848930nteger @ X12 @ Y12 ) @ ( times_3573771949741848930nteger @ X23 @ Y23 ) ) )
% 0.34/0.68 => ( ( finite_finite_nat @ S )
% 0.34/0.68 => ( ! [X3: nat] :
% 0.34/0.68 ( ( member_nat @ X3 @ S )
% 0.34/0.68 => ( R3 @ ( H2 @ X3 ) @ ( G @ X3 ) ) )
% 0.34/0.68 => ( R3 @ ( groups3455450783089532116nteger @ H2 @ S ) @ ( groups3455450783089532116nteger @ G @ S ) ) ) ) ) ) ).
% 0.34/0.68
% 0.34/0.68 % prod.related
% 0.34/0.68 thf(fact_7125_prod_Orelated,axiom,
% 0.34/0.68 ! [R3: code_integer > code_integer > $o,S: set_Code_integer,H2: code_integer > code_integer,G: code_integer > code_integer] :
% 0.34/0.68 ( ( R3 @ one_one_Code_integer @ one_one_Code_integer )
% 0.34/0.68 => ( ! [X12: code_integer,Y12: code_integer,X23: code_integer,Y23: code_integer] :
% 0.34/0.68 ( ( ( R3 @ X12 @ X23 )
% 0.34/0.68 & ( R3 @ Y12 @ Y23 ) )
% 0.34/0.68 => ( R3 @ ( times_3573771949741848930nteger @ X12 @ Y12 ) @ ( times_3573771949741848930nteger @ X23 @ Y23 ) ) )
% 0.34/0.68 => ( ( finite6017078050557962740nteger @ S )
% 0.34/0.68 => ( ! [X3: code_integer] :
% 0.34/0.68 ( ( member_Code_integer @ X3 @ S )
% 0.34/0.68 => ( R3 @ ( H2 @ X3 ) @ ( G @ X3 ) ) )
% 0.34/0.68 => ( R3 @ ( groups3674199335183972705nteger @ H2 @ S ) @ ( groups3674199335183972705nteger @ G @ S ) ) ) ) ) ) ).
% 0.34/0.68
% 0.34/0.68 % prod.related
% 0.34/0.68 thf(fact_7126_prod_Orelated,axiom,
% 0.34/0.68 ! [R3: code_natural > code_natural > $o,S: set_nat,H2: nat > code_natural,G: nat > code_natural] :
% 0.34/0.68 ( ( R3 @ one_one_Code_natural @ one_one_Code_natural )
% 0.34/0.68 => ( ! [X12: code_natural,Y12: code_natural,X23: code_natural,Y23: code_natural] :
% 0.34/0.68 ( ( ( R3 @ X12 @ X23 )
% 0.34/0.68 & ( R3 @ Y12 @ Y23 ) )
% 0.34/0.68 => ( R3 @ ( times_2397367101498566445atural @ X12 @ Y12 ) @ ( times_2397367101498566445atural @ X23 @ Y23 ) ) )
% 0.34/0.68 => ( ( finite_finite_nat @ S )
% 0.34/0.68 => ( ! [X3: nat] :
% 0.34/0.68 ( ( member_nat @ X3 @ S )
% 0.34/0.68 => ( R3 @ ( H2 @ X3 ) @ ( G @ X3 ) ) )
% 0.34/0.68 => ( R3 @ ( groups2279045934846249631atural @ H2 @ S ) @ ( groups2279045934846249631atural @ G @ S ) ) ) ) ) ) ).
% 0.34/0.68
% 0.34/0.68 % prod.related
% 0.34/0.68 thf(fact_7127_prod_Orelated,axiom,
% 0.34/0.68 ! [R3: code_natural > code_natural > $o,S: set_Code_integer,H2: code_integer > code_natural,G: code_integer > code_natural] :
% 0.34/0.68 ( ( R3 @ one_one_Code_natural @ one_one_Code_natural )
% 0.34/0.68 => ( ! [X12: code_natural,Y12: code_natural,X23: code_natural,Y23: code_natural] :
% 0.34/0.68 ( ( ( R3 @ X12 @ X23 )
% 0.34/0.68 & ( R3 @ Y12 @ Y23 ) )
% 0.34/0.68 => ( R3 @ ( times_2397367101498566445atural @ X12 @ Y12 ) @ ( times_2397367101498566445atural @ X23 @ Y23 ) ) )
% 0.34/0.68 => ( ( finite6017078050557962740nteger @ S )
% 0.34/0.68 => ( ! [X3: code_integer] :
% 0.34/0.68 ( ( member_Code_integer @ X3 @ S )
% 0.34/0.68 => ( R3 @ ( H2 @ X3 ) @ ( G @ X3 ) ) )
% 0.34/0.68 => ( R3 @ ( groups2497794486940690220atural @ H2 @ S ) @ ( groups2497794486940690220atural @ G @ S ) ) ) ) ) ) ).
% 0.34/0.68
% 0.34/0.68 % prod.related
% 0.34/0.68 thf(fact_7128_prod_Orelated,axiom,
% 0.34/0.68 ! [R3: assn > assn > $o,S: set_nat,H2: nat > assn,G: nat > assn] :
% 0.34/0.68 ( ( R3 @ one_one_assn @ one_one_assn )
% 0.34/0.68 => ( ! [X12: assn,Y12: assn,X23: assn,Y23: assn] :
% 0.34/0.68 ( ( ( R3 @ X12 @ X23 )
% 0.34/0.68 & ( R3 @ Y12 @ Y23 ) )
% 0.34/0.68 => ( R3 @ ( times_times_assn @ X12 @ Y12 ) @ ( times_times_assn @ X23 @ Y23 ) ) )
% 0.34/0.68 => ( ( finite_finite_nat @ S )
% 0.34/0.68 => ( ! [X3: nat] :
% 0.34/0.68 ( ( member_nat @ X3 @ S )
% 0.34/0.68 => ( R3 @ ( H2 @ X3 ) @ ( G @ X3 ) ) )
% 0.34/0.68 => ( R3 @ ( groups6906906614972039071t_assn @ H2 @ S ) @ ( groups6906906614972039071t_assn @ G @ S ) ) ) ) ) ) ).
% 0.34/0.68
% 0.34/0.68 % prod.related
% 0.34/0.68 thf(fact_7129_prod_Orelated,axiom,
% 0.34/0.68 ! [R3: assn > assn > $o,S: set_Code_integer,H2: code_integer > assn,G: code_integer > assn] :
% 0.34/0.68 ( ( R3 @ one_one_assn @ one_one_assn )
% 0.34/0.68 => ( ! [X12: assn,Y12: assn,X23: assn,Y23: assn] :
% 0.34/0.68 ( ( ( R3 @ X12 @ X23 )
% 0.34/0.68 & ( R3 @ Y12 @ Y23 ) )
% 0.34/0.68 => ( R3 @ ( times_times_assn @ X12 @ Y12 ) @ ( times_times_assn @ X23 @ Y23 ) ) )
% 0.34/0.68 => ( ( finite6017078050557962740nteger @ S )
% 0.34/0.68 => ( ! [X3: code_integer] :
% 0.34/0.68 ( ( member_Code_integer @ X3 @ S )
% 0.34/0.68 => ( R3 @ ( H2 @ X3 ) @ ( G @ X3 ) ) )
% 0.34/0.68 => ( R3 @ ( groups1304777262505850412r_assn @ H2 @ S ) @ ( groups1304777262505850412r_assn @ G @ S ) ) ) ) ) ) ).
% 0.34/0.68
% 0.34/0.68 % prod.related
% 0.34/0.68 thf(fact_7130_prod_Orelated,axiom,
% 0.34/0.68 ! [R3: rat > rat > $o,S: set_nat,H2: nat > rat,G: nat > rat] :
% 0.34/0.68 ( ( R3 @ one_one_rat @ one_one_rat )
% 0.34/0.68 => ( ! [X12: rat,Y12: rat,X23: rat,Y23: rat] :
% 0.34/0.68 ( ( ( R3 @ X12 @ X23 )
% 0.34/0.68 & ( R3 @ Y12 @ Y23 ) )
% 0.34/0.68 => ( R3 @ ( times_times_rat @ X12 @ Y12 ) @ ( times_times_rat @ X23 @ Y23 ) ) )
% 0.34/0.68 => ( ( finite_finite_nat @ S )
% 0.34/0.68 => ( ! [X3: nat] :
% 0.34/0.68 ( ( member_nat @ X3 @ S )
% 0.34/0.68 => ( R3 @ ( H2 @ X3 ) @ ( G @ X3 ) ) )
% 0.34/0.68 => ( R3 @ ( groups73079841787564623at_rat @ H2 @ S ) @ ( groups73079841787564623at_rat @ G @ S ) ) ) ) ) ) ).
% 0.34/0.68
% 0.34/0.68 % prod.related
% 0.34/0.68 thf(fact_7131_prod_Orelated,axiom,
% 0.34/0.68 ! [R3: rat > rat > $o,S: set_Code_integer,H2: code_integer > rat,G: code_integer > rat] :
% 0.34/0.68 ( ( R3 @ one_one_rat @ one_one_rat )
% 0.34/0.68 => ( ! [X12: rat,Y12: rat,X23: rat,Y23: rat] :
% 0.34/0.68 ( ( ( R3 @ X12 @ X23 )
% 0.34/0.68 & ( R3 @ Y12 @ Y23 ) )
% 0.34/0.68 => ( R3 @ ( times_times_rat @ X12 @ Y12 ) @ ( times_times_rat @ X23 @ Y23 ) ) )
% 0.34/0.68 => ( ( finite6017078050557962740nteger @ S )
% 0.34/0.68 => ( ! [X3: code_integer] :
% 0.34/0.68 ( ( member_Code_integer @ X3 @ S )
% 0.34/0.68 => ( R3 @ ( H2 @ X3 ) @ ( G @ X3 ) ) )
% 0.34/0.68 => ( R3 @ ( groups2555765274223993564er_rat @ H2 @ S ) @ ( groups2555765274223993564er_rat @ G @ S ) ) ) ) ) ) ).
% 0.34/0.68
% 0.34/0.68 % prod.related
% 0.34/0.68 thf(fact_7132_prod_Orelated,axiom,
% 0.34/0.68 ! [R3: nat > nat > $o,S: set_Code_integer,H2: code_integer > nat,G: code_integer > nat] :
% 0.34/0.68 ( ( R3 @ one_one_nat @ one_one_nat )
% 0.34/0.68 => ( ! [X12: nat,Y12: nat,X23: nat,Y23: nat] :
% 0.34/0.68 ( ( ( R3 @ X12 @ X23 )
% 0.34/0.68 & ( R3 @ Y12 @ Y23 ) )
% 0.34/0.68 => ( R3 @ ( times_times_nat @ X12 @ Y12 ) @ ( times_times_nat @ X23 @ Y23 ) ) )
% 0.34/0.68 => ( ( finite6017078050557962740nteger @ S )
% 0.34/0.68 => ( ! [X3: code_integer] :
% 0.34/0.68 ( ( member_Code_integer @ X3 @ S )
% 0.34/0.68 => ( R3 @ ( H2 @ X3 ) @ ( G @ X3 ) ) )
% 0.34/0.68 => ( R3 @ ( groups3190895334310489300er_nat @ H2 @ S ) @ ( groups3190895334310489300er_nat @ G @ S ) ) ) ) ) ) ).
% 0.34/0.68
% 0.34/0.68 % prod.related
% 0.34/0.68 thf(fact_7133_prod_Orelated,axiom,
% 0.34/0.68 ! [R3: int > int > $o,S: set_nat,H2: nat > int,G: nat > int] :
% 0.34/0.68 ( ( R3 @ one_one_int @ one_one_int )
% 0.34/0.68 => ( ! [X12: int,Y12: int,X23: int,Y23: int] :
% 0.34/0.68 ( ( ( R3 @ X12 @ X23 )
% 0.34/0.68 & ( R3 @ Y12 @ Y23 ) )
% 0.34/0.68 => ( R3 @ ( times_times_int @ X12 @ Y12 ) @ ( times_times_int @ X23 @ Y23 ) ) )
% 0.34/0.68 => ( ( finite_finite_nat @ S )
% 0.34/0.68 => ( ! [X3: nat] :
% 0.34/0.68 ( ( member_nat @ X3 @ S )
% 0.34/0.68 => ( R3 @ ( H2 @ X3 ) @ ( G @ X3 ) ) )
% 0.34/0.68 => ( R3 @ ( groups705719431365010083at_int @ H2 @ S ) @ ( groups705719431365010083at_int @ G @ S ) ) ) ) ) ) ).
% 0.34/0.68
% 0.34/0.68 % prod.related
% 0.34/0.68 thf(fact_7134_prod_Oinsert__if,axiom,
% 0.34/0.68 ! [A: set_int,X: int,G: int > assn] :
% 0.34/0.68 ( ( finite_finite_int @ A )
% 0.34/0.68 => ( ( ( member_int @ X @ A )
% 0.34/0.68 => ( ( groups7882442080178216443t_assn @ G @ ( insert_int @ X @ A ) )
% 0.34/0.68 = ( groups7882442080178216443t_assn @ G @ A ) ) )
% 0.34/0.68 & ( ~ ( member_int @ X @ A )
% 0.34/0.68 => ( ( groups7882442080178216443t_assn @ G @ ( insert_int @ X @ A ) )
% 0.34/0.68 = ( times_times_assn @ ( G @ X ) @ ( groups7882442080178216443t_assn @ G @ A ) ) ) ) ) ) ).
% 0.34/0.68
% 0.34/0.68 % prod.insert_if
% 0.34/0.68 thf(fact_7135_prod_Oinsert__if,axiom,
% 0.34/0.68 ! [A: set_o,X: $o,G: $o > assn] :
% 0.34/0.68 ( ( finite_finite_o @ A )
% 0.34/0.68 => ( ( ( member_o @ X @ A )
% 0.34/0.68 => ( ( groups5301882518646026715o_assn @ G @ ( insert_o @ X @ A ) )
% 0.34/0.68 = ( groups5301882518646026715o_assn @ G @ A ) ) )
% 0.34/0.68 & ( ~ ( member_o @ X @ A )
% 0.34/0.68 => ( ( groups5301882518646026715o_assn @ G @ ( insert_o @ X @ A ) )
% 0.34/0.68 = ( times_times_assn @ ( G @ X ) @ ( groups5301882518646026715o_assn @ G @ A ) ) ) ) ) ) ).
% 0.34/0.68
% 0.34/0.68 % prod.insert_if
% 0.34/0.68 thf(fact_7136_prod_Oinsert__if,axiom,
% 0.34/0.68 ! [A: set_Product_unit,X: product_unit,G: product_unit > assn] :
% 0.34/0.68 ( ( finite4290736615968046902t_unit @ A )
% 0.34/0.68 => ( ( ( member_Product_unit @ X @ A )
% 0.34/0.68 => ( ( groups5360233303959380816t_assn @ G @ ( insert_Product_unit @ X @ A ) )
% 0.34/0.68 = ( groups5360233303959380816t_assn @ G @ A ) ) )
% 0.34/0.68 & ( ~ ( member_Product_unit @ X @ A )
% 0.34/0.68 => ( ( groups5360233303959380816t_assn @ G @ ( insert_Product_unit @ X @ A ) )
% 0.34/0.68 = ( times_times_assn @ ( G @ X ) @ ( groups5360233303959380816t_assn @ G @ A ) ) ) ) ) ) ).
% 0.34/0.68
% 0.34/0.68 % prod.insert_if
% 0.34/0.68 thf(fact_7137_prod_Oinsert__if,axiom,
% 0.34/0.68 ! [A: set_nat,X: nat,G: nat > assn] :
% 0.34/0.68 ( ( finite_finite_nat @ A )
% 0.34/0.68 => ( ( ( member_nat @ X @ A )
% 0.34/0.68 => ( ( groups6906906614972039071t_assn @ G @ ( insert_nat @ X @ A ) )
% 0.34/0.68 = ( groups6906906614972039071t_assn @ G @ A ) ) )
% 0.34/0.68 & ( ~ ( member_nat @ X @ A )
% 0.34/0.68 => ( ( groups6906906614972039071t_assn @ G @ ( insert_nat @ X @ A ) )
% 0.34/0.68 = ( times_times_assn @ ( G @ X ) @ ( groups6906906614972039071t_assn @ G @ A ) ) ) ) ) ) ).
% 0.34/0.68
% 0.34/0.68 % prod.insert_if
% 0.34/0.68 thf(fact_7138_prod_Oinsert__if,axiom,
% 0.34/0.68 ! [A: set_Code_integer,X: code_integer,G: code_integer > assn] :
% 0.34/0.68 ( ( finite6017078050557962740nteger @ A )
% 0.34/0.68 => ( ( ( member_Code_integer @ X @ A )
% 0.34/0.68 => ( ( groups1304777262505850412r_assn @ G @ ( insert_Code_integer @ X @ A ) )
% 0.34/0.68 = ( groups1304777262505850412r_assn @ G @ A ) ) )
% 0.34/0.68 & ( ~ ( member_Code_integer @ X @ A )
% 0.34/0.68 => ( ( groups1304777262505850412r_assn @ G @ ( insert_Code_integer @ X @ A ) )
% 0.34/0.68 = ( times_times_assn @ ( G @ X ) @ ( groups1304777262505850412r_assn @ G @ A ) ) ) ) ) ) ).
% 0.34/0.68
% 0.34/0.68 % prod.insert_if
% 0.34/0.68 thf(fact_7139_prod_Oinsert__if,axiom,
% 0.34/0.68 ! [A: set_int,X: int,G: int > rat] :
% 0.34/0.68 ( ( finite_finite_int @ A )
% 0.34/0.68 => ( ( ( member_int @ X @ A )
% 0.34/0.68 => ( ( groups1072433553688619179nt_rat @ G @ ( insert_int @ X @ A ) )
% 0.34/0.68 = ( groups1072433553688619179nt_rat @ G @ A ) ) )
% 0.34/0.68 & ( ~ ( member_int @ X @ A )
% 0.34/0.68 => ( ( groups1072433553688619179nt_rat @ G @ ( insert_int @ X @ A ) )
% 0.34/0.68 = ( times_times_rat @ ( G @ X ) @ ( groups1072433553688619179nt_rat @ G @ A ) ) ) ) ) ) ).
% 0.34/0.68
% 0.34/0.68 % prod.insert_if
% 0.34/0.68 thf(fact_7140_prod_Oinsert__if,axiom,
% 0.34/0.68 ! [A: set_o,X: $o,G: $o > rat] :
% 0.34/0.68 ( ( finite_finite_o @ A )
% 0.34/0.68 => ( ( ( member_o @ X @ A )
% 0.34/0.68 => ( ( groups2869687844427037835_o_rat @ G @ ( insert_o @ X @ A ) )
% 0.34/0.68 = ( groups2869687844427037835_o_rat @ G @ A ) ) )
% 0.34/0.68 & ( ~ ( member_o @ X @ A )
% 0.34/0.68 => ( ( groups2869687844427037835_o_rat @ G @ ( insert_o @ X @ A ) )
% 0.34/0.68 = ( times_times_rat @ ( G @ X ) @ ( groups2869687844427037835_o_rat @ G @ A ) ) ) ) ) ) ).
% 0.34/0.68
% 0.34/0.68 % prod.insert_if
% 0.34/0.68 thf(fact_7141_prod_Oinsert__if,axiom,
% 0.34/0.68 ! [A: set_Product_unit,X: product_unit,G: product_unit > rat] :
% 0.34/0.68 ( ( finite4290736615968046902t_unit @ A )
% 0.34/0.68 => ( ( ( member_Product_unit @ X @ A )
% 0.34/0.68 => ( ( groups8984663225794451456it_rat @ G @ ( insert_Product_unit @ X @ A ) )
% 0.34/0.68 = ( groups8984663225794451456it_rat @ G @ A ) ) )
% 0.34/0.68 & ( ~ ( member_Product_unit @ X @ A )
% 0.34/0.68 => ( ( groups8984663225794451456it_rat @ G @ ( insert_Product_unit @ X @ A ) )
% 0.34/0.68 = ( times_times_rat @ ( G @ X ) @ ( groups8984663225794451456it_rat @ G @ A ) ) ) ) ) ) ).
% 0.34/0.68
% 0.34/0.68 % prod.insert_if
% 0.34/0.68 thf(fact_7142_prod_Oinsert__if,axiom,
% 0.34/0.68 ! [A: set_nat,X: nat,G: nat > rat] :
% 0.34/0.68 ( ( finite_finite_nat @ A )
% 0.34/0.68 => ( ( ( member_nat @ X @ A )
% 0.34/0.68 => ( ( groups73079841787564623at_rat @ G @ ( insert_nat @ X @ A ) )
% 0.34/0.68 = ( groups73079841787564623at_rat @ G @ A ) ) )
% 0.34/0.68 & ( ~ ( member_nat @ X @ A )
% 0.34/0.68 => ( ( groups73079841787564623at_rat @ G @ ( insert_nat @ X @ A ) )
% 0.34/0.68 = ( times_times_rat @ ( G @ X ) @ ( groups73079841787564623at_rat @ G @ A ) ) ) ) ) ) ).
% 0.34/0.68
% 0.34/0.68 % prod.insert_if
% 0.34/0.68 thf(fact_7143_prod_Oinsert__if,axiom,
% 0.34/0.68 ! [A: set_Code_integer,X: code_integer,G: code_integer > rat] :
% 0.34/0.68 ( ( finite6017078050557962740nteger @ A )
% 0.34/0.68 => ( ( ( member_Code_integer @ X @ A )
% 0.34/0.68 => ( ( groups2555765274223993564er_rat @ G @ ( insert_Code_integer @ X @ A ) )
% 0.34/0.68 = ( groups2555765274223993564er_rat @ G @ A ) ) )
% 0.34/0.68 & ( ~ ( member_Code_integer @ X @ A )
% 0.34/0.68 => ( ( groups2555765274223993564er_rat @ G @ ( insert_Code_integer @ X @ A ) )
% 0.34/0.68 = ( times_times_rat @ ( G @ X ) @ ( groups2555765274223993564er_rat @ G @ A ) ) ) ) ) ) ).
% 0.34/0.68
% 0.34/0.68 % prod.insert_if
% 0.34/0.68 thf(fact_7144_prod_Oreindex__bij__witness__not__neutral,axiom,
% 0.34/0.68 ! [S4: set_int,T5: set_int,S: set_int,I2: int > int,J: int > int,T2: set_int,G: int > assn,H2: int > assn] :
% 0.34/0.68 ( ( finite_finite_int @ S4 )
% 0.34/0.68 => ( ( finite_finite_int @ T5 )
% 0.34/0.68 => ( ! [A4: int] :
% 0.34/0.68 ( ( member_int @ A4 @ ( minus_minus_set_int @ S @ S4 ) )
% 0.34/0.68 => ( ( I2 @ ( J @ A4 ) )
% 0.34/0.68 = A4 ) )
% 0.34/0.68 => ( ! [A4: int] :
% 0.34/0.68 ( ( member_int @ A4 @ ( minus_minus_set_int @ S @ S4 ) )
% 0.34/0.68 => ( member_int @ ( J @ A4 ) @ ( minus_minus_set_int @ T2 @ T5 ) ) )
% 0.34/0.68 => ( ! [B2: int] :
% 0.34/0.68 ( ( member_int @ B2 @ ( minus_minus_set_int @ T2 @ T5 ) )
% 0.34/0.68 => ( ( J @ ( I2 @ B2 ) )
% 0.34/0.68 = B2 ) )
% 0.34/0.68 => ( ! [B2: int] :
% 0.34/0.68 ( ( member_int @ B2 @ ( minus_minus_set_int @ T2 @ T5 ) )
% 0.34/0.68 => ( member_int @ ( I2 @ B2 ) @ ( minus_minus_set_int @ S @ S4 ) ) )
% 0.34/0.68 => ( ! [A4: int] :
% 0.34/0.68 ( ( member_int @ A4 @ S4 )
% 0.34/0.68 => ( ( G @ A4 )
% 0.34/0.68 = one_one_assn ) )
% 0.34/0.68 => ( ! [B2: int] :
% 0.34/0.68 ( ( member_int @ B2 @ T5 )
% 0.34/0.68 => ( ( H2 @ B2 )
% 0.34/0.68 = one_one_assn ) )
% 0.34/0.68 => ( ! [A4: int] :
% 0.34/0.68 ( ( member_int @ A4 @ S )
% 0.34/0.68 => ( ( H2 @ ( J @ A4 ) )
% 0.34/0.68 = ( G @ A4 ) ) )
% 0.34/0.68 => ( ( groups7882442080178216443t_assn @ G @ S )
% 0.34/0.68 = ( groups7882442080178216443t_assn @ H2 @ T2 ) ) ) ) ) ) ) ) ) ) ) ).
% 0.34/0.68
% 0.34/0.68 % prod.reindex_bij_witness_not_neutral
% 0.34/0.68 thf(fact_7145_prod_Oreindex__bij__witness__not__neutral,axiom,
% 0.34/0.68 ! [S4: set_int,T5: set_o,S: set_int,I2: $o > int,J: int > $o,T2: set_o,G: int > assn,H2: $o > assn] :
% 0.34/0.68 ( ( finite_finite_int @ S4 )
% 0.34/0.68 => ( ( finite_finite_o @ T5 )
% 0.34/0.68 => ( ! [A4: int] :
% 0.34/0.68 ( ( member_int @ A4 @ ( minus_minus_set_int @ S @ S4 ) )
% 0.34/0.68 => ( ( I2 @ ( J @ A4 ) )
% 0.34/0.68 = A4 ) )
% 0.34/0.68 => ( ! [A4: int] :
% 0.34/0.68 ( ( member_int @ A4 @ ( minus_minus_set_int @ S @ S4 ) )
% 0.34/0.68 => ( member_o @ ( J @ A4 ) @ ( minus_minus_set_o @ T2 @ T5 ) ) )
% 0.34/0.68 => ( ! [B2: $o] :
% 0.34/0.68 ( ( member_o @ B2 @ ( minus_minus_set_o @ T2 @ T5 ) )
% 0.34/0.68 => ( ( J @ ( I2 @ B2 ) )
% 0.34/0.68 = B2 ) )
% 0.34/0.68 => ( ! [B2: $o] :
% 0.34/0.68 ( ( member_o @ B2 @ ( minus_minus_set_o @ T2 @ T5 ) )
% 0.34/0.68 => ( member_int @ ( I2 @ B2 ) @ ( minus_minus_set_int @ S @ S4 ) ) )
% 0.34/0.68 => ( ! [A4: int] :
% 0.34/0.68 ( ( member_int @ A4 @ S4 )
% 0.34/0.68 => ( ( G @ A4 )
% 0.34/0.68 = one_one_assn ) )
% 0.34/0.68 => ( ! [B2: $o] :
% 0.34/0.68 ( ( member_o @ B2 @ T5 )
% 0.34/0.68 => ( ( H2 @ B2 )
% 0.34/0.68 = one_one_assn ) )
% 0.34/0.68 => ( ! [A4: int] :
% 0.34/0.68 ( ( member_int @ A4 @ S )
% 0.34/0.68 => ( ( H2 @ ( J @ A4 ) )
% 0.34/0.68 = ( G @ A4 ) ) )
% 0.34/0.68 => ( ( groups7882442080178216443t_assn @ G @ S )
% 0.34/0.68 = ( groups5301882518646026715o_assn @ H2 @ T2 ) ) ) ) ) ) ) ) ) ) ) ).
% 0.34/0.68
% 0.34/0.68 % prod.reindex_bij_witness_not_neutral
% 0.34/0.68 thf(fact_7146_prod_Oreindex__bij__witness__not__neutral,axiom,
% 0.34/0.68 ! [S4: set_int,T5: set_Product_unit,S: set_int,I2: product_unit > int,J: int > product_unit,T2: set_Product_unit,G: int > assn,H2: product_unit > assn] :
% 0.34/0.68 ( ( finite_finite_int @ S4 )
% 0.34/0.68 => ( ( finite4290736615968046902t_unit @ T5 )
% 0.34/0.68 => ( ! [A4: int] :
% 0.34/0.68 ( ( member_int @ A4 @ ( minus_minus_set_int @ S @ S4 ) )
% 0.34/0.68 => ( ( I2 @ ( J @ A4 ) )
% 0.34/0.68 = A4 ) )
% 0.34/0.68 => ( ! [A4: int] :
% 0.34/0.68 ( ( member_int @ A4 @ ( minus_minus_set_int @ S @ S4 ) )
% 0.34/0.68 => ( member_Product_unit @ ( J @ A4 ) @ ( minus_6452836326544984404t_unit @ T2 @ T5 ) ) )
% 0.34/0.68 => ( ! [B2: product_unit] :
% 0.34/0.68 ( ( member_Product_unit @ B2 @ ( minus_6452836326544984404t_unit @ T2 @ T5 ) )
% 0.34/0.68 => ( ( J @ ( I2 @ B2 ) )
% 0.34/0.68 = B2 ) )
% 0.34/0.68 => ( ! [B2: product_unit] :
% 0.34/0.68 ( ( member_Product_unit @ B2 @ ( minus_6452836326544984404t_unit @ T2 @ T5 ) )
% 0.34/0.68 => ( member_int @ ( I2 @ B2 ) @ ( minus_minus_set_int @ S @ S4 ) ) )
% 0.34/0.68 => ( ! [A4: int] :
% 0.34/0.68 ( ( member_int @ A4 @ S4 )
% 0.34/0.68 => ( ( G @ A4 )
% 0.34/0.68 = one_one_assn ) )
% 0.34/0.68 => ( ! [B2: product_unit] :
% 0.34/0.68 ( ( member_Product_unit @ B2 @ T5 )
% 0.34/0.68 => ( ( H2 @ B2 )
% 0.34/0.68 = one_one_assn ) )
% 0.34/0.68 => ( ! [A4: int] :
% 0.34/0.68 ( ( member_int @ A4 @ S )
% 0.34/0.68 => ( ( H2 @ ( J @ A4 ) )
% 0.34/0.68 = ( G @ A4 ) ) )
% 0.34/0.68 => ( ( groups7882442080178216443t_assn @ G @ S )
% 0.34/0.68 = ( groups5360233303959380816t_assn @ H2 @ T2 ) ) ) ) ) ) ) ) ) ) ) ).
% 0.34/0.68
% 0.34/0.68 % prod.reindex_bij_witness_not_neutral
% 0.34/0.68 thf(fact_7147_prod_Oreindex__bij__witness__not__neutral,axiom,
% 0.34/0.68 ! [S4: set_o,T5: set_int,S: set_o,I2: int > $o,J: $o > int,T2: set_int,G: $o > assn,H2: int > assn] :
% 0.34/0.68 ( ( finite_finite_o @ S4 )
% 0.34/0.68 => ( ( finite_finite_int @ T5 )
% 0.34/0.68 => ( ! [A4: $o] :
% 0.34/0.68 ( ( member_o @ A4 @ ( minus_minus_set_o @ S @ S4 ) )
% 0.34/0.68 => ( ( I2 @ ( J @ A4 ) )
% 0.34/0.68 = A4 ) )
% 0.34/0.68 => ( ! [A4: $o] :
% 0.34/0.68 ( ( member_o @ A4 @ ( minus_minus_set_o @ S @ S4 ) )
% 0.34/0.68 => ( member_int @ ( J @ A4 ) @ ( minus_minus_set_int @ T2 @ T5 ) ) )
% 0.34/0.68 => ( ! [B2: int] :
% 0.34/0.68 ( ( member_int @ B2 @ ( minus_minus_set_int @ T2 @ T5 ) )
% 0.34/0.68 => ( ( J @ ( I2 @ B2 ) )
% 0.34/0.68 = B2 ) )
% 0.34/0.68 => ( ! [B2: int] :
% 0.34/0.68 ( ( member_int @ B2 @ ( minus_minus_set_int @ T2 @ T5 ) )
% 0.34/0.68 => ( member_o @ ( I2 @ B2 ) @ ( minus_minus_set_o @ S @ S4 ) ) )
% 0.34/0.68 => ( ! [A4: $o] :
% 0.34/0.68 ( ( member_o @ A4 @ S4 )
% 0.34/0.68 => ( ( G @ A4 )
% 0.34/0.68 = one_one_assn ) )
% 0.34/0.68 => ( ! [B2: int] :
% 0.34/0.68 ( ( member_int @ B2 @ T5 )
% 0.34/0.68 => ( ( H2 @ B2 )
% 0.34/0.68 = one_one_assn ) )
% 0.34/0.68 => ( ! [A4: $o] :
% 0.34/0.68 ( ( member_o @ A4 @ S )
% 0.34/0.68 => ( ( H2 @ ( J @ A4 ) )
% 0.34/0.68 = ( G @ A4 ) ) )
% 0.34/0.68 => ( ( groups5301882518646026715o_assn @ G @ S )
% 0.34/0.68 = ( groups7882442080178216443t_assn @ H2 @ T2 ) ) ) ) ) ) ) ) ) ) ) ).
% 0.34/0.68
% 0.34/0.68 % prod.reindex_bij_witness_not_neutral
% 0.34/0.68 thf(fact_7148_prod_Oreindex__bij__witness__not__neutral,axiom,
% 0.34/0.68 ! [S4: set_o,T5: set_o,S: set_o,I2: $o > $o,J: $o > $o,T2: set_o,G: $o > assn,H2: $o > assn] :
% 0.34/0.68 ( ( finite_finite_o @ S4 )
% 0.34/0.68 => ( ( finite_finite_o @ T5 )
% 0.34/0.68 => ( ! [A4: $o] :
% 0.34/0.68 ( ( member_o @ A4 @ ( minus_minus_set_o @ S @ S4 ) )
% 0.34/0.68 => ( ( I2 @ ( J @ A4 ) )
% 0.34/0.68 = A4 ) )
% 0.34/0.68 => ( ! [A4: $o] :
% 0.34/0.68 ( ( member_o @ A4 @ ( minus_minus_set_o @ S @ S4 ) )
% 0.34/0.68 => ( member_o @ ( J @ A4 ) @ ( minus_minus_set_o @ T2 @ T5 ) ) )
% 0.34/0.68 => ( ! [B2: $o] :
% 0.34/0.68 ( ( member_o @ B2 @ ( minus_minus_set_o @ T2 @ T5 ) )
% 0.34/0.68 => ( ( J @ ( I2 @ B2 ) )
% 0.34/0.68 = B2 ) )
% 0.34/0.68 => ( ! [B2: $o] :
% 0.34/0.68 ( ( member_o @ B2 @ ( minus_minus_set_o @ T2 @ T5 ) )
% 0.34/0.68 => ( member_o @ ( I2 @ B2 ) @ ( minus_minus_set_o @ S @ S4 ) ) )
% 0.34/0.68 => ( ! [A4: $o] :
% 0.34/0.68 ( ( member_o @ A4 @ S4 )
% 0.34/0.68 => ( ( G @ A4 )
% 0.34/0.68 = one_one_assn ) )
% 0.34/0.68 => ( ! [B2: $o] :
% 0.34/0.68 ( ( member_o @ B2 @ T5 )
% 0.34/0.68 => ( ( H2 @ B2 )
% 0.34/0.68 = one_one_assn ) )
% 0.34/0.68 => ( ! [A4: $o] :
% 0.34/0.68 ( ( member_o @ A4 @ S )
% 0.34/0.68 => ( ( H2 @ ( J @ A4 ) )
% 0.34/0.68 = ( G @ A4 ) ) )
% 0.34/0.68 => ( ( groups5301882518646026715o_assn @ G @ S )
% 0.34/0.68 = ( groups5301882518646026715o_assn @ H2 @ T2 ) ) ) ) ) ) ) ) ) ) ) ).
% 0.34/0.68
% 0.34/0.68 % prod.reindex_bij_witness_not_neutral
% 0.34/0.68 thf(fact_7149_prod_Oreindex__bij__witness__not__neutral,axiom,
% 0.34/0.68 ! [S4: set_o,T5: set_Product_unit,S: set_o,I2: product_unit > $o,J: $o > product_unit,T2: set_Product_unit,G: $o > assn,H2: product_unit > assn] :
% 0.34/0.68 ( ( finite_finite_o @ S4 )
% 0.34/0.68 => ( ( finite4290736615968046902t_unit @ T5 )
% 0.34/0.68 => ( ! [A4: $o] :
% 0.34/0.68 ( ( member_o @ A4 @ ( minus_minus_set_o @ S @ S4 ) )
% 0.34/0.68 => ( ( I2 @ ( J @ A4 ) )
% 0.34/0.68 = A4 ) )
% 0.34/0.68 => ( ! [A4: $o] :
% 0.34/0.68 ( ( member_o @ A4 @ ( minus_minus_set_o @ S @ S4 ) )
% 0.34/0.68 => ( member_Product_unit @ ( J @ A4 ) @ ( minus_6452836326544984404t_unit @ T2 @ T5 ) ) )
% 0.34/0.68 => ( ! [B2: product_unit] :
% 0.34/0.68 ( ( member_Product_unit @ B2 @ ( minus_6452836326544984404t_unit @ T2 @ T5 ) )
% 0.34/0.68 => ( ( J @ ( I2 @ B2 ) )
% 0.34/0.68 = B2 ) )
% 0.34/0.68 => ( ! [B2: product_unit] :
% 0.34/0.68 ( ( member_Product_unit @ B2 @ ( minus_6452836326544984404t_unit @ T2 @ T5 ) )
% 0.34/0.68 => ( member_o @ ( I2 @ B2 ) @ ( minus_minus_set_o @ S @ S4 ) ) )
% 0.34/0.68 => ( ! [A4: $o] :
% 0.34/0.68 ( ( member_o @ A4 @ S4 )
% 0.34/0.68 => ( ( G @ A4 )
% 0.34/0.68 = one_one_assn ) )
% 0.34/0.68 => ( ! [B2: product_unit] :
% 0.34/0.68 ( ( member_Product_unit @ B2 @ T5 )
% 0.34/0.68 => ( ( H2 @ B2 )
% 0.34/0.68 = one_one_assn ) )
% 0.34/0.68 => ( ! [A4: $o] :
% 0.34/0.68 ( ( member_o @ A4 @ S )
% 0.34/0.68 => ( ( H2 @ ( J @ A4 ) )
% 0.34/0.68 = ( G @ A4 ) ) )
% 0.34/0.68 => ( ( groups5301882518646026715o_assn @ G @ S )
% 0.34/0.68 = ( groups5360233303959380816t_assn @ H2 @ T2 ) ) ) ) ) ) ) ) ) ) ) ).
% 0.34/0.68
% 0.34/0.68 % prod.reindex_bij_witness_not_neutral
% 0.34/0.68 thf(fact_7150_prod_Oreindex__bij__witness__not__neutral,axiom,
% 0.34/0.68 ! [S4: set_Product_unit,T5: set_int,S: set_Product_unit,I2: int > product_unit,J: product_unit > int,T2: set_int,G: product_unit > assn,H2: int > assn] :
% 0.34/0.68 ( ( finite4290736615968046902t_unit @ S4 )
% 0.34/0.68 => ( ( finite_finite_int @ T5 )
% 0.34/0.68 => ( ! [A4: product_unit] :
% 0.34/0.68 ( ( member_Product_unit @ A4 @ ( minus_6452836326544984404t_unit @ S @ S4 ) )
% 0.34/0.68 => ( ( I2 @ ( J @ A4 ) )
% 0.34/0.68 = A4 ) )
% 0.34/0.68 => ( ! [A4: product_unit] :
% 0.34/0.68 ( ( member_Product_unit @ A4 @ ( minus_6452836326544984404t_unit @ S @ S4 ) )
% 0.34/0.68 => ( member_int @ ( J @ A4 ) @ ( minus_minus_set_int @ T2 @ T5 ) ) )
% 0.34/0.68 => ( ! [B2: int] :
% 0.34/0.68 ( ( member_int @ B2 @ ( minus_minus_set_int @ T2 @ T5 ) )
% 0.34/0.68 => ( ( J @ ( I2 @ B2 ) )
% 0.34/0.68 = B2 ) )
% 0.34/0.68 => ( ! [B2: int] :
% 0.34/0.68 ( ( member_int @ B2 @ ( minus_minus_set_int @ T2 @ T5 ) )
% 0.34/0.68 => ( member_Product_unit @ ( I2 @ B2 ) @ ( minus_6452836326544984404t_unit @ S @ S4 ) ) )
% 0.34/0.68 => ( ! [A4: product_unit] :
% 0.34/0.68 ( ( member_Product_unit @ A4 @ S4 )
% 0.34/0.68 => ( ( G @ A4 )
% 0.34/0.68 = one_one_assn ) )
% 0.34/0.68 => ( ! [B2: int] :
% 0.34/0.68 ( ( member_int @ B2 @ T5 )
% 0.34/0.68 => ( ( H2 @ B2 )
% 0.34/0.68 = one_one_assn ) )
% 0.34/0.68 => ( ! [A4: product_unit] :
% 0.34/0.68 ( ( member_Product_unit @ A4 @ S )
% 0.34/0.68 => ( ( H2 @ ( J @ A4 ) )
% 0.34/0.68 = ( G @ A4 ) ) )
% 0.34/0.68 => ( ( groups5360233303959380816t_assn @ G @ S )
% 0.34/0.68 = ( groups7882442080178216443t_assn @ H2 @ T2 ) ) ) ) ) ) ) ) ) ) ) ).
% 0.34/0.68
% 0.34/0.68 % prod.reindex_bij_witness_not_neutral
% 0.34/0.68 thf(fact_7151_prod_Oreindex__bij__witness__not__neutral,axiom,
% 0.34/0.68 ! [S4: set_Product_unit,T5: set_o,S: set_Product_unit,I2: $o > product_unit,J: product_unit > $o,T2: set_o,G: product_unit > assn,H2: $o > assn] :
% 0.34/0.68 ( ( finite4290736615968046902t_unit @ S4 )
% 0.34/0.68 => ( ( finite_finite_o @ T5 )
% 0.34/0.68 => ( ! [A4: product_unit] :
% 0.34/0.68 ( ( member_Product_unit @ A4 @ ( minus_6452836326544984404t_unit @ S @ S4 ) )
% 0.34/0.68 => ( ( I2 @ ( J @ A4 ) )
% 0.34/0.68 = A4 ) )
% 0.34/0.68 => ( ! [A4: product_unit] :
% 0.34/0.68 ( ( member_Product_unit @ A4 @ ( minus_6452836326544984404t_unit @ S @ S4 ) )
% 0.34/0.68 => ( member_o @ ( J @ A4 ) @ ( minus_minus_set_o @ T2 @ T5 ) ) )
% 0.34/0.68 => ( ! [B2: $o] :
% 0.34/0.68 ( ( member_o @ B2 @ ( minus_minus_set_o @ T2 @ T5 ) )
% 0.34/0.68 => ( ( J @ ( I2 @ B2 ) )
% 0.34/0.68 = B2 ) )
% 0.34/0.68 => ( ! [B2: $o] :
% 0.34/0.68 ( ( member_o @ B2 @ ( minus_minus_set_o @ T2 @ T5 ) )
% 0.34/0.68 => ( member_Product_unit @ ( I2 @ B2 ) @ ( minus_6452836326544984404t_unit @ S @ S4 ) ) )
% 0.34/0.68 => ( ! [A4: product_unit] :
% 0.34/0.68 ( ( member_Product_unit @ A4 @ S4 )
% 0.34/0.68 => ( ( G @ A4 )
% 0.34/0.68 = one_one_assn ) )
% 0.34/0.68 => ( ! [B2: $o] :
% 0.34/0.68 ( ( member_o @ B2 @ T5 )
% 0.34/0.68 => ( ( H2 @ B2 )
% 0.34/0.68 = one_one_assn ) )
% 0.34/0.68 => ( ! [A4: product_unit] :
% 0.34/0.68 ( ( member_Product_unit @ A4 @ S )
% 0.34/0.68 => ( ( H2 @ ( J @ A4 ) )
% 0.34/0.68 = ( G @ A4 ) ) )
% 0.34/0.68 => ( ( groups5360233303959380816t_assn @ G @ S )
% 0.34/0.68 = ( groups5301882518646026715o_assn @ H2 @ T2 ) ) ) ) ) ) ) ) ) ) ) ).
% 0.34/0.68
% 0.34/0.68 % prod.reindex_bij_witness_not_neutral
% 0.34/0.68 thf(fact_7152_prod_Oreindex__bij__witness__not__neutral,axiom,
% 0.34/0.68 ! [S4: set_Product_unit,T5: set_Product_unit,S: set_Product_unit,I2: product_unit > product_unit,J: product_unit > product_unit,T2: set_Product_unit,G: product_unit > assn,H2: product_unit > assn] :
% 0.34/0.68 ( ( finite4290736615968046902t_unit @ S4 )
% 0.34/0.68 => ( ( finite4290736615968046902t_unit @ T5 )
% 0.34/0.68 => ( ! [A4: product_unit] :
% 0.34/0.68 ( ( member_Product_unit @ A4 @ ( minus_6452836326544984404t_unit @ S @ S4 ) )
% 0.34/0.68 => ( ( I2 @ ( J @ A4 ) )
% 0.34/0.68 = A4 ) )
% 0.34/0.68 => ( ! [A4: product_unit] :
% 0.34/0.68 ( ( member_Product_unit @ A4 @ ( minus_6452836326544984404t_unit @ S @ S4 ) )
% 0.34/0.68 => ( member_Product_unit @ ( J @ A4 ) @ ( minus_6452836326544984404t_unit @ T2 @ T5 ) ) )
% 0.34/0.68 => ( ! [B2: product_unit] :
% 0.34/0.68 ( ( member_Product_unit @ B2 @ ( minus_6452836326544984404t_unit @ T2 @ T5 ) )
% 0.34/0.68 => ( ( J @ ( I2 @ B2 ) )
% 0.34/0.68 = B2 ) )
% 0.34/0.68 => ( ! [B2: product_unit] :
% 0.34/0.68 ( ( member_Product_unit @ B2 @ ( minus_6452836326544984404t_unit @ T2 @ T5 ) )
% 0.34/0.68 => ( member_Product_unit @ ( I2 @ B2 ) @ ( minus_6452836326544984404t_unit @ S @ S4 ) ) )
% 0.34/0.68 => ( ! [A4: product_unit] :
% 0.34/0.68 ( ( member_Product_unit @ A4 @ S4 )
% 0.34/0.68 => ( ( G @ A4 )
% 0.34/0.68 = one_one_assn ) )
% 0.34/0.68 => ( ! [B2: product_unit] :
% 0.34/0.68 ( ( member_Product_unit @ B2 @ T5 )
% 0.34/0.68 => ( ( H2 @ B2 )
% 0.34/0.68 = one_one_assn ) )
% 0.34/0.68 => ( ! [A4: product_unit] :
% 0.34/0.68 ( ( member_Product_unit @ A4 @ S )
% 0.34/0.68 => ( ( H2 @ ( J @ A4 ) )
% 0.34/0.68 = ( G @ A4 ) ) )
% 0.34/0.68 => ( ( groups5360233303959380816t_assn @ G @ S )
% 0.34/0.68 = ( groups5360233303959380816t_assn @ H2 @ T2 ) ) ) ) ) ) ) ) ) ) ) ).
% 0.34/0.68
% 0.34/0.68 % prod.reindex_bij_witness_not_neutral
% 0.34/0.68 thf(fact_7153_prod_Oreindex__bij__witness__not__neutral,axiom,
% 0.34/0.68 ! [S4: set_int,T5: set_Code_integer,S: set_int,I2: code_integer > int,J: int > code_integer,T2: set_Code_integer,G: int > assn,H2: code_integer > assn] :
% 0.34/0.68 ( ( finite_finite_int @ S4 )
% 0.34/0.68 => ( ( finite6017078050557962740nteger @ T5 )
% 0.34/0.68 => ( ! [A4: int] :
% 0.34/0.68 ( ( member_int @ A4 @ ( minus_minus_set_int @ S @ S4 ) )
% 0.34/0.68 => ( ( I2 @ ( J @ A4 ) )
% 0.34/0.68 = A4 ) )
% 0.34/0.68 => ( ! [A4: int] :
% 0.34/0.68 ( ( member_int @ A4 @ ( minus_minus_set_int @ S @ S4 ) )
% 0.34/0.68 => ( member_Code_integer @ ( J @ A4 ) @ ( minus_2355218937544613996nteger @ T2 @ T5 ) ) )
% 0.34/0.68 => ( ! [B2: code_integer] :
% 0.34/0.68 ( ( member_Code_integer @ B2 @ ( minus_2355218937544613996nteger @ T2 @ T5 ) )
% 0.34/0.68 => ( ( J @ ( I2 @ B2 ) )
% 0.34/0.68 = B2 ) )
% 0.34/0.68 => ( ! [B2: code_integer] :
% 0.34/0.68 ( ( member_Code_integer @ B2 @ ( minus_2355218937544613996nteger @ T2 @ T5 ) )
% 0.34/0.68 => ( member_int @ ( I2 @ B2 ) @ ( minus_minus_set_int @ S @ S4 ) ) )
% 0.34/0.68 => ( ! [A4: int] :
% 0.34/0.68 ( ( member_int @ A4 @ S4 )
% 0.34/0.68 => ( ( G @ A4 )
% 0.34/0.68 = one_one_assn ) )
% 0.34/0.68 => ( ! [B2: code_integer] :
% 0.34/0.68 ( ( member_Code_integer @ B2 @ T5 )
% 0.34/0.68 => ( ( H2 @ B2 )
% 0.34/0.68 = one_one_assn ) )
% 0.34/0.68 => ( ! [A4: int] :
% 0.34/0.68 ( ( member_int @ A4 @ S )
% 0.34/0.68 => ( ( H2 @ ( J @ A4 ) )
% 0.34/0.68 = ( G @ A4 ) ) )
% 0.34/0.68 => ( ( groups7882442080178216443t_assn @ G @ S )
% 0.34/0.68 = ( groups1304777262505850412r_assn @ H2 @ T2 ) ) ) ) ) ) ) ) ) ) ) ).
% 0.34/0.68
% 0.34/0.68 % prod.reindex_bij_witness_not_neutral
% 0.34/0.68 thf(fact_7154_gbinomial__minus,axiom,
% 0.34/0.68 ! [A2: rat,K3: nat] :
% 0.34/0.68 ( ( gbinomial_rat @ ( uminus_uminus_rat @ A2 ) @ K3 )
% 0.34/0.68 = ( times_times_rat @ ( power_power_rat @ ( uminus_uminus_rat @ one_one_rat ) @ K3 ) @ ( gbinomial_rat @ ( minus_minus_rat @ ( plus_plus_rat @ A2 @ ( semiri681578069525770553at_rat @ K3 ) ) @ one_one_rat ) @ K3 ) ) ) ).
% 0.34/0.68
% 0.34/0.68 % gbinomial_minus
% 0.34/0.68 thf(fact_7155_le__minus__one__simps_I3_J,axiom,
% 0.34/0.68 ~ ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) ) ).
% 0.34/0.68
% 0.34/0.68 % le_minus_one_simps(3)
% 0.34/0.68 thf(fact_7156_le__minus__one__simps_I3_J,axiom,
% 0.34/0.68 ~ ( ord_less_eq_rat @ zero_zero_rat @ ( uminus_uminus_rat @ one_one_rat ) ) ).
% 0.34/0.68
% 0.34/0.68 % le_minus_one_simps(3)
% 0.34/0.68 thf(fact_7157_le__minus__one__simps_I3_J,axiom,
% 0.34/0.68 ~ ( ord_less_eq_int @ zero_zero_int @ ( uminus_uminus_int @ one_one_int ) ) ).
% 0.34/0.68
% 0.34/0.68 % le_minus_one_simps(3)
% 0.34/0.68 thf(fact_7158_le__minus__one__simps_I1_J,axiom,
% 0.34/0.68 ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ zero_z3403309356797280102nteger ).
% 0.34/0.68
% 0.34/0.68 % le_minus_one_simps(1)
% 0.34/0.68 thf(fact_7159_le__minus__one__simps_I1_J,axiom,
% 0.34/0.68 ord_less_eq_rat @ ( uminus_uminus_rat @ one_one_rat ) @ zero_zero_rat ).
% 0.34/0.68
% 0.34/0.68 % le_minus_one_simps(1)
% 0.34/0.68 thf(fact_7160_le__minus__one__simps_I1_J,axiom,
% 0.34/0.68 ord_less_eq_int @ ( uminus_uminus_int @ one_one_int ) @ zero_zero_int ).
% 0.34/0.68
% 0.34/0.68 % le_minus_one_simps(1)
% 0.34/0.68 thf(fact_7161_less__minus__one__simps_I1_J,axiom,
% 0.34/0.68 ord_less_rat @ ( uminus_uminus_rat @ one_one_rat ) @ zero_zero_rat ).
% 0.34/0.68
% 0.34/0.68 % less_minus_one_simps(1)
% 0.34/0.68 thf(fact_7162_less__minus__one__simps_I1_J,axiom,
% 0.34/0.68 ord_less_int @ ( uminus_uminus_int @ one_one_int ) @ zero_zero_int ).
% 0.34/0.68
% 0.34/0.68 % less_minus_one_simps(1)
% 0.34/0.68 thf(fact_7163_less__minus__one__simps_I1_J,axiom,
% 0.34/0.68 ord_le6747313008572928689nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ zero_z3403309356797280102nteger ).
% 0.34/0.68
% 0.34/0.68 % less_minus_one_simps(1)
% 0.34/0.68 thf(fact_7164_less__minus__one__simps_I3_J,axiom,
% 0.34/0.68 ~ ( ord_less_rat @ zero_zero_rat @ ( uminus_uminus_rat @ one_one_rat ) ) ).
% 0.34/0.68
% 0.34/0.68 % less_minus_one_simps(3)
% 0.34/0.68 thf(fact_7165_less__minus__one__simps_I3_J,axiom,
% 0.34/0.68 ~ ( ord_less_int @ zero_zero_int @ ( uminus_uminus_int @ one_one_int ) ) ).
% 0.34/0.68
% 0.34/0.68 % less_minus_one_simps(3)
% 0.34/0.68 thf(fact_7166_less__minus__one__simps_I3_J,axiom,
% 0.34/0.68 ~ ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) ) ).
% 0.34/0.68
% 0.34/0.68 % less_minus_one_simps(3)
% 0.34/0.68 thf(fact_7167_inf__shunt,axiom,
% 0.34/0.68 ! [X: set_Product_unit,Y: set_Product_unit] :
% 0.34/0.68 ( ( ( inf_in4660618365625256667t_unit @ X @ Y )
% 0.34/0.68 = bot_bo3957492148770167129t_unit )
% 0.34/0.68 = ( ord_le3507040750410214029t_unit @ X @ ( uminus5944136376168626660t_unit @ Y ) ) ) ).
% 0.34/0.68
% 0.34/0.68 % inf_shunt
% 0.34/0.68 thf(fact_7168_inf__shunt,axiom,
% 0.34/0.68 ! [X: set_o,Y: set_o] :
% 0.34/0.68 ( ( ( inf_inf_set_o @ X @ Y )
% 0.34/0.68 = bot_bot_set_o )
% 0.34/0.68 = ( ord_less_eq_set_o @ X @ ( uminus_uminus_set_o @ Y ) ) ) ).
% 0.34/0.68
% 0.34/0.68 % inf_shunt
% 0.34/0.68 thf(fact_7169_inf__shunt,axiom,
% 0.34/0.68 ! [X: set_int,Y: set_int] :
% 0.34/0.68 ( ( ( inf_inf_set_int @ X @ Y )
% 0.34/0.68 = bot_bot_set_int )
% 0.34/0.68 = ( ord_less_eq_set_int @ X @ ( uminus1532241313380277803et_int @ Y ) ) ) ).
% 0.34/0.68
% 0.34/0.68 % inf_shunt
% 0.34/0.68 thf(fact_7170_inf__shunt,axiom,
% 0.34/0.68 ! [X: set_Pr4532377907799695533_nat_o,Y: set_Pr4532377907799695533_nat_o] :
% 0.34/0.68 ( ( ( inf_in1906310914598751387_nat_o @ X @ Y )
% 0.34/0.68 = bot_bo7824918357723582233_nat_o )
% 0.34/0.68 = ( ord_le2965882846123202637_nat_o @ X @ ( uminus5254974436814262692_nat_o @ Y ) ) ) ).
% 0.34/0.68
% 0.34/0.68 % inf_shunt
% 0.34/0.68 thf(fact_7171_inf__shunt,axiom,
% 0.34/0.68 ! [X: assn,Y: assn] :
% 0.34/0.68 ( ( ( inf_inf_assn @ X @ Y )
% 0.34/0.68 = bot_bot_assn )
% 0.34/0.68 = ( ord_less_eq_assn @ X @ ( uminus_uminus_assn @ Y ) ) ) ).
% 0.34/0.68
% 0.34/0.68 % inf_shunt
% 0.34/0.68 thf(fact_7172_inf__shunt,axiom,
% 0.34/0.68 ! [X: product_unit,Y: product_unit] :
% 0.34/0.68 ( ( ( inf_inf_Product_unit @ X @ Y )
% 0.34/0.68 = bot_bot_Product_unit )
% 0.34/0.68 = ( ord_le3221252021190050221t_unit @ X @ ( uminus2952777764628376836t_unit @ Y ) ) ) ).
% 0.34/0.68
% 0.34/0.68 % inf_shunt
% 0.34/0.68 thf(fact_7173_inf__shunt,axiom,
% 0.34/0.68 ! [X: set_nat,Y: set_nat] :
% 0.34/0.68 ( ( ( inf_inf_set_nat @ X @ Y )
% 0.34/0.68 = bot_bot_set_nat )
% 0.34/0.68 = ( ord_less_eq_set_nat @ X @ ( uminus5710092332889474511et_nat @ Y ) ) ) ).
% 0.34/0.68
% 0.34/0.68 % inf_shunt
% 0.34/0.68 thf(fact_7174_power__minus,axiom,
% 0.34/0.68 ! [A2: rat,N2: nat] :
% 0.34/0.68 ( ( power_power_rat @ ( uminus_uminus_rat @ A2 ) @ N2 )
% 0.34/0.68 = ( times_times_rat @ ( power_power_rat @ ( uminus_uminus_rat @ one_one_rat ) @ N2 ) @ ( power_power_rat @ A2 @ N2 ) ) ) ).
% 0.34/0.68
% 0.34/0.68 % power_minus
% 0.34/0.68 thf(fact_7175_power__minus,axiom,
% 0.34/0.68 ! [A2: int,N2: nat] :
% 0.34/0.68 ( ( power_power_int @ ( uminus_uminus_int @ A2 ) @ N2 )
% 0.34/0.68 = ( times_times_int @ ( power_power_int @ ( uminus_uminus_int @ one_one_int ) @ N2 ) @ ( power_power_int @ A2 @ N2 ) ) ) ).
% 0.34/0.68
% 0.34/0.68 % power_minus
% 0.34/0.68 thf(fact_7176_power__minus,axiom,
% 0.34/0.68 ! [A2: code_integer,N2: nat] :
% 0.34/0.68 ( ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ A2 ) @ N2 )
% 0.34/0.68 = ( times_3573771949741848930nteger @ ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ N2 ) @ ( power_8256067586552552935nteger @ A2 @ N2 ) ) ) ).
% 0.34/0.68
% 0.34/0.68 % power_minus
% 0.34/0.68 thf(fact_7177_disjoint__eq__subset__Compl,axiom,
% 0.34/0.68 ! [A: set_Product_unit,B4: set_Product_unit] :
% 0.34/0.68 ( ( ( inf_in4660618365625256667t_unit @ A @ B4 )
% 0.34/0.68 = bot_bo3957492148770167129t_unit )
% 0.34/0.68 = ( ord_le3507040750410214029t_unit @ A @ ( uminus5944136376168626660t_unit @ B4 ) ) ) ).
% 0.34/0.68
% 0.34/0.68 % disjoint_eq_subset_Compl
% 0.34/0.68 thf(fact_7178_disjoint__eq__subset__Compl,axiom,
% 0.34/0.68 ! [A: set_o,B4: set_o] :
% 0.34/0.68 ( ( ( inf_inf_set_o @ A @ B4 )
% 0.34/0.68 = bot_bot_set_o )
% 0.34/0.68 = ( ord_less_eq_set_o @ A @ ( uminus_uminus_set_o @ B4 ) ) ) ).
% 0.34/0.68
% 0.34/0.68 % disjoint_eq_subset_Compl
% 0.34/0.68 thf(fact_7179_disjoint__eq__subset__Compl,axiom,
% 0.34/0.68 ! [A: set_int,B4: set_int] :
% 0.34/0.68 ( ( ( inf_inf_set_int @ A @ B4 )
% 0.34/0.68 = bot_bot_set_int )
% 0.34/0.68 = ( ord_less_eq_set_int @ A @ ( uminus1532241313380277803et_int @ B4 ) ) ) ).
% 0.34/0.68
% 0.34/0.68 % disjoint_eq_subset_Compl
% 0.34/0.68 thf(fact_7180_disjoint__eq__subset__Compl,axiom,
% 0.34/0.68 ! [A: set_Pr4532377907799695533_nat_o,B4: set_Pr4532377907799695533_nat_o] :
% 0.34/0.68 ( ( ( inf_in1906310914598751387_nat_o @ A @ B4 )
% 0.34/0.68 = bot_bo7824918357723582233_nat_o )
% 0.34/0.68 = ( ord_le2965882846123202637_nat_o @ A @ ( uminus5254974436814262692_nat_o @ B4 ) ) ) ).
% 0.34/0.68
% 0.34/0.68 % disjoint_eq_subset_Compl
% 0.34/0.68 thf(fact_7181_disjoint__eq__subset__Compl,axiom,
% 0.34/0.68 ! [A: set_nat,B4: set_nat] :
% 0.34/0.68 ( ( ( inf_inf_set_nat @ A @ B4 )
% 0.34/0.68 = bot_bot_set_nat )
% 0.34/0.68 = ( ord_less_eq_set_nat @ A @ ( uminus5710092332889474511et_nat @ B4 ) ) ) ).
% 0.34/0.68
% 0.34/0.68 % disjoint_eq_subset_Compl
% 0.34/0.68 thf(fact_7182_Compl__insert,axiom,
% 0.34/0.68 ! [X: product_unit,A: set_Product_unit] :
% 0.34/0.68 ( ( uminus5944136376168626660t_unit @ ( insert_Product_unit @ X @ A ) )
% 0.34/0.68 = ( minus_6452836326544984404t_unit @ ( uminus5944136376168626660t_unit @ A ) @ ( insert_Product_unit @ X @ bot_bo3957492148770167129t_unit ) ) ) ).
% 0.34/0.68
% 0.34/0.68 % Compl_insert
% 0.34/0.68 thf(fact_7183_Compl__insert,axiom,
% 0.34/0.68 ! [X: $o,A: set_o] :
% 0.34/0.68 ( ( uminus_uminus_set_o @ ( insert_o @ X @ A ) )
% 0.34/0.68 = ( minus_minus_set_o @ ( uminus_uminus_set_o @ A ) @ ( insert_o @ X @ bot_bot_set_o ) ) ) ).
% 0.34/0.68
% 0.34/0.68 % Compl_insert
% 0.34/0.68 thf(fact_7184_Compl__insert,axiom,
% 0.34/0.68 ! [X: int,A: set_int] :
% 0.34/0.68 ( ( uminus1532241313380277803et_int @ ( insert_int @ X @ A ) )
% 0.34/0.68 = ( minus_minus_set_int @ ( uminus1532241313380277803et_int @ A ) @ ( insert_int @ X @ bot_bot_set_int ) ) ) ).
% 0.34/0.68
% 0.34/0.68 % Compl_insert
% 0.34/0.68 thf(fact_7185_Compl__insert,axiom,
% 0.34/0.68 ! [X: produc3658429121746597890et_nat > $o,A: set_Pr4532377907799695533_nat_o] :
% 0.34/0.68 ( ( uminus5254974436814262692_nat_o @ ( insert5175938949040314269_nat_o @ X @ A ) )
% 0.34/0.68 = ( minus_1801376950450012436_nat_o @ ( uminus5254974436814262692_nat_o @ A ) @ ( insert5175938949040314269_nat_o @ X @ bot_bo7824918357723582233_nat_o ) ) ) ).
% 0.34/0.68
% 0.34/0.68 % Compl_insert
% 0.34/0.68 thf(fact_7186_Compl__insert,axiom,
% 0.34/0.68 ! [X: nat,A: set_nat] :
% 0.34/0.68 ( ( uminus5710092332889474511et_nat @ ( insert_nat @ X @ A ) )
% 0.34/0.68 = ( minus_minus_set_nat @ ( uminus5710092332889474511et_nat @ A ) @ ( insert_nat @ X @ bot_bot_set_nat ) ) ) ).
% 0.34/0.68
% 0.34/0.68 % Compl_insert
% 0.34/0.68 thf(fact_7187_Image__subset__eq,axiom,
% 0.34/0.68 ! [R2: set_Pr1261947904930325089at_nat,A: set_nat,B4: set_nat] :
% 0.34/0.68 ( ( ord_less_eq_set_nat @ ( image_nat_nat @ R2 @ A ) @ B4 )
% 0.34/0.68 = ( ord_less_eq_set_nat @ A @ ( uminus5710092332889474511et_nat @ ( image_nat_nat @ ( converse_nat_nat @ R2 ) @ ( uminus5710092332889474511et_nat @ B4 ) ) ) ) ) ).
% 0.34/0.68
% 0.34/0.68 % Image_subset_eq
% 0.34/0.68 thf(fact_7188_less__1__prod2,axiom,
% 0.34/0.68 ! [I: set_int,I2: int,F: int > code_integer] :
% 0.34/0.68 ( ( finite_finite_int @ I )
% 0.34/0.68 => ( ( member_int @ I2 @ I )
% 0.34/0.68 => ( ( ord_le6747313008572928689nteger @ one_one_Code_integer @ ( F @ I2 ) )
% 0.34/0.68 => ( ! [I3: int] :
% 0.34/0.68 ( ( member_int @ I3 @ I )
% 0.34/0.68 => ( ord_le3102999989581377725nteger @ one_one_Code_integer @ ( F @ I3 ) ) )
% 0.34/0.68 => ( ord_le6747313008572928689nteger @ one_one_Code_integer @ ( groups3827104343326376752nteger @ F @ I ) ) ) ) ) ) ).
% 0.34/0.68
% 0.34/0.68 % less_1_prod2
% 0.34/0.68 thf(fact_7189_less__1__prod2,axiom,
% 0.34/0.68 ! [I: set_o,I2: $o,F: $o > code_integer] :
% 0.34/0.68 ( ( finite_finite_o @ I )
% 0.34/0.68 => ( ( member_o @ I2 @ I )
% 0.34/0.68 => ( ( ord_le6747313008572928689nteger @ one_one_Code_integer @ ( F @ I2 ) )
% 0.34/0.68 => ( ! [I3: $o] :
% 0.34/0.68 ( ( member_o @ I3 @ I )
% 0.34/0.68 => ( ord_le3102999989581377725nteger @ one_one_Code_integer @ ( F @ I3 ) ) )
% 0.34/0.68 => ( ord_le6747313008572928689nteger @ one_one_Code_integer @ ( groups7694694392188491536nteger @ F @ I ) ) ) ) ) ) ).
% 0.34/0.68
% 0.34/0.68 % less_1_prod2
% 0.34/0.68 thf(fact_7190_less__1__prod2,axiom,
% 0.34/0.68 ! [I: set_Product_unit,I2: product_unit,F: product_unit > code_integer] :
% 0.34/0.68 ( ( finite4290736615968046902t_unit @ I )
% 0.34/0.68 => ( ( member_Product_unit @ I2 @ I )
% 0.34/0.68 => ( ( ord_le6747313008572928689nteger @ one_one_Code_integer @ ( F @ I2 ) )
% 0.34/0.68 => ( ! [I3: product_unit] :
% 0.34/0.68 ( ( member_Product_unit @ I3 @ I )
% 0.34/0.68 => ( ord_le3102999989581377725nteger @ one_one_Code_integer @ ( F @ I3 ) ) )
% 0.34/0.68 => ( ord_le6747313008572928689nteger @ one_one_Code_integer @ ( groups8185359811944078981nteger @ F @ I ) ) ) ) ) ) ).
% 0.34/0.68
% 0.34/0.68 % less_1_prod2
% 0.34/0.68 thf(fact_7191_less__1__prod2,axiom,
% 0.34/0.68 ! [I: set_nat,I2: nat,F: nat > code_integer] :
% 0.34/0.68 ( ( finite_finite_nat @ I )
% 0.34/0.68 => ( ( member_nat @ I2 @ I )
% 0.34/0.68 => ( ( ord_le6747313008572928689nteger @ one_one_Code_integer @ ( F @ I2 ) )
% 0.34/0.68 => ( ! [I3: nat] :
% 0.34/0.68 ( ( member_nat @ I3 @ I )
% 0.34/0.68 => ( ord_le3102999989581377725nteger @ one_one_Code_integer @ ( F @ I3 ) ) )
% 0.34/0.68 => ( ord_le6747313008572928689nteger @ one_one_Code_integer @ ( groups3455450783089532116nteger @ F @ I ) ) ) ) ) ) ).
% 0.34/0.68
% 0.34/0.68 % less_1_prod2
% 0.34/0.68 thf(fact_7192_less__1__prod2,axiom,
% 0.34/0.68 ! [I: set_Code_integer,I2: code_integer,F: code_integer > code_integer] :
% 0.34/0.68 ( ( finite6017078050557962740nteger @ I )
% 0.34/0.68 => ( ( member_Code_integer @ I2 @ I )
% 0.34/0.68 => ( ( ord_le6747313008572928689nteger @ one_one_Code_integer @ ( F @ I2 ) )
% 0.34/0.68 => ( ! [I3: code_integer] :
% 0.34/0.68 ( ( member_Code_integer @ I3 @ I )
% 0.34/0.68 => ( ord_le3102999989581377725nteger @ one_one_Code_integer @ ( F @ I3 ) ) )
% 0.34/0.68 => ( ord_le6747313008572928689nteger @ one_one_Code_integer @ ( groups3674199335183972705nteger @ F @ I ) ) ) ) ) ) ).
% 0.34/0.68
% 0.34/0.68 % less_1_prod2
% 0.34/0.68 thf(fact_7193_less__1__prod2,axiom,
% 0.34/0.68 ! [I: set_int,I2: int,F: int > rat] :
% 0.34/0.68 ( ( finite_finite_int @ I )
% 0.34/0.68 => ( ( member_int @ I2 @ I )
% 0.34/0.68 => ( ( ord_less_rat @ one_one_rat @ ( F @ I2 ) )
% 0.34/0.68 => ( ! [I3: int] :
% 0.34/0.68 ( ( member_int @ I3 @ I )
% 0.34/0.68 => ( ord_less_eq_rat @ one_one_rat @ ( F @ I3 ) ) )
% 0.34/0.68 => ( ord_less_rat @ one_one_rat @ ( groups1072433553688619179nt_rat @ F @ I ) ) ) ) ) ) ).
% 0.34/0.68
% 0.34/0.68 % less_1_prod2
% 0.34/0.68 thf(fact_7194_less__1__prod2,axiom,
% 0.34/0.68 ! [I: set_o,I2: $o,F: $o > rat] :
% 0.34/0.68 ( ( finite_finite_o @ I )
% 0.34/0.68 => ( ( member_o @ I2 @ I )
% 0.34/0.68 => ( ( ord_less_rat @ one_one_rat @ ( F @ I2 ) )
% 0.34/0.68 => ( ! [I3: $o] :
% 0.34/0.68 ( ( member_o @ I3 @ I )
% 0.34/0.68 => ( ord_less_eq_rat @ one_one_rat @ ( F @ I3 ) ) )
% 0.34/0.68 => ( ord_less_rat @ one_one_rat @ ( groups2869687844427037835_o_rat @ F @ I ) ) ) ) ) ) ).
% 0.34/0.68
% 0.34/0.68 % less_1_prod2
% 0.34/0.68 thf(fact_7195_less__1__prod2,axiom,
% 0.34/0.68 ! [I: set_Product_unit,I2: product_unit,F: product_unit > rat] :
% 0.34/0.68 ( ( finite4290736615968046902t_unit @ I )
% 0.34/0.68 => ( ( member_Product_unit @ I2 @ I )
% 0.34/0.68 => ( ( ord_less_rat @ one_one_rat @ ( F @ I2 ) )
% 0.34/0.68 => ( ! [I3: product_unit] :
% 0.34/0.68 ( ( member_Product_unit @ I3 @ I )
% 0.34/0.68 => ( ord_less_eq_rat @ one_one_rat @ ( F @ I3 ) ) )
% 0.34/0.68 => ( ord_less_rat @ one_one_rat @ ( groups8984663225794451456it_rat @ F @ I ) ) ) ) ) ) ).
% 0.34/0.68
% 0.34/0.68 % less_1_prod2
% 0.34/0.68 thf(fact_7196_less__1__prod2,axiom,
% 0.34/0.68 ! [I: set_nat,I2: nat,F: nat > rat] :
% 0.34/0.68 ( ( finite_finite_nat @ I )
% 0.34/0.68 => ( ( member_nat @ I2 @ I )
% 0.34/0.68 => ( ( ord_less_rat @ one_one_rat @ ( F @ I2 ) )
% 0.34/0.68 => ( ! [I3: nat] :
% 0.34/0.68 ( ( member_nat @ I3 @ I )
% 0.34/0.68 => ( ord_less_eq_rat @ one_one_rat @ ( F @ I3 ) ) )
% 0.34/0.68 => ( ord_less_rat @ one_one_rat @ ( groups73079841787564623at_rat @ F @ I ) ) ) ) ) ) ).
% 0.34/0.68
% 0.34/0.68 % less_1_prod2
% 0.34/0.68 thf(fact_7197_less__1__prod2,axiom,
% 0.34/0.68 ! [I: set_Code_integer,I2: code_integer,F: code_integer > rat] :
% 0.34/0.68 ( ( finite6017078050557962740nteger @ I )
% 0.34/0.68 => ( ( member_Code_integer @ I2 @ I )
% 0.34/0.68 => ( ( ord_less_rat @ one_one_rat @ ( F @ I2 ) )
% 0.34/0.68 => ( ! [I3: code_integer] :
% 0.34/0.68 ( ( member_Code_integer @ I3 @ I )
% 0.34/0.68 => ( ord_less_eq_rat @ one_one_rat @ ( F @ I3 ) ) )
% 0.34/0.68 => ( ord_less_rat @ one_one_rat @ ( groups2555765274223993564er_rat @ F @ I ) ) ) ) ) ) ).
% 0.34/0.68
% 0.34/0.68 % less_1_prod2
% 0.34/0.68 thf(fact_7198_less__1__prod,axiom,
% 0.34/0.68 ! [I: set_Code_integer,F: code_integer > int] :
% 0.34/0.68 ( ( finite6017078050557962740nteger @ I )
% 0.34/0.68 => ( ( I != bot_bo3990330152332043303nteger )
% 0.34/0.68 => ( ! [I3: code_integer] :
% 0.34/0.68 ( ( member_Code_integer @ I3 @ I )
% 0.34/0.68 => ( ord_less_int @ one_one_int @ ( F @ I3 ) ) )
% 0.34/0.68 => ( ord_less_int @ one_one_int @ ( groups3188404863801439024er_int @ F @ I ) ) ) ) ) ).
% 0.34/0.68
% 0.34/0.68 % less_1_prod
% 0.34/0.68 thf(fact_7199_less__1__prod,axiom,
% 0.34/0.68 ! [I: set_Product_unit,F: product_unit > int] :
% 0.34/0.68 ( ( finite4290736615968046902t_unit @ I )
% 0.34/0.68 => ( ( I != bot_bo3957492148770167129t_unit )
% 0.34/0.68 => ( ! [I3: product_unit] :
% 0.34/0.68 ( ( member_Product_unit @ I3 @ I )
% 0.34/0.68 => ( ord_less_int @ one_one_int @ ( F @ I3 ) ) )
% 0.34/0.68 => ( ord_less_int @ one_one_int @ ( groups393930778517121108it_int @ F @ I ) ) ) ) ) ).
% 0.34/0.68
% 0.34/0.68 % less_1_prod
% 0.34/0.68 thf(fact_7200_less__1__prod,axiom,
% 0.34/0.68 ! [I: set_o,F: $o > int] :
% 0.34/0.68 ( ( finite_finite_o @ I )
% 0.34/0.68 => ( ( I != bot_bot_set_o )
% 0.34/0.68 => ( ! [I3: $o] :
% 0.34/0.68 ( ( member_o @ I3 @ I )
% 0.34/0.68 => ( ord_less_int @ one_one_int @ ( F @ I3 ) ) )
% 0.34/0.68 => ( ord_less_int @ one_one_int @ ( groups3502327434004483295_o_int @ F @ I ) ) ) ) ) ).
% 0.34/0.68
% 0.34/0.68 % less_1_prod
% 0.34/0.68 thf(fact_7201_less__1__prod,axiom,
% 0.34/0.68 ! [I: set_nat,F: nat > int] :
% 0.34/0.68 ( ( finite_finite_nat @ I )
% 0.34/0.68 => ( ( I != bot_bot_set_nat )
% 0.34/0.68 => ( ! [I3: nat] :
% 0.34/0.68 ( ( member_nat @ I3 @ I )
% 0.34/0.68 => ( ord_less_int @ one_one_int @ ( F @ I3 ) ) )
% 0.34/0.68 => ( ord_less_int @ one_one_int @ ( groups705719431365010083at_int @ F @ I ) ) ) ) ) ).
% 0.34/0.68
% 0.34/0.68 % less_1_prod
% 0.34/0.68 thf(fact_7202_less__1__prod,axiom,
% 0.34/0.68 ! [I: set_int,F: int > int] :
% 0.34/0.68 ( ( finite_finite_int @ I )
% 0.34/0.68 => ( ( I != bot_bot_set_int )
% 0.34/0.68 => ( ! [I3: int] :
% 0.34/0.68 ( ( member_int @ I3 @ I )
% 0.34/0.68 => ( ord_less_int @ one_one_int @ ( F @ I3 ) ) )
% 0.34/0.68 => ( ord_less_int @ one_one_int @ ( groups1705073143266064639nt_int @ F @ I ) ) ) ) ) ).
% 0.34/0.68
% 0.34/0.68 % less_1_prod
% 0.34/0.68 thf(fact_7203_less__1__prod,axiom,
% 0.34/0.68 ! [I: set_Code_integer,F: code_integer > code_integer] :
% 0.34/0.68 ( ( finite6017078050557962740nteger @ I )
% 0.34/0.68 => ( ( I != bot_bo3990330152332043303nteger )
% 0.34/0.68 => ( ! [I3: code_integer] :
% 0.34/0.68 ( ( member_Code_integer @ I3 @ I )
% 0.34/0.68 => ( ord_le6747313008572928689nteger @ one_one_Code_integer @ ( F @ I3 ) ) )
% 0.34/0.68 => ( ord_le6747313008572928689nteger @ one_one_Code_integer @ ( groups3674199335183972705nteger @ F @ I ) ) ) ) ) ).
% 0.34/0.68
% 0.34/0.68 % less_1_prod
% 0.34/0.68 thf(fact_7204_less__1__prod,axiom,
% 0.34/0.68 ! [I: set_Product_unit,F: product_unit > code_integer] :
% 0.34/0.68 ( ( finite4290736615968046902t_unit @ I )
% 0.34/0.68 => ( ( I != bot_bo3957492148770167129t_unit )
% 0.34/0.68 => ( ! [I3: product_unit] :
% 0.34/0.68 ( ( member_Product_unit @ I3 @ I )
% 0.34/0.68 => ( ord_le6747313008572928689nteger @ one_one_Code_integer @ ( F @ I3 ) ) )
% 0.34/0.68 => ( ord_le6747313008572928689nteger @ one_one_Code_integer @ ( groups8185359811944078981nteger @ F @ I ) ) ) ) ) ).
% 0.34/0.68
% 0.34/0.68 % less_1_prod
% 0.34/0.68 thf(fact_7205_less__1__prod,axiom,
% 0.34/0.68 ! [I: set_o,F: $o > code_integer] :
% 0.34/0.68 ( ( finite_finite_o @ I )
% 0.34/0.68 => ( ( I != bot_bot_set_o )
% 0.34/0.68 => ( ! [I3: $o] :
% 0.34/0.68 ( ( member_o @ I3 @ I )
% 0.34/0.68 => ( ord_le6747313008572928689nteger @ one_one_Code_integer @ ( F @ I3 ) ) )
% 0.34/0.68 => ( ord_le6747313008572928689nteger @ one_one_Code_integer @ ( groups7694694392188491536nteger @ F @ I ) ) ) ) ) ).
% 0.34/0.68
% 0.34/0.68 % less_1_prod
% 0.34/0.68 thf(fact_7206_less__1__prod,axiom,
% 0.34/0.68 ! [I: set_nat,F: nat > code_integer] :
% 0.34/0.68 ( ( finite_finite_nat @ I )
% 0.34/0.68 => ( ( I != bot_bot_set_nat )
% 0.34/0.68 => ( ! [I3: nat] :
% 0.34/0.68 ( ( member_nat @ I3 @ I )
% 0.34/0.68 => ( ord_le6747313008572928689nteger @ one_one_Code_integer @ ( F @ I3 ) ) )
% 0.34/0.68 => ( ord_le6747313008572928689nteger @ one_one_Code_integer @ ( groups3455450783089532116nteger @ F @ I ) ) ) ) ) ).
% 0.34/0.68
% 0.34/0.68 % less_1_prod
% 0.34/0.68 thf(fact_7207_less__1__prod,axiom,
% 0.34/0.68 ! [I: set_int,F: int > code_integer] :
% 0.34/0.68 ( ( finite_finite_int @ I )
% 0.34/0.68 => ( ( I != bot_bot_set_int )
% 0.34/0.68 => ( ! [I3: int] :
% 0.34/0.68 ( ( member_int @ I3 @ I )
% 0.34/0.68 => ( ord_le6747313008572928689nteger @ one_one_Code_integer @ ( F @ I3 ) ) )
% 0.34/0.68 => ( ord_le6747313008572928689nteger @ one_one_Code_integer @ ( groups3827104343326376752nteger @ F @ I ) ) ) ) ) ).
% 0.34/0.68
% 0.34/0.68 % less_1_prod
% 0.34/0.68 thf(fact_7208_prod_Oreindex__nontrivial,axiom,
% 0.34/0.68 ! [A: set_nat,H2: nat > nat,G: nat > assn] :
% 0.34/0.68 ( ( finite_finite_nat @ A )
% 0.34/0.68 => ( ! [X3: nat,Y2: nat] :
% 0.34/0.68 ( ( member_nat @ X3 @ A )
% 0.34/0.68 => ( ( member_nat @ Y2 @ A )
% 0.34/0.68 => ( ( X3 != Y2 )
% 0.34/0.68 => ( ( ( H2 @ X3 )
% 0.34/0.68 = ( H2 @ Y2 ) )
% 0.34/0.68 => ( ( G @ ( H2 @ X3 ) )
% 0.34/0.68 = one_one_assn ) ) ) ) )
% 0.34/0.68 => ( ( groups6906906614972039071t_assn @ G @ ( image_nat_nat2 @ H2 @ A ) )
% 0.34/0.68 = ( groups6906906614972039071t_assn @ ( comp_nat_assn_nat @ G @ H2 ) @ A ) ) ) ) ).
% 0.34/0.68
% 0.34/0.68 % prod.reindex_nontrivial
% 0.34/0.68 thf(fact_7209_prod_Oreindex__nontrivial,axiom,
% 0.34/0.68 ! [A: set_nat,H2: nat > code_integer,G: code_integer > assn] :
% 0.34/0.68 ( ( finite_finite_nat @ A )
% 0.34/0.68 => ( ! [X3: nat,Y2: nat] :
% 0.34/0.68 ( ( member_nat @ X3 @ A )
% 0.34/0.68 => ( ( member_nat @ Y2 @ A )
% 0.34/0.68 => ( ( X3 != Y2 )
% 0.34/0.68 => ( ( ( H2 @ X3 )
% 0.34/0.68 = ( H2 @ Y2 ) )
% 0.34/0.68 => ( ( G @ ( H2 @ X3 ) )
% 0.34/0.68 = one_one_assn ) ) ) ) )
% 0.34/0.68 => ( ( groups1304777262505850412r_assn @ G @ ( image_1215581382706833972nteger @ H2 @ A ) )
% 0.34/0.68 = ( groups6906906614972039071t_assn @ ( comp_C8462447471145125178sn_nat @ G @ H2 ) @ A ) ) ) ) ).
% 0.34/0.68
% 0.34/0.68 % prod.reindex_nontrivial
% 0.34/0.68 thf(fact_7210_prod_Oreindex__nontrivial,axiom,
% 0.34/0.68 ! [A: set_Code_integer,H2: code_integer > code_integer,G: code_integer > assn] :
% 0.34/0.68 ( ( finite6017078050557962740nteger @ A )
% 0.34/0.68 => ( ! [X3: code_integer,Y2: code_integer] :
% 0.34/0.68 ( ( member_Code_integer @ X3 @ A )
% 0.34/0.68 => ( ( member_Code_integer @ Y2 @ A )
% 0.34/0.68 => ( ( X3 != Y2 )
% 0.34/0.68 => ( ( ( H2 @ X3 )
% 0.34/0.68 = ( H2 @ Y2 ) )
% 0.34/0.68 => ( ( G @ ( H2 @ X3 ) )
% 0.34/0.68 = one_one_assn ) ) ) ) )
% 0.34/0.68 => ( ( groups1304777262505850412r_assn @ G @ ( image_4470545334726330049nteger @ H2 @ A ) )
% 0.34/0.68 = ( groups1304777262505850412r_assn @ ( comp_C3026223900822616519nteger @ G @ H2 ) @ A ) ) ) ) ).
% 0.34/0.68
% 0.34/0.68 % prod.reindex_nontrivial
% 0.34/0.68 thf(fact_7211_prod_Oreindex__nontrivial,axiom,
% 0.34/0.68 ! [A: set_Code_integer,H2: code_integer > code_integer,G: code_integer > nat] :
% 0.34/0.68 ( ( finite6017078050557962740nteger @ A )
% 0.34/0.68 => ( ! [X3: code_integer,Y2: code_integer] :
% 0.34/0.68 ( ( member_Code_integer @ X3 @ A )
% 0.34/0.68 => ( ( member_Code_integer @ Y2 @ A )
% 0.34/0.68 => ( ( X3 != Y2 )
% 0.34/0.68 => ( ( ( H2 @ X3 )
% 0.34/0.68 = ( H2 @ Y2 ) )
% 0.34/0.68 => ( ( G @ ( H2 @ X3 ) )
% 0.34/0.68 = one_one_nat ) ) ) ) )
% 0.34/0.68 => ( ( groups3190895334310489300er_nat @ G @ ( image_4470545334726330049nteger @ H2 @ A ) )
% 0.34/0.68 = ( groups3190895334310489300er_nat @ ( comp_C7238605732685392495nteger @ G @ H2 ) @ A ) ) ) ) ).
% 0.34/0.68
% 0.34/0.68 % prod.reindex_nontrivial
% 0.34/0.68 thf(fact_7212_prod_Oreindex__nontrivial,axiom,
% 0.34/0.68 ! [A: set_nat,H2: nat > nat,G: nat > int] :
% 0.34/0.68 ( ( finite_finite_nat @ A )
% 0.34/0.68 => ( ! [X3: nat,Y2: nat] :
% 0.34/0.68 ( ( member_nat @ X3 @ A )
% 0.34/0.68 => ( ( member_nat @ Y2 @ A )
% 0.34/0.68 => ( ( X3 != Y2 )
% 0.34/0.68 => ( ( ( H2 @ X3 )
% 0.34/0.68 = ( H2 @ Y2 ) )
% 0.34/0.68 => ( ( G @ ( H2 @ X3 ) )
% 0.34/0.68 = one_one_int ) ) ) ) )
% 0.34/0.68 => ( ( groups705719431365010083at_int @ G @ ( image_nat_nat2 @ H2 @ A ) )
% 0.34/0.68 = ( groups705719431365010083at_int @ ( comp_nat_int_nat @ G @ H2 ) @ A ) ) ) ) ).
% 0.34/0.68
% 0.34/0.68 % prod.reindex_nontrivial
% 0.34/0.68 thf(fact_7213_prod_Oreindex__nontrivial,axiom,
% 0.34/0.68 ! [A: set_nat,H2: nat > code_integer,G: code_integer > int] :
% 0.34/0.68 ( ( finite_finite_nat @ A )
% 0.34/0.68 => ( ! [X3: nat,Y2: nat] :
% 0.34/0.68 ( ( member_nat @ X3 @ A )
% 0.34/0.68 => ( ( member_nat @ Y2 @ A )
% 0.34/0.68 => ( ( X3 != Y2 )
% 0.34/0.68 => ( ( ( H2 @ X3 )
% 0.34/0.68 = ( H2 @ Y2 ) )
% 0.34/0.68 => ( ( G @ ( H2 @ X3 ) )
% 0.34/0.68 = one_one_int ) ) ) ) )
% 0.34/0.68 => ( ( groups3188404863801439024er_int @ G @ ( image_1215581382706833972nteger @ H2 @ A ) )
% 0.34/0.68 = ( groups705719431365010083at_int @ ( comp_C5048910307564104766nt_nat @ G @ H2 ) @ A ) ) ) ) ).
% 0.34/0.68
% 0.34/0.68 % prod.reindex_nontrivial
% 0.34/0.68 thf(fact_7214_prod_Oreindex__nontrivial,axiom,
% 0.34/0.68 ! [A: set_Code_integer,H2: code_integer > code_integer,G: code_integer > int] :
% 0.34/0.68 ( ( finite6017078050557962740nteger @ A )
% 0.34/0.68 => ( ! [X3: code_integer,Y2: code_integer] :
% 0.34/0.68 ( ( member_Code_integer @ X3 @ A )
% 0.34/0.68 => ( ( member_Code_integer @ Y2 @ A )
% 0.34/0.68 => ( ( X3 != Y2 )
% 0.34/0.68 => ( ( ( H2 @ X3 )
% 0.34/0.68 = ( H2 @ Y2 ) )
% 0.34/0.68 => ( ( G @ ( H2 @ X3 ) )
% 0.34/0.68 = one_one_int ) ) ) ) )
% 0.34/0.68 => ( ( groups3188404863801439024er_int @ G @ ( image_4470545334726330049nteger @ H2 @ A ) )
% 0.34/0.68 = ( groups3188404863801439024er_int @ ( comp_C7610259292922237131nteger @ G @ H2 ) @ A ) ) ) ) ).
% 0.34/0.68
% 0.34/0.68 % prod.reindex_nontrivial
% 0.34/0.68 thf(fact_7215_prod_Oreindex__nontrivial,axiom,
% 0.34/0.68 ! [A: set_nat,H2: nat > nat,G: nat > code_integer] :
% 0.34/0.68 ( ( finite_finite_nat @ A )
% 0.34/0.68 => ( ! [X3: nat,Y2: nat] :
% 0.34/0.68 ( ( member_nat @ X3 @ A )
% 0.34/0.68 => ( ( member_nat @ Y2 @ A )
% 0.34/0.68 => ( ( X3 != Y2 )
% 0.34/0.68 => ( ( ( H2 @ X3 )
% 0.34/0.68 = ( H2 @ Y2 ) )
% 0.34/0.68 => ( ( G @ ( H2 @ X3 ) )
% 0.34/0.68 = one_one_Code_integer ) ) ) ) )
% 0.34/0.68 => ( ( groups3455450783089532116nteger @ G @ ( image_nat_nat2 @ H2 @ A ) )
% 0.34/0.68 = ( groups3455450783089532116nteger @ ( comp_n3898172953802868194er_nat @ G @ H2 ) @ A ) ) ) ) ).
% 0.34/0.68
% 0.34/0.68 % prod.reindex_nontrivial
% 0.34/0.68 thf(fact_7216_prod_Oreindex__nontrivial,axiom,
% 0.34/0.68 ! [A: set_nat,H2: nat > code_integer,G: code_integer > code_integer] :
% 0.34/0.68 ( ( finite_finite_nat @ A )
% 0.34/0.68 => ( ! [X3: nat,Y2: nat] :
% 0.34/0.68 ( ( member_nat @ X3 @ A )
% 0.34/0.68 => ( ( member_nat @ Y2 @ A )
% 0.34/0.68 => ( ( X3 != Y2 )
% 0.34/0.68 => ( ( ( H2 @ X3 )
% 0.34/0.68 = ( H2 @ Y2 ) )
% 0.34/0.68 => ( ( G @ ( H2 @ X3 ) )
% 0.34/0.68 = one_one_Code_integer ) ) ) ) )
% 0.34/0.68 => ( ( groups3674199335183972705nteger @ G @ ( image_1215581382706833972nteger @ H2 @ A ) )
% 0.34/0.68 = ( groups3455450783089532116nteger @ ( comp_C6974050283906349679er_nat @ G @ H2 ) @ A ) ) ) ) ).
% 0.34/0.68
% 0.34/0.68 % prod.reindex_nontrivial
% 0.34/0.68 thf(fact_7217_prod_Oreindex__nontrivial,axiom,
% 0.34/0.68 ! [A: set_Code_integer,H2: code_integer > code_integer,G: code_integer > code_integer] :
% 0.34/0.68 ( ( finite6017078050557962740nteger @ A )
% 0.34/0.68 => ( ! [X3: code_integer,Y2: code_integer] :
% 0.34/0.68 ( ( member_Code_integer @ X3 @ A )
% 0.34/0.68 => ( ( member_Code_integer @ Y2 @ A )
% 0.34/0.68 => ( ( X3 != Y2 )
% 0.34/0.68 => ( ( ( H2 @ X3 )
% 0.34/0.68 = ( H2 @ Y2 ) )
% 0.34/0.68 => ( ( G @ ( H2 @ X3 ) )
% 0.34/0.68 = one_one_Code_integer ) ) ) ) )
% 0.34/0.68 => ( ( groups3674199335183972705nteger @ G @ ( image_4470545334726330049nteger @ H2 @ A ) )
% 0.34/0.68 = ( groups3674199335183972705nteger @ ( comp_C7449957260575251196nteger @ G @ H2 ) @ A ) ) ) ) ).
% 0.34/0.68
% 0.34/0.68 % prod.reindex_nontrivial
% 0.34/0.68 thf(fact_7218_prod_Osubset__diff,axiom,
% 0.34/0.68 ! [B4: set_Code_integer,A: set_Code_integer,G: code_integer > assn] :
% 0.34/0.68 ( ( ord_le7084787975880047091nteger @ B4 @ A )
% 0.34/0.68 => ( ( finite6017078050557962740nteger @ A )
% 0.34/0.68 => ( ( groups1304777262505850412r_assn @ G @ A )
% 0.34/0.68 = ( times_times_assn @ ( groups1304777262505850412r_assn @ G @ ( minus_2355218937544613996nteger @ A @ B4 ) ) @ ( groups1304777262505850412r_assn @ G @ B4 ) ) ) ) ) ).
% 0.34/0.68
% 0.34/0.68 % prod.subset_diff
% 0.34/0.68 thf(fact_7219_prod_Osubset__diff,axiom,
% 0.34/0.68 ! [B4: set_Code_integer,A: set_Code_integer,G: code_integer > rat] :
% 0.34/0.68 ( ( ord_le7084787975880047091nteger @ B4 @ A )
% 0.34/0.68 => ( ( finite6017078050557962740nteger @ A )
% 0.34/0.68 => ( ( groups2555765274223993564er_rat @ G @ A )
% 0.34/0.68 = ( times_times_rat @ ( groups2555765274223993564er_rat @ G @ ( minus_2355218937544613996nteger @ A @ B4 ) ) @ ( groups2555765274223993564er_rat @ G @ B4 ) ) ) ) ) ).
% 0.34/0.68
% 0.34/0.68 % prod.subset_diff
% 0.34/0.68 thf(fact_7220_prod_Osubset__diff,axiom,
% 0.34/0.68 ! [B4: set_Code_integer,A: set_Code_integer,G: code_integer > nat] :
% 0.34/0.68 ( ( ord_le7084787975880047091nteger @ B4 @ A )
% 0.34/0.68 => ( ( finite6017078050557962740nteger @ A )
% 0.34/0.68 => ( ( groups3190895334310489300er_nat @ G @ A )
% 0.34/0.68 = ( times_times_nat @ ( groups3190895334310489300er_nat @ G @ ( minus_2355218937544613996nteger @ A @ B4 ) ) @ ( groups3190895334310489300er_nat @ G @ B4 ) ) ) ) ) ).
% 0.34/0.68
% 0.34/0.68 % prod.subset_diff
% 0.34/0.68 thf(fact_7221_prod_Osubset__diff,axiom,
% 0.34/0.68 ! [B4: set_Code_integer,A: set_Code_integer,G: code_integer > int] :
% 0.34/0.68 ( ( ord_le7084787975880047091nteger @ B4 @ A )
% 0.34/0.68 => ( ( finite6017078050557962740nteger @ A )
% 0.34/0.68 => ( ( groups3188404863801439024er_int @ G @ A )
% 0.34/0.68 = ( times_times_int @ ( groups3188404863801439024er_int @ G @ ( minus_2355218937544613996nteger @ A @ B4 ) ) @ ( groups3188404863801439024er_int @ G @ B4 ) ) ) ) ) ).
% 0.34/0.68
% 0.34/0.68 % prod.subset_diff
% 0.34/0.68 thf(fact_7222_prod_Osubset__diff,axiom,
% 0.34/0.68 ! [B4: set_nat,A: set_nat,G: nat > assn] :
% 0.34/0.68 ( ( ord_less_eq_set_nat @ B4 @ A )
% 0.34/0.68 => ( ( finite_finite_nat @ A )
% 0.34/0.68 => ( ( groups6906906614972039071t_assn @ G @ A )
% 0.34/0.68 = ( times_times_assn @ ( groups6906906614972039071t_assn @ G @ ( minus_minus_set_nat @ A @ B4 ) ) @ ( groups6906906614972039071t_assn @ G @ B4 ) ) ) ) ) ).
% 0.34/0.68
% 0.34/0.68 % prod.subset_diff
% 0.34/0.68 thf(fact_7223_prod_Osubset__diff,axiom,
% 0.34/0.68 ! [B4: set_nat,A: set_nat,G: nat > rat] :
% 0.34/0.68 ( ( ord_less_eq_set_nat @ B4 @ A )
% 0.34/0.68 => ( ( finite_finite_nat @ A )
% 0.34/0.68 => ( ( groups73079841787564623at_rat @ G @ A )
% 0.34/0.68 = ( times_times_rat @ ( groups73079841787564623at_rat @ G @ ( minus_minus_set_nat @ A @ B4 ) ) @ ( groups73079841787564623at_rat @ G @ B4 ) ) ) ) ) ).
% 0.34/0.68
% 0.34/0.68 % prod.subset_diff
% 0.34/0.68 thf(fact_7224_prod_Osubset__diff,axiom,
% 0.34/0.68 ! [B4: set_nat,A: set_nat,G: nat > int] :
% 0.34/0.68 ( ( ord_less_eq_set_nat @ B4 @ A )
% 0.34/0.68 => ( ( finite_finite_nat @ A )
% 0.34/0.68 => ( ( groups705719431365010083at_int @ G @ A )
% 0.34/0.68 = ( times_times_int @ ( groups705719431365010083at_int @ G @ ( minus_minus_set_nat @ A @ B4 ) ) @ ( groups705719431365010083at_int @ G @ B4 ) ) ) ) ) ).
% 0.34/0.68
% 0.34/0.68 % prod.subset_diff
% 0.34/0.68 thf(fact_7225_prod_Osubset__diff,axiom,
% 0.34/0.68 ! [B4: set_nat,A: set_nat,G: nat > nat] :
% 0.34/0.68 ( ( ord_less_eq_set_nat @ B4 @ A )
% 0.34/0.68 => ( ( finite_finite_nat @ A )
% 0.34/0.68 => ( ( groups708209901874060359at_nat @ G @ A )
% 0.34/0.68 = ( times_times_nat @ ( groups708209901874060359at_nat @ G @ ( minus_minus_set_nat @ A @ B4 ) ) @ ( groups708209901874060359at_nat @ G @ B4 ) ) ) ) ) ).
% 0.34/0.68
% 0.34/0.68 % prod.subset_diff
% 0.34/0.68 thf(fact_7226_prod_Osame__carrier,axiom,
% 0.34/0.68 ! [C5: set_int,A: set_int,B4: set_int,G: int > assn,H2: int > assn] :
% 0.34/0.68 ( ( finite_finite_int @ C5 )
% 0.34/0.68 => ( ( ord_less_eq_set_int @ A @ C5 )
% 0.34/0.68 => ( ( ord_less_eq_set_int @ B4 @ C5 )
% 0.34/0.68 => ( ! [A4: int] :
% 0.34/0.68 ( ( member_int @ A4 @ ( minus_minus_set_int @ C5 @ A ) )
% 0.34/0.68 => ( ( G @ A4 )
% 0.34/0.68 = one_one_assn ) )
% 0.34/0.68 => ( ! [B2: int] :
% 0.34/0.68 ( ( member_int @ B2 @ ( minus_minus_set_int @ C5 @ B4 ) )
% 0.34/0.68 => ( ( H2 @ B2 )
% 0.34/0.68 = one_one_assn ) )
% 0.34/0.68 => ( ( ( groups7882442080178216443t_assn @ G @ A )
% 0.34/0.68 = ( groups7882442080178216443t_assn @ H2 @ B4 ) )
% 0.34/0.68 = ( ( groups7882442080178216443t_assn @ G @ C5 )
% 0.34/0.68 = ( groups7882442080178216443t_assn @ H2 @ C5 ) ) ) ) ) ) ) ) ).
% 0.34/0.68
% 0.34/0.68 % prod.same_carrier
% 0.34/0.68 thf(fact_7227_prod_Osame__carrier,axiom,
% 0.34/0.68 ! [C5: set_o,A: set_o,B4: set_o,G: $o > assn,H2: $o > assn] :
% 0.34/0.68 ( ( finite_finite_o @ C5 )
% 0.34/0.68 => ( ( ord_less_eq_set_o @ A @ C5 )
% 0.34/0.68 => ( ( ord_less_eq_set_o @ B4 @ C5 )
% 0.34/0.68 => ( ! [A4: $o] :
% 0.34/0.68 ( ( member_o @ A4 @ ( minus_minus_set_o @ C5 @ A ) )
% 0.34/0.68 => ( ( G @ A4 )
% 0.34/0.68 = one_one_assn ) )
% 0.34/0.68 => ( ! [B2: $o] :
% 0.34/0.68 ( ( member_o @ B2 @ ( minus_minus_set_o @ C5 @ B4 ) )
% 0.34/0.68 => ( ( H2 @ B2 )
% 0.34/0.68 = one_one_assn ) )
% 0.34/0.68 => ( ( ( groups5301882518646026715o_assn @ G @ A )
% 0.34/0.68 = ( groups5301882518646026715o_assn @ H2 @ B4 ) )
% 0.34/0.68 = ( ( groups5301882518646026715o_assn @ G @ C5 )
% 0.34/0.68 = ( groups5301882518646026715o_assn @ H2 @ C5 ) ) ) ) ) ) ) ) ).
% 0.34/0.68
% 0.34/0.68 % prod.same_carrier
% 0.34/0.68 thf(fact_7228_prod_Osame__carrier,axiom,
% 0.34/0.68 ! [C5: set_Product_unit,A: set_Product_unit,B4: set_Product_unit,G: product_unit > assn,H2: product_unit > assn] :
% 0.34/0.68 ( ( finite4290736615968046902t_unit @ C5 )
% 0.34/0.68 => ( ( ord_le3507040750410214029t_unit @ A @ C5 )
% 0.34/0.68 => ( ( ord_le3507040750410214029t_unit @ B4 @ C5 )
% 0.34/0.68 => ( ! [A4: product_unit] :
% 0.34/0.68 ( ( member_Product_unit @ A4 @ ( minus_6452836326544984404t_unit @ C5 @ A ) )
% 0.34/0.68 => ( ( G @ A4 )
% 0.34/0.68 = one_one_assn ) )
% 0.34/0.68 => ( ! [B2: product_unit] :
% 0.34/0.68 ( ( member_Product_unit @ B2 @ ( minus_6452836326544984404t_unit @ C5 @ B4 ) )
% 0.34/0.68 => ( ( H2 @ B2 )
% 0.34/0.68 = one_one_assn ) )
% 0.34/0.68 => ( ( ( groups5360233303959380816t_assn @ G @ A )
% 0.34/0.68 = ( groups5360233303959380816t_assn @ H2 @ B4 ) )
% 0.34/0.68 = ( ( groups5360233303959380816t_assn @ G @ C5 )
% 0.34/0.68 = ( groups5360233303959380816t_assn @ H2 @ C5 ) ) ) ) ) ) ) ) ).
% 0.34/0.68
% 0.34/0.68 % prod.same_carrier
% 0.34/0.68 thf(fact_7229_prod_Osame__carrier,axiom,
% 0.34/0.68 ! [C5: set_Code_integer,A: set_Code_integer,B4: set_Code_integer,G: code_integer > assn,H2: code_integer > assn] :
% 0.34/0.68 ( ( finite6017078050557962740nteger @ C5 )
% 0.34/0.68 => ( ( ord_le7084787975880047091nteger @ A @ C5 )
% 0.34/0.68 => ( ( ord_le7084787975880047091nteger @ B4 @ C5 )
% 0.34/0.68 => ( ! [A4: code_integer] :
% 0.34/0.68 ( ( member_Code_integer @ A4 @ ( minus_2355218937544613996nteger @ C5 @ A ) )
% 0.34/0.68 => ( ( G @ A4 )
% 0.34/0.68 = one_one_assn ) )
% 0.34/0.68 => ( ! [B2: code_integer] :
% 0.34/0.68 ( ( member_Code_integer @ B2 @ ( minus_2355218937544613996nteger @ C5 @ B4 ) )
% 0.34/0.68 => ( ( H2 @ B2 )
% 0.34/0.68 = one_one_assn ) )
% 0.34/0.68 => ( ( ( groups1304777262505850412r_assn @ G @ A )
% 0.34/0.68 = ( groups1304777262505850412r_assn @ H2 @ B4 ) )
% 0.34/0.68 = ( ( groups1304777262505850412r_assn @ G @ C5 )
% 0.34/0.68 = ( groups1304777262505850412r_assn @ H2 @ C5 ) ) ) ) ) ) ) ) ).
% 0.34/0.68
% 0.34/0.68 % prod.same_carrier
% 0.34/0.68 thf(fact_7230_prod_Osame__carrier,axiom,
% 0.34/0.68 ! [C5: set_int,A: set_int,B4: set_int,G: int > nat,H2: int > nat] :
% 0.34/0.68 ( ( finite_finite_int @ C5 )
% 0.34/0.68 => ( ( ord_less_eq_set_int @ A @ C5 )
% 0.34/0.68 => ( ( ord_less_eq_set_int @ B4 @ C5 )
% 0.34/0.68 => ( ! [A4: int] :
% 0.34/0.68 ( ( member_int @ A4 @ ( minus_minus_set_int @ C5 @ A ) )
% 0.34/0.68 => ( ( G @ A4 )
% 0.34/0.68 = one_one_nat ) )
% 0.34/0.68 => ( ! [B2: int] :
% 0.34/0.68 ( ( member_int @ B2 @ ( minus_minus_set_int @ C5 @ B4 ) )
% 0.34/0.68 => ( ( H2 @ B2 )
% 0.34/0.68 = one_one_nat ) )
% 0.34/0.68 => ( ( ( groups1707563613775114915nt_nat @ G @ A )
% 0.34/0.68 = ( groups1707563613775114915nt_nat @ H2 @ B4 ) )
% 0.34/0.68 = ( ( groups1707563613775114915nt_nat @ G @ C5 )
% 0.34/0.68 = ( groups1707563613775114915nt_nat @ H2 @ C5 ) ) ) ) ) ) ) ) ).
% 0.34/0.68
% 0.34/0.68 % prod.same_carrier
% 0.34/0.68 thf(fact_7231_prod_Osame__carrier,axiom,
% 0.34/0.68 ! [C5: set_o,A: set_o,B4: set_o,G: $o > nat,H2: $o > nat] :
% 0.34/0.68 ( ( finite_finite_o @ C5 )
% 0.34/0.68 => ( ( ord_less_eq_set_o @ A @ C5 )
% 0.34/0.68 => ( ( ord_less_eq_set_o @ B4 @ C5 )
% 0.34/0.68 => ( ! [A4: $o] :
% 0.34/0.68 ( ( member_o @ A4 @ ( minus_minus_set_o @ C5 @ A ) )
% 0.34/0.68 => ( ( G @ A4 )
% 0.34/0.68 = one_one_nat ) )
% 0.34/0.68 => ( ! [B2: $o] :
% 0.34/0.68 ( ( member_o @ B2 @ ( minus_minus_set_o @ C5 @ B4 ) )
% 0.34/0.68 => ( ( H2 @ B2 )
% 0.34/0.68 = one_one_nat ) )
% 0.34/0.68 => ( ( ( groups3504817904513533571_o_nat @ G @ A )
% 0.34/0.68 = ( groups3504817904513533571_o_nat @ H2 @ B4 ) )
% 0.34/0.68 = ( ( groups3504817904513533571_o_nat @ G @ C5 )
% 0.34/0.68 = ( groups3504817904513533571_o_nat @ H2 @ C5 ) ) ) ) ) ) ) ) ).
% 0.34/0.68
% 0.34/0.68 % prod.same_carrier
% 0.34/0.68 thf(fact_7232_prod_Osame__carrier,axiom,
% 0.34/0.68 ! [C5: set_Product_unit,A: set_Product_unit,B4: set_Product_unit,G: product_unit > nat,H2: product_unit > nat] :
% 0.34/0.68 ( ( finite4290736615968046902t_unit @ C5 )
% 0.34/0.68 => ( ( ord_le3507040750410214029t_unit @ A @ C5 )
% 0.34/0.68 => ( ( ord_le3507040750410214029t_unit @ B4 @ C5 )
% 0.34/0.68 => ( ! [A4: product_unit] :
% 0.34/0.68 ( ( member_Product_unit @ A4 @ ( minus_6452836326544984404t_unit @ C5 @ A ) )
% 0.34/0.68 => ( ( G @ A4 )
% 0.34/0.68 = one_one_nat ) )
% 0.34/0.68 => ( ! [B2: product_unit] :
% 0.34/0.68 ( ( member_Product_unit @ B2 @ ( minus_6452836326544984404t_unit @ C5 @ B4 ) )
% 0.34/0.68 => ( ( H2 @ B2 )
% 0.34/0.68 = one_one_nat ) )
% 0.34/0.68 => ( ( ( groups396421249026171384it_nat @ G @ A )
% 0.34/0.68 = ( groups396421249026171384it_nat @ H2 @ B4 ) )
% 0.34/0.68 = ( ( groups396421249026171384it_nat @ G @ C5 )
% 0.34/0.68 = ( groups396421249026171384it_nat @ H2 @ C5 ) ) ) ) ) ) ) ) ).
% 0.34/0.68
% 0.34/0.68 % prod.same_carrier
% 0.34/0.68 thf(fact_7233_prod_Osame__carrier,axiom,
% 0.34/0.68 ! [C5: set_Code_integer,A: set_Code_integer,B4: set_Code_integer,G: code_integer > nat,H2: code_integer > nat] :
% 0.34/0.68 ( ( finite6017078050557962740nteger @ C5 )
% 0.34/0.68 => ( ( ord_le7084787975880047091nteger @ A @ C5 )
% 0.34/0.68 => ( ( ord_le7084787975880047091nteger @ B4 @ C5 )
% 0.34/0.68 => ( ! [A4: code_integer] :
% 0.34/0.68 ( ( member_Code_integer @ A4 @ ( minus_2355218937544613996nteger @ C5 @ A ) )
% 0.34/0.68 => ( ( G @ A4 )
% 0.34/0.68 = one_one_nat ) )
% 0.34/0.68 => ( ! [B2: code_integer] :
% 0.34/0.68 ( ( member_Code_integer @ B2 @ ( minus_2355218937544613996nteger @ C5 @ B4 ) )
% 0.34/0.68 => ( ( H2 @ B2 )
% 0.34/0.68 = one_one_nat ) )
% 0.34/0.68 => ( ( ( groups3190895334310489300er_nat @ G @ A )
% 0.34/0.68 = ( groups3190895334310489300er_nat @ H2 @ B4 ) )
% 0.34/0.68 = ( ( groups3190895334310489300er_nat @ G @ C5 )
% 0.34/0.68 = ( groups3190895334310489300er_nat @ H2 @ C5 ) ) ) ) ) ) ) ) ).
% 0.34/0.68
% 0.34/0.68 % prod.same_carrier
% 0.34/0.68 thf(fact_7234_prod_Osame__carrier,axiom,
% 0.34/0.68 ! [C5: set_int,A: set_int,B4: set_int,G: int > int,H2: int > int] :
% 0.34/0.68 ( ( finite_finite_int @ C5 )
% 0.34/0.68 => ( ( ord_less_eq_set_int @ A @ C5 )
% 0.34/0.68 => ( ( ord_less_eq_set_int @ B4 @ C5 )
% 0.34/0.68 => ( ! [A4: int] :
% 0.34/0.68 ( ( member_int @ A4 @ ( minus_minus_set_int @ C5 @ A ) )
% 0.34/0.68 => ( ( G @ A4 )
% 0.34/0.68 = one_one_int ) )
% 0.34/0.68 => ( ! [B2: int] :
% 0.34/0.68 ( ( member_int @ B2 @ ( minus_minus_set_int @ C5 @ B4 ) )
% 0.34/0.68 => ( ( H2 @ B2 )
% 0.34/0.68 = one_one_int ) )
% 0.34/0.68 => ( ( ( groups1705073143266064639nt_int @ G @ A )
% 0.34/0.68 = ( groups1705073143266064639nt_int @ H2 @ B4 ) )
% 0.34/0.68 = ( ( groups1705073143266064639nt_int @ G @ C5 )
% 0.34/0.68 = ( groups1705073143266064639nt_int @ H2 @ C5 ) ) ) ) ) ) ) ) ).
% 0.34/0.68
% 0.34/0.68 % prod.same_carrier
% 0.34/0.68 thf(fact_7235_prod_Osame__carrier,axiom,
% 0.34/0.68 ! [C5: set_o,A: set_o,B4: set_o,G: $o > int,H2: $o > int] :
% 0.34/0.68 ( ( finite_finite_o @ C5 )
% 0.34/0.68 => ( ( ord_less_eq_set_o @ A @ C5 )
% 0.34/0.68 => ( ( ord_less_eq_set_o @ B4 @ C5 )
% 0.34/0.68 => ( ! [A4: $o] :
% 0.34/0.68 ( ( member_o @ A4 @ ( minus_minus_set_o @ C5 @ A ) )
% 0.34/0.68 => ( ( G @ A4 )
% 0.34/0.68 = one_one_int ) )
% 0.34/0.68 => ( ! [B2: $o] :
% 0.34/0.68 ( ( member_o @ B2 @ ( minus_minus_set_o @ C5 @ B4 ) )
% 0.34/0.68 => ( ( H2 @ B2 )
% 0.34/0.68 = one_one_int ) )
% 0.34/0.68 => ( ( ( groups3502327434004483295_o_int @ G @ A )
% 0.34/0.68 = ( groups3502327434004483295_o_int @ H2 @ B4 ) )
% 0.34/0.68 = ( ( groups3502327434004483295_o_int @ G @ C5 )
% 0.34/0.68 = ( groups3502327434004483295_o_int @ H2 @ C5 ) ) ) ) ) ) ) ) ).
% 0.34/0.68
% 0.34/0.68 % prod.same_carrier
% 0.34/0.68 thf(fact_7236_prod_Osame__carrierI,axiom,
% 0.34/0.68 ! [C5: set_int,A: set_int,B4: set_int,G: int > assn,H2: int > assn] :
% 0.34/0.68 ( ( finite_finite_int @ C5 )
% 0.34/0.68 => ( ( ord_less_eq_set_int @ A @ C5 )
% 0.34/0.68 => ( ( ord_less_eq_set_int @ B4 @ C5 )
% 0.34/0.68 => ( ! [A4: int] :
% 0.34/0.68 ( ( member_int @ A4 @ ( minus_minus_set_int @ C5 @ A ) )
% 0.34/0.68 => ( ( G @ A4 )
% 0.34/0.68 = one_one_assn ) )
% 0.34/0.68 => ( ! [B2: int] :
% 0.34/0.68 ( ( member_int @ B2 @ ( minus_minus_set_int @ C5 @ B4 ) )
% 0.34/0.68 => ( ( H2 @ B2 )
% 0.34/0.68 = one_one_assn ) )
% 0.34/0.68 => ( ( ( groups7882442080178216443t_assn @ G @ C5 )
% 0.34/0.68 = ( groups7882442080178216443t_assn @ H2 @ C5 ) )
% 0.34/0.68 => ( ( groups7882442080178216443t_assn @ G @ A )
% 0.34/0.68 = ( groups7882442080178216443t_assn @ H2 @ B4 ) ) ) ) ) ) ) ) ).
% 0.34/0.68
% 0.34/0.68 % prod.same_carrierI
% 0.34/0.68 thf(fact_7237_prod_Osame__carrierI,axiom,
% 0.34/0.68 ! [C5: set_o,A: set_o,B4: set_o,G: $o > assn,H2: $o > assn] :
% 0.34/0.68 ( ( finite_finite_o @ C5 )
% 0.34/0.68 => ( ( ord_less_eq_set_o @ A @ C5 )
% 0.34/0.68 => ( ( ord_less_eq_set_o @ B4 @ C5 )
% 0.34/0.68 => ( ! [A4: $o] :
% 0.34/0.68 ( ( member_o @ A4 @ ( minus_minus_set_o @ C5 @ A ) )
% 0.34/0.68 => ( ( G @ A4 )
% 0.34/0.68 = one_one_assn ) )
% 0.34/0.68 => ( ! [B2: $o] :
% 0.34/0.68 ( ( member_o @ B2 @ ( minus_minus_set_o @ C5 @ B4 ) )
% 0.34/0.68 => ( ( H2 @ B2 )
% 0.34/0.68 = one_one_assn ) )
% 0.34/0.68 => ( ( ( groups5301882518646026715o_assn @ G @ C5 )
% 0.34/0.68 = ( groups5301882518646026715o_assn @ H2 @ C5 ) )
% 0.34/0.68 => ( ( groups5301882518646026715o_assn @ G @ A )
% 0.34/0.68 = ( groups5301882518646026715o_assn @ H2 @ B4 ) ) ) ) ) ) ) ) ).
% 0.34/0.68
% 0.34/0.68 % prod.same_carrierI
% 0.34/0.68 thf(fact_7238_prod_Osame__carrierI,axiom,
% 0.34/0.68 ! [C5: set_Product_unit,A: set_Product_unit,B4: set_Product_unit,G: product_unit > assn,H2: product_unit > assn] :
% 0.34/0.68 ( ( finite4290736615968046902t_unit @ C5 )
% 0.34/0.68 => ( ( ord_le3507040750410214029t_unit @ A @ C5 )
% 0.34/0.68 => ( ( ord_le3507040750410214029t_unit @ B4 @ C5 )
% 0.34/0.68 => ( ! [A4: product_unit] :
% 0.34/0.68 ( ( member_Product_unit @ A4 @ ( minus_6452836326544984404t_unit @ C5 @ A ) )
% 0.34/0.68 => ( ( G @ A4 )
% 0.34/0.68 = one_one_assn ) )
% 0.34/0.68 => ( ! [B2: product_unit] :
% 0.34/0.68 ( ( member_Product_unit @ B2 @ ( minus_6452836326544984404t_unit @ C5 @ B4 ) )
% 0.34/0.68 => ( ( H2 @ B2 )
% 0.34/0.68 = one_one_assn ) )
% 0.34/0.68 => ( ( ( groups5360233303959380816t_assn @ G @ C5 )
% 0.34/0.68 = ( groups5360233303959380816t_assn @ H2 @ C5 ) )
% 0.34/0.68 => ( ( groups5360233303959380816t_assn @ G @ A )
% 0.34/0.68 = ( groups5360233303959380816t_assn @ H2 @ B4 ) ) ) ) ) ) ) ) ).
% 0.34/0.68
% 0.34/0.68 % prod.same_carrierI
% 0.34/0.68 thf(fact_7239_prod_Osame__carrierI,axiom,
% 0.34/0.68 ! [C5: set_Code_integer,A: set_Code_integer,B4: set_Code_integer,G: code_integer > assn,H2: code_integer > assn] :
% 0.34/0.68 ( ( finite6017078050557962740nteger @ C5 )
% 0.34/0.68 => ( ( ord_le7084787975880047091nteger @ A @ C5 )
% 0.34/0.68 => ( ( ord_le7084787975880047091nteger @ B4 @ C5 )
% 0.34/0.68 => ( ! [A4: code_integer] :
% 0.34/0.68 ( ( member_Code_integer @ A4 @ ( minus_2355218937544613996nteger @ C5 @ A ) )
% 0.34/0.68 => ( ( G @ A4 )
% 0.34/0.68 = one_one_assn ) )
% 0.34/0.68 => ( ! [B2: code_integer] :
% 0.34/0.68 ( ( member_Code_integer @ B2 @ ( minus_2355218937544613996nteger @ C5 @ B4 ) )
% 0.34/0.68 => ( ( H2 @ B2 )
% 0.34/0.68 = one_one_assn ) )
% 0.34/0.68 => ( ( ( groups1304777262505850412r_assn @ G @ C5 )
% 0.34/0.68 = ( groups1304777262505850412r_assn @ H2 @ C5 ) )
% 0.34/0.68 => ( ( groups1304777262505850412r_assn @ G @ A )
% 0.34/0.68 = ( groups1304777262505850412r_assn @ H2 @ B4 ) ) ) ) ) ) ) ) ).
% 0.34/0.68
% 0.34/0.68 % prod.same_carrierI
% 0.34/0.68 thf(fact_7240_prod_Osame__carrierI,axiom,
% 0.34/0.68 ! [C5: set_int,A: set_int,B4: set_int,G: int > nat,H2: int > nat] :
% 0.34/0.68 ( ( finite_finite_int @ C5 )
% 0.34/0.68 => ( ( ord_less_eq_set_int @ A @ C5 )
% 0.34/0.68 => ( ( ord_less_eq_set_int @ B4 @ C5 )
% 0.34/0.68 => ( ! [A4: int] :
% 0.34/0.68 ( ( member_int @ A4 @ ( minus_minus_set_int @ C5 @ A ) )
% 0.34/0.68 => ( ( G @ A4 )
% 0.34/0.68 = one_one_nat ) )
% 0.34/0.68 => ( ! [B2: int] :
% 0.34/0.68 ( ( member_int @ B2 @ ( minus_minus_set_int @ C5 @ B4 ) )
% 0.34/0.68 => ( ( H2 @ B2 )
% 0.34/0.68 = one_one_nat ) )
% 0.34/0.68 => ( ( ( groups1707563613775114915nt_nat @ G @ C5 )
% 0.34/0.68 = ( groups1707563613775114915nt_nat @ H2 @ C5 ) )
% 0.34/0.68 => ( ( groups1707563613775114915nt_nat @ G @ A )
% 0.34/0.68 = ( groups1707563613775114915nt_nat @ H2 @ B4 ) ) ) ) ) ) ) ) ).
% 0.34/0.68
% 0.34/0.68 % prod.same_carrierI
% 0.34/0.68 thf(fact_7241_prod_Osame__carrierI,axiom,
% 0.34/0.68 ! [C5: set_o,A: set_o,B4: set_o,G: $o > nat,H2: $o > nat] :
% 0.34/0.68 ( ( finite_finite_o @ C5 )
% 0.34/0.68 => ( ( ord_less_eq_set_o @ A @ C5 )
% 0.34/0.68 => ( ( ord_less_eq_set_o @ B4 @ C5 )
% 0.34/0.68 => ( ! [A4: $o] :
% 0.34/0.68 ( ( member_o @ A4 @ ( minus_minus_set_o @ C5 @ A ) )
% 0.34/0.68 => ( ( G @ A4 )
% 0.34/0.68 = one_one_nat ) )
% 0.34/0.68 => ( ! [B2: $o] :
% 0.34/0.68 ( ( member_o @ B2 @ ( minus_minus_set_o @ C5 @ B4 ) )
% 0.34/0.68 => ( ( H2 @ B2 )
% 0.34/0.68 = one_one_nat ) )
% 0.34/0.68 => ( ( ( groups3504817904513533571_o_nat @ G @ C5 )
% 0.34/0.68 = ( groups3504817904513533571_o_nat @ H2 @ C5 ) )
% 0.34/0.68 => ( ( groups3504817904513533571_o_nat @ G @ A )
% 0.34/0.68 = ( groups3504817904513533571_o_nat @ H2 @ B4 ) ) ) ) ) ) ) ) ).
% 0.34/0.68
% 0.34/0.68 % prod.same_carrierI
% 0.34/0.68 thf(fact_7242_prod_Osame__carrierI,axiom,
% 0.34/0.68 ! [C5: set_Product_unit,A: set_Product_unit,B4: set_Product_unit,G: product_unit > nat,H2: product_unit > nat] :
% 0.34/0.68 ( ( finite4290736615968046902t_unit @ C5 )
% 0.34/0.68 => ( ( ord_le3507040750410214029t_unit @ A @ C5 )
% 0.34/0.68 => ( ( ord_le3507040750410214029t_unit @ B4 @ C5 )
% 0.34/0.68 => ( ! [A4: product_unit] :
% 0.34/0.68 ( ( member_Product_unit @ A4 @ ( minus_6452836326544984404t_unit @ C5 @ A ) )
% 0.34/0.68 => ( ( G @ A4 )
% 0.34/0.68 = one_one_nat ) )
% 0.34/0.68 => ( ! [B2: product_unit] :
% 0.34/0.68 ( ( member_Product_unit @ B2 @ ( minus_6452836326544984404t_unit @ C5 @ B4 ) )
% 0.34/0.68 => ( ( H2 @ B2 )
% 0.34/0.68 = one_one_nat ) )
% 0.34/0.68 => ( ( ( groups396421249026171384it_nat @ G @ C5 )
% 0.34/0.68 = ( groups396421249026171384it_nat @ H2 @ C5 ) )
% 0.34/0.68 => ( ( groups396421249026171384it_nat @ G @ A )
% 0.34/0.68 = ( groups396421249026171384it_nat @ H2 @ B4 ) ) ) ) ) ) ) ) ).
% 0.34/0.68
% 0.34/0.68 % prod.same_carrierI
% 0.34/0.68 thf(fact_7243_prod_Osame__carrierI,axiom,
% 0.34/0.68 ! [C5: set_Code_integer,A: set_Code_integer,B4: set_Code_integer,G: code_integer > nat,H2: code_integer > nat] :
% 0.34/0.68 ( ( finite6017078050557962740nteger @ C5 )
% 0.34/0.68 => ( ( ord_le7084787975880047091nteger @ A @ C5 )
% 0.34/0.68 => ( ( ord_le7084787975880047091nteger @ B4 @ C5 )
% 0.34/0.68 => ( ! [A4: code_integer] :
% 0.34/0.68 ( ( member_Code_integer @ A4 @ ( minus_2355218937544613996nteger @ C5 @ A ) )
% 0.34/0.68 => ( ( G @ A4 )
% 0.34/0.68 = one_one_nat ) )
% 0.34/0.68 => ( ! [B2: code_integer] :
% 0.34/0.68 ( ( member_Code_integer @ B2 @ ( minus_2355218937544613996nteger @ C5 @ B4 ) )
% 0.34/0.68 => ( ( H2 @ B2 )
% 0.34/0.68 = one_one_nat ) )
% 0.34/0.68 => ( ( ( groups3190895334310489300er_nat @ G @ C5 )
% 0.34/0.68 = ( groups3190895334310489300er_nat @ H2 @ C5 ) )
% 0.34/0.68 => ( ( groups3190895334310489300er_nat @ G @ A )
% 0.34/0.68 = ( groups3190895334310489300er_nat @ H2 @ B4 ) ) ) ) ) ) ) ) ).
% 0.34/0.68
% 0.34/0.68 % prod.same_carrierI
% 0.34/0.68 thf(fact_7244_prod_Osame__carrierI,axiom,
% 0.34/0.68 ! [C5: set_int,A: set_int,B4: set_int,G: int > int,H2: int > int] :
% 0.34/0.68 ( ( finite_finite_int @ C5 )
% 0.34/0.68 => ( ( ord_less_eq_set_int @ A @ C5 )
% 0.34/0.68 => ( ( ord_less_eq_set_int @ B4 @ C5 )
% 0.34/0.68 => ( ! [A4: int] :
% 0.34/0.68 ( ( member_int @ A4 @ ( minus_minus_set_int @ C5 @ A ) )
% 0.34/0.68 => ( ( G @ A4 )
% 0.34/0.68 = one_one_int ) )
% 0.34/0.68 => ( ! [B2: int] :
% 0.34/0.68 ( ( member_int @ B2 @ ( minus_minus_set_int @ C5 @ B4 ) )
% 0.34/0.68 => ( ( H2 @ B2 )
% 0.34/0.68 = one_one_int ) )
% 0.34/0.68 => ( ( ( groups1705073143266064639nt_int @ G @ C5 )
% 0.34/0.68 = ( groups1705073143266064639nt_int @ H2 @ C5 ) )
% 0.34/0.68 => ( ( groups1705073143266064639nt_int @ G @ A )
% 0.34/0.68 = ( groups1705073143266064639nt_int @ H2 @ B4 ) ) ) ) ) ) ) ) ).
% 0.34/0.68
% 0.34/0.68 % prod.same_carrierI
% 0.34/0.68 thf(fact_7245_prod_Osame__carrierI,axiom,
% 0.34/0.68 ! [C5: set_o,A: set_o,B4: set_o,G: $o > int,H2: $o > int] :
% 0.34/0.68 ( ( finite_finite_o @ C5 )
% 0.34/0.68 => ( ( ord_less_eq_set_o @ A @ C5 )
% 0.34/0.68 => ( ( ord_less_eq_set_o @ B4 @ C5 )
% 0.34/0.68 => ( ! [A4: $o] :
% 0.34/0.68 ( ( member_o @ A4 @ ( minus_minus_set_o @ C5 @ A ) )
% 0.34/0.68 => ( ( G @ A4 )
% 0.34/0.68 = one_one_int ) )
% 0.34/0.68 => ( ! [B2: $o] :
% 0.34/0.68 ( ( member_o @ B2 @ ( minus_minus_set_o @ C5 @ B4 ) )
% 0.34/0.68 => ( ( H2 @ B2 )
% 0.34/0.68 = one_one_int ) )
% 0.34/0.68 => ( ( ( groups3502327434004483295_o_int @ G @ C5 )
% 0.34/0.68 = ( groups3502327434004483295_o_int @ H2 @ C5 ) )
% 0.34/0.68 => ( ( groups3502327434004483295_o_int @ G @ A )
% 0.34/0.68 = ( groups3502327434004483295_o_int @ H2 @ B4 ) ) ) ) ) ) ) ) ).
% 0.34/0.68
% 0.34/0.68 % prod.same_carrierI
% 0.34/0.68 thf(fact_7246_prod_Omono__neutral__left,axiom,
% 0.34/0.68 ! [T2: set_Code_integer,S: set_Code_integer,G: code_integer > assn] :
% 0.34/0.68 ( ( finite6017078050557962740nteger @ T2 )
% 0.34/0.68 => ( ( ord_le7084787975880047091nteger @ S @ T2 )
% 0.34/0.68 => ( ! [X3: code_integer] :
% 0.34/0.68 ( ( member_Code_integer @ X3 @ ( minus_2355218937544613996nteger @ T2 @ S ) )
% 0.34/0.68 => ( ( G @ X3 )
% 0.34/0.68 = one_one_assn ) )
% 0.34/0.68 => ( ( groups1304777262505850412r_assn @ G @ S )
% 0.34/0.68 = ( groups1304777262505850412r_assn @ G @ T2 ) ) ) ) ) ).
% 0.34/0.68
% 0.34/0.68 % prod.mono_neutral_left
% 0.34/0.68 thf(fact_7247_prod_Omono__neutral__left,axiom,
% 0.34/0.68 ! [T2: set_Code_integer,S: set_Code_integer,G: code_integer > nat] :
% 0.34/0.68 ( ( finite6017078050557962740nteger @ T2 )
% 0.34/0.68 => ( ( ord_le7084787975880047091nteger @ S @ T2 )
% 0.34/0.68 => ( ! [X3: code_integer] :
% 0.34/0.68 ( ( member_Code_integer @ X3 @ ( minus_2355218937544613996nteger @ T2 @ S ) )
% 0.34/0.68 => ( ( G @ X3 )
% 0.34/0.68 = one_one_nat ) )
% 0.34/0.68 => ( ( groups3190895334310489300er_nat @ G @ S )
% 0.34/0.68 = ( groups3190895334310489300er_nat @ G @ T2 ) ) ) ) ) ).
% 0.34/0.68
% 0.34/0.68 % prod.mono_neutral_left
% 0.34/0.68 thf(fact_7248_prod_Omono__neutral__left,axiom,
% 0.34/0.68 ! [T2: set_Code_integer,S: set_Code_integer,G: code_integer > int] :
% 0.34/0.68 ( ( finite6017078050557962740nteger @ T2 )
% 0.34/0.68 => ( ( ord_le7084787975880047091nteger @ S @ T2 )
% 0.34/0.68 => ( ! [X3: code_integer] :
% 0.34/0.68 ( ( member_Code_integer @ X3 @ ( minus_2355218937544613996nteger @ T2 @ S ) )
% 0.34/0.68 => ( ( G @ X3 )
% 0.34/0.68 = one_one_int ) )
% 0.34/0.68 => ( ( groups3188404863801439024er_int @ G @ S )
% 0.34/0.68 = ( groups3188404863801439024er_int @ G @ T2 ) ) ) ) ) ).
% 0.34/0.68
% 0.34/0.68 % prod.mono_neutral_left
% 0.34/0.68 thf(fact_7249_prod_Omono__neutral__left,axiom,
% 0.34/0.68 ! [T2: set_Code_integer,S: set_Code_integer,G: code_integer > code_integer] :
% 0.34/0.68 ( ( finite6017078050557962740nteger @ T2 )
% 0.34/0.68 => ( ( ord_le7084787975880047091nteger @ S @ T2 )
% 0.34/0.68 => ( ! [X3: code_integer] :
% 0.34/0.68 ( ( member_Code_integer @ X3 @ ( minus_2355218937544613996nteger @ T2 @ S ) )
% 0.34/0.68 => ( ( G @ X3 )
% 0.34/0.68 = one_one_Code_integer ) )
% 0.34/0.68 => ( ( groups3674199335183972705nteger @ G @ S )
% 0.34/0.68 = ( groups3674199335183972705nteger @ G @ T2 ) ) ) ) ) ).
% 0.34/0.68
% 0.34/0.68 % prod.mono_neutral_left
% 0.34/0.68 thf(fact_7250_prod_Omono__neutral__left,axiom,
% 0.34/0.68 ! [T2: set_Code_integer,S: set_Code_integer,G: code_integer > code_natural] :
% 0.34/0.68 ( ( finite6017078050557962740nteger @ T2 )
% 0.34/0.68 => ( ( ord_le7084787975880047091nteger @ S @ T2 )
% 0.34/0.68 => ( ! [X3: code_integer] :
% 0.34/0.68 ( ( member_Code_integer @ X3 @ ( minus_2355218937544613996nteger @ T2 @ S ) )
% 0.34/0.68 => ( ( G @ X3 )
% 0.34/0.68 = one_one_Code_natural ) )
% 0.34/0.68 => ( ( groups2497794486940690220atural @ G @ S )
% 0.34/0.68 = ( groups2497794486940690220atural @ G @ T2 ) ) ) ) ) ).
% 0.34/0.68
% 0.34/0.68 % prod.mono_neutral_left
% 0.34/0.68 thf(fact_7251_prod_Omono__neutral__left,axiom,
% 0.34/0.68 ! [T2: set_nat,S: set_nat,G: nat > assn] :
% 0.34/0.68 ( ( finite_finite_nat @ T2 )
% 0.34/0.68 => ( ( ord_less_eq_set_nat @ S @ T2 )
% 0.34/0.68 => ( ! [X3: nat] :
% 0.34/0.68 ( ( member_nat @ X3 @ ( minus_minus_set_nat @ T2 @ S ) )
% 0.34/0.68 => ( ( G @ X3 )
% 0.34/0.68 = one_one_assn ) )
% 0.34/0.68 => ( ( groups6906906614972039071t_assn @ G @ S )
% 0.34/0.68 = ( groups6906906614972039071t_assn @ G @ T2 ) ) ) ) ) ).
% 0.34/0.68
% 0.34/0.68 % prod.mono_neutral_left
% 0.34/0.68 thf(fact_7252_prod_Omono__neutral__left,axiom,
% 0.34/0.68 ! [T2: set_nat,S: set_nat,G: nat > int] :
% 0.34/0.68 ( ( finite_finite_nat @ T2 )
% 0.34/0.68 => ( ( ord_less_eq_set_nat @ S @ T2 )
% 0.34/0.68 => ( ! [X3: nat] :
% 0.34/0.68 ( ( member_nat @ X3 @ ( minus_minus_set_nat @ T2 @ S ) )
% 0.34/0.68 => ( ( G @ X3 )
% 0.34/0.68 = one_one_int ) )
% 0.34/0.68 => ( ( groups705719431365010083at_int @ G @ S )
% 0.34/0.68 = ( groups705719431365010083at_int @ G @ T2 ) ) ) ) ) ).
% 0.34/0.68
% 0.34/0.68 % prod.mono_neutral_left
% 0.34/0.68 thf(fact_7253_prod_Omono__neutral__left,axiom,
% 0.34/0.68 ! [T2: set_nat,S: set_nat,G: nat > code_integer] :
% 0.34/0.68 ( ( finite_finite_nat @ T2 )
% 0.34/0.68 => ( ( ord_less_eq_set_nat @ S @ T2 )
% 0.34/0.68 => ( ! [X3: nat] :
% 0.34/0.68 ( ( member_nat @ X3 @ ( minus_minus_set_nat @ T2 @ S ) )
% 0.34/0.68 => ( ( G @ X3 )
% 0.34/0.68 = one_one_Code_integer ) )
% 0.34/0.68 => ( ( groups3455450783089532116nteger @ G @ S )
% 0.34/0.68 = ( groups3455450783089532116nteger @ G @ T2 ) ) ) ) ) ).
% 0.34/0.68
% 0.34/0.68 % prod.mono_neutral_left
% 0.34/0.68 thf(fact_7254_prod_Omono__neutral__left,axiom,
% 0.34/0.68 ! [T2: set_nat,S: set_nat,G: nat > code_natural] :
% 0.34/0.68 ( ( finite_finite_nat @ T2 )
% 0.34/0.68 => ( ( ord_less_eq_set_nat @ S @ T2 )
% 0.34/0.68 => ( ! [X3: nat] :
% 0.34/0.68 ( ( member_nat @ X3 @ ( minus_minus_set_nat @ T2 @ S ) )
% 0.34/0.68 => ( ( G @ X3 )
% 0.34/0.68 = one_one_Code_natural ) )
% 0.34/0.68 => ( ( groups2279045934846249631atural @ G @ S )
% 0.34/0.68 = ( groups2279045934846249631atural @ G @ T2 ) ) ) ) ) ).
% 0.34/0.68
% 0.34/0.68 % prod.mono_neutral_left
% 0.34/0.68 thf(fact_7255_prod_Omono__neutral__left,axiom,
% 0.34/0.68 ! [T2: set_nat,S: set_nat,G: nat > nat] :
% 0.34/0.68 ( ( finite_finite_nat @ T2 )
% 0.34/0.68 => ( ( ord_less_eq_set_nat @ S @ T2 )
% 0.34/0.68 => ( ! [X3: nat] :
% 0.34/0.68 ( ( member_nat @ X3 @ ( minus_minus_set_nat @ T2 @ S ) )
% 0.34/0.68 => ( ( G @ X3 )
% 0.34/0.68 = one_one_nat ) )
% 0.34/0.68 => ( ( groups708209901874060359at_nat @ G @ S )
% 0.34/0.68 = ( groups708209901874060359at_nat @ G @ T2 ) ) ) ) ) ).
% 0.34/0.68
% 0.34/0.68 % prod.mono_neutral_left
% 0.34/0.68 thf(fact_7256_prod_Omono__neutral__right,axiom,
% 0.34/0.68 ! [T2: set_Code_integer,S: set_Code_integer,G: code_integer > assn] :
% 0.34/0.68 ( ( finite6017078050557962740nteger @ T2 )
% 0.34/0.68 => ( ( ord_le7084787975880047091nteger @ S @ T2 )
% 0.34/0.68 => ( ! [X3: code_integer] :
% 0.34/0.68 ( ( member_Code_integer @ X3 @ ( minus_2355218937544613996nteger @ T2 @ S ) )
% 0.34/0.68 => ( ( G @ X3 )
% 0.34/0.68 = one_one_assn ) )
% 0.34/0.68 => ( ( groups1304777262505850412r_assn @ G @ T2 )
% 0.34/0.68 = ( groups1304777262505850412r_assn @ G @ S ) ) ) ) ) ).
% 0.34/0.68
% 0.34/0.68 % prod.mono_neutral_right
% 0.34/0.68 thf(fact_7257_prod_Omono__neutral__right,axiom,
% 0.34/0.68 ! [T2: set_Code_integer,S: set_Code_integer,G: code_integer > nat] :
% 0.34/0.68 ( ( finite6017078050557962740nteger @ T2 )
% 0.34/0.68 => ( ( ord_le7084787975880047091nteger @ S @ T2 )
% 0.34/0.68 => ( ! [X3: code_integer] :
% 0.34/0.68 ( ( member_Code_integer @ X3 @ ( minus_2355218937544613996nteger @ T2 @ S ) )
% 0.34/0.68 => ( ( G @ X3 )
% 0.34/0.68 = one_one_nat ) )
% 0.34/0.68 => ( ( groups3190895334310489300er_nat @ G @ T2 )
% 0.34/0.68 = ( groups3190895334310489300er_nat @ G @ S ) ) ) ) ) ).
% 0.34/0.68
% 0.34/0.68 % prod.mono_neutral_right
% 0.34/0.68 thf(fact_7258_prod_Omono__neutral__right,axiom,
% 0.34/0.68 ! [T2: set_Code_integer,S: set_Code_integer,G: code_integer > int] :
% 0.34/0.68 ( ( finite6017078050557962740nteger @ T2 )
% 0.34/0.68 => ( ( ord_le7084787975880047091nteger @ S @ T2 )
% 0.34/0.68 => ( ! [X3: code_integer] :
% 0.34/0.68 ( ( member_Code_integer @ X3 @ ( minus_2355218937544613996nteger @ T2 @ S ) )
% 0.34/0.68 => ( ( G @ X3 )
% 0.34/0.68 = one_one_int ) )
% 0.34/0.68 => ( ( groups3188404863801439024er_int @ G @ T2 )
% 0.34/0.68 = ( groups3188404863801439024er_int @ G @ S ) ) ) ) ) ).
% 0.34/0.68
% 0.34/0.68 % prod.mono_neutral_right
% 0.34/0.68 thf(fact_7259_prod_Omono__neutral__right,axiom,
% 0.34/0.68 ! [T2: set_Code_integer,S: set_Code_integer,G: code_integer > code_integer] :
% 0.34/0.68 ( ( finite6017078050557962740nteger @ T2 )
% 0.34/0.68 => ( ( ord_le7084787975880047091nteger @ S @ T2 )
% 0.34/0.68 => ( ! [X3: code_integer] :
% 0.34/0.68 ( ( member_Code_integer @ X3 @ ( minus_2355218937544613996nteger @ T2 @ S ) )
% 0.34/0.68 => ( ( G @ X3 )
% 0.34/0.68 = one_one_Code_integer ) )
% 0.34/0.68 => ( ( groups3674199335183972705nteger @ G @ T2 )
% 0.34/0.68 = ( groups3674199335183972705nteger @ G @ S ) ) ) ) ) ).
% 0.34/0.68
% 0.34/0.68 % prod.mono_neutral_right
% 0.34/0.68 thf(fact_7260_prod_Omono__neutral__right,axiom,
% 0.34/0.68 ! [T2: set_Code_integer,S: set_Code_integer,G: code_integer > code_natural] :
% 0.34/0.68 ( ( finite6017078050557962740nteger @ T2 )
% 0.34/0.68 => ( ( ord_le7084787975880047091nteger @ S @ T2 )
% 0.34/0.68 => ( ! [X3: code_integer] :
% 0.34/0.68 ( ( member_Code_integer @ X3 @ ( minus_2355218937544613996nteger @ T2 @ S ) )
% 0.34/0.68 => ( ( G @ X3 )
% 0.34/0.68 = one_one_Code_natural ) )
% 0.34/0.68 => ( ( groups2497794486940690220atural @ G @ T2 )
% 0.34/0.68 = ( groups2497794486940690220atural @ G @ S ) ) ) ) ) ).
% 0.34/0.68
% 0.34/0.68 % prod.mono_neutral_right
% 0.34/0.68 thf(fact_7261_prod_Omono__neutral__right,axiom,
% 0.34/0.68 ! [T2: set_nat,S: set_nat,G: nat > assn] :
% 0.34/0.68 ( ( finite_finite_nat @ T2 )
% 0.34/0.68 => ( ( ord_less_eq_set_nat @ S @ T2 )
% 0.34/0.68 => ( ! [X3: nat] :
% 0.34/0.68 ( ( member_nat @ X3 @ ( minus_minus_set_nat @ T2 @ S ) )
% 0.34/0.68 => ( ( G @ X3 )
% 0.34/0.68 = one_one_assn ) )
% 0.34/0.68 => ( ( groups6906906614972039071t_assn @ G @ T2 )
% 0.34/0.68 = ( groups6906906614972039071t_assn @ G @ S ) ) ) ) ) ).
% 0.34/0.68
% 0.34/0.68 % prod.mono_neutral_right
% 0.34/0.68 thf(fact_7262_prod_Omono__neutral__right,axiom,
% 0.34/0.68 ! [T2: set_nat,S: set_nat,G: nat > int] :
% 0.34/0.68 ( ( finite_finite_nat @ T2 )
% 0.34/0.68 => ( ( ord_less_eq_set_nat @ S @ T2 )
% 0.34/0.68 => ( ! [X3: nat] :
% 0.34/0.68 ( ( member_nat @ X3 @ ( minus_minus_set_nat @ T2 @ S ) )
% 0.34/0.68 => ( ( G @ X3 )
% 0.34/0.68 = one_one_int ) )
% 0.34/0.68 => ( ( groups705719431365010083at_int @ G @ T2 )
% 0.34/0.68 = ( groups705719431365010083at_int @ G @ S ) ) ) ) ) ).
% 0.34/0.68
% 0.34/0.68 % prod.mono_neutral_right
% 0.34/0.68 thf(fact_7263_prod_Omono__neutral__right,axiom,
% 0.34/0.68 ! [T2: set_nat,S: set_nat,G: nat > code_integer] :
% 0.34/0.68 ( ( finite_finite_nat @ T2 )
% 0.34/0.68 => ( ( ord_less_eq_set_nat @ S @ T2 )
% 0.34/0.68 => ( ! [X3: nat] :
% 0.34/0.68 ( ( member_nat @ X3 @ ( minus_minus_set_nat @ T2 @ S ) )
% 0.34/0.68 => ( ( G @ X3 )
% 0.34/0.68 = one_one_Code_integer ) )
% 0.34/0.68 => ( ( groups3455450783089532116nteger @ G @ T2 )
% 0.34/0.68 = ( groups3455450783089532116nteger @ G @ S ) ) ) ) ) ).
% 0.34/0.68
% 0.34/0.68 % prod.mono_neutral_right
% 0.34/0.68 thf(fact_7264_prod_Omono__neutral__right,axiom,
% 0.34/0.68 ! [T2: set_nat,S: set_nat,G: nat > code_natural] :
% 0.34/0.68 ( ( finite_finite_nat @ T2 )
% 0.34/0.68 => ( ( ord_less_eq_set_nat @ S @ T2 )
% 0.34/0.68 => ( ! [X3: nat] :
% 0.34/0.68 ( ( member_nat @ X3 @ ( minus_minus_set_nat @ T2 @ S ) )
% 0.34/0.68 => ( ( G @ X3 )
% 0.34/0.68 = one_one_Code_natural ) )
% 0.34/0.68 => ( ( groups2279045934846249631atural @ G @ T2 )
% 0.34/0.68 = ( groups2279045934846249631atural @ G @ S ) ) ) ) ) ).
% 0.34/0.68
% 0.34/0.68 % prod.mono_neutral_right
% 0.34/0.68 thf(fact_7265_prod_Omono__neutral__right,axiom,
% 0.34/0.68 ! [T2: set_nat,S: set_nat,G: nat > nat] :
% 0.34/0.68 ( ( finite_finite_nat @ T2 )
% 0.34/0.68 => ( ( ord_less_eq_set_nat @ S @ T2 )
% 0.34/0.68 => ( ! [X3: nat] :
% 0.34/0.68 ( ( member_nat @ X3 @ ( minus_minus_set_nat @ T2 @ S ) )
% 0.34/0.68 => ( ( G @ X3 )
% 0.34/0.68 = one_one_nat ) )
% 0.34/0.68 => ( ( groups708209901874060359at_nat @ G @ T2 )
% 0.34/0.68 = ( groups708209901874060359at_nat @ G @ S ) ) ) ) ) ).
% 0.34/0.68
% 0.34/0.68 % prod.mono_neutral_right
% 0.34/0.68 thf(fact_7266_prod_Omono__neutral__cong__left,axiom,
% 0.34/0.68 ! [T2: set_int,S: set_int,H2: int > assn,G: int > assn] :
% 0.34/0.68 ( ( finite_finite_int @ T2 )
% 0.34/0.68 => ( ( ord_less_eq_set_int @ S @ T2 )
% 0.34/0.68 => ( ! [X3: int] :
% 0.34/0.68 ( ( member_int @ X3 @ ( minus_minus_set_int @ T2 @ S ) )
% 0.34/0.68 => ( ( H2 @ X3 )
% 0.34/0.68 = one_one_assn ) )
% 0.34/0.68 => ( ! [X3: int] :
% 0.34/0.68 ( ( member_int @ X3 @ S )
% 0.34/0.68 => ( ( G @ X3 )
% 0.34/0.68 = ( H2 @ X3 ) ) )
% 0.34/0.68 => ( ( groups7882442080178216443t_assn @ G @ S )
% 0.34/0.68 = ( groups7882442080178216443t_assn @ H2 @ T2 ) ) ) ) ) ) ).
% 0.34/0.68
% 0.34/0.68 % prod.mono_neutral_cong_left
% 0.34/0.68 thf(fact_7267_prod_Omono__neutral__cong__left,axiom,
% 0.34/0.68 ! [T2: set_o,S: set_o,H2: $o > assn,G: $o > assn] :
% 0.34/0.68 ( ( finite_finite_o @ T2 )
% 0.34/0.68 => ( ( ord_less_eq_set_o @ S @ T2 )
% 0.34/0.68 => ( ! [X3: $o] :
% 0.34/0.68 ( ( member_o @ X3 @ ( minus_minus_set_o @ T2 @ S ) )
% 0.34/0.68 => ( ( H2 @ X3 )
% 0.34/0.68 = one_one_assn ) )
% 0.34/0.68 => ( ! [X3: $o] :
% 0.34/0.68 ( ( member_o @ X3 @ S )
% 0.34/0.68 => ( ( G @ X3 )
% 0.34/0.68 = ( H2 @ X3 ) ) )
% 0.34/0.68 => ( ( groups5301882518646026715o_assn @ G @ S )
% 0.34/0.68 = ( groups5301882518646026715o_assn @ H2 @ T2 ) ) ) ) ) ) ).
% 0.34/0.68
% 0.34/0.68 % prod.mono_neutral_cong_left
% 0.34/0.68 thf(fact_7268_prod_Omono__neutral__cong__left,axiom,
% 0.34/0.68 ! [T2: set_Product_unit,S: set_Product_unit,H2: product_unit > assn,G: product_unit > assn] :
% 0.34/0.68 ( ( finite4290736615968046902t_unit @ T2 )
% 0.34/0.68 => ( ( ord_le3507040750410214029t_unit @ S @ T2 )
% 0.34/0.68 => ( ! [X3: product_unit] :
% 0.34/0.68 ( ( member_Product_unit @ X3 @ ( minus_6452836326544984404t_unit @ T2 @ S ) )
% 0.34/0.68 => ( ( H2 @ X3 )
% 0.34/0.68 = one_one_assn ) )
% 0.34/0.68 => ( ! [X3: product_unit] :
% 0.34/0.68 ( ( member_Product_unit @ X3 @ S )
% 0.34/0.68 => ( ( G @ X3 )
% 0.34/0.68 = ( H2 @ X3 ) ) )
% 0.34/0.68 => ( ( groups5360233303959380816t_assn @ G @ S )
% 0.34/0.68 = ( groups5360233303959380816t_assn @ H2 @ T2 ) ) ) ) ) ) ).
% 0.34/0.68
% 0.34/0.68 % prod.mono_neutral_cong_left
% 0.34/0.68 thf(fact_7269_prod_Omono__neutral__cong__left,axiom,
% 0.34/0.68 ! [T2: set_Code_integer,S: set_Code_integer,H2: code_integer > assn,G: code_integer > assn] :
% 0.34/0.68 ( ( finite6017078050557962740nteger @ T2 )
% 0.34/0.68 => ( ( ord_le7084787975880047091nteger @ S @ T2 )
% 0.34/0.68 => ( ! [X3: code_integer] :
% 0.34/0.68 ( ( member_Code_integer @ X3 @ ( minus_2355218937544613996nteger @ T2 @ S ) )
% 0.34/0.68 => ( ( H2 @ X3 )
% 0.34/0.68 = one_one_assn ) )
% 0.34/0.68 => ( ! [X3: code_integer] :
% 0.34/0.68 ( ( member_Code_integer @ X3 @ S )
% 0.34/0.68 => ( ( G @ X3 )
% 0.34/0.68 = ( H2 @ X3 ) ) )
% 0.34/0.68 => ( ( groups1304777262505850412r_assn @ G @ S )
% 0.34/0.68 = ( groups1304777262505850412r_assn @ H2 @ T2 ) ) ) ) ) ) ).
% 0.34/0.68
% 0.34/0.68 % prod.mono_neutral_cong_left
% 0.34/0.68 thf(fact_7270_prod_Omono__neutral__cong__left,axiom,
% 0.34/0.68 ! [T2: set_int,S: set_int,H2: int > nat,G: int > nat] :
% 0.34/0.68 ( ( finite_finite_int @ T2 )
% 0.34/0.68 => ( ( ord_less_eq_set_int @ S @ T2 )
% 0.34/0.68 => ( ! [X3: int] :
% 0.34/0.68 ( ( member_int @ X3 @ ( minus_minus_set_int @ T2 @ S ) )
% 0.34/0.68 => ( ( H2 @ X3 )
% 0.34/0.68 = one_one_nat ) )
% 0.34/0.68 => ( ! [X3: int] :
% 0.34/0.68 ( ( member_int @ X3 @ S )
% 0.34/0.68 => ( ( G @ X3 )
% 0.34/0.68 = ( H2 @ X3 ) ) )
% 0.34/0.68 => ( ( groups1707563613775114915nt_nat @ G @ S )
% 0.34/0.68 = ( groups1707563613775114915nt_nat @ H2 @ T2 ) ) ) ) ) ) ).
% 0.34/0.68
% 0.34/0.68 % prod.mono_neutral_cong_left
% 0.34/0.68 thf(fact_7271_prod_Omono__neutral__cong__left,axiom,
% 0.34/0.68 ! [T2: set_o,S: set_o,H2: $o > nat,G: $o > nat] :
% 0.34/0.68 ( ( finite_finite_o @ T2 )
% 0.34/0.68 => ( ( ord_less_eq_set_o @ S @ T2 )
% 0.34/0.68 => ( ! [X3: $o] :
% 0.34/0.68 ( ( member_o @ X3 @ ( minus_minus_set_o @ T2 @ S ) )
% 0.34/0.68 => ( ( H2 @ X3 )
% 0.34/0.68 = one_one_nat ) )
% 0.34/0.68 => ( ! [X3: $o] :
% 0.34/0.68 ( ( member_o @ X3 @ S )
% 0.34/0.68 => ( ( G @ X3 )
% 0.34/0.68 = ( H2 @ X3 ) ) )
% 0.34/0.68 => ( ( groups3504817904513533571_o_nat @ G @ S )
% 0.34/0.68 = ( groups3504817904513533571_o_nat @ H2 @ T2 ) ) ) ) ) ) ).
% 0.34/0.68
% 0.34/0.68 % prod.mono_neutral_cong_left
% 0.34/0.68 thf(fact_7272_prod_Omono__neutral__cong__left,axiom,
% 0.34/0.68 ! [T2: set_Product_unit,S: set_Product_unit,H2: product_unit > nat,G: product_unit > nat] :
% 0.34/0.68 ( ( finite4290736615968046902t_unit @ T2 )
% 0.34/0.68 => ( ( ord_le3507040750410214029t_unit @ S @ T2 )
% 0.34/0.68 => ( ! [X3: product_unit] :
% 0.34/0.68 ( ( member_Product_unit @ X3 @ ( minus_6452836326544984404t_unit @ T2 @ S ) )
% 0.34/0.68 => ( ( H2 @ X3 )
% 0.34/0.68 = one_one_nat ) )
% 0.34/0.68 => ( ! [X3: product_unit] :
% 0.34/0.68 ( ( member_Product_unit @ X3 @ S )
% 0.34/0.68 => ( ( G @ X3 )
% 0.34/0.68 = ( H2 @ X3 ) ) )
% 0.34/0.68 => ( ( groups396421249026171384it_nat @ G @ S )
% 0.34/0.68 = ( groups396421249026171384it_nat @ H2 @ T2 ) ) ) ) ) ) ).
% 0.34/0.68
% 0.34/0.68 % prod.mono_neutral_cong_left
% 0.34/0.68 thf(fact_7273_prod_Omono__neutral__cong__left,axiom,
% 0.34/0.68 ! [T2: set_Code_integer,S: set_Code_integer,H2: code_integer > nat,G: code_integer > nat] :
% 0.34/0.68 ( ( finite6017078050557962740nteger @ T2 )
% 0.34/0.68 => ( ( ord_le7084787975880047091nteger @ S @ T2 )
% 0.34/0.68 => ( ! [X3: code_integer] :
% 0.34/0.68 ( ( member_Code_integer @ X3 @ ( minus_2355218937544613996nteger @ T2 @ S ) )
% 0.34/0.68 => ( ( H2 @ X3 )
% 0.34/0.68 = one_one_nat ) )
% 0.34/0.68 => ( ! [X3: code_integer] :
% 0.34/0.68 ( ( member_Code_integer @ X3 @ S )
% 0.34/0.68 => ( ( G @ X3 )
% 0.34/0.68 = ( H2 @ X3 ) ) )
% 0.34/0.68 => ( ( groups3190895334310489300er_nat @ G @ S )
% 0.34/0.68 = ( groups3190895334310489300er_nat @ H2 @ T2 ) ) ) ) ) ) ).
% 0.34/0.68
% 0.34/0.68 % prod.mono_neutral_cong_left
% 0.34/0.68 thf(fact_7274_prod_Omono__neutral__cong__left,axiom,
% 0.34/0.68 ! [T2: set_int,S: set_int,H2: int > int,G: int > int] :
% 0.34/0.68 ( ( finite_finite_int @ T2 )
% 0.34/0.68 => ( ( ord_less_eq_set_int @ S @ T2 )
% 0.34/0.68 => ( ! [X3: int] :
% 0.34/0.68 ( ( member_int @ X3 @ ( minus_minus_set_int @ T2 @ S ) )
% 0.34/0.68 => ( ( H2 @ X3 )
% 0.34/0.68 = one_one_int ) )
% 0.34/0.68 => ( ! [X3: int] :
% 0.34/0.68 ( ( member_int @ X3 @ S )
% 0.34/0.68 => ( ( G @ X3 )
% 0.34/0.68 = ( H2 @ X3 ) ) )
% 0.34/0.68 => ( ( groups1705073143266064639nt_int @ G @ S )
% 0.34/0.68 = ( groups1705073143266064639nt_int @ H2 @ T2 ) ) ) ) ) ) ).
% 0.34/0.68
% 0.34/0.68 % prod.mono_neutral_cong_left
% 0.34/0.68 thf(fact_7275_prod_Omono__neutral__cong__left,axiom,
% 0.34/0.68 ! [T2: set_o,S: set_o,H2: $o > int,G: $o > int] :
% 0.34/0.68 ( ( finite_finite_o @ T2 )
% 0.34/0.68 => ( ( ord_less_eq_set_o @ S @ T2 )
% 0.34/0.68 => ( ! [X3: $o] :
% 0.34/0.68 ( ( member_o @ X3 @ ( minus_minus_set_o @ T2 @ S ) )
% 0.34/0.68 => ( ( H2 @ X3 )
% 0.34/0.68 = one_one_int ) )
% 0.34/0.68 => ( ! [X3: $o] :
% 0.34/0.68 ( ( member_o @ X3 @ S )
% 0.34/0.68 => ( ( G @ X3 )
% 0.34/0.68 = ( H2 @ X3 ) ) )
% 0.34/0.68 => ( ( groups3502327434004483295_o_int @ G @ S )
% 0.34/0.68 = ( groups3502327434004483295_o_int @ H2 @ T2 ) ) ) ) ) ) ).
% 0.34/0.68
% 0.34/0.68 % prod.mono_neutral_cong_left
% 0.34/0.68 thf(fact_7276_prod_Omono__neutral__cong__right,axiom,
% 0.34/0.68 ! [T2: set_int,S: set_int,G: int > assn,H2: int > assn] :
% 0.34/0.68 ( ( finite_finite_int @ T2 )
% 0.34/0.68 => ( ( ord_less_eq_set_int @ S @ T2 )
% 0.34/0.68 => ( ! [X3: int] :
% 0.34/0.68 ( ( member_int @ X3 @ ( minus_minus_set_int @ T2 @ S ) )
% 0.34/0.68 => ( ( G @ X3 )
% 0.34/0.68 = one_one_assn ) )
% 0.34/0.68 => ( ! [X3: int] :
% 0.34/0.68 ( ( member_int @ X3 @ S )
% 0.34/0.68 => ( ( G @ X3 )
% 0.34/0.68 = ( H2 @ X3 ) ) )
% 0.34/0.68 => ( ( groups7882442080178216443t_assn @ G @ T2 )
% 0.34/0.68 = ( groups7882442080178216443t_assn @ H2 @ S ) ) ) ) ) ) ).
% 0.34/0.68
% 0.34/0.68 % prod.mono_neutral_cong_right
% 0.34/0.68 thf(fact_7277_prod_Omono__neutral__cong__right,axiom,
% 0.34/0.68 ! [T2: set_o,S: set_o,G: $o > assn,H2: $o > assn] :
% 0.34/0.68 ( ( finite_finite_o @ T2 )
% 0.34/0.68 => ( ( ord_less_eq_set_o @ S @ T2 )
% 0.34/0.68 => ( ! [X3: $o] :
% 0.34/0.68 ( ( member_o @ X3 @ ( minus_minus_set_o @ T2 @ S ) )
% 0.34/0.68 => ( ( G @ X3 )
% 0.34/0.68 = one_one_assn ) )
% 0.34/0.68 => ( ! [X3: $o] :
% 0.34/0.68 ( ( member_o @ X3 @ S )
% 0.34/0.68 => ( ( G @ X3 )
% 0.34/0.68 = ( H2 @ X3 ) ) )
% 0.34/0.68 => ( ( groups5301882518646026715o_assn @ G @ T2 )
% 0.34/0.68 = ( groups5301882518646026715o_assn @ H2 @ S ) ) ) ) ) ) ).
% 0.34/0.68
% 0.34/0.68 % prod.mono_neutral_cong_right
% 0.34/0.68 thf(fact_7278_prod_Omono__neutral__cong__right,axiom,
% 0.34/0.68 ! [T2: set_Product_unit,S: set_Product_unit,G: product_unit > assn,H2: product_unit > assn] :
% 0.34/0.68 ( ( finite4290736615968046902t_unit @ T2 )
% 0.34/0.68 => ( ( ord_le3507040750410214029t_unit @ S @ T2 )
% 0.34/0.68 => ( ! [X3: product_unit] :
% 0.34/0.68 ( ( member_Product_unit @ X3 @ ( minus_6452836326544984404t_unit @ T2 @ S ) )
% 0.34/0.68 => ( ( G @ X3 )
% 0.34/0.68 = one_one_assn ) )
% 0.34/0.68 => ( ! [X3: product_unit] :
% 0.34/0.68 ( ( member_Product_unit @ X3 @ S )
% 0.34/0.68 => ( ( G @ X3 )
% 0.34/0.68 = ( H2 @ X3 ) ) )
% 0.34/0.68 => ( ( groups5360233303959380816t_assn @ G @ T2 )
% 0.34/0.68 = ( groups5360233303959380816t_assn @ H2 @ S ) ) ) ) ) ) ).
% 0.34/0.68
% 0.34/0.68 % prod.mono_neutral_cong_right
% 0.34/0.68 thf(fact_7279_prod_Omono__neutral__cong__right,axiom,
% 0.34/0.68 ! [T2: set_Code_integer,S: set_Code_integer,G: code_integer > assn,H2: code_integer > assn] :
% 0.34/0.68 ( ( finite6017078050557962740nteger @ T2 )
% 0.34/0.68 => ( ( ord_le7084787975880047091nteger @ S @ T2 )
% 0.34/0.68 => ( ! [X3: code_integer] :
% 0.34/0.68 ( ( member_Code_integer @ X3 @ ( minus_2355218937544613996nteger @ T2 @ S ) )
% 0.34/0.68 => ( ( G @ X3 )
% 0.34/0.68 = one_one_assn ) )
% 0.34/0.68 => ( ! [X3: code_integer] :
% 0.34/0.68 ( ( member_Code_integer @ X3 @ S )
% 0.34/0.68 => ( ( G @ X3 )
% 0.34/0.68 = ( H2 @ X3 ) ) )
% 0.34/0.68 => ( ( groups1304777262505850412r_assn @ G @ T2 )
% 0.34/0.68 = ( groups1304777262505850412r_assn @ H2 @ S ) ) ) ) ) ) ).
% 0.34/0.68
% 0.34/0.68 % prod.mono_neutral_cong_right
% 0.34/0.68 thf(fact_7280_prod_Omono__neutral__cong__right,axiom,
% 0.34/0.68 ! [T2: set_int,S: set_int,G: int > nat,H2: int > nat] :
% 0.34/0.68 ( ( finite_finite_int @ T2 )
% 0.34/0.68 => ( ( ord_less_eq_set_int @ S @ T2 )
% 0.34/0.68 => ( ! [X3: int] :
% 0.34/0.68 ( ( member_int @ X3 @ ( minus_minus_set_int @ T2 @ S ) )
% 0.34/0.68 => ( ( G @ X3 )
% 0.34/0.68 = one_one_nat ) )
% 0.34/0.68 => ( ! [X3: int] :
% 0.34/0.68 ( ( member_int @ X3 @ S )
% 0.34/0.68 => ( ( G @ X3 )
% 0.34/0.68 = ( H2 @ X3 ) ) )
% 0.34/0.68 => ( ( groups1707563613775114915nt_nat @ G @ T2 )
% 0.34/0.68 = ( groups1707563613775114915nt_nat @ H2 @ S ) ) ) ) ) ) ).
% 0.34/0.68
% 0.34/0.68 % prod.mono_neutral_cong_right
% 0.34/0.68 thf(fact_7281_prod_Omono__neutral__cong__right,axiom,
% 0.34/0.68 ! [T2: set_o,S: set_o,G: $o > nat,H2: $o > nat] :
% 0.34/0.68 ( ( finite_finite_o @ T2 )
% 0.34/0.68 => ( ( ord_less_eq_set_o @ S @ T2 )
% 0.34/0.68 => ( ! [X3: $o] :
% 0.34/0.68 ( ( member_o @ X3 @ ( minus_minus_set_o @ T2 @ S ) )
% 0.34/0.68 => ( ( G @ X3 )
% 0.34/0.68 = one_one_nat ) )
% 0.34/0.68 => ( ! [X3: $o] :
% 0.34/0.68 ( ( member_o @ X3 @ S )
% 0.34/0.68 => ( ( G @ X3 )
% 0.34/0.68 = ( H2 @ X3 ) ) )
% 0.34/0.68 => ( ( groups3504817904513533571_o_nat @ G @ T2 )
% 0.34/0.68 = ( groups3504817904513533571_o_nat @ H2 @ S ) ) ) ) ) ) ).
% 0.34/0.68
% 0.34/0.68 % prod.mono_neutral_cong_right
% 0.34/0.68 thf(fact_7282_prod_Omono__neutral__cong__right,axiom,
% 0.34/0.68 ! [T2: set_Product_unit,S: set_Product_unit,G: product_unit > nat,H2: product_unit > nat] :
% 0.34/0.68 ( ( finite4290736615968046902t_unit @ T2 )
% 0.34/0.68 => ( ( ord_le3507040750410214029t_unit @ S @ T2 )
% 0.34/0.68 => ( ! [X3: product_unit] :
% 0.34/0.68 ( ( member_Product_unit @ X3 @ ( minus_6452836326544984404t_unit @ T2 @ S ) )
% 0.34/0.68 => ( ( G @ X3 )
% 0.34/0.68 = one_one_nat ) )
% 0.34/0.68 => ( ! [X3: product_unit] :
% 0.34/0.68 ( ( member_Product_unit @ X3 @ S )
% 0.34/0.68 => ( ( G @ X3 )
% 0.34/0.68 = ( H2 @ X3 ) ) )
% 0.34/0.68 => ( ( groups396421249026171384it_nat @ G @ T2 )
% 0.34/0.68 = ( groups396421249026171384it_nat @ H2 @ S ) ) ) ) ) ) ).
% 0.34/0.68
% 0.34/0.68 % prod.mono_neutral_cong_right
% 0.34/0.68 thf(fact_7283_prod_Omono__neutral__cong__right,axiom,
% 0.34/0.68 ! [T2: set_Code_integer,S: set_Code_integer,G: code_integer > nat,H2: code_integer > nat] :
% 0.34/0.68 ( ( finite6017078050557962740nteger @ T2 )
% 0.34/0.68 => ( ( ord_le7084787975880047091nteger @ S @ T2 )
% 0.34/0.68 => ( ! [X3: code_integer] :
% 0.34/0.68 ( ( member_Code_integer @ X3 @ ( minus_2355218937544613996nteger @ T2 @ S ) )
% 0.34/0.68 => ( ( G @ X3 )
% 0.34/0.68 = one_one_nat ) )
% 0.34/0.68 => ( ! [X3: code_integer] :
% 0.34/0.68 ( ( member_Code_integer @ X3 @ S )
% 0.34/0.68 => ( ( G @ X3 )
% 0.34/0.68 = ( H2 @ X3 ) ) )
% 0.34/0.68 => ( ( groups3190895334310489300er_nat @ G @ T2 )
% 0.34/0.68 = ( groups3190895334310489300er_nat @ H2 @ S ) ) ) ) ) ) ).
% 0.34/0.68
% 0.34/0.68 % prod.mono_neutral_cong_right
% 0.34/0.68 thf(fact_7284_prod_Omono__neutral__cong__right,axiom,
% 0.34/0.68 ! [T2: set_int,S: set_int,G: int > int,H2: int > int] :
% 0.34/0.68 ( ( finite_finite_int @ T2 )
% 0.34/0.68 => ( ( ord_less_eq_set_int @ S @ T2 )
% 0.34/0.68 => ( ! [X3: int] :
% 0.34/0.68 ( ( member_int @ X3 @ ( minus_minus_set_int @ T2 @ S ) )
% 0.34/0.68 => ( ( G @ X3 )
% 0.34/0.68 = one_one_int ) )
% 0.34/0.68 => ( ! [X3: int] :
% 0.34/0.68 ( ( member_int @ X3 @ S )
% 0.34/0.68 => ( ( G @ X3 )
% 0.34/0.68 = ( H2 @ X3 ) ) )
% 0.34/0.68 => ( ( groups1705073143266064639nt_int @ G @ T2 )
% 0.34/0.68 = ( groups1705073143266064639nt_int @ H2 @ S ) ) ) ) ) ) ).
% 0.34/0.68
% 0.34/0.68 % prod.mono_neutral_cong_right
% 0.34/0.68 thf(fact_7285_prod_Omono__neutral__cong__right,axiom,
% 0.34/0.68 ! [T2: set_o,S: set_o,G: $o > int,H2: $o > int] :
% 0.34/0.68 ( ( finite_finite_o @ T2 )
% 0.34/0.68 => ( ( ord_less_eq_set_o @ S @ T2 )
% 0.34/0.68 => ( ! [X3: $o] :
% 0.34/0.68 ( ( member_o @ X3 @ ( minus_minus_set_o @ T2 @ S ) )
% 0.34/0.68 => ( ( G @ X3 )
% 0.34/0.68 = one_one_int ) )
% 0.34/0.68 => ( ! [X3: $o] :
% 0.34/0.68 ( ( member_o @ X3 @ S )
% 0.34/0.68 => ( ( G @ X3 )
% 0.34/0.68 = ( H2 @ X3 ) ) )
% 0.34/0.68 => ( ( groups3502327434004483295_o_int @ G @ T2 )
% 0.34/0.68 = ( groups3502327434004483295_o_int @ H2 @ S ) ) ) ) ) ) ).
% 0.34/0.68
% 0.34/0.68 % prod.mono_neutral_cong_right
% 0.34/0.68 thf(fact_7286_prod_Ounion__inter,axiom,
% 0.34/0.68 ! [A: set_Code_integer,B4: set_Code_integer,G: code_integer > assn] :
% 0.34/0.68 ( ( finite6017078050557962740nteger @ A )
% 0.34/0.68 => ( ( finite6017078050557962740nteger @ B4 )
% 0.34/0.68 => ( ( times_times_assn @ ( groups1304777262505850412r_assn @ G @ ( sup_su848401254843788991nteger @ A @ B4 ) ) @ ( groups1304777262505850412r_assn @ G @ ( inf_in1364745209274528805nteger @ A @ B4 ) ) )
% 0.34/0.68 = ( times_times_assn @ ( groups1304777262505850412r_assn @ G @ A ) @ ( groups1304777262505850412r_assn @ G @ B4 ) ) ) ) ) ).
% 0.34/0.68
% 0.34/0.68 % prod.union_inter
% 0.34/0.68 thf(fact_7287_prod_Ounion__inter,axiom,
% 0.34/0.68 ! [A: set_nat,B4: set_nat,G: nat > assn] :
% 0.34/0.68 ( ( finite_finite_nat @ A )
% 0.34/0.68 => ( ( finite_finite_nat @ B4 )
% 0.34/0.68 => ( ( times_times_assn @ ( groups6906906614972039071t_assn @ G @ ( sup_sup_set_nat @ A @ B4 ) ) @ ( groups6906906614972039071t_assn @ G @ ( inf_inf_set_nat @ A @ B4 ) ) )
% 0.34/0.68 = ( times_times_assn @ ( groups6906906614972039071t_assn @ G @ A ) @ ( groups6906906614972039071t_assn @ G @ B4 ) ) ) ) ) ).
% 0.34/0.68
% 0.34/0.68 % prod.union_inter
% 0.34/0.68 thf(fact_7288_prod_Ounion__inter,axiom,
% 0.34/0.68 ! [A: set_Code_integer,B4: set_Code_integer,G: code_integer > rat] :
% 0.34/0.68 ( ( finite6017078050557962740nteger @ A )
% 0.34/0.68 => ( ( finite6017078050557962740nteger @ B4 )
% 0.34/0.68 => ( ( times_times_rat @ ( groups2555765274223993564er_rat @ G @ ( sup_su848401254843788991nteger @ A @ B4 ) ) @ ( groups2555765274223993564er_rat @ G @ ( inf_in1364745209274528805nteger @ A @ B4 ) ) )
% 0.34/0.68 = ( times_times_rat @ ( groups2555765274223993564er_rat @ G @ A ) @ ( groups2555765274223993564er_rat @ G @ B4 ) ) ) ) ) ).
% 0.34/0.68
% 0.34/0.68 % prod.union_inter
% 0.34/0.68 thf(fact_7289_prod_Ounion__inter,axiom,
% 0.34/0.68 ! [A: set_nat,B4: set_nat,G: nat > rat] :
% 0.34/0.68 ( ( finite_finite_nat @ A )
% 0.34/0.68 => ( ( finite_finite_nat @ B4 )
% 0.34/0.68 => ( ( times_times_rat @ ( groups73079841787564623at_rat @ G @ ( sup_sup_set_nat @ A @ B4 ) ) @ ( groups73079841787564623at_rat @ G @ ( inf_inf_set_nat @ A @ B4 ) ) )
% 0.34/0.68 = ( times_times_rat @ ( groups73079841787564623at_rat @ G @ A ) @ ( groups73079841787564623at_rat @ G @ B4 ) ) ) ) ) ).
% 0.34/0.68
% 0.34/0.68 % prod.union_inter
% 0.34/0.68 thf(fact_7290_prod_Ounion__inter,axiom,
% 0.34/0.68 ! [A: set_Code_integer,B4: set_Code_integer,G: code_integer > nat] :
% 0.34/0.68 ( ( finite6017078050557962740nteger @ A )
% 0.34/0.68 => ( ( finite6017078050557962740nteger @ B4 )
% 0.34/0.68 => ( ( times_times_nat @ ( groups3190895334310489300er_nat @ G @ ( sup_su848401254843788991nteger @ A @ B4 ) ) @ ( groups3190895334310489300er_nat @ G @ ( inf_in1364745209274528805nteger @ A @ B4 ) ) )
% 0.34/0.68 = ( times_times_nat @ ( groups3190895334310489300er_nat @ G @ A ) @ ( groups3190895334310489300er_nat @ G @ B4 ) ) ) ) ) ).
% 0.34/0.68
% 0.34/0.68 % prod.union_inter
% 0.34/0.68 thf(fact_7291_prod_Ounion__inter,axiom,
% 0.34/0.68 ! [A: set_Code_integer,B4: set_Code_integer,G: code_integer > int] :
% 0.34/0.68 ( ( finite6017078050557962740nteger @ A )
% 0.34/0.68 => ( ( finite6017078050557962740nteger @ B4 )
% 0.34/0.68 => ( ( times_times_int @ ( groups3188404863801439024er_int @ G @ ( sup_su848401254843788991nteger @ A @ B4 ) ) @ ( groups3188404863801439024er_int @ G @ ( inf_in1364745209274528805nteger @ A @ B4 ) ) )
% 0.34/0.68 = ( times_times_int @ ( groups3188404863801439024er_int @ G @ A ) @ ( groups3188404863801439024er_int @ G @ B4 ) ) ) ) ) ).
% 0.34/0.68
% 0.34/0.68 % prod.union_inter
% 0.34/0.68 thf(fact_7292_prod_Ounion__inter,axiom,
% 0.34/0.68 ! [A: set_nat,B4: set_nat,G: nat > int] :
% 0.34/0.68 ( ( finite_finite_nat @ A )
% 0.34/0.68 => ( ( finite_finite_nat @ B4 )
% 0.34/0.68 => ( ( times_times_int @ ( groups705719431365010083at_int @ G @ ( sup_sup_set_nat @ A @ B4 ) ) @ ( groups705719431365010083at_int @ G @ ( inf_inf_set_nat @ A @ B4 ) ) )
% 0.34/0.68 = ( times_times_int @ ( groups705719431365010083at_int @ G @ A ) @ ( groups705719431365010083at_int @ G @ B4 ) ) ) ) ) ).
% 0.34/0.68
% 0.34/0.68 % prod.union_inter
% 0.34/0.68 thf(fact_7293_prod_Ounion__inter,axiom,
% 0.34/0.68 ! [A: set_nat,B4: set_nat,G: nat > nat] :
% 0.34/0.68 ( ( finite_finite_nat @ A )
% 0.34/0.68 => ( ( finite_finite_nat @ B4 )
% 0.34/0.68 => ( ( times_times_nat @ ( groups708209901874060359at_nat @ G @ ( sup_sup_set_nat @ A @ B4 ) ) @ ( groups708209901874060359at_nat @ G @ ( inf_inf_set_nat @ A @ B4 ) ) )
% 0.34/0.68 = ( times_times_nat @ ( groups708209901874060359at_nat @ G @ A ) @ ( groups708209901874060359at_nat @ G @ B4 ) ) ) ) ) ).
% 0.34/0.68
% 0.34/0.68 % prod.union_inter
% 0.34/0.68 thf(fact_7294_prod_OInt__Diff,axiom,
% 0.34/0.68 ! [A: set_Code_integer,G: code_integer > assn,B4: set_Code_integer] :
% 0.34/0.68 ( ( finite6017078050557962740nteger @ A )
% 0.34/0.68 => ( ( groups1304777262505850412r_assn @ G @ A )
% 0.34/0.68 = ( times_times_assn @ ( groups1304777262505850412r_assn @ G @ ( inf_in1364745209274528805nteger @ A @ B4 ) ) @ ( groups1304777262505850412r_assn @ G @ ( minus_2355218937544613996nteger @ A @ B4 ) ) ) ) ) ).
% 0.34/0.68
% 0.34/0.68 % prod.Int_Diff
% 0.34/0.68 thf(fact_7295_prod_OInt__Diff,axiom,
% 0.34/0.68 ! [A: set_nat,G: nat > assn,B4: set_nat] :
% 0.34/0.68 ( ( finite_finite_nat @ A )
% 0.34/0.68 => ( ( groups6906906614972039071t_assn @ G @ A )
% 0.34/0.68 = ( times_times_assn @ ( groups6906906614972039071t_assn @ G @ ( inf_inf_set_nat @ A @ B4 ) ) @ ( groups6906906614972039071t_assn @ G @ ( minus_minus_set_nat @ A @ B4 ) ) ) ) ) ).
% 0.34/0.68
% 0.34/0.68 % prod.Int_Diff
% 0.34/0.68 thf(fact_7296_prod_OInt__Diff,axiom,
% 0.34/0.68 ! [A: set_Code_integer,G: code_integer > rat,B4: set_Code_integer] :
% 0.34/0.68 ( ( finite6017078050557962740nteger @ A )
% 0.34/0.68 => ( ( groups2555765274223993564er_rat @ G @ A )
% 0.34/0.68 = ( times_times_rat @ ( groups2555765274223993564er_rat @ G @ ( inf_in1364745209274528805nteger @ A @ B4 ) ) @ ( groups2555765274223993564er_rat @ G @ ( minus_2355218937544613996nteger @ A @ B4 ) ) ) ) ) ).
% 0.34/0.68
% 0.34/0.68 % prod.Int_Diff
% 0.34/0.68 thf(fact_7297_prod_OInt__Diff,axiom,
% 0.34/0.68 ! [A: set_nat,G: nat > rat,B4: set_nat] :
% 0.34/0.68 ( ( finite_finite_nat @ A )
% 0.34/0.68 => ( ( groups73079841787564623at_rat @ G @ A )
% 0.34/0.68 = ( times_times_rat @ ( groups73079841787564623at_rat @ G @ ( inf_inf_set_nat @ A @ B4 ) ) @ ( groups73079841787564623at_rat @ G @ ( minus_minus_set_nat @ A @ B4 ) ) ) ) ) ).
% 0.34/0.68
% 0.34/0.68 % prod.Int_Diff
% 0.34/0.68 thf(fact_7298_prod_OInt__Diff,axiom,
% 0.34/0.68 ! [A: set_Code_integer,G: code_integer > nat,B4: set_Code_integer] :
% 0.34/0.68 ( ( finite6017078050557962740nteger @ A )
% 0.34/0.68 => ( ( groups3190895334310489300er_nat @ G @ A )
% 0.34/0.68 = ( times_times_nat @ ( groups3190895334310489300er_nat @ G @ ( inf_in1364745209274528805nteger @ A @ B4 ) ) @ ( groups3190895334310489300er_nat @ G @ ( minus_2355218937544613996nteger @ A @ B4 ) ) ) ) ) ).
% 0.34/0.68
% 0.34/0.68 % prod.Int_Diff
% 0.34/0.68 thf(fact_7299_prod_OInt__Diff,axiom,
% 0.34/0.68 ! [A: set_Code_integer,G: code_integer > int,B4: set_Code_integer] :
% 0.34/0.68 ( ( finite6017078050557962740nteger @ A )
% 0.34/0.68 => ( ( groups3188404863801439024er_int @ G @ A )
% 0.34/0.68 = ( times_times_int @ ( groups3188404863801439024er_int @ G @ ( inf_in1364745209274528805nteger @ A @ B4 ) ) @ ( groups3188404863801439024er_int @ G @ ( minus_2355218937544613996nteger @ A @ B4 ) ) ) ) ) ).
% 0.34/0.68
% 0.34/0.68 % prod.Int_Diff
% 0.34/0.68 thf(fact_7300_prod_OInt__Diff,axiom,
% 0.34/0.68 ! [A: set_nat,G: nat > int,B4: set_nat] :
% 0.34/0.68 ( ( finite_finite_nat @ A )
% 0.34/0.68 => ( ( groups705719431365010083at_int @ G @ A )
% 0.34/0.68 = ( times_times_int @ ( groups705719431365010083at_int @ G @ ( inf_inf_set_nat @ A @ B4 ) ) @ ( groups705719431365010083at_int @ G @ ( minus_minus_set_nat @ A @ B4 ) ) ) ) ) ).
% 0.34/0.68
% 0.34/0.68 % prod.Int_Diff
% 0.34/0.68 thf(fact_7301_prod_OInt__Diff,axiom,
% 0.34/0.68 ! [A: set_nat,G: nat > nat,B4: set_nat] :
% 0.34/0.68 ( ( finite_finite_nat @ A )
% 0.34/0.68 => ( ( groups708209901874060359at_nat @ G @ A )
% 0.34/0.68 = ( times_times_nat @ ( groups708209901874060359at_nat @ G @ ( inf_inf_set_nat @ A @ B4 ) ) @ ( groups708209901874060359at_nat @ G @ ( minus_minus_set_nat @ A @ B4 ) ) ) ) ) ).
% 0.34/0.68
% 0.34/0.68 % prod.Int_Diff
% 0.34/0.68 thf(fact_7302_prod_Omono__neutral__cong,axiom,
% 0.34/0.68 ! [T2: set_int,S: set_int,H2: int > assn,G: int > assn] :
% 0.34/0.68 ( ( finite_finite_int @ T2 )
% 0.34/0.68 => ( ( finite_finite_int @ S )
% 0.34/0.68 => ( ! [I3: int] :
% 0.34/0.68 ( ( member_int @ I3 @ ( minus_minus_set_int @ T2 @ S ) )
% 0.34/0.68 => ( ( H2 @ I3 )
% 0.34/0.68 = one_one_assn ) )
% 0.34/0.68 => ( ! [I3: int] :
% 0.34/0.68 ( ( member_int @ I3 @ ( minus_minus_set_int @ S @ T2 ) )
% 0.34/0.68 => ( ( G @ I3 )
% 0.34/0.68 = one_one_assn ) )
% 0.34/0.68 => ( ! [X3: int] :
% 0.34/0.68 ( ( member_int @ X3 @ ( inf_inf_set_int @ S @ T2 ) )
% 0.34/0.68 => ( ( G @ X3 )
% 0.34/0.68 = ( H2 @ X3 ) ) )
% 0.34/0.68 => ( ( groups7882442080178216443t_assn @ G @ S )
% 0.34/0.68 = ( groups7882442080178216443t_assn @ H2 @ T2 ) ) ) ) ) ) ) ).
% 0.34/0.68
% 0.34/0.68 % prod.mono_neutral_cong
% 0.34/0.68 thf(fact_7303_prod_Omono__neutral__cong,axiom,
% 0.34/0.68 ! [T2: set_o,S: set_o,H2: $o > assn,G: $o > assn] :
% 0.34/0.68 ( ( finite_finite_o @ T2 )
% 0.34/0.68 => ( ( finite_finite_o @ S )
% 0.34/0.68 => ( ! [I3: $o] :
% 0.34/0.68 ( ( member_o @ I3 @ ( minus_minus_set_o @ T2 @ S ) )
% 0.34/0.68 => ( ( H2 @ I3 )
% 0.34/0.68 = one_one_assn ) )
% 0.34/0.68 => ( ! [I3: $o] :
% 0.34/0.68 ( ( member_o @ I3 @ ( minus_minus_set_o @ S @ T2 ) )
% 0.34/0.68 => ( ( G @ I3 )
% 0.34/0.68 = one_one_assn ) )
% 0.34/0.68 => ( ! [X3: $o] :
% 0.34/0.68 ( ( member_o @ X3 @ ( inf_inf_set_o @ S @ T2 ) )
% 0.34/0.68 => ( ( G @ X3 )
% 0.34/0.68 = ( H2 @ X3 ) ) )
% 0.34/0.68 => ( ( groups5301882518646026715o_assn @ G @ S )
% 0.34/0.68 = ( groups5301882518646026715o_assn @ H2 @ T2 ) ) ) ) ) ) ) ).
% 0.34/0.68
% 0.34/0.68 % prod.mono_neutral_cong
% 0.34/0.68 thf(fact_7304_prod_Omono__neutral__cong,axiom,
% 0.34/0.68 ! [T2: set_Product_unit,S: set_Product_unit,H2: product_unit > assn,G: product_unit > assn] :
% 0.34/0.68 ( ( finite4290736615968046902t_unit @ T2 )
% 0.34/0.68 => ( ( finite4290736615968046902t_unit @ S )
% 0.34/0.68 => ( ! [I3: product_unit] :
% 0.34/0.68 ( ( member_Product_unit @ I3 @ ( minus_6452836326544984404t_unit @ T2 @ S ) )
% 0.34/0.68 => ( ( H2 @ I3 )
% 0.34/0.68 = one_one_assn ) )
% 0.34/0.68 => ( ! [I3: product_unit] :
% 0.34/0.68 ( ( member_Product_unit @ I3 @ ( minus_6452836326544984404t_unit @ S @ T2 ) )
% 0.34/0.68 => ( ( G @ I3 )
% 0.34/0.68 = one_one_assn ) )
% 0.34/0.68 => ( ! [X3: product_unit] :
% 0.34/0.68 ( ( member_Product_unit @ X3 @ ( inf_in4660618365625256667t_unit @ S @ T2 ) )
% 0.34/0.68 => ( ( G @ X3 )
% 0.34/0.68 = ( H2 @ X3 ) ) )
% 0.34/0.68 => ( ( groups5360233303959380816t_assn @ G @ S )
% 0.34/0.68 = ( groups5360233303959380816t_assn @ H2 @ T2 ) ) ) ) ) ) ) ).
% 0.34/0.68
% 0.34/0.68 % prod.mono_neutral_cong
% 0.34/0.68 thf(fact_7305_prod_Omono__neutral__cong,axiom,
% 0.34/0.68 ! [T2: set_Code_integer,S: set_Code_integer,H2: code_integer > assn,G: code_integer > assn] :
% 0.34/0.68 ( ( finite6017078050557962740nteger @ T2 )
% 0.34/0.68 => ( ( finite6017078050557962740nteger @ S )
% 0.34/0.68 => ( ! [I3: code_integer] :
% 0.34/0.68 ( ( member_Code_integer @ I3 @ ( minus_2355218937544613996nteger @ T2 @ S ) )
% 0.34/0.68 => ( ( H2 @ I3 )
% 0.34/0.68 = one_one_assn ) )
% 0.34/0.68 => ( ! [I3: code_integer] :
% 0.34/0.68 ( ( member_Code_integer @ I3 @ ( minus_2355218937544613996nteger @ S @ T2 ) )
% 0.34/0.68 => ( ( G @ I3 )
% 0.34/0.68 = one_one_assn ) )
% 0.34/0.68 => ( ! [X3: code_integer] :
% 0.34/0.68 ( ( member_Code_integer @ X3 @ ( inf_in1364745209274528805nteger @ S @ T2 ) )
% 0.34/0.68 => ( ( G @ X3 )
% 0.34/0.68 = ( H2 @ X3 ) ) )
% 0.34/0.68 => ( ( groups1304777262505850412r_assn @ G @ S )
% 0.34/0.68 = ( groups1304777262505850412r_assn @ H2 @ T2 ) ) ) ) ) ) ) ).
% 0.34/0.68
% 0.34/0.68 % prod.mono_neutral_cong
% 0.34/0.68 thf(fact_7306_prod_Omono__neutral__cong,axiom,
% 0.34/0.68 ! [T2: set_int,S: set_int,H2: int > nat,G: int > nat] :
% 0.34/0.68 ( ( finite_finite_int @ T2 )
% 0.34/0.68 => ( ( finite_finite_int @ S )
% 0.34/0.68 => ( ! [I3: int] :
% 0.34/0.68 ( ( member_int @ I3 @ ( minus_minus_set_int @ T2 @ S ) )
% 0.34/0.68 => ( ( H2 @ I3 )
% 0.34/0.68 = one_one_nat ) )
% 0.34/0.68 => ( ! [I3: int] :
% 0.34/0.68 ( ( member_int @ I3 @ ( minus_minus_set_int @ S @ T2 ) )
% 0.34/0.68 => ( ( G @ I3 )
% 0.34/0.68 = one_one_nat ) )
% 0.34/0.68 => ( ! [X3: int] :
% 0.34/0.68 ( ( member_int @ X3 @ ( inf_inf_set_int @ S @ T2 ) )
% 0.34/0.68 => ( ( G @ X3 )
% 0.34/0.68 = ( H2 @ X3 ) ) )
% 0.34/0.68 => ( ( groups1707563613775114915nt_nat @ G @ S )
% 0.34/0.68 = ( groups1707563613775114915nt_nat @ H2 @ T2 ) ) ) ) ) ) ) ).
% 0.34/0.68
% 0.34/0.68 % prod.mono_neutral_cong
% 0.34/0.68 thf(fact_7307_prod_Omono__neutral__cong,axiom,
% 0.34/0.68 ! [T2: set_o,S: set_o,H2: $o > nat,G: $o > nat] :
% 0.34/0.68 ( ( finite_finite_o @ T2 )
% 0.34/0.68 => ( ( finite_finite_o @ S )
% 0.34/0.68 => ( ! [I3: $o] :
% 0.34/0.68 ( ( member_o @ I3 @ ( minus_minus_set_o @ T2 @ S ) )
% 0.34/0.68 => ( ( H2 @ I3 )
% 0.34/0.68 = one_one_nat ) )
% 0.34/0.68 => ( ! [I3: $o] :
% 0.34/0.68 ( ( member_o @ I3 @ ( minus_minus_set_o @ S @ T2 ) )
% 0.34/0.68 => ( ( G @ I3 )
% 0.34/0.68 = one_one_nat ) )
% 0.34/0.68 => ( ! [X3: $o] :
% 0.34/0.68 ( ( member_o @ X3 @ ( inf_inf_set_o @ S @ T2 ) )
% 0.34/0.68 => ( ( G @ X3 )
% 0.34/0.68 = ( H2 @ X3 ) ) )
% 0.34/0.68 => ( ( groups3504817904513533571_o_nat @ G @ S )
% 0.34/0.68 = ( groups3504817904513533571_o_nat @ H2 @ T2 ) ) ) ) ) ) ) ).
% 0.34/0.68
% 0.34/0.68 % prod.mono_neutral_cong
% 0.34/0.68 thf(fact_7308_prod_Omono__neutral__cong,axiom,
% 0.34/0.68 ! [T2: set_Product_unit,S: set_Product_unit,H2: product_unit > nat,G: product_unit > nat] :
% 0.34/0.68 ( ( finite4290736615968046902t_unit @ T2 )
% 0.34/0.68 => ( ( finite4290736615968046902t_unit @ S )
% 0.34/0.68 => ( ! [I3: product_unit] :
% 0.34/0.68 ( ( member_Product_unit @ I3 @ ( minus_6452836326544984404t_unit @ T2 @ S ) )
% 0.34/0.68 => ( ( H2 @ I3 )
% 0.34/0.68 = one_one_nat ) )
% 0.34/0.68 => ( ! [I3: product_unit] :
% 0.34/0.68 ( ( member_Product_unit @ I3 @ ( minus_6452836326544984404t_unit @ S @ T2 ) )
% 0.34/0.68 => ( ( G @ I3 )
% 0.34/0.68 = one_one_nat ) )
% 0.34/0.68 => ( ! [X3: product_unit] :
% 0.34/0.68 ( ( member_Product_unit @ X3 @ ( inf_in4660618365625256667t_unit @ S @ T2 ) )
% 0.34/0.68 => ( ( G @ X3 )
% 0.34/0.68 = ( H2 @ X3 ) ) )
% 0.34/0.68 => ( ( groups396421249026171384it_nat @ G @ S )
% 0.34/0.68 = ( groups396421249026171384it_nat @ H2 @ T2 ) ) ) ) ) ) ) ).
% 0.34/0.68
% 0.34/0.68 % prod.mono_neutral_cong
% 0.34/0.68 thf(fact_7309_prod_Omono__neutral__cong,axiom,
% 0.34/0.68 ! [T2: set_Code_integer,S: set_Code_integer,H2: code_integer > nat,G: code_integer > nat] :
% 0.34/0.68 ( ( finite6017078050557962740nteger @ T2 )
% 0.34/0.68 => ( ( finite6017078050557962740nteger @ S )
% 0.34/0.68 => ( ! [I3: code_integer] :
% 0.34/0.68 ( ( member_Code_integer @ I3 @ ( minus_2355218937544613996nteger @ T2 @ S ) )
% 0.34/0.68 => ( ( H2 @ I3 )
% 0.34/0.68 = one_one_nat ) )
% 0.34/0.68 => ( ! [I3: code_integer] :
% 0.34/0.68 ( ( member_Code_integer @ I3 @ ( minus_2355218937544613996nteger @ S @ T2 ) )
% 0.34/0.68 => ( ( G @ I3 )
% 0.34/0.68 = one_one_nat ) )
% 0.34/0.68 => ( ! [X3: code_integer] :
% 0.34/0.68 ( ( member_Code_integer @ X3 @ ( inf_in1364745209274528805nteger @ S @ T2 ) )
% 0.34/0.68 => ( ( G @ X3 )
% 0.34/0.68 = ( H2 @ X3 ) ) )
% 0.34/0.68 => ( ( groups3190895334310489300er_nat @ G @ S )
% 0.34/0.68 = ( groups3190895334310489300er_nat @ H2 @ T2 ) ) ) ) ) ) ) ).
% 0.34/0.68
% 0.34/0.68 % prod.mono_neutral_cong
% 0.34/0.68 thf(fact_7310_prod_Omono__neutral__cong,axiom,
% 0.34/0.68 ! [T2: set_int,S: set_int,H2: int > int,G: int > int] :
% 0.34/0.68 ( ( finite_finite_int @ T2 )
% 0.34/0.68 => ( ( finite_finite_int @ S )
% 0.34/0.68 => ( ! [I3: int] :
% 0.34/0.68 ( ( member_int @ I3 @ ( minus_minus_set_int @ T2 @ S ) )
% 0.34/0.68 => ( ( H2 @ I3 )
% 0.34/0.68 = one_one_int ) )
% 0.34/0.68 => ( ! [I3: int] :
% 0.34/0.68 ( ( member_int @ I3 @ ( minus_minus_set_int @ S @ T2 ) )
% 0.34/0.68 => ( ( G @ I3 )
% 0.34/0.68 = one_one_int ) )
% 0.34/0.68 => ( ! [X3: int] :
% 0.34/0.68 ( ( member_int @ X3 @ ( inf_inf_set_int @ S @ T2 ) )
% 0.34/0.68 => ( ( G @ X3 )
% 0.34/0.68 = ( H2 @ X3 ) ) )
% 0.34/0.68 => ( ( groups1705073143266064639nt_int @ G @ S )
% 0.34/0.68 = ( groups1705073143266064639nt_int @ H2 @ T2 ) ) ) ) ) ) ) ).
% 0.34/0.68
% 0.34/0.68 % prod.mono_neutral_cong
% 0.34/0.68 thf(fact_7311_prod_Omono__neutral__cong,axiom,
% 0.34/0.68 ! [T2: set_o,S: set_o,H2: $o > int,G: $o > int] :
% 0.34/0.68 ( ( finite_finite_o @ T2 )
% 0.34/0.68 => ( ( finite_finite_o @ S )
% 0.34/0.68 => ( ! [I3: $o] :
% 0.34/0.68 ( ( member_o @ I3 @ ( minus_minus_set_o @ T2 @ S ) )
% 0.34/0.68 => ( ( H2 @ I3 )
% 0.34/0.68 = one_one_int ) )
% 0.34/0.68 => ( ! [I3: $o] :
% 0.34/0.68 ( ( member_o @ I3 @ ( minus_minus_set_o @ S @ T2 ) )
% 0.34/0.68 => ( ( G @ I3 )
% 0.34/0.68 = one_one_int ) )
% 0.34/0.68 => ( ! [X3: $o] :
% 0.34/0.68 ( ( member_o @ X3 @ ( inf_inf_set_o @ S @ T2 ) )
% 0.34/0.68 => ( ( G @ X3 )
% 0.34/0.68 = ( H2 @ X3 ) ) )
% 0.34/0.68 => ( ( groups3502327434004483295_o_int @ G @ S )
% 0.34/0.68 = ( groups3502327434004483295_o_int @ H2 @ T2 ) ) ) ) ) ) ) ).
% 0.34/0.68
% 0.34/0.68 % prod.mono_neutral_cong
% 0.34/0.68 thf(fact_7312_prod_Oeq__fold,axiom,
% 0.34/0.68 ( groups708209901874060359at_nat
% 0.34/0.68 = ( ^ [G3: nat > nat] : ( finite_fold_nat_nat @ ( comp_nat_nat_nat_nat @ times_times_nat @ G3 ) @ one_one_nat ) ) ) ).
% 0.34/0.68
% 0.34/0.68 % prod.eq_fold
% 0.34/0.68 thf(fact_7313_sum__bounded__above,axiom,
% 0.34/0.68 ! [A: set_nat,F: nat > code_integer,K4: code_integer] :
% 0.34/0.68 ( ! [I3: nat] :
% 0.34/0.68 ( ( member_nat @ I3 @ A )
% 0.34/0.68 => ( ord_le3102999989581377725nteger @ ( F @ I3 ) @ K4 ) )
% 0.34/0.68 => ( ord_le3102999989581377725nteger @ ( groups7501900531339628137nteger @ F @ A ) @ ( times_3573771949741848930nteger @ ( semiri4939895301339042750nteger @ ( finite_card_nat @ A ) ) @ K4 ) ) ) ).
% 0.34/0.69
% 0.34/0.69 % sum_bounded_above
% 0.34/0.69 thf(fact_7314_sum__bounded__above,axiom,
% 0.34/0.69 ! [A: set_int,F: int > code_integer,K4: code_integer] :
% 0.34/0.69 ( ! [I3: int] :
% 0.34/0.69 ( ( member_int @ I3 @ A )
% 0.34/0.69 => ( ord_le3102999989581377725nteger @ ( F @ I3 ) @ K4 ) )
% 0.34/0.69 => ( ord_le3102999989581377725nteger @ ( groups7873554091576472773nteger @ F @ A ) @ ( times_3573771949741848930nteger @ ( semiri4939895301339042750nteger @ ( finite_card_int @ A ) ) @ K4 ) ) ) ).
% 0.34/0.69
% 0.34/0.69 % sum_bounded_above
% 0.34/0.69 thf(fact_7315_sum__bounded__above,axiom,
% 0.34/0.69 ! [A: set_o,F: $o > code_integer,K4: code_integer] :
% 0.34/0.69 ( ! [I3: $o] :
% 0.34/0.69 ( ( member_o @ I3 @ A )
% 0.34/0.69 => ( ord_le3102999989581377725nteger @ ( F @ I3 ) @ K4 ) )
% 0.34/0.69 => ( ord_le3102999989581377725nteger @ ( groups4406642042086082107nteger @ F @ A ) @ ( times_3573771949741848930nteger @ ( semiri4939895301339042750nteger @ ( finite_card_o @ A ) ) @ K4 ) ) ) ).
% 0.34/0.69
% 0.34/0.69 % sum_bounded_above
% 0.34/0.69 thf(fact_7316_sum__bounded__above,axiom,
% 0.34/0.69 ! [A: set_Product_unit,F: product_unit > code_integer,K4: code_integer] :
% 0.34/0.69 ( ! [I3: product_unit] :
% 0.34/0.69 ( ( member_Product_unit @ I3 @ A )
% 0.34/0.69 => ( ord_le3102999989581377725nteger @ ( F @ I3 ) @ K4 ) )
% 0.34/0.69 => ( ord_le3102999989581377725nteger @ ( groups5393439102872367408nteger @ F @ A ) @ ( times_3573771949741848930nteger @ ( semiri4939895301339042750nteger @ ( finite410649719033368117t_unit @ A ) ) @ K4 ) ) ) ).
% 0.34/0.69
% 0.34/0.69 % sum_bounded_above
% 0.34/0.69 thf(fact_7317_sum__bounded__above,axiom,
% 0.34/0.69 ! [A: set_nat,F: nat > rat,K4: rat] :
% 0.34/0.69 ( ! [I3: nat] :
% 0.34/0.69 ( ( member_nat @ I3 @ A )
% 0.34/0.69 => ( ord_less_eq_rat @ ( F @ I3 ) @ K4 ) )
% 0.34/0.69 => ( ord_less_eq_rat @ ( groups2906978787729119204at_rat @ F @ A ) @ ( times_times_rat @ ( semiri681578069525770553at_rat @ ( finite_card_nat @ A ) ) @ K4 ) ) ) ).
% 0.34/0.69
% 0.34/0.69 % sum_bounded_above
% 0.34/0.69 thf(fact_7318_sum__bounded__above,axiom,
% 0.34/0.69 ! [A: set_int,F: int > rat,K4: rat] :
% 0.34/0.69 ( ! [I3: int] :
% 0.34/0.69 ( ( member_int @ I3 @ A )
% 0.34/0.69 => ( ord_less_eq_rat @ ( F @ I3 ) @ K4 ) )
% 0.34/0.69 => ( ord_less_eq_rat @ ( groups3906332499630173760nt_rat @ F @ A ) @ ( times_times_rat @ ( semiri681578069525770553at_rat @ ( finite_card_int @ A ) ) @ K4 ) ) ) ).
% 0.34/0.69
% 0.34/0.69 % sum_bounded_above
% 0.34/0.69 thf(fact_7319_sum__bounded__above,axiom,
% 0.34/0.69 ! [A: set_o,F: $o > rat,K4: rat] :
% 0.34/0.69 ( ! [I3: $o] :
% 0.34/0.69 ( ( member_o @ I3 @ A )
% 0.34/0.69 => ( ord_less_eq_rat @ ( F @ I3 ) @ K4 ) )
% 0.34/0.69 => ( ord_less_eq_rat @ ( groups7872700643590313910_o_rat @ F @ A ) @ ( times_times_rat @ ( semiri681578069525770553at_rat @ ( finite_card_o @ A ) ) @ K4 ) ) ) ).
% 0.34/0.69
% 0.34/0.69 % sum_bounded_above
% 0.34/0.69 thf(fact_7320_sum__bounded__above,axiom,
% 0.34/0.69 ! [A: set_Product_unit,F: product_unit > rat,K4: rat] :
% 0.34/0.69 ( ! [I3: product_unit] :
% 0.34/0.69 ( ( member_Product_unit @ I3 @ A )
% 0.34/0.69 => ( ord_less_eq_rat @ ( F @ I3 ) @ K4 ) )
% 0.34/0.69 => ( ord_less_eq_rat @ ( groups8131667117488461483it_rat @ F @ A ) @ ( times_times_rat @ ( semiri681578069525770553at_rat @ ( finite410649719033368117t_unit @ A ) ) @ K4 ) ) ) ).
% 0.34/0.69
% 0.34/0.69 % sum_bounded_above
% 0.34/0.69 thf(fact_7321_sum__bounded__above,axiom,
% 0.34/0.69 ! [A: set_int,F: int > nat,K4: nat] :
% 0.34/0.69 ( ! [I3: int] :
% 0.34/0.69 ( ( member_int @ I3 @ A )
% 0.34/0.69 => ( ord_less_eq_nat @ ( F @ I3 ) @ K4 ) )
% 0.34/0.69 => ( ord_less_eq_nat @ ( groups4541462559716669496nt_nat @ F @ A ) @ ( times_times_nat @ ( semiri1316708129612266289at_nat @ ( finite_card_int @ A ) ) @ K4 ) ) ) ).
% 0.34/0.69
% 0.34/0.69 % sum_bounded_above
% 0.34/0.69 thf(fact_7322_sum__bounded__above,axiom,
% 0.34/0.69 ! [A: set_o,F: $o > nat,K4: nat] :
% 0.34/0.69 ( ! [I3: $o] :
% 0.34/0.69 ( ( member_o @ I3 @ A )
% 0.34/0.69 => ( ord_less_eq_nat @ ( F @ I3 ) @ K4 ) )
% 0.34/0.69 => ( ord_less_eq_nat @ ( groups8507830703676809646_o_nat @ F @ A ) @ ( times_times_nat @ ( semiri1316708129612266289at_nat @ ( finite_card_o @ A ) ) @ K4 ) ) ) ).
% 0.34/0.69
% 0.34/0.69 % sum_bounded_above
% 0.34/0.69 thf(fact_7323_sum__bounded__below,axiom,
% 0.34/0.69 ! [A: set_nat,K4: code_integer,F: nat > code_integer] :
% 0.34/0.69 ( ! [I3: nat] :
% 0.34/0.69 ( ( member_nat @ I3 @ A )
% 0.34/0.69 => ( ord_le3102999989581377725nteger @ K4 @ ( F @ I3 ) ) )
% 0.34/0.69 => ( ord_le3102999989581377725nteger @ ( times_3573771949741848930nteger @ ( semiri4939895301339042750nteger @ ( finite_card_nat @ A ) ) @ K4 ) @ ( groups7501900531339628137nteger @ F @ A ) ) ) ).
% 0.34/0.69
% 0.34/0.69 % sum_bounded_below
% 0.34/0.69 thf(fact_7324_sum__bounded__below,axiom,
% 0.34/0.69 ! [A: set_int,K4: code_integer,F: int > code_integer] :
% 0.34/0.69 ( ! [I3: int] :
% 0.34/0.69 ( ( member_int @ I3 @ A )
% 0.34/0.69 => ( ord_le3102999989581377725nteger @ K4 @ ( F @ I3 ) ) )
% 0.34/0.69 => ( ord_le3102999989581377725nteger @ ( times_3573771949741848930nteger @ ( semiri4939895301339042750nteger @ ( finite_card_int @ A ) ) @ K4 ) @ ( groups7873554091576472773nteger @ F @ A ) ) ) ).
% 0.34/0.69
% 0.34/0.69 % sum_bounded_below
% 0.34/0.69 thf(fact_7325_sum__bounded__below,axiom,
% 0.34/0.69 ! [A: set_o,K4: code_integer,F: $o > code_integer] :
% 0.34/0.69 ( ! [I3: $o] :
% 0.34/0.69 ( ( member_o @ I3 @ A )
% 0.34/0.69 => ( ord_le3102999989581377725nteger @ K4 @ ( F @ I3 ) ) )
% 0.34/0.69 => ( ord_le3102999989581377725nteger @ ( times_3573771949741848930nteger @ ( semiri4939895301339042750nteger @ ( finite_card_o @ A ) ) @ K4 ) @ ( groups4406642042086082107nteger @ F @ A ) ) ) ).
% 0.34/0.69
% 0.34/0.69 % sum_bounded_below
% 0.34/0.69 thf(fact_7326_sum__bounded__below,axiom,
% 0.34/0.69 ! [A: set_Product_unit,K4: code_integer,F: product_unit > code_integer] :
% 0.34/0.69 ( ! [I3: product_unit] :
% 0.34/0.69 ( ( member_Product_unit @ I3 @ A )
% 0.34/0.69 => ( ord_le3102999989581377725nteger @ K4 @ ( F @ I3 ) ) )
% 0.34/0.69 => ( ord_le3102999989581377725nteger @ ( times_3573771949741848930nteger @ ( semiri4939895301339042750nteger @ ( finite410649719033368117t_unit @ A ) ) @ K4 ) @ ( groups5393439102872367408nteger @ F @ A ) ) ) ).
% 0.34/0.69
% 0.34/0.69 % sum_bounded_below
% 0.34/0.69 thf(fact_7327_sum__bounded__below,axiom,
% 0.34/0.69 ! [A: set_nat,K4: rat,F: nat > rat] :
% 0.34/0.69 ( ! [I3: nat] :
% 0.34/0.69 ( ( member_nat @ I3 @ A )
% 0.34/0.69 => ( ord_less_eq_rat @ K4 @ ( F @ I3 ) ) )
% 0.34/0.69 => ( ord_less_eq_rat @ ( times_times_rat @ ( semiri681578069525770553at_rat @ ( finite_card_nat @ A ) ) @ K4 ) @ ( groups2906978787729119204at_rat @ F @ A ) ) ) ).
% 0.34/0.69
% 0.34/0.69 % sum_bounded_below
% 0.34/0.69 thf(fact_7328_sum__bounded__below,axiom,
% 0.34/0.69 ! [A: set_int,K4: rat,F: int > rat] :
% 0.34/0.69 ( ! [I3: int] :
% 0.34/0.69 ( ( member_int @ I3 @ A )
% 0.34/0.69 => ( ord_less_eq_rat @ K4 @ ( F @ I3 ) ) )
% 0.34/0.69 => ( ord_less_eq_rat @ ( times_times_rat @ ( semiri681578069525770553at_rat @ ( finite_card_int @ A ) ) @ K4 ) @ ( groups3906332499630173760nt_rat @ F @ A ) ) ) ).
% 0.34/0.69
% 0.34/0.69 % sum_bounded_below
% 0.34/0.69 thf(fact_7329_sum__bounded__below,axiom,
% 0.34/0.69 ! [A: set_o,K4: rat,F: $o > rat] :
% 0.34/0.69 ( ! [I3: $o] :
% 0.34/0.69 ( ( member_o @ I3 @ A )
% 0.34/0.69 => ( ord_less_eq_rat @ K4 @ ( F @ I3 ) ) )
% 0.34/0.69 => ( ord_less_eq_rat @ ( times_times_rat @ ( semiri681578069525770553at_rat @ ( finite_card_o @ A ) ) @ K4 ) @ ( groups7872700643590313910_o_rat @ F @ A ) ) ) ).
% 0.34/0.69
% 0.34/0.69 % sum_bounded_below
% 0.34/0.69 thf(fact_7330_sum__bounded__below,axiom,
% 0.34/0.69 ! [A: set_Product_unit,K4: rat,F: product_unit > rat] :
% 0.34/0.69 ( ! [I3: product_unit] :
% 0.34/0.69 ( ( member_Product_unit @ I3 @ A )
% 0.34/0.69 => ( ord_less_eq_rat @ K4 @ ( F @ I3 ) ) )
% 0.34/0.69 => ( ord_less_eq_rat @ ( times_times_rat @ ( semiri681578069525770553at_rat @ ( finite410649719033368117t_unit @ A ) ) @ K4 ) @ ( groups8131667117488461483it_rat @ F @ A ) ) ) ).
% 0.34/0.69
% 0.34/0.69 % sum_bounded_below
% 0.34/0.69 thf(fact_7331_sum__bounded__below,axiom,
% 0.34/0.69 ! [A: set_int,K4: nat,F: int > nat] :
% 0.34/0.69 ( ! [I3: int] :
% 0.34/0.69 ( ( member_int @ I3 @ A )
% 0.34/0.69 => ( ord_less_eq_nat @ K4 @ ( F @ I3 ) ) )
% 0.34/0.69 => ( ord_less_eq_nat @ ( times_times_nat @ ( semiri1316708129612266289at_nat @ ( finite_card_int @ A ) ) @ K4 ) @ ( groups4541462559716669496nt_nat @ F @ A ) ) ) ).
% 0.34/0.69
% 0.34/0.69 % sum_bounded_below
% 0.34/0.69 thf(fact_7332_sum__bounded__below,axiom,
% 0.34/0.69 ! [A: set_o,K4: nat,F: $o > nat] :
% 0.34/0.69 ( ! [I3: $o] :
% 0.34/0.69 ( ( member_o @ I3 @ A )
% 0.34/0.69 => ( ord_less_eq_nat @ K4 @ ( F @ I3 ) ) )
% 0.34/0.69 => ( ord_less_eq_nat @ ( times_times_nat @ ( semiri1316708129612266289at_nat @ ( finite_card_o @ A ) ) @ K4 ) @ ( groups8507830703676809646_o_nat @ F @ A ) ) ) ).
% 0.34/0.69
% 0.34/0.69 % sum_bounded_below
% 0.34/0.69 thf(fact_7333_gbinomial__absorb__comp,axiom,
% 0.34/0.69 ! [A2: rat,K3: nat] :
% 0.34/0.69 ( ( times_times_rat @ ( minus_minus_rat @ A2 @ ( semiri681578069525770553at_rat @ K3 ) ) @ ( gbinomial_rat @ A2 @ K3 ) )
% 0.34/0.69 = ( times_times_rat @ A2 @ ( gbinomial_rat @ ( minus_minus_rat @ A2 @ one_one_rat ) @ K3 ) ) ) ).
% 0.34/0.69
% 0.34/0.69 % gbinomial_absorb_comp
% 0.34/0.69 thf(fact_7334_gbinomial__mult__1,axiom,
% 0.34/0.69 ! [A2: rat,K3: nat] :
% 0.34/0.69 ( ( times_times_rat @ A2 @ ( gbinomial_rat @ A2 @ K3 ) )
% 0.34/0.69 = ( plus_plus_rat @ ( times_times_rat @ ( semiri681578069525770553at_rat @ K3 ) @ ( gbinomial_rat @ A2 @ K3 ) ) @ ( times_times_rat @ ( semiri681578069525770553at_rat @ ( suc @ K3 ) ) @ ( gbinomial_rat @ A2 @ ( suc @ K3 ) ) ) ) ) ).
% 0.34/0.69
% 0.34/0.69 % gbinomial_mult_1
% 0.34/0.69 thf(fact_7335_gbinomial__mult__1_H,axiom,
% 0.34/0.69 ! [A2: rat,K3: nat] :
% 0.34/0.69 ( ( times_times_rat @ ( gbinomial_rat @ A2 @ K3 ) @ A2 )
% 0.34/0.69 = ( plus_plus_rat @ ( times_times_rat @ ( semiri681578069525770553at_rat @ K3 ) @ ( gbinomial_rat @ A2 @ K3 ) ) @ ( times_times_rat @ ( semiri681578069525770553at_rat @ ( suc @ K3 ) ) @ ( gbinomial_rat @ A2 @ ( suc @ K3 ) ) ) ) ) ).
% 0.34/0.69
% 0.34/0.69 % gbinomial_mult_1'
% 0.34/0.69 thf(fact_7336_boolean__algebra__class_Oboolean__algebra_Ocompl__unique,axiom,
% 0.34/0.69 ! [X: set_Pr4532377907799695533_nat_o,Y: set_Pr4532377907799695533_nat_o] :
% 0.34/0.69 ( ( ( inf_in1906310914598751387_nat_o @ X @ Y )
% 0.34/0.69 = bot_bo7824918357723582233_nat_o )
% 0.34/0.69 => ( ( ( sup_su5209123915105501825_nat_o @ X @ Y )
% 0.34/0.69 = top_to4869887016220417533_nat_o )
% 0.34/0.69 => ( ( uminus5254974436814262692_nat_o @ X )
% 0.34/0.69 = Y ) ) ) ).
% 0.34/0.69
% 0.34/0.69 % boolean_algebra_class.boolean_algebra.compl_unique
% 0.34/0.69 thf(fact_7337_boolean__algebra__class_Oboolean__algebra_Ocompl__unique,axiom,
% 0.34/0.69 ! [X: set_Product_unit,Y: set_Product_unit] :
% 0.34/0.69 ( ( ( inf_in4660618365625256667t_unit @ X @ Y )
% 0.34/0.69 = bot_bo3957492148770167129t_unit )
% 0.34/0.69 => ( ( ( sup_su793286257634532545t_unit @ X @ Y )
% 0.34/0.69 = top_to1996260823553986621t_unit )
% 0.34/0.69 => ( ( uminus5944136376168626660t_unit @ X )
% 0.34/0.69 = Y ) ) ) ).
% 0.34/0.69
% 0.34/0.69 % boolean_algebra_class.boolean_algebra.compl_unique
% 0.34/0.69 thf(fact_7338_boolean__algebra__class_Oboolean__algebra_Ocompl__unique,axiom,
% 0.34/0.69 ! [X: set_o,Y: set_o] :
% 0.34/0.69 ( ( ( inf_inf_set_o @ X @ Y )
% 0.34/0.69 = bot_bot_set_o )
% 0.34/0.69 => ( ( ( sup_sup_set_o @ X @ Y )
% 0.34/0.69 = top_top_set_o )
% 0.34/0.69 => ( ( uminus_uminus_set_o @ X )
% 0.34/0.69 = Y ) ) ) ).
% 0.34/0.69
% 0.34/0.69 % boolean_algebra_class.boolean_algebra.compl_unique
% 0.34/0.69 thf(fact_7339_boolean__algebra__class_Oboolean__algebra_Ocompl__unique,axiom,
% 0.34/0.69 ! [X: set_nat,Y: set_nat] :
% 0.34/0.69 ( ( ( inf_inf_set_nat @ X @ Y )
% 0.34/0.69 = bot_bot_set_nat )
% 0.34/0.69 => ( ( ( sup_sup_set_nat @ X @ Y )
% 0.34/0.69 = top_top_set_nat )
% 0.34/0.69 => ( ( uminus5710092332889474511et_nat @ X )
% 0.34/0.69 = Y ) ) ) ).
% 0.34/0.69
% 0.34/0.69 % boolean_algebra_class.boolean_algebra.compl_unique
% 0.34/0.69 thf(fact_7340_boolean__algebra__class_Oboolean__algebra_Ocompl__unique,axiom,
% 0.34/0.69 ! [X: set_int,Y: set_int] :
% 0.34/0.69 ( ( ( inf_inf_set_int @ X @ Y )
% 0.34/0.69 = bot_bot_set_int )
% 0.34/0.69 => ( ( ( sup_sup_set_int @ X @ Y )
% 0.34/0.69 = top_top_set_int )
% 0.34/0.69 => ( ( uminus1532241313380277803et_int @ X )
% 0.34/0.69 = Y ) ) ) ).
% 0.34/0.69
% 0.34/0.69 % boolean_algebra_class.boolean_algebra.compl_unique
% 0.34/0.69 thf(fact_7341_boolean__algebra__class_Oboolean__algebra_Ocompl__unique,axiom,
% 0.34/0.69 ! [X: assn,Y: assn] :
% 0.34/0.69 ( ( ( inf_inf_assn @ X @ Y )
% 0.34/0.69 = bot_bot_assn )
% 0.34/0.69 => ( ( ( sup_sup_assn @ X @ Y )
% 0.34/0.69 = top_top_assn )
% 0.34/0.69 => ( ( uminus_uminus_assn @ X )
% 0.34/0.69 = Y ) ) ) ).
% 0.34/0.69
% 0.34/0.69 % boolean_algebra_class.boolean_algebra.compl_unique
% 0.34/0.69 thf(fact_7342_boolean__algebra__class_Oboolean__algebra_Ocompl__unique,axiom,
% 0.34/0.69 ! [X: product_unit,Y: product_unit] :
% 0.34/0.69 ( ( ( inf_inf_Product_unit @ X @ Y )
% 0.34/0.69 = bot_bot_Product_unit )
% 0.34/0.69 => ( ( ( sup_sup_Product_unit @ X @ Y )
% 0.34/0.69 = top_top_Product_unit )
% 0.34/0.69 => ( ( uminus2952777764628376836t_unit @ X )
% 0.34/0.69 = Y ) ) ) ).
% 0.34/0.69
% 0.34/0.69 % boolean_algebra_class.boolean_algebra.compl_unique
% 0.34/0.69 thf(fact_7343_prod__mono__strict,axiom,
% 0.34/0.69 ! [A: set_Code_integer,F: code_integer > code_integer,G: code_integer > code_integer] :
% 0.34/0.69 ( ( finite6017078050557962740nteger @ A )
% 0.34/0.69 => ( ! [I3: code_integer] :
% 0.34/0.69 ( ( member_Code_integer @ I3 @ A )
% 0.34/0.69 => ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ ( F @ I3 ) )
% 0.34/0.69 & ( ord_le6747313008572928689nteger @ ( F @ I3 ) @ ( G @ I3 ) ) ) )
% 0.34/0.69 => ( ( A != bot_bo3990330152332043303nteger )
% 0.34/0.69 => ( ord_le6747313008572928689nteger @ ( groups3674199335183972705nteger @ F @ A ) @ ( groups3674199335183972705nteger @ G @ A ) ) ) ) ) ).
% 0.34/0.69
% 0.34/0.69 % prod_mono_strict
% 0.34/0.69 thf(fact_7344_prod__mono__strict,axiom,
% 0.34/0.69 ! [A: set_Product_unit,F: product_unit > code_integer,G: product_unit > code_integer] :
% 0.34/0.69 ( ( finite4290736615968046902t_unit @ A )
% 0.34/0.69 => ( ! [I3: product_unit] :
% 0.34/0.69 ( ( member_Product_unit @ I3 @ A )
% 0.34/0.69 => ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ ( F @ I3 ) )
% 0.34/0.69 & ( ord_le6747313008572928689nteger @ ( F @ I3 ) @ ( G @ I3 ) ) ) )
% 0.34/0.69 => ( ( A != bot_bo3957492148770167129t_unit )
% 0.34/0.69 => ( ord_le6747313008572928689nteger @ ( groups8185359811944078981nteger @ F @ A ) @ ( groups8185359811944078981nteger @ G @ A ) ) ) ) ) ).
% 0.34/0.69
% 0.34/0.69 % prod_mono_strict
% 0.34/0.69 thf(fact_7345_prod__mono__strict,axiom,
% 0.34/0.69 ! [A: set_o,F: $o > code_integer,G: $o > code_integer] :
% 0.34/0.69 ( ( finite_finite_o @ A )
% 0.34/0.69 => ( ! [I3: $o] :
% 0.34/0.69 ( ( member_o @ I3 @ A )
% 0.34/0.69 => ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ ( F @ I3 ) )
% 0.34/0.69 & ( ord_le6747313008572928689nteger @ ( F @ I3 ) @ ( G @ I3 ) ) ) )
% 0.34/0.69 => ( ( A != bot_bot_set_o )
% 0.34/0.69 => ( ord_le6747313008572928689nteger @ ( groups7694694392188491536nteger @ F @ A ) @ ( groups7694694392188491536nteger @ G @ A ) ) ) ) ) ).
% 0.34/0.69
% 0.34/0.69 % prod_mono_strict
% 0.34/0.69 thf(fact_7346_prod__mono__strict,axiom,
% 0.34/0.69 ! [A: set_nat,F: nat > code_integer,G: nat > code_integer] :
% 0.34/0.69 ( ( finite_finite_nat @ A )
% 0.34/0.69 => ( ! [I3: nat] :
% 0.34/0.69 ( ( member_nat @ I3 @ A )
% 0.34/0.69 => ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ ( F @ I3 ) )
% 0.34/0.69 & ( ord_le6747313008572928689nteger @ ( F @ I3 ) @ ( G @ I3 ) ) ) )
% 0.34/0.69 => ( ( A != bot_bot_set_nat )
% 0.34/0.69 => ( ord_le6747313008572928689nteger @ ( groups3455450783089532116nteger @ F @ A ) @ ( groups3455450783089532116nteger @ G @ A ) ) ) ) ) ).
% 0.34/0.69
% 0.34/0.69 % prod_mono_strict
% 0.34/0.69 thf(fact_7347_prod__mono__strict,axiom,
% 0.34/0.69 ! [A: set_int,F: int > code_integer,G: int > code_integer] :
% 0.34/0.69 ( ( finite_finite_int @ A )
% 0.34/0.69 => ( ! [I3: int] :
% 0.34/0.69 ( ( member_int @ I3 @ A )
% 0.34/0.69 => ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ ( F @ I3 ) )
% 0.34/0.69 & ( ord_le6747313008572928689nteger @ ( F @ I3 ) @ ( G @ I3 ) ) ) )
% 0.34/0.69 => ( ( A != bot_bot_set_int )
% 0.34/0.69 => ( ord_le6747313008572928689nteger @ ( groups3827104343326376752nteger @ F @ A ) @ ( groups3827104343326376752nteger @ G @ A ) ) ) ) ) ).
% 0.34/0.69
% 0.34/0.69 % prod_mono_strict
% 0.34/0.69 thf(fact_7348_prod__mono__strict,axiom,
% 0.34/0.69 ! [A: set_Code_integer,F: code_integer > rat,G: code_integer > rat] :
% 0.34/0.69 ( ( finite6017078050557962740nteger @ A )
% 0.34/0.69 => ( ! [I3: code_integer] :
% 0.34/0.69 ( ( member_Code_integer @ I3 @ A )
% 0.34/0.69 => ( ( ord_less_eq_rat @ zero_zero_rat @ ( F @ I3 ) )
% 0.34/0.69 & ( ord_less_rat @ ( F @ I3 ) @ ( G @ I3 ) ) ) )
% 0.34/0.69 => ( ( A != bot_bo3990330152332043303nteger )
% 0.34/0.69 => ( ord_less_rat @ ( groups2555765274223993564er_rat @ F @ A ) @ ( groups2555765274223993564er_rat @ G @ A ) ) ) ) ) ).
% 0.34/0.69
% 0.34/0.69 % prod_mono_strict
% 0.34/0.69 thf(fact_7349_prod__mono__strict,axiom,
% 0.34/0.69 ! [A: set_Product_unit,F: product_unit > rat,G: product_unit > rat] :
% 0.34/0.69 ( ( finite4290736615968046902t_unit @ A )
% 0.34/0.69 => ( ! [I3: product_unit] :
% 0.34/0.69 ( ( member_Product_unit @ I3 @ A )
% 0.34/0.69 => ( ( ord_less_eq_rat @ zero_zero_rat @ ( F @ I3 ) )
% 0.34/0.69 & ( ord_less_rat @ ( F @ I3 ) @ ( G @ I3 ) ) ) )
% 0.34/0.69 => ( ( A != bot_bo3957492148770167129t_unit )
% 0.34/0.69 => ( ord_less_rat @ ( groups8984663225794451456it_rat @ F @ A ) @ ( groups8984663225794451456it_rat @ G @ A ) ) ) ) ) ).
% 0.34/0.69
% 0.34/0.69 % prod_mono_strict
% 0.34/0.69 thf(fact_7350_prod__mono__strict,axiom,
% 0.34/0.69 ! [A: set_o,F: $o > rat,G: $o > rat] :
% 0.34/0.69 ( ( finite_finite_o @ A )
% 0.34/0.69 => ( ! [I3: $o] :
% 0.34/0.69 ( ( member_o @ I3 @ A )
% 0.34/0.69 => ( ( ord_less_eq_rat @ zero_zero_rat @ ( F @ I3 ) )
% 0.34/0.69 & ( ord_less_rat @ ( F @ I3 ) @ ( G @ I3 ) ) ) )
% 0.34/0.69 => ( ( A != bot_bot_set_o )
% 0.34/0.69 => ( ord_less_rat @ ( groups2869687844427037835_o_rat @ F @ A ) @ ( groups2869687844427037835_o_rat @ G @ A ) ) ) ) ) ).
% 0.34/0.69
% 0.34/0.69 % prod_mono_strict
% 0.34/0.69 thf(fact_7351_prod__mono__strict,axiom,
% 0.34/0.69 ! [A: set_nat,F: nat > rat,G: nat > rat] :
% 0.34/0.69 ( ( finite_finite_nat @ A )
% 0.34/0.69 => ( ! [I3: nat] :
% 0.34/0.69 ( ( member_nat @ I3 @ A )
% 0.34/0.69 => ( ( ord_less_eq_rat @ zero_zero_rat @ ( F @ I3 ) )
% 0.34/0.69 & ( ord_less_rat @ ( F @ I3 ) @ ( G @ I3 ) ) ) )
% 0.34/0.69 => ( ( A != bot_bot_set_nat )
% 0.34/0.69 => ( ord_less_rat @ ( groups73079841787564623at_rat @ F @ A ) @ ( groups73079841787564623at_rat @ G @ A ) ) ) ) ) ).
% 0.34/0.69
% 0.34/0.69 % prod_mono_strict
% 0.34/0.69 thf(fact_7352_prod__mono__strict,axiom,
% 0.34/0.69 ! [A: set_int,F: int > rat,G: int > rat] :
% 0.34/0.69 ( ( finite_finite_int @ A )
% 0.34/0.69 => ( ! [I3: int] :
% 0.34/0.69 ( ( member_int @ I3 @ A )
% 0.34/0.69 => ( ( ord_less_eq_rat @ zero_zero_rat @ ( F @ I3 ) )
% 0.34/0.69 & ( ord_less_rat @ ( F @ I3 ) @ ( G @ I3 ) ) ) )
% 0.34/0.69 => ( ( A != bot_bot_set_int )
% 0.34/0.69 => ( ord_less_rat @ ( groups1072433553688619179nt_rat @ F @ A ) @ ( groups1072433553688619179nt_rat @ G @ A ) ) ) ) ) ).
% 0.34/0.69
% 0.34/0.69 % prod_mono_strict
% 0.34/0.69 thf(fact_7353_prod_Oinsert__remove,axiom,
% 0.34/0.69 ! [A: set_Code_integer,G: code_integer > assn,X: code_integer] :
% 0.34/0.69 ( ( finite6017078050557962740nteger @ A )
% 0.34/0.69 => ( ( groups1304777262505850412r_assn @ G @ ( insert_Code_integer @ X @ A ) )
% 0.34/0.69 = ( times_times_assn @ ( G @ X ) @ ( groups1304777262505850412r_assn @ G @ ( minus_2355218937544613996nteger @ A @ ( insert_Code_integer @ X @ bot_bo3990330152332043303nteger ) ) ) ) ) ) ).
% 0.34/0.69
% 0.34/0.69 % prod.insert_remove
% 0.34/0.69 thf(fact_7354_prod_Oinsert__remove,axiom,
% 0.34/0.69 ! [A: set_Product_unit,G: product_unit > assn,X: product_unit] :
% 0.34/0.69 ( ( finite4290736615968046902t_unit @ A )
% 0.34/0.69 => ( ( groups5360233303959380816t_assn @ G @ ( insert_Product_unit @ X @ A ) )
% 0.34/0.69 = ( times_times_assn @ ( G @ X ) @ ( groups5360233303959380816t_assn @ G @ ( minus_6452836326544984404t_unit @ A @ ( insert_Product_unit @ X @ bot_bo3957492148770167129t_unit ) ) ) ) ) ) ).
% 0.34/0.69
% 0.34/0.69 % prod.insert_remove
% 0.34/0.69 thf(fact_7355_prod_Oinsert__remove,axiom,
% 0.34/0.69 ! [A: set_o,G: $o > assn,X: $o] :
% 0.34/0.69 ( ( finite_finite_o @ A )
% 0.34/0.69 => ( ( groups5301882518646026715o_assn @ G @ ( insert_o @ X @ A ) )
% 0.34/0.69 = ( times_times_assn @ ( G @ X ) @ ( groups5301882518646026715o_assn @ G @ ( minus_minus_set_o @ A @ ( insert_o @ X @ bot_bot_set_o ) ) ) ) ) ) ).
% 0.34/0.69
% 0.34/0.69 % prod.insert_remove
% 0.34/0.69 thf(fact_7356_prod_Oinsert__remove,axiom,
% 0.34/0.69 ! [A: set_int,G: int > assn,X: int] :
% 0.34/0.69 ( ( finite_finite_int @ A )
% 0.34/0.69 => ( ( groups7882442080178216443t_assn @ G @ ( insert_int @ X @ A ) )
% 0.34/0.69 = ( times_times_assn @ ( G @ X ) @ ( groups7882442080178216443t_assn @ G @ ( minus_minus_set_int @ A @ ( insert_int @ X @ bot_bot_set_int ) ) ) ) ) ) ).
% 0.34/0.69
% 0.34/0.69 % prod.insert_remove
% 0.34/0.69 thf(fact_7357_prod_Oinsert__remove,axiom,
% 0.34/0.69 ! [A: set_nat,G: nat > assn,X: nat] :
% 0.34/0.69 ( ( finite_finite_nat @ A )
% 0.34/0.69 => ( ( groups6906906614972039071t_assn @ G @ ( insert_nat @ X @ A ) )
% 0.34/0.69 = ( times_times_assn @ ( G @ X ) @ ( groups6906906614972039071t_assn @ G @ ( minus_minus_set_nat @ A @ ( insert_nat @ X @ bot_bot_set_nat ) ) ) ) ) ) ).
% 0.34/0.69
% 0.34/0.69 % prod.insert_remove
% 0.34/0.69 thf(fact_7358_prod_Oinsert__remove,axiom,
% 0.34/0.69 ! [A: set_Code_integer,G: code_integer > rat,X: code_integer] :
% 0.34/0.69 ( ( finite6017078050557962740nteger @ A )
% 0.34/0.69 => ( ( groups2555765274223993564er_rat @ G @ ( insert_Code_integer @ X @ A ) )
% 0.34/0.69 = ( times_times_rat @ ( G @ X ) @ ( groups2555765274223993564er_rat @ G @ ( minus_2355218937544613996nteger @ A @ ( insert_Code_integer @ X @ bot_bo3990330152332043303nteger ) ) ) ) ) ) ).
% 0.34/0.69
% 0.34/0.69 % prod.insert_remove
% 0.34/0.69 thf(fact_7359_prod_Oinsert__remove,axiom,
% 0.34/0.69 ! [A: set_Product_unit,G: product_unit > rat,X: product_unit] :
% 0.34/0.69 ( ( finite4290736615968046902t_unit @ A )
% 0.34/0.69 => ( ( groups8984663225794451456it_rat @ G @ ( insert_Product_unit @ X @ A ) )
% 0.34/0.69 = ( times_times_rat @ ( G @ X ) @ ( groups8984663225794451456it_rat @ G @ ( minus_6452836326544984404t_unit @ A @ ( insert_Product_unit @ X @ bot_bo3957492148770167129t_unit ) ) ) ) ) ) ).
% 0.34/0.69
% 0.34/0.69 % prod.insert_remove
% 0.34/0.69 thf(fact_7360_prod_Oinsert__remove,axiom,
% 0.34/0.69 ! [A: set_o,G: $o > rat,X: $o] :
% 0.34/0.69 ( ( finite_finite_o @ A )
% 0.34/0.69 => ( ( groups2869687844427037835_o_rat @ G @ ( insert_o @ X @ A ) )
% 0.34/0.69 = ( times_times_rat @ ( G @ X ) @ ( groups2869687844427037835_o_rat @ G @ ( minus_minus_set_o @ A @ ( insert_o @ X @ bot_bot_set_o ) ) ) ) ) ) ).
% 0.34/0.69
% 0.34/0.69 % prod.insert_remove
% 0.34/0.69 thf(fact_7361_prod_Oinsert__remove,axiom,
% 0.34/0.69 ! [A: set_int,G: int > rat,X: int] :
% 0.34/0.69 ( ( finite_finite_int @ A )
% 0.34/0.69 => ( ( groups1072433553688619179nt_rat @ G @ ( insert_int @ X @ A ) )
% 0.34/0.69 = ( times_times_rat @ ( G @ X ) @ ( groups1072433553688619179nt_rat @ G @ ( minus_minus_set_int @ A @ ( insert_int @ X @ bot_bot_set_int ) ) ) ) ) ) ).
% 0.34/0.69
% 0.34/0.69 % prod.insert_remove
% 0.34/0.69 thf(fact_7362_prod_Oinsert__remove,axiom,
% 0.34/0.69 ! [A: set_nat,G: nat > rat,X: nat] :
% 0.34/0.69 ( ( finite_finite_nat @ A )
% 0.34/0.69 => ( ( groups73079841787564623at_rat @ G @ ( insert_nat @ X @ A ) )
% 0.34/0.69 = ( times_times_rat @ ( G @ X ) @ ( groups73079841787564623at_rat @ G @ ( minus_minus_set_nat @ A @ ( insert_nat @ X @ bot_bot_set_nat ) ) ) ) ) ) ).
% 0.34/0.69
% 0.34/0.69 % prod.insert_remove
% 0.34/0.69 thf(fact_7363_prod_Oremove,axiom,
% 0.34/0.69 ! [A: set_Code_integer,X: code_integer,G: code_integer > assn] :
% 0.34/0.69 ( ( finite6017078050557962740nteger @ A )
% 0.34/0.69 => ( ( member_Code_integer @ X @ A )
% 0.34/0.69 => ( ( groups1304777262505850412r_assn @ G @ A )
% 0.34/0.69 = ( times_times_assn @ ( G @ X ) @ ( groups1304777262505850412r_assn @ G @ ( minus_2355218937544613996nteger @ A @ ( insert_Code_integer @ X @ bot_bo3990330152332043303nteger ) ) ) ) ) ) ) ).
% 0.34/0.69
% 0.34/0.69 % prod.remove
% 0.34/0.69 thf(fact_7364_prod_Oremove,axiom,
% 0.34/0.69 ! [A: set_Product_unit,X: product_unit,G: product_unit > assn] :
% 0.34/0.69 ( ( finite4290736615968046902t_unit @ A )
% 0.34/0.69 => ( ( member_Product_unit @ X @ A )
% 0.34/0.69 => ( ( groups5360233303959380816t_assn @ G @ A )
% 0.34/0.69 = ( times_times_assn @ ( G @ X ) @ ( groups5360233303959380816t_assn @ G @ ( minus_6452836326544984404t_unit @ A @ ( insert_Product_unit @ X @ bot_bo3957492148770167129t_unit ) ) ) ) ) ) ) ).
% 0.34/0.69
% 0.34/0.69 % prod.remove
% 0.34/0.69 thf(fact_7365_prod_Oremove,axiom,
% 0.34/0.69 ! [A: set_o,X: $o,G: $o > assn] :
% 0.34/0.69 ( ( finite_finite_o @ A )
% 0.34/0.69 => ( ( member_o @ X @ A )
% 0.34/0.69 => ( ( groups5301882518646026715o_assn @ G @ A )
% 0.34/0.69 = ( times_times_assn @ ( G @ X ) @ ( groups5301882518646026715o_assn @ G @ ( minus_minus_set_o @ A @ ( insert_o @ X @ bot_bot_set_o ) ) ) ) ) ) ) ).
% 0.34/0.69
% 0.34/0.69 % prod.remove
% 0.34/0.69 thf(fact_7366_prod_Oremove,axiom,
% 0.34/0.69 ! [A: set_int,X: int,G: int > assn] :
% 0.34/0.69 ( ( finite_finite_int @ A )
% 0.34/0.69 => ( ( member_int @ X @ A )
% 0.34/0.69 => ( ( groups7882442080178216443t_assn @ G @ A )
% 0.34/0.69 = ( times_times_assn @ ( G @ X ) @ ( groups7882442080178216443t_assn @ G @ ( minus_minus_set_int @ A @ ( insert_int @ X @ bot_bot_set_int ) ) ) ) ) ) ) ).
% 0.34/0.69
% 0.34/0.69 % prod.remove
% 0.34/0.69 thf(fact_7367_prod_Oremove,axiom,
% 0.34/0.69 ! [A: set_nat,X: nat,G: nat > assn] :
% 0.34/0.69 ( ( finite_finite_nat @ A )
% 0.34/0.69 => ( ( member_nat @ X @ A )
% 0.34/0.69 => ( ( groups6906906614972039071t_assn @ G @ A )
% 0.34/0.69 = ( times_times_assn @ ( G @ X ) @ ( groups6906906614972039071t_assn @ G @ ( minus_minus_set_nat @ A @ ( insert_nat @ X @ bot_bot_set_nat ) ) ) ) ) ) ) ).
% 0.34/0.69
% 0.34/0.69 % prod.remove
% 0.34/0.69 thf(fact_7368_prod_Oremove,axiom,
% 0.34/0.69 ! [A: set_Code_integer,X: code_integer,G: code_integer > rat] :
% 0.34/0.69 ( ( finite6017078050557962740nteger @ A )
% 0.34/0.69 => ( ( member_Code_integer @ X @ A )
% 0.34/0.69 => ( ( groups2555765274223993564er_rat @ G @ A )
% 0.34/0.69 = ( times_times_rat @ ( G @ X ) @ ( groups2555765274223993564er_rat @ G @ ( minus_2355218937544613996nteger @ A @ ( insert_Code_integer @ X @ bot_bo3990330152332043303nteger ) ) ) ) ) ) ) ).
% 0.34/0.69
% 0.34/0.69 % prod.remove
% 0.34/0.69 thf(fact_7369_prod_Oremove,axiom,
% 0.34/0.69 ! [A: set_Product_unit,X: product_unit,G: product_unit > rat] :
% 0.34/0.69 ( ( finite4290736615968046902t_unit @ A )
% 0.34/0.69 => ( ( member_Product_unit @ X @ A )
% 0.34/0.69 => ( ( groups8984663225794451456it_rat @ G @ A )
% 0.34/0.69 = ( times_times_rat @ ( G @ X ) @ ( groups8984663225794451456it_rat @ G @ ( minus_6452836326544984404t_unit @ A @ ( insert_Product_unit @ X @ bot_bo3957492148770167129t_unit ) ) ) ) ) ) ) ).
% 0.34/0.69
% 0.34/0.69 % prod.remove
% 0.34/0.69 thf(fact_7370_prod_Oremove,axiom,
% 0.34/0.69 ! [A: set_o,X: $o,G: $o > rat] :
% 0.34/0.69 ( ( finite_finite_o @ A )
% 0.34/0.69 => ( ( member_o @ X @ A )
% 0.34/0.69 => ( ( groups2869687844427037835_o_rat @ G @ A )
% 0.34/0.69 = ( times_times_rat @ ( G @ X ) @ ( groups2869687844427037835_o_rat @ G @ ( minus_minus_set_o @ A @ ( insert_o @ X @ bot_bot_set_o ) ) ) ) ) ) ) ).
% 0.34/0.69
% 0.34/0.69 % prod.remove
% 0.34/0.69 thf(fact_7371_prod_Oremove,axiom,
% 0.34/0.69 ! [A: set_int,X: int,G: int > rat] :
% 0.34/0.69 ( ( finite_finite_int @ A )
% 0.34/0.69 => ( ( member_int @ X @ A )
% 0.34/0.69 => ( ( groups1072433553688619179nt_rat @ G @ A )
% 0.34/0.69 = ( times_times_rat @ ( G @ X ) @ ( groups1072433553688619179nt_rat @ G @ ( minus_minus_set_int @ A @ ( insert_int @ X @ bot_bot_set_int ) ) ) ) ) ) ) ).
% 0.34/0.69
% 0.34/0.69 % prod.remove
% 0.34/0.69 thf(fact_7372_prod_Oremove,axiom,
% 0.34/0.69 ! [A: set_nat,X: nat,G: nat > rat] :
% 0.34/0.69 ( ( finite_finite_nat @ A )
% 0.34/0.69 => ( ( member_nat @ X @ A )
% 0.34/0.69 => ( ( groups73079841787564623at_rat @ G @ A )
% 0.34/0.69 = ( times_times_rat @ ( G @ X ) @ ( groups73079841787564623at_rat @ G @ ( minus_minus_set_nat @ A @ ( insert_nat @ X @ bot_bot_set_nat ) ) ) ) ) ) ) ).
% 0.34/0.69
% 0.34/0.69 % prod.remove
% 0.34/0.69 thf(fact_7373_prod_Ounion__inter__neutral,axiom,
% 0.34/0.69 ! [A: set_Code_integer,B4: set_Code_integer,G: code_integer > code_integer] :
% 0.34/0.69 ( ( finite6017078050557962740nteger @ A )
% 0.34/0.69 => ( ( finite6017078050557962740nteger @ B4 )
% 0.34/0.69 => ( ! [X3: code_integer] :
% 0.34/0.69 ( ( member_Code_integer @ X3 @ ( inf_in1364745209274528805nteger @ A @ B4 ) )
% 0.34/0.69 => ( ( G @ X3 )
% 0.34/0.69 = one_one_Code_integer ) )
% 0.34/0.69 => ( ( groups3674199335183972705nteger @ G @ ( sup_su848401254843788991nteger @ A @ B4 ) )
% 0.34/0.69 = ( times_3573771949741848930nteger @ ( groups3674199335183972705nteger @ G @ A ) @ ( groups3674199335183972705nteger @ G @ B4 ) ) ) ) ) ) ).
% 0.34/0.69
% 0.34/0.69 % prod.union_inter_neutral
% 0.34/0.69 thf(fact_7374_prod_Ounion__inter__neutral,axiom,
% 0.34/0.69 ! [A: set_Code_integer,B4: set_Code_integer,G: code_integer > code_natural] :
% 0.34/0.69 ( ( finite6017078050557962740nteger @ A )
% 0.34/0.69 => ( ( finite6017078050557962740nteger @ B4 )
% 0.34/0.69 => ( ! [X3: code_integer] :
% 0.34/0.69 ( ( member_Code_integer @ X3 @ ( inf_in1364745209274528805nteger @ A @ B4 ) )
% 0.34/0.69 => ( ( G @ X3 )
% 0.34/0.69 = one_one_Code_natural ) )
% 0.34/0.69 => ( ( groups2497794486940690220atural @ G @ ( sup_su848401254843788991nteger @ A @ B4 ) )
% 0.34/0.69 = ( times_2397367101498566445atural @ ( groups2497794486940690220atural @ G @ A ) @ ( groups2497794486940690220atural @ G @ B4 ) ) ) ) ) ) ).
% 0.34/0.69
% 0.34/0.69 % prod.union_inter_neutral
% 0.34/0.69 thf(fact_7375_prod_Ounion__inter__neutral,axiom,
% 0.34/0.69 ! [A: set_nat,B4: set_nat,G: nat > code_integer] :
% 0.34/0.69 ( ( finite_finite_nat @ A )
% 0.34/0.69 => ( ( finite_finite_nat @ B4 )
% 0.34/0.69 => ( ! [X3: nat] :
% 0.34/0.69 ( ( member_nat @ X3 @ ( inf_inf_set_nat @ A @ B4 ) )
% 0.34/0.69 => ( ( G @ X3 )
% 0.34/0.69 = one_one_Code_integer ) )
% 0.34/0.69 => ( ( groups3455450783089532116nteger @ G @ ( sup_sup_set_nat @ A @ B4 ) )
% 0.34/0.69 = ( times_3573771949741848930nteger @ ( groups3455450783089532116nteger @ G @ A ) @ ( groups3455450783089532116nteger @ G @ B4 ) ) ) ) ) ) ).
% 0.34/0.69
% 0.34/0.69 % prod.union_inter_neutral
% 0.34/0.69 thf(fact_7376_prod_Ounion__inter__neutral,axiom,
% 0.34/0.69 ! [A: set_nat,B4: set_nat,G: nat > code_natural] :
% 0.34/0.69 ( ( finite_finite_nat @ A )
% 0.34/0.69 => ( ( finite_finite_nat @ B4 )
% 0.34/0.69 => ( ! [X3: nat] :
% 0.34/0.69 ( ( member_nat @ X3 @ ( inf_inf_set_nat @ A @ B4 ) )
% 0.34/0.69 => ( ( G @ X3 )
% 0.34/0.69 = one_one_Code_natural ) )
% 0.34/0.69 => ( ( groups2279045934846249631atural @ G @ ( sup_sup_set_nat @ A @ B4 ) )
% 0.34/0.69 = ( times_2397367101498566445atural @ ( groups2279045934846249631atural @ G @ A ) @ ( groups2279045934846249631atural @ G @ B4 ) ) ) ) ) ) ).
% 0.34/0.69
% 0.34/0.69 % prod.union_inter_neutral
% 0.34/0.69 thf(fact_7377_prod_Ounion__inter__neutral,axiom,
% 0.34/0.69 ! [A: set_Code_integer,B4: set_Code_integer,G: code_integer > assn] :
% 0.34/0.69 ( ( finite6017078050557962740nteger @ A )
% 0.34/0.69 => ( ( finite6017078050557962740nteger @ B4 )
% 0.34/0.69 => ( ! [X3: code_integer] :
% 0.34/0.69 ( ( member_Code_integer @ X3 @ ( inf_in1364745209274528805nteger @ A @ B4 ) )
% 0.34/0.69 => ( ( G @ X3 )
% 0.34/0.69 = one_one_assn ) )
% 0.34/0.69 => ( ( groups1304777262505850412r_assn @ G @ ( sup_su848401254843788991nteger @ A @ B4 ) )
% 0.34/0.69 = ( times_times_assn @ ( groups1304777262505850412r_assn @ G @ A ) @ ( groups1304777262505850412r_assn @ G @ B4 ) ) ) ) ) ) ).
% 0.34/0.69
% 0.34/0.69 % prod.union_inter_neutral
% 0.34/0.69 thf(fact_7378_prod_Ounion__inter__neutral,axiom,
% 0.34/0.69 ! [A: set_nat,B4: set_nat,G: nat > assn] :
% 0.34/0.69 ( ( finite_finite_nat @ A )
% 0.34/0.69 => ( ( finite_finite_nat @ B4 )
% 0.34/0.69 => ( ! [X3: nat] :
% 0.34/0.69 ( ( member_nat @ X3 @ ( inf_inf_set_nat @ A @ B4 ) )
% 0.34/0.69 => ( ( G @ X3 )
% 0.34/0.69 = one_one_assn ) )
% 0.34/0.69 => ( ( groups6906906614972039071t_assn @ G @ ( sup_sup_set_nat @ A @ B4 ) )
% 0.34/0.69 = ( times_times_assn @ ( groups6906906614972039071t_assn @ G @ A ) @ ( groups6906906614972039071t_assn @ G @ B4 ) ) ) ) ) ) ).
% 0.34/0.69
% 0.34/0.69 % prod.union_inter_neutral
% 0.34/0.69 thf(fact_7379_prod_Ounion__inter__neutral,axiom,
% 0.34/0.69 ! [A: set_Code_integer,B4: set_Code_integer,G: code_integer > rat] :
% 0.34/0.69 ( ( finite6017078050557962740nteger @ A )
% 0.34/0.69 => ( ( finite6017078050557962740nteger @ B4 )
% 0.34/0.69 => ( ! [X3: code_integer] :
% 0.34/0.69 ( ( member_Code_integer @ X3 @ ( inf_in1364745209274528805nteger @ A @ B4 ) )
% 0.34/0.69 => ( ( G @ X3 )
% 0.34/0.69 = one_one_rat ) )
% 0.34/0.69 => ( ( groups2555765274223993564er_rat @ G @ ( sup_su848401254843788991nteger @ A @ B4 ) )
% 0.34/0.69 = ( times_times_rat @ ( groups2555765274223993564er_rat @ G @ A ) @ ( groups2555765274223993564er_rat @ G @ B4 ) ) ) ) ) ) ).
% 0.34/0.69
% 0.34/0.69 % prod.union_inter_neutral
% 0.34/0.69 thf(fact_7380_prod_Ounion__inter__neutral,axiom,
% 0.34/0.69 ! [A: set_nat,B4: set_nat,G: nat > rat] :
% 0.34/0.69 ( ( finite_finite_nat @ A )
% 0.34/0.69 => ( ( finite_finite_nat @ B4 )
% 0.34/0.69 => ( ! [X3: nat] :
% 0.34/0.69 ( ( member_nat @ X3 @ ( inf_inf_set_nat @ A @ B4 ) )
% 0.34/0.69 => ( ( G @ X3 )
% 0.34/0.69 = one_one_rat ) )
% 0.34/0.69 => ( ( groups73079841787564623at_rat @ G @ ( sup_sup_set_nat @ A @ B4 ) )
% 0.34/0.69 = ( times_times_rat @ ( groups73079841787564623at_rat @ G @ A ) @ ( groups73079841787564623at_rat @ G @ B4 ) ) ) ) ) ) ).
% 0.34/0.69
% 0.34/0.69 % prod.union_inter_neutral
% 0.34/0.69 thf(fact_7381_prod_Ounion__inter__neutral,axiom,
% 0.34/0.69 ! [A: set_Code_integer,B4: set_Code_integer,G: code_integer > nat] :
% 0.34/0.69 ( ( finite6017078050557962740nteger @ A )
% 0.34/0.69 => ( ( finite6017078050557962740nteger @ B4 )
% 0.34/0.69 => ( ! [X3: code_integer] :
% 0.34/0.69 ( ( member_Code_integer @ X3 @ ( inf_in1364745209274528805nteger @ A @ B4 ) )
% 0.34/0.69 => ( ( G @ X3 )
% 0.34/0.69 = one_one_nat ) )
% 0.34/0.69 => ( ( groups3190895334310489300er_nat @ G @ ( sup_su848401254843788991nteger @ A @ B4 ) )
% 0.34/0.69 = ( times_times_nat @ ( groups3190895334310489300er_nat @ G @ A ) @ ( groups3190895334310489300er_nat @ G @ B4 ) ) ) ) ) ) ).
% 0.34/0.69
% 0.34/0.69 % prod.union_inter_neutral
% 0.34/0.69 thf(fact_7382_prod_Ounion__inter__neutral,axiom,
% 0.34/0.69 ! [A: set_Code_integer,B4: set_Code_integer,G: code_integer > int] :
% 0.34/0.69 ( ( finite6017078050557962740nteger @ A )
% 0.34/0.69 => ( ( finite6017078050557962740nteger @ B4 )
% 0.34/0.69 => ( ! [X3: code_integer] :
% 0.34/0.69 ( ( member_Code_integer @ X3 @ ( inf_in1364745209274528805nteger @ A @ B4 ) )
% 0.34/0.69 => ( ( G @ X3 )
% 0.34/0.69 = one_one_int ) )
% 0.34/0.69 => ( ( groups3188404863801439024er_int @ G @ ( sup_su848401254843788991nteger @ A @ B4 ) )
% 0.34/0.69 = ( times_times_int @ ( groups3188404863801439024er_int @ G @ A ) @ ( groups3188404863801439024er_int @ G @ B4 ) ) ) ) ) ) ).
% 0.34/0.69
% 0.34/0.69 % prod.union_inter_neutral
% 0.34/0.69 thf(fact_7383_prod_Ounion__disjoint,axiom,
% 0.34/0.69 ! [A: set_Code_integer,B4: set_Code_integer,G: code_integer > assn] :
% 0.34/0.69 ( ( finite6017078050557962740nteger @ A )
% 0.34/0.69 => ( ( finite6017078050557962740nteger @ B4 )
% 0.34/0.69 => ( ( ( inf_in1364745209274528805nteger @ A @ B4 )
% 0.34/0.69 = bot_bo3990330152332043303nteger )
% 0.34/0.69 => ( ( groups1304777262505850412r_assn @ G @ ( sup_su848401254843788991nteger @ A @ B4 ) )
% 0.34/0.69 = ( times_times_assn @ ( groups1304777262505850412r_assn @ G @ A ) @ ( groups1304777262505850412r_assn @ G @ B4 ) ) ) ) ) ) ).
% 0.34/0.69
% 0.34/0.69 % prod.union_disjoint
% 0.34/0.69 thf(fact_7384_prod_Ounion__disjoint,axiom,
% 0.34/0.69 ! [A: set_Product_unit,B4: set_Product_unit,G: product_unit > assn] :
% 0.34/0.69 ( ( finite4290736615968046902t_unit @ A )
% 0.34/0.69 => ( ( finite4290736615968046902t_unit @ B4 )
% 0.34/0.69 => ( ( ( inf_in4660618365625256667t_unit @ A @ B4 )
% 0.34/0.69 = bot_bo3957492148770167129t_unit )
% 0.34/0.69 => ( ( groups5360233303959380816t_assn @ G @ ( sup_su793286257634532545t_unit @ A @ B4 ) )
% 0.34/0.69 = ( times_times_assn @ ( groups5360233303959380816t_assn @ G @ A ) @ ( groups5360233303959380816t_assn @ G @ B4 ) ) ) ) ) ) ).
% 0.34/0.69
% 0.34/0.69 % prod.union_disjoint
% 0.34/0.69 thf(fact_7385_prod_Ounion__disjoint,axiom,
% 0.34/0.69 ! [A: set_o,B4: set_o,G: $o > assn] :
% 0.34/0.69 ( ( finite_finite_o @ A )
% 0.34/0.69 => ( ( finite_finite_o @ B4 )
% 0.34/0.69 => ( ( ( inf_inf_set_o @ A @ B4 )
% 0.34/0.69 = bot_bot_set_o )
% 0.34/0.69 => ( ( groups5301882518646026715o_assn @ G @ ( sup_sup_set_o @ A @ B4 ) )
% 0.34/0.69 = ( times_times_assn @ ( groups5301882518646026715o_assn @ G @ A ) @ ( groups5301882518646026715o_assn @ G @ B4 ) ) ) ) ) ) ).
% 0.34/0.69
% 0.34/0.69 % prod.union_disjoint
% 0.34/0.69 thf(fact_7386_prod_Ounion__disjoint,axiom,
% 0.34/0.69 ! [A: set_nat,B4: set_nat,G: nat > assn] :
% 0.34/0.69 ( ( finite_finite_nat @ A )
% 0.34/0.69 => ( ( finite_finite_nat @ B4 )
% 0.34/0.69 => ( ( ( inf_inf_set_nat @ A @ B4 )
% 0.34/0.69 = bot_bot_set_nat )
% 0.34/0.69 => ( ( groups6906906614972039071t_assn @ G @ ( sup_sup_set_nat @ A @ B4 ) )
% 0.34/0.69 = ( times_times_assn @ ( groups6906906614972039071t_assn @ G @ A ) @ ( groups6906906614972039071t_assn @ G @ B4 ) ) ) ) ) ) ).
% 0.34/0.69
% 0.34/0.69 % prod.union_disjoint
% 0.34/0.69 thf(fact_7387_prod_Ounion__disjoint,axiom,
% 0.34/0.69 ! [A: set_int,B4: set_int,G: int > assn] :
% 0.34/0.69 ( ( finite_finite_int @ A )
% 0.34/0.69 => ( ( finite_finite_int @ B4 )
% 0.34/0.69 => ( ( ( inf_inf_set_int @ A @ B4 )
% 0.34/0.69 = bot_bot_set_int )
% 0.34/0.69 => ( ( groups7882442080178216443t_assn @ G @ ( sup_sup_set_int @ A @ B4 ) )
% 0.34/0.69 = ( times_times_assn @ ( groups7882442080178216443t_assn @ G @ A ) @ ( groups7882442080178216443t_assn @ G @ B4 ) ) ) ) ) ) ).
% 0.34/0.69
% 0.34/0.69 % prod.union_disjoint
% 0.34/0.69 thf(fact_7388_prod_Ounion__disjoint,axiom,
% 0.34/0.69 ! [A: set_Code_integer,B4: set_Code_integer,G: code_integer > rat] :
% 0.34/0.69 ( ( finite6017078050557962740nteger @ A )
% 0.34/0.69 => ( ( finite6017078050557962740nteger @ B4 )
% 0.34/0.69 => ( ( ( inf_in1364745209274528805nteger @ A @ B4 )
% 0.34/0.69 = bot_bo3990330152332043303nteger )
% 0.34/0.69 => ( ( groups2555765274223993564er_rat @ G @ ( sup_su848401254843788991nteger @ A @ B4 ) )
% 0.34/0.69 = ( times_times_rat @ ( groups2555765274223993564er_rat @ G @ A ) @ ( groups2555765274223993564er_rat @ G @ B4 ) ) ) ) ) ) ).
% 0.34/0.69
% 0.34/0.69 % prod.union_disjoint
% 0.34/0.69 thf(fact_7389_prod_Ounion__disjoint,axiom,
% 0.34/0.69 ! [A: set_Product_unit,B4: set_Product_unit,G: product_unit > rat] :
% 0.34/0.69 ( ( finite4290736615968046902t_unit @ A )
% 0.34/0.69 => ( ( finite4290736615968046902t_unit @ B4 )
% 0.34/0.69 => ( ( ( inf_in4660618365625256667t_unit @ A @ B4 )
% 0.34/0.69 = bot_bo3957492148770167129t_unit )
% 0.34/0.69 => ( ( groups8984663225794451456it_rat @ G @ ( sup_su793286257634532545t_unit @ A @ B4 ) )
% 0.34/0.69 = ( times_times_rat @ ( groups8984663225794451456it_rat @ G @ A ) @ ( groups8984663225794451456it_rat @ G @ B4 ) ) ) ) ) ) ).
% 0.34/0.69
% 0.34/0.69 % prod.union_disjoint
% 0.34/0.69 thf(fact_7390_prod_Ounion__disjoint,axiom,
% 0.34/0.69 ! [A: set_o,B4: set_o,G: $o > rat] :
% 0.34/0.69 ( ( finite_finite_o @ A )
% 0.34/0.69 => ( ( finite_finite_o @ B4 )
% 0.34/0.69 => ( ( ( inf_inf_set_o @ A @ B4 )
% 0.34/0.69 = bot_bot_set_o )
% 0.34/0.69 => ( ( groups2869687844427037835_o_rat @ G @ ( sup_sup_set_o @ A @ B4 ) )
% 0.34/0.69 = ( times_times_rat @ ( groups2869687844427037835_o_rat @ G @ A ) @ ( groups2869687844427037835_o_rat @ G @ B4 ) ) ) ) ) ) ).
% 0.34/0.69
% 0.34/0.69 % prod.union_disjoint
% 0.34/0.69 thf(fact_7391_prod_Ounion__disjoint,axiom,
% 0.34/0.69 ! [A: set_nat,B4: set_nat,G: nat > rat] :
% 0.34/0.69 ( ( finite_finite_nat @ A )
% 0.34/0.69 => ( ( finite_finite_nat @ B4 )
% 0.34/0.69 => ( ( ( inf_inf_set_nat @ A @ B4 )
% 0.34/0.69 = bot_bot_set_nat )
% 0.34/0.69 => ( ( groups73079841787564623at_rat @ G @ ( sup_sup_set_nat @ A @ B4 ) )
% 0.34/0.69 = ( times_times_rat @ ( groups73079841787564623at_rat @ G @ A ) @ ( groups73079841787564623at_rat @ G @ B4 ) ) ) ) ) ) ).
% 0.34/0.69
% 0.34/0.69 % prod.union_disjoint
% 0.34/0.69 thf(fact_7392_prod_Ounion__disjoint,axiom,
% 0.34/0.69 ! [A: set_int,B4: set_int,G: int > rat] :
% 0.34/0.69 ( ( finite_finite_int @ A )
% 0.34/0.69 => ( ( finite_finite_int @ B4 )
% 0.34/0.69 => ( ( ( inf_inf_set_int @ A @ B4 )
% 0.34/0.69 = bot_bot_set_int )
% 0.34/0.69 => ( ( groups1072433553688619179nt_rat @ G @ ( sup_sup_set_int @ A @ B4 ) )
% 0.34/0.69 = ( times_times_rat @ ( groups1072433553688619179nt_rat @ G @ A ) @ ( groups1072433553688619179nt_rat @ G @ B4 ) ) ) ) ) ) ).
% 0.34/0.69
% 0.34/0.69 % prod.union_disjoint
% 0.34/0.69 thf(fact_7393_prod_Ounion__diff2,axiom,
% 0.34/0.69 ! [A: set_Code_integer,B4: set_Code_integer,G: code_integer > assn] :
% 0.34/0.69 ( ( finite6017078050557962740nteger @ A )
% 0.34/0.69 => ( ( finite6017078050557962740nteger @ B4 )
% 0.34/0.69 => ( ( groups1304777262505850412r_assn @ G @ ( sup_su848401254843788991nteger @ A @ B4 ) )
% 0.34/0.69 = ( times_times_assn @ ( times_times_assn @ ( groups1304777262505850412r_assn @ G @ ( minus_2355218937544613996nteger @ A @ B4 ) ) @ ( groups1304777262505850412r_assn @ G @ ( minus_2355218937544613996nteger @ B4 @ A ) ) ) @ ( groups1304777262505850412r_assn @ G @ ( inf_in1364745209274528805nteger @ A @ B4 ) ) ) ) ) ) ).
% 0.34/0.69
% 0.34/0.69 % prod.union_diff2
% 0.34/0.69 thf(fact_7394_prod_Ounion__diff2,axiom,
% 0.34/0.69 ! [A: set_nat,B4: set_nat,G: nat > assn] :
% 0.34/0.69 ( ( finite_finite_nat @ A )
% 0.34/0.69 => ( ( finite_finite_nat @ B4 )
% 0.34/0.69 => ( ( groups6906906614972039071t_assn @ G @ ( sup_sup_set_nat @ A @ B4 ) )
% 0.34/0.69 = ( times_times_assn @ ( times_times_assn @ ( groups6906906614972039071t_assn @ G @ ( minus_minus_set_nat @ A @ B4 ) ) @ ( groups6906906614972039071t_assn @ G @ ( minus_minus_set_nat @ B4 @ A ) ) ) @ ( groups6906906614972039071t_assn @ G @ ( inf_inf_set_nat @ A @ B4 ) ) ) ) ) ) ).
% 0.34/0.69
% 0.34/0.69 % prod.union_diff2
% 0.34/0.69 thf(fact_7395_prod_Ounion__diff2,axiom,
% 0.34/0.69 ! [A: set_Code_integer,B4: set_Code_integer,G: code_integer > rat] :
% 0.34/0.69 ( ( finite6017078050557962740nteger @ A )
% 0.34/0.69 => ( ( finite6017078050557962740nteger @ B4 )
% 0.34/0.69 => ( ( groups2555765274223993564er_rat @ G @ ( sup_su848401254843788991nteger @ A @ B4 ) )
% 0.34/0.69 = ( times_times_rat @ ( times_times_rat @ ( groups2555765274223993564er_rat @ G @ ( minus_2355218937544613996nteger @ A @ B4 ) ) @ ( groups2555765274223993564er_rat @ G @ ( minus_2355218937544613996nteger @ B4 @ A ) ) ) @ ( groups2555765274223993564er_rat @ G @ ( inf_in1364745209274528805nteger @ A @ B4 ) ) ) ) ) ) ).
% 0.34/0.69
% 0.34/0.69 % prod.union_diff2
% 0.34/0.69 thf(fact_7396_prod_Ounion__diff2,axiom,
% 0.34/0.69 ! [A: set_nat,B4: set_nat,G: nat > rat] :
% 0.34/0.69 ( ( finite_finite_nat @ A )
% 0.34/0.69 => ( ( finite_finite_nat @ B4 )
% 0.34/0.69 => ( ( groups73079841787564623at_rat @ G @ ( sup_sup_set_nat @ A @ B4 ) )
% 0.34/0.69 = ( times_times_rat @ ( times_times_rat @ ( groups73079841787564623at_rat @ G @ ( minus_minus_set_nat @ A @ B4 ) ) @ ( groups73079841787564623at_rat @ G @ ( minus_minus_set_nat @ B4 @ A ) ) ) @ ( groups73079841787564623at_rat @ G @ ( inf_inf_set_nat @ A @ B4 ) ) ) ) ) ) ).
% 0.34/0.69
% 0.34/0.69 % prod.union_diff2
% 0.34/0.69 thf(fact_7397_prod_Ounion__diff2,axiom,
% 0.34/0.69 ! [A: set_Code_integer,B4: set_Code_integer,G: code_integer > nat] :
% 0.34/0.69 ( ( finite6017078050557962740nteger @ A )
% 0.34/0.69 => ( ( finite6017078050557962740nteger @ B4 )
% 0.34/0.69 => ( ( groups3190895334310489300er_nat @ G @ ( sup_su848401254843788991nteger @ A @ B4 ) )
% 0.34/0.69 = ( times_times_nat @ ( times_times_nat @ ( groups3190895334310489300er_nat @ G @ ( minus_2355218937544613996nteger @ A @ B4 ) ) @ ( groups3190895334310489300er_nat @ G @ ( minus_2355218937544613996nteger @ B4 @ A ) ) ) @ ( groups3190895334310489300er_nat @ G @ ( inf_in1364745209274528805nteger @ A @ B4 ) ) ) ) ) ) ).
% 0.34/0.69
% 0.34/0.69 % prod.union_diff2
% 0.34/0.69 thf(fact_7398_prod_Ounion__diff2,axiom,
% 0.34/0.69 ! [A: set_Code_integer,B4: set_Code_integer,G: code_integer > int] :
% 0.34/0.69 ( ( finite6017078050557962740nteger @ A )
% 0.34/0.69 => ( ( finite6017078050557962740nteger @ B4 )
% 0.34/0.69 => ( ( groups3188404863801439024er_int @ G @ ( sup_su848401254843788991nteger @ A @ B4 ) )
% 0.34/0.69 = ( times_times_int @ ( times_times_int @ ( groups3188404863801439024er_int @ G @ ( minus_2355218937544613996nteger @ A @ B4 ) ) @ ( groups3188404863801439024er_int @ G @ ( minus_2355218937544613996nteger @ B4 @ A ) ) ) @ ( groups3188404863801439024er_int @ G @ ( inf_in1364745209274528805nteger @ A @ B4 ) ) ) ) ) ) ).
% 0.34/0.69
% 0.34/0.69 % prod.union_diff2
% 0.34/0.69 thf(fact_7399_prod_Ounion__diff2,axiom,
% 0.34/0.69 ! [A: set_nat,B4: set_nat,G: nat > int] :
% 0.34/0.69 ( ( finite_finite_nat @ A )
% 0.34/0.69 => ( ( finite_finite_nat @ B4 )
% 0.34/0.69 => ( ( groups705719431365010083at_int @ G @ ( sup_sup_set_nat @ A @ B4 ) )
% 0.34/0.69 = ( times_times_int @ ( times_times_int @ ( groups705719431365010083at_int @ G @ ( minus_minus_set_nat @ A @ B4 ) ) @ ( groups705719431365010083at_int @ G @ ( minus_minus_set_nat @ B4 @ A ) ) ) @ ( groups705719431365010083at_int @ G @ ( inf_inf_set_nat @ A @ B4 ) ) ) ) ) ) ).
% 0.34/0.69
% 0.34/0.69 % prod.union_diff2
% 0.34/0.69 thf(fact_7400_prod_Ounion__diff2,axiom,
% 0.34/0.69 ! [A: set_nat,B4: set_nat,G: nat > nat] :
% 0.34/0.69 ( ( finite_finite_nat @ A )
% 0.34/0.69 => ( ( finite_finite_nat @ B4 )
% 0.34/0.69 => ( ( groups708209901874060359at_nat @ G @ ( sup_sup_set_nat @ A @ B4 ) )
% 0.34/0.69 = ( times_times_nat @ ( times_times_nat @ ( groups708209901874060359at_nat @ G @ ( minus_minus_set_nat @ A @ B4 ) ) @ ( groups708209901874060359at_nat @ G @ ( minus_minus_set_nat @ B4 @ A ) ) ) @ ( groups708209901874060359at_nat @ G @ ( inf_inf_set_nat @ A @ B4 ) ) ) ) ) ) ).
% 0.34/0.69
% 0.34/0.69 % prod.union_diff2
% 0.34/0.69 thf(fact_7401_Suc__times__gbinomial,axiom,
% 0.34/0.69 ! [K3: nat,A2: rat] :
% 0.34/0.69 ( ( times_times_rat @ ( semiri681578069525770553at_rat @ ( suc @ K3 ) ) @ ( gbinomial_rat @ ( plus_plus_rat @ A2 @ one_one_rat ) @ ( suc @ K3 ) ) )
% 0.34/0.69 = ( times_times_rat @ ( plus_plus_rat @ A2 @ one_one_rat ) @ ( gbinomial_rat @ A2 @ K3 ) ) ) ).
% 0.34/0.69
% 0.34/0.69 % Suc_times_gbinomial
% 0.34/0.69 thf(fact_7402_gbinomial__absorption,axiom,
% 0.34/0.69 ! [K3: nat,A2: rat] :
% 0.34/0.69 ( ( times_times_rat @ ( semiri681578069525770553at_rat @ ( suc @ K3 ) ) @ ( gbinomial_rat @ A2 @ ( suc @ K3 ) ) )
% 0.34/0.69 = ( times_times_rat @ A2 @ ( gbinomial_rat @ ( minus_minus_rat @ A2 @ one_one_rat ) @ K3 ) ) ) ).
% 0.34/0.69
% 0.34/0.69 % gbinomial_absorption
% 0.34/0.69 thf(fact_7403_gbinomial__trinomial__revision,axiom,
% 0.34/0.69 ! [K3: nat,M4: nat,A2: rat] :
% 0.34/0.69 ( ( ord_less_eq_nat @ K3 @ M4 )
% 0.34/0.69 => ( ( times_times_rat @ ( gbinomial_rat @ A2 @ M4 ) @ ( gbinomial_rat @ ( semiri681578069525770553at_rat @ M4 ) @ K3 ) )
% 0.34/0.69 = ( times_times_rat @ ( gbinomial_rat @ A2 @ K3 ) @ ( gbinomial_rat @ ( minus_minus_rat @ A2 @ ( semiri681578069525770553at_rat @ K3 ) ) @ ( minus_minus_nat @ M4 @ K3 ) ) ) ) ) ).
% 0.34/0.69
% 0.34/0.69 % gbinomial_trinomial_revision
% 0.34/0.69 thf(fact_7404_neg__one__power__add__eq__neg__one__power__diff,axiom,
% 0.34/0.69 ! [K3: nat,N2: nat] :
% 0.34/0.69 ( ( ord_less_eq_nat @ K3 @ N2 )
% 0.34/0.69 => ( ( power_power_int @ ( uminus_uminus_int @ one_one_int ) @ ( plus_plus_nat @ N2 @ K3 ) )
% 0.34/0.69 = ( power_power_int @ ( uminus_uminus_int @ one_one_int ) @ ( minus_minus_nat @ N2 @ K3 ) ) ) ) ).
% 0.34/0.69
% 0.34/0.69 % neg_one_power_add_eq_neg_one_power_diff
% 0.34/0.69 thf(fact_7405_neg__one__power__add__eq__neg__one__power__diff,axiom,
% 0.34/0.69 ! [K3: nat,N2: nat] :
% 0.34/0.69 ( ( ord_less_eq_nat @ K3 @ N2 )
% 0.34/0.69 => ( ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ ( plus_plus_nat @ N2 @ K3 ) )
% 0.34/0.69 = ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ ( minus_minus_nat @ N2 @ K3 ) ) ) ) ).
% 0.34/0.69
% 0.34/0.69 % neg_one_power_add_eq_neg_one_power_diff
% 0.34/0.69 thf(fact_7406_ex__less__of__nat__mult,axiom,
% 0.34/0.69 ! [X: rat,Y: rat] :
% 0.34/0.69 ( ( ord_less_rat @ zero_zero_rat @ X )
% 0.34/0.69 => ? [N3: nat] : ( ord_less_rat @ Y @ ( times_times_rat @ ( semiri681578069525770553at_rat @ N3 ) @ X ) ) ) ).
% 0.34/0.69
% 0.34/0.69 % ex_less_of_nat_mult
% 0.34/0.69 thf(fact_7407_gbinomial__absorption_H,axiom,
% 0.34/0.69 ! [K3: nat,A2: rat] :
% 0.34/0.69 ( ( ord_less_nat @ zero_zero_nat @ K3 )
% 0.34/0.69 => ( ( gbinomial_rat @ A2 @ K3 )
% 0.34/0.69 = ( times_times_rat @ ( divide_divide_rat @ A2 @ ( semiri681578069525770553at_rat @ K3 ) ) @ ( gbinomial_rat @ ( minus_minus_rat @ A2 @ one_one_rat ) @ ( minus_minus_nat @ K3 @ one_one_nat ) ) ) ) ) ).
% 0.34/0.69
% 0.34/0.69 % gbinomial_absorption'
% 0.34/0.69 thf(fact_7408_pochhammer__minus,axiom,
% 0.34/0.69 ! [B: rat,K3: nat] :
% 0.34/0.69 ( ( comm_s4028243227959126397er_rat @ ( uminus_uminus_rat @ B ) @ K3 )
% 0.34/0.69 = ( times_times_rat @ ( power_power_rat @ ( uminus_uminus_rat @ one_one_rat ) @ K3 ) @ ( comm_s4028243227959126397er_rat @ ( plus_plus_rat @ ( minus_minus_rat @ B @ ( semiri681578069525770553at_rat @ K3 ) ) @ one_one_rat ) @ K3 ) ) ) ).
% 0.34/0.69
% 0.34/0.69 % pochhammer_minus
% 0.34/0.69 thf(fact_7409_pochhammer__minus,axiom,
% 0.34/0.69 ! [B: int,K3: nat] :
% 0.34/0.69 ( ( comm_s4660882817536571857er_int @ ( uminus_uminus_int @ B ) @ K3 )
% 0.34/0.69 = ( times_times_int @ ( power_power_int @ ( uminus_uminus_int @ one_one_int ) @ K3 ) @ ( comm_s4660882817536571857er_int @ ( plus_plus_int @ ( minus_minus_int @ B @ ( semiri1314217659103216013at_int @ K3 ) ) @ one_one_int ) @ K3 ) ) ) ).
% 0.34/0.69
% 0.34/0.69 % pochhammer_minus
% 0.34/0.69 thf(fact_7410_pochhammer__minus,axiom,
% 0.34/0.69 ! [B: code_integer,K3: nat] :
% 0.34/0.69 ( ( comm_s8582702949713902594nteger @ ( uminus1351360451143612070nteger @ B ) @ K3 )
% 0.34/0.69 = ( times_3573771949741848930nteger @ ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ K3 ) @ ( comm_s8582702949713902594nteger @ ( plus_p5714425477246183910nteger @ ( minus_8373710615458151222nteger @ B @ ( semiri4939895301339042750nteger @ K3 ) ) @ one_one_Code_integer ) @ K3 ) ) ) ).
% 0.34/0.69
% 0.34/0.69 % pochhammer_minus
% 0.34/0.69 thf(fact_7411_pochhammer__minus_H,axiom,
% 0.34/0.69 ! [B: rat,K3: nat] :
% 0.34/0.69 ( ( comm_s4028243227959126397er_rat @ ( plus_plus_rat @ ( minus_minus_rat @ B @ ( semiri681578069525770553at_rat @ K3 ) ) @ one_one_rat ) @ K3 )
% 0.34/0.69 = ( times_times_rat @ ( power_power_rat @ ( uminus_uminus_rat @ one_one_rat ) @ K3 ) @ ( comm_s4028243227959126397er_rat @ ( uminus_uminus_rat @ B ) @ K3 ) ) ) ).
% 0.34/0.69
% 0.34/0.69 % pochhammer_minus'
% 0.34/0.69 thf(fact_7412_pochhammer__minus_H,axiom,
% 0.34/0.69 ! [B: int,K3: nat] :
% 0.34/0.69 ( ( comm_s4660882817536571857er_int @ ( plus_plus_int @ ( minus_minus_int @ B @ ( semiri1314217659103216013at_int @ K3 ) ) @ one_one_int ) @ K3 )
% 0.34/0.69 = ( times_times_int @ ( power_power_int @ ( uminus_uminus_int @ one_one_int ) @ K3 ) @ ( comm_s4660882817536571857er_int @ ( uminus_uminus_int @ B ) @ K3 ) ) ) ).
% 0.34/0.69
% 0.34/0.69 % pochhammer_minus'
% 0.34/0.69 thf(fact_7413_pochhammer__minus_H,axiom,
% 0.34/0.69 ! [B: code_integer,K3: nat] :
% 0.34/0.69 ( ( comm_s8582702949713902594nteger @ ( plus_p5714425477246183910nteger @ ( minus_8373710615458151222nteger @ B @ ( semiri4939895301339042750nteger @ K3 ) ) @ one_one_Code_integer ) @ K3 )
% 0.34/0.69 = ( times_3573771949741848930nteger @ ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ K3 ) @ ( comm_s8582702949713902594nteger @ ( uminus1351360451143612070nteger @ B ) @ K3 ) ) ) ).
% 0.34/0.69
% 0.34/0.69 % pochhammer_minus'
% 0.34/0.69 thf(fact_7414_zmult__zless__mono2__lemma,axiom,
% 0.34/0.69 ! [I2: int,J: int,K3: nat] :
% 0.34/0.69 ( ( ord_less_int @ I2 @ J )
% 0.34/0.69 => ( ( ord_less_nat @ zero_zero_nat @ K3 )
% 0.34/0.69 => ( ord_less_int @ ( times_times_int @ ( semiri1314217659103216013at_int @ K3 ) @ I2 ) @ ( times_times_int @ ( semiri1314217659103216013at_int @ K3 ) @ J ) ) ) ) ).
% 0.34/0.69
% 0.34/0.69 % zmult_zless_mono2_lemma
% 0.34/0.69 thf(fact_7415_nat__mult__div__cancel__disj,axiom,
% 0.34/0.69 ! [K3: nat,M4: nat,N2: nat] :
% 0.34/0.69 ( ( ( K3 = zero_zero_nat )
% 0.34/0.69 => ( ( divide_divide_nat @ ( times_times_nat @ K3 @ M4 ) @ ( times_times_nat @ K3 @ N2 ) )
% 0.34/0.69 = zero_zero_nat ) )
% 0.34/0.69 & ( ( K3 != zero_zero_nat )
% 0.34/0.69 => ( ( divide_divide_nat @ ( times_times_nat @ K3 @ M4 ) @ ( times_times_nat @ K3 @ N2 ) )
% 0.34/0.69 = ( divide_divide_nat @ M4 @ N2 ) ) ) ) ).
% 0.34/0.69
% 0.34/0.69 % nat_mult_div_cancel_disj
% 0.34/0.69 thf(fact_7416_pos__zmult__eq__1__iff__lemma,axiom,
% 0.34/0.69 ! [M4: int,N2: int] :
% 0.34/0.69 ( ( ( times_times_int @ M4 @ N2 )
% 0.34/0.69 = one_one_int )
% 0.34/0.69 => ( ( M4 = one_one_int )
% 0.34/0.69 | ( M4
% 0.34/0.69 = ( uminus_uminus_int @ one_one_int ) ) ) ) ).
% 0.34/0.69
% 0.34/0.69 % pos_zmult_eq_1_iff_lemma
% 0.34/0.69 thf(fact_7417_zmult__eq__1__iff,axiom,
% 0.34/0.69 ! [M4: int,N2: int] :
% 0.34/0.69 ( ( ( times_times_int @ M4 @ N2 )
% 0.34/0.69 = one_one_int )
% 0.34/0.69 = ( ( ( M4 = one_one_int )
% 0.34/0.69 & ( N2 = one_one_int ) )
% 0.34/0.69 | ( ( M4
% 0.34/0.69 = ( uminus_uminus_int @ one_one_int ) )
% 0.34/0.69 & ( N2
% 0.34/0.69 = ( uminus_uminus_int @ one_one_int ) ) ) ) ) ).
% 0.34/0.69
% 0.34/0.69 % zmult_eq_1_iff
% 0.34/0.69 thf(fact_7418_zdvd__mono,axiom,
% 0.34/0.69 ! [K3: int,M4: int,T4: int] :
% 0.34/0.69 ( ( K3 != zero_zero_int )
% 0.34/0.69 => ( ( dvd_dvd_int @ M4 @ T4 )
% 0.34/0.69 = ( dvd_dvd_int @ ( times_times_int @ K3 @ M4 ) @ ( times_times_int @ K3 @ T4 ) ) ) ) ).
% 0.34/0.69
% 0.34/0.69 % zdvd_mono
% 0.34/0.69 thf(fact_7419_zdvd__mult__cancel,axiom,
% 0.34/0.69 ! [K3: int,M4: int,N2: int] :
% 0.34/0.69 ( ( dvd_dvd_int @ ( times_times_int @ K3 @ M4 ) @ ( times_times_int @ K3 @ N2 ) )
% 0.34/0.69 => ( ( K3 != zero_zero_int )
% 0.34/0.69 => ( dvd_dvd_int @ M4 @ N2 ) ) ) ).
% 0.34/0.69
% 0.34/0.69 % zdvd_mult_cancel
% 0.34/0.69 thf(fact_7420_zdvd__reduce,axiom,
% 0.34/0.69 ! [K3: int,N2: int,M4: int] :
% 0.34/0.69 ( ( dvd_dvd_int @ K3 @ ( plus_plus_int @ N2 @ ( times_times_int @ K3 @ M4 ) ) )
% 0.34/0.69 = ( dvd_dvd_int @ K3 @ N2 ) ) ).
% 0.34/0.69
% 0.34/0.69 % zdvd_reduce
% 0.34/0.69 thf(fact_7421_zdvd__period,axiom,
% 0.34/0.69 ! [A2: int,D2: int,X: int,T4: int,C: int] :
% 0.34/0.69 ( ( dvd_dvd_int @ A2 @ D2 )
% 0.34/0.69 => ( ( dvd_dvd_int @ A2 @ ( plus_plus_int @ X @ T4 ) )
% 0.34/0.69 = ( dvd_dvd_int @ A2 @ ( plus_plus_int @ ( plus_plus_int @ X @ ( times_times_int @ C @ D2 ) ) @ T4 ) ) ) ) ).
% 0.34/0.69
% 0.34/0.69 % zdvd_period
% 0.34/0.69 thf(fact_7422_unique__quotient__lemma__neg,axiom,
% 0.34/0.69 ! [B: int,Q5: int,R6: int,Q4: int,R2: int] :
% 0.34/0.69 ( ( ord_less_eq_int @ ( plus_plus_int @ ( times_times_int @ B @ Q5 ) @ R6 ) @ ( plus_plus_int @ ( times_times_int @ B @ Q4 ) @ R2 ) )
% 0.34/0.69 => ( ( ord_less_eq_int @ R2 @ zero_zero_int )
% 0.34/0.69 => ( ( ord_less_int @ B @ R2 )
% 0.34/0.69 => ( ( ord_less_int @ B @ R6 )
% 0.34/0.69 => ( ord_less_eq_int @ Q4 @ Q5 ) ) ) ) ) ).
% 0.34/0.69
% 0.34/0.69 % unique_quotient_lemma_neg
% 0.34/0.69 thf(fact_7423_unique__quotient__lemma,axiom,
% 0.34/0.69 ! [B: int,Q5: int,R6: int,Q4: int,R2: int] :
% 0.34/0.69 ( ( ord_less_eq_int @ ( plus_plus_int @ ( times_times_int @ B @ Q5 ) @ R6 ) @ ( plus_plus_int @ ( times_times_int @ B @ Q4 ) @ R2 ) )
% 0.34/0.69 => ( ( ord_less_eq_int @ zero_zero_int @ R6 )
% 0.34/0.69 => ( ( ord_less_int @ R6 @ B )
% 0.34/0.69 => ( ( ord_less_int @ R2 @ B )
% 0.34/0.69 => ( ord_less_eq_int @ Q5 @ Q4 ) ) ) ) ) ).
% 0.34/0.69
% 0.34/0.69 % unique_quotient_lemma
% 0.34/0.69 thf(fact_7424_zdiv__mono2__neg__lemma,axiom,
% 0.34/0.69 ! [B: int,Q4: int,R2: int,B3: int,Q5: int,R6: int] :
% 0.34/0.69 ( ( ( plus_plus_int @ ( times_times_int @ B @ Q4 ) @ R2 )
% 0.34/0.69 = ( plus_plus_int @ ( times_times_int @ B3 @ Q5 ) @ R6 ) )
% 0.34/0.69 => ( ( ord_less_int @ ( plus_plus_int @ ( times_times_int @ B3 @ Q5 ) @ R6 ) @ zero_zero_int )
% 0.34/0.69 => ( ( ord_less_int @ R2 @ B )
% 0.34/0.69 => ( ( ord_less_eq_int @ zero_zero_int @ R6 )
% 0.34/0.69 => ( ( ord_less_int @ zero_zero_int @ B3 )
% 0.34/0.69 => ( ( ord_less_eq_int @ B3 @ B )
% 0.34/0.69 => ( ord_less_eq_int @ Q5 @ Q4 ) ) ) ) ) ) ) ).
% 0.34/0.69
% 0.34/0.69 % zdiv_mono2_neg_lemma
% 0.34/0.69 thf(fact_7425_zdiv__mono2__lemma,axiom,
% 0.34/0.69 ! [B: int,Q4: int,R2: int,B3: int,Q5: int,R6: int] :
% 0.34/0.69 ( ( ( plus_plus_int @ ( times_times_int @ B @ Q4 ) @ R2 )
% 0.34/0.69 = ( plus_plus_int @ ( times_times_int @ B3 @ Q5 ) @ R6 ) )
% 0.34/0.69 => ( ( ord_less_eq_int @ zero_zero_int @ ( plus_plus_int @ ( times_times_int @ B3 @ Q5 ) @ R6 ) )
% 0.34/0.69 => ( ( ord_less_int @ R6 @ B3 )
% 0.34/0.69 => ( ( ord_less_eq_int @ zero_zero_int @ R2 )
% 0.34/0.69 => ( ( ord_less_int @ zero_zero_int @ B3 )
% 0.34/0.69 => ( ( ord_less_eq_int @ B3 @ B )
% 0.34/0.69 => ( ord_less_eq_int @ Q4 @ Q5 ) ) ) ) ) ) ) ).
% 0.34/0.69
% 0.34/0.69 % zdiv_mono2_lemma
% 0.34/0.69 thf(fact_7426_int__div__pos__eq,axiom,
% 0.34/0.69 ! [A2: int,B: int,Q4: int,R2: int] :
% 0.34/0.69 ( ( A2
% 0.34/0.69 = ( plus_plus_int @ ( times_times_int @ B @ Q4 ) @ R2 ) )
% 0.34/0.69 => ( ( ord_less_eq_int @ zero_zero_int @ R2 )
% 0.34/0.69 => ( ( ord_less_int @ R2 @ B )
% 0.34/0.69 => ( ( divide_divide_int @ A2 @ B )
% 0.34/0.69 = Q4 ) ) ) ) ).
% 0.34/0.69
% 0.34/0.69 % int_div_pos_eq
% 0.34/0.69 thf(fact_7427_int__div__neg__eq,axiom,
% 0.34/0.69 ! [A2: int,B: int,Q4: int,R2: int] :
% 0.34/0.69 ( ( A2
% 0.34/0.69 = ( plus_plus_int @ ( times_times_int @ B @ Q4 ) @ R2 ) )
% 0.34/0.69 => ( ( ord_less_eq_int @ R2 @ zero_zero_int )
% 0.34/0.69 => ( ( ord_less_int @ B @ R2 )
% 0.34/0.69 => ( ( divide_divide_int @ A2 @ B )
% 0.34/0.69 = Q4 ) ) ) ) ).
% 0.34/0.69
% 0.34/0.69 % int_div_neg_eq
% 0.34/0.69 thf(fact_7428_q__pos__lemma,axiom,
% 0.34/0.69 ! [B3: int,Q5: int,R6: int] :
% 0.34/0.69 ( ( ord_less_eq_int @ zero_zero_int @ ( plus_plus_int @ ( times_times_int @ B3 @ Q5 ) @ R6 ) )
% 0.34/0.69 => ( ( ord_less_int @ R6 @ B3 )
% 0.34/0.69 => ( ( ord_less_int @ zero_zero_int @ B3 )
% 0.34/0.69 => ( ord_less_eq_int @ zero_zero_int @ Q5 ) ) ) ) ).
% 0.34/0.69
% 0.34/0.69 % q_pos_lemma
% 0.34/0.69 thf(fact_7429_split__zdiv,axiom,
% 0.34/0.69 ! [P: int > $o,N2: int,K3: int] :
% 0.34/0.69 ( ( P @ ( divide_divide_int @ N2 @ K3 ) )
% 0.34/0.69 = ( ( ( K3 = zero_zero_int )
% 0.34/0.69 => ( P @ zero_zero_int ) )
% 0.34/0.69 & ( ( ord_less_int @ zero_zero_int @ K3 )
% 0.34/0.69 => ! [I4: int,J3: int] :
% 0.34/0.69 ( ( ( ord_less_eq_int @ zero_zero_int @ J3 )
% 0.34/0.69 & ( ord_less_int @ J3 @ K3 )
% 0.34/0.69 & ( N2
% 0.34/0.69 = ( plus_plus_int @ ( times_times_int @ K3 @ I4 ) @ J3 ) ) )
% 0.34/0.69 => ( P @ I4 ) ) )
% 0.34/0.69 & ( ( ord_less_int @ K3 @ zero_zero_int )
% 0.34/0.69 => ! [I4: int,J3: int] :
% 0.34/0.69 ( ( ( ord_less_int @ K3 @ J3 )
% 0.34/0.69 & ( ord_less_eq_int @ J3 @ zero_zero_int )
% 0.34/0.69 & ( N2
% 0.34/0.69 = ( plus_plus_int @ ( times_times_int @ K3 @ I4 ) @ J3 ) ) )
% 0.34/0.69 => ( P @ I4 ) ) ) ) ) ).
% 0.34/0.69
% 0.34/0.69 % split_zdiv
% 0.34/0.69 thf(fact_7430_incr__mult__lemma,axiom,
% 0.34/0.69 ! [D2: int,P: int > $o,K3: int] :
% 0.34/0.69 ( ( ord_less_int @ zero_zero_int @ D2 )
% 0.34/0.69 => ( ! [X3: int] :
% 0.34/0.69 ( ( P @ X3 )
% 0.34/0.69 => ( P @ ( plus_plus_int @ X3 @ D2 ) ) )
% 0.34/0.69 => ( ( ord_less_eq_int @ zero_zero_int @ K3 )
% 0.34/0.69 => ! [X6: int] :
% 0.34/0.69 ( ( P @ X6 )
% 0.34/0.69 => ( P @ ( plus_plus_int @ X6 @ ( times_times_int @ K3 @ D2 ) ) ) ) ) ) ) ).
% 0.34/0.69
% 0.34/0.69 % incr_mult_lemma
% 0.34/0.69 thf(fact_7431_pos__zmult__eq__1__iff,axiom,
% 0.34/0.69 ! [M4: int,N2: int] :
% 0.34/0.69 ( ( ord_less_int @ zero_zero_int @ M4 )
% 0.34/0.69 => ( ( ( times_times_int @ M4 @ N2 )
% 0.34/0.69 = one_one_int )
% 0.34/0.69 = ( ( M4 = one_one_int )
% 0.34/0.69 & ( N2 = one_one_int ) ) ) ) ).
% 0.34/0.69
% 0.34/0.69 % pos_zmult_eq_1_iff
% 0.34/0.69 thf(fact_7432_zmult__zless__mono2,axiom,
% 0.34/0.69 ! [I2: int,J: int,K3: int] :
% 0.34/0.69 ( ( ord_less_int @ I2 @ J )
% 0.34/0.69 => ( ( ord_less_int @ zero_zero_int @ K3 )
% 0.34/0.69 => ( ord_less_int @ ( times_times_int @ K3 @ I2 ) @ ( times_times_int @ K3 @ J ) ) ) ) ).
% 0.34/0.69
% 0.34/0.69 % zmult_zless_mono2
% 0.34/0.69 thf(fact_7433_minusinfinity,axiom,
% 0.34/0.69 ! [D2: int,P1: int > $o,P: int > $o] :
% 0.34/0.69 ( ( ord_less_int @ zero_zero_int @ D2 )
% 0.34/0.69 => ( ! [X3: int,K: int] :
% 0.34/0.69 ( ( P1 @ X3 )
% 0.34/0.69 = ( P1 @ ( minus_minus_int @ X3 @ ( times_times_int @ K @ D2 ) ) ) )
% 0.34/0.69 => ( ? [Z6: int] :
% 0.34/0.69 ! [X3: int] :
% 0.34/0.69 ( ( ord_less_int @ X3 @ Z6 )
% 0.34/0.69 => ( ( P @ X3 )
% 0.34/0.69 = ( P1 @ X3 ) ) )
% 0.34/0.69 => ( ? [X_12: int] : ( P1 @ X_12 )
% 0.34/0.69 => ? [X_1: int] : ( P @ X_1 ) ) ) ) ) ).
% 0.34/0.69
% 0.34/0.69 % minusinfinity
% 0.34/0.69 thf(fact_7434_plusinfinity,axiom,
% 0.34/0.69 ! [D2: int,P7: int > $o,P: int > $o] :
% 0.34/0.69 ( ( ord_less_int @ zero_zero_int @ D2 )
% 0.34/0.69 => ( ! [X3: int,K: int] :
% 0.34/0.69 ( ( P7 @ X3 )
% 0.34/0.69 = ( P7 @ ( minus_minus_int @ X3 @ ( times_times_int @ K @ D2 ) ) ) )
% 0.34/0.69 => ( ? [Z6: int] :
% 0.34/0.69 ! [X3: int] :
% 0.34/0.69 ( ( ord_less_int @ Z6 @ X3 )
% 0.34/0.69 => ( ( P @ X3 )
% 0.34/0.69 = ( P7 @ X3 ) ) )
% 0.34/0.69 => ( ? [X_12: int] : ( P7 @ X_12 )
% 0.34/0.69 => ? [X_1: int] : ( P @ X_1 ) ) ) ) ) ).
% 0.34/0.69
% 0.34/0.69 % plusinfinity
% 0.34/0.69 thf(fact_7435_decr__mult__lemma,axiom,
% 0.34/0.69 ! [D2: int,P: int > $o,K3: int] :
% 0.34/0.69 ( ( ord_less_int @ zero_zero_int @ D2 )
% 0.34/0.69 => ( ! [X3: int] :
% 0.34/0.69 ( ( P @ X3 )
% 0.34/0.69 => ( P @ ( minus_minus_int @ X3 @ D2 ) ) )
% 0.34/0.69 => ( ( ord_less_eq_int @ zero_zero_int @ K3 )
% 0.34/0.69 => ! [X6: int] :
% 0.34/0.69 ( ( P @ X6 )
% 0.34/0.69 => ( P @ ( minus_minus_int @ X6 @ ( times_times_int @ K3 @ D2 ) ) ) ) ) ) ) ).
% 0.34/0.69
% 0.34/0.69 % decr_mult_lemma
% 0.34/0.69 thf(fact_7436_times__int__code_I2_J,axiom,
% 0.34/0.69 ! [L: int] :
% 0.34/0.69 ( ( times_times_int @ zero_zero_int @ L )
% 0.34/0.69 = zero_zero_int ) ).
% 0.34/0.69
% 0.34/0.69 % times_int_code(2)
% 0.34/0.69 thf(fact_7437_times__int__code_I1_J,axiom,
% 0.34/0.69 ! [K3: int] :
% 0.34/0.69 ( ( times_times_int @ K3 @ zero_zero_int )
% 0.34/0.69 = zero_zero_int ) ).
% 0.34/0.69
% 0.34/0.69 % times_int_code(1)
% 0.34/0.69 thf(fact_7438_int__distrib_I1_J,axiom,
% 0.34/0.69 ! [Z1: int,Z22: int,W: int] :
% 0.34/0.69 ( ( times_times_int @ ( plus_plus_int @ Z1 @ Z22 ) @ W )
% 0.34/0.69 = ( plus_plus_int @ ( times_times_int @ Z1 @ W ) @ ( times_times_int @ Z22 @ W ) ) ) ).
% 0.34/0.69
% 0.34/0.69 % int_distrib(1)
% 0.34/0.69 thf(fact_7439_int__distrib_I2_J,axiom,
% 0.34/0.69 ! [W: int,Z1: int,Z22: int] :
% 0.34/0.69 ( ( times_times_int @ W @ ( plus_plus_int @ Z1 @ Z22 ) )
% 0.34/0.69 = ( plus_plus_int @ ( times_times_int @ W @ Z1 ) @ ( times_times_int @ W @ Z22 ) ) ) ).
% 0.34/0.69
% 0.34/0.69 % int_distrib(2)
% 0.34/0.69 thf(fact_7440_int__distrib_I4_J,axiom,
% 0.34/0.69 ! [W: int,Z1: int,Z22: int] :
% 0.34/0.69 ( ( times_times_int @ W @ ( minus_minus_int @ Z1 @ Z22 ) )
% 0.34/0.69 = ( minus_minus_int @ ( times_times_int @ W @ Z1 ) @ ( times_times_int @ W @ Z22 ) ) ) ).
% 0.34/0.69
% 0.34/0.69 % int_distrib(4)
% 0.34/0.69 thf(fact_7441_int__distrib_I3_J,axiom,
% 0.34/0.69 ! [Z1: int,Z22: int,W: int] :
% 0.34/0.69 ( ( times_times_int @ ( minus_minus_int @ Z1 @ Z22 ) @ W )
% 0.34/0.69 = ( minus_minus_int @ ( times_times_int @ Z1 @ W ) @ ( times_times_int @ Z22 @ W ) ) ) ).
% 0.34/0.69
% 0.34/0.69 % int_distrib(3)
% 0.34/0.69 thf(fact_7442_nat__mult__div__cancel1,axiom,
% 0.34/0.69 ! [K3: nat,M4: nat,N2: nat] :
% 0.34/0.69 ( ( ord_less_nat @ zero_zero_nat @ K3 )
% 0.34/0.69 => ( ( divide_divide_nat @ ( times_times_nat @ K3 @ M4 ) @ ( times_times_nat @ K3 @ N2 ) )
% 0.34/0.69 = ( divide_divide_nat @ M4 @ N2 ) ) ) ).
% 0.34/0.69
% 0.34/0.69 % nat_mult_div_cancel1
% 0.34/0.69 thf(fact_7443_div__mult__self__is__m,axiom,
% 0.34/0.69 ! [N2: nat,M4: nat] :
% 0.34/0.69 ( ( ord_less_nat @ zero_zero_nat @ N2 )
% 0.34/0.69 => ( ( divide_divide_nat @ ( times_times_nat @ M4 @ N2 ) @ N2 )
% 0.34/0.69 = M4 ) ) ).
% 0.34/0.69
% 0.34/0.69 % div_mult_self_is_m
% 0.34/0.69 thf(fact_7444_div__mult__self1__is__m,axiom,
% 0.34/0.69 ! [N2: nat,M4: nat] :
% 0.34/0.69 ( ( ord_less_nat @ zero_zero_nat @ N2 )
% 0.34/0.69 => ( ( divide_divide_nat @ ( times_times_nat @ N2 @ M4 ) @ N2 )
% 0.34/0.69 = M4 ) ) ).
% 0.34/0.69
% 0.34/0.69 % div_mult_self1_is_m
% 0.34/0.69 thf(fact_7445_div__mult2__eq,axiom,
% 0.34/0.69 ! [M4: nat,N2: nat,Q4: nat] :
% 0.34/0.69 ( ( divide_divide_nat @ M4 @ ( times_times_nat @ N2 @ Q4 ) )
% 0.34/0.69 = ( divide_divide_nat @ ( divide_divide_nat @ M4 @ N2 ) @ Q4 ) ) ).
% 0.34/0.69
% 0.34/0.69 % div_mult2_eq
% 0.34/0.69 thf(fact_7446_zdiv__zmult2__eq,axiom,
% 0.34/0.69 ! [C: int,A2: int,B: int] :
% 0.34/0.69 ( ( ord_less_eq_int @ zero_zero_int @ C )
% 0.34/0.69 => ( ( divide_divide_int @ A2 @ ( times_times_int @ B @ C ) )
% 0.34/0.69 = ( divide_divide_int @ ( divide_divide_int @ A2 @ B ) @ C ) ) ) ).
% 0.34/0.69
% 0.34/0.69 % zdiv_zmult2_eq
% 0.34/0.69 thf(fact_7447_less__mult__imp__div__less,axiom,
% 0.34/0.69 ! [M4: nat,I2: nat,N2: nat] :
% 0.34/0.69 ( ( ord_less_nat @ M4 @ ( times_times_nat @ I2 @ N2 ) )
% 0.34/0.69 => ( ord_less_nat @ ( divide_divide_nat @ M4 @ N2 ) @ I2 ) ) ).
% 0.34/0.69
% 0.34/0.69 % less_mult_imp_div_less
% 0.34/0.69 thf(fact_7448_div__times__less__eq__dividend,axiom,
% 0.34/0.69 ! [M4: nat,N2: nat] : ( ord_less_eq_nat @ ( times_times_nat @ ( divide_divide_nat @ M4 @ N2 ) @ N2 ) @ M4 ) ).
% 0.34/0.69
% 0.34/0.69 % div_times_less_eq_dividend
% 0.34/0.69 thf(fact_7449_times__div__less__eq__dividend,axiom,
% 0.34/0.69 ! [N2: nat,M4: nat] : ( ord_less_eq_nat @ ( times_times_nat @ N2 @ ( divide_divide_nat @ M4 @ N2 ) ) @ M4 ) ).
% 0.34/0.69
% 0.34/0.69 % times_div_less_eq_dividend
% 0.34/0.69 thf(fact_7450_div__less__iff__less__mult,axiom,
% 0.34/0.69 ! [Q4: nat,M4: nat,N2: nat] :
% 0.34/0.69 ( ( ord_less_nat @ zero_zero_nat @ Q4 )
% 0.34/0.69 => ( ( ord_less_nat @ ( divide_divide_nat @ M4 @ Q4 ) @ N2 )
% 0.34/0.69 = ( ord_less_nat @ M4 @ ( times_times_nat @ N2 @ Q4 ) ) ) ) ).
% 0.34/0.69
% 0.34/0.69 % div_less_iff_less_mult
% 0.34/0.69 thf(fact_7451_less__eq__div__iff__mult__less__eq,axiom,
% 0.34/0.69 ! [Q4: nat,M4: nat,N2: nat] :
% 0.34/0.69 ( ( ord_less_nat @ zero_zero_nat @ Q4 )
% 0.34/0.69 => ( ( ord_less_eq_nat @ M4 @ ( divide_divide_nat @ N2 @ Q4 ) )
% 0.34/0.69 = ( ord_less_eq_nat @ ( times_times_nat @ M4 @ Q4 ) @ N2 ) ) ) ).
% 0.34/0.69
% 0.34/0.69 % less_eq_div_iff_mult_less_eq
% 0.34/0.69 thf(fact_7452_div__nat__eqI,axiom,
% 0.34/0.69 ! [N2: nat,Q4: nat,M4: nat] :
% 0.34/0.69 ( ( ord_less_eq_nat @ ( times_times_nat @ N2 @ Q4 ) @ M4 )
% 0.34/0.69 => ( ( ord_less_nat @ M4 @ ( times_times_nat @ N2 @ ( suc @ Q4 ) ) )
% 0.34/0.69 => ( ( divide_divide_nat @ M4 @ N2 )
% 0.34/0.69 = Q4 ) ) ) ).
% 0.34/0.69
% 0.34/0.69 % div_nat_eqI
% 0.34/0.69 thf(fact_7453_dividend__less__times__div,axiom,
% 0.34/0.69 ! [N2: nat,M4: nat] :
% 0.34/0.69 ( ( ord_less_nat @ zero_zero_nat @ N2 )
% 0.34/0.69 => ( ord_less_nat @ M4 @ ( plus_plus_nat @ N2 @ ( times_times_nat @ N2 @ ( divide_divide_nat @ M4 @ N2 ) ) ) ) ) ).
% 0.34/0.69
% 0.34/0.69 % dividend_less_times_div
% 0.34/0.69 thf(fact_7454_dividend__less__div__times,axiom,
% 0.34/0.69 ! [N2: nat,M4: nat] :
% 0.34/0.69 ( ( ord_less_nat @ zero_zero_nat @ N2 )
% 0.34/0.69 => ( ord_less_nat @ M4 @ ( plus_plus_nat @ N2 @ ( times_times_nat @ ( divide_divide_nat @ M4 @ N2 ) @ N2 ) ) ) ) ).
% 0.34/0.69
% 0.34/0.69 % dividend_less_div_times
% 0.34/0.69 thf(fact_7455_split__div,axiom,
% 0.34/0.69 ! [P: nat > $o,M4: nat,N2: nat] :
% 0.34/0.69 ( ( P @ ( divide_divide_nat @ M4 @ N2 ) )
% 0.34/0.69 = ( ( ( N2 = zero_zero_nat )
% 0.34/0.69 => ( P @ zero_zero_nat ) )
% 0.34/0.69 & ( ( N2 != zero_zero_nat )
% 0.34/0.69 => ! [I4: nat,J3: nat] :
% 0.34/0.69 ( ( ord_less_nat @ J3 @ N2 )
% 0.34/0.69 => ( ( M4
% 0.34/0.69 = ( plus_plus_nat @ ( times_times_nat @ N2 @ I4 ) @ J3 ) )
% 0.34/0.69 => ( P @ I4 ) ) ) ) ) ) ).
% 0.34/0.69
% 0.34/0.69 % split_div
% 0.34/0.69 thf(fact_7456_split__div_H,axiom,
% 0.34/0.69 ! [P: nat > $o,M4: nat,N2: nat] :
% 0.34/0.69 ( ( P @ ( divide_divide_nat @ M4 @ N2 ) )
% 0.34/0.69 = ( ( ( N2 = zero_zero_nat )
% 0.34/0.69 & ( P @ zero_zero_nat ) )
% 0.34/0.69 | ? [Q6: nat] :
% 0.34/0.69 ( ( ord_less_eq_nat @ ( times_times_nat @ N2 @ Q6 ) @ M4 )
% 0.34/0.69 & ( ord_less_nat @ M4 @ ( times_times_nat @ N2 @ ( suc @ Q6 ) ) )
% 0.34/0.69 & ( P @ Q6 ) ) ) ) ).
% 0.34/0.69
% 0.34/0.69 % split_div'
% 0.34/0.69 thf(fact_7457_int__ops_I7_J,axiom,
% 0.34/0.69 ! [A2: nat,B: nat] :
% 0.34/0.69 ( ( semiri1314217659103216013at_int @ ( times_times_nat @ A2 @ B ) )
% 0.34/0.69 = ( times_times_int @ ( semiri1314217659103216013at_int @ A2 ) @ ( semiri1314217659103216013at_int @ B ) ) ) ).
% 0.34/0.69
% 0.34/0.69 % int_ops(7)
% 0.34/0.69 thf(fact_7458_eucl__rel__int__iff,axiom,
% 0.34/0.69 ! [K3: int,L: int,Q4: int,R2: int] :
% 0.34/0.69 ( ( eucl_rel_int @ K3 @ L @ ( product_Pair_int_int @ Q4 @ R2 ) )
% 0.34/0.69 = ( ( K3
% 0.34/0.69 = ( plus_plus_int @ ( times_times_int @ L @ Q4 ) @ R2 ) )
% 0.34/0.69 & ( ( ord_less_int @ zero_zero_int @ L )
% 0.34/0.69 => ( ( ord_less_eq_int @ zero_zero_int @ R2 )
% 0.34/0.69 & ( ord_less_int @ R2 @ L ) ) )
% 0.34/0.69 & ( ~ ( ord_less_int @ zero_zero_int @ L )
% 0.34/0.69 => ( ( ( ord_less_int @ L @ zero_zero_int )
% 0.34/0.69 => ( ( ord_less_int @ L @ R2 )
% 0.34/0.69 & ( ord_less_eq_int @ R2 @ zero_zero_int ) ) )
% 0.34/0.69 & ( ~ ( ord_less_int @ L @ zero_zero_int )
% 0.34/0.69 => ( Q4 = zero_zero_int ) ) ) ) ) ) ).
% 0.34/0.69
% 0.34/0.69 % eucl_rel_int_iff
% 0.34/0.69 thf(fact_7459_eucl__rel__int__dividesI,axiom,
% 0.34/0.69 ! [L: int,K3: int,Q4: int] :
% 0.34/0.69 ( ( L != zero_zero_int )
% 0.34/0.69 => ( ( K3
% 0.34/0.69 = ( times_times_int @ Q4 @ L ) )
% 0.34/0.69 => ( eucl_rel_int @ K3 @ L @ ( product_Pair_int_int @ Q4 @ zero_zero_int ) ) ) ) ).
% 0.34/0.69
% 0.34/0.69 % eucl_rel_int_dividesI
% 0.34/0.69 thf(fact_7460_split__pos__lemma,axiom,
% 0.34/0.69 ! [K3: int,P: int > int > $o,N2: int] :
% 0.34/0.69 ( ( ord_less_int @ zero_zero_int @ K3 )
% 0.34/0.69 => ( ( P @ ( divide_divide_int @ N2 @ K3 ) @ ( modulo_modulo_int @ N2 @ K3 ) )
% 0.34/0.69 = ( ! [I4: int,J3: int] :
% 0.34/0.69 ( ( ( ord_less_eq_int @ zero_zero_int @ J3 )
% 0.34/0.69 & ( ord_less_int @ J3 @ K3 )
% 0.34/0.69 & ( N2
% 0.34/0.69 = ( plus_plus_int @ ( times_times_int @ K3 @ I4 ) @ J3 ) ) )
% 0.34/0.69 => ( P @ I4 @ J3 ) ) ) ) ) ).
% 0.34/0.69
% 0.34/0.69 % split_pos_lemma
% 0.34/0.69 thf(fact_7461_split__neg__lemma,axiom,
% 0.34/0.69 ! [K3: int,P: int > int > $o,N2: int] :
% 0.34/0.69 ( ( ord_less_int @ K3 @ zero_zero_int )
% 0.34/0.69 => ( ( P @ ( divide_divide_int @ N2 @ K3 ) @ ( modulo_modulo_int @ N2 @ K3 ) )
% 0.34/0.69 = ( ! [I4: int,J3: int] :
% 0.34/0.69 ( ( ( ord_less_int @ K3 @ J3 )
% 0.34/0.69 & ( ord_less_eq_int @ J3 @ zero_zero_int )
% 0.34/0.69 & ( N2
% 0.34/0.69 = ( plus_plus_int @ ( times_times_int @ K3 @ I4 ) @ J3 ) ) )
% 0.34/0.69 => ( P @ I4 @ J3 ) ) ) ) ) ).
% 0.34/0.69
% 0.34/0.69 % split_neg_lemma
% 0.34/0.69 thf(fact_7462_decr__lemma,axiom,
% 0.34/0.69 ! [D2: int,X: int,Z: int] :
% 0.34/0.69 ( ( ord_less_int @ zero_zero_int @ D2 )
% 0.34/0.69 => ( ord_less_int @ ( minus_minus_int @ X @ ( times_times_int @ ( plus_plus_int @ ( abs_abs_int @ ( minus_minus_int @ X @ Z ) ) @ one_one_int ) @ D2 ) ) @ Z ) ) ).
% 0.34/0.69
% 0.34/0.69 % decr_lemma
% 0.34/0.69 thf(fact_7463_incr__lemma,axiom,
% 0.34/0.69 ! [D2: int,Z: int,X: int] :
% 0.34/0.69 ( ( ord_less_int @ zero_zero_int @ D2 )
% 0.34/0.69 => ( ord_less_int @ Z @ ( plus_plus_int @ X @ ( times_times_int @ ( plus_plus_int @ ( abs_abs_int @ ( minus_minus_int @ X @ Z ) ) @ one_one_int ) @ D2 ) ) ) ) ).
% 0.34/0.69
% 0.34/0.69 % incr_lemma
% 0.34/0.69 thf(fact_7464_zmod__eq__0__iff,axiom,
% 0.34/0.69 ! [M4: int,D2: int] :
% 0.34/0.69 ( ( ( modulo_modulo_int @ M4 @ D2 )
% 0.34/0.69 = zero_zero_int )
% 0.34/0.69 = ( ? [Q6: int] :
% 0.34/0.69 ( M4
% 0.34/0.69 = ( times_times_int @ D2 @ Q6 ) ) ) ) ).
% 0.34/0.69
% 0.34/0.69 % zmod_eq_0_iff
% 0.34/0.69 thf(fact_7465_zmod__eq__0D,axiom,
% 0.34/0.69 ! [M4: int,D2: int] :
% 0.34/0.69 ( ( ( modulo_modulo_int @ M4 @ D2 )
% 0.34/0.69 = zero_zero_int )
% 0.34/0.69 => ? [Q7: int] :
% 0.34/0.69 ( M4
% 0.34/0.69 = ( times_times_int @ D2 @ Q7 ) ) ) ).
% 0.34/0.69
% 0.34/0.69 % zmod_eq_0D
% 0.34/0.69 thf(fact_7466_abs__zmult__eq__1,axiom,
% 0.34/0.69 ! [M4: int,N2: int] :
% 0.34/0.69 ( ( ( abs_abs_int @ ( times_times_int @ M4 @ N2 ) )
% 0.34/0.69 = one_one_int )
% 0.34/0.69 => ( ( abs_abs_int @ M4 )
% 0.34/0.69 = one_one_int ) ) ).
% 0.34/0.69
% 0.34/0.69 % abs_zmult_eq_1
% 0.34/0.69 thf(fact_7467_div__mod__decomp__int,axiom,
% 0.34/0.69 ! [A: int,N2: int] :
% 0.34/0.69 ( A
% 0.34/0.69 = ( plus_plus_int @ ( times_times_int @ ( divide_divide_int @ A @ N2 ) @ N2 ) @ ( modulo_modulo_int @ A @ N2 ) ) ) ).
% 0.34/0.69
% 0.34/0.69 % div_mod_decomp_int
% 0.34/0.69 thf(fact_7468_zdvd__mult__cancel1,axiom,
% 0.34/0.69 ! [M4: int,N2: int] :
% 0.34/0.69 ( ( M4 != zero_zero_int )
% 0.34/0.69 => ( ( dvd_dvd_int @ ( times_times_int @ M4 @ N2 ) @ M4 )
% 0.34/0.69 = ( ( abs_abs_int @ N2 )
% 0.34/0.69 = one_one_int ) ) ) ).
% 0.34/0.69
% 0.34/0.69 % zdvd_mult_cancel1
% 0.34/0.69 thf(fact_7469_split__zmod,axiom,
% 0.34/0.69 ! [P: int > $o,N2: int,K3: int] :
% 0.34/0.69 ( ( P @ ( modulo_modulo_int @ N2 @ K3 ) )
% 0.34/0.69 = ( ( ( K3 = zero_zero_int )
% 0.34/0.69 => ( P @ N2 ) )
% 0.34/0.69 & ( ( ord_less_int @ zero_zero_int @ K3 )
% 0.34/0.69 => ! [I4: int,J3: int] :
% 0.34/0.69 ( ( ( ord_less_eq_int @ zero_zero_int @ J3 )
% 0.34/0.69 & ( ord_less_int @ J3 @ K3 )
% 0.34/0.69 & ( N2
% 0.34/0.69 = ( plus_plus_int @ ( times_times_int @ K3 @ I4 ) @ J3 ) ) )
% 0.34/0.69 => ( P @ J3 ) ) )
% 0.34/0.69 & ( ( ord_less_int @ K3 @ zero_zero_int )
% 0.34/0.69 => ! [I4: int,J3: int] :
% 0.34/0.69 ( ( ( ord_less_int @ K3 @ J3 )
% 0.34/0.69 & ( ord_less_eq_int @ J3 @ zero_zero_int )
% 0.34/0.69 & ( N2
% 0.34/0.69 = ( plus_plus_int @ ( times_times_int @ K3 @ I4 ) @ J3 ) ) )
% 0.34/0.69 => ( P @ J3 ) ) ) ) ) ).
% 0.34/0.69
% 0.34/0.69 % split_zmod
% 0.34/0.69 thf(fact_7470_int__mod__neg__eq,axiom,
% 0.34/0.69 ! [A2: int,B: int,Q4: int,R2: int] :
% 0.34/0.69 ( ( A2
% 0.34/0.69 = ( plus_plus_int @ ( times_times_int @ B @ Q4 ) @ R2 ) )
% 0.34/0.69 => ( ( ord_less_eq_int @ R2 @ zero_zero_int )
% 0.34/0.69 => ( ( ord_less_int @ B @ R2 )
% 0.34/0.69 => ( ( modulo_modulo_int @ A2 @ B )
% 0.34/0.69 = R2 ) ) ) ) ).
% 0.34/0.69
% 0.34/0.69 % int_mod_neg_eq
% 0.34/0.69 thf(fact_7471_int__mod__pos__eq,axiom,
% 0.34/0.69 ! [A2: int,B: int,Q4: int,R2: int] :
% 0.34/0.69 ( ( A2
% 0.34/0.69 = ( plus_plus_int @ ( times_times_int @ B @ Q4 ) @ R2 ) )
% 0.34/0.69 => ( ( ord_less_eq_int @ zero_zero_int @ R2 )
% 0.34/0.69 => ( ( ord_less_int @ R2 @ B )
% 0.34/0.69 => ( ( modulo_modulo_int @ A2 @ B )
% 0.34/0.69 = R2 ) ) ) ) ).
% 0.34/0.69
% 0.34/0.69 % int_mod_pos_eq
% 0.34/0.69 thf(fact_7472_zmod__zmult2__eq,axiom,
% 0.34/0.69 ! [C: int,A2: int,B: int] :
% 0.34/0.69 ( ( ord_less_eq_int @ zero_zero_int @ C )
% 0.34/0.69 => ( ( modulo_modulo_int @ A2 @ ( times_times_int @ B @ C ) )
% 0.34/0.69 = ( plus_plus_int @ ( times_times_int @ B @ ( modulo_modulo_int @ ( divide_divide_int @ A2 @ B ) @ C ) ) @ ( modulo_modulo_int @ A2 @ B ) ) ) ) ).
% 0.34/0.69
% 0.34/0.69 % zmod_zmult2_eq
% 0.34/0.69 thf(fact_7473_eucl__rel__int_Osimps,axiom,
% 0.34/0.69 ( eucl_rel_int
% 0.34/0.69 = ( ^ [A13: int,A24: int,A32: product_prod_int_int] :
% 0.34/0.69 ( ? [K5: int] :
% 0.34/0.69 ( ( A13 = K5 )
% 0.34/0.69 & ( A24 = zero_zero_int )
% 0.34/0.69 & ( A32
% 0.34/0.69 = ( product_Pair_int_int @ zero_zero_int @ K5 ) ) )
% 0.34/0.69 | ? [L2: int,K5: int,Q6: int] :
% 0.34/0.69 ( ( A13 = K5 )
% 0.34/0.69 & ( A24 = L2 )
% 0.34/0.69 & ( A32
% 0.34/0.69 = ( product_Pair_int_int @ Q6 @ zero_zero_int ) )
% 0.34/0.69 & ( L2 != zero_zero_int )
% 0.34/0.69 & ( K5
% 0.34/0.69 = ( times_times_int @ Q6 @ L2 ) ) )
% 0.34/0.69 | ? [R4: int,L2: int,K5: int,Q6: int] :
% 0.34/0.69 ( ( A13 = K5 )
% 0.34/0.69 & ( A24 = L2 )
% 0.34/0.69 & ( A32
% 0.34/0.69 = ( product_Pair_int_int @ Q6 @ R4 ) )
% 0.34/0.69 & ( ( sgn_sgn_int @ R4 )
% 0.34/0.69 = ( sgn_sgn_int @ L2 ) )
% 0.34/0.69 & ( ord_less_int @ ( abs_abs_int @ R4 ) @ ( abs_abs_int @ L2 ) )
% 0.34/0.69 & ( K5
% 0.34/0.69 = ( plus_plus_int @ ( times_times_int @ Q6 @ L2 ) @ R4 ) ) ) ) ) ) ).
% 0.34/0.69
% 0.34/0.69 % eucl_rel_int.simps
% 0.34/0.69 thf(fact_7474_Suc__mod__mult__self4,axiom,
% 0.34/0.69 ! [N2: nat,K3: nat,M4: nat] :
% 0.34/0.69 ( ( modulo_modulo_nat @ ( suc @ ( plus_plus_nat @ ( times_times_nat @ N2 @ K3 ) @ M4 ) ) @ N2 )
% 0.34/0.69 = ( modulo_modulo_nat @ ( suc @ M4 ) @ N2 ) ) ).
% 0.34/0.69
% 0.34/0.69 % Suc_mod_mult_self4
% 0.34/0.69 thf(fact_7475_Suc__mod__mult__self3,axiom,
% 0.34/0.69 ! [K3: nat,N2: nat,M4: nat] :
% 0.34/0.69 ( ( modulo_modulo_nat @ ( suc @ ( plus_plus_nat @ ( times_times_nat @ K3 @ N2 ) @ M4 ) ) @ N2 )
% 0.34/0.69 = ( modulo_modulo_nat @ ( suc @ M4 ) @ N2 ) ) ).
% 0.34/0.69
% 0.34/0.69 % Suc_mod_mult_self3
% 0.34/0.69 thf(fact_7476_Suc__mod__mult__self2,axiom,
% 0.34/0.69 ! [M4: nat,N2: nat,K3: nat] :
% 0.34/0.69 ( ( modulo_modulo_nat @ ( suc @ ( plus_plus_nat @ M4 @ ( times_times_nat @ N2 @ K3 ) ) ) @ N2 )
% 0.34/0.69 = ( modulo_modulo_nat @ ( suc @ M4 ) @ N2 ) ) ).
% 0.34/0.69
% 0.34/0.69 % Suc_mod_mult_self2
% 0.34/0.69 thf(fact_7477_Suc__mod__mult__self1,axiom,
% 0.34/0.69 ! [M4: nat,K3: nat,N2: nat] :
% 0.34/0.69 ( ( modulo_modulo_nat @ ( suc @ ( plus_plus_nat @ M4 @ ( times_times_nat @ K3 @ N2 ) ) ) @ N2 )
% 0.34/0.69 = ( modulo_modulo_nat @ ( suc @ M4 ) @ N2 ) ) ).
% 0.34/0.69
% 0.34/0.69 % Suc_mod_mult_self1
% 0.34/0.69 thf(fact_7478_Suc__times__numeral__mod__eq,axiom,
% 0.34/0.69 ! [K3: num,N2: nat] :
% 0.34/0.69 ( ( ( numeral_numeral_nat @ K3 )
% 0.34/0.69 != one_one_nat )
% 0.34/0.69 => ( ( modulo_modulo_nat @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ K3 ) @ N2 ) ) @ ( numeral_numeral_nat @ K3 ) )
% 0.34/0.69 = one_one_nat ) ) ).
% 0.34/0.69
% 0.34/0.69 % Suc_times_numeral_mod_eq
% 0.34/0.69 thf(fact_7479_dvd__mult__sgn__iff,axiom,
% 0.34/0.69 ! [L: int,K3: int,R2: int] :
% 0.34/0.69 ( ( dvd_dvd_int @ L @ ( times_times_int @ K3 @ ( sgn_sgn_int @ R2 ) ) )
% 0.34/0.69 = ( ( dvd_dvd_int @ L @ K3 )
% 0.34/0.69 | ( R2 = zero_zero_int ) ) ) ).
% 0.34/0.69
% 0.34/0.69 % dvd_mult_sgn_iff
% 0.34/0.69 thf(fact_7480_dvd__sgn__mult__iff,axiom,
% 0.34/0.69 ! [L: int,R2: int,K3: int] :
% 0.34/0.69 ( ( dvd_dvd_int @ L @ ( times_times_int @ ( sgn_sgn_int @ R2 ) @ K3 ) )
% 0.34/0.69 = ( ( dvd_dvd_int @ L @ K3 )
% 0.34/0.69 | ( R2 = zero_zero_int ) ) ) ).
% 0.34/0.69
% 0.34/0.69 % dvd_sgn_mult_iff
% 0.34/0.69 thf(fact_7481_mult__sgn__dvd__iff,axiom,
% 0.34/0.69 ! [L: int,R2: int,K3: int] :
% 0.34/0.69 ( ( dvd_dvd_int @ ( times_times_int @ L @ ( sgn_sgn_int @ R2 ) ) @ K3 )
% 0.34/0.69 = ( ( dvd_dvd_int @ L @ K3 )
% 0.34/0.69 & ( ( R2 = zero_zero_int )
% 0.34/0.69 => ( K3 = zero_zero_int ) ) ) ) ).
% 0.34/0.69
% 0.34/0.69 % mult_sgn_dvd_iff
% 0.34/0.69 thf(fact_7482_sgn__mult__dvd__iff,axiom,
% 0.34/0.69 ! [R2: int,L: int,K3: int] :
% 0.34/0.69 ( ( dvd_dvd_int @ ( times_times_int @ ( sgn_sgn_int @ R2 ) @ L ) @ K3 )
% 0.34/0.69 = ( ( dvd_dvd_int @ L @ K3 )
% 0.34/0.69 & ( ( R2 = zero_zero_int )
% 0.34/0.69 => ( K3 = zero_zero_int ) ) ) ) ).
% 0.34/0.69
% 0.34/0.69 % sgn_mult_dvd_iff
% 0.34/0.69 thf(fact_7483_int__sgnE,axiom,
% 0.34/0.69 ! [K3: int] :
% 0.34/0.69 ~ ! [N3: nat,L3: int] :
% 0.34/0.69 ( K3
% 0.34/0.69 != ( times_times_int @ ( sgn_sgn_int @ L3 ) @ ( semiri1314217659103216013at_int @ N3 ) ) ) ).
% 0.34/0.69
% 0.34/0.69 % int_sgnE
% 0.34/0.69 thf(fact_7484_mod__eq__0D,axiom,
% 0.34/0.69 ! [M4: nat,D2: nat] :
% 0.34/0.69 ( ( ( modulo_modulo_nat @ M4 @ D2 )
% 0.34/0.69 = zero_zero_nat )
% 0.34/0.69 => ? [Q7: nat] :
% 0.34/0.69 ( M4
% 0.34/0.69 = ( times_times_nat @ D2 @ Q7 ) ) ) ).
% 0.34/0.69
% 0.34/0.69 % mod_eq_0D
% 0.34/0.69 thf(fact_7485_nat__mod__eq__iff,axiom,
% 0.34/0.69 ! [X: nat,N2: nat,Y: nat] :
% 0.34/0.69 ( ( ( modulo_modulo_nat @ X @ N2 )
% 0.34/0.69 = ( modulo_modulo_nat @ Y @ N2 ) )
% 0.34/0.69 = ( ? [Q1: nat,Q22: nat] :
% 0.34/0.69 ( ( plus_plus_nat @ X @ ( times_times_nat @ N2 @ Q1 ) )
% 0.34/0.69 = ( plus_plus_nat @ Y @ ( times_times_nat @ N2 @ Q22 ) ) ) ) ) ).
% 0.34/0.69
% 0.34/0.69 % nat_mod_eq_iff
% 0.34/0.69 thf(fact_7486_nat__mod__eq__lemma,axiom,
% 0.34/0.69 ! [X: nat,N2: nat,Y: nat] :
% 0.34/0.69 ( ( ( modulo_modulo_nat @ X @ N2 )
% 0.34/0.69 = ( modulo_modulo_nat @ Y @ N2 ) )
% 0.34/0.69 => ( ( ord_less_eq_nat @ Y @ X )
% 0.34/0.69 => ? [Q7: nat] :
% 0.34/0.69 ( X
% 0.34/0.69 = ( plus_plus_nat @ Y @ ( times_times_nat @ N2 @ Q7 ) ) ) ) ) ).
% 0.34/0.69
% 0.34/0.69 % nat_mod_eq_lemma
% 0.34/0.69 thf(fact_7487_mod__eq__nat2E,axiom,
% 0.34/0.69 ! [M4: nat,Q4: nat,N2: nat] :
% 0.34/0.69 ( ( ( modulo_modulo_nat @ M4 @ Q4 )
% 0.34/0.69 = ( modulo_modulo_nat @ N2 @ Q4 ) )
% 0.34/0.69 => ( ( ord_less_eq_nat @ M4 @ N2 )
% 0.34/0.69 => ~ ! [S8: nat] :
% 0.34/0.69 ( N2
% 0.34/0.69 != ( plus_plus_nat @ M4 @ ( times_times_nat @ Q4 @ S8 ) ) ) ) ) ).
% 0.34/0.69
% 0.34/0.69 % mod_eq_nat2E
% 0.34/0.69 thf(fact_7488_mod__eq__nat1E,axiom,
% 0.34/0.69 ! [M4: nat,Q4: nat,N2: nat] :
% 0.34/0.69 ( ( ( modulo_modulo_nat @ M4 @ Q4 )
% 0.34/0.69 = ( modulo_modulo_nat @ N2 @ Q4 ) )
% 0.34/0.69 => ( ( ord_less_eq_nat @ N2 @ M4 )
% 0.34/0.69 => ~ ! [S8: nat] :
% 0.34/0.69 ( M4
% 0.34/0.69 != ( plus_plus_nat @ N2 @ ( times_times_nat @ Q4 @ S8 ) ) ) ) ) ).
% 0.34/0.69
% 0.34/0.69 % mod_eq_nat1E
% 0.34/0.69 thf(fact_7489_div__mod__decomp,axiom,
% 0.34/0.69 ! [A: nat,N2: nat] :
% 0.34/0.69 ( A
% 0.34/0.69 = ( plus_plus_nat @ ( times_times_nat @ ( divide_divide_nat @ A @ N2 ) @ N2 ) @ ( modulo_modulo_nat @ A @ N2 ) ) ) ).
% 0.34/0.69
% 0.34/0.69 % div_mod_decomp
% 0.34/0.69 thf(fact_7490_mod__mult2__eq,axiom,
% 0.34/0.69 ! [M4: nat,N2: nat,Q4: nat] :
% 0.34/0.69 ( ( modulo_modulo_nat @ M4 @ ( times_times_nat @ N2 @ Q4 ) )
% 0.34/0.69 = ( plus_plus_nat @ ( times_times_nat @ N2 @ ( modulo_modulo_nat @ ( divide_divide_nat @ M4 @ N2 ) @ Q4 ) ) @ ( modulo_modulo_nat @ M4 @ N2 ) ) ) ).
% 0.34/0.69
% 0.34/0.69 % mod_mult2_eq
% 0.34/0.69 thf(fact_7491_modulo__nat__def,axiom,
% 0.34/0.69 ( modulo_modulo_nat
% 0.34/0.69 = ( ^ [M3: nat,N4: nat] : ( minus_minus_nat @ M3 @ ( times_times_nat @ ( divide_divide_nat @ M3 @ N4 ) @ N4 ) ) ) ) ).
% 0.34/0.69
% 0.34/0.69 % modulo_nat_def
% 0.34/0.69 thf(fact_7492_div__sgn__abs__cancel,axiom,
% 0.34/0.69 ! [V: int,K3: int,L: int] :
% 0.34/0.69 ( ( V != zero_zero_int )
% 0.34/0.69 => ( ( divide_divide_int @ ( times_times_int @ ( sgn_sgn_int @ V ) @ ( abs_abs_int @ K3 ) ) @ ( times_times_int @ ( sgn_sgn_int @ V ) @ ( abs_abs_int @ L ) ) )
% 0.34/0.69 = ( divide_divide_int @ ( abs_abs_int @ K3 ) @ ( abs_abs_int @ L ) ) ) ) ).
% 0.34/0.69
% 0.34/0.69 % div_sgn_abs_cancel
% 0.34/0.69 thf(fact_7493_div__dvd__sgn__abs,axiom,
% 0.34/0.69 ! [L: int,K3: int] :
% 0.34/0.69 ( ( dvd_dvd_int @ L @ K3 )
% 0.34/0.69 => ( ( divide_divide_int @ K3 @ L )
% 0.34/0.69 = ( times_times_int @ ( times_times_int @ ( sgn_sgn_int @ K3 ) @ ( sgn_sgn_int @ L ) ) @ ( divide_divide_int @ ( abs_abs_int @ K3 ) @ ( abs_abs_int @ L ) ) ) ) ) ).
% 0.34/0.69
% 0.34/0.69 % div_dvd_sgn_abs
% 0.34/0.69 thf(fact_7494_split__mod,axiom,
% 0.34/0.69 ! [P: nat > $o,M4: nat,N2: nat] :
% 0.34/0.69 ( ( P @ ( modulo_modulo_nat @ M4 @ N2 ) )
% 0.34/0.69 = ( ( ( N2 = zero_zero_nat )
% 0.34/0.69 => ( P @ M4 ) )
% 0.34/0.69 & ( ( N2 != zero_zero_nat )
% 0.34/0.69 => ! [I4: nat,J3: nat] :
% 0.34/0.69 ( ( ord_less_nat @ J3 @ N2 )
% 0.34/0.69 => ( ( M4
% 0.34/0.69 = ( plus_plus_nat @ ( times_times_nat @ N2 @ I4 ) @ J3 ) )
% 0.34/0.69 => ( P @ J3 ) ) ) ) ) ) ).
% 0.34/0.69
% 0.34/0.69 % split_mod
% 0.34/0.69 thf(fact_7495_Suc__times__mod__eq,axiom,
% 0.34/0.69 ! [M4: nat,N2: nat] :
% 0.34/0.69 ( ( ord_less_nat @ ( suc @ zero_zero_nat ) @ M4 )
% 0.34/0.69 => ( ( modulo_modulo_nat @ ( suc @ ( times_times_nat @ M4 @ N2 ) ) @ M4 )
% 0.34/0.69 = one_one_nat ) ) ).
% 0.34/0.69
% 0.34/0.69 % Suc_times_mod_eq
% 0.34/0.69 thf(fact_7496_eucl__rel__int__remainderI,axiom,
% 0.34/0.69 ! [R2: int,L: int,K3: int,Q4: int] :
% 0.34/0.69 ( ( ( sgn_sgn_int @ R2 )
% 0.34/0.69 = ( sgn_sgn_int @ L ) )
% 0.34/0.69 => ( ( ord_less_int @ ( abs_abs_int @ R2 ) @ ( abs_abs_int @ L ) )
% 0.34/0.69 => ( ( K3
% 0.34/0.69 = ( plus_plus_int @ ( times_times_int @ Q4 @ L ) @ R2 ) )
% 0.34/0.69 => ( eucl_rel_int @ K3 @ L @ ( product_Pair_int_int @ Q4 @ R2 ) ) ) ) ) ).
% 0.34/0.69
% 0.34/0.69 % eucl_rel_int_remainderI
% 0.34/0.69 thf(fact_7497_eucl__rel__int_Ocases,axiom,
% 0.34/0.69 ! [A1: int,A22: int,A33: product_prod_int_int] :
% 0.34/0.69 ( ( eucl_rel_int @ A1 @ A22 @ A33 )
% 0.34/0.69 => ( ( ( A22 = zero_zero_int )
% 0.34/0.69 => ( A33
% 0.34/0.69 != ( product_Pair_int_int @ zero_zero_int @ A1 ) ) )
% 0.34/0.69 => ( ! [Q7: int] :
% 0.34/0.69 ( ( A33
% 0.34/0.69 = ( product_Pair_int_int @ Q7 @ zero_zero_int ) )
% 0.34/0.69 => ( ( A22 != zero_zero_int )
% 0.34/0.69 => ( A1
% 0.34/0.69 != ( times_times_int @ Q7 @ A22 ) ) ) )
% 0.34/0.69 => ~ ! [R7: int,Q7: int] :
% 0.34/0.69 ( ( A33
% 0.34/0.69 = ( product_Pair_int_int @ Q7 @ R7 ) )
% 0.34/0.69 => ( ( ( sgn_sgn_int @ R7 )
% 0.34/0.69 = ( sgn_sgn_int @ A22 ) )
% 0.34/0.69 => ( ( ord_less_int @ ( abs_abs_int @ R7 ) @ ( abs_abs_int @ A22 ) )
% 0.34/0.69 => ( A1
% 0.34/0.69 != ( plus_plus_int @ ( times_times_int @ Q7 @ A22 ) @ R7 ) ) ) ) ) ) ) ) ).
% 0.34/0.69
% 0.34/0.69 % eucl_rel_int.cases
% 0.34/0.69 thf(fact_7498_bezw_Oelims,axiom,
% 0.34/0.69 ! [X: nat,Xa: nat,Y: product_prod_int_int] :
% 0.34/0.69 ( ( ( bezw @ X @ Xa )
% 0.34/0.69 = Y )
% 0.34/0.69 => ( ( ( Xa = zero_zero_nat )
% 0.34/0.69 => ( Y
% 0.34/0.69 = ( product_Pair_int_int @ one_one_int @ zero_zero_int ) ) )
% 0.34/0.69 & ( ( Xa != zero_zero_nat )
% 0.34/0.69 => ( Y
% 0.34/0.69 = ( product_Pair_int_int @ ( product_snd_int_int @ ( bezw @ Xa @ ( modulo_modulo_nat @ X @ Xa ) ) ) @ ( minus_minus_int @ ( product_fst_int_int @ ( bezw @ Xa @ ( modulo_modulo_nat @ X @ Xa ) ) ) @ ( times_times_int @ ( product_snd_int_int @ ( bezw @ Xa @ ( modulo_modulo_nat @ X @ Xa ) ) ) @ ( semiri1314217659103216013at_int @ ( divide_divide_nat @ X @ Xa ) ) ) ) ) ) ) ) ) ).
% 0.34/0.69
% 0.34/0.69 % bezw.elims
% 0.34/0.69 thf(fact_7499_bezw_Osimps,axiom,
% 0.34/0.69 ( bezw
% 0.34/0.69 = ( ^ [X2: nat,Y3: nat] : ( if_Pro3027730157355071871nt_int @ ( Y3 = zero_zero_nat ) @ ( product_Pair_int_int @ one_one_int @ zero_zero_int ) @ ( product_Pair_int_int @ ( product_snd_int_int @ ( bezw @ Y3 @ ( modulo_modulo_nat @ X2 @ Y3 ) ) ) @ ( minus_minus_int @ ( product_fst_int_int @ ( bezw @ Y3 @ ( modulo_modulo_nat @ X2 @ Y3 ) ) ) @ ( times_times_int @ ( product_snd_int_int @ ( bezw @ Y3 @ ( modulo_modulo_nat @ X2 @ Y3 ) ) ) @ ( semiri1314217659103216013at_int @ ( divide_divide_nat @ X2 @ Y3 ) ) ) ) ) ) ) ) ).
% 0.34/0.69
% 0.34/0.69 % bezw.simps
% 0.34/0.69 thf(fact_7500_bezw__non__0,axiom,
% 0.34/0.69 ! [Y: nat,X: nat] :
% 0.34/0.69 ( ( ord_less_nat @ zero_zero_nat @ Y )
% 0.34/0.69 => ( ( bezw @ X @ Y )
% 0.34/0.69 = ( product_Pair_int_int @ ( product_snd_int_int @ ( bezw @ Y @ ( modulo_modulo_nat @ X @ Y ) ) ) @ ( minus_minus_int @ ( product_fst_int_int @ ( bezw @ Y @ ( modulo_modulo_nat @ X @ Y ) ) ) @ ( times_times_int @ ( product_snd_int_int @ ( bezw @ Y @ ( modulo_modulo_nat @ X @ Y ) ) ) @ ( semiri1314217659103216013at_int @ ( divide_divide_nat @ X @ Y ) ) ) ) ) ) ) ).
% 0.34/0.69
% 0.34/0.69 % bezw_non_0
% 0.34/0.69 thf(fact_7501_divide__int__unfold,axiom,
% 0.34/0.69 ! [L: int,K3: int,N2: nat,M4: nat] :
% 0.34/0.69 ( ( ( ( ( sgn_sgn_int @ L )
% 0.34/0.69 = zero_zero_int )
% 0.34/0.69 | ( ( sgn_sgn_int @ K3 )
% 0.34/0.69 = zero_zero_int )
% 0.34/0.69 | ( N2 = zero_zero_nat ) )
% 0.34/0.69 => ( ( divide_divide_int @ ( times_times_int @ ( sgn_sgn_int @ K3 ) @ ( semiri1314217659103216013at_int @ M4 ) ) @ ( times_times_int @ ( sgn_sgn_int @ L ) @ ( semiri1314217659103216013at_int @ N2 ) ) )
% 0.34/0.69 = zero_zero_int ) )
% 0.34/0.69 & ( ~ ( ( ( sgn_sgn_int @ L )
% 0.34/0.69 = zero_zero_int )
% 0.34/0.69 | ( ( sgn_sgn_int @ K3 )
% 0.34/0.69 = zero_zero_int )
% 0.34/0.69 | ( N2 = zero_zero_nat ) )
% 0.34/0.69 => ( ( ( ( sgn_sgn_int @ K3 )
% 0.34/0.69 = ( sgn_sgn_int @ L ) )
% 0.34/0.69 => ( ( divide_divide_int @ ( times_times_int @ ( sgn_sgn_int @ K3 ) @ ( semiri1314217659103216013at_int @ M4 ) ) @ ( times_times_int @ ( sgn_sgn_int @ L ) @ ( semiri1314217659103216013at_int @ N2 ) ) )
% 0.34/0.69 = ( semiri1314217659103216013at_int @ ( divide_divide_nat @ M4 @ N2 ) ) ) )
% 0.34/0.69 & ( ( ( sgn_sgn_int @ K3 )
% 0.34/0.69 != ( sgn_sgn_int @ L ) )
% 0.34/0.69 => ( ( divide_divide_int @ ( times_times_int @ ( sgn_sgn_int @ K3 ) @ ( semiri1314217659103216013at_int @ M4 ) ) @ ( times_times_int @ ( sgn_sgn_int @ L ) @ ( semiri1314217659103216013at_int @ N2 ) ) )
% 0.34/0.69 = ( uminus_uminus_int
% 0.34/0.69 @ ( semiri1314217659103216013at_int
% 0.34/0.69 @ ( plus_plus_nat @ ( divide_divide_nat @ M4 @ N2 )
% 0.34/0.69 @ ( zero_n2687167440665602831ol_nat
% 0.34/0.69 @ ~ ( dvd_dvd_nat @ N2 @ M4 ) ) ) ) ) ) ) ) ) ) ).
% 0.34/0.69
% 0.34/0.69 % divide_int_unfold
% 0.34/0.69 thf(fact_7502_modulo__int__unfold,axiom,
% 0.34/0.69 ! [L: int,K3: int,N2: nat,M4: nat] :
% 0.34/0.69 ( ( ( ( ( sgn_sgn_int @ L )
% 0.34/0.69 = zero_zero_int )
% 0.34/0.69 | ( ( sgn_sgn_int @ K3 )
% 0.34/0.69 = zero_zero_int )
% 0.34/0.69 | ( N2 = zero_zero_nat ) )
% 0.34/0.69 => ( ( modulo_modulo_int @ ( times_times_int @ ( sgn_sgn_int @ K3 ) @ ( semiri1314217659103216013at_int @ M4 ) ) @ ( times_times_int @ ( sgn_sgn_int @ L ) @ ( semiri1314217659103216013at_int @ N2 ) ) )
% 0.34/0.69 = ( times_times_int @ ( sgn_sgn_int @ K3 ) @ ( semiri1314217659103216013at_int @ M4 ) ) ) )
% 0.34/0.69 & ( ~ ( ( ( sgn_sgn_int @ L )
% 0.34/0.69 = zero_zero_int )
% 0.34/0.69 | ( ( sgn_sgn_int @ K3 )
% 0.34/0.69 = zero_zero_int )
% 0.34/0.69 | ( N2 = zero_zero_nat ) )
% 0.34/0.69 => ( ( ( ( sgn_sgn_int @ K3 )
% 0.34/0.69 = ( sgn_sgn_int @ L ) )
% 0.34/0.69 => ( ( modulo_modulo_int @ ( times_times_int @ ( sgn_sgn_int @ K3 ) @ ( semiri1314217659103216013at_int @ M4 ) ) @ ( times_times_int @ ( sgn_sgn_int @ L ) @ ( semiri1314217659103216013at_int @ N2 ) ) )
% 0.34/0.69 = ( times_times_int @ ( sgn_sgn_int @ L ) @ ( semiri1314217659103216013at_int @ ( modulo_modulo_nat @ M4 @ N2 ) ) ) ) )
% 0.34/0.69 & ( ( ( sgn_sgn_int @ K3 )
% 0.34/0.69 != ( sgn_sgn_int @ L ) )
% 0.34/0.69 => ( ( modulo_modulo_int @ ( times_times_int @ ( sgn_sgn_int @ K3 ) @ ( semiri1314217659103216013at_int @ M4 ) ) @ ( times_times_int @ ( sgn_sgn_int @ L ) @ ( semiri1314217659103216013at_int @ N2 ) ) )
% 0.34/0.69 = ( times_times_int @ ( sgn_sgn_int @ L )
% 0.34/0.69 @ ( minus_minus_int
% 0.34/0.69 @ ( semiri1314217659103216013at_int
% 0.34/0.69 @ ( times_times_nat @ N2
% 0.34/0.69 @ ( zero_n2687167440665602831ol_nat
% 0.34/0.69 @ ~ ( dvd_dvd_nat @ N2 @ M4 ) ) ) )
% 0.34/0.69 @ ( semiri1314217659103216013at_int @ ( modulo_modulo_nat @ M4 @ N2 ) ) ) ) ) ) ) ) ) ).
% 0.34/0.69
% 0.34/0.69 % modulo_int_unfold
% 0.34/0.69 thf(fact_7503_bezw_Opelims,axiom,
% 0.34/0.69 ! [X: nat,Xa: nat,Y: product_prod_int_int] :
% 0.34/0.69 ( ( ( bezw @ X @ Xa )
% 0.34/0.69 = Y )
% 0.34/0.69 => ( ( accp_P4275260045618599050at_nat @ bezw_rel @ ( product_Pair_nat_nat @ X @ Xa ) )
% 0.34/0.69 => ~ ( ( ( ( Xa = zero_zero_nat )
% 0.34/0.69 => ( Y
% 0.34/0.69 = ( product_Pair_int_int @ one_one_int @ zero_zero_int ) ) )
% 0.34/0.69 & ( ( Xa != zero_zero_nat )
% 0.34/0.69 => ( Y
% 0.34/0.69 = ( product_Pair_int_int @ ( product_snd_int_int @ ( bezw @ Xa @ ( modulo_modulo_nat @ X @ Xa ) ) ) @ ( minus_minus_int @ ( product_fst_int_int @ ( bezw @ Xa @ ( modulo_modulo_nat @ X @ Xa ) ) ) @ ( times_times_int @ ( product_snd_int_int @ ( bezw @ Xa @ ( modulo_modulo_nat @ X @ Xa ) ) ) @ ( semiri1314217659103216013at_int @ ( divide_divide_nat @ X @ Xa ) ) ) ) ) ) ) )
% 0.34/0.69 => ~ ( accp_P4275260045618599050at_nat @ bezw_rel @ ( product_Pair_nat_nat @ X @ Xa ) ) ) ) ) ).
% 0.34/0.69
% 0.34/0.69 % bezw.pelims
% 0.34/0.69 thf(fact_7504_semiring__norm_I12_J,axiom,
% 0.34/0.69 ! [N2: num] :
% 0.34/0.69 ( ( times_times_num @ one @ N2 )
% 0.34/0.69 = N2 ) ).
% 0.34/0.69
% 0.34/0.69 % semiring_norm(12)
% 0.34/0.69 thf(fact_7505_semiring__norm_I11_J,axiom,
% 0.34/0.69 ! [M4: num] :
% 0.34/0.69 ( ( times_times_num @ M4 @ one )
% 0.34/0.69 = M4 ) ).
% 0.34/0.69
% 0.34/0.69 % semiring_norm(11)
% 0.34/0.69 thf(fact_7506_neg__eucl__rel__int__mult__2,axiom,
% 0.34/0.69 ! [B: int,A2: int,Q4: int,R2: int] :
% 0.34/0.69 ( ( ord_less_eq_int @ B @ zero_zero_int )
% 0.34/0.69 => ( ( eucl_rel_int @ ( plus_plus_int @ A2 @ one_one_int ) @ B @ ( product_Pair_int_int @ Q4 @ R2 ) )
% 0.34/0.69 => ( eucl_rel_int @ ( plus_plus_int @ one_one_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A2 ) ) @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B ) @ ( product_Pair_int_int @ Q4 @ ( minus_minus_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ R2 ) @ one_one_int ) ) ) ) ) ).
% 0.34/0.69
% 0.34/0.69 % neg_eucl_rel_int_mult_2
% 0.34/0.69 thf(fact_7507_semiring__norm_I13_J,axiom,
% 0.34/0.69 ! [M4: num,N2: num] :
% 0.34/0.69 ( ( times_times_num @ ( bit0 @ M4 ) @ ( bit0 @ N2 ) )
% 0.34/0.69 = ( bit0 @ ( bit0 @ ( times_times_num @ M4 @ N2 ) ) ) ) ).
% 0.34/0.69
% 0.34/0.69 % semiring_norm(13)
% 0.34/0.69 thf(fact_7508_num__double,axiom,
% 0.34/0.69 ! [N2: num] :
% 0.34/0.69 ( ( times_times_num @ ( bit0 @ one ) @ N2 )
% 0.34/0.69 = ( bit0 @ N2 ) ) ).
% 0.34/0.69
% 0.34/0.69 % num_double
% 0.34/0.69 thf(fact_7509_zmod__numeral__Bit0,axiom,
% 0.34/0.69 ! [V: num,W: num] :
% 0.34/0.69 ( ( modulo_modulo_int @ ( numeral_numeral_int @ ( bit0 @ V ) ) @ ( numeral_numeral_int @ ( bit0 @ W ) ) )
% 0.34/0.69 = ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( modulo_modulo_int @ ( numeral_numeral_int @ V ) @ ( numeral_numeral_int @ W ) ) ) ) ).
% 0.34/0.69
% 0.34/0.69 % zmod_numeral_Bit0
% 0.34/0.69 thf(fact_7510_odd__two__times__div__two__nat,axiom,
% 0.34/0.69 ! [N2: nat] :
% 0.34/0.69 ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
% 0.34/0.69 => ( ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ N2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 0.34/0.69 = ( minus_minus_nat @ N2 @ one_one_nat ) ) ) ).
% 0.34/0.69
% 0.34/0.69 % odd_two_times_div_two_nat
% 0.34/0.69 thf(fact_7511_Suc__double__not__eq__double,axiom,
% 0.34/0.69 ! [M4: nat,N2: nat] :
% 0.34/0.69 ( ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M4 ) )
% 0.34/0.69 != ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) ).
% 0.34/0.69
% 0.34/0.69 % Suc_double_not_eq_double
% 0.34/0.69 thf(fact_7512_double__not__eq__Suc__double,axiom,
% 0.34/0.69 ! [M4: nat,N2: nat] :
% 0.34/0.69 ( ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M4 )
% 0.34/0.69 != ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) ) ).
% 0.34/0.69
% 0.34/0.69 % double_not_eq_Suc_double
% 0.34/0.69 thf(fact_7513_nat__bit__induct,axiom,
% 0.34/0.69 ! [P: nat > $o,N2: nat] :
% 0.34/0.69 ( ( P @ zero_zero_nat )
% 0.34/0.69 => ( ! [N3: nat] :
% 0.34/0.69 ( ( P @ N3 )
% 0.34/0.69 => ( ( ord_less_nat @ zero_zero_nat @ N3 )
% 0.34/0.69 => ( P @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 ) ) ) )
% 0.34/0.69 => ( ! [N3: nat] :
% 0.34/0.69 ( ( P @ N3 )
% 0.34/0.69 => ( P @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 ) ) ) )
% 0.34/0.69 => ( P @ N2 ) ) ) ) ).
% 0.34/0.69
% 0.34/0.69 % nat_bit_induct
% 0.34/0.69 thf(fact_7514_int__bit__induct,axiom,
% 0.34/0.69 ! [P: int > $o,K3: int] :
% 0.34/0.69 ( ( P @ zero_zero_int )
% 0.34/0.69 => ( ( P @ ( uminus_uminus_int @ one_one_int ) )
% 0.34/0.69 => ( ! [K: int] :
% 0.34/0.69 ( ( P @ K )
% 0.34/0.69 => ( ( K != zero_zero_int )
% 0.34/0.69 => ( P @ ( times_times_int @ K @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) )
% 0.34/0.69 => ( ! [K: int] :
% 0.34/0.69 ( ( P @ K )
% 0.34/0.69 => ( ( K
% 0.34/0.69 != ( uminus_uminus_int @ one_one_int ) )
% 0.34/0.69 => ( P @ ( plus_plus_int @ one_one_int @ ( times_times_int @ K @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ) )
% 0.34/0.69 => ( P @ K3 ) ) ) ) ) ).
% 0.34/0.69
% 0.34/0.69 % int_bit_induct
% 0.34/0.69 thf(fact_7515_pos__zdiv__mult__2,axiom,
% 0.34/0.69 ! [A2: int,B: int] :
% 0.34/0.69 ( ( ord_less_eq_int @ zero_zero_int @ A2 )
% 0.34/0.69 => ( ( divide_divide_int @ ( plus_plus_int @ one_one_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B ) ) @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A2 ) )
% 0.34/0.69 = ( divide_divide_int @ B @ A2 ) ) ) ).
% 0.34/0.69
% 0.34/0.69 % pos_zdiv_mult_2
% 0.34/0.69 thf(fact_7516_neg__zdiv__mult__2,axiom,
% 0.34/0.69 ! [A2: int,B: int] :
% 0.34/0.69 ( ( ord_less_eq_int @ A2 @ zero_zero_int )
% 0.34/0.69 => ( ( divide_divide_int @ ( plus_plus_int @ one_one_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B ) ) @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A2 ) )
% 0.34/0.69 = ( divide_divide_int @ ( plus_plus_int @ B @ one_one_int ) @ A2 ) ) ) ).
% 0.34/0.69
% 0.34/0.69 % neg_zdiv_mult_2
% 0.34/0.69 thf(fact_7517_pos__zmod__mult__2,axiom,
% 0.34/0.69 ! [A2: int,B: int] :
% 0.34/0.69 ( ( ord_less_eq_int @ zero_zero_int @ A2 )
% 0.34/0.69 => ( ( modulo_modulo_int @ ( plus_plus_int @ one_one_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B ) ) @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A2 ) )
% 0.34/0.69 = ( plus_plus_int @ one_one_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( modulo_modulo_int @ B @ A2 ) ) ) ) ) ).
% 0.34/0.69
% 0.34/0.69 % pos_zmod_mult_2
% 0.34/0.69 thf(fact_7518_neg__zmod__mult__2,axiom,
% 0.34/0.69 ! [A2: int,B: int] :
% 0.34/0.69 ( ( ord_less_eq_int @ A2 @ zero_zero_int )
% 0.34/0.69 => ( ( modulo_modulo_int @ ( plus_plus_int @ one_one_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B ) ) @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A2 ) )
% 0.34/0.69 = ( minus_minus_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( modulo_modulo_int @ ( plus_plus_int @ B @ one_one_int ) @ A2 ) ) @ one_one_int ) ) ) ).
% 0.34/0.69
% 0.34/0.69 % neg_zmod_mult_2
% 0.34/0.69 thf(fact_7519_pos__eucl__rel__int__mult__2,axiom,
% 0.34/0.69 ! [B: int,A2: int,Q4: int,R2: int] :
% 0.34/0.69 ( ( ord_less_eq_int @ zero_zero_int @ B )
% 0.34/0.69 => ( ( eucl_rel_int @ A2 @ B @ ( product_Pair_int_int @ Q4 @ R2 ) )
% 0.34/0.69 => ( eucl_rel_int @ ( plus_plus_int @ one_one_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A2 ) ) @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B ) @ ( product_Pair_int_int @ Q4 @ ( plus_plus_int @ one_one_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ R2 ) ) ) ) ) ) ).
% 0.34/0.69
% 0.34/0.69 % pos_eucl_rel_int_mult_2
% 0.34/0.69 thf(fact_7520_signed__take__bit__Suc__bit0,axiom,
% 0.34/0.69 ! [N2: nat,K3: num] :
% 0.34/0.69 ( ( bit_ri631733984087533419it_int @ ( suc @ N2 ) @ ( numeral_numeral_int @ ( bit0 @ K3 ) ) )
% 0.34/0.69 = ( times_times_int @ ( bit_ri631733984087533419it_int @ N2 @ ( numeral_numeral_int @ K3 ) ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ).
% 0.34/0.69
% 0.34/0.69 % signed_take_bit_Suc_bit0
% 0.34/0.69 thf(fact_7521_signed__take__bit__Suc__minus__bit0,axiom,
% 0.34/0.69 ! [N2: nat,K3: num] :
% 0.34/0.69 ( ( bit_ri631733984087533419it_int @ ( suc @ N2 ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit0 @ K3 ) ) ) )
% 0.34/0.69 = ( times_times_int @ ( bit_ri631733984087533419it_int @ N2 @ ( uminus_uminus_int @ ( numeral_numeral_int @ K3 ) ) ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ).
% 0.34/0.69
% 0.34/0.69 % signed_take_bit_Suc_minus_bit0
% 0.34/0.69 thf(fact_7522_signed__take__bit__mult,axiom,
% 0.34/0.69 ! [N2: nat,K3: int,L: int] :
% 0.34/0.69 ( ( bit_ri631733984087533419it_int @ N2 @ ( times_times_int @ ( bit_ri631733984087533419it_int @ N2 @ K3 ) @ ( bit_ri631733984087533419it_int @ N2 @ L ) ) )
% 0.34/0.69 = ( bit_ri631733984087533419it_int @ N2 @ ( times_times_int @ K3 @ L ) ) ) ).
% 0.34/0.69
% 0.34/0.69 % signed_take_bit_mult
% 0.34/0.69 thf(fact_7523_and__int_Oelims,axiom,
% 0.34/0.69 ! [X: int,Xa: int,Y: int] :
% 0.34/0.69 ( ( ( bit_se725231765392027082nd_int @ X @ Xa )
% 0.34/0.69 = Y )
% 0.34/0.69 => ( ( ( ( member_int @ X @ ( insert_int @ zero_zero_int @ ( insert_int @ ( uminus_uminus_int @ one_one_int ) @ bot_bot_set_int ) ) )
% 0.34/0.69 & ( member_int @ Xa @ ( insert_int @ zero_zero_int @ ( insert_int @ ( uminus_uminus_int @ one_one_int ) @ bot_bot_set_int ) ) ) )
% 0.34/0.69 => ( Y
% 0.34/0.69 = ( uminus_uminus_int
% 0.34/0.69 @ ( zero_n2684676970156552555ol_int
% 0.34/0.69 @ ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ X )
% 0.34/0.69 & ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ Xa ) ) ) ) ) )
% 0.34/0.69 & ( ~ ( ( member_int @ X @ ( insert_int @ zero_zero_int @ ( insert_int @ ( uminus_uminus_int @ one_one_int ) @ bot_bot_set_int ) ) )
% 0.34/0.69 & ( member_int @ Xa @ ( insert_int @ zero_zero_int @ ( insert_int @ ( uminus_uminus_int @ one_one_int ) @ bot_bot_set_int ) ) ) )
% 0.34/0.69 => ( Y
% 0.34/0.69 = ( plus_plus_int
% 0.34/0.69 @ ( zero_n2684676970156552555ol_int
% 0.34/0.69 @ ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ X )
% 0.34/0.69 & ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ Xa ) ) )
% 0.34/0.69 @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se725231765392027082nd_int @ ( divide_divide_int @ X @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ ( divide_divide_int @ Xa @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) ).
% 0.34/0.69
% 0.34/0.69 % and_int.elims
% 0.34/0.69 thf(fact_7524_and__int_Osimps,axiom,
% 0.34/0.69 ( bit_se725231765392027082nd_int
% 0.34/0.69 = ( ^ [K5: int,L2: int] :
% 0.34/0.69 ( if_int
% 0.34/0.69 @ ( ( member_int @ K5 @ ( insert_int @ zero_zero_int @ ( insert_int @ ( uminus_uminus_int @ one_one_int ) @ bot_bot_set_int ) ) )
% 0.34/0.69 & ( member_int @ L2 @ ( insert_int @ zero_zero_int @ ( insert_int @ ( uminus_uminus_int @ one_one_int ) @ bot_bot_set_int ) ) ) )
% 0.34/0.69 @ ( uminus_uminus_int
% 0.34/0.69 @ ( zero_n2684676970156552555ol_int
% 0.34/0.69 @ ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ K5 )
% 0.34/0.69 & ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ L2 ) ) ) )
% 0.34/0.69 @ ( plus_plus_int
% 0.34/0.69 @ ( zero_n2684676970156552555ol_int
% 0.34/0.69 @ ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ K5 )
% 0.34/0.69 & ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ L2 ) ) )
% 0.34/0.69 @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se725231765392027082nd_int @ ( divide_divide_int @ K5 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ ( divide_divide_int @ L2 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ).
% 0.34/0.69
% 0.34/0.69 % and_int.simps
% 0.34/0.69 thf(fact_7525_signed__take__bit__Suc__minus__bit1,axiom,
% 0.34/0.69 ! [N2: nat,K3: num] :
% 0.34/0.69 ( ( bit_ri631733984087533419it_int @ ( suc @ N2 ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit1 @ K3 ) ) ) )
% 0.34/0.69 = ( plus_plus_int @ ( times_times_int @ ( bit_ri631733984087533419it_int @ N2 @ ( minus_minus_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ K3 ) ) @ one_one_int ) ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ one_one_int ) ) ).
% 0.34/0.69
% 0.34/0.69 % signed_take_bit_Suc_minus_bit1
% 0.34/0.69 thf(fact_7526_concat__bit__Suc,axiom,
% 0.34/0.69 ! [N2: nat,K3: int,L: int] :
% 0.34/0.69 ( ( bit_concat_bit @ ( suc @ N2 ) @ K3 @ L )
% 0.34/0.69 = ( plus_plus_int @ ( modulo_modulo_int @ K3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_concat_bit @ N2 @ ( divide_divide_int @ K3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ L ) ) ) ) ).
% 0.34/0.69
% 0.34/0.69 % concat_bit_Suc
% 0.34/0.69 thf(fact_7527_semiring__norm_I14_J,axiom,
% 0.34/0.69 ! [M4: num,N2: num] :
% 0.34/0.69 ( ( times_times_num @ ( bit0 @ M4 ) @ ( bit1 @ N2 ) )
% 0.34/0.69 = ( bit0 @ ( times_times_num @ M4 @ ( bit1 @ N2 ) ) ) ) ).
% 0.34/0.69
% 0.34/0.69 % semiring_norm(14)
% 0.34/0.69 thf(fact_7528_semiring__norm_I15_J,axiom,
% 0.34/0.69 ! [M4: num,N2: num] :
% 0.34/0.69 ( ( times_times_num @ ( bit1 @ M4 ) @ ( bit0 @ N2 ) )
% 0.34/0.69 = ( bit0 @ ( times_times_num @ ( bit1 @ M4 ) @ N2 ) ) ) ).
% 0.34/0.69
% 0.34/0.69 % semiring_norm(15)
% 0.34/0.69 thf(fact_7529_semiring__norm_I16_J,axiom,
% 0.34/0.69 ! [M4: num,N2: num] :
% 0.34/0.69 ( ( times_times_num @ ( bit1 @ M4 ) @ ( bit1 @ N2 ) )
% 0.34/0.69 = ( bit1 @ ( plus_plus_num @ ( plus_plus_num @ M4 @ N2 ) @ ( bit0 @ ( times_times_num @ M4 @ N2 ) ) ) ) ) ).
% 0.34/0.69
% 0.34/0.69 % semiring_norm(16)
% 0.34/0.69 thf(fact_7530_zmod__numeral__Bit1,axiom,
% 0.34/0.69 ! [V: num,W: num] :
% 0.34/0.69 ( ( modulo_modulo_int @ ( numeral_numeral_int @ ( bit1 @ V ) ) @ ( numeral_numeral_int @ ( bit0 @ W ) ) )
% 0.34/0.69 = ( plus_plus_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( modulo_modulo_int @ ( numeral_numeral_int @ V ) @ ( numeral_numeral_int @ W ) ) ) @ one_one_int ) ) ).
% 0.34/0.69
% 0.34/0.69 % zmod_numeral_Bit1
% 0.34/0.69 thf(fact_7531_signed__take__bit__Suc__bit1,axiom,
% 0.34/0.69 ! [N2: nat,K3: num] :
% 0.34/0.69 ( ( bit_ri631733984087533419it_int @ ( suc @ N2 ) @ ( numeral_numeral_int @ ( bit1 @ K3 ) ) )
% 0.34/0.69 = ( plus_plus_int @ ( times_times_int @ ( bit_ri631733984087533419it_int @ N2 @ ( numeral_numeral_int @ K3 ) ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ one_one_int ) ) ).
% 0.34/0.69
% 0.34/0.69 % signed_take_bit_Suc_bit1
% 0.34/0.69 thf(fact_7532_take__bit__mult,axiom,
% 0.34/0.69 ! [N2: nat,K3: int,L: int] :
% 0.34/0.69 ( ( bit_se2923211474154528505it_int @ N2 @ ( times_times_int @ ( bit_se2923211474154528505it_int @ N2 @ K3 ) @ ( bit_se2923211474154528505it_int @ N2 @ L ) ) )
% 0.34/0.69 = ( bit_se2923211474154528505it_int @ N2 @ ( times_times_int @ K3 @ L ) ) ) ).
% 0.34/0.69
% 0.34/0.69 % take_bit_mult
% 0.34/0.69 thf(fact_7533_take__bit__Suc__minus__bit0,axiom,
% 0.34/0.69 ! [N2: nat,K3: num] :
% 0.34/0.69 ( ( bit_se2923211474154528505it_int @ ( suc @ N2 ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit0 @ K3 ) ) ) )
% 0.34/0.69 = ( times_times_int @ ( bit_se2923211474154528505it_int @ N2 @ ( uminus_uminus_int @ ( numeral_numeral_int @ K3 ) ) ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ).
% 0.34/0.69
% 0.34/0.69 % take_bit_Suc_minus_bit0
% 0.34/0.69 thf(fact_7534_and__int__rec,axiom,
% 0.34/0.69 ( bit_se725231765392027082nd_int
% 0.34/0.69 = ( ^ [K5: int,L2: int] :
% 0.34/0.69 ( plus_plus_int
% 0.34/0.69 @ ( zero_n2684676970156552555ol_int
% 0.34/0.69 @ ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ K5 )
% 0.34/0.69 & ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ L2 ) ) )
% 0.34/0.69 @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se725231765392027082nd_int @ ( divide_divide_int @ K5 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ ( divide_divide_int @ L2 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ) ) ) ).
% 0.34/0.69
% 0.34/0.69 % and_int_rec
% 0.34/0.69 thf(fact_7535_and__int__unfold,axiom,
% 0.34/0.69 ( bit_se725231765392027082nd_int
% 0.34/0.69 = ( ^ [K5: int,L2: int] :
% 0.34/0.69 ( if_int
% 0.34/0.69 @ ( ( K5 = zero_zero_int )
% 0.34/0.69 | ( L2 = zero_zero_int ) )
% 0.34/0.69 @ zero_zero_int
% 0.34/0.69 @ ( if_int
% 0.34/0.69 @ ( K5
% 0.34/0.69 = ( uminus_uminus_int @ one_one_int ) )
% 0.34/0.69 @ L2
% 0.34/0.69 @ ( if_int
% 0.34/0.69 @ ( L2
% 0.34/0.69 = ( uminus_uminus_int @ one_one_int ) )
% 0.34/0.69 @ K5
% 0.34/0.69 @ ( plus_plus_int @ ( times_times_int @ ( modulo_modulo_int @ K5 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ ( modulo_modulo_int @ L2 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se725231765392027082nd_int @ ( divide_divide_int @ K5 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ ( divide_divide_int @ L2 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) ) ).
% 0.34/0.69
% 0.34/0.69 % and_int_unfold
% 0.34/0.69 thf(fact_7536_signed__take__bit__numeral__minus__bit1,axiom,
% 0.34/0.69 ! [L: num,K3: num] :
% 0.34/0.69 ( ( bit_ri631733984087533419it_int @ ( numeral_numeral_nat @ L ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit1 @ K3 ) ) ) )
% 0.34/0.69 = ( plus_plus_int @ ( times_times_int @ ( bit_ri631733984087533419it_int @ ( pred_numeral @ L ) @ ( minus_minus_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ K3 ) ) @ one_one_int ) ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ one_one_int ) ) ).
% 0.34/0.69
% 0.34/0.69 % signed_take_bit_numeral_minus_bit1
% 0.34/0.69 thf(fact_7537_signed__take__bit__numeral__bit1,axiom,
% 0.34/0.69 ! [L: num,K3: num] :
% 0.34/0.69 ( ( bit_ri631733984087533419it_int @ ( numeral_numeral_nat @ L ) @ ( numeral_numeral_int @ ( bit1 @ K3 ) ) )
% 0.34/0.69 = ( plus_plus_int @ ( times_times_int @ ( bit_ri631733984087533419it_int @ ( pred_numeral @ L ) @ ( numeral_numeral_int @ K3 ) ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ one_one_int ) ) ).
% 0.34/0.69
% 0.34/0.69 % signed_take_bit_numeral_bit1
% 0.34/0.69 thf(fact_7538_and__int_Opsimps,axiom,
% 0.34/0.69 ! [K3: int,L: int] :
% 0.34/0.69 ( ( accp_P1096762738010456898nt_int @ bit_and_int_rel @ ( product_Pair_int_int @ K3 @ L ) )
% 0.34/0.69 => ( ( ( ( member_int @ K3 @ ( insert_int @ zero_zero_int @ ( insert_int @ ( uminus_uminus_int @ one_one_int ) @ bot_bot_set_int ) ) )
% 0.34/0.69 & ( member_int @ L @ ( insert_int @ zero_zero_int @ ( insert_int @ ( uminus_uminus_int @ one_one_int ) @ bot_bot_set_int ) ) ) )
% 0.34/0.69 => ( ( bit_se725231765392027082nd_int @ K3 @ L )
% 0.34/0.69 = ( uminus_uminus_int
% 0.34/0.69 @ ( zero_n2684676970156552555ol_int
% 0.34/0.69 @ ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ K3 )
% 0.34/0.69 & ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ L ) ) ) ) ) )
% 0.34/0.69 & ( ~ ( ( member_int @ K3 @ ( insert_int @ zero_zero_int @ ( insert_int @ ( uminus_uminus_int @ one_one_int ) @ bot_bot_set_int ) ) )
% 0.34/0.69 & ( member_int @ L @ ( insert_int @ zero_zero_int @ ( insert_int @ ( uminus_uminus_int @ one_one_int ) @ bot_bot_set_int ) ) ) )
% 0.34/0.69 => ( ( bit_se725231765392027082nd_int @ K3 @ L )
% 0.34/0.69 = ( plus_plus_int
% 0.34/0.69 @ ( zero_n2684676970156552555ol_int
% 0.34/0.69 @ ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ K3 )
% 0.34/0.69 & ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ L ) ) )
% 0.34/0.69 @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se725231765392027082nd_int @ ( divide_divide_int @ K3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ ( divide_divide_int @ L @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) ).
% 0.34/0.69
% 0.34/0.69 % and_int.psimps
% 0.34/0.69 thf(fact_7539_and__int_Opelims,axiom,
% 0.34/0.69 ! [X: int,Xa: int,Y: int] :
% 0.34/0.69 ( ( ( bit_se725231765392027082nd_int @ X @ Xa )
% 0.34/0.69 = Y )
% 0.34/0.69 => ( ( accp_P1096762738010456898nt_int @ bit_and_int_rel @ ( product_Pair_int_int @ X @ Xa ) )
% 0.34/0.69 => ~ ( ( ( ( ( member_int @ X @ ( insert_int @ zero_zero_int @ ( insert_int @ ( uminus_uminus_int @ one_one_int ) @ bot_bot_set_int ) ) )
% 0.34/0.69 & ( member_int @ Xa @ ( insert_int @ zero_zero_int @ ( insert_int @ ( uminus_uminus_int @ one_one_int ) @ bot_bot_set_int ) ) ) )
% 0.34/0.69 => ( Y
% 0.34/0.69 = ( uminus_uminus_int
% 0.34/0.69 @ ( zero_n2684676970156552555ol_int
% 0.34/0.69 @ ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ X )
% 0.34/0.69 & ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ Xa ) ) ) ) ) )
% 0.34/0.69 & ( ~ ( ( member_int @ X @ ( insert_int @ zero_zero_int @ ( insert_int @ ( uminus_uminus_int @ one_one_int ) @ bot_bot_set_int ) ) )
% 0.34/0.69 & ( member_int @ Xa @ ( insert_int @ zero_zero_int @ ( insert_int @ ( uminus_uminus_int @ one_one_int ) @ bot_bot_set_int ) ) ) )
% 0.34/0.69 => ( Y
% 0.34/0.69 = ( plus_plus_int
% 0.34/0.69 @ ( zero_n2684676970156552555ol_int
% 0.34/0.69 @ ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ X )
% 0.34/0.69 & ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ Xa ) ) )
% 0.34/0.69 @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se725231765392027082nd_int @ ( divide_divide_int @ X @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ ( divide_divide_int @ Xa @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ) ) ) )
% 0.34/0.69 => ~ ( accp_P1096762738010456898nt_int @ bit_and_int_rel @ ( product_Pair_int_int @ X @ Xa ) ) ) ) ) ).
% 0.34/0.69
% 0.34/0.69 % and_int.pelims
% 0.34/0.69 thf(fact_7540_signed__take__bit__numeral__bit0,axiom,
% 0.34/0.69 ! [L: num,K3: num] :
% 0.34/0.69 ( ( bit_ri631733984087533419it_int @ ( numeral_numeral_nat @ L ) @ ( numeral_numeral_int @ ( bit0 @ K3 ) ) )
% 0.34/0.69 = ( times_times_int @ ( bit_ri631733984087533419it_int @ ( pred_numeral @ L ) @ ( numeral_numeral_int @ K3 ) ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ).
% 0.34/0.69
% 0.34/0.69 % signed_take_bit_numeral_bit0
% 0.34/0.69 thf(fact_7541_signed__take__bit__numeral__minus__bit0,axiom,
% 0.34/0.69 ! [L: num,K3: num] :
% 0.34/0.69 ( ( bit_ri631733984087533419it_int @ ( numeral_numeral_nat @ L ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit0 @ K3 ) ) ) )
% 0.34/0.69 = ( times_times_int @ ( bit_ri631733984087533419it_int @ ( pred_numeral @ L ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ K3 ) ) ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ).
% 0.34/0.69
% 0.34/0.69 % signed_take_bit_numeral_minus_bit0
% 0.34/0.69 thf(fact_7542_take__bit__numeral__minus__bit0,axiom,
% 0.34/0.69 ! [L: num,K3: num] :
% 0.34/0.69 ( ( bit_se2923211474154528505it_int @ ( numeral_numeral_nat @ L ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit0 @ K3 ) ) ) )
% 0.34/0.69 = ( times_times_int @ ( bit_se2923211474154528505it_int @ ( pred_numeral @ L ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ K3 ) ) ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ).
% 0.34/0.69
% 0.34/0.69 % take_bit_numeral_minus_bit0
% 0.34/0.69 thf(fact_7543_and__nat__unfold,axiom,
% 0.34/0.69 ( bit_se727722235901077358nd_nat
% 0.34/0.69 = ( ^ [M3: nat,N4: nat] :
% 0.34/0.69 ( if_nat
% 0.34/0.69 @ ( ( M3 = zero_zero_nat )
% 0.34/0.69 | ( N4 = zero_zero_nat ) )
% 0.34/0.69 @ zero_zero_nat
% 0.34/0.69 @ ( plus_plus_nat @ ( times_times_nat @ ( modulo_modulo_nat @ M3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( modulo_modulo_nat @ N4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( bit_se727722235901077358nd_nat @ ( divide_divide_nat @ M3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( divide_divide_nat @ N4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ).
% 0.34/0.69
% 0.34/0.69 % and_nat_unfold
% 0.34/0.69 thf(fact_7544_and__nat__rec,axiom,
% 0.34/0.69 ( bit_se727722235901077358nd_nat
% 0.34/0.69 = ( ^ [M3: nat,N4: nat] :
% 0.34/0.69 ( plus_plus_nat
% 0.34/0.69 @ ( zero_n2687167440665602831ol_nat
% 0.34/0.69 @ ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M3 )
% 0.34/0.69 & ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N4 ) ) )
% 0.34/0.69 @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( bit_se727722235901077358nd_nat @ ( divide_divide_nat @ M3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( divide_divide_nat @ N4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ).
% 0.34/0.69
% 0.34/0.69 % and_nat_rec
% 0.34/0.69 thf(fact_7545_and__int_Opinduct,axiom,
% 0.34/0.69 ! [A0: int,A1: int,P: int > int > $o] :
% 0.34/0.69 ( ( accp_P1096762738010456898nt_int @ bit_and_int_rel @ ( product_Pair_int_int @ A0 @ A1 ) )
% 0.34/0.69 => ( ! [K: int,L3: int] :
% 0.34/0.69 ( ( accp_P1096762738010456898nt_int @ bit_and_int_rel @ ( product_Pair_int_int @ K @ L3 ) )
% 0.34/0.69 => ( ( ~ ( ( member_int @ K @ ( insert_int @ zero_zero_int @ ( insert_int @ ( uminus_uminus_int @ one_one_int ) @ bot_bot_set_int ) ) )
% 0.34/0.69 & ( member_int @ L3 @ ( insert_int @ zero_zero_int @ ( insert_int @ ( uminus_uminus_int @ one_one_int ) @ bot_bot_set_int ) ) ) )
% 0.34/0.69 => ( P @ ( divide_divide_int @ K @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ ( divide_divide_int @ L3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) )
% 0.34/0.69 => ( P @ K @ L3 ) ) )
% 0.34/0.69 => ( P @ A0 @ A1 ) ) ) ).
% 0.34/0.69
% 0.34/0.69 % and_int.pinduct
% 0.34/0.69 thf(fact_7546_take__bit__numeral__minus__bit1,axiom,
% 0.34/0.69 ! [L: num,K3: num] :
% 0.34/0.69 ( ( bit_se2923211474154528505it_int @ ( numeral_numeral_nat @ L ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit1 @ K3 ) ) ) )
% 0.34/0.69 = ( plus_plus_int @ ( times_times_int @ ( bit_se2923211474154528505it_int @ ( pred_numeral @ L ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( inc @ K3 ) ) ) ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ one_one_int ) ) ).
% 0.34/0.69
% 0.34/0.69 % take_bit_numeral_minus_bit1
% 0.34/0.69 thf(fact_7547_take__bit__Suc__minus__bit1,axiom,
% 0.34/0.69 ! [N2: nat,K3: num] :
% 0.34/0.69 ( ( bit_se2923211474154528505it_int @ ( suc @ N2 ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit1 @ K3 ) ) ) )
% 0.34/0.69 = ( plus_plus_int @ ( times_times_int @ ( bit_se2923211474154528505it_int @ N2 @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( inc @ K3 ) ) ) ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ one_one_int ) ) ).
% 0.34/0.69
% 0.34/0.69 % take_bit_Suc_minus_bit1
% 0.34/0.69 thf(fact_7548_binomial__code,axiom,
% 0.34/0.69 ( binomial
% 0.34/0.69 = ( ^ [N4: nat,K5: nat] : ( if_nat @ ( ord_less_nat @ N4 @ K5 ) @ zero_zero_nat @ ( if_nat @ ( ord_less_nat @ N4 @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ K5 ) ) @ ( binomial @ N4 @ ( minus_minus_nat @ N4 @ K5 ) ) @ ( divide_divide_nat @ ( set_fo2584398358068434914at_nat @ times_times_nat @ ( plus_plus_nat @ ( minus_minus_nat @ N4 @ K5 ) @ one_one_nat ) @ N4 @ one_one_nat ) @ ( semiri1408675320244567234ct_nat @ K5 ) ) ) ) ) ) ).
% 0.34/0.69
% 0.34/0.69 % binomial_code
% 0.34/0.69 thf(fact_7549_modulo__int__def,axiom,
% 0.34/0.69 ( modulo_modulo_int
% 0.34/0.69 = ( ^ [K5: int,L2: int] :
% 0.34/0.69 ( if_int @ ( L2 = zero_zero_int ) @ K5
% 0.34/0.69 @ ( if_int
% 0.34/0.69 @ ( ( sgn_sgn_int @ K5 )
% 0.34/0.69 = ( sgn_sgn_int @ L2 ) )
% 0.34/0.69 @ ( times_times_int @ ( sgn_sgn_int @ L2 ) @ ( semiri1314217659103216013at_int @ ( modulo_modulo_nat @ ( nat2 @ ( abs_abs_int @ K5 ) ) @ ( nat2 @ ( abs_abs_int @ L2 ) ) ) ) )
% 0.34/0.69 @ ( times_times_int @ ( sgn_sgn_int @ L2 )
% 0.34/0.69 @ ( minus_minus_int
% 0.34/0.69 @ ( times_times_int @ ( abs_abs_int @ L2 )
% 0.34/0.69 @ ( zero_n2684676970156552555ol_int
% 0.34/0.69 @ ~ ( dvd_dvd_int @ L2 @ K5 ) ) )
% 0.34/0.69 @ ( semiri1314217659103216013at_int @ ( modulo_modulo_nat @ ( nat2 @ ( abs_abs_int @ K5 ) ) @ ( nat2 @ ( abs_abs_int @ L2 ) ) ) ) ) ) ) ) ) ) ).
% 0.34/0.69
% 0.34/0.69 % modulo_int_def
% 0.34/0.69 thf(fact_7550_or__int__unfold,axiom,
% 0.34/0.69 ( bit_se1409905431419307370or_int
% 0.34/0.69 = ( ^ [K5: int,L2: int] :
% 0.34/0.69 ( if_int
% 0.34/0.69 @ ( ( K5
% 0.34/0.69 = ( uminus_uminus_int @ one_one_int ) )
% 0.34/0.69 | ( L2
% 0.34/0.69 = ( uminus_uminus_int @ one_one_int ) ) )
% 0.34/0.69 @ ( uminus_uminus_int @ one_one_int )
% 0.34/0.69 @ ( if_int @ ( K5 = zero_zero_int ) @ L2 @ ( if_int @ ( L2 = zero_zero_int ) @ K5 @ ( plus_plus_int @ ( ord_max_int @ ( modulo_modulo_int @ K5 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ ( modulo_modulo_int @ L2 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se1409905431419307370or_int @ ( divide_divide_int @ K5 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ ( divide_divide_int @ L2 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) ) ).
% 0.34/0.69
% 0.34/0.69 % or_int_unfold
% 0.34/0.69 thf(fact_7551_Suc__times__binomial__eq,axiom,
% 0.34/0.69 ! [N2: nat,K3: nat] :
% 0.34/0.69 ( ( times_times_nat @ ( suc @ N2 ) @ ( binomial @ N2 @ K3 ) )
% 0.34/0.69 = ( times_times_nat @ ( binomial @ ( suc @ N2 ) @ ( suc @ K3 ) ) @ ( suc @ K3 ) ) ) ).
% 0.34/0.69
% 0.34/0.69 % Suc_times_binomial_eq
% 0.34/0.69 thf(fact_7552_Suc__times__binomial,axiom,
% 0.34/0.69 ! [K3: nat,N2: nat] :
% 0.34/0.69 ( ( times_times_nat @ ( suc @ K3 ) @ ( binomial @ ( suc @ N2 ) @ ( suc @ K3 ) ) )
% 0.34/0.69 = ( times_times_nat @ ( suc @ N2 ) @ ( binomial @ N2 @ K3 ) ) ) ).
% 0.34/0.69
% 0.34/0.69 % Suc_times_binomial
% 0.34/0.69 thf(fact_7553_choose__mult__lemma,axiom,
% 0.34/0.69 ! [M4: nat,R2: nat,K3: nat] :
% 0.34/0.69 ( ( times_times_nat @ ( binomial @ ( plus_plus_nat @ ( plus_plus_nat @ M4 @ R2 ) @ K3 ) @ ( plus_plus_nat @ M4 @ K3 ) ) @ ( binomial @ ( plus_plus_nat @ M4 @ K3 ) @ K3 ) )
% 0.34/0.69 = ( times_times_nat @ ( binomial @ ( plus_plus_nat @ ( plus_plus_nat @ M4 @ R2 ) @ K3 ) @ K3 ) @ ( binomial @ ( plus_plus_nat @ M4 @ R2 ) @ M4 ) ) ) ).
% 0.34/0.69
% 0.34/0.69 % choose_mult_lemma
% 0.34/0.69 thf(fact_7554_Suc__times__binomial__add,axiom,
% 0.34/0.69 ! [A2: nat,B: nat] :
% 0.34/0.69 ( ( times_times_nat @ ( suc @ A2 ) @ ( binomial @ ( suc @ ( plus_plus_nat @ A2 @ B ) ) @ ( suc @ A2 ) ) )
% 0.34/0.69 = ( times_times_nat @ ( suc @ B ) @ ( binomial @ ( suc @ ( plus_plus_nat @ A2 @ B ) ) @ A2 ) ) ) ).
% 0.34/0.69
% 0.34/0.69 % Suc_times_binomial_add
% 0.34/0.69 thf(fact_7555_binomial__fact__lemma,axiom,
% 0.34/0.69 ! [K3: nat,N2: nat] :
% 0.34/0.69 ( ( ord_less_eq_nat @ K3 @ N2 )
% 0.34/0.69 => ( ( times_times_nat @ ( times_times_nat @ ( semiri1408675320244567234ct_nat @ K3 ) @ ( semiri1408675320244567234ct_nat @ ( minus_minus_nat @ N2 @ K3 ) ) ) @ ( binomial @ N2 @ K3 ) )
% 0.34/0.69 = ( semiri1408675320244567234ct_nat @ N2 ) ) ) ).
% 0.34/0.69
% 0.34/0.69 % binomial_fact_lemma
% 0.34/0.69 thf(fact_7556_choose__mult,axiom,
% 0.34/0.69 ! [K3: nat,M4: nat,N2: nat] :
% 0.34/0.69 ( ( ord_less_eq_nat @ K3 @ M4 )
% 0.34/0.69 => ( ( ord_less_eq_nat @ M4 @ N2 )
% 0.34/0.69 => ( ( times_times_nat @ ( binomial @ N2 @ M4 ) @ ( binomial @ M4 @ K3 ) )
% 0.34/0.69 = ( times_times_nat @ ( binomial @ N2 @ K3 ) @ ( binomial @ ( minus_minus_nat @ N2 @ K3 ) @ ( minus_minus_nat @ M4 @ K3 ) ) ) ) ) ) ).
% 0.34/0.69
% 0.34/0.69 % choose_mult
% 0.34/0.69 thf(fact_7557_binomial__absorb__comp,axiom,
% 0.34/0.69 ! [N2: nat,K3: nat] :
% 0.34/0.69 ( ( times_times_nat @ ( minus_minus_nat @ N2 @ K3 ) @ ( binomial @ N2 @ K3 ) )
% 0.34/0.69 = ( times_times_nat @ N2 @ ( binomial @ ( minus_minus_nat @ N2 @ one_one_nat ) @ K3 ) ) ) ).
% 0.34/0.69
% 0.34/0.69 % binomial_absorb_comp
% 0.34/0.69 thf(fact_7558_binomial__Suc__Suc__eq__times,axiom,
% 0.34/0.69 ! [N2: nat,K3: nat] :
% 0.34/0.69 ( ( binomial @ ( suc @ N2 ) @ ( suc @ K3 ) )
% 0.34/0.69 = ( divide_divide_nat @ ( times_times_nat @ ( suc @ N2 ) @ ( binomial @ N2 @ K3 ) ) @ ( suc @ K3 ) ) ) ).
% 0.34/0.69
% 0.34/0.69 % binomial_Suc_Suc_eq_times
% 0.34/0.69 thf(fact_7559_nat__abs__mult__distrib,axiom,
% 0.34/0.69 ! [W: int,Z: int] :
% 0.34/0.69 ( ( nat2 @ ( abs_abs_int @ ( times_times_int @ W @ Z ) ) )
% 0.34/0.69 = ( times_times_nat @ ( nat2 @ ( abs_abs_int @ W ) ) @ ( nat2 @ ( abs_abs_int @ Z ) ) ) ) ).
% 0.34/0.69
% 0.34/0.69 % nat_abs_mult_distrib
% 0.34/0.69 thf(fact_7560_mult__inc,axiom,
% 0.34/0.69 ! [X: num,Y: num] :
% 0.34/0.69 ( ( times_times_num @ X @ ( inc @ Y ) )
% 0.34/0.69 = ( plus_plus_num @ ( times_times_num @ X @ Y ) @ X ) ) ).
% 0.34/0.69
% 0.34/0.69 % mult_inc
% 0.34/0.69 thf(fact_7561_binomial__absorption,axiom,
% 0.34/0.69 ! [K3: nat,N2: nat] :
% 0.34/0.69 ( ( times_times_nat @ ( suc @ K3 ) @ ( binomial @ N2 @ ( suc @ K3 ) ) )
% 0.34/0.69 = ( times_times_nat @ N2 @ ( binomial @ ( minus_minus_nat @ N2 @ one_one_nat ) @ K3 ) ) ) ).
% 0.34/0.69
% 0.34/0.69 % binomial_absorption
% 0.34/0.69 thf(fact_7562_nat__mult__distrib,axiom,
% 0.34/0.69 ! [Z: int,Z7: int] :
% 0.34/0.69 ( ( ord_less_eq_int @ zero_zero_int @ Z )
% 0.34/0.69 => ( ( nat2 @ ( times_times_int @ Z @ Z7 ) )
% 0.34/0.69 = ( times_times_nat @ ( nat2 @ Z ) @ ( nat2 @ Z7 ) ) ) ) ).
% 0.34/0.69
% 0.34/0.69 % nat_mult_distrib
% 0.34/0.69 thf(fact_7563_binomial__altdef__nat,axiom,
% 0.34/0.69 ! [K3: nat,N2: nat] :
% 0.34/0.69 ( ( ord_less_eq_nat @ K3 @ N2 )
% 0.34/0.69 => ( ( binomial @ N2 @ K3 )
% 0.34/0.69 = ( divide_divide_nat @ ( semiri1408675320244567234ct_nat @ N2 ) @ ( times_times_nat @ ( semiri1408675320244567234ct_nat @ K3 ) @ ( semiri1408675320244567234ct_nat @ ( minus_minus_nat @ N2 @ K3 ) ) ) ) ) ) ).
% 0.34/0.69
% 0.34/0.69 % binomial_altdef_nat
% 0.34/0.69 thf(fact_7564_times__binomial__minus1__eq,axiom,
% 0.34/0.69 ! [K3: nat,N2: nat] :
% 0.34/0.69 ( ( ord_less_nat @ zero_zero_nat @ K3 )
% 0.34/0.69 => ( ( times_times_nat @ K3 @ ( binomial @ N2 @ K3 ) )
% 0.34/0.69 = ( times_times_nat @ N2 @ ( binomial @ ( minus_minus_nat @ N2 @ one_one_nat ) @ ( minus_minus_nat @ K3 @ one_one_nat ) ) ) ) ) ).
% 0.34/0.69
% 0.34/0.69 % times_binomial_minus1_eq
% 0.34/0.69 thf(fact_7565_nat__mult__distrib__neg,axiom,
% 0.34/0.69 ! [Z: int,Z7: int] :
% 0.34/0.69 ( ( ord_less_eq_int @ Z @ zero_zero_int )
% 0.34/0.69 => ( ( nat2 @ ( times_times_int @ Z @ Z7 ) )
% 0.34/0.69 = ( times_times_nat @ ( nat2 @ ( uminus_uminus_int @ Z ) ) @ ( nat2 @ ( uminus_uminus_int @ Z7 ) ) ) ) ) ).
% 0.34/0.69
% 0.34/0.69 % nat_mult_distrib_neg
% 0.34/0.69 thf(fact_7566_choose__two,axiom,
% 0.34/0.69 ! [N2: nat] :
% 0.34/0.69 ( ( binomial @ N2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.34/0.69 = ( divide_divide_nat @ ( times_times_nat @ N2 @ ( minus_minus_nat @ N2 @ one_one_nat ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% 0.34/0.69
% 0.34/0.69 % choose_two
% 0.34/0.69 thf(fact_7567_or__int__rec,axiom,
% 0.34/0.69 ( bit_se1409905431419307370or_int
% 0.34/0.69 = ( ^ [K5: int,L2: int] :
% 0.34/0.69 ( plus_plus_int
% 0.34/0.69 @ ( zero_n2684676970156552555ol_int
% 0.34/0.69 @ ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ K5 )
% 0.34/0.69 | ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ L2 ) ) )
% 0.34/0.69 @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se1409905431419307370or_int @ ( divide_divide_int @ K5 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ ( divide_divide_int @ L2 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ) ) ) ).
% 0.34/0.69
% 0.34/0.69 % or_int_rec
% 0.34/0.69 thf(fact_7568_signed__take__bit__eq__take__bit__minus,axiom,
% 0.34/0.69 ( bit_ri631733984087533419it_int
% 0.34/0.69 = ( ^ [N4: nat,K5: int] : ( minus_minus_int @ ( bit_se2923211474154528505it_int @ ( suc @ N4 ) @ K5 ) @ ( times_times_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( suc @ N4 ) ) @ ( zero_n2684676970156552555ol_int @ ( bit_se1146084159140164899it_int @ K5 @ N4 ) ) ) ) ) ) ).
% 0.34/0.69
% 0.34/0.69 % signed_take_bit_eq_take_bit_minus
% 0.34/0.69 thf(fact_7569_or__nat__unfold,axiom,
% 0.34/0.69 ( bit_se1412395901928357646or_nat
% 0.34/0.69 = ( ^ [M3: nat,N4: nat] : ( if_nat @ ( M3 = zero_zero_nat ) @ N4 @ ( if_nat @ ( N4 = zero_zero_nat ) @ M3 @ ( plus_plus_nat @ ( ord_max_nat @ ( modulo_modulo_nat @ M3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( modulo_modulo_nat @ N4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( bit_se1412395901928357646or_nat @ ( divide_divide_nat @ M3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( divide_divide_nat @ N4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) ).
% 0.34/0.69
% 0.34/0.69 % or_nat_unfold
% 0.34/0.69 thf(fact_7570_push__bit__int__def,axiom,
% 0.34/0.69 ( bit_se545348938243370406it_int
% 0.34/0.69 = ( ^ [N4: nat,K5: int] : ( times_times_int @ K5 @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N4 ) ) ) ) ).
% 0.34/0.69
% 0.34/0.69 % push_bit_int_def
% 0.34/0.69 thf(fact_7571_push__bit__nat__def,axiom,
% 0.34/0.69 ( bit_se547839408752420682it_nat
% 0.34/0.69 = ( ^ [N4: nat,M3: nat] : ( times_times_nat @ M3 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N4 ) ) ) ) ).
% 0.34/0.69
% 0.34/0.69 % push_bit_nat_def
% 0.34/0.69 thf(fact_7572_set__bit__eq,axiom,
% 0.34/0.69 ( bit_se7879613467334960850it_int
% 0.34/0.69 = ( ^ [N4: nat,K5: int] :
% 0.34/0.69 ( plus_plus_int @ K5
% 0.34/0.69 @ ( times_times_int
% 0.34/0.69 @ ( zero_n2684676970156552555ol_int
% 0.34/0.69 @ ~ ( bit_se1146084159140164899it_int @ K5 @ N4 ) )
% 0.34/0.69 @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N4 ) ) ) ) ) ).
% 0.34/0.69
% 0.34/0.69 % set_bit_eq
% 0.34/0.69 thf(fact_7573_unset__bit__eq,axiom,
% 0.34/0.69 ( bit_se4203085406695923979it_int
% 0.34/0.69 = ( ^ [N4: nat,K5: int] : ( minus_minus_int @ K5 @ ( times_times_int @ ( zero_n2684676970156552555ol_int @ ( bit_se1146084159140164899it_int @ K5 @ N4 ) ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N4 ) ) ) ) ) ).
% 0.34/0.69
% 0.34/0.69 % unset_bit_eq
% 0.34/0.69 thf(fact_7574_or__nat__rec,axiom,
% 0.34/0.69 ( bit_se1412395901928357646or_nat
% 0.34/0.69 = ( ^ [M3: nat,N4: nat] :
% 0.34/0.69 ( plus_plus_nat
% 0.34/0.69 @ ( zero_n2687167440665602831ol_nat
% 0.34/0.69 @ ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M3 )
% 0.34/0.69 | ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N4 ) ) )
% 0.34/0.69 @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( bit_se1412395901928357646or_nat @ ( divide_divide_nat @ M3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( divide_divide_nat @ N4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ).
% 0.34/0.69
% 0.34/0.69 % or_nat_rec
% 0.34/0.69 thf(fact_7575_xor__nat__unfold,axiom,
% 0.34/0.69 ( bit_se6528837805403552850or_nat
% 0.34/0.69 = ( ^ [M3: nat,N4: nat] : ( if_nat @ ( M3 = zero_zero_nat ) @ N4 @ ( if_nat @ ( N4 = zero_zero_nat ) @ M3 @ ( plus_plus_nat @ ( modulo_modulo_nat @ ( plus_plus_nat @ ( modulo_modulo_nat @ M3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( modulo_modulo_nat @ N4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( bit_se6528837805403552850or_nat @ ( divide_divide_nat @ M3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( divide_divide_nat @ N4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) ).
% 0.34/0.69
% 0.34/0.69 % xor_nat_unfold
% 0.34/0.69 thf(fact_7576_take__bit__Suc__from__most,axiom,
% 0.34/0.69 ! [N2: nat,K3: int] :
% 0.34/0.69 ( ( bit_se2923211474154528505it_int @ ( suc @ N2 ) @ K3 )
% 0.34/0.69 = ( plus_plus_int @ ( times_times_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N2 ) @ ( zero_n2684676970156552555ol_int @ ( bit_se1146084159140164899it_int @ K3 @ N2 ) ) ) @ ( bit_se2923211474154528505it_int @ N2 @ K3 ) ) ) ).
% 0.34/0.69
% 0.34/0.69 % take_bit_Suc_from_most
% 0.34/0.69 thf(fact_7577_xor__int__rec,axiom,
% 0.34/0.69 ( bit_se6526347334894502574or_int
% 0.34/0.69 = ( ^ [K5: int,L2: int] :
% 0.34/0.69 ( plus_plus_int
% 0.34/0.69 @ ( zero_n2684676970156552555ol_int
% 0.34/0.69 @ ( ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ K5 ) )
% 0.34/0.69 != ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ L2 ) ) ) )
% 0.34/0.69 @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se6526347334894502574or_int @ ( divide_divide_int @ K5 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ ( divide_divide_int @ L2 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ) ) ) ).
% 0.34/0.69
% 0.34/0.69 % xor_int_rec
% 0.34/0.69 thf(fact_7578_xor__nat__rec,axiom,
% 0.34/0.69 ( bit_se6528837805403552850or_nat
% 0.34/0.69 = ( ^ [M3: nat,N4: nat] :
% 0.34/0.69 ( plus_plus_nat
% 0.34/0.69 @ ( zero_n2687167440665602831ol_nat
% 0.34/0.69 @ ( ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M3 ) )
% 0.34/0.69 != ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N4 ) ) ) )
% 0.34/0.69 @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( bit_se6528837805403552850or_nat @ ( divide_divide_nat @ M3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( divide_divide_nat @ N4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ).
% 0.34/0.69
% 0.34/0.69 % xor_nat_rec
% 0.34/0.69 thf(fact_7579_xor__int__unfold,axiom,
% 0.34/0.69 ( bit_se6526347334894502574or_int
% 0.34/0.69 = ( ^ [K5: int,L2: int] :
% 0.34/0.69 ( if_int
% 0.34/0.69 @ ( K5
% 0.34/0.69 = ( uminus_uminus_int @ one_one_int ) )
% 0.34/0.69 @ ( bit_ri7919022796975470100ot_int @ L2 )
% 0.34/0.69 @ ( if_int
% 0.34/0.69 @ ( L2
% 0.34/0.69 = ( uminus_uminus_int @ one_one_int ) )
% 0.34/0.69 @ ( bit_ri7919022796975470100ot_int @ K5 )
% 0.34/0.69 @ ( if_int @ ( K5 = zero_zero_int ) @ L2 @ ( if_int @ ( L2 = zero_zero_int ) @ K5 @ ( plus_plus_int @ ( abs_abs_int @ ( minus_minus_int @ ( modulo_modulo_int @ K5 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ ( modulo_modulo_int @ L2 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se6526347334894502574or_int @ ( divide_divide_int @ K5 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ ( divide_divide_int @ L2 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) ) ) ).
% 0.34/0.69
% 0.34/0.69 % xor_int_unfold
% 0.34/0.69 thf(fact_7580_not__int__rec,axiom,
% 0.34/0.69 ( bit_ri7919022796975470100ot_int
% 0.34/0.69 = ( ^ [K5: int] : ( plus_plus_int @ ( zero_n2684676970156552555ol_int @ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ K5 ) ) @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_ri7919022796975470100ot_int @ ( divide_divide_int @ K5 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ) ) ) ).
% 0.34/0.69
% 0.34/0.69 % not_int_rec
% 0.34/0.69 thf(fact_7581_or__not__numerals_I8_J,axiom,
% 0.34/0.69 ! [M4: num,N2: num] :
% 0.34/0.69 ( ( bit_se1409905431419307370or_int @ ( numeral_numeral_int @ ( bit1 @ M4 ) ) @ ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ ( bit0 @ N2 ) ) ) )
% 0.34/0.69 = ( plus_plus_int @ one_one_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se1409905431419307370or_int @ ( numeral_numeral_int @ M4 ) @ ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ N2 ) ) ) ) ) ) ).
% 0.34/0.69
% 0.34/0.69 % or_not_numerals(8)
% 0.34/0.69 thf(fact_7582_or__not__numerals_I9_J,axiom,
% 0.34/0.69 ! [M4: num,N2: num] :
% 0.34/0.69 ( ( bit_se1409905431419307370or_int @ ( numeral_numeral_int @ ( bit1 @ M4 ) ) @ ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ ( bit1 @ N2 ) ) ) )
% 0.34/0.69 = ( plus_plus_int @ one_one_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se1409905431419307370or_int @ ( numeral_numeral_int @ M4 ) @ ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ N2 ) ) ) ) ) ) ).
% 0.34/0.69
% 0.34/0.69 % or_not_numerals(9)
% 0.34/0.69 thf(fact_7583_and__not__numerals_I8_J,axiom,
% 0.34/0.69 ! [M4: num,N2: num] :
% 0.34/0.69 ( ( bit_se725231765392027082nd_int @ ( numeral_numeral_int @ ( bit1 @ M4 ) ) @ ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ ( bit0 @ N2 ) ) ) )
% 0.34/0.69 = ( plus_plus_int @ one_one_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se725231765392027082nd_int @ ( numeral_numeral_int @ M4 ) @ ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ N2 ) ) ) ) ) ) ).
% 0.34/0.69
% 0.34/0.69 % and_not_numerals(8)
% 0.34/0.69 thf(fact_7584_bool__assn__proper_I4_J,axiom,
% 0.34/0.69 ! [P: produc3658429121746597890et_nat > $o,Q2: produc3658429121746597890et_nat > $o] :
% 0.34/0.69 ( ( proper @ P )
% 0.34/0.69 => ( ( proper @ Q2 )
% 0.34/0.69 => ( proper
% 0.34/0.69 @ ^ [H4: produc3658429121746597890et_nat] :
% 0.34/0.69 ( ( P @ H4 )
% 0.34/0.69 & ( Q2 @ H4 ) ) ) ) ) ).
% 0.34/0.69
% 0.34/0.69 % bool_assn_proper(4)
% 0.34/0.69 thf(fact_7585_bool__assn__proper_I3_J,axiom,
% 0.34/0.69 ! [P: produc3658429121746597890et_nat > $o,Q2: produc3658429121746597890et_nat > $o] :
% 0.34/0.69 ( ( proper @ P )
% 0.34/0.69 => ( ( proper @ Q2 )
% 0.34/0.69 => ( proper
% 0.34/0.69 @ ^ [H4: produc3658429121746597890et_nat] :
% 0.34/0.69 ( ( P @ H4 )
% 0.34/0.69 | ( Q2 @ H4 ) ) ) ) ) ).
% 0.34/0.69
% 0.34/0.69 % bool_assn_proper(3)
% 0.34/0.69 thf(fact_7586_bool__assn__proper_I2_J,axiom,
% 0.34/0.69 ( proper
% 0.34/0.69 @ ^ [Uu: produc3658429121746597890et_nat] : $false ) ).
% 0.34/0.69
% 0.34/0.69 % bool_assn_proper(2)
% 0.34/0.69 thf(fact_7587_bool__assn__proper_I5_J,axiom,
% 0.34/0.69 ! [P: produc3658429121746597890et_nat > $o] :
% 0.34/0.69 ( ( proper @ P )
% 0.34/0.69 => ( proper
% 0.34/0.69 @ ^ [H4: produc3658429121746597890et_nat] :
% 0.34/0.69 ( ( in_range @ H4 )
% 0.34/0.69 & ~ ( P @ H4 ) ) ) ) ).
% 0.34/0.69
% 0.34/0.69 % bool_assn_proper(5)
% 0.34/0.69 thf(fact_7588_bot__assn__def,axiom,
% 0.34/0.69 ( bot_bot_assn
% 0.34/0.69 = ( abs_assn
% 0.34/0.69 @ ^ [Uu: produc3658429121746597890et_nat] : $false ) ) ).
% 0.34/0.69
% 0.34/0.69 % bot_assn_def
% 0.34/0.69 thf(fact_7589_sup__assn__def,axiom,
% 0.34/0.69 ( sup_sup_assn
% 0.34/0.69 = ( ^ [P2: assn,Q: assn] :
% 0.34/0.69 ( abs_assn
% 0.34/0.69 @ ^ [H4: produc3658429121746597890et_nat] :
% 0.34/0.69 ( ( rep_assn @ P2 @ H4 )
% 0.34/0.69 | ( rep_assn @ Q @ H4 ) ) ) ) ) ).
% 0.34/0.69
% 0.34/0.69 % sup_assn_def
% 0.34/0.69 thf(fact_7590_inf__assn__def,axiom,
% 0.34/0.69 ( inf_inf_assn
% 0.34/0.69 = ( ^ [P2: assn,Q: assn] :
% 0.34/0.69 ( abs_assn
% 0.34/0.69 @ ^ [H4: produc3658429121746597890et_nat] :
% 0.34/0.69 ( ( rep_assn @ P2 @ H4 )
% 0.34/0.69 & ( rep_assn @ Q @ H4 ) ) ) ) ) ).
% 0.34/0.69
% 0.34/0.69 % inf_assn_def
% 0.34/0.69 thf(fact_7591_nat__times__as__int,axiom,
% 0.34/0.69 ( times_times_nat
% 0.34/0.69 = ( ^ [A6: nat,B6: nat] : ( nat2 @ ( times_times_int @ ( semiri1314217659103216013at_int @ A6 ) @ ( semiri1314217659103216013at_int @ B6 ) ) ) ) ) ).
% 0.34/0.69
% 0.34/0.69 % nat_times_as_int
% 0.34/0.69 thf(fact_7592_divide__nat__def,axiom,
% 0.34/0.69 ( divide_divide_nat
% 0.34/0.69 = ( ^ [M3: nat,N4: nat] :
% 0.34/0.69 ( if_nat @ ( N4 = zero_zero_nat ) @ zero_zero_nat
% 0.34/0.69 @ ( lattic8265883725875713057ax_nat
% 0.34/0.69 @ ( collect_nat
% 0.34/0.69 @ ^ [K5: nat] : ( ord_less_eq_nat @ ( times_times_nat @ K5 @ N4 ) @ M3 ) ) ) ) ) ) ).
% 0.34/0.69
% 0.34/0.69 % divide_nat_def
% 0.34/0.69 thf(fact_7593_uminus__assn__def,axiom,
% 0.34/0.69 ( uminus_uminus_assn
% 0.34/0.69 = ( ^ [P2: assn] :
% 0.34/0.69 ( abs_assn
% 0.34/0.69 @ ^ [H4: produc3658429121746597890et_nat] :
% 0.34/0.69 ( ( in_range @ H4 )
% 0.34/0.69 & ~ ( rep_assn @ P2 @ H4 ) ) ) ) ) ).
% 0.34/0.69
% 0.34/0.69 % uminus_assn_def
% 0.34/0.69 thf(fact_7594_Gcd__eq__Max,axiom,
% 0.34/0.69 ! [M: set_nat] :
% 0.34/0.69 ( ( finite_finite_nat @ M )
% 0.34/0.69 => ( ( M != bot_bot_set_nat )
% 0.34/0.69 => ( ~ ( member_nat @ zero_zero_nat @ M )
% 0.34/0.69 => ( ( gcd_Gcd_nat @ M )
% 0.34/0.69 = ( lattic8265883725875713057ax_nat
% 0.34/0.69 @ ( comple7806235888213564991et_nat
% 0.34/0.69 @ ( image_nat_set_nat
% 0.34/0.69 @ ^ [M3: nat] :
% 0.34/0.69 ( collect_nat
% 0.34/0.69 @ ^ [D4: nat] : ( dvd_dvd_nat @ D4 @ M3 ) )
% 0.34/0.69 @ M ) ) ) ) ) ) ) ).
% 0.34/0.69
% 0.34/0.69 % Gcd_eq_Max
% 0.34/0.69 thf(fact_7595_and__not__numerals_I5_J,axiom,
% 0.34/0.69 ! [M4: num,N2: num] :
% 0.34/0.69 ( ( bit_se725231765392027082nd_int @ ( numeral_numeral_int @ ( bit0 @ M4 ) ) @ ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ ( bit0 @ N2 ) ) ) )
% 0.34/0.69 = ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se725231765392027082nd_int @ ( numeral_numeral_int @ M4 ) @ ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ N2 ) ) ) ) ) ).
% 0.34/0.69
% 0.34/0.69 % and_not_numerals(5)
% 0.34/0.69 thf(fact_7596_and__not__numerals_I6_J,axiom,
% 0.34/0.69 ! [M4: num,N2: num] :
% 0.34/0.69 ( ( bit_se725231765392027082nd_int @ ( numeral_numeral_int @ ( bit0 @ M4 ) ) @ ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ ( bit1 @ N2 ) ) ) )
% 0.34/0.69 = ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se725231765392027082nd_int @ ( numeral_numeral_int @ M4 ) @ ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ N2 ) ) ) ) ) ).
% 0.34/0.69
% 0.34/0.69 % and_not_numerals(6)
% 0.34/0.69 thf(fact_7597_and__not__numerals_I9_J,axiom,
% 0.34/0.69 ! [M4: num,N2: num] :
% 0.34/0.69 ( ( bit_se725231765392027082nd_int @ ( numeral_numeral_int @ ( bit1 @ M4 ) ) @ ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ ( bit1 @ N2 ) ) ) )
% 0.34/0.69 = ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se725231765392027082nd_int @ ( numeral_numeral_int @ M4 ) @ ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ N2 ) ) ) ) ) ).
% 0.34/0.69
% 0.34/0.69 % and_not_numerals(9)
% 0.34/0.69 thf(fact_7598_or__not__numerals_I6_J,axiom,
% 0.34/0.69 ! [M4: num,N2: num] :
% 0.34/0.69 ( ( bit_se1409905431419307370or_int @ ( numeral_numeral_int @ ( bit0 @ M4 ) ) @ ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ ( bit1 @ N2 ) ) ) )
% 0.34/0.69 = ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se1409905431419307370or_int @ ( numeral_numeral_int @ M4 ) @ ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ N2 ) ) ) ) ) ).
% 0.34/0.69
% 0.34/0.69 % or_not_numerals(6)
% 0.34/0.69 thf(fact_7599_or__not__numerals_I5_J,axiom,
% 0.34/0.69 ! [M4: num,N2: num] :
% 0.34/0.69 ( ( bit_se1409905431419307370or_int @ ( numeral_numeral_int @ ( bit0 @ M4 ) ) @ ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ ( bit0 @ N2 ) ) ) )
% 0.34/0.69 = ( plus_plus_int @ one_one_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se1409905431419307370or_int @ ( numeral_numeral_int @ M4 ) @ ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ N2 ) ) ) ) ) ) ).
% 0.34/0.69
% 0.34/0.69 % or_not_numerals(5)
% 0.34/0.69 thf(fact_7600_atMost__0,axiom,
% 0.34/0.69 ( ( set_ord_atMost_nat @ zero_zero_nat )
% 0.34/0.69 = ( insert_nat @ zero_zero_nat @ bot_bot_set_nat ) ) ).
% 0.34/0.69
% 0.34/0.69 % atMost_0
% 0.34/0.69 thf(fact_7601_atLeast1__atMost__eq__remove0,axiom,
% 0.34/0.69 ! [N2: nat] :
% 0.34/0.69 ( ( set_or1269000886237332187st_nat @ ( suc @ zero_zero_nat ) @ N2 )
% 0.34/0.69 = ( minus_minus_set_nat @ ( set_ord_atMost_nat @ N2 ) @ ( insert_nat @ zero_zero_nat @ bot_bot_set_nat ) ) ) ).
% 0.34/0.69
% 0.34/0.69 % atLeast1_atMost_eq_remove0
% 0.34/0.69 thf(fact_7602_vandermonde,axiom,
% 0.34/0.69 ! [M4: nat,N2: nat,R2: nat] :
% 0.34/0.69 ( ( groups3542108847815614940at_nat
% 0.34/0.69 @ ^ [K5: nat] : ( times_times_nat @ ( binomial @ M4 @ K5 ) @ ( binomial @ N2 @ ( minus_minus_nat @ R2 @ K5 ) ) )
% 0.34/0.69 @ ( set_ord_atMost_nat @ R2 ) )
% 0.34/0.69 = ( binomial @ ( plus_plus_nat @ M4 @ N2 ) @ R2 ) ) ).
% 0.34/0.69
% 0.34/0.69 % vandermonde
% 0.34/0.69 thf(fact_7603_fact__eq__fact__times,axiom,
% 0.34/0.69 ! [N2: nat,M4: nat] :
% 0.34/0.69 ( ( ord_less_eq_nat @ N2 @ M4 )
% 0.34/0.69 => ( ( semiri1408675320244567234ct_nat @ M4 )
% 0.34/0.69 = ( times_times_nat @ ( semiri1408675320244567234ct_nat @ N2 )
% 0.34/0.69 @ ( groups708209901874060359at_nat
% 0.34/0.69 @ ^ [X2: nat] : X2
% 0.34/0.69 @ ( set_or1269000886237332187st_nat @ ( suc @ N2 ) @ M4 ) ) ) ) ) ).
% 0.34/0.69
% 0.34/0.69 % fact_eq_fact_times
% 0.34/0.69 thf(fact_7604_binomial,axiom,
% 0.34/0.69 ! [A2: nat,B: nat,N2: nat] :
% 0.34/0.69 ( ( power_power_nat @ ( plus_plus_nat @ A2 @ B ) @ N2 )
% 0.34/0.69 = ( groups3542108847815614940at_nat
% 0.34/0.69 @ ^ [K5: nat] : ( times_times_nat @ ( times_times_nat @ ( semiri1316708129612266289at_nat @ ( binomial @ N2 @ K5 ) ) @ ( power_power_nat @ A2 @ K5 ) ) @ ( power_power_nat @ B @ ( minus_minus_nat @ N2 @ K5 ) ) )
% 0.34/0.69 @ ( set_ord_atMost_nat @ N2 ) ) ) ).
% 0.34/0.69
% 0.34/0.69 % binomial
% 0.34/0.69 thf(fact_7605_choose__square__sum,axiom,
% 0.34/0.69 ! [N2: nat] :
% 0.34/0.69 ( ( groups3542108847815614940at_nat
% 0.34/0.69 @ ^ [K5: nat] : ( power_power_nat @ ( binomial @ N2 @ K5 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.34/0.69 @ ( set_ord_atMost_nat @ N2 ) )
% 0.34/0.69 = ( binomial @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) @ N2 ) ) ).
% 0.34/0.69
% 0.34/0.69 % choose_square_sum
% 0.34/0.69 thf(fact_7606_gauss__sum__nat,axiom,
% 0.34/0.69 ! [N2: nat] :
% 0.34/0.69 ( ( groups3542108847815614940at_nat
% 0.34/0.69 @ ^ [X2: nat] : X2
% 0.34/0.69 @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N2 ) )
% 0.34/0.69 = ( divide_divide_nat @ ( times_times_nat @ N2 @ ( suc @ N2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% 0.34/0.69
% 0.34/0.69 % gauss_sum_nat
% 0.34/0.69 thf(fact_7607_binomial__r__part__sum,axiom,
% 0.34/0.69 ! [M4: nat] :
% 0.34/0.69 ( ( groups3542108847815614940at_nat @ ( binomial @ ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M4 ) @ one_one_nat ) ) @ ( set_ord_atMost_nat @ M4 ) )
% 0.34/0.69 = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M4 ) ) ) ).
% 0.34/0.69
% 0.34/0.69 % binomial_r_part_sum
% 0.34/0.69 thf(fact_7608_choose__linear__sum,axiom,
% 0.34/0.69 ! [N2: nat] :
% 0.34/0.69 ( ( groups3542108847815614940at_nat
% 0.34/0.69 @ ^ [I4: nat] : ( times_times_nat @ I4 @ ( binomial @ N2 @ I4 ) )
% 0.34/0.69 @ ( set_ord_atMost_nat @ N2 ) )
% 0.34/0.69 = ( times_times_nat @ N2 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( minus_minus_nat @ N2 @ one_one_nat ) ) ) ) ).
% 0.34/0.69
% 0.34/0.69 % choose_linear_sum
% 0.34/0.69 thf(fact_7609_arith__series__nat,axiom,
% 0.34/0.69 ! [A2: nat,D2: nat,N2: nat] :
% 0.34/0.69 ( ( groups3542108847815614940at_nat
% 0.34/0.69 @ ^ [I4: nat] : ( plus_plus_nat @ A2 @ ( times_times_nat @ I4 @ D2 ) )
% 0.34/0.69 @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N2 ) )
% 0.34/0.69 = ( divide_divide_nat @ ( times_times_nat @ ( suc @ N2 ) @ ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A2 ) @ ( times_times_nat @ N2 @ D2 ) ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% 0.34/0.69
% 0.34/0.69 % arith_series_nat
% 0.34/0.69 thf(fact_7610_Sum__Icc__nat,axiom,
% 0.34/0.69 ! [M4: nat,N2: nat] :
% 0.34/0.69 ( ( groups3542108847815614940at_nat
% 0.34/0.69 @ ^ [X2: nat] : X2
% 0.34/0.69 @ ( set_or1269000886237332187st_nat @ M4 @ N2 ) )
% 0.34/0.69 = ( divide_divide_nat @ ( minus_minus_nat @ ( times_times_nat @ N2 @ ( plus_plus_nat @ N2 @ one_one_nat ) ) @ ( times_times_nat @ M4 @ ( minus_minus_nat @ M4 @ one_one_nat ) ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% 0.34/0.69
% 0.34/0.69 % Sum_Icc_nat
% 0.34/0.69 thf(fact_7611_periodic__finite__ex,axiom,
% 0.34/0.69 ! [D2: int,P: int > $o] :
% 0.34/0.69 ( ( ord_less_int @ zero_zero_int @ D2 )
% 0.34/0.69 => ( ! [X3: int,K: int] :
% 0.34/0.69 ( ( P @ X3 )
% 0.34/0.69 = ( P @ ( minus_minus_int @ X3 @ ( times_times_int @ K @ D2 ) ) ) )
% 0.34/0.69 => ( ( ? [X7: int] : ( P @ X7 ) )
% 0.34/0.69 = ( ? [X2: int] :
% 0.34/0.69 ( ( member_int @ X2 @ ( set_or1266510415728281911st_int @ one_one_int @ D2 ) )
% 0.34/0.69 & ( P @ X2 ) ) ) ) ) ) ).
% 0.34/0.69
% 0.34/0.69 % periodic_finite_ex
% 0.34/0.69 thf(fact_7612_simp__from__to,axiom,
% 0.34/0.69 ( set_or1266510415728281911st_int
% 0.34/0.69 = ( ^ [I4: int,J3: int] : ( if_set_int @ ( ord_less_int @ J3 @ I4 ) @ bot_bot_set_int @ ( insert_int @ I4 @ ( set_or1266510415728281911st_int @ ( plus_plus_int @ I4 @ one_one_int ) @ J3 ) ) ) ) ) ).
% 0.34/0.69
% 0.34/0.69 % simp_from_to
% 0.34/0.69 thf(fact_7613_cpmi,axiom,
% 0.34/0.69 ! [D3: int,P: int > $o,P7: int > $o,B4: set_int] :
% 0.34/0.69 ( ( ord_less_int @ zero_zero_int @ D3 )
% 0.34/0.69 => ( ? [Z6: int] :
% 0.34/0.69 ! [X3: int] :
% 0.34/0.69 ( ( ord_less_int @ X3 @ Z6 )
% 0.34/0.69 => ( ( P @ X3 )
% 0.34/0.69 = ( P7 @ X3 ) ) )
% 0.34/0.69 => ( ! [X3: int] :
% 0.34/0.69 ( ! [Xa2: int] :
% 0.34/0.69 ( ( member_int @ Xa2 @ ( set_or1266510415728281911st_int @ one_one_int @ D3 ) )
% 0.34/0.69 => ! [Xb2: int] :
% 0.34/0.69 ( ( member_int @ Xb2 @ B4 )
% 0.34/0.69 => ( X3
% 0.34/0.69 != ( plus_plus_int @ Xb2 @ Xa2 ) ) ) )
% 0.34/0.69 => ( ( P @ X3 )
% 0.34/0.69 => ( P @ ( minus_minus_int @ X3 @ D3 ) ) ) )
% 0.34/0.69 => ( ! [X3: int,K: int] :
% 0.34/0.69 ( ( P7 @ X3 )
% 0.34/0.69 = ( P7 @ ( minus_minus_int @ X3 @ ( times_times_int @ K @ D3 ) ) ) )
% 0.34/0.69 => ( ( ? [X7: int] : ( P @ X7 ) )
% 0.34/0.69 = ( ? [X2: int] :
% 0.34/0.69 ( ( member_int @ X2 @ ( set_or1266510415728281911st_int @ one_one_int @ D3 ) )
% 0.34/0.69 & ( P7 @ X2 ) )
% 0.34/0.69 | ? [X2: int] :
% 0.34/0.69 ( ( member_int @ X2 @ ( set_or1266510415728281911st_int @ one_one_int @ D3 ) )
% 0.34/0.69 & ? [Y3: int] :
% 0.34/0.69 ( ( member_int @ Y3 @ B4 )
% 0.34/0.69 & ( P @ ( plus_plus_int @ Y3 @ X2 ) ) ) ) ) ) ) ) ) ) ).
% 0.34/0.69
% 0.34/0.69 % cpmi
% 0.34/0.69 thf(fact_7614_cppi,axiom,
% 0.34/0.69 ! [D3: int,P: int > $o,P7: int > $o,A: set_int] :
% 0.34/0.69 ( ( ord_less_int @ zero_zero_int @ D3 )
% 0.34/0.69 => ( ? [Z6: int] :
% 0.34/0.69 ! [X3: int] :
% 0.34/0.69 ( ( ord_less_int @ Z6 @ X3 )
% 0.34/0.69 => ( ( P @ X3 )
% 0.34/0.69 = ( P7 @ X3 ) ) )
% 0.34/0.69 => ( ! [X3: int] :
% 0.34/0.69 ( ! [Xa2: int] :
% 0.34/0.69 ( ( member_int @ Xa2 @ ( set_or1266510415728281911st_int @ one_one_int @ D3 ) )
% 0.34/0.69 => ! [Xb2: int] :
% 0.34/0.69 ( ( member_int @ Xb2 @ A )
% 0.34/0.69 => ( X3
% 0.34/0.69 != ( minus_minus_int @ Xb2 @ Xa2 ) ) ) )
% 0.34/0.69 => ( ( P @ X3 )
% 0.34/0.69 => ( P @ ( plus_plus_int @ X3 @ D3 ) ) ) )
% 0.34/0.69 => ( ! [X3: int,K: int] :
% 0.34/0.69 ( ( P7 @ X3 )
% 0.34/0.69 = ( P7 @ ( minus_minus_int @ X3 @ ( times_times_int @ K @ D3 ) ) ) )
% 0.34/0.69 => ( ( ? [X7: int] : ( P @ X7 ) )
% 0.34/0.69 = ( ? [X2: int] :
% 0.34/0.69 ( ( member_int @ X2 @ ( set_or1266510415728281911st_int @ one_one_int @ D3 ) )
% 0.34/0.69 & ( P7 @ X2 ) )
% 0.34/0.69 | ? [X2: int] :
% 0.34/0.69 ( ( member_int @ X2 @ ( set_or1266510415728281911st_int @ one_one_int @ D3 ) )
% 0.34/0.69 & ? [Y3: int] :
% 0.34/0.69 ( ( member_int @ Y3 @ A )
% 0.34/0.69 & ( P @ ( minus_minus_int @ Y3 @ X2 ) ) ) ) ) ) ) ) ) ) ).
% 0.34/0.69
% 0.34/0.69 % cppi
% 0.34/0.69 thf(fact_7615_divmod__step__nat__def,axiom,
% 0.34/0.69 ( unique5026877609467782581ep_nat
% 0.34/0.69 = ( ^ [L2: num] :
% 0.34/0.69 ( produc2626176000494625587at_nat
% 0.34/0.69 @ ^ [Q6: nat,R4: nat] : ( if_Pro6206227464963214023at_nat @ ( ord_less_eq_nat @ ( numeral_numeral_nat @ L2 ) @ R4 ) @ ( product_Pair_nat_nat @ ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Q6 ) @ one_one_nat ) @ ( minus_minus_nat @ R4 @ ( numeral_numeral_nat @ L2 ) ) ) @ ( product_Pair_nat_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Q6 ) @ R4 ) ) ) ) ) ).
% 0.34/0.69
% 0.34/0.69 % divmod_step_nat_def
% 0.34/0.69 thf(fact_7616_divmod__step__int__def,axiom,
% 0.34/0.69 ( unique5024387138958732305ep_int
% 0.34/0.69 = ( ^ [L2: num] :
% 0.34/0.69 ( produc4245557441103728435nt_int
% 0.34/0.69 @ ^ [Q6: int,R4: int] : ( if_Pro3027730157355071871nt_int @ ( ord_less_eq_int @ ( numeral_numeral_int @ L2 ) @ R4 ) @ ( product_Pair_int_int @ ( plus_plus_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ Q6 ) @ one_one_int ) @ ( minus_minus_int @ R4 @ ( numeral_numeral_int @ L2 ) ) ) @ ( product_Pair_int_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ Q6 ) @ R4 ) ) ) ) ) ).
% 0.34/0.69
% 0.34/0.69 % divmod_step_int_def
% 0.34/0.69 thf(fact_7617_Sum__Icc__int,axiom,
% 0.34/0.69 ! [M4: int,N2: int] :
% 0.34/0.69 ( ( ord_less_eq_int @ M4 @ N2 )
% 0.34/0.69 => ( ( groups4538972089207619220nt_int
% 0.34/0.69 @ ^ [X2: int] : X2
% 0.34/0.69 @ ( set_or1266510415728281911st_int @ M4 @ N2 ) )
% 0.34/0.69 = ( divide_divide_int @ ( minus_minus_int @ ( times_times_int @ N2 @ ( plus_plus_int @ N2 @ one_one_int ) ) @ ( times_times_int @ M4 @ ( minus_minus_int @ M4 @ one_one_int ) ) ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ).
% 0.34/0.69
% 0.34/0.69 % Sum_Icc_int
% 0.34/0.69 thf(fact_7618_Sum__Ico__nat,axiom,
% 0.34/0.69 ! [M4: nat,N2: nat] :
% 0.34/0.69 ( ( groups3542108847815614940at_nat
% 0.34/0.69 @ ^ [X2: nat] : X2
% 0.34/0.69 @ ( set_or4665077453230672383an_nat @ M4 @ N2 ) )
% 0.34/0.69 = ( divide_divide_nat @ ( minus_minus_nat @ ( times_times_nat @ N2 @ ( minus_minus_nat @ N2 @ one_one_nat ) ) @ ( times_times_nat @ M4 @ ( minus_minus_nat @ M4 @ one_one_nat ) ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% 0.34/0.69
% 0.34/0.69 % Sum_Ico_nat
% 0.34/0.69 thf(fact_7619_atLeastLessThan__singleton,axiom,
% 0.34/0.69 ! [M4: nat] :
% 0.34/0.69 ( ( set_or4665077453230672383an_nat @ M4 @ ( suc @ M4 ) )
% 0.34/0.69 = ( insert_nat @ M4 @ bot_bot_set_nat ) ) ).
% 0.34/0.69
% 0.34/0.69 % atLeastLessThan_singleton
% 0.34/0.69 thf(fact_7620_atLeastLessThan0,axiom,
% 0.34/0.69 ! [M4: nat] :
% 0.34/0.69 ( ( set_or4665077453230672383an_nat @ M4 @ zero_zero_nat )
% 0.34/0.69 = bot_bot_set_nat ) ).
% 0.34/0.69
% 0.34/0.69 % atLeastLessThan0
% 0.34/0.69 thf(fact_7621_atLeastLessThanSuc,axiom,
% 0.34/0.69 ! [M4: nat,N2: nat] :
% 0.34/0.69 ( ( ( ord_less_eq_nat @ M4 @ N2 )
% 0.34/0.69 => ( ( set_or4665077453230672383an_nat @ M4 @ ( suc @ N2 ) )
% 0.34/0.69 = ( insert_nat @ N2 @ ( set_or4665077453230672383an_nat @ M4 @ N2 ) ) ) )
% 0.34/0.69 & ( ~ ( ord_less_eq_nat @ M4 @ N2 )
% 0.34/0.69 => ( ( set_or4665077453230672383an_nat @ M4 @ ( suc @ N2 ) )
% 0.34/0.69 = bot_bot_set_nat ) ) ) ).
% 0.34/0.69
% 0.34/0.69 % atLeastLessThanSuc
% 0.34/0.69 thf(fact_7622_atLeastLessThan__nat__numeral,axiom,
% 0.34/0.69 ! [M4: nat,K3: num] :
% 0.34/0.69 ( ( ( ord_less_eq_nat @ M4 @ ( pred_numeral @ K3 ) )
% 0.34/0.69 => ( ( set_or4665077453230672383an_nat @ M4 @ ( numeral_numeral_nat @ K3 ) )
% 0.34/0.69 = ( insert_nat @ ( pred_numeral @ K3 ) @ ( set_or4665077453230672383an_nat @ M4 @ ( pred_numeral @ K3 ) ) ) ) )
% 0.34/0.69 & ( ~ ( ord_less_eq_nat @ M4 @ ( pred_numeral @ K3 ) )
% 0.34/0.69 => ( ( set_or4665077453230672383an_nat @ M4 @ ( numeral_numeral_nat @ K3 ) )
% 0.34/0.69 = bot_bot_set_nat ) ) ) ).
% 0.34/0.69
% 0.34/0.69 % atLeastLessThan_nat_numeral
% 0.34/0.69 thf(fact_7623_image__minus__const__atLeastLessThan__nat,axiom,
% 0.34/0.69 ! [C: nat,Y: nat,X: nat] :
% 0.34/0.69 ( ( ( ord_less_nat @ C @ Y )
% 0.34/0.69 => ( ( image_nat_nat2
% 0.34/0.69 @ ^ [I4: nat] : ( minus_minus_nat @ I4 @ C )
% 0.34/0.69 @ ( set_or4665077453230672383an_nat @ X @ Y ) )
% 0.34/0.69 = ( set_or4665077453230672383an_nat @ ( minus_minus_nat @ X @ C ) @ ( minus_minus_nat @ Y @ C ) ) ) )
% 0.34/0.69 & ( ~ ( ord_less_nat @ C @ Y )
% 0.34/0.69 => ( ( ( ord_less_nat @ X @ Y )
% 0.34/0.69 => ( ( image_nat_nat2
% 0.34/0.69 @ ^ [I4: nat] : ( minus_minus_nat @ I4 @ C )
% 0.34/0.69 @ ( set_or4665077453230672383an_nat @ X @ Y ) )
% 0.34/0.69 = ( insert_nat @ zero_zero_nat @ bot_bot_set_nat ) ) )
% 0.34/0.69 & ( ~ ( ord_less_nat @ X @ Y )
% 0.34/0.69 => ( ( image_nat_nat2
% 0.34/0.69 @ ^ [I4: nat] : ( minus_minus_nat @ I4 @ C )
% 0.34/0.69 @ ( set_or4665077453230672383an_nat @ X @ Y ) )
% 0.34/0.69 = bot_bot_set_nat ) ) ) ) ) ).
% 0.34/0.69
% 0.34/0.69 % image_minus_const_atLeastLessThan_nat
% 0.34/0.69 thf(fact_7624_divmod__step__integer__def,axiom,
% 0.34/0.69 ( unique4921790084139445826nteger
% 0.34/0.69 = ( ^ [L2: num] :
% 0.34/0.69 ( produc6916734918728496179nteger
% 0.34/0.69 @ ^ [Q6: code_integer,R4: code_integer] : ( if_Pro6119634080678213985nteger @ ( ord_le3102999989581377725nteger @ ( numera6620942414471956472nteger @ L2 ) @ R4 ) @ ( produc1086072967326762835nteger @ ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ Q6 ) @ one_one_Code_integer ) @ ( minus_8373710615458151222nteger @ R4 @ ( numera6620942414471956472nteger @ L2 ) ) ) @ ( produc1086072967326762835nteger @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ Q6 ) @ R4 ) ) ) ) ) ).
% 0.34/0.69
% 0.34/0.69 % divmod_step_integer_def
% 0.34/0.69 thf(fact_7625_triangle__def,axiom,
% 0.34/0.69 ( nat_triangle
% 0.34/0.69 = ( ^ [N4: nat] : ( divide_divide_nat @ ( times_times_nat @ N4 @ ( suc @ N4 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% 0.34/0.69
% 0.34/0.69 % triangle_def
% 0.34/0.69 thf(fact_7626_image__add__integer__atLeastLessThan,axiom,
% 0.34/0.69 ! [L: code_integer,U: code_integer] :
% 0.34/0.69 ( ( image_4470545334726330049nteger
% 0.34/0.69 @ ^ [X2: code_integer] : ( plus_p5714425477246183910nteger @ X2 @ L )
% 0.34/0.69 @ ( set_or8404916559141939852nteger @ zero_z3403309356797280102nteger @ ( minus_8373710615458151222nteger @ U @ L ) ) )
% 0.34/0.69 = ( set_or8404916559141939852nteger @ L @ U ) ) ).
% 0.34/0.69
% 0.34/0.69 % image_add_integer_atLeastLessThan
% 0.34/0.69 thf(fact_7627_atLeastLessThanPlusOne__atLeastAtMost__integer,axiom,
% 0.34/0.69 ! [L: code_integer,U: code_integer] :
% 0.34/0.69 ( ( set_or8404916559141939852nteger @ L @ ( plus_p5714425477246183910nteger @ U @ one_one_Code_integer ) )
% 0.34/0.69 = ( set_or189985376899183464nteger @ L @ U ) ) ).
% 0.34/0.69
% 0.34/0.69 % atLeastLessThanPlusOne_atLeastAtMost_integer
% 0.34/0.69 thf(fact_7628_times__integer__code_I2_J,axiom,
% 0.34/0.69 ! [L: code_integer] :
% 0.34/0.69 ( ( times_3573771949741848930nteger @ zero_z3403309356797280102nteger @ L )
% 0.34/0.69 = zero_z3403309356797280102nteger ) ).
% 0.34/0.69
% 0.34/0.69 % times_integer_code(2)
% 0.34/0.69 thf(fact_7629_times__integer__code_I1_J,axiom,
% 0.34/0.69 ! [K3: code_integer] :
% 0.34/0.69 ( ( times_3573771949741848930nteger @ K3 @ zero_z3403309356797280102nteger )
% 0.34/0.69 = zero_z3403309356797280102nteger ) ).
% 0.34/0.69
% 0.34/0.69 % times_integer_code(1)
% 0.34/0.69 thf(fact_7630_integer__of__int__code,axiom,
% 0.34/0.69 ( code_integer_of_int
% 0.34/0.69 = ( ^ [K5: int] :
% 0.34/0.69 ( if_Code_integer @ ( ord_less_int @ K5 @ zero_zero_int ) @ ( uminus1351360451143612070nteger @ ( code_integer_of_int @ ( uminus_uminus_int @ K5 ) ) )
% 0.34/0.69 @ ( if_Code_integer @ ( K5 = zero_zero_int ) @ zero_z3403309356797280102nteger
% 0.34/0.69 @ ( if_Code_integer
% 0.34/0.69 @ ( ( modulo_modulo_int @ K5 @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
% 0.34/0.69 = zero_zero_int )
% 0.34/0.69 @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( code_integer_of_int @ ( divide_divide_int @ K5 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) )
% 0.34/0.69 @ ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( code_integer_of_int @ ( divide_divide_int @ K5 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) @ one_one_Code_integer ) ) ) ) ) ) ).
% 0.34/0.69
% 0.34/0.69 % integer_of_int_code
% 0.34/0.69 thf(fact_7631_finite__atLeastLessThan__integer,axiom,
% 0.34/0.69 ! [L: code_integer,U: code_integer] : ( finite6017078050557962740nteger @ ( set_or8404916559141939852nteger @ L @ U ) ) ).
% 0.34/0.69
% 0.34/0.69 % finite_atLeastLessThan_integer
% 0.34/0.69 thf(fact_7632_finite__atLeastAtMost__integer,axiom,
% 0.34/0.69 ! [L: code_integer,U: code_integer] : ( finite6017078050557962740nteger @ ( set_or189985376899183464nteger @ L @ U ) ) ).
% 0.34/0.69
% 0.34/0.69 % finite_atLeastAtMost_integer
% 0.34/0.69 thf(fact_7633_finite__atLeastZeroLessThan__integer,axiom,
% 0.34/0.69 ! [U: code_integer] : ( finite6017078050557962740nteger @ ( set_or8404916559141939852nteger @ zero_z3403309356797280102nteger @ U ) ) ).
% 0.34/0.69
% 0.34/0.69 % finite_atLeastZeroLessThan_integer
% 0.34/0.69 thf(fact_7634_times__integer_Orsp,axiom,
% 0.34/0.69 ( bNF_re711492959462206631nt_int
% 0.34/0.69 @ ^ [Y4: int,Z4: int] : ( Y4 = Z4 )
% 0.34/0.69 @ ( bNF_re4712519889275205905nt_int
% 0.34/0.69 @ ^ [Y4: int,Z4: int] : ( Y4 = Z4 )
% 0.34/0.69 @ ^ [Y4: int,Z4: int] : ( Y4 = Z4 ) )
% 0.34/0.69 @ times_times_int
% 0.34/0.69 @ times_times_int ) ).
% 0.34/0.69
% 0.34/0.69 % times_integer.rsp
% 0.34/0.69 thf(fact_7635_times__natural_Orsp,axiom,
% 0.34/0.69 ( bNF_re1345281282404953727at_nat
% 0.34/0.69 @ ^ [Y4: nat,Z4: nat] : ( Y4 = Z4 )
% 0.34/0.69 @ ( bNF_re5653821019739307937at_nat
% 0.34/0.69 @ ^ [Y4: nat,Z4: nat] : ( Y4 = Z4 )
% 0.34/0.69 @ ^ [Y4: nat,Z4: nat] : ( Y4 = Z4 ) )
% 0.34/0.69 @ times_times_nat
% 0.34/0.69 @ times_times_nat ) ).
% 0.34/0.69
% 0.34/0.69 % times_natural.rsp
% 0.34/0.69 thf(fact_7636_times__integer_Oabs__eq,axiom,
% 0.34/0.69 ! [Xa: int,X: int] :
% 0.34/0.69 ( ( times_3573771949741848930nteger @ ( code_integer_of_int @ Xa ) @ ( code_integer_of_int @ X ) )
% 0.34/0.69 = ( code_integer_of_int @ ( times_times_int @ Xa @ X ) ) ) ).
% 0.34/0.69
% 0.34/0.69 % times_integer.abs_eq
% 0.34/0.69 thf(fact_7637_divmod__integer__eq__cases,axiom,
% 0.34/0.69 ( code_divmod_integer
% 0.34/0.69 = ( ^ [K5: code_integer,L2: code_integer] :
% 0.34/0.69 ( if_Pro6119634080678213985nteger @ ( K5 = zero_z3403309356797280102nteger ) @ ( produc1086072967326762835nteger @ zero_z3403309356797280102nteger @ zero_z3403309356797280102nteger )
% 0.34/0.69 @ ( if_Pro6119634080678213985nteger @ ( L2 = zero_z3403309356797280102nteger ) @ ( produc1086072967326762835nteger @ zero_z3403309356797280102nteger @ K5 )
% 0.34/0.69 @ ( comp_C1593894019821074884nteger @ ( comp_C8797469213163452608nteger @ produc6499014454317279255nteger @ times_3573771949741848930nteger ) @ sgn_sgn_Code_integer @ L2
% 0.34/0.69 @ ( if_Pro6119634080678213985nteger
% 0.34/0.69 @ ( ( sgn_sgn_Code_integer @ K5 )
% 0.34/0.69 = ( sgn_sgn_Code_integer @ L2 ) )
% 0.34/0.69 @ ( code_divmod_abs @ K5 @ L2 )
% 0.34/0.69 @ ( produc6916734918728496179nteger
% 0.34/0.69 @ ^ [R4: code_integer,S7: code_integer] : ( if_Pro6119634080678213985nteger @ ( S7 = zero_z3403309356797280102nteger ) @ ( produc1086072967326762835nteger @ ( uminus1351360451143612070nteger @ R4 ) @ zero_z3403309356797280102nteger ) @ ( produc1086072967326762835nteger @ ( minus_8373710615458151222nteger @ ( uminus1351360451143612070nteger @ R4 ) @ one_one_Code_integer ) @ ( minus_8373710615458151222nteger @ ( abs_abs_Code_integer @ L2 ) @ S7 ) ) )
% 0.34/0.69 @ ( code_divmod_abs @ K5 @ L2 ) ) ) ) ) ) ) ) ).
% 0.34/0.69
% 0.34/0.69 % divmod_integer_eq_cases
% 0.34/0.69 thf(fact_7638_plus__rat_Otransfer,axiom,
% 0.34/0.69 ( bNF_re7627151682743391978at_rat @ pcr_rat @ ( bNF_re8279943556446156061nt_rat @ pcr_rat @ pcr_rat )
% 0.34/0.69 @ ^ [X2: product_prod_int_int,Y3: product_prod_int_int] : ( product_Pair_int_int @ ( plus_plus_int @ ( times_times_int @ ( product_fst_int_int @ X2 ) @ ( product_snd_int_int @ Y3 ) ) @ ( times_times_int @ ( product_fst_int_int @ Y3 ) @ ( product_snd_int_int @ X2 ) ) ) @ ( times_times_int @ ( product_snd_int_int @ X2 ) @ ( product_snd_int_int @ Y3 ) ) )
% 0.34/0.69 @ plus_plus_rat ) ).
% 0.34/0.69
% 0.34/0.69 % plus_rat.transfer
% 0.34/0.69 thf(fact_7639_times__rat_Otransfer,axiom,
% 0.34/0.69 ( bNF_re7627151682743391978at_rat @ pcr_rat @ ( bNF_re8279943556446156061nt_rat @ pcr_rat @ pcr_rat )
% 0.34/0.69 @ ^ [X2: product_prod_int_int,Y3: product_prod_int_int] : ( product_Pair_int_int @ ( times_times_int @ ( product_fst_int_int @ X2 ) @ ( product_fst_int_int @ Y3 ) ) @ ( times_times_int @ ( product_snd_int_int @ X2 ) @ ( product_snd_int_int @ Y3 ) ) )
% 0.34/0.69 @ times_times_rat ) ).
% 0.34/0.69
% 0.34/0.69 % times_rat.transfer
% 0.34/0.69 thf(fact_7640_int__of__integer__code,axiom,
% 0.34/0.69 ( code_int_of_integer
% 0.34/0.69 = ( ^ [K5: code_integer] :
% 0.34/0.69 ( if_int @ ( ord_le6747313008572928689nteger @ K5 @ zero_z3403309356797280102nteger ) @ ( uminus_uminus_int @ ( code_int_of_integer @ ( uminus1351360451143612070nteger @ K5 ) ) )
% 0.34/0.69 @ ( if_int @ ( K5 = zero_z3403309356797280102nteger ) @ zero_zero_int
% 0.34/0.69 @ ( produc1553301316500091796er_int
% 0.34/0.69 @ ^ [L2: code_integer,J3: code_integer] : ( if_int @ ( J3 = zero_z3403309356797280102nteger ) @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( code_int_of_integer @ L2 ) ) @ ( plus_plus_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( code_int_of_integer @ L2 ) ) @ one_one_int ) )
% 0.34/0.69 @ ( code_divmod_integer @ K5 @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) ) ) ) ) ).
% 0.34/0.69
% 0.34/0.69 % int_of_integer_code
% 0.34/0.69 thf(fact_7641_positive_Otransfer,axiom,
% 0.34/0.69 ( bNF_re1494630372529172596at_o_o @ pcr_rat
% 0.34/0.69 @ ^ [Y4: $o,Z4: $o] : ( Y4 = Z4 )
% 0.34/0.69 @ ^ [X2: product_prod_int_int] : ( ord_less_int @ zero_zero_int @ ( times_times_int @ ( product_fst_int_int @ X2 ) @ ( product_snd_int_int @ X2 ) ) )
% 0.34/0.69 @ positive ) ).
% 0.34/0.69
% 0.34/0.69 % positive.transfer
% 0.34/0.69 thf(fact_7642_times__int_Otransfer,axiom,
% 0.34/0.69 ( bNF_re7408651293131936558nt_int @ pcr_int @ ( bNF_re7400052026677387805at_int @ pcr_int @ pcr_int )
% 0.34/0.69 @ ( produc27273713700761075at_nat
% 0.34/0.69 @ ^ [X2: nat,Y3: nat] :
% 0.34/0.69 ( produc2626176000494625587at_nat
% 0.34/0.69 @ ^ [U3: nat,V4: nat] : ( product_Pair_nat_nat @ ( plus_plus_nat @ ( times_times_nat @ X2 @ U3 ) @ ( times_times_nat @ Y3 @ V4 ) ) @ ( plus_plus_nat @ ( times_times_nat @ X2 @ V4 ) @ ( times_times_nat @ Y3 @ U3 ) ) ) ) )
% 0.34/0.69 @ times_times_int ) ).
% 0.34/0.69
% 0.34/0.69 % times_int.transfer
% 0.34/0.69 thf(fact_7643_times__int_Oabs__eq,axiom,
% 0.34/0.69 ! [Xa: product_prod_nat_nat,X: product_prod_nat_nat] :
% 0.34/0.69 ( ( times_times_int @ ( abs_Integ @ Xa ) @ ( abs_Integ @ X ) )
% 0.34/0.69 = ( abs_Integ
% 0.34/0.69 @ ( produc27273713700761075at_nat
% 0.34/0.69 @ ^ [X2: nat,Y3: nat] :
% 0.34/0.69 ( produc2626176000494625587at_nat
% 0.34/0.69 @ ^ [U3: nat,V4: nat] : ( product_Pair_nat_nat @ ( plus_plus_nat @ ( times_times_nat @ X2 @ U3 ) @ ( times_times_nat @ Y3 @ V4 ) ) @ ( plus_plus_nat @ ( times_times_nat @ X2 @ V4 ) @ ( times_times_nat @ Y3 @ U3 ) ) ) )
% 0.34/0.69 @ Xa
% 0.34/0.69 @ X ) ) ) ).
% 0.34/0.69
% 0.34/0.69 % times_int.abs_eq
% 0.34/0.69 thf(fact_7644_times__integer_Orep__eq,axiom,
% 0.34/0.69 ! [X: code_integer,Xa: code_integer] :
% 0.34/0.69 ( ( code_int_of_integer @ ( times_3573771949741848930nteger @ X @ Xa ) )
% 0.34/0.69 = ( times_times_int @ ( code_int_of_integer @ X ) @ ( code_int_of_integer @ Xa ) ) ) ).
% 0.34/0.69
% 0.34/0.69 % times_integer.rep_eq
% 0.34/0.69 thf(fact_7645_positive__mult,axiom,
% 0.34/0.69 ! [X: rat,Y: rat] :
% 0.34/0.69 ( ( positive @ X )
% 0.34/0.69 => ( ( positive @ Y )
% 0.34/0.69 => ( positive @ ( times_times_rat @ X @ Y ) ) ) ) ).
% 0.34/0.69
% 0.34/0.69 % positive_mult
% 0.34/0.69 thf(fact_7646_int__of__integer__less__iff,axiom,
% 0.34/0.69 ! [X: code_integer,Y: code_integer] :
% 0.34/0.69 ( ( ord_less_int @ ( code_int_of_integer @ X ) @ ( code_int_of_integer @ Y ) )
% 0.34/0.69 = ( ord_le6747313008572928689nteger @ X @ Y ) ) ).
% 0.34/0.69
% 0.34/0.69 % int_of_integer_less_iff
% 0.34/0.69 thf(fact_7647_type__definition__integer,axiom,
% 0.34/0.69 type_d8366093980585677751er_int @ code_int_of_integer @ code_integer_of_int @ top_top_set_int ).
% 0.34/0.69
% 0.34/0.69 % type_definition_integer
% 0.34/0.69 thf(fact_7648_positive_Orep__eq,axiom,
% 0.34/0.70 ( positive
% 0.34/0.70 = ( ^ [X2: rat] : ( ord_less_int @ zero_zero_int @ ( times_times_int @ ( product_fst_int_int @ ( rep_Rat @ X2 ) ) @ ( product_snd_int_int @ ( rep_Rat @ X2 ) ) ) ) ) ) ).
% 0.34/0.70
% 0.34/0.70 % positive.rep_eq
% 0.34/0.70 thf(fact_7649_nat__of__integer__less__iff,axiom,
% 0.34/0.70 ! [X: code_integer,Y: code_integer] :
% 0.34/0.70 ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ X )
% 0.34/0.70 => ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ Y )
% 0.34/0.70 => ( ( ord_less_nat @ ( code_nat_of_integer @ X ) @ ( code_nat_of_integer @ Y ) )
% 0.34/0.70 = ( ord_le6747313008572928689nteger @ X @ Y ) ) ) ) ).
% 0.34/0.70
% 0.34/0.70 % nat_of_integer_less_iff
% 0.34/0.70 thf(fact_7650_times__int_Orsp,axiom,
% 0.34/0.70 ( bNF_re3099431351363272937at_nat @ intrel @ ( bNF_re2241393799969408733at_nat @ intrel @ intrel )
% 0.34/0.70 @ ( produc27273713700761075at_nat
% 0.34/0.70 @ ^ [X2: nat,Y3: nat] :
% 0.34/0.70 ( produc2626176000494625587at_nat
% 0.34/0.70 @ ^ [U3: nat,V4: nat] : ( product_Pair_nat_nat @ ( plus_plus_nat @ ( times_times_nat @ X2 @ U3 ) @ ( times_times_nat @ Y3 @ V4 ) ) @ ( plus_plus_nat @ ( times_times_nat @ X2 @ V4 ) @ ( times_times_nat @ Y3 @ U3 ) ) ) ) )
% 0.34/0.70 @ ( produc27273713700761075at_nat
% 0.34/0.70 @ ^ [X2: nat,Y3: nat] :
% 0.34/0.70 ( produc2626176000494625587at_nat
% 0.34/0.70 @ ^ [U3: nat,V4: nat] : ( product_Pair_nat_nat @ ( plus_plus_nat @ ( times_times_nat @ X2 @ U3 ) @ ( times_times_nat @ Y3 @ V4 ) ) @ ( plus_plus_nat @ ( times_times_nat @ X2 @ V4 ) @ ( times_times_nat @ Y3 @ U3 ) ) ) ) ) ) ).
% 0.34/0.70
% 0.34/0.70 % times_int.rsp
% 0.34/0.70 thf(fact_7651_atLeast1__lessThan__eq__remove0,axiom,
% 0.34/0.70 ! [N2: nat] :
% 0.34/0.70 ( ( set_or4665077453230672383an_nat @ ( suc @ zero_zero_nat ) @ N2 )
% 0.34/0.70 = ( minus_minus_set_nat @ ( set_ord_lessThan_nat @ N2 ) @ ( insert_nat @ zero_zero_nat @ bot_bot_set_nat ) ) ) ).
% 0.34/0.70
% 0.34/0.70 % atLeast1_lessThan_eq_remove0
% 0.34/0.70 thf(fact_7652_min__Suc__gt_I1_J,axiom,
% 0.34/0.70 ! [A2: nat,B: nat] :
% 0.34/0.70 ( ( ord_less_nat @ A2 @ B )
% 0.34/0.70 => ( ( ord_min_nat @ ( suc @ A2 ) @ B )
% 0.34/0.70 = ( suc @ A2 ) ) ) ).
% 0.34/0.70
% 0.34/0.70 % min_Suc_gt(1)
% 0.34/0.70 thf(fact_7653_min__Suc__gt_I2_J,axiom,
% 0.34/0.70 ! [A2: nat,B: nat] :
% 0.34/0.70 ( ( ord_less_nat @ A2 @ B )
% 0.34/0.70 => ( ( ord_min_nat @ B @ ( suc @ A2 ) )
% 0.34/0.70 = ( suc @ A2 ) ) ) ).
% 0.34/0.70
% 0.34/0.70 % min_Suc_gt(2)
% 0.34/0.70 thf(fact_7654_lessThan__0,axiom,
% 0.34/0.70 ( ( set_ord_lessThan_nat @ zero_zero_nat )
% 0.34/0.70 = bot_bot_set_nat ) ).
% 0.34/0.70
% 0.34/0.70 % lessThan_0
% 0.34/0.70 thf(fact_7655_nat__mult__min__right,axiom,
% 0.34/0.70 ! [M4: nat,N2: nat,Q4: nat] :
% 0.34/0.70 ( ( times_times_nat @ M4 @ ( ord_min_nat @ N2 @ Q4 ) )
% 0.34/0.70 = ( ord_min_nat @ ( times_times_nat @ M4 @ N2 ) @ ( times_times_nat @ M4 @ Q4 ) ) ) ).
% 0.34/0.70
% 0.34/0.70 % nat_mult_min_right
% 0.34/0.70 thf(fact_7656_nat__mult__min__left,axiom,
% 0.34/0.70 ! [M4: nat,N2: nat,Q4: nat] :
% 0.34/0.70 ( ( times_times_nat @ ( ord_min_nat @ M4 @ N2 ) @ Q4 )
% 0.34/0.70 = ( ord_min_nat @ ( times_times_nat @ M4 @ Q4 ) @ ( times_times_nat @ N2 @ Q4 ) ) ) ).
% 0.34/0.70
% 0.34/0.70 % nat_mult_min_left
% 0.34/0.70 thf(fact_7657_lessThan__empty__iff,axiom,
% 0.34/0.70 ! [N2: nat] :
% 0.34/0.70 ( ( ( set_ord_lessThan_nat @ N2 )
% 0.34/0.70 = bot_bot_set_nat )
% 0.34/0.70 = ( N2 = zero_zero_nat ) ) ).
% 0.34/0.70
% 0.34/0.70 % lessThan_empty_iff
% 0.34/0.70 thf(fact_7658_image__atLeastZeroLessThan__integer,axiom,
% 0.34/0.70 ! [U: code_integer] :
% 0.34/0.70 ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ U )
% 0.34/0.70 => ( ( set_or8404916559141939852nteger @ zero_z3403309356797280102nteger @ U )
% 0.34/0.70 = ( image_1215581382706833972nteger @ semiri4939895301339042750nteger @ ( set_ord_lessThan_nat @ ( code_nat_of_integer @ U ) ) ) ) ) ).
% 0.34/0.70
% 0.34/0.70 % image_atLeastZeroLessThan_integer
% 0.34/0.70 thf(fact_7659_bezw__aux,axiom,
% 0.34/0.70 ! [X: nat,Y: nat] :
% 0.34/0.70 ( ( semiri1314217659103216013at_int @ ( gcd_gcd_nat @ X @ Y ) )
% 0.34/0.70 = ( plus_plus_int @ ( times_times_int @ ( product_fst_int_int @ ( bezw @ X @ Y ) ) @ ( semiri1314217659103216013at_int @ X ) ) @ ( times_times_int @ ( product_snd_int_int @ ( bezw @ X @ Y ) ) @ ( semiri1314217659103216013at_int @ Y ) ) ) ) ).
% 0.34/0.70
% 0.34/0.70 % bezw_aux
% 0.34/0.70 thf(fact_7660_INT__greaterThan__UNIV,axiom,
% 0.34/0.70 ( ( comple7806235888213564991et_nat @ ( image_nat_set_nat @ set_or1210151606488870762an_nat @ top_top_set_nat ) )
% 0.34/0.70 = bot_bot_set_nat ) ).
% 0.34/0.70
% 0.34/0.70 % INT_greaterThan_UNIV
% 0.34/0.70 thf(fact_7661_gcd__mult__distrib__nat,axiom,
% 0.34/0.70 ! [K3: nat,M4: nat,N2: nat] :
% 0.34/0.70 ( ( times_times_nat @ K3 @ ( gcd_gcd_nat @ M4 @ N2 ) )
% 0.34/0.70 = ( gcd_gcd_nat @ ( times_times_nat @ K3 @ M4 ) @ ( times_times_nat @ K3 @ N2 ) ) ) ).
% 0.34/0.70
% 0.34/0.70 % gcd_mult_distrib_nat
% 0.34/0.70 thf(fact_7662_Gcd__in,axiom,
% 0.34/0.70 ! [A: set_nat] :
% 0.34/0.70 ( ! [A4: nat,B2: nat] :
% 0.34/0.70 ( ( member_nat @ A4 @ A )
% 0.34/0.70 => ( ( member_nat @ B2 @ A )
% 0.34/0.70 => ( member_nat @ ( gcd_gcd_nat @ A4 @ B2 ) @ A ) ) )
% 0.34/0.70 => ( ( A != bot_bot_set_nat )
% 0.34/0.70 => ( member_nat @ ( gcd_Gcd_nat @ A ) @ A ) ) ) ).
% 0.34/0.70
% 0.34/0.70 % Gcd_in
% 0.34/0.70 thf(fact_7663_bezout__nat,axiom,
% 0.34/0.70 ! [A2: nat,B: nat] :
% 0.34/0.70 ( ( A2 != zero_zero_nat )
% 0.34/0.70 => ? [X3: nat,Y2: nat] :
% 0.34/0.70 ( ( times_times_nat @ A2 @ X3 )
% 0.34/0.70 = ( plus_plus_nat @ ( times_times_nat @ B @ Y2 ) @ ( gcd_gcd_nat @ A2 @ B ) ) ) ) ).
% 0.34/0.70
% 0.34/0.70 % bezout_nat
% 0.34/0.70 thf(fact_7664_bezout__gcd__nat_H,axiom,
% 0.34/0.70 ! [B: nat,A2: nat] :
% 0.34/0.70 ? [X3: nat,Y2: nat] :
% 0.34/0.70 ( ( ( ord_less_eq_nat @ ( times_times_nat @ B @ Y2 ) @ ( times_times_nat @ A2 @ X3 ) )
% 0.34/0.70 & ( ( minus_minus_nat @ ( times_times_nat @ A2 @ X3 ) @ ( times_times_nat @ B @ Y2 ) )
% 0.34/0.70 = ( gcd_gcd_nat @ A2 @ B ) ) )
% 0.34/0.70 | ( ( ord_less_eq_nat @ ( times_times_nat @ A2 @ Y2 ) @ ( times_times_nat @ B @ X3 ) )
% 0.34/0.70 & ( ( minus_minus_nat @ ( times_times_nat @ B @ X3 ) @ ( times_times_nat @ A2 @ Y2 ) )
% 0.34/0.70 = ( gcd_gcd_nat @ A2 @ B ) ) ) ) ).
% 0.34/0.70
% 0.34/0.70 % bezout_gcd_nat'
% 0.34/0.70 thf(fact_7665_greaterThan__Suc,axiom,
% 0.34/0.70 ! [K3: nat] :
% 0.34/0.70 ( ( set_or1210151606488870762an_nat @ ( suc @ K3 ) )
% 0.34/0.70 = ( minus_minus_set_nat @ ( set_or1210151606488870762an_nat @ K3 ) @ ( insert_nat @ ( suc @ K3 ) @ bot_bot_set_nat ) ) ) ).
% 0.34/0.70
% 0.34/0.70 % greaterThan_Suc
% 0.34/0.70 thf(fact_7666_plus__rat_Orsp,axiom,
% 0.34/0.70 ( bNF_re5228765855967844073nt_int @ ratrel @ ( bNF_re7145576690424134365nt_int @ ratrel @ ratrel )
% 0.34/0.70 @ ^ [X2: product_prod_int_int,Y3: product_prod_int_int] : ( product_Pair_int_int @ ( plus_plus_int @ ( times_times_int @ ( product_fst_int_int @ X2 ) @ ( product_snd_int_int @ Y3 ) ) @ ( times_times_int @ ( product_fst_int_int @ Y3 ) @ ( product_snd_int_int @ X2 ) ) ) @ ( times_times_int @ ( product_snd_int_int @ X2 ) @ ( product_snd_int_int @ Y3 ) ) )
% 0.34/0.70 @ ^ [X2: product_prod_int_int,Y3: product_prod_int_int] : ( product_Pair_int_int @ ( plus_plus_int @ ( times_times_int @ ( product_fst_int_int @ X2 ) @ ( product_snd_int_int @ Y3 ) ) @ ( times_times_int @ ( product_fst_int_int @ Y3 ) @ ( product_snd_int_int @ X2 ) ) ) @ ( times_times_int @ ( product_snd_int_int @ X2 ) @ ( product_snd_int_int @ Y3 ) ) ) ) ).
% 0.34/0.70
% 0.34/0.70 % plus_rat.rsp
% 0.34/0.70 thf(fact_7667_positive_Orsp,axiom,
% 0.34/0.70 ( bNF_re8699439704749558557nt_o_o @ ratrel
% 0.34/0.70 @ ^ [Y4: $o,Z4: $o] : ( Y4 = Z4 )
% 0.34/0.70 @ ^ [X2: product_prod_int_int] : ( ord_less_int @ zero_zero_int @ ( times_times_int @ ( product_fst_int_int @ X2 ) @ ( product_snd_int_int @ X2 ) ) )
% 0.34/0.70 @ ^ [X2: product_prod_int_int] : ( ord_less_int @ zero_zero_int @ ( times_times_int @ ( product_fst_int_int @ X2 ) @ ( product_snd_int_int @ X2 ) ) ) ) ).
% 0.34/0.70
% 0.34/0.70 % positive.rsp
% 0.34/0.70 thf(fact_7668_ratrel__iff,axiom,
% 0.34/0.70 ( ratrel
% 0.34/0.70 = ( ^ [X2: product_prod_int_int,Y3: product_prod_int_int] :
% 0.34/0.70 ( ( ( product_snd_int_int @ X2 )
% 0.34/0.70 != zero_zero_int )
% 0.34/0.70 & ( ( product_snd_int_int @ Y3 )
% 0.34/0.70 != zero_zero_int )
% 0.34/0.70 & ( ( times_times_int @ ( product_fst_int_int @ X2 ) @ ( product_snd_int_int @ Y3 ) )
% 0.34/0.70 = ( times_times_int @ ( product_fst_int_int @ Y3 ) @ ( product_snd_int_int @ X2 ) ) ) ) ) ) ).
% 0.34/0.70
% 0.34/0.70 % ratrel_iff
% 0.34/0.70 thf(fact_7669_bezout__int,axiom,
% 0.34/0.70 ! [X: int,Y: int] :
% 0.34/0.70 ? [U4: int,V3: int] :
% 0.34/0.70 ( ( plus_plus_int @ ( times_times_int @ U4 @ X ) @ ( times_times_int @ V3 @ Y ) )
% 0.34/0.70 = ( gcd_gcd_int @ X @ Y ) ) ).
% 0.34/0.70
% 0.34/0.70 % bezout_int
% 0.34/0.70 thf(fact_7670_gcd__mult__distrib__int,axiom,
% 0.34/0.70 ! [K3: int,M4: int,N2: int] :
% 0.34/0.70 ( ( times_times_int @ ( abs_abs_int @ K3 ) @ ( gcd_gcd_int @ M4 @ N2 ) )
% 0.34/0.70 = ( gcd_gcd_int @ ( times_times_int @ K3 @ M4 ) @ ( times_times_int @ K3 @ N2 ) ) ) ).
% 0.34/0.70
% 0.34/0.70 % gcd_mult_distrib_int
% 0.34/0.70 thf(fact_7671_ratrel__def,axiom,
% 0.34/0.70 ( ratrel
% 0.34/0.70 = ( ^ [X2: product_prod_int_int,Y3: product_prod_int_int] :
% 0.34/0.70 ( ( ( product_snd_int_int @ X2 )
% 0.34/0.70 != zero_zero_int )
% 0.34/0.70 & ( ( product_snd_int_int @ Y3 )
% 0.34/0.70 != zero_zero_int )
% 0.34/0.70 & ( ( times_times_int @ ( product_fst_int_int @ X2 ) @ ( product_snd_int_int @ Y3 ) )
% 0.34/0.70 = ( times_times_int @ ( product_fst_int_int @ Y3 ) @ ( product_snd_int_int @ X2 ) ) ) ) ) ) ).
% 0.34/0.70
% 0.34/0.70 % ratrel_def
% 0.34/0.70 thf(fact_7672_times__rat_Orsp,axiom,
% 0.34/0.70 ( bNF_re5228765855967844073nt_int @ ratrel @ ( bNF_re7145576690424134365nt_int @ ratrel @ ratrel )
% 0.34/0.70 @ ^ [X2: product_prod_int_int,Y3: product_prod_int_int] : ( product_Pair_int_int @ ( times_times_int @ ( product_fst_int_int @ X2 ) @ ( product_fst_int_int @ Y3 ) ) @ ( times_times_int @ ( product_snd_int_int @ X2 ) @ ( product_snd_int_int @ Y3 ) ) )
% 0.34/0.70 @ ^ [X2: product_prod_int_int,Y3: product_prod_int_int] : ( product_Pair_int_int @ ( times_times_int @ ( product_fst_int_int @ X2 ) @ ( product_fst_int_int @ Y3 ) ) @ ( times_times_int @ ( product_snd_int_int @ X2 ) @ ( product_snd_int_int @ Y3 ) ) ) ) ).
% 0.34/0.70
% 0.34/0.70 % times_rat.rsp
% 0.34/0.70 thf(fact_7673_plus__rat_Oabs__eq,axiom,
% 0.34/0.70 ! [Xa: product_prod_int_int,X: product_prod_int_int] :
% 0.34/0.70 ( ( ratrel @ Xa @ Xa )
% 0.34/0.70 => ( ( ratrel @ X @ X )
% 0.34/0.70 => ( ( plus_plus_rat @ ( abs_Rat @ Xa ) @ ( abs_Rat @ X ) )
% 0.34/0.70 = ( abs_Rat @ ( product_Pair_int_int @ ( plus_plus_int @ ( times_times_int @ ( product_fst_int_int @ Xa ) @ ( product_snd_int_int @ X ) ) @ ( times_times_int @ ( product_fst_int_int @ X ) @ ( product_snd_int_int @ Xa ) ) ) @ ( times_times_int @ ( product_snd_int_int @ Xa ) @ ( product_snd_int_int @ X ) ) ) ) ) ) ) ).
% 0.34/0.70
% 0.34/0.70 % plus_rat.abs_eq
% 0.34/0.70 thf(fact_7674_positive_Oabs__eq,axiom,
% 0.34/0.70 ! [X: product_prod_int_int] :
% 0.34/0.70 ( ( ratrel @ X @ X )
% 0.34/0.70 => ( ( positive @ ( abs_Rat @ X ) )
% 0.34/0.70 = ( ord_less_int @ zero_zero_int @ ( times_times_int @ ( product_fst_int_int @ X ) @ ( product_snd_int_int @ X ) ) ) ) ) ).
% 0.34/0.70
% 0.34/0.70 % positive.abs_eq
% 0.34/0.70 thf(fact_7675_times__rat_Oabs__eq,axiom,
% 0.34/0.70 ! [Xa: product_prod_int_int,X: product_prod_int_int] :
% 0.34/0.70 ( ( ratrel @ Xa @ Xa )
% 0.34/0.70 => ( ( ratrel @ X @ X )
% 0.34/0.70 => ( ( times_times_rat @ ( abs_Rat @ Xa ) @ ( abs_Rat @ X ) )
% 0.34/0.70 = ( abs_Rat @ ( product_Pair_int_int @ ( times_times_int @ ( product_fst_int_int @ Xa ) @ ( product_fst_int_int @ X ) ) @ ( times_times_int @ ( product_snd_int_int @ Xa ) @ ( product_snd_int_int @ X ) ) ) ) ) ) ) ).
% 0.34/0.70
% 0.34/0.70 % times_rat.abs_eq
% 0.34/0.70 thf(fact_7676_finite__greaterThanLessThan__integer,axiom,
% 0.34/0.70 ! [L: code_integer,U: code_integer] : ( finite6017078050557962740nteger @ ( set_or4266950643985792945nteger @ L @ U ) ) ).
% 0.34/0.70
% 0.34/0.70 % finite_greaterThanLessThan_integer
% 0.34/0.70 thf(fact_7677_atLeastPlusOneLessThan__greaterThanLessThan__integer,axiom,
% 0.34/0.70 ! [L: code_integer,U: code_integer] :
% 0.34/0.70 ( ( set_or8404916559141939852nteger @ ( plus_p5714425477246183910nteger @ L @ one_one_Code_integer ) @ U )
% 0.34/0.70 = ( set_or4266950643985792945nteger @ L @ U ) ) ).
% 0.34/0.70
% 0.34/0.70 % atLeastPlusOneLessThan_greaterThanLessThan_integer
% 0.34/0.70 thf(fact_7678_positive__def,axiom,
% 0.34/0.70 ( positive
% 0.34/0.70 = ( map_fu898904425404107465nt_o_o @ rep_Rat @ id_o
% 0.34/0.70 @ ^ [X2: product_prod_int_int] : ( ord_less_int @ zero_zero_int @ ( times_times_int @ ( product_fst_int_int @ X2 ) @ ( product_snd_int_int @ X2 ) ) ) ) ) ).
% 0.34/0.70
% 0.34/0.70 % positive_def
% 0.34/0.70 thf(fact_7679_finite__greaterThanAtMost__integer,axiom,
% 0.34/0.70 ! [L: code_integer,U: code_integer] : ( finite6017078050557962740nteger @ ( set_or2715278749043346189nteger @ L @ U ) ) ).
% 0.34/0.70
% 0.34/0.70 % finite_greaterThanAtMost_integer
% 0.34/0.70 thf(fact_7680_atLeastPlusOneAtMost__greaterThanAtMost__integer,axiom,
% 0.34/0.70 ! [L: code_integer,U: code_integer] :
% 0.34/0.70 ( ( set_or189985376899183464nteger @ ( plus_p5714425477246183910nteger @ L @ one_one_Code_integer ) @ U )
% 0.34/0.70 = ( set_or2715278749043346189nteger @ L @ U ) ) ).
% 0.34/0.70
% 0.34/0.70 % atLeastPlusOneAtMost_greaterThanAtMost_integer
% 0.34/0.70 thf(fact_7681_atLeast__Suc,axiom,
% 0.34/0.70 ! [K3: nat] :
% 0.34/0.70 ( ( set_ord_atLeast_nat @ ( suc @ K3 ) )
% 0.34/0.70 = ( minus_minus_set_nat @ ( set_ord_atLeast_nat @ K3 ) @ ( insert_nat @ K3 @ bot_bot_set_nat ) ) ) ).
% 0.34/0.70
% 0.34/0.70 % atLeast_Suc
% 0.34/0.70 thf(fact_7682_eventually__sequentially__Suc,axiom,
% 0.34/0.70 ! [P: nat > $o] :
% 0.34/0.70 ( ( eventually_nat
% 0.34/0.70 @ ^ [I4: nat] : ( P @ ( suc @ I4 ) )
% 0.34/0.70 @ at_top_nat )
% 0.34/0.70 = ( eventually_nat @ P @ at_top_nat ) ) ).
% 0.34/0.70
% 0.34/0.70 % eventually_sequentially_Suc
% 0.34/0.70 thf(fact_7683_eventually__sequentially__seg,axiom,
% 0.34/0.70 ! [P: nat > $o,K3: nat] :
% 0.34/0.70 ( ( eventually_nat
% 0.34/0.70 @ ^ [N4: nat] : ( P @ ( plus_plus_nat @ N4 @ K3 ) )
% 0.34/0.70 @ at_top_nat )
% 0.34/0.70 = ( eventually_nat @ P @ at_top_nat ) ) ).
% 0.34/0.70
% 0.34/0.70 % eventually_sequentially_seg
% 0.34/0.70 thf(fact_7684_filterlim__Suc,axiom,
% 0.34/0.70 filterlim_nat_nat @ suc @ at_top_nat @ at_top_nat ).
% 0.34/0.70
% 0.34/0.70 % filterlim_Suc
% 0.34/0.70 thf(fact_7685_eventually__sequentiallyI,axiom,
% 0.34/0.70 ! [C: nat,P: nat > $o] :
% 0.34/0.70 ( ! [X3: nat] :
% 0.34/0.70 ( ( ord_less_eq_nat @ C @ X3 )
% 0.34/0.70 => ( P @ X3 ) )
% 0.34/0.70 => ( eventually_nat @ P @ at_top_nat ) ) ).
% 0.34/0.70
% 0.34/0.70 % eventually_sequentiallyI
% 0.34/0.70 thf(fact_7686_eventually__sequentially,axiom,
% 0.34/0.70 ! [P: nat > $o] :
% 0.34/0.70 ( ( eventually_nat @ P @ at_top_nat )
% 0.34/0.70 = ( ? [N7: nat] :
% 0.34/0.70 ! [N4: nat] :
% 0.34/0.70 ( ( ord_less_eq_nat @ N7 @ N4 )
% 0.34/0.70 => ( P @ N4 ) ) ) ) ).
% 0.34/0.70
% 0.34/0.70 % eventually_sequentially
% 0.34/0.70 thf(fact_7687_le__sequentially,axiom,
% 0.34/0.70 ! [F2: filter_nat] :
% 0.34/0.70 ( ( ord_le2510731241096832064er_nat @ F2 @ at_top_nat )
% 0.34/0.70 = ( ! [N7: nat] : ( eventually_nat @ ( ord_less_eq_nat @ N7 ) @ F2 ) ) ) ).
% 0.34/0.70
% 0.34/0.70 % le_sequentially
% 0.34/0.70 thf(fact_7688_filterlim__atMost__at__top,axiom,
% 0.34/0.70 filter3212408913953519116et_nat @ set_ord_atMost_nat @ ( finite3254316476582989077op_nat @ top_top_set_nat ) @ at_top_nat ).
% 0.34/0.70
% 0.34/0.70 % filterlim_atMost_at_top
% 0.34/0.70 thf(fact_7689_filterlim__lessThan__at__top,axiom,
% 0.34/0.70 filter3212408913953519116et_nat @ set_ord_lessThan_nat @ ( finite3254316476582989077op_nat @ top_top_set_nat ) @ at_top_nat ).
% 0.34/0.70
% 0.34/0.70 % filterlim_lessThan_at_top
% 0.34/0.70 thf(fact_7690_trivial__limit__sequentially,axiom,
% 0.34/0.70 at_top_nat != bot_bot_filter_nat ).
% 0.34/0.70
% 0.34/0.70 % trivial_limit_sequentially
% 0.34/0.70 thf(fact_7691_eventually__False__sequentially,axiom,
% 0.34/0.70 ~ ( eventually_nat
% 0.34/0.70 @ ^ [N4: nat] : $false
% 0.34/0.70 @ at_top_nat ) ).
% 0.34/0.70
% 0.34/0.70 % eventually_False_sequentially
% 0.34/0.70 thf(fact_7692_plus__rat__def,axiom,
% 0.34/0.70 ( plus_plus_rat
% 0.34/0.70 = ( map_fu4333342158222067775at_rat @ rep_Rat @ ( map_fu5673905371560938248nt_rat @ rep_Rat @ abs_Rat )
% 0.34/0.70 @ ^ [X2: product_prod_int_int,Y3: product_prod_int_int] : ( product_Pair_int_int @ ( plus_plus_int @ ( times_times_int @ ( product_fst_int_int @ X2 ) @ ( product_snd_int_int @ Y3 ) ) @ ( times_times_int @ ( product_fst_int_int @ Y3 ) @ ( product_snd_int_int @ X2 ) ) ) @ ( times_times_int @ ( product_snd_int_int @ X2 ) @ ( product_snd_int_int @ Y3 ) ) ) ) ) ).
% 0.34/0.70
% 0.34/0.70 % plus_rat_def
% 0.34/0.70 thf(fact_7693_eventually__prod__sequentially,axiom,
% 0.34/0.70 ! [P: product_prod_nat_nat > $o] :
% 0.34/0.70 ( ( eventu1038000079068216329at_nat @ P @ ( prod_filter_nat_nat @ at_top_nat @ at_top_nat ) )
% 0.34/0.70 = ( ? [N7: nat] :
% 0.34/0.70 ! [M3: nat] :
% 0.34/0.70 ( ( ord_less_eq_nat @ N7 @ M3 )
% 0.34/0.70 => ! [N4: nat] :
% 0.34/0.70 ( ( ord_less_eq_nat @ N7 @ N4 )
% 0.34/0.70 => ( P @ ( product_Pair_nat_nat @ N4 @ M3 ) ) ) ) ) ) ).
% 0.34/0.70
% 0.34/0.70 % eventually_prod_sequentially
% 0.34/0.70 thf(fact_7694_times__rat__def,axiom,
% 0.34/0.70 ( times_times_rat
% 0.34/0.70 = ( map_fu4333342158222067775at_rat @ rep_Rat @ ( map_fu5673905371560938248nt_rat @ rep_Rat @ abs_Rat )
% 0.34/0.70 @ ^ [X2: product_prod_int_int,Y3: product_prod_int_int] : ( product_Pair_int_int @ ( times_times_int @ ( product_fst_int_int @ X2 ) @ ( product_fst_int_int @ Y3 ) ) @ ( times_times_int @ ( product_snd_int_int @ X2 ) @ ( product_snd_int_int @ Y3 ) ) ) ) ) ).
% 0.34/0.70
% 0.34/0.70 % times_rat_def
% 0.34/0.70 thf(fact_7695_add__rat,axiom,
% 0.34/0.70 ! [B: int,D2: int,A2: int,C: int] :
% 0.34/0.70 ( ( B != zero_zero_int )
% 0.34/0.70 => ( ( D2 != zero_zero_int )
% 0.34/0.70 => ( ( plus_plus_rat @ ( fract @ A2 @ B ) @ ( fract @ C @ D2 ) )
% 0.34/0.70 = ( fract @ ( plus_plus_int @ ( times_times_int @ A2 @ D2 ) @ ( times_times_int @ C @ B ) ) @ ( times_times_int @ B @ D2 ) ) ) ) ) ).
% 0.34/0.70
% 0.34/0.70 % add_rat
% 0.34/0.70 thf(fact_7696_divide__rat,axiom,
% 0.34/0.70 ! [A2: int,B: int,C: int,D2: int] :
% 0.34/0.70 ( ( divide_divide_rat @ ( fract @ A2 @ B ) @ ( fract @ C @ D2 ) )
% 0.34/0.70 = ( fract @ ( times_times_int @ A2 @ D2 ) @ ( times_times_int @ B @ C ) ) ) ).
% 0.34/0.70
% 0.34/0.70 % divide_rat
% 0.34/0.70 thf(fact_7697_sgn__rat,axiom,
% 0.34/0.70 ! [A2: int,B: int] :
% 0.34/0.70 ( ( sgn_sgn_rat @ ( fract @ A2 @ B ) )
% 0.34/0.70 = ( ring_1_of_int_rat @ ( times_times_int @ ( sgn_sgn_int @ A2 ) @ ( sgn_sgn_int @ B ) ) ) ) ).
% 0.34/0.70
% 0.34/0.70 % sgn_rat
% 0.34/0.70 thf(fact_7698_mult__rat,axiom,
% 0.34/0.70 ! [A2: int,B: int,C: int,D2: int] :
% 0.34/0.70 ( ( times_times_rat @ ( fract @ A2 @ B ) @ ( fract @ C @ D2 ) )
% 0.34/0.70 = ( fract @ ( times_times_int @ A2 @ C ) @ ( times_times_int @ B @ D2 ) ) ) ).
% 0.34/0.70
% 0.34/0.70 % mult_rat
% 0.34/0.70 thf(fact_7699_le__rat,axiom,
% 0.34/0.70 ! [B: int,D2: int,A2: int,C: int] :
% 0.34/0.70 ( ( B != zero_zero_int )
% 0.34/0.70 => ( ( D2 != zero_zero_int )
% 0.34/0.70 => ( ( ord_less_eq_rat @ ( fract @ A2 @ B ) @ ( fract @ C @ D2 ) )
% 0.34/0.70 = ( ord_less_eq_int @ ( times_times_int @ ( times_times_int @ A2 @ D2 ) @ ( times_times_int @ B @ D2 ) ) @ ( times_times_int @ ( times_times_int @ C @ B ) @ ( times_times_int @ B @ D2 ) ) ) ) ) ) ).
% 0.34/0.70
% 0.34/0.70 % le_rat
% 0.34/0.70 thf(fact_7700_less__rat,axiom,
% 0.34/0.70 ! [B: int,D2: int,A2: int,C: int] :
% 0.34/0.70 ( ( B != zero_zero_int )
% 0.34/0.70 => ( ( D2 != zero_zero_int )
% 0.34/0.70 => ( ( ord_less_rat @ ( fract @ A2 @ B ) @ ( fract @ C @ D2 ) )
% 0.34/0.70 = ( ord_less_int @ ( times_times_int @ ( times_times_int @ A2 @ D2 ) @ ( times_times_int @ B @ D2 ) ) @ ( times_times_int @ ( times_times_int @ C @ B ) @ ( times_times_int @ B @ D2 ) ) ) ) ) ) ).
% 0.34/0.70
% 0.34/0.70 % less_rat
% 0.34/0.70 thf(fact_7701_diff__rat,axiom,
% 0.34/0.70 ! [B: int,D2: int,A2: int,C: int] :
% 0.34/0.70 ( ( B != zero_zero_int )
% 0.34/0.70 => ( ( D2 != zero_zero_int )
% 0.34/0.70 => ( ( minus_minus_rat @ ( fract @ A2 @ B ) @ ( fract @ C @ D2 ) )
% 0.34/0.70 = ( fract @ ( minus_minus_int @ ( times_times_int @ A2 @ D2 ) @ ( times_times_int @ C @ B ) ) @ ( times_times_int @ B @ D2 ) ) ) ) ) ).
% 0.34/0.70
% 0.34/0.70 % diff_rat
% 0.34/0.70 thf(fact_7702_eq__rat_I1_J,axiom,
% 0.34/0.70 ! [B: int,D2: int,A2: int,C: int] :
% 0.34/0.70 ( ( B != zero_zero_int )
% 0.34/0.70 => ( ( D2 != zero_zero_int )
% 0.34/0.70 => ( ( ( fract @ A2 @ B )
% 0.34/0.70 = ( fract @ C @ D2 ) )
% 0.34/0.70 = ( ( times_times_int @ A2 @ D2 )
% 0.34/0.70 = ( times_times_int @ C @ B ) ) ) ) ) ).
% 0.34/0.70
% 0.34/0.70 % eq_rat(1)
% 0.34/0.70 thf(fact_7703_mult__rat__cancel,axiom,
% 0.34/0.70 ! [C: int,A2: int,B: int] :
% 0.34/0.70 ( ( C != zero_zero_int )
% 0.34/0.70 => ( ( fract @ ( times_times_int @ C @ A2 ) @ ( times_times_int @ C @ B ) )
% 0.34/0.70 = ( fract @ A2 @ B ) ) ) ).
% 0.34/0.70
% 0.34/0.70 % mult_rat_cancel
% 0.34/0.70 thf(fact_7704_mono__times__nat,axiom,
% 0.34/0.70 ! [N2: nat] :
% 0.34/0.70 ( ( ord_less_nat @ zero_zero_nat @ N2 )
% 0.34/0.70 => ( order_mono_nat_nat @ ( times_times_nat @ N2 ) ) ) ).
% 0.34/0.70
% 0.34/0.70 % mono_times_nat
% 0.34/0.70 thf(fact_7705_positive__rat,axiom,
% 0.34/0.70 ! [A2: int,B: int] :
% 0.34/0.70 ( ( positive @ ( fract @ A2 @ B ) )
% 0.34/0.70 = ( ord_less_int @ zero_zero_int @ ( times_times_int @ A2 @ B ) ) ) ).
% 0.34/0.70
% 0.34/0.70 % positive_rat
% 0.34/0.70 thf(fact_7706_set__encode__empty,axiom,
% 0.34/0.70 ( ( nat_set_encode @ bot_bot_set_nat )
% 0.34/0.70 = zero_zero_nat ) ).
% 0.34/0.70
% 0.34/0.70 % set_encode_empty
% 0.34/0.70 thf(fact_7707_normalize__crossproduct,axiom,
% 0.34/0.70 ! [Q4: int,S2: int,P3: int,R2: int] :
% 0.34/0.70 ( ( Q4 != zero_zero_int )
% 0.34/0.70 => ( ( S2 != zero_zero_int )
% 0.34/0.70 => ( ( ( normalize @ ( product_Pair_int_int @ P3 @ Q4 ) )
% 0.34/0.70 = ( normalize @ ( product_Pair_int_int @ R2 @ S2 ) ) )
% 0.34/0.70 => ( ( times_times_int @ P3 @ S2 )
% 0.34/0.70 = ( times_times_int @ R2 @ Q4 ) ) ) ) ) ).
% 0.34/0.70
% 0.34/0.70 % normalize_crossproduct
% 0.34/0.70 thf(fact_7708_rat__minus__code,axiom,
% 0.34/0.70 ! [P3: rat,Q4: rat] :
% 0.34/0.70 ( ( quotient_of @ ( minus_minus_rat @ P3 @ Q4 ) )
% 0.34/0.70 = ( produc4245557441103728435nt_int
% 0.34/0.70 @ ^ [A6: int,C4: int] :
% 0.34/0.70 ( produc4245557441103728435nt_int
% 0.34/0.70 @ ^ [B6: int,D4: int] : ( normalize @ ( product_Pair_int_int @ ( minus_minus_int @ ( times_times_int @ A6 @ D4 ) @ ( times_times_int @ B6 @ C4 ) ) @ ( times_times_int @ C4 @ D4 ) ) )
% 0.34/0.70 @ ( quotient_of @ Q4 ) )
% 0.34/0.70 @ ( quotient_of @ P3 ) ) ) ).
% 0.34/0.70
% 0.34/0.70 % rat_minus_code
% 0.34/0.70 thf(fact_7709_rat__plus__code,axiom,
% 0.34/0.70 ! [P3: rat,Q4: rat] :
% 0.34/0.70 ( ( quotient_of @ ( plus_plus_rat @ P3 @ Q4 ) )
% 0.34/0.70 = ( produc4245557441103728435nt_int
% 0.34/0.70 @ ^ [A6: int,C4: int] :
% 0.34/0.70 ( produc4245557441103728435nt_int
% 0.34/0.70 @ ^ [B6: int,D4: int] : ( normalize @ ( product_Pair_int_int @ ( plus_plus_int @ ( times_times_int @ A6 @ D4 ) @ ( times_times_int @ B6 @ C4 ) ) @ ( times_times_int @ C4 @ D4 ) ) )
% 0.34/0.70 @ ( quotient_of @ Q4 ) )
% 0.34/0.70 @ ( quotient_of @ P3 ) ) ) ).
% 0.34/0.70
% 0.34/0.70 % rat_plus_code
% 0.34/0.70 thf(fact_7710_rat__times__code,axiom,
% 0.34/0.70 ! [P3: rat,Q4: rat] :
% 0.36/0.70 ( ( quotient_of @ ( times_times_rat @ P3 @ Q4 ) )
% 0.36/0.70 = ( produc4245557441103728435nt_int
% 0.36/0.70 @ ^ [A6: int,C4: int] :
% 0.36/0.70 ( produc4245557441103728435nt_int
% 0.36/0.70 @ ^ [B6: int,D4: int] : ( normalize @ ( product_Pair_int_int @ ( times_times_int @ A6 @ B6 ) @ ( times_times_int @ C4 @ D4 ) ) )
% 0.36/0.70 @ ( quotient_of @ Q4 ) )
% 0.36/0.70 @ ( quotient_of @ P3 ) ) ) ).
% 0.36/0.70
% 0.36/0.70 % rat_times_code
% 0.36/0.70 thf(fact_7711_rat__less__code,axiom,
% 0.36/0.70 ( ord_less_rat
% 0.36/0.70 = ( ^ [P5: rat,Q6: rat] :
% 0.36/0.70 ( produc4947309494688390418_int_o
% 0.36/0.70 @ ^ [A6: int,C4: int] :
% 0.36/0.70 ( produc4947309494688390418_int_o
% 0.36/0.70 @ ^ [B6: int,D4: int] : ( ord_less_int @ ( times_times_int @ A6 @ D4 ) @ ( times_times_int @ C4 @ B6 ) )
% 0.36/0.70 @ ( quotient_of @ Q6 ) )
% 0.36/0.70 @ ( quotient_of @ P5 ) ) ) ) ).
% 0.36/0.70
% 0.36/0.70 % rat_less_code
% 0.36/0.70 thf(fact_7712_rat__less__eq__code,axiom,
% 0.36/0.70 ( ord_less_eq_rat
% 0.36/0.70 = ( ^ [P5: rat,Q6: rat] :
% 0.36/0.70 ( produc4947309494688390418_int_o
% 0.36/0.70 @ ^ [A6: int,C4: int] :
% 0.36/0.70 ( produc4947309494688390418_int_o
% 0.36/0.70 @ ^ [B6: int,D4: int] : ( ord_less_eq_int @ ( times_times_int @ A6 @ D4 ) @ ( times_times_int @ C4 @ B6 ) )
% 0.36/0.70 @ ( quotient_of @ Q6 ) )
% 0.36/0.70 @ ( quotient_of @ P5 ) ) ) ) ).
% 0.36/0.70
% 0.36/0.70 % rat_less_eq_code
% 0.36/0.70 thf(fact_7713_rat__divide__code,axiom,
% 0.36/0.70 ! [P3: rat,Q4: rat] :
% 0.36/0.70 ( ( quotient_of @ ( divide_divide_rat @ P3 @ Q4 ) )
% 0.36/0.70 = ( produc4245557441103728435nt_int
% 0.36/0.70 @ ^ [A6: int,C4: int] :
% 0.36/0.70 ( produc4245557441103728435nt_int
% 0.36/0.70 @ ^ [B6: int,D4: int] : ( normalize @ ( product_Pair_int_int @ ( times_times_int @ A6 @ D4 ) @ ( times_times_int @ C4 @ B6 ) ) )
% 0.36/0.70 @ ( quotient_of @ Q4 ) )
% 0.36/0.70 @ ( quotient_of @ P3 ) ) ) ).
% 0.36/0.70
% 0.36/0.70 % rat_divide_code
% 0.36/0.70 thf(fact_7714_rat__inverse__code,axiom,
% 0.36/0.70 ! [P3: rat] :
% 0.36/0.70 ( ( quotient_of @ ( inverse_inverse_rat @ P3 ) )
% 0.36/0.70 = ( produc4245557441103728435nt_int
% 0.36/0.70 @ ^ [A6: int,B6: int] : ( if_Pro3027730157355071871nt_int @ ( A6 = zero_zero_int ) @ ( product_Pair_int_int @ zero_zero_int @ one_one_int ) @ ( product_Pair_int_int @ ( times_times_int @ ( sgn_sgn_int @ A6 ) @ B6 ) @ ( abs_abs_int @ A6 ) ) )
% 0.36/0.70 @ ( quotient_of @ P3 ) ) ) ).
% 0.36/0.70
% 0.36/0.70 % rat_inverse_code
% 0.36/0.70 thf(fact_7715_divide__rat__def,axiom,
% 0.36/0.70 ( divide_divide_rat
% 0.36/0.70 = ( ^ [Q6: rat,R4: rat] : ( times_times_rat @ Q6 @ ( inverse_inverse_rat @ R4 ) ) ) ) ).
% 0.36/0.70
% 0.36/0.70 % divide_rat_def
% 0.36/0.70 thf(fact_7716_sub__BitM__One__eq,axiom,
% 0.36/0.70 ! [N2: num] :
% 0.36/0.70 ( ( neg_numeral_sub_int @ ( bitM @ N2 ) @ one )
% 0.36/0.70 = ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( neg_numeral_sub_int @ N2 @ one ) ) ) ).
% 0.36/0.70
% 0.36/0.70 % sub_BitM_One_eq
% 0.36/0.70 thf(fact_7717_set__decode__zero,axiom,
% 0.36/0.70 ( ( nat_set_decode @ zero_zero_nat )
% 0.36/0.70 = bot_bot_set_nat ) ).
% 0.36/0.70
% 0.36/0.70 % set_decode_zero
% 0.36/0.70 thf(fact_7718_cofinite__eq__sequentially,axiom,
% 0.36/0.70 cofinite_nat = at_top_nat ).
% 0.36/0.70
% 0.36/0.70 % cofinite_eq_sequentially
% 0.36/0.70 thf(fact_7719_INFM__nat__inductI,axiom,
% 0.36/0.70 ! [P: nat > $o,Q2: nat > $o] :
% 0.36/0.70 ( ( P @ zero_zero_nat )
% 0.36/0.70 => ( ! [I3: nat] :
% 0.36/0.70 ( ( P @ I3 )
% 0.36/0.70 => ? [J4: nat] :
% 0.36/0.70 ( ( ord_less_nat @ I3 @ J4 )
% 0.36/0.70 & ( P @ J4 )
% 0.36/0.70 & ( Q2 @ J4 ) ) )
% 0.36/0.70 => ( frequently_nat @ Q2 @ cofinite_nat ) ) ) ).
% 0.36/0.70
% 0.36/0.70 % INFM_nat_inductI
% 0.36/0.70 thf(fact_7720_lcm__altdef__int,axiom,
% 0.36/0.70 ( gcd_lcm_int
% 0.36/0.70 = ( ^ [A6: int,B6: int] : ( divide_divide_int @ ( times_times_int @ ( abs_abs_int @ A6 ) @ ( abs_abs_int @ B6 ) ) @ ( gcd_gcd_int @ A6 @ B6 ) ) ) ) ).
% 0.36/0.70
% 0.36/0.70 % lcm_altdef_int
% 0.36/0.70 thf(fact_7721_times__num__def,axiom,
% 0.36/0.70 ( times_times_num
% 0.36/0.70 = ( ^ [M3: num,N4: num] : ( num_of_nat @ ( times_times_nat @ ( nat_of_num @ M3 ) @ ( nat_of_num @ N4 ) ) ) ) ) ).
% 0.36/0.70
% 0.36/0.70 % times_num_def
% 0.36/0.70 thf(fact_7722_nat__of__num__mult,axiom,
% 0.36/0.70 ! [X: num,Y: num] :
% 0.36/0.70 ( ( nat_of_num @ ( times_times_num @ X @ Y ) )
% 0.36/0.70 = ( times_times_nat @ ( nat_of_num @ X ) @ ( nat_of_num @ Y ) ) ) ).
% 0.36/0.70
% 0.36/0.70 % nat_of_num_mult
% 0.36/0.70 thf(fact_7723_prod__gcd__lcm__int,axiom,
% 0.36/0.70 ! [M4: int,N2: int] :
% 0.36/0.70 ( ( times_times_int @ ( abs_abs_int @ M4 ) @ ( abs_abs_int @ N2 ) )
% 0.36/0.70 = ( times_times_int @ ( gcd_gcd_int @ M4 @ N2 ) @ ( gcd_lcm_int @ M4 @ N2 ) ) ) ).
% 0.36/0.70
% 0.36/0.70 % prod_gcd_lcm_int
% 0.36/0.70 thf(fact_7724_prod__gcd__lcm__nat,axiom,
% 0.36/0.70 ( times_times_nat
% 0.36/0.70 = ( ^ [M3: nat,N4: nat] : ( times_times_nat @ ( gcd_gcd_nat @ M3 @ N4 ) @ ( gcd_lcm_nat @ M3 @ N4 ) ) ) ) ).
% 0.36/0.70
% 0.36/0.70 % prod_gcd_lcm_nat
% 0.36/0.70 thf(fact_7725_lcm__code__integer,axiom,
% 0.36/0.70 ( gcd_lcm_Code_integer
% 0.36/0.70 = ( ^ [A6: code_integer,B6: code_integer] : ( divide6298287555418463151nteger @ ( times_3573771949741848930nteger @ ( abs_abs_Code_integer @ A6 ) @ ( abs_abs_Code_integer @ B6 ) ) @ ( gcd_gcd_Code_integer @ A6 @ B6 ) ) ) ) ).
% 0.36/0.70
% 0.36/0.70 % lcm_code_integer
% 0.36/0.70 thf(fact_7726_lcm__nat__def,axiom,
% 0.36/0.70 ( gcd_lcm_nat
% 0.36/0.70 = ( ^ [X2: nat,Y3: nat] : ( divide_divide_nat @ ( times_times_nat @ X2 @ Y3 ) @ ( gcd_gcd_nat @ X2 @ Y3 ) ) ) ) ).
% 0.36/0.70
% 0.36/0.70 % lcm_nat_def
% 0.36/0.70 thf(fact_7727_Lcm__eq__Max__nat,axiom,
% 0.36/0.70 ! [M: set_nat] :
% 0.36/0.70 ( ( finite_finite_nat @ M )
% 0.36/0.70 => ( ( M != bot_bot_set_nat )
% 0.36/0.70 => ( ~ ( member_nat @ zero_zero_nat @ M )
% 0.36/0.70 => ( ! [M2: nat,N3: nat] :
% 0.36/0.70 ( ( member_nat @ M2 @ M )
% 0.36/0.70 => ( ( member_nat @ N3 @ M )
% 0.36/0.70 => ( member_nat @ ( gcd_lcm_nat @ M2 @ N3 ) @ M ) ) )
% 0.36/0.70 => ( ( gcd_Lcm_nat @ M )
% 0.36/0.70 = ( lattic8265883725875713057ax_nat @ M ) ) ) ) ) ) ).
% 0.36/0.70
% 0.36/0.70 % Lcm_eq_Max_nat
% 0.36/0.70 thf(fact_7728_nat__of__num__sqr,axiom,
% 0.36/0.70 ! [X: num] :
% 0.36/0.70 ( ( nat_of_num @ ( sqr @ X ) )
% 0.36/0.70 = ( times_times_nat @ ( nat_of_num @ X ) @ ( nat_of_num @ X ) ) ) ).
% 0.36/0.70
% 0.36/0.70 % nat_of_num_sqr
% 0.36/0.70 thf(fact_7729_Lcm__nat__empty,axiom,
% 0.36/0.70 ( ( gcd_Lcm_nat @ bot_bot_set_nat )
% 0.36/0.70 = one_one_nat ) ).
% 0.36/0.70
% 0.36/0.70 % Lcm_nat_empty
% 0.36/0.70 thf(fact_7730_sqr__conv__mult,axiom,
% 0.36/0.70 ( sqr
% 0.36/0.70 = ( ^ [X2: num] : ( times_times_num @ X2 @ X2 ) ) ) ).
% 0.36/0.70
% 0.36/0.70 % sqr_conv_mult
% 0.36/0.70 thf(fact_7731_Lcm__in__lcm__closed__set__nat,axiom,
% 0.36/0.70 ! [M: set_nat] :
% 0.36/0.70 ( ( finite_finite_nat @ M )
% 0.36/0.70 => ( ( M != bot_bot_set_nat )
% 0.36/0.70 => ( ! [M2: nat,N3: nat] :
% 0.36/0.70 ( ( member_nat @ M2 @ M )
% 0.36/0.70 => ( ( member_nat @ N3 @ M )
% 0.36/0.70 => ( member_nat @ ( gcd_lcm_nat @ M2 @ N3 ) @ M ) ) )
% 0.36/0.70 => ( member_nat @ ( gcd_Lcm_nat @ M ) @ M ) ) ) ) ).
% 0.36/0.70
% 0.36/0.70 % Lcm_in_lcm_closed_set_nat
% 0.36/0.70 thf(fact_7732_pow_Osimps_I3_J,axiom,
% 0.36/0.70 ! [X: num,Y: num] :
% 0.36/0.70 ( ( pow @ X @ ( bit1 @ Y ) )
% 0.36/0.70 = ( times_times_num @ ( sqr @ ( pow @ X @ Y ) ) @ X ) ) ).
% 0.36/0.70
% 0.36/0.70 % pow.simps(3)
% 0.36/0.70 thf(fact_7733_coprime__crossproduct__nat,axiom,
% 0.36/0.70 ! [A2: nat,D2: nat,B: nat,C: nat] :
% 0.36/0.70 ( ( algebr934650988132801477me_nat @ A2 @ D2 )
% 0.36/0.70 => ( ( algebr934650988132801477me_nat @ B @ C )
% 0.36/0.70 => ( ( ( times_times_nat @ A2 @ C )
% 0.36/0.70 = ( times_times_nat @ B @ D2 ) )
% 0.36/0.70 = ( ( A2 = B )
% 0.36/0.70 & ( C = D2 ) ) ) ) ) ).
% 0.36/0.70
% 0.36/0.70 % coprime_crossproduct_nat
% 0.36/0.70 thf(fact_7734_coprime__crossproduct__int,axiom,
% 0.36/0.70 ! [A2: int,D2: int,B: int,C: int] :
% 0.36/0.70 ( ( algebr932160517623751201me_int @ A2 @ D2 )
% 0.36/0.70 => ( ( algebr932160517623751201me_int @ B @ C )
% 0.36/0.70 => ( ( ( times_times_int @ ( abs_abs_int @ A2 ) @ ( abs_abs_int @ C ) )
% 0.36/0.70 = ( times_times_int @ ( abs_abs_int @ B ) @ ( abs_abs_int @ D2 ) ) )
% 0.36/0.70 = ( ( ( abs_abs_int @ A2 )
% 0.36/0.70 = ( abs_abs_int @ B ) )
% 0.36/0.70 & ( ( abs_abs_int @ C )
% 0.36/0.70 = ( abs_abs_int @ D2 ) ) ) ) ) ) ).
% 0.36/0.70
% 0.36/0.70 % coprime_crossproduct_int
% 0.36/0.70 thf(fact_7735_in__range_Oelims_I3_J,axiom,
% 0.36/0.70 ! [X: produc3658429121746597890et_nat] :
% 0.36/0.70 ( ~ ( in_range @ X )
% 0.36/0.70 => ~ ! [H: heap_e7401611519738050253t_unit,As2: set_nat] :
% 0.36/0.70 ( ( X
% 0.36/0.70 = ( produc7507926704131184380et_nat @ H @ As2 ) )
% 0.36/0.70 => ! [X3: nat] :
% 0.36/0.70 ( ( member_nat @ X3 @ As2 )
% 0.36/0.70 => ( ord_less_nat @ X3 @ ( lim_Product_unit @ H ) ) ) ) ) ).
% 0.36/0.70
% 0.36/0.70 % in_range.elims(3)
% 0.36/0.70 thf(fact_7736_in__range_Osimps,axiom,
% 0.36/0.70 ! [H2: heap_e7401611519738050253t_unit,As: set_nat] :
% 0.36/0.70 ( ( in_range @ ( produc7507926704131184380et_nat @ H2 @ As ) )
% 0.36/0.70 = ( ! [X2: nat] :
% 0.36/0.70 ( ( member_nat @ X2 @ As )
% 0.36/0.70 => ( ord_less_nat @ X2 @ ( lim_Product_unit @ H2 ) ) ) ) ) ).
% 0.36/0.70
% 0.36/0.70 % in_range.simps
% 0.36/0.70 thf(fact_7737_in__range_Oelims_I1_J,axiom,
% 0.36/0.70 ! [X: produc3658429121746597890et_nat,Y: $o] :
% 0.36/0.70 ( ( ( in_range @ X )
% 0.36/0.70 = Y )
% 0.36/0.70 => ~ ! [H: heap_e7401611519738050253t_unit,As2: set_nat] :
% 0.36/0.70 ( ( X
% 0.36/0.70 = ( produc7507926704131184380et_nat @ H @ As2 ) )
% 0.36/0.70 => ( Y
% 0.36/0.70 = ( ~ ! [X2: nat] :
% 0.36/0.70 ( ( member_nat @ X2 @ As2 )
% 0.36/0.70 => ( ord_less_nat @ X2 @ ( lim_Product_unit @ H ) ) ) ) ) ) ) ).
% 0.36/0.70
% 0.36/0.70 % in_range.elims(1)
% 0.36/0.70 thf(fact_7738_in__range_Oelims_I2_J,axiom,
% 0.36/0.70 ! [X: produc3658429121746597890et_nat] :
% 0.36/0.70 ( ( in_range @ X )
% 0.36/0.70 => ~ ! [H: heap_e7401611519738050253t_unit,As2: set_nat] :
% 0.36/0.70 ( ( X
% 0.36/0.70 = ( produc7507926704131184380et_nat @ H @ As2 ) )
% 0.36/0.70 => ~ ! [X6: nat] :
% 0.36/0.70 ( ( member_nat @ X6 @ As2 )
% 0.36/0.70 => ( ord_less_nat @ X6 @ ( lim_Product_unit @ H ) ) ) ) ) ).
% 0.36/0.70
% 0.36/0.70 % in_range.elims(2)
% 0.36/0.70 thf(fact_7739_UNIV__bool,axiom,
% 0.36/0.70 ( top_top_set_o
% 0.36/0.70 = ( insert_o @ $false @ ( insert_o @ $true @ bot_bot_set_o ) ) ) ).
% 0.36/0.70
% 0.36/0.70 % UNIV_bool
% 0.36/0.70 thf(fact_7740_Rep__unit__induct,axiom,
% 0.36/0.70 ! [Y: $o,P: $o > $o] :
% 0.36/0.70 ( ( member_o @ Y @ ( insert_o @ $true @ bot_bot_set_o ) )
% 0.36/0.70 => ( ! [X3: product_unit] : ( P @ ( product_Rep_unit @ X3 ) )
% 0.36/0.70 => ( P @ Y ) ) ) ).
% 0.36/0.70
% 0.36/0.70 % Rep_unit_induct
% 0.36/0.70 thf(fact_7741_Abs__unit__inject,axiom,
% 0.36/0.70 ! [X: $o,Y: $o] :
% 0.36/0.70 ( ( member_o @ X @ ( insert_o @ $true @ bot_bot_set_o ) )
% 0.36/0.70 => ( ( member_o @ Y @ ( insert_o @ $true @ bot_bot_set_o ) )
% 0.36/0.70 => ( ( ( product_Abs_unit @ X )
% 0.36/0.70 = ( product_Abs_unit @ Y ) )
% 0.36/0.70 = ( X = Y ) ) ) ) ).
% 0.36/0.70
% 0.36/0.70 % Abs_unit_inject
% 0.36/0.70 thf(fact_7742_Rep__unit__inject,axiom,
% 0.36/0.70 ! [X: product_unit,Y: product_unit] :
% 0.36/0.70 ( ( ( product_Rep_unit @ X )
% 0.36/0.70 = ( product_Rep_unit @ Y ) )
% 0.36/0.70 = ( X = Y ) ) ).
% 0.36/0.70
% 0.36/0.70 % Rep_unit_inject
% 0.36/0.70 thf(fact_7743_Rep__unit__inverse,axiom,
% 0.36/0.70 ! [X: product_unit] :
% 0.36/0.70 ( ( product_Abs_unit @ ( product_Rep_unit @ X ) )
% 0.36/0.70 = X ) ).
% 0.36/0.70
% 0.36/0.70 % Rep_unit_inverse
% 0.36/0.70 thf(fact_7744_Abs__unit__inverse,axiom,
% 0.36/0.70 ! [Y: $o] :
% 0.36/0.70 ( ( member_o @ Y @ ( insert_o @ $true @ bot_bot_set_o ) )
% 0.36/0.70 => ( ( product_Rep_unit @ ( product_Abs_unit @ Y ) )
% 0.36/0.70 = Y ) ) ).
% 0.36/0.70
% 0.36/0.70 % Abs_unit_inverse
% 0.36/0.70 thf(fact_7745_type__definition__unit,axiom,
% 0.36/0.70 type_d6188575255521822967unit_o @ product_Rep_unit @ product_Abs_unit @ ( insert_o @ $true @ bot_bot_set_o ) ).
% 0.36/0.70
% 0.36/0.70 % type_definition_unit
% 0.36/0.70 thf(fact_7746_Rep__unit,axiom,
% 0.36/0.70 ! [X: product_unit] : ( member_o @ ( product_Rep_unit @ X ) @ ( insert_o @ $true @ bot_bot_set_o ) ) ).
% 0.36/0.70
% 0.36/0.70 % Rep_unit
% 0.36/0.70 thf(fact_7747_Abs__unit__cases,axiom,
% 0.36/0.70 ! [X: product_unit] :
% 0.36/0.70 ~ ! [Y2: $o] :
% 0.36/0.70 ( ( X
% 0.36/0.70 = ( product_Abs_unit @ Y2 ) )
% 0.36/0.70 => ~ ( member_o @ Y2 @ ( insert_o @ $true @ bot_bot_set_o ) ) ) ).
% 0.36/0.70
% 0.36/0.70 % Abs_unit_cases
% 0.36/0.70 thf(fact_7748_Rep__unit__cases,axiom,
% 0.36/0.70 ! [Y: $o] :
% 0.36/0.70 ( ( member_o @ Y @ ( insert_o @ $true @ bot_bot_set_o ) )
% 0.36/0.70 => ~ ! [X3: product_unit] :
% 0.36/0.70 ( Y
% 0.36/0.70 = ( ~ ( product_Rep_unit @ X3 ) ) ) ) ).
% 0.36/0.70
% 0.36/0.70 % Rep_unit_cases
% 0.36/0.70 thf(fact_7749_Abs__unit__induct,axiom,
% 0.36/0.70 ! [P: product_unit > $o,X: product_unit] :
% 0.36/0.70 ( ! [Y2: $o] :
% 0.36/0.70 ( ( member_o @ Y2 @ ( insert_o @ $true @ bot_bot_set_o ) )
% 0.36/0.70 => ( P @ ( product_Abs_unit @ Y2 ) ) )
% 0.36/0.70 => ( P @ X ) ) ).
% 0.36/0.70
% 0.36/0.70 % Abs_unit_induct
% 0.36/0.70 thf(fact_7750_integer__of__char__code,axiom,
% 0.36/0.70 ! [B0: $o,B1: $o,B22: $o,B32: $o,B42: $o,B52: $o,B62: $o,B72: $o] :
% 0.36/0.70 ( ( integer_of_char @ ( char2 @ B0 @ B1 @ B22 @ B32 @ B42 @ B52 @ B62 @ B72 ) )
% 0.36/0.70 = ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ ( zero_n356916108424825756nteger @ B72 ) @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) @ ( zero_n356916108424825756nteger @ B62 ) ) @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) @ ( zero_n356916108424825756nteger @ B52 ) ) @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) @ ( zero_n356916108424825756nteger @ B42 ) ) @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) @ ( zero_n356916108424825756nteger @ B32 ) ) @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) @ ( zero_n356916108424825756nteger @ B22 ) ) @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) @ ( zero_n356916108424825756nteger @ B1 ) ) @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) @ ( zero_n356916108424825756nteger @ B0 ) ) ) ).
% 0.36/0.70
% 0.36/0.70 % integer_of_char_code
% 0.36/0.70 thf(fact_7751_sum__encode__def,axiom,
% 0.36/0.70 ( nat_sum_encode
% 0.36/0.70 = ( sum_ca6763686470577984908at_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.36/0.70 @ ^ [B6: nat] : ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B6 ) ) ) ) ).
% 0.36/0.70
% 0.36/0.70 % sum_encode_def
% 0.36/0.70 thf(fact_7752_upt__0__eq__Nil__conv,axiom,
% 0.36/0.70 ! [J: nat] :
% 0.36/0.70 ( ( ( upt @ zero_zero_nat @ J )
% 0.36/0.70 = nil_nat )
% 0.36/0.70 = ( J = zero_zero_nat ) ) ).
% 0.36/0.70
% 0.36/0.70 % upt_0_eq_Nil_conv
% 0.36/0.70 thf(fact_7753_upt__merge,axiom,
% 0.36/0.70 ! [I2: nat,J: nat,K3: nat] :
% 0.36/0.70 ( ( ( ord_less_eq_nat @ I2 @ J )
% 0.36/0.70 & ( ord_less_eq_nat @ J @ K3 ) )
% 0.36/0.70 => ( ( append_nat @ ( upt @ I2 @ J ) @ ( upt @ J @ K3 ) )
% 0.36/0.70 = ( upt @ I2 @ K3 ) ) ) ).
% 0.36/0.70
% 0.36/0.70 % upt_merge
% 0.36/0.70 thf(fact_7754_map__add__upt_H,axiom,
% 0.36/0.70 ! [Ofs: nat,A2: nat,B: nat] :
% 0.36/0.70 ( ( map_nat_nat
% 0.36/0.70 @ ^ [I4: nat] : ( plus_plus_nat @ I4 @ Ofs )
% 0.36/0.70 @ ( upt @ A2 @ B ) )
% 0.36/0.70 = ( upt @ ( plus_plus_nat @ A2 @ Ofs ) @ ( plus_plus_nat @ B @ Ofs ) ) ) ).
% 0.36/0.70
% 0.36/0.70 % map_add_upt'
% 0.36/0.70 thf(fact_7755_upt__eq__append__conv,axiom,
% 0.36/0.70 ! [I2: nat,J: nat,Xs: list_nat,Ys: list_nat] :
% 0.36/0.70 ( ( ord_less_eq_nat @ I2 @ J )
% 0.36/0.70 => ( ( ( upt @ I2 @ J )
% 0.36/0.70 = ( append_nat @ Xs @ Ys ) )
% 0.36/0.70 = ( ? [K5: nat] :
% 0.36/0.70 ( ( ord_less_eq_nat @ I2 @ K5 )
% 0.36/0.70 & ( ord_less_eq_nat @ K5 @ J )
% 0.36/0.70 & ( ( upt @ I2 @ K5 )
% 0.36/0.70 = Xs )
% 0.36/0.70 & ( ( upt @ K5 @ J )
% 0.36/0.70 = Ys ) ) ) ) ) ).
% 0.36/0.70
% 0.36/0.70 % upt_eq_append_conv
% 0.36/0.70 thf(fact_7756_upt__append,axiom,
% 0.36/0.70 ! [I2: nat,J: nat] :
% 0.36/0.70 ( ( ord_less_nat @ I2 @ J )
% 0.36/0.70 => ( ( append_nat @ ( upt @ zero_zero_nat @ I2 ) @ ( upt @ I2 @ J ) )
% 0.36/0.70 = ( upt @ zero_zero_nat @ J ) ) ) ).
% 0.36/0.70
% 0.36/0.70 % upt_append
% 0.36/0.70 thf(fact_7757_butlast__upt,axiom,
% 0.36/0.70 ! [M4: nat,N2: nat] :
% 0.36/0.70 ( ( butlast_nat @ ( upt @ M4 @ N2 ) )
% 0.36/0.70 = ( upt @ M4 @ ( minus_minus_nat @ N2 @ one_one_nat ) ) ) ).
% 0.36/0.70
% 0.36/0.70 % butlast_upt
% 0.36/0.70 thf(fact_7758_upt__eq__lel__conv,axiom,
% 0.36/0.70 ! [L: nat,H2: nat,Is1: list_nat,I2: nat,Is2: list_nat] :
% 0.36/0.70 ( ( ( upt @ L @ H2 )
% 0.36/0.70 = ( append_nat @ Is1 @ ( cons_nat @ I2 @ Is2 ) ) )
% 0.36/0.70 = ( ( Is1
% 0.36/0.70 = ( upt @ L @ I2 ) )
% 0.36/0.70 & ( Is2
% 0.36/0.70 = ( upt @ ( suc @ I2 ) @ H2 ) )
% 0.36/0.70 & ( ord_less_eq_nat @ L @ I2 )
% 0.36/0.70 & ( ord_less_nat @ I2 @ H2 ) ) ) ).
% 0.36/0.70
% 0.36/0.70 % upt_eq_lel_conv
% 0.36/0.70 thf(fact_7759_upt__filter__extend,axiom,
% 0.36/0.70 ! [U: nat,U5: nat,P: nat > $o] :
% 0.36/0.70 ( ( ord_less_eq_nat @ U @ U5 )
% 0.36/0.70 => ( ! [I3: nat] :
% 0.36/0.70 ( ( ( ord_less_eq_nat @ U @ I3 )
% 0.36/0.70 & ( ord_less_nat @ I3 @ U5 ) )
% 0.36/0.70 => ~ ( P @ I3 ) )
% 0.36/0.70 => ( ( filter_nat2 @ P @ ( upt @ zero_zero_nat @ U ) )
% 0.36/0.70 = ( filter_nat2 @ P @ ( upt @ zero_zero_nat @ U5 ) ) ) ) ) ).
% 0.36/0.70
% 0.36/0.70 % upt_filter_extend
% 0.36/0.70 thf(fact_7760_in__range_Opelims_I3_J,axiom,
% 0.36/0.70 ! [X: produc3658429121746597890et_nat] :
% 0.36/0.70 ( ~ ( in_range @ X )
% 0.36/0.70 => ( ( accp_P5801069581201407417et_nat @ in_range_rel @ X )
% 0.36/0.70 => ~ ! [H: heap_e7401611519738050253t_unit,As2: set_nat] :
% 0.36/0.70 ( ( X
% 0.36/0.70 = ( produc7507926704131184380et_nat @ H @ As2 ) )
% 0.36/0.70 => ( ( accp_P5801069581201407417et_nat @ in_range_rel @ ( produc7507926704131184380et_nat @ H @ As2 ) )
% 0.36/0.70 => ! [X3: nat] :
% 0.36/0.70 ( ( member_nat @ X3 @ As2 )
% 0.36/0.70 => ( ord_less_nat @ X3 @ ( lim_Product_unit @ H ) ) ) ) ) ) ) ).
% 0.36/0.70
% 0.36/0.70 % in_range.pelims(3)
% 0.36/0.70 thf(fact_7761_in__range_Opelims_I1_J,axiom,
% 0.36/0.70 ! [X: produc3658429121746597890et_nat,Y: $o] :
% 0.36/0.70 ( ( ( in_range @ X )
% 0.36/0.70 = Y )
% 0.36/0.70 => ( ( accp_P5801069581201407417et_nat @ in_range_rel @ X )
% 0.36/0.70 => ~ ! [H: heap_e7401611519738050253t_unit,As2: set_nat] :
% 0.36/0.70 ( ( X
% 0.36/0.70 = ( produc7507926704131184380et_nat @ H @ As2 ) )
% 0.36/0.70 => ( ( Y
% 0.36/0.70 = ( ! [X2: nat] :
% 0.36/0.70 ( ( member_nat @ X2 @ As2 )
% 0.36/0.70 => ( ord_less_nat @ X2 @ ( lim_Product_unit @ H ) ) ) ) )
% 0.36/0.70 => ~ ( accp_P5801069581201407417et_nat @ in_range_rel @ ( produc7507926704131184380et_nat @ H @ As2 ) ) ) ) ) ) ).
% 0.36/0.70
% 0.36/0.70 % in_range.pelims(1)
% 0.36/0.70 thf(fact_7762_in__range_Opelims_I2_J,axiom,
% 0.36/0.70 ! [X: produc3658429121746597890et_nat] :
% 0.36/0.70 ( ( in_range @ X )
% 0.36/0.70 => ( ( accp_P5801069581201407417et_nat @ in_range_rel @ X )
% 0.36/0.70 => ~ ! [H: heap_e7401611519738050253t_unit,As2: set_nat] :
% 0.36/0.70 ( ( X
% 0.36/0.70 = ( produc7507926704131184380et_nat @ H @ As2 ) )
% 0.36/0.70 => ( ( accp_P5801069581201407417et_nat @ in_range_rel @ ( produc7507926704131184380et_nat @ H @ As2 ) )
% 0.36/0.70 => ~ ! [X6: nat] :
% 0.36/0.70 ( ( member_nat @ X6 @ As2 )
% 0.36/0.70 => ( ord_less_nat @ X6 @ ( lim_Product_unit @ H ) ) ) ) ) ) ) ).
% 0.36/0.70
% 0.36/0.70 % in_range.pelims(2)
% 0.36/0.70 thf(fact_7763_one__assn__raw_Opelims_I3_J,axiom,
% 0.36/0.70 ! [X: produc3658429121746597890et_nat] :
% 0.36/0.70 ( ~ ( one_assn_raw @ X )
% 0.36/0.70 => ( ( accp_P5801069581201407417et_nat @ one_assn_raw_rel @ X )
% 0.36/0.70 => ~ ! [H: heap_e7401611519738050253t_unit,As2: set_nat] :
% 0.36/0.70 ( ( X
% 0.36/0.70 = ( produc7507926704131184380et_nat @ H @ As2 ) )
% 0.36/0.70 => ( ( accp_P5801069581201407417et_nat @ one_assn_raw_rel @ ( produc7507926704131184380et_nat @ H @ As2 ) )
% 0.36/0.70 => ( As2 = bot_bot_set_nat ) ) ) ) ) ).
% 0.36/0.70
% 0.36/0.70 % one_assn_raw.pelims(3)
% 0.36/0.70 thf(fact_7764_one__assn__raw_Opelims_I2_J,axiom,
% 0.36/0.70 ! [X: produc3658429121746597890et_nat] :
% 0.36/0.70 ( ( one_assn_raw @ X )
% 0.36/0.70 => ( ( accp_P5801069581201407417et_nat @ one_assn_raw_rel @ X )
% 0.36/0.70 => ~ ! [H: heap_e7401611519738050253t_unit,As2: set_nat] :
% 0.36/0.70 ( ( X
% 0.36/0.70 = ( produc7507926704131184380et_nat @ H @ As2 ) )
% 0.36/0.70 => ( ( accp_P5801069581201407417et_nat @ one_assn_raw_rel @ ( produc7507926704131184380et_nat @ H @ As2 ) )
% 0.36/0.70 => ( As2 != bot_bot_set_nat ) ) ) ) ) ).
% 0.36/0.70
% 0.36/0.70 % one_assn_raw.pelims(2)
% 0.36/0.70 thf(fact_7765_one__assn__raw_Opelims_I1_J,axiom,
% 0.36/0.70 ! [X: produc3658429121746597890et_nat,Y: $o] :
% 0.36/0.70 ( ( ( one_assn_raw @ X )
% 0.36/0.70 = Y )
% 0.36/0.70 => ( ( accp_P5801069581201407417et_nat @ one_assn_raw_rel @ X )
% 0.36/0.70 => ~ ! [H: heap_e7401611519738050253t_unit,As2: set_nat] :
% 0.36/0.70 ( ( X
% 0.36/0.70 = ( produc7507926704131184380et_nat @ H @ As2 ) )
% 0.36/0.70 => ( ( Y
% 0.36/0.70 = ( As2 = bot_bot_set_nat ) )
% 0.36/0.70 => ~ ( accp_P5801069581201407417et_nat @ one_assn_raw_rel @ ( produc7507926704131184380et_nat @ H @ As2 ) ) ) ) ) ) ).
% 0.36/0.70
% 0.36/0.70 % one_assn_raw.pelims(1)
% 0.36/0.70 thf(fact_7766_times__int__def,axiom,
% 0.36/0.70 ( times_times_int
% 0.36/0.70 = ( map_fu4960017516451851995nt_int @ rep_Integ @ ( map_fu3667384564859982768at_int @ rep_Integ @ abs_Integ )
% 0.36/0.70 @ ( produc27273713700761075at_nat
% 0.36/0.70 @ ^ [X2: nat,Y3: nat] :
% 0.36/0.70 ( produc2626176000494625587at_nat
% 0.36/0.70 @ ^ [U3: nat,V4: nat] : ( product_Pair_nat_nat @ ( plus_plus_nat @ ( times_times_nat @ X2 @ U3 ) @ ( times_times_nat @ Y3 @ V4 ) ) @ ( plus_plus_nat @ ( times_times_nat @ X2 @ V4 ) @ ( times_times_nat @ Y3 @ U3 ) ) ) ) ) ) ) ).
% 0.36/0.70
% 0.36/0.70 % times_int_def
% 0.36/0.70 thf(fact_7767_next_Osimps,axiom,
% 0.36/0.70 ! [V: code_natural,W: code_natural] :
% 0.36/0.70 ( ( next @ ( produc3574140220909816553atural @ V @ W ) )
% 0.36/0.70 = ( produc6639722614265839536atural @ ( plus_p4538020629002901425atural @ ( minus_shift @ ( numera5444537566228673987atural @ ( bit0 @ ( bit1 @ ( bit0 @ ( bit1 @ ( bit0 @ ( bit1 @ ( bit0 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ one ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) @ ( minus_shift @ ( numera5444537566228673987atural @ ( bit1 @ ( bit1 @ ( bit0 @ ( bit1 @ ( bit0 @ ( bit1 @ ( bit0 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ one ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) @ ( times_2397367101498566445atural @ ( modulo8411746178871703098atural @ V @ ( numera5444537566228673987atural @ ( bit0 @ ( bit0 @ ( bit1 @ ( bit0 @ ( bit0 @ ( bit1 @ ( bit0 @ ( bit1 @ ( bit1 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit1 @ ( bit0 @ ( bit1 @ one ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) @ ( numera5444537566228673987atural @ ( bit0 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit0 @ ( bit0 @ ( bit1 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit0 @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) @ ( times_2397367101498566445atural @ ( divide5121882707175180666atural @ V @ ( numera5444537566228673987atural @ ( bit0 @ ( bit0 @ ( bit1 @ ( bit0 @ ( bit0 @ ( bit1 @ ( bit0 @ ( bit1 @ ( bit1 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit1 @ ( bit0 @ ( bit1 @ one ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) @ ( numera5444537566228673987atural @ ( bit1 @ ( bit1 @ ( bit0 @ ( bit0 @ ( bit1 @ ( bit1 @ ( bit0 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) @ ( plus_p4538020629002901425atural @ ( minus_shift @ ( numera5444537566228673987atural @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ one ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) @ ( times_2397367101498566445atural @ ( modulo8411746178871703098atural @ W @ ( numera5444537566228673987atural @ ( bit0 @ ( bit1 @ ( bit1 @ ( bit0 @ ( bit0 @ ( bit1 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit0 @ ( bit0 @ ( bit1 @ one ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) @ ( numera5444537566228673987atural @ ( bit0 @ ( bit0 @ ( bit1 @ ( bit0 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit0 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit0 @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) @ ( times_2397367101498566445atural @ ( divide5121882707175180666atural @ W @ ( numera5444537566228673987atural @ ( bit0 @ ( bit1 @ ( bit1 @ ( bit0 @ ( bit0 @ ( bit1 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit0 @ ( bit0 @ ( bit1 @ one ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) @ ( numera5444537566228673987atural @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit0 @ ( bit0 @ ( bit1 @ ( bit1 @ ( bit0 @ ( bit1 @ ( bit1 @ one ) ) ) ) ) ) ) ) ) ) ) ) ) ) @ one_one_Code_natural ) ) @ one_one_Code_natural ) @ ( produc3574140220909816553atural @ ( minus_shift @ ( numera5444537566228673987atural @ ( bit1 @ ( bit1 @ ( bit0 @ ( bit1 @ ( bit0 @ ( bit1 @ ( bit0 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ one ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) @ ( times_2397367101498566445atural @ ( modulo8411746178871703098atural @ V @ ( numera5444537566228673987atural @ ( bit0 @ ( bit0 @ ( bit1 @ ( bit0 @ ( bit0 @ ( bit1 @ ( bit0 @ ( bit1 @ ( bit1 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit1 @ ( bit0 @ ( bit1 @ one ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) @ ( numera5444537566228673987atural @ ( bit0 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit0 @ ( bit0 @ ( bit1 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit0 @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) @ ( times_2397367101498566445atural @ ( divide5121882707175180666atural @ V @ ( numera5444537566228673987atural @ ( bit0 @ ( bit0 @ ( bit1 @ ( bit0 @ ( bit0 @ ( bit1 @ ( bit0 @ ( bit1 @ ( bit1 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit1 @ ( bit0 @ ( bit1 @ one ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) @ ( numera5444537566228673987atural @ ( bit1 @ ( bit1 @ ( bit0 @ ( bit0 @ ( bit1 @ ( bit1 @ ( bit0 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) @ ( minus_shift @ ( numera5444537566228673987atural @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ one ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) @ ( times_2397367101498566445atural @ ( modulo8411746178871703098atural @ W @ ( numera5444537566228673987atural @ ( bit0 @ ( bit1 @ ( bit1 @ ( bit0 @ ( bit0 @ ( bit1 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit0 @ ( bit0 @ ( bit1 @ one ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) @ ( numera5444537566228673987atural @ ( bit0 @ ( bit0 @ ( bit1 @ ( bit0 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit0 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit0 @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) @ ( times_2397367101498566445atural @ ( divide5121882707175180666atural @ W @ ( numera5444537566228673987atural @ ( bit0 @ ( bit1 @ ( bit1 @ ( bit0 @ ( bit0 @ ( bit1 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit0 @ ( bit0 @ ( bit1 @ one ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) @ ( numera5444537566228673987atural @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit0 @ ( bit0 @ ( bit1 @ ( bit1 @ ( bit0 @ ( bit1 @ ( bit1 @ one ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% 0.36/0.70
% 0.36/0.70 % next.simps
% 0.36/0.70 thf(fact_7768_Random_Orange__def,axiom,
% 0.36/0.70 ( range
% 0.36/0.70 = ( ^ [K5: code_natural] :
% 0.36/0.70 ( produc5538323210962509403atural
% 0.36/0.70 @ ( iterat8892046348760725948atural @ ( log @ ( numera5444537566228673987atural @ ( bit1 @ ( bit0 @ ( bit0 @ ( bit1 @ ( bit0 @ ( bit1 @ ( bit0 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ one ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) @ K5 )
% 0.36/0.70 @ ^ [L2: code_natural] :
% 0.36/0.70 ( produc5538323210962509403atural @ next
% 0.36/0.70 @ ^ [V4: code_natural] : ( produc6639722614265839536atural @ ( plus_p4538020629002901425atural @ V4 @ ( times_2397367101498566445atural @ L2 @ ( numera5444537566228673987atural @ ( bit1 @ ( bit0 @ ( bit0 @ ( bit1 @ ( bit0 @ ( bit1 @ ( bit0 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ one ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) )
% 0.36/0.70 @ one_one_Code_natural )
% 0.36/0.70 @ ^ [V4: code_natural] : ( produc6639722614265839536atural @ ( modulo8411746178871703098atural @ V4 @ K5 ) ) ) ) ) ).
% 0.36/0.70
% 0.36/0.70 % Random.range_def
% 0.36/0.70 thf(fact_7769_bot__unit__def,axiom,
% 0.36/0.70 bot_bot_Product_unit = product_Unity ).
% 0.36/0.70
% 0.36/0.70 % bot_unit_def
% 0.36/0.70 thf(fact_7770_sup__unit__def,axiom,
% 0.36/0.70 ( sup_sup_Product_unit
% 0.36/0.70 = ( ^ [Uu2: product_unit,Uv: product_unit] : product_Unity ) ) ).
% 0.36/0.70
% 0.36/0.70 % sup_unit_def
% 0.36/0.70 thf(fact_7771_old_Ounit_Oexhaust,axiom,
% 0.36/0.70 ! [Y: product_unit] : ( Y = product_Unity ) ).
% 0.36/0.70
% 0.36/0.70 % old.unit.exhaust
% 0.36/0.70 thf(fact_7772_Inf__unit__def,axiom,
% 0.36/0.70 ( comple2584293577114468500t_unit
% 0.36/0.70 = ( ^ [Uu2: set_Product_unit] : product_Unity ) ) ).
% 0.36/0.70
% 0.36/0.70 % Inf_unit_def
% 0.36/0.70 thf(fact_7773_uminus__unit__def,axiom,
% 0.36/0.70 ( uminus2952777764628376836t_unit
% 0.36/0.70 = ( ^ [Uu2: product_unit] : product_Unity ) ) ).
% 0.36/0.70
% 0.36/0.70 % uminus_unit_def
% 0.36/0.70 thf(fact_7774_top__unit__def,axiom,
% 0.36/0.70 top_top_Product_unit = product_Unity ).
% 0.36/0.70
% 0.36/0.70 % top_unit_def
% 0.36/0.70 thf(fact_7775_Unity__def,axiom,
% 0.36/0.70 ( product_Unity
% 0.36/0.70 = ( product_Abs_unit @ $true ) ) ).
% 0.36/0.70
% 0.36/0.70 % Unity_def
% 0.36/0.70 thf(fact_7776_Sup__unit__def,axiom,
% 0.36/0.70 ( comple4687483117567863418t_unit
% 0.36/0.70 = ( ^ [Uu2: set_Product_unit] : product_Unity ) ) ).
% 0.36/0.70
% 0.36/0.70 % Sup_unit_def
% 0.36/0.70 thf(fact_7777_inf__unit__def,axiom,
% 0.36/0.70 ( inf_inf_Product_unit
% 0.36/0.70 = ( ^ [Uu2: product_unit,Uv: product_unit] : product_Unity ) ) ).
% 0.36/0.70
% 0.36/0.70 % inf_unit_def
% 0.36/0.70 thf(fact_7778_UNIV__unit,axiom,
% 0.36/0.70 ( top_to1996260823553986621t_unit
% 0.36/0.70 = ( insert_Product_unit @ product_Unity @ bot_bo3957492148770167129t_unit ) ) ).
% 0.36/0.70
% 0.36/0.70 % UNIV_unit
% 0.36/0.70 thf(fact_7779_default__unit__def,axiom,
% 0.36/0.70 defaul566961228789861419t_unit = product_Unity ).
% 0.36/0.70
% 0.36/0.70 % default_unit_def
% 0.36/0.70 thf(fact_7780_cone__def,axiom,
% 0.36/0.70 ( bNF_Cardinal_cone
% 0.36/0.70 = ( bNF_Ca7083678892733797993t_unit @ ( insert_Product_unit @ product_Unity @ bot_bo3957492148770167129t_unit ) ) ) ).
% 0.36/0.70
% 0.36/0.70 % cone_def
% 0.36/0.70 thf(fact_7781_times__integer__code_I3_J,axiom,
% 0.36/0.70 ! [M4: num,N2: num] :
% 0.36/0.70 ( ( times_3573771949741848930nteger @ ( code_Pos @ M4 ) @ ( code_Pos @ N2 ) )
% 0.36/0.70 = ( code_Pos @ ( times_times_num @ M4 @ N2 ) ) ) ).
% 0.36/0.70
% 0.36/0.70 % times_integer_code(3)
% 0.36/0.70 thf(fact_7782_times__integer__code_I6_J,axiom,
% 0.36/0.70 ! [M4: num,N2: num] :
% 0.36/0.70 ( ( times_3573771949741848930nteger @ ( code_Neg @ M4 ) @ ( code_Neg @ N2 ) )
% 0.36/0.70 = ( code_Pos @ ( times_times_num @ M4 @ N2 ) ) ) ).
% 0.36/0.70
% 0.36/0.70 % times_integer_code(6)
% 0.36/0.70 thf(fact_7783_times__integer__code_I5_J,axiom,
% 0.36/0.70 ! [M4: num,N2: num] :
% 0.36/0.70 ( ( times_3573771949741848930nteger @ ( code_Neg @ M4 ) @ ( code_Pos @ N2 ) )
% 0.36/0.70 = ( code_Neg @ ( times_times_num @ M4 @ N2 ) ) ) ).
% 0.36/0.70
% 0.36/0.70 % times_integer_code(5)
% 0.36/0.70 thf(fact_7784_times__integer__code_I4_J,axiom,
% 0.36/0.70 ! [M4: num,N2: num] :
% 0.36/0.70 ( ( times_3573771949741848930nteger @ ( code_Pos @ M4 ) @ ( code_Neg @ N2 ) )
% 0.36/0.70 = ( code_Neg @ ( times_times_num @ M4 @ N2 ) ) ) ).
% 0.36/0.70
% 0.36/0.70 % times_integer_code(4)
% 0.36/0.70 thf(fact_7785_times__natural_Orep__eq,axiom,
% 0.36/0.70 ! [X: code_natural,Xa: code_natural] :
% 0.36/0.70 ( ( code_nat_of_natural @ ( times_2397367101498566445atural @ X @ Xa ) )
% 0.36/0.70 = ( times_times_nat @ ( code_nat_of_natural @ X ) @ ( code_nat_of_natural @ Xa ) ) ) ).
% 0.36/0.70
% 0.36/0.70 % times_natural.rep_eq
% 0.36/0.70 thf(fact_7786_times__int__code_I3_J,axiom,
% 0.36/0.70 ! [M4: num,N2: num] :
% 0.36/0.70 ( ( times_times_int @ ( pos @ M4 ) @ ( pos @ N2 ) )
% 0.36/0.70 = ( pos @ ( times_times_num @ M4 @ N2 ) ) ) ).
% 0.36/0.70
% 0.36/0.70 % times_int_code(3)
% 0.36/0.70 thf(fact_7787_times__int__code_I4_J,axiom,
% 0.36/0.70 ! [M4: num,N2: num] :
% 0.36/0.70 ( ( times_times_int @ ( pos @ M4 ) @ ( neg @ N2 ) )
% 0.36/0.70 = ( neg @ ( times_times_num @ M4 @ N2 ) ) ) ).
% 0.36/0.70
% 0.36/0.70 % times_int_code(4)
% 0.36/0.70 thf(fact_7788_times__int__code_I5_J,axiom,
% 0.36/0.70 ! [M4: num,N2: num] :
% 0.36/0.70 ( ( times_times_int @ ( neg @ M4 ) @ ( pos @ N2 ) )
% 0.36/0.70 = ( neg @ ( times_times_num @ M4 @ N2 ) ) ) ).
% 0.36/0.70
% 0.36/0.70 % times_int_code(5)
% 0.36/0.70 thf(fact_7789_times__int__code_I6_J,axiom,
% 0.36/0.70 ! [M4: num,N2: num] :
% 0.36/0.70 ( ( times_times_int @ ( neg @ M4 ) @ ( neg @ N2 ) )
% 0.36/0.70 = ( pos @ ( times_times_num @ M4 @ N2 ) ) ) ).
% 0.36/0.70
% 0.36/0.70 % times_int_code(6)
% 0.36/0.70 thf(fact_7790_times__natural_Otransfer,axiom,
% 0.36/0.70 bNF_re88643428490162567atural @ code_pcr_natural @ ( bNF_re3704215830270325841atural @ code_pcr_natural @ code_pcr_natural ) @ times_times_nat @ times_2397367101498566445atural ).
% 0.36/0.70
% 0.36/0.70 % times_natural.transfer
% 0.36/0.70 thf(fact_7791_times__natural_Oabs__eq,axiom,
% 0.36/0.70 ! [Xa: nat,X: nat] :
% 0.36/0.70 ( ( times_2397367101498566445atural @ ( code_natural_of_nat @ Xa ) @ ( code_natural_of_nat @ X ) )
% 0.36/0.70 = ( code_natural_of_nat @ ( times_times_nat @ Xa @ X ) ) ) ).
% 0.36/0.70
% 0.36/0.70 % times_natural.abs_eq
% 0.36/0.70 thf(fact_7792_type__definition__natural,axiom,
% 0.36/0.70 type_d4410041424927559462al_nat @ code_nat_of_natural @ code_natural_of_nat @ top_top_set_nat ).
% 0.36/0.70
% 0.36/0.70 % type_definition_natural
% 0.36/0.70 thf(fact_7793_times__natural__def,axiom,
% 0.36/0.70 ( times_2397367101498566445atural
% 0.36/0.70 = ( map_fu6549440983881763648atural @ code_nat_of_natural @ ( map_fu1239815594074539274atural @ code_nat_of_natural @ code_natural_of_nat ) @ times_times_nat ) ) ).
% 0.36/0.70
% 0.36/0.70 % times_natural_def
% 0.36/0.70 thf(fact_7794_times__integer_Otransfer,axiom,
% 0.36/0.70 bNF_re398004352372739002nteger @ code_pcr_integer @ ( bNF_re3379532845092657523nteger @ code_pcr_integer @ code_pcr_integer ) @ times_times_int @ times_3573771949741848930nteger ).
% 0.36/0.70
% 0.36/0.70 % times_integer.transfer
% 0.36/0.70 thf(fact_7795_times__integer__def,axiom,
% 0.36/0.70 ( times_3573771949741848930nteger
% 0.36/0.70 = ( map_fu8272188784021352819nteger @ code_int_of_integer @ ( map_fu2599414010547811884nteger @ code_int_of_integer @ code_integer_of_int ) @ times_times_int ) ) ).
% 0.36/0.70
% 0.36/0.70 % times_integer_def
% 0.36/0.70
% 0.36/0.70 % Helper facts (29)
% 0.36/0.70 thf(help_If_2_1_If_001t__Int__Oint_T,axiom,
% 0.36/0.70 ! [X: int,Y: int] :
% 0.36/0.70 ( ( if_int @ $false @ X @ Y )
% 0.36/0.70 = Y ) ).
% 0.36/0.70
% 0.36/0.70 thf(help_If_1_1_If_001t__Int__Oint_T,axiom,
% 0.36/0.70 ! [X: int,Y: int] :
% 0.36/0.70 ( ( if_int @ $true @ X @ Y )
% 0.36/0.70 = X ) ).
% 0.36/0.70
% 0.36/0.70 thf(help_If_2_1_If_001t__Nat__Onat_T,axiom,
% 0.36/0.70 ! [X: nat,Y: nat] :
% 0.36/0.70 ( ( if_nat @ $false @ X @ Y )
% 0.36/0.70 = Y ) ).
% 0.36/0.70
% 0.36/0.70 thf(help_If_1_1_If_001t__Nat__Onat_T,axiom,
% 0.36/0.70 ! [X: nat,Y: nat] :
% 0.36/0.70 ( ( if_nat @ $true @ X @ Y )
% 0.36/0.70 = X ) ).
% 0.36/0.70
% 0.36/0.70 thf(help_If_2_1_If_001t__Rat__Orat_T,axiom,
% 0.36/0.70 ! [X: rat,Y: rat] :
% 0.36/0.70 ( ( if_rat @ $false @ X @ Y )
% 0.36/0.70 = Y ) ).
% 0.36/0.70
% 0.36/0.70 thf(help_If_1_1_If_001t__Rat__Orat_T,axiom,
% 0.36/0.70 ! [X: rat,Y: rat] :
% 0.36/0.70 ( ( if_rat @ $true @ X @ Y )
% 0.36/0.70 = X ) ).
% 0.36/0.70
% 0.36/0.70 thf(help_If_2_1_If_001t__Set__Oset_I_Eo_J_T,axiom,
% 0.36/0.70 ! [X: set_o,Y: set_o] :
% 0.36/0.70 ( ( if_set_o @ $false @ X @ Y )
% 0.36/0.70 = Y ) ).
% 0.36/0.70
% 0.36/0.70 thf(help_If_1_1_If_001t__Set__Oset_I_Eo_J_T,axiom,
% 0.36/0.70 ! [X: set_o,Y: set_o] :
% 0.36/0.70 ( ( if_set_o @ $true @ X @ Y )
% 0.36/0.70 = X ) ).
% 0.36/0.70
% 0.36/0.70 thf(help_If_2_1_If_001t__Assertions__Oassn_T,axiom,
% 0.36/0.70 ! [X: assn,Y: assn] :
% 0.36/0.70 ( ( if_assn @ $false @ X @ Y )
% 0.36/0.70 = Y ) ).
% 0.36/0.70
% 0.36/0.70 thf(help_If_1_1_If_001t__Assertions__Oassn_T,axiom,
% 0.36/0.70 ! [X: assn,Y: assn] :
% 0.36/0.70 ( ( if_assn @ $true @ X @ Y )
% 0.36/0.70 = X ) ).
% 0.36/0.70
% 0.36/0.70 thf(help_If_2_1_If_001t__Code____Numeral__Ointeger_T,axiom,
% 0.36/0.70 ! [X: code_integer,Y: code_integer] :
% 0.36/0.70 ( ( if_Code_integer @ $false @ X @ Y )
% 0.36/0.70 = Y ) ).
% 0.36/0.70
% 0.36/0.70 thf(help_If_1_1_If_001t__Code____Numeral__Ointeger_T,axiom,
% 0.36/0.70 ! [X: code_integer,Y: code_integer] :
% 0.36/0.70 ( ( if_Code_integer @ $true @ X @ Y )
% 0.36/0.70 = X ) ).
% 0.36/0.70
% 0.36/0.70 thf(help_If_2_1_If_001t__Code____Numeral__Onatural_T,axiom,
% 0.36/0.70 ! [X: code_natural,Y: code_natural] :
% 0.36/0.70 ( ( if_Code_natural @ $false @ X @ Y )
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% 14.17/7.98 thf(sy_c_BNF__Def_Orel__fun_001t__Int__Oint_001t__Int__Oint_001_062_It__Int__Oint_Mt__Int__Oint_J_001_062_It__Int__Oint_Mt__Int__Oint_J,type,
% 14.17/7.98 bNF_re711492959462206631nt_int: ( int > int > $o ) > ( ( int > int ) > ( int > int ) > $o ) > ( int > int > int ) > ( int > int > int ) > $o ).
% 14.17/7.98
% 14.17/7.98 thf(sy_c_BNF__Def_Orel__fun_001t__Int__Oint_001t__Int__Oint_001t__Int__Oint_001t__Int__Oint,type,
% 14.17/7.98 bNF_re4712519889275205905nt_int: ( int > int > $o ) > ( int > int > $o ) > ( int > int ) > ( int > int ) > $o ).
% 14.17/7.98
% 14.17/7.98 thf(sy_c_BNF__Def_Orel__fun_001t__Nat__Onat_001t__Code____Numeral__Onatural_001_062_It__Nat__Onat_Mt__Nat__Onat_J_001_062_It__Code____Numeral__Onatural_Mt__Code____Numeral__Onatural_J,type,
% 14.17/7.98 bNF_re88643428490162567atural: ( nat > code_natural > $o ) > ( ( nat > nat ) > ( code_natural > code_natural ) > $o ) > ( nat > nat > nat ) > ( code_natural > code_natural > code_natural ) > $o ).
% 14.17/7.98
% 14.17/7.98 thf(sy_c_BNF__Def_Orel__fun_001t__Nat__Onat_001t__Code____Numeral__Onatural_001t__Nat__Onat_001t__Code____Numeral__Onatural,type,
% 14.17/7.98 bNF_re3704215830270325841atural: ( nat > code_natural > $o ) > ( nat > code_natural > $o ) > ( nat > nat ) > ( code_natural > code_natural ) > $o ).
% 14.17/7.98
% 14.17/7.98 thf(sy_c_BNF__Def_Orel__fun_001t__Nat__Onat_001t__Nat__Onat_001_062_It__Nat__Onat_Mt__Nat__Onat_J_001_062_It__Nat__Onat_Mt__Nat__Onat_J,type,
% 14.17/7.98 bNF_re1345281282404953727at_nat: ( nat > nat > $o ) > ( ( nat > nat ) > ( nat > nat ) > $o ) > ( nat > nat > nat ) > ( nat > nat > nat ) > $o ).
% 14.17/7.98
% 14.17/7.98 thf(sy_c_BNF__Def_Orel__fun_001t__Nat__Onat_001t__Nat__Onat_001t__Nat__Onat_001t__Nat__Onat,type,
% 14.17/7.98 bNF_re5653821019739307937at_nat: ( nat > nat > $o ) > ( nat > nat > $o ) > ( nat > nat ) > ( nat > nat ) > $o ).
% 14.17/7.98
% 14.17/7.98 thf(sy_c_BNF__Def_Orel__fun_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_001_062_It__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_Mt__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_J_001_062_It__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_Mt__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_J,type,
% 14.17/7.98 bNF_re5228765855967844073nt_int: ( product_prod_int_int > product_prod_int_int > $o ) > ( ( product_prod_int_int > product_prod_int_int ) > ( product_prod_int_int > product_prod_int_int ) > $o ) > ( product_prod_int_int > product_prod_int_int > product_prod_int_int ) > ( product_prod_int_int > product_prod_int_int > product_prod_int_int ) > $o ).
% 14.17/7.98
% 14.17/7.98 thf(sy_c_BNF__Def_Orel__fun_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_001_Eo_001_Eo,type,
% 14.17/7.98 bNF_re8699439704749558557nt_o_o: ( product_prod_int_int > product_prod_int_int > $o ) > ( $o > $o > $o ) > ( product_prod_int_int > $o ) > ( product_prod_int_int > $o ) > $o ).
% 14.17/7.98
% 14.17/7.98 thf(sy_c_BNF__Def_Orel__fun_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J,type,
% 14.17/7.98 bNF_re7145576690424134365nt_int: ( product_prod_int_int > product_prod_int_int > $o ) > ( product_prod_int_int > product_prod_int_int > $o ) > ( product_prod_int_int > product_prod_int_int ) > ( product_prod_int_int > product_prod_int_int ) > $o ).
% 14.17/7.98
% 14.17/7.98 thf(sy_c_BNF__Def_Orel__fun_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_001t__Rat__Orat_001_062_It__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_Mt__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_J_001_062_It__Rat__Orat_Mt__Rat__Orat_J,type,
% 14.17/7.98 bNF_re7627151682743391978at_rat: ( product_prod_int_int > rat > $o ) > ( ( product_prod_int_int > product_prod_int_int ) > ( rat > rat ) > $o ) > ( product_prod_int_int > product_prod_int_int > product_prod_int_int ) > ( rat > rat > rat ) > $o ).
% 14.17/7.98
% 14.17/7.98 thf(sy_c_BNF__Def_Orel__fun_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_001t__Rat__Orat_001_Eo_001_Eo,type,
% 14.17/7.98 bNF_re1494630372529172596at_o_o: ( product_prod_int_int > rat > $o ) > ( $o > $o > $o ) > ( product_prod_int_int > $o ) > ( rat > $o ) > $o ).
% 14.17/7.98
% 14.17/7.98 thf(sy_c_BNF__Def_Orel__fun_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_001t__Rat__Orat_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_001t__Rat__Orat,type,
% 14.17/7.98 bNF_re8279943556446156061nt_rat: ( product_prod_int_int > rat > $o ) > ( product_prod_int_int > rat > $o ) > ( product_prod_int_int > product_prod_int_int ) > ( rat > rat ) > $o ).
% 14.17/7.98
% 14.17/7.98 thf(sy_c_BNF__Def_Orel__fun_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_001t__Int__Oint_001_062_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_Mt__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_001_062_It__Int__Oint_Mt__Int__Oint_J,type,
% 14.17/7.98 bNF_re7408651293131936558nt_int: ( product_prod_nat_nat > int > $o ) > ( ( product_prod_nat_nat > product_prod_nat_nat ) > ( int > int ) > $o ) > ( product_prod_nat_nat > product_prod_nat_nat > product_prod_nat_nat ) > ( int > int > int ) > $o ).
% 14.17/7.98
% 14.17/7.98 thf(sy_c_BNF__Def_Orel__fun_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_001t__Int__Oint_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_001t__Int__Oint,type,
% 14.17/7.98 bNF_re7400052026677387805at_int: ( product_prod_nat_nat > int > $o ) > ( product_prod_nat_nat > int > $o ) > ( product_prod_nat_nat > product_prod_nat_nat ) > ( int > int ) > $o ).
% 14.17/7.98
% 14.17/7.98 thf(sy_c_BNF__Def_Orel__fun_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_001_062_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_Mt__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_001_062_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_Mt__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
% 14.17/7.98 bNF_re3099431351363272937at_nat: ( product_prod_nat_nat > product_prod_nat_nat > $o ) > ( ( product_prod_nat_nat > product_prod_nat_nat ) > ( product_prod_nat_nat > product_prod_nat_nat ) > $o ) > ( product_prod_nat_nat > product_prod_nat_nat > product_prod_nat_nat ) > ( product_prod_nat_nat > product_prod_nat_nat > product_prod_nat_nat ) > $o ).
% 14.17/7.98
% 14.17/7.98 thf(sy_c_BNF__Def_Orel__fun_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
% 14.17/7.98 bNF_re2241393799969408733at_nat: ( product_prod_nat_nat > product_prod_nat_nat > $o ) > ( product_prod_nat_nat > product_prod_nat_nat > $o ) > ( product_prod_nat_nat > product_prod_nat_nat ) > ( product_prod_nat_nat > product_prod_nat_nat ) > $o ).
% 14.17/7.98
% 14.17/7.98 thf(sy_c_BNF__Def_OsndOp_001t__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J_001t__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J_001t__Set__Oset_It__Nat__Onat_J,type,
% 14.17/7.98 bNF_sn4039337771197282234et_nat: ( heap_e7401611519738050253t_unit > heap_e7401611519738050253t_unit > $o ) > ( heap_e7401611519738050253t_unit > set_nat > $o ) > produc3658429121746597890et_nat > produc3658429121746597890et_nat ).
% 14.17/7.98
% 14.17/7.98 thf(sy_c_BNF__Def_OsndOp_001t__Int__Oint_001t__Int__Oint_001t__Int__Oint,type,
% 14.17/7.98 bNF_sn1062102010912252026nt_int: ( int > int > $o ) > ( int > int > $o ) > product_prod_int_int > product_prod_int_int ).
% 14.17/7.98
% 14.17/7.98 thf(sy_c_BNF__Wellorder__Constructions_Obsqr_001t__Int__Oint,type,
% 14.17/7.98 bNF_We1730399552001240015qr_int: set_Pr958786334691620121nt_int > set_Pr2560585780119916871nt_int ).
% 14.17/7.98
% 14.17/7.98 thf(sy_c_BNF__Wellorder__Constructions_Obsqr_001t__Nat__Onat,type,
% 14.17/7.98 bNF_We1732890022510290291qr_nat: set_Pr1261947904930325089at_nat > set_Pr8693737435421807431at_nat ).
% 14.17/7.98
% 14.17/7.98 thf(sy_c_BNF__Wellorder__Relation_Owo__rel_001_062_It__Product____Type__Oprod_It__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J_Mt__Set__Oset_It__Nat__Onat_J_J_M_Eo_J,type,
% 14.17/7.98 bNF_We740377018556551609_nat_o: set_Pr2161125870931222855_nat_o > $o ).
% 14.17/7.98
% 14.17/7.98 thf(sy_c_BNF__Wellorder__Relation_Owo__rel_001_Eo,type,
% 14.17/7.98 bNF_We6437023980930612612_rel_o: set_Product_prod_o_o > $o ).
% 14.17/7.98
% 14.17/7.98 thf(sy_c_BNF__Wellorder__Relation_Owo__rel_001t__Int__Oint,type,
% 14.17/7.98 bNF_We3815749466139970368el_int: set_Pr958786334691620121nt_int > $o ).
% 14.17/7.98
% 14.17/7.98 thf(sy_c_BNF__Wellorder__Relation_Owo__rel_001t__Nat__Onat,type,
% 14.17/7.98 bNF_We3818239936649020644el_nat: set_Pr1261947904930325089at_nat > $o ).
% 14.17/7.98
% 14.17/7.98 thf(sy_c_BNF__Wellorder__Relation_Owo__rel_001t__Product____Type__Ounit,type,
% 14.17/7.98 bNF_We2091578341578594809t_unit: set_Pr5094982260447487303t_unit > $o ).
% 14.17/7.98
% 14.17/7.98 thf(sy_c_BNF__Wellorder__Relation_Owo__rel_OisMinim_001_062_It__Product____Type__Oprod_It__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J_Mt__Set__Oset_It__Nat__Onat_J_J_M_Eo_J,type,
% 14.17/7.98 bNF_We5608535359669600131_nat_o: set_Pr2161125870931222855_nat_o > set_Pr4532377907799695533_nat_o > ( produc3658429121746597890et_nat > $o ) > $o ).
% 14.17/7.98
% 14.17/7.98 thf(sy_c_BNF__Wellorder__Relation_Owo__rel_OisMinim_001_Eo,type,
% 14.17/7.98 bNF_We4647864369647134286inim_o: set_Product_prod_o_o > set_o > $o > $o ).
% 14.17/7.98
% 14.17/7.98 thf(sy_c_BNF__Wellorder__Relation_Owo__rel_OisMinim_001t__Int__Oint,type,
% 14.17/7.98 bNF_We6511469253196982518im_int: set_Pr958786334691620121nt_int > set_int > int > $o ).
% 14.17/7.98
% 14.17/7.98 thf(sy_c_BNF__Wellorder__Relation_Owo__rel_OisMinim_001t__Nat__Onat,type,
% 14.17/7.98 bNF_We6513959723706032794im_nat: set_Pr1261947904930325089at_nat > set_nat > nat > $o ).
% 14.17/7.98
% 14.17/7.98 thf(sy_c_BNF__Wellorder__Relation_Owo__rel_OisMinim_001t__Product____Type__Ounit,type,
% 14.17/7.98 bNF_We909621271159000003t_unit: set_Pr5094982260447487303t_unit > set_Product_unit > product_unit > $o ).
% 14.17/7.98
% 14.17/7.98 thf(sy_c_BNF__Wellorder__Relation_Owo__rel_Omax2_001_062_It__Product____Type__Oprod_It__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J_Mt__Set__Oset_It__Nat__Onat_J_J_M_Eo_J,type,
% 14.17/7.98 bNF_We6400040844883348763_nat_o: set_Pr2161125870931222855_nat_o > ( produc3658429121746597890et_nat > $o ) > ( produc3658429121746597890et_nat > $o ) > produc3658429121746597890et_nat > $o ).
% 14.17/7.98
% 14.17/7.98 thf(sy_c_BNF__Wellorder__Relation_Owo__rel_Omax2_001_Eo,type,
% 14.17/7.98 bNF_We3104086342435940326max2_o: set_Product_prod_o_o > $o > $o > $o ).
% 14.17/7.98
% 14.17/7.98 thf(sy_c_BNF__Wellorder__Relation_Owo__rel_Omax2_001t__Int__Oint,type,
% 14.17/7.98 bNF_We2378448595015613790x2_int: set_Pr958786334691620121nt_int > int > int > int ).
% 14.17/7.98
% 14.17/7.98 thf(sy_c_BNF__Wellorder__Relation_Owo__rel_Omax2_001t__Nat__Onat,type,
% 14.17/7.98 bNF_We2380939065524664066x2_nat: set_Pr1261947904930325089at_nat > nat > nat > nat ).
% 14.17/7.98
% 14.17/7.98 thf(sy_c_BNF__Wellorder__Relation_Owo__rel_Omax2_001t__Product____Type__Ounit,type,
% 14.17/7.98 bNF_We7204996500647184731t_unit: set_Pr5094982260447487303t_unit > product_unit > product_unit > product_unit ).
% 14.17/7.98
% 14.17/7.98 thf(sy_c_BNF__Wellorder__Relation_Owo__rel_Ominim_001_062_It__Product____Type__Oprod_It__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J_Mt__Set__Oset_It__Nat__Onat_J_J_M_Eo_J,type,
% 14.17/7.98 bNF_We8236795211600914329_nat_o: set_Pr2161125870931222855_nat_o > set_Pr4532377907799695533_nat_o > produc3658429121746597890et_nat > $o ).
% 14.17/7.98
% 14.17/7.98 thf(sy_c_BNF__Wellorder__Relation_Owo__rel_Ominim_001_Eo,type,
% 14.17/7.98 bNF_We537455116209922916inim_o: set_Product_prod_o_o > set_o > $o ).
% 14.17/7.98
% 14.17/7.98 thf(sy_c_BNF__Wellorder__Relation_Owo__rel_Ominim_001t__Int__Oint,type,
% 14.17/7.98 bNF_We3272633801743244640im_int: set_Pr958786334691620121nt_int > set_int > int ).
% 14.17/7.98
% 14.17/7.98 thf(sy_c_BNF__Wellorder__Relation_Owo__rel_Ominim_001t__Nat__Onat,type,
% 14.17/7.98 bNF_We3275124272252294916im_nat: set_Pr1261947904930325089at_nat > set_nat > nat ).
% 14.17/7.98
% 14.17/7.98 thf(sy_c_BNF__Wellorder__Relation_Owo__rel_Ominim_001t__Product____Type__Ounit,type,
% 14.17/7.98 bNF_We7185707985517244377t_unit: set_Pr5094982260447487303t_unit > set_Product_unit > product_unit ).
% 14.17/7.98
% 14.17/7.98 thf(sy_c_Basic__BNF__LFPs_Oprod_Osize__prod_001t__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J_001t__Set__Oset_It__Nat__Onat_J,type,
% 14.17/7.98 basic_5567952622663070111et_nat: ( heap_e7401611519738050253t_unit > nat ) > ( set_nat > nat ) > produc3658429121746597890et_nat > nat ).
% 14.17/7.98
% 14.17/7.98 thf(sy_c_Basic__BNF__LFPs_Oprod_Osize__prod_001t__Int__Oint_001t__Int__Oint,type,
% 14.17/7.98 basic_1872990034501187214nt_int: ( int > nat ) > ( int > nat ) > product_prod_int_int > nat ).
% 14.17/7.98
% 14.17/7.98 thf(sy_c_Basic__BNFs_Ofsts_001_062_It__Product____Type__Oprod_It__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J_Mt__Set__Oset_It__Nat__Onat_J_J_M_Eo_J_001t__Product____Type__Oprod_I_062_It__Product____Type__Oprod_It__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J_Mt__Set__Oset_It__Nat__Onat_J_J_M_Eo_J_Mt__Product____Type__Oprod_It__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J_Mt__Set__Oset_It__Nat__Onat_J_J_J,type,
% 14.17/7.98 basic_3966130907827183364et_nat: produc2732055786443039994et_nat > set_Pr4532377907799695533_nat_o ).
% 14.17/7.98
% 14.17/7.98 thf(sy_c_Basic__BNFs_Ofsts_001_062_It__Product____Type__Oprod_It__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J_Mt__Set__Oset_It__Nat__Onat_J_J_M_Eo_J_001t__Product____Type__Oprod_It__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J_Mt__Set__Oset_It__Nat__Onat_J_J,type,
% 14.17/7.98 basic_8654530971723826568et_nat: produc3925858234332021118et_nat > set_Pr4532377907799695533_nat_o ).
% 14.17/7.98
% 14.17/7.98 thf(sy_c_Basic__BNFs_Ofsts_001t__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J_001t__Set__Oset_It__Nat__Onat_J,type,
% 14.17/7.98 basic_7833836874476830220et_nat: produc3658429121746597890et_nat > set_he8644174446132254637t_unit ).
% 14.17/7.98
% 14.17/7.98 thf(sy_c_Basic__BNFs_Ofsts_001t__Int__Oint_001t__Int__Oint,type,
% 14.17/7.98 basic_fsts_int_int: product_prod_int_int > set_int ).
% 14.17/7.98
% 14.17/7.98 thf(sy_c_Basic__BNFs_Ofsts_001t__Nat__Onat_001t__Nat__Onat,type,
% 14.17/7.98 basic_fsts_nat_nat: product_prod_nat_nat > set_nat ).
% 14.17/7.98
% 14.17/7.98 thf(sy_c_Basic__BNFs_Osnds_001_062_It__Product____Type__Oprod_It__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J_Mt__Set__Oset_It__Nat__Onat_J_J_M_Eo_J_001t__Product____Type__Oprod_I_062_It__Product____Type__Oprod_It__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J_Mt__Set__Oset_It__Nat__Onat_J_J_M_Eo_J_Mt__Product____Type__Oprod_It__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J_Mt__Set__Oset_It__Nat__Onat_J_J_J,type,
% 14.17/7.98 basic_2768200282441914438et_nat: produc2732055786443039994et_nat > set_Pr3286484037609594932et_nat ).
% 14.17/7.98
% 14.17/7.98 thf(sy_c_Basic__BNFs_Osnds_001_062_It__Product____Type__Oprod_It__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J_Mt__Set__Oset_It__Nat__Onat_J_J_M_Eo_J_001t__Product____Type__Oprod_It__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J_Mt__Set__Oset_It__Nat__Onat_J_J,type,
% 14.17/7.98 basic_8479756702517612234et_nat: produc3925858234332021118et_nat > set_Pr3948176798113811640et_nat ).
% 14.17/7.98
% 14.17/7.98 thf(sy_c_Basic__BNFs_Osnds_001t__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J_001t__Set__Oset_It__Nat__Onat_J,type,
% 14.17/7.98 basic_2698842327037663054et_nat: produc3658429121746597890et_nat > set_set_nat ).
% 14.17/7.98
% 14.17/7.98 thf(sy_c_Basic__BNFs_Osnds_001t__Int__Oint_001t__Int__Oint,type,
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% 14.17/7.98 finite6602901031542678131t_unit: ( nat > product_unit > product_unit ) > product_unit > set_nat > product_unit ).
% 14.17/7.98
% 14.17/7.98 thf(sy_c_Finite__Set_Ofold_001t__Nat__Onat_001t__Set__Oset_I_Eo_J,type,
% 14.17/7.98 finite3217087857726763998_set_o: ( nat > set_o > set_o ) > set_o > set_nat > set_o ).
% 14.17/7.98
% 14.17/7.98 thf(sy_c_Finite__Set_Ofold_001t__Nat__Onat_001t__Set__Oset_It__Int__Oint_J,type,
% 14.17/7.98 finite1351632015609375740et_int: ( nat > set_int > set_int ) > set_int > set_nat > set_int ).
% 14.17/7.98
% 14.17/7.98 thf(sy_c_Finite__Set_Ofold_001t__Nat__Onat_001t__Set__Oset_It__Nat__Onat_J,type,
% 14.17/7.98 finite5529483035118572448et_nat: ( nat > set_nat > set_nat ) > set_nat > set_nat > set_nat ).
% 14.17/7.98
% 14.17/7.98 thf(sy_c_Finite__Set_Ofold_001t__Nat__Onat_001t__Set__Oset_It__Product____Type__Ounit_J,type,
% 14.17/7.98 finite9145781422887121747t_unit: ( nat > set_Product_unit > set_Product_unit ) > set_Product_unit > set_nat > set_Product_unit ).
% 14.17/7.98
% 14.17/7.98 thf(sy_c_Finite__Set_Ofold_001t__Product____Type__Ounit_001t__Product____Type__Ounit,type,
% 14.17/7.98 finite8834431295148555784t_unit: ( product_unit > product_unit > product_unit ) > product_unit > set_Product_unit > product_unit ).
% 14.17/7.98
% 14.17/7.98 thf(sy_c_Finite__Set_Ofold_001t__Product____Type__Ounit_001t__Set__Oset_It__Product____Type__Ounit_J,type,
% 14.17/7.98 finite2740360166580943592t_unit: ( product_unit > set_Product_unit > set_Product_unit ) > set_Product_unit > set_Product_unit > set_Product_unit ).
% 14.17/7.98
% 14.17/7.98 thf(sy_c_Finite__Set_Ofold_001t__Set__Oset_I_062_It__Product____Type__Oprod_It__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J_Mt__Set__Oset_It__Nat__Onat_J_J_M_Eo_J_J_001t__Set__Oset_I_062_It__Product____Type__Oprod_It__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J_Mt__Set__Oset_It__Nat__Onat_J_J_M_Eo_J_J,type,
% 14.17/7.98 finite8456856254070787592_nat_o: ( set_Pr4532377907799695533_nat_o > set_Pr4532377907799695533_nat_o > set_Pr4532377907799695533_nat_o ) > set_Pr4532377907799695533_nat_o > set_se1353816476878259085_nat_o > set_Pr4532377907799695533_nat_o ).
% 14.17/7.98
% 14.17/7.98 thf(sy_c_Finite__Set_Ofold_001t__Set__Oset_I_Eo_J_001t__Set__Oset_I_Eo_J,type,
% 14.17/7.98 finite4337638375924247368_set_o: ( set_o > set_o > set_o ) > set_o > set_set_o > set_o ).
% 14.17/7.98
% 14.17/7.98 thf(sy_c_Finite__Set_Ofold_001t__Set__Oset_It__Int__Oint_J_001t__Set__Oset_It__Int__Oint_J,type,
% 14.17/7.98 finite2508883933289561102et_int: ( set_int > set_int > set_int ) > set_int > set_set_int > set_int ).
% 14.17/7.98
% 14.17/7.98 thf(sy_c_Finite__Set_Ofold_001t__Set__Oset_It__Nat__Onat_J_001t__Set__Oset_It__Nat__Onat_J,type,
% 14.17/7.98 finite677925301803182934et_nat: ( set_nat > set_nat > set_nat ) > set_nat > set_set_nat > set_nat ).
% 14.17/7.98
% 14.17/7.98 thf(sy_c_Finite__Set_Ofold_001t__Set__Oset_It__Product____Type__Ounit_J_001t__Set__Oset_It__Product____Type__Ounit_J,type,
% 14.17/7.98 finite4243028863286625800t_unit: ( set_Product_unit > set_Product_unit > set_Product_unit ) > set_Product_unit > set_set_Product_unit > set_Product_unit ).
% 14.17/7.98
% 14.17/7.98 thf(sy_c_Fun_Ocomp_001_062_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J_001_062_It__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J_Mt__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J_J_001t__Code____Numeral__Ointeger,type,
% 14.17/7.98 comp_C8797469213163452608nteger: ( ( code_integer > code_integer ) > produc8923325533196201883nteger > produc8923325533196201883nteger ) > ( code_integer > code_integer > code_integer ) > code_integer > produc8923325533196201883nteger > produc8923325533196201883nteger ).
% 14.17/7.98
% 14.17/7.98 thf(sy_c_Fun_Ocomp_001_062_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J_001t__Code____Numeral__Ointeger_001t__Code____Numeral__Ointeger,type,
% 14.17/7.98 comp_C5076291335850163704nteger: ( ( code_integer > code_integer ) > code_integer ) > ( code_integer > code_integer > code_integer ) > code_integer > code_integer ).
% 14.17/7.98
% 14.17/7.98 thf(sy_c_Fun_Ocomp_001_062_It__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J_Mt__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J_J_001t__Set__Oset_It__Code____Numeral__Ointeger_J_001t__Code____Numeral__Ointeger,type,
% 14.17/7.98 comp_P4885729792186999034nteger: ( ( produc8923325533196201883nteger > produc8923325533196201883nteger ) > set_Code_integer ) > ( code_integer > produc8923325533196201883nteger > produc8923325533196201883nteger ) > code_integer > set_Code_integer ).
% 14.17/7.98
% 14.17/7.98 thf(sy_c_Fun_Ocomp_001_062_It__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J_Mt__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J_J_001t__Set__Oset_It__Nat__Onat_J_001t__Code____Numeral__Ointeger,type,
% 14.17/7.98 comp_P8262219321096090349nteger: ( ( produc8923325533196201883nteger > produc8923325533196201883nteger ) > set_nat ) > ( code_integer > produc8923325533196201883nteger > produc8923325533196201883nteger ) > code_integer > set_nat ).
% 14.17/7.98
% 14.17/7.98 thf(sy_c_Fun_Ocomp_001_062_It__Set__Oset_I_Eo_J_Mt__Assertions__Oassn_J_001_062_It__Set__Oset_It__Set__Oset_I_Eo_J_J_Mt__Assertions__Oassn_J_001_062_I_Eo_Mt__Assertions__Oassn_J,type,
% 14.17/7.98 comp_s518380907012905326o_assn: ( ( set_o > assn ) > set_set_o > assn ) > ( ( $o > assn ) > set_o > assn ) > ( $o > assn ) > set_set_o > assn ).
% 14.17/7.98
% 14.17/7.98 thf(sy_c_Fun_Ocomp_001_062_It__Set__Oset_I_Eo_J_Mt__Nat__Onat_J_001_062_It__Set__Oset_It__Set__Oset_I_Eo_J_J_Mt__Nat__Onat_J_001_062_I_Eo_Mt__Nat__Onat_J,type,
% 14.17/7.98 comp_s6084864194472624150_o_nat: ( ( set_o > nat ) > set_set_o > nat ) > ( ( $o > nat ) > set_o > nat ) > ( $o > nat ) > set_set_o > nat ).
% 14.17/7.98
% 14.17/7.98 thf(sy_c_Fun_Ocomp_001_062_It__Set__Oset_It__Code____Numeral__Ointeger_J_Mt__Assertions__Oassn_J_001_062_It__Set__Oset_It__Set__Oset_It__Code____Numeral__Ointeger_J_J_Mt__Assertions__Oassn_J_001_062_It__Code____Numeral__Ointeger_Mt__Assertions__Oassn_J,type,
% 14.17/7.98 comp_s3531641609786562035r_assn: ( ( set_Code_integer > assn ) > set_set_Code_integer > assn ) > ( ( code_integer > assn ) > set_Code_integer > assn ) > ( code_integer > assn ) > set_set_Code_integer > assn ).
% 14.17/7.98
% 14.17/7.98 thf(sy_c_Fun_Ocomp_001_062_It__Set__Oset_It__Code____Numeral__Ointeger_J_Mt__Nat__Onat_J_001_062_It__Set__Oset_It__Set__Oset_It__Code____Numeral__Ointeger_J_J_Mt__Nat__Onat_J_001_062_It__Code____Numeral__Ointeger_Mt__Nat__Onat_J,type,
% 14.17/7.98 comp_s8759123648413916139er_nat: ( ( set_Code_integer > nat ) > set_set_Code_integer > nat ) > ( ( code_integer > nat ) > set_Code_integer > nat ) > ( code_integer > nat ) > set_set_Code_integer > nat ).
% 14.17/7.98
% 14.17/7.98 thf(sy_c_Fun_Ocomp_001_062_It__Set__Oset_It__Int__Oint_J_Mt__Assertions__Oassn_J_001_062_It__Set__Oset_It__Set__Oset_It__Int__Oint_J_J_Mt__Assertions__Oassn_J_001_062_It__Int__Oint_Mt__Assertions__Oassn_J,type,
% 14.17/7.98 comp_s4595925057662634336t_assn: ( ( set_int > assn ) > set_set_int > assn ) > ( ( int > assn ) > set_int > assn ) > ( int > assn ) > set_set_int > assn ).
% 14.17/7.98
% 14.17/7.98 thf(sy_c_Fun_Ocomp_001_062_It__Set__Oset_It__Int__Oint_J_Mt__Int__Oint_J_001_062_It__Set__Oset_It__Set__Oset_It__Int__Oint_J_J_Mt__Int__Oint_J_001_062_It__Int__Oint_Mt__Int__Oint_J,type,
% 14.17/7.98 comp_s9098231955081793132nt_int: ( ( set_int > int ) > set_set_int > int ) > ( ( int > int ) > set_int > int ) > ( int > int ) > set_set_int > int ).
% 14.17/7.98
% 14.17/7.98 thf(sy_c_Fun_Ocomp_001_062_It__Set__Oset_It__Int__Oint_J_Mt__Nat__Onat_J_001_062_It__Set__Oset_It__Set__Oset_It__Int__Oint_J_J_Mt__Nat__Onat_J_001_062_It__Int__Oint_Mt__Nat__Onat_J,type,
% 14.17/7.98 comp_s3237642481825588056nt_nat: ( ( set_int > nat ) > set_set_int > nat ) > ( ( int > nat ) > set_int > nat ) > ( int > nat ) > set_set_int > nat ).
% 14.17/7.98
% 14.17/7.98 thf(sy_c_Fun_Ocomp_001_062_It__Set__Oset_It__Nat__Onat_J_Mt__Assertions__Oassn_J_001_062_It__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J_Mt__Assertions__Oassn_J_001_062_It__Nat__Onat_Mt__Assertions__Oassn_J,type,
% 14.17/7.98 comp_s2451231738238893644t_assn: ( ( set_nat > assn ) > set_set_nat > assn ) > ( ( nat > assn ) > set_nat > assn ) > ( nat > assn ) > set_set_nat > assn ).
% 14.17/7.98
% 14.17/7.98 thf(sy_c_Fun_Ocomp_001_062_It__Set__Oset_It__Nat__Onat_J_Mt__Nat__Onat_J_001_062_It__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J_Mt__Nat__Onat_J_001_062_It__Nat__Onat_Mt__Nat__Onat_J,type,
% 14.17/7.98 comp_s5456809700045850692at_nat: ( ( set_nat > nat ) > set_set_nat > nat ) > ( ( nat > nat ) > set_nat > nat ) > ( nat > nat ) > set_set_nat > nat ).
% 14.17/7.98
% 14.17/7.98 thf(sy_c_Fun_Ocomp_001_062_It__Set__Oset_It__Product____Type__Ounit_J_Mt__Assertions__Oassn_J_001_062_It__Set__Oset_It__Set__Oset_It__Product____Type__Ounit_J_J_Mt__Assertions__Oassn_J_001_062_It__Product____Type__Ounit_Mt__Assertions__Oassn_J,type,
% 14.17/7.98 comp_s8247530669214396195t_assn: ( ( set_Product_unit > assn ) > set_set_Product_unit > assn ) > ( ( product_unit > assn ) > set_Product_unit > assn ) > ( product_unit > assn ) > set_set_Product_unit > assn ).
% 14.17/7.98
% 14.17/7.98 thf(sy_c_Fun_Ocomp_001_062_It__Set__Oset_It__Product____Type__Ounit_J_Mt__Nat__Onat_J_001_062_It__Set__Oset_It__Set__Oset_It__Product____Type__Ounit_J_J_Mt__Nat__Onat_J_001_062_It__Product____Type__Ounit_Mt__Nat__Onat_J,type,
% 14.17/7.98 comp_s8928436226432208331it_nat: ( ( set_Product_unit > nat ) > set_set_Product_unit > nat ) > ( ( product_unit > nat ) > set_Product_unit > nat ) > ( product_unit > nat ) > set_set_Product_unit > nat ).
% 14.17/7.98
% 14.17/7.98 thf(sy_c_Fun_Ocomp_001_Eo_001_062_It__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J_Mt__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J_J_001t__Code____Numeral__Ointeger,type,
% 14.17/7.98 comp_o8922910837708301817nteger: ( $o > produc8923325533196201883nteger > produc8923325533196201883nteger ) > ( code_integer > $o ) > code_integer > produc8923325533196201883nteger > produc8923325533196201883nteger ).
% 14.17/7.98
% 14.17/7.98 thf(sy_c_Fun_Ocomp_001_Eo_001_062_It__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J_Mt__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J_J_001t__Product____Type__Ounit,type,
% 14.17/7.98 comp_o6961622296474427249t_unit: ( $o > produc8923325533196201883nteger > produc8923325533196201883nteger ) > ( product_unit > $o ) > product_unit > produc8923325533196201883nteger > produc8923325533196201883nteger ).
% 14.17/7.98
% 14.17/7.98 thf(sy_c_Fun_Ocomp_001_Eo_001t__Code____Numeral__Ointeger_001t__Code____Numeral__Ointeger,type,
% 14.17/7.98 comp_o1085144040659434033nteger: ( $o > code_integer ) > ( code_integer > $o ) > code_integer > code_integer ).
% 14.17/7.98
% 14.17/7.98 thf(sy_c_Fun_Ocomp_001_Eo_001t__Code____Numeral__Ointeger_001t__Product____Type__Ounit,type,
% 14.17/7.98 comp_o780160699970281785t_unit: ( $o > code_integer ) > ( product_unit > $o ) > product_unit > code_integer ).
% 14.17/7.98
% 14.17/7.98 thf(sy_c_Fun_Ocomp_001_Eo_001t__Code____Numeral__Onatural_001t__Product____Type__Ounit,type,
% 14.17/7.98 comp_o2643362028263920430t_unit: ( $o > code_natural ) > ( product_unit > $o ) > product_unit > code_natural ).
% 14.17/7.98
% 14.17/7.98 thf(sy_c_Fun_Ocomp_001_Eo_001t__Int__Oint_001t__Int__Oint,type,
% 14.17/7.98 comp_o_int_int: ( $o > int ) > ( int > $o ) > int > int ).
% 14.17/7.98
% 14.17/7.98 thf(sy_c_Fun_Ocomp_001_Eo_001t__Product____Type__Ounit_001_Eo,type,
% 14.17/7.98 comp_o3916225744453444672unit_o: ( $o > product_unit ) > ( $o > $o ) > $o > product_unit ).
% 14.17/7.98
% 14.17/7.98 thf(sy_c_Fun_Ocomp_001_Eo_001t__Product____Type__Ounit_001t__Code____Numeral__Ointeger,type,
% 14.17/7.98 comp_o6635994240230588853nteger: ( $o > product_unit ) > ( code_integer > $o ) > code_integer > product_unit ).
% 14.17/7.98
% 14.17/7.98 thf(sy_c_Fun_Ocomp_001_Eo_001t__Product____Type__Ounit_001t__Int__Oint,type,
% 14.17/7.98 comp_o7049022088938261380it_int: ( $o > product_unit ) > ( int > $o ) > int > product_unit ).
% 14.17/7.98
% 14.17/7.98 thf(sy_c_Fun_Ocomp_001_Eo_001t__Product____Type__Ounit_001t__Nat__Onat,type,
% 14.17/7.98 comp_o7051512559447311656it_nat: ( $o > product_unit ) > ( nat > $o ) > nat > product_unit ).
% 14.17/7.98
% 14.17/7.98 thf(sy_c_Fun_Ocomp_001_Eo_001t__Product____Type__Ounit_001t__Product____Type__Ounit,type,
% 14.17/7.98 comp_o7724735231917782709t_unit: ( $o > product_unit ) > ( product_unit > $o ) > product_unit > product_unit ).
% 14.17/7.98
% 14.17/7.98 thf(sy_c_Fun_Ocomp_001_Eo_001t__Set__Oset_I_Eo_J_001t__Product____Type__Ounit,type,
% 14.17/7.98 comp_o6895741221744486154t_unit: ( $o > set_o ) > ( product_unit > $o ) > product_unit > set_o ).
% 14.17/7.98
% 14.17/7.98 thf(sy_c_Fun_Ocomp_001_Eo_001t__Set__Oset_It__Code____Numeral__Ointeger_J_001_062_It__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J_Mt__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J_J,type,
% 14.17/7.98 comp_o7929101295636871983nteger: ( $o > set_Code_integer ) > ( ( produc8923325533196201883nteger > produc8923325533196201883nteger ) > $o ) > ( produc8923325533196201883nteger > produc8923325533196201883nteger ) > set_Code_integer ).
% 14.17/7.98
% 14.17/7.98 thf(sy_c_Fun_Ocomp_001_Eo_001t__Set__Oset_It__Code____Numeral__Ointeger_J_001t__Code____Numeral__Ointeger,type,
% 14.17/7.98 comp_o4291842030903409511nteger: ( $o > set_Code_integer ) > ( code_integer > $o ) > code_integer > set_Code_integer ).
% 14.17/7.98
% 14.17/7.98 thf(sy_c_Fun_Ocomp_001_Eo_001t__Set__Oset_It__Int__Oint_J_001t__Product____Type__Ounit,type,
% 14.17/7.98 comp_o1521574261646630388t_unit: ( $o > set_int ) > ( product_unit > $o ) > product_unit > set_int ).
% 14.17/7.98
% 14.17/7.98 thf(sy_c_Fun_Ocomp_001_Eo_001t__Set__Oset_It__Nat__Onat_J_001_062_It__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J_Mt__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J_J,type,
% 14.17/7.98 comp_o1289984117834210338nteger: ( $o > set_nat ) > ( ( produc8923325533196201883nteger > produc8923325533196201883nteger ) > $o ) > ( produc8923325533196201883nteger > produc8923325533196201883nteger ) > set_nat ).
% 14.17/7.98
% 14.17/7.98 thf(sy_c_Fun_Ocomp_001_Eo_001t__Set__Oset_It__Nat__Onat_J_001t__Code____Numeral__Ointeger,type,
% 14.17/7.98 comp_o172754640513974618nteger: ( $o > set_nat ) > ( code_integer > $o ) > code_integer > set_nat ).
% 14.17/7.98
% 14.17/7.98 thf(sy_c_Fun_Ocomp_001_Eo_001t__Set__Oset_It__Nat__Onat_J_001t__Product____Type__Ounit,type,
% 14.17/7.98 comp_o9190160103024335696t_unit: ( $o > set_nat ) > ( product_unit > $o ) > product_unit > set_nat ).
% 14.17/7.98
% 14.17/7.98 thf(sy_c_Fun_Ocomp_001_Eo_001t__Set__Oset_It__Product____Type__Ounit_J_001t__Product____Type__Ounit,type,
% 14.17/7.98 comp_o2086126948152544725t_unit: ( $o > set_Product_unit ) > ( product_unit > $o ) > product_unit > set_Product_unit ).
% 14.17/7.98
% 14.17/7.98 thf(sy_c_Fun_Ocomp_001t__Code____Numeral__Ointeger_001_062_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J_001t__Code____Numeral__Ointeger,type,
% 14.17/7.98 comp_C2931745410781319416nteger: ( code_integer > code_integer > code_integer ) > ( code_integer > code_integer ) > code_integer > code_integer > code_integer ).
% 14.17/7.98
% 14.17/7.98 thf(sy_c_Fun_Ocomp_001t__Code____Numeral__Ointeger_001_062_It__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J_Mt__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J_J_001_062_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J,type,
% 14.17/7.98 comp_C95226695241045696nteger: ( code_integer > produc8923325533196201883nteger > produc8923325533196201883nteger ) > ( ( code_integer > code_integer ) > code_integer ) > ( code_integer > code_integer ) > produc8923325533196201883nteger > produc8923325533196201883nteger ).
% 14.17/7.98
% 14.17/7.98 thf(sy_c_Fun_Ocomp_001t__Code____Numeral__Ointeger_001_062_It__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J_Mt__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J_J_001_Eo,type,
% 14.17/7.98 comp_C5733264393235242609eger_o: ( code_integer > produc8923325533196201883nteger > produc8923325533196201883nteger ) > ( $o > code_integer ) > $o > produc8923325533196201883nteger > produc8923325533196201883nteger ).
% 14.17/7.98
% 14.17/7.98 thf(sy_c_Fun_Ocomp_001t__Code____Numeral__Ointeger_001_062_It__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J_Mt__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J_J_001t__Code____Numeral__Ointeger,type,
% 14.17/7.98 comp_C1593894019821074884nteger: ( code_integer > produc8923325533196201883nteger > produc8923325533196201883nteger ) > ( code_integer > code_integer ) > code_integer > produc8923325533196201883nteger > produc8923325533196201883nteger ).
% 14.17/7.98
% 14.17/7.98 thf(sy_c_Fun_Ocomp_001t__Code____Numeral__Ointeger_001_062_It__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J_Mt__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J_J_001t__Int__Oint,type,
% 14.17/7.98 comp_C834242911430131603er_int: ( code_integer > produc8923325533196201883nteger > produc8923325533196201883nteger ) > ( int > code_integer ) > int > produc8923325533196201883nteger > produc8923325533196201883nteger ).
% 14.17/7.98
% 14.17/7.98 thf(sy_c_Fun_Ocomp_001t__Code____Numeral__Ointeger_001_062_It__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J_Mt__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J_J_001t__Nat__Onat,type,
% 14.17/7.98 comp_C836733381939181879er_nat: ( code_integer > produc8923325533196201883nteger > produc8923325533196201883nteger ) > ( nat > code_integer ) > nat > produc8923325533196201883nteger > produc8923325533196201883nteger ).
% 14.17/7.98
% 14.17/7.98 thf(sy_c_Fun_Ocomp_001t__Code____Numeral__Ointeger_001_062_It__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J_Mt__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J_J_001t__Product____Type__Ounit,type,
% 14.17/7.98 comp_C5854136495726678886t_unit: ( code_integer > produc8923325533196201883nteger > produc8923325533196201883nteger ) > ( product_unit > code_integer ) > product_unit > produc8923325533196201883nteger > produc8923325533196201883nteger ).
% 14.17/7.98
% 14.17/7.98 thf(sy_c_Fun_Ocomp_001t__Code____Numeral__Ointeger_001t__Assertions__Oassn_001t__Code____Numeral__Ointeger,type,
% 14.17/7.98 comp_C3026223900822616519nteger: ( code_integer > assn ) > ( code_integer > code_integer ) > code_integer > assn ).
% 14.17/7.98
% 14.17/7.98 thf(sy_c_Fun_Ocomp_001t__Code____Numeral__Ointeger_001t__Assertions__Oassn_001t__Nat__Onat,type,
% 14.17/7.98 comp_C8462447471145125178sn_nat: ( code_integer > assn ) > ( nat > code_integer ) > nat > assn ).
% 14.17/7.98
% 14.17/7.98 thf(sy_c_Fun_Ocomp_001t__Code____Numeral__Ointeger_001t__Code____Numeral__Ointeger_001t__Code____Numeral__Ointeger,type,
% 14.17/7.98 comp_C7449957260575251196nteger: ( code_integer > code_integer ) > ( code_integer > code_integer ) > code_integer > code_integer ).
% 14.17/7.98
% 14.17/7.98 thf(sy_c_Fun_Ocomp_001t__Code____Numeral__Ointeger_001t__Code____Numeral__Ointeger_001t__Code____Numeral__Onatural,type,
% 14.17/7.98 comp_C6273552412331968711atural: ( code_integer > code_integer ) > ( code_natural > code_integer ) > code_natural > code_integer ).
% 14.17/7.98
% 14.17/7.98 thf(sy_c_Fun_Ocomp_001t__Code____Numeral__Ointeger_001t__Code____Numeral__Ointeger_001t__Nat__Onat,type,
% 14.17/7.98 comp_C6974050283906349679er_nat: ( code_integer > code_integer ) > ( nat > code_integer ) > nat > code_integer ).
% 14.17/7.98
% 14.17/7.98 thf(sy_c_Fun_Ocomp_001t__Code____Numeral__Ointeger_001t__Code____Numeral__Ointeger_001t__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J,type,
% 14.17/7.98 comp_C1043215000170864188nteger: ( code_integer > code_integer ) > ( produc8923325533196201883nteger > code_integer ) > produc8923325533196201883nteger > code_integer ).
% 14.17/7.98
% 14.17/7.98 thf(sy_c_Fun_Ocomp_001t__Code____Numeral__Ointeger_001t__Code____Numeral__Ointeger_001t__Product____Type__Ounit,type,
% 14.17/7.98 comp_C2290379119555921966t_unit: ( code_integer > code_integer ) > ( product_unit > code_integer ) > product_unit > code_integer ).
% 14.17/7.98
% 14.17/7.98 thf(sy_c_Fun_Ocomp_001t__Code____Numeral__Ointeger_001t__Code____Numeral__Onatural_001t__Code____Numeral__Ointeger,type,
% 14.17/7.98 comp_C1891057325546811591nteger: ( code_integer > code_natural ) > ( code_integer > code_integer ) > code_integer > code_natural ).
% 14.17/7.98
% 14.17/7.98 thf(sy_c_Fun_Ocomp_001t__Code____Numeral__Ointeger_001t__Code____Numeral__Onatural_001t__Code____Numeral__Onatural,type,
% 14.17/7.98 comp_C714652477303529106atural: ( code_integer > code_natural ) > ( code_natural > code_integer ) > code_natural > code_natural ).
% 14.17/7.98
% 14.17/7.98 thf(sy_c_Fun_Ocomp_001t__Code____Numeral__Ointeger_001t__Code____Numeral__Onatural_001t__Product____Type__Ounit,type,
% 14.17/7.98 comp_C4153580447849560611t_unit: ( code_integer > code_natural ) > ( product_unit > code_integer ) > product_unit > code_natural ).
% 14.17/7.98
% 14.17/7.98 thf(sy_c_Fun_Ocomp_001t__Code____Numeral__Ointeger_001t__Int__Oint_001t__Code____Numeral__Ointeger,type,
% 14.17/7.98 comp_C7610259292922237131nteger: ( code_integer > int ) > ( code_integer > code_integer ) > code_integer > int ).
% 14.17/7.98
% 14.17/7.98 thf(sy_c_Fun_Ocomp_001t__Code____Numeral__Ointeger_001t__Int__Oint_001t__Nat__Onat,type,
% 14.17/7.98 comp_C5048910307564104766nt_nat: ( code_integer > int ) > ( nat > code_integer ) > nat > int ).
% 14.17/7.98
% 14.17/7.98 thf(sy_c_Fun_Ocomp_001t__Code____Numeral__Ointeger_001t__Nat__Onat_001t__Code____Numeral__Ointeger,type,
% 14.17/7.98 comp_C7238605732685392495nteger: ( code_integer > nat ) > ( code_integer > code_integer ) > code_integer > nat ).
% 14.17/7.98
% 14.17/7.98 thf(sy_c_Fun_Ocomp_001t__Code____Numeral__Ointeger_001t__Product____Type__Ounit_001t__Code____Numeral__Ointeger,type,
% 14.17/7.98 comp_C8146212659816229034nteger: ( code_integer > product_unit ) > ( code_integer > code_integer ) > code_integer > product_unit ).
% 14.17/7.98
% 14.17/7.98 thf(sy_c_Fun_Ocomp_001t__Code____Numeral__Ointeger_001t__Product____Type__Ounit_001t__Code____Numeral__Onatural,type,
% 14.17/7.98 comp_C6969807811572946549atural: ( code_integer > product_unit ) > ( code_natural > code_integer ) > code_natural > product_unit ).
% 14.17/7.98
% 14.17/7.98 thf(sy_c_Fun_Ocomp_001t__Code____Numeral__Ointeger_001t__Product____Type__Ounit_001t__Product____Type__Ounit,type,
% 14.17/7.98 comp_C2408090164605745152t_unit: ( code_integer > product_unit ) > ( product_unit > code_integer ) > product_unit > product_unit ).
% 14.17/7.98
% 14.17/7.98 thf(sy_c_Fun_Ocomp_001t__Code____Numeral__Ointeger_001t__Set__Oset_It__Code____Numeral__Ointeger_J_001_062_It__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J_Mt__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J_J,type,
% 14.17/7.98 comp_C6344129587964912378nteger: ( code_integer > set_Code_integer ) > ( ( produc8923325533196201883nteger > produc8923325533196201883nteger ) > code_integer ) > ( produc8923325533196201883nteger > produc8923325533196201883nteger ) > set_Code_integer ).
% 14.17/7.98
% 14.17/7.98 thf(sy_c_Fun_Ocomp_001t__Code____Numeral__Ointeger_001t__Set__Oset_It__Code____Numeral__Ointeger_J_001t__Code____Numeral__Ointeger,type,
% 14.17/7.98 comp_C8186931911283765298nteger: ( code_integer > set_Code_integer ) > ( code_integer > code_integer ) > code_integer > set_Code_integer ).
% 14.17/7.98
% 14.17/7.98 thf(sy_c_Fun_Ocomp_001t__Code____Numeral__Ointeger_001t__Set__Oset_It__Nat__Onat_J_001_062_It__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J_Mt__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J_J,type,
% 14.17/7.98 comp_C3184339336801759213nteger: ( code_integer > set_nat ) > ( ( produc8923325533196201883nteger > produc8923325533196201883nteger ) > code_integer ) > ( produc8923325533196201883nteger > produc8923325533196201883nteger ) > set_nat ).
% 14.17/7.98
% 14.17/7.98 thf(sy_c_Fun_Ocomp_001t__Code____Numeral__Ointeger_001t__Set__Oset_It__Nat__Onat_J_001t__Code____Numeral__Ointeger,type,
% 14.17/7.98 comp_C6537567860429791781nteger: ( code_integer > set_nat ) > ( code_integer > code_integer ) > code_integer > set_nat ).
% 14.17/7.98
% 14.17/7.98 thf(sy_c_Fun_Ocomp_001t__Code____Numeral__Onatural_001_062_It__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J_Mt__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J_J_001t__Code____Numeral__Ointeger,type,
% 14.17/7.98 comp_C456341964420721295nteger: ( code_natural > produc8923325533196201883nteger > produc8923325533196201883nteger ) > ( code_integer > code_natural ) > code_integer > produc8923325533196201883nteger > produc8923325533196201883nteger ).
% 14.17/7.98
% 14.17/7.98 thf(sy_c_Fun_Ocomp_001t__Int__Oint_001_062_It__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J_Mt__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J_J_001t__Code____Numeral__Ointeger,type,
% 14.17/7.98 comp_i2874689474467862163nteger: ( int > produc8923325533196201883nteger > produc8923325533196201883nteger ) > ( code_integer > int ) > code_integer > produc8923325533196201883nteger > produc8923325533196201883nteger ).
% 14.17/7.98
% 14.17/7.98 thf(sy_c_Fun_Ocomp_001t__Int__Oint_001_Eo_001_Eo,type,
% 14.17/7.98 comp_int_o_o: ( int > $o ) > ( $o > int ) > $o > $o ).
% 14.17/7.98
% 14.17/7.98 thf(sy_c_Fun_Ocomp_001t__Int__Oint_001_Eo_001t__Int__Oint,type,
% 14.17/7.98 comp_int_o_int: ( int > $o ) > ( int > int ) > int > $o ).
% 14.17/7.98
% 14.17/7.98 thf(sy_c_Fun_Ocomp_001t__Int__Oint_001_Eo_001t__Nat__Onat,type,
% 14.17/7.98 comp_int_o_nat: ( int > $o ) > ( nat > int ) > nat > $o ).
% 14.17/7.98
% 14.17/7.98 thf(sy_c_Fun_Ocomp_001t__Int__Oint_001t__Code____Numeral__Ointeger_001_Eo,type,
% 14.17/7.98 comp_i5286552343147518890eger_o: ( int > code_integer ) > ( $o > int ) > $o > code_integer ).
% 14.17/7.98
% 14.17/7.98 thf(sy_c_Fun_Ocomp_001t__Int__Oint_001t__Code____Numeral__Ointeger_001t__Code____Numeral__Ointeger,type,
% 14.17/7.98 comp_i6014793192326455499nteger: ( int > code_integer ) > ( code_integer > int ) > code_integer > code_integer ).
% 14.17/7.98
% 14.17/7.98 thf(sy_c_Fun_Ocomp_001t__Int__Oint_001t__Code____Numeral__Ointeger_001t__Int__Oint,type,
% 14.17/7.98 comp_i1585864551200866970er_int: ( int > code_integer ) > ( int > int ) > int > code_integer ).
% 14.17/7.98
% 14.17/7.98 thf(sy_c_Fun_Ocomp_001t__Int__Oint_001t__Code____Numeral__Ointeger_001t__Nat__Onat,type,
% 14.17/7.98 comp_i1588355021709917246er_nat: ( int > code_integer ) > ( nat > int ) > nat > code_integer ).
% 14.17/7.98
% 14.17/7.98 thf(sy_c_Fun_Ocomp_001t__Int__Oint_001t__Code____Numeral__Onatural_001t__Code____Numeral__Ointeger,type,
% 14.17/7.98 comp_i455893257298015894nteger: ( int > code_natural ) > ( code_integer > int ) > code_integer > code_natural ).
% 14.17/7.98
% 14.17/7.98 thf(sy_c_Fun_Ocomp_001t__Int__Oint_001t__Int__Oint_001_Eo,type,
% 14.17/7.98 comp_int_int_o: ( int > int ) > ( $o > int ) > $o > int ).
% 14.17/7.98
% 14.17/7.98 thf(sy_c_Fun_Ocomp_001t__Int__Oint_001t__Int__Oint_001t__Int__Oint,type,
% 14.17/7.98 comp_int_int_int: ( int > int ) > ( int > int ) > int > int ).
% 14.17/7.98
% 14.17/7.98 thf(sy_c_Fun_Ocomp_001t__Int__Oint_001t__Int__Oint_001t__Nat__Onat,type,
% 14.17/7.98 comp_int_int_nat: ( int > int ) > ( nat > int ) > nat > int ).
% 14.17/7.98
% 14.17/7.98 thf(sy_c_Fun_Ocomp_001t__Int__Oint_001t__Int__Oint_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J,type,
% 14.17/7.98 comp_i4330622015127193916nt_int: ( int > int ) > ( product_prod_int_int > int ) > product_prod_int_int > int ).
% 14.17/7.98
% 14.17/7.98 thf(sy_c_Fun_Ocomp_001t__Int__Oint_001t__Nat__Onat_001_Eo,type,
% 14.17/7.98 comp_int_nat_o: ( int > nat ) > ( $o > int ) > $o > nat ).
% 14.17/7.98
% 14.17/7.98 thf(sy_c_Fun_Ocomp_001t__Int__Oint_001t__Nat__Onat_001t__Int__Oint,type,
% 14.17/7.98 comp_int_nat_int: ( int > nat ) > ( int > int ) > int > nat ).
% 14.17/7.98
% 14.17/7.98 thf(sy_c_Fun_Ocomp_001t__Int__Oint_001t__Nat__Onat_001t__Nat__Onat,type,
% 14.17/7.98 comp_int_nat_nat: ( int > nat ) > ( nat > int ) > nat > nat ).
% 14.17/7.98
% 14.17/7.98 thf(sy_c_Fun_Ocomp_001t__Int__Oint_001t__Product____Type__Ounit_001t__Code____Numeral__Ointeger,type,
% 14.17/7.98 comp_i8439899303502915483nteger: ( int > product_unit ) > ( code_integer > int ) > code_integer > product_unit ).
% 14.17/7.98
% 14.17/7.98 thf(sy_c_Fun_Ocomp_001t__Int__Oint_001t__Set__Oset_I_Eo_J_001t__Code____Numeral__Ointeger,type,
% 14.17/7.98 comp_i5479505385629679622nteger: ( int > set_o ) > ( code_integer > int ) > code_integer > set_o ).
% 14.17/7.98
% 14.17/7.98 thf(sy_c_Fun_Ocomp_001t__Int__Oint_001t__Set__Oset_It__Int__Oint_J_001t__Code____Numeral__Ointeger,type,
% 14.17/7.98 comp_i1887841406321795152nteger: ( int > set_int ) > ( code_integer > int ) > code_integer > set_int ).
% 14.17/7.98
% 14.17/7.98 thf(sy_c_Fun_Ocomp_001t__Int__Oint_001t__Set__Oset_It__Int__Oint_J_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J,type,
% 14.17/7.98 comp_i5103216804979976198nt_int: ( int > set_int ) > ( product_prod_int_int > int ) > product_prod_int_int > set_int ).
% 14.17/7.98
% 14.17/7.98 thf(sy_c_Fun_Ocomp_001t__Int__Oint_001t__Set__Oset_It__Nat__Onat_J_001t__Code____Numeral__Ointeger,type,
% 14.17/7.98 comp_i5102403792180996084nteger: ( int > set_nat ) > ( code_integer > int ) > code_integer > set_nat ).
% 14.17/7.98
% 14.17/7.98 thf(sy_c_Fun_Ocomp_001t__Int__Oint_001t__Set__Oset_It__Nat__Onat_J_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J,type,
% 14.17/7.98 comp_i6200299785623974754nt_int: ( int > set_nat ) > ( product_prod_int_int > int ) > product_prod_int_int > set_nat ).
% 14.17/7.98
% 14.17/7.98 thf(sy_c_Fun_Ocomp_001t__Int__Oint_001t__Set__Oset_It__Product____Type__Ounit_J_001t__Code____Numeral__Ointeger,type,
% 14.17/7.98 comp_i3319814986458681467nteger: ( int > set_Product_unit ) > ( code_integer > int ) > code_integer > set_Product_unit ).
% 14.17/7.98
% 14.17/7.98 thf(sy_c_Fun_Ocomp_001t__Nat__Onat_001_062_It__Nat__Onat_Mt__Nat__Onat_J_001t__Nat__Onat,type,
% 14.17/7.98 comp_nat_nat_nat_nat: ( nat > nat > nat ) > ( nat > nat ) > nat > nat > nat ).
% 14.17/7.98
% 14.17/7.98 thf(sy_c_Fun_Ocomp_001t__Nat__Onat_001_062_It__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J_Mt__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J_J_001t__Code____Numeral__Ointeger,type,
% 14.17/7.98 comp_n2033352843805742135nteger: ( nat > produc8923325533196201883nteger > produc8923325533196201883nteger ) > ( code_integer > nat ) > code_integer > produc8923325533196201883nteger > produc8923325533196201883nteger ).
% 14.17/7.98
% 14.17/7.98 thf(sy_c_Fun_Ocomp_001t__Nat__Onat_001_062_It__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J_Mt__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J_J_001t__Code____Numeral__Onatural,type,
% 14.17/7.98 comp_n856947995562459650atural: ( nat > produc8923325533196201883nteger > produc8923325533196201883nteger ) > ( code_natural > nat ) > code_natural > produc8923325533196201883nteger > produc8923325533196201883nteger ).
% 14.17/7.98
% 14.17/7.98 thf(sy_c_Fun_Ocomp_001t__Nat__Onat_001t__Assertions__Oassn_001t__Nat__Onat,type,
% 14.17/7.98 comp_nat_assn_nat: ( nat > assn ) > ( nat > nat ) > nat > assn ).
% 14.17/7.98
% 14.17/7.98 thf(sy_c_Fun_Ocomp_001t__Nat__Onat_001t__Code____Numeral__Ointeger_001t__Code____Numeral__Ointeger,type,
% 14.17/7.98 comp_n6876073255676942959nteger: ( nat > code_integer ) > ( code_integer > nat ) > code_integer > code_integer ).
% 14.17/7.98
% 14.17/7.98 thf(sy_c_Fun_Ocomp_001t__Nat__Onat_001t__Code____Numeral__Ointeger_001t__Code____Numeral__Onatural,type,
% 14.17/7.98 comp_n5699668407433660474atural: ( nat > code_integer ) > ( code_natural > nat ) > code_natural > code_integer ).
% 14.17/7.98
% 14.17/7.98 thf(sy_c_Fun_Ocomp_001t__Nat__Onat_001t__Code____Numeral__Ointeger_001t__Nat__Onat,type,
% 14.17/7.98 comp_n3898172953802868194er_nat: ( nat > code_integer ) > ( nat > nat ) > nat > code_integer ).
% 14.17/7.98
% 14.17/7.98 thf(sy_c_Fun_Ocomp_001t__Nat__Onat_001t__Code____Numeral__Onatural_001_Eo,type,
% 14.17/7.98 comp_n2857367870785791611ural_o: ( nat > code_natural ) > ( $o > nat ) > $o > code_natural ).
% 14.17/7.98
% 14.17/7.98 thf(sy_c_Fun_Ocomp_001t__Nat__Onat_001t__Code____Numeral__Onatural_001t__Code____Numeral__Ointeger,type,
% 14.17/7.98 comp_n1317173320648503354nteger: ( nat > code_natural ) > ( code_integer > nat ) > code_integer > code_natural ).
% 14.17/7.98
% 14.17/7.98 thf(sy_c_Fun_Ocomp_001t__Nat__Onat_001t__Code____Numeral__Onatural_001t__Code____Numeral__Onatural,type,
% 14.17/7.98 comp_n140768472405220869atural: ( nat > code_natural ) > ( code_natural > nat ) > code_natural > code_natural ).
% 14.17/7.98
% 14.17/7.98 thf(sy_c_Fun_Ocomp_001t__Nat__Onat_001t__Code____Numeral__Onatural_001t__Int__Oint,type,
% 14.17/7.98 comp_n9160511493981425161al_int: ( nat > code_natural ) > ( int > nat ) > int > code_natural ).
% 14.17/7.98
% 14.17/7.98 thf(sy_c_Fun_Ocomp_001t__Nat__Onat_001t__Code____Numeral__Onatural_001t__Nat__Onat,type,
% 14.17/7.98 comp_n9163001964490475437al_nat: ( nat > code_natural ) > ( nat > nat ) > nat > code_natural ).
% 14.17/7.98
% 14.17/7.98 thf(sy_c_Fun_Ocomp_001t__Nat__Onat_001t__Int__Oint_001t__Nat__Onat,type,
% 14.17/7.98 comp_nat_int_nat: ( nat > int ) > ( nat > nat ) > nat > int ).
% 14.17/7.98
% 14.17/7.98 thf(sy_c_Fun_Ocomp_001t__Nat__Onat_001t__Product____Type__Ounit_001t__Code____Numeral__Onatural,type,
% 14.17/7.98 comp_n8360577435903631554atural: ( nat > product_unit ) > ( code_natural > nat ) > code_natural > product_unit ).
% 14.17/7.98
% 14.17/7.98 thf(sy_c_Fun_Ocomp_001t__Nat__Onat_001t__Set__Oset_It__Int__Oint_J_001t__Code____Numeral__Onatural,type,
% 14.17/7.98 comp_n1572716621429000127atural: ( nat > set_int ) > ( code_natural > nat ) > code_natural > set_int ).
% 14.17/7.98
% 14.17/7.98 thf(sy_c_Fun_Ocomp_001t__Nat__Onat_001t__Set__Oset_It__Nat__Onat_J_001_062_It__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J_Mt__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J_J,type,
% 14.17/7.98 comp_n3623798160786426464nteger: ( nat > set_nat ) > ( ( produc8923325533196201883nteger > produc8923325533196201883nteger ) > nat ) > ( produc8923325533196201883nteger > produc8923325533196201883nteger ) > set_nat ).
% 14.17/7.98
% 14.17/7.98 thf(sy_c_Fun_Ocomp_001t__Nat__Onat_001t__Set__Oset_It__Nat__Onat_J_001t__Code____Numeral__Ointeger,type,
% 14.17/7.98 comp_n5963683855531483544nteger: ( nat > set_nat ) > ( code_integer > nat ) > code_integer > set_nat ).
% 14.17/7.98
% 14.17/7.98 thf(sy_c_Fun_Ocomp_001t__Nat__Onat_001t__Set__Oset_It__Nat__Onat_J_001t__Code____Numeral__Onatural,type,
% 14.17/7.98 comp_n4787279007288201059atural: ( nat > set_nat ) > ( code_natural > nat ) > code_natural > set_nat ).
% 14.17/7.98
% 14.17/7.98 thf(sy_c_Fun_Ocomp_001t__Product____Type__Oprod_I_062_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J_M_062_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J_J_001t__Product____Type__Oprod_I_062_It__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J_Mt__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J_J_M_062_It__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J_Mt__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J_J_J_001t__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J,type,
% 14.17/7.98 comp_P5544053077415136032nteger: ( produc723880050079548563nteger > produc101387210554329899nteger ) > ( produc8923325533196201883nteger > produc723880050079548563nteger ) > produc8923325533196201883nteger > produc101387210554329899nteger ).
% 14.17/7.98
% 14.17/7.98 thf(sy_c_Fun_Ocomp_001t__Product____Type__Oprod_I_062_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J_Mt__Code____Numeral__Ointeger_J_001t__Product____Type__Oprod_I_062_It__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J_Mt__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J_J_M_062_It__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J_Mt__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J_J_J_001t__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J,type,
% 14.17/7.98 comp_P1414880308642389412nteger: ( produc2221115339011613079nteger > produc101387210554329899nteger ) > ( produc8923325533196201883nteger > produc2221115339011613079nteger ) > produc8923325533196201883nteger > produc101387210554329899nteger ).
% 14.17/7.98
% 14.17/7.98 thf(sy_c_Fun_Ocomp_001t__Product____Type__Oprod_It__Code____Numeral__Ointeger_M_062_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J_J_001t__Product____Type__Oprod_I_062_It__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J_Mt__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J_J_M_062_It__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J_Mt__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J_J_J_001t__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J,type,
% 14.17/7.98 comp_P6312383820422015140nteger: ( produc1397831737260518551nteger > produc101387210554329899nteger ) > ( produc8923325533196201883nteger > produc1397831737260518551nteger ) > produc8923325533196201883nteger > produc101387210554329899nteger ).
% 14.17/7.98
% 14.17/7.98 thf(sy_c_Fun_Ocomp_001t__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J_001t__Code____Numeral__Ointeger_001t__Code____Numeral__Ointeger,type,
% 14.17/7.98 comp_P7111796090210262324nteger: ( produc8923325533196201883nteger > code_integer ) > ( code_integer > produc8923325533196201883nteger ) > code_integer > code_integer ).
% 14.17/7.98
% 14.17/7.98 thf(sy_c_Fun_Ocomp_001t__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J_001t__Code____Numeral__Ointeger_001t__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J,type,
% 14.17/7.98 comp_P3428181540658402820nteger: ( produc8923325533196201883nteger > code_integer ) > ( produc8923325533196201883nteger > produc8923325533196201883nteger ) > produc8923325533196201883nteger > code_integer ).
% 14.17/7.98
% 14.17/7.98 thf(sy_c_Fun_Ocomp_001t__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J_001t__Code____Numeral__Ointeger_001t__Product____Type__Ounit,type,
% 14.17/7.98 comp_P659088277599099894t_unit: ( produc8923325533196201883nteger > code_integer ) > ( product_unit > produc8923325533196201883nteger ) > product_unit > code_integer ).
% 14.17/7.98
% 14.17/7.98 thf(sy_c_Fun_Ocomp_001t__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J_001t__Product____Type__Oprod_I_062_It__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J_Mt__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J_J_M_062_It__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J_Mt__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J_J_J_001t__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J,type,
% 14.17/7.98 comp_P234064951390806312nteger: ( produc8923325533196201883nteger > produc101387210554329899nteger ) > ( produc8923325533196201883nteger > produc8923325533196201883nteger ) > produc8923325533196201883nteger > produc101387210554329899nteger ).
% 14.17/7.98
% 14.17/7.98 thf(sy_c_Fun_Ocomp_001t__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__Nat__Onat_J_001t__Code____Numeral__Ointeger_001t__Code____Numeral__Ointeger,type,
% 14.17/7.98 comp_P7101464685127059239nteger: ( produc6433947425880249742er_nat > code_integer ) > ( code_integer > produc6433947425880249742er_nat ) > code_integer > code_integer ).
% 14.17/7.98
% 14.17/7.98 thf(sy_c_Fun_Ocomp_001t__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__Nat__Onat_J_001t__Code____Numeral__Ointeger_001t__Product____Type__Ounit,type,
% 14.17/7.98 comp_P7581484221039074243t_unit: ( produc6433947425880249742er_nat > code_integer ) > ( product_unit > produc6433947425880249742er_nat ) > product_unit > code_integer ).
% 14.17/7.98
% 14.17/7.98 thf(sy_c_Fun_Ocomp_001t__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__Nat__Onat_J_001t__Nat__Onat_001t__Code____Numeral__Ointeger,type,
% 14.17/7.98 comp_P5964733648194011802nteger: ( produc6433947425880249742er_nat > nat ) > ( code_integer > produc6433947425880249742er_nat ) > code_integer > nat ).
% 14.17/7.98
% 14.17/7.98 thf(sy_c_Fun_Ocomp_001t__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__Nat__Onat_J_001t__Nat__Onat_001t__Product____Type__Ounit,type,
% 14.17/7.98 comp_P6821003776282389008t_unit: ( produc6433947425880249742er_nat > nat ) > ( product_unit > produc6433947425880249742er_nat ) > product_unit > nat ).
% 14.17/7.98
% 14.17/7.98 thf(sy_c_Fun_Ocomp_001t__Product____Type__Oprod_It__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J_Mt__Set__Oset_It__Nat__Onat_J_J_001t__Set__Oset_It__Nat__Onat_J_001t__Product____Type__Oprod_It__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J_Mt__Set__Oset_It__Nat__Onat_J_J,type,
% 14.17/7.98 comp_P2930521774885306735et_nat: ( produc3658429121746597890et_nat > set_nat ) > ( produc3658429121746597890et_nat > produc3658429121746597890et_nat ) > produc3658429121746597890et_nat > set_nat ).
% 14.17/7.98
% 14.17/7.98 thf(sy_c_Fun_Ocomp_001t__Product____Type__Oprod_It__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J_Mt__Set__Oset_It__Nat__Onat_J_J_001t__Set__Oset_It__Nat__Onat_J_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J,type,
% 14.17/7.98 comp_P8606889026643445772nt_int: ( produc3658429121746597890et_nat > set_nat ) > ( product_prod_int_int > produc3658429121746597890et_nat ) > product_prod_int_int > set_nat ).
% 14.17/7.98
% 14.17/7.98 thf(sy_c_Fun_Ocomp_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_001t__Int__Oint_001t__Product____Type__Oprod_It__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J_Mt__Set__Oset_It__Nat__Onat_J_J,type,
% 14.17/7.98 comp_P8308861049717174982et_nat: ( product_prod_int_int > int ) > ( produc3658429121746597890et_nat > product_prod_int_int ) > produc3658429121746597890et_nat > int ).
% 14.17/7.98
% 14.17/7.98 thf(sy_c_Fun_Ocomp_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_001t__Int__Oint_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J,type,
% 14.17/7.98 comp_P5356232565778667509nt_int: ( product_prod_int_int > int ) > ( product_prod_int_int > product_prod_int_int ) > product_prod_int_int > int ).
% 14.17/7.98
% 14.17/7.98 thf(sy_c_Fun_Ocomp_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_001t__Int__Oint_001t__Product____Type__Ounit,type,
% 14.17/7.98 comp_P6796073005956588617t_unit: ( product_prod_int_int > int ) > ( product_unit > product_prod_int_int ) > product_unit > int ).
% 14.17/7.98
% 14.17/7.98 thf(sy_c_Fun_Ocomp_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Code____Numeral__Ointeger_J_001t__Code____Numeral__Ointeger_001_Eo,type,
% 14.17/7.98 comp_P7177471617286032590eger_o: ( produc2591615631378489102nteger > code_integer ) > ( $o > produc2591615631378489102nteger ) > $o > code_integer ).
% 14.17/7.98
% 14.17/7.98 thf(sy_c_Fun_Ocomp_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Code____Numeral__Ointeger_J_001t__Code____Numeral__Ointeger_001t__Code____Numeral__Ointeger,type,
% 14.17/7.98 comp_P6122662217367900327nteger: ( produc2591615631378489102nteger > code_integer ) > ( code_integer > produc2591615631378489102nteger ) > code_integer > code_integer ).
% 14.17/7.98
% 14.17/7.98 thf(sy_c_Fun_Ocomp_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Code____Numeral__Ointeger_J_001t__Code____Numeral__Ointeger_001t__Product____Type__Ounit,type,
% 14.17/7.98 comp_P7320554561093743171t_unit: ( produc2591615631378489102nteger > code_integer ) > ( product_unit > produc2591615631378489102nteger ) > product_unit > code_integer ).
% 14.17/7.98
% 14.17/7.98 thf(sy_c_Fun_Ocomp_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Code____Numeral__Ointeger_J_001t__Nat__Onat_001_Eo,type,
% 14.17/7.98 comp_P7179991235626930843_nat_o: ( produc2591615631378489102nteger > nat ) > ( $o > produc2591615631378489102nteger ) > $o > nat ).
% 14.17/7.98
% 14.17/7.98 thf(sy_c_Fun_Ocomp_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Code____Numeral__Ointeger_J_001t__Nat__Onat_001t__Code____Numeral__Ointeger,type,
% 14.17/7.98 comp_P5725896914063526938nteger: ( produc2591615631378489102nteger > nat ) > ( code_integer > produc2591615631378489102nteger ) > code_integer > nat ).
% 14.17/7.98
% 14.17/7.98 thf(sy_c_Fun_Ocomp_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Code____Numeral__Ointeger_J_001t__Nat__Onat_001t__Product____Type__Ounit,type,
% 14.17/7.98 comp_P7621350044865347728t_unit: ( produc2591615631378489102nteger > nat ) > ( product_unit > produc2591615631378489102nteger ) > product_unit > nat ).
% 14.17/7.98
% 14.17/7.98 thf(sy_c_Fun_Ocomp_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_001t__Nat__Onat_001_Eo,type,
% 14.17/7.98 comp_P4394204175247225320_nat_o: ( product_prod_nat_nat > nat ) > ( $o > product_prod_nat_nat ) > $o > nat ).
% 14.17/7.98
% 14.17/7.98 thf(sy_c_Fun_Ocomp_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_001t__Nat__Onat_001t__Product____Type__Ounit,type,
% 14.17/7.98 comp_P2980510830910040157t_unit: ( product_prod_nat_nat > nat ) > ( product_unit > product_prod_nat_nat ) > product_unit > nat ).
% 14.17/7.98
% 14.17/7.98 thf(sy_c_Fun_Ocomp_001t__Product____Type__Ounit_001_062_It__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J_Mt__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J_J_001t__Code____Numeral__Ointeger,type,
% 14.17/7.98 comp_P4511551582283981294nteger: ( product_unit > produc8923325533196201883nteger > produc8923325533196201883nteger ) > ( code_integer > product_unit ) > code_integer > produc8923325533196201883nteger > produc8923325533196201883nteger ).
% 14.17/7.98
% 14.17/7.98 thf(sy_c_Fun_Ocomp_001t__Product____Type__Ounit_001_062_It__Product____Type__Ounit_Mt__Product____Type__Ounit_J_001t__Code____Numeral__Ointeger,type,
% 14.17/7.98 comp_P4558236313141964142nteger: ( product_unit > product_unit > product_unit ) > ( code_integer > product_unit ) > code_integer > product_unit > product_unit ).
% 14.17/7.98
% 14.17/7.98 thf(sy_c_Fun_Ocomp_001t__Product____Type__Ounit_001_062_It__Product____Type__Ounit_Mt__Product____Type__Ounit_J_001t__Nat__Onat,type,
% 14.17/7.98 comp_P6204500334276302049it_nat: ( product_unit > product_unit > product_unit ) > ( nat > product_unit ) > nat > product_unit > product_unit ).
% 14.17/7.98
% 14.17/7.98 thf(sy_c_Fun_Ocomp_001t__Product____Type__Ounit_001t__Code____Numeral__Ointeger_001t__Code____Numeral__Ointeger,type,
% 14.17/7.98 comp_P6790823133229852966nteger: ( product_unit > code_integer ) > ( code_integer > product_unit ) > code_integer > code_integer ).
% 14.17/7.98
% 14.17/7.98 thf(sy_c_Fun_Ocomp_001t__Product____Type__Ounit_001t__Set__Oset_It__Code____Numeral__Ointeger_J_001_062_It__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J_Mt__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J_J,type,
% 14.17/7.98 comp_P4635212033913086756nteger: ( product_unit > set_Code_integer ) > ( ( produc8923325533196201883nteger > produc8923325533196201883nteger ) > product_unit ) > ( produc8923325533196201883nteger > produc8923325533196201883nteger ) > set_Code_integer ).
% 14.17/7.98
% 14.17/7.98 thf(sy_c_Fun_Ocomp_001t__Product____Type__Ounit_001t__Set__Oset_It__Code____Numeral__Ointeger_J_001t__Code____Numeral__Ointeger,type,
% 14.17/7.98 comp_P4767411901762164316nteger: ( product_unit > set_Code_integer ) > ( code_integer > product_unit ) > code_integer > set_Code_integer ).
% 14.17/7.98
% 14.17/7.98 thf(sy_c_Fun_Ocomp_001t__Product____Type__Ounit_001t__Set__Oset_It__Nat__Onat_J_001_062_It__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J_Mt__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J_J,type,
% 14.17/7.98 comp_P6101996899264665623nteger: ( product_unit > set_nat ) > ( ( produc8923325533196201883nteger > produc8923325533196201883nteger ) > product_unit ) > ( produc8923325533196201883nteger > produc8923325533196201883nteger ) > set_nat ).
% 14.17/7.98
% 14.17/7.98 thf(sy_c_Fun_Ocomp_001t__Product____Type__Ounit_001t__Set__Oset_It__Nat__Onat_J_001t__Code____Numeral__Ointeger,type,
% 14.17/7.98 comp_P5878433733084393551nteger: ( product_unit > set_nat ) > ( code_integer > product_unit ) > code_integer > set_nat ).
% 14.17/7.98
% 14.17/7.98 thf(sy_c_Fun_Ocomp_001t__Set__Oset_I_Eo_J_001_062_It__Set__Oset_I_Eo_J_Mt__Set__Oset_I_Eo_J_J_001t__Code____Numeral__Ointeger,type,
% 14.17/7.98 comp_s90868338361139993nteger: ( set_o > set_o > set_o ) > ( code_integer > set_o ) > code_integer > set_o > set_o ).
% 14.17/7.98
% 14.17/7.98 thf(sy_c_Fun_Ocomp_001t__Set__Oset_I_Eo_J_001_062_It__Set__Oset_I_Eo_J_Mt__Set__Oset_I_Eo_J_J_001t__Nat__Onat,type,
% 14.17/7.98 comp_s4424444925706540684_o_nat: ( set_o > set_o > set_o ) > ( nat > set_o ) > nat > set_o > set_o ).
% 14.17/7.98
% 14.17/7.98 thf(sy_c_Fun_Ocomp_001t__Set__Oset_It__Code____Numeral__Ointeger_J_001t__Set__Oset_It__Code____Numeral__Ointeger_J_001t__Code____Numeral__Ointeger,type,
% 14.17/7.98 comp_s956550612328592744nteger: ( set_Code_integer > set_Code_integer ) > ( code_integer > set_Code_integer ) > code_integer > set_Code_integer ).
% 14.17/7.98
% 14.17/7.98 thf(sy_c_Fun_Ocomp_001t__Set__Oset_It__Int__Oint_J_001_062_It__Set__Oset_It__Int__Oint_J_Mt__Set__Oset_It__Int__Oint_J_J_001t__Code____Numeral__Ointeger,type,
% 14.17/7.98 comp_s3634918272309710215nteger: ( set_int > set_int > set_int ) > ( code_integer > set_int ) > code_integer > set_int > set_int ).
% 14.17/7.98
% 14.17/7.98 thf(sy_c_Fun_Ocomp_001t__Set__Oset_It__Int__Oint_J_001_062_It__Set__Oset_It__Int__Oint_J_Mt__Set__Oset_It__Int__Oint_J_J_001t__Nat__Onat,type,
% 14.17/7.98 comp_s6689493721604790522nt_nat: ( set_int > set_int > set_int ) > ( nat > set_int ) > nat > set_int > set_int ).
% 14.17/7.98
% 14.17/7.98 thf(sy_c_Fun_Ocomp_001t__Set__Oset_It__Nat__Onat_J_001_062_It__Set__Oset_It__Nat__Onat_J_Mt__Set__Oset_It__Nat__Onat_J_J_001t__Code____Numeral__Ointeger,type,
% 14.17/7.98 comp_s3169385134670675059nteger: ( set_nat > set_nat > set_nat ) > ( code_integer > set_nat ) > code_integer > set_nat > set_nat ).
% 14.17/7.98
% 14.17/7.98 thf(sy_c_Fun_Ocomp_001t__Set__Oset_It__Nat__Onat_J_001_062_It__Set__Oset_It__Nat__Onat_J_Mt__Set__Oset_It__Nat__Onat_J_J_001t__Nat__Onat,type,
% 14.17/7.98 comp_s2351873599094224870at_nat: ( set_nat > set_nat > set_nat ) > ( nat > set_nat ) > nat > set_nat > set_nat ).
% 14.17/7.98
% 14.17/7.98 thf(sy_c_Fun_Ocomp_001t__Set__Oset_It__Nat__Onat_J_001t__Int__Oint_001t__Product____Type__Oprod_It__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J_Mt__Set__Oset_It__Nat__Onat_J_J,type,
% 14.17/7.98 comp_s1601710177542341017et_nat: ( set_nat > int ) > ( produc3658429121746597890et_nat > set_nat ) > produc3658429121746597890et_nat > int ).
% 14.17/7.98
% 14.17/7.98 thf(sy_c_Fun_Ocomp_001t__Set__Oset_It__Nat__Onat_J_001t__Set__Oset_It__Code____Numeral__Ointeger_J_001t__Code____Numeral__Ointeger,type,
% 14.17/7.98 comp_s8292494440407567195nteger: ( set_nat > set_Code_integer ) > ( code_integer > set_nat ) > code_integer > set_Code_integer ).
% 14.17/7.98
% 14.17/7.98 thf(sy_c_Fun_Ocomp_001t__Set__Oset_It__Nat__Onat_J_001t__Set__Oset_It__Nat__Onat_J_001t__Code____Numeral__Ointeger,type,
% 14.17/7.98 comp_s4605850120322998606nteger: ( set_nat > set_nat ) > ( code_integer > set_nat ) > code_integer > set_nat ).
% 14.17/7.98
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% 14.17/7.99 produc9020447175693601247nteger: ( code_integer > produc8923325533196201883nteger > produc8923325533196201883nteger ) > produc8923325533196201883nteger > produc5966032636589952099nteger ).
% 14.17/7.99
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% 14.17/7.99
% 14.17/7.99 thf(sy_c_Product__Type_Oapsnd_001t__Int__Oint_001t__Int__Oint_001t__Int__Oint,type,
% 14.17/7.99 produc4463282112584876420nt_int: ( int > int ) > product_prod_int_int > product_prod_int_int ).
% 14.17/7.99
% 14.17/7.99 thf(sy_c_Product__Type_Oapsnd_001t__Int__Oint_001t__Nat__Onat_001t__Int__Oint,type,
% 14.17/7.99 produc3463928400683821864at_int: ( int > nat ) > product_prod_int_int > product_prod_int_nat ).
% 14.17/7.99
% 14.17/7.99 thf(sy_c_Product__Type_Oapsnd_001t__Int__Oint_001t__Nat__Onat_001t__Nat__Onat,type,
% 14.17/7.99 produc3466418871192872140at_nat: ( int > nat ) > product_prod_nat_int > product_prod_nat_nat ).
% 14.17/7.99
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% 14.17/7.99
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% 14.17/7.99 produc2878475440847162070at_int: ( int > produc3925858234332021118et_nat ) > product_prod_int_int > produc1049655590333717003et_nat ).
% 14.17/7.99
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% 14.17/7.99 produc3447581998352395295_nat_o: ( int > produc3658429121746597890et_nat ) > produc3776562785724475560_o_int > produc3925858234332021118et_nat ).
% 14.17/7.99
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% 14.17/7.99 produc9121118195053341018at_int: ( int > produc3658429121746597890et_nat ) > product_prod_int_int > produc8065447618272157327et_nat ).
% 14.17/7.99
% 14.17/7.99 thf(sy_c_Product__Type_Oapsnd_001t__Int__Oint_001t__Set__Oset_It__Nat__Onat_J_001t__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J,type,
% 14.17/7.99 produc2122153897363312795t_unit: ( int > set_nat ) > produc2475246329834808232it_int > produc3658429121746597890et_nat ).
% 14.17/7.99
% 14.17/7.99 thf(sy_c_Product__Type_Oapsnd_001t__Int__Oint_001t__Set__Oset_It__Nat__Onat_J_001t__Int__Oint,type,
% 14.17/7.99 produc1253519502038700766at_int: ( int > set_nat ) > product_prod_int_int > produc2771989624626745363et_nat ).
% 14.17/7.99
% 14.17/7.99 thf(sy_c_Product__Type_Oapsnd_001t__Nat__Onat_001t__Nat__Onat_001t__Nat__Onat,type,
% 14.17/7.99 produc3094765310956027504at_nat: ( nat > nat ) > product_prod_nat_nat > product_prod_nat_nat ).
% 14.17/7.99
% 14.17/7.99 thf(sy_c_Product__Type_Oapsnd_001t__Product____Type__Oprod_I_062_It__Product____Type__Oprod_It__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J_Mt__Set__Oset_It__Nat__Onat_J_J_M_Eo_J_Mt__Product____Type__Oprod_It__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J_Mt__Set__Oset_It__Nat__Onat_J_J_J_001t__Product____Type__Oprod_I_062_It__Product____Type__Oprod_It__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J_Mt__Set__Oset_It__Nat__Onat_J_J_M_Eo_J_Mt__Product____Type__Oprod_It__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J_Mt__Set__Oset_It__Nat__Onat_J_J_J_001_062_It__Product____Type__Oprod_It__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J_Mt__Set__Oset_It__Nat__Onat_J_J_M_Eo_J,type,
% 14.17/7.99 produc969530845752564945_nat_o: ( produc3925858234332021118et_nat > produc3925858234332021118et_nat ) > produc2732055786443039994et_nat > produc2732055786443039994et_nat ).
% 14.17/7.99
% 14.17/7.99 thf(sy_c_Product__Type_Oapsnd_001t__Product____Type__Oprod_I_062_It__Product____Type__Oprod_It__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J_Mt__Set__Oset_It__Nat__Onat_J_J_M_Eo_J_Mt__Product____Type__Oprod_It__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J_Mt__Set__Oset_It__Nat__Onat_J_J_J_001t__Product____Type__Oprod_It__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J_Mt__Set__Oset_It__Nat__Onat_J_J_001_062_It__Product____Type__Oprod_It__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J_Mt__Set__Oset_It__Nat__Onat_J_J_M_Eo_J,type,
% 14.17/7.99 produc1515462096303866701_nat_o: ( produc3925858234332021118et_nat > produc3658429121746597890et_nat ) > produc2732055786443039994et_nat > produc3925858234332021118et_nat ).
% 14.17/7.99
% 14.17/7.99 thf(sy_c_Product__Type_Oapsnd_001t__Product____Type__Oprod_It__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J_Mt__Set__Oset_It__Nat__Onat_J_J_001t__Product____Type__Oprod_I_062_It__Product____Type__Oprod_It__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J_Mt__Set__Oset_It__Nat__Onat_J_J_M_Eo_J_Mt__Product____Type__Oprod_It__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J_Mt__Set__Oset_It__Nat__Onat_J_J_J_001_062_It__Product____Type__Oprod_It__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J_Mt__Set__Oset_It__Nat__Onat_J_J_M_Eo_J,type,
% 14.17/7.99 produc7714581247149323085_nat_o: ( produc3658429121746597890et_nat > produc3925858234332021118et_nat ) > produc3925858234332021118et_nat > produc2732055786443039994et_nat ).
% 14.17/7.99
% 14.17/7.99 thf(sy_c_Product__Type_Oapsnd_001t__Product____Type__Oprod_It__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J_Mt__Set__Oset_It__Nat__Onat_J_J_001t__Product____Type__Oprod_It__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J_Mt__Set__Oset_It__Nat__Onat_J_J_001_062_It__Product____Type__Oprod_It__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J_Mt__Set__Oset_It__Nat__Onat_J_J_M_Eo_J,type,
% 14.17/7.99 produc8750854537940449737_nat_o: ( produc3658429121746597890et_nat > produc3658429121746597890et_nat ) > produc3925858234332021118et_nat > produc3925858234332021118et_nat ).
% 14.17/7.99
% 14.17/7.99 thf(sy_c_Product__Type_Oapsnd_001t__Set__Oset_It__Nat__Onat_J_001t__Int__Oint_001t__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J,type,
% 14.17/7.99 produc7345612820240450715t_unit: ( set_nat > int ) > produc3658429121746597890et_nat > produc2475246329834808232it_int ).
% 14.17/7.99
% 14.17/7.99 thf(sy_c_Product__Type_Oapsnd_001t__Set__Oset_It__Nat__Onat_J_001t__Int__Oint_001t__Int__Oint,type,
% 14.17/7.99 produc2744032961877892830nt_int: ( set_nat > int ) > produc2771989624626745363et_nat > product_prod_int_int ).
% 14.17/7.99
% 14.17/7.99 thf(sy_c_Product__Type_Oapsnd_001t__Set__Oset_It__Nat__Onat_J_001t__Nat__Onat_001t__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J,type,
% 14.17/7.99 produc8442695800884449271t_unit: ( set_nat > nat ) > produc3658429121746597890et_nat > produc6653097349344004940it_nat ).
% 14.17/7.99
% 14.17/7.99 thf(sy_c_Product__Type_Oapsnd_001t__Set__Oset_It__Nat__Onat_J_001t__Nat__Onat_001t__Nat__Onat,type,
% 14.17/7.99 produc1747169720485888550at_nat: ( set_nat > nat ) > produc2400336064389900727et_nat > product_prod_nat_nat ).
% 14.17/7.99
% 14.17/7.99 thf(sy_c_Product__Type_Oapsnd_001t__Set__Oset_It__Nat__Onat_J_001t__Product____Type__Oprod_I_062_It__Product____Type__Oprod_It__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J_Mt__Set__Oset_It__Nat__Onat_J_J_M_Eo_J_Mt__Product____Type__Oprod_It__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J_Mt__Set__Oset_It__Nat__Onat_J_J_J_001_062_It__Product____Type__Oprod_It__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J_Mt__Set__Oset_It__Nat__Onat_J_J_M_Eo_J,type,
% 14.17/7.99 produc3009152965630989513_nat_o: ( set_nat > produc3925858234332021118et_nat ) > produc6500295350249520386et_nat > produc2732055786443039994et_nat ).
% 14.17/7.99
% 14.17/7.99 thf(sy_c_Product__Type_Oapsnd_001t__Set__Oset_It__Nat__Onat_J_001t__Product____Type__Oprod_I_062_It__Product____Type__Oprod_It__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J_Mt__Set__Oset_It__Nat__Onat_J_J_M_Eo_J_Mt__Product____Type__Oprod_It__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J_Mt__Set__Oset_It__Nat__Onat_J_J_J_001t__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J,type,
% 14.17/7.99 produc7176435795764513225t_unit: ( set_nat > produc3925858234332021118et_nat ) > produc3658429121746597890et_nat > produc7936767299998605306et_nat ).
% 14.17/7.99
% 14.17/7.99 thf(sy_c_Product__Type_Oapsnd_001t__Set__Oset_It__Nat__Onat_J_001t__Product____Type__Oprod_It__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J_Mt__Set__Oset_It__Nat__Onat_J_J_001_062_It__Product____Type__Oprod_It__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J_Mt__Set__Oset_It__Nat__Onat_J_J_M_Eo_J,type,
% 14.17/7.99 produc1718959742329256773_nat_o: ( set_nat > produc3658429121746597890et_nat ) > produc6500295350249520386et_nat > produc3925858234332021118et_nat ).
% 14.17/7.99
% 14.17/7.99 thf(sy_c_Product__Type_Oapsnd_001t__Set__Oset_It__Nat__Onat_J_001t__Product____Type__Oprod_It__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J_Mt__Set__Oset_It__Nat__Onat_J_J_001t__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J,type,
% 14.17/7.99 produc7441988914112407621t_unit: ( set_nat > produc3658429121746597890et_nat ) > produc3658429121746597890et_nat > produc5697189838369619070et_nat ).
% 14.17/7.99
% 14.17/7.99 thf(sy_c_Product__Type_Oapsnd_001t__Set__Oset_It__Nat__Onat_J_001t__Set__Oset_It__Nat__Onat_J_001t__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J,type,
% 14.17/7.99 produc3286488329990770625t_unit: ( set_nat > set_nat ) > produc3658429121746597890et_nat > produc3658429121746597890et_nat ).
% 14.17/7.99
% 14.17/7.99 thf(sy_c_Product__Type_Ocurry_001_062_It__Product____Type__Oprod_It__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J_Mt__Set__Oset_It__Nat__Onat_J_J_M_Eo_J_001t__Product____Type__Oprod_I_062_It__Product____Type__Oprod_It__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J_Mt__Set__Oset_It__Nat__Onat_J_J_M_Eo_J_Mt__Product____Type__Oprod_It__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J_Mt__Set__Oset_It__Nat__Onat_J_J_J_001_Eo,type,
% 14.17/7.99 produc5101573711933517782_nat_o: ( produc2732055786443039994et_nat > $o ) > ( produc3658429121746597890et_nat > $o ) > produc3925858234332021118et_nat > $o ).
% 14.17/7.99
% 14.17/7.99 thf(sy_c_Product__Type_Ocurry_001_062_It__Product____Type__Oprod_It__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J_Mt__Set__Oset_It__Nat__Onat_J_J_M_Eo_J_001t__Product____Type__Oprod_It__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J_Mt__Set__Oset_It__Nat__Onat_J_J_001_Eo,type,
% 14.17/7.99 produc6216949301066131538_nat_o: ( produc3925858234332021118et_nat > $o ) > ( produc3658429121746597890et_nat > $o ) > produc3658429121746597890et_nat > $o ).
% 14.17/7.99
% 14.17/7.99 thf(sy_c_Product__Type_Ocurry_001t__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J_001t__Set__Oset_It__Nat__Onat_J_001_Eo,type,
% 14.17/7.99 produc1807123492089482958_nat_o: ( produc3658429121746597890et_nat > $o ) > heap_e7401611519738050253t_unit > set_nat > $o ).
% 14.17/7.99
% 14.17/7.99 thf(sy_c_Product__Type_Ocurry_001t__Int__Oint_001t__Int__Oint_001_Eo,type,
% 14.17/7.99 produc175634133007206835_int_o: ( product_prod_int_int > $o ) > int > int > $o ).
% 14.17/7.99
% 14.17/7.99 thf(sy_c_Product__Type_Ocurry_001t__Nat__Onat_001t__Nat__Onat_001_Eo,type,
% 14.17/7.99 produc1310100445399344235_nat_o: ( product_prod_nat_nat > $o ) > nat > nat > $o ).
% 14.17/7.99
% 14.17/7.99 thf(sy_c_Product__Type_Omap__prod_001_062_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J_001_062_It__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J_Mt__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J_J_001_062_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J_001_062_It__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J_Mt__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J_J,type,
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% 14.17/7.99
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% 14.17/7.99 produc6158254626097298938nteger: ( ( code_integer > code_integer ) > produc8923325533196201883nteger > produc8923325533196201883nteger ) > ( code_integer > produc8923325533196201883nteger > produc8923325533196201883nteger ) > produc2221115339011613079nteger > produc101387210554329899nteger ).
% 14.17/7.99
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% 14.17/7.99 produc2784369760062397435at_nat: ( ( produc3658429121746597890et_nat > $o ) > nat ) > ( produc3658429121746597890et_nat > nat ) > produc3925858234332021118et_nat > product_prod_nat_nat ).
% 14.17/7.99
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% 14.17/7.99 produc7818937854452556646nteger: ( code_integer > code_integer > code_integer ) > ( code_integer > code_integer > code_integer ) > produc8923325533196201883nteger > produc723880050079548563nteger ).
% 14.17/7.99
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% 14.17/7.99 produc3789967387199524714nteger: ( code_integer > code_integer > code_integer ) > ( code_integer > code_integer ) > produc8923325533196201883nteger > produc2221115339011613079nteger ).
% 14.17/7.99
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% 14.17/7.99
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% 14.17/7.99
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% 14.17/7.99 thf(sy_c_Product__Type_Oprod_Ocase__prod_001t__Int__Oint_001t__Int__Oint_001_Eo,type,
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% 14.17/7.99
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% 14.17/7.99 thf(sy_c_Set_Oinsert_001t__Rat__Orat,type,
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% 14.17/7.99 type_d5515310227294319076nteger: ( ( code_integer > code_integer ) > code_integer ) > ( code_integer > code_integer > code_integer ) > set_Code_integer > $o ).
% 14.17/7.99
% 14.17/7.99 thf(sy_c_Typedef_Otype__definition_001_062_It__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J_Mt__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J_J_001_Eo,type,
% 14.17/7.99 type_d5661033022651575429eger_o: ( ( produc8923325533196201883nteger > produc8923325533196201883nteger ) > $o ) > ( $o > produc8923325533196201883nteger > produc8923325533196201883nteger ) > set_o > $o ).
% 14.17/7.99
% 14.17/7.99 thf(sy_c_Typedef_Otype__definition_001_062_It__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J_Mt__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J_J_001t__Code____Numeral__Ointeger,type,
% 14.17/7.99 type_d6544512618915850416nteger: ( ( produc8923325533196201883nteger > produc8923325533196201883nteger ) > code_integer ) > ( code_integer > produc8923325533196201883nteger > produc8923325533196201883nteger ) > set_Code_integer > $o ).
% 14.17/7.99
% 14.17/7.99 thf(sy_c_Typedef_Otype__definition_001_062_It__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J_Mt__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J_J_001t__Nat__Onat,type,
% 14.17/7.99 type_d6278479724763174947er_nat: ( ( produc8923325533196201883nteger > produc8923325533196201883nteger ) > nat ) > ( nat > produc8923325533196201883nteger > produc8923325533196201883nteger ) > set_nat > $o ).
% 14.17/7.99
% 14.17/7.99 thf(sy_c_Typedef_Otype__definition_001_062_It__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J_Mt__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J_J_001t__Product____Type__Ounit,type,
% 14.17/7.99 type_d6996789741066924922t_unit: ( ( produc8923325533196201883nteger > produc8923325533196201883nteger ) > product_unit ) > ( product_unit > produc8923325533196201883nteger > produc8923325533196201883nteger ) > set_Product_unit > $o ).
% 14.17/7.99
% 14.17/7.99 thf(sy_c_Typedef_Otype__definition_001t__Assertions__Oassn_001_062_It__Product____Type__Oprod_It__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J_Mt__Set__Oset_It__Nat__Onat_J_J_M_Eo_J,type,
% 14.17/7.99 type_d3909072315231072503_nat_o: ( assn > produc3658429121746597890et_nat > $o ) > ( ( produc3658429121746597890et_nat > $o ) > assn ) > set_Pr4532377907799695533_nat_o > $o ).
% 14.17/7.99
% 14.17/7.99 thf(sy_c_Typedef_Otype__definition_001t__Code____Numeral__Ointeger_001_062_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J,type,
% 14.17/7.99 type_d669523880323286244nteger: ( code_integer > code_integer > code_integer ) > ( ( code_integer > code_integer ) > code_integer ) > set_Co726186806682698383nteger > $o ).
% 14.17/7.99
% 14.17/7.99 thf(sy_c_Typedef_Otype__definition_001t__Code____Numeral__Ointeger_001_Eo,type,
% 14.17/7.99 type_d6812736842639505741eger_o: ( code_integer > $o ) > ( $o > code_integer ) > set_o > $o ).
% 14.17/7.99
% 14.17/7.99 thf(sy_c_Typedef_Otype__definition_001t__Code____Numeral__Ointeger_001t__Code____Numeral__Ointeger,type,
% 14.17/7.99 type_d4796611474492816872nteger: ( code_integer > code_integer ) > ( code_integer > code_integer ) > set_Code_integer > $o ).
% 14.17/7.99
% 14.17/7.99 thf(sy_c_Typedef_Otype__definition_001t__Code____Numeral__Ointeger_001t__Int__Oint,type,
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% 14.17/7.99
% 14.17/7.99 thf(sy_c_Typedef_Otype__definition_001t__Code____Numeral__Ointeger_001t__Nat__Onat,type,
% 14.17/7.99 type_d8368584451094728027er_nat: ( code_integer > nat ) > ( nat > code_integer ) > set_nat > $o ).
% 14.17/7.99
% 14.17/7.99 thf(sy_c_Typedef_Otype__definition_001t__Code____Numeral__Ointeger_001t__Product____Type__Ounit,type,
% 14.17/7.99 type_d9006111508436661314t_unit: ( code_integer > product_unit ) > ( product_unit > code_integer ) > set_Product_unit > $o ).
% 14.17/7.99
% 14.17/7.99 thf(sy_c_Typedef_Otype__definition_001t__Code____Numeral__Onatural_001t__Nat__Onat,type,
% 14.17/7.99 type_d4410041424927559462al_nat: ( code_natural > nat ) > ( nat > code_natural ) > set_nat > $o ).
% 14.17/7.99
% 14.17/7.99 thf(sy_c_Typedef_Otype__definition_001t__Nat__Onat_001t__Code____Numeral__Ointeger,type,
% 14.17/7.99 type_d8633139899873770843nteger: ( nat > code_integer ) > ( code_integer > nat ) > set_Code_integer > $o ).
% 14.17/7.99
% 14.17/7.99 thf(sy_c_Typedef_Otype__definition_001t__Nat__Onat_001t__Nat__Onat,type,
% 14.17/7.99 type_d6250493948777748686at_nat: ( nat > nat ) > ( nat > nat ) > set_nat > $o ).
% 14.17/7.99
% 14.17/7.99 thf(sy_c_Typedef_Otype__definition_001t__Nat__Onat_001t__Set__Oset_It__Nat__Onat_J,type,
% 14.17/7.99 type_d3768499744764857476et_nat: ( nat > set_nat ) > ( set_nat > nat ) > set_set_nat > $o ).
% 14.17/7.99
% 14.17/7.99 thf(sy_c_Typedef_Otype__definition_001t__Product____Type__Ounit_001_Eo,type,
% 14.17/7.99 type_d6188575255521822967unit_o: ( product_unit > $o ) > ( $o > product_unit ) > set_o > $o ).
% 14.17/7.99
% 14.17/7.99 thf(sy_c_Typedef_Otype__definition_001t__Set__Oset_It__Nat__Onat_J_001t__Nat__Onat,type,
% 14.17/7.99 type_d202864840109488260at_nat: ( set_nat > nat ) > ( nat > set_nat ) > set_nat > $o ).
% 14.17/7.99
% 14.17/7.99 thf(sy_c_Wellfounded_Oaccp_001t__Product____Type__Oprod_I_062_It__Int__Oint_Mt__Int__Oint_J_Mt__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_J,type,
% 14.17/7.99 accp_P4507279754199951662nt_int: ( produc5030937667867769463nt_int > produc5030937667867769463nt_int > $o ) > produc5030937667867769463nt_int > $o ).
% 14.17/7.99
% 14.17/7.99 thf(sy_c_Wellfounded_Oaccp_001t__Product____Type__Oprod_I_062_It__Int__Oint_Mt__Nat__Onat_J_Mt__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_J,type,
% 14.17/7.99 accp_P5604362734843950218nt_int: ( produc6128020648511768019nt_int > produc6128020648511768019nt_int > $o ) > produc6128020648511768019nt_int > $o ).
% 14.17/7.99
% 14.17/7.99 thf(sy_c_Wellfounded_Oaccp_001t__Product____Type__Oprod_I_062_It__Nat__Onat_Mt__Int__Oint_J_Mt__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
% 14.17/7.99 accp_P3050404660685545426at_nat: ( produc3574062574353363227at_nat > produc3574062574353363227at_nat > $o ) > produc3574062574353363227at_nat > $o ).
% 14.17/7.99
% 14.17/7.99 thf(sy_c_Wellfounded_Oaccp_001t__Product____Type__Oprod_I_062_It__Nat__Onat_Mt__Nat__Onat_J_Mt__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
% 14.17/7.99 accp_P4147487641329543982at_nat: ( produc4671145554997361783at_nat > produc4671145554997361783at_nat > $o ) > produc4671145554997361783at_nat > $o ).
% 14.17/7.99
% 14.17/7.99 thf(sy_c_Wellfounded_Oaccp_001t__Product____Type__Oprod_I_062_It__Product____Type__Oprod_It__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J_Mt__Set__Oset_It__Nat__Onat_J_J_M_Eo_J_Mt__Product____Type__Oprod_I_062_It__Product____Type__Oprod_It__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J_Mt__Set__Oset_It__Nat__Onat_J_J_M_Eo_J_Mt__Product____Type__Oprod_It__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J_Mt__Set__Oset_It__Nat__Onat_J_J_J_J,type,
% 14.17/7.99 accp_P1862375125659990705et_nat: ( produc2732055786443039994et_nat > produc2732055786443039994et_nat > $o ) > produc2732055786443039994et_nat > $o ).
% 14.17/7.99
% 14.17/7.99 thf(sy_c_Wellfounded_Oaccp_001t__Product____Type__Oprod_I_Eo_Mt__Product____Type__Oprod_It__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J_Mt__Set__Oset_It__Nat__Onat_J_J_J,type,
% 14.17/7.99 accp_P8458817951426537472et_nat: ( produc6197004810343482825et_nat > produc6197004810343482825et_nat > $o ) > produc6197004810343482825et_nat > $o ).
% 14.17/7.99
% 14.17/7.99 thf(sy_c_Wellfounded_Oaccp_001t__Product____Type__Oprod_It__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J_Mt__Set__Oset_It__Nat__Onat_J_J,type,
% 14.17/7.99 accp_P5801069581201407417et_nat: ( produc3658429121746597890et_nat > produc3658429121746597890et_nat > $o ) > produc3658429121746597890et_nat > $o ).
% 14.17/7.99
% 14.17/7.99 thf(sy_c_Wellfounded_Oaccp_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J,type,
% 14.17/7.99 accp_P1096762738010456898nt_int: ( product_prod_int_int > product_prod_int_int > $o ) > product_prod_int_int > $o ).
% 14.17/7.99
% 14.17/7.99 thf(sy_c_Wellfounded_Oaccp_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
% 14.17/7.99 accp_P4275260045618599050at_nat: ( product_prod_nat_nat > product_prod_nat_nat > $o ) > product_prod_nat_nat > $o ).
% 14.17/7.99
% 14.17/7.99 thf(sy_c_Wellfounded_Olex__prod_001_062_It__Product____Type__Oprod_It__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J_Mt__Set__Oset_It__Nat__Onat_J_J_M_Eo_J_001t__Product____Type__Oprod_I_062_It__Product____Type__Oprod_It__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J_Mt__Set__Oset_It__Nat__Onat_J_J_M_Eo_J_Mt__Product____Type__Oprod_It__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J_Mt__Set__Oset_It__Nat__Onat_J_J_J,type,
% 14.17/7.99 lex_pr4722427456421979612et_nat: set_Pr2161125870931222855_nat_o > set_Pr7928877670098842301et_nat > set_Pr3444600963470892981et_nat ).
% 14.17/7.99
% 14.17/7.99 thf(sy_c_Wellfounded_Olex__prod_001_062_It__Product____Type__Oprod_It__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J_Mt__Set__Oset_It__Nat__Onat_J_J_M_Eo_J_001t__Product____Type__Oprod_It__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J_Mt__Set__Oset_It__Nat__Onat_J_J,type,
% 14.17/7.99 lex_pr5340572901959109728et_nat: set_Pr2161125870931222855_nat_o > set_Pr719794911490849221et_nat > set_Pr7928877670098842301et_nat ).
% 14.17/7.99
% 14.17/7.99 thf(sy_c_Wellfounded_Olex__prod_001t__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J_001t__Set__Oset_It__Nat__Onat_J,type,
% 14.17/7.99 lex_pr538634069021852900et_nat: set_Pr1776176089890104647t_unit > set_Pr5488025237498180813et_nat > set_Pr719794911490849221et_nat ).
% 14.17/7.99
% 14.17/7.99 thf(sy_c_Wellfounded_Olex__prod_001t__Int__Oint_001t__Int__Oint,type,
% 14.17/7.99 lex_prod_int_int: set_Pr958786334691620121nt_int > set_Pr958786334691620121nt_int > set_Pr2560585780119916871nt_int ).
% 14.17/7.99
% 14.17/7.99 thf(sy_c_Wellfounded_Olex__prod_001t__Int__Oint_001t__Nat__Onat,type,
% 14.17/7.99 lex_prod_int_nat: set_Pr958786334691620121nt_int > set_Pr1261947904930325089at_nat > set_Pr5696620967126086215nt_nat ).
% 14.17/7.99
% 14.17/7.99 thf(sy_c_Wellfounded_Olex__prod_001t__Nat__Onat_001t__Int__Oint,type,
% 14.17/7.99 lex_prod_nat_int: set_Pr1261947904930325089at_nat > set_Pr958786334691620121nt_int > set_Pr5557702248415638087at_int ).
% 14.17/7.99
% 14.17/7.99 thf(sy_c_Wellfounded_Olex__prod_001t__Nat__Onat_001t__Nat__Onat,type,
% 14.17/7.99 lex_prod_nat_nat: set_Pr1261947904930325089at_nat > set_Pr1261947904930325089at_nat > set_Pr8693737435421807431at_nat ).
% 14.17/7.99
% 14.17/7.99 thf(sy_c_Wellfounded_Omax__ext_001_062_It__Product____Type__Oprod_It__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J_Mt__Set__Oset_It__Nat__Onat_J_J_M_Eo_J,type,
% 14.17/7.99 max_ex5265246208886859912_nat_o: set_Pr2161125870931222855_nat_o > set_Pr3813038829465207111_nat_o ).
% 14.17/7.99
% 14.17/7.99 thf(sy_c_Wellfounded_Omax__ext_001_Eo,type,
% 14.17/7.99 max_ext_o: set_Product_prod_o_o > set_Pr4577759397028426247_set_o ).
% 14.17/7.99
% 14.17/7.99 thf(sy_c_Wellfounded_Omax__ext_001t__Code____Numeral__Ointeger,type,
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% 14.17/7.99
% 14.17/7.99 thf(sy_c_Wellfounded_Omax__ext_001t__Int__Oint,type,
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% 14.17/7.99
% 14.17/7.99 thf(sy_c_Wellfounded_Omax__ext_001t__Nat__Onat,type,
% 14.17/7.99 max_ext_nat: set_Pr1261947904930325089at_nat > set_Pr5488025237498180813et_nat ).
% 14.17/7.99
% 14.17/7.99 thf(sy_c_Wellfounded_Omax__ext_001t__Product____Type__Ounit,type,
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% 14.17/7.99
% 14.17/7.99 thf(sy_c_Wellfounded_Omeasure_001t__Int__Oint,type,
% 14.17/7.99 measure_int: ( int > nat ) > set_Pr958786334691620121nt_int ).
% 14.17/7.99
% 14.17/7.99 thf(sy_c_Wellfounded_Omeasure_001t__Nat__Onat,type,
% 14.17/7.99 measure_nat: ( nat > nat ) > set_Pr1261947904930325089at_nat ).
% 14.17/7.99
% 14.17/7.99 thf(sy_c_Wellfounded_Omin__ext_001_062_It__Product____Type__Oprod_It__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J_Mt__Set__Oset_It__Nat__Onat_J_J_M_Eo_J,type,
% 14.17/7.99 min_ex3730736537596016502_nat_o: set_Pr2161125870931222855_nat_o > set_Pr3813038829465207111_nat_o ).
% 14.17/7.99
% 14.17/7.99 thf(sy_c_Wellfounded_Omin__ext_001_Eo,type,
% 14.17/7.99 min_ext_o: set_Product_prod_o_o > set_Pr4577759397028426247_set_o ).
% 14.17/7.99
% 14.17/7.99 thf(sy_c_Wellfounded_Omin__ext_001t__Int__Oint,type,
% 14.17/7.99 min_ext_int: set_Pr958786334691620121nt_int > set_Pr2522554150109002629et_int ).
% 14.17/7.99
% 14.17/7.99 thf(sy_c_Wellfounded_Omin__ext_001t__Nat__Onat,type,
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% 14.17/7.99
% 14.17/7.99 thf(sy_c_Wellfounded_Omin__ext_001t__Product____Type__Ounit,type,
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% 14.17/7.99
% 14.17/7.99 thf(sy_c_Wellfounded_Owf_001_062_It__Product____Type__Oprod_It__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J_Mt__Set__Oset_It__Nat__Onat_J_J_M_Eo_J,type,
% 14.17/7.99 wf_Pro8787509627421085110_nat_o: set_Pr2161125870931222855_nat_o > $o ).
% 14.17/7.99
% 14.17/7.99 thf(sy_c_Wellfounded_Owf_001_Eo,type,
% 14.17/7.99 wf_o: set_Product_prod_o_o > $o ).
% 14.17/7.99
% 14.17/7.99 thf(sy_c_Wellfounded_Owf_001t__Int__Oint,type,
% 14.17/7.99 wf_int: set_Pr958786334691620121nt_int > $o ).
% 14.17/7.99
% 14.17/7.99 thf(sy_c_Wellfounded_Owf_001t__Nat__Onat,type,
% 14.17/7.99 wf_nat: set_Pr1261947904930325089at_nat > $o ).
% 14.17/7.99
% 14.17/7.99 thf(sy_c_Wellfounded_Owf_001t__Product____Type__Ounit,type,
% 14.17/7.99 wf_Product_unit: set_Pr5094982260447487303t_unit > $o ).
% 14.17/7.99
% 14.17/7.99 thf(sy_c_Zorn_OChains_001t__Int__Oint,type,
% 14.17/7.99 chains_int: set_Pr958786334691620121nt_int > set_set_int ).
% 14.17/7.99
% 14.17/7.99 thf(sy_c_Zorn_OChains_001t__Nat__Onat,type,
% 14.17/7.99 chains_nat: set_Pr1261947904930325089at_nat > set_set_nat ).
% 14.17/7.99
% 14.17/7.99 thf(sy_c_Zorn_Ochains_001t__Nat__Onat,type,
% 14.17/7.99 chains_nat2: set_set_nat > set_set_set_nat ).
% 14.17/7.99
% 14.17/7.99 thf(sy_c_Zorn_Opred__on_Ochain_001_062_It__Product____Type__Oprod_It__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J_Mt__Set__Oset_It__Nat__Onat_J_J_M_Eo_J,type,
% 14.17/8.00 pred_c177362287423414904_nat_o: set_Pr4532377907799695533_nat_o > ( ( produc3658429121746597890et_nat > $o ) > ( produc3658429121746597890et_nat > $o ) > $o ) > set_Pr4532377907799695533_nat_o > $o ).
% 14.17/8.00
% 14.17/8.00 thf(sy_c_Zorn_Opred__on_Ochain_001_Eo,type,
% 14.17/8.00 pred_chain_o: set_o > ( $o > $o > $o ) > set_o > $o ).
% 14.17/8.00
% 14.17/8.00 thf(sy_c_Zorn_Opred__on_Ochain_001t__Int__Oint,type,
% 14.17/8.00 pred_chain_int: set_int > ( int > int > $o ) > set_int > $o ).
% 14.17/8.00
% 14.17/8.00 thf(sy_c_Zorn_Opred__on_Ochain_001t__Nat__Onat,type,
% 14.17/8.00 pred_chain_nat: set_nat > ( nat > nat > $o ) > set_nat > $o ).
% 14.17/8.00
% 14.17/8.00 thf(sy_c_Zorn_Opred__on_Ochain_001t__Product____Type__Ounit,type,
% 14.17/8.00 pred_c6915491785556283576t_unit: set_Product_unit > ( product_unit > product_unit > $o ) > set_Product_unit > $o ).
% 14.17/8.00
% 14.17/8.00 thf(sy_c_Zorn_Opred__on_Ochain_001t__Set__Oset_It__Code____Numeral__Ointeger_J,type,
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% 14.17/8.00
% 14.17/8.00 thf(sy_c_Zorn_Opred__on_Ochain_001t__Set__Oset_It__Nat__Onat_J,type,
% 14.17/8.00 pred_chain_set_nat: set_set_nat > ( set_nat > set_nat > $o ) > set_set_nat > $o ).
% 14.17/8.00
% 14.17/8.00 thf(sy_c_member_001_062_It__Product____Type__Oprod_It__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J_Mt__Set__Oset_It__Nat__Onat_J_J_M_Eo_J,type,
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% 14.17/8.00 thf(sy_c_member_001t__Set__Oset_It__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_J,type,
% 14.17/8.00 member2340774599025711042nt_int: set_Pr958786334691620121nt_int > set_se6260736226359567993nt_int > $o ).
% 14.17/8.00
% 14.17/8.00 thf(sy_c_member_001t__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
% 14.17/8.00 member2643936169264416010at_nat: set_Pr1261947904930325089at_nat > set_se7855581050983116737at_nat > $o ).
% 14.17/8.00
% 14.17/8.00 thf(sy_c_member_001t__Set__Oset_It__Product____Type__Oprod_It__Product____Type__Ounit_Mt__Product____Type__Ounit_J_J,type,
% 14.17/8.00 member1248447801810324848t_unit: set_Pr5094982260447487303t_unit > set_se8994758365886191399t_unit > $o ).
% 14.17/8.00
% 14.17/8.00 thf(sy_c_member_001t__Set__Oset_It__Product____Type__Ounit_J,type,
% 14.17/8.00 member5877623283571906838t_unit: set_Product_unit > set_set_Product_unit > $o ).
% 14.17/8.00
% 14.17/8.00 thf(sy_c_member_001t__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J,type,
% 14.17/8.00 member_set_set_nat: set_set_nat > set_set_set_nat > $o ).
% 14.17/8.00
% 14.17/8.00 thf(sy_v_h,type,
% 14.17/8.00 h: produc3658429121746597890et_nat ).
% 14.17/8.00
% 14.17/8.00 thf(fact_0_Rep__assn__inject,axiom,
% 14.17/8.00 ! [X: assn,Y: assn] :
% 14.17/8.00 ( ( ( rep_assn @ X )
% 14.17/8.00 = ( rep_assn @ Y ) )
% 14.17/8.00 = ( X = Y ) ) ).
% 14.17/8.00
% 14.17/8.00 thf(fact_1_Rep__assn__inverse,axiom,
% 14.17/8.00 ! [X: assn] :
% 14.17/8.00 ( ( abs_assn @ ( rep_assn @ X ) )
% 14.17/8.00 = X ) ).
% 14.17/8.00
% 14.17/8.00 thf(fact_2_Abs__assn__eqI_I2_J,axiom,
% 14.17/8.00 ! [P: produc3658429121746597890et_nat > $o,Pr: assn] :
% 14.17/8.00 ( ! [H: produc3658429121746597890et_nat] :
% 14.17/8.00 ( ( P @ H )
% 14.17/8.00 = ( rep_assn @ Pr @ H ) )
% 14.17/8.00 => ( Pr
% 14.17/8.00 = ( abs_assn @ P ) ) ) ).
% 14.17/8.00
% 14.17/8.00 thf(fact_3_Abs__assn__eqI_I1_J,axiom,
% 14.17/8.00 ! [P: produc3658429121746597890et_nat > $o,Pr: assn] :
% 14.17/8.00 ( ! [H: produc3658429121746597890et_nat] :
% 14.17/8.00 ( ( P @ H )
% 14.17/8.00 = ( rep_assn @ Pr @ H ) )
% 14.17/8.00 => ( ( abs_assn @ P )
% 14.17/8.00 = Pr ) ) ).
% 14.17/8.00
% 14.17/8.00 thf(fact_4_empty__iff,axiom,
% 14.17/8.00 ! [C: produc3658429121746597890et_nat > $o] :
% 14.17/8.00 ~ ( member6576561426505652726_nat_o @ C @ bot_bo7824918357723582233_nat_o ) ).
% 14.17/8.00
% 14.17/8.00 thf(fact_5_empty__iff,axiom,
% 14.17/8.00 ! [C: product_unit] :
% 14.17/8.00 ~ ( member_Product_unit @ C @ bot_bo3957492148770167129t_unit ) ).
% 14.17/8.00
% 14.17/8.00 thf(fact_6_empty__iff,axiom,
% 14.17/8.00 ! [C: $o] :
% 14.17/8.00 ~ ( member_o @ C @ bot_bot_set_o ) ).
% 14.17/8.00
% 14.17/8.00 thf(fact_7_empty__iff,axiom,
% 14.17/8.00 ! [C: nat] :
% 14.17/8.00 ~ ( member_nat @ C @ bot_bot_set_nat ) ).
% 14.17/8.00
% 14.17/8.00 thf(fact_8_empty__iff,axiom,
% 14.17/8.00 ! [C: int] :
% 14.17/8.00 ~ ( member_int @ C @ bot_bot_set_int ) ).
% 14.17/8.00
% 14.17/8.00 thf(fact_9_all__not__in__conv,axiom,
% 14.17/8.00 ! [A: set_Pr4532377907799695533_nat_o] :
% 14.17/8.00 ( ( ! [X2: produc3658429121746597890et_nat > $o] :
% 14.17/8.00 ~ ( member6576561426505652726_nat_o @ X2 @ A ) )
% 14.17/8.00 = ( A = bot_bo7824918357723582233_nat_o ) ) ).
% 14.17/8.00
% 14.17/8.00 thf(fact_10_all__not__in__conv,axiom,
% 14.17/8.00 ! [A: set_Product_unit] :
% 14.17/8.00 ( ( ! [X2: product_unit] :
% 14.17/8.00 ~ ( member_Product_unit @ X2 @ A ) )
% 14.17/8.00 = ( A = bot_bo3957492148770167129t_unit ) ) ).
% 14.17/8.00
% 14.17/8.00 thf(fact_11_all__not__in__conv,axiom,
% 14.17/8.00 ! [A: set_o] :
% 14.17/8.00 ( ( ! [X2: $o] :
% 14.17/8.00 ~ ( member_o @ X2 @ A ) )
% 14.17/8.00 = ( A = bot_bot_set_o ) ) ).
% 14.17/8.00
% 14.17/8.00 thf(fact_12_all__not__in__conv,axiom,
% 14.17/8.00 ! [A: set_nat] :
% 14.17/8.00 ( ( ! [X2: nat] :
% 14.17/8.00 ~ ( member_nat @ X2 @ A ) )
% 14.17/8.00 = ( A = bot_bot_set_nat ) ) ).
% 14.17/8.00
% 14.17/8.00 thf(fact_13_all__not__in__conv,axiom,
% 14.17/8.00 ! [A: set_int] :
% 14.17/8.00 ( ( ! [X2: int] :
% 14.17/8.00 ~ ( member_int @ X2 @ A ) )
% 14.17/8.00 = ( A = bot_bot_set_int ) ) ).
% 14.17/8.00
% 14.17/8.00 thf(fact_14_Collect__empty__eq,axiom,
% 14.17/8.00 ! [P: ( produc3658429121746597890et_nat > $o ) > $o] :
% 14.17/8.00 ( ( ( collec939566748876313656_nat_o @ P )
% 14.17/8.00 = bot_bo7824918357723582233_nat_o )
% 14.17/8.00 = ( ! [X2: produc3658429121746597890et_nat > $o] :
% 14.17/8.00 ~ ( P @ X2 ) ) ) ).
% 14.17/8.00
% 14.17/8.00 thf(fact_15_Collect__empty__eq,axiom,
% 14.17/8.00 ! [P: product_unit > $o] :
% 14.17/8.00 ( ( ( collect_Product_unit @ P )
% 14.17/8.00 = bot_bo3957492148770167129t_unit )
% 14.17/8.00 = ( ! [X2: product_unit] :
% 14.17/8.00 ~ ( P @ X2 ) ) ) ).
% 14.17/8.00
% 14.17/8.00 thf(fact_16_Collect__empty__eq,axiom,
% 14.17/8.00 ! [P: $o > $o] :
% 14.17/8.00 ( ( ( collect_o @ P )
% 14.17/8.00 = bot_bot_set_o )
% 14.17/8.00 = ( ! [X2: $o] :
% 14.17/8.00 ~ ( P @ X2 ) ) ) ).
% 14.17/8.00
% 14.17/8.00 thf(fact_17_Collect__empty__eq,axiom,
% 14.17/8.00 ! [P: nat > $o] :
% 14.17/8.00 ( ( ( collect_nat @ P )
% 14.17/8.00 = bot_bot_set_nat )
% 14.17/8.00 = ( ! [X2: nat] :
% 14.17/8.00 ~ ( P @ X2 ) ) ) ).
% 14.17/8.00
% 14.17/8.00 thf(fact_18_Collect__empty__eq,axiom,
% 14.17/8.00 ! [P: int > $o] :
% 14.17/8.00 ( ( ( collect_int @ P )
% 14.17/8.00 = bot_bot_set_int )
% 14.17/8.00 = ( ! [X2: int] :
% 14.17/8.00 ~ ( P @ X2 ) ) ) ).
% 14.17/8.00
% 14.17/8.00 thf(fact_19_empty__Collect__eq,axiom,
% 14.17/8.00 ! [P: ( produc3658429121746597890et_nat > $o ) > $o] :
% 14.17/8.00 ( ( bot_bo7824918357723582233_nat_o
% 14.17/8.00 = ( collec939566748876313656_nat_o @ P ) )
% 14.17/8.00 = ( ! [X2: produc3658429121746597890et_nat > $o] :
% 14.17/8.00 ~ ( P @ X2 ) ) ) ).
% 14.17/8.00
% 14.17/8.00 thf(fact_20_empty__Collect__eq,axiom,
% 14.17/8.00 ! [P: product_unit > $o] :
% 14.17/8.00 ( ( bot_bo3957492148770167129t_unit
% 14.17/8.00 = ( collect_Product_unit @ P ) )
% 14.17/8.00 = ( ! [X2: product_unit] :
% 14.17/8.00 ~ ( P @ X2 ) ) ) ).
% 14.17/8.00
% 14.17/8.00 thf(fact_21_empty__Collect__eq,axiom,
% 14.17/8.00 ! [P: $o > $o] :
% 14.17/8.00 ( ( bot_bot_set_o
% 14.17/8.00 = ( collect_o @ P ) )
% 14.17/8.00 = ( ! [X2: $o] :
% 14.17/8.00 ~ ( P @ X2 ) ) ) ).
% 14.17/8.00
% 14.17/8.00 thf(fact_22_empty__Collect__eq,axiom,
% 14.17/8.00 ! [P: nat > $o] :
% 14.17/8.00 ( ( bot_bot_set_nat
% 14.17/8.00 = ( collect_nat @ P ) )
% 14.17/8.00 = ( ! [X2: nat] :
% 14.17/8.00 ~ ( P @ X2 ) ) ) ).
% 14.17/8.00
% 14.17/8.00 thf(fact_23_empty__Collect__eq,axiom,
% 14.17/8.00 ! [P: int > $o] :
% 14.17/8.00 ( ( bot_bot_set_int
% 14.17/8.00 = ( collect_int @ P ) )
% 14.17/8.00 = ( ! [X2: int] :
% 14.17/8.00 ~ ( P @ X2 ) ) ) ).
% 14.17/8.00
% 14.17/8.00 thf(fact_24_emptyE,axiom,
% 14.17/8.00 ! [A2: produc3658429121746597890et_nat > $o] :
% 14.17/8.00 ~ ( member6576561426505652726_nat_o @ A2 @ bot_bo7824918357723582233_nat_o ) ).
% 14.17/8.00
% 14.17/8.00 thf(fact_25_emptyE,axiom,
% 14.17/8.00 ! [A2: product_unit] :
% 14.17/8.00 ~ ( member_Product_unit @ A2 @ bot_bo3957492148770167129t_unit ) ).
% 14.17/8.00
% 14.17/8.00 thf(fact_26_emptyE,axiom,
% 14.17/8.00 ! [A2: $o] :
% 14.17/8.00 ~ ( member_o @ A2 @ bot_bot_set_o ) ).
% 14.17/8.00
% 14.17/8.00 thf(fact_27_emptyE,axiom,
% 14.17/8.00 ! [A2: nat] :
% 14.17/8.00 ~ ( member_nat @ A2 @ bot_bot_set_nat ) ).
% 14.17/8.00
% 14.17/8.00 thf(fact_28_emptyE,axiom,
% 14.17/8.00 ! [A2: int] :
% 14.17/8.00 ~ ( member_int @ A2 @ bot_bot_set_int ) ).
% 14.17/8.00
% 14.17/8.00 thf(fact_29_equals0D,axiom,
% 14.17/8.00 ! [A: set_Product_unit,A2: product_unit] :
% 14.17/8.00 ( ( A = bot_bo3957492148770167129t_unit )
% 14.17/8.00 => ~ ( member_Product_unit @ A2 @ A ) ) ).
% 14.17/8.00
% 14.17/8.00 thf(fact_30_equals0D,axiom,
% 14.17/8.00 ! [A: set_o,A2: $o] :
% 14.17/8.00 ( ( A = bot_bot_set_o )
% 14.17/8.00 => ~ ( member_o @ A2 @ A ) ) ).
% 14.17/8.00
% 14.17/8.00 thf(fact_31_equals0D,axiom,
% 14.17/8.00 ! [A: set_nat,A2: nat] :
% 14.17/8.00 ( ( A = bot_bot_set_nat )
% 14.17/8.00 => ~ ( member_nat @ A2 @ A ) ) ).
% 14.17/8.00
% 14.17/8.00 thf(fact_32_equals0D,axiom,
% 14.17/8.00 ! [A: set_int,A2: int] :
% 14.17/8.00 ( ( A = bot_bot_set_int )
% 14.17/8.00 => ~ ( member_int @ A2 @ A ) ) ).
% 14.17/8.00
% 14.17/8.00 thf(fact_33_equals0D,axiom,
% 14.17/8.00 ! [A: set_Pr4532377907799695533_nat_o,A2: produc3658429121746597890et_nat > $o] :
% 14.17/8.00 ( ( A = bot_bo7824918357723582233_nat_o )
% 14.17/8.00 => ~ ( member6576561426505652726_nat_o @ A2 @ A ) ) ).
% 14.17/8.00
% 14.17/8.00 thf(fact_34_equals0I,axiom,
% 14.17/8.00 ! [A: set_Product_unit] :
% 14.17/8.00 ( ! [Y2: product_unit] :
% 14.17/8.00 ~ ( member_Product_unit @ Y2 @ A )
% 14.17/8.00 => ( A = bot_bo3957492148770167129t_unit ) ) ).
% 14.17/8.00
% 14.17/8.00 thf(fact_35_equals0I,axiom,
% 14.17/8.00 ! [A: set_o] :
% 14.17/8.00 ( ! [Y2: $o] :
% 14.17/8.00 ~ ( member_o @ Y2 @ A )
% 14.17/8.00 => ( A = bot_bot_set_o ) ) ).
% 14.17/8.00
% 14.17/8.00 thf(fact_36_equals0I,axiom,
% 14.17/8.00 ! [A: set_nat] :
% 14.17/8.00 ( ! [Y2: nat] :
% 14.17/8.00 ~ ( member_nat @ Y2 @ A )
% 14.17/8.00 => ( A = bot_bot_set_nat ) ) ).
% 14.17/8.00
% 14.17/8.00 thf(fact_37_equals0I,axiom,
% 14.17/8.00 ! [A: set_int] :
% 14.17/8.00 ( ! [Y2: int] :
% 14.17/8.00 ~ ( member_int @ Y2 @ A )
% 14.17/8.00 => ( A = bot_bot_set_int ) ) ).
% 14.17/8.00
% 14.17/8.00 thf(fact_38_equals0I,axiom,
% 14.17/8.00 ! [A: set_Pr4532377907799695533_nat_o] :
% 14.17/8.00 ( ! [Y2: produc3658429121746597890et_nat > $o] :
% 14.17/8.00 ~ ( member6576561426505652726_nat_o @ Y2 @ A )
% 14.17/8.00 => ( A = bot_bo7824918357723582233_nat_o ) ) ).
% 14.17/8.00
% 14.17/8.00 thf(fact_39_ex__in__conv,axiom,
% 14.17/8.00 ! [A: set_Product_unit] :
% 14.17/8.00 ( ( ? [X2: product_unit] : ( member_Product_unit @ X2 @ A ) )
% 14.17/8.00 = ( A != bot_bo3957492148770167129t_unit ) ) ).
% 14.17/8.00
% 14.17/8.00 thf(fact_40_ex__in__conv,axiom,
% 14.17/8.00 ! [A: set_o] :
% 14.17/8.00 ( ( ? [X2: $o] : ( member_o @ X2 @ A ) )
% 14.17/8.00 = ( A != bot_bot_set_o ) ) ).
% 14.17/8.00
% 14.17/8.00 thf(fact_41_ex__in__conv,axiom,
% 14.17/8.00 ! [A: set_nat] :
% 14.17/8.00 ( ( ? [X2: nat] : ( member_nat @ X2 @ A ) )
% 14.17/8.00 = ( A != bot_bot_set_nat ) ) ).
% 14.17/8.00
% 14.17/8.00 thf(fact_42_ex__in__conv,axiom,
% 14.17/8.00 ! [A: set_int] :
% 14.17/8.00 ( ( ? [X2: int] : ( member_int @ X2 @ A ) )
% 14.17/8.00 = ( A != bot_bot_set_int ) ) ).
% 14.17/8.00
% 14.17/8.00 thf(fact_43_ex__in__conv,axiom,
% 14.17/8.00 ! [A: set_Pr4532377907799695533_nat_o] :
% 14.17/8.00 ( ( ? [X2: produc3658429121746597890et_nat > $o] : ( member6576561426505652726_nat_o @ X2 @ A ) )
% 14.17/8.00 = ( A != bot_bo7824918357723582233_nat_o ) ) ).
% 14.17/8.00
% 14.17/8.00 thf(fact_44_set__notEmptyE,axiom,
% 14.17/8.00 ! [S: set_Product_unit] :
% 14.17/8.00 ( ( S != bot_bo3957492148770167129t_unit )
% 14.17/8.00 => ~ ! [X3: product_unit] :
% 14.17/8.00 ~ ( member_Product_unit @ X3 @ S ) ) ).
% 14.17/8.00
% 14.17/8.00 thf(fact_45_set__notEmptyE,axiom,
% 14.17/8.00 ! [S: set_o] :
% 14.17/8.00 ( ( S != bot_bot_set_o )
% 14.17/8.00 => ~ ! [X3: $o] :
% 14.17/8.00 ~ ( member_o @ X3 @ S ) ) ).
% 14.17/8.00
% 14.17/8.00 thf(fact_46_set__notEmptyE,axiom,
% 14.17/8.00 ! [S: set_nat] :
% 14.17/8.00 ( ( S != bot_bot_set_nat )
% 14.17/8.00 => ~ ! [X3: nat] :
% 14.17/8.00 ~ ( member_nat @ X3 @ S ) ) ).
% 14.17/8.00
% 14.17/8.00 thf(fact_47_set__notEmptyE,axiom,
% 14.17/8.00 ! [S: set_int] :
% 14.17/8.00 ( ( S != bot_bot_set_int )
% 14.17/8.00 => ~ ! [X3: int] :
% 14.17/8.00 ~ ( member_int @ X3 @ S ) ) ).
% 14.17/8.00
% 14.17/8.00 thf(fact_48_set__notEmptyE,axiom,
% 14.17/8.00 ! [S: set_Pr4532377907799695533_nat_o] :
% 14.17/8.00 ( ( S != bot_bo7824918357723582233_nat_o )
% 14.17/8.00 => ~ ! [X3: produc3658429121746597890et_nat > $o] :
% 14.17/8.00 ~ ( member6576561426505652726_nat_o @ X3 @ S ) ) ).
% 14.17/8.00
% 14.17/8.00 thf(fact_49_memb__imp__not__empty,axiom,
% 14.17/8.00 ! [X: product_unit,S: set_Product_unit] :
% 14.17/8.00 ( ( member_Product_unit @ X @ S )
% 14.17/8.00 => ( S != bot_bo3957492148770167129t_unit ) ) ).
% 14.17/8.00
% 14.17/8.00 thf(fact_50_memb__imp__not__empty,axiom,
% 14.17/8.00 ! [X: $o,S: set_o] :
% 14.17/8.00 ( ( member_o @ X @ S )
% 14.17/8.00 => ( S != bot_bot_set_o ) ) ).
% 14.17/8.00
% 14.17/8.00 thf(fact_51_memb__imp__not__empty,axiom,
% 14.17/8.00 ! [X: nat,S: set_nat] :
% 14.17/8.00 ( ( member_nat @ X @ S )
% 14.17/8.00 => ( S != bot_bot_set_nat ) ) ).
% 14.17/8.00
% 14.17/8.00 thf(fact_52_memb__imp__not__empty,axiom,
% 14.17/8.00 ! [X: int,S: set_int] :
% 14.17/8.00 ( ( member_int @ X @ S )
% 14.17/8.00 => ( S != bot_bot_set_int ) ) ).
% 14.17/8.00
% 14.17/8.00 thf(fact_53_memb__imp__not__empty,axiom,
% 14.17/8.00 ! [X: produc3658429121746597890et_nat > $o,S: set_Pr4532377907799695533_nat_o] :
% 14.17/8.00 ( ( member6576561426505652726_nat_o @ X @ S )
% 14.17/8.00 => ( S != bot_bo7824918357723582233_nat_o ) ) ).
% 14.17/8.00
% 14.17/8.00 thf(fact_54_bot__set__def,axiom,
% 14.17/8.00 ( bot_bo3957492148770167129t_unit
% 14.17/8.00 = ( collect_Product_unit @ bot_bo4642748612307482820unit_o ) ) ).
% 14.17/8.00
% 14.17/8.00 thf(fact_55_bot__set__def,axiom,
% 14.17/8.00 ( bot_bot_set_o
% 14.17/8.00 = ( collect_o @ bot_bot_o_o ) ) ).
% 14.17/8.00
% 14.17/8.00 thf(fact_56_bot__set__def,axiom,
% 14.17/8.00 ( bot_bot_set_nat
% 14.17/8.00 = ( collect_nat @ bot_bot_nat_o ) ) ).
% 14.17/8.00
% 14.17/8.00 thf(fact_57_bot__set__def,axiom,
% 14.17/8.00 ( bot_bot_set_int
% 14.17/8.00 = ( collect_int @ bot_bot_int_o ) ) ).
% 14.17/8.00
% 14.17/8.00 thf(fact_58_bot__set__def,axiom,
% 14.17/8.00 ( bot_bo7824918357723582233_nat_o
% 14.17/8.00 = ( collec939566748876313656_nat_o @ bot_bo7963750851167320836at_o_o ) ) ).
% 14.17/8.00
% 14.17/8.00 thf(fact_59_mod__false,axiom,
% 14.17/8.00 ! [H2: produc3658429121746597890et_nat] :
% 14.17/8.00 ~ ( rep_assn @ bot_bot_assn @ H2 ) ).
% 14.17/8.00
% 14.17/8.00 thf(fact_60_Set_Ois__empty__def,axiom,
% 14.17/8.00 ( is_emp5529367068699929823t_unit
% 14.17/8.00 = ( ^ [A3: set_Product_unit] : ( A3 = bot_bo3957492148770167129t_unit ) ) ) ).
% 14.17/8.00
% 14.17/8.00 thf(fact_61_Set_Ois__empty__def,axiom,
% 14.17/8.00 ( is_empty_o
% 14.17/8.00 = ( ^ [A3: set_o] : ( A3 = bot_bot_set_o ) ) ) ).
% 14.17/8.00
% 14.17/8.00 thf(fact_62_Set_Ois__empty__def,axiom,
% 14.17/8.00 ( is_empty_nat
% 14.17/8.00 = ( ^ [A3: set_nat] : ( A3 = bot_bot_set_nat ) ) ) ).
% 14.17/8.00
% 14.17/8.00 thf(fact_63_Set_Ois__empty__def,axiom,
% 14.17/8.00 ( is_empty_int
% 14.17/8.00 = ( ^ [A3: set_int] : ( A3 = bot_bot_set_int ) ) ) ).
% 14.17/8.00
% 14.17/8.00 thf(fact_64_Set_Ois__empty__def,axiom,
% 14.17/8.00 ( is_emp6650082560925436063_nat_o
% 14.17/8.00 = ( ^ [A3: set_Pr4532377907799695533_nat_o] : ( A3 = bot_bo7824918357723582233_nat_o ) ) ) ).
% 14.17/8.00
% 14.17/8.00 thf(fact_65_Collect__empty__eq__bot,axiom,
% 14.17/8.00 ! [P: product_unit > $o] :
% 14.17/8.00 ( ( ( collect_Product_unit @ P )
% 14.17/8.00 = bot_bo3957492148770167129t_unit )
% 14.17/8.00 = ( P = bot_bo4642748612307482820unit_o ) ) ).
% 14.17/8.00
% 14.17/8.00 thf(fact_66_Collect__empty__eq__bot,axiom,
% 14.17/8.00 ! [P: $o > $o] :
% 14.17/8.00 ( ( ( collect_o @ P )
% 14.17/8.00 = bot_bot_set_o )
% 14.17/8.00 = ( P = bot_bot_o_o ) ) ).
% 14.17/8.00
% 14.17/8.00 thf(fact_67_Collect__empty__eq__bot,axiom,
% 14.17/8.00 ! [P: nat > $o] :
% 14.17/8.00 ( ( ( collect_nat @ P )
% 14.17/8.00 = bot_bot_set_nat )
% 14.17/8.00 = ( P = bot_bot_nat_o ) ) ).
% 14.17/8.00
% 14.17/8.00 thf(fact_68_Collect__empty__eq__bot,axiom,
% 14.17/8.00 ! [P: int > $o] :
% 14.17/8.00 ( ( ( collect_int @ P )
% 14.17/8.00 = bot_bot_set_int )
% 14.17/8.00 = ( P = bot_bot_int_o ) ) ).
% 14.17/8.00
% 14.17/8.00 thf(fact_69_Collect__empty__eq__bot,axiom,
% 14.17/8.00 ! [P: ( produc3658429121746597890et_nat > $o ) > $o] :
% 14.17/8.00 ( ( ( collec939566748876313656_nat_o @ P )
% 14.17/8.00 = bot_bo7824918357723582233_nat_o )
% 14.17/8.00 = ( P = bot_bo7963750851167320836at_o_o ) ) ).
% 14.17/8.00
% 14.17/8.00 thf(fact_70_bot__empty__eq,axiom,
% 14.17/8.00 ( bot_bo4642748612307482820unit_o
% 14.17/8.00 = ( ^ [X2: product_unit] : ( member_Product_unit @ X2 @ bot_bo3957492148770167129t_unit ) ) ) ).
% 14.17/8.00
% 14.17/8.00 thf(fact_71_bot__empty__eq,axiom,
% 14.17/8.00 ( bot_bot_o_o
% 14.17/8.00 = ( ^ [X2: $o] : ( member_o @ X2 @ bot_bot_set_o ) ) ) ).
% 14.17/8.00
% 14.17/8.00 thf(fact_72_bot__empty__eq,axiom,
% 14.17/8.00 ( bot_bot_nat_o
% 14.17/8.00 = ( ^ [X2: nat] : ( member_nat @ X2 @ bot_bot_set_nat ) ) ) ).
% 14.17/8.00
% 14.17/8.00 thf(fact_73_bot__empty__eq,axiom,
% 14.17/8.00 ( bot_bot_int_o
% 14.17/8.00 = ( ^ [X2: int] : ( member_int @ X2 @ bot_bot_set_int ) ) ) ).
% 14.17/8.00
% 14.17/8.00 thf(fact_74_bot__empty__eq,axiom,
% 14.17/8.00 ( bot_bo7963750851167320836at_o_o
% 14.17/8.00 = ( ^ [X2: produc3658429121746597890et_nat > $o] : ( member6576561426505652726_nat_o @ X2 @ bot_bo7824918357723582233_nat_o ) ) ) ).
% 14.17/8.00
% 14.17/8.00 thf(fact_75_wand__assn__def,axiom,
% 14.17/8.00 ( wand_assn
% 14.17/8.00 = ( ^ [P2: assn,Q: assn] : ( abs_assn @ ( wand_raw @ ( rep_assn @ P2 ) @ ( rep_assn @ Q ) ) ) ) ) ).
% 14.17/8.00
% 14.17/8.00 thf(fact_76_one__assn__def,axiom,
% 14.17/8.00 ( one_one_assn
% 14.17/8.00 = ( abs_assn @ one_assn_raw ) ) ).
% 14.17/8.00
% 14.17/8.00 thf(fact_77_Abs__assn__inverse,axiom,
% 14.17/8.00 ! [Y: produc3658429121746597890et_nat > $o] :
% 14.17/8.00 ( ( member6576561426505652726_nat_o @ Y @ ( collec939566748876313656_nat_o @ proper ) )
% 14.17/8.00 => ( ( rep_assn @ ( abs_assn @ Y ) )
% 14.17/8.00 = Y ) ) ).
% 14.17/8.00
% 14.17/8.00 thf(fact_78_one__assn__raw_Osimps,axiom,
% 14.17/8.00 ! [H2: heap_e7401611519738050253t_unit,As: set_nat] :
% 14.17/8.00 ( ( one_assn_raw @ ( produc7507926704131184380et_nat @ H2 @ As ) )
% 14.17/8.00 = ( As = bot_bot_set_nat ) ) ).
% 14.17/8.00
% 14.17/8.00 thf(fact_79_one__assn__raw_Oelims_I1_J,axiom,
% 14.17/8.00 ! [X: produc3658429121746597890et_nat,Y: $o] :
% 14.17/8.00 ( ( ( one_assn_raw @ X )
% 14.17/8.00 = Y )
% 14.17/8.00 => ~ ! [H: heap_e7401611519738050253t_unit,As2: set_nat] :
% 14.17/8.00 ( ( X
% 14.17/8.00 = ( produc7507926704131184380et_nat @ H @ As2 ) )
% 14.17/8.00 => ( Y
% 14.17/8.00 = ( As2 != bot_bot_set_nat ) ) ) ) ).
% 14.17/8.00
% 14.17/8.00 thf(fact_80_one__assn__raw_Oelims_I2_J,axiom,
% 14.17/8.00 ! [X: produc3658429121746597890et_nat] :
% 14.17/8.00 ( ( one_assn_raw @ X )
% 14.17/8.00 => ~ ! [H: heap_e7401611519738050253t_unit,As2: set_nat] :
% 14.17/8.00 ( ( X
% 14.17/8.00 = ( produc7507926704131184380et_nat @ H @ As2 ) )
% 14.17/8.00 => ( As2 != bot_bot_set_nat ) ) ) ).
% 14.17/8.00
% 14.17/8.00 thf(fact_81_one__assn__raw_Oelims_I3_J,axiom,
% 14.17/8.00 ! [X: produc3658429121746597890et_nat] :
% 14.17/8.00 ( ~ ( one_assn_raw @ X )
% 14.17/8.00 => ~ ! [H: heap_e7401611519738050253t_unit,As2: set_nat] :
% 14.17/8.00 ( ( X
% 14.17/8.00 = ( produc7507926704131184380et_nat @ H @ As2 ) )
% 14.17/8.00 => ( As2 = bot_bot_set_nat ) ) ) ).
% 14.17/8.00
% 14.17/8.00 thf(fact_82_is__singletonI_H,axiom,
% 14.17/8.00 ! [A: set_Product_unit] :
% 14.17/8.00 ( ( A != bot_bo3957492148770167129t_unit )
% 14.17/8.00 => ( ! [X3: product_unit,Y2: product_unit] :
% 14.17/8.00 ( ( member_Product_unit @ X3 @ A )
% 14.17/8.00 => ( ( member_Product_unit @ Y2 @ A )
% 14.17/8.00 => ( X3 = Y2 ) ) )
% 14.17/8.00 => ( is_sin2160648248035936513t_unit @ A ) ) ) ).
% 14.17/8.00
% 14.17/8.00 thf(fact_83_is__singletonI_H,axiom,
% 14.17/8.00 ! [A: set_o] :
% 14.17/8.00 ( ( A != bot_bot_set_o )
% 14.17/8.00 => ( ! [X3: $o,Y2: $o] :
% 14.17/8.00 ( ( member_o @ X3 @ A )
% 14.17/8.00 => ( ( member_o @ Y2 @ A )
% 14.17/8.00 => ( X3 = Y2 ) ) )
% 14.17/8.00 => ( is_singleton_o @ A ) ) ) ).
% 14.17/8.00
% 14.17/8.00 thf(fact_84_is__singletonI_H,axiom,
% 14.17/8.00 ! [A: set_nat] :
% 14.17/8.00 ( ( A != bot_bot_set_nat )
% 14.17/8.00 => ( ! [X3: nat,Y2: nat] :
% 14.17/8.00 ( ( member_nat @ X3 @ A )
% 14.17/8.00 => ( ( member_nat @ Y2 @ A )
% 14.17/8.00 => ( X3 = Y2 ) ) )
% 14.17/8.00 => ( is_singleton_nat @ A ) ) ) ).
% 14.17/8.00
% 14.17/8.00 thf(fact_85_is__singletonI_H,axiom,
% 14.17/8.00 ! [A: set_int] :
% 14.17/8.00 ( ( A != bot_bot_set_int )
% 14.17/8.00 => ( ! [X3: int,Y2: int] :
% 14.17/8.00 ( ( member_int @ X3 @ A )
% 14.17/8.00 => ( ( member_int @ Y2 @ A )
% 14.17/8.00 => ( X3 = Y2 ) ) )
% 14.17/8.00 => ( is_singleton_int @ A ) ) ) ).
% 14.17/8.00
% 14.17/8.00 thf(fact_86_is__singletonI_H,axiom,
% 14.17/8.00 ! [A: set_Pr4532377907799695533_nat_o] :
% 14.17/8.00 ( ( A != bot_bo7824918357723582233_nat_o )
% 14.17/8.00 => ( ! [X3: produc3658429121746597890et_nat > $o,Y2: produc3658429121746597890et_nat > $o] :
% 14.17/8.00 ( ( member6576561426505652726_nat_o @ X3 @ A )
% 14.17/8.00 => ( ( member6576561426505652726_nat_o @ Y2 @ A )
% 14.17/8.00 => ( X3 = Y2 ) ) )
% 14.17/8.00 => ( is_sin5180296473474724033_nat_o @ A ) ) ) ).
% 14.17/8.00
% 14.17/8.00 thf(fact_87_wand__proper,axiom,
% 14.17/8.00 ! [P: produc3658429121746597890et_nat > $o,Q2: produc3658429121746597890et_nat > $o] : ( proper @ ( wand_raw @ P @ Q2 ) ) ).
% 14.17/8.00
% 14.17/8.00 thf(fact_88_one__assn__proper,axiom,
% 14.17/8.00 proper @ one_assn_raw ).
% 14.17/8.00
% 14.17/8.00 thf(fact_89_assn__basic__inequalities_I3_J,axiom,
% 14.17/8.00 bot_bot_assn != one_one_assn ).
% 14.17/8.00
% 14.17/8.00 thf(fact_90_bex2I,axiom,
% 14.17/8.00 ! [A2: heap_e7401611519738050253t_unit,B: set_nat,S: set_Pr3948176798113811640et_nat,P: heap_e7401611519738050253t_unit > set_nat > $o] :
% 14.17/8.00 ( ( member6260224972018164377et_nat @ ( produc7507926704131184380et_nat @ A2 @ B ) @ S )
% 14.17/8.00 => ( ( ( member6260224972018164377et_nat @ ( produc7507926704131184380et_nat @ A2 @ B ) @ S )
% 14.17/8.00 => ( P @ A2 @ B ) )
% 14.17/8.00 => ? [A4: heap_e7401611519738050253t_unit,B2: set_nat] :
% 14.17/8.00 ( ( member6260224972018164377et_nat @ ( produc7507926704131184380et_nat @ A4 @ B2 ) @ S )
% 14.17/8.00 & ( P @ A4 @ B2 ) ) ) ) ).
% 14.17/8.00
% 14.17/8.00 thf(fact_91_bex2I,axiom,
% 14.17/8.00 ! [A2: nat,B: nat,S: set_Pr1261947904930325089at_nat,P: nat > nat > $o] :
% 14.17/8.00 ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A2 @ B ) @ S )
% 14.17/8.00 => ( ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A2 @ B ) @ S )
% 14.17/8.00 => ( P @ A2 @ B ) )
% 14.17/8.00 => ? [A4: nat,B2: nat] :
% 14.17/8.00 ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A4 @ B2 ) @ S )
% 14.17/8.00 & ( P @ A4 @ B2 ) ) ) ) ).
% 14.17/8.00
% 14.17/8.00 thf(fact_92_bex2I,axiom,
% 14.17/8.00 ! [A2: int,B: int,S: set_Pr958786334691620121nt_int,P: int > int > $o] :
% 14.17/8.00 ( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ A2 @ B ) @ S )
% 14.17/8.00 => ( ( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ A2 @ B ) @ S )
% 14.17/8.00 => ( P @ A2 @ B ) )
% 14.17/8.00 => ? [A4: int,B2: int] :
% 14.17/8.00 ( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ A4 @ B2 ) @ S )
% 14.17/8.00 & ( P @ A4 @ B2 ) ) ) ) ).
% 14.17/8.00
% 14.17/8.00 thf(fact_93_bex2I,axiom,
% 14.17/8.00 ! [A2: produc3658429121746597890et_nat > $o,B: produc3658429121746597890et_nat,S: set_Pr3286484037609594932et_nat,P: ( produc3658429121746597890et_nat > $o ) > produc3658429121746597890et_nat > $o] :
% 14.17/8.00 ( ( member1996754912294343701et_nat @ ( produc5001842942810119800et_nat @ A2 @ B ) @ S )
% 14.17/8.00 => ( ( ( member1996754912294343701et_nat @ ( produc5001842942810119800et_nat @ A2 @ B ) @ S )
% 14.17/8.00 => ( P @ A2 @ B ) )
% 14.17/8.00 => ? [A4: produc3658429121746597890et_nat > $o,B2: produc3658429121746597890et_nat] :
% 14.17/8.00 ( ( member1996754912294343701et_nat @ ( produc5001842942810119800et_nat @ A4 @ B2 ) @ S )
% 14.17/8.00 & ( P @ A4 @ B2 ) ) ) ) ).
% 14.17/8.00
% 14.17/8.00 thf(fact_94_bex2I,axiom,
% 14.17/8.00 ! [A2: produc3658429121746597890et_nat > $o,B: produc3925858234332021118et_nat,S: set_Pr8536935166611901872et_nat,P: ( produc3658429121746597890et_nat > $o ) > produc3925858234332021118et_nat > $o] :
% 14.17/8.00 ( ( member6124377750444531601et_nat @ ( produc2245416461498447860et_nat @ A2 @ B ) @ S )
% 14.17/8.00 => ( ( ( member6124377750444531601et_nat @ ( produc2245416461498447860et_nat @ A2 @ B ) @ S )
% 14.17/8.00 => ( P @ A2 @ B ) )
% 14.17/8.00 => ? [A4: produc3658429121746597890et_nat > $o,B2: produc3925858234332021118et_nat] :
% 14.17/8.00 ( ( member6124377750444531601et_nat @ ( produc2245416461498447860et_nat @ A4 @ B2 ) @ S )
% 14.17/8.00 & ( P @ A4 @ B2 ) ) ) ) ).
% 14.17/8.00
% 14.17/8.00 thf(fact_95_one__assn__raw_Ocases,axiom,
% 14.17/8.00 ! [X: produc3658429121746597890et_nat] :
% 14.17/8.00 ~ ! [H: heap_e7401611519738050253t_unit,As2: set_nat] :
% 14.17/8.00 ( X
% 14.17/8.00 != ( produc7507926704131184380et_nat @ H @ As2 ) ) ).
% 14.17/8.00
% 14.17/8.00 thf(fact_96_sndE,axiom,
% 14.17/8.00 ! [X: product_prod_nat_nat,A2: nat,B: nat,P: nat > $o] :
% 14.17/8.00 ( ( X
% 14.17/8.00 = ( product_Pair_nat_nat @ A2 @ B ) )
% 14.17/8.00 => ( ( P @ ( product_snd_nat_nat @ X ) )
% 14.17/8.00 => ( P @ B ) ) ) ).
% 14.17/8.00
% 14.17/8.00 thf(fact_97_sndE,axiom,
% 14.17/8.00 ! [X: produc3925858234332021118et_nat,A2: produc3658429121746597890et_nat > $o,B: produc3658429121746597890et_nat,P: produc3658429121746597890et_nat > $o] :
% 14.17/8.00 ( ( X
% 14.17/8.00 = ( produc5001842942810119800et_nat @ A2 @ B ) )
% 14.17/8.00 => ( ( P @ ( produc4011572625026189258et_nat @ X ) )
% 14.17/8.00 => ( P @ B ) ) ) ).
% 14.17/8.00
% 14.17/8.00 thf(fact_98_sndE,axiom,
% 14.17/8.00 ! [X: produc2732055786443039994et_nat,A2: produc3658429121746597890et_nat > $o,B: produc3925858234332021118et_nat,P: produc3925858234332021118et_nat > $o] :
% 14.17/8.00 ( ( X
% 14.17/8.00 = ( produc2245416461498447860et_nat @ A2 @ B ) )
% 14.17/8.00 => ( ( P @ ( produc5374455773327741254et_nat @ X ) )
% 14.17/8.00 => ( P @ B ) ) ) ).
% 14.17/8.00
% 14.17/8.00 thf(fact_99_sndE,axiom,
% 14.17/8.00 ! [X: produc3658429121746597890et_nat,A2: heap_e7401611519738050253t_unit,B: set_nat,P: set_nat > $o] :
% 14.17/8.00 ( ( X
% 14.17/8.00 = ( produc7507926704131184380et_nat @ A2 @ B ) )
% 14.17/8.00 => ( ( P @ ( produc8586169260539613262et_nat @ X ) )
% 14.17/8.00 => ( P @ B ) ) ) ).
% 14.17/8.00
% 14.17/8.00 thf(fact_100_sndE,axiom,
% 14.17/8.00 ! [X: product_prod_int_int,A2: int,B: int,P: int > $o] :
% 14.17/8.00 ( ( X
% 14.17/8.00 = ( product_Pair_int_int @ A2 @ B ) )
% 14.17/8.00 => ( ( P @ ( product_snd_int_int @ X ) )
% 14.17/8.00 => ( P @ B ) ) ) ).
% 14.17/8.00
% 14.17/8.00 thf(fact_101_Rep__assn,axiom,
% 14.17/8.00 ! [X: assn] : ( member6576561426505652726_nat_o @ ( rep_assn @ X ) @ ( collec939566748876313656_nat_o @ proper ) ) ).
% 14.17/8.00
% 14.17/8.00 thf(fact_102_mem__Collect__eq,axiom,
% 14.17/8.00 ! [A2: produc3658429121746597890et_nat > $o,P: ( produc3658429121746597890et_nat > $o ) > $o] :
% 14.17/8.00 ( ( member6576561426505652726_nat_o @ A2 @ ( collec939566748876313656_nat_o @ P ) )
% 14.17/8.00 = ( P @ A2 ) ) ).
% 14.17/8.00
% 14.17/8.00 thf(fact_103_mem__Collect__eq,axiom,
% 14.17/8.00 ! [A2: nat,P: nat > $o] :
% 14.17/8.00 ( ( member_nat @ A2 @ ( collect_nat @ P ) )
% 14.17/8.00 = ( P @ A2 ) ) ).
% 14.17/8.00
% 14.17/8.00 thf(fact_104_mem__Collect__eq,axiom,
% 14.17/8.00 ! [A2: int,P: int > $o] :
% 14.17/8.00 ( ( member_int @ A2 @ ( collect_int @ P ) )
% 14.17/8.00 = ( P @ A2 ) ) ).
% 14.17/8.00
% 14.17/8.00 thf(fact_105_mem__Collect__eq,axiom,
% 14.17/8.00 ! [A2: $o,P: $o > $o] :
% 14.17/8.00 ( ( member_o @ A2 @ ( collect_o @ P ) )
% 14.17/8.00 = ( P @ A2 ) ) ).
% 14.17/8.00
% 14.17/8.00 thf(fact_106_mem__Collect__eq,axiom,
% 14.17/8.00 ! [A2: product_unit,P: product_unit > $o] :
% 14.17/8.00 ( ( member_Product_unit @ A2 @ ( collect_Product_unit @ P ) )
% 14.17/8.00 = ( P @ A2 ) ) ).
% 14.17/8.00
% 14.17/8.00 thf(fact_107_Collect__mem__eq,axiom,
% 14.17/8.00 ! [A: set_Pr4532377907799695533_nat_o] :
% 14.17/8.00 ( ( collec939566748876313656_nat_o
% 14.17/8.00 @ ^ [X2: produc3658429121746597890et_nat > $o] : ( member6576561426505652726_nat_o @ X2 @ A ) )
% 14.17/8.00 = A ) ).
% 14.17/8.00
% 14.17/8.00 thf(fact_108_Collect__mem__eq,axiom,
% 14.17/8.00 ! [A: set_nat] :
% 14.17/8.00 ( ( collect_nat
% 14.17/8.00 @ ^ [X2: nat] : ( member_nat @ X2 @ A ) )
% 14.17/8.00 = A ) ).
% 14.17/8.00
% 14.17/8.00 thf(fact_109_Collect__mem__eq,axiom,
% 14.17/8.00 ! [A: set_int] :
% 14.17/8.00 ( ( collect_int
% 14.17/8.00 @ ^ [X2: int] : ( member_int @ X2 @ A ) )
% 14.17/8.00 = A ) ).
% 14.17/8.00
% 14.17/8.00 thf(fact_110_Collect__mem__eq,axiom,
% 14.17/8.00 ! [A: set_o] :
% 14.17/8.00 ( ( collect_o
% 14.17/8.00 @ ^ [X2: $o] : ( member_o @ X2 @ A ) )
% 14.17/8.00 = A ) ).
% 14.17/8.00
% 14.17/8.00 thf(fact_111_Collect__mem__eq,axiom,
% 14.17/8.00 ! [A: set_Product_unit] :
% 14.17/8.00 ( ( collect_Product_unit
% 14.17/8.00 @ ^ [X2: product_unit] : ( member_Product_unit @ X2 @ A ) )
% 14.17/8.00 = A ) ).
% 14.17/8.00
% 14.17/8.00 thf(fact_112_Collect__cong,axiom,
% 14.17/8.00 ! [P: ( produc3658429121746597890et_nat > $o ) > $o,Q2: ( produc3658429121746597890et_nat > $o ) > $o] :
% 14.17/8.00 ( ! [X3: produc3658429121746597890et_nat > $o] :
% 14.17/8.00 ( ( P @ X3 )
% 14.17/8.00 = ( Q2 @ X3 ) )
% 14.17/8.00 => ( ( collec939566748876313656_nat_o @ P )
% 14.17/8.00 = ( collec939566748876313656_nat_o @ Q2 ) ) ) ).
% 14.17/8.00
% 14.17/8.00 thf(fact_113_Collect__cong,axiom,
% 14.17/8.00 ! [P: nat > $o,Q2: nat > $o] :
% 14.17/8.00 ( ! [X3: nat] :
% 14.17/8.00 ( ( P @ X3 )
% 14.17/8.00 = ( Q2 @ X3 ) )
% 14.17/8.00 => ( ( collect_nat @ P )
% 14.17/8.00 = ( collect_nat @ Q2 ) ) ) ).
% 14.17/8.00
% 14.17/8.00 thf(fact_114_Collect__cong,axiom,
% 14.17/8.00 ! [P: int > $o,Q2: int > $o] :
% 14.17/8.00 ( ! [X3: int] :
% 14.17/8.00 ( ( P @ X3 )
% 14.17/8.00 = ( Q2 @ X3 ) )
% 14.17/8.00 => ( ( collect_int @ P )
% 14.17/8.00 = ( collect_int @ Q2 ) ) ) ).
% 14.17/8.00
% 14.17/8.00 thf(fact_115_Collect__cong,axiom,
% 14.17/8.00 ! [P: $o > $o,Q2: $o > $o] :
% 14.17/8.00 ( ! [X3: $o] :
% 14.17/8.00 ( ( P @ X3 )
% 14.17/8.00 = ( Q2 @ X3 ) )
% 14.17/8.00 => ( ( collect_o @ P )
% 14.17/8.00 = ( collect_o @ Q2 ) ) ) ).
% 14.17/8.00
% 14.17/8.00 thf(fact_116_Collect__cong,axiom,
% 14.17/8.00 ! [P: product_unit > $o,Q2: product_unit > $o] :
% 14.17/8.00 ( ! [X3: product_unit] :
% 14.17/8.00 ( ( P @ X3 )
% 14.17/8.00 = ( Q2 @ X3 ) )
% 14.17/8.00 => ( ( collect_Product_unit @ P )
% 14.17/8.00 = ( collect_Product_unit @ Q2 ) ) ) ).
% 14.17/8.00
% 14.17/8.00 thf(fact_117_Rep__assn__cases,axiom,
% 14.17/8.00 ! [Y: produc3658429121746597890et_nat > $o] :
% 14.17/8.00 ( ( member6576561426505652726_nat_o @ Y @ ( collec939566748876313656_nat_o @ proper ) )
% 14.17/8.00 => ~ ! [X3: assn] :
% 14.17/8.00 ( Y
% 14.17/8.00 != ( rep_assn @ X3 ) ) ) ).
% 14.17/8.00
% 14.17/8.00 thf(fact_118_Rep__assn__induct,axiom,
% 14.17/8.00 ! [Y: produc3658429121746597890et_nat > $o,P: ( produc3658429121746597890et_nat > $o ) > $o] :
% 14.17/8.00 ( ( member6576561426505652726_nat_o @ Y @ ( collec939566748876313656_nat_o @ proper ) )
% 14.17/8.00 => ( ! [X3: assn] : ( P @ ( rep_assn @ X3 ) )
% 14.17/8.00 => ( P @ Y ) ) ) ).
% 14.17/8.00
% 14.17/8.00 thf(fact_119_Abs__assn__cases,axiom,
% 14.17/8.00 ! [X: assn] :
% 14.17/8.00 ~ ! [Y2: produc3658429121746597890et_nat > $o] :
% 14.17/8.00 ( ( X
% 14.17/8.00 = ( abs_assn @ Y2 ) )
% 14.17/8.00 => ~ ( member6576561426505652726_nat_o @ Y2 @ ( collec939566748876313656_nat_o @ proper ) ) ) ).
% 14.17/8.00
% 14.17/8.00 thf(fact_120_Abs__assn__induct,axiom,
% 14.17/8.00 ! [P: assn > $o,X: assn] :
% 14.17/8.00 ( ! [Y2: produc3658429121746597890et_nat > $o] :
% 14.17/8.00 ( ( member6576561426505652726_nat_o @ Y2 @ ( collec939566748876313656_nat_o @ proper ) )
% 14.17/8.00 => ( P @ ( abs_assn @ Y2 ) ) )
% 14.17/8.00 => ( P @ X ) ) ).
% 14.17/8.00
% 14.17/8.00 thf(fact_121_Abs__assn__inject,axiom,
% 14.17/8.00 ! [X: produc3658429121746597890et_nat > $o,Y: produc3658429121746597890et_nat > $o] :
% 14.17/8.00 ( ( member6576561426505652726_nat_o @ X @ ( collec939566748876313656_nat_o @ proper ) )
% 14.17/8.00 => ( ( member6576561426505652726_nat_o @ Y @ ( collec939566748876313656_nat_o @ proper ) )
% 14.17/8.00 => ( ( ( abs_assn @ X )
% 14.17/8.00 = ( abs_assn @ Y ) )
% 14.17/8.00 = ( X = Y ) ) ) ) ).
% 14.17/8.00
% 14.17/8.00 thf(fact_122_mod__h__bot__indep,axiom,
% 14.17/8.00 ! [P: assn,H2: heap_e7401611519738050253t_unit,H3: heap_e7401611519738050253t_unit] :
% 14.17/8.00 ( ( rep_assn @ P @ ( produc7507926704131184380et_nat @ H2 @ bot_bot_set_nat ) )
% 14.17/8.00 = ( rep_assn @ P @ ( produc7507926704131184380et_nat @ H3 @ bot_bot_set_nat ) ) ) ).
% 14.17/8.00
% 14.17/8.00 thf(fact_123_old_Oprod_Oinject,axiom,
% 14.17/8.00 ! [A2: heap_e7401611519738050253t_unit,B: set_nat,A5: heap_e7401611519738050253t_unit,B3: set_nat] :
% 14.17/8.00 ( ( ( produc7507926704131184380et_nat @ A2 @ B )
% 14.17/8.00 = ( produc7507926704131184380et_nat @ A5 @ B3 ) )
% 14.17/8.00 = ( ( A2 = A5 )
% 14.17/8.00 & ( B = B3 ) ) ) ).
% 14.17/8.00
% 14.17/8.00 thf(fact_124_old_Oprod_Oinject,axiom,
% 14.17/8.00 ! [A2: nat,B: nat,A5: nat,B3: nat] :
% 14.17/8.00 ( ( ( product_Pair_nat_nat @ A2 @ B )
% 14.17/8.00 = ( product_Pair_nat_nat @ A5 @ B3 ) )
% 14.17/8.00 = ( ( A2 = A5 )
% 14.17/8.00 & ( B = B3 ) ) ) ).
% 14.17/8.00
% 14.17/8.00 thf(fact_125_old_Oprod_Oinject,axiom,
% 14.17/8.00 ! [A2: int,B: int,A5: int,B3: int] :
% 14.17/8.00 ( ( ( product_Pair_int_int @ A2 @ B )
% 14.17/8.00 = ( product_Pair_int_int @ A5 @ B3 ) )
% 14.17/8.00 = ( ( A2 = A5 )
% 14.17/8.00 & ( B = B3 ) ) ) ).
% 14.17/8.00
% 14.17/8.00 thf(fact_126_old_Oprod_Oinject,axiom,
% 14.17/8.00 ! [A2: produc3658429121746597890et_nat > $o,B: produc3658429121746597890et_nat,A5: produc3658429121746597890et_nat > $o,B3: produc3658429121746597890et_nat] :
% 14.17/8.00 ( ( ( produc5001842942810119800et_nat @ A2 @ B )
% 14.17/8.00 = ( produc5001842942810119800et_nat @ A5 @ B3 ) )
% 14.17/8.00 = ( ( A2 = A5 )
% 14.17/8.00 & ( B = B3 ) ) ) ).
% 14.17/8.00
% 14.17/8.00 thf(fact_127_old_Oprod_Oinject,axiom,
% 14.17/8.00 ! [A2: produc3658429121746597890et_nat > $o,B: produc3925858234332021118et_nat,A5: produc3658429121746597890et_nat > $o,B3: produc3925858234332021118et_nat] :
% 14.17/8.00 ( ( ( produc2245416461498447860et_nat @ A2 @ B )
% 14.17/8.00 = ( produc2245416461498447860et_nat @ A5 @ B3 ) )
% 14.17/8.00 = ( ( A2 = A5 )
% 14.17/8.00 & ( B = B3 ) ) ) ).
% 14.17/8.00
% 14.17/8.00 thf(fact_128_prod_Oinject,axiom,
% 14.17/8.00 ! [X1: heap_e7401611519738050253t_unit,X22: set_nat,Y1: heap_e7401611519738050253t_unit,Y22: set_nat] :
% 14.17/8.00 ( ( ( produc7507926704131184380et_nat @ X1 @ X22 )
% 14.17/8.00 = ( produc7507926704131184380et_nat @ Y1 @ Y22 ) )
% 14.17/8.00 = ( ( X1 = Y1 )
% 14.17/8.00 & ( X22 = Y22 ) ) ) ).
% 14.17/8.00
% 14.17/8.00 thf(fact_129_prod_Oinject,axiom,
% 14.17/8.00 ! [X1: nat,X22: nat,Y1: nat,Y22: nat] :
% 14.17/8.00 ( ( ( product_Pair_nat_nat @ X1 @ X22 )
% 14.17/8.00 = ( product_Pair_nat_nat @ Y1 @ Y22 ) )
% 14.17/8.00 = ( ( X1 = Y1 )
% 14.17/8.00 & ( X22 = Y22 ) ) ) ).
% 14.17/8.00
% 14.17/8.00 thf(fact_130_prod_Oinject,axiom,
% 14.17/8.00 ! [X1: int,X22: int,Y1: int,Y22: int] :
% 14.17/8.00 ( ( ( product_Pair_int_int @ X1 @ X22 )
% 14.17/8.00 = ( product_Pair_int_int @ Y1 @ Y22 ) )
% 14.17/8.00 = ( ( X1 = Y1 )
% 14.17/8.00 & ( X22 = Y22 ) ) ) ).
% 14.17/8.00
% 14.17/8.00 thf(fact_131_prod_Oinject,axiom,
% 14.17/8.00 ! [X1: produc3658429121746597890et_nat > $o,X22: produc3658429121746597890et_nat,Y1: produc3658429121746597890et_nat > $o,Y22: produc3658429121746597890et_nat] :
% 14.17/8.00 ( ( ( produc5001842942810119800et_nat @ X1 @ X22 )
% 14.17/8.00 = ( produc5001842942810119800et_nat @ Y1 @ Y22 ) )
% 14.17/8.00 = ( ( X1 = Y1 )
% 14.17/8.00 & ( X22 = Y22 ) ) ) ).
% 14.17/8.00
% 14.17/8.00 thf(fact_132_prod_Oinject,axiom,
% 14.17/8.00 ! [X1: produc3658429121746597890et_nat > $o,X22: produc3925858234332021118et_nat,Y1: produc3658429121746597890et_nat > $o,Y22: produc3925858234332021118et_nat] :
% 14.17/8.00 ( ( ( produc2245416461498447860et_nat @ X1 @ X22 )
% 14.17/8.00 = ( produc2245416461498447860et_nat @ Y1 @ Y22 ) )
% 14.17/8.00 = ( ( X1 = Y1 )
% 14.17/8.00 & ( X22 = Y22 ) ) ) ).
% 14.17/8.00
% 14.17/8.00 thf(fact_133_sndI,axiom,
% 14.17/8.00 ! [X: product_prod_nat_nat,Y: nat,Z: nat] :
% 14.17/8.00 ( ( X
% 14.17/8.00 = ( product_Pair_nat_nat @ Y @ Z ) )
% 14.17/8.00 => ( ( product_snd_nat_nat @ X )
% 14.17/8.00 = Z ) ) ).
% 14.17/8.00
% 14.17/8.00 thf(fact_134_sndI,axiom,
% 14.17/8.00 ! [X: produc3925858234332021118et_nat,Y: produc3658429121746597890et_nat > $o,Z: produc3658429121746597890et_nat] :
% 14.17/8.00 ( ( X
% 14.17/8.00 = ( produc5001842942810119800et_nat @ Y @ Z ) )
% 14.17/8.00 => ( ( produc4011572625026189258et_nat @ X )
% 14.17/8.00 = Z ) ) ).
% 14.17/8.00
% 14.17/8.00 thf(fact_135_sndI,axiom,
% 14.17/8.00 ! [X: produc2732055786443039994et_nat,Y: produc3658429121746597890et_nat > $o,Z: produc3925858234332021118et_nat] :
% 14.17/8.00 ( ( X
% 14.17/8.00 = ( produc2245416461498447860et_nat @ Y @ Z ) )
% 14.17/8.00 => ( ( produc5374455773327741254et_nat @ X )
% 14.17/8.00 = Z ) ) ).
% 14.17/8.00
% 14.17/8.00 thf(fact_136_sndI,axiom,
% 14.17/8.00 ! [X: produc3658429121746597890et_nat,Y: heap_e7401611519738050253t_unit,Z: set_nat] :
% 14.17/8.00 ( ( X
% 14.17/8.00 = ( produc7507926704131184380et_nat @ Y @ Z ) )
% 14.17/8.00 => ( ( produc8586169260539613262et_nat @ X )
% 14.17/8.00 = Z ) ) ).
% 14.17/8.00
% 14.17/8.00 thf(fact_137_sndI,axiom,
% 14.17/8.00 ! [X: product_prod_int_int,Y: int,Z: int] :
% 14.17/8.00 ( ( X
% 14.17/8.00 = ( product_Pair_int_int @ Y @ Z ) )
% 14.17/8.00 => ( ( product_snd_int_int @ X )
% 14.17/8.00 = Z ) ) ).
% 14.17/8.00
% 14.17/8.00 thf(fact_138_snd__eqD,axiom,
% 14.17/8.00 ! [X: nat,Y: nat,A2: nat] :
% 14.17/8.00 ( ( ( product_snd_nat_nat @ ( product_Pair_nat_nat @ X @ Y ) )
% 14.17/8.00 = A2 )
% 14.17/8.00 => ( Y = A2 ) ) ).
% 14.17/8.00
% 14.17/8.00 thf(fact_139_snd__eqD,axiom,
% 14.17/8.00 ! [X: produc3658429121746597890et_nat > $o,Y: produc3658429121746597890et_nat,A2: produc3658429121746597890et_nat] :
% 14.17/8.00 ( ( ( produc4011572625026189258et_nat @ ( produc5001842942810119800et_nat @ X @ Y ) )
% 14.17/8.00 = A2 )
% 14.17/8.00 => ( Y = A2 ) ) ).
% 14.17/8.00
% 14.17/8.00 thf(fact_140_snd__eqD,axiom,
% 14.17/8.00 ! [X: produc3658429121746597890et_nat > $o,Y: produc3925858234332021118et_nat,A2: produc3925858234332021118et_nat] :
% 14.17/8.00 ( ( ( produc5374455773327741254et_nat @ ( produc2245416461498447860et_nat @ X @ Y ) )
% 14.17/8.00 = A2 )
% 14.17/8.00 => ( Y = A2 ) ) ).
% 14.17/8.00
% 14.17/8.00 thf(fact_141_snd__eqD,axiom,
% 14.17/8.00 ! [X: heap_e7401611519738050253t_unit,Y: set_nat,A2: set_nat] :
% 14.17/8.00 ( ( ( produc8586169260539613262et_nat @ ( produc7507926704131184380et_nat @ X @ Y ) )
% 14.17/8.00 = A2 )
% 14.17/8.00 => ( Y = A2 ) ) ).
% 14.17/8.00
% 14.17/8.00 thf(fact_142_snd__eqD,axiom,
% 14.17/8.00 ! [X: int,Y: int,A2: int] :
% 14.17/8.00 ( ( ( product_snd_int_int @ ( product_Pair_int_int @ X @ Y ) )
% 14.17/8.00 = A2 )
% 14.17/8.00 => ( Y = A2 ) ) ).
% 14.17/8.00
% 14.17/8.00 thf(fact_143_snd__conv,axiom,
% 14.17/8.00 ! [X1: nat,X22: nat] :
% 14.17/8.00 ( ( product_snd_nat_nat @ ( product_Pair_nat_nat @ X1 @ X22 ) )
% 14.17/8.00 = X22 ) ).
% 14.17/8.00
% 14.17/8.00 thf(fact_144_snd__conv,axiom,
% 14.17/8.00 ! [X1: produc3658429121746597890et_nat > $o,X22: produc3658429121746597890et_nat] :
% 14.17/8.00 ( ( produc4011572625026189258et_nat @ ( produc5001842942810119800et_nat @ X1 @ X22 ) )
% 14.17/8.00 = X22 ) ).
% 14.17/8.00
% 14.17/8.00 thf(fact_145_snd__conv,axiom,
% 14.17/8.00 ! [X1: produc3658429121746597890et_nat > $o,X22: produc3925858234332021118et_nat] :
% 14.17/8.00 ( ( produc5374455773327741254et_nat @ ( produc2245416461498447860et_nat @ X1 @ X22 ) )
% 14.17/8.00 = X22 ) ).
% 14.17/8.00
% 14.17/8.00 thf(fact_146_snd__conv,axiom,
% 14.17/8.00 ! [X1: heap_e7401611519738050253t_unit,X22: set_nat] :
% 14.17/8.00 ( ( produc8586169260539613262et_nat @ ( produc7507926704131184380et_nat @ X1 @ X22 ) )
% 14.17/8.00 = X22 ) ).
% 14.17/8.00
% 14.17/8.00 thf(fact_147_snd__conv,axiom,
% 14.17/8.00 ! [X1: int,X22: int] :
% 14.17/8.00 ( ( product_snd_int_int @ ( product_Pair_int_int @ X1 @ X22 ) )
% 14.17/8.00 = X22 ) ).
% 14.17/8.00
% 14.17/8.00 thf(fact_148_eq__snd__iff,axiom,
% 14.17/8.00 ! [B: nat,P3: product_prod_nat_nat] :
% 14.17/8.00 ( ( B
% 14.17/8.00 = ( product_snd_nat_nat @ P3 ) )
% 14.17/8.00 = ( ? [A6: nat] :
% 14.17/8.00 ( P3
% 14.17/8.00 = ( product_Pair_nat_nat @ A6 @ B ) ) ) ) ).
% 14.17/8.00
% 14.17/8.00 thf(fact_149_eq__snd__iff,axiom,
% 14.17/8.00 ! [B: produc3658429121746597890et_nat,P3: produc3925858234332021118et_nat] :
% 14.17/8.00 ( ( B
% 14.17/8.00 = ( produc4011572625026189258et_nat @ P3 ) )
% 14.17/8.00 = ( ? [A6: produc3658429121746597890et_nat > $o] :
% 14.17/8.00 ( P3
% 14.17/8.00 = ( produc5001842942810119800et_nat @ A6 @ B ) ) ) ) ).
% 14.17/8.00
% 14.17/8.00 thf(fact_150_eq__snd__iff,axiom,
% 14.17/8.00 ! [B: produc3925858234332021118et_nat,P3: produc2732055786443039994et_nat] :
% 14.17/8.00 ( ( B
% 14.17/8.00 = ( produc5374455773327741254et_nat @ P3 ) )
% 14.17/8.00 = ( ? [A6: produc3658429121746597890et_nat > $o] :
% 14.17/8.00 ( P3
% 14.17/8.00 = ( produc2245416461498447860et_nat @ A6 @ B ) ) ) ) ).
% 14.17/8.00
% 14.17/8.00 thf(fact_151_eq__snd__iff,axiom,
% 14.17/8.00 ! [B: set_nat,P3: produc3658429121746597890et_nat] :
% 14.17/8.00 ( ( B
% 14.17/8.00 = ( produc8586169260539613262et_nat @ P3 ) )
% 14.17/8.00 = ( ? [A6: heap_e7401611519738050253t_unit] :
% 14.17/8.00 ( P3
% 14.17/8.00 = ( produc7507926704131184380et_nat @ A6 @ B ) ) ) ) ).
% 14.17/8.00
% 14.17/8.00 thf(fact_152_eq__snd__iff,axiom,
% 14.17/8.00 ! [B: int,P3: product_prod_int_int] :
% 14.17/8.00 ( ( B
% 14.17/8.00 = ( product_snd_int_int @ P3 ) )
% 14.17/8.00 = ( ? [A6: int] :
% 14.17/8.00 ( P3
% 14.17/8.00 = ( product_Pair_int_int @ A6 @ B ) ) ) ) ).
% 14.17/8.00
% 14.17/8.00 thf(fact_153_pure__assn__raw_Oelims_I3_J,axiom,
% 14.17/8.00 ! [X: $o,Xa: produc3658429121746597890et_nat] :
% 14.17/8.00 ( ~ ( pure_a825153325127701367it_nat @ X @ Xa )
% 14.17/8.00 => ~ ! [H: heap_e7401611519738050253t_unit,As2: set_nat] :
% 14.17/8.00 ( ( Xa
% 14.17/8.00 = ( produc7507926704131184380et_nat @ H @ As2 ) )
% 14.17/8.00 => ( ( As2 = bot_bot_set_nat )
% 14.17/8.00 & X ) ) ) ).
% 14.17/8.00
% 14.17/8.00 thf(fact_154_pure__assn__raw_Oelims_I2_J,axiom,
% 14.17/8.00 ! [X: $o,Xa: produc3658429121746597890et_nat] :
% 14.17/8.00 ( ( pure_a825153325127701367it_nat @ X @ Xa )
% 14.17/8.00 => ~ ! [H: heap_e7401611519738050253t_unit,As2: set_nat] :
% 14.17/8.00 ( ( Xa
% 14.17/8.00 = ( produc7507926704131184380et_nat @ H @ As2 ) )
% 14.17/8.00 => ~ ( ( As2 = bot_bot_set_nat )
% 14.17/8.00 & X ) ) ) ).
% 14.17/8.00
% 14.17/8.00 thf(fact_155_pure__assn__raw_Oelims_I1_J,axiom,
% 14.17/8.00 ! [X: $o,Xa: produc3658429121746597890et_nat,Y: $o] :
% 14.17/8.00 ( ( ( pure_a825153325127701367it_nat @ X @ Xa )
% 14.17/8.00 = Y )
% 14.17/8.00 => ~ ! [H: heap_e7401611519738050253t_unit,As2: set_nat] :
% 14.17/8.00 ( ( Xa
% 14.17/8.00 = ( produc7507926704131184380et_nat @ H @ As2 ) )
% 14.17/8.00 => ( Y
% 14.17/8.00 = ( ~ ( ( As2 = bot_bot_set_nat )
% 14.17/8.00 & X ) ) ) ) ) ).
% 14.17/8.00
% 14.17/8.00 thf(fact_156_pure__assn__raw_Osimps,axiom,
% 14.17/8.00 ! [B: $o,H2: heap_e7401611519738050253t_unit,As: set_nat] :
% 14.17/8.00 ( ( pure_a825153325127701367it_nat @ B @ ( produc7507926704131184380et_nat @ H2 @ As ) )
% 14.17/8.00 = ( ( As = bot_bot_set_nat )
% 14.17/8.00 & B ) ) ).
% 14.17/8.00
% 14.17/8.00 thf(fact_157_pure__assn__proper,axiom,
% 14.17/8.00 ! [B: $o] : ( proper @ ( pure_a825153325127701367it_nat @ B ) ) ).
% 14.17/8.00
% 14.17/8.00 thf(fact_158_times__assn__raw_Ocases,axiom,
% 14.17/8.00 ! [X: produc2732055786443039994et_nat] :
% 14.17/8.00 ~ ! [P4: produc3658429121746597890et_nat > $o,Q3: produc3658429121746597890et_nat > $o,H: heap_e7401611519738050253t_unit,As2: set_nat] :
% 14.17/8.00 ( X
% 14.17/8.00 != ( produc2245416461498447860et_nat @ P4 @ ( produc5001842942810119800et_nat @ Q3 @ ( produc7507926704131184380et_nat @ H @ As2 ) ) ) ) ).
% 14.17/8.00
% 14.17/8.00 thf(fact_159_pure__assn__raw_Ocases,axiom,
% 14.17/8.00 ! [X: produc6197004810343482825et_nat] :
% 14.17/8.00 ~ ! [B2: $o,H: heap_e7401611519738050253t_unit,As2: set_nat] :
% 14.17/8.00 ( X
% 14.17/8.00 != ( produc3762353314782720579et_nat @ B2 @ ( produc7507926704131184380et_nat @ H @ As2 ) ) ) ).
% 14.17/8.00
% 14.17/8.00 thf(fact_160_old_Oprod_Oexhaust,axiom,
% 14.17/8.00 ! [Y: produc3658429121746597890et_nat] :
% 14.17/8.00 ~ ! [A4: heap_e7401611519738050253t_unit,B2: set_nat] :
% 14.17/8.00 ( Y
% 14.17/8.00 != ( produc7507926704131184380et_nat @ A4 @ B2 ) ) ).
% 14.17/8.00
% 14.17/8.00 thf(fact_161_old_Oprod_Oexhaust,axiom,
% 14.17/8.00 ! [Y: product_prod_nat_nat] :
% 14.17/8.00 ~ ! [A4: nat,B2: nat] :
% 14.17/8.00 ( Y
% 14.17/8.00 != ( product_Pair_nat_nat @ A4 @ B2 ) ) ).
% 14.17/8.00
% 14.17/8.00 thf(fact_162_old_Oprod_Oexhaust,axiom,
% 14.17/8.00 ! [Y: product_prod_int_int] :
% 14.17/8.00 ~ ! [A4: int,B2: int] :
% 14.17/8.00 ( Y
% 14.17/8.00 != ( product_Pair_int_int @ A4 @ B2 ) ) ).
% 14.17/8.00
% 14.17/8.00 thf(fact_163_old_Oprod_Oexhaust,axiom,
% 14.17/8.00 ! [Y: produc3925858234332021118et_nat] :
% 14.17/8.00 ~ ! [A4: produc3658429121746597890et_nat > $o,B2: produc3658429121746597890et_nat] :
% 14.17/8.00 ( Y
% 14.17/8.00 != ( produc5001842942810119800et_nat @ A4 @ B2 ) ) ).
% 14.17/8.00
% 14.17/8.00 thf(fact_164_old_Oprod_Oexhaust,axiom,
% 14.17/8.00 ! [Y: produc2732055786443039994et_nat] :
% 14.17/8.00 ~ ! [A4: produc3658429121746597890et_nat > $o,B2: produc3925858234332021118et_nat] :
% 14.17/8.00 ( Y
% 14.17/8.00 != ( produc2245416461498447860et_nat @ A4 @ B2 ) ) ).
% 14.17/8.00
% 14.17/8.00 thf(fact_165_surj__pair,axiom,
% 14.17/8.00 ! [P3: produc3658429121746597890et_nat] :
% 14.17/8.00 ? [X3: heap_e7401611519738050253t_unit,Y2: set_nat] :
% 14.17/8.00 ( P3
% 14.17/8.00 = ( produc7507926704131184380et_nat @ X3 @ Y2 ) ) ).
% 14.17/8.00
% 14.17/8.00 thf(fact_166_surj__pair,axiom,
% 14.17/8.00 ! [P3: product_prod_nat_nat] :
% 14.17/8.00 ? [X3: nat,Y2: nat] :
% 14.17/8.00 ( P3
% 14.17/8.00 = ( product_Pair_nat_nat @ X3 @ Y2 ) ) ).
% 14.17/8.00
% 14.17/8.00 thf(fact_167_surj__pair,axiom,
% 14.17/8.00 ! [P3: product_prod_int_int] :
% 14.17/8.00 ? [X3: int,Y2: int] :
% 14.17/8.00 ( P3
% 14.17/8.00 = ( product_Pair_int_int @ X3 @ Y2 ) ) ).
% 14.17/8.00
% 14.17/8.00 thf(fact_168_surj__pair,axiom,
% 14.17/8.00 ! [P3: produc3925858234332021118et_nat] :
% 14.17/8.00 ? [X3: produc3658429121746597890et_nat > $o,Y2: produc3658429121746597890et_nat] :
% 14.17/8.00 ( P3
% 14.17/8.00 = ( produc5001842942810119800et_nat @ X3 @ Y2 ) ) ).
% 14.17/8.00
% 14.17/8.00 thf(fact_169_surj__pair,axiom,
% 14.17/8.00 ! [P3: produc2732055786443039994et_nat] :
% 14.17/8.00 ? [X3: produc3658429121746597890et_nat > $o,Y2: produc3925858234332021118et_nat] :
% 14.17/8.00 ( P3
% 14.17/8.00 = ( produc2245416461498447860et_nat @ X3 @ Y2 ) ) ).
% 14.17/8.00
% 14.17/8.00 thf(fact_170_prod__cases,axiom,
% 14.17/8.00 ! [P: produc3658429121746597890et_nat > $o,P3: produc3658429121746597890et_nat] :
% 14.17/8.00 ( ! [A4: heap_e7401611519738050253t_unit,B2: set_nat] : ( P @ ( produc7507926704131184380et_nat @ A4 @ B2 ) )
% 14.17/8.00 => ( P @ P3 ) ) ).
% 14.17/8.00
% 14.17/8.00 thf(fact_171_prod__cases,axiom,
% 14.17/8.00 ! [P: product_prod_nat_nat > $o,P3: product_prod_nat_nat] :
% 14.17/8.00 ( ! [A4: nat,B2: nat] : ( P @ ( product_Pair_nat_nat @ A4 @ B2 ) )
% 14.17/8.00 => ( P @ P3 ) ) ).
% 14.17/8.00
% 14.17/8.00 thf(fact_172_prod__cases,axiom,
% 14.17/8.00 ! [P: product_prod_int_int > $o,P3: product_prod_int_int] :
% 14.17/8.00 ( ! [A4: int,B2: int] : ( P @ ( product_Pair_int_int @ A4 @ B2 ) )
% 14.17/8.00 => ( P @ P3 ) ) ).
% 14.17/8.00
% 14.17/8.00 thf(fact_173_prod__cases,axiom,
% 14.17/8.00 ! [P: produc3925858234332021118et_nat > $o,P3: produc3925858234332021118et_nat] :
% 14.17/8.00 ( ! [A4: produc3658429121746597890et_nat > $o,B2: produc3658429121746597890et_nat] : ( P @ ( produc5001842942810119800et_nat @ A4 @ B2 ) )
% 14.17/8.00 => ( P @ P3 ) ) ).
% 14.17/8.00
% 14.17/8.00 thf(fact_174_prod__cases,axiom,
% 14.17/8.00 ! [P: produc2732055786443039994et_nat > $o,P3: produc2732055786443039994et_nat] :
% 14.17/8.00 ( ! [A4: produc3658429121746597890et_nat > $o,B2: produc3925858234332021118et_nat] : ( P @ ( produc2245416461498447860et_nat @ A4 @ B2 ) )
% 14.17/8.00 => ( P @ P3 ) ) ).
% 14.17/8.00
% 14.17/8.00 thf(fact_175_Pair__inject,axiom,
% 14.17/8.00 ! [A2: heap_e7401611519738050253t_unit,B: set_nat,A5: heap_e7401611519738050253t_unit,B3: set_nat] :
% 14.17/8.00 ( ( ( produc7507926704131184380et_nat @ A2 @ B )
% 14.17/8.00 = ( produc7507926704131184380et_nat @ A5 @ B3 ) )
% 14.17/8.00 => ~ ( ( A2 = A5 )
% 14.17/8.00 => ( B != B3 ) ) ) ).
% 14.17/8.00
% 14.17/8.00 thf(fact_176_Pair__inject,axiom,
% 14.17/8.00 ! [A2: nat,B: nat,A5: nat,B3: nat] :
% 14.17/8.00 ( ( ( product_Pair_nat_nat @ A2 @ B )
% 14.17/8.00 = ( product_Pair_nat_nat @ A5 @ B3 ) )
% 14.17/8.00 => ~ ( ( A2 = A5 )
% 14.17/8.00 => ( B != B3 ) ) ) ).
% 14.17/8.00
% 14.17/8.00 thf(fact_177_Pair__inject,axiom,
% 14.17/8.00 ! [A2: int,B: int,A5: int,B3: int] :
% 14.17/8.00 ( ( ( product_Pair_int_int @ A2 @ B )
% 14.17/8.00 = ( product_Pair_int_int @ A5 @ B3 ) )
% 14.17/8.00 => ~ ( ( A2 = A5 )
% 14.17/8.00 => ( B != B3 ) ) ) ).
% 14.17/8.00
% 14.17/8.00 thf(fact_178_Pair__inject,axiom,
% 14.17/8.00 ! [A2: produc3658429121746597890et_nat > $o,B: produc3658429121746597890et_nat,A5: produc3658429121746597890et_nat > $o,B3: produc3658429121746597890et_nat] :
% 14.17/8.00 ( ( ( produc5001842942810119800et_nat @ A2 @ B )
% 14.17/8.00 = ( produc5001842942810119800et_nat @ A5 @ B3 ) )
% 14.17/8.00 => ~ ( ( A2 = A5 )
% 14.17/8.00 => ( B != B3 ) ) ) ).
% 14.17/8.00
% 14.17/8.00 thf(fact_179_Pair__inject,axiom,
% 14.17/8.00 ! [A2: produc3658429121746597890et_nat > $o,B: produc3925858234332021118et_nat,A5: produc3658429121746597890et_nat > $o,B3: produc3925858234332021118et_nat] :
% 14.17/8.00 ( ( ( produc2245416461498447860et_nat @ A2 @ B )
% 14.17/8.00 = ( produc2245416461498447860et_nat @ A5 @ B3 ) )
% 14.17/8.00 => ~ ( ( A2 = A5 )
% 14.17/8.00 => ( B != B3 ) ) ) ).
% 14.17/8.00
% 14.17/8.00 thf(fact_180_prod__cases3,axiom,
% 14.17/8.00 ! [Y: produc3925858234332021118et_nat] :
% 14.17/8.00 ~ ! [A4: produc3658429121746597890et_nat > $o,B2: heap_e7401611519738050253t_unit,C2: set_nat] :
% 14.17/8.00 ( Y
% 14.17/8.00 != ( produc5001842942810119800et_nat @ A4 @ ( produc7507926704131184380et_nat @ B2 @ C2 ) ) ) ).
% 14.17/8.00
% 14.17/8.00 thf(fact_181_prod__cases3,axiom,
% 14.17/8.00 ! [Y: produc2732055786443039994et_nat] :
% 14.17/8.00 ~ ! [A4: produc3658429121746597890et_nat > $o,B2: produc3658429121746597890et_nat > $o,C2: produc3658429121746597890et_nat] :
% 14.17/8.00 ( Y
% 14.17/8.00 != ( produc2245416461498447860et_nat @ A4 @ ( produc5001842942810119800et_nat @ B2 @ C2 ) ) ) ).
% 14.17/8.00
% 14.17/8.00 thf(fact_182_prod__cases4,axiom,
% 14.17/8.00 ! [Y: produc2732055786443039994et_nat] :
% 14.17/8.00 ~ ! [A4: produc3658429121746597890et_nat > $o,B2: produc3658429121746597890et_nat > $o,C2: heap_e7401611519738050253t_unit,D: set_nat] :
% 14.17/8.00 ( Y
% 14.17/8.00 != ( produc2245416461498447860et_nat @ A4 @ ( produc5001842942810119800et_nat @ B2 @ ( produc7507926704131184380et_nat @ C2 @ D ) ) ) ) ).
% 14.17/8.00
% 14.17/8.00 thf(fact_183_prod__induct3,axiom,
% 14.17/8.00 ! [P: produc3925858234332021118et_nat > $o,X: produc3925858234332021118et_nat] :
% 14.17/8.00 ( ! [A4: produc3658429121746597890et_nat > $o,B2: heap_e7401611519738050253t_unit,C2: set_nat] : ( P @ ( produc5001842942810119800et_nat @ A4 @ ( produc7507926704131184380et_nat @ B2 @ C2 ) ) )
% 14.17/8.00 => ( P @ X ) ) ).
% 14.17/8.00
% 14.17/8.00 thf(fact_184_prod__induct3,axiom,
% 14.17/8.00 ! [P: produc2732055786443039994et_nat > $o,X: produc2732055786443039994et_nat] :
% 14.17/8.00 ( ! [A4: produc3658429121746597890et_nat > $o,B2: produc3658429121746597890et_nat > $o,C2: produc3658429121746597890et_nat] : ( P @ ( produc2245416461498447860et_nat @ A4 @ ( produc5001842942810119800et_nat @ B2 @ C2 ) ) )
% 14.17/8.00 => ( P @ X ) ) ).
% 14.17/8.00
% 14.17/8.00 thf(fact_185_prod__induct4,axiom,
% 14.17/8.00 ! [P: produc2732055786443039994et_nat > $o,X: produc2732055786443039994et_nat] :
% 14.17/8.00 ( ! [A4: produc3658429121746597890et_nat > $o,B2: produc3658429121746597890et_nat > $o,C2: heap_e7401611519738050253t_unit,D: set_nat] : ( P @ ( produc2245416461498447860et_nat @ A4 @ ( produc5001842942810119800et_nat @ B2 @ ( produc7507926704131184380et_nat @ C2 @ D ) ) ) )
% 14.17/8.00 => ( P @ X ) ) ).
% 14.17/8.00
% 14.17/8.00 thf(fact_186_one__reorient,axiom,
% 14.17/8.00 ! [X: assn] :
% 14.17/8.00 ( ( one_one_assn = X )
% 14.17/8.00 = ( X = one_one_assn ) ) ).
% 14.17/8.00
% 14.17/8.00 thf(fact_187_one__reorient,axiom,
% 14.17/8.00 ! [X: nat] :
% 14.17/8.00 ( ( one_one_nat = X )
% 14.17/8.00 = ( X = one_one_nat ) ) ).
% 14.17/8.00
% 14.17/8.00 thf(fact_188_one__reorient,axiom,
% 14.17/8.00 ! [X: int] :
% 14.17/8.00 ( ( one_one_int = X )
% 14.17/8.00 = ( X = one_one_int ) ) ).
% 14.17/8.00
% 14.17/8.00 thf(fact_189_one__reorient,axiom,
% 14.17/8.00 ! [X: code_integer] :
% 14.17/8.00 ( ( one_one_Code_integer = X )
% 14.17/8.00 = ( X = one_one_Code_integer ) ) ).
% 14.17/8.00
% 14.17/8.00 thf(fact_190_one__reorient,axiom,
% 14.17/8.00 ! [X: code_natural] :
% 14.17/8.00 ( ( one_one_Code_natural = X )
% 14.17/8.00 = ( X = one_one_Code_natural ) ) ).
% 14.17/8.00
% 14.17/8.00 thf(fact_191_type__definition__assn,axiom,
% 14.17/8.00 type_d3909072315231072503_nat_o @ rep_assn @ abs_assn @ ( collec939566748876313656_nat_o @ proper ) ).
% 14.17/8.00
% 14.17/8.00 thf(fact_192_times__assn__proper,axiom,
% 14.17/8.00 ! [P: produc3658429121746597890et_nat > $o,Q2: produc3658429121746597890et_nat > $o] :
% 14.17/8.00 ( ( proper @ P )
% 14.17/8.00 => ( ( proper @ Q2 )
% 14.17/8.00 => ( proper @ ( times_assn_raw @ P @ Q2 ) ) ) ) ).
% 14.17/8.00
% 14.17/8.00 thf(fact_193_is__pure__assn__basic__simps_I2_J,axiom,
% 14.17/8.00 is_pure_assn @ one_one_assn ).
% 14.17/8.00
% 14.17/8.00 thf(fact_194_bijective__def,axiom,
% 14.17/8.00 ( biject2219996528949639174et_nat
% 14.17/8.00 = ( ^ [R: set_Pr3948176798113811640et_nat] :
% 14.17/8.00 ( ! [X2: heap_e7401611519738050253t_unit,Y3: set_nat,Z2: set_nat] :
% 14.17/8.00 ( ( ( member6260224972018164377et_nat @ ( produc7507926704131184380et_nat @ X2 @ Y3 ) @ R )
% 14.17/8.00 & ( member6260224972018164377et_nat @ ( produc7507926704131184380et_nat @ X2 @ Z2 ) @ R ) )
% 14.17/8.00 => ( Y3 = Z2 ) )
% 14.17/8.00 & ! [X2: heap_e7401611519738050253t_unit,Y3: heap_e7401611519738050253t_unit,Z2: set_nat] :
% 14.17/8.00 ( ( ( member6260224972018164377et_nat @ ( produc7507926704131184380et_nat @ X2 @ Z2 ) @ R )
% 14.17/8.00 & ( member6260224972018164377et_nat @ ( produc7507926704131184380et_nat @ Y3 @ Z2 ) @ R ) )
% 14.17/8.00 => ( X2 = Y3 ) ) ) ) ) ).
% 14.17/8.00
% 14.17/8.00 thf(fact_195_bijective__def,axiom,
% 14.17/8.00 ( bijective_nat_nat
% 14.17/8.00 = ( ^ [R: set_Pr1261947904930325089at_nat] :
% 14.17/8.00 ( ! [X2: nat,Y3: nat,Z2: nat] :
% 14.17/8.00 ( ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X2 @ Y3 ) @ R )
% 14.17/8.00 & ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X2 @ Z2 ) @ R ) )
% 14.17/8.00 => ( Y3 = Z2 ) )
% 14.17/8.00 & ! [X2: nat,Y3: nat,Z2: nat] :
% 14.17/8.00 ( ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X2 @ Z2 ) @ R )
% 14.17/8.00 & ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ Y3 @ Z2 ) @ R ) )
% 14.17/8.00 => ( X2 = Y3 ) ) ) ) ) ).
% 14.17/8.00
% 14.17/8.00 thf(fact_196_bijective__def,axiom,
% 14.17/8.00 ( bijective_int_int
% 14.17/8.00 = ( ^ [R: set_Pr958786334691620121nt_int] :
% 14.17/8.00 ( ! [X2: int,Y3: int,Z2: int] :
% 14.17/8.00 ( ( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ X2 @ Y3 ) @ R )
% 14.17/8.00 & ( member5262025264175285858nt_int @ ( product_Pair_int_int @ X2 @ Z2 ) @ R ) )
% 14.17/8.00 => ( Y3 = Z2 ) )
% 14.17/8.00 & ! [X2: int,Y3: int,Z2: int] :
% 14.17/8.00 ( ( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ X2 @ Z2 ) @ R )
% 14.17/8.00 & ( member5262025264175285858nt_int @ ( product_Pair_int_int @ Y3 @ Z2 ) @ R ) )
% 14.17/8.00 => ( X2 = Y3 ) ) ) ) ) ).
% 14.17/8.00
% 14.17/8.00 thf(fact_197_bijective__def,axiom,
% 14.17/8.00 ( biject2615096655818420098et_nat
% 14.17/8.00 = ( ^ [R: set_Pr3286484037609594932et_nat] :
% 14.17/8.00 ( ! [X2: produc3658429121746597890et_nat > $o,Y3: produc3658429121746597890et_nat,Z2: produc3658429121746597890et_nat] :
% 14.17/8.00 ( ( ( member1996754912294343701et_nat @ ( produc5001842942810119800et_nat @ X2 @ Y3 ) @ R )
% 14.17/8.00 & ( member1996754912294343701et_nat @ ( produc5001842942810119800et_nat @ X2 @ Z2 ) @ R ) )
% 14.17/8.00 => ( Y3 = Z2 ) )
% 14.17/8.00 & ! [X2: produc3658429121746597890et_nat > $o,Y3: produc3658429121746597890et_nat > $o,Z2: produc3658429121746597890et_nat] :
% 14.17/8.00 ( ( ( member1996754912294343701et_nat @ ( produc5001842942810119800et_nat @ X2 @ Z2 ) @ R )
% 14.17/8.00 & ( member1996754912294343701et_nat @ ( produc5001842942810119800et_nat @ Y3 @ Z2 ) @ R ) )
% 14.17/8.00 => ( X2 = Y3 ) ) ) ) ) ).
% 14.17/8.00
% 14.17/8.00 thf(fact_198_bijective__def,axiom,
% 14.17/8.00 ( biject1468766312547416318et_nat
% 14.17/8.00 = ( ^ [R: set_Pr8536935166611901872et_nat] :
% 14.17/8.00 ( ! [X2: produc3658429121746597890et_nat > $o,Y3: produc3925858234332021118et_nat,Z2: produc3925858234332021118et_nat] :
% 14.17/8.00 ( ( ( member6124377750444531601et_nat @ ( produc2245416461498447860et_nat @ X2 @ Y3 ) @ R )
% 14.17/8.00 & ( member6124377750444531601et_nat @ ( produc2245416461498447860et_nat @ X2 @ Z2 ) @ R ) )
% 14.17/8.00 => ( Y3 = Z2 ) )
% 14.17/8.00 & ! [X2: produc3658429121746597890et_nat > $o,Y3: produc3658429121746597890et_nat > $o,Z2: produc3925858234332021118et_nat] :
% 14.17/8.00 ( ( ( member6124377750444531601et_nat @ ( produc2245416461498447860et_nat @ X2 @ Z2 ) @ R )
% 14.17/8.00 & ( member6124377750444531601et_nat @ ( produc2245416461498447860et_nat @ Y3 @ Z2 ) @ R ) )
% 14.17/8.00 => ( X2 = Y3 ) ) ) ) ) ).
% 14.17/8.00
% 14.17/8.00 thf(fact_199_ssubst__Pair__rhs,axiom,
% 14.17/8.00 ! [R2: heap_e7401611519738050253t_unit,S2: set_nat,R3: set_Pr3948176798113811640et_nat,S3: set_nat] :
% 14.17/8.00 ( ( member6260224972018164377et_nat @ ( produc7507926704131184380et_nat @ R2 @ S2 ) @ R3 )
% 14.17/8.00 => ( ( S3 = S2 )
% 14.17/8.00 => ( member6260224972018164377et_nat @ ( produc7507926704131184380et_nat @ R2 @ S3 ) @ R3 ) ) ) ).
% 14.17/8.00
% 14.17/8.00 thf(fact_200_ssubst__Pair__rhs,axiom,
% 14.17/8.00 ! [R2: nat,S2: nat,R3: set_Pr1261947904930325089at_nat,S3: nat] :
% 14.17/8.00 ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ R2 @ S2 ) @ R3 )
% 14.17/8.00 => ( ( S3 = S2 )
% 14.17/8.00 => ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ R2 @ S3 ) @ R3 ) ) ) ).
% 14.17/8.00
% 14.17/8.00 thf(fact_201_ssubst__Pair__rhs,axiom,
% 14.17/8.00 ! [R2: int,S2: int,R3: set_Pr958786334691620121nt_int,S3: int] :
% 14.17/8.00 ( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ R2 @ S2 ) @ R3 )
% 14.17/8.00 => ( ( S3 = S2 )
% 14.17/8.00 => ( member5262025264175285858nt_int @ ( product_Pair_int_int @ R2 @ S3 ) @ R3 ) ) ) ).
% 14.17/8.00
% 14.17/8.00 thf(fact_202_ssubst__Pair__rhs,axiom,
% 14.17/8.00 ! [R2: produc3658429121746597890et_nat > $o,S2: produc3658429121746597890et_nat,R3: set_Pr3286484037609594932et_nat,S3: produc3658429121746597890et_nat] :
% 14.17/8.00 ( ( member1996754912294343701et_nat @ ( produc5001842942810119800et_nat @ R2 @ S2 ) @ R3 )
% 14.17/8.00 => ( ( S3 = S2 )
% 14.17/8.00 => ( member1996754912294343701et_nat @ ( produc5001842942810119800et_nat @ R2 @ S3 ) @ R3 ) ) ) ).
% 14.17/8.00
% 14.17/8.00 thf(fact_203_ssubst__Pair__rhs,axiom,
% 14.17/8.00 ! [R2: produc3658429121746597890et_nat > $o,S2: produc3925858234332021118et_nat,R3: set_Pr8536935166611901872et_nat,S3: produc3925858234332021118et_nat] :
% 14.17/8.00 ( ( member6124377750444531601et_nat @ ( produc2245416461498447860et_nat @ R2 @ S2 ) @ R3 )
% 14.17/8.00 => ( ( S3 = S2 )
% 14.17/8.00 => ( member6124377750444531601et_nat @ ( produc2245416461498447860et_nat @ R2 @ S3 ) @ R3 ) ) ) ).
% 14.17/8.00
% 14.17/8.00 thf(fact_204_is__singletonI,axiom,
% 14.17/8.00 ! [X: product_unit] : ( is_sin2160648248035936513t_unit @ ( insert_Product_unit @ X @ bot_bo3957492148770167129t_unit ) ) ).
% 14.17/8.00
% 14.17/8.00 thf(fact_205_is__singletonI,axiom,
% 14.17/8.00 ! [X: $o] : ( is_singleton_o @ ( insert_o @ X @ bot_bot_set_o ) ) ).
% 14.17/8.00
% 14.17/8.00 thf(fact_206_is__singletonI,axiom,
% 14.17/8.00 ! [X: nat] : ( is_singleton_nat @ ( insert_nat @ X @ bot_bot_set_nat ) ) ).
% 14.17/8.00
% 14.17/8.00 thf(fact_207_is__singletonI,axiom,
% 14.17/8.00 ! [X: int] : ( is_singleton_int @ ( insert_int @ X @ bot_bot_set_int ) ) ).
% 14.17/8.00
% 14.17/8.00 thf(fact_208_is__singletonI,axiom,
% 14.17/8.00 ! [X: produc3658429121746597890et_nat > $o] : ( is_sin5180296473474724033_nat_o @ ( insert5175938949040314269_nat_o @ X @ bot_bo7824918357723582233_nat_o ) ) ).
% 14.17/8.00
% 14.17/8.00 thf(fact_209_curryI,axiom,
% 14.17/8.00 ! [F: produc3658429121746597890et_nat > $o,A2: heap_e7401611519738050253t_unit,B: set_nat] :
% 14.17/8.00 ( ( F @ ( produc7507926704131184380et_nat @ A2 @ B ) )
% 14.17/8.00 => ( produc1807123492089482958_nat_o @ F @ A2 @ B ) ) ).
% 14.17/8.00
% 14.17/8.00 thf(fact_210_curryI,axiom,
% 14.17/8.00 ! [F: product_prod_nat_nat > $o,A2: nat,B: nat] :
% 14.17/8.00 ( ( F @ ( product_Pair_nat_nat @ A2 @ B ) )
% 14.17/8.00 => ( produc1310100445399344235_nat_o @ F @ A2 @ B ) ) ).
% 14.17/8.00
% 14.17/8.00 thf(fact_211_curryI,axiom,
% 14.17/8.00 ! [F: product_prod_int_int > $o,A2: int,B: int] :
% 14.17/8.00 ( ( F @ ( product_Pair_int_int @ A2 @ B ) )
% 14.17/8.00 => ( produc175634133007206835_int_o @ F @ A2 @ B ) ) ).
% 14.17/8.00
% 14.17/8.00 thf(fact_212_curryI,axiom,
% 14.17/8.00 ! [F: produc3925858234332021118et_nat > $o,A2: produc3658429121746597890et_nat > $o,B: produc3658429121746597890et_nat] :
% 14.17/8.00 ( ( F @ ( produc5001842942810119800et_nat @ A2 @ B ) )
% 14.17/8.00 => ( produc6216949301066131538_nat_o @ F @ A2 @ B ) ) ).
% 14.17/8.00
% 14.17/8.00 thf(fact_213_curryI,axiom,
% 14.17/8.00 ! [F: produc2732055786443039994et_nat > $o,A2: produc3658429121746597890et_nat > $o,B: produc3925858234332021118et_nat] :
% 14.17/8.00 ( ( F @ ( produc2245416461498447860et_nat @ A2 @ B ) )
% 14.17/8.00 => ( produc5101573711933517782_nat_o @ F @ A2 @ B ) ) ).
% 14.17/8.00
% 14.17/8.00 thf(fact_214_insertCI,axiom,
% 14.17/8.00 ! [A2: produc3658429121746597890et_nat > $o,B4: set_Pr4532377907799695533_nat_o,B: produc3658429121746597890et_nat > $o] :
% 14.17/8.00 ( ( ~ ( member6576561426505652726_nat_o @ A2 @ B4 )
% 14.17/8.00 => ( A2 = B ) )
% 14.17/8.00 => ( member6576561426505652726_nat_o @ A2 @ ( insert5175938949040314269_nat_o @ B @ B4 ) ) ) ).
% 14.17/8.00
% 14.17/8.00 thf(fact_215_insertCI,axiom,
% 14.17/8.00 ! [A2: nat,B4: set_nat,B: nat] :
% 14.17/8.00 ( ( ~ ( member_nat @ A2 @ B4 )
% 14.17/8.00 => ( A2 = B ) )
% 14.17/8.00 => ( member_nat @ A2 @ ( insert_nat @ B @ B4 ) ) ) ).
% 14.17/8.00
% 14.17/8.00 thf(fact_216_insertCI,axiom,
% 14.17/8.00 ! [A2: int,B4: set_int,B: int] :
% 14.17/8.00 ( ( ~ ( member_int @ A2 @ B4 )
% 14.17/8.00 => ( A2 = B ) )
% 14.17/8.00 => ( member_int @ A2 @ ( insert_int @ B @ B4 ) ) ) ).
% 14.17/8.00
% 14.17/8.00 thf(fact_217_insertCI,axiom,
% 14.17/8.00 ! [A2: $o,B4: set_o,B: $o] :
% 14.17/8.00 ( ( ~ ( member_o @ A2 @ B4 )
% 14.17/8.00 => ( A2 = B ) )
% 14.17/8.00 => ( member_o @ A2 @ ( insert_o @ B @ B4 ) ) ) ).
% 14.17/8.00
% 14.17/8.00 thf(fact_218_insertCI,axiom,
% 14.17/8.00 ! [A2: product_unit,B4: set_Product_unit,B: product_unit] :
% 14.17/8.00 ( ( ~ ( member_Product_unit @ A2 @ B4 )
% 14.17/8.00 => ( A2 = B ) )
% 14.17/8.00 => ( member_Product_unit @ A2 @ ( insert_Product_unit @ B @ B4 ) ) ) ).
% 14.17/8.00
% 14.17/8.00 thf(fact_219_insert__iff,axiom,
% 14.17/8.00 ! [A2: produc3658429121746597890et_nat > $o,B: produc3658429121746597890et_nat > $o,A: set_Pr4532377907799695533_nat_o] :
% 14.17/8.00 ( ( member6576561426505652726_nat_o @ A2 @ ( insert5175938949040314269_nat_o @ B @ A ) )
% 14.17/8.00 = ( ( A2 = B )
% 14.17/8.00 | ( member6576561426505652726_nat_o @ A2 @ A ) ) ) ).
% 14.17/8.00
% 14.17/8.00 thf(fact_220_insert__iff,axiom,
% 14.17/8.00 ! [A2: nat,B: nat,A: set_nat] :
% 14.17/8.00 ( ( member_nat @ A2 @ ( insert_nat @ B @ A ) )
% 14.17/8.00 = ( ( A2 = B )
% 14.17/8.00 | ( member_nat @ A2 @ A ) ) ) ).
% 14.17/8.00
% 14.17/8.00 thf(fact_221_insert__iff,axiom,
% 14.17/8.00 ! [A2: int,B: int,A: set_int] :
% 14.17/8.00 ( ( member_int @ A2 @ ( insert_int @ B @ A ) )
% 14.17/8.00 = ( ( A2 = B )
% 14.17/8.00 | ( member_int @ A2 @ A ) ) ) ).
% 14.17/8.00
% 14.17/8.00 thf(fact_222_insert__iff,axiom,
% 14.17/8.00 ! [A2: $o,B: $o,A: set_o] :
% 14.17/8.00 ( ( member_o @ A2 @ ( insert_o @ B @ A ) )
% 14.17/8.00 = ( ( A2 = B )
% 14.17/8.00 | ( member_o @ A2 @ A ) ) ) ).
% 14.17/8.00
% 14.17/8.00 thf(fact_223_insert__iff,axiom,
% 14.17/8.00 ! [A2: product_unit,B: product_unit,A: set_Product_unit] :
% 14.17/8.00 ( ( member_Product_unit @ A2 @ ( insert_Product_unit @ B @ A ) )
% 14.17/8.00 = ( ( A2 = B )
% 14.17/8.00 | ( member_Product_unit @ A2 @ A ) ) ) ).
% 14.17/8.00
% 14.17/8.00 thf(fact_224_insert__absorb2,axiom,
% 14.17/8.00 ! [X: nat,A: set_nat] :
% 14.17/8.00 ( ( insert_nat @ X @ ( insert_nat @ X @ A ) )
% 14.17/8.00 = ( insert_nat @ X @ A ) ) ).
% 14.17/8.00
% 14.17/8.00 thf(fact_225_insert__absorb2,axiom,
% 14.17/8.00 ! [X: int,A: set_int] :
% 14.17/8.00 ( ( insert_int @ X @ ( insert_int @ X @ A ) )
% 14.17/8.00 = ( insert_int @ X @ A ) ) ).
% 14.17/8.00
% 14.17/8.00 thf(fact_226_insert__absorb2,axiom,
% 14.17/8.00 ! [X: $o,A: set_o] :
% 14.17/8.00 ( ( insert_o @ X @ ( insert_o @ X @ A ) )
% 14.17/8.00 = ( insert_o @ X @ A ) ) ).
% 14.17/8.00
% 14.17/8.00 thf(fact_227_insert__absorb2,axiom,
% 14.17/8.00 ! [X: product_unit,A: set_Product_unit] :
% 14.17/8.00 ( ( insert_Product_unit @ X @ ( insert_Product_unit @ X @ A ) )
% 14.17/8.00 = ( insert_Product_unit @ X @ A ) ) ).
% 14.17/8.00
% 14.17/8.00 thf(fact_228_singletonI,axiom,
% 14.17/8.00 ! [A2: product_unit] : ( member_Product_unit @ A2 @ ( insert_Product_unit @ A2 @ bot_bo3957492148770167129t_unit ) ) ).
% 14.17/8.00
% 14.17/8.00 thf(fact_229_singletonI,axiom,
% 14.17/8.00 ! [A2: $o] : ( member_o @ A2 @ ( insert_o @ A2 @ bot_bot_set_o ) ) ).
% 14.17/8.00
% 14.17/8.00 thf(fact_230_singletonI,axiom,
% 14.17/8.00 ! [A2: nat] : ( member_nat @ A2 @ ( insert_nat @ A2 @ bot_bot_set_nat ) ) ).
% 14.17/8.00
% 14.17/8.00 thf(fact_231_singletonI,axiom,
% 14.17/8.00 ! [A2: int] : ( member_int @ A2 @ ( insert_int @ A2 @ bot_bot_set_int ) ) ).
% 14.17/8.00
% 14.17/8.00 thf(fact_232_singletonI,axiom,
% 14.17/8.00 ! [A2: produc3658429121746597890et_nat > $o] : ( member6576561426505652726_nat_o @ A2 @ ( insert5175938949040314269_nat_o @ A2 @ bot_bo7824918357723582233_nat_o ) ) ).
% 14.17/8.00
% 14.17/8.00 thf(fact_233_insertE,axiom,
% 14.17/8.00 ! [A2: produc3658429121746597890et_nat > $o,B: produc3658429121746597890et_nat > $o,A: set_Pr4532377907799695533_nat_o] :
% 14.17/8.00 ( ( member6576561426505652726_nat_o @ A2 @ ( insert5175938949040314269_nat_o @ B @ A ) )
% 14.17/8.00 => ( ( A2 != B )
% 14.17/8.00 => ( member6576561426505652726_nat_o @ A2 @ A ) ) ) ).
% 14.17/8.00
% 14.17/8.00 thf(fact_234_insertE,axiom,
% 14.17/8.00 ! [A2: nat,B: nat,A: set_nat] :
% 14.17/8.00 ( ( member_nat @ A2 @ ( insert_nat @ B @ A ) )
% 14.17/8.00 => ( ( A2 != B )
% 14.17/8.00 => ( member_nat @ A2 @ A ) ) ) ).
% 14.17/8.00
% 14.17/8.00 thf(fact_235_insertE,axiom,
% 14.17/8.00 ! [A2: int,B: int,A: set_int] :
% 14.17/8.00 ( ( member_int @ A2 @ ( insert_int @ B @ A ) )
% 14.17/8.00 => ( ( A2 != B )
% 14.17/8.00 => ( member_int @ A2 @ A ) ) ) ).
% 14.17/8.00
% 14.17/8.00 thf(fact_236_insertE,axiom,
% 14.17/8.00 ! [A2: $o,B: $o,A: set_o] :
% 14.17/8.00 ( ( member_o @ A2 @ ( insert_o @ B @ A ) )
% 14.17/8.00 => ( ( A2 = ~ B )
% 14.17/8.00 => ( member_o @ A2 @ A ) ) ) ).
% 14.17/8.00
% 14.17/8.00 thf(fact_237_insertE,axiom,
% 14.17/8.00 ! [A2: product_unit,B: product_unit,A: set_Product_unit] :
% 14.17/8.00 ( ( member_Product_unit @ A2 @ ( insert_Product_unit @ B @ A ) )
% 14.17/8.00 => ( ( A2 != B )
% 14.17/8.00 => ( member_Product_unit @ A2 @ A ) ) ) ).
% 14.17/8.00
% 14.17/8.00 thf(fact_238_insertI1,axiom,
% 14.17/8.00 ! [A2: produc3658429121746597890et_nat > $o,B4: set_Pr4532377907799695533_nat_o] : ( member6576561426505652726_nat_o @ A2 @ ( insert5175938949040314269_nat_o @ A2 @ B4 ) ) ).
% 14.17/8.00
% 14.17/8.00 thf(fact_239_insertI1,axiom,
% 14.17/8.00 ! [A2: nat,B4: set_nat] : ( member_nat @ A2 @ ( insert_nat @ A2 @ B4 ) ) ).
% 14.17/8.00
% 14.17/8.00 thf(fact_240_insertI1,axiom,
% 14.17/8.00 ! [A2: int,B4: set_int] : ( member_int @ A2 @ ( insert_int @ A2 @ B4 ) ) ).
% 14.17/8.00
% 14.17/8.00 thf(fact_241_insertI1,axiom,
% 14.17/8.00 ! [A2: $o,B4: set_o] : ( member_o @ A2 @ ( insert_o @ A2 @ B4 ) ) ).
% 14.17/8.00
% 14.17/8.00 thf(fact_242_insertI1,axiom,
% 14.17/8.00 ! [A2: product_unit,B4: set_Product_unit] : ( member_Product_unit @ A2 @ ( insert_Product_unit @ A2 @ B4 ) ) ).
% 14.17/8.00
% 14.17/8.00 thf(fact_243_insertI2,axiom,
% 14.17/8.00 ! [A2: produc3658429121746597890et_nat > $o,B4: set_Pr4532377907799695533_nat_o,B: produc3658429121746597890et_nat > $o] :
% 14.17/8.00 ( ( member6576561426505652726_nat_o @ A2 @ B4 )
% 14.17/8.00 => ( member6576561426505652726_nat_o @ A2 @ ( insert5175938949040314269_nat_o @ B @ B4 ) ) ) ).
% 14.17/8.00
% 14.17/8.00 thf(fact_244_insertI2,axiom,
% 14.17/8.00 ! [A2: nat,B4: set_nat,B: nat] :
% 14.17/8.00 ( ( member_nat @ A2 @ B4 )
% 14.17/8.00 => ( member_nat @ A2 @ ( insert_nat @ B @ B4 ) ) ) ).
% 14.17/8.00
% 14.17/8.00 thf(fact_245_insertI2,axiom,
% 14.17/8.00 ! [A2: int,B4: set_int,B: int] :
% 14.17/8.00 ( ( member_int @ A2 @ B4 )
% 14.17/8.00 => ( member_int @ A2 @ ( insert_int @ B @ B4 ) ) ) ).
% 14.17/8.00
% 14.17/8.00 thf(fact_246_insertI2,axiom,
% 14.17/8.00 ! [A2: $o,B4: set_o,B: $o] :
% 14.17/8.00 ( ( member_o @ A2 @ B4 )
% 14.17/8.00 => ( member_o @ A2 @ ( insert_o @ B @ B4 ) ) ) ).
% 14.17/8.00
% 14.17/8.00 thf(fact_247_insertI2,axiom,
% 14.17/8.00 ! [A2: product_unit,B4: set_Product_unit,B: product_unit] :
% 14.17/8.00 ( ( member_Product_unit @ A2 @ B4 )
% 14.17/8.00 => ( member_Product_unit @ A2 @ ( insert_Product_unit @ B @ B4 ) ) ) ).
% 14.17/8.00
% 14.17/8.00 thf(fact_248_Set_Oset__insert,axiom,
% 14.17/8.00 ! [X: produc3658429121746597890et_nat > $o,A: set_Pr4532377907799695533_nat_o] :
% 14.17/8.00 ( ( member6576561426505652726_nat_o @ X @ A )
% 14.17/8.00 => ~ ! [B5: set_Pr4532377907799695533_nat_o] :
% 14.17/8.00 ( ( A
% 14.17/8.00 = ( insert5175938949040314269_nat_o @ X @ B5 ) )
% 14.17/8.00 => ( member6576561426505652726_nat_o @ X @ B5 ) ) ) ).
% 14.17/8.00
% 14.17/8.00 thf(fact_249_Set_Oset__insert,axiom,
% 14.17/8.00 ! [X: nat,A: set_nat] :
% 14.17/8.00 ( ( member_nat @ X @ A )
% 14.17/8.00 => ~ ! [B5: set_nat] :
% 14.17/8.00 ( ( A
% 14.17/8.00 = ( insert_nat @ X @ B5 ) )
% 14.17/8.00 => ( member_nat @ X @ B5 ) ) ) ).
% 14.17/8.00
% 14.17/8.00 thf(fact_250_Set_Oset__insert,axiom,
% 14.17/8.00 ! [X: int,A: set_int] :
% 14.17/8.00 ( ( member_int @ X @ A )
% 14.17/8.00 => ~ ! [B5: set_int] :
% 14.17/8.00 ( ( A
% 14.17/8.00 = ( insert_int @ X @ B5 ) )
% 14.17/8.00 => ( member_int @ X @ B5 ) ) ) ).
% 14.17/8.00
% 14.17/8.00 thf(fact_251_Set_Oset__insert,axiom,
% 14.17/8.00 ! [X: $o,A: set_o] :
% 14.17/8.00 ( ( member_o @ X @ A )
% 14.17/8.00 => ~ ! [B5: set_o] :
% 14.17/8.00 ( ( A
% 14.17/8.00 = ( insert_o @ X @ B5 ) )
% 14.17/8.00 => ( member_o @ X @ B5 ) ) ) ).
% 14.17/8.00
% 14.17/8.00 thf(fact_252_Set_Oset__insert,axiom,
% 14.17/8.00 ! [X: product_unit,A: set_Product_unit] :
% 14.17/8.00 ( ( member_Product_unit @ X @ A )
% 14.17/8.00 => ~ ! [B5: set_Product_unit] :
% 14.17/8.00 ( ( A
% 14.17/8.00 = ( insert_Product_unit @ X @ B5 ) )
% 14.17/8.00 => ( member_Product_unit @ X @ B5 ) ) ) ).
% 14.17/8.00
% 14.17/8.00 thf(fact_253_insert__ident,axiom,
% 14.17/8.00 ! [X: produc3658429121746597890et_nat > $o,A: set_Pr4532377907799695533_nat_o,B4: set_Pr4532377907799695533_nat_o] :
% 14.17/8.00 ( ~ ( member6576561426505652726_nat_o @ X @ A )
% 14.17/8.00 => ( ~ ( member6576561426505652726_nat_o @ X @ B4 )
% 14.17/8.00 => ( ( ( insert5175938949040314269_nat_o @ X @ A )
% 14.17/8.00 = ( insert5175938949040314269_nat_o @ X @ B4 ) )
% 14.17/8.00 = ( A = B4 ) ) ) ) ).
% 14.17/8.00
% 14.17/8.00 thf(fact_254_insert__ident,axiom,
% 14.17/8.00 ! [X: nat,A: set_nat,B4: set_nat] :
% 14.17/8.00 ( ~ ( member_nat @ X @ A )
% 14.17/8.00 => ( ~ ( member_nat @ X @ B4 )
% 14.17/8.00 => ( ( ( insert_nat @ X @ A )
% 14.17/8.00 = ( insert_nat @ X @ B4 ) )
% 14.17/8.00 = ( A = B4 ) ) ) ) ).
% 14.17/8.00
% 14.17/8.00 thf(fact_255_insert__ident,axiom,
% 14.17/8.00 ! [X: int,A: set_int,B4: set_int] :
% 14.17/8.00 ( ~ ( member_int @ X @ A )
% 14.17/8.00 => ( ~ ( member_int @ X @ B4 )
% 14.17/8.00 => ( ( ( insert_int @ X @ A )
% 14.17/8.00 = ( insert_int @ X @ B4 ) )
% 14.17/8.00 = ( A = B4 ) ) ) ) ).
% 14.17/8.00
% 14.17/8.00 thf(fact_256_insert__ident,axiom,
% 14.17/8.00 ! [X: $o,A: set_o,B4: set_o] :
% 14.17/8.00 ( ~ ( member_o @ X @ A )
% 14.17/8.00 => ( ~ ( member_o @ X @ B4 )
% 14.17/8.00 => ( ( ( insert_o @ X @ A )
% 14.17/8.00 = ( insert_o @ X @ B4 ) )
% 14.17/8.00 = ( A = B4 ) ) ) ) ).
% 14.17/8.00
% 14.17/8.00 thf(fact_257_insert__ident,axiom,
% 14.17/8.00 ! [X: product_unit,A: set_Product_unit,B4: set_Product_unit] :
% 14.17/8.00 ( ~ ( member_Product_unit @ X @ A )
% 14.17/8.00 => ( ~ ( member_Product_unit @ X @ B4 )
% 14.17/8.00 => ( ( ( insert_Product_unit @ X @ A )
% 14.17/8.00 = ( insert_Product_unit @ X @ B4 ) )
% 14.17/8.00 = ( A = B4 ) ) ) ) ).
% 14.17/8.00
% 14.17/8.00 thf(fact_258_insert__absorb,axiom,
% 14.17/8.00 ! [A2: produc3658429121746597890et_nat > $o,A: set_Pr4532377907799695533_nat_o] :
% 14.17/8.00 ( ( member6576561426505652726_nat_o @ A2 @ A )
% 14.17/8.00 => ( ( insert5175938949040314269_nat_o @ A2 @ A )
% 14.17/8.00 = A ) ) ).
% 14.17/8.00
% 14.17/8.00 thf(fact_259_insert__absorb,axiom,
% 14.17/8.00 ! [A2: nat,A: set_nat] :
% 14.17/8.00 ( ( member_nat @ A2 @ A )
% 14.17/8.00 => ( ( insert_nat @ A2 @ A )
% 14.17/8.00 = A ) ) ).
% 14.17/8.00
% 14.17/8.00 thf(fact_260_insert__absorb,axiom,
% 14.17/8.00 ! [A2: int,A: set_int] :
% 14.17/8.00 ( ( member_int @ A2 @ A )
% 14.17/8.00 => ( ( insert_int @ A2 @ A )
% 14.17/8.00 = A ) ) ).
% 14.17/8.00
% 14.17/8.00 thf(fact_261_insert__absorb,axiom,
% 14.17/8.00 ! [A2: $o,A: set_o] :
% 14.17/8.00 ( ( member_o @ A2 @ A )
% 14.17/8.00 => ( ( insert_o @ A2 @ A )
% 14.17/8.00 = A ) ) ).
% 14.17/8.00
% 14.17/8.00 thf(fact_262_insert__absorb,axiom,
% 14.17/8.00 ! [A2: product_unit,A: set_Product_unit] :
% 14.17/8.00 ( ( member_Product_unit @ A2 @ A )
% 14.17/8.00 => ( ( insert_Product_unit @ A2 @ A )
% 14.17/8.00 = A ) ) ).
% 14.17/8.00
% 14.17/8.00 thf(fact_263_insert__eq__iff,axiom,
% 14.17/8.00 ! [A2: produc3658429121746597890et_nat > $o,A: set_Pr4532377907799695533_nat_o,B: produc3658429121746597890et_nat > $o,B4: set_Pr4532377907799695533_nat_o] :
% 14.17/8.00 ( ~ ( member6576561426505652726_nat_o @ A2 @ A )
% 14.17/8.00 => ( ~ ( member6576561426505652726_nat_o @ B @ B4 )
% 14.17/8.00 => ( ( ( insert5175938949040314269_nat_o @ A2 @ A )
% 14.17/8.00 = ( insert5175938949040314269_nat_o @ B @ B4 ) )
% 14.17/8.00 = ( ( ( A2 = B )
% 14.17/8.00 => ( A = B4 ) )
% 14.17/8.00 & ( ( A2 != B )
% 14.17/8.00 => ? [C3: set_Pr4532377907799695533_nat_o] :
% 14.17/8.00 ( ( A
% 14.17/8.00 = ( insert5175938949040314269_nat_o @ B @ C3 ) )
% 14.17/8.00 & ~ ( member6576561426505652726_nat_o @ B @ C3 )
% 14.17/8.00 & ( B4
% 14.17/8.00 = ( insert5175938949040314269_nat_o @ A2 @ C3 ) )
% 14.17/8.00 & ~ ( member6576561426505652726_nat_o @ A2 @ C3 ) ) ) ) ) ) ) ).
% 14.17/8.00
% 14.17/8.00 thf(fact_264_insert__eq__iff,axiom,
% 14.17/8.00 ! [A2: nat,A: set_nat,B: nat,B4: set_nat] :
% 14.17/8.00 ( ~ ( member_nat @ A2 @ A )
% 14.17/8.00 => ( ~ ( member_nat @ B @ B4 )
% 14.17/8.00 => ( ( ( insert_nat @ A2 @ A )
% 14.17/8.00 = ( insert_nat @ B @ B4 ) )
% 14.17/8.00 = ( ( ( A2 = B )
% 14.17/8.00 => ( A = B4 ) )
% 14.17/8.00 & ( ( A2 != B )
% 14.17/8.00 => ? [C3: set_nat] :
% 14.17/8.00 ( ( A
% 14.17/8.00 = ( insert_nat @ B @ C3 ) )
% 14.17/8.00 & ~ ( member_nat @ B @ C3 )
% 14.17/8.00 & ( B4
% 14.17/8.00 = ( insert_nat @ A2 @ C3 ) )
% 14.17/8.00 & ~ ( member_nat @ A2 @ C3 ) ) ) ) ) ) ) ).
% 14.17/8.00
% 14.17/8.00 thf(fact_265_insert__eq__iff,axiom,
% 14.17/8.00 ! [A2: int,A: set_int,B: int,B4: set_int] :
% 14.17/8.00 ( ~ ( member_int @ A2 @ A )
% 14.17/8.00 => ( ~ ( member_int @ B @ B4 )
% 14.17/8.00 => ( ( ( insert_int @ A2 @ A )
% 14.17/8.00 = ( insert_int @ B @ B4 ) )
% 14.17/8.00 = ( ( ( A2 = B )
% 14.17/8.00 => ( A = B4 ) )
% 14.17/8.00 & ( ( A2 != B )
% 14.17/8.00 => ? [C3: set_int] :
% 14.17/8.00 ( ( A
% 14.17/8.00 = ( insert_int @ B @ C3 ) )
% 14.17/8.00 & ~ ( member_int @ B @ C3 )
% 14.17/8.00 & ( B4
% 14.17/8.00 = ( insert_int @ A2 @ C3 ) )
% 14.17/8.00 & ~ ( member_int @ A2 @ C3 ) ) ) ) ) ) ) ).
% 14.17/8.00
% 14.17/8.00 thf(fact_266_insert__eq__iff,axiom,
% 14.17/8.00 ! [A2: $o,A: set_o,B: $o,B4: set_o] :
% 14.17/8.00 ( ~ ( member_o @ A2 @ A )
% 14.17/8.00 => ( ~ ( member_o @ B @ B4 )
% 14.17/8.00 => ( ( ( insert_o @ A2 @ A )
% 14.17/8.00 = ( insert_o @ B @ B4 ) )
% 14.17/8.00 = ( ( ( A2 = B )
% 14.17/8.00 => ( A = B4 ) )
% 14.17/8.00 & ( ( A2 = ~ B )
% 14.17/8.00 => ? [C3: set_o] :
% 14.17/8.00 ( ( A
% 14.17/8.00 = ( insert_o @ B @ C3 ) )
% 14.17/8.00 & ~ ( member_o @ B @ C3 )
% 14.17/8.00 & ( B4
% 14.17/8.00 = ( insert_o @ A2 @ C3 ) )
% 14.17/8.00 & ~ ( member_o @ A2 @ C3 ) ) ) ) ) ) ) ).
% 14.17/8.00
% 14.17/8.00 thf(fact_267_insert__eq__iff,axiom,
% 14.17/8.00 ! [A2: product_unit,A: set_Product_unit,B: product_unit,B4: set_Product_unit] :
% 14.17/8.00 ( ~ ( member_Product_unit @ A2 @ A )
% 14.17/8.00 => ( ~ ( member_Product_unit @ B @ B4 )
% 14.17/8.00 => ( ( ( insert_Product_unit @ A2 @ A )
% 14.17/8.00 = ( insert_Product_unit @ B @ B4 ) )
% 14.17/8.00 = ( ( ( A2 = B )
% 14.17/8.00 => ( A = B4 ) )
% 14.17/8.00 & ( ( A2 != B )
% 14.17/8.00 => ? [C3: set_Product_unit] :
% 14.17/8.00 ( ( A
% 14.17/8.00 = ( insert_Product_unit @ B @ C3 ) )
% 14.17/8.00 & ~ ( member_Product_unit @ B @ C3 )
% 14.17/8.00 & ( B4
% 14.17/8.00 = ( insert_Product_unit @ A2 @ C3 ) )
% 14.17/8.00 & ~ ( member_Product_unit @ A2 @ C3 ) ) ) ) ) ) ) ).
% 14.17/8.00
% 14.17/8.00 thf(fact_268_insert__commute,axiom,
% 14.17/8.00 ! [X: nat,Y: nat,A: set_nat] :
% 14.17/8.00 ( ( insert_nat @ X @ ( insert_nat @ Y @ A ) )
% 14.17/8.00 = ( insert_nat @ Y @ ( insert_nat @ X @ A ) ) ) ).
% 14.17/8.00
% 14.17/8.00 thf(fact_269_insert__commute,axiom,
% 14.17/8.00 ! [X: int,Y: int,A: set_int] :
% 14.17/8.00 ( ( insert_int @ X @ ( insert_int @ Y @ A ) )
% 14.17/8.00 = ( insert_int @ Y @ ( insert_int @ X @ A ) ) ) ).
% 14.17/8.00
% 14.17/8.00 thf(fact_270_insert__commute,axiom,
% 14.17/8.00 ! [X: $o,Y: $o,A: set_o] :
% 14.17/8.00 ( ( insert_o @ X @ ( insert_o @ Y @ A ) )
% 14.17/8.00 = ( insert_o @ Y @ ( insert_o @ X @ A ) ) ) ).
% 14.17/8.00
% 14.17/8.00 thf(fact_271_insert__commute,axiom,
% 14.17/8.00 ! [X: product_unit,Y: product_unit,A: set_Product_unit] :
% 14.17/8.00 ( ( insert_Product_unit @ X @ ( insert_Product_unit @ Y @ A ) )
% 14.17/8.00 = ( insert_Product_unit @ Y @ ( insert_Product_unit @ X @ A ) ) ) ).
% 14.17/8.00
% 14.17/8.00 thf(fact_272_mk__disjoint__insert,axiom,
% 14.17/8.00 ! [A2: produc3658429121746597890et_nat > $o,A: set_Pr4532377907799695533_nat_o] :
% 14.17/8.00 ( ( member6576561426505652726_nat_o @ A2 @ A )
% 14.17/8.00 => ? [B5: set_Pr4532377907799695533_nat_o] :
% 14.17/8.00 ( ( A
% 14.17/8.00 = ( insert5175938949040314269_nat_o @ A2 @ B5 ) )
% 14.17/8.00 & ~ ( member6576561426505652726_nat_o @ A2 @ B5 ) ) ) ).
% 14.17/8.00
% 14.17/8.00 thf(fact_273_mk__disjoint__insert,axiom,
% 14.17/8.00 ! [A2: nat,A: set_nat] :
% 14.17/8.00 ( ( member_nat @ A2 @ A )
% 14.17/8.00 => ? [B5: set_nat] :
% 14.17/8.00 ( ( A
% 14.17/8.00 = ( insert_nat @ A2 @ B5 ) )
% 14.17/8.00 & ~ ( member_nat @ A2 @ B5 ) ) ) ).
% 14.17/8.00
% 14.17/8.00 thf(fact_274_mk__disjoint__insert,axiom,
% 14.17/8.00 ! [A2: int,A: set_int] :
% 14.17/8.00 ( ( member_int @ A2 @ A )
% 14.17/8.00 => ? [B5: set_int] :
% 14.17/8.00 ( ( A
% 14.17/8.00 = ( insert_int @ A2 @ B5 ) )
% 14.17/8.00 & ~ ( member_int @ A2 @ B5 ) ) ) ).
% 14.17/8.00
% 14.17/8.00 thf(fact_275_mk__disjoint__insert,axiom,
% 14.17/8.00 ! [A2: $o,A: set_o] :
% 14.17/8.00 ( ( member_o @ A2 @ A )
% 14.17/8.00 => ? [B5: set_o] :
% 14.17/8.00 ( ( A
% 14.17/8.00 = ( insert_o @ A2 @ B5 ) )
% 14.17/8.00 & ~ ( member_o @ A2 @ B5 ) ) ) ).
% 14.17/8.00
% 14.17/8.00 thf(fact_276_mk__disjoint__insert,axiom,
% 14.17/8.00 ! [A2: product_unit,A: set_Product_unit] :
% 14.17/8.00 ( ( member_Product_unit @ A2 @ A )
% 14.17/8.00 => ? [B5: set_Product_unit] :
% 14.17/8.00 ( ( A
% 14.17/8.00 = ( insert_Product_unit @ A2 @ B5 ) )
% 14.17/8.00 & ~ ( member_Product_unit @ A2 @ B5 ) ) ) ).
% 14.17/8.00
% 14.17/8.00 thf(fact_277_singletonD,axiom,
% 14.17/8.00 ! [B: product_unit,A2: product_unit] :
% 14.17/8.00 ( ( member_Product_unit @ B @ ( insert_Product_unit @ A2 @ bot_bo3957492148770167129t_unit ) )
% 14.17/8.00 => ( B = A2 ) ) ).
% 14.17/8.00
% 14.17/8.00 thf(fact_278_singletonD,axiom,
% 14.17/8.00 ! [B: $o,A2: $o] :
% 14.17/8.00 ( ( member_o @ B @ ( insert_o @ A2 @ bot_bot_set_o ) )
% 14.17/8.00 => ( B = A2 ) ) ).
% 14.17/8.00
% 14.17/8.00 thf(fact_279_singletonD,axiom,
% 14.17/8.00 ! [B: nat,A2: nat] :
% 14.17/8.00 ( ( member_nat @ B @ ( insert_nat @ A2 @ bot_bot_set_nat ) )
% 14.17/8.00 => ( B = A2 ) ) ).
% 14.17/8.00
% 14.17/8.00 thf(fact_280_singletonD,axiom,
% 14.17/8.00 ! [B: int,A2: int] :
% 14.17/8.00 ( ( member_int @ B @ ( insert_int @ A2 @ bot_bot_set_int ) )
% 14.17/8.00 => ( B = A2 ) ) ).
% 14.17/8.00
% 14.17/8.00 thf(fact_281_singletonD,axiom,
% 14.17/8.00 ! [B: produc3658429121746597890et_nat > $o,A2: produc3658429121746597890et_nat > $o] :
% 14.17/8.00 ( ( member6576561426505652726_nat_o @ B @ ( insert5175938949040314269_nat_o @ A2 @ bot_bo7824918357723582233_nat_o ) )
% 14.17/8.00 => ( B = A2 ) ) ).
% 14.17/8.00
% 14.17/8.00 thf(fact_282_singleton__iff,axiom,
% 14.17/8.00 ! [B: product_unit,A2: product_unit] :
% 14.17/8.00 ( ( member_Product_unit @ B @ ( insert_Product_unit @ A2 @ bot_bo3957492148770167129t_unit ) )
% 14.17/8.00 = ( B = A2 ) ) ).
% 14.17/8.00
% 14.17/8.00 thf(fact_283_singleton__iff,axiom,
% 14.17/8.00 ! [B: $o,A2: $o] :
% 14.17/8.00 ( ( member_o @ B @ ( insert_o @ A2 @ bot_bot_set_o ) )
% 14.17/8.00 = ( B = A2 ) ) ).
% 14.17/8.00
% 14.17/8.00 thf(fact_284_singleton__iff,axiom,
% 14.17/8.00 ! [B: nat,A2: nat] :
% 14.17/8.00 ( ( member_nat @ B @ ( insert_nat @ A2 @ bot_bot_set_nat ) )
% 14.17/8.00 = ( B = A2 ) ) ).
% 14.17/8.00
% 14.17/8.00 thf(fact_285_singleton__iff,axiom,
% 14.17/8.00 ! [B: int,A2: int] :
% 14.17/8.00 ( ( member_int @ B @ ( insert_int @ A2 @ bot_bot_set_int ) )
% 14.17/8.00 = ( B = A2 ) ) ).
% 14.17/8.00
% 14.17/8.00 thf(fact_286_singleton__iff,axiom,
% 14.17/8.00 ! [B: produc3658429121746597890et_nat > $o,A2: produc3658429121746597890et_nat > $o] :
% 14.17/8.00 ( ( member6576561426505652726_nat_o @ B @ ( insert5175938949040314269_nat_o @ A2 @ bot_bo7824918357723582233_nat_o ) )
% 14.17/8.00 = ( B = A2 ) ) ).
% 14.17/8.00
% 14.17/8.00 thf(fact_287_doubleton__eq__iff,axiom,
% 14.17/8.00 ! [A2: product_unit,B: product_unit,C: product_unit,D2: product_unit] :
% 14.17/8.00 ( ( ( insert_Product_unit @ A2 @ ( insert_Product_unit @ B @ bot_bo3957492148770167129t_unit ) )
% 14.17/8.00 = ( insert_Product_unit @ C @ ( insert_Product_unit @ D2 @ bot_bo3957492148770167129t_unit ) ) )
% 14.17/8.00 = ( ( ( A2 = C )
% 14.17/8.00 & ( B = D2 ) )
% 14.17/8.00 | ( ( A2 = D2 )
% 14.17/8.00 & ( B = C ) ) ) ) ).
% 14.17/8.00
% 14.17/8.00 thf(fact_288_doubleton__eq__iff,axiom,
% 14.17/8.00 ! [A2: $o,B: $o,C: $o,D2: $o] :
% 14.17/8.00 ( ( ( insert_o @ A2 @ ( insert_o @ B @ bot_bot_set_o ) )
% 14.17/8.00 = ( insert_o @ C @ ( insert_o @ D2 @ bot_bot_set_o ) ) )
% 14.17/8.00 = ( ( ( A2 = C )
% 14.17/8.00 & ( B = D2 ) )
% 14.17/8.00 | ( ( A2 = D2 )
% 14.17/8.00 & ( B = C ) ) ) ) ).
% 14.17/8.00
% 14.17/8.00 thf(fact_289_doubleton__eq__iff,axiom,
% 14.17/8.00 ! [A2: nat,B: nat,C: nat,D2: nat] :
% 14.17/8.00 ( ( ( insert_nat @ A2 @ ( insert_nat @ B @ bot_bot_set_nat ) )
% 14.17/8.00 = ( insert_nat @ C @ ( insert_nat @ D2 @ bot_bot_set_nat ) ) )
% 14.17/8.00 = ( ( ( A2 = C )
% 14.17/8.00 & ( B = D2 ) )
% 14.17/8.00 | ( ( A2 = D2 )
% 14.17/8.00 & ( B = C ) ) ) ) ).
% 14.17/8.00
% 14.17/8.00 thf(fact_290_doubleton__eq__iff,axiom,
% 14.17/8.00 ! [A2: int,B: int,C: int,D2: int] :
% 14.17/8.00 ( ( ( insert_int @ A2 @ ( insert_int @ B @ bot_bot_set_int ) )
% 14.17/8.00 = ( insert_int @ C @ ( insert_int @ D2 @ bot_bot_set_int ) ) )
% 14.17/8.00 = ( ( ( A2 = C )
% 14.17/8.00 & ( B = D2 ) )
% 14.17/8.00 | ( ( A2 = D2 )
% 14.17/8.00 & ( B = C ) ) ) ) ).
% 14.17/8.00
% 14.17/8.00 thf(fact_291_doubleton__eq__iff,axiom,
% 14.17/8.00 ! [A2: produc3658429121746597890et_nat > $o,B: produc3658429121746597890et_nat > $o,C: produc3658429121746597890et_nat > $o,D2: produc3658429121746597890et_nat > $o] :
% 14.17/8.00 ( ( ( insert5175938949040314269_nat_o @ A2 @ ( insert5175938949040314269_nat_o @ B @ bot_bo7824918357723582233_nat_o ) )
% 14.17/8.00 = ( insert5175938949040314269_nat_o @ C @ ( insert5175938949040314269_nat_o @ D2 @ bot_bo7824918357723582233_nat_o ) ) )
% 14.17/8.00 = ( ( ( A2 = C )
% 14.17/8.00 & ( B = D2 ) )
% 14.17/8.00 | ( ( A2 = D2 )
% 14.17/8.00 & ( B = C ) ) ) ) ).
% 14.17/8.00
% 14.17/8.00 thf(fact_292_insert__not__empty,axiom,
% 14.17/8.00 ! [A2: product_unit,A: set_Product_unit] :
% 14.17/8.00 ( ( insert_Product_unit @ A2 @ A )
% 14.17/8.00 != bot_bo3957492148770167129t_unit ) ).
% 14.17/8.00
% 14.17/8.00 thf(fact_293_insert__not__empty,axiom,
% 14.17/8.00 ! [A2: $o,A: set_o] :
% 14.17/8.00 ( ( insert_o @ A2 @ A )
% 14.17/8.00 != bot_bot_set_o ) ).
% 14.17/8.00
% 14.17/8.00 thf(fact_294_insert__not__empty,axiom,
% 14.17/8.00 ! [A2: nat,A: set_nat] :
% 14.17/8.00 ( ( insert_nat @ A2 @ A )
% 14.17/8.00 != bot_bot_set_nat ) ).
% 14.17/8.00
% 14.17/8.00 thf(fact_295_insert__not__empty,axiom,
% 14.17/8.00 ! [A2: int,A: set_int] :
% 14.17/8.00 ( ( insert_int @ A2 @ A )
% 14.17/8.00 != bot_bot_set_int ) ).
% 14.17/8.00
% 14.17/8.00 thf(fact_296_insert__not__empty,axiom,
% 14.17/8.00 ! [A2: produc3658429121746597890et_nat > $o,A: set_Pr4532377907799695533_nat_o] :
% 14.17/8.00 ( ( insert5175938949040314269_nat_o @ A2 @ A )
% 14.17/8.00 != bot_bo7824918357723582233_nat_o ) ).
% 14.17/8.00
% 14.17/8.00 thf(fact_297_singleton__inject,axiom,
% 14.17/8.00 ! [A2: product_unit,B: product_unit] :
% 14.17/8.00 ( ( ( insert_Product_unit @ A2 @ bot_bo3957492148770167129t_unit )
% 14.17/8.00 = ( insert_Product_unit @ B @ bot_bo3957492148770167129t_unit ) )
% 14.17/8.00 => ( A2 = B ) ) ).
% 14.17/8.00
% 14.17/8.00 thf(fact_298_singleton__inject,axiom,
% 14.17/8.00 ! [A2: $o,B: $o] :
% 14.17/8.00 ( ( ( insert_o @ A2 @ bot_bot_set_o )
% 14.17/8.00 = ( insert_o @ B @ bot_bot_set_o ) )
% 14.17/8.00 => ( A2 = B ) ) ).
% 14.17/8.00
% 14.17/8.00 thf(fact_299_singleton__inject,axiom,
% 14.17/8.00 ! [A2: nat,B: nat] :
% 14.17/8.00 ( ( ( insert_nat @ A2 @ bot_bot_set_nat )
% 14.17/8.00 = ( insert_nat @ B @ bot_bot_set_nat ) )
% 14.17/8.00 => ( A2 = B ) ) ).
% 14.17/8.00
% 14.17/8.00 thf(fact_300_singleton__inject,axiom,
% 14.17/8.00 ! [A2: int,B: int] :
% 14.17/8.00 ( ( ( insert_int @ A2 @ bot_bot_set_int )
% 14.17/8.00 = ( insert_int @ B @ bot_bot_set_int ) )
% 14.17/8.00 => ( A2 = B ) ) ).
% 14.17/8.00
% 14.17/8.00 thf(fact_301_singleton__inject,axiom,
% 14.17/8.00 ! [A2: produc3658429121746597890et_nat > $o,B: produc3658429121746597890et_nat > $o] :
% 14.17/8.00 ( ( ( insert5175938949040314269_nat_o @ A2 @ bot_bo7824918357723582233_nat_o )
% 14.17/8.00 = ( insert5175938949040314269_nat_o @ B @ bot_bo7824918357723582233_nat_o ) )
% 14.17/8.00 => ( A2 = B ) ) ).
% 14.17/8.00
% 14.17/8.00 thf(fact_302_curryE,axiom,
% 14.17/8.00 ! [F: produc3658429121746597890et_nat > $o,A2: heap_e7401611519738050253t_unit,B: set_nat] :
% 14.17/8.00 ( ( produc1807123492089482958_nat_o @ F @ A2 @ B )
% 14.17/8.00 => ( F @ ( produc7507926704131184380et_nat @ A2 @ B ) ) ) ).
% 14.17/8.00
% 14.17/8.00 thf(fact_303_curryE,axiom,
% 14.17/8.00 ! [F: product_prod_nat_nat > $o,A2: nat,B: nat] :
% 14.17/8.00 ( ( produc1310100445399344235_nat_o @ F @ A2 @ B )
% 14.17/8.00 => ( F @ ( product_Pair_nat_nat @ A2 @ B ) ) ) ).
% 14.17/8.00
% 14.17/8.00 thf(fact_304_curryE,axiom,
% 14.17/8.00 ! [F: product_prod_int_int > $o,A2: int,B: int] :
% 14.17/8.00 ( ( produc175634133007206835_int_o @ F @ A2 @ B )
% 14.17/8.00 => ( F @ ( product_Pair_int_int @ A2 @ B ) ) ) ).
% 14.17/8.00
% 14.17/8.00 thf(fact_305_curryE,axiom,
% 14.17/8.00 ! [F: produc3925858234332021118et_nat > $o,A2: produc3658429121746597890et_nat > $o,B: produc3658429121746597890et_nat] :
% 14.17/8.00 ( ( produc6216949301066131538_nat_o @ F @ A2 @ B )
% 14.17/8.00 => ( F @ ( produc5001842942810119800et_nat @ A2 @ B ) ) ) ).
% 14.17/8.00
% 14.17/8.00 thf(fact_306_curryE,axiom,
% 14.17/8.00 ! [F: produc2732055786443039994et_nat > $o,A2: produc3658429121746597890et_nat > $o,B: produc3925858234332021118et_nat] :
% 14.17/8.00 ( ( produc5101573711933517782_nat_o @ F @ A2 @ B )
% 14.17/8.00 => ( F @ ( produc2245416461498447860et_nat @ A2 @ B ) ) ) ).
% 14.17/8.00
% 14.17/8.00 thf(fact_307_curryD,axiom,
% 14.17/8.00 ! [F: produc3658429121746597890et_nat > $o,A2: heap_e7401611519738050253t_unit,B: set_nat] :
% 14.17/8.00 ( ( produc1807123492089482958_nat_o @ F @ A2 @ B )
% 14.17/8.00 => ( F @ ( produc7507926704131184380et_nat @ A2 @ B ) ) ) ).
% 14.17/8.00
% 14.17/8.00 thf(fact_308_curryD,axiom,
% 14.17/8.00 ! [F: product_prod_nat_nat > $o,A2: nat,B: nat] :
% 14.17/8.00 ( ( produc1310100445399344235_nat_o @ F @ A2 @ B )
% 14.17/8.00 => ( F @ ( product_Pair_nat_nat @ A2 @ B ) ) ) ).
% 14.17/8.00
% 14.17/8.00 thf(fact_309_curryD,axiom,
% 14.17/8.00 ! [F: product_prod_int_int > $o,A2: int,B: int] :
% 14.17/8.00 ( ( produc175634133007206835_int_o @ F @ A2 @ B )
% 14.17/8.00 => ( F @ ( product_Pair_int_int @ A2 @ B ) ) ) ).
% 14.17/8.00
% 14.17/8.00 thf(fact_310_curryD,axiom,
% 14.17/8.00 ! [F: produc3925858234332021118et_nat > $o,A2: produc3658429121746597890et_nat > $o,B: produc3658429121746597890et_nat] :
% 14.17/8.00 ( ( produc6216949301066131538_nat_o @ F @ A2 @ B )
% 14.17/8.00 => ( F @ ( produc5001842942810119800et_nat @ A2 @ B ) ) ) ).
% 14.17/8.00
% 14.17/8.00 thf(fact_311_curryD,axiom,
% 14.17/8.00 ! [F: produc2732055786443039994et_nat > $o,A2: produc3658429121746597890et_nat > $o,B: produc3925858234332021118et_nat] :
% 14.17/8.00 ( ( produc5101573711933517782_nat_o @ F @ A2 @ B )
% 14.17/8.00 => ( F @ ( produc2245416461498447860et_nat @ A2 @ B ) ) ) ).
% 14.17/8.00
% 14.17/8.00 thf(fact_312_is__singleton__def,axiom,
% 14.17/8.00 ( is_sin2160648248035936513t_unit
% 14.17/8.00 = ( ^ [A3: set_Product_unit] :
% 14.17/8.00 ? [X2: product_unit] :
% 14.17/8.00 ( A3
% 14.17/8.00 = ( insert_Product_unit @ X2 @ bot_bo3957492148770167129t_unit ) ) ) ) ).
% 14.17/8.00
% 14.17/8.00 thf(fact_313_is__singleton__def,axiom,
% 14.17/8.00 ( is_singleton_o
% 14.17/8.00 = ( ^ [A3: set_o] :
% 14.17/8.00 ? [X2: $o] :
% 14.17/8.00 ( A3
% 14.17/8.00 = ( insert_o @ X2 @ bot_bot_set_o ) ) ) ) ).
% 14.17/8.00
% 14.17/8.00 thf(fact_314_is__singleton__def,axiom,
% 14.17/8.00 ( is_singleton_nat
% 14.17/8.00 = ( ^ [A3: set_nat] :
% 14.17/8.00 ? [X2: nat] :
% 14.17/8.00 ( A3
% 14.17/8.00 = ( insert_nat @ X2 @ bot_bot_set_nat ) ) ) ) ).
% 14.17/8.00
% 14.17/8.00 thf(fact_315_is__singleton__def,axiom,
% 14.17/8.00 ( is_singleton_int
% 14.17/8.00 = ( ^ [A3: set_int] :
% 14.17/8.00 ? [X2: int] :
% 14.17/8.00 ( A3
% 14.17/8.00 = ( insert_int @ X2 @ bot_bot_set_int ) ) ) ) ).
% 14.17/8.00
% 14.17/8.00 thf(fact_316_is__singleton__def,axiom,
% 14.17/8.00 ( is_sin5180296473474724033_nat_o
% 14.17/8.00 = ( ^ [A3: set_Pr4532377907799695533_nat_o] :
% 14.17/8.00 ? [X2: produc3658429121746597890et_nat > $o] :
% 14.17/8.00 ( A3
% 14.17/8.00 = ( insert5175938949040314269_nat_o @ X2 @ bot_bo7824918357723582233_nat_o ) ) ) ) ).
% 14.17/8.00
% 14.17/8.00 thf(fact_317_is__singletonE,axiom,
% 14.17/8.00 ! [A: set_Product_unit] :
% 14.17/8.00 ( ( is_sin2160648248035936513t_unit @ A )
% 14.17/8.00 => ~ ! [X3: product_unit] :
% 14.17/8.00 ( A
% 14.17/8.00 != ( insert_Product_unit @ X3 @ bot_bo3957492148770167129t_unit ) ) ) ).
% 14.17/8.00
% 14.17/8.00 thf(fact_318_is__singletonE,axiom,
% 14.17/8.00 ! [A: set_o] :
% 14.17/8.00 ( ( is_singleton_o @ A )
% 14.17/8.00 => ~ ! [X3: $o] :
% 14.17/8.00 ( A
% 14.17/8.00 != ( insert_o @ X3 @ bot_bot_set_o ) ) ) ).
% 14.17/8.00
% 14.17/8.00 thf(fact_319_is__singletonE,axiom,
% 14.17/8.00 ! [A: set_nat] :
% 14.17/8.00 ( ( is_singleton_nat @ A )
% 14.17/8.00 => ~ ! [X3: nat] :
% 14.17/8.00 ( A
% 14.17/8.00 != ( insert_nat @ X3 @ bot_bot_set_nat ) ) ) ).
% 14.17/8.00
% 14.17/8.00 thf(fact_320_is__singletonE,axiom,
% 14.17/8.00 ! [A: set_int] :
% 14.17/8.00 ( ( is_singleton_int @ A )
% 14.17/8.00 => ~ ! [X3: int] :
% 14.17/8.00 ( A
% 14.17/8.00 != ( insert_int @ X3 @ bot_bot_set_int ) ) ) ).
% 14.17/8.00
% 14.17/8.00 thf(fact_321_is__singletonE,axiom,
% 14.17/8.00 ! [A: set_Pr4532377907799695533_nat_o] :
% 14.17/8.00 ( ( is_sin5180296473474724033_nat_o @ A )
% 14.17/8.00 => ~ ! [X3: produc3658429121746597890et_nat > $o] :
% 14.17/8.00 ( A
% 14.17/8.00 != ( insert5175938949040314269_nat_o @ X3 @ bot_bo7824918357723582233_nat_o ) ) ) ).
% 14.17/8.00
% 14.17/8.00 thf(fact_322_is__pure__assn__basic__simps_I1_J,axiom,
% 14.17/8.00 is_pure_assn @ bot_bot_assn ).
% 14.17/8.00
% 14.17/8.00 thf(fact_323_is__singleton__the__elem,axiom,
% 14.17/8.00 ( is_sin2160648248035936513t_unit
% 14.17/8.00 = ( ^ [A3: set_Product_unit] :
% 14.17/8.00 ( A3
% 14.17/8.00 = ( insert_Product_unit @ ( the_el608902216710161154t_unit @ A3 ) @ bot_bo3957492148770167129t_unit ) ) ) ) ).
% 14.17/8.00
% 14.17/8.00 thf(fact_324_is__singleton__the__elem,axiom,
% 14.17/8.00 ( is_singleton_o
% 14.17/8.00 = ( ^ [A3: set_o] :
% 14.17/8.00 ( A3
% 14.17/8.00 = ( insert_o @ ( the_elem_o @ A3 ) @ bot_bot_set_o ) ) ) ) ).
% 14.17/8.00
% 14.17/8.00 thf(fact_325_is__singleton__the__elem,axiom,
% 14.17/8.00 ( is_singleton_nat
% 14.17/8.00 = ( ^ [A3: set_nat] :
% 14.17/8.00 ( A3
% 14.17/8.00 = ( insert_nat @ ( the_elem_nat @ A3 ) @ bot_bot_set_nat ) ) ) ) ).
% 14.17/8.00
% 14.17/8.00 thf(fact_326_is__singleton__the__elem,axiom,
% 14.17/8.00 ( is_singleton_int
% 14.17/8.00 = ( ^ [A3: set_int] :
% 14.17/8.00 ( A3
% 14.17/8.00 = ( insert_int @ ( the_elem_int @ A3 ) @ bot_bot_set_int ) ) ) ) ).
% 14.17/8.00
% 14.17/8.00 thf(fact_327_is__singleton__the__elem,axiom,
% 14.17/8.00 ( is_sin5180296473474724033_nat_o
% 14.17/8.00 = ( ^ [A3: set_Pr4532377907799695533_nat_o] :
% 14.17/8.00 ( A3
% 14.17/8.00 = ( insert5175938949040314269_nat_o @ ( the_el5959152443890766018_nat_o @ A3 ) @ bot_bo7824918357723582233_nat_o ) ) ) ) ).
% 14.17/8.00
% 14.17/8.00 thf(fact_328_the__elem__eq,axiom,
% 14.17/8.00 ! [X: product_unit] :
% 14.17/8.00 ( ( the_el608902216710161154t_unit @ ( insert_Product_unit @ X @ bot_bo3957492148770167129t_unit ) )
% 14.17/8.00 = X ) ).
% 14.17/8.00
% 14.17/8.00 thf(fact_329_the__elem__eq,axiom,
% 14.17/8.00 ! [X: $o] :
% 14.17/8.00 ( ( the_elem_o @ ( insert_o @ X @ bot_bot_set_o ) )
% 14.17/8.00 = X ) ).
% 14.17/8.00
% 14.17/8.00 thf(fact_330_the__elem__eq,axiom,
% 14.17/8.00 ! [X: nat] :
% 14.17/8.00 ( ( the_elem_nat @ ( insert_nat @ X @ bot_bot_set_nat ) )
% 14.17/8.00 = X ) ).
% 14.17/8.00
% 14.17/8.00 thf(fact_331_the__elem__eq,axiom,
% 14.17/8.00 ! [X: int] :
% 14.17/8.00 ( ( the_elem_int @ ( insert_int @ X @ bot_bot_set_int ) )
% 14.17/8.00 = X ) ).
% 14.17/8.00
% 14.17/8.00 thf(fact_332_the__elem__eq,axiom,
% 14.17/8.00 ! [X: produc3658429121746597890et_nat > $o] :
% 14.17/8.00 ( ( the_el5959152443890766018_nat_o @ ( insert5175938949040314269_nat_o @ X @ bot_bo7824918357723582233_nat_o ) )
% 14.17/8.00 = X ) ).
% 14.17/8.00
% 14.17/8.00 thf(fact_333_pure__assn__def,axiom,
% 14.17/8.00 ( pure_assn
% 14.17/8.00 = ( ^ [B6: $o] : ( abs_assn @ ( pure_a825153325127701367it_nat @ B6 ) ) ) ) ).
% 14.17/8.00
% 14.17/8.00 thf(fact_334_times__assn__def,axiom,
% 14.17/8.00 ( times_times_assn
% 14.17/8.00 = ( ^ [P2: assn,Q: assn] : ( abs_assn @ ( times_assn_raw @ ( rep_assn @ P2 ) @ ( rep_assn @ Q ) ) ) ) ) ).
% 14.17/8.00
% 14.17/8.00 thf(fact_335_type__definition__def,axiom,
% 14.17/8.00 ( type_d3909072315231072503_nat_o
% 14.17/8.00 = ( ^ [Rep: assn > produc3658429121746597890et_nat > $o,Abs: ( produc3658429121746597890et_nat > $o ) > assn,A3: set_Pr4532377907799695533_nat_o] :
% 14.17/8.00 ( ! [X2: assn] : ( member6576561426505652726_nat_o @ ( Rep @ X2 ) @ A3 )
% 14.17/8.00 & ! [X2: assn] :
% 14.17/8.00 ( ( Abs @ ( Rep @ X2 ) )
% 14.17/8.00 = X2 )
% 14.17/8.00 & ! [Y3: produc3658429121746597890et_nat > $o] :
% 14.17/8.00 ( ( member6576561426505652726_nat_o @ Y3 @ A3 )
% 14.17/8.00 => ( ( Rep @ ( Abs @ Y3 ) )
% 14.17/8.00 = Y3 ) ) ) ) ) ).
% 14.17/8.00
% 14.17/8.00 thf(fact_336_type__definition__def,axiom,
% 14.17/8.00 ( type_d8366093980585677751er_int
% 14.17/8.00 = ( ^ [Rep: code_integer > int,Abs: int > code_integer,A3: set_int] :
% 14.17/8.00 ( ! [X2: code_integer] : ( member_int @ ( Rep @ X2 ) @ A3 )
% 14.17/8.00 & ! [X2: code_integer] :
% 14.17/8.00 ( ( Abs @ ( Rep @ X2 ) )
% 14.17/8.00 = X2 )
% 14.17/8.00 & ! [Y3: int] :
% 14.17/8.00 ( ( member_int @ Y3 @ A3 )
% 14.17/8.00 => ( ( Rep @ ( Abs @ Y3 ) )
% 14.17/8.00 = Y3 ) ) ) ) ) ).
% 14.17/8.00
% 14.17/8.00 thf(fact_337_type__definition__def,axiom,
% 14.17/8.00 ( type_d6188575255521822967unit_o
% 14.17/8.00 = ( ^ [Rep: product_unit > $o,Abs: $o > product_unit,A3: set_o] :
% 14.17/8.00 ( ! [X2: product_unit] : ( member_o @ ( Rep @ X2 ) @ A3 )
% 14.17/8.00 & ! [X2: product_unit] :
% 14.17/8.00 ( ( Abs @ ( Rep @ X2 ) )
% 14.17/8.00 = X2 )
% 14.17/8.00 & ! [Y3: $o] :
% 14.17/8.00 ( ( member_o @ Y3 @ A3 )
% 14.17/8.00 => ( ( Rep @ ( Abs @ Y3 ) )
% 14.17/8.00 = Y3 ) ) ) ) ) ).
% 14.17/8.00
% 14.17/8.00 thf(fact_338_type__definition__def,axiom,
% 14.17/8.00 ( type_d4410041424927559462al_nat
% 14.17/8.00 = ( ^ [Rep: code_natural > nat,Abs: nat > code_natural,A3: set_nat] :
% 14.17/8.00 ( ! [X2: code_natural] : ( member_nat @ ( Rep @ X2 ) @ A3 )
% 14.17/8.00 & ! [X2: code_natural] :
% 14.17/8.00 ( ( Abs @ ( Rep @ X2 ) )
% 14.17/8.00 = X2 )
% 14.17/8.00 & ! [Y3: nat] :
% 14.17/8.00 ( ( member_nat @ Y3 @ A3 )
% 14.17/8.00 => ( ( Rep @ ( Abs @ Y3 ) )
% 14.17/8.00 = Y3 ) ) ) ) ) ).
% 14.17/8.00
% 14.17/8.00 thf(fact_339_type__definition_ORep__inverse,axiom,
% 14.17/8.00 ! [Rep2: assn > produc3658429121746597890et_nat > $o,Abs2: ( produc3658429121746597890et_nat > $o ) > assn,A: set_Pr4532377907799695533_nat_o,X: assn] :
% 14.17/8.00 ( ( type_d3909072315231072503_nat_o @ Rep2 @ Abs2 @ A )
% 14.17/8.00 => ( ( Abs2 @ ( Rep2 @ X ) )
% 14.17/8.00 = X ) ) ).
% 14.17/8.00
% 14.17/8.00 thf(fact_340_type__definition_ORep__inverse,axiom,
% 14.17/8.00 ! [Rep2: code_integer > int,Abs2: int > code_integer,A: set_int,X: code_integer] :
% 14.17/8.00 ( ( type_d8366093980585677751er_int @ Rep2 @ Abs2 @ A )
% 14.17/8.00 => ( ( Abs2 @ ( Rep2 @ X ) )
% 14.17/8.00 = X ) ) ).
% 14.17/8.00
% 14.17/8.00 thf(fact_341_type__definition_ORep__inverse,axiom,
% 14.17/8.00 ! [Rep2: product_unit > $o,Abs2: $o > product_unit,A: set_o,X: product_unit] :
% 14.17/8.00 ( ( type_d6188575255521822967unit_o @ Rep2 @ Abs2 @ A )
% 14.17/8.00 => ( ( Abs2 @ ( Rep2 @ X ) )
% 14.17/8.00 = X ) ) ).
% 14.17/8.00
% 14.17/8.00 thf(fact_342_type__definition_ORep__inverse,axiom,
% 14.17/8.00 ! [Rep2: code_natural > nat,Abs2: nat > code_natural,A: set_nat,X: code_natural] :
% 14.17/8.00 ( ( type_d4410041424927559462al_nat @ Rep2 @ Abs2 @ A )
% 14.17/8.00 => ( ( Abs2 @ ( Rep2 @ X ) )
% 14.17/8.00 = X ) ) ).
% 14.17/8.00
% 14.17/8.00 thf(fact_343_type__definition_OAbs__inverse,axiom,
% 14.17/8.00 ! [Rep2: assn > produc3658429121746597890et_nat > $o,Abs2: ( produc3658429121746597890et_nat > $o ) > assn,A: set_Pr4532377907799695533_nat_o,Y: produc3658429121746597890et_nat > $o] :
% 14.17/8.00 ( ( type_d3909072315231072503_nat_o @ Rep2 @ Abs2 @ A )
% 14.17/8.00 => ( ( member6576561426505652726_nat_o @ Y @ A )
% 14.17/8.00 => ( ( Rep2 @ ( Abs2 @ Y ) )
% 14.17/8.00 = Y ) ) ) ).
% 14.17/8.00
% 14.17/8.00 thf(fact_344_type__definition_OAbs__inverse,axiom,
% 14.17/8.00 ! [Rep2: code_integer > int,Abs2: int > code_integer,A: set_int,Y: int] :
% 14.17/8.00 ( ( type_d8366093980585677751er_int @ Rep2 @ Abs2 @ A )
% 14.17/8.00 => ( ( member_int @ Y @ A )
% 14.17/8.00 => ( ( Rep2 @ ( Abs2 @ Y ) )
% 14.17/8.00 = Y ) ) ) ).
% 14.17/8.00
% 14.17/8.00 thf(fact_345_type__definition_OAbs__inverse,axiom,
% 14.17/8.00 ! [Rep2: product_unit > $o,Abs2: $o > product_unit,A: set_o,Y: $o] :
% 14.17/8.00 ( ( type_d6188575255521822967unit_o @ Rep2 @ Abs2 @ A )
% 14.17/8.00 => ( ( member_o @ Y @ A )
% 14.17/8.00 => ( ( Rep2 @ ( Abs2 @ Y ) )
% 14.17/8.00 = Y ) ) ) ).
% 14.17/8.00
% 14.17/8.00 thf(fact_346_type__definition_OAbs__inverse,axiom,
% 14.17/8.00 ! [Rep2: code_natural > nat,Abs2: nat > code_natural,A: set_nat,Y: nat] :
% 14.17/8.00 ( ( type_d4410041424927559462al_nat @ Rep2 @ Abs2 @ A )
% 14.17/8.00 => ( ( member_nat @ Y @ A )
% 14.17/8.00 => ( ( Rep2 @ ( Abs2 @ Y ) )
% 14.17/8.00 = Y ) ) ) ).
% 14.17/8.00
% 14.17/8.00 thf(fact_347_type__definition_ORep__inject,axiom,
% 14.17/8.00 ! [Rep2: assn > produc3658429121746597890et_nat > $o,Abs2: ( produc3658429121746597890et_nat > $o ) > assn,A: set_Pr4532377907799695533_nat_o,X: assn,Y: assn] :
% 14.17/8.00 ( ( type_d3909072315231072503_nat_o @ Rep2 @ Abs2 @ A )
% 14.17/8.00 => ( ( ( Rep2 @ X )
% 14.17/8.00 = ( Rep2 @ Y ) )
% 14.17/8.00 = ( X = Y ) ) ) ).
% 14.17/8.00
% 14.17/8.00 thf(fact_348_type__definition_ORep__inject,axiom,
% 14.17/8.00 ! [Rep2: code_integer > int,Abs2: int > code_integer,A: set_int,X: code_integer,Y: code_integer] :
% 14.17/8.00 ( ( type_d8366093980585677751er_int @ Rep2 @ Abs2 @ A )
% 14.17/8.00 => ( ( ( Rep2 @ X )
% 14.17/8.00 = ( Rep2 @ Y ) )
% 14.17/8.00 = ( X = Y ) ) ) ).
% 14.17/8.00
% 14.17/8.00 thf(fact_349_type__definition_ORep__inject,axiom,
% 14.17/8.00 ! [Rep2: product_unit > $o,Abs2: $o > product_unit,A: set_o,X: product_unit,Y: product_unit] :
% 14.17/8.00 ( ( type_d6188575255521822967unit_o @ Rep2 @ Abs2 @ A )
% 14.17/8.00 => ( ( ( Rep2 @ X )
% 14.17/8.00 = ( Rep2 @ Y ) )
% 14.17/8.00 = ( X = Y ) ) ) ).
% 14.17/8.00
% 14.17/8.00 thf(fact_350_type__definition_ORep__inject,axiom,
% 14.17/8.00 ! [Rep2: code_natural > nat,Abs2: nat > code_natural,A: set_nat,X: code_natural,Y: code_natural] :
% 14.17/8.00 ( ( type_d4410041424927559462al_nat @ Rep2 @ Abs2 @ A )
% 14.17/8.00 => ( ( ( Rep2 @ X )
% 14.17/8.00 = ( Rep2 @ Y ) )
% 14.17/8.00 = ( X = Y ) ) ) ).
% 14.17/8.00
% 14.17/8.00 thf(fact_351_type__definition_ORep__induct,axiom,
% 14.17/8.00 ! [Rep2: assn > produc3658429121746597890et_nat > $o,Abs2: ( produc3658429121746597890et_nat > $o ) > assn,A: set_Pr4532377907799695533_nat_o,Y: produc3658429121746597890et_nat > $o,P: ( produc3658429121746597890et_nat > $o ) > $o] :
% 14.17/8.00 ( ( type_d3909072315231072503_nat_o @ Rep2 @ Abs2 @ A )
% 14.17/8.00 => ( ( member6576561426505652726_nat_o @ Y @ A )
% 14.17/8.00 => ( ! [X3: assn] : ( P @ ( Rep2 @ X3 ) )
% 14.17/8.00 => ( P @ Y ) ) ) ) ).
% 14.17/8.00
% 14.17/8.00 thf(fact_352_type__definition_ORep__induct,axiom,
% 14.17/8.00 ! [Rep2: code_integer > int,Abs2: int > code_integer,A: set_int,Y: int,P: int > $o] :
% 14.17/8.00 ( ( type_d8366093980585677751er_int @ Rep2 @ Abs2 @ A )
% 14.17/8.00 => ( ( member_int @ Y @ A )
% 14.17/8.00 => ( ! [X3: code_integer] : ( P @ ( Rep2 @ X3 ) )
% 14.17/8.00 => ( P @ Y ) ) ) ) ).
% 14.17/8.00
% 14.17/8.00 thf(fact_353_type__definition_ORep__induct,axiom,
% 14.17/8.00 ! [Rep2: product_unit > $o,Abs2: $o > product_unit,A: set_o,Y: $o,P: $o > $o] :
% 14.17/8.00 ( ( type_d6188575255521822967unit_o @ Rep2 @ Abs2 @ A )
% 14.17/8.00 => ( ( member_o @ Y @ A )
% 14.17/8.00 => ( ! [X3: product_unit] : ( P @ ( Rep2 @ X3 ) )
% 14.17/8.00 => ( P @ Y ) ) ) ) ).
% 14.17/8.00
% 14.17/8.00 thf(fact_354_type__definition_ORep__induct,axiom,
% 14.17/8.00 ! [Rep2: code_natural > nat,Abs2: nat > code_natural,A: set_nat,Y: nat,P: nat > $o] :
% 14.17/8.00 ( ( type_d4410041424927559462al_nat @ Rep2 @ Abs2 @ A )
% 14.17/8.00 => ( ( member_nat @ Y @ A )
% 14.17/8.00 => ( ! [X3: code_natural] : ( P @ ( Rep2 @ X3 ) )
% 14.17/8.00 => ( P @ Y ) ) ) ) ).
% 14.17/8.00
% 14.17/8.00 thf(fact_355_type__definition_OAbs__inject,axiom,
% 14.17/8.00 ! [Rep2: assn > produc3658429121746597890et_nat > $o,Abs2: ( produc3658429121746597890et_nat > $o ) > assn,A: set_Pr4532377907799695533_nat_o,X: produc3658429121746597890et_nat > $o,Y: produc3658429121746597890et_nat > $o] :
% 14.17/8.00 ( ( type_d3909072315231072503_nat_o @ Rep2 @ Abs2 @ A )
% 14.17/8.00 => ( ( member6576561426505652726_nat_o @ X @ A )
% 14.17/8.00 => ( ( member6576561426505652726_nat_o @ Y @ A )
% 14.17/8.00 => ( ( ( Abs2 @ X )
% 14.17/8.00 = ( Abs2 @ Y ) )
% 14.17/8.00 = ( X = Y ) ) ) ) ) ).
% 14.17/8.00
% 14.17/8.00 thf(fact_356_type__definition_OAbs__inject,axiom,
% 14.17/8.00 ! [Rep2: code_integer > int,Abs2: int > code_integer,A: set_int,X: int,Y: int] :
% 14.17/8.00 ( ( type_d8366093980585677751er_int @ Rep2 @ Abs2 @ A )
% 14.17/8.00 => ( ( member_int @ X @ A )
% 14.17/8.00 => ( ( member_int @ Y @ A )
% 14.17/8.00 => ( ( ( Abs2 @ X )
% 14.17/8.00 = ( Abs2 @ Y ) )
% 14.17/8.00 = ( X = Y ) ) ) ) ) ).
% 14.17/8.00
% 14.17/8.00 thf(fact_357_type__definition_OAbs__inject,axiom,
% 14.17/8.00 ! [Rep2: product_unit > $o,Abs2: $o > product_unit,A: set_o,X: $o,Y: $o] :
% 14.17/8.00 ( ( type_d6188575255521822967unit_o @ Rep2 @ Abs2 @ A )
% 14.17/8.00 => ( ( member_o @ X @ A )
% 14.17/8.00 => ( ( member_o @ Y @ A )
% 14.17/8.00 => ( ( ( Abs2 @ X )
% 14.17/8.00 = ( Abs2 @ Y ) )
% 14.17/8.00 = ( X = Y ) ) ) ) ) ).
% 14.17/8.00
% 14.17/8.00 thf(fact_358_type__definition_OAbs__inject,axiom,
% 14.17/8.00 ! [Rep2: code_natural > nat,Abs2: nat > code_natural,A: set_nat,X: nat,Y: nat] :
% 14.17/8.00 ( ( type_d4410041424927559462al_nat @ Rep2 @ Abs2 @ A )
% 14.17/8.00 => ( ( member_nat @ X @ A )
% 14.17/8.00 => ( ( member_nat @ Y @ A )
% 14.17/8.00 => ( ( ( Abs2 @ X )
% 14.17/8.00 = ( Abs2 @ Y ) )
% 14.17/8.00 = ( X = Y ) ) ) ) ) ).
% 14.17/8.00
% 14.17/8.00 thf(fact_359_pure__assn__eq__conv,axiom,
% 14.17/8.00 ! [P: $o,Q2: $o] :
% 14.17/8.00 ( ( ( pure_assn @ P )
% 14.17/8.00 = ( pure_assn @ Q2 ) )
% 14.17/8.00 = ( P = Q2 ) ) ).
% 14.17/8.00
% 14.17/8.00 thf(fact_360_mult_Oright__neutral,axiom,
% 14.17/8.00 ! [A2: code_integer] :
% 14.17/8.00 ( ( times_3573771949741848930nteger @ A2 @ one_one_Code_integer )
% 14.17/8.00 = A2 ) ).
% 14.17/8.00
% 14.17/8.00 thf(fact_361_mult_Oright__neutral,axiom,
% 14.17/8.00 ! [A2: code_natural] :
% 14.17/8.00 ( ( times_2397367101498566445atural @ A2 @ one_one_Code_natural )
% 14.17/8.00 = A2 ) ).
% 14.17/8.00
% 14.17/8.00 thf(fact_362_mult_Oright__neutral,axiom,
% 14.17/8.00 ! [A2: assn] :
% 14.17/8.00 ( ( times_times_assn @ A2 @ one_one_assn )
% 14.17/8.00 = A2 ) ).
% 14.17/8.00
% 14.17/8.00 thf(fact_363_mult_Oright__neutral,axiom,
% 14.17/8.00 ! [A2: rat] :
% 14.17/8.00 ( ( times_times_rat @ A2 @ one_one_rat )
% 14.17/8.00 = A2 ) ).
% 14.17/8.00
% 14.17/8.00 thf(fact_364_mult_Oright__neutral,axiom,
% 14.17/8.00 ! [A2: nat] :
% 14.17/8.00 ( ( times_times_nat @ A2 @ one_one_nat )
% 14.17/8.00 = A2 ) ).
% 14.17/8.00
% 14.17/8.00 thf(fact_365_mult_Oright__neutral,axiom,
% 14.17/8.00 ! [A2: int] :
% 14.17/8.00 ( ( times_times_int @ A2 @ one_one_int )
% 14.17/8.00 = A2 ) ).
% 14.17/8.00
% 14.17/8.00 thf(fact_366_mult__1,axiom,
% 14.17/8.00 ! [A2: code_integer] :
% 14.17/8.00 ( ( times_3573771949741848930nteger @ one_one_Code_integer @ A2 )
% 14.17/8.00 = A2 ) ).
% 14.17/8.00
% 14.17/8.00 thf(fact_367_mult__1,axiom,
% 14.17/8.00 ! [A2: code_natural] :
% 14.17/8.00 ( ( times_2397367101498566445atural @ one_one_Code_natural @ A2 )
% 14.17/8.00 = A2 ) ).
% 14.17/8.00
% 14.17/8.00 thf(fact_368_mult__1,axiom,
% 14.17/8.00 ! [A2: assn] :
% 14.17/8.00 ( ( times_times_assn @ one_one_assn @ A2 )
% 14.17/8.00 = A2 ) ).
% 14.17/8.00
% 14.17/8.00 thf(fact_369_mult__1,axiom,
% 14.17/8.00 ! [A2: rat] :
% 14.17/8.00 ( ( times_times_rat @ one_one_rat @ A2 )
% 14.17/8.00 = A2 ) ).
% 14.17/8.00
% 14.17/8.00 thf(fact_370_mult__1,axiom,
% 14.17/8.00 ! [A2: nat] :
% 14.17/8.00 ( ( times_times_nat @ one_one_nat @ A2 )
% 14.17/8.00 = A2 ) ).
% 14.17/8.00
% 14.17/8.00 thf(fact_371_mult__1,axiom,
% 14.17/8.00 ! [A2: int] :
% 14.17/8.00 ( ( times_times_int @ one_one_int @ A2 )
% 14.17/8.00 = A2 ) ).
% 14.17/8.00
% 14.17/8.00 thf(fact_372_merge__pure__star,axiom,
% 14.17/8.00 ! [A2: $o,B: $o] :
% 14.17/8.00 ( ( times_times_assn @ ( pure_assn @ A2 ) @ ( pure_assn @ B ) )
% 14.17/8.00 = ( pure_assn
% 14.17/8.00 @ ( A2
% 14.17/8.00 & B ) ) ) ).
% 14.17/8.00
% 14.17/8.00 thf(fact_373_star__false__right,axiom,
% 14.17/8.00 ! [P: assn] :
% 14.17/8.00 ( ( times_times_assn @ P @ bot_bot_assn )
% 14.17/8.00 = bot_bot_assn ) ).
% 14.17/8.00
% 14.17/8.00 thf(fact_374_star__false__left,axiom,
% 14.17/8.00 ! [P: assn] :
% 14.17/8.00 ( ( times_times_assn @ bot_bot_assn @ P )
% 14.17/8.00 = bot_bot_assn ) ).
% 14.17/8.00
% 14.17/8.00 thf(fact_375_pure__false,axiom,
% 14.17/8.00 ( ( pure_assn @ $false )
% 14.17/8.00 = bot_bot_assn ) ).
% 14.17/8.00
% 14.17/8.00 thf(fact_376_pure__assn__eq__false__iff,axiom,
% 14.17/8.00 ! [P: $o] :
% 14.17/8.00 ( ( ( pure_assn @ P )
% 14.17/8.00 = bot_bot_assn )
% 14.17/8.00 = ~ P ) ).
% 14.17/8.00
% 14.17/8.00 thf(fact_377_is__pure__assn__starI,axiom,
% 14.17/8.00 ! [A2: assn,B: assn] :
% 14.17/8.00 ( ( is_pure_assn @ A2 )
% 14.17/8.00 => ( ( is_pure_assn @ B )
% 14.17/8.00 => ( is_pure_assn @ ( times_times_assn @ A2 @ B ) ) ) ) ).
% 14.17/8.00
% 14.17/8.00 thf(fact_378_is__pure__assn__pure,axiom,
% 14.17/8.00 ! [P: $o] : ( is_pure_assn @ ( pure_assn @ P ) ) ).
% 14.17/8.00
% 14.17/8.00 thf(fact_379_pure__true,axiom,
% 14.17/8.00 ( ( pure_assn @ $true )
% 14.17/8.00 = one_one_assn ) ).
% 14.17/8.00
% 14.17/8.00 thf(fact_380_pure__assn__eq__emp__iff,axiom,
% 14.17/8.00 ! [P: $o] :
% 14.17/8.00 ( ( ( pure_assn @ P )
% 14.17/8.00 = one_one_assn )
% 14.17/8.00 = P ) ).
% 14.17/8.00
% 14.17/8.00 thf(fact_381_assn__times__comm,axiom,
% 14.17/8.00 ( times_times_assn
% 14.17/8.00 = ( ^ [P2: assn,Q: assn] : ( times_times_assn @ Q @ P2 ) ) ) ).
% 14.17/8.00
% 14.17/8.00 thf(fact_382_assn__times__assoc,axiom,
% 14.17/8.00 ! [P: assn,Q2: assn,R3: assn] :
% 14.17/8.00 ( ( times_times_assn @ ( times_times_assn @ P @ Q2 ) @ R3 )
% 14.17/8.00 = ( times_times_assn @ P @ ( times_times_assn @ Q2 @ R3 ) ) ) ).
% 14.17/8.00
% 14.17/8.00 thf(fact_383_ab__semigroup__mult__class_Omult_Oleft__commute,axiom,
% 14.17/8.00 ! [B: assn,A2: assn,C: assn] :
% 14.17/8.00 ( ( times_times_assn @ B @ ( times_times_assn @ A2 @ C ) )
% 14.17/8.00 = ( times_times_assn @ A2 @ ( times_times_assn @ B @ C ) ) ) ).
% 14.17/8.00
% 14.17/8.00 thf(fact_384_ab__semigroup__mult__class_Omult_Oleft__commute,axiom,
% 14.17/8.00 ! [B: rat,A2: rat,C: rat] :
% 14.17/8.00 ( ( times_times_rat @ B @ ( times_times_rat @ A2 @ C ) )
% 14.17/8.00 = ( times_times_rat @ A2 @ ( times_times_rat @ B @ C ) ) ) ).
% 14.17/8.00
% 14.17/8.00 thf(fact_385_ab__semigroup__mult__class_Omult_Oleft__commute,axiom,
% 14.17/8.00 ! [B: nat,A2: nat,C: nat] :
% 14.17/8.00 ( ( times_times_nat @ B @ ( times_times_nat @ A2 @ C ) )
% 14.17/8.00 = ( times_times_nat @ A2 @ ( times_times_nat @ B @ C ) ) ) ).
% 14.17/8.00
% 14.17/8.00 thf(fact_386_ab__semigroup__mult__class_Omult_Oleft__commute,axiom,
% 14.17/8.00 ! [B: int,A2: int,C: int] :
% 14.17/8.00 ( ( times_times_int @ B @ ( times_times_int @ A2 @ C ) )
% 14.17/8.00 = ( times_times_int @ A2 @ ( times_times_int @ B @ C ) ) ) ).
% 14.17/8.00
% 14.17/8.00 thf(fact_387_ab__semigroup__mult__class_Omult_Ocommute,axiom,
% 14.17/8.00 ( times_times_assn
% 14.17/8.00 = ( ^ [A6: assn,B6: assn] : ( times_times_assn @ B6 @ A6 ) ) ) ).
% 14.17/8.00
% 14.17/8.00 thf(fact_388_ab__semigroup__mult__class_Omult_Ocommute,axiom,
% 14.17/8.00 ( times_times_rat
% 14.17/8.00 = ( ^ [A6: rat,B6: rat] : ( times_times_rat @ B6 @ A6 ) ) ) ).
% 14.17/8.00
% 14.17/8.00 thf(fact_389_ab__semigroup__mult__class_Omult_Ocommute,axiom,
% 14.17/8.00 ( times_times_nat
% 14.17/8.00 = ( ^ [A6: nat,B6: nat] : ( times_times_nat @ B6 @ A6 ) ) ) ).
% 14.17/8.00
% 14.17/8.00 thf(fact_390_ab__semigroup__mult__class_Omult_Ocommute,axiom,
% 14.17/8.00 ( times_times_int
% 14.17/8.00 = ( ^ [A6: int,B6: int] : ( times_times_int @ B6 @ A6 ) ) ) ).
% 14.17/8.00
% 14.17/8.00 thf(fact_391_mult_Oassoc,axiom,
% 14.17/8.00 ! [A2: assn,B: assn,C: assn] :
% 14.17/8.00 ( ( times_times_assn @ ( times_times_assn @ A2 @ B ) @ C )
% 14.17/8.00 = ( times_times_assn @ A2 @ ( times_times_assn @ B @ C ) ) ) ).
% 14.17/8.00
% 14.17/8.00 thf(fact_392_mult_Oassoc,axiom,
% 14.17/8.00 ! [A2: rat,B: rat,C: rat] :
% 14.17/8.00 ( ( times_times_rat @ ( times_times_rat @ A2 @ B ) @ C )
% 14.17/8.00 = ( times_times_rat @ A2 @ ( times_times_rat @ B @ C ) ) ) ).
% 14.17/8.00
% 14.17/8.00 thf(fact_393_mult_Oassoc,axiom,
% 14.17/8.00 ! [A2: nat,B: nat,C: nat] :
% 14.17/8.00 ( ( times_times_nat @ ( times_times_nat @ A2 @ B ) @ C )
% 14.17/8.00 = ( times_times_nat @ A2 @ ( times_times_nat @ B @ C ) ) ) ).
% 14.17/8.00
% 14.17/8.00 thf(fact_394_mult_Oassoc,axiom,
% 14.17/8.00 ! [A2: int,B: int,C: int] :
% 14.17/8.00 ( ( times_times_int @ ( times_times_int @ A2 @ B ) @ C )
% 14.17/8.00 = ( times_times_int @ A2 @ ( times_times_int @ B @ C ) ) ) ).
% 14.17/8.00
% 14.17/8.00 thf(fact_395_comm__monoid__mult__class_Omult__1,axiom,
% 14.17/8.00 ! [A2: code_integer] :
% 14.17/8.00 ( ( times_3573771949741848930nteger @ one_one_Code_integer @ A2 )
% 14.17/8.00 = A2 ) ).
% 14.17/8.00
% 14.17/8.00 thf(fact_396_comm__monoid__mult__class_Omult__1,axiom,
% 14.17/8.00 ! [A2: code_natural] :
% 14.17/8.00 ( ( times_2397367101498566445atural @ one_one_Code_natural @ A2 )
% 14.17/8.00 = A2 ) ).
% 14.17/8.00
% 14.17/8.00 thf(fact_397_comm__monoid__mult__class_Omult__1,axiom,
% 14.17/8.00 ! [A2: assn] :
% 14.17/8.00 ( ( times_times_assn @ one_one_assn @ A2 )
% 14.17/8.00 = A2 ) ).
% 14.17/8.00
% 14.17/8.00 thf(fact_398_comm__monoid__mult__class_Omult__1,axiom,
% 14.17/8.00 ! [A2: rat] :
% 14.17/8.00 ( ( times_times_rat @ one_one_rat @ A2 )
% 14.17/8.00 = A2 ) ).
% 14.17/8.00
% 14.17/8.00 thf(fact_399_comm__monoid__mult__class_Omult__1,axiom,
% 14.17/8.00 ! [A2: nat] :
% 14.17/8.00 ( ( times_times_nat @ one_one_nat @ A2 )
% 14.17/8.00 = A2 ) ).
% 14.17/8.00
% 14.17/8.00 thf(fact_400_comm__monoid__mult__class_Omult__1,axiom,
% 14.17/8.00 ! [A2: int] :
% 14.17/8.00 ( ( times_times_int @ one_one_int @ A2 )
% 14.17/8.00 = A2 ) ).
% 14.17/8.00
% 14.17/8.00 thf(fact_401_mult_Ocomm__neutral,axiom,
% 14.17/8.00 ! [A2: code_integer] :
% 14.17/8.00 ( ( times_3573771949741848930nteger @ A2 @ one_one_Code_integer )
% 14.17/8.00 = A2 ) ).
% 14.17/8.00
% 14.17/8.00 thf(fact_402_mult_Ocomm__neutral,axiom,
% 14.17/8.00 ! [A2: code_natural] :
% 14.17/8.00 ( ( times_2397367101498566445atural @ A2 @ one_one_Code_natural )
% 14.17/8.00 = A2 ) ).
% 14.17/8.00
% 14.17/8.00 thf(fact_403_mult_Ocomm__neutral,axiom,
% 14.17/8.00 ! [A2: assn] :
% 14.17/8.00 ( ( times_times_assn @ A2 @ one_one_assn )
% 14.17/8.00 = A2 ) ).
% 14.17/8.00
% 14.17/8.00 thf(fact_404_mult_Ocomm__neutral,axiom,
% 14.17/8.00 ! [A2: rat] :
% 14.17/8.00 ( ( times_times_rat @ A2 @ one_one_rat )
% 14.17/8.00 = A2 ) ).
% 14.17/8.00
% 14.17/8.00 thf(fact_405_mult_Ocomm__neutral,axiom,
% 14.17/8.00 ! [A2: nat] :
% 14.17/8.00 ( ( times_times_nat @ A2 @ one_one_nat )
% 14.17/8.00 = A2 ) ).
% 14.17/8.00
% 14.17/8.00 thf(fact_406_mult_Ocomm__neutral,axiom,
% 14.17/8.00 ! [A2: int] :
% 14.17/8.00 ( ( times_times_int @ A2 @ one_one_int )
% 14.17/8.00 = A2 ) ).
% 14.17/8.00
% 14.17/8.00 thf(fact_407_mod__starE,axiom,
% 14.17/8.00 ! [A2: assn,B: assn,H2: produc3658429121746597890et_nat] :
% 14.17/8.00 ( ( rep_assn @ ( times_times_assn @ A2 @ B ) @ H2 )
% 14.17/8.00 => ~ ( ? [X_1: produc3658429121746597890et_nat] : ( rep_assn @ A2 @ X_1 )
% 14.17/8.00 => ! [H_2: produc3658429121746597890et_nat] :
% 14.17/8.00 ~ ( rep_assn @ B @ H_2 ) ) ) ).
% 14.17/8.00
% 14.17/8.00 thf(fact_408_mod__starD,axiom,
% 14.17/8.00 ! [A: assn,B4: assn,H2: produc3658429121746597890et_nat] :
% 14.17/8.00 ( ( rep_assn @ ( times_times_assn @ A @ B4 ) @ H2 )
% 14.17/8.00 => ? [H1: produc3658429121746597890et_nat,H22: produc3658429121746597890et_nat] :
% 14.17/8.00 ( ( rep_assn @ A @ H1 )
% 14.17/8.00 & ( rep_assn @ B4 @ H22 ) ) ) ).
% 14.17/8.00
% 14.17/8.00 thf(fact_409_assn__one__left,axiom,
% 14.17/8.00 ! [P: assn] :
% 14.17/8.00 ( ( times_times_assn @ one_one_assn @ P )
% 14.17/8.00 = P ) ).
% 14.17/8.00
% 14.17/8.00 thf(fact_410_is__pure__assnE,axiom,
% 14.17/8.00 ! [A2: assn] :
% 14.17/8.00 ( ( is_pure_assn @ A2 )
% 14.17/8.00 => ~ ! [P4: $o] :
% 14.17/8.00 ( A2
% 14.17/8.00 != ( pure_assn @ P4 ) ) ) ).
% 14.17/8.00
% 14.17/8.00 thf(fact_411_is__pure__assn__def,axiom,
% 14.17/8.00 ( is_pure_assn
% 14.17/8.00 = ( ^ [A6: assn] :
% 14.17/8.00 ? [P2: $o] :
% 14.17/8.00 ( A6
% 14.17/8.00 = ( pure_assn @ P2 ) ) ) ) ).
% 14.17/8.00
% 14.17/8.00 thf(fact_412_type__definition_ORep,axiom,
% 14.17/8.00 ! [Rep2: assn > produc3658429121746597890et_nat > $o,Abs2: ( produc3658429121746597890et_nat > $o ) > assn,A: set_Pr4532377907799695533_nat_o,X: assn] :
% 14.17/8.00 ( ( type_d3909072315231072503_nat_o @ Rep2 @ Abs2 @ A )
% 14.17/8.00 => ( member6576561426505652726_nat_o @ ( Rep2 @ X ) @ A ) ) ).
% 14.17/8.00
% 14.17/8.00 thf(fact_413_type__definition_ORep,axiom,
% 14.17/8.00 ! [Rep2: code_integer > int,Abs2: int > code_integer,A: set_int,X: code_integer] :
% 14.17/8.00 ( ( type_d8366093980585677751er_int @ Rep2 @ Abs2 @ A )
% 14.17/8.00 => ( member_int @ ( Rep2 @ X ) @ A ) ) ).
% 14.17/8.00
% 14.17/8.00 thf(fact_414_type__definition_ORep,axiom,
% 14.17/8.00 ! [Rep2: product_unit > $o,Abs2: $o > product_unit,A: set_o,X: product_unit] :
% 14.17/8.00 ( ( type_d6188575255521822967unit_o @ Rep2 @ Abs2 @ A )
% 14.17/8.00 => ( member_o @ ( Rep2 @ X ) @ A ) ) ).
% 14.17/8.00
% 14.17/8.00 thf(fact_415_type__definition_ORep,axiom,
% 14.17/8.00 ! [Rep2: code_natural > nat,Abs2: nat > code_natural,A: set_nat,X: code_natural] :
% 14.17/8.00 ( ( type_d4410041424927559462al_nat @ Rep2 @ Abs2 @ A )
% 14.17/8.00 => ( member_nat @ ( Rep2 @ X ) @ A ) ) ).
% 14.17/8.00
% 14.17/8.00 thf(fact_416_type__definition_Ointro,axiom,
% 14.17/8.00 ! [Rep2: assn > produc3658429121746597890et_nat > $o,A: set_Pr4532377907799695533_nat_o,Abs2: ( produc3658429121746597890et_nat > $o ) > assn] :
% 14.17/8.00 ( ! [X3: assn] : ( member6576561426505652726_nat_o @ ( Rep2 @ X3 ) @ A )
% 14.17/8.00 => ( ! [X3: assn] :
% 14.17/8.00 ( ( Abs2 @ ( Rep2 @ X3 ) )
% 14.17/8.00 = X3 )
% 14.17/8.00 => ( ! [Y2: produc3658429121746597890et_nat > $o] :
% 14.17/8.00 ( ( member6576561426505652726_nat_o @ Y2 @ A )
% 14.17/8.00 => ( ( Rep2 @ ( Abs2 @ Y2 ) )
% 14.17/8.00 = Y2 ) )
% 14.17/8.00 => ( type_d3909072315231072503_nat_o @ Rep2 @ Abs2 @ A ) ) ) ) ).
% 14.17/8.00
% 14.17/8.00 thf(fact_417_type__definition_Ointro,axiom,
% 14.17/8.00 ! [Rep2: code_integer > int,A: set_int,Abs2: int > code_integer] :
% 14.17/8.00 ( ! [X3: code_integer] : ( member_int @ ( Rep2 @ X3 ) @ A )
% 14.17/8.00 => ( ! [X3: code_integer] :
% 14.17/8.00 ( ( Abs2 @ ( Rep2 @ X3 ) )
% 14.17/8.00 = X3 )
% 14.17/8.00 => ( ! [Y2: int] :
% 14.17/8.00 ( ( member_int @ Y2 @ A )
% 14.17/8.00 => ( ( Rep2 @ ( Abs2 @ Y2 ) )
% 14.17/8.00 = Y2 ) )
% 14.17/8.00 => ( type_d8366093980585677751er_int @ Rep2 @ Abs2 @ A ) ) ) ) ).
% 14.17/8.00
% 14.17/8.00 thf(fact_418_type__definition_Ointro,axiom,
% 14.17/8.00 ! [Rep2: product_unit > $o,A: set_o,Abs2: $o > product_unit] :
% 14.17/8.00 ( ! [X3: product_unit] : ( member_o @ ( Rep2 @ X3 ) @ A )
% 14.17/8.00 => ( ! [X3: product_unit] :
% 14.17/8.00 ( ( Abs2 @ ( Rep2 @ X3 ) )
% 14.17/8.00 = X3 )
% 14.17/8.00 => ( ! [Y2: $o] :
% 14.17/8.00 ( ( member_o @ Y2 @ A )
% 14.17/8.00 => ( ( Rep2 @ ( Abs2 @ Y2 ) )
% 14.17/8.00 = Y2 ) )
% 14.17/8.00 => ( type_d6188575255521822967unit_o @ Rep2 @ Abs2 @ A ) ) ) ) ).
% 14.17/8.00
% 14.17/8.00 thf(fact_419_type__definition_Ointro,axiom,
% 14.17/8.00 ! [Rep2: code_natural > nat,A: set_nat,Abs2: nat > code_natural] :
% 14.17/8.00 ( ! [X3: code_natural] : ( member_nat @ ( Rep2 @ X3 ) @ A )
% 14.17/8.00 => ( ! [X3: code_natural] :
% 14.17/8.00 ( ( Abs2 @ ( Rep2 @ X3 ) )
% 14.17/8.00 = X3 )
% 14.17/8.00 => ( ! [Y2: nat] :
% 14.17/8.00 ( ( member_nat @ Y2 @ A )
% 14.17/8.00 => ( ( Rep2 @ ( Abs2 @ Y2 ) )
% 14.17/8.00 = Y2 ) )
% 14.17/8.00 => ( type_d4410041424927559462al_nat @ Rep2 @ Abs2 @ A ) ) ) ) ).
% 14.17/8.00
% 14.17/8.00 thf(fact_420_type__definition_OAbs__cases,axiom,
% 14.17/8.00 ! [Rep2: assn > produc3658429121746597890et_nat > $o,Abs2: ( produc3658429121746597890et_nat > $o ) > assn,A: set_Pr4532377907799695533_nat_o,X: assn] :
% 14.17/8.00 ( ( type_d3909072315231072503_nat_o @ Rep2 @ Abs2 @ A )
% 14.17/8.00 => ~ ! [Y2: produc3658429121746597890et_nat > $o] :
% 14.17/8.00 ( ( X
% 14.17/8.00 = ( Abs2 @ Y2 ) )
% 14.17/8.00 => ~ ( member6576561426505652726_nat_o @ Y2 @ A ) ) ) ).
% 14.17/8.00
% 14.17/8.00 thf(fact_421_type__definition_OAbs__cases,axiom,
% 14.17/8.00 ! [Rep2: code_integer > int,Abs2: int > code_integer,A: set_int,X: code_integer] :
% 14.17/8.00 ( ( type_d8366093980585677751er_int @ Rep2 @ Abs2 @ A )
% 14.17/8.00 => ~ ! [Y2: int] :
% 14.17/8.00 ( ( X
% 14.17/8.00 = ( Abs2 @ Y2 ) )
% 14.17/8.00 => ~ ( member_int @ Y2 @ A ) ) ) ).
% 14.17/8.00
% 14.17/8.00 thf(fact_422_type__definition_OAbs__cases,axiom,
% 14.17/8.00 ! [Rep2: product_unit > $o,Abs2: $o > product_unit,A: set_o,X: product_unit] :
% 14.17/8.00 ( ( type_d6188575255521822967unit_o @ Rep2 @ Abs2 @ A )
% 14.17/8.00 => ~ ! [Y2: $o] :
% 14.17/8.00 ( ( X
% 14.17/8.00 = ( Abs2 @ Y2 ) )
% 14.17/8.00 => ~ ( member_o @ Y2 @ A ) ) ) ).
% 14.17/8.00
% 14.17/8.00 thf(fact_423_type__definition_OAbs__cases,axiom,
% 14.17/8.00 ! [Rep2: code_natural > nat,Abs2: nat > code_natural,A: set_nat,X: code_natural] :
% 14.17/8.00 ( ( type_d4410041424927559462al_nat @ Rep2 @ Abs2 @ A )
% 14.17/8.00 => ~ ! [Y2: nat] :
% 14.17/8.00 ( ( X
% 14.17/8.00 = ( Abs2 @ Y2 ) )
% 14.17/8.00 => ~ ( member_nat @ Y2 @ A ) ) ) ).
% 14.17/8.00
% 14.17/8.00 thf(fact_424_type__definition_ORep__cases,axiom,
% 14.17/8.00 ! [Rep2: assn > produc3658429121746597890et_nat > $o,Abs2: ( produc3658429121746597890et_nat > $o ) > assn,A: set_Pr4532377907799695533_nat_o,Y: produc3658429121746597890et_nat > $o] :
% 14.17/8.00 ( ( type_d3909072315231072503_nat_o @ Rep2 @ Abs2 @ A )
% 14.17/8.00 => ( ( member6576561426505652726_nat_o @ Y @ A )
% 14.17/8.00 => ~ ! [X3: assn] :
% 14.17/8.00 ( Y
% 14.17/8.00 != ( Rep2 @ X3 ) ) ) ) ).
% 14.17/8.00
% 14.17/8.00 thf(fact_425_type__definition_ORep__cases,axiom,
% 14.17/8.00 ! [Rep2: code_integer > int,Abs2: int > code_integer,A: set_int,Y: int] :
% 14.17/8.00 ( ( type_d8366093980585677751er_int @ Rep2 @ Abs2 @ A )
% 14.17/8.00 => ( ( member_int @ Y @ A )
% 14.17/8.00 => ~ ! [X3: code_integer] :
% 14.17/8.00 ( Y
% 14.17/8.00 != ( Rep2 @ X3 ) ) ) ) ).
% 14.17/8.00
% 14.17/8.00 thf(fact_426_type__definition_ORep__cases,axiom,
% 14.17/8.00 ! [Rep2: product_unit > $o,Abs2: $o > product_unit,A: set_o,Y: $o] :
% 14.17/8.00 ( ( type_d6188575255521822967unit_o @ Rep2 @ Abs2 @ A )
% 14.17/8.00 => ( ( member_o @ Y @ A )
% 14.17/8.00 => ~ ! [X3: product_unit] :
% 14.17/8.00 ( Y
% 14.17/8.00 = ( ~ ( Rep2 @ X3 ) ) ) ) ) ).
% 14.17/8.00
% 14.17/8.00 thf(fact_427_type__definition_ORep__cases,axiom,
% 14.17/8.00 ! [Rep2: code_natural > nat,Abs2: nat > code_natural,A: set_nat,Y: nat] :
% 14.17/8.00 ( ( type_d4410041424927559462al_nat @ Rep2 @ Abs2 @ A )
% 14.17/8.00 => ( ( member_nat @ Y @ A )
% 14.17/8.00 => ~ ! [X3: code_natural] :
% 14.17/8.00 ( Y
% 14.17/8.00 != ( Rep2 @ X3 ) ) ) ) ).
% 14.17/8.00
% 14.17/8.00 thf(fact_428_type__definition_OAbs__induct,axiom,
% 14.17/8.00 ! [Rep2: assn > produc3658429121746597890et_nat > $o,Abs2: ( produc3658429121746597890et_nat > $o ) > assn,A: set_Pr4532377907799695533_nat_o,P: assn > $o,X: assn] :
% 14.17/8.00 ( ( type_d3909072315231072503_nat_o @ Rep2 @ Abs2 @ A )
% 14.17/8.00 => ( ! [Y2: produc3658429121746597890et_nat > $o] :
% 14.17/8.00 ( ( member6576561426505652726_nat_o @ Y2 @ A )
% 14.17/8.00 => ( P @ ( Abs2 @ Y2 ) ) )
% 14.17/8.00 => ( P @ X ) ) ) ).
% 14.17/8.00
% 14.17/8.00 thf(fact_429_type__definition_OAbs__induct,axiom,
% 14.17/8.00 ! [Rep2: code_integer > int,Abs2: int > code_integer,A: set_int,P: code_integer > $o,X: code_integer] :
% 14.17/8.00 ( ( type_d8366093980585677751er_int @ Rep2 @ Abs2 @ A )
% 14.17/8.00 => ( ! [Y2: int] :
% 14.17/8.00 ( ( member_int @ Y2 @ A )
% 14.17/8.00 => ( P @ ( Abs2 @ Y2 ) ) )
% 14.17/8.00 => ( P @ X ) ) ) ).
% 14.17/8.00
% 14.17/8.00 thf(fact_430_type__definition_OAbs__induct,axiom,
% 14.17/8.00 ! [Rep2: product_unit > $o,Abs2: $o > product_unit,A: set_o,P: product_unit > $o,X: product_unit] :
% 14.17/8.00 ( ( type_d6188575255521822967unit_o @ Rep2 @ Abs2 @ A )
% 14.17/8.00 => ( ! [Y2: $o] :
% 14.17/8.00 ( ( member_o @ Y2 @ A )
% 14.17/8.00 => ( P @ ( Abs2 @ Y2 ) ) )
% 14.17/8.00 => ( P @ X ) ) ) ).
% 14.17/8.00
% 14.17/8.00 thf(fact_431_type__definition_OAbs__induct,axiom,
% 14.17/8.00 ! [Rep2: code_natural > nat,Abs2: nat > code_natural,A: set_nat,P: code_natural > $o,X: code_natural] :
% 14.17/8.00 ( ( type_d4410041424927559462al_nat @ Rep2 @ Abs2 @ A )
% 14.17/8.00 => ( ! [Y2: nat] :
% 14.17/8.00 ( ( member_nat @ Y2 @ A )
% 14.17/8.00 => ( P @ ( Abs2 @ Y2 ) ) )
% 14.17/8.00 => ( P @ X ) ) ) ).
% 14.17/8.00
% 14.17/8.00 thf(fact_432_total__on__singleton,axiom,
% 14.17/8.00 ! [X: product_unit] : ( total_5665700600833547665t_unit @ ( insert_Product_unit @ X @ bot_bo3957492148770167129t_unit ) @ ( insert4616891024742550839t_unit @ ( produc5692694829384537815t_unit @ X @ X ) @ bot_bo1654022762305692595t_unit ) ) ).
% 14.17/8.00
% 14.17/8.00 thf(fact_433_total__on__singleton,axiom,
% 14.17/8.00 ! [X: $o] : ( total_on_o @ ( insert_o @ X @ bot_bot_set_o ) @ ( insert6201435330877294327od_o_o @ ( product_Pair_o_o @ X @ X ) @ bot_bo7073875226086086771od_o_o ) ) ).
% 14.17/8.00
% 14.17/8.00 thf(fact_434_total__on__singleton,axiom,
% 14.17/8.00 ! [X: nat] : ( total_on_nat @ ( insert_nat @ X @ bot_bot_set_nat ) @ ( insert8211810215607154385at_nat @ ( product_Pair_nat_nat @ X @ X ) @ bot_bo2099793752762293965at_nat ) ) ).
% 14.17/8.00
% 14.17/8.00 thf(fact_435_total__on__singleton,axiom,
% 14.17/8.00 ! [X: int] : ( total_on_int @ ( insert_int @ X @ bot_bot_set_int ) @ ( insert5033312907999012233nt_int @ ( product_Pair_int_int @ X @ X ) @ bot_bo1796632182523588997nt_int ) ) ).
% 14.17/8.00
% 14.17/8.00 thf(fact_436_total__on__singleton,axiom,
% 14.17/8.00 ! [X: produc3658429121746597890et_nat > $o] : ( total_7429123086324524369_nat_o @ ( insert5175938949040314269_nat_o @ X @ bot_bo7824918357723582233_nat_o ) @ ( insert5309882203591314743_nat_o @ ( produc7368190662567826135_nat_o @ X @ X ) @ bot_bo3657187498359500723_nat_o ) ) ).
% 14.17/8.00
% 14.17/8.00 thf(fact_437_refl__on__singleton,axiom,
% 14.17/8.00 ! [X: product_unit] : ( refl_on_Product_unit @ ( insert_Product_unit @ X @ bot_bo3957492148770167129t_unit ) @ ( insert4616891024742550839t_unit @ ( produc5692694829384537815t_unit @ X @ X ) @ bot_bo1654022762305692595t_unit ) ) ).
% 14.17/8.00
% 14.17/8.00 thf(fact_438_refl__on__singleton,axiom,
% 14.17/8.00 ! [X: $o] : ( refl_on_o @ ( insert_o @ X @ bot_bot_set_o ) @ ( insert6201435330877294327od_o_o @ ( product_Pair_o_o @ X @ X ) @ bot_bo7073875226086086771od_o_o ) ) ).
% 14.17/8.00
% 14.17/8.00 thf(fact_439_refl__on__singleton,axiom,
% 14.17/8.00 ! [X: nat] : ( refl_on_nat @ ( insert_nat @ X @ bot_bot_set_nat ) @ ( insert8211810215607154385at_nat @ ( product_Pair_nat_nat @ X @ X ) @ bot_bo2099793752762293965at_nat ) ) ).
% 14.17/8.00
% 14.17/8.00 thf(fact_440_refl__on__singleton,axiom,
% 14.17/8.00 ! [X: int] : ( refl_on_int @ ( insert_int @ X @ bot_bot_set_int ) @ ( insert5033312907999012233nt_int @ ( product_Pair_int_int @ X @ X ) @ bot_bo1796632182523588997nt_int ) ) ).
% 14.17/8.00
% 14.17/8.00 thf(fact_441_refl__on__singleton,axiom,
% 14.17/8.00 ! [X: produc3658429121746597890et_nat > $o] : ( refl_o1075864025857821388_nat_o @ ( insert5175938949040314269_nat_o @ X @ bot_bo7824918357723582233_nat_o ) @ ( insert5309882203591314743_nat_o @ ( produc7368190662567826135_nat_o @ X @ X ) @ bot_bo3657187498359500723_nat_o ) ) ).
% 14.17/8.00
% 14.17/8.00 thf(fact_442_Id__on__empty,axiom,
% 14.17/8.00 ( ( id_on_Product_unit @ bot_bo3957492148770167129t_unit )
% 14.17/8.00 = bot_bo1654022762305692595t_unit ) ).
% 14.17/8.00
% 14.17/8.00 thf(fact_443_Id__on__empty,axiom,
% 14.17/8.00 ( ( id_on_o @ bot_bot_set_o )
% 14.17/8.00 = bot_bo7073875226086086771od_o_o ) ).
% 14.17/8.00
% 14.17/8.00 thf(fact_444_Id__on__empty,axiom,
% 14.17/8.00 ( ( id_on_nat @ bot_bot_set_nat )
% 14.17/8.00 = bot_bo2099793752762293965at_nat ) ).
% 14.17/8.00
% 14.17/8.00 thf(fact_445_Id__on__empty,axiom,
% 14.17/8.00 ( ( id_on_int @ bot_bot_set_int )
% 14.17/8.00 = bot_bo1796632182523588997nt_int ) ).
% 14.17/8.00
% 14.17/8.00 thf(fact_446_Id__on__empty,axiom,
% 14.17/8.00 ( ( id_on_7438695062414216554_nat_o @ bot_bo7824918357723582233_nat_o )
% 14.17/8.00 = bot_bo3657187498359500723_nat_o ) ).
% 14.17/8.00
% 14.17/8.00 thf(fact_447_linear__order__on__singleton,axiom,
% 14.17/8.00 ! [X: product_unit] : ( order_888282469287196226t_unit @ ( insert_Product_unit @ X @ bot_bo3957492148770167129t_unit ) @ ( insert4616891024742550839t_unit @ ( produc5692694829384537815t_unit @ X @ X ) @ bot_bo1654022762305692595t_unit ) ) ).
% 14.17/8.00
% 14.17/8.00 thf(fact_448_linear__order__on__singleton,axiom,
% 14.17/8.00 ! [X: $o] : ( order_6238756238976269133r_on_o @ ( insert_o @ X @ bot_bot_set_o ) @ ( insert6201435330877294327od_o_o @ ( product_Pair_o_o @ X @ X ) @ bot_bo7073875226086086771od_o_o ) ) ).
% 14.17/8.00
% 14.17/8.00 thf(fact_449_linear__order__on__singleton,axiom,
% 14.17/8.00 ! [X: nat] : ( order_4473980167227706203on_nat @ ( insert_nat @ X @ bot_bot_set_nat ) @ ( insert8211810215607154385at_nat @ ( product_Pair_nat_nat @ X @ X ) @ bot_bo2099793752762293965at_nat ) ) ).
% 14.17/8.00
% 14.17/8.00 thf(fact_450_linear__order__on__singleton,axiom,
% 14.17/8.00 ! [X: int] : ( order_4471489696718655927on_int @ ( insert_int @ X @ bot_bot_set_int ) @ ( insert5033312907999012233nt_int @ ( product_Pair_int_int @ X @ X ) @ bot_bo1796632182523588997nt_int ) ) ).
% 14.17/8.00
% 14.17/8.00 thf(fact_451_linear__order__on__singleton,axiom,
% 14.17/8.00 ! [X: produc3658429121746597890et_nat > $o] : ( order_1145710701291701250_nat_o @ ( insert5175938949040314269_nat_o @ X @ bot_bo7824918357723582233_nat_o ) @ ( insert5309882203591314743_nat_o @ ( produc7368190662567826135_nat_o @ X @ X ) @ bot_bo3657187498359500723_nat_o ) ) ).
% 14.17/8.00
% 14.17/8.00 thf(fact_452_pure__assn__raw_Opelims_I1_J,axiom,
% 14.17/8.00 ! [X: $o,Xa: produc3658429121746597890et_nat,Y: $o] :
% 14.17/8.00 ( ( ( pure_a825153325127701367it_nat @ X @ Xa )
% 14.17/8.00 = Y )
% 14.17/8.00 => ( ( accp_P8458817951426537472et_nat @ pure_a6022498039421069578it_nat @ ( produc3762353314782720579et_nat @ X @ Xa ) )
% 14.17/8.00 => ~ ! [H: heap_e7401611519738050253t_unit,As2: set_nat] :
% 14.17/8.00 ( ( Xa
% 14.17/8.00 = ( produc7507926704131184380et_nat @ H @ As2 ) )
% 14.17/8.00 => ( ( Y
% 14.17/8.00 = ( ( As2 = bot_bot_set_nat )
% 14.17/8.00 & X ) )
% 14.17/8.00 => ~ ( accp_P8458817951426537472et_nat @ pure_a6022498039421069578it_nat @ ( produc3762353314782720579et_nat @ X @ ( produc7507926704131184380et_nat @ H @ As2 ) ) ) ) ) ) ) ).
% 14.17/8.00
% 14.17/8.00 thf(fact_453_pure__assn__raw_Opelims_I2_J,axiom,
% 14.17/8.00 ! [X: $o,Xa: produc3658429121746597890et_nat] :
% 14.17/8.00 ( ( pure_a825153325127701367it_nat @ X @ Xa )
% 14.17/8.00 => ( ( accp_P8458817951426537472et_nat @ pure_a6022498039421069578it_nat @ ( produc3762353314782720579et_nat @ X @ Xa ) )
% 14.17/8.00 => ~ ! [H: heap_e7401611519738050253t_unit,As2: set_nat] :
% 14.17/8.00 ( ( Xa
% 14.17/8.00 = ( produc7507926704131184380et_nat @ H @ As2 ) )
% 14.17/8.00 => ( ( accp_P8458817951426537472et_nat @ pure_a6022498039421069578it_nat @ ( produc3762353314782720579et_nat @ X @ ( produc7507926704131184380et_nat @ H @ As2 ) ) )
% 14.17/8.00 => ~ ( ( As2 = bot_bot_set_nat )
% 14.17/8.00 & X ) ) ) ) ) ).
% 14.17/8.00
% 14.17/8.00 thf(fact_454_pure__assn__raw_Opelims_I3_J,axiom,
% 14.17/8.00 ! [X: $o,Xa: produc3658429121746597890et_nat] :
% 14.17/8.00 ( ~ ( pure_a825153325127701367it_nat @ X @ Xa )
% 14.17/8.00 => ( ( accp_P8458817951426537472et_nat @ pure_a6022498039421069578it_nat @ ( produc3762353314782720579et_nat @ X @ Xa ) )
% 14.17/8.00 => ~ ! [H: heap_e7401611519738050253t_unit,As2: set_nat] :
% 14.17/8.00 ( ( Xa
% 14.17/8.00 = ( produc7507926704131184380et_nat @ H @ As2 ) )
% 14.17/8.00 => ( ( accp_P8458817951426537472et_nat @ pure_a6022498039421069578it_nat @ ( produc3762353314782720579et_nat @ X @ ( produc7507926704131184380et_nat @ H @ As2 ) ) )
% 14.17/8.00 => ( ( As2 = bot_bot_set_nat )
% 14.17/8.00 & X ) ) ) ) ) ).
% 14.17/8.00
% 14.17/8.00 thf(fact_455_Image__singleton__iff,axiom,
% 14.17/8.00 ! [B: nat,R2: set_Pr1763845938948868674it_nat,A2: product_unit] :
% 14.17/8.00 ( ( member_nat @ B @ ( image_5657742817519485770it_nat @ R2 @ ( insert_Product_unit @ A2 @ bot_bo3957492148770167129t_unit ) ) )
% 14.17/8.00 = ( member423003724879500579it_nat @ ( produc4803288329813530246it_nat @ A2 @ B ) @ R2 ) ) ).
% 14.17/8.00
% 14.17/8.00 thf(fact_456_Image__singleton__iff,axiom,
% 14.17/8.00 ! [B: int,R2: set_Pr8497134830871917726it_int,A2: product_unit] :
% 14.17/8.00 ( ( member_int @ B @ ( image_5655252347010435494it_int @ R2 @ ( insert_Product_unit @ A2 @ bot_bo3957492148770167129t_unit ) ) )
% 14.17/8.00 = ( member5468524742225079679it_int @ ( produc4800797859304479970it_int @ A2 @ B ) @ R2 ) ) ).
% 14.17/8.00
% 14.17/8.00 thf(fact_457_Image__singleton__iff,axiom,
% 14.17/8.00 ! [B: $o,R2: set_Pr5464989305201766866unit_o,A2: product_unit] :
% 14.17/8.00 ( ( member_o @ B @ ( image_Product_unit_o @ R2 @ ( insert_Product_unit @ A2 @ bot_bo3957492148770167129t_unit ) ) )
% 14.17/8.00 = ( member3689400097658526619unit_o @ ( produc8487917018914037602unit_o @ A2 @ B ) @ R2 ) ) ).
% 14.17/8.00
% 14.17/8.00 thf(fact_458_Image__singleton__iff,axiom,
% 14.17/8.00 ! [B: product_unit,R2: set_Pr5094982260447487303t_unit,A2: product_unit] :
% 14.17/8.00 ( ( member_Product_unit @ B @ ( image_8431784415008483091t_unit @ R2 @ ( insert_Product_unit @ A2 @ bot_bo3957492148770167129t_unit ) ) )
% 14.17/8.00 = ( member7821138191923116944t_unit @ ( produc5692694829384537815t_unit @ A2 @ B ) @ R2 ) ) ).
% 14.17/8.00
% 14.17/8.00 thf(fact_459_Image__singleton__iff,axiom,
% 14.17/8.00 ! [B: nat,R2: set_Pr2101469702781467981_o_nat,A2: $o] :
% 14.17/8.00 ( ( member_nat @ B @ ( image_o_nat @ R2 @ ( insert_o @ A2 @ bot_bot_set_o ) ) )
% 14.17/8.00 = ( member2802428098988154798_o_nat @ ( product_Pair_o_nat @ A2 @ B ) @ R2 ) ) ).
% 14.17/8.00
% 14.17/8.00 thf(fact_460_Image__singleton__iff,axiom,
% 14.17/8.00 ! [B: int,R2: set_Pr8834758594704517033_o_int,A2: $o] :
% 14.17/8.00 ( ( member_int @ B @ ( image_o_int @ R2 @ ( insert_o @ A2 @ bot_bot_set_o ) ) )
% 14.17/8.00 = ( member7847949116333733898_o_int @ ( product_Pair_o_int @ A2 @ B ) @ R2 ) ) ).
% 14.17/8.00
% 14.17/8.00 thf(fact_461_Image__singleton__iff,axiom,
% 14.17/8.00 ! [B: $o,R2: set_Product_prod_o_o,A2: $o] :
% 14.17/8.00 ( ( member_o @ B @ ( image_o_o @ R2 @ ( insert_o @ A2 @ bot_bot_set_o ) ) )
% 14.17/8.00 = ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ A2 @ B ) @ R2 ) ) ).
% 14.17/8.00
% 14.17/8.00 thf(fact_462_Image__singleton__iff,axiom,
% 14.17/8.00 ! [B: product_unit,R2: set_Pr6671815714118170108t_unit,A2: $o] :
% 14.17/8.00 ( ( member_Product_unit @ B @ ( image_o_Product_unit @ R2 @ ( insert_o @ A2 @ bot_bot_set_o ) ) )
% 14.17/8.00 = ( member9215336023991391429t_unit @ ( produc3901946521647900940t_unit @ A2 @ B ) @ R2 ) ) ).
% 14.17/8.00
% 14.17/8.00 thf(fact_463_Image__singleton__iff,axiom,
% 14.17/8.00 ! [B: int,R2: set_Pr7995236796853374141at_int,A2: nat] :
% 14.17/8.00 ( ( member_int @ B @ ( image_nat_int @ R2 @ ( insert_nat @ A2 @ bot_bot_set_nat ) ) )
% 14.17/8.00 = ( member4262671552274231302at_int @ ( product_Pair_nat_int @ A2 @ B ) @ R2 ) ) ).
% 14.17/8.00
% 14.17/8.00 thf(fact_464_Image__singleton__iff,axiom,
% 14.17/8.00 ! [B: $o,R2: set_Pr3149072824959771635_nat_o,A2: nat] :
% 14.17/8.00 ( ( member_o @ B @ ( image_nat_o @ R2 @ ( insert_nat @ A2 @ bot_bot_set_nat ) ) )
% 14.17/8.00 = ( member6310962623043647828_nat_o @ ( product_Pair_nat_o @ A2 @ B ) @ R2 ) ) ).
% 14.17/8.00
% 14.17/8.00 thf(fact_465_mult_Oright__assoc,axiom,
% 14.17/8.00 ! [A2: assn,B: assn,C: assn] :
% 14.17/8.00 ( ( times_times_assn @ ( times_times_assn @ A2 @ B ) @ C )
% 14.17/8.00 = ( times_times_assn @ A2 @ ( times_times_assn @ B @ C ) ) ) ).
% 14.17/8.00
% 14.17/8.00 thf(fact_466_mult_Oright__assoc,axiom,
% 14.17/8.00 ! [A2: rat,B: rat,C: rat] :
% 14.17/8.00 ( ( times_times_rat @ ( times_times_rat @ A2 @ B ) @ C )
% 14.17/8.00 = ( times_times_rat @ A2 @ ( times_times_rat @ B @ C ) ) ) ).
% 14.17/8.00
% 14.17/8.00 thf(fact_467_mult_Oright__assoc,axiom,
% 14.17/8.00 ! [A2: nat,B: nat,C: nat] :
% 14.17/8.00 ( ( times_times_nat @ ( times_times_nat @ A2 @ B ) @ C )
% 14.17/8.00 = ( times_times_nat @ A2 @ ( times_times_nat @ B @ C ) ) ) ).
% 14.17/8.00
% 14.17/8.00 thf(fact_468_mult_Oright__assoc,axiom,
% 14.17/8.00 ! [A2: int,B: int,C: int] :
% 14.17/8.00 ( ( times_times_int @ ( times_times_int @ A2 @ B ) @ C )
% 14.17/8.00 = ( times_times_int @ A2 @ ( times_times_int @ B @ C ) ) ) ).
% 14.17/8.00
% 14.17/8.00 thf(fact_469_mult_Oright__commute,axiom,
% 14.17/8.00 ! [A2: assn,B: assn,C: assn] :
% 14.17/8.00 ( ( times_times_assn @ ( times_times_assn @ A2 @ B ) @ C )
% 14.17/8.00 = ( times_times_assn @ ( times_times_assn @ A2 @ C ) @ B ) ) ).
% 14.17/8.00
% 14.17/8.00 thf(fact_470_mult_Oright__commute,axiom,
% 14.17/8.00 ! [A2: rat,B: rat,C: rat] :
% 14.17/8.00 ( ( times_times_rat @ ( times_times_rat @ A2 @ B ) @ C )
% 14.17/8.00 = ( times_times_rat @ ( times_times_rat @ A2 @ C ) @ B ) ) ).
% 14.17/8.00
% 14.17/8.00 thf(fact_471_mult_Oright__commute,axiom,
% 14.17/8.00 ! [A2: nat,B: nat,C: nat] :
% 14.17/8.00 ( ( times_times_nat @ ( times_times_nat @ A2 @ B ) @ C )
% 14.17/8.00 = ( times_times_nat @ ( times_times_nat @ A2 @ C ) @ B ) ) ).
% 14.17/8.00
% 14.17/8.00 thf(fact_472_mult_Oright__commute,axiom,
% 14.17/8.00 ! [A2: int,B: int,C: int] :
% 14.17/8.00 ( ( times_times_int @ ( times_times_int @ A2 @ B ) @ C )
% 14.17/8.00 = ( times_times_int @ ( times_times_int @ A2 @ C ) @ B ) ) ).
% 14.17/8.00
% 14.17/8.00 thf(fact_473_in__range__empty,axiom,
% 14.17/8.00 ! [H2: heap_e7401611519738050253t_unit] : ( in_range @ ( produc7507926704131184380et_nat @ H2 @ bot_bot_set_nat ) ) ).
% 14.17/8.00
% 14.17/8.00 thf(fact_474_ImageI,axiom,
% 14.17/8.00 ! [A2: nat,B: int,R2: set_Pr7995236796853374141at_int,A: set_nat] :
% 14.17/8.00 ( ( member4262671552274231302at_int @ ( product_Pair_nat_int @ A2 @ B ) @ R2 )
% 14.17/8.00 => ( ( member_nat @ A2 @ A )
% 14.17/8.00 => ( member_int @ B @ ( image_nat_int @ R2 @ A ) ) ) ) ).
% 14.17/8.00
% 14.17/8.00 thf(fact_475_ImageI,axiom,
% 14.17/8.00 ! [A2: nat,B: $o,R2: set_Pr3149072824959771635_nat_o,A: set_nat] :
% 14.17/8.00 ( ( member6310962623043647828_nat_o @ ( product_Pair_nat_o @ A2 @ B ) @ R2 )
% 14.17/8.00 => ( ( member_nat @ A2 @ A )
% 14.17/8.00 => ( member_o @ B @ ( image_nat_o @ R2 @ A ) ) ) ) ).
% 14.17/8.00
% 14.17/8.00 thf(fact_476_ImageI,axiom,
% 14.17/8.00 ! [A2: nat,B: product_unit,R2: set_Pr4334478416066269672t_unit,A: set_nat] :
% 14.17/8.00 ( ( member348563419383025609t_unit @ ( produc3434450474625521892t_unit @ A2 @ B ) @ R2 )
% 14.17/8.00 => ( ( member_nat @ A2 @ A )
% 14.17/8.00 => ( member_Product_unit @ B @ ( image_4288904962331477416t_unit @ R2 @ A ) ) ) ) ).
% 14.17/8.00
% 14.17/8.00 thf(fact_477_ImageI,axiom,
% 14.17/8.00 ! [A2: int,B: nat,R2: set_Pr3448869479623346877nt_nat,A: set_int] :
% 14.17/8.00 ( ( member216504246829706758nt_nat @ ( product_Pair_int_nat @ A2 @ B ) @ R2 )
% 14.17/8.00 => ( ( member_int @ A2 @ A )
% 14.17/8.00 => ( member_nat @ B @ ( image_int_nat @ R2 @ A ) ) ) ) ).
% 14.17/8.00
% 14.17/8.00 thf(fact_478_ImageI,axiom,
% 14.17/8.00 ! [A2: int,B: $o,R2: set_Pr903927857289325719_int_o,A: set_int] :
% 14.17/8.00 ( ( member4489920277610959864_int_o @ ( product_Pair_int_o @ A2 @ B ) @ R2 )
% 14.17/8.00 => ( ( member_int @ A2 @ A )
% 14.17/8.00 => ( member_o @ B @ ( image_int_o @ R2 @ A ) ) ) ) ).
% 14.17/8.00
% 14.17/8.00 thf(fact_479_ImageI,axiom,
% 14.17/8.00 ! [A2: int,B: product_unit,R2: set_Pr5889264611543340172t_unit,A: set_int] :
% 14.17/8.00 ( ( member6688357403439981677t_unit @ ( produc550872421827702152t_unit @ A2 @ B ) @ R2 )
% 14.17/8.00 => ( ( member_int @ A2 @ A )
% 14.17/8.00 => ( member_Product_unit @ B @ ( image_1405326909533657676t_unit @ R2 @ A ) ) ) ) ).
% 14.17/8.00
% 14.17/8.00 thf(fact_480_ImageI,axiom,
% 14.17/8.00 ! [A2: $o,B: nat,R2: set_Pr2101469702781467981_o_nat,A: set_o] :
% 14.17/8.00 ( ( member2802428098988154798_o_nat @ ( product_Pair_o_nat @ A2 @ B ) @ R2 )
% 14.17/8.00 => ( ( member_o @ A2 @ A )
% 14.17/8.00 => ( member_nat @ B @ ( image_o_nat @ R2 @ A ) ) ) ) ).
% 14.17/8.00
% 14.17/8.00 thf(fact_481_ImageI,axiom,
% 14.17/8.00 ! [A2: $o,B: int,R2: set_Pr8834758594704517033_o_int,A: set_o] :
% 14.17/8.00 ( ( member7847949116333733898_o_int @ ( product_Pair_o_int @ A2 @ B ) @ R2 )
% 14.17/8.00 => ( ( member_o @ A2 @ A )
% 14.17/8.00 => ( member_int @ B @ ( image_o_int @ R2 @ A ) ) ) ) ).
% 14.17/8.00
% 14.17/8.00 thf(fact_482_ImageI,axiom,
% 14.17/8.00 ! [A2: $o,B: $o,R2: set_Product_prod_o_o,A: set_o] :
% 14.17/8.00 ( ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ A2 @ B ) @ R2 )
% 14.17/8.00 => ( ( member_o @ A2 @ A )
% 14.17/8.00 => ( member_o @ B @ ( image_o_o @ R2 @ A ) ) ) ) ).
% 14.17/8.00
% 14.17/8.00 thf(fact_483_ImageI,axiom,
% 14.17/8.00 ! [A2: $o,B: product_unit,R2: set_Pr6671815714118170108t_unit,A: set_o] :
% 14.17/8.00 ( ( member9215336023991391429t_unit @ ( produc3901946521647900940t_unit @ A2 @ B ) @ R2 )
% 14.17/8.00 => ( ( member_o @ A2 @ A )
% 14.17/8.00 => ( member_Product_unit @ B @ ( image_o_Product_unit @ R2 @ A ) ) ) ) ).
% 14.17/8.00
% 14.17/8.00 thf(fact_484_Image__empty2,axiom,
% 14.17/8.00 ! [R3: set_Pr5094982260447487303t_unit] :
% 14.17/8.00 ( ( image_8431784415008483091t_unit @ R3 @ bot_bo3957492148770167129t_unit )
% 14.17/8.00 = bot_bo3957492148770167129t_unit ) ).
% 14.17/8.00
% 14.17/8.00 thf(fact_485_Image__empty2,axiom,
% 14.17/8.00 ! [R3: set_Pr5464989305201766866unit_o] :
% 14.17/8.00 ( ( image_Product_unit_o @ R3 @ bot_bo3957492148770167129t_unit )
% 14.17/8.00 = bot_bot_set_o ) ).
% 14.17/8.00
% 14.17/8.00 thf(fact_486_Image__empty2,axiom,
% 14.17/8.00 ! [R3: set_Pr1763845938948868674it_nat] :
% 14.17/8.00 ( ( image_5657742817519485770it_nat @ R3 @ bot_bo3957492148770167129t_unit )
% 14.17/8.00 = bot_bot_set_nat ) ).
% 14.17/8.00
% 14.17/8.00 thf(fact_487_Image__empty2,axiom,
% 14.17/8.00 ! [R3: set_Pr8497134830871917726it_int] :
% 14.17/8.00 ( ( image_5655252347010435494it_int @ R3 @ bot_bo3957492148770167129t_unit )
% 14.17/8.00 = bot_bot_set_int ) ).
% 14.17/8.00
% 14.17/8.00 thf(fact_488_Image__empty2,axiom,
% 14.17/8.00 ! [R3: set_Pr6671815714118170108t_unit] :
% 14.17/8.00 ( ( image_o_Product_unit @ R3 @ bot_bot_set_o )
% 14.17/8.00 = bot_bo3957492148770167129t_unit ) ).
% 14.17/8.00
% 14.17/8.00 thf(fact_489_Image__empty2,axiom,
% 14.17/8.00 ! [R3: set_Product_prod_o_o] :
% 14.17/8.00 ( ( image_o_o @ R3 @ bot_bot_set_o )
% 14.17/8.00 = bot_bot_set_o ) ).
% 14.17/8.00
% 14.17/8.00 thf(fact_490_Image__empty2,axiom,
% 14.17/8.00 ! [R3: set_Pr2101469702781467981_o_nat] :
% 14.17/8.00 ( ( image_o_nat @ R3 @ bot_bot_set_o )
% 14.17/8.00 = bot_bot_set_nat ) ).
% 14.17/8.00
% 14.17/8.00 thf(fact_491_Image__empty2,axiom,
% 14.17/8.00 ! [R3: set_Pr8834758594704517033_o_int] :
% 14.17/8.00 ( ( image_o_int @ R3 @ bot_bot_set_o )
% 14.17/8.00 = bot_bot_set_int ) ).
% 14.17/8.00
% 14.17/8.00 thf(fact_492_Image__empty2,axiom,
% 14.17/8.00 ! [R3: set_Pr4334478416066269672t_unit] :
% 14.17/8.00 ( ( image_4288904962331477416t_unit @ R3 @ bot_bot_set_nat )
% 14.17/8.00 = bot_bo3957492148770167129t_unit ) ).
% 14.17/8.00
% 14.17/8.00 thf(fact_493_Image__empty2,axiom,
% 14.17/8.00 ! [R3: set_Pr3149072824959771635_nat_o] :
% 14.17/8.00 ( ( image_nat_o @ R3 @ bot_bot_set_nat )
% 14.17/8.00 = bot_bot_set_o ) ).
% 14.17/8.00
% 14.17/8.00 thf(fact_494_Id__onI,axiom,
% 14.17/8.00 ! [A2: produc3658429121746597890et_nat > $o,A: set_Pr4532377907799695533_nat_o] :
% 14.17/8.00 ( ( member6576561426505652726_nat_o @ A2 @ A )
% 14.17/8.00 => ( member8781333585448626064_nat_o @ ( produc7368190662567826135_nat_o @ A2 @ A2 ) @ ( id_on_7438695062414216554_nat_o @ A ) ) ) ).
% 14.17/8.00
% 14.17/8.00 thf(fact_495_Id__onI,axiom,
% 14.17/8.00 ! [A2: $o,A: set_o] :
% 14.17/8.00 ( ( member_o @ A2 @ A )
% 14.17/8.00 => ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ A2 @ A2 ) @ ( id_on_o @ A ) ) ) ).
% 14.17/8.00
% 14.17/8.00 thf(fact_496_Id__onI,axiom,
% 14.17/8.00 ! [A2: product_unit,A: set_Product_unit] :
% 14.17/8.00 ( ( member_Product_unit @ A2 @ A )
% 14.17/8.00 => ( member7821138191923116944t_unit @ ( produc5692694829384537815t_unit @ A2 @ A2 ) @ ( id_on_Product_unit @ A ) ) ) ).
% 14.17/8.00
% 14.17/8.00 thf(fact_497_Id__onI,axiom,
% 14.17/8.00 ! [A2: nat,A: set_nat] :
% 14.17/8.00 ( ( member_nat @ A2 @ A )
% 14.17/8.00 => ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A2 @ A2 ) @ ( id_on_nat @ A ) ) ) ).
% 14.17/8.00
% 14.17/8.00 thf(fact_498_Id__onI,axiom,
% 14.17/8.00 ! [A2: int,A: set_int] :
% 14.17/8.00 ( ( member_int @ A2 @ A )
% 14.17/8.00 => ( member5262025264175285858nt_int @ ( product_Pair_int_int @ A2 @ A2 ) @ ( id_on_int @ A ) ) ) ).
% 14.17/8.00
% 14.17/8.00 thf(fact_499_bool__assn__proper_I1_J,axiom,
% 14.17/8.00 proper @ in_range ).
% 14.17/8.00
% 14.17/8.00 thf(fact_500_refl__on__domain,axiom,
% 14.17/8.00 ! [A: set_Pr4532377907799695533_nat_o,R2: set_Pr2161125870931222855_nat_o,A2: produc3658429121746597890et_nat > $o,B: produc3658429121746597890et_nat > $o] :
% 14.17/8.00 ( ( refl_o1075864025857821388_nat_o @ A @ R2 )
% 14.17/8.00 => ( ( member8781333585448626064_nat_o @ ( produc7368190662567826135_nat_o @ A2 @ B ) @ R2 )
% 14.17/8.00 => ( ( member6576561426505652726_nat_o @ A2 @ A )
% 14.17/8.00 & ( member6576561426505652726_nat_o @ B @ A ) ) ) ) ).
% 14.17/8.00
% 14.17/8.00 thf(fact_501_refl__on__domain,axiom,
% 14.17/8.00 ! [A: set_o,R2: set_Product_prod_o_o,A2: $o,B: $o] :
% 14.17/8.00 ( ( refl_on_o @ A @ R2 )
% 14.17/8.00 => ( ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ A2 @ B ) @ R2 )
% 14.17/8.00 => ( ( member_o @ A2 @ A )
% 14.17/8.00 & ( member_o @ B @ A ) ) ) ) ).
% 14.17/8.00
% 14.17/8.00 thf(fact_502_refl__on__domain,axiom,
% 14.17/8.00 ! [A: set_Product_unit,R2: set_Pr5094982260447487303t_unit,A2: product_unit,B: product_unit] :
% 14.17/8.00 ( ( refl_on_Product_unit @ A @ R2 )
% 14.17/8.00 => ( ( member7821138191923116944t_unit @ ( produc5692694829384537815t_unit @ A2 @ B ) @ R2 )
% 14.17/8.00 => ( ( member_Product_unit @ A2 @ A )
% 14.17/8.00 & ( member_Product_unit @ B @ A ) ) ) ) ).
% 14.17/8.00
% 14.17/8.00 thf(fact_503_refl__on__domain,axiom,
% 14.17/8.00 ! [A: set_nat,R2: set_Pr1261947904930325089at_nat,A2: nat,B: nat] :
% 14.17/8.00 ( ( refl_on_nat @ A @ R2 )
% 14.17/8.00 => ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A2 @ B ) @ R2 )
% 14.17/8.00 => ( ( member_nat @ A2 @ A )
% 14.17/8.00 & ( member_nat @ B @ A ) ) ) ) ).
% 14.17/8.00
% 14.17/8.00 thf(fact_504_refl__on__domain,axiom,
% 14.17/8.00 ! [A: set_int,R2: set_Pr958786334691620121nt_int,A2: int,B: int] :
% 14.17/8.00 ( ( refl_on_int @ A @ R2 )
% 14.17/8.00 => ( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ A2 @ B ) @ R2 )
% 14.17/8.00 => ( ( member_int @ A2 @ A )
% 14.17/8.00 & ( member_int @ B @ A ) ) ) ) ).
% 14.17/8.00
% 14.17/8.00 thf(fact_505_rev__ImageI,axiom,
% 14.17/8.00 ! [A2: nat,A: set_nat,B: int,R2: set_Pr7995236796853374141at_int] :
% 14.17/8.00 ( ( member_nat @ A2 @ A )
% 14.17/8.00 => ( ( member4262671552274231302at_int @ ( product_Pair_nat_int @ A2 @ B ) @ R2 )
% 14.17/8.00 => ( member_int @ B @ ( image_nat_int @ R2 @ A ) ) ) ) ).
% 14.17/8.00
% 14.17/8.00 thf(fact_506_rev__ImageI,axiom,
% 14.17/8.00 ! [A2: nat,A: set_nat,B: $o,R2: set_Pr3149072824959771635_nat_o] :
% 14.17/8.00 ( ( member_nat @ A2 @ A )
% 14.17/8.00 => ( ( member6310962623043647828_nat_o @ ( product_Pair_nat_o @ A2 @ B ) @ R2 )
% 14.17/8.00 => ( member_o @ B @ ( image_nat_o @ R2 @ A ) ) ) ) ).
% 14.17/8.00
% 14.17/8.00 thf(fact_507_rev__ImageI,axiom,
% 14.17/8.00 ! [A2: nat,A: set_nat,B: product_unit,R2: set_Pr4334478416066269672t_unit] :
% 14.17/8.00 ( ( member_nat @ A2 @ A )
% 14.17/8.00 => ( ( member348563419383025609t_unit @ ( produc3434450474625521892t_unit @ A2 @ B ) @ R2 )
% 14.17/8.00 => ( member_Product_unit @ B @ ( image_4288904962331477416t_unit @ R2 @ A ) ) ) ) ).
% 14.17/8.00
% 14.17/8.00 thf(fact_508_rev__ImageI,axiom,
% 14.17/8.00 ! [A2: int,A: set_int,B: nat,R2: set_Pr3448869479623346877nt_nat] :
% 14.17/8.00 ( ( member_int @ A2 @ A )
% 14.17/8.00 => ( ( member216504246829706758nt_nat @ ( product_Pair_int_nat @ A2 @ B ) @ R2 )
% 14.17/8.00 => ( member_nat @ B @ ( image_int_nat @ R2 @ A ) ) ) ) ).
% 14.17/8.00
% 14.17/8.00 thf(fact_509_rev__ImageI,axiom,
% 14.17/8.00 ! [A2: int,A: set_int,B: $o,R2: set_Pr903927857289325719_int_o] :
% 14.17/8.00 ( ( member_int @ A2 @ A )
% 14.17/8.00 => ( ( member4489920277610959864_int_o @ ( product_Pair_int_o @ A2 @ B ) @ R2 )
% 14.17/8.00 => ( member_o @ B @ ( image_int_o @ R2 @ A ) ) ) ) ).
% 14.17/8.00
% 14.17/8.00 thf(fact_510_rev__ImageI,axiom,
% 14.17/8.00 ! [A2: int,A: set_int,B: product_unit,R2: set_Pr5889264611543340172t_unit] :
% 14.17/8.00 ( ( member_int @ A2 @ A )
% 14.17/8.00 => ( ( member6688357403439981677t_unit @ ( produc550872421827702152t_unit @ A2 @ B ) @ R2 )
% 14.17/8.00 => ( member_Product_unit @ B @ ( image_1405326909533657676t_unit @ R2 @ A ) ) ) ) ).
% 14.17/8.00
% 14.17/8.00 thf(fact_511_rev__ImageI,axiom,
% 14.17/8.00 ! [A2: $o,A: set_o,B: nat,R2: set_Pr2101469702781467981_o_nat] :
% 14.17/8.00 ( ( member_o @ A2 @ A )
% 14.17/8.00 => ( ( member2802428098988154798_o_nat @ ( product_Pair_o_nat @ A2 @ B ) @ R2 )
% 14.17/8.00 => ( member_nat @ B @ ( image_o_nat @ R2 @ A ) ) ) ) ).
% 14.17/8.00
% 14.17/8.00 thf(fact_512_rev__ImageI,axiom,
% 14.17/8.00 ! [A2: $o,A: set_o,B: int,R2: set_Pr8834758594704517033_o_int] :
% 14.17/8.00 ( ( member_o @ A2 @ A )
% 14.17/8.00 => ( ( member7847949116333733898_o_int @ ( product_Pair_o_int @ A2 @ B ) @ R2 )
% 14.17/8.00 => ( member_int @ B @ ( image_o_int @ R2 @ A ) ) ) ) ).
% 14.17/8.00
% 14.17/8.00 thf(fact_513_rev__ImageI,axiom,
% 14.17/8.00 ! [A2: $o,A: set_o,B: $o,R2: set_Product_prod_o_o] :
% 14.17/8.00 ( ( member_o @ A2 @ A )
% 14.17/8.00 => ( ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ A2 @ B ) @ R2 )
% 14.17/8.00 => ( member_o @ B @ ( image_o_o @ R2 @ A ) ) ) ) ).
% 14.17/8.00
% 14.17/8.00 thf(fact_514_rev__ImageI,axiom,
% 14.17/8.00 ! [A2: $o,A: set_o,B: product_unit,R2: set_Pr6671815714118170108t_unit] :
% 14.17/8.00 ( ( member_o @ A2 @ A )
% 14.17/8.00 => ( ( member9215336023991391429t_unit @ ( produc3901946521647900940t_unit @ A2 @ B ) @ R2 )
% 14.17/8.00 => ( member_Product_unit @ B @ ( image_o_Product_unit @ R2 @ A ) ) ) ) ).
% 14.17/8.00
% 14.17/8.00 thf(fact_515_Image__iff,axiom,
% 14.17/8.00 ! [B: set_nat,R2: set_Pr3948176798113811640et_nat,A: set_he8644174446132254637t_unit] :
% 14.17/8.00 ( ( member_set_nat @ B @ ( image_3942343239045813184et_nat @ R2 @ A ) )
% 14.17/8.00 = ( ? [X2: heap_e7401611519738050253t_unit] :
% 14.17/8.00 ( ( member242389114213480694t_unit @ X2 @ A )
% 14.17/8.00 & ( member6260224972018164377et_nat @ ( produc7507926704131184380et_nat @ X2 @ B ) @ R2 ) ) ) ) ).
% 14.17/8.00
% 14.17/8.00 thf(fact_516_Image__iff,axiom,
% 14.17/8.00 ! [B: nat,R2: set_Pr1261947904930325089at_nat,A: set_nat] :
% 14.17/8.00 ( ( member_nat @ B @ ( image_nat_nat @ R2 @ A ) )
% 14.17/8.00 = ( ? [X2: nat] :
% 14.17/8.00 ( ( member_nat @ X2 @ A )
% 14.17/8.00 & ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X2 @ B ) @ R2 ) ) ) ) ).
% 14.17/8.00
% 14.17/8.00 thf(fact_517_Image__iff,axiom,
% 14.17/8.00 ! [B: int,R2: set_Pr958786334691620121nt_int,A: set_int] :
% 14.17/8.00 ( ( member_int @ B @ ( image_int_int @ R2 @ A ) )
% 14.17/8.00 = ( ? [X2: int] :
% 14.17/8.00 ( ( member_int @ X2 @ A )
% 14.17/8.00 & ( member5262025264175285858nt_int @ ( product_Pair_int_int @ X2 @ B ) @ R2 ) ) ) ) ).
% 14.17/8.00
% 14.17/8.00 thf(fact_518_Image__iff,axiom,
% 14.17/8.00 ! [B: produc3658429121746597890et_nat,R2: set_Pr3286484037609594932et_nat,A: set_Pr4532377907799695533_nat_o] :
% 14.17/8.00 ( ( member6260224972018164377et_nat @ B @ ( image_1814916760386648892et_nat @ R2 @ A ) )
% 14.17/8.00 = ( ? [X2: produc3658429121746597890et_nat > $o] :
% 14.17/8.00 ( ( member6576561426505652726_nat_o @ X2 @ A )
% 14.17/8.00 & ( member1996754912294343701et_nat @ ( produc5001842942810119800et_nat @ X2 @ B ) @ R2 ) ) ) ) ).
% 14.17/8.00
% 14.17/8.00 thf(fact_519_Image__iff,axiom,
% 14.17/8.00 ! [B: produc3925858234332021118et_nat,R2: set_Pr8536935166611901872et_nat,A: set_Pr4532377907799695533_nat_o] :
% 14.17/8.00 ( ( member1996754912294343701et_nat @ B @ ( image_7240710759097644216et_nat @ R2 @ A ) )
% 14.17/8.00 = ( ? [X2: produc3658429121746597890et_nat > $o] :
% 14.17/8.00 ( ( member6576561426505652726_nat_o @ X2 @ A )
% 14.17/8.00 & ( member6124377750444531601et_nat @ ( produc2245416461498447860et_nat @ X2 @ B ) @ R2 ) ) ) ) ).
% 14.17/8.00
% 14.17/8.00 thf(fact_520_ImageE,axiom,
% 14.17/8.00 ! [B: nat,R2: set_Pr3448869479623346877nt_nat,A: set_int] :
% 14.17/8.00 ( ( member_nat @ B @ ( image_int_nat @ R2 @ A ) )
% 14.17/8.00 => ~ ! [X3: int] :
% 14.17/8.00 ( ( member216504246829706758nt_nat @ ( product_Pair_int_nat @ X3 @ B ) @ R2 )
% 14.17/8.00 => ~ ( member_int @ X3 @ A ) ) ) ).
% 14.17/8.00
% 14.17/8.00 thf(fact_521_ImageE,axiom,
% 14.17/8.00 ! [B: nat,R2: set_Pr2101469702781467981_o_nat,A: set_o] :
% 14.17/8.00 ( ( member_nat @ B @ ( image_o_nat @ R2 @ A ) )
% 14.17/8.00 => ~ ! [X3: $o] :
% 14.17/8.00 ( ( member2802428098988154798_o_nat @ ( product_Pair_o_nat @ X3 @ B ) @ R2 )
% 14.17/8.00 => ~ ( member_o @ X3 @ A ) ) ) ).
% 14.17/8.00
% 14.17/8.00 thf(fact_522_ImageE,axiom,
% 14.17/8.00 ! [B: nat,R2: set_Pr1763845938948868674it_nat,A: set_Product_unit] :
% 14.17/8.00 ( ( member_nat @ B @ ( image_5657742817519485770it_nat @ R2 @ A ) )
% 14.17/8.00 => ~ ! [X3: product_unit] :
% 14.17/8.00 ( ( member423003724879500579it_nat @ ( produc4803288329813530246it_nat @ X3 @ B ) @ R2 )
% 14.17/8.00 => ~ ( member_Product_unit @ X3 @ A ) ) ) ).
% 14.17/8.00
% 14.17/8.00 thf(fact_523_ImageE,axiom,
% 14.17/8.00 ! [B: int,R2: set_Pr7995236796853374141at_int,A: set_nat] :
% 14.17/8.00 ( ( member_int @ B @ ( image_nat_int @ R2 @ A ) )
% 14.17/8.00 => ~ ! [X3: nat] :
% 14.17/8.00 ( ( member4262671552274231302at_int @ ( product_Pair_nat_int @ X3 @ B ) @ R2 )
% 14.17/8.00 => ~ ( member_nat @ X3 @ A ) ) ) ).
% 14.17/8.00
% 14.17/8.00 thf(fact_524_ImageE,axiom,
% 14.17/8.00 ! [B: int,R2: set_Pr8834758594704517033_o_int,A: set_o] :
% 14.17/8.00 ( ( member_int @ B @ ( image_o_int @ R2 @ A ) )
% 14.17/8.00 => ~ ! [X3: $o] :
% 14.17/8.00 ( ( member7847949116333733898_o_int @ ( product_Pair_o_int @ X3 @ B ) @ R2 )
% 14.17/8.00 => ~ ( member_o @ X3 @ A ) ) ) ).
% 14.17/8.00
% 14.17/8.00 thf(fact_525_ImageE,axiom,
% 14.17/8.00 ! [B: int,R2: set_Pr8497134830871917726it_int,A: set_Product_unit] :
% 14.17/8.00 ( ( member_int @ B @ ( image_5655252347010435494it_int @ R2 @ A ) )
% 14.17/8.00 => ~ ! [X3: product_unit] :
% 14.17/8.00 ( ( member5468524742225079679it_int @ ( produc4800797859304479970it_int @ X3 @ B ) @ R2 )
% 14.17/8.00 => ~ ( member_Product_unit @ X3 @ A ) ) ) ).
% 14.17/8.00
% 14.17/8.00 thf(fact_526_ImageE,axiom,
% 14.17/8.00 ! [B: $o,R2: set_Pr3149072824959771635_nat_o,A: set_nat] :
% 14.17/8.00 ( ( member_o @ B @ ( image_nat_o @ R2 @ A ) )
% 14.17/8.00 => ~ ! [X3: nat] :
% 14.17/8.00 ( ( member6310962623043647828_nat_o @ ( product_Pair_nat_o @ X3 @ B ) @ R2 )
% 14.17/8.00 => ~ ( member_nat @ X3 @ A ) ) ) ).
% 14.17/8.00
% 14.17/8.00 thf(fact_527_ImageE,axiom,
% 14.17/8.00 ! [B: $o,R2: set_Pr903927857289325719_int_o,A: set_int] :
% 14.17/8.00 ( ( member_o @ B @ ( image_int_o @ R2 @ A ) )
% 14.17/8.00 => ~ ! [X3: int] :
% 14.17/8.00 ( ( member4489920277610959864_int_o @ ( product_Pair_int_o @ X3 @ B ) @ R2 )
% 14.17/8.00 => ~ ( member_int @ X3 @ A ) ) ) ).
% 14.17/8.00
% 14.17/8.00 thf(fact_528_ImageE,axiom,
% 14.17/8.00 ! [B: $o,R2: set_Product_prod_o_o,A: set_o] :
% 14.17/8.00 ( ( member_o @ B @ ( image_o_o @ R2 @ A ) )
% 14.17/8.00 => ~ ! [X3: $o] :
% 14.17/8.00 ( ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ X3 @ B ) @ R2 )
% 14.17/8.00 => ~ ( member_o @ X3 @ A ) ) ) ).
% 14.17/8.00
% 14.17/8.00 thf(fact_529_ImageE,axiom,
% 14.17/8.00 ! [B: $o,R2: set_Pr5464989305201766866unit_o,A: set_Product_unit] :
% 14.17/8.00 ( ( member_o @ B @ ( image_Product_unit_o @ R2 @ A ) )
% 14.17/8.00 => ~ ! [X3: product_unit] :
% 14.17/8.00 ( ( member3689400097658526619unit_o @ ( produc8487917018914037602unit_o @ X3 @ B ) @ R2 )
% 14.17/8.00 => ~ ( member_Product_unit @ X3 @ A ) ) ) ).
% 14.17/8.00
% 14.17/8.00 thf(fact_530_refl__onD2,axiom,
% 14.17/8.00 ! [A: set_Pr4532377907799695533_nat_o,R2: set_Pr2161125870931222855_nat_o,X: produc3658429121746597890et_nat > $o,Y: produc3658429121746597890et_nat > $o] :
% 14.17/8.00 ( ( refl_o1075864025857821388_nat_o @ A @ R2 )
% 14.17/8.00 => ( ( member8781333585448626064_nat_o @ ( produc7368190662567826135_nat_o @ X @ Y ) @ R2 )
% 14.17/8.00 => ( member6576561426505652726_nat_o @ Y @ A ) ) ) ).
% 14.17/8.00
% 14.17/8.00 thf(fact_531_refl__onD2,axiom,
% 14.17/8.00 ! [A: set_o,R2: set_Product_prod_o_o,X: $o,Y: $o] :
% 14.17/8.00 ( ( refl_on_o @ A @ R2 )
% 14.17/8.00 => ( ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ X @ Y ) @ R2 )
% 14.17/8.00 => ( member_o @ Y @ A ) ) ) ).
% 14.17/8.00
% 14.17/8.00 thf(fact_532_refl__onD2,axiom,
% 14.17/8.00 ! [A: set_Product_unit,R2: set_Pr5094982260447487303t_unit,X: product_unit,Y: product_unit] :
% 14.17/8.00 ( ( refl_on_Product_unit @ A @ R2 )
% 14.17/8.00 => ( ( member7821138191923116944t_unit @ ( produc5692694829384537815t_unit @ X @ Y ) @ R2 )
% 14.17/8.00 => ( member_Product_unit @ Y @ A ) ) ) ).
% 14.17/8.00
% 14.17/8.00 thf(fact_533_refl__onD2,axiom,
% 14.17/8.00 ! [A: set_nat,R2: set_Pr1261947904930325089at_nat,X: nat,Y: nat] :
% 14.17/8.00 ( ( refl_on_nat @ A @ R2 )
% 14.17/8.00 => ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ Y ) @ R2 )
% 14.17/8.00 => ( member_nat @ Y @ A ) ) ) ).
% 14.17/8.00
% 14.17/8.00 thf(fact_534_refl__onD2,axiom,
% 14.17/8.00 ! [A: set_int,R2: set_Pr958786334691620121nt_int,X: int,Y: int] :
% 14.17/8.00 ( ( refl_on_int @ A @ R2 )
% 14.17/8.00 => ( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ X @ Y ) @ R2 )
% 14.17/8.00 => ( member_int @ Y @ A ) ) ) ).
% 14.17/8.00
% 14.17/8.00 thf(fact_535_refl__onD1,axiom,
% 14.17/8.00 ! [A: set_Pr4532377907799695533_nat_o,R2: set_Pr2161125870931222855_nat_o,X: produc3658429121746597890et_nat > $o,Y: produc3658429121746597890et_nat > $o] :
% 14.17/8.00 ( ( refl_o1075864025857821388_nat_o @ A @ R2 )
% 14.17/8.00 => ( ( member8781333585448626064_nat_o @ ( produc7368190662567826135_nat_o @ X @ Y ) @ R2 )
% 14.17/8.00 => ( member6576561426505652726_nat_o @ X @ A ) ) ) ).
% 14.17/8.00
% 14.17/8.00 thf(fact_536_refl__onD1,axiom,
% 14.17/8.00 ! [A: set_o,R2: set_Product_prod_o_o,X: $o,Y: $o] :
% 14.17/8.00 ( ( refl_on_o @ A @ R2 )
% 14.17/8.00 => ( ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ X @ Y ) @ R2 )
% 14.17/8.00 => ( member_o @ X @ A ) ) ) ).
% 14.17/8.00
% 14.17/8.00 thf(fact_537_refl__onD1,axiom,
% 14.17/8.00 ! [A: set_Product_unit,R2: set_Pr5094982260447487303t_unit,X: product_unit,Y: product_unit] :
% 14.17/8.00 ( ( refl_on_Product_unit @ A @ R2 )
% 14.17/8.00 => ( ( member7821138191923116944t_unit @ ( produc5692694829384537815t_unit @ X @ Y ) @ R2 )
% 14.17/8.00 => ( member_Product_unit @ X @ A ) ) ) ).
% 14.17/8.00
% 14.17/8.00 thf(fact_538_refl__onD1,axiom,
% 14.17/8.00 ! [A: set_nat,R2: set_Pr1261947904930325089at_nat,X: nat,Y: nat] :
% 14.17/8.00 ( ( refl_on_nat @ A @ R2 )
% 14.17/8.00 => ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ Y ) @ R2 )
% 14.17/8.00 => ( member_nat @ X @ A ) ) ) ).
% 14.17/8.00
% 14.17/8.00 thf(fact_539_refl__onD1,axiom,
% 14.17/8.00 ! [A: set_int,R2: set_Pr958786334691620121nt_int,X: int,Y: int] :
% 14.17/8.00 ( ( refl_on_int @ A @ R2 )
% 14.17/8.00 => ( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ X @ Y ) @ R2 )
% 14.17/8.00 => ( member_int @ X @ A ) ) ) ).
% 14.17/8.00
% 14.17/8.00 thf(fact_540_refl__onD,axiom,
% 14.17/8.00 ! [A: set_Pr4532377907799695533_nat_o,R2: set_Pr2161125870931222855_nat_o,A2: produc3658429121746597890et_nat > $o] :
% 14.17/8.00 ( ( refl_o1075864025857821388_nat_o @ A @ R2 )
% 14.17/8.00 => ( ( member6576561426505652726_nat_o @ A2 @ A )
% 14.17/8.00 => ( member8781333585448626064_nat_o @ ( produc7368190662567826135_nat_o @ A2 @ A2 ) @ R2 ) ) ) ).
% 14.17/8.00
% 14.17/8.00 thf(fact_541_refl__onD,axiom,
% 14.17/8.00 ! [A: set_o,R2: set_Product_prod_o_o,A2: $o] :
% 14.17/8.00 ( ( refl_on_o @ A @ R2 )
% 14.17/8.00 => ( ( member_o @ A2 @ A )
% 14.17/8.00 => ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ A2 @ A2 ) @ R2 ) ) ) ).
% 14.17/8.00
% 14.17/8.00 thf(fact_542_refl__onD,axiom,
% 14.17/8.00 ! [A: set_Product_unit,R2: set_Pr5094982260447487303t_unit,A2: product_unit] :
% 14.17/8.00 ( ( refl_on_Product_unit @ A @ R2 )
% 14.17/8.00 => ( ( member_Product_unit @ A2 @ A )
% 14.17/8.00 => ( member7821138191923116944t_unit @ ( produc5692694829384537815t_unit @ A2 @ A2 ) @ R2 ) ) ) ).
% 14.17/8.00
% 14.17/8.00 thf(fact_543_refl__onD,axiom,
% 14.17/8.00 ! [A: set_nat,R2: set_Pr1261947904930325089at_nat,A2: nat] :
% 14.17/8.00 ( ( refl_on_nat @ A @ R2 )
% 14.17/8.00 => ( ( member_nat @ A2 @ A )
% 14.17/8.00 => ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A2 @ A2 ) @ R2 ) ) ) ).
% 14.17/8.00
% 14.17/8.00 thf(fact_544_refl__onD,axiom,
% 14.17/8.00 ! [A: set_int,R2: set_Pr958786334691620121nt_int,A2: int] :
% 14.17/8.00 ( ( refl_on_int @ A @ R2 )
% 14.17/8.00 => ( ( member_int @ A2 @ A )
% 14.17/8.00 => ( member5262025264175285858nt_int @ ( product_Pair_int_int @ A2 @ A2 ) @ R2 ) ) ) ).
% 14.17/8.00
% 14.17/8.00 thf(fact_545_total__on__def,axiom,
% 14.17/8.00 ( total_on_nat
% 14.17/8.00 = ( ^ [A3: set_nat,R4: set_Pr1261947904930325089at_nat] :
% 14.17/8.00 ! [X2: nat] :
% 14.17/8.00 ( ( member_nat @ X2 @ A3 )
% 14.17/8.00 => ! [Y3: nat] :
% 14.17/8.00 ( ( member_nat @ Y3 @ A3 )
% 14.17/8.00 => ( ( X2 != Y3 )
% 14.17/8.00 => ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X2 @ Y3 ) @ R4 )
% 14.17/8.00 | ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ Y3 @ X2 ) @ R4 ) ) ) ) ) ) ) ).
% 14.17/8.00
% 14.17/8.00 thf(fact_546_total__on__def,axiom,
% 14.17/8.00 ( total_on_int
% 14.17/8.00 = ( ^ [A3: set_int,R4: set_Pr958786334691620121nt_int] :
% 14.17/8.00 ! [X2: int] :
% 14.17/8.00 ( ( member_int @ X2 @ A3 )
% 14.17/8.00 => ! [Y3: int] :
% 14.17/8.00 ( ( member_int @ Y3 @ A3 )
% 14.17/8.00 => ( ( X2 != Y3 )
% 14.17/8.00 => ( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ X2 @ Y3 ) @ R4 )
% 14.17/8.00 | ( member5262025264175285858nt_int @ ( product_Pair_int_int @ Y3 @ X2 ) @ R4 ) ) ) ) ) ) ) ).
% 14.17/8.00
% 14.17/8.00 thf(fact_547_total__onI,axiom,
% 14.17/8.00 ! [A: set_Pr4532377907799695533_nat_o,R2: set_Pr2161125870931222855_nat_o] :
% 14.17/8.00 ( ! [X3: produc3658429121746597890et_nat > $o,Y2: produc3658429121746597890et_nat > $o] :
% 14.17/8.00 ( ( member6576561426505652726_nat_o @ X3 @ A )
% 14.17/8.00 => ( ( member6576561426505652726_nat_o @ Y2 @ A )
% 14.17/8.00 => ( ( X3 != Y2 )
% 14.17/8.00 => ( ( member8781333585448626064_nat_o @ ( produc7368190662567826135_nat_o @ X3 @ Y2 ) @ R2 )
% 14.17/8.00 | ( member8781333585448626064_nat_o @ ( produc7368190662567826135_nat_o @ Y2 @ X3 ) @ R2 ) ) ) ) )
% 14.17/8.00 => ( total_7429123086324524369_nat_o @ A @ R2 ) ) ).
% 14.17/8.00
% 14.17/8.00 thf(fact_548_total__onI,axiom,
% 14.17/8.00 ! [A: set_o,R2: set_Product_prod_o_o] :
% 14.17/8.00 ( ! [X3: $o,Y2: $o] :
% 14.17/8.00 ( ( member_o @ X3 @ A )
% 14.17/8.00 => ( ( member_o @ Y2 @ A )
% 14.17/8.00 => ( ( X3 != Y2 )
% 14.17/8.00 => ( ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ X3 @ Y2 ) @ R2 )
% 14.17/8.00 | ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ Y2 @ X3 ) @ R2 ) ) ) ) )
% 14.17/8.00 => ( total_on_o @ A @ R2 ) ) ).
% 14.17/8.00
% 14.17/8.00 thf(fact_549_total__onI,axiom,
% 14.17/8.00 ! [A: set_Product_unit,R2: set_Pr5094982260447487303t_unit] :
% 14.17/8.00 ( ! [X3: product_unit,Y2: product_unit] :
% 14.17/8.00 ( ( member_Product_unit @ X3 @ A )
% 14.17/8.00 => ( ( member_Product_unit @ Y2 @ A )
% 14.17/8.00 => ( ( X3 != Y2 )
% 14.17/8.00 => ( ( member7821138191923116944t_unit @ ( produc5692694829384537815t_unit @ X3 @ Y2 ) @ R2 )
% 14.17/8.00 | ( member7821138191923116944t_unit @ ( produc5692694829384537815t_unit @ Y2 @ X3 ) @ R2 ) ) ) ) )
% 14.17/8.00 => ( total_5665700600833547665t_unit @ A @ R2 ) ) ).
% 14.17/8.00
% 14.17/8.00 thf(fact_550_total__onI,axiom,
% 14.17/8.00 ! [A: set_nat,R2: set_Pr1261947904930325089at_nat] :
% 14.17/8.00 ( ! [X3: nat,Y2: nat] :
% 14.17/8.00 ( ( member_nat @ X3 @ A )
% 14.17/8.00 => ( ( member_nat @ Y2 @ A )
% 14.17/8.00 => ( ( X3 != Y2 )
% 14.17/8.00 => ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X3 @ Y2 ) @ R2 )
% 14.17/8.00 | ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ Y2 @ X3 ) @ R2 ) ) ) ) )
% 14.17/8.00 => ( total_on_nat @ A @ R2 ) ) ).
% 14.17/8.00
% 14.17/8.00 thf(fact_551_total__onI,axiom,
% 14.17/8.00 ! [A: set_int,R2: set_Pr958786334691620121nt_int] :
% 14.17/8.00 ( ! [X3: int,Y2: int] :
% 14.17/8.00 ( ( member_int @ X3 @ A )
% 14.17/8.00 => ( ( member_int @ Y2 @ A )
% 14.17/8.00 => ( ( X3 != Y2 )
% 14.17/8.00 => ( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ X3 @ Y2 ) @ R2 )
% 14.17/8.00 | ( member5262025264175285858nt_int @ ( product_Pair_int_int @ Y2 @ X3 ) @ R2 ) ) ) ) )
% 14.17/8.00 => ( total_on_int @ A @ R2 ) ) ).
% 14.17/8.00
% 14.17/8.00 thf(fact_552_total__on__empty,axiom,
% 14.17/8.00 ! [R2: set_Pr5094982260447487303t_unit] : ( total_5665700600833547665t_unit @ bot_bo3957492148770167129t_unit @ R2 ) ).
% 14.17/8.00
% 14.17/8.00 thf(fact_553_total__on__empty,axiom,
% 14.17/8.00 ! [R2: set_Product_prod_o_o] : ( total_on_o @ bot_bot_set_o @ R2 ) ).
% 14.17/8.00
% 14.17/8.00 thf(fact_554_total__on__empty,axiom,
% 14.17/8.00 ! [R2: set_Pr1261947904930325089at_nat] : ( total_on_nat @ bot_bot_set_nat @ R2 ) ).
% 14.17/8.00
% 14.17/8.00 thf(fact_555_total__on__empty,axiom,
% 14.17/8.00 ! [R2: set_Pr958786334691620121nt_int] : ( total_on_int @ bot_bot_set_int @ R2 ) ).
% 14.17/8.00
% 14.17/8.00 thf(fact_556_total__on__empty,axiom,
% 14.17/8.00 ! [R2: set_Pr2161125870931222855_nat_o] : ( total_7429123086324524369_nat_o @ bot_bo7824918357723582233_nat_o @ R2 ) ).
% 14.17/8.00
% 14.17/8.00 thf(fact_557_models__in__range,axiom,
% 14.17/8.00 ! [P: assn,H2: produc3658429121746597890et_nat] :
% 14.17/8.00 ( ( rep_assn @ P @ H2 )
% 14.17/8.00 => ( in_range @ H2 ) ) ).
% 14.17/8.00
% 14.17/8.00 thf(fact_558_Id__on__iff,axiom,
% 14.17/8.00 ! [X: produc3658429121746597890et_nat > $o,Y: produc3658429121746597890et_nat > $o,A: set_Pr4532377907799695533_nat_o] :
% 14.17/8.00 ( ( member8781333585448626064_nat_o @ ( produc7368190662567826135_nat_o @ X @ Y ) @ ( id_on_7438695062414216554_nat_o @ A ) )
% 14.17/8.00 = ( ( X = Y )
% 14.17/8.00 & ( member6576561426505652726_nat_o @ X @ A ) ) ) ).
% 14.17/8.00
% 14.17/8.00 thf(fact_559_Id__on__iff,axiom,
% 14.17/8.00 ! [X: $o,Y: $o,A: set_o] :
% 14.17/8.00 ( ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ X @ Y ) @ ( id_on_o @ A ) )
% 14.17/8.00 = ( ( X = Y )
% 14.17/8.00 & ( member_o @ X @ A ) ) ) ).
% 14.17/8.00
% 14.17/8.00 thf(fact_560_Id__on__iff,axiom,
% 14.17/8.00 ! [X: product_unit,Y: product_unit,A: set_Product_unit] :
% 14.17/8.00 ( ( member7821138191923116944t_unit @ ( produc5692694829384537815t_unit @ X @ Y ) @ ( id_on_Product_unit @ A ) )
% 14.17/8.00 = ( ( X = Y )
% 14.17/8.00 & ( member_Product_unit @ X @ A ) ) ) ).
% 14.17/8.00
% 14.17/8.00 thf(fact_561_Id__on__iff,axiom,
% 14.17/8.00 ! [X: nat,Y: nat,A: set_nat] :
% 14.17/8.00 ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ Y ) @ ( id_on_nat @ A ) )
% 14.17/8.00 = ( ( X = Y )
% 14.17/8.00 & ( member_nat @ X @ A ) ) ) ).
% 14.17/8.00
% 14.17/8.00 thf(fact_562_Id__on__iff,axiom,
% 14.17/8.00 ! [X: int,Y: int,A: set_int] :
% 14.17/8.00 ( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ X @ Y ) @ ( id_on_int @ A ) )
% 14.17/8.00 = ( ( X = Y )
% 14.17/8.00 & ( member_int @ X @ A ) ) ) ).
% 14.17/8.01
% 14.17/8.01 thf(fact_563_Id__on__eqI,axiom,
% 14.17/8.01 ! [A2: produc3658429121746597890et_nat > $o,B: produc3658429121746597890et_nat > $o,A: set_Pr4532377907799695533_nat_o] :
% 14.17/8.01 ( ( A2 = B )
% 14.17/8.01 => ( ( member6576561426505652726_nat_o @ A2 @ A )
% 14.17/8.01 => ( member8781333585448626064_nat_o @ ( produc7368190662567826135_nat_o @ A2 @ B ) @ ( id_on_7438695062414216554_nat_o @ A ) ) ) ) ).
% 14.17/8.01
% 14.17/8.01 thf(fact_564_Id__on__eqI,axiom,
% 14.17/8.01 ! [A2: $o,B: $o,A: set_o] :
% 14.17/8.01 ( ( A2 = B )
% 14.17/8.01 => ( ( member_o @ A2 @ A )
% 14.17/8.01 => ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ A2 @ B ) @ ( id_on_o @ A ) ) ) ) ).
% 14.17/8.01
% 14.17/8.01 thf(fact_565_Id__on__eqI,axiom,
% 14.17/8.01 ! [A2: product_unit,B: product_unit,A: set_Product_unit] :
% 14.17/8.01 ( ( A2 = B )
% 14.17/8.01 => ( ( member_Product_unit @ A2 @ A )
% 14.17/8.01 => ( member7821138191923116944t_unit @ ( produc5692694829384537815t_unit @ A2 @ B ) @ ( id_on_Product_unit @ A ) ) ) ) ).
% 14.17/8.01
% 14.17/8.01 thf(fact_566_Id__on__eqI,axiom,
% 14.17/8.01 ! [A2: nat,B: nat,A: set_nat] :
% 14.17/8.01 ( ( A2 = B )
% 14.17/8.01 => ( ( member_nat @ A2 @ A )
% 14.17/8.01 => ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A2 @ B ) @ ( id_on_nat @ A ) ) ) ) ).
% 14.17/8.01
% 14.17/8.01 thf(fact_567_Id__on__eqI,axiom,
% 14.17/8.01 ! [A2: int,B: int,A: set_int] :
% 14.17/8.01 ( ( A2 = B )
% 14.17/8.01 => ( ( member_int @ A2 @ A )
% 14.17/8.01 => ( member5262025264175285858nt_int @ ( product_Pair_int_int @ A2 @ B ) @ ( id_on_int @ A ) ) ) ) ).
% 14.17/8.01
% 14.17/8.01 thf(fact_568_Id__onE,axiom,
% 14.17/8.01 ! [C: produc4928098042776334183_nat_o,A: set_Pr4532377907799695533_nat_o] :
% 14.17/8.01 ( ( member8781333585448626064_nat_o @ C @ ( id_on_7438695062414216554_nat_o @ A ) )
% 14.17/8.01 => ~ ! [X3: produc3658429121746597890et_nat > $o] :
% 14.17/8.01 ( ( member6576561426505652726_nat_o @ X3 @ A )
% 14.17/8.01 => ( C
% 14.17/8.01 != ( produc7368190662567826135_nat_o @ X3 @ X3 ) ) ) ) ).
% 14.17/8.01
% 14.17/8.01 thf(fact_569_Id__onE,axiom,
% 14.17/8.01 ! [C: product_prod_o_o,A: set_o] :
% 14.17/8.01 ( ( member7466972457876170832od_o_o @ C @ ( id_on_o @ A ) )
% 14.17/8.01 => ~ ! [X3: $o] :
% 14.17/8.01 ( ( member_o @ X3 @ A )
% 14.17/8.01 => ( C
% 14.17/8.01 != ( product_Pair_o_o @ X3 @ X3 ) ) ) ) ).
% 14.17/8.01
% 14.17/8.01 thf(fact_570_Id__onE,axiom,
% 14.17/8.01 ! [C: produc8459935480633519975t_unit,A: set_Product_unit] :
% 14.17/8.01 ( ( member7821138191923116944t_unit @ C @ ( id_on_Product_unit @ A ) )
% 14.17/8.01 => ~ ! [X3: product_unit] :
% 14.17/8.01 ( ( member_Product_unit @ X3 @ A )
% 14.17/8.01 => ( C
% 14.17/8.01 != ( produc5692694829384537815t_unit @ X3 @ X3 ) ) ) ) ).
% 14.17/8.01
% 14.17/8.01 thf(fact_571_Id__onE,axiom,
% 14.17/8.01 ! [C: product_prod_nat_nat,A: set_nat] :
% 14.17/8.01 ( ( member8440522571783428010at_nat @ C @ ( id_on_nat @ A ) )
% 14.17/8.01 => ~ ! [X3: nat] :
% 14.17/8.01 ( ( member_nat @ X3 @ A )
% 14.17/8.01 => ( C
% 14.17/8.01 != ( product_Pair_nat_nat @ X3 @ X3 ) ) ) ) ).
% 14.17/8.01
% 14.17/8.01 thf(fact_572_Id__onE,axiom,
% 14.17/8.01 ! [C: product_prod_int_int,A: set_int] :
% 14.17/8.01 ( ( member5262025264175285858nt_int @ C @ ( id_on_int @ A ) )
% 14.17/8.01 => ~ ! [X3: int] :
% 14.17/8.01 ( ( member_int @ X3 @ A )
% 14.17/8.01 => ( C
% 14.17/8.01 != ( product_Pair_int_int @ X3 @ X3 ) ) ) ) ).
% 14.17/8.01
% 14.17/8.01 thf(fact_573_lnear__order__on__empty,axiom,
% 14.17/8.01 order_888282469287196226t_unit @ bot_bo3957492148770167129t_unit @ bot_bo1654022762305692595t_unit ).
% 14.17/8.01
% 14.17/8.01 thf(fact_574_lnear__order__on__empty,axiom,
% 14.17/8.01 order_6238756238976269133r_on_o @ bot_bot_set_o @ bot_bo7073875226086086771od_o_o ).
% 14.17/8.01
% 14.17/8.01 thf(fact_575_lnear__order__on__empty,axiom,
% 14.17/8.01 order_4473980167227706203on_nat @ bot_bot_set_nat @ bot_bo2099793752762293965at_nat ).
% 14.17/8.01
% 14.17/8.01 thf(fact_576_lnear__order__on__empty,axiom,
% 14.17/8.01 order_4471489696718655927on_int @ bot_bot_set_int @ bot_bo1796632182523588997nt_int ).
% 14.17/8.01
% 14.17/8.01 thf(fact_577_lnear__order__on__empty,axiom,
% 14.17/8.01 order_1145710701291701250_nat_o @ bot_bo7824918357723582233_nat_o @ bot_bo3657187498359500723_nat_o ).
% 14.17/8.01
% 14.17/8.01 thf(fact_578_refl__on__empty,axiom,
% 14.17/8.01 refl_on_Product_unit @ bot_bo3957492148770167129t_unit @ bot_bo1654022762305692595t_unit ).
% 14.17/8.01
% 14.17/8.01 thf(fact_579_refl__on__empty,axiom,
% 14.17/8.01 refl_on_o @ bot_bot_set_o @ bot_bo7073875226086086771od_o_o ).
% 14.17/8.01
% 14.17/8.01 thf(fact_580_refl__on__empty,axiom,
% 14.17/8.01 refl_on_nat @ bot_bot_set_nat @ bot_bo2099793752762293965at_nat ).
% 14.17/8.01
% 14.17/8.01 thf(fact_581_refl__on__empty,axiom,
% 14.17/8.01 refl_on_int @ bot_bot_set_int @ bot_bo1796632182523588997nt_int ).
% 14.17/8.01
% 14.17/8.01 thf(fact_582_refl__on__empty,axiom,
% 14.17/8.01 refl_o1075864025857821388_nat_o @ bot_bo7824918357723582233_nat_o @ bot_bo3657187498359500723_nat_o ).
% 14.17/8.01
% 14.17/8.01 thf(fact_583_properD1,axiom,
% 14.17/8.01 ! [P: produc3658429121746597890et_nat > $o,H2: heap_e7401611519738050253t_unit,As: set_nat] :
% 14.17/8.01 ( ( proper @ P )
% 14.17/8.01 => ( ( P @ ( produc7507926704131184380et_nat @ H2 @ As ) )
% 14.17/8.01 => ( in_range @ ( produc7507926704131184380et_nat @ H2 @ As ) ) ) ) ).
% 14.17/8.01
% 14.17/8.01 thf(fact_584_proper__iff,axiom,
% 14.17/8.01 ! [P: produc3658429121746597890et_nat > $o,As: set_nat,H2: heap_e7401611519738050253t_unit,H3: heap_e7401611519738050253t_unit] :
% 14.17/8.01 ( ( proper @ P )
% 14.17/8.01 => ( ( relH @ As @ H2 @ H3 )
% 14.17/8.01 => ( ( in_range @ ( produc7507926704131184380et_nat @ H3 @ As ) )
% 14.17/8.01 => ( ( P @ ( produc7507926704131184380et_nat @ H2 @ As ) )
% 14.17/8.01 = ( P @ ( produc7507926704131184380et_nat @ H3 @ As ) ) ) ) ) ) ).
% 14.17/8.01
% 14.17/8.01 thf(fact_585_proper__def,axiom,
% 14.17/8.01 ( proper
% 14.17/8.01 = ( ^ [P2: produc3658429121746597890et_nat > $o] :
% 14.17/8.01 ! [H4: heap_e7401611519738050253t_unit,H5: heap_e7401611519738050253t_unit,As3: set_nat] :
% 14.17/8.01 ( ( ( P2 @ ( produc7507926704131184380et_nat @ H4 @ As3 ) )
% 14.17/8.01 => ( in_range @ ( produc7507926704131184380et_nat @ H4 @ As3 ) ) )
% 14.17/8.01 & ( ( ( P2 @ ( produc7507926704131184380et_nat @ H4 @ As3 ) )
% 14.17/8.01 & ( relH @ As3 @ H4 @ H5 )
% 14.17/8.01 & ( in_range @ ( produc7507926704131184380et_nat @ H5 @ As3 ) ) )
% 14.17/8.01 => ( P2 @ ( produc7507926704131184380et_nat @ H5 @ As3 ) ) ) ) ) ) ).
% 14.17/8.01
% 14.17/8.01 thf(fact_586_properD2,axiom,
% 14.17/8.01 ! [P: produc3658429121746597890et_nat > $o,H2: heap_e7401611519738050253t_unit,As: set_nat,H3: heap_e7401611519738050253t_unit] :
% 14.17/8.01 ( ( proper @ P )
% 14.17/8.01 => ( ( P @ ( produc7507926704131184380et_nat @ H2 @ As ) )
% 14.17/8.01 => ( ( relH @ As @ H2 @ H3 )
% 14.17/8.01 => ( ( in_range @ ( produc7507926704131184380et_nat @ H3 @ As ) )
% 14.17/8.01 => ( P @ ( produc7507926704131184380et_nat @ H3 @ As ) ) ) ) ) ) ).
% 14.17/8.01
% 14.17/8.01 thf(fact_587_properI,axiom,
% 14.17/8.01 ! [P: produc3658429121746597890et_nat > $o] :
% 14.17/8.01 ( ! [As2: set_nat,H: heap_e7401611519738050253t_unit] :
% 14.17/8.01 ( ( P @ ( produc7507926704131184380et_nat @ H @ As2 ) )
% 14.17/8.01 => ( in_range @ ( produc7507926704131184380et_nat @ H @ As2 ) ) )
% 14.17/8.01 => ( ! [As2: set_nat,H: heap_e7401611519738050253t_unit,H6: heap_e7401611519738050253t_unit] :
% 14.17/8.01 ( ( P @ ( produc7507926704131184380et_nat @ H @ As2 ) )
% 14.17/8.01 => ( ( relH @ As2 @ H @ H6 )
% 14.17/8.01 => ( ( in_range @ ( produc7507926704131184380et_nat @ H6 @ As2 ) )
% 14.17/8.01 => ( P @ ( produc7507926704131184380et_nat @ H6 @ As2 ) ) ) ) )
% 14.17/8.01 => ( proper @ P ) ) ) ).
% 14.17/8.01
% 14.17/8.01 thf(fact_588_pair__vimage__is__Image,axiom,
% 14.17/8.01 ! [U: heap_e7401611519738050253t_unit,E: set_Pr3948176798113811640et_nat] :
% 14.17/8.01 ( ( vimage5412618565253405765et_nat @ ( produc7507926704131184380et_nat @ U ) @ E )
% 14.17/8.01 = ( image_3942343239045813184et_nat @ E @ ( insert8563697753282147485t_unit @ U @ bot_bo3686245532977126937t_unit ) ) ) ).
% 14.17/8.01
% 14.17/8.01 thf(fact_589_pair__vimage__is__Image,axiom,
% 14.17/8.01 ! [U: nat,E: set_Pr1261947904930325089at_nat] :
% 14.17/8.01 ( ( vimage8013328719654469172at_nat @ ( product_Pair_nat_nat @ U ) @ E )
% 14.17/8.01 = ( image_nat_nat @ E @ ( insert_nat @ U @ bot_bot_set_nat ) ) ) ).
% 14.17/8.01
% 14.17/8.01 thf(fact_590_pair__vimage__is__Image,axiom,
% 14.17/8.01 ! [U: int,E: set_Pr958786334691620121nt_int] :
% 14.17/8.01 ( ( vimage7872673496510460816nt_int @ ( product_Pair_int_int @ U ) @ E )
% 14.17/8.01 = ( image_int_int @ E @ ( insert_int @ U @ bot_bot_set_int ) ) ) ).
% 14.17/8.01
% 14.17/8.01 thf(fact_591_pair__vimage__is__Image,axiom,
% 14.17/8.01 ! [U: produc3658429121746597890et_nat > $o,E: set_Pr3286484037609594932et_nat] :
% 14.17/8.01 ( ( vimage1990136491886319165et_nat @ ( produc5001842942810119800et_nat @ U ) @ E )
% 14.17/8.01 = ( image_1814916760386648892et_nat @ E @ ( insert5175938949040314269_nat_o @ U @ bot_bo7824918357723582233_nat_o ) ) ) ).
% 14.17/8.01
% 14.17/8.01 thf(fact_592_pair__vimage__is__Image,axiom,
% 14.17/8.01 ! [U: produc3658429121746597890et_nat > $o,E: set_Pr8536935166611901872et_nat] :
% 14.17/8.01 ( ( vimage7160910655466636085et_nat @ ( produc2245416461498447860et_nat @ U ) @ E )
% 14.17/8.01 = ( image_7240710759097644216et_nat @ E @ ( insert5175938949040314269_nat_o @ U @ bot_bo7824918357723582233_nat_o ) ) ) ).
% 14.17/8.01
% 14.17/8.01 thf(fact_593_Range__insert,axiom,
% 14.17/8.01 ! [A2: heap_e7401611519738050253t_unit,B: set_nat,R2: set_Pr3948176798113811640et_nat] :
% 14.17/8.01 ( ( range_9132311542094983838et_nat @ ( insert1271470778893645362et_nat @ ( produc7507926704131184380et_nat @ A2 @ B ) @ R2 ) )
% 14.17/8.01 = ( insert_set_nat @ B @ ( range_9132311542094983838et_nat @ R2 ) ) ) ).
% 14.17/8.01
% 14.17/8.01 thf(fact_594_Range__insert,axiom,
% 14.17/8.01 ! [A2: nat,B: nat,R2: set_Pr1261947904930325089at_nat] :
% 14.17/8.01 ( ( range_nat_nat @ ( insert8211810215607154385at_nat @ ( product_Pair_nat_nat @ A2 @ B ) @ R2 ) )
% 14.17/8.01 = ( insert_nat @ B @ ( range_nat_nat @ R2 ) ) ) ).
% 14.17/8.01
% 14.17/8.01 thf(fact_595_Range__insert,axiom,
% 14.17/8.01 ! [A2: int,B: int,R2: set_Pr958786334691620121nt_int] :
% 14.17/8.01 ( ( range_int_int @ ( insert5033312907999012233nt_int @ ( product_Pair_int_int @ A2 @ B ) @ R2 ) )
% 14.17/8.01 = ( insert_int @ B @ ( range_int_int @ R2 ) ) ) ).
% 14.17/8.01
% 14.17/8.01 thf(fact_596_Range__insert,axiom,
% 14.17/8.01 ! [A2: produc3658429121746597890et_nat > $o,B: produc3658429121746597890et_nat,R2: set_Pr3286484037609594932et_nat] :
% 14.17/8.01 ( ( range_3398330936903821850et_nat @ ( insert3111075074545270702et_nat @ ( produc5001842942810119800et_nat @ A2 @ B ) @ R2 ) )
% 14.17/8.01 = ( insert1271470778893645362et_nat @ B @ ( range_3398330936903821850et_nat @ R2 ) ) ) ).
% 14.17/8.01
% 14.17/8.01 thf(fact_597_Range__insert,axiom,
% 14.17/8.01 ! [A2: produc3658429121746597890et_nat > $o,B: produc3925858234332021118et_nat,R2: set_Pr8536935166611901872et_nat] :
% 14.17/8.01 ( ( range_7260058354244137878et_nat @ ( insert8297122186708525866et_nat @ ( produc2245416461498447860et_nat @ A2 @ B ) @ R2 ) )
% 14.17/8.01 = ( insert3111075074545270702et_nat @ B @ ( range_7260058354244137878et_nat @ R2 ) ) ) ).
% 14.17/8.01
% 14.17/8.01 thf(fact_598_Domain__insert,axiom,
% 14.17/8.01 ! [A2: heap_e7401611519738050253t_unit,B: set_nat,R2: set_Pr3948176798113811640et_nat] :
% 14.17/8.01 ( ( domain2007548326387822151et_nat @ ( insert1271470778893645362et_nat @ ( produc7507926704131184380et_nat @ A2 @ B ) @ R2 ) )
% 14.17/8.01 = ( insert8563697753282147485t_unit @ A2 @ ( domain2007548326387822151et_nat @ R2 ) ) ) ).
% 14.17/8.01
% 14.17/8.01 thf(fact_599_Domain__insert,axiom,
% 14.17/8.01 ! [A2: nat,B: nat,R2: set_Pr1261947904930325089at_nat] :
% 14.17/8.01 ( ( domain_nat_nat @ ( insert8211810215607154385at_nat @ ( product_Pair_nat_nat @ A2 @ B ) @ R2 ) )
% 14.17/8.01 = ( insert_nat @ A2 @ ( domain_nat_nat @ R2 ) ) ) ).
% 14.17/8.01
% 14.17/8.01 thf(fact_600_Domain__insert,axiom,
% 14.17/8.01 ! [A2: int,B: int,R2: set_Pr958786334691620121nt_int] :
% 14.17/8.01 ( ( domain_int_int @ ( insert5033312907999012233nt_int @ ( product_Pair_int_int @ A2 @ B ) @ R2 ) )
% 14.17/8.01 = ( insert_int @ A2 @ ( domain_int_int @ R2 ) ) ) ).
% 14.17/8.01
% 14.17/8.01 thf(fact_601_Domain__insert,axiom,
% 14.17/8.01 ! [A2: produc3658429121746597890et_nat > $o,B: produc3658429121746597890et_nat,R2: set_Pr3286484037609594932et_nat] :
% 14.17/8.01 ( ( domain2358961613337345475et_nat @ ( insert3111075074545270702et_nat @ ( produc5001842942810119800et_nat @ A2 @ B ) @ R2 ) )
% 14.17/8.01 = ( insert5175938949040314269_nat_o @ A2 @ ( domain2358961613337345475et_nat @ R2 ) ) ) ).
% 14.17/8.01
% 14.17/8.01 thf(fact_602_Domain__insert,axiom,
% 14.17/8.01 ! [A2: produc3658429121746597890et_nat > $o,B: produc3925858234332021118et_nat,R2: set_Pr8536935166611901872et_nat] :
% 14.17/8.01 ( ( domain3618434998481624895et_nat @ ( insert8297122186708525866et_nat @ ( produc2245416461498447860et_nat @ A2 @ B ) @ R2 ) )
% 14.17/8.01 = ( insert5175938949040314269_nat_o @ A2 @ ( domain3618434998481624895et_nat @ R2 ) ) ) ).
% 14.17/8.01
% 14.17/8.01 thf(fact_603_preorder__on__empty,axiom,
% 14.17/8.01 order_7534304569044981540t_unit @ bot_bo3957492148770167129t_unit @ bot_bo1654022762305692595t_unit ).
% 14.17/8.01
% 14.17/8.01 thf(fact_604_preorder__on__empty,axiom,
% 14.17/8.01 order_preorder_on_o @ bot_bot_set_o @ bot_bo7073875226086086771od_o_o ).
% 14.17/8.01
% 14.17/8.01 thf(fact_605_preorder__on__empty,axiom,
% 14.17/8.01 order_4861654808422542329on_nat @ bot_bot_set_nat @ bot_bo2099793752762293965at_nat ).
% 14.17/8.01
% 14.17/8.01 thf(fact_606_preorder__on__empty,axiom,
% 14.17/8.01 order_4859164337913492053on_int @ bot_bot_set_int @ bot_bo1796632182523588997nt_int ).
% 14.17/8.01
% 14.17/8.01 thf(fact_607_preorder__on__empty,axiom,
% 14.17/8.01 order_174704566861754596_nat_o @ bot_bo7824918357723582233_nat_o @ bot_bo3657187498359500723_nat_o ).
% 14.17/8.01
% 14.17/8.01 thf(fact_608_trancl__single,axiom,
% 14.17/8.01 ! [A2: nat,B: nat] :
% 14.17/8.01 ( ( transi6264000038957366511cl_nat @ ( insert8211810215607154385at_nat @ ( product_Pair_nat_nat @ A2 @ B ) @ bot_bo2099793752762293965at_nat ) )
% 14.17/8.01 = ( insert8211810215607154385at_nat @ ( product_Pair_nat_nat @ A2 @ B ) @ bot_bo2099793752762293965at_nat ) ) ).
% 14.17/8.01
% 14.17/8.01 thf(fact_609_trancl__single,axiom,
% 14.17/8.01 ! [A2: int,B: int] :
% 14.17/8.01 ( ( transi6261509568448316235cl_int @ ( insert5033312907999012233nt_int @ ( product_Pair_int_int @ A2 @ B ) @ bot_bo1796632182523588997nt_int ) )
% 14.17/8.01 = ( insert5033312907999012233nt_int @ ( product_Pair_int_int @ A2 @ B ) @ bot_bo1796632182523588997nt_int ) ) ).
% 14.17/8.01
% 14.17/8.01 thf(fact_610_trans__singleton,axiom,
% 14.17/8.01 ! [A2: nat] : ( trans_nat @ ( insert8211810215607154385at_nat @ ( product_Pair_nat_nat @ A2 @ A2 ) @ bot_bo2099793752762293965at_nat ) ) ).
% 14.17/8.01
% 14.17/8.01 thf(fact_611_trans__singleton,axiom,
% 14.17/8.01 ! [A2: int] : ( trans_int @ ( insert5033312907999012233nt_int @ ( product_Pair_int_int @ A2 @ A2 ) @ bot_bo1796632182523588997nt_int ) ) ).
% 14.17/8.01
% 14.17/8.01 thf(fact_612_mult_Osafe__commute,axiom,
% 14.17/8.01 ! [X: assn,Y: assn,A2: assn,B: assn] :
% 14.17/8.01 ( ( syntax7398250324933576852n_assn @ ( times_times_assn @ X @ Y ) @ A2 )
% 14.17/8.01 => ( ( times_times_assn @ A2 @ B )
% 14.17/8.01 = ( times_times_assn @ B @ A2 ) ) ) ).
% 14.17/8.01
% 14.17/8.01 thf(fact_613_mult_Osafe__commute,axiom,
% 14.17/8.01 ! [X: rat,Y: rat,A2: rat,B: rat] :
% 14.17/8.01 ( ( syntax3730441303064801268at_rat @ ( times_times_rat @ X @ Y ) @ A2 )
% 14.17/8.01 => ( ( times_times_rat @ A2 @ B )
% 14.17/8.01 = ( times_times_rat @ B @ A2 ) ) ) ).
% 14.17/8.01
% 14.17/8.01 thf(fact_614_mult_Osafe__commute,axiom,
% 14.17/8.01 ! [X: nat,Y: nat,A2: nat,B: nat] :
% 14.17/8.01 ( ( syntax4682126007086162916at_nat @ ( times_times_nat @ X @ Y ) @ A2 )
% 14.17/8.01 => ( ( times_times_nat @ A2 @ B )
% 14.17/8.01 = ( times_times_nat @ B @ A2 ) ) ) ).
% 14.17/8.01
% 14.17/8.01 thf(fact_615_mult_Osafe__commute,axiom,
% 14.17/8.01 ! [X: int,Y: int,A2: int,B: int] :
% 14.17/8.01 ( ( syntax5678989248478167196nt_int @ ( times_times_int @ X @ Y ) @ A2 )
% 14.17/8.01 => ( ( times_times_int @ A2 @ B )
% 14.17/8.01 = ( times_times_int @ B @ A2 ) ) ) ).
% 14.17/8.01
% 14.17/8.01 thf(fact_616_vimage__eq,axiom,
% 14.17/8.01 ! [A2: nat,F: nat > nat,B4: set_nat] :
% 14.17/8.01 ( ( member_nat @ A2 @ ( vimage_nat_nat @ F @ B4 ) )
% 14.17/8.01 = ( member_nat @ ( F @ A2 ) @ B4 ) ) ).
% 14.17/8.01
% 14.17/8.01 thf(fact_617_vimage__eq,axiom,
% 14.17/8.01 ! [A2: nat,F: nat > int,B4: set_int] :
% 14.17/8.01 ( ( member_nat @ A2 @ ( vimage_nat_int @ F @ B4 ) )
% 14.17/8.01 = ( member_int @ ( F @ A2 ) @ B4 ) ) ).
% 14.17/8.01
% 14.17/8.01 thf(fact_618_vimage__eq,axiom,
% 14.17/8.01 ! [A2: nat,F: nat > $o,B4: set_o] :
% 14.17/8.01 ( ( member_nat @ A2 @ ( vimage_nat_o @ F @ B4 ) )
% 14.17/8.01 = ( member_o @ ( F @ A2 ) @ B4 ) ) ).
% 14.17/8.01
% 14.17/8.01 thf(fact_619_vimage__eq,axiom,
% 14.17/8.01 ! [A2: nat,F: nat > product_unit,B4: set_Product_unit] :
% 14.17/8.01 ( ( member_nat @ A2 @ ( vimage4884490618288580032t_unit @ F @ B4 ) )
% 14.17/8.01 = ( member_Product_unit @ ( F @ A2 ) @ B4 ) ) ).
% 14.17/8.01
% 14.17/8.01 thf(fact_620_vimage__eq,axiom,
% 14.17/8.01 ! [A2: int,F: int > nat,B4: set_nat] :
% 14.17/8.01 ( ( member_int @ A2 @ ( vimage_int_nat @ F @ B4 ) )
% 14.17/8.01 = ( member_nat @ ( F @ A2 ) @ B4 ) ) ).
% 14.17/8.01
% 14.17/8.01 thf(fact_621_vimage__eq,axiom,
% 14.17/8.01 ! [A2: int,F: int > int,B4: set_int] :
% 14.17/8.01 ( ( member_int @ A2 @ ( vimage_int_int @ F @ B4 ) )
% 14.17/8.01 = ( member_int @ ( F @ A2 ) @ B4 ) ) ).
% 14.17/8.01
% 14.17/8.01 thf(fact_622_vimage__eq,axiom,
% 14.17/8.01 ! [A2: int,F: int > $o,B4: set_o] :
% 14.17/8.01 ( ( member_int @ A2 @ ( vimage_int_o @ F @ B4 ) )
% 14.17/8.01 = ( member_o @ ( F @ A2 ) @ B4 ) ) ).
% 14.17/8.01
% 14.17/8.01 thf(fact_623_vimage__eq,axiom,
% 14.17/8.01 ! [A2: int,F: int > product_unit,B4: set_Product_unit] :
% 14.17/8.01 ( ( member_int @ A2 @ ( vimage2000912565490760292t_unit @ F @ B4 ) )
% 14.17/8.01 = ( member_Product_unit @ ( F @ A2 ) @ B4 ) ) ).
% 14.17/8.01
% 14.17/8.01 thf(fact_624_vimage__eq,axiom,
% 14.17/8.01 ! [A2: $o,F: $o > nat,B4: set_nat] :
% 14.17/8.01 ( ( member_o @ A2 @ ( vimage_o_nat @ F @ B4 ) )
% 14.17/8.01 = ( member_nat @ ( F @ A2 ) @ B4 ) ) ).
% 14.17/8.01
% 14.17/8.01 thf(fact_625_vimage__eq,axiom,
% 14.17/8.01 ! [A2: $o,F: $o > int,B4: set_int] :
% 14.17/8.01 ( ( member_o @ A2 @ ( vimage_o_int @ F @ B4 ) )
% 14.17/8.01 = ( member_int @ ( F @ A2 ) @ B4 ) ) ).
% 14.17/8.01
% 14.17/8.01 thf(fact_626_vimageI,axiom,
% 14.17/8.01 ! [F: nat > nat,A2: nat,B: nat,B4: set_nat] :
% 14.17/8.01 ( ( ( F @ A2 )
% 14.17/8.01 = B )
% 14.17/8.01 => ( ( member_nat @ B @ B4 )
% 14.17/8.01 => ( member_nat @ A2 @ ( vimage_nat_nat @ F @ B4 ) ) ) ) ).
% 14.17/8.01
% 14.17/8.01 thf(fact_627_vimageI,axiom,
% 14.17/8.01 ! [F: int > nat,A2: int,B: nat,B4: set_nat] :
% 14.17/8.01 ( ( ( F @ A2 )
% 14.17/8.01 = B )
% 14.17/8.01 => ( ( member_nat @ B @ B4 )
% 14.17/8.01 => ( member_int @ A2 @ ( vimage_int_nat @ F @ B4 ) ) ) ) ).
% 14.17/8.01
% 14.17/8.01 thf(fact_628_vimageI,axiom,
% 14.17/8.01 ! [F: $o > nat,A2: $o,B: nat,B4: set_nat] :
% 14.17/8.01 ( ( ( F @ A2 )
% 14.17/8.01 = B )
% 14.17/8.01 => ( ( member_nat @ B @ B4 )
% 14.17/8.01 => ( member_o @ A2 @ ( vimage_o_nat @ F @ B4 ) ) ) ) ).
% 14.17/8.01
% 14.17/8.01 thf(fact_629_vimageI,axiom,
% 14.17/8.01 ! [F: product_unit > nat,A2: product_unit,B: nat,B4: set_nat] :
% 14.17/8.01 ( ( ( F @ A2 )
% 14.17/8.01 = B )
% 14.17/8.01 => ( ( member_nat @ B @ B4 )
% 14.17/8.01 => ( member_Product_unit @ A2 @ ( vimage6253328473476588386it_nat @ F @ B4 ) ) ) ) ).
% 14.17/8.01
% 14.17/8.01 thf(fact_630_vimageI,axiom,
% 14.17/8.01 ! [F: nat > int,A2: nat,B: int,B4: set_int] :
% 14.17/8.01 ( ( ( F @ A2 )
% 14.17/8.01 = B )
% 14.17/8.01 => ( ( member_int @ B @ B4 )
% 14.17/8.01 => ( member_nat @ A2 @ ( vimage_nat_int @ F @ B4 ) ) ) ) ).
% 14.17/8.01
% 14.17/8.01 thf(fact_631_vimageI,axiom,
% 14.17/8.01 ! [F: int > int,A2: int,B: int,B4: set_int] :
% 14.17/8.01 ( ( ( F @ A2 )
% 14.17/8.01 = B )
% 14.17/8.01 => ( ( member_int @ B @ B4 )
% 14.17/8.01 => ( member_int @ A2 @ ( vimage_int_int @ F @ B4 ) ) ) ) ).
% 14.17/8.01
% 14.17/8.01 thf(fact_632_vimageI,axiom,
% 14.17/8.01 ! [F: $o > int,A2: $o,B: int,B4: set_int] :
% 14.17/8.01 ( ( ( F @ A2 )
% 14.17/8.01 = B )
% 14.17/8.01 => ( ( member_int @ B @ B4 )
% 14.17/8.01 => ( member_o @ A2 @ ( vimage_o_int @ F @ B4 ) ) ) ) ).
% 14.17/8.01
% 14.17/8.01 thf(fact_633_vimageI,axiom,
% 14.17/8.01 ! [F: product_unit > int,A2: product_unit,B: int,B4: set_int] :
% 14.17/8.01 ( ( ( F @ A2 )
% 14.17/8.01 = B )
% 14.17/8.01 => ( ( member_int @ B @ B4 )
% 14.17/8.01 => ( member_Product_unit @ A2 @ ( vimage6250838002967538110it_int @ F @ B4 ) ) ) ) ).
% 14.17/8.01
% 14.17/8.01 thf(fact_634_vimageI,axiom,
% 14.17/8.01 ! [F: nat > $o,A2: nat,B: $o,B4: set_o] :
% 14.17/8.01 ( ( ( F @ A2 )
% 14.17/8.01 = B )
% 14.17/8.01 => ( ( member_o @ B @ B4 )
% 14.17/8.01 => ( member_nat @ A2 @ ( vimage_nat_o @ F @ B4 ) ) ) ) ).
% 14.17/8.01
% 14.17/8.01 thf(fact_635_vimageI,axiom,
% 14.17/8.01 ! [F: int > $o,A2: int,B: $o,B4: set_o] :
% 14.17/8.01 ( ( ( F @ A2 )
% 14.17/8.01 = B )
% 14.17/8.01 => ( ( member_o @ B @ B4 )
% 14.17/8.01 => ( member_int @ A2 @ ( vimage_int_o @ F @ B4 ) ) ) ) ).
% 14.17/8.01
% 14.17/8.01 thf(fact_636_vimage__empty,axiom,
% 14.17/8.01 ! [F: product_unit > product_unit] :
% 14.17/8.01 ( ( vimage7995052115951654139t_unit @ F @ bot_bo3957492148770167129t_unit )
% 14.17/8.01 = bot_bo3957492148770167129t_unit ) ).
% 14.17/8.01
% 14.17/8.01 thf(fact_637_vimage__empty,axiom,
% 14.17/8.01 ! [F: $o > product_unit] :
% 14.17/8.01 ( ( vimage4222377557387472176t_unit @ F @ bot_bo3957492148770167129t_unit )
% 14.17/8.01 = bot_bot_set_o ) ).
% 14.17/8.01
% 14.17/8.01 thf(fact_638_vimage__empty,axiom,
% 14.17/8.01 ! [F: nat > product_unit] :
% 14.17/8.01 ( ( vimage4884490618288580032t_unit @ F @ bot_bo3957492148770167129t_unit )
% 14.17/8.01 = bot_bot_set_nat ) ).
% 14.17/8.01
% 14.17/8.01 thf(fact_639_vimage__empty,axiom,
% 14.17/8.01 ! [F: int > product_unit] :
% 14.17/8.01 ( ( vimage2000912565490760292t_unit @ F @ bot_bo3957492148770167129t_unit )
% 14.17/8.01 = bot_bot_set_int ) ).
% 14.17/8.01
% 14.17/8.01 thf(fact_640_vimage__empty,axiom,
% 14.17/8.01 ! [F: product_unit > $o] :
% 14.17/8.01 ( ( vimage8808348054653608838unit_o @ F @ bot_bot_set_o )
% 14.17/8.01 = bot_bo3957492148770167129t_unit ) ).
% 14.17/8.01
% 14.17/8.01 thf(fact_641_vimage__empty,axiom,
% 14.17/8.01 ! [F: $o > $o] :
% 14.17/8.01 ( ( vimage_o_o @ F @ bot_bot_set_o )
% 14.17/8.01 = bot_bot_set_o ) ).
% 14.17/8.01
% 14.17/8.01 thf(fact_642_vimage__empty,axiom,
% 14.17/8.01 ! [F: nat > $o] :
% 14.17/8.01 ( ( vimage_nat_o @ F @ bot_bot_set_o )
% 14.17/8.01 = bot_bot_set_nat ) ).
% 14.17/8.01
% 14.17/8.01 thf(fact_643_vimage__empty,axiom,
% 14.17/8.01 ! [F: int > $o] :
% 14.17/8.01 ( ( vimage_int_o @ F @ bot_bot_set_o )
% 14.17/8.01 = bot_bot_set_int ) ).
% 14.17/8.01
% 14.17/8.01 thf(fact_644_vimage__empty,axiom,
% 14.17/8.01 ! [F: product_unit > nat] :
% 14.17/8.01 ( ( vimage6253328473476588386it_nat @ F @ bot_bot_set_nat )
% 14.17/8.01 = bot_bo3957492148770167129t_unit ) ).
% 14.17/8.01
% 14.17/8.01 thf(fact_645_vimage__empty,axiom,
% 14.17/8.01 ! [F: $o > nat] :
% 14.17/8.01 ( ( vimage_o_nat @ F @ bot_bot_set_nat )
% 14.17/8.01 = bot_bot_set_o ) ).
% 14.17/8.01
% 14.17/8.01 thf(fact_646_relH__sym,axiom,
% 14.17/8.01 ! [As: set_nat,H2: heap_e7401611519738050253t_unit,H3: heap_e7401611519738050253t_unit] :
% 14.17/8.01 ( ( relH @ As @ H2 @ H3 )
% 14.17/8.01 => ( relH @ As @ H3 @ H2 ) ) ).
% 14.17/8.01
% 14.17/8.01 thf(fact_647_relH__trans,axiom,
% 14.17/8.01 ! [As: set_nat,H12: heap_e7401611519738050253t_unit,H23: heap_e7401611519738050253t_unit,H32: heap_e7401611519738050253t_unit] :
% 14.17/8.01 ( ( relH @ As @ H12 @ H23 )
% 14.17/8.01 => ( ( relH @ As @ H23 @ H32 )
% 14.17/8.01 => ( relH @ As @ H12 @ H32 ) ) ) ).
% 14.17/8.01
% 14.17/8.01 thf(fact_648_vimage__Collect,axiom,
% 14.17/8.01 ! [P: nat > $o,F: nat > nat,Q2: nat > $o] :
% 14.17/8.01 ( ! [X3: nat] :
% 14.17/8.01 ( ( P @ ( F @ X3 ) )
% 14.17/8.01 = ( Q2 @ X3 ) )
% 14.17/8.01 => ( ( vimage_nat_nat @ F @ ( collect_nat @ P ) )
% 14.17/8.01 = ( collect_nat @ Q2 ) ) ) ).
% 14.17/8.01
% 14.17/8.01 thf(fact_649_vimage__Collect,axiom,
% 14.17/8.01 ! [P: nat > $o,F: int > nat,Q2: int > $o] :
% 14.17/8.01 ( ! [X3: int] :
% 14.17/8.01 ( ( P @ ( F @ X3 ) )
% 14.17/8.01 = ( Q2 @ X3 ) )
% 14.17/8.01 => ( ( vimage_int_nat @ F @ ( collect_nat @ P ) )
% 14.17/8.01 = ( collect_int @ Q2 ) ) ) ).
% 14.17/8.01
% 14.17/8.01 thf(fact_650_vimage__Collect,axiom,
% 14.17/8.01 ! [P: nat > $o,F: $o > nat,Q2: $o > $o] :
% 14.17/8.01 ( ! [X3: $o] :
% 14.17/8.01 ( ( P @ ( F @ X3 ) )
% 14.17/8.01 = ( Q2 @ X3 ) )
% 14.17/8.01 => ( ( vimage_o_nat @ F @ ( collect_nat @ P ) )
% 14.17/8.01 = ( collect_o @ Q2 ) ) ) ).
% 14.17/8.01
% 14.17/8.01 thf(fact_651_vimage__Collect,axiom,
% 14.17/8.01 ! [P: nat > $o,F: product_unit > nat,Q2: product_unit > $o] :
% 14.17/8.01 ( ! [X3: product_unit] :
% 14.17/8.01 ( ( P @ ( F @ X3 ) )
% 14.17/8.01 = ( Q2 @ X3 ) )
% 14.17/8.01 => ( ( vimage6253328473476588386it_nat @ F @ ( collect_nat @ P ) )
% 14.17/8.01 = ( collect_Product_unit @ Q2 ) ) ) ).
% 14.17/8.01
% 14.17/8.01 thf(fact_652_vimage__Collect,axiom,
% 14.17/8.01 ! [P: int > $o,F: nat > int,Q2: nat > $o] :
% 14.17/8.01 ( ! [X3: nat] :
% 14.17/8.01 ( ( P @ ( F @ X3 ) )
% 14.17/8.01 = ( Q2 @ X3 ) )
% 14.17/8.01 => ( ( vimage_nat_int @ F @ ( collect_int @ P ) )
% 14.17/8.01 = ( collect_nat @ Q2 ) ) ) ).
% 14.17/8.01
% 14.17/8.01 thf(fact_653_vimage__Collect,axiom,
% 14.17/8.01 ! [P: int > $o,F: int > int,Q2: int > $o] :
% 14.17/8.01 ( ! [X3: int] :
% 14.17/8.01 ( ( P @ ( F @ X3 ) )
% 14.17/8.01 = ( Q2 @ X3 ) )
% 14.17/8.01 => ( ( vimage_int_int @ F @ ( collect_int @ P ) )
% 14.17/8.01 = ( collect_int @ Q2 ) ) ) ).
% 14.17/8.01
% 14.17/8.01 thf(fact_654_vimage__Collect,axiom,
% 14.17/8.01 ! [P: int > $o,F: $o > int,Q2: $o > $o] :
% 14.17/8.01 ( ! [X3: $o] :
% 14.17/8.01 ( ( P @ ( F @ X3 ) )
% 14.17/8.01 = ( Q2 @ X3 ) )
% 14.17/8.01 => ( ( vimage_o_int @ F @ ( collect_int @ P ) )
% 14.17/8.01 = ( collect_o @ Q2 ) ) ) ).
% 14.17/8.01
% 14.17/8.01 thf(fact_655_vimage__Collect,axiom,
% 14.17/8.01 ! [P: int > $o,F: product_unit > int,Q2: product_unit > $o] :
% 14.17/8.01 ( ! [X3: product_unit] :
% 14.17/8.01 ( ( P @ ( F @ X3 ) )
% 14.17/8.01 = ( Q2 @ X3 ) )
% 14.17/8.01 => ( ( vimage6250838002967538110it_int @ F @ ( collect_int @ P ) )
% 14.17/8.01 = ( collect_Product_unit @ Q2 ) ) ) ).
% 14.17/8.01
% 14.17/8.01 thf(fact_656_vimage__Collect,axiom,
% 14.17/8.01 ! [P: $o > $o,F: nat > $o,Q2: nat > $o] :
% 14.17/8.01 ( ! [X3: nat] :
% 14.17/8.01 ( ( P @ ( F @ X3 ) )
% 14.17/8.01 = ( Q2 @ X3 ) )
% 14.17/8.01 => ( ( vimage_nat_o @ F @ ( collect_o @ P ) )
% 14.17/8.01 = ( collect_nat @ Q2 ) ) ) ).
% 14.17/8.01
% 14.17/8.01 thf(fact_657_vimage__Collect,axiom,
% 14.17/8.01 ! [P: $o > $o,F: int > $o,Q2: int > $o] :
% 14.17/8.01 ( ! [X3: int] :
% 14.17/8.01 ( ( P @ ( F @ X3 ) )
% 14.17/8.01 = ( Q2 @ X3 ) )
% 14.17/8.01 => ( ( vimage_int_o @ F @ ( collect_o @ P ) )
% 14.17/8.01 = ( collect_int @ Q2 ) ) ) ).
% 14.17/8.01
% 14.17/8.01 thf(fact_658_vimageI2,axiom,
% 14.17/8.01 ! [F: nat > nat,A2: nat,A: set_nat] :
% 14.17/8.01 ( ( member_nat @ ( F @ A2 ) @ A )
% 14.17/8.01 => ( member_nat @ A2 @ ( vimage_nat_nat @ F @ A ) ) ) ).
% 14.17/8.01
% 14.17/8.01 thf(fact_659_vimageI2,axiom,
% 14.17/8.01 ! [F: int > nat,A2: int,A: set_nat] :
% 14.17/8.01 ( ( member_nat @ ( F @ A2 ) @ A )
% 14.17/8.01 => ( member_int @ A2 @ ( vimage_int_nat @ F @ A ) ) ) ).
% 14.17/8.01
% 14.17/8.01 thf(fact_660_vimageI2,axiom,
% 14.17/8.01 ! [F: $o > nat,A2: $o,A: set_nat] :
% 14.17/8.01 ( ( member_nat @ ( F @ A2 ) @ A )
% 14.17/8.01 => ( member_o @ A2 @ ( vimage_o_nat @ F @ A ) ) ) ).
% 14.17/8.01
% 14.17/8.01 thf(fact_661_vimageI2,axiom,
% 14.17/8.01 ! [F: product_unit > nat,A2: product_unit,A: set_nat] :
% 14.17/8.01 ( ( member_nat @ ( F @ A2 ) @ A )
% 14.17/8.01 => ( member_Product_unit @ A2 @ ( vimage6253328473476588386it_nat @ F @ A ) ) ) ).
% 14.17/8.01
% 14.17/8.01 thf(fact_662_vimageI2,axiom,
% 14.17/8.01 ! [F: nat > int,A2: nat,A: set_int] :
% 14.17/8.01 ( ( member_int @ ( F @ A2 ) @ A )
% 14.17/8.01 => ( member_nat @ A2 @ ( vimage_nat_int @ F @ A ) ) ) ).
% 14.17/8.01
% 14.17/8.01 thf(fact_663_vimageI2,axiom,
% 14.17/8.01 ! [F: int > int,A2: int,A: set_int] :
% 14.17/8.01 ( ( member_int @ ( F @ A2 ) @ A )
% 14.17/8.01 => ( member_int @ A2 @ ( vimage_int_int @ F @ A ) ) ) ).
% 14.17/8.01
% 14.17/8.01 thf(fact_664_vimageI2,axiom,
% 14.17/8.01 ! [F: $o > int,A2: $o,A: set_int] :
% 14.17/8.01 ( ( member_int @ ( F @ A2 ) @ A )
% 14.17/8.01 => ( member_o @ A2 @ ( vimage_o_int @ F @ A ) ) ) ).
% 14.17/8.01
% 14.17/8.01 thf(fact_665_vimageI2,axiom,
% 14.17/8.01 ! [F: product_unit > int,A2: product_unit,A: set_int] :
% 14.17/8.01 ( ( member_int @ ( F @ A2 ) @ A )
% 14.17/8.01 => ( member_Product_unit @ A2 @ ( vimage6250838002967538110it_int @ F @ A ) ) ) ).
% 14.17/8.01
% 14.17/8.01 thf(fact_666_vimageI2,axiom,
% 14.17/8.01 ! [F: nat > $o,A2: nat,A: set_o] :
% 14.17/8.01 ( ( member_o @ ( F @ A2 ) @ A )
% 14.17/8.01 => ( member_nat @ A2 @ ( vimage_nat_o @ F @ A ) ) ) ).
% 14.17/8.01
% 14.17/8.01 thf(fact_667_vimageI2,axiom,
% 14.17/8.01 ! [F: int > $o,A2: int,A: set_o] :
% 14.17/8.01 ( ( member_o @ ( F @ A2 ) @ A )
% 14.17/8.01 => ( member_int @ A2 @ ( vimage_int_o @ F @ A ) ) ) ).
% 14.17/8.01
% 14.17/8.01 thf(fact_668_vimageE,axiom,
% 14.17/8.01 ! [A2: nat,F: nat > nat,B4: set_nat] :
% 14.17/8.01 ( ( member_nat @ A2 @ ( vimage_nat_nat @ F @ B4 ) )
% 14.17/8.01 => ( member_nat @ ( F @ A2 ) @ B4 ) ) ).
% 14.17/8.01
% 14.17/8.01 thf(fact_669_vimageE,axiom,
% 14.17/8.01 ! [A2: nat,F: nat > int,B4: set_int] :
% 14.17/8.01 ( ( member_nat @ A2 @ ( vimage_nat_int @ F @ B4 ) )
% 14.17/8.01 => ( member_int @ ( F @ A2 ) @ B4 ) ) ).
% 14.17/8.01
% 14.17/8.01 thf(fact_670_vimageE,axiom,
% 14.17/8.01 ! [A2: nat,F: nat > $o,B4: set_o] :
% 14.17/8.01 ( ( member_nat @ A2 @ ( vimage_nat_o @ F @ B4 ) )
% 14.17/8.01 => ( member_o @ ( F @ A2 ) @ B4 ) ) ).
% 14.17/8.01
% 14.17/8.01 thf(fact_671_vimageE,axiom,
% 14.17/8.01 ! [A2: nat,F: nat > product_unit,B4: set_Product_unit] :
% 14.17/8.01 ( ( member_nat @ A2 @ ( vimage4884490618288580032t_unit @ F @ B4 ) )
% 14.17/8.01 => ( member_Product_unit @ ( F @ A2 ) @ B4 ) ) ).
% 14.17/8.01
% 14.17/8.01 thf(fact_672_vimageE,axiom,
% 14.17/8.01 ! [A2: int,F: int > nat,B4: set_nat] :
% 14.17/8.01 ( ( member_int @ A2 @ ( vimage_int_nat @ F @ B4 ) )
% 14.17/8.01 => ( member_nat @ ( F @ A2 ) @ B4 ) ) ).
% 14.17/8.01
% 14.17/8.01 thf(fact_673_vimageE,axiom,
% 14.17/8.01 ! [A2: int,F: int > int,B4: set_int] :
% 14.17/8.01 ( ( member_int @ A2 @ ( vimage_int_int @ F @ B4 ) )
% 14.17/8.01 => ( member_int @ ( F @ A2 ) @ B4 ) ) ).
% 14.17/8.01
% 14.17/8.01 thf(fact_674_vimageE,axiom,
% 14.17/8.01 ! [A2: int,F: int > $o,B4: set_o] :
% 14.17/8.01 ( ( member_int @ A2 @ ( vimage_int_o @ F @ B4 ) )
% 14.17/8.01 => ( member_o @ ( F @ A2 ) @ B4 ) ) ).
% 14.17/8.01
% 14.17/8.01 thf(fact_675_vimageE,axiom,
% 14.17/8.01 ! [A2: int,F: int > product_unit,B4: set_Product_unit] :
% 14.17/8.01 ( ( member_int @ A2 @ ( vimage2000912565490760292t_unit @ F @ B4 ) )
% 14.17/8.01 => ( member_Product_unit @ ( F @ A2 ) @ B4 ) ) ).
% 14.17/8.01
% 14.17/8.01 thf(fact_676_vimageE,axiom,
% 14.17/8.01 ! [A2: $o,F: $o > nat,B4: set_nat] :
% 14.17/8.01 ( ( member_o @ A2 @ ( vimage_o_nat @ F @ B4 ) )
% 14.17/8.01 => ( member_nat @ ( F @ A2 ) @ B4 ) ) ).
% 14.17/8.01
% 14.17/8.01 thf(fact_677_vimageE,axiom,
% 14.17/8.01 ! [A2: $o,F: $o > int,B4: set_int] :
% 14.17/8.01 ( ( member_o @ A2 @ ( vimage_o_int @ F @ B4 ) )
% 14.17/8.01 => ( member_int @ ( F @ A2 ) @ B4 ) ) ).
% 14.17/8.01
% 14.17/8.01 thf(fact_678_vimageD,axiom,
% 14.17/8.01 ! [A2: nat,F: nat > nat,A: set_nat] :
% 14.17/8.01 ( ( member_nat @ A2 @ ( vimage_nat_nat @ F @ A ) )
% 14.17/8.01 => ( member_nat @ ( F @ A2 ) @ A ) ) ).
% 14.17/8.01
% 14.17/8.01 thf(fact_679_vimageD,axiom,
% 14.17/8.01 ! [A2: nat,F: nat > int,A: set_int] :
% 14.17/8.01 ( ( member_nat @ A2 @ ( vimage_nat_int @ F @ A ) )
% 14.17/8.01 => ( member_int @ ( F @ A2 ) @ A ) ) ).
% 14.17/8.01
% 14.17/8.01 thf(fact_680_vimageD,axiom,
% 14.17/8.01 ! [A2: nat,F: nat > $o,A: set_o] :
% 14.17/8.01 ( ( member_nat @ A2 @ ( vimage_nat_o @ F @ A ) )
% 14.17/8.01 => ( member_o @ ( F @ A2 ) @ A ) ) ).
% 14.17/8.01
% 14.17/8.01 thf(fact_681_vimageD,axiom,
% 14.17/8.01 ! [A2: nat,F: nat > product_unit,A: set_Product_unit] :
% 14.17/8.01 ( ( member_nat @ A2 @ ( vimage4884490618288580032t_unit @ F @ A ) )
% 14.17/8.01 => ( member_Product_unit @ ( F @ A2 ) @ A ) ) ).
% 14.17/8.01
% 14.17/8.01 thf(fact_682_vimageD,axiom,
% 14.17/8.01 ! [A2: int,F: int > nat,A: set_nat] :
% 14.17/8.01 ( ( member_int @ A2 @ ( vimage_int_nat @ F @ A ) )
% 14.17/8.01 => ( member_nat @ ( F @ A2 ) @ A ) ) ).
% 14.17/8.01
% 14.17/8.01 thf(fact_683_vimageD,axiom,
% 14.17/8.01 ! [A2: int,F: int > int,A: set_int] :
% 14.17/8.01 ( ( member_int @ A2 @ ( vimage_int_int @ F @ A ) )
% 14.17/8.01 => ( member_int @ ( F @ A2 ) @ A ) ) ).
% 14.17/8.01
% 14.17/8.01 thf(fact_684_vimageD,axiom,
% 14.17/8.01 ! [A2: int,F: int > $o,A: set_o] :
% 14.17/8.01 ( ( member_int @ A2 @ ( vimage_int_o @ F @ A ) )
% 14.17/8.01 => ( member_o @ ( F @ A2 ) @ A ) ) ).
% 14.17/8.01
% 14.17/8.01 thf(fact_685_vimageD,axiom,
% 14.17/8.01 ! [A2: int,F: int > product_unit,A: set_Product_unit] :
% 14.17/8.01 ( ( member_int @ A2 @ ( vimage2000912565490760292t_unit @ F @ A ) )
% 14.17/8.01 => ( member_Product_unit @ ( F @ A2 ) @ A ) ) ).
% 14.17/8.01
% 14.17/8.01 thf(fact_686_vimageD,axiom,
% 14.17/8.01 ! [A2: $o,F: $o > nat,A: set_nat] :
% 14.17/8.01 ( ( member_o @ A2 @ ( vimage_o_nat @ F @ A ) )
% 14.17/8.01 => ( member_nat @ ( F @ A2 ) @ A ) ) ).
% 14.17/8.01
% 14.17/8.01 thf(fact_687_vimageD,axiom,
% 14.17/8.01 ! [A2: $o,F: $o > int,A: set_int] :
% 14.17/8.01 ( ( member_o @ A2 @ ( vimage_o_int @ F @ A ) )
% 14.17/8.01 => ( member_int @ ( F @ A2 ) @ A ) ) ).
% 14.17/8.01
% 14.17/8.01 thf(fact_688_transD,axiom,
% 14.17/8.01 ! [R2: set_Pr1261947904930325089at_nat,X: nat,Y: nat,Z: nat] :
% 14.17/8.01 ( ( trans_nat @ R2 )
% 14.17/8.01 => ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ Y ) @ R2 )
% 14.17/8.01 => ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ Y @ Z ) @ R2 )
% 14.17/8.01 => ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ Z ) @ R2 ) ) ) ) ).
% 14.17/8.01
% 14.17/8.01 thf(fact_689_transD,axiom,
% 14.17/8.01 ! [R2: set_Pr958786334691620121nt_int,X: int,Y: int,Z: int] :
% 14.17/8.01 ( ( trans_int @ R2 )
% 14.17/8.01 => ( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ X @ Y ) @ R2 )
% 14.17/8.01 => ( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ Y @ Z ) @ R2 )
% 14.17/8.01 => ( member5262025264175285858nt_int @ ( product_Pair_int_int @ X @ Z ) @ R2 ) ) ) ) ).
% 14.17/8.01
% 14.17/8.01 thf(fact_690_transE,axiom,
% 14.17/8.01 ! [R2: set_Pr1261947904930325089at_nat,X: nat,Y: nat,Z: nat] :
% 14.17/8.01 ( ( trans_nat @ R2 )
% 14.17/8.01 => ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ Y ) @ R2 )
% 14.17/8.01 => ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ Y @ Z ) @ R2 )
% 14.17/8.01 => ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ Z ) @ R2 ) ) ) ) ).
% 14.17/8.01
% 14.17/8.01 thf(fact_691_transE,axiom,
% 14.17/8.01 ! [R2: set_Pr958786334691620121nt_int,X: int,Y: int,Z: int] :
% 14.17/8.01 ( ( trans_int @ R2 )
% 14.17/8.01 => ( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ X @ Y ) @ R2 )
% 14.17/8.01 => ( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ Y @ Z ) @ R2 )
% 14.17/8.01 => ( member5262025264175285858nt_int @ ( product_Pair_int_int @ X @ Z ) @ R2 ) ) ) ) ).
% 14.17/8.01
% 14.17/8.01 thf(fact_692_transI,axiom,
% 14.17/8.01 ! [R2: set_Pr1261947904930325089at_nat] :
% 14.17/8.01 ( ! [X3: nat,Y2: nat,Z3: nat] :
% 14.17/8.01 ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X3 @ Y2 ) @ R2 )
% 14.17/8.01 => ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ Y2 @ Z3 ) @ R2 )
% 14.17/8.01 => ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X3 @ Z3 ) @ R2 ) ) )
% 14.17/8.01 => ( trans_nat @ R2 ) ) ).
% 14.17/8.01
% 14.17/8.01 thf(fact_693_transI,axiom,
% 14.17/8.01 ! [R2: set_Pr958786334691620121nt_int] :
% 14.17/8.01 ( ! [X3: int,Y2: int,Z3: int] :
% 14.17/8.01 ( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ X3 @ Y2 ) @ R2 )
% 14.17/8.01 => ( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ Y2 @ Z3 ) @ R2 )
% 14.17/8.01 => ( member5262025264175285858nt_int @ ( product_Pair_int_int @ X3 @ Z3 ) @ R2 ) ) )
% 14.17/8.01 => ( trans_int @ R2 ) ) ).
% 14.17/8.01
% 14.17/8.01 thf(fact_694_trans__def,axiom,
% 14.17/8.01 ( trans_nat
% 14.17/8.01 = ( ^ [R4: set_Pr1261947904930325089at_nat] :
% 14.17/8.01 ! [X2: nat,Y3: nat,Z2: nat] :
% 14.17/8.01 ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X2 @ Y3 ) @ R4 )
% 14.17/8.01 => ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ Y3 @ Z2 ) @ R4 )
% 14.17/8.01 => ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X2 @ Z2 ) @ R4 ) ) ) ) ) ).
% 14.17/8.01
% 14.17/8.01 thf(fact_695_trans__def,axiom,
% 14.17/8.01 ( trans_int
% 14.17/8.01 = ( ^ [R4: set_Pr958786334691620121nt_int] :
% 14.17/8.01 ! [X2: int,Y3: int,Z2: int] :
% 14.17/8.01 ( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ X2 @ Y3 ) @ R4 )
% 14.17/8.01 => ( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ Y3 @ Z2 ) @ R4 )
% 14.17/8.01 => ( member5262025264175285858nt_int @ ( product_Pair_int_int @ X2 @ Z2 ) @ R4 ) ) ) ) ) ).
% 14.17/8.01
% 14.17/8.01 thf(fact_696_Domain_Ocases,axiom,
% 14.17/8.01 ! [A2: heap_e7401611519738050253t_unit,R2: set_Pr3948176798113811640et_nat] :
% 14.17/8.01 ( ( member242389114213480694t_unit @ A2 @ ( domain2007548326387822151et_nat @ R2 ) )
% 14.17/8.01 => ~ ! [B2: set_nat] :
% 14.17/8.01 ~ ( member6260224972018164377et_nat @ ( produc7507926704131184380et_nat @ A2 @ B2 ) @ R2 ) ) ).
% 14.17/8.01
% 14.17/8.01 thf(fact_697_Domain_Ocases,axiom,
% 14.17/8.01 ! [A2: nat,R2: set_Pr1261947904930325089at_nat] :
% 14.17/8.01 ( ( member_nat @ A2 @ ( domain_nat_nat @ R2 ) )
% 14.17/8.01 => ~ ! [B2: nat] :
% 14.17/8.01 ~ ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A2 @ B2 ) @ R2 ) ) ).
% 14.17/8.01
% 14.17/8.01 thf(fact_698_Domain_Ocases,axiom,
% 14.17/8.01 ! [A2: int,R2: set_Pr958786334691620121nt_int] :
% 14.17/8.01 ( ( member_int @ A2 @ ( domain_int_int @ R2 ) )
% 14.17/8.01 => ~ ! [B2: int] :
% 14.17/8.01 ~ ( member5262025264175285858nt_int @ ( product_Pair_int_int @ A2 @ B2 ) @ R2 ) ) ).
% 14.17/8.01
% 14.17/8.01 thf(fact_699_Domain_Ocases,axiom,
% 14.17/8.01 ! [A2: produc3658429121746597890et_nat > $o,R2: set_Pr3286484037609594932et_nat] :
% 14.17/8.01 ( ( member6576561426505652726_nat_o @ A2 @ ( domain2358961613337345475et_nat @ R2 ) )
% 14.17/8.01 => ~ ! [B2: produc3658429121746597890et_nat] :
% 14.17/8.01 ~ ( member1996754912294343701et_nat @ ( produc5001842942810119800et_nat @ A2 @ B2 ) @ R2 ) ) ).
% 14.17/8.01
% 14.17/8.01 thf(fact_700_Domain_Ocases,axiom,
% 14.17/8.01 ! [A2: produc3658429121746597890et_nat > $o,R2: set_Pr8536935166611901872et_nat] :
% 14.17/8.01 ( ( member6576561426505652726_nat_o @ A2 @ ( domain3618434998481624895et_nat @ R2 ) )
% 14.17/8.01 => ~ ! [B2: produc3925858234332021118et_nat] :
% 14.17/8.01 ~ ( member6124377750444531601et_nat @ ( produc2245416461498447860et_nat @ A2 @ B2 ) @ R2 ) ) ).
% 14.17/8.01
% 14.17/8.01 thf(fact_701_Domain_Osimps,axiom,
% 14.17/8.01 ! [A2: heap_e7401611519738050253t_unit,R2: set_Pr3948176798113811640et_nat] :
% 14.17/8.01 ( ( member242389114213480694t_unit @ A2 @ ( domain2007548326387822151et_nat @ R2 ) )
% 14.17/8.01 = ( ? [A6: heap_e7401611519738050253t_unit,B6: set_nat] :
% 14.17/8.01 ( ( A2 = A6 )
% 14.17/8.01 & ( member6260224972018164377et_nat @ ( produc7507926704131184380et_nat @ A6 @ B6 ) @ R2 ) ) ) ) ).
% 14.17/8.01
% 14.17/8.01 thf(fact_702_Domain_Osimps,axiom,
% 14.17/8.01 ! [A2: nat,R2: set_Pr1261947904930325089at_nat] :
% 14.17/8.01 ( ( member_nat @ A2 @ ( domain_nat_nat @ R2 ) )
% 14.17/8.01 = ( ? [A6: nat,B6: nat] :
% 14.17/8.01 ( ( A2 = A6 )
% 14.17/8.01 & ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A6 @ B6 ) @ R2 ) ) ) ) ).
% 14.17/8.01
% 14.17/8.01 thf(fact_703_Domain_Osimps,axiom,
% 14.17/8.01 ! [A2: int,R2: set_Pr958786334691620121nt_int] :
% 14.17/8.01 ( ( member_int @ A2 @ ( domain_int_int @ R2 ) )
% 14.17/8.01 = ( ? [A6: int,B6: int] :
% 14.17/8.01 ( ( A2 = A6 )
% 14.17/8.01 & ( member5262025264175285858nt_int @ ( product_Pair_int_int @ A6 @ B6 ) @ R2 ) ) ) ) ).
% 14.17/8.01
% 14.17/8.01 thf(fact_704_Domain_Osimps,axiom,
% 14.17/8.01 ! [A2: produc3658429121746597890et_nat > $o,R2: set_Pr3286484037609594932et_nat] :
% 14.17/8.01 ( ( member6576561426505652726_nat_o @ A2 @ ( domain2358961613337345475et_nat @ R2 ) )
% 14.17/8.01 = ( ? [A6: produc3658429121746597890et_nat > $o,B6: produc3658429121746597890et_nat] :
% 14.17/8.01 ( ( A2 = A6 )
% 14.17/8.01 & ( member1996754912294343701et_nat @ ( produc5001842942810119800et_nat @ A6 @ B6 ) @ R2 ) ) ) ) ).
% 14.17/8.01
% 14.17/8.01 thf(fact_705_Domain_Osimps,axiom,
% 14.17/8.01 ! [A2: produc3658429121746597890et_nat > $o,R2: set_Pr8536935166611901872et_nat] :
% 14.17/8.01 ( ( member6576561426505652726_nat_o @ A2 @ ( domain3618434998481624895et_nat @ R2 ) )
% 14.17/8.01 = ( ? [A6: produc3658429121746597890et_nat > $o,B6: produc3925858234332021118et_nat] :
% 14.17/8.01 ( ( A2 = A6 )
% 14.17/8.01 & ( member6124377750444531601et_nat @ ( produc2245416461498447860et_nat @ A6 @ B6 ) @ R2 ) ) ) ) ).
% 14.17/8.01
% 14.17/8.01 thf(fact_706_Domain_ODomainI,axiom,
% 14.17/8.01 ! [A2: heap_e7401611519738050253t_unit,B: set_nat,R2: set_Pr3948176798113811640et_nat] :
% 14.17/8.01 ( ( member6260224972018164377et_nat @ ( produc7507926704131184380et_nat @ A2 @ B ) @ R2 )
% 14.17/8.01 => ( member242389114213480694t_unit @ A2 @ ( domain2007548326387822151et_nat @ R2 ) ) ) ).
% 14.17/8.01
% 14.17/8.01 thf(fact_707_Domain_ODomainI,axiom,
% 14.17/8.01 ! [A2: nat,B: nat,R2: set_Pr1261947904930325089at_nat] :
% 14.17/8.01 ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A2 @ B ) @ R2 )
% 14.17/8.01 => ( member_nat @ A2 @ ( domain_nat_nat @ R2 ) ) ) ).
% 14.17/8.01
% 14.17/8.01 thf(fact_708_Domain_ODomainI,axiom,
% 14.17/8.01 ! [A2: int,B: int,R2: set_Pr958786334691620121nt_int] :
% 14.17/8.01 ( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ A2 @ B ) @ R2 )
% 14.17/8.01 => ( member_int @ A2 @ ( domain_int_int @ R2 ) ) ) ).
% 14.17/8.01
% 14.17/8.01 thf(fact_709_Domain_ODomainI,axiom,
% 14.17/8.01 ! [A2: produc3658429121746597890et_nat > $o,B: produc3658429121746597890et_nat,R2: set_Pr3286484037609594932et_nat] :
% 14.17/8.01 ( ( member1996754912294343701et_nat @ ( produc5001842942810119800et_nat @ A2 @ B ) @ R2 )
% 14.17/8.01 => ( member6576561426505652726_nat_o @ A2 @ ( domain2358961613337345475et_nat @ R2 ) ) ) ).
% 14.17/8.01
% 14.17/8.01 thf(fact_710_Domain_ODomainI,axiom,
% 14.17/8.01 ! [A2: produc3658429121746597890et_nat > $o,B: produc3925858234332021118et_nat,R2: set_Pr8536935166611901872et_nat] :
% 14.17/8.01 ( ( member6124377750444531601et_nat @ ( produc2245416461498447860et_nat @ A2 @ B ) @ R2 )
% 14.17/8.01 => ( member6576561426505652726_nat_o @ A2 @ ( domain3618434998481624895et_nat @ R2 ) ) ) ).
% 14.17/8.01
% 14.17/8.01 thf(fact_711_DomainE,axiom,
% 14.17/8.01 ! [A2: heap_e7401611519738050253t_unit,R2: set_Pr3948176798113811640et_nat] :
% 14.17/8.01 ( ( member242389114213480694t_unit @ A2 @ ( domain2007548326387822151et_nat @ R2 ) )
% 14.17/8.01 => ~ ! [B2: set_nat] :
% 14.17/8.01 ~ ( member6260224972018164377et_nat @ ( produc7507926704131184380et_nat @ A2 @ B2 ) @ R2 ) ) ).
% 14.17/8.01
% 14.17/8.01 thf(fact_712_DomainE,axiom,
% 14.17/8.01 ! [A2: nat,R2: set_Pr1261947904930325089at_nat] :
% 14.17/8.01 ( ( member_nat @ A2 @ ( domain_nat_nat @ R2 ) )
% 14.17/8.01 => ~ ! [B2: nat] :
% 14.17/8.01 ~ ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A2 @ B2 ) @ R2 ) ) ).
% 14.17/8.01
% 14.17/8.01 thf(fact_713_DomainE,axiom,
% 14.17/8.01 ! [A2: int,R2: set_Pr958786334691620121nt_int] :
% 14.17/8.01 ( ( member_int @ A2 @ ( domain_int_int @ R2 ) )
% 14.17/8.01 => ~ ! [B2: int] :
% 14.17/8.01 ~ ( member5262025264175285858nt_int @ ( product_Pair_int_int @ A2 @ B2 ) @ R2 ) ) ).
% 14.17/8.01
% 14.17/8.01 thf(fact_714_DomainE,axiom,
% 14.17/8.01 ! [A2: produc3658429121746597890et_nat > $o,R2: set_Pr3286484037609594932et_nat] :
% 14.17/8.01 ( ( member6576561426505652726_nat_o @ A2 @ ( domain2358961613337345475et_nat @ R2 ) )
% 14.17/8.01 => ~ ! [B2: produc3658429121746597890et_nat] :
% 14.17/8.01 ~ ( member1996754912294343701et_nat @ ( produc5001842942810119800et_nat @ A2 @ B2 ) @ R2 ) ) ).
% 14.17/8.01
% 14.17/8.01 thf(fact_715_DomainE,axiom,
% 14.17/8.01 ! [A2: produc3658429121746597890et_nat > $o,R2: set_Pr8536935166611901872et_nat] :
% 14.17/8.01 ( ( member6576561426505652726_nat_o @ A2 @ ( domain3618434998481624895et_nat @ R2 ) )
% 14.17/8.01 => ~ ! [B2: produc3925858234332021118et_nat] :
% 14.17/8.01 ~ ( member6124377750444531601et_nat @ ( produc2245416461498447860et_nat @ A2 @ B2 ) @ R2 ) ) ).
% 14.17/8.01
% 14.17/8.01 thf(fact_716_Domain__iff,axiom,
% 14.17/8.01 ! [A2: heap_e7401611519738050253t_unit,R2: set_Pr3948176798113811640et_nat] :
% 14.17/8.01 ( ( member242389114213480694t_unit @ A2 @ ( domain2007548326387822151et_nat @ R2 ) )
% 14.17/8.01 = ( ? [Y3: set_nat] : ( member6260224972018164377et_nat @ ( produc7507926704131184380et_nat @ A2 @ Y3 ) @ R2 ) ) ) ).
% 14.17/8.01
% 14.17/8.01 thf(fact_717_Domain__iff,axiom,
% 14.17/8.01 ! [A2: nat,R2: set_Pr1261947904930325089at_nat] :
% 14.17/8.01 ( ( member_nat @ A2 @ ( domain_nat_nat @ R2 ) )
% 14.17/8.01 = ( ? [Y3: nat] : ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A2 @ Y3 ) @ R2 ) ) ) ).
% 14.17/8.01
% 14.17/8.01 thf(fact_718_Domain__iff,axiom,
% 14.17/8.01 ! [A2: int,R2: set_Pr958786334691620121nt_int] :
% 14.17/8.01 ( ( member_int @ A2 @ ( domain_int_int @ R2 ) )
% 14.17/8.01 = ( ? [Y3: int] : ( member5262025264175285858nt_int @ ( product_Pair_int_int @ A2 @ Y3 ) @ R2 ) ) ) ).
% 14.17/8.01
% 14.17/8.01 thf(fact_719_Domain__iff,axiom,
% 14.17/8.01 ! [A2: produc3658429121746597890et_nat > $o,R2: set_Pr3286484037609594932et_nat] :
% 14.17/8.01 ( ( member6576561426505652726_nat_o @ A2 @ ( domain2358961613337345475et_nat @ R2 ) )
% 14.17/8.01 = ( ? [Y3: produc3658429121746597890et_nat] : ( member1996754912294343701et_nat @ ( produc5001842942810119800et_nat @ A2 @ Y3 ) @ R2 ) ) ) ).
% 14.17/8.01
% 14.17/8.01 thf(fact_720_Domain__iff,axiom,
% 14.17/8.01 ! [A2: produc3658429121746597890et_nat > $o,R2: set_Pr8536935166611901872et_nat] :
% 14.17/8.01 ( ( member6576561426505652726_nat_o @ A2 @ ( domain3618434998481624895et_nat @ R2 ) )
% 14.17/8.01 = ( ? [Y3: produc3925858234332021118et_nat] : ( member6124377750444531601et_nat @ ( produc2245416461498447860et_nat @ A2 @ Y3 ) @ R2 ) ) ) ).
% 14.17/8.01
% 14.17/8.01 thf(fact_721_Range_Ocases,axiom,
% 14.17/8.01 ! [A2: set_nat,R2: set_Pr3948176798113811640et_nat] :
% 14.17/8.01 ( ( member_set_nat @ A2 @ ( range_9132311542094983838et_nat @ R2 ) )
% 14.17/8.01 => ~ ! [A4: heap_e7401611519738050253t_unit] :
% 14.17/8.01 ~ ( member6260224972018164377et_nat @ ( produc7507926704131184380et_nat @ A4 @ A2 ) @ R2 ) ) ).
% 14.17/8.01
% 14.17/8.01 thf(fact_722_Range_Ocases,axiom,
% 14.17/8.01 ! [A2: nat,R2: set_Pr1261947904930325089at_nat] :
% 14.17/8.01 ( ( member_nat @ A2 @ ( range_nat_nat @ R2 ) )
% 14.17/8.01 => ~ ! [A4: nat] :
% 14.17/8.01 ~ ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A4 @ A2 ) @ R2 ) ) ).
% 14.17/8.01
% 14.17/8.01 thf(fact_723_Range_Ocases,axiom,
% 14.17/8.01 ! [A2: int,R2: set_Pr958786334691620121nt_int] :
% 14.17/8.01 ( ( member_int @ A2 @ ( range_int_int @ R2 ) )
% 14.17/8.01 => ~ ! [A4: int] :
% 14.17/8.01 ~ ( member5262025264175285858nt_int @ ( product_Pair_int_int @ A4 @ A2 ) @ R2 ) ) ).
% 14.17/8.01
% 14.17/8.01 thf(fact_724_Range_Ocases,axiom,
% 14.17/8.01 ! [A2: produc3658429121746597890et_nat,R2: set_Pr3286484037609594932et_nat] :
% 14.17/8.01 ( ( member6260224972018164377et_nat @ A2 @ ( range_3398330936903821850et_nat @ R2 ) )
% 14.17/8.01 => ~ ! [A4: produc3658429121746597890et_nat > $o] :
% 14.17/8.01 ~ ( member1996754912294343701et_nat @ ( produc5001842942810119800et_nat @ A4 @ A2 ) @ R2 ) ) ).
% 14.17/8.01
% 14.17/8.01 thf(fact_725_Range_Ocases,axiom,
% 14.17/8.01 ! [A2: produc3925858234332021118et_nat,R2: set_Pr8536935166611901872et_nat] :
% 14.17/8.01 ( ( member1996754912294343701et_nat @ A2 @ ( range_7260058354244137878et_nat @ R2 ) )
% 14.17/8.01 => ~ ! [A4: produc3658429121746597890et_nat > $o] :
% 14.17/8.01 ~ ( member6124377750444531601et_nat @ ( produc2245416461498447860et_nat @ A4 @ A2 ) @ R2 ) ) ).
% 14.17/8.01
% 14.17/8.01 thf(fact_726_Range_Osimps,axiom,
% 14.17/8.01 ! [A2: set_nat,R2: set_Pr3948176798113811640et_nat] :
% 14.17/8.01 ( ( member_set_nat @ A2 @ ( range_9132311542094983838et_nat @ R2 ) )
% 14.17/8.01 = ( ? [A6: heap_e7401611519738050253t_unit,B6: set_nat] :
% 14.17/8.01 ( ( A2 = B6 )
% 14.17/8.01 & ( member6260224972018164377et_nat @ ( produc7507926704131184380et_nat @ A6 @ B6 ) @ R2 ) ) ) ) ).
% 14.17/8.01
% 14.17/8.01 thf(fact_727_Range_Osimps,axiom,
% 14.17/8.01 ! [A2: nat,R2: set_Pr1261947904930325089at_nat] :
% 14.17/8.01 ( ( member_nat @ A2 @ ( range_nat_nat @ R2 ) )
% 14.17/8.01 = ( ? [A6: nat,B6: nat] :
% 14.17/8.01 ( ( A2 = B6 )
% 14.17/8.01 & ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A6 @ B6 ) @ R2 ) ) ) ) ).
% 14.17/8.01
% 14.17/8.01 thf(fact_728_Range_Osimps,axiom,
% 14.17/8.01 ! [A2: int,R2: set_Pr958786334691620121nt_int] :
% 14.17/8.01 ( ( member_int @ A2 @ ( range_int_int @ R2 ) )
% 14.17/8.01 = ( ? [A6: int,B6: int] :
% 14.17/8.01 ( ( A2 = B6 )
% 14.17/8.01 & ( member5262025264175285858nt_int @ ( product_Pair_int_int @ A6 @ B6 ) @ R2 ) ) ) ) ).
% 14.17/8.01
% 14.17/8.01 thf(fact_729_Range_Osimps,axiom,
% 14.17/8.01 ! [A2: produc3658429121746597890et_nat,R2: set_Pr3286484037609594932et_nat] :
% 14.17/8.01 ( ( member6260224972018164377et_nat @ A2 @ ( range_3398330936903821850et_nat @ R2 ) )
% 14.17/8.01 = ( ? [A6: produc3658429121746597890et_nat > $o,B6: produc3658429121746597890et_nat] :
% 14.17/8.01 ( ( A2 = B6 )
% 14.17/8.01 & ( member1996754912294343701et_nat @ ( produc5001842942810119800et_nat @ A6 @ B6 ) @ R2 ) ) ) ) ).
% 14.17/8.01
% 14.17/8.01 thf(fact_730_Range_Osimps,axiom,
% 14.17/8.01 ! [A2: produc3925858234332021118et_nat,R2: set_Pr8536935166611901872et_nat] :
% 14.17/8.01 ( ( member1996754912294343701et_nat @ A2 @ ( range_7260058354244137878et_nat @ R2 ) )
% 14.17/8.01 = ( ? [A6: produc3658429121746597890et_nat > $o,B6: produc3925858234332021118et_nat] :
% 14.17/8.01 ( ( A2 = B6 )
% 14.17/8.01 & ( member6124377750444531601et_nat @ ( produc2245416461498447860et_nat @ A6 @ B6 ) @ R2 ) ) ) ) ).
% 14.17/8.01
% 14.17/8.01 thf(fact_731_Range_Ointros,axiom,
% 14.17/8.01 ! [A2: heap_e7401611519738050253t_unit,B: set_nat,R2: set_Pr3948176798113811640et_nat] :
% 14.17/8.01 ( ( member6260224972018164377et_nat @ ( produc7507926704131184380et_nat @ A2 @ B ) @ R2 )
% 14.17/8.01 => ( member_set_nat @ B @ ( range_9132311542094983838et_nat @ R2 ) ) ) ).
% 14.17/8.01
% 14.17/8.01 thf(fact_732_Range_Ointros,axiom,
% 14.17/8.01 ! [A2: nat,B: nat,R2: set_Pr1261947904930325089at_nat] :
% 14.17/8.01 ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A2 @ B ) @ R2 )
% 14.17/8.01 => ( member_nat @ B @ ( range_nat_nat @ R2 ) ) ) ).
% 14.17/8.01
% 14.17/8.01 thf(fact_733_Range_Ointros,axiom,
% 14.17/8.01 ! [A2: int,B: int,R2: set_Pr958786334691620121nt_int] :
% 14.17/8.01 ( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ A2 @ B ) @ R2 )
% 14.17/8.01 => ( member_int @ B @ ( range_int_int @ R2 ) ) ) ).
% 14.17/8.01
% 14.17/8.01 thf(fact_734_Range_Ointros,axiom,
% 14.17/8.01 ! [A2: produc3658429121746597890et_nat > $o,B: produc3658429121746597890et_nat,R2: set_Pr3286484037609594932et_nat] :
% 14.17/8.01 ( ( member1996754912294343701et_nat @ ( produc5001842942810119800et_nat @ A2 @ B ) @ R2 )
% 14.17/8.01 => ( member6260224972018164377et_nat @ B @ ( range_3398330936903821850et_nat @ R2 ) ) ) ).
% 14.17/8.01
% 14.17/8.01 thf(fact_735_Range_Ointros,axiom,
% 14.17/8.01 ! [A2: produc3658429121746597890et_nat > $o,B: produc3925858234332021118et_nat,R2: set_Pr8536935166611901872et_nat] :
% 14.17/8.01 ( ( member6124377750444531601et_nat @ ( produc2245416461498447860et_nat @ A2 @ B ) @ R2 )
% 14.17/8.01 => ( member1996754912294343701et_nat @ B @ ( range_7260058354244137878et_nat @ R2 ) ) ) ).
% 14.17/8.01
% 14.17/8.01 thf(fact_736_RangeE,axiom,
% 14.17/8.01 ! [B: set_nat,R2: set_Pr3948176798113811640et_nat] :
% 14.17/8.01 ( ( member_set_nat @ B @ ( range_9132311542094983838et_nat @ R2 ) )
% 14.17/8.01 => ~ ! [A4: heap_e7401611519738050253t_unit] :
% 14.17/8.01 ~ ( member6260224972018164377et_nat @ ( produc7507926704131184380et_nat @ A4 @ B ) @ R2 ) ) ).
% 14.17/8.01
% 14.17/8.01 thf(fact_737_RangeE,axiom,
% 14.17/8.01 ! [B: nat,R2: set_Pr1261947904930325089at_nat] :
% 14.17/8.01 ( ( member_nat @ B @ ( range_nat_nat @ R2 ) )
% 14.17/8.01 => ~ ! [A4: nat] :
% 14.17/8.01 ~ ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A4 @ B ) @ R2 ) ) ).
% 14.17/8.01
% 14.17/8.01 thf(fact_738_RangeE,axiom,
% 14.17/8.01 ! [B: int,R2: set_Pr958786334691620121nt_int] :
% 14.17/8.01 ( ( member_int @ B @ ( range_int_int @ R2 ) )
% 14.17/8.01 => ~ ! [A4: int] :
% 14.17/8.01 ~ ( member5262025264175285858nt_int @ ( product_Pair_int_int @ A4 @ B ) @ R2 ) ) ).
% 14.17/8.01
% 14.17/8.01 thf(fact_739_RangeE,axiom,
% 14.17/8.01 ! [B: produc3658429121746597890et_nat,R2: set_Pr3286484037609594932et_nat] :
% 14.17/8.01 ( ( member6260224972018164377et_nat @ B @ ( range_3398330936903821850et_nat @ R2 ) )
% 14.17/8.01 => ~ ! [A4: produc3658429121746597890et_nat > $o] :
% 14.17/8.01 ~ ( member1996754912294343701et_nat @ ( produc5001842942810119800et_nat @ A4 @ B ) @ R2 ) ) ).
% 14.17/8.01
% 14.17/8.01 thf(fact_740_RangeE,axiom,
% 14.17/8.01 ! [B: produc3925858234332021118et_nat,R2: set_Pr8536935166611901872et_nat] :
% 14.17/8.01 ( ( member1996754912294343701et_nat @ B @ ( range_7260058354244137878et_nat @ R2 ) )
% 14.17/8.01 => ~ ! [A4: produc3658429121746597890et_nat > $o] :
% 14.17/8.01 ~ ( member6124377750444531601et_nat @ ( produc2245416461498447860et_nat @ A4 @ B ) @ R2 ) ) ).
% 14.17/8.01
% 14.17/8.01 thf(fact_741_Range__iff,axiom,
% 14.17/8.01 ! [A2: set_nat,R2: set_Pr3948176798113811640et_nat] :
% 14.17/8.01 ( ( member_set_nat @ A2 @ ( range_9132311542094983838et_nat @ R2 ) )
% 14.17/8.01 = ( ? [Y3: heap_e7401611519738050253t_unit] : ( member6260224972018164377et_nat @ ( produc7507926704131184380et_nat @ Y3 @ A2 ) @ R2 ) ) ) ).
% 14.17/8.01
% 14.17/8.01 thf(fact_742_Range__iff,axiom,
% 14.17/8.01 ! [A2: nat,R2: set_Pr1261947904930325089at_nat] :
% 14.17/8.01 ( ( member_nat @ A2 @ ( range_nat_nat @ R2 ) )
% 14.17/8.01 = ( ? [Y3: nat] : ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ Y3 @ A2 ) @ R2 ) ) ) ).
% 14.17/8.01
% 14.17/8.01 thf(fact_743_Range__iff,axiom,
% 14.17/8.01 ! [A2: int,R2: set_Pr958786334691620121nt_int] :
% 14.17/8.01 ( ( member_int @ A2 @ ( range_int_int @ R2 ) )
% 14.17/8.01 = ( ? [Y3: int] : ( member5262025264175285858nt_int @ ( product_Pair_int_int @ Y3 @ A2 ) @ R2 ) ) ) ).
% 14.17/8.01
% 14.17/8.01 thf(fact_744_Range__iff,axiom,
% 14.17/8.01 ! [A2: produc3658429121746597890et_nat,R2: set_Pr3286484037609594932et_nat] :
% 14.17/8.01 ( ( member6260224972018164377et_nat @ A2 @ ( range_3398330936903821850et_nat @ R2 ) )
% 14.17/8.01 = ( ? [Y3: produc3658429121746597890et_nat > $o] : ( member1996754912294343701et_nat @ ( produc5001842942810119800et_nat @ Y3 @ A2 ) @ R2 ) ) ) ).
% 14.17/8.01
% 14.17/8.01 thf(fact_745_Range__iff,axiom,
% 14.17/8.01 ! [A2: produc3925858234332021118et_nat,R2: set_Pr8536935166611901872et_nat] :
% 14.17/8.01 ( ( member1996754912294343701et_nat @ A2 @ ( range_7260058354244137878et_nat @ R2 ) )
% 14.17/8.01 = ( ? [Y3: produc3658429121746597890et_nat > $o] : ( member6124377750444531601et_nat @ ( produc2245416461498447860et_nat @ Y3 @ A2 ) @ R2 ) ) ) ).
% 14.17/8.01
% 14.17/8.01 thf(fact_746_vimage__singleton__eq,axiom,
% 14.17/8.01 ! [A2: nat,F: nat > product_unit,B: product_unit] :
% 14.17/8.01 ( ( member_nat @ A2 @ ( vimage4884490618288580032t_unit @ F @ ( insert_Product_unit @ B @ bot_bo3957492148770167129t_unit ) ) )
% 14.17/8.01 = ( ( F @ A2 )
% 14.17/8.01 = B ) ) ).
% 14.17/8.01
% 14.17/8.01 thf(fact_747_vimage__singleton__eq,axiom,
% 14.17/8.01 ! [A2: int,F: int > product_unit,B: product_unit] :
% 14.17/8.01 ( ( member_int @ A2 @ ( vimage2000912565490760292t_unit @ F @ ( insert_Product_unit @ B @ bot_bo3957492148770167129t_unit ) ) )
% 14.17/8.01 = ( ( F @ A2 )
% 14.17/8.01 = B ) ) ).
% 14.17/8.01
% 14.17/8.01 thf(fact_748_vimage__singleton__eq,axiom,
% 14.17/8.01 ! [A2: $o,F: $o > product_unit,B: product_unit] :
% 14.17/8.01 ( ( member_o @ A2 @ ( vimage4222377557387472176t_unit @ F @ ( insert_Product_unit @ B @ bot_bo3957492148770167129t_unit ) ) )
% 14.17/8.01 = ( ( F @ A2 )
% 14.17/8.01 = B ) ) ).
% 14.17/8.01
% 14.17/8.01 thf(fact_749_vimage__singleton__eq,axiom,
% 14.17/8.01 ! [A2: product_unit,F: product_unit > product_unit,B: product_unit] :
% 14.17/8.01 ( ( member_Product_unit @ A2 @ ( vimage7995052115951654139t_unit @ F @ ( insert_Product_unit @ B @ bot_bo3957492148770167129t_unit ) ) )
% 14.17/8.01 = ( ( F @ A2 )
% 14.17/8.01 = B ) ) ).
% 14.17/8.01
% 14.17/8.01 thf(fact_750_vimage__singleton__eq,axiom,
% 14.17/8.01 ! [A2: nat,F: nat > $o,B: $o] :
% 14.17/8.01 ( ( member_nat @ A2 @ ( vimage_nat_o @ F @ ( insert_o @ B @ bot_bot_set_o ) ) )
% 14.17/8.01 = ( ( F @ A2 )
% 14.17/8.01 = B ) ) ).
% 14.17/8.01
% 14.17/8.01 thf(fact_751_vimage__singleton__eq,axiom,
% 14.17/8.01 ! [A2: int,F: int > $o,B: $o] :
% 14.17/8.01 ( ( member_int @ A2 @ ( vimage_int_o @ F @ ( insert_o @ B @ bot_bot_set_o ) ) )
% 14.17/8.01 = ( ( F @ A2 )
% 14.17/8.01 = B ) ) ).
% 14.17/8.01
% 14.17/8.01 thf(fact_752_vimage__singleton__eq,axiom,
% 14.17/8.01 ! [A2: $o,F: $o > $o,B: $o] :
% 14.17/8.01 ( ( member_o @ A2 @ ( vimage_o_o @ F @ ( insert_o @ B @ bot_bot_set_o ) ) )
% 14.17/8.01 = ( ( F @ A2 )
% 14.17/8.01 = B ) ) ).
% 14.17/8.01
% 14.17/8.01 thf(fact_753_vimage__singleton__eq,axiom,
% 14.17/8.01 ! [A2: product_unit,F: product_unit > $o,B: $o] :
% 14.17/8.01 ( ( member_Product_unit @ A2 @ ( vimage8808348054653608838unit_o @ F @ ( insert_o @ B @ bot_bot_set_o ) ) )
% 14.17/8.01 = ( ( F @ A2 )
% 14.17/8.01 = B ) ) ).
% 14.17/8.01
% 14.17/8.01 thf(fact_754_vimage__singleton__eq,axiom,
% 14.17/8.01 ! [A2: nat,F: nat > nat,B: nat] :
% 14.17/8.01 ( ( member_nat @ A2 @ ( vimage_nat_nat @ F @ ( insert_nat @ B @ bot_bot_set_nat ) ) )
% 14.17/8.01 = ( ( F @ A2 )
% 14.17/8.01 = B ) ) ).
% 14.17/8.01
% 14.17/8.01 thf(fact_755_vimage__singleton__eq,axiom,
% 14.17/8.01 ! [A2: int,F: int > nat,B: nat] :
% 14.17/8.01 ( ( member_int @ A2 @ ( vimage_int_nat @ F @ ( insert_nat @ B @ bot_bot_set_nat ) ) )
% 14.17/8.01 = ( ( F @ A2 )
% 14.17/8.01 = B ) ) ).
% 14.17/8.01
% 14.17/8.01 thf(fact_756_relH__in__rangeI_I2_J,axiom,
% 14.17/8.01 ! [As: set_nat,H2: heap_e7401611519738050253t_unit,H3: heap_e7401611519738050253t_unit] :
% 14.17/8.01 ( ( relH @ As @ H2 @ H3 )
% 14.17/8.01 => ( in_range @ ( produc7507926704131184380et_nat @ H3 @ As ) ) ) ).
% 14.17/8.01
% 14.17/8.01 thf(fact_757_relH__in__rangeI_I1_J,axiom,
% 14.17/8.01 ! [As: set_nat,H2: heap_e7401611519738050253t_unit,H3: heap_e7401611519738050253t_unit] :
% 14.17/8.01 ( ( relH @ As @ H2 @ H3 )
% 14.17/8.01 => ( in_range @ ( produc7507926704131184380et_nat @ H2 @ As ) ) ) ).
% 14.17/8.01
% 14.17/8.01 thf(fact_758_relH__refl,axiom,
% 14.17/8.01 ! [H2: heap_e7401611519738050253t_unit,As: set_nat] :
% 14.17/8.01 ( ( in_range @ ( produc7507926704131184380et_nat @ H2 @ As ) )
% 14.17/8.01 => ( relH @ As @ H2 @ H2 ) ) ).
% 14.17/8.01
% 14.17/8.01 thf(fact_759_Image__empty__trancl__Image__empty,axiom,
% 14.17/8.01 ! [R3: set_Pr5094982260447487303t_unit,V: product_unit] :
% 14.17/8.01 ( ( ( image_8431784415008483091t_unit @ R3 @ ( insert_Product_unit @ V @ bot_bo3957492148770167129t_unit ) )
% 14.17/8.01 = bot_bo3957492148770167129t_unit )
% 14.17/8.01 => ( ( image_8431784415008483091t_unit @ ( transi430408012122777006t_unit @ R3 ) @ ( insert_Product_unit @ V @ bot_bo3957492148770167129t_unit ) )
% 14.17/8.01 = bot_bo3957492148770167129t_unit ) ) ).
% 14.17/8.01
% 14.17/8.01 thf(fact_760_Image__empty__trancl__Image__empty,axiom,
% 14.17/8.01 ! [R3: set_Product_prod_o_o,V: $o] :
% 14.17/8.01 ( ( ( image_o_o @ R3 @ ( insert_o @ V @ bot_bot_set_o ) )
% 14.17/8.01 = bot_bot_set_o )
% 14.17/8.01 => ( ( image_o_o @ ( transitive_trancl_o @ R3 ) @ ( insert_o @ V @ bot_bot_set_o ) )
% 14.17/8.01 = bot_bot_set_o ) ) ).
% 14.17/8.01
% 14.17/8.01 thf(fact_761_Image__empty__trancl__Image__empty,axiom,
% 14.17/8.01 ! [R3: set_Pr1261947904930325089at_nat,V: nat] :
% 14.17/8.01 ( ( ( image_nat_nat @ R3 @ ( insert_nat @ V @ bot_bot_set_nat ) )
% 14.17/8.01 = bot_bot_set_nat )
% 14.17/8.01 => ( ( image_nat_nat @ ( transi6264000038957366511cl_nat @ R3 ) @ ( insert_nat @ V @ bot_bot_set_nat ) )
% 14.17/8.01 = bot_bot_set_nat ) ) ).
% 14.17/8.01
% 14.17/8.01 thf(fact_762_Image__empty__trancl__Image__empty,axiom,
% 14.17/8.01 ! [R3: set_Pr958786334691620121nt_int,V: int] :
% 14.17/8.01 ( ( ( image_int_int @ R3 @ ( insert_int @ V @ bot_bot_set_int ) )
% 14.17/8.01 = bot_bot_set_int )
% 14.17/8.01 => ( ( image_int_int @ ( transi6261509568448316235cl_int @ R3 ) @ ( insert_int @ V @ bot_bot_set_int ) )
% 14.17/8.01 = bot_bot_set_int ) ) ).
% 14.17/8.01
% 14.17/8.01 thf(fact_763_Image__empty__trancl__Image__empty,axiom,
% 14.17/8.01 ! [R3: set_Pr2161125870931222855_nat_o,V: produc3658429121746597890et_nat > $o] :
% 14.17/8.01 ( ( ( image_2232984160439939859_nat_o @ R3 @ ( insert5175938949040314269_nat_o @ V @ bot_bo7824918357723582233_nat_o ) )
% 14.17/8.01 = bot_bo7824918357723582233_nat_o )
% 14.17/8.01 => ( ( image_2232984160439939859_nat_o @ ( transi4009562729418357614_nat_o @ R3 ) @ ( insert5175938949040314269_nat_o @ V @ bot_bo7824918357723582233_nat_o ) )
% 14.17/8.01 = bot_bo7824918357723582233_nat_o ) ) ).
% 14.17/8.01
% 14.17/8.01 thf(fact_764_mod__relH,axiom,
% 14.17/8.01 ! [As: set_nat,H2: heap_e7401611519738050253t_unit,H3: heap_e7401611519738050253t_unit,P: assn] :
% 14.17/8.01 ( ( relH @ As @ H2 @ H3 )
% 14.17/8.01 => ( ( rep_assn @ P @ ( produc7507926704131184380et_nat @ H2 @ As ) )
% 14.17/8.01 = ( rep_assn @ P @ ( produc7507926704131184380et_nat @ H3 @ As ) ) ) ) ).
% 14.17/8.01
% 14.17/8.01 thf(fact_765_for__in__RI,axiom,
% 14.17/8.01 ! [X: heap_e7401611519738050253t_unit,R3: set_Pr3948176798113811640et_nat] :
% 14.17/8.01 ( ( member242389114213480694t_unit @ X @ ( domain2007548326387822151et_nat @ R3 ) )
% 14.17/8.01 => ( member6260224972018164377et_nat @ ( produc7507926704131184380et_nat @ X @ ( fun_of5878246503291251064et_nat @ R3 @ X ) ) @ R3 ) ) ).
% 14.17/8.01
% 14.17/8.01 thf(fact_766_for__in__RI,axiom,
% 14.17/8.01 ! [X: nat,R3: set_Pr1261947904930325089at_nat] :
% 14.17/8.01 ( ( member_nat @ X @ ( domain_nat_nat @ R3 ) )
% 14.17/8.01 => ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ ( fun_of_rel_nat_nat @ R3 @ X ) ) @ R3 ) ) ).
% 14.17/8.01
% 14.17/8.01 thf(fact_767_for__in__RI,axiom,
% 14.17/8.01 ! [X: int,R3: set_Pr958786334691620121nt_int] :
% 14.17/8.01 ( ( member_int @ X @ ( domain_int_int @ R3 ) )
% 14.17/8.01 => ( member5262025264175285858nt_int @ ( product_Pair_int_int @ X @ ( fun_of_rel_int_int @ R3 @ X ) ) @ R3 ) ) ).
% 14.17/8.01
% 14.17/8.01 thf(fact_768_for__in__RI,axiom,
% 14.17/8.01 ! [X: produc3658429121746597890et_nat > $o,R3: set_Pr3286484037609594932et_nat] :
% 14.17/8.01 ( ( member6576561426505652726_nat_o @ X @ ( domain2358961613337345475et_nat @ R3 ) )
% 14.17/8.01 => ( member1996754912294343701et_nat @ ( produc5001842942810119800et_nat @ X @ ( fun_of3673525222065131764et_nat @ R3 @ X ) ) @ R3 ) ) ).
% 14.17/8.01
% 14.17/8.01 thf(fact_769_for__in__RI,axiom,
% 14.17/8.01 ! [X: produc3658429121746597890et_nat > $o,R3: set_Pr8536935166611901872et_nat] :
% 14.17/8.01 ( ( member6576561426505652726_nat_o @ X @ ( domain3618434998481624895et_nat @ R3 ) )
% 14.17/8.01 => ( member6124377750444531601et_nat @ ( produc2245416461498447860et_nat @ X @ ( fun_of384177504262425200et_nat @ R3 @ X ) ) @ R3 ) ) ).
% 14.17/8.01
% 14.17/8.01 thf(fact_770_trancl_Ocases,axiom,
% 14.17/8.01 ! [A1: nat,A22: nat,R2: set_Pr1261947904930325089at_nat] :
% 14.17/8.01 ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A1 @ A22 ) @ ( transi6264000038957366511cl_nat @ R2 ) )
% 14.17/8.01 => ( ~ ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A1 @ A22 ) @ R2 )
% 14.17/8.01 => ~ ! [B2: nat] :
% 14.17/8.01 ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A1 @ B2 ) @ ( transi6264000038957366511cl_nat @ R2 ) )
% 14.17/8.01 => ~ ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ B2 @ A22 ) @ R2 ) ) ) ) ).
% 14.17/8.01
% 14.17/8.01 thf(fact_771_trancl_Ocases,axiom,
% 14.17/8.01 ! [A1: int,A22: int,R2: set_Pr958786334691620121nt_int] :
% 14.17/8.01 ( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ A1 @ A22 ) @ ( transi6261509568448316235cl_int @ R2 ) )
% 14.17/8.01 => ( ~ ( member5262025264175285858nt_int @ ( product_Pair_int_int @ A1 @ A22 ) @ R2 )
% 14.17/8.01 => ~ ! [B2: int] :
% 14.17/8.01 ( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ A1 @ B2 ) @ ( transi6261509568448316235cl_int @ R2 ) )
% 14.17/8.01 => ~ ( member5262025264175285858nt_int @ ( product_Pair_int_int @ B2 @ A22 ) @ R2 ) ) ) ) ).
% 14.17/8.01
% 14.17/8.01 thf(fact_772_trancl_Osimps,axiom,
% 14.17/8.01 ! [A1: nat,A22: nat,R2: set_Pr1261947904930325089at_nat] :
% 14.17/8.01 ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A1 @ A22 ) @ ( transi6264000038957366511cl_nat @ R2 ) )
% 14.17/8.01 = ( ? [A6: nat,B6: nat] :
% 14.17/8.01 ( ( A1 = A6 )
% 14.17/8.01 & ( A22 = B6 )
% 14.17/8.01 & ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A6 @ B6 ) @ R2 ) )
% 14.17/8.01 | ? [A6: nat,B6: nat,C4: nat] :
% 14.17/8.01 ( ( A1 = A6 )
% 14.17/8.01 & ( A22 = C4 )
% 14.17/8.01 & ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A6 @ B6 ) @ ( transi6264000038957366511cl_nat @ R2 ) )
% 14.17/8.01 & ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ B6 @ C4 ) @ R2 ) ) ) ) ).
% 14.17/8.01
% 14.17/8.01 thf(fact_773_trancl_Osimps,axiom,
% 14.17/8.01 ! [A1: int,A22: int,R2: set_Pr958786334691620121nt_int] :
% 14.17/8.01 ( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ A1 @ A22 ) @ ( transi6261509568448316235cl_int @ R2 ) )
% 14.17/8.01 = ( ? [A6: int,B6: int] :
% 14.17/8.01 ( ( A1 = A6 )
% 14.17/8.01 & ( A22 = B6 )
% 14.17/8.01 & ( member5262025264175285858nt_int @ ( product_Pair_int_int @ A6 @ B6 ) @ R2 ) )
% 14.17/8.01 | ? [A6: int,B6: int,C4: int] :
% 14.17/8.01 ( ( A1 = A6 )
% 14.17/8.01 & ( A22 = C4 )
% 14.17/8.01 & ( member5262025264175285858nt_int @ ( product_Pair_int_int @ A6 @ B6 ) @ ( transi6261509568448316235cl_int @ R2 ) )
% 14.17/8.01 & ( member5262025264175285858nt_int @ ( product_Pair_int_int @ B6 @ C4 ) @ R2 ) ) ) ) ).
% 14.17/8.01
% 14.17/8.01 thf(fact_774_trancl_Or__into__trancl,axiom,
% 14.17/8.01 ! [A2: nat,B: nat,R2: set_Pr1261947904930325089at_nat] :
% 14.17/8.01 ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A2 @ B ) @ R2 )
% 14.17/8.01 => ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A2 @ B ) @ ( transi6264000038957366511cl_nat @ R2 ) ) ) ).
% 14.17/8.01
% 14.17/8.01 thf(fact_775_trancl_Or__into__trancl,axiom,
% 14.17/8.01 ! [A2: int,B: int,R2: set_Pr958786334691620121nt_int] :
% 14.17/8.01 ( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ A2 @ B ) @ R2 )
% 14.17/8.01 => ( member5262025264175285858nt_int @ ( product_Pair_int_int @ A2 @ B ) @ ( transi6261509568448316235cl_int @ R2 ) ) ) ).
% 14.17/8.01
% 14.17/8.01 thf(fact_776_tranclE,axiom,
% 14.17/8.01 ! [A2: nat,B: nat,R2: set_Pr1261947904930325089at_nat] :
% 14.17/8.01 ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A2 @ B ) @ ( transi6264000038957366511cl_nat @ R2 ) )
% 14.17/8.01 => ( ~ ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A2 @ B ) @ R2 )
% 14.17/8.01 => ~ ! [C2: nat] :
% 14.17/8.01 ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A2 @ C2 ) @ ( transi6264000038957366511cl_nat @ R2 ) )
% 14.17/8.01 => ~ ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ C2 @ B ) @ R2 ) ) ) ) ).
% 14.17/8.01
% 14.17/8.01 thf(fact_777_tranclE,axiom,
% 14.17/8.01 ! [A2: int,B: int,R2: set_Pr958786334691620121nt_int] :
% 14.17/8.01 ( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ A2 @ B ) @ ( transi6261509568448316235cl_int @ R2 ) )
% 14.17/8.01 => ( ~ ( member5262025264175285858nt_int @ ( product_Pair_int_int @ A2 @ B ) @ R2 )
% 14.17/8.01 => ~ ! [C2: int] :
% 14.17/8.01 ( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ A2 @ C2 ) @ ( transi6261509568448316235cl_int @ R2 ) )
% 14.17/8.01 => ~ ( member5262025264175285858nt_int @ ( product_Pair_int_int @ C2 @ B ) @ R2 ) ) ) ) ).
% 14.17/8.01
% 14.17/8.01 thf(fact_778_trancl__trans,axiom,
% 14.17/8.01 ! [X: nat,Y: nat,R2: set_Pr1261947904930325089at_nat,Z: nat] :
% 14.17/8.01 ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ Y ) @ ( transi6264000038957366511cl_nat @ R2 ) )
% 14.17/8.01 => ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ Y @ Z ) @ ( transi6264000038957366511cl_nat @ R2 ) )
% 14.17/8.01 => ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ Z ) @ ( transi6264000038957366511cl_nat @ R2 ) ) ) ) ).
% 14.17/8.01
% 14.17/8.01 thf(fact_779_trancl__trans,axiom,
% 14.17/8.01 ! [X: int,Y: int,R2: set_Pr958786334691620121nt_int,Z: int] :
% 14.17/8.01 ( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ X @ Y ) @ ( transi6261509568448316235cl_int @ R2 ) )
% 14.17/8.01 => ( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ Y @ Z ) @ ( transi6261509568448316235cl_int @ R2 ) )
% 14.17/8.01 => ( member5262025264175285858nt_int @ ( product_Pair_int_int @ X @ Z ) @ ( transi6261509568448316235cl_int @ R2 ) ) ) ) ).
% 14.17/8.01
% 14.17/8.01 thf(fact_780_trancl__induct,axiom,
% 14.17/8.01 ! [A2: nat,B: nat,R2: set_Pr1261947904930325089at_nat,P: nat > $o] :
% 14.17/8.01 ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A2 @ B ) @ ( transi6264000038957366511cl_nat @ R2 ) )
% 14.17/8.01 => ( ! [Y2: nat] :
% 14.17/8.01 ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A2 @ Y2 ) @ R2 )
% 14.17/8.01 => ( P @ Y2 ) )
% 14.17/8.01 => ( ! [Y2: nat,Z3: nat] :
% 14.17/8.01 ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A2 @ Y2 ) @ ( transi6264000038957366511cl_nat @ R2 ) )
% 14.17/8.01 => ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ Y2 @ Z3 ) @ R2 )
% 14.17/8.01 => ( ( P @ Y2 )
% 14.17/8.01 => ( P @ Z3 ) ) ) )
% 14.17/8.01 => ( P @ B ) ) ) ) ).
% 14.17/8.01
% 14.17/8.01 thf(fact_781_trancl__induct,axiom,
% 14.17/8.01 ! [A2: int,B: int,R2: set_Pr958786334691620121nt_int,P: int > $o] :
% 14.17/8.01 ( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ A2 @ B ) @ ( transi6261509568448316235cl_int @ R2 ) )
% 14.17/8.01 => ( ! [Y2: int] :
% 14.17/8.01 ( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ A2 @ Y2 ) @ R2 )
% 14.17/8.01 => ( P @ Y2 ) )
% 14.17/8.01 => ( ! [Y2: int,Z3: int] :
% 14.17/8.01 ( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ A2 @ Y2 ) @ ( transi6261509568448316235cl_int @ R2 ) )
% 14.17/8.01 => ( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ Y2 @ Z3 ) @ R2 )
% 14.17/8.01 => ( ( P @ Y2 )
% 14.17/8.01 => ( P @ Z3 ) ) ) )
% 14.17/8.01 => ( P @ B ) ) ) ) ).
% 14.17/8.01
% 14.17/8.01 thf(fact_782_r__r__into__trancl,axiom,
% 14.17/8.01 ! [A2: nat,B: nat,R3: set_Pr1261947904930325089at_nat,C: nat] :
% 14.17/8.01 ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A2 @ B ) @ R3 )
% 14.17/8.01 => ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ B @ C ) @ R3 )
% 14.17/8.01 => ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A2 @ C ) @ ( transi6264000038957366511cl_nat @ R3 ) ) ) ) ).
% 14.17/8.01
% 14.17/8.01 thf(fact_783_r__r__into__trancl,axiom,
% 14.17/8.01 ! [A2: int,B: int,R3: set_Pr958786334691620121nt_int,C: int] :
% 14.17/8.01 ( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ A2 @ B ) @ R3 )
% 14.17/8.01 => ( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ B @ C ) @ R3 )
% 14.17/8.01 => ( member5262025264175285858nt_int @ ( product_Pair_int_int @ A2 @ C ) @ ( transi6261509568448316235cl_int @ R3 ) ) ) ) ).
% 14.17/8.01
% 14.17/8.01 thf(fact_784_converse__tranclE,axiom,
% 14.17/8.01 ! [X: nat,Z: nat,R2: set_Pr1261947904930325089at_nat] :
% 14.17/8.01 ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ Z ) @ ( transi6264000038957366511cl_nat @ R2 ) )
% 14.17/8.01 => ( ~ ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ Z ) @ R2 )
% 14.17/8.01 => ~ ! [Y2: nat] :
% 14.17/8.01 ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ Y2 ) @ R2 )
% 14.17/8.01 => ~ ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ Y2 @ Z ) @ ( transi6264000038957366511cl_nat @ R2 ) ) ) ) ) ).
% 14.17/8.01
% 14.17/8.01 thf(fact_785_converse__tranclE,axiom,
% 14.17/8.01 ! [X: int,Z: int,R2: set_Pr958786334691620121nt_int] :
% 14.17/8.01 ( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ X @ Z ) @ ( transi6261509568448316235cl_int @ R2 ) )
% 14.17/8.01 => ( ~ ( member5262025264175285858nt_int @ ( product_Pair_int_int @ X @ Z ) @ R2 )
% 14.17/8.01 => ~ ! [Y2: int] :
% 14.17/8.01 ( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ X @ Y2 ) @ R2 )
% 14.17/8.01 => ~ ( member5262025264175285858nt_int @ ( product_Pair_int_int @ Y2 @ Z ) @ ( transi6261509568448316235cl_int @ R2 ) ) ) ) ) ).
% 14.17/8.01
% 14.17/8.01 thf(fact_786_trancl__induct2,axiom,
% 14.17/8.01 ! [Ax: heap_e7401611519738050253t_unit,Ay: set_nat,Bx: heap_e7401611519738050253t_unit,By: set_nat,R2: set_Pr719794911490849221et_nat,P: heap_e7401611519738050253t_unit > set_nat > $o] :
% 14.17/8.01 ( ( member6099555550032318734et_nat @ ( produc8199053930788261021et_nat @ ( produc7507926704131184380et_nat @ Ax @ Ay ) @ ( produc7507926704131184380et_nat @ Bx @ By ) ) @ ( transi4137310414928362273et_nat @ R2 ) )
% 14.17/8.01 => ( ! [A4: heap_e7401611519738050253t_unit,B2: set_nat] :
% 14.17/8.01 ( ( member6099555550032318734et_nat @ ( produc8199053930788261021et_nat @ ( produc7507926704131184380et_nat @ Ax @ Ay ) @ ( produc7507926704131184380et_nat @ A4 @ B2 ) ) @ R2 )
% 14.17/8.01 => ( P @ A4 @ B2 ) )
% 14.17/8.01 => ( ! [A4: heap_e7401611519738050253t_unit,B2: set_nat,Aa: heap_e7401611519738050253t_unit,Ba: set_nat] :
% 14.17/8.01 ( ( member6099555550032318734et_nat @ ( produc8199053930788261021et_nat @ ( produc7507926704131184380et_nat @ Ax @ Ay ) @ ( produc7507926704131184380et_nat @ A4 @ B2 ) ) @ ( transi4137310414928362273et_nat @ R2 ) )
% 14.17/8.01 => ( ( member6099555550032318734et_nat @ ( produc8199053930788261021et_nat @ ( produc7507926704131184380et_nat @ A4 @ B2 ) @ ( produc7507926704131184380et_nat @ Aa @ Ba ) ) @ R2 )
% 14.17/8.01 => ( ( P @ A4 @ B2 )
% 14.17/8.01 => ( P @ Aa @ Ba ) ) ) )
% 14.17/8.01 => ( P @ Bx @ By ) ) ) ) ).
% 14.17/8.01
% 14.17/8.01 thf(fact_787_trancl__induct2,axiom,
% 14.17/8.01 ! [Ax: nat,Ay: nat,Bx: nat,By: nat,R2: set_Pr8693737435421807431at_nat,P: nat > nat > $o] :
% 14.17/8.01 ( ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ ( product_Pair_nat_nat @ Ax @ Ay ) @ ( product_Pair_nat_nat @ Bx @ By ) ) @ ( transi243908449541399842at_nat @ R2 ) )
% 14.17/8.01 => ( ! [A4: nat,B2: nat] :
% 14.17/8.01 ( ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ ( product_Pair_nat_nat @ Ax @ Ay ) @ ( product_Pair_nat_nat @ A4 @ B2 ) ) @ R2 )
% 14.17/8.01 => ( P @ A4 @ B2 ) )
% 14.17/8.01 => ( ! [A4: nat,B2: nat,Aa: nat,Ba: nat] :
% 14.17/8.01 ( ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ ( product_Pair_nat_nat @ Ax @ Ay ) @ ( product_Pair_nat_nat @ A4 @ B2 ) ) @ ( transi243908449541399842at_nat @ R2 ) )
% 14.17/8.01 => ( ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ ( product_Pair_nat_nat @ A4 @ B2 ) @ ( product_Pair_nat_nat @ Aa @ Ba ) ) @ R2 )
% 14.17/8.01 => ( ( P @ A4 @ B2 )
% 14.17/8.01 => ( P @ Aa @ Ba ) ) ) )
% 14.17/8.01 => ( P @ Bx @ By ) ) ) ) ).
% 14.17/8.01
% 14.17/8.01 thf(fact_788_trancl__induct2,axiom,
% 14.17/8.01 ! [Ax: int,Ay: int,Bx: int,By: int,R2: set_Pr2560585780119916871nt_int,P: int > int > $o] :
% 14.17/8.01 ( ( member8566619992076573584nt_int @ ( produc3646306378393792727nt_int @ ( product_Pair_int_int @ Ax @ Ay ) @ ( product_Pair_int_int @ Bx @ By ) ) @ ( transi6288783178788033498nt_int @ R2 ) )
% 14.17/8.01 => ( ! [A4: int,B2: int] :
% 14.17/8.01 ( ( member8566619992076573584nt_int @ ( produc3646306378393792727nt_int @ ( product_Pair_int_int @ Ax @ Ay ) @ ( product_Pair_int_int @ A4 @ B2 ) ) @ R2 )
% 14.17/8.01 => ( P @ A4 @ B2 ) )
% 14.17/8.01 => ( ! [A4: int,B2: int,Aa: int,Ba: int] :
% 14.17/8.01 ( ( member8566619992076573584nt_int @ ( produc3646306378393792727nt_int @ ( product_Pair_int_int @ Ax @ Ay ) @ ( product_Pair_int_int @ A4 @ B2 ) ) @ ( transi6288783178788033498nt_int @ R2 ) )
% 14.17/8.01 => ( ( member8566619992076573584nt_int @ ( produc3646306378393792727nt_int @ ( product_Pair_int_int @ A4 @ B2 ) @ ( product_Pair_int_int @ Aa @ Ba ) ) @ R2 )
% 14.17/8.01 => ( ( P @ A4 @ B2 )
% 14.17/8.01 => ( P @ Aa @ Ba ) ) ) )
% 14.17/8.01 => ( P @ Bx @ By ) ) ) ) ).
% 14.17/8.01
% 14.17/8.01 thf(fact_789_trancl__induct2,axiom,
% 14.17/8.01 ! [Ax: produc3658429121746597890et_nat > $o,Ay: produc3658429121746597890et_nat,Bx: produc3658429121746597890et_nat > $o,By: produc3658429121746597890et_nat,R2: set_Pr7928877670098842301et_nat,P: ( produc3658429121746597890et_nat > $o ) > produc3658429121746597890et_nat > $o] :
% 14.17/8.01 ( ( member4763271486408492550et_nat @ ( produc8599840265553166229et_nat @ ( produc5001842942810119800et_nat @ Ax @ Ay ) @ ( produc5001842942810119800et_nat @ Bx @ By ) ) @ ( transi3145040225084697757et_nat @ R2 ) )
% 14.17/8.01 => ( ! [A4: produc3658429121746597890et_nat > $o,B2: produc3658429121746597890et_nat] :
% 14.17/8.01 ( ( member4763271486408492550et_nat @ ( produc8599840265553166229et_nat @ ( produc5001842942810119800et_nat @ Ax @ Ay ) @ ( produc5001842942810119800et_nat @ A4 @ B2 ) ) @ R2 )
% 14.17/8.01 => ( P @ A4 @ B2 ) )
% 14.17/8.01 => ( ! [A4: produc3658429121746597890et_nat > $o,B2: produc3658429121746597890et_nat,Aa: produc3658429121746597890et_nat > $o,Ba: produc3658429121746597890et_nat] :
% 14.17/8.01 ( ( member4763271486408492550et_nat @ ( produc8599840265553166229et_nat @ ( produc5001842942810119800et_nat @ Ax @ Ay ) @ ( produc5001842942810119800et_nat @ A4 @ B2 ) ) @ ( transi3145040225084697757et_nat @ R2 ) )
% 14.17/8.01 => ( ( member4763271486408492550et_nat @ ( produc8599840265553166229et_nat @ ( produc5001842942810119800et_nat @ A4 @ B2 ) @ ( produc5001842942810119800et_nat @ Aa @ Ba ) ) @ R2 )
% 14.17/8.01 => ( ( P @ A4 @ B2 )
% 14.17/8.01 => ( P @ Aa @ Ba ) ) ) )
% 14.17/8.01 => ( P @ Bx @ By ) ) ) ) ).
% 14.17/8.01
% 14.17/8.01 thf(fact_790_trancl__induct2,axiom,
% 14.17/8.01 ! [Ax: produc3658429121746597890et_nat > $o,Ay: produc3925858234332021118et_nat,Bx: produc3658429121746597890et_nat > $o,By: produc3925858234332021118et_nat,R2: set_Pr3444600963470892981et_nat,P: ( produc3658429121746597890et_nat > $o ) > produc3925858234332021118et_nat > $o] :
% 14.17/8.01 ( ( member6341495586645257982et_nat @ ( produc1940133919992309389et_nat @ ( produc2245416461498447860et_nat @ Ax @ Ay ) @ ( produc2245416461498447860et_nat @ Bx @ By ) ) @ ( transi5221092739591632921et_nat @ R2 ) )
% 14.17/8.01 => ( ! [A4: produc3658429121746597890et_nat > $o,B2: produc3925858234332021118et_nat] :
% 14.17/8.01 ( ( member6341495586645257982et_nat @ ( produc1940133919992309389et_nat @ ( produc2245416461498447860et_nat @ Ax @ Ay ) @ ( produc2245416461498447860et_nat @ A4 @ B2 ) ) @ R2 )
% 14.17/8.01 => ( P @ A4 @ B2 ) )
% 14.17/8.01 => ( ! [A4: produc3658429121746597890et_nat > $o,B2: produc3925858234332021118et_nat,Aa: produc3658429121746597890et_nat > $o,Ba: produc3925858234332021118et_nat] :
% 14.17/8.01 ( ( member6341495586645257982et_nat @ ( produc1940133919992309389et_nat @ ( produc2245416461498447860et_nat @ Ax @ Ay ) @ ( produc2245416461498447860et_nat @ A4 @ B2 ) ) @ ( transi5221092739591632921et_nat @ R2 ) )
% 14.17/8.01 => ( ( member6341495586645257982et_nat @ ( produc1940133919992309389et_nat @ ( produc2245416461498447860et_nat @ A4 @ B2 ) @ ( produc2245416461498447860et_nat @ Aa @ Ba ) ) @ R2 )
% 14.17/8.01 => ( ( P @ A4 @ B2 )
% 14.17/8.01 => ( P @ Aa @ Ba ) ) ) )
% 14.17/8.01 => ( P @ Bx @ By ) ) ) ) ).
% 14.17/8.01
% 14.17/8.01 thf(fact_791_converse__trancl__induct,axiom,
% 14.17/8.01 ! [A2: nat,B: nat,R2: set_Pr1261947904930325089at_nat,P: nat > $o] :
% 14.17/8.01 ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A2 @ B ) @ ( transi6264000038957366511cl_nat @ R2 ) )
% 14.17/8.01 => ( ! [Y2: nat] :
% 14.17/8.01 ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ Y2 @ B ) @ R2 )
% 14.17/8.01 => ( P @ Y2 ) )
% 14.17/8.01 => ( ! [Y2: nat,Z3: nat] :
% 14.17/8.01 ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ Y2 @ Z3 ) @ R2 )
% 14.17/8.01 => ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ Z3 @ B ) @ ( transi6264000038957366511cl_nat @ R2 ) )
% 14.17/8.01 => ( ( P @ Z3 )
% 14.17/8.01 => ( P @ Y2 ) ) ) )
% 14.17/8.01 => ( P @ A2 ) ) ) ) ).
% 14.17/8.01
% 14.17/8.01 thf(fact_792_converse__trancl__induct,axiom,
% 14.17/8.01 ! [A2: int,B: int,R2: set_Pr958786334691620121nt_int,P: int > $o] :
% 14.17/8.01 ( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ A2 @ B ) @ ( transi6261509568448316235cl_int @ R2 ) )
% 14.17/8.01 => ( ! [Y2: int] :
% 14.17/8.01 ( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ Y2 @ B ) @ R2 )
% 14.17/8.01 => ( P @ Y2 ) )
% 14.17/8.01 => ( ! [Y2: int,Z3: int] :
% 14.17/8.01 ( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ Y2 @ Z3 ) @ R2 )
% 14.17/8.01 => ( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ Z3 @ B ) @ ( transi6261509568448316235cl_int @ R2 ) )
% 14.17/8.01 => ( ( P @ Z3 )
% 14.17/8.01 => ( P @ Y2 ) ) ) )
% 14.17/8.01 => ( P @ A2 ) ) ) ) ).
% 14.17/8.01
% 14.17/8.01 thf(fact_793_trancl__trans__induct,axiom,
% 14.17/8.01 ! [X: nat,Y: nat,R2: set_Pr1261947904930325089at_nat,P: nat > nat > $o] :
% 14.17/8.01 ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ Y ) @ ( transi6264000038957366511cl_nat @ R2 ) )
% 14.17/8.01 => ( ! [X3: nat,Y2: nat] :
% 14.17/8.01 ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X3 @ Y2 ) @ R2 )
% 14.17/8.01 => ( P @ X3 @ Y2 ) )
% 14.17/8.01 => ( ! [X3: nat,Y2: nat,Z3: nat] :
% 14.17/8.01 ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X3 @ Y2 ) @ ( transi6264000038957366511cl_nat @ R2 ) )
% 14.17/8.01 => ( ( P @ X3 @ Y2 )
% 14.17/8.01 => ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ Y2 @ Z3 ) @ ( transi6264000038957366511cl_nat @ R2 ) )
% 14.17/8.01 => ( ( P @ Y2 @ Z3 )
% 14.17/8.01 => ( P @ X3 @ Z3 ) ) ) ) )
% 14.17/8.01 => ( P @ X @ Y ) ) ) ) ).
% 14.17/8.01
% 14.17/8.01 thf(fact_794_trancl__trans__induct,axiom,
% 14.17/8.01 ! [X: int,Y: int,R2: set_Pr958786334691620121nt_int,P: int > int > $o] :
% 14.17/8.01 ( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ X @ Y ) @ ( transi6261509568448316235cl_int @ R2 ) )
% 14.17/8.01 => ( ! [X3: int,Y2: int] :
% 14.17/8.01 ( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ X3 @ Y2 ) @ R2 )
% 14.17/8.01 => ( P @ X3 @ Y2 ) )
% 14.17/8.01 => ( ! [X3: int,Y2: int,Z3: int] :
% 14.17/8.01 ( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ X3 @ Y2 ) @ ( transi6261509568448316235cl_int @ R2 ) )
% 14.17/8.01 => ( ( P @ X3 @ Y2 )
% 14.17/8.01 => ( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ Y2 @ Z3 ) @ ( transi6261509568448316235cl_int @ R2 ) )
% 14.17/8.01 => ( ( P @ Y2 @ Z3 )
% 14.17/8.01 => ( P @ X3 @ Z3 ) ) ) ) )
% 14.17/8.01 => ( P @ X @ Y ) ) ) ) ).
% 14.17/8.01
% 14.17/8.01 thf(fact_795_trancl__into__trancl2,axiom,
% 14.17/8.01 ! [A2: nat,B: nat,R2: set_Pr1261947904930325089at_nat,C: nat] :
% 14.17/8.01 ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A2 @ B ) @ R2 )
% 14.17/8.01 => ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ B @ C ) @ ( transi6264000038957366511cl_nat @ R2 ) )
% 14.17/8.01 => ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A2 @ C ) @ ( transi6264000038957366511cl_nat @ R2 ) ) ) ) ).
% 14.17/8.01
% 14.17/8.01 thf(fact_796_trancl__into__trancl2,axiom,
% 14.17/8.01 ! [A2: int,B: int,R2: set_Pr958786334691620121nt_int,C: int] :
% 14.17/8.01 ( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ A2 @ B ) @ R2 )
% 14.17/8.01 => ( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ B @ C ) @ ( transi6261509568448316235cl_int @ R2 ) )
% 14.17/8.01 => ( member5262025264175285858nt_int @ ( product_Pair_int_int @ A2 @ C ) @ ( transi6261509568448316235cl_int @ R2 ) ) ) ) ).
% 14.17/8.01
% 14.17/8.01 thf(fact_797_Transitive__Closure_Otrancl__into__trancl,axiom,
% 14.17/8.01 ! [A2: nat,B: nat,R2: set_Pr1261947904930325089at_nat,C: nat] :
% 14.17/8.01 ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A2 @ B ) @ ( transi6264000038957366511cl_nat @ R2 ) )
% 14.17/8.01 => ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ B @ C ) @ R2 )
% 14.17/8.01 => ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A2 @ C ) @ ( transi6264000038957366511cl_nat @ R2 ) ) ) ) ).
% 14.17/8.01
% 14.17/8.01 thf(fact_798_Transitive__Closure_Otrancl__into__trancl,axiom,
% 14.17/8.01 ! [A2: int,B: int,R2: set_Pr958786334691620121nt_int,C: int] :
% 14.17/8.01 ( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ A2 @ B ) @ ( transi6261509568448316235cl_int @ R2 ) )
% 14.17/8.01 => ( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ B @ C ) @ R2 )
% 14.17/8.01 => ( member5262025264175285858nt_int @ ( product_Pair_int_int @ A2 @ C ) @ ( transi6261509568448316235cl_int @ R2 ) ) ) ) ).
% 14.17/8.01
% 14.17/8.01 thf(fact_799_irrefl__trancl__rD,axiom,
% 14.17/8.01 ! [R2: set_Pr1261947904930325089at_nat,X: nat,Y: nat] :
% 14.17/8.01 ( ! [X3: nat] :
% 14.17/8.01 ~ ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X3 @ X3 ) @ ( transi6264000038957366511cl_nat @ R2 ) )
% 14.17/8.01 => ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ Y ) @ R2 )
% 14.17/8.01 => ( X != Y ) ) ) ).
% 14.17/8.01
% 14.17/8.01 thf(fact_800_irrefl__trancl__rD,axiom,
% 14.17/8.01 ! [R2: set_Pr958786334691620121nt_int,X: int,Y: int] :
% 14.17/8.01 ( ! [X3: int] :
% 14.17/8.01 ~ ( member5262025264175285858nt_int @ ( product_Pair_int_int @ X3 @ X3 ) @ ( transi6261509568448316235cl_int @ R2 ) )
% 14.17/8.01 => ( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ X @ Y ) @ R2 )
% 14.17/8.01 => ( X != Y ) ) ) ).
% 14.17/8.01
% 14.17/8.01 thf(fact_801_trancl__Image__advance__ss,axiom,
% 14.17/8.01 ! [U: product_unit,V: product_unit,E: set_Pr5094982260447487303t_unit] :
% 14.17/8.01 ( ( member7821138191923116944t_unit @ ( produc5692694829384537815t_unit @ U @ V ) @ E )
% 14.17/8.01 => ( ord_le3507040750410214029t_unit @ ( image_8431784415008483091t_unit @ ( transi430408012122777006t_unit @ E ) @ ( insert_Product_unit @ V @ bot_bo3957492148770167129t_unit ) ) @ ( image_8431784415008483091t_unit @ ( transi430408012122777006t_unit @ E ) @ ( insert_Product_unit @ U @ bot_bo3957492148770167129t_unit ) ) ) ) ).
% 14.17/8.01
% 14.17/8.01 thf(fact_802_trancl__Image__advance__ss,axiom,
% 14.17/8.01 ! [U: $o,V: $o,E: set_Product_prod_o_o] :
% 14.17/8.01 ( ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ U @ V ) @ E )
% 14.17/8.01 => ( ord_less_eq_set_o @ ( image_o_o @ ( transitive_trancl_o @ E ) @ ( insert_o @ V @ bot_bot_set_o ) ) @ ( image_o_o @ ( transitive_trancl_o @ E ) @ ( insert_o @ U @ bot_bot_set_o ) ) ) ) ).
% 14.17/8.01
% 14.17/8.01 thf(fact_803_trancl__Image__advance__ss,axiom,
% 14.17/8.01 ! [U: int,V: int,E: set_Pr958786334691620121nt_int] :
% 14.17/8.01 ( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ U @ V ) @ E )
% 14.17/8.01 => ( ord_less_eq_set_int @ ( image_int_int @ ( transi6261509568448316235cl_int @ E ) @ ( insert_int @ V @ bot_bot_set_int ) ) @ ( image_int_int @ ( transi6261509568448316235cl_int @ E ) @ ( insert_int @ U @ bot_bot_set_int ) ) ) ) ).
% 14.17/8.01
% 14.17/8.01 thf(fact_804_trancl__Image__advance__ss,axiom,
% 14.17/8.01 ! [U: produc3658429121746597890et_nat > $o,V: produc3658429121746597890et_nat > $o,E: set_Pr2161125870931222855_nat_o] :
% 14.17/8.01 ( ( member8781333585448626064_nat_o @ ( produc7368190662567826135_nat_o @ U @ V ) @ E )
% 14.17/8.01 => ( ord_le2965882846123202637_nat_o @ ( image_2232984160439939859_nat_o @ ( transi4009562729418357614_nat_o @ E ) @ ( insert5175938949040314269_nat_o @ V @ bot_bo7824918357723582233_nat_o ) ) @ ( image_2232984160439939859_nat_o @ ( transi4009562729418357614_nat_o @ E ) @ ( insert5175938949040314269_nat_o @ U @ bot_bo7824918357723582233_nat_o ) ) ) ) ).
% 14.17/8.01
% 14.17/8.01 thf(fact_805_trancl__Image__advance__ss,axiom,
% 14.17/8.01 ! [U: nat,V: nat,E: set_Pr1261947904930325089at_nat] :
% 14.17/8.01 ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ U @ V ) @ E )
% 14.17/8.01 => ( ord_less_eq_set_nat @ ( image_nat_nat @ ( transi6264000038957366511cl_nat @ E ) @ ( insert_nat @ V @ bot_bot_set_nat ) ) @ ( image_nat_nat @ ( transi6264000038957366511cl_nat @ E ) @ ( insert_nat @ U @ bot_bot_set_nat ) ) ) ) ).
% 14.17/8.01
% 14.17/8.01 thf(fact_806_wand__assnI,axiom,
% 14.17/8.01 ! [H2: heap_e7401611519738050253t_unit,As: set_nat,Q2: assn,R3: assn] :
% 14.17/8.01 ( ( in_range @ ( produc7507926704131184380et_nat @ H2 @ As ) )
% 14.17/8.01 => ( ! [H6: heap_e7401611519738050253t_unit,As4: set_nat] :
% 14.17/8.01 ( ( ( inf_inf_set_nat @ As @ As4 )
% 14.17/8.01 = bot_bot_set_nat )
% 14.17/8.01 => ( ( relH @ As @ H2 @ H6 )
% 14.17/8.01 => ( ( in_range @ ( produc7507926704131184380et_nat @ H6 @ As ) )
% 14.17/8.01 => ( ( rep_assn @ Q2 @ ( produc7507926704131184380et_nat @ H6 @ As4 ) )
% 14.17/8.01 => ( rep_assn @ R3 @ ( produc7507926704131184380et_nat @ H6 @ ( sup_sup_set_nat @ As @ As4 ) ) ) ) ) ) )
% 14.17/8.01 => ( rep_assn @ ( wand_assn @ Q2 @ R3 ) @ ( produc7507926704131184380et_nat @ H2 @ As ) ) ) ) ).
% 14.17/8.01
% 14.17/8.01 thf(fact_807_trancl__Image__in__Range,axiom,
% 14.17/8.01 ! [R3: set_Pr1261947904930325089at_nat,V2: set_nat] : ( ord_less_eq_set_nat @ ( image_nat_nat @ ( transi6264000038957366511cl_nat @ R3 ) @ V2 ) @ ( range_nat_nat @ R3 ) ) ).
% 14.17/8.01
% 14.17/8.01 thf(fact_808_Pair__vimage__Sigma,axiom,
% 14.17/8.01 ! [X: nat,A: set_nat,F: nat > set_Product_unit] :
% 14.17/8.01 ( ( ( member_nat @ X @ A )
% 14.17/8.01 => ( ( vimage7305857734363866308t_unit @ ( produc3434450474625521892t_unit @ X ) @ ( produc5506539147108004339t_unit @ A @ F ) )
% 14.17/8.01 = ( F @ X ) ) )
% 14.17/8.01 & ( ~ ( member_nat @ X @ A )
% 14.17/8.01 => ( ( vimage7305857734363866308t_unit @ ( produc3434450474625521892t_unit @ X ) @ ( produc5506539147108004339t_unit @ A @ F ) )
% 14.17/8.01 = bot_bo3957492148770167129t_unit ) ) ) ).
% 14.17/8.01
% 14.17/8.01 thf(fact_809_Pair__vimage__Sigma,axiom,
% 14.17/8.01 ! [X: int,A: set_int,F: int > set_Product_unit] :
% 14.17/8.01 ( ( ( member_int @ X @ A )
% 14.17/8.01 => ( ( vimage4422279681566046568t_unit @ ( produc550872421827702152t_unit @ X ) @ ( produc2622961094310184599t_unit @ A @ F ) )
% 14.17/8.01 = ( F @ X ) ) )
% 14.17/8.01 & ( ~ ( member_int @ X @ A )
% 14.17/8.01 => ( ( vimage4422279681566046568t_unit @ ( produc550872421827702152t_unit @ X ) @ ( produc2622961094310184599t_unit @ A @ F ) )
% 14.17/8.01 = bot_bo3957492148770167129t_unit ) ) ) ).
% 14.17/8.01
% 14.17/8.01 thf(fact_810_Pair__vimage__Sigma,axiom,
% 14.17/8.01 ! [X: $o,A: set_o,F: $o > set_Product_unit] :
% 14.17/8.01 ( ( ( member_o @ X @ A )
% 14.17/8.01 => ( ( vimage508909887551888138t_unit @ ( produc3901946521647900940t_unit @ X ) @ ( produc411009820408249917t_unit @ A @ F ) )
% 14.17/8.01 = ( F @ X ) ) )
% 14.17/8.01 & ( ~ ( member_o @ X @ A )
% 14.17/8.01 => ( ( vimage508909887551888138t_unit @ ( produc3901946521647900940t_unit @ X ) @ ( produc411009820408249917t_unit @ A @ F ) )
% 14.17/8.01 = bot_bo3957492148770167129t_unit ) ) ) ).
% 14.17/8.01
% 14.17/8.01 thf(fact_811_Pair__vimage__Sigma,axiom,
% 14.17/8.01 ! [X: product_unit,A: set_Product_unit,F: product_unit > set_Product_unit] :
% 14.17/8.01 ( ( ( member_Product_unit @ X @ A )
% 14.17/8.01 => ( ( vimage8114940889160087125t_unit @ ( produc5692694829384537815t_unit @ X ) @ ( produc6121607364029222536t_unit @ A @ F ) )
% 14.17/8.01 = ( F @ X ) ) )
% 14.17/8.01 & ( ~ ( member_Product_unit @ X @ A )
% 14.17/8.01 => ( ( vimage8114940889160087125t_unit @ ( produc5692694829384537815t_unit @ X ) @ ( produc6121607364029222536t_unit @ A @ F ) )
% 14.17/8.01 = bot_bo3957492148770167129t_unit ) ) ) ).
% 14.17/8.01
% 14.17/8.01 thf(fact_812_Pair__vimage__Sigma,axiom,
% 14.17/8.01 ! [X: nat,A: set_nat,F: nat > set_o] :
% 14.17/8.01 ( ( ( member_nat @ X @ A )
% 14.17/8.01 => ( ( vimage6125388955539849434_nat_o @ ( product_Pair_nat_o @ X ) @ ( product_Sigma_nat_o @ A @ F ) )
% 14.17/8.01 = ( F @ X ) ) )
% 14.17/8.01 & ( ~ ( member_nat @ X @ A )
% 14.17/8.01 => ( ( vimage6125388955539849434_nat_o @ ( product_Pair_nat_o @ X ) @ ( product_Sigma_nat_o @ A @ F ) )
% 14.17/8.01 = bot_bot_set_o ) ) ) ).
% 14.17/8.01
% 14.17/8.01 thf(fact_813_Pair__vimage__Sigma,axiom,
% 14.17/8.01 ! [X: int,A: set_int,F: int > set_o] :
% 14.17/8.01 ( ( ( member_int @ X @ A )
% 14.17/8.01 => ( ( vimage4304346610107161470_int_o @ ( product_Pair_int_o @ X ) @ ( product_Sigma_int_o @ A @ F ) )
% 14.17/8.01 = ( F @ X ) ) )
% 14.17/8.01 & ( ~ ( member_int @ X @ A )
% 14.17/8.01 => ( ( vimage4304346610107161470_int_o @ ( product_Pair_int_o @ X ) @ ( product_Sigma_int_o @ A @ F ) )
% 14.17/8.01 = bot_bot_set_o ) ) ) ).
% 14.17/8.01
% 14.17/8.01 thf(fact_814_Pair__vimage__Sigma,axiom,
% 14.17/8.01 ! [X: $o,A: set_o,F: $o > set_o] :
% 14.17/8.01 ( ( ( member_o @ X @ A )
% 14.17/8.01 => ( ( vimage8945963521958007626od_o_o @ ( product_Pair_o_o @ X ) @ ( product_Sigma_o_o @ A @ F ) )
% 14.17/8.01 = ( F @ X ) ) )
% 14.17/8.01 & ( ~ ( member_o @ X @ A )
% 14.17/8.01 => ( ( vimage8945963521958007626od_o_o @ ( product_Pair_o_o @ X ) @ ( product_Sigma_o_o @ A @ F ) )
% 14.17/8.01 = bot_bot_set_o ) ) ) ).
% 14.17/8.01
% 14.17/8.01 thf(fact_815_Pair__vimage__Sigma,axiom,
% 14.17/8.01 ! [X: product_unit,A: set_Product_unit,F: product_unit > set_o] :
% 14.17/8.01 ( ( ( member_Product_unit @ X @ A )
% 14.17/8.01 => ( ( vimage9156843197228287893unit_o @ ( produc8487917018914037602unit_o @ X ) @ ( produc4996980317674386579unit_o @ A @ F ) )
% 14.17/8.01 = ( F @ X ) ) )
% 14.17/8.01 & ( ~ ( member_Product_unit @ X @ A )
% 14.17/8.01 => ( ( vimage9156843197228287893unit_o @ ( produc8487917018914037602unit_o @ X ) @ ( produc4996980317674386579unit_o @ A @ F ) )
% 14.17/8.01 = bot_bot_set_o ) ) ) ).
% 14.17/8.01
% 14.17/8.01 thf(fact_816_Pair__vimage__Sigma,axiom,
% 14.17/8.01 ! [X: int,A: set_int,F: int > set_nat] :
% 14.17/8.01 ( ( ( member_int @ X @ A )
% 14.17/8.01 => ( ( vimage9012682431555523728nt_nat @ ( product_Pair_int_nat @ X ) @ ( produc1456381018704787142nt_nat @ A @ F ) )
% 14.17/8.01 = ( F @ X ) ) )
% 14.17/8.01 & ( ~ ( member_int @ X @ A )
% 14.17/8.01 => ( ( vimage9012682431555523728nt_nat @ ( product_Pair_int_nat @ X ) @ ( produc1456381018704787142nt_nat @ A @ F ) )
% 14.17/8.01 = bot_bot_set_nat ) ) ) ).
% 14.17/8.01
% 14.17/8.01 thf(fact_817_Pair__vimage__Sigma,axiom,
% 14.17/8.01 ! [X: $o,A: set_o,F: $o > set_nat] :
% 14.17/8.01 ( ( ( member_o @ X @ A )
% 14.17/8.01 => ( ( vimage5656379182745576100_o_nat @ ( product_Pair_o_nat @ X ) @ ( product_Sigma_o_nat @ A @ F ) )
% 14.17/8.01 = ( F @ X ) ) )
% 14.17/8.01 & ( ~ ( member_o @ X @ A )
% 14.17/8.01 => ( ( vimage5656379182745576100_o_nat @ ( product_Pair_o_nat @ X ) @ ( product_Sigma_o_nat @ A @ F ) )
% 14.17/8.01 = bot_bot_set_nat ) ) ) ).
% 14.17/8.01
% 14.17/8.01 thf(fact_818_relH__def,axiom,
% 14.17/8.01 ( relH
% 14.17/8.01 = ( ^ [As3: set_nat,H4: heap_e7401611519738050253t_unit,H5: heap_e7401611519738050253t_unit] :
% 14.17/8.01 ( ( in_range @ ( produc7507926704131184380et_nat @ H4 @ As3 ) )
% 14.17/8.01 & ( in_range @ ( produc7507926704131184380et_nat @ H5 @ As3 ) )
% 14.17/8.01 & ! [T: typerep,X2: nat] :
% 14.17/8.01 ( ( member_nat @ X2 @ As3 )
% 14.17/8.01 => ( ( ( refs_Product_unit @ H4 @ T @ X2 )
% 14.17/8.01 = ( refs_Product_unit @ H5 @ T @ X2 ) )
% 14.17/8.01 & ( ( arrays_Product_unit @ H4 @ T @ X2 )
% 14.17/8.01 = ( arrays_Product_unit @ H5 @ T @ X2 ) ) ) ) ) ) ) ).
% 14.17/8.01
% 14.17/8.01 thf(fact_819_vimage__insert,axiom,
% 14.17/8.01 ! [F: nat > product_unit,A2: product_unit,B4: set_Product_unit] :
% 14.17/8.01 ( ( vimage4884490618288580032t_unit @ F @ ( insert_Product_unit @ A2 @ B4 ) )
% 14.17/8.01 = ( sup_sup_set_nat @ ( vimage4884490618288580032t_unit @ F @ ( insert_Product_unit @ A2 @ bot_bo3957492148770167129t_unit ) ) @ ( vimage4884490618288580032t_unit @ F @ B4 ) ) ) ).
% 14.17/8.01
% 14.17/8.01 thf(fact_820_vimage__insert,axiom,
% 14.17/8.01 ! [F: nat > $o,A2: $o,B4: set_o] :
% 14.17/8.01 ( ( vimage_nat_o @ F @ ( insert_o @ A2 @ B4 ) )
% 14.17/8.01 = ( sup_sup_set_nat @ ( vimage_nat_o @ F @ ( insert_o @ A2 @ bot_bot_set_o ) ) @ ( vimage_nat_o @ F @ B4 ) ) ) ).
% 14.17/8.01
% 14.17/8.01 thf(fact_821_vimage__insert,axiom,
% 14.17/8.01 ! [F: nat > nat,A2: nat,B4: set_nat] :
% 14.17/8.01 ( ( vimage_nat_nat @ F @ ( insert_nat @ A2 @ B4 ) )
% 14.17/8.01 = ( sup_sup_set_nat @ ( vimage_nat_nat @ F @ ( insert_nat @ A2 @ bot_bot_set_nat ) ) @ ( vimage_nat_nat @ F @ B4 ) ) ) ).
% 14.17/8.01
% 14.17/8.01 thf(fact_822_vimage__insert,axiom,
% 14.17/8.01 ! [F: nat > int,A2: int,B4: set_int] :
% 14.17/8.01 ( ( vimage_nat_int @ F @ ( insert_int @ A2 @ B4 ) )
% 14.17/8.01 = ( sup_sup_set_nat @ ( vimage_nat_int @ F @ ( insert_int @ A2 @ bot_bot_set_int ) ) @ ( vimage_nat_int @ F @ B4 ) ) ) ).
% 14.17/8.01
% 14.17/8.01 thf(fact_823_vimage__insert,axiom,
% 14.17/8.01 ! [F: nat > produc3658429121746597890et_nat > $o,A2: produc3658429121746597890et_nat > $o,B4: set_Pr4532377907799695533_nat_o] :
% 14.17/8.01 ( ( vimage8201027140005291904_nat_o @ F @ ( insert5175938949040314269_nat_o @ A2 @ B4 ) )
% 14.17/8.01 = ( sup_sup_set_nat @ ( vimage8201027140005291904_nat_o @ F @ ( insert5175938949040314269_nat_o @ A2 @ bot_bo7824918357723582233_nat_o ) ) @ ( vimage8201027140005291904_nat_o @ F @ B4 ) ) ) ).
% 14.17/8.01
% 14.17/8.01 thf(fact_824_trancl__over__edgeE,axiom,
% 14.17/8.01 ! [U: nat,W: nat,V1: nat,V22: nat,E: set_Pr1261947904930325089at_nat] :
% 14.17/8.01 ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ U @ W ) @ ( transi6264000038957366511cl_nat @ ( insert8211810215607154385at_nat @ ( product_Pair_nat_nat @ V1 @ V22 ) @ E ) ) )
% 14.17/8.01 => ( ~ ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ U @ W ) @ ( transi6264000038957366511cl_nat @ E ) )
% 14.17/8.01 => ~ ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ U @ V1 ) @ ( transi2905341329935302413cl_nat @ E ) )
% 14.17/8.01 => ~ ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ V22 @ W ) @ ( transi2905341329935302413cl_nat @ E ) ) ) ) ) ).
% 14.17/8.01
% 14.17/8.01 thf(fact_825_trancl__over__edgeE,axiom,
% 14.17/8.01 ! [U: int,W: int,V1: int,V22: int,E: set_Pr958786334691620121nt_int] :
% 14.17/8.01 ( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ U @ W ) @ ( transi6261509568448316235cl_int @ ( insert5033312907999012233nt_int @ ( product_Pair_int_int @ V1 @ V22 ) @ E ) ) )
% 14.17/8.01 => ( ~ ( member5262025264175285858nt_int @ ( product_Pair_int_int @ U @ W ) @ ( transi6261509568448316235cl_int @ E ) )
% 14.17/8.01 => ~ ( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ U @ V1 ) @ ( transi2902850859426252137cl_int @ E ) )
% 14.17/8.01 => ~ ( member5262025264175285858nt_int @ ( product_Pair_int_int @ V22 @ W ) @ ( transi2902850859426252137cl_int @ E ) ) ) ) ) ).
% 14.17/8.01
% 14.17/8.01 thf(fact_826_partial__order__on__empty,axiom,
% 14.17/8.01 order_2505591618835473410t_unit @ bot_bo3957492148770167129t_unit @ bot_bo1654022762305692595t_unit ).
% 14.17/8.01
% 14.17/8.01 thf(fact_827_partial__order__on__empty,axiom,
% 14.17/8.01 order_8538205470752950541r_on_o @ bot_bot_set_o @ bot_bo7073875226086086771od_o_o ).
% 14.17/8.01
% 14.17/8.01 thf(fact_828_partial__order__on__empty,axiom,
% 14.17/8.01 order_5251275573222108571on_nat @ bot_bot_set_nat @ bot_bo2099793752762293965at_nat ).
% 14.17/8.01
% 14.17/8.01 thf(fact_829_partial__order__on__empty,axiom,
% 14.17/8.01 order_5248785102713058295on_int @ bot_bot_set_int @ bot_bo1796632182523588997nt_int ).
% 14.17/8.01
% 14.17/8.01 thf(fact_830_partial__order__on__empty,axiom,
% 14.17/8.01 order_3690095022144090562_nat_o @ bot_bo7824918357723582233_nat_o @ bot_bo3657187498359500723_nat_o ).
% 14.17/8.01
% 14.17/8.01 thf(fact_831_prod__set__simps_I2_J,axiom,
% 14.17/8.01 ! [X: heap_e7401611519738050253t_unit,Y: set_nat] :
% 14.17/8.01 ( ( basic_2698842327037663054et_nat @ ( produc7507926704131184380et_nat @ X @ Y ) )
% 14.17/8.01 = ( insert_set_nat @ Y @ bot_bot_set_set_nat ) ) ).
% 14.17/8.01
% 14.17/8.01 thf(fact_832_prod__set__simps_I2_J,axiom,
% 14.17/8.01 ! [X: produc3658429121746597890et_nat > $o,Y: produc3658429121746597890et_nat] :
% 14.17/8.01 ( ( basic_8479756702517612234et_nat @ ( produc5001842942810119800et_nat @ X @ Y ) )
% 14.17/8.01 = ( insert1271470778893645362et_nat @ Y @ bot_bo3771712512152092492et_nat ) ) ).
% 14.17/8.01
% 14.17/8.01 thf(fact_833_prod__set__simps_I2_J,axiom,
% 14.17/8.01 ! [X: produc3658429121746597890et_nat > $o,Y: produc3925858234332021118et_nat] :
% 14.17/8.01 ( ( basic_2768200282441914438et_nat @ ( produc2245416461498447860et_nat @ X @ Y ) )
% 14.17/8.01 = ( insert3111075074545270702et_nat @ Y @ bot_bo1481135142794719944et_nat ) ) ).
% 14.17/8.01
% 14.17/8.01 thf(fact_834_prod__set__simps_I2_J,axiom,
% 14.17/8.01 ! [X: nat,Y: nat] :
% 14.17/8.01 ( ( basic_snds_nat_nat @ ( product_Pair_nat_nat @ X @ Y ) )
% 14.17/8.01 = ( insert_nat @ Y @ bot_bot_set_nat ) ) ).
% 14.17/8.01
% 14.17/8.01 thf(fact_835_prod__set__simps_I2_J,axiom,
% 14.17/8.01 ! [X: int,Y: int] :
% 14.17/8.01 ( ( basic_snds_int_int @ ( product_Pair_int_int @ X @ Y ) )
% 14.17/8.01 = ( insert_int @ Y @ bot_bot_set_int ) ) ).
% 14.17/8.01
% 14.17/8.01 thf(fact_836_dual__order_Orefl,axiom,
% 14.17/8.01 ! [A2: filter_nat] : ( ord_le2510731241096832064er_nat @ A2 @ A2 ) ).
% 14.17/8.01
% 14.17/8.01 thf(fact_837_dual__order_Orefl,axiom,
% 14.17/8.01 ! [A2: set_nat] : ( ord_less_eq_set_nat @ A2 @ A2 ) ).
% 14.17/8.01
% 14.17/8.01 thf(fact_838_dual__order_Orefl,axiom,
% 14.17/8.01 ! [A2: rat] : ( ord_less_eq_rat @ A2 @ A2 ) ).
% 14.17/8.01
% 14.17/8.01 thf(fact_839_dual__order_Orefl,axiom,
% 14.17/8.01 ! [A2: nat] : ( ord_less_eq_nat @ A2 @ A2 ) ).
% 14.17/8.01
% 14.17/8.01 thf(fact_840_dual__order_Orefl,axiom,
% 14.17/8.01 ! [A2: int] : ( ord_less_eq_int @ A2 @ A2 ) ).
% 14.17/8.01
% 14.17/8.01 thf(fact_841_order__refl,axiom,
% 14.17/8.01 ! [X: filter_nat] : ( ord_le2510731241096832064er_nat @ X @ X ) ).
% 14.17/8.01
% 14.17/8.01 thf(fact_842_order__refl,axiom,
% 14.17/8.01 ! [X: set_nat] : ( ord_less_eq_set_nat @ X @ X ) ).
% 14.17/8.01
% 14.17/8.01 thf(fact_843_order__refl,axiom,
% 14.17/8.01 ! [X: rat] : ( ord_less_eq_rat @ X @ X ) ).
% 14.17/8.01
% 14.17/8.01 thf(fact_844_order__refl,axiom,
% 14.17/8.01 ! [X: nat] : ( ord_less_eq_nat @ X @ X ) ).
% 14.17/8.01
% 14.17/8.01 thf(fact_845_order__refl,axiom,
% 14.17/8.01 ! [X: int] : ( ord_less_eq_int @ X @ X ) ).
% 14.17/8.01
% 14.17/8.01 thf(fact_846_subset__antisym,axiom,
% 14.17/8.01 ! [A: set_nat,B4: set_nat] :
% 14.17/8.01 ( ( ord_less_eq_set_nat @ A @ B4 )
% 14.17/8.01 => ( ( ord_less_eq_set_nat @ B4 @ A )
% 14.17/8.01 => ( A = B4 ) ) ) ).
% 14.17/8.01
% 14.17/8.01 thf(fact_847_subsetI,axiom,
% 14.17/8.01 ! [A: set_Pr4532377907799695533_nat_o,B4: set_Pr4532377907799695533_nat_o] :
% 14.17/8.01 ( ! [X3: produc3658429121746597890et_nat > $o] :
% 14.17/8.01 ( ( member6576561426505652726_nat_o @ X3 @ A )
% 14.17/8.01 => ( member6576561426505652726_nat_o @ X3 @ B4 ) )
% 14.17/8.01 => ( ord_le2965882846123202637_nat_o @ A @ B4 ) ) ).
% 14.17/8.01
% 14.17/8.01 thf(fact_848_subsetI,axiom,
% 14.17/8.01 ! [A: set_int,B4: set_int] :
% 14.17/8.01 ( ! [X3: int] :
% 14.17/8.01 ( ( member_int @ X3 @ A )
% 14.17/8.01 => ( member_int @ X3 @ B4 ) )
% 14.17/8.01 => ( ord_less_eq_set_int @ A @ B4 ) ) ).
% 14.17/8.01
% 14.17/8.01 thf(fact_849_subsetI,axiom,
% 14.17/8.01 ! [A: set_o,B4: set_o] :
% 14.17/8.01 ( ! [X3: $o] :
% 14.17/8.01 ( ( member_o @ X3 @ A )
% 14.17/8.01 => ( member_o @ X3 @ B4 ) )
% 14.17/8.01 => ( ord_less_eq_set_o @ A @ B4 ) ) ).
% 14.17/8.01
% 14.17/8.01 thf(fact_850_subsetI,axiom,
% 14.17/8.01 ! [A: set_Product_unit,B4: set_Product_unit] :
% 14.17/8.01 ( ! [X3: product_unit] :
% 14.17/8.01 ( ( member_Product_unit @ X3 @ A )
% 14.17/8.01 => ( member_Product_unit @ X3 @ B4 ) )
% 14.17/8.01 => ( ord_le3507040750410214029t_unit @ A @ B4 ) ) ).
% 14.17/8.01
% 14.17/8.01 thf(fact_851_subsetI,axiom,
% 14.17/8.01 ! [A: set_nat,B4: set_nat] :
% 14.17/8.01 ( ! [X3: nat] :
% 14.17/8.01 ( ( member_nat @ X3 @ A )
% 14.17/8.01 => ( member_nat @ X3 @ B4 ) )
% 14.17/8.01 => ( ord_less_eq_set_nat @ A @ B4 ) ) ).
% 14.17/8.01
% 14.17/8.01 thf(fact_852_Int__iff,axiom,
% 14.17/8.01 ! [C: produc3658429121746597890et_nat > $o,A: set_Pr4532377907799695533_nat_o,B4: set_Pr4532377907799695533_nat_o] :
% 14.17/8.01 ( ( member6576561426505652726_nat_o @ C @ ( inf_in1906310914598751387_nat_o @ A @ B4 ) )
% 14.17/8.01 = ( ( member6576561426505652726_nat_o @ C @ A )
% 14.17/8.01 & ( member6576561426505652726_nat_o @ C @ B4 ) ) ) ).
% 14.17/8.01
% 14.17/8.01 thf(fact_853_Int__iff,axiom,
% 14.17/8.01 ! [C: int,A: set_int,B4: set_int] :
% 14.17/8.01 ( ( member_int @ C @ ( inf_inf_set_int @ A @ B4 ) )
% 14.17/8.01 = ( ( member_int @ C @ A )
% 14.17/8.01 & ( member_int @ C @ B4 ) ) ) ).
% 14.17/8.01
% 14.17/8.01 thf(fact_854_Int__iff,axiom,
% 14.17/8.01 ! [C: $o,A: set_o,B4: set_o] :
% 14.17/8.01 ( ( member_o @ C @ ( inf_inf_set_o @ A @ B4 ) )
% 14.17/8.01 = ( ( member_o @ C @ A )
% 14.17/8.01 & ( member_o @ C @ B4 ) ) ) ).
% 14.17/8.01
% 14.17/8.01 thf(fact_855_Int__iff,axiom,
% 14.17/8.01 ! [C: product_unit,A: set_Product_unit,B4: set_Product_unit] :
% 14.17/8.01 ( ( member_Product_unit @ C @ ( inf_in4660618365625256667t_unit @ A @ B4 ) )
% 14.17/8.01 = ( ( member_Product_unit @ C @ A )
% 14.17/8.01 & ( member_Product_unit @ C @ B4 ) ) ) ).
% 14.17/8.01
% 14.17/8.01 thf(fact_856_Int__iff,axiom,
% 14.17/8.01 ! [C: nat,A: set_nat,B4: set_nat] :
% 14.17/8.01 ( ( member_nat @ C @ ( inf_inf_set_nat @ A @ B4 ) )
% 14.17/8.01 = ( ( member_nat @ C @ A )
% 14.17/8.01 & ( member_nat @ C @ B4 ) ) ) ).
% 14.17/8.01
% 14.17/8.01 thf(fact_857_IntI,axiom,
% 14.17/8.01 ! [C: produc3658429121746597890et_nat > $o,A: set_Pr4532377907799695533_nat_o,B4: set_Pr4532377907799695533_nat_o] :
% 14.17/8.01 ( ( member6576561426505652726_nat_o @ C @ A )
% 14.17/8.01 => ( ( member6576561426505652726_nat_o @ C @ B4 )
% 14.17/8.01 => ( member6576561426505652726_nat_o @ C @ ( inf_in1906310914598751387_nat_o @ A @ B4 ) ) ) ) ).
% 14.17/8.01
% 14.17/8.01 thf(fact_858_IntI,axiom,
% 14.17/8.01 ! [C: int,A: set_int,B4: set_int] :
% 14.17/8.01 ( ( member_int @ C @ A )
% 14.17/8.01 => ( ( member_int @ C @ B4 )
% 14.17/8.01 => ( member_int @ C @ ( inf_inf_set_int @ A @ B4 ) ) ) ) ).
% 14.17/8.01
% 14.17/8.01 thf(fact_859_IntI,axiom,
% 14.17/8.01 ! [C: $o,A: set_o,B4: set_o] :
% 14.17/8.01 ( ( member_o @ C @ A )
% 14.17/8.01 => ( ( member_o @ C @ B4 )
% 14.17/8.01 => ( member_o @ C @ ( inf_inf_set_o @ A @ B4 ) ) ) ) ).
% 14.17/8.01
% 14.17/8.01 thf(fact_860_IntI,axiom,
% 14.17/8.01 ! [C: product_unit,A: set_Product_unit,B4: set_Product_unit] :
% 14.17/8.01 ( ( member_Product_unit @ C @ A )
% 14.17/8.01 => ( ( member_Product_unit @ C @ B4 )
% 14.17/8.01 => ( member_Product_unit @ C @ ( inf_in4660618365625256667t_unit @ A @ B4 ) ) ) ) ).
% 14.17/8.01
% 14.17/8.01 thf(fact_861_IntI,axiom,
% 14.17/8.01 ! [C: nat,A: set_nat,B4: set_nat] :
% 14.17/8.01 ( ( member_nat @ C @ A )
% 14.17/8.01 => ( ( member_nat @ C @ B4 )
% 14.17/8.01 => ( member_nat @ C @ ( inf_inf_set_nat @ A @ B4 ) ) ) ) ).
% 14.17/8.01
% 14.17/8.01 thf(fact_862_Un__iff,axiom,
% 14.17/8.01 ! [C: produc3658429121746597890et_nat > $o,A: set_Pr4532377907799695533_nat_o,B4: set_Pr4532377907799695533_nat_o] :
% 14.17/8.01 ( ( member6576561426505652726_nat_o @ C @ ( sup_su5209123915105501825_nat_o @ A @ B4 ) )
% 14.17/8.01 = ( ( member6576561426505652726_nat_o @ C @ A )
% 14.17/8.01 | ( member6576561426505652726_nat_o @ C @ B4 ) ) ) ).
% 14.17/8.01
% 14.17/8.01 thf(fact_863_Un__iff,axiom,
% 14.17/8.01 ! [C: int,A: set_int,B4: set_int] :
% 14.17/8.01 ( ( member_int @ C @ ( sup_sup_set_int @ A @ B4 ) )
% 14.17/8.01 = ( ( member_int @ C @ A )
% 14.17/8.01 | ( member_int @ C @ B4 ) ) ) ).
% 14.17/8.01
% 14.17/8.01 thf(fact_864_Un__iff,axiom,
% 14.17/8.01 ! [C: $o,A: set_o,B4: set_o] :
% 14.17/8.01 ( ( member_o @ C @ ( sup_sup_set_o @ A @ B4 ) )
% 14.17/8.01 = ( ( member_o @ C @ A )
% 14.17/8.01 | ( member_o @ C @ B4 ) ) ) ).
% 14.17/8.01
% 14.17/8.01 thf(fact_865_Un__iff,axiom,
% 14.17/8.01 ! [C: product_unit,A: set_Product_unit,B4: set_Product_unit] :
% 14.17/8.01 ( ( member_Product_unit @ C @ ( sup_su793286257634532545t_unit @ A @ B4 ) )
% 14.17/8.01 = ( ( member_Product_unit @ C @ A )
% 14.17/8.01 | ( member_Product_unit @ C @ B4 ) ) ) ).
% 14.17/8.01
% 14.17/8.01 thf(fact_866_Un__iff,axiom,
% 14.17/8.01 ! [C: nat,A: set_nat,B4: set_nat] :
% 14.17/8.01 ( ( member_nat @ C @ ( sup_sup_set_nat @ A @ B4 ) )
% 14.17/8.01 = ( ( member_nat @ C @ A )
% 14.17/8.01 | ( member_nat @ C @ B4 ) ) ) ).
% 14.17/8.01
% 14.17/8.01 thf(fact_867_UnCI,axiom,
% 14.17/8.01 ! [C: produc3658429121746597890et_nat > $o,B4: set_Pr4532377907799695533_nat_o,A: set_Pr4532377907799695533_nat_o] :
% 14.17/8.01 ( ( ~ ( member6576561426505652726_nat_o @ C @ B4 )
% 14.17/8.01 => ( member6576561426505652726_nat_o @ C @ A ) )
% 14.17/8.01 => ( member6576561426505652726_nat_o @ C @ ( sup_su5209123915105501825_nat_o @ A @ B4 ) ) ) ).
% 14.17/8.01
% 14.17/8.01 thf(fact_868_UnCI,axiom,
% 14.17/8.01 ! [C: int,B4: set_int,A: set_int] :
% 14.17/8.01 ( ( ~ ( member_int @ C @ B4 )
% 14.17/8.01 => ( member_int @ C @ A ) )
% 14.17/8.01 => ( member_int @ C @ ( sup_sup_set_int @ A @ B4 ) ) ) ).
% 14.17/8.01
% 14.17/8.01 thf(fact_869_UnCI,axiom,
% 14.17/8.01 ! [C: $o,B4: set_o,A: set_o] :
% 14.17/8.01 ( ( ~ ( member_o @ C @ B4 )
% 14.17/8.01 => ( member_o @ C @ A ) )
% 14.17/8.01 => ( member_o @ C @ ( sup_sup_set_o @ A @ B4 ) ) ) ).
% 14.17/8.01
% 14.17/8.01 thf(fact_870_UnCI,axiom,
% 14.17/8.01 ! [C: product_unit,B4: set_Product_unit,A: set_Product_unit] :
% 14.17/8.01 ( ( ~ ( member_Product_unit @ C @ B4 )
% 14.17/8.01 => ( member_Product_unit @ C @ A ) )
% 14.17/8.01 => ( member_Product_unit @ C @ ( sup_su793286257634532545t_unit @ A @ B4 ) ) ) ).
% 14.17/8.01
% 14.17/8.01 thf(fact_871_UnCI,axiom,
% 14.17/8.01 ! [C: nat,B4: set_nat,A: set_nat] :
% 14.17/8.01 ( ( ~ ( member_nat @ C @ B4 )
% 14.17/8.01 => ( member_nat @ C @ A ) )
% 14.17/8.01 => ( member_nat @ C @ ( sup_sup_set_nat @ A @ B4 ) ) ) ).
% 14.17/8.01
% 14.17/8.01 thf(fact_872_empty__subsetI,axiom,
% 14.17/8.01 ! [A: set_Product_unit] : ( ord_le3507040750410214029t_unit @ bot_bo3957492148770167129t_unit @ A ) ).
% 14.17/8.01
% 14.17/8.01 thf(fact_873_empty__subsetI,axiom,
% 14.17/8.01 ! [A: set_o] : ( ord_less_eq_set_o @ bot_bot_set_o @ A ) ).
% 14.17/8.01
% 14.17/8.01 thf(fact_874_empty__subsetI,axiom,
% 14.17/8.01 ! [A: set_int] : ( ord_less_eq_set_int @ bot_bot_set_int @ A ) ).
% 14.17/8.01
% 14.17/8.01 thf(fact_875_empty__subsetI,axiom,
% 14.17/8.01 ! [A: set_Pr4532377907799695533_nat_o] : ( ord_le2965882846123202637_nat_o @ bot_bo7824918357723582233_nat_o @ A ) ).
% 14.17/8.01
% 14.17/8.01 thf(fact_876_empty__subsetI,axiom,
% 14.17/8.01 ! [A: set_nat] : ( ord_less_eq_set_nat @ bot_bot_set_nat @ A ) ).
% 14.17/8.01
% 14.17/8.01 thf(fact_877_subset__empty,axiom,
% 14.17/8.01 ! [A: set_Product_unit] :
% 14.17/8.01 ( ( ord_le3507040750410214029t_unit @ A @ bot_bo3957492148770167129t_unit )
% 14.17/8.01 = ( A = bot_bo3957492148770167129t_unit ) ) ).
% 14.17/8.01
% 14.17/8.01 thf(fact_878_subset__empty,axiom,
% 14.17/8.01 ! [A: set_o] :
% 14.17/8.01 ( ( ord_less_eq_set_o @ A @ bot_bot_set_o )
% 14.17/8.01 = ( A = bot_bot_set_o ) ) ).
% 14.17/8.01
% 14.17/8.01 thf(fact_879_subset__empty,axiom,
% 14.17/8.01 ! [A: set_int] :
% 14.17/8.01 ( ( ord_less_eq_set_int @ A @ bot_bot_set_int )
% 14.17/8.01 = ( A = bot_bot_set_int ) ) ).
% 14.17/8.01
% 14.17/8.01 thf(fact_880_subset__empty,axiom,
% 14.17/8.01 ! [A: set_Pr4532377907799695533_nat_o] :
% 14.17/8.01 ( ( ord_le2965882846123202637_nat_o @ A @ bot_bo7824918357723582233_nat_o )
% 14.17/8.01 = ( A = bot_bo7824918357723582233_nat_o ) ) ).
% 14.17/8.01
% 14.17/8.01 thf(fact_881_subset__empty,axiom,
% 14.17/8.01 ! [A: set_nat] :
% 14.17/8.01 ( ( ord_less_eq_set_nat @ A @ bot_bot_set_nat )
% 14.17/8.01 = ( A = bot_bot_set_nat ) ) ).
% 14.17/8.01
% 14.17/8.01 thf(fact_882_insert__subset,axiom,
% 14.17/8.01 ! [X: produc3658429121746597890et_nat > $o,A: set_Pr4532377907799695533_nat_o,B4: set_Pr4532377907799695533_nat_o] :
% 14.17/8.01 ( ( ord_le2965882846123202637_nat_o @ ( insert5175938949040314269_nat_o @ X @ A ) @ B4 )
% 14.17/8.01 = ( ( member6576561426505652726_nat_o @ X @ B4 )
% 14.17/8.01 & ( ord_le2965882846123202637_nat_o @ A @ B4 ) ) ) ).
% 14.17/8.01
% 14.17/8.01 thf(fact_883_insert__subset,axiom,
% 14.17/8.01 ! [X: int,A: set_int,B4: set_int] :
% 14.17/8.01 ( ( ord_less_eq_set_int @ ( insert_int @ X @ A ) @ B4 )
% 14.17/8.01 = ( ( member_int @ X @ B4 )
% 14.17/8.01 & ( ord_less_eq_set_int @ A @ B4 ) ) ) ).
% 14.17/8.01
% 14.17/8.01 thf(fact_884_insert__subset,axiom,
% 14.17/8.01 ! [X: $o,A: set_o,B4: set_o] :
% 14.17/8.01 ( ( ord_less_eq_set_o @ ( insert_o @ X @ A ) @ B4 )
% 14.17/8.01 = ( ( member_o @ X @ B4 )
% 14.17/8.01 & ( ord_less_eq_set_o @ A @ B4 ) ) ) ).
% 14.17/8.01
% 14.17/8.01 thf(fact_885_insert__subset,axiom,
% 14.17/8.01 ! [X: product_unit,A: set_Product_unit,B4: set_Product_unit] :
% 14.17/8.01 ( ( ord_le3507040750410214029t_unit @ ( insert_Product_unit @ X @ A ) @ B4 )
% 14.17/8.01 = ( ( member_Product_unit @ X @ B4 )
% 14.17/8.01 & ( ord_le3507040750410214029t_unit @ A @ B4 ) ) ) ).
% 14.17/8.01
% 14.17/8.01 thf(fact_886_insert__subset,axiom,
% 14.17/8.01 ! [X: nat,A: set_nat,B4: set_nat] :
% 14.17/8.01 ( ( ord_less_eq_set_nat @ ( insert_nat @ X @ A ) @ B4 )
% 14.17/8.01 = ( ( member_nat @ X @ B4 )
% 14.17/8.01 & ( ord_less_eq_set_nat @ A @ B4 ) ) ) ).
% 14.17/8.01
% 14.17/8.01 thf(fact_887_SigmaI,axiom,
% 14.17/8.01 ! [A2: nat,A: set_nat,B: int,B4: nat > set_int] :
% 14.17/8.01 ( ( member_nat @ A2 @ A )
% 14.17/8.01 => ( ( member_int @ B @ ( B4 @ A2 ) )
% 14.17/8.01 => ( member4262671552274231302at_int @ ( product_Pair_nat_int @ A2 @ B ) @ ( produc454536836294682310at_int @ A @ B4 ) ) ) ) ).
% 14.17/8.01
% 14.17/8.01 thf(fact_888_SigmaI,axiom,
% 14.17/8.01 ! [A2: nat,A: set_nat,B: $o,B4: nat > set_o] :
% 14.17/8.01 ( ( member_nat @ A2 @ A )
% 14.17/8.01 => ( ( member_o @ B @ ( B4 @ A2 ) )
% 14.17/8.01 => ( member6310962623043647828_nat_o @ ( product_Pair_nat_o @ A2 @ B ) @ ( product_Sigma_nat_o @ A @ B4 ) ) ) ) ).
% 14.17/8.01
% 14.17/8.01 thf(fact_889_SigmaI,axiom,
% 14.17/8.01 ! [A2: nat,A: set_nat,B: product_unit,B4: nat > set_Product_unit] :
% 14.17/8.01 ( ( member_nat @ A2 @ A )
% 14.17/8.01 => ( ( member_Product_unit @ B @ ( B4 @ A2 ) )
% 14.17/8.01 => ( member348563419383025609t_unit @ ( produc3434450474625521892t_unit @ A2 @ B ) @ ( produc5506539147108004339t_unit @ A @ B4 ) ) ) ) ).
% 14.17/8.01
% 14.17/8.01 thf(fact_890_SigmaI,axiom,
% 14.17/8.01 ! [A2: int,A: set_int,B: nat,B4: int > set_nat] :
% 14.17/8.01 ( ( member_int @ A2 @ A )
% 14.17/8.01 => ( ( member_nat @ B @ ( B4 @ A2 ) )
% 14.17/8.01 => ( member216504246829706758nt_nat @ ( product_Pair_int_nat @ A2 @ B ) @ ( produc1456381018704787142nt_nat @ A @ B4 ) ) ) ) ).
% 14.17/8.01
% 14.17/8.01 thf(fact_891_SigmaI,axiom,
% 14.17/8.01 ! [A2: int,A: set_int,B: $o,B4: int > set_o] :
% 14.17/8.01 ( ( member_int @ A2 @ A )
% 14.17/8.01 => ( ( member_o @ B @ ( B4 @ A2 ) )
% 14.17/8.01 => ( member4489920277610959864_int_o @ ( product_Pair_int_o @ A2 @ B ) @ ( product_Sigma_int_o @ A @ B4 ) ) ) ) ).
% 14.17/8.01
% 14.17/8.01 thf(fact_892_SigmaI,axiom,
% 14.17/8.01 ! [A2: int,A: set_int,B: product_unit,B4: int > set_Product_unit] :
% 14.17/8.01 ( ( member_int @ A2 @ A )
% 14.17/8.01 => ( ( member_Product_unit @ B @ ( B4 @ A2 ) )
% 14.17/8.01 => ( member6688357403439981677t_unit @ ( produc550872421827702152t_unit @ A2 @ B ) @ ( produc2622961094310184599t_unit @ A @ B4 ) ) ) ) ).
% 14.17/8.01
% 14.17/8.01 thf(fact_893_SigmaI,axiom,
% 14.17/8.01 ! [A2: $o,A: set_o,B: nat,B4: $o > set_nat] :
% 14.17/8.01 ( ( member_o @ A2 @ A )
% 14.17/8.01 => ( ( member_nat @ B @ ( B4 @ A2 ) )
% 14.17/8.01 => ( member2802428098988154798_o_nat @ ( product_Pair_o_nat @ A2 @ B ) @ ( product_Sigma_o_nat @ A @ B4 ) ) ) ) ).
% 14.17/8.01
% 14.17/8.01 thf(fact_894_SigmaI,axiom,
% 14.17/8.01 ! [A2: $o,A: set_o,B: int,B4: $o > set_int] :
% 14.17/8.01 ( ( member_o @ A2 @ A )
% 14.17/8.01 => ( ( member_int @ B @ ( B4 @ A2 ) )
% 14.17/8.01 => ( member7847949116333733898_o_int @ ( product_Pair_o_int @ A2 @ B ) @ ( product_Sigma_o_int @ A @ B4 ) ) ) ) ).
% 14.17/8.01
% 14.17/8.01 thf(fact_895_SigmaI,axiom,
% 14.17/8.01 ! [A2: $o,A: set_o,B: $o,B4: $o > set_o] :
% 14.17/8.01 ( ( member_o @ A2 @ A )
% 14.17/8.01 => ( ( member_o @ B @ ( B4 @ A2 ) )
% 14.17/8.01 => ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ A2 @ B ) @ ( product_Sigma_o_o @ A @ B4 ) ) ) ) ).
% 14.17/8.01
% 14.17/8.01 thf(fact_896_SigmaI,axiom,
% 14.17/8.01 ! [A2: $o,A: set_o,B: product_unit,B4: $o > set_Product_unit] :
% 14.17/8.01 ( ( member_o @ A2 @ A )
% 14.17/8.01 => ( ( member_Product_unit @ B @ ( B4 @ A2 ) )
% 14.17/8.01 => ( member9215336023991391429t_unit @ ( produc3901946521647900940t_unit @ A2 @ B ) @ ( produc411009820408249917t_unit @ A @ B4 ) ) ) ) ).
% 14.17/8.01
% 14.17/8.01 thf(fact_897_mem__Sigma__iff,axiom,
% 14.17/8.01 ! [A2: nat,B: int,A: set_nat,B4: nat > set_int] :
% 14.17/8.01 ( ( member4262671552274231302at_int @ ( product_Pair_nat_int @ A2 @ B ) @ ( produc454536836294682310at_int @ A @ B4 ) )
% 14.17/8.01 = ( ( member_nat @ A2 @ A )
% 14.17/8.01 & ( member_int @ B @ ( B4 @ A2 ) ) ) ) ).
% 14.17/8.01
% 14.17/8.01 thf(fact_898_mem__Sigma__iff,axiom,
% 14.17/8.01 ! [A2: nat,B: $o,A: set_nat,B4: nat > set_o] :
% 14.17/8.01 ( ( member6310962623043647828_nat_o @ ( product_Pair_nat_o @ A2 @ B ) @ ( product_Sigma_nat_o @ A @ B4 ) )
% 14.17/8.01 = ( ( member_nat @ A2 @ A )
% 14.17/8.01 & ( member_o @ B @ ( B4 @ A2 ) ) ) ) ).
% 14.17/8.01
% 14.17/8.01 thf(fact_899_mem__Sigma__iff,axiom,
% 14.17/8.01 ! [A2: nat,B: product_unit,A: set_nat,B4: nat > set_Product_unit] :
% 14.17/8.01 ( ( member348563419383025609t_unit @ ( produc3434450474625521892t_unit @ A2 @ B ) @ ( produc5506539147108004339t_unit @ A @ B4 ) )
% 14.17/8.01 = ( ( member_nat @ A2 @ A )
% 14.17/8.01 & ( member_Product_unit @ B @ ( B4 @ A2 ) ) ) ) ).
% 14.17/8.01
% 14.17/8.01 thf(fact_900_mem__Sigma__iff,axiom,
% 14.17/8.01 ! [A2: int,B: nat,A: set_int,B4: int > set_nat] :
% 14.17/8.01 ( ( member216504246829706758nt_nat @ ( product_Pair_int_nat @ A2 @ B ) @ ( produc1456381018704787142nt_nat @ A @ B4 ) )
% 14.17/8.01 = ( ( member_int @ A2 @ A )
% 14.17/8.01 & ( member_nat @ B @ ( B4 @ A2 ) ) ) ) ).
% 14.17/8.01
% 14.17/8.01 thf(fact_901_mem__Sigma__iff,axiom,
% 14.17/8.01 ! [A2: int,B: $o,A: set_int,B4: int > set_o] :
% 14.17/8.01 ( ( member4489920277610959864_int_o @ ( product_Pair_int_o @ A2 @ B ) @ ( product_Sigma_int_o @ A @ B4 ) )
% 14.17/8.01 = ( ( member_int @ A2 @ A )
% 14.17/8.01 & ( member_o @ B @ ( B4 @ A2 ) ) ) ) ).
% 14.17/8.01
% 14.17/8.01 thf(fact_902_mem__Sigma__iff,axiom,
% 14.17/8.01 ! [A2: int,B: product_unit,A: set_int,B4: int > set_Product_unit] :
% 14.17/8.01 ( ( member6688357403439981677t_unit @ ( produc550872421827702152t_unit @ A2 @ B ) @ ( produc2622961094310184599t_unit @ A @ B4 ) )
% 14.17/8.01 = ( ( member_int @ A2 @ A )
% 14.17/8.01 & ( member_Product_unit @ B @ ( B4 @ A2 ) ) ) ) ).
% 14.17/8.01
% 14.17/8.01 thf(fact_903_mem__Sigma__iff,axiom,
% 14.17/8.01 ! [A2: $o,B: nat,A: set_o,B4: $o > set_nat] :
% 14.17/8.01 ( ( member2802428098988154798_o_nat @ ( product_Pair_o_nat @ A2 @ B ) @ ( product_Sigma_o_nat @ A @ B4 ) )
% 14.17/8.01 = ( ( member_o @ A2 @ A )
% 14.17/8.01 & ( member_nat @ B @ ( B4 @ A2 ) ) ) ) ).
% 14.17/8.01
% 14.17/8.01 thf(fact_904_mem__Sigma__iff,axiom,
% 14.17/8.01 ! [A2: $o,B: int,A: set_o,B4: $o > set_int] :
% 14.17/8.01 ( ( member7847949116333733898_o_int @ ( product_Pair_o_int @ A2 @ B ) @ ( product_Sigma_o_int @ A @ B4 ) )
% 14.17/8.01 = ( ( member_o @ A2 @ A )
% 14.17/8.01 & ( member_int @ B @ ( B4 @ A2 ) ) ) ) ).
% 14.17/8.01
% 14.17/8.01 thf(fact_905_mem__Sigma__iff,axiom,
% 14.17/8.01 ! [A2: $o,B: $o,A: set_o,B4: $o > set_o] :
% 14.17/8.01 ( ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ A2 @ B ) @ ( product_Sigma_o_o @ A @ B4 ) )
% 14.17/8.01 = ( ( member_o @ A2 @ A )
% 14.17/8.01 & ( member_o @ B @ ( B4 @ A2 ) ) ) ) ).
% 14.17/8.01
% 14.17/8.01 thf(fact_906_mem__Sigma__iff,axiom,
% 14.17/8.01 ! [A2: $o,B: product_unit,A: set_o,B4: $o > set_Product_unit] :
% 14.17/8.01 ( ( member9215336023991391429t_unit @ ( produc3901946521647900940t_unit @ A2 @ B ) @ ( produc411009820408249917t_unit @ A @ B4 ) )
% 14.17/8.01 = ( ( member_o @ A2 @ A )
% 14.17/8.01 & ( member_Product_unit @ B @ ( B4 @ A2 ) ) ) ) ).
% 14.17/8.01
% 14.17/8.01 thf(fact_907_Un__empty,axiom,
% 14.17/8.01 ! [A: set_Product_unit,B4: set_Product_unit] :
% 14.17/8.01 ( ( ( sup_su793286257634532545t_unit @ A @ B4 )
% 14.17/8.01 = bot_bo3957492148770167129t_unit )
% 14.17/8.01 = ( ( A = bot_bo3957492148770167129t_unit )
% 14.17/8.01 & ( B4 = bot_bo3957492148770167129t_unit ) ) ) ).
% 14.17/8.01
% 14.17/8.01 thf(fact_908_Un__empty,axiom,
% 14.17/8.01 ! [A: set_o,B4: set_o] :
% 14.17/8.01 ( ( ( sup_sup_set_o @ A @ B4 )
% 14.17/8.01 = bot_bot_set_o )
% 14.17/8.01 = ( ( A = bot_bot_set_o )
% 14.17/8.01 & ( B4 = bot_bot_set_o ) ) ) ).
% 14.17/8.01
% 14.17/8.01 thf(fact_909_Un__empty,axiom,
% 14.17/8.01 ! [A: set_nat,B4: set_nat] :
% 14.17/8.01 ( ( ( sup_sup_set_nat @ A @ B4 )
% 14.17/8.01 = bot_bot_set_nat )
% 14.17/8.01 = ( ( A = bot_bot_set_nat )
% 14.17/8.01 & ( B4 = bot_bot_set_nat ) ) ) ).
% 14.17/8.01
% 14.17/8.01 thf(fact_910_Un__empty,axiom,
% 14.17/8.01 ! [A: set_int,B4: set_int] :
% 14.17/8.01 ( ( ( sup_sup_set_int @ A @ B4 )
% 14.17/8.01 = bot_bot_set_int )
% 14.17/8.01 = ( ( A = bot_bot_set_int )
% 14.17/8.01 & ( B4 = bot_bot_set_int ) ) ) ).
% 14.17/8.01
% 14.17/8.01 thf(fact_911_Un__empty,axiom,
% 14.17/8.01 ! [A: set_Pr4532377907799695533_nat_o,B4: set_Pr4532377907799695533_nat_o] :
% 14.17/8.01 ( ( ( sup_su5209123915105501825_nat_o @ A @ B4 )
% 14.17/8.01 = bot_bo7824918357723582233_nat_o )
% 14.17/8.01 = ( ( A = bot_bo7824918357723582233_nat_o )
% 14.17/8.01 & ( B4 = bot_bo7824918357723582233_nat_o ) ) ) ).
% 14.17/8.01
% 14.17/8.01 thf(fact_912_Int__subset__iff,axiom,
% 14.17/8.01 ! [C5: set_nat,A: set_nat,B4: set_nat] :
% 14.17/8.01 ( ( ord_less_eq_set_nat @ C5 @ ( inf_inf_set_nat @ A @ B4 ) )
% 14.17/8.01 = ( ( ord_less_eq_set_nat @ C5 @ A )
% 14.17/8.01 & ( ord_less_eq_set_nat @ C5 @ B4 ) ) ) ).
% 14.17/8.01
% 14.17/8.01 thf(fact_913_Int__insert__right__if1,axiom,
% 14.17/8.01 ! [A2: produc3658429121746597890et_nat > $o,A: set_Pr4532377907799695533_nat_o,B4: set_Pr4532377907799695533_nat_o] :
% 14.17/8.01 ( ( member6576561426505652726_nat_o @ A2 @ A )
% 14.17/8.01 => ( ( inf_in1906310914598751387_nat_o @ A @ ( insert5175938949040314269_nat_o @ A2 @ B4 ) )
% 14.17/8.01 = ( insert5175938949040314269_nat_o @ A2 @ ( inf_in1906310914598751387_nat_o @ A @ B4 ) ) ) ) ).
% 14.17/8.01
% 14.17/8.01 thf(fact_914_Int__insert__right__if1,axiom,
% 14.17/8.01 ! [A2: int,A: set_int,B4: set_int] :
% 14.17/8.01 ( ( member_int @ A2 @ A )
% 14.17/8.01 => ( ( inf_inf_set_int @ A @ ( insert_int @ A2 @ B4 ) )
% 14.17/8.01 = ( insert_int @ A2 @ ( inf_inf_set_int @ A @ B4 ) ) ) ) ).
% 14.17/8.01
% 14.17/8.01 thf(fact_915_Int__insert__right__if1,axiom,
% 14.17/8.01 ! [A2: $o,A: set_o,B4: set_o] :
% 14.17/8.01 ( ( member_o @ A2 @ A )
% 14.17/8.01 => ( ( inf_inf_set_o @ A @ ( insert_o @ A2 @ B4 ) )
% 14.17/8.01 = ( insert_o @ A2 @ ( inf_inf_set_o @ A @ B4 ) ) ) ) ).
% 14.17/8.01
% 14.17/8.01 thf(fact_916_Int__insert__right__if1,axiom,
% 14.17/8.01 ! [A2: product_unit,A: set_Product_unit,B4: set_Product_unit] :
% 14.17/8.01 ( ( member_Product_unit @ A2 @ A )
% 14.17/8.01 => ( ( inf_in4660618365625256667t_unit @ A @ ( insert_Product_unit @ A2 @ B4 ) )
% 14.17/8.01 = ( insert_Product_unit @ A2 @ ( inf_in4660618365625256667t_unit @ A @ B4 ) ) ) ) ).
% 14.17/8.01
% 14.17/8.01 thf(fact_917_Int__insert__right__if1,axiom,
% 14.17/8.01 ! [A2: nat,A: set_nat,B4: set_nat] :
% 14.17/8.01 ( ( member_nat @ A2 @ A )
% 14.17/8.01 => ( ( inf_inf_set_nat @ A @ ( insert_nat @ A2 @ B4 ) )
% 14.17/8.01 = ( insert_nat @ A2 @ ( inf_inf_set_nat @ A @ B4 ) ) ) ) ).
% 14.17/8.01
% 14.17/8.01 thf(fact_918_Int__insert__right__if0,axiom,
% 14.17/8.01 ! [A2: produc3658429121746597890et_nat > $o,A: set_Pr4532377907799695533_nat_o,B4: set_Pr4532377907799695533_nat_o] :
% 14.17/8.01 ( ~ ( member6576561426505652726_nat_o @ A2 @ A )
% 14.17/8.01 => ( ( inf_in1906310914598751387_nat_o @ A @ ( insert5175938949040314269_nat_o @ A2 @ B4 ) )
% 14.17/8.01 = ( inf_in1906310914598751387_nat_o @ A @ B4 ) ) ) ).
% 14.17/8.01
% 14.17/8.01 thf(fact_919_Int__insert__right__if0,axiom,
% 14.17/8.01 ! [A2: int,A: set_int,B4: set_int] :
% 14.17/8.01 ( ~ ( member_int @ A2 @ A )
% 14.17/8.01 => ( ( inf_inf_set_int @ A @ ( insert_int @ A2 @ B4 ) )
% 14.17/8.01 = ( inf_inf_set_int @ A @ B4 ) ) ) ).
% 14.17/8.01
% 14.17/8.01 thf(fact_920_Int__insert__right__if0,axiom,
% 14.17/8.01 ! [A2: $o,A: set_o,B4: set_o] :
% 14.17/8.01 ( ~ ( member_o @ A2 @ A )
% 14.17/8.01 => ( ( inf_inf_set_o @ A @ ( insert_o @ A2 @ B4 ) )
% 14.17/8.01 = ( inf_inf_set_o @ A @ B4 ) ) ) ).
% 14.17/8.01
% 14.17/8.01 thf(fact_921_Int__insert__right__if0,axiom,
% 14.17/8.01 ! [A2: product_unit,A: set_Product_unit,B4: set_Product_unit] :
% 14.17/8.01 ( ~ ( member_Product_unit @ A2 @ A )
% 14.17/8.01 => ( ( inf_in4660618365625256667t_unit @ A @ ( insert_Product_unit @ A2 @ B4 ) )
% 14.17/8.01 = ( inf_in4660618365625256667t_unit @ A @ B4 ) ) ) ).
% 14.17/8.01
% 14.17/8.01 thf(fact_922_Int__insert__right__if0,axiom,
% 14.17/8.01 ! [A2: nat,A: set_nat,B4: set_nat] :
% 14.17/8.01 ( ~ ( member_nat @ A2 @ A )
% 14.17/8.01 => ( ( inf_inf_set_nat @ A @ ( insert_nat @ A2 @ B4 ) )
% 14.17/8.01 = ( inf_inf_set_nat @ A @ B4 ) ) ) ).
% 14.17/8.01
% 14.17/8.01 thf(fact_923_insert__inter__insert,axiom,
% 14.17/8.01 ! [A2: int,A: set_int,B4: set_int] :
% 14.17/8.01 ( ( inf_inf_set_int @ ( insert_int @ A2 @ A ) @ ( insert_int @ A2 @ B4 ) )
% 14.17/8.01 = ( insert_int @ A2 @ ( inf_inf_set_int @ A @ B4 ) ) ) ).
% 14.17/8.01
% 14.17/8.01 thf(fact_924_insert__inter__insert,axiom,
% 14.17/8.01 ! [A2: $o,A: set_o,B4: set_o] :
% 14.17/8.01 ( ( inf_inf_set_o @ ( insert_o @ A2 @ A ) @ ( insert_o @ A2 @ B4 ) )
% 14.17/8.01 = ( insert_o @ A2 @ ( inf_inf_set_o @ A @ B4 ) ) ) ).
% 14.17/8.01
% 14.17/8.01 thf(fact_925_insert__inter__insert,axiom,
% 14.17/8.01 ! [A2: product_unit,A: set_Product_unit,B4: set_Product_unit] :
% 14.17/8.01 ( ( inf_in4660618365625256667t_unit @ ( insert_Product_unit @ A2 @ A ) @ ( insert_Product_unit @ A2 @ B4 ) )
% 14.17/8.01 = ( insert_Product_unit @ A2 @ ( inf_in4660618365625256667t_unit @ A @ B4 ) ) ) ).
% 14.17/8.01
% 14.17/8.01 thf(fact_926_insert__inter__insert,axiom,
% 14.17/8.01 ! [A2: nat,A: set_nat,B4: set_nat] :
% 14.17/8.01 ( ( inf_inf_set_nat @ ( insert_nat @ A2 @ A ) @ ( insert_nat @ A2 @ B4 ) )
% 14.17/8.01 = ( insert_nat @ A2 @ ( inf_inf_set_nat @ A @ B4 ) ) ) ).
% 14.17/8.01
% 14.17/8.01 thf(fact_927_Int__insert__left__if1,axiom,
% 14.17/8.01 ! [A2: produc3658429121746597890et_nat > $o,C5: set_Pr4532377907799695533_nat_o,B4: set_Pr4532377907799695533_nat_o] :
% 14.17/8.01 ( ( member6576561426505652726_nat_o @ A2 @ C5 )
% 14.17/8.01 => ( ( inf_in1906310914598751387_nat_o @ ( insert5175938949040314269_nat_o @ A2 @ B4 ) @ C5 )
% 14.17/8.01 = ( insert5175938949040314269_nat_o @ A2 @ ( inf_in1906310914598751387_nat_o @ B4 @ C5 ) ) ) ) ).
% 14.17/8.01
% 14.17/8.01 thf(fact_928_Int__insert__left__if1,axiom,
% 14.17/8.01 ! [A2: int,C5: set_int,B4: set_int] :
% 14.17/8.01 ( ( member_int @ A2 @ C5 )
% 14.17/8.01 => ( ( inf_inf_set_int @ ( insert_int @ A2 @ B4 ) @ C5 )
% 14.17/8.01 = ( insert_int @ A2 @ ( inf_inf_set_int @ B4 @ C5 ) ) ) ) ).
% 14.17/8.01
% 14.17/8.01 thf(fact_929_Int__insert__left__if1,axiom,
% 14.17/8.01 ! [A2: $o,C5: set_o,B4: set_o] :
% 14.17/8.01 ( ( member_o @ A2 @ C5 )
% 14.17/8.01 => ( ( inf_inf_set_o @ ( insert_o @ A2 @ B4 ) @ C5 )
% 14.17/8.01 = ( insert_o @ A2 @ ( inf_inf_set_o @ B4 @ C5 ) ) ) ) ).
% 14.17/8.01
% 14.17/8.01 thf(fact_930_Int__insert__left__if1,axiom,
% 14.17/8.01 ! [A2: product_unit,C5: set_Product_unit,B4: set_Product_unit] :
% 14.17/8.01 ( ( member_Product_unit @ A2 @ C5 )
% 14.17/8.01 => ( ( inf_in4660618365625256667t_unit @ ( insert_Product_unit @ A2 @ B4 ) @ C5 )
% 14.17/8.01 = ( insert_Product_unit @ A2 @ ( inf_in4660618365625256667t_unit @ B4 @ C5 ) ) ) ) ).
% 14.17/8.01
% 14.17/8.01 thf(fact_931_Int__insert__left__if1,axiom,
% 14.17/8.01 ! [A2: nat,C5: set_nat,B4: set_nat] :
% 14.17/8.01 ( ( member_nat @ A2 @ C5 )
% 14.17/8.01 => ( ( inf_inf_set_nat @ ( insert_nat @ A2 @ B4 ) @ C5 )
% 14.17/8.01 = ( insert_nat @ A2 @ ( inf_inf_set_nat @ B4 @ C5 ) ) ) ) ).
% 14.17/8.01
% 14.17/8.01 thf(fact_932_Int__insert__left__if0,axiom,
% 14.17/8.01 ! [A2: produc3658429121746597890et_nat > $o,C5: set_Pr4532377907799695533_nat_o,B4: set_Pr4532377907799695533_nat_o] :
% 14.17/8.01 ( ~ ( member6576561426505652726_nat_o @ A2 @ C5 )
% 14.17/8.01 => ( ( inf_in1906310914598751387_nat_o @ ( insert5175938949040314269_nat_o @ A2 @ B4 ) @ C5 )
% 14.17/8.01 = ( inf_in1906310914598751387_nat_o @ B4 @ C5 ) ) ) ).
% 14.17/8.01
% 14.17/8.01 thf(fact_933_Int__insert__left__if0,axiom,
% 14.17/8.01 ! [A2: int,C5: set_int,B4: set_int] :
% 14.17/8.01 ( ~ ( member_int @ A2 @ C5 )
% 14.17/8.01 => ( ( inf_inf_set_int @ ( insert_int @ A2 @ B4 ) @ C5 )
% 14.17/8.01 = ( inf_inf_set_int @ B4 @ C5 ) ) ) ).
% 14.17/8.01
% 14.17/8.01 thf(fact_934_Int__insert__left__if0,axiom,
% 14.17/8.01 ! [A2: $o,C5: set_o,B4: set_o] :
% 14.17/8.01 ( ~ ( member_o @ A2 @ C5 )
% 14.17/8.01 => ( ( inf_inf_set_o @ ( insert_o @ A2 @ B4 ) @ C5 )
% 14.17/8.01 = ( inf_inf_set_o @ B4 @ C5 ) ) ) ).
% 14.17/8.01
% 14.17/8.01 thf(fact_935_Int__insert__left__if0,axiom,
% 14.17/8.01 ! [A2: product_unit,C5: set_Product_unit,B4: set_Product_unit] :
% 14.17/8.01 ( ~ ( member_Product_unit @ A2 @ C5 )
% 14.17/8.01 => ( ( inf_in4660618365625256667t_unit @ ( insert_Product_unit @ A2 @ B4 ) @ C5 )
% 14.17/8.01 = ( inf_in4660618365625256667t_unit @ B4 @ C5 ) ) ) ).
% 14.17/8.01
% 14.17/8.01 thf(fact_936_Int__insert__left__if0,axiom,
% 14.17/8.01 ! [A2: nat,C5: set_nat,B4: set_nat] :
% 14.17/8.01 ( ~ ( member_nat @ A2 @ C5 )
% 14.17/8.01 => ( ( inf_inf_set_nat @ ( insert_nat @ A2 @ B4 ) @ C5 )
% 14.17/8.01 = ( inf_inf_set_nat @ B4 @ C5 ) ) ) ).
% 14.17/8.01
% 14.17/8.01 thf(fact_937_Un__subset__iff,axiom,
% 14.17/8.01 ! [A: set_nat,B4: set_nat,C5: set_nat] :
% 14.17/8.01 ( ( ord_less_eq_set_nat @ ( sup_sup_set_nat @ A @ B4 ) @ C5 )
% 14.17/8.01 = ( ( ord_less_eq_set_nat @ A @ C5 )
% 14.17/8.01 & ( ord_less_eq_set_nat @ B4 @ C5 ) ) ) ).
% 14.17/8.01
% 14.17/8.01 thf(fact_938_Un__insert__right,axiom,
% 14.17/8.01 ! [A: set_int,A2: int,B4: set_int] :
% 14.17/8.01 ( ( sup_sup_set_int @ A @ ( insert_int @ A2 @ B4 ) )
% 14.17/8.01 = ( insert_int @ A2 @ ( sup_sup_set_int @ A @ B4 ) ) ) ).
% 14.17/8.01
% 14.17/8.01 thf(fact_939_Un__insert__right,axiom,
% 14.17/8.01 ! [A: set_o,A2: $o,B4: set_o] :
% 14.17/8.01 ( ( sup_sup_set_o @ A @ ( insert_o @ A2 @ B4 ) )
% 14.17/8.01 = ( insert_o @ A2 @ ( sup_sup_set_o @ A @ B4 ) ) ) ).
% 14.17/8.01
% 14.17/8.01 thf(fact_940_Un__insert__right,axiom,
% 14.17/8.01 ! [A: set_Product_unit,A2: product_unit,B4: set_Product_unit] :
% 14.17/8.01 ( ( sup_su793286257634532545t_unit @ A @ ( insert_Product_unit @ A2 @ B4 ) )
% 14.17/8.01 = ( insert_Product_unit @ A2 @ ( sup_su793286257634532545t_unit @ A @ B4 ) ) ) ).
% 14.17/8.01
% 14.17/8.01 thf(fact_941_Un__insert__right,axiom,
% 14.17/8.01 ! [A: set_nat,A2: nat,B4: set_nat] :
% 14.17/8.01 ( ( sup_sup_set_nat @ A @ ( insert_nat @ A2 @ B4 ) )
% 14.17/8.01 = ( insert_nat @ A2 @ ( sup_sup_set_nat @ A @ B4 ) ) ) ).
% 14.17/8.01
% 14.17/8.01 thf(fact_942_Un__insert__left,axiom,
% 14.17/8.01 ! [A2: int,B4: set_int,C5: set_int] :
% 14.17/8.01 ( ( sup_sup_set_int @ ( insert_int @ A2 @ B4 ) @ C5 )
% 14.17/8.01 = ( insert_int @ A2 @ ( sup_sup_set_int @ B4 @ C5 ) ) ) ).
% 14.17/8.01
% 14.17/8.01 thf(fact_943_Un__insert__left,axiom,
% 14.17/8.01 ! [A2: $o,B4: set_o,C5: set_o] :
% 14.17/8.01 ( ( sup_sup_set_o @ ( insert_o @ A2 @ B4 ) @ C5 )
% 14.17/8.01 = ( insert_o @ A2 @ ( sup_sup_set_o @ B4 @ C5 ) ) ) ).
% 14.17/8.01
% 14.17/8.01 thf(fact_944_Un__insert__left,axiom,
% 14.17/8.01 ! [A2: product_unit,B4: set_Product_unit,C5: set_Product_unit] :
% 14.17/8.01 ( ( sup_su793286257634532545t_unit @ ( insert_Product_unit @ A2 @ B4 ) @ C5 )
% 14.17/8.01 = ( insert_Product_unit @ A2 @ ( sup_su793286257634532545t_unit @ B4 @ C5 ) ) ) ).
% 14.17/8.01
% 14.17/8.01 thf(fact_945_Un__insert__left,axiom,
% 14.17/8.01 ! [A2: nat,B4: set_nat,C5: set_nat] :
% 14.17/8.01 ( ( sup_sup_set_nat @ ( insert_nat @ A2 @ B4 ) @ C5 )
% 14.17/8.01 = ( insert_nat @ A2 @ ( sup_sup_set_nat @ B4 @ C5 ) ) ) ).
% 14.17/8.01
% 14.17/8.01 thf(fact_946_Un__Int__eq_I1_J,axiom,
% 14.17/8.01 ! [S: set_nat,T2: set_nat] :
% 14.17/8.01 ( ( inf_inf_set_nat @ ( sup_sup_set_nat @ S @ T2 ) @ S )
% 14.17/8.01 = S ) ).
% 14.17/8.01
% 14.17/8.01 thf(fact_947_Un__Int__eq_I2_J,axiom,
% 14.17/8.01 ! [S: set_nat,T2: set_nat] :
% 14.17/8.01 ( ( inf_inf_set_nat @ ( sup_sup_set_nat @ S @ T2 ) @ T2 )
% 14.17/8.01 = T2 ) ).
% 14.17/8.01
% 14.17/8.01 thf(fact_948_Un__Int__eq_I3_J,axiom,
% 14.17/8.01 ! [S: set_nat,T2: set_nat] :
% 14.17/8.01 ( ( inf_inf_set_nat @ S @ ( sup_sup_set_nat @ S @ T2 ) )
% 14.17/8.01 = S ) ).
% 14.17/8.01
% 14.17/8.01 thf(fact_949_Un__Int__eq_I4_J,axiom,
% 14.17/8.01 ! [T2: set_nat,S: set_nat] :
% 14.17/8.01 ( ( inf_inf_set_nat @ T2 @ ( sup_sup_set_nat @ S @ T2 ) )
% 14.17/8.01 = T2 ) ).
% 14.17/8.01
% 14.17/8.01 thf(fact_950_Int__Un__eq_I1_J,axiom,
% 14.17/8.01 ! [S: set_nat,T2: set_nat] :
% 14.17/8.01 ( ( sup_sup_set_nat @ ( inf_inf_set_nat @ S @ T2 ) @ S )
% 14.17/8.01 = S ) ).
% 14.17/8.01
% 14.17/8.01 thf(fact_951_Int__Un__eq_I2_J,axiom,
% 14.17/8.01 ! [S: set_nat,T2: set_nat] :
% 14.17/8.01 ( ( sup_sup_set_nat @ ( inf_inf_set_nat @ S @ T2 ) @ T2 )
% 14.17/8.01 = T2 ) ).
% 14.17/8.01
% 14.17/8.01 thf(fact_952_Int__Un__eq_I3_J,axiom,
% 14.17/8.01 ! [S: set_nat,T2: set_nat] :
% 14.17/8.01 ( ( sup_sup_set_nat @ S @ ( inf_inf_set_nat @ S @ T2 ) )
% 14.17/8.01 = S ) ).
% 14.17/8.01
% 14.17/8.01 thf(fact_953_Int__Un__eq_I4_J,axiom,
% 14.17/8.01 ! [T2: set_nat,S: set_nat] :
% 14.17/8.01 ( ( sup_sup_set_nat @ T2 @ ( inf_inf_set_nat @ S @ T2 ) )
% 14.17/8.01 = T2 ) ).
% 14.17/8.01
% 14.17/8.01 thf(fact_954_vimage__Int,axiom,
% 14.17/8.01 ! [F: nat > nat,A: set_nat,B4: set_nat] :
% 14.17/8.01 ( ( vimage_nat_nat @ F @ ( inf_inf_set_nat @ A @ B4 ) )
% 14.17/8.01 = ( inf_inf_set_nat @ ( vimage_nat_nat @ F @ A ) @ ( vimage_nat_nat @ F @ B4 ) ) ) ).
% 14.17/8.01
% 14.17/8.01 thf(fact_955_vimage__Un,axiom,
% 14.17/8.01 ! [F: nat > nat,A: set_nat,B4: set_nat] :
% 14.17/8.01 ( ( vimage_nat_nat @ F @ ( sup_sup_set_nat @ A @ B4 ) )
% 14.17/8.01 = ( sup_sup_set_nat @ ( vimage_nat_nat @ F @ A ) @ ( vimage_nat_nat @ F @ B4 ) ) ) ).
% 14.17/8.01
% 14.17/8.01 thf(fact_956_relH__dist__union,axiom,
% 14.17/8.01 ! [As: set_nat,As5: set_nat,H2: heap_e7401611519738050253t_unit,H3: heap_e7401611519738050253t_unit] :
% 14.17/8.01 ( ( relH @ ( sup_sup_set_nat @ As @ As5 ) @ H2 @ H3 )
% 14.17/8.01 = ( ( relH @ As @ H2 @ H3 )
% 14.17/8.01 & ( relH @ As5 @ H2 @ H3 ) ) ) ).
% 14.17/8.01
% 14.17/8.01 thf(fact_957_singleton__insert__inj__eq,axiom,
% 14.17/8.01 ! [B: product_unit,A2: product_unit,A: set_Product_unit] :
% 14.17/8.01 ( ( ( insert_Product_unit @ B @ bot_bo3957492148770167129t_unit )
% 14.17/8.01 = ( insert_Product_unit @ A2 @ A ) )
% 14.17/8.01 = ( ( A2 = B )
% 14.17/8.01 & ( ord_le3507040750410214029t_unit @ A @ ( insert_Product_unit @ B @ bot_bo3957492148770167129t_unit ) ) ) ) ).
% 14.17/8.01
% 14.17/8.01 thf(fact_958_singleton__insert__inj__eq,axiom,
% 14.17/8.01 ! [B: $o,A2: $o,A: set_o] :
% 14.17/8.01 ( ( ( insert_o @ B @ bot_bot_set_o )
% 14.17/8.01 = ( insert_o @ A2 @ A ) )
% 14.17/8.01 = ( ( A2 = B )
% 14.17/8.01 & ( ord_less_eq_set_o @ A @ ( insert_o @ B @ bot_bot_set_o ) ) ) ) ).
% 14.17/8.01
% 14.17/8.01 thf(fact_959_singleton__insert__inj__eq,axiom,
% 14.17/8.01 ! [B: int,A2: int,A: set_int] :
% 14.17/8.01 ( ( ( insert_int @ B @ bot_bot_set_int )
% 14.17/8.01 = ( insert_int @ A2 @ A ) )
% 14.17/8.01 = ( ( A2 = B )
% 14.17/8.01 & ( ord_less_eq_set_int @ A @ ( insert_int @ B @ bot_bot_set_int ) ) ) ) ).
% 14.17/8.01
% 14.17/8.01 thf(fact_960_singleton__insert__inj__eq,axiom,
% 14.17/8.01 ! [B: produc3658429121746597890et_nat > $o,A2: produc3658429121746597890et_nat > $o,A: set_Pr4532377907799695533_nat_o] :
% 14.17/8.01 ( ( ( insert5175938949040314269_nat_o @ B @ bot_bo7824918357723582233_nat_o )
% 14.17/8.01 = ( insert5175938949040314269_nat_o @ A2 @ A ) )
% 14.17/8.01 = ( ( A2 = B )
% 14.17/8.01 & ( ord_le2965882846123202637_nat_o @ A @ ( insert5175938949040314269_nat_o @ B @ bot_bo7824918357723582233_nat_o ) ) ) ) ).
% 14.17/8.01
% 14.17/8.01 thf(fact_961_singleton__insert__inj__eq,axiom,
% 14.17/8.01 ! [B: nat,A2: nat,A: set_nat] :
% 14.17/8.01 ( ( ( insert_nat @ B @ bot_bot_set_nat )
% 14.17/8.01 = ( insert_nat @ A2 @ A ) )
% 14.17/8.01 = ( ( A2 = B )
% 14.17/8.01 & ( ord_less_eq_set_nat @ A @ ( insert_nat @ B @ bot_bot_set_nat ) ) ) ) ).
% 14.17/8.01
% 14.17/8.01 thf(fact_962_singleton__insert__inj__eq_H,axiom,
% 14.17/8.01 ! [A2: product_unit,A: set_Product_unit,B: product_unit] :
% 14.17/8.01 ( ( ( insert_Product_unit @ A2 @ A )
% 14.17/8.01 = ( insert_Product_unit @ B @ bot_bo3957492148770167129t_unit ) )
% 14.17/8.01 = ( ( A2 = B )
% 14.17/8.01 & ( ord_le3507040750410214029t_unit @ A @ ( insert_Product_unit @ B @ bot_bo3957492148770167129t_unit ) ) ) ) ).
% 14.17/8.01
% 14.17/8.01 thf(fact_963_singleton__insert__inj__eq_H,axiom,
% 14.17/8.01 ! [A2: $o,A: set_o,B: $o] :
% 14.17/8.01 ( ( ( insert_o @ A2 @ A )
% 14.17/8.01 = ( insert_o @ B @ bot_bot_set_o ) )
% 14.17/8.01 = ( ( A2 = B )
% 14.17/8.01 & ( ord_less_eq_set_o @ A @ ( insert_o @ B @ bot_bot_set_o ) ) ) ) ).
% 14.17/8.01
% 14.17/8.01 thf(fact_964_singleton__insert__inj__eq_H,axiom,
% 14.17/8.01 ! [A2: int,A: set_int,B: int] :
% 14.17/8.01 ( ( ( insert_int @ A2 @ A )
% 14.17/8.01 = ( insert_int @ B @ bot_bot_set_int ) )
% 14.17/8.01 = ( ( A2 = B )
% 14.17/8.01 & ( ord_less_eq_set_int @ A @ ( insert_int @ B @ bot_bot_set_int ) ) ) ) ).
% 14.17/8.01
% 14.17/8.01 thf(fact_965_singleton__insert__inj__eq_H,axiom,
% 14.17/8.01 ! [A2: produc3658429121746597890et_nat > $o,A: set_Pr4532377907799695533_nat_o,B: produc3658429121746597890et_nat > $o] :
% 14.17/8.01 ( ( ( insert5175938949040314269_nat_o @ A2 @ A )
% 14.17/8.01 = ( insert5175938949040314269_nat_o @ B @ bot_bo7824918357723582233_nat_o ) )
% 14.17/8.01 = ( ( A2 = B )
% 14.17/8.01 & ( ord_le2965882846123202637_nat_o @ A @ ( insert5175938949040314269_nat_o @ B @ bot_bo7824918357723582233_nat_o ) ) ) ) ).
% 14.17/8.01
% 14.17/8.01 thf(fact_966_singleton__insert__inj__eq_H,axiom,
% 14.17/8.01 ! [A2: nat,A: set_nat,B: nat] :
% 14.17/8.01 ( ( ( insert_nat @ A2 @ A )
% 14.17/8.01 = ( insert_nat @ B @ bot_bot_set_nat ) )
% 14.17/8.01 = ( ( A2 = B )
% 14.17/8.01 & ( ord_less_eq_set_nat @ A @ ( insert_nat @ B @ bot_bot_set_nat ) ) ) ) ).
% 14.17/8.01
% 14.17/8.01 thf(fact_967_disjoint__insert_I2_J,axiom,
% 14.17/8.01 ! [A: set_Product_unit,B: product_unit,B4: set_Product_unit] :
% 14.17/8.01 ( ( bot_bo3957492148770167129t_unit
% 14.17/8.01 = ( inf_in4660618365625256667t_unit @ A @ ( insert_Product_unit @ B @ B4 ) ) )
% 14.17/8.01 = ( ~ ( member_Product_unit @ B @ A )
% 14.17/8.01 & ( bot_bo3957492148770167129t_unit
% 14.17/8.01 = ( inf_in4660618365625256667t_unit @ A @ B4 ) ) ) ) ).
% 14.17/8.01
% 14.17/8.01 thf(fact_968_disjoint__insert_I2_J,axiom,
% 14.17/8.01 ! [A: set_o,B: $o,B4: set_o] :
% 14.17/8.01 ( ( bot_bot_set_o
% 14.17/8.01 = ( inf_inf_set_o @ A @ ( insert_o @ B @ B4 ) ) )
% 14.17/8.01 = ( ~ ( member_o @ B @ A )
% 14.17/8.01 & ( bot_bot_set_o
% 14.17/8.01 = ( inf_inf_set_o @ A @ B4 ) ) ) ) ).
% 14.17/8.01
% 14.17/8.01 thf(fact_969_disjoint__insert_I2_J,axiom,
% 14.17/8.01 ! [A: set_nat,B: nat,B4: set_nat] :
% 14.17/8.01 ( ( bot_bot_set_nat
% 14.17/8.01 = ( inf_inf_set_nat @ A @ ( insert_nat @ B @ B4 ) ) )
% 14.17/8.01 = ( ~ ( member_nat @ B @ A )
% 14.17/8.01 & ( bot_bot_set_nat
% 14.17/8.01 = ( inf_inf_set_nat @ A @ B4 ) ) ) ) ).
% 14.17/8.01
% 14.17/8.01 thf(fact_970_disjoint__insert_I2_J,axiom,
% 14.17/8.01 ! [A: set_int,B: int,B4: set_int] :
% 14.17/8.01 ( ( bot_bot_set_int
% 14.17/8.01 = ( inf_inf_set_int @ A @ ( insert_int @ B @ B4 ) ) )
% 14.17/8.01 = ( ~ ( member_int @ B @ A )
% 14.17/8.01 & ( bot_bot_set_int
% 14.17/8.01 = ( inf_inf_set_int @ A @ B4 ) ) ) ) ).
% 14.17/8.01
% 14.17/8.01 thf(fact_971_disjoint__insert_I2_J,axiom,
% 14.17/8.01 ! [A: set_Pr4532377907799695533_nat_o,B: produc3658429121746597890et_nat > $o,B4: set_Pr4532377907799695533_nat_o] :
% 14.17/8.01 ( ( bot_bo7824918357723582233_nat_o
% 14.17/8.01 = ( inf_in1906310914598751387_nat_o @ A @ ( insert5175938949040314269_nat_o @ B @ B4 ) ) )
% 14.17/8.01 = ( ~ ( member6576561426505652726_nat_o @ B @ A )
% 14.17/8.01 & ( bot_bo7824918357723582233_nat_o
% 14.17/8.01 = ( inf_in1906310914598751387_nat_o @ A @ B4 ) ) ) ) ).
% 14.17/8.01
% 14.17/8.01 thf(fact_972_disjoint__insert_I1_J,axiom,
% 14.17/8.01 ! [B4: set_Product_unit,A2: product_unit,A: set_Product_unit] :
% 14.17/8.01 ( ( ( inf_in4660618365625256667t_unit @ B4 @ ( insert_Product_unit @ A2 @ A ) )
% 14.17/8.01 = bot_bo3957492148770167129t_unit )
% 14.17/8.01 = ( ~ ( member_Product_unit @ A2 @ B4 )
% 14.17/8.01 & ( ( inf_in4660618365625256667t_unit @ B4 @ A )
% 14.17/8.01 = bot_bo3957492148770167129t_unit ) ) ) ).
% 14.17/8.01
% 14.17/8.01 thf(fact_973_disjoint__insert_I1_J,axiom,
% 14.17/8.01 ! [B4: set_o,A2: $o,A: set_o] :
% 14.17/8.01 ( ( ( inf_inf_set_o @ B4 @ ( insert_o @ A2 @ A ) )
% 14.17/8.01 = bot_bot_set_o )
% 14.17/8.01 = ( ~ ( member_o @ A2 @ B4 )
% 14.17/8.01 & ( ( inf_inf_set_o @ B4 @ A )
% 14.17/8.01 = bot_bot_set_o ) ) ) ).
% 14.17/8.01
% 14.17/8.01 thf(fact_974_disjoint__insert_I1_J,axiom,
% 14.17/8.01 ! [B4: set_nat,A2: nat,A: set_nat] :
% 14.17/8.01 ( ( ( inf_inf_set_nat @ B4 @ ( insert_nat @ A2 @ A ) )
% 14.17/8.01 = bot_bot_set_nat )
% 14.17/8.01 = ( ~ ( member_nat @ A2 @ B4 )
% 14.17/8.01 & ( ( inf_inf_set_nat @ B4 @ A )
% 14.17/8.01 = bot_bot_set_nat ) ) ) ).
% 14.17/8.01
% 14.17/8.01 thf(fact_975_disjoint__insert_I1_J,axiom,
% 14.17/8.01 ! [B4: set_int,A2: int,A: set_int] :
% 14.17/8.01 ( ( ( inf_inf_set_int @ B4 @ ( insert_int @ A2 @ A ) )
% 14.17/8.01 = bot_bot_set_int )
% 14.17/8.01 = ( ~ ( member_int @ A2 @ B4 )
% 14.17/8.01 & ( ( inf_inf_set_int @ B4 @ A )
% 14.17/8.01 = bot_bot_set_int ) ) ) ).
% 14.17/8.01
% 14.17/8.01 thf(fact_976_disjoint__insert_I1_J,axiom,
% 14.17/8.01 ! [B4: set_Pr4532377907799695533_nat_o,A2: produc3658429121746597890et_nat > $o,A: set_Pr4532377907799695533_nat_o] :
% 14.17/8.01 ( ( ( inf_in1906310914598751387_nat_o @ B4 @ ( insert5175938949040314269_nat_o @ A2 @ A ) )
% 14.17/8.01 = bot_bo7824918357723582233_nat_o )
% 14.17/8.01 = ( ~ ( member6576561426505652726_nat_o @ A2 @ B4 )
% 14.17/8.01 & ( ( inf_in1906310914598751387_nat_o @ B4 @ A )
% 14.17/8.01 = bot_bo7824918357723582233_nat_o ) ) ) ).
% 14.17/8.01
% 14.17/8.01 thf(fact_977_insert__disjoint_I2_J,axiom,
% 14.17/8.01 ! [A2: product_unit,A: set_Product_unit,B4: set_Product_unit] :
% 14.17/8.01 ( ( bot_bo3957492148770167129t_unit
% 14.17/8.01 = ( inf_in4660618365625256667t_unit @ ( insert_Product_unit @ A2 @ A ) @ B4 ) )
% 14.17/8.01 = ( ~ ( member_Product_unit @ A2 @ B4 )
% 14.17/8.01 & ( bot_bo3957492148770167129t_unit
% 14.17/8.01 = ( inf_in4660618365625256667t_unit @ A @ B4 ) ) ) ) ).
% 14.17/8.01
% 14.17/8.01 thf(fact_978_insert__disjoint_I2_J,axiom,
% 14.17/8.01 ! [A2: $o,A: set_o,B4: set_o] :
% 14.17/8.01 ( ( bot_bot_set_o
% 14.17/8.01 = ( inf_inf_set_o @ ( insert_o @ A2 @ A ) @ B4 ) )
% 14.17/8.01 = ( ~ ( member_o @ A2 @ B4 )
% 14.17/8.01 & ( bot_bot_set_o
% 14.17/8.01 = ( inf_inf_set_o @ A @ B4 ) ) ) ) ).
% 14.17/8.01
% 14.17/8.01 thf(fact_979_insert__disjoint_I2_J,axiom,
% 14.17/8.01 ! [A2: nat,A: set_nat,B4: set_nat] :
% 14.17/8.01 ( ( bot_bot_set_nat
% 14.17/8.01 = ( inf_inf_set_nat @ ( insert_nat @ A2 @ A ) @ B4 ) )
% 14.17/8.01 = ( ~ ( member_nat @ A2 @ B4 )
% 14.17/8.01 & ( bot_bot_set_nat
% 14.17/8.01 = ( inf_inf_set_nat @ A @ B4 ) ) ) ) ).
% 14.17/8.01
% 14.17/8.01 thf(fact_980_insert__disjoint_I2_J,axiom,
% 14.17/8.01 ! [A2: int,A: set_int,B4: set_int] :
% 14.17/8.01 ( ( bot_bot_set_int
% 14.17/8.01 = ( inf_inf_set_int @ ( insert_int @ A2 @ A ) @ B4 ) )
% 14.17/8.01 = ( ~ ( member_int @ A2 @ B4 )
% 14.17/8.01 & ( bot_bot_set_int
% 14.17/8.01 = ( inf_inf_set_int @ A @ B4 ) ) ) ) ).
% 14.17/8.01
% 14.17/8.01 thf(fact_981_insert__disjoint_I2_J,axiom,
% 14.17/8.01 ! [A2: produc3658429121746597890et_nat > $o,A: set_Pr4532377907799695533_nat_o,B4: set_Pr4532377907799695533_nat_o] :
% 14.17/8.01 ( ( bot_bo7824918357723582233_nat_o
% 14.17/8.01 = ( inf_in1906310914598751387_nat_o @ ( insert5175938949040314269_nat_o @ A2 @ A ) @ B4 ) )
% 14.17/8.01 = ( ~ ( member6576561426505652726_nat_o @ A2 @ B4 )
% 14.17/8.01 & ( bot_bo7824918357723582233_nat_o
% 14.17/8.01 = ( inf_in1906310914598751387_nat_o @ A @ B4 ) ) ) ) ).
% 14.17/8.01
% 14.17/8.01 thf(fact_982_insert__disjoint_I1_J,axiom,
% 14.17/8.01 ! [A2: product_unit,A: set_Product_unit,B4: set_Product_unit] :
% 14.17/8.01 ( ( ( inf_in4660618365625256667t_unit @ ( insert_Product_unit @ A2 @ A ) @ B4 )
% 14.17/8.01 = bot_bo3957492148770167129t_unit )
% 14.17/8.01 = ( ~ ( member_Product_unit @ A2 @ B4 )
% 14.17/8.01 & ( ( inf_in4660618365625256667t_unit @ A @ B4 )
% 14.17/8.01 = bot_bo3957492148770167129t_unit ) ) ) ).
% 14.17/8.01
% 14.17/8.01 thf(fact_983_insert__disjoint_I1_J,axiom,
% 14.17/8.01 ! [A2: $o,A: set_o,B4: set_o] :
% 14.17/8.01 ( ( ( inf_inf_set_o @ ( insert_o @ A2 @ A ) @ B4 )
% 14.17/8.01 = bot_bot_set_o )
% 14.17/8.01 = ( ~ ( member_o @ A2 @ B4 )
% 14.17/8.01 & ( ( inf_inf_set_o @ A @ B4 )
% 14.17/8.01 = bot_bot_set_o ) ) ) ).
% 14.17/8.01
% 14.17/8.01 thf(fact_984_insert__disjoint_I1_J,axiom,
% 14.17/8.01 ! [A2: nat,A: set_nat,B4: set_nat] :
% 14.17/8.01 ( ( ( inf_inf_set_nat @ ( insert_nat @ A2 @ A ) @ B4 )
% 14.17/8.01 = bot_bot_set_nat )
% 14.17/8.01 = ( ~ ( member_nat @ A2 @ B4 )
% 14.17/8.01 & ( ( inf_inf_set_nat @ A @ B4 )
% 14.17/8.01 = bot_bot_set_nat ) ) ) ).
% 14.17/8.01
% 14.17/8.01 thf(fact_985_insert__disjoint_I1_J,axiom,
% 14.17/8.01 ! [A2: int,A: set_int,B4: set_int] :
% 14.17/8.01 ( ( ( inf_inf_set_int @ ( insert_int @ A2 @ A ) @ B4 )
% 14.17/8.01 = bot_bot_set_int )
% 14.17/8.01 = ( ~ ( member_int @ A2 @ B4 )
% 14.17/8.01 & ( ( inf_inf_set_int @ A @ B4 )
% 14.17/8.01 = bot_bot_set_int ) ) ) ).
% 14.17/8.01
% 14.17/8.01 thf(fact_986_insert__disjoint_I1_J,axiom,
% 14.17/8.01 ! [A2: produc3658429121746597890et_nat > $o,A: set_Pr4532377907799695533_nat_o,B4: set_Pr4532377907799695533_nat_o] :
% 14.17/8.01 ( ( ( inf_in1906310914598751387_nat_o @ ( insert5175938949040314269_nat_o @ A2 @ A ) @ B4 )
% 14.17/8.01 = bot_bo7824918357723582233_nat_o )
% 14.17/8.01 = ( ~ ( member6576561426505652726_nat_o @ A2 @ B4 )
% 14.17/8.01 & ( ( inf_in1906310914598751387_nat_o @ A @ B4 )
% 14.17/8.01 = bot_bo7824918357723582233_nat_o ) ) ) ).
% 14.17/8.01
% 14.17/8.01 thf(fact_987_Image__Id__on,axiom,
% 14.17/8.01 ! [A: set_nat,B4: set_nat] :
% 14.17/8.01 ( ( image_nat_nat @ ( id_on_nat @ A ) @ B4 )
% 14.17/8.01 = ( inf_inf_set_nat @ A @ B4 ) ) ).
% 14.17/8.01
% 14.17/8.01 thf(fact_988_in__range__dist__union,axiom,
% 14.17/8.01 ! [H2: heap_e7401611519738050253t_unit,As: set_nat,As5: set_nat] :
% 14.17/8.01 ( ( in_range @ ( produc7507926704131184380et_nat @ H2 @ ( sup_sup_set_nat @ As @ As5 ) ) )
% 14.17/8.01 = ( ( in_range @ ( produc7507926704131184380et_nat @ H2 @ As ) )
% 14.17/8.01 & ( in_range @ ( produc7507926704131184380et_nat @ H2 @ As5 ) ) ) ) ).
% 14.17/8.01
% 14.17/8.01 thf(fact_989_reachable__mono,axiom,
% 14.17/8.01 ! [R3: set_Pr1261947904930325089at_nat,R5: set_Pr1261947904930325089at_nat,X4: set_nat,X5: set_nat] :
% 14.17/8.01 ( ( ord_le3146513528884898305at_nat @ R3 @ R5 )
% 14.17/8.01 => ( ( ord_less_eq_set_nat @ X4 @ X5 )
% 14.17/8.01 => ( ord_less_eq_set_nat @ ( image_nat_nat @ ( transi2905341329935302413cl_nat @ R3 ) @ X4 ) @ ( image_nat_nat @ ( transi2905341329935302413cl_nat @ R5 ) @ X5 ) ) ) ) ).
% 14.17/8.01
% 14.17/8.01 thf(fact_990_order__antisym__conv,axiom,
% 14.17/8.01 ! [Y: filter_nat,X: filter_nat] :
% 14.17/8.01 ( ( ord_le2510731241096832064er_nat @ Y @ X )
% 14.17/8.01 => ( ( ord_le2510731241096832064er_nat @ X @ Y )
% 14.17/8.01 = ( X = Y ) ) ) ).
% 14.17/8.01
% 14.17/8.01 thf(fact_991_order__antisym__conv,axiom,
% 14.17/8.01 ! [Y: set_nat,X: set_nat] :
% 14.17/8.01 ( ( ord_less_eq_set_nat @ Y @ X )
% 14.17/8.01 => ( ( ord_less_eq_set_nat @ X @ Y )
% 14.17/8.01 = ( X = Y ) ) ) ).
% 14.17/8.01
% 14.17/8.01 thf(fact_992_order__antisym__conv,axiom,
% 14.17/8.01 ! [Y: rat,X: rat] :
% 14.17/8.01 ( ( ord_less_eq_rat @ Y @ X )
% 14.17/8.01 => ( ( ord_less_eq_rat @ X @ Y )
% 14.17/8.01 = ( X = Y ) ) ) ).
% 14.17/8.01
% 14.17/8.01 thf(fact_993_order__antisym__conv,axiom,
% 14.17/8.01 ! [Y: nat,X: nat] :
% 14.17/8.01 ( ( ord_less_eq_nat @ Y @ X )
% 14.17/8.01 => ( ( ord_less_eq_nat @ X @ Y )
% 14.17/8.01 = ( X = Y ) ) ) ).
% 14.17/8.01
% 14.17/8.01 thf(fact_994_order__antisym__conv,axiom,
% 14.17/8.01 ! [Y: int,X: int] :
% 14.17/8.01 ( ( ord_less_eq_int @ Y @ X )
% 14.17/8.01 => ( ( ord_less_eq_int @ X @ Y )
% 14.17/8.01 = ( X = Y ) ) ) ).
% 14.17/8.01
% 14.17/8.01 thf(fact_995_linorder__le__cases,axiom,
% 14.17/8.01 ! [X: rat,Y: rat] :
% 14.17/8.01 ( ~ ( ord_less_eq_rat @ X @ Y )
% 14.17/8.01 => ( ord_less_eq_rat @ Y @ X ) ) ).
% 14.17/8.01
% 14.17/8.01 thf(fact_996_linorder__le__cases,axiom,
% 14.17/8.01 ! [X: nat,Y: nat] :
% 14.17/8.01 ( ~ ( ord_less_eq_nat @ X @ Y )
% 14.17/8.01 => ( ord_less_eq_nat @ Y @ X ) ) ).
% 14.17/8.01
% 14.17/8.01 thf(fact_997_linorder__le__cases,axiom,
% 14.17/8.01 ! [X: int,Y: int] :
% 14.17/8.01 ( ~ ( ord_less_eq_int @ X @ Y )
% 14.17/8.01 => ( ord_less_eq_int @ Y @ X ) ) ).
% 14.17/8.01
% 14.17/8.01 thf(fact_998_ord__le__eq__subst,axiom,
% 14.17/8.01 ! [A2: rat,B: rat,F: rat > rat,C: rat] :
% 14.17/8.01 ( ( ord_less_eq_rat @ A2 @ B )
% 14.17/8.01 => ( ( ( F @ B )
% 14.17/8.01 = C )
% 14.17/8.01 => ( ! [X3: rat,Y2: rat] :
% 14.17/8.01 ( ( ord_less_eq_rat @ X3 @ Y2 )
% 14.17/8.01 => ( ord_less_eq_rat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
% 14.17/8.01 => ( ord_less_eq_rat @ ( F @ A2 ) @ C ) ) ) ) ).
% 14.17/8.01
% 14.17/8.01 thf(fact_999_ord__le__eq__subst,axiom,
% 14.17/8.01 ! [A2: rat,B: rat,F: rat > nat,C: nat] :
% 14.17/8.01 ( ( ord_less_eq_rat @ A2 @ B )
% 14.17/8.01 => ( ( ( F @ B )
% 14.17/8.01 = C )
% 14.17/8.01 => ( ! [X3: rat,Y2: rat] :
% 14.17/8.01 ( ( ord_less_eq_rat @ X3 @ Y2 )
% 14.17/8.01 => ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
% 14.17/8.01 => ( ord_less_eq_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% 14.17/8.01
% 14.17/8.01 thf(fact_1000_ord__le__eq__subst,axiom,
% 14.17/8.01 ! [A2: rat,B: rat,F: rat > int,C: int] :
% 14.17/8.01 ( ( ord_less_eq_rat @ A2 @ B )
% 14.17/8.01 => ( ( ( F @ B )
% 14.17/8.01 = C )
% 14.17/8.01 => ( ! [X3: rat,Y2: rat] :
% 14.17/8.01 ( ( ord_less_eq_rat @ X3 @ Y2 )
% 14.17/8.01 => ( ord_less_eq_int @ ( F @ X3 ) @ ( F @ Y2 ) ) )
% 14.17/8.01 => ( ord_less_eq_int @ ( F @ A2 ) @ C ) ) ) ) ).
% 14.17/8.01
% 14.17/8.01 thf(fact_1001_ord__le__eq__subst,axiom,
% 14.17/8.01 ! [A2: nat,B: nat,F: nat > rat,C: rat] :
% 14.17/8.01 ( ( ord_less_eq_nat @ A2 @ B )
% 14.17/8.01 => ( ( ( F @ B )
% 14.17/8.01 = C )
% 14.17/8.01 => ( ! [X3: nat,Y2: nat] :
% 14.17/8.01 ( ( ord_less_eq_nat @ X3 @ Y2 )
% 14.17/8.01 => ( ord_less_eq_rat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
% 14.17/8.01 => ( ord_less_eq_rat @ ( F @ A2 ) @ C ) ) ) ) ).
% 14.17/8.01
% 14.17/8.01 thf(fact_1002_ord__le__eq__subst,axiom,
% 14.17/8.01 ! [A2: nat,B: nat,F: nat > nat,C: nat] :
% 14.17/8.01 ( ( ord_less_eq_nat @ A2 @ B )
% 14.17/8.01 => ( ( ( F @ B )
% 14.17/8.01 = C )
% 14.17/8.01 => ( ! [X3: nat,Y2: nat] :
% 14.17/8.01 ( ( ord_less_eq_nat @ X3 @ Y2 )
% 14.17/8.01 => ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
% 14.17/8.01 => ( ord_less_eq_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% 14.17/8.01
% 14.17/8.01 thf(fact_1003_ord__le__eq__subst,axiom,
% 14.17/8.01 ! [A2: nat,B: nat,F: nat > int,C: int] :
% 14.17/8.01 ( ( ord_less_eq_nat @ A2 @ B )
% 14.17/8.01 => ( ( ( F @ B )
% 14.17/8.01 = C )
% 14.17/8.01 => ( ! [X3: nat,Y2: nat] :
% 14.17/8.01 ( ( ord_less_eq_nat @ X3 @ Y2 )
% 14.17/8.01 => ( ord_less_eq_int @ ( F @ X3 ) @ ( F @ Y2 ) ) )
% 14.17/8.01 => ( ord_less_eq_int @ ( F @ A2 ) @ C ) ) ) ) ).
% 14.17/8.01
% 14.17/8.01 thf(fact_1004_ord__le__eq__subst,axiom,
% 14.17/8.01 ! [A2: int,B: int,F: int > rat,C: rat] :
% 14.17/8.01 ( ( ord_less_eq_int @ A2 @ B )
% 14.17/8.01 => ( ( ( F @ B )
% 14.17/8.01 = C )
% 14.17/8.01 => ( ! [X3: int,Y2: int] :
% 14.17/8.01 ( ( ord_less_eq_int @ X3 @ Y2 )
% 14.17/8.01 => ( ord_less_eq_rat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
% 14.17/8.01 => ( ord_less_eq_rat @ ( F @ A2 ) @ C ) ) ) ) ).
% 14.17/8.01
% 14.17/8.01 thf(fact_1005_ord__le__eq__subst,axiom,
% 14.17/8.01 ! [A2: int,B: int,F: int > nat,C: nat] :
% 14.17/8.01 ( ( ord_less_eq_int @ A2 @ B )
% 14.17/8.01 => ( ( ( F @ B )
% 14.17/8.01 = C )
% 14.17/8.01 => ( ! [X3: int,Y2: int] :
% 14.17/8.01 ( ( ord_less_eq_int @ X3 @ Y2 )
% 14.17/8.01 => ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
% 14.17/8.01 => ( ord_less_eq_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% 14.17/8.01
% 14.17/8.01 thf(fact_1006_ord__le__eq__subst,axiom,
% 14.17/8.01 ! [A2: int,B: int,F: int > int,C: int] :
% 14.17/8.01 ( ( ord_less_eq_int @ A2 @ B )
% 14.17/8.01 => ( ( ( F @ B )
% 14.17/8.01 = C )
% 14.17/8.01 => ( ! [X3: int,Y2: int] :
% 14.17/8.01 ( ( ord_less_eq_int @ X3 @ Y2 )
% 14.17/8.01 => ( ord_less_eq_int @ ( F @ X3 ) @ ( F @ Y2 ) ) )
% 14.17/8.01 => ( ord_less_eq_int @ ( F @ A2 ) @ C ) ) ) ) ).
% 14.17/8.01
% 14.17/8.01 thf(fact_1007_ord__le__eq__subst,axiom,
% 14.17/8.01 ! [A2: filter_nat,B: filter_nat,F: filter_nat > rat,C: rat] :
% 14.17/8.01 ( ( ord_le2510731241096832064er_nat @ A2 @ B )
% 14.17/8.01 => ( ( ( F @ B )
% 14.17/8.01 = C )
% 14.17/8.01 => ( ! [X3: filter_nat,Y2: filter_nat] :
% 14.17/8.01 ( ( ord_le2510731241096832064er_nat @ X3 @ Y2 )
% 14.17/8.01 => ( ord_less_eq_rat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
% 14.17/8.01 => ( ord_less_eq_rat @ ( F @ A2 ) @ C ) ) ) ) ).
% 14.17/8.01
% 14.17/8.01 thf(fact_1008_ord__eq__le__subst,axiom,
% 14.17/8.01 ! [A2: rat,F: rat > rat,B: rat,C: rat] :
% 14.17/8.01 ( ( A2
% 14.17/8.01 = ( F @ B ) )
% 14.17/8.01 => ( ( ord_less_eq_rat @ B @ C )
% 14.17/8.01 => ( ! [X3: rat,Y2: rat] :
% 14.17/8.01 ( ( ord_less_eq_rat @ X3 @ Y2 )
% 14.17/8.01 => ( ord_less_eq_rat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
% 14.17/8.01 => ( ord_less_eq_rat @ A2 @ ( F @ C ) ) ) ) ) ).
% 14.17/8.01
% 14.17/8.01 thf(fact_1009_ord__eq__le__subst,axiom,
% 14.17/8.01 ! [A2: nat,F: rat > nat,B: rat,C: rat] :
% 14.17/8.01 ( ( A2
% 14.17/8.01 = ( F @ B ) )
% 14.17/8.01 => ( ( ord_less_eq_rat @ B @ C )
% 14.17/8.01 => ( ! [X3: rat,Y2: rat] :
% 14.17/8.01 ( ( ord_less_eq_rat @ X3 @ Y2 )
% 14.17/8.01 => ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
% 14.17/8.01 => ( ord_less_eq_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% 14.17/8.01
% 14.17/8.01 thf(fact_1010_ord__eq__le__subst,axiom,
% 14.17/8.01 ! [A2: int,F: rat > int,B: rat,C: rat] :
% 14.17/8.01 ( ( A2
% 14.17/8.01 = ( F @ B ) )
% 14.17/8.01 => ( ( ord_less_eq_rat @ B @ C )
% 14.17/8.01 => ( ! [X3: rat,Y2: rat] :
% 14.17/8.01 ( ( ord_less_eq_rat @ X3 @ Y2 )
% 14.17/8.01 => ( ord_less_eq_int @ ( F @ X3 ) @ ( F @ Y2 ) ) )
% 14.17/8.01 => ( ord_less_eq_int @ A2 @ ( F @ C ) ) ) ) ) ).
% 14.17/8.01
% 14.17/8.01 thf(fact_1011_ord__eq__le__subst,axiom,
% 14.17/8.01 ! [A2: rat,F: nat > rat,B: nat,C: nat] :
% 14.17/8.01 ( ( A2
% 14.17/8.01 = ( F @ B ) )
% 14.17/8.01 => ( ( ord_less_eq_nat @ B @ C )
% 14.17/8.01 => ( ! [X3: nat,Y2: nat] :
% 14.17/8.01 ( ( ord_less_eq_nat @ X3 @ Y2 )
% 14.17/8.01 => ( ord_less_eq_rat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
% 14.17/8.01 => ( ord_less_eq_rat @ A2 @ ( F @ C ) ) ) ) ) ).
% 14.17/8.01
% 14.17/8.01 thf(fact_1012_ord__eq__le__subst,axiom,
% 14.17/8.01 ! [A2: nat,F: nat > nat,B: nat,C: nat] :
% 14.17/8.01 ( ( A2
% 14.17/8.01 = ( F @ B ) )
% 14.17/8.01 => ( ( ord_less_eq_nat @ B @ C )
% 14.17/8.01 => ( ! [X3: nat,Y2: nat] :
% 14.17/8.01 ( ( ord_less_eq_nat @ X3 @ Y2 )
% 14.17/8.01 => ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
% 14.17/8.01 => ( ord_less_eq_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% 14.17/8.01
% 14.17/8.01 thf(fact_1013_ord__eq__le__subst,axiom,
% 14.17/8.01 ! [A2: int,F: nat > int,B: nat,C: nat] :
% 14.17/8.01 ( ( A2
% 14.17/8.01 = ( F @ B ) )
% 14.17/8.01 => ( ( ord_less_eq_nat @ B @ C )
% 14.17/8.01 => ( ! [X3: nat,Y2: nat] :
% 14.17/8.01 ( ( ord_less_eq_nat @ X3 @ Y2 )
% 14.17/8.01 => ( ord_less_eq_int @ ( F @ X3 ) @ ( F @ Y2 ) ) )
% 14.17/8.01 => ( ord_less_eq_int @ A2 @ ( F @ C ) ) ) ) ) ).
% 14.17/8.01
% 14.17/8.01 thf(fact_1014_ord__eq__le__subst,axiom,
% 14.17/8.01 ! [A2: rat,F: int > rat,B: int,C: int] :
% 14.17/8.01 ( ( A2
% 14.17/8.01 = ( F @ B ) )
% 14.17/8.01 => ( ( ord_less_eq_int @ B @ C )
% 14.17/8.01 => ( ! [X3: int,Y2: int] :
% 14.17/8.01 ( ( ord_less_eq_int @ X3 @ Y2 )
% 14.17/8.01 => ( ord_less_eq_rat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
% 14.17/8.01 => ( ord_less_eq_rat @ A2 @ ( F @ C ) ) ) ) ) ).
% 14.17/8.01
% 14.17/8.01 thf(fact_1015_ord__eq__le__subst,axiom,
% 14.17/8.01 ! [A2: nat,F: int > nat,B: int,C: int] :
% 14.17/8.01 ( ( A2
% 14.17/8.01 = ( F @ B ) )
% 14.17/8.01 => ( ( ord_less_eq_int @ B @ C )
% 14.17/8.01 => ( ! [X3: int,Y2: int] :
% 14.17/8.01 ( ( ord_less_eq_int @ X3 @ Y2 )
% 14.17/8.01 => ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
% 14.17/8.01 => ( ord_less_eq_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% 14.17/8.01
% 14.17/8.01 thf(fact_1016_ord__eq__le__subst,axiom,
% 14.17/8.01 ! [A2: int,F: int > int,B: int,C: int] :
% 14.17/8.01 ( ( A2
% 14.17/8.01 = ( F @ B ) )
% 14.17/8.01 => ( ( ord_less_eq_int @ B @ C )
% 14.17/8.01 => ( ! [X3: int,Y2: int] :
% 14.17/8.01 ( ( ord_less_eq_int @ X3 @ Y2 )
% 14.17/8.01 => ( ord_less_eq_int @ ( F @ X3 ) @ ( F @ Y2 ) ) )
% 14.17/8.01 => ( ord_less_eq_int @ A2 @ ( F @ C ) ) ) ) ) ).
% 14.17/8.01
% 14.17/8.01 thf(fact_1017_ord__eq__le__subst,axiom,
% 14.17/8.01 ! [A2: rat,F: filter_nat > rat,B: filter_nat,C: filter_nat] :
% 14.17/8.01 ( ( A2
% 14.17/8.01 = ( F @ B ) )
% 14.17/8.01 => ( ( ord_le2510731241096832064er_nat @ B @ C )
% 14.17/8.01 => ( ! [X3: filter_nat,Y2: filter_nat] :
% 14.17/8.01 ( ( ord_le2510731241096832064er_nat @ X3 @ Y2 )
% 14.17/8.01 => ( ord_less_eq_rat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
% 14.17/8.01 => ( ord_less_eq_rat @ A2 @ ( F @ C ) ) ) ) ) ).
% 14.17/8.01
% 14.17/8.01 thf(fact_1018_linorder__linear,axiom,
% 14.17/8.01 ! [X: rat,Y: rat] :
% 14.17/8.01 ( ( ord_less_eq_rat @ X @ Y )
% 14.17/8.01 | ( ord_less_eq_rat @ Y @ X ) ) ).
% 14.17/8.01
% 14.17/8.01 thf(fact_1019_linorder__linear,axiom,
% 14.17/8.01 ! [X: nat,Y: nat] :
% 14.17/8.01 ( ( ord_less_eq_nat @ X @ Y )
% 14.17/8.01 | ( ord_less_eq_nat @ Y @ X ) ) ).
% 14.17/8.01
% 14.17/8.01 thf(fact_1020_linorder__linear,axiom,
% 14.17/8.01 ! [X: int,Y: int] :
% 14.17/8.01 ( ( ord_less_eq_int @ X @ Y )
% 14.17/8.01 | ( ord_less_eq_int @ Y @ X ) ) ).
% 14.17/8.01
% 14.17/8.01 thf(fact_1021_subset__Collect__conv,axiom,
% 14.17/8.01 ! [S: set_Pr4532377907799695533_nat_o,P: ( produc3658429121746597890et_nat > $o ) > $o] :
% 14.17/8.01 ( ( ord_le2965882846123202637_nat_o @ S @ ( collec939566748876313656_nat_o @ P ) )
% 14.17/8.01 = ( ! [X2: produc3658429121746597890et_nat > $o] :
% 14.17/8.01 ( ( member6576561426505652726_nat_o @ X2 @ S )
% 14.17/8.01 => ( P @ X2 ) ) ) ) ).
% 14.17/8.01
% 14.17/8.01 thf(fact_1022_subset__Collect__conv,axiom,
% 14.17/8.01 ! [S: set_int,P: int > $o] :
% 14.17/8.01 ( ( ord_less_eq_set_int @ S @ ( collect_int @ P ) )
% 14.17/8.01 = ( ! [X2: int] :
% 14.17/8.01 ( ( member_int @ X2 @ S )
% 14.17/8.01 => ( P @ X2 ) ) ) ) ).
% 14.17/8.01
% 14.17/8.01 thf(fact_1023_subset__Collect__conv,axiom,
% 14.17/8.01 ! [S: set_o,P: $o > $o] :
% 14.17/8.01 ( ( ord_less_eq_set_o @ S @ ( collect_o @ P ) )
% 14.17/8.01 = ( ! [X2: $o] :
% 14.17/8.01 ( ( member_o @ X2 @ S )
% 14.17/8.01 => ( P @ X2 ) ) ) ) ).
% 14.17/8.01
% 14.17/8.01 thf(fact_1024_subset__Collect__conv,axiom,
% 14.17/8.01 ! [S: set_Product_unit,P: product_unit > $o] :
% 14.17/8.01 ( ( ord_le3507040750410214029t_unit @ S @ ( collect_Product_unit @ P ) )
% 14.17/8.01 = ( ! [X2: product_unit] :
% 14.17/8.01 ( ( member_Product_unit @ X2 @ S )
% 14.17/8.01 => ( P @ X2 ) ) ) ) ).
% 14.17/8.01
% 14.17/8.01 thf(fact_1025_subset__Collect__conv,axiom,
% 14.17/8.01 ! [S: set_nat,P: nat > $o] :
% 14.17/8.01 ( ( ord_less_eq_set_nat @ S @ ( collect_nat @ P ) )
% 14.17/8.01 = ( ! [X2: nat] :
% 14.17/8.01 ( ( member_nat @ X2 @ S )
% 14.17/8.01 => ( P @ X2 ) ) ) ) ).
% 14.17/8.01
% 14.17/8.01 thf(fact_1026_order__eq__refl,axiom,
% 14.17/8.01 ! [X: filter_nat,Y: filter_nat] :
% 14.17/8.01 ( ( X = Y )
% 14.17/8.01 => ( ord_le2510731241096832064er_nat @ X @ Y ) ) ).
% 14.17/8.01
% 14.17/8.01 thf(fact_1027_order__eq__refl,axiom,
% 14.17/8.01 ! [X: set_nat,Y: set_nat] :
% 14.17/8.01 ( ( X = Y )
% 14.17/8.01 => ( ord_less_eq_set_nat @ X @ Y ) ) ).
% 14.17/8.01
% 14.17/8.01 thf(fact_1028_order__eq__refl,axiom,
% 14.17/8.01 ! [X: rat,Y: rat] :
% 14.17/8.01 ( ( X = Y )
% 14.17/8.01 => ( ord_less_eq_rat @ X @ Y ) ) ).
% 14.17/8.01
% 14.17/8.01 thf(fact_1029_order__eq__refl,axiom,
% 14.17/8.01 ! [X: nat,Y: nat] :
% 14.17/8.01 ( ( X = Y )
% 14.17/8.01 => ( ord_less_eq_nat @ X @ Y ) ) ).
% 14.17/8.01
% 14.17/8.01 thf(fact_1030_order__eq__refl,axiom,
% 14.17/8.01 ! [X: int,Y: int] :
% 14.17/8.01 ( ( X = Y )
% 14.17/8.01 => ( ord_less_eq_int @ X @ Y ) ) ).
% 14.17/8.01
% 14.17/8.01 thf(fact_1031_ord__eq__le__eq__trans,axiom,
% 14.17/8.01 ! [A2: filter_nat,B: filter_nat,C: filter_nat,D2: filter_nat] :
% 14.17/8.01 ( ( A2 = B )
% 14.17/8.01 => ( ( ord_le2510731241096832064er_nat @ B @ C )
% 14.17/8.01 => ( ( C = D2 )
% 14.17/8.01 => ( ord_le2510731241096832064er_nat @ A2 @ D2 ) ) ) ) ).
% 14.17/8.01
% 14.17/8.01 thf(fact_1032_ord__eq__le__eq__trans,axiom,
% 14.17/8.01 ! [A2: set_nat,B: set_nat,C: set_nat,D2: set_nat] :
% 14.17/8.01 ( ( A2 = B )
% 14.17/8.01 => ( ( ord_less_eq_set_nat @ B @ C )
% 14.17/8.01 => ( ( C = D2 )
% 14.17/8.01 => ( ord_less_eq_set_nat @ A2 @ D2 ) ) ) ) ).
% 14.17/8.01
% 14.17/8.01 thf(fact_1033_ord__eq__le__eq__trans,axiom,
% 14.17/8.01 ! [A2: rat,B: rat,C: rat,D2: rat] :
% 14.17/8.01 ( ( A2 = B )
% 14.17/8.01 => ( ( ord_less_eq_rat @ B @ C )
% 14.17/8.01 => ( ( C = D2 )
% 14.17/8.01 => ( ord_less_eq_rat @ A2 @ D2 ) ) ) ) ).
% 14.17/8.01
% 14.17/8.01 thf(fact_1034_ord__eq__le__eq__trans,axiom,
% 14.17/8.01 ! [A2: nat,B: nat,C: nat,D2: nat] :
% 14.17/8.01 ( ( A2 = B )
% 14.17/8.01 => ( ( ord_less_eq_nat @ B @ C )
% 14.17/8.01 => ( ( C = D2 )
% 14.17/8.01 => ( ord_less_eq_nat @ A2 @ D2 ) ) ) ) ).
% 14.17/8.01
% 14.17/8.01 thf(fact_1035_ord__eq__le__eq__trans,axiom,
% 14.17/8.01 ! [A2: int,B: int,C: int,D2: int] :
% 14.17/8.01 ( ( A2 = B )
% 14.17/8.01 => ( ( ord_less_eq_int @ B @ C )
% 14.17/8.01 => ( ( C = D2 )
% 14.17/8.01 => ( ord_less_eq_int @ A2 @ D2 ) ) ) ) ).
% 14.17/8.01
% 14.17/8.01 thf(fact_1036_order__subst2,axiom,
% 14.17/8.01 ! [A2: rat,B: rat,F: rat > rat,C: rat] :
% 14.17/8.01 ( ( ord_less_eq_rat @ A2 @ B )
% 14.17/8.01 => ( ( ord_less_eq_rat @ ( F @ B ) @ C )
% 14.17/8.01 => ( ! [X3: rat,Y2: rat] :
% 14.17/8.01 ( ( ord_less_eq_rat @ X3 @ Y2 )
% 14.17/8.01 => ( ord_less_eq_rat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
% 14.17/8.01 => ( ord_less_eq_rat @ ( F @ A2 ) @ C ) ) ) ) ).
% 14.17/8.01
% 14.17/8.01 thf(fact_1037_order__subst2,axiom,
% 14.17/8.01 ! [A2: rat,B: rat,F: rat > nat,C: nat] :
% 14.17/8.01 ( ( ord_less_eq_rat @ A2 @ B )
% 14.17/8.01 => ( ( ord_less_eq_nat @ ( F @ B ) @ C )
% 14.17/8.01 => ( ! [X3: rat,Y2: rat] :
% 14.17/8.01 ( ( ord_less_eq_rat @ X3 @ Y2 )
% 14.17/8.01 => ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
% 14.17/8.01 => ( ord_less_eq_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% 14.17/8.01
% 14.17/8.01 thf(fact_1038_order__subst2,axiom,
% 14.17/8.01 ! [A2: rat,B: rat,F: rat > int,C: int] :
% 14.17/8.01 ( ( ord_less_eq_rat @ A2 @ B )
% 14.17/8.01 => ( ( ord_less_eq_int @ ( F @ B ) @ C )
% 14.17/8.01 => ( ! [X3: rat,Y2: rat] :
% 14.17/8.01 ( ( ord_less_eq_rat @ X3 @ Y2 )
% 14.17/8.01 => ( ord_less_eq_int @ ( F @ X3 ) @ ( F @ Y2 ) ) )
% 14.17/8.01 => ( ord_less_eq_int @ ( F @ A2 ) @ C ) ) ) ) ).
% 14.17/8.01
% 14.17/8.01 thf(fact_1039_order__subst2,axiom,
% 14.17/8.01 ! [A2: nat,B: nat,F: nat > rat,C: rat] :
% 14.17/8.01 ( ( ord_less_eq_nat @ A2 @ B )
% 14.17/8.01 => ( ( ord_less_eq_rat @ ( F @ B ) @ C )
% 14.17/8.01 => ( ! [X3: nat,Y2: nat] :
% 14.17/8.01 ( ( ord_less_eq_nat @ X3 @ Y2 )
% 14.17/8.01 => ( ord_less_eq_rat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
% 14.17/8.01 => ( ord_less_eq_rat @ ( F @ A2 ) @ C ) ) ) ) ).
% 14.17/8.01
% 14.17/8.01 thf(fact_1040_order__subst2,axiom,
% 14.17/8.01 ! [A2: nat,B: nat,F: nat > nat,C: nat] :
% 14.17/8.01 ( ( ord_less_eq_nat @ A2 @ B )
% 14.17/8.01 => ( ( ord_less_eq_nat @ ( F @ B ) @ C )
% 14.17/8.01 => ( ! [X3: nat,Y2: nat] :
% 14.17/8.01 ( ( ord_less_eq_nat @ X3 @ Y2 )
% 14.17/8.01 => ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
% 14.17/8.01 => ( ord_less_eq_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% 14.17/8.01
% 14.17/8.01 thf(fact_1041_order__subst2,axiom,
% 14.17/8.01 ! [A2: nat,B: nat,F: nat > int,C: int] :
% 14.17/8.01 ( ( ord_less_eq_nat @ A2 @ B )
% 14.17/8.01 => ( ( ord_less_eq_int @ ( F @ B ) @ C )
% 14.17/8.01 => ( ! [X3: nat,Y2: nat] :
% 14.17/8.01 ( ( ord_less_eq_nat @ X3 @ Y2 )
% 14.17/8.01 => ( ord_less_eq_int @ ( F @ X3 ) @ ( F @ Y2 ) ) )
% 14.17/8.01 => ( ord_less_eq_int @ ( F @ A2 ) @ C ) ) ) ) ).
% 14.17/8.01
% 14.17/8.01 thf(fact_1042_order__subst2,axiom,
% 14.17/8.01 ! [A2: int,B: int,F: int > rat,C: rat] :
% 14.17/8.01 ( ( ord_less_eq_int @ A2 @ B )
% 14.17/8.01 => ( ( ord_less_eq_rat @ ( F @ B ) @ C )
% 14.17/8.01 => ( ! [X3: int,Y2: int] :
% 14.17/8.01 ( ( ord_less_eq_int @ X3 @ Y2 )
% 14.17/8.01 => ( ord_less_eq_rat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
% 14.17/8.01 => ( ord_less_eq_rat @ ( F @ A2 ) @ C ) ) ) ) ).
% 14.17/8.01
% 14.17/8.01 thf(fact_1043_order__subst2,axiom,
% 14.17/8.01 ! [A2: int,B: int,F: int > nat,C: nat] :
% 14.17/8.01 ( ( ord_less_eq_int @ A2 @ B )
% 14.17/8.01 => ( ( ord_less_eq_nat @ ( F @ B ) @ C )
% 14.17/8.01 => ( ! [X3: int,Y2: int] :
% 14.17/8.01 ( ( ord_less_eq_int @ X3 @ Y2 )
% 14.17/8.01 => ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
% 14.17/8.01 => ( ord_less_eq_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% 14.17/8.01
% 14.17/8.01 thf(fact_1044_order__subst2,axiom,
% 14.17/8.01 ! [A2: int,B: int,F: int > int,C: int] :
% 14.17/8.01 ( ( ord_less_eq_int @ A2 @ B )
% 14.17/8.01 => ( ( ord_less_eq_int @ ( F @ B ) @ C )
% 14.17/8.01 => ( ! [X3: int,Y2: int] :
% 14.17/8.01 ( ( ord_less_eq_int @ X3 @ Y2 )
% 14.17/8.01 => ( ord_less_eq_int @ ( F @ X3 ) @ ( F @ Y2 ) ) )
% 14.17/8.01 => ( ord_less_eq_int @ ( F @ A2 ) @ C ) ) ) ) ).
% 14.17/8.01
% 14.17/8.01 thf(fact_1045_order__subst2,axiom,
% 14.17/8.01 ! [A2: filter_nat,B: filter_nat,F: filter_nat > rat,C: rat] :
% 14.17/8.01 ( ( ord_le2510731241096832064er_nat @ A2 @ B )
% 14.17/8.01 => ( ( ord_less_eq_rat @ ( F @ B ) @ C )
% 14.17/8.01 => ( ! [X3: filter_nat,Y2: filter_nat] :
% 14.17/8.01 ( ( ord_le2510731241096832064er_nat @ X3 @ Y2 )
% 14.17/8.01 => ( ord_less_eq_rat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
% 14.17/8.01 => ( ord_less_eq_rat @ ( F @ A2 ) @ C ) ) ) ) ).
% 14.17/8.01
% 14.17/8.01 thf(fact_1046_order__subst1,axiom,
% 14.17/8.01 ! [A2: rat,F: rat > rat,B: rat,C: rat] :
% 14.17/8.01 ( ( ord_less_eq_rat @ A2 @ ( F @ B ) )
% 14.17/8.01 => ( ( ord_less_eq_rat @ B @ C )
% 14.17/8.01 => ( ! [X3: rat,Y2: rat] :
% 14.17/8.01 ( ( ord_less_eq_rat @ X3 @ Y2 )
% 14.17/8.01 => ( ord_less_eq_rat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
% 14.17/8.01 => ( ord_less_eq_rat @ A2 @ ( F @ C ) ) ) ) ) ).
% 14.17/8.01
% 14.17/8.01 thf(fact_1047_order__subst1,axiom,
% 14.17/8.01 ! [A2: rat,F: nat > rat,B: nat,C: nat] :
% 14.17/8.01 ( ( ord_less_eq_rat @ A2 @ ( F @ B ) )
% 14.17/8.01 => ( ( ord_less_eq_nat @ B @ C )
% 14.17/8.01 => ( ! [X3: nat,Y2: nat] :
% 14.17/8.01 ( ( ord_less_eq_nat @ X3 @ Y2 )
% 14.17/8.01 => ( ord_less_eq_rat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
% 14.17/8.01 => ( ord_less_eq_rat @ A2 @ ( F @ C ) ) ) ) ) ).
% 14.17/8.01
% 14.17/8.01 thf(fact_1048_order__subst1,axiom,
% 14.17/8.01 ! [A2: rat,F: int > rat,B: int,C: int] :
% 14.17/8.01 ( ( ord_less_eq_rat @ A2 @ ( F @ B ) )
% 14.17/8.01 => ( ( ord_less_eq_int @ B @ C )
% 14.17/8.01 => ( ! [X3: int,Y2: int] :
% 14.17/8.01 ( ( ord_less_eq_int @ X3 @ Y2 )
% 14.17/8.01 => ( ord_less_eq_rat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
% 14.17/8.01 => ( ord_less_eq_rat @ A2 @ ( F @ C ) ) ) ) ) ).
% 14.17/8.01
% 14.17/8.01 thf(fact_1049_order__subst1,axiom,
% 14.17/8.01 ! [A2: nat,F: rat > nat,B: rat,C: rat] :
% 14.17/8.01 ( ( ord_less_eq_nat @ A2 @ ( F @ B ) )
% 14.17/8.01 => ( ( ord_less_eq_rat @ B @ C )
% 14.17/8.01 => ( ! [X3: rat,Y2: rat] :
% 14.17/8.01 ( ( ord_less_eq_rat @ X3 @ Y2 )
% 14.17/8.01 => ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
% 14.17/8.01 => ( ord_less_eq_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% 14.17/8.01
% 14.17/8.01 thf(fact_1050_order__subst1,axiom,
% 14.17/8.01 ! [A2: nat,F: nat > nat,B: nat,C: nat] :
% 14.17/8.01 ( ( ord_less_eq_nat @ A2 @ ( F @ B ) )
% 14.17/8.01 => ( ( ord_less_eq_nat @ B @ C )
% 14.17/8.01 => ( ! [X3: nat,Y2: nat] :
% 14.17/8.01 ( ( ord_less_eq_nat @ X3 @ Y2 )
% 14.17/8.01 => ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
% 14.17/8.01 => ( ord_less_eq_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% 14.17/8.01
% 14.17/8.01 thf(fact_1051_order__subst1,axiom,
% 14.17/8.01 ! [A2: nat,F: int > nat,B: int,C: int] :
% 14.17/8.01 ( ( ord_less_eq_nat @ A2 @ ( F @ B ) )
% 14.17/8.01 => ( ( ord_less_eq_int @ B @ C )
% 14.17/8.01 => ( ! [X3: int,Y2: int] :
% 14.17/8.01 ( ( ord_less_eq_int @ X3 @ Y2 )
% 14.17/8.01 => ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
% 14.17/8.01 => ( ord_less_eq_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% 14.17/8.01
% 14.17/8.01 thf(fact_1052_order__subst1,axiom,
% 14.17/8.01 ! [A2: int,F: rat > int,B: rat,C: rat] :
% 14.17/8.01 ( ( ord_less_eq_int @ A2 @ ( F @ B ) )
% 14.17/8.01 => ( ( ord_less_eq_rat @ B @ C )
% 14.17/8.01 => ( ! [X3: rat,Y2: rat] :
% 14.17/8.01 ( ( ord_less_eq_rat @ X3 @ Y2 )
% 14.17/8.01 => ( ord_less_eq_int @ ( F @ X3 ) @ ( F @ Y2 ) ) )
% 14.17/8.01 => ( ord_less_eq_int @ A2 @ ( F @ C ) ) ) ) ) ).
% 14.17/8.01
% 14.17/8.01 thf(fact_1053_order__subst1,axiom,
% 14.17/8.01 ! [A2: int,F: nat > int,B: nat,C: nat] :
% 14.17/8.01 ( ( ord_less_eq_int @ A2 @ ( F @ B ) )
% 14.17/8.01 => ( ( ord_less_eq_nat @ B @ C )
% 14.17/8.01 => ( ! [X3: nat,Y2: nat] :
% 14.17/8.01 ( ( ord_less_eq_nat @ X3 @ Y2 )
% 14.17/8.01 => ( ord_less_eq_int @ ( F @ X3 ) @ ( F @ Y2 ) ) )
% 14.17/8.01 => ( ord_less_eq_int @ A2 @ ( F @ C ) ) ) ) ) ).
% 14.17/8.01
% 14.17/8.01 thf(fact_1054_order__subst1,axiom,
% 14.17/8.01 ! [A2: int,F: int > int,B: int,C: int] :
% 14.17/8.01 ( ( ord_less_eq_int @ A2 @ ( F @ B ) )
% 14.17/8.01 => ( ( ord_less_eq_int @ B @ C )
% 14.17/8.01 => ( ! [X3: int,Y2: int] :
% 14.17/8.01 ( ( ord_less_eq_int @ X3 @ Y2 )
% 14.17/8.01 => ( ord_less_eq_int @ ( F @ X3 ) @ ( F @ Y2 ) ) )
% 14.17/8.01 => ( ord_less_eq_int @ A2 @ ( F @ C ) ) ) ) ) ).
% 14.17/8.01
% 14.17/8.01 thf(fact_1055_order__subst1,axiom,
% 14.17/8.01 ! [A2: filter_nat,F: rat > filter_nat,B: rat,C: rat] :
% 14.17/8.01 ( ( ord_le2510731241096832064er_nat @ A2 @ ( F @ B ) )
% 14.17/8.01 => ( ( ord_less_eq_rat @ B @ C )
% 14.17/8.01 => ( ! [X3: rat,Y2: rat] :
% 14.17/8.01 ( ( ord_less_eq_rat @ X3 @ Y2 )
% 14.17/8.01 => ( ord_le2510731241096832064er_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
% 14.17/8.01 => ( ord_le2510731241096832064er_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% 14.17/8.01
% 14.17/8.01 thf(fact_1056_Orderings_Oorder__eq__iff,axiom,
% 14.17/8.01 ( ( ^ [Y4: filter_nat,Z4: filter_nat] : ( Y4 = Z4 ) )
% 14.17/8.01 = ( ^ [A6: filter_nat,B6: filter_nat] :
% 14.17/8.01 ( ( ord_le2510731241096832064er_nat @ A6 @ B6 )
% 14.17/8.01 & ( ord_le2510731241096832064er_nat @ B6 @ A6 ) ) ) ) ).
% 14.17/8.01
% 14.17/8.01 thf(fact_1057_Orderings_Oorder__eq__iff,axiom,
% 14.17/8.01 ( ( ^ [Y4: set_nat,Z4: set_nat] : ( Y4 = Z4 ) )
% 14.17/8.01 = ( ^ [A6: set_nat,B6: set_nat] :
% 14.17/8.01 ( ( ord_less_eq_set_nat @ A6 @ B6 )
% 14.17/8.01 & ( ord_less_eq_set_nat @ B6 @ A6 ) ) ) ) ).
% 14.17/8.01
% 14.17/8.01 thf(fact_1058_Orderings_Oorder__eq__iff,axiom,
% 14.17/8.01 ( ( ^ [Y4: rat,Z4: rat] : ( Y4 = Z4 ) )
% 14.17/8.01 = ( ^ [A6: rat,B6: rat] :
% 14.17/8.01 ( ( ord_less_eq_rat @ A6 @ B6 )
% 14.17/8.01 & ( ord_less_eq_rat @ B6 @ A6 ) ) ) ) ).
% 14.17/8.01
% 14.17/8.01 thf(fact_1059_Orderings_Oorder__eq__iff,axiom,
% 14.17/8.01 ( ( ^ [Y4: nat,Z4: nat] : ( Y4 = Z4 ) )
% 14.17/8.01 = ( ^ [A6: nat,B6: nat] :
% 14.17/8.01 ( ( ord_less_eq_nat @ A6 @ B6 )
% 14.17/8.01 & ( ord_less_eq_nat @ B6 @ A6 ) ) ) ) ).
% 14.17/8.01
% 14.17/8.01 thf(fact_1060_Orderings_Oorder__eq__iff,axiom,
% 14.17/8.01 ( ( ^ [Y4: int,Z4: int] : ( Y4 = Z4 ) )
% 14.17/8.01 = ( ^ [A6: int,B6: int] :
% 14.17/8.01 ( ( ord_less_eq_int @ A6 @ B6 )
% 14.17/8.01 & ( ord_less_eq_int @ B6 @ A6 ) ) ) ) ).
% 14.17/8.01
% 14.17/8.01 thf(fact_1061_inter__eq__subsetI,axiom,
% 14.17/8.01 ! [S: set_nat,S4: set_nat,A: set_nat,B4: set_nat] :
% 14.17/8.01 ( ( ord_less_eq_set_nat @ S @ S4 )
% 14.17/8.01 => ( ( ( inf_inf_set_nat @ A @ S4 )
% 14.17/8.01 = ( inf_inf_set_nat @ B4 @ S4 ) )
% 14.17/8.01 => ( ( inf_inf_set_nat @ A @ S )
% 14.17/8.01 = ( inf_inf_set_nat @ B4 @ S ) ) ) ) ).
% 14.17/8.01
% 14.17/8.01 thf(fact_1062_Int__left__commute,axiom,
% 14.17/8.01 ! [A: set_nat,B4: set_nat,C5: set_nat] :
% 14.17/8.01 ( ( inf_inf_set_nat @ A @ ( inf_inf_set_nat @ B4 @ C5 ) )
% 14.17/8.01 = ( inf_inf_set_nat @ B4 @ ( inf_inf_set_nat @ A @ C5 ) ) ) ).
% 14.17/8.01
% 14.17/8.01 thf(fact_1063_Int__Collect__mono,axiom,
% 14.17/8.01 ! [A: set_Pr4532377907799695533_nat_o,B4: set_Pr4532377907799695533_nat_o,P: ( produc3658429121746597890et_nat > $o ) > $o,Q2: ( produc3658429121746597890et_nat > $o ) > $o] :
% 14.17/8.01 ( ( ord_le2965882846123202637_nat_o @ A @ B4 )
% 14.17/8.01 => ( ! [X3: produc3658429121746597890et_nat > $o] :
% 14.17/8.01 ( ( member6576561426505652726_nat_o @ X3 @ A )
% 14.17/8.01 => ( ( P @ X3 )
% 14.17/8.01 => ( Q2 @ X3 ) ) )
% 14.17/8.01 => ( ord_le2965882846123202637_nat_o @ ( inf_in1906310914598751387_nat_o @ A @ ( collec939566748876313656_nat_o @ P ) ) @ ( inf_in1906310914598751387_nat_o @ B4 @ ( collec939566748876313656_nat_o @ Q2 ) ) ) ) ) ).
% 14.17/8.01
% 14.17/8.01 thf(fact_1064_Int__Collect__mono,axiom,
% 14.17/8.01 ! [A: set_int,B4: set_int,P: int > $o,Q2: int > $o] :
% 14.17/8.01 ( ( ord_less_eq_set_int @ A @ B4 )
% 14.17/8.01 => ( ! [X3: int] :
% 14.17/8.01 ( ( member_int @ X3 @ A )
% 14.17/8.01 => ( ( P @ X3 )
% 14.17/8.01 => ( Q2 @ X3 ) ) )
% 14.17/8.01 => ( ord_less_eq_set_int @ ( inf_inf_set_int @ A @ ( collect_int @ P ) ) @ ( inf_inf_set_int @ B4 @ ( collect_int @ Q2 ) ) ) ) ) ).
% 14.17/8.01
% 14.17/8.01 thf(fact_1065_Int__Collect__mono,axiom,
% 14.17/8.01 ! [A: set_o,B4: set_o,P: $o > $o,Q2: $o > $o] :
% 14.17/8.01 ( ( ord_less_eq_set_o @ A @ B4 )
% 14.17/8.01 => ( ! [X3: $o] :
% 14.17/8.01 ( ( member_o @ X3 @ A )
% 14.17/8.01 => ( ( P @ X3 )
% 14.17/8.01 => ( Q2 @ X3 ) ) )
% 14.17/8.01 => ( ord_less_eq_set_o @ ( inf_inf_set_o @ A @ ( collect_o @ P ) ) @ ( inf_inf_set_o @ B4 @ ( collect_o @ Q2 ) ) ) ) ) ).
% 14.17/8.01
% 14.17/8.01 thf(fact_1066_Int__Collect__mono,axiom,
% 14.17/8.01 ! [A: set_Product_unit,B4: set_Product_unit,P: product_unit > $o,Q2: product_unit > $o] :
% 14.17/8.01 ( ( ord_le3507040750410214029t_unit @ A @ B4 )
% 14.17/8.01 => ( ! [X3: product_unit] :
% 14.17/8.01 ( ( member_Product_unit @ X3 @ A )
% 14.17/8.01 => ( ( P @ X3 )
% 14.17/8.01 => ( Q2 @ X3 ) ) )
% 14.17/8.01 => ( ord_le3507040750410214029t_unit @ ( inf_in4660618365625256667t_unit @ A @ ( collect_Product_unit @ P ) ) @ ( inf_in4660618365625256667t_unit @ B4 @ ( collect_Product_unit @ Q2 ) ) ) ) ) ).
% 14.17/8.01
% 14.17/8.01 thf(fact_1067_Int__Collect__mono,axiom,
% 14.17/8.01 ! [A: set_nat,B4: set_nat,P: nat > $o,Q2: nat > $o] :
% 14.17/8.01 ( ( ord_less_eq_set_nat @ A @ B4 )
% 14.17/8.01 => ( ! [X3: nat] :
% 14.17/8.01 ( ( member_nat @ X3 @ A )
% 14.17/8.01 => ( ( P @ X3 )
% 14.17/8.01 => ( Q2 @ X3 ) ) )
% 14.17/8.01 => ( ord_less_eq_set_nat @ ( inf_inf_set_nat @ A @ ( collect_nat @ P ) ) @ ( inf_inf_set_nat @ B4 @ ( collect_nat @ Q2 ) ) ) ) ) ).
% 14.17/8.01
% 14.17/8.01 thf(fact_1068_Collect__mono__iff,axiom,
% 14.17/8.01 ! [P: ( produc3658429121746597890et_nat > $o ) > $o,Q2: ( produc3658429121746597890et_nat > $o ) > $o] :
% 14.17/8.01 ( ( ord_le2965882846123202637_nat_o @ ( collec939566748876313656_nat_o @ P ) @ ( collec939566748876313656_nat_o @ Q2 ) )
% 14.17/8.01 = ( ! [X2: produc3658429121746597890et_nat > $o] :
% 14.17/8.01 ( ( P @ X2 )
% 14.17/8.01 => ( Q2 @ X2 ) ) ) ) ).
% 14.17/8.01
% 14.17/8.01 thf(fact_1069_Collect__mono__iff,axiom,
% 14.17/8.01 ! [P: int > $o,Q2: int > $o] :
% 14.17/8.01 ( ( ord_less_eq_set_int @ ( collect_int @ P ) @ ( collect_int @ Q2 ) )
% 14.17/8.01 = ( ! [X2: int] :
% 14.17/8.01 ( ( P @ X2 )
% 14.17/8.01 => ( Q2 @ X2 ) ) ) ) ).
% 14.17/8.01
% 14.17/8.01 thf(fact_1070_Collect__mono__iff,axiom,
% 14.17/8.01 ! [P: $o > $o,Q2: $o > $o] :
% 14.17/8.01 ( ( ord_less_eq_set_o @ ( collect_o @ P ) @ ( collect_o @ Q2 ) )
% 14.17/8.01 = ( ! [X2: $o] :
% 14.17/8.01 ( ( P @ X2 )
% 14.17/8.01 => ( Q2 @ X2 ) ) ) ) ).
% 14.17/8.01
% 14.17/8.01 thf(fact_1071_Collect__mono__iff,axiom,
% 14.17/8.01 ! [P: product_unit > $o,Q2: product_unit > $o] :
% 14.17/8.01 ( ( ord_le3507040750410214029t_unit @ ( collect_Product_unit @ P ) @ ( collect_Product_unit @ Q2 ) )
% 14.17/8.01 = ( ! [X2: product_unit] :
% 14.17/8.01 ( ( P @ X2 )
% 14.17/8.01 => ( Q2 @ X2 ) ) ) ) ).
% 14.17/8.01
% 14.17/8.01 thf(fact_1072_Collect__mono__iff,axiom,
% 14.17/8.01 ! [P: nat > $o,Q2: nat > $o] :
% 14.17/8.01 ( ( ord_less_eq_set_nat @ ( collect_nat @ P ) @ ( collect_nat @ Q2 ) )
% 14.17/8.01 = ( ! [X2: nat] :
% 14.17/8.01 ( ( P @ X2 )
% 14.17/8.01 => ( Q2 @ X2 ) ) ) ) ).
% 14.17/8.01
% 14.17/8.01 thf(fact_1073_Un__left__commute,axiom,
% 14.17/8.01 ! [A: set_nat,B4: set_nat,C5: set_nat] :
% 14.17/8.01 ( ( sup_sup_set_nat @ A @ ( sup_sup_set_nat @ B4 @ C5 ) )
% 14.17/8.01 = ( sup_sup_set_nat @ B4 @ ( sup_sup_set_nat @ A @ C5 ) ) ) ).
% 14.17/8.01
% 14.17/8.01 thf(fact_1074_Un__Int__distrib2,axiom,
% 14.17/8.01 ! [B4: set_nat,C5: set_nat,A: set_nat] :
% 14.17/8.01 ( ( sup_sup_set_nat @ ( inf_inf_set_nat @ B4 @ C5 ) @ A )
% 14.17/8.01 = ( inf_inf_set_nat @ ( sup_sup_set_nat @ B4 @ A ) @ ( sup_sup_set_nat @ C5 @ A ) ) ) ).
% 14.17/8.01
% 14.17/8.01 thf(fact_1075_Un__Int__assoc__eq,axiom,
% 14.17/8.01 ! [A: set_nat,B4: set_nat,C5: set_nat] :
% 14.17/8.01 ( ( ( sup_sup_set_nat @ ( inf_inf_set_nat @ A @ B4 ) @ C5 )
% 14.17/8.01 = ( inf_inf_set_nat @ A @ ( sup_sup_set_nat @ B4 @ C5 ) ) )
% 14.17/8.01 = ( ord_less_eq_set_nat @ C5 @ A ) ) ).
% 14.17/8.01
% 14.17/8.01 thf(fact_1076_Int__left__absorb,axiom,
% 14.17/8.01 ! [A: set_nat,B4: set_nat] :
% 14.17/8.01 ( ( inf_inf_set_nat @ A @ ( inf_inf_set_nat @ A @ B4 ) )
% 14.17/8.01 = ( inf_inf_set_nat @ A @ B4 ) ) ).
% 14.17/8.01
% 14.17/8.01 thf(fact_1077_Int__Un__distrib2,axiom,
% 14.17/8.01 ! [B4: set_nat,C5: set_nat,A: set_nat] :
% 14.17/8.01 ( ( inf_inf_set_nat @ ( sup_sup_set_nat @ B4 @ C5 ) @ A )
% 14.17/8.01 = ( sup_sup_set_nat @ ( inf_inf_set_nat @ B4 @ A ) @ ( inf_inf_set_nat @ C5 @ A ) ) ) ).
% 14.17/8.01
% 14.17/8.01 thf(fact_1078_Un__left__absorb,axiom,
% 14.17/8.01 ! [A: set_nat,B4: set_nat] :
% 14.17/8.01 ( ( sup_sup_set_nat @ A @ ( sup_sup_set_nat @ A @ B4 ) )
% 14.17/8.01 = ( sup_sup_set_nat @ A @ B4 ) ) ).
% 14.17/8.01
% 14.17/8.01 thf(fact_1079_Un__Int__distrib,axiom,
% 14.17/8.01 ! [A: set_nat,B4: set_nat,C5: set_nat] :
% 14.17/8.01 ( ( sup_sup_set_nat @ A @ ( inf_inf_set_nat @ B4 @ C5 ) )
% 14.17/8.01 = ( inf_inf_set_nat @ ( sup_sup_set_nat @ A @ B4 ) @ ( sup_sup_set_nat @ A @ C5 ) ) ) ).
% 14.17/8.01
% 14.17/8.01 thf(fact_1080_Int__Un__distrib,axiom,
% 14.17/8.01 ! [A: set_nat,B4: set_nat,C5: set_nat] :
% 14.17/8.01 ( ( inf_inf_set_nat @ A @ ( sup_sup_set_nat @ B4 @ C5 ) )
% 14.17/8.01 = ( sup_sup_set_nat @ ( inf_inf_set_nat @ A @ B4 ) @ ( inf_inf_set_nat @ A @ C5 ) ) ) ).
% 14.17/8.01
% 14.17/8.01 thf(fact_1081_set__eq__subset,axiom,
% 14.17/8.01 ( ( ^ [Y4: set_nat,Z4: set_nat] : ( Y4 = Z4 ) )
% 14.17/8.01 = ( ^ [A3: set_nat,B7: set_nat] :
% 14.17/8.01 ( ( ord_less_eq_set_nat @ A3 @ B7 )
% 14.17/8.01 & ( ord_less_eq_set_nat @ B7 @ A3 ) ) ) ) ).
% 14.17/8.01
% 14.17/8.01 thf(fact_1082_antisym,axiom,
% 14.17/8.01 ! [A2: filter_nat,B: filter_nat] :
% 14.17/8.01 ( ( ord_le2510731241096832064er_nat @ A2 @ B )
% 14.17/8.01 => ( ( ord_le2510731241096832064er_nat @ B @ A2 )
% 14.17/8.01 => ( A2 = B ) ) ) ).
% 14.17/8.01
% 14.17/8.01 thf(fact_1083_antisym,axiom,
% 14.17/8.01 ! [A2: set_nat,B: set_nat] :
% 14.17/8.01 ( ( ord_less_eq_set_nat @ A2 @ B )
% 14.17/8.01 => ( ( ord_less_eq_set_nat @ B @ A2 )
% 14.17/8.01 => ( A2 = B ) ) ) ).
% 14.17/8.01
% 14.17/8.01 thf(fact_1084_antisym,axiom,
% 14.17/8.01 ! [A2: rat,B: rat] :
% 14.17/8.01 ( ( ord_less_eq_rat @ A2 @ B )
% 14.17/8.01 => ( ( ord_less_eq_rat @ B @ A2 )
% 14.17/8.01 => ( A2 = B ) ) ) ).
% 14.17/8.01
% 14.17/8.01 thf(fact_1085_antisym,axiom,
% 14.17/8.01 ! [A2: nat,B: nat] :
% 14.17/8.01 ( ( ord_less_eq_nat @ A2 @ B )
% 14.17/8.01 => ( ( ord_less_eq_nat @ B @ A2 )
% 14.17/8.01 => ( A2 = B ) ) ) ).
% 14.17/8.01
% 14.17/8.01 thf(fact_1086_antisym,axiom,
% 14.17/8.01 ! [A2: int,B: int] :
% 14.17/8.01 ( ( ord_less_eq_int @ A2 @ B )
% 14.17/8.01 => ( ( ord_less_eq_int @ B @ A2 )
% 14.17/8.01 => ( A2 = B ) ) ) ).
% 14.17/8.01
% 14.17/8.01 thf(fact_1087_subset__trans,axiom,
% 14.17/8.01 ! [A: set_nat,B4: set_nat,C5: set_nat] :
% 14.17/8.01 ( ( ord_less_eq_set_nat @ A @ B4 )
% 14.17/8.01 => ( ( ord_less_eq_set_nat @ B4 @ C5 )
% 14.17/8.01 => ( ord_less_eq_set_nat @ A @ C5 ) ) ) ).
% 14.17/8.01
% 14.17/8.01 thf(fact_1088_subset__Un__eq,axiom,
% 14.17/8.01 ( ord_less_eq_set_nat
% 14.17/8.01 = ( ^ [A3: set_nat,B7: set_nat] :
% 14.17/8.01 ( ( sup_sup_set_nat @ A3 @ B7 )
% 14.17/8.01 = B7 ) ) ) ).
% 14.17/8.01
% 14.17/8.01 thf(fact_1089_Un__Int__crazy,axiom,
% 14.17/8.01 ! [A: set_nat,B4: set_nat,C5: set_nat] :
% 14.17/8.01 ( ( sup_sup_set_nat @ ( sup_sup_set_nat @ ( inf_inf_set_nat @ A @ B4 ) @ ( inf_inf_set_nat @ B4 @ C5 ) ) @ ( inf_inf_set_nat @ C5 @ A ) )
% 14.17/8.01 = ( inf_inf_set_nat @ ( inf_inf_set_nat @ ( sup_sup_set_nat @ A @ B4 ) @ ( sup_sup_set_nat @ B4 @ C5 ) ) @ ( sup_sup_set_nat @ C5 @ A ) ) ) ).
% 14.17/8.01
% 14.17/8.01 thf(fact_1090_Int__greatest,axiom,
% 14.17/8.01 ! [C5: set_nat,A: set_nat,B4: set_nat] :
% 14.17/8.01 ( ( ord_less_eq_set_nat @ C5 @ A )
% 14.17/8.01 => ( ( ord_less_eq_set_nat @ C5 @ B4 )
% 14.17/8.01 => ( ord_less_eq_set_nat @ C5 @ ( inf_inf_set_nat @ A @ B4 ) ) ) ) ).
% 14.17/8.01
% 14.17/8.01 thf(fact_1091_Collect__mono,axiom,
% 14.17/8.01 ! [P: ( produc3658429121746597890et_nat > $o ) > $o,Q2: ( produc3658429121746597890et_nat > $o ) > $o] :
% 14.17/8.01 ( ! [X3: produc3658429121746597890et_nat > $o] :
% 14.17/8.01 ( ( P @ X3 )
% 14.17/8.01 => ( Q2 @ X3 ) )
% 14.17/8.01 => ( ord_le2965882846123202637_nat_o @ ( collec939566748876313656_nat_o @ P ) @ ( collec939566748876313656_nat_o @ Q2 ) ) ) ).
% 14.17/8.01
% 14.17/8.01 thf(fact_1092_Collect__mono,axiom,
% 14.17/8.01 ! [P: int > $o,Q2: int > $o] :
% 14.17/8.01 ( ! [X3: int] :
% 14.17/8.01 ( ( P @ X3 )
% 14.17/8.01 => ( Q2 @ X3 ) )
% 14.17/8.01 => ( ord_less_eq_set_int @ ( collect_int @ P ) @ ( collect_int @ Q2 ) ) ) ).
% 14.17/8.01
% 14.17/8.01 thf(fact_1093_Collect__mono,axiom,
% 14.17/8.01 ! [P: $o > $o,Q2: $o > $o] :
% 14.17/8.01 ( ! [X3: $o] :
% 14.17/8.01 ( ( P @ X3 )
% 14.17/8.01 => ( Q2 @ X3 ) )
% 14.17/8.01 => ( ord_less_eq_set_o @ ( collect_o @ P ) @ ( collect_o @ Q2 ) ) ) ).
% 14.17/8.01
% 14.17/8.01 thf(fact_1094_Collect__mono,axiom,
% 14.17/8.01 ! [P: product_unit > $o,Q2: product_unit > $o] :
% 14.17/8.01 ( ! [X3: product_unit] :
% 14.17/8.01 ( ( P @ X3 )
% 14.17/8.01 => ( Q2 @ X3 ) )
% 14.17/8.01 => ( ord_le3507040750410214029t_unit @ ( collect_Product_unit @ P ) @ ( collect_Product_unit @ Q2 ) ) ) ).
% 14.17/8.01
% 14.17/8.01 thf(fact_1095_Collect__mono,axiom,
% 14.17/8.01 ! [P: nat > $o,Q2: nat > $o] :
% 14.17/8.01 ( ! [X3: nat] :
% 14.17/8.01 ( ( P @ X3 )
% 14.17/8.01 => ( Q2 @ X3 ) )
% 14.17/8.01 => ( ord_less_eq_set_nat @ ( collect_nat @ P ) @ ( collect_nat @ Q2 ) ) ) ).
% 14.17/8.01
% 14.17/8.01 thf(fact_1096_subset__refl,axiom,
% 14.17/8.01 ! [A: set_nat] : ( ord_less_eq_set_nat @ A @ A ) ).
% 14.17/8.01
% 14.17/8.01 thf(fact_1097_Int__commute,axiom,
% 14.17/8.01 ( inf_inf_set_nat
% 14.17/8.01 = ( ^ [A3: set_nat,B7: set_nat] : ( inf_inf_set_nat @ B7 @ A3 ) ) ) ).
% 14.17/8.01
% 14.17/8.01 thf(fact_1098_Int__absorb2,axiom,
% 14.17/8.01 ! [A: set_nat,B4: set_nat] :
% 14.17/8.01 ( ( ord_less_eq_set_nat @ A @ B4 )
% 14.17/8.01 => ( ( inf_inf_set_nat @ A @ B4 )
% 14.17/8.01 = A ) ) ).
% 14.17/8.01
% 14.17/8.01 thf(fact_1099_Int__absorb1,axiom,
% 14.17/8.01 ! [B4: set_nat,A: set_nat] :
% 14.17/8.01 ( ( ord_less_eq_set_nat @ B4 @ A )
% 14.17/8.01 => ( ( inf_inf_set_nat @ A @ B4 )
% 14.17/8.01 = B4 ) ) ).
% 14.17/8.01
% 14.17/8.01 thf(fact_1100_subset__iff,axiom,
% 14.17/8.01 ( ord_le2965882846123202637_nat_o
% 14.17/8.01 = ( ^ [A3: set_Pr4532377907799695533_nat_o,B7: set_Pr4532377907799695533_nat_o] :
% 14.17/8.01 ! [T: produc3658429121746597890et_nat > $o] :
% 14.17/8.01 ( ( member6576561426505652726_nat_o @ T @ A3 )
% 14.17/8.01 => ( member6576561426505652726_nat_o @ T @ B7 ) ) ) ) ).
% 14.17/8.01
% 14.17/8.01 thf(fact_1101_subset__iff,axiom,
% 14.17/8.01 ( ord_less_eq_set_int
% 14.17/8.01 = ( ^ [A3: set_int,B7: set_int] :
% 14.17/8.01 ! [T: int] :
% 14.17/8.01 ( ( member_int @ T @ A3 )
% 14.17/8.01 => ( member_int @ T @ B7 ) ) ) ) ).
% 14.17/8.01
% 14.17/8.01 thf(fact_1102_subset__iff,axiom,
% 14.17/8.01 ( ord_less_eq_set_o
% 14.17/8.01 = ( ^ [A3: set_o,B7: set_o] :
% 14.17/8.01 ! [T: $o] :
% 14.17/8.01 ( ( member_o @ T @ A3 )
% 14.17/8.01 => ( member_o @ T @ B7 ) ) ) ) ).
% 14.17/8.01
% 14.17/8.01 thf(fact_1103_subset__iff,axiom,
% 14.17/8.01 ( ord_le3507040750410214029t_unit
% 14.17/8.01 = ( ^ [A3: set_Product_unit,B7: set_Product_unit] :
% 14.17/8.01 ! [T: product_unit] :
% 14.17/8.01 ( ( member_Product_unit @ T @ A3 )
% 14.17/8.01 => ( member_Product_unit @ T @ B7 ) ) ) ) ).
% 14.17/8.01
% 14.17/8.01 thf(fact_1104_subset__iff,axiom,
% 14.17/8.01 ( ord_less_eq_set_nat
% 14.17/8.01 = ( ^ [A3: set_nat,B7: set_nat] :
% 14.17/8.01 ! [T: nat] :
% 14.17/8.01 ( ( member_nat @ T @ A3 )
% 14.17/8.01 => ( member_nat @ T @ B7 ) ) ) ) ).
% 14.17/8.01
% 14.17/8.01 thf(fact_1105_subset__UnE,axiom,
% 14.17/8.01 ! [C5: set_nat,A: set_nat,B4: set_nat] :
% 14.17/8.01 ( ( ord_less_eq_set_nat @ C5 @ ( sup_sup_set_nat @ A @ B4 ) )
% 14.17/8.01 => ~ ! [A7: set_nat] :
% 14.17/8.01 ( ( ord_less_eq_set_nat @ A7 @ A )
% 14.17/8.01 => ! [B8: set_nat] :
% 14.17/8.01 ( ( ord_less_eq_set_nat @ B8 @ B4 )
% 14.17/8.01 => ( C5
% 14.17/8.01 != ( sup_sup_set_nat @ A7 @ B8 ) ) ) ) ) ).
% 14.17/8.01
% 14.17/8.01 thf(fact_1106_equalityD2,axiom,
% 14.17/8.01 ! [A: set_nat,B4: set_nat] :
% 14.17/8.01 ( ( A = B4 )
% 14.17/8.01 => ( ord_less_eq_set_nat @ B4 @ A ) ) ).
% 14.17/8.01
% 14.17/8.01 thf(fact_1107_equalityD1,axiom,
% 14.17/8.01 ! [A: set_nat,B4: set_nat] :
% 14.17/8.01 ( ( A = B4 )
% 14.17/8.01 => ( ord_less_eq_set_nat @ A @ B4 ) ) ).
% 14.17/8.01
% 14.17/8.01 thf(fact_1108_Un__commute,axiom,
% 14.17/8.01 ( sup_sup_set_nat
% 14.17/8.01 = ( ^ [A3: set_nat,B7: set_nat] : ( sup_sup_set_nat @ B7 @ A3 ) ) ) ).
% 14.17/8.01
% 14.17/8.01 thf(fact_1109_Un__absorb2,axiom,
% 14.17/8.01 ! [B4: set_nat,A: set_nat] :
% 14.17/8.01 ( ( ord_less_eq_set_nat @ B4 @ A )
% 14.17/8.01 => ( ( sup_sup_set_nat @ A @ B4 )
% 14.17/8.01 = A ) ) ).
% 14.17/8.01
% 14.17/8.01 thf(fact_1110_Un__absorb1,axiom,
% 14.17/8.01 ! [A: set_nat,B4: set_nat] :
% 14.17/8.01 ( ( ord_less_eq_set_nat @ A @ B4 )
% 14.17/8.01 => ( ( sup_sup_set_nat @ A @ B4 )
% 14.17/8.01 = B4 ) ) ).
% 14.17/8.01
% 14.17/8.01 thf(fact_1111_Int__lower2,axiom,
% 14.17/8.01 ! [A: set_nat,B4: set_nat] : ( ord_less_eq_set_nat @ ( inf_inf_set_nat @ A @ B4 ) @ B4 ) ).
% 14.17/8.01
% 14.17/8.01 thf(fact_1112_Int__lower1,axiom,
% 14.17/8.01 ! [A: set_nat,B4: set_nat] : ( ord_less_eq_set_nat @ ( inf_inf_set_nat @ A @ B4 ) @ A ) ).
% 14.17/8.01
% 14.17/8.01 thf(fact_1113_Int__absorb,axiom,
% 14.17/8.01 ! [A: set_nat] :
% 14.17/8.01 ( ( inf_inf_set_nat @ A @ A )
% 14.17/8.01 = A ) ).
% 14.17/8.01
% 14.17/8.01 thf(fact_1114_subset__eq,axiom,
% 14.17/8.01 ( ord_le2965882846123202637_nat_o
% 14.17/8.01 = ( ^ [A3: set_Pr4532377907799695533_nat_o,B7: set_Pr4532377907799695533_nat_o] :
% 14.17/8.01 ! [X2: produc3658429121746597890et_nat > $o] :
% 14.17/8.01 ( ( member6576561426505652726_nat_o @ X2 @ A3 )
% 14.17/8.01 => ( member6576561426505652726_nat_o @ X2 @ B7 ) ) ) ) ).
% 14.17/8.01
% 14.17/8.01 thf(fact_1115_subset__eq,axiom,
% 14.17/8.01 ( ord_less_eq_set_int
% 14.17/8.01 = ( ^ [A3: set_int,B7: set_int] :
% 14.17/8.01 ! [X2: int] :
% 14.17/8.01 ( ( member_int @ X2 @ A3 )
% 14.17/8.01 => ( member_int @ X2 @ B7 ) ) ) ) ).
% 14.17/8.01
% 14.17/8.01 thf(fact_1116_subset__eq,axiom,
% 14.17/8.01 ( ord_less_eq_set_o
% 14.17/8.01 = ( ^ [A3: set_o,B7: set_o] :
% 14.17/8.01 ! [X2: $o] :
% 14.17/8.01 ( ( member_o @ X2 @ A3 )
% 14.17/8.01 => ( member_o @ X2 @ B7 ) ) ) ) ).
% 14.17/8.01
% 14.17/8.01 thf(fact_1117_subset__eq,axiom,
% 14.17/8.01 ( ord_le3507040750410214029t_unit
% 14.17/8.01 = ( ^ [A3: set_Product_unit,B7: set_Product_unit] :
% 14.17/8.01 ! [X2: product_unit] :
% 14.17/8.01 ( ( member_Product_unit @ X2 @ A3 )
% 14.17/8.01 => ( member_Product_unit @ X2 @ B7 ) ) ) ) ).
% 14.17/8.01
% 14.17/8.01 thf(fact_1118_subset__eq,axiom,
% 14.17/8.01 ( ord_less_eq_set_nat
% 14.17/8.01 = ( ^ [A3: set_nat,B7: set_nat] :
% 14.17/8.01 ! [X2: nat] :
% 14.17/8.01 ( ( member_nat @ X2 @ A3 )
% 14.17/8.01 => ( member_nat @ X2 @ B7 ) ) ) ) ).
% 14.17/8.01
% 14.17/8.01 thf(fact_1119_equalityE,axiom,
% 14.17/8.01 ! [A: set_nat,B4: set_nat] :
% 14.17/8.01 ( ( A = B4 )
% 14.17/8.01 => ~ ( ( ord_less_eq_set_nat @ A @ B4 )
% 14.17/8.01 => ~ ( ord_less_eq_set_nat @ B4 @ A ) ) ) ).
% 14.17/8.01
% 14.17/8.01 thf(fact_1120_Un__upper2,axiom,
% 14.17/8.01 ! [B4: set_nat,A: set_nat] : ( ord_less_eq_set_nat @ B4 @ ( sup_sup_set_nat @ A @ B4 ) ) ).
% 14.17/8.01
% 14.17/8.01 thf(fact_1121_Un__upper1,axiom,
% 14.17/8.01 ! [A: set_nat,B4: set_nat] : ( ord_less_eq_set_nat @ A @ ( sup_sup_set_nat @ A @ B4 ) ) ).
% 14.17/8.01
% 14.17/8.01 thf(fact_1122_Un__absorb,axiom,
% 14.17/8.01 ! [A: set_nat] :
% 14.17/8.01 ( ( sup_sup_set_nat @ A @ A )
% 14.17/8.01 = A ) ).
% 14.17/8.01
% 14.17/8.01 thf(fact_1123_Int__assoc,axiom,
% 14.17/8.01 ! [A: set_nat,B4: set_nat,C5: set_nat] :
% 14.17/8.01 ( ( inf_inf_set_nat @ ( inf_inf_set_nat @ A @ B4 ) @ C5 )
% 14.17/8.01 = ( inf_inf_set_nat @ A @ ( inf_inf_set_nat @ B4 @ C5 ) ) ) ).
% 14.17/8.01
% 14.17/8.01 thf(fact_1124_Un__least,axiom,
% 14.17/8.01 ! [A: set_nat,C5: set_nat,B4: set_nat] :
% 14.17/8.01 ( ( ord_less_eq_set_nat @ A @ C5 )
% 14.17/8.01 => ( ( ord_less_eq_set_nat @ B4 @ C5 )
% 14.17/8.01 => ( ord_less_eq_set_nat @ ( sup_sup_set_nat @ A @ B4 ) @ C5 ) ) ) ).
% 14.17/8.01
% 14.17/8.01 thf(fact_1125_Un__assoc,axiom,
% 14.17/8.01 ! [A: set_nat,B4: set_nat,C5: set_nat] :
% 14.17/8.02 ( ( sup_sup_set_nat @ ( sup_sup_set_nat @ A @ B4 ) @ C5 )
% 14.17/8.02 = ( sup_sup_set_nat @ A @ ( sup_sup_set_nat @ B4 @ C5 ) ) ) ).
% 14.17/8.02
% 14.17/8.02 thf(fact_1126_Int__mono,axiom,
% 14.17/8.02 ! [A: set_nat,C5: set_nat,B4: set_nat,D3: set_nat] :
% 14.17/8.02 ( ( ord_less_eq_set_nat @ A @ C5 )
% 14.17/8.02 => ( ( ord_less_eq_set_nat @ B4 @ D3 )
% 14.17/8.02 => ( ord_less_eq_set_nat @ ( inf_inf_set_nat @ A @ B4 ) @ ( inf_inf_set_nat @ C5 @ D3 ) ) ) ) ).
% 14.17/8.02
% 14.17/8.02 thf(fact_1127_subsetD,axiom,
% 14.17/8.02 ! [A: set_Pr4532377907799695533_nat_o,B4: set_Pr4532377907799695533_nat_o,C: produc3658429121746597890et_nat > $o] :
% 14.17/8.02 ( ( ord_le2965882846123202637_nat_o @ A @ B4 )
% 14.17/8.02 => ( ( member6576561426505652726_nat_o @ C @ A )
% 14.17/8.02 => ( member6576561426505652726_nat_o @ C @ B4 ) ) ) ).
% 14.17/8.02
% 14.17/8.02 thf(fact_1128_subsetD,axiom,
% 14.17/8.02 ! [A: set_int,B4: set_int,C: int] :
% 14.17/8.02 ( ( ord_less_eq_set_int @ A @ B4 )
% 14.17/8.02 => ( ( member_int @ C @ A )
% 14.17/8.02 => ( member_int @ C @ B4 ) ) ) ).
% 14.17/8.02
% 14.17/8.02 thf(fact_1129_subsetD,axiom,
% 14.17/8.02 ! [A: set_o,B4: set_o,C: $o] :
% 14.17/8.02 ( ( ord_less_eq_set_o @ A @ B4 )
% 14.17/8.02 => ( ( member_o @ C @ A )
% 14.17/8.02 => ( member_o @ C @ B4 ) ) ) ).
% 14.17/8.02
% 14.17/8.02 thf(fact_1130_subsetD,axiom,
% 14.17/8.02 ! [A: set_Product_unit,B4: set_Product_unit,C: product_unit] :
% 14.17/8.02 ( ( ord_le3507040750410214029t_unit @ A @ B4 )
% 14.17/8.02 => ( ( member_Product_unit @ C @ A )
% 14.17/8.02 => ( member_Product_unit @ C @ B4 ) ) ) ).
% 14.17/8.02
% 14.17/8.02 thf(fact_1131_subsetD,axiom,
% 14.17/8.02 ! [A: set_nat,B4: set_nat,C: nat] :
% 14.17/8.02 ( ( ord_less_eq_set_nat @ A @ B4 )
% 14.17/8.02 => ( ( member_nat @ C @ A )
% 14.17/8.02 => ( member_nat @ C @ B4 ) ) ) ).
% 14.17/8.02
% 14.17/8.02 thf(fact_1132_in__mono,axiom,
% 14.17/8.02 ! [A: set_Pr4532377907799695533_nat_o,B4: set_Pr4532377907799695533_nat_o,X: produc3658429121746597890et_nat > $o] :
% 14.17/8.02 ( ( ord_le2965882846123202637_nat_o @ A @ B4 )
% 14.17/8.02 => ( ( member6576561426505652726_nat_o @ X @ A )
% 14.17/8.02 => ( member6576561426505652726_nat_o @ X @ B4 ) ) ) ).
% 14.17/8.02
% 14.17/8.02 thf(fact_1133_in__mono,axiom,
% 14.17/8.02 ! [A: set_int,B4: set_int,X: int] :
% 14.17/8.02 ( ( ord_less_eq_set_int @ A @ B4 )
% 14.17/8.02 => ( ( member_int @ X @ A )
% 14.17/8.02 => ( member_int @ X @ B4 ) ) ) ).
% 14.17/8.02
% 14.17/8.02 thf(fact_1134_in__mono,axiom,
% 14.17/8.02 ! [A: set_o,B4: set_o,X: $o] :
% 14.17/8.02 ( ( ord_less_eq_set_o @ A @ B4 )
% 14.17/8.02 => ( ( member_o @ X @ A )
% 14.17/8.02 => ( member_o @ X @ B4 ) ) ) ).
% 14.17/8.02
% 14.17/8.02 thf(fact_1135_in__mono,axiom,
% 14.17/8.02 ! [A: set_Product_unit,B4: set_Product_unit,X: product_unit] :
% 14.17/8.02 ( ( ord_le3507040750410214029t_unit @ A @ B4 )
% 14.17/8.02 => ( ( member_Product_unit @ X @ A )
% 14.17/8.02 => ( member_Product_unit @ X @ B4 ) ) ) ).
% 14.17/8.02
% 14.17/8.02 thf(fact_1136_in__mono,axiom,
% 14.17/8.02 ! [A: set_nat,B4: set_nat,X: nat] :
% 14.17/8.02 ( ( ord_less_eq_set_nat @ A @ B4 )
% 14.17/8.02 => ( ( member_nat @ X @ A )
% 14.17/8.02 => ( member_nat @ X @ B4 ) ) ) ).
% 14.17/8.02
% 14.17/8.02 thf(fact_1137_ball__Un,axiom,
% 14.17/8.02 ! [A: set_nat,B4: set_nat,P: nat > $o] :
% 14.17/8.02 ( ( ! [X2: nat] :
% 14.17/8.02 ( ( member_nat @ X2 @ ( sup_sup_set_nat @ A @ B4 ) )
% 14.17/8.02 => ( P @ X2 ) ) )
% 14.17/8.02 = ( ! [X2: nat] :
% 14.17/8.02 ( ( member_nat @ X2 @ A )
% 14.17/8.02 => ( P @ X2 ) )
% 14.17/8.02 & ! [X2: nat] :
% 14.17/8.02 ( ( member_nat @ X2 @ B4 )
% 14.17/8.02 => ( P @ X2 ) ) ) ) ).
% 14.17/8.02
% 14.17/8.02 thf(fact_1138_Un__mono,axiom,
% 14.17/8.02 ! [A: set_nat,C5: set_nat,B4: set_nat,D3: set_nat] :
% 14.17/8.02 ( ( ord_less_eq_set_nat @ A @ C5 )
% 14.17/8.02 => ( ( ord_less_eq_set_nat @ B4 @ D3 )
% 14.17/8.02 => ( ord_less_eq_set_nat @ ( sup_sup_set_nat @ A @ B4 ) @ ( sup_sup_set_nat @ C5 @ D3 ) ) ) ) ).
% 14.17/8.02
% 14.17/8.02 thf(fact_1139_bex__Un,axiom,
% 14.17/8.02 ! [A: set_nat,B4: set_nat,P: nat > $o] :
% 14.17/8.02 ( ( ? [X2: nat] :
% 14.17/8.02 ( ( member_nat @ X2 @ ( sup_sup_set_nat @ A @ B4 ) )
% 14.17/8.02 & ( P @ X2 ) ) )
% 14.17/8.02 = ( ? [X2: nat] :
% 14.17/8.02 ( ( member_nat @ X2 @ A )
% 14.17/8.02 & ( P @ X2 ) )
% 14.17/8.02 | ? [X2: nat] :
% 14.17/8.02 ( ( member_nat @ X2 @ B4 )
% 14.17/8.02 & ( P @ X2 ) ) ) ) ).
% 14.17/8.02
% 14.17/8.02 thf(fact_1140_IntD2,axiom,
% 14.17/8.02 ! [C: produc3658429121746597890et_nat > $o,A: set_Pr4532377907799695533_nat_o,B4: set_Pr4532377907799695533_nat_o] :
% 14.17/8.02 ( ( member6576561426505652726_nat_o @ C @ ( inf_in1906310914598751387_nat_o @ A @ B4 ) )
% 14.17/8.02 => ( member6576561426505652726_nat_o @ C @ B4 ) ) ).
% 14.17/8.02
% 14.17/8.02 thf(fact_1141_IntD2,axiom,
% 14.17/8.02 ! [C: int,A: set_int,B4: set_int] :
% 14.17/8.02 ( ( member_int @ C @ ( inf_inf_set_int @ A @ B4 ) )
% 14.17/8.02 => ( member_int @ C @ B4 ) ) ).
% 14.17/8.02
% 14.17/8.02 thf(fact_1142_IntD2,axiom,
% 14.17/8.02 ! [C: $o,A: set_o,B4: set_o] :
% 14.17/8.02 ( ( member_o @ C @ ( inf_inf_set_o @ A @ B4 ) )
% 14.17/8.02 => ( member_o @ C @ B4 ) ) ).
% 14.17/8.02
% 14.17/8.02 thf(fact_1143_IntD2,axiom,
% 14.17/8.02 ! [C: product_unit,A: set_Product_unit,B4: set_Product_unit] :
% 14.17/8.02 ( ( member_Product_unit @ C @ ( inf_in4660618365625256667t_unit @ A @ B4 ) )
% 14.17/8.02 => ( member_Product_unit @ C @ B4 ) ) ).
% 14.17/8.02
% 14.17/8.02 thf(fact_1144_IntD2,axiom,
% 14.17/8.02 ! [C: nat,A: set_nat,B4: set_nat] :
% 14.17/8.02 ( ( member_nat @ C @ ( inf_inf_set_nat @ A @ B4 ) )
% 14.17/8.02 => ( member_nat @ C @ B4 ) ) ).
% 14.17/8.02
% 14.17/8.02 thf(fact_1145_IntD1,axiom,
% 14.17/8.02 ! [C: produc3658429121746597890et_nat > $o,A: set_Pr4532377907799695533_nat_o,B4: set_Pr4532377907799695533_nat_o] :
% 14.17/8.02 ( ( member6576561426505652726_nat_o @ C @ ( inf_in1906310914598751387_nat_o @ A @ B4 ) )
% 14.17/8.02 => ( member6576561426505652726_nat_o @ C @ A ) ) ).
% 14.17/8.02
% 14.17/8.02 thf(fact_1146_IntD1,axiom,
% 14.17/8.02 ! [C: int,A: set_int,B4: set_int] :
% 14.17/8.02 ( ( member_int @ C @ ( inf_inf_set_int @ A @ B4 ) )
% 14.17/8.02 => ( member_int @ C @ A ) ) ).
% 14.17/8.02
% 14.17/8.02 thf(fact_1147_IntD1,axiom,
% 14.17/8.02 ! [C: $o,A: set_o,B4: set_o] :
% 14.17/8.02 ( ( member_o @ C @ ( inf_inf_set_o @ A @ B4 ) )
% 14.17/8.02 => ( member_o @ C @ A ) ) ).
% 14.17/8.02
% 14.17/8.02 thf(fact_1148_IntD1,axiom,
% 14.17/8.02 ! [C: product_unit,A: set_Product_unit,B4: set_Product_unit] :
% 14.17/8.02 ( ( member_Product_unit @ C @ ( inf_in4660618365625256667t_unit @ A @ B4 ) )
% 14.17/8.02 => ( member_Product_unit @ C @ A ) ) ).
% 14.17/8.02
% 14.17/8.02 thf(fact_1149_IntD1,axiom,
% 14.17/8.02 ! [C: nat,A: set_nat,B4: set_nat] :
% 14.17/8.02 ( ( member_nat @ C @ ( inf_inf_set_nat @ A @ B4 ) )
% 14.17/8.02 => ( member_nat @ C @ A ) ) ).
% 14.17/8.02
% 14.17/8.02 thf(fact_1150_UnI2,axiom,
% 14.17/8.02 ! [C: produc3658429121746597890et_nat > $o,B4: set_Pr4532377907799695533_nat_o,A: set_Pr4532377907799695533_nat_o] :
% 14.17/8.02 ( ( member6576561426505652726_nat_o @ C @ B4 )
% 14.17/8.02 => ( member6576561426505652726_nat_o @ C @ ( sup_su5209123915105501825_nat_o @ A @ B4 ) ) ) ).
% 14.17/8.02
% 14.17/8.02 thf(fact_1151_UnI2,axiom,
% 14.17/8.02 ! [C: int,B4: set_int,A: set_int] :
% 14.17/8.02 ( ( member_int @ C @ B4 )
% 14.17/8.02 => ( member_int @ C @ ( sup_sup_set_int @ A @ B4 ) ) ) ).
% 14.17/8.02
% 14.17/8.02 thf(fact_1152_UnI2,axiom,
% 14.17/8.02 ! [C: $o,B4: set_o,A: set_o] :
% 14.17/8.02 ( ( member_o @ C @ B4 )
% 14.17/8.02 => ( member_o @ C @ ( sup_sup_set_o @ A @ B4 ) ) ) ).
% 14.17/8.02
% 14.17/8.02 thf(fact_1153_UnI2,axiom,
% 14.17/8.02 ! [C: product_unit,B4: set_Product_unit,A: set_Product_unit] :
% 14.17/8.02 ( ( member_Product_unit @ C @ B4 )
% 14.17/8.02 => ( member_Product_unit @ C @ ( sup_su793286257634532545t_unit @ A @ B4 ) ) ) ).
% 14.17/8.02
% 14.17/8.02 thf(fact_1154_UnI2,axiom,
% 14.17/8.02 ! [C: nat,B4: set_nat,A: set_nat] :
% 14.17/8.02 ( ( member_nat @ C @ B4 )
% 14.17/8.02 => ( member_nat @ C @ ( sup_sup_set_nat @ A @ B4 ) ) ) ).
% 14.17/8.02
% 14.17/8.02 thf(fact_1155_UnI1,axiom,
% 14.17/8.02 ! [C: produc3658429121746597890et_nat > $o,A: set_Pr4532377907799695533_nat_o,B4: set_Pr4532377907799695533_nat_o] :
% 14.17/8.02 ( ( member6576561426505652726_nat_o @ C @ A )
% 14.17/8.02 => ( member6576561426505652726_nat_o @ C @ ( sup_su5209123915105501825_nat_o @ A @ B4 ) ) ) ).
% 14.17/8.02
% 14.17/8.02 thf(fact_1156_UnI1,axiom,
% 14.17/8.02 ! [C: int,A: set_int,B4: set_int] :
% 14.17/8.02 ( ( member_int @ C @ A )
% 14.17/8.02 => ( member_int @ C @ ( sup_sup_set_int @ A @ B4 ) ) ) ).
% 14.17/8.02
% 14.17/8.02 thf(fact_1157_UnI1,axiom,
% 14.17/8.02 ! [C: $o,A: set_o,B4: set_o] :
% 14.17/8.02 ( ( member_o @ C @ A )
% 14.17/8.02 => ( member_o @ C @ ( sup_sup_set_o @ A @ B4 ) ) ) ).
% 14.17/8.02
% 14.17/8.02 thf(fact_1158_UnI1,axiom,
% 14.17/8.02 ! [C: product_unit,A: set_Product_unit,B4: set_Product_unit] :
% 14.17/8.02 ( ( member_Product_unit @ C @ A )
% 14.17/8.02 => ( member_Product_unit @ C @ ( sup_su793286257634532545t_unit @ A @ B4 ) ) ) ).
% 14.17/8.02
% 14.17/8.02 thf(fact_1159_UnI1,axiom,
% 14.17/8.02 ! [C: nat,A: set_nat,B4: set_nat] :
% 14.17/8.02 ( ( member_nat @ C @ A )
% 14.17/8.02 => ( member_nat @ C @ ( sup_sup_set_nat @ A @ B4 ) ) ) ).
% 14.17/8.02
% 14.17/8.02 thf(fact_1160_IntE,axiom,
% 14.17/8.02 ! [C: produc3658429121746597890et_nat > $o,A: set_Pr4532377907799695533_nat_o,B4: set_Pr4532377907799695533_nat_o] :
% 14.17/8.02 ( ( member6576561426505652726_nat_o @ C @ ( inf_in1906310914598751387_nat_o @ A @ B4 ) )
% 14.17/8.02 => ~ ( ( member6576561426505652726_nat_o @ C @ A )
% 14.17/8.02 => ~ ( member6576561426505652726_nat_o @ C @ B4 ) ) ) ).
% 14.17/8.02
% 14.17/8.02 thf(fact_1161_IntE,axiom,
% 14.17/8.02 ! [C: int,A: set_int,B4: set_int] :
% 14.17/8.02 ( ( member_int @ C @ ( inf_inf_set_int @ A @ B4 ) )
% 14.17/8.02 => ~ ( ( member_int @ C @ A )
% 14.17/8.02 => ~ ( member_int @ C @ B4 ) ) ) ).
% 14.17/8.02
% 14.17/8.02 thf(fact_1162_IntE,axiom,
% 14.17/8.02 ! [C: $o,A: set_o,B4: set_o] :
% 14.17/8.02 ( ( member_o @ C @ ( inf_inf_set_o @ A @ B4 ) )
% 14.17/8.02 => ~ ( ( member_o @ C @ A )
% 14.17/8.02 => ~ ( member_o @ C @ B4 ) ) ) ).
% 14.17/8.02
% 14.17/8.02 thf(fact_1163_IntE,axiom,
% 14.17/8.02 ! [C: product_unit,A: set_Product_unit,B4: set_Product_unit] :
% 14.17/8.02 ( ( member_Product_unit @ C @ ( inf_in4660618365625256667t_unit @ A @ B4 ) )
% 14.17/8.02 => ~ ( ( member_Product_unit @ C @ A )
% 14.17/8.02 => ~ ( member_Product_unit @ C @ B4 ) ) ) ).
% 14.17/8.02
% 14.17/8.02 thf(fact_1164_IntE,axiom,
% 14.17/8.02 ! [C: nat,A: set_nat,B4: set_nat] :
% 14.17/8.02 ( ( member_nat @ C @ ( inf_inf_set_nat @ A @ B4 ) )
% 14.17/8.02 => ~ ( ( member_nat @ C @ A )
% 14.17/8.02 => ~ ( member_nat @ C @ B4 ) ) ) ).
% 14.17/8.02
% 14.17/8.02 thf(fact_1165_UnE,axiom,
% 14.17/8.02 ! [C: produc3658429121746597890et_nat > $o,A: set_Pr4532377907799695533_nat_o,B4: set_Pr4532377907799695533_nat_o] :
% 14.17/8.02 ( ( member6576561426505652726_nat_o @ C @ ( sup_su5209123915105501825_nat_o @ A @ B4 ) )
% 14.17/8.02 => ( ~ ( member6576561426505652726_nat_o @ C @ A )
% 14.17/8.02 => ( member6576561426505652726_nat_o @ C @ B4 ) ) ) ).
% 14.17/8.02
% 14.17/8.02 thf(fact_1166_UnE,axiom,
% 14.17/8.02 ! [C: int,A: set_int,B4: set_int] :
% 14.17/8.02 ( ( member_int @ C @ ( sup_sup_set_int @ A @ B4 ) )
% 14.17/8.02 => ( ~ ( member_int @ C @ A )
% 14.17/8.02 => ( member_int @ C @ B4 ) ) ) ).
% 14.17/8.02
% 14.17/8.02 thf(fact_1167_UnE,axiom,
% 14.17/8.02 ! [C: $o,A: set_o,B4: set_o] :
% 14.17/8.02 ( ( member_o @ C @ ( sup_sup_set_o @ A @ B4 ) )
% 14.17/8.02 => ( ~ ( member_o @ C @ A )
% 14.17/8.02 => ( member_o @ C @ B4 ) ) ) ).
% 14.17/8.02
% 14.17/8.02 thf(fact_1168_UnE,axiom,
% 14.17/8.02 ! [C: product_unit,A: set_Product_unit,B4: set_Product_unit] :
% 14.17/8.02 ( ( member_Product_unit @ C @ ( sup_su793286257634532545t_unit @ A @ B4 ) )
% 14.17/8.02 => ( ~ ( member_Product_unit @ C @ A )
% 14.17/8.02 => ( member_Product_unit @ C @ B4 ) ) ) ).
% 14.17/8.02
% 14.17/8.02 thf(fact_1169_UnE,axiom,
% 14.17/8.02 ! [C: nat,A: set_nat,B4: set_nat] :
% 14.17/8.02 ( ( member_nat @ C @ ( sup_sup_set_nat @ A @ B4 ) )
% 14.17/8.02 => ( ~ ( member_nat @ C @ A )
% 14.17/8.02 => ( member_nat @ C @ B4 ) ) ) ).
% 14.17/8.02
% 14.17/8.02 thf(fact_1170_dual__order_Otrans,axiom,
% 14.17/8.02 ! [B: filter_nat,A2: filter_nat,C: filter_nat] :
% 14.17/8.02 ( ( ord_le2510731241096832064er_nat @ B @ A2 )
% 14.17/8.02 => ( ( ord_le2510731241096832064er_nat @ C @ B )
% 14.17/8.02 => ( ord_le2510731241096832064er_nat @ C @ A2 ) ) ) ).
% 14.17/8.02
% 14.17/8.02 thf(fact_1171_dual__order_Otrans,axiom,
% 14.17/8.02 ! [B: set_nat,A2: set_nat,C: set_nat] :
% 14.17/8.02 ( ( ord_less_eq_set_nat @ B @ A2 )
% 14.17/8.02 => ( ( ord_less_eq_set_nat @ C @ B )
% 14.17/8.02 => ( ord_less_eq_set_nat @ C @ A2 ) ) ) ).
% 14.17/8.02
% 14.17/8.02 thf(fact_1172_dual__order_Otrans,axiom,
% 14.17/8.02 ! [B: rat,A2: rat,C: rat] :
% 14.17/8.02 ( ( ord_less_eq_rat @ B @ A2 )
% 14.17/8.02 => ( ( ord_less_eq_rat @ C @ B )
% 14.17/8.02 => ( ord_less_eq_rat @ C @ A2 ) ) ) ).
% 14.17/8.02
% 14.17/8.02 thf(fact_1173_dual__order_Otrans,axiom,
% 14.17/8.02 ! [B: nat,A2: nat,C: nat] :
% 14.17/8.02 ( ( ord_less_eq_nat @ B @ A2 )
% 14.17/8.02 => ( ( ord_less_eq_nat @ C @ B )
% 14.17/8.02 => ( ord_less_eq_nat @ C @ A2 ) ) ) ).
% 14.17/8.02
% 14.17/8.02 thf(fact_1174_dual__order_Otrans,axiom,
% 14.17/8.02 ! [B: int,A2: int,C: int] :
% 14.17/8.02 ( ( ord_less_eq_int @ B @ A2 )
% 14.17/8.02 => ( ( ord_less_eq_int @ C @ B )
% 14.17/8.02 => ( ord_less_eq_int @ C @ A2 ) ) ) ).
% 14.17/8.02
% 14.17/8.02 thf(fact_1175_dual__order_Oantisym,axiom,
% 14.17/8.02 ! [B: filter_nat,A2: filter_nat] :
% 14.17/8.02 ( ( ord_le2510731241096832064er_nat @ B @ A2 )
% 14.17/8.02 => ( ( ord_le2510731241096832064er_nat @ A2 @ B )
% 14.17/8.02 => ( A2 = B ) ) ) ).
% 14.17/8.02
% 14.17/8.02 thf(fact_1176_dual__order_Oantisym,axiom,
% 14.17/8.02 ! [B: set_nat,A2: set_nat] :
% 14.17/8.02 ( ( ord_less_eq_set_nat @ B @ A2 )
% 14.17/8.02 => ( ( ord_less_eq_set_nat @ A2 @ B )
% 14.17/8.02 => ( A2 = B ) ) ) ).
% 14.17/8.02
% 14.17/8.02 thf(fact_1177_dual__order_Oantisym,axiom,
% 14.17/8.02 ! [B: rat,A2: rat] :
% 14.17/8.02 ( ( ord_less_eq_rat @ B @ A2 )
% 14.17/8.02 => ( ( ord_less_eq_rat @ A2 @ B )
% 14.17/8.02 => ( A2 = B ) ) ) ).
% 14.17/8.02
% 14.17/8.02 thf(fact_1178_dual__order_Oantisym,axiom,
% 14.17/8.02 ! [B: nat,A2: nat] :
% 14.17/8.02 ( ( ord_less_eq_nat @ B @ A2 )
% 14.17/8.02 => ( ( ord_less_eq_nat @ A2 @ B )
% 14.17/8.02 => ( A2 = B ) ) ) ).
% 14.17/8.02
% 14.17/8.02 thf(fact_1179_dual__order_Oantisym,axiom,
% 14.17/8.02 ! [B: int,A2: int] :
% 14.17/8.02 ( ( ord_less_eq_int @ B @ A2 )
% 14.17/8.02 => ( ( ord_less_eq_int @ A2 @ B )
% 14.17/8.02 => ( A2 = B ) ) ) ).
% 14.17/8.02
% 14.17/8.02 thf(fact_1180_dual__order_Oeq__iff,axiom,
% 14.17/8.02 ( ( ^ [Y4: filter_nat,Z4: filter_nat] : ( Y4 = Z4 ) )
% 14.17/8.02 = ( ^ [A6: filter_nat,B6: filter_nat] :
% 14.17/8.02 ( ( ord_le2510731241096832064er_nat @ B6 @ A6 )
% 14.17/8.02 & ( ord_le2510731241096832064er_nat @ A6 @ B6 ) ) ) ) ).
% 14.17/8.02
% 14.17/8.02 thf(fact_1181_dual__order_Oeq__iff,axiom,
% 14.17/8.02 ( ( ^ [Y4: set_nat,Z4: set_nat] : ( Y4 = Z4 ) )
% 14.17/8.02 = ( ^ [A6: set_nat,B6: set_nat] :
% 14.17/8.02 ( ( ord_less_eq_set_nat @ B6 @ A6 )
% 14.17/8.02 & ( ord_less_eq_set_nat @ A6 @ B6 ) ) ) ) ).
% 14.17/8.02
% 14.17/8.02 thf(fact_1182_dual__order_Oeq__iff,axiom,
% 14.17/8.02 ( ( ^ [Y4: rat,Z4: rat] : ( Y4 = Z4 ) )
% 14.17/8.02 = ( ^ [A6: rat,B6: rat] :
% 14.17/8.02 ( ( ord_less_eq_rat @ B6 @ A6 )
% 14.17/8.02 & ( ord_less_eq_rat @ A6 @ B6 ) ) ) ) ).
% 14.17/8.02
% 14.17/8.02 thf(fact_1183_dual__order_Oeq__iff,axiom,
% 14.17/8.02 ( ( ^ [Y4: nat,Z4: nat] : ( Y4 = Z4 ) )
% 14.17/8.02 = ( ^ [A6: nat,B6: nat] :
% 14.17/8.02 ( ( ord_less_eq_nat @ B6 @ A6 )
% 14.17/8.02 & ( ord_less_eq_nat @ A6 @ B6 ) ) ) ) ).
% 14.17/8.02
% 14.17/8.02 thf(fact_1184_dual__order_Oeq__iff,axiom,
% 14.17/8.02 ( ( ^ [Y4: int,Z4: int] : ( Y4 = Z4 ) )
% 14.17/8.02 = ( ^ [A6: int,B6: int] :
% 14.17/8.02 ( ( ord_less_eq_int @ B6 @ A6 )
% 14.17/8.02 & ( ord_less_eq_int @ A6 @ B6 ) ) ) ) ).
% 14.17/8.02
% 14.17/8.02 thf(fact_1185_linorder__wlog,axiom,
% 14.17/8.02 ! [P: rat > rat > $o,A2: rat,B: rat] :
% 14.17/8.02 ( ! [A4: rat,B2: rat] :
% 14.17/8.02 ( ( ord_less_eq_rat @ A4 @ B2 )
% 14.17/8.02 => ( P @ A4 @ B2 ) )
% 14.17/8.02 => ( ! [A4: rat,B2: rat] :
% 14.17/8.02 ( ( P @ B2 @ A4 )
% 14.17/8.02 => ( P @ A4 @ B2 ) )
% 14.17/8.02 => ( P @ A2 @ B ) ) ) ).
% 14.17/8.02
% 14.17/8.02 thf(fact_1186_linorder__wlog,axiom,
% 14.17/8.02 ! [P: nat > nat > $o,A2: nat,B: nat] :
% 14.17/8.02 ( ! [A4: nat,B2: nat] :
% 14.17/8.02 ( ( ord_less_eq_nat @ A4 @ B2 )
% 14.17/8.02 => ( P @ A4 @ B2 ) )
% 14.17/8.02 => ( ! [A4: nat,B2: nat] :
% 14.17/8.02 ( ( P @ B2 @ A4 )
% 14.17/8.02 => ( P @ A4 @ B2 ) )
% 14.17/8.02 => ( P @ A2 @ B ) ) ) ).
% 14.17/8.02
% 14.17/8.02 thf(fact_1187_linorder__wlog,axiom,
% 14.17/8.02 ! [P: int > int > $o,A2: int,B: int] :
% 14.17/8.02 ( ! [A4: int,B2: int] :
% 14.17/8.02 ( ( ord_less_eq_int @ A4 @ B2 )
% 14.17/8.02 => ( P @ A4 @ B2 ) )
% 14.17/8.02 => ( ! [A4: int,B2: int] :
% 14.17/8.02 ( ( P @ B2 @ A4 )
% 14.17/8.02 => ( P @ A4 @ B2 ) )
% 14.17/8.02 => ( P @ A2 @ B ) ) ) ).
% 14.17/8.02
% 14.17/8.02 thf(fact_1188_order__trans,axiom,
% 14.17/8.02 ! [X: filter_nat,Y: filter_nat,Z: filter_nat] :
% 14.17/8.02 ( ( ord_le2510731241096832064er_nat @ X @ Y )
% 14.17/8.02 => ( ( ord_le2510731241096832064er_nat @ Y @ Z )
% 14.17/8.02 => ( ord_le2510731241096832064er_nat @ X @ Z ) ) ) ).
% 14.17/8.02
% 14.17/8.02 thf(fact_1189_order__trans,axiom,
% 14.17/8.02 ! [X: set_nat,Y: set_nat,Z: set_nat] :
% 14.17/8.02 ( ( ord_less_eq_set_nat @ X @ Y )
% 14.17/8.02 => ( ( ord_less_eq_set_nat @ Y @ Z )
% 14.17/8.02 => ( ord_less_eq_set_nat @ X @ Z ) ) ) ).
% 14.17/8.02
% 14.17/8.02 thf(fact_1190_order__trans,axiom,
% 14.17/8.02 ! [X: rat,Y: rat,Z: rat] :
% 14.17/8.02 ( ( ord_less_eq_rat @ X @ Y )
% 14.17/8.02 => ( ( ord_less_eq_rat @ Y @ Z )
% 14.17/8.02 => ( ord_less_eq_rat @ X @ Z ) ) ) ).
% 14.17/8.02
% 14.17/8.02 thf(fact_1191_order__trans,axiom,
% 14.17/8.02 ! [X: nat,Y: nat,Z: nat] :
% 14.17/8.02 ( ( ord_less_eq_nat @ X @ Y )
% 14.17/8.02 => ( ( ord_less_eq_nat @ Y @ Z )
% 14.17/8.02 => ( ord_less_eq_nat @ X @ Z ) ) ) ).
% 14.17/8.02
% 14.17/8.02 thf(fact_1192_order__trans,axiom,
% 14.17/8.02 ! [X: int,Y: int,Z: int] :
% 14.17/8.02 ( ( ord_less_eq_int @ X @ Y )
% 14.17/8.02 => ( ( ord_less_eq_int @ Y @ Z )
% 14.17/8.02 => ( ord_less_eq_int @ X @ Z ) ) ) ).
% 14.17/8.02
% 14.17/8.02 thf(fact_1193_order_Otrans,axiom,
% 14.17/8.02 ! [A2: filter_nat,B: filter_nat,C: filter_nat] :
% 14.17/8.02 ( ( ord_le2510731241096832064er_nat @ A2 @ B )
% 14.17/8.02 => ( ( ord_le2510731241096832064er_nat @ B @ C )
% 14.17/8.02 => ( ord_le2510731241096832064er_nat @ A2 @ C ) ) ) ).
% 14.17/8.02
% 14.17/8.02 thf(fact_1194_order_Otrans,axiom,
% 14.17/8.02 ! [A2: set_nat,B: set_nat,C: set_nat] :
% 14.17/8.02 ( ( ord_less_eq_set_nat @ A2 @ B )
% 14.17/8.02 => ( ( ord_less_eq_set_nat @ B @ C )
% 14.17/8.02 => ( ord_less_eq_set_nat @ A2 @ C ) ) ) ).
% 14.17/8.02
% 14.17/8.02 thf(fact_1195_order_Otrans,axiom,
% 14.17/8.02 ! [A2: rat,B: rat,C: rat] :
% 14.17/8.02 ( ( ord_less_eq_rat @ A2 @ B )
% 14.17/8.02 => ( ( ord_less_eq_rat @ B @ C )
% 14.17/8.02 => ( ord_less_eq_rat @ A2 @ C ) ) ) ).
% 14.17/8.02
% 14.17/8.02 thf(fact_1196_order_Otrans,axiom,
% 14.17/8.02 ! [A2: nat,B: nat,C: nat] :
% 14.17/8.02 ( ( ord_less_eq_nat @ A2 @ B )
% 14.17/8.02 => ( ( ord_less_eq_nat @ B @ C )
% 14.17/8.02 => ( ord_less_eq_nat @ A2 @ C ) ) ) ).
% 14.17/8.02
% 14.17/8.02 thf(fact_1197_order_Otrans,axiom,
% 14.17/8.02 ! [A2: int,B: int,C: int] :
% 14.17/8.02 ( ( ord_less_eq_int @ A2 @ B )
% 14.17/8.02 => ( ( ord_less_eq_int @ B @ C )
% 14.17/8.02 => ( ord_less_eq_int @ A2 @ C ) ) ) ).
% 14.17/8.02
% 14.17/8.02 thf(fact_1198_order__antisym,axiom,
% 14.17/8.02 ! [X: filter_nat,Y: filter_nat] :
% 14.17/8.02 ( ( ord_le2510731241096832064er_nat @ X @ Y )
% 14.17/8.02 => ( ( ord_le2510731241096832064er_nat @ Y @ X )
% 14.17/8.02 => ( X = Y ) ) ) ).
% 14.17/8.02
% 14.17/8.02 thf(fact_1199_order__antisym,axiom,
% 14.17/8.02 ! [X: set_nat,Y: set_nat] :
% 14.17/8.02 ( ( ord_less_eq_set_nat @ X @ Y )
% 14.17/8.02 => ( ( ord_less_eq_set_nat @ Y @ X )
% 14.17/8.02 => ( X = Y ) ) ) ).
% 14.17/8.02
% 14.17/8.02 thf(fact_1200_order__antisym,axiom,
% 14.17/8.02 ! [X: rat,Y: rat] :
% 14.17/8.02 ( ( ord_less_eq_rat @ X @ Y )
% 14.17/8.02 => ( ( ord_less_eq_rat @ Y @ X )
% 14.17/8.02 => ( X = Y ) ) ) ).
% 14.17/8.02
% 14.17/8.02 thf(fact_1201_order__antisym,axiom,
% 14.17/8.02 ! [X: nat,Y: nat] :
% 14.17/8.02 ( ( ord_less_eq_nat @ X @ Y )
% 14.17/8.02 => ( ( ord_less_eq_nat @ Y @ X )
% 14.17/8.02 => ( X = Y ) ) ) ).
% 14.17/8.02
% 14.17/8.02 thf(fact_1202_order__antisym,axiom,
% 14.17/8.02 ! [X: int,Y: int] :
% 14.17/8.02 ( ( ord_less_eq_int @ X @ Y )
% 14.17/8.02 => ( ( ord_less_eq_int @ Y @ X )
% 14.17/8.02 => ( X = Y ) ) ) ).
% 14.17/8.02
% 14.17/8.02 thf(fact_1203_ord__le__eq__trans,axiom,
% 14.17/8.02 ! [A2: filter_nat,B: filter_nat,C: filter_nat] :
% 14.17/8.02 ( ( ord_le2510731241096832064er_nat @ A2 @ B )
% 14.17/8.02 => ( ( B = C )
% 14.17/8.02 => ( ord_le2510731241096832064er_nat @ A2 @ C ) ) ) ).
% 14.17/8.02
% 14.17/8.02 thf(fact_1204_ord__le__eq__trans,axiom,
% 14.17/8.02 ! [A2: set_nat,B: set_nat,C: set_nat] :
% 14.17/8.02 ( ( ord_less_eq_set_nat @ A2 @ B )
% 14.17/8.02 => ( ( B = C )
% 14.17/8.02 => ( ord_less_eq_set_nat @ A2 @ C ) ) ) ).
% 14.17/8.02
% 14.17/8.02 thf(fact_1205_ord__le__eq__trans,axiom,
% 14.17/8.02 ! [A2: rat,B: rat,C: rat] :
% 14.17/8.02 ( ( ord_less_eq_rat @ A2 @ B )
% 14.17/8.02 => ( ( B = C )
% 14.17/8.02 => ( ord_less_eq_rat @ A2 @ C ) ) ) ).
% 14.17/8.02
% 14.17/8.02 thf(fact_1206_ord__le__eq__trans,axiom,
% 14.17/8.02 ! [A2: nat,B: nat,C: nat] :
% 14.17/8.02 ( ( ord_less_eq_nat @ A2 @ B )
% 14.17/8.02 => ( ( B = C )
% 14.17/8.02 => ( ord_less_eq_nat @ A2 @ C ) ) ) ).
% 14.17/8.02
% 14.17/8.02 thf(fact_1207_ord__le__eq__trans,axiom,
% 14.17/8.02 ! [A2: int,B: int,C: int] :
% 14.17/8.02 ( ( ord_less_eq_int @ A2 @ B )
% 14.17/8.02 => ( ( B = C )
% 14.17/8.02 => ( ord_less_eq_int @ A2 @ C ) ) ) ).
% 14.17/8.02
% 14.17/8.02 thf(fact_1208_ord__eq__le__trans,axiom,
% 14.17/8.02 ! [A2: filter_nat,B: filter_nat,C: filter_nat] :
% 14.17/8.02 ( ( A2 = B )
% 14.17/8.02 => ( ( ord_le2510731241096832064er_nat @ B @ C )
% 14.17/8.02 => ( ord_le2510731241096832064er_nat @ A2 @ C ) ) ) ).
% 14.17/8.02
% 14.17/8.02 thf(fact_1209_ord__eq__le__trans,axiom,
% 14.17/8.02 ! [A2: set_nat,B: set_nat,C: set_nat] :
% 14.17/8.02 ( ( A2 = B )
% 14.17/8.02 => ( ( ord_less_eq_set_nat @ B @ C )
% 14.17/8.02 => ( ord_less_eq_set_nat @ A2 @ C ) ) ) ).
% 14.17/8.02
% 14.17/8.02 thf(fact_1210_ord__eq__le__trans,axiom,
% 14.17/8.02 ! [A2: rat,B: rat,C: rat] :
% 14.17/8.02 ( ( A2 = B )
% 14.17/8.02 => ( ( ord_less_eq_rat @ B @ C )
% 14.17/8.02 => ( ord_less_eq_rat @ A2 @ C ) ) ) ).
% 14.17/8.02
% 14.17/8.02 thf(fact_1211_ord__eq__le__trans,axiom,
% 14.17/8.02 ! [A2: nat,B: nat,C: nat] :
% 14.17/8.02 ( ( A2 = B )
% 14.17/8.02 => ( ( ord_less_eq_nat @ B @ C )
% 14.17/8.02 => ( ord_less_eq_nat @ A2 @ C ) ) ) ).
% 14.17/8.02
% 14.17/8.02 thf(fact_1212_ord__eq__le__trans,axiom,
% 14.17/8.02 ! [A2: int,B: int,C: int] :
% 14.17/8.02 ( ( A2 = B )
% 14.17/8.02 => ( ( ord_less_eq_int @ B @ C )
% 14.17/8.02 => ( ord_less_eq_int @ A2 @ C ) ) ) ).
% 14.17/8.02
% 14.17/8.02 thf(fact_1213_order__class_Oorder__eq__iff,axiom,
% 14.17/8.02 ( ( ^ [Y4: filter_nat,Z4: filter_nat] : ( Y4 = Z4 ) )
% 14.17/8.02 = ( ^ [X2: filter_nat,Y3: filter_nat] :
% 14.17/8.02 ( ( ord_le2510731241096832064er_nat @ X2 @ Y3 )
% 14.17/8.02 & ( ord_le2510731241096832064er_nat @ Y3 @ X2 ) ) ) ) ).
% 14.17/8.02
% 14.17/8.02 thf(fact_1214_order__class_Oorder__eq__iff,axiom,
% 14.17/8.02 ( ( ^ [Y4: set_nat,Z4: set_nat] : ( Y4 = Z4 ) )
% 14.17/8.02 = ( ^ [X2: set_nat,Y3: set_nat] :
% 14.17/8.02 ( ( ord_less_eq_set_nat @ X2 @ Y3 )
% 14.17/8.02 & ( ord_less_eq_set_nat @ Y3 @ X2 ) ) ) ) ).
% 14.17/8.02
% 14.17/8.02 thf(fact_1215_order__class_Oorder__eq__iff,axiom,
% 14.17/8.02 ( ( ^ [Y4: rat,Z4: rat] : ( Y4 = Z4 ) )
% 14.17/8.02 = ( ^ [X2: rat,Y3: rat] :
% 14.17/8.02 ( ( ord_less_eq_rat @ X2 @ Y3 )
% 14.17/8.02 & ( ord_less_eq_rat @ Y3 @ X2 ) ) ) ) ).
% 14.17/8.02
% 14.17/8.02 thf(fact_1216_order__class_Oorder__eq__iff,axiom,
% 14.17/8.02 ( ( ^ [Y4: nat,Z4: nat] : ( Y4 = Z4 ) )
% 14.17/8.02 = ( ^ [X2: nat,Y3: nat] :
% 14.17/8.02 ( ( ord_less_eq_nat @ X2 @ Y3 )
% 14.17/8.02 & ( ord_less_eq_nat @ Y3 @ X2 ) ) ) ) ).
% 14.17/8.02
% 14.17/8.02 thf(fact_1217_order__class_Oorder__eq__iff,axiom,
% 14.17/8.02 ( ( ^ [Y4: int,Z4: int] : ( Y4 = Z4 ) )
% 14.17/8.02 = ( ^ [X2: int,Y3: int] :
% 14.17/8.02 ( ( ord_less_eq_int @ X2 @ Y3 )
% 14.17/8.02 & ( ord_less_eq_int @ Y3 @ X2 ) ) ) ) ).
% 14.17/8.02
% 14.17/8.02 thf(fact_1218_le__cases3,axiom,
% 14.17/8.02 ! [X: rat,Y: rat,Z: rat] :
% 14.17/8.02 ( ( ( ord_less_eq_rat @ X @ Y )
% 14.17/8.02 => ~ ( ord_less_eq_rat @ Y @ Z ) )
% 14.17/8.02 => ( ( ( ord_less_eq_rat @ Y @ X )
% 14.17/8.02 => ~ ( ord_less_eq_rat @ X @ Z ) )
% 14.17/8.02 => ( ( ( ord_less_eq_rat @ X @ Z )
% 14.17/8.02 => ~ ( ord_less_eq_rat @ Z @ Y ) )
% 14.17/8.02 => ( ( ( ord_less_eq_rat @ Z @ Y )
% 14.17/8.02 => ~ ( ord_less_eq_rat @ Y @ X ) )
% 14.17/8.02 => ( ( ( ord_less_eq_rat @ Y @ Z )
% 14.17/8.02 => ~ ( ord_less_eq_rat @ Z @ X ) )
% 14.17/8.02 => ~ ( ( ord_less_eq_rat @ Z @ X )
% 14.17/8.02 => ~ ( ord_less_eq_rat @ X @ Y ) ) ) ) ) ) ) ).
% 14.17/8.02
% 14.17/8.02 thf(fact_1219_le__cases3,axiom,
% 14.17/8.02 ! [X: nat,Y: nat,Z: nat] :
% 14.17/8.02 ( ( ( ord_less_eq_nat @ X @ Y )
% 14.17/8.02 => ~ ( ord_less_eq_nat @ Y @ Z ) )
% 14.17/8.02 => ( ( ( ord_less_eq_nat @ Y @ X )
% 14.17/8.02 => ~ ( ord_less_eq_nat @ X @ Z ) )
% 14.17/8.02 => ( ( ( ord_less_eq_nat @ X @ Z )
% 14.17/8.02 => ~ ( ord_less_eq_nat @ Z @ Y ) )
% 14.17/8.02 => ( ( ( ord_less_eq_nat @ Z @ Y )
% 14.17/8.02 => ~ ( ord_less_eq_nat @ Y @ X ) )
% 14.17/8.02 => ( ( ( ord_less_eq_nat @ Y @ Z )
% 14.17/8.02 => ~ ( ord_less_eq_nat @ Z @ X ) )
% 14.17/8.02 => ~ ( ( ord_less_eq_nat @ Z @ X )
% 14.17/8.02 => ~ ( ord_less_eq_nat @ X @ Y ) ) ) ) ) ) ) ).
% 14.17/8.02
% 14.17/8.02 thf(fact_1220_le__cases3,axiom,
% 14.17/8.02 ! [X: int,Y: int,Z: int] :
% 14.17/8.02 ( ( ( ord_less_eq_int @ X @ Y )
% 14.17/8.02 => ~ ( ord_less_eq_int @ Y @ Z ) )
% 14.17/8.02 => ( ( ( ord_less_eq_int @ Y @ X )
% 14.17/8.02 => ~ ( ord_less_eq_int @ X @ Z ) )
% 14.17/8.02 => ( ( ( ord_less_eq_int @ X @ Z )
% 14.17/8.02 => ~ ( ord_less_eq_int @ Z @ Y ) )
% 14.17/8.02 => ( ( ( ord_less_eq_int @ Z @ Y )
% 14.17/8.02 => ~ ( ord_less_eq_int @ Y @ X ) )
% 14.17/8.02 => ( ( ( ord_less_eq_int @ Y @ Z )
% 14.17/8.02 => ~ ( ord_less_eq_int @ Z @ X ) )
% 14.17/8.02 => ~ ( ( ord_less_eq_int @ Z @ X )
% 14.17/8.02 => ~ ( ord_less_eq_int @ X @ Y ) ) ) ) ) ) ) ).
% 14.17/8.02
% 14.17/8.02 thf(fact_1221_nle__le,axiom,
% 14.17/8.02 ! [A2: rat,B: rat] :
% 14.17/8.02 ( ( ~ ( ord_less_eq_rat @ A2 @ B ) )
% 14.17/8.02 = ( ( ord_less_eq_rat @ B @ A2 )
% 14.17/8.02 & ( B != A2 ) ) ) ).
% 14.17/8.02
% 14.17/8.02 thf(fact_1222_nle__le,axiom,
% 14.17/8.02 ! [A2: nat,B: nat] :
% 14.17/8.02 ( ( ~ ( ord_less_eq_nat @ A2 @ B ) )
% 14.17/8.02 = ( ( ord_less_eq_nat @ B @ A2 )
% 14.17/8.02 & ( B != A2 ) ) ) ).
% 14.17/8.02
% 14.17/8.02 thf(fact_1223_nle__le,axiom,
% 14.17/8.02 ! [A2: int,B: int] :
% 14.17/8.02 ( ( ~ ( ord_less_eq_int @ A2 @ B ) )
% 14.17/8.02 = ( ( ord_less_eq_int @ B @ A2 )
% 14.17/8.02 & ( B != A2 ) ) ) ).
% 14.17/8.02
% 14.17/8.02 thf(fact_1224_Sigma__mono,axiom,
% 14.17/8.02 ! [A: set_Pr4532377907799695533_nat_o,C5: set_Pr4532377907799695533_nat_o,B4: ( produc3658429121746597890et_nat > $o ) > set_nat,D3: ( produc3658429121746597890et_nat > $o ) > set_nat] :
% 14.17/8.02 ( ( ord_le2965882846123202637_nat_o @ A @ C5 )
% 14.17/8.02 => ( ! [X3: produc3658429121746597890et_nat > $o] :
% 14.17/8.02 ( ( member6576561426505652726_nat_o @ X3 @ A )
% 14.17/8.02 => ( ord_less_eq_set_nat @ ( B4 @ X3 ) @ ( D3 @ X3 ) ) )
% 14.17/8.02 => ( ord_le4460016228128491106_o_nat @ ( produc3142209312014666069_o_nat @ A @ B4 ) @ ( produc3142209312014666069_o_nat @ C5 @ D3 ) ) ) ) ).
% 14.17/8.02
% 14.17/8.02 thf(fact_1225_Sigma__mono,axiom,
% 14.17/8.02 ! [A: set_int,C5: set_int,B4: int > set_nat,D3: int > set_nat] :
% 14.17/8.02 ( ( ord_less_eq_set_int @ A @ C5 )
% 14.17/8.02 => ( ! [X3: int] :
% 14.17/8.02 ( ( member_int @ X3 @ A )
% 14.17/8.02 => ( ord_less_eq_set_nat @ ( B4 @ X3 ) @ ( D3 @ X3 ) ) )
% 14.17/8.02 => ( ord_le5333435103577920093nt_nat @ ( produc1456381018704787142nt_nat @ A @ B4 ) @ ( produc1456381018704787142nt_nat @ C5 @ D3 ) ) ) ) ).
% 14.17/8.02
% 14.17/8.02 thf(fact_1226_Sigma__mono,axiom,
% 14.17/8.02 ! [A: set_o,C5: set_o,B4: $o > set_nat,D3: $o > set_nat] :
% 14.17/8.02 ( ( ord_less_eq_set_o @ A @ C5 )
% 14.17/8.02 => ( ! [X3: $o] :
% 14.17/8.02 ( ( member_o @ X3 @ A )
% 14.17/8.02 => ( ord_less_eq_set_nat @ ( B4 @ X3 ) @ ( D3 @ X3 ) ) )
% 14.17/8.02 => ( ord_le6472378536984071597_o_nat @ ( product_Sigma_o_nat @ A @ B4 ) @ ( product_Sigma_o_nat @ C5 @ D3 ) ) ) ) ).
% 14.17/8.02
% 14.17/8.02 thf(fact_1227_Sigma__mono,axiom,
% 14.17/8.02 ! [A: set_Product_unit,C5: set_Product_unit,B4: product_unit > set_nat,D3: product_unit > set_nat] :
% 14.17/8.02 ( ( ord_le3507040750410214029t_unit @ A @ C5 )
% 14.17/8.02 => ( ! [X3: product_unit] :
% 14.17/8.02 ( ( member_Product_unit @ X3 @ A )
% 14.17/8.02 => ( ord_less_eq_set_nat @ ( B4 @ X3 ) @ ( D3 @ X3 ) ) )
% 14.17/8.02 => ( ord_le8121013740822022818it_nat @ ( produc6875377002296012693it_nat @ A @ B4 ) @ ( produc6875377002296012693it_nat @ C5 @ D3 ) ) ) ) ).
% 14.17/8.02
% 14.17/8.02 thf(fact_1228_Sigma__mono,axiom,
% 14.17/8.02 ! [A: set_nat,C5: set_nat,B4: nat > set_nat,D3: nat > set_nat] :
% 14.17/8.02 ( ( ord_less_eq_set_nat @ A @ C5 )
% 14.17/8.02 => ( ! [X3: nat] :
% 14.17/8.02 ( ( member_nat @ X3 @ A )
% 14.17/8.02 => ( ord_less_eq_set_nat @ ( B4 @ X3 ) @ ( D3 @ X3 ) ) )
% 14.17/8.02 => ( ord_le3146513528884898305at_nat @ ( produc457027306803732586at_nat @ A @ B4 ) @ ( produc457027306803732586at_nat @ C5 @ D3 ) ) ) ) ).
% 14.17/8.02
% 14.17/8.02 thf(fact_1229_rtrancl__Un__separator__converseE,axiom,
% 14.17/8.02 ! [A2: nat,B: nat,P: set_Pr1261947904930325089at_nat,Q2: set_Pr1261947904930325089at_nat] :
% 14.17/8.02 ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A2 @ B ) @ ( transi2905341329935302413cl_nat @ ( sup_su6327502436637775413at_nat @ P @ Q2 ) ) )
% 14.17/8.02 => ( ! [X3: nat,Y2: nat] :
% 14.17/8.02 ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X3 @ B ) @ ( transi2905341329935302413cl_nat @ P ) )
% 14.17/8.02 => ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ Y2 @ X3 ) @ Q2 )
% 14.17/8.02 => ( Y2 = X3 ) ) )
% 14.17/8.02 => ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A2 @ B ) @ ( transi2905341329935302413cl_nat @ P ) ) ) ) ).
% 14.17/8.02
% 14.17/8.02 thf(fact_1230_rtrancl__Un__separator__converseE,axiom,
% 14.17/8.02 ! [A2: int,B: int,P: set_Pr958786334691620121nt_int,Q2: set_Pr958786334691620121nt_int] :
% 14.17/8.02 ( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ A2 @ B ) @ ( transi2902850859426252137cl_int @ ( sup_su6024340866399070445nt_int @ P @ Q2 ) ) )
% 14.17/8.02 => ( ! [X3: int,Y2: int] :
% 14.17/8.02 ( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ X3 @ B ) @ ( transi2902850859426252137cl_int @ P ) )
% 14.17/8.02 => ( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ Y2 @ X3 ) @ Q2 )
% 14.17/8.02 => ( Y2 = X3 ) ) )
% 14.17/8.02 => ( member5262025264175285858nt_int @ ( product_Pair_int_int @ A2 @ B ) @ ( transi2902850859426252137cl_int @ P ) ) ) ) ).
% 14.17/8.02
% 14.17/8.02 thf(fact_1231_rtrancl__Un__separatorE,axiom,
% 14.17/8.02 ! [A2: nat,B: nat,P: set_Pr1261947904930325089at_nat,Q2: set_Pr1261947904930325089at_nat] :
% 14.17/8.02 ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A2 @ B ) @ ( transi2905341329935302413cl_nat @ ( sup_su6327502436637775413at_nat @ P @ Q2 ) ) )
% 14.17/8.02 => ( ! [X3: nat,Y2: nat] :
% 14.17/8.02 ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A2 @ X3 ) @ ( transi2905341329935302413cl_nat @ P ) )
% 14.17/8.02 => ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X3 @ Y2 ) @ Q2 )
% 14.17/8.02 => ( X3 = Y2 ) ) )
% 14.17/8.02 => ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A2 @ B ) @ ( transi2905341329935302413cl_nat @ P ) ) ) ) ).
% 14.17/8.02
% 14.17/8.02 thf(fact_1232_rtrancl__Un__separatorE,axiom,
% 14.17/8.02 ! [A2: int,B: int,P: set_Pr958786334691620121nt_int,Q2: set_Pr958786334691620121nt_int] :
% 14.17/8.02 ( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ A2 @ B ) @ ( transi2902850859426252137cl_int @ ( sup_su6024340866399070445nt_int @ P @ Q2 ) ) )
% 14.17/8.02 => ( ! [X3: int,Y2: int] :
% 14.17/8.02 ( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ A2 @ X3 ) @ ( transi2902850859426252137cl_int @ P ) )
% 14.17/8.02 => ( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ X3 @ Y2 ) @ Q2 )
% 14.17/8.02 => ( X3 = Y2 ) ) )
% 14.17/8.02 => ( member5262025264175285858nt_int @ ( product_Pair_int_int @ A2 @ B ) @ ( transi2902850859426252137cl_int @ P ) ) ) ) ).
% 14.17/8.02
% 14.17/8.02 thf(fact_1233_disjoint__mono,axiom,
% 14.17/8.02 ! [A2: set_Product_unit,A5: set_Product_unit,B: set_Product_unit,B3: set_Product_unit] :
% 14.17/8.02 ( ( ord_le3507040750410214029t_unit @ A2 @ A5 )
% 14.17/8.02 => ( ( ord_le3507040750410214029t_unit @ B @ B3 )
% 14.17/8.02 => ( ( ( inf_in4660618365625256667t_unit @ A5 @ B3 )
% 14.17/8.02 = bot_bo3957492148770167129t_unit )
% 14.17/8.02 => ( ( inf_in4660618365625256667t_unit @ A2 @ B )
% 14.17/8.02 = bot_bo3957492148770167129t_unit ) ) ) ) ).
% 14.17/8.02
% 14.17/8.02 thf(fact_1234_disjoint__mono,axiom,
% 14.17/8.02 ! [A2: set_o,A5: set_o,B: set_o,B3: set_o] :
% 14.17/8.02 ( ( ord_less_eq_set_o @ A2 @ A5 )
% 14.17/8.02 => ( ( ord_less_eq_set_o @ B @ B3 )
% 14.17/8.02 => ( ( ( inf_inf_set_o @ A5 @ B3 )
% 14.17/8.02 = bot_bot_set_o )
% 14.17/8.02 => ( ( inf_inf_set_o @ A2 @ B )
% 14.17/8.02 = bot_bot_set_o ) ) ) ) ).
% 14.17/8.02
% 14.17/8.02 thf(fact_1235_disjoint__mono,axiom,
% 14.17/8.02 ! [A2: set_int,A5: set_int,B: set_int,B3: set_int] :
% 14.17/8.02 ( ( ord_less_eq_set_int @ A2 @ A5 )
% 14.17/8.02 => ( ( ord_less_eq_set_int @ B @ B3 )
% 14.17/8.02 => ( ( ( inf_inf_set_int @ A5 @ B3 )
% 14.17/8.02 = bot_bot_set_int )
% 14.17/8.02 => ( ( inf_inf_set_int @ A2 @ B )
% 14.17/8.02 = bot_bot_set_int ) ) ) ) ).
% 14.17/8.02
% 14.17/8.02 thf(fact_1236_disjoint__mono,axiom,
% 14.17/8.02 ! [A2: set_Pr4532377907799695533_nat_o,A5: set_Pr4532377907799695533_nat_o,B: set_Pr4532377907799695533_nat_o,B3: set_Pr4532377907799695533_nat_o] :
% 14.17/8.02 ( ( ord_le2965882846123202637_nat_o @ A2 @ A5 )
% 14.17/8.02 => ( ( ord_le2965882846123202637_nat_o @ B @ B3 )
% 14.17/8.02 => ( ( ( inf_in1906310914598751387_nat_o @ A5 @ B3 )
% 14.17/8.02 = bot_bo7824918357723582233_nat_o )
% 14.17/8.02 => ( ( inf_in1906310914598751387_nat_o @ A2 @ B )
% 14.17/8.02 = bot_bo7824918357723582233_nat_o ) ) ) ) ).
% 14.17/8.02
% 14.17/8.02 thf(fact_1237_disjoint__mono,axiom,
% 14.17/8.02 ! [A2: set_nat,A5: set_nat,B: set_nat,B3: set_nat] :
% 14.17/8.02 ( ( ord_less_eq_set_nat @ A2 @ A5 )
% 14.17/8.02 => ( ( ord_less_eq_set_nat @ B @ B3 )
% 14.17/8.02 => ( ( ( inf_inf_set_nat @ A5 @ B3 )
% 14.17/8.02 = bot_bot_set_nat )
% 14.17/8.02 => ( ( inf_inf_set_nat @ A2 @ B )
% 14.17/8.02 = bot_bot_set_nat ) ) ) ) ).
% 14.17/8.02
% 14.17/8.02 thf(fact_1238_rtrancl__image__unfold__right,axiom,
% 14.17/8.02 ! [E: set_Pr1261947904930325089at_nat,V2: set_nat] : ( ord_less_eq_set_nat @ ( image_nat_nat @ E @ ( image_nat_nat @ ( transi2905341329935302413cl_nat @ E ) @ V2 ) ) @ ( image_nat_nat @ ( transi2905341329935302413cl_nat @ E ) @ V2 ) ) ).
% 14.17/8.02
% 14.17/8.02 thf(fact_1239_rtrancl__reachable__induct,axiom,
% 14.17/8.02 ! [I: set_nat,INV: set_nat,E: set_Pr1261947904930325089at_nat] :
% 14.17/8.02 ( ( ord_less_eq_set_nat @ I @ INV )
% 14.17/8.02 => ( ( ord_less_eq_set_nat @ ( image_nat_nat @ E @ INV ) @ INV )
% 14.17/8.02 => ( ord_less_eq_set_nat @ ( image_nat_nat @ ( transi2905341329935302413cl_nat @ E ) @ I ) @ INV ) ) ) ).
% 14.17/8.02
% 14.17/8.02 thf(fact_1240_Image__Int__subset,axiom,
% 14.17/8.02 ! [R3: set_Pr1261947904930325089at_nat,A: set_nat,B4: set_nat] : ( ord_less_eq_set_nat @ ( image_nat_nat @ R3 @ ( inf_inf_set_nat @ A @ B4 ) ) @ ( inf_inf_set_nat @ ( image_nat_nat @ R3 @ A ) @ ( image_nat_nat @ R3 @ B4 ) ) ) ).
% 14.17/8.02
% 14.17/8.02 thf(fact_1241_Image__mono,axiom,
% 14.17/8.02 ! [R6: set_Pr1261947904930325089at_nat,R2: set_Pr1261947904930325089at_nat,A8: set_nat,A: set_nat] :
% 14.17/8.02 ( ( ord_le3146513528884898305at_nat @ R6 @ R2 )
% 14.17/8.02 => ( ( ord_less_eq_set_nat @ A8 @ A )
% 14.17/8.02 => ( ord_less_eq_set_nat @ ( image_nat_nat @ R6 @ A8 ) @ ( image_nat_nat @ R2 @ A ) ) ) ) ).
% 14.17/8.02
% 14.17/8.02 thf(fact_1242_refl__on__Un,axiom,
% 14.17/8.02 ! [A: set_nat,R2: set_Pr1261947904930325089at_nat,B4: set_nat,S2: set_Pr1261947904930325089at_nat] :
% 14.17/8.02 ( ( refl_on_nat @ A @ R2 )
% 14.17/8.02 => ( ( refl_on_nat @ B4 @ S2 )
% 14.17/8.02 => ( refl_on_nat @ ( sup_sup_set_nat @ A @ B4 ) @ ( sup_su6327502436637775413at_nat @ R2 @ S2 ) ) ) ) ).
% 14.17/8.02
% 14.17/8.02 thf(fact_1243_refl__on__Int,axiom,
% 14.17/8.02 ! [A: set_nat,R2: set_Pr1261947904930325089at_nat,B4: set_nat,S2: set_Pr1261947904930325089at_nat] :
% 14.17/8.02 ( ( refl_on_nat @ A @ R2 )
% 14.17/8.02 => ( ( refl_on_nat @ B4 @ S2 )
% 14.17/8.02 => ( refl_on_nat @ ( inf_inf_set_nat @ A @ B4 ) @ ( inf_in2572325071724192079at_nat @ R2 @ S2 ) ) ) ) ).
% 14.17/8.02
% 14.17/8.02 thf(fact_1244_trancl__union__outside,axiom,
% 14.17/8.02 ! [V: nat,W: nat,E: set_Pr1261947904930325089at_nat,U2: set_Pr1261947904930325089at_nat] :
% 14.17/8.02 ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ V @ W ) @ ( transi6264000038957366511cl_nat @ ( sup_su6327502436637775413at_nat @ E @ U2 ) ) )
% 14.17/8.02 => ( ~ ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ V @ W ) @ ( transi6264000038957366511cl_nat @ E ) )
% 14.17/8.02 => ? [X3: nat,Y2: nat] :
% 14.17/8.02 ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ V @ X3 ) @ ( transi2905341329935302413cl_nat @ ( sup_su6327502436637775413at_nat @ E @ U2 ) ) )
% 14.17/8.02 & ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X3 @ Y2 ) @ U2 )
% 14.17/8.02 & ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ Y2 @ W ) @ ( transi2905341329935302413cl_nat @ ( sup_su6327502436637775413at_nat @ E @ U2 ) ) ) ) ) ) ).
% 14.17/8.02
% 14.17/8.02 thf(fact_1245_trancl__union__outside,axiom,
% 14.17/8.02 ! [V: int,W: int,E: set_Pr958786334691620121nt_int,U2: set_Pr958786334691620121nt_int] :
% 14.17/8.02 ( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ V @ W ) @ ( transi6261509568448316235cl_int @ ( sup_su6024340866399070445nt_int @ E @ U2 ) ) )
% 14.17/8.02 => ( ~ ( member5262025264175285858nt_int @ ( product_Pair_int_int @ V @ W ) @ ( transi6261509568448316235cl_int @ E ) )
% 14.17/8.02 => ? [X3: int,Y2: int] :
% 14.17/8.02 ( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ V @ X3 ) @ ( transi2902850859426252137cl_int @ ( sup_su6024340866399070445nt_int @ E @ U2 ) ) )
% 14.17/8.02 & ( member5262025264175285858nt_int @ ( product_Pair_int_int @ X3 @ Y2 ) @ U2 )
% 14.17/8.02 & ( member5262025264175285858nt_int @ ( product_Pair_int_int @ Y2 @ W ) @ ( transi2902850859426252137cl_int @ ( sup_su6024340866399070445nt_int @ E @ U2 ) ) ) ) ) ) ).
% 14.17/8.02
% 14.17/8.02 thf(fact_1246_SigmaE,axiom,
% 14.17/8.02 ! [C: product_prod_nat_int,A: set_nat,B4: nat > set_int] :
% 14.17/8.02 ( ( member4262671552274231302at_int @ C @ ( produc454536836294682310at_int @ A @ B4 ) )
% 14.17/8.02 => ~ ! [X3: nat] :
% 14.17/8.02 ( ( member_nat @ X3 @ A )
% 14.17/8.02 => ! [Y2: int] :
% 14.17/8.02 ( ( member_int @ Y2 @ ( B4 @ X3 ) )
% 14.17/8.02 => ( C
% 14.17/8.02 != ( product_Pair_nat_int @ X3 @ Y2 ) ) ) ) ) ).
% 14.17/8.02
% 14.17/8.02 thf(fact_1247_SigmaE,axiom,
% 14.17/8.02 ! [C: product_prod_nat_o,A: set_nat,B4: nat > set_o] :
% 14.17/8.02 ( ( member6310962623043647828_nat_o @ C @ ( product_Sigma_nat_o @ A @ B4 ) )
% 14.17/8.02 => ~ ! [X3: nat] :
% 14.17/8.02 ( ( member_nat @ X3 @ A )
% 14.17/8.02 => ! [Y2: $o] :
% 14.17/8.02 ( ( member_o @ Y2 @ ( B4 @ X3 ) )
% 14.17/8.02 => ( C
% 14.17/8.02 != ( product_Pair_nat_o @ X3 @ Y2 ) ) ) ) ) ).
% 14.17/8.02
% 14.17/8.02 thf(fact_1248_SigmaE,axiom,
% 14.17/8.02 ! [C: produc526117512892780850t_unit,A: set_nat,B4: nat > set_Product_unit] :
% 14.17/8.02 ( ( member348563419383025609t_unit @ C @ ( produc5506539147108004339t_unit @ A @ B4 ) )
% 14.17/8.02 => ~ ! [X3: nat] :
% 14.17/8.02 ( ( member_nat @ X3 @ A )
% 14.17/8.02 => ! [Y2: product_unit] :
% 14.17/8.02 ( ( member_Product_unit @ Y2 @ ( B4 @ X3 ) )
% 14.17/8.02 => ( C
% 14.17/8.02 != ( produc3434450474625521892t_unit @ X3 @ Y2 ) ) ) ) ) ).
% 14.17/8.02
% 14.17/8.02 thf(fact_1249_SigmaE,axiom,
% 14.17/8.02 ! [C: product_prod_int_nat,A: set_int,B4: int > set_nat] :
% 14.17/8.02 ( ( member216504246829706758nt_nat @ C @ ( produc1456381018704787142nt_nat @ A @ B4 ) )
% 14.17/8.02 => ~ ! [X3: int] :
% 14.17/8.02 ( ( member_int @ X3 @ A )
% 14.17/8.02 => ! [Y2: nat] :
% 14.17/8.02 ( ( member_nat @ Y2 @ ( B4 @ X3 ) )
% 14.17/8.02 => ( C
% 14.17/8.02 != ( product_Pair_int_nat @ X3 @ Y2 ) ) ) ) ) ).
% 14.17/8.02
% 14.17/8.02 thf(fact_1250_SigmaE,axiom,
% 14.17/8.02 ! [C: product_prod_int_o,A: set_int,B4: int > set_o] :
% 14.17/8.02 ( ( member4489920277610959864_int_o @ C @ ( product_Sigma_int_o @ A @ B4 ) )
% 14.17/8.02 => ~ ! [X3: int] :
% 14.17/8.02 ( ( member_int @ X3 @ A )
% 14.17/8.02 => ! [Y2: $o] :
% 14.17/8.02 ( ( member_o @ Y2 @ ( B4 @ X3 ) )
% 14.17/8.02 => ( C
% 14.17/8.02 != ( product_Pair_int_o @ X3 @ Y2 ) ) ) ) ) ).
% 14.17/8.02
% 14.17/8.02 thf(fact_1251_SigmaE,axiom,
% 14.17/8.02 ! [C: produc6865911496949736918t_unit,A: set_int,B4: int > set_Product_unit] :
% 14.17/8.02 ( ( member6688357403439981677t_unit @ C @ ( produc2622961094310184599t_unit @ A @ B4 ) )
% 14.17/8.02 => ~ ! [X3: int] :
% 14.17/8.02 ( ( member_int @ X3 @ A )
% 14.17/8.02 => ! [Y2: product_unit] :
% 14.17/8.02 ( ( member_Product_unit @ Y2 @ ( B4 @ X3 ) )
% 14.17/8.02 => ( C
% 14.17/8.02 != ( produc550872421827702152t_unit @ X3 @ Y2 ) ) ) ) ) ).
% 14.17/8.02
% 14.17/8.02 thf(fact_1252_SigmaE,axiom,
% 14.17/8.02 ! [C: product_prod_o_nat,A: set_o,B4: $o > set_nat] :
% 14.17/8.02 ( ( member2802428098988154798_o_nat @ C @ ( product_Sigma_o_nat @ A @ B4 ) )
% 14.17/8.02 => ~ ! [X3: $o] :
% 14.17/8.02 ( ( member_o @ X3 @ A )
% 14.17/8.02 => ! [Y2: nat] :
% 14.17/8.02 ( ( member_nat @ Y2 @ ( B4 @ X3 ) )
% 14.17/8.02 => ( C
% 14.17/8.02 != ( product_Pair_o_nat @ X3 @ Y2 ) ) ) ) ) ).
% 14.17/8.02
% 14.17/8.02 thf(fact_1253_SigmaE,axiom,
% 14.17/8.02 ! [C: product_prod_o_int,A: set_o,B4: $o > set_int] :
% 14.17/8.02 ( ( member7847949116333733898_o_int @ C @ ( product_Sigma_o_int @ A @ B4 ) )
% 14.17/8.02 => ~ ! [X3: $o] :
% 14.17/8.02 ( ( member_o @ X3 @ A )
% 14.17/8.02 => ! [Y2: int] :
% 14.17/8.02 ( ( member_int @ Y2 @ ( B4 @ X3 ) )
% 14.17/8.02 => ( C
% 14.17/8.02 != ( product_Pair_o_int @ X3 @ Y2 ) ) ) ) ) ).
% 14.17/8.02
% 14.17/8.02 thf(fact_1254_SigmaE,axiom,
% 14.17/8.02 ! [C: product_prod_o_o,A: set_o,B4: $o > set_o] :
% 14.17/8.02 ( ( member7466972457876170832od_o_o @ C @ ( product_Sigma_o_o @ A @ B4 ) )
% 14.17/8.02 => ~ ! [X3: $o] :
% 14.17/8.02 ( ( member_o @ X3 @ A )
% 14.17/8.02 => ! [Y2: $o] :
% 14.17/8.02 ( ( member_o @ Y2 @ ( B4 @ X3 ) )
% 14.17/8.02 => ( C
% 14.17/8.02 != ( product_Pair_o_o @ X3 @ Y2 ) ) ) ) ) ).
% 14.17/8.02
% 14.17/8.02 thf(fact_1255_SigmaE,axiom,
% 14.17/8.02 ! [C: produc3788275437394581276t_unit,A: set_o,B4: $o > set_Product_unit] :
% 14.17/8.02 ( ( member9215336023991391429t_unit @ C @ ( produc411009820408249917t_unit @ A @ B4 ) )
% 14.17/8.02 => ~ ! [X3: $o] :
% 14.17/8.02 ( ( member_o @ X3 @ A )
% 14.17/8.02 => ! [Y2: product_unit] :
% 14.17/8.02 ( ( member_Product_unit @ Y2 @ ( B4 @ X3 ) )
% 14.17/8.02 => ( C
% 14.17/8.02 != ( produc3901946521647900940t_unit @ X3 @ Y2 ) ) ) ) ) ).
% 14.17/8.02
% 14.17/8.02 thf(fact_1256_SigmaD1,axiom,
% 14.17/8.02 ! [A2: heap_e7401611519738050253t_unit,B: set_nat,A: set_he8644174446132254637t_unit,B4: heap_e7401611519738050253t_unit > set_set_nat] :
% 14.17/8.02 ( ( member6260224972018164377et_nat @ ( produc7507926704131184380et_nat @ A2 @ B ) @ ( produc5260988393495149579et_nat @ A @ B4 ) )
% 14.17/8.02 => ( member242389114213480694t_unit @ A2 @ A ) ) ).
% 14.17/8.02
% 14.17/8.02 thf(fact_1257_SigmaD1,axiom,
% 14.17/8.02 ! [A2: nat,B: nat,A: set_nat,B4: nat > set_nat] :
% 14.17/8.02 ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A2 @ B ) @ ( produc457027306803732586at_nat @ A @ B4 ) )
% 14.17/8.02 => ( member_nat @ A2 @ A ) ) ).
% 14.17/8.02
% 14.17/8.02 thf(fact_1258_SigmaD1,axiom,
% 14.17/8.02 ! [A2: int,B: int,A: set_int,B4: int > set_int] :
% 14.17/8.02 ( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ A2 @ B ) @ ( produc1453890548195736866nt_int @ A @ B4 ) )
% 14.17/8.02 => ( member_int @ A2 @ A ) ) ).
% 14.17/8.02
% 14.17/8.02 thf(fact_1259_SigmaD1,axiom,
% 14.17/8.02 ! [A2: produc3658429121746597890et_nat > $o,B: produc3658429121746597890et_nat,A: set_Pr4532377907799695533_nat_o,B4: ( produc3658429121746597890et_nat > $o ) > set_Pr3948176798113811640et_nat] :
% 14.17/8.02 ( ( member1996754912294343701et_nat @ ( produc5001842942810119800et_nat @ A2 @ B ) @ ( produc8052021730408606599et_nat @ A @ B4 ) )
% 14.17/8.02 => ( member6576561426505652726_nat_o @ A2 @ A ) ) ).
% 14.17/8.02
% 14.17/8.02 thf(fact_1260_SigmaD1,axiom,
% 14.17/8.02 ! [A2: produc3658429121746597890et_nat > $o,B: produc3925858234332021118et_nat,A: set_Pr4532377907799695533_nat_o,B4: ( produc3658429121746597890et_nat > $o ) > set_Pr3286484037609594932et_nat] :
% 14.17/8.02 ( ( member6124377750444531601et_nat @ ( produc2245416461498447860et_nat @ A2 @ B ) @ ( produc7016705354093686019et_nat @ A @ B4 ) )
% 14.17/8.02 => ( member6576561426505652726_nat_o @ A2 @ A ) ) ).
% 14.17/8.02
% 14.17/8.02 thf(fact_1261_SigmaD2,axiom,
% 14.17/8.02 ! [A2: heap_e7401611519738050253t_unit,B: set_nat,A: set_he8644174446132254637t_unit,B4: heap_e7401611519738050253t_unit > set_set_nat] :
% 14.17/8.02 ( ( member6260224972018164377et_nat @ ( produc7507926704131184380et_nat @ A2 @ B ) @ ( produc5260988393495149579et_nat @ A @ B4 ) )
% 14.17/8.02 => ( member_set_nat @ B @ ( B4 @ A2 ) ) ) ).
% 14.17/8.02
% 14.17/8.02 thf(fact_1262_SigmaD2,axiom,
% 14.17/8.02 ! [A2: nat,B: nat,A: set_nat,B4: nat > set_nat] :
% 14.17/8.02 ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A2 @ B ) @ ( produc457027306803732586at_nat @ A @ B4 ) )
% 14.17/8.02 => ( member_nat @ B @ ( B4 @ A2 ) ) ) ).
% 14.17/8.02
% 14.17/8.02 thf(fact_1263_SigmaD2,axiom,
% 14.17/8.02 ! [A2: int,B: int,A: set_int,B4: int > set_int] :
% 14.17/8.02 ( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ A2 @ B ) @ ( produc1453890548195736866nt_int @ A @ B4 ) )
% 14.17/8.02 => ( member_int @ B @ ( B4 @ A2 ) ) ) ).
% 14.17/8.02
% 14.17/8.02 thf(fact_1264_SigmaD2,axiom,
% 14.17/8.02 ! [A2: produc3658429121746597890et_nat > $o,B: produc3658429121746597890et_nat,A: set_Pr4532377907799695533_nat_o,B4: ( produc3658429121746597890et_nat > $o ) > set_Pr3948176798113811640et_nat] :
% 14.17/8.02 ( ( member1996754912294343701et_nat @ ( produc5001842942810119800et_nat @ A2 @ B ) @ ( produc8052021730408606599et_nat @ A @ B4 ) )
% 14.17/8.02 => ( member6260224972018164377et_nat @ B @ ( B4 @ A2 ) ) ) ).
% 14.17/8.02
% 14.17/8.02 thf(fact_1265_SigmaD2,axiom,
% 14.17/8.02 ! [A2: produc3658429121746597890et_nat > $o,B: produc3925858234332021118et_nat,A: set_Pr4532377907799695533_nat_o,B4: ( produc3658429121746597890et_nat > $o ) > set_Pr3286484037609594932et_nat] :
% 14.17/8.02 ( ( member6124377750444531601et_nat @ ( produc2245416461498447860et_nat @ A2 @ B ) @ ( produc7016705354093686019et_nat @ A @ B4 ) )
% 14.17/8.02 => ( member1996754912294343701et_nat @ B @ ( B4 @ A2 ) ) ) ).
% 14.17/8.02
% 14.17/8.02 thf(fact_1266_SigmaE2,axiom,
% 14.17/8.02 ! [A2: nat,B: int,A: set_nat,B4: nat > set_int] :
% 14.17/8.02 ( ( member4262671552274231302at_int @ ( product_Pair_nat_int @ A2 @ B ) @ ( produc454536836294682310at_int @ A @ B4 ) )
% 14.17/8.02 => ~ ( ( member_nat @ A2 @ A )
% 14.17/8.02 => ~ ( member_int @ B @ ( B4 @ A2 ) ) ) ) ).
% 14.17/8.02
% 14.17/8.02 thf(fact_1267_SigmaE2,axiom,
% 14.17/8.02 ! [A2: nat,B: $o,A: set_nat,B4: nat > set_o] :
% 14.17/8.02 ( ( member6310962623043647828_nat_o @ ( product_Pair_nat_o @ A2 @ B ) @ ( product_Sigma_nat_o @ A @ B4 ) )
% 14.17/8.02 => ~ ( ( member_nat @ A2 @ A )
% 14.17/8.02 => ~ ( member_o @ B @ ( B4 @ A2 ) ) ) ) ).
% 14.17/8.02
% 14.17/8.02 thf(fact_1268_SigmaE2,axiom,
% 14.17/8.02 ! [A2: nat,B: product_unit,A: set_nat,B4: nat > set_Product_unit] :
% 14.17/8.02 ( ( member348563419383025609t_unit @ ( produc3434450474625521892t_unit @ A2 @ B ) @ ( produc5506539147108004339t_unit @ A @ B4 ) )
% 14.17/8.02 => ~ ( ( member_nat @ A2 @ A )
% 14.17/8.02 => ~ ( member_Product_unit @ B @ ( B4 @ A2 ) ) ) ) ).
% 14.17/8.02
% 14.17/8.02 thf(fact_1269_SigmaE2,axiom,
% 14.17/8.02 ! [A2: int,B: nat,A: set_int,B4: int > set_nat] :
% 14.17/8.02 ( ( member216504246829706758nt_nat @ ( product_Pair_int_nat @ A2 @ B ) @ ( produc1456381018704787142nt_nat @ A @ B4 ) )
% 14.17/8.02 => ~ ( ( member_int @ A2 @ A )
% 14.17/8.02 => ~ ( member_nat @ B @ ( B4 @ A2 ) ) ) ) ).
% 14.17/8.02
% 14.17/8.02 thf(fact_1270_SigmaE2,axiom,
% 14.17/8.02 ! [A2: int,B: $o,A: set_int,B4: int > set_o] :
% 14.17/8.02 ( ( member4489920277610959864_int_o @ ( product_Pair_int_o @ A2 @ B ) @ ( product_Sigma_int_o @ A @ B4 ) )
% 14.17/8.02 => ~ ( ( member_int @ A2 @ A )
% 14.17/8.02 => ~ ( member_o @ B @ ( B4 @ A2 ) ) ) ) ).
% 14.17/8.02
% 14.17/8.02 thf(fact_1271_SigmaE2,axiom,
% 14.17/8.02 ! [A2: int,B: product_unit,A: set_int,B4: int > set_Product_unit] :
% 14.17/8.02 ( ( member6688357403439981677t_unit @ ( produc550872421827702152t_unit @ A2 @ B ) @ ( produc2622961094310184599t_unit @ A @ B4 ) )
% 14.17/8.02 => ~ ( ( member_int @ A2 @ A )
% 14.17/8.02 => ~ ( member_Product_unit @ B @ ( B4 @ A2 ) ) ) ) ).
% 14.17/8.02
% 14.17/8.02 thf(fact_1272_SigmaE2,axiom,
% 14.17/8.02 ! [A2: $o,B: nat,A: set_o,B4: $o > set_nat] :
% 14.17/8.02 ( ( member2802428098988154798_o_nat @ ( product_Pair_o_nat @ A2 @ B ) @ ( product_Sigma_o_nat @ A @ B4 ) )
% 14.17/8.02 => ~ ( ( member_o @ A2 @ A )
% 14.17/8.02 => ~ ( member_nat @ B @ ( B4 @ A2 ) ) ) ) ).
% 14.17/8.02
% 14.17/8.02 thf(fact_1273_SigmaE2,axiom,
% 14.17/8.02 ! [A2: $o,B: int,A: set_o,B4: $o > set_int] :
% 14.17/8.02 ( ( member7847949116333733898_o_int @ ( product_Pair_o_int @ A2 @ B ) @ ( product_Sigma_o_int @ A @ B4 ) )
% 14.17/8.02 => ~ ( ( member_o @ A2 @ A )
% 14.17/8.02 => ~ ( member_int @ B @ ( B4 @ A2 ) ) ) ) ).
% 14.17/8.02
% 14.17/8.02 thf(fact_1274_SigmaE2,axiom,
% 14.17/8.02 ! [A2: $o,B: $o,A: set_o,B4: $o > set_o] :
% 14.17/8.02 ( ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ A2 @ B ) @ ( product_Sigma_o_o @ A @ B4 ) )
% 14.17/8.02 => ~ ( ( member_o @ A2 @ A )
% 14.17/8.02 => ~ ( member_o @ B @ ( B4 @ A2 ) ) ) ) ).
% 14.17/8.02
% 14.17/8.02 thf(fact_1275_SigmaE2,axiom,
% 14.17/8.02 ! [A2: $o,B: product_unit,A: set_o,B4: $o > set_Product_unit] :
% 14.17/8.02 ( ( member9215336023991391429t_unit @ ( produc3901946521647900940t_unit @ A2 @ B ) @ ( produc411009820408249917t_unit @ A @ B4 ) )
% 14.17/8.02 => ~ ( ( member_o @ A2 @ A )
% 14.17/8.02 => ~ ( member_Product_unit @ B @ ( B4 @ A2 ) ) ) ) ).
% 14.17/8.02
% 14.17/8.02 thf(fact_1276_trancl__image__by__rtrancl,axiom,
% 14.17/8.02 ! [E: set_Pr1261947904930325089at_nat,Vi: set_nat] :
% 14.17/8.02 ( ( sup_sup_set_nat @ ( image_nat_nat @ ( transi6264000038957366511cl_nat @ E ) @ Vi ) @ Vi )
% 14.17/8.02 = ( image_nat_nat @ ( transi2905341329935302413cl_nat @ E ) @ Vi ) ) ).
% 14.17/8.02
% 14.17/8.02 thf(fact_1277_converse__rtranclE_H,axiom,
% 14.17/8.02 ! [U: nat,V: nat,R3: set_Pr1261947904930325089at_nat] :
% 14.17/8.02 ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ U @ V ) @ ( transi2905341329935302413cl_nat @ R3 ) )
% 14.17/8.02 => ( ( U != V )
% 14.17/8.02 => ~ ! [Vh: nat] :
% 14.17/8.02 ( ( U != Vh )
% 14.17/8.02 => ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ U @ Vh ) @ R3 )
% 14.17/8.02 => ~ ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ Vh @ V ) @ ( transi2905341329935302413cl_nat @ R3 ) ) ) ) ) ) ).
% 14.17/8.02
% 14.17/8.02 thf(fact_1278_converse__rtranclE_H,axiom,
% 14.17/8.02 ! [U: int,V: int,R3: set_Pr958786334691620121nt_int] :
% 14.17/8.02 ( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ U @ V ) @ ( transi2902850859426252137cl_int @ R3 ) )
% 14.17/8.02 => ( ( U != V )
% 14.17/8.02 => ~ ! [Vh: int] :
% 14.17/8.02 ( ( U != Vh )
% 14.17/8.02 => ( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ U @ Vh ) @ R3 )
% 14.17/8.02 => ~ ( member5262025264175285858nt_int @ ( product_Pair_int_int @ Vh @ V ) @ ( transi2902850859426252137cl_int @ R3 ) ) ) ) ) ) ).
% 14.17/8.02
% 14.17/8.02 thf(fact_1279_rtrancl_Ocases,axiom,
% 14.17/8.02 ! [A1: nat,A22: nat,R2: set_Pr1261947904930325089at_nat] :
% 14.17/8.02 ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A1 @ A22 ) @ ( transi2905341329935302413cl_nat @ R2 ) )
% 14.17/8.02 => ( ( A22 != A1 )
% 14.17/8.02 => ~ ! [B2: nat] :
% 14.17/8.02 ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A1 @ B2 ) @ ( transi2905341329935302413cl_nat @ R2 ) )
% 14.17/8.02 => ~ ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ B2 @ A22 ) @ R2 ) ) ) ) ).
% 14.17/8.02
% 14.17/8.02 thf(fact_1280_rtrancl_Ocases,axiom,
% 14.17/8.02 ! [A1: int,A22: int,R2: set_Pr958786334691620121nt_int] :
% 14.17/8.02 ( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ A1 @ A22 ) @ ( transi2902850859426252137cl_int @ R2 ) )
% 14.17/8.02 => ( ( A22 != A1 )
% 14.17/8.02 => ~ ! [B2: int] :
% 14.17/8.02 ( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ A1 @ B2 ) @ ( transi2902850859426252137cl_int @ R2 ) )
% 14.17/8.02 => ~ ( member5262025264175285858nt_int @ ( product_Pair_int_int @ B2 @ A22 ) @ R2 ) ) ) ) ).
% 14.17/8.02
% 14.17/8.02 thf(fact_1281_rtrancl_Osimps,axiom,
% 14.17/8.02 ! [A1: nat,A22: nat,R2: set_Pr1261947904930325089at_nat] :
% 14.17/8.02 ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A1 @ A22 ) @ ( transi2905341329935302413cl_nat @ R2 ) )
% 14.17/8.02 = ( ? [A6: nat] :
% 14.17/8.02 ( ( A1 = A6 )
% 14.17/8.02 & ( A22 = A6 ) )
% 14.17/8.02 | ? [A6: nat,B6: nat,C4: nat] :
% 14.17/8.02 ( ( A1 = A6 )
% 14.17/8.02 & ( A22 = C4 )
% 14.17/8.02 & ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A6 @ B6 ) @ ( transi2905341329935302413cl_nat @ R2 ) )
% 14.17/8.02 & ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ B6 @ C4 ) @ R2 ) ) ) ) ).
% 14.17/8.02
% 14.17/8.02 thf(fact_1282_rtrancl_Osimps,axiom,
% 14.17/8.02 ! [A1: int,A22: int,R2: set_Pr958786334691620121nt_int] :
% 14.17/8.02 ( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ A1 @ A22 ) @ ( transi2902850859426252137cl_int @ R2 ) )
% 14.17/8.02 = ( ? [A6: int] :
% 14.17/8.02 ( ( A1 = A6 )
% 14.17/8.02 & ( A22 = A6 ) )
% 14.17/8.02 | ? [A6: int,B6: int,C4: int] :
% 14.17/8.02 ( ( A1 = A6 )
% 14.17/8.02 & ( A22 = C4 )
% 14.17/8.02 & ( member5262025264175285858nt_int @ ( product_Pair_int_int @ A6 @ B6 ) @ ( transi2902850859426252137cl_int @ R2 ) )
% 14.17/8.02 & ( member5262025264175285858nt_int @ ( product_Pair_int_int @ B6 @ C4 ) @ R2 ) ) ) ) ).
% 14.17/8.02
% 14.17/8.02 thf(fact_1283_rtrancl_Ortrancl__refl,axiom,
% 14.17/8.02 ! [A2: nat,R2: set_Pr1261947904930325089at_nat] : ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A2 @ A2 ) @ ( transi2905341329935302413cl_nat @ R2 ) ) ).
% 14.17/8.02
% 14.17/8.02 thf(fact_1284_rtrancl_Ortrancl__refl,axiom,
% 14.17/8.02 ! [A2: int,R2: set_Pr958786334691620121nt_int] : ( member5262025264175285858nt_int @ ( product_Pair_int_int @ A2 @ A2 ) @ ( transi2902850859426252137cl_int @ R2 ) ) ).
% 14.17/8.02
% 14.17/8.02 thf(fact_1285_rtrancl_Ortrancl__into__rtrancl,axiom,
% 14.17/8.02 ! [A2: nat,B: nat,R2: set_Pr1261947904930325089at_nat,C: nat] :
% 14.17/8.02 ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A2 @ B ) @ ( transi2905341329935302413cl_nat @ R2 ) )
% 14.17/8.02 => ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ B @ C ) @ R2 )
% 14.17/8.02 => ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A2 @ C ) @ ( transi2905341329935302413cl_nat @ R2 ) ) ) ) ).
% 14.17/8.02
% 14.17/8.02 thf(fact_1286_rtrancl_Ortrancl__into__rtrancl,axiom,
% 14.17/8.02 ! [A2: int,B: int,R2: set_Pr958786334691620121nt_int,C: int] :
% 14.17/8.02 ( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ A2 @ B ) @ ( transi2902850859426252137cl_int @ R2 ) )
% 14.17/8.02 => ( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ B @ C ) @ R2 )
% 14.17/8.02 => ( member5262025264175285858nt_int @ ( product_Pair_int_int @ A2 @ C ) @ ( transi2902850859426252137cl_int @ R2 ) ) ) ) ).
% 14.17/8.02
% 14.17/8.02 thf(fact_1287_rtranclE,axiom,
% 14.17/8.02 ! [A2: nat,B: nat,R2: set_Pr1261947904930325089at_nat] :
% 14.17/8.02 ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A2 @ B ) @ ( transi2905341329935302413cl_nat @ R2 ) )
% 14.17/8.02 => ( ( A2 != B )
% 14.17/8.02 => ~ ! [Y2: nat] :
% 14.17/8.02 ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A2 @ Y2 ) @ ( transi2905341329935302413cl_nat @ R2 ) )
% 14.17/8.02 => ~ ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ Y2 @ B ) @ R2 ) ) ) ) ).
% 14.17/8.02
% 14.17/8.02 thf(fact_1288_rtranclE,axiom,
% 14.17/8.02 ! [A2: int,B: int,R2: set_Pr958786334691620121nt_int] :
% 14.17/8.02 ( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ A2 @ B ) @ ( transi2902850859426252137cl_int @ R2 ) )
% 14.17/8.02 => ( ( A2 != B )
% 14.17/8.02 => ~ ! [Y2: int] :
% 14.17/8.02 ( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ A2 @ Y2 ) @ ( transi2902850859426252137cl_int @ R2 ) )
% 14.17/8.02 => ~ ( member5262025264175285858nt_int @ ( product_Pair_int_int @ Y2 @ B ) @ R2 ) ) ) ) ).
% 14.17/8.02
% 14.17/8.02 thf(fact_1289_rtrancl__trans,axiom,
% 14.17/8.02 ! [X: nat,Y: nat,R2: set_Pr1261947904930325089at_nat,Z: nat] :
% 14.17/8.02 ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ Y ) @ ( transi2905341329935302413cl_nat @ R2 ) )
% 14.17/8.02 => ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ Y @ Z ) @ ( transi2905341329935302413cl_nat @ R2 ) )
% 14.17/8.02 => ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ Z ) @ ( transi2905341329935302413cl_nat @ R2 ) ) ) ) ).
% 14.17/8.02
% 14.17/8.02 thf(fact_1290_rtrancl__trans,axiom,
% 14.17/8.02 ! [X: int,Y: int,R2: set_Pr958786334691620121nt_int,Z: int] :
% 14.17/8.02 ( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ X @ Y ) @ ( transi2902850859426252137cl_int @ R2 ) )
% 14.17/8.02 => ( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ Y @ Z ) @ ( transi2902850859426252137cl_int @ R2 ) )
% 14.17/8.02 => ( member5262025264175285858nt_int @ ( product_Pair_int_int @ X @ Z ) @ ( transi2902850859426252137cl_int @ R2 ) ) ) ) ).
% 14.17/8.02
% 14.17/8.02 thf(fact_1291_rtrancl__induct,axiom,
% 14.17/8.02 ! [A2: nat,B: nat,R2: set_Pr1261947904930325089at_nat,P: nat > $o] :
% 14.17/8.02 ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A2 @ B ) @ ( transi2905341329935302413cl_nat @ R2 ) )
% 14.17/8.02 => ( ( P @ A2 )
% 14.17/8.02 => ( ! [Y2: nat,Z3: nat] :
% 14.17/8.02 ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A2 @ Y2 ) @ ( transi2905341329935302413cl_nat @ R2 ) )
% 14.17/8.02 => ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ Y2 @ Z3 ) @ R2 )
% 14.17/8.02 => ( ( P @ Y2 )
% 14.17/8.02 => ( P @ Z3 ) ) ) )
% 14.17/8.02 => ( P @ B ) ) ) ) ).
% 14.17/8.02
% 14.17/8.02 thf(fact_1292_rtrancl__induct,axiom,
% 14.17/8.02 ! [A2: int,B: int,R2: set_Pr958786334691620121nt_int,P: int > $o] :
% 14.17/8.02 ( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ A2 @ B ) @ ( transi2902850859426252137cl_int @ R2 ) )
% 14.17/8.02 => ( ( P @ A2 )
% 14.17/8.02 => ( ! [Y2: int,Z3: int] :
% 14.17/8.02 ( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ A2 @ Y2 ) @ ( transi2902850859426252137cl_int @ R2 ) )
% 14.17/8.02 => ( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ Y2 @ Z3 ) @ R2 )
% 14.17/8.02 => ( ( P @ Y2 )
% 14.17/8.02 => ( P @ Z3 ) ) ) )
% 14.17/8.02 => ( P @ B ) ) ) ) ).
% 14.17/8.02
% 14.17/8.02 thf(fact_1293_converse__rtranclE,axiom,
% 14.17/8.02 ! [X: nat,Z: nat,R2: set_Pr1261947904930325089at_nat] :
% 14.17/8.02 ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ Z ) @ ( transi2905341329935302413cl_nat @ R2 ) )
% 14.17/8.02 => ( ( X != Z )
% 14.17/8.02 => ~ ! [Y2: nat] :
% 14.17/8.02 ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ Y2 ) @ R2 )
% 14.17/8.02 => ~ ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ Y2 @ Z ) @ ( transi2905341329935302413cl_nat @ R2 ) ) ) ) ) ).
% 14.17/8.02
% 14.17/8.02 thf(fact_1294_converse__rtranclE,axiom,
% 14.17/8.02 ! [X: int,Z: int,R2: set_Pr958786334691620121nt_int] :
% 14.17/8.02 ( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ X @ Z ) @ ( transi2902850859426252137cl_int @ R2 ) )
% 14.17/8.02 => ( ( X != Z )
% 14.17/8.02 => ~ ! [Y2: int] :
% 14.17/8.02 ( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ X @ Y2 ) @ R2 )
% 14.17/8.02 => ~ ( member5262025264175285858nt_int @ ( product_Pair_int_int @ Y2 @ Z ) @ ( transi2902850859426252137cl_int @ R2 ) ) ) ) ) ).
% 14.17/8.02
% 14.17/8.02 thf(fact_1295_converse__rtrancl__induct,axiom,
% 14.17/8.02 ! [A2: nat,B: nat,R2: set_Pr1261947904930325089at_nat,P: nat > $o] :
% 14.17/8.02 ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A2 @ B ) @ ( transi2905341329935302413cl_nat @ R2 ) )
% 14.17/8.02 => ( ( P @ B )
% 14.17/8.02 => ( ! [Y2: nat,Z3: nat] :
% 14.17/8.02 ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ Y2 @ Z3 ) @ R2 )
% 14.17/8.02 => ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ Z3 @ B ) @ ( transi2905341329935302413cl_nat @ R2 ) )
% 14.17/8.02 => ( ( P @ Z3 )
% 14.17/8.02 => ( P @ Y2 ) ) ) )
% 14.17/8.02 => ( P @ A2 ) ) ) ) ).
% 14.17/8.02
% 14.17/8.02 thf(fact_1296_converse__rtrancl__induct,axiom,
% 14.17/8.02 ! [A2: int,B: int,R2: set_Pr958786334691620121nt_int,P: int > $o] :
% 14.17/8.02 ( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ A2 @ B ) @ ( transi2902850859426252137cl_int @ R2 ) )
% 14.17/8.02 => ( ( P @ B )
% 14.17/8.02 => ( ! [Y2: int,Z3: int] :
% 14.17/8.02 ( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ Y2 @ Z3 ) @ R2 )
% 14.17/8.02 => ( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ Z3 @ B ) @ ( transi2902850859426252137cl_int @ R2 ) )
% 14.17/8.02 => ( ( P @ Z3 )
% 14.17/8.02 => ( P @ Y2 ) ) ) )
% 14.17/8.02 => ( P @ A2 ) ) ) ) ).
% 14.17/8.02
% 14.17/8.02 thf(fact_1297_converse__rtrancl__into__rtrancl,axiom,
% 14.17/8.02 ! [A2: nat,B: nat,R2: set_Pr1261947904930325089at_nat,C: nat] :
% 14.17/8.02 ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A2 @ B ) @ R2 )
% 14.17/8.02 => ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ B @ C ) @ ( transi2905341329935302413cl_nat @ R2 ) )
% 14.17/8.02 => ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A2 @ C ) @ ( transi2905341329935302413cl_nat @ R2 ) ) ) ) ).
% 14.17/8.02
% 14.17/8.02 thf(fact_1298_converse__rtrancl__into__rtrancl,axiom,
% 14.17/8.02 ! [A2: int,B: int,R2: set_Pr958786334691620121nt_int,C: int] :
% 14.17/8.02 ( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ A2 @ B ) @ R2 )
% 14.17/8.02 => ( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ B @ C ) @ ( transi2902850859426252137cl_int @ R2 ) )
% 14.17/8.02 => ( member5262025264175285858nt_int @ ( product_Pair_int_int @ A2 @ C ) @ ( transi2902850859426252137cl_int @ R2 ) ) ) ) ).
% 14.17/8.02
% 14.17/8.02 thf(fact_1299_rtrancl__induct2,axiom,
% 14.17/8.02 ! [Ax: heap_e7401611519738050253t_unit,Ay: set_nat,Bx: heap_e7401611519738050253t_unit,By: set_nat,R2: set_Pr719794911490849221et_nat,P: heap_e7401611519738050253t_unit > set_nat > $o] :
% 14.17/8.02 ( ( member6099555550032318734et_nat @ ( produc8199053930788261021et_nat @ ( produc7507926704131184380et_nat @ Ax @ Ay ) @ ( produc7507926704131184380et_nat @ Bx @ By ) ) @ ( transi6919426408697524543et_nat @ R2 ) )
% 14.17/8.02 => ( ( P @ Ax @ Ay )
% 14.17/8.02 => ( ! [A4: heap_e7401611519738050253t_unit,B2: set_nat,Aa: heap_e7401611519738050253t_unit,Ba: set_nat] :
% 14.17/8.02 ( ( member6099555550032318734et_nat @ ( produc8199053930788261021et_nat @ ( produc7507926704131184380et_nat @ Ax @ Ay ) @ ( produc7507926704131184380et_nat @ A4 @ B2 ) ) @ ( transi6919426408697524543et_nat @ R2 ) )
% 14.17/8.02 => ( ( member6099555550032318734et_nat @ ( produc8199053930788261021et_nat @ ( produc7507926704131184380et_nat @ A4 @ B2 ) @ ( produc7507926704131184380et_nat @ Aa @ Ba ) ) @ R2 )
% 14.17/8.02 => ( ( P @ A4 @ B2 )
% 14.17/8.02 => ( P @ Aa @ Ba ) ) ) )
% 14.17/8.02 => ( P @ Bx @ By ) ) ) ) ).
% 14.17/8.02
% 14.17/8.02 thf(fact_1300_rtrancl__induct2,axiom,
% 14.17/8.02 ! [Ax: nat,Ay: nat,Bx: nat,By: nat,R2: set_Pr8693737435421807431at_nat,P: nat > nat > $o] :
% 14.17/8.02 ( ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ ( product_Pair_nat_nat @ Ax @ Ay ) @ ( product_Pair_nat_nat @ Bx @ By ) ) @ ( transi8609417484261754244at_nat @ R2 ) )
% 14.17/8.02 => ( ( P @ Ax @ Ay )
% 14.17/8.02 => ( ! [A4: nat,B2: nat,Aa: nat,Ba: nat] :
% 14.17/8.02 ( ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ ( product_Pair_nat_nat @ Ax @ Ay ) @ ( product_Pair_nat_nat @ A4 @ B2 ) ) @ ( transi8609417484261754244at_nat @ R2 ) )
% 14.17/8.02 => ( ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ ( product_Pair_nat_nat @ A4 @ B2 ) @ ( product_Pair_nat_nat @ Aa @ Ba ) ) @ R2 )
% 14.17/8.02 => ( ( P @ A4 @ B2 )
% 14.17/8.02 => ( P @ Aa @ Ba ) ) ) )
% 14.17/8.02 => ( P @ Bx @ By ) ) ) ) ).
% 14.17/8.02
% 14.17/8.02 thf(fact_1301_rtrancl__induct2,axiom,
% 14.17/8.02 ! [Ax: int,Ay: int,Bx: int,By: int,R2: set_Pr2560585780119916871nt_int,P: int > int > $o] :
% 14.17/8.02 ( ( member8566619992076573584nt_int @ ( produc3646306378393792727nt_int @ ( product_Pair_int_int @ Ax @ Ay ) @ ( product_Pair_int_int @ Bx @ By ) ) @ ( transi5430920176653612092nt_int @ R2 ) )
% 14.17/8.02 => ( ( P @ Ax @ Ay )
% 14.17/8.02 => ( ! [A4: int,B2: int,Aa: int,Ba: int] :
% 14.17/8.02 ( ( member8566619992076573584nt_int @ ( produc3646306378393792727nt_int @ ( product_Pair_int_int @ Ax @ Ay ) @ ( product_Pair_int_int @ A4 @ B2 ) ) @ ( transi5430920176653612092nt_int @ R2 ) )
% 14.17/8.02 => ( ( member8566619992076573584nt_int @ ( produc3646306378393792727nt_int @ ( product_Pair_int_int @ A4 @ B2 ) @ ( product_Pair_int_int @ Aa @ Ba ) ) @ R2 )
% 14.17/8.02 => ( ( P @ A4 @ B2 )
% 14.17/8.02 => ( P @ Aa @ Ba ) ) ) )
% 14.17/8.02 => ( P @ Bx @ By ) ) ) ) ).
% 14.17/8.02
% 14.17/8.02 thf(fact_1302_rtrancl__induct2,axiom,
% 14.17/8.02 ! [Ax: produc3658429121746597890et_nat > $o,Ay: produc3658429121746597890et_nat,Bx: produc3658429121746597890et_nat > $o,By: produc3658429121746597890et_nat,R2: set_Pr7928877670098842301et_nat,P: ( produc3658429121746597890et_nat > $o ) > produc3658429121746597890et_nat > $o] :
% 14.17/8.02 ( ( member4763271486408492550et_nat @ ( produc8599840265553166229et_nat @ ( produc5001842942810119800et_nat @ Ax @ Ay ) @ ( produc5001842942810119800et_nat @ Bx @ By ) ) @ ( transi2414916928880341691et_nat @ R2 ) )
% 14.17/8.02 => ( ( P @ Ax @ Ay )
% 14.17/8.02 => ( ! [A4: produc3658429121746597890et_nat > $o,B2: produc3658429121746597890et_nat,Aa: produc3658429121746597890et_nat > $o,Ba: produc3658429121746597890et_nat] :
% 14.17/8.02 ( ( member4763271486408492550et_nat @ ( produc8599840265553166229et_nat @ ( produc5001842942810119800et_nat @ Ax @ Ay ) @ ( produc5001842942810119800et_nat @ A4 @ B2 ) ) @ ( transi2414916928880341691et_nat @ R2 ) )
% 14.17/8.02 => ( ( member4763271486408492550et_nat @ ( produc8599840265553166229et_nat @ ( produc5001842942810119800et_nat @ A4 @ B2 ) @ ( produc5001842942810119800et_nat @ Aa @ Ba ) ) @ R2 )
% 14.17/8.02 => ( ( P @ A4 @ B2 )
% 14.17/8.02 => ( P @ Aa @ Ba ) ) ) )
% 14.17/8.02 => ( P @ Bx @ By ) ) ) ) ).
% 14.17/8.02
% 14.17/8.02 thf(fact_1303_rtrancl__induct2,axiom,
% 14.17/8.02 ! [Ax: produc3658429121746597890et_nat > $o,Ay: produc3925858234332021118et_nat,Bx: produc3658429121746597890et_nat > $o,By: produc3925858234332021118et_nat,R2: set_Pr3444600963470892981et_nat,P: ( produc3658429121746597890et_nat > $o ) > produc3925858234332021118et_nat > $o] :
% 14.17/8.02 ( ( member6341495586645257982et_nat @ ( produc1940133919992309389et_nat @ ( produc2245416461498447860et_nat @ Ax @ Ay ) @ ( produc2245416461498447860et_nat @ Bx @ By ) ) @ ( transi344104063936286263et_nat @ R2 ) )
% 14.17/8.02 => ( ( P @ Ax @ Ay )
% 14.17/8.02 => ( ! [A4: produc3658429121746597890et_nat > $o,B2: produc3925858234332021118et_nat,Aa: produc3658429121746597890et_nat > $o,Ba: produc3925858234332021118et_nat] :
% 14.17/8.02 ( ( member6341495586645257982et_nat @ ( produc1940133919992309389et_nat @ ( produc2245416461498447860et_nat @ Ax @ Ay ) @ ( produc2245416461498447860et_nat @ A4 @ B2 ) ) @ ( transi344104063936286263et_nat @ R2 ) )
% 14.17/8.02 => ( ( member6341495586645257982et_nat @ ( produc1940133919992309389et_nat @ ( produc2245416461498447860et_nat @ A4 @ B2 ) @ ( produc2245416461498447860et_nat @ Aa @ Ba ) ) @ R2 )
% 14.17/8.02 => ( ( P @ A4 @ B2 )
% 14.17/8.02 => ( P @ Aa @ Ba ) ) ) )
% 14.17/8.02 => ( P @ Bx @ By ) ) ) ) ).
% 14.17/8.02
% 14.17/8.02 thf(fact_1304_converse__rtranclE2,axiom,
% 14.17/8.02 ! [Xa: heap_e7401611519738050253t_unit,Xb: set_nat,Za: heap_e7401611519738050253t_unit,Zb: set_nat,R2: set_Pr719794911490849221et_nat] :
% 14.17/8.02 ( ( member6099555550032318734et_nat @ ( produc8199053930788261021et_nat @ ( produc7507926704131184380et_nat @ Xa @ Xb ) @ ( produc7507926704131184380et_nat @ Za @ Zb ) ) @ ( transi6919426408697524543et_nat @ R2 ) )
% 14.17/8.02 => ( ( ( produc7507926704131184380et_nat @ Xa @ Xb )
% 14.17/8.02 != ( produc7507926704131184380et_nat @ Za @ Zb ) )
% 14.17/8.02 => ~ ! [A4: heap_e7401611519738050253t_unit,B2: set_nat] :
% 14.17/8.02 ( ( member6099555550032318734et_nat @ ( produc8199053930788261021et_nat @ ( produc7507926704131184380et_nat @ Xa @ Xb ) @ ( produc7507926704131184380et_nat @ A4 @ B2 ) ) @ R2 )
% 14.17/8.02 => ~ ( member6099555550032318734et_nat @ ( produc8199053930788261021et_nat @ ( produc7507926704131184380et_nat @ A4 @ B2 ) @ ( produc7507926704131184380et_nat @ Za @ Zb ) ) @ ( transi6919426408697524543et_nat @ R2 ) ) ) ) ) ).
% 14.17/8.02
% 14.17/8.02 thf(fact_1305_converse__rtranclE2,axiom,
% 14.17/8.02 ! [Xa: nat,Xb: nat,Za: nat,Zb: nat,R2: set_Pr8693737435421807431at_nat] :
% 14.17/8.02 ( ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ ( product_Pair_nat_nat @ Xa @ Xb ) @ ( product_Pair_nat_nat @ Za @ Zb ) ) @ ( transi8609417484261754244at_nat @ R2 ) )
% 14.17/8.02 => ( ( ( product_Pair_nat_nat @ Xa @ Xb )
% 14.17/8.02 != ( product_Pair_nat_nat @ Za @ Zb ) )
% 14.17/8.02 => ~ ! [A4: nat,B2: nat] :
% 14.17/8.02 ( ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ ( product_Pair_nat_nat @ Xa @ Xb ) @ ( product_Pair_nat_nat @ A4 @ B2 ) ) @ R2 )
% 14.17/8.02 => ~ ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ ( product_Pair_nat_nat @ A4 @ B2 ) @ ( product_Pair_nat_nat @ Za @ Zb ) ) @ ( transi8609417484261754244at_nat @ R2 ) ) ) ) ) ).
% 14.17/8.02
% 14.17/8.02 thf(fact_1306_converse__rtranclE2,axiom,
% 14.17/8.02 ! [Xa: int,Xb: int,Za: int,Zb: int,R2: set_Pr2560585780119916871nt_int] :
% 14.17/8.02 ( ( member8566619992076573584nt_int @ ( produc3646306378393792727nt_int @ ( product_Pair_int_int @ Xa @ Xb ) @ ( product_Pair_int_int @ Za @ Zb ) ) @ ( transi5430920176653612092nt_int @ R2 ) )
% 14.17/8.02 => ( ( ( product_Pair_int_int @ Xa @ Xb )
% 14.17/8.02 != ( product_Pair_int_int @ Za @ Zb ) )
% 14.17/8.02 => ~ ! [A4: int,B2: int] :
% 14.17/8.02 ( ( member8566619992076573584nt_int @ ( produc3646306378393792727nt_int @ ( product_Pair_int_int @ Xa @ Xb ) @ ( product_Pair_int_int @ A4 @ B2 ) ) @ R2 )
% 14.17/8.02 => ~ ( member8566619992076573584nt_int @ ( produc3646306378393792727nt_int @ ( product_Pair_int_int @ A4 @ B2 ) @ ( product_Pair_int_int @ Za @ Zb ) ) @ ( transi5430920176653612092nt_int @ R2 ) ) ) ) ) ).
% 14.17/8.02
% 14.17/8.02 thf(fact_1307_converse__rtranclE2,axiom,
% 14.17/8.02 ! [Xa: produc3658429121746597890et_nat > $o,Xb: produc3658429121746597890et_nat,Za: produc3658429121746597890et_nat > $o,Zb: produc3658429121746597890et_nat,R2: set_Pr7928877670098842301et_nat] :
% 14.17/8.02 ( ( member4763271486408492550et_nat @ ( produc8599840265553166229et_nat @ ( produc5001842942810119800et_nat @ Xa @ Xb ) @ ( produc5001842942810119800et_nat @ Za @ Zb ) ) @ ( transi2414916928880341691et_nat @ R2 ) )
% 14.17/8.02 => ( ( ( produc5001842942810119800et_nat @ Xa @ Xb )
% 14.17/8.02 != ( produc5001842942810119800et_nat @ Za @ Zb ) )
% 14.17/8.02 => ~ ! [A4: produc3658429121746597890et_nat > $o,B2: produc3658429121746597890et_nat] :
% 14.17/8.02 ( ( member4763271486408492550et_nat @ ( produc8599840265553166229et_nat @ ( produc5001842942810119800et_nat @ Xa @ Xb ) @ ( produc5001842942810119800et_nat @ A4 @ B2 ) ) @ R2 )
% 14.17/8.02 => ~ ( member4763271486408492550et_nat @ ( produc8599840265553166229et_nat @ ( produc5001842942810119800et_nat @ A4 @ B2 ) @ ( produc5001842942810119800et_nat @ Za @ Zb ) ) @ ( transi2414916928880341691et_nat @ R2 ) ) ) ) ) ).
% 14.17/8.02
% 14.17/8.02 thf(fact_1308_converse__rtranclE2,axiom,
% 14.17/8.02 ! [Xa: produc3658429121746597890et_nat > $o,Xb: produc3925858234332021118et_nat,Za: produc3658429121746597890et_nat > $o,Zb: produc3925858234332021118et_nat,R2: set_Pr3444600963470892981et_nat] :
% 14.17/8.02 ( ( member6341495586645257982et_nat @ ( produc1940133919992309389et_nat @ ( produc2245416461498447860et_nat @ Xa @ Xb ) @ ( produc2245416461498447860et_nat @ Za @ Zb ) ) @ ( transi344104063936286263et_nat @ R2 ) )
% 14.17/8.02 => ( ( ( produc2245416461498447860et_nat @ Xa @ Xb )
% 14.17/8.02 != ( produc2245416461498447860et_nat @ Za @ Zb ) )
% 14.17/8.02 => ~ ! [A4: produc3658429121746597890et_nat > $o,B2: produc3925858234332021118et_nat] :
% 14.17/8.02 ( ( member6341495586645257982et_nat @ ( produc1940133919992309389et_nat @ ( produc2245416461498447860et_nat @ Xa @ Xb ) @ ( produc2245416461498447860et_nat @ A4 @ B2 ) ) @ R2 )
% 14.17/8.02 => ~ ( member6341495586645257982et_nat @ ( produc1940133919992309389et_nat @ ( produc2245416461498447860et_nat @ A4 @ B2 ) @ ( produc2245416461498447860et_nat @ Za @ Zb ) ) @ ( transi344104063936286263et_nat @ R2 ) ) ) ) ) ).
% 14.17/8.02
% 14.17/8.02 thf(fact_1309_converse__rtrancl__induct2,axiom,
% 14.17/8.02 ! [Ax: heap_e7401611519738050253t_unit,Ay: set_nat,Bx: heap_e7401611519738050253t_unit,By: set_nat,R2: set_Pr719794911490849221et_nat,P: heap_e7401611519738050253t_unit > set_nat > $o] :
% 14.17/8.02 ( ( member6099555550032318734et_nat @ ( produc8199053930788261021et_nat @ ( produc7507926704131184380et_nat @ Ax @ Ay ) @ ( produc7507926704131184380et_nat @ Bx @ By ) ) @ ( transi6919426408697524543et_nat @ R2 ) )
% 14.17/8.02 => ( ( P @ Bx @ By )
% 14.17/8.02 => ( ! [A4: heap_e7401611519738050253t_unit,B2: set_nat,Aa: heap_e7401611519738050253t_unit,Ba: set_nat] :
% 14.17/8.02 ( ( member6099555550032318734et_nat @ ( produc8199053930788261021et_nat @ ( produc7507926704131184380et_nat @ A4 @ B2 ) @ ( produc7507926704131184380et_nat @ Aa @ Ba ) ) @ R2 )
% 14.17/8.02 => ( ( member6099555550032318734et_nat @ ( produc8199053930788261021et_nat @ ( produc7507926704131184380et_nat @ Aa @ Ba ) @ ( produc7507926704131184380et_nat @ Bx @ By ) ) @ ( transi6919426408697524543et_nat @ R2 ) )
% 14.17/8.02 => ( ( P @ Aa @ Ba )
% 14.17/8.02 => ( P @ A4 @ B2 ) ) ) )
% 14.17/8.02 => ( P @ Ax @ Ay ) ) ) ) ).
% 14.17/8.02
% 14.17/8.02 thf(fact_1310_converse__rtrancl__induct2,axiom,
% 14.17/8.02 ! [Ax: nat,Ay: nat,Bx: nat,By: nat,R2: set_Pr8693737435421807431at_nat,P: nat > nat > $o] :
% 14.17/8.02 ( ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ ( product_Pair_nat_nat @ Ax @ Ay ) @ ( product_Pair_nat_nat @ Bx @ By ) ) @ ( transi8609417484261754244at_nat @ R2 ) )
% 14.17/8.02 => ( ( P @ Bx @ By )
% 14.17/8.02 => ( ! [A4: nat,B2: nat,Aa: nat,Ba: nat] :
% 14.17/8.02 ( ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ ( product_Pair_nat_nat @ A4 @ B2 ) @ ( product_Pair_nat_nat @ Aa @ Ba ) ) @ R2 )
% 14.17/8.02 => ( ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ ( product_Pair_nat_nat @ Aa @ Ba ) @ ( product_Pair_nat_nat @ Bx @ By ) ) @ ( transi8609417484261754244at_nat @ R2 ) )
% 14.17/8.02 => ( ( P @ Aa @ Ba )
% 14.17/8.02 => ( P @ A4 @ B2 ) ) ) )
% 14.17/8.02 => ( P @ Ax @ Ay ) ) ) ) ).
% 14.17/8.02
% 14.17/8.02 thf(fact_1311_converse__rtrancl__induct2,axiom,
% 14.17/8.02 ! [Ax: int,Ay: int,Bx: int,By: int,R2: set_Pr2560585780119916871nt_int,P: int > int > $o] :
% 14.17/8.02 ( ( member8566619992076573584nt_int @ ( produc3646306378393792727nt_int @ ( product_Pair_int_int @ Ax @ Ay ) @ ( product_Pair_int_int @ Bx @ By ) ) @ ( transi5430920176653612092nt_int @ R2 ) )
% 14.17/8.02 => ( ( P @ Bx @ By )
% 14.17/8.02 => ( ! [A4: int,B2: int,Aa: int,Ba: int] :
% 14.17/8.02 ( ( member8566619992076573584nt_int @ ( produc3646306378393792727nt_int @ ( product_Pair_int_int @ A4 @ B2 ) @ ( product_Pair_int_int @ Aa @ Ba ) ) @ R2 )
% 14.17/8.02 => ( ( member8566619992076573584nt_int @ ( produc3646306378393792727nt_int @ ( product_Pair_int_int @ Aa @ Ba ) @ ( product_Pair_int_int @ Bx @ By ) ) @ ( transi5430920176653612092nt_int @ R2 ) )
% 14.17/8.02 => ( ( P @ Aa @ Ba )
% 14.17/8.02 => ( P @ A4 @ B2 ) ) ) )
% 14.17/8.02 => ( P @ Ax @ Ay ) ) ) ) ).
% 14.17/8.02
% 14.17/8.02 thf(fact_1312_converse__rtrancl__induct2,axiom,
% 14.17/8.02 ! [Ax: produc3658429121746597890et_nat > $o,Ay: produc3658429121746597890et_nat,Bx: produc3658429121746597890et_nat > $o,By: produc3658429121746597890et_nat,R2: set_Pr7928877670098842301et_nat,P: ( produc3658429121746597890et_nat > $o ) > produc3658429121746597890et_nat > $o] :
% 14.17/8.02 ( ( member4763271486408492550et_nat @ ( produc8599840265553166229et_nat @ ( produc5001842942810119800et_nat @ Ax @ Ay ) @ ( produc5001842942810119800et_nat @ Bx @ By ) ) @ ( transi2414916928880341691et_nat @ R2 ) )
% 14.17/8.02 => ( ( P @ Bx @ By )
% 14.17/8.02 => ( ! [A4: produc3658429121746597890et_nat > $o,B2: produc3658429121746597890et_nat,Aa: produc3658429121746597890et_nat > $o,Ba: produc3658429121746597890et_nat] :
% 14.17/8.02 ( ( member4763271486408492550et_nat @ ( produc8599840265553166229et_nat @ ( produc5001842942810119800et_nat @ A4 @ B2 ) @ ( produc5001842942810119800et_nat @ Aa @ Ba ) ) @ R2 )
% 14.17/8.02 => ( ( member4763271486408492550et_nat @ ( produc8599840265553166229et_nat @ ( produc5001842942810119800et_nat @ Aa @ Ba ) @ ( produc5001842942810119800et_nat @ Bx @ By ) ) @ ( transi2414916928880341691et_nat @ R2 ) )
% 14.17/8.02 => ( ( P @ Aa @ Ba )
% 14.17/8.02 => ( P @ A4 @ B2 ) ) ) )
% 14.17/8.02 => ( P @ Ax @ Ay ) ) ) ) ).
% 14.17/8.02
% 14.17/8.02 thf(fact_1313_converse__rtrancl__induct2,axiom,
% 14.17/8.02 ! [Ax: produc3658429121746597890et_nat > $o,Ay: produc3925858234332021118et_nat,Bx: produc3658429121746597890et_nat > $o,By: produc3925858234332021118et_nat,R2: set_Pr3444600963470892981et_nat,P: ( produc3658429121746597890et_nat > $o ) > produc3925858234332021118et_nat > $o] :
% 14.17/8.02 ( ( member6341495586645257982et_nat @ ( produc1940133919992309389et_nat @ ( produc2245416461498447860et_nat @ Ax @ Ay ) @ ( produc2245416461498447860et_nat @ Bx @ By ) ) @ ( transi344104063936286263et_nat @ R2 ) )
% 14.17/8.02 => ( ( P @ Bx @ By )
% 14.17/8.02 => ( ! [A4: produc3658429121746597890et_nat > $o,B2: produc3925858234332021118et_nat,Aa: produc3658429121746597890et_nat > $o,Ba: produc3925858234332021118et_nat] :
% 14.17/8.02 ( ( member6341495586645257982et_nat @ ( produc1940133919992309389et_nat @ ( produc2245416461498447860et_nat @ A4 @ B2 ) @ ( produc2245416461498447860et_nat @ Aa @ Ba ) ) @ R2 )
% 14.17/8.02 => ( ( member6341495586645257982et_nat @ ( produc1940133919992309389et_nat @ ( produc2245416461498447860et_nat @ Aa @ Ba ) @ ( produc2245416461498447860et_nat @ Bx @ By ) ) @ ( transi344104063936286263et_nat @ R2 ) )
% 14.17/8.02 => ( ( P @ Aa @ Ba )
% 14.17/8.02 => ( P @ A4 @ B2 ) ) ) )
% 14.17/8.02 => ( P @ Ax @ Ay ) ) ) ) ).
% 14.17/8.02
% 14.17/8.02 thf(fact_1314_Un__empty__right,axiom,
% 14.17/8.02 ! [A: set_Product_unit] :
% 14.17/8.02 ( ( sup_su793286257634532545t_unit @ A @ bot_bo3957492148770167129t_unit )
% 14.17/8.02 = A ) ).
% 14.17/8.02
% 14.17/8.02 thf(fact_1315_Un__empty__right,axiom,
% 14.17/8.02 ! [A: set_o] :
% 14.17/8.02 ( ( sup_sup_set_o @ A @ bot_bot_set_o )
% 14.17/8.02 = A ) ).
% 14.17/8.02
% 14.17/8.02 thf(fact_1316_Un__empty__right,axiom,
% 14.17/8.02 ! [A: set_nat] :
% 14.17/8.02 ( ( sup_sup_set_nat @ A @ bot_bot_set_nat )
% 14.17/8.02 = A ) ).
% 14.17/8.02
% 14.17/8.02 thf(fact_1317_Un__empty__right,axiom,
% 14.17/8.02 ! [A: set_int] :
% 14.17/8.02 ( ( sup_sup_set_int @ A @ bot_bot_set_int )
% 14.17/8.02 = A ) ).
% 14.17/8.02
% 14.17/8.02 thf(fact_1318_Un__empty__right,axiom,
% 14.17/8.02 ! [A: set_Pr4532377907799695533_nat_o] :
% 14.17/8.02 ( ( sup_su5209123915105501825_nat_o @ A @ bot_bo7824918357723582233_nat_o )
% 14.17/8.02 = A ) ).
% 14.17/8.02
% 14.17/8.02 thf(fact_1319_Un__empty__left,axiom,
% 14.17/8.02 ! [B4: set_Product_unit] :
% 14.17/8.02 ( ( sup_su793286257634532545t_unit @ bot_bo3957492148770167129t_unit @ B4 )
% 14.17/8.02 = B4 ) ).
% 14.17/8.02
% 14.17/8.02 thf(fact_1320_Un__empty__left,axiom,
% 14.17/8.02 ! [B4: set_o] :
% 14.17/8.02 ( ( sup_sup_set_o @ bot_bot_set_o @ B4 )
% 14.17/8.02 = B4 ) ).
% 14.17/8.02
% 14.17/8.02 thf(fact_1321_Un__empty__left,axiom,
% 14.17/8.02 ! [B4: set_nat] :
% 14.17/8.02 ( ( sup_sup_set_nat @ bot_bot_set_nat @ B4 )
% 14.17/8.02 = B4 ) ).
% 14.17/8.02
% 14.17/8.02 thf(fact_1322_Un__empty__left,axiom,
% 14.17/8.02 ! [B4: set_int] :
% 14.17/8.02 ( ( sup_sup_set_int @ bot_bot_set_int @ B4 )
% 14.17/8.02 = B4 ) ).
% 14.17/8.02
% 14.17/8.02 thf(fact_1323_Un__empty__left,axiom,
% 14.17/8.02 ! [B4: set_Pr4532377907799695533_nat_o] :
% 14.17/8.02 ( ( sup_su5209123915105501825_nat_o @ bot_bo7824918357723582233_nat_o @ B4 )
% 14.17/8.02 = B4 ) ).
% 14.17/8.02
% 14.17/8.02 thf(fact_1324_disjoint__iff__not__equal,axiom,
% 14.17/8.02 ! [A: set_Product_unit,B4: set_Product_unit] :
% 14.17/8.02 ( ( ( inf_in4660618365625256667t_unit @ A @ B4 )
% 14.17/8.02 = bot_bo3957492148770167129t_unit )
% 14.17/8.02 = ( ! [X2: product_unit] :
% 14.17/8.02 ( ( member_Product_unit @ X2 @ A )
% 14.17/8.02 => ! [Y3: product_unit] :
% 14.17/8.02 ( ( member_Product_unit @ Y3 @ B4 )
% 14.17/8.02 => ( X2 != Y3 ) ) ) ) ) ).
% 14.17/8.02
% 14.17/8.02 thf(fact_1325_disjoint__iff__not__equal,axiom,
% 14.17/8.02 ! [A: set_o,B4: set_o] :
% 14.17/8.02 ( ( ( inf_inf_set_o @ A @ B4 )
% 14.17/8.02 = bot_bot_set_o )
% 14.17/8.02 = ( ! [X2: $o] :
% 14.17/8.02 ( ( member_o @ X2 @ A )
% 14.17/8.02 => ! [Y3: $o] :
% 14.17/8.02 ( ( member_o @ Y3 @ B4 )
% 14.17/8.02 => ( X2 = ~ Y3 ) ) ) ) ) ).
% 14.17/8.02
% 14.17/8.02 thf(fact_1326_disjoint__iff__not__equal,axiom,
% 14.17/8.02 ! [A: set_nat,B4: set_nat] :
% 14.17/8.02 ( ( ( inf_inf_set_nat @ A @ B4 )
% 14.17/8.02 = bot_bot_set_nat )
% 14.17/8.02 = ( ! [X2: nat] :
% 14.17/8.02 ( ( member_nat @ X2 @ A )
% 14.17/8.02 => ! [Y3: nat] :
% 14.17/8.02 ( ( member_nat @ Y3 @ B4 )
% 14.17/8.02 => ( X2 != Y3 ) ) ) ) ) ).
% 14.17/8.02
% 14.17/8.02 thf(fact_1327_disjoint__iff__not__equal,axiom,
% 14.17/8.02 ! [A: set_int,B4: set_int] :
% 14.17/8.02 ( ( ( inf_inf_set_int @ A @ B4 )
% 14.17/8.02 = bot_bot_set_int )
% 14.17/8.02 = ( ! [X2: int] :
% 14.17/8.02 ( ( member_int @ X2 @ A )
% 14.17/8.02 => ! [Y3: int] :
% 14.17/8.02 ( ( member_int @ Y3 @ B4 )
% 14.17/8.02 => ( X2 != Y3 ) ) ) ) ) ).
% 14.17/8.02
% 14.17/8.02 thf(fact_1328_disjoint__iff__not__equal,axiom,
% 14.17/8.02 ! [A: set_Pr4532377907799695533_nat_o,B4: set_Pr4532377907799695533_nat_o] :
% 14.17/8.02 ( ( ( inf_in1906310914598751387_nat_o @ A @ B4 )
% 14.17/8.02 = bot_bo7824918357723582233_nat_o )
% 14.17/8.02 = ( ! [X2: produc3658429121746597890et_nat > $o] :
% 14.17/8.02 ( ( member6576561426505652726_nat_o @ X2 @ A )
% 14.17/8.02 => ! [Y3: produc3658429121746597890et_nat > $o] :
% 14.17/8.02 ( ( member6576561426505652726_nat_o @ Y3 @ B4 )
% 14.17/8.02 => ( X2 != Y3 ) ) ) ) ) ).
% 14.17/8.02
% 14.17/8.02 thf(fact_1329_Int__empty__right,axiom,
% 14.17/8.02 ! [A: set_Product_unit] :
% 14.17/8.02 ( ( inf_in4660618365625256667t_unit @ A @ bot_bo3957492148770167129t_unit )
% 14.17/8.02 = bot_bo3957492148770167129t_unit ) ).
% 14.17/8.02
% 14.17/8.02 thf(fact_1330_Int__empty__right,axiom,
% 14.17/8.02 ! [A: set_o] :
% 14.17/8.02 ( ( inf_inf_set_o @ A @ bot_bot_set_o )
% 14.17/8.02 = bot_bot_set_o ) ).
% 14.17/8.02
% 14.17/8.02 thf(fact_1331_Int__empty__right,axiom,
% 14.17/8.02 ! [A: set_nat] :
% 14.17/8.02 ( ( inf_inf_set_nat @ A @ bot_bot_set_nat )
% 14.17/8.02 = bot_bot_set_nat ) ).
% 14.17/8.02
% 14.17/8.02 thf(fact_1332_Int__empty__right,axiom,
% 14.17/8.02 ! [A: set_int] :
% 14.17/8.02 ( ( inf_inf_set_int @ A @ bot_bot_set_int )
% 14.17/8.02 = bot_bot_set_int ) ).
% 14.17/8.02
% 14.17/8.02 thf(fact_1333_Int__empty__right,axiom,
% 14.17/8.02 ! [A: set_Pr4532377907799695533_nat_o] :
% 14.17/8.02 ( ( inf_in1906310914598751387_nat_o @ A @ bot_bo7824918357723582233_nat_o )
% 14.17/8.02 = bot_bo7824918357723582233_nat_o ) ).
% 14.17/8.02
% 14.17/8.02 thf(fact_1334_Int__empty__left,axiom,
% 14.17/8.02 ! [B4: set_Product_unit] :
% 14.17/8.02 ( ( inf_in4660618365625256667t_unit @ bot_bo3957492148770167129t_unit @ B4 )
% 14.17/8.02 = bot_bo3957492148770167129t_unit ) ).
% 14.17/8.02
% 14.17/8.02 thf(fact_1335_Int__empty__left,axiom,
% 14.17/8.02 ! [B4: set_o] :
% 14.17/8.02 ( ( inf_inf_set_o @ bot_bot_set_o @ B4 )
% 14.17/8.02 = bot_bot_set_o ) ).
% 14.17/8.02
% 14.17/8.02 thf(fact_1336_Int__empty__left,axiom,
% 14.17/8.02 ! [B4: set_nat] :
% 14.17/8.02 ( ( inf_inf_set_nat @ bot_bot_set_nat @ B4 )
% 14.17/8.02 = bot_bot_set_nat ) ).
% 14.17/8.02
% 14.17/8.02 thf(fact_1337_Int__empty__left,axiom,
% 14.17/8.02 ! [B4: set_int] :
% 14.17/8.02 ( ( inf_inf_set_int @ bot_bot_set_int @ B4 )
% 14.17/8.02 = bot_bot_set_int ) ).
% 14.17/8.02
% 14.17/8.02 thf(fact_1338_Int__empty__left,axiom,
% 14.17/8.02 ! [B4: set_Pr4532377907799695533_nat_o] :
% 14.17/8.02 ( ( inf_in1906310914598751387_nat_o @ bot_bo7824918357723582233_nat_o @ B4 )
% 14.17/8.02 = bot_bo7824918357723582233_nat_o ) ).
% 14.17/8.02
% 14.17/8.02 thf(fact_1339_disjoint__iff,axiom,
% 14.17/8.02 ! [A: set_Product_unit,B4: set_Product_unit] :
% 14.17/8.02 ( ( ( inf_in4660618365625256667t_unit @ A @ B4 )
% 14.17/8.02 = bot_bo3957492148770167129t_unit )
% 14.17/8.02 = ( ! [X2: product_unit] :
% 14.17/8.02 ( ( member_Product_unit @ X2 @ A )
% 14.17/8.02 => ~ ( member_Product_unit @ X2 @ B4 ) ) ) ) ).
% 14.17/8.02
% 14.17/8.02 thf(fact_1340_disjoint__iff,axiom,
% 14.17/8.02 ! [A: set_o,B4: set_o] :
% 14.17/8.02 ( ( ( inf_inf_set_o @ A @ B4 )
% 14.17/8.02 = bot_bot_set_o )
% 14.17/8.02 = ( ! [X2: $o] :
% 14.17/8.02 ( ( member_o @ X2 @ A )
% 14.17/8.02 => ~ ( member_o @ X2 @ B4 ) ) ) ) ).
% 14.17/8.02
% 14.17/8.02 thf(fact_1341_disjoint__iff,axiom,
% 14.17/8.02 ! [A: set_nat,B4: set_nat] :
% 14.17/8.02 ( ( ( inf_inf_set_nat @ A @ B4 )
% 14.17/8.02 = bot_bot_set_nat )
% 14.17/8.02 = ( ! [X2: nat] :
% 14.17/8.02 ( ( member_nat @ X2 @ A )
% 14.17/8.02 => ~ ( member_nat @ X2 @ B4 ) ) ) ) ).
% 14.17/8.02
% 14.17/8.02 thf(fact_1342_disjoint__iff,axiom,
% 14.17/8.02 ! [A: set_int,B4: set_int] :
% 14.17/8.02 ( ( ( inf_inf_set_int @ A @ B4 )
% 14.17/8.02 = bot_bot_set_int )
% 14.17/8.02 = ( ! [X2: int] :
% 14.17/8.02 ( ( member_int @ X2 @ A )
% 14.17/8.02 => ~ ( member_int @ X2 @ B4 ) ) ) ) ).
% 14.17/8.02
% 14.17/8.02 thf(fact_1343_disjoint__iff,axiom,
% 14.17/8.02 ! [A: set_Pr4532377907799695533_nat_o,B4: set_Pr4532377907799695533_nat_o] :
% 14.17/8.02 ( ( ( inf_in1906310914598751387_nat_o @ A @ B4 )
% 14.17/8.02 = bot_bo7824918357723582233_nat_o )
% 14.17/8.02 = ( ! [X2: produc3658429121746597890et_nat > $o] :
% 14.17/8.02 ( ( member6576561426505652726_nat_o @ X2 @ A )
% 14.17/8.02 => ~ ( member6576561426505652726_nat_o @ X2 @ B4 ) ) ) ) ).
% 14.17/8.02
% 14.17/8.02 thf(fact_1344_Int__emptyI,axiom,
% 14.17/8.02 ! [A: set_Product_unit,B4: set_Product_unit] :
% 14.17/8.02 ( ! [X3: product_unit] :
% 14.17/8.02 ( ( member_Product_unit @ X3 @ A )
% 14.17/8.02 => ~ ( member_Product_unit @ X3 @ B4 ) )
% 14.17/8.02 => ( ( inf_in4660618365625256667t_unit @ A @ B4 )
% 14.17/8.02 = bot_bo3957492148770167129t_unit ) ) ).
% 14.17/8.02
% 14.17/8.02 thf(fact_1345_Int__emptyI,axiom,
% 14.17/8.02 ! [A: set_o,B4: set_o] :
% 14.17/8.02 ( ! [X3: $o] :
% 14.17/8.02 ( ( member_o @ X3 @ A )
% 14.17/8.02 => ~ ( member_o @ X3 @ B4 ) )
% 14.17/8.02 => ( ( inf_inf_set_o @ A @ B4 )
% 14.17/8.02 = bot_bot_set_o ) ) ).
% 14.17/8.02
% 14.17/8.02 thf(fact_1346_Int__emptyI,axiom,
% 14.17/8.02 ! [A: set_nat,B4: set_nat] :
% 14.17/8.02 ( ! [X3: nat] :
% 14.17/8.02 ( ( member_nat @ X3 @ A )
% 14.17/8.02 => ~ ( member_nat @ X3 @ B4 ) )
% 14.17/8.02 => ( ( inf_inf_set_nat @ A @ B4 )
% 14.17/8.02 = bot_bot_set_nat ) ) ).
% 14.17/8.02
% 14.17/8.02 thf(fact_1347_Int__emptyI,axiom,
% 14.17/8.02 ! [A: set_int,B4: set_int] :
% 14.17/8.02 ( ! [X3: int] :
% 14.17/8.02 ( ( member_int @ X3 @ A )
% 14.17/8.02 => ~ ( member_int @ X3 @ B4 ) )
% 14.17/8.02 => ( ( inf_inf_set_int @ A @ B4 )
% 14.17/8.02 = bot_bot_set_int ) ) ).
% 14.17/8.02
% 14.17/8.02 thf(fact_1348_Int__emptyI,axiom,
% 14.17/8.02 ! [A: set_Pr4532377907799695533_nat_o,B4: set_Pr4532377907799695533_nat_o] :
% 14.17/8.02 ( ! [X3: produc3658429121746597890et_nat > $o] :
% 14.17/8.02 ( ( member6576561426505652726_nat_o @ X3 @ A )
% 14.17/8.02 => ~ ( member6576561426505652726_nat_o @ X3 @ B4 ) )
% 14.17/8.02 => ( ( inf_in1906310914598751387_nat_o @ A @ B4 )
% 14.17/8.02 = bot_bo7824918357723582233_nat_o ) ) ).
% 14.17/8.02
% 14.17/8.02 thf(fact_1349_disjointI,axiom,
% 14.17/8.02 ! [A2: set_Product_unit,B: set_Product_unit] :
% 14.17/8.02 ( ! [X3: product_unit] :
% 14.17/8.02 ( ( member_Product_unit @ X3 @ A2 )
% 14.17/8.02 => ~ ( member_Product_unit @ X3 @ B ) )
% 14.17/8.02 => ( ( inf_in4660618365625256667t_unit @ A2 @ B )
% 14.17/8.02 = bot_bo3957492148770167129t_unit ) ) ).
% 14.17/8.02
% 14.17/8.02 thf(fact_1350_disjointI,axiom,
% 14.17/8.02 ! [A2: set_o,B: set_o] :
% 14.17/8.02 ( ! [X3: $o] :
% 14.17/8.02 ( ( member_o @ X3 @ A2 )
% 14.17/8.02 => ~ ( member_o @ X3 @ B ) )
% 14.17/8.02 => ( ( inf_inf_set_o @ A2 @ B )
% 14.17/8.02 = bot_bot_set_o ) ) ).
% 14.17/8.02
% 14.17/8.02 thf(fact_1351_disjointI,axiom,
% 14.17/8.02 ! [A2: set_nat,B: set_nat] :
% 14.17/8.02 ( ! [X3: nat] :
% 14.17/8.02 ( ( member_nat @ X3 @ A2 )
% 14.17/8.02 => ~ ( member_nat @ X3 @ B ) )
% 14.17/8.02 => ( ( inf_inf_set_nat @ A2 @ B )
% 14.17/8.02 = bot_bot_set_nat ) ) ).
% 14.17/8.02
% 14.17/8.02 thf(fact_1352_disjointI,axiom,
% 14.17/8.02 ! [A2: set_int,B: set_int] :
% 14.17/8.02 ( ! [X3: int] :
% 14.17/8.02 ( ( member_int @ X3 @ A2 )
% 14.17/8.02 => ~ ( member_int @ X3 @ B ) )
% 14.17/8.02 => ( ( inf_inf_set_int @ A2 @ B )
% 14.17/8.02 = bot_bot_set_int ) ) ).
% 14.17/8.02
% 14.17/8.02 thf(fact_1353_disjointI,axiom,
% 14.17/8.02 ! [A2: set_Pr4532377907799695533_nat_o,B: set_Pr4532377907799695533_nat_o] :
% 14.17/8.02 ( ! [X3: produc3658429121746597890et_nat > $o] :
% 14.17/8.02 ( ( member6576561426505652726_nat_o @ X3 @ A2 )
% 14.17/8.02 => ~ ( member6576561426505652726_nat_o @ X3 @ B ) )
% 14.17/8.02 => ( ( inf_in1906310914598751387_nat_o @ A2 @ B )
% 14.17/8.02 = bot_bo7824918357723582233_nat_o ) ) ).
% 14.17/8.02
% 14.17/8.02 thf(fact_1354_Int__insert__right,axiom,
% 14.17/8.02 ! [A2: produc3658429121746597890et_nat > $o,A: set_Pr4532377907799695533_nat_o,B4: set_Pr4532377907799695533_nat_o] :
% 14.17/8.02 ( ( ( member6576561426505652726_nat_o @ A2 @ A )
% 14.17/8.02 => ( ( inf_in1906310914598751387_nat_o @ A @ ( insert5175938949040314269_nat_o @ A2 @ B4 ) )
% 14.17/8.02 = ( insert5175938949040314269_nat_o @ A2 @ ( inf_in1906310914598751387_nat_o @ A @ B4 ) ) ) )
% 14.17/8.02 & ( ~ ( member6576561426505652726_nat_o @ A2 @ A )
% 14.17/8.02 => ( ( inf_in1906310914598751387_nat_o @ A @ ( insert5175938949040314269_nat_o @ A2 @ B4 ) )
% 14.17/8.02 = ( inf_in1906310914598751387_nat_o @ A @ B4 ) ) ) ) ).
% 14.17/8.02
% 14.17/8.02 thf(fact_1355_Int__insert__right,axiom,
% 14.17/8.02 ! [A2: int,A: set_int,B4: set_int] :
% 14.17/8.02 ( ( ( member_int @ A2 @ A )
% 14.17/8.02 => ( ( inf_inf_set_int @ A @ ( insert_int @ A2 @ B4 ) )
% 14.17/8.02 = ( insert_int @ A2 @ ( inf_inf_set_int @ A @ B4 ) ) ) )
% 14.17/8.02 & ( ~ ( member_int @ A2 @ A )
% 14.17/8.02 => ( ( inf_inf_set_int @ A @ ( insert_int @ A2 @ B4 ) )
% 14.17/8.02 = ( inf_inf_set_int @ A @ B4 ) ) ) ) ).
% 14.17/8.02
% 14.17/8.02 thf(fact_1356_Int__insert__right,axiom,
% 14.17/8.02 ! [A2: $o,A: set_o,B4: set_o] :
% 14.17/8.02 ( ( ( member_o @ A2 @ A )
% 14.17/8.02 => ( ( inf_inf_set_o @ A @ ( insert_o @ A2 @ B4 ) )
% 14.17/8.02 = ( insert_o @ A2 @ ( inf_inf_set_o @ A @ B4 ) ) ) )
% 14.17/8.02 & ( ~ ( member_o @ A2 @ A )
% 14.17/8.02 => ( ( inf_inf_set_o @ A @ ( insert_o @ A2 @ B4 ) )
% 14.17/8.02 = ( inf_inf_set_o @ A @ B4 ) ) ) ) ).
% 14.17/8.02
% 14.17/8.02 thf(fact_1357_Int__insert__right,axiom,
% 14.17/8.02 ! [A2: product_unit,A: set_Product_unit,B4: set_Product_unit] :
% 14.17/8.02 ( ( ( member_Product_unit @ A2 @ A )
% 14.17/8.02 => ( ( inf_in4660618365625256667t_unit @ A @ ( insert_Product_unit @ A2 @ B4 ) )
% 14.17/8.02 = ( insert_Product_unit @ A2 @ ( inf_in4660618365625256667t_unit @ A @ B4 ) ) ) )
% 14.17/8.02 & ( ~ ( member_Product_unit @ A2 @ A )
% 14.17/8.02 => ( ( inf_in4660618365625256667t_unit @ A @ ( insert_Product_unit @ A2 @ B4 ) )
% 14.17/8.02 = ( inf_in4660618365625256667t_unit @ A @ B4 ) ) ) ) ).
% 14.17/8.02
% 14.17/8.02 thf(fact_1358_Int__insert__right,axiom,
% 14.17/8.02 ! [A2: nat,A: set_nat,B4: set_nat] :
% 14.17/8.02 ( ( ( member_nat @ A2 @ A )
% 14.17/8.02 => ( ( inf_inf_set_nat @ A @ ( insert_nat @ A2 @ B4 ) )
% 14.17/8.02 = ( insert_nat @ A2 @ ( inf_inf_set_nat @ A @ B4 ) ) ) )
% 14.17/8.02 & ( ~ ( member_nat @ A2 @ A )
% 14.17/8.02 => ( ( inf_inf_set_nat @ A @ ( insert_nat @ A2 @ B4 ) )
% 14.17/8.02 = ( inf_inf_set_nat @ A @ B4 ) ) ) ) ).
% 14.17/8.02
% 14.17/8.02 thf(fact_1359_Int__insert__left,axiom,
% 14.17/8.02 ! [A2: produc3658429121746597890et_nat > $o,C5: set_Pr4532377907799695533_nat_o,B4: set_Pr4532377907799695533_nat_o] :
% 14.17/8.02 ( ( ( member6576561426505652726_nat_o @ A2 @ C5 )
% 14.17/8.02 => ( ( inf_in1906310914598751387_nat_o @ ( insert5175938949040314269_nat_o @ A2 @ B4 ) @ C5 )
% 14.17/8.02 = ( insert5175938949040314269_nat_o @ A2 @ ( inf_in1906310914598751387_nat_o @ B4 @ C5 ) ) ) )
% 14.17/8.02 & ( ~ ( member6576561426505652726_nat_o @ A2 @ C5 )
% 14.17/8.02 => ( ( inf_in1906310914598751387_nat_o @ ( insert5175938949040314269_nat_o @ A2 @ B4 ) @ C5 )
% 14.17/8.02 = ( inf_in1906310914598751387_nat_o @ B4 @ C5 ) ) ) ) ).
% 14.17/8.02
% 14.17/8.02 thf(fact_1360_Int__insert__left,axiom,
% 14.17/8.02 ! [A2: int,C5: set_int,B4: set_int] :
% 14.17/8.02 ( ( ( member_int @ A2 @ C5 )
% 14.17/8.02 => ( ( inf_inf_set_int @ ( insert_int @ A2 @ B4 ) @ C5 )
% 14.17/8.02 = ( insert_int @ A2 @ ( inf_inf_set_int @ B4 @ C5 ) ) ) )
% 14.17/8.02 & ( ~ ( member_int @ A2 @ C5 )
% 14.17/8.02 => ( ( inf_inf_set_int @ ( insert_int @ A2 @ B4 ) @ C5 )
% 14.17/8.02 = ( inf_inf_set_int @ B4 @ C5 ) ) ) ) ).
% 14.17/8.02
% 14.17/8.02 thf(fact_1361_Int__insert__left,axiom,
% 14.17/8.02 ! [A2: $o,C5: set_o,B4: set_o] :
% 14.17/8.02 ( ( ( member_o @ A2 @ C5 )
% 14.17/8.02 => ( ( inf_inf_set_o @ ( insert_o @ A2 @ B4 ) @ C5 )
% 14.17/8.02 = ( insert_o @ A2 @ ( inf_inf_set_o @ B4 @ C5 ) ) ) )
% 14.17/8.02 & ( ~ ( member_o @ A2 @ C5 )
% 14.17/8.02 => ( ( inf_inf_set_o @ ( insert_o @ A2 @ B4 ) @ C5 )
% 14.17/8.02 = ( inf_inf_set_o @ B4 @ C5 ) ) ) ) ).
% 14.17/8.02
% 14.17/8.02 thf(fact_1362_Int__insert__left,axiom,
% 14.17/8.02 ! [A2: product_unit,C5: set_Product_unit,B4: set_Product_unit] :
% 14.17/8.02 ( ( ( member_Product_unit @ A2 @ C5 )
% 14.17/8.02 => ( ( inf_in4660618365625256667t_unit @ ( insert_Product_unit @ A2 @ B4 ) @ C5 )
% 14.17/8.02 = ( insert_Product_unit @ A2 @ ( inf_in4660618365625256667t_unit @ B4 @ C5 ) ) ) )
% 14.17/8.02 & ( ~ ( member_Product_unit @ A2 @ C5 )
% 14.17/8.02 => ( ( inf_in4660618365625256667t_unit @ ( insert_Product_unit @ A2 @ B4 ) @ C5 )
% 14.17/8.02 = ( inf_in4660618365625256667t_unit @ B4 @ C5 ) ) ) ) ).
% 14.17/8.02
% 14.17/8.02 thf(fact_1363_Int__insert__left,axiom,
% 14.17/8.02 ! [A2: nat,C5: set_nat,B4: set_nat] :
% 14.17/8.02 ( ( ( member_nat @ A2 @ C5 )
% 14.17/8.02 => ( ( inf_inf_set_nat @ ( insert_nat @ A2 @ B4 ) @ C5 )
% 14.17/8.02 = ( insert_nat @ A2 @ ( inf_inf_set_nat @ B4 @ C5 ) ) ) )
% 14.17/8.02 & ( ~ ( member_nat @ A2 @ C5 )
% 14.17/8.02 => ( ( inf_inf_set_nat @ ( insert_nat @ A2 @ B4 ) @ C5 )
% 14.17/8.02 = ( inf_inf_set_nat @ B4 @ C5 ) ) ) ) ).
% 14.17/8.02
% 14.17/8.02 thf(fact_1364_subrelI,axiom,
% 14.17/8.02 ! [R2: set_Pr3948176798113811640et_nat,S2: set_Pr3948176798113811640et_nat] :
% 14.17/8.02 ( ! [X3: heap_e7401611519738050253t_unit,Y2: set_nat] :
% 14.17/8.02 ( ( member6260224972018164377et_nat @ ( produc7507926704131184380et_nat @ X3 @ Y2 ) @ R2 )
% 14.17/8.02 => ( member6260224972018164377et_nat @ ( produc7507926704131184380et_nat @ X3 @ Y2 ) @ S2 ) )
% 14.17/8.02 => ( ord_le8090042543978450200et_nat @ R2 @ S2 ) ) ).
% 14.17/8.02
% 14.17/8.02 thf(fact_1365_subrelI,axiom,
% 14.17/8.02 ! [R2: set_Pr1261947904930325089at_nat,S2: set_Pr1261947904930325089at_nat] :
% 14.17/8.02 ( ! [X3: nat,Y2: nat] :
% 14.17/8.02 ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X3 @ Y2 ) @ R2 )
% 14.17/8.02 => ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X3 @ Y2 ) @ S2 ) )
% 14.17/8.02 => ( ord_le3146513528884898305at_nat @ R2 @ S2 ) ) ).
% 14.17/8.02
% 14.17/8.02 thf(fact_1366_subrelI,axiom,
% 14.17/8.02 ! [R2: set_Pr958786334691620121nt_int,S2: set_Pr958786334691620121nt_int] :
% 14.17/8.02 ( ! [X3: int,Y2: int] :
% 14.17/8.02 ( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ X3 @ Y2 ) @ R2 )
% 14.17/8.02 => ( member5262025264175285858nt_int @ ( product_Pair_int_int @ X3 @ Y2 ) @ S2 ) )
% 14.17/8.02 => ( ord_le2843351958646193337nt_int @ R2 @ S2 ) ) ).
% 14.17/8.02
% 14.17/8.02 thf(fact_1367_subrelI,axiom,
% 14.17/8.02 ! [R2: set_Pr3286484037609594932et_nat,S2: set_Pr3286484037609594932et_nat] :
% 14.17/8.02 ( ! [X3: produc3658429121746597890et_nat > $o,Y2: produc3658429121746597890et_nat] :
% 14.17/8.02 ( ( member1996754912294343701et_nat @ ( produc5001842942810119800et_nat @ X3 @ Y2 ) @ R2 )
% 14.17/8.02 => ( member1996754912294343701et_nat @ ( produc5001842942810119800et_nat @ X3 @ Y2 ) @ S2 ) )
% 14.17/8.02 => ( ord_le5966269811547037844et_nat @ R2 @ S2 ) ) ).
% 14.17/8.02
% 14.17/8.02 thf(fact_1368_subrelI,axiom,
% 14.17/8.02 ! [R2: set_Pr8536935166611901872et_nat,S2: set_Pr8536935166611901872et_nat] :
% 14.17/8.02 ( ! [X3: produc3658429121746597890et_nat > $o,Y2: produc3925858234332021118et_nat] :
% 14.17/8.02 ( ( member6124377750444531601et_nat @ ( produc2245416461498447860et_nat @ X3 @ Y2 ) @ R2 )
% 14.17/8.02 => ( member6124377750444531601et_nat @ ( produc2245416461498447860et_nat @ X3 @ Y2 ) @ S2 ) )
% 14.17/8.02 => ( ord_le4763372923235995152et_nat @ R2 @ S2 ) ) ).
% 14.17/8.02
% 14.17/8.02 thf(fact_1369_bot_Oextremum__uniqueI,axiom,
% 14.17/8.02 ! [A2: set_Product_unit] :
% 14.17/8.02 ( ( ord_le3507040750410214029t_unit @ A2 @ bot_bo3957492148770167129t_unit )
% 14.17/8.02 => ( A2 = bot_bo3957492148770167129t_unit ) ) ).
% 14.17/8.02
% 14.17/8.02 thf(fact_1370_bot_Oextremum__uniqueI,axiom,
% 14.17/8.02 ! [A2: set_o] :
% 14.17/8.02 ( ( ord_less_eq_set_o @ A2 @ bot_bot_set_o )
% 14.17/8.02 => ( A2 = bot_bot_set_o ) ) ).
% 14.17/8.02
% 14.17/8.02 thf(fact_1371_bot_Oextremum__uniqueI,axiom,
% 14.17/8.02 ! [A2: set_int] :
% 14.17/8.02 ( ( ord_less_eq_set_int @ A2 @ bot_bot_set_int )
% 14.17/8.02 => ( A2 = bot_bot_set_int ) ) ).
% 14.17/8.02
% 14.17/8.02 thf(fact_1372_bot_Oextremum__uniqueI,axiom,
% 14.17/8.02 ! [A2: set_Pr4532377907799695533_nat_o] :
% 14.17/8.02 ( ( ord_le2965882846123202637_nat_o @ A2 @ bot_bo7824918357723582233_nat_o )
% 14.17/8.02 => ( A2 = bot_bo7824918357723582233_nat_o ) ) ).
% 14.17/8.02
% 14.17/8.02 thf(fact_1373_bot_Oextremum__uniqueI,axiom,
% 14.17/8.02 ! [A2: filter_nat] :
% 14.17/8.02 ( ( ord_le2510731241096832064er_nat @ A2 @ bot_bot_filter_nat )
% 14.17/8.02 => ( A2 = bot_bot_filter_nat ) ) ).
% 14.17/8.02
% 14.17/8.02 thf(fact_1374_bot_Oextremum__uniqueI,axiom,
% 14.17/8.02 ! [A2: set_nat] :
% 14.17/8.02 ( ( ord_less_eq_set_nat @ A2 @ bot_bot_set_nat )
% 14.17/8.02 => ( A2 = bot_bot_set_nat ) ) ).
% 14.17/8.02
% 14.17/8.02 thf(fact_1375_bot_Oextremum__uniqueI,axiom,
% 14.17/8.02 ! [A2: nat] :
% 14.17/8.02 ( ( ord_less_eq_nat @ A2 @ bot_bot_nat )
% 14.17/8.02 => ( A2 = bot_bot_nat ) ) ).
% 14.17/8.02
% 14.17/8.02 thf(fact_1376_bot_Oextremum__unique,axiom,
% 14.17/8.02 ! [A2: set_Product_unit] :
% 14.17/8.02 ( ( ord_le3507040750410214029t_unit @ A2 @ bot_bo3957492148770167129t_unit )
% 14.17/8.02 = ( A2 = bot_bo3957492148770167129t_unit ) ) ).
% 14.17/8.02
% 14.17/8.02 thf(fact_1377_bot_Oextremum__unique,axiom,
% 14.17/8.02 ! [A2: set_o] :
% 14.17/8.02 ( ( ord_less_eq_set_o @ A2 @ bot_bot_set_o )
% 14.17/8.02 = ( A2 = bot_bot_set_o ) ) ).
% 14.17/8.02
% 14.17/8.02 thf(fact_1378_bot_Oextremum__unique,axiom,
% 14.17/8.02 ! [A2: set_int] :
% 14.17/8.02 ( ( ord_less_eq_set_int @ A2 @ bot_bot_set_int )
% 14.17/8.02 = ( A2 = bot_bot_set_int ) ) ).
% 14.17/8.02
% 14.17/8.02 thf(fact_1379_bot_Oextremum__unique,axiom,
% 14.17/8.02 ! [A2: set_Pr4532377907799695533_nat_o] :
% 14.17/8.02 ( ( ord_le2965882846123202637_nat_o @ A2 @ bot_bo7824918357723582233_nat_o )
% 14.17/8.02 = ( A2 = bot_bo7824918357723582233_nat_o ) ) ).
% 14.17/8.02
% 14.17/8.02 thf(fact_1380_bot_Oextremum__unique,axiom,
% 14.17/8.02 ! [A2: filter_nat] :
% 14.17/8.02 ( ( ord_le2510731241096832064er_nat @ A2 @ bot_bot_filter_nat )
% 14.17/8.02 = ( A2 = bot_bot_filter_nat ) ) ).
% 14.17/8.02
% 14.17/8.02 thf(fact_1381_bot_Oextremum__unique,axiom,
% 14.17/8.02 ! [A2: set_nat] :
% 14.17/8.02 ( ( ord_less_eq_set_nat @ A2 @ bot_bot_set_nat )
% 14.17/8.02 = ( A2 = bot_bot_set_nat ) ) ).
% 14.17/8.02
% 14.17/8.02 thf(fact_1382_bot_Oextremum__unique,axiom,
% 14.17/8.02 ! [A2: nat] :
% 14.17/8.02 ( ( ord_less_eq_nat @ A2 @ bot_bot_nat )
% 14.17/8.02 = ( A2 = bot_bot_nat ) ) ).
% 14.17/8.02
% 14.17/8.02 thf(fact_1383_bot_Oextremum,axiom,
% 14.17/8.02 ! [A2: set_Product_unit] : ( ord_le3507040750410214029t_unit @ bot_bo3957492148770167129t_unit @ A2 ) ).
% 14.17/8.02
% 14.17/8.02 thf(fact_1384_bot_Oextremum,axiom,
% 14.17/8.02 ! [A2: set_o] : ( ord_less_eq_set_o @ bot_bot_set_o @ A2 ) ).
% 14.17/8.02
% 14.17/8.02 thf(fact_1385_bot_Oextremum,axiom,
% 14.17/8.02 ! [A2: set_int] : ( ord_less_eq_set_int @ bot_bot_set_int @ A2 ) ).
% 14.17/8.02
% 14.17/8.02 thf(fact_1386_bot_Oextremum,axiom,
% 14.17/8.02 ! [A2: set_Pr4532377907799695533_nat_o] : ( ord_le2965882846123202637_nat_o @ bot_bo7824918357723582233_nat_o @ A2 ) ).
% 14.17/8.02
% 14.17/8.02 thf(fact_1387_bot_Oextremum,axiom,
% 14.17/8.02 ! [A2: filter_nat] : ( ord_le2510731241096832064er_nat @ bot_bot_filter_nat @ A2 ) ).
% 14.17/8.02
% 14.17/8.02 thf(fact_1388_bot_Oextremum,axiom,
% 14.17/8.02 ! [A2: set_nat] : ( ord_less_eq_set_nat @ bot_bot_set_nat @ A2 ) ).
% 14.17/8.02
% 14.17/8.02 thf(fact_1389_bot_Oextremum,axiom,
% 14.17/8.02 ! [A2: nat] : ( ord_less_eq_nat @ bot_bot_nat @ A2 ) ).
% 14.17/8.02
% 14.17/8.02 thf(fact_1390_subset__emptyI,axiom,
% 14.17/8.02 ! [A: set_Product_unit] :
% 14.17/8.02 ( ! [X3: product_unit] :
% 14.17/8.02 ~ ( member_Product_unit @ X3 @ A )
% 14.17/8.02 => ( ord_le3507040750410214029t_unit @ A @ bot_bo3957492148770167129t_unit ) ) ).
% 14.17/8.02
% 14.17/8.02 thf(fact_1391_subset__emptyI,axiom,
% 14.17/8.02 ! [A: set_o] :
% 14.17/8.02 ( ! [X3: $o] :
% 14.17/8.02 ~ ( member_o @ X3 @ A )
% 14.17/8.02 => ( ord_less_eq_set_o @ A @ bot_bot_set_o ) ) ).
% 14.17/8.02
% 14.17/8.02 thf(fact_1392_subset__emptyI,axiom,
% 14.17/8.02 ! [A: set_int] :
% 14.17/8.02 ( ! [X3: int] :
% 14.17/8.02 ~ ( member_int @ X3 @ A )
% 14.17/8.02 => ( ord_less_eq_set_int @ A @ bot_bot_set_int ) ) ).
% 14.17/8.02
% 14.17/8.02 thf(fact_1393_subset__emptyI,axiom,
% 14.17/8.02 ! [A: set_Pr4532377907799695533_nat_o] :
% 14.17/8.02 ( ! [X3: produc3658429121746597890et_nat > $o] :
% 14.17/8.02 ~ ( member6576561426505652726_nat_o @ X3 @ A )
% 14.17/8.02 => ( ord_le2965882846123202637_nat_o @ A @ bot_bo7824918357723582233_nat_o ) ) ).
% 14.17/8.02
% 14.17/8.02 thf(fact_1394_subset__emptyI,axiom,
% 14.17/8.02 ! [A: set_nat] :
% 14.17/8.02 ( ! [X3: nat] :
% 14.17/8.02 ~ ( member_nat @ X3 @ A )
% 14.17/8.02 => ( ord_less_eq_set_nat @ A @ bot_bot_set_nat ) ) ).
% 14.17/8.02
% 14.17/8.02 thf(fact_1395_insert__subsetI,axiom,
% 14.17/8.02 ! [X: produc3658429121746597890et_nat > $o,A: set_Pr4532377907799695533_nat_o,X4: set_Pr4532377907799695533_nat_o] :
% 14.17/8.02 ( ( member6576561426505652726_nat_o @ X @ A )
% 14.17/8.02 => ( ( ord_le2965882846123202637_nat_o @ X4 @ A )
% 14.17/8.02 => ( ord_le2965882846123202637_nat_o @ ( insert5175938949040314269_nat_o @ X @ X4 ) @ A ) ) ) ).
% 14.17/8.02
% 14.17/8.02 thf(fact_1396_insert__subsetI,axiom,
% 14.17/8.02 ! [X: int,A: set_int,X4: set_int] :
% 14.17/8.02 ( ( member_int @ X @ A )
% 14.17/8.02 => ( ( ord_less_eq_set_int @ X4 @ A )
% 14.17/8.02 => ( ord_less_eq_set_int @ ( insert_int @ X @ X4 ) @ A ) ) ) ).
% 14.17/8.02
% 14.17/8.02 thf(fact_1397_insert__subsetI,axiom,
% 14.17/8.02 ! [X: $o,A: set_o,X4: set_o] :
% 14.17/8.02 ( ( member_o @ X @ A )
% 14.17/8.02 => ( ( ord_less_eq_set_o @ X4 @ A )
% 14.17/8.02 => ( ord_less_eq_set_o @ ( insert_o @ X @ X4 ) @ A ) ) ) ).
% 14.17/8.02
% 14.17/8.02 thf(fact_1398_insert__subsetI,axiom,
% 14.17/8.02 ! [X: product_unit,A: set_Product_unit,X4: set_Product_unit] :
% 14.17/8.02 ( ( member_Product_unit @ X @ A )
% 14.17/8.02 => ( ( ord_le3507040750410214029t_unit @ X4 @ A )
% 14.17/8.02 => ( ord_le3507040750410214029t_unit @ ( insert_Product_unit @ X @ X4 ) @ A ) ) ) ).
% 14.17/8.02
% 14.17/8.02 thf(fact_1399_insert__subsetI,axiom,
% 14.17/8.02 ! [X: nat,A: set_nat,X4: set_nat] :
% 14.17/8.02 ( ( member_nat @ X @ A )
% 14.17/8.02 => ( ( ord_less_eq_set_nat @ X4 @ A )
% 14.17/8.02 => ( ord_less_eq_set_nat @ ( insert_nat @ X @ X4 ) @ A ) ) ) ).
% 14.17/8.02
% 14.17/8.02 thf(fact_1400_subset__insertI2,axiom,
% 14.17/8.02 ! [A: set_int,B4: set_int,B: int] :
% 14.17/8.02 ( ( ord_less_eq_set_int @ A @ B4 )
% 14.17/8.02 => ( ord_less_eq_set_int @ A @ ( insert_int @ B @ B4 ) ) ) ).
% 14.17/8.02
% 14.17/8.02 thf(fact_1401_subset__insertI2,axiom,
% 14.17/8.02 ! [A: set_o,B4: set_o,B: $o] :
% 14.17/8.02 ( ( ord_less_eq_set_o @ A @ B4 )
% 14.17/8.02 => ( ord_less_eq_set_o @ A @ ( insert_o @ B @ B4 ) ) ) ).
% 14.17/8.02
% 14.17/8.02 thf(fact_1402_subset__insertI2,axiom,
% 14.17/8.02 ! [A: set_Product_unit,B4: set_Product_unit,B: product_unit] :
% 14.17/8.02 ( ( ord_le3507040750410214029t_unit @ A @ B4 )
% 14.17/8.02 => ( ord_le3507040750410214029t_unit @ A @ ( insert_Product_unit @ B @ B4 ) ) ) ).
% 14.17/8.02
% 14.17/8.02 thf(fact_1403_subset__insertI2,axiom,
% 14.17/8.02 ! [A: set_nat,B4: set_nat,B: nat] :
% 14.17/8.02 ( ( ord_less_eq_set_nat @ A @ B4 )
% 14.17/8.02 => ( ord_less_eq_set_nat @ A @ ( insert_nat @ B @ B4 ) ) ) ).
% 14.17/8.02
% 14.17/8.02 thf(fact_1404_subset__insertI,axiom,
% 14.17/8.02 ! [B4: set_int,A2: int] : ( ord_less_eq_set_int @ B4 @ ( insert_int @ A2 @ B4 ) ) ).
% 14.17/8.02
% 14.17/8.02 thf(fact_1405_subset__insertI,axiom,
% 14.17/8.02 ! [B4: set_o,A2: $o] : ( ord_less_eq_set_o @ B4 @ ( insert_o @ A2 @ B4 ) ) ).
% 14.17/8.02
% 14.17/8.02 thf(fact_1406_subset__insertI,axiom,
% 14.17/8.02 ! [B4: set_Product_unit,A2: product_unit] : ( ord_le3507040750410214029t_unit @ B4 @ ( insert_Product_unit @ A2 @ B4 ) ) ).
% 14.17/8.02
% 14.17/8.02 thf(fact_1407_subset__insertI,axiom,
% 14.17/8.02 ! [B4: set_nat,A2: nat] : ( ord_less_eq_set_nat @ B4 @ ( insert_nat @ A2 @ B4 ) ) ).
% 14.17/8.02
% 14.17/8.02 thf(fact_1408_subset__insert,axiom,
% 14.17/8.02 ! [X: produc3658429121746597890et_nat > $o,A: set_Pr4532377907799695533_nat_o,B4: set_Pr4532377907799695533_nat_o] :
% 14.17/8.02 ( ~ ( member6576561426505652726_nat_o @ X @ A )
% 14.17/8.02 => ( ( ord_le2965882846123202637_nat_o @ A @ ( insert5175938949040314269_nat_o @ X @ B4 ) )
% 14.17/8.02 = ( ord_le2965882846123202637_nat_o @ A @ B4 ) ) ) ).
% 14.17/8.02
% 14.17/8.02 thf(fact_1409_subset__insert,axiom,
% 14.17/8.02 ! [X: int,A: set_int,B4: set_int] :
% 14.17/8.02 ( ~ ( member_int @ X @ A )
% 14.17/8.02 => ( ( ord_less_eq_set_int @ A @ ( insert_int @ X @ B4 ) )
% 14.17/8.02 = ( ord_less_eq_set_int @ A @ B4 ) ) ) ).
% 14.17/8.02
% 14.17/8.02 thf(fact_1410_subset__insert,axiom,
% 14.17/8.02 ! [X: $o,A: set_o,B4: set_o] :
% 14.17/8.02 ( ~ ( member_o @ X @ A )
% 14.17/8.02 => ( ( ord_less_eq_set_o @ A @ ( insert_o @ X @ B4 ) )
% 14.17/8.02 = ( ord_less_eq_set_o @ A @ B4 ) ) ) ).
% 14.17/8.02
% 14.17/8.02 thf(fact_1411_subset__insert,axiom,
% 14.17/8.02 ! [X: product_unit,A: set_Product_unit,B4: set_Product_unit] :
% 14.17/8.02 ( ~ ( member_Product_unit @ X @ A )
% 14.17/8.02 => ( ( ord_le3507040750410214029t_unit @ A @ ( insert_Product_unit @ X @ B4 ) )
% 14.17/8.02 = ( ord_le3507040750410214029t_unit @ A @ B4 ) ) ) ).
% 14.17/8.02
% 14.17/8.02 thf(fact_1412_subset__insert,axiom,
% 14.17/8.02 ! [X: nat,A: set_nat,B4: set_nat] :
% 14.17/8.02 ( ~ ( member_nat @ X @ A )
% 14.17/8.02 => ( ( ord_less_eq_set_nat @ A @ ( insert_nat @ X @ B4 ) )
% 14.17/8.02 = ( ord_less_eq_set_nat @ A @ B4 ) ) ) ).
% 14.17/8.02
% 14.17/8.02 thf(fact_1413_insert__mono,axiom,
% 14.17/8.02 ! [C5: set_int,D3: set_int,A2: int] :
% 14.17/8.02 ( ( ord_less_eq_set_int @ C5 @ D3 )
% 14.17/8.02 => ( ord_less_eq_set_int @ ( insert_int @ A2 @ C5 ) @ ( insert_int @ A2 @ D3 ) ) ) ).
% 14.17/8.02
% 14.17/8.02 thf(fact_1414_insert__mono,axiom,
% 14.17/8.02 ! [C5: set_o,D3: set_o,A2: $o] :
% 14.17/8.02 ( ( ord_less_eq_set_o @ C5 @ D3 )
% 14.17/8.02 => ( ord_less_eq_set_o @ ( insert_o @ A2 @ C5 ) @ ( insert_o @ A2 @ D3 ) ) ) ).
% 14.17/8.02
% 14.17/8.02 thf(fact_1415_insert__mono,axiom,
% 14.17/8.02 ! [C5: set_Product_unit,D3: set_Product_unit,A2: product_unit] :
% 14.17/8.02 ( ( ord_le3507040750410214029t_unit @ C5 @ D3 )
% 14.17/8.02 => ( ord_le3507040750410214029t_unit @ ( insert_Product_unit @ A2 @ C5 ) @ ( insert_Product_unit @ A2 @ D3 ) ) ) ).
% 14.17/8.02
% 14.17/8.02 thf(fact_1416_insert__mono,axiom,
% 14.17/8.02 ! [C5: set_nat,D3: set_nat,A2: nat] :
% 14.17/8.02 ( ( ord_less_eq_set_nat @ C5 @ D3 )
% 14.17/8.02 => ( ord_less_eq_set_nat @ ( insert_nat @ A2 @ C5 ) @ ( insert_nat @ A2 @ D3 ) ) ) ).
% 14.17/8.02
% 14.17/8.02 thf(fact_1417_Image__Un,axiom,
% 14.17/8.02 ! [R3: set_Pr1261947904930325089at_nat,A: set_nat,B4: set_nat] :
% 14.17/8.02 ( ( image_nat_nat @ R3 @ ( sup_sup_set_nat @ A @ B4 ) )
% 14.17/8.02 = ( sup_sup_set_nat @ ( image_nat_nat @ R3 @ A ) @ ( image_nat_nat @ R3 @ B4 ) ) ) ).
% 14.17/8.02
% 14.17/8.02 thf(fact_1418_subset__vimage__iff,axiom,
% 14.17/8.02 ! [A: set_nat,F: nat > produc3658429121746597890et_nat > $o,B4: set_Pr4532377907799695533_nat_o] :
% 14.17/8.02 ( ( ord_less_eq_set_nat @ A @ ( vimage8201027140005291904_nat_o @ F @ B4 ) )
% 14.17/8.02 = ( ! [X2: nat] :
% 14.17/8.02 ( ( member_nat @ X2 @ A )
% 14.17/8.02 => ( member6576561426505652726_nat_o @ ( F @ X2 ) @ B4 ) ) ) ) ).
% 14.17/8.02
% 14.17/8.02 thf(fact_1419_subset__vimage__iff,axiom,
% 14.17/8.02 ! [A: set_nat,F: nat > nat,B4: set_nat] :
% 14.17/8.02 ( ( ord_less_eq_set_nat @ A @ ( vimage_nat_nat @ F @ B4 ) )
% 14.17/8.02 = ( ! [X2: nat] :
% 14.17/8.02 ( ( member_nat @ X2 @ A )
% 14.17/8.02 => ( member_nat @ ( F @ X2 ) @ B4 ) ) ) ) ).
% 14.17/8.02
% 14.17/8.02 thf(fact_1420_subset__vimage__iff,axiom,
% 14.17/8.02 ! [A: set_nat,F: nat > int,B4: set_int] :
% 14.17/8.02 ( ( ord_less_eq_set_nat @ A @ ( vimage_nat_int @ F @ B4 ) )
% 14.17/8.02 = ( ! [X2: nat] :
% 14.17/8.02 ( ( member_nat @ X2 @ A )
% 14.17/8.02 => ( member_int @ ( F @ X2 ) @ B4 ) ) ) ) ).
% 14.17/8.02
% 14.17/8.02 thf(fact_1421_subset__vimage__iff,axiom,
% 14.17/8.02 ! [A: set_nat,F: nat > $o,B4: set_o] :
% 14.17/8.02 ( ( ord_less_eq_set_nat @ A @ ( vimage_nat_o @ F @ B4 ) )
% 14.17/8.02 = ( ! [X2: nat] :
% 14.17/8.02 ( ( member_nat @ X2 @ A )
% 14.17/8.02 => ( member_o @ ( F @ X2 ) @ B4 ) ) ) ) ).
% 14.17/8.02
% 14.17/8.02 thf(fact_1422_subset__vimage__iff,axiom,
% 14.17/8.02 ! [A: set_nat,F: nat > product_unit,B4: set_Product_unit] :
% 14.17/8.02 ( ( ord_less_eq_set_nat @ A @ ( vimage4884490618288580032t_unit @ F @ B4 ) )
% 14.17/8.02 = ( ! [X2: nat] :
% 14.17/8.02 ( ( member_nat @ X2 @ A )
% 14.17/8.02 => ( member_Product_unit @ ( F @ X2 ) @ B4 ) ) ) ) ).
% 14.17/8.02
% 14.17/8.02 thf(fact_1423_vimage__mono,axiom,
% 14.17/8.02 ! [A: set_nat,B4: set_nat,F: nat > nat] :
% 14.17/8.02 ( ( ord_less_eq_set_nat @ A @ B4 )
% 14.17/8.02 => ( ord_less_eq_set_nat @ ( vimage_nat_nat @ F @ A ) @ ( vimage_nat_nat @ F @ B4 ) ) ) ).
% 14.17/8.02
% 14.17/8.02 thf(fact_1424_relH__subset,axiom,
% 14.17/8.02 ! [Bs: set_nat,H2: heap_e7401611519738050253t_unit,H3: heap_e7401611519738050253t_unit,As: set_nat] :
% 14.17/8.02 ( ( relH @ Bs @ H2 @ H3 )
% 14.17/8.02 => ( ( ord_less_eq_set_nat @ As @ Bs )
% 14.17/8.02 => ( relH @ As @ H2 @ H3 ) ) ) ).
% 14.17/8.02
% 14.17/8.02 thf(fact_1425_rtrancl__apply__insert,axiom,
% 14.17/8.02 ! [R3: set_Pr5094982260447487303t_unit,X: product_unit,S: set_Product_unit] :
% 14.17/8.02 ( ( image_8431784415008483091t_unit @ ( transi6481887417741755664t_unit @ R3 ) @ ( insert_Product_unit @ X @ S ) )
% 14.17/8.02 = ( insert_Product_unit @ X @ ( image_8431784415008483091t_unit @ ( transi6481887417741755664t_unit @ R3 ) @ ( sup_su793286257634532545t_unit @ S @ ( image_8431784415008483091t_unit @ R3 @ ( insert_Product_unit @ X @ bot_bo3957492148770167129t_unit ) ) ) ) ) ) ).
% 14.17/8.02
% 14.17/8.02 thf(fact_1426_rtrancl__apply__insert,axiom,
% 14.17/8.02 ! [R3: set_Product_prod_o_o,X: $o,S: set_o] :
% 14.17/8.02 ( ( image_o_o @ ( transitive_rtrancl_o @ R3 ) @ ( insert_o @ X @ S ) )
% 14.17/8.02 = ( insert_o @ X @ ( image_o_o @ ( transitive_rtrancl_o @ R3 ) @ ( sup_sup_set_o @ S @ ( image_o_o @ R3 @ ( insert_o @ X @ bot_bot_set_o ) ) ) ) ) ) ).
% 14.17/8.02
% 14.17/8.02 thf(fact_1427_rtrancl__apply__insert,axiom,
% 14.17/8.02 ! [R3: set_Pr1261947904930325089at_nat,X: nat,S: set_nat] :
% 14.17/8.02 ( ( image_nat_nat @ ( transi2905341329935302413cl_nat @ R3 ) @ ( insert_nat @ X @ S ) )
% 14.17/8.02 = ( insert_nat @ X @ ( image_nat_nat @ ( transi2905341329935302413cl_nat @ R3 ) @ ( sup_sup_set_nat @ S @ ( image_nat_nat @ R3 @ ( insert_nat @ X @ bot_bot_set_nat ) ) ) ) ) ) ).
% 14.17/8.02
% 14.17/8.02 thf(fact_1428_rtrancl__apply__insert,axiom,
% 14.17/8.02 ! [R3: set_Pr958786334691620121nt_int,X: int,S: set_int] :
% 14.17/8.02 ( ( image_int_int @ ( transi2902850859426252137cl_int @ R3 ) @ ( insert_int @ X @ S ) )
% 14.17/8.02 = ( insert_int @ X @ ( image_int_int @ ( transi2902850859426252137cl_int @ R3 ) @ ( sup_sup_set_int @ S @ ( image_int_int @ R3 @ ( insert_int @ X @ bot_bot_set_int ) ) ) ) ) ) ).
% 14.17/8.02
% 14.17/8.02 thf(fact_1429_rtrancl__apply__insert,axiom,
% 14.17/8.02 ! [R3: set_Pr2161125870931222855_nat_o,X: produc3658429121746597890et_nat > $o,S: set_Pr4532377907799695533_nat_o] :
% 14.17/8.02 ( ( image_2232984160439939859_nat_o @ ( transi712980445759334096_nat_o @ R3 ) @ ( insert5175938949040314269_nat_o @ X @ S ) )
% 14.17/8.02 = ( insert5175938949040314269_nat_o @ X @ ( image_2232984160439939859_nat_o @ ( transi712980445759334096_nat_o @ R3 ) @ ( sup_su5209123915105501825_nat_o @ S @ ( image_2232984160439939859_nat_o @ R3 @ ( insert5175938949040314269_nat_o @ X @ bot_bo7824918357723582233_nat_o ) ) ) ) ) ) ).
% 14.17/8.02
% 14.17/8.02 thf(fact_1430_antisymD,axiom,
% 14.17/8.02 ! [R2: set_Pr1261947904930325089at_nat,A2: nat,B: nat] :
% 14.17/8.02 ( ( antisym_nat @ R2 )
% 14.17/8.02 => ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A2 @ B ) @ R2 )
% 14.17/8.02 => ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ B @ A2 ) @ R2 )
% 14.17/8.02 => ( A2 = B ) ) ) ) ).
% 14.17/8.02
% 14.17/8.02 thf(fact_1431_antisymD,axiom,
% 14.17/8.02 ! [R2: set_Pr958786334691620121nt_int,A2: int,B: int] :
% 14.17/8.02 ( ( antisym_int @ R2 )
% 14.17/8.02 => ( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ A2 @ B ) @ R2 )
% 14.17/8.02 => ( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ B @ A2 ) @ R2 )
% 14.17/8.02 => ( A2 = B ) ) ) ) ).
% 14.17/8.02
% 14.17/8.02 thf(fact_1432_antisymI,axiom,
% 14.17/8.02 ! [R2: set_Pr1261947904930325089at_nat] :
% 14.17/8.02 ( ! [X3: nat,Y2: nat] :
% 14.17/8.02 ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X3 @ Y2 ) @ R2 )
% 14.17/8.02 => ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ Y2 @ X3 ) @ R2 )
% 14.17/8.02 => ( X3 = Y2 ) ) )
% 14.17/8.02 => ( antisym_nat @ R2 ) ) ).
% 14.17/8.02
% 14.17/8.02 thf(fact_1433_antisymI,axiom,
% 14.17/8.02 ! [R2: set_Pr958786334691620121nt_int] :
% 14.17/8.02 ( ! [X3: int,Y2: int] :
% 14.17/8.02 ( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ X3 @ Y2 ) @ R2 )
% 14.17/8.02 => ( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ Y2 @ X3 ) @ R2 )
% 14.17/8.02 => ( X3 = Y2 ) ) )
% 14.17/8.02 => ( antisym_int @ R2 ) ) ).
% 14.17/8.02
% 14.17/8.02 thf(fact_1434_antisym__def,axiom,
% 14.17/8.02 ( antisym_nat
% 14.17/8.02 = ( ^ [R4: set_Pr1261947904930325089at_nat] :
% 14.17/8.02 ! [X2: nat,Y3: nat] :
% 14.17/8.02 ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X2 @ Y3 ) @ R4 )
% 14.17/8.02 => ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ Y3 @ X2 ) @ R4 )
% 14.17/8.02 => ( X2 = Y3 ) ) ) ) ) ).
% 14.17/8.02
% 14.17/8.02 thf(fact_1435_antisym__def,axiom,
% 14.17/8.02 ( antisym_int
% 14.17/8.02 = ( ^ [R4: set_Pr958786334691620121nt_int] :
% 14.17/8.02 ! [X2: int,Y3: int] :
% 14.17/8.02 ( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ X2 @ Y3 ) @ R4 )
% 14.17/8.02 => ( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ Y3 @ X2 ) @ R4 )
% 14.17/8.02 => ( X2 = Y3 ) ) ) ) ) ).
% 14.17/8.02
% 14.17/8.02 thf(fact_1436_Image__absorb__rtrancl,axiom,
% 14.17/8.02 ! [A: set_Pr1261947904930325089at_nat,B4: set_nat,C5: set_nat] :
% 14.17/8.02 ( ( trans_nat @ A )
% 14.17/8.02 => ( ( refl_on_nat @ B4 @ A )
% 14.17/8.02 => ( ( ord_less_eq_set_nat @ C5 @ B4 )
% 14.17/8.02 => ( ( image_nat_nat @ ( transi2905341329935302413cl_nat @ A ) @ C5 )
% 14.17/8.02 = ( image_nat_nat @ A @ C5 ) ) ) ) ) ).
% 14.17/8.02
% 14.17/8.02 thf(fact_1437_times__assn__raw_Osimps,axiom,
% 14.17/8.02 ! [P: produc3658429121746597890et_nat > $o,Q2: produc3658429121746597890et_nat > $o,H2: heap_e7401611519738050253t_unit,As: set_nat] :
% 14.17/8.02 ( ( times_assn_raw @ P @ Q2 @ ( produc7507926704131184380et_nat @ H2 @ As ) )
% 14.17/8.02 = ( ? [As1: set_nat,As22: set_nat] :
% 14.17/8.02 ( ( As
% 14.17/8.02 = ( sup_sup_set_nat @ As1 @ As22 ) )
% 14.17/8.02 & ( ( inf_inf_set_nat @ As1 @ As22 )
% 14.17/8.02 = bot_bot_set_nat )
% 14.17/8.02 & ( P @ ( produc7507926704131184380et_nat @ H2 @ As1 ) )
% 14.17/8.02 & ( Q2 @ ( produc7507926704131184380et_nat @ H2 @ As22 ) ) ) ) ) ).
% 14.17/8.02
% 14.17/8.02 thf(fact_1438_times__assn__raw_Oelims_I1_J,axiom,
% 14.17/8.02 ! [X: produc3658429121746597890et_nat > $o,Xa: produc3658429121746597890et_nat > $o,Xb: produc3658429121746597890et_nat,Y: $o] :
% 14.17/8.02 ( ( ( times_assn_raw @ X @ Xa @ Xb )
% 14.17/8.02 = Y )
% 14.17/8.02 => ~ ! [H: heap_e7401611519738050253t_unit,As2: set_nat] :
% 14.17/8.02 ( ( Xb
% 14.17/8.02 = ( produc7507926704131184380et_nat @ H @ As2 ) )
% 14.17/8.02 => ( Y
% 14.17/8.02 = ( ~ ? [As1: set_nat,As22: set_nat] :
% 14.17/8.02 ( ( As2
% 14.17/8.02 = ( sup_sup_set_nat @ As1 @ As22 ) )
% 14.17/8.02 & ( ( inf_inf_set_nat @ As1 @ As22 )
% 14.17/8.02 = bot_bot_set_nat )
% 14.17/8.02 & ( X @ ( produc7507926704131184380et_nat @ H @ As1 ) )
% 14.17/8.02 & ( Xa @ ( produc7507926704131184380et_nat @ H @ As22 ) ) ) ) ) ) ) ).
% 14.17/8.02
% 14.17/8.02 thf(fact_1439_times__assn__raw_Oelims_I2_J,axiom,
% 14.17/8.02 ! [X: produc3658429121746597890et_nat > $o,Xa: produc3658429121746597890et_nat > $o,Xb: produc3658429121746597890et_nat] :
% 14.17/8.02 ( ( times_assn_raw @ X @ Xa @ Xb )
% 14.17/8.02 => ~ ! [H: heap_e7401611519738050253t_unit,As2: set_nat] :
% 14.17/8.02 ( ( Xb
% 14.17/8.02 = ( produc7507926704131184380et_nat @ H @ As2 ) )
% 14.17/8.02 => ~ ? [As12: set_nat,As23: set_nat] :
% 14.17/8.02 ( ( As2
% 14.17/8.02 = ( sup_sup_set_nat @ As12 @ As23 ) )
% 14.17/8.02 & ( ( inf_inf_set_nat @ As12 @ As23 )
% 14.17/8.02 = bot_bot_set_nat )
% 14.17/8.02 & ( X @ ( produc7507926704131184380et_nat @ H @ As12 ) )
% 14.17/8.02 & ( Xa @ ( produc7507926704131184380et_nat @ H @ As23 ) ) ) ) ) ).
% 14.17/8.02
% 14.17/8.02 thf(fact_1440_times__assn__raw_Oelims_I3_J,axiom,
% 14.17/8.02 ! [X: produc3658429121746597890et_nat > $o,Xa: produc3658429121746597890et_nat > $o,Xb: produc3658429121746597890et_nat] :
% 14.17/8.02 ( ~ ( times_assn_raw @ X @ Xa @ Xb )
% 14.17/8.02 => ~ ! [H: heap_e7401611519738050253t_unit,As2: set_nat] :
% 14.17/8.02 ( ( Xb
% 14.17/8.02 = ( produc7507926704131184380et_nat @ H @ As2 ) )
% 14.17/8.02 => ? [As13: set_nat,As24: set_nat] :
% 14.17/8.02 ( ( As2
% 14.17/8.02 = ( sup_sup_set_nat @ As13 @ As24 ) )
% 14.17/8.02 & ( ( inf_inf_set_nat @ As13 @ As24 )
% 14.17/8.02 = bot_bot_set_nat )
% 14.17/8.02 & ( X @ ( produc7507926704131184380et_nat @ H @ As13 ) )
% 14.17/8.02 & ( Xa @ ( produc7507926704131184380et_nat @ H @ As24 ) ) ) ) ) ).
% 14.17/8.02
% 14.17/8.02 thf(fact_1441_rtrancl__Image__advance__ss,axiom,
% 14.17/8.02 ! [U: product_unit,V: product_unit,E: set_Pr5094982260447487303t_unit] :
% 14.17/8.02 ( ( member7821138191923116944t_unit @ ( produc5692694829384537815t_unit @ U @ V ) @ E )
% 14.17/8.02 => ( ord_le3507040750410214029t_unit @ ( image_8431784415008483091t_unit @ ( transi6481887417741755664t_unit @ E ) @ ( insert_Product_unit @ V @ bot_bo3957492148770167129t_unit ) ) @ ( image_8431784415008483091t_unit @ ( transi6481887417741755664t_unit @ E ) @ ( insert_Product_unit @ U @ bot_bo3957492148770167129t_unit ) ) ) ) ).
% 14.17/8.02
% 14.17/8.02 thf(fact_1442_rtrancl__Image__advance__ss,axiom,
% 14.17/8.02 ! [U: $o,V: $o,E: set_Product_prod_o_o] :
% 14.17/8.02 ( ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ U @ V ) @ E )
% 14.17/8.02 => ( ord_less_eq_set_o @ ( image_o_o @ ( transitive_rtrancl_o @ E ) @ ( insert_o @ V @ bot_bot_set_o ) ) @ ( image_o_o @ ( transitive_rtrancl_o @ E ) @ ( insert_o @ U @ bot_bot_set_o ) ) ) ) ).
% 14.17/8.02
% 14.17/8.02 thf(fact_1443_rtrancl__Image__advance__ss,axiom,
% 14.17/8.02 ! [U: int,V: int,E: set_Pr958786334691620121nt_int] :
% 14.17/8.02 ( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ U @ V ) @ E )
% 14.17/8.02 => ( ord_less_eq_set_int @ ( image_int_int @ ( transi2902850859426252137cl_int @ E ) @ ( insert_int @ V @ bot_bot_set_int ) ) @ ( image_int_int @ ( transi2902850859426252137cl_int @ E ) @ ( insert_int @ U @ bot_bot_set_int ) ) ) ) ).
% 14.17/8.02
% 14.17/8.02 thf(fact_1444_rtrancl__Image__advance__ss,axiom,
% 14.17/8.02 ! [U: produc3658429121746597890et_nat > $o,V: produc3658429121746597890et_nat > $o,E: set_Pr2161125870931222855_nat_o] :
% 14.17/8.02 ( ( member8781333585448626064_nat_o @ ( produc7368190662567826135_nat_o @ U @ V ) @ E )
% 14.17/8.02 => ( ord_le2965882846123202637_nat_o @ ( image_2232984160439939859_nat_o @ ( transi712980445759334096_nat_o @ E ) @ ( insert5175938949040314269_nat_o @ V @ bot_bo7824918357723582233_nat_o ) ) @ ( image_2232984160439939859_nat_o @ ( transi712980445759334096_nat_o @ E ) @ ( insert5175938949040314269_nat_o @ U @ bot_bo7824918357723582233_nat_o ) ) ) ) ).
% 14.17/8.02
% 14.17/8.02 thf(fact_1445_rtrancl__Image__advance__ss,axiom,
% 14.17/8.02 ! [U: nat,V: nat,E: set_Pr1261947904930325089at_nat] :
% 14.17/8.02 ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ U @ V ) @ E )
% 14.17/8.02 => ( ord_less_eq_set_nat @ ( image_nat_nat @ ( transi2905341329935302413cl_nat @ E ) @ ( insert_nat @ V @ bot_bot_set_nat ) ) @ ( image_nat_nat @ ( transi2905341329935302413cl_nat @ E ) @ ( insert_nat @ U @ bot_bot_set_nat ) ) ) ) ).
% 14.17/8.02
% 14.17/8.02 thf(fact_1446_rtrancl__image__advance__rtrancl,axiom,
% 14.17/8.02 ! [Q4: produc3658429121746597890et_nat > $o,R3: set_Pr2161125870931222855_nat_o,Q0: set_Pr4532377907799695533_nat_o,X: produc3658429121746597890et_nat > $o] :
% 14.17/8.02 ( ( member6576561426505652726_nat_o @ Q4 @ ( image_2232984160439939859_nat_o @ ( transi712980445759334096_nat_o @ R3 ) @ Q0 ) )
% 14.17/8.02 => ( ( member8781333585448626064_nat_o @ ( produc7368190662567826135_nat_o @ Q4 @ X ) @ ( transi712980445759334096_nat_o @ R3 ) )
% 14.17/8.02 => ( member6576561426505652726_nat_o @ X @ ( image_2232984160439939859_nat_o @ ( transi712980445759334096_nat_o @ R3 ) @ Q0 ) ) ) ) ).
% 14.17/8.02
% 14.17/8.02 thf(fact_1447_rtrancl__image__advance__rtrancl,axiom,
% 14.17/8.02 ! [Q4: $o,R3: set_Product_prod_o_o,Q0: set_o,X: $o] :
% 14.17/8.02 ( ( member_o @ Q4 @ ( image_o_o @ ( transitive_rtrancl_o @ R3 ) @ Q0 ) )
% 14.17/8.02 => ( ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ Q4 @ X ) @ ( transitive_rtrancl_o @ R3 ) )
% 14.17/8.02 => ( member_o @ X @ ( image_o_o @ ( transitive_rtrancl_o @ R3 ) @ Q0 ) ) ) ) ).
% 14.17/8.02
% 14.17/8.02 thf(fact_1448_rtrancl__image__advance__rtrancl,axiom,
% 14.17/8.02 ! [Q4: product_unit,R3: set_Pr5094982260447487303t_unit,Q0: set_Product_unit,X: product_unit] :
% 14.17/8.02 ( ( member_Product_unit @ Q4 @ ( image_8431784415008483091t_unit @ ( transi6481887417741755664t_unit @ R3 ) @ Q0 ) )
% 14.17/8.02 => ( ( member7821138191923116944t_unit @ ( produc5692694829384537815t_unit @ Q4 @ X ) @ ( transi6481887417741755664t_unit @ R3 ) )
% 14.17/8.02 => ( member_Product_unit @ X @ ( image_8431784415008483091t_unit @ ( transi6481887417741755664t_unit @ R3 ) @ Q0 ) ) ) ) ).
% 14.17/8.02
% 14.17/8.02 thf(fact_1449_rtrancl__image__advance__rtrancl,axiom,
% 14.17/8.02 ! [Q4: nat,R3: set_Pr1261947904930325089at_nat,Q0: set_nat,X: nat] :
% 14.17/8.02 ( ( member_nat @ Q4 @ ( image_nat_nat @ ( transi2905341329935302413cl_nat @ R3 ) @ Q0 ) )
% 14.17/8.02 => ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ Q4 @ X ) @ ( transi2905341329935302413cl_nat @ R3 ) )
% 14.17/8.02 => ( member_nat @ X @ ( image_nat_nat @ ( transi2905341329935302413cl_nat @ R3 ) @ Q0 ) ) ) ) ).
% 14.17/8.02
% 14.17/8.02 thf(fact_1450_rtrancl__image__advance__rtrancl,axiom,
% 14.17/8.02 ! [Q4: int,R3: set_Pr958786334691620121nt_int,Q0: set_int,X: int] :
% 14.17/8.02 ( ( member_int @ Q4 @ ( image_int_int @ ( transi2902850859426252137cl_int @ R3 ) @ Q0 ) )
% 14.17/8.02 => ( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ Q4 @ X ) @ ( transi2902850859426252137cl_int @ R3 ) )
% 14.17/8.02 => ( member_int @ X @ ( image_int_int @ ( transi2902850859426252137cl_int @ R3 ) @ Q0 ) ) ) ) ).
% 14.17/8.02
% 14.17/8.02 thf(fact_1451_rtrancl__image__advance,axiom,
% 14.17/8.02 ! [Q4: produc3658429121746597890et_nat > $o,R3: set_Pr2161125870931222855_nat_o,Q0: set_Pr4532377907799695533_nat_o,X: produc3658429121746597890et_nat > $o] :
% 14.17/8.02 ( ( member6576561426505652726_nat_o @ Q4 @ ( image_2232984160439939859_nat_o @ ( transi712980445759334096_nat_o @ R3 ) @ Q0 ) )
% 14.17/8.02 => ( ( member8781333585448626064_nat_o @ ( produc7368190662567826135_nat_o @ Q4 @ X ) @ R3 )
% 14.17/8.02 => ( member6576561426505652726_nat_o @ X @ ( image_2232984160439939859_nat_o @ ( transi712980445759334096_nat_o @ R3 ) @ Q0 ) ) ) ) ).
% 14.17/8.02
% 14.17/8.02 thf(fact_1452_rtrancl__image__advance,axiom,
% 14.17/8.02 ! [Q4: $o,R3: set_Product_prod_o_o,Q0: set_o,X: $o] :
% 14.17/8.02 ( ( member_o @ Q4 @ ( image_o_o @ ( transitive_rtrancl_o @ R3 ) @ Q0 ) )
% 14.17/8.02 => ( ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ Q4 @ X ) @ R3 )
% 14.17/8.02 => ( member_o @ X @ ( image_o_o @ ( transitive_rtrancl_o @ R3 ) @ Q0 ) ) ) ) ).
% 14.17/8.02
% 14.17/8.02 thf(fact_1453_rtrancl__image__advance,axiom,
% 14.17/8.02 ! [Q4: product_unit,R3: set_Pr5094982260447487303t_unit,Q0: set_Product_unit,X: product_unit] :
% 14.17/8.02 ( ( member_Product_unit @ Q4 @ ( image_8431784415008483091t_unit @ ( transi6481887417741755664t_unit @ R3 ) @ Q0 ) )
% 14.17/8.02 => ( ( member7821138191923116944t_unit @ ( produc5692694829384537815t_unit @ Q4 @ X ) @ R3 )
% 14.17/8.02 => ( member_Product_unit @ X @ ( image_8431784415008483091t_unit @ ( transi6481887417741755664t_unit @ R3 ) @ Q0 ) ) ) ) ).
% 14.17/8.02
% 14.17/8.02 thf(fact_1454_rtrancl__image__advance,axiom,
% 14.17/8.02 ! [Q4: nat,R3: set_Pr1261947904930325089at_nat,Q0: set_nat,X: nat] :
% 14.17/8.02 ( ( member_nat @ Q4 @ ( image_nat_nat @ ( transi2905341329935302413cl_nat @ R3 ) @ Q0 ) )
% 14.17/8.02 => ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ Q4 @ X ) @ R3 )
% 14.17/8.02 => ( member_nat @ X @ ( image_nat_nat @ ( transi2905341329935302413cl_nat @ R3 ) @ Q0 ) ) ) ) ).
% 14.17/8.02
% 14.17/8.02 thf(fact_1455_rtrancl__image__advance,axiom,
% 14.17/8.02 ! [Q4: int,R3: set_Pr958786334691620121nt_int,Q0: set_int,X: int] :
% 14.17/8.02 ( ( member_int @ Q4 @ ( image_int_int @ ( transi2902850859426252137cl_int @ R3 ) @ Q0 ) )
% 14.17/8.02 => ( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ Q4 @ X ) @ R3 )
% 14.17/8.02 => ( member_int @ X @ ( image_int_int @ ( transi2902850859426252137cl_int @ R3 ) @ Q0 ) ) ) ) ).
% 14.17/8.02
% 14.17/8.02 thf(fact_1456_trancl__rtrancl__trancl,axiom,
% 14.17/8.02 ! [A2: nat,B: nat,R2: set_Pr1261947904930325089at_nat,C: nat] :
% 14.17/8.02 ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A2 @ B ) @ ( transi6264000038957366511cl_nat @ R2 ) )
% 14.17/8.02 => ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ B @ C ) @ ( transi2905341329935302413cl_nat @ R2 ) )
% 14.17/8.02 => ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A2 @ C ) @ ( transi6264000038957366511cl_nat @ R2 ) ) ) ) ).
% 14.17/8.02
% 14.17/8.02 thf(fact_1457_trancl__rtrancl__trancl,axiom,
% 14.17/8.02 ! [A2: int,B: int,R2: set_Pr958786334691620121nt_int,C: int] :
% 14.17/8.02 ( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ A2 @ B ) @ ( transi6261509568448316235cl_int @ R2 ) )
% 14.17/8.02 => ( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ B @ C ) @ ( transi2902850859426252137cl_int @ R2 ) )
% 14.17/8.02 => ( member5262025264175285858nt_int @ ( product_Pair_int_int @ A2 @ C ) @ ( transi6261509568448316235cl_int @ R2 ) ) ) ) ).
% 14.17/8.02
% 14.17/8.02 thf(fact_1458_rtrancl__trancl__trancl,axiom,
% 14.17/8.02 ! [X: nat,Y: nat,R2: set_Pr1261947904930325089at_nat,Z: nat] :
% 14.17/8.02 ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ Y ) @ ( transi2905341329935302413cl_nat @ R2 ) )
% 14.17/8.02 => ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ Y @ Z ) @ ( transi6264000038957366511cl_nat @ R2 ) )
% 14.17/8.02 => ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ Z ) @ ( transi6264000038957366511cl_nat @ R2 ) ) ) ) ).
% 14.17/8.02
% 14.17/8.02 thf(fact_1459_rtrancl__trancl__trancl,axiom,
% 14.17/8.02 ! [X: int,Y: int,R2: set_Pr958786334691620121nt_int,Z: int] :
% 14.17/8.02 ( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ X @ Y ) @ ( transi2902850859426252137cl_int @ R2 ) )
% 14.17/8.02 => ( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ Y @ Z ) @ ( transi6261509568448316235cl_int @ R2 ) )
% 14.17/8.02 => ( member5262025264175285858nt_int @ ( product_Pair_int_int @ X @ Z ) @ ( transi6261509568448316235cl_int @ R2 ) ) ) ) ).
% 14.17/8.02
% 14.17/8.02 thf(fact_1460_rtrancl__into__trancl2,axiom,
% 14.17/8.02 ! [A2: nat,B: nat,R2: set_Pr1261947904930325089at_nat,C: nat] :
% 14.17/8.02 ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A2 @ B ) @ R2 )
% 14.17/8.02 => ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ B @ C ) @ ( transi2905341329935302413cl_nat @ R2 ) )
% 14.17/8.02 => ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A2 @ C ) @ ( transi6264000038957366511cl_nat @ R2 ) ) ) ) ).
% 14.17/8.02
% 14.17/8.02 thf(fact_1461_rtrancl__into__trancl2,axiom,
% 14.17/8.02 ! [A2: int,B: int,R2: set_Pr958786334691620121nt_int,C: int] :
% 14.17/8.02 ( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ A2 @ B ) @ R2 )
% 14.17/8.02 => ( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ B @ C ) @ ( transi2902850859426252137cl_int @ R2 ) )
% 14.17/8.02 => ( member5262025264175285858nt_int @ ( product_Pair_int_int @ A2 @ C ) @ ( transi6261509568448316235cl_int @ R2 ) ) ) ) ).
% 14.17/8.02
% 14.17/8.02 thf(fact_1462_rtrancl__into__trancl1,axiom,
% 14.17/8.02 ! [A2: nat,B: nat,R2: set_Pr1261947904930325089at_nat,C: nat] :
% 14.17/8.02 ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A2 @ B ) @ ( transi2905341329935302413cl_nat @ R2 ) )
% 14.17/8.02 => ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ B @ C ) @ R2 )
% 14.17/8.02 => ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A2 @ C ) @ ( transi6264000038957366511cl_nat @ R2 ) ) ) ) ).
% 14.17/8.02
% 14.17/8.02 thf(fact_1463_rtrancl__into__trancl1,axiom,
% 14.17/8.02 ! [A2: int,B: int,R2: set_Pr958786334691620121nt_int,C: int] :
% 14.17/8.02 ( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ A2 @ B ) @ ( transi2902850859426252137cl_int @ R2 ) )
% 14.17/8.02 => ( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ B @ C ) @ R2 )
% 14.17/8.02 => ( member5262025264175285858nt_int @ ( product_Pair_int_int @ A2 @ C ) @ ( transi6261509568448316235cl_int @ R2 ) ) ) ) ).
% 14.17/8.02
% 14.17/8.02 thf(fact_1464_rtrancl__eq__or__trancl,axiom,
% 14.17/8.02 ! [X: nat,Y: nat,R3: set_Pr1261947904930325089at_nat] :
% 14.17/8.02 ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ Y ) @ ( transi2905341329935302413cl_nat @ R3 ) )
% 14.17/8.02 = ( ( X = Y )
% 14.17/8.02 | ( ( X != Y )
% 14.17/8.02 & ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ Y ) @ ( transi6264000038957366511cl_nat @ R3 ) ) ) ) ) ).
% 14.17/8.02
% 14.17/8.02 thf(fact_1465_rtrancl__eq__or__trancl,axiom,
% 14.17/8.02 ! [X: int,Y: int,R3: set_Pr958786334691620121nt_int] :
% 14.17/8.02 ( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ X @ Y ) @ ( transi2902850859426252137cl_int @ R3 ) )
% 14.17/8.02 = ( ( X = Y )
% 14.17/8.02 | ( ( X != Y )
% 14.17/8.02 & ( member5262025264175285858nt_int @ ( product_Pair_int_int @ X @ Y ) @ ( transi6261509568448316235cl_int @ R3 ) ) ) ) ) ).
% 14.17/8.02
% 14.17/8.02 thf(fact_1466_trancl__into__rtrancl,axiom,
% 14.17/8.02 ! [A2: nat,B: nat,R2: set_Pr1261947904930325089at_nat] :
% 14.17/8.02 ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A2 @ B ) @ ( transi6264000038957366511cl_nat @ R2 ) )
% 14.17/8.02 => ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A2 @ B ) @ ( transi2905341329935302413cl_nat @ R2 ) ) ) ).
% 14.17/8.02
% 14.17/8.02 thf(fact_1467_trancl__into__rtrancl,axiom,
% 14.17/8.02 ! [A2: int,B: int,R2: set_Pr958786334691620121nt_int] :
% 14.17/8.02 ( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ A2 @ B ) @ ( transi6261509568448316235cl_int @ R2 ) )
% 14.17/8.02 => ( member5262025264175285858nt_int @ ( product_Pair_int_int @ A2 @ B ) @ ( transi2902850859426252137cl_int @ R2 ) ) ) ).
% 14.17/8.02
% 14.17/8.02 thf(fact_1468_tranclD2,axiom,
% 14.17/8.02 ! [X: nat,Y: nat,R3: set_Pr1261947904930325089at_nat] :
% 14.17/8.02 ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ Y ) @ ( transi6264000038957366511cl_nat @ R3 ) )
% 14.17/8.02 => ? [Z3: nat] :
% 14.17/8.02 ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ Z3 ) @ ( transi2905341329935302413cl_nat @ R3 ) )
% 14.17/8.02 & ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ Z3 @ Y ) @ R3 ) ) ) ).
% 14.17/8.02
% 14.17/8.02 thf(fact_1469_tranclD2,axiom,
% 14.17/8.02 ! [X: int,Y: int,R3: set_Pr958786334691620121nt_int] :
% 14.17/8.02 ( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ X @ Y ) @ ( transi6261509568448316235cl_int @ R3 ) )
% 14.17/8.02 => ? [Z3: int] :
% 14.17/8.02 ( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ X @ Z3 ) @ ( transi2902850859426252137cl_int @ R3 ) )
% 14.17/8.02 & ( member5262025264175285858nt_int @ ( product_Pair_int_int @ Z3 @ Y ) @ R3 ) ) ) ).
% 14.17/8.02
% 14.17/8.02 thf(fact_1470_rtranclD,axiom,
% 14.17/8.02 ! [A2: nat,B: nat,R3: set_Pr1261947904930325089at_nat] :
% 14.17/8.02 ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A2 @ B ) @ ( transi2905341329935302413cl_nat @ R3 ) )
% 14.17/8.02 => ( ( A2 = B )
% 14.17/8.02 | ( ( A2 != B )
% 14.17/8.02 & ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A2 @ B ) @ ( transi6264000038957366511cl_nat @ R3 ) ) ) ) ) ).
% 14.17/8.02
% 14.17/8.02 thf(fact_1471_rtranclD,axiom,
% 14.17/8.02 ! [A2: int,B: int,R3: set_Pr958786334691620121nt_int] :
% 14.17/8.02 ( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ A2 @ B ) @ ( transi2902850859426252137cl_int @ R3 ) )
% 14.17/8.02 => ( ( A2 = B )
% 14.17/8.02 | ( ( A2 != B )
% 14.17/8.02 & ( member5262025264175285858nt_int @ ( product_Pair_int_int @ A2 @ B ) @ ( transi6261509568448316235cl_int @ R3 ) ) ) ) ) ).
% 14.17/8.02
% 14.17/8.02 thf(fact_1472_tranclD,axiom,
% 14.17/8.02 ! [X: nat,Y: nat,R3: set_Pr1261947904930325089at_nat] :
% 14.17/8.02 ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ Y ) @ ( transi6264000038957366511cl_nat @ R3 ) )
% 14.17/8.02 => ? [Z3: nat] :
% 14.17/8.02 ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ Z3 ) @ R3 )
% 14.17/8.02 & ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ Z3 @ Y ) @ ( transi2905341329935302413cl_nat @ R3 ) ) ) ) ).
% 14.17/8.02
% 14.17/8.02 thf(fact_1473_tranclD,axiom,
% 14.17/8.02 ! [X: int,Y: int,R3: set_Pr958786334691620121nt_int] :
% 14.17/8.02 ( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ X @ Y ) @ ( transi6261509568448316235cl_int @ R3 ) )
% 14.17/8.02 => ? [Z3: int] :
% 14.17/8.02 ( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ X @ Z3 ) @ R3 )
% 14.17/8.02 & ( member5262025264175285858nt_int @ ( product_Pair_int_int @ Z3 @ Y ) @ ( transi2902850859426252137cl_int @ R3 ) ) ) ) ).
% 14.17/8.02
% 14.17/8.02 thf(fact_1474_insert__is__Un,axiom,
% 14.17/8.02 ( insert_Product_unit
% 14.17/8.02 = ( ^ [A6: product_unit] : ( sup_su793286257634532545t_unit @ ( insert_Product_unit @ A6 @ bot_bo3957492148770167129t_unit ) ) ) ) ).
% 14.17/8.02
% 14.17/8.02 thf(fact_1475_insert__is__Un,axiom,
% 14.17/8.02 ( insert_o
% 14.17/8.02 = ( ^ [A6: $o] : ( sup_sup_set_o @ ( insert_o @ A6 @ bot_bot_set_o ) ) ) ) ).
% 14.17/8.02
% 14.17/8.02 thf(fact_1476_insert__is__Un,axiom,
% 14.17/8.02 ( insert_nat
% 14.17/8.02 = ( ^ [A6: nat] : ( sup_sup_set_nat @ ( insert_nat @ A6 @ bot_bot_set_nat ) ) ) ) ).
% 14.17/8.02
% 14.17/8.02 thf(fact_1477_insert__is__Un,axiom,
% 14.17/8.02 ( insert_int
% 14.17/8.02 = ( ^ [A6: int] : ( sup_sup_set_int @ ( insert_int @ A6 @ bot_bot_set_int ) ) ) ) ).
% 14.17/8.02
% 14.17/8.02 thf(fact_1478_insert__is__Un,axiom,
% 14.17/8.02 ( insert5175938949040314269_nat_o
% 14.17/8.02 = ( ^ [A6: produc3658429121746597890et_nat > $o] : ( sup_su5209123915105501825_nat_o @ ( insert5175938949040314269_nat_o @ A6 @ bot_bo7824918357723582233_nat_o ) ) ) ) ).
% 14.17/8.02
% 14.17/8.02 thf(fact_1479_Un__singleton__iff,axiom,
% 14.17/8.02 ! [A: set_Product_unit,B4: set_Product_unit,X: product_unit] :
% 14.17/8.02 ( ( ( sup_su793286257634532545t_unit @ A @ B4 )
% 14.17/8.02 = ( insert_Product_unit @ X @ bot_bo3957492148770167129t_unit ) )
% 14.17/8.02 = ( ( ( A = bot_bo3957492148770167129t_unit )
% 14.17/8.02 & ( B4
% 14.17/8.02 = ( insert_Product_unit @ X @ bot_bo3957492148770167129t_unit ) ) )
% 14.17/8.02 | ( ( A
% 14.17/8.02 = ( insert_Product_unit @ X @ bot_bo3957492148770167129t_unit ) )
% 14.17/8.02 & ( B4 = bot_bo3957492148770167129t_unit ) )
% 14.17/8.02 | ( ( A
% 14.17/8.02 = ( insert_Product_unit @ X @ bot_bo3957492148770167129t_unit ) )
% 14.17/8.02 & ( B4
% 14.17/8.02 = ( insert_Product_unit @ X @ bot_bo3957492148770167129t_unit ) ) ) ) ) ).
% 14.17/8.02
% 14.17/8.02 thf(fact_1480_Un__singleton__iff,axiom,
% 14.17/8.02 ! [A: set_o,B4: set_o,X: $o] :
% 14.17/8.02 ( ( ( sup_sup_set_o @ A @ B4 )
% 14.17/8.02 = ( insert_o @ X @ bot_bot_set_o ) )
% 14.17/8.02 = ( ( ( A = bot_bot_set_o )
% 14.17/8.02 & ( B4
% 14.17/8.02 = ( insert_o @ X @ bot_bot_set_o ) ) )
% 14.17/8.02 | ( ( A
% 14.17/8.02 = ( insert_o @ X @ bot_bot_set_o ) )
% 14.17/8.02 & ( B4 = bot_bot_set_o ) )
% 14.17/8.02 | ( ( A
% 14.17/8.02 = ( insert_o @ X @ bot_bot_set_o ) )
% 14.17/8.02 & ( B4
% 14.17/8.02 = ( insert_o @ X @ bot_bot_set_o ) ) ) ) ) ).
% 14.17/8.02
% 14.17/8.02 thf(fact_1481_Un__singleton__iff,axiom,
% 14.17/8.02 ! [A: set_nat,B4: set_nat,X: nat] :
% 14.17/8.02 ( ( ( sup_sup_set_nat @ A @ B4 )
% 14.17/8.02 = ( insert_nat @ X @ bot_bot_set_nat ) )
% 14.17/8.02 = ( ( ( A = bot_bot_set_nat )
% 14.17/8.02 & ( B4
% 14.17/8.02 = ( insert_nat @ X @ bot_bot_set_nat ) ) )
% 14.17/8.02 | ( ( A
% 14.17/8.02 = ( insert_nat @ X @ bot_bot_set_nat ) )
% 14.17/8.02 & ( B4 = bot_bot_set_nat ) )
% 14.17/8.02 | ( ( A
% 14.17/8.02 = ( insert_nat @ X @ bot_bot_set_nat ) )
% 14.17/8.02 & ( B4
% 14.17/8.02 = ( insert_nat @ X @ bot_bot_set_nat ) ) ) ) ) ).
% 14.17/8.02
% 14.17/8.02 thf(fact_1482_Un__singleton__iff,axiom,
% 14.17/8.02 ! [A: set_int,B4: set_int,X: int] :
% 14.17/8.02 ( ( ( sup_sup_set_int @ A @ B4 )
% 14.17/8.02 = ( insert_int @ X @ bot_bot_set_int ) )
% 14.17/8.02 = ( ( ( A = bot_bot_set_int )
% 14.17/8.02 & ( B4
% 14.17/8.02 = ( insert_int @ X @ bot_bot_set_int ) ) )
% 14.17/8.02 | ( ( A
% 14.17/8.02 = ( insert_int @ X @ bot_bot_set_int ) )
% 14.17/8.02 & ( B4 = bot_bot_set_int ) )
% 14.17/8.02 | ( ( A
% 14.17/8.02 = ( insert_int @ X @ bot_bot_set_int ) )
% 14.17/8.02 & ( B4
% 14.17/8.02 = ( insert_int @ X @ bot_bot_set_int ) ) ) ) ) ).
% 14.17/8.02
% 14.17/8.02 thf(fact_1483_Un__singleton__iff,axiom,
% 14.17/8.02 ! [A: set_Pr4532377907799695533_nat_o,B4: set_Pr4532377907799695533_nat_o,X: produc3658429121746597890et_nat > $o] :
% 14.17/8.02 ( ( ( sup_su5209123915105501825_nat_o @ A @ B4 )
% 14.17/8.02 = ( insert5175938949040314269_nat_o @ X @ bot_bo7824918357723582233_nat_o ) )
% 14.17/8.02 = ( ( ( A = bot_bo7824918357723582233_nat_o )
% 14.17/8.02 & ( B4
% 14.17/8.02 = ( insert5175938949040314269_nat_o @ X @ bot_bo7824918357723582233_nat_o ) ) )
% 14.17/8.02 | ( ( A
% 14.17/8.02 = ( insert5175938949040314269_nat_o @ X @ bot_bo7824918357723582233_nat_o ) )
% 14.17/8.02 & ( B4 = bot_bo7824918357723582233_nat_o ) )
% 14.17/8.02 | ( ( A
% 14.17/8.02 = ( insert5175938949040314269_nat_o @ X @ bot_bo7824918357723582233_nat_o ) )
% 14.17/8.02 & ( B4
% 14.17/8.02 = ( insert5175938949040314269_nat_o @ X @ bot_bo7824918357723582233_nat_o ) ) ) ) ) ).
% 14.17/8.02
% 14.17/8.02 thf(fact_1484_singleton__Un__iff,axiom,
% 14.17/8.02 ! [X: product_unit,A: set_Product_unit,B4: set_Product_unit] :
% 14.17/8.02 ( ( ( insert_Product_unit @ X @ bot_bo3957492148770167129t_unit )
% 14.17/8.02 = ( sup_su793286257634532545t_unit @ A @ B4 ) )
% 14.17/8.02 = ( ( ( A = bot_bo3957492148770167129t_unit )
% 14.17/8.02 & ( B4
% 14.17/8.02 = ( insert_Product_unit @ X @ bot_bo3957492148770167129t_unit ) ) )
% 14.17/8.02 | ( ( A
% 14.17/8.02 = ( insert_Product_unit @ X @ bot_bo3957492148770167129t_unit ) )
% 14.17/8.02 & ( B4 = bot_bo3957492148770167129t_unit ) )
% 14.17/8.02 | ( ( A
% 14.17/8.02 = ( insert_Product_unit @ X @ bot_bo3957492148770167129t_unit ) )
% 14.17/8.02 & ( B4
% 14.17/8.02 = ( insert_Product_unit @ X @ bot_bo3957492148770167129t_unit ) ) ) ) ) ).
% 14.17/8.02
% 14.17/8.02 thf(fact_1485_singleton__Un__iff,axiom,
% 14.17/8.02 ! [X: $o,A: set_o,B4: set_o] :
% 14.17/8.02 ( ( ( insert_o @ X @ bot_bot_set_o )
% 14.17/8.02 = ( sup_sup_set_o @ A @ B4 ) )
% 14.17/8.02 = ( ( ( A = bot_bot_set_o )
% 14.17/8.02 & ( B4
% 14.17/8.02 = ( insert_o @ X @ bot_bot_set_o ) ) )
% 14.17/8.02 | ( ( A
% 14.17/8.02 = ( insert_o @ X @ bot_bot_set_o ) )
% 14.17/8.02 & ( B4 = bot_bot_set_o ) )
% 14.17/8.02 | ( ( A
% 14.17/8.02 = ( insert_o @ X @ bot_bot_set_o ) )
% 14.17/8.02 & ( B4
% 14.17/8.02 = ( insert_o @ X @ bot_bot_set_o ) ) ) ) ) ).
% 14.17/8.02
% 14.17/8.02 thf(fact_1486_singleton__Un__iff,axiom,
% 14.17/8.02 ! [X: nat,A: set_nat,B4: set_nat] :
% 14.17/8.02 ( ( ( insert_nat @ X @ bot_bot_set_nat )
% 14.17/8.02 = ( sup_sup_set_nat @ A @ B4 ) )
% 14.17/8.02 = ( ( ( A = bot_bot_set_nat )
% 14.17/8.02 & ( B4
% 14.17/8.02 = ( insert_nat @ X @ bot_bot_set_nat ) ) )
% 14.17/8.02 | ( ( A
% 14.17/8.02 = ( insert_nat @ X @ bot_bot_set_nat ) )
% 14.17/8.02 & ( B4 = bot_bot_set_nat ) )
% 14.17/8.02 | ( ( A
% 14.17/8.02 = ( insert_nat @ X @ bot_bot_set_nat ) )
% 14.17/8.02 & ( B4
% 14.17/8.02 = ( insert_nat @ X @ bot_bot_set_nat ) ) ) ) ) ).
% 14.17/8.02
% 14.17/8.02 thf(fact_1487_singleton__Un__iff,axiom,
% 14.17/8.02 ! [X: int,A: set_int,B4: set_int] :
% 14.17/8.02 ( ( ( insert_int @ X @ bot_bot_set_int )
% 14.17/8.02 = ( sup_sup_set_int @ A @ B4 ) )
% 14.17/8.02 = ( ( ( A = bot_bot_set_int )
% 14.17/8.02 & ( B4
% 14.17/8.02 = ( insert_int @ X @ bot_bot_set_int ) ) )
% 14.17/8.02 | ( ( A
% 14.17/8.02 = ( insert_int @ X @ bot_bot_set_int ) )
% 14.17/8.02 & ( B4 = bot_bot_set_int ) )
% 14.17/8.02 | ( ( A
% 14.17/8.02 = ( insert_int @ X @ bot_bot_set_int ) )
% 14.17/8.02 & ( B4
% 14.17/8.02 = ( insert_int @ X @ bot_bot_set_int ) ) ) ) ) ).
% 14.17/8.02
% 14.17/8.02 thf(fact_1488_singleton__Un__iff,axiom,
% 14.17/8.02 ! [X: produc3658429121746597890et_nat > $o,A: set_Pr4532377907799695533_nat_o,B4: set_Pr4532377907799695533_nat_o] :
% 14.17/8.02 ( ( ( insert5175938949040314269_nat_o @ X @ bot_bo7824918357723582233_nat_o )
% 14.17/8.02 = ( sup_su5209123915105501825_nat_o @ A @ B4 ) )
% 14.17/8.02 = ( ( ( A = bot_bo7824918357723582233_nat_o )
% 14.17/8.02 & ( B4
% 14.17/8.02 = ( insert5175938949040314269_nat_o @ X @ bot_bo7824918357723582233_nat_o ) ) )
% 14.17/8.02 | ( ( A
% 14.17/8.02 = ( insert5175938949040314269_nat_o @ X @ bot_bo7824918357723582233_nat_o ) )
% 14.17/8.02 & ( B4 = bot_bo7824918357723582233_nat_o ) )
% 14.17/8.02 | ( ( A
% 14.17/8.02 = ( insert5175938949040314269_nat_o @ X @ bot_bo7824918357723582233_nat_o ) )
% 14.17/8.02 & ( B4
% 14.17/8.02 = ( insert5175938949040314269_nat_o @ X @ bot_bo7824918357723582233_nat_o ) ) ) ) ) ).
% 14.17/8.02
% 14.17/8.02 thf(fact_1489_Not__Domain__rtrancl,axiom,
% 14.17/8.02 ! [X: produc3658429121746597890et_nat > $o,R3: set_Pr2161125870931222855_nat_o,Y: produc3658429121746597890et_nat > $o] :
% 14.17/8.02 ( ~ ( member6576561426505652726_nat_o @ X @ ( domain4881241903163524940_nat_o @ R3 ) )
% 14.17/8.02 => ( ( member8781333585448626064_nat_o @ ( produc7368190662567826135_nat_o @ X @ Y ) @ ( transi712980445759334096_nat_o @ R3 ) )
% 14.17/8.02 = ( X = Y ) ) ) ).
% 14.17/8.02
% 14.17/8.02 thf(fact_1490_Not__Domain__rtrancl,axiom,
% 14.17/8.02 ! [X: $o,R3: set_Product_prod_o_o,Y: $o] :
% 14.17/8.02 ( ~ ( member_o @ X @ ( domain_o_o @ R3 ) )
% 14.17/8.02 => ( ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ X @ Y ) @ ( transitive_rtrancl_o @ R3 ) )
% 14.17/8.02 = ( X = Y ) ) ) ).
% 14.17/8.02
% 14.17/8.02 thf(fact_1491_Not__Domain__rtrancl,axiom,
% 14.17/8.02 ! [X: product_unit,R3: set_Pr5094982260447487303t_unit,Y: product_unit] :
% 14.17/8.02 ( ~ ( member_Product_unit @ X @ ( domain4292024440186670924t_unit @ R3 ) )
% 14.17/8.02 => ( ( member7821138191923116944t_unit @ ( produc5692694829384537815t_unit @ X @ Y ) @ ( transi6481887417741755664t_unit @ R3 ) )
% 14.17/8.02 = ( X = Y ) ) ) ).
% 14.17/8.02
% 14.17/8.02 thf(fact_1492_Not__Domain__rtrancl,axiom,
% 14.17/8.02 ! [X: nat,R3: set_Pr1261947904930325089at_nat,Y: nat] :
% 14.17/8.02 ( ~ ( member_nat @ X @ ( domain_nat_nat @ R3 ) )
% 14.17/8.02 => ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ Y ) @ ( transi2905341329935302413cl_nat @ R3 ) )
% 14.17/8.02 = ( X = Y ) ) ) ).
% 14.17/8.02
% 14.17/8.02 thf(fact_1493_Not__Domain__rtrancl,axiom,
% 14.17/8.02 ! [X: int,R3: set_Pr958786334691620121nt_int,Y: int] :
% 14.17/8.02 ( ~ ( member_int @ X @ ( domain_int_int @ R3 ) )
% 14.17/8.02 => ( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ X @ Y ) @ ( transi2902850859426252137cl_int @ R3 ) )
% 14.17/8.02 = ( X = Y ) ) ) ).
% 14.17/8.02
% 14.17/8.02 thf(fact_1494_subset__singletonD,axiom,
% 14.17/8.02 ! [A: set_Product_unit,X: product_unit] :
% 14.17/8.02 ( ( ord_le3507040750410214029t_unit @ A @ ( insert_Product_unit @ X @ bot_bo3957492148770167129t_unit ) )
% 14.17/8.02 => ( ( A = bot_bo3957492148770167129t_unit )
% 14.17/8.02 | ( A
% 14.17/8.02 = ( insert_Product_unit @ X @ bot_bo3957492148770167129t_unit ) ) ) ) ).
% 14.17/8.02
% 14.17/8.02 thf(fact_1495_subset__singletonD,axiom,
% 14.17/8.02 ! [A: set_o,X: $o] :
% 14.17/8.02 ( ( ord_less_eq_set_o @ A @ ( insert_o @ X @ bot_bot_set_o ) )
% 14.17/8.02 => ( ( A = bot_bot_set_o )
% 14.17/8.02 | ( A
% 14.17/8.02 = ( insert_o @ X @ bot_bot_set_o ) ) ) ) ).
% 14.17/8.02
% 14.17/8.02 thf(fact_1496_subset__singletonD,axiom,
% 14.17/8.02 ! [A: set_int,X: int] :
% 14.17/8.02 ( ( ord_less_eq_set_int @ A @ ( insert_int @ X @ bot_bot_set_int ) )
% 14.17/8.02 => ( ( A = bot_bot_set_int )
% 14.17/8.02 | ( A
% 14.17/8.02 = ( insert_int @ X @ bot_bot_set_int ) ) ) ) ).
% 14.17/8.02
% 14.17/8.02 thf(fact_1497_subset__singletonD,axiom,
% 14.17/8.02 ! [A: set_Pr4532377907799695533_nat_o,X: produc3658429121746597890et_nat > $o] :
% 14.17/8.02 ( ( ord_le2965882846123202637_nat_o @ A @ ( insert5175938949040314269_nat_o @ X @ bot_bo7824918357723582233_nat_o ) )
% 14.17/8.02 => ( ( A = bot_bo7824918357723582233_nat_o )
% 14.17/8.02 | ( A
% 14.17/8.02 = ( insert5175938949040314269_nat_o @ X @ bot_bo7824918357723582233_nat_o ) ) ) ) ).
% 14.17/8.02
% 14.17/8.02 thf(fact_1498_subset__singletonD,axiom,
% 14.17/8.02 ! [A: set_nat,X: nat] :
% 14.17/8.02 ( ( ord_less_eq_set_nat @ A @ ( insert_nat @ X @ bot_bot_set_nat ) )
% 14.17/8.02 => ( ( A = bot_bot_set_nat )
% 14.17/8.02 | ( A
% 14.17/8.02 = ( insert_nat @ X @ bot_bot_set_nat ) ) ) ) ).
% 14.17/8.02
% 14.17/8.02 thf(fact_1499_subset__singleton__iff,axiom,
% 14.17/8.02 ! [X4: set_Product_unit,A2: product_unit] :
% 14.17/8.02 ( ( ord_le3507040750410214029t_unit @ X4 @ ( insert_Product_unit @ A2 @ bot_bo3957492148770167129t_unit ) )
% 14.17/8.02 = ( ( X4 = bot_bo3957492148770167129t_unit )
% 14.17/8.02 | ( X4
% 14.17/8.02 = ( insert_Product_unit @ A2 @ bot_bo3957492148770167129t_unit ) ) ) ) ).
% 14.17/8.02
% 14.17/8.02 thf(fact_1500_subset__singleton__iff,axiom,
% 14.17/8.02 ! [X4: set_o,A2: $o] :
% 14.17/8.02 ( ( ord_less_eq_set_o @ X4 @ ( insert_o @ A2 @ bot_bot_set_o ) )
% 14.17/8.02 = ( ( X4 = bot_bot_set_o )
% 14.17/8.02 | ( X4
% 14.17/8.02 = ( insert_o @ A2 @ bot_bot_set_o ) ) ) ) ).
% 14.17/8.02
% 14.17/8.02 thf(fact_1501_subset__singleton__iff,axiom,
% 14.17/8.02 ! [X4: set_int,A2: int] :
% 14.17/8.02 ( ( ord_less_eq_set_int @ X4 @ ( insert_int @ A2 @ bot_bot_set_int ) )
% 14.17/8.02 = ( ( X4 = bot_bot_set_int )
% 14.17/8.02 | ( X4
% 14.17/8.02 = ( insert_int @ A2 @ bot_bot_set_int ) ) ) ) ).
% 14.17/8.02
% 14.17/8.02 thf(fact_1502_subset__singleton__iff,axiom,
% 14.17/8.02 ! [X4: set_Pr4532377907799695533_nat_o,A2: produc3658429121746597890et_nat > $o] :
% 14.17/8.02 ( ( ord_le2965882846123202637_nat_o @ X4 @ ( insert5175938949040314269_nat_o @ A2 @ bot_bo7824918357723582233_nat_o ) )
% 14.17/8.02 = ( ( X4 = bot_bo7824918357723582233_nat_o )
% 14.17/8.02 | ( X4
% 14.17/8.02 = ( insert5175938949040314269_nat_o @ A2 @ bot_bo7824918357723582233_nat_o ) ) ) ) ).
% 14.17/8.02
% 14.17/8.02 thf(fact_1503_subset__singleton__iff,axiom,
% 14.17/8.02 ! [X4: set_nat,A2: nat] :
% 14.17/8.02 ( ( ord_less_eq_set_nat @ X4 @ ( insert_nat @ A2 @ bot_bot_set_nat ) )
% 14.17/8.02 = ( ( X4 = bot_bot_set_nat )
% 14.17/8.02 | ( X4
% 14.17/8.02 = ( insert_nat @ A2 @ bot_bot_set_nat ) ) ) ) ).
% 14.17/8.02
% 14.17/8.02 thf(fact_1504_mod__star__conv,axiom,
% 14.17/8.02 ! [A: assn,B4: assn,H2: produc3658429121746597890et_nat] :
% 14.17/8.02 ( ( rep_assn @ ( times_times_assn @ A @ B4 ) @ H2 )
% 14.17/8.02 = ( ? [Hr: heap_e7401611519738050253t_unit,As1: set_nat,As22: set_nat] :
% 14.17/8.02 ( ( H2
% 14.17/8.02 = ( produc7507926704131184380et_nat @ Hr @ ( sup_sup_set_nat @ As1 @ As22 ) ) )
% 14.17/8.02 & ( ( inf_inf_set_nat @ As1 @ As22 )
% 14.17/8.02 = bot_bot_set_nat )
% 14.17/8.02 & ( rep_assn @ A @ ( produc7507926704131184380et_nat @ Hr @ As1 ) )
% 14.17/8.02 & ( rep_assn @ B4 @ ( produc7507926704131184380et_nat @ Hr @ As22 ) ) ) ) ) ).
% 14.17/8.02
% 14.17/8.02 thf(fact_1505_star__assnI,axiom,
% 14.17/8.02 ! [P: assn,H2: heap_e7401611519738050253t_unit,As: set_nat,Q2: assn,As5: set_nat] :
% 14.17/8.02 ( ( rep_assn @ P @ ( produc7507926704131184380et_nat @ H2 @ As ) )
% 14.17/8.02 => ( ( rep_assn @ Q2 @ ( produc7507926704131184380et_nat @ H2 @ As5 ) )
% 14.17/8.02 => ( ( ( inf_inf_set_nat @ As @ As5 )
% 14.17/8.02 = bot_bot_set_nat )
% 14.17/8.02 => ( rep_assn @ ( times_times_assn @ P @ Q2 ) @ ( produc7507926704131184380et_nat @ H2 @ ( sup_sup_set_nat @ As @ As5 ) ) ) ) ) ) ).
% 14.17/8.02
% 14.17/8.02 thf(fact_1506_in__range__subset,axiom,
% 14.17/8.02 ! [As: set_nat,As5: set_nat,H2: heap_e7401611519738050253t_unit] :
% 14.17/8.02 ( ( ord_less_eq_set_nat @ As @ As5 )
% 14.17/8.02 => ( ( in_range @ ( produc7507926704131184380et_nat @ H2 @ As5 ) )
% 14.17/8.02 => ( in_range @ ( produc7507926704131184380et_nat @ H2 @ As ) ) ) ) ).
% 14.17/8.02
% 14.17/8.02 thf(fact_1507_wand__raw_Osimps,axiom,
% 14.17/8.02 ! [P: produc3658429121746597890et_nat > $o,Q2: produc3658429121746597890et_nat > $o,H2: heap_e7401611519738050253t_unit,As: set_nat] :
% 14.17/8.02 ( ( wand_raw @ P @ Q2 @ ( produc7507926704131184380et_nat @ H2 @ As ) )
% 14.17/8.02 = ( ( in_range @ ( produc7507926704131184380et_nat @ H2 @ As ) )
% 14.17/8.02 & ! [H5: heap_e7401611519738050253t_unit,As6: set_nat] :
% 14.17/8.02 ( ( ( ( inf_inf_set_nat @ As @ As6 )
% 14.17/8.02 = bot_bot_set_nat )
% 14.17/8.02 & ( relH @ As @ H2 @ H5 )
% 14.17/8.02 & ( in_range @ ( produc7507926704131184380et_nat @ H5 @ As ) )
% 14.17/8.02 & ( P @ ( produc7507926704131184380et_nat @ H5 @ As6 ) ) )
% 14.17/8.02 => ( Q2 @ ( produc7507926704131184380et_nat @ H5 @ ( sup_sup_set_nat @ As @ As6 ) ) ) ) ) ) ).
% 14.17/8.02
% 14.17/8.02 thf(fact_1508_wand__raw_Oelims_I1_J,axiom,
% 14.17/8.02 ! [X: produc3658429121746597890et_nat > $o,Xa: produc3658429121746597890et_nat > $o,Xb: produc3658429121746597890et_nat,Y: $o] :
% 14.17/8.02 ( ( ( wand_raw @ X @ Xa @ Xb )
% 14.17/8.02 = Y )
% 14.17/8.02 => ~ ! [H: heap_e7401611519738050253t_unit,As2: set_nat] :
% 14.17/8.02 ( ( Xb
% 14.17/8.02 = ( produc7507926704131184380et_nat @ H @ As2 ) )
% 14.17/8.02 => ( Y
% 14.17/8.02 = ( ~ ( ( in_range @ ( produc7507926704131184380et_nat @ H @ As2 ) )
% 14.17/8.02 & ! [H5: heap_e7401611519738050253t_unit,As6: set_nat] :
% 14.17/8.02 ( ( ( ( inf_inf_set_nat @ As2 @ As6 )
% 14.17/8.02 = bot_bot_set_nat )
% 14.17/8.02 & ( relH @ As2 @ H @ H5 )
% 14.17/8.02 & ( in_range @ ( produc7507926704131184380et_nat @ H5 @ As2 ) )
% 14.17/8.02 & ( X @ ( produc7507926704131184380et_nat @ H5 @ As6 ) ) )
% 14.17/8.02 => ( Xa @ ( produc7507926704131184380et_nat @ H5 @ ( sup_sup_set_nat @ As2 @ As6 ) ) ) ) ) ) ) ) ) ).
% 14.17/8.02
% 14.17/8.02 thf(fact_1509_wand__raw_Oelims_I2_J,axiom,
% 14.17/8.02 ! [X: produc3658429121746597890et_nat > $o,Xa: produc3658429121746597890et_nat > $o,Xb: produc3658429121746597890et_nat] :
% 14.17/8.02 ( ( wand_raw @ X @ Xa @ Xb )
% 14.17/8.02 => ~ ! [H: heap_e7401611519738050253t_unit,As2: set_nat] :
% 14.17/8.02 ( ( Xb
% 14.17/8.02 = ( produc7507926704131184380et_nat @ H @ As2 ) )
% 14.17/8.02 => ~ ( ( in_range @ ( produc7507926704131184380et_nat @ H @ As2 ) )
% 14.17/8.02 & ! [H7: heap_e7401611519738050253t_unit,As7: set_nat] :
% 14.17/8.02 ( ( ( ( inf_inf_set_nat @ As2 @ As7 )
% 14.17/8.02 = bot_bot_set_nat )
% 14.17/8.02 & ( relH @ As2 @ H @ H7 )
% 14.17/8.02 & ( in_range @ ( produc7507926704131184380et_nat @ H7 @ As2 ) )
% 14.17/8.02 & ( X @ ( produc7507926704131184380et_nat @ H7 @ As7 ) ) )
% 14.17/8.02 => ( Xa @ ( produc7507926704131184380et_nat @ H7 @ ( sup_sup_set_nat @ As2 @ As7 ) ) ) ) ) ) ) ).
% 14.17/8.02
% 14.17/8.02 thf(fact_1510_wand__raw_Oelims_I3_J,axiom,
% 14.17/8.02 ! [X: produc3658429121746597890et_nat > $o,Xa: produc3658429121746597890et_nat > $o,Xb: produc3658429121746597890et_nat] :
% 14.17/8.02 ( ~ ( wand_raw @ X @ Xa @ Xb )
% 14.17/8.02 => ~ ! [H: heap_e7401611519738050253t_unit,As2: set_nat] :
% 14.17/8.02 ( ( Xb
% 14.17/8.02 = ( produc7507926704131184380et_nat @ H @ As2 ) )
% 14.17/8.02 => ( ( in_range @ ( produc7507926704131184380et_nat @ H @ As2 ) )
% 14.17/8.02 & ! [H6: heap_e7401611519738050253t_unit,As4: set_nat] :
% 14.17/8.02 ( ( ( ( inf_inf_set_nat @ As2 @ As4 )
% 14.17/8.02 = bot_bot_set_nat )
% 14.17/8.02 & ( relH @ As2 @ H @ H6 )
% 14.17/8.02 & ( in_range @ ( produc7507926704131184380et_nat @ H6 @ As2 ) )
% 14.17/8.02 & ( X @ ( produc7507926704131184380et_nat @ H6 @ As4 ) ) )
% 14.17/8.02 => ( Xa @ ( produc7507926704131184380et_nat @ H6 @ ( sup_sup_set_nat @ As2 @ As4 ) ) ) ) ) ) ) ).
% 14.17/8.02
% 14.17/8.02 thf(fact_1511_Image__empty__rtrancl__Image__id,axiom,
% 14.17/8.02 ! [R3: set_Pr5094982260447487303t_unit,V: product_unit] :
% 14.17/8.02 ( ( ( image_8431784415008483091t_unit @ R3 @ ( insert_Product_unit @ V @ bot_bo3957492148770167129t_unit ) )
% 14.17/8.02 = bot_bo3957492148770167129t_unit )
% 14.17/8.02 => ( ( image_8431784415008483091t_unit @ ( transi6481887417741755664t_unit @ R3 ) @ ( insert_Product_unit @ V @ bot_bo3957492148770167129t_unit ) )
% 14.17/8.02 = ( insert_Product_unit @ V @ bot_bo3957492148770167129t_unit ) ) ) ).
% 14.17/8.02
% 14.17/8.02 thf(fact_1512_Image__empty__rtrancl__Image__id,axiom,
% 14.17/8.02 ! [R3: set_Product_prod_o_o,V: $o] :
% 14.17/8.02 ( ( ( image_o_o @ R3 @ ( insert_o @ V @ bot_bot_set_o ) )
% 14.17/8.02 = bot_bot_set_o )
% 14.17/8.02 => ( ( image_o_o @ ( transitive_rtrancl_o @ R3 ) @ ( insert_o @ V @ bot_bot_set_o ) )
% 14.17/8.02 = ( insert_o @ V @ bot_bot_set_o ) ) ) ).
% 14.17/8.02
% 14.17/8.02 thf(fact_1513_Image__empty__rtrancl__Image__id,axiom,
% 14.17/8.02 ! [R3: set_Pr1261947904930325089at_nat,V: nat] :
% 14.17/8.02 ( ( ( image_nat_nat @ R3 @ ( insert_nat @ V @ bot_bot_set_nat ) )
% 14.17/8.02 = bot_bot_set_nat )
% 14.17/8.02 => ( ( image_nat_nat @ ( transi2905341329935302413cl_nat @ R3 ) @ ( insert_nat @ V @ bot_bot_set_nat ) )
% 14.17/8.02 = ( insert_nat @ V @ bot_bot_set_nat ) ) ) ).
% 14.17/8.02
% 14.17/8.02 thf(fact_1514_Image__empty__rtrancl__Image__id,axiom,
% 14.17/8.02 ! [R3: set_Pr958786334691620121nt_int,V: int] :
% 14.17/8.02 ( ( ( image_int_int @ R3 @ ( insert_int @ V @ bot_bot_set_int ) )
% 14.17/8.02 = bot_bot_set_int )
% 14.17/8.02 => ( ( image_int_int @ ( transi2902850859426252137cl_int @ R3 ) @ ( insert_int @ V @ bot_bot_set_int ) )
% 14.17/8.02 = ( insert_int @ V @ bot_bot_set_int ) ) ) ).
% 14.17/8.02
% 14.17/8.02 thf(fact_1515_Image__empty__rtrancl__Image__id,axiom,
% 14.17/8.02 ! [R3: set_Pr2161125870931222855_nat_o,V: produc3658429121746597890et_nat > $o] :
% 14.17/8.02 ( ( ( image_2232984160439939859_nat_o @ R3 @ ( insert5175938949040314269_nat_o @ V @ bot_bo7824918357723582233_nat_o ) )
% 14.17/8.02 = bot_bo7824918357723582233_nat_o )
% 14.17/8.02 => ( ( image_2232984160439939859_nat_o @ ( transi712980445759334096_nat_o @ R3 ) @ ( insert5175938949040314269_nat_o @ V @ bot_bo7824918357723582233_nat_o ) )
% 14.17/8.02 = ( insert5175938949040314269_nat_o @ V @ bot_bo7824918357723582233_nat_o ) ) ) ).
% 14.17/8.02
% 14.17/8.02 thf(fact_1516_sup__bot_Oright__neutral,axiom,
% 14.17/8.02 ! [A2: assn] :
% 14.17/8.02 ( ( sup_sup_assn @ A2 @ bot_bot_assn )
% 14.17/8.02 = A2 ) ).
% 14.17/8.02
% 14.17/8.02 thf(fact_1517_sup__bot_Oright__neutral,axiom,
% 14.17/8.02 ! [A2: product_unit] :
% 14.17/8.02 ( ( sup_sup_Product_unit @ A2 @ bot_bot_Product_unit )
% 14.17/8.02 = A2 ) ).
% 14.17/8.02
% 14.17/8.02 thf(fact_1518_sup__bot_Oright__neutral,axiom,
% 14.17/8.02 ! [A2: set_Product_unit] :
% 14.17/8.02 ( ( sup_su793286257634532545t_unit @ A2 @ bot_bo3957492148770167129t_unit )
% 14.17/8.02 = A2 ) ).
% 14.17/8.02
% 14.17/8.02 thf(fact_1519_sup__bot_Oright__neutral,axiom,
% 14.17/8.02 ! [A2: set_o] :
% 14.17/8.02 ( ( sup_sup_set_o @ A2 @ bot_bot_set_o )
% 14.17/8.02 = A2 ) ).
% 14.17/8.02
% 14.17/8.02 thf(fact_1520_sup__bot_Oright__neutral,axiom,
% 14.17/8.02 ! [A2: set_nat] :
% 14.17/8.02 ( ( sup_sup_set_nat @ A2 @ bot_bot_set_nat )
% 14.17/8.02 = A2 ) ).
% 14.17/8.02
% 14.17/8.02 thf(fact_1521_sup__bot_Oright__neutral,axiom,
% 14.17/8.02 ! [A2: set_int] :
% 14.17/8.02 ( ( sup_sup_set_int @ A2 @ bot_bot_set_int )
% 14.17/8.02 = A2 ) ).
% 14.17/8.02
% 14.17/8.02 thf(fact_1522_sup__bot_Oright__neutral,axiom,
% 14.17/8.02 ! [A2: set_Pr4532377907799695533_nat_o] :
% 14.17/8.02 ( ( sup_su5209123915105501825_nat_o @ A2 @ bot_bo7824918357723582233_nat_o )
% 14.17/8.02 = A2 ) ).
% 14.17/8.02
% 14.17/8.02 thf(fact_1523_sup__bot_Oneutr__eq__iff,axiom,
% 14.17/8.02 ! [A2: assn,B: assn] :
% 14.17/8.02 ( ( bot_bot_assn
% 14.17/8.02 = ( sup_sup_assn @ A2 @ B ) )
% 14.17/8.02 = ( ( A2 = bot_bot_assn )
% 14.17/8.02 & ( B = bot_bot_assn ) ) ) ).
% 14.17/8.02
% 14.17/8.02 thf(fact_1524_sup__bot_Oneutr__eq__iff,axiom,
% 14.17/8.02 ! [A2: product_unit,B: product_unit] :
% 14.17/8.02 ( ( bot_bot_Product_unit
% 14.17/8.02 = ( sup_sup_Product_unit @ A2 @ B ) )
% 14.17/8.02 = ( ( A2 = bot_bot_Product_unit )
% 14.17/8.02 & ( B = bot_bot_Product_unit ) ) ) ).
% 14.17/8.02
% 14.17/8.02 thf(fact_1525_sup__bot_Oneutr__eq__iff,axiom,
% 14.17/8.02 ! [A2: set_Product_unit,B: set_Product_unit] :
% 14.17/8.02 ( ( bot_bo3957492148770167129t_unit
% 14.17/8.02 = ( sup_su793286257634532545t_unit @ A2 @ B ) )
% 14.17/8.02 = ( ( A2 = bot_bo3957492148770167129t_unit )
% 14.17/8.02 & ( B = bot_bo3957492148770167129t_unit ) ) ) ).
% 14.17/8.02
% 14.17/8.02 thf(fact_1526_sup__bot_Oneutr__eq__iff,axiom,
% 14.17/8.02 ! [A2: set_o,B: set_o] :
% 14.17/8.02 ( ( bot_bot_set_o
% 14.17/8.02 = ( sup_sup_set_o @ A2 @ B ) )
% 14.17/8.02 = ( ( A2 = bot_bot_set_o )
% 14.17/8.02 & ( B = bot_bot_set_o ) ) ) ).
% 14.17/8.02
% 14.17/8.02 thf(fact_1527_sup__bot_Oneutr__eq__iff,axiom,
% 14.17/8.02 ! [A2: set_nat,B: set_nat] :
% 14.17/8.02 ( ( bot_bot_set_nat
% 14.17/8.02 = ( sup_sup_set_nat @ A2 @ B ) )
% 14.17/8.02 = ( ( A2 = bot_bot_set_nat )
% 14.17/8.02 & ( B = bot_bot_set_nat ) ) ) ).
% 14.17/8.02
% 14.17/8.02 thf(fact_1528_sup__bot_Oneutr__eq__iff,axiom,
% 14.17/8.02 ! [A2: set_int,B: set_int] :
% 14.17/8.02 ( ( bot_bot_set_int
% 14.17/8.02 = ( sup_sup_set_int @ A2 @ B ) )
% 14.17/8.02 = ( ( A2 = bot_bot_set_int )
% 14.17/8.02 & ( B = bot_bot_set_int ) ) ) ).
% 14.17/8.02
% 14.17/8.02 thf(fact_1529_sup__bot_Oneutr__eq__iff,axiom,
% 14.17/8.02 ! [A2: set_Pr4532377907799695533_nat_o,B: set_Pr4532377907799695533_nat_o] :
% 14.17/8.02 ( ( bot_bo7824918357723582233_nat_o
% 14.17/8.02 = ( sup_su5209123915105501825_nat_o @ A2 @ B ) )
% 14.17/8.02 = ( ( A2 = bot_bo7824918357723582233_nat_o )
% 14.17/8.02 & ( B = bot_bo7824918357723582233_nat_o ) ) ) ).
% 14.17/8.02
% 14.17/8.02 thf(fact_1530_sup__bot_Oleft__neutral,axiom,
% 14.17/8.02 ! [A2: assn] :
% 14.17/8.02 ( ( sup_sup_assn @ bot_bot_assn @ A2 )
% 14.17/8.02 = A2 ) ).
% 14.17/8.02
% 14.17/8.02 thf(fact_1531_sup__bot_Oleft__neutral,axiom,
% 14.17/8.02 ! [A2: product_unit] :
% 14.17/8.02 ( ( sup_sup_Product_unit @ bot_bot_Product_unit @ A2 )
% 14.17/8.02 = A2 ) ).
% 14.17/8.02
% 14.17/8.02 thf(fact_1532_sup__bot_Oleft__neutral,axiom,
% 14.17/8.02 ! [A2: set_Product_unit] :
% 14.17/8.02 ( ( sup_su793286257634532545t_unit @ bot_bo3957492148770167129t_unit @ A2 )
% 14.17/8.02 = A2 ) ).
% 14.17/8.02
% 14.17/8.02 thf(fact_1533_sup__bot_Oleft__neutral,axiom,
% 14.17/8.02 ! [A2: set_o] :
% 14.17/8.02 ( ( sup_sup_set_o @ bot_bot_set_o @ A2 )
% 14.17/8.02 = A2 ) ).
% 14.17/8.02
% 14.17/8.02 thf(fact_1534_sup__bot_Oleft__neutral,axiom,
% 14.17/8.02 ! [A2: set_nat] :
% 14.17/8.02 ( ( sup_sup_set_nat @ bot_bot_set_nat @ A2 )
% 14.17/8.02 = A2 ) ).
% 14.17/8.02
% 14.17/8.02 thf(fact_1535_sup__bot_Oleft__neutral,axiom,
% 14.17/8.02 ! [A2: set_int] :
% 14.17/8.02 ( ( sup_sup_set_int @ bot_bot_set_int @ A2 )
% 14.17/8.02 = A2 ) ).
% 14.17/8.02
% 14.17/8.02 thf(fact_1536_sup__bot_Oleft__neutral,axiom,
% 14.17/8.02 ! [A2: set_Pr4532377907799695533_nat_o] :
% 14.17/8.02 ( ( sup_su5209123915105501825_nat_o @ bot_bo7824918357723582233_nat_o @ A2 )
% 14.17/8.02 = A2 ) ).
% 14.17/8.02
% 14.17/8.02 thf(fact_1537_sup__bot_Oeq__neutr__iff,axiom,
% 14.17/8.02 ! [A2: assn,B: assn] :
% 14.17/8.02 ( ( ( sup_sup_assn @ A2 @ B )
% 14.17/8.02 = bot_bot_assn )
% 14.17/8.02 = ( ( A2 = bot_bot_assn )
% 14.17/8.02 & ( B = bot_bot_assn ) ) ) ).
% 14.17/8.02
% 14.17/8.02 thf(fact_1538_sup__bot_Oeq__neutr__iff,axiom,
% 14.17/8.02 ! [A2: product_unit,B: product_unit] :
% 14.17/8.02 ( ( ( sup_sup_Product_unit @ A2 @ B )
% 14.17/8.02 = bot_bot_Product_unit )
% 14.17/8.02 = ( ( A2 = bot_bot_Product_unit )
% 14.17/8.02 & ( B = bot_bot_Product_unit ) ) ) ).
% 14.17/8.02
% 14.17/8.02 thf(fact_1539_sup__bot_Oeq__neutr__iff,axiom,
% 14.17/8.02 ! [A2: set_Product_unit,B: set_Product_unit] :
% 14.17/8.02 ( ( ( sup_su793286257634532545t_unit @ A2 @ B )
% 14.17/8.02 = bot_bo3957492148770167129t_unit )
% 14.17/8.02 = ( ( A2 = bot_bo3957492148770167129t_unit )
% 14.17/8.02 & ( B = bot_bo3957492148770167129t_unit ) ) ) ).
% 14.17/8.02
% 14.17/8.02 thf(fact_1540_sup__bot_Oeq__neutr__iff,axiom,
% 14.17/8.02 ! [A2: set_o,B: set_o] :
% 14.17/8.02 ( ( ( sup_sup_set_o @ A2 @ B )
% 14.17/8.02 = bot_bot_set_o )
% 14.17/8.02 = ( ( A2 = bot_bot_set_o )
% 14.17/8.02 & ( B = bot_bot_set_o ) ) ) ).
% 14.17/8.02
% 14.17/8.02 thf(fact_1541_sup__bot_Oeq__neutr__iff,axiom,
% 14.17/8.02 ! [A2: set_nat,B: set_nat] :
% 14.17/8.02 ( ( ( sup_sup_set_nat @ A2 @ B )
% 14.17/8.02 = bot_bot_set_nat )
% 14.17/8.02 = ( ( A2 = bot_bot_set_nat )
% 14.17/8.02 & ( B = bot_bot_set_nat ) ) ) ).
% 14.17/8.02
% 14.17/8.02 thf(fact_1542_sup__bot_Oeq__neutr__iff,axiom,
% 14.17/8.02 ! [A2: set_int,B: set_int] :
% 14.17/8.02 ( ( ( sup_sup_set_int @ A2 @ B )
% 14.17/8.02 = bot_bot_set_int )
% 14.17/8.02 = ( ( A2 = bot_bot_set_int )
% 14.17/8.02 & ( B = bot_bot_set_int ) ) ) ).
% 14.17/8.02
% 14.17/8.02 thf(fact_1543_sup__bot_Oeq__neutr__iff,axiom,
% 14.17/8.02 ! [A2: set_Pr4532377907799695533_nat_o,B: set_Pr4532377907799695533_nat_o] :
% 14.17/8.02 ( ( ( sup_su5209123915105501825_nat_o @ A2 @ B )
% 14.17/8.02 = bot_bo7824918357723582233_nat_o )
% 14.17/8.02 = ( ( A2 = bot_bo7824918357723582233_nat_o )
% 14.17/8.02 & ( B = bot_bo7824918357723582233_nat_o ) ) ) ).
% 14.17/8.02
% 14.17/8.02 thf(fact_1544_sup__eq__bot__iff,axiom,
% 14.17/8.02 ! [X: assn,Y: assn] :
% 14.17/8.02 ( ( ( sup_sup_assn @ X @ Y )
% 14.17/8.02 = bot_bot_assn )
% 14.17/8.02 = ( ( X = bot_bot_assn )
% 14.17/8.02 & ( Y = bot_bot_assn ) ) ) ).
% 14.17/8.02
% 14.17/8.02 thf(fact_1545_sup__eq__bot__iff,axiom,
% 14.17/8.02 ! [X: product_unit,Y: product_unit] :
% 14.17/8.02 ( ( ( sup_sup_Product_unit @ X @ Y )
% 14.17/8.02 = bot_bot_Product_unit )
% 14.17/8.02 = ( ( X = bot_bot_Product_unit )
% 14.17/8.02 & ( Y = bot_bot_Product_unit ) ) ) ).
% 14.17/8.02
% 14.17/8.02 thf(fact_1546_sup__eq__bot__iff,axiom,
% 14.17/8.02 ! [X: set_Product_unit,Y: set_Product_unit] :
% 14.17/8.02 ( ( ( sup_su793286257634532545t_unit @ X @ Y )
% 14.17/8.02 = bot_bo3957492148770167129t_unit )
% 14.17/8.02 = ( ( X = bot_bo3957492148770167129t_unit )
% 14.17/8.02 & ( Y = bot_bo3957492148770167129t_unit ) ) ) ).
% 14.17/8.02
% 14.17/8.02 thf(fact_1547_sup__eq__bot__iff,axiom,
% 14.17/8.02 ! [X: set_o,Y: set_o] :
% 14.17/8.02 ( ( ( sup_sup_set_o @ X @ Y )
% 14.17/8.02 = bot_bot_set_o )
% 14.17/8.02 = ( ( X = bot_bot_set_o )
% 14.17/8.02 & ( Y = bot_bot_set_o ) ) ) ).
% 14.17/8.02
% 14.17/8.02 thf(fact_1548_sup__eq__bot__iff,axiom,
% 14.17/8.02 ! [X: set_nat,Y: set_nat] :
% 14.17/8.02 ( ( ( sup_sup_set_nat @ X @ Y )
% 14.17/8.02 = bot_bot_set_nat )
% 14.17/8.02 = ( ( X = bot_bot_set_nat )
% 14.17/8.02 & ( Y = bot_bot_set_nat ) ) ) ).
% 14.17/8.02
% 14.17/8.02 thf(fact_1549_sup__eq__bot__iff,axiom,
% 14.17/8.02 ! [X: set_int,Y: set_int] :
% 14.17/8.02 ( ( ( sup_sup_set_int @ X @ Y )
% 14.17/8.02 = bot_bot_set_int )
% 14.17/8.02 = ( ( X = bot_bot_set_int )
% 14.17/8.02 & ( Y = bot_bot_set_int ) ) ) ).
% 14.17/8.02
% 14.17/8.02 thf(fact_1550_sup__eq__bot__iff,axiom,
% 14.17/8.02 ! [X: set_Pr4532377907799695533_nat_o,Y: set_Pr4532377907799695533_nat_o] :
% 14.17/8.02 ( ( ( sup_su5209123915105501825_nat_o @ X @ Y )
% 14.17/8.02 = bot_bo7824918357723582233_nat_o )
% 14.17/8.02 = ( ( X = bot_bo7824918357723582233_nat_o )
% 14.17/8.02 & ( Y = bot_bo7824918357723582233_nat_o ) ) ) ).
% 14.17/8.02
% 14.17/8.02 thf(fact_1551_bot__eq__sup__iff,axiom,
% 14.17/8.02 ! [X: assn,Y: assn] :
% 14.17/8.02 ( ( bot_bot_assn
% 14.17/8.02 = ( sup_sup_assn @ X @ Y ) )
% 14.17/8.02 = ( ( X = bot_bot_assn )
% 14.17/8.02 & ( Y = bot_bot_assn ) ) ) ).
% 14.17/8.02
% 14.17/8.02 thf(fact_1552_bot__eq__sup__iff,axiom,
% 14.17/8.02 ! [X: product_unit,Y: product_unit] :
% 14.17/8.02 ( ( bot_bot_Product_unit
% 14.17/8.02 = ( sup_sup_Product_unit @ X @ Y ) )
% 14.17/8.02 = ( ( X = bot_bot_Product_unit )
% 14.17/8.02 & ( Y = bot_bot_Product_unit ) ) ) ).
% 14.17/8.02
% 14.17/8.02 thf(fact_1553_bot__eq__sup__iff,axiom,
% 14.17/8.02 ! [X: set_Product_unit,Y: set_Product_unit] :
% 14.17/8.02 ( ( bot_bo3957492148770167129t_unit
% 14.17/8.02 = ( sup_su793286257634532545t_unit @ X @ Y ) )
% 14.17/8.02 = ( ( X = bot_bo3957492148770167129t_unit )
% 14.17/8.02 & ( Y = bot_bo3957492148770167129t_unit ) ) ) ).
% 14.17/8.02
% 14.17/8.02 thf(fact_1554_bot__eq__sup__iff,axiom,
% 14.17/8.02 ! [X: set_o,Y: set_o] :
% 14.17/8.02 ( ( bot_bot_set_o
% 14.17/8.02 = ( sup_sup_set_o @ X @ Y ) )
% 14.17/8.02 = ( ( X = bot_bot_set_o )
% 14.17/8.02 & ( Y = bot_bot_set_o ) ) ) ).
% 14.17/8.02
% 14.17/8.02 thf(fact_1555_bot__eq__sup__iff,axiom,
% 14.17/8.02 ! [X: set_nat,Y: set_nat] :
% 14.17/8.02 ( ( bot_bot_set_nat
% 14.17/8.02 = ( sup_sup_set_nat @ X @ Y ) )
% 14.17/8.02 = ( ( X = bot_bot_set_nat )
% 14.17/8.02 & ( Y = bot_bot_set_nat ) ) ) ).
% 14.17/8.02
% 14.17/8.02 thf(fact_1556_bot__eq__sup__iff,axiom,
% 14.17/8.02 ! [X: set_int,Y: set_int] :
% 14.17/8.02 ( ( bot_bot_set_int
% 14.17/8.02 = ( sup_sup_set_int @ X @ Y ) )
% 14.17/8.02 = ( ( X = bot_bot_set_int )
% 14.17/8.02 & ( Y = bot_bot_set_int ) ) ) ).
% 14.17/8.02
% 14.17/8.02 thf(fact_1557_bot__eq__sup__iff,axiom,
% 14.17/8.02 ! [X: set_Pr4532377907799695533_nat_o,Y: set_Pr4532377907799695533_nat_o] :
% 14.17/8.02 ( ( bot_bo7824918357723582233_nat_o
% 14.17/8.02 = ( sup_su5209123915105501825_nat_o @ X @ Y ) )
% 14.17/8.02 = ( ( X = bot_bo7824918357723582233_nat_o )
% 14.17/8.02 & ( Y = bot_bo7824918357723582233_nat_o ) ) ) ).
% 14.17/8.02
% 14.17/8.02 thf(fact_1558_sup__bot__right,axiom,
% 14.17/8.02 ! [X: assn] :
% 14.17/8.02 ( ( sup_sup_assn @ X @ bot_bot_assn )
% 14.17/8.02 = X ) ).
% 14.17/8.02
% 14.17/8.02 thf(fact_1559_sup__bot__right,axiom,
% 14.17/8.02 ! [X: product_unit] :
% 14.17/8.02 ( ( sup_sup_Product_unit @ X @ bot_bot_Product_unit )
% 14.17/8.02 = X ) ).
% 14.17/8.02
% 14.17/8.02 thf(fact_1560_sup__bot__right,axiom,
% 14.17/8.02 ! [X: set_Product_unit] :
% 14.17/8.02 ( ( sup_su793286257634532545t_unit @ X @ bot_bo3957492148770167129t_unit )
% 14.17/8.02 = X ) ).
% 14.17/8.02
% 14.17/8.02 thf(fact_1561_sup__bot__right,axiom,
% 14.17/8.02 ! [X: set_o] :
% 14.17/8.02 ( ( sup_sup_set_o @ X @ bot_bot_set_o )
% 14.17/8.02 = X ) ).
% 14.17/8.02
% 14.17/8.02 thf(fact_1562_sup__bot__right,axiom,
% 14.17/8.02 ! [X: set_nat] :
% 14.17/8.02 ( ( sup_sup_set_nat @ X @ bot_bot_set_nat )
% 14.17/8.02 = X ) ).
% 14.17/8.02
% 14.17/8.02 thf(fact_1563_sup__bot__right,axiom,
% 14.17/8.02 ! [X: set_int] :
% 14.17/8.02 ( ( sup_sup_set_int @ X @ bot_bot_set_int )
% 14.17/8.02 = X ) ).
% 14.17/8.02
% 14.17/8.02 thf(fact_1564_sup__bot__right,axiom,
% 14.17/8.02 ! [X: set_Pr4532377907799695533_nat_o] :
% 14.17/8.02 ( ( sup_su5209123915105501825_nat_o @ X @ bot_bo7824918357723582233_nat_o )
% 14.17/8.02 = X ) ).
% 14.17/8.02
% 14.17/8.02 thf(fact_1565_sup__bot__left,axiom,
% 14.17/8.02 ! [X: assn] :
% 14.17/8.02 ( ( sup_sup_assn @ bot_bot_assn @ X )
% 14.17/8.02 = X ) ).
% 14.17/8.02
% 14.17/8.02 thf(fact_1566_sup__bot__left,axiom,
% 14.17/8.02 ! [X: product_unit] :
% 14.17/8.02 ( ( sup_sup_Product_unit @ bot_bot_Product_unit @ X )
% 14.17/8.02 = X ) ).
% 14.17/8.02
% 14.17/8.02 thf(fact_1567_sup__bot__left,axiom,
% 14.17/8.02 ! [X: set_Product_unit] :
% 14.17/8.02 ( ( sup_su793286257634532545t_unit @ bot_bo3957492148770167129t_unit @ X )
% 14.17/8.02 = X ) ).
% 14.17/8.02
% 14.17/8.02 thf(fact_1568_sup__bot__left,axiom,
% 14.17/8.02 ! [X: set_o] :
% 14.17/8.02 ( ( sup_sup_set_o @ bot_bot_set_o @ X )
% 14.17/8.02 = X ) ).
% 14.17/8.02
% 14.17/8.02 thf(fact_1569_sup__bot__left,axiom,
% 14.17/8.02 ! [X: set_nat] :
% 14.17/8.02 ( ( sup_sup_set_nat @ bot_bot_set_nat @ X )
% 14.17/8.02 = X ) ).
% 14.17/8.02
% 14.17/8.02 thf(fact_1570_sup__bot__left,axiom,
% 14.17/8.02 ! [X: set_int] :
% 14.17/8.02 ( ( sup_sup_set_int @ bot_bot_set_int @ X )
% 14.17/8.02 = X ) ).
% 14.17/8.02
% 14.17/8.02 thf(fact_1571_sup__bot__left,axiom,
% 14.17/8.02 ! [X: set_Pr4532377907799695533_nat_o] :
% 14.17/8.02 ( ( sup_su5209123915105501825_nat_o @ bot_bo7824918357723582233_nat_o @ X )
% 14.17/8.02 = X ) ).
% 14.17/8.02
% 14.17/8.02 thf(fact_1572_boolean__algebra_Oconj__zero__right,axiom,
% 14.17/8.02 ! [X: assn] :
% 14.17/8.02 ( ( inf_inf_assn @ X @ bot_bot_assn )
% 14.17/8.02 = bot_bot_assn ) ).
% 14.17/8.02
% 14.17/8.02 thf(fact_1573_boolean__algebra_Oconj__zero__right,axiom,
% 14.17/8.02 ! [X: product_unit] :
% 14.17/8.02 ( ( inf_inf_Product_unit @ X @ bot_bot_Product_unit )
% 14.17/8.02 = bot_bot_Product_unit ) ).
% 14.17/8.02
% 14.17/8.02 thf(fact_1574_boolean__algebra_Oconj__zero__right,axiom,
% 14.17/8.02 ! [X: set_Product_unit] :
% 14.17/8.02 ( ( inf_in4660618365625256667t_unit @ X @ bot_bo3957492148770167129t_unit )
% 14.17/8.02 = bot_bo3957492148770167129t_unit ) ).
% 14.17/8.02
% 14.17/8.02 thf(fact_1575_boolean__algebra_Oconj__zero__right,axiom,
% 14.17/8.02 ! [X: set_o] :
% 14.17/8.02 ( ( inf_inf_set_o @ X @ bot_bot_set_o )
% 14.17/8.02 = bot_bot_set_o ) ).
% 14.17/8.02
% 14.17/8.02 thf(fact_1576_boolean__algebra_Oconj__zero__right,axiom,
% 14.17/8.02 ! [X: set_nat] :
% 14.17/8.02 ( ( inf_inf_set_nat @ X @ bot_bot_set_nat )
% 14.17/8.02 = bot_bot_set_nat ) ).
% 14.17/8.02
% 14.17/8.02 thf(fact_1577_boolean__algebra_Oconj__zero__right,axiom,
% 14.17/8.02 ! [X: set_int] :
% 14.17/8.02 ( ( inf_inf_set_int @ X @ bot_bot_set_int )
% 14.17/8.02 = bot_bot_set_int ) ).
% 14.17/8.02
% 14.17/8.02 thf(fact_1578_boolean__algebra_Oconj__zero__right,axiom,
% 14.17/8.02 ! [X: set_Pr4532377907799695533_nat_o] :
% 14.17/8.02 ( ( inf_in1906310914598751387_nat_o @ X @ bot_bo7824918357723582233_nat_o )
% 14.17/8.02 = bot_bo7824918357723582233_nat_o ) ).
% 14.17/8.02
% 14.17/8.02 thf(fact_1579_boolean__algebra_Oconj__zero__left,axiom,
% 14.17/8.02 ! [X: assn] :
% 14.17/8.02 ( ( inf_inf_assn @ bot_bot_assn @ X )
% 14.17/8.02 = bot_bot_assn ) ).
% 14.17/8.02
% 14.17/8.02 thf(fact_1580_boolean__algebra_Oconj__zero__left,axiom,
% 14.17/8.02 ! [X: product_unit] :
% 14.17/8.02 ( ( inf_inf_Product_unit @ bot_bot_Product_unit @ X )
% 14.17/8.02 = bot_bot_Product_unit ) ).
% 14.17/8.02
% 14.17/8.02 thf(fact_1581_boolean__algebra_Oconj__zero__left,axiom,
% 14.17/8.02 ! [X: set_Product_unit] :
% 14.17/8.02 ( ( inf_in4660618365625256667t_unit @ bot_bo3957492148770167129t_unit @ X )
% 14.17/8.02 = bot_bo3957492148770167129t_unit ) ).
% 14.17/8.02
% 14.17/8.02 thf(fact_1582_boolean__algebra_Oconj__zero__left,axiom,
% 14.17/8.02 ! [X: set_o] :
% 14.17/8.02 ( ( inf_inf_set_o @ bot_bot_set_o @ X )
% 14.17/8.02 = bot_bot_set_o ) ).
% 14.17/8.02
% 14.17/8.02 thf(fact_1583_boolean__algebra_Oconj__zero__left,axiom,
% 14.17/8.02 ! [X: set_nat] :
% 14.17/8.02 ( ( inf_inf_set_nat @ bot_bot_set_nat @ X )
% 14.17/8.02 = bot_bot_set_nat ) ).
% 14.17/8.02
% 14.17/8.02 thf(fact_1584_boolean__algebra_Oconj__zero__left,axiom,
% 14.17/8.02 ! [X: set_int] :
% 14.17/8.02 ( ( inf_inf_set_int @ bot_bot_set_int @ X )
% 14.17/8.02 = bot_bot_set_int ) ).
% 14.17/8.02
% 14.17/8.02 thf(fact_1585_boolean__algebra_Oconj__zero__left,axiom,
% 14.17/8.02 ! [X: set_Pr4532377907799695533_nat_o] :
% 14.17/8.02 ( ( inf_in1906310914598751387_nat_o @ bot_bo7824918357723582233_nat_o @ X )
% 14.17/8.02 = bot_bo7824918357723582233_nat_o ) ).
% 14.17/8.02
% 14.17/8.02 thf(fact_1586_inf__bot__right,axiom,
% 14.17/8.02 ! [X: assn] :
% 14.17/8.02 ( ( inf_inf_assn @ X @ bot_bot_assn )
% 14.17/8.02 = bot_bot_assn ) ).
% 14.17/8.02
% 14.17/8.02 thf(fact_1587_inf__bot__right,axiom,
% 14.17/8.02 ! [X: product_unit] :
% 14.17/8.02 ( ( inf_inf_Product_unit @ X @ bot_bot_Product_unit )
% 14.17/8.02 = bot_bot_Product_unit ) ).
% 14.17/8.02
% 14.17/8.02 thf(fact_1588_inf__bot__right,axiom,
% 14.17/8.02 ! [X: set_Product_unit] :
% 14.17/8.02 ( ( inf_in4660618365625256667t_unit @ X @ bot_bo3957492148770167129t_unit )
% 14.17/8.02 = bot_bo3957492148770167129t_unit ) ).
% 14.17/8.02
% 14.17/8.02 thf(fact_1589_inf__bot__right,axiom,
% 14.17/8.02 ! [X: set_o] :
% 14.17/8.02 ( ( inf_inf_set_o @ X @ bot_bot_set_o )
% 14.17/8.02 = bot_bot_set_o ) ).
% 14.17/8.02
% 14.17/8.02 thf(fact_1590_inf__bot__right,axiom,
% 14.17/8.02 ! [X: set_nat] :
% 14.17/8.02 ( ( inf_inf_set_nat @ X @ bot_bot_set_nat )
% 14.17/8.02 = bot_bot_set_nat ) ).
% 14.17/8.02
% 14.17/8.02 thf(fact_1591_inf__bot__right,axiom,
% 14.17/8.02 ! [X: set_int] :
% 14.17/8.02 ( ( inf_inf_set_int @ X @ bot_bot_set_int )
% 14.17/8.02 = bot_bot_set_int ) ).
% 14.17/8.02
% 14.17/8.02 thf(fact_1592_inf__bot__right,axiom,
% 14.17/8.02 ! [X: set_Pr4532377907799695533_nat_o] :
% 14.17/8.02 ( ( inf_in1906310914598751387_nat_o @ X @ bot_bo7824918357723582233_nat_o )
% 14.17/8.02 = bot_bo7824918357723582233_nat_o ) ).
% 14.17/8.02
% 14.17/8.02 thf(fact_1593_merge__pure__or,axiom,
% 14.17/8.02 ! [A2: $o,B: $o] :
% 14.17/8.02 ( ( sup_sup_assn @ ( pure_assn @ A2 ) @ ( pure_assn @ B ) )
% 14.17/8.02 = ( pure_assn
% 14.17/8.02 @ ( A2
% 14.17/8.02 | B ) ) ) ).
% 14.17/8.02
% 14.17/8.02 thf(fact_1594_merge__pure__and,axiom,
% 14.17/8.02 ! [A2: $o,B: $o] :
% 14.17/8.02 ( ( inf_inf_assn @ ( pure_assn @ A2 ) @ ( pure_assn @ B ) )
% 14.17/8.02 = ( pure_assn
% 14.17/8.02 @ ( A2
% 14.17/8.02 & B ) ) ) ).
% 14.17/8.02
% 14.17/8.02 thf(fact_1595_inf__bot__left,axiom,
% 14.17/8.02 ! [X: assn] :
% 14.17/8.02 ( ( inf_inf_assn @ bot_bot_assn @ X )
% 14.17/8.02 = bot_bot_assn ) ).
% 14.17/8.02
% 14.17/8.02 thf(fact_1596_inf__bot__left,axiom,
% 14.17/8.02 ! [X: product_unit] :
% 14.17/8.02 ( ( inf_inf_Product_unit @ bot_bot_Product_unit @ X )
% 14.17/8.02 = bot_bot_Product_unit ) ).
% 14.17/8.02
% 14.17/8.02 thf(fact_1597_inf__bot__left,axiom,
% 14.17/8.02 ! [X: set_Product_unit] :
% 14.17/8.02 ( ( inf_in4660618365625256667t_unit @ bot_bo3957492148770167129t_unit @ X )
% 14.17/8.02 = bot_bo3957492148770167129t_unit ) ).
% 14.17/8.02
% 14.17/8.02 thf(fact_1598_inf__bot__left,axiom,
% 14.17/8.02 ! [X: set_o] :
% 14.17/8.02 ( ( inf_inf_set_o @ bot_bot_set_o @ X )
% 14.17/8.02 = bot_bot_set_o ) ).
% 14.17/8.02
% 14.17/8.02 thf(fact_1599_inf__bot__left,axiom,
% 14.17/8.02 ! [X: set_nat] :
% 14.17/8.02 ( ( inf_inf_set_nat @ bot_bot_set_nat @ X )
% 14.17/8.02 = bot_bot_set_nat ) ).
% 14.17/8.02
% 14.17/8.02 thf(fact_1600_inf__bot__left,axiom,
% 14.17/8.02 ! [X: set_int] :
% 14.17/8.02 ( ( inf_inf_set_int @ bot_bot_set_int @ X )
% 14.17/8.02 = bot_bot_set_int ) ).
% 14.17/8.02
% 14.17/8.02 thf(fact_1601_inf__bot__left,axiom,
% 14.17/8.02 ! [X: set_Pr4532377907799695533_nat_o] :
% 14.17/8.02 ( ( inf_in1906310914598751387_nat_o @ bot_bo7824918357723582233_nat_o @ X )
% 14.17/8.02 = bot_bo7824918357723582233_nat_o ) ).
% 14.17/8.02
% 14.17/8.02 thf(fact_1602_boolean__algebra_Odisj__zero__right,axiom,
% 14.17/8.02 ! [X: assn] :
% 14.17/8.02 ( ( sup_sup_assn @ X @ bot_bot_assn )
% 14.17/8.02 = X ) ).
% 14.17/8.02
% 14.17/8.02 thf(fact_1603_boolean__algebra_Odisj__zero__right,axiom,
% 14.17/8.02 ! [X: product_unit] :
% 14.17/8.02 ( ( sup_sup_Product_unit @ X @ bot_bot_Product_unit )
% 14.17/8.02 = X ) ).
% 14.17/8.02
% 14.17/8.02 thf(fact_1604_boolean__algebra_Odisj__zero__right,axiom,
% 14.17/8.02 ! [X: set_Product_unit] :
% 14.17/8.02 ( ( sup_su793286257634532545t_unit @ X @ bot_bo3957492148770167129t_unit )
% 14.17/8.02 = X ) ).
% 14.17/8.02
% 14.17/8.02 thf(fact_1605_boolean__algebra_Odisj__zero__right,axiom,
% 14.17/8.02 ! [X: set_o] :
% 14.17/8.02 ( ( sup_sup_set_o @ X @ bot_bot_set_o )
% 14.17/8.02 = X ) ).
% 14.17/8.02
% 14.17/8.02 thf(fact_1606_boolean__algebra_Odisj__zero__right,axiom,
% 14.17/8.02 ! [X: set_nat] :
% 14.17/8.02 ( ( sup_sup_set_nat @ X @ bot_bot_set_nat )
% 14.17/8.02 = X ) ).
% 14.17/8.02
% 14.17/8.02 thf(fact_1607_boolean__algebra_Odisj__zero__right,axiom,
% 14.17/8.02 ! [X: set_int] :
% 14.17/8.02 ( ( sup_sup_set_int @ X @ bot_bot_set_int )
% 14.17/8.02 = X ) ).
% 14.17/8.02
% 14.17/8.02 thf(fact_1608_boolean__algebra_Odisj__zero__right,axiom,
% 14.17/8.02 ! [X: set_Pr4532377907799695533_nat_o] :
% 14.17/8.02 ( ( sup_su5209123915105501825_nat_o @ X @ bot_bo7824918357723582233_nat_o )
% 14.17/8.02 = X ) ).
% 14.17/8.02
% 14.17/8.02 thf(fact_1609_prod__set__defs_I2_J,axiom,
% 14.17/8.02 ( basic_2698842327037663054et_nat
% 14.17/8.02 = ( ^ [P5: produc3658429121746597890et_nat] : ( insert_set_nat @ ( produc8586169260539613262et_nat @ P5 ) @ bot_bot_set_set_nat ) ) ) ).
% 14.17/8.02
% 14.17/8.02 thf(fact_1610_prod__set__defs_I2_J,axiom,
% 14.17/8.02 ( basic_snds_int_int
% 14.17/8.02 = ( ^ [P5: product_prod_int_int] : ( insert_int @ ( product_snd_int_int @ P5 ) @ bot_bot_set_int ) ) ) ).
% 14.17/8.02
% 14.17/8.02 thf(fact_1611_snds_Ocases,axiom,
% 14.17/8.02 ! [A2: set_nat,P3: produc3658429121746597890et_nat] :
% 14.17/8.02 ( ( member_set_nat @ A2 @ ( basic_2698842327037663054et_nat @ P3 ) )
% 14.17/8.02 => ( A2
% 14.17/8.02 = ( produc8586169260539613262et_nat @ P3 ) ) ) ).
% 14.17/8.02
% 14.17/8.02 thf(fact_1612_snds_Ocases,axiom,
% 14.17/8.02 ! [A2: int,P3: product_prod_int_int] :
% 14.17/8.02 ( ( member_int @ A2 @ ( basic_snds_int_int @ P3 ) )
% 14.17/8.02 => ( A2
% 14.17/8.02 = ( product_snd_int_int @ P3 ) ) ) ).
% 14.17/8.02
% 14.17/8.02 thf(fact_1613_snds_Osimps,axiom,
% 14.17/8.02 ! [A2: set_nat,P3: produc3658429121746597890et_nat] :
% 14.17/8.02 ( ( member_set_nat @ A2 @ ( basic_2698842327037663054et_nat @ P3 ) )
% 14.17/8.02 = ( A2
% 14.17/8.02 = ( produc8586169260539613262et_nat @ P3 ) ) ) ).
% 14.17/8.02
% 14.17/8.02 thf(fact_1614_snds_Osimps,axiom,
% 14.17/8.02 ! [A2: int,P3: product_prod_int_int] :
% 14.17/8.02 ( ( member_int @ A2 @ ( basic_snds_int_int @ P3 ) )
% 14.17/8.02 = ( A2
% 14.17/8.02 = ( product_snd_int_int @ P3 ) ) ) ).
% 14.17/8.02
% 14.17/8.02 thf(fact_1615_snds_Ointros,axiom,
% 14.17/8.02 ! [P3: produc3658429121746597890et_nat] : ( member_set_nat @ ( produc8586169260539613262et_nat @ P3 ) @ ( basic_2698842327037663054et_nat @ P3 ) ) ).
% 14.17/8.02
% 14.17/8.02 thf(fact_1616_snds_Ointros,axiom,
% 14.17/8.02 ! [P3: product_prod_int_int] : ( member_int @ ( product_snd_int_int @ P3 ) @ ( basic_snds_int_int @ P3 ) ) ).
% 14.17/8.02
% 14.17/8.02 thf(fact_1617_wand__raw_Opelims_I1_J,axiom,
% 14.17/8.02 ! [X: produc3658429121746597890et_nat > $o,Xa: produc3658429121746597890et_nat > $o,Xb: produc3658429121746597890et_nat,Y: $o] :
% 14.17/8.02 ( ( ( wand_raw @ X @ Xa @ Xb )
% 14.17/8.02 = Y )
% 14.17/8.02 => ( ( accp_P1862375125659990705et_nat @ wand_raw_rel @ ( produc2245416461498447860et_nat @ X @ ( produc5001842942810119800et_nat @ Xa @ Xb ) ) )
% 14.17/8.02 => ~ ! [H: heap_e7401611519738050253t_unit,As2: set_nat] :
% 14.17/8.02 ( ( Xb
% 14.17/8.02 = ( produc7507926704131184380et_nat @ H @ As2 ) )
% 14.17/8.02 => ( ( Y
% 14.17/8.02 = ( ( in_range @ ( produc7507926704131184380et_nat @ H @ As2 ) )
% 14.17/8.02 & ! [H5: heap_e7401611519738050253t_unit,As6: set_nat] :
% 14.17/8.02 ( ( ( ( inf_inf_set_nat @ As2 @ As6 )
% 14.17/8.02 = bot_bot_set_nat )
% 14.17/8.02 & ( relH @ As2 @ H @ H5 )
% 14.17/8.02 & ( in_range @ ( produc7507926704131184380et_nat @ H5 @ As2 ) )
% 14.17/8.02 & ( X @ ( produc7507926704131184380et_nat @ H5 @ As6 ) ) )
% 14.17/8.02 => ( Xa @ ( produc7507926704131184380et_nat @ H5 @ ( sup_sup_set_nat @ As2 @ As6 ) ) ) ) ) )
% 14.17/8.02 => ~ ( accp_P1862375125659990705et_nat @ wand_raw_rel @ ( produc2245416461498447860et_nat @ X @ ( produc5001842942810119800et_nat @ Xa @ ( produc7507926704131184380et_nat @ H @ As2 ) ) ) ) ) ) ) ) ).
% 14.17/8.02
% 14.17/8.02 thf(fact_1618_wand__raw_Opelims_I2_J,axiom,
% 14.17/8.02 ! [X: produc3658429121746597890et_nat > $o,Xa: produc3658429121746597890et_nat > $o,Xb: produc3658429121746597890et_nat] :
% 14.17/8.02 ( ( wand_raw @ X @ Xa @ Xb )
% 14.17/8.02 => ( ( accp_P1862375125659990705et_nat @ wand_raw_rel @ ( produc2245416461498447860et_nat @ X @ ( produc5001842942810119800et_nat @ Xa @ Xb ) ) )
% 14.17/8.02 => ~ ! [H: heap_e7401611519738050253t_unit,As2: set_nat] :
% 14.17/8.02 ( ( Xb
% 14.17/8.02 = ( produc7507926704131184380et_nat @ H @ As2 ) )
% 14.17/8.02 => ( ( accp_P1862375125659990705et_nat @ wand_raw_rel @ ( produc2245416461498447860et_nat @ X @ ( produc5001842942810119800et_nat @ Xa @ ( produc7507926704131184380et_nat @ H @ As2 ) ) ) )
% 14.17/8.02 => ~ ( ( in_range @ ( produc7507926704131184380et_nat @ H @ As2 ) )
% 14.17/8.02 & ! [H7: heap_e7401611519738050253t_unit,As7: set_nat] :
% 14.17/8.02 ( ( ( ( inf_inf_set_nat @ As2 @ As7 )
% 14.17/8.02 = bot_bot_set_nat )
% 14.17/8.02 & ( relH @ As2 @ H @ H7 )
% 14.17/8.02 & ( in_range @ ( produc7507926704131184380et_nat @ H7 @ As2 ) )
% 14.17/8.02 & ( X @ ( produc7507926704131184380et_nat @ H7 @ As7 ) ) )
% 14.17/8.02 => ( Xa @ ( produc7507926704131184380et_nat @ H7 @ ( sup_sup_set_nat @ As2 @ As7 ) ) ) ) ) ) ) ) ) ).
% 14.17/8.02
% 14.17/8.02 thf(fact_1619_wand__raw_Opelims_I3_J,axiom,
% 14.17/8.02 ! [X: produc3658429121746597890et_nat > $o,Xa: produc3658429121746597890et_nat > $o,Xb: produc3658429121746597890et_nat] :
% 14.17/8.02 ( ~ ( wand_raw @ X @ Xa @ Xb )
% 14.17/8.02 => ( ( accp_P1862375125659990705et_nat @ wand_raw_rel @ ( produc2245416461498447860et_nat @ X @ ( produc5001842942810119800et_nat @ Xa @ Xb ) ) )
% 14.17/8.02 => ~ ! [H: heap_e7401611519738050253t_unit,As2: set_nat] :
% 14.17/8.02 ( ( Xb
% 14.17/8.02 = ( produc7507926704131184380et_nat @ H @ As2 ) )
% 14.17/8.02 => ( ( accp_P1862375125659990705et_nat @ wand_raw_rel @ ( produc2245416461498447860et_nat @ X @ ( produc5001842942810119800et_nat @ Xa @ ( produc7507926704131184380et_nat @ H @ As2 ) ) ) )
% 14.17/8.02 => ( ( in_range @ ( produc7507926704131184380et_nat @ H @ As2 ) )
% 14.17/8.02 & ! [H6: heap_e7401611519738050253t_unit,As4: set_nat] :
% 14.17/8.02 ( ( ( ( inf_inf_set_nat @ As2 @ As4 )
% 14.17/8.02 = bot_bot_set_nat )
% 14.17/8.02 & ( relH @ As2 @ H @ H6 )
% 14.17/8.02 & ( in_range @ ( produc7507926704131184380et_nat @ H6 @ As2 ) )
% 14.17/8.02 & ( X @ ( produc7507926704131184380et_nat @ H6 @ As4 ) ) )
% 14.17/8.02 => ( Xa @ ( produc7507926704131184380et_nat @ H6 @ ( sup_sup_set_nat @ As2 @ As4 ) ) ) ) ) ) ) ) ) ).
% 14.17/8.02
% 14.17/8.02 thf(fact_1620_le__numeral__extra_I4_J,axiom,
% 14.17/8.02 ord_le3102999989581377725nteger @ one_one_Code_integer @ one_one_Code_integer ).
% 14.17/8.02
% 14.17/8.02 thf(fact_1621_le__numeral__extra_I4_J,axiom,
% 14.17/8.02 ord_le1926595141338095240atural @ one_one_Code_natural @ one_one_Code_natural ).
% 14.17/8.02
% 14.17/8.02 thf(fact_1622_le__numeral__extra_I4_J,axiom,
% 14.17/8.02 ord_less_eq_rat @ one_one_rat @ one_one_rat ).
% 14.17/8.02
% 14.17/8.02 thf(fact_1623_le__numeral__extra_I4_J,axiom,
% 14.17/8.02 ord_less_eq_nat @ one_one_nat @ one_one_nat ).
% 14.17/8.02
% 14.17/8.02 thf(fact_1624_le__numeral__extra_I4_J,axiom,
% 14.17/8.02 ord_less_eq_int @ one_one_int @ one_one_int ).
% 14.17/8.02
% 14.17/8.02 thf(fact_1625_less__eq__assn__def,axiom,
% 14.17/8.02 ( ord_less_eq_assn
% 14.17/8.02 = ( ^ [A6: assn,B6: assn] :
% 14.17/8.02 ( A6
% 14.17/8.02 = ( inf_inf_assn @ A6 @ B6 ) ) ) ) ).
% 14.17/8.02
% 14.17/8.02 thf(fact_1626_times__assn__raw_Opelims_I3_J,axiom,
% 14.17/8.02 ! [X: produc3658429121746597890et_nat > $o,Xa: produc3658429121746597890et_nat > $o,Xb: produc3658429121746597890et_nat] :
% 14.17/8.02 ( ~ ( times_assn_raw @ X @ Xa @ Xb )
% 14.17/8.02 => ( ( accp_P1862375125659990705et_nat @ times_assn_raw_rel @ ( produc2245416461498447860et_nat @ X @ ( produc5001842942810119800et_nat @ Xa @ Xb ) ) )
% 14.17/8.02 => ~ ! [H: heap_e7401611519738050253t_unit,As2: set_nat] :
% 14.17/8.02 ( ( Xb
% 14.17/8.02 = ( produc7507926704131184380et_nat @ H @ As2 ) )
% 14.17/8.02 => ( ( accp_P1862375125659990705et_nat @ times_assn_raw_rel @ ( produc2245416461498447860et_nat @ X @ ( produc5001842942810119800et_nat @ Xa @ ( produc7507926704131184380et_nat @ H @ As2 ) ) ) )
% 14.17/8.02 => ? [As13: set_nat,As24: set_nat] :
% 14.17/8.02 ( ( As2
% 14.17/8.02 = ( sup_sup_set_nat @ As13 @ As24 ) )
% 14.17/8.02 & ( ( inf_inf_set_nat @ As13 @ As24 )
% 14.17/8.02 = bot_bot_set_nat )
% 14.17/8.02 & ( X @ ( produc7507926704131184380et_nat @ H @ As13 ) )
% 14.17/8.02 & ( Xa @ ( produc7507926704131184380et_nat @ H @ As24 ) ) ) ) ) ) ) ).
% 14.17/8.02
% 14.17/8.02 thf(fact_1627_times__assn__raw_Opelims_I2_J,axiom,
% 14.17/8.02 ! [X: produc3658429121746597890et_nat > $o,Xa: produc3658429121746597890et_nat > $o,Xb: produc3658429121746597890et_nat] :
% 14.17/8.02 ( ( times_assn_raw @ X @ Xa @ Xb )
% 14.17/8.02 => ( ( accp_P1862375125659990705et_nat @ times_assn_raw_rel @ ( produc2245416461498447860et_nat @ X @ ( produc5001842942810119800et_nat @ Xa @ Xb ) ) )
% 14.17/8.02 => ~ ! [H: heap_e7401611519738050253t_unit,As2: set_nat] :
% 14.17/8.02 ( ( Xb
% 14.17/8.02 = ( produc7507926704131184380et_nat @ H @ As2 ) )
% 14.17/8.02 => ( ( accp_P1862375125659990705et_nat @ times_assn_raw_rel @ ( produc2245416461498447860et_nat @ X @ ( produc5001842942810119800et_nat @ Xa @ ( produc7507926704131184380et_nat @ H @ As2 ) ) ) )
% 14.17/8.02 => ~ ? [As12: set_nat,As23: set_nat] :
% 14.17/8.02 ( ( As2
% 14.17/8.02 = ( sup_sup_set_nat @ As12 @ As23 ) )
% 14.17/8.02 & ( ( inf_inf_set_nat @ As12 @ As23 )
% 14.17/8.02 = bot_bot_set_nat )
% 14.17/8.02 & ( X @ ( produc7507926704131184380et_nat @ H @ As12 ) )
% 14.17/8.02 & ( Xa @ ( produc7507926704131184380et_nat @ H @ As23 ) ) ) ) ) ) ) ).
% 14.17/8.02
% 14.17/8.02 thf(fact_1628_times__assn__raw_Opelims_I1_J,axiom,
% 14.17/8.02 ! [X: produc3658429121746597890et_nat > $o,Xa: produc3658429121746597890et_nat > $o,Xb: produc3658429121746597890et_nat,Y: $o] :
% 14.17/8.02 ( ( ( times_assn_raw @ X @ Xa @ Xb )
% 14.17/8.02 = Y )
% 14.17/8.02 => ( ( accp_P1862375125659990705et_nat @ times_assn_raw_rel @ ( produc2245416461498447860et_nat @ X @ ( produc5001842942810119800et_nat @ Xa @ Xb ) ) )
% 14.17/8.02 => ~ ! [H: heap_e7401611519738050253t_unit,As2: set_nat] :
% 14.17/8.02 ( ( Xb
% 14.17/8.02 = ( produc7507926704131184380et_nat @ H @ As2 ) )
% 14.17/8.02 => ( ( Y
% 14.17/8.02 = ( ? [As1: set_nat,As22: set_nat] :
% 14.17/8.02 ( ( As2
% 14.17/8.02 = ( sup_sup_set_nat @ As1 @ As22 ) )
% 14.17/8.02 & ( ( inf_inf_set_nat @ As1 @ As22 )
% 14.17/8.02 = bot_bot_set_nat )
% 14.17/8.02 & ( X @ ( produc7507926704131184380et_nat @ H @ As1 ) )
% 14.17/8.02 & ( Xa @ ( produc7507926704131184380et_nat @ H @ As22 ) ) ) ) )
% 14.17/8.02 => ~ ( accp_P1862375125659990705et_nat @ times_assn_raw_rel @ ( produc2245416461498447860et_nat @ X @ ( produc5001842942810119800et_nat @ Xa @ ( produc7507926704131184380et_nat @ H @ As2 ) ) ) ) ) ) ) ) ).
% 14.17/8.02
% 14.17/8.02 thf(fact_1629_subset__Image1__Image1__iff,axiom,
% 14.17/8.02 ! [R2: set_Pr5094982260447487303t_unit,A2: product_unit,B: product_unit] :
% 14.17/8.02 ( ( order_7534304569044981540t_unit @ ( field_Product_unit @ R2 ) @ R2 )
% 14.17/8.02 => ( ( member_Product_unit @ A2 @ ( field_Product_unit @ R2 ) )
% 14.17/8.02 => ( ( member_Product_unit @ B @ ( field_Product_unit @ R2 ) )
% 14.17/8.02 => ( ( ord_le3507040750410214029t_unit @ ( image_8431784415008483091t_unit @ R2 @ ( insert_Product_unit @ A2 @ bot_bo3957492148770167129t_unit ) ) @ ( image_8431784415008483091t_unit @ R2 @ ( insert_Product_unit @ B @ bot_bo3957492148770167129t_unit ) ) )
% 14.17/8.02 = ( member7821138191923116944t_unit @ ( produc5692694829384537815t_unit @ B @ A2 ) @ R2 ) ) ) ) ) ).
% 14.17/8.02
% 14.17/8.02 thf(fact_1630_subset__Image1__Image1__iff,axiom,
% 14.17/8.02 ! [R2: set_Product_prod_o_o,A2: $o,B: $o] :
% 14.17/8.02 ( ( order_preorder_on_o @ ( field_o @ R2 ) @ R2 )
% 14.17/8.02 => ( ( member_o @ A2 @ ( field_o @ R2 ) )
% 14.17/8.02 => ( ( member_o @ B @ ( field_o @ R2 ) )
% 14.17/8.02 => ( ( ord_less_eq_set_o @ ( image_o_o @ R2 @ ( insert_o @ A2 @ bot_bot_set_o ) ) @ ( image_o_o @ R2 @ ( insert_o @ B @ bot_bot_set_o ) ) )
% 14.17/8.02 = ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ B @ A2 ) @ R2 ) ) ) ) ) ).
% 14.17/8.02
% 14.17/8.02 thf(fact_1631_subset__Image1__Image1__iff,axiom,
% 14.17/8.02 ! [R2: set_Pr958786334691620121nt_int,A2: int,B: int] :
% 14.17/8.02 ( ( order_4859164337913492053on_int @ ( field_int @ R2 ) @ R2 )
% 14.17/8.02 => ( ( member_int @ A2 @ ( field_int @ R2 ) )
% 14.17/8.02 => ( ( member_int @ B @ ( field_int @ R2 ) )
% 14.17/8.02 => ( ( ord_less_eq_set_int @ ( image_int_int @ R2 @ ( insert_int @ A2 @ bot_bot_set_int ) ) @ ( image_int_int @ R2 @ ( insert_int @ B @ bot_bot_set_int ) ) )
% 14.17/8.02 = ( member5262025264175285858nt_int @ ( product_Pair_int_int @ B @ A2 ) @ R2 ) ) ) ) ) ).
% 14.17/8.02
% 14.17/8.02 thf(fact_1632_subset__Image1__Image1__iff,axiom,
% 14.17/8.02 ! [R2: set_Pr2161125870931222855_nat_o,A2: produc3658429121746597890et_nat > $o,B: produc3658429121746597890et_nat > $o] :
% 14.17/8.02 ( ( order_174704566861754596_nat_o @ ( field_81821745747897882_nat_o @ R2 ) @ R2 )
% 14.17/8.02 => ( ( member6576561426505652726_nat_o @ A2 @ ( field_81821745747897882_nat_o @ R2 ) )
% 14.17/8.02 => ( ( member6576561426505652726_nat_o @ B @ ( field_81821745747897882_nat_o @ R2 ) )
% 14.17/8.02 => ( ( ord_le2965882846123202637_nat_o @ ( image_2232984160439939859_nat_o @ R2 @ ( insert5175938949040314269_nat_o @ A2 @ bot_bo7824918357723582233_nat_o ) ) @ ( image_2232984160439939859_nat_o @ R2 @ ( insert5175938949040314269_nat_o @ B @ bot_bo7824918357723582233_nat_o ) ) )
% 14.17/8.02 = ( member8781333585448626064_nat_o @ ( produc7368190662567826135_nat_o @ B @ A2 ) @ R2 ) ) ) ) ) ).
% 14.17/8.02
% 14.17/8.02 thf(fact_1633_subset__Image1__Image1__iff,axiom,
% 14.17/8.02 ! [R2: set_Pr1261947904930325089at_nat,A2: nat,B: nat] :
% 14.17/8.02 ( ( order_4861654808422542329on_nat @ ( field_nat @ R2 ) @ R2 )
% 14.17/8.02 => ( ( member_nat @ A2 @ ( field_nat @ R2 ) )
% 14.17/8.02 => ( ( member_nat @ B @ ( field_nat @ R2 ) )
% 14.17/8.02 => ( ( ord_less_eq_set_nat @ ( image_nat_nat @ R2 @ ( insert_nat @ A2 @ bot_bot_set_nat ) ) @ ( image_nat_nat @ R2 @ ( insert_nat @ B @ bot_bot_set_nat ) ) )
% 14.17/8.02 = ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ B @ A2 ) @ R2 ) ) ) ) ) ).
% 14.17/8.02
% 14.17/8.02 thf(fact_1634_Field__insert,axiom,
% 14.17/8.02 ! [A2: product_unit,B: product_unit,R2: set_Pr5094982260447487303t_unit] :
% 14.17/8.02 ( ( field_Product_unit @ ( insert4616891024742550839t_unit @ ( produc5692694829384537815t_unit @ A2 @ B ) @ R2 ) )
% 14.17/8.02 = ( sup_su793286257634532545t_unit @ ( insert_Product_unit @ A2 @ ( insert_Product_unit @ B @ bot_bo3957492148770167129t_unit ) ) @ ( field_Product_unit @ R2 ) ) ) ).
% 14.17/8.02
% 14.17/8.02 thf(fact_1635_Field__insert,axiom,
% 14.17/8.02 ! [A2: $o,B: $o,R2: set_Product_prod_o_o] :
% 14.17/8.02 ( ( field_o @ ( insert6201435330877294327od_o_o @ ( product_Pair_o_o @ A2 @ B ) @ R2 ) )
% 14.17/8.02 = ( sup_sup_set_o @ ( insert_o @ A2 @ ( insert_o @ B @ bot_bot_set_o ) ) @ ( field_o @ R2 ) ) ) ).
% 14.17/8.02
% 14.17/8.02 thf(fact_1636_Field__insert,axiom,
% 14.17/8.02 ! [A2: nat,B: nat,R2: set_Pr1261947904930325089at_nat] :
% 14.17/8.02 ( ( field_nat @ ( insert8211810215607154385at_nat @ ( product_Pair_nat_nat @ A2 @ B ) @ R2 ) )
% 14.17/8.02 = ( sup_sup_set_nat @ ( insert_nat @ A2 @ ( insert_nat @ B @ bot_bot_set_nat ) ) @ ( field_nat @ R2 ) ) ) ).
% 14.17/8.02
% 14.17/8.02 thf(fact_1637_Field__insert,axiom,
% 14.17/8.02 ! [A2: int,B: int,R2: set_Pr958786334691620121nt_int] :
% 14.17/8.02 ( ( field_int @ ( insert5033312907999012233nt_int @ ( product_Pair_int_int @ A2 @ B ) @ R2 ) )
% 14.17/8.02 = ( sup_sup_set_int @ ( insert_int @ A2 @ ( insert_int @ B @ bot_bot_set_int ) ) @ ( field_int @ R2 ) ) ) ).
% 14.17/8.02
% 14.17/8.02 thf(fact_1638_Field__insert,axiom,
% 14.17/8.02 ! [A2: produc3658429121746597890et_nat > $o,B: produc3658429121746597890et_nat > $o,R2: set_Pr2161125870931222855_nat_o] :
% 14.17/8.02 ( ( field_81821745747897882_nat_o @ ( insert5309882203591314743_nat_o @ ( produc7368190662567826135_nat_o @ A2 @ B ) @ R2 ) )
% 14.17/8.02 = ( sup_su5209123915105501825_nat_o @ ( insert5175938949040314269_nat_o @ A2 @ ( insert5175938949040314269_nat_o @ B @ bot_bo7824918357723582233_nat_o ) ) @ ( field_81821745747897882_nat_o @ R2 ) ) ) ).
% 14.17/8.02
% 14.17/8.02 thf(fact_1639_dom__ran__disj__comp,axiom,
% 14.17/8.02 ! [R3: set_Pr5094982260447487303t_unit] :
% 14.17/8.02 ( ( ( inf_in4660618365625256667t_unit @ ( domain4292024440186670924t_unit @ R3 ) @ ( range_6617724647562260917t_unit @ R3 ) )
% 14.17/8.02 = bot_bo3957492148770167129t_unit )
% 14.17/8.02 => ( ( relcom848357998638694028t_unit @ R3 @ R3 )
% 14.17/8.02 = bot_bo1654022762305692595t_unit ) ) ).
% 14.17/8.02
% 14.17/8.02 thf(fact_1640_dom__ran__disj__comp,axiom,
% 14.17/8.02 ! [R3: set_Product_prod_o_o] :
% 14.17/8.02 ( ( ( inf_inf_set_o @ ( domain_o_o @ R3 ) @ ( range_o_o @ R3 ) )
% 14.17/8.02 = bot_bot_set_o )
% 14.17/8.02 => ( ( relcomp_o_o_o @ R3 @ R3 )
% 14.17/8.02 = bot_bo7073875226086086771od_o_o ) ) ).
% 14.17/8.02
% 14.17/8.02 thf(fact_1641_dom__ran__disj__comp,axiom,
% 14.17/8.02 ! [R3: set_Pr1261947904930325089at_nat] :
% 14.17/8.02 ( ( ( inf_inf_set_nat @ ( domain_nat_nat @ R3 ) @ ( range_nat_nat @ R3 ) )
% 14.17/8.02 = bot_bot_set_nat )
% 14.17/8.02 => ( ( relcomp_nat_nat_nat @ R3 @ R3 )
% 14.17/8.02 = bot_bo2099793752762293965at_nat ) ) ).
% 14.17/8.02
% 14.17/8.02 thf(fact_1642_dom__ran__disj__comp,axiom,
% 14.17/8.02 ! [R3: set_Pr958786334691620121nt_int] :
% 14.17/8.02 ( ( ( inf_inf_set_int @ ( domain_int_int @ R3 ) @ ( range_int_int @ R3 ) )
% 14.17/8.02 = bot_bot_set_int )
% 14.17/8.02 => ( ( relcomp_int_int_int @ R3 @ R3 )
% 14.17/8.02 = bot_bo1796632182523588997nt_int ) ) ).
% 14.17/8.02
% 14.17/8.02 thf(fact_1643_dom__ran__disj__comp,axiom,
% 14.17/8.02 ! [R3: set_Pr2161125870931222855_nat_o] :
% 14.17/8.02 ( ( ( inf_in1906310914598751387_nat_o @ ( domain4881241903163524940_nat_o @ R3 ) @ ( range_5289318983786108341_nat_o @ R3 ) )
% 14.17/8.02 = bot_bo7824918357723582233_nat_o )
% 14.17/8.02 => ( ( relcom6251035627269349452_nat_o @ R3 @ R3 )
% 14.17/8.02 = bot_bo3657187498359500723_nat_o ) ) ).
% 14.17/8.02
% 14.17/8.02 thf(fact_1644_proj__iff,axiom,
% 14.17/8.02 ! [A: set_Product_unit,R2: set_Pr5094982260447487303t_unit,X: product_unit,Y: product_unit] :
% 14.17/8.02 ( ( equiv_4189356295871601603t_unit @ A @ R2 )
% 14.17/8.02 => ( ( ord_le3507040750410214029t_unit @ ( insert_Product_unit @ X @ ( insert_Product_unit @ Y @ bot_bo3957492148770167129t_unit ) ) @ A )
% 14.17/8.02 => ( ( ( equiv_5763582929720877514t_unit @ R2 @ X )
% 14.17/8.02 = ( equiv_5763582929720877514t_unit @ R2 @ Y ) )
% 14.17/8.02 = ( member7821138191923116944t_unit @ ( produc5692694829384537815t_unit @ X @ Y ) @ R2 ) ) ) ) ).
% 14.17/8.02
% 14.17/8.02 thf(fact_1645_proj__iff,axiom,
% 14.17/8.02 ! [A: set_o,R2: set_Product_prod_o_o,X: $o,Y: $o] :
% 14.17/8.02 ( ( equiv_equiv_o @ A @ R2 )
% 14.17/8.02 => ( ( ord_less_eq_set_o @ ( insert_o @ X @ ( insert_o @ Y @ bot_bot_set_o ) ) @ A )
% 14.17/8.02 => ( ( ( equiv_proj_o_o @ R2 @ X )
% 14.17/8.02 = ( equiv_proj_o_o @ R2 @ Y ) )
% 14.17/8.02 = ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ X @ Y ) @ R2 ) ) ) ) ).
% 14.17/8.02
% 14.17/8.02 thf(fact_1646_proj__iff,axiom,
% 14.17/8.02 ! [A: set_int,R2: set_Pr958786334691620121nt_int,X: int,Y: int] :
% 14.17/8.02 ( ( equiv_equiv_int @ A @ R2 )
% 14.17/8.02 => ( ( ord_less_eq_set_int @ ( insert_int @ X @ ( insert_int @ Y @ bot_bot_set_int ) ) @ A )
% 14.17/8.02 => ( ( ( equiv_proj_int_int @ R2 @ X )
% 14.17/8.02 = ( equiv_proj_int_int @ R2 @ Y ) )
% 14.17/8.02 = ( member5262025264175285858nt_int @ ( product_Pair_int_int @ X @ Y ) @ R2 ) ) ) ) ).
% 14.17/8.02
% 14.17/8.02 thf(fact_1647_proj__iff,axiom,
% 14.17/8.02 ! [A: set_Pr4532377907799695533_nat_o,R2: set_Pr2161125870931222855_nat_o,X: produc3658429121746597890et_nat > $o,Y: produc3658429121746597890et_nat > $o] :
% 14.17/8.02 ( ( equiv_6726761681920839555_nat_o @ A @ R2 )
% 14.17/8.02 => ( ( ord_le2965882846123202637_nat_o @ ( insert5175938949040314269_nat_o @ X @ ( insert5175938949040314269_nat_o @ Y @ bot_bo7824918357723582233_nat_o ) ) @ A )
% 14.17/8.02 => ( ( ( equiv_3235788788526563786_nat_o @ R2 @ X )
% 14.17/8.02 = ( equiv_3235788788526563786_nat_o @ R2 @ Y ) )
% 14.17/8.02 = ( member8781333585448626064_nat_o @ ( produc7368190662567826135_nat_o @ X @ Y ) @ R2 ) ) ) ) ).
% 14.17/8.02
% 14.17/8.02 thf(fact_1648_proj__iff,axiom,
% 14.17/8.02 ! [A: set_nat,R2: set_Pr1261947904930325089at_nat,X: nat,Y: nat] :
% 14.17/8.02 ( ( equiv_equiv_nat @ A @ R2 )
% 14.17/8.02 => ( ( ord_less_eq_set_nat @ ( insert_nat @ X @ ( insert_nat @ Y @ bot_bot_set_nat ) ) @ A )
% 14.17/8.02 => ( ( ( equiv_proj_nat_nat @ R2 @ X )
% 14.17/8.02 = ( equiv_proj_nat_nat @ R2 @ Y ) )
% 14.17/8.02 = ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ Y ) @ R2 ) ) ) ) ).
% 14.17/8.02
% 14.17/8.02 thf(fact_1649_Refl__antisym__eq__Image1__Image1__iff,axiom,
% 14.17/8.02 ! [R2: set_Pr5094982260447487303t_unit,A2: product_unit,B: product_unit] :
% 14.17/8.02 ( ( refl_on_Product_unit @ ( field_Product_unit @ R2 ) @ R2 )
% 14.17/8.02 => ( ( antisym_Product_unit @ R2 )
% 14.17/8.02 => ( ( member_Product_unit @ A2 @ ( field_Product_unit @ R2 ) )
% 14.17/8.02 => ( ( member_Product_unit @ B @ ( field_Product_unit @ R2 ) )
% 14.17/8.02 => ( ( ( image_8431784415008483091t_unit @ R2 @ ( insert_Product_unit @ A2 @ bot_bo3957492148770167129t_unit ) )
% 14.17/8.02 = ( image_8431784415008483091t_unit @ R2 @ ( insert_Product_unit @ B @ bot_bo3957492148770167129t_unit ) ) )
% 14.17/8.02 = ( A2 = B ) ) ) ) ) ) ).
% 14.17/8.02
% 14.17/8.02 thf(fact_1650_Refl__antisym__eq__Image1__Image1__iff,axiom,
% 14.17/8.02 ! [R2: set_Product_prod_o_o,A2: $o,B: $o] :
% 14.17/8.02 ( ( refl_on_o @ ( field_o @ R2 ) @ R2 )
% 14.17/8.02 => ( ( antisym_o @ R2 )
% 14.17/8.02 => ( ( member_o @ A2 @ ( field_o @ R2 ) )
% 14.17/8.02 => ( ( member_o @ B @ ( field_o @ R2 ) )
% 14.17/8.02 => ( ( ( image_o_o @ R2 @ ( insert_o @ A2 @ bot_bot_set_o ) )
% 14.17/8.02 = ( image_o_o @ R2 @ ( insert_o @ B @ bot_bot_set_o ) ) )
% 14.17/8.02 = ( A2 = B ) ) ) ) ) ) ).
% 14.17/8.02
% 14.17/8.02 thf(fact_1651_Refl__antisym__eq__Image1__Image1__iff,axiom,
% 14.17/8.02 ! [R2: set_Pr1261947904930325089at_nat,A2: nat,B: nat] :
% 14.17/8.02 ( ( refl_on_nat @ ( field_nat @ R2 ) @ R2 )
% 14.17/8.02 => ( ( antisym_nat @ R2 )
% 14.17/8.02 => ( ( member_nat @ A2 @ ( field_nat @ R2 ) )
% 14.17/8.02 => ( ( member_nat @ B @ ( field_nat @ R2 ) )
% 14.17/8.02 => ( ( ( image_nat_nat @ R2 @ ( insert_nat @ A2 @ bot_bot_set_nat ) )
% 14.17/8.02 = ( image_nat_nat @ R2 @ ( insert_nat @ B @ bot_bot_set_nat ) ) )
% 14.17/8.02 = ( A2 = B ) ) ) ) ) ) ).
% 14.17/8.02
% 14.17/8.02 thf(fact_1652_Refl__antisym__eq__Image1__Image1__iff,axiom,
% 14.17/8.02 ! [R2: set_Pr958786334691620121nt_int,A2: int,B: int] :
% 14.17/8.02 ( ( refl_on_int @ ( field_int @ R2 ) @ R2 )
% 14.17/8.02 => ( ( antisym_int @ R2 )
% 14.17/8.02 => ( ( member_int @ A2 @ ( field_int @ R2 ) )
% 14.17/8.02 => ( ( member_int @ B @ ( field_int @ R2 ) )
% 14.17/8.02 => ( ( ( image_int_int @ R2 @ ( insert_int @ A2 @ bot_bot_set_int ) )
% 14.17/8.02 = ( image_int_int @ R2 @ ( insert_int @ B @ bot_bot_set_int ) ) )
% 14.17/8.02 = ( A2 = B ) ) ) ) ) ) ).
% 14.17/8.02
% 14.17/8.02 thf(fact_1653_Refl__antisym__eq__Image1__Image1__iff,axiom,
% 14.17/8.02 ! [R2: set_Pr2161125870931222855_nat_o,A2: produc3658429121746597890et_nat > $o,B: produc3658429121746597890et_nat > $o] :
% 14.17/8.02 ( ( refl_o1075864025857821388_nat_o @ ( field_81821745747897882_nat_o @ R2 ) @ R2 )
% 14.17/8.02 => ( ( antisy9190549550370998639_nat_o @ R2 )
% 14.17/8.02 => ( ( member6576561426505652726_nat_o @ A2 @ ( field_81821745747897882_nat_o @ R2 ) )
% 14.17/8.02 => ( ( member6576561426505652726_nat_o @ B @ ( field_81821745747897882_nat_o @ R2 ) )
% 14.17/8.03 => ( ( ( image_2232984160439939859_nat_o @ R2 @ ( insert5175938949040314269_nat_o @ A2 @ bot_bo7824918357723582233_nat_o ) )
% 14.17/8.03 = ( image_2232984160439939859_nat_o @ R2 @ ( insert5175938949040314269_nat_o @ B @ bot_bo7824918357723582233_nat_o ) ) )
% 14.17/8.03 = ( A2 = B ) ) ) ) ) ) ).
% 14.17/8.03
% 14.17/8.03 thf(fact_1654_equiv__class__nondisjoint,axiom,
% 14.17/8.03 ! [A: set_Product_unit,R2: set_Pr5094982260447487303t_unit,X: product_unit,A2: product_unit,B: product_unit] :
% 14.17/8.03 ( ( equiv_4189356295871601603t_unit @ A @ R2 )
% 14.17/8.03 => ( ( member_Product_unit @ X @ ( inf_in4660618365625256667t_unit @ ( image_8431784415008483091t_unit @ R2 @ ( insert_Product_unit @ A2 @ bot_bo3957492148770167129t_unit ) ) @ ( image_8431784415008483091t_unit @ R2 @ ( insert_Product_unit @ B @ bot_bo3957492148770167129t_unit ) ) ) )
% 14.17/8.03 => ( member7821138191923116944t_unit @ ( produc5692694829384537815t_unit @ A2 @ B ) @ R2 ) ) ) ).
% 14.17/8.03
% 14.17/8.03 thf(fact_1655_equiv__class__nondisjoint,axiom,
% 14.17/8.03 ! [A: set_o,R2: set_Product_prod_o_o,X: $o,A2: $o,B: $o] :
% 14.17/8.03 ( ( equiv_equiv_o @ A @ R2 )
% 14.17/8.03 => ( ( member_o @ X @ ( inf_inf_set_o @ ( image_o_o @ R2 @ ( insert_o @ A2 @ bot_bot_set_o ) ) @ ( image_o_o @ R2 @ ( insert_o @ B @ bot_bot_set_o ) ) ) )
% 14.17/8.03 => ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ A2 @ B ) @ R2 ) ) ) ).
% 14.17/8.03
% 14.17/8.03 thf(fact_1656_equiv__class__nondisjoint,axiom,
% 14.17/8.03 ! [A: set_nat,R2: set_Pr1261947904930325089at_nat,X: nat,A2: nat,B: nat] :
% 14.17/8.03 ( ( equiv_equiv_nat @ A @ R2 )
% 14.17/8.03 => ( ( member_nat @ X @ ( inf_inf_set_nat @ ( image_nat_nat @ R2 @ ( insert_nat @ A2 @ bot_bot_set_nat ) ) @ ( image_nat_nat @ R2 @ ( insert_nat @ B @ bot_bot_set_nat ) ) ) )
% 14.17/8.03 => ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A2 @ B ) @ R2 ) ) ) ).
% 14.17/8.03
% 14.17/8.03 thf(fact_1657_equiv__class__nondisjoint,axiom,
% 14.17/8.03 ! [A: set_int,R2: set_Pr958786334691620121nt_int,X: int,A2: int,B: int] :
% 14.17/8.03 ( ( equiv_equiv_int @ A @ R2 )
% 14.17/8.03 => ( ( member_int @ X @ ( inf_inf_set_int @ ( image_int_int @ R2 @ ( insert_int @ A2 @ bot_bot_set_int ) ) @ ( image_int_int @ R2 @ ( insert_int @ B @ bot_bot_set_int ) ) ) )
% 14.17/8.03 => ( member5262025264175285858nt_int @ ( product_Pair_int_int @ A2 @ B ) @ R2 ) ) ) ).
% 14.17/8.03
% 14.17/8.03 thf(fact_1658_equiv__class__nondisjoint,axiom,
% 14.17/8.03 ! [A: set_Pr4532377907799695533_nat_o,R2: set_Pr2161125870931222855_nat_o,X: produc3658429121746597890et_nat > $o,A2: produc3658429121746597890et_nat > $o,B: produc3658429121746597890et_nat > $o] :
% 14.17/8.03 ( ( equiv_6726761681920839555_nat_o @ A @ R2 )
% 14.17/8.03 => ( ( member6576561426505652726_nat_o @ X @ ( inf_in1906310914598751387_nat_o @ ( image_2232984160439939859_nat_o @ R2 @ ( insert5175938949040314269_nat_o @ A2 @ bot_bo7824918357723582233_nat_o ) ) @ ( image_2232984160439939859_nat_o @ R2 @ ( insert5175938949040314269_nat_o @ B @ bot_bo7824918357723582233_nat_o ) ) ) )
% 14.17/8.03 => ( member8781333585448626064_nat_o @ ( produc7368190662567826135_nat_o @ A2 @ B ) @ R2 ) ) ) ).
% 14.17/8.03
% 14.17/8.03 thf(fact_1659_finite__reachable__advance,axiom,
% 14.17/8.03 ! [E: set_Pr4811707699266497531nteger,V0: code_integer,V: code_integer] :
% 14.17/8.03 ( ( finite6017078050557962740nteger @ ( image_4147621952751189391nteger @ ( transi3984306202085910426nteger @ E ) @ ( insert_Code_integer @ V0 @ bot_bo3990330152332043303nteger ) ) )
% 14.17/8.03 => ( ( member157494554546826820nteger @ ( produc1086072967326762835nteger @ V0 @ V ) @ ( transi3984306202085910426nteger @ E ) )
% 14.17/8.03 => ( finite6017078050557962740nteger @ ( image_4147621952751189391nteger @ ( transi3984306202085910426nteger @ E ) @ ( insert_Code_integer @ V @ bot_bo3990330152332043303nteger ) ) ) ) ) ).
% 14.17/8.03
% 14.17/8.03 thf(fact_1660_finite__reachable__advance,axiom,
% 14.17/8.03 ! [E: set_Pr5094982260447487303t_unit,V0: product_unit,V: product_unit] :
% 14.17/8.03 ( ( finite4290736615968046902t_unit @ ( image_8431784415008483091t_unit @ ( transi6481887417741755664t_unit @ E ) @ ( insert_Product_unit @ V0 @ bot_bo3957492148770167129t_unit ) ) )
% 14.17/8.03 => ( ( member7821138191923116944t_unit @ ( produc5692694829384537815t_unit @ V0 @ V ) @ ( transi6481887417741755664t_unit @ E ) )
% 14.17/8.03 => ( finite4290736615968046902t_unit @ ( image_8431784415008483091t_unit @ ( transi6481887417741755664t_unit @ E ) @ ( insert_Product_unit @ V @ bot_bo3957492148770167129t_unit ) ) ) ) ) ).
% 14.17/8.03
% 14.17/8.03 thf(fact_1661_finite__reachable__advance,axiom,
% 14.17/8.03 ! [E: set_Product_prod_o_o,V0: $o,V: $o] :
% 14.17/8.03 ( ( finite_finite_o @ ( image_o_o @ ( transitive_rtrancl_o @ E ) @ ( insert_o @ V0 @ bot_bot_set_o ) ) )
% 14.17/8.03 => ( ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ V0 @ V ) @ ( transitive_rtrancl_o @ E ) )
% 14.17/8.03 => ( finite_finite_o @ ( image_o_o @ ( transitive_rtrancl_o @ E ) @ ( insert_o @ V @ bot_bot_set_o ) ) ) ) ) ).
% 14.17/8.03
% 14.17/8.03 thf(fact_1662_finite__reachable__advance,axiom,
% 14.17/8.03 ! [E: set_Pr1261947904930325089at_nat,V0: nat,V: nat] :
% 14.17/8.03 ( ( finite_finite_nat @ ( image_nat_nat @ ( transi2905341329935302413cl_nat @ E ) @ ( insert_nat @ V0 @ bot_bot_set_nat ) ) )
% 14.17/8.03 => ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ V0 @ V ) @ ( transi2905341329935302413cl_nat @ E ) )
% 14.17/8.03 => ( finite_finite_nat @ ( image_nat_nat @ ( transi2905341329935302413cl_nat @ E ) @ ( insert_nat @ V @ bot_bot_set_nat ) ) ) ) ) ).
% 14.17/8.03
% 14.17/8.03 thf(fact_1663_finite__reachable__advance,axiom,
% 14.17/8.03 ! [E: set_Pr958786334691620121nt_int,V0: int,V: int] :
% 14.17/8.03 ( ( finite_finite_int @ ( image_int_int @ ( transi2902850859426252137cl_int @ E ) @ ( insert_int @ V0 @ bot_bot_set_int ) ) )
% 14.17/8.03 => ( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ V0 @ V ) @ ( transi2902850859426252137cl_int @ E ) )
% 14.17/8.03 => ( finite_finite_int @ ( image_int_int @ ( transi2902850859426252137cl_int @ E ) @ ( insert_int @ V @ bot_bot_set_int ) ) ) ) ) ).
% 14.17/8.03
% 14.17/8.03 thf(fact_1664_finite__reachable__advance,axiom,
% 14.17/8.03 ! [E: set_Pr2161125870931222855_nat_o,V0: produc3658429121746597890et_nat > $o,V: produc3658429121746597890et_nat > $o] :
% 14.17/8.03 ( ( finite3252695134891459830_nat_o @ ( image_2232984160439939859_nat_o @ ( transi712980445759334096_nat_o @ E ) @ ( insert5175938949040314269_nat_o @ V0 @ bot_bo7824918357723582233_nat_o ) ) )
% 14.17/8.03 => ( ( member8781333585448626064_nat_o @ ( produc7368190662567826135_nat_o @ V0 @ V ) @ ( transi712980445759334096_nat_o @ E ) )
% 14.17/8.03 => ( finite3252695134891459830_nat_o @ ( image_2232984160439939859_nat_o @ ( transi712980445759334096_nat_o @ E ) @ ( insert5175938949040314269_nat_o @ V @ bot_bo7824918357723582233_nat_o ) ) ) ) ) ).
% 14.17/8.03
% 14.17/8.03 thf(fact_1665_finite__Field__eq__finite,axiom,
% 14.17/8.03 ! [R3: set_Pr1261947904930325089at_nat] :
% 14.17/8.03 ( ( finite_finite_nat @ ( field_nat @ R3 ) )
% 14.17/8.03 = ( finite6177210948735845034at_nat @ R3 ) ) ).
% 14.17/8.03
% 14.17/8.03 thf(fact_1666_finite__Field__eq__finite,axiom,
% 14.17/8.03 ! [R3: set_Pr4811707699266497531nteger] :
% 14.17/8.03 ( ( finite6017078050557962740nteger @ ( field_Code_integer @ R3 ) )
% 14.17/8.03 = ( finite2154694888287295812nteger @ R3 ) ) ).
% 14.17/8.03
% 14.17/8.03 thf(fact_1667_Field__empty,axiom,
% 14.17/8.03 ( ( field_Product_unit @ bot_bo1654022762305692595t_unit )
% 14.17/8.03 = bot_bo3957492148770167129t_unit ) ).
% 14.17/8.03
% 14.17/8.03 thf(fact_1668_Field__empty,axiom,
% 14.17/8.03 ( ( field_o @ bot_bo7073875226086086771od_o_o )
% 14.17/8.03 = bot_bot_set_o ) ).
% 14.17/8.03
% 14.17/8.03 thf(fact_1669_Field__empty,axiom,
% 14.17/8.03 ( ( field_nat @ bot_bo2099793752762293965at_nat )
% 14.17/8.03 = bot_bot_set_nat ) ).
% 14.17/8.03
% 14.17/8.03 thf(fact_1670_Field__empty,axiom,
% 14.17/8.03 ( ( field_int @ bot_bo1796632182523588997nt_int )
% 14.17/8.03 = bot_bot_set_int ) ).
% 14.17/8.03
% 14.17/8.03 thf(fact_1671_Field__empty,axiom,
% 14.17/8.03 ( ( field_81821745747897882_nat_o @ bot_bo3657187498359500723_nat_o )
% 14.17/8.03 = bot_bo7824918357723582233_nat_o ) ).
% 14.17/8.03
% 14.17/8.03 thf(fact_1672_Field__Un,axiom,
% 14.17/8.03 ! [R2: set_Pr1261947904930325089at_nat,S2: set_Pr1261947904930325089at_nat] :
% 14.17/8.03 ( ( field_nat @ ( sup_su6327502436637775413at_nat @ R2 @ S2 ) )
% 14.17/8.03 = ( sup_sup_set_nat @ ( field_nat @ R2 ) @ ( field_nat @ S2 ) ) ) ).
% 14.17/8.03
% 14.17/8.03 thf(fact_1673_finite__Field,axiom,
% 14.17/8.03 ! [R2: set_Pr1261947904930325089at_nat] :
% 14.17/8.03 ( ( finite6177210948735845034at_nat @ R2 )
% 14.17/8.03 => ( finite_finite_nat @ ( field_nat @ R2 ) ) ) ).
% 14.17/8.03
% 14.17/8.03 thf(fact_1674_finite__Field,axiom,
% 14.17/8.03 ! [R2: set_Pr4811707699266497531nteger] :
% 14.17/8.03 ( ( finite2154694888287295812nteger @ R2 )
% 14.17/8.03 => ( finite6017078050557962740nteger @ ( field_Code_integer @ R2 ) ) ) ).
% 14.17/8.03
% 14.17/8.03 thf(fact_1675_relcompEpair,axiom,
% 14.17/8.03 ! [A2: heap_e7401611519738050253t_unit,C: set_nat,R2: set_Pr1776176089890104647t_unit,S2: set_Pr3948176798113811640et_nat] :
% 14.17/8.03 ( ( member6260224972018164377et_nat @ ( produc7507926704131184380et_nat @ A2 @ C ) @ ( relcom4930750947919833671et_nat @ R2 @ S2 ) )
% 14.17/8.03 => ~ ! [B2: heap_e7401611519738050253t_unit] :
% 14.17/8.03 ( ( member5046970461208427408t_unit @ ( produc8256076736985659095t_unit @ A2 @ B2 ) @ R2 )
% 14.17/8.03 => ~ ( member6260224972018164377et_nat @ ( produc7507926704131184380et_nat @ B2 @ C ) @ S2 ) ) ) ).
% 14.17/8.03
% 14.17/8.03 thf(fact_1676_relcompEpair,axiom,
% 14.17/8.03 ! [A2: heap_e7401611519738050253t_unit,C: set_nat,R2: set_Pr3948176798113811640et_nat,S2: set_Pr5488025237498180813et_nat] :
% 14.17/8.03 ( ( member6260224972018164377et_nat @ ( produc7507926704131184380et_nat @ A2 @ C ) @ ( relcom8881708762719574682et_nat @ R2 @ S2 ) )
% 14.17/8.03 => ~ ! [B2: set_nat] :
% 14.17/8.03 ( ( member6260224972018164377et_nat @ ( produc7507926704131184380et_nat @ A2 @ B2 ) @ R2 )
% 14.17/8.03 => ~ ( member8277197624267554838et_nat @ ( produc4532415448927165861et_nat @ B2 @ C ) @ S2 ) ) ) ).
% 14.17/8.03
% 14.17/8.03 thf(fact_1677_relcompEpair,axiom,
% 14.17/8.03 ! [A2: nat,C: nat,R2: set_Pr1261947904930325089at_nat,S2: set_Pr1261947904930325089at_nat] :
% 14.17/8.03 ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A2 @ C ) @ ( relcomp_nat_nat_nat @ R2 @ S2 ) )
% 14.17/8.03 => ~ ! [B2: nat] :
% 14.17/8.03 ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A2 @ B2 ) @ R2 )
% 14.17/8.03 => ~ ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ B2 @ C ) @ S2 ) ) ) ).
% 14.17/8.03
% 14.17/8.03 thf(fact_1678_relcompEpair,axiom,
% 14.17/8.03 ! [A2: int,C: int,R2: set_Pr958786334691620121nt_int,S2: set_Pr958786334691620121nt_int] :
% 14.17/8.03 ( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ A2 @ C ) @ ( relcomp_int_int_int @ R2 @ S2 ) )
% 14.17/8.03 => ~ ! [B2: int] :
% 14.17/8.03 ( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ A2 @ B2 ) @ R2 )
% 14.17/8.03 => ~ ( member5262025264175285858nt_int @ ( product_Pair_int_int @ B2 @ C ) @ S2 ) ) ) ).
% 14.17/8.03
% 14.17/8.03 thf(fact_1679_relcompEpair,axiom,
% 14.17/8.03 ! [A2: produc3658429121746597890et_nat > $o,C: produc3658429121746597890et_nat,R2: set_Pr2161125870931222855_nat_o,S2: set_Pr3286484037609594932et_nat] :
% 14.17/8.03 ( ( member1996754912294343701et_nat @ ( produc5001842942810119800et_nat @ A2 @ C ) @ ( relcom6619595054755144899et_nat @ R2 @ S2 ) )
% 14.17/8.03 => ~ ! [B2: produc3658429121746597890et_nat > $o] :
% 14.17/8.03 ( ( member8781333585448626064_nat_o @ ( produc7368190662567826135_nat_o @ A2 @ B2 ) @ R2 )
% 14.17/8.03 => ~ ( member1996754912294343701et_nat @ ( produc5001842942810119800et_nat @ B2 @ C ) @ S2 ) ) ) ).
% 14.17/8.03
% 14.17/8.03 thf(fact_1680_relcompEpair,axiom,
% 14.17/8.03 ! [A2: produc3658429121746597890et_nat > $o,C: produc3658429121746597890et_nat,R2: set_Pr3286484037609594932et_nat,S2: set_Pr719794911490849221et_nat] :
% 14.17/8.03 ( ( member1996754912294343701et_nat @ ( produc5001842942810119800et_nat @ A2 @ C ) @ ( relcom5975814107203797650et_nat @ R2 @ S2 ) )
% 14.17/8.03 => ~ ! [B2: produc3658429121746597890et_nat] :
% 14.17/8.03 ( ( member1996754912294343701et_nat @ ( produc5001842942810119800et_nat @ A2 @ B2 ) @ R2 )
% 14.17/8.03 => ~ ( member6099555550032318734et_nat @ ( produc8199053930788261021et_nat @ B2 @ C ) @ S2 ) ) ) ).
% 14.17/8.03
% 14.17/8.03 thf(fact_1681_relcompEpair,axiom,
% 14.17/8.03 ! [A2: produc3658429121746597890et_nat > $o,C: produc3658429121746597890et_nat,R2: set_Pr8536935166611901872et_nat,S2: set_Pr3721596426557623617et_nat] :
% 14.17/8.03 ( ( member1996754912294343701et_nat @ ( produc5001842942810119800et_nat @ A2 @ C ) @ ( relcom430014394250001422et_nat @ R2 @ S2 ) )
% 14.17/8.03 => ~ ! [B2: produc3925858234332021118et_nat] :
% 14.17/8.03 ( ( member6124377750444531601et_nat @ ( produc2245416461498447860et_nat @ A2 @ B2 ) @ R2 )
% 14.17/8.03 => ~ ( member6613709072892554et_nat @ ( produc5530789758353788441et_nat @ B2 @ C ) @ S2 ) ) ) ).
% 14.17/8.03
% 14.17/8.03 thf(fact_1682_relcompEpair,axiom,
% 14.17/8.03 ! [A2: produc3658429121746597890et_nat > $o,C: produc3925858234332021118et_nat,R2: set_Pr2161125870931222855_nat_o,S2: set_Pr8536935166611901872et_nat] :
% 14.17/8.03 ( ( member6124377750444531601et_nat @ ( produc2245416461498447860et_nat @ A2 @ C ) @ ( relcom2608444778321954367et_nat @ R2 @ S2 ) )
% 14.17/8.03 => ~ ! [B2: produc3658429121746597890et_nat > $o] :
% 14.17/8.03 ( ( member8781333585448626064_nat_o @ ( produc7368190662567826135_nat_o @ A2 @ B2 ) @ R2 )
% 14.17/8.03 => ~ ( member6124377750444531601et_nat @ ( produc2245416461498447860et_nat @ B2 @ C ) @ S2 ) ) ) ).
% 14.17/8.03
% 14.17/8.03 thf(fact_1683_relcompEpair,axiom,
% 14.17/8.03 ! [A2: produc3658429121746597890et_nat > $o,C: produc3925858234332021118et_nat,R2: set_Pr3286484037609594932et_nat,S2: set_Pr8807100098903490369et_nat] :
% 14.17/8.03 ( ( member6124377750444531601et_nat @ ( produc2245416461498447860et_nat @ A2 @ C ) @ ( relcom8031709186682293262et_nat @ R2 @ S2 ) )
% 14.17/8.03 => ~ ! [B2: produc3658429121746597890et_nat] :
% 14.17/8.03 ( ( member1996754912294343701et_nat @ ( produc5001842942810119800et_nat @ A2 @ B2 ) @ R2 )
% 14.17/8.03 => ~ ( member2737525317995836554et_nat @ ( produc3909112513931304473et_nat @ B2 @ C ) @ S2 ) ) ) ).
% 14.17/8.03
% 14.17/8.03 thf(fact_1684_relcompEpair,axiom,
% 14.17/8.03 ! [A2: produc3658429121746597890et_nat > $o,C: produc3925858234332021118et_nat,R2: set_Pr8536935166611901872et_nat,S2: set_Pr7928877670098842301et_nat] :
% 14.17/8.03 ( ( member6124377750444531601et_nat @ ( produc2245416461498447860et_nat @ A2 @ C ) @ ( relcom8303140135939483018et_nat @ R2 @ S2 ) )
% 14.17/8.03 => ~ ! [B2: produc3925858234332021118et_nat] :
% 14.17/8.03 ( ( member6124377750444531601et_nat @ ( produc2245416461498447860et_nat @ A2 @ B2 ) @ R2 )
% 14.17/8.03 => ~ ( member4763271486408492550et_nat @ ( produc8599840265553166229et_nat @ B2 @ C ) @ S2 ) ) ) ).
% 14.17/8.03
% 14.17/8.03 thf(fact_1685_relcompE,axiom,
% 14.17/8.03 ! [Xz: produc3658429121746597890et_nat,R2: set_Pr1776176089890104647t_unit,S2: set_Pr3948176798113811640et_nat] :
% 14.17/8.03 ( ( member6260224972018164377et_nat @ Xz @ ( relcom4930750947919833671et_nat @ R2 @ S2 ) )
% 14.17/8.03 => ~ ! [X3: heap_e7401611519738050253t_unit,Y2: heap_e7401611519738050253t_unit,Z3: set_nat] :
% 14.17/8.03 ( ( Xz
% 14.17/8.03 = ( produc7507926704131184380et_nat @ X3 @ Z3 ) )
% 14.17/8.03 => ( ( member5046970461208427408t_unit @ ( produc8256076736985659095t_unit @ X3 @ Y2 ) @ R2 )
% 14.17/8.03 => ~ ( member6260224972018164377et_nat @ ( produc7507926704131184380et_nat @ Y2 @ Z3 ) @ S2 ) ) ) ) ).
% 14.17/8.03
% 14.17/8.03 thf(fact_1686_relcompE,axiom,
% 14.17/8.03 ! [Xz: produc3658429121746597890et_nat,R2: set_Pr3948176798113811640et_nat,S2: set_Pr5488025237498180813et_nat] :
% 14.17/8.03 ( ( member6260224972018164377et_nat @ Xz @ ( relcom8881708762719574682et_nat @ R2 @ S2 ) )
% 14.17/8.03 => ~ ! [X3: heap_e7401611519738050253t_unit,Y2: set_nat,Z3: set_nat] :
% 14.17/8.03 ( ( Xz
% 14.17/8.03 = ( produc7507926704131184380et_nat @ X3 @ Z3 ) )
% 14.17/8.03 => ( ( member6260224972018164377et_nat @ ( produc7507926704131184380et_nat @ X3 @ Y2 ) @ R2 )
% 14.17/8.03 => ~ ( member8277197624267554838et_nat @ ( produc4532415448927165861et_nat @ Y2 @ Z3 ) @ S2 ) ) ) ) ).
% 14.17/8.03
% 14.17/8.03 thf(fact_1687_relcompE,axiom,
% 14.17/8.03 ! [Xz: product_prod_nat_nat,R2: set_Pr1261947904930325089at_nat,S2: set_Pr1261947904930325089at_nat] :
% 14.17/8.03 ( ( member8440522571783428010at_nat @ Xz @ ( relcomp_nat_nat_nat @ R2 @ S2 ) )
% 14.17/8.03 => ~ ! [X3: nat,Y2: nat,Z3: nat] :
% 14.17/8.03 ( ( Xz
% 14.17/8.03 = ( product_Pair_nat_nat @ X3 @ Z3 ) )
% 14.17/8.03 => ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X3 @ Y2 ) @ R2 )
% 14.17/8.03 => ~ ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ Y2 @ Z3 ) @ S2 ) ) ) ) ).
% 14.17/8.03
% 14.17/8.03 thf(fact_1688_relcompE,axiom,
% 14.17/8.03 ! [Xz: product_prod_int_int,R2: set_Pr958786334691620121nt_int,S2: set_Pr958786334691620121nt_int] :
% 14.17/8.03 ( ( member5262025264175285858nt_int @ Xz @ ( relcomp_int_int_int @ R2 @ S2 ) )
% 14.17/8.03 => ~ ! [X3: int,Y2: int,Z3: int] :
% 14.17/8.03 ( ( Xz
% 14.17/8.03 = ( product_Pair_int_int @ X3 @ Z3 ) )
% 14.17/8.03 => ( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ X3 @ Y2 ) @ R2 )
% 14.17/8.03 => ~ ( member5262025264175285858nt_int @ ( product_Pair_int_int @ Y2 @ Z3 ) @ S2 ) ) ) ) ).
% 14.17/8.03
% 14.17/8.03 thf(fact_1689_relcompE,axiom,
% 14.17/8.03 ! [Xz: produc3925858234332021118et_nat,R2: set_Pr2161125870931222855_nat_o,S2: set_Pr3286484037609594932et_nat] :
% 14.17/8.03 ( ( member1996754912294343701et_nat @ Xz @ ( relcom6619595054755144899et_nat @ R2 @ S2 ) )
% 14.17/8.03 => ~ ! [X3: produc3658429121746597890et_nat > $o,Y2: produc3658429121746597890et_nat > $o,Z3: produc3658429121746597890et_nat] :
% 14.17/8.03 ( ( Xz
% 14.17/8.03 = ( produc5001842942810119800et_nat @ X3 @ Z3 ) )
% 14.17/8.03 => ( ( member8781333585448626064_nat_o @ ( produc7368190662567826135_nat_o @ X3 @ Y2 ) @ R2 )
% 14.17/8.03 => ~ ( member1996754912294343701et_nat @ ( produc5001842942810119800et_nat @ Y2 @ Z3 ) @ S2 ) ) ) ) ).
% 14.17/8.03
% 14.17/8.03 thf(fact_1690_relcompE,axiom,
% 14.17/8.03 ! [Xz: produc3925858234332021118et_nat,R2: set_Pr3286484037609594932et_nat,S2: set_Pr719794911490849221et_nat] :
% 14.17/8.03 ( ( member1996754912294343701et_nat @ Xz @ ( relcom5975814107203797650et_nat @ R2 @ S2 ) )
% 14.17/8.03 => ~ ! [X3: produc3658429121746597890et_nat > $o,Y2: produc3658429121746597890et_nat,Z3: produc3658429121746597890et_nat] :
% 14.17/8.03 ( ( Xz
% 14.17/8.03 = ( produc5001842942810119800et_nat @ X3 @ Z3 ) )
% 14.17/8.03 => ( ( member1996754912294343701et_nat @ ( produc5001842942810119800et_nat @ X3 @ Y2 ) @ R2 )
% 14.17/8.03 => ~ ( member6099555550032318734et_nat @ ( produc8199053930788261021et_nat @ Y2 @ Z3 ) @ S2 ) ) ) ) ).
% 14.17/8.03
% 14.17/8.03 thf(fact_1691_relcompE,axiom,
% 14.17/8.03 ! [Xz: produc3925858234332021118et_nat,R2: set_Pr8536935166611901872et_nat,S2: set_Pr3721596426557623617et_nat] :
% 14.17/8.03 ( ( member1996754912294343701et_nat @ Xz @ ( relcom430014394250001422et_nat @ R2 @ S2 ) )
% 14.17/8.03 => ~ ! [X3: produc3658429121746597890et_nat > $o,Y2: produc3925858234332021118et_nat,Z3: produc3658429121746597890et_nat] :
% 14.17/8.03 ( ( Xz
% 14.17/8.03 = ( produc5001842942810119800et_nat @ X3 @ Z3 ) )
% 14.17/8.03 => ( ( member6124377750444531601et_nat @ ( produc2245416461498447860et_nat @ X3 @ Y2 ) @ R2 )
% 14.17/8.03 => ~ ( member6613709072892554et_nat @ ( produc5530789758353788441et_nat @ Y2 @ Z3 ) @ S2 ) ) ) ) ).
% 14.17/8.03
% 14.17/8.03 thf(fact_1692_relcompE,axiom,
% 14.17/8.03 ! [Xz: produc2732055786443039994et_nat,R2: set_Pr2161125870931222855_nat_o,S2: set_Pr8536935166611901872et_nat] :
% 14.17/8.03 ( ( member6124377750444531601et_nat @ Xz @ ( relcom2608444778321954367et_nat @ R2 @ S2 ) )
% 14.17/8.03 => ~ ! [X3: produc3658429121746597890et_nat > $o,Y2: produc3658429121746597890et_nat > $o,Z3: produc3925858234332021118et_nat] :
% 14.17/8.03 ( ( Xz
% 14.17/8.03 = ( produc2245416461498447860et_nat @ X3 @ Z3 ) )
% 14.17/8.03 => ( ( member8781333585448626064_nat_o @ ( produc7368190662567826135_nat_o @ X3 @ Y2 ) @ R2 )
% 14.17/8.03 => ~ ( member6124377750444531601et_nat @ ( produc2245416461498447860et_nat @ Y2 @ Z3 ) @ S2 ) ) ) ) ).
% 14.17/8.03
% 14.17/8.03 thf(fact_1693_relcompE,axiom,
% 14.17/8.03 ! [Xz: produc2732055786443039994et_nat,R2: set_Pr3286484037609594932et_nat,S2: set_Pr8807100098903490369et_nat] :
% 14.17/8.03 ( ( member6124377750444531601et_nat @ Xz @ ( relcom8031709186682293262et_nat @ R2 @ S2 ) )
% 14.17/8.03 => ~ ! [X3: produc3658429121746597890et_nat > $o,Y2: produc3658429121746597890et_nat,Z3: produc3925858234332021118et_nat] :
% 14.17/8.03 ( ( Xz
% 14.17/8.03 = ( produc2245416461498447860et_nat @ X3 @ Z3 ) )
% 14.17/8.03 => ( ( member1996754912294343701et_nat @ ( produc5001842942810119800et_nat @ X3 @ Y2 ) @ R2 )
% 14.17/8.03 => ~ ( member2737525317995836554et_nat @ ( produc3909112513931304473et_nat @ Y2 @ Z3 ) @ S2 ) ) ) ) ).
% 14.17/8.03
% 14.17/8.03 thf(fact_1694_relcompE,axiom,
% 14.17/8.03 ! [Xz: produc2732055786443039994et_nat,R2: set_Pr8536935166611901872et_nat,S2: set_Pr7928877670098842301et_nat] :
% 14.17/8.03 ( ( member6124377750444531601et_nat @ Xz @ ( relcom8303140135939483018et_nat @ R2 @ S2 ) )
% 14.17/8.03 => ~ ! [X3: produc3658429121746597890et_nat > $o,Y2: produc3925858234332021118et_nat,Z3: produc3925858234332021118et_nat] :
% 14.17/8.03 ( ( Xz
% 14.17/8.03 = ( produc2245416461498447860et_nat @ X3 @ Z3 ) )
% 14.17/8.03 => ( ( member6124377750444531601et_nat @ ( produc2245416461498447860et_nat @ X3 @ Y2 ) @ R2 )
% 14.17/8.03 => ~ ( member4763271486408492550et_nat @ ( produc8599840265553166229et_nat @ Y2 @ Z3 ) @ S2 ) ) ) ) ).
% 14.17/8.03
% 14.17/8.03 thf(fact_1695_relcomp_OrelcompI,axiom,
% 14.17/8.03 ! [A2: heap_e7401611519738050253t_unit,B: heap_e7401611519738050253t_unit,R2: set_Pr1776176089890104647t_unit,C: set_nat,S2: set_Pr3948176798113811640et_nat] :
% 14.17/8.03 ( ( member5046970461208427408t_unit @ ( produc8256076736985659095t_unit @ A2 @ B ) @ R2 )
% 14.17/8.03 => ( ( member6260224972018164377et_nat @ ( produc7507926704131184380et_nat @ B @ C ) @ S2 )
% 14.17/8.03 => ( member6260224972018164377et_nat @ ( produc7507926704131184380et_nat @ A2 @ C ) @ ( relcom4930750947919833671et_nat @ R2 @ S2 ) ) ) ) ).
% 14.17/8.03
% 14.17/8.03 thf(fact_1696_relcomp_OrelcompI,axiom,
% 14.17/8.03 ! [A2: produc3658429121746597890et_nat > $o,B: produc3658429121746597890et_nat > $o,R2: set_Pr2161125870931222855_nat_o,C: produc3658429121746597890et_nat,S2: set_Pr3286484037609594932et_nat] :
% 14.17/8.03 ( ( member8781333585448626064_nat_o @ ( produc7368190662567826135_nat_o @ A2 @ B ) @ R2 )
% 14.17/8.03 => ( ( member1996754912294343701et_nat @ ( produc5001842942810119800et_nat @ B @ C ) @ S2 )
% 14.17/8.03 => ( member1996754912294343701et_nat @ ( produc5001842942810119800et_nat @ A2 @ C ) @ ( relcom6619595054755144899et_nat @ R2 @ S2 ) ) ) ) ).
% 14.17/8.03
% 14.17/8.03 thf(fact_1697_relcomp_OrelcompI,axiom,
% 14.17/8.03 ! [A2: produc3658429121746597890et_nat > $o,B: produc3658429121746597890et_nat > $o,R2: set_Pr2161125870931222855_nat_o,C: produc3925858234332021118et_nat,S2: set_Pr8536935166611901872et_nat] :
% 14.17/8.03 ( ( member8781333585448626064_nat_o @ ( produc7368190662567826135_nat_o @ A2 @ B ) @ R2 )
% 14.17/8.03 => ( ( member6124377750444531601et_nat @ ( produc2245416461498447860et_nat @ B @ C ) @ S2 )
% 14.17/8.03 => ( member6124377750444531601et_nat @ ( produc2245416461498447860et_nat @ A2 @ C ) @ ( relcom2608444778321954367et_nat @ R2 @ S2 ) ) ) ) ).
% 14.17/8.03
% 14.17/8.03 thf(fact_1698_relcomp_OrelcompI,axiom,
% 14.17/8.03 ! [A2: heap_e7401611519738050253t_unit,B: set_nat,R2: set_Pr3948176798113811640et_nat,C: set_nat,S2: set_Pr5488025237498180813et_nat] :
% 14.17/8.03 ( ( member6260224972018164377et_nat @ ( produc7507926704131184380et_nat @ A2 @ B ) @ R2 )
% 14.17/8.03 => ( ( member8277197624267554838et_nat @ ( produc4532415448927165861et_nat @ B @ C ) @ S2 )
% 14.17/8.03 => ( member6260224972018164377et_nat @ ( produc7507926704131184380et_nat @ A2 @ C ) @ ( relcom8881708762719574682et_nat @ R2 @ S2 ) ) ) ) ).
% 14.17/8.03
% 14.17/8.03 thf(fact_1699_relcomp_OrelcompI,axiom,
% 14.17/8.03 ! [A2: nat,B: nat,R2: set_Pr1261947904930325089at_nat,C: nat,S2: set_Pr1261947904930325089at_nat] :
% 14.17/8.03 ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A2 @ B ) @ R2 )
% 14.17/8.03 => ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ B @ C ) @ S2 )
% 14.17/8.03 => ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A2 @ C ) @ ( relcomp_nat_nat_nat @ R2 @ S2 ) ) ) ) ).
% 14.17/8.03
% 14.17/8.03 thf(fact_1700_relcomp_OrelcompI,axiom,
% 14.17/8.03 ! [A2: int,B: int,R2: set_Pr958786334691620121nt_int,C: int,S2: set_Pr958786334691620121nt_int] :
% 14.17/8.03 ( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ A2 @ B ) @ R2 )
% 14.17/8.03 => ( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ B @ C ) @ S2 )
% 14.17/8.03 => ( member5262025264175285858nt_int @ ( product_Pair_int_int @ A2 @ C ) @ ( relcomp_int_int_int @ R2 @ S2 ) ) ) ) ).
% 14.17/8.03
% 14.17/8.03 thf(fact_1701_relcomp_OrelcompI,axiom,
% 14.17/8.03 ! [A2: produc3658429121746597890et_nat > $o,B: produc3658429121746597890et_nat,R2: set_Pr3286484037609594932et_nat,C: produc3658429121746597890et_nat,S2: set_Pr719794911490849221et_nat] :
% 14.17/8.03 ( ( member1996754912294343701et_nat @ ( produc5001842942810119800et_nat @ A2 @ B ) @ R2 )
% 14.17/8.03 => ( ( member6099555550032318734et_nat @ ( produc8199053930788261021et_nat @ B @ C ) @ S2 )
% 14.17/8.03 => ( member1996754912294343701et_nat @ ( produc5001842942810119800et_nat @ A2 @ C ) @ ( relcom5975814107203797650et_nat @ R2 @ S2 ) ) ) ) ).
% 14.17/8.03
% 14.17/8.03 thf(fact_1702_relcomp_OrelcompI,axiom,
% 14.17/8.03 ! [A2: produc3658429121746597890et_nat > $o,B: produc3658429121746597890et_nat,R2: set_Pr3286484037609594932et_nat,C: produc3925858234332021118et_nat,S2: set_Pr8807100098903490369et_nat] :
% 14.17/8.03 ( ( member1996754912294343701et_nat @ ( produc5001842942810119800et_nat @ A2 @ B ) @ R2 )
% 14.17/8.03 => ( ( member2737525317995836554et_nat @ ( produc3909112513931304473et_nat @ B @ C ) @ S2 )
% 14.17/8.03 => ( member6124377750444531601et_nat @ ( produc2245416461498447860et_nat @ A2 @ C ) @ ( relcom8031709186682293262et_nat @ R2 @ S2 ) ) ) ) ).
% 14.17/8.03
% 14.17/8.03 thf(fact_1703_relcomp_OrelcompI,axiom,
% 14.17/8.03 ! [A2: produc3658429121746597890et_nat > $o,B: produc3925858234332021118et_nat,R2: set_Pr8536935166611901872et_nat,C: produc3658429121746597890et_nat,S2: set_Pr3721596426557623617et_nat] :
% 14.17/8.03 ( ( member6124377750444531601et_nat @ ( produc2245416461498447860et_nat @ A2 @ B ) @ R2 )
% 14.17/8.03 => ( ( member6613709072892554et_nat @ ( produc5530789758353788441et_nat @ B @ C ) @ S2 )
% 14.17/8.03 => ( member1996754912294343701et_nat @ ( produc5001842942810119800et_nat @ A2 @ C ) @ ( relcom430014394250001422et_nat @ R2 @ S2 ) ) ) ) ).
% 14.17/8.03
% 14.17/8.03 thf(fact_1704_relcomp_OrelcompI,axiom,
% 14.17/8.03 ! [A2: produc3658429121746597890et_nat > $o,B: produc3925858234332021118et_nat,R2: set_Pr8536935166611901872et_nat,C: produc3925858234332021118et_nat,S2: set_Pr7928877670098842301et_nat] :
% 14.17/8.03 ( ( member6124377750444531601et_nat @ ( produc2245416461498447860et_nat @ A2 @ B ) @ R2 )
% 14.17/8.03 => ( ( member4763271486408492550et_nat @ ( produc8599840265553166229et_nat @ B @ C ) @ S2 )
% 14.17/8.03 => ( member6124377750444531601et_nat @ ( produc2245416461498447860et_nat @ A2 @ C ) @ ( relcom8303140135939483018et_nat @ R2 @ S2 ) ) ) ) ).
% 14.17/8.03
% 14.17/8.03 thf(fact_1705_relcomp_Osimps,axiom,
% 14.17/8.03 ! [A1: heap_e7401611519738050253t_unit,A22: set_nat,R2: set_Pr1776176089890104647t_unit,S2: set_Pr3948176798113811640et_nat] :
% 14.17/8.03 ( ( member6260224972018164377et_nat @ ( produc7507926704131184380et_nat @ A1 @ A22 ) @ ( relcom4930750947919833671et_nat @ R2 @ S2 ) )
% 14.17/8.03 = ( ? [A6: heap_e7401611519738050253t_unit,B6: heap_e7401611519738050253t_unit,C4: set_nat] :
% 14.17/8.03 ( ( A1 = A6 )
% 14.17/8.03 & ( A22 = C4 )
% 14.17/8.03 & ( member5046970461208427408t_unit @ ( produc8256076736985659095t_unit @ A6 @ B6 ) @ R2 )
% 14.17/8.03 & ( member6260224972018164377et_nat @ ( produc7507926704131184380et_nat @ B6 @ C4 ) @ S2 ) ) ) ) ).
% 14.17/8.03
% 14.17/8.03 thf(fact_1706_relcomp_Osimps,axiom,
% 14.17/8.03 ! [A1: heap_e7401611519738050253t_unit,A22: set_nat,R2: set_Pr3948176798113811640et_nat,S2: set_Pr5488025237498180813et_nat] :
% 14.17/8.03 ( ( member6260224972018164377et_nat @ ( produc7507926704131184380et_nat @ A1 @ A22 ) @ ( relcom8881708762719574682et_nat @ R2 @ S2 ) )
% 14.17/8.03 = ( ? [A6: heap_e7401611519738050253t_unit,B6: set_nat,C4: set_nat] :
% 14.17/8.03 ( ( A1 = A6 )
% 14.17/8.03 & ( A22 = C4 )
% 14.17/8.03 & ( member6260224972018164377et_nat @ ( produc7507926704131184380et_nat @ A6 @ B6 ) @ R2 )
% 14.17/8.03 & ( member8277197624267554838et_nat @ ( produc4532415448927165861et_nat @ B6 @ C4 ) @ S2 ) ) ) ) ).
% 14.17/8.03
% 14.17/8.03 thf(fact_1707_relcomp_Osimps,axiom,
% 14.17/8.03 ! [A1: nat,A22: nat,R2: set_Pr1261947904930325089at_nat,S2: set_Pr1261947904930325089at_nat] :
% 14.17/8.03 ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A1 @ A22 ) @ ( relcomp_nat_nat_nat @ R2 @ S2 ) )
% 14.17/8.03 = ( ? [A6: nat,B6: nat,C4: nat] :
% 14.17/8.03 ( ( A1 = A6 )
% 14.17/8.03 & ( A22 = C4 )
% 14.17/8.03 & ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A6 @ B6 ) @ R2 )
% 14.17/8.03 & ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ B6 @ C4 ) @ S2 ) ) ) ) ).
% 14.17/8.03
% 14.17/8.03 thf(fact_1708_relcomp_Osimps,axiom,
% 14.17/8.03 ! [A1: int,A22: int,R2: set_Pr958786334691620121nt_int,S2: set_Pr958786334691620121nt_int] :
% 14.17/8.03 ( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ A1 @ A22 ) @ ( relcomp_int_int_int @ R2 @ S2 ) )
% 14.17/8.03 = ( ? [A6: int,B6: int,C4: int] :
% 14.17/8.03 ( ( A1 = A6 )
% 14.17/8.03 & ( A22 = C4 )
% 14.17/8.03 & ( member5262025264175285858nt_int @ ( product_Pair_int_int @ A6 @ B6 ) @ R2 )
% 14.17/8.03 & ( member5262025264175285858nt_int @ ( product_Pair_int_int @ B6 @ C4 ) @ S2 ) ) ) ) ).
% 14.17/8.03
% 14.17/8.03 thf(fact_1709_relcomp_Osimps,axiom,
% 14.17/8.03 ! [A1: produc3658429121746597890et_nat > $o,A22: produc3658429121746597890et_nat,R2: set_Pr2161125870931222855_nat_o,S2: set_Pr3286484037609594932et_nat] :
% 14.17/8.03 ( ( member1996754912294343701et_nat @ ( produc5001842942810119800et_nat @ A1 @ A22 ) @ ( relcom6619595054755144899et_nat @ R2 @ S2 ) )
% 14.17/8.03 = ( ? [A6: produc3658429121746597890et_nat > $o,B6: produc3658429121746597890et_nat > $o,C4: produc3658429121746597890et_nat] :
% 14.17/8.03 ( ( A1 = A6 )
% 14.17/8.03 & ( A22 = C4 )
% 14.17/8.03 & ( member8781333585448626064_nat_o @ ( produc7368190662567826135_nat_o @ A6 @ B6 ) @ R2 )
% 14.17/8.03 & ( member1996754912294343701et_nat @ ( produc5001842942810119800et_nat @ B6 @ C4 ) @ S2 ) ) ) ) ).
% 14.17/8.03
% 14.17/8.03 thf(fact_1710_relcomp_Osimps,axiom,
% 14.17/8.03 ! [A1: produc3658429121746597890et_nat > $o,A22: produc3658429121746597890et_nat,R2: set_Pr3286484037609594932et_nat,S2: set_Pr719794911490849221et_nat] :
% 14.17/8.03 ( ( member1996754912294343701et_nat @ ( produc5001842942810119800et_nat @ A1 @ A22 ) @ ( relcom5975814107203797650et_nat @ R2 @ S2 ) )
% 14.17/8.03 = ( ? [A6: produc3658429121746597890et_nat > $o,B6: produc3658429121746597890et_nat,C4: produc3658429121746597890et_nat] :
% 14.17/8.03 ( ( A1 = A6 )
% 14.17/8.03 & ( A22 = C4 )
% 14.17/8.03 & ( member1996754912294343701et_nat @ ( produc5001842942810119800et_nat @ A6 @ B6 ) @ R2 )
% 14.17/8.03 & ( member6099555550032318734et_nat @ ( produc8199053930788261021et_nat @ B6 @ C4 ) @ S2 ) ) ) ) ).
% 14.17/8.03
% 14.17/8.03 thf(fact_1711_relcomp_Osimps,axiom,
% 14.17/8.03 ! [A1: produc3658429121746597890et_nat > $o,A22: produc3658429121746597890et_nat,R2: set_Pr8536935166611901872et_nat,S2: set_Pr3721596426557623617et_nat] :
% 14.17/8.03 ( ( member1996754912294343701et_nat @ ( produc5001842942810119800et_nat @ A1 @ A22 ) @ ( relcom430014394250001422et_nat @ R2 @ S2 ) )
% 14.17/8.03 = ( ? [A6: produc3658429121746597890et_nat > $o,B6: produc3925858234332021118et_nat,C4: produc3658429121746597890et_nat] :
% 14.17/8.03 ( ( A1 = A6 )
% 14.17/8.03 & ( A22 = C4 )
% 14.17/8.03 & ( member6124377750444531601et_nat @ ( produc2245416461498447860et_nat @ A6 @ B6 ) @ R2 )
% 14.17/8.03 & ( member6613709072892554et_nat @ ( produc5530789758353788441et_nat @ B6 @ C4 ) @ S2 ) ) ) ) ).
% 14.17/8.03
% 14.17/8.03 thf(fact_1712_relcomp_Osimps,axiom,
% 14.17/8.03 ! [A1: produc3658429121746597890et_nat > $o,A22: produc3925858234332021118et_nat,R2: set_Pr2161125870931222855_nat_o,S2: set_Pr8536935166611901872et_nat] :
% 14.17/8.03 ( ( member6124377750444531601et_nat @ ( produc2245416461498447860et_nat @ A1 @ A22 ) @ ( relcom2608444778321954367et_nat @ R2 @ S2 ) )
% 14.17/8.03 = ( ? [A6: produc3658429121746597890et_nat > $o,B6: produc3658429121746597890et_nat > $o,C4: produc3925858234332021118et_nat] :
% 14.17/8.03 ( ( A1 = A6 )
% 14.17/8.03 & ( A22 = C4 )
% 14.17/8.03 & ( member8781333585448626064_nat_o @ ( produc7368190662567826135_nat_o @ A6 @ B6 ) @ R2 )
% 14.17/8.03 & ( member6124377750444531601et_nat @ ( produc2245416461498447860et_nat @ B6 @ C4 ) @ S2 ) ) ) ) ).
% 14.17/8.03
% 14.17/8.03 thf(fact_1713_relcomp_Osimps,axiom,
% 14.17/8.03 ! [A1: produc3658429121746597890et_nat > $o,A22: produc3925858234332021118et_nat,R2: set_Pr3286484037609594932et_nat,S2: set_Pr8807100098903490369et_nat] :
% 14.17/8.03 ( ( member6124377750444531601et_nat @ ( produc2245416461498447860et_nat @ A1 @ A22 ) @ ( relcom8031709186682293262et_nat @ R2 @ S2 ) )
% 14.17/8.03 = ( ? [A6: produc3658429121746597890et_nat > $o,B6: produc3658429121746597890et_nat,C4: produc3925858234332021118et_nat] :
% 14.17/8.03 ( ( A1 = A6 )
% 14.17/8.03 & ( A22 = C4 )
% 14.17/8.03 & ( member1996754912294343701et_nat @ ( produc5001842942810119800et_nat @ A6 @ B6 ) @ R2 )
% 14.17/8.03 & ( member2737525317995836554et_nat @ ( produc3909112513931304473et_nat @ B6 @ C4 ) @ S2 ) ) ) ) ).
% 14.17/8.03
% 14.17/8.03 thf(fact_1714_relcomp_Osimps,axiom,
% 14.17/8.03 ! [A1: produc3658429121746597890et_nat > $o,A22: produc3925858234332021118et_nat,R2: set_Pr8536935166611901872et_nat,S2: set_Pr7928877670098842301et_nat] :
% 14.17/8.03 ( ( member6124377750444531601et_nat @ ( produc2245416461498447860et_nat @ A1 @ A22 ) @ ( relcom8303140135939483018et_nat @ R2 @ S2 ) )
% 14.17/8.03 = ( ? [A6: produc3658429121746597890et_nat > $o,B6: produc3925858234332021118et_nat,C4: produc3925858234332021118et_nat] :
% 14.17/8.03 ( ( A1 = A6 )
% 14.17/8.03 & ( A22 = C4 )
% 14.17/8.03 & ( member6124377750444531601et_nat @ ( produc2245416461498447860et_nat @ A6 @ B6 ) @ R2 )
% 14.17/8.03 & ( member4763271486408492550et_nat @ ( produc8599840265553166229et_nat @ B6 @ C4 ) @ S2 ) ) ) ) ).
% 14.17/8.03
% 14.17/8.03 thf(fact_1715_relcomp_Ocases,axiom,
% 14.17/8.03 ! [A1: heap_e7401611519738050253t_unit,A22: set_nat,R2: set_Pr1776176089890104647t_unit,S2: set_Pr3948176798113811640et_nat] :
% 14.17/8.03 ( ( member6260224972018164377et_nat @ ( produc7507926704131184380et_nat @ A1 @ A22 ) @ ( relcom4930750947919833671et_nat @ R2 @ S2 ) )
% 14.17/8.03 => ~ ! [B2: heap_e7401611519738050253t_unit] :
% 14.17/8.03 ( ( member5046970461208427408t_unit @ ( produc8256076736985659095t_unit @ A1 @ B2 ) @ R2 )
% 14.17/8.03 => ~ ( member6260224972018164377et_nat @ ( produc7507926704131184380et_nat @ B2 @ A22 ) @ S2 ) ) ) ).
% 14.17/8.03
% 14.17/8.03 thf(fact_1716_relcomp_Ocases,axiom,
% 14.17/8.03 ! [A1: heap_e7401611519738050253t_unit,A22: set_nat,R2: set_Pr3948176798113811640et_nat,S2: set_Pr5488025237498180813et_nat] :
% 14.17/8.03 ( ( member6260224972018164377et_nat @ ( produc7507926704131184380et_nat @ A1 @ A22 ) @ ( relcom8881708762719574682et_nat @ R2 @ S2 ) )
% 14.17/8.03 => ~ ! [B2: set_nat] :
% 14.17/8.03 ( ( member6260224972018164377et_nat @ ( produc7507926704131184380et_nat @ A1 @ B2 ) @ R2 )
% 14.17/8.03 => ~ ( member8277197624267554838et_nat @ ( produc4532415448927165861et_nat @ B2 @ A22 ) @ S2 ) ) ) ).
% 14.17/8.03
% 14.17/8.03 thf(fact_1717_relcomp_Ocases,axiom,
% 14.17/8.03 ! [A1: nat,A22: nat,R2: set_Pr1261947904930325089at_nat,S2: set_Pr1261947904930325089at_nat] :
% 14.17/8.03 ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A1 @ A22 ) @ ( relcomp_nat_nat_nat @ R2 @ S2 ) )
% 14.17/8.03 => ~ ! [B2: nat] :
% 14.17/8.03 ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A1 @ B2 ) @ R2 )
% 14.17/8.03 => ~ ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ B2 @ A22 ) @ S2 ) ) ) ).
% 14.17/8.03
% 14.17/8.03 thf(fact_1718_relcomp_Ocases,axiom,
% 14.17/8.03 ! [A1: int,A22: int,R2: set_Pr958786334691620121nt_int,S2: set_Pr958786334691620121nt_int] :
% 14.17/8.03 ( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ A1 @ A22 ) @ ( relcomp_int_int_int @ R2 @ S2 ) )
% 14.17/8.03 => ~ ! [B2: int] :
% 14.17/8.03 ( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ A1 @ B2 ) @ R2 )
% 14.17/8.03 => ~ ( member5262025264175285858nt_int @ ( product_Pair_int_int @ B2 @ A22 ) @ S2 ) ) ) ).
% 14.17/8.03
% 14.17/8.03 thf(fact_1719_relcomp_Ocases,axiom,
% 14.17/8.03 ! [A1: produc3658429121746597890et_nat > $o,A22: produc3658429121746597890et_nat,R2: set_Pr2161125870931222855_nat_o,S2: set_Pr3286484037609594932et_nat] :
% 14.17/8.03 ( ( member1996754912294343701et_nat @ ( produc5001842942810119800et_nat @ A1 @ A22 ) @ ( relcom6619595054755144899et_nat @ R2 @ S2 ) )
% 14.17/8.03 => ~ ! [B2: produc3658429121746597890et_nat > $o] :
% 14.17/8.03 ( ( member8781333585448626064_nat_o @ ( produc7368190662567826135_nat_o @ A1 @ B2 ) @ R2 )
% 14.17/8.03 => ~ ( member1996754912294343701et_nat @ ( produc5001842942810119800et_nat @ B2 @ A22 ) @ S2 ) ) ) ).
% 14.17/8.03
% 14.17/8.03 thf(fact_1720_relcomp_Ocases,axiom,
% 14.17/8.03 ! [A1: produc3658429121746597890et_nat > $o,A22: produc3658429121746597890et_nat,R2: set_Pr3286484037609594932et_nat,S2: set_Pr719794911490849221et_nat] :
% 14.17/8.03 ( ( member1996754912294343701et_nat @ ( produc5001842942810119800et_nat @ A1 @ A22 ) @ ( relcom5975814107203797650et_nat @ R2 @ S2 ) )
% 14.17/8.03 => ~ ! [B2: produc3658429121746597890et_nat] :
% 14.17/8.03 ( ( member1996754912294343701et_nat @ ( produc5001842942810119800et_nat @ A1 @ B2 ) @ R2 )
% 14.17/8.03 => ~ ( member6099555550032318734et_nat @ ( produc8199053930788261021et_nat @ B2 @ A22 ) @ S2 ) ) ) ).
% 14.17/8.03
% 14.17/8.03 thf(fact_1721_relcomp_Ocases,axiom,
% 14.17/8.03 ! [A1: produc3658429121746597890et_nat > $o,A22: produc3658429121746597890et_nat,R2: set_Pr8536935166611901872et_nat,S2: set_Pr3721596426557623617et_nat] :
% 14.17/8.03 ( ( member1996754912294343701et_nat @ ( produc5001842942810119800et_nat @ A1 @ A22 ) @ ( relcom430014394250001422et_nat @ R2 @ S2 ) )
% 14.17/8.03 => ~ ! [B2: produc3925858234332021118et_nat] :
% 14.17/8.03 ( ( member6124377750444531601et_nat @ ( produc2245416461498447860et_nat @ A1 @ B2 ) @ R2 )
% 14.17/8.03 => ~ ( member6613709072892554et_nat @ ( produc5530789758353788441et_nat @ B2 @ A22 ) @ S2 ) ) ) ).
% 14.17/8.03
% 14.17/8.03 thf(fact_1722_relcomp_Ocases,axiom,
% 14.17/8.03 ! [A1: produc3658429121746597890et_nat > $o,A22: produc3925858234332021118et_nat,R2: set_Pr2161125870931222855_nat_o,S2: set_Pr8536935166611901872et_nat] :
% 14.17/8.03 ( ( member6124377750444531601et_nat @ ( produc2245416461498447860et_nat @ A1 @ A22 ) @ ( relcom2608444778321954367et_nat @ R2 @ S2 ) )
% 14.17/8.03 => ~ ! [B2: produc3658429121746597890et_nat > $o] :
% 14.17/8.03 ( ( member8781333585448626064_nat_o @ ( produc7368190662567826135_nat_o @ A1 @ B2 ) @ R2 )
% 14.17/8.03 => ~ ( member6124377750444531601et_nat @ ( produc2245416461498447860et_nat @ B2 @ A22 ) @ S2 ) ) ) ).
% 14.17/8.03
% 14.17/8.03 thf(fact_1723_relcomp_Ocases,axiom,
% 14.17/8.03 ! [A1: produc3658429121746597890et_nat > $o,A22: produc3925858234332021118et_nat,R2: set_Pr3286484037609594932et_nat,S2: set_Pr8807100098903490369et_nat] :
% 14.17/8.03 ( ( member6124377750444531601et_nat @ ( produc2245416461498447860et_nat @ A1 @ A22 ) @ ( relcom8031709186682293262et_nat @ R2 @ S2 ) )
% 14.17/8.03 => ~ ! [B2: produc3658429121746597890et_nat] :
% 14.17/8.03 ( ( member1996754912294343701et_nat @ ( produc5001842942810119800et_nat @ A1 @ B2 ) @ R2 )
% 14.17/8.03 => ~ ( member2737525317995836554et_nat @ ( produc3909112513931304473et_nat @ B2 @ A22 ) @ S2 ) ) ) ).
% 14.17/8.03
% 14.17/8.03 thf(fact_1724_relcomp_Ocases,axiom,
% 14.17/8.03 ! [A1: produc3658429121746597890et_nat > $o,A22: produc3925858234332021118et_nat,R2: set_Pr8536935166611901872et_nat,S2: set_Pr7928877670098842301et_nat] :
% 14.17/8.03 ( ( member6124377750444531601et_nat @ ( produc2245416461498447860et_nat @ A1 @ A22 ) @ ( relcom8303140135939483018et_nat @ R2 @ S2 ) )
% 14.17/8.03 => ~ ! [B2: produc3925858234332021118et_nat] :
% 14.17/8.03 ( ( member6124377750444531601et_nat @ ( produc2245416461498447860et_nat @ A1 @ B2 ) @ R2 )
% 14.17/8.03 => ~ ( member4763271486408492550et_nat @ ( produc8599840265553166229et_nat @ B2 @ A22 ) @ S2 ) ) ) ).
% 14.17/8.03
% 14.17/8.03 thf(fact_1725_FieldI2,axiom,
% 14.17/8.03 ! [I2: produc3658429121746597890et_nat > $o,J: produc3658429121746597890et_nat > $o,R3: set_Pr2161125870931222855_nat_o] :
% 14.17/8.03 ( ( member8781333585448626064_nat_o @ ( produc7368190662567826135_nat_o @ I2 @ J ) @ R3 )
% 14.17/8.03 => ( member6576561426505652726_nat_o @ J @ ( field_81821745747897882_nat_o @ R3 ) ) ) ).
% 14.17/8.03
% 14.17/8.03 thf(fact_1726_FieldI2,axiom,
% 14.17/8.03 ! [I2: $o,J: $o,R3: set_Product_prod_o_o] :
% 14.17/8.03 ( ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ I2 @ J ) @ R3 )
% 14.17/8.03 => ( member_o @ J @ ( field_o @ R3 ) ) ) ).
% 14.17/8.03
% 14.17/8.03 thf(fact_1727_FieldI2,axiom,
% 14.17/8.03 ! [I2: product_unit,J: product_unit,R3: set_Pr5094982260447487303t_unit] :
% 14.17/8.03 ( ( member7821138191923116944t_unit @ ( produc5692694829384537815t_unit @ I2 @ J ) @ R3 )
% 14.17/8.03 => ( member_Product_unit @ J @ ( field_Product_unit @ R3 ) ) ) ).
% 14.17/8.03
% 14.17/8.03 thf(fact_1728_FieldI2,axiom,
% 14.17/8.03 ! [I2: nat,J: nat,R3: set_Pr1261947904930325089at_nat] :
% 14.17/8.03 ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ I2 @ J ) @ R3 )
% 14.17/8.03 => ( member_nat @ J @ ( field_nat @ R3 ) ) ) ).
% 14.17/8.03
% 14.17/8.03 thf(fact_1729_FieldI2,axiom,
% 14.17/8.03 ! [I2: int,J: int,R3: set_Pr958786334691620121nt_int] :
% 14.17/8.03 ( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ I2 @ J ) @ R3 )
% 14.17/8.03 => ( member_int @ J @ ( field_int @ R3 ) ) ) ).
% 14.17/8.03
% 14.17/8.03 thf(fact_1730_FieldI1,axiom,
% 14.17/8.03 ! [I2: produc3658429121746597890et_nat > $o,J: produc3658429121746597890et_nat > $o,R3: set_Pr2161125870931222855_nat_o] :
% 14.17/8.03 ( ( member8781333585448626064_nat_o @ ( produc7368190662567826135_nat_o @ I2 @ J ) @ R3 )
% 14.17/8.03 => ( member6576561426505652726_nat_o @ I2 @ ( field_81821745747897882_nat_o @ R3 ) ) ) ).
% 14.17/8.03
% 14.17/8.03 thf(fact_1731_FieldI1,axiom,
% 14.17/8.03 ! [I2: $o,J: $o,R3: set_Product_prod_o_o] :
% 14.17/8.03 ( ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ I2 @ J ) @ R3 )
% 14.17/8.03 => ( member_o @ I2 @ ( field_o @ R3 ) ) ) ).
% 14.17/8.03
% 14.17/8.03 thf(fact_1732_FieldI1,axiom,
% 14.17/8.03 ! [I2: product_unit,J: product_unit,R3: set_Pr5094982260447487303t_unit] :
% 14.17/8.03 ( ( member7821138191923116944t_unit @ ( produc5692694829384537815t_unit @ I2 @ J ) @ R3 )
% 14.17/8.03 => ( member_Product_unit @ I2 @ ( field_Product_unit @ R3 ) ) ) ).
% 14.17/8.03
% 14.17/8.03 thf(fact_1733_FieldI1,axiom,
% 14.17/8.03 ! [I2: nat,J: nat,R3: set_Pr1261947904930325089at_nat] :
% 14.17/8.03 ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ I2 @ J ) @ R3 )
% 14.17/8.03 => ( member_nat @ I2 @ ( field_nat @ R3 ) ) ) ).
% 14.17/8.03
% 14.17/8.03 thf(fact_1734_FieldI1,axiom,
% 14.17/8.03 ! [I2: int,J: int,R3: set_Pr958786334691620121nt_int] :
% 14.17/8.03 ( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ I2 @ J ) @ R3 )
% 14.17/8.03 => ( member_int @ I2 @ ( field_int @ R3 ) ) ) ).
% 14.17/8.03
% 14.17/8.03 thf(fact_1735_mono__Field,axiom,
% 14.17/8.03 ! [R2: set_Pr1261947904930325089at_nat,S2: set_Pr1261947904930325089at_nat] :
% 14.17/8.03 ( ( ord_le3146513528884898305at_nat @ R2 @ S2 )
% 14.17/8.03 => ( ord_less_eq_set_nat @ ( field_nat @ R2 ) @ ( field_nat @ S2 ) ) ) ).
% 14.17/8.03
% 14.17/8.03 thf(fact_1736_equiv__class__self,axiom,
% 14.17/8.03 ! [A: set_Product_unit,R2: set_Pr5094982260447487303t_unit,A2: product_unit] :
% 14.17/8.03 ( ( equiv_4189356295871601603t_unit @ A @ R2 )
% 14.17/8.03 => ( ( member_Product_unit @ A2 @ A )
% 14.17/8.03 => ( member_Product_unit @ A2 @ ( image_8431784415008483091t_unit @ R2 @ ( insert_Product_unit @ A2 @ bot_bo3957492148770167129t_unit ) ) ) ) ) ).
% 14.17/8.03
% 14.17/8.03 thf(fact_1737_equiv__class__self,axiom,
% 14.17/8.03 ! [A: set_o,R2: set_Product_prod_o_o,A2: $o] :
% 14.17/8.03 ( ( equiv_equiv_o @ A @ R2 )
% 14.17/8.03 => ( ( member_o @ A2 @ A )
% 14.17/8.03 => ( member_o @ A2 @ ( image_o_o @ R2 @ ( insert_o @ A2 @ bot_bot_set_o ) ) ) ) ) ).
% 14.17/8.03
% 14.17/8.03 thf(fact_1738_equiv__class__self,axiom,
% 14.17/8.03 ! [A: set_nat,R2: set_Pr1261947904930325089at_nat,A2: nat] :
% 14.17/8.03 ( ( equiv_equiv_nat @ A @ R2 )
% 14.17/8.03 => ( ( member_nat @ A2 @ A )
% 14.17/8.03 => ( member_nat @ A2 @ ( image_nat_nat @ R2 @ ( insert_nat @ A2 @ bot_bot_set_nat ) ) ) ) ) ).
% 14.17/8.03
% 14.17/8.03 thf(fact_1739_equiv__class__self,axiom,
% 14.17/8.03 ! [A: set_int,R2: set_Pr958786334691620121nt_int,A2: int] :
% 14.17/8.03 ( ( equiv_equiv_int @ A @ R2 )
% 14.17/8.03 => ( ( member_int @ A2 @ A )
% 14.17/8.03 => ( member_int @ A2 @ ( image_int_int @ R2 @ ( insert_int @ A2 @ bot_bot_set_int ) ) ) ) ) ).
% 14.17/8.03
% 14.17/8.03 thf(fact_1740_equiv__class__self,axiom,
% 14.17/8.03 ! [A: set_Pr4532377907799695533_nat_o,R2: set_Pr2161125870931222855_nat_o,A2: produc3658429121746597890et_nat > $o] :
% 14.17/8.03 ( ( equiv_6726761681920839555_nat_o @ A @ R2 )
% 14.17/8.03 => ( ( member6576561426505652726_nat_o @ A2 @ A )
% 14.17/8.03 => ( member6576561426505652726_nat_o @ A2 @ ( image_2232984160439939859_nat_o @ R2 @ ( insert5175938949040314269_nat_o @ A2 @ bot_bo7824918357723582233_nat_o ) ) ) ) ) ).
% 14.17/8.03
% 14.17/8.03 thf(fact_1741_Field__def,axiom,
% 14.17/8.03 ( field_nat
% 14.17/8.03 = ( ^ [R4: set_Pr1261947904930325089at_nat] : ( sup_sup_set_nat @ ( domain_nat_nat @ R4 ) @ ( range_nat_nat @ R4 ) ) ) ) ).
% 14.17/8.03
% 14.17/8.03 thf(fact_1742_equiv__class__eq__iff,axiom,
% 14.17/8.03 ! [A: set_Product_unit,R2: set_Pr5094982260447487303t_unit,X: product_unit,Y: product_unit] :
% 14.17/8.03 ( ( equiv_4189356295871601603t_unit @ A @ R2 )
% 14.17/8.03 => ( ( member7821138191923116944t_unit @ ( produc5692694829384537815t_unit @ X @ Y ) @ R2 )
% 14.17/8.03 = ( ( ( image_8431784415008483091t_unit @ R2 @ ( insert_Product_unit @ X @ bot_bo3957492148770167129t_unit ) )
% 14.17/8.03 = ( image_8431784415008483091t_unit @ R2 @ ( insert_Product_unit @ Y @ bot_bo3957492148770167129t_unit ) ) )
% 14.17/8.03 & ( member_Product_unit @ X @ A )
% 14.17/8.03 & ( member_Product_unit @ Y @ A ) ) ) ) ).
% 14.17/8.03
% 14.17/8.03 thf(fact_1743_equiv__class__eq__iff,axiom,
% 14.17/8.03 ! [A: set_o,R2: set_Product_prod_o_o,X: $o,Y: $o] :
% 14.17/8.03 ( ( equiv_equiv_o @ A @ R2 )
% 14.17/8.03 => ( ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ X @ Y ) @ R2 )
% 14.17/8.03 = ( ( ( image_o_o @ R2 @ ( insert_o @ X @ bot_bot_set_o ) )
% 14.17/8.03 = ( image_o_o @ R2 @ ( insert_o @ Y @ bot_bot_set_o ) ) )
% 14.17/8.03 & ( member_o @ X @ A )
% 14.17/8.03 & ( member_o @ Y @ A ) ) ) ) ).
% 14.17/8.03
% 14.17/8.03 thf(fact_1744_equiv__class__eq__iff,axiom,
% 14.17/8.03 ! [A: set_nat,R2: set_Pr1261947904930325089at_nat,X: nat,Y: nat] :
% 14.17/8.03 ( ( equiv_equiv_nat @ A @ R2 )
% 14.17/8.03 => ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ Y ) @ R2 )
% 14.17/8.03 = ( ( ( image_nat_nat @ R2 @ ( insert_nat @ X @ bot_bot_set_nat ) )
% 14.17/8.03 = ( image_nat_nat @ R2 @ ( insert_nat @ Y @ bot_bot_set_nat ) ) )
% 14.17/8.03 & ( member_nat @ X @ A )
% 14.17/8.03 & ( member_nat @ Y @ A ) ) ) ) ).
% 14.17/8.03
% 14.17/8.03 thf(fact_1745_equiv__class__eq__iff,axiom,
% 14.17/8.03 ! [A: set_int,R2: set_Pr958786334691620121nt_int,X: int,Y: int] :
% 14.17/8.03 ( ( equiv_equiv_int @ A @ R2 )
% 14.17/8.03 => ( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ X @ Y ) @ R2 )
% 14.17/8.03 = ( ( ( image_int_int @ R2 @ ( insert_int @ X @ bot_bot_set_int ) )
% 14.17/8.03 = ( image_int_int @ R2 @ ( insert_int @ Y @ bot_bot_set_int ) ) )
% 14.17/8.03 & ( member_int @ X @ A )
% 14.17/8.03 & ( member_int @ Y @ A ) ) ) ) ).
% 14.17/8.03
% 14.17/8.03 thf(fact_1746_equiv__class__eq__iff,axiom,
% 14.17/8.03 ! [A: set_Pr4532377907799695533_nat_o,R2: set_Pr2161125870931222855_nat_o,X: produc3658429121746597890et_nat > $o,Y: produc3658429121746597890et_nat > $o] :
% 14.17/8.03 ( ( equiv_6726761681920839555_nat_o @ A @ R2 )
% 14.17/8.03 => ( ( member8781333585448626064_nat_o @ ( produc7368190662567826135_nat_o @ X @ Y ) @ R2 )
% 14.17/8.03 = ( ( ( image_2232984160439939859_nat_o @ R2 @ ( insert5175938949040314269_nat_o @ X @ bot_bo7824918357723582233_nat_o ) )
% 14.17/8.03 = ( image_2232984160439939859_nat_o @ R2 @ ( insert5175938949040314269_nat_o @ Y @ bot_bo7824918357723582233_nat_o ) ) )
% 14.17/8.03 & ( member6576561426505652726_nat_o @ X @ A )
% 14.17/8.03 & ( member6576561426505652726_nat_o @ Y @ A ) ) ) ) ).
% 14.17/8.03
% 14.17/8.03 thf(fact_1747_eq__equiv__class__iff,axiom,
% 14.17/8.03 ! [A: set_Product_unit,R2: set_Pr5094982260447487303t_unit,X: product_unit,Y: product_unit] :
% 14.17/8.03 ( ( equiv_4189356295871601603t_unit @ A @ R2 )
% 14.17/8.03 => ( ( member_Product_unit @ X @ A )
% 14.17/8.03 => ( ( member_Product_unit @ Y @ A )
% 14.17/8.03 => ( ( ( image_8431784415008483091t_unit @ R2 @ ( insert_Product_unit @ X @ bot_bo3957492148770167129t_unit ) )
% 14.17/8.03 = ( image_8431784415008483091t_unit @ R2 @ ( insert_Product_unit @ Y @ bot_bo3957492148770167129t_unit ) ) )
% 14.17/8.03 = ( member7821138191923116944t_unit @ ( produc5692694829384537815t_unit @ X @ Y ) @ R2 ) ) ) ) ) ).
% 14.17/8.03
% 14.17/8.03 thf(fact_1748_eq__equiv__class__iff,axiom,
% 14.17/8.03 ! [A: set_o,R2: set_Product_prod_o_o,X: $o,Y: $o] :
% 14.17/8.03 ( ( equiv_equiv_o @ A @ R2 )
% 14.17/8.03 => ( ( member_o @ X @ A )
% 14.17/8.03 => ( ( member_o @ Y @ A )
% 14.17/8.03 => ( ( ( image_o_o @ R2 @ ( insert_o @ X @ bot_bot_set_o ) )
% 14.17/8.03 = ( image_o_o @ R2 @ ( insert_o @ Y @ bot_bot_set_o ) ) )
% 14.17/8.03 = ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ X @ Y ) @ R2 ) ) ) ) ) ).
% 14.17/8.03
% 14.17/8.03 thf(fact_1749_eq__equiv__class__iff,axiom,
% 14.17/8.03 ! [A: set_nat,R2: set_Pr1261947904930325089at_nat,X: nat,Y: nat] :
% 14.17/8.03 ( ( equiv_equiv_nat @ A @ R2 )
% 14.17/8.03 => ( ( member_nat @ X @ A )
% 14.17/8.03 => ( ( member_nat @ Y @ A )
% 14.17/8.03 => ( ( ( image_nat_nat @ R2 @ ( insert_nat @ X @ bot_bot_set_nat ) )
% 14.17/8.03 = ( image_nat_nat @ R2 @ ( insert_nat @ Y @ bot_bot_set_nat ) ) )
% 14.17/8.03 = ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ Y ) @ R2 ) ) ) ) ) ).
% 14.17/8.03
% 14.17/8.03 thf(fact_1750_eq__equiv__class__iff,axiom,
% 14.17/8.03 ! [A: set_int,R2: set_Pr958786334691620121nt_int,X: int,Y: int] :
% 14.17/8.03 ( ( equiv_equiv_int @ A @ R2 )
% 14.17/8.03 => ( ( member_int @ X @ A )
% 14.17/8.03 => ( ( member_int @ Y @ A )
% 14.17/8.03 => ( ( ( image_int_int @ R2 @ ( insert_int @ X @ bot_bot_set_int ) )
% 14.17/8.03 = ( image_int_int @ R2 @ ( insert_int @ Y @ bot_bot_set_int ) ) )
% 14.17/8.03 = ( member5262025264175285858nt_int @ ( product_Pair_int_int @ X @ Y ) @ R2 ) ) ) ) ) ).
% 14.17/8.03
% 14.17/8.03 thf(fact_1751_eq__equiv__class__iff,axiom,
% 14.17/8.03 ! [A: set_Pr4532377907799695533_nat_o,R2: set_Pr2161125870931222855_nat_o,X: produc3658429121746597890et_nat > $o,Y: produc3658429121746597890et_nat > $o] :
% 14.17/8.03 ( ( equiv_6726761681920839555_nat_o @ A @ R2 )
% 14.17/8.03 => ( ( member6576561426505652726_nat_o @ X @ A )
% 14.17/8.03 => ( ( member6576561426505652726_nat_o @ Y @ A )
% 14.17/8.03 => ( ( ( image_2232984160439939859_nat_o @ R2 @ ( insert5175938949040314269_nat_o @ X @ bot_bo7824918357723582233_nat_o ) )
% 14.17/8.03 = ( image_2232984160439939859_nat_o @ R2 @ ( insert5175938949040314269_nat_o @ Y @ bot_bo7824918357723582233_nat_o ) ) )
% 14.17/8.03 = ( member8781333585448626064_nat_o @ ( produc7368190662567826135_nat_o @ X @ Y ) @ R2 ) ) ) ) ) ).
% 14.17/8.03
% 14.17/8.03 thf(fact_1752_equiv__class__eq,axiom,
% 14.17/8.03 ! [A: set_Product_unit,R2: set_Pr5094982260447487303t_unit,A2: product_unit,B: product_unit] :
% 14.17/8.03 ( ( equiv_4189356295871601603t_unit @ A @ R2 )
% 14.17/8.03 => ( ( member7821138191923116944t_unit @ ( produc5692694829384537815t_unit @ A2 @ B ) @ R2 )
% 14.17/8.03 => ( ( image_8431784415008483091t_unit @ R2 @ ( insert_Product_unit @ A2 @ bot_bo3957492148770167129t_unit ) )
% 14.17/8.03 = ( image_8431784415008483091t_unit @ R2 @ ( insert_Product_unit @ B @ bot_bo3957492148770167129t_unit ) ) ) ) ) ).
% 14.17/8.03
% 14.17/8.03 thf(fact_1753_equiv__class__eq,axiom,
% 14.17/8.03 ! [A: set_o,R2: set_Product_prod_o_o,A2: $o,B: $o] :
% 14.17/8.03 ( ( equiv_equiv_o @ A @ R2 )
% 14.17/8.03 => ( ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ A2 @ B ) @ R2 )
% 14.17/8.03 => ( ( image_o_o @ R2 @ ( insert_o @ A2 @ bot_bot_set_o ) )
% 14.17/8.03 = ( image_o_o @ R2 @ ( insert_o @ B @ bot_bot_set_o ) ) ) ) ) ).
% 14.17/8.03
% 14.17/8.03 thf(fact_1754_equiv__class__eq,axiom,
% 14.17/8.03 ! [A: set_nat,R2: set_Pr1261947904930325089at_nat,A2: nat,B: nat] :
% 14.17/8.03 ( ( equiv_equiv_nat @ A @ R2 )
% 14.17/8.03 => ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A2 @ B ) @ R2 )
% 14.17/8.03 => ( ( image_nat_nat @ R2 @ ( insert_nat @ A2 @ bot_bot_set_nat ) )
% 14.17/8.03 = ( image_nat_nat @ R2 @ ( insert_nat @ B @ bot_bot_set_nat ) ) ) ) ) ).
% 14.17/8.03
% 14.17/8.03 thf(fact_1755_equiv__class__eq,axiom,
% 14.17/8.03 ! [A: set_int,R2: set_Pr958786334691620121nt_int,A2: int,B: int] :
% 14.17/8.03 ( ( equiv_equiv_int @ A @ R2 )
% 14.17/8.03 => ( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ A2 @ B ) @ R2 )
% 14.17/8.03 => ( ( image_int_int @ R2 @ ( insert_int @ A2 @ bot_bot_set_int ) )
% 14.17/8.03 = ( image_int_int @ R2 @ ( insert_int @ B @ bot_bot_set_int ) ) ) ) ) ).
% 14.17/8.03
% 14.17/8.03 thf(fact_1756_equiv__class__eq,axiom,
% 14.17/8.03 ! [A: set_Pr4532377907799695533_nat_o,R2: set_Pr2161125870931222855_nat_o,A2: produc3658429121746597890et_nat > $o,B: produc3658429121746597890et_nat > $o] :
% 14.17/8.03 ( ( equiv_6726761681920839555_nat_o @ A @ R2 )
% 14.17/8.03 => ( ( member8781333585448626064_nat_o @ ( produc7368190662567826135_nat_o @ A2 @ B ) @ R2 )
% 14.17/8.03 => ( ( image_2232984160439939859_nat_o @ R2 @ ( insert5175938949040314269_nat_o @ A2 @ bot_bo7824918357723582233_nat_o ) )
% 14.17/8.03 = ( image_2232984160439939859_nat_o @ R2 @ ( insert5175938949040314269_nat_o @ B @ bot_bo7824918357723582233_nat_o ) ) ) ) ) ).
% 14.17/8.03
% 14.17/8.03 thf(fact_1757_eq__equiv__class,axiom,
% 14.17/8.03 ! [R2: set_Pr5094982260447487303t_unit,A2: product_unit,B: product_unit,A: set_Product_unit] :
% 14.17/8.03 ( ( ( image_8431784415008483091t_unit @ R2 @ ( insert_Product_unit @ A2 @ bot_bo3957492148770167129t_unit ) )
% 14.17/8.03 = ( image_8431784415008483091t_unit @ R2 @ ( insert_Product_unit @ B @ bot_bo3957492148770167129t_unit ) ) )
% 14.17/8.03 => ( ( equiv_4189356295871601603t_unit @ A @ R2 )
% 14.17/8.03 => ( ( member_Product_unit @ B @ A )
% 14.17/8.03 => ( member7821138191923116944t_unit @ ( produc5692694829384537815t_unit @ A2 @ B ) @ R2 ) ) ) ) ).
% 14.17/8.03
% 14.17/8.03 thf(fact_1758_eq__equiv__class,axiom,
% 14.17/8.03 ! [R2: set_Product_prod_o_o,A2: $o,B: $o,A: set_o] :
% 14.17/8.03 ( ( ( image_o_o @ R2 @ ( insert_o @ A2 @ bot_bot_set_o ) )
% 14.17/8.03 = ( image_o_o @ R2 @ ( insert_o @ B @ bot_bot_set_o ) ) )
% 14.17/8.03 => ( ( equiv_equiv_o @ A @ R2 )
% 14.17/8.03 => ( ( member_o @ B @ A )
% 14.17/8.03 => ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ A2 @ B ) @ R2 ) ) ) ) ).
% 14.17/8.03
% 14.17/8.03 thf(fact_1759_eq__equiv__class,axiom,
% 14.17/8.03 ! [R2: set_Pr1261947904930325089at_nat,A2: nat,B: nat,A: set_nat] :
% 14.17/8.03 ( ( ( image_nat_nat @ R2 @ ( insert_nat @ A2 @ bot_bot_set_nat ) )
% 14.17/8.03 = ( image_nat_nat @ R2 @ ( insert_nat @ B @ bot_bot_set_nat ) ) )
% 14.17/8.03 => ( ( equiv_equiv_nat @ A @ R2 )
% 14.17/8.03 => ( ( member_nat @ B @ A )
% 14.17/8.03 => ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A2 @ B ) @ R2 ) ) ) ) ).
% 14.17/8.03
% 14.17/8.03 thf(fact_1760_eq__equiv__class,axiom,
% 14.17/8.03 ! [R2: set_Pr958786334691620121nt_int,A2: int,B: int,A: set_int] :
% 14.17/8.03 ( ( ( image_int_int @ R2 @ ( insert_int @ A2 @ bot_bot_set_int ) )
% 14.17/8.03 = ( image_int_int @ R2 @ ( insert_int @ B @ bot_bot_set_int ) ) )
% 14.17/8.03 => ( ( equiv_equiv_int @ A @ R2 )
% 14.17/8.03 => ( ( member_int @ B @ A )
% 14.17/8.03 => ( member5262025264175285858nt_int @ ( product_Pair_int_int @ A2 @ B ) @ R2 ) ) ) ) ).
% 14.17/8.03
% 14.17/8.03 thf(fact_1761_eq__equiv__class,axiom,
% 14.17/8.03 ! [R2: set_Pr2161125870931222855_nat_o,A2: produc3658429121746597890et_nat > $o,B: produc3658429121746597890et_nat > $o,A: set_Pr4532377907799695533_nat_o] :
% 14.17/8.03 ( ( ( image_2232984160439939859_nat_o @ R2 @ ( insert5175938949040314269_nat_o @ A2 @ bot_bo7824918357723582233_nat_o ) )
% 14.17/8.03 = ( image_2232984160439939859_nat_o @ R2 @ ( insert5175938949040314269_nat_o @ B @ bot_bo7824918357723582233_nat_o ) ) )
% 14.17/8.03 => ( ( equiv_6726761681920839555_nat_o @ A @ R2 )
% 14.17/8.03 => ( ( member6576561426505652726_nat_o @ B @ A )
% 14.17/8.03 => ( member8781333585448626064_nat_o @ ( produc7368190662567826135_nat_o @ A2 @ B ) @ R2 ) ) ) ) ).
% 14.17/8.03
% 14.17/8.03 thf(fact_1762_rtrancl__Image__in__Field,axiom,
% 14.17/8.03 ! [R3: set_Pr1261947904930325089at_nat,V2: set_nat] : ( ord_less_eq_set_nat @ ( image_nat_nat @ ( transi2905341329935302413cl_nat @ R3 ) @ V2 ) @ ( sup_sup_set_nat @ ( field_nat @ R3 ) @ V2 ) ) ).
% 14.17/8.03
% 14.17/8.03 thf(fact_1763_refines__equiv__class__eq2,axiom,
% 14.17/8.03 ! [R3: set_Pr5094982260447487303t_unit,S: set_Pr5094982260447487303t_unit,A: set_Product_unit,A2: product_unit] :
% 14.17/8.03 ( ( ord_le930853871177014503t_unit @ R3 @ S )
% 14.17/8.03 => ( ( equiv_4189356295871601603t_unit @ A @ R3 )
% 14.17/8.03 => ( ( equiv_4189356295871601603t_unit @ A @ S )
% 14.17/8.03 => ( ( image_8431784415008483091t_unit @ S @ ( image_8431784415008483091t_unit @ R3 @ ( insert_Product_unit @ A2 @ bot_bo3957492148770167129t_unit ) ) )
% 14.17/8.03 = ( image_8431784415008483091t_unit @ S @ ( insert_Product_unit @ A2 @ bot_bo3957492148770167129t_unit ) ) ) ) ) ) ).
% 14.17/8.03
% 14.17/8.03 thf(fact_1764_refines__equiv__class__eq2,axiom,
% 14.17/8.03 ! [R3: set_Product_prod_o_o,S: set_Product_prod_o_o,A: set_o,A2: $o] :
% 14.17/8.03 ( ( ord_le192195778054999463od_o_o @ R3 @ S )
% 14.17/8.03 => ( ( equiv_equiv_o @ A @ R3 )
% 14.17/8.03 => ( ( equiv_equiv_o @ A @ S )
% 14.17/8.03 => ( ( image_o_o @ S @ ( image_o_o @ R3 @ ( insert_o @ A2 @ bot_bot_set_o ) ) )
% 14.17/8.03 = ( image_o_o @ S @ ( insert_o @ A2 @ bot_bot_set_o ) ) ) ) ) ) ).
% 14.17/8.03
% 14.17/8.03 thf(fact_1765_refines__equiv__class__eq2,axiom,
% 14.17/8.03 ! [R3: set_Pr1261947904930325089at_nat,S: set_Pr1261947904930325089at_nat,A: set_nat,A2: nat] :
% 14.17/8.03 ( ( ord_le3146513528884898305at_nat @ R3 @ S )
% 14.17/8.03 => ( ( equiv_equiv_nat @ A @ R3 )
% 14.17/8.03 => ( ( equiv_equiv_nat @ A @ S )
% 14.17/8.03 => ( ( image_nat_nat @ S @ ( image_nat_nat @ R3 @ ( insert_nat @ A2 @ bot_bot_set_nat ) ) )
% 14.17/8.03 = ( image_nat_nat @ S @ ( insert_nat @ A2 @ bot_bot_set_nat ) ) ) ) ) ) ).
% 14.17/8.03
% 14.17/8.03 thf(fact_1766_refines__equiv__class__eq2,axiom,
% 14.17/8.03 ! [R3: set_Pr958786334691620121nt_int,S: set_Pr958786334691620121nt_int,A: set_int,A2: int] :
% 14.17/8.03 ( ( ord_le2843351958646193337nt_int @ R3 @ S )
% 14.17/8.03 => ( ( equiv_equiv_int @ A @ R3 )
% 14.17/8.03 => ( ( equiv_equiv_int @ A @ S )
% 14.17/8.03 => ( ( image_int_int @ S @ ( image_int_int @ R3 @ ( insert_int @ A2 @ bot_bot_set_int ) ) )
% 14.17/8.03 = ( image_int_int @ S @ ( insert_int @ A2 @ bot_bot_set_int ) ) ) ) ) ) ).
% 14.17/8.03
% 14.17/8.03 thf(fact_1767_refines__equiv__class__eq2,axiom,
% 14.17/8.03 ! [R3: set_Pr2161125870931222855_nat_o,S: set_Pr2161125870931222855_nat_o,A: set_Pr4532377907799695533_nat_o,A2: produc3658429121746597890et_nat > $o] :
% 14.17/8.03 ( ( ord_le6659818918159358183_nat_o @ R3 @ S )
% 14.17/8.03 => ( ( equiv_6726761681920839555_nat_o @ A @ R3 )
% 14.17/8.03 => ( ( equiv_6726761681920839555_nat_o @ A @ S )
% 14.17/8.03 => ( ( image_2232984160439939859_nat_o @ S @ ( image_2232984160439939859_nat_o @ R3 @ ( insert5175938949040314269_nat_o @ A2 @ bot_bo7824918357723582233_nat_o ) ) )
% 14.17/8.03 = ( image_2232984160439939859_nat_o @ S @ ( insert5175938949040314269_nat_o @ A2 @ bot_bo7824918357723582233_nat_o ) ) ) ) ) ) ).
% 14.17/8.03
% 14.17/8.03 thf(fact_1768_refines__equiv__class__eq,axiom,
% 14.17/8.03 ! [R3: set_Pr5094982260447487303t_unit,S: set_Pr5094982260447487303t_unit,A: set_Product_unit,A2: product_unit] :
% 14.17/8.03 ( ( ord_le930853871177014503t_unit @ R3 @ S )
% 14.17/8.03 => ( ( equiv_4189356295871601603t_unit @ A @ R3 )
% 14.17/8.03 => ( ( equiv_4189356295871601603t_unit @ A @ S )
% 14.17/8.03 => ( ( image_8431784415008483091t_unit @ R3 @ ( image_8431784415008483091t_unit @ S @ ( insert_Product_unit @ A2 @ bot_bo3957492148770167129t_unit ) ) )
% 14.17/8.03 = ( image_8431784415008483091t_unit @ S @ ( insert_Product_unit @ A2 @ bot_bo3957492148770167129t_unit ) ) ) ) ) ) ).
% 14.17/8.03
% 14.17/8.03 thf(fact_1769_refines__equiv__class__eq,axiom,
% 14.17/8.03 ! [R3: set_Product_prod_o_o,S: set_Product_prod_o_o,A: set_o,A2: $o] :
% 14.17/8.03 ( ( ord_le192195778054999463od_o_o @ R3 @ S )
% 14.17/8.03 => ( ( equiv_equiv_o @ A @ R3 )
% 14.17/8.03 => ( ( equiv_equiv_o @ A @ S )
% 14.17/8.03 => ( ( image_o_o @ R3 @ ( image_o_o @ S @ ( insert_o @ A2 @ bot_bot_set_o ) ) )
% 14.17/8.03 = ( image_o_o @ S @ ( insert_o @ A2 @ bot_bot_set_o ) ) ) ) ) ) ).
% 14.17/8.03
% 14.17/8.03 thf(fact_1770_refines__equiv__class__eq,axiom,
% 14.17/8.03 ! [R3: set_Pr1261947904930325089at_nat,S: set_Pr1261947904930325089at_nat,A: set_nat,A2: nat] :
% 14.17/8.03 ( ( ord_le3146513528884898305at_nat @ R3 @ S )
% 14.17/8.03 => ( ( equiv_equiv_nat @ A @ R3 )
% 14.17/8.03 => ( ( equiv_equiv_nat @ A @ S )
% 14.17/8.03 => ( ( image_nat_nat @ R3 @ ( image_nat_nat @ S @ ( insert_nat @ A2 @ bot_bot_set_nat ) ) )
% 14.17/8.03 = ( image_nat_nat @ S @ ( insert_nat @ A2 @ bot_bot_set_nat ) ) ) ) ) ) ).
% 14.17/8.03
% 14.17/8.03 thf(fact_1771_refines__equiv__class__eq,axiom,
% 14.17/8.03 ! [R3: set_Pr958786334691620121nt_int,S: set_Pr958786334691620121nt_int,A: set_int,A2: int] :
% 14.17/8.03 ( ( ord_le2843351958646193337nt_int @ R3 @ S )
% 14.17/8.03 => ( ( equiv_equiv_int @ A @ R3 )
% 14.17/8.03 => ( ( equiv_equiv_int @ A @ S )
% 14.17/8.03 => ( ( image_int_int @ R3 @ ( image_int_int @ S @ ( insert_int @ A2 @ bot_bot_set_int ) ) )
% 14.17/8.03 = ( image_int_int @ S @ ( insert_int @ A2 @ bot_bot_set_int ) ) ) ) ) ) ).
% 14.17/8.03
% 14.17/8.03 thf(fact_1772_refines__equiv__class__eq,axiom,
% 14.17/8.03 ! [R3: set_Pr2161125870931222855_nat_o,S: set_Pr2161125870931222855_nat_o,A: set_Pr4532377907799695533_nat_o,A2: produc3658429121746597890et_nat > $o] :
% 14.17/8.03 ( ( ord_le6659818918159358183_nat_o @ R3 @ S )
% 14.17/8.03 => ( ( equiv_6726761681920839555_nat_o @ A @ R3 )
% 14.17/8.03 => ( ( equiv_6726761681920839555_nat_o @ A @ S )
% 14.17/8.03 => ( ( image_2232984160439939859_nat_o @ R3 @ ( image_2232984160439939859_nat_o @ S @ ( insert5175938949040314269_nat_o @ A2 @ bot_bo7824918357723582233_nat_o ) ) )
% 14.17/8.03 = ( image_2232984160439939859_nat_o @ S @ ( insert5175938949040314269_nat_o @ A2 @ bot_bo7824918357723582233_nat_o ) ) ) ) ) ) ).
% 14.17/8.03
% 14.17/8.03 thf(fact_1773_Partial__order__eq__Image1__Image1__iff,axiom,
% 14.17/8.03 ! [R2: set_Pr5094982260447487303t_unit,A2: product_unit,B: product_unit] :
% 14.17/8.03 ( ( order_2505591618835473410t_unit @ ( field_Product_unit @ R2 ) @ R2 )
% 14.17/8.03 => ( ( member_Product_unit @ A2 @ ( field_Product_unit @ R2 ) )
% 14.17/8.03 => ( ( member_Product_unit @ B @ ( field_Product_unit @ R2 ) )
% 14.17/8.03 => ( ( ( image_8431784415008483091t_unit @ R2 @ ( insert_Product_unit @ A2 @ bot_bo3957492148770167129t_unit ) )
% 14.17/8.03 = ( image_8431784415008483091t_unit @ R2 @ ( insert_Product_unit @ B @ bot_bo3957492148770167129t_unit ) ) )
% 14.17/8.03 = ( A2 = B ) ) ) ) ) ).
% 14.17/8.03
% 14.17/8.03 thf(fact_1774_Partial__order__eq__Image1__Image1__iff,axiom,
% 14.17/8.03 ! [R2: set_Product_prod_o_o,A2: $o,B: $o] :
% 14.17/8.03 ( ( order_8538205470752950541r_on_o @ ( field_o @ R2 ) @ R2 )
% 14.17/8.03 => ( ( member_o @ A2 @ ( field_o @ R2 ) )
% 14.17/8.03 => ( ( member_o @ B @ ( field_o @ R2 ) )
% 14.17/8.03 => ( ( ( image_o_o @ R2 @ ( insert_o @ A2 @ bot_bot_set_o ) )
% 14.17/8.03 = ( image_o_o @ R2 @ ( insert_o @ B @ bot_bot_set_o ) ) )
% 14.17/8.03 = ( A2 = B ) ) ) ) ) ).
% 14.17/8.03
% 14.17/8.03 thf(fact_1775_Partial__order__eq__Image1__Image1__iff,axiom,
% 14.17/8.03 ! [R2: set_Pr1261947904930325089at_nat,A2: nat,B: nat] :
% 14.17/8.03 ( ( order_5251275573222108571on_nat @ ( field_nat @ R2 ) @ R2 )
% 14.17/8.03 => ( ( member_nat @ A2 @ ( field_nat @ R2 ) )
% 14.17/8.03 => ( ( member_nat @ B @ ( field_nat @ R2 ) )
% 14.17/8.03 => ( ( ( image_nat_nat @ R2 @ ( insert_nat @ A2 @ bot_bot_set_nat ) )
% 14.17/8.03 = ( image_nat_nat @ R2 @ ( insert_nat @ B @ bot_bot_set_nat ) ) )
% 14.17/8.03 = ( A2 = B ) ) ) ) ) ).
% 14.17/8.03
% 14.17/8.03 thf(fact_1776_Partial__order__eq__Image1__Image1__iff,axiom,
% 14.17/8.03 ! [R2: set_Pr958786334691620121nt_int,A2: int,B: int] :
% 14.17/8.03 ( ( order_5248785102713058295on_int @ ( field_int @ R2 ) @ R2 )
% 14.17/8.03 => ( ( member_int @ A2 @ ( field_int @ R2 ) )
% 14.17/8.03 => ( ( member_int @ B @ ( field_int @ R2 ) )
% 14.17/8.03 => ( ( ( image_int_int @ R2 @ ( insert_int @ A2 @ bot_bot_set_int ) )
% 14.17/8.03 = ( image_int_int @ R2 @ ( insert_int @ B @ bot_bot_set_int ) ) )
% 14.17/8.03 = ( A2 = B ) ) ) ) ) ).
% 14.17/8.03
% 14.17/8.03 thf(fact_1777_Partial__order__eq__Image1__Image1__iff,axiom,
% 14.17/8.03 ! [R2: set_Pr2161125870931222855_nat_o,A2: produc3658429121746597890et_nat > $o,B: produc3658429121746597890et_nat > $o] :
% 14.17/8.03 ( ( order_3690095022144090562_nat_o @ ( field_81821745747897882_nat_o @ R2 ) @ R2 )
% 14.17/8.03 => ( ( member6576561426505652726_nat_o @ A2 @ ( field_81821745747897882_nat_o @ R2 ) )
% 14.17/8.03 => ( ( member6576561426505652726_nat_o @ B @ ( field_81821745747897882_nat_o @ R2 ) )
% 14.17/8.03 => ( ( ( image_2232984160439939859_nat_o @ R2 @ ( insert5175938949040314269_nat_o @ A2 @ bot_bo7824918357723582233_nat_o ) )
% 14.17/8.03 = ( image_2232984160439939859_nat_o @ R2 @ ( insert5175938949040314269_nat_o @ B @ bot_bo7824918357723582233_nat_o ) ) )
% 14.17/8.03 = ( A2 = B ) ) ) ) ) ).
% 14.17/8.03
% 14.17/8.03 thf(fact_1778_finite__reachable__restrictedI,axiom,
% 14.17/8.03 ! [Q2: set_Code_integer,I: set_Code_integer,E: set_Pr4811707699266497531nteger] :
% 14.17/8.03 ( ( finite6017078050557962740nteger @ Q2 )
% 14.17/8.03 => ( ( ord_le7084787975880047091nteger @ I @ Q2 )
% 14.17/8.03 => ( ( ord_le7084787975880047091nteger @ ( range_1907724884897625393nteger @ E ) @ Q2 )
% 14.17/8.03 => ( finite6017078050557962740nteger @ ( image_4147621952751189391nteger @ ( transi3984306202085910426nteger @ E ) @ I ) ) ) ) ) ).
% 14.17/8.03
% 14.17/8.03 thf(fact_1779_finite__reachable__restrictedI,axiom,
% 14.17/8.03 ! [Q2: set_nat,I: set_nat,E: set_Pr1261947904930325089at_nat] :
% 14.17/8.03 ( ( finite_finite_nat @ Q2 )
% 14.17/8.03 => ( ( ord_less_eq_set_nat @ I @ Q2 )
% 14.17/8.03 => ( ( ord_less_eq_set_nat @ ( range_nat_nat @ E ) @ Q2 )
% 14.17/8.03 => ( finite_finite_nat @ ( image_nat_nat @ ( transi2905341329935302413cl_nat @ E ) @ I ) ) ) ) ) ).
% 14.17/8.03
% 14.17/8.03 thf(fact_1780_subset__Image__Image__iff,axiom,
% 14.17/8.03 ! [R2: set_Pr958786334691620121nt_int,A: set_int,B4: set_int] :
% 14.17/8.03 ( ( order_4859164337913492053on_int @ ( field_int @ R2 ) @ R2 )
% 14.17/8.03 => ( ( ord_less_eq_set_int @ A @ ( field_int @ R2 ) )
% 14.17/8.03 => ( ( ord_less_eq_set_int @ B4 @ ( field_int @ R2 ) )
% 14.17/8.03 => ( ( ord_less_eq_set_int @ ( image_int_int @ R2 @ A ) @ ( image_int_int @ R2 @ B4 ) )
% 14.17/8.03 = ( ! [X2: int] :
% 14.17/8.03 ( ( member_int @ X2 @ A )
% 14.17/8.03 => ? [Y3: int] :
% 14.17/8.03 ( ( member_int @ Y3 @ B4 )
% 14.17/8.03 & ( member5262025264175285858nt_int @ ( product_Pair_int_int @ Y3 @ X2 ) @ R2 ) ) ) ) ) ) ) ) ).
% 14.17/8.03
% 14.17/8.03 thf(fact_1781_subset__Image__Image__iff,axiom,
% 14.17/8.03 ! [R2: set_Pr1261947904930325089at_nat,A: set_nat,B4: set_nat] :
% 14.17/8.03 ( ( order_4861654808422542329on_nat @ ( field_nat @ R2 ) @ R2 )
% 14.17/8.03 => ( ( ord_less_eq_set_nat @ A @ ( field_nat @ R2 ) )
% 14.17/8.03 => ( ( ord_less_eq_set_nat @ B4 @ ( field_nat @ R2 ) )
% 14.17/8.03 => ( ( ord_less_eq_set_nat @ ( image_nat_nat @ R2 @ A ) @ ( image_nat_nat @ R2 @ B4 ) )
% 14.17/8.03 = ( ! [X2: nat] :
% 14.17/8.03 ( ( member_nat @ X2 @ A )
% 14.17/8.03 => ? [Y3: nat] :
% 14.17/8.03 ( ( member_nat @ Y3 @ B4 )
% 14.17/8.03 & ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ Y3 @ X2 ) @ R2 ) ) ) ) ) ) ) ) ).
% 14.17/8.03
% 14.17/8.03 thf(fact_1782_equiv__class__subset,axiom,
% 14.17/8.03 ! [A: set_Product_unit,R2: set_Pr5094982260447487303t_unit,A2: product_unit,B: product_unit] :
% 14.17/8.03 ( ( equiv_4189356295871601603t_unit @ A @ R2 )
% 14.17/8.03 => ( ( member7821138191923116944t_unit @ ( produc5692694829384537815t_unit @ A2 @ B ) @ R2 )
% 14.17/8.03 => ( ord_le3507040750410214029t_unit @ ( image_8431784415008483091t_unit @ R2 @ ( insert_Product_unit @ A2 @ bot_bo3957492148770167129t_unit ) ) @ ( image_8431784415008483091t_unit @ R2 @ ( insert_Product_unit @ B @ bot_bo3957492148770167129t_unit ) ) ) ) ) ).
% 14.17/8.03
% 14.17/8.03 thf(fact_1783_equiv__class__subset,axiom,
% 14.17/8.03 ! [A: set_o,R2: set_Product_prod_o_o,A2: $o,B: $o] :
% 14.17/8.03 ( ( equiv_equiv_o @ A @ R2 )
% 14.17/8.03 => ( ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ A2 @ B ) @ R2 )
% 14.17/8.03 => ( ord_less_eq_set_o @ ( image_o_o @ R2 @ ( insert_o @ A2 @ bot_bot_set_o ) ) @ ( image_o_o @ R2 @ ( insert_o @ B @ bot_bot_set_o ) ) ) ) ) ).
% 14.17/8.03
% 14.17/8.03 thf(fact_1784_equiv__class__subset,axiom,
% 14.17/8.03 ! [A: set_int,R2: set_Pr958786334691620121nt_int,A2: int,B: int] :
% 14.17/8.03 ( ( equiv_equiv_int @ A @ R2 )
% 14.17/8.03 => ( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ A2 @ B ) @ R2 )
% 14.17/8.03 => ( ord_less_eq_set_int @ ( image_int_int @ R2 @ ( insert_int @ A2 @ bot_bot_set_int ) ) @ ( image_int_int @ R2 @ ( insert_int @ B @ bot_bot_set_int ) ) ) ) ) ).
% 14.17/8.03
% 14.17/8.03 thf(fact_1785_equiv__class__subset,axiom,
% 14.17/8.03 ! [A: set_Pr4532377907799695533_nat_o,R2: set_Pr2161125870931222855_nat_o,A2: produc3658429121746597890et_nat > $o,B: produc3658429121746597890et_nat > $o] :
% 14.17/8.03 ( ( equiv_6726761681920839555_nat_o @ A @ R2 )
% 14.17/8.03 => ( ( member8781333585448626064_nat_o @ ( produc7368190662567826135_nat_o @ A2 @ B ) @ R2 )
% 14.17/8.03 => ( ord_le2965882846123202637_nat_o @ ( image_2232984160439939859_nat_o @ R2 @ ( insert5175938949040314269_nat_o @ A2 @ bot_bo7824918357723582233_nat_o ) ) @ ( image_2232984160439939859_nat_o @ R2 @ ( insert5175938949040314269_nat_o @ B @ bot_bo7824918357723582233_nat_o ) ) ) ) ) ).
% 14.17/8.03
% 14.17/8.03 thf(fact_1786_equiv__class__subset,axiom,
% 14.17/8.03 ! [A: set_nat,R2: set_Pr1261947904930325089at_nat,A2: nat,B: nat] :
% 14.17/8.03 ( ( equiv_equiv_nat @ A @ R2 )
% 14.17/8.03 => ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A2 @ B ) @ R2 )
% 14.17/8.03 => ( ord_less_eq_set_nat @ ( image_nat_nat @ R2 @ ( insert_nat @ A2 @ bot_bot_set_nat ) ) @ ( image_nat_nat @ R2 @ ( insert_nat @ B @ bot_bot_set_nat ) ) ) ) ) ).
% 14.17/8.03
% 14.17/8.03 thf(fact_1787_subset__equiv__class,axiom,
% 14.17/8.03 ! [A: set_Product_unit,R2: set_Pr5094982260447487303t_unit,B: product_unit,A2: product_unit] :
% 14.17/8.03 ( ( equiv_4189356295871601603t_unit @ A @ R2 )
% 14.17/8.03 => ( ( ord_le3507040750410214029t_unit @ ( image_8431784415008483091t_unit @ R2 @ ( insert_Product_unit @ B @ bot_bo3957492148770167129t_unit ) ) @ ( image_8431784415008483091t_unit @ R2 @ ( insert_Product_unit @ A2 @ bot_bo3957492148770167129t_unit ) ) )
% 14.17/8.03 => ( ( member_Product_unit @ B @ A )
% 14.17/8.03 => ( member7821138191923116944t_unit @ ( produc5692694829384537815t_unit @ A2 @ B ) @ R2 ) ) ) ) ).
% 14.17/8.03
% 14.17/8.03 thf(fact_1788_subset__equiv__class,axiom,
% 14.17/8.03 ! [A: set_o,R2: set_Product_prod_o_o,B: $o,A2: $o] :
% 14.17/8.03 ( ( equiv_equiv_o @ A @ R2 )
% 14.17/8.03 => ( ( ord_less_eq_set_o @ ( image_o_o @ R2 @ ( insert_o @ B @ bot_bot_set_o ) ) @ ( image_o_o @ R2 @ ( insert_o @ A2 @ bot_bot_set_o ) ) )
% 14.17/8.03 => ( ( member_o @ B @ A )
% 14.17/8.03 => ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ A2 @ B ) @ R2 ) ) ) ) ).
% 14.17/8.03
% 14.17/8.03 thf(fact_1789_subset__equiv__class,axiom,
% 14.17/8.03 ! [A: set_int,R2: set_Pr958786334691620121nt_int,B: int,A2: int] :
% 14.17/8.03 ( ( equiv_equiv_int @ A @ R2 )
% 14.17/8.03 => ( ( ord_less_eq_set_int @ ( image_int_int @ R2 @ ( insert_int @ B @ bot_bot_set_int ) ) @ ( image_int_int @ R2 @ ( insert_int @ A2 @ bot_bot_set_int ) ) )
% 14.17/8.03 => ( ( member_int @ B @ A )
% 14.17/8.03 => ( member5262025264175285858nt_int @ ( product_Pair_int_int @ A2 @ B ) @ R2 ) ) ) ) ).
% 14.17/8.03
% 14.17/8.03 thf(fact_1790_subset__equiv__class,axiom,
% 14.17/8.03 ! [A: set_Pr4532377907799695533_nat_o,R2: set_Pr2161125870931222855_nat_o,B: produc3658429121746597890et_nat > $o,A2: produc3658429121746597890et_nat > $o] :
% 14.17/8.03 ( ( equiv_6726761681920839555_nat_o @ A @ R2 )
% 14.17/8.03 => ( ( ord_le2965882846123202637_nat_o @ ( image_2232984160439939859_nat_o @ R2 @ ( insert5175938949040314269_nat_o @ B @ bot_bo7824918357723582233_nat_o ) ) @ ( image_2232984160439939859_nat_o @ R2 @ ( insert5175938949040314269_nat_o @ A2 @ bot_bo7824918357723582233_nat_o ) ) )
% 14.17/8.03 => ( ( member6576561426505652726_nat_o @ B @ A )
% 14.17/8.03 => ( member8781333585448626064_nat_o @ ( produc7368190662567826135_nat_o @ A2 @ B ) @ R2 ) ) ) ) ).
% 14.17/8.03
% 14.17/8.03 thf(fact_1791_subset__equiv__class,axiom,
% 14.17/8.03 ! [A: set_nat,R2: set_Pr1261947904930325089at_nat,B: nat,A2: nat] :
% 14.17/8.03 ( ( equiv_equiv_nat @ A @ R2 )
% 14.17/8.03 => ( ( ord_less_eq_set_nat @ ( image_nat_nat @ R2 @ ( insert_nat @ B @ bot_bot_set_nat ) ) @ ( image_nat_nat @ R2 @ ( insert_nat @ A2 @ bot_bot_set_nat ) ) )
% 14.17/8.03 => ( ( member_nat @ B @ A )
% 14.17/8.03 => ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A2 @ B ) @ R2 ) ) ) ) ).
% 14.17/8.03
% 14.17/8.03 thf(fact_1792_finite__finite__vimage__IntI,axiom,
% 14.17/8.03 ! [F2: set_Code_integer,H2: code_integer > code_integer,A: set_Code_integer] :
% 14.17/8.03 ( ( finite6017078050557962740nteger @ F2 )
% 14.17/8.03 => ( ! [Y2: code_integer] :
% 14.17/8.03 ( ( member_Code_integer @ Y2 @ F2 )
% 14.17/8.03 => ( finite6017078050557962740nteger @ ( inf_in1364745209274528805nteger @ ( vimage1318793397690208119nteger @ H2 @ ( insert_Code_integer @ Y2 @ bot_bo3990330152332043303nteger ) ) @ A ) ) )
% 14.17/8.03 => ( finite6017078050557962740nteger @ ( inf_in1364745209274528805nteger @ ( vimage1318793397690208119nteger @ H2 @ F2 ) @ A ) ) ) ) ).
% 14.17/8.03
% 14.17/8.03 thf(fact_1793_finite__finite__vimage__IntI,axiom,
% 14.17/8.03 ! [F2: set_Code_integer,H2: nat > code_integer,A: set_nat] :
% 14.17/8.03 ( ( finite6017078050557962740nteger @ F2 )
% 14.17/8.03 => ( ! [Y2: code_integer] :
% 14.17/8.03 ( ( member_Code_integer @ Y2 @ F2 )
% 14.17/8.03 => ( finite_finite_nat @ ( inf_inf_set_nat @ ( vimage6104394758324456170nteger @ H2 @ ( insert_Code_integer @ Y2 @ bot_bo3990330152332043303nteger ) ) @ A ) ) )
% 14.17/8.03 => ( finite_finite_nat @ ( inf_inf_set_nat @ ( vimage6104394758324456170nteger @ H2 @ F2 ) @ A ) ) ) ) ).
% 14.17/8.03
% 14.17/8.03 thf(fact_1794_finite__finite__vimage__IntI,axiom,
% 14.17/8.03 ! [F2: set_Product_unit,H2: code_integer > product_unit,A: set_Code_integer] :
% 14.17/8.03 ( ( finite4290736615968046902t_unit @ F2 )
% 14.17/8.03 => ( ! [Y2: product_unit] :
% 14.17/8.03 ( ( member_Product_unit @ Y2 @ F2 )
% 14.17/8.03 => ( finite6017078050557962740nteger @ ( inf_in1364745209274528805nteger @ ( vimage5651688891684837235t_unit @ H2 @ ( insert_Product_unit @ Y2 @ bot_bo3957492148770167129t_unit ) ) @ A ) ) )
% 14.17/8.03 => ( finite6017078050557962740nteger @ ( inf_in1364745209274528805nteger @ ( vimage5651688891684837235t_unit @ H2 @ F2 ) @ A ) ) ) ) ).
% 14.17/8.03
% 14.17/8.03 thf(fact_1795_finite__finite__vimage__IntI,axiom,
% 14.17/8.03 ! [F2: set_Product_unit,H2: nat > product_unit,A: set_nat] :
% 14.17/8.03 ( ( finite4290736615968046902t_unit @ F2 )
% 14.17/8.03 => ( ! [Y2: product_unit] :
% 14.17/8.03 ( ( member_Product_unit @ Y2 @ F2 )
% 14.17/8.03 => ( finite_finite_nat @ ( inf_inf_set_nat @ ( vimage4884490618288580032t_unit @ H2 @ ( insert_Product_unit @ Y2 @ bot_bo3957492148770167129t_unit ) ) @ A ) ) )
% 14.17/8.03 => ( finite_finite_nat @ ( inf_inf_set_nat @ ( vimage4884490618288580032t_unit @ H2 @ F2 ) @ A ) ) ) ) ).
% 14.17/8.03
% 14.17/8.03 thf(fact_1796_finite__finite__vimage__IntI,axiom,
% 14.17/8.03 ! [F2: set_o,H2: code_integer > $o,A: set_Code_integer] :
% 14.17/8.03 ( ( finite_finite_o @ F2 )
% 14.17/8.03 => ( ! [Y2: $o] :
% 14.17/8.03 ( ( member_o @ Y2 @ F2 )
% 14.17/8.03 => ( finite6017078050557962740nteger @ ( inf_in1364745209274528805nteger @ ( vimage3005800680117336574eger_o @ H2 @ ( insert_o @ Y2 @ bot_bot_set_o ) ) @ A ) ) )
% 14.17/8.03 => ( finite6017078050557962740nteger @ ( inf_in1364745209274528805nteger @ ( vimage3005800680117336574eger_o @ H2 @ F2 ) @ A ) ) ) ) ).
% 14.17/8.03
% 14.17/8.03 thf(fact_1797_finite__finite__vimage__IntI,axiom,
% 14.17/8.03 ! [F2: set_o,H2: nat > $o,A: set_nat] :
% 14.17/8.03 ( ( finite_finite_o @ F2 )
% 14.17/8.03 => ( ! [Y2: $o] :
% 14.17/8.03 ( ( member_o @ Y2 @ F2 )
% 14.17/8.03 => ( finite_finite_nat @ ( inf_inf_set_nat @ ( vimage_nat_o @ H2 @ ( insert_o @ Y2 @ bot_bot_set_o ) ) @ A ) ) )
% 14.17/8.03 => ( finite_finite_nat @ ( inf_inf_set_nat @ ( vimage_nat_o @ H2 @ F2 ) @ A ) ) ) ) ).
% 14.17/8.03
% 14.17/8.03 thf(fact_1798_finite__finite__vimage__IntI,axiom,
% 14.17/8.03 ! [F2: set_nat,H2: code_integer > nat,A: set_Code_integer] :
% 14.17/8.03 ( ( finite_finite_nat @ F2 )
% 14.17/8.03 => ( ! [Y2: nat] :
% 14.17/8.03 ( ( member_nat @ Y2 @ F2 )
% 14.17/8.03 => ( finite6017078050557962740nteger @ ( inf_in1364745209274528805nteger @ ( vimage5839839309545413354er_nat @ H2 @ ( insert_nat @ Y2 @ bot_bot_set_nat ) ) @ A ) ) )
% 14.17/8.03 => ( finite6017078050557962740nteger @ ( inf_in1364745209274528805nteger @ ( vimage5839839309545413354er_nat @ H2 @ F2 ) @ A ) ) ) ) ).
% 14.17/8.03
% 14.17/8.03 thf(fact_1799_finite__finite__vimage__IntI,axiom,
% 14.17/8.03 ! [F2: set_nat,H2: nat > nat,A: set_nat] :
% 14.17/8.03 ( ( finite_finite_nat @ F2 )
% 14.17/8.03 => ( ! [Y2: nat] :
% 14.17/8.03 ( ( member_nat @ Y2 @ F2 )
% 14.17/8.03 => ( finite_finite_nat @ ( inf_inf_set_nat @ ( vimage_nat_nat @ H2 @ ( insert_nat @ Y2 @ bot_bot_set_nat ) ) @ A ) ) )
% 14.17/8.03 => ( finite_finite_nat @ ( inf_inf_set_nat @ ( vimage_nat_nat @ H2 @ F2 ) @ A ) ) ) ) ).
% 14.17/8.03
% 14.17/8.03 thf(fact_1800_finite__finite__vimage__IntI,axiom,
% 14.17/8.03 ! [F2: set_int,H2: code_integer > int,A: set_Code_integer] :
% 14.17/8.03 ( ( finite_finite_int @ F2 )
% 14.17/8.03 => ( ! [Y2: int] :
% 14.17/8.03 ( ( member_int @ Y2 @ F2 )
% 14.17/8.03 => ( finite6017078050557962740nteger @ ( inf_in1364745209274528805nteger @ ( vimage5837348839036363078er_int @ H2 @ ( insert_int @ Y2 @ bot_bot_set_int ) ) @ A ) ) )
% 14.17/8.03 => ( finite6017078050557962740nteger @ ( inf_in1364745209274528805nteger @ ( vimage5837348839036363078er_int @ H2 @ F2 ) @ A ) ) ) ) ).
% 14.17/8.03
% 14.17/8.03 thf(fact_1801_finite__finite__vimage__IntI,axiom,
% 14.17/8.03 ! [F2: set_int,H2: nat > int,A: set_nat] :
% 14.17/8.03 ( ( finite_finite_int @ F2 )
% 14.17/8.03 => ( ! [Y2: int] :
% 14.17/8.03 ( ( member_int @ Y2 @ F2 )
% 14.17/8.03 => ( finite_finite_nat @ ( inf_inf_set_nat @ ( vimage_nat_int @ H2 @ ( insert_int @ Y2 @ bot_bot_set_int ) ) @ A ) ) )
% 14.17/8.03 => ( finite_finite_nat @ ( inf_inf_set_nat @ ( vimage_nat_int @ H2 @ F2 ) @ A ) ) ) ) ).
% 14.17/8.03
% 14.17/8.03 thf(fact_1802_finite__insert,axiom,
% 14.17/8.03 ! [A2: int,A: set_int] :
% 14.17/8.03 ( ( finite_finite_int @ ( insert_int @ A2 @ A ) )
% 14.17/8.03 = ( finite_finite_int @ A ) ) ).
% 14.17/8.03
% 14.17/8.03 thf(fact_1803_finite__insert,axiom,
% 14.17/8.03 ! [A2: $o,A: set_o] :
% 14.17/8.03 ( ( finite_finite_o @ ( insert_o @ A2 @ A ) )
% 14.17/8.03 = ( finite_finite_o @ A ) ) ).
% 14.17/8.03
% 14.17/8.03 thf(fact_1804_finite__insert,axiom,
% 14.17/8.03 ! [A2: product_unit,A: set_Product_unit] :
% 14.17/8.03 ( ( finite4290736615968046902t_unit @ ( insert_Product_unit @ A2 @ A ) )
% 14.17/8.03 = ( finite4290736615968046902t_unit @ A ) ) ).
% 14.17/8.03
% 14.17/8.03 thf(fact_1805_finite__insert,axiom,
% 14.17/8.03 ! [A2: nat,A: set_nat] :
% 14.17/8.03 ( ( finite_finite_nat @ ( insert_nat @ A2 @ A ) )
% 14.17/8.03 = ( finite_finite_nat @ A ) ) ).
% 14.17/8.03
% 14.17/8.03 thf(fact_1806_finite__insert,axiom,
% 14.17/8.03 ! [A2: code_integer,A: set_Code_integer] :
% 14.17/8.03 ( ( finite6017078050557962740nteger @ ( insert_Code_integer @ A2 @ A ) )
% 14.17/8.03 = ( finite6017078050557962740nteger @ A ) ) ).
% 14.17/8.03
% 14.17/8.03 thf(fact_1807_finite__subset__induct_H,axiom,
% 14.17/8.03 ! [F2: set_Code_integer,A: set_Code_integer,P: set_Code_integer > $o] :
% 14.17/8.03 ( ( finite6017078050557962740nteger @ F2 )
% 14.17/8.03 => ( ( ord_le7084787975880047091nteger @ F2 @ A )
% 14.17/8.03 => ( ( P @ bot_bo3990330152332043303nteger )
% 14.17/8.03 => ( ! [A4: code_integer,F3: set_Code_integer] :
% 14.17/8.03 ( ( finite6017078050557962740nteger @ F3 )
% 14.17/8.03 => ( ( member_Code_integer @ A4 @ A )
% 14.17/8.03 => ( ( ord_le7084787975880047091nteger @ F3 @ A )
% 14.17/8.03 => ( ~ ( member_Code_integer @ A4 @ F3 )
% 14.17/8.03 => ( ( P @ F3 )
% 14.17/8.03 => ( P @ ( insert_Code_integer @ A4 @ F3 ) ) ) ) ) ) )
% 14.17/8.03 => ( P @ F2 ) ) ) ) ) ).
% 14.17/8.03
% 14.17/8.03 thf(fact_1808_finite__subset__induct_H,axiom,
% 14.17/8.03 ! [F2: set_Product_unit,A: set_Product_unit,P: set_Product_unit > $o] :
% 14.17/8.03 ( ( finite4290736615968046902t_unit @ F2 )
% 14.17/8.03 => ( ( ord_le3507040750410214029t_unit @ F2 @ A )
% 14.17/8.03 => ( ( P @ bot_bo3957492148770167129t_unit )
% 14.17/8.03 => ( ! [A4: product_unit,F3: set_Product_unit] :
% 14.17/8.03 ( ( finite4290736615968046902t_unit @ F3 )
% 14.17/8.03 => ( ( member_Product_unit @ A4 @ A )
% 14.17/8.03 => ( ( ord_le3507040750410214029t_unit @ F3 @ A )
% 14.17/8.03 => ( ~ ( member_Product_unit @ A4 @ F3 )
% 14.17/8.03 => ( ( P @ F3 )
% 14.17/8.03 => ( P @ ( insert_Product_unit @ A4 @ F3 ) ) ) ) ) ) )
% 14.17/8.03 => ( P @ F2 ) ) ) ) ) ).
% 14.17/8.03
% 14.17/8.03 thf(fact_1809_finite__subset__induct_H,axiom,
% 14.17/8.03 ! [F2: set_o,A: set_o,P: set_o > $o] :
% 14.17/8.03 ( ( finite_finite_o @ F2 )
% 14.17/8.03 => ( ( ord_less_eq_set_o @ F2 @ A )
% 14.17/8.03 => ( ( P @ bot_bot_set_o )
% 14.17/8.03 => ( ! [A4: $o,F3: set_o] :
% 14.17/8.03 ( ( finite_finite_o @ F3 )
% 14.17/8.03 => ( ( member_o @ A4 @ A )
% 14.17/8.03 => ( ( ord_less_eq_set_o @ F3 @ A )
% 14.17/8.03 => ( ~ ( member_o @ A4 @ F3 )
% 14.17/8.03 => ( ( P @ F3 )
% 14.17/8.03 => ( P @ ( insert_o @ A4 @ F3 ) ) ) ) ) ) )
% 14.17/8.03 => ( P @ F2 ) ) ) ) ) ).
% 14.17/8.03
% 14.17/8.03 thf(fact_1810_finite__subset__induct_H,axiom,
% 14.17/8.03 ! [F2: set_int,A: set_int,P: set_int > $o] :
% 14.17/8.03 ( ( finite_finite_int @ F2 )
% 14.17/8.03 => ( ( ord_less_eq_set_int @ F2 @ A )
% 14.17/8.03 => ( ( P @ bot_bot_set_int )
% 14.17/8.03 => ( ! [A4: int,F3: set_int] :
% 14.17/8.03 ( ( finite_finite_int @ F3 )
% 14.17/8.03 => ( ( member_int @ A4 @ A )
% 14.17/8.03 => ( ( ord_less_eq_set_int @ F3 @ A )
% 14.17/8.03 => ( ~ ( member_int @ A4 @ F3 )
% 14.17/8.03 => ( ( P @ F3 )
% 14.17/8.03 => ( P @ ( insert_int @ A4 @ F3 ) ) ) ) ) ) )
% 14.17/8.03 => ( P @ F2 ) ) ) ) ) ).
% 14.17/8.03
% 14.17/8.03 thf(fact_1811_finite__subset__induct_H,axiom,
% 14.17/8.03 ! [F2: set_Pr4532377907799695533_nat_o,A: set_Pr4532377907799695533_nat_o,P: set_Pr4532377907799695533_nat_o > $o] :
% 14.17/8.03 ( ( finite3252695134891459830_nat_o @ F2 )
% 14.17/8.03 => ( ( ord_le2965882846123202637_nat_o @ F2 @ A )
% 14.17/8.03 => ( ( P @ bot_bo7824918357723582233_nat_o )
% 14.17/8.03 => ( ! [A4: produc3658429121746597890et_nat > $o,F3: set_Pr4532377907799695533_nat_o] :
% 14.17/8.03 ( ( finite3252695134891459830_nat_o @ F3 )
% 14.17/8.03 => ( ( member6576561426505652726_nat_o @ A4 @ A )
% 14.17/8.03 => ( ( ord_le2965882846123202637_nat_o @ F3 @ A )
% 14.17/8.03 => ( ~ ( member6576561426505652726_nat_o @ A4 @ F3 )
% 14.17/8.03 => ( ( P @ F3 )
% 14.17/8.03 => ( P @ ( insert5175938949040314269_nat_o @ A4 @ F3 ) ) ) ) ) ) )
% 14.17/8.03 => ( P @ F2 ) ) ) ) ) ).
% 14.17/8.03
% 14.17/8.03 thf(fact_1812_finite__subset__induct_H,axiom,
% 14.17/8.03 ! [F2: set_nat,A: set_nat,P: set_nat > $o] :
% 14.17/8.03 ( ( finite_finite_nat @ F2 )
% 14.17/8.03 => ( ( ord_less_eq_set_nat @ F2 @ A )
% 14.17/8.03 => ( ( P @ bot_bot_set_nat )
% 14.17/8.03 => ( ! [A4: nat,F3: set_nat] :
% 14.17/8.03 ( ( finite_finite_nat @ F3 )
% 14.17/8.03 => ( ( member_nat @ A4 @ A )
% 14.17/8.03 => ( ( ord_less_eq_set_nat @ F3 @ A )
% 14.17/8.03 => ( ~ ( member_nat @ A4 @ F3 )
% 14.17/8.03 => ( ( P @ F3 )
% 14.17/8.03 => ( P @ ( insert_nat @ A4 @ F3 ) ) ) ) ) ) )
% 14.17/8.03 => ( P @ F2 ) ) ) ) ) ).
% 14.17/8.03
% 14.17/8.03 thf(fact_1813_finite__subset__induct,axiom,
% 14.17/8.03 ! [F2: set_Code_integer,A: set_Code_integer,P: set_Code_integer > $o] :
% 14.17/8.03 ( ( finite6017078050557962740nteger @ F2 )
% 14.17/8.03 => ( ( ord_le7084787975880047091nteger @ F2 @ A )
% 14.17/8.03 => ( ( P @ bot_bo3990330152332043303nteger )
% 14.17/8.03 => ( ! [A4: code_integer,F3: set_Code_integer] :
% 14.17/8.03 ( ( finite6017078050557962740nteger @ F3 )
% 14.17/8.03 => ( ( member_Code_integer @ A4 @ A )
% 14.17/8.03 => ( ~ ( member_Code_integer @ A4 @ F3 )
% 14.17/8.03 => ( ( P @ F3 )
% 14.17/8.03 => ( P @ ( insert_Code_integer @ A4 @ F3 ) ) ) ) ) )
% 14.17/8.03 => ( P @ F2 ) ) ) ) ) ).
% 14.17/8.03
% 14.17/8.03 thf(fact_1814_finite__subset__induct,axiom,
% 14.17/8.03 ! [F2: set_Product_unit,A: set_Product_unit,P: set_Product_unit > $o] :
% 14.17/8.03 ( ( finite4290736615968046902t_unit @ F2 )
% 14.17/8.03 => ( ( ord_le3507040750410214029t_unit @ F2 @ A )
% 14.17/8.03 => ( ( P @ bot_bo3957492148770167129t_unit )
% 14.17/8.03 => ( ! [A4: product_unit,F3: set_Product_unit] :
% 14.17/8.03 ( ( finite4290736615968046902t_unit @ F3 )
% 14.17/8.03 => ( ( member_Product_unit @ A4 @ A )
% 14.17/8.03 => ( ~ ( member_Product_unit @ A4 @ F3 )
% 14.17/8.03 => ( ( P @ F3 )
% 14.17/8.03 => ( P @ ( insert_Product_unit @ A4 @ F3 ) ) ) ) ) )
% 14.17/8.03 => ( P @ F2 ) ) ) ) ) ).
% 14.17/8.03
% 14.17/8.03 thf(fact_1815_finite__subset__induct,axiom,
% 14.17/8.03 ! [F2: set_o,A: set_o,P: set_o > $o] :
% 14.17/8.03 ( ( finite_finite_o @ F2 )
% 14.17/8.03 => ( ( ord_less_eq_set_o @ F2 @ A )
% 14.17/8.03 => ( ( P @ bot_bot_set_o )
% 14.17/8.03 => ( ! [A4: $o,F3: set_o] :
% 14.17/8.03 ( ( finite_finite_o @ F3 )
% 14.17/8.03 => ( ( member_o @ A4 @ A )
% 14.17/8.03 => ( ~ ( member_o @ A4 @ F3 )
% 14.17/8.03 => ( ( P @ F3 )
% 14.17/8.03 => ( P @ ( insert_o @ A4 @ F3 ) ) ) ) ) )
% 14.17/8.03 => ( P @ F2 ) ) ) ) ) ).
% 14.17/8.03
% 14.17/8.03 thf(fact_1816_finite__subset__induct,axiom,
% 14.17/8.03 ! [F2: set_int,A: set_int,P: set_int > $o] :
% 14.17/8.03 ( ( finite_finite_int @ F2 )
% 14.17/8.03 => ( ( ord_less_eq_set_int @ F2 @ A )
% 14.17/8.03 => ( ( P @ bot_bot_set_int )
% 14.17/8.03 => ( ! [A4: int,F3: set_int] :
% 14.17/8.03 ( ( finite_finite_int @ F3 )
% 14.17/8.03 => ( ( member_int @ A4 @ A )
% 14.17/8.03 => ( ~ ( member_int @ A4 @ F3 )
% 14.17/8.03 => ( ( P @ F3 )
% 14.17/8.03 => ( P @ ( insert_int @ A4 @ F3 ) ) ) ) ) )
% 14.17/8.03 => ( P @ F2 ) ) ) ) ) ).
% 14.17/8.03
% 14.17/8.03 thf(fact_1817_finite__subset__induct,axiom,
% 14.17/8.03 ! [F2: set_Pr4532377907799695533_nat_o,A: set_Pr4532377907799695533_nat_o,P: set_Pr4532377907799695533_nat_o > $o] :
% 14.17/8.03 ( ( finite3252695134891459830_nat_o @ F2 )
% 14.17/8.03 => ( ( ord_le2965882846123202637_nat_o @ F2 @ A )
% 14.17/8.03 => ( ( P @ bot_bo7824918357723582233_nat_o )
% 14.17/8.03 => ( ! [A4: produc3658429121746597890et_nat > $o,F3: set_Pr4532377907799695533_nat_o] :
% 14.17/8.03 ( ( finite3252695134891459830_nat_o @ F3 )
% 14.17/8.03 => ( ( member6576561426505652726_nat_o @ A4 @ A )
% 14.17/8.03 => ( ~ ( member6576561426505652726_nat_o @ A4 @ F3 )
% 14.17/8.03 => ( ( P @ F3 )
% 14.17/8.03 => ( P @ ( insert5175938949040314269_nat_o @ A4 @ F3 ) ) ) ) ) )
% 14.17/8.03 => ( P @ F2 ) ) ) ) ) ).
% 14.17/8.03
% 14.17/8.03 thf(fact_1818_finite__subset__induct,axiom,
% 14.17/8.03 ! [F2: set_nat,A: set_nat,P: set_nat > $o] :
% 14.17/8.03 ( ( finite_finite_nat @ F2 )
% 14.17/8.03 => ( ( ord_less_eq_set_nat @ F2 @ A )
% 14.17/8.03 => ( ( P @ bot_bot_set_nat )
% 14.17/8.03 => ( ! [A4: nat,F3: set_nat] :
% 14.17/8.03 ( ( finite_finite_nat @ F3 )
% 14.17/8.03 => ( ( member_nat @ A4 @ A )
% 14.17/8.03 => ( ~ ( member_nat @ A4 @ F3 )
% 14.17/8.03 => ( ( P @ F3 )
% 14.17/8.03 => ( P @ ( insert_nat @ A4 @ F3 ) ) ) ) ) )
% 14.17/8.03 => ( P @ F2 ) ) ) ) ) ).
% 14.17/8.03
% 14.17/8.03 thf(fact_1819_finite__ranking__induct,axiom,
% 14.17/8.03 ! [S: set_Code_integer,P: set_Code_integer > $o,F: code_integer > rat] :
% 14.17/8.03 ( ( finite6017078050557962740nteger @ S )
% 14.17/8.03 => ( ( P @ bot_bo3990330152332043303nteger )
% 14.17/8.03 => ( ! [X3: code_integer,S5: set_Code_integer] :
% 14.17/8.03 ( ( finite6017078050557962740nteger @ S5 )
% 14.17/8.03 => ( ! [Y5: code_integer] :
% 14.17/8.03 ( ( member_Code_integer @ Y5 @ S5 )
% 14.17/8.03 => ( ord_less_eq_rat @ ( F @ Y5 ) @ ( F @ X3 ) ) )
% 14.17/8.03 => ( ( P @ S5 )
% 14.17/8.03 => ( P @ ( insert_Code_integer @ X3 @ S5 ) ) ) ) )
% 14.17/8.03 => ( P @ S ) ) ) ) ).
% 14.17/8.03
% 14.17/8.03 thf(fact_1820_finite__ranking__induct,axiom,
% 14.17/8.03 ! [S: set_Product_unit,P: set_Product_unit > $o,F: product_unit > rat] :
% 14.17/8.03 ( ( finite4290736615968046902t_unit @ S )
% 14.17/8.03 => ( ( P @ bot_bo3957492148770167129t_unit )
% 14.17/8.03 => ( ! [X3: product_unit,S5: set_Product_unit] :
% 14.17/8.03 ( ( finite4290736615968046902t_unit @ S5 )
% 14.17/8.03 => ( ! [Y5: product_unit] :
% 14.17/8.03 ( ( member_Product_unit @ Y5 @ S5 )
% 14.17/8.03 => ( ord_less_eq_rat @ ( F @ Y5 ) @ ( F @ X3 ) ) )
% 14.17/8.03 => ( ( P @ S5 )
% 14.17/8.03 => ( P @ ( insert_Product_unit @ X3 @ S5 ) ) ) ) )
% 14.17/8.03 => ( P @ S ) ) ) ) ).
% 14.17/8.03
% 14.17/8.03 thf(fact_1821_finite__ranking__induct,axiom,
% 14.17/8.03 ! [S: set_o,P: set_o > $o,F: $o > rat] :
% 14.17/8.03 ( ( finite_finite_o @ S )
% 14.17/8.03 => ( ( P @ bot_bot_set_o )
% 14.17/8.03 => ( ! [X3: $o,S5: set_o] :
% 14.17/8.03 ( ( finite_finite_o @ S5 )
% 14.17/8.03 => ( ! [Y5: $o] :
% 14.17/8.03 ( ( member_o @ Y5 @ S5 )
% 14.17/8.03 => ( ord_less_eq_rat @ ( F @ Y5 ) @ ( F @ X3 ) ) )
% 14.17/8.03 => ( ( P @ S5 )
% 14.17/8.03 => ( P @ ( insert_o @ X3 @ S5 ) ) ) ) )
% 14.17/8.03 => ( P @ S ) ) ) ) ).
% 14.17/8.03
% 14.17/8.03 thf(fact_1822_finite__ranking__induct,axiom,
% 14.17/8.03 ! [S: set_nat,P: set_nat > $o,F: nat > rat] :
% 14.17/8.03 ( ( finite_finite_nat @ S )
% 14.17/8.03 => ( ( P @ bot_bot_set_nat )
% 14.17/8.03 => ( ! [X3: nat,S5: set_nat] :
% 14.17/8.03 ( ( finite_finite_nat @ S5 )
% 14.17/8.03 => ( ! [Y5: nat] :
% 14.17/8.03 ( ( member_nat @ Y5 @ S5 )
% 14.17/8.03 => ( ord_less_eq_rat @ ( F @ Y5 ) @ ( F @ X3 ) ) )
% 14.17/8.03 => ( ( P @ S5 )
% 14.17/8.03 => ( P @ ( insert_nat @ X3 @ S5 ) ) ) ) )
% 14.17/8.03 => ( P @ S ) ) ) ) ).
% 14.17/8.03
% 14.17/8.03 thf(fact_1823_finite__ranking__induct,axiom,
% 14.17/8.03 ! [S: set_int,P: set_int > $o,F: int > rat] :
% 14.17/8.03 ( ( finite_finite_int @ S )
% 14.17/8.03 => ( ( P @ bot_bot_set_int )
% 14.17/8.03 => ( ! [X3: int,S5: set_int] :
% 14.17/8.03 ( ( finite_finite_int @ S5 )
% 14.17/8.03 => ( ! [Y5: int] :
% 14.17/8.03 ( ( member_int @ Y5 @ S5 )
% 14.17/8.03 => ( ord_less_eq_rat @ ( F @ Y5 ) @ ( F @ X3 ) ) )
% 14.17/8.03 => ( ( P @ S5 )
% 14.17/8.03 => ( P @ ( insert_int @ X3 @ S5 ) ) ) ) )
% 14.17/8.03 => ( P @ S ) ) ) ) ).
% 14.17/8.03
% 14.17/8.03 thf(fact_1824_finite__ranking__induct,axiom,
% 14.17/8.03 ! [S: set_Code_integer,P: set_Code_integer > $o,F: code_integer > nat] :
% 14.17/8.03 ( ( finite6017078050557962740nteger @ S )
% 14.17/8.03 => ( ( P @ bot_bo3990330152332043303nteger )
% 14.17/8.03 => ( ! [X3: code_integer,S5: set_Code_integer] :
% 14.17/8.03 ( ( finite6017078050557962740nteger @ S5 )
% 14.17/8.03 => ( ! [Y5: code_integer] :
% 14.17/8.03 ( ( member_Code_integer @ Y5 @ S5 )
% 14.17/8.03 => ( ord_less_eq_nat @ ( F @ Y5 ) @ ( F @ X3 ) ) )
% 14.17/8.03 => ( ( P @ S5 )
% 14.17/8.03 => ( P @ ( insert_Code_integer @ X3 @ S5 ) ) ) ) )
% 14.17/8.03 => ( P @ S ) ) ) ) ).
% 14.17/8.03
% 14.17/8.03 thf(fact_1825_finite__ranking__induct,axiom,
% 14.17/8.03 ! [S: set_Product_unit,P: set_Product_unit > $o,F: product_unit > nat] :
% 14.17/8.03 ( ( finite4290736615968046902t_unit @ S )
% 14.17/8.03 => ( ( P @ bot_bo3957492148770167129t_unit )
% 14.17/8.03 => ( ! [X3: product_unit,S5: set_Product_unit] :
% 14.17/8.03 ( ( finite4290736615968046902t_unit @ S5 )
% 14.17/8.03 => ( ! [Y5: product_unit] :
% 14.17/8.03 ( ( member_Product_unit @ Y5 @ S5 )
% 14.17/8.03 => ( ord_less_eq_nat @ ( F @ Y5 ) @ ( F @ X3 ) ) )
% 14.17/8.03 => ( ( P @ S5 )
% 14.17/8.03 => ( P @ ( insert_Product_unit @ X3 @ S5 ) ) ) ) )
% 14.17/8.03 => ( P @ S ) ) ) ) ).
% 14.17/8.03
% 14.17/8.03 thf(fact_1826_finite__ranking__induct,axiom,
% 14.17/8.03 ! [S: set_o,P: set_o > $o,F: $o > nat] :
% 14.17/8.03 ( ( finite_finite_o @ S )
% 14.17/8.03 => ( ( P @ bot_bot_set_o )
% 14.17/8.03 => ( ! [X3: $o,S5: set_o] :
% 14.17/8.03 ( ( finite_finite_o @ S5 )
% 14.17/8.03 => ( ! [Y5: $o] :
% 14.17/8.03 ( ( member_o @ Y5 @ S5 )
% 14.17/8.03 => ( ord_less_eq_nat @ ( F @ Y5 ) @ ( F @ X3 ) ) )
% 14.17/8.03 => ( ( P @ S5 )
% 14.17/8.03 => ( P @ ( insert_o @ X3 @ S5 ) ) ) ) )
% 14.17/8.03 => ( P @ S ) ) ) ) ).
% 14.17/8.03
% 14.17/8.03 thf(fact_1827_finite__ranking__induct,axiom,
% 14.17/8.03 ! [S: set_nat,P: set_nat > $o,F: nat > nat] :
% 14.17/8.03 ( ( finite_finite_nat @ S )
% 14.17/8.03 => ( ( P @ bot_bot_set_nat )
% 14.17/8.03 => ( ! [X3: nat,S5: set_nat] :
% 14.17/8.03 ( ( finite_finite_nat @ S5 )
% 14.17/8.03 => ( ! [Y5: nat] :
% 14.17/8.03 ( ( member_nat @ Y5 @ S5 )
% 14.17/8.03 => ( ord_less_eq_nat @ ( F @ Y5 ) @ ( F @ X3 ) ) )
% 14.17/8.03 => ( ( P @ S5 )
% 14.17/8.03 => ( P @ ( insert_nat @ X3 @ S5 ) ) ) ) )
% 14.17/8.03 => ( P @ S ) ) ) ) ).
% 14.17/8.03
% 14.17/8.03 thf(fact_1828_finite__ranking__induct,axiom,
% 14.17/8.03 ! [S: set_int,P: set_int > $o,F: int > nat] :
% 14.17/8.03 ( ( finite_finite_int @ S )
% 14.17/8.03 => ( ( P @ bot_bot_set_int )
% 14.17/8.03 => ( ! [X3: int,S5: set_int] :
% 14.17/8.03 ( ( finite_finite_int @ S5 )
% 14.17/8.03 => ( ! [Y5: int] :
% 14.17/8.03 ( ( member_int @ Y5 @ S5 )
% 14.17/8.03 => ( ord_less_eq_nat @ ( F @ Y5 ) @ ( F @ X3 ) ) )
% 14.17/8.03 => ( ( P @ S5 )
% 14.17/8.03 => ( P @ ( insert_int @ X3 @ S5 ) ) ) ) )
% 14.17/8.03 => ( P @ S ) ) ) ) ).
% 14.17/8.03
% 14.17/8.03 thf(fact_1829_infinite__finite__induct,axiom,
% 14.17/8.03 ! [P: set_Code_integer > $o,A: set_Code_integer] :
% 14.17/8.03 ( ! [A9: set_Code_integer] :
% 14.17/8.03 ( ~ ( finite6017078050557962740nteger @ A9 )
% 14.17/8.03 => ( P @ A9 ) )
% 14.17/8.03 => ( ( P @ bot_bo3990330152332043303nteger )
% 14.17/8.03 => ( ! [X3: code_integer,F3: set_Code_integer] :
% 14.17/8.03 ( ( finite6017078050557962740nteger @ F3 )
% 14.17/8.03 => ( ~ ( member_Code_integer @ X3 @ F3 )
% 14.17/8.03 => ( ( P @ F3 )
% 14.17/8.03 => ( P @ ( insert_Code_integer @ X3 @ F3 ) ) ) ) )
% 14.17/8.03 => ( P @ A ) ) ) ) ).
% 14.17/8.03
% 14.17/8.03 thf(fact_1830_infinite__finite__induct,axiom,
% 14.17/8.03 ! [P: set_Product_unit > $o,A: set_Product_unit] :
% 14.17/8.03 ( ! [A9: set_Product_unit] :
% 14.17/8.03 ( ~ ( finite4290736615968046902t_unit @ A9 )
% 14.17/8.03 => ( P @ A9 ) )
% 14.17/8.03 => ( ( P @ bot_bo3957492148770167129t_unit )
% 14.17/8.03 => ( ! [X3: product_unit,F3: set_Product_unit] :
% 14.17/8.03 ( ( finite4290736615968046902t_unit @ F3 )
% 14.17/8.03 => ( ~ ( member_Product_unit @ X3 @ F3 )
% 14.17/8.03 => ( ( P @ F3 )
% 14.17/8.03 => ( P @ ( insert_Product_unit @ X3 @ F3 ) ) ) ) )
% 14.17/8.03 => ( P @ A ) ) ) ) ).
% 14.17/8.03
% 14.17/8.03 thf(fact_1831_infinite__finite__induct,axiom,
% 14.17/8.03 ! [P: set_o > $o,A: set_o] :
% 14.17/8.03 ( ! [A9: set_o] :
% 14.17/8.03 ( ~ ( finite_finite_o @ A9 )
% 14.17/8.03 => ( P @ A9 ) )
% 14.17/8.03 => ( ( P @ bot_bot_set_o )
% 14.17/8.03 => ( ! [X3: $o,F3: set_o] :
% 14.17/8.03 ( ( finite_finite_o @ F3 )
% 14.17/8.03 => ( ~ ( member_o @ X3 @ F3 )
% 14.17/8.03 => ( ( P @ F3 )
% 14.17/8.03 => ( P @ ( insert_o @ X3 @ F3 ) ) ) ) )
% 14.17/8.03 => ( P @ A ) ) ) ) ).
% 14.17/8.03
% 14.17/8.03 thf(fact_1832_infinite__finite__induct,axiom,
% 14.17/8.03 ! [P: set_nat > $o,A: set_nat] :
% 14.17/8.03 ( ! [A9: set_nat] :
% 14.17/8.03 ( ~ ( finite_finite_nat @ A9 )
% 14.17/8.03 => ( P @ A9 ) )
% 14.17/8.03 => ( ( P @ bot_bot_set_nat )
% 14.17/8.03 => ( ! [X3: nat,F3: set_nat] :
% 14.17/8.03 ( ( finite_finite_nat @ F3 )
% 14.17/8.03 => ( ~ ( member_nat @ X3 @ F3 )
% 14.17/8.03 => ( ( P @ F3 )
% 14.17/8.03 => ( P @ ( insert_nat @ X3 @ F3 ) ) ) ) )
% 14.17/8.03 => ( P @ A ) ) ) ) ).
% 14.17/8.03
% 14.17/8.03 thf(fact_1833_infinite__finite__induct,axiom,
% 14.17/8.03 ! [P: set_int > $o,A: set_int] :
% 14.17/8.03 ( ! [A9: set_int] :
% 14.17/8.03 ( ~ ( finite_finite_int @ A9 )
% 14.17/8.03 => ( P @ A9 ) )
% 14.17/8.03 => ( ( P @ bot_bot_set_int )
% 14.17/8.03 => ( ! [X3: int,F3: set_int] :
% 14.17/8.03 ( ( finite_finite_int @ F3 )
% 14.17/8.03 => ( ~ ( member_int @ X3 @ F3 )
% 14.17/8.03 => ( ( P @ F3 )
% 14.17/8.03 => ( P @ ( insert_int @ X3 @ F3 ) ) ) ) )
% 14.17/8.03 => ( P @ A ) ) ) ) ).
% 14.17/8.03
% 14.17/8.03 thf(fact_1834_infinite__finite__induct,axiom,
% 14.17/8.03 ! [P: set_Pr4532377907799695533_nat_o > $o,A: set_Pr4532377907799695533_nat_o] :
% 14.17/8.03 ( ! [A9: set_Pr4532377907799695533_nat_o] :
% 14.17/8.03 ( ~ ( finite3252695134891459830_nat_o @ A9 )
% 14.17/8.03 => ( P @ A9 ) )
% 14.17/8.03 => ( ( P @ bot_bo7824918357723582233_nat_o )
% 14.17/8.03 => ( ! [X3: produc3658429121746597890et_nat > $o,F3: set_Pr4532377907799695533_nat_o] :
% 14.17/8.03 ( ( finite3252695134891459830_nat_o @ F3 )
% 14.17/8.03 => ( ~ ( member6576561426505652726_nat_o @ X3 @ F3 )
% 14.17/8.03 => ( ( P @ F3 )
% 14.17/8.03 => ( P @ ( insert5175938949040314269_nat_o @ X3 @ F3 ) ) ) ) )
% 14.17/8.03 => ( P @ A ) ) ) ) ).
% 14.17/8.03
% 14.17/8.03 thf(fact_1835_finite__ne__induct,axiom,
% 14.17/8.03 ! [F2: set_Code_integer,P: set_Code_integer > $o] :
% 14.17/8.03 ( ( finite6017078050557962740nteger @ F2 )
% 14.17/8.03 => ( ( F2 != bot_bo3990330152332043303nteger )
% 14.17/8.03 => ( ! [X3: code_integer] : ( P @ ( insert_Code_integer @ X3 @ bot_bo3990330152332043303nteger ) )
% 14.17/8.03 => ( ! [X3: code_integer,F3: set_Code_integer] :
% 14.17/8.03 ( ( finite6017078050557962740nteger @ F3 )
% 14.17/8.03 => ( ( F3 != bot_bo3990330152332043303nteger )
% 14.17/8.03 => ( ~ ( member_Code_integer @ X3 @ F3 )
% 14.17/8.03 => ( ( P @ F3 )
% 14.17/8.03 => ( P @ ( insert_Code_integer @ X3 @ F3 ) ) ) ) ) )
% 14.17/8.03 => ( P @ F2 ) ) ) ) ) ).
% 14.17/8.03
% 14.17/8.03 thf(fact_1836_finite__ne__induct,axiom,
% 14.17/8.03 ! [F2: set_Product_unit,P: set_Product_unit > $o] :
% 14.17/8.03 ( ( finite4290736615968046902t_unit @ F2 )
% 14.17/8.03 => ( ( F2 != bot_bo3957492148770167129t_unit )
% 14.17/8.03 => ( ! [X3: product_unit] : ( P @ ( insert_Product_unit @ X3 @ bot_bo3957492148770167129t_unit ) )
% 14.17/8.03 => ( ! [X3: product_unit,F3: set_Product_unit] :
% 14.17/8.03 ( ( finite4290736615968046902t_unit @ F3 )
% 14.17/8.03 => ( ( F3 != bot_bo3957492148770167129t_unit )
% 14.17/8.03 => ( ~ ( member_Product_unit @ X3 @ F3 )
% 14.17/8.03 => ( ( P @ F3 )
% 14.17/8.03 => ( P @ ( insert_Product_unit @ X3 @ F3 ) ) ) ) ) )
% 14.17/8.03 => ( P @ F2 ) ) ) ) ) ).
% 14.17/8.03
% 14.17/8.03 thf(fact_1837_finite__ne__induct,axiom,
% 14.17/8.03 ! [F2: set_o,P: set_o > $o] :
% 14.17/8.03 ( ( finite_finite_o @ F2 )
% 14.17/8.03 => ( ( F2 != bot_bot_set_o )
% 14.17/8.03 => ( ! [X3: $o] : ( P @ ( insert_o @ X3 @ bot_bot_set_o ) )
% 14.17/8.03 => ( ! [X3: $o,F3: set_o] :
% 14.17/8.03 ( ( finite_finite_o @ F3 )
% 14.17/8.03 => ( ( F3 != bot_bot_set_o )
% 14.17/8.03 => ( ~ ( member_o @ X3 @ F3 )
% 14.17/8.03 => ( ( P @ F3 )
% 14.17/8.03 => ( P @ ( insert_o @ X3 @ F3 ) ) ) ) ) )
% 14.17/8.03 => ( P @ F2 ) ) ) ) ) ).
% 14.17/8.03
% 14.17/8.03 thf(fact_1838_finite__ne__induct,axiom,
% 14.17/8.03 ! [F2: set_nat,P: set_nat > $o] :
% 14.17/8.03 ( ( finite_finite_nat @ F2 )
% 14.17/8.03 => ( ( F2 != bot_bot_set_nat )
% 14.17/8.03 => ( ! [X3: nat] : ( P @ ( insert_nat @ X3 @ bot_bot_set_nat ) )
% 14.17/8.03 => ( ! [X3: nat,F3: set_nat] :
% 14.17/8.03 ( ( finite_finite_nat @ F3 )
% 14.17/8.03 => ( ( F3 != bot_bot_set_nat )
% 14.17/8.03 => ( ~ ( member_nat @ X3 @ F3 )
% 14.17/8.03 => ( ( P @ F3 )
% 14.17/8.03 => ( P @ ( insert_nat @ X3 @ F3 ) ) ) ) ) )
% 14.17/8.03 => ( P @ F2 ) ) ) ) ) ).
% 14.17/8.03
% 14.17/8.03 thf(fact_1839_finite__ne__induct,axiom,
% 14.17/8.03 ! [F2: set_int,P: set_int > $o] :
% 14.17/8.03 ( ( finite_finite_int @ F2 )
% 14.17/8.03 => ( ( F2 != bot_bot_set_int )
% 14.17/8.03 => ( ! [X3: int] : ( P @ ( insert_int @ X3 @ bot_bot_set_int ) )
% 14.17/8.03 => ( ! [X3: int,F3: set_int] :
% 14.17/8.03 ( ( finite_finite_int @ F3 )
% 14.17/8.03 => ( ( F3 != bot_bot_set_int )
% 14.17/8.03 => ( ~ ( member_int @ X3 @ F3 )
% 14.17/8.03 => ( ( P @ F3 )
% 14.17/8.03 => ( P @ ( insert_int @ X3 @ F3 ) ) ) ) ) )
% 14.17/8.03 => ( P @ F2 ) ) ) ) ) ).
% 14.17/8.03
% 14.17/8.03 thf(fact_1840_finite__ne__induct,axiom,
% 14.17/8.03 ! [F2: set_Pr4532377907799695533_nat_o,P: set_Pr4532377907799695533_nat_o > $o] :
% 14.17/8.03 ( ( finite3252695134891459830_nat_o @ F2 )
% 14.17/8.03 => ( ( F2 != bot_bo7824918357723582233_nat_o )
% 14.17/8.03 => ( ! [X3: produc3658429121746597890et_nat > $o] : ( P @ ( insert5175938949040314269_nat_o @ X3 @ bot_bo7824918357723582233_nat_o ) )
% 14.17/8.03 => ( ! [X3: produc3658429121746597890et_nat > $o,F3: set_Pr4532377907799695533_nat_o] :
% 14.17/8.03 ( ( finite3252695134891459830_nat_o @ F3 )
% 14.17/8.03 => ( ( F3 != bot_bo7824918357723582233_nat_o )
% 14.17/8.03 => ( ~ ( member6576561426505652726_nat_o @ X3 @ F3 )
% 14.17/8.03 => ( ( P @ F3 )
% 14.17/8.03 => ( P @ ( insert5175938949040314269_nat_o @ X3 @ F3 ) ) ) ) ) )
% 14.17/8.03 => ( P @ F2 ) ) ) ) ) ).
% 14.17/8.03
% 14.17/8.03 thf(fact_1841_finite__induct,axiom,
% 14.17/8.03 ! [F2: set_Code_integer,P: set_Code_integer > $o] :
% 14.17/8.03 ( ( finite6017078050557962740nteger @ F2 )
% 14.17/8.03 => ( ( P @ bot_bo3990330152332043303nteger )
% 14.17/8.03 => ( ! [X3: code_integer,F3: set_Code_integer] :
% 14.17/8.03 ( ( finite6017078050557962740nteger @ F3 )
% 14.17/8.03 => ( ~ ( member_Code_integer @ X3 @ F3 )
% 14.17/8.03 => ( ( P @ F3 )
% 14.17/8.03 => ( P @ ( insert_Code_integer @ X3 @ F3 ) ) ) ) )
% 14.17/8.03 => ( P @ F2 ) ) ) ) ).
% 14.17/8.03
% 14.17/8.03 thf(fact_1842_finite__induct,axiom,
% 14.17/8.03 ! [F2: set_Product_unit,P: set_Product_unit > $o] :
% 14.17/8.03 ( ( finite4290736615968046902t_unit @ F2 )
% 14.17/8.03 => ( ( P @ bot_bo3957492148770167129t_unit )
% 14.17/8.03 => ( ! [X3: product_unit,F3: set_Product_unit] :
% 14.17/8.03 ( ( finite4290736615968046902t_unit @ F3 )
% 14.17/8.03 => ( ~ ( member_Product_unit @ X3 @ F3 )
% 14.17/8.03 => ( ( P @ F3 )
% 14.17/8.03 => ( P @ ( insert_Product_unit @ X3 @ F3 ) ) ) ) )
% 14.17/8.03 => ( P @ F2 ) ) ) ) ).
% 14.17/8.03
% 14.17/8.03 thf(fact_1843_finite__induct,axiom,
% 14.17/8.03 ! [F2: set_o,P: set_o > $o] :
% 14.17/8.03 ( ( finite_finite_o @ F2 )
% 14.17/8.03 => ( ( P @ bot_bot_set_o )
% 14.17/8.03 => ( ! [X3: $o,F3: set_o] :
% 14.17/8.03 ( ( finite_finite_o @ F3 )
% 14.17/8.03 => ( ~ ( member_o @ X3 @ F3 )
% 14.17/8.03 => ( ( P @ F3 )
% 14.17/8.03 => ( P @ ( insert_o @ X3 @ F3 ) ) ) ) )
% 14.17/8.03 => ( P @ F2 ) ) ) ) ).
% 14.17/8.03
% 14.17/8.03 thf(fact_1844_finite__induct,axiom,
% 14.17/8.03 ! [F2: set_nat,P: set_nat > $o] :
% 14.17/8.03 ( ( finite_finite_nat @ F2 )
% 14.17/8.03 => ( ( P @ bot_bot_set_nat )
% 14.17/8.03 => ( ! [X3: nat,F3: set_nat] :
% 14.17/8.03 ( ( finite_finite_nat @ F3 )
% 14.17/8.03 => ( ~ ( member_nat @ X3 @ F3 )
% 14.17/8.03 => ( ( P @ F3 )
% 14.17/8.03 => ( P @ ( insert_nat @ X3 @ F3 ) ) ) ) )
% 14.17/8.03 => ( P @ F2 ) ) ) ) ).
% 14.17/8.03
% 14.17/8.03 thf(fact_1845_finite__induct,axiom,
% 14.17/8.03 ! [F2: set_int,P: set_int > $o] :
% 14.17/8.03 ( ( finite_finite_int @ F2 )
% 14.17/8.03 => ( ( P @ bot_bot_set_int )
% 14.17/8.03 => ( ! [X3: int,F3: set_int] :
% 14.17/8.03 ( ( finite_finite_int @ F3 )
% 14.17/8.03 => ( ~ ( member_int @ X3 @ F3 )
% 14.17/8.03 => ( ( P @ F3 )
% 14.17/8.03 => ( P @ ( insert_int @ X3 @ F3 ) ) ) ) )
% 14.17/8.03 => ( P @ F2 ) ) ) ) ).
% 14.17/8.03
% 14.17/8.03 thf(fact_1846_finite__induct,axiom,
% 14.17/8.03 ! [F2: set_Pr4532377907799695533_nat_o,P: set_Pr4532377907799695533_nat_o > $o] :
% 14.17/8.03 ( ( finite3252695134891459830_nat_o @ F2 )
% 14.17/8.03 => ( ( P @ bot_bo7824918357723582233_nat_o )
% 14.17/8.03 => ( ! [X3: produc3658429121746597890et_nat > $o,F3: set_Pr4532377907799695533_nat_o] :
% 14.17/8.03 ( ( finite3252695134891459830_nat_o @ F3 )
% 14.17/8.03 => ( ~ ( member6576561426505652726_nat_o @ X3 @ F3 )
% 14.17/8.03 => ( ( P @ F3 )
% 14.17/8.03 => ( P @ ( insert5175938949040314269_nat_o @ X3 @ F3 ) ) ) ) )
% 14.17/8.03 => ( P @ F2 ) ) ) ) ).
% 14.17/8.03
% 14.17/8.03 thf(fact_1847_finite_Osimps,axiom,
% 14.17/8.03 ( finite6017078050557962740nteger
% 14.17/8.03 = ( ^ [A6: set_Code_integer] :
% 14.17/8.03 ( ( A6 = bot_bo3990330152332043303nteger )
% 14.17/8.03 | ? [A3: set_Code_integer,B6: code_integer] :
% 14.17/8.03 ( ( A6
% 14.17/8.03 = ( insert_Code_integer @ B6 @ A3 ) )
% 14.17/8.03 & ( finite6017078050557962740nteger @ A3 ) ) ) ) ) ).
% 14.17/8.03
% 14.17/8.03 thf(fact_1848_finite_Osimps,axiom,
% 14.17/8.03 ( finite4290736615968046902t_unit
% 14.17/8.03 = ( ^ [A6: set_Product_unit] :
% 14.17/8.03 ( ( A6 = bot_bo3957492148770167129t_unit )
% 14.17/8.03 | ? [A3: set_Product_unit,B6: product_unit] :
% 14.17/8.03 ( ( A6
% 14.17/8.03 = ( insert_Product_unit @ B6 @ A3 ) )
% 14.17/8.03 & ( finite4290736615968046902t_unit @ A3 ) ) ) ) ) ).
% 14.17/8.03
% 14.17/8.03 thf(fact_1849_finite_Osimps,axiom,
% 14.17/8.03 ( finite_finite_o
% 14.17/8.03 = ( ^ [A6: set_o] :
% 14.17/8.03 ( ( A6 = bot_bot_set_o )
% 14.17/8.03 | ? [A3: set_o,B6: $o] :
% 14.17/8.03 ( ( A6
% 14.17/8.03 = ( insert_o @ B6 @ A3 ) )
% 14.17/8.03 & ( finite_finite_o @ A3 ) ) ) ) ) ).
% 14.17/8.03
% 14.17/8.03 thf(fact_1850_finite_Osimps,axiom,
% 14.17/8.03 ( finite_finite_nat
% 14.17/8.03 = ( ^ [A6: set_nat] :
% 14.17/8.03 ( ( A6 = bot_bot_set_nat )
% 14.17/8.03 | ? [A3: set_nat,B6: nat] :
% 14.17/8.03 ( ( A6
% 14.17/8.03 = ( insert_nat @ B6 @ A3 ) )
% 14.17/8.03 & ( finite_finite_nat @ A3 ) ) ) ) ) ).
% 14.17/8.03
% 14.17/8.03 thf(fact_1851_finite_Osimps,axiom,
% 14.17/8.03 ( finite_finite_int
% 14.17/8.03 = ( ^ [A6: set_int] :
% 14.17/8.03 ( ( A6 = bot_bot_set_int )
% 14.17/8.03 | ? [A3: set_int,B6: int] :
% 14.17/8.03 ( ( A6
% 14.17/8.03 = ( insert_int @ B6 @ A3 ) )
% 14.17/8.03 & ( finite_finite_int @ A3 ) ) ) ) ) ).
% 14.17/8.03
% 14.17/8.03 thf(fact_1852_finite_Osimps,axiom,
% 14.17/8.03 ( finite3252695134891459830_nat_o
% 14.17/8.03 = ( ^ [A6: set_Pr4532377907799695533_nat_o] :
% 14.17/8.03 ( ( A6 = bot_bo7824918357723582233_nat_o )
% 14.17/8.03 | ? [A3: set_Pr4532377907799695533_nat_o,B6: produc3658429121746597890et_nat > $o] :
% 14.17/8.03 ( ( A6
% 14.17/8.03 = ( insert5175938949040314269_nat_o @ B6 @ A3 ) )
% 14.17/8.03 & ( finite3252695134891459830_nat_o @ A3 ) ) ) ) ) ).
% 14.17/8.03
% 14.17/8.03 thf(fact_1853_infinite__imp__nonempty,axiom,
% 14.17/8.03 ! [S: set_Code_integer] :
% 14.17/8.03 ( ~ ( finite6017078050557962740nteger @ S )
% 14.17/8.03 => ( S != bot_bo3990330152332043303nteger ) ) ).
% 14.17/8.03
% 14.17/8.03 thf(fact_1854_infinite__imp__nonempty,axiom,
% 14.17/8.03 ! [S: set_Product_unit] :
% 14.17/8.03 ( ~ ( finite4290736615968046902t_unit @ S )
% 14.17/8.03 => ( S != bot_bo3957492148770167129t_unit ) ) ).
% 14.17/8.03
% 14.17/8.03 thf(fact_1855_infinite__imp__nonempty,axiom,
% 14.17/8.03 ! [S: set_o] :
% 14.17/8.03 ( ~ ( finite_finite_o @ S )
% 14.17/8.03 => ( S != bot_bot_set_o ) ) ).
% 14.17/8.03
% 14.17/8.03 thf(fact_1856_infinite__imp__nonempty,axiom,
% 14.17/8.03 ! [S: set_nat] :
% 14.17/8.03 ( ~ ( finite_finite_nat @ S )
% 14.17/8.03 => ( S != bot_bot_set_nat ) ) ).
% 14.17/8.03
% 14.17/8.03 thf(fact_1857_infinite__imp__nonempty,axiom,
% 14.17/8.03 ! [S: set_int] :
% 14.17/8.03 ( ~ ( finite_finite_int @ S )
% 14.17/8.03 => ( S != bot_bot_set_int ) ) ).
% 14.17/8.03
% 14.17/8.03 thf(fact_1858_infinite__imp__nonempty,axiom,
% 14.17/8.03 ! [S: set_Pr4532377907799695533_nat_o] :
% 14.17/8.03 ( ~ ( finite3252695134891459830_nat_o @ S )
% 14.17/8.03 => ( S != bot_bo7824918357723582233_nat_o ) ) ).
% 14.17/8.03
% 14.17/8.03 thf(fact_1859_finite_OemptyI,axiom,
% 14.17/8.03 finite6017078050557962740nteger @ bot_bo3990330152332043303nteger ).
% 14.17/8.03
% 14.17/8.03 thf(fact_1860_finite_OemptyI,axiom,
% 14.17/8.03 finite4290736615968046902t_unit @ bot_bo3957492148770167129t_unit ).
% 14.17/8.03
% 14.17/8.03 thf(fact_1861_finite_OemptyI,axiom,
% 14.17/8.03 finite_finite_o @ bot_bot_set_o ).
% 14.17/8.03
% 14.17/8.03 thf(fact_1862_finite_OemptyI,axiom,
% 14.17/8.03 finite_finite_nat @ bot_bot_set_nat ).
% 14.17/8.03
% 14.17/8.03 thf(fact_1863_finite_OemptyI,axiom,
% 14.17/8.03 finite_finite_int @ bot_bot_set_int ).
% 14.17/8.03
% 14.17/8.03 thf(fact_1864_finite_OemptyI,axiom,
% 14.17/8.03 finite3252695134891459830_nat_o @ bot_bo7824918357723582233_nat_o ).
% 14.17/8.03
% 14.17/8.03 thf(fact_1865_finite_OinsertI,axiom,
% 14.17/8.03 ! [A: set_int,A2: int] :
% 14.17/8.03 ( ( finite_finite_int @ A )
% 14.17/8.03 => ( finite_finite_int @ ( insert_int @ A2 @ A ) ) ) ).
% 14.17/8.03
% 14.17/8.03 thf(fact_1866_finite_OinsertI,axiom,
% 14.17/8.03 ! [A: set_o,A2: $o] :
% 14.17/8.03 ( ( finite_finite_o @ A )
% 14.17/8.03 => ( finite_finite_o @ ( insert_o @ A2 @ A ) ) ) ).
% 14.17/8.03
% 14.17/8.03 thf(fact_1867_finite_OinsertI,axiom,
% 14.17/8.03 ! [A: set_Product_unit,A2: product_unit] :
% 14.17/8.03 ( ( finite4290736615968046902t_unit @ A )
% 14.17/8.03 => ( finite4290736615968046902t_unit @ ( insert_Product_unit @ A2 @ A ) ) ) ).
% 14.17/8.03
% 14.17/8.03 thf(fact_1868_finite_OinsertI,axiom,
% 14.17/8.03 ! [A: set_nat,A2: nat] :
% 14.17/8.03 ( ( finite_finite_nat @ A )
% 14.17/8.03 => ( finite_finite_nat @ ( insert_nat @ A2 @ A ) ) ) ).
% 14.17/8.03
% 14.17/8.03 thf(fact_1869_finite_OinsertI,axiom,
% 14.17/8.03 ! [A: set_Code_integer,A2: code_integer] :
% 14.17/8.03 ( ( finite6017078050557962740nteger @ A )
% 14.17/8.03 => ( finite6017078050557962740nteger @ ( insert_Code_integer @ A2 @ A ) ) ) ).
% 14.17/8.03
% 14.17/8.03 thf(fact_1870_finite__has__minimal,axiom,
% 14.17/8.03 ! [A: set_Code_integer] :
% 14.17/8.03 ( ( finite6017078050557962740nteger @ A )
% 14.17/8.03 => ( ( A != bot_bo3990330152332043303nteger )
% 14.17/8.03 => ? [X3: code_integer] :
% 14.17/8.03 ( ( member_Code_integer @ X3 @ A )
% 14.17/8.03 & ! [Xa2: code_integer] :
% 14.17/8.03 ( ( member_Code_integer @ Xa2 @ A )
% 14.17/8.03 => ( ( ord_le3102999989581377725nteger @ Xa2 @ X3 )
% 14.17/8.03 => ( X3 = Xa2 ) ) ) ) ) ) ).
% 14.17/8.03
% 14.17/8.03 thf(fact_1871_finite__has__minimal,axiom,
% 14.17/8.03 ! [A: set_Product_unit] :
% 14.17/8.03 ( ( finite4290736615968046902t_unit @ A )
% 14.17/8.03 => ( ( A != bot_bo3957492148770167129t_unit )
% 14.17/8.03 => ? [X3: product_unit] :
% 14.17/8.03 ( ( member_Product_unit @ X3 @ A )
% 14.17/8.03 & ! [Xa2: product_unit] :
% 14.17/8.03 ( ( member_Product_unit @ Xa2 @ A )
% 14.17/8.03 => ( ( ord_le3221252021190050221t_unit @ Xa2 @ X3 )
% 14.17/8.03 => ( X3 = Xa2 ) ) ) ) ) ) ).
% 14.17/8.03
% 14.17/8.03 thf(fact_1872_finite__has__minimal,axiom,
% 14.17/8.03 ! [A: set_o] :
% 14.17/8.03 ( ( finite_finite_o @ A )
% 14.17/8.03 => ( ( A != bot_bot_set_o )
% 14.17/8.03 => ? [X3: $o] :
% 14.17/8.03 ( ( member_o @ X3 @ A )
% 14.17/8.03 & ! [Xa2: $o] :
% 14.17/8.03 ( ( member_o @ Xa2 @ A )
% 14.17/8.03 => ( ( ord_less_eq_o @ Xa2 @ X3 )
% 14.17/8.03 => ( X3 = Xa2 ) ) ) ) ) ) ).
% 14.17/8.03
% 14.17/8.03 thf(fact_1873_finite__has__minimal,axiom,
% 14.17/8.03 ! [A: set_Pr4532377907799695533_nat_o] :
% 14.17/8.03 ( ( finite3252695134891459830_nat_o @ A )
% 14.25/8.03 => ( ( A != bot_bo7824918357723582233_nat_o )
% 14.25/8.03 => ? [X3: produc3658429121746597890et_nat > $o] :
% 14.25/8.03 ( ( member6576561426505652726_nat_o @ X3 @ A )
% 14.25/8.03 & ! [Xa2: produc3658429121746597890et_nat > $o] :
% 14.25/8.03 ( ( member6576561426505652726_nat_o @ Xa2 @ A )
% 14.25/8.03 => ( ( ord_le729326519192465773_nat_o @ Xa2 @ X3 )
% 14.25/8.03 => ( X3 = Xa2 ) ) ) ) ) ) ).
% 14.25/8.03
% 14.25/8.03 thf(fact_1874_finite__has__minimal,axiom,
% 14.25/8.03 ! [A: set_filter_nat] :
% 14.25/8.03 ( ( finite2119507909894593271er_nat @ A )
% 14.25/8.03 => ( ( A != bot_bo498966703094740906er_nat )
% 14.25/8.03 => ? [X3: filter_nat] :
% 14.25/8.03 ( ( member_filter_nat @ X3 @ A )
% 14.25/8.03 & ! [Xa2: filter_nat] :
% 14.25/8.03 ( ( member_filter_nat @ Xa2 @ A )
% 14.25/8.03 => ( ( ord_le2510731241096832064er_nat @ Xa2 @ X3 )
% 14.25/8.03 => ( X3 = Xa2 ) ) ) ) ) ) ).
% 14.25/8.03
% 14.25/8.03 thf(fact_1875_finite__has__minimal,axiom,
% 14.25/8.03 ! [A: set_set_nat] :
% 14.25/8.03 ( ( finite1152437895449049373et_nat @ A )
% 14.25/8.03 => ( ( A != bot_bot_set_set_nat )
% 14.25/8.03 => ? [X3: set_nat] :
% 14.25/8.03 ( ( member_set_nat @ X3 @ A )
% 14.25/8.03 & ! [Xa2: set_nat] :
% 14.25/8.03 ( ( member_set_nat @ Xa2 @ A )
% 14.25/8.03 => ( ( ord_less_eq_set_nat @ Xa2 @ X3 )
% 14.25/8.03 => ( X3 = Xa2 ) ) ) ) ) ) ).
% 14.25/8.03
% 14.25/8.03 thf(fact_1876_finite__has__minimal,axiom,
% 14.25/8.03 ! [A: set_rat] :
% 14.25/8.03 ( ( finite_finite_rat @ A )
% 14.25/8.03 => ( ( A != bot_bot_set_rat )
% 14.25/8.03 => ? [X3: rat] :
% 14.25/8.03 ( ( member_rat @ X3 @ A )
% 14.25/8.03 & ! [Xa2: rat] :
% 14.25/8.03 ( ( member_rat @ Xa2 @ A )
% 14.25/8.03 => ( ( ord_less_eq_rat @ Xa2 @ X3 )
% 14.25/8.03 => ( X3 = Xa2 ) ) ) ) ) ) ).
% 14.25/8.03
% 14.25/8.03 thf(fact_1877_finite__has__minimal,axiom,
% 14.25/8.03 ! [A: set_nat] :
% 14.25/8.03 ( ( finite_finite_nat @ A )
% 14.25/8.03 => ( ( A != bot_bot_set_nat )
% 14.25/8.03 => ? [X3: nat] :
% 14.25/8.03 ( ( member_nat @ X3 @ A )
% 14.25/8.03 & ! [Xa2: nat] :
% 14.25/8.03 ( ( member_nat @ Xa2 @ A )
% 14.25/8.03 => ( ( ord_less_eq_nat @ Xa2 @ X3 )
% 14.25/8.03 => ( X3 = Xa2 ) ) ) ) ) ) ).
% 14.25/8.03
% 14.25/8.03 thf(fact_1878_finite__has__minimal,axiom,
% 14.25/8.03 ! [A: set_int] :
% 14.25/8.03 ( ( finite_finite_int @ A )
% 14.25/8.03 => ( ( A != bot_bot_set_int )
% 14.25/8.03 => ? [X3: int] :
% 14.25/8.03 ( ( member_int @ X3 @ A )
% 14.25/8.03 & ! [Xa2: int] :
% 14.25/8.03 ( ( member_int @ Xa2 @ A )
% 14.25/8.03 => ( ( ord_less_eq_int @ Xa2 @ X3 )
% 14.25/8.03 => ( X3 = Xa2 ) ) ) ) ) ) ).
% 14.25/8.03
% 14.25/8.03 thf(fact_1879_finite__has__maximal,axiom,
% 14.25/8.03 ! [A: set_Code_integer] :
% 14.25/8.03 ( ( finite6017078050557962740nteger @ A )
% 14.25/8.03 => ( ( A != bot_bo3990330152332043303nteger )
% 14.25/8.03 => ? [X3: code_integer] :
% 14.25/8.03 ( ( member_Code_integer @ X3 @ A )
% 14.25/8.03 & ! [Xa2: code_integer] :
% 14.25/8.03 ( ( member_Code_integer @ Xa2 @ A )
% 14.25/8.03 => ( ( ord_le3102999989581377725nteger @ X3 @ Xa2 )
% 14.25/8.03 => ( X3 = Xa2 ) ) ) ) ) ) ).
% 14.25/8.03
% 14.25/8.03 thf(fact_1880_finite__has__maximal,axiom,
% 14.25/8.03 ! [A: set_Product_unit] :
% 14.25/8.03 ( ( finite4290736615968046902t_unit @ A )
% 14.25/8.03 => ( ( A != bot_bo3957492148770167129t_unit )
% 14.25/8.03 => ? [X3: product_unit] :
% 14.25/8.03 ( ( member_Product_unit @ X3 @ A )
% 14.25/8.03 & ! [Xa2: product_unit] :
% 14.25/8.03 ( ( member_Product_unit @ Xa2 @ A )
% 14.25/8.03 => ( ( ord_le3221252021190050221t_unit @ X3 @ Xa2 )
% 14.25/8.03 => ( X3 = Xa2 ) ) ) ) ) ) ).
% 14.25/8.03
% 14.25/8.03 thf(fact_1881_finite__has__maximal,axiom,
% 14.25/8.03 ! [A: set_o] :
% 14.25/8.03 ( ( finite_finite_o @ A )
% 14.25/8.03 => ( ( A != bot_bot_set_o )
% 14.25/8.03 => ? [X3: $o] :
% 14.25/8.03 ( ( member_o @ X3 @ A )
% 14.25/8.03 & ! [Xa2: $o] :
% 14.25/8.03 ( ( member_o @ Xa2 @ A )
% 14.25/8.03 => ( ( ord_less_eq_o @ X3 @ Xa2 )
% 14.25/8.03 => ( X3 = Xa2 ) ) ) ) ) ) ).
% 14.25/8.03
% 14.25/8.03 thf(fact_1882_finite__has__maximal,axiom,
% 14.25/8.03 ! [A: set_Pr4532377907799695533_nat_o] :
% 14.25/8.03 ( ( finite3252695134891459830_nat_o @ A )
% 14.25/8.03 => ( ( A != bot_bo7824918357723582233_nat_o )
% 14.25/8.03 => ? [X3: produc3658429121746597890et_nat > $o] :
% 14.25/8.03 ( ( member6576561426505652726_nat_o @ X3 @ A )
% 14.25/8.03 & ! [Xa2: produc3658429121746597890et_nat > $o] :
% 14.25/8.03 ( ( member6576561426505652726_nat_o @ Xa2 @ A )
% 14.25/8.03 => ( ( ord_le729326519192465773_nat_o @ X3 @ Xa2 )
% 14.25/8.03 => ( X3 = Xa2 ) ) ) ) ) ) ).
% 14.25/8.03
% 14.25/8.03 thf(fact_1883_finite__has__maximal,axiom,
% 14.25/8.03 ! [A: set_filter_nat] :
% 14.25/8.03 ( ( finite2119507909894593271er_nat @ A )
% 14.25/8.03 => ( ( A != bot_bo498966703094740906er_nat )
% 14.25/8.03 => ? [X3: filter_nat] :
% 14.25/8.03 ( ( member_filter_nat @ X3 @ A )
% 14.25/8.03 & ! [Xa2: filter_nat] :
% 14.25/8.03 ( ( member_filter_nat @ Xa2 @ A )
% 14.25/8.03 => ( ( ord_le2510731241096832064er_nat @ X3 @ Xa2 )
% 14.25/8.03 => ( X3 = Xa2 ) ) ) ) ) ) ).
% 14.25/8.03
% 14.25/8.03 thf(fact_1884_finite__has__maximal,axiom,
% 14.25/8.03 ! [A: set_set_nat] :
% 14.25/8.03 ( ( finite1152437895449049373et_nat @ A )
% 14.25/8.03 => ( ( A != bot_bot_set_set_nat )
% 14.25/8.03 => ? [X3: set_nat] :
% 14.25/8.03 ( ( member_set_nat @ X3 @ A )
% 14.25/8.03 & ! [Xa2: set_nat] :
% 14.25/8.03 ( ( member_set_nat @ Xa2 @ A )
% 14.25/8.03 => ( ( ord_less_eq_set_nat @ X3 @ Xa2 )
% 14.25/8.03 => ( X3 = Xa2 ) ) ) ) ) ) ).
% 14.25/8.03
% 14.25/8.03 thf(fact_1885_finite__has__maximal,axiom,
% 14.25/8.03 ! [A: set_rat] :
% 14.25/8.03 ( ( finite_finite_rat @ A )
% 14.25/8.03 => ( ( A != bot_bot_set_rat )
% 14.25/8.03 => ? [X3: rat] :
% 14.25/8.03 ( ( member_rat @ X3 @ A )
% 14.25/8.03 & ! [Xa2: rat] :
% 14.25/8.03 ( ( member_rat @ Xa2 @ A )
% 14.25/8.03 => ( ( ord_less_eq_rat @ X3 @ Xa2 )
% 14.25/8.03 => ( X3 = Xa2 ) ) ) ) ) ) ).
% 14.25/8.03
% 14.25/8.03 thf(fact_1886_finite__has__maximal,axiom,
% 14.25/8.03 ! [A: set_nat] :
% 14.25/8.03 ( ( finite_finite_nat @ A )
% 14.25/8.03 => ( ( A != bot_bot_set_nat )
% 14.25/8.03 => ? [X3: nat] :
% 14.25/8.03 ( ( member_nat @ X3 @ A )
% 14.25/8.03 & ! [Xa2: nat] :
% 14.25/8.03 ( ( member_nat @ Xa2 @ A )
% 14.25/8.03 => ( ( ord_less_eq_nat @ X3 @ Xa2 )
% 14.25/8.03 => ( X3 = Xa2 ) ) ) ) ) ) ).
% 14.25/8.03
% 14.25/8.03 thf(fact_1887_finite__has__maximal,axiom,
% 14.25/8.03 ! [A: set_int] :
% 14.25/8.03 ( ( finite_finite_int @ A )
% 14.25/8.03 => ( ( A != bot_bot_set_int )
% 14.25/8.03 => ? [X3: int] :
% 14.25/8.03 ( ( member_int @ X3 @ A )
% 14.25/8.03 & ! [Xa2: int] :
% 14.25/8.03 ( ( member_int @ Xa2 @ A )
% 14.25/8.03 => ( ( ord_less_eq_int @ X3 @ Xa2 )
% 14.25/8.03 => ( X3 = Xa2 ) ) ) ) ) ) ).
% 14.25/8.03
% 14.25/8.03 thf(fact_1888_finite_Ocases,axiom,
% 14.25/8.03 ! [A2: set_Code_integer] :
% 14.25/8.03 ( ( finite6017078050557962740nteger @ A2 )
% 14.25/8.03 => ( ( A2 != bot_bo3990330152332043303nteger )
% 14.25/8.03 => ~ ! [A9: set_Code_integer] :
% 14.25/8.03 ( ? [A4: code_integer] :
% 14.25/8.03 ( A2
% 14.25/8.03 = ( insert_Code_integer @ A4 @ A9 ) )
% 14.25/8.03 => ~ ( finite6017078050557962740nteger @ A9 ) ) ) ) ).
% 14.25/8.03
% 14.25/8.03 thf(fact_1889_finite_Ocases,axiom,
% 14.25/8.03 ! [A2: set_Product_unit] :
% 14.25/8.03 ( ( finite4290736615968046902t_unit @ A2 )
% 14.25/8.03 => ( ( A2 != bot_bo3957492148770167129t_unit )
% 14.25/8.03 => ~ ! [A9: set_Product_unit] :
% 14.25/8.03 ( ? [A4: product_unit] :
% 14.25/8.03 ( A2
% 14.25/8.03 = ( insert_Product_unit @ A4 @ A9 ) )
% 14.25/8.03 => ~ ( finite4290736615968046902t_unit @ A9 ) ) ) ) ).
% 14.25/8.03
% 14.25/8.03 thf(fact_1890_finite_Ocases,axiom,
% 14.25/8.03 ! [A2: set_o] :
% 14.25/8.03 ( ( finite_finite_o @ A2 )
% 14.25/8.03 => ( ( A2 != bot_bot_set_o )
% 14.25/8.03 => ~ ! [A9: set_o] :
% 14.25/8.03 ( ? [A4: $o] :
% 14.25/8.03 ( A2
% 14.25/8.03 = ( insert_o @ A4 @ A9 ) )
% 14.25/8.03 => ~ ( finite_finite_o @ A9 ) ) ) ) ).
% 14.25/8.03
% 14.25/8.03 thf(fact_1891_finite_Ocases,axiom,
% 14.25/8.03 ! [A2: set_nat] :
% 14.25/8.03 ( ( finite_finite_nat @ A2 )
% 14.25/8.03 => ( ( A2 != bot_bot_set_nat )
% 14.25/8.03 => ~ ! [A9: set_nat] :
% 14.25/8.03 ( ? [A4: nat] :
% 14.25/8.03 ( A2
% 14.25/8.03 = ( insert_nat @ A4 @ A9 ) )
% 14.25/8.03 => ~ ( finite_finite_nat @ A9 ) ) ) ) ).
% 14.25/8.03
% 14.25/8.03 thf(fact_1892_finite_Ocases,axiom,
% 14.25/8.03 ! [A2: set_int] :
% 14.25/8.03 ( ( finite_finite_int @ A2 )
% 14.25/8.03 => ( ( A2 != bot_bot_set_int )
% 14.25/8.03 => ~ ! [A9: set_int] :
% 14.25/8.03 ( ? [A4: int] :
% 14.25/8.03 ( A2
% 14.25/8.03 = ( insert_int @ A4 @ A9 ) )
% 14.25/8.03 => ~ ( finite_finite_int @ A9 ) ) ) ) ).
% 14.25/8.03
% 14.25/8.03 thf(fact_1893_finite_Ocases,axiom,
% 14.25/8.03 ! [A2: set_Pr4532377907799695533_nat_o] :
% 14.25/8.03 ( ( finite3252695134891459830_nat_o @ A2 )
% 14.25/8.03 => ( ( A2 != bot_bo7824918357723582233_nat_o )
% 14.25/8.03 => ~ ! [A9: set_Pr4532377907799695533_nat_o] :
% 14.25/8.03 ( ? [A4: produc3658429121746597890et_nat > $o] :
% 14.25/8.03 ( A2
% 14.25/8.03 = ( insert5175938949040314269_nat_o @ A4 @ A9 ) )
% 14.25/8.03 => ~ ( finite3252695134891459830_nat_o @ A9 ) ) ) ) ).
% 14.25/8.03
% 14.25/8.03 thf(fact_1894_min__ext__compat,axiom,
% 14.25/8.03 ! [R3: set_Pr5094982260447487303t_unit,S: set_Pr5094982260447487303t_unit] :
% 14.25/8.03 ( ( ord_le930853871177014503t_unit @ ( relcom848357998638694028t_unit @ R3 @ S ) @ R3 )
% 14.25/8.03 => ( ord_le8072594200992201959t_unit @ ( relcom7108816970757792620t_unit @ ( min_ext_Product_unit @ R3 ) @ ( sup_su3573767163274163995t_unit @ ( min_ext_Product_unit @ S ) @ ( insert6793119152790066487t_unit @ ( produc7269801343787132631t_unit @ bot_bo3957492148770167129t_unit @ bot_bo3957492148770167129t_unit ) @ bot_bo5113747506523071411t_unit ) ) ) @ ( min_ext_Product_unit @ R3 ) ) ) ).
% 14.25/8.03
% 14.25/8.03 thf(fact_1895_min__ext__compat,axiom,
% 14.25/8.03 ! [R3: set_Product_prod_o_o,S: set_Product_prod_o_o] :
% 14.25/8.03 ( ( ord_le192195778054999463od_o_o @ ( relcomp_o_o_o @ R3 @ S ) @ R3 )
% 14.25/8.03 => ( ord_le2885120122233735591_set_o @ ( relcom7534720040753365047_set_o @ ( min_ext_o @ R3 ) @ ( sup_su7766833992073560027_set_o @ ( min_ext_o @ S ) @ ( insert1494450971503307511_set_o @ ( produc5838405689764958487_set_o @ bot_bot_set_o @ bot_bot_set_o ) @ bot_bo8143257439550059635_set_o ) ) ) @ ( min_ext_o @ R3 ) ) ) ).
% 14.25/8.03
% 14.25/8.03 thf(fact_1896_min__ext__compat,axiom,
% 14.25/8.03 ! [R3: set_Pr1261947904930325089at_nat,S: set_Pr1261947904930325089at_nat] :
% 14.25/8.03 ( ( ord_le3146513528884898305at_nat @ ( relcomp_nat_nat_nat @ R3 @ S ) @ R3 )
% 14.25/8.03 => ( ord_le4402255868550298733et_nat @ ( relcom236561585145426837et_nat @ ( min_ext_nat @ R3 ) @ ( sup_su4251893418135750817et_nat @ ( min_ext_nat @ S ) @ ( insert3810226134351308605et_nat @ ( produc4532415448927165861et_nat @ bot_bot_set_nat @ bot_bot_set_nat ) @ bot_bo4952753636535259449et_nat ) ) ) @ ( min_ext_nat @ R3 ) ) ) ).
% 14.25/8.03
% 14.25/8.03 thf(fact_1897_min__ext__compat,axiom,
% 14.25/8.03 ! [R3: set_Pr958786334691620121nt_int,S: set_Pr958786334691620121nt_int] :
% 14.25/8.03 ( ( ord_le2843351958646193337nt_int @ ( relcomp_int_int_int @ R3 @ S ) @ R3 )
% 14.25/8.03 => ( ord_le1436784781161120549et_int @ ( relcom4034950705519254697et_int @ ( min_ext_int @ R3 ) @ ( sup_su1286422330746572633et_int @ ( min_ext_int @ S ) @ ( insert7328952640415156725et_int @ ( produc6363374080413544029et_int @ bot_bot_set_int @ bot_bot_set_int ) @ bot_bo1987282549146081265et_int ) ) ) @ ( min_ext_int @ R3 ) ) ) ).
% 14.25/8.03
% 14.25/8.03 thf(fact_1898_min__ext__compat,axiom,
% 14.25/8.03 ! [R3: set_Pr2161125870931222855_nat_o,S: set_Pr2161125870931222855_nat_o] :
% 14.25/8.03 ( ( ord_le6659818918159358183_nat_o @ ( relcom6251035627269349452_nat_o @ R3 @ S ) @ R3 )
% 14.25/8.03 => ( ord_le8001614231985977575_nat_o @ ( relcom7236094467932586796_nat_o @ ( min_ex3730736537596016502_nat_o @ R3 ) @ ( sup_su4863159966000489243_nat_o @ ( min_ex3730736537596016502_nat_o @ S ) @ ( insert3244108804717745463_nat_o @ ( produc5149316574908490455_nat_o @ bot_bo7824918357723582233_nat_o @ bot_bo7824918357723582233_nat_o ) @ bot_bo7535334947429593011_nat_o ) ) ) @ ( min_ex3730736537596016502_nat_o @ R3 ) ) ) ).
% 14.25/8.03
% 14.25/8.03 thf(fact_1899_arg__min__least,axiom,
% 14.25/8.03 ! [S: set_Code_integer,Y: code_integer,F: code_integer > rat] :
% 14.25/8.03 ( ( finite6017078050557962740nteger @ S )
% 14.25/8.03 => ( ( S != bot_bo3990330152332043303nteger )
% 14.25/8.03 => ( ( member_Code_integer @ Y @ S )
% 14.25/8.03 => ( ord_less_eq_rat @ ( F @ ( lattic5439806495466278992er_rat @ F @ S ) ) @ ( F @ Y ) ) ) ) ) ).
% 14.25/8.03
% 14.25/8.03 thf(fact_1900_arg__min__least,axiom,
% 14.25/8.03 ! [S: set_Product_unit,Y: product_unit,F: product_unit > rat] :
% 14.25/8.03 ( ( finite4290736615968046902t_unit @ S )
% 14.25/8.03 => ( ( S != bot_bo3957492148770167129t_unit )
% 14.25/8.03 => ( ( member_Product_unit @ Y @ S )
% 14.25/8.03 => ( ord_less_eq_rat @ ( F @ ( lattic725702332050428812it_rat @ F @ S ) ) @ ( F @ Y ) ) ) ) ) ).
% 14.25/8.03
% 14.25/8.03 thf(fact_1901_arg__min__least,axiom,
% 14.25/8.03 ! [S: set_o,Y: $o,F: $o > rat] :
% 14.25/8.03 ( ( finite_finite_o @ S )
% 14.25/8.03 => ( ( S != bot_bot_set_o )
% 14.25/8.03 => ( ( member_o @ Y @ S )
% 14.25/8.03 => ( ord_less_eq_rat @ ( F @ ( lattic2140725968369957399_o_rat @ F @ S ) ) @ ( F @ Y ) ) ) ) ) ).
% 14.25/8.03
% 14.25/8.03 thf(fact_1902_arg__min__least,axiom,
% 14.25/8.03 ! [S: set_nat,Y: nat,F: nat > rat] :
% 14.25/8.03 ( ( finite_finite_nat @ S )
% 14.25/8.03 => ( ( S != bot_bot_set_nat )
% 14.25/8.03 => ( ( member_nat @ Y @ S )
% 14.25/8.03 => ( ord_less_eq_rat @ ( F @ ( lattic6811802900495863747at_rat @ F @ S ) ) @ ( F @ Y ) ) ) ) ) ).
% 14.25/8.03
% 14.25/8.03 thf(fact_1903_arg__min__least,axiom,
% 14.25/8.03 ! [S: set_int,Y: int,F: int > rat] :
% 14.25/8.03 ( ( finite_finite_int @ S )
% 14.25/8.03 => ( ( S != bot_bot_set_int )
% 14.25/8.03 => ( ( member_int @ Y @ S )
% 14.25/8.03 => ( ord_less_eq_rat @ ( F @ ( lattic7811156612396918303nt_rat @ F @ S ) ) @ ( F @ Y ) ) ) ) ) ).
% 14.25/8.03
% 14.25/8.03 thf(fact_1904_arg__min__least,axiom,
% 14.25/8.03 ! [S: set_Code_integer,Y: code_integer,F: code_integer > nat] :
% 14.25/8.03 ( ( finite6017078050557962740nteger @ S )
% 14.25/8.03 => ( ( S != bot_bo3990330152332043303nteger )
% 14.25/8.03 => ( ( member_Code_integer @ Y @ S )
% 14.25/8.03 => ( ord_less_eq_nat @ ( F @ ( lattic6074936555552774728er_nat @ F @ S ) ) @ ( F @ Y ) ) ) ) ) ).
% 14.25/8.03
% 14.25/8.03 thf(fact_1905_arg__min__least,axiom,
% 14.25/8.03 ! [S: set_Product_unit,Y: product_unit,F: product_unit > nat] :
% 14.25/8.03 ( ( finite4290736615968046902t_unit @ S )
% 14.25/8.03 => ( ( S != bot_bo3957492148770167129t_unit )
% 14.25/8.03 => ( ( member_Product_unit @ Y @ S )
% 14.25/8.03 => ( ord_less_eq_nat @ ( F @ ( lattic1360832392136924548it_nat @ F @ S ) ) @ ( F @ Y ) ) ) ) ) ).
% 14.25/8.03
% 14.25/8.03 thf(fact_1906_arg__min__least,axiom,
% 14.25/8.03 ! [S: set_o,Y: $o,F: $o > nat] :
% 14.25/8.03 ( ( finite_finite_o @ S )
% 14.25/8.03 => ( ( S != bot_bot_set_o )
% 14.25/8.03 => ( ( member_o @ Y @ S )
% 14.25/8.03 => ( ord_less_eq_nat @ ( F @ ( lattic2775856028456453135_o_nat @ F @ S ) ) @ ( F @ Y ) ) ) ) ) ).
% 14.25/8.03
% 14.25/8.03 thf(fact_1907_arg__min__least,axiom,
% 14.25/8.03 ! [S: set_nat,Y: nat,F: nat > nat] :
% 14.25/8.03 ( ( finite_finite_nat @ S )
% 14.25/8.03 => ( ( S != bot_bot_set_nat )
% 14.25/8.03 => ( ( member_nat @ Y @ S )
% 14.25/8.03 => ( ord_less_eq_nat @ ( F @ ( lattic7446932960582359483at_nat @ F @ S ) ) @ ( F @ Y ) ) ) ) ) ).
% 14.25/8.03
% 14.25/8.03 thf(fact_1908_arg__min__least,axiom,
% 14.25/8.03 ! [S: set_int,Y: int,F: int > nat] :
% 14.25/8.03 ( ( finite_finite_int @ S )
% 14.25/8.03 => ( ( S != bot_bot_set_int )
% 14.25/8.03 => ( ( member_int @ Y @ S )
% 14.25/8.03 => ( ord_less_eq_nat @ ( F @ ( lattic8446286672483414039nt_nat @ F @ S ) ) @ ( F @ Y ) ) ) ) ) ).
% 14.25/8.03
% 14.25/8.03 thf(fact_1909_Total__subset__Id,axiom,
% 14.25/8.03 ! [R2: set_Pr1261947904930325089at_nat] :
% 14.25/8.03 ( ( total_on_nat @ ( field_nat @ R2 ) @ R2 )
% 14.25/8.03 => ( ( ord_le3146513528884898305at_nat @ R2 @ id_nat )
% 14.25/8.03 => ( ( R2 = bot_bo2099793752762293965at_nat )
% 14.25/8.03 | ? [A4: nat] :
% 14.25/8.03 ( R2
% 14.25/8.03 = ( insert8211810215607154385at_nat @ ( product_Pair_nat_nat @ A4 @ A4 ) @ bot_bo2099793752762293965at_nat ) ) ) ) ) ).
% 14.25/8.03
% 14.25/8.03 thf(fact_1910_Total__subset__Id,axiom,
% 14.25/8.03 ! [R2: set_Pr958786334691620121nt_int] :
% 14.25/8.03 ( ( total_on_int @ ( field_int @ R2 ) @ R2 )
% 14.25/8.03 => ( ( ord_le2843351958646193337nt_int @ R2 @ id_int )
% 14.25/8.03 => ( ( R2 = bot_bo1796632182523588997nt_int )
% 14.25/8.03 | ? [A4: int] :
% 14.25/8.03 ( R2
% 14.25/8.03 = ( insert5033312907999012233nt_int @ ( product_Pair_int_int @ A4 @ A4 ) @ bot_bo1796632182523588997nt_int ) ) ) ) ) ).
% 14.25/8.03
% 14.25/8.03 thf(fact_1911_finite__transitivity__chain,axiom,
% 14.25/8.03 ! [A: set_Code_integer,R3: code_integer > code_integer > $o] :
% 14.25/8.03 ( ( finite6017078050557962740nteger @ A )
% 14.25/8.03 => ( ! [X3: code_integer] :
% 14.25/8.03 ~ ( R3 @ X3 @ X3 )
% 14.25/8.03 => ( ! [X3: code_integer,Y2: code_integer,Z3: code_integer] :
% 14.25/8.03 ( ( R3 @ X3 @ Y2 )
% 14.25/8.03 => ( ( R3 @ Y2 @ Z3 )
% 14.25/8.03 => ( R3 @ X3 @ Z3 ) ) )
% 14.25/8.03 => ( ! [X3: code_integer] :
% 14.25/8.03 ( ( member_Code_integer @ X3 @ A )
% 14.25/8.03 => ? [Y5: code_integer] :
% 14.25/8.03 ( ( member_Code_integer @ Y5 @ A )
% 14.25/8.03 & ( R3 @ X3 @ Y5 ) ) )
% 14.25/8.03 => ( A = bot_bo3990330152332043303nteger ) ) ) ) ) ).
% 14.25/8.03
% 14.25/8.03 thf(fact_1912_finite__transitivity__chain,axiom,
% 14.25/8.03 ! [A: set_Product_unit,R3: product_unit > product_unit > $o] :
% 14.25/8.03 ( ( finite4290736615968046902t_unit @ A )
% 14.25/8.03 => ( ! [X3: product_unit] :
% 14.25/8.03 ~ ( R3 @ X3 @ X3 )
% 14.25/8.03 => ( ! [X3: product_unit,Y2: product_unit,Z3: product_unit] :
% 14.25/8.03 ( ( R3 @ X3 @ Y2 )
% 14.25/8.03 => ( ( R3 @ Y2 @ Z3 )
% 14.25/8.03 => ( R3 @ X3 @ Z3 ) ) )
% 14.25/8.03 => ( ! [X3: product_unit] :
% 14.25/8.03 ( ( member_Product_unit @ X3 @ A )
% 14.25/8.03 => ? [Y5: product_unit] :
% 14.25/8.03 ( ( member_Product_unit @ Y5 @ A )
% 14.25/8.03 & ( R3 @ X3 @ Y5 ) ) )
% 14.25/8.03 => ( A = bot_bo3957492148770167129t_unit ) ) ) ) ) ).
% 14.25/8.03
% 14.25/8.03 thf(fact_1913_finite__transitivity__chain,axiom,
% 14.25/8.03 ! [A: set_o,R3: $o > $o > $o] :
% 14.25/8.03 ( ( finite_finite_o @ A )
% 14.25/8.03 => ( ! [X3: $o] :
% 14.25/8.03 ~ ( R3 @ X3 @ X3 )
% 14.25/8.03 => ( ! [X3: $o,Y2: $o,Z3: $o] :
% 14.25/8.03 ( ( R3 @ X3 @ Y2 )
% 14.25/8.03 => ( ( R3 @ Y2 @ Z3 )
% 14.25/8.03 => ( R3 @ X3 @ Z3 ) ) )
% 14.25/8.03 => ( ! [X3: $o] :
% 14.25/8.03 ( ( member_o @ X3 @ A )
% 14.25/8.03 => ? [Y5: $o] :
% 14.25/8.03 ( ( member_o @ Y5 @ A )
% 14.25/8.03 & ( R3 @ X3 @ Y5 ) ) )
% 14.25/8.03 => ( A = bot_bot_set_o ) ) ) ) ) ).
% 14.25/8.03
% 14.25/8.03 thf(fact_1914_finite__transitivity__chain,axiom,
% 14.25/8.03 ! [A: set_nat,R3: nat > nat > $o] :
% 14.25/8.03 ( ( finite_finite_nat @ A )
% 14.25/8.03 => ( ! [X3: nat] :
% 14.25/8.03 ~ ( R3 @ X3 @ X3 )
% 14.25/8.03 => ( ! [X3: nat,Y2: nat,Z3: nat] :
% 14.25/8.03 ( ( R3 @ X3 @ Y2 )
% 14.25/8.03 => ( ( R3 @ Y2 @ Z3 )
% 14.25/8.03 => ( R3 @ X3 @ Z3 ) ) )
% 14.25/8.03 => ( ! [X3: nat] :
% 14.25/8.03 ( ( member_nat @ X3 @ A )
% 14.25/8.03 => ? [Y5: nat] :
% 14.25/8.03 ( ( member_nat @ Y5 @ A )
% 14.25/8.03 & ( R3 @ X3 @ Y5 ) ) )
% 14.25/8.03 => ( A = bot_bot_set_nat ) ) ) ) ) ).
% 14.25/8.03
% 14.25/8.03 thf(fact_1915_finite__transitivity__chain,axiom,
% 14.25/8.03 ! [A: set_int,R3: int > int > $o] :
% 14.25/8.03 ( ( finite_finite_int @ A )
% 14.25/8.03 => ( ! [X3: int] :
% 14.25/8.03 ~ ( R3 @ X3 @ X3 )
% 14.25/8.03 => ( ! [X3: int,Y2: int,Z3: int] :
% 14.25/8.03 ( ( R3 @ X3 @ Y2 )
% 14.25/8.03 => ( ( R3 @ Y2 @ Z3 )
% 14.25/8.03 => ( R3 @ X3 @ Z3 ) ) )
% 14.25/8.03 => ( ! [X3: int] :
% 14.25/8.03 ( ( member_int @ X3 @ A )
% 14.25/8.03 => ? [Y5: int] :
% 14.25/8.03 ( ( member_int @ Y5 @ A )
% 14.25/8.03 & ( R3 @ X3 @ Y5 ) ) )
% 14.25/8.03 => ( A = bot_bot_set_int ) ) ) ) ) ).
% 14.25/8.03
% 14.25/8.03 thf(fact_1916_finite__transitivity__chain,axiom,
% 14.25/8.03 ! [A: set_Pr4532377907799695533_nat_o,R3: ( produc3658429121746597890et_nat > $o ) > ( produc3658429121746597890et_nat > $o ) > $o] :
% 14.25/8.03 ( ( finite3252695134891459830_nat_o @ A )
% 14.25/8.03 => ( ! [X3: produc3658429121746597890et_nat > $o] :
% 14.25/8.03 ~ ( R3 @ X3 @ X3 )
% 14.25/8.03 => ( ! [X3: produc3658429121746597890et_nat > $o,Y2: produc3658429121746597890et_nat > $o,Z3: produc3658429121746597890et_nat > $o] :
% 14.25/8.03 ( ( R3 @ X3 @ Y2 )
% 14.25/8.03 => ( ( R3 @ Y2 @ Z3 )
% 14.25/8.03 => ( R3 @ X3 @ Z3 ) ) )
% 14.25/8.03 => ( ! [X3: produc3658429121746597890et_nat > $o] :
% 14.25/8.03 ( ( member6576561426505652726_nat_o @ X3 @ A )
% 14.25/8.03 => ? [Y5: produc3658429121746597890et_nat > $o] :
% 14.25/8.03 ( ( member6576561426505652726_nat_o @ Y5 @ A )
% 14.25/8.03 & ( R3 @ X3 @ Y5 ) ) )
% 14.25/8.03 => ( A = bot_bo7824918357723582233_nat_o ) ) ) ) ) ).
% 14.25/8.03
% 14.25/8.03 thf(fact_1917_disjnt__equiv__class,axiom,
% 14.25/8.03 ! [A: set_Product_unit,R2: set_Pr5094982260447487303t_unit,A2: product_unit,B: product_unit] :
% 14.25/8.03 ( ( equiv_4189356295871601603t_unit @ A @ R2 )
% 14.25/8.03 => ( ( disjnt_Product_unit @ ( image_8431784415008483091t_unit @ R2 @ ( insert_Product_unit @ A2 @ bot_bo3957492148770167129t_unit ) ) @ ( image_8431784415008483091t_unit @ R2 @ ( insert_Product_unit @ B @ bot_bo3957492148770167129t_unit ) ) )
% 14.25/8.03 = ( ~ ( member7821138191923116944t_unit @ ( produc5692694829384537815t_unit @ A2 @ B ) @ R2 ) ) ) ) ).
% 14.25/8.03
% 14.25/8.03 thf(fact_1918_disjnt__equiv__class,axiom,
% 14.25/8.03 ! [A: set_o,R2: set_Product_prod_o_o,A2: $o,B: $o] :
% 14.25/8.03 ( ( equiv_equiv_o @ A @ R2 )
% 14.25/8.03 => ( ( disjnt_o @ ( image_o_o @ R2 @ ( insert_o @ A2 @ bot_bot_set_o ) ) @ ( image_o_o @ R2 @ ( insert_o @ B @ bot_bot_set_o ) ) )
% 14.25/8.03 = ( ~ ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ A2 @ B ) @ R2 ) ) ) ) ).
% 14.25/8.03
% 14.25/8.03 thf(fact_1919_disjnt__equiv__class,axiom,
% 14.25/8.03 ! [A: set_nat,R2: set_Pr1261947904930325089at_nat,A2: nat,B: nat] :
% 14.25/8.03 ( ( equiv_equiv_nat @ A @ R2 )
% 14.25/8.03 => ( ( disjnt_nat @ ( image_nat_nat @ R2 @ ( insert_nat @ A2 @ bot_bot_set_nat ) ) @ ( image_nat_nat @ R2 @ ( insert_nat @ B @ bot_bot_set_nat ) ) )
% 14.25/8.03 = ( ~ ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A2 @ B ) @ R2 ) ) ) ) ).
% 14.25/8.03
% 14.25/8.03 thf(fact_1920_disjnt__equiv__class,axiom,
% 14.25/8.03 ! [A: set_int,R2: set_Pr958786334691620121nt_int,A2: int,B: int] :
% 14.25/8.03 ( ( equiv_equiv_int @ A @ R2 )
% 14.25/8.03 => ( ( disjnt_int @ ( image_int_int @ R2 @ ( insert_int @ A2 @ bot_bot_set_int ) ) @ ( image_int_int @ R2 @ ( insert_int @ B @ bot_bot_set_int ) ) )
% 14.25/8.03 = ( ~ ( member5262025264175285858nt_int @ ( product_Pair_int_int @ A2 @ B ) @ R2 ) ) ) ) ).
% 14.25/8.03
% 14.25/8.03 thf(fact_1921_disjnt__equiv__class,axiom,
% 14.25/8.03 ! [A: set_Pr4532377907799695533_nat_o,R2: set_Pr2161125870931222855_nat_o,A2: produc3658429121746597890et_nat > $o,B: produc3658429121746597890et_nat > $o] :
% 14.25/8.03 ( ( equiv_6726761681920839555_nat_o @ A @ R2 )
% 14.25/8.03 => ( ( disjnt7869448817988085652_nat_o @ ( image_2232984160439939859_nat_o @ R2 @ ( insert5175938949040314269_nat_o @ A2 @ bot_bo7824918357723582233_nat_o ) ) @ ( image_2232984160439939859_nat_o @ R2 @ ( insert5175938949040314269_nat_o @ B @ bot_bo7824918357723582233_nat_o ) ) )
% 14.25/8.03 = ( ~ ( member8781333585448626064_nat_o @ ( produc7368190662567826135_nat_o @ A2 @ B ) @ R2 ) ) ) ) ).
% 14.25/8.03
% 14.25/8.03 thf(fact_1922_Sup__fin_Oinsert,axiom,
% 14.25/8.03 ! [A: set_set_nat,X: set_nat] :
% 14.25/8.03 ( ( finite1152437895449049373et_nat @ A )
% 14.25/8.03 => ( ( A != bot_bot_set_set_nat )
% 14.25/8.03 => ( ( lattic3835124923745554447et_nat @ ( insert_set_nat @ X @ A ) )
% 14.25/8.03 = ( sup_sup_set_nat @ X @ ( lattic3835124923745554447et_nat @ A ) ) ) ) ) ).
% 14.25/8.03
% 14.25/8.03 thf(fact_1923_Sup__fin_Oinsert,axiom,
% 14.25/8.03 ! [A: set_assn,X: assn] :
% 14.25/8.03 ( ( finite_finite_assn @ A )
% 14.25/8.03 => ( ( A != bot_bot_set_assn )
% 14.25/8.03 => ( ( lattic2150320897289308081n_assn @ ( insert_assn @ X @ A ) )
% 14.25/8.03 = ( sup_sup_assn @ X @ ( lattic2150320897289308081n_assn @ A ) ) ) ) ) ).
% 14.25/8.03
% 14.25/8.03 thf(fact_1924_Sup__fin_Oinsert,axiom,
% 14.25/8.03 ! [A: set_Product_unit,X: product_unit] :
% 14.25/8.03 ( ( finite4290736615968046902t_unit @ A )
% 14.25/8.03 => ( ( A != bot_bo3957492148770167129t_unit )
% 14.25/8.03 => ( ( lattic5294303975357428420t_unit @ ( insert_Product_unit @ X @ A ) )
% 14.25/8.03 = ( sup_sup_Product_unit @ X @ ( lattic5294303975357428420t_unit @ A ) ) ) ) ) ).
% 14.25/8.03
% 14.25/8.03 thf(fact_1925_Sup__fin_Oinsert,axiom,
% 14.25/8.03 ! [A: set_o,X: $o] :
% 14.25/8.03 ( ( finite_finite_o @ A )
% 14.25/8.03 => ( ( A != bot_bot_set_o )
% 14.25/8.03 => ( ( lattic1508158080041050831_fin_o @ ( insert_o @ X @ A ) )
% 14.25/8.03 = ( sup_sup_o @ X @ ( lattic1508158080041050831_fin_o @ A ) ) ) ) ) ).
% 14.25/8.03
% 14.25/8.03 thf(fact_1926_Sup__fin_Oinsert,axiom,
% 14.25/8.03 ! [A: set_nat,X: nat] :
% 14.25/8.03 ( ( finite_finite_nat @ A )
% 14.25/8.03 => ( ( A != bot_bot_set_nat )
% 14.25/8.03 => ( ( lattic1093996805478795353in_nat @ ( insert_nat @ X @ A ) )
% 14.25/8.03 = ( sup_sup_nat @ X @ ( lattic1093996805478795353in_nat @ A ) ) ) ) ) ).
% 14.25/8.03
% 14.25/8.03 thf(fact_1927_Sup__fin_Oinsert,axiom,
% 14.25/8.03 ! [A: set_int,X: int] :
% 14.25/8.03 ( ( finite_finite_int @ A )
% 14.25/8.03 => ( ( A != bot_bot_set_int )
% 14.25/8.03 => ( ( lattic1091506334969745077in_int @ ( insert_int @ X @ A ) )
% 14.25/8.03 = ( sup_sup_int @ X @ ( lattic1091506334969745077in_int @ A ) ) ) ) ) ).
% 14.25/8.03
% 14.25/8.03 thf(fact_1928_Sup__fin_Oinsert,axiom,
% 14.25/8.03 ! [A: set_Pr4532377907799695533_nat_o,X: produc3658429121746597890et_nat > $o] :
% 14.25/8.03 ( ( finite3252695134891459830_nat_o @ A )
% 14.25/8.03 => ( ( A != bot_bo7824918357723582233_nat_o )
% 14.25/8.03 => ( ( lattic7320199455484906628_nat_o @ ( insert5175938949040314269_nat_o @ X @ A ) )
% 14.25/8.03 = ( sup_su5453871518329203617_nat_o @ X @ ( lattic7320199455484906628_nat_o @ A ) ) ) ) ) ).
% 14.25/8.03
% 14.25/8.03 thf(fact_1929_Inf__fin_Oinsert,axiom,
% 14.25/8.03 ! [A: set_set_nat,X: set_nat] :
% 14.25/8.03 ( ( finite1152437895449049373et_nat @ A )
% 14.25/8.03 => ( ( A != bot_bot_set_set_nat )
% 14.25/8.03 => ( ( lattic3014633134055518761et_nat @ ( insert_set_nat @ X @ A ) )
% 14.25/8.03 = ( inf_inf_set_nat @ X @ ( lattic3014633134055518761et_nat @ A ) ) ) ) ) ).
% 14.25/8.03
% 14.25/8.03 thf(fact_1930_Inf__fin_Oinsert,axiom,
% 14.25/8.03 ! [A: set_assn,X: assn] :
% 14.25/8.03 ( ( finite_finite_assn @ A )
% 14.25/8.03 => ( ( A != bot_bot_set_assn )
% 14.25/8.03 => ( ( lattic47131356835913163n_assn @ ( insert_assn @ X @ A ) )
% 14.25/8.03 = ( inf_inf_assn @ X @ ( lattic47131356835913163n_assn @ A ) ) ) ) ) ).
% 14.25/8.03
% 14.25/8.03 thf(fact_1931_Inf__fin_Oinsert,axiom,
% 14.25/8.03 ! [A: set_Product_unit,X: product_unit] :
% 14.25/8.03 ( ( finite4290736615968046902t_unit @ A )
% 14.25/8.03 => ( ( A != bot_bo3957492148770167129t_unit )
% 14.25/8.03 => ( ( lattic1263872656861969706t_unit @ ( insert_Product_unit @ X @ A ) )
% 14.25/8.03 = ( inf_inf_Product_unit @ X @ ( lattic1263872656861969706t_unit @ A ) ) ) ) ) ).
% 14.25/8.03
% 14.25/8.03 thf(fact_1932_Inf__fin_Oinsert,axiom,
% 14.25/8.03 ! [A: set_o,X: $o] :
% 14.25/8.03 ( ( finite_finite_o @ A )
% 14.25/8.03 => ( ( A != bot_bot_set_o )
% 14.25/8.03 => ( ( lattic4107685809792843317_fin_o @ ( insert_o @ X @ A ) )
% 14.25/8.03 = ( inf_inf_o @ X @ ( lattic4107685809792843317_fin_o @ A ) ) ) ) ) ).
% 14.25/8.03
% 14.25/8.03 thf(fact_1933_Inf__fin_Oinsert,axiom,
% 14.25/8.03 ! [A: set_nat,X: nat] :
% 14.25/8.03 ( ( finite_finite_nat @ A )
% 14.25/8.03 => ( ( A != bot_bot_set_nat )
% 14.25/8.03 => ( ( lattic5238388535129920115in_nat @ ( insert_nat @ X @ A ) )
% 14.25/8.03 = ( inf_inf_nat @ X @ ( lattic5238388535129920115in_nat @ A ) ) ) ) ) ).
% 14.25/8.03
% 14.25/8.03 thf(fact_1934_Inf__fin_Oinsert,axiom,
% 14.25/8.03 ! [A: set_int,X: int] :
% 14.25/8.03 ( ( finite_finite_int @ A )
% 14.25/8.03 => ( ( A != bot_bot_set_int )
% 14.25/8.03 => ( ( lattic5235898064620869839in_int @ ( insert_int @ X @ A ) )
% 14.25/8.03 = ( inf_inf_int @ X @ ( lattic5235898064620869839in_int @ A ) ) ) ) ) ).
% 14.25/8.03
% 14.25/8.03 thf(fact_1935_Inf__fin_Oinsert,axiom,
% 14.25/8.03 ! [A: set_Pr4532377907799695533_nat_o,X: produc3658429121746597890et_nat > $o] :
% 14.25/8.03 ( ( finite3252695134891459830_nat_o @ A )
% 14.25/8.03 => ( ( A != bot_bo7824918357723582233_nat_o )
% 14.25/8.03 => ( ( lattic956194824204696298_nat_o @ ( insert5175938949040314269_nat_o @ X @ A ) )
% 14.25/8.03 = ( inf_in1318976480646536635_nat_o @ X @ ( lattic956194824204696298_nat_o @ A ) ) ) ) ) ).
% 14.25/8.03
% 14.25/8.03 thf(fact_1936_in__quotient__imp__in__rel,axiom,
% 14.25/8.03 ! [A: set_Product_unit,R2: set_Pr5094982260447487303t_unit,X4: set_Product_unit,X: product_unit,Y: product_unit] :
% 14.25/8.03 ( ( equiv_4189356295871601603t_unit @ A @ R2 )
% 14.25/8.03 => ( ( member5877623283571906838t_unit @ X4 @ ( equiv_5172566880711946708t_unit @ A @ R2 ) )
% 14.25/8.03 => ( ( ord_le3507040750410214029t_unit @ ( insert_Product_unit @ X @ ( insert_Product_unit @ Y @ bot_bo3957492148770167129t_unit ) ) @ X4 )
% 14.25/8.03 => ( member7821138191923116944t_unit @ ( produc5692694829384537815t_unit @ X @ Y ) @ R2 ) ) ) ) ).
% 14.25/8.03
% 14.25/8.03 thf(fact_1937_in__quotient__imp__in__rel,axiom,
% 14.25/8.03 ! [A: set_o,R2: set_Product_prod_o_o,X4: set_o,X: $o,Y: $o] :
% 14.25/8.03 ( ( equiv_equiv_o @ A @ R2 )
% 14.25/8.03 => ( ( member_set_o @ X4 @ ( equiv_quotient_o @ A @ R2 ) )
% 14.25/8.03 => ( ( ord_less_eq_set_o @ ( insert_o @ X @ ( insert_o @ Y @ bot_bot_set_o ) ) @ X4 )
% 14.25/8.03 => ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ X @ Y ) @ R2 ) ) ) ) ).
% 14.25/8.03
% 14.25/8.03 thf(fact_1938_in__quotient__imp__in__rel,axiom,
% 14.25/8.03 ! [A: set_int,R2: set_Pr958786334691620121nt_int,X4: set_int,X: int,Y: int] :
% 14.25/8.03 ( ( equiv_equiv_int @ A @ R2 )
% 14.25/8.03 => ( ( member_set_int @ X4 @ ( equiv_quotient_int @ A @ R2 ) )
% 14.25/8.03 => ( ( ord_less_eq_set_int @ ( insert_int @ X @ ( insert_int @ Y @ bot_bot_set_int ) ) @ X4 )
% 14.25/8.03 => ( member5262025264175285858nt_int @ ( product_Pair_int_int @ X @ Y ) @ R2 ) ) ) ) ).
% 14.25/8.03
% 14.25/8.03 thf(fact_1939_in__quotient__imp__in__rel,axiom,
% 14.25/8.03 ! [A: set_Pr4532377907799695533_nat_o,R2: set_Pr2161125870931222855_nat_o,X4: set_Pr4532377907799695533_nat_o,X: produc3658429121746597890et_nat > $o,Y: produc3658429121746597890et_nat > $o] :
% 14.25/8.03 ( ( equiv_6726761681920839555_nat_o @ A @ R2 )
% 14.25/8.03 => ( ( member1452989533377600726_nat_o @ X4 @ ( equiv_7899955228204526484_nat_o @ A @ R2 ) )
% 14.25/8.03 => ( ( ord_le2965882846123202637_nat_o @ ( insert5175938949040314269_nat_o @ X @ ( insert5175938949040314269_nat_o @ Y @ bot_bo7824918357723582233_nat_o ) ) @ X4 )
% 14.25/8.03 => ( member8781333585448626064_nat_o @ ( produc7368190662567826135_nat_o @ X @ Y ) @ R2 ) ) ) ) ).
% 14.25/8.03
% 14.25/8.03 thf(fact_1940_in__quotient__imp__in__rel,axiom,
% 14.25/8.03 ! [A: set_nat,R2: set_Pr1261947904930325089at_nat,X4: set_nat,X: nat,Y: nat] :
% 14.25/8.03 ( ( equiv_equiv_nat @ A @ R2 )
% 14.25/8.03 => ( ( member_set_nat @ X4 @ ( equiv_quotient_nat @ A @ R2 ) )
% 14.25/8.03 => ( ( ord_less_eq_set_nat @ ( insert_nat @ X @ ( insert_nat @ Y @ bot_bot_set_nat ) ) @ X4 )
% 14.25/8.03 => ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ Y ) @ R2 ) ) ) ) ).
% 14.25/8.03
% 14.25/8.03 thf(fact_1941_prod__set__simps_I1_J,axiom,
% 14.25/8.03 ! [X: heap_e7401611519738050253t_unit,Y: set_nat] :
% 14.25/8.03 ( ( basic_7833836874476830220et_nat @ ( produc7507926704131184380et_nat @ X @ Y ) )
% 14.25/8.03 = ( insert8563697753282147485t_unit @ X @ bot_bo3686245532977126937t_unit ) ) ).
% 14.25/8.03
% 14.25/8.03 thf(fact_1942_prod__set__simps_I1_J,axiom,
% 14.25/8.03 ! [X: nat,Y: nat] :
% 14.25/8.03 ( ( basic_fsts_nat_nat @ ( product_Pair_nat_nat @ X @ Y ) )
% 14.25/8.03 = ( insert_nat @ X @ bot_bot_set_nat ) ) ).
% 14.25/8.03
% 14.25/8.03 thf(fact_1943_prod__set__simps_I1_J,axiom,
% 14.25/8.03 ! [X: int,Y: int] :
% 14.25/8.03 ( ( basic_fsts_int_int @ ( product_Pair_int_int @ X @ Y ) )
% 14.25/8.03 = ( insert_int @ X @ bot_bot_set_int ) ) ).
% 14.25/8.03
% 14.25/8.03 thf(fact_1944_prod__set__simps_I1_J,axiom,
% 14.25/8.03 ! [X: produc3658429121746597890et_nat > $o,Y: produc3658429121746597890et_nat] :
% 14.25/8.03 ( ( basic_8654530971723826568et_nat @ ( produc5001842942810119800et_nat @ X @ Y ) )
% 14.25/8.03 = ( insert5175938949040314269_nat_o @ X @ bot_bo7824918357723582233_nat_o ) ) ).
% 14.25/8.03
% 14.25/8.03 thf(fact_1945_prod__set__simps_I1_J,axiom,
% 14.25/8.03 ! [X: produc3658429121746597890et_nat > $o,Y: produc3925858234332021118et_nat] :
% 14.25/8.03 ( ( basic_3966130907827183364et_nat @ ( produc2245416461498447860et_nat @ X @ Y ) )
% 14.25/8.03 = ( insert5175938949040314269_nat_o @ X @ bot_bo7824918357723582233_nat_o ) ) ).
% 14.25/8.03
% 14.25/8.03 thf(fact_1946_rel__restrict__tranclI,axiom,
% 14.25/8.03 ! [X: produc3658429121746597890et_nat > $o,Y: produc3658429121746597890et_nat > $o,E: set_Pr2161125870931222855_nat_o,R3: set_Pr4532377907799695533_nat_o] :
% 14.25/8.03 ( ( member8781333585448626064_nat_o @ ( produc7368190662567826135_nat_o @ X @ Y ) @ ( transi4009562729418357614_nat_o @ E ) )
% 14.25/8.03 => ( ~ ( member6576561426505652726_nat_o @ X @ R3 )
% 14.25/8.03 => ( ~ ( member6576561426505652726_nat_o @ Y @ R3 )
% 14.25/8.03 => ( ( ord_le2965882846123202637_nat_o @ ( image_2232984160439939859_nat_o @ E @ R3 ) @ R3 )
% 14.25/8.03 => ( member8781333585448626064_nat_o @ ( produc7368190662567826135_nat_o @ X @ Y ) @ ( transi4009562729418357614_nat_o @ ( rel_re3331361160397388719_nat_o @ E @ R3 ) ) ) ) ) ) ) ).
% 14.25/8.03
% 14.25/8.03 thf(fact_1947_rel__restrict__tranclI,axiom,
% 14.25/8.03 ! [X: $o,Y: $o,E: set_Product_prod_o_o,R3: set_o] :
% 14.25/8.03 ( ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ X @ Y ) @ ( transitive_trancl_o @ E ) )
% 14.25/8.03 => ( ~ ( member_o @ X @ R3 )
% 14.25/8.03 => ( ~ ( member_o @ Y @ R3 )
% 14.25/8.03 => ( ( ord_less_eq_set_o @ ( image_o_o @ E @ R3 ) @ R3 )
% 14.25/8.03 => ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ X @ Y ) @ ( transitive_trancl_o @ ( rel_restrict_o @ E @ R3 ) ) ) ) ) ) ) ).
% 14.25/8.03
% 14.25/8.03 thf(fact_1948_rel__restrict__tranclI,axiom,
% 14.25/8.03 ! [X: product_unit,Y: product_unit,E: set_Pr5094982260447487303t_unit,R3: set_Product_unit] :
% 14.25/8.03 ( ( member7821138191923116944t_unit @ ( produc5692694829384537815t_unit @ X @ Y ) @ ( transi430408012122777006t_unit @ E ) )
% 14.25/8.03 => ( ~ ( member_Product_unit @ X @ R3 )
% 14.25/8.03 => ( ~ ( member_Product_unit @ Y @ R3 )
% 14.25/8.03 => ( ( ord_le3507040750410214029t_unit @ ( image_8431784415008483091t_unit @ E @ R3 ) @ R3 )
% 14.25/8.03 => ( member7821138191923116944t_unit @ ( produc5692694829384537815t_unit @ X @ Y ) @ ( transi430408012122777006t_unit @ ( rel_re9207996470947987439t_unit @ E @ R3 ) ) ) ) ) ) ) ).
% 14.25/8.03
% 14.25/8.03 thf(fact_1949_rel__restrict__tranclI,axiom,
% 14.25/8.03 ! [X: int,Y: int,E: set_Pr958786334691620121nt_int,R3: set_int] :
% 14.25/8.03 ( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ X @ Y ) @ ( transi6261509568448316235cl_int @ E ) )
% 14.25/8.03 => ( ~ ( member_int @ X @ R3 )
% 14.25/8.03 => ( ~ ( member_int @ Y @ R3 )
% 14.25/8.03 => ( ( ord_less_eq_set_int @ ( image_int_int @ E @ R3 ) @ R3 )
% 14.25/8.03 => ( member5262025264175285858nt_int @ ( product_Pair_int_int @ X @ Y ) @ ( transi6261509568448316235cl_int @ ( rel_restrict_int @ E @ R3 ) ) ) ) ) ) ) ).
% 14.25/8.03
% 14.25/8.03 thf(fact_1950_rel__restrict__tranclI,axiom,
% 14.25/8.03 ! [X: nat,Y: nat,E: set_Pr1261947904930325089at_nat,R3: set_nat] :
% 14.25/8.03 ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ Y ) @ ( transi6264000038957366511cl_nat @ E ) )
% 14.25/8.03 => ( ~ ( member_nat @ X @ R3 )
% 14.25/8.03 => ( ~ ( member_nat @ Y @ R3 )
% 14.25/8.03 => ( ( ord_less_eq_set_nat @ ( image_nat_nat @ E @ R3 ) @ R3 )
% 14.25/8.03 => ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ Y ) @ ( transi6264000038957366511cl_nat @ ( rel_restrict_nat @ E @ R3 ) ) ) ) ) ) ) ).
% 14.25/8.03
% 14.25/8.03 thf(fact_1951_pair__in__Id__conv,axiom,
% 14.25/8.03 ! [A2: nat,B: nat] :
% 14.25/8.03 ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A2 @ B ) @ id_nat )
% 14.25/8.03 = ( A2 = B ) ) ).
% 14.25/8.03
% 14.25/8.03 thf(fact_1952_pair__in__Id__conv,axiom,
% 14.25/8.03 ! [A2: int,B: int] :
% 14.25/8.03 ( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ A2 @ B ) @ id_int )
% 14.25/8.03 = ( A2 = B ) ) ).
% 14.25/8.03
% 14.25/8.03 thf(fact_1953_IdI,axiom,
% 14.25/8.03 ! [A2: nat] : ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A2 @ A2 ) @ id_nat ) ).
% 14.25/8.03
% 14.25/8.03 thf(fact_1954_IdI,axiom,
% 14.25/8.03 ! [A2: int] : ( member5262025264175285858nt_int @ ( product_Pair_int_int @ A2 @ A2 ) @ id_int ) ).
% 14.25/8.03
% 14.25/8.03 thf(fact_1955_quotient__empty,axiom,
% 14.25/8.03 ! [R2: set_Pr5094982260447487303t_unit] :
% 14.25/8.03 ( ( equiv_5172566880711946708t_unit @ bot_bo3957492148770167129t_unit @ R2 )
% 14.25/8.03 = bot_bo6284674557190688313t_unit ) ).
% 14.25/8.03
% 14.25/8.03 thf(fact_1956_quotient__empty,axiom,
% 14.25/8.03 ! [R2: set_Product_prod_o_o] :
% 14.25/8.03 ( ( equiv_quotient_o @ bot_bot_set_o @ R2 )
% 14.25/8.03 = bot_bot_set_set_o ) ).
% 14.25/8.03
% 14.25/8.03 thf(fact_1957_quotient__empty,axiom,
% 14.25/8.03 ! [R2: set_Pr1261947904930325089at_nat] :
% 14.25/8.03 ( ( equiv_quotient_nat @ bot_bot_set_nat @ R2 )
% 14.25/8.03 = bot_bot_set_set_nat ) ).
% 14.25/8.03
% 14.25/8.03 thf(fact_1958_quotient__empty,axiom,
% 14.25/8.03 ! [R2: set_Pr958786334691620121nt_int] :
% 14.25/8.03 ( ( equiv_quotient_int @ bot_bot_set_int @ R2 )
% 14.25/8.03 = bot_bot_set_set_int ) ).
% 14.25/8.03
% 14.25/8.03 thf(fact_1959_quotient__empty,axiom,
% 14.25/8.03 ! [R2: set_Pr2161125870931222855_nat_o] :
% 14.25/8.03 ( ( equiv_7899955228204526484_nat_o @ bot_bo7824918357723582233_nat_o @ R2 )
% 14.25/8.03 = bot_bo3736544561982865913_nat_o ) ).
% 14.25/8.03
% 14.25/8.03 thf(fact_1960_quotient__is__empty,axiom,
% 14.25/8.03 ! [A: set_Product_unit,R2: set_Pr5094982260447487303t_unit] :
% 14.25/8.03 ( ( ( equiv_5172566880711946708t_unit @ A @ R2 )
% 14.25/8.03 = bot_bo6284674557190688313t_unit )
% 14.25/8.03 = ( A = bot_bo3957492148770167129t_unit ) ) ).
% 14.25/8.03
% 14.25/8.03 thf(fact_1961_quotient__is__empty,axiom,
% 14.25/8.03 ! [A: set_o,R2: set_Product_prod_o_o] :
% 14.25/8.03 ( ( ( equiv_quotient_o @ A @ R2 )
% 14.25/8.03 = bot_bot_set_set_o )
% 14.25/8.03 = ( A = bot_bot_set_o ) ) ).
% 14.25/8.03
% 14.25/8.03 thf(fact_1962_quotient__is__empty,axiom,
% 14.25/8.03 ! [A: set_nat,R2: set_Pr1261947904930325089at_nat] :
% 14.25/8.03 ( ( ( equiv_quotient_nat @ A @ R2 )
% 14.25/8.03 = bot_bot_set_set_nat )
% 14.25/8.03 = ( A = bot_bot_set_nat ) ) ).
% 14.25/8.03
% 14.25/8.03 thf(fact_1963_quotient__is__empty,axiom,
% 14.25/8.03 ! [A: set_int,R2: set_Pr958786334691620121nt_int] :
% 14.25/8.03 ( ( ( equiv_quotient_int @ A @ R2 )
% 14.25/8.03 = bot_bot_set_set_int )
% 14.25/8.03 = ( A = bot_bot_set_int ) ) ).
% 14.25/8.03
% 14.25/8.03 thf(fact_1964_quotient__is__empty,axiom,
% 14.25/8.03 ! [A: set_Pr4532377907799695533_nat_o,R2: set_Pr2161125870931222855_nat_o] :
% 14.25/8.03 ( ( ( equiv_7899955228204526484_nat_o @ A @ R2 )
% 14.25/8.03 = bot_bo3736544561982865913_nat_o )
% 14.25/8.03 = ( A = bot_bo7824918357723582233_nat_o ) ) ).
% 14.25/8.03
% 14.25/8.03 thf(fact_1965_quotient__is__empty2,axiom,
% 14.25/8.03 ! [A: set_Product_unit,R2: set_Pr5094982260447487303t_unit] :
% 14.25/8.03 ( ( bot_bo6284674557190688313t_unit
% 14.25/8.03 = ( equiv_5172566880711946708t_unit @ A @ R2 ) )
% 14.25/8.03 = ( A = bot_bo3957492148770167129t_unit ) ) ).
% 14.25/8.03
% 14.25/8.03 thf(fact_1966_quotient__is__empty2,axiom,
% 14.25/8.03 ! [A: set_o,R2: set_Product_prod_o_o] :
% 14.25/8.03 ( ( bot_bot_set_set_o
% 14.25/8.03 = ( equiv_quotient_o @ A @ R2 ) )
% 14.25/8.03 = ( A = bot_bot_set_o ) ) ).
% 14.25/8.03
% 14.25/8.03 thf(fact_1967_quotient__is__empty2,axiom,
% 14.25/8.03 ! [A: set_nat,R2: set_Pr1261947904930325089at_nat] :
% 14.25/8.03 ( ( bot_bot_set_set_nat
% 14.25/8.03 = ( equiv_quotient_nat @ A @ R2 ) )
% 14.25/8.03 = ( A = bot_bot_set_nat ) ) ).
% 14.25/8.03
% 14.25/8.03 thf(fact_1968_quotient__is__empty2,axiom,
% 14.25/8.03 ! [A: set_int,R2: set_Pr958786334691620121nt_int] :
% 14.25/8.03 ( ( bot_bot_set_set_int
% 14.25/8.03 = ( equiv_quotient_int @ A @ R2 ) )
% 14.25/8.03 = ( A = bot_bot_set_int ) ) ).
% 14.25/8.03
% 14.25/8.03 thf(fact_1969_quotient__is__empty2,axiom,
% 14.25/8.03 ! [A: set_Pr4532377907799695533_nat_o,R2: set_Pr2161125870931222855_nat_o] :
% 14.25/8.03 ( ( bot_bo3736544561982865913_nat_o
% 14.25/8.03 = ( equiv_7899955228204526484_nat_o @ A @ R2 ) )
% 14.25/8.03 = ( A = bot_bo7824918357723582233_nat_o ) ) ).
% 14.25/8.03
% 14.25/8.03 thf(fact_1970_disjnt__self__iff__empty,axiom,
% 14.25/8.03 ! [S: set_Product_unit] :
% 14.25/8.03 ( ( disjnt_Product_unit @ S @ S )
% 14.25/8.03 = ( S = bot_bo3957492148770167129t_unit ) ) ).
% 14.25/8.03
% 14.25/8.03 thf(fact_1971_disjnt__self__iff__empty,axiom,
% 14.25/8.03 ! [S: set_o] :
% 14.25/8.03 ( ( disjnt_o @ S @ S )
% 14.25/8.03 = ( S = bot_bot_set_o ) ) ).
% 14.25/8.03
% 14.25/8.03 thf(fact_1972_disjnt__self__iff__empty,axiom,
% 14.25/8.03 ! [S: set_nat] :
% 14.25/8.03 ( ( disjnt_nat @ S @ S )
% 14.25/8.03 = ( S = bot_bot_set_nat ) ) ).
% 14.25/8.03
% 14.25/8.03 thf(fact_1973_disjnt__self__iff__empty,axiom,
% 14.25/8.03 ! [S: set_int] :
% 14.25/8.03 ( ( disjnt_int @ S @ S )
% 14.25/8.03 = ( S = bot_bot_set_int ) ) ).
% 14.25/8.03
% 14.25/8.03 thf(fact_1974_disjnt__self__iff__empty,axiom,
% 14.25/8.03 ! [S: set_Pr4532377907799695533_nat_o] :
% 14.25/8.03 ( ( disjnt7869448817988085652_nat_o @ S @ S )
% 14.25/8.03 = ( S = bot_bo7824918357723582233_nat_o ) ) ).
% 14.25/8.03
% 14.25/8.03 thf(fact_1975_disjnt__insert1,axiom,
% 14.25/8.03 ! [A2: produc3658429121746597890et_nat > $o,X4: set_Pr4532377907799695533_nat_o,Y6: set_Pr4532377907799695533_nat_o] :
% 14.25/8.03 ( ( disjnt7869448817988085652_nat_o @ ( insert5175938949040314269_nat_o @ A2 @ X4 ) @ Y6 )
% 14.25/8.03 = ( ~ ( member6576561426505652726_nat_o @ A2 @ Y6 )
% 14.25/8.03 & ( disjnt7869448817988085652_nat_o @ X4 @ Y6 ) ) ) ).
% 14.25/8.03
% 14.25/8.03 thf(fact_1976_disjnt__insert1,axiom,
% 14.25/8.03 ! [A2: nat,X4: set_nat,Y6: set_nat] :
% 14.25/8.03 ( ( disjnt_nat @ ( insert_nat @ A2 @ X4 ) @ Y6 )
% 14.25/8.03 = ( ~ ( member_nat @ A2 @ Y6 )
% 14.25/8.03 & ( disjnt_nat @ X4 @ Y6 ) ) ) ).
% 14.25/8.03
% 14.25/8.03 thf(fact_1977_disjnt__insert1,axiom,
% 14.25/8.03 ! [A2: int,X4: set_int,Y6: set_int] :
% 14.25/8.03 ( ( disjnt_int @ ( insert_int @ A2 @ X4 ) @ Y6 )
% 14.25/8.03 = ( ~ ( member_int @ A2 @ Y6 )
% 14.25/8.03 & ( disjnt_int @ X4 @ Y6 ) ) ) ).
% 14.25/8.03
% 14.25/8.03 thf(fact_1978_disjnt__insert1,axiom,
% 14.25/8.03 ! [A2: $o,X4: set_o,Y6: set_o] :
% 14.25/8.03 ( ( disjnt_o @ ( insert_o @ A2 @ X4 ) @ Y6 )
% 14.25/8.03 = ( ~ ( member_o @ A2 @ Y6 )
% 14.25/8.03 & ( disjnt_o @ X4 @ Y6 ) ) ) ).
% 14.25/8.03
% 14.25/8.03 thf(fact_1979_disjnt__insert1,axiom,
% 14.25/8.03 ! [A2: product_unit,X4: set_Product_unit,Y6: set_Product_unit] :
% 14.25/8.03 ( ( disjnt_Product_unit @ ( insert_Product_unit @ A2 @ X4 ) @ Y6 )
% 14.25/8.03 = ( ~ ( member_Product_unit @ A2 @ Y6 )
% 14.25/8.03 & ( disjnt_Product_unit @ X4 @ Y6 ) ) ) ).
% 14.25/8.03
% 14.25/8.03 thf(fact_1980_disjnt__insert2,axiom,
% 14.25/8.03 ! [Y6: set_Pr4532377907799695533_nat_o,A2: produc3658429121746597890et_nat > $o,X4: set_Pr4532377907799695533_nat_o] :
% 14.25/8.03 ( ( disjnt7869448817988085652_nat_o @ Y6 @ ( insert5175938949040314269_nat_o @ A2 @ X4 ) )
% 14.25/8.03 = ( ~ ( member6576561426505652726_nat_o @ A2 @ Y6 )
% 14.25/8.03 & ( disjnt7869448817988085652_nat_o @ Y6 @ X4 ) ) ) ).
% 14.25/8.03
% 14.25/8.03 thf(fact_1981_disjnt__insert2,axiom,
% 14.25/8.03 ! [Y6: set_nat,A2: nat,X4: set_nat] :
% 14.25/8.03 ( ( disjnt_nat @ Y6 @ ( insert_nat @ A2 @ X4 ) )
% 14.25/8.03 = ( ~ ( member_nat @ A2 @ Y6 )
% 14.25/8.03 & ( disjnt_nat @ Y6 @ X4 ) ) ) ).
% 14.25/8.03
% 14.25/8.03 thf(fact_1982_disjnt__insert2,axiom,
% 14.25/8.03 ! [Y6: set_int,A2: int,X4: set_int] :
% 14.25/8.03 ( ( disjnt_int @ Y6 @ ( insert_int @ A2 @ X4 ) )
% 14.25/8.03 = ( ~ ( member_int @ A2 @ Y6 )
% 14.25/8.03 & ( disjnt_int @ Y6 @ X4 ) ) ) ).
% 14.25/8.03
% 14.25/8.03 thf(fact_1983_disjnt__insert2,axiom,
% 14.25/8.03 ! [Y6: set_o,A2: $o,X4: set_o] :
% 14.25/8.03 ( ( disjnt_o @ Y6 @ ( insert_o @ A2 @ X4 ) )
% 14.25/8.03 = ( ~ ( member_o @ A2 @ Y6 )
% 14.25/8.03 & ( disjnt_o @ Y6 @ X4 ) ) ) ).
% 14.25/8.03
% 14.25/8.03 thf(fact_1984_disjnt__insert2,axiom,
% 14.25/8.03 ! [Y6: set_Product_unit,A2: product_unit,X4: set_Product_unit] :
% 14.25/8.03 ( ( disjnt_Product_unit @ Y6 @ ( insert_Product_unit @ A2 @ X4 ) )
% 14.25/8.03 = ( ~ ( member_Product_unit @ A2 @ Y6 )
% 14.25/8.03 & ( disjnt_Product_unit @ Y6 @ X4 ) ) ) ).
% 14.25/8.03
% 14.25/8.03 thf(fact_1985_rel__restrict__empty,axiom,
% 14.25/8.03 ! [R3: set_Pr5094982260447487303t_unit] :
% 14.25/8.03 ( ( rel_re9207996470947987439t_unit @ R3 @ bot_bo3957492148770167129t_unit )
% 14.25/8.03 = R3 ) ).
% 14.25/8.03
% 14.25/8.03 thf(fact_1986_rel__restrict__empty,axiom,
% 14.25/8.03 ! [R3: set_Product_prod_o_o] :
% 14.25/8.03 ( ( rel_restrict_o @ R3 @ bot_bot_set_o )
% 14.25/8.03 = R3 ) ).
% 14.25/8.03
% 14.25/8.03 thf(fact_1987_rel__restrict__empty,axiom,
% 14.25/8.03 ! [R3: set_Pr1261947904930325089at_nat] :
% 14.25/8.03 ( ( rel_restrict_nat @ R3 @ bot_bot_set_nat )
% 14.25/8.03 = R3 ) ).
% 14.25/8.03
% 14.25/8.03 thf(fact_1988_rel__restrict__empty,axiom,
% 14.25/8.03 ! [R3: set_Pr958786334691620121nt_int] :
% 14.25/8.03 ( ( rel_restrict_int @ R3 @ bot_bot_set_int )
% 14.25/8.03 = R3 ) ).
% 14.25/8.03
% 14.25/8.03 thf(fact_1989_rel__restrict__empty,axiom,
% 14.25/8.03 ! [R3: set_Pr2161125870931222855_nat_o] :
% 14.25/8.03 ( ( rel_re3331361160397388719_nat_o @ R3 @ bot_bo7824918357723582233_nat_o )
% 14.25/8.03 = R3 ) ).
% 14.25/8.03
% 14.25/8.03 thf(fact_1990_disjnt__Un1,axiom,
% 14.25/8.03 ! [A: set_nat,B4: set_nat,C5: set_nat] :
% 14.25/8.03 ( ( disjnt_nat @ ( sup_sup_set_nat @ A @ B4 ) @ C5 )
% 14.25/8.03 = ( ( disjnt_nat @ A @ C5 )
% 14.25/8.03 & ( disjnt_nat @ B4 @ C5 ) ) ) ).
% 14.25/8.03
% 14.25/8.03 thf(fact_1991_disjnt__Un2,axiom,
% 14.25/8.03 ! [C5: set_nat,A: set_nat,B4: set_nat] :
% 14.25/8.03 ( ( disjnt_nat @ C5 @ ( sup_sup_set_nat @ A @ B4 ) )
% 14.25/8.03 = ( ( disjnt_nat @ C5 @ A )
% 14.25/8.03 & ( disjnt_nat @ C5 @ B4 ) ) ) ).
% 14.25/8.03
% 14.25/8.03 thf(fact_1992_Sup__fin_Osingleton,axiom,
% 14.25/8.03 ! [X: product_unit] :
% 14.25/8.03 ( ( lattic5294303975357428420t_unit @ ( insert_Product_unit @ X @ bot_bo3957492148770167129t_unit ) )
% 14.25/8.03 = X ) ).
% 14.25/8.03
% 14.25/8.03 thf(fact_1993_Sup__fin_Osingleton,axiom,
% 14.25/8.03 ! [X: $o] :
% 14.25/8.03 ( ( lattic1508158080041050831_fin_o @ ( insert_o @ X @ bot_bot_set_o ) )
% 14.25/8.03 = X ) ).
% 14.25/8.03
% 14.25/8.03 thf(fact_1994_Sup__fin_Osingleton,axiom,
% 14.25/8.03 ! [X: nat] :
% 14.25/8.03 ( ( lattic1093996805478795353in_nat @ ( insert_nat @ X @ bot_bot_set_nat ) )
% 14.25/8.03 = X ) ).
% 14.25/8.03
% 14.25/8.03 thf(fact_1995_Sup__fin_Osingleton,axiom,
% 14.25/8.03 ! [X: int] :
% 14.25/8.03 ( ( lattic1091506334969745077in_int @ ( insert_int @ X @ bot_bot_set_int ) )
% 14.25/8.03 = X ) ).
% 14.25/8.03
% 14.25/8.03 thf(fact_1996_Sup__fin_Osingleton,axiom,
% 14.25/8.03 ! [X: produc3658429121746597890et_nat > $o] :
% 14.25/8.03 ( ( lattic7320199455484906628_nat_o @ ( insert5175938949040314269_nat_o @ X @ bot_bo7824918357723582233_nat_o ) )
% 14.25/8.03 = X ) ).
% 14.25/8.03
% 14.25/8.03 thf(fact_1997_Inf__fin_Osingleton,axiom,
% 14.25/8.03 ! [X: product_unit] :
% 14.25/8.03 ( ( lattic1263872656861969706t_unit @ ( insert_Product_unit @ X @ bot_bo3957492148770167129t_unit ) )
% 14.25/8.03 = X ) ).
% 14.25/8.03
% 14.25/8.03 thf(fact_1998_Inf__fin_Osingleton,axiom,
% 14.25/8.03 ! [X: $o] :
% 14.25/8.03 ( ( lattic4107685809792843317_fin_o @ ( insert_o @ X @ bot_bot_set_o ) )
% 14.25/8.03 = X ) ).
% 14.25/8.03
% 14.25/8.03 thf(fact_1999_Inf__fin_Osingleton,axiom,
% 14.25/8.03 ! [X: nat] :
% 14.25/8.03 ( ( lattic5238388535129920115in_nat @ ( insert_nat @ X @ bot_bot_set_nat ) )
% 14.25/8.03 = X ) ).
% 14.25/8.03
% 14.25/8.03 thf(fact_2000_Inf__fin_Osingleton,axiom,
% 14.25/8.03 ! [X: int] :
% 14.25/8.03 ( ( lattic5235898064620869839in_int @ ( insert_int @ X @ bot_bot_set_int ) )
% 14.25/8.03 = X ) ).
% 14.25/8.03
% 14.25/8.03 thf(fact_2001_Inf__fin_Osingleton,axiom,
% 14.25/8.03 ! [X: produc3658429121746597890et_nat > $o] :
% 14.25/8.03 ( ( lattic956194824204696298_nat_o @ ( insert5175938949040314269_nat_o @ X @ bot_bo7824918357723582233_nat_o ) )
% 14.25/8.03 = X ) ).
% 14.25/8.03
% 14.25/8.03 thf(fact_2002_disjnt__iff,axiom,
% 14.25/8.03 ( disjnt7869448817988085652_nat_o
% 14.25/8.03 = ( ^ [A3: set_Pr4532377907799695533_nat_o,B7: set_Pr4532377907799695533_nat_o] :
% 14.25/8.03 ! [X2: produc3658429121746597890et_nat > $o] :
% 14.25/8.03 ~ ( ( member6576561426505652726_nat_o @ X2 @ A3 )
% 14.25/8.03 & ( member6576561426505652726_nat_o @ X2 @ B7 ) ) ) ) ).
% 14.25/8.03
% 14.25/8.03 thf(fact_2003_disjnt__iff,axiom,
% 14.25/8.03 ( disjnt_nat
% 14.25/8.03 = ( ^ [A3: set_nat,B7: set_nat] :
% 14.25/8.03 ! [X2: nat] :
% 14.25/8.03 ~ ( ( member_nat @ X2 @ A3 )
% 14.25/8.03 & ( member_nat @ X2 @ B7 ) ) ) ) ).
% 14.25/8.03
% 14.25/8.03 thf(fact_2004_disjnt__iff,axiom,
% 14.25/8.03 ( disjnt_int
% 14.25/8.03 = ( ^ [A3: set_int,B7: set_int] :
% 14.25/8.03 ! [X2: int] :
% 14.25/8.03 ~ ( ( member_int @ X2 @ A3 )
% 14.25/8.03 & ( member_int @ X2 @ B7 ) ) ) ) ).
% 14.25/8.03
% 14.25/8.03 thf(fact_2005_disjnt__iff,axiom,
% 14.25/8.03 ( disjnt_o
% 14.25/8.03 = ( ^ [A3: set_o,B7: set_o] :
% 14.25/8.03 ! [X2: $o] :
% 14.25/8.03 ~ ( ( member_o @ X2 @ A3 )
% 14.25/8.03 & ( member_o @ X2 @ B7 ) ) ) ) ).
% 14.25/8.03
% 14.25/8.03 thf(fact_2006_disjnt__iff,axiom,
% 14.25/8.03 ( disjnt_Product_unit
% 14.25/8.03 = ( ^ [A3: set_Product_unit,B7: set_Product_unit] :
% 14.25/8.03 ! [X2: product_unit] :
% 14.25/8.03 ~ ( ( member_Product_unit @ X2 @ A3 )
% 14.25/8.03 & ( member_Product_unit @ X2 @ B7 ) ) ) ) ).
% 14.25/8.03
% 14.25/8.03 thf(fact_2007_IdE,axiom,
% 14.25/8.03 ! [P3: product_prod_nat_nat] :
% 14.25/8.03 ( ( member8440522571783428010at_nat @ P3 @ id_nat )
% 14.25/8.03 => ~ ! [X3: nat] :
% 14.25/8.03 ( P3
% 14.25/8.03 != ( product_Pair_nat_nat @ X3 @ X3 ) ) ) ).
% 14.25/8.03
% 14.25/8.03 thf(fact_2008_IdE,axiom,
% 14.25/8.03 ! [P3: product_prod_int_int] :
% 14.25/8.03 ( ( member5262025264175285858nt_int @ P3 @ id_int )
% 14.25/8.03 => ~ ! [X3: int] :
% 14.25/8.03 ( P3
% 14.25/8.03 != ( product_Pair_int_int @ X3 @ X3 ) ) ) ).
% 14.25/8.03
% 14.25/8.03 thf(fact_2009_disjnt__empty1,axiom,
% 14.25/8.03 ! [A: set_Product_unit] : ( disjnt_Product_unit @ bot_bo3957492148770167129t_unit @ A ) ).
% 14.25/8.03
% 14.25/8.03 thf(fact_2010_disjnt__empty1,axiom,
% 14.25/8.03 ! [A: set_o] : ( disjnt_o @ bot_bot_set_o @ A ) ).
% 14.25/8.03
% 14.25/8.03 thf(fact_2011_disjnt__empty1,axiom,
% 14.25/8.03 ! [A: set_nat] : ( disjnt_nat @ bot_bot_set_nat @ A ) ).
% 14.25/8.03
% 14.25/8.03 thf(fact_2012_disjnt__empty1,axiom,
% 14.25/8.03 ! [A: set_int] : ( disjnt_int @ bot_bot_set_int @ A ) ).
% 14.25/8.03
% 14.25/8.03 thf(fact_2013_disjnt__empty1,axiom,
% 14.25/8.03 ! [A: set_Pr4532377907799695533_nat_o] : ( disjnt7869448817988085652_nat_o @ bot_bo7824918357723582233_nat_o @ A ) ).
% 14.25/8.03
% 14.25/8.03 thf(fact_2014_disjnt__empty2,axiom,
% 14.25/8.03 ! [A: set_Product_unit] : ( disjnt_Product_unit @ A @ bot_bo3957492148770167129t_unit ) ).
% 14.25/8.03
% 14.25/8.03 thf(fact_2015_disjnt__empty2,axiom,
% 14.25/8.03 ! [A: set_o] : ( disjnt_o @ A @ bot_bot_set_o ) ).
% 14.25/8.03
% 14.25/8.03 thf(fact_2016_disjnt__empty2,axiom,
% 14.25/8.03 ! [A: set_nat] : ( disjnt_nat @ A @ bot_bot_set_nat ) ).
% 14.25/8.03
% 14.25/8.03 thf(fact_2017_disjnt__empty2,axiom,
% 14.25/8.03 ! [A: set_int] : ( disjnt_int @ A @ bot_bot_set_int ) ).
% 14.25/8.03
% 14.25/8.03 thf(fact_2018_disjnt__empty2,axiom,
% 14.25/8.03 ! [A: set_Pr4532377907799695533_nat_o] : ( disjnt7869448817988085652_nat_o @ A @ bot_bo7824918357723582233_nat_o ) ).
% 14.25/8.03
% 14.25/8.03 thf(fact_2019_disjnt__subset1,axiom,
% 14.25/8.03 ! [X4: set_nat,Y6: set_nat,Z5: set_nat] :
% 14.25/8.03 ( ( disjnt_nat @ X4 @ Y6 )
% 14.25/8.03 => ( ( ord_less_eq_set_nat @ Z5 @ X4 )
% 14.25/8.03 => ( disjnt_nat @ Z5 @ Y6 ) ) ) ).
% 14.25/8.03
% 14.25/8.03 thf(fact_2020_disjnt__subset2,axiom,
% 14.25/8.03 ! [X4: set_nat,Y6: set_nat,Z5: set_nat] :
% 14.25/8.03 ( ( disjnt_nat @ X4 @ Y6 )
% 14.25/8.03 => ( ( ord_less_eq_set_nat @ Z5 @ Y6 )
% 14.25/8.03 => ( disjnt_nat @ X4 @ Z5 ) ) ) ).
% 14.25/8.03
% 14.25/8.03 thf(fact_2021_disjnt__insert,axiom,
% 14.25/8.03 ! [X: produc3658429121746597890et_nat > $o,N: set_Pr4532377907799695533_nat_o,M: set_Pr4532377907799695533_nat_o] :
% 14.25/8.03 ( ~ ( member6576561426505652726_nat_o @ X @ N )
% 14.25/8.03 => ( ( disjnt7869448817988085652_nat_o @ M @ N )
% 14.25/8.03 => ( disjnt7869448817988085652_nat_o @ ( insert5175938949040314269_nat_o @ X @ M ) @ N ) ) ) ).
% 14.25/8.03
% 14.25/8.03 thf(fact_2022_disjnt__insert,axiom,
% 14.25/8.03 ! [X: nat,N: set_nat,M: set_nat] :
% 14.25/8.03 ( ~ ( member_nat @ X @ N )
% 14.25/8.03 => ( ( disjnt_nat @ M @ N )
% 14.25/8.03 => ( disjnt_nat @ ( insert_nat @ X @ M ) @ N ) ) ) ).
% 14.25/8.03
% 14.25/8.03 thf(fact_2023_disjnt__insert,axiom,
% 14.25/8.03 ! [X: int,N: set_int,M: set_int] :
% 14.25/8.03 ( ~ ( member_int @ X @ N )
% 14.25/8.03 => ( ( disjnt_int @ M @ N )
% 14.25/8.03 => ( disjnt_int @ ( insert_int @ X @ M ) @ N ) ) ) ).
% 14.25/8.03
% 14.25/8.03 thf(fact_2024_disjnt__insert,axiom,
% 14.25/8.03 ! [X: $o,N: set_o,M: set_o] :
% 14.25/8.03 ( ~ ( member_o @ X @ N )
% 14.25/8.03 => ( ( disjnt_o @ M @ N )
% 14.25/8.03 => ( disjnt_o @ ( insert_o @ X @ M ) @ N ) ) ) ).
% 14.25/8.03
% 14.25/8.03 thf(fact_2025_disjnt__insert,axiom,
% 14.25/8.03 ! [X: product_unit,N: set_Product_unit,M: set_Product_unit] :
% 14.25/8.03 ( ~ ( member_Product_unit @ X @ N )
% 14.25/8.03 => ( ( disjnt_Product_unit @ M @ N )
% 14.25/8.03 => ( disjnt_Product_unit @ ( insert_Product_unit @ X @ M ) @ N ) ) ) ).
% 14.25/8.03
% 14.25/8.03 thf(fact_2026_rel__restrict__notR_I2_J,axiom,
% 14.25/8.03 ! [X: produc3658429121746597890et_nat > $o,Y: produc3658429121746597890et_nat > $o,A: set_Pr2161125870931222855_nat_o,R3: set_Pr4532377907799695533_nat_o] :
% 14.25/8.03 ( ( member8781333585448626064_nat_o @ ( produc7368190662567826135_nat_o @ X @ Y ) @ ( rel_re3331361160397388719_nat_o @ A @ R3 ) )
% 14.25/8.03 => ~ ( member6576561426505652726_nat_o @ Y @ R3 ) ) ).
% 14.25/8.03
% 14.25/8.03 thf(fact_2027_rel__restrict__notR_I2_J,axiom,
% 14.25/8.03 ! [X: $o,Y: $o,A: set_Product_prod_o_o,R3: set_o] :
% 14.25/8.03 ( ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ X @ Y ) @ ( rel_restrict_o @ A @ R3 ) )
% 14.25/8.03 => ~ ( member_o @ Y @ R3 ) ) ).
% 14.25/8.03
% 14.25/8.03 thf(fact_2028_rel__restrict__notR_I2_J,axiom,
% 14.25/8.03 ! [X: product_unit,Y: product_unit,A: set_Pr5094982260447487303t_unit,R3: set_Product_unit] :
% 14.25/8.03 ( ( member7821138191923116944t_unit @ ( produc5692694829384537815t_unit @ X @ Y ) @ ( rel_re9207996470947987439t_unit @ A @ R3 ) )
% 14.25/8.03 => ~ ( member_Product_unit @ Y @ R3 ) ) ).
% 14.25/8.03
% 14.25/8.03 thf(fact_2029_rel__restrict__notR_I2_J,axiom,
% 14.25/8.03 ! [X: nat,Y: nat,A: set_Pr1261947904930325089at_nat,R3: set_nat] :
% 14.25/8.03 ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ Y ) @ ( rel_restrict_nat @ A @ R3 ) )
% 14.25/8.03 => ~ ( member_nat @ Y @ R3 ) ) ).
% 14.25/8.03
% 14.25/8.03 thf(fact_2030_rel__restrict__notR_I2_J,axiom,
% 14.25/8.03 ! [X: int,Y: int,A: set_Pr958786334691620121nt_int,R3: set_int] :
% 14.25/8.03 ( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ X @ Y ) @ ( rel_restrict_int @ A @ R3 ) )
% 14.25/8.03 => ~ ( member_int @ Y @ R3 ) ) ).
% 14.25/8.03
% 14.25/8.03 thf(fact_2031_rel__restrict__notR_I1_J,axiom,
% 14.25/8.03 ! [X: produc3658429121746597890et_nat > $o,Y: produc3658429121746597890et_nat > $o,A: set_Pr2161125870931222855_nat_o,R3: set_Pr4532377907799695533_nat_o] :
% 14.25/8.03 ( ( member8781333585448626064_nat_o @ ( produc7368190662567826135_nat_o @ X @ Y ) @ ( rel_re3331361160397388719_nat_o @ A @ R3 ) )
% 14.25/8.03 => ~ ( member6576561426505652726_nat_o @ X @ R3 ) ) ).
% 14.25/8.03
% 14.25/8.03 thf(fact_2032_rel__restrict__notR_I1_J,axiom,
% 14.25/8.03 ! [X: $o,Y: $o,A: set_Product_prod_o_o,R3: set_o] :
% 14.25/8.03 ( ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ X @ Y ) @ ( rel_restrict_o @ A @ R3 ) )
% 14.25/8.03 => ~ ( member_o @ X @ R3 ) ) ).
% 14.25/8.03
% 14.25/8.03 thf(fact_2033_rel__restrict__notR_I1_J,axiom,
% 14.25/8.03 ! [X: product_unit,Y: product_unit,A: set_Pr5094982260447487303t_unit,R3: set_Product_unit] :
% 14.25/8.03 ( ( member7821138191923116944t_unit @ ( produc5692694829384537815t_unit @ X @ Y ) @ ( rel_re9207996470947987439t_unit @ A @ R3 ) )
% 14.25/8.03 => ~ ( member_Product_unit @ X @ R3 ) ) ).
% 14.25/8.03
% 14.25/8.03 thf(fact_2034_rel__restrict__notR_I1_J,axiom,
% 14.25/8.03 ! [X: nat,Y: nat,A: set_Pr1261947904930325089at_nat,R3: set_nat] :
% 14.25/8.03 ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ Y ) @ ( rel_restrict_nat @ A @ R3 ) )
% 14.25/8.03 => ~ ( member_nat @ X @ R3 ) ) ).
% 14.25/8.03
% 14.25/8.03 thf(fact_2035_rel__restrict__notR_I1_J,axiom,
% 14.25/8.03 ! [X: int,Y: int,A: set_Pr958786334691620121nt_int,R3: set_int] :
% 14.25/8.03 ( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ X @ Y ) @ ( rel_restrict_int @ A @ R3 ) )
% 14.25/8.03 => ~ ( member_int @ X @ R3 ) ) ).
% 14.25/8.03
% 14.25/8.03 thf(fact_2036_rel__restrictI,axiom,
% 14.25/8.03 ! [X: produc3658429121746597890et_nat > $o,R3: set_Pr4532377907799695533_nat_o,Y: produc3658429121746597890et_nat > $o,E: set_Pr2161125870931222855_nat_o] :
% 14.25/8.03 ( ~ ( member6576561426505652726_nat_o @ X @ R3 )
% 14.25/8.03 => ( ~ ( member6576561426505652726_nat_o @ Y @ R3 )
% 14.25/8.03 => ( ( member8781333585448626064_nat_o @ ( produc7368190662567826135_nat_o @ X @ Y ) @ E )
% 14.25/8.03 => ( member8781333585448626064_nat_o @ ( produc7368190662567826135_nat_o @ X @ Y ) @ ( rel_re3331361160397388719_nat_o @ E @ R3 ) ) ) ) ) ).
% 14.25/8.03
% 14.25/8.03 thf(fact_2037_rel__restrictI,axiom,
% 14.25/8.03 ! [X: $o,R3: set_o,Y: $o,E: set_Product_prod_o_o] :
% 14.25/8.03 ( ~ ( member_o @ X @ R3 )
% 14.25/8.03 => ( ~ ( member_o @ Y @ R3 )
% 14.25/8.03 => ( ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ X @ Y ) @ E )
% 14.25/8.03 => ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ X @ Y ) @ ( rel_restrict_o @ E @ R3 ) ) ) ) ) ).
% 14.25/8.03
% 14.25/8.03 thf(fact_2038_rel__restrictI,axiom,
% 14.25/8.03 ! [X: product_unit,R3: set_Product_unit,Y: product_unit,E: set_Pr5094982260447487303t_unit] :
% 14.25/8.03 ( ~ ( member_Product_unit @ X @ R3 )
% 14.25/8.03 => ( ~ ( member_Product_unit @ Y @ R3 )
% 14.25/8.03 => ( ( member7821138191923116944t_unit @ ( produc5692694829384537815t_unit @ X @ Y ) @ E )
% 14.25/8.03 => ( member7821138191923116944t_unit @ ( produc5692694829384537815t_unit @ X @ Y ) @ ( rel_re9207996470947987439t_unit @ E @ R3 ) ) ) ) ) ).
% 14.25/8.03
% 14.25/8.03 thf(fact_2039_rel__restrictI,axiom,
% 14.25/8.03 ! [X: nat,R3: set_nat,Y: nat,E: set_Pr1261947904930325089at_nat] :
% 14.25/8.03 ( ~ ( member_nat @ X @ R3 )
% 14.25/8.03 => ( ~ ( member_nat @ Y @ R3 )
% 14.25/8.03 => ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ Y ) @ E )
% 14.25/8.03 => ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ Y ) @ ( rel_restrict_nat @ E @ R3 ) ) ) ) ) ).
% 14.25/8.03
% 14.25/8.03 thf(fact_2040_rel__restrictI,axiom,
% 14.25/8.03 ! [X: int,R3: set_int,Y: int,E: set_Pr958786334691620121nt_int] :
% 14.25/8.03 ( ~ ( member_int @ X @ R3 )
% 14.25/8.03 => ( ~ ( member_int @ Y @ R3 )
% 14.25/8.03 => ( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ X @ Y ) @ E )
% 14.25/8.03 => ( member5262025264175285858nt_int @ ( product_Pair_int_int @ X @ Y ) @ ( rel_restrict_int @ E @ R3 ) ) ) ) ) ).
% 14.25/8.03
% 14.25/8.03 thf(fact_2041_rel__restrict__lift,axiom,
% 14.25/8.03 ! [X: nat,Y: nat,E: set_Pr1261947904930325089at_nat,R3: set_nat] :
% 14.25/8.03 ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ Y ) @ ( rel_restrict_nat @ E @ R3 ) )
% 14.25/8.03 => ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ Y ) @ E ) ) ).
% 14.25/8.03
% 14.25/8.03 thf(fact_2042_rel__restrict__lift,axiom,
% 14.25/8.03 ! [X: int,Y: int,E: set_Pr958786334691620121nt_int,R3: set_int] :
% 14.25/8.03 ( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ X @ Y ) @ ( rel_restrict_int @ E @ R3 ) )
% 14.25/8.03 => ( member5262025264175285858nt_int @ ( product_Pair_int_int @ X @ Y ) @ E ) ) ).
% 14.25/8.03
% 14.25/8.03 thf(fact_2043_rel__restrict__union,axiom,
% 14.25/8.03 ! [R3: set_Pr1261947904930325089at_nat,A: set_nat,B4: set_nat] :
% 14.25/8.03 ( ( rel_restrict_nat @ R3 @ ( sup_sup_set_nat @ A @ B4 ) )
% 14.25/8.03 = ( rel_restrict_nat @ ( rel_restrict_nat @ R3 @ A ) @ B4 ) ) ).
% 14.25/8.03
% 14.25/8.03 thf(fact_2044_Inf__fin__le__Sup__fin,axiom,
% 14.25/8.03 ! [A: set_Product_unit] :
% 14.25/8.03 ( ( finite4290736615968046902t_unit @ A )
% 14.25/8.03 => ( ( A != bot_bo3957492148770167129t_unit )
% 14.25/8.03 => ( ord_le3221252021190050221t_unit @ ( lattic1263872656861969706t_unit @ A ) @ ( lattic5294303975357428420t_unit @ A ) ) ) ) ).
% 14.25/8.03
% 14.25/8.03 thf(fact_2045_Inf__fin__le__Sup__fin,axiom,
% 14.25/8.03 ! [A: set_o] :
% 14.25/8.03 ( ( finite_finite_o @ A )
% 14.25/8.03 => ( ( A != bot_bot_set_o )
% 14.25/8.03 => ( ord_less_eq_o @ ( lattic4107685809792843317_fin_o @ A ) @ ( lattic1508158080041050831_fin_o @ A ) ) ) ) ).
% 14.25/8.03
% 14.25/8.03 thf(fact_2046_Inf__fin__le__Sup__fin,axiom,
% 14.25/8.03 ! [A: set_Pr4532377907799695533_nat_o] :
% 14.25/8.03 ( ( finite3252695134891459830_nat_o @ A )
% 14.25/8.03 => ( ( A != bot_bo7824918357723582233_nat_o )
% 14.25/8.03 => ( ord_le729326519192465773_nat_o @ ( lattic956194824204696298_nat_o @ A ) @ ( lattic7320199455484906628_nat_o @ A ) ) ) ) ).
% 14.25/8.03
% 14.25/8.03 thf(fact_2047_Inf__fin__le__Sup__fin,axiom,
% 14.25/8.03 ! [A: set_filter_nat] :
% 14.25/8.03 ( ( finite2119507909894593271er_nat @ A )
% 14.25/8.03 => ( ( A != bot_bo498966703094740906er_nat )
% 14.25/8.03 => ( ord_le2510731241096832064er_nat @ ( lattic9128708996678779395er_nat @ A ) @ ( lattic5930898082463196905er_nat @ A ) ) ) ) ).
% 14.25/8.03
% 14.25/8.03 thf(fact_2048_Inf__fin__le__Sup__fin,axiom,
% 14.25/8.03 ! [A: set_set_nat] :
% 14.25/8.03 ( ( finite1152437895449049373et_nat @ A )
% 14.25/8.03 => ( ( A != bot_bot_set_set_nat )
% 14.25/8.03 => ( ord_less_eq_set_nat @ ( lattic3014633134055518761et_nat @ A ) @ ( lattic3835124923745554447et_nat @ A ) ) ) ) ).
% 14.25/8.03
% 14.25/8.03 thf(fact_2049_Inf__fin__le__Sup__fin,axiom,
% 14.25/8.03 ! [A: set_rat] :
% 14.25/8.03 ( ( finite_finite_rat @ A )
% 14.25/8.03 => ( ( A != bot_bot_set_rat )
% 14.25/8.03 => ( ord_less_eq_rat @ ( lattic4603258475043424379in_rat @ A ) @ ( lattic458866745392299617in_rat @ A ) ) ) ) ).
% 14.25/8.03
% 14.25/8.03 thf(fact_2050_Inf__fin__le__Sup__fin,axiom,
% 14.25/8.03 ! [A: set_nat] :
% 14.25/8.03 ( ( finite_finite_nat @ A )
% 14.25/8.03 => ( ( A != bot_bot_set_nat )
% 14.25/8.03 => ( ord_less_eq_nat @ ( lattic5238388535129920115in_nat @ A ) @ ( lattic1093996805478795353in_nat @ A ) ) ) ) ).
% 14.25/8.03
% 14.25/8.03 thf(fact_2051_Inf__fin__le__Sup__fin,axiom,
% 14.25/8.03 ! [A: set_int] :
% 14.25/8.03 ( ( finite_finite_int @ A )
% 14.25/8.03 => ( ( A != bot_bot_set_int )
% 14.25/8.03 => ( ord_less_eq_int @ ( lattic5235898064620869839in_int @ A ) @ ( lattic1091506334969745077in_int @ A ) ) ) ) ).
% 14.25/8.03
% 14.25/8.03 thf(fact_2052_disjnt__Sigma__iff,axiom,
% 14.25/8.03 ! [A: set_nat,C5: nat > set_Product_unit,B4: set_nat] :
% 14.25/8.03 ( ( disjnt595863415355979947t_unit @ ( produc5506539147108004339t_unit @ A @ C5 ) @ ( produc5506539147108004339t_unit @ B4 @ C5 ) )
% 14.25/8.03 = ( ! [X2: nat] :
% 14.25/8.03 ( ( member_nat @ X2 @ ( inf_inf_set_nat @ A @ B4 ) )
% 14.25/8.03 => ( ( C5 @ X2 )
% 14.25/8.03 = bot_bo3957492148770167129t_unit ) )
% 14.25/8.03 | ( disjnt_nat @ A @ B4 ) ) ) ).
% 14.25/8.03
% 14.25/8.03 thf(fact_2053_disjnt__Sigma__iff,axiom,
% 14.25/8.03 ! [A: set_nat,C5: nat > set_o,B4: set_nat] :
% 14.25/8.03 ( ( disjnt5652777605541633334_nat_o @ ( product_Sigma_nat_o @ A @ C5 ) @ ( product_Sigma_nat_o @ B4 @ C5 ) )
% 14.25/8.03 = ( ! [X2: nat] :
% 14.25/8.03 ( ( member_nat @ X2 @ ( inf_inf_set_nat @ A @ B4 ) )
% 14.25/8.03 => ( ( C5 @ X2 )
% 14.25/8.03 = bot_bot_set_o ) )
% 14.25/8.03 | ( disjnt_nat @ A @ B4 ) ) ) ).
% 14.25/8.03
% 14.25/8.03 thf(fact_2054_disjnt__Sigma__iff,axiom,
% 14.25/8.03 ! [A: set_nat,C5: nat > set_nat,B4: set_nat] :
% 14.25/8.03 ( ( disjnt2713546992526078024at_nat @ ( produc457027306803732586at_nat @ A @ C5 ) @ ( produc457027306803732586at_nat @ B4 @ C5 ) )
% 14.25/8.03 = ( ! [X2: nat] :
% 14.25/8.03 ( ( member_nat @ X2 @ ( inf_inf_set_nat @ A @ B4 ) )
% 14.25/8.03 => ( ( C5 @ X2 )
% 14.25/8.03 = bot_bot_set_nat ) )
% 14.25/8.03 | ( disjnt_nat @ A @ B4 ) ) ) ).
% 14.25/8.03
% 14.25/8.03 thf(fact_2055_disjnt__Sigma__iff,axiom,
% 14.25/8.03 ! [A: set_nat,C5: nat > set_int,B4: set_nat] :
% 14.25/8.03 ( ( disjnt7759068009871657124at_int @ ( produc454536836294682310at_int @ A @ C5 ) @ ( produc454536836294682310at_int @ B4 @ C5 ) )
% 14.25/8.03 = ( ! [X2: nat] :
% 14.25/8.03 ( ( member_nat @ X2 @ ( inf_inf_set_nat @ A @ B4 ) )
% 14.25/8.03 => ( ( C5 @ X2 )
% 14.25/8.03 = bot_bot_set_int ) )
% 14.25/8.03 | ( disjnt_nat @ A @ B4 ) ) ) ).
% 14.25/8.03
% 14.25/8.03 thf(fact_2056_disjnt__Sigma__iff,axiom,
% 14.25/8.03 ! [A: set_nat,C5: nat > set_Pr4532377907799695533_nat_o,B4: set_nat] :
% 14.25/8.03 ( ( disjnt4476176636800211563_nat_o @ ( produc5650067098820117939_nat_o @ A @ C5 ) @ ( produc5650067098820117939_nat_o @ B4 @ C5 ) )
% 14.25/8.03 = ( ! [X2: nat] :
% 14.25/8.03 ( ( member_nat @ X2 @ ( inf_inf_set_nat @ A @ B4 ) )
% 14.25/8.03 => ( ( C5 @ X2 )
% 14.25/8.03 = bot_bo7824918357723582233_nat_o ) )
% 14.25/8.03 | ( disjnt_nat @ A @ B4 ) ) ) ).
% 14.25/8.03
% 14.25/8.03 thf(fact_2057_disjnt__def,axiom,
% 14.25/8.03 ( disjnt_Product_unit
% 14.25/8.03 = ( ^ [A3: set_Product_unit,B7: set_Product_unit] :
% 14.25/8.03 ( ( inf_in4660618365625256667t_unit @ A3 @ B7 )
% 14.25/8.03 = bot_bo3957492148770167129t_unit ) ) ) ).
% 14.25/8.03
% 14.25/8.03 thf(fact_2058_disjnt__def,axiom,
% 14.25/8.03 ( disjnt_o
% 14.25/8.03 = ( ^ [A3: set_o,B7: set_o] :
% 14.25/8.03 ( ( inf_inf_set_o @ A3 @ B7 )
% 14.25/8.03 = bot_bot_set_o ) ) ) ).
% 14.25/8.03
% 14.25/8.03 thf(fact_2059_disjnt__def,axiom,
% 14.25/8.03 ( disjnt_nat
% 14.25/8.03 = ( ^ [A3: set_nat,B7: set_nat] :
% 14.25/8.03 ( ( inf_inf_set_nat @ A3 @ B7 )
% 14.25/8.04 = bot_bot_set_nat ) ) ) ).
% 14.25/8.04
% 14.25/8.04 thf(fact_2060_disjnt__def,axiom,
% 14.25/8.04 ( disjnt_int
% 14.25/8.04 = ( ^ [A3: set_int,B7: set_int] :
% 14.25/8.04 ( ( inf_inf_set_int @ A3 @ B7 )
% 14.25/8.04 = bot_bot_set_int ) ) ) ).
% 14.25/8.04
% 14.25/8.04 thf(fact_2061_disjnt__def,axiom,
% 14.25/8.04 ( disjnt7869448817988085652_nat_o
% 14.25/8.04 = ( ^ [A3: set_Pr4532377907799695533_nat_o,B7: set_Pr4532377907799695533_nat_o] :
% 14.25/8.04 ( ( inf_in1906310914598751387_nat_o @ A3 @ B7 )
% 14.25/8.04 = bot_bo7824918357723582233_nat_o ) ) ) ).
% 14.25/8.04
% 14.25/8.04 thf(fact_2062_quotient__eqI,axiom,
% 14.25/8.04 ! [A: set_Pr4532377907799695533_nat_o,R2: set_Pr2161125870931222855_nat_o,X4: set_Pr4532377907799695533_nat_o,Y6: set_Pr4532377907799695533_nat_o,X: produc3658429121746597890et_nat > $o,Y: produc3658429121746597890et_nat > $o] :
% 14.25/8.04 ( ( equiv_6726761681920839555_nat_o @ A @ R2 )
% 14.25/8.04 => ( ( member1452989533377600726_nat_o @ X4 @ ( equiv_7899955228204526484_nat_o @ A @ R2 ) )
% 14.25/8.04 => ( ( member1452989533377600726_nat_o @ Y6 @ ( equiv_7899955228204526484_nat_o @ A @ R2 ) )
% 14.25/8.04 => ( ( member6576561426505652726_nat_o @ X @ X4 )
% 14.25/8.04 => ( ( member6576561426505652726_nat_o @ Y @ Y6 )
% 14.25/8.04 => ( ( member8781333585448626064_nat_o @ ( produc7368190662567826135_nat_o @ X @ Y ) @ R2 )
% 14.25/8.04 => ( X4 = Y6 ) ) ) ) ) ) ) ).
% 14.25/8.04
% 14.25/8.04 thf(fact_2063_quotient__eqI,axiom,
% 14.25/8.04 ! [A: set_o,R2: set_Product_prod_o_o,X4: set_o,Y6: set_o,X: $o,Y: $o] :
% 14.25/8.04 ( ( equiv_equiv_o @ A @ R2 )
% 14.25/8.04 => ( ( member_set_o @ X4 @ ( equiv_quotient_o @ A @ R2 ) )
% 14.25/8.04 => ( ( member_set_o @ Y6 @ ( equiv_quotient_o @ A @ R2 ) )
% 14.25/8.04 => ( ( member_o @ X @ X4 )
% 14.25/8.04 => ( ( member_o @ Y @ Y6 )
% 14.25/8.04 => ( ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ X @ Y ) @ R2 )
% 14.25/8.04 => ( X4 = Y6 ) ) ) ) ) ) ) ).
% 14.25/8.04
% 14.25/8.04 thf(fact_2064_quotient__eqI,axiom,
% 14.25/8.04 ! [A: set_Product_unit,R2: set_Pr5094982260447487303t_unit,X4: set_Product_unit,Y6: set_Product_unit,X: product_unit,Y: product_unit] :
% 14.25/8.04 ( ( equiv_4189356295871601603t_unit @ A @ R2 )
% 14.25/8.04 => ( ( member5877623283571906838t_unit @ X4 @ ( equiv_5172566880711946708t_unit @ A @ R2 ) )
% 14.25/8.04 => ( ( member5877623283571906838t_unit @ Y6 @ ( equiv_5172566880711946708t_unit @ A @ R2 ) )
% 14.25/8.04 => ( ( member_Product_unit @ X @ X4 )
% 14.25/8.04 => ( ( member_Product_unit @ Y @ Y6 )
% 14.25/8.04 => ( ( member7821138191923116944t_unit @ ( produc5692694829384537815t_unit @ X @ Y ) @ R2 )
% 14.25/8.04 => ( X4 = Y6 ) ) ) ) ) ) ) ).
% 14.25/8.04
% 14.25/8.04 thf(fact_2065_quotient__eqI,axiom,
% 14.25/8.04 ! [A: set_nat,R2: set_Pr1261947904930325089at_nat,X4: set_nat,Y6: set_nat,X: nat,Y: nat] :
% 14.25/8.04 ( ( equiv_equiv_nat @ A @ R2 )
% 14.25/8.04 => ( ( member_set_nat @ X4 @ ( equiv_quotient_nat @ A @ R2 ) )
% 14.25/8.04 => ( ( member_set_nat @ Y6 @ ( equiv_quotient_nat @ A @ R2 ) )
% 14.25/8.04 => ( ( member_nat @ X @ X4 )
% 14.25/8.04 => ( ( member_nat @ Y @ Y6 )
% 14.25/8.04 => ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ Y ) @ R2 )
% 14.25/8.04 => ( X4 = Y6 ) ) ) ) ) ) ) ).
% 14.25/8.04
% 14.25/8.04 thf(fact_2066_quotient__eqI,axiom,
% 14.25/8.04 ! [A: set_int,R2: set_Pr958786334691620121nt_int,X4: set_int,Y6: set_int,X: int,Y: int] :
% 14.25/8.04 ( ( equiv_equiv_int @ A @ R2 )
% 14.25/8.04 => ( ( member_set_int @ X4 @ ( equiv_quotient_int @ A @ R2 ) )
% 14.25/8.04 => ( ( member_set_int @ Y6 @ ( equiv_quotient_int @ A @ R2 ) )
% 14.25/8.04 => ( ( member_int @ X @ X4 )
% 14.25/8.04 => ( ( member_int @ Y @ Y6 )
% 14.25/8.04 => ( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ X @ Y ) @ R2 )
% 14.25/8.04 => ( X4 = Y6 ) ) ) ) ) ) ) ).
% 14.25/8.04
% 14.25/8.04 thf(fact_2067_quotient__eq__iff,axiom,
% 14.25/8.04 ! [A: set_Pr4532377907799695533_nat_o,R2: set_Pr2161125870931222855_nat_o,X4: set_Pr4532377907799695533_nat_o,Y6: set_Pr4532377907799695533_nat_o,X: produc3658429121746597890et_nat > $o,Y: produc3658429121746597890et_nat > $o] :
% 14.25/8.04 ( ( equiv_6726761681920839555_nat_o @ A @ R2 )
% 14.25/8.04 => ( ( member1452989533377600726_nat_o @ X4 @ ( equiv_7899955228204526484_nat_o @ A @ R2 ) )
% 14.25/8.04 => ( ( member1452989533377600726_nat_o @ Y6 @ ( equiv_7899955228204526484_nat_o @ A @ R2 ) )
% 14.25/8.04 => ( ( member6576561426505652726_nat_o @ X @ X4 )
% 14.25/8.04 => ( ( member6576561426505652726_nat_o @ Y @ Y6 )
% 14.25/8.04 => ( ( X4 = Y6 )
% 14.25/8.04 = ( member8781333585448626064_nat_o @ ( produc7368190662567826135_nat_o @ X @ Y ) @ R2 ) ) ) ) ) ) ) ).
% 14.25/8.04
% 14.25/8.04 thf(fact_2068_quotient__eq__iff,axiom,
% 14.25/8.04 ! [A: set_o,R2: set_Product_prod_o_o,X4: set_o,Y6: set_o,X: $o,Y: $o] :
% 14.25/8.04 ( ( equiv_equiv_o @ A @ R2 )
% 14.25/8.04 => ( ( member_set_o @ X4 @ ( equiv_quotient_o @ A @ R2 ) )
% 14.25/8.04 => ( ( member_set_o @ Y6 @ ( equiv_quotient_o @ A @ R2 ) )
% 14.25/8.04 => ( ( member_o @ X @ X4 )
% 14.25/8.04 => ( ( member_o @ Y @ Y6 )
% 14.25/8.04 => ( ( X4 = Y6 )
% 14.25/8.04 = ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ X @ Y ) @ R2 ) ) ) ) ) ) ) ).
% 14.25/8.04
% 14.25/8.04 thf(fact_2069_quotient__eq__iff,axiom,
% 14.25/8.04 ! [A: set_Product_unit,R2: set_Pr5094982260447487303t_unit,X4: set_Product_unit,Y6: set_Product_unit,X: product_unit,Y: product_unit] :
% 14.25/8.04 ( ( equiv_4189356295871601603t_unit @ A @ R2 )
% 14.25/8.04 => ( ( member5877623283571906838t_unit @ X4 @ ( equiv_5172566880711946708t_unit @ A @ R2 ) )
% 14.25/8.04 => ( ( member5877623283571906838t_unit @ Y6 @ ( equiv_5172566880711946708t_unit @ A @ R2 ) )
% 14.25/8.04 => ( ( member_Product_unit @ X @ X4 )
% 14.25/8.04 => ( ( member_Product_unit @ Y @ Y6 )
% 14.25/8.04 => ( ( X4 = Y6 )
% 14.25/8.04 = ( member7821138191923116944t_unit @ ( produc5692694829384537815t_unit @ X @ Y ) @ R2 ) ) ) ) ) ) ) ).
% 14.25/8.04
% 14.25/8.04 thf(fact_2070_quotient__eq__iff,axiom,
% 14.25/8.04 ! [A: set_nat,R2: set_Pr1261947904930325089at_nat,X4: set_nat,Y6: set_nat,X: nat,Y: nat] :
% 14.25/8.04 ( ( equiv_equiv_nat @ A @ R2 )
% 14.25/8.04 => ( ( member_set_nat @ X4 @ ( equiv_quotient_nat @ A @ R2 ) )
% 14.25/8.04 => ( ( member_set_nat @ Y6 @ ( equiv_quotient_nat @ A @ R2 ) )
% 14.25/8.04 => ( ( member_nat @ X @ X4 )
% 14.25/8.04 => ( ( member_nat @ Y @ Y6 )
% 14.25/8.04 => ( ( X4 = Y6 )
% 14.25/8.04 = ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ Y ) @ R2 ) ) ) ) ) ) ) ).
% 14.25/8.04
% 14.25/8.04 thf(fact_2071_quotient__eq__iff,axiom,
% 14.25/8.04 ! [A: set_int,R2: set_Pr958786334691620121nt_int,X4: set_int,Y6: set_int,X: int,Y: int] :
% 14.25/8.04 ( ( equiv_equiv_int @ A @ R2 )
% 14.25/8.04 => ( ( member_set_int @ X4 @ ( equiv_quotient_int @ A @ R2 ) )
% 14.25/8.04 => ( ( member_set_int @ Y6 @ ( equiv_quotient_int @ A @ R2 ) )
% 14.25/8.04 => ( ( member_int @ X @ X4 )
% 14.25/8.04 => ( ( member_int @ Y @ Y6 )
% 14.25/8.04 => ( ( X4 = Y6 )
% 14.25/8.04 = ( member5262025264175285858nt_int @ ( product_Pair_int_int @ X @ Y ) @ R2 ) ) ) ) ) ) ) ).
% 14.25/8.04
% 14.25/8.04 thf(fact_2072_in__quotient__imp__closed,axiom,
% 14.25/8.04 ! [A: set_Pr4532377907799695533_nat_o,R2: set_Pr2161125870931222855_nat_o,X4: set_Pr4532377907799695533_nat_o,X: produc3658429121746597890et_nat > $o,Y: produc3658429121746597890et_nat > $o] :
% 14.25/8.04 ( ( equiv_6726761681920839555_nat_o @ A @ R2 )
% 14.25/8.04 => ( ( member1452989533377600726_nat_o @ X4 @ ( equiv_7899955228204526484_nat_o @ A @ R2 ) )
% 14.25/8.04 => ( ( member6576561426505652726_nat_o @ X @ X4 )
% 14.25/8.04 => ( ( member8781333585448626064_nat_o @ ( produc7368190662567826135_nat_o @ X @ Y ) @ R2 )
% 14.25/8.04 => ( member6576561426505652726_nat_o @ Y @ X4 ) ) ) ) ) ).
% 14.25/8.04
% 14.25/8.04 thf(fact_2073_in__quotient__imp__closed,axiom,
% 14.25/8.04 ! [A: set_o,R2: set_Product_prod_o_o,X4: set_o,X: $o,Y: $o] :
% 14.25/8.04 ( ( equiv_equiv_o @ A @ R2 )
% 14.25/8.04 => ( ( member_set_o @ X4 @ ( equiv_quotient_o @ A @ R2 ) )
% 14.25/8.04 => ( ( member_o @ X @ X4 )
% 14.25/8.04 => ( ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ X @ Y ) @ R2 )
% 14.25/8.04 => ( member_o @ Y @ X4 ) ) ) ) ) ).
% 14.25/8.04
% 14.25/8.04 thf(fact_2074_in__quotient__imp__closed,axiom,
% 14.25/8.04 ! [A: set_Product_unit,R2: set_Pr5094982260447487303t_unit,X4: set_Product_unit,X: product_unit,Y: product_unit] :
% 14.25/8.04 ( ( equiv_4189356295871601603t_unit @ A @ R2 )
% 14.25/8.04 => ( ( member5877623283571906838t_unit @ X4 @ ( equiv_5172566880711946708t_unit @ A @ R2 ) )
% 14.25/8.04 => ( ( member_Product_unit @ X @ X4 )
% 14.25/8.04 => ( ( member7821138191923116944t_unit @ ( produc5692694829384537815t_unit @ X @ Y ) @ R2 )
% 14.25/8.04 => ( member_Product_unit @ Y @ X4 ) ) ) ) ) ).
% 14.25/8.04
% 14.25/8.04 thf(fact_2075_in__quotient__imp__closed,axiom,
% 14.25/8.04 ! [A: set_nat,R2: set_Pr1261947904930325089at_nat,X4: set_nat,X: nat,Y: nat] :
% 14.25/8.04 ( ( equiv_equiv_nat @ A @ R2 )
% 14.25/8.04 => ( ( member_set_nat @ X4 @ ( equiv_quotient_nat @ A @ R2 ) )
% 14.25/8.04 => ( ( member_nat @ X @ X4 )
% 14.25/8.04 => ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ Y ) @ R2 )
% 14.25/8.04 => ( member_nat @ Y @ X4 ) ) ) ) ) ).
% 14.25/8.04
% 14.25/8.04 thf(fact_2076_in__quotient__imp__closed,axiom,
% 14.25/8.04 ! [A: set_int,R2: set_Pr958786334691620121nt_int,X4: set_int,X: int,Y: int] :
% 14.25/8.04 ( ( equiv_equiv_int @ A @ R2 )
% 14.25/8.04 => ( ( member_set_int @ X4 @ ( equiv_quotient_int @ A @ R2 ) )
% 14.25/8.04 => ( ( member_int @ X @ X4 )
% 14.25/8.04 => ( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ X @ Y ) @ R2 )
% 14.25/8.04 => ( member_int @ Y @ X4 ) ) ) ) ) ).
% 14.25/8.04
% 14.25/8.04 thf(fact_2077_in__quotient__imp__non__empty,axiom,
% 14.25/8.04 ! [A: set_Product_unit,R2: set_Pr5094982260447487303t_unit,X4: set_Product_unit] :
% 14.25/8.04 ( ( equiv_4189356295871601603t_unit @ A @ R2 )
% 14.25/8.04 => ( ( member5877623283571906838t_unit @ X4 @ ( equiv_5172566880711946708t_unit @ A @ R2 ) )
% 14.25/8.04 => ( X4 != bot_bo3957492148770167129t_unit ) ) ) ).
% 14.25/8.04
% 14.25/8.04 thf(fact_2078_in__quotient__imp__non__empty,axiom,
% 14.25/8.04 ! [A: set_o,R2: set_Product_prod_o_o,X4: set_o] :
% 14.25/8.04 ( ( equiv_equiv_o @ A @ R2 )
% 14.25/8.04 => ( ( member_set_o @ X4 @ ( equiv_quotient_o @ A @ R2 ) )
% 14.25/8.04 => ( X4 != bot_bot_set_o ) ) ) ).
% 14.25/8.04
% 14.25/8.04 thf(fact_2079_in__quotient__imp__non__empty,axiom,
% 14.25/8.04 ! [A: set_nat,R2: set_Pr1261947904930325089at_nat,X4: set_nat] :
% 14.25/8.04 ( ( equiv_equiv_nat @ A @ R2 )
% 14.25/8.04 => ( ( member_set_nat @ X4 @ ( equiv_quotient_nat @ A @ R2 ) )
% 14.25/8.04 => ( X4 != bot_bot_set_nat ) ) ) ).
% 14.25/8.04
% 14.25/8.04 thf(fact_2080_in__quotient__imp__non__empty,axiom,
% 14.25/8.04 ! [A: set_int,R2: set_Pr958786334691620121nt_int,X4: set_int] :
% 14.25/8.04 ( ( equiv_equiv_int @ A @ R2 )
% 14.25/8.04 => ( ( member_set_int @ X4 @ ( equiv_quotient_int @ A @ R2 ) )
% 14.25/8.04 => ( X4 != bot_bot_set_int ) ) ) ).
% 14.25/8.04
% 14.25/8.04 thf(fact_2081_in__quotient__imp__non__empty,axiom,
% 14.25/8.04 ! [A: set_Pr4532377907799695533_nat_o,R2: set_Pr2161125870931222855_nat_o,X4: set_Pr4532377907799695533_nat_o] :
% 14.25/8.04 ( ( equiv_6726761681920839555_nat_o @ A @ R2 )
% 14.25/8.04 => ( ( member1452989533377600726_nat_o @ X4 @ ( equiv_7899955228204526484_nat_o @ A @ R2 ) )
% 14.25/8.04 => ( X4 != bot_bo7824918357723582233_nat_o ) ) ) ).
% 14.25/8.04
% 14.25/8.04 thf(fact_2082_rel__restrict__trancl__notR_I2_J,axiom,
% 14.25/8.04 ! [V: produc3658429121746597890et_nat > $o,W: produc3658429121746597890et_nat > $o,E: set_Pr2161125870931222855_nat_o,R3: set_Pr4532377907799695533_nat_o] :
% 14.25/8.04 ( ( member8781333585448626064_nat_o @ ( produc7368190662567826135_nat_o @ V @ W ) @ ( transi4009562729418357614_nat_o @ ( rel_re3331361160397388719_nat_o @ E @ R3 ) ) )
% 14.25/8.04 => ~ ( member6576561426505652726_nat_o @ W @ R3 ) ) ).
% 14.25/8.04
% 14.25/8.04 thf(fact_2083_rel__restrict__trancl__notR_I2_J,axiom,
% 14.25/8.04 ! [V: $o,W: $o,E: set_Product_prod_o_o,R3: set_o] :
% 14.25/8.04 ( ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ V @ W ) @ ( transitive_trancl_o @ ( rel_restrict_o @ E @ R3 ) ) )
% 14.25/8.04 => ~ ( member_o @ W @ R3 ) ) ).
% 14.25/8.04
% 14.25/8.04 thf(fact_2084_rel__restrict__trancl__notR_I2_J,axiom,
% 14.25/8.04 ! [V: product_unit,W: product_unit,E: set_Pr5094982260447487303t_unit,R3: set_Product_unit] :
% 14.25/8.04 ( ( member7821138191923116944t_unit @ ( produc5692694829384537815t_unit @ V @ W ) @ ( transi430408012122777006t_unit @ ( rel_re9207996470947987439t_unit @ E @ R3 ) ) )
% 14.25/8.04 => ~ ( member_Product_unit @ W @ R3 ) ) ).
% 14.25/8.04
% 14.25/8.04 thf(fact_2085_rel__restrict__trancl__notR_I2_J,axiom,
% 14.25/8.04 ! [V: nat,W: nat,E: set_Pr1261947904930325089at_nat,R3: set_nat] :
% 14.25/8.04 ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ V @ W ) @ ( transi6264000038957366511cl_nat @ ( rel_restrict_nat @ E @ R3 ) ) )
% 14.25/8.04 => ~ ( member_nat @ W @ R3 ) ) ).
% 14.25/8.04
% 14.25/8.04 thf(fact_2086_rel__restrict__trancl__notR_I2_J,axiom,
% 14.25/8.04 ! [V: int,W: int,E: set_Pr958786334691620121nt_int,R3: set_int] :
% 14.25/8.04 ( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ V @ W ) @ ( transi6261509568448316235cl_int @ ( rel_restrict_int @ E @ R3 ) ) )
% 14.25/8.04 => ~ ( member_int @ W @ R3 ) ) ).
% 14.25/8.04
% 14.25/8.04 thf(fact_2087_rel__restrict__trancl__notR_I1_J,axiom,
% 14.25/8.04 ! [V: produc3658429121746597890et_nat > $o,W: produc3658429121746597890et_nat > $o,E: set_Pr2161125870931222855_nat_o,R3: set_Pr4532377907799695533_nat_o] :
% 14.25/8.04 ( ( member8781333585448626064_nat_o @ ( produc7368190662567826135_nat_o @ V @ W ) @ ( transi4009562729418357614_nat_o @ ( rel_re3331361160397388719_nat_o @ E @ R3 ) ) )
% 14.25/8.04 => ~ ( member6576561426505652726_nat_o @ V @ R3 ) ) ).
% 14.25/8.04
% 14.25/8.04 thf(fact_2088_rel__restrict__trancl__notR_I1_J,axiom,
% 14.25/8.04 ! [V: $o,W: $o,E: set_Product_prod_o_o,R3: set_o] :
% 14.25/8.04 ( ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ V @ W ) @ ( transitive_trancl_o @ ( rel_restrict_o @ E @ R3 ) ) )
% 14.25/8.04 => ~ ( member_o @ V @ R3 ) ) ).
% 14.25/8.04
% 14.25/8.04 thf(fact_2089_rel__restrict__trancl__notR_I1_J,axiom,
% 14.25/8.04 ! [V: product_unit,W: product_unit,E: set_Pr5094982260447487303t_unit,R3: set_Product_unit] :
% 14.25/8.04 ( ( member7821138191923116944t_unit @ ( produc5692694829384537815t_unit @ V @ W ) @ ( transi430408012122777006t_unit @ ( rel_re9207996470947987439t_unit @ E @ R3 ) ) )
% 14.25/8.04 => ~ ( member_Product_unit @ V @ R3 ) ) ).
% 14.25/8.04
% 14.25/8.04 thf(fact_2090_rel__restrict__trancl__notR_I1_J,axiom,
% 14.25/8.04 ! [V: nat,W: nat,E: set_Pr1261947904930325089at_nat,R3: set_nat] :
% 14.25/8.04 ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ V @ W ) @ ( transi6264000038957366511cl_nat @ ( rel_restrict_nat @ E @ R3 ) ) )
% 14.25/8.04 => ~ ( member_nat @ V @ R3 ) ) ).
% 14.25/8.04
% 14.25/8.04 thf(fact_2091_rel__restrict__trancl__notR_I1_J,axiom,
% 14.25/8.04 ! [V: int,W: int,E: set_Pr958786334691620121nt_int,R3: set_int] :
% 14.25/8.04 ( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ V @ W ) @ ( transi6261509568448316235cl_int @ ( rel_restrict_int @ E @ R3 ) ) )
% 14.25/8.04 => ~ ( member_int @ V @ R3 ) ) ).
% 14.25/8.04
% 14.25/8.04 thf(fact_2092_rel__restrict__trancl__mem,axiom,
% 14.25/8.04 ! [A2: nat,B: nat,A: set_Pr1261947904930325089at_nat,R3: set_nat] :
% 14.25/8.04 ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A2 @ B ) @ ( transi6264000038957366511cl_nat @ ( rel_restrict_nat @ A @ R3 ) ) )
% 14.25/8.04 => ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A2 @ B ) @ ( rel_restrict_nat @ ( transi6264000038957366511cl_nat @ A ) @ R3 ) ) ) ).
% 14.25/8.04
% 14.25/8.04 thf(fact_2093_rel__restrict__trancl__mem,axiom,
% 14.25/8.04 ! [A2: int,B: int,A: set_Pr958786334691620121nt_int,R3: set_int] :
% 14.25/8.04 ( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ A2 @ B ) @ ( transi6261509568448316235cl_int @ ( rel_restrict_int @ A @ R3 ) ) )
% 14.25/8.04 => ( member5262025264175285858nt_int @ ( product_Pair_int_int @ A2 @ B ) @ ( rel_restrict_int @ ( transi6261509568448316235cl_int @ A ) @ R3 ) ) ) ).
% 14.25/8.04
% 14.25/8.04 thf(fact_2094_rel__restrict__mono2,axiom,
% 14.25/8.04 ! [R3: set_nat,S: set_nat,A: set_Pr1261947904930325089at_nat] :
% 14.25/8.04 ( ( ord_less_eq_set_nat @ R3 @ S )
% 14.25/8.04 => ( ord_le3146513528884898305at_nat @ ( rel_restrict_nat @ A @ S ) @ ( rel_restrict_nat @ A @ R3 ) ) ) ).
% 14.25/8.04
% 14.25/8.04 thf(fact_2095_singleton__quotient,axiom,
% 14.25/8.04 ! [X: product_unit,R2: set_Pr5094982260447487303t_unit] :
% 14.25/8.04 ( ( equiv_5172566880711946708t_unit @ ( insert_Product_unit @ X @ bot_bo3957492148770167129t_unit ) @ R2 )
% 14.25/8.04 = ( insert8986028188087627965t_unit @ ( image_8431784415008483091t_unit @ R2 @ ( insert_Product_unit @ X @ bot_bo3957492148770167129t_unit ) ) @ bot_bo6284674557190688313t_unit ) ) ).
% 14.25/8.04
% 14.25/8.04 thf(fact_2096_singleton__quotient,axiom,
% 14.25/8.04 ! [X: $o,R2: set_Product_prod_o_o] :
% 14.25/8.04 ( ( equiv_quotient_o @ ( insert_o @ X @ bot_bot_set_o ) @ R2 )
% 14.25/8.04 = ( insert_set_o @ ( image_o_o @ R2 @ ( insert_o @ X @ bot_bot_set_o ) ) @ bot_bot_set_set_o ) ) ).
% 14.25/8.04
% 14.25/8.04 thf(fact_2097_singleton__quotient,axiom,
% 14.25/8.04 ! [X: nat,R2: set_Pr1261947904930325089at_nat] :
% 14.25/8.04 ( ( equiv_quotient_nat @ ( insert_nat @ X @ bot_bot_set_nat ) @ R2 )
% 14.25/8.04 = ( insert_set_nat @ ( image_nat_nat @ R2 @ ( insert_nat @ X @ bot_bot_set_nat ) ) @ bot_bot_set_set_nat ) ) ).
% 14.25/8.04
% 14.25/8.04 thf(fact_2098_singleton__quotient,axiom,
% 14.25/8.04 ! [X: int,R2: set_Pr958786334691620121nt_int] :
% 14.25/8.04 ( ( equiv_quotient_int @ ( insert_int @ X @ bot_bot_set_int ) @ R2 )
% 14.25/8.04 = ( insert_set_int @ ( image_int_int @ R2 @ ( insert_int @ X @ bot_bot_set_int ) ) @ bot_bot_set_set_int ) ) ).
% 14.25/8.04
% 14.25/8.04 thf(fact_2099_singleton__quotient,axiom,
% 14.25/8.04 ! [X: produc3658429121746597890et_nat > $o,R2: set_Pr2161125870931222855_nat_o] :
% 14.25/8.04 ( ( equiv_7899955228204526484_nat_o @ ( insert5175938949040314269_nat_o @ X @ bot_bo7824918357723582233_nat_o ) @ R2 )
% 14.25/8.04 = ( insert2507855216428400253_nat_o @ ( image_2232984160439939859_nat_o @ R2 @ ( insert5175938949040314269_nat_o @ X @ bot_bo7824918357723582233_nat_o ) ) @ bot_bo3736544561982865913_nat_o ) ) ).
% 14.25/8.04
% 14.25/8.04 thf(fact_2100_quotientI,axiom,
% 14.25/8.04 ! [X: product_unit,A: set_Product_unit,R2: set_Pr5094982260447487303t_unit] :
% 14.25/8.04 ( ( member_Product_unit @ X @ A )
% 14.25/8.04 => ( member5877623283571906838t_unit @ ( image_8431784415008483091t_unit @ R2 @ ( insert_Product_unit @ X @ bot_bo3957492148770167129t_unit ) ) @ ( equiv_5172566880711946708t_unit @ A @ R2 ) ) ) ).
% 14.25/8.04
% 14.25/8.04 thf(fact_2101_quotientI,axiom,
% 14.25/8.04 ! [X: $o,A: set_o,R2: set_Product_prod_o_o] :
% 14.25/8.04 ( ( member_o @ X @ A )
% 14.25/8.04 => ( member_set_o @ ( image_o_o @ R2 @ ( insert_o @ X @ bot_bot_set_o ) ) @ ( equiv_quotient_o @ A @ R2 ) ) ) ).
% 14.25/8.04
% 14.25/8.04 thf(fact_2102_quotientI,axiom,
% 14.25/8.04 ! [X: nat,A: set_nat,R2: set_Pr1261947904930325089at_nat] :
% 14.25/8.04 ( ( member_nat @ X @ A )
% 14.25/8.04 => ( member_set_nat @ ( image_nat_nat @ R2 @ ( insert_nat @ X @ bot_bot_set_nat ) ) @ ( equiv_quotient_nat @ A @ R2 ) ) ) ).
% 14.25/8.04
% 14.25/8.04 thf(fact_2103_quotientI,axiom,
% 14.25/8.04 ! [X: int,A: set_int,R2: set_Pr958786334691620121nt_int] :
% 14.25/8.04 ( ( member_int @ X @ A )
% 14.25/8.04 => ( member_set_int @ ( image_int_int @ R2 @ ( insert_int @ X @ bot_bot_set_int ) ) @ ( equiv_quotient_int @ A @ R2 ) ) ) ).
% 14.25/8.04
% 14.25/8.04 thf(fact_2104_quotientI,axiom,
% 14.25/8.04 ! [X: produc3658429121746597890et_nat > $o,A: set_Pr4532377907799695533_nat_o,R2: set_Pr2161125870931222855_nat_o] :
% 14.25/8.04 ( ( member6576561426505652726_nat_o @ X @ A )
% 14.25/8.04 => ( member1452989533377600726_nat_o @ ( image_2232984160439939859_nat_o @ R2 @ ( insert5175938949040314269_nat_o @ X @ bot_bo7824918357723582233_nat_o ) ) @ ( equiv_7899955228204526484_nat_o @ A @ R2 ) ) ) ).
% 14.25/8.04
% 14.25/8.04 thf(fact_2105_quotientE,axiom,
% 14.25/8.04 ! [X4: set_Product_unit,A: set_Product_unit,R2: set_Pr5094982260447487303t_unit] :
% 14.25/8.04 ( ( member5877623283571906838t_unit @ X4 @ ( equiv_5172566880711946708t_unit @ A @ R2 ) )
% 14.25/8.04 => ~ ! [X3: product_unit] :
% 14.25/8.04 ( ( X4
% 14.25/8.04 = ( image_8431784415008483091t_unit @ R2 @ ( insert_Product_unit @ X3 @ bot_bo3957492148770167129t_unit ) ) )
% 14.25/8.04 => ~ ( member_Product_unit @ X3 @ A ) ) ) ).
% 14.25/8.04
% 14.25/8.04 thf(fact_2106_quotientE,axiom,
% 14.25/8.04 ! [X4: set_o,A: set_o,R2: set_Product_prod_o_o] :
% 14.25/8.04 ( ( member_set_o @ X4 @ ( equiv_quotient_o @ A @ R2 ) )
% 14.25/8.04 => ~ ! [X3: $o] :
% 14.25/8.04 ( ( X4
% 14.25/8.04 = ( image_o_o @ R2 @ ( insert_o @ X3 @ bot_bot_set_o ) ) )
% 14.25/8.04 => ~ ( member_o @ X3 @ A ) ) ) ).
% 14.25/8.04
% 14.25/8.04 thf(fact_2107_quotientE,axiom,
% 14.25/8.04 ! [X4: set_nat,A: set_nat,R2: set_Pr1261947904930325089at_nat] :
% 14.25/8.04 ( ( member_set_nat @ X4 @ ( equiv_quotient_nat @ A @ R2 ) )
% 14.25/8.04 => ~ ! [X3: nat] :
% 14.25/8.04 ( ( X4
% 14.25/8.04 = ( image_nat_nat @ R2 @ ( insert_nat @ X3 @ bot_bot_set_nat ) ) )
% 14.25/8.04 => ~ ( member_nat @ X3 @ A ) ) ) ).
% 14.25/8.04
% 14.25/8.04 thf(fact_2108_quotientE,axiom,
% 14.25/8.04 ! [X4: set_int,A: set_int,R2: set_Pr958786334691620121nt_int] :
% 14.25/8.04 ( ( member_set_int @ X4 @ ( equiv_quotient_int @ A @ R2 ) )
% 14.25/8.04 => ~ ! [X3: int] :
% 14.25/8.04 ( ( X4
% 14.25/8.04 = ( image_int_int @ R2 @ ( insert_int @ X3 @ bot_bot_set_int ) ) )
% 14.25/8.04 => ~ ( member_int @ X3 @ A ) ) ) ).
% 14.25/8.04
% 14.25/8.04 thf(fact_2109_quotientE,axiom,
% 14.25/8.04 ! [X4: set_Pr4532377907799695533_nat_o,A: set_Pr4532377907799695533_nat_o,R2: set_Pr2161125870931222855_nat_o] :
% 14.25/8.04 ( ( member1452989533377600726_nat_o @ X4 @ ( equiv_7899955228204526484_nat_o @ A @ R2 ) )
% 14.25/8.04 => ~ ! [X3: produc3658429121746597890et_nat > $o] :
% 14.25/8.04 ( ( X4
% 14.25/8.04 = ( image_2232984160439939859_nat_o @ R2 @ ( insert5175938949040314269_nat_o @ X3 @ bot_bo7824918357723582233_nat_o ) ) )
% 14.25/8.04 => ~ ( member6576561426505652726_nat_o @ X3 @ A ) ) ) ).
% 14.25/8.04
% 14.25/8.04 thf(fact_2110_Inf__fin_OboundedE,axiom,
% 14.25/8.04 ! [A: set_Product_unit,X: product_unit] :
% 14.25/8.04 ( ( finite4290736615968046902t_unit @ A )
% 14.25/8.04 => ( ( A != bot_bo3957492148770167129t_unit )
% 14.25/8.04 => ( ( ord_le3221252021190050221t_unit @ X @ ( lattic1263872656861969706t_unit @ A ) )
% 14.25/8.04 => ! [A10: product_unit] :
% 14.25/8.04 ( ( member_Product_unit @ A10 @ A )
% 14.25/8.04 => ( ord_le3221252021190050221t_unit @ X @ A10 ) ) ) ) ) ).
% 14.25/8.04
% 14.25/8.04 thf(fact_2111_Inf__fin_OboundedE,axiom,
% 14.25/8.04 ! [A: set_o,X: $o] :
% 14.25/8.04 ( ( finite_finite_o @ A )
% 14.25/8.04 => ( ( A != bot_bot_set_o )
% 14.25/8.04 => ( ( ord_less_eq_o @ X @ ( lattic4107685809792843317_fin_o @ A ) )
% 14.25/8.04 => ! [A10: $o] :
% 14.25/8.04 ( ( member_o @ A10 @ A )
% 14.25/8.04 => ( ord_less_eq_o @ X @ A10 ) ) ) ) ) ).
% 14.25/8.04
% 14.25/8.04 thf(fact_2112_Inf__fin_OboundedE,axiom,
% 14.25/8.04 ! [A: set_Pr4532377907799695533_nat_o,X: produc3658429121746597890et_nat > $o] :
% 14.25/8.04 ( ( finite3252695134891459830_nat_o @ A )
% 14.25/8.04 => ( ( A != bot_bo7824918357723582233_nat_o )
% 14.25/8.04 => ( ( ord_le729326519192465773_nat_o @ X @ ( lattic956194824204696298_nat_o @ A ) )
% 14.25/8.04 => ! [A10: produc3658429121746597890et_nat > $o] :
% 14.25/8.04 ( ( member6576561426505652726_nat_o @ A10 @ A )
% 14.25/8.04 => ( ord_le729326519192465773_nat_o @ X @ A10 ) ) ) ) ) ).
% 14.25/8.04
% 14.25/8.04 thf(fact_2113_Inf__fin_OboundedE,axiom,
% 14.25/8.04 ! [A: set_filter_nat,X: filter_nat] :
% 14.25/8.04 ( ( finite2119507909894593271er_nat @ A )
% 14.25/8.04 => ( ( A != bot_bo498966703094740906er_nat )
% 14.25/8.04 => ( ( ord_le2510731241096832064er_nat @ X @ ( lattic9128708996678779395er_nat @ A ) )
% 14.25/8.04 => ! [A10: filter_nat] :
% 14.25/8.04 ( ( member_filter_nat @ A10 @ A )
% 14.25/8.04 => ( ord_le2510731241096832064er_nat @ X @ A10 ) ) ) ) ) ).
% 14.25/8.04
% 14.25/8.04 thf(fact_2114_Inf__fin_OboundedE,axiom,
% 14.25/8.04 ! [A: set_set_nat,X: set_nat] :
% 14.25/8.04 ( ( finite1152437895449049373et_nat @ A )
% 14.25/8.04 => ( ( A != bot_bot_set_set_nat )
% 14.25/8.04 => ( ( ord_less_eq_set_nat @ X @ ( lattic3014633134055518761et_nat @ A ) )
% 14.25/8.04 => ! [A10: set_nat] :
% 14.25/8.04 ( ( member_set_nat @ A10 @ A )
% 14.25/8.04 => ( ord_less_eq_set_nat @ X @ A10 ) ) ) ) ) ).
% 14.25/8.04
% 14.25/8.04 thf(fact_2115_Inf__fin_OboundedE,axiom,
% 14.25/8.04 ! [A: set_rat,X: rat] :
% 14.25/8.04 ( ( finite_finite_rat @ A )
% 14.25/8.04 => ( ( A != bot_bot_set_rat )
% 14.25/8.04 => ( ( ord_less_eq_rat @ X @ ( lattic4603258475043424379in_rat @ A ) )
% 14.25/8.04 => ! [A10: rat] :
% 14.25/8.04 ( ( member_rat @ A10 @ A )
% 14.25/8.04 => ( ord_less_eq_rat @ X @ A10 ) ) ) ) ) ).
% 14.25/8.04
% 14.25/8.04 thf(fact_2116_Inf__fin_OboundedE,axiom,
% 14.25/8.04 ! [A: set_nat,X: nat] :
% 14.25/8.04 ( ( finite_finite_nat @ A )
% 14.25/8.04 => ( ( A != bot_bot_set_nat )
% 14.25/8.04 => ( ( ord_less_eq_nat @ X @ ( lattic5238388535129920115in_nat @ A ) )
% 14.25/8.04 => ! [A10: nat] :
% 14.25/8.04 ( ( member_nat @ A10 @ A )
% 14.25/8.04 => ( ord_less_eq_nat @ X @ A10 ) ) ) ) ) ).
% 14.25/8.04
% 14.25/8.04 thf(fact_2117_Inf__fin_OboundedE,axiom,
% 14.25/8.04 ! [A: set_int,X: int] :
% 14.25/8.04 ( ( finite_finite_int @ A )
% 14.25/8.04 => ( ( A != bot_bot_set_int )
% 14.25/8.04 => ( ( ord_less_eq_int @ X @ ( lattic5235898064620869839in_int @ A ) )
% 14.25/8.04 => ! [A10: int] :
% 14.25/8.04 ( ( member_int @ A10 @ A )
% 14.25/8.04 => ( ord_less_eq_int @ X @ A10 ) ) ) ) ) ).
% 14.25/8.04
% 14.25/8.04 thf(fact_2118_Inf__fin_OboundedI,axiom,
% 14.25/8.04 ! [A: set_Product_unit,X: product_unit] :
% 14.25/8.04 ( ( finite4290736615968046902t_unit @ A )
% 14.25/8.04 => ( ( A != bot_bo3957492148770167129t_unit )
% 14.25/8.04 => ( ! [A4: product_unit] :
% 14.25/8.04 ( ( member_Product_unit @ A4 @ A )
% 14.25/8.04 => ( ord_le3221252021190050221t_unit @ X @ A4 ) )
% 14.25/8.04 => ( ord_le3221252021190050221t_unit @ X @ ( lattic1263872656861969706t_unit @ A ) ) ) ) ) ).
% 14.25/8.04
% 14.25/8.04 thf(fact_2119_Inf__fin_OboundedI,axiom,
% 14.25/8.04 ! [A: set_o,X: $o] :
% 14.25/8.04 ( ( finite_finite_o @ A )
% 14.25/8.04 => ( ( A != bot_bot_set_o )
% 14.25/8.04 => ( ! [A4: $o] :
% 14.25/8.04 ( ( member_o @ A4 @ A )
% 14.25/8.04 => ( ord_less_eq_o @ X @ A4 ) )
% 14.25/8.04 => ( ord_less_eq_o @ X @ ( lattic4107685809792843317_fin_o @ A ) ) ) ) ) ).
% 14.25/8.04
% 14.25/8.04 thf(fact_2120_Inf__fin_OboundedI,axiom,
% 14.25/8.04 ! [A: set_Pr4532377907799695533_nat_o,X: produc3658429121746597890et_nat > $o] :
% 14.25/8.04 ( ( finite3252695134891459830_nat_o @ A )
% 14.25/8.04 => ( ( A != bot_bo7824918357723582233_nat_o )
% 14.25/8.04 => ( ! [A4: produc3658429121746597890et_nat > $o] :
% 14.25/8.04 ( ( member6576561426505652726_nat_o @ A4 @ A )
% 14.25/8.04 => ( ord_le729326519192465773_nat_o @ X @ A4 ) )
% 14.25/8.04 => ( ord_le729326519192465773_nat_o @ X @ ( lattic956194824204696298_nat_o @ A ) ) ) ) ) ).
% 14.25/8.04
% 14.25/8.04 thf(fact_2121_Inf__fin_OboundedI,axiom,
% 14.25/8.04 ! [A: set_filter_nat,X: filter_nat] :
% 14.25/8.04 ( ( finite2119507909894593271er_nat @ A )
% 14.25/8.04 => ( ( A != bot_bo498966703094740906er_nat )
% 14.25/8.04 => ( ! [A4: filter_nat] :
% 14.25/8.04 ( ( member_filter_nat @ A4 @ A )
% 14.25/8.04 => ( ord_le2510731241096832064er_nat @ X @ A4 ) )
% 14.25/8.04 => ( ord_le2510731241096832064er_nat @ X @ ( lattic9128708996678779395er_nat @ A ) ) ) ) ) ).
% 14.25/8.04
% 14.25/8.04 thf(fact_2122_Inf__fin_OboundedI,axiom,
% 14.25/8.04 ! [A: set_set_nat,X: set_nat] :
% 14.25/8.04 ( ( finite1152437895449049373et_nat @ A )
% 14.25/8.04 => ( ( A != bot_bot_set_set_nat )
% 14.25/8.04 => ( ! [A4: set_nat] :
% 14.25/8.04 ( ( member_set_nat @ A4 @ A )
% 14.25/8.04 => ( ord_less_eq_set_nat @ X @ A4 ) )
% 14.25/8.04 => ( ord_less_eq_set_nat @ X @ ( lattic3014633134055518761et_nat @ A ) ) ) ) ) ).
% 14.25/8.04
% 14.25/8.04 thf(fact_2123_Inf__fin_OboundedI,axiom,
% 14.25/8.04 ! [A: set_rat,X: rat] :
% 14.25/8.04 ( ( finite_finite_rat @ A )
% 14.25/8.04 => ( ( A != bot_bot_set_rat )
% 14.25/8.04 => ( ! [A4: rat] :
% 14.25/8.04 ( ( member_rat @ A4 @ A )
% 14.25/8.04 => ( ord_less_eq_rat @ X @ A4 ) )
% 14.25/8.04 => ( ord_less_eq_rat @ X @ ( lattic4603258475043424379in_rat @ A ) ) ) ) ) ).
% 14.25/8.04
% 14.25/8.04 thf(fact_2124_Inf__fin_OboundedI,axiom,
% 14.25/8.04 ! [A: set_nat,X: nat] :
% 14.25/8.04 ( ( finite_finite_nat @ A )
% 14.25/8.04 => ( ( A != bot_bot_set_nat )
% 14.25/8.04 => ( ! [A4: nat] :
% 14.25/8.04 ( ( member_nat @ A4 @ A )
% 14.25/8.04 => ( ord_less_eq_nat @ X @ A4 ) )
% 14.25/8.04 => ( ord_less_eq_nat @ X @ ( lattic5238388535129920115in_nat @ A ) ) ) ) ) ).
% 14.25/8.04
% 14.25/8.04 thf(fact_2125_Inf__fin_OboundedI,axiom,
% 14.25/8.04 ! [A: set_int,X: int] :
% 14.25/8.04 ( ( finite_finite_int @ A )
% 14.25/8.04 => ( ( A != bot_bot_set_int )
% 14.25/8.04 => ( ! [A4: int] :
% 14.25/8.04 ( ( member_int @ A4 @ A )
% 14.25/8.04 => ( ord_less_eq_int @ X @ A4 ) )
% 14.25/8.04 => ( ord_less_eq_int @ X @ ( lattic5235898064620869839in_int @ A ) ) ) ) ) ).
% 14.25/8.04
% 14.25/8.04 thf(fact_2126_Sup__fin_OboundedE,axiom,
% 14.25/8.04 ! [A: set_Product_unit,X: product_unit] :
% 14.25/8.04 ( ( finite4290736615968046902t_unit @ A )
% 14.25/8.04 => ( ( A != bot_bo3957492148770167129t_unit )
% 14.25/8.04 => ( ( ord_le3221252021190050221t_unit @ ( lattic5294303975357428420t_unit @ A ) @ X )
% 14.25/8.04 => ! [A10: product_unit] :
% 14.25/8.04 ( ( member_Product_unit @ A10 @ A )
% 14.25/8.04 => ( ord_le3221252021190050221t_unit @ A10 @ X ) ) ) ) ) ).
% 14.25/8.04
% 14.25/8.04 thf(fact_2127_Sup__fin_OboundedE,axiom,
% 14.25/8.04 ! [A: set_o,X: $o] :
% 14.25/8.04 ( ( finite_finite_o @ A )
% 14.25/8.04 => ( ( A != bot_bot_set_o )
% 14.25/8.04 => ( ( ord_less_eq_o @ ( lattic1508158080041050831_fin_o @ A ) @ X )
% 14.25/8.04 => ! [A10: $o] :
% 14.25/8.04 ( ( member_o @ A10 @ A )
% 14.25/8.04 => ( ord_less_eq_o @ A10 @ X ) ) ) ) ) ).
% 14.25/8.04
% 14.25/8.04 thf(fact_2128_Sup__fin_OboundedE,axiom,
% 14.25/8.04 ! [A: set_Pr4532377907799695533_nat_o,X: produc3658429121746597890et_nat > $o] :
% 14.25/8.04 ( ( finite3252695134891459830_nat_o @ A )
% 14.25/8.04 => ( ( A != bot_bo7824918357723582233_nat_o )
% 14.25/8.04 => ( ( ord_le729326519192465773_nat_o @ ( lattic7320199455484906628_nat_o @ A ) @ X )
% 14.25/8.04 => ! [A10: produc3658429121746597890et_nat > $o] :
% 14.25/8.04 ( ( member6576561426505652726_nat_o @ A10 @ A )
% 14.25/8.04 => ( ord_le729326519192465773_nat_o @ A10 @ X ) ) ) ) ) ).
% 14.25/8.04
% 14.25/8.04 thf(fact_2129_Sup__fin_OboundedE,axiom,
% 14.25/8.04 ! [A: set_filter_nat,X: filter_nat] :
% 14.25/8.04 ( ( finite2119507909894593271er_nat @ A )
% 14.25/8.04 => ( ( A != bot_bo498966703094740906er_nat )
% 14.25/8.04 => ( ( ord_le2510731241096832064er_nat @ ( lattic5930898082463196905er_nat @ A ) @ X )
% 14.25/8.04 => ! [A10: filter_nat] :
% 14.25/8.04 ( ( member_filter_nat @ A10 @ A )
% 14.25/8.04 => ( ord_le2510731241096832064er_nat @ A10 @ X ) ) ) ) ) ).
% 14.25/8.04
% 14.25/8.04 thf(fact_2130_Sup__fin_OboundedE,axiom,
% 14.25/8.04 ! [A: set_set_nat,X: set_nat] :
% 14.25/8.04 ( ( finite1152437895449049373et_nat @ A )
% 14.25/8.04 => ( ( A != bot_bot_set_set_nat )
% 14.25/8.04 => ( ( ord_less_eq_set_nat @ ( lattic3835124923745554447et_nat @ A ) @ X )
% 14.25/8.04 => ! [A10: set_nat] :
% 14.25/8.04 ( ( member_set_nat @ A10 @ A )
% 14.25/8.04 => ( ord_less_eq_set_nat @ A10 @ X ) ) ) ) ) ).
% 14.25/8.04
% 14.25/8.04 thf(fact_2131_Sup__fin_OboundedE,axiom,
% 14.25/8.04 ! [A: set_rat,X: rat] :
% 14.25/8.04 ( ( finite_finite_rat @ A )
% 14.25/8.04 => ( ( A != bot_bot_set_rat )
% 14.25/8.04 => ( ( ord_less_eq_rat @ ( lattic458866745392299617in_rat @ A ) @ X )
% 14.25/8.04 => ! [A10: rat] :
% 14.25/8.04 ( ( member_rat @ A10 @ A )
% 14.25/8.04 => ( ord_less_eq_rat @ A10 @ X ) ) ) ) ) ).
% 14.25/8.04
% 14.25/8.04 thf(fact_2132_Sup__fin_OboundedE,axiom,
% 14.25/8.04 ! [A: set_nat,X: nat] :
% 14.25/8.04 ( ( finite_finite_nat @ A )
% 14.25/8.04 => ( ( A != bot_bot_set_nat )
% 14.25/8.04 => ( ( ord_less_eq_nat @ ( lattic1093996805478795353in_nat @ A ) @ X )
% 14.25/8.04 => ! [A10: nat] :
% 14.25/8.04 ( ( member_nat @ A10 @ A )
% 14.25/8.04 => ( ord_less_eq_nat @ A10 @ X ) ) ) ) ) ).
% 14.25/8.04
% 14.25/8.04 thf(fact_2133_Sup__fin_OboundedE,axiom,
% 14.25/8.04 ! [A: set_int,X: int] :
% 14.25/8.04 ( ( finite_finite_int @ A )
% 14.25/8.04 => ( ( A != bot_bot_set_int )
% 14.25/8.04 => ( ( ord_less_eq_int @ ( lattic1091506334969745077in_int @ A ) @ X )
% 14.25/8.04 => ! [A10: int] :
% 14.25/8.04 ( ( member_int @ A10 @ A )
% 14.25/8.04 => ( ord_less_eq_int @ A10 @ X ) ) ) ) ) ).
% 14.25/8.04
% 14.25/8.04 thf(fact_2134_Sup__fin_OboundedI,axiom,
% 14.25/8.04 ! [A: set_Product_unit,X: product_unit] :
% 14.25/8.04 ( ( finite4290736615968046902t_unit @ A )
% 14.25/8.04 => ( ( A != bot_bo3957492148770167129t_unit )
% 14.25/8.04 => ( ! [A4: product_unit] :
% 14.25/8.04 ( ( member_Product_unit @ A4 @ A )
% 14.25/8.04 => ( ord_le3221252021190050221t_unit @ A4 @ X ) )
% 14.25/8.04 => ( ord_le3221252021190050221t_unit @ ( lattic5294303975357428420t_unit @ A ) @ X ) ) ) ) ).
% 14.25/8.04
% 14.25/8.04 thf(fact_2135_Sup__fin_OboundedI,axiom,
% 14.25/8.04 ! [A: set_o,X: $o] :
% 14.25/8.04 ( ( finite_finite_o @ A )
% 14.25/8.04 => ( ( A != bot_bot_set_o )
% 14.25/8.04 => ( ! [A4: $o] :
% 14.25/8.04 ( ( member_o @ A4 @ A )
% 14.25/8.04 => ( ord_less_eq_o @ A4 @ X ) )
% 14.25/8.04 => ( ord_less_eq_o @ ( lattic1508158080041050831_fin_o @ A ) @ X ) ) ) ) ).
% 14.25/8.04
% 14.25/8.04 thf(fact_2136_Sup__fin_OboundedI,axiom,
% 14.25/8.04 ! [A: set_Pr4532377907799695533_nat_o,X: produc3658429121746597890et_nat > $o] :
% 14.25/8.04 ( ( finite3252695134891459830_nat_o @ A )
% 14.25/8.04 => ( ( A != bot_bo7824918357723582233_nat_o )
% 14.25/8.04 => ( ! [A4: produc3658429121746597890et_nat > $o] :
% 14.25/8.04 ( ( member6576561426505652726_nat_o @ A4 @ A )
% 14.25/8.04 => ( ord_le729326519192465773_nat_o @ A4 @ X ) )
% 14.25/8.04 => ( ord_le729326519192465773_nat_o @ ( lattic7320199455484906628_nat_o @ A ) @ X ) ) ) ) ).
% 14.25/8.04
% 14.25/8.04 thf(fact_2137_Sup__fin_OboundedI,axiom,
% 14.25/8.04 ! [A: set_filter_nat,X: filter_nat] :
% 14.25/8.04 ( ( finite2119507909894593271er_nat @ A )
% 14.25/8.04 => ( ( A != bot_bo498966703094740906er_nat )
% 14.25/8.04 => ( ! [A4: filter_nat] :
% 14.25/8.04 ( ( member_filter_nat @ A4 @ A )
% 14.25/8.04 => ( ord_le2510731241096832064er_nat @ A4 @ X ) )
% 14.25/8.04 => ( ord_le2510731241096832064er_nat @ ( lattic5930898082463196905er_nat @ A ) @ X ) ) ) ) ).
% 14.25/8.04
% 14.25/8.04 thf(fact_2138_Sup__fin_OboundedI,axiom,
% 14.25/8.04 ! [A: set_set_nat,X: set_nat] :
% 14.25/8.04 ( ( finite1152437895449049373et_nat @ A )
% 14.25/8.04 => ( ( A != bot_bot_set_set_nat )
% 14.25/8.04 => ( ! [A4: set_nat] :
% 14.25/8.04 ( ( member_set_nat @ A4 @ A )
% 14.25/8.04 => ( ord_less_eq_set_nat @ A4 @ X ) )
% 14.25/8.04 => ( ord_less_eq_set_nat @ ( lattic3835124923745554447et_nat @ A ) @ X ) ) ) ) ).
% 14.25/8.04
% 14.25/8.04 thf(fact_2139_Sup__fin_OboundedI,axiom,
% 14.25/8.04 ! [A: set_rat,X: rat] :
% 14.25/8.04 ( ( finite_finite_rat @ A )
% 14.25/8.04 => ( ( A != bot_bot_set_rat )
% 14.25/8.04 => ( ! [A4: rat] :
% 14.25/8.04 ( ( member_rat @ A4 @ A )
% 14.25/8.04 => ( ord_less_eq_rat @ A4 @ X ) )
% 14.25/8.04 => ( ord_less_eq_rat @ ( lattic458866745392299617in_rat @ A ) @ X ) ) ) ) ).
% 14.25/8.04
% 14.25/8.04 thf(fact_2140_Sup__fin_OboundedI,axiom,
% 14.25/8.04 ! [A: set_nat,X: nat] :
% 14.25/8.04 ( ( finite_finite_nat @ A )
% 14.25/8.04 => ( ( A != bot_bot_set_nat )
% 14.25/8.04 => ( ! [A4: nat] :
% 14.25/8.04 ( ( member_nat @ A4 @ A )
% 14.25/8.04 => ( ord_less_eq_nat @ A4 @ X ) )
% 14.25/8.04 => ( ord_less_eq_nat @ ( lattic1093996805478795353in_nat @ A ) @ X ) ) ) ) ).
% 14.25/8.04
% 14.25/8.04 thf(fact_2141_Sup__fin_OboundedI,axiom,
% 14.25/8.04 ! [A: set_int,X: int] :
% 14.25/8.04 ( ( finite_finite_int @ A )
% 14.25/8.04 => ( ( A != bot_bot_set_int )
% 14.25/8.04 => ( ! [A4: int] :
% 14.25/8.04 ( ( member_int @ A4 @ A )
% 14.25/8.04 => ( ord_less_eq_int @ A4 @ X ) )
% 14.25/8.04 => ( ord_less_eq_int @ ( lattic1091506334969745077in_int @ A ) @ X ) ) ) ) ).
% 14.25/8.04
% 14.25/8.04 thf(fact_2142_Inf__fin_Obounded__iff,axiom,
% 14.25/8.04 ! [A: set_Product_unit,X: product_unit] :
% 14.25/8.04 ( ( finite4290736615968046902t_unit @ A )
% 14.25/8.04 => ( ( A != bot_bo3957492148770167129t_unit )
% 14.25/8.04 => ( ( ord_le3221252021190050221t_unit @ X @ ( lattic1263872656861969706t_unit @ A ) )
% 14.25/8.04 = ( ! [X2: product_unit] :
% 14.25/8.04 ( ( member_Product_unit @ X2 @ A )
% 14.25/8.04 => ( ord_le3221252021190050221t_unit @ X @ X2 ) ) ) ) ) ) ).
% 14.25/8.04
% 14.25/8.04 thf(fact_2143_Inf__fin_Obounded__iff,axiom,
% 14.25/8.04 ! [A: set_o,X: $o] :
% 14.25/8.04 ( ( finite_finite_o @ A )
% 14.25/8.04 => ( ( A != bot_bot_set_o )
% 14.25/8.04 => ( ( ord_less_eq_o @ X @ ( lattic4107685809792843317_fin_o @ A ) )
% 14.25/8.04 = ( ! [X2: $o] :
% 14.25/8.04 ( ( member_o @ X2 @ A )
% 14.25/8.04 => ( ord_less_eq_o @ X @ X2 ) ) ) ) ) ) ).
% 14.25/8.04
% 14.25/8.04 thf(fact_2144_Inf__fin_Obounded__iff,axiom,
% 14.25/8.04 ! [A: set_Pr4532377907799695533_nat_o,X: produc3658429121746597890et_nat > $o] :
% 14.25/8.04 ( ( finite3252695134891459830_nat_o @ A )
% 14.25/8.04 => ( ( A != bot_bo7824918357723582233_nat_o )
% 14.25/8.04 => ( ( ord_le729326519192465773_nat_o @ X @ ( lattic956194824204696298_nat_o @ A ) )
% 14.25/8.04 = ( ! [X2: produc3658429121746597890et_nat > $o] :
% 14.25/8.04 ( ( member6576561426505652726_nat_o @ X2 @ A )
% 14.25/8.04 => ( ord_le729326519192465773_nat_o @ X @ X2 ) ) ) ) ) ) ).
% 14.25/8.04
% 14.25/8.04 thf(fact_2145_Inf__fin_Obounded__iff,axiom,
% 14.25/8.04 ! [A: set_filter_nat,X: filter_nat] :
% 14.25/8.04 ( ( finite2119507909894593271er_nat @ A )
% 14.25/8.04 => ( ( A != bot_bo498966703094740906er_nat )
% 14.25/8.04 => ( ( ord_le2510731241096832064er_nat @ X @ ( lattic9128708996678779395er_nat @ A ) )
% 14.25/8.04 = ( ! [X2: filter_nat] :
% 14.25/8.04 ( ( member_filter_nat @ X2 @ A )
% 14.25/8.04 => ( ord_le2510731241096832064er_nat @ X @ X2 ) ) ) ) ) ) ).
% 14.25/8.04
% 14.25/8.04 thf(fact_2146_Inf__fin_Obounded__iff,axiom,
% 14.25/8.04 ! [A: set_set_nat,X: set_nat] :
% 14.25/8.04 ( ( finite1152437895449049373et_nat @ A )
% 14.25/8.04 => ( ( A != bot_bot_set_set_nat )
% 14.25/8.04 => ( ( ord_less_eq_set_nat @ X @ ( lattic3014633134055518761et_nat @ A ) )
% 14.25/8.04 = ( ! [X2: set_nat] :
% 14.25/8.04 ( ( member_set_nat @ X2 @ A )
% 14.25/8.04 => ( ord_less_eq_set_nat @ X @ X2 ) ) ) ) ) ) ).
% 14.25/8.04
% 14.25/8.04 thf(fact_2147_Inf__fin_Obounded__iff,axiom,
% 14.25/8.04 ! [A: set_rat,X: rat] :
% 14.25/8.04 ( ( finite_finite_rat @ A )
% 14.25/8.04 => ( ( A != bot_bot_set_rat )
% 14.25/8.04 => ( ( ord_less_eq_rat @ X @ ( lattic4603258475043424379in_rat @ A ) )
% 14.25/8.04 = ( ! [X2: rat] :
% 14.25/8.04 ( ( member_rat @ X2 @ A )
% 14.25/8.04 => ( ord_less_eq_rat @ X @ X2 ) ) ) ) ) ) ).
% 14.25/8.04
% 14.25/8.04 thf(fact_2148_Inf__fin_Obounded__iff,axiom,
% 14.25/8.04 ! [A: set_nat,X: nat] :
% 14.25/8.04 ( ( finite_finite_nat @ A )
% 14.25/8.04 => ( ( A != bot_bot_set_nat )
% 14.25/8.04 => ( ( ord_less_eq_nat @ X @ ( lattic5238388535129920115in_nat @ A ) )
% 14.25/8.04 = ( ! [X2: nat] :
% 14.25/8.04 ( ( member_nat @ X2 @ A )
% 14.25/8.04 => ( ord_less_eq_nat @ X @ X2 ) ) ) ) ) ) ).
% 14.25/8.04
% 14.25/8.04 thf(fact_2149_Inf__fin_Obounded__iff,axiom,
% 14.25/8.04 ! [A: set_int,X: int] :
% 14.25/8.04 ( ( finite_finite_int @ A )
% 14.25/8.04 => ( ( A != bot_bot_set_int )
% 14.25/8.04 => ( ( ord_less_eq_int @ X @ ( lattic5235898064620869839in_int @ A ) )
% 14.25/8.04 = ( ! [X2: int] :
% 14.25/8.04 ( ( member_int @ X2 @ A )
% 14.25/8.04 => ( ord_less_eq_int @ X @ X2 ) ) ) ) ) ) ).
% 14.25/8.04
% 14.25/8.04 thf(fact_2150_Sup__fin_Obounded__iff,axiom,
% 14.25/8.04 ! [A: set_Product_unit,X: product_unit] :
% 14.25/8.04 ( ( finite4290736615968046902t_unit @ A )
% 14.25/8.04 => ( ( A != bot_bo3957492148770167129t_unit )
% 14.25/8.04 => ( ( ord_le3221252021190050221t_unit @ ( lattic5294303975357428420t_unit @ A ) @ X )
% 14.25/8.04 = ( ! [X2: product_unit] :
% 14.25/8.04 ( ( member_Product_unit @ X2 @ A )
% 14.25/8.04 => ( ord_le3221252021190050221t_unit @ X2 @ X ) ) ) ) ) ) ).
% 14.25/8.04
% 14.25/8.04 thf(fact_2151_Sup__fin_Obounded__iff,axiom,
% 14.25/8.04 ! [A: set_o,X: $o] :
% 14.25/8.04 ( ( finite_finite_o @ A )
% 14.25/8.04 => ( ( A != bot_bot_set_o )
% 14.25/8.04 => ( ( ord_less_eq_o @ ( lattic1508158080041050831_fin_o @ A ) @ X )
% 14.25/8.04 = ( ! [X2: $o] :
% 14.25/8.04 ( ( member_o @ X2 @ A )
% 14.25/8.04 => ( ord_less_eq_o @ X2 @ X ) ) ) ) ) ) ).
% 14.25/8.04
% 14.25/8.04 thf(fact_2152_Sup__fin_Obounded__iff,axiom,
% 14.25/8.04 ! [A: set_Pr4532377907799695533_nat_o,X: produc3658429121746597890et_nat > $o] :
% 14.25/8.04 ( ( finite3252695134891459830_nat_o @ A )
% 14.25/8.04 => ( ( A != bot_bo7824918357723582233_nat_o )
% 14.25/8.04 => ( ( ord_le729326519192465773_nat_o @ ( lattic7320199455484906628_nat_o @ A ) @ X )
% 14.25/8.04 = ( ! [X2: produc3658429121746597890et_nat > $o] :
% 14.25/8.04 ( ( member6576561426505652726_nat_o @ X2 @ A )
% 14.25/8.04 => ( ord_le729326519192465773_nat_o @ X2 @ X ) ) ) ) ) ) ).
% 14.25/8.04
% 14.25/8.04 thf(fact_2153_Sup__fin_Obounded__iff,axiom,
% 14.25/8.04 ! [A: set_filter_nat,X: filter_nat] :
% 14.25/8.04 ( ( finite2119507909894593271er_nat @ A )
% 14.25/8.04 => ( ( A != bot_bo498966703094740906er_nat )
% 14.25/8.04 => ( ( ord_le2510731241096832064er_nat @ ( lattic5930898082463196905er_nat @ A ) @ X )
% 14.25/8.04 = ( ! [X2: filter_nat] :
% 14.25/8.04 ( ( member_filter_nat @ X2 @ A )
% 14.25/8.04 => ( ord_le2510731241096832064er_nat @ X2 @ X ) ) ) ) ) ) ).
% 14.25/8.04
% 14.25/8.04 thf(fact_2154_Sup__fin_Obounded__iff,axiom,
% 14.25/8.04 ! [A: set_set_nat,X: set_nat] :
% 14.25/8.04 ( ( finite1152437895449049373et_nat @ A )
% 14.25/8.04 => ( ( A != bot_bot_set_set_nat )
% 14.25/8.04 => ( ( ord_less_eq_set_nat @ ( lattic3835124923745554447et_nat @ A ) @ X )
% 14.25/8.04 = ( ! [X2: set_nat] :
% 14.25/8.04 ( ( member_set_nat @ X2 @ A )
% 14.25/8.04 => ( ord_less_eq_set_nat @ X2 @ X ) ) ) ) ) ) ).
% 14.25/8.04
% 14.25/8.04 thf(fact_2155_Sup__fin_Obounded__iff,axiom,
% 14.25/8.04 ! [A: set_rat,X: rat] :
% 14.25/8.04 ( ( finite_finite_rat @ A )
% 14.25/8.04 => ( ( A != bot_bot_set_rat )
% 14.25/8.04 => ( ( ord_less_eq_rat @ ( lattic458866745392299617in_rat @ A ) @ X )
% 14.25/8.04 = ( ! [X2: rat] :
% 14.25/8.04 ( ( member_rat @ X2 @ A )
% 14.25/8.04 => ( ord_less_eq_rat @ X2 @ X ) ) ) ) ) ) ).
% 14.25/8.04
% 14.25/8.04 thf(fact_2156_Sup__fin_Obounded__iff,axiom,
% 14.25/8.04 ! [A: set_nat,X: nat] :
% 14.25/8.04 ( ( finite_finite_nat @ A )
% 14.25/8.04 => ( ( A != bot_bot_set_nat )
% 14.25/8.04 => ( ( ord_less_eq_nat @ ( lattic1093996805478795353in_nat @ A ) @ X )
% 14.25/8.04 = ( ! [X2: nat] :
% 14.25/8.04 ( ( member_nat @ X2 @ A )
% 14.25/8.04 => ( ord_less_eq_nat @ X2 @ X ) ) ) ) ) ) ).
% 14.25/8.04
% 14.25/8.04 thf(fact_2157_Sup__fin_Obounded__iff,axiom,
% 14.25/8.04 ! [A: set_int,X: int] :
% 14.25/8.04 ( ( finite_finite_int @ A )
% 14.25/8.04 => ( ( A != bot_bot_set_int )
% 14.25/8.04 => ( ( ord_less_eq_int @ ( lattic1091506334969745077in_int @ A ) @ X )
% 14.25/8.04 = ( ! [X2: int] :
% 14.25/8.04 ( ( member_int @ X2 @ A )
% 14.25/8.04 => ( ord_less_eq_int @ X2 @ X ) ) ) ) ) ) ).
% 14.25/8.04
% 14.25/8.04 thf(fact_2158_quotient__disj,axiom,
% 14.25/8.04 ! [A: set_Product_unit,R2: set_Pr5094982260447487303t_unit,X4: set_Product_unit,Y6: set_Product_unit] :
% 14.25/8.04 ( ( equiv_4189356295871601603t_unit @ A @ R2 )
% 14.25/8.04 => ( ( member5877623283571906838t_unit @ X4 @ ( equiv_5172566880711946708t_unit @ A @ R2 ) )
% 14.25/8.04 => ( ( member5877623283571906838t_unit @ Y6 @ ( equiv_5172566880711946708t_unit @ A @ R2 ) )
% 14.25/8.04 => ( ( X4 = Y6 )
% 14.25/8.04 | ( ( inf_in4660618365625256667t_unit @ X4 @ Y6 )
% 14.25/8.04 = bot_bo3957492148770167129t_unit ) ) ) ) ) ).
% 14.25/8.04
% 14.25/8.04 thf(fact_2159_quotient__disj,axiom,
% 14.25/8.04 ! [A: set_o,R2: set_Product_prod_o_o,X4: set_o,Y6: set_o] :
% 14.25/8.04 ( ( equiv_equiv_o @ A @ R2 )
% 14.25/8.04 => ( ( member_set_o @ X4 @ ( equiv_quotient_o @ A @ R2 ) )
% 14.25/8.04 => ( ( member_set_o @ Y6 @ ( equiv_quotient_o @ A @ R2 ) )
% 14.25/8.04 => ( ( X4 = Y6 )
% 14.25/8.04 | ( ( inf_inf_set_o @ X4 @ Y6 )
% 14.25/8.04 = bot_bot_set_o ) ) ) ) ) ).
% 14.25/8.04
% 14.25/8.04 thf(fact_2160_quotient__disj,axiom,
% 14.25/8.04 ! [A: set_nat,R2: set_Pr1261947904930325089at_nat,X4: set_nat,Y6: set_nat] :
% 14.25/8.04 ( ( equiv_equiv_nat @ A @ R2 )
% 14.25/8.04 => ( ( member_set_nat @ X4 @ ( equiv_quotient_nat @ A @ R2 ) )
% 14.25/8.04 => ( ( member_set_nat @ Y6 @ ( equiv_quotient_nat @ A @ R2 ) )
% 14.25/8.04 => ( ( X4 = Y6 )
% 14.25/8.04 | ( ( inf_inf_set_nat @ X4 @ Y6 )
% 14.25/8.04 = bot_bot_set_nat ) ) ) ) ) ).
% 14.25/8.04
% 14.25/8.04 thf(fact_2161_quotient__disj,axiom,
% 14.25/8.04 ! [A: set_int,R2: set_Pr958786334691620121nt_int,X4: set_int,Y6: set_int] :
% 14.25/8.04 ( ( equiv_equiv_int @ A @ R2 )
% 14.25/8.04 => ( ( member_set_int @ X4 @ ( equiv_quotient_int @ A @ R2 ) )
% 14.25/8.04 => ( ( member_set_int @ Y6 @ ( equiv_quotient_int @ A @ R2 ) )
% 14.25/8.04 => ( ( X4 = Y6 )
% 14.25/8.04 | ( ( inf_inf_set_int @ X4 @ Y6 )
% 14.25/8.04 = bot_bot_set_int ) ) ) ) ) ).
% 14.25/8.04
% 14.25/8.04 thf(fact_2162_quotient__disj,axiom,
% 14.25/8.04 ! [A: set_Pr4532377907799695533_nat_o,R2: set_Pr2161125870931222855_nat_o,X4: set_Pr4532377907799695533_nat_o,Y6: set_Pr4532377907799695533_nat_o] :
% 14.25/8.04 ( ( equiv_6726761681920839555_nat_o @ A @ R2 )
% 14.25/8.04 => ( ( member1452989533377600726_nat_o @ X4 @ ( equiv_7899955228204526484_nat_o @ A @ R2 ) )
% 14.25/8.04 => ( ( member1452989533377600726_nat_o @ Y6 @ ( equiv_7899955228204526484_nat_o @ A @ R2 ) )
% 14.25/8.04 => ( ( X4 = Y6 )
% 14.25/8.04 | ( ( inf_in1906310914598751387_nat_o @ X4 @ Y6 )
% 14.25/8.04 = bot_bo7824918357723582233_nat_o ) ) ) ) ) ).
% 14.25/8.04
% 14.25/8.04 thf(fact_2163_rel__restrict__Int__empty,axiom,
% 14.25/8.04 ! [A: set_Product_unit,R3: set_Pr5094982260447487303t_unit] :
% 14.25/8.04 ( ( ( inf_in4660618365625256667t_unit @ A @ ( field_Product_unit @ R3 ) )
% 14.25/8.04 = bot_bo3957492148770167129t_unit )
% 14.25/8.04 => ( ( rel_re9207996470947987439t_unit @ R3 @ A )
% 14.25/8.04 = R3 ) ) ).
% 14.25/8.04
% 14.25/8.04 thf(fact_2164_rel__restrict__Int__empty,axiom,
% 14.25/8.04 ! [A: set_o,R3: set_Product_prod_o_o] :
% 14.25/8.04 ( ( ( inf_inf_set_o @ A @ ( field_o @ R3 ) )
% 14.25/8.04 = bot_bot_set_o )
% 14.25/8.04 => ( ( rel_restrict_o @ R3 @ A )
% 14.25/8.04 = R3 ) ) ).
% 14.25/8.04
% 14.25/8.04 thf(fact_2165_rel__restrict__Int__empty,axiom,
% 14.25/8.04 ! [A: set_nat,R3: set_Pr1261947904930325089at_nat] :
% 14.25/8.04 ( ( ( inf_inf_set_nat @ A @ ( field_nat @ R3 ) )
% 14.25/8.04 = bot_bot_set_nat )
% 14.25/8.04 => ( ( rel_restrict_nat @ R3 @ A )
% 14.25/8.04 = R3 ) ) ).
% 14.25/8.04
% 14.25/8.04 thf(fact_2166_rel__restrict__Int__empty,axiom,
% 14.25/8.04 ! [A: set_int,R3: set_Pr958786334691620121nt_int] :
% 14.25/8.04 ( ( ( inf_inf_set_int @ A @ ( field_int @ R3 ) )
% 14.25/8.04 = bot_bot_set_int )
% 14.25/8.04 => ( ( rel_restrict_int @ R3 @ A )
% 14.25/8.04 = R3 ) ) ).
% 14.25/8.04
% 14.25/8.04 thf(fact_2167_rel__restrict__Int__empty,axiom,
% 14.25/8.04 ! [A: set_Pr4532377907799695533_nat_o,R3: set_Pr2161125870931222855_nat_o] :
% 14.25/8.04 ( ( ( inf_in1906310914598751387_nat_o @ A @ ( field_81821745747897882_nat_o @ R3 ) )
% 14.25/8.04 = bot_bo7824918357723582233_nat_o )
% 14.25/8.04 => ( ( rel_re3331361160397388719_nat_o @ R3 @ A )
% 14.25/8.04 = R3 ) ) ).
% 14.25/8.04
% 14.25/8.04 thf(fact_2168_eq__equiv__class__iff2,axiom,
% 14.25/8.04 ! [A: set_Product_unit,R2: set_Pr5094982260447487303t_unit,X: product_unit,Y: product_unit] :
% 14.25/8.04 ( ( equiv_4189356295871601603t_unit @ A @ R2 )
% 14.25/8.04 => ( ( member_Product_unit @ X @ A )
% 14.25/8.04 => ( ( member_Product_unit @ Y @ A )
% 14.25/8.04 => ( ( ( equiv_5172566880711946708t_unit @ ( insert_Product_unit @ X @ bot_bo3957492148770167129t_unit ) @ R2 )
% 14.25/8.04 = ( equiv_5172566880711946708t_unit @ ( insert_Product_unit @ Y @ bot_bo3957492148770167129t_unit ) @ R2 ) )
% 14.25/8.04 = ( member7821138191923116944t_unit @ ( produc5692694829384537815t_unit @ X @ Y ) @ R2 ) ) ) ) ) ).
% 14.25/8.04
% 14.25/8.04 thf(fact_2169_eq__equiv__class__iff2,axiom,
% 14.25/8.04 ! [A: set_o,R2: set_Product_prod_o_o,X: $o,Y: $o] :
% 14.25/8.04 ( ( equiv_equiv_o @ A @ R2 )
% 14.25/8.04 => ( ( member_o @ X @ A )
% 14.25/8.04 => ( ( member_o @ Y @ A )
% 14.25/8.04 => ( ( ( equiv_quotient_o @ ( insert_o @ X @ bot_bot_set_o ) @ R2 )
% 14.25/8.04 = ( equiv_quotient_o @ ( insert_o @ Y @ bot_bot_set_o ) @ R2 ) )
% 14.25/8.04 = ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ X @ Y ) @ R2 ) ) ) ) ) ).
% 14.25/8.04
% 14.25/8.04 thf(fact_2170_eq__equiv__class__iff2,axiom,
% 14.25/8.04 ! [A: set_nat,R2: set_Pr1261947904930325089at_nat,X: nat,Y: nat] :
% 14.25/8.04 ( ( equiv_equiv_nat @ A @ R2 )
% 14.25/8.04 => ( ( member_nat @ X @ A )
% 14.25/8.04 => ( ( member_nat @ Y @ A )
% 14.25/8.04 => ( ( ( equiv_quotient_nat @ ( insert_nat @ X @ bot_bot_set_nat ) @ R2 )
% 14.25/8.04 = ( equiv_quotient_nat @ ( insert_nat @ Y @ bot_bot_set_nat ) @ R2 ) )
% 14.25/8.04 = ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ Y ) @ R2 ) ) ) ) ) ).
% 14.25/8.04
% 14.25/8.04 thf(fact_2171_eq__equiv__class__iff2,axiom,
% 14.25/8.04 ! [A: set_int,R2: set_Pr958786334691620121nt_int,X: int,Y: int] :
% 14.25/8.04 ( ( equiv_equiv_int @ A @ R2 )
% 14.25/8.04 => ( ( member_int @ X @ A )
% 14.25/8.04 => ( ( member_int @ Y @ A )
% 14.25/8.04 => ( ( ( equiv_quotient_int @ ( insert_int @ X @ bot_bot_set_int ) @ R2 )
% 14.25/8.04 = ( equiv_quotient_int @ ( insert_int @ Y @ bot_bot_set_int ) @ R2 ) )
% 14.25/8.04 = ( member5262025264175285858nt_int @ ( product_Pair_int_int @ X @ Y ) @ R2 ) ) ) ) ) ).
% 14.25/8.04
% 14.25/8.04 thf(fact_2172_eq__equiv__class__iff2,axiom,
% 14.25/8.04 ! [A: set_Pr4532377907799695533_nat_o,R2: set_Pr2161125870931222855_nat_o,X: produc3658429121746597890et_nat > $o,Y: produc3658429121746597890et_nat > $o] :
% 14.25/8.04 ( ( equiv_6726761681920839555_nat_o @ A @ R2 )
% 14.25/8.04 => ( ( member6576561426505652726_nat_o @ X @ A )
% 14.25/8.04 => ( ( member6576561426505652726_nat_o @ Y @ A )
% 14.25/8.04 => ( ( ( equiv_7899955228204526484_nat_o @ ( insert5175938949040314269_nat_o @ X @ bot_bo7824918357723582233_nat_o ) @ R2 )
% 14.25/8.04 = ( equiv_7899955228204526484_nat_o @ ( insert5175938949040314269_nat_o @ Y @ bot_bo7824918357723582233_nat_o ) @ R2 ) )
% 14.25/8.04 = ( member8781333585448626064_nat_o @ ( produc7368190662567826135_nat_o @ X @ Y ) @ R2 ) ) ) ) ) ).
% 14.25/8.04
% 14.25/8.04 thf(fact_2173_Inf__fin_Osubset__imp,axiom,
% 14.25/8.04 ! [A: set_Product_unit,B4: set_Product_unit] :
% 14.25/8.04 ( ( ord_le3507040750410214029t_unit @ A @ B4 )
% 14.25/8.04 => ( ( A != bot_bo3957492148770167129t_unit )
% 14.25/8.04 => ( ( finite4290736615968046902t_unit @ B4 )
% 14.25/8.04 => ( ord_le3221252021190050221t_unit @ ( lattic1263872656861969706t_unit @ B4 ) @ ( lattic1263872656861969706t_unit @ A ) ) ) ) ) ).
% 14.25/8.04
% 14.25/8.04 thf(fact_2174_Inf__fin_Osubset__imp,axiom,
% 14.25/8.04 ! [A: set_o,B4: set_o] :
% 14.25/8.04 ( ( ord_less_eq_set_o @ A @ B4 )
% 14.25/8.04 => ( ( A != bot_bot_set_o )
% 14.25/8.04 => ( ( finite_finite_o @ B4 )
% 14.25/8.04 => ( ord_less_eq_o @ ( lattic4107685809792843317_fin_o @ B4 ) @ ( lattic4107685809792843317_fin_o @ A ) ) ) ) ) ).
% 14.25/8.04
% 14.25/8.04 thf(fact_2175_Inf__fin_Osubset__imp,axiom,
% 14.25/8.04 ! [A: set_Pr4532377907799695533_nat_o,B4: set_Pr4532377907799695533_nat_o] :
% 14.25/8.04 ( ( ord_le2965882846123202637_nat_o @ A @ B4 )
% 14.25/8.04 => ( ( A != bot_bo7824918357723582233_nat_o )
% 14.25/8.04 => ( ( finite3252695134891459830_nat_o @ B4 )
% 14.25/8.04 => ( ord_le729326519192465773_nat_o @ ( lattic956194824204696298_nat_o @ B4 ) @ ( lattic956194824204696298_nat_o @ A ) ) ) ) ) ).
% 14.25/8.04
% 14.25/8.04 thf(fact_2176_Inf__fin_Osubset__imp,axiom,
% 14.25/8.04 ! [A: set_filter_nat,B4: set_filter_nat] :
% 14.25/8.04 ( ( ord_le2426478655948331894er_nat @ A @ B4 )
% 14.25/8.04 => ( ( A != bot_bo498966703094740906er_nat )
% 14.25/8.04 => ( ( finite2119507909894593271er_nat @ B4 )
% 14.25/8.04 => ( ord_le2510731241096832064er_nat @ ( lattic9128708996678779395er_nat @ B4 ) @ ( lattic9128708996678779395er_nat @ A ) ) ) ) ) ).
% 14.25/8.04
% 14.25/8.04 thf(fact_2177_Inf__fin_Osubset__imp,axiom,
% 14.25/8.04 ! [A: set_set_nat,B4: set_set_nat] :
% 14.25/8.04 ( ( ord_le6893508408891458716et_nat @ A @ B4 )
% 14.25/8.04 => ( ( A != bot_bot_set_set_nat )
% 14.25/8.04 => ( ( finite1152437895449049373et_nat @ B4 )
% 14.25/8.04 => ( ord_less_eq_set_nat @ ( lattic3014633134055518761et_nat @ B4 ) @ ( lattic3014633134055518761et_nat @ A ) ) ) ) ) ).
% 14.25/8.04
% 14.25/8.04 thf(fact_2178_Inf__fin_Osubset__imp,axiom,
% 14.25/8.04 ! [A: set_rat,B4: set_rat] :
% 14.25/8.04 ( ( ord_less_eq_set_rat @ A @ B4 )
% 14.25/8.04 => ( ( A != bot_bot_set_rat )
% 14.25/8.04 => ( ( finite_finite_rat @ B4 )
% 14.25/8.04 => ( ord_less_eq_rat @ ( lattic4603258475043424379in_rat @ B4 ) @ ( lattic4603258475043424379in_rat @ A ) ) ) ) ) ).
% 14.25/8.04
% 14.25/8.04 thf(fact_2179_Inf__fin_Osubset__imp,axiom,
% 14.25/8.04 ! [A: set_int,B4: set_int] :
% 14.25/8.04 ( ( ord_less_eq_set_int @ A @ B4 )
% 14.25/8.04 => ( ( A != bot_bot_set_int )
% 14.25/8.04 => ( ( finite_finite_int @ B4 )
% 14.25/8.04 => ( ord_less_eq_int @ ( lattic5235898064620869839in_int @ B4 ) @ ( lattic5235898064620869839in_int @ A ) ) ) ) ) ).
% 14.25/8.04
% 14.25/8.04 thf(fact_2180_Inf__fin_Osubset__imp,axiom,
% 14.25/8.04 ! [A: set_nat,B4: set_nat] :
% 14.25/8.04 ( ( ord_less_eq_set_nat @ A @ B4 )
% 14.25/8.04 => ( ( A != bot_bot_set_nat )
% 14.25/8.04 => ( ( finite_finite_nat @ B4 )
% 14.25/8.04 => ( ord_less_eq_nat @ ( lattic5238388535129920115in_nat @ B4 ) @ ( lattic5238388535129920115in_nat @ A ) ) ) ) ) ).
% 14.25/8.04
% 14.25/8.04 thf(fact_2181_Sup__fin_Osubset__imp,axiom,
% 14.25/8.04 ! [A: set_Product_unit,B4: set_Product_unit] :
% 14.25/8.04 ( ( ord_le3507040750410214029t_unit @ A @ B4 )
% 14.25/8.04 => ( ( A != bot_bo3957492148770167129t_unit )
% 14.25/8.04 => ( ( finite4290736615968046902t_unit @ B4 )
% 14.25/8.04 => ( ord_le3221252021190050221t_unit @ ( lattic5294303975357428420t_unit @ A ) @ ( lattic5294303975357428420t_unit @ B4 ) ) ) ) ) ).
% 14.25/8.04
% 14.25/8.04 thf(fact_2182_Sup__fin_Osubset__imp,axiom,
% 14.25/8.04 ! [A: set_o,B4: set_o] :
% 14.25/8.04 ( ( ord_less_eq_set_o @ A @ B4 )
% 14.25/8.04 => ( ( A != bot_bot_set_o )
% 14.25/8.04 => ( ( finite_finite_o @ B4 )
% 14.25/8.04 => ( ord_less_eq_o @ ( lattic1508158080041050831_fin_o @ A ) @ ( lattic1508158080041050831_fin_o @ B4 ) ) ) ) ) ).
% 14.25/8.04
% 14.25/8.04 thf(fact_2183_Sup__fin_Osubset__imp,axiom,
% 14.25/8.04 ! [A: set_Pr4532377907799695533_nat_o,B4: set_Pr4532377907799695533_nat_o] :
% 14.25/8.04 ( ( ord_le2965882846123202637_nat_o @ A @ B4 )
% 14.25/8.04 => ( ( A != bot_bo7824918357723582233_nat_o )
% 14.25/8.04 => ( ( finite3252695134891459830_nat_o @ B4 )
% 14.25/8.04 => ( ord_le729326519192465773_nat_o @ ( lattic7320199455484906628_nat_o @ A ) @ ( lattic7320199455484906628_nat_o @ B4 ) ) ) ) ) ).
% 14.25/8.04
% 14.25/8.04 thf(fact_2184_Sup__fin_Osubset__imp,axiom,
% 14.25/8.04 ! [A: set_filter_nat,B4: set_filter_nat] :
% 14.25/8.04 ( ( ord_le2426478655948331894er_nat @ A @ B4 )
% 14.25/8.04 => ( ( A != bot_bo498966703094740906er_nat )
% 14.25/8.04 => ( ( finite2119507909894593271er_nat @ B4 )
% 14.25/8.04 => ( ord_le2510731241096832064er_nat @ ( lattic5930898082463196905er_nat @ A ) @ ( lattic5930898082463196905er_nat @ B4 ) ) ) ) ) ).
% 14.25/8.04
% 14.25/8.04 thf(fact_2185_Sup__fin_Osubset__imp,axiom,
% 14.25/8.04 ! [A: set_set_nat,B4: set_set_nat] :
% 14.25/8.04 ( ( ord_le6893508408891458716et_nat @ A @ B4 )
% 14.25/8.04 => ( ( A != bot_bot_set_set_nat )
% 14.25/8.04 => ( ( finite1152437895449049373et_nat @ B4 )
% 14.25/8.04 => ( ord_less_eq_set_nat @ ( lattic3835124923745554447et_nat @ A ) @ ( lattic3835124923745554447et_nat @ B4 ) ) ) ) ) ).
% 14.25/8.04
% 14.25/8.04 thf(fact_2186_Sup__fin_Osubset__imp,axiom,
% 14.25/8.04 ! [A: set_rat,B4: set_rat] :
% 14.25/8.04 ( ( ord_less_eq_set_rat @ A @ B4 )
% 14.25/8.04 => ( ( A != bot_bot_set_rat )
% 14.25/8.04 => ( ( finite_finite_rat @ B4 )
% 14.25/8.04 => ( ord_less_eq_rat @ ( lattic458866745392299617in_rat @ A ) @ ( lattic458866745392299617in_rat @ B4 ) ) ) ) ) ).
% 14.25/8.04
% 14.25/8.04 thf(fact_2187_Sup__fin_Osubset__imp,axiom,
% 14.25/8.04 ! [A: set_int,B4: set_int] :
% 14.25/8.04 ( ( ord_less_eq_set_int @ A @ B4 )
% 14.25/8.04 => ( ( A != bot_bot_set_int )
% 14.25/8.04 => ( ( finite_finite_int @ B4 )
% 14.25/8.04 => ( ord_less_eq_int @ ( lattic1091506334969745077in_int @ A ) @ ( lattic1091506334969745077in_int @ B4 ) ) ) ) ) ).
% 14.25/8.04
% 14.25/8.04 thf(fact_2188_Sup__fin_Osubset__imp,axiom,
% 14.25/8.04 ! [A: set_nat,B4: set_nat] :
% 14.25/8.04 ( ( ord_less_eq_set_nat @ A @ B4 )
% 14.25/8.04 => ( ( A != bot_bot_set_nat )
% 14.25/8.04 => ( ( finite_finite_nat @ B4 )
% 14.25/8.04 => ( ord_less_eq_nat @ ( lattic1093996805478795353in_nat @ A ) @ ( lattic1093996805478795353in_nat @ B4 ) ) ) ) ) ).
% 14.25/8.04
% 14.25/8.04 thf(fact_2189_Inf__fin_Osubset,axiom,
% 14.25/8.04 ! [A: set_set_nat,B4: set_set_nat] :
% 14.25/8.04 ( ( finite1152437895449049373et_nat @ A )
% 14.25/8.04 => ( ( B4 != bot_bot_set_set_nat )
% 14.25/8.04 => ( ( ord_le6893508408891458716et_nat @ B4 @ A )
% 14.25/8.04 => ( ( inf_inf_set_nat @ ( lattic3014633134055518761et_nat @ B4 ) @ ( lattic3014633134055518761et_nat @ A ) )
% 14.25/8.04 = ( lattic3014633134055518761et_nat @ A ) ) ) ) ) ).
% 14.25/8.04
% 14.25/8.04 thf(fact_2190_Inf__fin_Osubset,axiom,
% 14.25/8.04 ! [A: set_assn,B4: set_assn] :
% 14.25/8.04 ( ( finite_finite_assn @ A )
% 14.25/8.04 => ( ( B4 != bot_bot_set_assn )
% 14.25/8.04 => ( ( ord_less_eq_set_assn @ B4 @ A )
% 14.25/8.04 => ( ( inf_inf_assn @ ( lattic47131356835913163n_assn @ B4 ) @ ( lattic47131356835913163n_assn @ A ) )
% 14.25/8.04 = ( lattic47131356835913163n_assn @ A ) ) ) ) ) ).
% 14.25/8.04
% 14.25/8.04 thf(fact_2191_Inf__fin_Osubset,axiom,
% 14.25/8.04 ! [A: set_Product_unit,B4: set_Product_unit] :
% 14.25/8.04 ( ( finite4290736615968046902t_unit @ A )
% 14.25/8.04 => ( ( B4 != bot_bo3957492148770167129t_unit )
% 14.25/8.04 => ( ( ord_le3507040750410214029t_unit @ B4 @ A )
% 14.25/8.04 => ( ( inf_inf_Product_unit @ ( lattic1263872656861969706t_unit @ B4 ) @ ( lattic1263872656861969706t_unit @ A ) )
% 14.25/8.04 = ( lattic1263872656861969706t_unit @ A ) ) ) ) ) ).
% 14.25/8.04
% 14.25/8.04 thf(fact_2192_Inf__fin_Osubset,axiom,
% 14.25/8.04 ! [A: set_o,B4: set_o] :
% 14.25/8.04 ( ( finite_finite_o @ A )
% 14.25/8.04 => ( ( B4 != bot_bot_set_o )
% 14.25/8.04 => ( ( ord_less_eq_set_o @ B4 @ A )
% 14.25/8.04 => ( ( inf_inf_o @ ( lattic4107685809792843317_fin_o @ B4 ) @ ( lattic4107685809792843317_fin_o @ A ) )
% 14.25/8.04 = ( lattic4107685809792843317_fin_o @ A ) ) ) ) ) ).
% 14.25/8.04
% 14.25/8.04 thf(fact_2193_Inf__fin_Osubset,axiom,
% 14.25/8.04 ! [A: set_int,B4: set_int] :
% 14.25/8.04 ( ( finite_finite_int @ A )
% 14.25/8.04 => ( ( B4 != bot_bot_set_int )
% 14.25/8.04 => ( ( ord_less_eq_set_int @ B4 @ A )
% 14.25/8.04 => ( ( inf_inf_int @ ( lattic5235898064620869839in_int @ B4 ) @ ( lattic5235898064620869839in_int @ A ) )
% 14.25/8.04 = ( lattic5235898064620869839in_int @ A ) ) ) ) ) ).
% 14.25/8.04
% 14.25/8.04 thf(fact_2194_Inf__fin_Osubset,axiom,
% 14.25/8.04 ! [A: set_Pr4532377907799695533_nat_o,B4: set_Pr4532377907799695533_nat_o] :
% 14.25/8.04 ( ( finite3252695134891459830_nat_o @ A )
% 14.25/8.04 => ( ( B4 != bot_bo7824918357723582233_nat_o )
% 14.25/8.04 => ( ( ord_le2965882846123202637_nat_o @ B4 @ A )
% 14.25/8.04 => ( ( inf_in1318976480646536635_nat_o @ ( lattic956194824204696298_nat_o @ B4 ) @ ( lattic956194824204696298_nat_o @ A ) )
% 14.25/8.04 = ( lattic956194824204696298_nat_o @ A ) ) ) ) ) ).
% 14.25/8.04
% 14.25/8.04 thf(fact_2195_Inf__fin_Osubset,axiom,
% 14.25/8.04 ! [A: set_nat,B4: set_nat] :
% 14.25/8.04 ( ( finite_finite_nat @ A )
% 14.25/8.04 => ( ( B4 != bot_bot_set_nat )
% 14.25/8.04 => ( ( ord_less_eq_set_nat @ B4 @ A )
% 14.25/8.04 => ( ( inf_inf_nat @ ( lattic5238388535129920115in_nat @ B4 ) @ ( lattic5238388535129920115in_nat @ A ) )
% 14.25/8.04 = ( lattic5238388535129920115in_nat @ A ) ) ) ) ) ).
% 14.25/8.04
% 14.25/8.04 thf(fact_2196_Sup__fin_Osubset,axiom,
% 14.25/8.04 ! [A: set_set_nat,B4: set_set_nat] :
% 14.25/8.04 ( ( finite1152437895449049373et_nat @ A )
% 14.25/8.04 => ( ( B4 != bot_bot_set_set_nat )
% 14.25/8.04 => ( ( ord_le6893508408891458716et_nat @ B4 @ A )
% 14.25/8.04 => ( ( sup_sup_set_nat @ ( lattic3835124923745554447et_nat @ B4 ) @ ( lattic3835124923745554447et_nat @ A ) )
% 14.25/8.04 = ( lattic3835124923745554447et_nat @ A ) ) ) ) ) ).
% 14.25/8.04
% 14.25/8.04 thf(fact_2197_Sup__fin_Osubset,axiom,
% 14.25/8.04 ! [A: set_assn,B4: set_assn] :
% 14.25/8.04 ( ( finite_finite_assn @ A )
% 14.25/8.04 => ( ( B4 != bot_bot_set_assn )
% 14.25/8.04 => ( ( ord_less_eq_set_assn @ B4 @ A )
% 14.25/8.04 => ( ( sup_sup_assn @ ( lattic2150320897289308081n_assn @ B4 ) @ ( lattic2150320897289308081n_assn @ A ) )
% 14.25/8.04 = ( lattic2150320897289308081n_assn @ A ) ) ) ) ) ).
% 14.25/8.04
% 14.25/8.04 thf(fact_2198_Sup__fin_Osubset,axiom,
% 14.25/8.04 ! [A: set_Product_unit,B4: set_Product_unit] :
% 14.25/8.04 ( ( finite4290736615968046902t_unit @ A )
% 14.25/8.04 => ( ( B4 != bot_bo3957492148770167129t_unit )
% 14.25/8.04 => ( ( ord_le3507040750410214029t_unit @ B4 @ A )
% 14.25/8.04 => ( ( sup_sup_Product_unit @ ( lattic5294303975357428420t_unit @ B4 ) @ ( lattic5294303975357428420t_unit @ A ) )
% 14.25/8.04 = ( lattic5294303975357428420t_unit @ A ) ) ) ) ) ).
% 14.25/8.04
% 14.25/8.04 thf(fact_2199_Sup__fin_Osubset,axiom,
% 14.25/8.04 ! [A: set_o,B4: set_o] :
% 14.25/8.04 ( ( finite_finite_o @ A )
% 14.25/8.04 => ( ( B4 != bot_bot_set_o )
% 14.25/8.04 => ( ( ord_less_eq_set_o @ B4 @ A )
% 14.25/8.04 => ( ( sup_sup_o @ ( lattic1508158080041050831_fin_o @ B4 ) @ ( lattic1508158080041050831_fin_o @ A ) )
% 14.25/8.04 = ( lattic1508158080041050831_fin_o @ A ) ) ) ) ) ).
% 14.25/8.04
% 14.25/8.04 thf(fact_2200_Sup__fin_Osubset,axiom,
% 14.25/8.04 ! [A: set_int,B4: set_int] :
% 14.25/8.04 ( ( finite_finite_int @ A )
% 14.25/8.04 => ( ( B4 != bot_bot_set_int )
% 14.25/8.04 => ( ( ord_less_eq_set_int @ B4 @ A )
% 14.25/8.04 => ( ( sup_sup_int @ ( lattic1091506334969745077in_int @ B4 ) @ ( lattic1091506334969745077in_int @ A ) )
% 14.25/8.04 = ( lattic1091506334969745077in_int @ A ) ) ) ) ) ).
% 14.25/8.04
% 14.25/8.04 thf(fact_2201_Sup__fin_Osubset,axiom,
% 14.25/8.04 ! [A: set_Pr4532377907799695533_nat_o,B4: set_Pr4532377907799695533_nat_o] :
% 14.25/8.04 ( ( finite3252695134891459830_nat_o @ A )
% 14.25/8.04 => ( ( B4 != bot_bo7824918357723582233_nat_o )
% 14.25/8.04 => ( ( ord_le2965882846123202637_nat_o @ B4 @ A )
% 14.25/8.04 => ( ( sup_su5453871518329203617_nat_o @ ( lattic7320199455484906628_nat_o @ B4 ) @ ( lattic7320199455484906628_nat_o @ A ) )
% 14.25/8.04 = ( lattic7320199455484906628_nat_o @ A ) ) ) ) ) ).
% 14.25/8.04
% 14.25/8.04 thf(fact_2202_Sup__fin_Osubset,axiom,
% 14.25/8.04 ! [A: set_nat,B4: set_nat] :
% 14.25/8.04 ( ( finite_finite_nat @ A )
% 14.25/8.04 => ( ( B4 != bot_bot_set_nat )
% 14.25/8.04 => ( ( ord_less_eq_set_nat @ B4 @ A )
% 14.25/8.04 => ( ( sup_sup_nat @ ( lattic1093996805478795353in_nat @ B4 ) @ ( lattic1093996805478795353in_nat @ A ) )
% 14.25/8.04 = ( lattic1093996805478795353in_nat @ A ) ) ) ) ) ).
% 14.25/8.04
% 14.25/8.04 thf(fact_2203_Inf__fin_Oclosed,axiom,
% 14.25/8.04 ! [A: set_set_nat] :
% 14.25/8.04 ( ( finite1152437895449049373et_nat @ A )
% 14.25/8.04 => ( ( A != bot_bot_set_set_nat )
% 14.25/8.04 => ( ! [X3: set_nat,Y2: set_nat] : ( member_set_nat @ ( inf_inf_set_nat @ X3 @ Y2 ) @ ( insert_set_nat @ X3 @ ( insert_set_nat @ Y2 @ bot_bot_set_set_nat ) ) )
% 14.25/8.04 => ( member_set_nat @ ( lattic3014633134055518761et_nat @ A ) @ A ) ) ) ) ).
% 14.25/8.04
% 14.25/8.04 thf(fact_2204_Inf__fin_Oclosed,axiom,
% 14.25/8.04 ! [A: set_assn] :
% 14.25/8.04 ( ( finite_finite_assn @ A )
% 14.25/8.04 => ( ( A != bot_bot_set_assn )
% 14.25/8.04 => ( ! [X3: assn,Y2: assn] : ( member_assn @ ( inf_inf_assn @ X3 @ Y2 ) @ ( insert_assn @ X3 @ ( insert_assn @ Y2 @ bot_bot_set_assn ) ) )
% 14.25/8.04 => ( member_assn @ ( lattic47131356835913163n_assn @ A ) @ A ) ) ) ) ).
% 14.25/8.04
% 14.25/8.04 thf(fact_2205_Inf__fin_Oclosed,axiom,
% 14.25/8.04 ! [A: set_Product_unit] :
% 14.25/8.04 ( ( finite4290736615968046902t_unit @ A )
% 14.25/8.04 => ( ( A != bot_bo3957492148770167129t_unit )
% 14.25/8.04 => ( ! [X3: product_unit,Y2: product_unit] : ( member_Product_unit @ ( inf_inf_Product_unit @ X3 @ Y2 ) @ ( insert_Product_unit @ X3 @ ( insert_Product_unit @ Y2 @ bot_bo3957492148770167129t_unit ) ) )
% 14.25/8.04 => ( member_Product_unit @ ( lattic1263872656861969706t_unit @ A ) @ A ) ) ) ) ).
% 14.25/8.04
% 14.25/8.04 thf(fact_2206_Inf__fin_Oclosed,axiom,
% 14.25/8.04 ! [A: set_o] :
% 14.25/8.04 ( ( finite_finite_o @ A )
% 14.25/8.04 => ( ( A != bot_bot_set_o )
% 14.25/8.04 => ( ! [X3: $o,Y2: $o] : ( member_o @ ( inf_inf_o @ X3 @ Y2 ) @ ( insert_o @ X3 @ ( insert_o @ Y2 @ bot_bot_set_o ) ) )
% 14.25/8.04 => ( member_o @ ( lattic4107685809792843317_fin_o @ A ) @ A ) ) ) ) ).
% 14.25/8.04
% 14.25/8.04 thf(fact_2207_Inf__fin_Oclosed,axiom,
% 14.25/8.04 ! [A: set_nat] :
% 14.25/8.04 ( ( finite_finite_nat @ A )
% 14.25/8.04 => ( ( A != bot_bot_set_nat )
% 14.25/8.04 => ( ! [X3: nat,Y2: nat] : ( member_nat @ ( inf_inf_nat @ X3 @ Y2 ) @ ( insert_nat @ X3 @ ( insert_nat @ Y2 @ bot_bot_set_nat ) ) )
% 14.25/8.04 => ( member_nat @ ( lattic5238388535129920115in_nat @ A ) @ A ) ) ) ) ).
% 14.25/8.04
% 14.25/8.04 thf(fact_2208_Inf__fin_Oclosed,axiom,
% 14.25/8.04 ! [A: set_int] :
% 14.25/8.04 ( ( finite_finite_int @ A )
% 14.25/8.04 => ( ( A != bot_bot_set_int )
% 14.25/8.04 => ( ! [X3: int,Y2: int] : ( member_int @ ( inf_inf_int @ X3 @ Y2 ) @ ( insert_int @ X3 @ ( insert_int @ Y2 @ bot_bot_set_int ) ) )
% 14.25/8.04 => ( member_int @ ( lattic5235898064620869839in_int @ A ) @ A ) ) ) ) ).
% 14.25/8.04
% 14.25/8.04 thf(fact_2209_Inf__fin_Oclosed,axiom,
% 14.25/8.04 ! [A: set_Pr4532377907799695533_nat_o] :
% 14.25/8.04 ( ( finite3252695134891459830_nat_o @ A )
% 14.25/8.04 => ( ( A != bot_bo7824918357723582233_nat_o )
% 14.25/8.04 => ( ! [X3: produc3658429121746597890et_nat > $o,Y2: produc3658429121746597890et_nat > $o] : ( member6576561426505652726_nat_o @ ( inf_in1318976480646536635_nat_o @ X3 @ Y2 ) @ ( insert5175938949040314269_nat_o @ X3 @ ( insert5175938949040314269_nat_o @ Y2 @ bot_bo7824918357723582233_nat_o ) ) )
% 14.25/8.04 => ( member6576561426505652726_nat_o @ ( lattic956194824204696298_nat_o @ A ) @ A ) ) ) ) ).
% 14.25/8.04
% 14.25/8.04 thf(fact_2210_Inf__fin_Oinsert__not__elem,axiom,
% 14.25/8.04 ! [A: set_set_nat,X: set_nat] :
% 14.25/8.04 ( ( finite1152437895449049373et_nat @ A )
% 14.25/8.04 => ( ~ ( member_set_nat @ X @ A )
% 14.25/8.04 => ( ( A != bot_bot_set_set_nat )
% 14.25/8.04 => ( ( lattic3014633134055518761et_nat @ ( insert_set_nat @ X @ A ) )
% 14.25/8.04 = ( inf_inf_set_nat @ X @ ( lattic3014633134055518761et_nat @ A ) ) ) ) ) ) ).
% 14.25/8.04
% 14.25/8.04 thf(fact_2211_Inf__fin_Oinsert__not__elem,axiom,
% 14.25/8.04 ! [A: set_assn,X: assn] :
% 14.25/8.04 ( ( finite_finite_assn @ A )
% 14.25/8.04 => ( ~ ( member_assn @ X @ A )
% 14.25/8.04 => ( ( A != bot_bot_set_assn )
% 14.25/8.04 => ( ( lattic47131356835913163n_assn @ ( insert_assn @ X @ A ) )
% 14.25/8.04 = ( inf_inf_assn @ X @ ( lattic47131356835913163n_assn @ A ) ) ) ) ) ) ).
% 14.25/8.04
% 14.25/8.04 thf(fact_2212_Inf__fin_Oinsert__not__elem,axiom,
% 14.25/8.04 ! [A: set_Product_unit,X: product_unit] :
% 14.25/8.04 ( ( finite4290736615968046902t_unit @ A )
% 14.25/8.04 => ( ~ ( member_Product_unit @ X @ A )
% 14.25/8.04 => ( ( A != bot_bo3957492148770167129t_unit )
% 14.25/8.04 => ( ( lattic1263872656861969706t_unit @ ( insert_Product_unit @ X @ A ) )
% 14.25/8.04 = ( inf_inf_Product_unit @ X @ ( lattic1263872656861969706t_unit @ A ) ) ) ) ) ) ).
% 14.25/8.04
% 14.25/8.04 thf(fact_2213_Inf__fin_Oinsert__not__elem,axiom,
% 14.25/8.04 ! [A: set_o,X: $o] :
% 14.25/8.04 ( ( finite_finite_o @ A )
% 14.25/8.04 => ( ~ ( member_o @ X @ A )
% 14.25/8.04 => ( ( A != bot_bot_set_o )
% 14.25/8.04 => ( ( lattic4107685809792843317_fin_o @ ( insert_o @ X @ A ) )
% 14.25/8.04 = ( inf_inf_o @ X @ ( lattic4107685809792843317_fin_o @ A ) ) ) ) ) ) ).
% 14.25/8.04
% 14.25/8.04 thf(fact_2214_Inf__fin_Oinsert__not__elem,axiom,
% 14.25/8.04 ! [A: set_nat,X: nat] :
% 14.25/8.04 ( ( finite_finite_nat @ A )
% 14.25/8.04 => ( ~ ( member_nat @ X @ A )
% 14.25/8.04 => ( ( A != bot_bot_set_nat )
% 14.25/8.04 => ( ( lattic5238388535129920115in_nat @ ( insert_nat @ X @ A ) )
% 14.25/8.04 = ( inf_inf_nat @ X @ ( lattic5238388535129920115in_nat @ A ) ) ) ) ) ) ).
% 14.25/8.04
% 14.25/8.04 thf(fact_2215_Inf__fin_Oinsert__not__elem,axiom,
% 14.25/8.04 ! [A: set_int,X: int] :
% 14.25/8.04 ( ( finite_finite_int @ A )
% 14.25/8.04 => ( ~ ( member_int @ X @ A )
% 14.25/8.04 => ( ( A != bot_bot_set_int )
% 14.25/8.04 => ( ( lattic5235898064620869839in_int @ ( insert_int @ X @ A ) )
% 14.25/8.04 = ( inf_inf_int @ X @ ( lattic5235898064620869839in_int @ A ) ) ) ) ) ) ).
% 14.25/8.04
% 14.25/8.04 thf(fact_2216_Inf__fin_Oinsert__not__elem,axiom,
% 14.25/8.04 ! [A: set_Pr4532377907799695533_nat_o,X: produc3658429121746597890et_nat > $o] :
% 14.25/8.04 ( ( finite3252695134891459830_nat_o @ A )
% 14.25/8.04 => ( ~ ( member6576561426505652726_nat_o @ X @ A )
% 14.25/8.04 => ( ( A != bot_bo7824918357723582233_nat_o )
% 14.25/8.04 => ( ( lattic956194824204696298_nat_o @ ( insert5175938949040314269_nat_o @ X @ A ) )
% 14.25/8.04 = ( inf_in1318976480646536635_nat_o @ X @ ( lattic956194824204696298_nat_o @ A ) ) ) ) ) ) ).
% 14.25/8.04
% 14.25/8.04 thf(fact_2217_Sup__fin_Oclosed,axiom,
% 14.25/8.04 ! [A: set_set_nat] :
% 14.25/8.04 ( ( finite1152437895449049373et_nat @ A )
% 14.25/8.04 => ( ( A != bot_bot_set_set_nat )
% 14.25/8.04 => ( ! [X3: set_nat,Y2: set_nat] : ( member_set_nat @ ( sup_sup_set_nat @ X3 @ Y2 ) @ ( insert_set_nat @ X3 @ ( insert_set_nat @ Y2 @ bot_bot_set_set_nat ) ) )
% 14.25/8.04 => ( member_set_nat @ ( lattic3835124923745554447et_nat @ A ) @ A ) ) ) ) ).
% 14.25/8.04
% 14.25/8.04 thf(fact_2218_Sup__fin_Oclosed,axiom,
% 14.25/8.04 ! [A: set_assn] :
% 14.25/8.04 ( ( finite_finite_assn @ A )
% 14.25/8.04 => ( ( A != bot_bot_set_assn )
% 14.25/8.04 => ( ! [X3: assn,Y2: assn] : ( member_assn @ ( sup_sup_assn @ X3 @ Y2 ) @ ( insert_assn @ X3 @ ( insert_assn @ Y2 @ bot_bot_set_assn ) ) )
% 14.25/8.04 => ( member_assn @ ( lattic2150320897289308081n_assn @ A ) @ A ) ) ) ) ).
% 14.25/8.04
% 14.25/8.04 thf(fact_2219_Sup__fin_Oclosed,axiom,
% 14.25/8.04 ! [A: set_Product_unit] :
% 14.25/8.04 ( ( finite4290736615968046902t_unit @ A )
% 14.25/8.04 => ( ( A != bot_bo3957492148770167129t_unit )
% 14.25/8.04 => ( ! [X3: product_unit,Y2: product_unit] : ( member_Product_unit @ ( sup_sup_Product_unit @ X3 @ Y2 ) @ ( insert_Product_unit @ X3 @ ( insert_Product_unit @ Y2 @ bot_bo3957492148770167129t_unit ) ) )
% 14.25/8.04 => ( member_Product_unit @ ( lattic5294303975357428420t_unit @ A ) @ A ) ) ) ) ).
% 14.25/8.04
% 14.25/8.04 thf(fact_2220_Sup__fin_Oclosed,axiom,
% 14.25/8.04 ! [A: set_o] :
% 14.25/8.04 ( ( finite_finite_o @ A )
% 14.25/8.04 => ( ( A != bot_bot_set_o )
% 14.25/8.04 => ( ! [X3: $o,Y2: $o] : ( member_o @ ( sup_sup_o @ X3 @ Y2 ) @ ( insert_o @ X3 @ ( insert_o @ Y2 @ bot_bot_set_o ) ) )
% 14.25/8.04 => ( member_o @ ( lattic1508158080041050831_fin_o @ A ) @ A ) ) ) ) ).
% 14.25/8.04
% 14.25/8.04 thf(fact_2221_Sup__fin_Oclosed,axiom,
% 14.25/8.04 ! [A: set_nat] :
% 14.25/8.04 ( ( finite_finite_nat @ A )
% 14.25/8.04 => ( ( A != bot_bot_set_nat )
% 14.25/8.04 => ( ! [X3: nat,Y2: nat] : ( member_nat @ ( sup_sup_nat @ X3 @ Y2 ) @ ( insert_nat @ X3 @ ( insert_nat @ Y2 @ bot_bot_set_nat ) ) )
% 14.25/8.04 => ( member_nat @ ( lattic1093996805478795353in_nat @ A ) @ A ) ) ) ) ).
% 14.25/8.04
% 14.25/8.04 thf(fact_2222_Sup__fin_Oclosed,axiom,
% 14.25/8.04 ! [A: set_int] :
% 14.25/8.04 ( ( finite_finite_int @ A )
% 14.25/8.04 => ( ( A != bot_bot_set_int )
% 14.25/8.04 => ( ! [X3: int,Y2: int] : ( member_int @ ( sup_sup_int @ X3 @ Y2 ) @ ( insert_int @ X3 @ ( insert_int @ Y2 @ bot_bot_set_int ) ) )
% 14.25/8.04 => ( member_int @ ( lattic1091506334969745077in_int @ A ) @ A ) ) ) ) ).
% 14.25/8.04
% 14.25/8.04 thf(fact_2223_Sup__fin_Oclosed,axiom,
% 14.25/8.04 ! [A: set_Pr4532377907799695533_nat_o] :
% 14.25/8.04 ( ( finite3252695134891459830_nat_o @ A )
% 14.25/8.04 => ( ( A != bot_bo7824918357723582233_nat_o )
% 14.25/8.04 => ( ! [X3: produc3658429121746597890et_nat > $o,Y2: produc3658429121746597890et_nat > $o] : ( member6576561426505652726_nat_o @ ( sup_su5453871518329203617_nat_o @ X3 @ Y2 ) @ ( insert5175938949040314269_nat_o @ X3 @ ( insert5175938949040314269_nat_o @ Y2 @ bot_bo7824918357723582233_nat_o ) ) )
% 14.25/8.04 => ( member6576561426505652726_nat_o @ ( lattic7320199455484906628_nat_o @ A ) @ A ) ) ) ) ).
% 14.25/8.04
% 14.25/8.04 thf(fact_2224_Sup__fin_Oinsert__not__elem,axiom,
% 14.25/8.04 ! [A: set_set_nat,X: set_nat] :
% 14.25/8.04 ( ( finite1152437895449049373et_nat @ A )
% 14.25/8.04 => ( ~ ( member_set_nat @ X @ A )
% 14.25/8.04 => ( ( A != bot_bot_set_set_nat )
% 14.25/8.04 => ( ( lattic3835124923745554447et_nat @ ( insert_set_nat @ X @ A ) )
% 14.25/8.04 = ( sup_sup_set_nat @ X @ ( lattic3835124923745554447et_nat @ A ) ) ) ) ) ) ).
% 14.25/8.04
% 14.25/8.04 thf(fact_2225_Sup__fin_Oinsert__not__elem,axiom,
% 14.25/8.04 ! [A: set_assn,X: assn] :
% 14.25/8.04 ( ( finite_finite_assn @ A )
% 14.25/8.04 => ( ~ ( member_assn @ X @ A )
% 14.25/8.04 => ( ( A != bot_bot_set_assn )
% 14.25/8.04 => ( ( lattic2150320897289308081n_assn @ ( insert_assn @ X @ A ) )
% 14.25/8.04 = ( sup_sup_assn @ X @ ( lattic2150320897289308081n_assn @ A ) ) ) ) ) ) ).
% 14.25/8.04
% 14.25/8.04 thf(fact_2226_Sup__fin_Oinsert__not__elem,axiom,
% 14.25/8.04 ! [A: set_Product_unit,X: product_unit] :
% 14.25/8.04 ( ( finite4290736615968046902t_unit @ A )
% 14.25/8.04 => ( ~ ( member_Product_unit @ X @ A )
% 14.25/8.04 => ( ( A != bot_bo3957492148770167129t_unit )
% 14.25/8.04 => ( ( lattic5294303975357428420t_unit @ ( insert_Product_unit @ X @ A ) )
% 14.25/8.04 = ( sup_sup_Product_unit @ X @ ( lattic5294303975357428420t_unit @ A ) ) ) ) ) ) ).
% 14.25/8.04
% 14.25/8.04 thf(fact_2227_Sup__fin_Oinsert__not__elem,axiom,
% 14.25/8.04 ! [A: set_o,X: $o] :
% 14.25/8.04 ( ( finite_finite_o @ A )
% 14.25/8.04 => ( ~ ( member_o @ X @ A )
% 14.25/8.04 => ( ( A != bot_bot_set_o )
% 14.25/8.04 => ( ( lattic1508158080041050831_fin_o @ ( insert_o @ X @ A ) )
% 14.25/8.04 = ( sup_sup_o @ X @ ( lattic1508158080041050831_fin_o @ A ) ) ) ) ) ) ).
% 14.25/8.04
% 14.25/8.04 thf(fact_2228_Sup__fin_Oinsert__not__elem,axiom,
% 14.25/8.04 ! [A: set_nat,X: nat] :
% 14.25/8.04 ( ( finite_finite_nat @ A )
% 14.25/8.04 => ( ~ ( member_nat @ X @ A )
% 14.25/8.04 => ( ( A != bot_bot_set_nat )
% 14.25/8.04 => ( ( lattic1093996805478795353in_nat @ ( insert_nat @ X @ A ) )
% 14.25/8.04 = ( sup_sup_nat @ X @ ( lattic1093996805478795353in_nat @ A ) ) ) ) ) ) ).
% 14.25/8.04
% 14.25/8.04 thf(fact_2229_Sup__fin_Oinsert__not__elem,axiom,
% 14.25/8.04 ! [A: set_int,X: int] :
% 14.25/8.04 ( ( finite_finite_int @ A )
% 14.25/8.04 => ( ~ ( member_int @ X @ A )
% 14.25/8.04 => ( ( A != bot_bot_set_int )
% 14.25/8.04 => ( ( lattic1091506334969745077in_int @ ( insert_int @ X @ A ) )
% 14.25/8.04 = ( sup_sup_int @ X @ ( lattic1091506334969745077in_int @ A ) ) ) ) ) ) ).
% 14.25/8.04
% 14.25/8.04 thf(fact_2230_Sup__fin_Oinsert__not__elem,axiom,
% 14.25/8.04 ! [A: set_Pr4532377907799695533_nat_o,X: produc3658429121746597890et_nat > $o] :
% 14.25/8.04 ( ( finite3252695134891459830_nat_o @ A )
% 14.25/8.04 => ( ~ ( member6576561426505652726_nat_o @ X @ A )
% 14.25/8.04 => ( ( A != bot_bo7824918357723582233_nat_o )
% 14.25/8.04 => ( ( lattic7320199455484906628_nat_o @ ( insert5175938949040314269_nat_o @ X @ A ) )
% 14.25/8.04 = ( sup_su5453871518329203617_nat_o @ X @ ( lattic7320199455484906628_nat_o @ A ) ) ) ) ) ) ).
% 14.25/8.04
% 14.25/8.04 thf(fact_2231_Inf__fin_Ounion,axiom,
% 14.25/8.04 ! [A: set_set_nat,B4: set_set_nat] :
% 14.25/8.04 ( ( finite1152437895449049373et_nat @ A )
% 14.25/8.04 => ( ( A != bot_bot_set_set_nat )
% 14.25/8.04 => ( ( finite1152437895449049373et_nat @ B4 )
% 14.25/8.04 => ( ( B4 != bot_bot_set_set_nat )
% 14.25/8.04 => ( ( lattic3014633134055518761et_nat @ ( sup_sup_set_set_nat @ A @ B4 ) )
% 14.25/8.04 = ( inf_inf_set_nat @ ( lattic3014633134055518761et_nat @ A ) @ ( lattic3014633134055518761et_nat @ B4 ) ) ) ) ) ) ) ).
% 14.25/8.04
% 14.25/8.04 thf(fact_2232_Inf__fin_Ounion,axiom,
% 14.25/8.04 ! [A: set_assn,B4: set_assn] :
% 14.25/8.04 ( ( finite_finite_assn @ A )
% 14.25/8.04 => ( ( A != bot_bot_set_assn )
% 14.25/8.04 => ( ( finite_finite_assn @ B4 )
% 14.25/8.04 => ( ( B4 != bot_bot_set_assn )
% 14.25/8.04 => ( ( lattic47131356835913163n_assn @ ( sup_sup_set_assn @ A @ B4 ) )
% 14.25/8.04 = ( inf_inf_assn @ ( lattic47131356835913163n_assn @ A ) @ ( lattic47131356835913163n_assn @ B4 ) ) ) ) ) ) ) ).
% 14.25/8.04
% 14.25/8.04 thf(fact_2233_Inf__fin_Ounion,axiom,
% 14.25/8.04 ! [A: set_Product_unit,B4: set_Product_unit] :
% 14.25/8.04 ( ( finite4290736615968046902t_unit @ A )
% 14.25/8.04 => ( ( A != bot_bo3957492148770167129t_unit )
% 14.25/8.04 => ( ( finite4290736615968046902t_unit @ B4 )
% 14.25/8.04 => ( ( B4 != bot_bo3957492148770167129t_unit )
% 14.25/8.04 => ( ( lattic1263872656861969706t_unit @ ( sup_su793286257634532545t_unit @ A @ B4 ) )
% 14.25/8.04 = ( inf_inf_Product_unit @ ( lattic1263872656861969706t_unit @ A ) @ ( lattic1263872656861969706t_unit @ B4 ) ) ) ) ) ) ) ).
% 14.25/8.04
% 14.25/8.04 thf(fact_2234_Inf__fin_Ounion,axiom,
% 14.25/8.04 ! [A: set_o,B4: set_o] :
% 14.25/8.04 ( ( finite_finite_o @ A )
% 14.25/8.04 => ( ( A != bot_bot_set_o )
% 14.25/8.04 => ( ( finite_finite_o @ B4 )
% 14.25/8.04 => ( ( B4 != bot_bot_set_o )
% 14.25/8.04 => ( ( lattic4107685809792843317_fin_o @ ( sup_sup_set_o @ A @ B4 ) )
% 14.25/8.04 = ( inf_inf_o @ ( lattic4107685809792843317_fin_o @ A ) @ ( lattic4107685809792843317_fin_o @ B4 ) ) ) ) ) ) ) ).
% 14.25/8.04
% 14.25/8.04 thf(fact_2235_Inf__fin_Ounion,axiom,
% 14.25/8.04 ! [A: set_nat,B4: set_nat] :
% 14.25/8.04 ( ( finite_finite_nat @ A )
% 14.25/8.04 => ( ( A != bot_bot_set_nat )
% 14.25/8.04 => ( ( finite_finite_nat @ B4 )
% 14.25/8.04 => ( ( B4 != bot_bot_set_nat )
% 14.25/8.04 => ( ( lattic5238388535129920115in_nat @ ( sup_sup_set_nat @ A @ B4 ) )
% 14.25/8.04 = ( inf_inf_nat @ ( lattic5238388535129920115in_nat @ A ) @ ( lattic5238388535129920115in_nat @ B4 ) ) ) ) ) ) ) ).
% 14.25/8.04
% 14.25/8.04 thf(fact_2236_Inf__fin_Ounion,axiom,
% 14.25/8.04 ! [A: set_int,B4: set_int] :
% 14.25/8.04 ( ( finite_finite_int @ A )
% 14.25/8.04 => ( ( A != bot_bot_set_int )
% 14.25/8.04 => ( ( finite_finite_int @ B4 )
% 14.25/8.04 => ( ( B4 != bot_bot_set_int )
% 14.25/8.04 => ( ( lattic5235898064620869839in_int @ ( sup_sup_set_int @ A @ B4 ) )
% 14.25/8.04 = ( inf_inf_int @ ( lattic5235898064620869839in_int @ A ) @ ( lattic5235898064620869839in_int @ B4 ) ) ) ) ) ) ) ).
% 14.25/8.04
% 14.25/8.04 thf(fact_2237_Inf__fin_Ounion,axiom,
% 14.25/8.04 ! [A: set_Pr4532377907799695533_nat_o,B4: set_Pr4532377907799695533_nat_o] :
% 14.25/8.04 ( ( finite3252695134891459830_nat_o @ A )
% 14.25/8.04 => ( ( A != bot_bo7824918357723582233_nat_o )
% 14.25/8.04 => ( ( finite3252695134891459830_nat_o @ B4 )
% 14.25/8.04 => ( ( B4 != bot_bo7824918357723582233_nat_o )
% 14.25/8.04 => ( ( lattic956194824204696298_nat_o @ ( sup_su5209123915105501825_nat_o @ A @ B4 ) )
% 14.25/8.04 = ( inf_in1318976480646536635_nat_o @ ( lattic956194824204696298_nat_o @ A ) @ ( lattic956194824204696298_nat_o @ B4 ) ) ) ) ) ) ) ).
% 14.25/8.04
% 14.25/8.04 thf(fact_2238_Sup__fin_Ounion,axiom,
% 14.25/8.04 ! [A: set_set_nat,B4: set_set_nat] :
% 14.25/8.04 ( ( finite1152437895449049373et_nat @ A )
% 14.25/8.04 => ( ( A != bot_bot_set_set_nat )
% 14.25/8.04 => ( ( finite1152437895449049373et_nat @ B4 )
% 14.25/8.04 => ( ( B4 != bot_bot_set_set_nat )
% 14.25/8.04 => ( ( lattic3835124923745554447et_nat @ ( sup_sup_set_set_nat @ A @ B4 ) )
% 14.25/8.04 = ( sup_sup_set_nat @ ( lattic3835124923745554447et_nat @ A ) @ ( lattic3835124923745554447et_nat @ B4 ) ) ) ) ) ) ) ).
% 14.25/8.04
% 14.25/8.04 thf(fact_2239_Sup__fin_Ounion,axiom,
% 14.25/8.04 ! [A: set_assn,B4: set_assn] :
% 14.25/8.04 ( ( finite_finite_assn @ A )
% 14.25/8.04 => ( ( A != bot_bot_set_assn )
% 14.25/8.04 => ( ( finite_finite_assn @ B4 )
% 14.25/8.04 => ( ( B4 != bot_bot_set_assn )
% 14.25/8.04 => ( ( lattic2150320897289308081n_assn @ ( sup_sup_set_assn @ A @ B4 ) )
% 14.25/8.04 = ( sup_sup_assn @ ( lattic2150320897289308081n_assn @ A ) @ ( lattic2150320897289308081n_assn @ B4 ) ) ) ) ) ) ) ).
% 14.25/8.04
% 14.25/8.04 thf(fact_2240_Sup__fin_Ounion,axiom,
% 14.25/8.04 ! [A: set_Product_unit,B4: set_Product_unit] :
% 14.25/8.04 ( ( finite4290736615968046902t_unit @ A )
% 14.25/8.04 => ( ( A != bot_bo3957492148770167129t_unit )
% 14.25/8.04 => ( ( finite4290736615968046902t_unit @ B4 )
% 14.25/8.04 => ( ( B4 != bot_bo3957492148770167129t_unit )
% 14.25/8.04 => ( ( lattic5294303975357428420t_unit @ ( sup_su793286257634532545t_unit @ A @ B4 ) )
% 14.25/8.04 = ( sup_sup_Product_unit @ ( lattic5294303975357428420t_unit @ A ) @ ( lattic5294303975357428420t_unit @ B4 ) ) ) ) ) ) ) ).
% 14.25/8.04
% 14.25/8.04 thf(fact_2241_Sup__fin_Ounion,axiom,
% 14.25/8.04 ! [A: set_o,B4: set_o] :
% 14.25/8.04 ( ( finite_finite_o @ A )
% 14.25/8.04 => ( ( A != bot_bot_set_o )
% 14.25/8.04 => ( ( finite_finite_o @ B4 )
% 14.25/8.04 => ( ( B4 != bot_bot_set_o )
% 14.25/8.04 => ( ( lattic1508158080041050831_fin_o @ ( sup_sup_set_o @ A @ B4 ) )
% 14.25/8.04 = ( sup_sup_o @ ( lattic1508158080041050831_fin_o @ A ) @ ( lattic1508158080041050831_fin_o @ B4 ) ) ) ) ) ) ) ).
% 14.25/8.04
% 14.25/8.04 thf(fact_2242_Sup__fin_Ounion,axiom,
% 14.25/8.04 ! [A: set_nat,B4: set_nat] :
% 14.25/8.04 ( ( finite_finite_nat @ A )
% 14.25/8.04 => ( ( A != bot_bot_set_nat )
% 14.25/8.04 => ( ( finite_finite_nat @ B4 )
% 14.25/8.04 => ( ( B4 != bot_bot_set_nat )
% 14.25/8.04 => ( ( lattic1093996805478795353in_nat @ ( sup_sup_set_nat @ A @ B4 ) )
% 14.25/8.04 = ( sup_sup_nat @ ( lattic1093996805478795353in_nat @ A ) @ ( lattic1093996805478795353in_nat @ B4 ) ) ) ) ) ) ) ).
% 14.25/8.04
% 14.25/8.04 thf(fact_2243_Sup__fin_Ounion,axiom,
% 14.25/8.04 ! [A: set_int,B4: set_int] :
% 14.25/8.04 ( ( finite_finite_int @ A )
% 14.25/8.04 => ( ( A != bot_bot_set_int )
% 14.25/8.04 => ( ( finite_finite_int @ B4 )
% 14.25/8.04 => ( ( B4 != bot_bot_set_int )
% 14.25/8.04 => ( ( lattic1091506334969745077in_int @ ( sup_sup_set_int @ A @ B4 ) )
% 14.25/8.04 = ( sup_sup_int @ ( lattic1091506334969745077in_int @ A ) @ ( lattic1091506334969745077in_int @ B4 ) ) ) ) ) ) ) ).
% 14.25/8.04
% 14.25/8.04 thf(fact_2244_Sup__fin_Ounion,axiom,
% 14.25/8.04 ! [A: set_Pr4532377907799695533_nat_o,B4: set_Pr4532377907799695533_nat_o] :
% 14.25/8.04 ( ( finite3252695134891459830_nat_o @ A )
% 14.25/8.04 => ( ( A != bot_bo7824918357723582233_nat_o )
% 14.25/8.04 => ( ( finite3252695134891459830_nat_o @ B4 )
% 14.25/8.04 => ( ( B4 != bot_bo7824918357723582233_nat_o )
% 14.25/8.04 => ( ( lattic7320199455484906628_nat_o @ ( sup_su5209123915105501825_nat_o @ A @ B4 ) )
% 14.25/8.04 = ( sup_su5453871518329203617_nat_o @ ( lattic7320199455484906628_nat_o @ A ) @ ( lattic7320199455484906628_nat_o @ B4 ) ) ) ) ) ) ) ).
% 14.25/8.04
% 14.25/8.04 thf(fact_2245_refl__on__reflcl__Image,axiom,
% 14.25/8.04 ! [B4: set_nat,A: set_Pr1261947904930325089at_nat,C5: set_nat] :
% 14.25/8.04 ( ( refl_on_nat @ B4 @ A )
% 14.25/8.04 => ( ( ord_less_eq_set_nat @ C5 @ B4 )
% 14.25/8.04 => ( ( image_nat_nat @ ( sup_su6327502436637775413at_nat @ A @ id_nat ) @ C5 )
% 14.25/8.04 = ( image_nat_nat @ A @ C5 ) ) ) ) ).
% 14.25/8.04
% 14.25/8.04 thf(fact_2246_E__closed__restr__reach__cases,axiom,
% 14.25/8.04 ! [U: produc3658429121746597890et_nat > $o,V: produc3658429121746597890et_nat > $o,E: set_Pr2161125870931222855_nat_o,R3: set_Pr4532377907799695533_nat_o] :
% 14.25/8.04 ( ( member8781333585448626064_nat_o @ ( produc7368190662567826135_nat_o @ U @ V ) @ ( transi712980445759334096_nat_o @ E ) )
% 14.25/8.04 => ( ( ord_le2965882846123202637_nat_o @ ( image_2232984160439939859_nat_o @ E @ R3 ) @ R3 )
% 14.25/8.04 => ( ~ ( member6576561426505652726_nat_o @ V @ R3 )
% 14.25/8.04 => ~ ( ~ ( member6576561426505652726_nat_o @ U @ R3 )
% 14.25/8.04 => ~ ( member8781333585448626064_nat_o @ ( produc7368190662567826135_nat_o @ U @ V ) @ ( transi712980445759334096_nat_o @ ( rel_re3331361160397388719_nat_o @ E @ R3 ) ) ) ) ) ) ) ).
% 14.25/8.04
% 14.25/8.04 thf(fact_2247_E__closed__restr__reach__cases,axiom,
% 14.25/8.04 ! [U: $o,V: $o,E: set_Product_prod_o_o,R3: set_o] :
% 14.25/8.04 ( ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ U @ V ) @ ( transitive_rtrancl_o @ E ) )
% 14.25/8.04 => ( ( ord_less_eq_set_o @ ( image_o_o @ E @ R3 ) @ R3 )
% 14.25/8.04 => ( ~ ( member_o @ V @ R3 )
% 14.25/8.04 => ~ ( ~ ( member_o @ U @ R3 )
% 14.25/8.04 => ~ ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ U @ V ) @ ( transitive_rtrancl_o @ ( rel_restrict_o @ E @ R3 ) ) ) ) ) ) ) ).
% 14.25/8.04
% 14.25/8.04 thf(fact_2248_E__closed__restr__reach__cases,axiom,
% 14.25/8.04 ! [U: product_unit,V: product_unit,E: set_Pr5094982260447487303t_unit,R3: set_Product_unit] :
% 14.25/8.04 ( ( member7821138191923116944t_unit @ ( produc5692694829384537815t_unit @ U @ V ) @ ( transi6481887417741755664t_unit @ E ) )
% 14.25/8.04 => ( ( ord_le3507040750410214029t_unit @ ( image_8431784415008483091t_unit @ E @ R3 ) @ R3 )
% 14.25/8.04 => ( ~ ( member_Product_unit @ V @ R3 )
% 14.25/8.04 => ~ ( ~ ( member_Product_unit @ U @ R3 )
% 14.25/8.04 => ~ ( member7821138191923116944t_unit @ ( produc5692694829384537815t_unit @ U @ V ) @ ( transi6481887417741755664t_unit @ ( rel_re9207996470947987439t_unit @ E @ R3 ) ) ) ) ) ) ) ).
% 14.25/8.04
% 14.25/8.04 thf(fact_2249_E__closed__restr__reach__cases,axiom,
% 14.25/8.04 ! [U: int,V: int,E: set_Pr958786334691620121nt_int,R3: set_int] :
% 14.25/8.04 ( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ U @ V ) @ ( transi2902850859426252137cl_int @ E ) )
% 14.25/8.04 => ( ( ord_less_eq_set_int @ ( image_int_int @ E @ R3 ) @ R3 )
% 14.25/8.04 => ( ~ ( member_int @ V @ R3 )
% 14.25/8.04 => ~ ( ~ ( member_int @ U @ R3 )
% 14.25/8.04 => ~ ( member5262025264175285858nt_int @ ( product_Pair_int_int @ U @ V ) @ ( transi2902850859426252137cl_int @ ( rel_restrict_int @ E @ R3 ) ) ) ) ) ) ) ).
% 14.25/8.04
% 14.25/8.04 thf(fact_2250_E__closed__restr__reach__cases,axiom,
% 14.25/8.04 ! [U: nat,V: nat,E: set_Pr1261947904930325089at_nat,R3: set_nat] :
% 14.25/8.04 ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ U @ V ) @ ( transi2905341329935302413cl_nat @ E ) )
% 14.25/8.04 => ( ( ord_less_eq_set_nat @ ( image_nat_nat @ E @ R3 ) @ R3 )
% 14.25/8.04 => ( ~ ( member_nat @ V @ R3 )
% 14.25/8.04 => ~ ( ~ ( member_nat @ U @ R3 )
% 14.25/8.04 => ~ ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ U @ V ) @ ( transi2905341329935302413cl_nat @ ( rel_restrict_nat @ E @ R3 ) ) ) ) ) ) ) ).
% 14.25/8.04
% 14.25/8.04 thf(fact_2251_max__ext__compat,axiom,
% 14.25/8.04 ! [R3: set_Pr5094982260447487303t_unit,S: set_Pr5094982260447487303t_unit] :
% 14.25/8.04 ( ( ord_le930853871177014503t_unit @ ( relcom848357998638694028t_unit @ R3 @ S ) @ R3 )
% 14.25/8.04 => ( ord_le8072594200992201959t_unit @ ( relcom7108816970757792620t_unit @ ( max_ext_Product_unit @ R3 ) @ ( sup_su3573767163274163995t_unit @ ( max_ext_Product_unit @ S ) @ ( insert6793119152790066487t_unit @ ( produc7269801343787132631t_unit @ bot_bo3957492148770167129t_unit @ bot_bo3957492148770167129t_unit ) @ bot_bo5113747506523071411t_unit ) ) ) @ ( max_ext_Product_unit @ R3 ) ) ) ).
% 14.25/8.04
% 14.25/8.04 thf(fact_2252_max__ext__compat,axiom,
% 14.25/8.04 ! [R3: set_Product_prod_o_o,S: set_Product_prod_o_o] :
% 14.25/8.04 ( ( ord_le192195778054999463od_o_o @ ( relcomp_o_o_o @ R3 @ S ) @ R3 )
% 14.25/8.04 => ( ord_le2885120122233735591_set_o @ ( relcom7534720040753365047_set_o @ ( max_ext_o @ R3 ) @ ( sup_su7766833992073560027_set_o @ ( max_ext_o @ S ) @ ( insert1494450971503307511_set_o @ ( produc5838405689764958487_set_o @ bot_bot_set_o @ bot_bot_set_o ) @ bot_bo8143257439550059635_set_o ) ) ) @ ( max_ext_o @ R3 ) ) ) ).
% 14.25/8.04
% 14.25/8.04 thf(fact_2253_max__ext__compat,axiom,
% 14.25/8.04 ! [R3: set_Pr1261947904930325089at_nat,S: set_Pr1261947904930325089at_nat] :
% 14.25/8.04 ( ( ord_le3146513528884898305at_nat @ ( relcomp_nat_nat_nat @ R3 @ S ) @ R3 )
% 14.25/8.04 => ( ord_le4402255868550298733et_nat @ ( relcom236561585145426837et_nat @ ( max_ext_nat @ R3 ) @ ( sup_su4251893418135750817et_nat @ ( max_ext_nat @ S ) @ ( insert3810226134351308605et_nat @ ( produc4532415448927165861et_nat @ bot_bot_set_nat @ bot_bot_set_nat ) @ bot_bo4952753636535259449et_nat ) ) ) @ ( max_ext_nat @ R3 ) ) ) ).
% 14.25/8.04
% 14.25/8.04 thf(fact_2254_max__ext__compat,axiom,
% 14.25/8.04 ! [R3: set_Pr958786334691620121nt_int,S: set_Pr958786334691620121nt_int] :
% 14.25/8.04 ( ( ord_le2843351958646193337nt_int @ ( relcomp_int_int_int @ R3 @ S ) @ R3 )
% 14.25/8.04 => ( ord_le1436784781161120549et_int @ ( relcom4034950705519254697et_int @ ( max_ext_int @ R3 ) @ ( sup_su1286422330746572633et_int @ ( max_ext_int @ S ) @ ( insert7328952640415156725et_int @ ( produc6363374080413544029et_int @ bot_bot_set_int @ bot_bot_set_int ) @ bot_bo1987282549146081265et_int ) ) ) @ ( max_ext_int @ R3 ) ) ) ).
% 14.25/8.04
% 14.25/8.04 thf(fact_2255_max__ext__compat,axiom,
% 14.25/8.04 ! [R3: set_Pr2161125870931222855_nat_o,S: set_Pr2161125870931222855_nat_o] :
% 14.25/8.04 ( ( ord_le6659818918159358183_nat_o @ ( relcom6251035627269349452_nat_o @ R3 @ S ) @ R3 )
% 14.25/8.04 => ( ord_le8001614231985977575_nat_o @ ( relcom7236094467932586796_nat_o @ ( max_ex5265246208886859912_nat_o @ R3 ) @ ( sup_su4863159966000489243_nat_o @ ( max_ex5265246208886859912_nat_o @ S ) @ ( insert3244108804717745463_nat_o @ ( produc5149316574908490455_nat_o @ bot_bo7824918357723582233_nat_o @ bot_bo7824918357723582233_nat_o ) @ bot_bo7535334947429593011_nat_o ) ) ) @ ( max_ex5265246208886859912_nat_o @ R3 ) ) ) ).
% 14.25/8.04
% 14.25/8.04 thf(fact_2256_BNF__Greatest__Fixpoint_OIdD,axiom,
% 14.25/8.04 ! [A2: nat,B: nat] :
% 14.25/8.04 ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A2 @ B ) @ id_nat )
% 14.25/8.04 => ( A2 = B ) ) ).
% 14.25/8.04
% 14.25/8.04 thf(fact_2257_BNF__Greatest__Fixpoint_OIdD,axiom,
% 14.25/8.04 ! [A2: int,B: int] :
% 14.25/8.04 ( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ A2 @ B ) @ id_int )
% 14.25/8.04 => ( A2 = B ) ) ).
% 14.25/8.04
% 14.25/8.04 thf(fact_2258_Sup__fin_Oremove,axiom,
% 14.25/8.04 ! [A: set_set_nat,X: set_nat] :
% 14.25/8.04 ( ( finite1152437895449049373et_nat @ A )
% 14.25/8.04 => ( ( member_set_nat @ X @ A )
% 14.25/8.04 => ( ( ( ( minus_2163939370556025621et_nat @ A @ ( insert_set_nat @ X @ bot_bot_set_set_nat ) )
% 14.25/8.04 = bot_bot_set_set_nat )
% 14.25/8.04 => ( ( lattic3835124923745554447et_nat @ A )
% 14.25/8.04 = X ) )
% 14.25/8.04 & ( ( ( minus_2163939370556025621et_nat @ A @ ( insert_set_nat @ X @ bot_bot_set_set_nat ) )
% 14.25/8.04 != bot_bot_set_set_nat )
% 14.25/8.04 => ( ( lattic3835124923745554447et_nat @ A )
% 14.25/8.04 = ( sup_sup_set_nat @ X @ ( lattic3835124923745554447et_nat @ ( minus_2163939370556025621et_nat @ A @ ( insert_set_nat @ X @ bot_bot_set_set_nat ) ) ) ) ) ) ) ) ) ).
% 14.25/8.04
% 14.25/8.04 thf(fact_2259_Sup__fin_Oremove,axiom,
% 14.25/8.04 ! [A: set_assn,X: assn] :
% 14.25/8.04 ( ( finite_finite_assn @ A )
% 14.25/8.04 => ( ( member_assn @ X @ A )
% 14.25/8.04 => ( ( ( ( minus_minus_set_assn @ A @ ( insert_assn @ X @ bot_bot_set_assn ) )
% 14.25/8.04 = bot_bot_set_assn )
% 14.25/8.04 => ( ( lattic2150320897289308081n_assn @ A )
% 14.25/8.04 = X ) )
% 14.25/8.04 & ( ( ( minus_minus_set_assn @ A @ ( insert_assn @ X @ bot_bot_set_assn ) )
% 14.25/8.04 != bot_bot_set_assn )
% 14.25/8.04 => ( ( lattic2150320897289308081n_assn @ A )
% 14.25/8.04 = ( sup_sup_assn @ X @ ( lattic2150320897289308081n_assn @ ( minus_minus_set_assn @ A @ ( insert_assn @ X @ bot_bot_set_assn ) ) ) ) ) ) ) ) ) ).
% 14.25/8.04
% 14.25/8.04 thf(fact_2260_Sup__fin_Oremove,axiom,
% 14.25/8.04 ! [A: set_Product_unit,X: product_unit] :
% 14.25/8.04 ( ( finite4290736615968046902t_unit @ A )
% 14.25/8.04 => ( ( member_Product_unit @ X @ A )
% 14.25/8.04 => ( ( ( ( minus_6452836326544984404t_unit @ A @ ( insert_Product_unit @ X @ bot_bo3957492148770167129t_unit ) )
% 14.25/8.04 = bot_bo3957492148770167129t_unit )
% 14.25/8.04 => ( ( lattic5294303975357428420t_unit @ A )
% 14.25/8.04 = X ) )
% 14.25/8.04 & ( ( ( minus_6452836326544984404t_unit @ A @ ( insert_Product_unit @ X @ bot_bo3957492148770167129t_unit ) )
% 14.25/8.04 != bot_bo3957492148770167129t_unit )
% 14.25/8.04 => ( ( lattic5294303975357428420t_unit @ A )
% 14.25/8.04 = ( sup_sup_Product_unit @ X @ ( lattic5294303975357428420t_unit @ ( minus_6452836326544984404t_unit @ A @ ( insert_Product_unit @ X @ bot_bo3957492148770167129t_unit ) ) ) ) ) ) ) ) ) ).
% 14.25/8.04
% 14.25/8.04 thf(fact_2261_Sup__fin_Oremove,axiom,
% 14.25/8.04 ! [A: set_o,X: $o] :
% 14.25/8.04 ( ( finite_finite_o @ A )
% 14.25/8.04 => ( ( member_o @ X @ A )
% 14.25/8.04 => ( ( lattic1508158080041050831_fin_o @ A )
% 14.25/8.04 = ( ( ( ( minus_minus_set_o @ A @ ( insert_o @ X @ bot_bot_set_o ) )
% 14.25/8.04 = bot_bot_set_o )
% 14.25/8.04 => X )
% 14.25/8.04 & ( ( ( minus_minus_set_o @ A @ ( insert_o @ X @ bot_bot_set_o ) )
% 14.25/8.04 != bot_bot_set_o )
% 14.25/8.04 => ( sup_sup_o @ X @ ( lattic1508158080041050831_fin_o @ ( minus_minus_set_o @ A @ ( insert_o @ X @ bot_bot_set_o ) ) ) ) ) ) ) ) ) ).
% 14.25/8.04
% 14.25/8.04 thf(fact_2262_Sup__fin_Oremove,axiom,
% 14.25/8.04 ! [A: set_int,X: int] :
% 14.25/8.04 ( ( finite_finite_int @ A )
% 14.25/8.04 => ( ( member_int @ X @ A )
% 14.25/8.04 => ( ( ( ( minus_minus_set_int @ A @ ( insert_int @ X @ bot_bot_set_int ) )
% 14.25/8.04 = bot_bot_set_int )
% 14.25/8.04 => ( ( lattic1091506334969745077in_int @ A )
% 14.25/8.04 = X ) )
% 14.25/8.04 & ( ( ( minus_minus_set_int @ A @ ( insert_int @ X @ bot_bot_set_int ) )
% 14.25/8.04 != bot_bot_set_int )
% 14.25/8.04 => ( ( lattic1091506334969745077in_int @ A )
% 14.25/8.04 = ( sup_sup_int @ X @ ( lattic1091506334969745077in_int @ ( minus_minus_set_int @ A @ ( insert_int @ X @ bot_bot_set_int ) ) ) ) ) ) ) ) ) ).
% 14.25/8.04
% 14.25/8.04 thf(fact_2263_Sup__fin_Oremove,axiom,
% 14.25/8.04 ! [A: set_Pr4532377907799695533_nat_o,X: produc3658429121746597890et_nat > $o] :
% 14.25/8.04 ( ( finite3252695134891459830_nat_o @ A )
% 14.25/8.04 => ( ( member6576561426505652726_nat_o @ X @ A )
% 14.25/8.04 => ( ( ( ( minus_1801376950450012436_nat_o @ A @ ( insert5175938949040314269_nat_o @ X @ bot_bo7824918357723582233_nat_o ) )
% 14.25/8.04 = bot_bo7824918357723582233_nat_o )
% 14.25/8.04 => ( ( lattic7320199455484906628_nat_o @ A )
% 14.25/8.04 = X ) )
% 14.25/8.04 & ( ( ( minus_1801376950450012436_nat_o @ A @ ( insert5175938949040314269_nat_o @ X @ bot_bo7824918357723582233_nat_o ) )
% 14.25/8.04 != bot_bo7824918357723582233_nat_o )
% 14.25/8.04 => ( ( lattic7320199455484906628_nat_o @ A )
% 14.25/8.04 = ( sup_su5453871518329203617_nat_o @ X @ ( lattic7320199455484906628_nat_o @ ( minus_1801376950450012436_nat_o @ A @ ( insert5175938949040314269_nat_o @ X @ bot_bo7824918357723582233_nat_o ) ) ) ) ) ) ) ) ) ).
% 14.25/8.04
% 14.25/8.04 thf(fact_2264_Sup__fin_Oremove,axiom,
% 14.25/8.04 ! [A: set_nat,X: nat] :
% 14.25/8.04 ( ( finite_finite_nat @ A )
% 14.25/8.04 => ( ( member_nat @ X @ A )
% 14.25/8.04 => ( ( ( ( minus_minus_set_nat @ A @ ( insert_nat @ X @ bot_bot_set_nat ) )
% 14.25/8.04 = bot_bot_set_nat )
% 14.25/8.04 => ( ( lattic1093996805478795353in_nat @ A )
% 14.25/8.04 = X ) )
% 14.25/8.04 & ( ( ( minus_minus_set_nat @ A @ ( insert_nat @ X @ bot_bot_set_nat ) )
% 14.25/8.04 != bot_bot_set_nat )
% 14.25/8.04 => ( ( lattic1093996805478795353in_nat @ A )
% 14.25/8.04 = ( sup_sup_nat @ X @ ( lattic1093996805478795353in_nat @ ( minus_minus_set_nat @ A @ ( insert_nat @ X @ bot_bot_set_nat ) ) ) ) ) ) ) ) ) ).
% 14.25/8.04
% 14.25/8.04 thf(fact_2265_Sup__fin_Oinsert__remove,axiom,
% 14.25/8.04 ! [A: set_set_nat,X: set_nat] :
% 14.25/8.04 ( ( finite1152437895449049373et_nat @ A )
% 14.25/8.04 => ( ( ( ( minus_2163939370556025621et_nat @ A @ ( insert_set_nat @ X @ bot_bot_set_set_nat ) )
% 14.25/8.04 = bot_bot_set_set_nat )
% 14.25/8.04 => ( ( lattic3835124923745554447et_nat @ ( insert_set_nat @ X @ A ) )
% 14.25/8.04 = X ) )
% 14.25/8.04 & ( ( ( minus_2163939370556025621et_nat @ A @ ( insert_set_nat @ X @ bot_bot_set_set_nat ) )
% 14.25/8.04 != bot_bot_set_set_nat )
% 14.25/8.04 => ( ( lattic3835124923745554447et_nat @ ( insert_set_nat @ X @ A ) )
% 14.25/8.04 = ( sup_sup_set_nat @ X @ ( lattic3835124923745554447et_nat @ ( minus_2163939370556025621et_nat @ A @ ( insert_set_nat @ X @ bot_bot_set_set_nat ) ) ) ) ) ) ) ) ).
% 14.25/8.04
% 14.25/8.04 thf(fact_2266_Sup__fin_Oinsert__remove,axiom,
% 14.25/8.04 ! [A: set_assn,X: assn] :
% 14.25/8.04 ( ( finite_finite_assn @ A )
% 14.25/8.04 => ( ( ( ( minus_minus_set_assn @ A @ ( insert_assn @ X @ bot_bot_set_assn ) )
% 14.25/8.04 = bot_bot_set_assn )
% 14.25/8.04 => ( ( lattic2150320897289308081n_assn @ ( insert_assn @ X @ A ) )
% 14.25/8.04 = X ) )
% 14.25/8.04 & ( ( ( minus_minus_set_assn @ A @ ( insert_assn @ X @ bot_bot_set_assn ) )
% 14.25/8.04 != bot_bot_set_assn )
% 14.25/8.04 => ( ( lattic2150320897289308081n_assn @ ( insert_assn @ X @ A ) )
% 14.25/8.04 = ( sup_sup_assn @ X @ ( lattic2150320897289308081n_assn @ ( minus_minus_set_assn @ A @ ( insert_assn @ X @ bot_bot_set_assn ) ) ) ) ) ) ) ) ).
% 14.25/8.04
% 14.25/8.04 thf(fact_2267_Sup__fin_Oinsert__remove,axiom,
% 14.25/8.04 ! [A: set_Product_unit,X: product_unit] :
% 14.25/8.04 ( ( finite4290736615968046902t_unit @ A )
% 14.25/8.04 => ( ( ( ( minus_6452836326544984404t_unit @ A @ ( insert_Product_unit @ X @ bot_bo3957492148770167129t_unit ) )
% 14.25/8.04 = bot_bo3957492148770167129t_unit )
% 14.25/8.04 => ( ( lattic5294303975357428420t_unit @ ( insert_Product_unit @ X @ A ) )
% 14.25/8.04 = X ) )
% 14.25/8.04 & ( ( ( minus_6452836326544984404t_unit @ A @ ( insert_Product_unit @ X @ bot_bo3957492148770167129t_unit ) )
% 14.25/8.04 != bot_bo3957492148770167129t_unit )
% 14.25/8.04 => ( ( lattic5294303975357428420t_unit @ ( insert_Product_unit @ X @ A ) )
% 14.25/8.04 = ( sup_sup_Product_unit @ X @ ( lattic5294303975357428420t_unit @ ( minus_6452836326544984404t_unit @ A @ ( insert_Product_unit @ X @ bot_bo3957492148770167129t_unit ) ) ) ) ) ) ) ) ).
% 14.25/8.04
% 14.25/8.04 thf(fact_2268_Sup__fin_Oinsert__remove,axiom,
% 14.25/8.04 ! [A: set_o,X: $o] :
% 14.25/8.04 ( ( finite_finite_o @ A )
% 14.25/8.04 => ( ( lattic1508158080041050831_fin_o @ ( insert_o @ X @ A ) )
% 14.25/8.04 = ( ( ( ( minus_minus_set_o @ A @ ( insert_o @ X @ bot_bot_set_o ) )
% 14.25/8.04 = bot_bot_set_o )
% 14.25/8.04 => X )
% 14.25/8.04 & ( ( ( minus_minus_set_o @ A @ ( insert_o @ X @ bot_bot_set_o ) )
% 14.25/8.04 != bot_bot_set_o )
% 14.25/8.04 => ( sup_sup_o @ X @ ( lattic1508158080041050831_fin_o @ ( minus_minus_set_o @ A @ ( insert_o @ X @ bot_bot_set_o ) ) ) ) ) ) ) ) ).
% 14.25/8.04
% 14.25/8.04 thf(fact_2269_Sup__fin_Oinsert__remove,axiom,
% 14.25/8.04 ! [A: set_int,X: int] :
% 14.25/8.04 ( ( finite_finite_int @ A )
% 14.25/8.04 => ( ( ( ( minus_minus_set_int @ A @ ( insert_int @ X @ bot_bot_set_int ) )
% 14.25/8.04 = bot_bot_set_int )
% 14.25/8.04 => ( ( lattic1091506334969745077in_int @ ( insert_int @ X @ A ) )
% 14.25/8.04 = X ) )
% 14.25/8.04 & ( ( ( minus_minus_set_int @ A @ ( insert_int @ X @ bot_bot_set_int ) )
% 14.25/8.04 != bot_bot_set_int )
% 14.25/8.04 => ( ( lattic1091506334969745077in_int @ ( insert_int @ X @ A ) )
% 14.25/8.04 = ( sup_sup_int @ X @ ( lattic1091506334969745077in_int @ ( minus_minus_set_int @ A @ ( insert_int @ X @ bot_bot_set_int ) ) ) ) ) ) ) ) ).
% 14.25/8.04
% 14.25/8.04 thf(fact_2270_Sup__fin_Oinsert__remove,axiom,
% 14.25/8.04 ! [A: set_Pr4532377907799695533_nat_o,X: produc3658429121746597890et_nat > $o] :
% 14.25/8.04 ( ( finite3252695134891459830_nat_o @ A )
% 14.25/8.04 => ( ( ( ( minus_1801376950450012436_nat_o @ A @ ( insert5175938949040314269_nat_o @ X @ bot_bo7824918357723582233_nat_o ) )
% 14.25/8.04 = bot_bo7824918357723582233_nat_o )
% 14.25/8.04 => ( ( lattic7320199455484906628_nat_o @ ( insert5175938949040314269_nat_o @ X @ A ) )
% 14.25/8.04 = X ) )
% 14.25/8.04 & ( ( ( minus_1801376950450012436_nat_o @ A @ ( insert5175938949040314269_nat_o @ X @ bot_bo7824918357723582233_nat_o ) )
% 14.25/8.04 != bot_bo7824918357723582233_nat_o )
% 14.25/8.04 => ( ( lattic7320199455484906628_nat_o @ ( insert5175938949040314269_nat_o @ X @ A ) )
% 14.25/8.04 = ( sup_su5453871518329203617_nat_o @ X @ ( lattic7320199455484906628_nat_o @ ( minus_1801376950450012436_nat_o @ A @ ( insert5175938949040314269_nat_o @ X @ bot_bo7824918357723582233_nat_o ) ) ) ) ) ) ) ) ).
% 14.25/8.04
% 14.25/8.04 thf(fact_2271_Sup__fin_Oinsert__remove,axiom,
% 14.25/8.04 ! [A: set_nat,X: nat] :
% 14.25/8.04 ( ( finite_finite_nat @ A )
% 14.25/8.04 => ( ( ( ( minus_minus_set_nat @ A @ ( insert_nat @ X @ bot_bot_set_nat ) )
% 14.25/8.04 = bot_bot_set_nat )
% 14.25/8.04 => ( ( lattic1093996805478795353in_nat @ ( insert_nat @ X @ A ) )
% 14.25/8.04 = X ) )
% 14.25/8.04 & ( ( ( minus_minus_set_nat @ A @ ( insert_nat @ X @ bot_bot_set_nat ) )
% 14.25/8.04 != bot_bot_set_nat )
% 14.25/8.04 => ( ( lattic1093996805478795353in_nat @ ( insert_nat @ X @ A ) )
% 14.25/8.04 = ( sup_sup_nat @ X @ ( lattic1093996805478795353in_nat @ ( minus_minus_set_nat @ A @ ( insert_nat @ X @ bot_bot_set_nat ) ) ) ) ) ) ) ) ).
% 14.25/8.04
% 14.25/8.04 thf(fact_2272_Inf__fin_Oremove,axiom,
% 14.25/8.04 ! [A: set_set_nat,X: set_nat] :
% 14.25/8.04 ( ( finite1152437895449049373et_nat @ A )
% 14.25/8.04 => ( ( member_set_nat @ X @ A )
% 14.25/8.04 => ( ( ( ( minus_2163939370556025621et_nat @ A @ ( insert_set_nat @ X @ bot_bot_set_set_nat ) )
% 14.25/8.04 = bot_bot_set_set_nat )
% 14.25/8.04 => ( ( lattic3014633134055518761et_nat @ A )
% 14.25/8.04 = X ) )
% 14.25/8.04 & ( ( ( minus_2163939370556025621et_nat @ A @ ( insert_set_nat @ X @ bot_bot_set_set_nat ) )
% 14.25/8.04 != bot_bot_set_set_nat )
% 14.25/8.04 => ( ( lattic3014633134055518761et_nat @ A )
% 14.25/8.04 = ( inf_inf_set_nat @ X @ ( lattic3014633134055518761et_nat @ ( minus_2163939370556025621et_nat @ A @ ( insert_set_nat @ X @ bot_bot_set_set_nat ) ) ) ) ) ) ) ) ) ).
% 14.25/8.04
% 14.25/8.04 thf(fact_2273_Inf__fin_Oremove,axiom,
% 14.25/8.04 ! [A: set_assn,X: assn] :
% 14.25/8.04 ( ( finite_finite_assn @ A )
% 14.25/8.04 => ( ( member_assn @ X @ A )
% 14.25/8.04 => ( ( ( ( minus_minus_set_assn @ A @ ( insert_assn @ X @ bot_bot_set_assn ) )
% 14.25/8.04 = bot_bot_set_assn )
% 14.25/8.04 => ( ( lattic47131356835913163n_assn @ A )
% 14.25/8.04 = X ) )
% 14.25/8.04 & ( ( ( minus_minus_set_assn @ A @ ( insert_assn @ X @ bot_bot_set_assn ) )
% 14.25/8.04 != bot_bot_set_assn )
% 14.25/8.04 => ( ( lattic47131356835913163n_assn @ A )
% 14.25/8.04 = ( inf_inf_assn @ X @ ( lattic47131356835913163n_assn @ ( minus_minus_set_assn @ A @ ( insert_assn @ X @ bot_bot_set_assn ) ) ) ) ) ) ) ) ) ).
% 14.25/8.04
% 14.25/8.04 thf(fact_2274_Inf__fin_Oremove,axiom,
% 14.25/8.04 ! [A: set_Product_unit,X: product_unit] :
% 14.25/8.04 ( ( finite4290736615968046902t_unit @ A )
% 14.25/8.04 => ( ( member_Product_unit @ X @ A )
% 14.25/8.04 => ( ( ( ( minus_6452836326544984404t_unit @ A @ ( insert_Product_unit @ X @ bot_bo3957492148770167129t_unit ) )
% 14.25/8.04 = bot_bo3957492148770167129t_unit )
% 14.25/8.04 => ( ( lattic1263872656861969706t_unit @ A )
% 14.25/8.04 = X ) )
% 14.25/8.04 & ( ( ( minus_6452836326544984404t_unit @ A @ ( insert_Product_unit @ X @ bot_bo3957492148770167129t_unit ) )
% 14.25/8.04 != bot_bo3957492148770167129t_unit )
% 14.25/8.04 => ( ( lattic1263872656861969706t_unit @ A )
% 14.25/8.04 = ( inf_inf_Product_unit @ X @ ( lattic1263872656861969706t_unit @ ( minus_6452836326544984404t_unit @ A @ ( insert_Product_unit @ X @ bot_bo3957492148770167129t_unit ) ) ) ) ) ) ) ) ) ).
% 14.25/8.04
% 14.25/8.04 thf(fact_2275_Inf__fin_Oremove,axiom,
% 14.25/8.04 ! [A: set_o,X: $o] :
% 14.25/8.04 ( ( finite_finite_o @ A )
% 14.25/8.04 => ( ( member_o @ X @ A )
% 14.25/8.04 => ( ( lattic4107685809792843317_fin_o @ A )
% 14.25/8.04 = ( ( ( ( minus_minus_set_o @ A @ ( insert_o @ X @ bot_bot_set_o ) )
% 14.25/8.04 = bot_bot_set_o )
% 14.25/8.04 => X )
% 14.25/8.04 & ( ( ( minus_minus_set_o @ A @ ( insert_o @ X @ bot_bot_set_o ) )
% 14.25/8.04 != bot_bot_set_o )
% 14.25/8.04 => ( inf_inf_o @ X @ ( lattic4107685809792843317_fin_o @ ( minus_minus_set_o @ A @ ( insert_o @ X @ bot_bot_set_o ) ) ) ) ) ) ) ) ) ).
% 14.25/8.04
% 14.25/8.04 thf(fact_2276_Inf__fin_Oremove,axiom,
% 14.25/8.04 ! [A: set_int,X: int] :
% 14.25/8.04 ( ( finite_finite_int @ A )
% 14.25/8.04 => ( ( member_int @ X @ A )
% 14.25/8.04 => ( ( ( ( minus_minus_set_int @ A @ ( insert_int @ X @ bot_bot_set_int ) )
% 14.25/8.04 = bot_bot_set_int )
% 14.25/8.04 => ( ( lattic5235898064620869839in_int @ A )
% 14.25/8.04 = X ) )
% 14.25/8.04 & ( ( ( minus_minus_set_int @ A @ ( insert_int @ X @ bot_bot_set_int ) )
% 14.25/8.04 != bot_bot_set_int )
% 14.25/8.04 => ( ( lattic5235898064620869839in_int @ A )
% 14.25/8.04 = ( inf_inf_int @ X @ ( lattic5235898064620869839in_int @ ( minus_minus_set_int @ A @ ( insert_int @ X @ bot_bot_set_int ) ) ) ) ) ) ) ) ) ).
% 14.25/8.04
% 14.25/8.04 thf(fact_2277_Inf__fin_Oremove,axiom,
% 14.25/8.04 ! [A: set_Pr4532377907799695533_nat_o,X: produc3658429121746597890et_nat > $o] :
% 14.25/8.04 ( ( finite3252695134891459830_nat_o @ A )
% 14.25/8.04 => ( ( member6576561426505652726_nat_o @ X @ A )
% 14.25/8.04 => ( ( ( ( minus_1801376950450012436_nat_o @ A @ ( insert5175938949040314269_nat_o @ X @ bot_bo7824918357723582233_nat_o ) )
% 14.25/8.04 = bot_bo7824918357723582233_nat_o )
% 14.25/8.04 => ( ( lattic956194824204696298_nat_o @ A )
% 14.25/8.04 = X ) )
% 14.25/8.04 & ( ( ( minus_1801376950450012436_nat_o @ A @ ( insert5175938949040314269_nat_o @ X @ bot_bo7824918357723582233_nat_o ) )
% 14.25/8.04 != bot_bo7824918357723582233_nat_o )
% 14.25/8.04 => ( ( lattic956194824204696298_nat_o @ A )
% 14.25/8.04 = ( inf_in1318976480646536635_nat_o @ X @ ( lattic956194824204696298_nat_o @ ( minus_1801376950450012436_nat_o @ A @ ( insert5175938949040314269_nat_o @ X @ bot_bo7824918357723582233_nat_o ) ) ) ) ) ) ) ) ) ).
% 14.25/8.04
% 14.25/8.04 thf(fact_2278_Inf__fin_Oremove,axiom,
% 14.25/8.04 ! [A: set_nat,X: nat] :
% 14.25/8.04 ( ( finite_finite_nat @ A )
% 14.25/8.04 => ( ( member_nat @ X @ A )
% 14.25/8.04 => ( ( ( ( minus_minus_set_nat @ A @ ( insert_nat @ X @ bot_bot_set_nat ) )
% 14.25/8.04 = bot_bot_set_nat )
% 14.25/8.04 => ( ( lattic5238388535129920115in_nat @ A )
% 14.25/8.04 = X ) )
% 14.25/8.04 & ( ( ( minus_minus_set_nat @ A @ ( insert_nat @ X @ bot_bot_set_nat ) )
% 14.25/8.04 != bot_bot_set_nat )
% 14.25/8.04 => ( ( lattic5238388535129920115in_nat @ A )
% 14.25/8.04 = ( inf_inf_nat @ X @ ( lattic5238388535129920115in_nat @ ( minus_minus_set_nat @ A @ ( insert_nat @ X @ bot_bot_set_nat ) ) ) ) ) ) ) ) ) ).
% 14.25/8.04
% 14.25/8.04 thf(fact_2279_Inf__fin_Oinsert__remove,axiom,
% 14.25/8.04 ! [A: set_set_nat,X: set_nat] :
% 14.25/8.04 ( ( finite1152437895449049373et_nat @ A )
% 14.25/8.04 => ( ( ( ( minus_2163939370556025621et_nat @ A @ ( insert_set_nat @ X @ bot_bot_set_set_nat ) )
% 14.25/8.04 = bot_bot_set_set_nat )
% 14.25/8.04 => ( ( lattic3014633134055518761et_nat @ ( insert_set_nat @ X @ A ) )
% 14.25/8.04 = X ) )
% 14.25/8.04 & ( ( ( minus_2163939370556025621et_nat @ A @ ( insert_set_nat @ X @ bot_bot_set_set_nat ) )
% 14.25/8.04 != bot_bot_set_set_nat )
% 14.25/8.04 => ( ( lattic3014633134055518761et_nat @ ( insert_set_nat @ X @ A ) )
% 14.25/8.04 = ( inf_inf_set_nat @ X @ ( lattic3014633134055518761et_nat @ ( minus_2163939370556025621et_nat @ A @ ( insert_set_nat @ X @ bot_bot_set_set_nat ) ) ) ) ) ) ) ) ).
% 14.25/8.04
% 14.25/8.04 thf(fact_2280_Inf__fin_Oinsert__remove,axiom,
% 14.25/8.04 ! [A: set_assn,X: assn] :
% 14.25/8.04 ( ( finite_finite_assn @ A )
% 14.25/8.04 => ( ( ( ( minus_minus_set_assn @ A @ ( insert_assn @ X @ bot_bot_set_assn ) )
% 14.25/8.04 = bot_bot_set_assn )
% 14.25/8.04 => ( ( lattic47131356835913163n_assn @ ( insert_assn @ X @ A ) )
% 14.25/8.04 = X ) )
% 14.25/8.04 & ( ( ( minus_minus_set_assn @ A @ ( insert_assn @ X @ bot_bot_set_assn ) )
% 14.25/8.04 != bot_bot_set_assn )
% 14.25/8.04 => ( ( lattic47131356835913163n_assn @ ( insert_assn @ X @ A ) )
% 14.25/8.04 = ( inf_inf_assn @ X @ ( lattic47131356835913163n_assn @ ( minus_minus_set_assn @ A @ ( insert_assn @ X @ bot_bot_set_assn ) ) ) ) ) ) ) ) ).
% 14.25/8.04
% 14.25/8.04 thf(fact_2281_Inf__fin_Oinsert__remove,axiom,
% 14.25/8.04 ! [A: set_Product_unit,X: product_unit] :
% 14.25/8.04 ( ( finite4290736615968046902t_unit @ A )
% 14.25/8.04 => ( ( ( ( minus_6452836326544984404t_unit @ A @ ( insert_Product_unit @ X @ bot_bo3957492148770167129t_unit ) )
% 14.25/8.04 = bot_bo3957492148770167129t_unit )
% 14.25/8.04 => ( ( lattic1263872656861969706t_unit @ ( insert_Product_unit @ X @ A ) )
% 14.25/8.04 = X ) )
% 14.25/8.04 & ( ( ( minus_6452836326544984404t_unit @ A @ ( insert_Product_unit @ X @ bot_bo3957492148770167129t_unit ) )
% 14.25/8.04 != bot_bo3957492148770167129t_unit )
% 14.25/8.04 => ( ( lattic1263872656861969706t_unit @ ( insert_Product_unit @ X @ A ) )
% 14.25/8.04 = ( inf_inf_Product_unit @ X @ ( lattic1263872656861969706t_unit @ ( minus_6452836326544984404t_unit @ A @ ( insert_Product_unit @ X @ bot_bo3957492148770167129t_unit ) ) ) ) ) ) ) ) ).
% 14.25/8.04
% 14.25/8.04 thf(fact_2282_Inf__fin_Oinsert__remove,axiom,
% 14.25/8.04 ! [A: set_o,X: $o] :
% 14.25/8.04 ( ( finite_finite_o @ A )
% 14.25/8.04 => ( ( lattic4107685809792843317_fin_o @ ( insert_o @ X @ A ) )
% 14.25/8.04 = ( ( ( ( minus_minus_set_o @ A @ ( insert_o @ X @ bot_bot_set_o ) )
% 14.25/8.04 = bot_bot_set_o )
% 14.25/8.04 => X )
% 14.25/8.04 & ( ( ( minus_minus_set_o @ A @ ( insert_o @ X @ bot_bot_set_o ) )
% 14.25/8.04 != bot_bot_set_o )
% 14.25/8.04 => ( inf_inf_o @ X @ ( lattic4107685809792843317_fin_o @ ( minus_minus_set_o @ A @ ( insert_o @ X @ bot_bot_set_o ) ) ) ) ) ) ) ) ).
% 14.25/8.04
% 14.25/8.04 thf(fact_2283_Inf__fin_Oinsert__remove,axiom,
% 14.25/8.04 ! [A: set_int,X: int] :
% 14.25/8.04 ( ( finite_finite_int @ A )
% 14.25/8.04 => ( ( ( ( minus_minus_set_int @ A @ ( insert_int @ X @ bot_bot_set_int ) )
% 14.25/8.04 = bot_bot_set_int )
% 14.25/8.04 => ( ( lattic5235898064620869839in_int @ ( insert_int @ X @ A ) )
% 14.25/8.04 = X ) )
% 14.25/8.04 & ( ( ( minus_minus_set_int @ A @ ( insert_int @ X @ bot_bot_set_int ) )
% 14.25/8.04 != bot_bot_set_int )
% 14.25/8.04 => ( ( lattic5235898064620869839in_int @ ( insert_int @ X @ A ) )
% 14.25/8.04 = ( inf_inf_int @ X @ ( lattic5235898064620869839in_int @ ( minus_minus_set_int @ A @ ( insert_int @ X @ bot_bot_set_int ) ) ) ) ) ) ) ) ).
% 14.25/8.04
% 14.25/8.04 thf(fact_2284_Inf__fin_Oinsert__remove,axiom,
% 14.25/8.04 ! [A: set_Pr4532377907799695533_nat_o,X: produc3658429121746597890et_nat > $o] :
% 14.25/8.04 ( ( finite3252695134891459830_nat_o @ A )
% 14.25/8.04 => ( ( ( ( minus_1801376950450012436_nat_o @ A @ ( insert5175938949040314269_nat_o @ X @ bot_bo7824918357723582233_nat_o ) )
% 14.25/8.04 = bot_bo7824918357723582233_nat_o )
% 14.25/8.04 => ( ( lattic956194824204696298_nat_o @ ( insert5175938949040314269_nat_o @ X @ A ) )
% 14.25/8.04 = X ) )
% 14.25/8.04 & ( ( ( minus_1801376950450012436_nat_o @ A @ ( insert5175938949040314269_nat_o @ X @ bot_bo7824918357723582233_nat_o ) )
% 14.25/8.04 != bot_bo7824918357723582233_nat_o )
% 14.25/8.04 => ( ( lattic956194824204696298_nat_o @ ( insert5175938949040314269_nat_o @ X @ A ) )
% 14.25/8.04 = ( inf_in1318976480646536635_nat_o @ X @ ( lattic956194824204696298_nat_o @ ( minus_1801376950450012436_nat_o @ A @ ( insert5175938949040314269_nat_o @ X @ bot_bo7824918357723582233_nat_o ) ) ) ) ) ) ) ) ).
% 14.25/8.04
% 14.25/8.04 thf(fact_2285_Inf__fin_Oinsert__remove,axiom,
% 14.25/8.04 ! [A: set_nat,X: nat] :
% 14.25/8.04 ( ( finite_finite_nat @ A )
% 14.25/8.04 => ( ( ( ( minus_minus_set_nat @ A @ ( insert_nat @ X @ bot_bot_set_nat ) )
% 14.25/8.04 = bot_bot_set_nat )
% 14.25/8.04 => ( ( lattic5238388535129920115in_nat @ ( insert_nat @ X @ A ) )
% 14.25/8.04 = X ) )
% 14.25/8.04 & ( ( ( minus_minus_set_nat @ A @ ( insert_nat @ X @ bot_bot_set_nat ) )
% 14.25/8.04 != bot_bot_set_nat )
% 14.25/8.04 => ( ( lattic5238388535129920115in_nat @ ( insert_nat @ X @ A ) )
% 14.25/8.04 = ( inf_inf_nat @ X @ ( lattic5238388535129920115in_nat @ ( minus_minus_set_nat @ A @ ( insert_nat @ X @ bot_bot_set_nat ) ) ) ) ) ) ) ) ).
% 14.25/8.04
% 14.25/8.04 thf(fact_2286_acyclic__insert,axiom,
% 14.25/8.04 ! [Y: nat,X: nat,R2: set_Pr1261947904930325089at_nat] :
% 14.25/8.04 ( ( transi3139058337892597825ic_nat @ ( insert8211810215607154385at_nat @ ( product_Pair_nat_nat @ Y @ X ) @ R2 ) )
% 14.25/8.04 = ( ( transi3139058337892597825ic_nat @ R2 )
% 14.25/8.04 & ~ ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ Y ) @ ( transi2905341329935302413cl_nat @ R2 ) ) ) ) ).
% 14.25/8.04
% 14.25/8.04 thf(fact_2287_acyclic__insert,axiom,
% 14.25/8.04 ! [Y: int,X: int,R2: set_Pr958786334691620121nt_int] :
% 14.25/8.04 ( ( transi3136567867383547549ic_int @ ( insert5033312907999012233nt_int @ ( product_Pair_int_int @ Y @ X ) @ R2 ) )
% 14.25/8.04 = ( ( transi3136567867383547549ic_int @ R2 )
% 14.25/8.04 & ~ ( member5262025264175285858nt_int @ ( product_Pair_int_int @ X @ Y ) @ ( transi2902850859426252137cl_int @ R2 ) ) ) ) ).
% 14.25/8.04
% 14.25/8.04 thf(fact_2288_irrefl__tranclI,axiom,
% 14.25/8.04 ! [R2: set_Pr1261947904930325089at_nat,X: nat] :
% 14.25/8.04 ( ( ( inf_in2572325071724192079at_nat @ ( converse_nat_nat @ R2 ) @ ( transi2905341329935302413cl_nat @ R2 ) )
% 14.25/8.04 = bot_bo2099793752762293965at_nat )
% 14.25/8.04 => ~ ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ X ) @ ( transi6264000038957366511cl_nat @ R2 ) ) ) ).
% 14.25/8.04
% 14.25/8.04 thf(fact_2289_irrefl__tranclI,axiom,
% 14.25/8.04 ! [R2: set_Pr958786334691620121nt_int,X: int] :
% 14.25/8.04 ( ( ( inf_in2269163501485487111nt_int @ ( converse_int_int @ R2 ) @ ( transi2902850859426252137cl_int @ R2 ) )
% 14.25/8.04 = bot_bo1796632182523588997nt_int )
% 14.25/8.04 => ~ ( member5262025264175285858nt_int @ ( product_Pair_int_int @ X @ X ) @ ( transi6261509568448316235cl_int @ R2 ) ) ) ).
% 14.25/8.04
% 14.25/8.04 thf(fact_2290_inf__img__fin__dom_H,axiom,
% 14.25/8.04 ! [F: code_integer > code_integer,A: set_Code_integer] :
% 14.25/8.04 ( ( finite6017078050557962740nteger @ ( image_4470545334726330049nteger @ F @ A ) )
% 14.25/8.04 => ( ~ ( finite6017078050557962740nteger @ A )
% 14.25/8.04 => ? [X3: code_integer] :
% 14.25/8.04 ( ( member_Code_integer @ X3 @ ( image_4470545334726330049nteger @ F @ A ) )
% 14.25/8.04 & ~ ( finite6017078050557962740nteger @ ( inf_in1364745209274528805nteger @ ( vimage1318793397690208119nteger @ F @ ( insert_Code_integer @ X3 @ bot_bo3990330152332043303nteger ) ) @ A ) ) ) ) ) ).
% 14.25/8.04
% 14.25/8.04 thf(fact_2291_inf__img__fin__dom_H,axiom,
% 14.25/8.04 ! [F: nat > code_integer,A: set_nat] :
% 14.25/8.04 ( ( finite6017078050557962740nteger @ ( image_1215581382706833972nteger @ F @ A ) )
% 14.25/8.04 => ( ~ ( finite_finite_nat @ A )
% 14.25/8.04 => ? [X3: code_integer] :
% 14.25/8.04 ( ( member_Code_integer @ X3 @ ( image_1215581382706833972nteger @ F @ A ) )
% 14.25/8.04 & ~ ( finite_finite_nat @ ( inf_inf_set_nat @ ( vimage6104394758324456170nteger @ F @ ( insert_Code_integer @ X3 @ bot_bo3990330152332043303nteger ) ) @ A ) ) ) ) ) ).
% 14.25/8.04
% 14.25/8.04 thf(fact_2292_inf__img__fin__dom_H,axiom,
% 14.25/8.04 ! [F: code_integer > product_unit,A: set_Code_integer] :
% 14.25/8.04 ( ( finite4290736615968046902t_unit @ ( image_7182656606676592809t_unit @ F @ A ) )
% 14.25/8.04 => ( ~ ( finite6017078050557962740nteger @ A )
% 14.25/8.04 => ? [X3: product_unit] :
% 14.25/8.04 ( ( member_Product_unit @ X3 @ ( image_7182656606676592809t_unit @ F @ A ) )
% 14.25/8.04 & ~ ( finite6017078050557962740nteger @ ( inf_in1364745209274528805nteger @ ( vimage5651688891684837235t_unit @ F @ ( insert_Product_unit @ X3 @ bot_bo3957492148770167129t_unit ) ) @ A ) ) ) ) ) ).
% 14.25/8.04
% 14.25/8.04 thf(fact_2293_inf__img__fin__dom_H,axiom,
% 14.25/8.04 ! [F: nat > product_unit,A: set_nat] :
% 14.25/8.04 ( ( finite4290736615968046902t_unit @ ( image_8730104196221521654t_unit @ F @ A ) )
% 14.25/8.04 => ( ~ ( finite_finite_nat @ A )
% 14.25/8.04 => ? [X3: product_unit] :
% 14.25/8.04 ( ( member_Product_unit @ X3 @ ( image_8730104196221521654t_unit @ F @ A ) )
% 14.25/8.04 & ~ ( finite_finite_nat @ ( inf_inf_set_nat @ ( vimage4884490618288580032t_unit @ F @ ( insert_Product_unit @ X3 @ bot_bo3957492148770167129t_unit ) ) @ A ) ) ) ) ) ).
% 14.25/8.04
% 14.25/8.04 thf(fact_2294_inf__img__fin__dom_H,axiom,
% 14.25/8.04 ! [F: code_integer > $o,A: set_Code_integer] :
% 14.25/8.04 ( ( finite_finite_o @ ( image_Code_integer_o @ F @ A ) )
% 14.25/8.04 => ( ~ ( finite6017078050557962740nteger @ A )
% 14.25/8.04 => ? [X3: $o] :
% 14.25/8.04 ( ( member_o @ X3 @ ( image_Code_integer_o @ F @ A ) )
% 14.25/8.04 & ~ ( finite6017078050557962740nteger @ ( inf_in1364745209274528805nteger @ ( vimage3005800680117336574eger_o @ F @ ( insert_o @ X3 @ bot_bot_set_o ) ) @ A ) ) ) ) ) ).
% 14.25/8.04
% 14.25/8.04 thf(fact_2295_inf__img__fin__dom_H,axiom,
% 14.25/8.04 ! [F: nat > $o,A: set_nat] :
% 14.25/8.04 ( ( finite_finite_o @ ( image_nat_o2 @ F @ A ) )
% 14.25/8.04 => ( ~ ( finite_finite_nat @ A )
% 14.25/8.04 => ? [X3: $o] :
% 14.25/8.04 ( ( member_o @ X3 @ ( image_nat_o2 @ F @ A ) )
% 14.25/8.04 & ~ ( finite_finite_nat @ ( inf_inf_set_nat @ ( vimage_nat_o @ F @ ( insert_o @ X3 @ bot_bot_set_o ) ) @ A ) ) ) ) ) ).
% 14.25/8.04
% 14.25/8.04 thf(fact_2296_inf__img__fin__dom_H,axiom,
% 14.25/8.04 ! [F: code_integer > nat,A: set_Code_integer] :
% 14.25/8.04 ( ( finite_finite_nat @ ( image_951025933927791156er_nat @ F @ A ) )
% 14.25/8.04 => ( ~ ( finite6017078050557962740nteger @ A )
% 14.25/8.04 => ? [X3: nat] :
% 14.25/8.04 ( ( member_nat @ X3 @ ( image_951025933927791156er_nat @ F @ A ) )
% 14.25/8.04 & ~ ( finite6017078050557962740nteger @ ( inf_in1364745209274528805nteger @ ( vimage5839839309545413354er_nat @ F @ ( insert_nat @ X3 @ bot_bot_set_nat ) ) @ A ) ) ) ) ) ).
% 14.25/8.04
% 14.25/8.04 thf(fact_2297_inf__img__fin__dom_H,axiom,
% 14.25/8.04 ! [F: nat > nat,A: set_nat] :
% 14.25/8.04 ( ( finite_finite_nat @ ( image_nat_nat2 @ F @ A ) )
% 14.25/8.04 => ( ~ ( finite_finite_nat @ A )
% 14.25/8.04 => ? [X3: nat] :
% 14.25/8.04 ( ( member_nat @ X3 @ ( image_nat_nat2 @ F @ A ) )
% 14.25/8.04 & ~ ( finite_finite_nat @ ( inf_inf_set_nat @ ( vimage_nat_nat @ F @ ( insert_nat @ X3 @ bot_bot_set_nat ) ) @ A ) ) ) ) ) ).
% 14.25/8.04
% 14.25/8.04 thf(fact_2298_inf__img__fin__dom_H,axiom,
% 14.25/8.04 ! [F: code_integer > int,A: set_Code_integer] :
% 14.25/8.04 ( ( finite_finite_int @ ( image_948535463418740880er_int @ F @ A ) )
% 14.25/8.04 => ( ~ ( finite6017078050557962740nteger @ A )
% 14.25/8.04 => ? [X3: int] :
% 14.25/8.04 ( ( member_int @ X3 @ ( image_948535463418740880er_int @ F @ A ) )
% 14.25/8.04 & ~ ( finite6017078050557962740nteger @ ( inf_in1364745209274528805nteger @ ( vimage5837348839036363078er_int @ F @ ( insert_int @ X3 @ bot_bot_set_int ) ) @ A ) ) ) ) ) ).
% 14.25/8.04
% 14.25/8.04 thf(fact_2299_inf__img__fin__dom_H,axiom,
% 14.25/8.04 ! [F: nat > int,A: set_nat] :
% 14.25/8.04 ( ( finite_finite_int @ ( image_nat_int2 @ F @ A ) )
% 14.25/8.04 => ( ~ ( finite_finite_nat @ A )
% 14.25/8.04 => ? [X3: int] :
% 14.25/8.04 ( ( member_int @ X3 @ ( image_nat_int2 @ F @ A ) )
% 14.25/8.04 & ~ ( finite_finite_nat @ ( inf_inf_set_nat @ ( vimage_nat_int @ F @ ( insert_int @ X3 @ bot_bot_set_int ) ) @ A ) ) ) ) ) ).
% 14.25/8.04
% 14.25/8.04 thf(fact_2300_inf__img__fin__domE_H,axiom,
% 14.25/8.04 ! [F: code_integer > code_integer,A: set_Code_integer] :
% 14.25/8.04 ( ( finite6017078050557962740nteger @ ( image_4470545334726330049nteger @ F @ A ) )
% 14.25/8.04 => ( ~ ( finite6017078050557962740nteger @ A )
% 14.25/8.04 => ~ ! [Y2: code_integer] :
% 14.25/8.04 ( ( member_Code_integer @ Y2 @ ( image_4470545334726330049nteger @ F @ A ) )
% 14.25/8.04 => ( finite6017078050557962740nteger @ ( inf_in1364745209274528805nteger @ ( vimage1318793397690208119nteger @ F @ ( insert_Code_integer @ Y2 @ bot_bo3990330152332043303nteger ) ) @ A ) ) ) ) ) ).
% 14.25/8.04
% 14.25/8.04 thf(fact_2301_inf__img__fin__domE_H,axiom,
% 14.25/8.04 ! [F: nat > code_integer,A: set_nat] :
% 14.25/8.04 ( ( finite6017078050557962740nteger @ ( image_1215581382706833972nteger @ F @ A ) )
% 14.25/8.04 => ( ~ ( finite_finite_nat @ A )
% 14.25/8.04 => ~ ! [Y2: code_integer] :
% 14.25/8.04 ( ( member_Code_integer @ Y2 @ ( image_1215581382706833972nteger @ F @ A ) )
% 14.25/8.04 => ( finite_finite_nat @ ( inf_inf_set_nat @ ( vimage6104394758324456170nteger @ F @ ( insert_Code_integer @ Y2 @ bot_bo3990330152332043303nteger ) ) @ A ) ) ) ) ) ).
% 14.25/8.04
% 14.25/8.04 thf(fact_2302_inf__img__fin__domE_H,axiom,
% 14.25/8.04 ! [F: code_integer > product_unit,A: set_Code_integer] :
% 14.25/8.04 ( ( finite4290736615968046902t_unit @ ( image_7182656606676592809t_unit @ F @ A ) )
% 14.25/8.04 => ( ~ ( finite6017078050557962740nteger @ A )
% 14.25/8.04 => ~ ! [Y2: product_unit] :
% 14.25/8.04 ( ( member_Product_unit @ Y2 @ ( image_7182656606676592809t_unit @ F @ A ) )
% 14.25/8.04 => ( finite6017078050557962740nteger @ ( inf_in1364745209274528805nteger @ ( vimage5651688891684837235t_unit @ F @ ( insert_Product_unit @ Y2 @ bot_bo3957492148770167129t_unit ) ) @ A ) ) ) ) ) ).
% 14.25/8.04
% 14.25/8.04 thf(fact_2303_inf__img__fin__domE_H,axiom,
% 14.25/8.04 ! [F: nat > product_unit,A: set_nat] :
% 14.25/8.04 ( ( finite4290736615968046902t_unit @ ( image_8730104196221521654t_unit @ F @ A ) )
% 14.25/8.04 => ( ~ ( finite_finite_nat @ A )
% 14.25/8.04 => ~ ! [Y2: product_unit] :
% 14.25/8.04 ( ( member_Product_unit @ Y2 @ ( image_8730104196221521654t_unit @ F @ A ) )
% 14.25/8.04 => ( finite_finite_nat @ ( inf_inf_set_nat @ ( vimage4884490618288580032t_unit @ F @ ( insert_Product_unit @ Y2 @ bot_bo3957492148770167129t_unit ) ) @ A ) ) ) ) ) ).
% 14.25/8.04
% 14.25/8.04 thf(fact_2304_inf__img__fin__domE_H,axiom,
% 14.25/8.04 ! [F: code_integer > $o,A: set_Code_integer] :
% 14.25/8.04 ( ( finite_finite_o @ ( image_Code_integer_o @ F @ A ) )
% 14.25/8.04 => ( ~ ( finite6017078050557962740nteger @ A )
% 14.25/8.04 => ~ ! [Y2: $o] :
% 14.25/8.04 ( ( member_o @ Y2 @ ( image_Code_integer_o @ F @ A ) )
% 14.25/8.04 => ( finite6017078050557962740nteger @ ( inf_in1364745209274528805nteger @ ( vimage3005800680117336574eger_o @ F @ ( insert_o @ Y2 @ bot_bot_set_o ) ) @ A ) ) ) ) ) ).
% 14.25/8.04
% 14.25/8.04 thf(fact_2305_inf__img__fin__domE_H,axiom,
% 14.25/8.04 ! [F: nat > $o,A: set_nat] :
% 14.25/8.04 ( ( finite_finite_o @ ( image_nat_o2 @ F @ A ) )
% 14.25/8.04 => ( ~ ( finite_finite_nat @ A )
% 14.25/8.04 => ~ ! [Y2: $o] :
% 14.25/8.04 ( ( member_o @ Y2 @ ( image_nat_o2 @ F @ A ) )
% 14.25/8.04 => ( finite_finite_nat @ ( inf_inf_set_nat @ ( vimage_nat_o @ F @ ( insert_o @ Y2 @ bot_bot_set_o ) ) @ A ) ) ) ) ) ).
% 14.25/8.04
% 14.25/8.04 thf(fact_2306_inf__img__fin__domE_H,axiom,
% 14.25/8.04 ! [F: code_integer > nat,A: set_Code_integer] :
% 14.25/8.04 ( ( finite_finite_nat @ ( image_951025933927791156er_nat @ F @ A ) )
% 14.25/8.04 => ( ~ ( finite6017078050557962740nteger @ A )
% 14.25/8.04 => ~ ! [Y2: nat] :
% 14.25/8.04 ( ( member_nat @ Y2 @ ( image_951025933927791156er_nat @ F @ A ) )
% 14.25/8.04 => ( finite6017078050557962740nteger @ ( inf_in1364745209274528805nteger @ ( vimage5839839309545413354er_nat @ F @ ( insert_nat @ Y2 @ bot_bot_set_nat ) ) @ A ) ) ) ) ) ).
% 14.25/8.04
% 14.25/8.04 thf(fact_2307_inf__img__fin__domE_H,axiom,
% 14.25/8.04 ! [F: nat > nat,A: set_nat] :
% 14.25/8.04 ( ( finite_finite_nat @ ( image_nat_nat2 @ F @ A ) )
% 14.25/8.04 => ( ~ ( finite_finite_nat @ A )
% 14.25/8.04 => ~ ! [Y2: nat] :
% 14.25/8.04 ( ( member_nat @ Y2 @ ( image_nat_nat2 @ F @ A ) )
% 14.25/8.04 => ( finite_finite_nat @ ( inf_inf_set_nat @ ( vimage_nat_nat @ F @ ( insert_nat @ Y2 @ bot_bot_set_nat ) ) @ A ) ) ) ) ) ).
% 14.25/8.04
% 14.25/8.04 thf(fact_2308_inf__img__fin__domE_H,axiom,
% 14.25/8.04 ! [F: code_integer > int,A: set_Code_integer] :
% 14.25/8.04 ( ( finite_finite_int @ ( image_948535463418740880er_int @ F @ A ) )
% 14.25/8.04 => ( ~ ( finite6017078050557962740nteger @ A )
% 14.25/8.04 => ~ ! [Y2: int] :
% 14.25/8.04 ( ( member_int @ Y2 @ ( image_948535463418740880er_int @ F @ A ) )
% 14.25/8.04 => ( finite6017078050557962740nteger @ ( inf_in1364745209274528805nteger @ ( vimage5837348839036363078er_int @ F @ ( insert_int @ Y2 @ bot_bot_set_int ) ) @ A ) ) ) ) ) ).
% 14.25/8.04
% 14.25/8.04 thf(fact_2309_inf__img__fin__domE_H,axiom,
% 14.25/8.04 ! [F: nat > int,A: set_nat] :
% 14.25/8.04 ( ( finite_finite_int @ ( image_nat_int2 @ F @ A ) )
% 14.25/8.04 => ( ~ ( finite_finite_nat @ A )
% 14.25/8.04 => ~ ! [Y2: int] :
% 14.25/8.04 ( ( member_int @ Y2 @ ( image_nat_int2 @ F @ A ) )
% 14.25/8.04 => ( finite_finite_nat @ ( inf_inf_set_nat @ ( vimage_nat_int @ F @ ( insert_int @ Y2 @ bot_bot_set_int ) ) @ A ) ) ) ) ) ).
% 14.25/8.04
% 14.25/8.04 thf(fact_2310_image__eqI,axiom,
% 14.25/8.04 ! [B: code_integer,F: code_integer > code_integer,X: code_integer,A: set_Code_integer] :
% 14.25/8.04 ( ( B
% 14.25/8.04 = ( F @ X ) )
% 14.25/8.04 => ( ( member_Code_integer @ X @ A )
% 14.25/8.04 => ( member_Code_integer @ B @ ( image_4470545334726330049nteger @ F @ A ) ) ) ) ).
% 14.25/8.04
% 14.25/8.04 thf(fact_2311_image__eqI,axiom,
% 14.25/8.04 ! [B: code_integer,F: nat > code_integer,X: nat,A: set_nat] :
% 14.25/8.04 ( ( B
% 14.25/8.04 = ( F @ X ) )
% 14.25/8.04 => ( ( member_nat @ X @ A )
% 14.25/8.04 => ( member_Code_integer @ B @ ( image_1215581382706833972nteger @ F @ A ) ) ) ) ).
% 14.25/8.04
% 14.25/8.04 thf(fact_2312_image__eqI,axiom,
% 14.25/8.04 ! [B: nat,F: nat > nat,X: nat,A: set_nat] :
% 14.25/8.04 ( ( B
% 14.25/8.04 = ( F @ X ) )
% 14.25/8.04 => ( ( member_nat @ X @ A )
% 14.25/8.04 => ( member_nat @ B @ ( image_nat_nat2 @ F @ A ) ) ) ) ).
% 14.25/8.04
% 14.25/8.04 thf(fact_2313_image__eqI,axiom,
% 14.25/8.04 ! [B: int,F: nat > int,X: nat,A: set_nat] :
% 14.25/8.04 ( ( B
% 14.25/8.04 = ( F @ X ) )
% 14.25/8.04 => ( ( member_nat @ X @ A )
% 14.25/8.04 => ( member_int @ B @ ( image_nat_int2 @ F @ A ) ) ) ) ).
% 14.25/8.04
% 14.25/8.04 thf(fact_2314_image__eqI,axiom,
% 14.25/8.04 ! [B: $o,F: nat > $o,X: nat,A: set_nat] :
% 14.25/8.04 ( ( B
% 14.25/8.04 = ( F @ X ) )
% 14.25/8.04 => ( ( member_nat @ X @ A )
% 14.25/8.04 => ( member_o @ B @ ( image_nat_o2 @ F @ A ) ) ) ) ).
% 14.25/8.04
% 14.25/8.04 thf(fact_2315_image__eqI,axiom,
% 14.25/8.04 ! [B: product_unit,F: nat > product_unit,X: nat,A: set_nat] :
% 14.25/8.04 ( ( B
% 14.25/8.04 = ( F @ X ) )
% 14.25/8.04 => ( ( member_nat @ X @ A )
% 14.25/8.04 => ( member_Product_unit @ B @ ( image_8730104196221521654t_unit @ F @ A ) ) ) ) ).
% 14.25/8.04
% 14.25/8.04 thf(fact_2316_image__eqI,axiom,
% 14.25/8.04 ! [B: nat,F: int > nat,X: int,A: set_int] :
% 14.25/8.04 ( ( B
% 14.25/8.04 = ( F @ X ) )
% 14.25/8.04 => ( ( member_int @ X @ A )
% 14.25/8.04 => ( member_nat @ B @ ( image_int_nat2 @ F @ A ) ) ) ) ).
% 14.25/8.04
% 14.25/8.04 thf(fact_2317_image__eqI,axiom,
% 14.25/8.04 ! [B: int,F: int > int,X: int,A: set_int] :
% 14.25/8.04 ( ( B
% 14.25/8.04 = ( F @ X ) )
% 14.25/8.04 => ( ( member_int @ X @ A )
% 14.25/8.04 => ( member_int @ B @ ( image_int_int2 @ F @ A ) ) ) ) ).
% 14.25/8.04
% 14.25/8.04 thf(fact_2318_image__eqI,axiom,
% 14.25/8.04 ! [B: $o,F: int > $o,X: int,A: set_int] :
% 14.25/8.04 ( ( B
% 14.25/8.04 = ( F @ X ) )
% 14.25/8.04 => ( ( member_int @ X @ A )
% 14.25/8.04 => ( member_o @ B @ ( image_int_o2 @ F @ A ) ) ) ) ).
% 14.25/8.04
% 14.25/8.04 thf(fact_2319_image__eqI,axiom,
% 14.25/8.04 ! [B: product_unit,F: int > product_unit,X: int,A: set_int] :
% 14.25/8.04 ( ( B
% 14.25/8.04 = ( F @ X ) )
% 14.25/8.04 => ( ( member_int @ X @ A )
% 14.25/8.04 => ( member_Product_unit @ B @ ( image_5846526143423701914t_unit @ F @ A ) ) ) ) ).
% 14.25/8.04
% 14.25/8.04 thf(fact_2320_Diff__idemp,axiom,
% 14.25/8.04 ! [A: set_nat,B4: set_nat] :
% 14.25/8.04 ( ( minus_minus_set_nat @ ( minus_minus_set_nat @ A @ B4 ) @ B4 )
% 14.25/8.04 = ( minus_minus_set_nat @ A @ B4 ) ) ).
% 14.25/8.04
% 14.25/8.04 thf(fact_2321_Diff__iff,axiom,
% 14.25/8.04 ! [C: produc3658429121746597890et_nat > $o,A: set_Pr4532377907799695533_nat_o,B4: set_Pr4532377907799695533_nat_o] :
% 14.25/8.04 ( ( member6576561426505652726_nat_o @ C @ ( minus_1801376950450012436_nat_o @ A @ B4 ) )
% 14.25/8.04 = ( ( member6576561426505652726_nat_o @ C @ A )
% 14.25/8.04 & ~ ( member6576561426505652726_nat_o @ C @ B4 ) ) ) ).
% 14.25/8.04
% 14.25/8.04 thf(fact_2322_Diff__iff,axiom,
% 14.25/8.04 ! [C: int,A: set_int,B4: set_int] :
% 14.25/8.04 ( ( member_int @ C @ ( minus_minus_set_int @ A @ B4 ) )
% 14.25/8.04 = ( ( member_int @ C @ A )
% 14.25/8.04 & ~ ( member_int @ C @ B4 ) ) ) ).
% 14.25/8.04
% 14.25/8.04 thf(fact_2323_Diff__iff,axiom,
% 14.25/8.04 ! [C: $o,A: set_o,B4: set_o] :
% 14.25/8.04 ( ( member_o @ C @ ( minus_minus_set_o @ A @ B4 ) )
% 14.25/8.04 = ( ( member_o @ C @ A )
% 14.25/8.04 & ~ ( member_o @ C @ B4 ) ) ) ).
% 14.25/8.04
% 14.25/8.04 thf(fact_2324_Diff__iff,axiom,
% 14.25/8.04 ! [C: product_unit,A: set_Product_unit,B4: set_Product_unit] :
% 14.25/8.04 ( ( member_Product_unit @ C @ ( minus_6452836326544984404t_unit @ A @ B4 ) )
% 14.25/8.04 = ( ( member_Product_unit @ C @ A )
% 14.25/8.04 & ~ ( member_Product_unit @ C @ B4 ) ) ) ).
% 14.25/8.04
% 14.25/8.04 thf(fact_2325_Diff__iff,axiom,
% 14.25/8.04 ! [C: nat,A: set_nat,B4: set_nat] :
% 14.25/8.04 ( ( member_nat @ C @ ( minus_minus_set_nat @ A @ B4 ) )
% 14.25/8.04 = ( ( member_nat @ C @ A )
% 14.25/8.04 & ~ ( member_nat @ C @ B4 ) ) ) ).
% 14.25/8.04
% 14.25/8.04 thf(fact_2326_DiffI,axiom,
% 14.25/8.04 ! [C: produc3658429121746597890et_nat > $o,A: set_Pr4532377907799695533_nat_o,B4: set_Pr4532377907799695533_nat_o] :
% 14.25/8.04 ( ( member6576561426505652726_nat_o @ C @ A )
% 14.25/8.04 => ( ~ ( member6576561426505652726_nat_o @ C @ B4 )
% 14.25/8.04 => ( member6576561426505652726_nat_o @ C @ ( minus_1801376950450012436_nat_o @ A @ B4 ) ) ) ) ).
% 14.25/8.04
% 14.25/8.04 thf(fact_2327_DiffI,axiom,
% 14.25/8.04 ! [C: int,A: set_int,B4: set_int] :
% 14.25/8.04 ( ( member_int @ C @ A )
% 14.25/8.04 => ( ~ ( member_int @ C @ B4 )
% 14.25/8.04 => ( member_int @ C @ ( minus_minus_set_int @ A @ B4 ) ) ) ) ).
% 14.25/8.04
% 14.25/8.04 thf(fact_2328_DiffI,axiom,
% 14.25/8.04 ! [C: $o,A: set_o,B4: set_o] :
% 14.25/8.04 ( ( member_o @ C @ A )
% 14.25/8.04 => ( ~ ( member_o @ C @ B4 )
% 14.25/8.04 => ( member_o @ C @ ( minus_minus_set_o @ A @ B4 ) ) ) ) ).
% 14.25/8.04
% 14.25/8.04 thf(fact_2329_DiffI,axiom,
% 14.25/8.04 ! [C: product_unit,A: set_Product_unit,B4: set_Product_unit] :
% 14.25/8.04 ( ( member_Product_unit @ C @ A )
% 14.25/8.04 => ( ~ ( member_Product_unit @ C @ B4 )
% 14.25/8.04 => ( member_Product_unit @ C @ ( minus_6452836326544984404t_unit @ A @ B4 ) ) ) ) ).
% 14.25/8.04
% 14.25/8.04 thf(fact_2330_DiffI,axiom,
% 14.25/8.04 ! [C: nat,A: set_nat,B4: set_nat] :
% 14.25/8.04 ( ( member_nat @ C @ A )
% 14.25/8.04 => ( ~ ( member_nat @ C @ B4 )
% 14.25/8.04 => ( member_nat @ C @ ( minus_minus_set_nat @ A @ B4 ) ) ) ) ).
% 14.25/8.04
% 14.25/8.04 thf(fact_2331_image__is__empty,axiom,
% 14.25/8.04 ! [F: code_integer > code_integer,A: set_Code_integer] :
% 14.25/8.04 ( ( ( image_4470545334726330049nteger @ F @ A )
% 14.25/8.04 = bot_bo3990330152332043303nteger )
% 14.25/8.04 = ( A = bot_bo3990330152332043303nteger ) ) ).
% 14.25/8.04
% 14.25/8.04 thf(fact_2332_image__is__empty,axiom,
% 14.25/8.04 ! [F: nat > code_integer,A: set_nat] :
% 14.25/8.04 ( ( ( image_1215581382706833972nteger @ F @ A )
% 14.25/8.04 = bot_bo3990330152332043303nteger )
% 14.25/8.04 = ( A = bot_bot_set_nat ) ) ).
% 14.25/8.04
% 14.25/8.04 thf(fact_2333_image__is__empty,axiom,
% 14.25/8.04 ! [F: product_unit > product_unit,A: set_Product_unit] :
% 14.25/8.04 ( ( ( image_405062704495631173t_unit @ F @ A )
% 14.25/8.04 = bot_bo3957492148770167129t_unit )
% 14.25/8.04 = ( A = bot_bo3957492148770167129t_unit ) ) ).
% 14.25/8.04
% 14.25/8.04 thf(fact_2334_image__is__empty,axiom,
% 14.25/8.04 ! [F: $o > product_unit,A: set_o] :
% 14.25/8.04 ( ( ( image_o_Product_unit2 @ F @ A )
% 14.25/8.04 = bot_bo3957492148770167129t_unit )
% 14.25/8.04 = ( A = bot_bot_set_o ) ) ).
% 14.25/8.04
% 14.25/8.04 thf(fact_2335_image__is__empty,axiom,
% 14.25/8.04 ! [F: nat > product_unit,A: set_nat] :
% 14.25/8.04 ( ( ( image_8730104196221521654t_unit @ F @ A )
% 14.25/8.04 = bot_bo3957492148770167129t_unit )
% 14.25/8.04 = ( A = bot_bot_set_nat ) ) ).
% 14.25/8.04
% 14.25/8.04 thf(fact_2336_image__is__empty,axiom,
% 14.25/8.04 ! [F: int > product_unit,A: set_int] :
% 14.25/8.04 ( ( ( image_5846526143423701914t_unit @ F @ A )
% 14.25/8.04 = bot_bo3957492148770167129t_unit )
% 14.25/8.04 = ( A = bot_bot_set_int ) ) ).
% 14.25/8.04
% 14.25/8.04 thf(fact_2337_image__is__empty,axiom,
% 14.25/8.04 ! [F: product_unit > $o,A: set_Product_unit] :
% 14.25/8.04 ( ( ( image_Product_unit_o2 @ F @ A )
% 14.25/8.04 = bot_bot_set_o )
% 14.25/8.04 = ( A = bot_bo3957492148770167129t_unit ) ) ).
% 14.25/8.04
% 14.25/8.04 thf(fact_2338_image__is__empty,axiom,
% 14.25/8.04 ! [F: $o > $o,A: set_o] :
% 14.25/8.04 ( ( ( image_o_o2 @ F @ A )
% 14.25/8.04 = bot_bot_set_o )
% 14.25/8.04 = ( A = bot_bot_set_o ) ) ).
% 14.25/8.04
% 14.25/8.04 thf(fact_2339_image__is__empty,axiom,
% 14.25/8.04 ! [F: nat > $o,A: set_nat] :
% 14.25/8.04 ( ( ( image_nat_o2 @ F @ A )
% 14.25/8.04 = bot_bot_set_o )
% 14.25/8.04 = ( A = bot_bot_set_nat ) ) ).
% 14.25/8.04
% 14.25/8.04 thf(fact_2340_image__is__empty,axiom,
% 14.25/8.04 ! [F: int > $o,A: set_int] :
% 14.25/8.04 ( ( ( image_int_o2 @ F @ A )
% 14.25/8.04 = bot_bot_set_o )
% 14.25/8.04 = ( A = bot_bot_set_int ) ) ).
% 14.25/8.04
% 14.25/8.04 thf(fact_2341_empty__is__image,axiom,
% 14.25/8.04 ! [F: code_integer > code_integer,A: set_Code_integer] :
% 14.25/8.04 ( ( bot_bo3990330152332043303nteger
% 14.25/8.04 = ( image_4470545334726330049nteger @ F @ A ) )
% 14.25/8.04 = ( A = bot_bo3990330152332043303nteger ) ) ).
% 14.25/8.04
% 14.25/8.04 thf(fact_2342_empty__is__image,axiom,
% 14.25/8.04 ! [F: nat > code_integer,A: set_nat] :
% 14.25/8.04 ( ( bot_bo3990330152332043303nteger
% 14.25/8.04 = ( image_1215581382706833972nteger @ F @ A ) )
% 14.25/8.04 = ( A = bot_bot_set_nat ) ) ).
% 14.25/8.04
% 14.25/8.04 thf(fact_2343_empty__is__image,axiom,
% 14.25/8.04 ! [F: product_unit > product_unit,A: set_Product_unit] :
% 14.25/8.04 ( ( bot_bo3957492148770167129t_unit
% 14.25/8.04 = ( image_405062704495631173t_unit @ F @ A ) )
% 14.25/8.04 = ( A = bot_bo3957492148770167129t_unit ) ) ).
% 14.25/8.04
% 14.25/8.04 thf(fact_2344_empty__is__image,axiom,
% 14.25/8.04 ! [F: $o > product_unit,A: set_o] :
% 14.25/8.04 ( ( bot_bo3957492148770167129t_unit
% 14.25/8.04 = ( image_o_Product_unit2 @ F @ A ) )
% 14.25/8.04 = ( A = bot_bot_set_o ) ) ).
% 14.25/8.04
% 14.25/8.04 thf(fact_2345_empty__is__image,axiom,
% 14.25/8.04 ! [F: nat > product_unit,A: set_nat] :
% 14.25/8.04 ( ( bot_bo3957492148770167129t_unit
% 14.25/8.04 = ( image_8730104196221521654t_unit @ F @ A ) )
% 14.25/8.04 = ( A = bot_bot_set_nat ) ) ).
% 14.25/8.04
% 14.25/8.04 thf(fact_2346_empty__is__image,axiom,
% 14.25/8.04 ! [F: int > product_unit,A: set_int] :
% 14.25/8.04 ( ( bot_bo3957492148770167129t_unit
% 14.25/8.04 = ( image_5846526143423701914t_unit @ F @ A ) )
% 14.25/8.04 = ( A = bot_bot_set_int ) ) ).
% 14.25/8.04
% 14.25/8.04 thf(fact_2347_empty__is__image,axiom,
% 14.25/8.04 ! [F: product_unit > $o,A: set_Product_unit] :
% 14.25/8.04 ( ( bot_bot_set_o
% 14.25/8.04 = ( image_Product_unit_o2 @ F @ A ) )
% 14.25/8.04 = ( A = bot_bo3957492148770167129t_unit ) ) ).
% 14.25/8.04
% 14.25/8.04 thf(fact_2348_empty__is__image,axiom,
% 14.25/8.04 ! [F: $o > $o,A: set_o] :
% 14.25/8.04 ( ( bot_bot_set_o
% 14.25/8.04 = ( image_o_o2 @ F @ A ) )
% 14.25/8.04 = ( A = bot_bot_set_o ) ) ).
% 14.25/8.04
% 14.25/8.04 thf(fact_2349_empty__is__image,axiom,
% 14.25/8.04 ! [F: nat > $o,A: set_nat] :
% 14.25/8.04 ( ( bot_bot_set_o
% 14.25/8.04 = ( image_nat_o2 @ F @ A ) )
% 14.25/8.04 = ( A = bot_bot_set_nat ) ) ).
% 14.25/8.04
% 14.25/8.04 thf(fact_2350_empty__is__image,axiom,
% 14.25/8.04 ! [F: int > $o,A: set_int] :
% 14.25/8.04 ( ( bot_bot_set_o
% 14.25/8.04 = ( image_int_o2 @ F @ A ) )
% 14.25/8.04 = ( A = bot_bot_set_int ) ) ).
% 14.25/8.04
% 14.25/8.04 thf(fact_2351_image__empty,axiom,
% 14.25/8.04 ! [F: code_integer > code_integer] :
% 14.25/8.04 ( ( image_4470545334726330049nteger @ F @ bot_bo3990330152332043303nteger )
% 14.25/8.04 = bot_bo3990330152332043303nteger ) ).
% 14.25/8.04
% 14.25/8.04 thf(fact_2352_image__empty,axiom,
% 14.25/8.04 ! [F: product_unit > product_unit] :
% 14.25/8.04 ( ( image_405062704495631173t_unit @ F @ bot_bo3957492148770167129t_unit )
% 14.25/8.04 = bot_bo3957492148770167129t_unit ) ).
% 14.25/8.04
% 14.25/8.04 thf(fact_2353_image__empty,axiom,
% 14.25/8.04 ! [F: product_unit > $o] :
% 14.25/8.04 ( ( image_Product_unit_o2 @ F @ bot_bo3957492148770167129t_unit )
% 14.25/8.04 = bot_bot_set_o ) ).
% 14.25/8.04
% 14.25/8.04 thf(fact_2354_image__empty,axiom,
% 14.25/8.04 ! [F: product_unit > nat] :
% 14.25/8.04 ( ( image_875570014554754200it_nat @ F @ bot_bo3957492148770167129t_unit )
% 14.25/8.04 = bot_bot_set_nat ) ).
% 14.25/8.04
% 14.25/8.04 thf(fact_2355_image__empty,axiom,
% 14.25/8.04 ! [F: product_unit > int] :
% 14.25/8.04 ( ( image_873079544045703924it_int @ F @ bot_bo3957492148770167129t_unit )
% 14.25/8.04 = bot_bot_set_int ) ).
% 14.25/8.04
% 14.25/8.04 thf(fact_2356_image__empty,axiom,
% 14.25/8.04 ! [F: $o > product_unit] :
% 14.25/8.04 ( ( image_o_Product_unit2 @ F @ bot_bot_set_o )
% 14.25/8.04 = bot_bo3957492148770167129t_unit ) ).
% 14.25/8.04
% 14.25/8.04 thf(fact_2357_image__empty,axiom,
% 14.25/8.04 ! [F: $o > $o] :
% 14.25/8.04 ( ( image_o_o2 @ F @ bot_bot_set_o )
% 14.25/8.04 = bot_bot_set_o ) ).
% 14.25/8.04
% 14.25/8.04 thf(fact_2358_image__empty,axiom,
% 14.25/8.04 ! [F: $o > nat] :
% 14.25/8.04 ( ( image_o_nat2 @ F @ bot_bot_set_o )
% 14.25/8.04 = bot_bot_set_nat ) ).
% 14.25/8.04
% 14.25/8.04 thf(fact_2359_image__empty,axiom,
% 14.25/8.04 ! [F: $o > int] :
% 14.25/8.04 ( ( image_o_int2 @ F @ bot_bot_set_o )
% 14.25/8.04 = bot_bot_set_int ) ).
% 14.25/8.04
% 14.25/8.04 thf(fact_2360_image__empty,axiom,
% 14.25/8.04 ! [F: nat > code_integer] :
% 14.25/8.04 ( ( image_1215581382706833972nteger @ F @ bot_bot_set_nat )
% 14.25/8.04 = bot_bo3990330152332043303nteger ) ).
% 14.25/8.04
% 14.25/8.04 thf(fact_2361_insert__image,axiom,
% 14.25/8.04 ! [X: code_integer,A: set_Code_integer,F: code_integer > code_integer] :
% 14.25/8.04 ( ( member_Code_integer @ X @ A )
% 14.25/8.04 => ( ( insert_Code_integer @ ( F @ X ) @ ( image_4470545334726330049nteger @ F @ A ) )
% 14.25/8.04 = ( image_4470545334726330049nteger @ F @ A ) ) ) ).
% 14.25/8.04
% 14.25/8.04 thf(fact_2362_insert__image,axiom,
% 14.25/8.04 ! [X: nat,A: set_nat,F: nat > code_integer] :
% 14.25/8.04 ( ( member_nat @ X @ A )
% 14.25/8.04 => ( ( insert_Code_integer @ ( F @ X ) @ ( image_1215581382706833972nteger @ F @ A ) )
% 14.25/8.04 = ( image_1215581382706833972nteger @ F @ A ) ) ) ).
% 14.25/8.04
% 14.25/8.04 thf(fact_2363_insert__image,axiom,
% 14.25/8.04 ! [X: nat,A: set_nat,F: nat > nat] :
% 14.25/8.04 ( ( member_nat @ X @ A )
% 14.25/8.04 => ( ( insert_nat @ ( F @ X ) @ ( image_nat_nat2 @ F @ A ) )
% 14.25/8.04 = ( image_nat_nat2 @ F @ A ) ) ) ).
% 14.25/8.04
% 14.25/8.04 thf(fact_2364_insert__image,axiom,
% 14.25/8.04 ! [X: nat,A: set_nat,F: nat > int] :
% 14.25/8.04 ( ( member_nat @ X @ A )
% 14.25/8.04 => ( ( insert_int @ ( F @ X ) @ ( image_nat_int2 @ F @ A ) )
% 14.25/8.04 = ( image_nat_int2 @ F @ A ) ) ) ).
% 14.25/8.04
% 14.25/8.04 thf(fact_2365_insert__image,axiom,
% 14.25/8.04 ! [X: nat,A: set_nat,F: nat > $o] :
% 14.25/8.04 ( ( member_nat @ X @ A )
% 14.25/8.04 => ( ( insert_o @ ( F @ X ) @ ( image_nat_o2 @ F @ A ) )
% 14.25/8.04 = ( image_nat_o2 @ F @ A ) ) ) ).
% 14.25/8.04
% 14.25/8.04 thf(fact_2366_insert__image,axiom,
% 14.25/8.04 ! [X: nat,A: set_nat,F: nat > product_unit] :
% 14.25/8.04 ( ( member_nat @ X @ A )
% 14.25/8.04 => ( ( insert_Product_unit @ ( F @ X ) @ ( image_8730104196221521654t_unit @ F @ A ) )
% 14.25/8.04 = ( image_8730104196221521654t_unit @ F @ A ) ) ) ).
% 14.25/8.04
% 14.25/8.04 thf(fact_2367_insert__image,axiom,
% 14.25/8.04 ! [X: int,A: set_int,F: int > nat] :
% 14.25/8.04 ( ( member_int @ X @ A )
% 14.25/8.04 => ( ( insert_nat @ ( F @ X ) @ ( image_int_nat2 @ F @ A ) )
% 14.25/8.04 = ( image_int_nat2 @ F @ A ) ) ) ).
% 14.25/8.04
% 14.25/8.04 thf(fact_2368_insert__image,axiom,
% 14.25/8.04 ! [X: int,A: set_int,F: int > int] :
% 14.25/8.04 ( ( member_int @ X @ A )
% 14.25/8.04 => ( ( insert_int @ ( F @ X ) @ ( image_int_int2 @ F @ A ) )
% 14.25/8.04 = ( image_int_int2 @ F @ A ) ) ) ).
% 14.25/8.04
% 14.25/8.04 thf(fact_2369_insert__image,axiom,
% 14.25/8.04 ! [X: int,A: set_int,F: int > $o] :
% 14.25/8.04 ( ( member_int @ X @ A )
% 14.25/8.04 => ( ( insert_o @ ( F @ X ) @ ( image_int_o2 @ F @ A ) )
% 14.25/8.04 = ( image_int_o2 @ F @ A ) ) ) ).
% 14.25/8.04
% 14.25/8.04 thf(fact_2370_insert__image,axiom,
% 14.25/8.04 ! [X: int,A: set_int,F: int > product_unit] :
% 14.25/8.04 ( ( member_int @ X @ A )
% 14.25/8.04 => ( ( insert_Product_unit @ ( F @ X ) @ ( image_5846526143423701914t_unit @ F @ A ) )
% 14.25/8.04 = ( image_5846526143423701914t_unit @ F @ A ) ) ) ).
% 14.25/8.04
% 14.25/8.04 thf(fact_2371_image__insert,axiom,
% 14.25/8.04 ! [F: code_integer > code_integer,A2: code_integer,B4: set_Code_integer] :
% 14.25/8.04 ( ( image_4470545334726330049nteger @ F @ ( insert_Code_integer @ A2 @ B4 ) )
% 14.25/8.04 = ( insert_Code_integer @ ( F @ A2 ) @ ( image_4470545334726330049nteger @ F @ B4 ) ) ) ).
% 14.25/8.04
% 14.25/8.04 thf(fact_2372_image__insert,axiom,
% 14.25/8.04 ! [F: nat > code_integer,A2: nat,B4: set_nat] :
% 14.25/8.04 ( ( image_1215581382706833972nteger @ F @ ( insert_nat @ A2 @ B4 ) )
% 14.25/8.04 = ( insert_Code_integer @ ( F @ A2 ) @ ( image_1215581382706833972nteger @ F @ B4 ) ) ) ).
% 14.25/8.04
% 14.25/8.04 thf(fact_2373_image__insert,axiom,
% 14.25/8.04 ! [F: nat > nat,A2: nat,B4: set_nat] :
% 14.25/8.04 ( ( image_nat_nat2 @ F @ ( insert_nat @ A2 @ B4 ) )
% 14.25/8.04 = ( insert_nat @ ( F @ A2 ) @ ( image_nat_nat2 @ F @ B4 ) ) ) ).
% 14.25/8.04
% 14.25/8.04 thf(fact_2374_image__insert,axiom,
% 14.25/8.04 ! [F: nat > int,A2: nat,B4: set_nat] :
% 14.25/8.04 ( ( image_nat_int2 @ F @ ( insert_nat @ A2 @ B4 ) )
% 14.25/8.04 = ( insert_int @ ( F @ A2 ) @ ( image_nat_int2 @ F @ B4 ) ) ) ).
% 14.25/8.04
% 14.25/8.04 thf(fact_2375_image__insert,axiom,
% 14.25/8.04 ! [F: nat > $o,A2: nat,B4: set_nat] :
% 14.25/8.04 ( ( image_nat_o2 @ F @ ( insert_nat @ A2 @ B4 ) )
% 14.25/8.04 = ( insert_o @ ( F @ A2 ) @ ( image_nat_o2 @ F @ B4 ) ) ) ).
% 14.25/8.04
% 14.25/8.04 thf(fact_2376_image__insert,axiom,
% 14.25/8.04 ! [F: nat > product_unit,A2: nat,B4: set_nat] :
% 14.25/8.04 ( ( image_8730104196221521654t_unit @ F @ ( insert_nat @ A2 @ B4 ) )
% 14.25/8.04 = ( insert_Product_unit @ ( F @ A2 ) @ ( image_8730104196221521654t_unit @ F @ B4 ) ) ) ).
% 14.25/8.04
% 14.25/8.04 thf(fact_2377_image__insert,axiom,
% 14.25/8.04 ! [F: int > nat,A2: int,B4: set_int] :
% 14.25/8.04 ( ( image_int_nat2 @ F @ ( insert_int @ A2 @ B4 ) )
% 14.25/8.04 = ( insert_nat @ ( F @ A2 ) @ ( image_int_nat2 @ F @ B4 ) ) ) ).
% 14.25/8.04
% 14.25/8.04 thf(fact_2378_image__insert,axiom,
% 14.25/8.04 ! [F: int > int,A2: int,B4: set_int] :
% 14.25/8.04 ( ( image_int_int2 @ F @ ( insert_int @ A2 @ B4 ) )
% 14.25/8.04 = ( insert_int @ ( F @ A2 ) @ ( image_int_int2 @ F @ B4 ) ) ) ).
% 14.25/8.04
% 14.25/8.04 thf(fact_2379_image__insert,axiom,
% 14.25/8.04 ! [F: int > $o,A2: int,B4: set_int] :
% 14.25/8.04 ( ( image_int_o2 @ F @ ( insert_int @ A2 @ B4 ) )
% 14.25/8.04 = ( insert_o @ ( F @ A2 ) @ ( image_int_o2 @ F @ B4 ) ) ) ).
% 14.25/8.04
% 14.25/8.04 thf(fact_2380_image__insert,axiom,
% 14.25/8.04 ! [F: int > product_unit,A2: int,B4: set_int] :
% 14.25/8.04 ( ( image_5846526143423701914t_unit @ F @ ( insert_int @ A2 @ B4 ) )
% 14.25/8.04 = ( insert_Product_unit @ ( F @ A2 ) @ ( image_5846526143423701914t_unit @ F @ B4 ) ) ) ).
% 14.25/8.04
% 14.25/8.04 thf(fact_2381_img__snd,axiom,
% 14.25/8.04 ! [A2: nat,B: nat,S: set_Pr1261947904930325089at_nat] :
% 14.25/8.04 ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A2 @ B ) @ S )
% 14.25/8.04 => ( member_nat @ B @ ( image_2486076414777270412at_nat @ product_snd_nat_nat @ S ) ) ) ).
% 14.25/8.04
% 14.25/8.04 thf(fact_2382_img__snd,axiom,
% 14.25/8.04 ! [A2: produc3658429121746597890et_nat > $o,B: produc3658429121746597890et_nat,S: set_Pr3286484037609594932et_nat] :
% 14.25/8.04 ( ( member1996754912294343701et_nat @ ( produc5001842942810119800et_nat @ A2 @ B ) @ S )
% 14.25/8.04 => ( member6260224972018164377et_nat @ B @ ( image_4390311411407041927et_nat @ produc4011572625026189258et_nat @ S ) ) ) ).
% 14.25/8.04
% 14.25/8.04 thf(fact_2383_img__snd,axiom,
% 14.25/8.04 ! [A2: produc3658429121746597890et_nat > $o,B: produc3925858234332021118et_nat,S: set_Pr8536935166611901872et_nat] :
% 14.25/8.04 ( ( member6124377750444531601et_nat @ ( produc2245416461498447860et_nat @ A2 @ B ) @ S )
% 14.25/8.04 => ( member1996754912294343701et_nat @ B @ ( image_4557310310409863807et_nat @ produc5374455773327741254et_nat @ S ) ) ) ).
% 14.25/8.04
% 14.25/8.04 thf(fact_2384_img__snd,axiom,
% 14.25/8.04 ! [A2: heap_e7401611519738050253t_unit,B: set_nat,S: set_Pr3948176798113811640et_nat] :
% 14.25/8.04 ( ( member6260224972018164377et_nat @ ( produc7507926704131184380et_nat @ A2 @ B ) @ S )
% 14.25/8.04 => ( member_set_nat @ B @ ( image_6088997261112373647et_nat @ produc8586169260539613262et_nat @ S ) ) ) ).
% 14.25/8.04
% 14.25/8.04 thf(fact_2385_img__snd,axiom,
% 14.25/8.04 ! [A2: int,B: int,S: set_Pr958786334691620121nt_int] :
% 14.25/8.04 ( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ A2 @ B ) @ S )
% 14.25/8.04 => ( member_int @ B @ ( image_5042161079198086560nt_int @ product_snd_int_int @ S ) ) ) ).
% 14.25/8.04
% 14.25/8.04 thf(fact_2386_Diff__cancel,axiom,
% 14.25/8.04 ! [A: set_Product_unit] :
% 14.25/8.04 ( ( minus_6452836326544984404t_unit @ A @ A )
% 14.25/8.04 = bot_bo3957492148770167129t_unit ) ).
% 14.25/8.04
% 14.25/8.04 thf(fact_2387_Diff__cancel,axiom,
% 14.25/8.04 ! [A: set_o] :
% 14.25/8.04 ( ( minus_minus_set_o @ A @ A )
% 14.25/8.04 = bot_bot_set_o ) ).
% 14.25/8.04
% 14.25/8.04 thf(fact_2388_Diff__cancel,axiom,
% 14.25/8.04 ! [A: set_int] :
% 14.25/8.04 ( ( minus_minus_set_int @ A @ A )
% 14.25/8.04 = bot_bot_set_int ) ).
% 14.25/8.04
% 14.25/8.04 thf(fact_2389_Diff__cancel,axiom,
% 14.25/8.04 ! [A: set_Pr4532377907799695533_nat_o] :
% 14.25/8.04 ( ( minus_1801376950450012436_nat_o @ A @ A )
% 14.25/8.04 = bot_bo7824918357723582233_nat_o ) ).
% 14.25/8.04
% 14.25/8.04 thf(fact_2390_Diff__cancel,axiom,
% 14.25/8.04 ! [A: set_nat] :
% 14.25/8.04 ( ( minus_minus_set_nat @ A @ A )
% 14.25/8.04 = bot_bot_set_nat ) ).
% 14.25/8.04
% 14.25/8.04 thf(fact_2391_empty__Diff,axiom,
% 14.25/8.04 ! [A: set_Product_unit] :
% 14.25/8.04 ( ( minus_6452836326544984404t_unit @ bot_bo3957492148770167129t_unit @ A )
% 14.25/8.04 = bot_bo3957492148770167129t_unit ) ).
% 14.25/8.04
% 14.25/8.04 thf(fact_2392_empty__Diff,axiom,
% 14.25/8.04 ! [A: set_o] :
% 14.25/8.04 ( ( minus_minus_set_o @ bot_bot_set_o @ A )
% 14.25/8.04 = bot_bot_set_o ) ).
% 14.25/8.04
% 14.25/8.04 thf(fact_2393_empty__Diff,axiom,
% 14.25/8.04 ! [A: set_int] :
% 14.25/8.04 ( ( minus_minus_set_int @ bot_bot_set_int @ A )
% 14.25/8.04 = bot_bot_set_int ) ).
% 14.25/8.04
% 14.25/8.04 thf(fact_2394_empty__Diff,axiom,
% 14.25/8.04 ! [A: set_Pr4532377907799695533_nat_o] :
% 14.25/8.04 ( ( minus_1801376950450012436_nat_o @ bot_bo7824918357723582233_nat_o @ A )
% 14.25/8.04 = bot_bo7824918357723582233_nat_o ) ).
% 14.25/8.04
% 14.25/8.04 thf(fact_2395_empty__Diff,axiom,
% 14.25/8.04 ! [A: set_nat] :
% 14.25/8.04 ( ( minus_minus_set_nat @ bot_bot_set_nat @ A )
% 14.25/8.04 = bot_bot_set_nat ) ).
% 14.25/8.04
% 14.25/8.04 thf(fact_2396_Diff__empty,axiom,
% 14.25/8.04 ! [A: set_Product_unit] :
% 14.25/8.04 ( ( minus_6452836326544984404t_unit @ A @ bot_bo3957492148770167129t_unit )
% 14.25/8.04 = A ) ).
% 14.25/8.04
% 14.25/8.04 thf(fact_2397_Diff__empty,axiom,
% 14.25/8.04 ! [A: set_o] :
% 14.25/8.04 ( ( minus_minus_set_o @ A @ bot_bot_set_o )
% 14.25/8.04 = A ) ).
% 14.25/8.04
% 14.25/8.04 thf(fact_2398_Diff__empty,axiom,
% 14.25/8.04 ! [A: set_int] :
% 14.25/8.04 ( ( minus_minus_set_int @ A @ bot_bot_set_int )
% 14.25/8.04 = A ) ).
% 14.25/8.04
% 14.25/8.04 thf(fact_2399_Diff__empty,axiom,
% 14.25/8.04 ! [A: set_Pr4532377907799695533_nat_o] :
% 14.25/8.04 ( ( minus_1801376950450012436_nat_o @ A @ bot_bo7824918357723582233_nat_o )
% 14.25/8.04 = A ) ).
% 14.25/8.04
% 14.25/8.04 thf(fact_2400_Diff__empty,axiom,
% 14.25/8.04 ! [A: set_nat] :
% 14.25/8.04 ( ( minus_minus_set_nat @ A @ bot_bot_set_nat )
% 14.25/8.04 = A ) ).
% 14.25/8.04
% 14.25/8.04 thf(fact_2401_insert__Diff1,axiom,
% 14.25/8.04 ! [X: produc3658429121746597890et_nat > $o,B4: set_Pr4532377907799695533_nat_o,A: set_Pr4532377907799695533_nat_o] :
% 14.25/8.04 ( ( member6576561426505652726_nat_o @ X @ B4 )
% 14.25/8.04 => ( ( minus_1801376950450012436_nat_o @ ( insert5175938949040314269_nat_o @ X @ A ) @ B4 )
% 14.25/8.04 = ( minus_1801376950450012436_nat_o @ A @ B4 ) ) ) ).
% 14.25/8.04
% 14.25/8.04 thf(fact_2402_insert__Diff1,axiom,
% 14.25/8.04 ! [X: int,B4: set_int,A: set_int] :
% 14.25/8.04 ( ( member_int @ X @ B4 )
% 14.25/8.04 => ( ( minus_minus_set_int @ ( insert_int @ X @ A ) @ B4 )
% 14.25/8.04 = ( minus_minus_set_int @ A @ B4 ) ) ) ).
% 14.25/8.04
% 14.25/8.04 thf(fact_2403_insert__Diff1,axiom,
% 14.25/8.04 ! [X: $o,B4: set_o,A: set_o] :
% 14.25/8.04 ( ( member_o @ X @ B4 )
% 14.25/8.04 => ( ( minus_minus_set_o @ ( insert_o @ X @ A ) @ B4 )
% 14.25/8.04 = ( minus_minus_set_o @ A @ B4 ) ) ) ).
% 14.25/8.04
% 14.25/8.04 thf(fact_2404_insert__Diff1,axiom,
% 14.25/8.04 ! [X: product_unit,B4: set_Product_unit,A: set_Product_unit] :
% 14.25/8.04 ( ( member_Product_unit @ X @ B4 )
% 14.25/8.04 => ( ( minus_6452836326544984404t_unit @ ( insert_Product_unit @ X @ A ) @ B4 )
% 14.25/8.04 = ( minus_6452836326544984404t_unit @ A @ B4 ) ) ) ).
% 14.25/8.04
% 14.25/8.04 thf(fact_2405_insert__Diff1,axiom,
% 14.25/8.04 ! [X: nat,B4: set_nat,A: set_nat] :
% 14.25/8.04 ( ( member_nat @ X @ B4 )
% 14.25/8.04 => ( ( minus_minus_set_nat @ ( insert_nat @ X @ A ) @ B4 )
% 14.25/8.04 = ( minus_minus_set_nat @ A @ B4 ) ) ) ).
% 14.25/8.04
% 14.25/8.04 thf(fact_2406_Diff__insert0,axiom,
% 14.25/8.04 ! [X: produc3658429121746597890et_nat > $o,A: set_Pr4532377907799695533_nat_o,B4: set_Pr4532377907799695533_nat_o] :
% 14.25/8.04 ( ~ ( member6576561426505652726_nat_o @ X @ A )
% 14.25/8.04 => ( ( minus_1801376950450012436_nat_o @ A @ ( insert5175938949040314269_nat_o @ X @ B4 ) )
% 14.25/8.04 = ( minus_1801376950450012436_nat_o @ A @ B4 ) ) ) ).
% 14.25/8.04
% 14.25/8.04 thf(fact_2407_Diff__insert0,axiom,
% 14.25/8.04 ! [X: int,A: set_int,B4: set_int] :
% 14.25/8.04 ( ~ ( member_int @ X @ A )
% 14.25/8.04 => ( ( minus_minus_set_int @ A @ ( insert_int @ X @ B4 ) )
% 14.25/8.04 = ( minus_minus_set_int @ A @ B4 ) ) ) ).
% 14.25/8.04
% 14.25/8.04 thf(fact_2408_Diff__insert0,axiom,
% 14.25/8.04 ! [X: $o,A: set_o,B4: set_o] :
% 14.25/8.04 ( ~ ( member_o @ X @ A )
% 14.25/8.04 => ( ( minus_minus_set_o @ A @ ( insert_o @ X @ B4 ) )
% 14.25/8.04 = ( minus_minus_set_o @ A @ B4 ) ) ) ).
% 14.25/8.04
% 14.25/8.04 thf(fact_2409_Diff__insert0,axiom,
% 14.25/8.04 ! [X: product_unit,A: set_Product_unit,B4: set_Product_unit] :
% 14.25/8.04 ( ~ ( member_Product_unit @ X @ A )
% 14.25/8.04 => ( ( minus_6452836326544984404t_unit @ A @ ( insert_Product_unit @ X @ B4 ) )
% 14.25/8.04 = ( minus_6452836326544984404t_unit @ A @ B4 ) ) ) ).
% 14.25/8.04
% 14.25/8.04 thf(fact_2410_Diff__insert0,axiom,
% 14.25/8.04 ! [X: nat,A: set_nat,B4: set_nat] :
% 14.25/8.04 ( ~ ( member_nat @ X @ A )
% 14.25/8.04 => ( ( minus_minus_set_nat @ A @ ( insert_nat @ X @ B4 ) )
% 14.25/8.04 = ( minus_minus_set_nat @ A @ B4 ) ) ) ).
% 14.25/8.04
% 14.25/8.04 thf(fact_2411_Un__Diff__cancel2,axiom,
% 14.25/8.04 ! [B4: set_nat,A: set_nat] :
% 14.25/8.04 ( ( sup_sup_set_nat @ ( minus_minus_set_nat @ B4 @ A ) @ A )
% 14.25/8.04 = ( sup_sup_set_nat @ B4 @ A ) ) ).
% 14.25/8.04
% 14.25/8.04 thf(fact_2412_Un__Diff__cancel,axiom,
% 14.25/8.04 ! [A: set_nat,B4: set_nat] :
% 14.25/8.04 ( ( sup_sup_set_nat @ A @ ( minus_minus_set_nat @ B4 @ A ) )
% 14.25/8.04 = ( sup_sup_set_nat @ A @ B4 ) ) ).
% 14.25/8.04
% 14.25/8.04 thf(fact_2413_converse__iff,axiom,
% 14.25/8.04 ! [A2: set_nat,B: heap_e7401611519738050253t_unit,R2: set_Pr3948176798113811640et_nat] :
% 14.25/8.04 ( ( member727419437372278867t_unit @ ( produc2471974422597015918t_unit @ A2 @ B ) @ ( conver7272282955447398104et_nat @ R2 ) )
% 14.25/8.04 = ( member6260224972018164377et_nat @ ( produc7507926704131184380et_nat @ B @ A2 ) @ R2 ) ) ).
% 14.25/8.04
% 14.25/8.04 thf(fact_2414_converse__iff,axiom,
% 14.25/8.04 ! [A2: produc3658429121746597890et_nat,B: produc3658429121746597890et_nat > $o,R2: set_Pr3286484037609594932et_nat] :
% 14.25/8.04 ( ( member4978015717948814295_nat_o @ ( produc2829238748666232050_nat_o @ A2 @ B ) @ ( conver4003969491454187092et_nat @ R2 ) )
% 14.25/8.04 = ( member1996754912294343701et_nat @ ( produc5001842942810119800et_nat @ B @ A2 ) @ R2 ) ) ).
% 14.25/8.04
% 14.25/8.04 thf(fact_2415_converse__iff,axiom,
% 14.25/8.04 ! [A2: produc3925858234332021118et_nat,B: produc3658429121746597890et_nat > $o,R2: set_Pr8536935166611901872et_nat] :
% 14.25/8.04 ( ( member8354763867512313179_nat_o @ ( produc3674526432113299062_nat_o @ A2 @ B ) @ ( conver3577387295554834384et_nat @ R2 ) )
% 14.25/8.04 = ( member6124377750444531601et_nat @ ( produc2245416461498447860et_nat @ B @ A2 ) @ R2 ) ) ).
% 14.25/8.04
% 14.25/8.04 thf(fact_2416_converse__iff,axiom,
% 14.25/8.04 ! [A2: heap_e7401611519738050253t_unit,B: set_nat,R2: set_Pr4071100604818519410t_unit] :
% 14.25/8.04 ( ( member6260224972018164377et_nat @ ( produc7507926704131184380et_nat @ A2 @ B ) @ ( conver2236330673913229642t_unit @ R2 ) )
% 14.25/8.04 = ( member727419437372278867t_unit @ ( produc2471974422597015918t_unit @ B @ A2 ) @ R2 ) ) ).
% 14.25/8.04
% 14.25/8.04 thf(fact_2417_converse__iff,axiom,
% 14.25/8.04 ! [A2: nat,B: nat,R2: set_Pr1261947904930325089at_nat] :
% 14.25/8.04 ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A2 @ B ) @ ( converse_nat_nat @ R2 ) )
% 14.25/8.04 = ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ B @ A2 ) @ R2 ) ) ).
% 14.25/8.04
% 14.25/8.04 thf(fact_2418_converse__iff,axiom,
% 14.25/8.04 ! [A2: int,B: int,R2: set_Pr958786334691620121nt_int] :
% 14.25/8.04 ( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ A2 @ B ) @ ( converse_int_int @ R2 ) )
% 14.25/8.04 = ( member5262025264175285858nt_int @ ( product_Pair_int_int @ B @ A2 ) @ R2 ) ) ).
% 14.25/8.04
% 14.25/8.04 thf(fact_2419_converse__iff,axiom,
% 14.25/8.04 ! [A2: produc3658429121746597890et_nat > $o,B: produc3658429121746597890et_nat,R2: set_Pr4208978187861016310_nat_o] :
% 14.25/8.04 ( ( member1996754912294343701et_nat @ ( produc5001842942810119800et_nat @ A2 @ B ) @ ( conver1831365297310299342_nat_o @ R2 ) )
% 14.25/8.04 = ( member4978015717948814295_nat_o @ ( produc2829238748666232050_nat_o @ B @ A2 ) @ R2 ) ) ).
% 14.25/8.04
% 14.25/8.04 thf(fact_2420_converse__iff,axiom,
% 14.25/8.04 ! [A2: produc3658429121746597890et_nat > $o,B: produc3925858234332021118et_nat,R2: set_Pr7705059561495379578_nat_o] :
% 14.25/8.04 ( ( member6124377750444531601et_nat @ ( produc2245416461498447860et_nat @ A2 @ B ) @ ( conver5006497266169685586_nat_o @ R2 ) )
% 14.25/8.04 = ( member8354763867512313179_nat_o @ ( produc3674526432113299062_nat_o @ B @ A2 ) @ R2 ) ) ).
% 14.25/8.04
% 14.25/8.04 thf(fact_2421_Diff__eq__empty__iff,axiom,
% 14.25/8.04 ! [A: set_Product_unit,B4: set_Product_unit] :
% 14.25/8.04 ( ( ( minus_6452836326544984404t_unit @ A @ B4 )
% 14.25/8.04 = bot_bo3957492148770167129t_unit )
% 14.25/8.04 = ( ord_le3507040750410214029t_unit @ A @ B4 ) ) ).
% 14.25/8.04
% 14.25/8.04 thf(fact_2422_Diff__eq__empty__iff,axiom,
% 14.25/8.04 ! [A: set_o,B4: set_o] :
% 14.25/8.04 ( ( ( minus_minus_set_o @ A @ B4 )
% 14.25/8.04 = bot_bot_set_o )
% 14.25/8.04 = ( ord_less_eq_set_o @ A @ B4 ) ) ).
% 14.25/8.04
% 14.25/8.04 thf(fact_2423_Diff__eq__empty__iff,axiom,
% 14.25/8.04 ! [A: set_int,B4: set_int] :
% 14.25/8.04 ( ( ( minus_minus_set_int @ A @ B4 )
% 14.25/8.04 = bot_bot_set_int )
% 14.25/8.04 = ( ord_less_eq_set_int @ A @ B4 ) ) ).
% 14.25/8.04
% 14.25/8.04 thf(fact_2424_Diff__eq__empty__iff,axiom,
% 14.25/8.04 ! [A: set_Pr4532377907799695533_nat_o,B4: set_Pr4532377907799695533_nat_o] :
% 14.25/8.04 ( ( ( minus_1801376950450012436_nat_o @ A @ B4 )
% 14.25/8.04 = bot_bo7824918357723582233_nat_o )
% 14.25/8.04 = ( ord_le2965882846123202637_nat_o @ A @ B4 ) ) ).
% 14.25/8.04
% 14.25/8.04 thf(fact_2425_Diff__eq__empty__iff,axiom,
% 14.25/8.04 ! [A: set_nat,B4: set_nat] :
% 14.25/8.04 ( ( ( minus_minus_set_nat @ A @ B4 )
% 14.25/8.04 = bot_bot_set_nat )
% 14.25/8.04 = ( ord_less_eq_set_nat @ A @ B4 ) ) ).
% 14.25/8.04
% 14.25/8.04 thf(fact_2426_insert__Diff__single,axiom,
% 14.25/8.04 ! [A2: product_unit,A: set_Product_unit] :
% 14.25/8.04 ( ( insert_Product_unit @ A2 @ ( minus_6452836326544984404t_unit @ A @ ( insert_Product_unit @ A2 @ bot_bo3957492148770167129t_unit ) ) )
% 14.25/8.04 = ( insert_Product_unit @ A2 @ A ) ) ).
% 14.25/8.04
% 14.25/8.04 thf(fact_2427_insert__Diff__single,axiom,
% 14.25/8.04 ! [A2: $o,A: set_o] :
% 14.25/8.04 ( ( insert_o @ A2 @ ( minus_minus_set_o @ A @ ( insert_o @ A2 @ bot_bot_set_o ) ) )
% 14.25/8.04 = ( insert_o @ A2 @ A ) ) ).
% 14.25/8.04
% 14.25/8.04 thf(fact_2428_insert__Diff__single,axiom,
% 14.25/8.04 ! [A2: int,A: set_int] :
% 14.25/8.04 ( ( insert_int @ A2 @ ( minus_minus_set_int @ A @ ( insert_int @ A2 @ bot_bot_set_int ) ) )
% 14.25/8.04 = ( insert_int @ A2 @ A ) ) ).
% 14.25/8.04
% 14.25/8.04 thf(fact_2429_insert__Diff__single,axiom,
% 14.25/8.04 ! [A2: produc3658429121746597890et_nat > $o,A: set_Pr4532377907799695533_nat_o] :
% 14.25/8.04 ( ( insert5175938949040314269_nat_o @ A2 @ ( minus_1801376950450012436_nat_o @ A @ ( insert5175938949040314269_nat_o @ A2 @ bot_bo7824918357723582233_nat_o ) ) )
% 14.25/8.04 = ( insert5175938949040314269_nat_o @ A2 @ A ) ) ).
% 14.25/8.04
% 14.25/8.04 thf(fact_2430_insert__Diff__single,axiom,
% 14.25/8.04 ! [A2: nat,A: set_nat] :
% 14.25/8.04 ( ( insert_nat @ A2 @ ( minus_minus_set_nat @ A @ ( insert_nat @ A2 @ bot_bot_set_nat ) ) )
% 14.25/8.04 = ( insert_nat @ A2 @ A ) ) ).
% 14.25/8.04
% 14.25/8.04 thf(fact_2431_finite__Diff__insert,axiom,
% 14.25/8.04 ! [A: set_int,A2: int,B4: set_int] :
% 14.25/8.04 ( ( finite_finite_int @ ( minus_minus_set_int @ A @ ( insert_int @ A2 @ B4 ) ) )
% 14.25/8.04 = ( finite_finite_int @ ( minus_minus_set_int @ A @ B4 ) ) ) ).
% 14.25/8.04
% 14.25/8.04 thf(fact_2432_finite__Diff__insert,axiom,
% 14.25/8.04 ! [A: set_o,A2: $o,B4: set_o] :
% 14.25/8.04 ( ( finite_finite_o @ ( minus_minus_set_o @ A @ ( insert_o @ A2 @ B4 ) ) )
% 14.25/8.04 = ( finite_finite_o @ ( minus_minus_set_o @ A @ B4 ) ) ) ).
% 14.25/8.04
% 14.25/8.04 thf(fact_2433_finite__Diff__insert,axiom,
% 14.25/8.04 ! [A: set_Product_unit,A2: product_unit,B4: set_Product_unit] :
% 14.25/8.04 ( ( finite4290736615968046902t_unit @ ( minus_6452836326544984404t_unit @ A @ ( insert_Product_unit @ A2 @ B4 ) ) )
% 14.25/8.04 = ( finite4290736615968046902t_unit @ ( minus_6452836326544984404t_unit @ A @ B4 ) ) ) ).
% 14.25/8.04
% 14.25/8.04 thf(fact_2434_finite__Diff__insert,axiom,
% 14.25/8.04 ! [A: set_Code_integer,A2: code_integer,B4: set_Code_integer] :
% 14.25/8.04 ( ( finite6017078050557962740nteger @ ( minus_2355218937544613996nteger @ A @ ( insert_Code_integer @ A2 @ B4 ) ) )
% 14.25/8.04 = ( finite6017078050557962740nteger @ ( minus_2355218937544613996nteger @ A @ B4 ) ) ) ).
% 14.25/8.04
% 14.25/8.04 thf(fact_2435_finite__Diff__insert,axiom,
% 14.25/8.04 ! [A: set_nat,A2: nat,B4: set_nat] :
% 14.25/8.04 ( ( finite_finite_nat @ ( minus_minus_set_nat @ A @ ( insert_nat @ A2 @ B4 ) ) )
% 14.25/8.04 = ( finite_finite_nat @ ( minus_minus_set_nat @ A @ B4 ) ) ) ).
% 14.25/8.04
% 14.25/8.04 thf(fact_2436_Diff__disjoint,axiom,
% 14.25/8.04 ! [A: set_Product_unit,B4: set_Product_unit] :
% 14.25/8.04 ( ( inf_in4660618365625256667t_unit @ A @ ( minus_6452836326544984404t_unit @ B4 @ A ) )
% 14.25/8.04 = bot_bo3957492148770167129t_unit ) ).
% 14.25/8.04
% 14.25/8.04 thf(fact_2437_Diff__disjoint,axiom,
% 14.25/8.04 ! [A: set_o,B4: set_o] :
% 14.25/8.04 ( ( inf_inf_set_o @ A @ ( minus_minus_set_o @ B4 @ A ) )
% 14.25/8.04 = bot_bot_set_o ) ).
% 14.25/8.04
% 14.25/8.04 thf(fact_2438_Diff__disjoint,axiom,
% 14.25/8.04 ! [A: set_int,B4: set_int] :
% 14.25/8.04 ( ( inf_inf_set_int @ A @ ( minus_minus_set_int @ B4 @ A ) )
% 14.25/8.04 = bot_bot_set_int ) ).
% 14.25/8.04
% 14.25/8.04 thf(fact_2439_Diff__disjoint,axiom,
% 14.25/8.04 ! [A: set_Pr4532377907799695533_nat_o,B4: set_Pr4532377907799695533_nat_o] :
% 14.25/8.04 ( ( inf_in1906310914598751387_nat_o @ A @ ( minus_1801376950450012436_nat_o @ B4 @ A ) )
% 14.25/8.04 = bot_bo7824918357723582233_nat_o ) ).
% 14.25/8.04
% 14.25/8.04 thf(fact_2440_Diff__disjoint,axiom,
% 14.25/8.04 ! [A: set_nat,B4: set_nat] :
% 14.25/8.04 ( ( inf_inf_set_nat @ A @ ( minus_minus_set_nat @ B4 @ A ) )
% 14.25/8.04 = bot_bot_set_nat ) ).
% 14.25/8.04
% 14.25/8.04 thf(fact_2441_image__diff__subset,axiom,
% 14.25/8.04 ! [F: code_integer > code_integer,A: set_Code_integer,B4: set_Code_integer] : ( ord_le7084787975880047091nteger @ ( minus_2355218937544613996nteger @ ( image_4470545334726330049nteger @ F @ A ) @ ( image_4470545334726330049nteger @ F @ B4 ) ) @ ( image_4470545334726330049nteger @ F @ ( minus_2355218937544613996nteger @ A @ B4 ) ) ) ).
% 14.25/8.04
% 14.25/8.04 thf(fact_2442_image__diff__subset,axiom,
% 14.25/8.04 ! [F: nat > set_nat,A: set_nat,B4: set_nat] : ( ord_le6893508408891458716et_nat @ ( minus_2163939370556025621et_nat @ ( image_nat_set_nat @ F @ A ) @ ( image_nat_set_nat @ F @ B4 ) ) @ ( image_nat_set_nat @ F @ ( minus_minus_set_nat @ A @ B4 ) ) ) ).
% 14.25/8.04
% 14.25/8.04 thf(fact_2443_image__diff__subset,axiom,
% 14.25/8.04 ! [F: nat > code_integer,A: set_nat,B4: set_nat] : ( ord_le7084787975880047091nteger @ ( minus_2355218937544613996nteger @ ( image_1215581382706833972nteger @ F @ A ) @ ( image_1215581382706833972nteger @ F @ B4 ) ) @ ( image_1215581382706833972nteger @ F @ ( minus_minus_set_nat @ A @ B4 ) ) ) ).
% 14.25/8.04
% 14.25/8.04 thf(fact_2444_image__diff__subset,axiom,
% 14.25/8.04 ! [F: nat > nat,A: set_nat,B4: set_nat] : ( ord_less_eq_set_nat @ ( minus_minus_set_nat @ ( image_nat_nat2 @ F @ A ) @ ( image_nat_nat2 @ F @ B4 ) ) @ ( image_nat_nat2 @ F @ ( minus_minus_set_nat @ A @ B4 ) ) ) ).
% 14.25/8.04
% 14.25/8.04 thf(fact_2445_rev__image__eqI,axiom,
% 14.25/8.04 ! [X: code_integer,A: set_Code_integer,B: code_integer,F: code_integer > code_integer] :
% 14.25/8.04 ( ( member_Code_integer @ X @ A )
% 14.25/8.04 => ( ( B
% 14.25/8.04 = ( F @ X ) )
% 14.25/8.04 => ( member_Code_integer @ B @ ( image_4470545334726330049nteger @ F @ A ) ) ) ) ).
% 14.25/8.04
% 14.25/8.04 thf(fact_2446_rev__image__eqI,axiom,
% 14.25/8.04 ! [X: nat,A: set_nat,B: code_integer,F: nat > code_integer] :
% 14.25/8.04 ( ( member_nat @ X @ A )
% 14.25/8.04 => ( ( B
% 14.25/8.04 = ( F @ X ) )
% 14.25/8.04 => ( member_Code_integer @ B @ ( image_1215581382706833972nteger @ F @ A ) ) ) ) ).
% 14.25/8.04
% 14.25/8.04 thf(fact_2447_rev__image__eqI,axiom,
% 14.25/8.04 ! [X: nat,A: set_nat,B: nat,F: nat > nat] :
% 14.25/8.04 ( ( member_nat @ X @ A )
% 14.25/8.04 => ( ( B
% 14.25/8.04 = ( F @ X ) )
% 14.25/8.04 => ( member_nat @ B @ ( image_nat_nat2 @ F @ A ) ) ) ) ).
% 14.25/8.04
% 14.25/8.04 thf(fact_2448_rev__image__eqI,axiom,
% 14.25/8.04 ! [X: nat,A: set_nat,B: int,F: nat > int] :
% 14.25/8.04 ( ( member_nat @ X @ A )
% 14.25/8.04 => ( ( B
% 14.25/8.04 = ( F @ X ) )
% 14.25/8.04 => ( member_int @ B @ ( image_nat_int2 @ F @ A ) ) ) ) ).
% 14.25/8.04
% 14.25/8.04 thf(fact_2449_rev__image__eqI,axiom,
% 14.25/8.04 ! [X: nat,A: set_nat,B: $o,F: nat > $o] :
% 14.25/8.04 ( ( member_nat @ X @ A )
% 14.25/8.04 => ( ( B
% 14.25/8.04 = ( F @ X ) )
% 14.25/8.04 => ( member_o @ B @ ( image_nat_o2 @ F @ A ) ) ) ) ).
% 14.25/8.04
% 14.25/8.04 thf(fact_2450_rev__image__eqI,axiom,
% 14.25/8.04 ! [X: nat,A: set_nat,B: product_unit,F: nat > product_unit] :
% 14.25/8.04 ( ( member_nat @ X @ A )
% 14.25/8.04 => ( ( B
% 14.25/8.04 = ( F @ X ) )
% 14.25/8.04 => ( member_Product_unit @ B @ ( image_8730104196221521654t_unit @ F @ A ) ) ) ) ).
% 14.25/8.04
% 14.25/8.04 thf(fact_2451_rev__image__eqI,axiom,
% 14.25/8.04 ! [X: int,A: set_int,B: nat,F: int > nat] :
% 14.25/8.04 ( ( member_int @ X @ A )
% 14.25/8.04 => ( ( B
% 14.25/8.04 = ( F @ X ) )
% 14.25/8.04 => ( member_nat @ B @ ( image_int_nat2 @ F @ A ) ) ) ) ).
% 14.25/8.04
% 14.25/8.04 thf(fact_2452_rev__image__eqI,axiom,
% 14.25/8.04 ! [X: int,A: set_int,B: int,F: int > int] :
% 14.25/8.04 ( ( member_int @ X @ A )
% 14.25/8.04 => ( ( B
% 14.25/8.04 = ( F @ X ) )
% 14.25/8.04 => ( member_int @ B @ ( image_int_int2 @ F @ A ) ) ) ) ).
% 14.25/8.04
% 14.25/8.04 thf(fact_2453_rev__image__eqI,axiom,
% 14.25/8.04 ! [X: int,A: set_int,B: $o,F: int > $o] :
% 14.25/8.04 ( ( member_int @ X @ A )
% 14.25/8.04 => ( ( B
% 14.25/8.04 = ( F @ X ) )
% 14.25/8.04 => ( member_o @ B @ ( image_int_o2 @ F @ A ) ) ) ) ).
% 14.25/8.04
% 14.25/8.04 thf(fact_2454_rev__image__eqI,axiom,
% 14.25/8.04 ! [X: int,A: set_int,B: product_unit,F: int > product_unit] :
% 14.25/8.04 ( ( member_int @ X @ A )
% 14.25/8.04 => ( ( B
% 14.25/8.04 = ( F @ X ) )
% 14.25/8.04 => ( member_Product_unit @ B @ ( image_5846526143423701914t_unit @ F @ A ) ) ) ) ).
% 14.25/8.04
% 14.25/8.04 thf(fact_2455_ball__imageD,axiom,
% 14.25/8.04 ! [F: nat > set_nat,A: set_nat,P: set_nat > $o] :
% 14.25/8.04 ( ! [X3: set_nat] :
% 14.25/8.04 ( ( member_set_nat @ X3 @ ( image_nat_set_nat @ F @ A ) )
% 14.25/8.04 => ( P @ X3 ) )
% 14.25/8.04 => ! [X6: nat] :
% 14.25/8.04 ( ( member_nat @ X6 @ A )
% 14.25/8.04 => ( P @ ( F @ X6 ) ) ) ) ).
% 14.25/8.04
% 14.25/8.04 thf(fact_2456_ball__imageD,axiom,
% 14.25/8.04 ! [F: nat > nat,A: set_nat,P: nat > $o] :
% 14.25/8.04 ( ! [X3: nat] :
% 14.25/8.04 ( ( member_nat @ X3 @ ( image_nat_nat2 @ F @ A ) )
% 14.25/8.04 => ( P @ X3 ) )
% 14.25/8.04 => ! [X6: nat] :
% 14.25/8.04 ( ( member_nat @ X6 @ A )
% 14.25/8.04 => ( P @ ( F @ X6 ) ) ) ) ).
% 14.25/8.04
% 14.25/8.04 thf(fact_2457_ball__imageD,axiom,
% 14.25/8.04 ! [F: code_integer > code_integer,A: set_Code_integer,P: code_integer > $o] :
% 14.25/8.04 ( ! [X3: code_integer] :
% 14.25/8.04 ( ( member_Code_integer @ X3 @ ( image_4470545334726330049nteger @ F @ A ) )
% 14.25/8.04 => ( P @ X3 ) )
% 14.25/8.04 => ! [X6: code_integer] :
% 14.25/8.04 ( ( member_Code_integer @ X6 @ A )
% 14.25/8.04 => ( P @ ( F @ X6 ) ) ) ) ).
% 14.25/8.04
% 14.25/8.04 thf(fact_2458_ball__imageD,axiom,
% 14.25/8.04 ! [F: nat > code_integer,A: set_nat,P: code_integer > $o] :
% 14.25/8.04 ( ! [X3: code_integer] :
% 14.25/8.04 ( ( member_Code_integer @ X3 @ ( image_1215581382706833972nteger @ F @ A ) )
% 14.25/8.04 => ( P @ X3 ) )
% 14.25/8.04 => ! [X6: nat] :
% 14.25/8.04 ( ( member_nat @ X6 @ A )
% 14.25/8.04 => ( P @ ( F @ X6 ) ) ) ) ).
% 14.25/8.04
% 14.25/8.04 thf(fact_2459_image__cong,axiom,
% 14.25/8.04 ! [M: set_Code_integer,N: set_Code_integer,F: code_integer > code_integer,G: code_integer > code_integer] :
% 14.25/8.04 ( ( M = N )
% 14.25/8.04 => ( ! [X3: code_integer] :
% 14.25/8.04 ( ( member_Code_integer @ X3 @ N )
% 14.25/8.04 => ( ( F @ X3 )
% 14.25/8.04 = ( G @ X3 ) ) )
% 14.25/8.04 => ( ( image_4470545334726330049nteger @ F @ M )
% 14.25/8.04 = ( image_4470545334726330049nteger @ G @ N ) ) ) ) ).
% 14.25/8.04
% 14.25/8.04 thf(fact_2460_image__cong,axiom,
% 14.25/8.04 ! [M: set_nat,N: set_nat,F: nat > set_nat,G: nat > set_nat] :
% 14.25/8.04 ( ( M = N )
% 14.25/8.04 => ( ! [X3: nat] :
% 14.25/8.04 ( ( member_nat @ X3 @ N )
% 14.25/8.04 => ( ( F @ X3 )
% 14.25/8.04 = ( G @ X3 ) ) )
% 14.25/8.04 => ( ( image_nat_set_nat @ F @ M )
% 14.25/8.04 = ( image_nat_set_nat @ G @ N ) ) ) ) ).
% 14.25/8.04
% 14.25/8.04 thf(fact_2461_image__cong,axiom,
% 14.25/8.04 ! [M: set_nat,N: set_nat,F: nat > nat,G: nat > nat] :
% 14.25/8.04 ( ( M = N )
% 14.25/8.04 => ( ! [X3: nat] :
% 14.25/8.04 ( ( member_nat @ X3 @ N )
% 14.25/8.04 => ( ( F @ X3 )
% 14.25/8.04 = ( G @ X3 ) ) )
% 14.25/8.04 => ( ( image_nat_nat2 @ F @ M )
% 14.25/8.04 = ( image_nat_nat2 @ G @ N ) ) ) ) ).
% 14.25/8.04
% 14.25/8.04 thf(fact_2462_image__cong,axiom,
% 14.25/8.04 ! [M: set_nat,N: set_nat,F: nat > code_integer,G: nat > code_integer] :
% 14.25/8.04 ( ( M = N )
% 14.25/8.04 => ( ! [X3: nat] :
% 14.25/8.04 ( ( member_nat @ X3 @ N )
% 14.25/8.04 => ( ( F @ X3 )
% 14.25/8.04 = ( G @ X3 ) ) )
% 14.25/8.04 => ( ( image_1215581382706833972nteger @ F @ M )
% 14.25/8.04 = ( image_1215581382706833972nteger @ G @ N ) ) ) ) ).
% 14.25/8.04
% 14.25/8.04 thf(fact_2463_bex__imageD,axiom,
% 14.25/8.04 ! [F: nat > set_nat,A: set_nat,P: set_nat > $o] :
% 14.25/8.04 ( ? [X6: set_nat] :
% 14.25/8.04 ( ( member_set_nat @ X6 @ ( image_nat_set_nat @ F @ A ) )
% 14.25/8.04 & ( P @ X6 ) )
% 14.25/8.04 => ? [X3: nat] :
% 14.25/8.04 ( ( member_nat @ X3 @ A )
% 14.25/8.04 & ( P @ ( F @ X3 ) ) ) ) ).
% 14.25/8.04
% 14.25/8.04 thf(fact_2464_bex__imageD,axiom,
% 14.25/8.04 ! [F: nat > nat,A: set_nat,P: nat > $o] :
% 14.25/8.04 ( ? [X6: nat] :
% 14.25/8.04 ( ( member_nat @ X6 @ ( image_nat_nat2 @ F @ A ) )
% 14.25/8.04 & ( P @ X6 ) )
% 14.25/8.04 => ? [X3: nat] :
% 14.25/8.04 ( ( member_nat @ X3 @ A )
% 14.25/8.04 & ( P @ ( F @ X3 ) ) ) ) ).
% 14.25/8.04
% 14.25/8.04 thf(fact_2465_bex__imageD,axiom,
% 14.25/8.04 ! [F: code_integer > code_integer,A: set_Code_integer,P: code_integer > $o] :
% 14.25/8.04 ( ? [X6: code_integer] :
% 14.25/8.04 ( ( member_Code_integer @ X6 @ ( image_4470545334726330049nteger @ F @ A ) )
% 14.25/8.04 & ( P @ X6 ) )
% 14.25/8.04 => ? [X3: code_integer] :
% 14.25/8.04 ( ( member_Code_integer @ X3 @ A )
% 14.25/8.04 & ( P @ ( F @ X3 ) ) ) ) ).
% 14.25/8.04
% 14.25/8.04 thf(fact_2466_bex__imageD,axiom,
% 14.25/8.04 ! [F: nat > code_integer,A: set_nat,P: code_integer > $o] :
% 14.25/8.04 ( ? [X6: code_integer] :
% 14.25/8.04 ( ( member_Code_integer @ X6 @ ( image_1215581382706833972nteger @ F @ A ) )
% 14.25/8.04 & ( P @ X6 ) )
% 14.25/8.04 => ? [X3: nat] :
% 14.25/8.04 ( ( member_nat @ X3 @ A )
% 14.25/8.04 & ( P @ ( F @ X3 ) ) ) ) ).
% 14.25/8.04
% 14.25/8.04 thf(fact_2467_image__iff,axiom,
% 14.25/8.04 ! [Z: set_nat,F: nat > set_nat,A: set_nat] :
% 14.25/8.04 ( ( member_set_nat @ Z @ ( image_nat_set_nat @ F @ A ) )
% 14.25/8.04 = ( ? [X2: nat] :
% 14.25/8.04 ( ( member_nat @ X2 @ A )
% 14.25/8.04 & ( Z
% 14.25/8.04 = ( F @ X2 ) ) ) ) ) ).
% 14.25/8.04
% 14.25/8.04 thf(fact_2468_image__iff,axiom,
% 14.25/8.04 ! [Z: code_integer,F: code_integer > code_integer,A: set_Code_integer] :
% 14.25/8.04 ( ( member_Code_integer @ Z @ ( image_4470545334726330049nteger @ F @ A ) )
% 14.25/8.04 = ( ? [X2: code_integer] :
% 14.25/8.04 ( ( member_Code_integer @ X2 @ A )
% 14.25/8.04 & ( Z
% 14.25/8.04 = ( F @ X2 ) ) ) ) ) ).
% 14.25/8.04
% 14.25/8.04 thf(fact_2469_image__iff,axiom,
% 14.25/8.04 ! [Z: code_integer,F: nat > code_integer,A: set_nat] :
% 14.25/8.04 ( ( member_Code_integer @ Z @ ( image_1215581382706833972nteger @ F @ A ) )
% 14.25/8.04 = ( ? [X2: nat] :
% 14.25/8.04 ( ( member_nat @ X2 @ A )
% 14.25/8.04 & ( Z
% 14.25/8.04 = ( F @ X2 ) ) ) ) ) ).
% 14.25/8.04
% 14.25/8.04 thf(fact_2470_image__iff,axiom,
% 14.25/8.04 ! [Z: nat,F: nat > nat,A: set_nat] :
% 14.25/8.04 ( ( member_nat @ Z @ ( image_nat_nat2 @ F @ A ) )
% 14.25/8.04 = ( ? [X2: nat] :
% 14.25/8.04 ( ( member_nat @ X2 @ A )
% 14.25/8.04 & ( Z
% 14.25/8.04 = ( F @ X2 ) ) ) ) ) ).
% 14.25/8.04
% 14.25/8.04 thf(fact_2471_imageI,axiom,
% 14.25/8.04 ! [X: code_integer,A: set_Code_integer,F: code_integer > code_integer] :
% 14.25/8.04 ( ( member_Code_integer @ X @ A )
% 14.25/8.04 => ( member_Code_integer @ ( F @ X ) @ ( image_4470545334726330049nteger @ F @ A ) ) ) ).
% 14.25/8.04
% 14.25/8.04 thf(fact_2472_imageI,axiom,
% 14.25/8.04 ! [X: nat,A: set_nat,F: nat > code_integer] :
% 14.25/8.04 ( ( member_nat @ X @ A )
% 14.25/8.04 => ( member_Code_integer @ ( F @ X ) @ ( image_1215581382706833972nteger @ F @ A ) ) ) ).
% 14.25/8.04
% 14.25/8.04 thf(fact_2473_imageI,axiom,
% 14.25/8.04 ! [X: nat,A: set_nat,F: nat > nat] :
% 14.25/8.04 ( ( member_nat @ X @ A )
% 14.25/8.04 => ( member_nat @ ( F @ X ) @ ( image_nat_nat2 @ F @ A ) ) ) ).
% 14.25/8.04
% 14.25/8.04 thf(fact_2474_imageI,axiom,
% 14.25/8.04 ! [X: nat,A: set_nat,F: nat > int] :
% 14.25/8.04 ( ( member_nat @ X @ A )
% 14.25/8.04 => ( member_int @ ( F @ X ) @ ( image_nat_int2 @ F @ A ) ) ) ).
% 14.25/8.04
% 14.25/8.04 thf(fact_2475_imageI,axiom,
% 14.25/8.04 ! [X: nat,A: set_nat,F: nat > $o] :
% 14.25/8.04 ( ( member_nat @ X @ A )
% 14.25/8.04 => ( member_o @ ( F @ X ) @ ( image_nat_o2 @ F @ A ) ) ) ).
% 14.25/8.04
% 14.25/8.04 thf(fact_2476_imageI,axiom,
% 14.25/8.04 ! [X: nat,A: set_nat,F: nat > product_unit] :
% 14.25/8.04 ( ( member_nat @ X @ A )
% 14.25/8.04 => ( member_Product_unit @ ( F @ X ) @ ( image_8730104196221521654t_unit @ F @ A ) ) ) ).
% 14.25/8.04
% 14.25/8.04 thf(fact_2477_imageI,axiom,
% 14.25/8.04 ! [X: int,A: set_int,F: int > nat] :
% 14.25/8.04 ( ( member_int @ X @ A )
% 14.25/8.04 => ( member_nat @ ( F @ X ) @ ( image_int_nat2 @ F @ A ) ) ) ).
% 14.25/8.04
% 14.25/8.04 thf(fact_2478_imageI,axiom,
% 14.25/8.04 ! [X: int,A: set_int,F: int > int] :
% 14.25/8.04 ( ( member_int @ X @ A )
% 14.25/8.04 => ( member_int @ ( F @ X ) @ ( image_int_int2 @ F @ A ) ) ) ).
% 14.25/8.04
% 14.25/8.04 thf(fact_2479_imageI,axiom,
% 14.25/8.04 ! [X: int,A: set_int,F: int > $o] :
% 14.25/8.04 ( ( member_int @ X @ A )
% 14.25/8.04 => ( member_o @ ( F @ X ) @ ( image_int_o2 @ F @ A ) ) ) ).
% 14.25/8.04
% 14.25/8.04 thf(fact_2480_imageI,axiom,
% 14.25/8.04 ! [X: int,A: set_int,F: int > product_unit] :
% 14.25/8.04 ( ( member_int @ X @ A )
% 14.25/8.04 => ( member_Product_unit @ ( F @ X ) @ ( image_5846526143423701914t_unit @ F @ A ) ) ) ).
% 14.25/8.04
% 14.25/8.04 thf(fact_2481_DiffD2,axiom,
% 14.25/8.04 ! [C: produc3658429121746597890et_nat > $o,A: set_Pr4532377907799695533_nat_o,B4: set_Pr4532377907799695533_nat_o] :
% 14.25/8.04 ( ( member6576561426505652726_nat_o @ C @ ( minus_1801376950450012436_nat_o @ A @ B4 ) )
% 14.25/8.05 => ~ ( member6576561426505652726_nat_o @ C @ B4 ) ) ).
% 14.25/8.05
% 14.25/8.05 thf(fact_2482_DiffD2,axiom,
% 14.25/8.05 ! [C: int,A: set_int,B4: set_int] :
% 14.25/8.05 ( ( member_int @ C @ ( minus_minus_set_int @ A @ B4 ) )
% 14.25/8.05 => ~ ( member_int @ C @ B4 ) ) ).
% 14.25/8.05
% 14.25/8.05 thf(fact_2483_DiffD2,axiom,
% 14.25/8.05 ! [C: $o,A: set_o,B4: set_o] :
% 14.25/8.05 ( ( member_o @ C @ ( minus_minus_set_o @ A @ B4 ) )
% 14.25/8.05 => ~ ( member_o @ C @ B4 ) ) ).
% 14.25/8.05
% 14.25/8.05 thf(fact_2484_DiffD2,axiom,
% 14.25/8.05 ! [C: product_unit,A: set_Product_unit,B4: set_Product_unit] :
% 14.25/8.05 ( ( member_Product_unit @ C @ ( minus_6452836326544984404t_unit @ A @ B4 ) )
% 14.25/8.05 => ~ ( member_Product_unit @ C @ B4 ) ) ).
% 14.25/8.05
% 14.25/8.05 thf(fact_2485_DiffD2,axiom,
% 14.25/8.05 ! [C: nat,A: set_nat,B4: set_nat] :
% 14.25/8.05 ( ( member_nat @ C @ ( minus_minus_set_nat @ A @ B4 ) )
% 14.25/8.05 => ~ ( member_nat @ C @ B4 ) ) ).
% 14.25/8.05
% 14.25/8.05 thf(fact_2486_DiffD1,axiom,
% 14.25/8.05 ! [C: produc3658429121746597890et_nat > $o,A: set_Pr4532377907799695533_nat_o,B4: set_Pr4532377907799695533_nat_o] :
% 14.25/8.05 ( ( member6576561426505652726_nat_o @ C @ ( minus_1801376950450012436_nat_o @ A @ B4 ) )
% 14.25/8.05 => ( member6576561426505652726_nat_o @ C @ A ) ) ).
% 14.25/8.05
% 14.25/8.05 thf(fact_2487_DiffD1,axiom,
% 14.25/8.05 ! [C: int,A: set_int,B4: set_int] :
% 14.25/8.05 ( ( member_int @ C @ ( minus_minus_set_int @ A @ B4 ) )
% 14.25/8.05 => ( member_int @ C @ A ) ) ).
% 14.25/8.05
% 14.25/8.05 thf(fact_2488_DiffD1,axiom,
% 14.25/8.05 ! [C: $o,A: set_o,B4: set_o] :
% 14.25/8.05 ( ( member_o @ C @ ( minus_minus_set_o @ A @ B4 ) )
% 14.25/8.05 => ( member_o @ C @ A ) ) ).
% 14.25/8.05
% 14.25/8.05 thf(fact_2489_DiffD1,axiom,
% 14.25/8.05 ! [C: product_unit,A: set_Product_unit,B4: set_Product_unit] :
% 14.25/8.05 ( ( member_Product_unit @ C @ ( minus_6452836326544984404t_unit @ A @ B4 ) )
% 14.25/8.05 => ( member_Product_unit @ C @ A ) ) ).
% 14.25/8.05
% 14.25/8.05 thf(fact_2490_DiffD1,axiom,
% 14.25/8.05 ! [C: nat,A: set_nat,B4: set_nat] :
% 14.25/8.05 ( ( member_nat @ C @ ( minus_minus_set_nat @ A @ B4 ) )
% 14.25/8.05 => ( member_nat @ C @ A ) ) ).
% 14.25/8.05
% 14.25/8.05 thf(fact_2491_DiffE,axiom,
% 14.25/8.05 ! [C: produc3658429121746597890et_nat > $o,A: set_Pr4532377907799695533_nat_o,B4: set_Pr4532377907799695533_nat_o] :
% 14.25/8.05 ( ( member6576561426505652726_nat_o @ C @ ( minus_1801376950450012436_nat_o @ A @ B4 ) )
% 14.25/8.05 => ~ ( ( member6576561426505652726_nat_o @ C @ A )
% 14.25/8.05 => ( member6576561426505652726_nat_o @ C @ B4 ) ) ) ).
% 14.25/8.05
% 14.25/8.05 thf(fact_2492_DiffE,axiom,
% 14.25/8.05 ! [C: int,A: set_int,B4: set_int] :
% 14.25/8.05 ( ( member_int @ C @ ( minus_minus_set_int @ A @ B4 ) )
% 14.25/8.05 => ~ ( ( member_int @ C @ A )
% 14.25/8.05 => ( member_int @ C @ B4 ) ) ) ).
% 14.25/8.05
% 14.25/8.05 thf(fact_2493_DiffE,axiom,
% 14.25/8.05 ! [C: $o,A: set_o,B4: set_o] :
% 14.25/8.05 ( ( member_o @ C @ ( minus_minus_set_o @ A @ B4 ) )
% 14.25/8.05 => ~ ( ( member_o @ C @ A )
% 14.25/8.05 => ( member_o @ C @ B4 ) ) ) ).
% 14.25/8.05
% 14.25/8.05 thf(fact_2494_DiffE,axiom,
% 14.25/8.05 ! [C: product_unit,A: set_Product_unit,B4: set_Product_unit] :
% 14.25/8.05 ( ( member_Product_unit @ C @ ( minus_6452836326544984404t_unit @ A @ B4 ) )
% 14.25/8.05 => ~ ( ( member_Product_unit @ C @ A )
% 14.25/8.05 => ( member_Product_unit @ C @ B4 ) ) ) ).
% 14.25/8.05
% 14.25/8.05 thf(fact_2495_DiffE,axiom,
% 14.25/8.05 ! [C: nat,A: set_nat,B4: set_nat] :
% 14.25/8.05 ( ( member_nat @ C @ ( minus_minus_set_nat @ A @ B4 ) )
% 14.25/8.05 => ~ ( ( member_nat @ C @ A )
% 14.25/8.05 => ( member_nat @ C @ B4 ) ) ) ).
% 14.25/8.05
% 14.25/8.05 thf(fact_2496_cancel__ab__semigroup__add__class_Odiff__right__commute,axiom,
% 14.25/8.05 ! [A2: rat,C: rat,B: rat] :
% 14.25/8.05 ( ( minus_minus_rat @ ( minus_minus_rat @ A2 @ C ) @ B )
% 14.25/8.05 = ( minus_minus_rat @ ( minus_minus_rat @ A2 @ B ) @ C ) ) ).
% 14.25/8.05
% 14.25/8.05 thf(fact_2497_cancel__ab__semigroup__add__class_Odiff__right__commute,axiom,
% 14.25/8.05 ! [A2: nat,C: nat,B: nat] :
% 14.25/8.05 ( ( minus_minus_nat @ ( minus_minus_nat @ A2 @ C ) @ B )
% 14.25/8.05 = ( minus_minus_nat @ ( minus_minus_nat @ A2 @ B ) @ C ) ) ).
% 14.25/8.05
% 14.25/8.05 thf(fact_2498_cancel__ab__semigroup__add__class_Odiff__right__commute,axiom,
% 14.25/8.05 ! [A2: int,C: int,B: int] :
% 14.25/8.05 ( ( minus_minus_int @ ( minus_minus_int @ A2 @ C ) @ B )
% 14.25/8.05 = ( minus_minus_int @ ( minus_minus_int @ A2 @ B ) @ C ) ) ).
% 14.25/8.05
% 14.25/8.05 thf(fact_2499_diff__eq__diff__eq,axiom,
% 14.25/8.05 ! [A2: rat,B: rat,C: rat,D2: rat] :
% 14.25/8.05 ( ( ( minus_minus_rat @ A2 @ B )
% 14.25/8.05 = ( minus_minus_rat @ C @ D2 ) )
% 14.25/8.05 => ( ( A2 = B )
% 14.25/8.05 = ( C = D2 ) ) ) ).
% 14.25/8.05
% 14.25/8.05 thf(fact_2500_diff__eq__diff__eq,axiom,
% 14.25/8.05 ! [A2: int,B: int,C: int,D2: int] :
% 14.25/8.05 ( ( ( minus_minus_int @ A2 @ B )
% 14.25/8.05 = ( minus_minus_int @ C @ D2 ) )
% 14.25/8.05 => ( ( A2 = B )
% 14.25/8.05 = ( C = D2 ) ) ) ).
% 14.25/8.05
% 14.25/8.05 thf(fact_2501_in__image__insert__iff,axiom,
% 14.25/8.05 ! [B4: set_set_Product_unit,X: product_unit,A: set_Product_unit] :
% 14.25/8.05 ( ! [C6: set_Product_unit] :
% 14.25/8.05 ( ( member5877623283571906838t_unit @ C6 @ B4 )
% 14.25/8.05 => ~ ( member_Product_unit @ X @ C6 ) )
% 14.25/8.05 => ( ( member5877623283571906838t_unit @ A @ ( image_4019945301284682565t_unit @ ( insert_Product_unit @ X ) @ B4 ) )
% 14.25/8.05 = ( ( member_Product_unit @ X @ A )
% 14.25/8.05 & ( member5877623283571906838t_unit @ ( minus_6452836326544984404t_unit @ A @ ( insert_Product_unit @ X @ bot_bo3957492148770167129t_unit ) ) @ B4 ) ) ) ) ).
% 14.25/8.05
% 14.25/8.05 thf(fact_2502_in__image__insert__iff,axiom,
% 14.25/8.05 ! [B4: set_set_o,X: $o,A: set_o] :
% 14.25/8.05 ( ! [C6: set_o] :
% 14.25/8.05 ( ( member_set_o @ C6 @ B4 )
% 14.25/8.05 => ~ ( member_o @ X @ C6 ) )
% 14.25/8.05 => ( ( member_set_o @ A @ ( image_set_o_set_o @ ( insert_o @ X ) @ B4 ) )
% 14.25/8.05 = ( ( member_o @ X @ A )
% 14.25/8.05 & ( member_set_o @ ( minus_minus_set_o @ A @ ( insert_o @ X @ bot_bot_set_o ) ) @ B4 ) ) ) ) ).
% 14.25/8.05
% 14.25/8.05 thf(fact_2503_in__image__insert__iff,axiom,
% 14.25/8.05 ! [B4: set_set_int,X: int,A: set_int] :
% 14.25/8.05 ( ! [C6: set_int] :
% 14.25/8.05 ( ( member_set_int @ C6 @ B4 )
% 14.25/8.05 => ~ ( member_int @ X @ C6 ) )
% 14.25/8.05 => ( ( member_set_int @ A @ ( image_524474410958335435et_int @ ( insert_int @ X ) @ B4 ) )
% 14.25/8.05 = ( ( member_int @ X @ A )
% 14.25/8.05 & ( member_set_int @ ( minus_minus_set_int @ A @ ( insert_int @ X @ bot_bot_set_int ) ) @ B4 ) ) ) ) ).
% 14.25/8.05
% 14.25/8.05 thf(fact_2504_in__image__insert__iff,axiom,
% 14.25/8.05 ! [B4: set_se1353816476878259085_nat_o,X: produc3658429121746597890et_nat > $o,A: set_Pr4532377907799695533_nat_o] :
% 14.25/8.05 ( ! [C6: set_Pr4532377907799695533_nat_o] :
% 14.25/8.05 ( ( member1452989533377600726_nat_o @ C6 @ B4 )
% 14.25/8.05 => ~ ( member6576561426505652726_nat_o @ X @ C6 ) )
% 14.25/8.05 => ( ( member1452989533377600726_nat_o @ A @ ( image_8925030434418351941_nat_o @ ( insert5175938949040314269_nat_o @ X ) @ B4 ) )
% 14.25/8.05 = ( ( member6576561426505652726_nat_o @ X @ A )
% 14.25/8.05 & ( member1452989533377600726_nat_o @ ( minus_1801376950450012436_nat_o @ A @ ( insert5175938949040314269_nat_o @ X @ bot_bo7824918357723582233_nat_o ) ) @ B4 ) ) ) ) ).
% 14.25/8.05
% 14.25/8.05 thf(fact_2505_in__image__insert__iff,axiom,
% 14.25/8.05 ! [B4: set_set_nat,X: nat,A: set_nat] :
% 14.25/8.05 ( ! [C6: set_nat] :
% 14.25/8.05 ( ( member_set_nat @ C6 @ B4 )
% 14.25/8.05 => ~ ( member_nat @ X @ C6 ) )
% 14.25/8.05 => ( ( member_set_nat @ A @ ( image_7916887816326733075et_nat @ ( insert_nat @ X ) @ B4 ) )
% 14.25/8.05 = ( ( member_nat @ X @ A )
% 14.25/8.05 & ( member_set_nat @ ( minus_minus_set_nat @ A @ ( insert_nat @ X @ bot_bot_set_nat ) ) @ B4 ) ) ) ) ).
% 14.25/8.05
% 14.25/8.05 thf(fact_2506_converse_Ocases,axiom,
% 14.25/8.05 ! [A1: set_nat,A22: heap_e7401611519738050253t_unit,R2: set_Pr3948176798113811640et_nat] :
% 14.25/8.05 ( ( member727419437372278867t_unit @ ( produc2471974422597015918t_unit @ A1 @ A22 ) @ ( conver7272282955447398104et_nat @ R2 ) )
% 14.25/8.05 => ( member6260224972018164377et_nat @ ( produc7507926704131184380et_nat @ A22 @ A1 ) @ R2 ) ) ).
% 14.25/8.05
% 14.25/8.05 thf(fact_2507_converse_Ocases,axiom,
% 14.25/8.05 ! [A1: produc3658429121746597890et_nat,A22: produc3658429121746597890et_nat > $o,R2: set_Pr3286484037609594932et_nat] :
% 14.25/8.05 ( ( member4978015717948814295_nat_o @ ( produc2829238748666232050_nat_o @ A1 @ A22 ) @ ( conver4003969491454187092et_nat @ R2 ) )
% 14.25/8.05 => ( member1996754912294343701et_nat @ ( produc5001842942810119800et_nat @ A22 @ A1 ) @ R2 ) ) ).
% 14.25/8.05
% 14.25/8.05 thf(fact_2508_converse_Ocases,axiom,
% 14.25/8.05 ! [A1: produc3925858234332021118et_nat,A22: produc3658429121746597890et_nat > $o,R2: set_Pr8536935166611901872et_nat] :
% 14.25/8.05 ( ( member8354763867512313179_nat_o @ ( produc3674526432113299062_nat_o @ A1 @ A22 ) @ ( conver3577387295554834384et_nat @ R2 ) )
% 14.25/8.05 => ( member6124377750444531601et_nat @ ( produc2245416461498447860et_nat @ A22 @ A1 ) @ R2 ) ) ).
% 14.25/8.05
% 14.25/8.05 thf(fact_2509_converse_Ocases,axiom,
% 14.25/8.05 ! [A1: heap_e7401611519738050253t_unit,A22: set_nat,R2: set_Pr4071100604818519410t_unit] :
% 14.25/8.05 ( ( member6260224972018164377et_nat @ ( produc7507926704131184380et_nat @ A1 @ A22 ) @ ( conver2236330673913229642t_unit @ R2 ) )
% 14.25/8.05 => ( member727419437372278867t_unit @ ( produc2471974422597015918t_unit @ A22 @ A1 ) @ R2 ) ) ).
% 14.25/8.05
% 14.25/8.05 thf(fact_2510_converse_Ocases,axiom,
% 14.25/8.05 ! [A1: nat,A22: nat,R2: set_Pr1261947904930325089at_nat] :
% 14.25/8.05 ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A1 @ A22 ) @ ( converse_nat_nat @ R2 ) )
% 14.25/8.05 => ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A22 @ A1 ) @ R2 ) ) ).
% 14.25/8.05
% 14.25/8.05 thf(fact_2511_converse_Ocases,axiom,
% 14.25/8.05 ! [A1: int,A22: int,R2: set_Pr958786334691620121nt_int] :
% 14.25/8.05 ( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ A1 @ A22 ) @ ( converse_int_int @ R2 ) )
% 14.25/8.05 => ( member5262025264175285858nt_int @ ( product_Pair_int_int @ A22 @ A1 ) @ R2 ) ) ).
% 14.25/8.05
% 14.25/8.05 thf(fact_2512_converse_Ocases,axiom,
% 14.25/8.05 ! [A1: produc3658429121746597890et_nat > $o,A22: produc3658429121746597890et_nat,R2: set_Pr4208978187861016310_nat_o] :
% 14.25/8.05 ( ( member1996754912294343701et_nat @ ( produc5001842942810119800et_nat @ A1 @ A22 ) @ ( conver1831365297310299342_nat_o @ R2 ) )
% 14.25/8.05 => ( member4978015717948814295_nat_o @ ( produc2829238748666232050_nat_o @ A22 @ A1 ) @ R2 ) ) ).
% 14.25/8.05
% 14.25/8.05 thf(fact_2513_converse_Ocases,axiom,
% 14.25/8.05 ! [A1: produc3658429121746597890et_nat > $o,A22: produc3925858234332021118et_nat,R2: set_Pr7705059561495379578_nat_o] :
% 14.25/8.05 ( ( member6124377750444531601et_nat @ ( produc2245416461498447860et_nat @ A1 @ A22 ) @ ( conver5006497266169685586_nat_o @ R2 ) )
% 14.25/8.05 => ( member8354763867512313179_nat_o @ ( produc3674526432113299062_nat_o @ A22 @ A1 ) @ R2 ) ) ).
% 14.25/8.05
% 14.25/8.05 thf(fact_2514_converse_Osimps,axiom,
% 14.25/8.05 ! [A1: set_nat,A22: heap_e7401611519738050253t_unit,R2: set_Pr3948176798113811640et_nat] :
% 14.25/8.05 ( ( member727419437372278867t_unit @ ( produc2471974422597015918t_unit @ A1 @ A22 ) @ ( conver7272282955447398104et_nat @ R2 ) )
% 14.25/8.05 = ( ? [A6: heap_e7401611519738050253t_unit,B6: set_nat] :
% 14.25/8.05 ( ( A1 = B6 )
% 14.25/8.05 & ( A22 = A6 )
% 14.25/8.05 & ( member6260224972018164377et_nat @ ( produc7507926704131184380et_nat @ A6 @ B6 ) @ R2 ) ) ) ) ).
% 14.25/8.05
% 14.25/8.05 thf(fact_2515_converse_Osimps,axiom,
% 14.25/8.05 ! [A1: produc3658429121746597890et_nat,A22: produc3658429121746597890et_nat > $o,R2: set_Pr3286484037609594932et_nat] :
% 14.25/8.05 ( ( member4978015717948814295_nat_o @ ( produc2829238748666232050_nat_o @ A1 @ A22 ) @ ( conver4003969491454187092et_nat @ R2 ) )
% 14.25/8.05 = ( ? [A6: produc3658429121746597890et_nat > $o,B6: produc3658429121746597890et_nat] :
% 14.25/8.05 ( ( A1 = B6 )
% 14.25/8.05 & ( A22 = A6 )
% 14.25/8.05 & ( member1996754912294343701et_nat @ ( produc5001842942810119800et_nat @ A6 @ B6 ) @ R2 ) ) ) ) ).
% 14.25/8.05
% 14.25/8.05 thf(fact_2516_converse_Osimps,axiom,
% 14.25/8.05 ! [A1: produc3925858234332021118et_nat,A22: produc3658429121746597890et_nat > $o,R2: set_Pr8536935166611901872et_nat] :
% 14.25/8.05 ( ( member8354763867512313179_nat_o @ ( produc3674526432113299062_nat_o @ A1 @ A22 ) @ ( conver3577387295554834384et_nat @ R2 ) )
% 14.25/8.05 = ( ? [A6: produc3658429121746597890et_nat > $o,B6: produc3925858234332021118et_nat] :
% 14.25/8.05 ( ( A1 = B6 )
% 14.25/8.05 & ( A22 = A6 )
% 14.25/8.05 & ( member6124377750444531601et_nat @ ( produc2245416461498447860et_nat @ A6 @ B6 ) @ R2 ) ) ) ) ).
% 14.25/8.05
% 14.25/8.05 thf(fact_2517_converse_Osimps,axiom,
% 14.25/8.05 ! [A1: heap_e7401611519738050253t_unit,A22: set_nat,R2: set_Pr4071100604818519410t_unit] :
% 14.25/8.05 ( ( member6260224972018164377et_nat @ ( produc7507926704131184380et_nat @ A1 @ A22 ) @ ( conver2236330673913229642t_unit @ R2 ) )
% 14.25/8.05 = ( ? [A6: set_nat,B6: heap_e7401611519738050253t_unit] :
% 14.25/8.05 ( ( A1 = B6 )
% 14.25/8.05 & ( A22 = A6 )
% 14.25/8.05 & ( member727419437372278867t_unit @ ( produc2471974422597015918t_unit @ A6 @ B6 ) @ R2 ) ) ) ) ).
% 14.25/8.05
% 14.25/8.05 thf(fact_2518_converse_Osimps,axiom,
% 14.25/8.05 ! [A1: nat,A22: nat,R2: set_Pr1261947904930325089at_nat] :
% 14.25/8.05 ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A1 @ A22 ) @ ( converse_nat_nat @ R2 ) )
% 14.25/8.05 = ( ? [A6: nat,B6: nat] :
% 14.25/8.05 ( ( A1 = B6 )
% 14.25/8.05 & ( A22 = A6 )
% 14.25/8.05 & ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A6 @ B6 ) @ R2 ) ) ) ) ).
% 14.25/8.05
% 14.25/8.05 thf(fact_2519_converse_Osimps,axiom,
% 14.25/8.05 ! [A1: int,A22: int,R2: set_Pr958786334691620121nt_int] :
% 14.25/8.05 ( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ A1 @ A22 ) @ ( converse_int_int @ R2 ) )
% 14.25/8.05 = ( ? [A6: int,B6: int] :
% 14.25/8.05 ( ( A1 = B6 )
% 14.25/8.05 & ( A22 = A6 )
% 14.25/8.05 & ( member5262025264175285858nt_int @ ( product_Pair_int_int @ A6 @ B6 ) @ R2 ) ) ) ) ).
% 14.25/8.05
% 14.25/8.05 thf(fact_2520_converse_Osimps,axiom,
% 14.25/8.05 ! [A1: produc3658429121746597890et_nat > $o,A22: produc3658429121746597890et_nat,R2: set_Pr4208978187861016310_nat_o] :
% 14.25/8.05 ( ( member1996754912294343701et_nat @ ( produc5001842942810119800et_nat @ A1 @ A22 ) @ ( conver1831365297310299342_nat_o @ R2 ) )
% 14.25/8.05 = ( ? [A6: produc3658429121746597890et_nat,B6: produc3658429121746597890et_nat > $o] :
% 14.25/8.05 ( ( A1 = B6 )
% 14.25/8.05 & ( A22 = A6 )
% 14.25/8.05 & ( member4978015717948814295_nat_o @ ( produc2829238748666232050_nat_o @ A6 @ B6 ) @ R2 ) ) ) ) ).
% 14.25/8.05
% 14.25/8.05 thf(fact_2521_converse_Osimps,axiom,
% 14.25/8.05 ! [A1: produc3658429121746597890et_nat > $o,A22: produc3925858234332021118et_nat,R2: set_Pr7705059561495379578_nat_o] :
% 14.25/8.05 ( ( member6124377750444531601et_nat @ ( produc2245416461498447860et_nat @ A1 @ A22 ) @ ( conver5006497266169685586_nat_o @ R2 ) )
% 14.25/8.05 = ( ? [A6: produc3925858234332021118et_nat,B6: produc3658429121746597890et_nat > $o] :
% 14.25/8.05 ( ( A1 = B6 )
% 14.25/8.05 & ( A22 = A6 )
% 14.25/8.05 & ( member8354763867512313179_nat_o @ ( produc3674526432113299062_nat_o @ A6 @ B6 ) @ R2 ) ) ) ) ).
% 14.25/8.05
% 14.25/8.05 thf(fact_2522_converseD,axiom,
% 14.25/8.05 ! [A2: set_nat,B: heap_e7401611519738050253t_unit,R2: set_Pr3948176798113811640et_nat] :
% 14.25/8.05 ( ( member727419437372278867t_unit @ ( produc2471974422597015918t_unit @ A2 @ B ) @ ( conver7272282955447398104et_nat @ R2 ) )
% 14.25/8.05 => ( member6260224972018164377et_nat @ ( produc7507926704131184380et_nat @ B @ A2 ) @ R2 ) ) ).
% 14.25/8.05
% 14.25/8.05 thf(fact_2523_converseD,axiom,
% 14.25/8.05 ! [A2: produc3658429121746597890et_nat,B: produc3658429121746597890et_nat > $o,R2: set_Pr3286484037609594932et_nat] :
% 14.25/8.05 ( ( member4978015717948814295_nat_o @ ( produc2829238748666232050_nat_o @ A2 @ B ) @ ( conver4003969491454187092et_nat @ R2 ) )
% 14.25/8.05 => ( member1996754912294343701et_nat @ ( produc5001842942810119800et_nat @ B @ A2 ) @ R2 ) ) ).
% 14.25/8.05
% 14.25/8.05 thf(fact_2524_converseD,axiom,
% 14.25/8.05 ! [A2: produc3925858234332021118et_nat,B: produc3658429121746597890et_nat > $o,R2: set_Pr8536935166611901872et_nat] :
% 14.25/8.05 ( ( member8354763867512313179_nat_o @ ( produc3674526432113299062_nat_o @ A2 @ B ) @ ( conver3577387295554834384et_nat @ R2 ) )
% 14.25/8.05 => ( member6124377750444531601et_nat @ ( produc2245416461498447860et_nat @ B @ A2 ) @ R2 ) ) ).
% 14.25/8.05
% 14.25/8.05 thf(fact_2525_converseD,axiom,
% 14.25/8.05 ! [A2: heap_e7401611519738050253t_unit,B: set_nat,R2: set_Pr4071100604818519410t_unit] :
% 14.25/8.05 ( ( member6260224972018164377et_nat @ ( produc7507926704131184380et_nat @ A2 @ B ) @ ( conver2236330673913229642t_unit @ R2 ) )
% 14.25/8.05 => ( member727419437372278867t_unit @ ( produc2471974422597015918t_unit @ B @ A2 ) @ R2 ) ) ).
% 14.25/8.05
% 14.25/8.05 thf(fact_2526_converseD,axiom,
% 14.25/8.05 ! [A2: nat,B: nat,R2: set_Pr1261947904930325089at_nat] :
% 14.25/8.05 ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A2 @ B ) @ ( converse_nat_nat @ R2 ) )
% 14.25/8.05 => ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ B @ A2 ) @ R2 ) ) ).
% 14.25/8.05
% 14.25/8.05 thf(fact_2527_converseD,axiom,
% 14.25/8.05 ! [A2: int,B: int,R2: set_Pr958786334691620121nt_int] :
% 14.25/8.05 ( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ A2 @ B ) @ ( converse_int_int @ R2 ) )
% 14.25/8.05 => ( member5262025264175285858nt_int @ ( product_Pair_int_int @ B @ A2 ) @ R2 ) ) ).
% 14.25/8.05
% 14.25/8.05 thf(fact_2528_converseD,axiom,
% 14.25/8.05 ! [A2: produc3658429121746597890et_nat > $o,B: produc3658429121746597890et_nat,R2: set_Pr4208978187861016310_nat_o] :
% 14.25/8.05 ( ( member1996754912294343701et_nat @ ( produc5001842942810119800et_nat @ A2 @ B ) @ ( conver1831365297310299342_nat_o @ R2 ) )
% 14.25/8.05 => ( member4978015717948814295_nat_o @ ( produc2829238748666232050_nat_o @ B @ A2 ) @ R2 ) ) ).
% 14.25/8.05
% 14.25/8.05 thf(fact_2529_converseD,axiom,
% 14.25/8.05 ! [A2: produc3658429121746597890et_nat > $o,B: produc3925858234332021118et_nat,R2: set_Pr7705059561495379578_nat_o] :
% 14.25/8.05 ( ( member6124377750444531601et_nat @ ( produc2245416461498447860et_nat @ A2 @ B ) @ ( conver5006497266169685586_nat_o @ R2 ) )
% 14.25/8.05 => ( member8354763867512313179_nat_o @ ( produc3674526432113299062_nat_o @ B @ A2 ) @ R2 ) ) ).
% 14.25/8.05
% 14.25/8.05 thf(fact_2530_converseE,axiom,
% 14.25/8.05 ! [Yx: produc7348995623955488188t_unit,R2: set_Pr3948176798113811640et_nat] :
% 14.25/8.05 ( ( member727419437372278867t_unit @ Yx @ ( conver7272282955447398104et_nat @ R2 ) )
% 14.25/8.05 => ~ ! [X3: heap_e7401611519738050253t_unit,Y2: set_nat] :
% 14.25/8.05 ( ( Yx
% 14.25/8.05 = ( produc2471974422597015918t_unit @ Y2 @ X3 ) )
% 14.25/8.05 => ~ ( member6260224972018164377et_nat @ ( produc7507926704131184380et_nat @ X3 @ Y2 ) @ R2 ) ) ) ).
% 14.25/8.05
% 14.25/8.05 thf(fact_2531_converseE,axiom,
% 14.25/8.05 ! [Yx: produc6907119039986491712_nat_o,R2: set_Pr3286484037609594932et_nat] :
% 14.25/8.05 ( ( member4978015717948814295_nat_o @ Yx @ ( conver4003969491454187092et_nat @ R2 ) )
% 14.25/8.05 => ~ ! [X3: produc3658429121746597890et_nat > $o,Y2: produc3658429121746597890et_nat] :
% 14.25/8.05 ( ( Yx
% 14.25/8.05 = ( produc2829238748666232050_nat_o @ Y2 @ X3 ) )
% 14.25/8.05 => ~ ( member1996754912294343701et_nat @ ( produc5001842942810119800et_nat @ X3 @ Y2 ) @ R2 ) ) ) ).
% 14.25/8.05
% 14.25/8.05 thf(fact_2532_converseE,axiom,
% 14.25/8.05 ! [Yx: produc4962441903510821572_nat_o,R2: set_Pr8536935166611901872et_nat] :
% 14.25/8.05 ( ( member8354763867512313179_nat_o @ Yx @ ( conver3577387295554834384et_nat @ R2 ) )
% 14.25/8.05 => ~ ! [X3: produc3658429121746597890et_nat > $o,Y2: produc3925858234332021118et_nat] :
% 14.25/8.05 ( ( Yx
% 14.25/8.05 = ( produc3674526432113299062_nat_o @ Y2 @ X3 ) )
% 14.25/8.05 => ~ ( member6124377750444531601et_nat @ ( produc2245416461498447860et_nat @ X3 @ Y2 ) @ R2 ) ) ) ).
% 14.25/8.05
% 14.25/8.05 thf(fact_2533_converseE,axiom,
% 14.25/8.05 ! [Yx: produc3658429121746597890et_nat,R2: set_Pr4071100604818519410t_unit] :
% 14.25/8.05 ( ( member6260224972018164377et_nat @ Yx @ ( conver2236330673913229642t_unit @ R2 ) )
% 14.25/8.05 => ~ ! [X3: set_nat,Y2: heap_e7401611519738050253t_unit] :
% 14.25/8.05 ( ( Yx
% 14.25/8.05 = ( produc7507926704131184380et_nat @ Y2 @ X3 ) )
% 14.25/8.05 => ~ ( member727419437372278867t_unit @ ( produc2471974422597015918t_unit @ X3 @ Y2 ) @ R2 ) ) ) ).
% 14.25/8.05
% 14.25/8.05 thf(fact_2534_converseE,axiom,
% 14.25/8.05 ! [Yx: product_prod_nat_nat,R2: set_Pr1261947904930325089at_nat] :
% 14.25/8.05 ( ( member8440522571783428010at_nat @ Yx @ ( converse_nat_nat @ R2 ) )
% 14.25/8.05 => ~ ! [X3: nat,Y2: nat] :
% 14.25/8.05 ( ( Yx
% 14.25/8.05 = ( product_Pair_nat_nat @ Y2 @ X3 ) )
% 14.25/8.05 => ~ ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X3 @ Y2 ) @ R2 ) ) ) ).
% 14.25/8.05
% 14.25/8.05 thf(fact_2535_converseE,axiom,
% 14.25/8.05 ! [Yx: product_prod_int_int,R2: set_Pr958786334691620121nt_int] :
% 14.25/8.05 ( ( member5262025264175285858nt_int @ Yx @ ( converse_int_int @ R2 ) )
% 14.25/8.05 => ~ ! [X3: int,Y2: int] :
% 14.25/8.05 ( ( Yx
% 14.25/8.05 = ( product_Pair_int_int @ Y2 @ X3 ) )
% 14.25/8.05 => ~ ( member5262025264175285858nt_int @ ( product_Pair_int_int @ X3 @ Y2 ) @ R2 ) ) ) ).
% 14.25/8.05
% 14.25/8.05 thf(fact_2536_converseE,axiom,
% 14.25/8.05 ! [Yx: produc3925858234332021118et_nat,R2: set_Pr4208978187861016310_nat_o] :
% 14.25/8.05 ( ( member1996754912294343701et_nat @ Yx @ ( conver1831365297310299342_nat_o @ R2 ) )
% 14.25/8.05 => ~ ! [X3: produc3658429121746597890et_nat,Y2: produc3658429121746597890et_nat > $o] :
% 14.25/8.05 ( ( Yx
% 14.25/8.05 = ( produc5001842942810119800et_nat @ Y2 @ X3 ) )
% 14.25/8.05 => ~ ( member4978015717948814295_nat_o @ ( produc2829238748666232050_nat_o @ X3 @ Y2 ) @ R2 ) ) ) ).
% 14.25/8.05
% 14.25/8.05 thf(fact_2537_converseE,axiom,
% 14.25/8.05 ! [Yx: produc2732055786443039994et_nat,R2: set_Pr7705059561495379578_nat_o] :
% 14.25/8.05 ( ( member6124377750444531601et_nat @ Yx @ ( conver5006497266169685586_nat_o @ R2 ) )
% 14.25/8.05 => ~ ! [X3: produc3925858234332021118et_nat,Y2: produc3658429121746597890et_nat > $o] :
% 14.25/8.05 ( ( Yx
% 14.25/8.05 = ( produc2245416461498447860et_nat @ Y2 @ X3 ) )
% 14.25/8.05 => ~ ( member8354763867512313179_nat_o @ ( produc3674526432113299062_nat_o @ X3 @ Y2 ) @ R2 ) ) ) ).
% 14.25/8.05
% 14.25/8.05 thf(fact_2538_converseI,axiom,
% 14.25/8.05 ! [A2: set_nat,B: heap_e7401611519738050253t_unit,R2: set_Pr4071100604818519410t_unit] :
% 14.25/8.05 ( ( member727419437372278867t_unit @ ( produc2471974422597015918t_unit @ A2 @ B ) @ R2 )
% 14.25/8.05 => ( member6260224972018164377et_nat @ ( produc7507926704131184380et_nat @ B @ A2 ) @ ( conver2236330673913229642t_unit @ R2 ) ) ) ).
% 14.25/8.05
% 14.25/8.05 thf(fact_2539_converseI,axiom,
% 14.25/8.05 ! [A2: produc3658429121746597890et_nat,B: produc3658429121746597890et_nat > $o,R2: set_Pr4208978187861016310_nat_o] :
% 14.25/8.05 ( ( member4978015717948814295_nat_o @ ( produc2829238748666232050_nat_o @ A2 @ B ) @ R2 )
% 14.25/8.05 => ( member1996754912294343701et_nat @ ( produc5001842942810119800et_nat @ B @ A2 ) @ ( conver1831365297310299342_nat_o @ R2 ) ) ) ).
% 14.25/8.05
% 14.25/8.05 thf(fact_2540_converseI,axiom,
% 14.25/8.05 ! [A2: produc3925858234332021118et_nat,B: produc3658429121746597890et_nat > $o,R2: set_Pr7705059561495379578_nat_o] :
% 14.25/8.05 ( ( member8354763867512313179_nat_o @ ( produc3674526432113299062_nat_o @ A2 @ B ) @ R2 )
% 14.25/8.05 => ( member6124377750444531601et_nat @ ( produc2245416461498447860et_nat @ B @ A2 ) @ ( conver5006497266169685586_nat_o @ R2 ) ) ) ).
% 14.25/8.05
% 14.25/8.05 thf(fact_2541_converseI,axiom,
% 14.25/8.05 ! [A2: heap_e7401611519738050253t_unit,B: set_nat,R2: set_Pr3948176798113811640et_nat] :
% 14.25/8.05 ( ( member6260224972018164377et_nat @ ( produc7507926704131184380et_nat @ A2 @ B ) @ R2 )
% 14.25/8.05 => ( member727419437372278867t_unit @ ( produc2471974422597015918t_unit @ B @ A2 ) @ ( conver7272282955447398104et_nat @ R2 ) ) ) ).
% 14.25/8.05
% 14.25/8.05 thf(fact_2542_converseI,axiom,
% 14.25/8.05 ! [A2: nat,B: nat,R2: set_Pr1261947904930325089at_nat] :
% 14.25/8.05 ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A2 @ B ) @ R2 )
% 14.25/8.05 => ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ B @ A2 ) @ ( converse_nat_nat @ R2 ) ) ) ).
% 14.25/8.05
% 14.25/8.05 thf(fact_2543_converseI,axiom,
% 14.25/8.05 ! [A2: int,B: int,R2: set_Pr958786334691620121nt_int] :
% 14.25/8.05 ( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ A2 @ B ) @ R2 )
% 14.25/8.05 => ( member5262025264175285858nt_int @ ( product_Pair_int_int @ B @ A2 ) @ ( converse_int_int @ R2 ) ) ) ).
% 14.25/8.05
% 14.25/8.05 thf(fact_2544_converseI,axiom,
% 14.25/8.05 ! [A2: produc3658429121746597890et_nat > $o,B: produc3658429121746597890et_nat,R2: set_Pr3286484037609594932et_nat] :
% 14.25/8.05 ( ( member1996754912294343701et_nat @ ( produc5001842942810119800et_nat @ A2 @ B ) @ R2 )
% 14.25/8.05 => ( member4978015717948814295_nat_o @ ( produc2829238748666232050_nat_o @ B @ A2 ) @ ( conver4003969491454187092et_nat @ R2 ) ) ) ).
% 14.25/8.05
% 14.25/8.05 thf(fact_2545_converseI,axiom,
% 14.25/8.05 ! [A2: produc3658429121746597890et_nat > $o,B: produc3925858234332021118et_nat,R2: set_Pr8536935166611901872et_nat] :
% 14.25/8.05 ( ( member6124377750444531601et_nat @ ( produc2245416461498447860et_nat @ A2 @ B ) @ R2 )
% 14.25/8.05 => ( member8354763867512313179_nat_o @ ( produc3674526432113299062_nat_o @ B @ A2 ) @ ( conver3577387295554834384et_nat @ R2 ) ) ) ).
% 14.25/8.05
% 14.25/8.05 thf(fact_2546_diff__eq__diff__less__eq,axiom,
% 14.25/8.05 ! [A2: rat,B: rat,C: rat,D2: rat] :
% 14.25/8.05 ( ( ( minus_minus_rat @ A2 @ B )
% 14.25/8.05 = ( minus_minus_rat @ C @ D2 ) )
% 14.25/8.05 => ( ( ord_less_eq_rat @ A2 @ B )
% 14.25/8.05 = ( ord_less_eq_rat @ C @ D2 ) ) ) ).
% 14.25/8.05
% 14.25/8.05 thf(fact_2547_diff__eq__diff__less__eq,axiom,
% 14.25/8.05 ! [A2: int,B: int,C: int,D2: int] :
% 14.25/8.05 ( ( ( minus_minus_int @ A2 @ B )
% 14.25/8.05 = ( minus_minus_int @ C @ D2 ) )
% 14.25/8.05 => ( ( ord_less_eq_int @ A2 @ B )
% 14.25/8.05 = ( ord_less_eq_int @ C @ D2 ) ) ) ).
% 14.25/8.05
% 14.25/8.05 thf(fact_2548_diff__right__mono,axiom,
% 14.25/8.05 ! [A2: rat,B: rat,C: rat] :
% 14.25/8.05 ( ( ord_less_eq_rat @ A2 @ B )
% 14.25/8.05 => ( ord_less_eq_rat @ ( minus_minus_rat @ A2 @ C ) @ ( minus_minus_rat @ B @ C ) ) ) ).
% 14.25/8.05
% 14.25/8.05 thf(fact_2549_diff__right__mono,axiom,
% 14.25/8.05 ! [A2: int,B: int,C: int] :
% 14.25/8.05 ( ( ord_less_eq_int @ A2 @ B )
% 14.25/8.05 => ( ord_less_eq_int @ ( minus_minus_int @ A2 @ C ) @ ( minus_minus_int @ B @ C ) ) ) ).
% 14.25/8.05
% 14.25/8.05 thf(fact_2550_diff__left__mono,axiom,
% 14.25/8.05 ! [B: rat,A2: rat,C: rat] :
% 14.25/8.05 ( ( ord_less_eq_rat @ B @ A2 )
% 14.25/8.05 => ( ord_less_eq_rat @ ( minus_minus_rat @ C @ A2 ) @ ( minus_minus_rat @ C @ B ) ) ) ).
% 14.25/8.05
% 14.25/8.05 thf(fact_2551_diff__left__mono,axiom,
% 14.25/8.05 ! [B: int,A2: int,C: int] :
% 14.25/8.05 ( ( ord_less_eq_int @ B @ A2 )
% 14.25/8.05 => ( ord_less_eq_int @ ( minus_minus_int @ C @ A2 ) @ ( minus_minus_int @ C @ B ) ) ) ).
% 14.25/8.05
% 14.25/8.05 thf(fact_2552_diff__mono,axiom,
% 14.25/8.05 ! [A2: rat,B: rat,D2: rat,C: rat] :
% 14.25/8.05 ( ( ord_less_eq_rat @ A2 @ B )
% 14.25/8.05 => ( ( ord_less_eq_rat @ D2 @ C )
% 14.25/8.05 => ( ord_less_eq_rat @ ( minus_minus_rat @ A2 @ C ) @ ( minus_minus_rat @ B @ D2 ) ) ) ) ).
% 14.25/8.05
% 14.25/8.05 thf(fact_2553_diff__mono,axiom,
% 14.25/8.05 ! [A2: int,B: int,D2: int,C: int] :
% 14.25/8.05 ( ( ord_less_eq_int @ A2 @ B )
% 14.25/8.05 => ( ( ord_less_eq_int @ D2 @ C )
% 14.25/8.05 => ( ord_less_eq_int @ ( minus_minus_int @ A2 @ C ) @ ( minus_minus_int @ B @ D2 ) ) ) ) ).
% 14.25/8.05
% 14.25/8.05 thf(fact_2554_subset__image__iff,axiom,
% 14.25/8.05 ! [B4: set_Code_integer,F: code_integer > code_integer,A: set_Code_integer] :
% 14.25/8.05 ( ( ord_le7084787975880047091nteger @ B4 @ ( image_4470545334726330049nteger @ F @ A ) )
% 14.25/8.05 = ( ? [AA: set_Code_integer] :
% 14.25/8.05 ( ( ord_le7084787975880047091nteger @ AA @ A )
% 14.25/8.05 & ( B4
% 14.25/8.05 = ( image_4470545334726330049nteger @ F @ AA ) ) ) ) ) ).
% 14.25/8.05
% 14.25/8.05 thf(fact_2555_subset__image__iff,axiom,
% 14.25/8.05 ! [B4: set_set_nat,F: nat > set_nat,A: set_nat] :
% 14.25/8.05 ( ( ord_le6893508408891458716et_nat @ B4 @ ( image_nat_set_nat @ F @ A ) )
% 14.25/8.05 = ( ? [AA: set_nat] :
% 14.25/8.05 ( ( ord_less_eq_set_nat @ AA @ A )
% 14.25/8.05 & ( B4
% 14.25/8.05 = ( image_nat_set_nat @ F @ AA ) ) ) ) ) ).
% 14.25/8.05
% 14.25/8.05 thf(fact_2556_subset__image__iff,axiom,
% 14.25/8.05 ! [B4: set_Code_integer,F: nat > code_integer,A: set_nat] :
% 14.25/8.05 ( ( ord_le7084787975880047091nteger @ B4 @ ( image_1215581382706833972nteger @ F @ A ) )
% 14.25/8.05 = ( ? [AA: set_nat] :
% 14.25/8.05 ( ( ord_less_eq_set_nat @ AA @ A )
% 14.25/8.05 & ( B4
% 14.25/8.05 = ( image_1215581382706833972nteger @ F @ AA ) ) ) ) ) ).
% 14.25/8.05
% 14.25/8.05 thf(fact_2557_subset__image__iff,axiom,
% 14.25/8.05 ! [B4: set_nat,F: nat > nat,A: set_nat] :
% 14.25/8.05 ( ( ord_less_eq_set_nat @ B4 @ ( image_nat_nat2 @ F @ A ) )
% 14.25/8.05 = ( ? [AA: set_nat] :
% 14.25/8.05 ( ( ord_less_eq_set_nat @ AA @ A )
% 14.25/8.05 & ( B4
% 14.25/8.05 = ( image_nat_nat2 @ F @ AA ) ) ) ) ) ).
% 14.25/8.05
% 14.25/8.05 thf(fact_2558_image__subset__iff,axiom,
% 14.25/8.05 ! [F: nat > set_nat,A: set_nat,B4: set_set_nat] :
% 14.25/8.05 ( ( ord_le6893508408891458716et_nat @ ( image_nat_set_nat @ F @ A ) @ B4 )
% 14.25/8.05 = ( ! [X2: nat] :
% 14.25/8.05 ( ( member_nat @ X2 @ A )
% 14.25/8.05 => ( member_set_nat @ ( F @ X2 ) @ B4 ) ) ) ) ).
% 14.25/8.05
% 14.25/8.05 thf(fact_2559_image__subset__iff,axiom,
% 14.25/8.05 ! [F: code_integer > code_integer,A: set_Code_integer,B4: set_Code_integer] :
% 14.25/8.05 ( ( ord_le7084787975880047091nteger @ ( image_4470545334726330049nteger @ F @ A ) @ B4 )
% 14.25/8.05 = ( ! [X2: code_integer] :
% 14.25/8.05 ( ( member_Code_integer @ X2 @ A )
% 14.25/8.05 => ( member_Code_integer @ ( F @ X2 ) @ B4 ) ) ) ) ).
% 14.25/8.05
% 14.25/8.05 thf(fact_2560_image__subset__iff,axiom,
% 14.25/8.05 ! [F: nat > code_integer,A: set_nat,B4: set_Code_integer] :
% 14.25/8.05 ( ( ord_le7084787975880047091nteger @ ( image_1215581382706833972nteger @ F @ A ) @ B4 )
% 14.25/8.05 = ( ! [X2: nat] :
% 14.25/8.05 ( ( member_nat @ X2 @ A )
% 14.25/8.05 => ( member_Code_integer @ ( F @ X2 ) @ B4 ) ) ) ) ).
% 14.25/8.05
% 14.25/8.05 thf(fact_2561_image__subset__iff,axiom,
% 14.25/8.05 ! [F: nat > nat,A: set_nat,B4: set_nat] :
% 14.25/8.05 ( ( ord_less_eq_set_nat @ ( image_nat_nat2 @ F @ A ) @ B4 )
% 14.25/8.05 = ( ! [X2: nat] :
% 14.25/8.05 ( ( member_nat @ X2 @ A )
% 14.25/8.05 => ( member_nat @ ( F @ X2 ) @ B4 ) ) ) ) ).
% 14.25/8.05
% 14.25/8.05 thf(fact_2562_subset__imageE,axiom,
% 14.25/8.05 ! [B4: set_Code_integer,F: code_integer > code_integer,A: set_Code_integer] :
% 14.25/8.05 ( ( ord_le7084787975880047091nteger @ B4 @ ( image_4470545334726330049nteger @ F @ A ) )
% 14.25/8.05 => ~ ! [C6: set_Code_integer] :
% 14.25/8.05 ( ( ord_le7084787975880047091nteger @ C6 @ A )
% 14.25/8.05 => ( B4
% 14.25/8.05 != ( image_4470545334726330049nteger @ F @ C6 ) ) ) ) ).
% 14.25/8.05
% 14.25/8.05 thf(fact_2563_subset__imageE,axiom,
% 14.25/8.05 ! [B4: set_set_nat,F: nat > set_nat,A: set_nat] :
% 14.25/8.05 ( ( ord_le6893508408891458716et_nat @ B4 @ ( image_nat_set_nat @ F @ A ) )
% 14.25/8.05 => ~ ! [C6: set_nat] :
% 14.25/8.05 ( ( ord_less_eq_set_nat @ C6 @ A )
% 14.25/8.05 => ( B4
% 14.25/8.05 != ( image_nat_set_nat @ F @ C6 ) ) ) ) ).
% 14.25/8.05
% 14.25/8.05 thf(fact_2564_subset__imageE,axiom,
% 14.25/8.05 ! [B4: set_Code_integer,F: nat > code_integer,A: set_nat] :
% 14.25/8.05 ( ( ord_le7084787975880047091nteger @ B4 @ ( image_1215581382706833972nteger @ F @ A ) )
% 14.25/8.05 => ~ ! [C6: set_nat] :
% 14.25/8.05 ( ( ord_less_eq_set_nat @ C6 @ A )
% 14.25/8.05 => ( B4
% 14.25/8.05 != ( image_1215581382706833972nteger @ F @ C6 ) ) ) ) ).
% 14.25/8.05
% 14.25/8.05 thf(fact_2565_subset__imageE,axiom,
% 14.25/8.05 ! [B4: set_nat,F: nat > nat,A: set_nat] :
% 14.25/8.05 ( ( ord_less_eq_set_nat @ B4 @ ( image_nat_nat2 @ F @ A ) )
% 14.25/8.05 => ~ ! [C6: set_nat] :
% 14.25/8.05 ( ( ord_less_eq_set_nat @ C6 @ A )
% 14.25/8.05 => ( B4
% 14.25/8.05 != ( image_nat_nat2 @ F @ C6 ) ) ) ) ).
% 14.25/8.05
% 14.25/8.05 thf(fact_2566_image__subsetI,axiom,
% 14.25/8.05 ! [A: set_Code_integer,F: code_integer > code_integer,B4: set_Code_integer] :
% 14.25/8.05 ( ! [X3: code_integer] :
% 14.25/8.05 ( ( member_Code_integer @ X3 @ A )
% 14.25/8.05 => ( member_Code_integer @ ( F @ X3 ) @ B4 ) )
% 14.25/8.05 => ( ord_le7084787975880047091nteger @ ( image_4470545334726330049nteger @ F @ A ) @ B4 ) ) ).
% 14.25/8.05
% 14.25/8.05 thf(fact_2567_image__subsetI,axiom,
% 14.25/8.05 ! [A: set_nat,F: nat > code_integer,B4: set_Code_integer] :
% 14.25/8.05 ( ! [X3: nat] :
% 14.25/8.05 ( ( member_nat @ X3 @ A )
% 14.25/8.05 => ( member_Code_integer @ ( F @ X3 ) @ B4 ) )
% 14.25/8.05 => ( ord_le7084787975880047091nteger @ ( image_1215581382706833972nteger @ F @ A ) @ B4 ) ) ).
% 14.25/8.05
% 14.25/8.05 thf(fact_2568_image__subsetI,axiom,
% 14.25/8.05 ! [A: set_nat,F: nat > int,B4: set_int] :
% 14.25/8.05 ( ! [X3: nat] :
% 14.25/8.05 ( ( member_nat @ X3 @ A )
% 14.25/8.05 => ( member_int @ ( F @ X3 ) @ B4 ) )
% 14.25/8.05 => ( ord_less_eq_set_int @ ( image_nat_int2 @ F @ A ) @ B4 ) ) ).
% 14.25/8.05
% 14.25/8.05 thf(fact_2569_image__subsetI,axiom,
% 14.25/8.05 ! [A: set_nat,F: nat > $o,B4: set_o] :
% 14.25/8.05 ( ! [X3: nat] :
% 14.25/8.05 ( ( member_nat @ X3 @ A )
% 14.25/8.05 => ( member_o @ ( F @ X3 ) @ B4 ) )
% 14.25/8.05 => ( ord_less_eq_set_o @ ( image_nat_o2 @ F @ A ) @ B4 ) ) ).
% 14.25/8.05
% 14.25/8.05 thf(fact_2570_image__subsetI,axiom,
% 14.25/8.05 ! [A: set_nat,F: nat > product_unit,B4: set_Product_unit] :
% 14.25/8.05 ( ! [X3: nat] :
% 14.25/8.05 ( ( member_nat @ X3 @ A )
% 14.25/8.05 => ( member_Product_unit @ ( F @ X3 ) @ B4 ) )
% 14.25/8.05 => ( ord_le3507040750410214029t_unit @ ( image_8730104196221521654t_unit @ F @ A ) @ B4 ) ) ).
% 14.25/8.05
% 14.25/8.05 thf(fact_2571_image__subsetI,axiom,
% 14.25/8.05 ! [A: set_int,F: int > int,B4: set_int] :
% 14.25/8.05 ( ! [X3: int] :
% 14.25/8.05 ( ( member_int @ X3 @ A )
% 14.25/8.05 => ( member_int @ ( F @ X3 ) @ B4 ) )
% 14.25/8.05 => ( ord_less_eq_set_int @ ( image_int_int2 @ F @ A ) @ B4 ) ) ).
% 14.25/8.05
% 14.25/8.05 thf(fact_2572_image__subsetI,axiom,
% 14.25/8.05 ! [A: set_int,F: int > $o,B4: set_o] :
% 14.25/8.05 ( ! [X3: int] :
% 14.25/8.05 ( ( member_int @ X3 @ A )
% 14.25/8.05 => ( member_o @ ( F @ X3 ) @ B4 ) )
% 14.25/8.05 => ( ord_less_eq_set_o @ ( image_int_o2 @ F @ A ) @ B4 ) ) ).
% 14.25/8.05
% 14.25/8.05 thf(fact_2573_image__subsetI,axiom,
% 14.25/8.05 ! [A: set_int,F: int > product_unit,B4: set_Product_unit] :
% 14.25/8.05 ( ! [X3: int] :
% 14.25/8.05 ( ( member_int @ X3 @ A )
% 14.25/8.05 => ( member_Product_unit @ ( F @ X3 ) @ B4 ) )
% 14.25/8.05 => ( ord_le3507040750410214029t_unit @ ( image_5846526143423701914t_unit @ F @ A ) @ B4 ) ) ).
% 14.25/8.05
% 14.25/8.05 thf(fact_2574_image__subsetI,axiom,
% 14.25/8.05 ! [A: set_o,F: $o > int,B4: set_int] :
% 14.25/8.05 ( ! [X3: $o] :
% 14.25/8.05 ( ( member_o @ X3 @ A )
% 14.25/8.05 => ( member_int @ ( F @ X3 ) @ B4 ) )
% 14.25/8.05 => ( ord_less_eq_set_int @ ( image_o_int2 @ F @ A ) @ B4 ) ) ).
% 14.25/8.05
% 14.25/8.05 thf(fact_2575_image__subsetI,axiom,
% 14.25/8.05 ! [A: set_o,F: $o > $o,B4: set_o] :
% 14.25/8.05 ( ! [X3: $o] :
% 14.25/8.05 ( ( member_o @ X3 @ A )
% 14.25/8.05 => ( member_o @ ( F @ X3 ) @ B4 ) )
% 14.25/8.05 => ( ord_less_eq_set_o @ ( image_o_o2 @ F @ A ) @ B4 ) ) ).
% 14.25/8.05
% 14.25/8.05 thf(fact_2576_image__mono,axiom,
% 14.25/8.05 ! [A: set_Code_integer,B4: set_Code_integer,F: code_integer > code_integer] :
% 14.25/8.05 ( ( ord_le7084787975880047091nteger @ A @ B4 )
% 14.25/8.05 => ( ord_le7084787975880047091nteger @ ( image_4470545334726330049nteger @ F @ A ) @ ( image_4470545334726330049nteger @ F @ B4 ) ) ) ).
% 14.25/8.05
% 14.25/8.05 thf(fact_2577_image__mono,axiom,
% 14.25/8.05 ! [A: set_nat,B4: set_nat,F: nat > set_nat] :
% 14.25/8.05 ( ( ord_less_eq_set_nat @ A @ B4 )
% 14.25/8.05 => ( ord_le6893508408891458716et_nat @ ( image_nat_set_nat @ F @ A ) @ ( image_nat_set_nat @ F @ B4 ) ) ) ).
% 14.25/8.05
% 14.25/8.05 thf(fact_2578_image__mono,axiom,
% 14.25/8.05 ! [A: set_nat,B4: set_nat,F: nat > code_integer] :
% 14.25/8.05 ( ( ord_less_eq_set_nat @ A @ B4 )
% 14.25/8.05 => ( ord_le7084787975880047091nteger @ ( image_1215581382706833972nteger @ F @ A ) @ ( image_1215581382706833972nteger @ F @ B4 ) ) ) ).
% 14.25/8.05
% 14.25/8.05 thf(fact_2579_image__mono,axiom,
% 14.25/8.05 ! [A: set_nat,B4: set_nat,F: nat > nat] :
% 14.25/8.05 ( ( ord_less_eq_set_nat @ A @ B4 )
% 14.25/8.05 => ( ord_less_eq_set_nat @ ( image_nat_nat2 @ F @ A ) @ ( image_nat_nat2 @ F @ B4 ) ) ) ).
% 14.25/8.05
% 14.25/8.05 thf(fact_2580_image__Un,axiom,
% 14.25/8.05 ! [F: code_integer > code_integer,A: set_Code_integer,B4: set_Code_integer] :
% 14.25/8.05 ( ( image_4470545334726330049nteger @ F @ ( sup_su848401254843788991nteger @ A @ B4 ) )
% 14.25/8.05 = ( sup_su848401254843788991nteger @ ( image_4470545334726330049nteger @ F @ A ) @ ( image_4470545334726330049nteger @ F @ B4 ) ) ) ).
% 14.25/8.05
% 14.25/8.05 thf(fact_2581_image__Un,axiom,
% 14.25/8.05 ! [F: nat > set_nat,A: set_nat,B4: set_nat] :
% 14.25/8.05 ( ( image_nat_set_nat @ F @ ( sup_sup_set_nat @ A @ B4 ) )
% 14.25/8.05 = ( sup_sup_set_set_nat @ ( image_nat_set_nat @ F @ A ) @ ( image_nat_set_nat @ F @ B4 ) ) ) ).
% 14.25/8.05
% 14.25/8.05 thf(fact_2582_image__Un,axiom,
% 14.25/8.05 ! [F: nat > code_integer,A: set_nat,B4: set_nat] :
% 14.25/8.05 ( ( image_1215581382706833972nteger @ F @ ( sup_sup_set_nat @ A @ B4 ) )
% 14.25/8.05 = ( sup_su848401254843788991nteger @ ( image_1215581382706833972nteger @ F @ A ) @ ( image_1215581382706833972nteger @ F @ B4 ) ) ) ).
% 14.25/8.05
% 14.25/8.05 thf(fact_2583_image__Un,axiom,
% 14.25/8.05 ! [F: nat > nat,A: set_nat,B4: set_nat] :
% 14.25/8.05 ( ( image_nat_nat2 @ F @ ( sup_sup_set_nat @ A @ B4 ) )
% 14.25/8.05 = ( sup_sup_set_nat @ ( image_nat_nat2 @ F @ A ) @ ( image_nat_nat2 @ F @ B4 ) ) ) ).
% 14.25/8.05
% 14.25/8.05 thf(fact_2584_double__diff,axiom,
% 14.25/8.05 ! [A: set_nat,B4: set_nat,C5: set_nat] :
% 14.25/8.05 ( ( ord_less_eq_set_nat @ A @ B4 )
% 14.25/8.05 => ( ( ord_less_eq_set_nat @ B4 @ C5 )
% 14.25/8.05 => ( ( minus_minus_set_nat @ B4 @ ( minus_minus_set_nat @ C5 @ A ) )
% 14.25/8.05 = A ) ) ) ).
% 14.25/8.05
% 14.25/8.05 thf(fact_2585_Diff__subset,axiom,
% 14.25/8.05 ! [A: set_nat,B4: set_nat] : ( ord_less_eq_set_nat @ ( minus_minus_set_nat @ A @ B4 ) @ A ) ).
% 14.25/8.05
% 14.25/8.05 thf(fact_2586_Diff__mono,axiom,
% 14.25/8.05 ! [A: set_nat,C5: set_nat,D3: set_nat,B4: set_nat] :
% 14.25/8.05 ( ( ord_less_eq_set_nat @ A @ C5 )
% 14.25/8.05 => ( ( ord_less_eq_set_nat @ D3 @ B4 )
% 14.25/8.05 => ( ord_less_eq_set_nat @ ( minus_minus_set_nat @ A @ B4 ) @ ( minus_minus_set_nat @ C5 @ D3 ) ) ) ) ).
% 14.25/8.05
% 14.25/8.05 thf(fact_2587_insert__Diff__if,axiom,
% 14.25/8.05 ! [X: produc3658429121746597890et_nat > $o,B4: set_Pr4532377907799695533_nat_o,A: set_Pr4532377907799695533_nat_o] :
% 14.25/8.05 ( ( ( member6576561426505652726_nat_o @ X @ B4 )
% 14.25/8.05 => ( ( minus_1801376950450012436_nat_o @ ( insert5175938949040314269_nat_o @ X @ A ) @ B4 )
% 14.25/8.05 = ( minus_1801376950450012436_nat_o @ A @ B4 ) ) )
% 14.25/8.05 & ( ~ ( member6576561426505652726_nat_o @ X @ B4 )
% 14.25/8.05 => ( ( minus_1801376950450012436_nat_o @ ( insert5175938949040314269_nat_o @ X @ A ) @ B4 )
% 14.25/8.05 = ( insert5175938949040314269_nat_o @ X @ ( minus_1801376950450012436_nat_o @ A @ B4 ) ) ) ) ) ).
% 14.25/8.05
% 14.25/8.05 thf(fact_2588_insert__Diff__if,axiom,
% 14.25/8.05 ! [X: int,B4: set_int,A: set_int] :
% 14.25/8.05 ( ( ( member_int @ X @ B4 )
% 14.25/8.05 => ( ( minus_minus_set_int @ ( insert_int @ X @ A ) @ B4 )
% 14.25/8.05 = ( minus_minus_set_int @ A @ B4 ) ) )
% 14.25/8.05 & ( ~ ( member_int @ X @ B4 )
% 14.25/8.05 => ( ( minus_minus_set_int @ ( insert_int @ X @ A ) @ B4 )
% 14.25/8.05 = ( insert_int @ X @ ( minus_minus_set_int @ A @ B4 ) ) ) ) ) ).
% 14.25/8.05
% 14.25/8.05 thf(fact_2589_insert__Diff__if,axiom,
% 14.25/8.05 ! [X: $o,B4: set_o,A: set_o] :
% 14.25/8.05 ( ( ( member_o @ X @ B4 )
% 14.25/8.05 => ( ( minus_minus_set_o @ ( insert_o @ X @ A ) @ B4 )
% 14.25/8.05 = ( minus_minus_set_o @ A @ B4 ) ) )
% 14.25/8.05 & ( ~ ( member_o @ X @ B4 )
% 14.25/8.05 => ( ( minus_minus_set_o @ ( insert_o @ X @ A ) @ B4 )
% 14.25/8.05 = ( insert_o @ X @ ( minus_minus_set_o @ A @ B4 ) ) ) ) ) ).
% 14.25/8.05
% 14.25/8.05 thf(fact_2590_insert__Diff__if,axiom,
% 14.25/8.05 ! [X: product_unit,B4: set_Product_unit,A: set_Product_unit] :
% 14.25/8.05 ( ( ( member_Product_unit @ X @ B4 )
% 14.25/8.05 => ( ( minus_6452836326544984404t_unit @ ( insert_Product_unit @ X @ A ) @ B4 )
% 14.25/8.05 = ( minus_6452836326544984404t_unit @ A @ B4 ) ) )
% 14.25/8.05 & ( ~ ( member_Product_unit @ X @ B4 )
% 14.25/8.05 => ( ( minus_6452836326544984404t_unit @ ( insert_Product_unit @ X @ A ) @ B4 )
% 14.25/8.05 = ( insert_Product_unit @ X @ ( minus_6452836326544984404t_unit @ A @ B4 ) ) ) ) ) ).
% 14.25/8.05
% 14.25/8.05 thf(fact_2591_insert__Diff__if,axiom,
% 14.25/8.05 ! [X: nat,B4: set_nat,A: set_nat] :
% 14.25/8.05 ( ( ( member_nat @ X @ B4 )
% 14.25/8.05 => ( ( minus_minus_set_nat @ ( insert_nat @ X @ A ) @ B4 )
% 14.25/8.05 = ( minus_minus_set_nat @ A @ B4 ) ) )
% 14.25/8.05 & ( ~ ( member_nat @ X @ B4 )
% 14.25/8.05 => ( ( minus_minus_set_nat @ ( insert_nat @ X @ A ) @ B4 )
% 14.25/8.05 = ( insert_nat @ X @ ( minus_minus_set_nat @ A @ B4 ) ) ) ) ) ).
% 14.25/8.05
% 14.25/8.05 thf(fact_2592_Diff__Int__distrib2,axiom,
% 14.25/8.05 ! [A: set_nat,B4: set_nat,C5: set_nat] :
% 14.25/8.05 ( ( inf_inf_set_nat @ ( minus_minus_set_nat @ A @ B4 ) @ C5 )
% 14.25/8.05 = ( minus_minus_set_nat @ ( inf_inf_set_nat @ A @ C5 ) @ ( inf_inf_set_nat @ B4 @ C5 ) ) ) ).
% 14.25/8.05
% 14.25/8.05 thf(fact_2593_Diff__Int__distrib,axiom,
% 14.25/8.05 ! [C5: set_nat,A: set_nat,B4: set_nat] :
% 14.25/8.05 ( ( inf_inf_set_nat @ C5 @ ( minus_minus_set_nat @ A @ B4 ) )
% 14.25/8.05 = ( minus_minus_set_nat @ ( inf_inf_set_nat @ C5 @ A ) @ ( inf_inf_set_nat @ C5 @ B4 ) ) ) ).
% 14.25/8.05
% 14.25/8.05 thf(fact_2594_Diff__Diff__Int,axiom,
% 14.25/8.05 ! [A: set_nat,B4: set_nat] :
% 14.25/8.05 ( ( minus_minus_set_nat @ A @ ( minus_minus_set_nat @ A @ B4 ) )
% 14.25/8.05 = ( inf_inf_set_nat @ A @ B4 ) ) ).
% 14.25/8.05
% 14.25/8.05 thf(fact_2595_Diff__Int2,axiom,
% 14.25/8.05 ! [A: set_nat,C5: set_nat,B4: set_nat] :
% 14.25/8.05 ( ( minus_minus_set_nat @ ( inf_inf_set_nat @ A @ C5 ) @ ( inf_inf_set_nat @ B4 @ C5 ) )
% 14.25/8.05 = ( minus_minus_set_nat @ ( inf_inf_set_nat @ A @ C5 ) @ B4 ) ) ).
% 14.25/8.05
% 14.25/8.05 thf(fact_2596_Int__Diff,axiom,
% 14.25/8.05 ! [A: set_nat,B4: set_nat,C5: set_nat] :
% 14.25/8.05 ( ( minus_minus_set_nat @ ( inf_inf_set_nat @ A @ B4 ) @ C5 )
% 14.25/8.05 = ( inf_inf_set_nat @ A @ ( minus_minus_set_nat @ B4 @ C5 ) ) ) ).
% 14.25/8.05
% 14.25/8.05 thf(fact_2597_set__diff__diff__left,axiom,
% 14.25/8.05 ! [A: set_nat,B4: set_nat,C5: set_nat] :
% 14.25/8.05 ( ( minus_minus_set_nat @ ( minus_minus_set_nat @ A @ B4 ) @ C5 )
% 14.25/8.05 = ( minus_minus_set_nat @ A @ ( sup_sup_set_nat @ B4 @ C5 ) ) ) ).
% 14.25/8.05
% 14.25/8.05 thf(fact_2598_Un__Diff,axiom,
% 14.25/8.05 ! [A: set_nat,B4: set_nat,C5: set_nat] :
% 14.25/8.05 ( ( minus_minus_set_nat @ ( sup_sup_set_nat @ A @ B4 ) @ C5 )
% 14.25/8.05 = ( sup_sup_set_nat @ ( minus_minus_set_nat @ A @ C5 ) @ ( minus_minus_set_nat @ B4 @ C5 ) ) ) ).
% 14.25/8.05
% 14.25/8.05 thf(fact_2599_vimage__Diff,axiom,
% 14.25/8.05 ! [F: nat > nat,A: set_nat,B4: set_nat] :
% 14.25/8.05 ( ( vimage_nat_nat @ F @ ( minus_minus_set_nat @ A @ B4 ) )
% 14.25/8.05 = ( minus_minus_set_nat @ ( vimage_nat_nat @ F @ A ) @ ( vimage_nat_nat @ F @ B4 ) ) ) ).
% 14.25/8.05
% 14.25/8.05 thf(fact_2600_rtrancl__converseI,axiom,
% 14.25/8.05 ! [Y: nat,X: nat,R2: set_Pr1261947904930325089at_nat] :
% 14.25/8.05 ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ Y @ X ) @ ( transi2905341329935302413cl_nat @ R2 ) )
% 14.25/8.05 => ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ Y ) @ ( transi2905341329935302413cl_nat @ ( converse_nat_nat @ R2 ) ) ) ) ).
% 14.25/8.05
% 14.25/8.05 thf(fact_2601_rtrancl__converseI,axiom,
% 14.25/8.05 ! [Y: int,X: int,R2: set_Pr958786334691620121nt_int] :
% 14.25/8.05 ( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ Y @ X ) @ ( transi2902850859426252137cl_int @ R2 ) )
% 14.25/8.05 => ( member5262025264175285858nt_int @ ( product_Pair_int_int @ X @ Y ) @ ( transi2902850859426252137cl_int @ ( converse_int_int @ R2 ) ) ) ) ).
% 14.25/8.05
% 14.25/8.05 thf(fact_2602_rtrancl__converseD,axiom,
% 14.25/8.05 ! [X: nat,Y: nat,R2: set_Pr1261947904930325089at_nat] :
% 14.25/8.05 ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ Y ) @ ( transi2905341329935302413cl_nat @ ( converse_nat_nat @ R2 ) ) )
% 14.25/8.05 => ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ Y @ X ) @ ( transi2905341329935302413cl_nat @ R2 ) ) ) ).
% 14.25/8.05
% 14.25/8.05 thf(fact_2603_rtrancl__converseD,axiom,
% 14.25/8.05 ! [X: int,Y: int,R2: set_Pr958786334691620121nt_int] :
% 14.25/8.05 ( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ X @ Y ) @ ( transi2902850859426252137cl_int @ ( converse_int_int @ R2 ) ) )
% 14.25/8.05 => ( member5262025264175285858nt_int @ ( product_Pair_int_int @ Y @ X ) @ ( transi2902850859426252137cl_int @ R2 ) ) ) ).
% 14.25/8.05
% 14.25/8.05 thf(fact_2604_trancl__converseI,axiom,
% 14.25/8.05 ! [X: nat,Y: nat,R2: set_Pr1261947904930325089at_nat] :
% 14.25/8.05 ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ Y ) @ ( converse_nat_nat @ ( transi6264000038957366511cl_nat @ R2 ) ) )
% 14.25/8.05 => ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ Y ) @ ( transi6264000038957366511cl_nat @ ( converse_nat_nat @ R2 ) ) ) ) ).
% 14.25/8.05
% 14.25/8.05 thf(fact_2605_trancl__converseI,axiom,
% 14.25/8.05 ! [X: int,Y: int,R2: set_Pr958786334691620121nt_int] :
% 14.25/8.05 ( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ X @ Y ) @ ( converse_int_int @ ( transi6261509568448316235cl_int @ R2 ) ) )
% 14.25/8.05 => ( member5262025264175285858nt_int @ ( product_Pair_int_int @ X @ Y ) @ ( transi6261509568448316235cl_int @ ( converse_int_int @ R2 ) ) ) ) ).
% 14.25/8.05
% 14.25/8.05 thf(fact_2606_trancl__converseD,axiom,
% 14.25/8.05 ! [X: nat,Y: nat,R2: set_Pr1261947904930325089at_nat] :
% 14.25/8.05 ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ Y ) @ ( transi6264000038957366511cl_nat @ ( converse_nat_nat @ R2 ) ) )
% 14.25/8.05 => ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ Y ) @ ( converse_nat_nat @ ( transi6264000038957366511cl_nat @ R2 ) ) ) ) ).
% 14.25/8.05
% 14.25/8.05 thf(fact_2607_trancl__converseD,axiom,
% 14.25/8.05 ! [X: int,Y: int,R2: set_Pr958786334691620121nt_int] :
% 14.25/8.05 ( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ X @ Y ) @ ( transi6261509568448316235cl_int @ ( converse_int_int @ R2 ) ) )
% 14.25/8.05 => ( member5262025264175285858nt_int @ ( product_Pair_int_int @ X @ Y ) @ ( converse_int_int @ ( transi6261509568448316235cl_int @ R2 ) ) ) ) ).
% 14.25/8.05
% 14.25/8.05 thf(fact_2608_diff__shunt__var,axiom,
% 14.25/8.05 ! [X: set_Product_unit,Y: set_Product_unit] :
% 14.25/8.05 ( ( ( minus_6452836326544984404t_unit @ X @ Y )
% 14.25/8.05 = bot_bo3957492148770167129t_unit )
% 14.25/8.05 = ( ord_le3507040750410214029t_unit @ X @ Y ) ) ).
% 14.25/8.05
% 14.25/8.05 thf(fact_2609_diff__shunt__var,axiom,
% 14.25/8.05 ! [X: set_o,Y: set_o] :
% 14.25/8.05 ( ( ( minus_minus_set_o @ X @ Y )
% 14.25/8.05 = bot_bot_set_o )
% 14.25/8.05 = ( ord_less_eq_set_o @ X @ Y ) ) ).
% 14.25/8.05
% 14.25/8.05 thf(fact_2610_diff__shunt__var,axiom,
% 14.25/8.05 ! [X: set_int,Y: set_int] :
% 14.25/8.05 ( ( ( minus_minus_set_int @ X @ Y )
% 14.25/8.05 = bot_bot_set_int )
% 14.25/8.05 = ( ord_less_eq_set_int @ X @ Y ) ) ).
% 14.25/8.05
% 14.25/8.05 thf(fact_2611_diff__shunt__var,axiom,
% 14.25/8.05 ! [X: set_Pr4532377907799695533_nat_o,Y: set_Pr4532377907799695533_nat_o] :
% 14.25/8.05 ( ( ( minus_1801376950450012436_nat_o @ X @ Y )
% 14.25/8.05 = bot_bo7824918357723582233_nat_o )
% 14.25/8.05 = ( ord_le2965882846123202637_nat_o @ X @ Y ) ) ).
% 14.25/8.05
% 14.25/8.05 thf(fact_2612_diff__shunt__var,axiom,
% 14.25/8.05 ! [X: assn,Y: assn] :
% 14.25/8.05 ( ( ( minus_minus_assn @ X @ Y )
% 14.25/8.05 = bot_bot_assn )
% 14.25/8.05 = ( ord_less_eq_assn @ X @ Y ) ) ).
% 14.25/8.05
% 14.25/8.05 thf(fact_2613_diff__shunt__var,axiom,
% 14.25/8.05 ! [X: set_nat,Y: set_nat] :
% 14.25/8.05 ( ( ( minus_minus_set_nat @ X @ Y )
% 14.25/8.05 = bot_bot_set_nat )
% 14.25/8.05 = ( ord_less_eq_set_nat @ X @ Y ) ) ).
% 14.25/8.05
% 14.25/8.05 thf(fact_2614_in__snd__imageE,axiom,
% 14.25/8.05 ! [Y: nat,S: set_Pr1261947904930325089at_nat] :
% 14.25/8.05 ( ( member_nat @ Y @ ( image_2486076414777270412at_nat @ product_snd_nat_nat @ S ) )
% 14.25/8.05 => ~ ! [X3: nat] :
% 14.25/8.05 ~ ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X3 @ Y ) @ S ) ) ).
% 14.25/8.05
% 14.25/8.05 thf(fact_2615_in__snd__imageE,axiom,
% 14.25/8.05 ! [Y: produc3658429121746597890et_nat,S: set_Pr3286484037609594932et_nat] :
% 14.25/8.05 ( ( member6260224972018164377et_nat @ Y @ ( image_4390311411407041927et_nat @ produc4011572625026189258et_nat @ S ) )
% 14.25/8.05 => ~ ! [X3: produc3658429121746597890et_nat > $o] :
% 14.25/8.05 ~ ( member1996754912294343701et_nat @ ( produc5001842942810119800et_nat @ X3 @ Y ) @ S ) ) ).
% 14.25/8.05
% 14.25/8.05 thf(fact_2616_in__snd__imageE,axiom,
% 14.25/8.05 ! [Y: produc3925858234332021118et_nat,S: set_Pr8536935166611901872et_nat] :
% 14.25/8.05 ( ( member1996754912294343701et_nat @ Y @ ( image_4557310310409863807et_nat @ produc5374455773327741254et_nat @ S ) )
% 14.25/8.05 => ~ ! [X3: produc3658429121746597890et_nat > $o] :
% 14.25/8.05 ~ ( member6124377750444531601et_nat @ ( produc2245416461498447860et_nat @ X3 @ Y ) @ S ) ) ).
% 14.25/8.05
% 14.25/8.05 thf(fact_2617_in__snd__imageE,axiom,
% 14.25/8.05 ! [Y: set_nat,S: set_Pr3948176798113811640et_nat] :
% 14.25/8.05 ( ( member_set_nat @ Y @ ( image_6088997261112373647et_nat @ produc8586169260539613262et_nat @ S ) )
% 14.25/8.05 => ~ ! [X3: heap_e7401611519738050253t_unit] :
% 14.25/8.05 ~ ( member6260224972018164377et_nat @ ( produc7507926704131184380et_nat @ X3 @ Y ) @ S ) ) ).
% 14.25/8.05
% 14.25/8.05 thf(fact_2618_in__snd__imageE,axiom,
% 14.25/8.05 ! [Y: int,S: set_Pr958786334691620121nt_int] :
% 14.25/8.05 ( ( member_int @ Y @ ( image_5042161079198086560nt_int @ product_snd_int_int @ S ) )
% 14.25/8.05 => ~ ! [X3: int] :
% 14.25/8.05 ~ ( member5262025264175285858nt_int @ ( product_Pair_int_int @ X3 @ Y ) @ S ) ) ).
% 14.25/8.05
% 14.25/8.05 thf(fact_2619_image__Int__subset,axiom,
% 14.25/8.05 ! [F: code_integer > code_integer,A: set_Code_integer,B4: set_Code_integer] : ( ord_le7084787975880047091nteger @ ( image_4470545334726330049nteger @ F @ ( inf_in1364745209274528805nteger @ A @ B4 ) ) @ ( inf_in1364745209274528805nteger @ ( image_4470545334726330049nteger @ F @ A ) @ ( image_4470545334726330049nteger @ F @ B4 ) ) ) ).
% 14.25/8.05
% 14.25/8.05 thf(fact_2620_image__Int__subset,axiom,
% 14.25/8.05 ! [F: nat > set_nat,A: set_nat,B4: set_nat] : ( ord_le6893508408891458716et_nat @ ( image_nat_set_nat @ F @ ( inf_inf_set_nat @ A @ B4 ) ) @ ( inf_inf_set_set_nat @ ( image_nat_set_nat @ F @ A ) @ ( image_nat_set_nat @ F @ B4 ) ) ) ).
% 14.25/8.05
% 14.25/8.05 thf(fact_2621_image__Int__subset,axiom,
% 14.25/8.05 ! [F: nat > code_integer,A: set_nat,B4: set_nat] : ( ord_le7084787975880047091nteger @ ( image_1215581382706833972nteger @ F @ ( inf_inf_set_nat @ A @ B4 ) ) @ ( inf_in1364745209274528805nteger @ ( image_1215581382706833972nteger @ F @ A ) @ ( image_1215581382706833972nteger @ F @ B4 ) ) ) ).
% 14.25/8.05
% 14.25/8.05 thf(fact_2622_image__Int__subset,axiom,
% 14.25/8.05 ! [F: nat > nat,A: set_nat,B4: set_nat] : ( ord_less_eq_set_nat @ ( image_nat_nat2 @ F @ ( inf_inf_set_nat @ A @ B4 ) ) @ ( inf_inf_set_nat @ ( image_nat_nat2 @ F @ A ) @ ( image_nat_nat2 @ F @ B4 ) ) ) ).
% 14.25/8.05
% 14.25/8.05 thf(fact_2623_subset__minus__empty,axiom,
% 14.25/8.05 ! [A: set_Product_unit,B4: set_Product_unit] :
% 14.25/8.05 ( ( ord_le3507040750410214029t_unit @ A @ B4 )
% 14.25/8.05 => ( ( minus_6452836326544984404t_unit @ A @ B4 )
% 14.25/8.05 = bot_bo3957492148770167129t_unit ) ) ).
% 14.25/8.05
% 14.25/8.05 thf(fact_2624_subset__minus__empty,axiom,
% 14.25/8.05 ! [A: set_o,B4: set_o] :
% 14.25/8.05 ( ( ord_less_eq_set_o @ A @ B4 )
% 14.25/8.05 => ( ( minus_minus_set_o @ A @ B4 )
% 14.25/8.05 = bot_bot_set_o ) ) ).
% 14.25/8.05
% 14.25/8.05 thf(fact_2625_subset__minus__empty,axiom,
% 14.25/8.05 ! [A: set_int,B4: set_int] :
% 14.25/8.05 ( ( ord_less_eq_set_int @ A @ B4 )
% 14.25/8.05 => ( ( minus_minus_set_int @ A @ B4 )
% 14.25/8.05 = bot_bot_set_int ) ) ).
% 14.25/8.05
% 14.25/8.05 thf(fact_2626_subset__minus__empty,axiom,
% 14.25/8.05 ! [A: set_Pr4532377907799695533_nat_o,B4: set_Pr4532377907799695533_nat_o] :
% 14.25/8.05 ( ( ord_le2965882846123202637_nat_o @ A @ B4 )
% 14.25/8.05 => ( ( minus_1801376950450012436_nat_o @ A @ B4 )
% 14.25/8.05 = bot_bo7824918357723582233_nat_o ) ) ).
% 14.25/8.05
% 14.25/8.05 thf(fact_2627_subset__minus__empty,axiom,
% 14.25/8.05 ! [A: set_nat,B4: set_nat] :
% 14.25/8.05 ( ( ord_less_eq_set_nat @ A @ B4 )
% 14.25/8.05 => ( ( minus_minus_set_nat @ A @ B4 )
% 14.25/8.05 = bot_bot_set_nat ) ) ).
% 14.25/8.05
% 14.25/8.05 thf(fact_2628_image__subset__iff__subset__vimage,axiom,
% 14.25/8.05 ! [F: code_integer > code_integer,A: set_Code_integer,B4: set_Code_integer] :
% 14.25/8.05 ( ( ord_le7084787975880047091nteger @ ( image_4470545334726330049nteger @ F @ A ) @ B4 )
% 14.25/8.05 = ( ord_le7084787975880047091nteger @ A @ ( vimage1318793397690208119nteger @ F @ B4 ) ) ) ).
% 14.25/8.05
% 14.25/8.05 thf(fact_2629_image__subset__iff__subset__vimage,axiom,
% 14.25/8.05 ! [F: nat > set_nat,A: set_nat,B4: set_set_nat] :
% 14.25/8.05 ( ( ord_le6893508408891458716et_nat @ ( image_nat_set_nat @ F @ A ) @ B4 )
% 14.25/8.05 = ( ord_less_eq_set_nat @ A @ ( vimage_nat_set_nat @ F @ B4 ) ) ) ).
% 14.25/8.05
% 14.25/8.05 thf(fact_2630_image__subset__iff__subset__vimage,axiom,
% 14.25/8.05 ! [F: nat > code_integer,A: set_nat,B4: set_Code_integer] :
% 14.25/8.05 ( ( ord_le7084787975880047091nteger @ ( image_1215581382706833972nteger @ F @ A ) @ B4 )
% 14.25/8.05 = ( ord_less_eq_set_nat @ A @ ( vimage6104394758324456170nteger @ F @ B4 ) ) ) ).
% 14.25/8.05
% 14.25/8.05 thf(fact_2631_image__subset__iff__subset__vimage,axiom,
% 14.25/8.05 ! [F: nat > nat,A: set_nat,B4: set_nat] :
% 14.25/8.05 ( ( ord_less_eq_set_nat @ ( image_nat_nat2 @ F @ A ) @ B4 )
% 14.25/8.05 = ( ord_less_eq_set_nat @ A @ ( vimage_nat_nat @ F @ B4 ) ) ) ).
% 14.25/8.05
% 14.25/8.05 thf(fact_2632_image__vimage__subset,axiom,
% 14.25/8.05 ! [F: nat > set_nat,A: set_set_nat] : ( ord_le6893508408891458716et_nat @ ( image_nat_set_nat @ F @ ( vimage_nat_set_nat @ F @ A ) ) @ A ) ).
% 14.25/8.05
% 14.25/8.05 thf(fact_2633_image__vimage__subset,axiom,
% 14.25/8.05 ! [F: code_integer > code_integer,A: set_Code_integer] : ( ord_le7084787975880047091nteger @ ( image_4470545334726330049nteger @ F @ ( vimage1318793397690208119nteger @ F @ A ) ) @ A ) ).
% 14.25/8.05
% 14.25/8.05 thf(fact_2634_image__vimage__subset,axiom,
% 14.25/8.05 ! [F: nat > code_integer,A: set_Code_integer] : ( ord_le7084787975880047091nteger @ ( image_1215581382706833972nteger @ F @ ( vimage6104394758324456170nteger @ F @ A ) ) @ A ) ).
% 14.25/8.05
% 14.25/8.05 thf(fact_2635_image__vimage__subset,axiom,
% 14.25/8.05 ! [F: nat > nat,A: set_nat] : ( ord_less_eq_set_nat @ ( image_nat_nat2 @ F @ ( vimage_nat_nat @ F @ A ) ) @ A ) ).
% 14.25/8.05
% 14.25/8.05 thf(fact_2636_Diff__insert,axiom,
% 14.25/8.05 ! [A: set_Product_unit,A2: product_unit,B4: set_Product_unit] :
% 14.25/8.05 ( ( minus_6452836326544984404t_unit @ A @ ( insert_Product_unit @ A2 @ B4 ) )
% 14.25/8.05 = ( minus_6452836326544984404t_unit @ ( minus_6452836326544984404t_unit @ A @ B4 ) @ ( insert_Product_unit @ A2 @ bot_bo3957492148770167129t_unit ) ) ) ).
% 14.25/8.05
% 14.25/8.05 thf(fact_2637_Diff__insert,axiom,
% 14.25/8.05 ! [A: set_o,A2: $o,B4: set_o] :
% 14.25/8.05 ( ( minus_minus_set_o @ A @ ( insert_o @ A2 @ B4 ) )
% 14.25/8.05 = ( minus_minus_set_o @ ( minus_minus_set_o @ A @ B4 ) @ ( insert_o @ A2 @ bot_bot_set_o ) ) ) ).
% 14.25/8.05
% 14.25/8.05 thf(fact_2638_Diff__insert,axiom,
% 14.25/8.05 ! [A: set_int,A2: int,B4: set_int] :
% 14.25/8.05 ( ( minus_minus_set_int @ A @ ( insert_int @ A2 @ B4 ) )
% 14.25/8.05 = ( minus_minus_set_int @ ( minus_minus_set_int @ A @ B4 ) @ ( insert_int @ A2 @ bot_bot_set_int ) ) ) ).
% 14.25/8.05
% 14.25/8.05 thf(fact_2639_Diff__insert,axiom,
% 14.25/8.05 ! [A: set_Pr4532377907799695533_nat_o,A2: produc3658429121746597890et_nat > $o,B4: set_Pr4532377907799695533_nat_o] :
% 14.25/8.05 ( ( minus_1801376950450012436_nat_o @ A @ ( insert5175938949040314269_nat_o @ A2 @ B4 ) )
% 14.25/8.05 = ( minus_1801376950450012436_nat_o @ ( minus_1801376950450012436_nat_o @ A @ B4 ) @ ( insert5175938949040314269_nat_o @ A2 @ bot_bo7824918357723582233_nat_o ) ) ) ).
% 14.25/8.05
% 14.25/8.05 thf(fact_2640_Diff__insert,axiom,
% 14.25/8.05 ! [A: set_nat,A2: nat,B4: set_nat] :
% 14.25/8.05 ( ( minus_minus_set_nat @ A @ ( insert_nat @ A2 @ B4 ) )
% 14.25/8.05 = ( minus_minus_set_nat @ ( minus_minus_set_nat @ A @ B4 ) @ ( insert_nat @ A2 @ bot_bot_set_nat ) ) ) ).
% 14.25/8.05
% 14.25/8.05 thf(fact_2641_insert__Diff,axiom,
% 14.25/8.05 ! [A2: product_unit,A: set_Product_unit] :
% 14.25/8.05 ( ( member_Product_unit @ A2 @ A )
% 14.25/8.05 => ( ( insert_Product_unit @ A2 @ ( minus_6452836326544984404t_unit @ A @ ( insert_Product_unit @ A2 @ bot_bo3957492148770167129t_unit ) ) )
% 14.25/8.05 = A ) ) ).
% 14.25/8.05
% 14.25/8.05 thf(fact_2642_insert__Diff,axiom,
% 14.25/8.05 ! [A2: $o,A: set_o] :
% 14.25/8.05 ( ( member_o @ A2 @ A )
% 14.25/8.05 => ( ( insert_o @ A2 @ ( minus_minus_set_o @ A @ ( insert_o @ A2 @ bot_bot_set_o ) ) )
% 14.25/8.05 = A ) ) ).
% 14.25/8.05
% 14.25/8.05 thf(fact_2643_insert__Diff,axiom,
% 14.25/8.05 ! [A2: int,A: set_int] :
% 14.25/8.05 ( ( member_int @ A2 @ A )
% 14.25/8.05 => ( ( insert_int @ A2 @ ( minus_minus_set_int @ A @ ( insert_int @ A2 @ bot_bot_set_int ) ) )
% 14.25/8.05 = A ) ) ).
% 14.25/8.05
% 14.25/8.05 thf(fact_2644_insert__Diff,axiom,
% 14.25/8.05 ! [A2: produc3658429121746597890et_nat > $o,A: set_Pr4532377907799695533_nat_o] :
% 14.25/8.05 ( ( member6576561426505652726_nat_o @ A2 @ A )
% 14.25/8.05 => ( ( insert5175938949040314269_nat_o @ A2 @ ( minus_1801376950450012436_nat_o @ A @ ( insert5175938949040314269_nat_o @ A2 @ bot_bo7824918357723582233_nat_o ) ) )
% 14.25/8.05 = A ) ) ).
% 14.25/8.05
% 14.25/8.05 thf(fact_2645_insert__Diff,axiom,
% 14.25/8.05 ! [A2: nat,A: set_nat] :
% 14.25/8.05 ( ( member_nat @ A2 @ A )
% 14.25/8.05 => ( ( insert_nat @ A2 @ ( minus_minus_set_nat @ A @ ( insert_nat @ A2 @ bot_bot_set_nat ) ) )
% 14.25/8.05 = A ) ) ).
% 14.25/8.05
% 14.25/8.05 thf(fact_2646_Diff__insert2,axiom,
% 14.25/8.05 ! [A: set_Product_unit,A2: product_unit,B4: set_Product_unit] :
% 14.25/8.05 ( ( minus_6452836326544984404t_unit @ A @ ( insert_Product_unit @ A2 @ B4 ) )
% 14.25/8.05 = ( minus_6452836326544984404t_unit @ ( minus_6452836326544984404t_unit @ A @ ( insert_Product_unit @ A2 @ bot_bo3957492148770167129t_unit ) ) @ B4 ) ) ).
% 14.25/8.05
% 14.25/8.05 thf(fact_2647_Diff__insert2,axiom,
% 14.25/8.05 ! [A: set_o,A2: $o,B4: set_o] :
% 14.25/8.05 ( ( minus_minus_set_o @ A @ ( insert_o @ A2 @ B4 ) )
% 14.25/8.05 = ( minus_minus_set_o @ ( minus_minus_set_o @ A @ ( insert_o @ A2 @ bot_bot_set_o ) ) @ B4 ) ) ).
% 14.25/8.05
% 14.25/8.05 thf(fact_2648_Diff__insert2,axiom,
% 14.25/8.05 ! [A: set_int,A2: int,B4: set_int] :
% 14.25/8.05 ( ( minus_minus_set_int @ A @ ( insert_int @ A2 @ B4 ) )
% 14.25/8.05 = ( minus_minus_set_int @ ( minus_minus_set_int @ A @ ( insert_int @ A2 @ bot_bot_set_int ) ) @ B4 ) ) ).
% 14.25/8.05
% 14.25/8.05 thf(fact_2649_Diff__insert2,axiom,
% 14.25/8.05 ! [A: set_Pr4532377907799695533_nat_o,A2: produc3658429121746597890et_nat > $o,B4: set_Pr4532377907799695533_nat_o] :
% 14.25/8.05 ( ( minus_1801376950450012436_nat_o @ A @ ( insert5175938949040314269_nat_o @ A2 @ B4 ) )
% 14.25/8.05 = ( minus_1801376950450012436_nat_o @ ( minus_1801376950450012436_nat_o @ A @ ( insert5175938949040314269_nat_o @ A2 @ bot_bo7824918357723582233_nat_o ) ) @ B4 ) ) ).
% 14.25/8.05
% 14.25/8.05 thf(fact_2650_Diff__insert2,axiom,
% 14.25/8.05 ! [A: set_nat,A2: nat,B4: set_nat] :
% 14.25/8.05 ( ( minus_minus_set_nat @ A @ ( insert_nat @ A2 @ B4 ) )
% 14.25/8.05 = ( minus_minus_set_nat @ ( minus_minus_set_nat @ A @ ( insert_nat @ A2 @ bot_bot_set_nat ) ) @ B4 ) ) ).
% 14.25/8.05
% 14.25/8.05 thf(fact_2651_insert__minus__eq,axiom,
% 14.25/8.05 ! [X: product_unit,Y: product_unit,A: set_Product_unit] :
% 14.25/8.05 ( ( X != Y )
% 14.25/8.05 => ( ( minus_6452836326544984404t_unit @ ( insert_Product_unit @ X @ A ) @ ( insert_Product_unit @ Y @ bot_bo3957492148770167129t_unit ) )
% 14.25/8.05 = ( insert_Product_unit @ X @ ( minus_6452836326544984404t_unit @ A @ ( insert_Product_unit @ Y @ bot_bo3957492148770167129t_unit ) ) ) ) ) ).
% 14.25/8.05
% 14.25/8.05 thf(fact_2652_insert__minus__eq,axiom,
% 14.25/8.05 ! [X: $o,Y: $o,A: set_o] :
% 14.25/8.05 ( ( X != Y )
% 14.25/8.05 => ( ( minus_minus_set_o @ ( insert_o @ X @ A ) @ ( insert_o @ Y @ bot_bot_set_o ) )
% 14.25/8.05 = ( insert_o @ X @ ( minus_minus_set_o @ A @ ( insert_o @ Y @ bot_bot_set_o ) ) ) ) ) ).
% 14.25/8.05
% 14.25/8.05 thf(fact_2653_insert__minus__eq,axiom,
% 14.25/8.05 ! [X: int,Y: int,A: set_int] :
% 14.25/8.05 ( ( X != Y )
% 14.25/8.05 => ( ( minus_minus_set_int @ ( insert_int @ X @ A ) @ ( insert_int @ Y @ bot_bot_set_int ) )
% 14.25/8.05 = ( insert_int @ X @ ( minus_minus_set_int @ A @ ( insert_int @ Y @ bot_bot_set_int ) ) ) ) ) ).
% 14.25/8.05
% 14.25/8.05 thf(fact_2654_insert__minus__eq,axiom,
% 14.25/8.05 ! [X: produc3658429121746597890et_nat > $o,Y: produc3658429121746597890et_nat > $o,A: set_Pr4532377907799695533_nat_o] :
% 14.25/8.05 ( ( X != Y )
% 14.25/8.05 => ( ( minus_1801376950450012436_nat_o @ ( insert5175938949040314269_nat_o @ X @ A ) @ ( insert5175938949040314269_nat_o @ Y @ bot_bo7824918357723582233_nat_o ) )
% 14.25/8.05 = ( insert5175938949040314269_nat_o @ X @ ( minus_1801376950450012436_nat_o @ A @ ( insert5175938949040314269_nat_o @ Y @ bot_bo7824918357723582233_nat_o ) ) ) ) ) ).
% 14.25/8.05
% 14.25/8.05 thf(fact_2655_insert__minus__eq,axiom,
% 14.25/8.05 ! [X: nat,Y: nat,A: set_nat] :
% 14.25/8.05 ( ( X != Y )
% 14.25/8.05 => ( ( minus_minus_set_nat @ ( insert_nat @ X @ A ) @ ( insert_nat @ Y @ bot_bot_set_nat ) )
% 14.25/8.05 = ( insert_nat @ X @ ( minus_minus_set_nat @ A @ ( insert_nat @ Y @ bot_bot_set_nat ) ) ) ) ) ).
% 14.25/8.05
% 14.25/8.05 thf(fact_2656_Diff__insert__absorb,axiom,
% 14.25/8.05 ! [X: product_unit,A: set_Product_unit] :
% 14.25/8.05 ( ~ ( member_Product_unit @ X @ A )
% 14.25/8.05 => ( ( minus_6452836326544984404t_unit @ ( insert_Product_unit @ X @ A ) @ ( insert_Product_unit @ X @ bot_bo3957492148770167129t_unit ) )
% 14.25/8.05 = A ) ) ).
% 14.25/8.05
% 14.25/8.05 thf(fact_2657_Diff__insert__absorb,axiom,
% 14.25/8.05 ! [X: $o,A: set_o] :
% 14.25/8.05 ( ~ ( member_o @ X @ A )
% 14.25/8.05 => ( ( minus_minus_set_o @ ( insert_o @ X @ A ) @ ( insert_o @ X @ bot_bot_set_o ) )
% 14.25/8.05 = A ) ) ).
% 14.25/8.05
% 14.25/8.05 thf(fact_2658_Diff__insert__absorb,axiom,
% 14.25/8.05 ! [X: int,A: set_int] :
% 14.25/8.05 ( ~ ( member_int @ X @ A )
% 14.25/8.05 => ( ( minus_minus_set_int @ ( insert_int @ X @ A ) @ ( insert_int @ X @ bot_bot_set_int ) )
% 14.25/8.05 = A ) ) ).
% 14.25/8.05
% 14.25/8.05 thf(fact_2659_Diff__insert__absorb,axiom,
% 14.25/8.05 ! [X: produc3658429121746597890et_nat > $o,A: set_Pr4532377907799695533_nat_o] :
% 14.25/8.05 ( ~ ( member6576561426505652726_nat_o @ X @ A )
% 14.25/8.05 => ( ( minus_1801376950450012436_nat_o @ ( insert5175938949040314269_nat_o @ X @ A ) @ ( insert5175938949040314269_nat_o @ X @ bot_bo7824918357723582233_nat_o ) )
% 14.25/8.05 = A ) ) ).
% 14.25/8.05
% 14.25/8.05 thf(fact_2660_Diff__insert__absorb,axiom,
% 14.25/8.05 ! [X: nat,A: set_nat] :
% 14.25/8.05 ( ~ ( member_nat @ X @ A )
% 14.25/8.05 => ( ( minus_minus_set_nat @ ( insert_nat @ X @ A ) @ ( insert_nat @ X @ bot_bot_set_nat ) )
% 14.25/8.05 = A ) ) ).
% 14.25/8.05
% 14.25/8.05 thf(fact_2661_set__minus__singleton__eq,axiom,
% 14.25/8.05 ! [X: product_unit,X4: set_Product_unit] :
% 14.25/8.05 ( ~ ( member_Product_unit @ X @ X4 )
% 14.25/8.05 => ( ( minus_6452836326544984404t_unit @ X4 @ ( insert_Product_unit @ X @ bot_bo3957492148770167129t_unit ) )
% 14.25/8.05 = X4 ) ) ).
% 14.25/8.05
% 14.25/8.05 thf(fact_2662_set__minus__singleton__eq,axiom,
% 14.25/8.05 ! [X: $o,X4: set_o] :
% 14.25/8.05 ( ~ ( member_o @ X @ X4 )
% 14.25/8.05 => ( ( minus_minus_set_o @ X4 @ ( insert_o @ X @ bot_bot_set_o ) )
% 14.25/8.05 = X4 ) ) ).
% 14.25/8.05
% 14.25/8.05 thf(fact_2663_set__minus__singleton__eq,axiom,
% 14.25/8.05 ! [X: int,X4: set_int] :
% 14.25/8.05 ( ~ ( member_int @ X @ X4 )
% 14.25/8.05 => ( ( minus_minus_set_int @ X4 @ ( insert_int @ X @ bot_bot_set_int ) )
% 14.25/8.05 = X4 ) ) ).
% 14.25/8.05
% 14.25/8.05 thf(fact_2664_set__minus__singleton__eq,axiom,
% 14.25/8.05 ! [X: produc3658429121746597890et_nat > $o,X4: set_Pr4532377907799695533_nat_o] :
% 14.25/8.05 ( ~ ( member6576561426505652726_nat_o @ X @ X4 )
% 14.25/8.05 => ( ( minus_1801376950450012436_nat_o @ X4 @ ( insert5175938949040314269_nat_o @ X @ bot_bo7824918357723582233_nat_o ) )
% 14.25/8.05 = X4 ) ) ).
% 14.25/8.05
% 14.25/8.05 thf(fact_2665_set__minus__singleton__eq,axiom,
% 14.25/8.05 ! [X: nat,X4: set_nat] :
% 14.25/8.05 ( ~ ( member_nat @ X @ X4 )
% 14.25/8.05 => ( ( minus_minus_set_nat @ X4 @ ( insert_nat @ X @ bot_bot_set_nat ) )
% 14.25/8.05 = X4 ) ) ).
% 14.25/8.05
% 14.25/8.05 thf(fact_2666_subset__Diff__insert,axiom,
% 14.25/8.05 ! [A: set_Pr4532377907799695533_nat_o,B4: set_Pr4532377907799695533_nat_o,X: produc3658429121746597890et_nat > $o,C5: set_Pr4532377907799695533_nat_o] :
% 14.25/8.05 ( ( ord_le2965882846123202637_nat_o @ A @ ( minus_1801376950450012436_nat_o @ B4 @ ( insert5175938949040314269_nat_o @ X @ C5 ) ) )
% 14.25/8.05 = ( ( ord_le2965882846123202637_nat_o @ A @ ( minus_1801376950450012436_nat_o @ B4 @ C5 ) )
% 14.25/8.05 & ~ ( member6576561426505652726_nat_o @ X @ A ) ) ) ).
% 14.25/8.05
% 14.25/8.05 thf(fact_2667_subset__Diff__insert,axiom,
% 14.25/8.05 ! [A: set_int,B4: set_int,X: int,C5: set_int] :
% 14.25/8.05 ( ( ord_less_eq_set_int @ A @ ( minus_minus_set_int @ B4 @ ( insert_int @ X @ C5 ) ) )
% 14.25/8.05 = ( ( ord_less_eq_set_int @ A @ ( minus_minus_set_int @ B4 @ C5 ) )
% 14.25/8.05 & ~ ( member_int @ X @ A ) ) ) ).
% 14.25/8.05
% 14.25/8.05 thf(fact_2668_subset__Diff__insert,axiom,
% 14.25/8.05 ! [A: set_o,B4: set_o,X: $o,C5: set_o] :
% 14.25/8.05 ( ( ord_less_eq_set_o @ A @ ( minus_minus_set_o @ B4 @ ( insert_o @ X @ C5 ) ) )
% 14.25/8.05 = ( ( ord_less_eq_set_o @ A @ ( minus_minus_set_o @ B4 @ C5 ) )
% 14.25/8.05 & ~ ( member_o @ X @ A ) ) ) ).
% 14.25/8.05
% 14.25/8.05 thf(fact_2669_subset__Diff__insert,axiom,
% 14.25/8.05 ! [A: set_Product_unit,B4: set_Product_unit,X: product_unit,C5: set_Product_unit] :
% 14.25/8.05 ( ( ord_le3507040750410214029t_unit @ A @ ( minus_6452836326544984404t_unit @ B4 @ ( insert_Product_unit @ X @ C5 ) ) )
% 14.25/8.05 = ( ( ord_le3507040750410214029t_unit @ A @ ( minus_6452836326544984404t_unit @ B4 @ C5 ) )
% 14.25/8.05 & ~ ( member_Product_unit @ X @ A ) ) ) ).
% 14.25/8.05
% 14.25/8.05 thf(fact_2670_subset__Diff__insert,axiom,
% 14.25/8.05 ! [A: set_nat,B4: set_nat,X: nat,C5: set_nat] :
% 14.25/8.05 ( ( ord_less_eq_set_nat @ A @ ( minus_minus_set_nat @ B4 @ ( insert_nat @ X @ C5 ) ) )
% 14.25/8.05 = ( ( ord_less_eq_set_nat @ A @ ( minus_minus_set_nat @ B4 @ C5 ) )
% 14.25/8.05 & ~ ( member_nat @ X @ A ) ) ) ).
% 14.25/8.05
% 14.25/8.05 thf(fact_2671_Diff__triv,axiom,
% 14.25/8.05 ! [A: set_Product_unit,B4: set_Product_unit] :
% 14.25/8.05 ( ( ( inf_in4660618365625256667t_unit @ A @ B4 )
% 14.25/8.05 = bot_bo3957492148770167129t_unit )
% 14.25/8.05 => ( ( minus_6452836326544984404t_unit @ A @ B4 )
% 14.25/8.05 = A ) ) ).
% 14.25/8.05
% 14.25/8.05 thf(fact_2672_Diff__triv,axiom,
% 14.25/8.05 ! [A: set_o,B4: set_o] :
% 14.25/8.05 ( ( ( inf_inf_set_o @ A @ B4 )
% 14.25/8.05 = bot_bot_set_o )
% 14.25/8.05 => ( ( minus_minus_set_o @ A @ B4 )
% 14.25/8.05 = A ) ) ).
% 14.25/8.05
% 14.25/8.05 thf(fact_2673_Diff__triv,axiom,
% 14.25/8.05 ! [A: set_int,B4: set_int] :
% 14.25/8.05 ( ( ( inf_inf_set_int @ A @ B4 )
% 14.25/8.05 = bot_bot_set_int )
% 14.25/8.05 => ( ( minus_minus_set_int @ A @ B4 )
% 14.25/8.05 = A ) ) ).
% 14.25/8.05
% 14.25/8.05 thf(fact_2674_Diff__triv,axiom,
% 14.25/8.05 ! [A: set_Pr4532377907799695533_nat_o,B4: set_Pr4532377907799695533_nat_o] :
% 14.25/8.05 ( ( ( inf_in1906310914598751387_nat_o @ A @ B4 )
% 14.25/8.05 = bot_bo7824918357723582233_nat_o )
% 14.25/8.05 => ( ( minus_1801376950450012436_nat_o @ A @ B4 )
% 14.25/8.05 = A ) ) ).
% 14.25/8.05
% 14.25/8.05 thf(fact_2675_Diff__triv,axiom,
% 14.25/8.05 ! [A: set_nat,B4: set_nat] :
% 14.25/8.05 ( ( ( inf_inf_set_nat @ A @ B4 )
% 14.25/8.05 = bot_bot_set_nat )
% 14.25/8.05 => ( ( minus_minus_set_nat @ A @ B4 )
% 14.25/8.05 = A ) ) ).
% 14.25/8.05
% 14.25/8.05 thf(fact_2676_Int__Diff__disjoint,axiom,
% 14.25/8.05 ! [A: set_Product_unit,B4: set_Product_unit] :
% 14.25/8.05 ( ( inf_in4660618365625256667t_unit @ ( inf_in4660618365625256667t_unit @ A @ B4 ) @ ( minus_6452836326544984404t_unit @ A @ B4 ) )
% 14.25/8.05 = bot_bo3957492148770167129t_unit ) ).
% 14.25/8.05
% 14.25/8.05 thf(fact_2677_Int__Diff__disjoint,axiom,
% 14.25/8.05 ! [A: set_o,B4: set_o] :
% 14.25/8.05 ( ( inf_inf_set_o @ ( inf_inf_set_o @ A @ B4 ) @ ( minus_minus_set_o @ A @ B4 ) )
% 14.25/8.05 = bot_bot_set_o ) ).
% 14.25/8.05
% 14.25/8.05 thf(fact_2678_Int__Diff__disjoint,axiom,
% 14.25/8.05 ! [A: set_int,B4: set_int] :
% 14.25/8.05 ( ( inf_inf_set_int @ ( inf_inf_set_int @ A @ B4 ) @ ( minus_minus_set_int @ A @ B4 ) )
% 14.25/8.05 = bot_bot_set_int ) ).
% 14.25/8.05
% 14.25/8.05 thf(fact_2679_Int__Diff__disjoint,axiom,
% 14.25/8.05 ! [A: set_Pr4532377907799695533_nat_o,B4: set_Pr4532377907799695533_nat_o] :
% 14.25/8.05 ( ( inf_in1906310914598751387_nat_o @ ( inf_in1906310914598751387_nat_o @ A @ B4 ) @ ( minus_1801376950450012436_nat_o @ A @ B4 ) )
% 14.25/8.05 = bot_bo7824918357723582233_nat_o ) ).
% 14.25/8.05
% 14.25/8.05 thf(fact_2680_Int__Diff__disjoint,axiom,
% 14.25/8.05 ! [A: set_nat,B4: set_nat] :
% 14.25/8.05 ( ( inf_inf_set_nat @ ( inf_inf_set_nat @ A @ B4 ) @ ( minus_minus_set_nat @ A @ B4 ) )
% 14.25/8.05 = bot_bot_set_nat ) ).
% 14.25/8.05
% 14.25/8.05 thf(fact_2681_disjoint__alt__simp1,axiom,
% 14.25/8.05 ! [A: set_Product_unit,B4: set_Product_unit] :
% 14.25/8.05 ( ( ( minus_6452836326544984404t_unit @ A @ B4 )
% 14.25/8.05 = A )
% 14.25/8.05 = ( ( inf_in4660618365625256667t_unit @ A @ B4 )
% 14.25/8.05 = bot_bo3957492148770167129t_unit ) ) ).
% 14.25/8.05
% 14.25/8.05 thf(fact_2682_disjoint__alt__simp1,axiom,
% 14.25/8.05 ! [A: set_o,B4: set_o] :
% 14.25/8.05 ( ( ( minus_minus_set_o @ A @ B4 )
% 14.25/8.05 = A )
% 14.25/8.05 = ( ( inf_inf_set_o @ A @ B4 )
% 14.25/8.05 = bot_bot_set_o ) ) ).
% 14.25/8.05
% 14.25/8.05 thf(fact_2683_disjoint__alt__simp1,axiom,
% 14.25/8.05 ! [A: set_int,B4: set_int] :
% 14.25/8.05 ( ( ( minus_minus_set_int @ A @ B4 )
% 14.25/8.05 = A )
% 14.25/8.05 = ( ( inf_inf_set_int @ A @ B4 )
% 14.25/8.05 = bot_bot_set_int ) ) ).
% 14.25/8.05
% 14.25/8.05 thf(fact_2684_disjoint__alt__simp1,axiom,
% 14.25/8.05 ! [A: set_Pr4532377907799695533_nat_o,B4: set_Pr4532377907799695533_nat_o] :
% 14.25/8.05 ( ( ( minus_1801376950450012436_nat_o @ A @ B4 )
% 14.25/8.05 = A )
% 14.25/8.05 = ( ( inf_in1906310914598751387_nat_o @ A @ B4 )
% 14.25/8.05 = bot_bo7824918357723582233_nat_o ) ) ).
% 14.25/8.05
% 14.25/8.05 thf(fact_2685_disjoint__alt__simp1,axiom,
% 14.25/8.05 ! [A: set_nat,B4: set_nat] :
% 14.25/8.05 ( ( ( minus_minus_set_nat @ A @ B4 )
% 14.25/8.05 = A )
% 14.25/8.05 = ( ( inf_inf_set_nat @ A @ B4 )
% 14.25/8.05 = bot_bot_set_nat ) ) ).
% 14.25/8.05
% 14.25/8.05 thf(fact_2686_disjoint__alt__simp2,axiom,
% 14.25/8.05 ! [A: set_Product_unit,B4: set_Product_unit] :
% 14.25/8.05 ( ( ( minus_6452836326544984404t_unit @ A @ B4 )
% 14.25/8.05 != A )
% 14.25/8.05 = ( ( inf_in4660618365625256667t_unit @ A @ B4 )
% 14.25/8.05 != bot_bo3957492148770167129t_unit ) ) ).
% 14.25/8.05
% 14.25/8.05 thf(fact_2687_disjoint__alt__simp2,axiom,
% 14.25/8.05 ! [A: set_o,B4: set_o] :
% 14.25/8.05 ( ( ( minus_minus_set_o @ A @ B4 )
% 14.25/8.05 != A )
% 14.25/8.05 = ( ( inf_inf_set_o @ A @ B4 )
% 14.25/8.05 != bot_bot_set_o ) ) ).
% 14.25/8.05
% 14.25/8.05 thf(fact_2688_disjoint__alt__simp2,axiom,
% 14.25/8.05 ! [A: set_int,B4: set_int] :
% 14.25/8.05 ( ( ( minus_minus_set_int @ A @ B4 )
% 14.25/8.05 != A )
% 14.25/8.05 = ( ( inf_inf_set_int @ A @ B4 )
% 14.25/8.05 != bot_bot_set_int ) ) ).
% 14.25/8.05
% 14.25/8.05 thf(fact_2689_disjoint__alt__simp2,axiom,
% 14.25/8.05 ! [A: set_Pr4532377907799695533_nat_o,B4: set_Pr4532377907799695533_nat_o] :
% 14.25/8.05 ( ( ( minus_1801376950450012436_nat_o @ A @ B4 )
% 14.25/8.05 != A )
% 14.25/8.05 = ( ( inf_in1906310914598751387_nat_o @ A @ B4 )
% 14.25/8.05 != bot_bo7824918357723582233_nat_o ) ) ).
% 14.25/8.05
% 14.25/8.05 thf(fact_2690_disjoint__alt__simp2,axiom,
% 14.25/8.05 ! [A: set_nat,B4: set_nat] :
% 14.25/8.05 ( ( ( minus_minus_set_nat @ A @ B4 )
% 14.25/8.05 != A )
% 14.25/8.05 = ( ( inf_inf_set_nat @ A @ B4 )
% 14.25/8.05 != bot_bot_set_nat ) ) ).
% 14.25/8.05
% 14.25/8.05 thf(fact_2691_Diff__subset__conv,axiom,
% 14.25/8.05 ! [A: set_nat,B4: set_nat,C5: set_nat] :
% 14.25/8.05 ( ( ord_less_eq_set_nat @ ( minus_minus_set_nat @ A @ B4 ) @ C5 )
% 14.25/8.05 = ( ord_less_eq_set_nat @ A @ ( sup_sup_set_nat @ B4 @ C5 ) ) ) ).
% 14.25/8.05
% 14.25/8.05 thf(fact_2692_Diff__partition,axiom,
% 14.25/8.05 ! [A: set_nat,B4: set_nat] :
% 14.25/8.05 ( ( ord_less_eq_set_nat @ A @ B4 )
% 14.25/8.05 => ( ( sup_sup_set_nat @ A @ ( minus_minus_set_nat @ B4 @ A ) )
% 14.25/8.05 = B4 ) ) ).
% 14.25/8.05
% 14.25/8.05 thf(fact_2693_Un__Diff__Int,axiom,
% 14.25/8.05 ! [A: set_nat,B4: set_nat] :
% 14.25/8.05 ( ( sup_sup_set_nat @ ( minus_minus_set_nat @ A @ B4 ) @ ( inf_inf_set_nat @ A @ B4 ) )
% 14.25/8.05 = A ) ).
% 14.25/8.05
% 14.25/8.05 thf(fact_2694_Int__Diff__Un,axiom,
% 14.25/8.05 ! [A: set_nat,B4: set_nat] :
% 14.25/8.05 ( ( sup_sup_set_nat @ ( inf_inf_set_nat @ A @ B4 ) @ ( minus_minus_set_nat @ A @ B4 ) )
% 14.25/8.05 = A ) ).
% 14.25/8.05
% 14.25/8.05 thf(fact_2695_Diff__Int,axiom,
% 14.25/8.05 ! [A: set_nat,B4: set_nat,C5: set_nat] :
% 14.25/8.05 ( ( minus_minus_set_nat @ A @ ( inf_inf_set_nat @ B4 @ C5 ) )
% 14.25/8.05 = ( sup_sup_set_nat @ ( minus_minus_set_nat @ A @ B4 ) @ ( minus_minus_set_nat @ A @ C5 ) ) ) ).
% 14.25/8.05
% 14.25/8.05 thf(fact_2696_Diff__Un,axiom,
% 14.25/8.05 ! [A: set_nat,B4: set_nat,C5: set_nat] :
% 14.25/8.05 ( ( minus_minus_set_nat @ A @ ( sup_sup_set_nat @ B4 @ C5 ) )
% 14.25/8.05 = ( inf_inf_set_nat @ ( minus_minus_set_nat @ A @ B4 ) @ ( minus_minus_set_nat @ A @ C5 ) ) ) ).
% 14.25/8.05
% 14.25/8.05 thf(fact_2697_Range__snd,axiom,
% 14.25/8.05 ( range_9132311542094983838et_nat
% 14.25/8.05 = ( image_6088997261112373647et_nat @ produc8586169260539613262et_nat ) ) ).
% 14.25/8.05
% 14.25/8.05 thf(fact_2698_Range__snd,axiom,
% 14.25/8.05 ( range_int_int
% 14.25/8.05 = ( image_5042161079198086560nt_int @ product_snd_int_int ) ) ).
% 14.25/8.05
% 14.25/8.05 thf(fact_2699_snd__eq__Range,axiom,
% 14.25/8.05 ! [R3: set_Pr3948176798113811640et_nat] :
% 14.25/8.05 ( ( image_6088997261112373647et_nat @ produc8586169260539613262et_nat @ R3 )
% 14.25/8.05 = ( range_9132311542094983838et_nat @ R3 ) ) ).
% 14.25/8.05
% 14.25/8.05 thf(fact_2700_snd__eq__Range,axiom,
% 14.25/8.05 ! [R3: set_Pr958786334691620121nt_int] :
% 14.25/8.05 ( ( image_5042161079198086560nt_int @ product_snd_int_int @ R3 )
% 14.25/8.05 = ( range_int_int @ R3 ) ) ).
% 14.25/8.05
% 14.25/8.05 thf(fact_2701_the__elem__image__unique,axiom,
% 14.25/8.05 ! [A: set_Code_integer,F: code_integer > code_integer,X: code_integer] :
% 14.25/8.05 ( ( A != bot_bo3990330152332043303nteger )
% 14.25/8.05 => ( ! [Y2: code_integer] :
% 14.25/8.05 ( ( member_Code_integer @ Y2 @ A )
% 14.25/8.05 => ( ( F @ Y2 )
% 14.25/8.05 = ( F @ X ) ) )
% 14.25/8.05 => ( ( the_el4656377885804969768nteger @ ( image_4470545334726330049nteger @ F @ A ) )
% 14.25/8.05 = ( F @ X ) ) ) ) ).
% 14.25/8.05
% 14.25/8.05 thf(fact_2702_the__elem__image__unique,axiom,
% 14.25/8.05 ! [A: set_nat,F: nat > set_nat,X: nat] :
% 14.25/8.05 ( ( A != bot_bot_set_nat )
% 14.25/8.05 => ( ! [Y2: nat] :
% 14.25/8.05 ( ( member_nat @ Y2 @ A )
% 14.25/8.05 => ( ( F @ Y2 )
% 14.25/8.05 = ( F @ X ) ) )
% 14.25/8.05 => ( ( the_elem_set_nat @ ( image_nat_set_nat @ F @ A ) )
% 14.25/8.05 = ( F @ X ) ) ) ) ).
% 14.25/8.05
% 14.25/8.05 thf(fact_2703_the__elem__image__unique,axiom,
% 14.25/8.05 ! [A: set_nat,F: nat > nat,X: nat] :
% 14.25/8.05 ( ( A != bot_bot_set_nat )
% 14.25/8.05 => ( ! [Y2: nat] :
% 14.25/8.05 ( ( member_nat @ Y2 @ A )
% 14.25/8.05 => ( ( F @ Y2 )
% 14.25/8.05 = ( F @ X ) ) )
% 14.25/8.05 => ( ( the_elem_nat @ ( image_nat_nat2 @ F @ A ) )
% 14.25/8.05 = ( F @ X ) ) ) ) ).
% 14.25/8.05
% 14.25/8.05 thf(fact_2704_the__elem__image__unique,axiom,
% 14.25/8.05 ! [A: set_nat,F: nat > code_integer,X: nat] :
% 14.25/8.05 ( ( A != bot_bot_set_nat )
% 14.25/8.05 => ( ! [Y2: nat] :
% 14.25/8.05 ( ( member_nat @ Y2 @ A )
% 14.25/8.05 => ( ( F @ Y2 )
% 14.25/8.05 = ( F @ X ) ) )
% 14.25/8.05 => ( ( the_el4656377885804969768nteger @ ( image_1215581382706833972nteger @ F @ A ) )
% 14.25/8.05 = ( F @ X ) ) ) ) ).
% 14.25/8.05
% 14.25/8.05 thf(fact_2705_cyclicE,axiom,
% 14.25/8.05 ! [G: set_Pr1261947904930325089at_nat] :
% 14.25/8.05 ( ~ ( transi3139058337892597825ic_nat @ G )
% 14.25/8.05 => ~ ! [X3: nat] :
% 14.25/8.05 ~ ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X3 @ X3 ) @ ( transi6264000038957366511cl_nat @ G ) ) ) ).
% 14.25/8.05
% 14.25/8.05 thf(fact_2706_cyclicE,axiom,
% 14.25/8.05 ! [G: set_Pr958786334691620121nt_int] :
% 14.25/8.05 ( ~ ( transi3136567867383547549ic_int @ G )
% 14.25/8.05 => ~ ! [X3: int] :
% 14.25/8.05 ~ ( member5262025264175285858nt_int @ ( product_Pair_int_int @ X3 @ X3 ) @ ( transi6261509568448316235cl_int @ G ) ) ) ).
% 14.25/8.05
% 14.25/8.05 thf(fact_2707_acyclic__def,axiom,
% 14.25/8.05 ( transi3139058337892597825ic_nat
% 14.25/8.05 = ( ^ [R4: set_Pr1261947904930325089at_nat] :
% 14.25/8.05 ! [X2: nat] :
% 14.25/8.05 ~ ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X2 @ X2 ) @ ( transi6264000038957366511cl_nat @ R4 ) ) ) ) ).
% 14.25/8.05
% 14.25/8.05 thf(fact_2708_acyclic__def,axiom,
% 14.25/8.05 ( transi3136567867383547549ic_int
% 14.25/8.05 = ( ^ [R4: set_Pr958786334691620121nt_int] :
% 14.25/8.05 ! [X2: int] :
% 14.25/8.05 ~ ( member5262025264175285858nt_int @ ( product_Pair_int_int @ X2 @ X2 ) @ ( transi6261509568448316235cl_int @ R4 ) ) ) ) ).
% 14.25/8.05
% 14.25/8.05 thf(fact_2709_acyclicI,axiom,
% 14.25/8.05 ! [R2: set_Pr1261947904930325089at_nat] :
% 14.25/8.05 ( ! [X3: nat] :
% 14.25/8.05 ~ ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X3 @ X3 ) @ ( transi6264000038957366511cl_nat @ R2 ) )
% 14.25/8.05 => ( transi3139058337892597825ic_nat @ R2 ) ) ).
% 14.25/8.05
% 14.25/8.05 thf(fact_2710_acyclicI,axiom,
% 14.25/8.05 ! [R2: set_Pr958786334691620121nt_int] :
% 14.25/8.05 ( ! [X3: int] :
% 14.25/8.05 ~ ( member5262025264175285858nt_int @ ( product_Pair_int_int @ X3 @ X3 ) @ ( transi6261509568448316235cl_int @ R2 ) )
% 14.25/8.05 => ( transi3136567867383547549ic_int @ R2 ) ) ).
% 14.25/8.05
% 14.25/8.05 thf(fact_2711_snd__image__mp,axiom,
% 14.25/8.05 ! [A: set_Pr3286484037609594932et_nat,B4: set_Pr3948176798113811640et_nat,X: produc3658429121746597890et_nat > $o,Y: produc3658429121746597890et_nat] :
% 14.25/8.05 ( ( ord_le8090042543978450200et_nat @ ( image_4390311411407041927et_nat @ produc4011572625026189258et_nat @ A ) @ B4 )
% 14.25/8.05 => ( ( member1996754912294343701et_nat @ ( produc5001842942810119800et_nat @ X @ Y ) @ A )
% 14.25/8.05 => ( member6260224972018164377et_nat @ Y @ B4 ) ) ) ).
% 14.25/8.05
% 14.25/8.05 thf(fact_2712_snd__image__mp,axiom,
% 14.25/8.05 ! [A: set_Pr8536935166611901872et_nat,B4: set_Pr3286484037609594932et_nat,X: produc3658429121746597890et_nat > $o,Y: produc3925858234332021118et_nat] :
% 14.25/8.05 ( ( ord_le5966269811547037844et_nat @ ( image_4557310310409863807et_nat @ produc5374455773327741254et_nat @ A ) @ B4 )
% 14.25/8.05 => ( ( member6124377750444531601et_nat @ ( produc2245416461498447860et_nat @ X @ Y ) @ A )
% 14.25/8.05 => ( member1996754912294343701et_nat @ Y @ B4 ) ) ) ).
% 14.25/8.05
% 14.25/8.05 thf(fact_2713_snd__image__mp,axiom,
% 14.25/8.05 ! [A: set_Pr3948176798113811640et_nat,B4: set_set_nat,X: heap_e7401611519738050253t_unit,Y: set_nat] :
% 14.25/8.05 ( ( ord_le6893508408891458716et_nat @ ( image_6088997261112373647et_nat @ produc8586169260539613262et_nat @ A ) @ B4 )
% 14.25/8.05 => ( ( member6260224972018164377et_nat @ ( produc7507926704131184380et_nat @ X @ Y ) @ A )
% 14.25/8.05 => ( member_set_nat @ Y @ B4 ) ) ) ).
% 14.25/8.05
% 14.25/8.05 thf(fact_2714_snd__image__mp,axiom,
% 14.25/8.05 ! [A: set_Pr958786334691620121nt_int,B4: set_int,X: int,Y: int] :
% 14.25/8.05 ( ( ord_less_eq_set_int @ ( image_5042161079198086560nt_int @ product_snd_int_int @ A ) @ B4 )
% 14.25/8.05 => ( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ X @ Y ) @ A )
% 14.25/8.05 => ( member_int @ Y @ B4 ) ) ) ).
% 14.25/8.05
% 14.25/8.05 thf(fact_2715_snd__image__mp,axiom,
% 14.25/8.05 ! [A: set_Pr1261947904930325089at_nat,B4: set_nat,X: nat,Y: nat] :
% 14.25/8.05 ( ( ord_less_eq_set_nat @ ( image_2486076414777270412at_nat @ product_snd_nat_nat @ A ) @ B4 )
% 14.25/8.05 => ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ Y ) @ A )
% 14.25/8.05 => ( member_nat @ Y @ B4 ) ) ) ).
% 14.25/8.05
% 14.25/8.05 thf(fact_2716_snd__in__Field,axiom,
% 14.25/8.05 ! [R3: set_Pr958786334691620121nt_int] : ( ord_less_eq_set_int @ ( image_5042161079198086560nt_int @ product_snd_int_int @ R3 ) @ ( field_int @ R3 ) ) ).
% 14.25/8.05
% 14.25/8.05 thf(fact_2717_snd__in__Field,axiom,
% 14.25/8.05 ! [R3: set_Pr1261947904930325089at_nat] : ( ord_less_eq_set_nat @ ( image_2486076414777270412at_nat @ product_snd_nat_nat @ R3 ) @ ( field_nat @ R3 ) ) ).
% 14.25/8.05
% 14.25/8.05 thf(fact_2718_Image__subset__snd__image,axiom,
% 14.25/8.05 ! [A: set_Pr3948176798113811640et_nat,B4: set_he8644174446132254637t_unit] : ( ord_le6893508408891458716et_nat @ ( image_3942343239045813184et_nat @ A @ B4 ) @ ( image_6088997261112373647et_nat @ produc8586169260539613262et_nat @ A ) ) ).
% 14.25/8.05
% 14.25/8.05 thf(fact_2719_Image__subset__snd__image,axiom,
% 14.25/8.05 ! [A: set_Pr958786334691620121nt_int,B4: set_int] : ( ord_less_eq_set_int @ ( image_int_int @ A @ B4 ) @ ( image_5042161079198086560nt_int @ product_snd_int_int @ A ) ) ).
% 14.25/8.05
% 14.25/8.05 thf(fact_2720_infinite__remove,axiom,
% 14.25/8.05 ! [S: set_Code_integer,A2: code_integer] :
% 14.25/8.05 ( ~ ( finite6017078050557962740nteger @ S )
% 14.25/8.05 => ~ ( finite6017078050557962740nteger @ ( minus_2355218937544613996nteger @ S @ ( insert_Code_integer @ A2 @ bot_bo3990330152332043303nteger ) ) ) ) ).
% 14.25/8.05
% 14.25/8.05 thf(fact_2721_infinite__remove,axiom,
% 14.25/8.05 ! [S: set_Product_unit,A2: product_unit] :
% 14.25/8.05 ( ~ ( finite4290736615968046902t_unit @ S )
% 14.25/8.05 => ~ ( finite4290736615968046902t_unit @ ( minus_6452836326544984404t_unit @ S @ ( insert_Product_unit @ A2 @ bot_bo3957492148770167129t_unit ) ) ) ) ).
% 14.25/8.05
% 14.25/8.05 thf(fact_2722_infinite__remove,axiom,
% 14.25/8.05 ! [S: set_o,A2: $o] :
% 14.25/8.05 ( ~ ( finite_finite_o @ S )
% 14.25/8.05 => ~ ( finite_finite_o @ ( minus_minus_set_o @ S @ ( insert_o @ A2 @ bot_bot_set_o ) ) ) ) ).
% 14.25/8.05
% 14.25/8.05 thf(fact_2723_infinite__remove,axiom,
% 14.25/8.05 ! [S: set_int,A2: int] :
% 14.25/8.05 ( ~ ( finite_finite_int @ S )
% 14.25/8.05 => ~ ( finite_finite_int @ ( minus_minus_set_int @ S @ ( insert_int @ A2 @ bot_bot_set_int ) ) ) ) ).
% 14.25/8.05
% 14.25/8.05 thf(fact_2724_infinite__remove,axiom,
% 14.25/8.05 ! [S: set_Pr4532377907799695533_nat_o,A2: produc3658429121746597890et_nat > $o] :
% 14.25/8.05 ( ~ ( finite3252695134891459830_nat_o @ S )
% 14.25/8.05 => ~ ( finite3252695134891459830_nat_o @ ( minus_1801376950450012436_nat_o @ S @ ( insert5175938949040314269_nat_o @ A2 @ bot_bo7824918357723582233_nat_o ) ) ) ) ).
% 14.25/8.05
% 14.25/8.05 thf(fact_2725_infinite__remove,axiom,
% 14.25/8.05 ! [S: set_nat,A2: nat] :
% 14.25/8.05 ( ~ ( finite_finite_nat @ S )
% 14.25/8.05 => ~ ( finite_finite_nat @ ( minus_minus_set_nat @ S @ ( insert_nat @ A2 @ bot_bot_set_nat ) ) ) ) ).
% 14.25/8.05
% 14.25/8.05 thf(fact_2726_infinite__coinduct,axiom,
% 14.25/8.05 ! [X4: set_Code_integer > $o,A: set_Code_integer] :
% 14.25/8.05 ( ( X4 @ A )
% 14.25/8.05 => ( ! [A9: set_Code_integer] :
% 14.25/8.05 ( ( X4 @ A9 )
% 14.25/8.05 => ? [X6: code_integer] :
% 14.25/8.05 ( ( member_Code_integer @ X6 @ A9 )
% 14.25/8.05 & ( ( X4 @ ( minus_2355218937544613996nteger @ A9 @ ( insert_Code_integer @ X6 @ bot_bo3990330152332043303nteger ) ) )
% 14.25/8.05 | ~ ( finite6017078050557962740nteger @ ( minus_2355218937544613996nteger @ A9 @ ( insert_Code_integer @ X6 @ bot_bo3990330152332043303nteger ) ) ) ) ) )
% 14.25/8.05 => ~ ( finite6017078050557962740nteger @ A ) ) ) ).
% 14.25/8.05
% 14.25/8.05 thf(fact_2727_infinite__coinduct,axiom,
% 14.25/8.05 ! [X4: set_Product_unit > $o,A: set_Product_unit] :
% 14.25/8.05 ( ( X4 @ A )
% 14.25/8.05 => ( ! [A9: set_Product_unit] :
% 14.25/8.05 ( ( X4 @ A9 )
% 14.25/8.05 => ? [X6: product_unit] :
% 14.25/8.05 ( ( member_Product_unit @ X6 @ A9 )
% 14.25/8.05 & ( ( X4 @ ( minus_6452836326544984404t_unit @ A9 @ ( insert_Product_unit @ X6 @ bot_bo3957492148770167129t_unit ) ) )
% 14.25/8.05 | ~ ( finite4290736615968046902t_unit @ ( minus_6452836326544984404t_unit @ A9 @ ( insert_Product_unit @ X6 @ bot_bo3957492148770167129t_unit ) ) ) ) ) )
% 14.25/8.05 => ~ ( finite4290736615968046902t_unit @ A ) ) ) ).
% 14.25/8.05
% 14.25/8.05 thf(fact_2728_infinite__coinduct,axiom,
% 14.25/8.05 ! [X4: set_o > $o,A: set_o] :
% 14.25/8.05 ( ( X4 @ A )
% 14.25/8.05 => ( ! [A9: set_o] :
% 14.25/8.05 ( ( X4 @ A9 )
% 14.25/8.05 => ? [X6: $o] :
% 14.25/8.05 ( ( member_o @ X6 @ A9 )
% 14.25/8.05 & ( ( X4 @ ( minus_minus_set_o @ A9 @ ( insert_o @ X6 @ bot_bot_set_o ) ) )
% 14.25/8.05 | ~ ( finite_finite_o @ ( minus_minus_set_o @ A9 @ ( insert_o @ X6 @ bot_bot_set_o ) ) ) ) ) )
% 14.25/8.05 => ~ ( finite_finite_o @ A ) ) ) ).
% 14.25/8.05
% 14.25/8.05 thf(fact_2729_infinite__coinduct,axiom,
% 14.25/8.05 ! [X4: set_int > $o,A: set_int] :
% 14.25/8.05 ( ( X4 @ A )
% 14.25/8.05 => ( ! [A9: set_int] :
% 14.25/8.05 ( ( X4 @ A9 )
% 14.25/8.05 => ? [X6: int] :
% 14.25/8.05 ( ( member_int @ X6 @ A9 )
% 14.25/8.05 & ( ( X4 @ ( minus_minus_set_int @ A9 @ ( insert_int @ X6 @ bot_bot_set_int ) ) )
% 14.25/8.05 | ~ ( finite_finite_int @ ( minus_minus_set_int @ A9 @ ( insert_int @ X6 @ bot_bot_set_int ) ) ) ) ) )
% 14.25/8.05 => ~ ( finite_finite_int @ A ) ) ) ).
% 14.25/8.05
% 14.25/8.05 thf(fact_2730_infinite__coinduct,axiom,
% 14.25/8.05 ! [X4: set_Pr4532377907799695533_nat_o > $o,A: set_Pr4532377907799695533_nat_o] :
% 14.25/8.05 ( ( X4 @ A )
% 14.25/8.05 => ( ! [A9: set_Pr4532377907799695533_nat_o] :
% 14.25/8.05 ( ( X4 @ A9 )
% 14.25/8.05 => ? [X6: produc3658429121746597890et_nat > $o] :
% 14.25/8.05 ( ( member6576561426505652726_nat_o @ X6 @ A9 )
% 14.25/8.05 & ( ( X4 @ ( minus_1801376950450012436_nat_o @ A9 @ ( insert5175938949040314269_nat_o @ X6 @ bot_bo7824918357723582233_nat_o ) ) )
% 14.25/8.05 | ~ ( finite3252695134891459830_nat_o @ ( minus_1801376950450012436_nat_o @ A9 @ ( insert5175938949040314269_nat_o @ X6 @ bot_bo7824918357723582233_nat_o ) ) ) ) ) )
% 14.25/8.05 => ~ ( finite3252695134891459830_nat_o @ A ) ) ) ).
% 14.25/8.05
% 14.25/8.05 thf(fact_2731_infinite__coinduct,axiom,
% 14.25/8.05 ! [X4: set_nat > $o,A: set_nat] :
% 14.25/8.05 ( ( X4 @ A )
% 14.25/8.05 => ( ! [A9: set_nat] :
% 14.25/8.05 ( ( X4 @ A9 )
% 14.25/8.05 => ? [X6: nat] :
% 14.25/8.05 ( ( member_nat @ X6 @ A9 )
% 14.25/8.05 & ( ( X4 @ ( minus_minus_set_nat @ A9 @ ( insert_nat @ X6 @ bot_bot_set_nat ) ) )
% 14.25/8.05 | ~ ( finite_finite_nat @ ( minus_minus_set_nat @ A9 @ ( insert_nat @ X6 @ bot_bot_set_nat ) ) ) ) ) )
% 14.25/8.05 => ~ ( finite_finite_nat @ A ) ) ) ).
% 14.25/8.05
% 14.25/8.05 thf(fact_2732_finite__empty__induct,axiom,
% 14.25/8.05 ! [A: set_Code_integer,P: set_Code_integer > $o] :
% 14.25/8.05 ( ( finite6017078050557962740nteger @ A )
% 14.25/8.05 => ( ( P @ A )
% 14.25/8.05 => ( ! [A4: code_integer,A9: set_Code_integer] :
% 14.25/8.05 ( ( finite6017078050557962740nteger @ A9 )
% 14.25/8.05 => ( ( member_Code_integer @ A4 @ A9 )
% 14.25/8.05 => ( ( P @ A9 )
% 14.25/8.05 => ( P @ ( minus_2355218937544613996nteger @ A9 @ ( insert_Code_integer @ A4 @ bot_bo3990330152332043303nteger ) ) ) ) ) )
% 14.25/8.05 => ( P @ bot_bo3990330152332043303nteger ) ) ) ) ).
% 14.25/8.05
% 14.25/8.05 thf(fact_2733_finite__empty__induct,axiom,
% 14.25/8.05 ! [A: set_Product_unit,P: set_Product_unit > $o] :
% 14.25/8.05 ( ( finite4290736615968046902t_unit @ A )
% 14.25/8.05 => ( ( P @ A )
% 14.25/8.05 => ( ! [A4: product_unit,A9: set_Product_unit] :
% 14.25/8.05 ( ( finite4290736615968046902t_unit @ A9 )
% 14.25/8.05 => ( ( member_Product_unit @ A4 @ A9 )
% 14.25/8.05 => ( ( P @ A9 )
% 14.25/8.05 => ( P @ ( minus_6452836326544984404t_unit @ A9 @ ( insert_Product_unit @ A4 @ bot_bo3957492148770167129t_unit ) ) ) ) ) )
% 14.25/8.05 => ( P @ bot_bo3957492148770167129t_unit ) ) ) ) ).
% 14.25/8.05
% 14.25/8.05 thf(fact_2734_finite__empty__induct,axiom,
% 14.25/8.05 ! [A: set_o,P: set_o > $o] :
% 14.25/8.05 ( ( finite_finite_o @ A )
% 14.25/8.05 => ( ( P @ A )
% 14.25/8.05 => ( ! [A4: $o,A9: set_o] :
% 14.25/8.05 ( ( finite_finite_o @ A9 )
% 14.25/8.05 => ( ( member_o @ A4 @ A9 )
% 14.25/8.05 => ( ( P @ A9 )
% 14.25/8.05 => ( P @ ( minus_minus_set_o @ A9 @ ( insert_o @ A4 @ bot_bot_set_o ) ) ) ) ) )
% 14.25/8.05 => ( P @ bot_bot_set_o ) ) ) ) ).
% 14.25/8.05
% 14.25/8.05 thf(fact_2735_finite__empty__induct,axiom,
% 14.25/8.05 ! [A: set_int,P: set_int > $o] :
% 14.25/8.05 ( ( finite_finite_int @ A )
% 14.25/8.05 => ( ( P @ A )
% 14.25/8.05 => ( ! [A4: int,A9: set_int] :
% 14.25/8.05 ( ( finite_finite_int @ A9 )
% 14.25/8.05 => ( ( member_int @ A4 @ A9 )
% 14.25/8.05 => ( ( P @ A9 )
% 14.25/8.05 => ( P @ ( minus_minus_set_int @ A9 @ ( insert_int @ A4 @ bot_bot_set_int ) ) ) ) ) )
% 14.25/8.05 => ( P @ bot_bot_set_int ) ) ) ) ).
% 14.25/8.05
% 14.25/8.05 thf(fact_2736_finite__empty__induct,axiom,
% 14.25/8.05 ! [A: set_Pr4532377907799695533_nat_o,P: set_Pr4532377907799695533_nat_o > $o] :
% 14.25/8.05 ( ( finite3252695134891459830_nat_o @ A )
% 14.25/8.05 => ( ( P @ A )
% 14.25/8.05 => ( ! [A4: produc3658429121746597890et_nat > $o,A9: set_Pr4532377907799695533_nat_o] :
% 14.25/8.05 ( ( finite3252695134891459830_nat_o @ A9 )
% 14.25/8.05 => ( ( member6576561426505652726_nat_o @ A4 @ A9 )
% 14.25/8.05 => ( ( P @ A9 )
% 14.25/8.05 => ( P @ ( minus_1801376950450012436_nat_o @ A9 @ ( insert5175938949040314269_nat_o @ A4 @ bot_bo7824918357723582233_nat_o ) ) ) ) ) )
% 14.25/8.05 => ( P @ bot_bo7824918357723582233_nat_o ) ) ) ) ).
% 14.25/8.05
% 14.25/8.05 thf(fact_2737_finite__empty__induct,axiom,
% 14.25/8.05 ! [A: set_nat,P: set_nat > $o] :
% 14.25/8.05 ( ( finite_finite_nat @ A )
% 14.25/8.05 => ( ( P @ A )
% 14.25/8.05 => ( ! [A4: nat,A9: set_nat] :
% 14.25/8.05 ( ( finite_finite_nat @ A9 )
% 14.25/8.05 => ( ( member_nat @ A4 @ A9 )
% 14.25/8.05 => ( ( P @ A9 )
% 14.25/8.05 => ( P @ ( minus_minus_set_nat @ A9 @ ( insert_nat @ A4 @ bot_bot_set_nat ) ) ) ) ) )
% 14.25/8.05 => ( P @ bot_bot_set_nat ) ) ) ) ).
% 14.25/8.05
% 14.25/8.05 thf(fact_2738_subset__insert__iff,axiom,
% 14.25/8.05 ! [A: set_Product_unit,X: product_unit,B4: set_Product_unit] :
% 14.25/8.05 ( ( ord_le3507040750410214029t_unit @ A @ ( insert_Product_unit @ X @ B4 ) )
% 14.25/8.05 = ( ( ( member_Product_unit @ X @ A )
% 14.25/8.05 => ( ord_le3507040750410214029t_unit @ ( minus_6452836326544984404t_unit @ A @ ( insert_Product_unit @ X @ bot_bo3957492148770167129t_unit ) ) @ B4 ) )
% 14.25/8.05 & ( ~ ( member_Product_unit @ X @ A )
% 14.25/8.05 => ( ord_le3507040750410214029t_unit @ A @ B4 ) ) ) ) ).
% 14.25/8.05
% 14.25/8.05 thf(fact_2739_subset__insert__iff,axiom,
% 14.25/8.05 ! [A: set_o,X: $o,B4: set_o] :
% 14.25/8.05 ( ( ord_less_eq_set_o @ A @ ( insert_o @ X @ B4 ) )
% 14.25/8.05 = ( ( ( member_o @ X @ A )
% 14.25/8.05 => ( ord_less_eq_set_o @ ( minus_minus_set_o @ A @ ( insert_o @ X @ bot_bot_set_o ) ) @ B4 ) )
% 14.25/8.05 & ( ~ ( member_o @ X @ A )
% 14.25/8.05 => ( ord_less_eq_set_o @ A @ B4 ) ) ) ) ).
% 14.25/8.05
% 14.25/8.05 thf(fact_2740_subset__insert__iff,axiom,
% 14.25/8.05 ! [A: set_int,X: int,B4: set_int] :
% 14.25/8.05 ( ( ord_less_eq_set_int @ A @ ( insert_int @ X @ B4 ) )
% 14.25/8.05 = ( ( ( member_int @ X @ A )
% 14.25/8.05 => ( ord_less_eq_set_int @ ( minus_minus_set_int @ A @ ( insert_int @ X @ bot_bot_set_int ) ) @ B4 ) )
% 14.25/8.05 & ( ~ ( member_int @ X @ A )
% 14.25/8.05 => ( ord_less_eq_set_int @ A @ B4 ) ) ) ) ).
% 14.25/8.05
% 14.25/8.05 thf(fact_2741_subset__insert__iff,axiom,
% 14.25/8.05 ! [A: set_Pr4532377907799695533_nat_o,X: produc3658429121746597890et_nat > $o,B4: set_Pr4532377907799695533_nat_o] :
% 14.25/8.05 ( ( ord_le2965882846123202637_nat_o @ A @ ( insert5175938949040314269_nat_o @ X @ B4 ) )
% 14.25/8.05 = ( ( ( member6576561426505652726_nat_o @ X @ A )
% 14.25/8.05 => ( ord_le2965882846123202637_nat_o @ ( minus_1801376950450012436_nat_o @ A @ ( insert5175938949040314269_nat_o @ X @ bot_bo7824918357723582233_nat_o ) ) @ B4 ) )
% 14.25/8.05 & ( ~ ( member6576561426505652726_nat_o @ X @ A )
% 14.25/8.05 => ( ord_le2965882846123202637_nat_o @ A @ B4 ) ) ) ) ).
% 14.25/8.05
% 14.25/8.05 thf(fact_2742_subset__insert__iff,axiom,
% 14.25/8.05 ! [A: set_nat,X: nat,B4: set_nat] :
% 14.25/8.05 ( ( ord_less_eq_set_nat @ A @ ( insert_nat @ X @ B4 ) )
% 14.25/8.05 = ( ( ( member_nat @ X @ A )
% 14.25/8.05 => ( ord_less_eq_set_nat @ ( minus_minus_set_nat @ A @ ( insert_nat @ X @ bot_bot_set_nat ) ) @ B4 ) )
% 14.25/8.05 & ( ~ ( member_nat @ X @ A )
% 14.25/8.05 => ( ord_less_eq_set_nat @ A @ B4 ) ) ) ) ).
% 14.25/8.05
% 14.25/8.05 thf(fact_2743_Diff__single__insert,axiom,
% 14.25/8.05 ! [A: set_Product_unit,X: product_unit,B4: set_Product_unit] :
% 14.25/8.05 ( ( ord_le3507040750410214029t_unit @ ( minus_6452836326544984404t_unit @ A @ ( insert_Product_unit @ X @ bot_bo3957492148770167129t_unit ) ) @ B4 )
% 14.25/8.05 => ( ord_le3507040750410214029t_unit @ A @ ( insert_Product_unit @ X @ B4 ) ) ) ).
% 14.25/8.05
% 14.25/8.05 thf(fact_2744_Diff__single__insert,axiom,
% 14.25/8.05 ! [A: set_o,X: $o,B4: set_o] :
% 14.25/8.05 ( ( ord_less_eq_set_o @ ( minus_minus_set_o @ A @ ( insert_o @ X @ bot_bot_set_o ) ) @ B4 )
% 14.25/8.05 => ( ord_less_eq_set_o @ A @ ( insert_o @ X @ B4 ) ) ) ).
% 14.25/8.05
% 14.25/8.05 thf(fact_2745_Diff__single__insert,axiom,
% 14.25/8.05 ! [A: set_int,X: int,B4: set_int] :
% 14.25/8.05 ( ( ord_less_eq_set_int @ ( minus_minus_set_int @ A @ ( insert_int @ X @ bot_bot_set_int ) ) @ B4 )
% 14.25/8.05 => ( ord_less_eq_set_int @ A @ ( insert_int @ X @ B4 ) ) ) ).
% 14.25/8.05
% 14.25/8.05 thf(fact_2746_Diff__single__insert,axiom,
% 14.25/8.05 ! [A: set_Pr4532377907799695533_nat_o,X: produc3658429121746597890et_nat > $o,B4: set_Pr4532377907799695533_nat_o] :
% 14.25/8.05 ( ( ord_le2965882846123202637_nat_o @ ( minus_1801376950450012436_nat_o @ A @ ( insert5175938949040314269_nat_o @ X @ bot_bo7824918357723582233_nat_o ) ) @ B4 )
% 14.25/8.05 => ( ord_le2965882846123202637_nat_o @ A @ ( insert5175938949040314269_nat_o @ X @ B4 ) ) ) ).
% 14.25/8.05
% 14.25/8.05 thf(fact_2747_Diff__single__insert,axiom,
% 14.25/8.05 ! [A: set_nat,X: nat,B4: set_nat] :
% 14.25/8.05 ( ( ord_less_eq_set_nat @ ( minus_minus_set_nat @ A @ ( insert_nat @ X @ bot_bot_set_nat ) ) @ B4 )
% 14.25/8.05 => ( ord_less_eq_set_nat @ A @ ( insert_nat @ X @ B4 ) ) ) ).
% 14.25/8.05
% 14.25/8.05 thf(fact_2748_Field__rel__restrict,axiom,
% 14.25/8.05 ! [R3: set_Pr1261947904930325089at_nat,A: set_nat] : ( ord_less_eq_set_nat @ ( field_nat @ ( rel_restrict_nat @ R3 @ A ) ) @ ( minus_minus_set_nat @ ( field_nat @ R3 ) @ A ) ) ).
% 14.25/8.05
% 14.25/8.05 thf(fact_2749_Domain__rel__restrict,axiom,
% 14.25/8.05 ! [R3: set_Pr1261947904930325089at_nat,A: set_nat] : ( ord_less_eq_set_nat @ ( domain_nat_nat @ ( rel_restrict_nat @ R3 @ A ) ) @ ( minus_minus_set_nat @ ( domain_nat_nat @ R3 ) @ A ) ) ).
% 14.25/8.05
% 14.25/8.05 thf(fact_2750_Range__rel__restrict,axiom,
% 14.25/8.05 ! [R3: set_Pr1261947904930325089at_nat,A: set_nat] : ( ord_less_eq_set_nat @ ( range_nat_nat @ ( rel_restrict_nat @ R3 @ A ) ) @ ( minus_minus_set_nat @ ( range_nat_nat @ R3 ) @ A ) ) ).
% 14.25/8.05
% 14.25/8.05 thf(fact_2751_inf__img__fin__domE,axiom,
% 14.25/8.05 ! [F: nat > code_integer,A: set_nat] :
% 14.25/8.05 ( ( finite6017078050557962740nteger @ ( image_1215581382706833972nteger @ F @ A ) )
% 14.25/8.05 => ( ~ ( finite_finite_nat @ A )
% 14.25/8.05 => ~ ! [Y2: code_integer] :
% 14.25/8.05 ( ( member_Code_integer @ Y2 @ ( image_1215581382706833972nteger @ F @ A ) )
% 14.25/8.05 => ( finite_finite_nat @ ( vimage6104394758324456170nteger @ F @ ( insert_Code_integer @ Y2 @ bot_bo3990330152332043303nteger ) ) ) ) ) ) ).
% 14.25/8.05
% 14.25/8.05 thf(fact_2752_inf__img__fin__domE,axiom,
% 14.25/8.05 ! [F: code_integer > code_integer,A: set_Code_integer] :
% 14.25/8.05 ( ( finite6017078050557962740nteger @ ( image_4470545334726330049nteger @ F @ A ) )
% 14.25/8.05 => ( ~ ( finite6017078050557962740nteger @ A )
% 14.25/8.05 => ~ ! [Y2: code_integer] :
% 14.25/8.05 ( ( member_Code_integer @ Y2 @ ( image_4470545334726330049nteger @ F @ A ) )
% 14.25/8.05 => ( finite6017078050557962740nteger @ ( vimage1318793397690208119nteger @ F @ ( insert_Code_integer @ Y2 @ bot_bo3990330152332043303nteger ) ) ) ) ) ) ).
% 14.25/8.05
% 14.25/8.05 thf(fact_2753_inf__img__fin__domE,axiom,
% 14.25/8.05 ! [F: nat > product_unit,A: set_nat] :
% 14.25/8.05 ( ( finite4290736615968046902t_unit @ ( image_8730104196221521654t_unit @ F @ A ) )
% 14.25/8.05 => ( ~ ( finite_finite_nat @ A )
% 14.25/8.05 => ~ ! [Y2: product_unit] :
% 14.25/8.05 ( ( member_Product_unit @ Y2 @ ( image_8730104196221521654t_unit @ F @ A ) )
% 14.25/8.05 => ( finite_finite_nat @ ( vimage4884490618288580032t_unit @ F @ ( insert_Product_unit @ Y2 @ bot_bo3957492148770167129t_unit ) ) ) ) ) ) ).
% 14.25/8.05
% 14.25/8.05 thf(fact_2754_inf__img__fin__domE,axiom,
% 14.25/8.05 ! [F: code_integer > product_unit,A: set_Code_integer] :
% 14.25/8.05 ( ( finite4290736615968046902t_unit @ ( image_7182656606676592809t_unit @ F @ A ) )
% 14.25/8.05 => ( ~ ( finite6017078050557962740nteger @ A )
% 14.25/8.05 => ~ ! [Y2: product_unit] :
% 14.25/8.05 ( ( member_Product_unit @ Y2 @ ( image_7182656606676592809t_unit @ F @ A ) )
% 14.25/8.05 => ( finite6017078050557962740nteger @ ( vimage5651688891684837235t_unit @ F @ ( insert_Product_unit @ Y2 @ bot_bo3957492148770167129t_unit ) ) ) ) ) ) ).
% 14.25/8.05
% 14.25/8.05 thf(fact_2755_inf__img__fin__domE,axiom,
% 14.25/8.05 ! [F: nat > $o,A: set_nat] :
% 14.25/8.05 ( ( finite_finite_o @ ( image_nat_o2 @ F @ A ) )
% 14.25/8.05 => ( ~ ( finite_finite_nat @ A )
% 14.25/8.05 => ~ ! [Y2: $o] :
% 14.25/8.05 ( ( member_o @ Y2 @ ( image_nat_o2 @ F @ A ) )
% 14.25/8.05 => ( finite_finite_nat @ ( vimage_nat_o @ F @ ( insert_o @ Y2 @ bot_bot_set_o ) ) ) ) ) ) ).
% 14.25/8.05
% 14.25/8.05 thf(fact_2756_inf__img__fin__domE,axiom,
% 14.25/8.05 ! [F: code_integer > $o,A: set_Code_integer] :
% 14.25/8.05 ( ( finite_finite_o @ ( image_Code_integer_o @ F @ A ) )
% 14.25/8.05 => ( ~ ( finite6017078050557962740nteger @ A )
% 14.25/8.05 => ~ ! [Y2: $o] :
% 14.25/8.05 ( ( member_o @ Y2 @ ( image_Code_integer_o @ F @ A ) )
% 14.25/8.05 => ( finite6017078050557962740nteger @ ( vimage3005800680117336574eger_o @ F @ ( insert_o @ Y2 @ bot_bot_set_o ) ) ) ) ) ) ).
% 14.25/8.05
% 14.25/8.05 thf(fact_2757_inf__img__fin__domE,axiom,
% 14.25/8.05 ! [F: nat > nat,A: set_nat] :
% 14.25/8.05 ( ( finite_finite_nat @ ( image_nat_nat2 @ F @ A ) )
% 14.25/8.05 => ( ~ ( finite_finite_nat @ A )
% 14.25/8.05 => ~ ! [Y2: nat] :
% 14.25/8.05 ( ( member_nat @ Y2 @ ( image_nat_nat2 @ F @ A ) )
% 14.25/8.05 => ( finite_finite_nat @ ( vimage_nat_nat @ F @ ( insert_nat @ Y2 @ bot_bot_set_nat ) ) ) ) ) ) ).
% 14.25/8.05
% 14.25/8.05 thf(fact_2758_inf__img__fin__domE,axiom,
% 14.25/8.05 ! [F: code_integer > nat,A: set_Code_integer] :
% 14.25/8.05 ( ( finite_finite_nat @ ( image_951025933927791156er_nat @ F @ A ) )
% 14.25/8.05 => ( ~ ( finite6017078050557962740nteger @ A )
% 14.25/8.05 => ~ ! [Y2: nat] :
% 14.25/8.05 ( ( member_nat @ Y2 @ ( image_951025933927791156er_nat @ F @ A ) )
% 14.25/8.05 => ( finite6017078050557962740nteger @ ( vimage5839839309545413354er_nat @ F @ ( insert_nat @ Y2 @ bot_bot_set_nat ) ) ) ) ) ) ).
% 14.25/8.05
% 14.25/8.05 thf(fact_2759_inf__img__fin__domE,axiom,
% 14.25/8.05 ! [F: nat > int,A: set_nat] :
% 14.25/8.05 ( ( finite_finite_int @ ( image_nat_int2 @ F @ A ) )
% 14.25/8.05 => ( ~ ( finite_finite_nat @ A )
% 14.25/8.05 => ~ ! [Y2: int] :
% 14.25/8.05 ( ( member_int @ Y2 @ ( image_nat_int2 @ F @ A ) )
% 14.25/8.05 => ( finite_finite_nat @ ( vimage_nat_int @ F @ ( insert_int @ Y2 @ bot_bot_set_int ) ) ) ) ) ) ).
% 14.25/8.05
% 14.25/8.05 thf(fact_2760_inf__img__fin__domE,axiom,
% 14.25/8.05 ! [F: code_integer > int,A: set_Code_integer] :
% 14.25/8.05 ( ( finite_finite_int @ ( image_948535463418740880er_int @ F @ A ) )
% 14.25/8.05 => ( ~ ( finite6017078050557962740nteger @ A )
% 14.25/8.05 => ~ ! [Y2: int] :
% 14.25/8.05 ( ( member_int @ Y2 @ ( image_948535463418740880er_int @ F @ A ) )
% 14.25/8.05 => ( finite6017078050557962740nteger @ ( vimage5837348839036363078er_int @ F @ ( insert_int @ Y2 @ bot_bot_set_int ) ) ) ) ) ) ).
% 14.25/8.05
% 14.25/8.05 thf(fact_2761_inf__img__fin__dom,axiom,
% 14.25/8.05 ! [F: nat > code_integer,A: set_nat] :
% 14.25/8.05 ( ( finite6017078050557962740nteger @ ( image_1215581382706833972nteger @ F @ A ) )
% 14.25/8.05 => ( ~ ( finite_finite_nat @ A )
% 14.25/8.05 => ? [X3: code_integer] :
% 14.25/8.05 ( ( member_Code_integer @ X3 @ ( image_1215581382706833972nteger @ F @ A ) )
% 14.25/8.05 & ~ ( finite_finite_nat @ ( vimage6104394758324456170nteger @ F @ ( insert_Code_integer @ X3 @ bot_bo3990330152332043303nteger ) ) ) ) ) ) ).
% 14.25/8.05
% 14.25/8.05 thf(fact_2762_inf__img__fin__dom,axiom,
% 14.25/8.05 ! [F: code_integer > code_integer,A: set_Code_integer] :
% 14.25/8.05 ( ( finite6017078050557962740nteger @ ( image_4470545334726330049nteger @ F @ A ) )
% 14.25/8.05 => ( ~ ( finite6017078050557962740nteger @ A )
% 14.25/8.05 => ? [X3: code_integer] :
% 14.25/8.05 ( ( member_Code_integer @ X3 @ ( image_4470545334726330049nteger @ F @ A ) )
% 14.25/8.05 & ~ ( finite6017078050557962740nteger @ ( vimage1318793397690208119nteger @ F @ ( insert_Code_integer @ X3 @ bot_bo3990330152332043303nteger ) ) ) ) ) ) ).
% 14.25/8.05
% 14.25/8.05 thf(fact_2763_inf__img__fin__dom,axiom,
% 14.25/8.05 ! [F: nat > product_unit,A: set_nat] :
% 14.25/8.05 ( ( finite4290736615968046902t_unit @ ( image_8730104196221521654t_unit @ F @ A ) )
% 14.25/8.05 => ( ~ ( finite_finite_nat @ A )
% 14.25/8.05 => ? [X3: product_unit] :
% 14.25/8.05 ( ( member_Product_unit @ X3 @ ( image_8730104196221521654t_unit @ F @ A ) )
% 14.25/8.05 & ~ ( finite_finite_nat @ ( vimage4884490618288580032t_unit @ F @ ( insert_Product_unit @ X3 @ bot_bo3957492148770167129t_unit ) ) ) ) ) ) ).
% 14.25/8.05
% 14.25/8.05 thf(fact_2764_inf__img__fin__dom,axiom,
% 14.25/8.05 ! [F: code_integer > product_unit,A: set_Code_integer] :
% 14.25/8.05 ( ( finite4290736615968046902t_unit @ ( image_7182656606676592809t_unit @ F @ A ) )
% 14.25/8.05 => ( ~ ( finite6017078050557962740nteger @ A )
% 14.25/8.05 => ? [X3: product_unit] :
% 14.25/8.05 ( ( member_Product_unit @ X3 @ ( image_7182656606676592809t_unit @ F @ A ) )
% 14.25/8.05 & ~ ( finite6017078050557962740nteger @ ( vimage5651688891684837235t_unit @ F @ ( insert_Product_unit @ X3 @ bot_bo3957492148770167129t_unit ) ) ) ) ) ) ).
% 14.25/8.05
% 14.25/8.05 thf(fact_2765_inf__img__fin__dom,axiom,
% 14.25/8.05 ! [F: nat > $o,A: set_nat] :
% 14.25/8.05 ( ( finite_finite_o @ ( image_nat_o2 @ F @ A ) )
% 14.25/8.05 => ( ~ ( finite_finite_nat @ A )
% 14.25/8.05 => ? [X3: $o] :
% 14.25/8.05 ( ( member_o @ X3 @ ( image_nat_o2 @ F @ A ) )
% 14.25/8.05 & ~ ( finite_finite_nat @ ( vimage_nat_o @ F @ ( insert_o @ X3 @ bot_bot_set_o ) ) ) ) ) ) ).
% 14.25/8.05
% 14.25/8.05 thf(fact_2766_inf__img__fin__dom,axiom,
% 14.25/8.05 ! [F: code_integer > $o,A: set_Code_integer] :
% 14.25/8.05 ( ( finite_finite_o @ ( image_Code_integer_o @ F @ A ) )
% 14.25/8.05 => ( ~ ( finite6017078050557962740nteger @ A )
% 14.25/8.05 => ? [X3: $o] :
% 14.25/8.05 ( ( member_o @ X3 @ ( image_Code_integer_o @ F @ A ) )
% 14.25/8.05 & ~ ( finite6017078050557962740nteger @ ( vimage3005800680117336574eger_o @ F @ ( insert_o @ X3 @ bot_bot_set_o ) ) ) ) ) ) ).
% 14.25/8.05
% 14.25/8.05 thf(fact_2767_inf__img__fin__dom,axiom,
% 14.25/8.05 ! [F: nat > nat,A: set_nat] :
% 14.25/8.05 ( ( finite_finite_nat @ ( image_nat_nat2 @ F @ A ) )
% 14.25/8.05 => ( ~ ( finite_finite_nat @ A )
% 14.25/8.05 => ? [X3: nat] :
% 14.25/8.05 ( ( member_nat @ X3 @ ( image_nat_nat2 @ F @ A ) )
% 14.25/8.05 & ~ ( finite_finite_nat @ ( vimage_nat_nat @ F @ ( insert_nat @ X3 @ bot_bot_set_nat ) ) ) ) ) ) ).
% 14.25/8.05
% 14.25/8.05 thf(fact_2768_inf__img__fin__dom,axiom,
% 14.25/8.05 ! [F: code_integer > nat,A: set_Code_integer] :
% 14.25/8.05 ( ( finite_finite_nat @ ( image_951025933927791156er_nat @ F @ A ) )
% 14.25/8.05 => ( ~ ( finite6017078050557962740nteger @ A )
% 14.25/8.05 => ? [X3: nat] :
% 14.25/8.05 ( ( member_nat @ X3 @ ( image_951025933927791156er_nat @ F @ A ) )
% 14.25/8.05 & ~ ( finite6017078050557962740nteger @ ( vimage5839839309545413354er_nat @ F @ ( insert_nat @ X3 @ bot_bot_set_nat ) ) ) ) ) ) ).
% 14.25/8.05
% 14.25/8.05 thf(fact_2769_inf__img__fin__dom,axiom,
% 14.25/8.05 ! [F: nat > int,A: set_nat] :
% 14.25/8.05 ( ( finite_finite_int @ ( image_nat_int2 @ F @ A ) )
% 14.25/8.05 => ( ~ ( finite_finite_nat @ A )
% 14.25/8.05 => ? [X3: int] :
% 14.25/8.05 ( ( member_int @ X3 @ ( image_nat_int2 @ F @ A ) )
% 14.25/8.05 & ~ ( finite_finite_nat @ ( vimage_nat_int @ F @ ( insert_int @ X3 @ bot_bot_set_int ) ) ) ) ) ) ).
% 14.25/8.05
% 14.25/8.05 thf(fact_2770_inf__img__fin__dom,axiom,
% 14.25/8.05 ! [F: code_integer > int,A: set_Code_integer] :
% 14.25/8.05 ( ( finite_finite_int @ ( image_948535463418740880er_int @ F @ A ) )
% 14.25/8.05 => ( ~ ( finite6017078050557962740nteger @ A )
% 14.25/8.05 => ? [X3: int] :
% 14.25/8.05 ( ( member_int @ X3 @ ( image_948535463418740880er_int @ F @ A ) )
% 14.25/8.05 & ~ ( finite6017078050557962740nteger @ ( vimage5837348839036363078er_int @ F @ ( insert_int @ X3 @ bot_bot_set_int ) ) ) ) ) ) ).
% 14.25/8.05
% 14.25/8.05 thf(fact_2771_acyclic__insert__cyclic,axiom,
% 14.25/8.05 ! [G: set_Pr1261947904930325089at_nat,X: nat,Y: nat] :
% 14.25/8.05 ( ( transi3139058337892597825ic_nat @ G )
% 14.25/8.05 => ( ~ ( transi3139058337892597825ic_nat @ ( insert8211810215607154385at_nat @ ( product_Pair_nat_nat @ X @ Y ) @ G ) )
% 14.25/8.05 => ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ Y @ X ) @ ( transi2905341329935302413cl_nat @ G ) ) ) ) ).
% 14.25/8.05
% 14.25/8.05 thf(fact_2772_acyclic__insert__cyclic,axiom,
% 14.25/8.05 ! [G: set_Pr958786334691620121nt_int,X: int,Y: int] :
% 14.25/8.05 ( ( transi3136567867383547549ic_int @ G )
% 14.25/8.05 => ( ~ ( transi3136567867383547549ic_int @ ( insert5033312907999012233nt_int @ ( product_Pair_int_int @ X @ Y ) @ G ) )
% 14.25/8.05 => ( member5262025264175285858nt_int @ ( product_Pair_int_int @ Y @ X ) @ ( transi2902850859426252137cl_int @ G ) ) ) ) ).
% 14.25/8.05
% 14.25/8.05 thf(fact_2773_Inf__fin_Ohom__commute,axiom,
% 14.25/8.05 ! [H2: set_nat > set_nat,N: set_set_nat] :
% 14.25/8.05 ( ! [X3: set_nat,Y2: set_nat] :
% 14.25/8.05 ( ( H2 @ ( inf_inf_set_nat @ X3 @ Y2 ) )
% 14.25/8.05 = ( inf_inf_set_nat @ ( H2 @ X3 ) @ ( H2 @ Y2 ) ) )
% 14.25/8.05 => ( ( finite1152437895449049373et_nat @ N )
% 14.25/8.05 => ( ( N != bot_bot_set_set_nat )
% 14.25/8.05 => ( ( H2 @ ( lattic3014633134055518761et_nat @ N ) )
% 14.25/8.05 = ( lattic3014633134055518761et_nat @ ( image_7916887816326733075et_nat @ H2 @ N ) ) ) ) ) ) ).
% 14.25/8.05
% 14.25/8.05 thf(fact_2774_Inf__fin_Ohom__commute,axiom,
% 14.25/8.05 ! [H2: assn > assn,N: set_assn] :
% 14.25/8.05 ( ! [X3: assn,Y2: assn] :
% 14.25/8.05 ( ( H2 @ ( inf_inf_assn @ X3 @ Y2 ) )
% 14.25/8.05 = ( inf_inf_assn @ ( H2 @ X3 ) @ ( H2 @ Y2 ) ) )
% 14.25/8.05 => ( ( finite_finite_assn @ N )
% 14.25/8.05 => ( ( N != bot_bot_set_assn )
% 14.25/8.05 => ( ( H2 @ ( lattic47131356835913163n_assn @ N ) )
% 14.25/8.05 = ( lattic47131356835913163n_assn @ ( image_assn_assn @ H2 @ N ) ) ) ) ) ) ).
% 14.25/8.05
% 14.25/8.05 thf(fact_2775_Inf__fin_Ohom__commute,axiom,
% 14.25/8.05 ! [H2: product_unit > product_unit,N: set_Product_unit] :
% 14.25/8.05 ( ! [X3: product_unit,Y2: product_unit] :
% 14.25/8.05 ( ( H2 @ ( inf_inf_Product_unit @ X3 @ Y2 ) )
% 14.25/8.05 = ( inf_inf_Product_unit @ ( H2 @ X3 ) @ ( H2 @ Y2 ) ) )
% 14.25/8.05 => ( ( finite4290736615968046902t_unit @ N )
% 14.25/8.05 => ( ( N != bot_bo3957492148770167129t_unit )
% 14.25/8.05 => ( ( H2 @ ( lattic1263872656861969706t_unit @ N ) )
% 14.25/8.05 = ( lattic1263872656861969706t_unit @ ( image_405062704495631173t_unit @ H2 @ N ) ) ) ) ) ) ).
% 14.25/8.05
% 14.25/8.05 thf(fact_2776_Inf__fin_Ohom__commute,axiom,
% 14.25/8.05 ! [H2: $o > $o,N: set_o] :
% 14.25/8.05 ( ! [X3: $o,Y2: $o] :
% 14.25/8.05 ( ( H2 @ ( inf_inf_o @ X3 @ Y2 ) )
% 14.25/8.05 = ( inf_inf_o @ ( H2 @ X3 ) @ ( H2 @ Y2 ) ) )
% 14.25/8.05 => ( ( finite_finite_o @ N )
% 14.25/8.05 => ( ( N != bot_bot_set_o )
% 14.25/8.05 => ( ( H2 @ ( lattic4107685809792843317_fin_o @ N ) )
% 14.25/8.05 = ( lattic4107685809792843317_fin_o @ ( image_o_o2 @ H2 @ N ) ) ) ) ) ) ).
% 14.25/8.05
% 14.25/8.05 thf(fact_2777_Inf__fin_Ohom__commute,axiom,
% 14.25/8.05 ! [H2: nat > nat,N: set_nat] :
% 14.25/8.05 ( ! [X3: nat,Y2: nat] :
% 14.25/8.05 ( ( H2 @ ( inf_inf_nat @ X3 @ Y2 ) )
% 14.25/8.05 = ( inf_inf_nat @ ( H2 @ X3 ) @ ( H2 @ Y2 ) ) )
% 14.25/8.05 => ( ( finite_finite_nat @ N )
% 14.25/8.05 => ( ( N != bot_bot_set_nat )
% 14.25/8.05 => ( ( H2 @ ( lattic5238388535129920115in_nat @ N ) )
% 14.25/8.05 = ( lattic5238388535129920115in_nat @ ( image_nat_nat2 @ H2 @ N ) ) ) ) ) ) ).
% 14.25/8.05
% 14.25/8.05 thf(fact_2778_Inf__fin_Ohom__commute,axiom,
% 14.25/8.05 ! [H2: int > int,N: set_int] :
% 14.25/8.05 ( ! [X3: int,Y2: int] :
% 14.25/8.05 ( ( H2 @ ( inf_inf_int @ X3 @ Y2 ) )
% 14.25/8.05 = ( inf_inf_int @ ( H2 @ X3 ) @ ( H2 @ Y2 ) ) )
% 14.25/8.05 => ( ( finite_finite_int @ N )
% 14.25/8.05 => ( ( N != bot_bot_set_int )
% 14.25/8.05 => ( ( H2 @ ( lattic5235898064620869839in_int @ N ) )
% 14.25/8.05 = ( lattic5235898064620869839in_int @ ( image_int_int2 @ H2 @ N ) ) ) ) ) ) ).
% 14.25/8.05
% 14.25/8.05 thf(fact_2779_Inf__fin_Ohom__commute,axiom,
% 14.25/8.05 ! [H2: ( produc3658429121746597890et_nat > $o ) > produc3658429121746597890et_nat > $o,N: set_Pr4532377907799695533_nat_o] :
% 14.25/8.05 ( ! [X3: produc3658429121746597890et_nat > $o,Y2: produc3658429121746597890et_nat > $o] :
% 14.25/8.05 ( ( H2 @ ( inf_in1318976480646536635_nat_o @ X3 @ Y2 ) )
% 14.25/8.05 = ( inf_in1318976480646536635_nat_o @ ( H2 @ X3 ) @ ( H2 @ Y2 ) ) )
% 14.25/8.05 => ( ( finite3252695134891459830_nat_o @ N )
% 14.25/8.05 => ( ( N != bot_bo7824918357723582233_nat_o )
% 14.25/8.05 => ( ( H2 @ ( lattic956194824204696298_nat_o @ N ) )
% 14.25/8.05 = ( lattic956194824204696298_nat_o @ ( image_3399141053782081349_nat_o @ H2 @ N ) ) ) ) ) ) ).
% 14.25/8.05
% 14.25/8.05 thf(fact_2780_Sup__fin_Ohom__commute,axiom,
% 14.25/8.05 ! [H2: set_nat > set_nat,N: set_set_nat] :
% 14.25/8.05 ( ! [X3: set_nat,Y2: set_nat] :
% 14.25/8.05 ( ( H2 @ ( sup_sup_set_nat @ X3 @ Y2 ) )
% 14.25/8.05 = ( sup_sup_set_nat @ ( H2 @ X3 ) @ ( H2 @ Y2 ) ) )
% 14.25/8.05 => ( ( finite1152437895449049373et_nat @ N )
% 14.25/8.05 => ( ( N != bot_bot_set_set_nat )
% 14.25/8.05 => ( ( H2 @ ( lattic3835124923745554447et_nat @ N ) )
% 14.25/8.05 = ( lattic3835124923745554447et_nat @ ( image_7916887816326733075et_nat @ H2 @ N ) ) ) ) ) ) ).
% 14.25/8.05
% 14.25/8.05 thf(fact_2781_Sup__fin_Ohom__commute,axiom,
% 14.25/8.05 ! [H2: assn > assn,N: set_assn] :
% 14.25/8.05 ( ! [X3: assn,Y2: assn] :
% 14.25/8.05 ( ( H2 @ ( sup_sup_assn @ X3 @ Y2 ) )
% 14.25/8.05 = ( sup_sup_assn @ ( H2 @ X3 ) @ ( H2 @ Y2 ) ) )
% 14.25/8.05 => ( ( finite_finite_assn @ N )
% 14.25/8.05 => ( ( N != bot_bot_set_assn )
% 14.25/8.05 => ( ( H2 @ ( lattic2150320897289308081n_assn @ N ) )
% 14.25/8.05 = ( lattic2150320897289308081n_assn @ ( image_assn_assn @ H2 @ N ) ) ) ) ) ) ).
% 14.25/8.05
% 14.25/8.05 thf(fact_2782_Sup__fin_Ohom__commute,axiom,
% 14.25/8.05 ! [H2: product_unit > product_unit,N: set_Product_unit] :
% 14.25/8.05 ( ! [X3: product_unit,Y2: product_unit] :
% 14.25/8.05 ( ( H2 @ ( sup_sup_Product_unit @ X3 @ Y2 ) )
% 14.25/8.05 = ( sup_sup_Product_unit @ ( H2 @ X3 ) @ ( H2 @ Y2 ) ) )
% 14.25/8.05 => ( ( finite4290736615968046902t_unit @ N )
% 14.25/8.05 => ( ( N != bot_bo3957492148770167129t_unit )
% 14.25/8.05 => ( ( H2 @ ( lattic5294303975357428420t_unit @ N ) )
% 14.25/8.05 = ( lattic5294303975357428420t_unit @ ( image_405062704495631173t_unit @ H2 @ N ) ) ) ) ) ) ).
% 14.25/8.05
% 14.25/8.05 thf(fact_2783_Sup__fin_Ohom__commute,axiom,
% 14.25/8.05 ! [H2: $o > $o,N: set_o] :
% 14.25/8.05 ( ! [X3: $o,Y2: $o] :
% 14.25/8.05 ( ( H2 @ ( sup_sup_o @ X3 @ Y2 ) )
% 14.25/8.05 = ( sup_sup_o @ ( H2 @ X3 ) @ ( H2 @ Y2 ) ) )
% 14.25/8.05 => ( ( finite_finite_o @ N )
% 14.25/8.05 => ( ( N != bot_bot_set_o )
% 14.25/8.05 => ( ( H2 @ ( lattic1508158080041050831_fin_o @ N ) )
% 14.25/8.05 = ( lattic1508158080041050831_fin_o @ ( image_o_o2 @ H2 @ N ) ) ) ) ) ) ).
% 14.25/8.05
% 14.25/8.05 thf(fact_2784_Sup__fin_Ohom__commute,axiom,
% 14.25/8.05 ! [H2: nat > nat,N: set_nat] :
% 14.25/8.05 ( ! [X3: nat,Y2: nat] :
% 14.25/8.05 ( ( H2 @ ( sup_sup_nat @ X3 @ Y2 ) )
% 14.25/8.05 = ( sup_sup_nat @ ( H2 @ X3 ) @ ( H2 @ Y2 ) ) )
% 14.25/8.05 => ( ( finite_finite_nat @ N )
% 14.25/8.05 => ( ( N != bot_bot_set_nat )
% 14.25/8.05 => ( ( H2 @ ( lattic1093996805478795353in_nat @ N ) )
% 14.25/8.05 = ( lattic1093996805478795353in_nat @ ( image_nat_nat2 @ H2 @ N ) ) ) ) ) ) ).
% 14.25/8.05
% 14.25/8.05 thf(fact_2785_Sup__fin_Ohom__commute,axiom,
% 14.25/8.05 ! [H2: int > int,N: set_int] :
% 14.25/8.05 ( ! [X3: int,Y2: int] :
% 14.25/8.05 ( ( H2 @ ( sup_sup_int @ X3 @ Y2 ) )
% 14.25/8.05 = ( sup_sup_int @ ( H2 @ X3 ) @ ( H2 @ Y2 ) ) )
% 14.25/8.05 => ( ( finite_finite_int @ N )
% 14.25/8.05 => ( ( N != bot_bot_set_int )
% 14.25/8.05 => ( ( H2 @ ( lattic1091506334969745077in_int @ N ) )
% 14.25/8.05 = ( lattic1091506334969745077in_int @ ( image_int_int2 @ H2 @ N ) ) ) ) ) ) ).
% 14.25/8.05
% 14.25/8.05 thf(fact_2786_Sup__fin_Ohom__commute,axiom,
% 14.25/8.05 ! [H2: ( produc3658429121746597890et_nat > $o ) > produc3658429121746597890et_nat > $o,N: set_Pr4532377907799695533_nat_o] :
% 14.25/8.05 ( ! [X3: produc3658429121746597890et_nat > $o,Y2: produc3658429121746597890et_nat > $o] :
% 14.25/8.05 ( ( H2 @ ( sup_su5453871518329203617_nat_o @ X3 @ Y2 ) )
% 14.25/8.05 = ( sup_su5453871518329203617_nat_o @ ( H2 @ X3 ) @ ( H2 @ Y2 ) ) )
% 14.25/8.05 => ( ( finite3252695134891459830_nat_o @ N )
% 14.25/8.05 => ( ( N != bot_bo7824918357723582233_nat_o )
% 14.25/8.05 => ( ( H2 @ ( lattic7320199455484906628_nat_o @ N ) )
% 14.25/8.05 = ( lattic7320199455484906628_nat_o @ ( image_3399141053782081349_nat_o @ H2 @ N ) ) ) ) ) ) ).
% 14.25/8.05
% 14.25/8.05 thf(fact_2787_finite__remove__induct,axiom,
% 14.25/8.05 ! [B4: set_Code_integer,P: set_Code_integer > $o] :
% 14.25/8.05 ( ( finite6017078050557962740nteger @ B4 )
% 14.25/8.05 => ( ( P @ bot_bo3990330152332043303nteger )
% 14.25/8.05 => ( ! [A9: set_Code_integer] :
% 14.25/8.05 ( ( finite6017078050557962740nteger @ A9 )
% 14.25/8.05 => ( ( A9 != bot_bo3990330152332043303nteger )
% 14.25/8.05 => ( ( ord_le7084787975880047091nteger @ A9 @ B4 )
% 14.25/8.05 => ( ! [X6: code_integer] :
% 14.25/8.05 ( ( member_Code_integer @ X6 @ A9 )
% 14.25/8.05 => ( P @ ( minus_2355218937544613996nteger @ A9 @ ( insert_Code_integer @ X6 @ bot_bo3990330152332043303nteger ) ) ) )
% 14.25/8.05 => ( P @ A9 ) ) ) ) )
% 14.25/8.05 => ( P @ B4 ) ) ) ) ).
% 14.25/8.05
% 14.25/8.05 thf(fact_2788_finite__remove__induct,axiom,
% 14.25/8.05 ! [B4: set_Product_unit,P: set_Product_unit > $o] :
% 14.25/8.05 ( ( finite4290736615968046902t_unit @ B4 )
% 14.25/8.05 => ( ( P @ bot_bo3957492148770167129t_unit )
% 14.25/8.05 => ( ! [A9: set_Product_unit] :
% 14.25/8.05 ( ( finite4290736615968046902t_unit @ A9 )
% 14.25/8.05 => ( ( A9 != bot_bo3957492148770167129t_unit )
% 14.25/8.05 => ( ( ord_le3507040750410214029t_unit @ A9 @ B4 )
% 14.25/8.05 => ( ! [X6: product_unit] :
% 14.25/8.05 ( ( member_Product_unit @ X6 @ A9 )
% 14.25/8.05 => ( P @ ( minus_6452836326544984404t_unit @ A9 @ ( insert_Product_unit @ X6 @ bot_bo3957492148770167129t_unit ) ) ) )
% 14.25/8.05 => ( P @ A9 ) ) ) ) )
% 14.25/8.05 => ( P @ B4 ) ) ) ) ).
% 14.25/8.05
% 14.25/8.05 thf(fact_2789_finite__remove__induct,axiom,
% 14.25/8.05 ! [B4: set_o,P: set_o > $o] :
% 14.25/8.05 ( ( finite_finite_o @ B4 )
% 14.25/8.05 => ( ( P @ bot_bot_set_o )
% 14.25/8.05 => ( ! [A9: set_o] :
% 14.25/8.05 ( ( finite_finite_o @ A9 )
% 14.25/8.05 => ( ( A9 != bot_bot_set_o )
% 14.25/8.05 => ( ( ord_less_eq_set_o @ A9 @ B4 )
% 14.25/8.05 => ( ! [X6: $o] :
% 14.25/8.05 ( ( member_o @ X6 @ A9 )
% 14.25/8.05 => ( P @ ( minus_minus_set_o @ A9 @ ( insert_o @ X6 @ bot_bot_set_o ) ) ) )
% 14.25/8.05 => ( P @ A9 ) ) ) ) )
% 14.25/8.05 => ( P @ B4 ) ) ) ) ).
% 14.25/8.05
% 14.25/8.05 thf(fact_2790_finite__remove__induct,axiom,
% 14.25/8.05 ! [B4: set_int,P: set_int > $o] :
% 14.25/8.05 ( ( finite_finite_int @ B4 )
% 14.25/8.05 => ( ( P @ bot_bot_set_int )
% 14.25/8.05 => ( ! [A9: set_int] :
% 14.25/8.05 ( ( finite_finite_int @ A9 )
% 14.25/8.05 => ( ( A9 != bot_bot_set_int )
% 14.25/8.05 => ( ( ord_less_eq_set_int @ A9 @ B4 )
% 14.25/8.05 => ( ! [X6: int] :
% 14.25/8.05 ( ( member_int @ X6 @ A9 )
% 14.25/8.05 => ( P @ ( minus_minus_set_int @ A9 @ ( insert_int @ X6 @ bot_bot_set_int ) ) ) )
% 14.25/8.05 => ( P @ A9 ) ) ) ) )
% 14.25/8.05 => ( P @ B4 ) ) ) ) ).
% 14.25/8.05
% 14.25/8.05 thf(fact_2791_finite__remove__induct,axiom,
% 14.25/8.05 ! [B4: set_Pr4532377907799695533_nat_o,P: set_Pr4532377907799695533_nat_o > $o] :
% 14.25/8.05 ( ( finite3252695134891459830_nat_o @ B4 )
% 14.25/8.05 => ( ( P @ bot_bo7824918357723582233_nat_o )
% 14.25/8.05 => ( ! [A9: set_Pr4532377907799695533_nat_o] :
% 14.25/8.05 ( ( finite3252695134891459830_nat_o @ A9 )
% 14.25/8.05 => ( ( A9 != bot_bo7824918357723582233_nat_o )
% 14.25/8.05 => ( ( ord_le2965882846123202637_nat_o @ A9 @ B4 )
% 14.25/8.05 => ( ! [X6: produc3658429121746597890et_nat > $o] :
% 14.25/8.05 ( ( member6576561426505652726_nat_o @ X6 @ A9 )
% 14.25/8.05 => ( P @ ( minus_1801376950450012436_nat_o @ A9 @ ( insert5175938949040314269_nat_o @ X6 @ bot_bo7824918357723582233_nat_o ) ) ) )
% 14.25/8.05 => ( P @ A9 ) ) ) ) )
% 14.25/8.05 => ( P @ B4 ) ) ) ) ).
% 14.25/8.05
% 14.25/8.05 thf(fact_2792_finite__remove__induct,axiom,
% 14.25/8.05 ! [B4: set_nat,P: set_nat > $o] :
% 14.25/8.05 ( ( finite_finite_nat @ B4 )
% 14.25/8.05 => ( ( P @ bot_bot_set_nat )
% 14.25/8.05 => ( ! [A9: set_nat] :
% 14.25/8.05 ( ( finite_finite_nat @ A9 )
% 14.25/8.05 => ( ( A9 != bot_bot_set_nat )
% 14.25/8.05 => ( ( ord_less_eq_set_nat @ A9 @ B4 )
% 14.25/8.05 => ( ! [X6: nat] :
% 14.25/8.05 ( ( member_nat @ X6 @ A9 )
% 14.25/8.05 => ( P @ ( minus_minus_set_nat @ A9 @ ( insert_nat @ X6 @ bot_bot_set_nat ) ) ) )
% 14.25/8.05 => ( P @ A9 ) ) ) ) )
% 14.25/8.05 => ( P @ B4 ) ) ) ) ).
% 14.25/8.05
% 14.25/8.05 thf(fact_2793_remove__induct,axiom,
% 14.25/8.05 ! [P: set_Code_integer > $o,B4: set_Code_integer] :
% 14.25/8.05 ( ( P @ bot_bo3990330152332043303nteger )
% 14.25/8.05 => ( ( ~ ( finite6017078050557962740nteger @ B4 )
% 14.25/8.05 => ( P @ B4 ) )
% 14.25/8.05 => ( ! [A9: set_Code_integer] :
% 14.25/8.05 ( ( finite6017078050557962740nteger @ A9 )
% 14.25/8.05 => ( ( A9 != bot_bo3990330152332043303nteger )
% 14.25/8.05 => ( ( ord_le7084787975880047091nteger @ A9 @ B4 )
% 14.25/8.05 => ( ! [X6: code_integer] :
% 14.25/8.05 ( ( member_Code_integer @ X6 @ A9 )
% 14.25/8.05 => ( P @ ( minus_2355218937544613996nteger @ A9 @ ( insert_Code_integer @ X6 @ bot_bo3990330152332043303nteger ) ) ) )
% 14.25/8.05 => ( P @ A9 ) ) ) ) )
% 14.25/8.05 => ( P @ B4 ) ) ) ) ).
% 14.25/8.05
% 14.25/8.05 thf(fact_2794_remove__induct,axiom,
% 14.25/8.05 ! [P: set_Product_unit > $o,B4: set_Product_unit] :
% 14.25/8.05 ( ( P @ bot_bo3957492148770167129t_unit )
% 14.25/8.05 => ( ( ~ ( finite4290736615968046902t_unit @ B4 )
% 14.25/8.05 => ( P @ B4 ) )
% 14.25/8.05 => ( ! [A9: set_Product_unit] :
% 14.25/8.05 ( ( finite4290736615968046902t_unit @ A9 )
% 14.25/8.05 => ( ( A9 != bot_bo3957492148770167129t_unit )
% 14.25/8.05 => ( ( ord_le3507040750410214029t_unit @ A9 @ B4 )
% 14.25/8.05 => ( ! [X6: product_unit] :
% 14.25/8.05 ( ( member_Product_unit @ X6 @ A9 )
% 14.25/8.05 => ( P @ ( minus_6452836326544984404t_unit @ A9 @ ( insert_Product_unit @ X6 @ bot_bo3957492148770167129t_unit ) ) ) )
% 14.25/8.05 => ( P @ A9 ) ) ) ) )
% 14.25/8.05 => ( P @ B4 ) ) ) ) ).
% 14.25/8.05
% 14.25/8.05 thf(fact_2795_remove__induct,axiom,
% 14.25/8.05 ! [P: set_o > $o,B4: set_o] :
% 14.25/8.05 ( ( P @ bot_bot_set_o )
% 14.25/8.05 => ( ( ~ ( finite_finite_o @ B4 )
% 14.25/8.05 => ( P @ B4 ) )
% 14.25/8.05 => ( ! [A9: set_o] :
% 14.25/8.05 ( ( finite_finite_o @ A9 )
% 14.25/8.05 => ( ( A9 != bot_bot_set_o )
% 14.25/8.05 => ( ( ord_less_eq_set_o @ A9 @ B4 )
% 14.25/8.05 => ( ! [X6: $o] :
% 14.25/8.05 ( ( member_o @ X6 @ A9 )
% 14.25/8.05 => ( P @ ( minus_minus_set_o @ A9 @ ( insert_o @ X6 @ bot_bot_set_o ) ) ) )
% 14.25/8.05 => ( P @ A9 ) ) ) ) )
% 14.25/8.05 => ( P @ B4 ) ) ) ) ).
% 14.25/8.05
% 14.25/8.05 thf(fact_2796_remove__induct,axiom,
% 14.25/8.05 ! [P: set_int > $o,B4: set_int] :
% 14.25/8.05 ( ( P @ bot_bot_set_int )
% 14.25/8.05 => ( ( ~ ( finite_finite_int @ B4 )
% 14.25/8.05 => ( P @ B4 ) )
% 14.25/8.05 => ( ! [A9: set_int] :
% 14.25/8.05 ( ( finite_finite_int @ A9 )
% 14.25/8.05 => ( ( A9 != bot_bot_set_int )
% 14.25/8.05 => ( ( ord_less_eq_set_int @ A9 @ B4 )
% 14.25/8.05 => ( ! [X6: int] :
% 14.25/8.05 ( ( member_int @ X6 @ A9 )
% 14.25/8.05 => ( P @ ( minus_minus_set_int @ A9 @ ( insert_int @ X6 @ bot_bot_set_int ) ) ) )
% 14.25/8.05 => ( P @ A9 ) ) ) ) )
% 14.25/8.05 => ( P @ B4 ) ) ) ) ).
% 14.25/8.05
% 14.25/8.05 thf(fact_2797_remove__induct,axiom,
% 14.25/8.05 ! [P: set_Pr4532377907799695533_nat_o > $o,B4: set_Pr4532377907799695533_nat_o] :
% 14.25/8.05 ( ( P @ bot_bo7824918357723582233_nat_o )
% 14.25/8.05 => ( ( ~ ( finite3252695134891459830_nat_o @ B4 )
% 14.25/8.05 => ( P @ B4 ) )
% 14.25/8.05 => ( ! [A9: set_Pr4532377907799695533_nat_o] :
% 14.25/8.05 ( ( finite3252695134891459830_nat_o @ A9 )
% 14.25/8.05 => ( ( A9 != bot_bo7824918357723582233_nat_o )
% 14.25/8.05 => ( ( ord_le2965882846123202637_nat_o @ A9 @ B4 )
% 14.25/8.05 => ( ! [X6: produc3658429121746597890et_nat > $o] :
% 14.25/8.05 ( ( member6576561426505652726_nat_o @ X6 @ A9 )
% 14.25/8.05 => ( P @ ( minus_1801376950450012436_nat_o @ A9 @ ( insert5175938949040314269_nat_o @ X6 @ bot_bo7824918357723582233_nat_o ) ) ) )
% 14.25/8.05 => ( P @ A9 ) ) ) ) )
% 14.25/8.05 => ( P @ B4 ) ) ) ) ).
% 14.25/8.05
% 14.25/8.05 thf(fact_2798_remove__induct,axiom,
% 14.25/8.05 ! [P: set_nat > $o,B4: set_nat] :
% 14.25/8.05 ( ( P @ bot_bot_set_nat )
% 14.25/8.05 => ( ( ~ ( finite_finite_nat @ B4 )
% 14.25/8.05 => ( P @ B4 ) )
% 14.25/8.05 => ( ! [A9: set_nat] :
% 14.25/8.05 ( ( finite_finite_nat @ A9 )
% 14.25/8.05 => ( ( A9 != bot_bot_set_nat )
% 14.25/8.05 => ( ( ord_less_eq_set_nat @ A9 @ B4 )
% 14.25/8.05 => ( ! [X6: nat] :
% 14.25/8.05 ( ( member_nat @ X6 @ A9 )
% 14.25/8.05 => ( P @ ( minus_minus_set_nat @ A9 @ ( insert_nat @ X6 @ bot_bot_set_nat ) ) ) )
% 14.25/8.05 => ( P @ A9 ) ) ) ) )
% 14.25/8.05 => ( P @ B4 ) ) ) ) ).
% 14.25/8.05
% 14.25/8.05 thf(fact_2799_Linear__order__in__diff__Id,axiom,
% 14.25/8.05 ! [R2: set_Pr2161125870931222855_nat_o,A2: produc3658429121746597890et_nat > $o,B: produc3658429121746597890et_nat > $o] :
% 14.25/8.05 ( ( order_1145710701291701250_nat_o @ ( field_81821745747897882_nat_o @ R2 ) @ R2 )
% 14.25/8.05 => ( ( member6576561426505652726_nat_o @ A2 @ ( field_81821745747897882_nat_o @ R2 ) )
% 14.25/8.05 => ( ( member6576561426505652726_nat_o @ B @ ( field_81821745747897882_nat_o @ R2 ) )
% 14.25/8.05 => ( ( member8781333585448626064_nat_o @ ( produc7368190662567826135_nat_o @ A2 @ B ) @ R2 )
% 14.25/8.05 = ( ~ ( member8781333585448626064_nat_o @ ( produc7368190662567826135_nat_o @ B @ A2 ) @ ( minus_9189903262233110958_nat_o @ R2 @ id_Pro6775173067513470761_nat_o ) ) ) ) ) ) ) ).
% 14.25/8.05
% 14.25/8.05 thf(fact_2800_Linear__order__in__diff__Id,axiom,
% 14.25/8.05 ! [R2: set_Product_prod_o_o,A2: $o,B: $o] :
% 14.25/8.05 ( ( order_6238756238976269133r_on_o @ ( field_o @ R2 ) @ R2 )
% 14.25/8.05 => ( ( member_o @ A2 @ ( field_o @ R2 ) )
% 14.25/8.05 => ( ( member_o @ B @ ( field_o @ R2 ) )
% 14.25/8.05 => ( ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ A2 @ B ) @ R2 )
% 14.25/8.05 = ( ~ ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ B @ A2 ) @ ( minus_7052513330764642158od_o_o @ R2 @ id_o2 ) ) ) ) ) ) ) ).
% 14.25/8.05
% 14.25/8.05 thf(fact_2801_Linear__order__in__diff__Id,axiom,
% 14.25/8.05 ! [R2: set_Pr5094982260447487303t_unit,A2: product_unit,B: product_unit] :
% 14.25/8.05 ( ( order_888282469287196226t_unit @ ( field_Product_unit @ R2 ) @ R2 )
% 14.25/8.05 => ( ( member_Product_unit @ A2 @ ( field_Product_unit @ R2 ) )
% 14.25/8.05 => ( ( member_Product_unit @ B @ ( field_Product_unit @ R2 ) )
% 14.25/8.05 => ( ( member7821138191923116944t_unit @ ( produc5692694829384537815t_unit @ A2 @ B ) @ R2 )
% 14.25/8.05 = ( ~ ( member7821138191923116944t_unit @ ( produc5692694829384537815t_unit @ B @ A2 ) @ ( minus_6985556733573110190t_unit @ R2 @ id_Product_unit ) ) ) ) ) ) ) ).
% 14.25/8.05
% 14.25/8.05 thf(fact_2802_Linear__order__in__diff__Id,axiom,
% 14.25/8.05 ! [R2: set_Pr1261947904930325089at_nat,A2: nat,B: nat] :
% 14.25/8.05 ( ( order_4473980167227706203on_nat @ ( field_nat @ R2 ) @ R2 )
% 14.25/8.05 => ( ( member_nat @ A2 @ ( field_nat @ R2 ) )
% 14.25/8.05 => ( ( member_nat @ B @ ( field_nat @ R2 ) )
% 14.25/8.05 => ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A2 @ B ) @ R2 )
% 14.25/8.05 = ( ~ ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ B @ A2 ) @ ( minus_1356011639430497352at_nat @ R2 @ id_nat ) ) ) ) ) ) ) ).
% 14.25/8.05
% 14.25/8.05 thf(fact_2803_Linear__order__in__diff__Id,axiom,
% 14.25/8.05 ! [R2: set_Pr958786334691620121nt_int,A2: int,B: int] :
% 14.25/8.05 ( ( order_4471489696718655927on_int @ ( field_int @ R2 ) @ R2 )
% 14.25/8.05 => ( ( member_int @ A2 @ ( field_int @ R2 ) )
% 14.25/8.05 => ( ( member_int @ B @ ( field_int @ R2 ) )
% 14.25/8.05 => ( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ A2 @ B ) @ R2 )
% 14.25/8.05 = ( ~ ( member5262025264175285858nt_int @ ( product_Pair_int_int @ B @ A2 ) @ ( minus_1052850069191792384nt_int @ R2 @ id_int ) ) ) ) ) ) ) ).
% 14.25/8.05
% 14.25/8.05 thf(fact_2804_max__ext_Omax__extI,axiom,
% 14.25/8.05 ! [X4: set_Code_integer,Y6: set_Code_integer,R3: set_Pr4811707699266497531nteger] :
% 14.25/8.05 ( ( finite6017078050557962740nteger @ X4 )
% 14.25/8.05 => ( ( finite6017078050557962740nteger @ Y6 )
% 14.25/8.05 => ( ( Y6 != bot_bo3990330152332043303nteger )
% 14.25/8.05 => ( ! [X3: code_integer] :
% 14.25/8.05 ( ( member_Code_integer @ X3 @ X4 )
% 14.25/8.05 => ? [Xa2: code_integer] :
% 14.25/8.05 ( ( member_Code_integer @ Xa2 @ Y6 )
% 14.25/8.05 & ( member157494554546826820nteger @ ( produc1086072967326762835nteger @ X3 @ Xa2 ) @ R3 ) ) )
% 14.25/8.05 => ( member4307123515891402160nteger @ ( produc7443773368509356479nteger @ X4 @ Y6 ) @ ( max_ext_Code_integer @ R3 ) ) ) ) ) ) ).
% 14.25/8.05
% 14.25/8.05 thf(fact_2805_max__ext_Omax__extI,axiom,
% 14.25/8.05 ! [X4: set_Product_unit,Y6: set_Product_unit,R3: set_Pr5094982260447487303t_unit] :
% 14.25/8.05 ( ( finite4290736615968046902t_unit @ X4 )
% 14.25/8.05 => ( ( finite4290736615968046902t_unit @ Y6 )
% 14.25/8.05 => ( ( Y6 != bot_bo3957492148770167129t_unit )
% 14.25/8.05 => ( ! [X3: product_unit] :
% 14.25/8.05 ( ( member_Product_unit @ X3 @ X4 )
% 14.25/8.05 => ? [Xa2: product_unit] :
% 14.25/8.05 ( ( member_Product_unit @ Xa2 @ Y6 )
% 14.25/8.05 & ( member7821138191923116944t_unit @ ( produc5692694829384537815t_unit @ X3 @ Xa2 ) @ R3 ) ) )
% 14.25/8.05 => ( member4189953012975926160t_unit @ ( produc7269801343787132631t_unit @ X4 @ Y6 ) @ ( max_ext_Product_unit @ R3 ) ) ) ) ) ) ).
% 14.25/8.05
% 14.25/8.05 thf(fact_2806_max__ext_Omax__extI,axiom,
% 14.25/8.05 ! [X4: set_o,Y6: set_o,R3: set_Product_prod_o_o] :
% 14.25/8.05 ( ( finite_finite_o @ X4 )
% 14.25/8.05 => ( ( finite_finite_o @ Y6 )
% 14.25/8.05 => ( ( Y6 != bot_bot_set_o )
% 14.25/8.05 => ( ! [X3: $o] :
% 14.25/8.05 ( ( member_o @ X3 @ X4 )
% 14.25/8.05 => ? [Xa2: $o] :
% 14.25/8.05 ( ( member_o @ Xa2 @ Y6 )
% 14.25/8.05 & ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ X3 @ Xa2 ) @ R3 ) ) )
% 14.25/8.05 => ( member9116954335612470352_set_o @ ( produc5838405689764958487_set_o @ X4 @ Y6 ) @ ( max_ext_o @ R3 ) ) ) ) ) ) ).
% 14.25/8.05
% 14.25/8.05 thf(fact_2807_max__ext_Omax__extI,axiom,
% 14.25/8.05 ! [X4: set_nat,Y6: set_nat,R3: set_Pr1261947904930325089at_nat] :
% 14.25/8.05 ( ( finite_finite_nat @ X4 )
% 14.25/8.05 => ( ( finite_finite_nat @ Y6 )
% 14.25/8.05 => ( ( Y6 != bot_bot_set_nat )
% 14.25/8.05 => ( ! [X3: nat] :
% 14.25/8.05 ( ( member_nat @ X3 @ X4 )
% 14.25/8.05 => ? [Xa2: nat] :
% 14.25/8.05 ( ( member_nat @ Xa2 @ Y6 )
% 14.25/8.05 & ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X3 @ Xa2 ) @ R3 ) ) )
% 14.25/8.05 => ( member8277197624267554838et_nat @ ( produc4532415448927165861et_nat @ X4 @ Y6 ) @ ( max_ext_nat @ R3 ) ) ) ) ) ) ).
% 14.25/8.05
% 14.25/8.05 thf(fact_2808_max__ext_Omax__extI,axiom,
% 14.25/8.05 ! [X4: set_int,Y6: set_int,R3: set_Pr958786334691620121nt_int] :
% 14.25/8.05 ( ( finite_finite_int @ X4 )
% 14.25/8.05 => ( ( finite_finite_int @ Y6 )
% 14.25/8.05 => ( ( Y6 != bot_bot_set_int )
% 14.25/8.05 => ( ! [X3: int] :
% 14.25/8.05 ( ( member_int @ X3 @ X4 )
% 14.25/8.05 => ? [Xa2: int] :
% 14.25/8.05 ( ( member_int @ Xa2 @ Y6 )
% 14.25/8.05 & ( member5262025264175285858nt_int @ ( product_Pair_int_int @ X3 @ Xa2 ) @ R3 ) ) )
% 14.25/8.05 => ( member2572552093476627150et_int @ ( produc6363374080413544029et_int @ X4 @ Y6 ) @ ( max_ext_int @ R3 ) ) ) ) ) ) ).
% 14.25/8.05
% 14.25/8.05 thf(fact_2809_max__ext_Omax__extI,axiom,
% 14.25/8.05 ! [X4: set_Pr4532377907799695533_nat_o,Y6: set_Pr4532377907799695533_nat_o,R3: set_Pr2161125870931222855_nat_o] :
% 14.25/8.05 ( ( finite3252695134891459830_nat_o @ X4 )
% 14.25/8.05 => ( ( finite3252695134891459830_nat_o @ Y6 )
% 14.25/8.05 => ( ( Y6 != bot_bo7824918357723582233_nat_o )
% 14.25/8.05 => ( ! [X3: produc3658429121746597890et_nat > $o] :
% 14.25/8.05 ( ( member6576561426505652726_nat_o @ X3 @ X4 )
% 14.25/8.05 => ? [Xa2: produc3658429121746597890et_nat > $o] :
% 14.25/8.05 ( ( member6576561426505652726_nat_o @ Xa2 @ Y6 )
% 14.25/8.05 & ( member8781333585448626064_nat_o @ ( produc7368190662567826135_nat_o @ X3 @ Xa2 ) @ R3 ) ) )
% 14.25/8.05 => ( member6623172407746373520_nat_o @ ( produc5149316574908490455_nat_o @ X4 @ Y6 ) @ ( max_ex5265246208886859912_nat_o @ R3 ) ) ) ) ) ) ).
% 14.25/8.05
% 14.25/8.05 thf(fact_2810_max__ext_Osimps,axiom,
% 14.25/8.05 ! [A1: set_Code_integer,A22: set_Code_integer,R3: set_Pr4811707699266497531nteger] :
% 14.25/8.05 ( ( member4307123515891402160nteger @ ( produc7443773368509356479nteger @ A1 @ A22 ) @ ( max_ext_Code_integer @ R3 ) )
% 14.25/8.05 = ( ( finite6017078050557962740nteger @ A1 )
% 14.25/8.05 & ( finite6017078050557962740nteger @ A22 )
% 14.25/8.05 & ( A22 != bot_bo3990330152332043303nteger )
% 14.25/8.05 & ! [X2: code_integer] :
% 14.25/8.05 ( ( member_Code_integer @ X2 @ A1 )
% 14.25/8.05 => ? [Y3: code_integer] :
% 14.25/8.05 ( ( member_Code_integer @ Y3 @ A22 )
% 14.25/8.05 & ( member157494554546826820nteger @ ( produc1086072967326762835nteger @ X2 @ Y3 ) @ R3 ) ) ) ) ) ).
% 14.25/8.05
% 14.25/8.05 thf(fact_2811_max__ext_Osimps,axiom,
% 14.25/8.05 ! [A1: set_Product_unit,A22: set_Product_unit,R3: set_Pr5094982260447487303t_unit] :
% 14.25/8.05 ( ( member4189953012975926160t_unit @ ( produc7269801343787132631t_unit @ A1 @ A22 ) @ ( max_ext_Product_unit @ R3 ) )
% 14.25/8.05 = ( ( finite4290736615968046902t_unit @ A1 )
% 14.25/8.05 & ( finite4290736615968046902t_unit @ A22 )
% 14.25/8.05 & ( A22 != bot_bo3957492148770167129t_unit )
% 14.25/8.05 & ! [X2: product_unit] :
% 14.25/8.05 ( ( member_Product_unit @ X2 @ A1 )
% 14.25/8.05 => ? [Y3: product_unit] :
% 14.25/8.05 ( ( member_Product_unit @ Y3 @ A22 )
% 14.25/8.05 & ( member7821138191923116944t_unit @ ( produc5692694829384537815t_unit @ X2 @ Y3 ) @ R3 ) ) ) ) ) ).
% 14.25/8.05
% 14.25/8.05 thf(fact_2812_max__ext_Osimps,axiom,
% 14.25/8.05 ! [A1: set_o,A22: set_o,R3: set_Product_prod_o_o] :
% 14.25/8.05 ( ( member9116954335612470352_set_o @ ( produc5838405689764958487_set_o @ A1 @ A22 ) @ ( max_ext_o @ R3 ) )
% 14.25/8.05 = ( ( finite_finite_o @ A1 )
% 14.25/8.05 & ( finite_finite_o @ A22 )
% 14.25/8.05 & ( A22 != bot_bot_set_o )
% 14.25/8.05 & ! [X2: $o] :
% 14.25/8.05 ( ( member_o @ X2 @ A1 )
% 14.25/8.05 => ? [Y3: $o] :
% 14.25/8.05 ( ( member_o @ Y3 @ A22 )
% 14.25/8.05 & ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ X2 @ Y3 ) @ R3 ) ) ) ) ) ).
% 14.25/8.05
% 14.25/8.05 thf(fact_2813_max__ext_Osimps,axiom,
% 14.25/8.05 ! [A1: set_nat,A22: set_nat,R3: set_Pr1261947904930325089at_nat] :
% 14.25/8.05 ( ( member8277197624267554838et_nat @ ( produc4532415448927165861et_nat @ A1 @ A22 ) @ ( max_ext_nat @ R3 ) )
% 14.25/8.05 = ( ( finite_finite_nat @ A1 )
% 14.25/8.05 & ( finite_finite_nat @ A22 )
% 14.25/8.05 & ( A22 != bot_bot_set_nat )
% 14.25/8.05 & ! [X2: nat] :
% 14.25/8.05 ( ( member_nat @ X2 @ A1 )
% 14.25/8.05 => ? [Y3: nat] :
% 14.25/8.05 ( ( member_nat @ Y3 @ A22 )
% 14.25/8.05 & ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X2 @ Y3 ) @ R3 ) ) ) ) ) ).
% 14.25/8.05
% 14.25/8.05 thf(fact_2814_max__ext_Osimps,axiom,
% 14.25/8.05 ! [A1: set_int,A22: set_int,R3: set_Pr958786334691620121nt_int] :
% 14.25/8.05 ( ( member2572552093476627150et_int @ ( produc6363374080413544029et_int @ A1 @ A22 ) @ ( max_ext_int @ R3 ) )
% 14.25/8.05 = ( ( finite_finite_int @ A1 )
% 14.25/8.05 & ( finite_finite_int @ A22 )
% 14.25/8.05 & ( A22 != bot_bot_set_int )
% 14.25/8.05 & ! [X2: int] :
% 14.25/8.05 ( ( member_int @ X2 @ A1 )
% 14.25/8.05 => ? [Y3: int] :
% 14.25/8.05 ( ( member_int @ Y3 @ A22 )
% 14.25/8.05 & ( member5262025264175285858nt_int @ ( product_Pair_int_int @ X2 @ Y3 ) @ R3 ) ) ) ) ) ).
% 14.25/8.05
% 14.25/8.05 thf(fact_2815_max__ext_Osimps,axiom,
% 14.25/8.05 ! [A1: set_Pr4532377907799695533_nat_o,A22: set_Pr4532377907799695533_nat_o,R3: set_Pr2161125870931222855_nat_o] :
% 14.25/8.05 ( ( member6623172407746373520_nat_o @ ( produc5149316574908490455_nat_o @ A1 @ A22 ) @ ( max_ex5265246208886859912_nat_o @ R3 ) )
% 14.25/8.05 = ( ( finite3252695134891459830_nat_o @ A1 )
% 14.25/8.05 & ( finite3252695134891459830_nat_o @ A22 )
% 14.25/8.05 & ( A22 != bot_bo7824918357723582233_nat_o )
% 14.25/8.05 & ! [X2: produc3658429121746597890et_nat > $o] :
% 14.25/8.05 ( ( member6576561426505652726_nat_o @ X2 @ A1 )
% 14.25/8.05 => ? [Y3: produc3658429121746597890et_nat > $o] :
% 14.25/8.05 ( ( member6576561426505652726_nat_o @ Y3 @ A22 )
% 14.25/8.05 & ( member8781333585448626064_nat_o @ ( produc7368190662567826135_nat_o @ X2 @ Y3 ) @ R3 ) ) ) ) ) ).
% 14.25/8.05
% 14.25/8.05 thf(fact_2816_max__ext_Ocases,axiom,
% 14.25/8.05 ! [A1: set_Code_integer,A22: set_Code_integer,R3: set_Pr4811707699266497531nteger] :
% 14.25/8.05 ( ( member4307123515891402160nteger @ ( produc7443773368509356479nteger @ A1 @ A22 ) @ ( max_ext_Code_integer @ R3 ) )
% 14.25/8.05 => ~ ( ( finite6017078050557962740nteger @ A1 )
% 14.25/8.05 => ( ( finite6017078050557962740nteger @ A22 )
% 14.25/8.05 => ( ( A22 != bot_bo3990330152332043303nteger )
% 14.25/8.05 => ~ ! [X6: code_integer] :
% 14.25/8.05 ( ( member_Code_integer @ X6 @ A1 )
% 14.25/8.05 => ? [Xa3: code_integer] :
% 14.25/8.05 ( ( member_Code_integer @ Xa3 @ A22 )
% 14.25/8.05 & ( member157494554546826820nteger @ ( produc1086072967326762835nteger @ X6 @ Xa3 ) @ R3 ) ) ) ) ) ) ) ).
% 14.25/8.05
% 14.25/8.05 thf(fact_2817_max__ext_Ocases,axiom,
% 14.25/8.05 ! [A1: set_Product_unit,A22: set_Product_unit,R3: set_Pr5094982260447487303t_unit] :
% 14.25/8.05 ( ( member4189953012975926160t_unit @ ( produc7269801343787132631t_unit @ A1 @ A22 ) @ ( max_ext_Product_unit @ R3 ) )
% 14.25/8.05 => ~ ( ( finite4290736615968046902t_unit @ A1 )
% 14.25/8.05 => ( ( finite4290736615968046902t_unit @ A22 )
% 14.25/8.05 => ( ( A22 != bot_bo3957492148770167129t_unit )
% 14.25/8.05 => ~ ! [X6: product_unit] :
% 14.25/8.05 ( ( member_Product_unit @ X6 @ A1 )
% 14.25/8.05 => ? [Xa3: product_unit] :
% 14.25/8.05 ( ( member_Product_unit @ Xa3 @ A22 )
% 14.25/8.05 & ( member7821138191923116944t_unit @ ( produc5692694829384537815t_unit @ X6 @ Xa3 ) @ R3 ) ) ) ) ) ) ) ).
% 14.25/8.05
% 14.25/8.05 thf(fact_2818_max__ext_Ocases,axiom,
% 14.25/8.05 ! [A1: set_o,A22: set_o,R3: set_Product_prod_o_o] :
% 14.25/8.05 ( ( member9116954335612470352_set_o @ ( produc5838405689764958487_set_o @ A1 @ A22 ) @ ( max_ext_o @ R3 ) )
% 14.25/8.05 => ~ ( ( finite_finite_o @ A1 )
% 14.25/8.05 => ( ( finite_finite_o @ A22 )
% 14.25/8.05 => ( ( A22 != bot_bot_set_o )
% 14.25/8.05 => ~ ! [X6: $o] :
% 14.25/8.05 ( ( member_o @ X6 @ A1 )
% 14.25/8.05 => ? [Xa3: $o] :
% 14.25/8.05 ( ( member_o @ Xa3 @ A22 )
% 14.25/8.05 & ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ X6 @ Xa3 ) @ R3 ) ) ) ) ) ) ) ).
% 14.25/8.05
% 14.25/8.05 thf(fact_2819_max__ext_Ocases,axiom,
% 14.25/8.05 ! [A1: set_nat,A22: set_nat,R3: set_Pr1261947904930325089at_nat] :
% 14.25/8.05 ( ( member8277197624267554838et_nat @ ( produc4532415448927165861et_nat @ A1 @ A22 ) @ ( max_ext_nat @ R3 ) )
% 14.25/8.05 => ~ ( ( finite_finite_nat @ A1 )
% 14.25/8.05 => ( ( finite_finite_nat @ A22 )
% 14.25/8.05 => ( ( A22 != bot_bot_set_nat )
% 14.25/8.05 => ~ ! [X6: nat] :
% 14.25/8.05 ( ( member_nat @ X6 @ A1 )
% 14.25/8.05 => ? [Xa3: nat] :
% 14.25/8.05 ( ( member_nat @ Xa3 @ A22 )
% 14.25/8.05 & ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X6 @ Xa3 ) @ R3 ) ) ) ) ) ) ) ).
% 14.25/8.05
% 14.25/8.05 thf(fact_2820_max__ext_Ocases,axiom,
% 14.25/8.05 ! [A1: set_int,A22: set_int,R3: set_Pr958786334691620121nt_int] :
% 14.25/8.05 ( ( member2572552093476627150et_int @ ( produc6363374080413544029et_int @ A1 @ A22 ) @ ( max_ext_int @ R3 ) )
% 14.25/8.05 => ~ ( ( finite_finite_int @ A1 )
% 14.25/8.05 => ( ( finite_finite_int @ A22 )
% 14.25/8.05 => ( ( A22 != bot_bot_set_int )
% 14.25/8.05 => ~ ! [X6: int] :
% 14.25/8.05 ( ( member_int @ X6 @ A1 )
% 14.25/8.05 => ? [Xa3: int] :
% 14.25/8.05 ( ( member_int @ Xa3 @ A22 )
% 14.25/8.05 & ( member5262025264175285858nt_int @ ( product_Pair_int_int @ X6 @ Xa3 ) @ R3 ) ) ) ) ) ) ) ).
% 14.25/8.05
% 14.25/8.05 thf(fact_2821_max__ext_Ocases,axiom,
% 14.25/8.05 ! [A1: set_Pr4532377907799695533_nat_o,A22: set_Pr4532377907799695533_nat_o,R3: set_Pr2161125870931222855_nat_o] :
% 14.25/8.05 ( ( member6623172407746373520_nat_o @ ( produc5149316574908490455_nat_o @ A1 @ A22 ) @ ( max_ex5265246208886859912_nat_o @ R3 ) )
% 14.25/8.05 => ~ ( ( finite3252695134891459830_nat_o @ A1 )
% 14.25/8.05 => ( ( finite3252695134891459830_nat_o @ A22 )
% 14.25/8.05 => ( ( A22 != bot_bo7824918357723582233_nat_o )
% 14.25/8.05 => ~ ! [X6: produc3658429121746597890et_nat > $o] :
% 14.25/8.05 ( ( member6576561426505652726_nat_o @ X6 @ A1 )
% 14.25/8.05 => ? [Xa3: produc3658429121746597890et_nat > $o] :
% 14.25/8.05 ( ( member6576561426505652726_nat_o @ Xa3 @ A22 )
% 14.25/8.05 & ( member8781333585448626064_nat_o @ ( produc7368190662567826135_nat_o @ X6 @ Xa3 ) @ R3 ) ) ) ) ) ) ) ).
% 14.25/8.05
% 14.25/8.05 thf(fact_2822_inf__period_I2_J,axiom,
% 14.25/8.05 ! [P: rat > $o,D3: rat,Q2: rat > $o] :
% 14.25/8.05 ( ! [X3: rat,K: rat] :
% 14.25/8.05 ( ( P @ X3 )
% 14.25/8.05 = ( P @ ( minus_minus_rat @ X3 @ ( times_times_rat @ K @ D3 ) ) ) )
% 14.25/8.05 => ( ! [X3: rat,K: rat] :
% 14.25/8.05 ( ( Q2 @ X3 )
% 14.25/8.05 = ( Q2 @ ( minus_minus_rat @ X3 @ ( times_times_rat @ K @ D3 ) ) ) )
% 14.25/8.05 => ! [X6: rat,K2: rat] :
% 14.25/8.05 ( ( ( P @ X6 )
% 14.25/8.05 | ( Q2 @ X6 ) )
% 14.25/8.05 = ( ( P @ ( minus_minus_rat @ X6 @ ( times_times_rat @ K2 @ D3 ) ) )
% 14.25/8.05 | ( Q2 @ ( minus_minus_rat @ X6 @ ( times_times_rat @ K2 @ D3 ) ) ) ) ) ) ) ).
% 14.25/8.05
% 14.25/8.05 thf(fact_2823_inf__period_I2_J,axiom,
% 14.25/8.05 ! [P: int > $o,D3: int,Q2: int > $o] :
% 14.25/8.05 ( ! [X3: int,K: int] :
% 14.25/8.05 ( ( P @ X3 )
% 14.25/8.05 = ( P @ ( minus_minus_int @ X3 @ ( times_times_int @ K @ D3 ) ) ) )
% 14.25/8.05 => ( ! [X3: int,K: int] :
% 14.25/8.05 ( ( Q2 @ X3 )
% 14.25/8.05 = ( Q2 @ ( minus_minus_int @ X3 @ ( times_times_int @ K @ D3 ) ) ) )
% 14.25/8.05 => ! [X6: int,K2: int] :
% 14.25/8.05 ( ( ( P @ X6 )
% 14.25/8.05 | ( Q2 @ X6 ) )
% 14.25/8.05 = ( ( P @ ( minus_minus_int @ X6 @ ( times_times_int @ K2 @ D3 ) ) )
% 14.25/8.05 | ( Q2 @ ( minus_minus_int @ X6 @ ( times_times_int @ K2 @ D3 ) ) ) ) ) ) ) ).
% 14.25/8.05
% 14.25/8.05 thf(fact_2824_inf__period_I1_J,axiom,
% 14.25/8.05 ! [P: rat > $o,D3: rat,Q2: rat > $o] :
% 14.25/8.05 ( ! [X3: rat,K: rat] :
% 14.25/8.05 ( ( P @ X3 )
% 14.25/8.05 = ( P @ ( minus_minus_rat @ X3 @ ( times_times_rat @ K @ D3 ) ) ) )
% 14.25/8.05 => ( ! [X3: rat,K: rat] :
% 14.25/8.05 ( ( Q2 @ X3 )
% 14.25/8.05 = ( Q2 @ ( minus_minus_rat @ X3 @ ( times_times_rat @ K @ D3 ) ) ) )
% 14.25/8.05 => ! [X6: rat,K2: rat] :
% 14.25/8.05 ( ( ( P @ X6 )
% 14.25/8.05 & ( Q2 @ X6 ) )
% 14.25/8.05 = ( ( P @ ( minus_minus_rat @ X6 @ ( times_times_rat @ K2 @ D3 ) ) )
% 14.25/8.05 & ( Q2 @ ( minus_minus_rat @ X6 @ ( times_times_rat @ K2 @ D3 ) ) ) ) ) ) ) ).
% 14.25/8.05
% 14.25/8.05 thf(fact_2825_inf__period_I1_J,axiom,
% 14.25/8.05 ! [P: int > $o,D3: int,Q2: int > $o] :
% 14.25/8.05 ( ! [X3: int,K: int] :
% 14.25/8.05 ( ( P @ X3 )
% 14.25/8.05 = ( P @ ( minus_minus_int @ X3 @ ( times_times_int @ K @ D3 ) ) ) )
% 14.25/8.05 => ( ! [X3: int,K: int] :
% 14.25/8.05 ( ( Q2 @ X3 )
% 14.25/8.05 = ( Q2 @ ( minus_minus_int @ X3 @ ( times_times_int @ K @ D3 ) ) ) )
% 14.25/8.05 => ! [X6: int,K2: int] :
% 14.25/8.05 ( ( ( P @ X6 )
% 14.25/8.05 & ( Q2 @ X6 ) )
% 14.25/8.05 = ( ( P @ ( minus_minus_int @ X6 @ ( times_times_int @ K2 @ D3 ) ) )
% 14.25/8.05 & ( Q2 @ ( minus_minus_int @ X6 @ ( times_times_int @ K2 @ D3 ) ) ) ) ) ) ) ).
% 14.25/8.05
% 14.25/8.05 thf(fact_2826_left__diff__distrib,axiom,
% 14.25/8.05 ! [A2: rat,B: rat,C: rat] :
% 14.25/8.05 ( ( times_times_rat @ ( minus_minus_rat @ A2 @ B ) @ C )
% 14.25/8.05 = ( minus_minus_rat @ ( times_times_rat @ A2 @ C ) @ ( times_times_rat @ B @ C ) ) ) ).
% 14.25/8.05
% 14.25/8.05 thf(fact_2827_left__diff__distrib,axiom,
% 14.25/8.05 ! [A2: int,B: int,C: int] :
% 14.25/8.05 ( ( times_times_int @ ( minus_minus_int @ A2 @ B ) @ C )
% 14.25/8.05 = ( minus_minus_int @ ( times_times_int @ A2 @ C ) @ ( times_times_int @ B @ C ) ) ) ).
% 14.25/8.05
% 14.25/8.05 thf(fact_2828_right__diff__distrib,axiom,
% 14.25/8.05 ! [A2: rat,B: rat,C: rat] :
% 14.25/8.05 ( ( times_times_rat @ A2 @ ( minus_minus_rat @ B @ C ) )
% 14.25/8.05 = ( minus_minus_rat @ ( times_times_rat @ A2 @ B ) @ ( times_times_rat @ A2 @ C ) ) ) ).
% 14.25/8.05
% 14.25/8.05 thf(fact_2829_right__diff__distrib,axiom,
% 14.25/8.05 ! [A2: int,B: int,C: int] :
% 14.25/8.05 ( ( times_times_int @ A2 @ ( minus_minus_int @ B @ C ) )
% 14.25/8.05 = ( minus_minus_int @ ( times_times_int @ A2 @ B ) @ ( times_times_int @ A2 @ C ) ) ) ).
% 14.25/8.05
% 14.25/8.05 thf(fact_2830_left__diff__distrib_H,axiom,
% 14.25/8.05 ! [B: rat,C: rat,A2: rat] :
% 14.25/8.05 ( ( times_times_rat @ ( minus_minus_rat @ B @ C ) @ A2 )
% 14.25/8.05 = ( minus_minus_rat @ ( times_times_rat @ B @ A2 ) @ ( times_times_rat @ C @ A2 ) ) ) ).
% 14.25/8.05
% 14.25/8.05 thf(fact_2831_left__diff__distrib_H,axiom,
% 14.25/8.05 ! [B: nat,C: nat,A2: nat] :
% 14.25/8.05 ( ( times_times_nat @ ( minus_minus_nat @ B @ C ) @ A2 )
% 14.25/8.05 = ( minus_minus_nat @ ( times_times_nat @ B @ A2 ) @ ( times_times_nat @ C @ A2 ) ) ) ).
% 14.25/8.05
% 14.25/8.05 thf(fact_2832_left__diff__distrib_H,axiom,
% 14.25/8.05 ! [B: int,C: int,A2: int] :
% 14.25/8.05 ( ( times_times_int @ ( minus_minus_int @ B @ C ) @ A2 )
% 14.25/8.05 = ( minus_minus_int @ ( times_times_int @ B @ A2 ) @ ( times_times_int @ C @ A2 ) ) ) ).
% 14.25/8.05
% 14.25/8.05 thf(fact_2833_right__diff__distrib_H,axiom,
% 14.25/8.05 ! [A2: rat,B: rat,C: rat] :
% 14.25/8.05 ( ( times_times_rat @ A2 @ ( minus_minus_rat @ B @ C ) )
% 14.25/8.05 = ( minus_minus_rat @ ( times_times_rat @ A2 @ B ) @ ( times_times_rat @ A2 @ C ) ) ) ).
% 14.25/8.05
% 14.25/8.05 thf(fact_2834_right__diff__distrib_H,axiom,
% 14.25/8.05 ! [A2: nat,B: nat,C: nat] :
% 14.25/8.05 ( ( times_times_nat @ A2 @ ( minus_minus_nat @ B @ C ) )
% 14.25/8.05 = ( minus_minus_nat @ ( times_times_nat @ A2 @ B ) @ ( times_times_nat @ A2 @ C ) ) ) ).
% 14.25/8.05
% 14.25/8.05 thf(fact_2835_right__diff__distrib_H,axiom,
% 14.25/8.05 ! [A2: int,B: int,C: int] :
% 14.25/8.05 ( ( times_times_int @ A2 @ ( minus_minus_int @ B @ C ) )
% 14.25/8.05 = ( minus_minus_int @ ( times_times_int @ A2 @ B ) @ ( times_times_int @ A2 @ C ) ) ) ).
% 14.25/8.05
% 14.25/8.05 thf(fact_2836_in__lex__prod,axiom,
% 14.25/8.05 ! [A2: nat,B: int,A5: nat,B3: int,R2: set_Pr1261947904930325089at_nat,S2: set_Pr958786334691620121nt_int] :
% 14.25/8.05 ( ( member4619661753630440592at_int @ ( produc886916002739626967at_int @ ( product_Pair_nat_int @ A2 @ B ) @ ( product_Pair_nat_int @ A5 @ B3 ) ) @ ( lex_prod_nat_int @ R2 @ S2 ) )
% 14.25/8.05 = ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A2 @ A5 ) @ R2 )
% 14.25/8.05 | ( ( A2 = A5 )
% 14.25/8.05 & ( member5262025264175285858nt_int @ ( product_Pair_int_int @ B @ B3 ) @ S2 ) ) ) ) ).
% 14.25/8.05
% 14.25/8.05 thf(fact_2837_in__lex__prod,axiom,
% 14.25/8.05 ! [A2: int,B: nat,A5: int,B3: nat,R2: set_Pr958786334691620121nt_int,S2: set_Pr1261947904930325089at_nat] :
% 14.25/8.05 ( ( member2930414080797523088nt_nat @ ( produc8921240378546987991nt_nat @ ( product_Pair_int_nat @ A2 @ B ) @ ( product_Pair_int_nat @ A5 @ B3 ) ) @ ( lex_prod_int_nat @ R2 @ S2 ) )
% 14.25/8.05 = ( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ A2 @ A5 ) @ R2 )
% 14.25/8.05 | ( ( A2 = A5 )
% 14.25/8.05 & ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ B @ B3 ) @ S2 ) ) ) ) ).
% 14.25/8.05
% 14.25/8.05 thf(fact_2838_in__lex__prod,axiom,
% 14.25/8.05 ! [A2: heap_e7401611519738050253t_unit,B: set_nat,A5: heap_e7401611519738050253t_unit,B3: set_nat,R2: set_Pr1776176089890104647t_unit,S2: set_Pr5488025237498180813et_nat] :
% 14.25/8.05 ( ( member6099555550032318734et_nat @ ( produc8199053930788261021et_nat @ ( produc7507926704131184380et_nat @ A2 @ B ) @ ( produc7507926704131184380et_nat @ A5 @ B3 ) ) @ ( lex_pr538634069021852900et_nat @ R2 @ S2 ) )
% 14.25/8.05 = ( ( member5046970461208427408t_unit @ ( produc8256076736985659095t_unit @ A2 @ A5 ) @ R2 )
% 14.25/8.05 | ( ( A2 = A5 )
% 14.25/8.05 & ( member8277197624267554838et_nat @ ( produc4532415448927165861et_nat @ B @ B3 ) @ S2 ) ) ) ) ).
% 14.25/8.05
% 14.25/8.05 thf(fact_2839_in__lex__prod,axiom,
% 14.25/8.05 ! [A2: nat,B: nat,A5: nat,B3: nat,R2: set_Pr1261947904930325089at_nat,S2: set_Pr1261947904930325089at_nat] :
% 14.25/8.05 ( ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ ( product_Pair_nat_nat @ A2 @ B ) @ ( product_Pair_nat_nat @ A5 @ B3 ) ) @ ( lex_prod_nat_nat @ R2 @ S2 ) )
% 14.25/8.05 = ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A2 @ A5 ) @ R2 )
% 14.25/8.05 | ( ( A2 = A5 )
% 14.25/8.05 & ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ B @ B3 ) @ S2 ) ) ) ) ).
% 14.25/8.05
% 14.25/8.05 thf(fact_2840_in__lex__prod,axiom,
% 14.25/8.05 ! [A2: int,B: int,A5: int,B3: int,R2: set_Pr958786334691620121nt_int,S2: set_Pr958786334691620121nt_int] :
% 14.25/8.05 ( ( member8566619992076573584nt_int @ ( produc3646306378393792727nt_int @ ( product_Pair_int_int @ A2 @ B ) @ ( product_Pair_int_int @ A5 @ B3 ) ) @ ( lex_prod_int_int @ R2 @ S2 ) )
% 14.25/8.05 = ( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ A2 @ A5 ) @ R2 )
% 14.25/8.05 | ( ( A2 = A5 )
% 14.25/8.05 & ( member5262025264175285858nt_int @ ( product_Pair_int_int @ B @ B3 ) @ S2 ) ) ) ) ).
% 14.25/8.05
% 14.25/8.05 thf(fact_2841_in__lex__prod,axiom,
% 14.25/8.05 ! [A2: produc3658429121746597890et_nat > $o,B: produc3658429121746597890et_nat,A5: produc3658429121746597890et_nat > $o,B3: produc3658429121746597890et_nat,R2: set_Pr2161125870931222855_nat_o,S2: set_Pr719794911490849221et_nat] :
% 14.25/8.05 ( ( member4763271486408492550et_nat @ ( produc8599840265553166229et_nat @ ( produc5001842942810119800et_nat @ A2 @ B ) @ ( produc5001842942810119800et_nat @ A5 @ B3 ) ) @ ( lex_pr5340572901959109728et_nat @ R2 @ S2 ) )
% 14.25/8.05 = ( ( member8781333585448626064_nat_o @ ( produc7368190662567826135_nat_o @ A2 @ A5 ) @ R2 )
% 14.25/8.05 | ( ( A2 = A5 )
% 14.25/8.05 & ( member6099555550032318734et_nat @ ( produc8199053930788261021et_nat @ B @ B3 ) @ S2 ) ) ) ) ).
% 14.25/8.05
% 14.25/8.05 thf(fact_2842_in__lex__prod,axiom,
% 14.25/8.05 ! [A2: produc3658429121746597890et_nat > $o,B: produc3925858234332021118et_nat,A5: produc3658429121746597890et_nat > $o,B3: produc3925858234332021118et_nat,R2: set_Pr2161125870931222855_nat_o,S2: set_Pr7928877670098842301et_nat] :
% 14.25/8.05 ( ( member6341495586645257982et_nat @ ( produc1940133919992309389et_nat @ ( produc2245416461498447860et_nat @ A2 @ B ) @ ( produc2245416461498447860et_nat @ A5 @ B3 ) ) @ ( lex_pr4722427456421979612et_nat @ R2 @ S2 ) )
% 14.25/8.05 = ( ( member8781333585448626064_nat_o @ ( produc7368190662567826135_nat_o @ A2 @ A5 ) @ R2 )
% 14.25/8.05 | ( ( A2 = A5 )
% 14.25/8.05 & ( member4763271486408492550et_nat @ ( produc8599840265553166229et_nat @ B @ B3 ) @ S2 ) ) ) ) ).
% 14.25/8.05
% 14.25/8.05 thf(fact_2843_Linear__order__wf__diff__Id,axiom,
% 14.25/8.05 ! [R2: set_Pr5094982260447487303t_unit] :
% 14.25/8.05 ( ( order_888282469287196226t_unit @ ( field_Product_unit @ R2 ) @ R2 )
% 14.25/8.05 => ( ( wf_Product_unit @ ( minus_6985556733573110190t_unit @ R2 @ id_Product_unit ) )
% 14.25/8.05 = ( ! [A3: set_Product_unit] :
% 14.25/8.05 ( ( ord_le3507040750410214029t_unit @ A3 @ ( field_Product_unit @ R2 ) )
% 14.25/8.05 => ( ( A3 != bot_bo3957492148770167129t_unit )
% 14.25/8.05 => ? [X2: product_unit] :
% 14.25/8.05 ( ( member_Product_unit @ X2 @ A3 )
% 14.25/8.05 & ! [Y3: product_unit] :
% 14.25/8.05 ( ( member_Product_unit @ Y3 @ A3 )
% 14.25/8.05 => ( member7821138191923116944t_unit @ ( produc5692694829384537815t_unit @ X2 @ Y3 ) @ R2 ) ) ) ) ) ) ) ) ).
% 14.25/8.05
% 14.25/8.05 thf(fact_2844_Linear__order__wf__diff__Id,axiom,
% 14.25/8.05 ! [R2: set_Product_prod_o_o] :
% 14.25/8.05 ( ( order_6238756238976269133r_on_o @ ( field_o @ R2 ) @ R2 )
% 14.25/8.05 => ( ( wf_o @ ( minus_7052513330764642158od_o_o @ R2 @ id_o2 ) )
% 14.25/8.05 = ( ! [A3: set_o] :
% 14.25/8.05 ( ( ord_less_eq_set_o @ A3 @ ( field_o @ R2 ) )
% 14.25/8.05 => ( ( A3 != bot_bot_set_o )
% 14.25/8.05 => ? [X2: $o] :
% 14.25/8.05 ( ( member_o @ X2 @ A3 )
% 14.25/8.05 & ! [Y3: $o] :
% 14.25/8.05 ( ( member_o @ Y3 @ A3 )
% 14.25/8.05 => ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ X2 @ Y3 ) @ R2 ) ) ) ) ) ) ) ) ).
% 14.25/8.05
% 14.25/8.05 thf(fact_2845_Linear__order__wf__diff__Id,axiom,
% 14.25/8.05 ! [R2: set_Pr958786334691620121nt_int] :
% 14.25/8.05 ( ( order_4471489696718655927on_int @ ( field_int @ R2 ) @ R2 )
% 14.25/8.05 => ( ( wf_int @ ( minus_1052850069191792384nt_int @ R2 @ id_int ) )
% 14.25/8.05 = ( ! [A3: set_int] :
% 14.25/8.05 ( ( ord_less_eq_set_int @ A3 @ ( field_int @ R2 ) )
% 14.25/8.05 => ( ( A3 != bot_bot_set_int )
% 14.25/8.05 => ? [X2: int] :
% 14.25/8.05 ( ( member_int @ X2 @ A3 )
% 14.25/8.05 & ! [Y3: int] :
% 14.25/8.05 ( ( member_int @ Y3 @ A3 )
% 14.25/8.05 => ( member5262025264175285858nt_int @ ( product_Pair_int_int @ X2 @ Y3 ) @ R2 ) ) ) ) ) ) ) ) ).
% 14.25/8.05
% 14.25/8.05 thf(fact_2846_Linear__order__wf__diff__Id,axiom,
% 14.25/8.05 ! [R2: set_Pr2161125870931222855_nat_o] :
% 14.25/8.05 ( ( order_1145710701291701250_nat_o @ ( field_81821745747897882_nat_o @ R2 ) @ R2 )
% 14.25/8.05 => ( ( wf_Pro8787509627421085110_nat_o @ ( minus_9189903262233110958_nat_o @ R2 @ id_Pro6775173067513470761_nat_o ) )
% 14.25/8.05 = ( ! [A3: set_Pr4532377907799695533_nat_o] :
% 14.25/8.05 ( ( ord_le2965882846123202637_nat_o @ A3 @ ( field_81821745747897882_nat_o @ R2 ) )
% 14.25/8.05 => ( ( A3 != bot_bo7824918357723582233_nat_o )
% 14.25/8.05 => ? [X2: produc3658429121746597890et_nat > $o] :
% 14.25/8.05 ( ( member6576561426505652726_nat_o @ X2 @ A3 )
% 14.25/8.05 & ! [Y3: produc3658429121746597890et_nat > $o] :
% 14.25/8.05 ( ( member6576561426505652726_nat_o @ Y3 @ A3 )
% 14.25/8.05 => ( member8781333585448626064_nat_o @ ( produc7368190662567826135_nat_o @ X2 @ Y3 ) @ R2 ) ) ) ) ) ) ) ) ).
% 14.25/8.05
% 14.25/8.05 thf(fact_2847_Linear__order__wf__diff__Id,axiom,
% 14.25/8.05 ! [R2: set_Pr1261947904930325089at_nat] :
% 14.25/8.05 ( ( order_4473980167227706203on_nat @ ( field_nat @ R2 ) @ R2 )
% 14.25/8.05 => ( ( wf_nat @ ( minus_1356011639430497352at_nat @ R2 @ id_nat ) )
% 14.25/8.05 = ( ! [A3: set_nat] :
% 14.25/8.05 ( ( ord_less_eq_set_nat @ A3 @ ( field_nat @ R2 ) )
% 14.25/8.05 => ( ( A3 != bot_bot_set_nat )
% 14.25/8.05 => ? [X2: nat] :
% 14.25/8.05 ( ( member_nat @ X2 @ A3 )
% 14.25/8.05 & ! [Y3: nat] :
% 14.25/8.05 ( ( member_nat @ Y3 @ A3 )
% 14.25/8.05 => ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X2 @ Y3 ) @ R2 ) ) ) ) ) ) ) ) ).
% 14.25/8.05
% 14.25/8.05 thf(fact_2848_remove__def,axiom,
% 14.25/8.05 ( remove_Product_unit
% 14.25/8.05 = ( ^ [X2: product_unit,A3: set_Product_unit] : ( minus_6452836326544984404t_unit @ A3 @ ( insert_Product_unit @ X2 @ bot_bo3957492148770167129t_unit ) ) ) ) ).
% 14.25/8.05
% 14.25/8.05 thf(fact_2849_remove__def,axiom,
% 14.25/8.05 ( remove_o
% 14.25/8.05 = ( ^ [X2: $o,A3: set_o] : ( minus_minus_set_o @ A3 @ ( insert_o @ X2 @ bot_bot_set_o ) ) ) ) ).
% 14.25/8.05
% 14.25/8.05 thf(fact_2850_remove__def,axiom,
% 14.25/8.05 ( remove_int
% 14.25/8.05 = ( ^ [X2: int,A3: set_int] : ( minus_minus_set_int @ A3 @ ( insert_int @ X2 @ bot_bot_set_int ) ) ) ) ).
% 14.25/8.05
% 14.25/8.05 thf(fact_2851_remove__def,axiom,
% 14.25/8.05 ( remove4651630035290841522_nat_o
% 14.25/8.05 = ( ^ [X2: produc3658429121746597890et_nat > $o,A3: set_Pr4532377907799695533_nat_o] : ( minus_1801376950450012436_nat_o @ A3 @ ( insert5175938949040314269_nat_o @ X2 @ bot_bo7824918357723582233_nat_o ) ) ) ) ).
% 14.25/8.05
% 14.25/8.05 thf(fact_2852_remove__def,axiom,
% 14.25/8.05 ( remove_nat
% 14.25/8.05 = ( ^ [X2: nat,A3: set_nat] : ( minus_minus_set_nat @ A3 @ ( insert_nat @ X2 @ bot_bot_set_nat ) ) ) ) ).
% 14.25/8.05
% 14.25/8.05 thf(fact_2853_wo__rel_Ocases__Total3,axiom,
% 14.25/8.05 ! [R2: set_Pr5094982260447487303t_unit,A2: product_unit,B: product_unit,Phi: product_unit > product_unit > $o] :
% 14.25/8.05 ( ( bNF_We2091578341578594809t_unit @ R2 )
% 14.25/8.05 => ( ( ord_le3507040750410214029t_unit @ ( insert_Product_unit @ A2 @ ( insert_Product_unit @ B @ bot_bo3957492148770167129t_unit ) ) @ ( field_Product_unit @ R2 ) )
% 14.25/8.05 => ( ( ( ( member7821138191923116944t_unit @ ( produc5692694829384537815t_unit @ A2 @ B ) @ ( minus_6985556733573110190t_unit @ R2 @ id_Product_unit ) )
% 14.25/8.05 | ( member7821138191923116944t_unit @ ( produc5692694829384537815t_unit @ B @ A2 ) @ ( minus_6985556733573110190t_unit @ R2 @ id_Product_unit ) ) )
% 14.25/8.05 => ( Phi @ A2 @ B ) )
% 14.25/8.05 => ( ( ( A2 = B )
% 14.25/8.05 => ( Phi @ A2 @ B ) )
% 14.25/8.05 => ( Phi @ A2 @ B ) ) ) ) ) ).
% 14.25/8.05
% 14.25/8.05 thf(fact_2854_wo__rel_Ocases__Total3,axiom,
% 14.25/8.05 ! [R2: set_Product_prod_o_o,A2: $o,B: $o,Phi: $o > $o > $o] :
% 14.25/8.05 ( ( bNF_We6437023980930612612_rel_o @ R2 )
% 14.25/8.05 => ( ( ord_less_eq_set_o @ ( insert_o @ A2 @ ( insert_o @ B @ bot_bot_set_o ) ) @ ( field_o @ R2 ) )
% 14.25/8.05 => ( ( ( ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ A2 @ B ) @ ( minus_7052513330764642158od_o_o @ R2 @ id_o2 ) )
% 14.25/8.05 | ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ B @ A2 ) @ ( minus_7052513330764642158od_o_o @ R2 @ id_o2 ) ) )
% 14.25/8.05 => ( Phi @ A2 @ B ) )
% 14.25/8.05 => ( ( ( A2 = B )
% 14.25/8.05 => ( Phi @ A2 @ B ) )
% 14.25/8.05 => ( Phi @ A2 @ B ) ) ) ) ) ).
% 14.25/8.05
% 14.25/8.05 thf(fact_2855_wo__rel_Ocases__Total3,axiom,
% 14.25/8.05 ! [R2: set_Pr958786334691620121nt_int,A2: int,B: int,Phi: int > int > $o] :
% 14.25/8.05 ( ( bNF_We3815749466139970368el_int @ R2 )
% 14.25/8.05 => ( ( ord_less_eq_set_int @ ( insert_int @ A2 @ ( insert_int @ B @ bot_bot_set_int ) ) @ ( field_int @ R2 ) )
% 14.25/8.05 => ( ( ( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ A2 @ B ) @ ( minus_1052850069191792384nt_int @ R2 @ id_int ) )
% 14.25/8.05 | ( member5262025264175285858nt_int @ ( product_Pair_int_int @ B @ A2 ) @ ( minus_1052850069191792384nt_int @ R2 @ id_int ) ) )
% 14.25/8.05 => ( Phi @ A2 @ B ) )
% 14.25/8.05 => ( ( ( A2 = B )
% 14.25/8.05 => ( Phi @ A2 @ B ) )
% 14.25/8.05 => ( Phi @ A2 @ B ) ) ) ) ) ).
% 14.25/8.05
% 14.25/8.05 thf(fact_2856_wo__rel_Ocases__Total3,axiom,
% 14.25/8.05 ! [R2: set_Pr2161125870931222855_nat_o,A2: produc3658429121746597890et_nat > $o,B: produc3658429121746597890et_nat > $o,Phi: ( produc3658429121746597890et_nat > $o ) > ( produc3658429121746597890et_nat > $o ) > $o] :
% 14.25/8.05 ( ( bNF_We740377018556551609_nat_o @ R2 )
% 14.25/8.05 => ( ( ord_le2965882846123202637_nat_o @ ( insert5175938949040314269_nat_o @ A2 @ ( insert5175938949040314269_nat_o @ B @ bot_bo7824918357723582233_nat_o ) ) @ ( field_81821745747897882_nat_o @ R2 ) )
% 14.25/8.05 => ( ( ( ( member8781333585448626064_nat_o @ ( produc7368190662567826135_nat_o @ A2 @ B ) @ ( minus_9189903262233110958_nat_o @ R2 @ id_Pro6775173067513470761_nat_o ) )
% 14.25/8.05 | ( member8781333585448626064_nat_o @ ( produc7368190662567826135_nat_o @ B @ A2 ) @ ( minus_9189903262233110958_nat_o @ R2 @ id_Pro6775173067513470761_nat_o ) ) )
% 14.25/8.05 => ( Phi @ A2 @ B ) )
% 14.25/8.05 => ( ( ( A2 = B )
% 14.25/8.05 => ( Phi @ A2 @ B ) )
% 14.25/8.05 => ( Phi @ A2 @ B ) ) ) ) ) ).
% 14.25/8.05
% 14.25/8.05 thf(fact_2857_wo__rel_Ocases__Total3,axiom,
% 14.25/8.05 ! [R2: set_Pr1261947904930325089at_nat,A2: nat,B: nat,Phi: nat > nat > $o] :
% 14.25/8.05 ( ( bNF_We3818239936649020644el_nat @ R2 )
% 14.25/8.05 => ( ( ord_less_eq_set_nat @ ( insert_nat @ A2 @ ( insert_nat @ B @ bot_bot_set_nat ) ) @ ( field_nat @ R2 ) )
% 14.25/8.05 => ( ( ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A2 @ B ) @ ( minus_1356011639430497352at_nat @ R2 @ id_nat ) )
% 14.25/8.05 | ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ B @ A2 ) @ ( minus_1356011639430497352at_nat @ R2 @ id_nat ) ) )
% 14.25/8.05 => ( Phi @ A2 @ B ) )
% 14.25/8.05 => ( ( ( A2 = B )
% 14.25/8.05 => ( Phi @ A2 @ B ) )
% 14.25/8.05 => ( Phi @ A2 @ B ) ) ) ) ) ).
% 14.25/8.05
% 14.25/8.05 thf(fact_2858_wf__max,axiom,
% 14.25/8.05 ! [R3: set_Pr2161125870931222855_nat_o] :
% 14.25/8.05 ( ( wf_Pro8787509627421085110_nat_o @ ( conver2764731894203590139_nat_o @ R3 ) )
% 14.25/8.05 => ( ( R3 != bot_bo3657187498359500723_nat_o )
% 14.25/8.05 => ~ ! [M2: produc3658429121746597890et_nat > $o] :
% 14.25/8.05 ~ ( member6576561426505652726_nat_o @ M2 @ ( minus_1801376950450012436_nat_o @ ( range_5289318983786108341_nat_o @ R3 ) @ ( domain4881241903163524940_nat_o @ R3 ) ) ) ) ) ).
% 14.25/8.05
% 14.25/8.05 thf(fact_2859_wf__max,axiom,
% 14.25/8.05 ! [R3: set_Pr958786334691620121nt_int] :
% 14.25/8.05 ( ( wf_int @ ( converse_int_int @ R3 ) )
% 14.25/8.05 => ( ( R3 != bot_bo1796632182523588997nt_int )
% 14.25/8.05 => ~ ! [M2: int] :
% 14.25/8.05 ~ ( member_int @ M2 @ ( minus_minus_set_int @ ( range_int_int @ R3 ) @ ( domain_int_int @ R3 ) ) ) ) ) ).
% 14.25/8.05
% 14.25/8.05 thf(fact_2860_wf__max,axiom,
% 14.25/8.05 ! [R3: set_Product_prod_o_o] :
% 14.25/8.05 ( ( wf_o @ ( converse_o_o @ R3 ) )
% 14.25/8.05 => ( ( R3 != bot_bo7073875226086086771od_o_o )
% 14.25/8.05 => ~ ! [M2: $o] :
% 14.25/8.05 ~ ( member_o @ M2 @ ( minus_minus_set_o @ ( range_o_o @ R3 ) @ ( domain_o_o @ R3 ) ) ) ) ) ).
% 14.25/8.05
% 14.25/8.05 thf(fact_2861_wf__max,axiom,
% 14.25/8.05 ! [R3: set_Pr5094982260447487303t_unit] :
% 14.25/8.05 ( ( wf_Product_unit @ ( conver2495244158383847931t_unit @ R3 ) )
% 14.25/8.05 => ( ( R3 != bot_bo1654022762305692595t_unit )
% 14.25/8.05 => ~ ! [M2: product_unit] :
% 14.25/8.05 ~ ( member_Product_unit @ M2 @ ( minus_6452836326544984404t_unit @ ( range_6617724647562260917t_unit @ R3 ) @ ( domain4292024440186670924t_unit @ R3 ) ) ) ) ) ).
% 14.25/8.05
% 14.25/8.05 thf(fact_2862_wf__max,axiom,
% 14.25/8.05 ! [R3: set_Pr1261947904930325089at_nat] :
% 14.25/8.05 ( ( wf_nat @ ( converse_nat_nat @ R3 ) )
% 14.25/8.05 => ( ( R3 != bot_bo2099793752762293965at_nat )
% 14.25/8.05 => ~ ! [M2: nat] :
% 14.25/8.05 ~ ( member_nat @ M2 @ ( minus_minus_set_nat @ ( range_nat_nat @ R3 ) @ ( domain_nat_nat @ R3 ) ) ) ) ) ).
% 14.25/8.05
% 14.25/8.05 thf(fact_2863_wf__Un,axiom,
% 14.25/8.05 ! [R2: set_Pr5094982260447487303t_unit,S2: set_Pr5094982260447487303t_unit] :
% 14.25/8.05 ( ( wf_Product_unit @ R2 )
% 14.25/8.05 => ( ( wf_Product_unit @ S2 )
% 14.25/8.05 => ( ( ( inf_in4660618365625256667t_unit @ ( domain4292024440186670924t_unit @ R2 ) @ ( range_6617724647562260917t_unit @ S2 ) )
% 14.25/8.05 = bot_bo3957492148770167129t_unit )
% 14.25/8.05 => ( wf_Product_unit @ ( sup_su6481247232241170203t_unit @ R2 @ S2 ) ) ) ) ) ).
% 14.25/8.05
% 14.25/8.05 thf(fact_2864_wf__Un,axiom,
% 14.25/8.05 ! [R2: set_Product_prod_o_o,S2: set_Product_prod_o_o] :
% 14.25/8.05 ( ( wf_o @ R2 )
% 14.25/8.05 => ( ( wf_o @ S2 )
% 14.25/8.05 => ( ( ( inf_inf_set_o @ ( domain_o_o @ R2 ) @ ( range_o_o @ S2 ) )
% 14.25/8.05 = bot_bot_set_o )
% 14.25/8.05 => ( wf_o @ ( sup_su5769328420594410459od_o_o @ R2 @ S2 ) ) ) ) ) ).
% 14.25/8.05
% 14.25/8.05 thf(fact_2865_wf__Un,axiom,
% 14.25/8.05 ! [R2: set_Pr1261947904930325089at_nat,S2: set_Pr1261947904930325089at_nat] :
% 14.25/8.05 ( ( wf_nat @ R2 )
% 14.25/8.05 => ( ( wf_nat @ S2 )
% 14.25/8.05 => ( ( ( inf_inf_set_nat @ ( domain_nat_nat @ R2 ) @ ( range_nat_nat @ S2 ) )
% 14.25/8.05 = bot_bot_set_nat )
% 14.25/8.05 => ( wf_nat @ ( sup_su6327502436637775413at_nat @ R2 @ S2 ) ) ) ) ) ).
% 14.25/8.05
% 14.25/8.05 thf(fact_2866_wf__Un,axiom,
% 14.25/8.05 ! [R2: set_Pr958786334691620121nt_int,S2: set_Pr958786334691620121nt_int] :
% 14.25/8.05 ( ( wf_int @ R2 )
% 14.25/8.05 => ( ( wf_int @ S2 )
% 14.25/8.05 => ( ( ( inf_inf_set_int @ ( domain_int_int @ R2 ) @ ( range_int_int @ S2 ) )
% 14.25/8.05 = bot_bot_set_int )
% 14.25/8.05 => ( wf_int @ ( sup_su6024340866399070445nt_int @ R2 @ S2 ) ) ) ) ) ).
% 14.25/8.05
% 14.25/8.05 thf(fact_2867_wf__Un,axiom,
% 14.25/8.05 ! [R2: set_Pr2161125870931222855_nat_o,S2: set_Pr2161125870931222855_nat_o] :
% 14.25/8.05 ( ( wf_Pro8787509627421085110_nat_o @ R2 )
% 14.25/8.05 => ( ( wf_Pro8787509627421085110_nat_o @ S2 )
% 14.25/8.05 => ( ( ( inf_in1906310914598751387_nat_o @ ( domain4881241903163524940_nat_o @ R2 ) @ ( range_5289318983786108341_nat_o @ S2 ) )
% 14.25/8.05 = bot_bo7824918357723582233_nat_o )
% 14.25/8.05 => ( wf_Pro8787509627421085110_nat_o @ ( sup_su8328026650783933211_nat_o @ R2 @ S2 ) ) ) ) ) ).
% 14.25/8.05
% 14.25/8.05 thf(fact_2868_Linear__order__Well__order__iff,axiom,
% 14.25/8.05 ! [R2: set_Pr5094982260447487303t_unit] :
% 14.25/8.05 ( ( order_888282469287196226t_unit @ ( field_Product_unit @ R2 ) @ R2 )
% 14.25/8.05 => ( ( order_3164716736300062379t_unit @ ( field_Product_unit @ R2 ) @ R2 )
% 14.25/8.05 = ( ! [A3: set_Product_unit] :
% 14.25/8.05 ( ( ord_le3507040750410214029t_unit @ A3 @ ( field_Product_unit @ R2 ) )
% 14.25/8.05 => ( ( A3 != bot_bo3957492148770167129t_unit )
% 14.25/8.05 => ? [X2: product_unit] :
% 14.25/8.05 ( ( member_Product_unit @ X2 @ A3 )
% 14.25/8.05 & ! [Y3: product_unit] :
% 14.25/8.05 ( ( member_Product_unit @ Y3 @ A3 )
% 14.25/8.05 => ( member7821138191923116944t_unit @ ( produc5692694829384537815t_unit @ X2 @ Y3 ) @ R2 ) ) ) ) ) ) ) ) ).
% 14.25/8.05
% 14.25/8.05 thf(fact_2869_Linear__order__Well__order__iff,axiom,
% 14.25/8.05 ! [R2: set_Product_prod_o_o] :
% 14.25/8.05 ( ( order_6238756238976269133r_on_o @ ( field_o @ R2 ) @ R2 )
% 14.25/8.05 => ( ( order_2615775362358406454r_on_o @ ( field_o @ R2 ) @ R2 )
% 14.25/8.05 = ( ! [A3: set_o] :
% 14.25/8.05 ( ( ord_less_eq_set_o @ A3 @ ( field_o @ R2 ) )
% 14.25/8.05 => ( ( A3 != bot_bot_set_o )
% 14.25/8.05 => ? [X2: $o] :
% 14.25/8.05 ( ( member_o @ X2 @ A3 )
% 14.25/8.05 & ! [Y3: $o] :
% 14.25/8.05 ( ( member_o @ Y3 @ A3 )
% 14.25/8.05 => ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ X2 @ Y3 ) @ R2 ) ) ) ) ) ) ) ) ).
% 14.25/8.05
% 14.25/8.05 thf(fact_2870_Linear__order__Well__order__iff,axiom,
% 14.25/8.05 ! [R2: set_Pr958786334691620121nt_int] :
% 14.25/8.05 ( ( order_4471489696718655927on_int @ ( field_int @ R2 ) @ R2 )
% 14.25/8.05 => ( ( order_2886507596567047182on_int @ ( field_int @ R2 ) @ R2 )
% 14.25/8.05 = ( ! [A3: set_int] :
% 14.25/8.05 ( ( ord_less_eq_set_int @ A3 @ ( field_int @ R2 ) )
% 14.25/8.05 => ( ( A3 != bot_bot_set_int )
% 14.25/8.05 => ? [X2: int] :
% 14.25/8.05 ( ( member_int @ X2 @ A3 )
% 14.25/8.05 & ! [Y3: int] :
% 14.25/8.05 ( ( member_int @ Y3 @ A3 )
% 14.25/8.05 => ( member5262025264175285858nt_int @ ( product_Pair_int_int @ X2 @ Y3 ) @ R2 ) ) ) ) ) ) ) ) ).
% 14.25/8.05
% 14.25/8.05 thf(fact_2871_Linear__order__Well__order__iff,axiom,
% 14.25/8.05 ! [R2: set_Pr2161125870931222855_nat_o] :
% 14.25/8.05 ( ( order_1145710701291701250_nat_o @ ( field_81821745747897882_nat_o @ R2 ) @ R2 )
% 14.25/8.05 => ( ( order_2118566763991480939_nat_o @ ( field_81821745747897882_nat_o @ R2 ) @ R2 )
% 14.25/8.05 = ( ! [A3: set_Pr4532377907799695533_nat_o] :
% 14.25/8.05 ( ( ord_le2965882846123202637_nat_o @ A3 @ ( field_81821745747897882_nat_o @ R2 ) )
% 14.25/8.05 => ( ( A3 != bot_bo7824918357723582233_nat_o )
% 14.25/8.05 => ? [X2: produc3658429121746597890et_nat > $o] :
% 14.25/8.05 ( ( member6576561426505652726_nat_o @ X2 @ A3 )
% 14.25/8.05 & ! [Y3: produc3658429121746597890et_nat > $o] :
% 14.25/8.05 ( ( member6576561426505652726_nat_o @ Y3 @ A3 )
% 14.25/8.05 => ( member8781333585448626064_nat_o @ ( produc7368190662567826135_nat_o @ X2 @ Y3 ) @ R2 ) ) ) ) ) ) ) ) ).
% 14.25/8.05
% 14.25/8.05 thf(fact_2872_Linear__order__Well__order__iff,axiom,
% 14.25/8.05 ! [R2: set_Pr1261947904930325089at_nat] :
% 14.25/8.05 ( ( order_4473980167227706203on_nat @ ( field_nat @ R2 ) @ R2 )
% 14.25/8.05 => ( ( order_2888998067076097458on_nat @ ( field_nat @ R2 ) @ R2 )
% 14.25/8.05 = ( ! [A3: set_nat] :
% 14.25/8.05 ( ( ord_less_eq_set_nat @ A3 @ ( field_nat @ R2 ) )
% 14.25/8.05 => ( ( A3 != bot_bot_set_nat )
% 14.25/8.05 => ? [X2: nat] :
% 14.25/8.05 ( ( member_nat @ X2 @ A3 )
% 14.25/8.05 & ! [Y3: nat] :
% 14.25/8.05 ( ( member_nat @ Y3 @ A3 )
% 14.25/8.05 => ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X2 @ Y3 ) @ R2 ) ) ) ) ) ) ) ) ).
% 14.25/8.05
% 14.25/8.05 thf(fact_2873_underS__incr,axiom,
% 14.25/8.05 ! [R2: set_Pr958786334691620121nt_int,A2: int,B: int] :
% 14.25/8.05 ( ( trans_int @ R2 )
% 14.25/8.05 => ( ( antisym_int @ R2 )
% 14.25/8.05 => ( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ A2 @ B ) @ R2 )
% 14.25/8.05 => ( ord_less_eq_set_int @ ( order_underS_int @ R2 @ A2 ) @ ( order_underS_int @ R2 @ B ) ) ) ) ) ).
% 14.25/8.05
% 14.25/8.05 thf(fact_2874_underS__incr,axiom,
% 14.25/8.05 ! [R2: set_Pr1261947904930325089at_nat,A2: nat,B: nat] :
% 14.25/8.05 ( ( trans_nat @ R2 )
% 14.25/8.05 => ( ( antisym_nat @ R2 )
% 14.25/8.05 => ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A2 @ B ) @ R2 )
% 14.25/8.05 => ( ord_less_eq_set_nat @ ( order_underS_nat @ R2 @ A2 ) @ ( order_underS_nat @ R2 @ B ) ) ) ) ) ).
% 14.25/8.05
% 14.25/8.05 thf(fact_2875_member__remove,axiom,
% 14.25/8.05 ! [X: produc3658429121746597890et_nat > $o,Y: produc3658429121746597890et_nat > $o,A: set_Pr4532377907799695533_nat_o] :
% 14.25/8.05 ( ( member6576561426505652726_nat_o @ X @ ( remove4651630035290841522_nat_o @ Y @ A ) )
% 14.25/8.05 = ( ( member6576561426505652726_nat_o @ X @ A )
% 14.25/8.05 & ( X != Y ) ) ) ).
% 14.25/8.05
% 14.25/8.05 thf(fact_2876_member__remove,axiom,
% 14.25/8.05 ! [X: nat,Y: nat,A: set_nat] :
% 14.25/8.05 ( ( member_nat @ X @ ( remove_nat @ Y @ A ) )
% 14.25/8.05 = ( ( member_nat @ X @ A )
% 14.25/8.05 & ( X != Y ) ) ) ).
% 14.25/8.05
% 14.25/8.05 thf(fact_2877_member__remove,axiom,
% 14.25/8.05 ! [X: int,Y: int,A: set_int] :
% 14.25/8.05 ( ( member_int @ X @ ( remove_int @ Y @ A ) )
% 14.25/8.05 = ( ( member_int @ X @ A )
% 14.25/8.05 & ( X != Y ) ) ) ).
% 14.25/8.05
% 14.25/8.05 thf(fact_2878_member__remove,axiom,
% 14.25/8.05 ! [X: $o,Y: $o,A: set_o] :
% 14.25/8.05 ( ( member_o @ X @ ( remove_o @ Y @ A ) )
% 14.25/8.05 = ( ( member_o @ X @ A )
% 14.25/8.05 & ( X != Y ) ) ) ).
% 14.25/8.05
% 14.25/8.05 thf(fact_2879_member__remove,axiom,
% 14.25/8.05 ! [X: product_unit,Y: product_unit,A: set_Product_unit] :
% 14.25/8.05 ( ( member_Product_unit @ X @ ( remove_Product_unit @ Y @ A ) )
% 14.25/8.05 = ( ( member_Product_unit @ X @ A )
% 14.25/8.05 & ( X != Y ) ) ) ).
% 14.25/8.05
% 14.25/8.05 thf(fact_2880_wf__insert,axiom,
% 14.25/8.05 ! [Y: nat,X: nat,R2: set_Pr1261947904930325089at_nat] :
% 14.25/8.05 ( ( wf_nat @ ( insert8211810215607154385at_nat @ ( product_Pair_nat_nat @ Y @ X ) @ R2 ) )
% 14.25/8.05 = ( ( wf_nat @ R2 )
% 14.25/8.05 & ~ ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ Y ) @ ( transi2905341329935302413cl_nat @ R2 ) ) ) ) ).
% 14.25/8.05
% 14.25/8.05 thf(fact_2881_wf__insert,axiom,
% 14.25/8.05 ! [Y: int,X: int,R2: set_Pr958786334691620121nt_int] :
% 14.25/8.05 ( ( wf_int @ ( insert5033312907999012233nt_int @ ( product_Pair_int_int @ Y @ X ) @ R2 ) )
% 14.25/8.05 = ( ( wf_int @ R2 )
% 14.25/8.05 & ~ ( member5262025264175285858nt_int @ ( product_Pair_int_int @ X @ Y ) @ ( transi2902850859426252137cl_int @ R2 ) ) ) ) ).
% 14.25/8.05
% 14.25/8.05 thf(fact_2882_wo__rel_Owell__order__induct,axiom,
% 14.25/8.05 ! [R2: set_Pr1261947904930325089at_nat,P: nat > $o,A2: nat] :
% 14.25/8.05 ( ( bNF_We3818239936649020644el_nat @ R2 )
% 14.25/8.05 => ( ! [X3: nat] :
% 14.25/8.05 ( ! [Y5: nat] :
% 14.25/8.05 ( ( ( Y5 != X3 )
% 14.25/8.05 & ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ Y5 @ X3 ) @ R2 ) )
% 14.25/8.05 => ( P @ Y5 ) )
% 14.25/8.05 => ( P @ X3 ) )
% 14.25/8.05 => ( P @ A2 ) ) ) ).
% 14.25/8.05
% 14.25/8.05 thf(fact_2883_wo__rel_Owell__order__induct,axiom,
% 14.25/8.05 ! [R2: set_Pr958786334691620121nt_int,P: int > $o,A2: int] :
% 14.25/8.05 ( ( bNF_We3815749466139970368el_int @ R2 )
% 14.25/8.05 => ( ! [X3: int] :
% 14.25/8.05 ( ! [Y5: int] :
% 14.25/8.05 ( ( ( Y5 != X3 )
% 14.25/8.05 & ( member5262025264175285858nt_int @ ( product_Pair_int_int @ Y5 @ X3 ) @ R2 ) )
% 14.25/8.05 => ( P @ Y5 ) )
% 14.25/8.05 => ( P @ X3 ) )
% 14.25/8.05 => ( P @ A2 ) ) ) ).
% 14.25/8.05
% 14.25/8.05 thf(fact_2884_wf__induct__rule,axiom,
% 14.25/8.05 ! [R2: set_Pr1261947904930325089at_nat,P: nat > $o,A2: nat] :
% 14.25/8.05 ( ( wf_nat @ R2 )
% 14.25/8.05 => ( ! [X3: nat] :
% 14.25/8.05 ( ! [Y5: nat] :
% 14.25/8.05 ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ Y5 @ X3 ) @ R2 )
% 14.25/8.05 => ( P @ Y5 ) )
% 14.25/8.05 => ( P @ X3 ) )
% 14.25/8.05 => ( P @ A2 ) ) ) ).
% 14.25/8.05
% 14.25/8.05 thf(fact_2885_wf__induct__rule,axiom,
% 14.25/8.05 ! [R2: set_Pr958786334691620121nt_int,P: int > $o,A2: int] :
% 14.25/8.05 ( ( wf_int @ R2 )
% 14.25/8.05 => ( ! [X3: int] :
% 14.25/8.05 ( ! [Y5: int] :
% 14.25/8.05 ( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ Y5 @ X3 ) @ R2 )
% 14.25/8.05 => ( P @ Y5 ) )
% 14.25/8.05 => ( P @ X3 ) )
% 14.25/8.05 => ( P @ A2 ) ) ) ).
% 14.25/8.05
% 14.25/8.05 thf(fact_2886_wf__eq__minimal,axiom,
% 14.25/8.05 ( wf_Pro8787509627421085110_nat_o
% 14.25/8.05 = ( ^ [R4: set_Pr2161125870931222855_nat_o] :
% 14.25/8.05 ! [Q: set_Pr4532377907799695533_nat_o] :
% 14.25/8.05 ( ? [X2: produc3658429121746597890et_nat > $o] : ( member6576561426505652726_nat_o @ X2 @ Q )
% 14.25/8.05 => ? [X2: produc3658429121746597890et_nat > $o] :
% 14.25/8.05 ( ( member6576561426505652726_nat_o @ X2 @ Q )
% 14.25/8.05 & ! [Y3: produc3658429121746597890et_nat > $o] :
% 14.25/8.05 ( ( member8781333585448626064_nat_o @ ( produc7368190662567826135_nat_o @ Y3 @ X2 ) @ R4 )
% 14.25/8.05 => ~ ( member6576561426505652726_nat_o @ Y3 @ Q ) ) ) ) ) ) ).
% 14.25/8.05
% 14.25/8.05 thf(fact_2887_wf__eq__minimal,axiom,
% 14.25/8.05 ( wf_o
% 14.25/8.05 = ( ^ [R4: set_Product_prod_o_o] :
% 14.25/8.05 ! [Q: set_o] :
% 14.25/8.05 ( ? [X2: $o] : ( member_o @ X2 @ Q )
% 14.25/8.05 => ? [X2: $o] :
% 14.25/8.05 ( ( member_o @ X2 @ Q )
% 14.25/8.05 & ! [Y3: $o] :
% 14.25/8.05 ( ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ Y3 @ X2 ) @ R4 )
% 14.25/8.05 => ~ ( member_o @ Y3 @ Q ) ) ) ) ) ) ).
% 14.25/8.05
% 14.25/8.05 thf(fact_2888_wf__eq__minimal,axiom,
% 14.25/8.05 ( wf_Product_unit
% 14.25/8.05 = ( ^ [R4: set_Pr5094982260447487303t_unit] :
% 14.25/8.05 ! [Q: set_Product_unit] :
% 14.25/8.05 ( ? [X2: product_unit] : ( member_Product_unit @ X2 @ Q )
% 14.25/8.05 => ? [X2: product_unit] :
% 14.25/8.05 ( ( member_Product_unit @ X2 @ Q )
% 14.25/8.05 & ! [Y3: product_unit] :
% 14.25/8.05 ( ( member7821138191923116944t_unit @ ( produc5692694829384537815t_unit @ Y3 @ X2 ) @ R4 )
% 14.25/8.05 => ~ ( member_Product_unit @ Y3 @ Q ) ) ) ) ) ) ).
% 14.25/8.05
% 14.25/8.05 thf(fact_2889_wf__eq__minimal,axiom,
% 14.25/8.05 ( wf_nat
% 14.25/8.05 = ( ^ [R4: set_Pr1261947904930325089at_nat] :
% 14.25/8.05 ! [Q: set_nat] :
% 14.25/8.05 ( ? [X2: nat] : ( member_nat @ X2 @ Q )
% 14.25/8.05 => ? [X2: nat] :
% 14.25/8.05 ( ( member_nat @ X2 @ Q )
% 14.25/8.05 & ! [Y3: nat] :
% 14.25/8.05 ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ Y3 @ X2 ) @ R4 )
% 14.25/8.05 => ~ ( member_nat @ Y3 @ Q ) ) ) ) ) ) ).
% 14.25/8.05
% 14.25/8.05 thf(fact_2890_wf__eq__minimal,axiom,
% 14.25/8.05 ( wf_int
% 14.25/8.05 = ( ^ [R4: set_Pr958786334691620121nt_int] :
% 14.25/8.05 ! [Q: set_int] :
% 14.25/8.05 ( ? [X2: int] : ( member_int @ X2 @ Q )
% 14.25/8.05 => ? [X2: int] :
% 14.25/8.05 ( ( member_int @ X2 @ Q )
% 14.25/8.05 & ! [Y3: int] :
% 14.25/8.05 ( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ Y3 @ X2 ) @ R4 )
% 14.25/8.05 => ~ ( member_int @ Y3 @ Q ) ) ) ) ) ) ).
% 14.25/8.05
% 14.25/8.05 thf(fact_2891_wf__not__refl,axiom,
% 14.25/8.05 ! [R2: set_Pr1261947904930325089at_nat,A2: nat] :
% 14.25/8.05 ( ( wf_nat @ R2 )
% 14.25/8.05 => ~ ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A2 @ A2 ) @ R2 ) ) ).
% 14.25/8.05
% 14.25/8.05 thf(fact_2892_wf__not__refl,axiom,
% 14.25/8.05 ! [R2: set_Pr958786334691620121nt_int,A2: int] :
% 14.25/8.05 ( ( wf_int @ R2 )
% 14.25/8.05 => ~ ( member5262025264175285858nt_int @ ( product_Pair_int_int @ A2 @ A2 ) @ R2 ) ) ).
% 14.25/8.05
% 14.25/8.05 thf(fact_2893_wf__not__sym,axiom,
% 14.25/8.05 ! [R2: set_Pr1261947904930325089at_nat,A2: nat,X: nat] :
% 14.25/8.05 ( ( wf_nat @ R2 )
% 14.25/8.05 => ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A2 @ X ) @ R2 )
% 14.25/8.05 => ~ ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ A2 ) @ R2 ) ) ) ).
% 14.25/8.05
% 14.25/8.05 thf(fact_2894_wf__not__sym,axiom,
% 14.25/8.05 ! [R2: set_Pr958786334691620121nt_int,A2: int,X: int] :
% 14.25/8.05 ( ( wf_int @ R2 )
% 14.25/8.05 => ( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ A2 @ X ) @ R2 )
% 14.25/8.05 => ~ ( member5262025264175285858nt_int @ ( product_Pair_int_int @ X @ A2 ) @ R2 ) ) ) ).
% 14.25/8.05
% 14.25/8.05 thf(fact_2895_wf__irrefl,axiom,
% 14.25/8.05 ! [R2: set_Pr1261947904930325089at_nat,A2: nat] :
% 14.25/8.05 ( ( wf_nat @ R2 )
% 14.25/8.05 => ~ ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A2 @ A2 ) @ R2 ) ) ).
% 14.25/8.05
% 14.25/8.05 thf(fact_2896_wf__irrefl,axiom,
% 14.25/8.05 ! [R2: set_Pr958786334691620121nt_int,A2: int] :
% 14.25/8.05 ( ( wf_int @ R2 )
% 14.25/8.05 => ~ ( member5262025264175285858nt_int @ ( product_Pair_int_int @ A2 @ A2 ) @ R2 ) ) ).
% 14.25/8.05
% 14.25/8.05 thf(fact_2897_wf__induct,axiom,
% 14.25/8.05 ! [R2: set_Pr1261947904930325089at_nat,P: nat > $o,A2: nat] :
% 14.25/8.05 ( ( wf_nat @ R2 )
% 14.25/8.05 => ( ! [X3: nat] :
% 14.25/8.05 ( ! [Y5: nat] :
% 14.25/8.05 ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ Y5 @ X3 ) @ R2 )
% 14.25/8.05 => ( P @ Y5 ) )
% 14.25/8.05 => ( P @ X3 ) )
% 14.25/8.05 => ( P @ A2 ) ) ) ).
% 14.25/8.05
% 14.25/8.05 thf(fact_2898_wf__induct,axiom,
% 14.25/8.05 ! [R2: set_Pr958786334691620121nt_int,P: int > $o,A2: int] :
% 14.25/8.05 ( ( wf_int @ R2 )
% 14.25/8.05 => ( ! [X3: int] :
% 14.25/8.05 ( ! [Y5: int] :
% 14.25/8.05 ( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ Y5 @ X3 ) @ R2 )
% 14.25/8.05 => ( P @ Y5 ) )
% 14.25/8.05 => ( P @ X3 ) )
% 14.25/8.05 => ( P @ A2 ) ) ) ).
% 14.25/8.05
% 14.25/8.05 thf(fact_2899_wf__asym,axiom,
% 14.25/8.05 ! [R2: set_Pr1261947904930325089at_nat,A2: nat,X: nat] :
% 14.25/8.05 ( ( wf_nat @ R2 )
% 14.25/8.05 => ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A2 @ X ) @ R2 )
% 14.25/8.05 => ~ ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ A2 ) @ R2 ) ) ) ).
% 14.25/8.05
% 14.25/8.05 thf(fact_2900_wf__asym,axiom,
% 14.25/8.05 ! [R2: set_Pr958786334691620121nt_int,A2: int,X: int] :
% 14.25/8.05 ( ( wf_int @ R2 )
% 14.25/8.05 => ( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ A2 @ X ) @ R2 )
% 14.25/8.05 => ~ ( member5262025264175285858nt_int @ ( product_Pair_int_int @ X @ A2 ) @ R2 ) ) ) ).
% 14.25/8.05
% 14.25/8.05 thf(fact_2901_wfUNIVI,axiom,
% 14.25/8.05 ! [R2: set_Pr1261947904930325089at_nat] :
% 14.25/8.05 ( ! [P4: nat > $o,X3: nat] :
% 14.25/8.05 ( ! [Xa2: nat] :
% 14.25/8.05 ( ! [Y2: nat] :
% 14.25/8.05 ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ Y2 @ Xa2 ) @ R2 )
% 14.25/8.05 => ( P4 @ Y2 ) )
% 14.25/8.05 => ( P4 @ Xa2 ) )
% 14.25/8.05 => ( P4 @ X3 ) )
% 14.25/8.05 => ( wf_nat @ R2 ) ) ).
% 14.25/8.05
% 14.25/8.05 thf(fact_2902_wfUNIVI,axiom,
% 14.25/8.05 ! [R2: set_Pr958786334691620121nt_int] :
% 14.25/8.05 ( ! [P4: int > $o,X3: int] :
% 14.25/8.05 ( ! [Xa2: int] :
% 14.25/8.05 ( ! [Y2: int] :
% 14.25/8.05 ( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ Y2 @ Xa2 ) @ R2 )
% 14.25/8.05 => ( P4 @ Y2 ) )
% 14.25/8.05 => ( P4 @ Xa2 ) )
% 14.25/8.05 => ( P4 @ X3 ) )
% 14.25/8.05 => ( wf_int @ R2 ) ) ).
% 14.25/8.05
% 14.25/8.05 thf(fact_2903_wfI__min,axiom,
% 14.25/8.05 ! [R3: set_Pr2161125870931222855_nat_o] :
% 14.25/8.05 ( ! [X3: produc3658429121746597890et_nat > $o,Q3: set_Pr4532377907799695533_nat_o] :
% 14.25/8.05 ( ( member6576561426505652726_nat_o @ X3 @ Q3 )
% 14.25/8.05 => ? [Xa2: produc3658429121746597890et_nat > $o] :
% 14.25/8.05 ( ( member6576561426505652726_nat_o @ Xa2 @ Q3 )
% 14.25/8.05 & ! [Y2: produc3658429121746597890et_nat > $o] :
% 14.25/8.05 ( ( member8781333585448626064_nat_o @ ( produc7368190662567826135_nat_o @ Y2 @ Xa2 ) @ R3 )
% 14.25/8.05 => ~ ( member6576561426505652726_nat_o @ Y2 @ Q3 ) ) ) )
% 14.25/8.05 => ( wf_Pro8787509627421085110_nat_o @ R3 ) ) ).
% 14.25/8.05
% 14.25/8.05 thf(fact_2904_wfI__min,axiom,
% 14.25/8.05 ! [R3: set_Product_prod_o_o] :
% 14.25/8.05 ( ! [X3: $o,Q3: set_o] :
% 14.25/8.05 ( ( member_o @ X3 @ Q3 )
% 14.25/8.05 => ? [Xa2: $o] :
% 14.25/8.05 ( ( member_o @ Xa2 @ Q3 )
% 14.25/8.05 & ! [Y2: $o] :
% 14.25/8.05 ( ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ Y2 @ Xa2 ) @ R3 )
% 14.25/8.05 => ~ ( member_o @ Y2 @ Q3 ) ) ) )
% 14.25/8.05 => ( wf_o @ R3 ) ) ).
% 14.25/8.05
% 14.25/8.05 thf(fact_2905_wfI__min,axiom,
% 14.25/8.05 ! [R3: set_Pr5094982260447487303t_unit] :
% 14.25/8.05 ( ! [X3: product_unit,Q3: set_Product_unit] :
% 14.25/8.05 ( ( member_Product_unit @ X3 @ Q3 )
% 14.25/8.05 => ? [Xa2: product_unit] :
% 14.25/8.05 ( ( member_Product_unit @ Xa2 @ Q3 )
% 14.25/8.05 & ! [Y2: product_unit] :
% 14.25/8.05 ( ( member7821138191923116944t_unit @ ( produc5692694829384537815t_unit @ Y2 @ Xa2 ) @ R3 )
% 14.25/8.05 => ~ ( member_Product_unit @ Y2 @ Q3 ) ) ) )
% 14.25/8.05 => ( wf_Product_unit @ R3 ) ) ).
% 14.25/8.05
% 14.25/8.05 thf(fact_2906_wfI__min,axiom,
% 14.25/8.05 ! [R3: set_Pr1261947904930325089at_nat] :
% 14.25/8.05 ( ! [X3: nat,Q3: set_nat] :
% 14.25/8.05 ( ( member_nat @ X3 @ Q3 )
% 14.25/8.05 => ? [Xa2: nat] :
% 14.25/8.05 ( ( member_nat @ Xa2 @ Q3 )
% 14.25/8.05 & ! [Y2: nat] :
% 14.25/8.05 ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ Y2 @ Xa2 ) @ R3 )
% 14.25/8.05 => ~ ( member_nat @ Y2 @ Q3 ) ) ) )
% 14.25/8.05 => ( wf_nat @ R3 ) ) ).
% 14.25/8.05
% 14.25/8.05 thf(fact_2907_wfI__min,axiom,
% 14.25/8.05 ! [R3: set_Pr958786334691620121nt_int] :
% 14.25/8.05 ( ! [X3: int,Q3: set_int] :
% 14.25/8.05 ( ( member_int @ X3 @ Q3 )
% 14.25/8.05 => ? [Xa2: int] :
% 14.25/8.05 ( ( member_int @ Xa2 @ Q3 )
% 14.25/8.05 & ! [Y2: int] :
% 14.25/8.05 ( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ Y2 @ Xa2 ) @ R3 )
% 14.25/8.05 => ~ ( member_int @ Y2 @ Q3 ) ) ) )
% 14.25/8.05 => ( wf_int @ R3 ) ) ).
% 14.25/8.05
% 14.25/8.05 thf(fact_2908_wfE__min,axiom,
% 14.25/8.05 ! [R3: set_Pr2161125870931222855_nat_o,X: produc3658429121746597890et_nat > $o,Q2: set_Pr4532377907799695533_nat_o] :
% 14.25/8.05 ( ( wf_Pro8787509627421085110_nat_o @ R3 )
% 14.25/8.05 => ( ( member6576561426505652726_nat_o @ X @ Q2 )
% 14.25/8.05 => ~ ! [Z3: produc3658429121746597890et_nat > $o] :
% 14.25/8.05 ( ( member6576561426505652726_nat_o @ Z3 @ Q2 )
% 14.25/8.05 => ~ ! [Y5: produc3658429121746597890et_nat > $o] :
% 14.25/8.05 ( ( member8781333585448626064_nat_o @ ( produc7368190662567826135_nat_o @ Y5 @ Z3 ) @ R3 )
% 14.25/8.05 => ~ ( member6576561426505652726_nat_o @ Y5 @ Q2 ) ) ) ) ) ).
% 14.25/8.05
% 14.25/8.05 thf(fact_2909_wfE__min,axiom,
% 14.25/8.05 ! [R3: set_Product_prod_o_o,X: $o,Q2: set_o] :
% 14.25/8.05 ( ( wf_o @ R3 )
% 14.25/8.05 => ( ( member_o @ X @ Q2 )
% 14.25/8.05 => ~ ! [Z3: $o] :
% 14.25/8.05 ( ( member_o @ Z3 @ Q2 )
% 14.25/8.05 => ~ ! [Y5: $o] :
% 14.25/8.05 ( ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ Y5 @ Z3 ) @ R3 )
% 14.25/8.05 => ~ ( member_o @ Y5 @ Q2 ) ) ) ) ) ).
% 14.25/8.05
% 14.25/8.05 thf(fact_2910_wfE__min,axiom,
% 14.25/8.05 ! [R3: set_Pr5094982260447487303t_unit,X: product_unit,Q2: set_Product_unit] :
% 14.25/8.05 ( ( wf_Product_unit @ R3 )
% 14.25/8.05 => ( ( member_Product_unit @ X @ Q2 )
% 14.25/8.05 => ~ ! [Z3: product_unit] :
% 14.25/8.05 ( ( member_Product_unit @ Z3 @ Q2 )
% 14.25/8.05 => ~ ! [Y5: product_unit] :
% 14.25/8.05 ( ( member7821138191923116944t_unit @ ( produc5692694829384537815t_unit @ Y5 @ Z3 ) @ R3 )
% 14.25/8.05 => ~ ( member_Product_unit @ Y5 @ Q2 ) ) ) ) ) ).
% 14.25/8.05
% 14.25/8.05 thf(fact_2911_wfE__min,axiom,
% 14.25/8.05 ! [R3: set_Pr1261947904930325089at_nat,X: nat,Q2: set_nat] :
% 14.25/8.05 ( ( wf_nat @ R3 )
% 14.25/8.05 => ( ( member_nat @ X @ Q2 )
% 14.25/8.05 => ~ ! [Z3: nat] :
% 14.25/8.05 ( ( member_nat @ Z3 @ Q2 )
% 14.25/8.05 => ~ ! [Y5: nat] :
% 14.25/8.05 ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ Y5 @ Z3 ) @ R3 )
% 14.25/8.05 => ~ ( member_nat @ Y5 @ Q2 ) ) ) ) ) ).
% 14.25/8.05
% 14.25/8.05 thf(fact_2912_wfE__min,axiom,
% 14.25/8.05 ! [R3: set_Pr958786334691620121nt_int,X: int,Q2: set_int] :
% 14.25/8.05 ( ( wf_int @ R3 )
% 14.25/8.05 => ( ( member_int @ X @ Q2 )
% 14.25/8.05 => ~ ! [Z3: int] :
% 14.25/8.05 ( ( member_int @ Z3 @ Q2 )
% 14.25/8.05 => ~ ! [Y5: int] :
% 14.25/8.05 ( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ Y5 @ Z3 ) @ R3 )
% 14.25/8.05 => ~ ( member_int @ Y5 @ Q2 ) ) ) ) ) ).
% 14.25/8.05
% 14.25/8.05 thf(fact_2913_wf__def,axiom,
% 14.25/8.05 ( wf_nat
% 14.25/8.05 = ( ^ [R4: set_Pr1261947904930325089at_nat] :
% 14.25/8.05 ! [P2: nat > $o] :
% 14.25/8.05 ( ! [X2: nat] :
% 14.25/8.05 ( ! [Y3: nat] :
% 14.25/8.05 ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ Y3 @ X2 ) @ R4 )
% 14.25/8.05 => ( P2 @ Y3 ) )
% 14.25/8.05 => ( P2 @ X2 ) )
% 14.25/8.05 => ! [X7: nat] : ( P2 @ X7 ) ) ) ) ).
% 14.25/8.05
% 14.25/8.05 thf(fact_2914_wf__def,axiom,
% 14.25/8.05 ( wf_int
% 14.25/8.05 = ( ^ [R4: set_Pr958786334691620121nt_int] :
% 14.25/8.05 ! [P2: int > $o] :
% 14.25/8.05 ( ! [X2: int] :
% 14.25/8.05 ( ! [Y3: int] :
% 14.25/8.05 ( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ Y3 @ X2 ) @ R4 )
% 14.25/8.05 => ( P2 @ Y3 ) )
% 14.25/8.05 => ( P2 @ X2 ) )
% 14.25/8.05 => ! [X7: int] : ( P2 @ X7 ) ) ) ) ).
% 14.25/8.05
% 14.25/8.05 thf(fact_2915_well__order__on__domain,axiom,
% 14.25/8.05 ! [A: set_Pr4532377907799695533_nat_o,R2: set_Pr2161125870931222855_nat_o,A2: produc3658429121746597890et_nat > $o,B: produc3658429121746597890et_nat > $o] :
% 14.25/8.05 ( ( order_2118566763991480939_nat_o @ A @ R2 )
% 14.25/8.05 => ( ( member8781333585448626064_nat_o @ ( produc7368190662567826135_nat_o @ A2 @ B ) @ R2 )
% 14.25/8.05 => ( ( member6576561426505652726_nat_o @ A2 @ A )
% 14.25/8.05 & ( member6576561426505652726_nat_o @ B @ A ) ) ) ) ).
% 14.25/8.05
% 14.25/8.05 thf(fact_2916_well__order__on__domain,axiom,
% 14.25/8.05 ! [A: set_o,R2: set_Product_prod_o_o,A2: $o,B: $o] :
% 14.25/8.05 ( ( order_2615775362358406454r_on_o @ A @ R2 )
% 14.25/8.05 => ( ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ A2 @ B ) @ R2 )
% 14.25/8.05 => ( ( member_o @ A2 @ A )
% 14.25/8.05 & ( member_o @ B @ A ) ) ) ) ).
% 14.25/8.05
% 14.25/8.05 thf(fact_2917_well__order__on__domain,axiom,
% 14.25/8.05 ! [A: set_Product_unit,R2: set_Pr5094982260447487303t_unit,A2: product_unit,B: product_unit] :
% 14.25/8.05 ( ( order_3164716736300062379t_unit @ A @ R2 )
% 14.25/8.05 => ( ( member7821138191923116944t_unit @ ( produc5692694829384537815t_unit @ A2 @ B ) @ R2 )
% 14.25/8.05 => ( ( member_Product_unit @ A2 @ A )
% 14.25/8.05 & ( member_Product_unit @ B @ A ) ) ) ) ).
% 14.25/8.05
% 14.25/8.05 thf(fact_2918_well__order__on__domain,axiom,
% 14.25/8.05 ! [A: set_nat,R2: set_Pr1261947904930325089at_nat,A2: nat,B: nat] :
% 14.25/8.05 ( ( order_2888998067076097458on_nat @ A @ R2 )
% 14.25/8.05 => ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A2 @ B ) @ R2 )
% 14.25/8.05 => ( ( member_nat @ A2 @ A )
% 14.25/8.05 & ( member_nat @ B @ A ) ) ) ) ).
% 14.25/8.05
% 14.25/8.05 thf(fact_2919_well__order__on__domain,axiom,
% 14.25/8.05 ! [A: set_int,R2: set_Pr958786334691620121nt_int,A2: int,B: int] :
% 14.25/8.05 ( ( order_2886507596567047182on_int @ A @ R2 )
% 14.25/8.05 => ( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ A2 @ B ) @ R2 )
% 14.25/8.05 => ( ( member_int @ A2 @ A )
% 14.25/8.05 & ( member_int @ B @ A ) ) ) ) ).
% 14.25/8.05
% 14.25/8.05 thf(fact_2920_underS__E,axiom,
% 14.25/8.05 ! [I2: produc3658429121746597890et_nat > $o,R3: set_Pr2161125870931222855_nat_o,J: produc3658429121746597890et_nat > $o] :
% 14.25/8.05 ( ( member6576561426505652726_nat_o @ I2 @ ( order_7262550146012401043_nat_o @ R3 @ J ) )
% 14.25/8.05 => ( ( I2 != J )
% 14.25/8.05 & ( member8781333585448626064_nat_o @ ( produc7368190662567826135_nat_o @ I2 @ J ) @ R3 ) ) ) ).
% 14.25/8.05
% 14.25/8.05 thf(fact_2921_underS__E,axiom,
% 14.25/8.05 ! [I2: $o,R3: set_Product_prod_o_o,J: $o] :
% 14.25/8.05 ( ( member_o @ I2 @ ( order_underS_o @ R3 @ J ) )
% 14.25/8.05 => ( ( I2 != J )
% 14.25/8.05 & ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ I2 @ J ) @ R3 ) ) ) ).
% 14.25/8.05
% 14.25/8.05 thf(fact_2922_underS__E,axiom,
% 14.25/8.05 ! [I2: product_unit,R3: set_Pr5094982260447487303t_unit,J: product_unit] :
% 14.25/8.05 ( ( member_Product_unit @ I2 @ ( order_5809128134054296019t_unit @ R3 @ J ) )
% 14.25/8.05 => ( ( I2 != J )
% 14.25/8.05 & ( member7821138191923116944t_unit @ ( produc5692694829384537815t_unit @ I2 @ J ) @ R3 ) ) ) ).
% 14.25/8.05
% 14.25/8.05 thf(fact_2923_underS__E,axiom,
% 14.25/8.05 ! [I2: nat,R3: set_Pr1261947904930325089at_nat,J: nat] :
% 14.25/8.05 ( ( member_nat @ I2 @ ( order_underS_nat @ R3 @ J ) )
% 14.25/8.05 => ( ( I2 != J )
% 14.25/8.05 & ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ I2 @ J ) @ R3 ) ) ) ).
% 14.25/8.05
% 14.25/8.05 thf(fact_2924_underS__E,axiom,
% 14.25/8.05 ! [I2: int,R3: set_Pr958786334691620121nt_int,J: int] :
% 14.25/8.05 ( ( member_int @ I2 @ ( order_underS_int @ R3 @ J ) )
% 14.25/8.05 => ( ( I2 != J )
% 14.25/8.05 & ( member5262025264175285858nt_int @ ( product_Pair_int_int @ I2 @ J ) @ R3 ) ) ) ).
% 14.25/8.05
% 14.25/8.05 thf(fact_2925_underS__I,axiom,
% 14.25/8.05 ! [I2: produc3658429121746597890et_nat > $o,J: produc3658429121746597890et_nat > $o,R3: set_Pr2161125870931222855_nat_o] :
% 14.25/8.05 ( ( I2 != J )
% 14.25/8.05 => ( ( member8781333585448626064_nat_o @ ( produc7368190662567826135_nat_o @ I2 @ J ) @ R3 )
% 14.25/8.05 => ( member6576561426505652726_nat_o @ I2 @ ( order_7262550146012401043_nat_o @ R3 @ J ) ) ) ) ).
% 14.25/8.05
% 14.25/8.05 thf(fact_2926_underS__I,axiom,
% 14.25/8.05 ! [I2: $o,J: $o,R3: set_Product_prod_o_o] :
% 14.25/8.05 ( ( I2 != J )
% 14.25/8.05 => ( ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ I2 @ J ) @ R3 )
% 14.25/8.05 => ( member_o @ I2 @ ( order_underS_o @ R3 @ J ) ) ) ) ).
% 14.25/8.05
% 14.25/8.05 thf(fact_2927_underS__I,axiom,
% 14.25/8.05 ! [I2: product_unit,J: product_unit,R3: set_Pr5094982260447487303t_unit] :
% 14.25/8.05 ( ( I2 != J )
% 14.25/8.05 => ( ( member7821138191923116944t_unit @ ( produc5692694829384537815t_unit @ I2 @ J ) @ R3 )
% 14.25/8.05 => ( member_Product_unit @ I2 @ ( order_5809128134054296019t_unit @ R3 @ J ) ) ) ) ).
% 14.25/8.05
% 14.25/8.05 thf(fact_2928_underS__I,axiom,
% 14.25/8.05 ! [I2: nat,J: nat,R3: set_Pr1261947904930325089at_nat] :
% 14.25/8.05 ( ( I2 != J )
% 14.25/8.05 => ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ I2 @ J ) @ R3 )
% 14.25/8.05 => ( member_nat @ I2 @ ( order_underS_nat @ R3 @ J ) ) ) ) ).
% 14.25/8.05
% 14.25/8.05 thf(fact_2929_underS__I,axiom,
% 14.25/8.05 ! [I2: int,J: int,R3: set_Pr958786334691620121nt_int] :
% 14.25/8.05 ( ( I2 != J )
% 14.25/8.05 => ( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ I2 @ J ) @ R3 )
% 14.25/8.05 => ( member_int @ I2 @ ( order_underS_int @ R3 @ J ) ) ) ) ).
% 14.25/8.05
% 14.25/8.05 thf(fact_2930_BNF__Least__Fixpoint_OunderS__Field,axiom,
% 14.25/8.05 ! [I2: produc3658429121746597890et_nat > $o,R3: set_Pr2161125870931222855_nat_o,J: produc3658429121746597890et_nat > $o] :
% 14.25/8.05 ( ( member6576561426505652726_nat_o @ I2 @ ( order_7262550146012401043_nat_o @ R3 @ J ) )
% 14.25/8.05 => ( member6576561426505652726_nat_o @ I2 @ ( field_81821745747897882_nat_o @ R3 ) ) ) ).
% 14.25/8.05
% 14.25/8.05 thf(fact_2931_BNF__Least__Fixpoint_OunderS__Field,axiom,
% 14.25/8.05 ! [I2: nat,R3: set_Pr1261947904930325089at_nat,J: nat] :
% 14.25/8.05 ( ( member_nat @ I2 @ ( order_underS_nat @ R3 @ J ) )
% 14.25/8.05 => ( member_nat @ I2 @ ( field_nat @ R3 ) ) ) ).
% 14.25/8.05
% 14.25/8.05 thf(fact_2932_BNF__Least__Fixpoint_OunderS__Field,axiom,
% 14.25/8.05 ! [I2: int,R3: set_Pr958786334691620121nt_int,J: int] :
% 14.25/8.05 ( ( member_int @ I2 @ ( order_underS_int @ R3 @ J ) )
% 14.25/8.05 => ( member_int @ I2 @ ( field_int @ R3 ) ) ) ).
% 14.25/8.05
% 14.25/8.05 thf(fact_2933_BNF__Least__Fixpoint_OunderS__Field,axiom,
% 14.25/8.05 ! [I2: $o,R3: set_Product_prod_o_o,J: $o] :
% 14.25/8.05 ( ( member_o @ I2 @ ( order_underS_o @ R3 @ J ) )
% 14.25/8.05 => ( member_o @ I2 @ ( field_o @ R3 ) ) ) ).
% 14.25/8.05
% 14.25/8.05 thf(fact_2934_BNF__Least__Fixpoint_OunderS__Field,axiom,
% 14.25/8.05 ! [I2: product_unit,R3: set_Pr5094982260447487303t_unit,J: product_unit] :
% 14.25/8.05 ( ( member_Product_unit @ I2 @ ( order_5809128134054296019t_unit @ R3 @ J ) )
% 14.25/8.05 => ( member_Product_unit @ I2 @ ( field_Product_unit @ R3 ) ) ) ).
% 14.25/8.05
% 14.25/8.05 thf(fact_2935_wo__rel_OTOTALS,axiom,
% 14.25/8.05 ! [R2: set_Pr1261947904930325089at_nat] :
% 14.25/8.05 ( ( bNF_We3818239936649020644el_nat @ R2 )
% 14.25/8.05 => ! [X6: nat] :
% 14.25/8.05 ( ( member_nat @ X6 @ ( field_nat @ R2 ) )
% 14.25/8.05 => ! [Xa2: nat] :
% 14.25/8.05 ( ( member_nat @ Xa2 @ ( field_nat @ R2 ) )
% 14.25/8.05 => ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X6 @ Xa2 ) @ R2 )
% 14.25/8.05 | ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ Xa2 @ X6 ) @ R2 ) ) ) ) ) ).
% 14.25/8.05
% 14.25/8.05 thf(fact_2936_wo__rel_OTOTALS,axiom,
% 14.25/8.05 ! [R2: set_Pr958786334691620121nt_int] :
% 14.25/8.05 ( ( bNF_We3815749466139970368el_int @ R2 )
% 14.25/8.05 => ! [X6: int] :
% 14.25/8.05 ( ( member_int @ X6 @ ( field_int @ R2 ) )
% 14.25/8.05 => ! [Xa2: int] :
% 14.25/8.05 ( ( member_int @ Xa2 @ ( field_int @ R2 ) )
% 14.25/8.05 => ( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ X6 @ Xa2 ) @ R2 )
% 14.25/8.06 | ( member5262025264175285858nt_int @ ( product_Pair_int_int @ Xa2 @ X6 ) @ R2 ) ) ) ) ) ).
% 14.25/8.06
% 14.25/8.06 thf(fact_2937_wfE__min_H,axiom,
% 14.25/8.06 ! [R3: set_Pr5094982260447487303t_unit,Q2: set_Product_unit] :
% 14.25/8.06 ( ( wf_Product_unit @ R3 )
% 14.25/8.06 => ( ( Q2 != bot_bo3957492148770167129t_unit )
% 14.25/8.06 => ~ ! [Z3: product_unit] :
% 14.25/8.06 ( ( member_Product_unit @ Z3 @ Q2 )
% 14.25/8.06 => ~ ! [Y5: product_unit] :
% 14.25/8.06 ( ( member7821138191923116944t_unit @ ( produc5692694829384537815t_unit @ Y5 @ Z3 ) @ R3 )
% 14.25/8.06 => ~ ( member_Product_unit @ Y5 @ Q2 ) ) ) ) ) ).
% 14.25/8.06
% 14.25/8.06 thf(fact_2938_wfE__min_H,axiom,
% 14.25/8.06 ! [R3: set_Product_prod_o_o,Q2: set_o] :
% 14.25/8.06 ( ( wf_o @ R3 )
% 14.25/8.06 => ( ( Q2 != bot_bot_set_o )
% 14.25/8.06 => ~ ! [Z3: $o] :
% 14.25/8.06 ( ( member_o @ Z3 @ Q2 )
% 14.25/8.06 => ~ ! [Y5: $o] :
% 14.25/8.06 ( ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ Y5 @ Z3 ) @ R3 )
% 14.25/8.06 => ~ ( member_o @ Y5 @ Q2 ) ) ) ) ) ).
% 14.25/8.06
% 14.25/8.06 thf(fact_2939_wfE__min_H,axiom,
% 14.25/8.06 ! [R3: set_Pr1261947904930325089at_nat,Q2: set_nat] :
% 14.25/8.06 ( ( wf_nat @ R3 )
% 14.25/8.06 => ( ( Q2 != bot_bot_set_nat )
% 14.25/8.06 => ~ ! [Z3: nat] :
% 14.25/8.06 ( ( member_nat @ Z3 @ Q2 )
% 14.25/8.06 => ~ ! [Y5: nat] :
% 14.25/8.06 ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ Y5 @ Z3 ) @ R3 )
% 14.25/8.06 => ~ ( member_nat @ Y5 @ Q2 ) ) ) ) ) ).
% 14.25/8.06
% 14.25/8.06 thf(fact_2940_wfE__min_H,axiom,
% 14.25/8.06 ! [R3: set_Pr958786334691620121nt_int,Q2: set_int] :
% 14.25/8.06 ( ( wf_int @ R3 )
% 14.25/8.06 => ( ( Q2 != bot_bot_set_int )
% 14.25/8.06 => ~ ! [Z3: int] :
% 14.25/8.06 ( ( member_int @ Z3 @ Q2 )
% 14.25/8.06 => ~ ! [Y5: int] :
% 14.25/8.06 ( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ Y5 @ Z3 ) @ R3 )
% 14.25/8.06 => ~ ( member_int @ Y5 @ Q2 ) ) ) ) ) ).
% 14.25/8.06
% 14.25/8.06 thf(fact_2941_wfE__min_H,axiom,
% 14.25/8.06 ! [R3: set_Pr2161125870931222855_nat_o,Q2: set_Pr4532377907799695533_nat_o] :
% 14.25/8.06 ( ( wf_Pro8787509627421085110_nat_o @ R3 )
% 14.25/8.06 => ( ( Q2 != bot_bo7824918357723582233_nat_o )
% 14.25/8.06 => ~ ! [Z3: produc3658429121746597890et_nat > $o] :
% 14.25/8.06 ( ( member6576561426505652726_nat_o @ Z3 @ Q2 )
% 14.25/8.06 => ~ ! [Y5: produc3658429121746597890et_nat > $o] :
% 14.25/8.06 ( ( member8781333585448626064_nat_o @ ( produc7368190662567826135_nat_o @ Y5 @ Z3 ) @ R3 )
% 14.25/8.06 => ~ ( member6576561426505652726_nat_o @ Y5 @ Q2 ) ) ) ) ) ).
% 14.25/8.06
% 14.25/8.06 thf(fact_2942_well__order__on__empty,axiom,
% 14.25/8.06 order_3164716736300062379t_unit @ bot_bo3957492148770167129t_unit @ bot_bo1654022762305692595t_unit ).
% 14.25/8.06
% 14.25/8.06 thf(fact_2943_well__order__on__empty,axiom,
% 14.25/8.06 order_2615775362358406454r_on_o @ bot_bot_set_o @ bot_bo7073875226086086771od_o_o ).
% 14.25/8.06
% 14.25/8.06 thf(fact_2944_well__order__on__empty,axiom,
% 14.25/8.06 order_2888998067076097458on_nat @ bot_bot_set_nat @ bot_bo2099793752762293965at_nat ).
% 14.25/8.06
% 14.25/8.06 thf(fact_2945_well__order__on__empty,axiom,
% 14.25/8.06 order_2886507596567047182on_int @ bot_bot_set_int @ bot_bo1796632182523588997nt_int ).
% 14.25/8.06
% 14.25/8.06 thf(fact_2946_well__order__on__empty,axiom,
% 14.25/8.06 order_2118566763991480939_nat_o @ bot_bo7824918357723582233_nat_o @ bot_bo3657187498359500723_nat_o ).
% 14.25/8.06
% 14.25/8.06 thf(fact_2947_well__order__induct__imp,axiom,
% 14.25/8.06 ! [R2: set_Pr2161125870931222855_nat_o,P: ( produc3658429121746597890et_nat > $o ) > $o,A2: produc3658429121746597890et_nat > $o] :
% 14.25/8.06 ( ( bNF_We740377018556551609_nat_o @ R2 )
% 14.25/8.06 => ( ! [X3: produc3658429121746597890et_nat > $o] :
% 14.25/8.06 ( ! [Y5: produc3658429121746597890et_nat > $o] :
% 14.25/8.06 ( ( ( Y5 != X3 )
% 14.25/8.06 & ( member8781333585448626064_nat_o @ ( produc7368190662567826135_nat_o @ Y5 @ X3 ) @ R2 ) )
% 14.25/8.06 => ( ( member6576561426505652726_nat_o @ Y5 @ ( field_81821745747897882_nat_o @ R2 ) )
% 14.25/8.06 => ( P @ Y5 ) ) )
% 14.25/8.06 => ( ( member6576561426505652726_nat_o @ X3 @ ( field_81821745747897882_nat_o @ R2 ) )
% 14.25/8.06 => ( P @ X3 ) ) )
% 14.25/8.06 => ( ( member6576561426505652726_nat_o @ A2 @ ( field_81821745747897882_nat_o @ R2 ) )
% 14.25/8.06 => ( P @ A2 ) ) ) ) ).
% 14.25/8.06
% 14.25/8.06 thf(fact_2948_well__order__induct__imp,axiom,
% 14.25/8.06 ! [R2: set_Product_prod_o_o,P: $o > $o,A2: $o] :
% 14.25/8.06 ( ( bNF_We6437023980930612612_rel_o @ R2 )
% 14.25/8.06 => ( ! [X3: $o] :
% 14.25/8.06 ( ! [Y5: $o] :
% 14.25/8.06 ( ( ( Y5 != X3 )
% 14.25/8.06 & ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ Y5 @ X3 ) @ R2 ) )
% 14.25/8.06 => ( ( member_o @ Y5 @ ( field_o @ R2 ) )
% 14.25/8.06 => ( P @ Y5 ) ) )
% 14.25/8.06 => ( ( member_o @ X3 @ ( field_o @ R2 ) )
% 14.25/8.06 => ( P @ X3 ) ) )
% 14.25/8.06 => ( ( member_o @ A2 @ ( field_o @ R2 ) )
% 14.25/8.06 => ( P @ A2 ) ) ) ) ).
% 14.25/8.06
% 14.25/8.06 thf(fact_2949_well__order__induct__imp,axiom,
% 14.25/8.06 ! [R2: set_Pr5094982260447487303t_unit,P: product_unit > $o,A2: product_unit] :
% 14.25/8.06 ( ( bNF_We2091578341578594809t_unit @ R2 )
% 14.25/8.06 => ( ! [X3: product_unit] :
% 14.25/8.06 ( ! [Y5: product_unit] :
% 14.25/8.06 ( ( ( Y5 != X3 )
% 14.25/8.06 & ( member7821138191923116944t_unit @ ( produc5692694829384537815t_unit @ Y5 @ X3 ) @ R2 ) )
% 14.25/8.06 => ( ( member_Product_unit @ Y5 @ ( field_Product_unit @ R2 ) )
% 14.25/8.06 => ( P @ Y5 ) ) )
% 14.25/8.06 => ( ( member_Product_unit @ X3 @ ( field_Product_unit @ R2 ) )
% 14.25/8.06 => ( P @ X3 ) ) )
% 14.25/8.06 => ( ( member_Product_unit @ A2 @ ( field_Product_unit @ R2 ) )
% 14.25/8.06 => ( P @ A2 ) ) ) ) ).
% 14.25/8.06
% 14.25/8.06 thf(fact_2950_well__order__induct__imp,axiom,
% 14.25/8.06 ! [R2: set_Pr1261947904930325089at_nat,P: nat > $o,A2: nat] :
% 14.25/8.06 ( ( bNF_We3818239936649020644el_nat @ R2 )
% 14.25/8.06 => ( ! [X3: nat] :
% 14.25/8.06 ( ! [Y5: nat] :
% 14.25/8.06 ( ( ( Y5 != X3 )
% 14.25/8.06 & ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ Y5 @ X3 ) @ R2 ) )
% 14.25/8.06 => ( ( member_nat @ Y5 @ ( field_nat @ R2 ) )
% 14.25/8.06 => ( P @ Y5 ) ) )
% 14.25/8.06 => ( ( member_nat @ X3 @ ( field_nat @ R2 ) )
% 14.25/8.06 => ( P @ X3 ) ) )
% 14.25/8.06 => ( ( member_nat @ A2 @ ( field_nat @ R2 ) )
% 14.25/8.06 => ( P @ A2 ) ) ) ) ).
% 14.25/8.06
% 14.25/8.06 thf(fact_2951_well__order__induct__imp,axiom,
% 14.25/8.06 ! [R2: set_Pr958786334691620121nt_int,P: int > $o,A2: int] :
% 14.25/8.06 ( ( bNF_We3815749466139970368el_int @ R2 )
% 14.25/8.06 => ( ! [X3: int] :
% 14.25/8.06 ( ! [Y5: int] :
% 14.25/8.06 ( ( ( Y5 != X3 )
% 14.25/8.06 & ( member5262025264175285858nt_int @ ( product_Pair_int_int @ Y5 @ X3 ) @ R2 ) )
% 14.25/8.06 => ( ( member_int @ Y5 @ ( field_int @ R2 ) )
% 14.25/8.06 => ( P @ Y5 ) ) )
% 14.25/8.06 => ( ( member_int @ X3 @ ( field_int @ R2 ) )
% 14.25/8.06 => ( P @ X3 ) ) )
% 14.25/8.06 => ( ( member_int @ A2 @ ( field_int @ R2 ) )
% 14.25/8.06 => ( P @ A2 ) ) ) ) ).
% 14.25/8.06
% 14.25/8.06 thf(fact_2952_underS__empty,axiom,
% 14.25/8.06 ! [A2: product_unit,R2: set_Pr5094982260447487303t_unit] :
% 14.25/8.06 ( ~ ( member_Product_unit @ A2 @ ( field_Product_unit @ R2 ) )
% 14.25/8.06 => ( ( order_5809128134054296019t_unit @ R2 @ A2 )
% 14.25/8.06 = bot_bo3957492148770167129t_unit ) ) ).
% 14.25/8.06
% 14.25/8.06 thf(fact_2953_underS__empty,axiom,
% 14.25/8.06 ! [A2: $o,R2: set_Product_prod_o_o] :
% 14.25/8.06 ( ~ ( member_o @ A2 @ ( field_o @ R2 ) )
% 14.25/8.06 => ( ( order_underS_o @ R2 @ A2 )
% 14.25/8.06 = bot_bot_set_o ) ) ).
% 14.25/8.06
% 14.25/8.06 thf(fact_2954_underS__empty,axiom,
% 14.25/8.06 ! [A2: nat,R2: set_Pr1261947904930325089at_nat] :
% 14.25/8.06 ( ~ ( member_nat @ A2 @ ( field_nat @ R2 ) )
% 14.25/8.06 => ( ( order_underS_nat @ R2 @ A2 )
% 14.25/8.06 = bot_bot_set_nat ) ) ).
% 14.25/8.06
% 14.25/8.06 thf(fact_2955_underS__empty,axiom,
% 14.25/8.06 ! [A2: int,R2: set_Pr958786334691620121nt_int] :
% 14.25/8.06 ( ~ ( member_int @ A2 @ ( field_int @ R2 ) )
% 14.25/8.06 => ( ( order_underS_int @ R2 @ A2 )
% 14.25/8.06 = bot_bot_set_int ) ) ).
% 14.25/8.06
% 14.25/8.06 thf(fact_2956_underS__empty,axiom,
% 14.25/8.06 ! [A2: produc3658429121746597890et_nat > $o,R2: set_Pr2161125870931222855_nat_o] :
% 14.25/8.06 ( ~ ( member6576561426505652726_nat_o @ A2 @ ( field_81821745747897882_nat_o @ R2 ) )
% 14.25/8.06 => ( ( order_7262550146012401043_nat_o @ R2 @ A2 )
% 14.25/8.06 = bot_bo7824918357723582233_nat_o ) ) ).
% 14.25/8.06
% 14.25/8.06 thf(fact_2957_wfI__pf,axiom,
% 14.25/8.06 ! [R3: set_Pr5094982260447487303t_unit] :
% 14.25/8.06 ( ! [A9: set_Product_unit] :
% 14.25/8.06 ( ( ord_le3507040750410214029t_unit @ A9 @ ( image_8431784415008483091t_unit @ R3 @ A9 ) )
% 14.25/8.06 => ( A9 = bot_bo3957492148770167129t_unit ) )
% 14.25/8.06 => ( wf_Product_unit @ R3 ) ) ).
% 14.25/8.06
% 14.25/8.06 thf(fact_2958_wfI__pf,axiom,
% 14.25/8.06 ! [R3: set_Product_prod_o_o] :
% 14.25/8.06 ( ! [A9: set_o] :
% 14.25/8.06 ( ( ord_less_eq_set_o @ A9 @ ( image_o_o @ R3 @ A9 ) )
% 14.25/8.06 => ( A9 = bot_bot_set_o ) )
% 14.25/8.06 => ( wf_o @ R3 ) ) ).
% 14.25/8.06
% 14.25/8.06 thf(fact_2959_wfI__pf,axiom,
% 14.25/8.06 ! [R3: set_Pr958786334691620121nt_int] :
% 14.25/8.06 ( ! [A9: set_int] :
% 14.25/8.06 ( ( ord_less_eq_set_int @ A9 @ ( image_int_int @ R3 @ A9 ) )
% 14.25/8.06 => ( A9 = bot_bot_set_int ) )
% 14.25/8.06 => ( wf_int @ R3 ) ) ).
% 14.25/8.06
% 14.25/8.06 thf(fact_2960_wfI__pf,axiom,
% 14.25/8.06 ! [R3: set_Pr2161125870931222855_nat_o] :
% 14.25/8.06 ( ! [A9: set_Pr4532377907799695533_nat_o] :
% 14.25/8.06 ( ( ord_le2965882846123202637_nat_o @ A9 @ ( image_2232984160439939859_nat_o @ R3 @ A9 ) )
% 14.25/8.06 => ( A9 = bot_bo7824918357723582233_nat_o ) )
% 14.25/8.06 => ( wf_Pro8787509627421085110_nat_o @ R3 ) ) ).
% 14.25/8.06
% 14.25/8.06 thf(fact_2961_wfI__pf,axiom,
% 14.25/8.06 ! [R3: set_Pr1261947904930325089at_nat] :
% 14.25/8.06 ( ! [A9: set_nat] :
% 14.25/8.06 ( ( ord_less_eq_set_nat @ A9 @ ( image_nat_nat @ R3 @ A9 ) )
% 14.25/8.06 => ( A9 = bot_bot_set_nat ) )
% 14.25/8.06 => ( wf_nat @ R3 ) ) ).
% 14.25/8.06
% 14.25/8.06 thf(fact_2962_wfE__pf,axiom,
% 14.25/8.06 ! [R3: set_Pr5094982260447487303t_unit,A: set_Product_unit] :
% 14.25/8.06 ( ( wf_Product_unit @ R3 )
% 14.25/8.06 => ( ( ord_le3507040750410214029t_unit @ A @ ( image_8431784415008483091t_unit @ R3 @ A ) )
% 14.25/8.06 => ( A = bot_bo3957492148770167129t_unit ) ) ) ).
% 14.25/8.06
% 14.25/8.06 thf(fact_2963_wfE__pf,axiom,
% 14.25/8.06 ! [R3: set_Product_prod_o_o,A: set_o] :
% 14.25/8.06 ( ( wf_o @ R3 )
% 14.25/8.06 => ( ( ord_less_eq_set_o @ A @ ( image_o_o @ R3 @ A ) )
% 14.25/8.06 => ( A = bot_bot_set_o ) ) ) ).
% 14.25/8.06
% 14.25/8.06 thf(fact_2964_wfE__pf,axiom,
% 14.25/8.06 ! [R3: set_Pr958786334691620121nt_int,A: set_int] :
% 14.25/8.06 ( ( wf_int @ R3 )
% 14.25/8.06 => ( ( ord_less_eq_set_int @ A @ ( image_int_int @ R3 @ A ) )
% 14.25/8.06 => ( A = bot_bot_set_int ) ) ) ).
% 14.25/8.06
% 14.25/8.06 thf(fact_2965_wfE__pf,axiom,
% 14.25/8.06 ! [R3: set_Pr2161125870931222855_nat_o,A: set_Pr4532377907799695533_nat_o] :
% 14.25/8.06 ( ( wf_Pro8787509627421085110_nat_o @ R3 )
% 14.25/8.06 => ( ( ord_le2965882846123202637_nat_o @ A @ ( image_2232984160439939859_nat_o @ R3 @ A ) )
% 14.25/8.06 => ( A = bot_bo7824918357723582233_nat_o ) ) ) ).
% 14.25/8.06
% 14.25/8.06 thf(fact_2966_wfE__pf,axiom,
% 14.25/8.06 ! [R3: set_Pr1261947904930325089at_nat,A: set_nat] :
% 14.25/8.06 ( ( wf_nat @ R3 )
% 14.25/8.06 => ( ( ord_less_eq_set_nat @ A @ ( image_nat_nat @ R3 @ A ) )
% 14.25/8.06 => ( A = bot_bot_set_nat ) ) ) ).
% 14.25/8.06
% 14.25/8.06 thf(fact_2967_wf__eq__minimal2,axiom,
% 14.25/8.06 ( wf_Product_unit
% 14.25/8.06 = ( ^ [R4: set_Pr5094982260447487303t_unit] :
% 14.25/8.06 ! [A3: set_Product_unit] :
% 14.25/8.06 ( ( ( ord_le3507040750410214029t_unit @ A3 @ ( field_Product_unit @ R4 ) )
% 14.25/8.06 & ( A3 != bot_bo3957492148770167129t_unit ) )
% 14.25/8.06 => ? [X2: product_unit] :
% 14.25/8.06 ( ( member_Product_unit @ X2 @ A3 )
% 14.25/8.06 & ! [Y3: product_unit] :
% 14.25/8.06 ( ( member_Product_unit @ Y3 @ A3 )
% 14.25/8.06 => ~ ( member7821138191923116944t_unit @ ( produc5692694829384537815t_unit @ Y3 @ X2 ) @ R4 ) ) ) ) ) ) ).
% 14.25/8.06
% 14.25/8.06 thf(fact_2968_wf__eq__minimal2,axiom,
% 14.25/8.06 ( wf_o
% 14.25/8.06 = ( ^ [R4: set_Product_prod_o_o] :
% 14.25/8.06 ! [A3: set_o] :
% 14.25/8.06 ( ( ( ord_less_eq_set_o @ A3 @ ( field_o @ R4 ) )
% 14.25/8.06 & ( A3 != bot_bot_set_o ) )
% 14.25/8.06 => ? [X2: $o] :
% 14.25/8.06 ( ( member_o @ X2 @ A3 )
% 14.25/8.06 & ! [Y3: $o] :
% 14.25/8.06 ( ( member_o @ Y3 @ A3 )
% 14.25/8.06 => ~ ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ Y3 @ X2 ) @ R4 ) ) ) ) ) ) ).
% 14.25/8.06
% 14.25/8.06 thf(fact_2969_wf__eq__minimal2,axiom,
% 14.25/8.06 ( wf_int
% 14.25/8.06 = ( ^ [R4: set_Pr958786334691620121nt_int] :
% 14.25/8.06 ! [A3: set_int] :
% 14.25/8.06 ( ( ( ord_less_eq_set_int @ A3 @ ( field_int @ R4 ) )
% 14.25/8.06 & ( A3 != bot_bot_set_int ) )
% 14.25/8.06 => ? [X2: int] :
% 14.25/8.06 ( ( member_int @ X2 @ A3 )
% 14.25/8.06 & ! [Y3: int] :
% 14.25/8.06 ( ( member_int @ Y3 @ A3 )
% 14.25/8.06 => ~ ( member5262025264175285858nt_int @ ( product_Pair_int_int @ Y3 @ X2 ) @ R4 ) ) ) ) ) ) ).
% 14.25/8.06
% 14.25/8.06 thf(fact_2970_wf__eq__minimal2,axiom,
% 14.25/8.06 ( wf_Pro8787509627421085110_nat_o
% 14.25/8.06 = ( ^ [R4: set_Pr2161125870931222855_nat_o] :
% 14.25/8.06 ! [A3: set_Pr4532377907799695533_nat_o] :
% 14.25/8.06 ( ( ( ord_le2965882846123202637_nat_o @ A3 @ ( field_81821745747897882_nat_o @ R4 ) )
% 14.25/8.06 & ( A3 != bot_bo7824918357723582233_nat_o ) )
% 14.25/8.06 => ? [X2: produc3658429121746597890et_nat > $o] :
% 14.25/8.06 ( ( member6576561426505652726_nat_o @ X2 @ A3 )
% 14.25/8.06 & ! [Y3: produc3658429121746597890et_nat > $o] :
% 14.25/8.06 ( ( member6576561426505652726_nat_o @ Y3 @ A3 )
% 14.25/8.06 => ~ ( member8781333585448626064_nat_o @ ( produc7368190662567826135_nat_o @ Y3 @ X2 ) @ R4 ) ) ) ) ) ) ).
% 14.25/8.06
% 14.25/8.06 thf(fact_2971_wf__eq__minimal2,axiom,
% 14.25/8.06 ( wf_nat
% 14.25/8.06 = ( ^ [R4: set_Pr1261947904930325089at_nat] :
% 14.25/8.06 ! [A3: set_nat] :
% 14.25/8.06 ( ( ( ord_less_eq_set_nat @ A3 @ ( field_nat @ R4 ) )
% 14.25/8.06 & ( A3 != bot_bot_set_nat ) )
% 14.25/8.06 => ? [X2: nat] :
% 14.25/8.06 ( ( member_nat @ X2 @ A3 )
% 14.25/8.06 & ! [Y3: nat] :
% 14.25/8.06 ( ( member_nat @ Y3 @ A3 )
% 14.25/8.06 => ~ ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ Y3 @ X2 ) @ R4 ) ) ) ) ) ) ).
% 14.25/8.06
% 14.25/8.06 thf(fact_2972_wo__rel_Ocases__Total,axiom,
% 14.25/8.06 ! [R2: set_Pr5094982260447487303t_unit,A2: product_unit,B: product_unit,Phi: product_unit > product_unit > $o] :
% 14.25/8.06 ( ( bNF_We2091578341578594809t_unit @ R2 )
% 14.25/8.06 => ( ( ord_le3507040750410214029t_unit @ ( insert_Product_unit @ A2 @ ( insert_Product_unit @ B @ bot_bo3957492148770167129t_unit ) ) @ ( field_Product_unit @ R2 ) )
% 14.25/8.06 => ( ( ( member7821138191923116944t_unit @ ( produc5692694829384537815t_unit @ A2 @ B ) @ R2 )
% 14.25/8.06 => ( Phi @ A2 @ B ) )
% 14.25/8.06 => ( ( ( member7821138191923116944t_unit @ ( produc5692694829384537815t_unit @ B @ A2 ) @ R2 )
% 14.25/8.06 => ( Phi @ A2 @ B ) )
% 14.25/8.06 => ( Phi @ A2 @ B ) ) ) ) ) ).
% 14.25/8.06
% 14.25/8.06 thf(fact_2973_wo__rel_Ocases__Total,axiom,
% 14.25/8.06 ! [R2: set_Product_prod_o_o,A2: $o,B: $o,Phi: $o > $o > $o] :
% 14.25/8.06 ( ( bNF_We6437023980930612612_rel_o @ R2 )
% 14.25/8.06 => ( ( ord_less_eq_set_o @ ( insert_o @ A2 @ ( insert_o @ B @ bot_bot_set_o ) ) @ ( field_o @ R2 ) )
% 14.25/8.06 => ( ( ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ A2 @ B ) @ R2 )
% 14.25/8.06 => ( Phi @ A2 @ B ) )
% 14.25/8.06 => ( ( ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ B @ A2 ) @ R2 )
% 14.25/8.06 => ( Phi @ A2 @ B ) )
% 14.25/8.06 => ( Phi @ A2 @ B ) ) ) ) ) ).
% 14.25/8.06
% 14.25/8.06 thf(fact_2974_wo__rel_Ocases__Total,axiom,
% 14.25/8.06 ! [R2: set_Pr958786334691620121nt_int,A2: int,B: int,Phi: int > int > $o] :
% 14.25/8.06 ( ( bNF_We3815749466139970368el_int @ R2 )
% 14.25/8.06 => ( ( ord_less_eq_set_int @ ( insert_int @ A2 @ ( insert_int @ B @ bot_bot_set_int ) ) @ ( field_int @ R2 ) )
% 14.25/8.06 => ( ( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ A2 @ B ) @ R2 )
% 14.25/8.06 => ( Phi @ A2 @ B ) )
% 14.25/8.06 => ( ( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ B @ A2 ) @ R2 )
% 14.25/8.06 => ( Phi @ A2 @ B ) )
% 14.25/8.06 => ( Phi @ A2 @ B ) ) ) ) ) ).
% 14.25/8.06
% 14.25/8.06 thf(fact_2975_wo__rel_Ocases__Total,axiom,
% 14.25/8.06 ! [R2: set_Pr2161125870931222855_nat_o,A2: produc3658429121746597890et_nat > $o,B: produc3658429121746597890et_nat > $o,Phi: ( produc3658429121746597890et_nat > $o ) > ( produc3658429121746597890et_nat > $o ) > $o] :
% 14.25/8.06 ( ( bNF_We740377018556551609_nat_o @ R2 )
% 14.25/8.06 => ( ( ord_le2965882846123202637_nat_o @ ( insert5175938949040314269_nat_o @ A2 @ ( insert5175938949040314269_nat_o @ B @ bot_bo7824918357723582233_nat_o ) ) @ ( field_81821745747897882_nat_o @ R2 ) )
% 14.25/8.06 => ( ( ( member8781333585448626064_nat_o @ ( produc7368190662567826135_nat_o @ A2 @ B ) @ R2 )
% 14.25/8.06 => ( Phi @ A2 @ B ) )
% 14.25/8.06 => ( ( ( member8781333585448626064_nat_o @ ( produc7368190662567826135_nat_o @ B @ A2 ) @ R2 )
% 14.25/8.06 => ( Phi @ A2 @ B ) )
% 14.25/8.06 => ( Phi @ A2 @ B ) ) ) ) ) ).
% 14.25/8.06
% 14.25/8.06 thf(fact_2976_wo__rel_Ocases__Total,axiom,
% 14.25/8.06 ! [R2: set_Pr1261947904930325089at_nat,A2: nat,B: nat,Phi: nat > nat > $o] :
% 14.25/8.06 ( ( bNF_We3818239936649020644el_nat @ R2 )
% 14.25/8.06 => ( ( ord_less_eq_set_nat @ ( insert_nat @ A2 @ ( insert_nat @ B @ bot_bot_set_nat ) ) @ ( field_nat @ R2 ) )
% 14.25/8.06 => ( ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A2 @ B ) @ R2 )
% 14.25/8.06 => ( Phi @ A2 @ B ) )
% 14.25/8.06 => ( ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ B @ A2 ) @ R2 )
% 14.25/8.06 => ( Phi @ A2 @ B ) )
% 14.25/8.06 => ( Phi @ A2 @ B ) ) ) ) ) ).
% 14.25/8.06
% 14.25/8.06 thf(fact_2977_wf__no__path,axiom,
% 14.25/8.06 ! [R3: set_Pr5094982260447487303t_unit] :
% 14.25/8.06 ( ( ( inf_in4660618365625256667t_unit @ ( domain4292024440186670924t_unit @ R3 ) @ ( range_6617724647562260917t_unit @ R3 ) )
% 14.25/8.06 = bot_bo3957492148770167129t_unit )
% 14.25/8.06 => ( wf_Product_unit @ R3 ) ) ).
% 14.25/8.06
% 14.25/8.06 thf(fact_2978_wf__no__path,axiom,
% 14.25/8.06 ! [R3: set_Product_prod_o_o] :
% 14.25/8.06 ( ( ( inf_inf_set_o @ ( domain_o_o @ R3 ) @ ( range_o_o @ R3 ) )
% 14.25/8.06 = bot_bot_set_o )
% 14.25/8.06 => ( wf_o @ R3 ) ) ).
% 14.25/8.06
% 14.25/8.06 thf(fact_2979_wf__no__path,axiom,
% 14.25/8.06 ! [R3: set_Pr1261947904930325089at_nat] :
% 14.25/8.06 ( ( ( inf_inf_set_nat @ ( domain_nat_nat @ R3 ) @ ( range_nat_nat @ R3 ) )
% 14.25/8.06 = bot_bot_set_nat )
% 14.25/8.06 => ( wf_nat @ R3 ) ) ).
% 14.25/8.06
% 14.25/8.06 thf(fact_2980_wf__no__path,axiom,
% 14.25/8.06 ! [R3: set_Pr958786334691620121nt_int] :
% 14.25/8.06 ( ( ( inf_inf_set_int @ ( domain_int_int @ R3 ) @ ( range_int_int @ R3 ) )
% 14.25/8.06 = bot_bot_set_int )
% 14.25/8.06 => ( wf_int @ R3 ) ) ).
% 14.25/8.06
% 14.25/8.06 thf(fact_2981_wf__no__path,axiom,
% 14.25/8.06 ! [R3: set_Pr2161125870931222855_nat_o] :
% 14.25/8.06 ( ( ( inf_in1906310914598751387_nat_o @ ( domain4881241903163524940_nat_o @ R3 ) @ ( range_5289318983786108341_nat_o @ R3 ) )
% 14.25/8.06 = bot_bo7824918357723582233_nat_o )
% 14.25/8.06 => ( wf_Pro8787509627421085110_nat_o @ R3 ) ) ).
% 14.25/8.06
% 14.25/8.06 thf(fact_2982_wf__min,axiom,
% 14.25/8.06 ! [R3: set_Pr2161125870931222855_nat_o] :
% 14.25/8.06 ( ( wf_Pro8787509627421085110_nat_o @ R3 )
% 14.25/8.06 => ( ( R3 != bot_bo3657187498359500723_nat_o )
% 14.25/8.06 => ~ ! [M2: produc3658429121746597890et_nat > $o] :
% 14.25/8.06 ~ ( member6576561426505652726_nat_o @ M2 @ ( minus_1801376950450012436_nat_o @ ( domain4881241903163524940_nat_o @ R3 ) @ ( range_5289318983786108341_nat_o @ R3 ) ) ) ) ) ).
% 14.25/8.06
% 14.25/8.06 thf(fact_2983_wf__min,axiom,
% 14.25/8.06 ! [R3: set_Pr958786334691620121nt_int] :
% 14.25/8.06 ( ( wf_int @ R3 )
% 14.25/8.06 => ( ( R3 != bot_bo1796632182523588997nt_int )
% 14.25/8.06 => ~ ! [M2: int] :
% 14.25/8.06 ~ ( member_int @ M2 @ ( minus_minus_set_int @ ( domain_int_int @ R3 ) @ ( range_int_int @ R3 ) ) ) ) ) ).
% 14.25/8.06
% 14.25/8.06 thf(fact_2984_wf__min,axiom,
% 14.25/8.06 ! [R3: set_Product_prod_o_o] :
% 14.25/8.06 ( ( wf_o @ R3 )
% 14.25/8.06 => ( ( R3 != bot_bo7073875226086086771od_o_o )
% 14.25/8.06 => ~ ! [M2: $o] :
% 14.25/8.06 ~ ( member_o @ M2 @ ( minus_minus_set_o @ ( domain_o_o @ R3 ) @ ( range_o_o @ R3 ) ) ) ) ) ).
% 14.25/8.06
% 14.25/8.06 thf(fact_2985_wf__min,axiom,
% 14.25/8.06 ! [R3: set_Pr5094982260447487303t_unit] :
% 14.25/8.06 ( ( wf_Product_unit @ R3 )
% 14.25/8.06 => ( ( R3 != bot_bo1654022762305692595t_unit )
% 14.25/8.06 => ~ ! [M2: product_unit] :
% 14.25/8.06 ~ ( member_Product_unit @ M2 @ ( minus_6452836326544984404t_unit @ ( domain4292024440186670924t_unit @ R3 ) @ ( range_6617724647562260917t_unit @ R3 ) ) ) ) ) ).
% 14.25/8.06
% 14.25/8.06 thf(fact_2986_wf__min,axiom,
% 14.25/8.06 ! [R3: set_Pr1261947904930325089at_nat] :
% 14.25/8.06 ( ( wf_nat @ R3 )
% 14.25/8.06 => ( ( R3 != bot_bo2099793752762293965at_nat )
% 14.25/8.06 => ~ ! [M2: nat] :
% 14.25/8.06 ~ ( member_nat @ M2 @ ( minus_minus_set_nat @ ( domain_nat_nat @ R3 ) @ ( range_nat_nat @ R3 ) ) ) ) ) ).
% 14.25/8.06
% 14.25/8.06 thf(fact_2987_underS__incl__iff,axiom,
% 14.25/8.06 ! [R2: set_Pr2161125870931222855_nat_o,A2: produc3658429121746597890et_nat > $o,B: produc3658429121746597890et_nat > $o] :
% 14.25/8.06 ( ( order_1145710701291701250_nat_o @ ( field_81821745747897882_nat_o @ R2 ) @ R2 )
% 14.25/8.06 => ( ( member6576561426505652726_nat_o @ A2 @ ( field_81821745747897882_nat_o @ R2 ) )
% 14.25/8.06 => ( ( member6576561426505652726_nat_o @ B @ ( field_81821745747897882_nat_o @ R2 ) )
% 14.25/8.06 => ( ( ord_le2965882846123202637_nat_o @ ( order_7262550146012401043_nat_o @ R2 @ A2 ) @ ( order_7262550146012401043_nat_o @ R2 @ B ) )
% 14.25/8.06 = ( member8781333585448626064_nat_o @ ( produc7368190662567826135_nat_o @ A2 @ B ) @ R2 ) ) ) ) ) ).
% 14.25/8.06
% 14.25/8.06 thf(fact_2988_underS__incl__iff,axiom,
% 14.25/8.06 ! [R2: set_Product_prod_o_o,A2: $o,B: $o] :
% 14.25/8.06 ( ( order_6238756238976269133r_on_o @ ( field_o @ R2 ) @ R2 )
% 14.25/8.06 => ( ( member_o @ A2 @ ( field_o @ R2 ) )
% 14.25/8.06 => ( ( member_o @ B @ ( field_o @ R2 ) )
% 14.25/8.06 => ( ( ord_less_eq_set_o @ ( order_underS_o @ R2 @ A2 ) @ ( order_underS_o @ R2 @ B ) )
% 14.25/8.06 = ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ A2 @ B ) @ R2 ) ) ) ) ) ).
% 14.25/8.06
% 14.25/8.06 thf(fact_2989_underS__incl__iff,axiom,
% 14.25/8.06 ! [R2: set_Pr5094982260447487303t_unit,A2: product_unit,B: product_unit] :
% 14.25/8.06 ( ( order_888282469287196226t_unit @ ( field_Product_unit @ R2 ) @ R2 )
% 14.25/8.06 => ( ( member_Product_unit @ A2 @ ( field_Product_unit @ R2 ) )
% 14.25/8.06 => ( ( member_Product_unit @ B @ ( field_Product_unit @ R2 ) )
% 14.25/8.06 => ( ( ord_le3507040750410214029t_unit @ ( order_5809128134054296019t_unit @ R2 @ A2 ) @ ( order_5809128134054296019t_unit @ R2 @ B ) )
% 14.25/8.06 = ( member7821138191923116944t_unit @ ( produc5692694829384537815t_unit @ A2 @ B ) @ R2 ) ) ) ) ) ).
% 14.25/8.06
% 14.25/8.06 thf(fact_2990_underS__incl__iff,axiom,
% 14.25/8.06 ! [R2: set_Pr958786334691620121nt_int,A2: int,B: int] :
% 14.25/8.06 ( ( order_4471489696718655927on_int @ ( field_int @ R2 ) @ R2 )
% 14.25/8.06 => ( ( member_int @ A2 @ ( field_int @ R2 ) )
% 14.25/8.06 => ( ( member_int @ B @ ( field_int @ R2 ) )
% 14.25/8.06 => ( ( ord_less_eq_set_int @ ( order_underS_int @ R2 @ A2 ) @ ( order_underS_int @ R2 @ B ) )
% 14.25/8.06 = ( member5262025264175285858nt_int @ ( product_Pair_int_int @ A2 @ B ) @ R2 ) ) ) ) ) ).
% 14.25/8.06
% 14.25/8.06 thf(fact_2991_underS__incl__iff,axiom,
% 14.25/8.06 ! [R2: set_Pr1261947904930325089at_nat,A2: nat,B: nat] :
% 14.25/8.06 ( ( order_4473980167227706203on_nat @ ( field_nat @ R2 ) @ R2 )
% 14.25/8.06 => ( ( member_nat @ A2 @ ( field_nat @ R2 ) )
% 14.25/8.06 => ( ( member_nat @ B @ ( field_nat @ R2 ) )
% 14.25/8.06 => ( ( ord_less_eq_set_nat @ ( order_underS_nat @ R2 @ A2 ) @ ( order_underS_nat @ R2 @ B ) )
% 14.25/8.06 = ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A2 @ B ) @ R2 ) ) ) ) ) ).
% 14.25/8.06
% 14.25/8.06 thf(fact_2992_wo__rel_OWell__order__isMinim__exists,axiom,
% 14.25/8.06 ! [R2: set_Pr5094982260447487303t_unit,B4: set_Product_unit] :
% 14.25/8.06 ( ( bNF_We2091578341578594809t_unit @ R2 )
% 14.25/8.06 => ( ( ord_le3507040750410214029t_unit @ B4 @ ( field_Product_unit @ R2 ) )
% 14.25/8.06 => ( ( B4 != bot_bo3957492148770167129t_unit )
% 14.25/8.06 => ? [X_1: product_unit] : ( bNF_We909621271159000003t_unit @ R2 @ B4 @ X_1 ) ) ) ) ).
% 14.25/8.06
% 14.25/8.06 thf(fact_2993_wo__rel_OWell__order__isMinim__exists,axiom,
% 14.25/8.06 ! [R2: set_Product_prod_o_o,B4: set_o] :
% 14.25/8.06 ( ( bNF_We6437023980930612612_rel_o @ R2 )
% 14.25/8.06 => ( ( ord_less_eq_set_o @ B4 @ ( field_o @ R2 ) )
% 14.25/8.06 => ( ( B4 != bot_bot_set_o )
% 14.25/8.06 => ? [X_1: $o] : ( bNF_We4647864369647134286inim_o @ R2 @ B4 @ X_1 ) ) ) ) ).
% 14.25/8.06
% 14.25/8.06 thf(fact_2994_wo__rel_OWell__order__isMinim__exists,axiom,
% 14.25/8.06 ! [R2: set_Pr958786334691620121nt_int,B4: set_int] :
% 14.25/8.06 ( ( bNF_We3815749466139970368el_int @ R2 )
% 14.25/8.06 => ( ( ord_less_eq_set_int @ B4 @ ( field_int @ R2 ) )
% 14.25/8.06 => ( ( B4 != bot_bot_set_int )
% 14.25/8.06 => ? [X_1: int] : ( bNF_We6511469253196982518im_int @ R2 @ B4 @ X_1 ) ) ) ) ).
% 14.25/8.06
% 14.25/8.06 thf(fact_2995_wo__rel_OWell__order__isMinim__exists,axiom,
% 14.25/8.06 ! [R2: set_Pr2161125870931222855_nat_o,B4: set_Pr4532377907799695533_nat_o] :
% 14.25/8.06 ( ( bNF_We740377018556551609_nat_o @ R2 )
% 14.25/8.06 => ( ( ord_le2965882846123202637_nat_o @ B4 @ ( field_81821745747897882_nat_o @ R2 ) )
% 14.25/8.06 => ( ( B4 != bot_bo7824918357723582233_nat_o )
% 14.25/8.06 => ? [X_1: produc3658429121746597890et_nat > $o] : ( bNF_We5608535359669600131_nat_o @ R2 @ B4 @ X_1 ) ) ) ) ).
% 14.25/8.06
% 14.25/8.06 thf(fact_2996_wo__rel_OWell__order__isMinim__exists,axiom,
% 14.25/8.06 ! [R2: set_Pr1261947904930325089at_nat,B4: set_nat] :
% 14.25/8.06 ( ( bNF_We3818239936649020644el_nat @ R2 )
% 14.25/8.06 => ( ( ord_less_eq_set_nat @ B4 @ ( field_nat @ R2 ) )
% 14.25/8.06 => ( ( B4 != bot_bot_set_nat )
% 14.25/8.06 => ? [X_1: nat] : ( bNF_We6513959723706032794im_nat @ R2 @ B4 @ X_1 ) ) ) ) ).
% 14.25/8.06
% 14.25/8.06 thf(fact_2997_Refl__under__underS,axiom,
% 14.25/8.06 ! [R2: set_Pr5094982260447487303t_unit,A2: product_unit] :
% 14.25/8.06 ( ( refl_on_Product_unit @ ( field_Product_unit @ R2 ) @ R2 )
% 14.25/8.06 => ( ( member_Product_unit @ A2 @ ( field_Product_unit @ R2 ) )
% 14.25/8.06 => ( ( order_6949322460238004850t_unit @ R2 @ A2 )
% 14.25/8.06 = ( sup_su793286257634532545t_unit @ ( order_5809128134054296019t_unit @ R2 @ A2 ) @ ( insert_Product_unit @ A2 @ bot_bo3957492148770167129t_unit ) ) ) ) ) ).
% 14.25/8.06
% 14.25/8.06 thf(fact_2998_Refl__under__underS,axiom,
% 14.25/8.06 ! [R2: set_Product_prod_o_o,A2: $o] :
% 14.25/8.06 ( ( refl_on_o @ ( field_o @ R2 ) @ R2 )
% 14.25/8.06 => ( ( member_o @ A2 @ ( field_o @ R2 ) )
% 14.25/8.06 => ( ( order_under_o @ R2 @ A2 )
% 14.25/8.06 = ( sup_sup_set_o @ ( order_underS_o @ R2 @ A2 ) @ ( insert_o @ A2 @ bot_bot_set_o ) ) ) ) ) ).
% 14.25/8.06
% 14.25/8.06 thf(fact_2999_Refl__under__underS,axiom,
% 14.25/8.06 ! [R2: set_Pr1261947904930325089at_nat,A2: nat] :
% 14.25/8.06 ( ( refl_on_nat @ ( field_nat @ R2 ) @ R2 )
% 14.25/8.06 => ( ( member_nat @ A2 @ ( field_nat @ R2 ) )
% 14.25/8.06 => ( ( order_under_nat @ R2 @ A2 )
% 14.25/8.06 = ( sup_sup_set_nat @ ( order_underS_nat @ R2 @ A2 ) @ ( insert_nat @ A2 @ bot_bot_set_nat ) ) ) ) ) ).
% 14.25/8.06
% 14.25/8.06 thf(fact_3000_Refl__under__underS,axiom,
% 14.25/8.06 ! [R2: set_Pr958786334691620121nt_int,A2: int] :
% 14.25/8.06 ( ( refl_on_int @ ( field_int @ R2 ) @ R2 )
% 14.25/8.06 => ( ( member_int @ A2 @ ( field_int @ R2 ) )
% 14.25/8.06 => ( ( order_under_int @ R2 @ A2 )
% 14.25/8.06 = ( sup_sup_set_int @ ( order_underS_int @ R2 @ A2 ) @ ( insert_int @ A2 @ bot_bot_set_int ) ) ) ) ) ).
% 14.25/8.06
% 14.25/8.06 thf(fact_3001_Refl__under__underS,axiom,
% 14.25/8.06 ! [R2: set_Pr2161125870931222855_nat_o,A2: produc3658429121746597890et_nat > $o] :
% 14.25/8.06 ( ( refl_o1075864025857821388_nat_o @ ( field_81821745747897882_nat_o @ R2 ) @ R2 )
% 14.25/8.06 => ( ( member6576561426505652726_nat_o @ A2 @ ( field_81821745747897882_nat_o @ R2 ) )
% 14.25/8.06 => ( ( order_2885205318754725938_nat_o @ R2 @ A2 )
% 14.25/8.06 = ( sup_su5209123915105501825_nat_o @ ( order_7262550146012401043_nat_o @ R2 @ A2 ) @ ( insert5175938949040314269_nat_o @ A2 @ bot_bo7824918357723582233_nat_o ) ) ) ) ) ).
% 14.25/8.06
% 14.25/8.06 thf(fact_3002_wo__rel_Ominim__inField,axiom,
% 14.25/8.06 ! [R2: set_Pr5094982260447487303t_unit,B4: set_Product_unit] :
% 14.25/8.06 ( ( bNF_We2091578341578594809t_unit @ R2 )
% 14.25/8.06 => ( ( ord_le3507040750410214029t_unit @ B4 @ ( field_Product_unit @ R2 ) )
% 14.25/8.06 => ( ( B4 != bot_bo3957492148770167129t_unit )
% 14.25/8.06 => ( member_Product_unit @ ( bNF_We7185707985517244377t_unit @ R2 @ B4 ) @ ( field_Product_unit @ R2 ) ) ) ) ) ).
% 14.25/8.06
% 14.25/8.06 thf(fact_3003_wo__rel_Ominim__inField,axiom,
% 14.25/8.06 ! [R2: set_Product_prod_o_o,B4: set_o] :
% 14.25/8.06 ( ( bNF_We6437023980930612612_rel_o @ R2 )
% 14.25/8.06 => ( ( ord_less_eq_set_o @ B4 @ ( field_o @ R2 ) )
% 14.25/8.06 => ( ( B4 != bot_bot_set_o )
% 14.25/8.06 => ( member_o @ ( bNF_We537455116209922916inim_o @ R2 @ B4 ) @ ( field_o @ R2 ) ) ) ) ) ).
% 14.25/8.06
% 14.25/8.06 thf(fact_3004_wo__rel_Ominim__inField,axiom,
% 14.25/8.06 ! [R2: set_Pr958786334691620121nt_int,B4: set_int] :
% 14.25/8.06 ( ( bNF_We3815749466139970368el_int @ R2 )
% 14.25/8.06 => ( ( ord_less_eq_set_int @ B4 @ ( field_int @ R2 ) )
% 14.25/8.06 => ( ( B4 != bot_bot_set_int )
% 14.25/8.06 => ( member_int @ ( bNF_We3272633801743244640im_int @ R2 @ B4 ) @ ( field_int @ R2 ) ) ) ) ) ).
% 14.25/8.06
% 14.25/8.06 thf(fact_3005_wo__rel_Ominim__inField,axiom,
% 14.25/8.06 ! [R2: set_Pr2161125870931222855_nat_o,B4: set_Pr4532377907799695533_nat_o] :
% 14.25/8.06 ( ( bNF_We740377018556551609_nat_o @ R2 )
% 14.25/8.06 => ( ( ord_le2965882846123202637_nat_o @ B4 @ ( field_81821745747897882_nat_o @ R2 ) )
% 14.25/8.06 => ( ( B4 != bot_bo7824918357723582233_nat_o )
% 14.25/8.06 => ( member6576561426505652726_nat_o @ ( bNF_We8236795211600914329_nat_o @ R2 @ B4 ) @ ( field_81821745747897882_nat_o @ R2 ) ) ) ) ) ).
% 14.25/8.06
% 14.25/8.06 thf(fact_3006_wo__rel_Ominim__inField,axiom,
% 14.25/8.06 ! [R2: set_Pr1261947904930325089at_nat,B4: set_nat] :
% 14.25/8.06 ( ( bNF_We3818239936649020644el_nat @ R2 )
% 14.25/8.06 => ( ( ord_less_eq_set_nat @ B4 @ ( field_nat @ R2 ) )
% 14.25/8.06 => ( ( B4 != bot_bot_set_nat )
% 14.25/8.06 => ( member_nat @ ( bNF_We3275124272252294916im_nat @ R2 @ B4 ) @ ( field_nat @ R2 ) ) ) ) ) ).
% 14.25/8.06
% 14.25/8.06 thf(fact_3007_wo__rel_Ominim__in,axiom,
% 14.25/8.06 ! [R2: set_Pr5094982260447487303t_unit,B4: set_Product_unit] :
% 14.25/8.06 ( ( bNF_We2091578341578594809t_unit @ R2 )
% 14.25/8.06 => ( ( ord_le3507040750410214029t_unit @ B4 @ ( field_Product_unit @ R2 ) )
% 14.25/8.06 => ( ( B4 != bot_bo3957492148770167129t_unit )
% 14.25/8.06 => ( member_Product_unit @ ( bNF_We7185707985517244377t_unit @ R2 @ B4 ) @ B4 ) ) ) ) ).
% 14.25/8.06
% 14.25/8.06 thf(fact_3008_wo__rel_Ominim__in,axiom,
% 14.25/8.06 ! [R2: set_Product_prod_o_o,B4: set_o] :
% 14.25/8.06 ( ( bNF_We6437023980930612612_rel_o @ R2 )
% 14.25/8.06 => ( ( ord_less_eq_set_o @ B4 @ ( field_o @ R2 ) )
% 14.25/8.06 => ( ( B4 != bot_bot_set_o )
% 14.25/8.06 => ( member_o @ ( bNF_We537455116209922916inim_o @ R2 @ B4 ) @ B4 ) ) ) ) ).
% 14.25/8.06
% 14.25/8.06 thf(fact_3009_wo__rel_Ominim__in,axiom,
% 14.25/8.06 ! [R2: set_Pr958786334691620121nt_int,B4: set_int] :
% 14.25/8.06 ( ( bNF_We3815749466139970368el_int @ R2 )
% 14.25/8.06 => ( ( ord_less_eq_set_int @ B4 @ ( field_int @ R2 ) )
% 14.25/8.06 => ( ( B4 != bot_bot_set_int )
% 14.25/8.06 => ( member_int @ ( bNF_We3272633801743244640im_int @ R2 @ B4 ) @ B4 ) ) ) ) ).
% 14.25/8.06
% 14.25/8.06 thf(fact_3010_wo__rel_Ominim__in,axiom,
% 14.25/8.06 ! [R2: set_Pr2161125870931222855_nat_o,B4: set_Pr4532377907799695533_nat_o] :
% 14.25/8.06 ( ( bNF_We740377018556551609_nat_o @ R2 )
% 14.25/8.06 => ( ( ord_le2965882846123202637_nat_o @ B4 @ ( field_81821745747897882_nat_o @ R2 ) )
% 14.25/8.06 => ( ( B4 != bot_bo7824918357723582233_nat_o )
% 14.25/8.06 => ( member6576561426505652726_nat_o @ ( bNF_We8236795211600914329_nat_o @ R2 @ B4 ) @ B4 ) ) ) ) ).
% 14.25/8.06
% 14.25/8.06 thf(fact_3011_wo__rel_Ominim__in,axiom,
% 14.25/8.06 ! [R2: set_Pr1261947904930325089at_nat,B4: set_nat] :
% 14.25/8.06 ( ( bNF_We3818239936649020644el_nat @ R2 )
% 14.25/8.06 => ( ( ord_less_eq_set_nat @ B4 @ ( field_nat @ R2 ) )
% 14.25/8.06 => ( ( B4 != bot_bot_set_nat )
% 14.25/8.06 => ( member_nat @ ( bNF_We3275124272252294916im_nat @ R2 @ B4 ) @ B4 ) ) ) ) ).
% 14.25/8.06
% 14.25/8.06 thf(fact_3012_wo__rel_Oequals__minim,axiom,
% 14.25/8.06 ! [R2: set_Pr2161125870931222855_nat_o,B4: set_Pr4532377907799695533_nat_o,A2: produc3658429121746597890et_nat > $o] :
% 14.25/8.06 ( ( bNF_We740377018556551609_nat_o @ R2 )
% 14.25/8.06 => ( ( ord_le2965882846123202637_nat_o @ B4 @ ( field_81821745747897882_nat_o @ R2 ) )
% 14.25/8.06 => ( ( member6576561426505652726_nat_o @ A2 @ B4 )
% 14.25/8.06 => ( ! [B2: produc3658429121746597890et_nat > $o] :
% 14.25/8.06 ( ( member6576561426505652726_nat_o @ B2 @ B4 )
% 14.25/8.06 => ( member8781333585448626064_nat_o @ ( produc7368190662567826135_nat_o @ A2 @ B2 ) @ R2 ) )
% 14.25/8.06 => ( A2
% 14.25/8.06 = ( bNF_We8236795211600914329_nat_o @ R2 @ B4 ) ) ) ) ) ) ).
% 14.25/8.06
% 14.25/8.06 thf(fact_3013_wo__rel_Oequals__minim,axiom,
% 14.25/8.06 ! [R2: set_Product_prod_o_o,B4: set_o,A2: $o] :
% 14.25/8.06 ( ( bNF_We6437023980930612612_rel_o @ R2 )
% 14.25/8.06 => ( ( ord_less_eq_set_o @ B4 @ ( field_o @ R2 ) )
% 14.25/8.06 => ( ( member_o @ A2 @ B4 )
% 14.25/8.06 => ( ! [B2: $o] :
% 14.25/8.06 ( ( member_o @ B2 @ B4 )
% 14.25/8.06 => ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ A2 @ B2 ) @ R2 ) )
% 14.25/8.06 => ( A2
% 14.25/8.06 = ( bNF_We537455116209922916inim_o @ R2 @ B4 ) ) ) ) ) ) ).
% 14.25/8.06
% 14.25/8.06 thf(fact_3014_wo__rel_Oequals__minim,axiom,
% 14.25/8.06 ! [R2: set_Pr5094982260447487303t_unit,B4: set_Product_unit,A2: product_unit] :
% 14.25/8.06 ( ( bNF_We2091578341578594809t_unit @ R2 )
% 14.25/8.06 => ( ( ord_le3507040750410214029t_unit @ B4 @ ( field_Product_unit @ R2 ) )
% 14.25/8.06 => ( ( member_Product_unit @ A2 @ B4 )
% 14.25/8.06 => ( ! [B2: product_unit] :
% 14.25/8.06 ( ( member_Product_unit @ B2 @ B4 )
% 14.25/8.06 => ( member7821138191923116944t_unit @ ( produc5692694829384537815t_unit @ A2 @ B2 ) @ R2 ) )
% 14.25/8.06 => ( A2
% 14.25/8.06 = ( bNF_We7185707985517244377t_unit @ R2 @ B4 ) ) ) ) ) ) ).
% 14.25/8.06
% 14.25/8.06 thf(fact_3015_wo__rel_Oequals__minim,axiom,
% 14.25/8.06 ! [R2: set_Pr958786334691620121nt_int,B4: set_int,A2: int] :
% 14.25/8.06 ( ( bNF_We3815749466139970368el_int @ R2 )
% 14.25/8.06 => ( ( ord_less_eq_set_int @ B4 @ ( field_int @ R2 ) )
% 14.25/8.06 => ( ( member_int @ A2 @ B4 )
% 14.25/8.06 => ( ! [B2: int] :
% 14.25/8.06 ( ( member_int @ B2 @ B4 )
% 14.25/8.06 => ( member5262025264175285858nt_int @ ( product_Pair_int_int @ A2 @ B2 ) @ R2 ) )
% 14.25/8.06 => ( A2
% 14.25/8.06 = ( bNF_We3272633801743244640im_int @ R2 @ B4 ) ) ) ) ) ) ).
% 14.25/8.06
% 14.25/8.06 thf(fact_3016_wo__rel_Oequals__minim,axiom,
% 14.25/8.06 ! [R2: set_Pr1261947904930325089at_nat,B4: set_nat,A2: nat] :
% 14.25/8.06 ( ( bNF_We3818239936649020644el_nat @ R2 )
% 14.25/8.06 => ( ( ord_less_eq_set_nat @ B4 @ ( field_nat @ R2 ) )
% 14.25/8.06 => ( ( member_nat @ A2 @ B4 )
% 14.25/8.06 => ( ! [B2: nat] :
% 14.25/8.06 ( ( member_nat @ B2 @ B4 )
% 14.25/8.06 => ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A2 @ B2 ) @ R2 ) )
% 14.25/8.06 => ( A2
% 14.25/8.06 = ( bNF_We3275124272252294916im_nat @ R2 @ B4 ) ) ) ) ) ) ).
% 14.25/8.06
% 14.25/8.06 thf(fact_3017_wo__rel_Ominim__least,axiom,
% 14.25/8.06 ! [R2: set_Pr2161125870931222855_nat_o,B4: set_Pr4532377907799695533_nat_o,B: produc3658429121746597890et_nat > $o] :
% 14.25/8.06 ( ( bNF_We740377018556551609_nat_o @ R2 )
% 14.25/8.06 => ( ( ord_le2965882846123202637_nat_o @ B4 @ ( field_81821745747897882_nat_o @ R2 ) )
% 14.25/8.06 => ( ( member6576561426505652726_nat_o @ B @ B4 )
% 14.25/8.06 => ( member8781333585448626064_nat_o @ ( produc7368190662567826135_nat_o @ ( bNF_We8236795211600914329_nat_o @ R2 @ B4 ) @ B ) @ R2 ) ) ) ) ).
% 14.25/8.06
% 14.25/8.06 thf(fact_3018_wo__rel_Ominim__least,axiom,
% 14.25/8.06 ! [R2: set_Product_prod_o_o,B4: set_o,B: $o] :
% 14.25/8.06 ( ( bNF_We6437023980930612612_rel_o @ R2 )
% 14.25/8.06 => ( ( ord_less_eq_set_o @ B4 @ ( field_o @ R2 ) )
% 14.25/8.06 => ( ( member_o @ B @ B4 )
% 14.25/8.06 => ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ ( bNF_We537455116209922916inim_o @ R2 @ B4 ) @ B ) @ R2 ) ) ) ) ).
% 14.25/8.06
% 14.25/8.06 thf(fact_3019_wo__rel_Ominim__least,axiom,
% 14.25/8.06 ! [R2: set_Pr5094982260447487303t_unit,B4: set_Product_unit,B: product_unit] :
% 14.25/8.06 ( ( bNF_We2091578341578594809t_unit @ R2 )
% 14.25/8.06 => ( ( ord_le3507040750410214029t_unit @ B4 @ ( field_Product_unit @ R2 ) )
% 14.25/8.06 => ( ( member_Product_unit @ B @ B4 )
% 14.25/8.06 => ( member7821138191923116944t_unit @ ( produc5692694829384537815t_unit @ ( bNF_We7185707985517244377t_unit @ R2 @ B4 ) @ B ) @ R2 ) ) ) ) ).
% 14.25/8.06
% 14.25/8.06 thf(fact_3020_wo__rel_Ominim__least,axiom,
% 14.25/8.06 ! [R2: set_Pr958786334691620121nt_int,B4: set_int,B: int] :
% 14.25/8.06 ( ( bNF_We3815749466139970368el_int @ R2 )
% 14.25/8.06 => ( ( ord_less_eq_set_int @ B4 @ ( field_int @ R2 ) )
% 14.25/8.06 => ( ( member_int @ B @ B4 )
% 14.25/8.06 => ( member5262025264175285858nt_int @ ( product_Pair_int_int @ ( bNF_We3272633801743244640im_int @ R2 @ B4 ) @ B ) @ R2 ) ) ) ) ).
% 14.25/8.06
% 14.25/8.06 thf(fact_3021_wo__rel_Ominim__least,axiom,
% 14.25/8.06 ! [R2: set_Pr1261947904930325089at_nat,B4: set_nat,B: nat] :
% 14.25/8.06 ( ( bNF_We3818239936649020644el_nat @ R2 )
% 14.25/8.06 => ( ( ord_less_eq_set_nat @ B4 @ ( field_nat @ R2 ) )
% 14.25/8.06 => ( ( member_nat @ B @ B4 )
% 14.25/8.06 => ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ ( bNF_We3275124272252294916im_nat @ R2 @ B4 ) @ B ) @ R2 ) ) ) ) ).
% 14.25/8.06
% 14.25/8.06 thf(fact_3022_wo__rel_Omax2__greater__among,axiom,
% 14.25/8.06 ! [R2: set_Pr5094982260447487303t_unit,A2: product_unit,B: product_unit] :
% 14.25/8.06 ( ( bNF_We2091578341578594809t_unit @ R2 )
% 14.25/8.06 => ( ( member_Product_unit @ A2 @ ( field_Product_unit @ R2 ) )
% 14.25/8.06 => ( ( member_Product_unit @ B @ ( field_Product_unit @ R2 ) )
% 14.25/8.06 => ( ( member7821138191923116944t_unit @ ( produc5692694829384537815t_unit @ A2 @ ( bNF_We7204996500647184731t_unit @ R2 @ A2 @ B ) ) @ R2 )
% 14.25/8.06 & ( member7821138191923116944t_unit @ ( produc5692694829384537815t_unit @ B @ ( bNF_We7204996500647184731t_unit @ R2 @ A2 @ B ) ) @ R2 )
% 14.25/8.06 & ( member_Product_unit @ ( bNF_We7204996500647184731t_unit @ R2 @ A2 @ B ) @ ( insert_Product_unit @ A2 @ ( insert_Product_unit @ B @ bot_bo3957492148770167129t_unit ) ) ) ) ) ) ) ).
% 14.25/8.06
% 14.25/8.06 thf(fact_3023_wo__rel_Omax2__greater__among,axiom,
% 14.25/8.06 ! [R2: set_Product_prod_o_o,A2: $o,B: $o] :
% 14.25/8.06 ( ( bNF_We6437023980930612612_rel_o @ R2 )
% 14.25/8.06 => ( ( member_o @ A2 @ ( field_o @ R2 ) )
% 14.25/8.06 => ( ( member_o @ B @ ( field_o @ R2 ) )
% 14.25/8.06 => ( ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ A2 @ ( bNF_We3104086342435940326max2_o @ R2 @ A2 @ B ) ) @ R2 )
% 14.25/8.06 & ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ B @ ( bNF_We3104086342435940326max2_o @ R2 @ A2 @ B ) ) @ R2 )
% 14.25/8.06 & ( member_o @ ( bNF_We3104086342435940326max2_o @ R2 @ A2 @ B ) @ ( insert_o @ A2 @ ( insert_o @ B @ bot_bot_set_o ) ) ) ) ) ) ) ).
% 14.25/8.06
% 14.25/8.06 thf(fact_3024_wo__rel_Omax2__greater__among,axiom,
% 14.25/8.06 ! [R2: set_Pr1261947904930325089at_nat,A2: nat,B: nat] :
% 14.25/8.06 ( ( bNF_We3818239936649020644el_nat @ R2 )
% 14.25/8.06 => ( ( member_nat @ A2 @ ( field_nat @ R2 ) )
% 14.25/8.06 => ( ( member_nat @ B @ ( field_nat @ R2 ) )
% 14.25/8.06 => ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A2 @ ( bNF_We2380939065524664066x2_nat @ R2 @ A2 @ B ) ) @ R2 )
% 14.25/8.06 & ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ B @ ( bNF_We2380939065524664066x2_nat @ R2 @ A2 @ B ) ) @ R2 )
% 14.25/8.06 & ( member_nat @ ( bNF_We2380939065524664066x2_nat @ R2 @ A2 @ B ) @ ( insert_nat @ A2 @ ( insert_nat @ B @ bot_bot_set_nat ) ) ) ) ) ) ) ).
% 14.25/8.06
% 14.25/8.06 thf(fact_3025_wo__rel_Omax2__greater__among,axiom,
% 14.25/8.06 ! [R2: set_Pr958786334691620121nt_int,A2: int,B: int] :
% 14.25/8.06 ( ( bNF_We3815749466139970368el_int @ R2 )
% 14.25/8.06 => ( ( member_int @ A2 @ ( field_int @ R2 ) )
% 14.25/8.06 => ( ( member_int @ B @ ( field_int @ R2 ) )
% 14.25/8.06 => ( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ A2 @ ( bNF_We2378448595015613790x2_int @ R2 @ A2 @ B ) ) @ R2 )
% 14.25/8.06 & ( member5262025264175285858nt_int @ ( product_Pair_int_int @ B @ ( bNF_We2378448595015613790x2_int @ R2 @ A2 @ B ) ) @ R2 )
% 14.25/8.06 & ( member_int @ ( bNF_We2378448595015613790x2_int @ R2 @ A2 @ B ) @ ( insert_int @ A2 @ ( insert_int @ B @ bot_bot_set_int ) ) ) ) ) ) ) ).
% 14.25/8.06
% 14.25/8.06 thf(fact_3026_wo__rel_Omax2__greater__among,axiom,
% 14.25/8.06 ! [R2: set_Pr2161125870931222855_nat_o,A2: produc3658429121746597890et_nat > $o,B: produc3658429121746597890et_nat > $o] :
% 14.25/8.06 ( ( bNF_We740377018556551609_nat_o @ R2 )
% 14.25/8.06 => ( ( member6576561426505652726_nat_o @ A2 @ ( field_81821745747897882_nat_o @ R2 ) )
% 14.25/8.06 => ( ( member6576561426505652726_nat_o @ B @ ( field_81821745747897882_nat_o @ R2 ) )
% 14.25/8.06 => ( ( member8781333585448626064_nat_o @ ( produc7368190662567826135_nat_o @ A2 @ ( bNF_We6400040844883348763_nat_o @ R2 @ A2 @ B ) ) @ R2 )
% 14.25/8.06 & ( member8781333585448626064_nat_o @ ( produc7368190662567826135_nat_o @ B @ ( bNF_We6400040844883348763_nat_o @ R2 @ A2 @ B ) ) @ R2 )
% 14.25/8.06 & ( member6576561426505652726_nat_o @ ( bNF_We6400040844883348763_nat_o @ R2 @ A2 @ B ) @ ( insert5175938949040314269_nat_o @ A2 @ ( insert5175938949040314269_nat_o @ B @ bot_bo7824918357723582233_nat_o ) ) ) ) ) ) ) ).
% 14.25/8.06
% 14.25/8.06 thf(fact_3027_wo__rel_Omax2__def,axiom,
% 14.25/8.06 ! [R2: set_Pr1261947904930325089at_nat,A2: nat,B: nat] :
% 14.25/8.06 ( ( bNF_We3818239936649020644el_nat @ R2 )
% 14.25/8.06 => ( ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A2 @ B ) @ R2 )
% 14.25/8.06 => ( ( bNF_We2380939065524664066x2_nat @ R2 @ A2 @ B )
% 14.25/8.06 = B ) )
% 14.25/8.06 & ( ~ ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A2 @ B ) @ R2 )
% 14.25/8.06 => ( ( bNF_We2380939065524664066x2_nat @ R2 @ A2 @ B )
% 14.25/8.06 = A2 ) ) ) ) ).
% 14.25/8.06
% 14.25/8.06 thf(fact_3028_wo__rel_Omax2__def,axiom,
% 14.25/8.06 ! [R2: set_Pr958786334691620121nt_int,A2: int,B: int] :
% 14.25/8.06 ( ( bNF_We3815749466139970368el_int @ R2 )
% 14.25/8.06 => ( ( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ A2 @ B ) @ R2 )
% 14.25/8.06 => ( ( bNF_We2378448595015613790x2_int @ R2 @ A2 @ B )
% 14.25/8.06 = B ) )
% 14.25/8.06 & ( ~ ( member5262025264175285858nt_int @ ( product_Pair_int_int @ A2 @ B ) @ R2 )
% 14.25/8.06 => ( ( bNF_We2378448595015613790x2_int @ R2 @ A2 @ B )
% 14.25/8.06 = A2 ) ) ) ) ).
% 14.25/8.06
% 14.25/8.06 thf(fact_3029_under__incr,axiom,
% 14.25/8.06 ! [R2: set_Pr958786334691620121nt_int,A2: int,B: int] :
% 14.25/8.06 ( ( trans_int @ R2 )
% 14.25/8.06 => ( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ A2 @ B ) @ R2 )
% 14.25/8.06 => ( ord_less_eq_set_int @ ( order_under_int @ R2 @ A2 ) @ ( order_under_int @ R2 @ B ) ) ) ) ).
% 14.25/8.06
% 14.25/8.06 thf(fact_3030_under__incr,axiom,
% 14.25/8.06 ! [R2: set_Pr1261947904930325089at_nat,A2: nat,B: nat] :
% 14.25/8.06 ( ( trans_nat @ R2 )
% 14.25/8.06 => ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A2 @ B ) @ R2 )
% 14.25/8.06 => ( ord_less_eq_set_nat @ ( order_under_nat @ R2 @ A2 ) @ ( order_under_nat @ R2 @ B ) ) ) ) ).
% 14.25/8.06
% 14.25/8.06 thf(fact_3031_wo__rel_Ominim__isMinim,axiom,
% 14.25/8.06 ! [R2: set_Pr5094982260447487303t_unit,B4: set_Product_unit] :
% 14.25/8.06 ( ( bNF_We2091578341578594809t_unit @ R2 )
% 14.25/8.06 => ( ( ord_le3507040750410214029t_unit @ B4 @ ( field_Product_unit @ R2 ) )
% 14.25/8.06 => ( ( B4 != bot_bo3957492148770167129t_unit )
% 14.25/8.06 => ( bNF_We909621271159000003t_unit @ R2 @ B4 @ ( bNF_We7185707985517244377t_unit @ R2 @ B4 ) ) ) ) ) ).
% 14.25/8.06
% 14.25/8.06 thf(fact_3032_wo__rel_Ominim__isMinim,axiom,
% 14.25/8.06 ! [R2: set_Product_prod_o_o,B4: set_o] :
% 14.25/8.06 ( ( bNF_We6437023980930612612_rel_o @ R2 )
% 14.25/8.06 => ( ( ord_less_eq_set_o @ B4 @ ( field_o @ R2 ) )
% 14.25/8.06 => ( ( B4 != bot_bot_set_o )
% 14.25/8.06 => ( bNF_We4647864369647134286inim_o @ R2 @ B4 @ ( bNF_We537455116209922916inim_o @ R2 @ B4 ) ) ) ) ) ).
% 14.25/8.06
% 14.25/8.06 thf(fact_3033_wo__rel_Ominim__isMinim,axiom,
% 14.25/8.06 ! [R2: set_Pr958786334691620121nt_int,B4: set_int] :
% 14.25/8.06 ( ( bNF_We3815749466139970368el_int @ R2 )
% 14.25/8.06 => ( ( ord_less_eq_set_int @ B4 @ ( field_int @ R2 ) )
% 14.25/8.06 => ( ( B4 != bot_bot_set_int )
% 14.25/8.06 => ( bNF_We6511469253196982518im_int @ R2 @ B4 @ ( bNF_We3272633801743244640im_int @ R2 @ B4 ) ) ) ) ) ).
% 14.25/8.06
% 14.25/8.06 thf(fact_3034_wo__rel_Ominim__isMinim,axiom,
% 14.25/8.06 ! [R2: set_Pr2161125870931222855_nat_o,B4: set_Pr4532377907799695533_nat_o] :
% 14.25/8.06 ( ( bNF_We740377018556551609_nat_o @ R2 )
% 14.25/8.06 => ( ( ord_le2965882846123202637_nat_o @ B4 @ ( field_81821745747897882_nat_o @ R2 ) )
% 14.25/8.06 => ( ( B4 != bot_bo7824918357723582233_nat_o )
% 14.25/8.06 => ( bNF_We5608535359669600131_nat_o @ R2 @ B4 @ ( bNF_We8236795211600914329_nat_o @ R2 @ B4 ) ) ) ) ) ).
% 14.25/8.06
% 14.25/8.06 thf(fact_3035_wo__rel_Ominim__isMinim,axiom,
% 14.25/8.06 ! [R2: set_Pr1261947904930325089at_nat,B4: set_nat] :
% 14.25/8.06 ( ( bNF_We3818239936649020644el_nat @ R2 )
% 14.25/8.06 => ( ( ord_less_eq_set_nat @ B4 @ ( field_nat @ R2 ) )
% 14.25/8.06 => ( ( B4 != bot_bot_set_nat )
% 14.25/8.06 => ( bNF_We6513959723706032794im_nat @ R2 @ B4 @ ( bNF_We3275124272252294916im_nat @ R2 @ B4 ) ) ) ) ) ).
% 14.25/8.06
% 14.25/8.06 thf(fact_3036_wo__rel_OisMinim__def,axiom,
% 14.25/8.06 ! [R2: set_Pr2161125870931222855_nat_o,A: set_Pr4532377907799695533_nat_o,B: produc3658429121746597890et_nat > $o] :
% 14.25/8.06 ( ( bNF_We740377018556551609_nat_o @ R2 )
% 14.25/8.06 => ( ( bNF_We5608535359669600131_nat_o @ R2 @ A @ B )
% 14.25/8.06 = ( ( member6576561426505652726_nat_o @ B @ A )
% 14.25/8.06 & ! [X2: produc3658429121746597890et_nat > $o] :
% 14.25/8.06 ( ( member6576561426505652726_nat_o @ X2 @ A )
% 14.25/8.06 => ( member8781333585448626064_nat_o @ ( produc7368190662567826135_nat_o @ B @ X2 ) @ R2 ) ) ) ) ) ).
% 14.25/8.06
% 14.25/8.06 thf(fact_3037_wo__rel_OisMinim__def,axiom,
% 14.25/8.06 ! [R2: set_Product_prod_o_o,A: set_o,B: $o] :
% 14.25/8.06 ( ( bNF_We6437023980930612612_rel_o @ R2 )
% 14.25/8.06 => ( ( bNF_We4647864369647134286inim_o @ R2 @ A @ B )
% 14.25/8.06 = ( ( member_o @ B @ A )
% 14.25/8.06 & ! [X2: $o] :
% 14.25/8.06 ( ( member_o @ X2 @ A )
% 14.25/8.06 => ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ B @ X2 ) @ R2 ) ) ) ) ) ).
% 14.25/8.06
% 14.25/8.06 thf(fact_3038_wo__rel_OisMinim__def,axiom,
% 14.25/8.06 ! [R2: set_Pr5094982260447487303t_unit,A: set_Product_unit,B: product_unit] :
% 14.25/8.06 ( ( bNF_We2091578341578594809t_unit @ R2 )
% 14.25/8.06 => ( ( bNF_We909621271159000003t_unit @ R2 @ A @ B )
% 14.25/8.06 = ( ( member_Product_unit @ B @ A )
% 14.25/8.06 & ! [X2: product_unit] :
% 14.25/8.06 ( ( member_Product_unit @ X2 @ A )
% 14.25/8.06 => ( member7821138191923116944t_unit @ ( produc5692694829384537815t_unit @ B @ X2 ) @ R2 ) ) ) ) ) ).
% 14.25/8.06
% 14.25/8.06 thf(fact_3039_wo__rel_OisMinim__def,axiom,
% 14.25/8.06 ! [R2: set_Pr1261947904930325089at_nat,A: set_nat,B: nat] :
% 14.25/8.06 ( ( bNF_We3818239936649020644el_nat @ R2 )
% 14.25/8.06 => ( ( bNF_We6513959723706032794im_nat @ R2 @ A @ B )
% 14.25/8.06 = ( ( member_nat @ B @ A )
% 14.25/8.06 & ! [X2: nat] :
% 14.25/8.06 ( ( member_nat @ X2 @ A )
% 14.25/8.06 => ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ B @ X2 ) @ R2 ) ) ) ) ) ).
% 14.25/8.06
% 14.25/8.06 thf(fact_3040_wo__rel_OisMinim__def,axiom,
% 14.25/8.06 ! [R2: set_Pr958786334691620121nt_int,A: set_int,B: int] :
% 14.25/8.06 ( ( bNF_We3815749466139970368el_int @ R2 )
% 14.25/8.06 => ( ( bNF_We6511469253196982518im_int @ R2 @ A @ B )
% 14.25/8.06 = ( ( member_int @ B @ A )
% 14.25/8.06 & ! [X2: int] :
% 14.25/8.06 ( ( member_int @ X2 @ A )
% 14.25/8.06 => ( member5262025264175285858nt_int @ ( product_Pair_int_int @ B @ X2 ) @ R2 ) ) ) ) ) ).
% 14.25/8.06
% 14.25/8.06 thf(fact_3041_wo__rel_Omax2__greater,axiom,
% 14.25/8.06 ! [R2: set_Pr2161125870931222855_nat_o,A2: produc3658429121746597890et_nat > $o,B: produc3658429121746597890et_nat > $o] :
% 14.25/8.06 ( ( bNF_We740377018556551609_nat_o @ R2 )
% 14.25/8.06 => ( ( member6576561426505652726_nat_o @ A2 @ ( field_81821745747897882_nat_o @ R2 ) )
% 14.25/8.06 => ( ( member6576561426505652726_nat_o @ B @ ( field_81821745747897882_nat_o @ R2 ) )
% 14.25/8.06 => ( ( member8781333585448626064_nat_o @ ( produc7368190662567826135_nat_o @ A2 @ ( bNF_We6400040844883348763_nat_o @ R2 @ A2 @ B ) ) @ R2 )
% 14.25/8.06 & ( member8781333585448626064_nat_o @ ( produc7368190662567826135_nat_o @ B @ ( bNF_We6400040844883348763_nat_o @ R2 @ A2 @ B ) ) @ R2 ) ) ) ) ) ).
% 14.25/8.06
% 14.25/8.06 thf(fact_3042_wo__rel_Omax2__greater,axiom,
% 14.25/8.06 ! [R2: set_Product_prod_o_o,A2: $o,B: $o] :
% 14.25/8.06 ( ( bNF_We6437023980930612612_rel_o @ R2 )
% 14.25/8.06 => ( ( member_o @ A2 @ ( field_o @ R2 ) )
% 14.25/8.06 => ( ( member_o @ B @ ( field_o @ R2 ) )
% 14.25/8.06 => ( ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ A2 @ ( bNF_We3104086342435940326max2_o @ R2 @ A2 @ B ) ) @ R2 )
% 14.25/8.06 & ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ B @ ( bNF_We3104086342435940326max2_o @ R2 @ A2 @ B ) ) @ R2 ) ) ) ) ) ).
% 14.25/8.06
% 14.25/8.06 thf(fact_3043_wo__rel_Omax2__greater,axiom,
% 14.25/8.06 ! [R2: set_Pr5094982260447487303t_unit,A2: product_unit,B: product_unit] :
% 14.25/8.06 ( ( bNF_We2091578341578594809t_unit @ R2 )
% 14.25/8.06 => ( ( member_Product_unit @ A2 @ ( field_Product_unit @ R2 ) )
% 14.25/8.06 => ( ( member_Product_unit @ B @ ( field_Product_unit @ R2 ) )
% 14.25/8.06 => ( ( member7821138191923116944t_unit @ ( produc5692694829384537815t_unit @ A2 @ ( bNF_We7204996500647184731t_unit @ R2 @ A2 @ B ) ) @ R2 )
% 14.25/8.06 & ( member7821138191923116944t_unit @ ( produc5692694829384537815t_unit @ B @ ( bNF_We7204996500647184731t_unit @ R2 @ A2 @ B ) ) @ R2 ) ) ) ) ) ).
% 14.25/8.06
% 14.25/8.06 thf(fact_3044_wo__rel_Omax2__greater,axiom,
% 14.25/8.06 ! [R2: set_Pr1261947904930325089at_nat,A2: nat,B: nat] :
% 14.25/8.06 ( ( bNF_We3818239936649020644el_nat @ R2 )
% 14.25/8.06 => ( ( member_nat @ A2 @ ( field_nat @ R2 ) )
% 14.25/8.06 => ( ( member_nat @ B @ ( field_nat @ R2 ) )
% 14.25/8.06 => ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A2 @ ( bNF_We2380939065524664066x2_nat @ R2 @ A2 @ B ) ) @ R2 )
% 14.25/8.06 & ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ B @ ( bNF_We2380939065524664066x2_nat @ R2 @ A2 @ B ) ) @ R2 ) ) ) ) ) ).
% 14.25/8.06
% 14.25/8.06 thf(fact_3045_wo__rel_Omax2__greater,axiom,
% 14.25/8.06 ! [R2: set_Pr958786334691620121nt_int,A2: int,B: int] :
% 14.25/8.06 ( ( bNF_We3815749466139970368el_int @ R2 )
% 14.25/8.06 => ( ( member_int @ A2 @ ( field_int @ R2 ) )
% 14.25/8.06 => ( ( member_int @ B @ ( field_int @ R2 ) )
% 14.25/8.06 => ( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ A2 @ ( bNF_We2378448595015613790x2_int @ R2 @ A2 @ B ) ) @ R2 )
% 14.25/8.06 & ( member5262025264175285858nt_int @ ( product_Pair_int_int @ B @ ( bNF_We2378448595015613790x2_int @ R2 @ A2 @ B ) ) @ R2 ) ) ) ) ) ).
% 14.25/8.06
% 14.25/8.06 thf(fact_3046_wo__rel_Omax2__equals2,axiom,
% 14.25/8.06 ! [R2: set_Pr2161125870931222855_nat_o,A2: produc3658429121746597890et_nat > $o,B: produc3658429121746597890et_nat > $o] :
% 14.25/8.06 ( ( bNF_We740377018556551609_nat_o @ R2 )
% 14.25/8.06 => ( ( member6576561426505652726_nat_o @ A2 @ ( field_81821745747897882_nat_o @ R2 ) )
% 14.25/8.06 => ( ( member6576561426505652726_nat_o @ B @ ( field_81821745747897882_nat_o @ R2 ) )
% 14.25/8.06 => ( ( ( bNF_We6400040844883348763_nat_o @ R2 @ A2 @ B )
% 14.25/8.06 = B )
% 14.25/8.06 = ( member8781333585448626064_nat_o @ ( produc7368190662567826135_nat_o @ A2 @ B ) @ R2 ) ) ) ) ) ).
% 14.25/8.06
% 14.25/8.06 thf(fact_3047_wo__rel_Omax2__equals2,axiom,
% 14.25/8.06 ! [R2: set_Product_prod_o_o,A2: $o,B: $o] :
% 14.25/8.06 ( ( bNF_We6437023980930612612_rel_o @ R2 )
% 14.25/8.06 => ( ( member_o @ A2 @ ( field_o @ R2 ) )
% 14.25/8.06 => ( ( member_o @ B @ ( field_o @ R2 ) )
% 14.25/8.06 => ( ( ( bNF_We3104086342435940326max2_o @ R2 @ A2 @ B )
% 14.25/8.06 = B )
% 14.25/8.06 = ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ A2 @ B ) @ R2 ) ) ) ) ) ).
% 14.25/8.06
% 14.25/8.06 thf(fact_3048_wo__rel_Omax2__equals2,axiom,
% 14.25/8.06 ! [R2: set_Pr5094982260447487303t_unit,A2: product_unit,B: product_unit] :
% 14.25/8.06 ( ( bNF_We2091578341578594809t_unit @ R2 )
% 14.25/8.06 => ( ( member_Product_unit @ A2 @ ( field_Product_unit @ R2 ) )
% 14.25/8.06 => ( ( member_Product_unit @ B @ ( field_Product_unit @ R2 ) )
% 14.25/8.06 => ( ( ( bNF_We7204996500647184731t_unit @ R2 @ A2 @ B )
% 14.25/8.06 = B )
% 14.25/8.06 = ( member7821138191923116944t_unit @ ( produc5692694829384537815t_unit @ A2 @ B ) @ R2 ) ) ) ) ) ).
% 14.25/8.06
% 14.25/8.06 thf(fact_3049_wo__rel_Omax2__equals2,axiom,
% 14.25/8.06 ! [R2: set_Pr1261947904930325089at_nat,A2: nat,B: nat] :
% 14.25/8.06 ( ( bNF_We3818239936649020644el_nat @ R2 )
% 14.25/8.06 => ( ( member_nat @ A2 @ ( field_nat @ R2 ) )
% 14.25/8.06 => ( ( member_nat @ B @ ( field_nat @ R2 ) )
% 14.25/8.06 => ( ( ( bNF_We2380939065524664066x2_nat @ R2 @ A2 @ B )
% 14.25/8.06 = B )
% 14.25/8.06 = ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A2 @ B ) @ R2 ) ) ) ) ) ).
% 14.25/8.06
% 14.25/8.06 thf(fact_3050_wo__rel_Omax2__equals2,axiom,
% 14.25/8.06 ! [R2: set_Pr958786334691620121nt_int,A2: int,B: int] :
% 14.25/8.06 ( ( bNF_We3815749466139970368el_int @ R2 )
% 14.25/8.06 => ( ( member_int @ A2 @ ( field_int @ R2 ) )
% 14.25/8.06 => ( ( member_int @ B @ ( field_int @ R2 ) )
% 14.25/8.06 => ( ( ( bNF_We2378448595015613790x2_int @ R2 @ A2 @ B )
% 14.25/8.06 = B )
% 14.25/8.06 = ( member5262025264175285858nt_int @ ( product_Pair_int_int @ A2 @ B ) @ R2 ) ) ) ) ) ).
% 14.25/8.06
% 14.25/8.06 thf(fact_3051_wo__rel_Omax2__equals1,axiom,
% 14.25/8.06 ! [R2: set_Pr2161125870931222855_nat_o,A2: produc3658429121746597890et_nat > $o,B: produc3658429121746597890et_nat > $o] :
% 14.25/8.06 ( ( bNF_We740377018556551609_nat_o @ R2 )
% 14.25/8.06 => ( ( member6576561426505652726_nat_o @ A2 @ ( field_81821745747897882_nat_o @ R2 ) )
% 14.25/8.06 => ( ( member6576561426505652726_nat_o @ B @ ( field_81821745747897882_nat_o @ R2 ) )
% 14.25/8.06 => ( ( ( bNF_We6400040844883348763_nat_o @ R2 @ A2 @ B )
% 14.25/8.06 = A2 )
% 14.25/8.06 = ( member8781333585448626064_nat_o @ ( produc7368190662567826135_nat_o @ B @ A2 ) @ R2 ) ) ) ) ) ).
% 14.25/8.06
% 14.25/8.06 thf(fact_3052_wo__rel_Omax2__equals1,axiom,
% 14.25/8.06 ! [R2: set_Product_prod_o_o,A2: $o,B: $o] :
% 14.25/8.06 ( ( bNF_We6437023980930612612_rel_o @ R2 )
% 14.25/8.06 => ( ( member_o @ A2 @ ( field_o @ R2 ) )
% 14.25/8.06 => ( ( member_o @ B @ ( field_o @ R2 ) )
% 14.25/8.06 => ( ( ( bNF_We3104086342435940326max2_o @ R2 @ A2 @ B )
% 14.25/8.06 = A2 )
% 14.25/8.06 = ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ B @ A2 ) @ R2 ) ) ) ) ) ).
% 14.25/8.06
% 14.25/8.06 thf(fact_3053_wo__rel_Omax2__equals1,axiom,
% 14.25/8.06 ! [R2: set_Pr5094982260447487303t_unit,A2: product_unit,B: product_unit] :
% 14.25/8.06 ( ( bNF_We2091578341578594809t_unit @ R2 )
% 14.25/8.06 => ( ( member_Product_unit @ A2 @ ( field_Product_unit @ R2 ) )
% 14.25/8.06 => ( ( member_Product_unit @ B @ ( field_Product_unit @ R2 ) )
% 14.25/8.06 => ( ( ( bNF_We7204996500647184731t_unit @ R2 @ A2 @ B )
% 14.25/8.06 = A2 )
% 14.25/8.06 = ( member7821138191923116944t_unit @ ( produc5692694829384537815t_unit @ B @ A2 ) @ R2 ) ) ) ) ) ).
% 14.25/8.06
% 14.25/8.06 thf(fact_3054_wo__rel_Omax2__equals1,axiom,
% 14.25/8.06 ! [R2: set_Pr1261947904930325089at_nat,A2: nat,B: nat] :
% 14.25/8.06 ( ( bNF_We3818239936649020644el_nat @ R2 )
% 14.25/8.06 => ( ( member_nat @ A2 @ ( field_nat @ R2 ) )
% 14.25/8.06 => ( ( member_nat @ B @ ( field_nat @ R2 ) )
% 14.25/8.06 => ( ( ( bNF_We2380939065524664066x2_nat @ R2 @ A2 @ B )
% 14.25/8.06 = A2 )
% 14.25/8.06 = ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ B @ A2 ) @ R2 ) ) ) ) ) ).
% 14.25/8.06
% 14.25/8.06 thf(fact_3055_wo__rel_Omax2__equals1,axiom,
% 14.25/8.06 ! [R2: set_Pr958786334691620121nt_int,A2: int,B: int] :
% 14.25/8.06 ( ( bNF_We3815749466139970368el_int @ R2 )
% 14.25/8.06 => ( ( member_int @ A2 @ ( field_int @ R2 ) )
% 14.25/8.06 => ( ( member_int @ B @ ( field_int @ R2 ) )
% 14.25/8.06 => ( ( ( bNF_We2378448595015613790x2_int @ R2 @ A2 @ B )
% 14.25/8.06 = A2 )
% 14.25/8.06 = ( member5262025264175285858nt_int @ ( product_Pair_int_int @ B @ A2 ) @ R2 ) ) ) ) ) ).
% 14.25/8.06
% 14.25/8.06 thf(fact_3056_wo__rel_Omax2__among,axiom,
% 14.25/8.06 ! [R2: set_Pr5094982260447487303t_unit,A2: product_unit,B: product_unit] :
% 14.25/8.06 ( ( bNF_We2091578341578594809t_unit @ R2 )
% 14.25/8.06 => ( ( member_Product_unit @ A2 @ ( field_Product_unit @ R2 ) )
% 14.25/8.06 => ( ( member_Product_unit @ B @ ( field_Product_unit @ R2 ) )
% 14.25/8.06 => ( member_Product_unit @ ( bNF_We7204996500647184731t_unit @ R2 @ A2 @ B ) @ ( insert_Product_unit @ A2 @ ( insert_Product_unit @ B @ bot_bo3957492148770167129t_unit ) ) ) ) ) ) ).
% 14.25/8.06
% 14.25/8.06 thf(fact_3057_wo__rel_Omax2__among,axiom,
% 14.25/8.06 ! [R2: set_Product_prod_o_o,A2: $o,B: $o] :
% 14.25/8.06 ( ( bNF_We6437023980930612612_rel_o @ R2 )
% 14.25/8.06 => ( ( member_o @ A2 @ ( field_o @ R2 ) )
% 14.25/8.06 => ( ( member_o @ B @ ( field_o @ R2 ) )
% 14.25/8.06 => ( member_o @ ( bNF_We3104086342435940326max2_o @ R2 @ A2 @ B ) @ ( insert_o @ A2 @ ( insert_o @ B @ bot_bot_set_o ) ) ) ) ) ) ).
% 14.25/8.06
% 14.25/8.06 thf(fact_3058_wo__rel_Omax2__among,axiom,
% 14.25/8.06 ! [R2: set_Pr1261947904930325089at_nat,A2: nat,B: nat] :
% 14.25/8.06 ( ( bNF_We3818239936649020644el_nat @ R2 )
% 14.25/8.06 => ( ( member_nat @ A2 @ ( field_nat @ R2 ) )
% 14.25/8.06 => ( ( member_nat @ B @ ( field_nat @ R2 ) )
% 14.25/8.06 => ( member_nat @ ( bNF_We2380939065524664066x2_nat @ R2 @ A2 @ B ) @ ( insert_nat @ A2 @ ( insert_nat @ B @ bot_bot_set_nat ) ) ) ) ) ) ).
% 14.25/8.06
% 14.25/8.06 thf(fact_3059_wo__rel_Omax2__among,axiom,
% 14.25/8.06 ! [R2: set_Pr958786334691620121nt_int,A2: int,B: int] :
% 14.25/8.06 ( ( bNF_We3815749466139970368el_int @ R2 )
% 14.25/8.06 => ( ( member_int @ A2 @ ( field_int @ R2 ) )
% 14.25/8.06 => ( ( member_int @ B @ ( field_int @ R2 ) )
% 14.25/8.06 => ( member_int @ ( bNF_We2378448595015613790x2_int @ R2 @ A2 @ B ) @ ( insert_int @ A2 @ ( insert_int @ B @ bot_bot_set_int ) ) ) ) ) ) ).
% 14.25/8.06
% 14.25/8.06 thf(fact_3060_wo__rel_Omax2__among,axiom,
% 14.25/8.06 ! [R2: set_Pr2161125870931222855_nat_o,A2: produc3658429121746597890et_nat > $o,B: produc3658429121746597890et_nat > $o] :
% 14.25/8.06 ( ( bNF_We740377018556551609_nat_o @ R2 )
% 14.25/8.06 => ( ( member6576561426505652726_nat_o @ A2 @ ( field_81821745747897882_nat_o @ R2 ) )
% 14.25/8.06 => ( ( member6576561426505652726_nat_o @ B @ ( field_81821745747897882_nat_o @ R2 ) )
% 14.25/8.06 => ( member6576561426505652726_nat_o @ ( bNF_We6400040844883348763_nat_o @ R2 @ A2 @ B ) @ ( insert5175938949040314269_nat_o @ A2 @ ( insert5175938949040314269_nat_o @ B @ bot_bo7824918357723582233_nat_o ) ) ) ) ) ) ).
% 14.25/8.06
% 14.25/8.06 thf(fact_3061_chains__extend,axiom,
% 14.25/8.06 ! [C: set_set_nat,S: set_set_nat,Z: set_nat] :
% 14.25/8.06 ( ( member_set_set_nat @ C @ ( chains_nat2 @ S ) )
% 14.25/8.06 => ( ( member_set_nat @ Z @ S )
% 14.25/8.06 => ( ! [X3: set_nat] :
% 14.25/8.06 ( ( member_set_nat @ X3 @ C )
% 14.25/8.06 => ( ord_less_eq_set_nat @ X3 @ Z ) )
% 14.25/8.06 => ( member_set_set_nat @ ( sup_sup_set_set_nat @ ( insert_set_nat @ Z @ bot_bot_set_set_nat ) @ C ) @ ( chains_nat2 @ S ) ) ) ) ) ).
% 14.25/8.06
% 14.25/8.06 thf(fact_3062_Zorns__po__lemma,axiom,
% 14.25/8.06 ! [R2: set_Pr1261947904930325089at_nat] :
% 14.25/8.06 ( ( order_5251275573222108571on_nat @ ( field_nat @ R2 ) @ R2 )
% 14.25/8.06 => ( ! [C6: set_nat] :
% 14.25/8.06 ( ( member_set_nat @ C6 @ ( chains_nat @ R2 ) )
% 14.25/8.06 => ? [X6: nat] :
% 14.25/8.06 ( ( member_nat @ X6 @ ( field_nat @ R2 ) )
% 14.25/8.06 & ! [Xa3: nat] :
% 14.25/8.06 ( ( member_nat @ Xa3 @ C6 )
% 14.25/8.06 => ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ Xa3 @ X6 ) @ R2 ) ) ) )
% 14.25/8.06 => ? [X3: nat] :
% 14.25/8.06 ( ( member_nat @ X3 @ ( field_nat @ R2 ) )
% 14.25/8.06 & ! [Xa2: nat] :
% 14.25/8.06 ( ( member_nat @ Xa2 @ ( field_nat @ R2 ) )
% 14.25/8.06 => ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X3 @ Xa2 ) @ R2 )
% 14.25/8.06 => ( Xa2 = X3 ) ) ) ) ) ) ).
% 14.25/8.06
% 14.25/8.06 thf(fact_3063_Zorns__po__lemma,axiom,
% 14.25/8.06 ! [R2: set_Pr958786334691620121nt_int] :
% 14.25/8.06 ( ( order_5248785102713058295on_int @ ( field_int @ R2 ) @ R2 )
% 14.25/8.06 => ( ! [C6: set_int] :
% 14.25/8.06 ( ( member_set_int @ C6 @ ( chains_int @ R2 ) )
% 14.25/8.06 => ? [X6: int] :
% 14.25/8.06 ( ( member_int @ X6 @ ( field_int @ R2 ) )
% 14.25/8.06 & ! [Xa3: int] :
% 14.25/8.06 ( ( member_int @ Xa3 @ C6 )
% 14.25/8.06 => ( member5262025264175285858nt_int @ ( product_Pair_int_int @ Xa3 @ X6 ) @ R2 ) ) ) )
% 14.25/8.06 => ? [X3: int] :
% 14.25/8.06 ( ( member_int @ X3 @ ( field_int @ R2 ) )
% 14.25/8.06 & ! [Xa2: int] :
% 14.25/8.06 ( ( member_int @ Xa2 @ ( field_int @ R2 ) )
% 14.25/8.06 => ( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ X3 @ Xa2 ) @ R2 )
% 14.25/8.06 => ( Xa2 = X3 ) ) ) ) ) ) ).
% 14.25/8.06
% 14.25/8.06 thf(fact_3064_bsqr__max2,axiom,
% 14.25/8.06 ! [R2: set_Pr1261947904930325089at_nat,A1: nat,A22: nat,B1: nat,B22: nat] :
% 14.25/8.06 ( ( order_2888998067076097458on_nat @ ( field_nat @ R2 ) @ R2 )
% 14.25/8.06 => ( ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ ( product_Pair_nat_nat @ A1 @ A22 ) @ ( product_Pair_nat_nat @ B1 @ B22 ) ) @ ( bNF_We1732890022510290291qr_nat @ R2 ) )
% 14.25/8.06 => ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ ( bNF_We2380939065524664066x2_nat @ R2 @ A1 @ A22 ) @ ( bNF_We2380939065524664066x2_nat @ R2 @ B1 @ B22 ) ) @ R2 ) ) ) ).
% 14.25/8.06
% 14.25/8.06 thf(fact_3065_bsqr__max2,axiom,
% 14.25/8.06 ! [R2: set_Pr958786334691620121nt_int,A1: int,A22: int,B1: int,B22: int] :
% 14.25/8.06 ( ( order_2886507596567047182on_int @ ( field_int @ R2 ) @ R2 )
% 14.25/8.06 => ( ( member8566619992076573584nt_int @ ( produc3646306378393792727nt_int @ ( product_Pair_int_int @ A1 @ A22 ) @ ( product_Pair_int_int @ B1 @ B22 ) ) @ ( bNF_We1730399552001240015qr_int @ R2 ) )
% 14.25/8.06 => ( member5262025264175285858nt_int @ ( product_Pair_int_int @ ( bNF_We2378448595015613790x2_int @ R2 @ A1 @ A22 ) @ ( bNF_We2378448595015613790x2_int @ R2 @ B1 @ B22 ) ) @ R2 ) ) ) ).
% 14.25/8.06
% 14.25/8.06 thf(fact_3066_wf__Union,axiom,
% 14.25/8.06 ! [R3: set_se8994758365886191399t_unit] :
% 14.25/8.06 ( ! [X3: set_Pr5094982260447487303t_unit] :
% 14.25/8.06 ( ( member1248447801810324848t_unit @ X3 @ R3 )
% 14.25/8.06 => ( wf_Product_unit @ X3 ) )
% 14.25/8.06 => ( ! [X3: set_Pr5094982260447487303t_unit] :
% 14.25/8.06 ( ( member1248447801810324848t_unit @ X3 @ R3 )
% 14.25/8.06 => ! [Xa3: set_Pr5094982260447487303t_unit] :
% 14.25/8.06 ( ( member1248447801810324848t_unit @ Xa3 @ R3 )
% 14.25/8.06 => ( ( X3 != Xa3 )
% 14.25/8.06 => ( ( inf_in4660618365625256667t_unit @ ( domain4292024440186670924t_unit @ X3 ) @ ( range_6617724647562260917t_unit @ Xa3 ) )
% 14.25/8.06 = bot_bo3957492148770167129t_unit ) ) ) )
% 14.25/8.06 => ( wf_Product_unit @ ( comple2798140142350441396t_unit @ R3 ) ) ) ) ).
% 14.25/8.06
% 14.25/8.06 thf(fact_3067_wf__Union,axiom,
% 14.25/8.06 ! [R3: set_se491199571785101031od_o_o] :
% 14.25/8.06 ( ! [X3: set_Product_prod_o_o] :
% 14.25/8.06 ( ( member3157015407008874800od_o_o @ X3 @ R3 )
% 14.25/8.06 => ( wf_o @ X3 ) )
% 14.25/8.06 => ( ! [X3: set_Product_prod_o_o] :
% 14.25/8.06 ( ( member3157015407008874800od_o_o @ X3 @ R3 )
% 14.25/8.06 => ! [Xa3: set_Product_prod_o_o] :
% 14.25/8.06 ( ( member3157015407008874800od_o_o @ Xa3 @ R3 )
% 14.25/8.06 => ( ( X3 != Xa3 )
% 14.25/8.06 => ( ( inf_inf_set_o @ ( domain_o_o @ X3 ) @ ( range_o_o @ Xa3 ) )
% 14.25/8.06 = bot_bot_set_o ) ) ) )
% 14.25/8.06 => ( wf_o @ ( comple2567785453016644468od_o_o @ R3 ) ) ) ) ).
% 14.25/8.06
% 14.25/8.06 thf(fact_3068_wf__Union,axiom,
% 14.25/8.06 ! [R3: set_se7855581050983116737at_nat] :
% 14.25/8.06 ( ! [X3: set_Pr1261947904930325089at_nat] :
% 14.25/8.06 ( ( member2643936169264416010at_nat @ X3 @ R3 )
% 14.25/8.06 => ( wf_nat @ X3 ) )
% 14.25/8.06 => ( ! [X3: set_Pr1261947904930325089at_nat] :
% 14.25/8.06 ( ( member2643936169264416010at_nat @ X3 @ R3 )
% 14.25/8.06 => ! [Xa3: set_Pr1261947904930325089at_nat] :
% 14.25/8.06 ( ( member2643936169264416010at_nat @ Xa3 @ R3 )
% 14.25/8.06 => ( ( X3 != Xa3 )
% 14.25/8.06 => ( ( inf_inf_set_nat @ ( domain_nat_nat @ X3 ) @ ( range_nat_nat @ Xa3 ) )
% 14.25/8.06 = bot_bot_set_nat ) ) ) )
% 14.25/8.06 => ( wf_nat @ ( comple5685304695842803022at_nat @ R3 ) ) ) ) ).
% 14.25/8.06
% 14.25/8.06 thf(fact_3069_wf__Union,axiom,
% 14.25/8.06 ! [R3: set_se6260736226359567993nt_int] :
% 14.25/8.06 ( ! [X3: set_Pr958786334691620121nt_int] :
% 14.25/8.06 ( ( member2340774599025711042nt_int @ X3 @ R3 )
% 14.25/8.06 => ( wf_int @ X3 ) )
% 14.25/8.06 => ( ! [X3: set_Pr958786334691620121nt_int] :
% 14.25/8.06 ( ( member2340774599025711042nt_int @ X3 @ R3 )
% 14.25/8.06 => ! [Xa3: set_Pr958786334691620121nt_int] :
% 14.25/8.06 ( ( member2340774599025711042nt_int @ Xa3 @ R3 )
% 14.25/8.06 => ( ( X3 != Xa3 )
% 14.25/8.06 => ( ( inf_inf_set_int @ ( domain_int_int @ X3 ) @ ( range_int_int @ Xa3 ) )
% 14.25/8.06 = bot_bot_set_int ) ) ) )
% 14.25/8.06 => ( wf_int @ ( comple5382143125604098054nt_int @ R3 ) ) ) ) ).
% 14.25/8.06
% 14.25/8.06 thf(fact_3070_wf__Union,axiom,
% 14.25/8.06 ! [R3: set_se7663487083892625191_nat_o] :
% 14.25/8.06 ( ! [X3: set_Pr2161125870931222855_nat_o] :
% 14.25/8.06 ( ( member5475208728337331568_nat_o @ X3 @ R3 )
% 14.25/8.06 => ( wf_Pro8787509627421085110_nat_o @ X3 ) )
% 14.25/8.06 => ( ! [X3: set_Pr2161125870931222855_nat_o] :
% 14.25/8.06 ( ( member5475208728337331568_nat_o @ X3 @ R3 )
% 14.25/8.06 => ! [Xa3: set_Pr2161125870931222855_nat_o] :
% 14.25/8.06 ( ( member5475208728337331568_nat_o @ Xa3 @ R3 )
% 14.25/8.06 => ( ( X3 != Xa3 )
% 14.25/8.06 => ( ( inf_in1906310914598751387_nat_o @ ( domain4881241903163524940_nat_o @ X3 ) @ ( range_5289318983786108341_nat_o @ Xa3 ) )
% 14.25/8.06 = bot_bo7824918357723582233_nat_o ) ) ) )
% 14.25/8.06 => ( wf_Pro8787509627421085110_nat_o @ ( comple8164645552537203636_nat_o @ R3 ) ) ) ) ).
% 14.25/8.06
% 14.25/8.06 thf(fact_3071_Pow__singleton__iff,axiom,
% 14.25/8.06 ! [X4: set_Product_unit,Y6: set_Product_unit] :
% 14.25/8.06 ( ( ( pow_Product_unit @ X4 )
% 14.25/8.06 = ( insert8986028188087627965t_unit @ Y6 @ bot_bo6284674557190688313t_unit ) )
% 14.25/8.06 = ( ( X4 = bot_bo3957492148770167129t_unit )
% 14.25/8.06 & ( Y6 = bot_bo3957492148770167129t_unit ) ) ) ).
% 14.25/8.06
% 14.25/8.06 thf(fact_3072_Pow__singleton__iff,axiom,
% 14.25/8.06 ! [X4: set_o,Y6: set_o] :
% 14.25/8.06 ( ( ( pow_o @ X4 )
% 14.25/8.06 = ( insert_set_o @ Y6 @ bot_bot_set_set_o ) )
% 14.25/8.06 = ( ( X4 = bot_bot_set_o )
% 14.25/8.06 & ( Y6 = bot_bot_set_o ) ) ) ).
% 14.25/8.06
% 14.25/8.06 thf(fact_3073_Pow__singleton__iff,axiom,
% 14.25/8.06 ! [X4: set_nat,Y6: set_nat] :
% 14.25/8.06 ( ( ( pow_nat @ X4 )
% 14.25/8.06 = ( insert_set_nat @ Y6 @ bot_bot_set_set_nat ) )
% 14.25/8.06 = ( ( X4 = bot_bot_set_nat )
% 14.25/8.06 & ( Y6 = bot_bot_set_nat ) ) ) ).
% 14.25/8.06
% 14.25/8.06 thf(fact_3074_Pow__singleton__iff,axiom,
% 14.25/8.06 ! [X4: set_int,Y6: set_int] :
% 14.25/8.06 ( ( ( pow_int @ X4 )
% 14.25/8.06 = ( insert_set_int @ Y6 @ bot_bot_set_set_int ) )
% 14.25/8.06 = ( ( X4 = bot_bot_set_int )
% 14.25/8.06 & ( Y6 = bot_bot_set_int ) ) ) ).
% 14.25/8.06
% 14.25/8.06 thf(fact_3075_Pow__singleton__iff,axiom,
% 14.25/8.06 ! [X4: set_Pr4532377907799695533_nat_o,Y6: set_Pr4532377907799695533_nat_o] :
% 14.25/8.06 ( ( ( pow_Pr7829143134171777386_nat_o @ X4 )
% 14.25/8.06 = ( insert2507855216428400253_nat_o @ Y6 @ bot_bo3736544561982865913_nat_o ) )
% 14.25/8.06 = ( ( X4 = bot_bo7824918357723582233_nat_o )
% 14.25/8.06 & ( Y6 = bot_bo7824918357723582233_nat_o ) ) ) ).
% 14.25/8.06
% 14.25/8.06 thf(fact_3076_Pow__empty,axiom,
% 14.25/8.06 ( ( pow_Product_unit @ bot_bo3957492148770167129t_unit )
% 14.25/8.06 = ( insert8986028188087627965t_unit @ bot_bo3957492148770167129t_unit @ bot_bo6284674557190688313t_unit ) ) ).
% 14.25/8.06
% 14.25/8.06 thf(fact_3077_Pow__empty,axiom,
% 14.25/8.06 ( ( pow_o @ bot_bot_set_o )
% 14.25/8.06 = ( insert_set_o @ bot_bot_set_o @ bot_bot_set_set_o ) ) ).
% 14.25/8.06
% 14.25/8.06 thf(fact_3078_Pow__empty,axiom,
% 14.25/8.06 ( ( pow_nat @ bot_bot_set_nat )
% 14.25/8.06 = ( insert_set_nat @ bot_bot_set_nat @ bot_bot_set_set_nat ) ) ).
% 14.25/8.06
% 14.25/8.06 thf(fact_3079_Pow__empty,axiom,
% 14.25/8.06 ( ( pow_int @ bot_bot_set_int )
% 14.25/8.06 = ( insert_set_int @ bot_bot_set_int @ bot_bot_set_set_int ) ) ).
% 14.25/8.06
% 14.25/8.06 thf(fact_3080_Pow__empty,axiom,
% 14.25/8.06 ( ( pow_Pr7829143134171777386_nat_o @ bot_bo7824918357723582233_nat_o )
% 14.25/8.06 = ( insert2507855216428400253_nat_o @ bot_bo7824918357723582233_nat_o @ bot_bo3736544561982865913_nat_o ) ) ).
% 14.25/8.06
% 14.25/8.06 thf(fact_3081_rtrancl__mapI,axiom,
% 14.25/8.06 ! [A2: nat,B: nat,E: set_Pr1261947904930325089at_nat,F: nat > nat] :
% 14.25/8.06 ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A2 @ B ) @ ( transi2905341329935302413cl_nat @ E ) )
% 14.25/8.06 => ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ ( F @ A2 ) @ ( F @ B ) ) @ ( transi2905341329935302413cl_nat @ ( image_5168914502847457605at_nat @ ( pairself_nat_nat @ F ) @ E ) ) ) ) ).
% 14.25/8.06
% 14.25/8.06 thf(fact_3082_rtrancl__mapI,axiom,
% 14.25/8.06 ! [A2: nat,B: nat,E: set_Pr1261947904930325089at_nat,F: nat > int] :
% 14.25/8.06 ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A2 @ B ) @ ( transi2905341329935302413cl_nat @ E ) )
% 14.25/8.06 => ( member5262025264175285858nt_int @ ( product_Pair_int_int @ ( F @ A2 ) @ ( F @ B ) ) @ ( transi2902850859426252137cl_int @ ( image_1990417195239315453nt_int @ ( pairself_nat_int @ F ) @ E ) ) ) ) ).
% 14.25/8.06
% 14.25/8.06 thf(fact_3083_rtrancl__mapI,axiom,
% 14.25/8.06 ! [A2: int,B: int,E: set_Pr958786334691620121nt_int,F: int > nat] :
% 14.25/8.06 ( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ A2 @ B ) @ ( transi2902850859426252137cl_int @ E ) )
% 14.25/8.06 => ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ ( F @ A2 ) @ ( F @ B ) ) @ ( transi2905341329935302413cl_nat @ ( image_5831868185956570253at_nat @ ( pairself_int_nat @ F ) @ E ) ) ) ) ).
% 14.25/8.06
% 14.25/8.06 thf(fact_3084_rtrancl__mapI,axiom,
% 14.25/8.06 ! [A2: int,B: int,E: set_Pr958786334691620121nt_int,F: int > int] :
% 14.25/8.06 ( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ A2 @ B ) @ ( transi2902850859426252137cl_int @ E ) )
% 14.25/8.06 => ( member5262025264175285858nt_int @ ( product_Pair_int_int @ ( F @ A2 ) @ ( F @ B ) ) @ ( transi2902850859426252137cl_int @ ( image_2653370878348428101nt_int @ ( pairself_int_int @ F ) @ E ) ) ) ) ).
% 14.25/8.06
% 14.25/8.06 thf(fact_3085_disjoint__image__subset,axiom,
% 14.25/8.06 ! [A11: set_set_nat,F: set_nat > set_nat] :
% 14.25/8.06 ( ( pairwise_set_nat @ disjnt_nat @ A11 )
% 14.25/8.06 => ( ! [X8: set_nat] :
% 14.25/8.06 ( ( member_set_nat @ X8 @ A11 )
% 14.25/8.06 => ( ord_less_eq_set_nat @ ( F @ X8 ) @ X8 ) )
% 14.25/8.06 => ( pairwise_set_nat @ disjnt_nat @ ( image_7916887816326733075et_nat @ F @ A11 ) ) ) ) ).
% 14.25/8.06
% 14.25/8.06 thf(fact_3086_pairself_Opelims,axiom,
% 14.25/8.06 ! [X: nat > nat,Xa: product_prod_nat_nat,Y: product_prod_nat_nat] :
% 14.25/8.06 ( ( ( pairself_nat_nat @ X @ Xa )
% 14.25/8.06 = Y )
% 14.25/8.06 => ( ( accp_P4147487641329543982at_nat @ pairself_rel_nat_nat @ ( produc4237282728380940073at_nat @ X @ Xa ) )
% 14.25/8.06 => ~ ! [A4: nat,B2: nat] :
% 14.25/8.06 ( ( Xa
% 14.25/8.06 = ( product_Pair_nat_nat @ A4 @ B2 ) )
% 14.25/8.06 => ( ( Y
% 14.25/8.06 = ( product_Pair_nat_nat @ ( X @ A4 ) @ ( X @ B2 ) ) )
% 14.25/8.06 => ~ ( accp_P4147487641329543982at_nat @ pairself_rel_nat_nat @ ( produc4237282728380940073at_nat @ X @ ( product_Pair_nat_nat @ A4 @ B2 ) ) ) ) ) ) ) ).
% 14.25/8.06
% 14.25/8.06 thf(fact_3087_pairself_Opelims,axiom,
% 14.25/8.06 ! [X: nat > int,Xa: product_prod_nat_nat,Y: product_prod_int_int] :
% 14.25/8.06 ( ( ( pairself_nat_int @ X @ Xa )
% 14.25/8.06 = Y )
% 14.25/8.06 => ( ( accp_P3050404660685545426at_nat @ pairself_rel_nat_int @ ( produc3140199747736941517at_nat @ X @ Xa ) )
% 14.25/8.06 => ~ ! [A4: nat,B2: nat] :
% 14.25/8.06 ( ( Xa
% 14.25/8.06 = ( product_Pair_nat_nat @ A4 @ B2 ) )
% 14.25/8.06 => ( ( Y
% 14.25/8.06 = ( product_Pair_int_int @ ( X @ A4 ) @ ( X @ B2 ) ) )
% 14.25/8.06 => ~ ( accp_P3050404660685545426at_nat @ pairself_rel_nat_int @ ( produc3140199747736941517at_nat @ X @ ( product_Pair_nat_nat @ A4 @ B2 ) ) ) ) ) ) ) ).
% 14.25/8.06
% 14.25/8.06 thf(fact_3088_pairself_Opelims,axiom,
% 14.25/8.06 ! [X: int > nat,Xa: product_prod_int_int,Y: product_prod_nat_nat] :
% 14.25/8.06 ( ( ( pairself_int_nat @ X @ Xa )
% 14.25/8.06 = Y )
% 14.25/8.06 => ( ( accp_P5604362734843950218nt_int @ pairself_rel_int_nat @ ( produc2818822084525909125nt_int @ X @ Xa ) )
% 14.25/8.06 => ~ ! [A4: int,B2: int] :
% 14.25/8.06 ( ( Xa
% 14.25/8.06 = ( product_Pair_int_int @ A4 @ B2 ) )
% 14.25/8.06 => ( ( Y
% 14.25/8.06 = ( product_Pair_nat_nat @ ( X @ A4 ) @ ( X @ B2 ) ) )
% 14.25/8.06 => ~ ( accp_P5604362734843950218nt_int @ pairself_rel_int_nat @ ( produc2818822084525909125nt_int @ X @ ( product_Pair_int_int @ A4 @ B2 ) ) ) ) ) ) ) ).
% 14.25/8.06
% 14.25/8.06 thf(fact_3089_pairself_Opelims,axiom,
% 14.25/8.06 ! [X: int > int,Xa: product_prod_int_int,Y: product_prod_int_int] :
% 14.25/8.06 ( ( ( pairself_int_int @ X @ Xa )
% 14.25/8.06 = Y )
% 14.25/8.06 => ( ( accp_P4507279754199951662nt_int @ pairself_rel_int_int @ ( produc1721739103881910569nt_int @ X @ Xa ) )
% 14.25/8.06 => ~ ! [A4: int,B2: int] :
% 14.25/8.06 ( ( Xa
% 14.25/8.06 = ( product_Pair_int_int @ A4 @ B2 ) )
% 14.25/8.06 => ( ( Y
% 14.25/8.06 = ( product_Pair_int_int @ ( X @ A4 ) @ ( X @ B2 ) ) )
% 14.25/8.06 => ~ ( accp_P4507279754199951662nt_int @ pairself_rel_int_int @ ( produc1721739103881910569nt_int @ X @ ( product_Pair_int_int @ A4 @ B2 ) ) ) ) ) ) ) ).
% 14.25/8.06
% 14.25/8.06 thf(fact_3090_Pow__iff,axiom,
% 14.25/8.06 ! [A: set_nat,B4: set_nat] :
% 14.25/8.06 ( ( member_set_nat @ A @ ( pow_nat @ B4 ) )
% 14.25/8.06 = ( ord_less_eq_set_nat @ A @ B4 ) ) ).
% 14.25/8.06
% 14.25/8.06 thf(fact_3091_PowI,axiom,
% 14.25/8.06 ! [A: set_nat,B4: set_nat] :
% 14.25/8.06 ( ( ord_less_eq_set_nat @ A @ B4 )
% 14.25/8.06 => ( member_set_nat @ A @ ( pow_nat @ B4 ) ) ) ).
% 14.25/8.06
% 14.25/8.06 thf(fact_3092_Pow__Int__eq,axiom,
% 14.25/8.06 ! [A: set_nat,B4: set_nat] :
% 14.25/8.06 ( ( pow_nat @ ( inf_inf_set_nat @ A @ B4 ) )
% 14.25/8.06 = ( inf_inf_set_set_nat @ ( pow_nat @ A ) @ ( pow_nat @ B4 ) ) ) ).
% 14.25/8.06
% 14.25/8.06 thf(fact_3093_pairwiseI,axiom,
% 14.25/8.06 ! [S: set_Pr4532377907799695533_nat_o,R3: ( produc3658429121746597890et_nat > $o ) > ( produc3658429121746597890et_nat > $o ) > $o] :
% 14.25/8.06 ( ! [X3: produc3658429121746597890et_nat > $o,Y2: produc3658429121746597890et_nat > $o] :
% 14.25/8.06 ( ( member6576561426505652726_nat_o @ X3 @ S )
% 14.25/8.06 => ( ( member6576561426505652726_nat_o @ Y2 @ S )
% 14.25/8.06 => ( ( X3 != Y2 )
% 14.25/8.06 => ( R3 @ X3 @ Y2 ) ) ) )
% 14.25/8.06 => ( pairwi2432143822833250616_nat_o @ R3 @ S ) ) ).
% 14.25/8.06
% 14.25/8.06 thf(fact_3094_pairwiseI,axiom,
% 14.25/8.06 ! [S: set_nat,R3: nat > nat > $o] :
% 14.25/8.06 ( ! [X3: nat,Y2: nat] :
% 14.25/8.06 ( ( member_nat @ X3 @ S )
% 14.25/8.06 => ( ( member_nat @ Y2 @ S )
% 14.25/8.06 => ( ( X3 != Y2 )
% 14.25/8.06 => ( R3 @ X3 @ Y2 ) ) ) )
% 14.25/8.06 => ( pairwise_nat @ R3 @ S ) ) ).
% 14.25/8.06
% 14.25/8.06 thf(fact_3095_pairwiseI,axiom,
% 14.25/8.06 ! [S: set_int,R3: int > int > $o] :
% 14.25/8.06 ( ! [X3: int,Y2: int] :
% 14.25/8.06 ( ( member_int @ X3 @ S )
% 14.25/8.06 => ( ( member_int @ Y2 @ S )
% 14.25/8.06 => ( ( X3 != Y2 )
% 14.25/8.06 => ( R3 @ X3 @ Y2 ) ) ) )
% 14.25/8.06 => ( pairwise_int @ R3 @ S ) ) ).
% 14.25/8.06
% 14.25/8.06 thf(fact_3096_pairwiseI,axiom,
% 14.25/8.06 ! [S: set_o,R3: $o > $o > $o] :
% 14.25/8.06 ( ! [X3: $o,Y2: $o] :
% 14.25/8.06 ( ( member_o @ X3 @ S )
% 14.25/8.06 => ( ( member_o @ Y2 @ S )
% 14.25/8.06 => ( ( X3 != Y2 )
% 14.25/8.06 => ( R3 @ X3 @ Y2 ) ) ) )
% 14.25/8.06 => ( pairwise_o @ R3 @ S ) ) ).
% 14.25/8.06
% 14.25/8.06 thf(fact_3097_pairwiseI,axiom,
% 14.25/8.06 ! [S: set_Product_unit,R3: product_unit > product_unit > $o] :
% 14.25/8.06 ( ! [X3: product_unit,Y2: product_unit] :
% 14.25/8.06 ( ( member_Product_unit @ X3 @ S )
% 14.25/8.06 => ( ( member_Product_unit @ Y2 @ S )
% 14.25/8.06 => ( ( X3 != Y2 )
% 14.25/8.06 => ( R3 @ X3 @ Y2 ) ) ) )
% 14.25/8.06 => ( pairwi7919127943033184120t_unit @ R3 @ S ) ) ).
% 14.25/8.06
% 14.25/8.06 thf(fact_3098_pairwiseD,axiom,
% 14.25/8.06 ! [R3: ( produc3658429121746597890et_nat > $o ) > ( produc3658429121746597890et_nat > $o ) > $o,S: set_Pr4532377907799695533_nat_o,X: produc3658429121746597890et_nat > $o,Y: produc3658429121746597890et_nat > $o] :
% 14.25/8.06 ( ( pairwi2432143822833250616_nat_o @ R3 @ S )
% 14.25/8.06 => ( ( member6576561426505652726_nat_o @ X @ S )
% 14.25/8.06 => ( ( member6576561426505652726_nat_o @ Y @ S )
% 14.25/8.06 => ( ( X != Y )
% 14.25/8.06 => ( R3 @ X @ Y ) ) ) ) ) ).
% 14.25/8.06
% 14.25/8.06 thf(fact_3099_pairwiseD,axiom,
% 14.25/8.06 ! [R3: nat > nat > $o,S: set_nat,X: nat,Y: nat] :
% 14.25/8.06 ( ( pairwise_nat @ R3 @ S )
% 14.25/8.06 => ( ( member_nat @ X @ S )
% 14.25/8.06 => ( ( member_nat @ Y @ S )
% 14.25/8.06 => ( ( X != Y )
% 14.25/8.06 => ( R3 @ X @ Y ) ) ) ) ) ).
% 14.25/8.06
% 14.25/8.06 thf(fact_3100_pairwiseD,axiom,
% 14.25/8.06 ! [R3: int > int > $o,S: set_int,X: int,Y: int] :
% 14.25/8.06 ( ( pairwise_int @ R3 @ S )
% 14.25/8.06 => ( ( member_int @ X @ S )
% 14.25/8.06 => ( ( member_int @ Y @ S )
% 14.25/8.06 => ( ( X != Y )
% 14.25/8.06 => ( R3 @ X @ Y ) ) ) ) ) ).
% 14.25/8.06
% 14.25/8.06 thf(fact_3101_pairwiseD,axiom,
% 14.25/8.06 ! [R3: $o > $o > $o,S: set_o,X: $o,Y: $o] :
% 14.25/8.06 ( ( pairwise_o @ R3 @ S )
% 14.25/8.06 => ( ( member_o @ X @ S )
% 14.25/8.06 => ( ( member_o @ Y @ S )
% 14.25/8.06 => ( ( X != Y )
% 14.25/8.06 => ( R3 @ X @ Y ) ) ) ) ) ).
% 14.25/8.06
% 14.25/8.06 thf(fact_3102_pairwiseD,axiom,
% 14.25/8.06 ! [R3: product_unit > product_unit > $o,S: set_Product_unit,X: product_unit,Y: product_unit] :
% 14.25/8.06 ( ( pairwi7919127943033184120t_unit @ R3 @ S )
% 14.25/8.06 => ( ( member_Product_unit @ X @ S )
% 14.25/8.06 => ( ( member_Product_unit @ Y @ S )
% 14.25/8.06 => ( ( X != Y )
% 14.25/8.06 => ( R3 @ X @ Y ) ) ) ) ) ).
% 14.25/8.06
% 14.25/8.06 thf(fact_3103_refl__on__UNION,axiom,
% 14.25/8.06 ! [S: set_nat,A: nat > set_nat,R2: nat > set_Pr1261947904930325089at_nat] :
% 14.25/8.06 ( ! [X3: nat] :
% 14.25/8.06 ( ( member_nat @ X3 @ S )
% 14.25/8.06 => ( refl_on_nat @ ( A @ X3 ) @ ( R2 @ X3 ) ) )
% 14.25/8.06 => ( refl_on_nat @ ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ A @ S ) ) @ ( comple5685304695842803022at_nat @ ( image_7178329752028323786at_nat @ R2 @ S ) ) ) ) ).
% 14.25/8.06
% 14.25/8.06 thf(fact_3104_Pow__bottom,axiom,
% 14.25/8.06 ! [B4: set_Product_unit] : ( member5877623283571906838t_unit @ bot_bo3957492148770167129t_unit @ ( pow_Product_unit @ B4 ) ) ).
% 14.25/8.06
% 14.25/8.06 thf(fact_3105_Pow__bottom,axiom,
% 14.25/8.06 ! [B4: set_o] : ( member_set_o @ bot_bot_set_o @ ( pow_o @ B4 ) ) ).
% 14.25/8.06
% 14.25/8.06 thf(fact_3106_Pow__bottom,axiom,
% 14.25/8.06 ! [B4: set_nat] : ( member_set_nat @ bot_bot_set_nat @ ( pow_nat @ B4 ) ) ).
% 14.25/8.06
% 14.25/8.06 thf(fact_3107_Pow__bottom,axiom,
% 14.25/8.06 ! [B4: set_int] : ( member_set_int @ bot_bot_set_int @ ( pow_int @ B4 ) ) ).
% 14.25/8.06
% 14.25/8.06 thf(fact_3108_Pow__bottom,axiom,
% 14.25/8.06 ! [B4: set_Pr4532377907799695533_nat_o] : ( member1452989533377600726_nat_o @ bot_bo7824918357723582233_nat_o @ ( pow_Pr7829143134171777386_nat_o @ B4 ) ) ).
% 14.25/8.06
% 14.25/8.06 thf(fact_3109_Pow__mono,axiom,
% 14.25/8.06 ! [A: set_nat,B4: set_nat] :
% 14.25/8.06 ( ( ord_less_eq_set_nat @ A @ B4 )
% 14.25/8.06 => ( ord_le6893508408891458716et_nat @ ( pow_nat @ A ) @ ( pow_nat @ B4 ) ) ) ).
% 14.25/8.06
% 14.25/8.06 thf(fact_3110_PowD,axiom,
% 14.25/8.06 ! [A: set_nat,B4: set_nat] :
% 14.25/8.06 ( ( member_set_nat @ A @ ( pow_nat @ B4 ) )
% 14.25/8.06 => ( ord_less_eq_set_nat @ A @ B4 ) ) ).
% 14.25/8.06
% 14.25/8.06 thf(fact_3111_Un__Pow__subset,axiom,
% 14.25/8.06 ! [A: set_nat,B4: set_nat] : ( ord_le6893508408891458716et_nat @ ( sup_sup_set_set_nat @ ( pow_nat @ A ) @ ( pow_nat @ B4 ) ) @ ( pow_nat @ ( sup_sup_set_nat @ A @ B4 ) ) ) ).
% 14.25/8.06
% 14.25/8.06 thf(fact_3112_pairwise__imageI,axiom,
% 14.25/8.06 ! [A: set_Code_integer,F: code_integer > code_integer,P: code_integer > code_integer > $o] :
% 14.25/8.06 ( ! [X3: code_integer,Y2: code_integer] :
% 14.25/8.06 ( ( member_Code_integer @ X3 @ A )
% 14.25/8.06 => ( ( member_Code_integer @ Y2 @ A )
% 14.25/8.06 => ( ( X3 != Y2 )
% 14.25/8.06 => ( ( ( F @ X3 )
% 14.25/8.06 != ( F @ Y2 ) )
% 14.25/8.06 => ( P @ ( F @ X3 ) @ ( F @ Y2 ) ) ) ) ) )
% 14.25/8.06 => ( pairwi4741737654671186738nteger @ P @ ( image_4470545334726330049nteger @ F @ A ) ) ) ).
% 14.25/8.06
% 14.25/8.06 thf(fact_3113_pairwise__imageI,axiom,
% 14.25/8.06 ! [A: set_nat,F: nat > set_nat,P: set_nat > set_nat > $o] :
% 14.25/8.06 ( ! [X3: nat,Y2: nat] :
% 14.25/8.06 ( ( member_nat @ X3 @ A )
% 14.25/8.06 => ( ( member_nat @ Y2 @ A )
% 14.25/8.06 => ( ( X3 != Y2 )
% 14.25/8.06 => ( ( ( F @ X3 )
% 14.25/8.06 != ( F @ Y2 ) )
% 14.25/8.06 => ( P @ ( F @ X3 ) @ ( F @ Y2 ) ) ) ) ) )
% 14.25/8.06 => ( pairwise_set_nat @ P @ ( image_nat_set_nat @ F @ A ) ) ) ).
% 14.25/8.06
% 14.25/8.06 thf(fact_3114_pairwise__imageI,axiom,
% 14.25/8.06 ! [A: set_nat,F: nat > nat,P: nat > nat > $o] :
% 14.25/8.06 ( ! [X3: nat,Y2: nat] :
% 14.25/8.06 ( ( member_nat @ X3 @ A )
% 14.25/8.06 => ( ( member_nat @ Y2 @ A )
% 14.25/8.06 => ( ( X3 != Y2 )
% 14.25/8.06 => ( ( ( F @ X3 )
% 14.25/8.06 != ( F @ Y2 ) )
% 14.25/8.06 => ( P @ ( F @ X3 ) @ ( F @ Y2 ) ) ) ) ) )
% 14.25/8.06 => ( pairwise_nat @ P @ ( image_nat_nat2 @ F @ A ) ) ) ).
% 14.25/8.06
% 14.25/8.06 thf(fact_3115_pairwise__imageI,axiom,
% 14.25/8.06 ! [A: set_nat,F: nat > code_integer,P: code_integer > code_integer > $o] :
% 14.25/8.06 ( ! [X3: nat,Y2: nat] :
% 14.25/8.06 ( ( member_nat @ X3 @ A )
% 14.25/8.06 => ( ( member_nat @ Y2 @ A )
% 14.25/8.06 => ( ( X3 != Y2 )
% 14.25/8.06 => ( ( ( F @ X3 )
% 14.25/8.06 != ( F @ Y2 ) )
% 14.25/8.06 => ( P @ ( F @ X3 ) @ ( F @ Y2 ) ) ) ) ) )
% 14.25/8.06 => ( pairwi4741737654671186738nteger @ P @ ( image_1215581382706833972nteger @ F @ A ) ) ) ).
% 14.25/8.06
% 14.25/8.06 thf(fact_3116_pairwise__empty,axiom,
% 14.25/8.06 ! [P: product_unit > product_unit > $o] : ( pairwi7919127943033184120t_unit @ P @ bot_bo3957492148770167129t_unit ) ).
% 14.25/8.06
% 14.25/8.06 thf(fact_3117_pairwise__empty,axiom,
% 14.25/8.06 ! [P: $o > $o > $o] : ( pairwise_o @ P @ bot_bot_set_o ) ).
% 14.25/8.06
% 14.25/8.06 thf(fact_3118_pairwise__empty,axiom,
% 14.25/8.06 ! [P: nat > nat > $o] : ( pairwise_nat @ P @ bot_bot_set_nat ) ).
% 14.25/8.06
% 14.25/8.06 thf(fact_3119_pairwise__empty,axiom,
% 14.25/8.06 ! [P: int > int > $o] : ( pairwise_int @ P @ bot_bot_set_int ) ).
% 14.25/8.06
% 14.25/8.06 thf(fact_3120_pairwise__empty,axiom,
% 14.25/8.06 ! [P: ( produc3658429121746597890et_nat > $o ) > ( produc3658429121746597890et_nat > $o ) > $o] : ( pairwi2432143822833250616_nat_o @ P @ bot_bo7824918357723582233_nat_o ) ).
% 14.25/8.06
% 14.25/8.06 thf(fact_3121_pairwise__subset,axiom,
% 14.25/8.06 ! [P: nat > nat > $o,S: set_nat,T2: set_nat] :
% 14.25/8.06 ( ( pairwise_nat @ P @ S )
% 14.25/8.06 => ( ( ord_less_eq_set_nat @ T2 @ S )
% 14.25/8.06 => ( pairwise_nat @ P @ T2 ) ) ) ).
% 14.25/8.06
% 14.25/8.06 thf(fact_3122_pairwise__mono,axiom,
% 14.25/8.06 ! [P: nat > nat > $o,A: set_nat,Q2: nat > nat > $o,B4: set_nat] :
% 14.25/8.06 ( ( pairwise_nat @ P @ A )
% 14.25/8.06 => ( ! [X3: nat,Y2: nat] :
% 14.25/8.06 ( ( P @ X3 @ Y2 )
% 14.25/8.06 => ( Q2 @ X3 @ Y2 ) )
% 14.25/8.06 => ( ( ord_less_eq_set_nat @ B4 @ A )
% 14.25/8.06 => ( pairwise_nat @ Q2 @ B4 ) ) ) ) ).
% 14.25/8.06
% 14.25/8.06 thf(fact_3123_pairwise__insert,axiom,
% 14.25/8.06 ! [R2: ( produc3658429121746597890et_nat > $o ) > ( produc3658429121746597890et_nat > $o ) > $o,X: produc3658429121746597890et_nat > $o,S2: set_Pr4532377907799695533_nat_o] :
% 14.25/8.06 ( ( pairwi2432143822833250616_nat_o @ R2 @ ( insert5175938949040314269_nat_o @ X @ S2 ) )
% 14.25/8.06 = ( ! [Y3: produc3658429121746597890et_nat > $o] :
% 14.25/8.06 ( ( ( member6576561426505652726_nat_o @ Y3 @ S2 )
% 14.25/8.06 & ( Y3 != X ) )
% 14.25/8.06 => ( ( R2 @ X @ Y3 )
% 14.25/8.06 & ( R2 @ Y3 @ X ) ) )
% 14.25/8.06 & ( pairwi2432143822833250616_nat_o @ R2 @ S2 ) ) ) ).
% 14.25/8.06
% 14.25/8.06 thf(fact_3124_pairwise__insert,axiom,
% 14.25/8.06 ! [R2: nat > nat > $o,X: nat,S2: set_nat] :
% 14.25/8.06 ( ( pairwise_nat @ R2 @ ( insert_nat @ X @ S2 ) )
% 14.25/8.06 = ( ! [Y3: nat] :
% 14.25/8.06 ( ( ( member_nat @ Y3 @ S2 )
% 14.25/8.06 & ( Y3 != X ) )
% 14.25/8.06 => ( ( R2 @ X @ Y3 )
% 14.25/8.06 & ( R2 @ Y3 @ X ) ) )
% 14.25/8.06 & ( pairwise_nat @ R2 @ S2 ) ) ) ).
% 14.25/8.06
% 14.25/8.06 thf(fact_3125_pairwise__insert,axiom,
% 14.25/8.06 ! [R2: int > int > $o,X: int,S2: set_int] :
% 14.25/8.06 ( ( pairwise_int @ R2 @ ( insert_int @ X @ S2 ) )
% 14.25/8.06 = ( ! [Y3: int] :
% 14.25/8.06 ( ( ( member_int @ Y3 @ S2 )
% 14.25/8.06 & ( Y3 != X ) )
% 14.25/8.06 => ( ( R2 @ X @ Y3 )
% 14.25/8.06 & ( R2 @ Y3 @ X ) ) )
% 14.25/8.06 & ( pairwise_int @ R2 @ S2 ) ) ) ).
% 14.25/8.06
% 14.25/8.06 thf(fact_3126_pairwise__insert,axiom,
% 14.25/8.06 ! [R2: $o > $o > $o,X: $o,S2: set_o] :
% 14.25/8.06 ( ( pairwise_o @ R2 @ ( insert_o @ X @ S2 ) )
% 14.25/8.06 = ( ! [Y3: $o] :
% 14.25/8.06 ( ( ( member_o @ Y3 @ S2 )
% 14.25/8.06 & ( Y3 != X ) )
% 14.25/8.06 => ( ( R2 @ X @ Y3 )
% 14.25/8.06 & ( R2 @ Y3 @ X ) ) )
% 14.25/8.06 & ( pairwise_o @ R2 @ S2 ) ) ) ).
% 14.25/8.06
% 14.25/8.06 thf(fact_3127_pairwise__insert,axiom,
% 14.25/8.06 ! [R2: product_unit > product_unit > $o,X: product_unit,S2: set_Product_unit] :
% 14.25/8.06 ( ( pairwi7919127943033184120t_unit @ R2 @ ( insert_Product_unit @ X @ S2 ) )
% 14.25/8.06 = ( ! [Y3: product_unit] :
% 14.25/8.06 ( ( ( member_Product_unit @ Y3 @ S2 )
% 14.25/8.06 & ( Y3 != X ) )
% 14.25/8.06 => ( ( R2 @ X @ Y3 )
% 14.25/8.06 & ( R2 @ Y3 @ X ) ) )
% 14.25/8.06 & ( pairwi7919127943033184120t_unit @ R2 @ S2 ) ) ) ).
% 14.25/8.06
% 14.25/8.06 thf(fact_3128_finite__UNION__then__finite,axiom,
% 14.25/8.06 ! [B4: ( produc3658429121746597890et_nat > $o ) > set_nat,A: set_Pr4532377907799695533_nat_o,A2: produc3658429121746597890et_nat > $o] :
% 14.25/8.06 ( ( finite_finite_nat @ ( comple7399068483239264473et_nat @ ( image_6888641473154130958et_nat @ B4 @ A ) ) )
% 14.25/8.06 => ( ( member6576561426505652726_nat_o @ A2 @ A )
% 14.25/8.06 => ( finite_finite_nat @ ( B4 @ A2 ) ) ) ) ).
% 14.25/8.06
% 14.25/8.06 thf(fact_3129_finite__UNION__then__finite,axiom,
% 14.25/8.06 ! [B4: nat > set_nat,A: set_nat,A2: nat] :
% 14.25/8.06 ( ( finite_finite_nat @ ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ B4 @ A ) ) )
% 14.25/8.06 => ( ( member_nat @ A2 @ A )
% 14.25/8.06 => ( finite_finite_nat @ ( B4 @ A2 ) ) ) ) ).
% 14.25/8.06
% 14.25/8.06 thf(fact_3130_finite__UNION__then__finite,axiom,
% 14.25/8.06 ! [B4: int > set_nat,A: set_int,A2: int] :
% 14.25/8.06 ( ( finite_finite_nat @ ( comple7399068483239264473et_nat @ ( image_int_set_nat @ B4 @ A ) ) )
% 14.25/8.06 => ( ( member_int @ A2 @ A )
% 14.25/8.06 => ( finite_finite_nat @ ( B4 @ A2 ) ) ) ) ).
% 14.25/8.06
% 14.25/8.06 thf(fact_3131_finite__UNION__then__finite,axiom,
% 14.25/8.06 ! [B4: $o > set_nat,A: set_o,A2: $o] :
% 14.25/8.06 ( ( finite_finite_nat @ ( comple7399068483239264473et_nat @ ( image_o_set_nat @ B4 @ A ) ) )
% 14.25/8.06 => ( ( member_o @ A2 @ A )
% 14.25/8.06 => ( finite_finite_nat @ ( B4 @ A2 ) ) ) ) ).
% 14.25/8.06
% 14.25/8.06 thf(fact_3132_finite__UNION__then__finite,axiom,
% 14.25/8.06 ! [B4: product_unit > set_nat,A: set_Product_unit,A2: product_unit] :
% 14.25/8.06 ( ( finite_finite_nat @ ( comple7399068483239264473et_nat @ ( image_8173849991827986510et_nat @ B4 @ A ) ) )
% 14.25/8.06 => ( ( member_Product_unit @ A2 @ A )
% 14.25/8.06 => ( finite_finite_nat @ ( B4 @ A2 ) ) ) ) ).
% 14.25/8.06
% 14.25/8.06 thf(fact_3133_finite__UNION__then__finite,axiom,
% 14.25/8.06 ! [B4: ( produc3658429121746597890et_nat > $o ) > set_Code_integer,A: set_Pr4532377907799695533_nat_o,A2: produc3658429121746597890et_nat > $o] :
% 14.25/8.06 ( ( finite6017078050557962740nteger @ ( comple739944243200306918nteger @ ( image_8657708916238040603nteger @ B4 @ A ) ) )
% 14.25/8.06 => ( ( member6576561426505652726_nat_o @ A2 @ A )
% 14.25/8.06 => ( finite6017078050557962740nteger @ ( B4 @ A2 ) ) ) ) ).
% 14.25/8.06
% 14.25/8.06 thf(fact_3134_finite__UNION__then__finite,axiom,
% 14.25/8.06 ! [B4: nat > set_Code_integer,A: set_nat,A2: nat] :
% 14.25/8.06 ( ( finite6017078050557962740nteger @ ( comple739944243200306918nteger @ ( image_2385391725447763818nteger @ B4 @ A ) ) )
% 14.25/8.06 => ( ( member_nat @ A2 @ A )
% 14.25/8.06 => ( finite6017078050557962740nteger @ ( B4 @ A2 ) ) ) ) ).
% 14.25/8.06
% 14.25/8.06 thf(fact_3135_finite__UNION__then__finite,axiom,
% 14.25/8.06 ! [B4: int > set_Code_integer,A: set_int,A2: int] :
% 14.25/8.06 ( ( finite6017078050557962740nteger @ ( comple739944243200306918nteger @ ( image_75573793354812870nteger @ B4 @ A ) ) )
% 14.25/8.06 => ( ( member_int @ A2 @ A )
% 14.25/8.06 => ( finite6017078050557962740nteger @ ( B4 @ A2 ) ) ) ) ).
% 14.25/8.06
% 14.25/8.06 thf(fact_3136_finite__UNION__then__finite,axiom,
% 14.25/8.06 ! [B4: $o > set_Code_integer,A: set_o,A2: $o] :
% 14.25/8.06 ( ( finite6017078050557962740nteger @ ( comple739944243200306918nteger @ ( image_6491638074533506790nteger @ B4 @ A ) ) )
% 14.25/8.06 => ( ( member_o @ A2 @ A )
% 14.25/8.06 => ( finite6017078050557962740nteger @ ( B4 @ A2 ) ) ) ) ).
% 14.25/8.06
% 14.25/8.06 thf(fact_3137_finite__UNION__then__finite,axiom,
% 14.25/8.06 ! [B4: product_unit > set_Code_integer,A: set_Product_unit,A2: product_unit] :
% 14.25/8.06 ( ( finite6017078050557962740nteger @ ( comple739944243200306918nteger @ ( image_4571654851662262875nteger @ B4 @ A ) ) )
% 14.25/8.06 => ( ( member_Product_unit @ A2 @ A )
% 14.25/8.06 => ( finite6017078050557962740nteger @ ( B4 @ A2 ) ) ) ) ).
% 14.25/8.06
% 14.25/8.06 thf(fact_3138_image__Pow__surj,axiom,
% 14.25/8.06 ! [F: nat > set_nat,A: set_nat,B4: set_set_nat] :
% 14.25/8.06 ( ( ( image_nat_set_nat @ F @ A )
% 14.25/8.06 = B4 )
% 14.25/8.06 => ( ( image_6725021117256019401et_nat @ ( image_nat_set_nat @ F ) @ ( pow_nat @ A ) )
% 14.25/8.06 = ( pow_set_nat @ B4 ) ) ) ).
% 14.25/8.06
% 14.25/8.06 thf(fact_3139_image__Pow__surj,axiom,
% 14.25/8.06 ! [F: nat > nat,A: set_nat,B4: set_nat] :
% 14.25/8.06 ( ( ( image_nat_nat2 @ F @ A )
% 14.25/8.06 = B4 )
% 14.25/8.06 => ( ( image_7916887816326733075et_nat @ ( image_nat_nat2 @ F ) @ ( pow_nat @ A ) )
% 14.25/8.06 = ( pow_nat @ B4 ) ) ) ).
% 14.25/8.06
% 14.25/8.06 thf(fact_3140_image__Pow__surj,axiom,
% 14.25/8.06 ! [F: code_integer > code_integer,A: set_Code_integer,B4: set_Code_integer] :
% 14.25/8.06 ( ( ( image_4470545334726330049nteger @ F @ A )
% 14.25/8.06 = B4 )
% 14.25/8.06 => ( ( image_6270750266955655469nteger @ ( image_4470545334726330049nteger @ F ) @ ( pow_Code_integer @ A ) )
% 14.25/8.06 = ( pow_Code_integer @ B4 ) ) ) ).
% 14.25/8.06
% 14.25/8.06 thf(fact_3141_image__Pow__surj,axiom,
% 14.25/8.06 ! [F: nat > code_integer,A: set_nat,B4: set_Code_integer] :
% 14.25/8.06 ( ( ( image_1215581382706833972nteger @ F @ A )
% 14.25/8.06 = B4 )
% 14.25/8.06 => ( ( image_6916300684244607776nteger @ ( image_1215581382706833972nteger @ F ) @ ( pow_nat @ A ) )
% 14.25/8.06 = ( pow_Code_integer @ B4 ) ) ) ).
% 14.25/8.06
% 14.25/8.06 thf(fact_3142_Pow__insert,axiom,
% 14.25/8.06 ! [A2: nat,A: set_nat] :
% 14.25/8.06 ( ( pow_nat @ ( insert_nat @ A2 @ A ) )
% 14.25/8.06 = ( sup_sup_set_set_nat @ ( pow_nat @ A ) @ ( image_7916887816326733075et_nat @ ( insert_nat @ A2 ) @ ( pow_nat @ A ) ) ) ) ).
% 14.25/8.06
% 14.25/8.06 thf(fact_3143_Pow__insert,axiom,
% 14.25/8.06 ! [A2: int,A: set_int] :
% 14.25/8.06 ( ( pow_int @ ( insert_int @ A2 @ A ) )
% 14.25/8.06 = ( sup_sup_set_set_int @ ( pow_int @ A ) @ ( image_524474410958335435et_int @ ( insert_int @ A2 ) @ ( pow_int @ A ) ) ) ) ).
% 14.25/8.06
% 14.25/8.06 thf(fact_3144_Pow__insert,axiom,
% 14.25/8.06 ! [A2: $o,A: set_o] :
% 14.25/8.06 ( ( pow_o @ ( insert_o @ A2 @ A ) )
% 14.25/8.06 = ( sup_sup_set_set_o @ ( pow_o @ A ) @ ( image_set_o_set_o @ ( insert_o @ A2 ) @ ( pow_o @ A ) ) ) ) ).
% 14.25/8.06
% 14.25/8.06 thf(fact_3145_Pow__insert,axiom,
% 14.25/8.06 ! [A2: product_unit,A: set_Product_unit] :
% 14.25/8.06 ( ( pow_Product_unit @ ( insert_Product_unit @ A2 @ A ) )
% 14.25/8.06 = ( sup_su5975557051335651745t_unit @ ( pow_Product_unit @ A ) @ ( image_4019945301284682565t_unit @ ( insert_Product_unit @ A2 ) @ ( pow_Product_unit @ A ) ) ) ) ).
% 14.25/8.06
% 14.25/8.06 thf(fact_3146_pairself_Osimps,axiom,
% 14.25/8.06 ! [F: nat > nat,A2: nat,B: nat] :
% 14.25/8.06 ( ( pairself_nat_nat @ F @ ( product_Pair_nat_nat @ A2 @ B ) )
% 14.25/8.06 = ( product_Pair_nat_nat @ ( F @ A2 ) @ ( F @ B ) ) ) ).
% 14.25/8.06
% 14.25/8.06 thf(fact_3147_pairself_Osimps,axiom,
% 14.25/8.06 ! [F: nat > int,A2: nat,B: nat] :
% 14.25/8.06 ( ( pairself_nat_int @ F @ ( product_Pair_nat_nat @ A2 @ B ) )
% 14.25/8.06 = ( product_Pair_int_int @ ( F @ A2 ) @ ( F @ B ) ) ) ).
% 14.25/8.06
% 14.25/8.06 thf(fact_3148_pairself_Osimps,axiom,
% 14.25/8.06 ! [F: int > nat,A2: int,B: int] :
% 14.25/8.06 ( ( pairself_int_nat @ F @ ( product_Pair_int_int @ A2 @ B ) )
% 14.25/8.06 = ( product_Pair_nat_nat @ ( F @ A2 ) @ ( F @ B ) ) ) ).
% 14.25/8.06
% 14.25/8.06 thf(fact_3149_pairself_Osimps,axiom,
% 14.25/8.06 ! [F: int > int,A2: int,B: int] :
% 14.25/8.06 ( ( pairself_int_int @ F @ ( product_Pair_int_int @ A2 @ B ) )
% 14.25/8.06 = ( product_Pair_int_int @ ( F @ A2 ) @ ( F @ B ) ) ) ).
% 14.25/8.06
% 14.25/8.06 thf(fact_3150_pairself_Oelims,axiom,
% 14.25/8.06 ! [X: nat > nat,Xa: product_prod_nat_nat,Y: product_prod_nat_nat] :
% 14.25/8.06 ( ( ( pairself_nat_nat @ X @ Xa )
% 14.25/8.06 = Y )
% 14.25/8.06 => ~ ! [A4: nat,B2: nat] :
% 14.25/8.06 ( ( Xa
% 14.25/8.06 = ( product_Pair_nat_nat @ A4 @ B2 ) )
% 14.25/8.06 => ( Y
% 14.25/8.06 != ( product_Pair_nat_nat @ ( X @ A4 ) @ ( X @ B2 ) ) ) ) ) ).
% 14.25/8.06
% 14.25/8.06 thf(fact_3151_pairself_Oelims,axiom,
% 14.25/8.06 ! [X: nat > int,Xa: product_prod_nat_nat,Y: product_prod_int_int] :
% 14.25/8.06 ( ( ( pairself_nat_int @ X @ Xa )
% 14.25/8.06 = Y )
% 14.25/8.06 => ~ ! [A4: nat,B2: nat] :
% 14.25/8.06 ( ( Xa
% 14.25/8.06 = ( product_Pair_nat_nat @ A4 @ B2 ) )
% 14.25/8.06 => ( Y
% 14.25/8.06 != ( product_Pair_int_int @ ( X @ A4 ) @ ( X @ B2 ) ) ) ) ) ).
% 14.25/8.06
% 14.25/8.06 thf(fact_3152_pairself_Oelims,axiom,
% 14.25/8.06 ! [X: int > nat,Xa: product_prod_int_int,Y: product_prod_nat_nat] :
% 14.25/8.06 ( ( ( pairself_int_nat @ X @ Xa )
% 14.25/8.06 = Y )
% 14.25/8.06 => ~ ! [A4: int,B2: int] :
% 14.25/8.06 ( ( Xa
% 14.25/8.06 = ( product_Pair_int_int @ A4 @ B2 ) )
% 14.25/8.06 => ( Y
% 14.25/8.06 != ( product_Pair_nat_nat @ ( X @ A4 ) @ ( X @ B2 ) ) ) ) ) ).
% 14.25/8.06
% 14.25/8.06 thf(fact_3153_pairself_Oelims,axiom,
% 14.25/8.06 ! [X: int > int,Xa: product_prod_int_int,Y: product_prod_int_int] :
% 14.25/8.06 ( ( ( pairself_int_int @ X @ Xa )
% 14.25/8.06 = Y )
% 14.25/8.06 => ~ ! [A4: int,B2: int] :
% 14.25/8.06 ( ( Xa
% 14.25/8.06 = ( product_Pair_int_int @ A4 @ B2 ) )
% 14.25/8.06 => ( Y
% 14.25/8.06 != ( product_Pair_int_int @ ( X @ A4 ) @ ( X @ B2 ) ) ) ) ) ).
% 14.25/8.06
% 14.25/8.06 thf(fact_3154_pairwise__singleton,axiom,
% 14.25/8.06 ! [P: product_unit > product_unit > $o,A: product_unit] : ( pairwi7919127943033184120t_unit @ P @ ( insert_Product_unit @ A @ bot_bo3957492148770167129t_unit ) ) ).
% 14.25/8.06
% 14.25/8.06 thf(fact_3155_pairwise__singleton,axiom,
% 14.25/8.06 ! [P: $o > $o > $o,A: $o] : ( pairwise_o @ P @ ( insert_o @ A @ bot_bot_set_o ) ) ).
% 14.25/8.06
% 14.25/8.06 thf(fact_3156_pairwise__singleton,axiom,
% 14.25/8.06 ! [P: nat > nat > $o,A: nat] : ( pairwise_nat @ P @ ( insert_nat @ A @ bot_bot_set_nat ) ) ).
% 14.25/8.06
% 14.25/8.06 thf(fact_3157_pairwise__singleton,axiom,
% 14.25/8.06 ! [P: int > int > $o,A: int] : ( pairwise_int @ P @ ( insert_int @ A @ bot_bot_set_int ) ) ).
% 14.25/8.06
% 14.25/8.06 thf(fact_3158_pairwise__singleton,axiom,
% 14.25/8.06 ! [P: ( produc3658429121746597890et_nat > $o ) > ( produc3658429121746597890et_nat > $o ) > $o,A: produc3658429121746597890et_nat > $o] : ( pairwi2432143822833250616_nat_o @ P @ ( insert5175938949040314269_nat_o @ A @ bot_bo7824918357723582233_nat_o ) ) ).
% 14.25/8.06
% 14.25/8.06 thf(fact_3159_finite__Sup__in,axiom,
% 14.25/8.06 ! [A: set_set_nat] :
% 14.25/8.06 ( ( finite1152437895449049373et_nat @ A )
% 14.25/8.06 => ( ( A != bot_bot_set_set_nat )
% 14.25/8.06 => ( ! [X3: set_nat,Y2: set_nat] :
% 14.25/8.06 ( ( member_set_nat @ X3 @ A )
% 14.25/8.06 => ( ( member_set_nat @ Y2 @ A )
% 14.25/8.06 => ( member_set_nat @ ( sup_sup_set_nat @ X3 @ Y2 ) @ A ) ) )
% 14.25/8.06 => ( member_set_nat @ ( comple7399068483239264473et_nat @ A ) @ A ) ) ) ) ).
% 14.25/8.06
% 14.25/8.06 thf(fact_3160_finite__Sup__in,axiom,
% 14.25/8.06 ! [A: set_o] :
% 14.25/8.06 ( ( finite_finite_o @ A )
% 14.25/8.06 => ( ( A != bot_bot_set_o )
% 14.25/8.06 => ( ! [X3: $o,Y2: $o] :
% 14.25/8.06 ( ( member_o @ X3 @ A )
% 14.25/8.06 => ( ( member_o @ Y2 @ A )
% 14.25/8.06 => ( member_o @ ( sup_sup_o @ X3 @ Y2 ) @ A ) ) )
% 14.25/8.06 => ( member_o @ ( complete_Sup_Sup_o @ A ) @ A ) ) ) ) ).
% 14.25/8.06
% 14.25/8.06 thf(fact_3161_finite__Sup__in,axiom,
% 14.25/8.06 ! [A: set_Pr4532377907799695533_nat_o] :
% 14.25/8.06 ( ( finite3252695134891459830_nat_o @ A )
% 14.25/8.06 => ( ( A != bot_bo7824918357723582233_nat_o )
% 14.25/8.06 => ( ! [X3: produc3658429121746597890et_nat > $o,Y2: produc3658429121746597890et_nat > $o] :
% 14.25/8.06 ( ( member6576561426505652726_nat_o @ X3 @ A )
% 14.25/8.06 => ( ( member6576561426505652726_nat_o @ Y2 @ A )
% 14.25/8.06 => ( member6576561426505652726_nat_o @ ( sup_su5453871518329203617_nat_o @ X3 @ Y2 ) @ A ) ) )
% 14.25/8.06 => ( member6576561426505652726_nat_o @ ( comple4225350640907012154_nat_o @ A ) @ A ) ) ) ) ).
% 14.25/8.06
% 14.25/8.06 thf(fact_3162_finite__Sup__in,axiom,
% 14.25/8.06 ! [A: set_Product_unit] :
% 14.25/8.06 ( ( finite4290736615968046902t_unit @ A )
% 14.25/8.06 => ( ( A != bot_bo3957492148770167129t_unit )
% 14.25/8.06 => ( ! [X3: product_unit,Y2: product_unit] :
% 14.25/8.06 ( ( member_Product_unit @ X3 @ A )
% 14.25/8.06 => ( ( member_Product_unit @ Y2 @ A )
% 14.25/8.06 => ( member_Product_unit @ ( sup_sup_Product_unit @ X3 @ Y2 ) @ A ) ) )
% 14.25/8.06 => ( member_Product_unit @ ( comple4687483117567863418t_unit @ A ) @ A ) ) ) ) ).
% 14.25/8.06
% 14.25/8.06 thf(fact_3163_Sup__fin__Sup,axiom,
% 14.25/8.06 ! [A: set_o] :
% 14.25/8.06 ( ( finite_finite_o @ A )
% 14.25/8.06 => ( ( A != bot_bot_set_o )
% 14.25/8.06 => ( ( lattic1508158080041050831_fin_o @ A )
% 14.25/8.06 = ( complete_Sup_Sup_o @ A ) ) ) ) ).
% 14.25/8.06
% 14.25/8.06 thf(fact_3164_Sup__fin__Sup,axiom,
% 14.25/8.06 ! [A: set_Pr4532377907799695533_nat_o] :
% 14.25/8.06 ( ( finite3252695134891459830_nat_o @ A )
% 14.25/8.06 => ( ( A != bot_bo7824918357723582233_nat_o )
% 14.25/8.06 => ( ( lattic7320199455484906628_nat_o @ A )
% 14.25/8.06 = ( comple4225350640907012154_nat_o @ A ) ) ) ) ).
% 14.25/8.06
% 14.25/8.06 thf(fact_3165_Sup__fin__Sup,axiom,
% 14.25/8.06 ! [A: set_Product_unit] :
% 14.25/8.06 ( ( finite4290736615968046902t_unit @ A )
% 14.25/8.06 => ( ( A != bot_bo3957492148770167129t_unit )
% 14.25/8.06 => ( ( lattic5294303975357428420t_unit @ A )
% 14.25/8.06 = ( comple4687483117567863418t_unit @ A ) ) ) ) ).
% 14.25/8.06
% 14.25/8.06 thf(fact_3166_insert__partition,axiom,
% 14.25/8.06 ! [X: set_Product_unit,F2: set_set_Product_unit] :
% 14.25/8.06 ( ~ ( member5877623283571906838t_unit @ X @ F2 )
% 14.25/8.06 => ( ! [X3: set_Product_unit] :
% 14.25/8.06 ( ( member5877623283571906838t_unit @ X3 @ ( insert8986028188087627965t_unit @ X @ F2 ) )
% 14.25/8.06 => ! [Xa3: set_Product_unit] :
% 14.25/8.06 ( ( member5877623283571906838t_unit @ Xa3 @ ( insert8986028188087627965t_unit @ X @ F2 ) )
% 14.25/8.06 => ( ( X3 != Xa3 )
% 14.25/8.06 => ( ( inf_in4660618365625256667t_unit @ X3 @ Xa3 )
% 14.25/8.06 = bot_bo3957492148770167129t_unit ) ) ) )
% 14.25/8.06 => ( ( inf_in4660618365625256667t_unit @ X @ ( comple3349607820504768346t_unit @ F2 ) )
% 14.25/8.06 = bot_bo3957492148770167129t_unit ) ) ) ).
% 14.25/8.06
% 14.25/8.06 thf(fact_3167_insert__partition,axiom,
% 14.25/8.06 ! [X: set_o,F2: set_set_o] :
% 14.25/8.06 ( ~ ( member_set_o @ X @ F2 )
% 14.25/8.06 => ( ! [X3: set_o] :
% 14.25/8.06 ( ( member_set_o @ X3 @ ( insert_set_o @ X @ F2 ) )
% 14.25/8.06 => ! [Xa3: set_o] :
% 14.25/8.06 ( ( member_set_o @ Xa3 @ ( insert_set_o @ X @ F2 ) )
% 14.25/8.06 => ( ( X3 != Xa3 )
% 14.25/8.06 => ( ( inf_inf_set_o @ X3 @ Xa3 )
% 14.25/8.06 = bot_bot_set_o ) ) ) )
% 14.25/8.06 => ( ( inf_inf_set_o @ X @ ( comple90263536869209701_set_o @ F2 ) )
% 14.25/8.06 = bot_bot_set_o ) ) ) ).
% 14.25/8.06
% 14.25/8.06 thf(fact_3168_insert__partition,axiom,
% 14.25/8.06 ! [X: set_nat,F2: set_set_nat] :
% 14.25/8.06 ( ~ ( member_set_nat @ X @ F2 )
% 14.25/8.06 => ( ! [X3: set_nat] :
% 14.25/8.06 ( ( member_set_nat @ X3 @ ( insert_set_nat @ X @ F2 ) )
% 14.25/8.06 => ! [Xa3: set_nat] :
% 14.25/8.06 ( ( member_set_nat @ Xa3 @ ( insert_set_nat @ X @ F2 ) )
% 14.25/8.06 => ( ( X3 != Xa3 )
% 14.25/8.06 => ( ( inf_inf_set_nat @ X3 @ Xa3 )
% 14.25/8.06 = bot_bot_set_nat ) ) ) )
% 14.25/8.06 => ( ( inf_inf_set_nat @ X @ ( comple7399068483239264473et_nat @ F2 ) )
% 14.25/8.06 = bot_bot_set_nat ) ) ) ).
% 14.25/8.06
% 14.25/8.06 thf(fact_3169_insert__partition,axiom,
% 14.25/8.06 ! [X: set_int,F2: set_set_int] :
% 14.25/8.06 ( ~ ( member_set_int @ X @ F2 )
% 14.25/8.06 => ( ! [X3: set_int] :
% 14.25/8.06 ( ( member_set_int @ X3 @ ( insert_set_int @ X @ F2 ) )
% 14.25/8.06 => ! [Xa3: set_int] :
% 14.25/8.06 ( ( member_set_int @ Xa3 @ ( insert_set_int @ X @ F2 ) )
% 14.25/8.06 => ( ( X3 != Xa3 )
% 14.25/8.06 => ( ( inf_inf_set_int @ X3 @ Xa3 )
% 14.25/8.06 = bot_bot_set_int ) ) ) )
% 14.25/8.06 => ( ( inf_inf_set_int @ X @ ( comple3221217463730067765et_int @ F2 ) )
% 14.25/8.06 = bot_bot_set_int ) ) ) ).
% 14.25/8.06
% 14.25/8.06 thf(fact_3170_insert__partition,axiom,
% 14.25/8.06 ! [X: set_Pr4532377907799695533_nat_o,F2: set_se1353816476878259085_nat_o] :
% 14.25/8.06 ( ~ ( member1452989533377600726_nat_o @ X @ F2 )
% 14.25/8.06 => ( ! [X3: set_Pr4532377907799695533_nat_o] :
% 14.25/8.06 ( ( member1452989533377600726_nat_o @ X3 @ ( insert2507855216428400253_nat_o @ X @ F2 ) )
% 14.25/8.06 => ! [Xa3: set_Pr4532377907799695533_nat_o] :
% 14.25/8.06 ( ( member1452989533377600726_nat_o @ Xa3 @ ( insert2507855216428400253_nat_o @ X @ F2 ) )
% 14.25/8.06 => ( ( X3 != Xa3 )
% 14.25/8.06 => ( ( inf_in1906310914598751387_nat_o @ X3 @ Xa3 )
% 14.25/8.06 = bot_bo7824918357723582233_nat_o ) ) ) )
% 14.25/8.06 => ( ( inf_in1906310914598751387_nat_o @ X @ ( comple3382087578465172250_nat_o @ F2 ) )
% 14.25/8.06 = bot_bo7824918357723582233_nat_o ) ) ) ).
% 14.25/8.06
% 14.25/8.06 thf(fact_3171_image__Pow__mono,axiom,
% 14.25/8.06 ! [F: nat > set_nat,A: set_nat,B4: set_set_nat] :
% 14.25/8.06 ( ( ord_le6893508408891458716et_nat @ ( image_nat_set_nat @ F @ A ) @ B4 )
% 14.25/8.06 => ( ord_le9131159989063066194et_nat @ ( image_6725021117256019401et_nat @ ( image_nat_set_nat @ F ) @ ( pow_nat @ A ) ) @ ( pow_set_nat @ B4 ) ) ) ).
% 14.25/8.06
% 14.25/8.06 thf(fact_3172_image__Pow__mono,axiom,
% 14.25/8.06 ! [F: code_integer > code_integer,A: set_Code_integer,B4: set_Code_integer] :
% 14.25/8.06 ( ( ord_le7084787975880047091nteger @ ( image_4470545334726330049nteger @ F @ A ) @ B4 )
% 14.25/8.06 => ( ord_le1914454125413604393nteger @ ( image_6270750266955655469nteger @ ( image_4470545334726330049nteger @ F ) @ ( pow_Code_integer @ A ) ) @ ( pow_Code_integer @ B4 ) ) ) ).
% 14.25/8.06
% 14.25/8.06 thf(fact_3173_image__Pow__mono,axiom,
% 14.25/8.06 ! [F: nat > code_integer,A: set_nat,B4: set_Code_integer] :
% 14.25/8.06 ( ( ord_le7084787975880047091nteger @ ( image_1215581382706833972nteger @ F @ A ) @ B4 )
% 14.25/8.06 => ( ord_le1914454125413604393nteger @ ( image_6916300684244607776nteger @ ( image_1215581382706833972nteger @ F ) @ ( pow_nat @ A ) ) @ ( pow_Code_integer @ B4 ) ) ) ).
% 14.25/8.06
% 14.25/8.06 thf(fact_3174_image__Pow__mono,axiom,
% 14.25/8.06 ! [F: nat > nat,A: set_nat,B4: set_nat] :
% 14.25/8.06 ( ( ord_less_eq_set_nat @ ( image_nat_nat2 @ F @ A ) @ B4 )
% 14.25/8.06 => ( ord_le6893508408891458716et_nat @ ( image_7916887816326733075et_nat @ ( image_nat_nat2 @ F ) @ ( pow_nat @ A ) ) @ ( pow_nat @ B4 ) ) ) ).
% 14.25/8.06
% 14.25/8.06 thf(fact_3175_pairwise__alt,axiom,
% 14.25/8.06 ( pairwi7919127943033184120t_unit
% 14.25/8.06 = ( ^ [R: product_unit > product_unit > $o,S6: set_Product_unit] :
% 14.25/8.06 ! [X2: product_unit] :
% 14.25/8.06 ( ( member_Product_unit @ X2 @ S6 )
% 14.25/8.06 => ! [Y3: product_unit] :
% 14.25/8.06 ( ( member_Product_unit @ Y3 @ ( minus_6452836326544984404t_unit @ S6 @ ( insert_Product_unit @ X2 @ bot_bo3957492148770167129t_unit ) ) )
% 14.25/8.06 => ( R @ X2 @ Y3 ) ) ) ) ) ).
% 14.25/8.06
% 14.25/8.06 thf(fact_3176_pairwise__alt,axiom,
% 14.25/8.06 ( pairwise_o
% 14.25/8.06 = ( ^ [R: $o > $o > $o,S6: set_o] :
% 14.25/8.06 ! [X2: $o] :
% 14.25/8.06 ( ( member_o @ X2 @ S6 )
% 14.25/8.06 => ! [Y3: $o] :
% 14.25/8.06 ( ( member_o @ Y3 @ ( minus_minus_set_o @ S6 @ ( insert_o @ X2 @ bot_bot_set_o ) ) )
% 14.25/8.06 => ( R @ X2 @ Y3 ) ) ) ) ) ).
% 14.25/8.06
% 14.25/8.06 thf(fact_3177_pairwise__alt,axiom,
% 14.25/8.06 ( pairwise_int
% 14.25/8.06 = ( ^ [R: int > int > $o,S6: set_int] :
% 14.25/8.06 ! [X2: int] :
% 14.25/8.06 ( ( member_int @ X2 @ S6 )
% 14.25/8.06 => ! [Y3: int] :
% 14.25/8.06 ( ( member_int @ Y3 @ ( minus_minus_set_int @ S6 @ ( insert_int @ X2 @ bot_bot_set_int ) ) )
% 14.25/8.06 => ( R @ X2 @ Y3 ) ) ) ) ) ).
% 14.25/8.06
% 14.25/8.06 thf(fact_3178_pairwise__alt,axiom,
% 14.25/8.06 ( pairwi2432143822833250616_nat_o
% 14.25/8.06 = ( ^ [R: ( produc3658429121746597890et_nat > $o ) > ( produc3658429121746597890et_nat > $o ) > $o,S6: set_Pr4532377907799695533_nat_o] :
% 14.25/8.06 ! [X2: produc3658429121746597890et_nat > $o] :
% 14.25/8.06 ( ( member6576561426505652726_nat_o @ X2 @ S6 )
% 14.25/8.06 => ! [Y3: produc3658429121746597890et_nat > $o] :
% 14.25/8.06 ( ( member6576561426505652726_nat_o @ Y3 @ ( minus_1801376950450012436_nat_o @ S6 @ ( insert5175938949040314269_nat_o @ X2 @ bot_bo7824918357723582233_nat_o ) ) )
% 14.25/8.06 => ( R @ X2 @ Y3 ) ) ) ) ) ).
% 14.25/8.06
% 14.25/8.06 thf(fact_3179_pairwise__alt,axiom,
% 14.25/8.06 ( pairwise_nat
% 14.25/8.06 = ( ^ [R: nat > nat > $o,S6: set_nat] :
% 14.25/8.06 ! [X2: nat] :
% 14.25/8.06 ( ( member_nat @ X2 @ S6 )
% 14.25/8.06 => ! [Y3: nat] :
% 14.25/8.06 ( ( member_nat @ Y3 @ ( minus_minus_set_nat @ S6 @ ( insert_nat @ X2 @ bot_bot_set_nat ) ) )
% 14.25/8.06 => ( R @ X2 @ Y3 ) ) ) ) ) ).
% 14.25/8.06
% 14.25/8.06 thf(fact_3180_Sup__insert,axiom,
% 14.25/8.06 ! [A2: $o,A: set_o] :
% 14.25/8.06 ( ( complete_Sup_Sup_o @ ( insert_o @ A2 @ A ) )
% 14.25/8.06 = ( sup_sup_o @ A2 @ ( complete_Sup_Sup_o @ A ) ) ) ).
% 14.25/8.06
% 14.25/8.06 thf(fact_3181_Sup__insert,axiom,
% 14.25/8.06 ! [A2: set_nat,A: set_set_nat] :
% 14.25/8.06 ( ( comple7399068483239264473et_nat @ ( insert_set_nat @ A2 @ A ) )
% 14.25/8.06 = ( sup_sup_set_nat @ A2 @ ( comple7399068483239264473et_nat @ A ) ) ) ).
% 14.25/8.06
% 14.25/8.06 thf(fact_3182_Sup__insert,axiom,
% 14.25/8.06 ! [A2: product_unit,A: set_Product_unit] :
% 14.25/8.06 ( ( comple4687483117567863418t_unit @ ( insert_Product_unit @ A2 @ A ) )
% 14.25/8.06 = ( sup_sup_Product_unit @ A2 @ ( comple4687483117567863418t_unit @ A ) ) ) ).
% 14.25/8.06
% 14.25/8.06 thf(fact_3183_ccpo__Sup__singleton,axiom,
% 14.25/8.06 ! [X: $o] :
% 14.25/8.06 ( ( complete_Sup_Sup_o @ ( insert_o @ X @ bot_bot_set_o ) )
% 14.25/8.06 = X ) ).
% 14.25/8.06
% 14.25/8.06 thf(fact_3184_ccpo__Sup__singleton,axiom,
% 14.25/8.06 ! [X: produc3658429121746597890et_nat > $o] :
% 14.25/8.06 ( ( comple4225350640907012154_nat_o @ ( insert5175938949040314269_nat_o @ X @ bot_bo7824918357723582233_nat_o ) )
% 14.25/8.06 = X ) ).
% 14.25/8.06
% 14.25/8.06 thf(fact_3185_ccpo__Sup__singleton,axiom,
% 14.25/8.06 ! [X: product_unit] :
% 14.25/8.06 ( ( comple4687483117567863418t_unit @ ( insert_Product_unit @ X @ bot_bo3957492148770167129t_unit ) )
% 14.25/8.06 = X ) ).
% 14.25/8.06
% 14.25/8.06 thf(fact_3186_cSup__singleton,axiom,
% 14.25/8.06 ! [X: $o] :
% 14.25/8.06 ( ( complete_Sup_Sup_o @ ( insert_o @ X @ bot_bot_set_o ) )
% 14.25/8.06 = X ) ).
% 14.25/8.06
% 14.25/8.06 thf(fact_3187_cSup__singleton,axiom,
% 14.25/8.06 ! [X: int] :
% 14.25/8.06 ( ( complete_Sup_Sup_int @ ( insert_int @ X @ bot_bot_set_int ) )
% 14.25/8.06 = X ) ).
% 14.25/8.06
% 14.25/8.06 thf(fact_3188_cSup__singleton,axiom,
% 14.25/8.06 ! [X: produc3658429121746597890et_nat > $o] :
% 14.25/8.06 ( ( comple4225350640907012154_nat_o @ ( insert5175938949040314269_nat_o @ X @ bot_bo7824918357723582233_nat_o ) )
% 14.25/8.06 = X ) ).
% 14.25/8.06
% 14.25/8.06 thf(fact_3189_cSup__singleton,axiom,
% 14.25/8.06 ! [X: nat] :
% 14.25/8.06 ( ( complete_Sup_Sup_nat @ ( insert_nat @ X @ bot_bot_set_nat ) )
% 14.25/8.06 = X ) ).
% 14.25/8.06
% 14.25/8.06 thf(fact_3190_cSup__singleton,axiom,
% 14.25/8.06 ! [X: product_unit] :
% 14.25/8.06 ( ( comple4687483117567863418t_unit @ ( insert_Product_unit @ X @ bot_bo3957492148770167129t_unit ) )
% 14.25/8.06 = X ) ).
% 14.25/8.06
% 14.25/8.06 thf(fact_3191_Sup__empty,axiom,
% 14.25/8.06 ( ( comple3349607820504768346t_unit @ bot_bo6284674557190688313t_unit )
% 14.25/8.06 = bot_bo3957492148770167129t_unit ) ).
% 14.25/8.06
% 14.25/8.06 thf(fact_3192_Sup__empty,axiom,
% 14.25/8.06 ( ( comple90263536869209701_set_o @ bot_bot_set_set_o )
% 14.25/8.06 = bot_bot_set_o ) ).
% 14.25/8.06
% 14.25/8.06 thf(fact_3193_Sup__empty,axiom,
% 14.25/8.06 ( ( comple7399068483239264473et_nat @ bot_bot_set_set_nat )
% 14.25/8.06 = bot_bot_set_nat ) ).
% 14.25/8.06
% 14.25/8.06 thf(fact_3194_Sup__empty,axiom,
% 14.25/8.06 ( ( comple3221217463730067765et_int @ bot_bot_set_set_int )
% 14.25/8.06 = bot_bot_set_int ) ).
% 14.25/8.06
% 14.25/8.06 thf(fact_3195_Sup__empty,axiom,
% 14.25/8.06 ( ( comple3382087578465172250_nat_o @ bot_bo3736544561982865913_nat_o )
% 14.25/8.06 = bot_bo7824918357723582233_nat_o ) ).
% 14.25/8.06
% 14.25/8.06 thf(fact_3196_Sup__empty,axiom,
% 14.25/8.06 ( ( complete_Sup_Sup_o @ bot_bot_set_o )
% 14.25/8.06 = bot_bot_o ) ).
% 14.25/8.06
% 14.25/8.06 thf(fact_3197_Sup__empty,axiom,
% 14.25/8.06 ( ( comple4225350640907012154_nat_o @ bot_bo7824918357723582233_nat_o )
% 14.25/8.06 = bot_bo6993851861578111033_nat_o ) ).
% 14.25/8.06
% 14.25/8.06 thf(fact_3198_Sup__empty,axiom,
% 14.25/8.06 ( ( comple4687483117567863418t_unit @ bot_bo3957492148770167129t_unit )
% 14.25/8.06 = bot_bot_Product_unit ) ).
% 14.25/8.06
% 14.25/8.06 thf(fact_3199_Sup__bot__conv_I1_J,axiom,
% 14.25/8.06 ! [A: set_set_Product_unit] :
% 14.25/8.06 ( ( ( comple3349607820504768346t_unit @ A )
% 14.25/8.06 = bot_bo3957492148770167129t_unit )
% 14.25/8.06 = ( ! [X2: set_Product_unit] :
% 14.25/8.06 ( ( member5877623283571906838t_unit @ X2 @ A )
% 14.25/8.06 => ( X2 = bot_bo3957492148770167129t_unit ) ) ) ) ).
% 14.25/8.06
% 14.25/8.06 thf(fact_3200_Sup__bot__conv_I1_J,axiom,
% 14.25/8.06 ! [A: set_set_o] :
% 14.25/8.06 ( ( ( comple90263536869209701_set_o @ A )
% 14.25/8.06 = bot_bot_set_o )
% 14.25/8.06 = ( ! [X2: set_o] :
% 14.25/8.06 ( ( member_set_o @ X2 @ A )
% 14.25/8.06 => ( X2 = bot_bot_set_o ) ) ) ) ).
% 14.25/8.06
% 14.25/8.06 thf(fact_3201_Sup__bot__conv_I1_J,axiom,
% 14.25/8.06 ! [A: set_set_nat] :
% 14.25/8.06 ( ( ( comple7399068483239264473et_nat @ A )
% 14.25/8.06 = bot_bot_set_nat )
% 14.25/8.06 = ( ! [X2: set_nat] :
% 14.25/8.06 ( ( member_set_nat @ X2 @ A )
% 14.25/8.06 => ( X2 = bot_bot_set_nat ) ) ) ) ).
% 14.25/8.06
% 14.25/8.06 thf(fact_3202_Sup__bot__conv_I1_J,axiom,
% 14.25/8.06 ! [A: set_set_int] :
% 14.25/8.06 ( ( ( comple3221217463730067765et_int @ A )
% 14.25/8.06 = bot_bot_set_int )
% 14.25/8.06 = ( ! [X2: set_int] :
% 14.25/8.06 ( ( member_set_int @ X2 @ A )
% 14.25/8.06 => ( X2 = bot_bot_set_int ) ) ) ) ).
% 14.25/8.06
% 14.25/8.06 thf(fact_3203_Sup__bot__conv_I1_J,axiom,
% 14.25/8.06 ! [A: set_se1353816476878259085_nat_o] :
% 14.25/8.06 ( ( ( comple3382087578465172250_nat_o @ A )
% 14.25/8.06 = bot_bo7824918357723582233_nat_o )
% 14.25/8.06 = ( ! [X2: set_Pr4532377907799695533_nat_o] :
% 14.25/8.06 ( ( member1452989533377600726_nat_o @ X2 @ A )
% 14.25/8.06 => ( X2 = bot_bo7824918357723582233_nat_o ) ) ) ) ).
% 14.25/8.06
% 14.25/8.06 thf(fact_3204_Sup__bot__conv_I1_J,axiom,
% 14.25/8.06 ! [A: set_Product_unit] :
% 14.25/8.06 ( ( ( comple4687483117567863418t_unit @ A )
% 14.25/8.06 = bot_bot_Product_unit )
% 14.25/8.06 = ( ! [X2: product_unit] :
% 14.25/8.06 ( ( member_Product_unit @ X2 @ A )
% 14.25/8.06 => ( X2 = bot_bot_Product_unit ) ) ) ) ).
% 14.25/8.06
% 14.25/8.06 thf(fact_3205_Sup__bot__conv_I2_J,axiom,
% 14.25/8.06 ! [A: set_set_Product_unit] :
% 14.25/8.06 ( ( bot_bo3957492148770167129t_unit
% 14.25/8.06 = ( comple3349607820504768346t_unit @ A ) )
% 14.25/8.06 = ( ! [X2: set_Product_unit] :
% 14.25/8.06 ( ( member5877623283571906838t_unit @ X2 @ A )
% 14.25/8.06 => ( X2 = bot_bo3957492148770167129t_unit ) ) ) ) ).
% 14.25/8.06
% 14.25/8.06 thf(fact_3206_Sup__bot__conv_I2_J,axiom,
% 14.25/8.06 ! [A: set_set_o] :
% 14.25/8.06 ( ( bot_bot_set_o
% 14.25/8.06 = ( comple90263536869209701_set_o @ A ) )
% 14.25/8.06 = ( ! [X2: set_o] :
% 14.25/8.06 ( ( member_set_o @ X2 @ A )
% 14.25/8.06 => ( X2 = bot_bot_set_o ) ) ) ) ).
% 14.25/8.06
% 14.25/8.06 thf(fact_3207_Sup__bot__conv_I2_J,axiom,
% 14.25/8.06 ! [A: set_set_nat] :
% 14.25/8.06 ( ( bot_bot_set_nat
% 14.25/8.06 = ( comple7399068483239264473et_nat @ A ) )
% 14.25/8.06 = ( ! [X2: set_nat] :
% 14.25/8.06 ( ( member_set_nat @ X2 @ A )
% 14.25/8.06 => ( X2 = bot_bot_set_nat ) ) ) ) ).
% 14.25/8.06
% 14.25/8.06 thf(fact_3208_Sup__bot__conv_I2_J,axiom,
% 14.25/8.06 ! [A: set_set_int] :
% 14.25/8.06 ( ( bot_bot_set_int
% 14.25/8.06 = ( comple3221217463730067765et_int @ A ) )
% 14.25/8.06 = ( ! [X2: set_int] :
% 14.25/8.06 ( ( member_set_int @ X2 @ A )
% 14.25/8.06 => ( X2 = bot_bot_set_int ) ) ) ) ).
% 14.25/8.06
% 14.25/8.06 thf(fact_3209_Sup__bot__conv_I2_J,axiom,
% 14.25/8.06 ! [A: set_se1353816476878259085_nat_o] :
% 14.25/8.06 ( ( bot_bo7824918357723582233_nat_o
% 14.25/8.06 = ( comple3382087578465172250_nat_o @ A ) )
% 14.25/8.06 = ( ! [X2: set_Pr4532377907799695533_nat_o] :
% 14.25/8.06 ( ( member1452989533377600726_nat_o @ X2 @ A )
% 14.25/8.06 => ( X2 = bot_bo7824918357723582233_nat_o ) ) ) ) ).
% 14.25/8.06
% 14.25/8.06 thf(fact_3210_Sup__bot__conv_I2_J,axiom,
% 14.25/8.06 ! [A: set_Product_unit] :
% 14.25/8.06 ( ( bot_bot_Product_unit
% 14.25/8.06 = ( comple4687483117567863418t_unit @ A ) )
% 14.25/8.06 = ( ! [X2: product_unit] :
% 14.25/8.06 ( ( member_Product_unit @ X2 @ A )
% 14.25/8.06 => ( X2 = bot_bot_Product_unit ) ) ) ) ).
% 14.25/8.06
% 14.25/8.06 thf(fact_3211_cSup__eq__Sup__fin,axiom,
% 14.25/8.06 ! [X4: set_o] :
% 14.25/8.06 ( ( finite_finite_o @ X4 )
% 14.25/8.06 => ( ( X4 != bot_bot_set_o )
% 14.25/8.06 => ( ( complete_Sup_Sup_o @ X4 )
% 14.25/8.06 = ( lattic1508158080041050831_fin_o @ X4 ) ) ) ) ).
% 14.25/8.06
% 14.25/8.06 thf(fact_3212_cSup__eq__Sup__fin,axiom,
% 14.25/8.06 ! [X4: set_int] :
% 14.25/8.06 ( ( finite_finite_int @ X4 )
% 14.25/8.06 => ( ( X4 != bot_bot_set_int )
% 14.25/8.06 => ( ( complete_Sup_Sup_int @ X4 )
% 14.25/8.06 = ( lattic1091506334969745077in_int @ X4 ) ) ) ) ).
% 14.25/8.06
% 14.25/8.06 thf(fact_3213_cSup__eq__Sup__fin,axiom,
% 14.25/8.06 ! [X4: set_Pr4532377907799695533_nat_o] :
% 14.25/8.06 ( ( finite3252695134891459830_nat_o @ X4 )
% 14.25/8.06 => ( ( X4 != bot_bo7824918357723582233_nat_o )
% 14.25/8.06 => ( ( comple4225350640907012154_nat_o @ X4 )
% 14.25/8.06 = ( lattic7320199455484906628_nat_o @ X4 ) ) ) ) ).
% 14.25/8.06
% 14.25/8.06 thf(fact_3214_cSup__eq__Sup__fin,axiom,
% 14.25/8.06 ! [X4: set_nat] :
% 14.25/8.06 ( ( finite_finite_nat @ X4 )
% 14.25/8.06 => ( ( X4 != bot_bot_set_nat )
% 14.25/8.06 => ( ( complete_Sup_Sup_nat @ X4 )
% 14.25/8.06 = ( lattic1093996805478795353in_nat @ X4 ) ) ) ) ).
% 14.25/8.06
% 14.25/8.06 thf(fact_3215_cSup__eq__Sup__fin,axiom,
% 14.25/8.06 ! [X4: set_Product_unit] :
% 14.25/8.06 ( ( finite4290736615968046902t_unit @ X4 )
% 14.25/8.06 => ( ( X4 != bot_bo3957492148770167129t_unit )
% 14.25/8.06 => ( ( comple4687483117567863418t_unit @ X4 )
% 14.25/8.06 = ( lattic5294303975357428420t_unit @ X4 ) ) ) ) ).
% 14.25/8.06
% 14.25/8.06 thf(fact_3216_Union__image__insert,axiom,
% 14.25/8.06 ! [F: nat > set_nat,A2: nat,B4: set_nat] :
% 14.25/8.06 ( ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ F @ ( insert_nat @ A2 @ B4 ) ) )
% 14.25/8.06 = ( sup_sup_set_nat @ ( F @ A2 ) @ ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ F @ B4 ) ) ) ) ).
% 14.25/8.06
% 14.25/8.06 thf(fact_3217_Union__image__insert,axiom,
% 14.25/8.06 ! [F: int > set_nat,A2: int,B4: set_int] :
% 14.25/8.06 ( ( comple7399068483239264473et_nat @ ( image_int_set_nat @ F @ ( insert_int @ A2 @ B4 ) ) )
% 14.25/8.06 = ( sup_sup_set_nat @ ( F @ A2 ) @ ( comple7399068483239264473et_nat @ ( image_int_set_nat @ F @ B4 ) ) ) ) ).
% 14.25/8.06
% 14.25/8.06 thf(fact_3218_Union__image__insert,axiom,
% 14.25/8.06 ! [F: $o > set_nat,A2: $o,B4: set_o] :
% 14.25/8.06 ( ( comple7399068483239264473et_nat @ ( image_o_set_nat @ F @ ( insert_o @ A2 @ B4 ) ) )
% 14.25/8.06 = ( sup_sup_set_nat @ ( F @ A2 ) @ ( comple7399068483239264473et_nat @ ( image_o_set_nat @ F @ B4 ) ) ) ) ).
% 14.25/8.06
% 14.25/8.06 thf(fact_3219_Union__image__insert,axiom,
% 14.25/8.06 ! [F: product_unit > set_nat,A2: product_unit,B4: set_Product_unit] :
% 14.25/8.06 ( ( comple7399068483239264473et_nat @ ( image_8173849991827986510et_nat @ F @ ( insert_Product_unit @ A2 @ B4 ) ) )
% 14.25/8.06 = ( sup_sup_set_nat @ ( F @ A2 ) @ ( comple7399068483239264473et_nat @ ( image_8173849991827986510et_nat @ F @ B4 ) ) ) ) ).
% 14.25/8.06
% 14.25/8.06 thf(fact_3220_Union__image__empty,axiom,
% 14.25/8.06 ! [A: set_nat,F: product_unit > set_nat] :
% 14.25/8.06 ( ( sup_sup_set_nat @ A @ ( comple7399068483239264473et_nat @ ( image_8173849991827986510et_nat @ F @ bot_bo3957492148770167129t_unit ) ) )
% 14.25/8.06 = A ) ).
% 14.25/8.06
% 14.25/8.06 thf(fact_3221_Union__image__empty,axiom,
% 14.25/8.06 ! [A: set_nat,F: $o > set_nat] :
% 14.25/8.06 ( ( sup_sup_set_nat @ A @ ( comple7399068483239264473et_nat @ ( image_o_set_nat @ F @ bot_bot_set_o ) ) )
% 14.25/8.06 = A ) ).
% 14.25/8.06
% 14.25/8.06 thf(fact_3222_Union__image__empty,axiom,
% 14.25/8.06 ! [A: set_nat,F: nat > set_nat] :
% 14.25/8.06 ( ( sup_sup_set_nat @ A @ ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ F @ bot_bot_set_nat ) ) )
% 14.25/8.06 = A ) ).
% 14.25/8.06
% 14.25/8.06 thf(fact_3223_Union__image__empty,axiom,
% 14.25/8.06 ! [A: set_nat,F: int > set_nat] :
% 14.25/8.06 ( ( sup_sup_set_nat @ A @ ( comple7399068483239264473et_nat @ ( image_int_set_nat @ F @ bot_bot_set_int ) ) )
% 14.25/8.06 = A ) ).
% 14.25/8.06
% 14.25/8.06 thf(fact_3224_Union__image__empty,axiom,
% 14.25/8.06 ! [A: set_nat,F: ( produc3658429121746597890et_nat > $o ) > set_nat] :
% 14.25/8.06 ( ( sup_sup_set_nat @ A @ ( comple7399068483239264473et_nat @ ( image_6888641473154130958et_nat @ F @ bot_bo7824918357723582233_nat_o ) ) )
% 14.25/8.06 = A ) ).
% 14.25/8.06
% 14.25/8.06 thf(fact_3225_Sup__SUP__eq,axiom,
% 14.25/8.06 ( comple3590721164184202819at_o_o
% 14.25/8.06 = ( ^ [S6: set_Pr6409761941060870310at_o_o,X2: produc3658429121746597890et_nat > $o] : ( member6576561426505652726_nat_o @ X2 @ ( comple3382087578465172250_nat_o @ ( image_8639947063239330806_nat_o @ collec939566748876313656_nat_o @ S6 ) ) ) ) ) ).
% 14.25/8.06
% 14.25/8.06 thf(fact_3226_Sup__SUP__eq,axiom,
% 14.25/8.06 ( comple8317665133742190828_nat_o
% 14.25/8.06 = ( ^ [S6: set_nat_o,X2: nat] : ( member_nat @ X2 @ ( comple7399068483239264473et_nat @ ( image_nat_o_set_nat @ collect_nat @ S6 ) ) ) ) ) ).
% 14.25/8.06
% 14.25/8.06 thf(fact_3227_Sup__SUP__eq,axiom,
% 14.25/8.06 ( comple6496622788309502864_int_o
% 14.25/8.06 = ( ^ [S6: set_int_o,X2: int] : ( member_int @ X2 @ ( comple3221217463730067765et_int @ ( image_int_o_set_int @ collect_int @ S6 ) ) ) ) ) ).
% 14.25/8.06
% 14.25/8.06 thf(fact_3228_Sup__SUP__eq,axiom,
% 14.25/8.06 ( complete_Sup_Sup_o_o
% 14.25/8.06 = ( ^ [S6: set_o_o,X2: $o] : ( member_o @ X2 @ ( comple90263536869209701_set_o @ ( image_o_o_set_o @ collect_o @ S6 ) ) ) ) ) ).
% 14.25/8.06
% 14.25/8.06 thf(fact_3229_Sup__SUP__eq,axiom,
% 14.25/8.06 ( comple4773641153059738115unit_o
% 14.25/8.06 = ( ^ [S6: set_Product_unit_o,X2: product_unit] : ( member_Product_unit @ X2 @ ( comple3349607820504768346t_unit @ ( image_5138372075374782070t_unit @ collect_Product_unit @ S6 ) ) ) ) ) ).
% 14.25/8.06
% 14.25/8.06 thf(fact_3230_empty__Union__conv,axiom,
% 14.25/8.06 ! [A: set_set_Product_unit] :
% 14.25/8.06 ( ( bot_bo3957492148770167129t_unit
% 14.25/8.06 = ( comple3349607820504768346t_unit @ A ) )
% 14.25/8.06 = ( ! [X2: set_Product_unit] :
% 14.25/8.06 ( ( member5877623283571906838t_unit @ X2 @ A )
% 14.25/8.06 => ( X2 = bot_bo3957492148770167129t_unit ) ) ) ) ).
% 14.25/8.06
% 14.25/8.06 thf(fact_3231_empty__Union__conv,axiom,
% 14.25/8.06 ! [A: set_set_o] :
% 14.25/8.06 ( ( bot_bot_set_o
% 14.25/8.06 = ( comple90263536869209701_set_o @ A ) )
% 14.25/8.06 = ( ! [X2: set_o] :
% 14.25/8.06 ( ( member_set_o @ X2 @ A )
% 14.25/8.06 => ( X2 = bot_bot_set_o ) ) ) ) ).
% 14.25/8.06
% 14.25/8.06 thf(fact_3232_empty__Union__conv,axiom,
% 14.25/8.06 ! [A: set_set_nat] :
% 14.25/8.06 ( ( bot_bot_set_nat
% 14.25/8.06 = ( comple7399068483239264473et_nat @ A ) )
% 14.25/8.06 = ( ! [X2: set_nat] :
% 14.25/8.06 ( ( member_set_nat @ X2 @ A )
% 14.25/8.06 => ( X2 = bot_bot_set_nat ) ) ) ) ).
% 14.25/8.06
% 14.25/8.06 thf(fact_3233_empty__Union__conv,axiom,
% 14.25/8.06 ! [A: set_set_int] :
% 14.25/8.06 ( ( bot_bot_set_int
% 14.25/8.06 = ( comple3221217463730067765et_int @ A ) )
% 14.25/8.06 = ( ! [X2: set_int] :
% 14.25/8.06 ( ( member_set_int @ X2 @ A )
% 14.25/8.06 => ( X2 = bot_bot_set_int ) ) ) ) ).
% 14.25/8.06
% 14.25/8.06 thf(fact_3234_empty__Union__conv,axiom,
% 14.25/8.06 ! [A: set_se1353816476878259085_nat_o] :
% 14.25/8.06 ( ( bot_bo7824918357723582233_nat_o
% 14.25/8.06 = ( comple3382087578465172250_nat_o @ A ) )
% 14.25/8.06 = ( ! [X2: set_Pr4532377907799695533_nat_o] :
% 14.25/8.06 ( ( member1452989533377600726_nat_o @ X2 @ A )
% 14.25/8.06 => ( X2 = bot_bo7824918357723582233_nat_o ) ) ) ) ).
% 14.25/8.06
% 14.25/8.06 thf(fact_3235_Union__empty__conv,axiom,
% 14.25/8.06 ! [A: set_set_Product_unit] :
% 14.25/8.06 ( ( ( comple3349607820504768346t_unit @ A )
% 14.25/8.06 = bot_bo3957492148770167129t_unit )
% 14.25/8.06 = ( ! [X2: set_Product_unit] :
% 14.25/8.06 ( ( member5877623283571906838t_unit @ X2 @ A )
% 14.25/8.06 => ( X2 = bot_bo3957492148770167129t_unit ) ) ) ) ).
% 14.25/8.06
% 14.25/8.06 thf(fact_3236_Union__empty__conv,axiom,
% 14.25/8.06 ! [A: set_set_o] :
% 14.25/8.06 ( ( ( comple90263536869209701_set_o @ A )
% 14.25/8.06 = bot_bot_set_o )
% 14.25/8.06 = ( ! [X2: set_o] :
% 14.25/8.06 ( ( member_set_o @ X2 @ A )
% 14.25/8.06 => ( X2 = bot_bot_set_o ) ) ) ) ).
% 14.25/8.06
% 14.25/8.06 thf(fact_3237_Union__empty__conv,axiom,
% 14.25/8.06 ! [A: set_set_nat] :
% 14.25/8.06 ( ( ( comple7399068483239264473et_nat @ A )
% 14.25/8.06 = bot_bot_set_nat )
% 14.25/8.06 = ( ! [X2: set_nat] :
% 14.25/8.06 ( ( member_set_nat @ X2 @ A )
% 14.25/8.06 => ( X2 = bot_bot_set_nat ) ) ) ) ).
% 14.25/8.06
% 14.25/8.06 thf(fact_3238_Union__empty__conv,axiom,
% 14.25/8.06 ! [A: set_set_int] :
% 14.25/8.06 ( ( ( comple3221217463730067765et_int @ A )
% 14.25/8.06 = bot_bot_set_int )
% 14.25/8.06 = ( ! [X2: set_int] :
% 14.25/8.06 ( ( member_set_int @ X2 @ A )
% 14.25/8.06 => ( X2 = bot_bot_set_int ) ) ) ) ).
% 14.25/8.06
% 14.25/8.06 thf(fact_3239_Union__empty__conv,axiom,
% 14.25/8.06 ! [A: set_se1353816476878259085_nat_o] :
% 14.25/8.06 ( ( ( comple3382087578465172250_nat_o @ A )
% 14.25/8.06 = bot_bo7824918357723582233_nat_o )
% 14.25/8.06 = ( ! [X2: set_Pr4532377907799695533_nat_o] :
% 14.25/8.06 ( ( member1452989533377600726_nat_o @ X2 @ A )
% 14.25/8.06 => ( X2 = bot_bo7824918357723582233_nat_o ) ) ) ) ).
% 14.25/8.06
% 14.25/8.06 thf(fact_3240_less__eq__Sup,axiom,
% 14.25/8.06 ! [A: set_o,U: $o] :
% 14.25/8.06 ( ! [V3: $o] :
% 14.25/8.06 ( ( member_o @ V3 @ A )
% 14.25/8.06 => ( ord_less_eq_o @ U @ V3 ) )
% 14.25/8.06 => ( ( A != bot_bot_set_o )
% 14.25/8.06 => ( ord_less_eq_o @ U @ ( complete_Sup_Sup_o @ A ) ) ) ) ).
% 14.25/8.06
% 14.25/8.06 thf(fact_3241_less__eq__Sup,axiom,
% 14.25/8.06 ! [A: set_Pr4532377907799695533_nat_o,U: produc3658429121746597890et_nat > $o] :
% 14.25/8.06 ( ! [V3: produc3658429121746597890et_nat > $o] :
% 14.25/8.06 ( ( member6576561426505652726_nat_o @ V3 @ A )
% 14.25/8.06 => ( ord_le729326519192465773_nat_o @ U @ V3 ) )
% 14.25/8.06 => ( ( A != bot_bo7824918357723582233_nat_o )
% 14.25/8.06 => ( ord_le729326519192465773_nat_o @ U @ ( comple4225350640907012154_nat_o @ A ) ) ) ) ).
% 14.25/8.06
% 14.25/8.06 thf(fact_3242_less__eq__Sup,axiom,
% 14.25/8.06 ! [A: set_filter_nat,U: filter_nat] :
% 14.25/8.06 ( ! [V3: filter_nat] :
% 14.25/8.06 ( ( member_filter_nat @ V3 @ A )
% 14.25/8.06 => ( ord_le2510731241096832064er_nat @ U @ V3 ) )
% 14.25/8.06 => ( ( A != bot_bo498966703094740906er_nat )
% 14.25/8.06 => ( ord_le2510731241096832064er_nat @ U @ ( comple2454491259644266675er_nat @ A ) ) ) ) ).
% 14.25/8.06
% 14.25/8.06 thf(fact_3243_less__eq__Sup,axiom,
% 14.25/8.06 ! [A: set_set_nat,U: set_nat] :
% 14.25/8.06 ( ! [V3: set_nat] :
% 14.25/8.06 ( ( member_set_nat @ V3 @ A )
% 14.25/8.06 => ( ord_less_eq_set_nat @ U @ V3 ) )
% 14.25/8.06 => ( ( A != bot_bot_set_set_nat )
% 14.25/8.06 => ( ord_less_eq_set_nat @ U @ ( comple7399068483239264473et_nat @ A ) ) ) ) ).
% 14.25/8.06
% 14.25/8.06 thf(fact_3244_less__eq__Sup,axiom,
% 14.25/8.06 ! [A: set_Product_unit,U: product_unit] :
% 14.25/8.06 ( ! [V3: product_unit] :
% 14.25/8.06 ( ( member_Product_unit @ V3 @ A )
% 14.25/8.06 => ( ord_le3221252021190050221t_unit @ U @ V3 ) )
% 14.25/8.06 => ( ( A != bot_bo3957492148770167129t_unit )
% 14.25/8.06 => ( ord_le3221252021190050221t_unit @ U @ ( comple4687483117567863418t_unit @ A ) ) ) ) ).
% 14.25/8.06
% 14.25/8.06 thf(fact_3245_cSup__least,axiom,
% 14.25/8.06 ! [X4: set_o,Z: $o] :
% 14.25/8.06 ( ( X4 != bot_bot_set_o )
% 14.25/8.06 => ( ! [X3: $o] :
% 14.25/8.06 ( ( member_o @ X3 @ X4 )
% 14.25/8.06 => ( ord_less_eq_o @ X3 @ Z ) )
% 14.25/8.06 => ( ord_less_eq_o @ ( complete_Sup_Sup_o @ X4 ) @ Z ) ) ) ).
% 14.25/8.06
% 14.25/8.06 thf(fact_3246_cSup__least,axiom,
% 14.25/8.06 ! [X4: set_Pr4532377907799695533_nat_o,Z: produc3658429121746597890et_nat > $o] :
% 14.25/8.06 ( ( X4 != bot_bo7824918357723582233_nat_o )
% 14.25/8.06 => ( ! [X3: produc3658429121746597890et_nat > $o] :
% 14.25/8.06 ( ( member6576561426505652726_nat_o @ X3 @ X4 )
% 14.25/8.06 => ( ord_le729326519192465773_nat_o @ X3 @ Z ) )
% 14.25/8.06 => ( ord_le729326519192465773_nat_o @ ( comple4225350640907012154_nat_o @ X4 ) @ Z ) ) ) ).
% 14.25/8.06
% 14.25/8.06 thf(fact_3247_cSup__least,axiom,
% 14.25/8.06 ! [X4: set_filter_nat,Z: filter_nat] :
% 14.25/8.06 ( ( X4 != bot_bo498966703094740906er_nat )
% 14.25/8.06 => ( ! [X3: filter_nat] :
% 14.25/8.06 ( ( member_filter_nat @ X3 @ X4 )
% 14.25/8.06 => ( ord_le2510731241096832064er_nat @ X3 @ Z ) )
% 14.25/8.06 => ( ord_le2510731241096832064er_nat @ ( comple2454491259644266675er_nat @ X4 ) @ Z ) ) ) ).
% 14.25/8.06
% 14.25/8.06 thf(fact_3248_cSup__least,axiom,
% 14.25/8.06 ! [X4: set_set_nat,Z: set_nat] :
% 14.25/8.06 ( ( X4 != bot_bot_set_set_nat )
% 14.25/8.06 => ( ! [X3: set_nat] :
% 14.25/8.06 ( ( member_set_nat @ X3 @ X4 )
% 14.25/8.06 => ( ord_less_eq_set_nat @ X3 @ Z ) )
% 14.25/8.06 => ( ord_less_eq_set_nat @ ( comple7399068483239264473et_nat @ X4 ) @ Z ) ) ) ).
% 14.25/8.06
% 14.25/8.06 thf(fact_3249_cSup__least,axiom,
% 14.25/8.06 ! [X4: set_int,Z: int] :
% 14.25/8.06 ( ( X4 != bot_bot_set_int )
% 14.25/8.06 => ( ! [X3: int] :
% 14.25/8.06 ( ( member_int @ X3 @ X4 )
% 14.25/8.06 => ( ord_less_eq_int @ X3 @ Z ) )
% 14.25/8.06 => ( ord_less_eq_int @ ( complete_Sup_Sup_int @ X4 ) @ Z ) ) ) ).
% 14.25/8.06
% 14.25/8.06 thf(fact_3250_cSup__least,axiom,
% 14.25/8.06 ! [X4: set_nat,Z: nat] :
% 14.25/8.06 ( ( X4 != bot_bot_set_nat )
% 14.25/8.06 => ( ! [X3: nat] :
% 14.25/8.06 ( ( member_nat @ X3 @ X4 )
% 14.25/8.06 => ( ord_less_eq_nat @ X3 @ Z ) )
% 14.25/8.06 => ( ord_less_eq_nat @ ( complete_Sup_Sup_nat @ X4 ) @ Z ) ) ) ).
% 14.25/8.06
% 14.25/8.06 thf(fact_3251_cSup__least,axiom,
% 14.25/8.06 ! [X4: set_Product_unit,Z: product_unit] :
% 14.25/8.06 ( ( X4 != bot_bo3957492148770167129t_unit )
% 14.25/8.06 => ( ! [X3: product_unit] :
% 14.25/8.06 ( ( member_Product_unit @ X3 @ X4 )
% 14.25/8.06 => ( ord_le3221252021190050221t_unit @ X3 @ Z ) )
% 14.25/8.06 => ( ord_le3221252021190050221t_unit @ ( comple4687483117567863418t_unit @ X4 ) @ Z ) ) ) ).
% 14.25/8.06
% 14.25/8.06 thf(fact_3252_cSup__eq__non__empty,axiom,
% 14.25/8.06 ! [X4: set_o,A2: $o] :
% 14.25/8.06 ( ( X4 != bot_bot_set_o )
% 14.25/8.06 => ( ! [X3: $o] :
% 14.25/8.06 ( ( member_o @ X3 @ X4 )
% 14.25/8.06 => ( ord_less_eq_o @ X3 @ A2 ) )
% 14.25/8.06 => ( ! [Y2: $o] :
% 14.25/8.06 ( ! [X6: $o] :
% 14.25/8.06 ( ( member_o @ X6 @ X4 )
% 14.25/8.06 => ( ord_less_eq_o @ X6 @ Y2 ) )
% 14.25/8.06 => ( ord_less_eq_o @ A2 @ Y2 ) )
% 14.25/8.06 => ( ( complete_Sup_Sup_o @ X4 )
% 14.25/8.06 = A2 ) ) ) ) ).
% 14.25/8.06
% 14.25/8.06 thf(fact_3253_cSup__eq__non__empty,axiom,
% 14.25/8.06 ! [X4: set_Pr4532377907799695533_nat_o,A2: produc3658429121746597890et_nat > $o] :
% 14.25/8.06 ( ( X4 != bot_bo7824918357723582233_nat_o )
% 14.25/8.06 => ( ! [X3: produc3658429121746597890et_nat > $o] :
% 14.25/8.06 ( ( member6576561426505652726_nat_o @ X3 @ X4 )
% 14.25/8.06 => ( ord_le729326519192465773_nat_o @ X3 @ A2 ) )
% 14.25/8.06 => ( ! [Y2: produc3658429121746597890et_nat > $o] :
% 14.25/8.06 ( ! [X6: produc3658429121746597890et_nat > $o] :
% 14.25/8.06 ( ( member6576561426505652726_nat_o @ X6 @ X4 )
% 14.25/8.06 => ( ord_le729326519192465773_nat_o @ X6 @ Y2 ) )
% 14.25/8.06 => ( ord_le729326519192465773_nat_o @ A2 @ Y2 ) )
% 14.25/8.06 => ( ( comple4225350640907012154_nat_o @ X4 )
% 14.25/8.06 = A2 ) ) ) ) ).
% 14.25/8.06
% 14.25/8.06 thf(fact_3254_cSup__eq__non__empty,axiom,
% 14.25/8.06 ! [X4: set_filter_nat,A2: filter_nat] :
% 14.25/8.06 ( ( X4 != bot_bo498966703094740906er_nat )
% 14.25/8.06 => ( ! [X3: filter_nat] :
% 14.25/8.06 ( ( member_filter_nat @ X3 @ X4 )
% 14.25/8.06 => ( ord_le2510731241096832064er_nat @ X3 @ A2 ) )
% 14.25/8.06 => ( ! [Y2: filter_nat] :
% 14.25/8.06 ( ! [X6: filter_nat] :
% 14.25/8.06 ( ( member_filter_nat @ X6 @ X4 )
% 14.25/8.06 => ( ord_le2510731241096832064er_nat @ X6 @ Y2 ) )
% 14.25/8.06 => ( ord_le2510731241096832064er_nat @ A2 @ Y2 ) )
% 14.25/8.06 => ( ( comple2454491259644266675er_nat @ X4 )
% 14.25/8.06 = A2 ) ) ) ) ).
% 14.25/8.06
% 14.25/8.06 thf(fact_3255_cSup__eq__non__empty,axiom,
% 14.25/8.06 ! [X4: set_set_nat,A2: set_nat] :
% 14.25/8.06 ( ( X4 != bot_bot_set_set_nat )
% 14.25/8.06 => ( ! [X3: set_nat] :
% 14.25/8.06 ( ( member_set_nat @ X3 @ X4 )
% 14.25/8.06 => ( ord_less_eq_set_nat @ X3 @ A2 ) )
% 14.25/8.06 => ( ! [Y2: set_nat] :
% 14.25/8.06 ( ! [X6: set_nat] :
% 14.25/8.06 ( ( member_set_nat @ X6 @ X4 )
% 14.25/8.06 => ( ord_less_eq_set_nat @ X6 @ Y2 ) )
% 14.25/8.06 => ( ord_less_eq_set_nat @ A2 @ Y2 ) )
% 14.25/8.06 => ( ( comple7399068483239264473et_nat @ X4 )
% 14.25/8.06 = A2 ) ) ) ) ).
% 14.25/8.06
% 14.25/8.06 thf(fact_3256_cSup__eq__non__empty,axiom,
% 14.25/8.06 ! [X4: set_int,A2: int] :
% 14.25/8.06 ( ( X4 != bot_bot_set_int )
% 14.25/8.06 => ( ! [X3: int] :
% 14.25/8.06 ( ( member_int @ X3 @ X4 )
% 14.25/8.06 => ( ord_less_eq_int @ X3 @ A2 ) )
% 14.25/8.06 => ( ! [Y2: int] :
% 14.25/8.06 ( ! [X6: int] :
% 14.25/8.06 ( ( member_int @ X6 @ X4 )
% 14.25/8.06 => ( ord_less_eq_int @ X6 @ Y2 ) )
% 14.25/8.06 => ( ord_less_eq_int @ A2 @ Y2 ) )
% 14.25/8.06 => ( ( complete_Sup_Sup_int @ X4 )
% 14.25/8.06 = A2 ) ) ) ) ).
% 14.25/8.06
% 14.25/8.06 thf(fact_3257_cSup__eq__non__empty,axiom,
% 14.25/8.06 ! [X4: set_nat,A2: nat] :
% 14.25/8.06 ( ( X4 != bot_bot_set_nat )
% 14.25/8.06 => ( ! [X3: nat] :
% 14.25/8.06 ( ( member_nat @ X3 @ X4 )
% 14.25/8.06 => ( ord_less_eq_nat @ X3 @ A2 ) )
% 14.25/8.06 => ( ! [Y2: nat] :
% 14.25/8.06 ( ! [X6: nat] :
% 14.25/8.06 ( ( member_nat @ X6 @ X4 )
% 14.25/8.06 => ( ord_less_eq_nat @ X6 @ Y2 ) )
% 14.25/8.06 => ( ord_less_eq_nat @ A2 @ Y2 ) )
% 14.25/8.06 => ( ( complete_Sup_Sup_nat @ X4 )
% 14.25/8.06 = A2 ) ) ) ) ).
% 14.25/8.06
% 14.25/8.06 thf(fact_3258_cSup__eq__non__empty,axiom,
% 14.25/8.06 ! [X4: set_Product_unit,A2: product_unit] :
% 14.25/8.06 ( ( X4 != bot_bo3957492148770167129t_unit )
% 14.25/8.06 => ( ! [X3: product_unit] :
% 14.25/8.06 ( ( member_Product_unit @ X3 @ X4 )
% 14.25/8.06 => ( ord_le3221252021190050221t_unit @ X3 @ A2 ) )
% 14.25/8.06 => ( ! [Y2: product_unit] :
% 14.25/8.06 ( ! [X6: product_unit] :
% 14.25/8.06 ( ( member_Product_unit @ X6 @ X4 )
% 14.25/8.06 => ( ord_le3221252021190050221t_unit @ X6 @ Y2 ) )
% 14.25/8.06 => ( ord_le3221252021190050221t_unit @ A2 @ Y2 ) )
% 14.25/8.06 => ( ( comple4687483117567863418t_unit @ X4 )
% 14.25/8.06 = A2 ) ) ) ) ).
% 14.25/8.06
% 14.25/8.06 thf(fact_3259_SUP__eq__const,axiom,
% 14.25/8.06 ! [I: set_nat,F: nat > set_nat,X: set_nat] :
% 14.25/8.06 ( ( I != bot_bot_set_nat )
% 14.25/8.06 => ( ! [I3: nat] :
% 14.25/8.06 ( ( member_nat @ I3 @ I )
% 14.25/8.06 => ( ( F @ I3 )
% 14.25/8.06 = X ) )
% 14.25/8.06 => ( ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ F @ I ) )
% 14.25/8.06 = X ) ) ) ).
% 14.25/8.06
% 14.25/8.06 thf(fact_3260_SUP__eq__const,axiom,
% 14.25/8.06 ! [I: set_Product_unit,F: product_unit > product_unit,X: product_unit] :
% 14.25/8.06 ( ( I != bot_bo3957492148770167129t_unit )
% 14.25/8.06 => ( ! [I3: product_unit] :
% 14.25/8.06 ( ( member_Product_unit @ I3 @ I )
% 14.25/8.06 => ( ( F @ I3 )
% 14.25/8.06 = X ) )
% 14.25/8.06 => ( ( comple4687483117567863418t_unit @ ( image_405062704495631173t_unit @ F @ I ) )
% 14.25/8.06 = X ) ) ) ).
% 14.25/8.06
% 14.25/8.06 thf(fact_3261_SUP__eq__const,axiom,
% 14.25/8.06 ! [I: set_o,F: $o > product_unit,X: product_unit] :
% 14.25/8.06 ( ( I != bot_bot_set_o )
% 14.25/8.06 => ( ! [I3: $o] :
% 14.25/8.06 ( ( member_o @ I3 @ I )
% 14.25/8.06 => ( ( F @ I3 )
% 14.25/8.06 = X ) )
% 14.25/8.06 => ( ( comple4687483117567863418t_unit @ ( image_o_Product_unit2 @ F @ I ) )
% 14.25/8.06 = X ) ) ) ).
% 14.25/8.06
% 14.25/8.06 thf(fact_3262_SUP__eq__const,axiom,
% 14.25/8.06 ! [I: set_nat,F: nat > product_unit,X: product_unit] :
% 14.25/8.06 ( ( I != bot_bot_set_nat )
% 14.25/8.06 => ( ! [I3: nat] :
% 14.25/8.06 ( ( member_nat @ I3 @ I )
% 14.25/8.06 => ( ( F @ I3 )
% 14.25/8.06 = X ) )
% 14.25/8.06 => ( ( comple4687483117567863418t_unit @ ( image_8730104196221521654t_unit @ F @ I ) )
% 14.25/8.06 = X ) ) ) ).
% 14.25/8.06
% 14.25/8.06 thf(fact_3263_SUP__eq__const,axiom,
% 14.25/8.06 ! [I: set_int,F: int > product_unit,X: product_unit] :
% 14.25/8.06 ( ( I != bot_bot_set_int )
% 14.25/8.06 => ( ! [I3: int] :
% 14.25/8.06 ( ( member_int @ I3 @ I )
% 14.25/8.06 => ( ( F @ I3 )
% 14.25/8.06 = X ) )
% 14.25/8.06 => ( ( comple4687483117567863418t_unit @ ( image_5846526143423701914t_unit @ F @ I ) )
% 14.25/8.06 = X ) ) ) ).
% 14.25/8.06
% 14.25/8.06 thf(fact_3264_SUP__eq__const,axiom,
% 14.25/8.06 ! [I: set_Pr4532377907799695533_nat_o,F: ( produc3658429121746597890et_nat > $o ) > product_unit,X: product_unit] :
% 14.25/8.06 ( ( I != bot_bo7824918357723582233_nat_o )
% 14.25/8.06 => ( ! [I3: produc3658429121746597890et_nat > $o] :
% 14.25/8.06 ( ( member6576561426505652726_nat_o @ I3 @ I )
% 14.25/8.06 => ( ( F @ I3 )
% 14.25/8.06 = X ) )
% 14.25/8.06 => ( ( comple4687483117567863418t_unit @ ( image_8451000719036923781t_unit @ F @ I ) )
% 14.25/8.06 = X ) ) ) ).
% 14.25/8.06
% 14.25/8.06 thf(fact_3265_Union__disjoint,axiom,
% 14.25/8.06 ! [C5: set_set_Product_unit,A: set_Product_unit] :
% 14.25/8.06 ( ( ( inf_in4660618365625256667t_unit @ ( comple3349607820504768346t_unit @ C5 ) @ A )
% 14.25/8.06 = bot_bo3957492148770167129t_unit )
% 14.25/8.06 = ( ! [X2: set_Product_unit] :
% 14.25/8.06 ( ( member5877623283571906838t_unit @ X2 @ C5 )
% 14.25/8.06 => ( ( inf_in4660618365625256667t_unit @ X2 @ A )
% 14.25/8.06 = bot_bo3957492148770167129t_unit ) ) ) ) ).
% 14.25/8.06
% 14.25/8.06 thf(fact_3266_Union__disjoint,axiom,
% 14.25/8.06 ! [C5: set_set_o,A: set_o] :
% 14.25/8.06 ( ( ( inf_inf_set_o @ ( comple90263536869209701_set_o @ C5 ) @ A )
% 14.25/8.06 = bot_bot_set_o )
% 14.25/8.06 = ( ! [X2: set_o] :
% 14.25/8.06 ( ( member_set_o @ X2 @ C5 )
% 14.25/8.06 => ( ( inf_inf_set_o @ X2 @ A )
% 14.25/8.06 = bot_bot_set_o ) ) ) ) ).
% 14.25/8.06
% 14.25/8.06 thf(fact_3267_Union__disjoint,axiom,
% 14.25/8.06 ! [C5: set_set_nat,A: set_nat] :
% 14.25/8.06 ( ( ( inf_inf_set_nat @ ( comple7399068483239264473et_nat @ C5 ) @ A )
% 14.25/8.06 = bot_bot_set_nat )
% 14.25/8.06 = ( ! [X2: set_nat] :
% 14.25/8.06 ( ( member_set_nat @ X2 @ C5 )
% 14.25/8.06 => ( ( inf_inf_set_nat @ X2 @ A )
% 14.25/8.06 = bot_bot_set_nat ) ) ) ) ).
% 14.25/8.06
% 14.25/8.06 thf(fact_3268_Union__disjoint,axiom,
% 14.25/8.06 ! [C5: set_set_int,A: set_int] :
% 14.25/8.06 ( ( ( inf_inf_set_int @ ( comple3221217463730067765et_int @ C5 ) @ A )
% 14.25/8.06 = bot_bot_set_int )
% 14.25/8.06 = ( ! [X2: set_int] :
% 14.25/8.06 ( ( member_set_int @ X2 @ C5 )
% 14.25/8.06 => ( ( inf_inf_set_int @ X2 @ A )
% 14.25/8.06 = bot_bot_set_int ) ) ) ) ).
% 14.25/8.06
% 14.25/8.06 thf(fact_3269_Union__disjoint,axiom,
% 14.25/8.06 ! [C5: set_se1353816476878259085_nat_o,A: set_Pr4532377907799695533_nat_o] :
% 14.25/8.06 ( ( ( inf_in1906310914598751387_nat_o @ ( comple3382087578465172250_nat_o @ C5 ) @ A )
% 14.25/8.06 = bot_bo7824918357723582233_nat_o )
% 14.25/8.06 = ( ! [X2: set_Pr4532377907799695533_nat_o] :
% 14.25/8.06 ( ( member1452989533377600726_nat_o @ X2 @ C5 )
% 14.25/8.06 => ( ( inf_in1906310914598751387_nat_o @ X2 @ A )
% 14.25/8.06 = bot_bo7824918357723582233_nat_o ) ) ) ) ).
% 14.25/8.06
% 14.25/8.06 thf(fact_3270_Union__empty,axiom,
% 14.25/8.06 ( ( comple3349607820504768346t_unit @ bot_bo6284674557190688313t_unit )
% 14.25/8.06 = bot_bo3957492148770167129t_unit ) ).
% 14.25/8.06
% 14.25/8.06 thf(fact_3271_Union__empty,axiom,
% 14.25/8.06 ( ( comple90263536869209701_set_o @ bot_bot_set_set_o )
% 14.25/8.06 = bot_bot_set_o ) ).
% 14.25/8.06
% 14.25/8.06 thf(fact_3272_Union__empty,axiom,
% 14.25/8.06 ( ( comple7399068483239264473et_nat @ bot_bot_set_set_nat )
% 14.25/8.06 = bot_bot_set_nat ) ).
% 14.25/8.06
% 14.25/8.06 thf(fact_3273_Union__empty,axiom,
% 14.25/8.06 ( ( comple3221217463730067765et_int @ bot_bot_set_set_int )
% 14.25/8.06 = bot_bot_set_int ) ).
% 14.25/8.06
% 14.25/8.06 thf(fact_3274_Union__empty,axiom,
% 14.25/8.06 ( ( comple3382087578465172250_nat_o @ bot_bo3736544561982865913_nat_o )
% 14.25/8.06 = bot_bo7824918357723582233_nat_o ) ).
% 14.25/8.06
% 14.25/8.06 thf(fact_3275_SUP__eq__iff,axiom,
% 14.25/8.06 ! [I: set_Product_unit,C: product_unit,F: product_unit > product_unit] :
% 14.25/8.06 ( ( I != bot_bo3957492148770167129t_unit )
% 14.25/8.06 => ( ! [I3: product_unit] :
% 14.25/8.06 ( ( member_Product_unit @ I3 @ I )
% 14.25/8.06 => ( ord_le3221252021190050221t_unit @ C @ ( F @ I3 ) ) )
% 14.25/8.06 => ( ( ( comple4687483117567863418t_unit @ ( image_405062704495631173t_unit @ F @ I ) )
% 14.25/8.06 = C )
% 14.25/8.06 = ( ! [X2: product_unit] :
% 14.25/8.06 ( ( member_Product_unit @ X2 @ I )
% 14.25/8.06 => ( ( F @ X2 )
% 14.25/8.06 = C ) ) ) ) ) ) ).
% 14.25/8.06
% 14.25/8.06 thf(fact_3276_SUP__eq__iff,axiom,
% 14.25/8.06 ! [I: set_o,C: product_unit,F: $o > product_unit] :
% 14.25/8.06 ( ( I != bot_bot_set_o )
% 14.25/8.06 => ( ! [I3: $o] :
% 14.25/8.06 ( ( member_o @ I3 @ I )
% 14.25/8.06 => ( ord_le3221252021190050221t_unit @ C @ ( F @ I3 ) ) )
% 14.25/8.06 => ( ( ( comple4687483117567863418t_unit @ ( image_o_Product_unit2 @ F @ I ) )
% 14.25/8.06 = C )
% 14.25/8.06 = ( ! [X2: $o] :
% 14.25/8.06 ( ( member_o @ X2 @ I )
% 14.25/8.06 => ( ( F @ X2 )
% 14.25/8.06 = C ) ) ) ) ) ) ).
% 14.25/8.06
% 14.25/8.06 thf(fact_3277_SUP__eq__iff,axiom,
% 14.25/8.06 ! [I: set_nat,C: product_unit,F: nat > product_unit] :
% 14.25/8.06 ( ( I != bot_bot_set_nat )
% 14.25/8.06 => ( ! [I3: nat] :
% 14.25/8.06 ( ( member_nat @ I3 @ I )
% 14.25/8.06 => ( ord_le3221252021190050221t_unit @ C @ ( F @ I3 ) ) )
% 14.25/8.06 => ( ( ( comple4687483117567863418t_unit @ ( image_8730104196221521654t_unit @ F @ I ) )
% 14.25/8.06 = C )
% 14.25/8.06 = ( ! [X2: nat] :
% 14.25/8.06 ( ( member_nat @ X2 @ I )
% 14.25/8.06 => ( ( F @ X2 )
% 14.25/8.06 = C ) ) ) ) ) ) ).
% 14.25/8.06
% 14.25/8.06 thf(fact_3278_SUP__eq__iff,axiom,
% 14.25/8.06 ! [I: set_int,C: product_unit,F: int > product_unit] :
% 14.25/8.06 ( ( I != bot_bot_set_int )
% 14.25/8.06 => ( ! [I3: int] :
% 14.25/8.06 ( ( member_int @ I3 @ I )
% 14.25/8.06 => ( ord_le3221252021190050221t_unit @ C @ ( F @ I3 ) ) )
% 14.25/8.06 => ( ( ( comple4687483117567863418t_unit @ ( image_5846526143423701914t_unit @ F @ I ) )
% 14.25/8.06 = C )
% 14.25/8.06 = ( ! [X2: int] :
% 14.25/8.06 ( ( member_int @ X2 @ I )
% 14.25/8.06 => ( ( F @ X2 )
% 14.25/8.06 = C ) ) ) ) ) ) ).
% 14.25/8.06
% 14.25/8.06 thf(fact_3279_SUP__eq__iff,axiom,
% 14.25/8.06 ! [I: set_Product_unit,C: filter_nat,F: product_unit > filter_nat] :
% 14.25/8.06 ( ( I != bot_bo3957492148770167129t_unit )
% 14.25/8.06 => ( ! [I3: product_unit] :
% 14.25/8.06 ( ( member_Product_unit @ I3 @ I )
% 14.25/8.06 => ( ord_le2510731241096832064er_nat @ C @ ( F @ I3 ) ) )
% 14.25/8.06 => ( ( ( comple2454491259644266675er_nat @ ( image_1853478764106424488er_nat @ F @ I ) )
% 14.25/8.06 = C )
% 14.25/8.06 = ( ! [X2: product_unit] :
% 14.25/8.06 ( ( member_Product_unit @ X2 @ I )
% 14.25/8.06 => ( ( F @ X2 )
% 14.25/8.06 = C ) ) ) ) ) ) ).
% 14.25/8.06
% 14.25/8.06 thf(fact_3280_SUP__eq__iff,axiom,
% 14.25/8.06 ! [I: set_o,C: filter_nat,F: $o > filter_nat] :
% 14.25/8.06 ( ( I != bot_bot_set_o )
% 14.25/8.06 => ( ! [I3: $o] :
% 14.25/8.06 ( ( member_o @ I3 @ I )
% 14.25/8.06 => ( ord_le2510731241096832064er_nat @ C @ ( F @ I3 ) ) )
% 14.25/8.06 => ( ( ( comple2454491259644266675er_nat @ ( image_o_filter_nat @ F @ I ) )
% 14.25/8.06 = C )
% 14.25/8.06 = ( ! [X2: $o] :
% 14.25/8.06 ( ( member_o @ X2 @ I )
% 14.25/8.06 => ( ( F @ X2 )
% 14.25/8.06 = C ) ) ) ) ) ) ).
% 14.25/8.06
% 14.25/8.06 thf(fact_3281_SUP__eq__iff,axiom,
% 14.25/8.06 ! [I: set_nat,C: filter_nat,F: nat > filter_nat] :
% 14.25/8.06 ( ( I != bot_bot_set_nat )
% 14.25/8.06 => ( ! [I3: nat] :
% 14.25/8.06 ( ( member_nat @ I3 @ I )
% 14.25/8.06 => ( ord_le2510731241096832064er_nat @ C @ ( F @ I3 ) ) )
% 14.25/8.06 => ( ( ( comple2454491259644266675er_nat @ ( image_nat_filter_nat @ F @ I ) )
% 14.25/8.06 = C )
% 14.25/8.06 = ( ! [X2: nat] :
% 14.25/8.06 ( ( member_nat @ X2 @ I )
% 14.25/8.06 => ( ( F @ X2 )
% 14.25/8.06 = C ) ) ) ) ) ) ).
% 14.25/8.06
% 14.25/8.06 thf(fact_3282_SUP__eq__iff,axiom,
% 14.25/8.06 ! [I: set_int,C: filter_nat,F: int > filter_nat] :
% 14.25/8.06 ( ( I != bot_bot_set_int )
% 14.25/8.06 => ( ! [I3: int] :
% 14.25/8.06 ( ( member_int @ I3 @ I )
% 14.25/8.06 => ( ord_le2510731241096832064er_nat @ C @ ( F @ I3 ) ) )
% 14.25/8.06 => ( ( ( comple2454491259644266675er_nat @ ( image_int_filter_nat @ F @ I ) )
% 14.25/8.06 = C )
% 14.25/8.06 = ( ! [X2: int] :
% 14.25/8.06 ( ( member_int @ X2 @ I )
% 14.25/8.06 => ( ( F @ X2 )
% 14.25/8.06 = C ) ) ) ) ) ) ).
% 14.25/8.06
% 14.25/8.06 thf(fact_3283_SUP__eq__iff,axiom,
% 14.25/8.06 ! [I: set_Product_unit,C: set_nat,F: product_unit > set_nat] :
% 14.25/8.06 ( ( I != bot_bo3957492148770167129t_unit )
% 14.25/8.06 => ( ! [I3: product_unit] :
% 14.25/8.06 ( ( member_Product_unit @ I3 @ I )
% 14.25/8.06 => ( ord_less_eq_set_nat @ C @ ( F @ I3 ) ) )
% 14.25/8.06 => ( ( ( comple7399068483239264473et_nat @ ( image_8173849991827986510et_nat @ F @ I ) )
% 14.25/8.06 = C )
% 14.25/8.06 = ( ! [X2: product_unit] :
% 14.25/8.06 ( ( member_Product_unit @ X2 @ I )
% 14.25/8.06 => ( ( F @ X2 )
% 14.25/8.06 = C ) ) ) ) ) ) ).
% 14.25/8.06
% 14.25/8.06 thf(fact_3284_SUP__eq__iff,axiom,
% 14.25/8.06 ! [I: set_o,C: set_nat,F: $o > set_nat] :
% 14.25/8.06 ( ( I != bot_bot_set_o )
% 14.25/8.06 => ( ! [I3: $o] :
% 14.25/8.06 ( ( member_o @ I3 @ I )
% 14.25/8.06 => ( ord_less_eq_set_nat @ C @ ( F @ I3 ) ) )
% 14.25/8.06 => ( ( ( comple7399068483239264473et_nat @ ( image_o_set_nat @ F @ I ) )
% 14.25/8.06 = C )
% 14.25/8.06 = ( ! [X2: $o] :
% 14.25/8.06 ( ( member_o @ X2 @ I )
% 14.25/8.06 => ( ( F @ X2 )
% 14.25/8.06 = C ) ) ) ) ) ) ).
% 14.25/8.06
% 14.25/8.06 thf(fact_3285_cSUP__least,axiom,
% 14.25/8.06 ! [A: set_Product_unit,F: product_unit > int,M: int] :
% 14.25/8.06 ( ( A != bot_bo3957492148770167129t_unit )
% 14.25/8.06 => ( ! [X3: product_unit] :
% 14.25/8.06 ( ( member_Product_unit @ X3 @ A )
% 14.25/8.06 => ( ord_less_eq_int @ ( F @ X3 ) @ M ) )
% 14.25/8.06 => ( ord_less_eq_int @ ( complete_Sup_Sup_int @ ( image_873079544045703924it_int @ F @ A ) ) @ M ) ) ) ).
% 14.25/8.06
% 14.25/8.06 thf(fact_3286_cSUP__least,axiom,
% 14.25/8.06 ! [A: set_o,F: $o > int,M: int] :
% 14.25/8.06 ( ( A != bot_bot_set_o )
% 14.25/8.06 => ( ! [X3: $o] :
% 14.25/8.06 ( ( member_o @ X3 @ A )
% 14.25/8.06 => ( ord_less_eq_int @ ( F @ X3 ) @ M ) )
% 14.25/8.06 => ( ord_less_eq_int @ ( complete_Sup_Sup_int @ ( image_o_int2 @ F @ A ) ) @ M ) ) ) ).
% 14.25/8.06
% 14.25/8.06 thf(fact_3287_cSUP__least,axiom,
% 14.25/8.06 ! [A: set_nat,F: nat > int,M: int] :
% 14.25/8.06 ( ( A != bot_bot_set_nat )
% 14.25/8.06 => ( ! [X3: nat] :
% 14.25/8.06 ( ( member_nat @ X3 @ A )
% 14.25/8.06 => ( ord_less_eq_int @ ( F @ X3 ) @ M ) )
% 14.25/8.06 => ( ord_less_eq_int @ ( complete_Sup_Sup_int @ ( image_nat_int2 @ F @ A ) ) @ M ) ) ) ).
% 14.25/8.06
% 14.25/8.06 thf(fact_3288_cSUP__least,axiom,
% 14.25/8.06 ! [A: set_int,F: int > int,M: int] :
% 14.25/8.06 ( ( A != bot_bot_set_int )
% 14.25/8.06 => ( ! [X3: int] :
% 14.25/8.06 ( ( member_int @ X3 @ A )
% 14.25/8.06 => ( ord_less_eq_int @ ( F @ X3 ) @ M ) )
% 14.25/8.06 => ( ord_less_eq_int @ ( complete_Sup_Sup_int @ ( image_int_int2 @ F @ A ) ) @ M ) ) ) ).
% 14.25/8.06
% 14.25/8.06 thf(fact_3289_cSUP__least,axiom,
% 14.25/8.06 ! [A: set_Product_unit,F: product_unit > nat,M: nat] :
% 14.25/8.06 ( ( A != bot_bo3957492148770167129t_unit )
% 14.25/8.06 => ( ! [X3: product_unit] :
% 14.25/8.06 ( ( member_Product_unit @ X3 @ A )
% 14.25/8.06 => ( ord_less_eq_nat @ ( F @ X3 ) @ M ) )
% 14.25/8.06 => ( ord_less_eq_nat @ ( complete_Sup_Sup_nat @ ( image_875570014554754200it_nat @ F @ A ) ) @ M ) ) ) ).
% 14.25/8.06
% 14.25/8.06 thf(fact_3290_cSUP__least,axiom,
% 14.25/8.06 ! [A: set_o,F: $o > nat,M: nat] :
% 14.25/8.06 ( ( A != bot_bot_set_o )
% 14.25/8.06 => ( ! [X3: $o] :
% 14.25/8.06 ( ( member_o @ X3 @ A )
% 14.25/8.06 => ( ord_less_eq_nat @ ( F @ X3 ) @ M ) )
% 14.25/8.06 => ( ord_less_eq_nat @ ( complete_Sup_Sup_nat @ ( image_o_nat2 @ F @ A ) ) @ M ) ) ) ).
% 14.25/8.06
% 14.25/8.06 thf(fact_3291_cSUP__least,axiom,
% 14.25/8.06 ! [A: set_nat,F: nat > nat,M: nat] :
% 14.25/8.06 ( ( A != bot_bot_set_nat )
% 14.25/8.06 => ( ! [X3: nat] :
% 14.25/8.06 ( ( member_nat @ X3 @ A )
% 14.25/8.06 => ( ord_less_eq_nat @ ( F @ X3 ) @ M ) )
% 14.25/8.06 => ( ord_less_eq_nat @ ( complete_Sup_Sup_nat @ ( image_nat_nat2 @ F @ A ) ) @ M ) ) ) ).
% 14.25/8.06
% 14.25/8.06 thf(fact_3292_cSUP__least,axiom,
% 14.25/8.06 ! [A: set_int,F: int > nat,M: nat] :
% 14.25/8.06 ( ( A != bot_bot_set_int )
% 14.25/8.06 => ( ! [X3: int] :
% 14.25/8.06 ( ( member_int @ X3 @ A )
% 14.25/8.06 => ( ord_less_eq_nat @ ( F @ X3 ) @ M ) )
% 14.25/8.06 => ( ord_less_eq_nat @ ( complete_Sup_Sup_nat @ ( image_int_nat2 @ F @ A ) ) @ M ) ) ) ).
% 14.25/8.06
% 14.25/8.06 thf(fact_3293_cSUP__least,axiom,
% 14.25/8.06 ! [A: set_Product_unit,F: product_unit > product_unit,M: product_unit] :
% 14.25/8.06 ( ( A != bot_bo3957492148770167129t_unit )
% 14.25/8.06 => ( ! [X3: product_unit] :
% 14.25/8.06 ( ( member_Product_unit @ X3 @ A )
% 14.25/8.06 => ( ord_le3221252021190050221t_unit @ ( F @ X3 ) @ M ) )
% 14.25/8.06 => ( ord_le3221252021190050221t_unit @ ( comple4687483117567863418t_unit @ ( image_405062704495631173t_unit @ F @ A ) ) @ M ) ) ) ).
% 14.25/8.06
% 14.25/8.06 thf(fact_3294_cSUP__least,axiom,
% 14.25/8.06 ! [A: set_o,F: $o > product_unit,M: product_unit] :
% 14.25/8.06 ( ( A != bot_bot_set_o )
% 14.25/8.06 => ( ! [X3: $o] :
% 14.25/8.06 ( ( member_o @ X3 @ A )
% 14.25/8.06 => ( ord_le3221252021190050221t_unit @ ( F @ X3 ) @ M ) )
% 14.25/8.06 => ( ord_le3221252021190050221t_unit @ ( comple4687483117567863418t_unit @ ( image_o_Product_unit2 @ F @ A ) ) @ M ) ) ) ).
% 14.25/8.06
% 14.25/8.06 thf(fact_3295_Sup__inf__eq__bot__iff,axiom,
% 14.25/8.06 ! [B4: set_set_Product_unit,A2: set_Product_unit] :
% 14.25/8.06 ( ( ( inf_in4660618365625256667t_unit @ ( comple3349607820504768346t_unit @ B4 ) @ A2 )
% 14.25/8.06 = bot_bo3957492148770167129t_unit )
% 14.25/8.06 = ( ! [X2: set_Product_unit] :
% 14.25/8.06 ( ( member5877623283571906838t_unit @ X2 @ B4 )
% 14.25/8.06 => ( ( inf_in4660618365625256667t_unit @ X2 @ A2 )
% 14.25/8.06 = bot_bo3957492148770167129t_unit ) ) ) ) ).
% 14.25/8.06
% 14.25/8.06 thf(fact_3296_Sup__inf__eq__bot__iff,axiom,
% 14.25/8.06 ! [B4: set_set_o,A2: set_o] :
% 14.25/8.06 ( ( ( inf_inf_set_o @ ( comple90263536869209701_set_o @ B4 ) @ A2 )
% 14.25/8.06 = bot_bot_set_o )
% 14.25/8.06 = ( ! [X2: set_o] :
% 14.25/8.06 ( ( member_set_o @ X2 @ B4 )
% 14.25/8.06 => ( ( inf_inf_set_o @ X2 @ A2 )
% 14.25/8.06 = bot_bot_set_o ) ) ) ) ).
% 14.25/8.06
% 14.25/8.06 thf(fact_3297_Sup__inf__eq__bot__iff,axiom,
% 14.25/8.06 ! [B4: set_set_nat,A2: set_nat] :
% 14.25/8.06 ( ( ( inf_inf_set_nat @ ( comple7399068483239264473et_nat @ B4 ) @ A2 )
% 14.25/8.06 = bot_bot_set_nat )
% 14.25/8.06 = ( ! [X2: set_nat] :
% 14.25/8.06 ( ( member_set_nat @ X2 @ B4 )
% 14.25/8.06 => ( ( inf_inf_set_nat @ X2 @ A2 )
% 14.25/8.06 = bot_bot_set_nat ) ) ) ) ).
% 14.25/8.06
% 14.25/8.06 thf(fact_3298_Sup__inf__eq__bot__iff,axiom,
% 14.25/8.06 ! [B4: set_set_int,A2: set_int] :
% 14.25/8.06 ( ( ( inf_inf_set_int @ ( comple3221217463730067765et_int @ B4 ) @ A2 )
% 14.25/8.06 = bot_bot_set_int )
% 14.25/8.06 = ( ! [X2: set_int] :
% 14.25/8.06 ( ( member_set_int @ X2 @ B4 )
% 14.25/8.06 => ( ( inf_inf_set_int @ X2 @ A2 )
% 14.25/8.06 = bot_bot_set_int ) ) ) ) ).
% 14.25/8.06
% 14.25/8.06 thf(fact_3299_Sup__inf__eq__bot__iff,axiom,
% 14.25/8.06 ! [B4: set_se1353816476878259085_nat_o,A2: set_Pr4532377907799695533_nat_o] :
% 14.25/8.06 ( ( ( inf_in1906310914598751387_nat_o @ ( comple3382087578465172250_nat_o @ B4 ) @ A2 )
% 14.25/8.06 = bot_bo7824918357723582233_nat_o )
% 14.25/8.06 = ( ! [X2: set_Pr4532377907799695533_nat_o] :
% 14.25/8.06 ( ( member1452989533377600726_nat_o @ X2 @ B4 )
% 14.25/8.06 => ( ( inf_in1906310914598751387_nat_o @ X2 @ A2 )
% 14.25/8.06 = bot_bo7824918357723582233_nat_o ) ) ) ) ).
% 14.25/8.06
% 14.25/8.06 thf(fact_3300_Sup__inf__eq__bot__iff,axiom,
% 14.25/8.06 ! [B4: set_Product_unit,A2: product_unit] :
% 14.25/8.06 ( ( ( inf_inf_Product_unit @ ( comple4687483117567863418t_unit @ B4 ) @ A2 )
% 14.25/8.06 = bot_bot_Product_unit )
% 14.25/8.06 = ( ! [X2: product_unit] :
% 14.25/8.06 ( ( member_Product_unit @ X2 @ B4 )
% 14.25/8.06 => ( ( inf_inf_Product_unit @ X2 @ A2 )
% 14.25/8.06 = bot_bot_Product_unit ) ) ) ) ).
% 14.25/8.06
% 14.25/8.06 thf(fact_3301_UNION__fun__upd,axiom,
% 14.25/8.06 ! [A: product_unit > set_Product_unit,I2: product_unit,B4: set_Product_unit,J2: set_Product_unit] :
% 14.25/8.06 ( ( comple3349607820504768346t_unit @ ( image_7778121751606633509t_unit @ ( fun_up488859312337516701t_unit @ A @ I2 @ B4 ) @ J2 ) )
% 14.25/8.06 = ( sup_su793286257634532545t_unit @ ( comple3349607820504768346t_unit @ ( image_7778121751606633509t_unit @ A @ ( minus_6452836326544984404t_unit @ J2 @ ( insert_Product_unit @ I2 @ bot_bo3957492148770167129t_unit ) ) ) ) @ ( if_set_Product_unit @ ( member_Product_unit @ I2 @ J2 ) @ B4 @ bot_bo3957492148770167129t_unit ) ) ) ).
% 14.25/8.06
% 14.25/8.06 thf(fact_3302_UNION__fun__upd,axiom,
% 14.25/8.06 ! [A: product_unit > set_o,I2: product_unit,B4: set_o,J2: set_Product_unit] :
% 14.25/8.06 ( ( comple90263536869209701_set_o @ ( image_6930536745178679216_set_o @ ( fun_up1762901246852280360_set_o @ A @ I2 @ B4 ) @ J2 ) )
% 14.25/8.06 = ( sup_sup_set_o @ ( comple90263536869209701_set_o @ ( image_6930536745178679216_set_o @ A @ ( minus_6452836326544984404t_unit @ J2 @ ( insert_Product_unit @ I2 @ bot_bo3957492148770167129t_unit ) ) ) ) @ ( if_set_o @ ( member_Product_unit @ I2 @ J2 ) @ B4 @ bot_bot_set_o ) ) ) ).
% 14.25/8.06
% 14.25/8.06 thf(fact_3303_UNION__fun__upd,axiom,
% 14.25/8.06 ! [A: product_unit > set_nat,I2: product_unit,B4: set_nat,J2: set_Product_unit] :
% 14.25/8.06 ( ( comple7399068483239264473et_nat @ ( image_8173849991827986510et_nat @ ( fun_up4569834200710200790et_nat @ A @ I2 @ B4 ) @ J2 ) )
% 14.25/8.06 = ( sup_sup_set_nat @ ( comple7399068483239264473et_nat @ ( image_8173849991827986510et_nat @ A @ ( minus_6452836326544984404t_unit @ J2 @ ( insert_Product_unit @ I2 @ bot_bo3957492148770167129t_unit ) ) ) ) @ ( if_set_nat @ ( member_Product_unit @ I2 @ J2 ) @ B4 @ bot_bot_set_nat ) ) ) ).
% 14.25/8.06
% 14.25/8.06 thf(fact_3304_UNION__fun__upd,axiom,
% 14.25/8.06 ! [A: product_unit > set_int,I2: product_unit,B4: set_int,J2: set_Product_unit] :
% 14.25/8.06 ( ( comple3221217463730067765et_int @ ( image_3995998972318789802et_int @ ( fun_up391983181201004082et_int @ A @ I2 @ B4 ) @ J2 ) )
% 14.25/8.06 = ( sup_sup_set_int @ ( comple3221217463730067765et_int @ ( image_3995998972318789802et_int @ A @ ( minus_6452836326544984404t_unit @ J2 @ ( insert_Product_unit @ I2 @ bot_bo3957492148770167129t_unit ) ) ) ) @ ( if_set_int @ ( member_Product_unit @ I2 @ J2 ) @ B4 @ bot_bot_set_int ) ) ) ).
% 14.25/8.06
% 14.25/8.06 thf(fact_3305_UNION__fun__upd,axiom,
% 14.25/8.06 ! [A: $o > set_Product_unit,I2: $o,B4: set_Product_unit,J2: set_o] :
% 14.25/8.06 ( ( comple3349607820504768346t_unit @ ( image_663195431845683034t_unit @ ( fun_up4718931970374059986t_unit @ A @ I2 @ B4 ) @ J2 ) )
% 14.25/8.06 = ( sup_su793286257634532545t_unit @ ( comple3349607820504768346t_unit @ ( image_663195431845683034t_unit @ A @ ( minus_minus_set_o @ J2 @ ( insert_o @ I2 @ bot_bot_set_o ) ) ) ) @ ( if_set_Product_unit @ ( member_o @ I2 @ J2 ) @ B4 @ bot_bo3957492148770167129t_unit ) ) ) ).
% 14.25/8.06
% 14.25/8.06 thf(fact_3306_UNION__fun__upd,axiom,
% 14.25/8.06 ! [A: $o > set_o,I2: $o,B4: set_o,J2: set_o] :
% 14.25/8.06 ( ( comple90263536869209701_set_o @ ( image_o_set_o @ ( fun_upd_o_set_o @ A @ I2 @ B4 ) @ J2 ) )
% 14.25/8.06 = ( sup_sup_set_o @ ( comple90263536869209701_set_o @ ( image_o_set_o @ A @ ( minus_minus_set_o @ J2 @ ( insert_o @ I2 @ bot_bot_set_o ) ) ) ) @ ( if_set_o @ ( member_o @ I2 @ J2 ) @ B4 @ bot_bot_set_o ) ) ) ).
% 14.25/8.06
% 14.25/8.06 thf(fact_3307_UNION__fun__upd,axiom,
% 14.25/8.06 ! [A: $o > set_nat,I2: $o,B4: set_nat,J2: set_o] :
% 14.25/8.06 ( ( comple7399068483239264473et_nat @ ( image_o_set_nat @ ( fun_upd_o_set_nat @ A @ I2 @ B4 ) @ J2 ) )
% 14.25/8.06 = ( sup_sup_set_nat @ ( comple7399068483239264473et_nat @ ( image_o_set_nat @ A @ ( minus_minus_set_o @ J2 @ ( insert_o @ I2 @ bot_bot_set_o ) ) ) ) @ ( if_set_nat @ ( member_o @ I2 @ J2 ) @ B4 @ bot_bot_set_nat ) ) ) ).
% 14.25/8.06
% 14.25/8.06 thf(fact_3308_UNION__fun__upd,axiom,
% 14.25/8.06 ! [A: $o > set_int,I2: $o,B4: set_int,J2: set_o] :
% 14.25/8.06 ( ( comple3221217463730067765et_int @ ( image_o_set_int @ ( fun_upd_o_set_int @ A @ I2 @ B4 ) @ J2 ) )
% 14.25/8.06 = ( sup_sup_set_int @ ( comple3221217463730067765et_int @ ( image_o_set_int @ A @ ( minus_minus_set_o @ J2 @ ( insert_o @ I2 @ bot_bot_set_o ) ) ) ) @ ( if_set_int @ ( member_o @ I2 @ J2 ) @ B4 @ bot_bot_set_int ) ) ) ).
% 14.25/8.06
% 14.25/8.06 thf(fact_3309_UNION__fun__upd,axiom,
% 14.25/8.06 ! [A: int > set_Product_unit,I2: int,B4: set_Product_unit,J2: set_int] :
% 14.25/8.06 ( ( comple3349607820504768346t_unit @ ( image_8658272795163236474t_unit @ ( fun_up5054257004045450754t_unit @ A @ I2 @ B4 ) @ J2 ) )
% 14.25/8.06 = ( sup_su793286257634532545t_unit @ ( comple3349607820504768346t_unit @ ( image_8658272795163236474t_unit @ A @ ( minus_minus_set_int @ J2 @ ( insert_int @ I2 @ bot_bot_set_int ) ) ) ) @ ( if_set_Product_unit @ ( member_int @ I2 @ J2 ) @ B4 @ bot_bo3957492148770167129t_unit ) ) ) ).
% 14.25/8.06
% 14.25/8.06 thf(fact_3310_UNION__fun__upd,axiom,
% 14.25/8.06 ! [A: int > set_o,I2: int,B4: set_o,J2: set_int] :
% 14.25/8.06 ( ( comple90263536869209701_set_o @ ( image_int_set_o @ ( fun_upd_int_set_o @ A @ I2 @ B4 ) @ J2 ) )
% 14.25/8.06 = ( sup_sup_set_o @ ( comple90263536869209701_set_o @ ( image_int_set_o @ A @ ( minus_minus_set_int @ J2 @ ( insert_int @ I2 @ bot_bot_set_int ) ) ) ) @ ( if_set_o @ ( member_int @ I2 @ J2 ) @ B4 @ bot_bot_set_o ) ) ) ).
% 14.25/8.06
% 14.25/8.06 thf(fact_3311_cSUP__union,axiom,
% 14.25/8.06 ! [A: set_Product_unit,F: product_unit > nat,B4: set_Product_unit] :
% 14.25/8.06 ( ( A != bot_bo3957492148770167129t_unit )
% 14.25/8.06 => ( ( condit2214826472909112428ve_nat @ ( image_875570014554754200it_nat @ F @ A ) )
% 14.25/8.06 => ( ( B4 != bot_bo3957492148770167129t_unit )
% 14.25/8.06 => ( ( condit2214826472909112428ve_nat @ ( image_875570014554754200it_nat @ F @ B4 ) )
% 14.25/8.06 => ( ( complete_Sup_Sup_nat @ ( image_875570014554754200it_nat @ F @ ( sup_su793286257634532545t_unit @ A @ B4 ) ) )
% 14.25/8.06 = ( sup_sup_nat @ ( complete_Sup_Sup_nat @ ( image_875570014554754200it_nat @ F @ A ) ) @ ( complete_Sup_Sup_nat @ ( image_875570014554754200it_nat @ F @ B4 ) ) ) ) ) ) ) ) ).
% 14.25/8.06
% 14.25/8.06 thf(fact_3312_cSUP__union,axiom,
% 14.25/8.06 ! [A: set_o,F: $o > nat,B4: set_o] :
% 14.25/8.06 ( ( A != bot_bot_set_o )
% 14.25/8.06 => ( ( condit2214826472909112428ve_nat @ ( image_o_nat2 @ F @ A ) )
% 14.25/8.06 => ( ( B4 != bot_bot_set_o )
% 14.25/8.06 => ( ( condit2214826472909112428ve_nat @ ( image_o_nat2 @ F @ B4 ) )
% 14.25/8.06 => ( ( complete_Sup_Sup_nat @ ( image_o_nat2 @ F @ ( sup_sup_set_o @ A @ B4 ) ) )
% 14.25/8.06 = ( sup_sup_nat @ ( complete_Sup_Sup_nat @ ( image_o_nat2 @ F @ A ) ) @ ( complete_Sup_Sup_nat @ ( image_o_nat2 @ F @ B4 ) ) ) ) ) ) ) ) ).
% 14.25/8.06
% 14.25/8.06 thf(fact_3313_cSUP__union,axiom,
% 14.25/8.06 ! [A: set_nat,F: nat > nat,B4: set_nat] :
% 14.25/8.06 ( ( A != bot_bot_set_nat )
% 14.25/8.06 => ( ( condit2214826472909112428ve_nat @ ( image_nat_nat2 @ F @ A ) )
% 14.25/8.06 => ( ( B4 != bot_bot_set_nat )
% 14.25/8.06 => ( ( condit2214826472909112428ve_nat @ ( image_nat_nat2 @ F @ B4 ) )
% 14.25/8.06 => ( ( complete_Sup_Sup_nat @ ( image_nat_nat2 @ F @ ( sup_sup_set_nat @ A @ B4 ) ) )
% 14.25/8.06 = ( sup_sup_nat @ ( complete_Sup_Sup_nat @ ( image_nat_nat2 @ F @ A ) ) @ ( complete_Sup_Sup_nat @ ( image_nat_nat2 @ F @ B4 ) ) ) ) ) ) ) ) ).
% 14.25/8.06
% 14.25/8.06 thf(fact_3314_cSUP__union,axiom,
% 14.25/8.06 ! [A: set_int,F: int > nat,B4: set_int] :
% 14.25/8.06 ( ( A != bot_bot_set_int )
% 14.25/8.06 => ( ( condit2214826472909112428ve_nat @ ( image_int_nat2 @ F @ A ) )
% 14.25/8.06 => ( ( B4 != bot_bot_set_int )
% 14.25/8.06 => ( ( condit2214826472909112428ve_nat @ ( image_int_nat2 @ F @ B4 ) )
% 14.25/8.06 => ( ( complete_Sup_Sup_nat @ ( image_int_nat2 @ F @ ( sup_sup_set_int @ A @ B4 ) ) )
% 14.25/8.06 = ( sup_sup_nat @ ( complete_Sup_Sup_nat @ ( image_int_nat2 @ F @ A ) ) @ ( complete_Sup_Sup_nat @ ( image_int_nat2 @ F @ B4 ) ) ) ) ) ) ) ) ).
% 14.25/8.06
% 14.25/8.06 thf(fact_3315_cSUP__union,axiom,
% 14.25/8.06 ! [A: set_Product_unit,F: product_unit > product_unit,B4: set_Product_unit] :
% 14.25/8.06 ( ( A != bot_bo3957492148770167129t_unit )
% 14.25/8.06 => ( ( condit2080475194861589617t_unit @ ( image_405062704495631173t_unit @ F @ A ) )
% 14.25/8.06 => ( ( B4 != bot_bo3957492148770167129t_unit )
% 14.25/8.06 => ( ( condit2080475194861589617t_unit @ ( image_405062704495631173t_unit @ F @ B4 ) )
% 14.25/8.06 => ( ( comple4687483117567863418t_unit @ ( image_405062704495631173t_unit @ F @ ( sup_su793286257634532545t_unit @ A @ B4 ) ) )
% 14.25/8.06 = ( sup_sup_Product_unit @ ( comple4687483117567863418t_unit @ ( image_405062704495631173t_unit @ F @ A ) ) @ ( comple4687483117567863418t_unit @ ( image_405062704495631173t_unit @ F @ B4 ) ) ) ) ) ) ) ) ).
% 14.25/8.06
% 14.25/8.06 thf(fact_3316_cSUP__union,axiom,
% 14.25/8.06 ! [A: set_o,F: $o > product_unit,B4: set_o] :
% 14.25/8.06 ( ( A != bot_bot_set_o )
% 14.25/8.06 => ( ( condit2080475194861589617t_unit @ ( image_o_Product_unit2 @ F @ A ) )
% 14.25/8.06 => ( ( B4 != bot_bot_set_o )
% 14.25/8.06 => ( ( condit2080475194861589617t_unit @ ( image_o_Product_unit2 @ F @ B4 ) )
% 14.25/8.06 => ( ( comple4687483117567863418t_unit @ ( image_o_Product_unit2 @ F @ ( sup_sup_set_o @ A @ B4 ) ) )
% 14.25/8.06 = ( sup_sup_Product_unit @ ( comple4687483117567863418t_unit @ ( image_o_Product_unit2 @ F @ A ) ) @ ( comple4687483117567863418t_unit @ ( image_o_Product_unit2 @ F @ B4 ) ) ) ) ) ) ) ) ).
% 14.25/8.06
% 14.25/8.06 thf(fact_3317_cSUP__union,axiom,
% 14.25/8.06 ! [A: set_nat,F: nat > product_unit,B4: set_nat] :
% 14.25/8.06 ( ( A != bot_bot_set_nat )
% 14.25/8.06 => ( ( condit2080475194861589617t_unit @ ( image_8730104196221521654t_unit @ F @ A ) )
% 14.25/8.06 => ( ( B4 != bot_bot_set_nat )
% 14.25/8.06 => ( ( condit2080475194861589617t_unit @ ( image_8730104196221521654t_unit @ F @ B4 ) )
% 14.25/8.06 => ( ( comple4687483117567863418t_unit @ ( image_8730104196221521654t_unit @ F @ ( sup_sup_set_nat @ A @ B4 ) ) )
% 14.25/8.06 = ( sup_sup_Product_unit @ ( comple4687483117567863418t_unit @ ( image_8730104196221521654t_unit @ F @ A ) ) @ ( comple4687483117567863418t_unit @ ( image_8730104196221521654t_unit @ F @ B4 ) ) ) ) ) ) ) ) ).
% 14.25/8.06
% 14.25/8.06 thf(fact_3318_cSUP__union,axiom,
% 14.25/8.06 ! [A: set_int,F: int > product_unit,B4: set_int] :
% 14.25/8.06 ( ( A != bot_bot_set_int )
% 14.25/8.06 => ( ( condit2080475194861589617t_unit @ ( image_5846526143423701914t_unit @ F @ A ) )
% 14.25/8.06 => ( ( B4 != bot_bot_set_int )
% 14.25/8.06 => ( ( condit2080475194861589617t_unit @ ( image_5846526143423701914t_unit @ F @ B4 ) )
% 14.25/8.06 => ( ( comple4687483117567863418t_unit @ ( image_5846526143423701914t_unit @ F @ ( sup_sup_set_int @ A @ B4 ) ) )
% 14.25/8.06 = ( sup_sup_Product_unit @ ( comple4687483117567863418t_unit @ ( image_5846526143423701914t_unit @ F @ A ) ) @ ( comple4687483117567863418t_unit @ ( image_5846526143423701914t_unit @ F @ B4 ) ) ) ) ) ) ) ) ).
% 14.25/8.06
% 14.25/8.06 thf(fact_3319_cSUP__union,axiom,
% 14.25/8.06 ! [A: set_Product_unit,F: product_unit > set_nat,B4: set_Product_unit] :
% 14.25/8.06 ( ( A != bot_bo3957492148770167129t_unit )
% 14.25/8.06 => ( ( condit5477540289124974626et_nat @ ( image_8173849991827986510et_nat @ F @ A ) )
% 14.25/8.06 => ( ( B4 != bot_bo3957492148770167129t_unit )
% 14.25/8.06 => ( ( condit5477540289124974626et_nat @ ( image_8173849991827986510et_nat @ F @ B4 ) )
% 14.25/8.06 => ( ( comple7399068483239264473et_nat @ ( image_8173849991827986510et_nat @ F @ ( sup_su793286257634532545t_unit @ A @ B4 ) ) )
% 14.25/8.06 = ( sup_sup_set_nat @ ( comple7399068483239264473et_nat @ ( image_8173849991827986510et_nat @ F @ A ) ) @ ( comple7399068483239264473et_nat @ ( image_8173849991827986510et_nat @ F @ B4 ) ) ) ) ) ) ) ) ).
% 14.25/8.06
% 14.25/8.06 thf(fact_3320_cSUP__union,axiom,
% 14.25/8.06 ! [A: set_o,F: $o > set_nat,B4: set_o] :
% 14.25/8.06 ( ( A != bot_bot_set_o )
% 14.25/8.06 => ( ( condit5477540289124974626et_nat @ ( image_o_set_nat @ F @ A ) )
% 14.25/8.06 => ( ( B4 != bot_bot_set_o )
% 14.25/8.06 => ( ( condit5477540289124974626et_nat @ ( image_o_set_nat @ F @ B4 ) )
% 14.25/8.06 => ( ( comple7399068483239264473et_nat @ ( image_o_set_nat @ F @ ( sup_sup_set_o @ A @ B4 ) ) )
% 14.25/8.06 = ( sup_sup_set_nat @ ( comple7399068483239264473et_nat @ ( image_o_set_nat @ F @ A ) ) @ ( comple7399068483239264473et_nat @ ( image_o_set_nat @ F @ B4 ) ) ) ) ) ) ) ) ).
% 14.25/8.06
% 14.25/8.06 thf(fact_3321_pair__in__swap__image,axiom,
% 14.25/8.06 ! [Y: set_nat,X: heap_e7401611519738050253t_unit,A: set_Pr3948176798113811640et_nat] :
% 14.25/8.06 ( ( member727419437372278867t_unit @ ( produc2471974422597015918t_unit @ Y @ X ) @ ( image_1893317993577548229t_unit @ produc6300697153377358172et_nat @ A ) )
% 14.25/8.06 = ( member6260224972018164377et_nat @ ( produc7507926704131184380et_nat @ X @ Y ) @ A ) ) ).
% 14.25/8.06
% 14.25/8.06 thf(fact_3322_pair__in__swap__image,axiom,
% 14.25/8.06 ! [Y: produc3658429121746597890et_nat,X: produc3658429121746597890et_nat > $o,A: set_Pr3286484037609594932et_nat] :
% 14.25/8.06 ( ( member4978015717948814295_nat_o @ ( produc2829238748666232050_nat_o @ Y @ X ) @ ( image_6229134628169569989_nat_o @ produc3251101204053596376et_nat @ A ) )
% 14.25/8.06 = ( member1996754912294343701et_nat @ ( produc5001842942810119800et_nat @ X @ Y ) @ A ) ) ).
% 14.25/8.06
% 14.25/8.06 thf(fact_3323_pair__in__swap__image,axiom,
% 14.25/8.06 ! [Y: produc3925858234332021118et_nat,X: produc3658429121746597890et_nat > $o,A: set_Pr8536935166611901872et_nat] :
% 14.25/8.06 ( ( member8354763867512313179_nat_o @ ( produc3674526432113299062_nat_o @ Y @ X ) @ ( image_4071802983491761093_nat_o @ produc1151471490888544852et_nat @ A ) )
% 14.25/8.06 = ( member6124377750444531601et_nat @ ( produc2245416461498447860et_nat @ X @ Y ) @ A ) ) ).
% 14.25/8.06
% 14.25/8.06 thf(fact_3324_pair__in__swap__image,axiom,
% 14.25/8.06 ! [Y: heap_e7401611519738050253t_unit,X: set_nat,A: set_Pr4071100604818519410t_unit] :
% 14.25/8.06 ( ( member6260224972018164377et_nat @ ( produc7507926704131184380et_nat @ Y @ X ) @ ( image_5838825082199883205et_nat @ produc1264744871843189710t_unit @ A ) )
% 14.25/8.06 = ( member727419437372278867t_unit @ ( produc2471974422597015918t_unit @ X @ Y ) @ A ) ) ).
% 14.25/8.06
% 14.25/8.06 thf(fact_3325_pair__in__swap__image,axiom,
% 14.25/8.06 ! [Y: nat,X: nat,A: set_Pr1261947904930325089at_nat] :
% 14.25/8.06 ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ Y @ X ) @ ( image_5168914502847457605at_nat @ product_swap_nat_nat @ A ) )
% 14.25/8.06 = ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ Y ) @ A ) ) ).
% 14.25/8.06
% 14.25/8.06 thf(fact_3326_pair__in__swap__image,axiom,
% 14.25/8.06 ! [Y: int,X: int,A: set_Pr958786334691620121nt_int] :
% 14.25/8.06 ( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ Y @ X ) @ ( image_2653370878348428101nt_int @ product_swap_int_int @ A ) )
% 14.25/8.06 = ( member5262025264175285858nt_int @ ( product_Pair_int_int @ X @ Y ) @ A ) ) ).
% 14.25/8.06
% 14.25/8.06 thf(fact_3327_pair__in__swap__image,axiom,
% 14.25/8.06 ! [Y: produc3658429121746597890et_nat > $o,X: produc3658429121746597890et_nat,A: set_Pr4208978187861016310_nat_o] :
% 14.25/8.06 ( ( member1996754912294343701et_nat @ ( produc5001842942810119800et_nat @ Y @ X ) @ ( image_4501728140019976901et_nat @ produc1078497009909708626_nat_o @ A ) )
% 14.25/8.06 = ( member4978015717948814295_nat_o @ ( produc2829238748666232050_nat_o @ X @ Y ) @ A ) ) ).
% 14.25/8.06
% 14.25/8.06 thf(fact_3328_pair__in__swap__image,axiom,
% 14.25/8.06 ! [Y: produc3658429121746597890et_nat > $o,X: produc3925858234332021118et_nat,A: set_Pr7705059561495379578_nat_o] :
% 14.25/8.06 ( ( member6124377750444531601et_nat @ ( produc2245416461498447860et_nat @ Y @ X ) @ ( image_7255662321850427333et_nat @ produc2580581461503396054_nat_o @ A ) )
% 14.25/8.06 = ( member8354763867512313179_nat_o @ ( produc3674526432113299062_nat_o @ X @ Y ) @ A ) ) ).
% 14.25/8.06
% 14.25/8.06 thf(fact_3329_cSUP__insert,axiom,
% 14.25/8.06 ! [A: set_Product_unit,F: product_unit > nat,A2: product_unit] :
% 14.25/8.06 ( ( A != bot_bo3957492148770167129t_unit )
% 14.25/8.06 => ( ( condit2214826472909112428ve_nat @ ( image_875570014554754200it_nat @ F @ A ) )
% 14.25/8.06 => ( ( complete_Sup_Sup_nat @ ( image_875570014554754200it_nat @ F @ ( insert_Product_unit @ A2 @ A ) ) )
% 14.25/8.06 = ( sup_sup_nat @ ( F @ A2 ) @ ( complete_Sup_Sup_nat @ ( image_875570014554754200it_nat @ F @ A ) ) ) ) ) ) ).
% 14.25/8.06
% 14.25/8.06 thf(fact_3330_cSUP__insert,axiom,
% 14.25/8.06 ! [A: set_o,F: $o > nat,A2: $o] :
% 14.25/8.06 ( ( A != bot_bot_set_o )
% 14.25/8.06 => ( ( condit2214826472909112428ve_nat @ ( image_o_nat2 @ F @ A ) )
% 14.25/8.06 => ( ( complete_Sup_Sup_nat @ ( image_o_nat2 @ F @ ( insert_o @ A2 @ A ) ) )
% 14.25/8.06 = ( sup_sup_nat @ ( F @ A2 ) @ ( complete_Sup_Sup_nat @ ( image_o_nat2 @ F @ A ) ) ) ) ) ) ).
% 14.25/8.06
% 14.25/8.06 thf(fact_3331_cSUP__insert,axiom,
% 14.25/8.06 ! [A: set_nat,F: nat > nat,A2: nat] :
% 14.25/8.06 ( ( A != bot_bot_set_nat )
% 14.25/8.06 => ( ( condit2214826472909112428ve_nat @ ( image_nat_nat2 @ F @ A ) )
% 14.25/8.06 => ( ( complete_Sup_Sup_nat @ ( image_nat_nat2 @ F @ ( insert_nat @ A2 @ A ) ) )
% 14.25/8.06 = ( sup_sup_nat @ ( F @ A2 ) @ ( complete_Sup_Sup_nat @ ( image_nat_nat2 @ F @ A ) ) ) ) ) ) ).
% 14.25/8.06
% 14.25/8.06 thf(fact_3332_cSUP__insert,axiom,
% 14.25/8.06 ! [A: set_int,F: int > nat,A2: int] :
% 14.25/8.06 ( ( A != bot_bot_set_int )
% 14.25/8.06 => ( ( condit2214826472909112428ve_nat @ ( image_int_nat2 @ F @ A ) )
% 14.25/8.06 => ( ( complete_Sup_Sup_nat @ ( image_int_nat2 @ F @ ( insert_int @ A2 @ A ) ) )
% 14.25/8.06 = ( sup_sup_nat @ ( F @ A2 ) @ ( complete_Sup_Sup_nat @ ( image_int_nat2 @ F @ A ) ) ) ) ) ) ).
% 14.25/8.06
% 14.25/8.06 thf(fact_3333_cSUP__insert,axiom,
% 14.25/8.06 ! [A: set_Product_unit,F: product_unit > product_unit,A2: product_unit] :
% 14.25/8.06 ( ( A != bot_bo3957492148770167129t_unit )
% 14.25/8.06 => ( ( condit2080475194861589617t_unit @ ( image_405062704495631173t_unit @ F @ A ) )
% 14.25/8.06 => ( ( comple4687483117567863418t_unit @ ( image_405062704495631173t_unit @ F @ ( insert_Product_unit @ A2 @ A ) ) )
% 14.25/8.06 = ( sup_sup_Product_unit @ ( F @ A2 ) @ ( comple4687483117567863418t_unit @ ( image_405062704495631173t_unit @ F @ A ) ) ) ) ) ) ).
% 14.25/8.06
% 14.25/8.06 thf(fact_3334_cSUP__insert,axiom,
% 14.25/8.06 ! [A: set_o,F: $o > product_unit,A2: $o] :
% 14.25/8.06 ( ( A != bot_bot_set_o )
% 14.25/8.06 => ( ( condit2080475194861589617t_unit @ ( image_o_Product_unit2 @ F @ A ) )
% 14.25/8.06 => ( ( comple4687483117567863418t_unit @ ( image_o_Product_unit2 @ F @ ( insert_o @ A2 @ A ) ) )
% 14.25/8.06 = ( sup_sup_Product_unit @ ( F @ A2 ) @ ( comple4687483117567863418t_unit @ ( image_o_Product_unit2 @ F @ A ) ) ) ) ) ) ).
% 14.25/8.06
% 14.25/8.06 thf(fact_3335_cSUP__insert,axiom,
% 14.25/8.06 ! [A: set_nat,F: nat > product_unit,A2: nat] :
% 14.25/8.06 ( ( A != bot_bot_set_nat )
% 14.25/8.06 => ( ( condit2080475194861589617t_unit @ ( image_8730104196221521654t_unit @ F @ A ) )
% 14.25/8.06 => ( ( comple4687483117567863418t_unit @ ( image_8730104196221521654t_unit @ F @ ( insert_nat @ A2 @ A ) ) )
% 14.25/8.06 = ( sup_sup_Product_unit @ ( F @ A2 ) @ ( comple4687483117567863418t_unit @ ( image_8730104196221521654t_unit @ F @ A ) ) ) ) ) ) ).
% 14.25/8.06
% 14.25/8.06 thf(fact_3336_cSUP__insert,axiom,
% 14.25/8.06 ! [A: set_int,F: int > product_unit,A2: int] :
% 14.25/8.06 ( ( A != bot_bot_set_int )
% 14.25/8.06 => ( ( condit2080475194861589617t_unit @ ( image_5846526143423701914t_unit @ F @ A ) )
% 14.25/8.06 => ( ( comple4687483117567863418t_unit @ ( image_5846526143423701914t_unit @ F @ ( insert_int @ A2 @ A ) ) )
% 14.25/8.06 = ( sup_sup_Product_unit @ ( F @ A2 ) @ ( comple4687483117567863418t_unit @ ( image_5846526143423701914t_unit @ F @ A ) ) ) ) ) ) ).
% 14.25/8.06
% 14.25/8.06 thf(fact_3337_cSUP__insert,axiom,
% 14.25/8.06 ! [A: set_Product_unit,F: product_unit > set_nat,A2: product_unit] :
% 14.25/8.06 ( ( A != bot_bo3957492148770167129t_unit )
% 14.25/8.06 => ( ( condit5477540289124974626et_nat @ ( image_8173849991827986510et_nat @ F @ A ) )
% 14.25/8.06 => ( ( comple7399068483239264473et_nat @ ( image_8173849991827986510et_nat @ F @ ( insert_Product_unit @ A2 @ A ) ) )
% 14.25/8.06 = ( sup_sup_set_nat @ ( F @ A2 ) @ ( comple7399068483239264473et_nat @ ( image_8173849991827986510et_nat @ F @ A ) ) ) ) ) ) ).
% 14.25/8.06
% 14.25/8.06 thf(fact_3338_cSUP__insert,axiom,
% 14.25/8.06 ! [A: set_o,F: $o > set_nat,A2: $o] :
% 14.25/8.06 ( ( A != bot_bot_set_o )
% 14.25/8.06 => ( ( condit5477540289124974626et_nat @ ( image_o_set_nat @ F @ A ) )
% 14.25/8.06 => ( ( comple7399068483239264473et_nat @ ( image_o_set_nat @ F @ ( insert_o @ A2 @ A ) ) )
% 14.25/8.06 = ( sup_sup_set_nat @ ( F @ A2 ) @ ( comple7399068483239264473et_nat @ ( image_o_set_nat @ F @ A ) ) ) ) ) ) ).
% 14.25/8.06
% 14.25/8.06 thf(fact_3339_cSup__inter__less__eq,axiom,
% 14.25/8.06 ! [A: set_o,B4: set_o] :
% 14.25/8.06 ( ( condit5488710616941104124bove_o @ A )
% 14.25/8.06 => ( ( condit5488710616941104124bove_o @ B4 )
% 14.25/8.06 => ( ( ( inf_inf_set_o @ A @ B4 )
% 14.25/8.06 != bot_bot_set_o )
% 14.25/8.06 => ( ord_less_eq_o @ ( complete_Sup_Sup_o @ ( inf_inf_set_o @ A @ B4 ) ) @ ( sup_sup_o @ ( complete_Sup_Sup_o @ A ) @ ( complete_Sup_Sup_o @ B4 ) ) ) ) ) ) ).
% 14.25/8.06
% 14.25/8.06 thf(fact_3340_cSup__inter__less__eq,axiom,
% 14.25/8.06 ! [A: set_Pr4532377907799695533_nat_o,B4: set_Pr4532377907799695533_nat_o] :
% 14.25/8.06 ( ( condit4352657709669162033_nat_o @ A )
% 14.25/8.06 => ( ( condit4352657709669162033_nat_o @ B4 )
% 14.25/8.06 => ( ( ( inf_in1906310914598751387_nat_o @ A @ B4 )
% 14.25/8.06 != bot_bo7824918357723582233_nat_o )
% 14.25/8.06 => ( ord_le729326519192465773_nat_o @ ( comple4225350640907012154_nat_o @ ( inf_in1906310914598751387_nat_o @ A @ B4 ) ) @ ( sup_su5453871518329203617_nat_o @ ( comple4225350640907012154_nat_o @ A ) @ ( comple4225350640907012154_nat_o @ B4 ) ) ) ) ) ) ).
% 14.25/8.06
% 14.25/8.06 thf(fact_3341_cSup__inter__less__eq,axiom,
% 14.25/8.06 ! [A: set_filter_nat,B4: set_filter_nat] :
% 14.25/8.06 ( ( condit4029664723990530684er_nat @ A )
% 14.25/8.06 => ( ( condit4029664723990530684er_nat @ B4 )
% 14.25/8.06 => ( ( ( inf_in7395780696554941096er_nat @ A @ B4 )
% 14.25/8.06 != bot_bo498966703094740906er_nat )
% 14.25/8.06 => ( ord_le2510731241096832064er_nat @ ( comple2454491259644266675er_nat @ ( inf_in7395780696554941096er_nat @ A @ B4 ) ) @ ( sup_sup_filter_nat @ ( comple2454491259644266675er_nat @ A ) @ ( comple2454491259644266675er_nat @ B4 ) ) ) ) ) ) ).
% 14.25/8.06
% 14.25/8.06 thf(fact_3342_cSup__inter__less__eq,axiom,
% 14.25/8.06 ! [A: set_set_nat,B4: set_set_nat] :
% 14.25/8.06 ( ( condit5477540289124974626et_nat @ A )
% 14.25/8.06 => ( ( condit5477540289124974626et_nat @ B4 )
% 14.25/8.06 => ( ( ( inf_inf_set_set_nat @ A @ B4 )
% 14.25/8.06 != bot_bot_set_set_nat )
% 14.25/8.06 => ( ord_less_eq_set_nat @ ( comple7399068483239264473et_nat @ ( inf_inf_set_set_nat @ A @ B4 ) ) @ ( sup_sup_set_nat @ ( comple7399068483239264473et_nat @ A ) @ ( comple7399068483239264473et_nat @ B4 ) ) ) ) ) ) ).
% 14.25/8.06
% 14.25/8.06 thf(fact_3343_cSup__inter__less__eq,axiom,
% 14.25/8.06 ! [A: set_int,B4: set_int] :
% 14.25/8.06 ( ( condit2212336002400062152ve_int @ A )
% 14.25/8.06 => ( ( condit2212336002400062152ve_int @ B4 )
% 14.25/8.06 => ( ( ( inf_inf_set_int @ A @ B4 )
% 14.25/8.06 != bot_bot_set_int )
% 14.25/8.06 => ( ord_less_eq_int @ ( complete_Sup_Sup_int @ ( inf_inf_set_int @ A @ B4 ) ) @ ( sup_sup_int @ ( complete_Sup_Sup_int @ A ) @ ( complete_Sup_Sup_int @ B4 ) ) ) ) ) ) ).
% 14.25/8.06
% 14.25/8.06 thf(fact_3344_cSup__inter__less__eq,axiom,
% 14.25/8.06 ! [A: set_nat,B4: set_nat] :
% 14.25/8.06 ( ( condit2214826472909112428ve_nat @ A )
% 14.25/8.06 => ( ( condit2214826472909112428ve_nat @ B4 )
% 14.25/8.06 => ( ( ( inf_inf_set_nat @ A @ B4 )
% 14.25/8.06 != bot_bot_set_nat )
% 14.25/8.06 => ( ord_less_eq_nat @ ( complete_Sup_Sup_nat @ ( inf_inf_set_nat @ A @ B4 ) ) @ ( sup_sup_nat @ ( complete_Sup_Sup_nat @ A ) @ ( complete_Sup_Sup_nat @ B4 ) ) ) ) ) ) ).
% 14.25/8.06
% 14.25/8.06 thf(fact_3345_cSup__inter__less__eq,axiom,
% 14.25/8.06 ! [A: set_Product_unit,B4: set_Product_unit] :
% 14.25/8.06 ( ( condit2080475194861589617t_unit @ A )
% 14.25/8.06 => ( ( condit2080475194861589617t_unit @ B4 )
% 14.25/8.06 => ( ( ( inf_in4660618365625256667t_unit @ A @ B4 )
% 14.25/8.06 != bot_bo3957492148770167129t_unit )
% 14.25/8.06 => ( ord_le3221252021190050221t_unit @ ( comple4687483117567863418t_unit @ ( inf_in4660618365625256667t_unit @ A @ B4 ) ) @ ( sup_sup_Product_unit @ ( comple4687483117567863418t_unit @ A ) @ ( comple4687483117567863418t_unit @ B4 ) ) ) ) ) ) ).
% 14.25/8.06
% 14.25/8.06 thf(fact_3346_cSUP__subset__mono,axiom,
% 14.25/8.06 ! [A: set_Product_unit,G: product_unit > int,B4: set_Product_unit,F: product_unit > int] :
% 14.25/8.06 ( ( A != bot_bo3957492148770167129t_unit )
% 14.25/8.06 => ( ( condit2212336002400062152ve_int @ ( image_873079544045703924it_int @ G @ B4 ) )
% 14.25/8.06 => ( ( ord_le3507040750410214029t_unit @ A @ B4 )
% 14.25/8.06 => ( ! [X3: product_unit] :
% 14.25/8.06 ( ( member_Product_unit @ X3 @ A )
% 14.25/8.06 => ( ord_less_eq_int @ ( F @ X3 ) @ ( G @ X3 ) ) )
% 14.25/8.06 => ( ord_less_eq_int @ ( complete_Sup_Sup_int @ ( image_873079544045703924it_int @ F @ A ) ) @ ( complete_Sup_Sup_int @ ( image_873079544045703924it_int @ G @ B4 ) ) ) ) ) ) ) ).
% 14.25/8.06
% 14.25/8.06 thf(fact_3347_cSUP__subset__mono,axiom,
% 14.25/8.06 ! [A: set_o,G: $o > int,B4: set_o,F: $o > int] :
% 14.25/8.06 ( ( A != bot_bot_set_o )
% 14.25/8.06 => ( ( condit2212336002400062152ve_int @ ( image_o_int2 @ G @ B4 ) )
% 14.25/8.06 => ( ( ord_less_eq_set_o @ A @ B4 )
% 14.25/8.06 => ( ! [X3: $o] :
% 14.25/8.06 ( ( member_o @ X3 @ A )
% 14.25/8.06 => ( ord_less_eq_int @ ( F @ X3 ) @ ( G @ X3 ) ) )
% 14.25/8.06 => ( ord_less_eq_int @ ( complete_Sup_Sup_int @ ( image_o_int2 @ F @ A ) ) @ ( complete_Sup_Sup_int @ ( image_o_int2 @ G @ B4 ) ) ) ) ) ) ) ).
% 14.25/8.06
% 14.25/8.06 thf(fact_3348_cSUP__subset__mono,axiom,
% 14.25/8.06 ! [A: set_int,G: int > int,B4: set_int,F: int > int] :
% 14.25/8.06 ( ( A != bot_bot_set_int )
% 14.25/8.06 => ( ( condit2212336002400062152ve_int @ ( image_int_int2 @ G @ B4 ) )
% 14.25/8.06 => ( ( ord_less_eq_set_int @ A @ B4 )
% 14.25/8.06 => ( ! [X3: int] :
% 14.25/8.06 ( ( member_int @ X3 @ A )
% 14.25/8.06 => ( ord_less_eq_int @ ( F @ X3 ) @ ( G @ X3 ) ) )
% 14.25/8.06 => ( ord_less_eq_int @ ( complete_Sup_Sup_int @ ( image_int_int2 @ F @ A ) ) @ ( complete_Sup_Sup_int @ ( image_int_int2 @ G @ B4 ) ) ) ) ) ) ) ).
% 14.25/8.06
% 14.25/8.06 thf(fact_3349_cSUP__subset__mono,axiom,
% 14.25/8.06 ! [A: set_nat,G: nat > int,B4: set_nat,F: nat > int] :
% 14.25/8.06 ( ( A != bot_bot_set_nat )
% 14.25/8.06 => ( ( condit2212336002400062152ve_int @ ( image_nat_int2 @ G @ B4 ) )
% 14.25/8.06 => ( ( ord_less_eq_set_nat @ A @ B4 )
% 14.25/8.06 => ( ! [X3: nat] :
% 14.25/8.06 ( ( member_nat @ X3 @ A )
% 14.25/8.06 => ( ord_less_eq_int @ ( F @ X3 ) @ ( G @ X3 ) ) )
% 14.25/8.06 => ( ord_less_eq_int @ ( complete_Sup_Sup_int @ ( image_nat_int2 @ F @ A ) ) @ ( complete_Sup_Sup_int @ ( image_nat_int2 @ G @ B4 ) ) ) ) ) ) ) ).
% 14.25/8.06
% 14.25/8.06 thf(fact_3350_cSUP__subset__mono,axiom,
% 14.25/8.06 ! [A: set_Product_unit,G: product_unit > nat,B4: set_Product_unit,F: product_unit > nat] :
% 14.25/8.06 ( ( A != bot_bo3957492148770167129t_unit )
% 14.25/8.06 => ( ( condit2214826472909112428ve_nat @ ( image_875570014554754200it_nat @ G @ B4 ) )
% 14.25/8.06 => ( ( ord_le3507040750410214029t_unit @ A @ B4 )
% 14.25/8.06 => ( ! [X3: product_unit] :
% 14.25/8.06 ( ( member_Product_unit @ X3 @ A )
% 14.25/8.06 => ( ord_less_eq_nat @ ( F @ X3 ) @ ( G @ X3 ) ) )
% 14.25/8.06 => ( ord_less_eq_nat @ ( complete_Sup_Sup_nat @ ( image_875570014554754200it_nat @ F @ A ) ) @ ( complete_Sup_Sup_nat @ ( image_875570014554754200it_nat @ G @ B4 ) ) ) ) ) ) ) ).
% 14.25/8.06
% 14.25/8.06 thf(fact_3351_cSUP__subset__mono,axiom,
% 14.25/8.06 ! [A: set_o,G: $o > nat,B4: set_o,F: $o > nat] :
% 14.25/8.06 ( ( A != bot_bot_set_o )
% 14.25/8.06 => ( ( condit2214826472909112428ve_nat @ ( image_o_nat2 @ G @ B4 ) )
% 14.25/8.06 => ( ( ord_less_eq_set_o @ A @ B4 )
% 14.25/8.06 => ( ! [X3: $o] :
% 14.25/8.06 ( ( member_o @ X3 @ A )
% 14.25/8.06 => ( ord_less_eq_nat @ ( F @ X3 ) @ ( G @ X3 ) ) )
% 14.25/8.06 => ( ord_less_eq_nat @ ( complete_Sup_Sup_nat @ ( image_o_nat2 @ F @ A ) ) @ ( complete_Sup_Sup_nat @ ( image_o_nat2 @ G @ B4 ) ) ) ) ) ) ) ).
% 14.25/8.06
% 14.25/8.06 thf(fact_3352_cSUP__subset__mono,axiom,
% 14.25/8.06 ! [A: set_int,G: int > nat,B4: set_int,F: int > nat] :
% 14.25/8.06 ( ( A != bot_bot_set_int )
% 14.25/8.06 => ( ( condit2214826472909112428ve_nat @ ( image_int_nat2 @ G @ B4 ) )
% 14.25/8.06 => ( ( ord_less_eq_set_int @ A @ B4 )
% 14.25/8.06 => ( ! [X3: int] :
% 14.25/8.06 ( ( member_int @ X3 @ A )
% 14.25/8.06 => ( ord_less_eq_nat @ ( F @ X3 ) @ ( G @ X3 ) ) )
% 14.25/8.06 => ( ord_less_eq_nat @ ( complete_Sup_Sup_nat @ ( image_int_nat2 @ F @ A ) ) @ ( complete_Sup_Sup_nat @ ( image_int_nat2 @ G @ B4 ) ) ) ) ) ) ) ).
% 14.25/8.06
% 14.25/8.06 thf(fact_3353_cSUP__subset__mono,axiom,
% 14.25/8.06 ! [A: set_nat,G: nat > nat,B4: set_nat,F: nat > nat] :
% 14.25/8.06 ( ( A != bot_bot_set_nat )
% 14.25/8.06 => ( ( condit2214826472909112428ve_nat @ ( image_nat_nat2 @ G @ B4 ) )
% 14.25/8.06 => ( ( ord_less_eq_set_nat @ A @ B4 )
% 14.25/8.06 => ( ! [X3: nat] :
% 14.25/8.06 ( ( member_nat @ X3 @ A )
% 14.25/8.06 => ( ord_less_eq_nat @ ( F @ X3 ) @ ( G @ X3 ) ) )
% 14.25/8.06 => ( ord_less_eq_nat @ ( complete_Sup_Sup_nat @ ( image_nat_nat2 @ F @ A ) ) @ ( complete_Sup_Sup_nat @ ( image_nat_nat2 @ G @ B4 ) ) ) ) ) ) ) ).
% 14.25/8.06
% 14.25/8.06 thf(fact_3354_cSUP__subset__mono,axiom,
% 14.25/8.06 ! [A: set_Product_unit,G: product_unit > product_unit,B4: set_Product_unit,F: product_unit > product_unit] :
% 14.25/8.06 ( ( A != bot_bo3957492148770167129t_unit )
% 14.25/8.06 => ( ( condit2080475194861589617t_unit @ ( image_405062704495631173t_unit @ G @ B4 ) )
% 14.25/8.06 => ( ( ord_le3507040750410214029t_unit @ A @ B4 )
% 14.25/8.06 => ( ! [X3: product_unit] :
% 14.25/8.06 ( ( member_Product_unit @ X3 @ A )
% 14.25/8.06 => ( ord_le3221252021190050221t_unit @ ( F @ X3 ) @ ( G @ X3 ) ) )
% 14.25/8.06 => ( ord_le3221252021190050221t_unit @ ( comple4687483117567863418t_unit @ ( image_405062704495631173t_unit @ F @ A ) ) @ ( comple4687483117567863418t_unit @ ( image_405062704495631173t_unit @ G @ B4 ) ) ) ) ) ) ) ).
% 14.25/8.06
% 14.25/8.06 thf(fact_3355_cSUP__subset__mono,axiom,
% 14.25/8.06 ! [A: set_o,G: $o > product_unit,B4: set_o,F: $o > product_unit] :
% 14.25/8.06 ( ( A != bot_bot_set_o )
% 14.25/8.06 => ( ( condit2080475194861589617t_unit @ ( image_o_Product_unit2 @ G @ B4 ) )
% 14.25/8.06 => ( ( ord_less_eq_set_o @ A @ B4 )
% 14.25/8.06 => ( ! [X3: $o] :
% 14.25/8.06 ( ( member_o @ X3 @ A )
% 14.25/8.06 => ( ord_le3221252021190050221t_unit @ ( F @ X3 ) @ ( G @ X3 ) ) )
% 14.25/8.06 => ( ord_le3221252021190050221t_unit @ ( comple4687483117567863418t_unit @ ( image_o_Product_unit2 @ F @ A ) ) @ ( comple4687483117567863418t_unit @ ( image_o_Product_unit2 @ G @ B4 ) ) ) ) ) ) ) ).
% 14.25/8.06
% 14.25/8.06 thf(fact_3356_Image__Int__eq,axiom,
% 14.25/8.06 ! [R3: set_Pr1261947904930325089at_nat,A: set_nat,B4: set_nat] :
% 14.25/8.06 ( ( single5507084439971879973at_nat @ ( converse_nat_nat @ R3 ) )
% 14.25/8.06 => ( ( image_nat_nat @ R3 @ ( inf_inf_set_nat @ A @ B4 ) )
% 14.25/8.06 = ( inf_inf_set_nat @ ( image_nat_nat @ R3 @ A ) @ ( image_nat_nat @ R3 @ B4 ) ) ) ) ).
% 14.25/8.06
% 14.25/8.06 thf(fact_3357_image__update,axiom,
% 14.25/8.06 ! [X: code_integer,A: set_Code_integer,F: code_integer > code_integer,N2: code_integer] :
% 14.25/8.06 ( ~ ( member_Code_integer @ X @ A )
% 14.25/8.06 => ( ( image_4470545334726330049nteger @ ( fun_up8355007323143964473nteger @ F @ X @ N2 ) @ A )
% 14.25/8.06 = ( image_4470545334726330049nteger @ F @ A ) ) ) ).
% 14.25/8.06
% 14.25/8.06 thf(fact_3358_image__update,axiom,
% 14.25/8.06 ! [X: nat,A: set_nat,F: nat > set_nat,N2: set_nat] :
% 14.25/8.06 ( ~ ( member_nat @ X @ A )
% 14.25/8.06 => ( ( image_nat_set_nat @ ( fun_upd_nat_set_nat @ F @ X @ N2 ) @ A )
% 14.25/8.06 = ( image_nat_set_nat @ F @ A ) ) ) ).
% 14.25/8.06
% 14.25/8.06 thf(fact_3359_image__update,axiom,
% 14.25/8.06 ! [X: nat,A: set_nat,F: nat > nat,N2: nat] :
% 14.25/8.06 ( ~ ( member_nat @ X @ A )
% 14.25/8.06 => ( ( image_nat_nat2 @ ( fun_upd_nat_nat @ F @ X @ N2 ) @ A )
% 14.25/8.06 = ( image_nat_nat2 @ F @ A ) ) ) ).
% 14.25/8.06
% 14.25/8.06 thf(fact_3360_image__update,axiom,
% 14.25/8.06 ! [X: nat,A: set_nat,F: nat > code_integer,N2: code_integer] :
% 14.25/8.06 ( ~ ( member_nat @ X @ A )
% 14.25/8.06 => ( ( image_1215581382706833972nteger @ ( fun_up3904115110581602988nteger @ F @ X @ N2 ) @ A )
% 14.25/8.06 = ( image_1215581382706833972nteger @ F @ A ) ) ) ).
% 14.25/8.06
% 14.25/8.06 thf(fact_3361_bdd__above__empty,axiom,
% 14.25/8.06 condit2080475194861589617t_unit @ bot_bo3957492148770167129t_unit ).
% 14.25/8.06
% 14.25/8.06 thf(fact_3362_bdd__above__empty,axiom,
% 14.25/8.06 condit5488710616941104124bove_o @ bot_bot_set_o ).
% 14.25/8.06
% 14.25/8.06 thf(fact_3363_bdd__above__empty,axiom,
% 14.25/8.06 condit2214826472909112428ve_nat @ bot_bot_set_nat ).
% 14.25/8.06
% 14.25/8.06 thf(fact_3364_bdd__above__empty,axiom,
% 14.25/8.06 condit2212336002400062152ve_int @ bot_bot_set_int ).
% 14.25/8.06
% 14.25/8.06 thf(fact_3365_bdd__above__empty,axiom,
% 14.25/8.06 condit4352657709669162033_nat_o @ bot_bo7824918357723582233_nat_o ).
% 14.25/8.06
% 14.25/8.06 thf(fact_3366_bdd__above__insert,axiom,
% 14.25/8.06 ! [A2: nat,A: set_nat] :
% 14.25/8.06 ( ( condit2214826472909112428ve_nat @ ( insert_nat @ A2 @ A ) )
% 14.25/8.06 = ( condit2214826472909112428ve_nat @ A ) ) ).
% 14.25/8.06
% 14.25/8.06 thf(fact_3367_bdd__above__insert,axiom,
% 14.25/8.06 ! [A2: int,A: set_int] :
% 14.25/8.06 ( ( condit2212336002400062152ve_int @ ( insert_int @ A2 @ A ) )
% 14.25/8.06 = ( condit2212336002400062152ve_int @ A ) ) ).
% 14.25/8.06
% 14.25/8.06 thf(fact_3368_bdd__above__insert,axiom,
% 14.25/8.06 ! [A2: $o,A: set_o] :
% 14.25/8.06 ( ( condit5488710616941104124bove_o @ ( insert_o @ A2 @ A ) )
% 14.25/8.06 = ( condit5488710616941104124bove_o @ A ) ) ).
% 14.25/8.06
% 14.25/8.06 thf(fact_3369_bdd__above__insert,axiom,
% 14.25/8.06 ! [A2: product_unit,A: set_Product_unit] :
% 14.25/8.06 ( ( condit2080475194861589617t_unit @ ( insert_Product_unit @ A2 @ A ) )
% 14.25/8.06 = ( condit2080475194861589617t_unit @ A ) ) ).
% 14.25/8.06
% 14.25/8.06 thf(fact_3370_swap__simp,axiom,
% 14.25/8.06 ! [X: set_nat,Y: heap_e7401611519738050253t_unit] :
% 14.25/8.06 ( ( produc1264744871843189710t_unit @ ( produc2471974422597015918t_unit @ X @ Y ) )
% 14.25/8.06 = ( produc7507926704131184380et_nat @ Y @ X ) ) ).
% 14.25/8.06
% 14.25/8.06 thf(fact_3371_swap__simp,axiom,
% 14.25/8.06 ! [X: produc3658429121746597890et_nat,Y: produc3658429121746597890et_nat > $o] :
% 14.25/8.06 ( ( produc1078497009909708626_nat_o @ ( produc2829238748666232050_nat_o @ X @ Y ) )
% 14.25/8.06 = ( produc5001842942810119800et_nat @ Y @ X ) ) ).
% 14.25/8.06
% 14.25/8.06 thf(fact_3372_swap__simp,axiom,
% 14.25/8.06 ! [X: produc3925858234332021118et_nat,Y: produc3658429121746597890et_nat > $o] :
% 14.25/8.06 ( ( produc2580581461503396054_nat_o @ ( produc3674526432113299062_nat_o @ X @ Y ) )
% 14.25/8.06 = ( produc2245416461498447860et_nat @ Y @ X ) ) ).
% 14.25/8.06
% 14.25/8.06 thf(fact_3373_swap__simp,axiom,
% 14.25/8.06 ! [X: heap_e7401611519738050253t_unit,Y: set_nat] :
% 14.25/8.06 ( ( produc6300697153377358172et_nat @ ( produc7507926704131184380et_nat @ X @ Y ) )
% 14.25/8.06 = ( produc2471974422597015918t_unit @ Y @ X ) ) ).
% 14.25/8.06
% 14.25/8.06 thf(fact_3374_swap__simp,axiom,
% 14.25/8.06 ! [X: nat,Y: nat] :
% 14.25/8.06 ( ( product_swap_nat_nat @ ( product_Pair_nat_nat @ X @ Y ) )
% 14.25/8.06 = ( product_Pair_nat_nat @ Y @ X ) ) ).
% 14.25/8.06
% 14.25/8.06 thf(fact_3375_swap__simp,axiom,
% 14.25/8.06 ! [X: int,Y: int] :
% 14.25/8.06 ( ( product_swap_int_int @ ( product_Pair_int_int @ X @ Y ) )
% 14.25/8.06 = ( product_Pair_int_int @ Y @ X ) ) ).
% 14.25/8.06
% 14.25/8.06 thf(fact_3376_swap__simp,axiom,
% 14.25/8.06 ! [X: produc3658429121746597890et_nat > $o,Y: produc3658429121746597890et_nat] :
% 14.25/8.06 ( ( produc3251101204053596376et_nat @ ( produc5001842942810119800et_nat @ X @ Y ) )
% 14.25/8.06 = ( produc2829238748666232050_nat_o @ Y @ X ) ) ).
% 14.25/8.06
% 14.25/8.06 thf(fact_3377_swap__simp,axiom,
% 14.25/8.06 ! [X: produc3658429121746597890et_nat > $o,Y: produc3925858234332021118et_nat] :
% 14.25/8.06 ( ( produc1151471490888544852et_nat @ ( produc2245416461498447860et_nat @ X @ Y ) )
% 14.25/8.06 = ( produc3674526432113299062_nat_o @ Y @ X ) ) ).
% 14.25/8.06
% 14.25/8.06 thf(fact_3378_single__valuedD,axiom,
% 14.25/8.06 ! [R2: set_Pr3948176798113811640et_nat,X: heap_e7401611519738050253t_unit,Y: set_nat,Z: set_nat] :
% 14.25/8.06 ( ( single6620584844202787856et_nat @ R2 )
% 14.25/8.06 => ( ( member6260224972018164377et_nat @ ( produc7507926704131184380et_nat @ X @ Y ) @ R2 )
% 14.25/8.06 => ( ( member6260224972018164377et_nat @ ( produc7507926704131184380et_nat @ X @ Z ) @ R2 )
% 14.25/8.06 => ( Y = Z ) ) ) ) ).
% 14.25/8.06
% 14.25/8.06 thf(fact_3379_single__valuedD,axiom,
% 14.25/8.06 ! [R2: set_Pr1261947904930325089at_nat,X: nat,Y: nat,Z: nat] :
% 14.25/8.06 ( ( single5507084439971879973at_nat @ R2 )
% 14.25/8.06 => ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ Y ) @ R2 )
% 14.25/8.06 => ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ Z ) @ R2 )
% 14.25/8.06 => ( Y = Z ) ) ) ) ).
% 14.25/8.06
% 14.25/8.06 thf(fact_3380_single__valuedD,axiom,
% 14.25/8.06 ! [R2: set_Pr958786334691620121nt_int,X: int,Y: int,Z: int] :
% 14.25/8.06 ( ( single6503947681363884253nt_int @ R2 )
% 14.25/8.06 => ( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ X @ Y ) @ R2 )
% 14.25/8.06 => ( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ X @ Z ) @ R2 )
% 14.25/8.06 => ( Y = Z ) ) ) ) ).
% 14.25/8.06
% 14.25/8.06 thf(fact_3381_single__valuedD,axiom,
% 14.25/8.06 ! [R2: set_Pr3286484037609594932et_nat,X: produc3658429121746597890et_nat > $o,Y: produc3658429121746597890et_nat,Z: produc3658429121746597890et_nat] :
% 14.25/8.06 ( ( single7715245794981374860et_nat @ R2 )
% 14.25/8.06 => ( ( member1996754912294343701et_nat @ ( produc5001842942810119800et_nat @ X @ Y ) @ R2 )
% 14.25/8.06 => ( ( member1996754912294343701et_nat @ ( produc5001842942810119800et_nat @ X @ Z ) @ R2 )
% 14.25/8.06 => ( Y = Z ) ) ) ) ).
% 14.25/8.06
% 14.25/8.06 thf(fact_3382_single__valuedD,axiom,
% 14.25/8.06 ! [R2: set_Pr8536935166611901872et_nat,X: produc3658429121746597890et_nat > $o,Y: produc3925858234332021118et_nat,Z: produc3925858234332021118et_nat] :
% 14.25/8.06 ( ( single261872189793494280et_nat @ R2 )
% 14.25/8.06 => ( ( member6124377750444531601et_nat @ ( produc2245416461498447860et_nat @ X @ Y ) @ R2 )
% 14.25/8.06 => ( ( member6124377750444531601et_nat @ ( produc2245416461498447860et_nat @ X @ Z ) @ R2 )
% 14.25/8.06 => ( Y = Z ) ) ) ) ).
% 14.25/8.06
% 14.25/8.06 thf(fact_3383_single__valuedI,axiom,
% 14.25/8.06 ! [R2: set_Pr3948176798113811640et_nat] :
% 14.25/8.06 ( ! [X3: heap_e7401611519738050253t_unit,Y2: set_nat,Z3: set_nat] :
% 14.25/8.06 ( ( member6260224972018164377et_nat @ ( produc7507926704131184380et_nat @ X3 @ Y2 ) @ R2 )
% 14.25/8.06 => ( ( member6260224972018164377et_nat @ ( produc7507926704131184380et_nat @ X3 @ Z3 ) @ R2 )
% 14.25/8.06 => ( Y2 = Z3 ) ) )
% 14.25/8.06 => ( single6620584844202787856et_nat @ R2 ) ) ).
% 14.25/8.06
% 14.25/8.06 thf(fact_3384_single__valuedI,axiom,
% 14.25/8.06 ! [R2: set_Pr1261947904930325089at_nat] :
% 14.25/8.06 ( ! [X3: nat,Y2: nat,Z3: nat] :
% 14.25/8.06 ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X3 @ Y2 ) @ R2 )
% 14.25/8.06 => ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X3 @ Z3 ) @ R2 )
% 14.25/8.06 => ( Y2 = Z3 ) ) )
% 14.25/8.06 => ( single5507084439971879973at_nat @ R2 ) ) ).
% 14.25/8.06
% 14.25/8.06 thf(fact_3385_single__valuedI,axiom,
% 14.25/8.06 ! [R2: set_Pr958786334691620121nt_int] :
% 14.25/8.06 ( ! [X3: int,Y2: int,Z3: int] :
% 14.25/8.06 ( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ X3 @ Y2 ) @ R2 )
% 14.25/8.06 => ( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ X3 @ Z3 ) @ R2 )
% 14.25/8.06 => ( Y2 = Z3 ) ) )
% 14.25/8.06 => ( single6503947681363884253nt_int @ R2 ) ) ).
% 14.25/8.06
% 14.25/8.06 thf(fact_3386_single__valuedI,axiom,
% 14.25/8.06 ! [R2: set_Pr3286484037609594932et_nat] :
% 14.25/8.06 ( ! [X3: produc3658429121746597890et_nat > $o,Y2: produc3658429121746597890et_nat,Z3: produc3658429121746597890et_nat] :
% 14.25/8.06 ( ( member1996754912294343701et_nat @ ( produc5001842942810119800et_nat @ X3 @ Y2 ) @ R2 )
% 14.25/8.06 => ( ( member1996754912294343701et_nat @ ( produc5001842942810119800et_nat @ X3 @ Z3 ) @ R2 )
% 14.25/8.06 => ( Y2 = Z3 ) ) )
% 14.25/8.06 => ( single7715245794981374860et_nat @ R2 ) ) ).
% 14.25/8.06
% 14.25/8.06 thf(fact_3387_single__valuedI,axiom,
% 14.25/8.06 ! [R2: set_Pr8536935166611901872et_nat] :
% 14.25/8.06 ( ! [X3: produc3658429121746597890et_nat > $o,Y2: produc3925858234332021118et_nat,Z3: produc3925858234332021118et_nat] :
% 14.25/8.06 ( ( member6124377750444531601et_nat @ ( produc2245416461498447860et_nat @ X3 @ Y2 ) @ R2 )
% 14.25/8.06 => ( ( member6124377750444531601et_nat @ ( produc2245416461498447860et_nat @ X3 @ Z3 ) @ R2 )
% 14.25/8.06 => ( Y2 = Z3 ) ) )
% 14.25/8.06 => ( single261872189793494280et_nat @ R2 ) ) ).
% 14.25/8.06
% 14.25/8.06 thf(fact_3388_single__valued__def,axiom,
% 14.25/8.06 ( single6620584844202787856et_nat
% 14.25/8.06 = ( ^ [R4: set_Pr3948176798113811640et_nat] :
% 14.25/8.06 ! [X2: heap_e7401611519738050253t_unit,Y3: set_nat] :
% 14.25/8.06 ( ( member6260224972018164377et_nat @ ( produc7507926704131184380et_nat @ X2 @ Y3 ) @ R4 )
% 14.25/8.06 => ! [Z2: set_nat] :
% 14.25/8.06 ( ( member6260224972018164377et_nat @ ( produc7507926704131184380et_nat @ X2 @ Z2 ) @ R4 )
% 14.25/8.06 => ( Y3 = Z2 ) ) ) ) ) ).
% 14.25/8.06
% 14.25/8.06 thf(fact_3389_single__valued__def,axiom,
% 14.25/8.06 ( single5507084439971879973at_nat
% 14.25/8.06 = ( ^ [R4: set_Pr1261947904930325089at_nat] :
% 14.25/8.06 ! [X2: nat,Y3: nat] :
% 14.25/8.06 ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X2 @ Y3 ) @ R4 )
% 14.25/8.06 => ! [Z2: nat] :
% 14.25/8.06 ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X2 @ Z2 ) @ R4 )
% 14.25/8.06 => ( Y3 = Z2 ) ) ) ) ) ).
% 14.25/8.06
% 14.25/8.06 thf(fact_3390_single__valued__def,axiom,
% 14.25/8.06 ( single6503947681363884253nt_int
% 14.25/8.06 = ( ^ [R4: set_Pr958786334691620121nt_int] :
% 14.25/8.06 ! [X2: int,Y3: int] :
% 14.25/8.06 ( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ X2 @ Y3 ) @ R4 )
% 14.25/8.07 => ! [Z2: int] :
% 14.25/8.07 ( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ X2 @ Z2 ) @ R4 )
% 14.25/8.07 => ( Y3 = Z2 ) ) ) ) ) ).
% 14.25/8.07
% 14.25/8.07 thf(fact_3391_single__valued__def,axiom,
% 14.25/8.07 ( single7715245794981374860et_nat
% 14.25/8.07 = ( ^ [R4: set_Pr3286484037609594932et_nat] :
% 14.25/8.07 ! [X2: produc3658429121746597890et_nat > $o,Y3: produc3658429121746597890et_nat] :
% 14.25/8.07 ( ( member1996754912294343701et_nat @ ( produc5001842942810119800et_nat @ X2 @ Y3 ) @ R4 )
% 14.25/8.07 => ! [Z2: produc3658429121746597890et_nat] :
% 14.25/8.07 ( ( member1996754912294343701et_nat @ ( produc5001842942810119800et_nat @ X2 @ Z2 ) @ R4 )
% 14.25/8.07 => ( Y3 = Z2 ) ) ) ) ) ).
% 14.25/8.07
% 14.25/8.07 thf(fact_3392_single__valued__def,axiom,
% 14.25/8.07 ( single261872189793494280et_nat
% 14.25/8.07 = ( ^ [R4: set_Pr8536935166611901872et_nat] :
% 14.25/8.07 ! [X2: produc3658429121746597890et_nat > $o,Y3: produc3925858234332021118et_nat] :
% 14.25/8.07 ( ( member6124377750444531601et_nat @ ( produc2245416461498447860et_nat @ X2 @ Y3 ) @ R4 )
% 14.25/8.07 => ! [Z2: produc3925858234332021118et_nat] :
% 14.25/8.07 ( ( member6124377750444531601et_nat @ ( produc2245416461498447860et_nat @ X2 @ Z2 ) @ R4 )
% 14.25/8.07 => ( Y3 = Z2 ) ) ) ) ) ).
% 14.25/8.07
% 14.25/8.07 thf(fact_3393_single__valued__confluent,axiom,
% 14.25/8.07 ! [R2: set_Pr1261947904930325089at_nat,X: nat,Y: nat,Z: nat] :
% 14.25/8.07 ( ( single5507084439971879973at_nat @ R2 )
% 14.25/8.07 => ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ Y ) @ ( transi2905341329935302413cl_nat @ R2 ) )
% 14.25/8.07 => ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ Z ) @ ( transi2905341329935302413cl_nat @ R2 ) )
% 14.25/8.07 => ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ Y @ Z ) @ ( transi2905341329935302413cl_nat @ R2 ) )
% 14.25/8.07 | ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ Z @ Y ) @ ( transi2905341329935302413cl_nat @ R2 ) ) ) ) ) ) ).
% 14.25/8.07
% 14.25/8.07 thf(fact_3394_single__valued__confluent,axiom,
% 14.25/8.07 ! [R2: set_Pr958786334691620121nt_int,X: int,Y: int,Z: int] :
% 14.25/8.07 ( ( single6503947681363884253nt_int @ R2 )
% 14.25/8.07 => ( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ X @ Y ) @ ( transi2902850859426252137cl_int @ R2 ) )
% 14.25/8.07 => ( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ X @ Z ) @ ( transi2902850859426252137cl_int @ R2 ) )
% 14.25/8.07 => ( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ Y @ Z ) @ ( transi2902850859426252137cl_int @ R2 ) )
% 14.25/8.07 | ( member5262025264175285858nt_int @ ( product_Pair_int_int @ Z @ Y ) @ ( transi2902850859426252137cl_int @ R2 ) ) ) ) ) ) ).
% 14.25/8.07
% 14.25/8.07 thf(fact_3395_cSup__le__iff,axiom,
% 14.25/8.07 ! [S: set_o,A2: $o] :
% 14.25/8.07 ( ( S != bot_bot_set_o )
% 14.25/8.07 => ( ( condit5488710616941104124bove_o @ S )
% 14.25/8.07 => ( ( ord_less_eq_o @ ( complete_Sup_Sup_o @ S ) @ A2 )
% 14.25/8.07 = ( ! [X2: $o] :
% 14.25/8.07 ( ( member_o @ X2 @ S )
% 14.25/8.07 => ( ord_less_eq_o @ X2 @ A2 ) ) ) ) ) ) ).
% 14.25/8.07
% 14.25/8.07 thf(fact_3396_cSup__le__iff,axiom,
% 14.25/8.07 ! [S: set_Pr4532377907799695533_nat_o,A2: produc3658429121746597890et_nat > $o] :
% 14.25/8.07 ( ( S != bot_bo7824918357723582233_nat_o )
% 14.25/8.07 => ( ( condit4352657709669162033_nat_o @ S )
% 14.25/8.07 => ( ( ord_le729326519192465773_nat_o @ ( comple4225350640907012154_nat_o @ S ) @ A2 )
% 14.25/8.07 = ( ! [X2: produc3658429121746597890et_nat > $o] :
% 14.25/8.07 ( ( member6576561426505652726_nat_o @ X2 @ S )
% 14.25/8.07 => ( ord_le729326519192465773_nat_o @ X2 @ A2 ) ) ) ) ) ) ).
% 14.25/8.07
% 14.25/8.07 thf(fact_3397_cSup__le__iff,axiom,
% 14.25/8.07 ! [S: set_filter_nat,A2: filter_nat] :
% 14.25/8.07 ( ( S != bot_bo498966703094740906er_nat )
% 14.25/8.07 => ( ( condit4029664723990530684er_nat @ S )
% 14.25/8.07 => ( ( ord_le2510731241096832064er_nat @ ( comple2454491259644266675er_nat @ S ) @ A2 )
% 14.25/8.07 = ( ! [X2: filter_nat] :
% 14.25/8.07 ( ( member_filter_nat @ X2 @ S )
% 14.25/8.07 => ( ord_le2510731241096832064er_nat @ X2 @ A2 ) ) ) ) ) ) ).
% 14.25/8.07
% 14.25/8.07 thf(fact_3398_cSup__le__iff,axiom,
% 14.25/8.07 ! [S: set_set_nat,A2: set_nat] :
% 14.25/8.07 ( ( S != bot_bot_set_set_nat )
% 14.25/8.07 => ( ( condit5477540289124974626et_nat @ S )
% 14.25/8.07 => ( ( ord_less_eq_set_nat @ ( comple7399068483239264473et_nat @ S ) @ A2 )
% 14.25/8.07 = ( ! [X2: set_nat] :
% 14.25/8.07 ( ( member_set_nat @ X2 @ S )
% 14.25/8.07 => ( ord_less_eq_set_nat @ X2 @ A2 ) ) ) ) ) ) ).
% 14.25/8.07
% 14.25/8.07 thf(fact_3399_cSup__le__iff,axiom,
% 14.25/8.07 ! [S: set_int,A2: int] :
% 14.25/8.07 ( ( S != bot_bot_set_int )
% 14.25/8.07 => ( ( condit2212336002400062152ve_int @ S )
% 14.25/8.07 => ( ( ord_less_eq_int @ ( complete_Sup_Sup_int @ S ) @ A2 )
% 14.25/8.07 = ( ! [X2: int] :
% 14.25/8.07 ( ( member_int @ X2 @ S )
% 14.25/8.07 => ( ord_less_eq_int @ X2 @ A2 ) ) ) ) ) ) ).
% 14.25/8.07
% 14.25/8.07 thf(fact_3400_cSup__le__iff,axiom,
% 14.25/8.07 ! [S: set_nat,A2: nat] :
% 14.25/8.07 ( ( S != bot_bot_set_nat )
% 14.25/8.07 => ( ( condit2214826472909112428ve_nat @ S )
% 14.25/8.07 => ( ( ord_less_eq_nat @ ( complete_Sup_Sup_nat @ S ) @ A2 )
% 14.25/8.07 = ( ! [X2: nat] :
% 14.25/8.07 ( ( member_nat @ X2 @ S )
% 14.25/8.07 => ( ord_less_eq_nat @ X2 @ A2 ) ) ) ) ) ) ).
% 14.25/8.07
% 14.25/8.07 thf(fact_3401_cSup__le__iff,axiom,
% 14.25/8.07 ! [S: set_Product_unit,A2: product_unit] :
% 14.25/8.07 ( ( S != bot_bo3957492148770167129t_unit )
% 14.25/8.07 => ( ( condit2080475194861589617t_unit @ S )
% 14.25/8.07 => ( ( ord_le3221252021190050221t_unit @ ( comple4687483117567863418t_unit @ S ) @ A2 )
% 14.25/8.07 = ( ! [X2: product_unit] :
% 14.25/8.07 ( ( member_Product_unit @ X2 @ S )
% 14.25/8.07 => ( ord_le3221252021190050221t_unit @ X2 @ A2 ) ) ) ) ) ) ).
% 14.25/8.07
% 14.25/8.07 thf(fact_3402_cSup__mono,axiom,
% 14.25/8.07 ! [B4: set_o,A: set_o] :
% 14.25/8.07 ( ( B4 != bot_bot_set_o )
% 14.25/8.07 => ( ( condit5488710616941104124bove_o @ A )
% 14.25/8.07 => ( ! [B2: $o] :
% 14.25/8.07 ( ( member_o @ B2 @ B4 )
% 14.25/8.07 => ? [X6: $o] :
% 14.25/8.07 ( ( member_o @ X6 @ A )
% 14.25/8.07 & ( ord_less_eq_o @ B2 @ X6 ) ) )
% 14.25/8.07 => ( ord_less_eq_o @ ( complete_Sup_Sup_o @ B4 ) @ ( complete_Sup_Sup_o @ A ) ) ) ) ) ).
% 14.25/8.07
% 14.25/8.07 thf(fact_3403_cSup__mono,axiom,
% 14.25/8.07 ! [B4: set_Pr4532377907799695533_nat_o,A: set_Pr4532377907799695533_nat_o] :
% 14.25/8.07 ( ( B4 != bot_bo7824918357723582233_nat_o )
% 14.25/8.07 => ( ( condit4352657709669162033_nat_o @ A )
% 14.25/8.07 => ( ! [B2: produc3658429121746597890et_nat > $o] :
% 14.25/8.07 ( ( member6576561426505652726_nat_o @ B2 @ B4 )
% 14.25/8.07 => ? [X6: produc3658429121746597890et_nat > $o] :
% 14.25/8.07 ( ( member6576561426505652726_nat_o @ X6 @ A )
% 14.25/8.07 & ( ord_le729326519192465773_nat_o @ B2 @ X6 ) ) )
% 14.25/8.07 => ( ord_le729326519192465773_nat_o @ ( comple4225350640907012154_nat_o @ B4 ) @ ( comple4225350640907012154_nat_o @ A ) ) ) ) ) ).
% 14.25/8.07
% 14.25/8.07 thf(fact_3404_cSup__mono,axiom,
% 14.25/8.07 ! [B4: set_filter_nat,A: set_filter_nat] :
% 14.25/8.07 ( ( B4 != bot_bo498966703094740906er_nat )
% 14.25/8.07 => ( ( condit4029664723990530684er_nat @ A )
% 14.25/8.07 => ( ! [B2: filter_nat] :
% 14.25/8.07 ( ( member_filter_nat @ B2 @ B4 )
% 14.25/8.07 => ? [X6: filter_nat] :
% 14.25/8.07 ( ( member_filter_nat @ X6 @ A )
% 14.25/8.07 & ( ord_le2510731241096832064er_nat @ B2 @ X6 ) ) )
% 14.25/8.07 => ( ord_le2510731241096832064er_nat @ ( comple2454491259644266675er_nat @ B4 ) @ ( comple2454491259644266675er_nat @ A ) ) ) ) ) ).
% 14.25/8.07
% 14.25/8.07 thf(fact_3405_cSup__mono,axiom,
% 14.25/8.07 ! [B4: set_set_nat,A: set_set_nat] :
% 14.25/8.07 ( ( B4 != bot_bot_set_set_nat )
% 14.25/8.07 => ( ( condit5477540289124974626et_nat @ A )
% 14.25/8.07 => ( ! [B2: set_nat] :
% 14.25/8.07 ( ( member_set_nat @ B2 @ B4 )
% 14.25/8.07 => ? [X6: set_nat] :
% 14.25/8.07 ( ( member_set_nat @ X6 @ A )
% 14.25/8.07 & ( ord_less_eq_set_nat @ B2 @ X6 ) ) )
% 14.25/8.07 => ( ord_less_eq_set_nat @ ( comple7399068483239264473et_nat @ B4 ) @ ( comple7399068483239264473et_nat @ A ) ) ) ) ) ).
% 14.25/8.07
% 14.25/8.07 thf(fact_3406_cSup__mono,axiom,
% 14.25/8.07 ! [B4: set_int,A: set_int] :
% 14.25/8.07 ( ( B4 != bot_bot_set_int )
% 14.25/8.07 => ( ( condit2212336002400062152ve_int @ A )
% 14.25/8.07 => ( ! [B2: int] :
% 14.25/8.07 ( ( member_int @ B2 @ B4 )
% 14.25/8.07 => ? [X6: int] :
% 14.25/8.07 ( ( member_int @ X6 @ A )
% 14.25/8.07 & ( ord_less_eq_int @ B2 @ X6 ) ) )
% 14.25/8.07 => ( ord_less_eq_int @ ( complete_Sup_Sup_int @ B4 ) @ ( complete_Sup_Sup_int @ A ) ) ) ) ) ).
% 14.25/8.07
% 14.25/8.07 thf(fact_3407_cSup__mono,axiom,
% 14.25/8.07 ! [B4: set_nat,A: set_nat] :
% 14.25/8.07 ( ( B4 != bot_bot_set_nat )
% 14.25/8.07 => ( ( condit2214826472909112428ve_nat @ A )
% 14.25/8.07 => ( ! [B2: nat] :
% 14.25/8.07 ( ( member_nat @ B2 @ B4 )
% 14.25/8.07 => ? [X6: nat] :
% 14.25/8.07 ( ( member_nat @ X6 @ A )
% 14.25/8.07 & ( ord_less_eq_nat @ B2 @ X6 ) ) )
% 14.25/8.07 => ( ord_less_eq_nat @ ( complete_Sup_Sup_nat @ B4 ) @ ( complete_Sup_Sup_nat @ A ) ) ) ) ) ).
% 14.25/8.07
% 14.25/8.07 thf(fact_3408_cSup__mono,axiom,
% 14.25/8.07 ! [B4: set_Product_unit,A: set_Product_unit] :
% 14.25/8.07 ( ( B4 != bot_bo3957492148770167129t_unit )
% 14.25/8.07 => ( ( condit2080475194861589617t_unit @ A )
% 14.25/8.07 => ( ! [B2: product_unit] :
% 14.25/8.07 ( ( member_Product_unit @ B2 @ B4 )
% 14.25/8.07 => ? [X6: product_unit] :
% 14.25/8.07 ( ( member_Product_unit @ X6 @ A )
% 14.25/8.07 & ( ord_le3221252021190050221t_unit @ B2 @ X6 ) ) )
% 14.25/8.07 => ( ord_le3221252021190050221t_unit @ ( comple4687483117567863418t_unit @ B4 ) @ ( comple4687483117567863418t_unit @ A ) ) ) ) ) ).
% 14.25/8.07
% 14.25/8.07 thf(fact_3409_cSUP__le__iff,axiom,
% 14.25/8.07 ! [A: set_Product_unit,F: product_unit > int,U: int] :
% 14.25/8.07 ( ( A != bot_bo3957492148770167129t_unit )
% 14.25/8.07 => ( ( condit2212336002400062152ve_int @ ( image_873079544045703924it_int @ F @ A ) )
% 14.25/8.07 => ( ( ord_less_eq_int @ ( complete_Sup_Sup_int @ ( image_873079544045703924it_int @ F @ A ) ) @ U )
% 14.25/8.07 = ( ! [X2: product_unit] :
% 14.25/8.07 ( ( member_Product_unit @ X2 @ A )
% 14.25/8.07 => ( ord_less_eq_int @ ( F @ X2 ) @ U ) ) ) ) ) ) ).
% 14.25/8.07
% 14.25/8.07 thf(fact_3410_cSUP__le__iff,axiom,
% 14.25/8.07 ! [A: set_o,F: $o > int,U: int] :
% 14.25/8.07 ( ( A != bot_bot_set_o )
% 14.25/8.07 => ( ( condit2212336002400062152ve_int @ ( image_o_int2 @ F @ A ) )
% 14.25/8.07 => ( ( ord_less_eq_int @ ( complete_Sup_Sup_int @ ( image_o_int2 @ F @ A ) ) @ U )
% 14.25/8.07 = ( ! [X2: $o] :
% 14.25/8.07 ( ( member_o @ X2 @ A )
% 14.25/8.07 => ( ord_less_eq_int @ ( F @ X2 ) @ U ) ) ) ) ) ) ).
% 14.25/8.07
% 14.25/8.07 thf(fact_3411_cSUP__le__iff,axiom,
% 14.25/8.07 ! [A: set_nat,F: nat > int,U: int] :
% 14.25/8.07 ( ( A != bot_bot_set_nat )
% 14.25/8.07 => ( ( condit2212336002400062152ve_int @ ( image_nat_int2 @ F @ A ) )
% 14.25/8.07 => ( ( ord_less_eq_int @ ( complete_Sup_Sup_int @ ( image_nat_int2 @ F @ A ) ) @ U )
% 14.25/8.07 = ( ! [X2: nat] :
% 14.25/8.07 ( ( member_nat @ X2 @ A )
% 14.25/8.07 => ( ord_less_eq_int @ ( F @ X2 ) @ U ) ) ) ) ) ) ).
% 14.25/8.07
% 14.25/8.07 thf(fact_3412_cSUP__le__iff,axiom,
% 14.25/8.07 ! [A: set_int,F: int > int,U: int] :
% 14.25/8.07 ( ( A != bot_bot_set_int )
% 14.25/8.07 => ( ( condit2212336002400062152ve_int @ ( image_int_int2 @ F @ A ) )
% 14.25/8.07 => ( ( ord_less_eq_int @ ( complete_Sup_Sup_int @ ( image_int_int2 @ F @ A ) ) @ U )
% 14.25/8.07 = ( ! [X2: int] :
% 14.25/8.07 ( ( member_int @ X2 @ A )
% 14.25/8.07 => ( ord_less_eq_int @ ( F @ X2 ) @ U ) ) ) ) ) ) ).
% 14.25/8.07
% 14.25/8.07 thf(fact_3413_cSUP__le__iff,axiom,
% 14.25/8.07 ! [A: set_Product_unit,F: product_unit > nat,U: nat] :
% 14.25/8.07 ( ( A != bot_bo3957492148770167129t_unit )
% 14.25/8.07 => ( ( condit2214826472909112428ve_nat @ ( image_875570014554754200it_nat @ F @ A ) )
% 14.25/8.07 => ( ( ord_less_eq_nat @ ( complete_Sup_Sup_nat @ ( image_875570014554754200it_nat @ F @ A ) ) @ U )
% 14.25/8.07 = ( ! [X2: product_unit] :
% 14.25/8.07 ( ( member_Product_unit @ X2 @ A )
% 14.25/8.07 => ( ord_less_eq_nat @ ( F @ X2 ) @ U ) ) ) ) ) ) ).
% 14.25/8.07
% 14.25/8.07 thf(fact_3414_cSUP__le__iff,axiom,
% 14.25/8.07 ! [A: set_o,F: $o > nat,U: nat] :
% 14.25/8.07 ( ( A != bot_bot_set_o )
% 14.25/8.07 => ( ( condit2214826472909112428ve_nat @ ( image_o_nat2 @ F @ A ) )
% 14.25/8.07 => ( ( ord_less_eq_nat @ ( complete_Sup_Sup_nat @ ( image_o_nat2 @ F @ A ) ) @ U )
% 14.25/8.07 = ( ! [X2: $o] :
% 14.25/8.07 ( ( member_o @ X2 @ A )
% 14.25/8.07 => ( ord_less_eq_nat @ ( F @ X2 ) @ U ) ) ) ) ) ) ).
% 14.25/8.07
% 14.25/8.07 thf(fact_3415_cSUP__le__iff,axiom,
% 14.25/8.07 ! [A: set_nat,F: nat > nat,U: nat] :
% 14.25/8.07 ( ( A != bot_bot_set_nat )
% 14.25/8.07 => ( ( condit2214826472909112428ve_nat @ ( image_nat_nat2 @ F @ A ) )
% 14.25/8.07 => ( ( ord_less_eq_nat @ ( complete_Sup_Sup_nat @ ( image_nat_nat2 @ F @ A ) ) @ U )
% 14.25/8.07 = ( ! [X2: nat] :
% 14.25/8.07 ( ( member_nat @ X2 @ A )
% 14.25/8.07 => ( ord_less_eq_nat @ ( F @ X2 ) @ U ) ) ) ) ) ) ).
% 14.25/8.07
% 14.25/8.07 thf(fact_3416_cSUP__le__iff,axiom,
% 14.25/8.07 ! [A: set_int,F: int > nat,U: nat] :
% 14.25/8.07 ( ( A != bot_bot_set_int )
% 14.25/8.07 => ( ( condit2214826472909112428ve_nat @ ( image_int_nat2 @ F @ A ) )
% 14.25/8.07 => ( ( ord_less_eq_nat @ ( complete_Sup_Sup_nat @ ( image_int_nat2 @ F @ A ) ) @ U )
% 14.25/8.07 = ( ! [X2: int] :
% 14.25/8.07 ( ( member_int @ X2 @ A )
% 14.25/8.07 => ( ord_less_eq_nat @ ( F @ X2 ) @ U ) ) ) ) ) ) ).
% 14.25/8.07
% 14.25/8.07 thf(fact_3417_cSUP__le__iff,axiom,
% 14.25/8.07 ! [A: set_Product_unit,F: product_unit > product_unit,U: product_unit] :
% 14.25/8.07 ( ( A != bot_bo3957492148770167129t_unit )
% 14.25/8.07 => ( ( condit2080475194861589617t_unit @ ( image_405062704495631173t_unit @ F @ A ) )
% 14.25/8.07 => ( ( ord_le3221252021190050221t_unit @ ( comple4687483117567863418t_unit @ ( image_405062704495631173t_unit @ F @ A ) ) @ U )
% 14.25/8.07 = ( ! [X2: product_unit] :
% 14.25/8.07 ( ( member_Product_unit @ X2 @ A )
% 14.25/8.07 => ( ord_le3221252021190050221t_unit @ ( F @ X2 ) @ U ) ) ) ) ) ) ).
% 14.25/8.07
% 14.25/8.07 thf(fact_3418_cSUP__le__iff,axiom,
% 14.25/8.07 ! [A: set_o,F: $o > product_unit,U: product_unit] :
% 14.25/8.07 ( ( A != bot_bot_set_o )
% 14.25/8.07 => ( ( condit2080475194861589617t_unit @ ( image_o_Product_unit2 @ F @ A ) )
% 14.25/8.07 => ( ( ord_le3221252021190050221t_unit @ ( comple4687483117567863418t_unit @ ( image_o_Product_unit2 @ F @ A ) ) @ U )
% 14.25/8.07 = ( ! [X2: $o] :
% 14.25/8.07 ( ( member_o @ X2 @ A )
% 14.25/8.07 => ( ord_le3221252021190050221t_unit @ ( F @ X2 ) @ U ) ) ) ) ) ) ).
% 14.25/8.07
% 14.25/8.07 thf(fact_3419_cSUP__mono,axiom,
% 14.25/8.07 ! [A: set_Product_unit,G: nat > set_nat,B4: set_nat,F: product_unit > set_nat] :
% 14.25/8.07 ( ( A != bot_bo3957492148770167129t_unit )
% 14.25/8.07 => ( ( condit5477540289124974626et_nat @ ( image_nat_set_nat @ G @ B4 ) )
% 14.25/8.07 => ( ! [N3: product_unit] :
% 14.25/8.07 ( ( member_Product_unit @ N3 @ A )
% 14.25/8.07 => ? [X6: nat] :
% 14.25/8.07 ( ( member_nat @ X6 @ B4 )
% 14.25/8.07 & ( ord_less_eq_set_nat @ ( F @ N3 ) @ ( G @ X6 ) ) ) )
% 14.25/8.07 => ( ord_less_eq_set_nat @ ( comple7399068483239264473et_nat @ ( image_8173849991827986510et_nat @ F @ A ) ) @ ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ G @ B4 ) ) ) ) ) ) ).
% 14.25/8.07
% 14.25/8.07 thf(fact_3420_cSUP__mono,axiom,
% 14.25/8.07 ! [A: set_o,G: nat > set_nat,B4: set_nat,F: $o > set_nat] :
% 14.25/8.07 ( ( A != bot_bot_set_o )
% 14.25/8.07 => ( ( condit5477540289124974626et_nat @ ( image_nat_set_nat @ G @ B4 ) )
% 14.25/8.07 => ( ! [N3: $o] :
% 14.25/8.07 ( ( member_o @ N3 @ A )
% 14.25/8.07 => ? [X6: nat] :
% 14.25/8.07 ( ( member_nat @ X6 @ B4 )
% 14.25/8.07 & ( ord_less_eq_set_nat @ ( F @ N3 ) @ ( G @ X6 ) ) ) )
% 14.25/8.07 => ( ord_less_eq_set_nat @ ( comple7399068483239264473et_nat @ ( image_o_set_nat @ F @ A ) ) @ ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ G @ B4 ) ) ) ) ) ) ).
% 14.25/8.07
% 14.25/8.07 thf(fact_3421_cSUP__mono,axiom,
% 14.25/8.07 ! [A: set_nat,G: nat > set_nat,B4: set_nat,F: nat > set_nat] :
% 14.25/8.07 ( ( A != bot_bot_set_nat )
% 14.25/8.07 => ( ( condit5477540289124974626et_nat @ ( image_nat_set_nat @ G @ B4 ) )
% 14.25/8.07 => ( ! [N3: nat] :
% 14.25/8.07 ( ( member_nat @ N3 @ A )
% 14.25/8.07 => ? [X6: nat] :
% 14.25/8.07 ( ( member_nat @ X6 @ B4 )
% 14.25/8.07 & ( ord_less_eq_set_nat @ ( F @ N3 ) @ ( G @ X6 ) ) ) )
% 14.25/8.07 => ( ord_less_eq_set_nat @ ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ F @ A ) ) @ ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ G @ B4 ) ) ) ) ) ) ).
% 14.25/8.07
% 14.25/8.07 thf(fact_3422_cSUP__mono,axiom,
% 14.25/8.07 ! [A: set_int,G: nat > set_nat,B4: set_nat,F: int > set_nat] :
% 14.25/8.07 ( ( A != bot_bot_set_int )
% 14.25/8.07 => ( ( condit5477540289124974626et_nat @ ( image_nat_set_nat @ G @ B4 ) )
% 14.25/8.07 => ( ! [N3: int] :
% 14.25/8.07 ( ( member_int @ N3 @ A )
% 14.25/8.07 => ? [X6: nat] :
% 14.25/8.07 ( ( member_nat @ X6 @ B4 )
% 14.25/8.07 & ( ord_less_eq_set_nat @ ( F @ N3 ) @ ( G @ X6 ) ) ) )
% 14.25/8.07 => ( ord_less_eq_set_nat @ ( comple7399068483239264473et_nat @ ( image_int_set_nat @ F @ A ) ) @ ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ G @ B4 ) ) ) ) ) ) ).
% 14.25/8.07
% 14.25/8.07 thf(fact_3423_cSUP__mono,axiom,
% 14.25/8.07 ! [A: set_Pr4532377907799695533_nat_o,G: nat > set_nat,B4: set_nat,F: ( produc3658429121746597890et_nat > $o ) > set_nat] :
% 14.25/8.07 ( ( A != bot_bo7824918357723582233_nat_o )
% 14.25/8.07 => ( ( condit5477540289124974626et_nat @ ( image_nat_set_nat @ G @ B4 ) )
% 14.25/8.07 => ( ! [N3: produc3658429121746597890et_nat > $o] :
% 14.25/8.07 ( ( member6576561426505652726_nat_o @ N3 @ A )
% 14.25/8.07 => ? [X6: nat] :
% 14.25/8.07 ( ( member_nat @ X6 @ B4 )
% 14.25/8.07 & ( ord_less_eq_set_nat @ ( F @ N3 ) @ ( G @ X6 ) ) ) )
% 14.25/8.07 => ( ord_less_eq_set_nat @ ( comple7399068483239264473et_nat @ ( image_6888641473154130958et_nat @ F @ A ) ) @ ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ G @ B4 ) ) ) ) ) ) ).
% 14.25/8.07
% 14.25/8.07 thf(fact_3424_cSUP__mono,axiom,
% 14.25/8.07 ! [A: set_Product_unit,G: nat > nat,B4: set_nat,F: product_unit > nat] :
% 14.25/8.07 ( ( A != bot_bo3957492148770167129t_unit )
% 14.25/8.07 => ( ( condit2214826472909112428ve_nat @ ( image_nat_nat2 @ G @ B4 ) )
% 14.25/8.07 => ( ! [N3: product_unit] :
% 14.25/8.07 ( ( member_Product_unit @ N3 @ A )
% 14.25/8.07 => ? [X6: nat] :
% 14.25/8.07 ( ( member_nat @ X6 @ B4 )
% 14.25/8.07 & ( ord_less_eq_nat @ ( F @ N3 ) @ ( G @ X6 ) ) ) )
% 14.25/8.07 => ( ord_less_eq_nat @ ( complete_Sup_Sup_nat @ ( image_875570014554754200it_nat @ F @ A ) ) @ ( complete_Sup_Sup_nat @ ( image_nat_nat2 @ G @ B4 ) ) ) ) ) ) ).
% 14.25/8.07
% 14.25/8.07 thf(fact_3425_cSUP__mono,axiom,
% 14.25/8.07 ! [A: set_o,G: nat > nat,B4: set_nat,F: $o > nat] :
% 14.25/8.07 ( ( A != bot_bot_set_o )
% 14.25/8.07 => ( ( condit2214826472909112428ve_nat @ ( image_nat_nat2 @ G @ B4 ) )
% 14.25/8.07 => ( ! [N3: $o] :
% 14.25/8.07 ( ( member_o @ N3 @ A )
% 14.25/8.07 => ? [X6: nat] :
% 14.25/8.07 ( ( member_nat @ X6 @ B4 )
% 14.25/8.07 & ( ord_less_eq_nat @ ( F @ N3 ) @ ( G @ X6 ) ) ) )
% 14.25/8.07 => ( ord_less_eq_nat @ ( complete_Sup_Sup_nat @ ( image_o_nat2 @ F @ A ) ) @ ( complete_Sup_Sup_nat @ ( image_nat_nat2 @ G @ B4 ) ) ) ) ) ) ).
% 14.25/8.07
% 14.25/8.07 thf(fact_3426_cSUP__mono,axiom,
% 14.25/8.07 ! [A: set_nat,G: nat > nat,B4: set_nat,F: nat > nat] :
% 14.25/8.07 ( ( A != bot_bot_set_nat )
% 14.25/8.07 => ( ( condit2214826472909112428ve_nat @ ( image_nat_nat2 @ G @ B4 ) )
% 14.25/8.07 => ( ! [N3: nat] :
% 14.25/8.07 ( ( member_nat @ N3 @ A )
% 14.25/8.07 => ? [X6: nat] :
% 14.25/8.07 ( ( member_nat @ X6 @ B4 )
% 14.25/8.07 & ( ord_less_eq_nat @ ( F @ N3 ) @ ( G @ X6 ) ) ) )
% 14.25/8.07 => ( ord_less_eq_nat @ ( complete_Sup_Sup_nat @ ( image_nat_nat2 @ F @ A ) ) @ ( complete_Sup_Sup_nat @ ( image_nat_nat2 @ G @ B4 ) ) ) ) ) ) ).
% 14.25/8.07
% 14.25/8.07 thf(fact_3427_cSUP__mono,axiom,
% 14.25/8.07 ! [A: set_int,G: nat > nat,B4: set_nat,F: int > nat] :
% 14.25/8.07 ( ( A != bot_bot_set_int )
% 14.25/8.07 => ( ( condit2214826472909112428ve_nat @ ( image_nat_nat2 @ G @ B4 ) )
% 14.25/8.07 => ( ! [N3: int] :
% 14.25/8.07 ( ( member_int @ N3 @ A )
% 14.25/8.07 => ? [X6: nat] :
% 14.25/8.07 ( ( member_nat @ X6 @ B4 )
% 14.25/8.07 & ( ord_less_eq_nat @ ( F @ N3 ) @ ( G @ X6 ) ) ) )
% 14.25/8.07 => ( ord_less_eq_nat @ ( complete_Sup_Sup_nat @ ( image_int_nat2 @ F @ A ) ) @ ( complete_Sup_Sup_nat @ ( image_nat_nat2 @ G @ B4 ) ) ) ) ) ) ).
% 14.25/8.07
% 14.25/8.07 thf(fact_3428_cSUP__mono,axiom,
% 14.25/8.07 ! [A: set_Pr4532377907799695533_nat_o,G: nat > nat,B4: set_nat,F: ( produc3658429121746597890et_nat > $o ) > nat] :
% 14.25/8.07 ( ( A != bot_bo7824918357723582233_nat_o )
% 14.25/8.07 => ( ( condit2214826472909112428ve_nat @ ( image_nat_nat2 @ G @ B4 ) )
% 14.25/8.07 => ( ! [N3: produc3658429121746597890et_nat > $o] :
% 14.25/8.07 ( ( member6576561426505652726_nat_o @ N3 @ A )
% 14.25/8.07 => ? [X6: nat] :
% 14.25/8.07 ( ( member_nat @ X6 @ B4 )
% 14.25/8.07 & ( ord_less_eq_nat @ ( F @ N3 ) @ ( G @ X6 ) ) ) )
% 14.25/8.07 => ( ord_less_eq_nat @ ( complete_Sup_Sup_nat @ ( image_2174381888560928856_o_nat @ F @ A ) ) @ ( complete_Sup_Sup_nat @ ( image_nat_nat2 @ G @ B4 ) ) ) ) ) ) ).
% 14.25/8.07
% 14.25/8.07 thf(fact_3429_cSup__subset__mono,axiom,
% 14.25/8.07 ! [A: set_o,B4: set_o] :
% 14.25/8.07 ( ( A != bot_bot_set_o )
% 14.25/8.07 => ( ( condit5488710616941104124bove_o @ B4 )
% 14.25/8.07 => ( ( ord_less_eq_set_o @ A @ B4 )
% 14.25/8.07 => ( ord_less_eq_o @ ( complete_Sup_Sup_o @ A ) @ ( complete_Sup_Sup_o @ B4 ) ) ) ) ) ).
% 14.25/8.07
% 14.25/8.07 thf(fact_3430_cSup__subset__mono,axiom,
% 14.25/8.07 ! [A: set_Pr4532377907799695533_nat_o,B4: set_Pr4532377907799695533_nat_o] :
% 14.25/8.07 ( ( A != bot_bo7824918357723582233_nat_o )
% 14.25/8.07 => ( ( condit4352657709669162033_nat_o @ B4 )
% 14.25/8.07 => ( ( ord_le2965882846123202637_nat_o @ A @ B4 )
% 14.25/8.07 => ( ord_le729326519192465773_nat_o @ ( comple4225350640907012154_nat_o @ A ) @ ( comple4225350640907012154_nat_o @ B4 ) ) ) ) ) ).
% 14.25/8.07
% 14.25/8.07 thf(fact_3431_cSup__subset__mono,axiom,
% 14.25/8.07 ! [A: set_filter_nat,B4: set_filter_nat] :
% 14.25/8.07 ( ( A != bot_bo498966703094740906er_nat )
% 14.25/8.07 => ( ( condit4029664723990530684er_nat @ B4 )
% 14.25/8.07 => ( ( ord_le2426478655948331894er_nat @ A @ B4 )
% 14.25/8.07 => ( ord_le2510731241096832064er_nat @ ( comple2454491259644266675er_nat @ A ) @ ( comple2454491259644266675er_nat @ B4 ) ) ) ) ) ).
% 14.25/8.07
% 14.25/8.07 thf(fact_3432_cSup__subset__mono,axiom,
% 14.25/8.07 ! [A: set_set_nat,B4: set_set_nat] :
% 14.25/8.07 ( ( A != bot_bot_set_set_nat )
% 14.25/8.07 => ( ( condit5477540289124974626et_nat @ B4 )
% 14.25/8.07 => ( ( ord_le6893508408891458716et_nat @ A @ B4 )
% 14.25/8.07 => ( ord_less_eq_set_nat @ ( comple7399068483239264473et_nat @ A ) @ ( comple7399068483239264473et_nat @ B4 ) ) ) ) ) ).
% 14.25/8.07
% 14.25/8.07 thf(fact_3433_cSup__subset__mono,axiom,
% 14.25/8.07 ! [A: set_int,B4: set_int] :
% 14.25/8.07 ( ( A != bot_bot_set_int )
% 14.25/8.07 => ( ( condit2212336002400062152ve_int @ B4 )
% 14.25/8.07 => ( ( ord_less_eq_set_int @ A @ B4 )
% 14.25/8.07 => ( ord_less_eq_int @ ( complete_Sup_Sup_int @ A ) @ ( complete_Sup_Sup_int @ B4 ) ) ) ) ) ).
% 14.25/8.07
% 14.25/8.07 thf(fact_3434_cSup__subset__mono,axiom,
% 14.25/8.07 ! [A: set_nat,B4: set_nat] :
% 14.25/8.07 ( ( A != bot_bot_set_nat )
% 14.25/8.07 => ( ( condit2214826472909112428ve_nat @ B4 )
% 14.25/8.07 => ( ( ord_less_eq_set_nat @ A @ B4 )
% 14.25/8.07 => ( ord_less_eq_nat @ ( complete_Sup_Sup_nat @ A ) @ ( complete_Sup_Sup_nat @ B4 ) ) ) ) ) ).
% 14.25/8.07
% 14.25/8.07 thf(fact_3435_cSup__subset__mono,axiom,
% 14.25/8.07 ! [A: set_Product_unit,B4: set_Product_unit] :
% 14.25/8.07 ( ( A != bot_bo3957492148770167129t_unit )
% 14.25/8.07 => ( ( condit2080475194861589617t_unit @ B4 )
% 14.25/8.07 => ( ( ord_le3507040750410214029t_unit @ A @ B4 )
% 14.25/8.07 => ( ord_le3221252021190050221t_unit @ ( comple4687483117567863418t_unit @ A ) @ ( comple4687483117567863418t_unit @ B4 ) ) ) ) ) ).
% 14.25/8.07
% 14.25/8.07 thf(fact_3436_cSup__insert__If,axiom,
% 14.25/8.07 ! [X4: set_set_nat,A2: set_nat] :
% 14.25/8.07 ( ( condit5477540289124974626et_nat @ X4 )
% 14.25/8.07 => ( ( ( X4 = bot_bot_set_set_nat )
% 14.25/8.07 => ( ( comple7399068483239264473et_nat @ ( insert_set_nat @ A2 @ X4 ) )
% 14.25/8.07 = A2 ) )
% 14.25/8.07 & ( ( X4 != bot_bot_set_set_nat )
% 14.25/8.07 => ( ( comple7399068483239264473et_nat @ ( insert_set_nat @ A2 @ X4 ) )
% 14.25/8.07 = ( sup_sup_set_nat @ A2 @ ( comple7399068483239264473et_nat @ X4 ) ) ) ) ) ) ).
% 14.25/8.07
% 14.25/8.07 thf(fact_3437_cSup__insert__If,axiom,
% 14.25/8.07 ! [X4: set_o,A2: $o] :
% 14.25/8.07 ( ( condit5488710616941104124bove_o @ X4 )
% 14.25/8.07 => ( ( complete_Sup_Sup_o @ ( insert_o @ A2 @ X4 ) )
% 14.25/8.07 = ( ( ( X4 = bot_bot_set_o )
% 14.25/8.07 => A2 )
% 14.25/8.07 & ( ( X4 != bot_bot_set_o )
% 14.25/8.07 => ( sup_sup_o @ A2 @ ( complete_Sup_Sup_o @ X4 ) ) ) ) ) ) ).
% 14.25/8.07
% 14.25/8.07 thf(fact_3438_cSup__insert__If,axiom,
% 14.25/8.07 ! [X4: set_int,A2: int] :
% 14.25/8.07 ( ( condit2212336002400062152ve_int @ X4 )
% 14.25/8.07 => ( ( ( X4 = bot_bot_set_int )
% 14.25/8.07 => ( ( complete_Sup_Sup_int @ ( insert_int @ A2 @ X4 ) )
% 14.25/8.07 = A2 ) )
% 14.25/8.07 & ( ( X4 != bot_bot_set_int )
% 14.25/8.07 => ( ( complete_Sup_Sup_int @ ( insert_int @ A2 @ X4 ) )
% 14.25/8.07 = ( sup_sup_int @ A2 @ ( complete_Sup_Sup_int @ X4 ) ) ) ) ) ) ).
% 14.25/8.07
% 14.25/8.07 thf(fact_3439_cSup__insert__If,axiom,
% 14.25/8.07 ! [X4: set_Pr4532377907799695533_nat_o,A2: produc3658429121746597890et_nat > $o] :
% 14.25/8.07 ( ( condit4352657709669162033_nat_o @ X4 )
% 14.25/8.07 => ( ( ( X4 = bot_bo7824918357723582233_nat_o )
% 14.25/8.07 => ( ( comple4225350640907012154_nat_o @ ( insert5175938949040314269_nat_o @ A2 @ X4 ) )
% 14.25/8.07 = A2 ) )
% 14.25/8.07 & ( ( X4 != bot_bo7824918357723582233_nat_o )
% 14.25/8.07 => ( ( comple4225350640907012154_nat_o @ ( insert5175938949040314269_nat_o @ A2 @ X4 ) )
% 14.25/8.07 = ( sup_su5453871518329203617_nat_o @ A2 @ ( comple4225350640907012154_nat_o @ X4 ) ) ) ) ) ) ).
% 14.25/8.07
% 14.25/8.07 thf(fact_3440_cSup__insert__If,axiom,
% 14.25/8.07 ! [X4: set_nat,A2: nat] :
% 14.25/8.07 ( ( condit2214826472909112428ve_nat @ X4 )
% 14.25/8.07 => ( ( ( X4 = bot_bot_set_nat )
% 14.25/8.07 => ( ( complete_Sup_Sup_nat @ ( insert_nat @ A2 @ X4 ) )
% 14.25/8.07 = A2 ) )
% 14.25/8.07 & ( ( X4 != bot_bot_set_nat )
% 14.25/8.07 => ( ( complete_Sup_Sup_nat @ ( insert_nat @ A2 @ X4 ) )
% 14.25/8.07 = ( sup_sup_nat @ A2 @ ( complete_Sup_Sup_nat @ X4 ) ) ) ) ) ) ).
% 14.25/8.07
% 14.25/8.07 thf(fact_3441_cSup__insert__If,axiom,
% 14.25/8.07 ! [X4: set_Product_unit,A2: product_unit] :
% 14.25/8.07 ( ( condit2080475194861589617t_unit @ X4 )
% 14.25/8.07 => ( ( ( X4 = bot_bo3957492148770167129t_unit )
% 14.25/8.07 => ( ( comple4687483117567863418t_unit @ ( insert_Product_unit @ A2 @ X4 ) )
% 14.25/8.07 = A2 ) )
% 14.25/8.07 & ( ( X4 != bot_bo3957492148770167129t_unit )
% 14.25/8.07 => ( ( comple4687483117567863418t_unit @ ( insert_Product_unit @ A2 @ X4 ) )
% 14.25/8.07 = ( sup_sup_Product_unit @ A2 @ ( comple4687483117567863418t_unit @ X4 ) ) ) ) ) ) ).
% 14.25/8.07
% 14.25/8.07 thf(fact_3442_cSup__insert,axiom,
% 14.25/8.07 ! [X4: set_set_nat,A2: set_nat] :
% 14.25/8.07 ( ( X4 != bot_bot_set_set_nat )
% 14.25/8.07 => ( ( condit5477540289124974626et_nat @ X4 )
% 14.25/8.07 => ( ( comple7399068483239264473et_nat @ ( insert_set_nat @ A2 @ X4 ) )
% 14.25/8.07 = ( sup_sup_set_nat @ A2 @ ( comple7399068483239264473et_nat @ X4 ) ) ) ) ) ).
% 14.25/8.07
% 14.25/8.07 thf(fact_3443_cSup__insert,axiom,
% 14.25/8.07 ! [X4: set_o,A2: $o] :
% 14.25/8.07 ( ( X4 != bot_bot_set_o )
% 14.25/8.07 => ( ( condit5488710616941104124bove_o @ X4 )
% 14.25/8.07 => ( ( complete_Sup_Sup_o @ ( insert_o @ A2 @ X4 ) )
% 14.25/8.07 = ( sup_sup_o @ A2 @ ( complete_Sup_Sup_o @ X4 ) ) ) ) ) ).
% 14.25/8.07
% 14.25/8.07 thf(fact_3444_cSup__insert,axiom,
% 14.25/8.07 ! [X4: set_int,A2: int] :
% 14.25/8.07 ( ( X4 != bot_bot_set_int )
% 14.25/8.07 => ( ( condit2212336002400062152ve_int @ X4 )
% 14.25/8.07 => ( ( complete_Sup_Sup_int @ ( insert_int @ A2 @ X4 ) )
% 14.25/8.07 = ( sup_sup_int @ A2 @ ( complete_Sup_Sup_int @ X4 ) ) ) ) ) ).
% 14.25/8.07
% 14.25/8.07 thf(fact_3445_cSup__insert,axiom,
% 14.25/8.07 ! [X4: set_Pr4532377907799695533_nat_o,A2: produc3658429121746597890et_nat > $o] :
% 14.25/8.07 ( ( X4 != bot_bo7824918357723582233_nat_o )
% 14.25/8.07 => ( ( condit4352657709669162033_nat_o @ X4 )
% 14.25/8.07 => ( ( comple4225350640907012154_nat_o @ ( insert5175938949040314269_nat_o @ A2 @ X4 ) )
% 14.25/8.07 = ( sup_su5453871518329203617_nat_o @ A2 @ ( comple4225350640907012154_nat_o @ X4 ) ) ) ) ) ).
% 14.25/8.07
% 14.25/8.07 thf(fact_3446_cSup__insert,axiom,
% 14.25/8.07 ! [X4: set_nat,A2: nat] :
% 14.25/8.07 ( ( X4 != bot_bot_set_nat )
% 14.25/8.07 => ( ( condit2214826472909112428ve_nat @ X4 )
% 14.25/8.07 => ( ( complete_Sup_Sup_nat @ ( insert_nat @ A2 @ X4 ) )
% 14.25/8.07 = ( sup_sup_nat @ A2 @ ( complete_Sup_Sup_nat @ X4 ) ) ) ) ) ).
% 14.25/8.07
% 14.25/8.07 thf(fact_3447_cSup__insert,axiom,
% 14.25/8.07 ! [X4: set_Product_unit,A2: product_unit] :
% 14.25/8.07 ( ( X4 != bot_bo3957492148770167129t_unit )
% 14.25/8.07 => ( ( condit2080475194861589617t_unit @ X4 )
% 14.25/8.07 => ( ( comple4687483117567863418t_unit @ ( insert_Product_unit @ A2 @ X4 ) )
% 14.25/8.07 = ( sup_sup_Product_unit @ A2 @ ( comple4687483117567863418t_unit @ X4 ) ) ) ) ) ).
% 14.25/8.07
% 14.25/8.07 thf(fact_3448_cSup__union__distrib,axiom,
% 14.25/8.07 ! [A: set_set_nat,B4: set_set_nat] :
% 14.25/8.07 ( ( A != bot_bot_set_set_nat )
% 14.25/8.07 => ( ( condit5477540289124974626et_nat @ A )
% 14.25/8.07 => ( ( B4 != bot_bot_set_set_nat )
% 14.25/8.07 => ( ( condit5477540289124974626et_nat @ B4 )
% 14.25/8.07 => ( ( comple7399068483239264473et_nat @ ( sup_sup_set_set_nat @ A @ B4 ) )
% 14.25/8.07 = ( sup_sup_set_nat @ ( comple7399068483239264473et_nat @ A ) @ ( comple7399068483239264473et_nat @ B4 ) ) ) ) ) ) ) ).
% 14.25/8.07
% 14.25/8.07 thf(fact_3449_cSup__union__distrib,axiom,
% 14.25/8.07 ! [A: set_o,B4: set_o] :
% 14.25/8.07 ( ( A != bot_bot_set_o )
% 14.25/8.07 => ( ( condit5488710616941104124bove_o @ A )
% 14.25/8.07 => ( ( B4 != bot_bot_set_o )
% 14.25/8.07 => ( ( condit5488710616941104124bove_o @ B4 )
% 14.25/8.07 => ( ( complete_Sup_Sup_o @ ( sup_sup_set_o @ A @ B4 ) )
% 14.25/8.07 = ( sup_sup_o @ ( complete_Sup_Sup_o @ A ) @ ( complete_Sup_Sup_o @ B4 ) ) ) ) ) ) ) ).
% 14.25/8.07
% 14.25/8.07 thf(fact_3450_cSup__union__distrib,axiom,
% 14.25/8.07 ! [A: set_int,B4: set_int] :
% 14.25/8.07 ( ( A != bot_bot_set_int )
% 14.25/8.07 => ( ( condit2212336002400062152ve_int @ A )
% 14.25/8.07 => ( ( B4 != bot_bot_set_int )
% 14.25/8.07 => ( ( condit2212336002400062152ve_int @ B4 )
% 14.25/8.07 => ( ( complete_Sup_Sup_int @ ( sup_sup_set_int @ A @ B4 ) )
% 14.25/8.07 = ( sup_sup_int @ ( complete_Sup_Sup_int @ A ) @ ( complete_Sup_Sup_int @ B4 ) ) ) ) ) ) ) ).
% 14.25/8.07
% 14.25/8.07 thf(fact_3451_cSup__union__distrib,axiom,
% 14.25/8.07 ! [A: set_Pr4532377907799695533_nat_o,B4: set_Pr4532377907799695533_nat_o] :
% 14.25/8.07 ( ( A != bot_bo7824918357723582233_nat_o )
% 14.25/8.07 => ( ( condit4352657709669162033_nat_o @ A )
% 14.25/8.07 => ( ( B4 != bot_bo7824918357723582233_nat_o )
% 14.25/8.07 => ( ( condit4352657709669162033_nat_o @ B4 )
% 14.25/8.07 => ( ( comple4225350640907012154_nat_o @ ( sup_su5209123915105501825_nat_o @ A @ B4 ) )
% 14.25/8.07 = ( sup_su5453871518329203617_nat_o @ ( comple4225350640907012154_nat_o @ A ) @ ( comple4225350640907012154_nat_o @ B4 ) ) ) ) ) ) ) ).
% 14.25/8.07
% 14.25/8.07 thf(fact_3452_cSup__union__distrib,axiom,
% 14.25/8.07 ! [A: set_nat,B4: set_nat] :
% 14.25/8.07 ( ( A != bot_bot_set_nat )
% 14.25/8.07 => ( ( condit2214826472909112428ve_nat @ A )
% 14.25/8.07 => ( ( B4 != bot_bot_set_nat )
% 14.25/8.07 => ( ( condit2214826472909112428ve_nat @ B4 )
% 14.25/8.07 => ( ( complete_Sup_Sup_nat @ ( sup_sup_set_nat @ A @ B4 ) )
% 14.25/8.07 = ( sup_sup_nat @ ( complete_Sup_Sup_nat @ A ) @ ( complete_Sup_Sup_nat @ B4 ) ) ) ) ) ) ) ).
% 14.25/8.07
% 14.25/8.07 thf(fact_3453_cSup__union__distrib,axiom,
% 14.25/8.07 ! [A: set_Product_unit,B4: set_Product_unit] :
% 14.25/8.07 ( ( A != bot_bo3957492148770167129t_unit )
% 14.25/8.07 => ( ( condit2080475194861589617t_unit @ A )
% 14.25/8.07 => ( ( B4 != bot_bo3957492148770167129t_unit )
% 14.25/8.07 => ( ( condit2080475194861589617t_unit @ B4 )
% 14.25/8.07 => ( ( comple4687483117567863418t_unit @ ( sup_su793286257634532545t_unit @ A @ B4 ) )
% 14.25/8.07 = ( sup_sup_Product_unit @ ( comple4687483117567863418t_unit @ A ) @ ( comple4687483117567863418t_unit @ B4 ) ) ) ) ) ) ) ).
% 14.25/8.07
% 14.25/8.07 thf(fact_3454_fun__upd__image,axiom,
% 14.25/8.07 ! [X: code_integer,A: set_Code_integer,F: code_integer > code_integer,Y: code_integer] :
% 14.25/8.07 ( ( ( member_Code_integer @ X @ A )
% 14.25/8.07 => ( ( image_4470545334726330049nteger @ ( fun_up8355007323143964473nteger @ F @ X @ Y ) @ A )
% 14.25/8.07 = ( insert_Code_integer @ Y @ ( image_4470545334726330049nteger @ F @ ( minus_2355218937544613996nteger @ A @ ( insert_Code_integer @ X @ bot_bo3990330152332043303nteger ) ) ) ) ) )
% 14.25/8.07 & ( ~ ( member_Code_integer @ X @ A )
% 14.25/8.07 => ( ( image_4470545334726330049nteger @ ( fun_up8355007323143964473nteger @ F @ X @ Y ) @ A )
% 14.25/8.07 = ( image_4470545334726330049nteger @ F @ A ) ) ) ) ).
% 14.25/8.07
% 14.25/8.07 thf(fact_3455_fun__upd__image,axiom,
% 14.25/8.07 ! [X: product_unit,A: set_Product_unit,F: product_unit > nat,Y: nat] :
% 14.25/8.07 ( ( ( member_Product_unit @ X @ A )
% 14.25/8.07 => ( ( image_875570014554754200it_nat @ ( fun_up6473050886840373792it_nat @ F @ X @ Y ) @ A )
% 14.25/8.07 = ( insert_nat @ Y @ ( image_875570014554754200it_nat @ F @ ( minus_6452836326544984404t_unit @ A @ ( insert_Product_unit @ X @ bot_bo3957492148770167129t_unit ) ) ) ) ) )
% 14.25/8.07 & ( ~ ( member_Product_unit @ X @ A )
% 14.25/8.07 => ( ( image_875570014554754200it_nat @ ( fun_up6473050886840373792it_nat @ F @ X @ Y ) @ A )
% 14.25/8.07 = ( image_875570014554754200it_nat @ F @ A ) ) ) ) ).
% 14.25/8.07
% 14.25/8.07 thf(fact_3456_fun__upd__image,axiom,
% 14.25/8.07 ! [X: product_unit,A: set_Product_unit,F: product_unit > int,Y: int] :
% 14.25/8.07 ( ( ( member_Product_unit @ X @ A )
% 14.25/8.07 => ( ( image_873079544045703924it_int @ ( fun_up6470560416331323516it_int @ F @ X @ Y ) @ A )
% 14.25/8.07 = ( insert_int @ Y @ ( image_873079544045703924it_int @ F @ ( minus_6452836326544984404t_unit @ A @ ( insert_Product_unit @ X @ bot_bo3957492148770167129t_unit ) ) ) ) ) )
% 14.25/8.07 & ( ~ ( member_Product_unit @ X @ A )
% 14.25/8.07 => ( ( image_873079544045703924it_int @ ( fun_up6470560416331323516it_int @ F @ X @ Y ) @ A )
% 14.25/8.07 = ( image_873079544045703924it_int @ F @ A ) ) ) ) ).
% 14.25/8.07
% 14.25/8.07 thf(fact_3457_fun__upd__image,axiom,
% 14.25/8.07 ! [X: product_unit,A: set_Product_unit,F: product_unit > $o,Y: $o] :
% 14.25/8.07 ( ( ( member_Product_unit @ X @ A )
% 14.25/8.07 => ( ( image_Product_unit_o2 @ ( fun_up5432024876597553480unit_o @ F @ X @ Y ) @ A )
% 14.25/8.07 = ( insert_o @ Y @ ( image_Product_unit_o2 @ F @ ( minus_6452836326544984404t_unit @ A @ ( insert_Product_unit @ X @ bot_bo3957492148770167129t_unit ) ) ) ) ) )
% 14.25/8.07 & ( ~ ( member_Product_unit @ X @ A )
% 14.25/8.07 => ( ( image_Product_unit_o2 @ ( fun_up5432024876597553480unit_o @ F @ X @ Y ) @ A )
% 14.25/8.07 = ( image_Product_unit_o2 @ F @ A ) ) ) ) ).
% 14.25/8.07
% 14.25/8.07 thf(fact_3458_fun__upd__image,axiom,
% 14.25/8.07 ! [X: product_unit,A: set_Product_unit,F: product_unit > product_unit,Y: product_unit] :
% 14.25/8.07 ( ( ( member_Product_unit @ X @ A )
% 14.25/8.07 => ( ( image_405062704495631173t_unit @ ( fun_up6875000483049295805t_unit @ F @ X @ Y ) @ A )
% 14.25/8.07 = ( insert_Product_unit @ Y @ ( image_405062704495631173t_unit @ F @ ( minus_6452836326544984404t_unit @ A @ ( insert_Product_unit @ X @ bot_bo3957492148770167129t_unit ) ) ) ) ) )
% 14.25/8.07 & ( ~ ( member_Product_unit @ X @ A )
% 14.25/8.07 => ( ( image_405062704495631173t_unit @ ( fun_up6875000483049295805t_unit @ F @ X @ Y ) @ A )
% 14.25/8.07 = ( image_405062704495631173t_unit @ F @ A ) ) ) ) ).
% 14.25/8.07
% 14.25/8.07 thf(fact_3459_fun__upd__image,axiom,
% 14.25/8.07 ! [X: $o,A: set_o,F: $o > nat,Y: nat] :
% 14.25/8.07 ( ( ( member_o @ X @ A )
% 14.25/8.07 => ( ( image_o_nat2 @ ( fun_upd_o_nat @ F @ X @ Y ) @ A )
% 14.25/8.07 = ( insert_nat @ Y @ ( image_o_nat2 @ F @ ( minus_minus_set_o @ A @ ( insert_o @ X @ bot_bot_set_o ) ) ) ) ) )
% 14.25/8.07 & ( ~ ( member_o @ X @ A )
% 14.25/8.07 => ( ( image_o_nat2 @ ( fun_upd_o_nat @ F @ X @ Y ) @ A )
% 14.25/8.07 = ( image_o_nat2 @ F @ A ) ) ) ) ).
% 14.25/8.07
% 14.25/8.07 thf(fact_3460_fun__upd__image,axiom,
% 14.25/8.07 ! [X: $o,A: set_o,F: $o > int,Y: int] :
% 14.25/8.07 ( ( ( member_o @ X @ A )
% 14.25/8.07 => ( ( image_o_int2 @ ( fun_upd_o_int @ F @ X @ Y ) @ A )
% 14.25/8.07 = ( insert_int @ Y @ ( image_o_int2 @ F @ ( minus_minus_set_o @ A @ ( insert_o @ X @ bot_bot_set_o ) ) ) ) ) )
% 14.25/8.07 & ( ~ ( member_o @ X @ A )
% 14.25/8.07 => ( ( image_o_int2 @ ( fun_upd_o_int @ F @ X @ Y ) @ A )
% 14.25/8.07 = ( image_o_int2 @ F @ A ) ) ) ) ).
% 14.25/8.07
% 14.25/8.07 thf(fact_3461_fun__upd__image,axiom,
% 14.25/8.07 ! [X: $o,A: set_o,F: $o > $o,Y: $o] :
% 14.25/8.07 ( ( ( member_o @ X @ A )
% 14.25/8.07 => ( ( image_o_o2 @ ( fun_upd_o_o @ F @ X @ Y ) @ A )
% 14.25/8.07 = ( insert_o @ Y @ ( image_o_o2 @ F @ ( minus_minus_set_o @ A @ ( insert_o @ X @ bot_bot_set_o ) ) ) ) ) )
% 14.25/8.07 & ( ~ ( member_o @ X @ A )
% 14.25/8.07 => ( ( image_o_o2 @ ( fun_upd_o_o @ F @ X @ Y ) @ A )
% 14.25/8.07 = ( image_o_o2 @ F @ A ) ) ) ) ).
% 14.25/8.07
% 14.25/8.07 thf(fact_3462_fun__upd__image,axiom,
% 14.25/8.07 ! [X: $o,A: set_o,F: $o > product_unit,Y: product_unit] :
% 14.25/8.07 ( ( ( member_o @ X @ A )
% 14.25/8.07 => ( ( image_o_Product_unit2 @ ( fun_up846054379331416818t_unit @ F @ X @ Y ) @ A )
% 14.25/8.07 = ( insert_Product_unit @ Y @ ( image_o_Product_unit2 @ F @ ( minus_minus_set_o @ A @ ( insert_o @ X @ bot_bot_set_o ) ) ) ) ) )
% 14.25/8.07 & ( ~ ( member_o @ X @ A )
% 14.25/8.07 => ( ( image_o_Product_unit2 @ ( fun_up846054379331416818t_unit @ F @ X @ Y ) @ A )
% 14.25/8.07 = ( image_o_Product_unit2 @ F @ A ) ) ) ) ).
% 14.25/8.07
% 14.25/8.07 thf(fact_3463_fun__upd__image,axiom,
% 14.25/8.07 ! [X: int,A: set_int,F: int > nat,Y: nat] :
% 14.25/8.07 ( ( ( member_int @ X @ A )
% 14.25/8.07 => ( ( image_int_nat2 @ ( fun_upd_int_nat @ F @ X @ Y ) @ A )
% 14.25/8.07 = ( insert_nat @ Y @ ( image_int_nat2 @ F @ ( minus_minus_set_int @ A @ ( insert_int @ X @ bot_bot_set_int ) ) ) ) ) )
% 14.25/8.07 & ( ~ ( member_int @ X @ A )
% 14.25/8.07 => ( ( image_int_nat2 @ ( fun_upd_int_nat @ F @ X @ Y ) @ A )
% 14.25/8.07 = ( image_int_nat2 @ F @ A ) ) ) ) ).
% 14.25/8.07
% 14.25/8.07 thf(fact_3464_Sup__fin_Oeq__fold,axiom,
% 14.25/8.07 ! [A: set_int,X: int] :
% 14.25/8.07 ( ( finite_finite_int @ A )
% 14.25/8.07 => ( ( lattic1091506334969745077in_int @ ( insert_int @ X @ A ) )
% 14.25/8.07 = ( finite_fold_int_int @ sup_sup_int @ X @ A ) ) ) ).
% 14.25/8.07
% 14.25/8.07 thf(fact_3465_Sup__fin_Oeq__fold,axiom,
% 14.25/8.07 ! [A: set_o,X: $o] :
% 14.25/8.07 ( ( finite_finite_o @ A )
% 14.25/8.07 => ( ( lattic1508158080041050831_fin_o @ ( insert_o @ X @ A ) )
% 14.25/8.07 = ( finite_fold_o_o @ sup_sup_o @ X @ A ) ) ) ).
% 14.25/8.07
% 14.25/8.07 thf(fact_3466_Sup__fin_Oeq__fold,axiom,
% 14.25/8.07 ! [A: set_nat,X: nat] :
% 14.25/8.07 ( ( finite_finite_nat @ A )
% 14.25/8.07 => ( ( lattic1093996805478795353in_nat @ ( insert_nat @ X @ A ) )
% 14.25/8.07 = ( finite_fold_nat_nat @ sup_sup_nat @ X @ A ) ) ) ).
% 14.25/8.07
% 14.25/8.07 thf(fact_3467_Sup__fin_Oeq__fold,axiom,
% 14.25/8.07 ! [A: set_set_nat,X: set_nat] :
% 14.25/8.07 ( ( finite1152437895449049373et_nat @ A )
% 14.25/8.07 => ( ( lattic3835124923745554447et_nat @ ( insert_set_nat @ X @ A ) )
% 14.25/8.07 = ( finite677925301803182934et_nat @ sup_sup_set_nat @ X @ A ) ) ) ).
% 14.25/8.07
% 14.25/8.07 thf(fact_3468_Sup__fin_Oeq__fold,axiom,
% 14.25/8.07 ! [A: set_assn,X: assn] :
% 14.25/8.07 ( ( finite_finite_assn @ A )
% 14.25/8.07 => ( ( lattic2150320897289308081n_assn @ ( insert_assn @ X @ A ) )
% 14.25/8.07 = ( finite1440822074540989594n_assn @ sup_sup_assn @ X @ A ) ) ) ).
% 14.25/8.07
% 14.25/8.07 thf(fact_3469_Sup__fin_Oeq__fold,axiom,
% 14.25/8.07 ! [A: set_Product_unit,X: product_unit] :
% 14.25/8.07 ( ( finite4290736615968046902t_unit @ A )
% 14.25/8.07 => ( ( lattic5294303975357428420t_unit @ ( insert_Product_unit @ X @ A ) )
% 14.25/8.07 = ( finite8834431295148555784t_unit @ sup_sup_Product_unit @ X @ A ) ) ) ).
% 14.25/8.07
% 14.25/8.07 thf(fact_3470_Inf__fin_Oeq__fold,axiom,
% 14.25/8.07 ! [A: set_int,X: int] :
% 14.25/8.07 ( ( finite_finite_int @ A )
% 14.25/8.07 => ( ( lattic5235898064620869839in_int @ ( insert_int @ X @ A ) )
% 14.25/8.07 = ( finite_fold_int_int @ inf_inf_int @ X @ A ) ) ) ).
% 14.25/8.07
% 14.25/8.07 thf(fact_3471_Inf__fin_Oeq__fold,axiom,
% 14.25/8.07 ! [A: set_o,X: $o] :
% 14.25/8.07 ( ( finite_finite_o @ A )
% 14.25/8.07 => ( ( lattic4107685809792843317_fin_o @ ( insert_o @ X @ A ) )
% 14.25/8.07 = ( finite_fold_o_o @ inf_inf_o @ X @ A ) ) ) ).
% 14.25/8.07
% 14.25/8.07 thf(fact_3472_Inf__fin_Oeq__fold,axiom,
% 14.25/8.07 ! [A: set_nat,X: nat] :
% 14.25/8.07 ( ( finite_finite_nat @ A )
% 14.25/8.07 => ( ( lattic5238388535129920115in_nat @ ( insert_nat @ X @ A ) )
% 14.25/8.07 = ( finite_fold_nat_nat @ inf_inf_nat @ X @ A ) ) ) ).
% 14.25/8.07
% 14.25/8.07 thf(fact_3473_Inf__fin_Oeq__fold,axiom,
% 14.25/8.07 ! [A: set_set_nat,X: set_nat] :
% 14.25/8.07 ( ( finite1152437895449049373et_nat @ A )
% 14.25/8.07 => ( ( lattic3014633134055518761et_nat @ ( insert_set_nat @ X @ A ) )
% 14.25/8.07 = ( finite677925301803182934et_nat @ inf_inf_set_nat @ X @ A ) ) ) ).
% 14.25/8.07
% 14.25/8.07 thf(fact_3474_Inf__fin_Oeq__fold,axiom,
% 14.25/8.07 ! [A: set_assn,X: assn] :
% 14.25/8.07 ( ( finite_finite_assn @ A )
% 14.25/8.07 => ( ( lattic47131356835913163n_assn @ ( insert_assn @ X @ A ) )
% 14.25/8.07 = ( finite1440822074540989594n_assn @ inf_inf_assn @ X @ A ) ) ) ).
% 14.25/8.07
% 14.25/8.07 thf(fact_3475_Inf__fin_Oeq__fold,axiom,
% 14.25/8.07 ! [A: set_Product_unit,X: product_unit] :
% 14.25/8.07 ( ( finite4290736615968046902t_unit @ A )
% 14.25/8.07 => ( ( lattic1263872656861969706t_unit @ ( insert_Product_unit @ X @ A ) )
% 14.25/8.07 = ( finite8834431295148555784t_unit @ inf_inf_Product_unit @ X @ A ) ) ) ).
% 14.25/8.07
% 14.25/8.07 thf(fact_3476_wf__bounded__set,axiom,
% 14.25/8.07 ! [R2: set_Pr1261947904930325089at_nat,Ub: nat > set_Code_integer,F: nat > set_Code_integer] :
% 14.25/8.07 ( ! [A4: nat,B2: nat] :
% 14.25/8.07 ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ B2 @ A4 ) @ R2 )
% 14.25/8.07 => ( ( finite6017078050557962740nteger @ ( Ub @ A4 ) )
% 14.25/8.07 & ( ord_le7084787975880047091nteger @ ( Ub @ B2 ) @ ( Ub @ A4 ) )
% 14.25/8.07 & ( ord_le7084787975880047091nteger @ ( F @ B2 ) @ ( Ub @ A4 ) )
% 14.25/8.07 & ( ord_le1307284697595431911nteger @ ( F @ A4 ) @ ( F @ B2 ) ) ) )
% 14.25/8.07 => ( wf_nat @ R2 ) ) ).
% 14.25/8.07
% 14.25/8.07 thf(fact_3477_wf__bounded__set,axiom,
% 14.25/8.07 ! [R2: set_Pr958786334691620121nt_int,Ub: int > set_Code_integer,F: int > set_Code_integer] :
% 14.25/8.07 ( ! [A4: int,B2: int] :
% 14.25/8.07 ( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ B2 @ A4 ) @ R2 )
% 14.25/8.07 => ( ( finite6017078050557962740nteger @ ( Ub @ A4 ) )
% 14.25/8.07 & ( ord_le7084787975880047091nteger @ ( Ub @ B2 ) @ ( Ub @ A4 ) )
% 14.25/8.07 & ( ord_le7084787975880047091nteger @ ( F @ B2 ) @ ( Ub @ A4 ) )
% 14.25/8.07 & ( ord_le1307284697595431911nteger @ ( F @ A4 ) @ ( F @ B2 ) ) ) )
% 14.25/8.07 => ( wf_int @ R2 ) ) ).
% 14.25/8.07
% 14.25/8.07 thf(fact_3478_wf__bounded__set,axiom,
% 14.25/8.07 ! [R2: set_Pr1261947904930325089at_nat,Ub: nat > set_nat,F: nat > set_nat] :
% 14.25/8.07 ( ! [A4: nat,B2: nat] :
% 14.25/8.07 ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ B2 @ A4 ) @ R2 )
% 14.25/8.07 => ( ( finite_finite_nat @ ( Ub @ A4 ) )
% 14.25/8.07 & ( ord_less_eq_set_nat @ ( Ub @ B2 ) @ ( Ub @ A4 ) )
% 14.25/8.07 & ( ord_less_eq_set_nat @ ( F @ B2 ) @ ( Ub @ A4 ) )
% 14.25/8.07 & ( ord_less_set_nat @ ( F @ A4 ) @ ( F @ B2 ) ) ) )
% 14.25/8.07 => ( wf_nat @ R2 ) ) ).
% 14.25/8.07
% 14.25/8.07 thf(fact_3479_wf__bounded__set,axiom,
% 14.25/8.07 ! [R2: set_Pr958786334691620121nt_int,Ub: int > set_nat,F: int > set_nat] :
% 14.25/8.07 ( ! [A4: int,B2: int] :
% 14.25/8.07 ( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ B2 @ A4 ) @ R2 )
% 14.25/8.07 => ( ( finite_finite_nat @ ( Ub @ A4 ) )
% 14.25/8.07 & ( ord_less_eq_set_nat @ ( Ub @ B2 ) @ ( Ub @ A4 ) )
% 14.25/8.07 & ( ord_less_eq_set_nat @ ( F @ B2 ) @ ( Ub @ A4 ) )
% 14.25/8.07 & ( ord_less_set_nat @ ( F @ A4 ) @ ( F @ B2 ) ) ) )
% 14.25/8.07 => ( wf_int @ R2 ) ) ).
% 14.25/8.07
% 14.25/8.07 thf(fact_3480_psubset__insert__iff,axiom,
% 14.25/8.07 ! [A: set_Product_unit,X: product_unit,B4: set_Product_unit] :
% 14.25/8.07 ( ( ord_le8056459307392131481t_unit @ A @ ( insert_Product_unit @ X @ B4 ) )
% 14.25/8.07 = ( ( ( member_Product_unit @ X @ B4 )
% 14.25/8.07 => ( ord_le8056459307392131481t_unit @ A @ B4 ) )
% 14.25/8.07 & ( ~ ( member_Product_unit @ X @ B4 )
% 14.25/8.07 => ( ( ( member_Product_unit @ X @ A )
% 14.25/8.07 => ( ord_le8056459307392131481t_unit @ ( minus_6452836326544984404t_unit @ A @ ( insert_Product_unit @ X @ bot_bo3957492148770167129t_unit ) ) @ B4 ) )
% 14.25/8.07 & ( ~ ( member_Product_unit @ X @ A )
% 14.25/8.07 => ( ord_le3507040750410214029t_unit @ A @ B4 ) ) ) ) ) ) ).
% 14.25/8.07
% 14.25/8.07 thf(fact_3481_psubset__insert__iff,axiom,
% 14.25/8.07 ! [A: set_o,X: $o,B4: set_o] :
% 14.25/8.07 ( ( ord_less_set_o @ A @ ( insert_o @ X @ B4 ) )
% 14.25/8.07 = ( ( ( member_o @ X @ B4 )
% 14.25/8.07 => ( ord_less_set_o @ A @ B4 ) )
% 14.25/8.07 & ( ~ ( member_o @ X @ B4 )
% 14.25/8.07 => ( ( ( member_o @ X @ A )
% 14.25/8.07 => ( ord_less_set_o @ ( minus_minus_set_o @ A @ ( insert_o @ X @ bot_bot_set_o ) ) @ B4 ) )
% 14.25/8.07 & ( ~ ( member_o @ X @ A )
% 14.25/8.07 => ( ord_less_eq_set_o @ A @ B4 ) ) ) ) ) ) ).
% 14.25/8.07
% 14.25/8.07 thf(fact_3482_psubset__insert__iff,axiom,
% 14.25/8.07 ! [A: set_int,X: int,B4: set_int] :
% 14.25/8.07 ( ( ord_less_set_int @ A @ ( insert_int @ X @ B4 ) )
% 14.25/8.07 = ( ( ( member_int @ X @ B4 )
% 14.25/8.07 => ( ord_less_set_int @ A @ B4 ) )
% 14.25/8.07 & ( ~ ( member_int @ X @ B4 )
% 14.25/8.07 => ( ( ( member_int @ X @ A )
% 14.25/8.07 => ( ord_less_set_int @ ( minus_minus_set_int @ A @ ( insert_int @ X @ bot_bot_set_int ) ) @ B4 ) )
% 14.25/8.07 & ( ~ ( member_int @ X @ A )
% 14.25/8.07 => ( ord_less_eq_set_int @ A @ B4 ) ) ) ) ) ) ).
% 14.25/8.07
% 14.25/8.07 thf(fact_3483_psubset__insert__iff,axiom,
% 14.25/8.07 ! [A: set_Pr4532377907799695533_nat_o,X: produc3658429121746597890et_nat > $o,B4: set_Pr4532377907799695533_nat_o] :
% 14.25/8.07 ( ( ord_le2453136405763929_nat_o @ A @ ( insert5175938949040314269_nat_o @ X @ B4 ) )
% 14.25/8.07 = ( ( ( member6576561426505652726_nat_o @ X @ B4 )
% 14.25/8.07 => ( ord_le2453136405763929_nat_o @ A @ B4 ) )
% 14.25/8.07 & ( ~ ( member6576561426505652726_nat_o @ X @ B4 )
% 14.25/8.07 => ( ( ( member6576561426505652726_nat_o @ X @ A )
% 14.25/8.07 => ( ord_le2453136405763929_nat_o @ ( minus_1801376950450012436_nat_o @ A @ ( insert5175938949040314269_nat_o @ X @ bot_bo7824918357723582233_nat_o ) ) @ B4 ) )
% 14.25/8.07 & ( ~ ( member6576561426505652726_nat_o @ X @ A )
% 14.25/8.07 => ( ord_le2965882846123202637_nat_o @ A @ B4 ) ) ) ) ) ) ).
% 14.25/8.07
% 14.25/8.07 thf(fact_3484_psubset__insert__iff,axiom,
% 14.25/8.07 ! [A: set_nat,X: nat,B4: set_nat] :
% 14.25/8.07 ( ( ord_less_set_nat @ A @ ( insert_nat @ X @ B4 ) )
% 14.25/8.07 = ( ( ( member_nat @ X @ B4 )
% 14.25/8.07 => ( ord_less_set_nat @ A @ B4 ) )
% 14.25/8.07 & ( ~ ( member_nat @ X @ B4 )
% 14.25/8.07 => ( ( ( member_nat @ X @ A )
% 14.25/8.07 => ( ord_less_set_nat @ ( minus_minus_set_nat @ A @ ( insert_nat @ X @ bot_bot_set_nat ) ) @ B4 ) )
% 14.25/8.07 & ( ~ ( member_nat @ X @ A )
% 14.25/8.07 => ( ord_less_eq_set_nat @ A @ B4 ) ) ) ) ) ) ).
% 14.25/8.07
% 14.25/8.07 thf(fact_3485_finite__induct__select,axiom,
% 14.25/8.07 ! [S: set_Code_integer,P: set_Code_integer > $o] :
% 14.25/8.07 ( ( finite6017078050557962740nteger @ S )
% 14.25/8.07 => ( ( P @ bot_bo3990330152332043303nteger )
% 14.25/8.07 => ( ! [T3: set_Code_integer] :
% 14.25/8.07 ( ( ord_le1307284697595431911nteger @ T3 @ S )
% 14.25/8.07 => ( ( P @ T3 )
% 14.25/8.07 => ? [X6: code_integer] :
% 14.25/8.07 ( ( member_Code_integer @ X6 @ ( minus_2355218937544613996nteger @ S @ T3 ) )
% 14.25/8.07 & ( P @ ( insert_Code_integer @ X6 @ T3 ) ) ) ) )
% 14.25/8.07 => ( P @ S ) ) ) ) ).
% 14.25/8.07
% 14.25/8.07 thf(fact_3486_finite__induct__select,axiom,
% 14.25/8.07 ! [S: set_Product_unit,P: set_Product_unit > $o] :
% 14.25/8.07 ( ( finite4290736615968046902t_unit @ S )
% 14.25/8.07 => ( ( P @ bot_bo3957492148770167129t_unit )
% 14.25/8.07 => ( ! [T3: set_Product_unit] :
% 14.25/8.07 ( ( ord_le8056459307392131481t_unit @ T3 @ S )
% 14.25/8.07 => ( ( P @ T3 )
% 14.25/8.07 => ? [X6: product_unit] :
% 14.25/8.07 ( ( member_Product_unit @ X6 @ ( minus_6452836326544984404t_unit @ S @ T3 ) )
% 14.25/8.07 & ( P @ ( insert_Product_unit @ X6 @ T3 ) ) ) ) )
% 14.25/8.07 => ( P @ S ) ) ) ) ).
% 14.25/8.07
% 14.25/8.07 thf(fact_3487_finite__induct__select,axiom,
% 14.25/8.07 ! [S: set_o,P: set_o > $o] :
% 14.25/8.07 ( ( finite_finite_o @ S )
% 14.25/8.07 => ( ( P @ bot_bot_set_o )
% 14.25/8.07 => ( ! [T3: set_o] :
% 14.25/8.07 ( ( ord_less_set_o @ T3 @ S )
% 14.25/8.07 => ( ( P @ T3 )
% 14.25/8.07 => ? [X6: $o] :
% 14.25/8.07 ( ( member_o @ X6 @ ( minus_minus_set_o @ S @ T3 ) )
% 14.25/8.07 & ( P @ ( insert_o @ X6 @ T3 ) ) ) ) )
% 14.25/8.07 => ( P @ S ) ) ) ) ).
% 14.25/8.07
% 14.25/8.07 thf(fact_3488_finite__induct__select,axiom,
% 14.25/8.07 ! [S: set_int,P: set_int > $o] :
% 14.25/8.07 ( ( finite_finite_int @ S )
% 14.25/8.07 => ( ( P @ bot_bot_set_int )
% 14.25/8.07 => ( ! [T3: set_int] :
% 14.25/8.07 ( ( ord_less_set_int @ T3 @ S )
% 14.25/8.07 => ( ( P @ T3 )
% 14.25/8.07 => ? [X6: int] :
% 14.25/8.07 ( ( member_int @ X6 @ ( minus_minus_set_int @ S @ T3 ) )
% 14.25/8.07 & ( P @ ( insert_int @ X6 @ T3 ) ) ) ) )
% 14.25/8.07 => ( P @ S ) ) ) ) ).
% 14.25/8.07
% 14.25/8.07 thf(fact_3489_finite__induct__select,axiom,
% 14.25/8.07 ! [S: set_Pr4532377907799695533_nat_o,P: set_Pr4532377907799695533_nat_o > $o] :
% 14.25/8.07 ( ( finite3252695134891459830_nat_o @ S )
% 14.25/8.07 => ( ( P @ bot_bo7824918357723582233_nat_o )
% 14.25/8.07 => ( ! [T3: set_Pr4532377907799695533_nat_o] :
% 14.25/8.07 ( ( ord_le2453136405763929_nat_o @ T3 @ S )
% 14.25/8.07 => ( ( P @ T3 )
% 14.25/8.07 => ? [X6: produc3658429121746597890et_nat > $o] :
% 14.25/8.07 ( ( member6576561426505652726_nat_o @ X6 @ ( minus_1801376950450012436_nat_o @ S @ T3 ) )
% 14.25/8.07 & ( P @ ( insert5175938949040314269_nat_o @ X6 @ T3 ) ) ) ) )
% 14.25/8.07 => ( P @ S ) ) ) ) ).
% 14.25/8.07
% 14.25/8.07 thf(fact_3490_finite__induct__select,axiom,
% 14.25/8.07 ! [S: set_nat,P: set_nat > $o] :
% 14.25/8.07 ( ( finite_finite_nat @ S )
% 14.25/8.07 => ( ( P @ bot_bot_set_nat )
% 14.25/8.07 => ( ! [T3: set_nat] :
% 14.25/8.07 ( ( ord_less_set_nat @ T3 @ S )
% 14.25/8.07 => ( ( P @ T3 )
% 14.25/8.07 => ? [X6: nat] :
% 14.25/8.07 ( ( member_nat @ X6 @ ( minus_minus_set_nat @ S @ T3 ) )
% 14.25/8.07 & ( P @ ( insert_nat @ X6 @ T3 ) ) ) ) )
% 14.25/8.07 => ( P @ S ) ) ) ) ).
% 14.25/8.07
% 14.25/8.07 thf(fact_3491_Max_Osubset__imp,axiom,
% 14.25/8.07 ! [A: set_Code_integer,B4: set_Code_integer] :
% 14.25/8.07 ( ( ord_le7084787975880047091nteger @ A @ B4 )
% 14.25/8.07 => ( ( A != bot_bo3990330152332043303nteger )
% 14.25/8.07 => ( ( finite6017078050557962740nteger @ B4 )
% 14.25/8.07 => ( ord_le3102999989581377725nteger @ ( lattic4901227151466704046nteger @ A ) @ ( lattic4901227151466704046nteger @ B4 ) ) ) ) ) ).
% 14.25/8.07
% 14.25/8.07 thf(fact_3492_Max_Osubset__imp,axiom,
% 14.25/8.07 ! [A: set_Product_unit,B4: set_Product_unit] :
% 14.25/8.07 ( ( ord_le3507040750410214029t_unit @ A @ B4 )
% 14.25/8.07 => ( ( A != bot_bo3957492148770167129t_unit )
% 14.25/8.07 => ( ( finite4290736615968046902t_unit @ B4 )
% 14.25/8.07 => ( ord_le3221252021190050221t_unit @ ( lattic1210392890313158908t_unit @ A ) @ ( lattic1210392890313158908t_unit @ B4 ) ) ) ) ) ).
% 14.25/8.07
% 14.25/8.07 thf(fact_3493_Max_Osubset__imp,axiom,
% 14.25/8.07 ! [A: set_o,B4: set_o] :
% 14.25/8.07 ( ( ord_less_eq_set_o @ A @ B4 )
% 14.25/8.07 => ( ( A != bot_bot_set_o )
% 14.25/8.07 => ( ( finite_finite_o @ B4 )
% 14.25/8.07 => ( ord_less_eq_o @ ( lattic1921953407002678535_Max_o @ A ) @ ( lattic1921953407002678535_Max_o @ B4 ) ) ) ) ) ).
% 14.25/8.07
% 14.25/8.07 thf(fact_3494_Max_Osubset__imp,axiom,
% 14.25/8.07 ! [A: set_rat,B4: set_rat] :
% 14.25/8.07 ( ( ord_less_eq_set_rat @ A @ B4 )
% 14.25/8.07 => ( ( A != bot_bot_set_rat )
% 14.25/8.07 => ( ( finite_finite_rat @ B4 )
% 14.25/8.07 => ( ord_less_eq_rat @ ( lattic7630753665789217321ax_rat @ A ) @ ( lattic7630753665789217321ax_rat @ B4 ) ) ) ) ) ).
% 14.25/8.07
% 14.25/8.07 thf(fact_3495_Max_Osubset__imp,axiom,
% 14.25/8.07 ! [A: set_int,B4: set_int] :
% 14.25/8.07 ( ( ord_less_eq_set_int @ A @ B4 )
% 14.25/8.07 => ( ( A != bot_bot_set_int )
% 14.25/8.07 => ( ( finite_finite_int @ B4 )
% 14.25/8.07 => ( ord_less_eq_int @ ( lattic8263393255366662781ax_int @ A ) @ ( lattic8263393255366662781ax_int @ B4 ) ) ) ) ) ).
% 14.25/8.07
% 14.25/8.07 thf(fact_3496_Max_Osubset__imp,axiom,
% 14.25/8.07 ! [A: set_nat,B4: set_nat] :
% 14.25/8.07 ( ( ord_less_eq_set_nat @ A @ B4 )
% 14.25/8.07 => ( ( A != bot_bot_set_nat )
% 14.25/8.07 => ( ( finite_finite_nat @ B4 )
% 14.25/8.07 => ( ord_less_eq_nat @ ( lattic8265883725875713057ax_nat @ A ) @ ( lattic8265883725875713057ax_nat @ B4 ) ) ) ) ) ).
% 14.25/8.07
% 14.25/8.07 thf(fact_3497_Max__mono,axiom,
% 14.25/8.07 ! [M: set_Code_integer,N: set_Code_integer] :
% 14.25/8.07 ( ( ord_le7084787975880047091nteger @ M @ N )
% 14.25/8.07 => ( ( M != bot_bo3990330152332043303nteger )
% 14.25/8.07 => ( ( finite6017078050557962740nteger @ N )
% 14.25/8.07 => ( ord_le3102999989581377725nteger @ ( lattic4901227151466704046nteger @ M ) @ ( lattic4901227151466704046nteger @ N ) ) ) ) ) ).
% 14.25/8.07
% 14.25/8.07 thf(fact_3498_Max__mono,axiom,
% 14.25/8.07 ! [M: set_Product_unit,N: set_Product_unit] :
% 14.25/8.07 ( ( ord_le3507040750410214029t_unit @ M @ N )
% 14.25/8.07 => ( ( M != bot_bo3957492148770167129t_unit )
% 14.25/8.07 => ( ( finite4290736615968046902t_unit @ N )
% 14.25/8.07 => ( ord_le3221252021190050221t_unit @ ( lattic1210392890313158908t_unit @ M ) @ ( lattic1210392890313158908t_unit @ N ) ) ) ) ) ).
% 14.25/8.07
% 14.25/8.07 thf(fact_3499_Max__mono,axiom,
% 14.25/8.07 ! [M: set_o,N: set_o] :
% 14.25/8.07 ( ( ord_less_eq_set_o @ M @ N )
% 14.25/8.07 => ( ( M != bot_bot_set_o )
% 14.25/8.07 => ( ( finite_finite_o @ N )
% 14.25/8.07 => ( ord_less_eq_o @ ( lattic1921953407002678535_Max_o @ M ) @ ( lattic1921953407002678535_Max_o @ N ) ) ) ) ) ).
% 14.25/8.07
% 14.25/8.07 thf(fact_3500_Max__mono,axiom,
% 14.25/8.07 ! [M: set_rat,N: set_rat] :
% 14.25/8.07 ( ( ord_less_eq_set_rat @ M @ N )
% 14.25/8.07 => ( ( M != bot_bot_set_rat )
% 14.25/8.07 => ( ( finite_finite_rat @ N )
% 14.25/8.07 => ( ord_less_eq_rat @ ( lattic7630753665789217321ax_rat @ M ) @ ( lattic7630753665789217321ax_rat @ N ) ) ) ) ) ).
% 14.25/8.07
% 14.25/8.07 thf(fact_3501_Max__mono,axiom,
% 14.25/8.07 ! [M: set_int,N: set_int] :
% 14.25/8.07 ( ( ord_less_eq_set_int @ M @ N )
% 14.25/8.07 => ( ( M != bot_bot_set_int )
% 14.25/8.07 => ( ( finite_finite_int @ N )
% 14.25/8.07 => ( ord_less_eq_int @ ( lattic8263393255366662781ax_int @ M ) @ ( lattic8263393255366662781ax_int @ N ) ) ) ) ) ).
% 14.25/8.07
% 14.25/8.07 thf(fact_3502_Max__mono,axiom,
% 14.25/8.07 ! [M: set_nat,N: set_nat] :
% 14.25/8.07 ( ( ord_less_eq_set_nat @ M @ N )
% 14.25/8.07 => ( ( M != bot_bot_set_nat )
% 14.25/8.07 => ( ( finite_finite_nat @ N )
% 14.25/8.07 => ( ord_less_eq_nat @ ( lattic8265883725875713057ax_nat @ M ) @ ( lattic8265883725875713057ax_nat @ N ) ) ) ) ) ).
% 14.25/8.07
% 14.25/8.07 thf(fact_3503_image__vimage__eq,axiom,
% 14.25/8.07 ! [F: code_integer > code_integer,A: set_Code_integer] :
% 14.25/8.07 ( ( image_4470545334726330049nteger @ F @ ( vimage1318793397690208119nteger @ F @ A ) )
% 14.25/8.07 = ( inf_in1364745209274528805nteger @ A @ ( image_4470545334726330049nteger @ F @ top_to4645266643341252675nteger ) ) ) ).
% 14.25/8.07
% 14.25/8.07 thf(fact_3504_image__vimage__eq,axiom,
% 14.25/8.07 ! [F: product_unit > nat,A: set_nat] :
% 14.25/8.07 ( ( image_875570014554754200it_nat @ F @ ( vimage6253328473476588386it_nat @ F @ A ) )
% 14.25/8.07 = ( inf_inf_set_nat @ A @ ( image_875570014554754200it_nat @ F @ top_to1996260823553986621t_unit ) ) ) ).
% 14.25/8.07
% 14.25/8.07 thf(fact_3505_image__vimage__eq,axiom,
% 14.25/8.07 ! [F: $o > nat,A: set_nat] :
% 14.25/8.07 ( ( image_o_nat2 @ F @ ( vimage_o_nat @ F @ A ) )
% 14.25/8.07 = ( inf_inf_set_nat @ A @ ( image_o_nat2 @ F @ top_top_set_o ) ) ) ).
% 14.25/8.07
% 14.25/8.07 thf(fact_3506_image__vimage__eq,axiom,
% 14.25/8.07 ! [F: nat > set_nat,A: set_set_nat] :
% 14.25/8.07 ( ( image_nat_set_nat @ F @ ( vimage_nat_set_nat @ F @ A ) )
% 14.25/8.07 = ( inf_inf_set_set_nat @ A @ ( image_nat_set_nat @ F @ top_top_set_nat ) ) ) ).
% 14.25/8.07
% 14.25/8.07 thf(fact_3507_image__vimage__eq,axiom,
% 14.25/8.07 ! [F: nat > code_integer,A: set_Code_integer] :
% 14.25/8.07 ( ( image_1215581382706833972nteger @ F @ ( vimage6104394758324456170nteger @ F @ A ) )
% 14.25/8.07 = ( inf_in1364745209274528805nteger @ A @ ( image_1215581382706833972nteger @ F @ top_top_set_nat ) ) ) ).
% 14.25/8.07
% 14.25/8.07 thf(fact_3508_image__vimage__eq,axiom,
% 14.25/8.07 ! [F: nat > nat,A: set_nat] :
% 14.25/8.07 ( ( image_nat_nat2 @ F @ ( vimage_nat_nat @ F @ A ) )
% 14.25/8.07 = ( inf_inf_set_nat @ A @ ( image_nat_nat2 @ F @ top_top_set_nat ) ) ) ).
% 14.25/8.07
% 14.25/8.07 thf(fact_3509_image__vimage__eq,axiom,
% 14.25/8.07 ! [F: int > nat,A: set_nat] :
% 14.25/8.07 ( ( image_int_nat2 @ F @ ( vimage_int_nat @ F @ A ) )
% 14.25/8.07 = ( inf_inf_set_nat @ A @ ( image_int_nat2 @ F @ top_top_set_int ) ) ) ).
% 14.25/8.07
% 14.25/8.07 thf(fact_3510_UNIV__I,axiom,
% 14.25/8.07 ! [X: produc3658429121746597890et_nat > $o] : ( member6576561426505652726_nat_o @ X @ top_to4869887016220417533_nat_o ) ).
% 14.25/8.07
% 14.25/8.07 thf(fact_3511_UNIV__I,axiom,
% 14.25/8.07 ! [X: product_unit] : ( member_Product_unit @ X @ top_to1996260823553986621t_unit ) ).
% 14.25/8.07
% 14.25/8.07 thf(fact_3512_UNIV__I,axiom,
% 14.25/8.07 ! [X: $o] : ( member_o @ X @ top_top_set_o ) ).
% 14.25/8.07
% 14.25/8.07 thf(fact_3513_UNIV__I,axiom,
% 14.25/8.07 ! [X: nat] : ( member_nat @ X @ top_top_set_nat ) ).
% 14.25/8.07
% 14.25/8.07 thf(fact_3514_UNIV__I,axiom,
% 14.25/8.07 ! [X: int] : ( member_int @ X @ top_top_set_int ) ).
% 14.25/8.07
% 14.25/8.07 thf(fact_3515_Int__UNIV,axiom,
% 14.25/8.07 ! [A: set_Product_unit,B4: set_Product_unit] :
% 14.25/8.07 ( ( ( inf_in4660618365625256667t_unit @ A @ B4 )
% 14.25/8.07 = top_to1996260823553986621t_unit )
% 14.25/8.07 = ( ( A = top_to1996260823553986621t_unit )
% 14.25/8.07 & ( B4 = top_to1996260823553986621t_unit ) ) ) ).
% 14.25/8.07
% 14.25/8.07 thf(fact_3516_Int__UNIV,axiom,
% 14.25/8.07 ! [A: set_o,B4: set_o] :
% 14.25/8.07 ( ( ( inf_inf_set_o @ A @ B4 )
% 14.25/8.07 = top_top_set_o )
% 14.25/8.07 = ( ( A = top_top_set_o )
% 14.25/8.07 & ( B4 = top_top_set_o ) ) ) ).
% 14.25/8.07
% 14.25/8.07 thf(fact_3517_Int__UNIV,axiom,
% 14.25/8.07 ! [A: set_nat,B4: set_nat] :
% 14.25/8.07 ( ( ( inf_inf_set_nat @ A @ B4 )
% 14.25/8.07 = top_top_set_nat )
% 14.25/8.07 = ( ( A = top_top_set_nat )
% 14.25/8.07 & ( B4 = top_top_set_nat ) ) ) ).
% 14.25/8.07
% 14.25/8.07 thf(fact_3518_Int__UNIV,axiom,
% 14.25/8.07 ! [A: set_int,B4: set_int] :
% 14.25/8.07 ( ( ( inf_inf_set_int @ A @ B4 )
% 14.25/8.07 = top_top_set_int )
% 14.25/8.07 = ( ( A = top_top_set_int )
% 14.25/8.07 & ( B4 = top_top_set_int ) ) ) ).
% 14.25/8.07
% 14.25/8.07 thf(fact_3519_psubsetI,axiom,
% 14.25/8.07 ! [A: set_nat,B4: set_nat] :
% 14.25/8.07 ( ( ord_less_eq_set_nat @ A @ B4 )
% 14.25/8.07 => ( ( A != B4 )
% 14.25/8.07 => ( ord_less_set_nat @ A @ B4 ) ) ) ).
% 14.25/8.07
% 14.25/8.07 thf(fact_3520_vimage__UNIV,axiom,
% 14.25/8.07 ! [F: product_unit > product_unit] :
% 14.25/8.07 ( ( vimage7995052115951654139t_unit @ F @ top_to1996260823553986621t_unit )
% 14.25/8.07 = top_to1996260823553986621t_unit ) ).
% 14.25/8.07
% 14.25/8.07 thf(fact_3521_vimage__UNIV,axiom,
% 14.25/8.07 ! [F: $o > product_unit] :
% 14.25/8.07 ( ( vimage4222377557387472176t_unit @ F @ top_to1996260823553986621t_unit )
% 14.25/8.07 = top_top_set_o ) ).
% 14.25/8.07
% 14.25/8.07 thf(fact_3522_vimage__UNIV,axiom,
% 14.25/8.07 ! [F: nat > product_unit] :
% 14.25/8.07 ( ( vimage4884490618288580032t_unit @ F @ top_to1996260823553986621t_unit )
% 14.25/8.07 = top_top_set_nat ) ).
% 14.25/8.07
% 14.25/8.07 thf(fact_3523_vimage__UNIV,axiom,
% 14.25/8.07 ! [F: int > product_unit] :
% 14.25/8.07 ( ( vimage2000912565490760292t_unit @ F @ top_to1996260823553986621t_unit )
% 14.25/8.07 = top_top_set_int ) ).
% 14.25/8.07
% 14.25/8.07 thf(fact_3524_vimage__UNIV,axiom,
% 14.25/8.07 ! [F: product_unit > $o] :
% 14.25/8.07 ( ( vimage8808348054653608838unit_o @ F @ top_top_set_o )
% 14.25/8.07 = top_to1996260823553986621t_unit ) ).
% 14.25/8.07
% 14.25/8.07 thf(fact_3525_vimage__UNIV,axiom,
% 14.25/8.07 ! [F: $o > $o] :
% 14.25/8.07 ( ( vimage_o_o @ F @ top_top_set_o )
% 14.25/8.07 = top_top_set_o ) ).
% 14.25/8.07
% 14.25/8.07 thf(fact_3526_vimage__UNIV,axiom,
% 14.25/8.07 ! [F: nat > $o] :
% 14.25/8.07 ( ( vimage_nat_o @ F @ top_top_set_o )
% 14.25/8.07 = top_top_set_nat ) ).
% 14.25/8.07
% 14.25/8.07 thf(fact_3527_vimage__UNIV,axiom,
% 14.25/8.07 ! [F: int > $o] :
% 14.25/8.07 ( ( vimage_int_o @ F @ top_top_set_o )
% 14.25/8.07 = top_top_set_int ) ).
% 14.25/8.07
% 14.25/8.07 thf(fact_3528_vimage__UNIV,axiom,
% 14.25/8.07 ! [F: product_unit > nat] :
% 14.25/8.07 ( ( vimage6253328473476588386it_nat @ F @ top_top_set_nat )
% 14.25/8.07 = top_to1996260823553986621t_unit ) ).
% 14.25/8.07
% 14.25/8.07 thf(fact_3529_vimage__UNIV,axiom,
% 14.25/8.07 ! [F: $o > nat] :
% 14.25/8.07 ( ( vimage_o_nat @ F @ top_top_set_nat )
% 14.25/8.07 = top_top_set_o ) ).
% 14.25/8.07
% 14.25/8.07 thf(fact_3530_Pow__UNIV,axiom,
% 14.25/8.07 ( ( pow_Product_unit @ top_to1996260823553986621t_unit )
% 14.25/8.07 = top_to1767297665138865437t_unit ) ).
% 14.25/8.07
% 14.25/8.07 thf(fact_3531_Pow__UNIV,axiom,
% 14.25/8.07 ( ( pow_o @ top_top_set_o )
% 14.25/8.07 = top_top_set_set_o ) ).
% 14.25/8.07
% 14.25/8.07 thf(fact_3532_Pow__UNIV,axiom,
% 14.25/8.07 ( ( pow_nat @ top_top_set_nat )
% 14.25/8.07 = top_top_set_set_nat ) ).
% 14.25/8.07
% 14.25/8.07 thf(fact_3533_Pow__UNIV,axiom,
% 14.25/8.07 ( ( pow_int @ top_top_set_int )
% 14.25/8.07 = top_top_set_set_int ) ).
% 14.25/8.07
% 14.25/8.07 thf(fact_3534_Diff__UNIV,axiom,
% 14.25/8.07 ! [A: set_Pr4532377907799695533_nat_o] :
% 14.25/8.07 ( ( minus_1801376950450012436_nat_o @ A @ top_to4869887016220417533_nat_o )
% 14.25/8.07 = bot_bo7824918357723582233_nat_o ) ).
% 14.25/8.07
% 14.25/8.07 thf(fact_3535_Diff__UNIV,axiom,
% 14.25/8.07 ! [A: set_Product_unit] :
% 14.25/8.07 ( ( minus_6452836326544984404t_unit @ A @ top_to1996260823553986621t_unit )
% 14.25/8.07 = bot_bo3957492148770167129t_unit ) ).
% 14.25/8.07
% 14.25/8.07 thf(fact_3536_Diff__UNIV,axiom,
% 14.25/8.07 ! [A: set_o] :
% 14.25/8.07 ( ( minus_minus_set_o @ A @ top_top_set_o )
% 14.25/8.07 = bot_bot_set_o ) ).
% 14.25/8.07
% 14.25/8.07 thf(fact_3537_Diff__UNIV,axiom,
% 14.25/8.07 ! [A: set_int] :
% 14.25/8.07 ( ( minus_minus_set_int @ A @ top_top_set_int )
% 14.25/8.07 = bot_bot_set_int ) ).
% 14.25/8.07
% 14.25/8.07 thf(fact_3538_Diff__UNIV,axiom,
% 14.25/8.07 ! [A: set_nat] :
% 14.25/8.07 ( ( minus_minus_set_nat @ A @ top_top_set_nat )
% 14.25/8.07 = bot_bot_set_nat ) ).
% 14.25/8.07
% 14.25/8.07 thf(fact_3539_Max__singleton,axiom,
% 14.25/8.07 ! [X: product_unit] :
% 14.25/8.07 ( ( lattic1210392890313158908t_unit @ ( insert_Product_unit @ X @ bot_bo3957492148770167129t_unit ) )
% 14.25/8.07 = X ) ).
% 14.25/8.07
% 14.25/8.07 thf(fact_3540_Max__singleton,axiom,
% 14.25/8.07 ! [X: $o] :
% 14.25/8.07 ( ( lattic1921953407002678535_Max_o @ ( insert_o @ X @ bot_bot_set_o ) )
% 14.25/8.07 = X ) ).
% 14.25/8.07
% 14.25/8.07 thf(fact_3541_Max__singleton,axiom,
% 14.25/8.07 ! [X: int] :
% 14.25/8.07 ( ( lattic8263393255366662781ax_int @ ( insert_int @ X @ bot_bot_set_int ) )
% 14.25/8.07 = X ) ).
% 14.25/8.07
% 14.25/8.07 thf(fact_3542_Max__singleton,axiom,
% 14.25/8.07 ! [X: nat] :
% 14.25/8.07 ( ( lattic8265883725875713057ax_nat @ ( insert_nat @ X @ bot_bot_set_nat ) )
% 14.25/8.07 = X ) ).
% 14.25/8.07
% 14.25/8.07 thf(fact_3543_range__snd,axiom,
% 14.25/8.07 ( ( image_6088997261112373647et_nat @ produc8586169260539613262et_nat @ top_to2524957550908328808et_nat )
% 14.25/8.07 = top_top_set_set_nat ) ).
% 14.25/8.07
% 14.25/8.07 thf(fact_3544_range__snd,axiom,
% 14.25/8.07 ( ( image_5042161079198086560nt_int @ product_snd_int_int @ top_to4366644338036079209nt_int )
% 14.25/8.07 = top_top_set_int ) ).
% 14.25/8.07
% 14.25/8.07 thf(fact_3545_Domain__Id,axiom,
% 14.25/8.07 ( ( domain4292024440186670924t_unit @ id_Product_unit )
% 14.25/8.07 = top_to1996260823553986621t_unit ) ).
% 14.25/8.07
% 14.25/8.07 thf(fact_3546_Domain__Id,axiom,
% 14.25/8.07 ( ( domain_o_o @ id_o2 )
% 14.25/8.07 = top_top_set_o ) ).
% 14.25/8.07
% 14.25/8.07 thf(fact_3547_Domain__Id,axiom,
% 14.25/8.07 ( ( domain_nat_nat @ id_nat )
% 14.25/8.07 = top_top_set_nat ) ).
% 14.25/8.07
% 14.25/8.07 thf(fact_3548_Domain__Id,axiom,
% 14.25/8.07 ( ( domain_int_int @ id_int )
% 14.25/8.07 = top_top_set_int ) ).
% 14.25/8.07
% 14.25/8.07 thf(fact_3549_Range__Id,axiom,
% 14.25/8.07 ( ( range_6617724647562260917t_unit @ id_Product_unit )
% 14.25/8.07 = top_to1996260823553986621t_unit ) ).
% 14.25/8.07
% 14.25/8.07 thf(fact_3550_Range__Id,axiom,
% 14.25/8.07 ( ( range_o_o @ id_o2 )
% 14.25/8.07 = top_top_set_o ) ).
% 14.25/8.07
% 14.25/8.07 thf(fact_3551_Range__Id,axiom,
% 14.25/8.07 ( ( range_nat_nat @ id_nat )
% 14.25/8.07 = top_top_set_nat ) ).
% 14.25/8.07
% 14.25/8.07 thf(fact_3552_Range__Id,axiom,
% 14.25/8.07 ( ( range_int_int @ id_int )
% 14.25/8.07 = top_top_set_int ) ).
% 14.25/8.07
% 14.25/8.07 thf(fact_3553_Max_Obounded__iff,axiom,
% 14.25/8.07 ! [A: set_Code_integer,X: code_integer] :
% 14.25/8.07 ( ( finite6017078050557962740nteger @ A )
% 14.25/8.07 => ( ( A != bot_bo3990330152332043303nteger )
% 14.25/8.07 => ( ( ord_le3102999989581377725nteger @ ( lattic4901227151466704046nteger @ A ) @ X )
% 14.25/8.07 = ( ! [X2: code_integer] :
% 14.25/8.07 ( ( member_Code_integer @ X2 @ A )
% 14.25/8.07 => ( ord_le3102999989581377725nteger @ X2 @ X ) ) ) ) ) ) ).
% 14.25/8.07
% 14.25/8.07 thf(fact_3554_Max_Obounded__iff,axiom,
% 14.25/8.07 ! [A: set_Product_unit,X: product_unit] :
% 14.25/8.07 ( ( finite4290736615968046902t_unit @ A )
% 14.25/8.07 => ( ( A != bot_bo3957492148770167129t_unit )
% 14.25/8.07 => ( ( ord_le3221252021190050221t_unit @ ( lattic1210392890313158908t_unit @ A ) @ X )
% 14.25/8.07 = ( ! [X2: product_unit] :
% 14.25/8.07 ( ( member_Product_unit @ X2 @ A )
% 14.25/8.07 => ( ord_le3221252021190050221t_unit @ X2 @ X ) ) ) ) ) ) ).
% 14.25/8.07
% 14.25/8.07 thf(fact_3555_Max_Obounded__iff,axiom,
% 14.25/8.07 ! [A: set_o,X: $o] :
% 14.25/8.07 ( ( finite_finite_o @ A )
% 14.25/8.07 => ( ( A != bot_bot_set_o )
% 14.25/8.07 => ( ( ord_less_eq_o @ ( lattic1921953407002678535_Max_o @ A ) @ X )
% 14.25/8.07 = ( ! [X2: $o] :
% 14.25/8.07 ( ( member_o @ X2 @ A )
% 14.25/8.07 => ( ord_less_eq_o @ X2 @ X ) ) ) ) ) ) ).
% 14.25/8.07
% 14.25/8.07 thf(fact_3556_Max_Obounded__iff,axiom,
% 14.25/8.07 ! [A: set_rat,X: rat] :
% 14.25/8.07 ( ( finite_finite_rat @ A )
% 14.25/8.07 => ( ( A != bot_bot_set_rat )
% 14.25/8.07 => ( ( ord_less_eq_rat @ ( lattic7630753665789217321ax_rat @ A ) @ X )
% 14.25/8.07 = ( ! [X2: rat] :
% 14.25/8.07 ( ( member_rat @ X2 @ A )
% 14.25/8.07 => ( ord_less_eq_rat @ X2 @ X ) ) ) ) ) ) ).
% 14.25/8.07
% 14.25/8.07 thf(fact_3557_Max_Obounded__iff,axiom,
% 14.25/8.07 ! [A: set_int,X: int] :
% 14.25/8.07 ( ( finite_finite_int @ A )
% 14.25/8.07 => ( ( A != bot_bot_set_int )
% 14.25/8.07 => ( ( ord_less_eq_int @ ( lattic8263393255366662781ax_int @ A ) @ X )
% 14.25/8.07 = ( ! [X2: int] :
% 14.25/8.07 ( ( member_int @ X2 @ A )
% 14.25/8.07 => ( ord_less_eq_int @ X2 @ X ) ) ) ) ) ) ).
% 14.25/8.07
% 14.25/8.07 thf(fact_3558_Max_Obounded__iff,axiom,
% 14.25/8.07 ! [A: set_nat,X: nat] :
% 14.25/8.07 ( ( finite_finite_nat @ A )
% 14.25/8.07 => ( ( A != bot_bot_set_nat )
% 14.25/8.07 => ( ( ord_less_eq_nat @ ( lattic8265883725875713057ax_nat @ A ) @ X )
% 14.25/8.07 = ( ! [X2: nat] :
% 14.25/8.07 ( ( member_nat @ X2 @ A )
% 14.25/8.07 => ( ord_less_eq_nat @ X2 @ X ) ) ) ) ) ) ).
% 14.25/8.07
% 14.25/8.07 thf(fact_3559_Max__less__iff,axiom,
% 14.25/8.07 ! [A: set_Product_unit,X: product_unit] :
% 14.25/8.07 ( ( finite4290736615968046902t_unit @ A )
% 14.25/8.07 => ( ( A != bot_bo3957492148770167129t_unit )
% 14.25/8.07 => ( ( ord_le361264281704409273t_unit @ ( lattic1210392890313158908t_unit @ A ) @ X )
% 14.25/8.07 = ( ! [X2: product_unit] :
% 14.25/8.07 ( ( member_Product_unit @ X2 @ A )
% 14.25/8.07 => ( ord_le361264281704409273t_unit @ X2 @ X ) ) ) ) ) ) ).
% 14.25/8.07
% 14.25/8.07 thf(fact_3560_Max__less__iff,axiom,
% 14.25/8.07 ! [A: set_o,X: $o] :
% 14.25/8.07 ( ( finite_finite_o @ A )
% 14.25/8.07 => ( ( A != bot_bot_set_o )
% 14.25/8.07 => ( ( ord_less_o @ ( lattic1921953407002678535_Max_o @ A ) @ X )
% 14.25/8.07 = ( ! [X2: $o] :
% 14.25/8.07 ( ( member_o @ X2 @ A )
% 14.25/8.07 => ( ord_less_o @ X2 @ X ) ) ) ) ) ) ).
% 14.25/8.07
% 14.25/8.07 thf(fact_3561_Max__less__iff,axiom,
% 14.25/8.07 ! [A: set_int,X: int] :
% 14.25/8.07 ( ( finite_finite_int @ A )
% 14.25/8.07 => ( ( A != bot_bot_set_int )
% 14.25/8.07 => ( ( ord_less_int @ ( lattic8263393255366662781ax_int @ A ) @ X )
% 14.25/8.07 = ( ! [X2: int] :
% 14.25/8.07 ( ( member_int @ X2 @ A )
% 14.25/8.07 => ( ord_less_int @ X2 @ X ) ) ) ) ) ) ).
% 14.25/8.07
% 14.25/8.07 thf(fact_3562_Max__less__iff,axiom,
% 14.25/8.07 ! [A: set_Code_integer,X: code_integer] :
% 14.25/8.07 ( ( finite6017078050557962740nteger @ A )
% 14.25/8.07 => ( ( A != bot_bo3990330152332043303nteger )
% 14.25/8.07 => ( ( ord_le6747313008572928689nteger @ ( lattic4901227151466704046nteger @ A ) @ X )
% 14.25/8.07 = ( ! [X2: code_integer] :
% 14.25/8.07 ( ( member_Code_integer @ X2 @ A )
% 14.25/8.07 => ( ord_le6747313008572928689nteger @ X2 @ X ) ) ) ) ) ) ).
% 14.25/8.07
% 14.25/8.07 thf(fact_3563_Max__less__iff,axiom,
% 14.25/8.07 ! [A: set_rat,X: rat] :
% 14.25/8.07 ( ( finite_finite_rat @ A )
% 14.25/8.07 => ( ( A != bot_bot_set_rat )
% 14.25/8.07 => ( ( ord_less_rat @ ( lattic7630753665789217321ax_rat @ A ) @ X )
% 14.25/8.07 = ( ! [X2: rat] :
% 14.25/8.07 ( ( member_rat @ X2 @ A )
% 14.25/8.07 => ( ord_less_rat @ X2 @ X ) ) ) ) ) ) ).
% 14.25/8.07
% 14.25/8.07 thf(fact_3564_Max__less__iff,axiom,
% 14.25/8.07 ! [A: set_nat,X: nat] :
% 14.25/8.07 ( ( finite_finite_nat @ A )
% 14.25/8.07 => ( ( A != bot_bot_set_nat )
% 14.25/8.07 => ( ( ord_less_nat @ ( lattic8265883725875713057ax_nat @ A ) @ X )
% 14.25/8.07 = ( ! [X2: nat] :
% 14.25/8.07 ( ( member_nat @ X2 @ A )
% 14.25/8.07 => ( ord_less_nat @ X2 @ X ) ) ) ) ) ) ).
% 14.25/8.07
% 14.25/8.07 thf(fact_3565_lt__ex,axiom,
% 14.25/8.07 ! [X: int] :
% 14.25/8.07 ? [Y2: int] : ( ord_less_int @ Y2 @ X ) ).
% 14.25/8.07
% 14.25/8.07 thf(fact_3566_lt__ex,axiom,
% 14.25/8.07 ! [X: rat] :
% 14.25/8.07 ? [Y2: rat] : ( ord_less_rat @ Y2 @ X ) ).
% 14.25/8.07
% 14.25/8.07 thf(fact_3567_gt__ex,axiom,
% 14.25/8.07 ! [X: nat] :
% 14.25/8.07 ? [X_1: nat] : ( ord_less_nat @ X @ X_1 ) ).
% 14.25/8.07
% 14.25/8.07 thf(fact_3568_gt__ex,axiom,
% 14.25/8.07 ! [X: int] :
% 14.25/8.07 ? [X_1: int] : ( ord_less_int @ X @ X_1 ) ).
% 14.25/8.07
% 14.25/8.07 thf(fact_3569_gt__ex,axiom,
% 14.25/8.07 ! [X: rat] :
% 14.25/8.07 ? [X_1: rat] : ( ord_less_rat @ X @ X_1 ) ).
% 14.25/8.07
% 14.25/8.07 thf(fact_3570_dense,axiom,
% 14.25/8.07 ! [X: rat,Y: rat] :
% 14.25/8.07 ( ( ord_less_rat @ X @ Y )
% 14.25/8.07 => ? [Z3: rat] :
% 14.25/8.07 ( ( ord_less_rat @ X @ Z3 )
% 14.25/8.07 & ( ord_less_rat @ Z3 @ Y ) ) ) ).
% 14.25/8.07
% 14.25/8.07 thf(fact_3571_less__imp__neq,axiom,
% 14.25/8.07 ! [X: assn,Y: assn] :
% 14.25/8.07 ( ( ord_less_assn @ X @ Y )
% 14.25/8.07 => ( X != Y ) ) ).
% 14.25/8.07
% 14.25/8.07 thf(fact_3572_less__imp__neq,axiom,
% 14.25/8.07 ! [X: nat,Y: nat] :
% 14.25/8.07 ( ( ord_less_nat @ X @ Y )
% 14.25/8.07 => ( X != Y ) ) ).
% 14.25/8.07
% 14.25/8.07 thf(fact_3573_less__imp__neq,axiom,
% 14.25/8.07 ! [X: int,Y: int] :
% 14.25/8.07 ( ( ord_less_int @ X @ Y )
% 14.25/8.07 => ( X != Y ) ) ).
% 14.25/8.07
% 14.25/8.07 thf(fact_3574_less__imp__neq,axiom,
% 14.25/8.07 ! [X: code_integer,Y: code_integer] :
% 14.25/8.07 ( ( ord_le6747313008572928689nteger @ X @ Y )
% 14.25/8.07 => ( X != Y ) ) ).
% 14.25/8.07
% 14.25/8.07 thf(fact_3575_less__imp__neq,axiom,
% 14.25/8.07 ! [X: rat,Y: rat] :
% 14.25/8.07 ( ( ord_less_rat @ X @ Y )
% 14.25/8.07 => ( X != Y ) ) ).
% 14.25/8.07
% 14.25/8.07 thf(fact_3576_order_Oasym,axiom,
% 14.25/8.07 ! [A2: assn,B: assn] :
% 14.25/8.07 ( ( ord_less_assn @ A2 @ B )
% 14.25/8.07 => ~ ( ord_less_assn @ B @ A2 ) ) ).
% 14.25/8.07
% 14.25/8.07 thf(fact_3577_order_Oasym,axiom,
% 14.25/8.07 ! [A2: nat,B: nat] :
% 14.25/8.07 ( ( ord_less_nat @ A2 @ B )
% 14.25/8.07 => ~ ( ord_less_nat @ B @ A2 ) ) ).
% 14.25/8.07
% 14.25/8.07 thf(fact_3578_order_Oasym,axiom,
% 14.25/8.07 ! [A2: int,B: int] :
% 14.25/8.07 ( ( ord_less_int @ A2 @ B )
% 14.25/8.07 => ~ ( ord_less_int @ B @ A2 ) ) ).
% 14.25/8.07
% 14.25/8.07 thf(fact_3579_order_Oasym,axiom,
% 14.25/8.07 ! [A2: code_integer,B: code_integer] :
% 14.25/8.07 ( ( ord_le6747313008572928689nteger @ A2 @ B )
% 14.25/8.07 => ~ ( ord_le6747313008572928689nteger @ B @ A2 ) ) ).
% 14.25/8.07
% 14.25/8.07 thf(fact_3580_order_Oasym,axiom,
% 14.25/8.07 ! [A2: rat,B: rat] :
% 14.25/8.07 ( ( ord_less_rat @ A2 @ B )
% 14.25/8.07 => ~ ( ord_less_rat @ B @ A2 ) ) ).
% 14.25/8.07
% 14.25/8.07 thf(fact_3581_ord__eq__less__trans,axiom,
% 14.25/8.07 ! [A2: assn,B: assn,C: assn] :
% 14.25/8.07 ( ( A2 = B )
% 14.25/8.07 => ( ( ord_less_assn @ B @ C )
% 14.25/8.07 => ( ord_less_assn @ A2 @ C ) ) ) ).
% 14.25/8.07
% 14.25/8.07 thf(fact_3582_ord__eq__less__trans,axiom,
% 14.25/8.07 ! [A2: nat,B: nat,C: nat] :
% 14.25/8.07 ( ( A2 = B )
% 14.25/8.07 => ( ( ord_less_nat @ B @ C )
% 14.25/8.07 => ( ord_less_nat @ A2 @ C ) ) ) ).
% 14.25/8.07
% 14.25/8.07 thf(fact_3583_ord__eq__less__trans,axiom,
% 14.25/8.07 ! [A2: int,B: int,C: int] :
% 14.25/8.07 ( ( A2 = B )
% 14.25/8.07 => ( ( ord_less_int @ B @ C )
% 14.25/8.07 => ( ord_less_int @ A2 @ C ) ) ) ).
% 14.25/8.07
% 14.25/8.07 thf(fact_3584_ord__eq__less__trans,axiom,
% 14.25/8.07 ! [A2: code_integer,B: code_integer,C: code_integer] :
% 14.25/8.07 ( ( A2 = B )
% 14.25/8.07 => ( ( ord_le6747313008572928689nteger @ B @ C )
% 14.25/8.07 => ( ord_le6747313008572928689nteger @ A2 @ C ) ) ) ).
% 14.25/8.07
% 14.25/8.07 thf(fact_3585_ord__eq__less__trans,axiom,
% 14.25/8.07 ! [A2: rat,B: rat,C: rat] :
% 14.25/8.07 ( ( A2 = B )
% 14.25/8.07 => ( ( ord_less_rat @ B @ C )
% 14.25/8.07 => ( ord_less_rat @ A2 @ C ) ) ) ).
% 14.25/8.07
% 14.25/8.07 thf(fact_3586_ord__less__eq__trans,axiom,
% 14.25/8.07 ! [A2: assn,B: assn,C: assn] :
% 14.25/8.07 ( ( ord_less_assn @ A2 @ B )
% 14.25/8.07 => ( ( B = C )
% 14.25/8.07 => ( ord_less_assn @ A2 @ C ) ) ) ).
% 14.25/8.07
% 14.25/8.07 thf(fact_3587_ord__less__eq__trans,axiom,
% 14.25/8.07 ! [A2: nat,B: nat,C: nat] :
% 14.25/8.07 ( ( ord_less_nat @ A2 @ B )
% 14.25/8.07 => ( ( B = C )
% 14.25/8.07 => ( ord_less_nat @ A2 @ C ) ) ) ).
% 14.25/8.07
% 14.25/8.07 thf(fact_3588_ord__less__eq__trans,axiom,
% 14.25/8.07 ! [A2: int,B: int,C: int] :
% 14.25/8.07 ( ( ord_less_int @ A2 @ B )
% 14.25/8.07 => ( ( B = C )
% 14.25/8.07 => ( ord_less_int @ A2 @ C ) ) ) ).
% 14.25/8.07
% 14.25/8.07 thf(fact_3589_ord__less__eq__trans,axiom,
% 14.25/8.07 ! [A2: code_integer,B: code_integer,C: code_integer] :
% 14.25/8.07 ( ( ord_le6747313008572928689nteger @ A2 @ B )
% 14.25/8.07 => ( ( B = C )
% 14.25/8.07 => ( ord_le6747313008572928689nteger @ A2 @ C ) ) ) ).
% 14.25/8.07
% 14.25/8.07 thf(fact_3590_ord__less__eq__trans,axiom,
% 14.25/8.07 ! [A2: rat,B: rat,C: rat] :
% 14.25/8.07 ( ( ord_less_rat @ A2 @ B )
% 14.25/8.07 => ( ( B = C )
% 14.25/8.07 => ( ord_less_rat @ A2 @ C ) ) ) ).
% 14.25/8.07
% 14.25/8.07 thf(fact_3591_less__induct,axiom,
% 14.25/8.07 ! [P: nat > $o,A2: nat] :
% 14.25/8.07 ( ! [X3: nat] :
% 14.25/8.07 ( ! [Y5: nat] :
% 14.25/8.07 ( ( ord_less_nat @ Y5 @ X3 )
% 14.25/8.07 => ( P @ Y5 ) )
% 14.25/8.07 => ( P @ X3 ) )
% 14.25/8.07 => ( P @ A2 ) ) ).
% 14.25/8.07
% 14.25/8.07 thf(fact_3592_antisym__conv3,axiom,
% 14.25/8.07 ! [Y: nat,X: nat] :
% 14.25/8.07 ( ~ ( ord_less_nat @ Y @ X )
% 14.25/8.07 => ( ( ~ ( ord_less_nat @ X @ Y ) )
% 14.25/8.07 = ( X = Y ) ) ) ).
% 14.25/8.07
% 14.25/8.07 thf(fact_3593_antisym__conv3,axiom,
% 14.25/8.07 ! [Y: int,X: int] :
% 14.25/8.07 ( ~ ( ord_less_int @ Y @ X )
% 14.25/8.07 => ( ( ~ ( ord_less_int @ X @ Y ) )
% 14.25/8.07 = ( X = Y ) ) ) ).
% 14.25/8.07
% 14.25/8.07 thf(fact_3594_antisym__conv3,axiom,
% 14.25/8.07 ! [Y: code_integer,X: code_integer] :
% 14.25/8.07 ( ~ ( ord_le6747313008572928689nteger @ Y @ X )
% 14.25/8.07 => ( ( ~ ( ord_le6747313008572928689nteger @ X @ Y ) )
% 14.25/8.07 = ( X = Y ) ) ) ).
% 14.25/8.07
% 14.25/8.07 thf(fact_3595_antisym__conv3,axiom,
% 14.25/8.07 ! [Y: rat,X: rat] :
% 14.25/8.07 ( ~ ( ord_less_rat @ Y @ X )
% 14.25/8.07 => ( ( ~ ( ord_less_rat @ X @ Y ) )
% 14.25/8.07 = ( X = Y ) ) ) ).
% 14.25/8.07
% 14.25/8.07 thf(fact_3596_linorder__cases,axiom,
% 14.25/8.07 ! [X: nat,Y: nat] :
% 14.25/8.07 ( ~ ( ord_less_nat @ X @ Y )
% 14.25/8.07 => ( ( X != Y )
% 14.25/8.07 => ( ord_less_nat @ Y @ X ) ) ) ).
% 14.25/8.07
% 14.25/8.07 thf(fact_3597_linorder__cases,axiom,
% 14.25/8.07 ! [X: int,Y: int] :
% 14.25/8.07 ( ~ ( ord_less_int @ X @ Y )
% 14.25/8.07 => ( ( X != Y )
% 14.25/8.07 => ( ord_less_int @ Y @ X ) ) ) ).
% 14.25/8.07
% 14.25/8.07 thf(fact_3598_linorder__cases,axiom,
% 14.25/8.07 ! [X: code_integer,Y: code_integer] :
% 14.25/8.07 ( ~ ( ord_le6747313008572928689nteger @ X @ Y )
% 14.25/8.07 => ( ( X != Y )
% 14.25/8.07 => ( ord_le6747313008572928689nteger @ Y @ X ) ) ) ).
% 14.25/8.07
% 14.25/8.07 thf(fact_3599_linorder__cases,axiom,
% 14.25/8.07 ! [X: rat,Y: rat] :
% 14.25/8.07 ( ~ ( ord_less_rat @ X @ Y )
% 14.25/8.07 => ( ( X != Y )
% 14.25/8.07 => ( ord_less_rat @ Y @ X ) ) ) ).
% 14.25/8.07
% 14.25/8.07 thf(fact_3600_dual__order_Oasym,axiom,
% 14.25/8.07 ! [B: assn,A2: assn] :
% 14.25/8.07 ( ( ord_less_assn @ B @ A2 )
% 14.25/8.07 => ~ ( ord_less_assn @ A2 @ B ) ) ).
% 14.25/8.07
% 14.25/8.07 thf(fact_3601_dual__order_Oasym,axiom,
% 14.25/8.07 ! [B: nat,A2: nat] :
% 14.25/8.07 ( ( ord_less_nat @ B @ A2 )
% 14.25/8.07 => ~ ( ord_less_nat @ A2 @ B ) ) ).
% 14.25/8.07
% 14.25/8.07 thf(fact_3602_dual__order_Oasym,axiom,
% 14.25/8.07 ! [B: int,A2: int] :
% 14.25/8.07 ( ( ord_less_int @ B @ A2 )
% 14.25/8.07 => ~ ( ord_less_int @ A2 @ B ) ) ).
% 14.25/8.07
% 14.25/8.07 thf(fact_3603_dual__order_Oasym,axiom,
% 14.25/8.07 ! [B: code_integer,A2: code_integer] :
% 14.25/8.07 ( ( ord_le6747313008572928689nteger @ B @ A2 )
% 14.25/8.07 => ~ ( ord_le6747313008572928689nteger @ A2 @ B ) ) ).
% 14.25/8.07
% 14.25/8.07 thf(fact_3604_dual__order_Oasym,axiom,
% 14.25/8.07 ! [B: rat,A2: rat] :
% 14.25/8.07 ( ( ord_less_rat @ B @ A2 )
% 14.25/8.07 => ~ ( ord_less_rat @ A2 @ B ) ) ).
% 14.25/8.07
% 14.25/8.07 thf(fact_3605_dual__order_Oirrefl,axiom,
% 14.25/8.07 ! [A2: assn] :
% 14.25/8.07 ~ ( ord_less_assn @ A2 @ A2 ) ).
% 14.25/8.07
% 14.25/8.07 thf(fact_3606_dual__order_Oirrefl,axiom,
% 14.25/8.07 ! [A2: nat] :
% 14.25/8.07 ~ ( ord_less_nat @ A2 @ A2 ) ).
% 14.25/8.07
% 14.25/8.07 thf(fact_3607_dual__order_Oirrefl,axiom,
% 14.25/8.07 ! [A2: int] :
% 14.25/8.07 ~ ( ord_less_int @ A2 @ A2 ) ).
% 14.25/8.07
% 14.25/8.07 thf(fact_3608_dual__order_Oirrefl,axiom,
% 14.25/8.07 ! [A2: code_integer] :
% 14.25/8.07 ~ ( ord_le6747313008572928689nteger @ A2 @ A2 ) ).
% 14.25/8.07
% 14.25/8.07 thf(fact_3609_dual__order_Oirrefl,axiom,
% 14.25/8.07 ! [A2: rat] :
% 14.25/8.07 ~ ( ord_less_rat @ A2 @ A2 ) ).
% 14.25/8.07
% 14.25/8.07 thf(fact_3610_exists__least__iff,axiom,
% 14.25/8.07 ( ( ^ [P6: nat > $o] :
% 14.25/8.07 ? [X9: nat] : ( P6 @ X9 ) )
% 14.25/8.07 = ( ^ [P2: nat > $o] :
% 14.25/8.07 ? [N4: nat] :
% 14.25/8.07 ( ( P2 @ N4 )
% 14.25/8.07 & ! [M3: nat] :
% 14.25/8.07 ( ( ord_less_nat @ M3 @ N4 )
% 14.25/8.07 => ~ ( P2 @ M3 ) ) ) ) ) ).
% 14.25/8.07
% 14.25/8.07 thf(fact_3611_linorder__less__wlog,axiom,
% 14.25/8.07 ! [P: nat > nat > $o,A2: nat,B: nat] :
% 14.25/8.07 ( ! [A4: nat,B2: nat] :
% 14.25/8.07 ( ( ord_less_nat @ A4 @ B2 )
% 14.25/8.07 => ( P @ A4 @ B2 ) )
% 14.25/8.07 => ( ! [A4: nat] : ( P @ A4 @ A4 )
% 14.25/8.07 => ( ! [A4: nat,B2: nat] :
% 14.25/8.07 ( ( P @ B2 @ A4 )
% 14.25/8.07 => ( P @ A4 @ B2 ) )
% 14.25/8.07 => ( P @ A2 @ B ) ) ) ) ).
% 14.25/8.07
% 14.25/8.07 thf(fact_3612_linorder__less__wlog,axiom,
% 14.25/8.07 ! [P: int > int > $o,A2: int,B: int] :
% 14.25/8.07 ( ! [A4: int,B2: int] :
% 14.25/8.07 ( ( ord_less_int @ A4 @ B2 )
% 14.25/8.07 => ( P @ A4 @ B2 ) )
% 14.25/8.07 => ( ! [A4: int] : ( P @ A4 @ A4 )
% 14.25/8.07 => ( ! [A4: int,B2: int] :
% 14.25/8.07 ( ( P @ B2 @ A4 )
% 14.25/8.07 => ( P @ A4 @ B2 ) )
% 14.25/8.07 => ( P @ A2 @ B ) ) ) ) ).
% 14.25/8.07
% 14.25/8.07 thf(fact_3613_linorder__less__wlog,axiom,
% 14.25/8.07 ! [P: code_integer > code_integer > $o,A2: code_integer,B: code_integer] :
% 14.25/8.07 ( ! [A4: code_integer,B2: code_integer] :
% 14.25/8.07 ( ( ord_le6747313008572928689nteger @ A4 @ B2 )
% 14.25/8.07 => ( P @ A4 @ B2 ) )
% 14.25/8.07 => ( ! [A4: code_integer] : ( P @ A4 @ A4 )
% 14.25/8.07 => ( ! [A4: code_integer,B2: code_integer] :
% 14.25/8.07 ( ( P @ B2 @ A4 )
% 14.25/8.07 => ( P @ A4 @ B2 ) )
% 14.25/8.07 => ( P @ A2 @ B ) ) ) ) ).
% 14.25/8.07
% 14.25/8.07 thf(fact_3614_linorder__less__wlog,axiom,
% 14.25/8.07 ! [P: rat > rat > $o,A2: rat,B: rat] :
% 14.25/8.07 ( ! [A4: rat,B2: rat] :
% 14.25/8.07 ( ( ord_less_rat @ A4 @ B2 )
% 14.25/8.07 => ( P @ A4 @ B2 ) )
% 14.25/8.07 => ( ! [A4: rat] : ( P @ A4 @ A4 )
% 14.25/8.07 => ( ! [A4: rat,B2: rat] :
% 14.25/8.07 ( ( P @ B2 @ A4 )
% 14.25/8.07 => ( P @ A4 @ B2 ) )
% 14.25/8.07 => ( P @ A2 @ B ) ) ) ) ).
% 14.25/8.07
% 14.25/8.07 thf(fact_3615_order_Ostrict__trans,axiom,
% 14.25/8.07 ! [A2: assn,B: assn,C: assn] :
% 14.25/8.07 ( ( ord_less_assn @ A2 @ B )
% 14.25/8.07 => ( ( ord_less_assn @ B @ C )
% 14.25/8.07 => ( ord_less_assn @ A2 @ C ) ) ) ).
% 14.25/8.07
% 14.25/8.07 thf(fact_3616_order_Ostrict__trans,axiom,
% 14.25/8.07 ! [A2: nat,B: nat,C: nat] :
% 14.25/8.07 ( ( ord_less_nat @ A2 @ B )
% 14.25/8.07 => ( ( ord_less_nat @ B @ C )
% 14.25/8.07 => ( ord_less_nat @ A2 @ C ) ) ) ).
% 14.25/8.07
% 14.25/8.07 thf(fact_3617_order_Ostrict__trans,axiom,
% 14.25/8.07 ! [A2: int,B: int,C: int] :
% 14.25/8.07 ( ( ord_less_int @ A2 @ B )
% 14.25/8.07 => ( ( ord_less_int @ B @ C )
% 14.25/8.07 => ( ord_less_int @ A2 @ C ) ) ) ).
% 14.25/8.07
% 14.25/8.07 thf(fact_3618_order_Ostrict__trans,axiom,
% 14.25/8.07 ! [A2: code_integer,B: code_integer,C: code_integer] :
% 14.25/8.07 ( ( ord_le6747313008572928689nteger @ A2 @ B )
% 14.25/8.07 => ( ( ord_le6747313008572928689nteger @ B @ C )
% 14.25/8.07 => ( ord_le6747313008572928689nteger @ A2 @ C ) ) ) ).
% 14.25/8.07
% 14.25/8.07 thf(fact_3619_order_Ostrict__trans,axiom,
% 14.25/8.07 ! [A2: rat,B: rat,C: rat] :
% 14.25/8.07 ( ( ord_less_rat @ A2 @ B )
% 14.25/8.07 => ( ( ord_less_rat @ B @ C )
% 14.25/8.07 => ( ord_less_rat @ A2 @ C ) ) ) ).
% 14.25/8.07
% 14.25/8.07 thf(fact_3620_top_Oextremum__strict,axiom,
% 14.25/8.07 ! [A2: set_Product_unit] :
% 14.25/8.07 ~ ( ord_le8056459307392131481t_unit @ top_to1996260823553986621t_unit @ A2 ) ).
% 14.25/8.07
% 14.25/8.07 thf(fact_3621_top_Oextremum__strict,axiom,
% 14.25/8.07 ! [A2: set_o] :
% 14.25/8.07 ~ ( ord_less_set_o @ top_top_set_o @ A2 ) ).
% 14.25/8.07
% 14.25/8.07 thf(fact_3622_top_Oextremum__strict,axiom,
% 14.25/8.07 ! [A2: set_nat] :
% 14.25/8.07 ~ ( ord_less_set_nat @ top_top_set_nat @ A2 ) ).
% 14.25/8.07
% 14.25/8.07 thf(fact_3623_top_Oextremum__strict,axiom,
% 14.25/8.07 ! [A2: set_int] :
% 14.25/8.07 ~ ( ord_less_set_int @ top_top_set_int @ A2 ) ).
% 14.25/8.07
% 14.25/8.07 thf(fact_3624_top_Oextremum__strict,axiom,
% 14.25/8.07 ! [A2: assn] :
% 14.25/8.07 ~ ( ord_less_assn @ top_top_assn @ A2 ) ).
% 14.25/8.07
% 14.25/8.07 thf(fact_3625_top_Onot__eq__extremum,axiom,
% 14.25/8.07 ! [A2: set_Product_unit] :
% 14.25/8.07 ( ( A2 != top_to1996260823553986621t_unit )
% 14.25/8.07 = ( ord_le8056459307392131481t_unit @ A2 @ top_to1996260823553986621t_unit ) ) ).
% 14.25/8.07
% 14.25/8.07 thf(fact_3626_top_Onot__eq__extremum,axiom,
% 14.25/8.07 ! [A2: set_o] :
% 14.25/8.07 ( ( A2 != top_top_set_o )
% 14.25/8.07 = ( ord_less_set_o @ A2 @ top_top_set_o ) ) ).
% 14.25/8.07
% 14.25/8.07 thf(fact_3627_top_Onot__eq__extremum,axiom,
% 14.25/8.07 ! [A2: set_nat] :
% 14.25/8.07 ( ( A2 != top_top_set_nat )
% 14.25/8.07 = ( ord_less_set_nat @ A2 @ top_top_set_nat ) ) ).
% 14.25/8.07
% 14.25/8.07 thf(fact_3628_top_Onot__eq__extremum,axiom,
% 14.25/8.07 ! [A2: set_int] :
% 14.25/8.07 ( ( A2 != top_top_set_int )
% 14.25/8.07 = ( ord_less_set_int @ A2 @ top_top_set_int ) ) ).
% 14.25/8.07
% 14.25/8.07 thf(fact_3629_top_Onot__eq__extremum,axiom,
% 14.25/8.07 ! [A2: assn] :
% 14.25/8.07 ( ( A2 != top_top_assn )
% 14.25/8.07 = ( ord_less_assn @ A2 @ top_top_assn ) ) ).
% 14.25/8.07
% 14.25/8.07 thf(fact_3630_not__less__iff__gr__or__eq,axiom,
% 14.25/8.07 ! [X: nat,Y: nat] :
% 14.25/8.07 ( ( ~ ( ord_less_nat @ X @ Y ) )
% 14.25/8.07 = ( ( ord_less_nat @ Y @ X )
% 14.25/8.07 | ( X = Y ) ) ) ).
% 14.25/8.07
% 14.25/8.07 thf(fact_3631_not__less__iff__gr__or__eq,axiom,
% 14.25/8.07 ! [X: int,Y: int] :
% 14.25/8.07 ( ( ~ ( ord_less_int @ X @ Y ) )
% 14.25/8.07 = ( ( ord_less_int @ Y @ X )
% 14.25/8.07 | ( X = Y ) ) ) ).
% 14.25/8.07
% 14.25/8.07 thf(fact_3632_not__less__iff__gr__or__eq,axiom,
% 14.25/8.07 ! [X: code_integer,Y: code_integer] :
% 14.25/8.07 ( ( ~ ( ord_le6747313008572928689nteger @ X @ Y ) )
% 14.25/8.07 = ( ( ord_le6747313008572928689nteger @ Y @ X )
% 14.25/8.07 | ( X = Y ) ) ) ).
% 14.25/8.07
% 14.25/8.07 thf(fact_3633_not__less__iff__gr__or__eq,axiom,
% 14.25/8.07 ! [X: rat,Y: rat] :
% 14.25/8.07 ( ( ~ ( ord_less_rat @ X @ Y ) )
% 14.25/8.07 = ( ( ord_less_rat @ Y @ X )
% 14.25/8.07 | ( X = Y ) ) ) ).
% 14.25/8.07
% 14.25/8.07 thf(fact_3634_dual__order_Ostrict__trans,axiom,
% 14.25/8.07 ! [B: assn,A2: assn,C: assn] :
% 14.25/8.07 ( ( ord_less_assn @ B @ A2 )
% 14.25/8.07 => ( ( ord_less_assn @ C @ B )
% 14.25/8.07 => ( ord_less_assn @ C @ A2 ) ) ) ).
% 14.25/8.07
% 14.25/8.07 thf(fact_3635_dual__order_Ostrict__trans,axiom,
% 14.25/8.07 ! [B: nat,A2: nat,C: nat] :
% 14.25/8.07 ( ( ord_less_nat @ B @ A2 )
% 14.25/8.07 => ( ( ord_less_nat @ C @ B )
% 14.25/8.07 => ( ord_less_nat @ C @ A2 ) ) ) ).
% 14.25/8.07
% 14.25/8.07 thf(fact_3636_dual__order_Ostrict__trans,axiom,
% 14.25/8.07 ! [B: int,A2: int,C: int] :
% 14.25/8.07 ( ( ord_less_int @ B @ A2 )
% 14.25/8.07 => ( ( ord_less_int @ C @ B )
% 14.25/8.07 => ( ord_less_int @ C @ A2 ) ) ) ).
% 14.25/8.07
% 14.25/8.07 thf(fact_3637_dual__order_Ostrict__trans,axiom,
% 14.25/8.07 ! [B: code_integer,A2: code_integer,C: code_integer] :
% 14.25/8.07 ( ( ord_le6747313008572928689nteger @ B @ A2 )
% 14.25/8.07 => ( ( ord_le6747313008572928689nteger @ C @ B )
% 14.25/8.07 => ( ord_le6747313008572928689nteger @ C @ A2 ) ) ) ).
% 14.25/8.07
% 14.25/8.07 thf(fact_3638_dual__order_Ostrict__trans,axiom,
% 14.25/8.07 ! [B: rat,A2: rat,C: rat] :
% 14.25/8.07 ( ( ord_less_rat @ B @ A2 )
% 14.25/8.07 => ( ( ord_less_rat @ C @ B )
% 14.25/8.07 => ( ord_less_rat @ C @ A2 ) ) ) ).
% 14.25/8.07
% 14.25/8.07 thf(fact_3639_order_Ostrict__implies__not__eq,axiom,
% 14.25/8.07 ! [A2: assn,B: assn] :
% 14.25/8.07 ( ( ord_less_assn @ A2 @ B )
% 14.25/8.07 => ( A2 != B ) ) ).
% 14.25/8.07
% 14.25/8.07 thf(fact_3640_order_Ostrict__implies__not__eq,axiom,
% 14.25/8.07 ! [A2: nat,B: nat] :
% 14.25/8.07 ( ( ord_less_nat @ A2 @ B )
% 14.25/8.07 => ( A2 != B ) ) ).
% 14.25/8.07
% 14.25/8.07 thf(fact_3641_order_Ostrict__implies__not__eq,axiom,
% 14.25/8.07 ! [A2: int,B: int] :
% 14.25/8.07 ( ( ord_less_int @ A2 @ B )
% 14.25/8.07 => ( A2 != B ) ) ).
% 14.25/8.07
% 14.25/8.07 thf(fact_3642_order_Ostrict__implies__not__eq,axiom,
% 14.25/8.07 ! [A2: code_integer,B: code_integer] :
% 14.25/8.07 ( ( ord_le6747313008572928689nteger @ A2 @ B )
% 14.25/8.07 => ( A2 != B ) ) ).
% 14.25/8.07
% 14.25/8.07 thf(fact_3643_order_Ostrict__implies__not__eq,axiom,
% 14.25/8.07 ! [A2: rat,B: rat] :
% 14.25/8.07 ( ( ord_less_rat @ A2 @ B )
% 14.25/8.07 => ( A2 != B ) ) ).
% 14.25/8.07
% 14.25/8.07 thf(fact_3644_dual__order_Ostrict__implies__not__eq,axiom,
% 14.25/8.07 ! [B: assn,A2: assn] :
% 14.25/8.07 ( ( ord_less_assn @ B @ A2 )
% 14.25/8.07 => ( A2 != B ) ) ).
% 14.25/8.07
% 14.25/8.07 thf(fact_3645_dual__order_Ostrict__implies__not__eq,axiom,
% 14.25/8.07 ! [B: nat,A2: nat] :
% 14.25/8.07 ( ( ord_less_nat @ B @ A2 )
% 14.25/8.07 => ( A2 != B ) ) ).
% 14.25/8.07
% 14.25/8.07 thf(fact_3646_dual__order_Ostrict__implies__not__eq,axiom,
% 14.25/8.07 ! [B: int,A2: int] :
% 14.25/8.07 ( ( ord_less_int @ B @ A2 )
% 14.25/8.07 => ( A2 != B ) ) ).
% 14.25/8.07
% 14.25/8.07 thf(fact_3647_dual__order_Ostrict__implies__not__eq,axiom,
% 14.25/8.07 ! [B: code_integer,A2: code_integer] :
% 14.25/8.07 ( ( ord_le6747313008572928689nteger @ B @ A2 )
% 14.25/8.07 => ( A2 != B ) ) ).
% 14.25/8.07
% 14.25/8.07 thf(fact_3648_dual__order_Ostrict__implies__not__eq,axiom,
% 14.25/8.07 ! [B: rat,A2: rat] :
% 14.25/8.07 ( ( ord_less_rat @ B @ A2 )
% 14.25/8.07 => ( A2 != B ) ) ).
% 14.25/8.07
% 14.25/8.07 thf(fact_3649_psubsetD,axiom,
% 14.25/8.07 ! [A: set_Pr4532377907799695533_nat_o,B4: set_Pr4532377907799695533_nat_o,C: produc3658429121746597890et_nat > $o] :
% 14.25/8.07 ( ( ord_le2453136405763929_nat_o @ A @ B4 )
% 14.25/8.07 => ( ( member6576561426505652726_nat_o @ C @ A )
% 14.25/8.07 => ( member6576561426505652726_nat_o @ C @ B4 ) ) ) ).
% 14.25/8.07
% 14.25/8.07 thf(fact_3650_psubsetD,axiom,
% 14.25/8.07 ! [A: set_nat,B4: set_nat,C: nat] :
% 14.25/8.07 ( ( ord_less_set_nat @ A @ B4 )
% 14.25/8.07 => ( ( member_nat @ C @ A )
% 14.25/8.07 => ( member_nat @ C @ B4 ) ) ) ).
% 14.25/8.07
% 14.25/8.07 thf(fact_3651_psubsetD,axiom,
% 14.25/8.07 ! [A: set_int,B4: set_int,C: int] :
% 14.25/8.07 ( ( ord_less_set_int @ A @ B4 )
% 14.25/8.07 => ( ( member_int @ C @ A )
% 14.25/8.07 => ( member_int @ C @ B4 ) ) ) ).
% 14.25/8.07
% 14.25/8.07 thf(fact_3652_psubsetD,axiom,
% 14.25/8.07 ! [A: set_o,B4: set_o,C: $o] :
% 14.25/8.07 ( ( ord_less_set_o @ A @ B4 )
% 14.25/8.07 => ( ( member_o @ C @ A )
% 14.25/8.07 => ( member_o @ C @ B4 ) ) ) ).
% 14.25/8.07
% 14.25/8.07 thf(fact_3653_psubsetD,axiom,
% 14.25/8.07 ! [A: set_Product_unit,B4: set_Product_unit,C: product_unit] :
% 14.25/8.07 ( ( ord_le8056459307392131481t_unit @ A @ B4 )
% 14.25/8.07 => ( ( member_Product_unit @ C @ A )
% 14.25/8.07 => ( member_Product_unit @ C @ B4 ) ) ) ).
% 14.25/8.07
% 14.25/8.07 thf(fact_3654_UNIV__eq__I,axiom,
% 14.25/8.07 ! [A: set_Pr4532377907799695533_nat_o] :
% 14.25/8.07 ( ! [X3: produc3658429121746597890et_nat > $o] : ( member6576561426505652726_nat_o @ X3 @ A )
% 14.25/8.07 => ( top_to4869887016220417533_nat_o = A ) ) ).
% 14.25/8.07
% 14.25/8.07 thf(fact_3655_UNIV__eq__I,axiom,
% 14.25/8.07 ! [A: set_Product_unit] :
% 14.25/8.07 ( ! [X3: product_unit] : ( member_Product_unit @ X3 @ A )
% 14.25/8.07 => ( top_to1996260823553986621t_unit = A ) ) ).
% 14.25/8.07
% 14.25/8.07 thf(fact_3656_UNIV__eq__I,axiom,
% 14.25/8.07 ! [A: set_o] :
% 14.25/8.07 ( ! [X3: $o] : ( member_o @ X3 @ A )
% 14.25/8.07 => ( top_top_set_o = A ) ) ).
% 14.25/8.07
% 14.25/8.07 thf(fact_3657_UNIV__eq__I,axiom,
% 14.25/8.07 ! [A: set_nat] :
% 14.25/8.07 ( ! [X3: nat] : ( member_nat @ X3 @ A )
% 14.25/8.07 => ( top_top_set_nat = A ) ) ).
% 14.25/8.07
% 14.25/8.07 thf(fact_3658_UNIV__eq__I,axiom,
% 14.25/8.07 ! [A: set_int] :
% 14.25/8.07 ( ! [X3: int] : ( member_int @ X3 @ A )
% 14.25/8.07 => ( top_top_set_int = A ) ) ).
% 14.25/8.07
% 14.25/8.07 thf(fact_3659_UNIV__witness,axiom,
% 14.25/8.07 ? [X3: produc3658429121746597890et_nat > $o] : ( member6576561426505652726_nat_o @ X3 @ top_to4869887016220417533_nat_o ) ).
% 14.25/8.07
% 14.25/8.07 thf(fact_3660_UNIV__witness,axiom,
% 14.25/8.07 ? [X3: product_unit] : ( member_Product_unit @ X3 @ top_to1996260823553986621t_unit ) ).
% 14.25/8.07
% 14.25/8.07 thf(fact_3661_UNIV__witness,axiom,
% 14.25/8.07 ? [X3: $o] : ( member_o @ X3 @ top_top_set_o ) ).
% 14.25/8.07
% 14.25/8.07 thf(fact_3662_UNIV__witness,axiom,
% 14.25/8.07 ? [X3: nat] : ( member_nat @ X3 @ top_top_set_nat ) ).
% 14.25/8.07
% 14.25/8.07 thf(fact_3663_UNIV__witness,axiom,
% 14.25/8.07 ? [X3: int] : ( member_int @ X3 @ top_top_set_int ) ).
% 14.25/8.07
% 14.25/8.07 thf(fact_3664_linorder__neqE,axiom,
% 14.25/8.07 ! [X: nat,Y: nat] :
% 14.25/8.07 ( ( X != Y )
% 14.25/8.07 => ( ~ ( ord_less_nat @ X @ Y )
% 14.25/8.07 => ( ord_less_nat @ Y @ X ) ) ) ).
% 14.25/8.07
% 14.25/8.07 thf(fact_3665_linorder__neqE,axiom,
% 14.25/8.07 ! [X: int,Y: int] :
% 14.25/8.07 ( ( X != Y )
% 14.25/8.07 => ( ~ ( ord_less_int @ X @ Y )
% 14.25/8.07 => ( ord_less_int @ Y @ X ) ) ) ).
% 14.25/8.07
% 14.25/8.07 thf(fact_3666_linorder__neqE,axiom,
% 14.25/8.07 ! [X: code_integer,Y: code_integer] :
% 14.25/8.07 ( ( X != Y )
% 14.25/8.07 => ( ~ ( ord_le6747313008572928689nteger @ X @ Y )
% 14.25/8.07 => ( ord_le6747313008572928689nteger @ Y @ X ) ) ) ).
% 14.25/8.07
% 14.25/8.07 thf(fact_3667_linorder__neqE,axiom,
% 14.25/8.07 ! [X: rat,Y: rat] :
% 14.25/8.07 ( ( X != Y )
% 14.25/8.07 => ( ~ ( ord_less_rat @ X @ Y )
% 14.25/8.07 => ( ord_less_rat @ Y @ X ) ) ) ).
% 14.25/8.07
% 14.25/8.07 thf(fact_3668_order__less__asym,axiom,
% 14.25/8.07 ! [X: assn,Y: assn] :
% 14.25/8.07 ( ( ord_less_assn @ X @ Y )
% 14.25/8.07 => ~ ( ord_less_assn @ Y @ X ) ) ).
% 14.25/8.07
% 14.25/8.07 thf(fact_3669_order__less__asym,axiom,
% 14.25/8.07 ! [X: nat,Y: nat] :
% 14.25/8.07 ( ( ord_less_nat @ X @ Y )
% 14.25/8.07 => ~ ( ord_less_nat @ Y @ X ) ) ).
% 14.25/8.07
% 14.25/8.07 thf(fact_3670_order__less__asym,axiom,
% 14.25/8.07 ! [X: int,Y: int] :
% 14.25/8.07 ( ( ord_less_int @ X @ Y )
% 14.25/8.07 => ~ ( ord_less_int @ Y @ X ) ) ).
% 14.25/8.07
% 14.25/8.07 thf(fact_3671_order__less__asym,axiom,
% 14.25/8.07 ! [X: code_integer,Y: code_integer] :
% 14.25/8.07 ( ( ord_le6747313008572928689nteger @ X @ Y )
% 14.25/8.07 => ~ ( ord_le6747313008572928689nteger @ Y @ X ) ) ).
% 14.25/8.07
% 14.25/8.07 thf(fact_3672_order__less__asym,axiom,
% 14.25/8.07 ! [X: rat,Y: rat] :
% 14.25/8.07 ( ( ord_less_rat @ X @ Y )
% 14.25/8.07 => ~ ( ord_less_rat @ Y @ X ) ) ).
% 14.25/8.07
% 14.25/8.07 thf(fact_3673_linorder__neq__iff,axiom,
% 14.25/8.07 ! [X: nat,Y: nat] :
% 14.25/8.07 ( ( X != Y )
% 14.25/8.07 = ( ( ord_less_nat @ X @ Y )
% 14.25/8.07 | ( ord_less_nat @ Y @ X ) ) ) ).
% 14.25/8.07
% 14.25/8.07 thf(fact_3674_linorder__neq__iff,axiom,
% 14.25/8.07 ! [X: int,Y: int] :
% 14.25/8.07 ( ( X != Y )
% 14.25/8.07 = ( ( ord_less_int @ X @ Y )
% 14.25/8.07 | ( ord_less_int @ Y @ X ) ) ) ).
% 14.25/8.07
% 14.25/8.07 thf(fact_3675_linorder__neq__iff,axiom,
% 14.25/8.07 ! [X: code_integer,Y: code_integer] :
% 14.25/8.07 ( ( X != Y )
% 14.25/8.07 = ( ( ord_le6747313008572928689nteger @ X @ Y )
% 14.25/8.07 | ( ord_le6747313008572928689nteger @ Y @ X ) ) ) ).
% 14.25/8.07
% 14.25/8.07 thf(fact_3676_linorder__neq__iff,axiom,
% 14.25/8.07 ! [X: rat,Y: rat] :
% 14.25/8.07 ( ( X != Y )
% 14.25/8.07 = ( ( ord_less_rat @ X @ Y )
% 14.25/8.07 | ( ord_less_rat @ Y @ X ) ) ) ).
% 14.25/8.07
% 14.25/8.07 thf(fact_3677_order__less__asym_H,axiom,
% 14.25/8.07 ! [A2: assn,B: assn] :
% 14.25/8.07 ( ( ord_less_assn @ A2 @ B )
% 14.25/8.07 => ~ ( ord_less_assn @ B @ A2 ) ) ).
% 14.25/8.07
% 14.25/8.07 thf(fact_3678_order__less__asym_H,axiom,
% 14.25/8.07 ! [A2: nat,B: nat] :
% 14.25/8.07 ( ( ord_less_nat @ A2 @ B )
% 14.25/8.07 => ~ ( ord_less_nat @ B @ A2 ) ) ).
% 14.25/8.07
% 14.25/8.07 thf(fact_3679_order__less__asym_H,axiom,
% 14.25/8.07 ! [A2: int,B: int] :
% 14.25/8.07 ( ( ord_less_int @ A2 @ B )
% 14.25/8.07 => ~ ( ord_less_int @ B @ A2 ) ) ).
% 14.25/8.07
% 14.25/8.07 thf(fact_3680_order__less__asym_H,axiom,
% 14.25/8.07 ! [A2: code_integer,B: code_integer] :
% 14.25/8.07 ( ( ord_le6747313008572928689nteger @ A2 @ B )
% 14.25/8.07 => ~ ( ord_le6747313008572928689nteger @ B @ A2 ) ) ).
% 14.25/8.07
% 14.25/8.07 thf(fact_3681_order__less__asym_H,axiom,
% 14.25/8.07 ! [A2: rat,B: rat] :
% 14.25/8.07 ( ( ord_less_rat @ A2 @ B )
% 14.25/8.07 => ~ ( ord_less_rat @ B @ A2 ) ) ).
% 14.25/8.07
% 14.25/8.07 thf(fact_3682_order__less__trans,axiom,
% 14.25/8.07 ! [X: assn,Y: assn,Z: assn] :
% 14.25/8.07 ( ( ord_less_assn @ X @ Y )
% 14.25/8.07 => ( ( ord_less_assn @ Y @ Z )
% 14.25/8.07 => ( ord_less_assn @ X @ Z ) ) ) ).
% 14.25/8.07
% 14.25/8.07 thf(fact_3683_order__less__trans,axiom,
% 14.25/8.07 ! [X: nat,Y: nat,Z: nat] :
% 14.25/8.07 ( ( ord_less_nat @ X @ Y )
% 14.25/8.07 => ( ( ord_less_nat @ Y @ Z )
% 14.25/8.07 => ( ord_less_nat @ X @ Z ) ) ) ).
% 14.25/8.07
% 14.25/8.07 thf(fact_3684_order__less__trans,axiom,
% 14.25/8.07 ! [X: int,Y: int,Z: int] :
% 14.25/8.07 ( ( ord_less_int @ X @ Y )
% 14.25/8.07 => ( ( ord_less_int @ Y @ Z )
% 14.25/8.07 => ( ord_less_int @ X @ Z ) ) ) ).
% 14.25/8.07
% 14.25/8.07 thf(fact_3685_order__less__trans,axiom,
% 14.25/8.07 ! [X: code_integer,Y: code_integer,Z: code_integer] :
% 14.25/8.07 ( ( ord_le6747313008572928689nteger @ X @ Y )
% 14.25/8.07 => ( ( ord_le6747313008572928689nteger @ Y @ Z )
% 14.25/8.07 => ( ord_le6747313008572928689nteger @ X @ Z ) ) ) ).
% 14.25/8.07
% 14.25/8.07 thf(fact_3686_order__less__trans,axiom,
% 14.25/8.07 ! [X: rat,Y: rat,Z: rat] :
% 14.25/8.07 ( ( ord_less_rat @ X @ Y )
% 14.25/8.07 => ( ( ord_less_rat @ Y @ Z )
% 14.25/8.07 => ( ord_less_rat @ X @ Z ) ) ) ).
% 14.25/8.07
% 14.25/8.07 thf(fact_3687_ord__eq__less__subst,axiom,
% 14.25/8.07 ! [A2: assn,F: assn > assn,B: assn,C: assn] :
% 14.25/8.07 ( ( A2
% 14.25/8.07 = ( F @ B ) )
% 14.25/8.07 => ( ( ord_less_assn @ B @ C )
% 14.25/8.07 => ( ! [X3: assn,Y2: assn] :
% 14.25/8.07 ( ( ord_less_assn @ X3 @ Y2 )
% 14.25/8.07 => ( ord_less_assn @ ( F @ X3 ) @ ( F @ Y2 ) ) )
% 14.25/8.07 => ( ord_less_assn @ A2 @ ( F @ C ) ) ) ) ) ).
% 14.25/8.07
% 14.25/8.07 thf(fact_3688_ord__eq__less__subst,axiom,
% 14.25/8.07 ! [A2: nat,F: assn > nat,B: assn,C: assn] :
% 14.25/8.07 ( ( A2
% 14.25/8.07 = ( F @ B ) )
% 14.25/8.07 => ( ( ord_less_assn @ B @ C )
% 14.25/8.07 => ( ! [X3: assn,Y2: assn] :
% 14.25/8.07 ( ( ord_less_assn @ X3 @ Y2 )
% 14.25/8.07 => ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
% 14.25/8.07 => ( ord_less_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% 14.25/8.07
% 14.25/8.07 thf(fact_3689_ord__eq__less__subst,axiom,
% 14.25/8.07 ! [A2: int,F: assn > int,B: assn,C: assn] :
% 14.25/8.07 ( ( A2
% 14.25/8.07 = ( F @ B ) )
% 14.25/8.07 => ( ( ord_less_assn @ B @ C )
% 14.25/8.07 => ( ! [X3: assn,Y2: assn] :
% 14.25/8.07 ( ( ord_less_assn @ X3 @ Y2 )
% 14.25/8.07 => ( ord_less_int @ ( F @ X3 ) @ ( F @ Y2 ) ) )
% 14.25/8.07 => ( ord_less_int @ A2 @ ( F @ C ) ) ) ) ) ).
% 14.25/8.07
% 14.25/8.07 thf(fact_3690_ord__eq__less__subst,axiom,
% 14.25/8.07 ! [A2: code_integer,F: assn > code_integer,B: assn,C: assn] :
% 14.25/8.07 ( ( A2
% 14.25/8.07 = ( F @ B ) )
% 14.25/8.07 => ( ( ord_less_assn @ B @ C )
% 14.25/8.07 => ( ! [X3: assn,Y2: assn] :
% 14.25/8.07 ( ( ord_less_assn @ X3 @ Y2 )
% 14.25/8.07 => ( ord_le6747313008572928689nteger @ ( F @ X3 ) @ ( F @ Y2 ) ) )
% 14.25/8.07 => ( ord_le6747313008572928689nteger @ A2 @ ( F @ C ) ) ) ) ) ).
% 14.25/8.07
% 14.25/8.07 thf(fact_3691_ord__eq__less__subst,axiom,
% 14.25/8.07 ! [A2: rat,F: assn > rat,B: assn,C: assn] :
% 14.25/8.07 ( ( A2
% 14.25/8.07 = ( F @ B ) )
% 14.25/8.07 => ( ( ord_less_assn @ B @ C )
% 14.25/8.07 => ( ! [X3: assn,Y2: assn] :
% 14.25/8.07 ( ( ord_less_assn @ X3 @ Y2 )
% 14.25/8.07 => ( ord_less_rat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
% 14.25/8.07 => ( ord_less_rat @ A2 @ ( F @ C ) ) ) ) ) ).
% 14.25/8.07
% 14.25/8.07 thf(fact_3692_ord__eq__less__subst,axiom,
% 14.25/8.07 ! [A2: assn,F: nat > assn,B: nat,C: nat] :
% 14.25/8.07 ( ( A2
% 14.25/8.07 = ( F @ B ) )
% 14.25/8.07 => ( ( ord_less_nat @ B @ C )
% 14.25/8.07 => ( ! [X3: nat,Y2: nat] :
% 14.25/8.07 ( ( ord_less_nat @ X3 @ Y2 )
% 14.25/8.07 => ( ord_less_assn @ ( F @ X3 ) @ ( F @ Y2 ) ) )
% 14.25/8.07 => ( ord_less_assn @ A2 @ ( F @ C ) ) ) ) ) ).
% 14.25/8.07
% 14.25/8.07 thf(fact_3693_ord__eq__less__subst,axiom,
% 14.25/8.07 ! [A2: nat,F: nat > nat,B: nat,C: nat] :
% 14.25/8.07 ( ( A2
% 14.25/8.07 = ( F @ B ) )
% 14.25/8.07 => ( ( ord_less_nat @ B @ C )
% 14.25/8.07 => ( ! [X3: nat,Y2: nat] :
% 14.25/8.07 ( ( ord_less_nat @ X3 @ Y2 )
% 14.25/8.07 => ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
% 14.25/8.07 => ( ord_less_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% 14.25/8.07
% 14.25/8.07 thf(fact_3694_ord__eq__less__subst,axiom,
% 14.25/8.07 ! [A2: int,F: nat > int,B: nat,C: nat] :
% 14.25/8.07 ( ( A2
% 14.25/8.07 = ( F @ B ) )
% 14.25/8.07 => ( ( ord_less_nat @ B @ C )
% 14.25/8.07 => ( ! [X3: nat,Y2: nat] :
% 14.25/8.07 ( ( ord_less_nat @ X3 @ Y2 )
% 14.25/8.07 => ( ord_less_int @ ( F @ X3 ) @ ( F @ Y2 ) ) )
% 14.25/8.07 => ( ord_less_int @ A2 @ ( F @ C ) ) ) ) ) ).
% 14.25/8.07
% 14.25/8.07 thf(fact_3695_ord__eq__less__subst,axiom,
% 14.25/8.07 ! [A2: code_integer,F: nat > code_integer,B: nat,C: nat] :
% 14.25/8.07 ( ( A2
% 14.25/8.07 = ( F @ B ) )
% 14.25/8.07 => ( ( ord_less_nat @ B @ C )
% 14.25/8.07 => ( ! [X3: nat,Y2: nat] :
% 14.25/8.07 ( ( ord_less_nat @ X3 @ Y2 )
% 14.25/8.07 => ( ord_le6747313008572928689nteger @ ( F @ X3 ) @ ( F @ Y2 ) ) )
% 14.25/8.07 => ( ord_le6747313008572928689nteger @ A2 @ ( F @ C ) ) ) ) ) ).
% 14.25/8.07
% 14.25/8.07 thf(fact_3696_ord__eq__less__subst,axiom,
% 14.25/8.07 ! [A2: rat,F: nat > rat,B: nat,C: nat] :
% 14.25/8.07 ( ( A2
% 14.25/8.07 = ( F @ B ) )
% 14.25/8.07 => ( ( ord_less_nat @ B @ C )
% 14.25/8.07 => ( ! [X3: nat,Y2: nat] :
% 14.25/8.07 ( ( ord_less_nat @ X3 @ Y2 )
% 14.25/8.07 => ( ord_less_rat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
% 14.25/8.07 => ( ord_less_rat @ A2 @ ( F @ C ) ) ) ) ) ).
% 14.25/8.07
% 14.25/8.07 thf(fact_3697_ord__less__eq__subst,axiom,
% 14.25/8.07 ! [A2: assn,B: assn,F: assn > assn,C: assn] :
% 14.25/8.07 ( ( ord_less_assn @ A2 @ B )
% 14.25/8.07 => ( ( ( F @ B )
% 14.25/8.07 = C )
% 14.25/8.07 => ( ! [X3: assn,Y2: assn] :
% 14.25/8.07 ( ( ord_less_assn @ X3 @ Y2 )
% 14.25/8.07 => ( ord_less_assn @ ( F @ X3 ) @ ( F @ Y2 ) ) )
% 14.25/8.07 => ( ord_less_assn @ ( F @ A2 ) @ C ) ) ) ) ).
% 14.25/8.07
% 14.25/8.07 thf(fact_3698_ord__less__eq__subst,axiom,
% 14.25/8.07 ! [A2: assn,B: assn,F: assn > nat,C: nat] :
% 14.25/8.07 ( ( ord_less_assn @ A2 @ B )
% 14.25/8.07 => ( ( ( F @ B )
% 14.25/8.07 = C )
% 14.25/8.07 => ( ! [X3: assn,Y2: assn] :
% 14.25/8.07 ( ( ord_less_assn @ X3 @ Y2 )
% 14.25/8.07 => ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
% 14.25/8.07 => ( ord_less_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% 14.25/8.07
% 14.25/8.07 thf(fact_3699_ord__less__eq__subst,axiom,
% 14.25/8.07 ! [A2: assn,B: assn,F: assn > int,C: int] :
% 14.25/8.07 ( ( ord_less_assn @ A2 @ B )
% 14.25/8.07 => ( ( ( F @ B )
% 14.25/8.07 = C )
% 14.25/8.07 => ( ! [X3: assn,Y2: assn] :
% 14.25/8.07 ( ( ord_less_assn @ X3 @ Y2 )
% 14.25/8.07 => ( ord_less_int @ ( F @ X3 ) @ ( F @ Y2 ) ) )
% 14.25/8.07 => ( ord_less_int @ ( F @ A2 ) @ C ) ) ) ) ).
% 14.25/8.07
% 14.25/8.07 thf(fact_3700_ord__less__eq__subst,axiom,
% 14.25/8.07 ! [A2: assn,B: assn,F: assn > code_integer,C: code_integer] :
% 14.25/8.07 ( ( ord_less_assn @ A2 @ B )
% 14.25/8.07 => ( ( ( F @ B )
% 14.25/8.07 = C )
% 14.25/8.07 => ( ! [X3: assn,Y2: assn] :
% 14.25/8.07 ( ( ord_less_assn @ X3 @ Y2 )
% 14.25/8.07 => ( ord_le6747313008572928689nteger @ ( F @ X3 ) @ ( F @ Y2 ) ) )
% 14.25/8.07 => ( ord_le6747313008572928689nteger @ ( F @ A2 ) @ C ) ) ) ) ).
% 14.25/8.07
% 14.25/8.07 thf(fact_3701_ord__less__eq__subst,axiom,
% 14.25/8.07 ! [A2: assn,B: assn,F: assn > rat,C: rat] :
% 14.25/8.07 ( ( ord_less_assn @ A2 @ B )
% 14.25/8.07 => ( ( ( F @ B )
% 14.25/8.07 = C )
% 14.25/8.07 => ( ! [X3: assn,Y2: assn] :
% 14.25/8.07 ( ( ord_less_assn @ X3 @ Y2 )
% 14.25/8.07 => ( ord_less_rat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
% 14.25/8.07 => ( ord_less_rat @ ( F @ A2 ) @ C ) ) ) ) ).
% 14.25/8.07
% 14.25/8.07 thf(fact_3702_ord__less__eq__subst,axiom,
% 14.25/8.07 ! [A2: nat,B: nat,F: nat > assn,C: assn] :
% 14.25/8.07 ( ( ord_less_nat @ A2 @ B )
% 14.25/8.07 => ( ( ( F @ B )
% 14.25/8.07 = C )
% 14.25/8.07 => ( ! [X3: nat,Y2: nat] :
% 14.25/8.07 ( ( ord_less_nat @ X3 @ Y2 )
% 14.25/8.07 => ( ord_less_assn @ ( F @ X3 ) @ ( F @ Y2 ) ) )
% 14.25/8.07 => ( ord_less_assn @ ( F @ A2 ) @ C ) ) ) ) ).
% 14.25/8.07
% 14.25/8.07 thf(fact_3703_ord__less__eq__subst,axiom,
% 14.25/8.07 ! [A2: nat,B: nat,F: nat > nat,C: nat] :
% 14.25/8.07 ( ( ord_less_nat @ A2 @ B )
% 14.25/8.07 => ( ( ( F @ B )
% 14.25/8.07 = C )
% 14.25/8.07 => ( ! [X3: nat,Y2: nat] :
% 14.25/8.07 ( ( ord_less_nat @ X3 @ Y2 )
% 14.25/8.07 => ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
% 14.25/8.07 => ( ord_less_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% 14.25/8.07
% 14.25/8.07 thf(fact_3704_ord__less__eq__subst,axiom,
% 14.25/8.07 ! [A2: nat,B: nat,F: nat > int,C: int] :
% 14.25/8.07 ( ( ord_less_nat @ A2 @ B )
% 14.25/8.07 => ( ( ( F @ B )
% 14.25/8.07 = C )
% 14.25/8.07 => ( ! [X3: nat,Y2: nat] :
% 14.25/8.07 ( ( ord_less_nat @ X3 @ Y2 )
% 14.25/8.07 => ( ord_less_int @ ( F @ X3 ) @ ( F @ Y2 ) ) )
% 14.25/8.07 => ( ord_less_int @ ( F @ A2 ) @ C ) ) ) ) ).
% 14.25/8.07
% 14.25/8.07 thf(fact_3705_ord__less__eq__subst,axiom,
% 14.25/8.07 ! [A2: nat,B: nat,F: nat > code_integer,C: code_integer] :
% 14.25/8.07 ( ( ord_less_nat @ A2 @ B )
% 14.25/8.07 => ( ( ( F @ B )
% 14.25/8.07 = C )
% 14.25/8.07 => ( ! [X3: nat,Y2: nat] :
% 14.25/8.07 ( ( ord_less_nat @ X3 @ Y2 )
% 14.25/8.07 => ( ord_le6747313008572928689nteger @ ( F @ X3 ) @ ( F @ Y2 ) ) )
% 14.25/8.07 => ( ord_le6747313008572928689nteger @ ( F @ A2 ) @ C ) ) ) ) ).
% 14.25/8.07
% 14.25/8.07 thf(fact_3706_ord__less__eq__subst,axiom,
% 14.25/8.07 ! [A2: nat,B: nat,F: nat > rat,C: rat] :
% 14.25/8.07 ( ( ord_less_nat @ A2 @ B )
% 14.25/8.07 => ( ( ( F @ B )
% 14.25/8.07 = C )
% 14.25/8.07 => ( ! [X3: nat,Y2: nat] :
% 14.25/8.07 ( ( ord_less_nat @ X3 @ Y2 )
% 14.25/8.07 => ( ord_less_rat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
% 14.25/8.07 => ( ord_less_rat @ ( F @ A2 ) @ C ) ) ) ) ).
% 14.25/8.07
% 14.25/8.07 thf(fact_3707_order__less__irrefl,axiom,
% 14.25/8.07 ! [X: assn] :
% 14.25/8.07 ~ ( ord_less_assn @ X @ X ) ).
% 14.25/8.07
% 14.25/8.07 thf(fact_3708_order__less__irrefl,axiom,
% 14.25/8.07 ! [X: nat] :
% 14.25/8.07 ~ ( ord_less_nat @ X @ X ) ).
% 14.25/8.07
% 14.25/8.07 thf(fact_3709_order__less__irrefl,axiom,
% 14.25/8.07 ! [X: int] :
% 14.25/8.07 ~ ( ord_less_int @ X @ X ) ).
% 14.25/8.07
% 14.25/8.07 thf(fact_3710_order__less__irrefl,axiom,
% 14.25/8.07 ! [X: code_integer] :
% 14.25/8.07 ~ ( ord_le6747313008572928689nteger @ X @ X ) ).
% 14.25/8.07
% 14.25/8.07 thf(fact_3711_order__less__irrefl,axiom,
% 14.25/8.07 ! [X: rat] :
% 14.25/8.07 ~ ( ord_less_rat @ X @ X ) ).
% 14.25/8.07
% 14.25/8.07 thf(fact_3712_order__less__subst1,axiom,
% 14.25/8.07 ! [A2: assn,F: assn > assn,B: assn,C: assn] :
% 14.25/8.07 ( ( ord_less_assn @ A2 @ ( F @ B ) )
% 14.25/8.07 => ( ( ord_less_assn @ B @ C )
% 14.25/8.07 => ( ! [X3: assn,Y2: assn] :
% 14.25/8.07 ( ( ord_less_assn @ X3 @ Y2 )
% 14.25/8.07 => ( ord_less_assn @ ( F @ X3 ) @ ( F @ Y2 ) ) )
% 14.25/8.07 => ( ord_less_assn @ A2 @ ( F @ C ) ) ) ) ) ).
% 14.25/8.07
% 14.25/8.07 thf(fact_3713_order__less__subst1,axiom,
% 14.25/8.07 ! [A2: assn,F: nat > assn,B: nat,C: nat] :
% 14.25/8.07 ( ( ord_less_assn @ A2 @ ( F @ B ) )
% 14.25/8.07 => ( ( ord_less_nat @ B @ C )
% 14.25/8.07 => ( ! [X3: nat,Y2: nat] :
% 14.25/8.07 ( ( ord_less_nat @ X3 @ Y2 )
% 14.25/8.07 => ( ord_less_assn @ ( F @ X3 ) @ ( F @ Y2 ) ) )
% 14.25/8.07 => ( ord_less_assn @ A2 @ ( F @ C ) ) ) ) ) ).
% 14.25/8.07
% 14.25/8.07 thf(fact_3714_order__less__subst1,axiom,
% 14.25/8.07 ! [A2: assn,F: int > assn,B: int,C: int] :
% 14.25/8.07 ( ( ord_less_assn @ A2 @ ( F @ B ) )
% 14.25/8.07 => ( ( ord_less_int @ B @ C )
% 14.25/8.07 => ( ! [X3: int,Y2: int] :
% 14.25/8.07 ( ( ord_less_int @ X3 @ Y2 )
% 14.25/8.07 => ( ord_less_assn @ ( F @ X3 ) @ ( F @ Y2 ) ) )
% 14.25/8.07 => ( ord_less_assn @ A2 @ ( F @ C ) ) ) ) ) ).
% 14.25/8.07
% 14.25/8.07 thf(fact_3715_order__less__subst1,axiom,
% 14.25/8.07 ! [A2: assn,F: code_integer > assn,B: code_integer,C: code_integer] :
% 14.25/8.07 ( ( ord_less_assn @ A2 @ ( F @ B ) )
% 14.25/8.07 => ( ( ord_le6747313008572928689nteger @ B @ C )
% 14.25/8.07 => ( ! [X3: code_integer,Y2: code_integer] :
% 14.25/8.07 ( ( ord_le6747313008572928689nteger @ X3 @ Y2 )
% 14.25/8.07 => ( ord_less_assn @ ( F @ X3 ) @ ( F @ Y2 ) ) )
% 14.25/8.07 => ( ord_less_assn @ A2 @ ( F @ C ) ) ) ) ) ).
% 14.25/8.07
% 14.25/8.07 thf(fact_3716_order__less__subst1,axiom,
% 14.25/8.07 ! [A2: assn,F: rat > assn,B: rat,C: rat] :
% 14.25/8.07 ( ( ord_less_assn @ A2 @ ( F @ B ) )
% 14.25/8.07 => ( ( ord_less_rat @ B @ C )
% 14.25/8.07 => ( ! [X3: rat,Y2: rat] :
% 14.25/8.07 ( ( ord_less_rat @ X3 @ Y2 )
% 14.25/8.07 => ( ord_less_assn @ ( F @ X3 ) @ ( F @ Y2 ) ) )
% 14.25/8.07 => ( ord_less_assn @ A2 @ ( F @ C ) ) ) ) ) ).
% 14.25/8.07
% 14.25/8.07 thf(fact_3717_order__less__subst1,axiom,
% 14.25/8.07 ! [A2: nat,F: assn > nat,B: assn,C: assn] :
% 14.25/8.07 ( ( ord_less_nat @ A2 @ ( F @ B ) )
% 14.25/8.07 => ( ( ord_less_assn @ B @ C )
% 14.25/8.07 => ( ! [X3: assn,Y2: assn] :
% 14.25/8.07 ( ( ord_less_assn @ X3 @ Y2 )
% 14.25/8.07 => ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
% 14.25/8.07 => ( ord_less_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% 14.25/8.07
% 14.25/8.07 thf(fact_3718_order__less__subst1,axiom,
% 14.25/8.07 ! [A2: nat,F: nat > nat,B: nat,C: nat] :
% 14.25/8.07 ( ( ord_less_nat @ A2 @ ( F @ B ) )
% 14.25/8.07 => ( ( ord_less_nat @ B @ C )
% 14.25/8.07 => ( ! [X3: nat,Y2: nat] :
% 14.25/8.07 ( ( ord_less_nat @ X3 @ Y2 )
% 14.25/8.07 => ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
% 14.25/8.07 => ( ord_less_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% 14.25/8.07
% 14.25/8.07 thf(fact_3719_order__less__subst1,axiom,
% 14.25/8.07 ! [A2: nat,F: int > nat,B: int,C: int] :
% 14.25/8.07 ( ( ord_less_nat @ A2 @ ( F @ B ) )
% 14.25/8.07 => ( ( ord_less_int @ B @ C )
% 14.25/8.07 => ( ! [X3: int,Y2: int] :
% 14.25/8.07 ( ( ord_less_int @ X3 @ Y2 )
% 14.25/8.07 => ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
% 14.25/8.07 => ( ord_less_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% 14.25/8.07
% 14.25/8.07 thf(fact_3720_order__less__subst1,axiom,
% 14.25/8.07 ! [A2: nat,F: code_integer > nat,B: code_integer,C: code_integer] :
% 14.25/8.07 ( ( ord_less_nat @ A2 @ ( F @ B ) )
% 14.25/8.07 => ( ( ord_le6747313008572928689nteger @ B @ C )
% 14.25/8.07 => ( ! [X3: code_integer,Y2: code_integer] :
% 14.25/8.07 ( ( ord_le6747313008572928689nteger @ X3 @ Y2 )
% 14.25/8.07 => ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
% 14.25/8.07 => ( ord_less_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% 14.25/8.07
% 14.25/8.07 thf(fact_3721_order__less__subst1,axiom,
% 14.25/8.07 ! [A2: nat,F: rat > nat,B: rat,C: rat] :
% 14.25/8.07 ( ( ord_less_nat @ A2 @ ( F @ B ) )
% 14.25/8.07 => ( ( ord_less_rat @ B @ C )
% 14.25/8.07 => ( ! [X3: rat,Y2: rat] :
% 14.25/8.07 ( ( ord_less_rat @ X3 @ Y2 )
% 14.25/8.07 => ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
% 14.25/8.07 => ( ord_less_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% 14.25/8.07
% 14.25/8.07 thf(fact_3722_order__less__subst2,axiom,
% 14.25/8.07 ! [A2: assn,B: assn,F: assn > assn,C: assn] :
% 14.25/8.07 ( ( ord_less_assn @ A2 @ B )
% 14.25/8.07 => ( ( ord_less_assn @ ( F @ B ) @ C )
% 14.25/8.07 => ( ! [X3: assn,Y2: assn] :
% 14.25/8.07 ( ( ord_less_assn @ X3 @ Y2 )
% 14.25/8.07 => ( ord_less_assn @ ( F @ X3 ) @ ( F @ Y2 ) ) )
% 14.25/8.07 => ( ord_less_assn @ ( F @ A2 ) @ C ) ) ) ) ).
% 14.25/8.07
% 14.25/8.07 thf(fact_3723_order__less__subst2,axiom,
% 14.25/8.07 ! [A2: assn,B: assn,F: assn > nat,C: nat] :
% 14.25/8.07 ( ( ord_less_assn @ A2 @ B )
% 14.25/8.07 => ( ( ord_less_nat @ ( F @ B ) @ C )
% 14.25/8.07 => ( ! [X3: assn,Y2: assn] :
% 14.25/8.07 ( ( ord_less_assn @ X3 @ Y2 )
% 14.25/8.07 => ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
% 14.25/8.07 => ( ord_less_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% 14.25/8.07
% 14.25/8.07 thf(fact_3724_order__less__subst2,axiom,
% 14.25/8.07 ! [A2: assn,B: assn,F: assn > int,C: int] :
% 14.25/8.07 ( ( ord_less_assn @ A2 @ B )
% 14.25/8.07 => ( ( ord_less_int @ ( F @ B ) @ C )
% 14.25/8.07 => ( ! [X3: assn,Y2: assn] :
% 14.25/8.07 ( ( ord_less_assn @ X3 @ Y2 )
% 14.25/8.07 => ( ord_less_int @ ( F @ X3 ) @ ( F @ Y2 ) ) )
% 14.25/8.07 => ( ord_less_int @ ( F @ A2 ) @ C ) ) ) ) ).
% 14.25/8.07
% 14.25/8.07 thf(fact_3725_order__less__subst2,axiom,
% 14.25/8.07 ! [A2: assn,B: assn,F: assn > code_integer,C: code_integer] :
% 14.25/8.07 ( ( ord_less_assn @ A2 @ B )
% 14.25/8.07 => ( ( ord_le6747313008572928689nteger @ ( F @ B ) @ C )
% 14.25/8.07 => ( ! [X3: assn,Y2: assn] :
% 14.25/8.07 ( ( ord_less_assn @ X3 @ Y2 )
% 14.25/8.07 => ( ord_le6747313008572928689nteger @ ( F @ X3 ) @ ( F @ Y2 ) ) )
% 14.25/8.07 => ( ord_le6747313008572928689nteger @ ( F @ A2 ) @ C ) ) ) ) ).
% 14.25/8.07
% 14.25/8.07 thf(fact_3726_order__less__subst2,axiom,
% 14.25/8.07 ! [A2: assn,B: assn,F: assn > rat,C: rat] :
% 14.25/8.07 ( ( ord_less_assn @ A2 @ B )
% 14.25/8.07 => ( ( ord_less_rat @ ( F @ B ) @ C )
% 14.25/8.07 => ( ! [X3: assn,Y2: assn] :
% 14.25/8.07 ( ( ord_less_assn @ X3 @ Y2 )
% 14.25/8.07 => ( ord_less_rat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
% 14.25/8.07 => ( ord_less_rat @ ( F @ A2 ) @ C ) ) ) ) ).
% 14.25/8.07
% 14.25/8.07 thf(fact_3727_order__less__subst2,axiom,
% 14.25/8.07 ! [A2: nat,B: nat,F: nat > assn,C: assn] :
% 14.25/8.07 ( ( ord_less_nat @ A2 @ B )
% 14.25/8.07 => ( ( ord_less_assn @ ( F @ B ) @ C )
% 14.25/8.07 => ( ! [X3: nat,Y2: nat] :
% 14.25/8.07 ( ( ord_less_nat @ X3 @ Y2 )
% 14.25/8.07 => ( ord_less_assn @ ( F @ X3 ) @ ( F @ Y2 ) ) )
% 14.25/8.07 => ( ord_less_assn @ ( F @ A2 ) @ C ) ) ) ) ).
% 14.25/8.07
% 14.25/8.07 thf(fact_3728_order__less__subst2,axiom,
% 14.25/8.07 ! [A2: nat,B: nat,F: nat > nat,C: nat] :
% 14.25/8.07 ( ( ord_less_nat @ A2 @ B )
% 14.25/8.07 => ( ( ord_less_nat @ ( F @ B ) @ C )
% 14.25/8.07 => ( ! [X3: nat,Y2: nat] :
% 14.25/8.07 ( ( ord_less_nat @ X3 @ Y2 )
% 14.25/8.07 => ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
% 14.25/8.07 => ( ord_less_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% 14.25/8.07
% 14.25/8.07 thf(fact_3729_order__less__subst2,axiom,
% 14.25/8.07 ! [A2: nat,B: nat,F: nat > int,C: int] :
% 14.25/8.07 ( ( ord_less_nat @ A2 @ B )
% 14.25/8.07 => ( ( ord_less_int @ ( F @ B ) @ C )
% 14.25/8.07 => ( ! [X3: nat,Y2: nat] :
% 14.25/8.07 ( ( ord_less_nat @ X3 @ Y2 )
% 14.25/8.07 => ( ord_less_int @ ( F @ X3 ) @ ( F @ Y2 ) ) )
% 14.25/8.07 => ( ord_less_int @ ( F @ A2 ) @ C ) ) ) ) ).
% 14.25/8.07
% 14.25/8.07 thf(fact_3730_order__less__subst2,axiom,
% 14.25/8.07 ! [A2: nat,B: nat,F: nat > code_integer,C: code_integer] :
% 14.25/8.07 ( ( ord_less_nat @ A2 @ B )
% 14.25/8.07 => ( ( ord_le6747313008572928689nteger @ ( F @ B ) @ C )
% 14.25/8.07 => ( ! [X3: nat,Y2: nat] :
% 14.25/8.07 ( ( ord_less_nat @ X3 @ Y2 )
% 14.25/8.07 => ( ord_le6747313008572928689nteger @ ( F @ X3 ) @ ( F @ Y2 ) ) )
% 14.25/8.07 => ( ord_le6747313008572928689nteger @ ( F @ A2 ) @ C ) ) ) ) ).
% 14.25/8.07
% 14.25/8.07 thf(fact_3731_order__less__subst2,axiom,
% 14.25/8.07 ! [A2: nat,B: nat,F: nat > rat,C: rat] :
% 14.25/8.07 ( ( ord_less_nat @ A2 @ B )
% 14.25/8.07 => ( ( ord_less_rat @ ( F @ B ) @ C )
% 14.25/8.07 => ( ! [X3: nat,Y2: nat] :
% 14.25/8.07 ( ( ord_less_nat @ X3 @ Y2 )
% 14.25/8.07 => ( ord_less_rat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
% 14.25/8.07 => ( ord_less_rat @ ( F @ A2 ) @ C ) ) ) ) ).
% 14.25/8.07
% 14.25/8.07 thf(fact_3732_order__less__not__sym,axiom,
% 14.25/8.07 ! [X: assn,Y: assn] :
% 14.25/8.07 ( ( ord_less_assn @ X @ Y )
% 14.25/8.07 => ~ ( ord_less_assn @ Y @ X ) ) ).
% 14.25/8.07
% 14.25/8.07 thf(fact_3733_order__less__not__sym,axiom,
% 14.25/8.07 ! [X: nat,Y: nat] :
% 14.25/8.07 ( ( ord_less_nat @ X @ Y )
% 14.25/8.07 => ~ ( ord_less_nat @ Y @ X ) ) ).
% 14.25/8.07
% 14.25/8.07 thf(fact_3734_order__less__not__sym,axiom,
% 14.25/8.07 ! [X: int,Y: int] :
% 14.25/8.07 ( ( ord_less_int @ X @ Y )
% 14.25/8.07 => ~ ( ord_less_int @ Y @ X ) ) ).
% 14.25/8.07
% 14.25/8.07 thf(fact_3735_order__less__not__sym,axiom,
% 14.25/8.07 ! [X: code_integer,Y: code_integer] :
% 14.25/8.07 ( ( ord_le6747313008572928689nteger @ X @ Y )
% 14.25/8.07 => ~ ( ord_le6747313008572928689nteger @ Y @ X ) ) ).
% 14.25/8.07
% 14.25/8.07 thf(fact_3736_order__less__not__sym,axiom,
% 14.25/8.07 ! [X: rat,Y: rat] :
% 14.25/8.07 ( ( ord_less_rat @ X @ Y )
% 14.25/8.07 => ~ ( ord_less_rat @ Y @ X ) ) ).
% 14.25/8.07
% 14.25/8.07 thf(fact_3737_order__less__imp__triv,axiom,
% 14.25/8.07 ! [X: assn,Y: assn,P: $o] :
% 14.25/8.07 ( ( ord_less_assn @ X @ Y )
% 14.25/8.07 => ( ( ord_less_assn @ Y @ X )
% 14.25/8.07 => P ) ) ).
% 14.25/8.07
% 14.25/8.07 thf(fact_3738_order__less__imp__triv,axiom,
% 14.25/8.07 ! [X: nat,Y: nat,P: $o] :
% 14.25/8.07 ( ( ord_less_nat @ X @ Y )
% 14.25/8.07 => ( ( ord_less_nat @ Y @ X )
% 14.25/8.07 => P ) ) ).
% 14.25/8.07
% 14.25/8.07 thf(fact_3739_order__less__imp__triv,axiom,
% 14.25/8.07 ! [X: int,Y: int,P: $o] :
% 14.25/8.07 ( ( ord_less_int @ X @ Y )
% 14.25/8.07 => ( ( ord_less_int @ Y @ X )
% 14.25/8.07 => P ) ) ).
% 14.25/8.07
% 14.25/8.07 thf(fact_3740_order__less__imp__triv,axiom,
% 14.25/8.07 ! [X: code_integer,Y: code_integer,P: $o] :
% 14.25/8.07 ( ( ord_le6747313008572928689nteger @ X @ Y )
% 14.25/8.07 => ( ( ord_le6747313008572928689nteger @ Y @ X )
% 14.25/8.07 => P ) ) ).
% 14.25/8.07
% 14.25/8.07 thf(fact_3741_order__less__imp__triv,axiom,
% 14.25/8.07 ! [X: rat,Y: rat,P: $o] :
% 14.25/8.07 ( ( ord_less_rat @ X @ Y )
% 14.25/8.07 => ( ( ord_less_rat @ Y @ X )
% 14.25/8.07 => P ) ) ).
% 14.25/8.07
% 14.25/8.07 thf(fact_3742_linorder__less__linear,axiom,
% 14.25/8.07 ! [X: nat,Y: nat] :
% 14.25/8.07 ( ( ord_less_nat @ X @ Y )
% 14.25/8.07 | ( X = Y )
% 14.25/8.07 | ( ord_less_nat @ Y @ X ) ) ).
% 14.25/8.07
% 14.25/8.07 thf(fact_3743_linorder__less__linear,axiom,
% 14.25/8.07 ! [X: int,Y: int] :
% 14.25/8.07 ( ( ord_less_int @ X @ Y )
% 14.25/8.07 | ( X = Y )
% 14.25/8.07 | ( ord_less_int @ Y @ X ) ) ).
% 14.25/8.07
% 14.25/8.07 thf(fact_3744_linorder__less__linear,axiom,
% 14.25/8.07 ! [X: code_integer,Y: code_integer] :
% 14.25/8.07 ( ( ord_le6747313008572928689nteger @ X @ Y )
% 14.25/8.07 | ( X = Y )
% 14.25/8.07 | ( ord_le6747313008572928689nteger @ Y @ X ) ) ).
% 14.25/8.07
% 14.25/8.07 thf(fact_3745_linorder__less__linear,axiom,
% 14.25/8.07 ! [X: rat,Y: rat] :
% 14.25/8.07 ( ( ord_less_rat @ X @ Y )
% 14.25/8.07 | ( X = Y )
% 14.25/8.07 | ( ord_less_rat @ Y @ X ) ) ).
% 14.25/8.07
% 14.25/8.07 thf(fact_3746_order__less__imp__not__eq,axiom,
% 14.25/8.07 ! [X: assn,Y: assn] :
% 14.25/8.07 ( ( ord_less_assn @ X @ Y )
% 14.25/8.07 => ( X != Y ) ) ).
% 14.25/8.07
% 14.25/8.07 thf(fact_3747_order__less__imp__not__eq,axiom,
% 14.25/8.07 ! [X: nat,Y: nat] :
% 14.25/8.07 ( ( ord_less_nat @ X @ Y )
% 14.25/8.07 => ( X != Y ) ) ).
% 14.25/8.07
% 14.25/8.07 thf(fact_3748_order__less__imp__not__eq,axiom,
% 14.25/8.07 ! [X: int,Y: int] :
% 14.25/8.07 ( ( ord_less_int @ X @ Y )
% 14.25/8.07 => ( X != Y ) ) ).
% 14.25/8.07
% 14.25/8.07 thf(fact_3749_order__less__imp__not__eq,axiom,
% 14.25/8.07 ! [X: code_integer,Y: code_integer] :
% 14.25/8.07 ( ( ord_le6747313008572928689nteger @ X @ Y )
% 14.25/8.07 => ( X != Y ) ) ).
% 14.25/8.07
% 14.25/8.07 thf(fact_3750_order__less__imp__not__eq,axiom,
% 14.25/8.07 ! [X: rat,Y: rat] :
% 14.25/8.07 ( ( ord_less_rat @ X @ Y )
% 14.25/8.07 => ( X != Y ) ) ).
% 14.25/8.07
% 14.25/8.07 thf(fact_3751_order__less__imp__not__eq2,axiom,
% 14.25/8.07 ! [X: assn,Y: assn] :
% 14.25/8.07 ( ( ord_less_assn @ X @ Y )
% 14.25/8.07 => ( Y != X ) ) ).
% 14.25/8.07
% 14.25/8.07 thf(fact_3752_order__less__imp__not__eq2,axiom,
% 14.25/8.07 ! [X: nat,Y: nat] :
% 14.25/8.07 ( ( ord_less_nat @ X @ Y )
% 14.25/8.07 => ( Y != X ) ) ).
% 14.25/8.07
% 14.25/8.07 thf(fact_3753_order__less__imp__not__eq2,axiom,
% 14.25/8.07 ! [X: int,Y: int] :
% 14.25/8.07 ( ( ord_less_int @ X @ Y )
% 14.25/8.07 => ( Y != X ) ) ).
% 14.25/8.07
% 14.25/8.07 thf(fact_3754_order__less__imp__not__eq2,axiom,
% 14.25/8.07 ! [X: code_integer,Y: code_integer] :
% 14.25/8.07 ( ( ord_le6747313008572928689nteger @ X @ Y )
% 14.25/8.07 => ( Y != X ) ) ).
% 14.25/8.07
% 14.25/8.07 thf(fact_3755_order__less__imp__not__eq2,axiom,
% 14.25/8.07 ! [X: rat,Y: rat] :
% 14.25/8.07 ( ( ord_less_rat @ X @ Y )
% 14.25/8.07 => ( Y != X ) ) ).
% 14.25/8.07
% 14.25/8.07 thf(fact_3756_order__less__imp__not__less,axiom,
% 14.25/8.07 ! [X: assn,Y: assn] :
% 14.25/8.07 ( ( ord_less_assn @ X @ Y )
% 14.25/8.07 => ~ ( ord_less_assn @ Y @ X ) ) ).
% 14.25/8.07
% 14.25/8.07 thf(fact_3757_order__less__imp__not__less,axiom,
% 14.25/8.07 ! [X: nat,Y: nat] :
% 14.25/8.07 ( ( ord_less_nat @ X @ Y )
% 14.25/8.07 => ~ ( ord_less_nat @ Y @ X ) ) ).
% 14.25/8.07
% 14.25/8.07 thf(fact_3758_order__less__imp__not__less,axiom,
% 14.25/8.07 ! [X: int,Y: int] :
% 14.25/8.07 ( ( ord_less_int @ X @ Y )
% 14.25/8.07 => ~ ( ord_less_int @ Y @ X ) ) ).
% 14.25/8.07
% 14.25/8.07 thf(fact_3759_order__less__imp__not__less,axiom,
% 14.25/8.07 ! [X: code_integer,Y: code_integer] :
% 14.25/8.07 ( ( ord_le6747313008572928689nteger @ X @ Y )
% 14.25/8.07 => ~ ( ord_le6747313008572928689nteger @ Y @ X ) ) ).
% 14.25/8.07
% 14.25/8.07 thf(fact_3760_order__less__imp__not__less,axiom,
% 14.25/8.07 ! [X: rat,Y: rat] :
% 14.25/8.07 ( ( ord_less_rat @ X @ Y )
% 14.25/8.07 => ~ ( ord_less_rat @ Y @ X ) ) ).
% 14.25/8.07
% 14.25/8.07 thf(fact_3761_Max__gr__iff,axiom,
% 14.25/8.07 ! [A: set_Product_unit,X: product_unit] :
% 14.25/8.07 ( ( finite4290736615968046902t_unit @ A )
% 14.25/8.07 => ( ( A != bot_bo3957492148770167129t_unit )
% 14.25/8.07 => ( ( ord_le361264281704409273t_unit @ X @ ( lattic1210392890313158908t_unit @ A ) )
% 14.25/8.07 = ( ? [X2: product_unit] :
% 14.25/8.07 ( ( member_Product_unit @ X2 @ A )
% 14.25/8.07 & ( ord_le361264281704409273t_unit @ X @ X2 ) ) ) ) ) ) ).
% 14.25/8.07
% 14.25/8.07 thf(fact_3762_Max__gr__iff,axiom,
% 14.25/8.07 ! [A: set_o,X: $o] :
% 14.25/8.07 ( ( finite_finite_o @ A )
% 14.25/8.07 => ( ( A != bot_bot_set_o )
% 14.25/8.07 => ( ( ord_less_o @ X @ ( lattic1921953407002678535_Max_o @ A ) )
% 14.25/8.07 = ( ? [X2: $o] :
% 14.25/8.07 ( ( member_o @ X2 @ A )
% 14.25/8.07 & ( ord_less_o @ X @ X2 ) ) ) ) ) ) ).
% 14.25/8.07
% 14.25/8.07 thf(fact_3763_Max__gr__iff,axiom,
% 14.25/8.07 ! [A: set_int,X: int] :
% 14.25/8.07 ( ( finite_finite_int @ A )
% 14.25/8.07 => ( ( A != bot_bot_set_int )
% 14.25/8.07 => ( ( ord_less_int @ X @ ( lattic8263393255366662781ax_int @ A ) )
% 14.25/8.07 = ( ? [X2: int] :
% 14.25/8.07 ( ( member_int @ X2 @ A )
% 14.25/8.07 & ( ord_less_int @ X @ X2 ) ) ) ) ) ) ).
% 14.25/8.07
% 14.25/8.07 thf(fact_3764_Max__gr__iff,axiom,
% 14.25/8.07 ! [A: set_Code_integer,X: code_integer] :
% 14.25/8.07 ( ( finite6017078050557962740nteger @ A )
% 14.25/8.07 => ( ( A != bot_bo3990330152332043303nteger )
% 14.25/8.07 => ( ( ord_le6747313008572928689nteger @ X @ ( lattic4901227151466704046nteger @ A ) )
% 14.25/8.07 = ( ? [X2: code_integer] :
% 14.25/8.07 ( ( member_Code_integer @ X2 @ A )
% 14.25/8.07 & ( ord_le6747313008572928689nteger @ X @ X2 ) ) ) ) ) ) ).
% 14.25/8.07
% 14.25/8.07 thf(fact_3765_Max__gr__iff,axiom,
% 14.25/8.07 ! [A: set_rat,X: rat] :
% 14.25/8.07 ( ( finite_finite_rat @ A )
% 14.25/8.07 => ( ( A != bot_bot_set_rat )
% 14.25/8.07 => ( ( ord_less_rat @ X @ ( lattic7630753665789217321ax_rat @ A ) )
% 14.25/8.07 = ( ? [X2: rat] :
% 14.25/8.07 ( ( member_rat @ X2 @ A )
% 14.25/8.07 & ( ord_less_rat @ X @ X2 ) ) ) ) ) ) ).
% 14.25/8.07
% 14.25/8.07 thf(fact_3766_Max__gr__iff,axiom,
% 14.25/8.07 ! [A: set_nat,X: nat] :
% 14.25/8.07 ( ( finite_finite_nat @ A )
% 14.25/8.07 => ( ( A != bot_bot_set_nat )
% 14.25/8.07 => ( ( ord_less_nat @ X @ ( lattic8265883725875713057ax_nat @ A ) )
% 14.25/8.07 = ( ? [X2: nat] :
% 14.25/8.07 ( ( member_nat @ X2 @ A )
% 14.25/8.07 & ( ord_less_nat @ X @ X2 ) ) ) ) ) ) ).
% 14.25/8.07
% 14.25/8.07 thf(fact_3767_top__greatest,axiom,
% 14.25/8.07 ! [A2: assn] : ( ord_less_eq_assn @ A2 @ top_top_assn ) ).
% 14.25/8.07
% 14.25/8.07 thf(fact_3768_top__greatest,axiom,
% 14.25/8.07 ! [A2: set_Product_unit] : ( ord_le3507040750410214029t_unit @ A2 @ top_to1996260823553986621t_unit ) ).
% 14.25/8.07
% 14.25/8.07 thf(fact_3769_top__greatest,axiom,
% 14.25/8.07 ! [A2: set_o] : ( ord_less_eq_set_o @ A2 @ top_top_set_o ) ).
% 14.25/8.07
% 14.25/8.07 thf(fact_3770_top__greatest,axiom,
% 14.25/8.07 ! [A2: set_int] : ( ord_less_eq_set_int @ A2 @ top_top_set_int ) ).
% 14.25/8.07
% 14.25/8.07 thf(fact_3771_top__greatest,axiom,
% 14.25/8.07 ! [A2: filter_nat] : ( ord_le2510731241096832064er_nat @ A2 @ top_top_filter_nat ) ).
% 14.25/8.07
% 14.25/8.07 thf(fact_3772_top__greatest,axiom,
% 14.25/8.07 ! [A2: set_nat] : ( ord_less_eq_set_nat @ A2 @ top_top_set_nat ) ).
% 14.25/8.07
% 14.25/8.07 thf(fact_3773_top_Oextremum__unique,axiom,
% 14.25/8.07 ! [A2: assn] :
% 14.25/8.07 ( ( ord_less_eq_assn @ top_top_assn @ A2 )
% 14.25/8.07 = ( A2 = top_top_assn ) ) ).
% 14.25/8.07
% 14.25/8.07 thf(fact_3774_top_Oextremum__unique,axiom,
% 14.25/8.07 ! [A2: set_Product_unit] :
% 14.25/8.07 ( ( ord_le3507040750410214029t_unit @ top_to1996260823553986621t_unit @ A2 )
% 14.25/8.07 = ( A2 = top_to1996260823553986621t_unit ) ) ).
% 14.25/8.07
% 14.25/8.07 thf(fact_3775_top_Oextremum__unique,axiom,
% 14.25/8.07 ! [A2: set_o] :
% 14.25/8.07 ( ( ord_less_eq_set_o @ top_top_set_o @ A2 )
% 14.25/8.07 = ( A2 = top_top_set_o ) ) ).
% 14.25/8.07
% 14.25/8.07 thf(fact_3776_top_Oextremum__unique,axiom,
% 14.25/8.07 ! [A2: set_int] :
% 14.25/8.07 ( ( ord_less_eq_set_int @ top_top_set_int @ A2 )
% 14.25/8.07 = ( A2 = top_top_set_int ) ) ).
% 14.25/8.07
% 14.25/8.07 thf(fact_3777_top_Oextremum__unique,axiom,
% 14.25/8.07 ! [A2: filter_nat] :
% 14.25/8.07 ( ( ord_le2510731241096832064er_nat @ top_top_filter_nat @ A2 )
% 14.25/8.07 = ( A2 = top_top_filter_nat ) ) ).
% 14.25/8.07
% 14.25/8.07 thf(fact_3778_top_Oextremum__unique,axiom,
% 14.25/8.07 ! [A2: set_nat] :
% 14.25/8.07 ( ( ord_less_eq_set_nat @ top_top_set_nat @ A2 )
% 14.25/8.07 = ( A2 = top_top_set_nat ) ) ).
% 14.25/8.07
% 14.25/8.07 thf(fact_3779_top_Oextremum__uniqueI,axiom,
% 14.25/8.07 ! [A2: assn] :
% 14.25/8.07 ( ( ord_less_eq_assn @ top_top_assn @ A2 )
% 14.25/8.07 => ( A2 = top_top_assn ) ) ).
% 14.25/8.07
% 14.25/8.07 thf(fact_3780_top_Oextremum__uniqueI,axiom,
% 14.25/8.07 ! [A2: set_Product_unit] :
% 14.25/8.07 ( ( ord_le3507040750410214029t_unit @ top_to1996260823553986621t_unit @ A2 )
% 14.25/8.07 => ( A2 = top_to1996260823553986621t_unit ) ) ).
% 14.25/8.07
% 14.25/8.07 thf(fact_3781_top_Oextremum__uniqueI,axiom,
% 14.25/8.07 ! [A2: set_o] :
% 14.25/8.07 ( ( ord_less_eq_set_o @ top_top_set_o @ A2 )
% 14.25/8.07 => ( A2 = top_top_set_o ) ) ).
% 14.25/8.07
% 14.25/8.07 thf(fact_3782_top_Oextremum__uniqueI,axiom,
% 14.25/8.07 ! [A2: set_int] :
% 14.25/8.07 ( ( ord_less_eq_set_int @ top_top_set_int @ A2 )
% 14.25/8.07 => ( A2 = top_top_set_int ) ) ).
% 14.25/8.07
% 14.25/8.07 thf(fact_3783_top_Oextremum__uniqueI,axiom,
% 14.25/8.07 ! [A2: filter_nat] :
% 14.25/8.07 ( ( ord_le2510731241096832064er_nat @ top_top_filter_nat @ A2 )
% 14.25/8.07 => ( A2 = top_top_filter_nat ) ) ).
% 14.25/8.07
% 14.25/8.07 thf(fact_3784_top_Oextremum__uniqueI,axiom,
% 14.25/8.07 ! [A2: set_nat] :
% 14.25/8.07 ( ( ord_less_eq_set_nat @ top_top_set_nat @ A2 )
% 14.25/8.07 => ( A2 = top_top_set_nat ) ) ).
% 14.25/8.07
% 14.25/8.07 thf(fact_3785_leD,axiom,
% 14.25/8.07 ! [Y: assn,X: assn] :
% 14.25/8.07 ( ( ord_less_eq_assn @ Y @ X )
% 14.25/8.07 => ~ ( ord_less_assn @ X @ Y ) ) ).
% 14.25/8.07
% 14.25/8.07 thf(fact_3786_leD,axiom,
% 14.25/8.07 ! [Y: code_integer,X: code_integer] :
% 14.25/8.07 ( ( ord_le3102999989581377725nteger @ Y @ X )
% 14.25/8.07 => ~ ( ord_le6747313008572928689nteger @ X @ Y ) ) ).
% 14.25/8.07
% 14.25/8.07 thf(fact_3787_leD,axiom,
% 14.25/8.07 ! [Y: filter_nat,X: filter_nat] :
% 14.25/8.07 ( ( ord_le2510731241096832064er_nat @ Y @ X )
% 14.25/8.07 => ~ ( ord_less_filter_nat @ X @ Y ) ) ).
% 14.25/8.07
% 14.25/8.07 thf(fact_3788_leD,axiom,
% 14.25/8.07 ! [Y: set_nat,X: set_nat] :
% 14.25/8.07 ( ( ord_less_eq_set_nat @ Y @ X )
% 14.25/8.07 => ~ ( ord_less_set_nat @ X @ Y ) ) ).
% 14.25/8.07
% 14.25/8.07 thf(fact_3789_leD,axiom,
% 14.25/8.07 ! [Y: rat,X: rat] :
% 14.25/8.07 ( ( ord_less_eq_rat @ Y @ X )
% 14.25/8.07 => ~ ( ord_less_rat @ X @ Y ) ) ).
% 14.25/8.07
% 14.25/8.07 thf(fact_3790_leD,axiom,
% 14.25/8.07 ! [Y: nat,X: nat] :
% 14.25/8.07 ( ( ord_less_eq_nat @ Y @ X )
% 14.25/8.07 => ~ ( ord_less_nat @ X @ Y ) ) ).
% 14.25/8.07
% 14.25/8.07 thf(fact_3791_leD,axiom,
% 14.25/8.07 ! [Y: int,X: int] :
% 14.25/8.07 ( ( ord_less_eq_int @ Y @ X )
% 14.25/8.07 => ~ ( ord_less_int @ X @ Y ) ) ).
% 14.25/8.07
% 14.25/8.07 thf(fact_3792_leI,axiom,
% 14.25/8.07 ! [X: code_integer,Y: code_integer] :
% 14.25/8.07 ( ~ ( ord_le6747313008572928689nteger @ X @ Y )
% 14.25/8.07 => ( ord_le3102999989581377725nteger @ Y @ X ) ) ).
% 14.25/8.07
% 14.25/8.07 thf(fact_3793_leI,axiom,
% 14.25/8.07 ! [X: rat,Y: rat] :
% 14.25/8.07 ( ~ ( ord_less_rat @ X @ Y )
% 14.25/8.07 => ( ord_less_eq_rat @ Y @ X ) ) ).
% 14.25/8.07
% 14.25/8.07 thf(fact_3794_leI,axiom,
% 14.25/8.07 ! [X: nat,Y: nat] :
% 14.25/8.07 ( ~ ( ord_less_nat @ X @ Y )
% 14.25/8.07 => ( ord_less_eq_nat @ Y @ X ) ) ).
% 14.25/8.07
% 14.25/8.07 thf(fact_3795_leI,axiom,
% 14.25/8.07 ! [X: int,Y: int] :
% 14.25/8.07 ( ~ ( ord_less_int @ X @ Y )
% 14.25/8.07 => ( ord_less_eq_int @ Y @ X ) ) ).
% 14.25/8.07
% 14.25/8.07 thf(fact_3796_nless__le,axiom,
% 14.25/8.07 ! [A2: assn,B: assn] :
% 14.25/8.07 ( ( ~ ( ord_less_assn @ A2 @ B ) )
% 14.25/8.07 = ( ~ ( ord_less_eq_assn @ A2 @ B )
% 14.25/8.07 | ( A2 = B ) ) ) ).
% 14.25/8.07
% 14.25/8.07 thf(fact_3797_nless__le,axiom,
% 14.25/8.07 ! [A2: code_integer,B: code_integer] :
% 14.25/8.07 ( ( ~ ( ord_le6747313008572928689nteger @ A2 @ B ) )
% 14.25/8.07 = ( ~ ( ord_le3102999989581377725nteger @ A2 @ B )
% 14.25/8.07 | ( A2 = B ) ) ) ).
% 14.25/8.07
% 14.25/8.07 thf(fact_3798_nless__le,axiom,
% 14.25/8.07 ! [A2: filter_nat,B: filter_nat] :
% 14.25/8.07 ( ( ~ ( ord_less_filter_nat @ A2 @ B ) )
% 14.25/8.07 = ( ~ ( ord_le2510731241096832064er_nat @ A2 @ B )
% 14.25/8.07 | ( A2 = B ) ) ) ).
% 14.25/8.07
% 14.25/8.07 thf(fact_3799_nless__le,axiom,
% 14.25/8.07 ! [A2: set_nat,B: set_nat] :
% 14.25/8.07 ( ( ~ ( ord_less_set_nat @ A2 @ B ) )
% 14.25/8.07 = ( ~ ( ord_less_eq_set_nat @ A2 @ B )
% 14.25/8.07 | ( A2 = B ) ) ) ).
% 14.25/8.07
% 14.25/8.07 thf(fact_3800_nless__le,axiom,
% 14.25/8.07 ! [A2: rat,B: rat] :
% 14.25/8.07 ( ( ~ ( ord_less_rat @ A2 @ B ) )
% 14.25/8.07 = ( ~ ( ord_less_eq_rat @ A2 @ B )
% 14.25/8.07 | ( A2 = B ) ) ) ).
% 14.25/8.07
% 14.25/8.07 thf(fact_3801_nless__le,axiom,
% 14.25/8.07 ! [A2: nat,B: nat] :
% 14.25/8.07 ( ( ~ ( ord_less_nat @ A2 @ B ) )
% 14.25/8.07 = ( ~ ( ord_less_eq_nat @ A2 @ B )
% 14.25/8.07 | ( A2 = B ) ) ) ).
% 14.25/8.07
% 14.25/8.07 thf(fact_3802_nless__le,axiom,
% 14.25/8.07 ! [A2: int,B: int] :
% 14.25/8.07 ( ( ~ ( ord_less_int @ A2 @ B ) )
% 14.25/8.07 = ( ~ ( ord_less_eq_int @ A2 @ B )
% 14.25/8.07 | ( A2 = B ) ) ) ).
% 14.25/8.07
% 14.25/8.07 thf(fact_3803_antisym__conv1,axiom,
% 14.25/8.07 ! [X: assn,Y: assn] :
% 14.25/8.07 ( ~ ( ord_less_assn @ X @ Y )
% 14.25/8.07 => ( ( ord_less_eq_assn @ X @ Y )
% 14.25/8.07 = ( X = Y ) ) ) ).
% 14.25/8.07
% 14.25/8.07 thf(fact_3804_antisym__conv1,axiom,
% 14.25/8.07 ! [X: code_integer,Y: code_integer] :
% 14.25/8.07 ( ~ ( ord_le6747313008572928689nteger @ X @ Y )
% 14.25/8.07 => ( ( ord_le3102999989581377725nteger @ X @ Y )
% 14.25/8.07 = ( X = Y ) ) ) ).
% 14.25/8.07
% 14.25/8.07 thf(fact_3805_antisym__conv1,axiom,
% 14.25/8.07 ! [X: filter_nat,Y: filter_nat] :
% 14.25/8.07 ( ~ ( ord_less_filter_nat @ X @ Y )
% 14.25/8.07 => ( ( ord_le2510731241096832064er_nat @ X @ Y )
% 14.25/8.07 = ( X = Y ) ) ) ).
% 14.25/8.07
% 14.25/8.07 thf(fact_3806_antisym__conv1,axiom,
% 14.25/8.07 ! [X: set_nat,Y: set_nat] :
% 14.25/8.07 ( ~ ( ord_less_set_nat @ X @ Y )
% 14.25/8.07 => ( ( ord_less_eq_set_nat @ X @ Y )
% 14.25/8.07 = ( X = Y ) ) ) ).
% 14.25/8.07
% 14.25/8.07 thf(fact_3807_antisym__conv1,axiom,
% 14.25/8.07 ! [X: rat,Y: rat] :
% 14.25/8.07 ( ~ ( ord_less_rat @ X @ Y )
% 14.25/8.07 => ( ( ord_less_eq_rat @ X @ Y )
% 14.25/8.07 = ( X = Y ) ) ) ).
% 14.25/8.07
% 14.25/8.07 thf(fact_3808_antisym__conv1,axiom,
% 14.25/8.07 ! [X: nat,Y: nat] :
% 14.25/8.07 ( ~ ( ord_less_nat @ X @ Y )
% 14.25/8.07 => ( ( ord_less_eq_nat @ X @ Y )
% 14.25/8.07 = ( X = Y ) ) ) ).
% 14.25/8.07
% 14.25/8.07 thf(fact_3809_antisym__conv1,axiom,
% 14.25/8.07 ! [X: int,Y: int] :
% 14.25/8.07 ( ~ ( ord_less_int @ X @ Y )
% 14.25/8.07 => ( ( ord_less_eq_int @ X @ Y )
% 14.25/8.07 = ( X = Y ) ) ) ).
% 14.25/8.07
% 14.25/8.07 thf(fact_3810_antisym__conv2,axiom,
% 14.25/8.07 ! [X: assn,Y: assn] :
% 14.25/8.07 ( ( ord_less_eq_assn @ X @ Y )
% 14.25/8.07 => ( ( ~ ( ord_less_assn @ X @ Y ) )
% 14.25/8.07 = ( X = Y ) ) ) ).
% 14.25/8.07
% 14.25/8.07 thf(fact_3811_antisym__conv2,axiom,
% 14.25/8.07 ! [X: code_integer,Y: code_integer] :
% 14.25/8.07 ( ( ord_le3102999989581377725nteger @ X @ Y )
% 14.25/8.07 => ( ( ~ ( ord_le6747313008572928689nteger @ X @ Y ) )
% 14.25/8.07 = ( X = Y ) ) ) ).
% 14.25/8.07
% 14.25/8.07 thf(fact_3812_antisym__conv2,axiom,
% 14.25/8.07 ! [X: filter_nat,Y: filter_nat] :
% 14.25/8.07 ( ( ord_le2510731241096832064er_nat @ X @ Y )
% 14.25/8.07 => ( ( ~ ( ord_less_filter_nat @ X @ Y ) )
% 14.25/8.07 = ( X = Y ) ) ) ).
% 14.25/8.07
% 14.25/8.07 thf(fact_3813_antisym__conv2,axiom,
% 14.25/8.07 ! [X: set_nat,Y: set_nat] :
% 14.25/8.07 ( ( ord_less_eq_set_nat @ X @ Y )
% 14.25/8.07 => ( ( ~ ( ord_less_set_nat @ X @ Y ) )
% 14.25/8.07 = ( X = Y ) ) ) ).
% 14.25/8.07
% 14.25/8.07 thf(fact_3814_antisym__conv2,axiom,
% 14.25/8.07 ! [X: rat,Y: rat] :
% 14.25/8.07 ( ( ord_less_eq_rat @ X @ Y )
% 14.25/8.07 => ( ( ~ ( ord_less_rat @ X @ Y ) )
% 14.25/8.07 = ( X = Y ) ) ) ).
% 14.25/8.07
% 14.25/8.07 thf(fact_3815_antisym__conv2,axiom,
% 14.25/8.07 ! [X: nat,Y: nat] :
% 14.25/8.07 ( ( ord_less_eq_nat @ X @ Y )
% 14.25/8.07 => ( ( ~ ( ord_less_nat @ X @ Y ) )
% 14.25/8.07 = ( X = Y ) ) ) ).
% 14.25/8.07
% 14.25/8.07 thf(fact_3816_antisym__conv2,axiom,
% 14.25/8.07 ! [X: int,Y: int] :
% 14.25/8.07 ( ( ord_less_eq_int @ X @ Y )
% 14.25/8.07 => ( ( ~ ( ord_less_int @ X @ Y ) )
% 14.25/8.07 = ( X = Y ) ) ) ).
% 14.25/8.07
% 14.25/8.07 thf(fact_3817_dense__ge,axiom,
% 14.25/8.07 ! [Z: rat,Y: rat] :
% 14.25/8.07 ( ! [X3: rat] :
% 14.25/8.07 ( ( ord_less_rat @ Z @ X3 )
% 14.25/8.07 => ( ord_less_eq_rat @ Y @ X3 ) )
% 14.25/8.07 => ( ord_less_eq_rat @ Y @ Z ) ) ).
% 14.25/8.07
% 14.25/8.07 thf(fact_3818_dense__le,axiom,
% 14.25/8.07 ! [Y: rat,Z: rat] :
% 14.25/8.07 ( ! [X3: rat] :
% 14.25/8.07 ( ( ord_less_rat @ X3 @ Y )
% 14.25/8.07 => ( ord_less_eq_rat @ X3 @ Z ) )
% 14.25/8.07 => ( ord_less_eq_rat @ Y @ Z ) ) ).
% 14.25/8.07
% 14.25/8.07 thf(fact_3819_less__le__not__le,axiom,
% 14.25/8.07 ( ord_less_assn
% 14.25/8.07 = ( ^ [X2: assn,Y3: assn] :
% 14.25/8.07 ( ( ord_less_eq_assn @ X2 @ Y3 )
% 14.25/8.07 & ~ ( ord_less_eq_assn @ Y3 @ X2 ) ) ) ) ).
% 14.25/8.07
% 14.25/8.07 thf(fact_3820_less__le__not__le,axiom,
% 14.25/8.07 ( ord_le6747313008572928689nteger
% 14.25/8.07 = ( ^ [X2: code_integer,Y3: code_integer] :
% 14.25/8.07 ( ( ord_le3102999989581377725nteger @ X2 @ Y3 )
% 14.25/8.07 & ~ ( ord_le3102999989581377725nteger @ Y3 @ X2 ) ) ) ) ).
% 14.25/8.07
% 14.25/8.07 thf(fact_3821_less__le__not__le,axiom,
% 14.25/8.07 ( ord_less_filter_nat
% 14.25/8.07 = ( ^ [X2: filter_nat,Y3: filter_nat] :
% 14.25/8.07 ( ( ord_le2510731241096832064er_nat @ X2 @ Y3 )
% 14.25/8.07 & ~ ( ord_le2510731241096832064er_nat @ Y3 @ X2 ) ) ) ) ).
% 14.25/8.07
% 14.25/8.07 thf(fact_3822_less__le__not__le,axiom,
% 14.25/8.07 ( ord_less_set_nat
% 14.25/8.07 = ( ^ [X2: set_nat,Y3: set_nat] :
% 14.25/8.07 ( ( ord_less_eq_set_nat @ X2 @ Y3 )
% 14.25/8.07 & ~ ( ord_less_eq_set_nat @ Y3 @ X2 ) ) ) ) ).
% 14.25/8.07
% 14.25/8.07 thf(fact_3823_less__le__not__le,axiom,
% 14.25/8.07 ( ord_less_rat
% 14.25/8.07 = ( ^ [X2: rat,Y3: rat] :
% 14.25/8.07 ( ( ord_less_eq_rat @ X2 @ Y3 )
% 14.25/8.07 & ~ ( ord_less_eq_rat @ Y3 @ X2 ) ) ) ) ).
% 14.25/8.07
% 14.25/8.07 thf(fact_3824_less__le__not__le,axiom,
% 14.25/8.07 ( ord_less_nat
% 14.25/8.07 = ( ^ [X2: nat,Y3: nat] :
% 14.25/8.07 ( ( ord_less_eq_nat @ X2 @ Y3 )
% 14.25/8.07 & ~ ( ord_less_eq_nat @ Y3 @ X2 ) ) ) ) ).
% 14.25/8.07
% 14.25/8.07 thf(fact_3825_less__le__not__le,axiom,
% 14.25/8.07 ( ord_less_int
% 14.25/8.07 = ( ^ [X2: int,Y3: int] :
% 14.25/8.07 ( ( ord_less_eq_int @ X2 @ Y3 )
% 14.25/8.07 & ~ ( ord_less_eq_int @ Y3 @ X2 ) ) ) ) ).
% 14.25/8.07
% 14.25/8.07 thf(fact_3826_not__le__imp__less,axiom,
% 14.25/8.07 ! [Y: code_integer,X: code_integer] :
% 14.25/8.07 ( ~ ( ord_le3102999989581377725nteger @ Y @ X )
% 14.25/8.07 => ( ord_le6747313008572928689nteger @ X @ Y ) ) ).
% 14.25/8.07
% 14.25/8.07 thf(fact_3827_not__le__imp__less,axiom,
% 14.25/8.07 ! [Y: rat,X: rat] :
% 14.25/8.07 ( ~ ( ord_less_eq_rat @ Y @ X )
% 14.25/8.07 => ( ord_less_rat @ X @ Y ) ) ).
% 14.25/8.07
% 14.25/8.07 thf(fact_3828_not__le__imp__less,axiom,
% 14.25/8.07 ! [Y: nat,X: nat] :
% 14.25/8.07 ( ~ ( ord_less_eq_nat @ Y @ X )
% 14.25/8.07 => ( ord_less_nat @ X @ Y ) ) ).
% 14.25/8.07
% 14.25/8.07 thf(fact_3829_not__le__imp__less,axiom,
% 14.25/8.07 ! [Y: int,X: int] :
% 14.25/8.07 ( ~ ( ord_less_eq_int @ Y @ X )
% 14.25/8.07 => ( ord_less_int @ X @ Y ) ) ).
% 14.25/8.07
% 14.25/8.07 thf(fact_3830_order_Oorder__iff__strict,axiom,
% 14.25/8.07 ( ord_less_eq_assn
% 14.25/8.07 = ( ^ [A6: assn,B6: assn] :
% 14.25/8.07 ( ( ord_less_assn @ A6 @ B6 )
% 14.25/8.07 | ( A6 = B6 ) ) ) ) ).
% 14.25/8.07
% 14.25/8.07 thf(fact_3831_order_Oorder__iff__strict,axiom,
% 14.25/8.07 ( ord_le3102999989581377725nteger
% 14.25/8.07 = ( ^ [A6: code_integer,B6: code_integer] :
% 14.25/8.07 ( ( ord_le6747313008572928689nteger @ A6 @ B6 )
% 14.25/8.07 | ( A6 = B6 ) ) ) ) ).
% 14.25/8.07
% 14.25/8.07 thf(fact_3832_order_Oorder__iff__strict,axiom,
% 14.25/8.07 ( ord_le2510731241096832064er_nat
% 14.25/8.07 = ( ^ [A6: filter_nat,B6: filter_nat] :
% 14.25/8.07 ( ( ord_less_filter_nat @ A6 @ B6 )
% 14.25/8.07 | ( A6 = B6 ) ) ) ) ).
% 14.25/8.07
% 14.25/8.07 thf(fact_3833_order_Oorder__iff__strict,axiom,
% 14.25/8.07 ( ord_less_eq_set_nat
% 14.25/8.07 = ( ^ [A6: set_nat,B6: set_nat] :
% 14.25/8.07 ( ( ord_less_set_nat @ A6 @ B6 )
% 14.25/8.07 | ( A6 = B6 ) ) ) ) ).
% 14.25/8.07
% 14.25/8.07 thf(fact_3834_order_Oorder__iff__strict,axiom,
% 14.25/8.07 ( ord_less_eq_rat
% 14.25/8.07 = ( ^ [A6: rat,B6: rat] :
% 14.25/8.07 ( ( ord_less_rat @ A6 @ B6 )
% 14.25/8.07 | ( A6 = B6 ) ) ) ) ).
% 14.25/8.07
% 14.25/8.07 thf(fact_3835_order_Oorder__iff__strict,axiom,
% 14.25/8.07 ( ord_less_eq_nat
% 14.25/8.07 = ( ^ [A6: nat,B6: nat] :
% 14.25/8.07 ( ( ord_less_nat @ A6 @ B6 )
% 14.25/8.07 | ( A6 = B6 ) ) ) ) ).
% 14.25/8.07
% 14.25/8.07 thf(fact_3836_order_Oorder__iff__strict,axiom,
% 14.25/8.07 ( ord_less_eq_int
% 14.25/8.07 = ( ^ [A6: int,B6: int] :
% 14.25/8.07 ( ( ord_less_int @ A6 @ B6 )
% 14.25/8.07 | ( A6 = B6 ) ) ) ) ).
% 14.25/8.07
% 14.25/8.07 thf(fact_3837_order_Ostrict__iff__order,axiom,
% 14.25/8.07 ( ord_less_assn
% 14.25/8.07 = ( ^ [A6: assn,B6: assn] :
% 14.25/8.07 ( ( ord_less_eq_assn @ A6 @ B6 )
% 14.25/8.07 & ( A6 != B6 ) ) ) ) ).
% 14.25/8.07
% 14.25/8.07 thf(fact_3838_order_Ostrict__iff__order,axiom,
% 14.25/8.07 ( ord_le6747313008572928689nteger
% 14.25/8.07 = ( ^ [A6: code_integer,B6: code_integer] :
% 14.25/8.07 ( ( ord_le3102999989581377725nteger @ A6 @ B6 )
% 14.25/8.07 & ( A6 != B6 ) ) ) ) ).
% 14.25/8.07
% 14.25/8.07 thf(fact_3839_order_Ostrict__iff__order,axiom,
% 14.25/8.07 ( ord_less_filter_nat
% 14.25/8.07 = ( ^ [A6: filter_nat,B6: filter_nat] :
% 14.25/8.07 ( ( ord_le2510731241096832064er_nat @ A6 @ B6 )
% 14.25/8.07 & ( A6 != B6 ) ) ) ) ).
% 14.25/8.07
% 14.25/8.07 thf(fact_3840_order_Ostrict__iff__order,axiom,
% 14.25/8.07 ( ord_less_set_nat
% 14.25/8.07 = ( ^ [A6: set_nat,B6: set_nat] :
% 14.25/8.07 ( ( ord_less_eq_set_nat @ A6 @ B6 )
% 14.25/8.07 & ( A6 != B6 ) ) ) ) ).
% 14.25/8.07
% 14.25/8.07 thf(fact_3841_order_Ostrict__iff__order,axiom,
% 14.25/8.07 ( ord_less_rat
% 14.25/8.07 = ( ^ [A6: rat,B6: rat] :
% 14.25/8.07 ( ( ord_less_eq_rat @ A6 @ B6 )
% 14.25/8.07 & ( A6 != B6 ) ) ) ) ).
% 14.25/8.07
% 14.25/8.07 thf(fact_3842_order_Ostrict__iff__order,axiom,
% 14.25/8.07 ( ord_less_nat
% 14.25/8.07 = ( ^ [A6: nat,B6: nat] :
% 14.25/8.07 ( ( ord_less_eq_nat @ A6 @ B6 )
% 14.25/8.07 & ( A6 != B6 ) ) ) ) ).
% 14.25/8.07
% 14.25/8.07 thf(fact_3843_order_Ostrict__iff__order,axiom,
% 14.25/8.07 ( ord_less_int
% 14.25/8.07 = ( ^ [A6: int,B6: int] :
% 14.25/8.07 ( ( ord_less_eq_int @ A6 @ B6 )
% 14.25/8.07 & ( A6 != B6 ) ) ) ) ).
% 14.25/8.07
% 14.25/8.07 thf(fact_3844_order_Ostrict__trans1,axiom,
% 14.25/8.07 ! [A2: assn,B: assn,C: assn] :
% 14.25/8.07 ( ( ord_less_eq_assn @ A2 @ B )
% 14.25/8.07 => ( ( ord_less_assn @ B @ C )
% 14.25/8.07 => ( ord_less_assn @ A2 @ C ) ) ) ).
% 14.25/8.07
% 14.25/8.07 thf(fact_3845_order_Ostrict__trans1,axiom,
% 14.25/8.07 ! [A2: code_integer,B: code_integer,C: code_integer] :
% 14.25/8.07 ( ( ord_le3102999989581377725nteger @ A2 @ B )
% 14.25/8.07 => ( ( ord_le6747313008572928689nteger @ B @ C )
% 14.25/8.07 => ( ord_le6747313008572928689nteger @ A2 @ C ) ) ) ).
% 14.25/8.07
% 14.25/8.07 thf(fact_3846_order_Ostrict__trans1,axiom,
% 14.25/8.07 ! [A2: filter_nat,B: filter_nat,C: filter_nat] :
% 14.25/8.07 ( ( ord_le2510731241096832064er_nat @ A2 @ B )
% 14.25/8.07 => ( ( ord_less_filter_nat @ B @ C )
% 14.25/8.07 => ( ord_less_filter_nat @ A2 @ C ) ) ) ).
% 14.25/8.07
% 14.25/8.07 thf(fact_3847_order_Ostrict__trans1,axiom,
% 14.25/8.07 ! [A2: set_nat,B: set_nat,C: set_nat] :
% 14.25/8.07 ( ( ord_less_eq_set_nat @ A2 @ B )
% 14.25/8.07 => ( ( ord_less_set_nat @ B @ C )
% 14.25/8.07 => ( ord_less_set_nat @ A2 @ C ) ) ) ).
% 14.25/8.07
% 14.25/8.07 thf(fact_3848_order_Ostrict__trans1,axiom,
% 14.25/8.07 ! [A2: rat,B: rat,C: rat] :
% 14.25/8.07 ( ( ord_less_eq_rat @ A2 @ B )
% 14.25/8.07 => ( ( ord_less_rat @ B @ C )
% 14.25/8.07 => ( ord_less_rat @ A2 @ C ) ) ) ).
% 14.25/8.07
% 14.25/8.07 thf(fact_3849_order_Ostrict__trans1,axiom,
% 14.25/8.07 ! [A2: nat,B: nat,C: nat] :
% 14.25/8.07 ( ( ord_less_eq_nat @ A2 @ B )
% 14.25/8.07 => ( ( ord_less_nat @ B @ C )
% 14.25/8.07 => ( ord_less_nat @ A2 @ C ) ) ) ).
% 14.25/8.07
% 14.25/8.07 thf(fact_3850_order_Ostrict__trans1,axiom,
% 14.25/8.07 ! [A2: int,B: int,C: int] :
% 14.25/8.07 ( ( ord_less_eq_int @ A2 @ B )
% 14.25/8.07 => ( ( ord_less_int @ B @ C )
% 14.25/8.07 => ( ord_less_int @ A2 @ C ) ) ) ).
% 14.25/8.07
% 14.25/8.07 thf(fact_3851_order_Ostrict__trans2,axiom,
% 14.25/8.07 ! [A2: assn,B: assn,C: assn] :
% 14.25/8.07 ( ( ord_less_assn @ A2 @ B )
% 14.25/8.07 => ( ( ord_less_eq_assn @ B @ C )
% 14.25/8.07 => ( ord_less_assn @ A2 @ C ) ) ) ).
% 14.25/8.07
% 14.25/8.07 thf(fact_3852_order_Ostrict__trans2,axiom,
% 14.25/8.07 ! [A2: code_integer,B: code_integer,C: code_integer] :
% 14.25/8.07 ( ( ord_le6747313008572928689nteger @ A2 @ B )
% 14.25/8.07 => ( ( ord_le3102999989581377725nteger @ B @ C )
% 14.25/8.07 => ( ord_le6747313008572928689nteger @ A2 @ C ) ) ) ).
% 14.25/8.07
% 14.25/8.07 thf(fact_3853_order_Ostrict__trans2,axiom,
% 14.25/8.07 ! [A2: filter_nat,B: filter_nat,C: filter_nat] :
% 14.25/8.07 ( ( ord_less_filter_nat @ A2 @ B )
% 14.25/8.07 => ( ( ord_le2510731241096832064er_nat @ B @ C )
% 14.25/8.07 => ( ord_less_filter_nat @ A2 @ C ) ) ) ).
% 14.25/8.07
% 14.25/8.07 thf(fact_3854_order_Ostrict__trans2,axiom,
% 14.25/8.07 ! [A2: set_nat,B: set_nat,C: set_nat] :
% 14.25/8.07 ( ( ord_less_set_nat @ A2 @ B )
% 14.25/8.07 => ( ( ord_less_eq_set_nat @ B @ C )
% 14.25/8.07 => ( ord_less_set_nat @ A2 @ C ) ) ) ).
% 14.25/8.07
% 14.25/8.07 thf(fact_3855_order_Ostrict__trans2,axiom,
% 14.25/8.07 ! [A2: rat,B: rat,C: rat] :
% 14.25/8.07 ( ( ord_less_rat @ A2 @ B )
% 14.25/8.07 => ( ( ord_less_eq_rat @ B @ C )
% 14.25/8.07 => ( ord_less_rat @ A2 @ C ) ) ) ).
% 14.25/8.07
% 14.25/8.07 thf(fact_3856_order_Ostrict__trans2,axiom,
% 14.25/8.07 ! [A2: nat,B: nat,C: nat] :
% 14.25/8.07 ( ( ord_less_nat @ A2 @ B )
% 14.25/8.07 => ( ( ord_less_eq_nat @ B @ C )
% 14.25/8.07 => ( ord_less_nat @ A2 @ C ) ) ) ).
% 14.25/8.07
% 14.25/8.07 thf(fact_3857_order_Ostrict__trans2,axiom,
% 14.25/8.07 ! [A2: int,B: int,C: int] :
% 14.25/8.07 ( ( ord_less_int @ A2 @ B )
% 14.25/8.07 => ( ( ord_less_eq_int @ B @ C )
% 14.25/8.07 => ( ord_less_int @ A2 @ C ) ) ) ).
% 14.25/8.07
% 14.25/8.07 thf(fact_3858_order_Ostrict__iff__not,axiom,
% 14.25/8.07 ( ord_less_assn
% 14.25/8.07 = ( ^ [A6: assn,B6: assn] :
% 14.25/8.07 ( ( ord_less_eq_assn @ A6 @ B6 )
% 14.25/8.07 & ~ ( ord_less_eq_assn @ B6 @ A6 ) ) ) ) ).
% 14.25/8.07
% 14.25/8.07 thf(fact_3859_order_Ostrict__iff__not,axiom,
% 14.25/8.07 ( ord_le6747313008572928689nteger
% 14.25/8.07 = ( ^ [A6: code_integer,B6: code_integer] :
% 14.25/8.07 ( ( ord_le3102999989581377725nteger @ A6 @ B6 )
% 14.25/8.07 & ~ ( ord_le3102999989581377725nteger @ B6 @ A6 ) ) ) ) ).
% 14.25/8.07
% 14.25/8.07 thf(fact_3860_order_Ostrict__iff__not,axiom,
% 14.25/8.07 ( ord_less_filter_nat
% 14.25/8.07 = ( ^ [A6: filter_nat,B6: filter_nat] :
% 14.25/8.07 ( ( ord_le2510731241096832064er_nat @ A6 @ B6 )
% 14.25/8.07 & ~ ( ord_le2510731241096832064er_nat @ B6 @ A6 ) ) ) ) ).
% 14.25/8.07
% 14.25/8.07 thf(fact_3861_order_Ostrict__iff__not,axiom,
% 14.25/8.07 ( ord_less_set_nat
% 14.25/8.07 = ( ^ [A6: set_nat,B6: set_nat] :
% 14.25/8.07 ( ( ord_less_eq_set_nat @ A6 @ B6 )
% 14.25/8.07 & ~ ( ord_less_eq_set_nat @ B6 @ A6 ) ) ) ) ).
% 14.25/8.07
% 14.25/8.07 thf(fact_3862_order_Ostrict__iff__not,axiom,
% 14.25/8.07 ( ord_less_rat
% 14.25/8.07 = ( ^ [A6: rat,B6: rat] :
% 14.25/8.07 ( ( ord_less_eq_rat @ A6 @ B6 )
% 14.25/8.07 & ~ ( ord_less_eq_rat @ B6 @ A6 ) ) ) ) ).
% 14.25/8.07
% 14.25/8.07 thf(fact_3863_order_Ostrict__iff__not,axiom,
% 14.25/8.07 ( ord_less_nat
% 14.25/8.07 = ( ^ [A6: nat,B6: nat] :
% 14.25/8.07 ( ( ord_less_eq_nat @ A6 @ B6 )
% 14.25/8.07 & ~ ( ord_less_eq_nat @ B6 @ A6 ) ) ) ) ).
% 14.25/8.07
% 14.25/8.07 thf(fact_3864_order_Ostrict__iff__not,axiom,
% 14.25/8.07 ( ord_less_int
% 14.25/8.07 = ( ^ [A6: int,B6: int] :
% 14.25/8.07 ( ( ord_less_eq_int @ A6 @ B6 )
% 14.25/8.07 & ~ ( ord_less_eq_int @ B6 @ A6 ) ) ) ) ).
% 14.25/8.07
% 14.25/8.07 thf(fact_3865_dense__ge__bounded,axiom,
% 14.25/8.07 ! [Z: rat,X: rat,Y: rat] :
% 14.25/8.07 ( ( ord_less_rat @ Z @ X )
% 14.25/8.07 => ( ! [W2: rat] :
% 14.25/8.07 ( ( ord_less_rat @ Z @ W2 )
% 14.25/8.07 => ( ( ord_less_rat @ W2 @ X )
% 14.25/8.07 => ( ord_less_eq_rat @ Y @ W2 ) ) )
% 14.25/8.07 => ( ord_less_eq_rat @ Y @ Z ) ) ) ).
% 14.25/8.07
% 14.25/8.07 thf(fact_3866_dense__le__bounded,axiom,
% 14.25/8.07 ! [X: rat,Y: rat,Z: rat] :
% 14.25/8.07 ( ( ord_less_rat @ X @ Y )
% 14.25/8.07 => ( ! [W2: rat] :
% 14.25/8.07 ( ( ord_less_rat @ X @ W2 )
% 14.25/8.07 => ( ( ord_less_rat @ W2 @ Y )
% 14.25/8.07 => ( ord_less_eq_rat @ W2 @ Z ) ) )
% 14.25/8.07 => ( ord_less_eq_rat @ Y @ Z ) ) ) ).
% 14.25/8.07
% 14.25/8.07 thf(fact_3867_dual__order_Oorder__iff__strict,axiom,
% 14.25/8.07 ( ord_less_eq_assn
% 14.25/8.07 = ( ^ [B6: assn,A6: assn] :
% 14.25/8.07 ( ( ord_less_assn @ B6 @ A6 )
% 14.25/8.07 | ( A6 = B6 ) ) ) ) ).
% 14.25/8.07
% 14.25/8.07 thf(fact_3868_dual__order_Oorder__iff__strict,axiom,
% 14.25/8.07 ( ord_le3102999989581377725nteger
% 14.25/8.07 = ( ^ [B6: code_integer,A6: code_integer] :
% 14.25/8.07 ( ( ord_le6747313008572928689nteger @ B6 @ A6 )
% 14.25/8.07 | ( A6 = B6 ) ) ) ) ).
% 14.25/8.07
% 14.25/8.07 thf(fact_3869_dual__order_Oorder__iff__strict,axiom,
% 14.25/8.07 ( ord_le2510731241096832064er_nat
% 14.25/8.07 = ( ^ [B6: filter_nat,A6: filter_nat] :
% 14.25/8.07 ( ( ord_less_filter_nat @ B6 @ A6 )
% 14.25/8.07 | ( A6 = B6 ) ) ) ) ).
% 14.25/8.07
% 14.25/8.07 thf(fact_3870_dual__order_Oorder__iff__strict,axiom,
% 14.25/8.07 ( ord_less_eq_set_nat
% 14.25/8.07 = ( ^ [B6: set_nat,A6: set_nat] :
% 14.25/8.07 ( ( ord_less_set_nat @ B6 @ A6 )
% 14.25/8.07 | ( A6 = B6 ) ) ) ) ).
% 14.25/8.07
% 14.25/8.07 thf(fact_3871_dual__order_Oorder__iff__strict,axiom,
% 14.25/8.07 ( ord_less_eq_rat
% 14.25/8.07 = ( ^ [B6: rat,A6: rat] :
% 14.25/8.07 ( ( ord_less_rat @ B6 @ A6 )
% 14.25/8.07 | ( A6 = B6 ) ) ) ) ).
% 14.25/8.07
% 14.25/8.07 thf(fact_3872_dual__order_Oorder__iff__strict,axiom,
% 14.25/8.07 ( ord_less_eq_nat
% 14.25/8.07 = ( ^ [B6: nat,A6: nat] :
% 14.25/8.07 ( ( ord_less_nat @ B6 @ A6 )
% 14.25/8.07 | ( A6 = B6 ) ) ) ) ).
% 14.25/8.07
% 14.25/8.07 thf(fact_3873_dual__order_Oorder__iff__strict,axiom,
% 14.25/8.07 ( ord_less_eq_int
% 14.25/8.07 = ( ^ [B6: int,A6: int] :
% 14.25/8.07 ( ( ord_less_int @ B6 @ A6 )
% 14.25/8.07 | ( A6 = B6 ) ) ) ) ).
% 14.25/8.07
% 14.25/8.07 thf(fact_3874_dual__order_Ostrict__iff__order,axiom,
% 14.25/8.07 ( ord_less_assn
% 14.25/8.07 = ( ^ [B6: assn,A6: assn] :
% 14.25/8.07 ( ( ord_less_eq_assn @ B6 @ A6 )
% 14.25/8.07 & ( A6 != B6 ) ) ) ) ).
% 14.25/8.07
% 14.25/8.07 thf(fact_3875_dual__order_Ostrict__iff__order,axiom,
% 14.25/8.07 ( ord_le6747313008572928689nteger
% 14.25/8.07 = ( ^ [B6: code_integer,A6: code_integer] :
% 14.25/8.07 ( ( ord_le3102999989581377725nteger @ B6 @ A6 )
% 14.25/8.07 & ( A6 != B6 ) ) ) ) ).
% 14.25/8.07
% 14.25/8.07 thf(fact_3876_dual__order_Ostrict__iff__order,axiom,
% 14.25/8.07 ( ord_less_filter_nat
% 14.25/8.07 = ( ^ [B6: filter_nat,A6: filter_nat] :
% 14.25/8.07 ( ( ord_le2510731241096832064er_nat @ B6 @ A6 )
% 14.25/8.07 & ( A6 != B6 ) ) ) ) ).
% 14.25/8.07
% 14.25/8.07 thf(fact_3877_dual__order_Ostrict__iff__order,axiom,
% 14.25/8.07 ( ord_less_set_nat
% 14.25/8.07 = ( ^ [B6: set_nat,A6: set_nat] :
% 14.25/8.07 ( ( ord_less_eq_set_nat @ B6 @ A6 )
% 14.25/8.07 & ( A6 != B6 ) ) ) ) ).
% 14.25/8.07
% 14.25/8.07 thf(fact_3878_dual__order_Ostrict__iff__order,axiom,
% 14.25/8.07 ( ord_less_rat
% 14.25/8.07 = ( ^ [B6: rat,A6: rat] :
% 14.25/8.07 ( ( ord_less_eq_rat @ B6 @ A6 )
% 14.25/8.07 & ( A6 != B6 ) ) ) ) ).
% 14.25/8.07
% 14.25/8.07 thf(fact_3879_dual__order_Ostrict__iff__order,axiom,
% 14.25/8.07 ( ord_less_nat
% 14.25/8.07 = ( ^ [B6: nat,A6: nat] :
% 14.25/8.07 ( ( ord_less_eq_nat @ B6 @ A6 )
% 14.25/8.07 & ( A6 != B6 ) ) ) ) ).
% 14.25/8.07
% 14.25/8.07 thf(fact_3880_dual__order_Ostrict__iff__order,axiom,
% 14.25/8.07 ( ord_less_int
% 14.25/8.07 = ( ^ [B6: int,A6: int] :
% 14.25/8.07 ( ( ord_less_eq_int @ B6 @ A6 )
% 14.25/8.07 & ( A6 != B6 ) ) ) ) ).
% 14.25/8.07
% 14.25/8.07 thf(fact_3881_dual__order_Ostrict__trans1,axiom,
% 14.25/8.07 ! [B: assn,A2: assn,C: assn] :
% 14.25/8.07 ( ( ord_less_eq_assn @ B @ A2 )
% 14.25/8.07 => ( ( ord_less_assn @ C @ B )
% 14.25/8.07 => ( ord_less_assn @ C @ A2 ) ) ) ).
% 14.25/8.07
% 14.25/8.07 thf(fact_3882_dual__order_Ostrict__trans1,axiom,
% 14.25/8.07 ! [B: code_integer,A2: code_integer,C: code_integer] :
% 14.25/8.07 ( ( ord_le3102999989581377725nteger @ B @ A2 )
% 14.25/8.07 => ( ( ord_le6747313008572928689nteger @ C @ B )
% 14.25/8.07 => ( ord_le6747313008572928689nteger @ C @ A2 ) ) ) ).
% 14.25/8.07
% 14.25/8.07 thf(fact_3883_dual__order_Ostrict__trans1,axiom,
% 14.25/8.07 ! [B: filter_nat,A2: filter_nat,C: filter_nat] :
% 14.25/8.07 ( ( ord_le2510731241096832064er_nat @ B @ A2 )
% 14.25/8.07 => ( ( ord_less_filter_nat @ C @ B )
% 14.25/8.07 => ( ord_less_filter_nat @ C @ A2 ) ) ) ).
% 14.25/8.07
% 14.25/8.07 thf(fact_3884_dual__order_Ostrict__trans1,axiom,
% 14.25/8.07 ! [B: set_nat,A2: set_nat,C: set_nat] :
% 14.25/8.07 ( ( ord_less_eq_set_nat @ B @ A2 )
% 14.25/8.07 => ( ( ord_less_set_nat @ C @ B )
% 14.25/8.07 => ( ord_less_set_nat @ C @ A2 ) ) ) ).
% 14.25/8.07
% 14.25/8.07 thf(fact_3885_dual__order_Ostrict__trans1,axiom,
% 14.25/8.07 ! [B: rat,A2: rat,C: rat] :
% 14.25/8.07 ( ( ord_less_eq_rat @ B @ A2 )
% 14.25/8.07 => ( ( ord_less_rat @ C @ B )
% 14.25/8.07 => ( ord_less_rat @ C @ A2 ) ) ) ).
% 14.25/8.07
% 14.25/8.07 thf(fact_3886_dual__order_Ostrict__trans1,axiom,
% 14.25/8.07 ! [B: nat,A2: nat,C: nat] :
% 14.25/8.07 ( ( ord_less_eq_nat @ B @ A2 )
% 14.25/8.07 => ( ( ord_less_nat @ C @ B )
% 14.25/8.07 => ( ord_less_nat @ C @ A2 ) ) ) ).
% 14.25/8.07
% 14.25/8.07 thf(fact_3887_dual__order_Ostrict__trans1,axiom,
% 14.25/8.07 ! [B: int,A2: int,C: int] :
% 14.25/8.07 ( ( ord_less_eq_int @ B @ A2 )
% 14.25/8.07 => ( ( ord_less_int @ C @ B )
% 14.25/8.07 => ( ord_less_int @ C @ A2 ) ) ) ).
% 14.25/8.07
% 14.25/8.07 thf(fact_3888_dual__order_Ostrict__trans2,axiom,
% 14.25/8.07 ! [B: assn,A2: assn,C: assn] :
% 14.25/8.07 ( ( ord_less_assn @ B @ A2 )
% 14.25/8.07 => ( ( ord_less_eq_assn @ C @ B )
% 14.25/8.07 => ( ord_less_assn @ C @ A2 ) ) ) ).
% 14.25/8.07
% 14.25/8.07 thf(fact_3889_dual__order_Ostrict__trans2,axiom,
% 14.25/8.07 ! [B: code_integer,A2: code_integer,C: code_integer] :
% 14.25/8.07 ( ( ord_le6747313008572928689nteger @ B @ A2 )
% 14.25/8.07 => ( ( ord_le3102999989581377725nteger @ C @ B )
% 14.25/8.07 => ( ord_le6747313008572928689nteger @ C @ A2 ) ) ) ).
% 14.25/8.07
% 14.25/8.07 thf(fact_3890_dual__order_Ostrict__trans2,axiom,
% 14.25/8.07 ! [B: filter_nat,A2: filter_nat,C: filter_nat] :
% 14.25/8.07 ( ( ord_less_filter_nat @ B @ A2 )
% 14.25/8.07 => ( ( ord_le2510731241096832064er_nat @ C @ B )
% 14.25/8.07 => ( ord_less_filter_nat @ C @ A2 ) ) ) ).
% 14.25/8.07
% 14.25/8.07 thf(fact_3891_dual__order_Ostrict__trans2,axiom,
% 14.25/8.07 ! [B: set_nat,A2: set_nat,C: set_nat] :
% 14.25/8.07 ( ( ord_less_set_nat @ B @ A2 )
% 14.25/8.07 => ( ( ord_less_eq_set_nat @ C @ B )
% 14.25/8.07 => ( ord_less_set_nat @ C @ A2 ) ) ) ).
% 14.25/8.07
% 14.25/8.07 thf(fact_3892_dual__order_Ostrict__trans2,axiom,
% 14.25/8.07 ! [B: rat,A2: rat,C: rat] :
% 14.25/8.07 ( ( ord_less_rat @ B @ A2 )
% 14.25/8.07 => ( ( ord_less_eq_rat @ C @ B )
% 14.25/8.07 => ( ord_less_rat @ C @ A2 ) ) ) ).
% 14.25/8.07
% 14.25/8.07 thf(fact_3893_dual__order_Ostrict__trans2,axiom,
% 14.25/8.07 ! [B: nat,A2: nat,C: nat] :
% 14.25/8.07 ( ( ord_less_nat @ B @ A2 )
% 14.25/8.07 => ( ( ord_less_eq_nat @ C @ B )
% 14.25/8.07 => ( ord_less_nat @ C @ A2 ) ) ) ).
% 14.25/8.07
% 14.25/8.07 thf(fact_3894_dual__order_Ostrict__trans2,axiom,
% 14.25/8.07 ! [B: int,A2: int,C: int] :
% 14.25/8.07 ( ( ord_less_int @ B @ A2 )
% 14.25/8.07 => ( ( ord_less_eq_int @ C @ B )
% 14.25/8.07 => ( ord_less_int @ C @ A2 ) ) ) ).
% 14.25/8.07
% 14.25/8.07 thf(fact_3895_dual__order_Ostrict__iff__not,axiom,
% 14.25/8.07 ( ord_less_assn
% 14.25/8.07 = ( ^ [B6: assn,A6: assn] :
% 14.25/8.07 ( ( ord_less_eq_assn @ B6 @ A6 )
% 14.25/8.07 & ~ ( ord_less_eq_assn @ A6 @ B6 ) ) ) ) ).
% 14.25/8.07
% 14.25/8.07 thf(fact_3896_dual__order_Ostrict__iff__not,axiom,
% 14.25/8.07 ( ord_le6747313008572928689nteger
% 14.25/8.07 = ( ^ [B6: code_integer,A6: code_integer] :
% 14.25/8.07 ( ( ord_le3102999989581377725nteger @ B6 @ A6 )
% 14.25/8.07 & ~ ( ord_le3102999989581377725nteger @ A6 @ B6 ) ) ) ) ).
% 14.25/8.07
% 14.25/8.07 thf(fact_3897_dual__order_Ostrict__iff__not,axiom,
% 14.25/8.07 ( ord_less_filter_nat
% 14.25/8.07 = ( ^ [B6: filter_nat,A6: filter_nat] :
% 14.25/8.07 ( ( ord_le2510731241096832064er_nat @ B6 @ A6 )
% 14.25/8.07 & ~ ( ord_le2510731241096832064er_nat @ A6 @ B6 ) ) ) ) ).
% 14.25/8.07
% 14.25/8.07 thf(fact_3898_dual__order_Ostrict__iff__not,axiom,
% 14.25/8.07 ( ord_less_set_nat
% 14.25/8.07 = ( ^ [B6: set_nat,A6: set_nat] :
% 14.25/8.07 ( ( ord_less_eq_set_nat @ B6 @ A6 )
% 14.25/8.07 & ~ ( ord_less_eq_set_nat @ A6 @ B6 ) ) ) ) ).
% 14.25/8.07
% 14.25/8.07 thf(fact_3899_dual__order_Ostrict__iff__not,axiom,
% 14.25/8.07 ( ord_less_rat
% 14.25/8.07 = ( ^ [B6: rat,A6: rat] :
% 14.25/8.07 ( ( ord_less_eq_rat @ B6 @ A6 )
% 14.25/8.07 & ~ ( ord_less_eq_rat @ A6 @ B6 ) ) ) ) ).
% 14.25/8.07
% 14.25/8.07 thf(fact_3900_dual__order_Ostrict__iff__not,axiom,
% 14.25/8.07 ( ord_less_nat
% 14.25/8.07 = ( ^ [B6: nat,A6: nat] :
% 14.25/8.07 ( ( ord_less_eq_nat @ B6 @ A6 )
% 14.25/8.07 & ~ ( ord_less_eq_nat @ A6 @ B6 ) ) ) ) ).
% 14.25/8.07
% 14.25/8.07 thf(fact_3901_dual__order_Ostrict__iff__not,axiom,
% 14.25/8.07 ( ord_less_int
% 14.25/8.07 = ( ^ [B6: int,A6: int] :
% 14.25/8.07 ( ( ord_less_eq_int @ B6 @ A6 )
% 14.25/8.07 & ~ ( ord_less_eq_int @ A6 @ B6 ) ) ) ) ).
% 14.25/8.07
% 14.25/8.07 thf(fact_3902_order_Ostrict__implies__order,axiom,
% 14.25/8.07 ! [A2: assn,B: assn] :
% 14.25/8.07 ( ( ord_less_assn @ A2 @ B )
% 14.25/8.07 => ( ord_less_eq_assn @ A2 @ B ) ) ).
% 14.25/8.07
% 14.25/8.07 thf(fact_3903_order_Ostrict__implies__order,axiom,
% 14.25/8.07 ! [A2: code_integer,B: code_integer] :
% 14.25/8.07 ( ( ord_le6747313008572928689nteger @ A2 @ B )
% 14.25/8.07 => ( ord_le3102999989581377725nteger @ A2 @ B ) ) ).
% 14.25/8.07
% 14.25/8.07 thf(fact_3904_order_Ostrict__implies__order,axiom,
% 14.25/8.07 ! [A2: filter_nat,B: filter_nat] :
% 14.25/8.07 ( ( ord_less_filter_nat @ A2 @ B )
% 14.25/8.07 => ( ord_le2510731241096832064er_nat @ A2 @ B ) ) ).
% 14.25/8.07
% 14.25/8.07 thf(fact_3905_order_Ostrict__implies__order,axiom,
% 14.25/8.07 ! [A2: set_nat,B: set_nat] :
% 14.25/8.07 ( ( ord_less_set_nat @ A2 @ B )
% 14.25/8.07 => ( ord_less_eq_set_nat @ A2 @ B ) ) ).
% 14.25/8.07
% 14.25/8.07 thf(fact_3906_order_Ostrict__implies__order,axiom,
% 14.25/8.07 ! [A2: rat,B: rat] :
% 14.25/8.07 ( ( ord_less_rat @ A2 @ B )
% 14.25/8.07 => ( ord_less_eq_rat @ A2 @ B ) ) ).
% 14.25/8.07
% 14.25/8.07 thf(fact_3907_order_Ostrict__implies__order,axiom,
% 14.25/8.07 ! [A2: nat,B: nat] :
% 14.25/8.07 ( ( ord_less_nat @ A2 @ B )
% 14.25/8.07 => ( ord_less_eq_nat @ A2 @ B ) ) ).
% 14.25/8.07
% 14.25/8.07 thf(fact_3908_order_Ostrict__implies__order,axiom,
% 14.25/8.07 ! [A2: int,B: int] :
% 14.25/8.07 ( ( ord_less_int @ A2 @ B )
% 14.25/8.07 => ( ord_less_eq_int @ A2 @ B ) ) ).
% 14.25/8.07
% 14.25/8.07 thf(fact_3909_dual__order_Ostrict__implies__order,axiom,
% 14.25/8.07 ! [B: assn,A2: assn] :
% 14.25/8.07 ( ( ord_less_assn @ B @ A2 )
% 14.25/8.07 => ( ord_less_eq_assn @ B @ A2 ) ) ).
% 14.25/8.07
% 14.25/8.07 thf(fact_3910_dual__order_Ostrict__implies__order,axiom,
% 14.25/8.07 ! [B: code_integer,A2: code_integer] :
% 14.25/8.07 ( ( ord_le6747313008572928689nteger @ B @ A2 )
% 14.25/8.07 => ( ord_le3102999989581377725nteger @ B @ A2 ) ) ).
% 14.25/8.07
% 14.25/8.07 thf(fact_3911_dual__order_Ostrict__implies__order,axiom,
% 14.25/8.07 ! [B: filter_nat,A2: filter_nat] :
% 14.25/8.07 ( ( ord_less_filter_nat @ B @ A2 )
% 14.25/8.07 => ( ord_le2510731241096832064er_nat @ B @ A2 ) ) ).
% 14.25/8.07
% 14.25/8.07 thf(fact_3912_dual__order_Ostrict__implies__order,axiom,
% 14.25/8.07 ! [B: set_nat,A2: set_nat] :
% 14.25/8.07 ( ( ord_less_set_nat @ B @ A2 )
% 14.25/8.07 => ( ord_less_eq_set_nat @ B @ A2 ) ) ).
% 14.25/8.07
% 14.25/8.07 thf(fact_3913_dual__order_Ostrict__implies__order,axiom,
% 14.25/8.07 ! [B: rat,A2: rat] :
% 14.25/8.07 ( ( ord_less_rat @ B @ A2 )
% 14.25/8.07 => ( ord_less_eq_rat @ B @ A2 ) ) ).
% 14.25/8.07
% 14.25/8.07 thf(fact_3914_dual__order_Ostrict__implies__order,axiom,
% 14.25/8.07 ! [B: nat,A2: nat] :
% 14.25/8.07 ( ( ord_less_nat @ B @ A2 )
% 14.25/8.07 => ( ord_less_eq_nat @ B @ A2 ) ) ).
% 14.25/8.07
% 14.25/8.07 thf(fact_3915_dual__order_Ostrict__implies__order,axiom,
% 14.25/8.07 ! [B: int,A2: int] :
% 14.25/8.07 ( ( ord_less_int @ B @ A2 )
% 14.25/8.07 => ( ord_less_eq_int @ B @ A2 ) ) ).
% 14.25/8.07
% 14.25/8.07 thf(fact_3916_order__le__less,axiom,
% 14.25/8.07 ( ord_less_eq_assn
% 14.25/8.07 = ( ^ [X2: assn,Y3: assn] :
% 14.25/8.07 ( ( ord_less_assn @ X2 @ Y3 )
% 14.25/8.07 | ( X2 = Y3 ) ) ) ) ).
% 14.25/8.07
% 14.25/8.07 thf(fact_3917_order__le__less,axiom,
% 14.25/8.07 ( ord_le3102999989581377725nteger
% 14.25/8.07 = ( ^ [X2: code_integer,Y3: code_integer] :
% 14.25/8.07 ( ( ord_le6747313008572928689nteger @ X2 @ Y3 )
% 14.25/8.07 | ( X2 = Y3 ) ) ) ) ).
% 14.25/8.07
% 14.25/8.07 thf(fact_3918_order__le__less,axiom,
% 14.25/8.07 ( ord_le2510731241096832064er_nat
% 14.25/8.07 = ( ^ [X2: filter_nat,Y3: filter_nat] :
% 14.25/8.07 ( ( ord_less_filter_nat @ X2 @ Y3 )
% 14.25/8.07 | ( X2 = Y3 ) ) ) ) ).
% 14.25/8.07
% 14.25/8.07 thf(fact_3919_order__le__less,axiom,
% 14.25/8.07 ( ord_less_eq_set_nat
% 14.25/8.07 = ( ^ [X2: set_nat,Y3: set_nat] :
% 14.25/8.07 ( ( ord_less_set_nat @ X2 @ Y3 )
% 14.25/8.07 | ( X2 = Y3 ) ) ) ) ).
% 14.25/8.07
% 14.25/8.07 thf(fact_3920_order__le__less,axiom,
% 14.25/8.07 ( ord_less_eq_rat
% 14.25/8.07 = ( ^ [X2: rat,Y3: rat] :
% 14.25/8.07 ( ( ord_less_rat @ X2 @ Y3 )
% 14.25/8.07 | ( X2 = Y3 ) ) ) ) ).
% 14.25/8.07
% 14.25/8.07 thf(fact_3921_order__le__less,axiom,
% 14.25/8.07 ( ord_less_eq_nat
% 14.25/8.07 = ( ^ [X2: nat,Y3: nat] :
% 14.25/8.07 ( ( ord_less_nat @ X2 @ Y3 )
% 14.25/8.07 | ( X2 = Y3 ) ) ) ) ).
% 14.25/8.07
% 14.25/8.07 thf(fact_3922_order__le__less,axiom,
% 14.25/8.07 ( ord_less_eq_int
% 14.25/8.07 = ( ^ [X2: int,Y3: int] :
% 14.25/8.07 ( ( ord_less_int @ X2 @ Y3 )
% 14.25/8.07 | ( X2 = Y3 ) ) ) ) ).
% 14.25/8.07
% 14.25/8.07 thf(fact_3923_order__less__le,axiom,
% 14.25/8.07 ( ord_less_assn
% 14.25/8.07 = ( ^ [X2: assn,Y3: assn] :
% 14.25/8.07 ( ( ord_less_eq_assn @ X2 @ Y3 )
% 14.25/8.07 & ( X2 != Y3 ) ) ) ) ).
% 14.25/8.07
% 14.25/8.07 thf(fact_3924_order__less__le,axiom,
% 14.25/8.07 ( ord_le6747313008572928689nteger
% 14.25/8.07 = ( ^ [X2: code_integer,Y3: code_integer] :
% 14.25/8.07 ( ( ord_le3102999989581377725nteger @ X2 @ Y3 )
% 14.25/8.07 & ( X2 != Y3 ) ) ) ) ).
% 14.25/8.07
% 14.25/8.07 thf(fact_3925_order__less__le,axiom,
% 14.25/8.07 ( ord_less_filter_nat
% 14.25/8.07 = ( ^ [X2: filter_nat,Y3: filter_nat] :
% 14.25/8.07 ( ( ord_le2510731241096832064er_nat @ X2 @ Y3 )
% 14.25/8.07 & ( X2 != Y3 ) ) ) ) ).
% 14.25/8.07
% 14.25/8.07 thf(fact_3926_order__less__le,axiom,
% 14.25/8.07 ( ord_less_set_nat
% 14.25/8.07 = ( ^ [X2: set_nat,Y3: set_nat] :
% 14.25/8.07 ( ( ord_less_eq_set_nat @ X2 @ Y3 )
% 14.25/8.07 & ( X2 != Y3 ) ) ) ) ).
% 14.25/8.07
% 14.25/8.07 thf(fact_3927_order__less__le,axiom,
% 14.25/8.07 ( ord_less_rat
% 14.25/8.07 = ( ^ [X2: rat,Y3: rat] :
% 14.25/8.07 ( ( ord_less_eq_rat @ X2 @ Y3 )
% 14.25/8.07 & ( X2 != Y3 ) ) ) ) ).
% 14.25/8.07
% 14.25/8.07 thf(fact_3928_order__less__le,axiom,
% 14.25/8.07 ( ord_less_nat
% 14.25/8.07 = ( ^ [X2: nat,Y3: nat] :
% 14.25/8.07 ( ( ord_less_eq_nat @ X2 @ Y3 )
% 14.25/8.07 & ( X2 != Y3 ) ) ) ) ).
% 14.25/8.07
% 14.25/8.07 thf(fact_3929_order__less__le,axiom,
% 14.25/8.07 ( ord_less_int
% 14.25/8.07 = ( ^ [X2: int,Y3: int] :
% 14.25/8.07 ( ( ord_less_eq_int @ X2 @ Y3 )
% 14.25/8.07 & ( X2 != Y3 ) ) ) ) ).
% 14.25/8.07
% 14.25/8.07 thf(fact_3930_linorder__not__le,axiom,
% 14.25/8.07 ! [X: code_integer,Y: code_integer] :
% 14.25/8.07 ( ( ~ ( ord_le3102999989581377725nteger @ X @ Y ) )
% 14.25/8.07 = ( ord_le6747313008572928689nteger @ Y @ X ) ) ).
% 14.25/8.07
% 14.25/8.07 thf(fact_3931_linorder__not__le,axiom,
% 14.25/8.07 ! [X: rat,Y: rat] :
% 14.25/8.07 ( ( ~ ( ord_less_eq_rat @ X @ Y ) )
% 14.25/8.07 = ( ord_less_rat @ Y @ X ) ) ).
% 14.25/8.07
% 14.25/8.07 thf(fact_3932_linorder__not__le,axiom,
% 14.25/8.07 ! [X: nat,Y: nat] :
% 14.25/8.07 ( ( ~ ( ord_less_eq_nat @ X @ Y ) )
% 14.25/8.07 = ( ord_less_nat @ Y @ X ) ) ).
% 14.25/8.07
% 14.25/8.07 thf(fact_3933_linorder__not__le,axiom,
% 14.25/8.07 ! [X: int,Y: int] :
% 14.25/8.07 ( ( ~ ( ord_less_eq_int @ X @ Y ) )
% 14.25/8.07 = ( ord_less_int @ Y @ X ) ) ).
% 14.25/8.07
% 14.25/8.07 thf(fact_3934_linorder__not__less,axiom,
% 14.25/8.07 ! [X: code_integer,Y: code_integer] :
% 14.25/8.07 ( ( ~ ( ord_le6747313008572928689nteger @ X @ Y ) )
% 14.25/8.07 = ( ord_le3102999989581377725nteger @ Y @ X ) ) ).
% 14.25/8.07
% 14.25/8.07 thf(fact_3935_linorder__not__less,axiom,
% 14.25/8.07 ! [X: rat,Y: rat] :
% 14.25/8.07 ( ( ~ ( ord_less_rat @ X @ Y ) )
% 14.25/8.07 = ( ord_less_eq_rat @ Y @ X ) ) ).
% 14.25/8.07
% 14.25/8.07 thf(fact_3936_linorder__not__less,axiom,
% 14.25/8.07 ! [X: nat,Y: nat] :
% 14.25/8.07 ( ( ~ ( ord_less_nat @ X @ Y ) )
% 14.25/8.07 = ( ord_less_eq_nat @ Y @ X ) ) ).
% 14.25/8.07
% 14.25/8.07 thf(fact_3937_linorder__not__less,axiom,
% 14.25/8.07 ! [X: int,Y: int] :
% 14.25/8.07 ( ( ~ ( ord_less_int @ X @ Y ) )
% 14.25/8.07 = ( ord_less_eq_int @ Y @ X ) ) ).
% 14.25/8.07
% 14.25/8.07 thf(fact_3938_order__less__imp__le,axiom,
% 14.25/8.07 ! [X: assn,Y: assn] :
% 14.25/8.07 ( ( ord_less_assn @ X @ Y )
% 14.25/8.07 => ( ord_less_eq_assn @ X @ Y ) ) ).
% 14.25/8.07
% 14.25/8.07 thf(fact_3939_order__less__imp__le,axiom,
% 14.25/8.07 ! [X: code_integer,Y: code_integer] :
% 14.25/8.07 ( ( ord_le6747313008572928689nteger @ X @ Y )
% 14.25/8.07 => ( ord_le3102999989581377725nteger @ X @ Y ) ) ).
% 14.25/8.07
% 14.25/8.07 thf(fact_3940_order__less__imp__le,axiom,
% 14.25/8.07 ! [X: filter_nat,Y: filter_nat] :
% 14.25/8.07 ( ( ord_less_filter_nat @ X @ Y )
% 14.25/8.07 => ( ord_le2510731241096832064er_nat @ X @ Y ) ) ).
% 14.25/8.07
% 14.25/8.07 thf(fact_3941_order__less__imp__le,axiom,
% 14.25/8.07 ! [X: set_nat,Y: set_nat] :
% 14.25/8.07 ( ( ord_less_set_nat @ X @ Y )
% 14.25/8.07 => ( ord_less_eq_set_nat @ X @ Y ) ) ).
% 14.25/8.07
% 14.25/8.07 thf(fact_3942_order__less__imp__le,axiom,
% 14.25/8.07 ! [X: rat,Y: rat] :
% 14.25/8.07 ( ( ord_less_rat @ X @ Y )
% 14.25/8.07 => ( ord_less_eq_rat @ X @ Y ) ) ).
% 14.25/8.07
% 14.25/8.07 thf(fact_3943_order__less__imp__le,axiom,
% 14.25/8.07 ! [X: nat,Y: nat] :
% 14.25/8.07 ( ( ord_less_nat @ X @ Y )
% 14.25/8.07 => ( ord_less_eq_nat @ X @ Y ) ) ).
% 14.25/8.07
% 14.25/8.07 thf(fact_3944_order__less__imp__le,axiom,
% 14.25/8.07 ! [X: int,Y: int] :
% 14.25/8.07 ( ( ord_less_int @ X @ Y )
% 14.25/8.07 => ( ord_less_eq_int @ X @ Y ) ) ).
% 14.25/8.07
% 14.25/8.07 thf(fact_3945_order__le__neq__trans,axiom,
% 14.25/8.07 ! [A2: assn,B: assn] :
% 14.25/8.07 ( ( ord_less_eq_assn @ A2 @ B )
% 14.25/8.07 => ( ( A2 != B )
% 14.25/8.07 => ( ord_less_assn @ A2 @ B ) ) ) ).
% 14.25/8.07
% 14.25/8.07 thf(fact_3946_order__le__neq__trans,axiom,
% 14.25/8.07 ! [A2: code_integer,B: code_integer] :
% 14.25/8.07 ( ( ord_le3102999989581377725nteger @ A2 @ B )
% 14.25/8.07 => ( ( A2 != B )
% 14.25/8.07 => ( ord_le6747313008572928689nteger @ A2 @ B ) ) ) ).
% 14.25/8.07
% 14.25/8.07 thf(fact_3947_order__le__neq__trans,axiom,
% 14.25/8.07 ! [A2: filter_nat,B: filter_nat] :
% 14.25/8.07 ( ( ord_le2510731241096832064er_nat @ A2 @ B )
% 14.25/8.07 => ( ( A2 != B )
% 14.25/8.07 => ( ord_less_filter_nat @ A2 @ B ) ) ) ).
% 14.25/8.07
% 14.25/8.07 thf(fact_3948_order__le__neq__trans,axiom,
% 14.25/8.07 ! [A2: set_nat,B: set_nat] :
% 14.25/8.07 ( ( ord_less_eq_set_nat @ A2 @ B )
% 14.25/8.07 => ( ( A2 != B )
% 14.25/8.07 => ( ord_less_set_nat @ A2 @ B ) ) ) ).
% 14.25/8.07
% 14.25/8.07 thf(fact_3949_order__le__neq__trans,axiom,
% 14.25/8.07 ! [A2: rat,B: rat] :
% 14.25/8.07 ( ( ord_less_eq_rat @ A2 @ B )
% 14.25/8.07 => ( ( A2 != B )
% 14.25/8.07 => ( ord_less_rat @ A2 @ B ) ) ) ).
% 14.25/8.07
% 14.25/8.07 thf(fact_3950_order__le__neq__trans,axiom,
% 14.25/8.07 ! [A2: nat,B: nat] :
% 14.25/8.07 ( ( ord_less_eq_nat @ A2 @ B )
% 14.25/8.07 => ( ( A2 != B )
% 14.25/8.07 => ( ord_less_nat @ A2 @ B ) ) ) ).
% 14.25/8.07
% 14.25/8.07 thf(fact_3951_order__le__neq__trans,axiom,
% 14.25/8.07 ! [A2: int,B: int] :
% 14.25/8.07 ( ( ord_less_eq_int @ A2 @ B )
% 14.25/8.07 => ( ( A2 != B )
% 14.25/8.07 => ( ord_less_int @ A2 @ B ) ) ) ).
% 14.25/8.07
% 14.25/8.07 thf(fact_3952_order__neq__le__trans,axiom,
% 14.25/8.07 ! [A2: assn,B: assn] :
% 14.25/8.07 ( ( A2 != B )
% 14.25/8.07 => ( ( ord_less_eq_assn @ A2 @ B )
% 14.25/8.07 => ( ord_less_assn @ A2 @ B ) ) ) ).
% 14.25/8.07
% 14.25/8.07 thf(fact_3953_order__neq__le__trans,axiom,
% 14.25/8.07 ! [A2: code_integer,B: code_integer] :
% 14.25/8.07 ( ( A2 != B )
% 14.25/8.07 => ( ( ord_le3102999989581377725nteger @ A2 @ B )
% 14.25/8.07 => ( ord_le6747313008572928689nteger @ A2 @ B ) ) ) ).
% 14.25/8.07
% 14.25/8.07 thf(fact_3954_order__neq__le__trans,axiom,
% 14.25/8.07 ! [A2: filter_nat,B: filter_nat] :
% 14.25/8.07 ( ( A2 != B )
% 14.25/8.07 => ( ( ord_le2510731241096832064er_nat @ A2 @ B )
% 14.25/8.07 => ( ord_less_filter_nat @ A2 @ B ) ) ) ).
% 14.25/8.07
% 14.25/8.07 thf(fact_3955_order__neq__le__trans,axiom,
% 14.25/8.07 ! [A2: set_nat,B: set_nat] :
% 14.25/8.07 ( ( A2 != B )
% 14.25/8.07 => ( ( ord_less_eq_set_nat @ A2 @ B )
% 14.25/8.07 => ( ord_less_set_nat @ A2 @ B ) ) ) ).
% 14.25/8.07
% 14.25/8.07 thf(fact_3956_order__neq__le__trans,axiom,
% 14.25/8.07 ! [A2: rat,B: rat] :
% 14.25/8.07 ( ( A2 != B )
% 14.25/8.07 => ( ( ord_less_eq_rat @ A2 @ B )
% 14.25/8.07 => ( ord_less_rat @ A2 @ B ) ) ) ).
% 14.25/8.07
% 14.25/8.07 thf(fact_3957_order__neq__le__trans,axiom,
% 14.25/8.07 ! [A2: nat,B: nat] :
% 14.25/8.07 ( ( A2 != B )
% 14.25/8.07 => ( ( ord_less_eq_nat @ A2 @ B )
% 14.25/8.07 => ( ord_less_nat @ A2 @ B ) ) ) ).
% 14.25/8.07
% 14.25/8.07 thf(fact_3958_order__neq__le__trans,axiom,
% 14.25/8.07 ! [A2: int,B: int] :
% 14.25/8.07 ( ( A2 != B )
% 14.25/8.07 => ( ( ord_less_eq_int @ A2 @ B )
% 14.25/8.07 => ( ord_less_int @ A2 @ B ) ) ) ).
% 14.25/8.07
% 14.25/8.07 thf(fact_3959_order__le__less__trans,axiom,
% 14.25/8.07 ! [X: assn,Y: assn,Z: assn] :
% 14.25/8.07 ( ( ord_less_eq_assn @ X @ Y )
% 14.25/8.07 => ( ( ord_less_assn @ Y @ Z )
% 14.25/8.07 => ( ord_less_assn @ X @ Z ) ) ) ).
% 14.25/8.07
% 14.25/8.07 thf(fact_3960_order__le__less__trans,axiom,
% 14.25/8.07 ! [X: code_integer,Y: code_integer,Z: code_integer] :
% 14.25/8.07 ( ( ord_le3102999989581377725nteger @ X @ Y )
% 14.25/8.07 => ( ( ord_le6747313008572928689nteger @ Y @ Z )
% 14.25/8.07 => ( ord_le6747313008572928689nteger @ X @ Z ) ) ) ).
% 14.25/8.07
% 14.25/8.07 thf(fact_3961_order__le__less__trans,axiom,
% 14.25/8.07 ! [X: filter_nat,Y: filter_nat,Z: filter_nat] :
% 14.25/8.07 ( ( ord_le2510731241096832064er_nat @ X @ Y )
% 14.25/8.07 => ( ( ord_less_filter_nat @ Y @ Z )
% 14.25/8.07 => ( ord_less_filter_nat @ X @ Z ) ) ) ).
% 14.25/8.07
% 14.25/8.07 thf(fact_3962_order__le__less__trans,axiom,
% 14.25/8.07 ! [X: set_nat,Y: set_nat,Z: set_nat] :
% 14.25/8.07 ( ( ord_less_eq_set_nat @ X @ Y )
% 14.25/8.07 => ( ( ord_less_set_nat @ Y @ Z )
% 14.25/8.07 => ( ord_less_set_nat @ X @ Z ) ) ) ).
% 14.25/8.07
% 14.25/8.07 thf(fact_3963_order__le__less__trans,axiom,
% 14.25/8.07 ! [X: rat,Y: rat,Z: rat] :
% 14.25/8.07 ( ( ord_less_eq_rat @ X @ Y )
% 14.25/8.07 => ( ( ord_less_rat @ Y @ Z )
% 14.25/8.07 => ( ord_less_rat @ X @ Z ) ) ) ).
% 14.25/8.07
% 14.25/8.07 thf(fact_3964_order__le__less__trans,axiom,
% 14.25/8.07 ! [X: nat,Y: nat,Z: nat] :
% 14.25/8.07 ( ( ord_less_eq_nat @ X @ Y )
% 14.25/8.07 => ( ( ord_less_nat @ Y @ Z )
% 14.25/8.07 => ( ord_less_nat @ X @ Z ) ) ) ).
% 14.25/8.07
% 14.25/8.07 thf(fact_3965_order__le__less__trans,axiom,
% 14.25/8.07 ! [X: int,Y: int,Z: int] :
% 14.25/8.07 ( ( ord_less_eq_int @ X @ Y )
% 14.25/8.07 => ( ( ord_less_int @ Y @ Z )
% 14.25/8.07 => ( ord_less_int @ X @ Z ) ) ) ).
% 14.25/8.07
% 14.25/8.07 thf(fact_3966_order__less__le__trans,axiom,
% 14.25/8.07 ! [X: assn,Y: assn,Z: assn] :
% 14.25/8.07 ( ( ord_less_assn @ X @ Y )
% 14.25/8.07 => ( ( ord_less_eq_assn @ Y @ Z )
% 14.25/8.07 => ( ord_less_assn @ X @ Z ) ) ) ).
% 14.25/8.07
% 14.25/8.07 thf(fact_3967_order__less__le__trans,axiom,
% 14.25/8.07 ! [X: code_integer,Y: code_integer,Z: code_integer] :
% 14.25/8.07 ( ( ord_le6747313008572928689nteger @ X @ Y )
% 14.25/8.07 => ( ( ord_le3102999989581377725nteger @ Y @ Z )
% 14.25/8.07 => ( ord_le6747313008572928689nteger @ X @ Z ) ) ) ).
% 14.25/8.07
% 14.25/8.07 thf(fact_3968_order__less__le__trans,axiom,
% 14.25/8.07 ! [X: filter_nat,Y: filter_nat,Z: filter_nat] :
% 14.25/8.07 ( ( ord_less_filter_nat @ X @ Y )
% 14.25/8.07 => ( ( ord_le2510731241096832064er_nat @ Y @ Z )
% 14.25/8.07 => ( ord_less_filter_nat @ X @ Z ) ) ) ).
% 14.25/8.07
% 14.25/8.07 thf(fact_3969_order__less__le__trans,axiom,
% 14.25/8.07 ! [X: set_nat,Y: set_nat,Z: set_nat] :
% 14.25/8.07 ( ( ord_less_set_nat @ X @ Y )
% 14.25/8.07 => ( ( ord_less_eq_set_nat @ Y @ Z )
% 14.25/8.07 => ( ord_less_set_nat @ X @ Z ) ) ) ).
% 14.25/8.07
% 14.25/8.07 thf(fact_3970_order__less__le__trans,axiom,
% 14.25/8.07 ! [X: rat,Y: rat,Z: rat] :
% 14.25/8.07 ( ( ord_less_rat @ X @ Y )
% 14.25/8.07 => ( ( ord_less_eq_rat @ Y @ Z )
% 14.25/8.07 => ( ord_less_rat @ X @ Z ) ) ) ).
% 14.25/8.07
% 14.25/8.07 thf(fact_3971_order__less__le__trans,axiom,
% 14.25/8.07 ! [X: nat,Y: nat,Z: nat] :
% 14.25/8.07 ( ( ord_less_nat @ X @ Y )
% 14.25/8.07 => ( ( ord_less_eq_nat @ Y @ Z )
% 14.25/8.07 => ( ord_less_nat @ X @ Z ) ) ) ).
% 14.25/8.07
% 14.25/8.07 thf(fact_3972_order__less__le__trans,axiom,
% 14.25/8.07 ! [X: int,Y: int,Z: int] :
% 14.25/8.07 ( ( ord_less_int @ X @ Y )
% 14.25/8.07 => ( ( ord_less_eq_int @ Y @ Z )
% 14.25/8.07 => ( ord_less_int @ X @ Z ) ) ) ).
% 14.25/8.07
% 14.25/8.07 thf(fact_3973_order__le__less__subst1,axiom,
% 14.25/8.07 ! [A2: assn,F: assn > assn,B: assn,C: assn] :
% 14.25/8.07 ( ( ord_less_eq_assn @ A2 @ ( F @ B ) )
% 14.25/8.07 => ( ( ord_less_assn @ B @ C )
% 14.25/8.07 => ( ! [X3: assn,Y2: assn] :
% 14.25/8.07 ( ( ord_less_assn @ X3 @ Y2 )
% 14.25/8.07 => ( ord_less_assn @ ( F @ X3 ) @ ( F @ Y2 ) ) )
% 14.25/8.07 => ( ord_less_assn @ A2 @ ( F @ C ) ) ) ) ) ).
% 14.25/8.07
% 14.25/8.07 thf(fact_3974_order__le__less__subst1,axiom,
% 14.25/8.07 ! [A2: code_integer,F: assn > code_integer,B: assn,C: assn] :
% 14.25/8.07 ( ( ord_le3102999989581377725nteger @ A2 @ ( F @ B ) )
% 14.25/8.07 => ( ( ord_less_assn @ B @ C )
% 14.25/8.07 => ( ! [X3: assn,Y2: assn] :
% 14.25/8.07 ( ( ord_less_assn @ X3 @ Y2 )
% 14.25/8.07 => ( ord_le6747313008572928689nteger @ ( F @ X3 ) @ ( F @ Y2 ) ) )
% 14.25/8.07 => ( ord_le6747313008572928689nteger @ A2 @ ( F @ C ) ) ) ) ) ).
% 14.25/8.07
% 14.25/8.07 thf(fact_3975_order__le__less__subst1,axiom,
% 14.25/8.07 ! [A2: assn,F: nat > assn,B: nat,C: nat] :
% 14.25/8.07 ( ( ord_less_eq_assn @ A2 @ ( F @ B ) )
% 14.25/8.07 => ( ( ord_less_nat @ B @ C )
% 14.25/8.07 => ( ! [X3: nat,Y2: nat] :
% 14.25/8.07 ( ( ord_less_nat @ X3 @ Y2 )
% 14.25/8.07 => ( ord_less_assn @ ( F @ X3 ) @ ( F @ Y2 ) ) )
% 14.25/8.07 => ( ord_less_assn @ A2 @ ( F @ C ) ) ) ) ) ).
% 14.25/8.07
% 14.25/8.07 thf(fact_3976_order__le__less__subst1,axiom,
% 14.25/8.07 ! [A2: code_integer,F: nat > code_integer,B: nat,C: nat] :
% 14.25/8.07 ( ( ord_le3102999989581377725nteger @ A2 @ ( F @ B ) )
% 14.25/8.07 => ( ( ord_less_nat @ B @ C )
% 14.25/8.07 => ( ! [X3: nat,Y2: nat] :
% 14.25/8.07 ( ( ord_less_nat @ X3 @ Y2 )
% 14.25/8.07 => ( ord_le6747313008572928689nteger @ ( F @ X3 ) @ ( F @ Y2 ) ) )
% 14.25/8.07 => ( ord_le6747313008572928689nteger @ A2 @ ( F @ C ) ) ) ) ) ).
% 14.25/8.07
% 14.25/8.07 thf(fact_3977_order__le__less__subst1,axiom,
% 14.25/8.07 ! [A2: assn,F: int > assn,B: int,C: int] :
% 14.25/8.07 ( ( ord_less_eq_assn @ A2 @ ( F @ B ) )
% 14.25/8.08 => ( ( ord_less_int @ B @ C )
% 14.25/8.08 => ( ! [X3: int,Y2: int] :
% 14.25/8.08 ( ( ord_less_int @ X3 @ Y2 )
% 14.25/8.08 => ( ord_less_assn @ ( F @ X3 ) @ ( F @ Y2 ) ) )
% 14.25/8.08 => ( ord_less_assn @ A2 @ ( F @ C ) ) ) ) ) ).
% 14.25/8.08
% 14.25/8.08 thf(fact_3978_order__le__less__subst1,axiom,
% 14.25/8.08 ! [A2: code_integer,F: int > code_integer,B: int,C: int] :
% 14.25/8.08 ( ( ord_le3102999989581377725nteger @ A2 @ ( F @ B ) )
% 14.25/8.08 => ( ( ord_less_int @ B @ C )
% 14.25/8.08 => ( ! [X3: int,Y2: int] :
% 14.25/8.08 ( ( ord_less_int @ X3 @ Y2 )
% 14.25/8.08 => ( ord_le6747313008572928689nteger @ ( F @ X3 ) @ ( F @ Y2 ) ) )
% 14.25/8.08 => ( ord_le6747313008572928689nteger @ A2 @ ( F @ C ) ) ) ) ) ).
% 14.25/8.08
% 14.25/8.08 thf(fact_3979_order__le__less__subst1,axiom,
% 14.25/8.08 ! [A2: assn,F: code_integer > assn,B: code_integer,C: code_integer] :
% 14.25/8.08 ( ( ord_less_eq_assn @ A2 @ ( F @ B ) )
% 14.25/8.08 => ( ( ord_le6747313008572928689nteger @ B @ C )
% 14.25/8.08 => ( ! [X3: code_integer,Y2: code_integer] :
% 14.25/8.08 ( ( ord_le6747313008572928689nteger @ X3 @ Y2 )
% 14.25/8.08 => ( ord_less_assn @ ( F @ X3 ) @ ( F @ Y2 ) ) )
% 14.25/8.08 => ( ord_less_assn @ A2 @ ( F @ C ) ) ) ) ) ).
% 14.25/8.08
% 14.25/8.08 thf(fact_3980_order__le__less__subst1,axiom,
% 14.25/8.08 ! [A2: code_integer,F: code_integer > code_integer,B: code_integer,C: code_integer] :
% 14.25/8.08 ( ( ord_le3102999989581377725nteger @ A2 @ ( F @ B ) )
% 14.25/8.08 => ( ( ord_le6747313008572928689nteger @ B @ C )
% 14.25/8.08 => ( ! [X3: code_integer,Y2: code_integer] :
% 14.25/8.08 ( ( ord_le6747313008572928689nteger @ X3 @ Y2 )
% 14.25/8.08 => ( ord_le6747313008572928689nteger @ ( F @ X3 ) @ ( F @ Y2 ) ) )
% 14.25/8.08 => ( ord_le6747313008572928689nteger @ A2 @ ( F @ C ) ) ) ) ) ).
% 14.25/8.08
% 14.25/8.08 thf(fact_3981_order__le__less__subst1,axiom,
% 14.25/8.08 ! [A2: assn,F: rat > assn,B: rat,C: rat] :
% 14.25/8.08 ( ( ord_less_eq_assn @ A2 @ ( F @ B ) )
% 14.25/8.08 => ( ( ord_less_rat @ B @ C )
% 14.25/8.08 => ( ! [X3: rat,Y2: rat] :
% 14.25/8.08 ( ( ord_less_rat @ X3 @ Y2 )
% 14.25/8.08 => ( ord_less_assn @ ( F @ X3 ) @ ( F @ Y2 ) ) )
% 14.25/8.08 => ( ord_less_assn @ A2 @ ( F @ C ) ) ) ) ) ).
% 14.25/8.08
% 14.25/8.08 thf(fact_3982_order__le__less__subst1,axiom,
% 14.25/8.08 ! [A2: code_integer,F: rat > code_integer,B: rat,C: rat] :
% 14.25/8.08 ( ( ord_le3102999989581377725nteger @ A2 @ ( F @ B ) )
% 14.25/8.08 => ( ( ord_less_rat @ B @ C )
% 14.25/8.08 => ( ! [X3: rat,Y2: rat] :
% 14.25/8.08 ( ( ord_less_rat @ X3 @ Y2 )
% 14.25/8.08 => ( ord_le6747313008572928689nteger @ ( F @ X3 ) @ ( F @ Y2 ) ) )
% 14.25/8.08 => ( ord_le6747313008572928689nteger @ A2 @ ( F @ C ) ) ) ) ) ).
% 14.25/8.08
% 14.25/8.08 thf(fact_3983_order__le__less__subst2,axiom,
% 14.25/8.08 ! [A2: rat,B: rat,F: rat > assn,C: assn] :
% 14.25/8.08 ( ( ord_less_eq_rat @ A2 @ B )
% 14.25/8.08 => ( ( ord_less_assn @ ( F @ B ) @ C )
% 14.25/8.08 => ( ! [X3: rat,Y2: rat] :
% 14.25/8.08 ( ( ord_less_eq_rat @ X3 @ Y2 )
% 14.25/8.08 => ( ord_less_eq_assn @ ( F @ X3 ) @ ( F @ Y2 ) ) )
% 14.25/8.08 => ( ord_less_assn @ ( F @ A2 ) @ C ) ) ) ) ).
% 14.25/8.08
% 14.25/8.08 thf(fact_3984_order__le__less__subst2,axiom,
% 14.25/8.08 ! [A2: rat,B: rat,F: rat > code_integer,C: code_integer] :
% 14.25/8.08 ( ( ord_less_eq_rat @ A2 @ B )
% 14.25/8.08 => ( ( ord_le6747313008572928689nteger @ ( F @ B ) @ C )
% 14.25/8.08 => ( ! [X3: rat,Y2: rat] :
% 14.25/8.08 ( ( ord_less_eq_rat @ X3 @ Y2 )
% 14.25/8.08 => ( ord_le3102999989581377725nteger @ ( F @ X3 ) @ ( F @ Y2 ) ) )
% 14.25/8.08 => ( ord_le6747313008572928689nteger @ ( F @ A2 ) @ C ) ) ) ) ).
% 14.25/8.08
% 14.25/8.08 thf(fact_3985_order__le__less__subst2,axiom,
% 14.25/8.08 ! [A2: rat,B: rat,F: rat > rat,C: rat] :
% 14.25/8.08 ( ( ord_less_eq_rat @ A2 @ B )
% 14.25/8.08 => ( ( ord_less_rat @ ( F @ B ) @ C )
% 14.25/8.08 => ( ! [X3: rat,Y2: rat] :
% 14.25/8.08 ( ( ord_less_eq_rat @ X3 @ Y2 )
% 14.25/8.08 => ( ord_less_eq_rat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
% 14.25/8.08 => ( ord_less_rat @ ( F @ A2 ) @ C ) ) ) ) ).
% 14.25/8.08
% 14.25/8.08 thf(fact_3986_order__le__less__subst2,axiom,
% 14.25/8.08 ! [A2: rat,B: rat,F: rat > nat,C: nat] :
% 14.25/8.08 ( ( ord_less_eq_rat @ A2 @ B )
% 14.25/8.08 => ( ( ord_less_nat @ ( F @ B ) @ C )
% 14.25/8.08 => ( ! [X3: rat,Y2: rat] :
% 14.25/8.08 ( ( ord_less_eq_rat @ X3 @ Y2 )
% 14.25/8.08 => ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
% 14.25/8.08 => ( ord_less_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% 14.25/8.08
% 14.25/8.08 thf(fact_3987_order__le__less__subst2,axiom,
% 14.25/8.08 ! [A2: rat,B: rat,F: rat > int,C: int] :
% 14.25/8.08 ( ( ord_less_eq_rat @ A2 @ B )
% 14.25/8.08 => ( ( ord_less_int @ ( F @ B ) @ C )
% 14.25/8.08 => ( ! [X3: rat,Y2: rat] :
% 14.25/8.08 ( ( ord_less_eq_rat @ X3 @ Y2 )
% 14.25/8.08 => ( ord_less_eq_int @ ( F @ X3 ) @ ( F @ Y2 ) ) )
% 14.25/8.08 => ( ord_less_int @ ( F @ A2 ) @ C ) ) ) ) ).
% 14.25/8.08
% 14.25/8.08 thf(fact_3988_order__le__less__subst2,axiom,
% 14.25/8.08 ! [A2: nat,B: nat,F: nat > assn,C: assn] :
% 14.25/8.08 ( ( ord_less_eq_nat @ A2 @ B )
% 14.25/8.08 => ( ( ord_less_assn @ ( F @ B ) @ C )
% 14.25/8.08 => ( ! [X3: nat,Y2: nat] :
% 14.25/8.08 ( ( ord_less_eq_nat @ X3 @ Y2 )
% 14.25/8.08 => ( ord_less_eq_assn @ ( F @ X3 ) @ ( F @ Y2 ) ) )
% 14.25/8.08 => ( ord_less_assn @ ( F @ A2 ) @ C ) ) ) ) ).
% 14.25/8.08
% 14.25/8.08 thf(fact_3989_order__le__less__subst2,axiom,
% 14.25/8.08 ! [A2: nat,B: nat,F: nat > code_integer,C: code_integer] :
% 14.25/8.08 ( ( ord_less_eq_nat @ A2 @ B )
% 14.25/8.08 => ( ( ord_le6747313008572928689nteger @ ( F @ B ) @ C )
% 14.25/8.08 => ( ! [X3: nat,Y2: nat] :
% 14.25/8.08 ( ( ord_less_eq_nat @ X3 @ Y2 )
% 14.25/8.08 => ( ord_le3102999989581377725nteger @ ( F @ X3 ) @ ( F @ Y2 ) ) )
% 14.25/8.08 => ( ord_le6747313008572928689nteger @ ( F @ A2 ) @ C ) ) ) ) ).
% 14.25/8.08
% 14.25/8.08 thf(fact_3990_order__le__less__subst2,axiom,
% 14.25/8.08 ! [A2: nat,B: nat,F: nat > rat,C: rat] :
% 14.25/8.08 ( ( ord_less_eq_nat @ A2 @ B )
% 14.25/8.08 => ( ( ord_less_rat @ ( F @ B ) @ C )
% 14.25/8.08 => ( ! [X3: nat,Y2: nat] :
% 14.25/8.08 ( ( ord_less_eq_nat @ X3 @ Y2 )
% 14.25/8.08 => ( ord_less_eq_rat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
% 14.25/8.08 => ( ord_less_rat @ ( F @ A2 ) @ C ) ) ) ) ).
% 14.25/8.08
% 14.25/8.08 thf(fact_3991_order__le__less__subst2,axiom,
% 14.25/8.08 ! [A2: nat,B: nat,F: nat > nat,C: nat] :
% 14.25/8.08 ( ( ord_less_eq_nat @ A2 @ B )
% 14.25/8.08 => ( ( ord_less_nat @ ( F @ B ) @ C )
% 14.25/8.08 => ( ! [X3: nat,Y2: nat] :
% 14.25/8.08 ( ( ord_less_eq_nat @ X3 @ Y2 )
% 14.25/8.08 => ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
% 14.25/8.08 => ( ord_less_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% 14.25/8.08
% 14.25/8.08 thf(fact_3992_order__le__less__subst2,axiom,
% 14.25/8.08 ! [A2: nat,B: nat,F: nat > int,C: int] :
% 14.25/8.08 ( ( ord_less_eq_nat @ A2 @ B )
% 14.25/8.08 => ( ( ord_less_int @ ( F @ B ) @ C )
% 14.25/8.08 => ( ! [X3: nat,Y2: nat] :
% 14.25/8.08 ( ( ord_less_eq_nat @ X3 @ Y2 )
% 14.25/8.08 => ( ord_less_eq_int @ ( F @ X3 ) @ ( F @ Y2 ) ) )
% 14.25/8.08 => ( ord_less_int @ ( F @ A2 ) @ C ) ) ) ) ).
% 14.25/8.08
% 14.25/8.08 thf(fact_3993_order__less__le__subst1,axiom,
% 14.25/8.08 ! [A2: assn,F: rat > assn,B: rat,C: rat] :
% 14.25/8.08 ( ( ord_less_assn @ A2 @ ( F @ B ) )
% 14.25/8.08 => ( ( ord_less_eq_rat @ B @ C )
% 14.25/8.08 => ( ! [X3: rat,Y2: rat] :
% 14.25/8.08 ( ( ord_less_eq_rat @ X3 @ Y2 )
% 14.25/8.08 => ( ord_less_eq_assn @ ( F @ X3 ) @ ( F @ Y2 ) ) )
% 14.25/8.08 => ( ord_less_assn @ A2 @ ( F @ C ) ) ) ) ) ).
% 14.25/8.08
% 14.25/8.08 thf(fact_3994_order__less__le__subst1,axiom,
% 14.25/8.08 ! [A2: code_integer,F: rat > code_integer,B: rat,C: rat] :
% 14.25/8.08 ( ( ord_le6747313008572928689nteger @ A2 @ ( F @ B ) )
% 14.25/8.08 => ( ( ord_less_eq_rat @ B @ C )
% 14.25/8.08 => ( ! [X3: rat,Y2: rat] :
% 14.25/8.08 ( ( ord_less_eq_rat @ X3 @ Y2 )
% 14.25/8.08 => ( ord_le3102999989581377725nteger @ ( F @ X3 ) @ ( F @ Y2 ) ) )
% 14.25/8.08 => ( ord_le6747313008572928689nteger @ A2 @ ( F @ C ) ) ) ) ) ).
% 14.25/8.08
% 14.25/8.08 thf(fact_3995_order__less__le__subst1,axiom,
% 14.25/8.08 ! [A2: rat,F: rat > rat,B: rat,C: rat] :
% 14.25/8.08 ( ( ord_less_rat @ A2 @ ( F @ B ) )
% 14.25/8.08 => ( ( ord_less_eq_rat @ B @ C )
% 14.25/8.08 => ( ! [X3: rat,Y2: rat] :
% 14.25/8.08 ( ( ord_less_eq_rat @ X3 @ Y2 )
% 14.25/8.08 => ( ord_less_eq_rat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
% 14.25/8.08 => ( ord_less_rat @ A2 @ ( F @ C ) ) ) ) ) ).
% 14.25/8.08
% 14.25/8.08 thf(fact_3996_order__less__le__subst1,axiom,
% 14.25/8.08 ! [A2: nat,F: rat > nat,B: rat,C: rat] :
% 14.25/8.08 ( ( ord_less_nat @ A2 @ ( F @ B ) )
% 14.25/8.08 => ( ( ord_less_eq_rat @ B @ C )
% 14.25/8.08 => ( ! [X3: rat,Y2: rat] :
% 14.25/8.08 ( ( ord_less_eq_rat @ X3 @ Y2 )
% 14.25/8.08 => ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
% 14.25/8.08 => ( ord_less_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% 14.25/8.08
% 14.25/8.08 thf(fact_3997_order__less__le__subst1,axiom,
% 14.25/8.08 ! [A2: int,F: rat > int,B: rat,C: rat] :
% 14.25/8.08 ( ( ord_less_int @ A2 @ ( F @ B ) )
% 14.25/8.08 => ( ( ord_less_eq_rat @ B @ C )
% 14.25/8.08 => ( ! [X3: rat,Y2: rat] :
% 14.25/8.08 ( ( ord_less_eq_rat @ X3 @ Y2 )
% 14.25/8.08 => ( ord_less_eq_int @ ( F @ X3 ) @ ( F @ Y2 ) ) )
% 14.25/8.08 => ( ord_less_int @ A2 @ ( F @ C ) ) ) ) ) ).
% 14.25/8.08
% 14.25/8.08 thf(fact_3998_order__less__le__subst1,axiom,
% 14.25/8.08 ! [A2: assn,F: nat > assn,B: nat,C: nat] :
% 14.25/8.08 ( ( ord_less_assn @ A2 @ ( F @ B ) )
% 14.25/8.08 => ( ( ord_less_eq_nat @ B @ C )
% 14.25/8.08 => ( ! [X3: nat,Y2: nat] :
% 14.25/8.08 ( ( ord_less_eq_nat @ X3 @ Y2 )
% 14.25/8.08 => ( ord_less_eq_assn @ ( F @ X3 ) @ ( F @ Y2 ) ) )
% 14.25/8.08 => ( ord_less_assn @ A2 @ ( F @ C ) ) ) ) ) ).
% 14.25/8.08
% 14.25/8.08 thf(fact_3999_order__less__le__subst1,axiom,
% 14.25/8.08 ! [A2: code_integer,F: nat > code_integer,B: nat,C: nat] :
% 14.25/8.08 ( ( ord_le6747313008572928689nteger @ A2 @ ( F @ B ) )
% 14.25/8.08 => ( ( ord_less_eq_nat @ B @ C )
% 14.25/8.08 => ( ! [X3: nat,Y2: nat] :
% 14.25/8.08 ( ( ord_less_eq_nat @ X3 @ Y2 )
% 14.25/8.08 => ( ord_le3102999989581377725nteger @ ( F @ X3 ) @ ( F @ Y2 ) ) )
% 14.25/8.08 => ( ord_le6747313008572928689nteger @ A2 @ ( F @ C ) ) ) ) ) ).
% 14.25/8.08
% 14.25/8.08 thf(fact_4000_order__less__le__subst1,axiom,
% 14.25/8.08 ! [A2: rat,F: nat > rat,B: nat,C: nat] :
% 14.25/8.08 ( ( ord_less_rat @ A2 @ ( F @ B ) )
% 14.25/8.08 => ( ( ord_less_eq_nat @ B @ C )
% 14.25/8.08 => ( ! [X3: nat,Y2: nat] :
% 14.25/8.08 ( ( ord_less_eq_nat @ X3 @ Y2 )
% 14.25/8.08 => ( ord_less_eq_rat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
% 14.25/8.08 => ( ord_less_rat @ A2 @ ( F @ C ) ) ) ) ) ).
% 14.25/8.08
% 14.25/8.08 thf(fact_4001_order__less__le__subst1,axiom,
% 14.25/8.08 ! [A2: nat,F: nat > nat,B: nat,C: nat] :
% 14.25/8.08 ( ( ord_less_nat @ A2 @ ( F @ B ) )
% 14.25/8.08 => ( ( ord_less_eq_nat @ B @ C )
% 14.25/8.08 => ( ! [X3: nat,Y2: nat] :
% 14.25/8.08 ( ( ord_less_eq_nat @ X3 @ Y2 )
% 14.25/8.08 => ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
% 14.25/8.08 => ( ord_less_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% 14.25/8.08
% 14.25/8.08 thf(fact_4002_order__less__le__subst1,axiom,
% 14.25/8.08 ! [A2: int,F: nat > int,B: nat,C: nat] :
% 14.25/8.08 ( ( ord_less_int @ A2 @ ( F @ B ) )
% 14.25/8.08 => ( ( ord_less_eq_nat @ B @ C )
% 14.25/8.08 => ( ! [X3: nat,Y2: nat] :
% 14.25/8.08 ( ( ord_less_eq_nat @ X3 @ Y2 )
% 14.25/8.08 => ( ord_less_eq_int @ ( F @ X3 ) @ ( F @ Y2 ) ) )
% 14.25/8.08 => ( ord_less_int @ A2 @ ( F @ C ) ) ) ) ) ).
% 14.25/8.08
% 14.25/8.08 thf(fact_4003_order__less__le__subst2,axiom,
% 14.25/8.08 ! [A2: assn,B: assn,F: assn > assn,C: assn] :
% 14.25/8.08 ( ( ord_less_assn @ A2 @ B )
% 14.25/8.08 => ( ( ord_less_eq_assn @ ( F @ B ) @ C )
% 14.25/8.08 => ( ! [X3: assn,Y2: assn] :
% 14.25/8.08 ( ( ord_less_assn @ X3 @ Y2 )
% 14.25/8.08 => ( ord_less_assn @ ( F @ X3 ) @ ( F @ Y2 ) ) )
% 14.25/8.08 => ( ord_less_assn @ ( F @ A2 ) @ C ) ) ) ) ).
% 14.25/8.08
% 14.25/8.08 thf(fact_4004_order__less__le__subst2,axiom,
% 14.25/8.08 ! [A2: assn,B: assn,F: assn > code_integer,C: code_integer] :
% 14.25/8.08 ( ( ord_less_assn @ A2 @ B )
% 14.25/8.08 => ( ( ord_le3102999989581377725nteger @ ( F @ B ) @ C )
% 14.25/8.08 => ( ! [X3: assn,Y2: assn] :
% 14.25/8.08 ( ( ord_less_assn @ X3 @ Y2 )
% 14.25/8.08 => ( ord_le6747313008572928689nteger @ ( F @ X3 ) @ ( F @ Y2 ) ) )
% 14.25/8.08 => ( ord_le6747313008572928689nteger @ ( F @ A2 ) @ C ) ) ) ) ).
% 14.25/8.08
% 14.25/8.08 thf(fact_4005_order__less__le__subst2,axiom,
% 14.25/8.08 ! [A2: nat,B: nat,F: nat > assn,C: assn] :
% 14.25/8.08 ( ( ord_less_nat @ A2 @ B )
% 14.25/8.08 => ( ( ord_less_eq_assn @ ( F @ B ) @ C )
% 14.25/8.08 => ( ! [X3: nat,Y2: nat] :
% 14.25/8.08 ( ( ord_less_nat @ X3 @ Y2 )
% 14.25/8.08 => ( ord_less_assn @ ( F @ X3 ) @ ( F @ Y2 ) ) )
% 14.25/8.08 => ( ord_less_assn @ ( F @ A2 ) @ C ) ) ) ) ).
% 14.25/8.08
% 14.25/8.08 thf(fact_4006_order__less__le__subst2,axiom,
% 14.25/8.08 ! [A2: nat,B: nat,F: nat > code_integer,C: code_integer] :
% 14.25/8.08 ( ( ord_less_nat @ A2 @ B )
% 14.25/8.08 => ( ( ord_le3102999989581377725nteger @ ( F @ B ) @ C )
% 14.25/8.08 => ( ! [X3: nat,Y2: nat] :
% 14.25/8.08 ( ( ord_less_nat @ X3 @ Y2 )
% 14.25/8.08 => ( ord_le6747313008572928689nteger @ ( F @ X3 ) @ ( F @ Y2 ) ) )
% 14.25/8.08 => ( ord_le6747313008572928689nteger @ ( F @ A2 ) @ C ) ) ) ) ).
% 14.25/8.08
% 14.25/8.08 thf(fact_4007_order__less__le__subst2,axiom,
% 14.25/8.08 ! [A2: int,B: int,F: int > assn,C: assn] :
% 14.25/8.08 ( ( ord_less_int @ A2 @ B )
% 14.25/8.08 => ( ( ord_less_eq_assn @ ( F @ B ) @ C )
% 14.25/8.08 => ( ! [X3: int,Y2: int] :
% 14.25/8.08 ( ( ord_less_int @ X3 @ Y2 )
% 14.25/8.08 => ( ord_less_assn @ ( F @ X3 ) @ ( F @ Y2 ) ) )
% 14.25/8.08 => ( ord_less_assn @ ( F @ A2 ) @ C ) ) ) ) ).
% 14.25/8.08
% 14.25/8.08 thf(fact_4008_order__less__le__subst2,axiom,
% 14.25/8.08 ! [A2: int,B: int,F: int > code_integer,C: code_integer] :
% 14.25/8.08 ( ( ord_less_int @ A2 @ B )
% 14.25/8.08 => ( ( ord_le3102999989581377725nteger @ ( F @ B ) @ C )
% 14.25/8.08 => ( ! [X3: int,Y2: int] :
% 14.25/8.08 ( ( ord_less_int @ X3 @ Y2 )
% 14.25/8.08 => ( ord_le6747313008572928689nteger @ ( F @ X3 ) @ ( F @ Y2 ) ) )
% 14.25/8.08 => ( ord_le6747313008572928689nteger @ ( F @ A2 ) @ C ) ) ) ) ).
% 14.25/8.08
% 14.25/8.08 thf(fact_4009_order__less__le__subst2,axiom,
% 14.25/8.08 ! [A2: code_integer,B: code_integer,F: code_integer > assn,C: assn] :
% 14.25/8.08 ( ( ord_le6747313008572928689nteger @ A2 @ B )
% 14.25/8.08 => ( ( ord_less_eq_assn @ ( F @ B ) @ C )
% 14.25/8.08 => ( ! [X3: code_integer,Y2: code_integer] :
% 14.25/8.08 ( ( ord_le6747313008572928689nteger @ X3 @ Y2 )
% 14.25/8.08 => ( ord_less_assn @ ( F @ X3 ) @ ( F @ Y2 ) ) )
% 14.25/8.08 => ( ord_less_assn @ ( F @ A2 ) @ C ) ) ) ) ).
% 14.25/8.08
% 14.25/8.08 thf(fact_4010_order__less__le__subst2,axiom,
% 14.25/8.08 ! [A2: code_integer,B: code_integer,F: code_integer > code_integer,C: code_integer] :
% 14.25/8.08 ( ( ord_le6747313008572928689nteger @ A2 @ B )
% 14.25/8.08 => ( ( ord_le3102999989581377725nteger @ ( F @ B ) @ C )
% 14.25/8.08 => ( ! [X3: code_integer,Y2: code_integer] :
% 14.25/8.08 ( ( ord_le6747313008572928689nteger @ X3 @ Y2 )
% 14.25/8.08 => ( ord_le6747313008572928689nteger @ ( F @ X3 ) @ ( F @ Y2 ) ) )
% 14.25/8.08 => ( ord_le6747313008572928689nteger @ ( F @ A2 ) @ C ) ) ) ) ).
% 14.25/8.08
% 14.25/8.08 thf(fact_4011_order__less__le__subst2,axiom,
% 14.25/8.08 ! [A2: rat,B: rat,F: rat > assn,C: assn] :
% 14.25/8.08 ( ( ord_less_rat @ A2 @ B )
% 14.25/8.08 => ( ( ord_less_eq_assn @ ( F @ B ) @ C )
% 14.25/8.08 => ( ! [X3: rat,Y2: rat] :
% 14.25/8.08 ( ( ord_less_rat @ X3 @ Y2 )
% 14.25/8.08 => ( ord_less_assn @ ( F @ X3 ) @ ( F @ Y2 ) ) )
% 14.25/8.08 => ( ord_less_assn @ ( F @ A2 ) @ C ) ) ) ) ).
% 14.25/8.08
% 14.25/8.08 thf(fact_4012_order__less__le__subst2,axiom,
% 14.25/8.08 ! [A2: rat,B: rat,F: rat > code_integer,C: code_integer] :
% 14.25/8.08 ( ( ord_less_rat @ A2 @ B )
% 14.25/8.08 => ( ( ord_le3102999989581377725nteger @ ( F @ B ) @ C )
% 14.25/8.08 => ( ! [X3: rat,Y2: rat] :
% 14.25/8.08 ( ( ord_less_rat @ X3 @ Y2 )
% 14.25/8.08 => ( ord_le6747313008572928689nteger @ ( F @ X3 ) @ ( F @ Y2 ) ) )
% 14.25/8.08 => ( ord_le6747313008572928689nteger @ ( F @ A2 ) @ C ) ) ) ) ).
% 14.25/8.08
% 14.25/8.08 thf(fact_4013_linorder__le__less__linear,axiom,
% 14.25/8.08 ! [X: code_integer,Y: code_integer] :
% 14.25/8.08 ( ( ord_le3102999989581377725nteger @ X @ Y )
% 14.25/8.08 | ( ord_le6747313008572928689nteger @ Y @ X ) ) ).
% 14.25/8.08
% 14.25/8.08 thf(fact_4014_linorder__le__less__linear,axiom,
% 14.25/8.08 ! [X: rat,Y: rat] :
% 14.25/8.08 ( ( ord_less_eq_rat @ X @ Y )
% 14.25/8.08 | ( ord_less_rat @ Y @ X ) ) ).
% 14.25/8.08
% 14.25/8.08 thf(fact_4015_linorder__le__less__linear,axiom,
% 14.25/8.08 ! [X: nat,Y: nat] :
% 14.25/8.08 ( ( ord_less_eq_nat @ X @ Y )
% 14.25/8.08 | ( ord_less_nat @ Y @ X ) ) ).
% 14.25/8.08
% 14.25/8.08 thf(fact_4016_linorder__le__less__linear,axiom,
% 14.25/8.08 ! [X: int,Y: int] :
% 14.25/8.08 ( ( ord_less_eq_int @ X @ Y )
% 14.25/8.08 | ( ord_less_int @ Y @ X ) ) ).
% 14.25/8.08
% 14.25/8.08 thf(fact_4017_order__le__imp__less__or__eq,axiom,
% 14.25/8.08 ! [X: assn,Y: assn] :
% 14.25/8.08 ( ( ord_less_eq_assn @ X @ Y )
% 14.25/8.08 => ( ( ord_less_assn @ X @ Y )
% 14.25/8.08 | ( X = Y ) ) ) ).
% 14.25/8.08
% 14.25/8.08 thf(fact_4018_order__le__imp__less__or__eq,axiom,
% 14.25/8.08 ! [X: code_integer,Y: code_integer] :
% 14.25/8.08 ( ( ord_le3102999989581377725nteger @ X @ Y )
% 14.25/8.08 => ( ( ord_le6747313008572928689nteger @ X @ Y )
% 14.25/8.08 | ( X = Y ) ) ) ).
% 14.25/8.08
% 14.25/8.08 thf(fact_4019_order__le__imp__less__or__eq,axiom,
% 14.25/8.08 ! [X: filter_nat,Y: filter_nat] :
% 14.25/8.08 ( ( ord_le2510731241096832064er_nat @ X @ Y )
% 14.25/8.08 => ( ( ord_less_filter_nat @ X @ Y )
% 14.25/8.08 | ( X = Y ) ) ) ).
% 14.25/8.08
% 14.25/8.08 thf(fact_4020_order__le__imp__less__or__eq,axiom,
% 14.25/8.08 ! [X: set_nat,Y: set_nat] :
% 14.25/8.08 ( ( ord_less_eq_set_nat @ X @ Y )
% 14.25/8.08 => ( ( ord_less_set_nat @ X @ Y )
% 14.25/8.08 | ( X = Y ) ) ) ).
% 14.25/8.08
% 14.25/8.08 thf(fact_4021_order__le__imp__less__or__eq,axiom,
% 14.25/8.08 ! [X: rat,Y: rat] :
% 14.25/8.08 ( ( ord_less_eq_rat @ X @ Y )
% 14.25/8.08 => ( ( ord_less_rat @ X @ Y )
% 14.25/8.08 | ( X = Y ) ) ) ).
% 14.25/8.08
% 14.25/8.08 thf(fact_4022_order__le__imp__less__or__eq,axiom,
% 14.25/8.08 ! [X: nat,Y: nat] :
% 14.25/8.08 ( ( ord_less_eq_nat @ X @ Y )
% 14.25/8.08 => ( ( ord_less_nat @ X @ Y )
% 14.25/8.08 | ( X = Y ) ) ) ).
% 14.25/8.08
% 14.25/8.08 thf(fact_4023_order__le__imp__less__or__eq,axiom,
% 14.25/8.08 ! [X: int,Y: int] :
% 14.25/8.08 ( ( ord_less_eq_int @ X @ Y )
% 14.25/8.08 => ( ( ord_less_int @ X @ Y )
% 14.25/8.08 | ( X = Y ) ) ) ).
% 14.25/8.08
% 14.25/8.08 thf(fact_4024_less__numeral__extra_I4_J,axiom,
% 14.25/8.08 ~ ( ord_le5570908160329646204atural @ one_one_Code_natural @ one_one_Code_natural ) ).
% 14.25/8.08
% 14.25/8.08 thf(fact_4025_less__numeral__extra_I4_J,axiom,
% 14.25/8.08 ~ ( ord_less_nat @ one_one_nat @ one_one_nat ) ).
% 14.25/8.08
% 14.25/8.08 thf(fact_4026_less__numeral__extra_I4_J,axiom,
% 14.25/8.08 ~ ( ord_less_int @ one_one_int @ one_one_int ) ).
% 14.25/8.08
% 14.25/8.08 thf(fact_4027_less__numeral__extra_I4_J,axiom,
% 14.25/8.08 ~ ( ord_le6747313008572928689nteger @ one_one_Code_integer @ one_one_Code_integer ) ).
% 14.25/8.08
% 14.25/8.08 thf(fact_4028_less__numeral__extra_I4_J,axiom,
% 14.25/8.08 ~ ( ord_less_rat @ one_one_rat @ one_one_rat ) ).
% 14.25/8.08
% 14.25/8.08 thf(fact_4029_diff__strict__mono,axiom,
% 14.25/8.08 ! [A2: int,B: int,D2: int,C: int] :
% 14.25/8.08 ( ( ord_less_int @ A2 @ B )
% 14.25/8.08 => ( ( ord_less_int @ D2 @ C )
% 14.25/8.08 => ( ord_less_int @ ( minus_minus_int @ A2 @ C ) @ ( minus_minus_int @ B @ D2 ) ) ) ) ).
% 14.25/8.08
% 14.25/8.08 thf(fact_4030_diff__strict__mono,axiom,
% 14.25/8.08 ! [A2: code_integer,B: code_integer,D2: code_integer,C: code_integer] :
% 14.25/8.08 ( ( ord_le6747313008572928689nteger @ A2 @ B )
% 14.25/8.08 => ( ( ord_le6747313008572928689nteger @ D2 @ C )
% 14.25/8.08 => ( ord_le6747313008572928689nteger @ ( minus_8373710615458151222nteger @ A2 @ C ) @ ( minus_8373710615458151222nteger @ B @ D2 ) ) ) ) ).
% 14.25/8.08
% 14.25/8.08 thf(fact_4031_diff__strict__mono,axiom,
% 14.25/8.08 ! [A2: rat,B: rat,D2: rat,C: rat] :
% 14.25/8.08 ( ( ord_less_rat @ A2 @ B )
% 14.25/8.08 => ( ( ord_less_rat @ D2 @ C )
% 14.25/8.08 => ( ord_less_rat @ ( minus_minus_rat @ A2 @ C ) @ ( minus_minus_rat @ B @ D2 ) ) ) ) ).
% 14.25/8.08
% 14.25/8.08 thf(fact_4032_diff__eq__diff__less,axiom,
% 14.25/8.08 ! [A2: int,B: int,C: int,D2: int] :
% 14.25/8.08 ( ( ( minus_minus_int @ A2 @ B )
% 14.25/8.08 = ( minus_minus_int @ C @ D2 ) )
% 14.25/8.08 => ( ( ord_less_int @ A2 @ B )
% 14.25/8.08 = ( ord_less_int @ C @ D2 ) ) ) ).
% 14.25/8.08
% 14.25/8.08 thf(fact_4033_diff__eq__diff__less,axiom,
% 14.25/8.08 ! [A2: code_integer,B: code_integer,C: code_integer,D2: code_integer] :
% 14.25/8.08 ( ( ( minus_8373710615458151222nteger @ A2 @ B )
% 14.25/8.08 = ( minus_8373710615458151222nteger @ C @ D2 ) )
% 14.25/8.08 => ( ( ord_le6747313008572928689nteger @ A2 @ B )
% 14.25/8.08 = ( ord_le6747313008572928689nteger @ C @ D2 ) ) ) ).
% 14.25/8.08
% 14.25/8.08 thf(fact_4034_diff__eq__diff__less,axiom,
% 14.25/8.08 ! [A2: rat,B: rat,C: rat,D2: rat] :
% 14.25/8.08 ( ( ( minus_minus_rat @ A2 @ B )
% 14.25/8.08 = ( minus_minus_rat @ C @ D2 ) )
% 14.25/8.08 => ( ( ord_less_rat @ A2 @ B )
% 14.25/8.08 = ( ord_less_rat @ C @ D2 ) ) ) ).
% 14.25/8.08
% 14.25/8.08 thf(fact_4035_diff__strict__left__mono,axiom,
% 14.25/8.08 ! [B: int,A2: int,C: int] :
% 14.25/8.08 ( ( ord_less_int @ B @ A2 )
% 14.25/8.08 => ( ord_less_int @ ( minus_minus_int @ C @ A2 ) @ ( minus_minus_int @ C @ B ) ) ) ).
% 14.25/8.08
% 14.25/8.08 thf(fact_4036_diff__strict__left__mono,axiom,
% 14.25/8.08 ! [B: code_integer,A2: code_integer,C: code_integer] :
% 14.25/8.08 ( ( ord_le6747313008572928689nteger @ B @ A2 )
% 14.25/8.08 => ( ord_le6747313008572928689nteger @ ( minus_8373710615458151222nteger @ C @ A2 ) @ ( minus_8373710615458151222nteger @ C @ B ) ) ) ).
% 14.25/8.08
% 14.25/8.08 thf(fact_4037_diff__strict__left__mono,axiom,
% 14.25/8.08 ! [B: rat,A2: rat,C: rat] :
% 14.25/8.08 ( ( ord_less_rat @ B @ A2 )
% 14.25/8.08 => ( ord_less_rat @ ( minus_minus_rat @ C @ A2 ) @ ( minus_minus_rat @ C @ B ) ) ) ).
% 14.25/8.08
% 14.25/8.08 thf(fact_4038_diff__strict__right__mono,axiom,
% 14.25/8.08 ! [A2: int,B: int,C: int] :
% 14.25/8.08 ( ( ord_less_int @ A2 @ B )
% 14.25/8.08 => ( ord_less_int @ ( minus_minus_int @ A2 @ C ) @ ( minus_minus_int @ B @ C ) ) ) ).
% 14.25/8.08
% 14.25/8.08 thf(fact_4039_diff__strict__right__mono,axiom,
% 14.25/8.08 ! [A2: code_integer,B: code_integer,C: code_integer] :
% 14.25/8.08 ( ( ord_le6747313008572928689nteger @ A2 @ B )
% 14.25/8.08 => ( ord_le6747313008572928689nteger @ ( minus_8373710615458151222nteger @ A2 @ C ) @ ( minus_8373710615458151222nteger @ B @ C ) ) ) ).
% 14.25/8.08
% 14.25/8.08 thf(fact_4040_diff__strict__right__mono,axiom,
% 14.25/8.08 ! [A2: rat,B: rat,C: rat] :
% 14.25/8.08 ( ( ord_less_rat @ A2 @ B )
% 14.25/8.08 => ( ord_less_rat @ ( minus_minus_rat @ A2 @ C ) @ ( minus_minus_rat @ B @ C ) ) ) ).
% 14.25/8.08
% 14.25/8.08 thf(fact_4041_bot_Oextremum__strict,axiom,
% 14.25/8.08 ! [A2: set_Product_unit] :
% 14.25/8.08 ~ ( ord_le8056459307392131481t_unit @ A2 @ bot_bo3957492148770167129t_unit ) ).
% 14.25/8.08
% 14.25/8.08 thf(fact_4042_bot_Oextremum__strict,axiom,
% 14.25/8.08 ! [A2: set_o] :
% 14.25/8.08 ~ ( ord_less_set_o @ A2 @ bot_bot_set_o ) ).
% 14.25/8.08
% 14.25/8.08 thf(fact_4043_bot_Oextremum__strict,axiom,
% 14.25/8.08 ! [A2: set_nat] :
% 14.25/8.08 ~ ( ord_less_set_nat @ A2 @ bot_bot_set_nat ) ).
% 14.25/8.08
% 14.25/8.08 thf(fact_4044_bot_Oextremum__strict,axiom,
% 14.25/8.08 ! [A2: set_int] :
% 14.25/8.08 ~ ( ord_less_set_int @ A2 @ bot_bot_set_int ) ).
% 14.25/8.08
% 14.25/8.08 thf(fact_4045_bot_Oextremum__strict,axiom,
% 14.25/8.08 ! [A2: set_Pr4532377907799695533_nat_o] :
% 14.25/8.08 ~ ( ord_le2453136405763929_nat_o @ A2 @ bot_bo7824918357723582233_nat_o ) ).
% 14.25/8.08
% 14.25/8.08 thf(fact_4046_bot_Oextremum__strict,axiom,
% 14.25/8.08 ! [A2: assn] :
% 14.25/8.08 ~ ( ord_less_assn @ A2 @ bot_bot_assn ) ).
% 14.25/8.08
% 14.25/8.08 thf(fact_4047_bot_Oextremum__strict,axiom,
% 14.25/8.08 ! [A2: nat] :
% 14.25/8.08 ~ ( ord_less_nat @ A2 @ bot_bot_nat ) ).
% 14.25/8.08
% 14.25/8.08 thf(fact_4048_bot_Onot__eq__extremum,axiom,
% 14.25/8.08 ! [A2: set_Product_unit] :
% 14.25/8.08 ( ( A2 != bot_bo3957492148770167129t_unit )
% 14.25/8.08 = ( ord_le8056459307392131481t_unit @ bot_bo3957492148770167129t_unit @ A2 ) ) ).
% 14.25/8.08
% 14.25/8.08 thf(fact_4049_bot_Onot__eq__extremum,axiom,
% 14.25/8.08 ! [A2: set_o] :
% 14.25/8.08 ( ( A2 != bot_bot_set_o )
% 14.25/8.08 = ( ord_less_set_o @ bot_bot_set_o @ A2 ) ) ).
% 14.25/8.08
% 14.25/8.08 thf(fact_4050_bot_Onot__eq__extremum,axiom,
% 14.25/8.08 ! [A2: set_nat] :
% 14.25/8.08 ( ( A2 != bot_bot_set_nat )
% 14.25/8.08 = ( ord_less_set_nat @ bot_bot_set_nat @ A2 ) ) ).
% 14.25/8.08
% 14.25/8.08 thf(fact_4051_bot_Onot__eq__extremum,axiom,
% 14.25/8.08 ! [A2: set_int] :
% 14.25/8.08 ( ( A2 != bot_bot_set_int )
% 14.25/8.08 = ( ord_less_set_int @ bot_bot_set_int @ A2 ) ) ).
% 14.25/8.08
% 14.25/8.08 thf(fact_4052_bot_Onot__eq__extremum,axiom,
% 14.25/8.08 ! [A2: set_Pr4532377907799695533_nat_o] :
% 14.25/8.08 ( ( A2 != bot_bo7824918357723582233_nat_o )
% 14.25/8.08 = ( ord_le2453136405763929_nat_o @ bot_bo7824918357723582233_nat_o @ A2 ) ) ).
% 14.25/8.08
% 14.25/8.08 thf(fact_4053_bot_Onot__eq__extremum,axiom,
% 14.25/8.08 ! [A2: assn] :
% 14.25/8.08 ( ( A2 != bot_bot_assn )
% 14.25/8.08 = ( ord_less_assn @ bot_bot_assn @ A2 ) ) ).
% 14.25/8.08
% 14.25/8.08 thf(fact_4054_bot_Onot__eq__extremum,axiom,
% 14.25/8.08 ! [A2: nat] :
% 14.25/8.08 ( ( A2 != bot_bot_nat )
% 14.25/8.08 = ( ord_less_nat @ bot_bot_nat @ A2 ) ) ).
% 14.25/8.08
% 14.25/8.08 thf(fact_4055_not__psubset__empty,axiom,
% 14.25/8.08 ! [A: set_Product_unit] :
% 14.25/8.08 ~ ( ord_le8056459307392131481t_unit @ A @ bot_bo3957492148770167129t_unit ) ).
% 14.25/8.08
% 14.25/8.08 thf(fact_4056_not__psubset__empty,axiom,
% 14.25/8.08 ! [A: set_o] :
% 14.25/8.08 ~ ( ord_less_set_o @ A @ bot_bot_set_o ) ).
% 14.25/8.08
% 14.25/8.08 thf(fact_4057_not__psubset__empty,axiom,
% 14.25/8.08 ! [A: set_nat] :
% 14.25/8.08 ~ ( ord_less_set_nat @ A @ bot_bot_set_nat ) ).
% 14.25/8.08
% 14.25/8.08 thf(fact_4058_not__psubset__empty,axiom,
% 14.25/8.08 ! [A: set_int] :
% 14.25/8.08 ~ ( ord_less_set_int @ A @ bot_bot_set_int ) ).
% 14.25/8.08
% 14.25/8.08 thf(fact_4059_not__psubset__empty,axiom,
% 14.25/8.08 ! [A: set_Pr4532377907799695533_nat_o] :
% 14.25/8.08 ~ ( ord_le2453136405763929_nat_o @ A @ bot_bo7824918357723582233_nat_o ) ).
% 14.25/8.08
% 14.25/8.08 thf(fact_4060_psubsetE,axiom,
% 14.25/8.08 ! [A: set_nat,B4: set_nat] :
% 14.25/8.08 ( ( ord_less_set_nat @ A @ B4 )
% 14.25/8.08 => ~ ( ( ord_less_eq_set_nat @ A @ B4 )
% 14.25/8.08 => ( ord_less_eq_set_nat @ B4 @ A ) ) ) ).
% 14.25/8.08
% 14.25/8.08 thf(fact_4061_psubset__eq,axiom,
% 14.25/8.08 ( ord_less_set_nat
% 14.25/8.08 = ( ^ [A3: set_nat,B7: set_nat] :
% 14.25/8.08 ( ( ord_less_eq_set_nat @ A3 @ B7 )
% 14.25/8.08 & ( A3 != B7 ) ) ) ) ).
% 14.25/8.08
% 14.25/8.08 thf(fact_4062_psubset__imp__subset,axiom,
% 14.25/8.08 ! [A: set_nat,B4: set_nat] :
% 14.25/8.08 ( ( ord_less_set_nat @ A @ B4 )
% 14.25/8.08 => ( ord_less_eq_set_nat @ A @ B4 ) ) ).
% 14.25/8.08
% 14.25/8.08 thf(fact_4063_psubset__subset__trans,axiom,
% 14.25/8.08 ! [A: set_nat,B4: set_nat,C5: set_nat] :
% 14.25/8.08 ( ( ord_less_set_nat @ A @ B4 )
% 14.25/8.08 => ( ( ord_less_eq_set_nat @ B4 @ C5 )
% 14.25/8.08 => ( ord_less_set_nat @ A @ C5 ) ) ) ).
% 14.25/8.08
% 14.25/8.08 thf(fact_4064_subset__not__subset__eq,axiom,
% 14.25/8.08 ( ord_less_set_nat
% 14.25/8.08 = ( ^ [A3: set_nat,B7: set_nat] :
% 14.25/8.08 ( ( ord_less_eq_set_nat @ A3 @ B7 )
% 14.25/8.08 & ~ ( ord_less_eq_set_nat @ B7 @ A3 ) ) ) ) ).
% 14.25/8.08
% 14.25/8.08 thf(fact_4065_subset__psubset__trans,axiom,
% 14.25/8.08 ! [A: set_nat,B4: set_nat,C5: set_nat] :
% 14.25/8.08 ( ( ord_less_eq_set_nat @ A @ B4 )
% 14.25/8.08 => ( ( ord_less_set_nat @ B4 @ C5 )
% 14.25/8.08 => ( ord_less_set_nat @ A @ C5 ) ) ) ).
% 14.25/8.08
% 14.25/8.08 thf(fact_4066_subset__iff__psubset__eq,axiom,
% 14.25/8.08 ( ord_less_eq_set_nat
% 14.25/8.08 = ( ^ [A3: set_nat,B7: set_nat] :
% 14.25/8.08 ( ( ord_less_set_nat @ A3 @ B7 )
% 14.25/8.08 | ( A3 = B7 ) ) ) ) ).
% 14.25/8.08
% 14.25/8.08 thf(fact_4067_rangeI,axiom,
% 14.25/8.08 ! [F: code_integer > code_integer,X: code_integer] : ( member_Code_integer @ ( F @ X ) @ ( image_4470545334726330049nteger @ F @ top_to4645266643341252675nteger ) ) ).
% 14.25/8.08
% 14.25/8.08 thf(fact_4068_rangeI,axiom,
% 14.25/8.08 ! [F: product_unit > nat,X: product_unit] : ( member_nat @ ( F @ X ) @ ( image_875570014554754200it_nat @ F @ top_to1996260823553986621t_unit ) ) ).
% 14.25/8.08
% 14.25/8.08 thf(fact_4069_rangeI,axiom,
% 14.25/8.08 ! [F: product_unit > int,X: product_unit] : ( member_int @ ( F @ X ) @ ( image_873079544045703924it_int @ F @ top_to1996260823553986621t_unit ) ) ).
% 14.25/8.08
% 14.25/8.08 thf(fact_4070_rangeI,axiom,
% 14.25/8.08 ! [F: product_unit > $o,X: product_unit] : ( member_o @ ( F @ X ) @ ( image_Product_unit_o2 @ F @ top_to1996260823553986621t_unit ) ) ).
% 14.25/8.08
% 14.25/8.08 thf(fact_4071_rangeI,axiom,
% 14.25/8.08 ! [F: product_unit > product_unit,X: product_unit] : ( member_Product_unit @ ( F @ X ) @ ( image_405062704495631173t_unit @ F @ top_to1996260823553986621t_unit ) ) ).
% 14.25/8.08
% 14.25/8.08 thf(fact_4072_rangeI,axiom,
% 14.25/8.08 ! [F: $o > nat,X: $o] : ( member_nat @ ( F @ X ) @ ( image_o_nat2 @ F @ top_top_set_o ) ) ).
% 14.25/8.08
% 14.25/8.08 thf(fact_4073_rangeI,axiom,
% 14.25/8.08 ! [F: $o > int,X: $o] : ( member_int @ ( F @ X ) @ ( image_o_int2 @ F @ top_top_set_o ) ) ).
% 14.25/8.08
% 14.25/8.08 thf(fact_4074_rangeI,axiom,
% 14.25/8.08 ! [F: $o > $o,X: $o] : ( member_o @ ( F @ X ) @ ( image_o_o2 @ F @ top_top_set_o ) ) ).
% 14.25/8.08
% 14.25/8.08 thf(fact_4075_rangeI,axiom,
% 14.25/8.08 ! [F: $o > product_unit,X: $o] : ( member_Product_unit @ ( F @ X ) @ ( image_o_Product_unit2 @ F @ top_top_set_o ) ) ).
% 14.25/8.08
% 14.25/8.08 thf(fact_4076_rangeI,axiom,
% 14.25/8.08 ! [F: nat > code_integer,X: nat] : ( member_Code_integer @ ( F @ X ) @ ( image_1215581382706833972nteger @ F @ top_top_set_nat ) ) ).
% 14.25/8.08
% 14.25/8.08 thf(fact_4077_range__eqI,axiom,
% 14.25/8.08 ! [B: code_integer,F: code_integer > code_integer,X: code_integer] :
% 14.25/8.08 ( ( B
% 14.25/8.08 = ( F @ X ) )
% 14.25/8.08 => ( member_Code_integer @ B @ ( image_4470545334726330049nteger @ F @ top_to4645266643341252675nteger ) ) ) ).
% 14.25/8.08
% 14.25/8.08 thf(fact_4078_range__eqI,axiom,
% 14.25/8.08 ! [B: nat,F: product_unit > nat,X: product_unit] :
% 14.25/8.08 ( ( B
% 14.25/8.08 = ( F @ X ) )
% 14.25/8.08 => ( member_nat @ B @ ( image_875570014554754200it_nat @ F @ top_to1996260823553986621t_unit ) ) ) ).
% 14.25/8.08
% 14.25/8.08 thf(fact_4079_range__eqI,axiom,
% 14.25/8.08 ! [B: int,F: product_unit > int,X: product_unit] :
% 14.25/8.08 ( ( B
% 14.25/8.08 = ( F @ X ) )
% 14.25/8.08 => ( member_int @ B @ ( image_873079544045703924it_int @ F @ top_to1996260823553986621t_unit ) ) ) ).
% 14.25/8.08
% 14.25/8.08 thf(fact_4080_range__eqI,axiom,
% 14.25/8.08 ! [B: $o,F: product_unit > $o,X: product_unit] :
% 14.25/8.08 ( ( B
% 14.25/8.08 = ( F @ X ) )
% 14.25/8.08 => ( member_o @ B @ ( image_Product_unit_o2 @ F @ top_to1996260823553986621t_unit ) ) ) ).
% 14.25/8.08
% 14.25/8.08 thf(fact_4081_range__eqI,axiom,
% 14.25/8.08 ! [B: product_unit,F: product_unit > product_unit,X: product_unit] :
% 14.25/8.08 ( ( B
% 14.25/8.08 = ( F @ X ) )
% 14.25/8.08 => ( member_Product_unit @ B @ ( image_405062704495631173t_unit @ F @ top_to1996260823553986621t_unit ) ) ) ).
% 14.25/8.08
% 14.25/8.08 thf(fact_4082_range__eqI,axiom,
% 14.25/8.08 ! [B: nat,F: $o > nat,X: $o] :
% 14.25/8.08 ( ( B
% 14.25/8.08 = ( F @ X ) )
% 14.25/8.08 => ( member_nat @ B @ ( image_o_nat2 @ F @ top_top_set_o ) ) ) ).
% 14.25/8.08
% 14.25/8.08 thf(fact_4083_range__eqI,axiom,
% 14.25/8.08 ! [B: int,F: $o > int,X: $o] :
% 14.25/8.08 ( ( B
% 14.25/8.08 = ( F @ X ) )
% 14.25/8.08 => ( member_int @ B @ ( image_o_int2 @ F @ top_top_set_o ) ) ) ).
% 14.25/8.08
% 14.25/8.08 thf(fact_4084_range__eqI,axiom,
% 14.25/8.08 ! [B: $o,F: $o > $o,X: $o] :
% 14.25/8.08 ( ( B
% 14.25/8.08 = ( F @ X ) )
% 14.25/8.08 => ( member_o @ B @ ( image_o_o2 @ F @ top_top_set_o ) ) ) ).
% 14.25/8.08
% 14.25/8.08 thf(fact_4085_range__eqI,axiom,
% 14.25/8.08 ! [B: product_unit,F: $o > product_unit,X: $o] :
% 14.25/8.08 ( ( B
% 14.25/8.08 = ( F @ X ) )
% 14.25/8.08 => ( member_Product_unit @ B @ ( image_o_Product_unit2 @ F @ top_top_set_o ) ) ) ).
% 14.25/8.08
% 14.25/8.08 thf(fact_4086_range__eqI,axiom,
% 14.25/8.08 ! [B: code_integer,F: nat > code_integer,X: nat] :
% 14.25/8.08 ( ( B
% 14.25/8.08 = ( F @ X ) )
% 14.25/8.08 => ( member_Code_integer @ B @ ( image_1215581382706833972nteger @ F @ top_top_set_nat ) ) ) ).
% 14.25/8.08
% 14.25/8.08 thf(fact_4087_empty__not__UNIV,axiom,
% 14.25/8.08 bot_bo7824918357723582233_nat_o != top_to4869887016220417533_nat_o ).
% 14.25/8.08
% 14.25/8.08 thf(fact_4088_empty__not__UNIV,axiom,
% 14.25/8.08 bot_bo3957492148770167129t_unit != top_to1996260823553986621t_unit ).
% 14.25/8.08
% 14.25/8.08 thf(fact_4089_empty__not__UNIV,axiom,
% 14.25/8.08 bot_bot_set_o != top_top_set_o ).
% 14.25/8.08
% 14.25/8.08 thf(fact_4090_empty__not__UNIV,axiom,
% 14.25/8.08 bot_bot_set_nat != top_top_set_nat ).
% 14.25/8.08
% 14.25/8.08 thf(fact_4091_empty__not__UNIV,axiom,
% 14.25/8.08 bot_bot_set_int != top_top_set_int ).
% 14.25/8.08
% 14.25/8.08 thf(fact_4092_subset__UNIV,axiom,
% 14.25/8.08 ! [A: set_Product_unit] : ( ord_le3507040750410214029t_unit @ A @ top_to1996260823553986621t_unit ) ).
% 14.25/8.08
% 14.25/8.08 thf(fact_4093_subset__UNIV,axiom,
% 14.25/8.08 ! [A: set_o] : ( ord_less_eq_set_o @ A @ top_top_set_o ) ).
% 14.25/8.08
% 14.25/8.08 thf(fact_4094_subset__UNIV,axiom,
% 14.25/8.08 ! [A: set_int] : ( ord_less_eq_set_int @ A @ top_top_set_int ) ).
% 14.25/8.08
% 14.25/8.08 thf(fact_4095_subset__UNIV,axiom,
% 14.25/8.08 ! [A: set_nat] : ( ord_less_eq_set_nat @ A @ top_top_set_nat ) ).
% 14.25/8.08
% 14.25/8.08 thf(fact_4096_psubset__imp__ex__mem,axiom,
% 14.25/8.08 ! [A: set_Pr4532377907799695533_nat_o,B4: set_Pr4532377907799695533_nat_o] :
% 14.25/8.08 ( ( ord_le2453136405763929_nat_o @ A @ B4 )
% 14.25/8.08 => ? [B2: produc3658429121746597890et_nat > $o] : ( member6576561426505652726_nat_o @ B2 @ ( minus_1801376950450012436_nat_o @ B4 @ A ) ) ) ).
% 14.25/8.08
% 14.25/8.08 thf(fact_4097_psubset__imp__ex__mem,axiom,
% 14.25/8.08 ! [A: set_int,B4: set_int] :
% 14.25/8.08 ( ( ord_less_set_int @ A @ B4 )
% 14.25/8.08 => ? [B2: int] : ( member_int @ B2 @ ( minus_minus_set_int @ B4 @ A ) ) ) ).
% 14.25/8.08
% 14.25/8.08 thf(fact_4098_psubset__imp__ex__mem,axiom,
% 14.25/8.08 ! [A: set_o,B4: set_o] :
% 14.25/8.08 ( ( ord_less_set_o @ A @ B4 )
% 14.25/8.08 => ? [B2: $o] : ( member_o @ B2 @ ( minus_minus_set_o @ B4 @ A ) ) ) ).
% 14.25/8.08
% 14.25/8.08 thf(fact_4099_psubset__imp__ex__mem,axiom,
% 14.25/8.08 ! [A: set_Product_unit,B4: set_Product_unit] :
% 14.25/8.08 ( ( ord_le8056459307392131481t_unit @ A @ B4 )
% 14.25/8.08 => ? [B2: product_unit] : ( member_Product_unit @ B2 @ ( minus_6452836326544984404t_unit @ B4 @ A ) ) ) ).
% 14.25/8.08
% 14.25/8.08 thf(fact_4100_psubset__imp__ex__mem,axiom,
% 14.25/8.08 ! [A: set_nat,B4: set_nat] :
% 14.25/8.08 ( ( ord_less_set_nat @ A @ B4 )
% 14.25/8.08 => ? [B2: nat] : ( member_nat @ B2 @ ( minus_minus_set_nat @ B4 @ A ) ) ) ).
% 14.25/8.08
% 14.25/8.08 thf(fact_4101_insert__UNIV,axiom,
% 14.25/8.08 ! [X: product_unit] :
% 14.25/8.08 ( ( insert_Product_unit @ X @ top_to1996260823553986621t_unit )
% 14.25/8.08 = top_to1996260823553986621t_unit ) ).
% 14.25/8.08
% 14.25/8.08 thf(fact_4102_insert__UNIV,axiom,
% 14.25/8.08 ! [X: $o] :
% 14.25/8.08 ( ( insert_o @ X @ top_top_set_o )
% 14.25/8.08 = top_top_set_o ) ).
% 14.25/8.08
% 14.25/8.08 thf(fact_4103_insert__UNIV,axiom,
% 14.25/8.08 ! [X: nat] :
% 14.25/8.08 ( ( insert_nat @ X @ top_top_set_nat )
% 14.25/8.08 = top_top_set_nat ) ).
% 14.25/8.08
% 14.25/8.08 thf(fact_4104_insert__UNIV,axiom,
% 14.25/8.08 ! [X: int] :
% 14.25/8.08 ( ( insert_int @ X @ top_top_set_int )
% 14.25/8.08 = top_top_set_int ) ).
% 14.25/8.08
% 14.25/8.08 thf(fact_4105_Int__UNIV__left,axiom,
% 14.25/8.08 ! [B4: set_Product_unit] :
% 14.25/8.08 ( ( inf_in4660618365625256667t_unit @ top_to1996260823553986621t_unit @ B4 )
% 14.25/8.08 = B4 ) ).
% 14.25/8.08
% 14.25/8.08 thf(fact_4106_Int__UNIV__left,axiom,
% 14.25/8.08 ! [B4: set_o] :
% 14.25/8.08 ( ( inf_inf_set_o @ top_top_set_o @ B4 )
% 14.25/8.08 = B4 ) ).
% 14.25/8.08
% 14.25/8.08 thf(fact_4107_Int__UNIV__left,axiom,
% 14.25/8.08 ! [B4: set_nat] :
% 14.25/8.08 ( ( inf_inf_set_nat @ top_top_set_nat @ B4 )
% 14.25/8.08 = B4 ) ).
% 14.25/8.08
% 14.25/8.08 thf(fact_4108_Int__UNIV__left,axiom,
% 14.25/8.08 ! [B4: set_int] :
% 14.25/8.08 ( ( inf_inf_set_int @ top_top_set_int @ B4 )
% 14.25/8.08 = B4 ) ).
% 14.25/8.08
% 14.25/8.08 thf(fact_4109_Int__UNIV__right,axiom,
% 14.25/8.08 ! [A: set_Product_unit] :
% 14.25/8.08 ( ( inf_in4660618365625256667t_unit @ A @ top_to1996260823553986621t_unit )
% 14.25/8.08 = A ) ).
% 14.25/8.08
% 14.25/8.08 thf(fact_4110_Int__UNIV__right,axiom,
% 14.25/8.08 ! [A: set_o] :
% 14.25/8.08 ( ( inf_inf_set_o @ A @ top_top_set_o )
% 14.25/8.08 = A ) ).
% 14.25/8.08
% 14.25/8.08 thf(fact_4111_Int__UNIV__right,axiom,
% 14.25/8.08 ! [A: set_nat] :
% 14.25/8.08 ( ( inf_inf_set_nat @ A @ top_top_set_nat )
% 14.25/8.08 = A ) ).
% 14.25/8.08
% 14.25/8.08 thf(fact_4112_Int__UNIV__right,axiom,
% 14.25/8.08 ! [A: set_int] :
% 14.25/8.08 ( ( inf_inf_set_int @ A @ top_top_set_int )
% 14.25/8.08 = A ) ).
% 14.25/8.08
% 14.25/8.08 thf(fact_4113_Un__UNIV__left,axiom,
% 14.25/8.08 ! [B4: set_Product_unit] :
% 14.25/8.08 ( ( sup_su793286257634532545t_unit @ top_to1996260823553986621t_unit @ B4 )
% 14.25/8.08 = top_to1996260823553986621t_unit ) ).
% 14.25/8.08
% 14.25/8.08 thf(fact_4114_Un__UNIV__left,axiom,
% 14.25/8.08 ! [B4: set_o] :
% 14.25/8.08 ( ( sup_sup_set_o @ top_top_set_o @ B4 )
% 14.25/8.08 = top_top_set_o ) ).
% 14.25/8.08
% 14.25/8.08 thf(fact_4115_Un__UNIV__left,axiom,
% 14.25/8.08 ! [B4: set_nat] :
% 14.25/8.08 ( ( sup_sup_set_nat @ top_top_set_nat @ B4 )
% 14.25/8.08 = top_top_set_nat ) ).
% 14.25/8.08
% 14.25/8.08 thf(fact_4116_Un__UNIV__left,axiom,
% 14.25/8.08 ! [B4: set_int] :
% 14.25/8.08 ( ( sup_sup_set_int @ top_top_set_int @ B4 )
% 14.25/8.08 = top_top_set_int ) ).
% 14.25/8.08
% 14.25/8.08 thf(fact_4117_Un__UNIV__right,axiom,
% 14.25/8.08 ! [A: set_Product_unit] :
% 14.25/8.08 ( ( sup_su793286257634532545t_unit @ A @ top_to1996260823553986621t_unit )
% 14.25/8.08 = top_to1996260823553986621t_unit ) ).
% 14.25/8.08
% 14.25/8.08 thf(fact_4118_Un__UNIV__right,axiom,
% 14.25/8.08 ! [A: set_o] :
% 14.25/8.08 ( ( sup_sup_set_o @ A @ top_top_set_o )
% 14.25/8.08 = top_top_set_o ) ).
% 14.25/8.08
% 14.25/8.08 thf(fact_4119_Un__UNIV__right,axiom,
% 14.25/8.08 ! [A: set_nat] :
% 14.25/8.08 ( ( sup_sup_set_nat @ A @ top_top_set_nat )
% 14.25/8.08 = top_top_set_nat ) ).
% 14.25/8.08
% 14.25/8.08 thf(fact_4120_Un__UNIV__right,axiom,
% 14.25/8.08 ! [A: set_int] :
% 14.25/8.08 ( ( sup_sup_set_int @ A @ top_top_set_int )
% 14.25/8.08 = top_top_set_int ) ).
% 14.25/8.08
% 14.25/8.08 thf(fact_4121_type__copy__ex__RepI,axiom,
% 14.25/8.08 ! [Rep2: assn > produc3658429121746597890et_nat > $o,Abs2: ( produc3658429121746597890et_nat > $o ) > assn,F2: ( produc3658429121746597890et_nat > $o ) > $o] :
% 14.25/8.08 ( ( type_d3909072315231072503_nat_o @ Rep2 @ Abs2 @ top_to4869887016220417533_nat_o )
% 14.25/8.08 => ( ( ? [X7: produc3658429121746597890et_nat > $o] : ( F2 @ X7 ) )
% 14.25/8.08 = ( ? [B6: assn] : ( F2 @ ( Rep2 @ B6 ) ) ) ) ) ).
% 14.25/8.08
% 14.25/8.08 thf(fact_4122_type__copy__ex__RepI,axiom,
% 14.25/8.08 ! [Rep2: code_integer > int,Abs2: int > code_integer,F2: int > $o] :
% 14.25/8.08 ( ( type_d8366093980585677751er_int @ Rep2 @ Abs2 @ top_top_set_int )
% 14.25/8.08 => ( ( ? [X7: int] : ( F2 @ X7 ) )
% 14.25/8.08 = ( ? [B6: code_integer] : ( F2 @ ( Rep2 @ B6 ) ) ) ) ) ).
% 14.25/8.08
% 14.25/8.08 thf(fact_4123_type__copy__ex__RepI,axiom,
% 14.25/8.08 ! [Rep2: product_unit > $o,Abs2: $o > product_unit,F2: $o > $o] :
% 14.25/8.08 ( ( type_d6188575255521822967unit_o @ Rep2 @ Abs2 @ top_top_set_o )
% 14.25/8.08 => ( ( ? [X7: $o] : ( F2 @ X7 ) )
% 14.25/8.08 = ( ? [B6: product_unit] : ( F2 @ ( Rep2 @ B6 ) ) ) ) ) ).
% 14.25/8.08
% 14.25/8.08 thf(fact_4124_type__copy__ex__RepI,axiom,
% 14.25/8.08 ! [Rep2: code_natural > nat,Abs2: nat > code_natural,F2: nat > $o] :
% 14.25/8.08 ( ( type_d4410041424927559462al_nat @ Rep2 @ Abs2 @ top_top_set_nat )
% 14.25/8.08 => ( ( ? [X7: nat] : ( F2 @ X7 ) )
% 14.25/8.08 = ( ? [B6: code_natural] : ( F2 @ ( Rep2 @ B6 ) ) ) ) ) ).
% 14.25/8.08
% 14.25/8.08 thf(fact_4125_type__copy__obj__one__point__absE,axiom,
% 14.25/8.08 ! [Rep2: assn > produc3658429121746597890et_nat > $o,Abs2: ( produc3658429121746597890et_nat > $o ) > assn,S2: assn] :
% 14.25/8.08 ( ( type_d3909072315231072503_nat_o @ Rep2 @ Abs2 @ top_to4869887016220417533_nat_o )
% 14.25/8.08 => ~ ! [X3: produc3658429121746597890et_nat > $o] :
% 14.25/8.08 ( S2
% 14.25/8.08 != ( Abs2 @ X3 ) ) ) ).
% 14.25/8.08
% 14.25/8.08 thf(fact_4126_type__copy__obj__one__point__absE,axiom,
% 14.25/8.08 ! [Rep2: code_integer > int,Abs2: int > code_integer,S2: code_integer] :
% 14.25/8.08 ( ( type_d8366093980585677751er_int @ Rep2 @ Abs2 @ top_top_set_int )
% 14.25/8.08 => ~ ! [X3: int] :
% 14.25/8.08 ( S2
% 14.25/8.08 != ( Abs2 @ X3 ) ) ) ).
% 14.25/8.08
% 14.25/8.08 thf(fact_4127_type__copy__obj__one__point__absE,axiom,
% 14.25/8.08 ! [Rep2: product_unit > $o,Abs2: $o > product_unit,S2: product_unit] :
% 14.25/8.08 ( ( type_d6188575255521822967unit_o @ Rep2 @ Abs2 @ top_top_set_o )
% 14.25/8.08 => ~ ! [X3: $o] :
% 14.25/8.08 ( S2
% 14.25/8.08 != ( Abs2 @ X3 ) ) ) ).
% 14.25/8.08
% 14.25/8.08 thf(fact_4128_type__copy__obj__one__point__absE,axiom,
% 14.25/8.08 ! [Rep2: code_natural > nat,Abs2: nat > code_natural,S2: code_natural] :
% 14.25/8.08 ( ( type_d4410041424927559462al_nat @ Rep2 @ Abs2 @ top_top_set_nat )
% 14.25/8.08 => ~ ! [X3: nat] :
% 14.25/8.08 ( S2
% 14.25/8.08 != ( Abs2 @ X3 ) ) ) ).
% 14.25/8.08
% 14.25/8.08 thf(fact_4129_less__1__mult,axiom,
% 14.25/8.08 ! [M4: code_natural,N2: code_natural] :
% 14.25/8.08 ( ( ord_le5570908160329646204atural @ one_one_Code_natural @ M4 )
% 14.25/8.08 => ( ( ord_le5570908160329646204atural @ one_one_Code_natural @ N2 )
% 14.25/8.08 => ( ord_le5570908160329646204atural @ one_one_Code_natural @ ( times_2397367101498566445atural @ M4 @ N2 ) ) ) ) ).
% 14.25/8.08
% 14.25/8.08 thf(fact_4130_less__1__mult,axiom,
% 14.25/8.08 ! [M4: nat,N2: nat] :
% 14.25/8.08 ( ( ord_less_nat @ one_one_nat @ M4 )
% 14.25/8.08 => ( ( ord_less_nat @ one_one_nat @ N2 )
% 14.25/8.08 => ( ord_less_nat @ one_one_nat @ ( times_times_nat @ M4 @ N2 ) ) ) ) ).
% 14.25/8.08
% 14.25/8.08 thf(fact_4131_less__1__mult,axiom,
% 14.25/8.08 ! [M4: int,N2: int] :
% 14.25/8.08 ( ( ord_less_int @ one_one_int @ M4 )
% 14.25/8.08 => ( ( ord_less_int @ one_one_int @ N2 )
% 14.25/8.08 => ( ord_less_int @ one_one_int @ ( times_times_int @ M4 @ N2 ) ) ) ) ).
% 14.25/8.08
% 14.25/8.08 thf(fact_4132_less__1__mult,axiom,
% 14.25/8.08 ! [M4: code_integer,N2: code_integer] :
% 14.25/8.08 ( ( ord_le6747313008572928689nteger @ one_one_Code_integer @ M4 )
% 14.25/8.08 => ( ( ord_le6747313008572928689nteger @ one_one_Code_integer @ N2 )
% 14.25/8.08 => ( ord_le6747313008572928689nteger @ one_one_Code_integer @ ( times_3573771949741848930nteger @ M4 @ N2 ) ) ) ) ).
% 14.25/8.08
% 14.25/8.08 thf(fact_4133_less__1__mult,axiom,
% 14.25/8.08 ! [M4: rat,N2: rat] :
% 14.25/8.08 ( ( ord_less_rat @ one_one_rat @ M4 )
% 14.25/8.08 => ( ( ord_less_rat @ one_one_rat @ N2 )
% 14.25/8.08 => ( ord_less_rat @ one_one_rat @ ( times_times_rat @ M4 @ N2 ) ) ) ) ).
% 14.25/8.08
% 14.25/8.08 thf(fact_4134_infinite__growing,axiom,
% 14.25/8.08 ! [X4: set_Product_unit] :
% 14.25/8.08 ( ( X4 != bot_bo3957492148770167129t_unit )
% 14.25/8.08 => ( ! [X3: product_unit] :
% 14.25/8.08 ( ( member_Product_unit @ X3 @ X4 )
% 14.25/8.08 => ? [Xa2: product_unit] :
% 14.25/8.08 ( ( member_Product_unit @ Xa2 @ X4 )
% 14.25/8.08 & ( ord_le361264281704409273t_unit @ X3 @ Xa2 ) ) )
% 14.25/8.08 => ~ ( finite4290736615968046902t_unit @ X4 ) ) ) ).
% 14.25/8.08
% 14.25/8.08 thf(fact_4135_infinite__growing,axiom,
% 14.25/8.08 ! [X4: set_o] :
% 14.25/8.08 ( ( X4 != bot_bot_set_o )
% 14.25/8.08 => ( ! [X3: $o] :
% 14.25/8.08 ( ( member_o @ X3 @ X4 )
% 14.25/8.08 => ? [Xa2: $o] :
% 14.25/8.08 ( ( member_o @ Xa2 @ X4 )
% 14.25/8.08 & ( ord_less_o @ X3 @ Xa2 ) ) )
% 14.25/8.08 => ~ ( finite_finite_o @ X4 ) ) ) ).
% 14.25/8.08
% 14.25/8.08 thf(fact_4136_infinite__growing,axiom,
% 14.25/8.08 ! [X4: set_nat] :
% 14.25/8.08 ( ( X4 != bot_bot_set_nat )
% 14.25/8.08 => ( ! [X3: nat] :
% 14.25/8.08 ( ( member_nat @ X3 @ X4 )
% 14.25/8.08 => ? [Xa2: nat] :
% 14.25/8.08 ( ( member_nat @ Xa2 @ X4 )
% 14.25/8.08 & ( ord_less_nat @ X3 @ Xa2 ) ) )
% 14.25/8.08 => ~ ( finite_finite_nat @ X4 ) ) ) ).
% 14.25/8.08
% 14.25/8.08 thf(fact_4137_infinite__growing,axiom,
% 14.25/8.08 ! [X4: set_int] :
% 14.25/8.08 ( ( X4 != bot_bot_set_int )
% 14.25/8.08 => ( ! [X3: int] :
% 14.25/8.08 ( ( member_int @ X3 @ X4 )
% 14.25/8.08 => ? [Xa2: int] :
% 14.25/8.08 ( ( member_int @ Xa2 @ X4 )
% 14.25/8.08 & ( ord_less_int @ X3 @ Xa2 ) ) )
% 14.25/8.08 => ~ ( finite_finite_int @ X4 ) ) ) ).
% 14.25/8.08
% 14.25/8.08 thf(fact_4138_infinite__growing,axiom,
% 14.25/8.08 ! [X4: set_Code_integer] :
% 14.25/8.08 ( ( X4 != bot_bo3990330152332043303nteger )
% 14.25/8.08 => ( ! [X3: code_integer] :
% 14.25/8.08 ( ( member_Code_integer @ X3 @ X4 )
% 14.25/8.08 => ? [Xa2: code_integer] :
% 14.25/8.08 ( ( member_Code_integer @ Xa2 @ X4 )
% 14.25/8.08 & ( ord_le6747313008572928689nteger @ X3 @ Xa2 ) ) )
% 14.25/8.08 => ~ ( finite6017078050557962740nteger @ X4 ) ) ) ).
% 14.25/8.08
% 14.25/8.08 thf(fact_4139_infinite__growing,axiom,
% 14.25/8.08 ! [X4: set_rat] :
% 14.25/8.08 ( ( X4 != bot_bot_set_rat )
% 14.25/8.08 => ( ! [X3: rat] :
% 14.25/8.08 ( ( member_rat @ X3 @ X4 )
% 14.25/8.08 => ? [Xa2: rat] :
% 14.25/8.08 ( ( member_rat @ Xa2 @ X4 )
% 14.25/8.08 & ( ord_less_rat @ X3 @ Xa2 ) ) )
% 14.25/8.08 => ~ ( finite_finite_rat @ X4 ) ) ) ).
% 14.25/8.08
% 14.25/8.08 thf(fact_4140_ex__min__if__finite,axiom,
% 14.25/8.08 ! [S: set_Product_unit] :
% 14.25/8.08 ( ( finite4290736615968046902t_unit @ S )
% 14.25/8.08 => ( ( S != bot_bo3957492148770167129t_unit )
% 14.25/8.08 => ? [X3: product_unit] :
% 14.25/8.08 ( ( member_Product_unit @ X3 @ S )
% 14.25/8.08 & ~ ? [Xa2: product_unit] :
% 14.25/8.08 ( ( member_Product_unit @ Xa2 @ S )
% 14.25/8.08 & ( ord_le361264281704409273t_unit @ Xa2 @ X3 ) ) ) ) ) ).
% 14.25/8.08
% 14.25/8.08 thf(fact_4141_ex__min__if__finite,axiom,
% 14.25/8.08 ! [S: set_o] :
% 14.25/8.08 ( ( finite_finite_o @ S )
% 14.25/8.08 => ( ( S != bot_bot_set_o )
% 14.25/8.08 => ? [X3: $o] :
% 14.25/8.08 ( ( member_o @ X3 @ S )
% 14.25/8.08 & ~ ? [Xa2: $o] :
% 14.25/8.08 ( ( member_o @ Xa2 @ S )
% 14.25/8.08 & ( ord_less_o @ Xa2 @ X3 ) ) ) ) ) ).
% 14.25/8.08
% 14.25/8.08 thf(fact_4142_ex__min__if__finite,axiom,
% 14.25/8.08 ! [S: set_Pr4532377907799695533_nat_o] :
% 14.25/8.08 ( ( finite3252695134891459830_nat_o @ S )
% 14.25/8.08 => ( ( S != bot_bo7824918357723582233_nat_o )
% 14.25/8.08 => ? [X3: produc3658429121746597890et_nat > $o] :
% 14.25/8.08 ( ( member6576561426505652726_nat_o @ X3 @ S )
% 14.25/8.08 & ~ ? [Xa2: produc3658429121746597890et_nat > $o] :
% 14.25/8.08 ( ( member6576561426505652726_nat_o @ Xa2 @ S )
% 14.25/8.08 & ( ord_le8176553561544679545_nat_o @ Xa2 @ X3 ) ) ) ) ) ).
% 14.25/8.08
% 14.25/8.08 thf(fact_4143_ex__min__if__finite,axiom,
% 14.25/8.08 ! [S: set_assn] :
% 14.25/8.08 ( ( finite_finite_assn @ S )
% 14.25/8.08 => ( ( S != bot_bot_set_assn )
% 14.25/8.08 => ? [X3: assn] :
% 14.25/8.08 ( ( member_assn @ X3 @ S )
% 14.25/8.08 & ~ ? [Xa2: assn] :
% 14.25/8.08 ( ( member_assn @ Xa2 @ S )
% 14.25/8.08 & ( ord_less_assn @ Xa2 @ X3 ) ) ) ) ) ).
% 14.25/8.08
% 14.25/8.08 thf(fact_4144_ex__min__if__finite,axiom,
% 14.25/8.08 ! [S: set_nat] :
% 14.25/8.08 ( ( finite_finite_nat @ S )
% 14.25/8.08 => ( ( S != bot_bot_set_nat )
% 14.25/8.08 => ? [X3: nat] :
% 14.25/8.08 ( ( member_nat @ X3 @ S )
% 14.25/8.08 & ~ ? [Xa2: nat] :
% 14.25/8.08 ( ( member_nat @ Xa2 @ S )
% 14.25/8.08 & ( ord_less_nat @ Xa2 @ X3 ) ) ) ) ) ).
% 14.25/8.08
% 14.25/8.08 thf(fact_4145_ex__min__if__finite,axiom,
% 14.25/8.08 ! [S: set_int] :
% 14.25/8.08 ( ( finite_finite_int @ S )
% 14.25/8.08 => ( ( S != bot_bot_set_int )
% 14.25/8.08 => ? [X3: int] :
% 14.25/8.08 ( ( member_int @ X3 @ S )
% 14.25/8.08 & ~ ? [Xa2: int] :
% 14.25/8.08 ( ( member_int @ Xa2 @ S )
% 14.25/8.08 & ( ord_less_int @ Xa2 @ X3 ) ) ) ) ) ).
% 14.25/8.08
% 14.25/8.08 thf(fact_4146_ex__min__if__finite,axiom,
% 14.25/8.08 ! [S: set_Code_integer] :
% 14.25/8.08 ( ( finite6017078050557962740nteger @ S )
% 14.25/8.08 => ( ( S != bot_bo3990330152332043303nteger )
% 14.25/8.08 => ? [X3: code_integer] :
% 14.25/8.08 ( ( member_Code_integer @ X3 @ S )
% 14.25/8.08 & ~ ? [Xa2: code_integer] :
% 14.25/8.08 ( ( member_Code_integer @ Xa2 @ S )
% 14.25/8.08 & ( ord_le6747313008572928689nteger @ Xa2 @ X3 ) ) ) ) ) ).
% 14.25/8.08
% 14.25/8.08 thf(fact_4147_ex__min__if__finite,axiom,
% 14.25/8.08 ! [S: set_rat] :
% 14.25/8.08 ( ( finite_finite_rat @ S )
% 14.25/8.08 => ( ( S != bot_bot_set_rat )
% 14.25/8.08 => ? [X3: rat] :
% 14.25/8.08 ( ( member_rat @ X3 @ S )
% 14.25/8.08 & ~ ? [Xa2: rat] :
% 14.25/8.08 ( ( member_rat @ Xa2 @ S )
% 14.25/8.08 & ( ord_less_rat @ Xa2 @ X3 ) ) ) ) ) ).
% 14.25/8.08
% 14.25/8.08 thf(fact_4148_less__cSupD,axiom,
% 14.25/8.08 ! [X4: set_int,Z: int] :
% 14.25/8.08 ( ( X4 != bot_bot_set_int )
% 14.25/8.08 => ( ( ord_less_int @ Z @ ( complete_Sup_Sup_int @ X4 ) )
% 14.25/8.08 => ? [X3: int] :
% 14.25/8.08 ( ( member_int @ X3 @ X4 )
% 14.25/8.08 & ( ord_less_int @ Z @ X3 ) ) ) ) ).
% 14.25/8.08
% 14.25/8.08 thf(fact_4149_less__cSupD,axiom,
% 14.25/8.08 ! [X4: set_nat,Z: nat] :
% 14.25/8.08 ( ( X4 != bot_bot_set_nat )
% 14.25/8.08 => ( ( ord_less_nat @ Z @ ( complete_Sup_Sup_nat @ X4 ) )
% 14.25/8.08 => ? [X3: nat] :
% 14.25/8.08 ( ( member_nat @ X3 @ X4 )
% 14.25/8.08 & ( ord_less_nat @ Z @ X3 ) ) ) ) ).
% 14.25/8.08
% 14.25/8.08 thf(fact_4150_less__cSupD,axiom,
% 14.25/8.08 ! [X4: set_Product_unit,Z: product_unit] :
% 14.25/8.08 ( ( X4 != bot_bo3957492148770167129t_unit )
% 14.25/8.08 => ( ( ord_le361264281704409273t_unit @ Z @ ( comple4687483117567863418t_unit @ X4 ) )
% 14.25/8.08 => ? [X3: product_unit] :
% 14.25/8.08 ( ( member_Product_unit @ X3 @ X4 )
% 14.25/8.08 & ( ord_le361264281704409273t_unit @ Z @ X3 ) ) ) ) ).
% 14.25/8.08
% 14.25/8.08 thf(fact_4151_less__cSupE,axiom,
% 14.25/8.08 ! [Y: int,X4: set_int] :
% 14.25/8.08 ( ( ord_less_int @ Y @ ( complete_Sup_Sup_int @ X4 ) )
% 14.25/8.08 => ( ( X4 != bot_bot_set_int )
% 14.25/8.08 => ~ ! [X3: int] :
% 14.25/8.08 ( ( member_int @ X3 @ X4 )
% 14.25/8.08 => ~ ( ord_less_int @ Y @ X3 ) ) ) ) ).
% 14.25/8.08
% 14.25/8.08 thf(fact_4152_less__cSupE,axiom,
% 14.25/8.08 ! [Y: nat,X4: set_nat] :
% 14.25/8.08 ( ( ord_less_nat @ Y @ ( complete_Sup_Sup_nat @ X4 ) )
% 14.25/8.08 => ( ( X4 != bot_bot_set_nat )
% 14.25/8.08 => ~ ! [X3: nat] :
% 14.25/8.08 ( ( member_nat @ X3 @ X4 )
% 14.25/8.08 => ~ ( ord_less_nat @ Y @ X3 ) ) ) ) ).
% 14.25/8.08
% 14.25/8.08 thf(fact_4153_less__cSupE,axiom,
% 14.25/8.08 ! [Y: product_unit,X4: set_Product_unit] :
% 14.25/8.08 ( ( ord_le361264281704409273t_unit @ Y @ ( comple4687483117567863418t_unit @ X4 ) )
% 14.25/8.08 => ( ( X4 != bot_bo3957492148770167129t_unit )
% 14.25/8.08 => ~ ! [X3: product_unit] :
% 14.25/8.08 ( ( member_Product_unit @ X3 @ X4 )
% 14.25/8.08 => ~ ( ord_le361264281704409273t_unit @ Y @ X3 ) ) ) ) ).
% 14.25/8.08
% 14.25/8.08 thf(fact_4154_Max__in,axiom,
% 14.25/8.08 ! [A: set_Code_integer] :
% 14.25/8.08 ( ( finite6017078050557962740nteger @ A )
% 14.25/8.08 => ( ( A != bot_bo3990330152332043303nteger )
% 14.25/8.08 => ( member_Code_integer @ ( lattic4901227151466704046nteger @ A ) @ A ) ) ) ).
% 14.25/8.08
% 14.25/8.08 thf(fact_4155_Max__in,axiom,
% 14.25/8.08 ! [A: set_Product_unit] :
% 14.25/8.08 ( ( finite4290736615968046902t_unit @ A )
% 14.25/8.08 => ( ( A != bot_bo3957492148770167129t_unit )
% 14.25/8.08 => ( member_Product_unit @ ( lattic1210392890313158908t_unit @ A ) @ A ) ) ) ).
% 14.25/8.08
% 14.25/8.08 thf(fact_4156_Max__in,axiom,
% 14.25/8.08 ! [A: set_o] :
% 14.25/8.08 ( ( finite_finite_o @ A )
% 14.25/8.08 => ( ( A != bot_bot_set_o )
% 14.25/8.08 => ( member_o @ ( lattic1921953407002678535_Max_o @ A ) @ A ) ) ) ).
% 14.25/8.08
% 14.25/8.08 thf(fact_4157_Max__in,axiom,
% 14.25/8.08 ! [A: set_int] :
% 14.25/8.08 ( ( finite_finite_int @ A )
% 14.25/8.08 => ( ( A != bot_bot_set_int )
% 14.25/8.08 => ( member_int @ ( lattic8263393255366662781ax_int @ A ) @ A ) ) ) ).
% 14.25/8.08
% 14.25/8.08 thf(fact_4158_Max__in,axiom,
% 14.25/8.08 ! [A: set_nat] :
% 14.25/8.08 ( ( finite_finite_nat @ A )
% 14.25/8.08 => ( ( A != bot_bot_set_nat )
% 14.25/8.08 => ( member_nat @ ( lattic8265883725875713057ax_nat @ A ) @ A ) ) ) ).
% 14.25/8.08
% 14.25/8.08 thf(fact_4159_range__subsetD,axiom,
% 14.25/8.08 ! [F: code_integer > code_integer,B4: set_Code_integer,I2: code_integer] :
% 14.25/8.08 ( ( ord_le7084787975880047091nteger @ ( image_4470545334726330049nteger @ F @ top_to4645266643341252675nteger ) @ B4 )
% 14.25/8.08 => ( member_Code_integer @ ( F @ I2 ) @ B4 ) ) ).
% 14.25/8.08
% 14.25/8.08 thf(fact_4160_range__subsetD,axiom,
% 14.25/8.08 ! [F: product_unit > int,B4: set_int,I2: product_unit] :
% 14.25/8.08 ( ( ord_less_eq_set_int @ ( image_873079544045703924it_int @ F @ top_to1996260823553986621t_unit ) @ B4 )
% 14.25/8.08 => ( member_int @ ( F @ I2 ) @ B4 ) ) ).
% 14.25/8.08
% 14.25/8.08 thf(fact_4161_range__subsetD,axiom,
% 14.25/8.08 ! [F: product_unit > $o,B4: set_o,I2: product_unit] :
% 14.25/8.08 ( ( ord_less_eq_set_o @ ( image_Product_unit_o2 @ F @ top_to1996260823553986621t_unit ) @ B4 )
% 14.25/8.08 => ( member_o @ ( F @ I2 ) @ B4 ) ) ).
% 14.25/8.08
% 14.25/8.08 thf(fact_4162_range__subsetD,axiom,
% 14.25/8.08 ! [F: product_unit > product_unit,B4: set_Product_unit,I2: product_unit] :
% 14.25/8.08 ( ( ord_le3507040750410214029t_unit @ ( image_405062704495631173t_unit @ F @ top_to1996260823553986621t_unit ) @ B4 )
% 14.25/8.08 => ( member_Product_unit @ ( F @ I2 ) @ B4 ) ) ).
% 14.25/8.08
% 14.25/8.08 thf(fact_4163_range__subsetD,axiom,
% 14.25/8.08 ! [F: $o > int,B4: set_int,I2: $o] :
% 14.25/8.08 ( ( ord_less_eq_set_int @ ( image_o_int2 @ F @ top_top_set_o ) @ B4 )
% 14.25/8.08 => ( member_int @ ( F @ I2 ) @ B4 ) ) ).
% 14.25/8.08
% 14.25/8.08 thf(fact_4164_range__subsetD,axiom,
% 14.25/8.08 ! [F: $o > $o,B4: set_o,I2: $o] :
% 14.25/8.08 ( ( ord_less_eq_set_o @ ( image_o_o2 @ F @ top_top_set_o ) @ B4 )
% 14.25/8.08 => ( member_o @ ( F @ I2 ) @ B4 ) ) ).
% 14.25/8.08
% 14.25/8.08 thf(fact_4165_range__subsetD,axiom,
% 14.25/8.08 ! [F: $o > product_unit,B4: set_Product_unit,I2: $o] :
% 14.25/8.08 ( ( ord_le3507040750410214029t_unit @ ( image_o_Product_unit2 @ F @ top_top_set_o ) @ B4 )
% 14.25/8.08 => ( member_Product_unit @ ( F @ I2 ) @ B4 ) ) ).
% 14.25/8.08
% 14.25/8.08 thf(fact_4166_range__subsetD,axiom,
% 14.25/8.08 ! [F: nat > code_integer,B4: set_Code_integer,I2: nat] :
% 14.25/8.08 ( ( ord_le7084787975880047091nteger @ ( image_1215581382706833972nteger @ F @ top_top_set_nat ) @ B4 )
% 14.25/8.08 => ( member_Code_integer @ ( F @ I2 ) @ B4 ) ) ).
% 14.25/8.08
% 14.25/8.08 thf(fact_4167_range__subsetD,axiom,
% 14.25/8.08 ! [F: nat > int,B4: set_int,I2: nat] :
% 14.25/8.08 ( ( ord_less_eq_set_int @ ( image_nat_int2 @ F @ top_top_set_nat ) @ B4 )
% 14.25/8.08 => ( member_int @ ( F @ I2 ) @ B4 ) ) ).
% 14.25/8.08
% 14.25/8.08 thf(fact_4168_range__subsetD,axiom,
% 14.25/8.08 ! [F: nat > $o,B4: set_o,I2: nat] :
% 14.25/8.08 ( ( ord_less_eq_set_o @ ( image_nat_o2 @ F @ top_top_set_nat ) @ B4 )
% 14.25/8.08 => ( member_o @ ( F @ I2 ) @ B4 ) ) ).
% 14.25/8.08
% 14.25/8.08 thf(fact_4169_surj__vimage__empty,axiom,
% 14.25/8.08 ! [F: code_integer > code_integer,A: set_Code_integer] :
% 14.25/8.08 ( ( ( image_4470545334726330049nteger @ F @ top_to4645266643341252675nteger )
% 14.25/8.08 = top_to4645266643341252675nteger )
% 14.25/8.08 => ( ( ( vimage1318793397690208119nteger @ F @ A )
% 14.25/8.08 = bot_bo3990330152332043303nteger )
% 14.25/8.08 = ( A = bot_bo3990330152332043303nteger ) ) ) ).
% 14.25/8.08
% 14.25/8.08 thf(fact_4170_surj__vimage__empty,axiom,
% 14.25/8.08 ! [F: product_unit > product_unit,A: set_Product_unit] :
% 14.25/8.08 ( ( ( image_405062704495631173t_unit @ F @ top_to1996260823553986621t_unit )
% 14.25/8.08 = top_to1996260823553986621t_unit )
% 14.25/8.08 => ( ( ( vimage7995052115951654139t_unit @ F @ A )
% 14.25/8.08 = bot_bo3957492148770167129t_unit )
% 14.25/8.08 = ( A = bot_bo3957492148770167129t_unit ) ) ) ).
% 14.25/8.08
% 14.25/8.08 thf(fact_4171_surj__vimage__empty,axiom,
% 14.25/8.08 ! [F: product_unit > $o,A: set_o] :
% 14.25/8.08 ( ( ( image_Product_unit_o2 @ F @ top_to1996260823553986621t_unit )
% 14.25/8.08 = top_top_set_o )
% 14.25/8.08 => ( ( ( vimage8808348054653608838unit_o @ F @ A )
% 14.25/8.08 = bot_bo3957492148770167129t_unit )
% 14.25/8.08 = ( A = bot_bot_set_o ) ) ) ).
% 14.25/8.08
% 14.25/8.08 thf(fact_4172_surj__vimage__empty,axiom,
% 14.25/8.08 ! [F: product_unit > nat,A: set_nat] :
% 14.25/8.08 ( ( ( image_875570014554754200it_nat @ F @ top_to1996260823553986621t_unit )
% 14.25/8.08 = top_top_set_nat )
% 14.25/8.08 => ( ( ( vimage6253328473476588386it_nat @ F @ A )
% 14.25/8.08 = bot_bo3957492148770167129t_unit )
% 14.25/8.08 = ( A = bot_bot_set_nat ) ) ) ).
% 14.25/8.08
% 14.25/8.08 thf(fact_4173_surj__vimage__empty,axiom,
% 14.25/8.08 ! [F: product_unit > int,A: set_int] :
% 14.25/8.08 ( ( ( image_873079544045703924it_int @ F @ top_to1996260823553986621t_unit )
% 14.25/8.08 = top_top_set_int )
% 14.25/8.08 => ( ( ( vimage6250838002967538110it_int @ F @ A )
% 14.25/8.08 = bot_bo3957492148770167129t_unit )
% 14.25/8.08 = ( A = bot_bot_set_int ) ) ) ).
% 14.25/8.08
% 14.25/8.08 thf(fact_4174_surj__vimage__empty,axiom,
% 14.25/8.08 ! [F: $o > product_unit,A: set_Product_unit] :
% 14.25/8.08 ( ( ( image_o_Product_unit2 @ F @ top_top_set_o )
% 14.25/8.08 = top_to1996260823553986621t_unit )
% 14.25/8.08 => ( ( ( vimage4222377557387472176t_unit @ F @ A )
% 14.25/8.08 = bot_bot_set_o )
% 14.25/8.08 = ( A = bot_bo3957492148770167129t_unit ) ) ) ).
% 14.25/8.08
% 14.25/8.08 thf(fact_4175_surj__vimage__empty,axiom,
% 14.25/8.08 ! [F: $o > $o,A: set_o] :
% 14.25/8.08 ( ( ( image_o_o2 @ F @ top_top_set_o )
% 14.25/8.08 = top_top_set_o )
% 14.25/8.08 => ( ( ( vimage_o_o @ F @ A )
% 14.25/8.08 = bot_bot_set_o )
% 14.25/8.08 = ( A = bot_bot_set_o ) ) ) ).
% 14.25/8.08
% 14.25/8.08 thf(fact_4176_surj__vimage__empty,axiom,
% 14.25/8.08 ! [F: $o > nat,A: set_nat] :
% 14.25/8.08 ( ( ( image_o_nat2 @ F @ top_top_set_o )
% 14.25/8.08 = top_top_set_nat )
% 14.25/8.08 => ( ( ( vimage_o_nat @ F @ A )
% 14.25/8.08 = bot_bot_set_o )
% 14.25/8.08 = ( A = bot_bot_set_nat ) ) ) ).
% 14.25/8.08
% 14.25/8.08 thf(fact_4177_surj__vimage__empty,axiom,
% 14.25/8.08 ! [F: $o > int,A: set_int] :
% 14.25/8.08 ( ( ( image_o_int2 @ F @ top_top_set_o )
% 14.25/8.08 = top_top_set_int )
% 14.25/8.08 => ( ( ( vimage_o_int @ F @ A )
% 14.25/8.08 = bot_bot_set_o )
% 14.25/8.08 = ( A = bot_bot_set_int ) ) ) ).
% 14.25/8.08
% 14.25/8.08 thf(fact_4178_surj__vimage__empty,axiom,
% 14.25/8.08 ! [F: nat > code_integer,A: set_Code_integer] :
% 14.25/8.08 ( ( ( image_1215581382706833972nteger @ F @ top_top_set_nat )
% 14.25/8.08 = top_to4645266643341252675nteger )
% 14.25/8.08 => ( ( ( vimage6104394758324456170nteger @ F @ A )
% 14.25/8.08 = bot_bot_set_nat )
% 14.25/8.08 = ( A = bot_bo3990330152332043303nteger ) ) ) ).
% 14.25/8.08
% 14.25/8.08 thf(fact_4179_acyclicI__order,axiom,
% 14.25/8.08 ! [R2: set_Pr1261947904930325089at_nat,F: nat > assn] :
% 14.25/8.08 ( ! [A4: nat,B2: nat] :
% 14.25/8.08 ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A4 @ B2 ) @ R2 )
% 14.25/8.08 => ( ord_less_assn @ ( F @ B2 ) @ ( F @ A4 ) ) )
% 14.25/8.08 => ( transi3139058337892597825ic_nat @ R2 ) ) ).
% 14.25/8.08
% 14.25/8.08 thf(fact_4180_acyclicI__order,axiom,
% 14.25/8.08 ! [R2: set_Pr958786334691620121nt_int,F: int > assn] :
% 14.25/8.08 ( ! [A4: int,B2: int] :
% 14.25/8.08 ( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ A4 @ B2 ) @ R2 )
% 14.25/8.08 => ( ord_less_assn @ ( F @ B2 ) @ ( F @ A4 ) ) )
% 14.25/8.08 => ( transi3136567867383547549ic_int @ R2 ) ) ).
% 14.25/8.08
% 14.25/8.08 thf(fact_4181_acyclicI__order,axiom,
% 14.25/8.08 ! [R2: set_Pr1261947904930325089at_nat,F: nat > nat] :
% 14.25/8.08 ( ! [A4: nat,B2: nat] :
% 14.25/8.08 ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A4 @ B2 ) @ R2 )
% 14.25/8.08 => ( ord_less_nat @ ( F @ B2 ) @ ( F @ A4 ) ) )
% 14.25/8.08 => ( transi3139058337892597825ic_nat @ R2 ) ) ).
% 14.25/8.08
% 14.25/8.08 thf(fact_4182_acyclicI__order,axiom,
% 14.25/8.08 ! [R2: set_Pr958786334691620121nt_int,F: int > nat] :
% 14.25/8.08 ( ! [A4: int,B2: int] :
% 14.25/8.08 ( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ A4 @ B2 ) @ R2 )
% 14.25/8.08 => ( ord_less_nat @ ( F @ B2 ) @ ( F @ A4 ) ) )
% 14.25/8.08 => ( transi3136567867383547549ic_int @ R2 ) ) ).
% 14.25/8.08
% 14.25/8.08 thf(fact_4183_acyclicI__order,axiom,
% 14.25/8.08 ! [R2: set_Pr1261947904930325089at_nat,F: nat > int] :
% 14.25/8.08 ( ! [A4: nat,B2: nat] :
% 14.25/8.08 ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A4 @ B2 ) @ R2 )
% 14.25/8.08 => ( ord_less_int @ ( F @ B2 ) @ ( F @ A4 ) ) )
% 14.25/8.08 => ( transi3139058337892597825ic_nat @ R2 ) ) ).
% 14.25/8.08
% 14.25/8.08 thf(fact_4184_acyclicI__order,axiom,
% 14.25/8.08 ! [R2: set_Pr958786334691620121nt_int,F: int > int] :
% 14.25/8.08 ( ! [A4: int,B2: int] :
% 14.25/8.08 ( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ A4 @ B2 ) @ R2 )
% 14.25/8.08 => ( ord_less_int @ ( F @ B2 ) @ ( F @ A4 ) ) )
% 14.25/8.08 => ( transi3136567867383547549ic_int @ R2 ) ) ).
% 14.25/8.08
% 14.25/8.08 thf(fact_4185_acyclicI__order,axiom,
% 14.25/8.08 ! [R2: set_Pr1261947904930325089at_nat,F: nat > code_integer] :
% 14.25/8.08 ( ! [A4: nat,B2: nat] :
% 14.25/8.08 ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A4 @ B2 ) @ R2 )
% 14.25/8.08 => ( ord_le6747313008572928689nteger @ ( F @ B2 ) @ ( F @ A4 ) ) )
% 14.25/8.08 => ( transi3139058337892597825ic_nat @ R2 ) ) ).
% 14.25/8.08
% 14.25/8.08 thf(fact_4186_acyclicI__order,axiom,
% 14.25/8.08 ! [R2: set_Pr958786334691620121nt_int,F: int > code_integer] :
% 14.25/8.08 ( ! [A4: int,B2: int] :
% 14.25/8.08 ( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ A4 @ B2 ) @ R2 )
% 14.25/8.08 => ( ord_le6747313008572928689nteger @ ( F @ B2 ) @ ( F @ A4 ) ) )
% 14.25/8.08 => ( transi3136567867383547549ic_int @ R2 ) ) ).
% 14.25/8.08
% 14.25/8.08 thf(fact_4187_acyclicI__order,axiom,
% 14.25/8.08 ! [R2: set_Pr1261947904930325089at_nat,F: nat > rat] :
% 14.25/8.08 ( ! [A4: nat,B2: nat] :
% 14.25/8.08 ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A4 @ B2 ) @ R2 )
% 14.25/8.08 => ( ord_less_rat @ ( F @ B2 ) @ ( F @ A4 ) ) )
% 14.25/8.08 => ( transi3139058337892597825ic_nat @ R2 ) ) ).
% 14.25/8.08
% 14.25/8.08 thf(fact_4188_acyclicI__order,axiom,
% 14.25/8.08 ! [R2: set_Pr958786334691620121nt_int,F: int > rat] :
% 14.25/8.08 ( ! [A4: int,B2: int] :
% 14.25/8.08 ( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ A4 @ B2 ) @ R2 )
% 14.25/8.08 => ( ord_less_rat @ ( F @ B2 ) @ ( F @ A4 ) ) )
% 14.25/8.08 => ( transi3136567867383547549ic_int @ R2 ) ) ).
% 14.25/8.08
% 14.25/8.08 thf(fact_4189_type__definition_OAbs__image,axiom,
% 14.25/8.08 ! [Rep2: set_nat > nat,Abs2: nat > set_nat,A: set_nat] :
% 14.25/8.08 ( ( type_d202864840109488260at_nat @ Rep2 @ Abs2 @ A )
% 14.25/8.08 => ( ( image_nat_set_nat @ Abs2 @ A )
% 14.25/8.08 = top_top_set_set_nat ) ) ).
% 14.25/8.08
% 14.25/8.08 thf(fact_4190_type__definition_OAbs__image,axiom,
% 14.25/8.08 ! [Rep2: code_integer > code_integer,Abs2: code_integer > code_integer,A: set_Code_integer] :
% 14.25/8.08 ( ( type_d4796611474492816872nteger @ Rep2 @ Abs2 @ A )
% 14.25/8.08 => ( ( image_4470545334726330049nteger @ Abs2 @ A )
% 14.25/8.08 = top_to4645266643341252675nteger ) ) ).
% 14.25/8.08
% 14.25/8.08 thf(fact_4191_type__definition_OAbs__image,axiom,
% 14.25/8.08 ! [Rep2: code_integer > nat,Abs2: nat > code_integer,A: set_nat] :
% 14.25/8.08 ( ( type_d8368584451094728027er_nat @ Rep2 @ Abs2 @ A )
% 14.25/8.08 => ( ( image_1215581382706833972nteger @ Abs2 @ A )
% 14.25/8.08 = top_to4645266643341252675nteger ) ) ).
% 14.25/8.08
% 14.25/8.08 thf(fact_4192_type__definition_OAbs__image,axiom,
% 14.25/8.08 ! [Rep2: nat > nat,Abs2: nat > nat,A: set_nat] :
% 14.25/8.08 ( ( type_d6250493948777748686at_nat @ Rep2 @ Abs2 @ A )
% 14.25/8.08 => ( ( image_nat_nat2 @ Abs2 @ A )
% 14.25/8.08 = top_top_set_nat ) ) ).
% 14.25/8.08
% 14.25/8.08 thf(fact_4193_type__definition_OAbs__image,axiom,
% 14.25/8.08 ! [Rep2: assn > produc3658429121746597890et_nat > $o,Abs2: ( produc3658429121746597890et_nat > $o ) > assn,A: set_Pr4532377907799695533_nat_o] :
% 14.25/8.08 ( ( type_d3909072315231072503_nat_o @ Rep2 @ Abs2 @ A )
% 14.25/8.08 => ( ( image_6451031022632655280o_assn @ Abs2 @ A )
% 14.25/8.08 = top_top_set_assn ) ) ).
% 14.25/8.08
% 14.25/8.08 thf(fact_4194_type__definition_OAbs__image,axiom,
% 14.25/8.08 ! [Rep2: code_integer > int,Abs2: int > code_integer,A: set_int] :
% 14.25/8.08 ( ( type_d8366093980585677751er_int @ Rep2 @ Abs2 @ A )
% 14.25/8.08 => ( ( image_1587234942943678608nteger @ Abs2 @ A )
% 14.25/8.08 = top_to4645266643341252675nteger ) ) ).
% 14.25/8.08
% 14.25/8.08 thf(fact_4195_type__definition_OAbs__image,axiom,
% 14.25/8.08 ! [Rep2: product_unit > $o,Abs2: $o > product_unit,A: set_o] :
% 14.25/8.08 ( ( type_d6188575255521822967unit_o @ Rep2 @ Abs2 @ A )
% 14.25/8.08 => ( ( image_o_Product_unit2 @ Abs2 @ A )
% 14.25/8.08 = top_to1996260823553986621t_unit ) ) ).
% 14.25/8.08
% 14.25/8.08 thf(fact_4196_type__definition_OAbs__image,axiom,
% 14.25/8.08 ! [Rep2: code_natural > nat,Abs2: nat > code_natural,A: set_nat] :
% 14.25/8.08 ( ( type_d4410041424927559462al_nat @ Rep2 @ Abs2 @ A )
% 14.25/8.08 => ( ( image_39176534463551487atural @ Abs2 @ A )
% 14.25/8.08 = top_to9103716463852441742atural ) ) ).
% 14.25/8.08
% 14.25/8.08 thf(fact_4197_type__definition_ORep__range,axiom,
% 14.25/8.08 ! [Rep2: code_integer > code_integer,Abs2: code_integer > code_integer,A: set_Code_integer] :
% 14.25/8.08 ( ( type_d4796611474492816872nteger @ Rep2 @ Abs2 @ A )
% 14.25/8.08 => ( ( image_4470545334726330049nteger @ Rep2 @ top_to4645266643341252675nteger )
% 14.25/8.08 = A ) ) ).
% 14.25/8.08
% 14.25/8.08 thf(fact_4198_type__definition_ORep__range,axiom,
% 14.25/8.08 ! [Rep2: nat > set_nat,Abs2: set_nat > nat,A: set_set_nat] :
% 14.25/8.08 ( ( type_d3768499744764857476et_nat @ Rep2 @ Abs2 @ A )
% 14.25/8.08 => ( ( image_nat_set_nat @ Rep2 @ top_top_set_nat )
% 14.25/8.08 = A ) ) ).
% 14.25/8.08
% 14.25/8.08 thf(fact_4199_type__definition_ORep__range,axiom,
% 14.25/8.08 ! [Rep2: nat > nat,Abs2: nat > nat,A: set_nat] :
% 14.25/8.08 ( ( type_d6250493948777748686at_nat @ Rep2 @ Abs2 @ A )
% 14.25/8.08 => ( ( image_nat_nat2 @ Rep2 @ top_top_set_nat )
% 14.25/8.08 = A ) ) ).
% 14.25/8.08
% 14.25/8.08 thf(fact_4200_type__definition_ORep__range,axiom,
% 14.25/8.08 ! [Rep2: nat > code_integer,Abs2: code_integer > nat,A: set_Code_integer] :
% 14.25/8.08 ( ( type_d8633139899873770843nteger @ Rep2 @ Abs2 @ A )
% 14.25/8.08 => ( ( image_1215581382706833972nteger @ Rep2 @ top_top_set_nat )
% 14.25/8.08 = A ) ) ).
% 14.25/8.08
% 14.25/8.08 thf(fact_4201_type__definition_ORep__range,axiom,
% 14.25/8.08 ! [Rep2: assn > produc3658429121746597890et_nat > $o,Abs2: ( produc3658429121746597890et_nat > $o ) > assn,A: set_Pr4532377907799695533_nat_o] :
% 14.25/8.08 ( ( type_d3909072315231072503_nat_o @ Rep2 @ Abs2 @ A )
% 14.25/8.08 => ( ( image_5465535159827709278_nat_o @ Rep2 @ top_top_set_assn )
% 14.25/8.08 = A ) ) ).
% 14.25/8.08
% 14.25/8.08 thf(fact_4202_type__definition_ORep__range,axiom,
% 14.25/8.08 ! [Rep2: code_integer > int,Abs2: int > code_integer,A: set_int] :
% 14.25/8.08 ( ( type_d8366093980585677751er_int @ Rep2 @ Abs2 @ A )
% 14.25/8.08 => ( ( image_948535463418740880er_int @ Rep2 @ top_to4645266643341252675nteger )
% 14.25/8.08 = A ) ) ).
% 14.25/8.08
% 14.25/8.08 thf(fact_4203_type__definition_ORep__range,axiom,
% 14.25/8.08 ! [Rep2: product_unit > $o,Abs2: $o > product_unit,A: set_o] :
% 14.25/8.08 ( ( type_d6188575255521822967unit_o @ Rep2 @ Abs2 @ A )
% 14.25/8.08 => ( ( image_Product_unit_o2 @ Rep2 @ top_to1996260823553986621t_unit )
% 14.25/8.08 = A ) ) ).
% 14.25/8.08
% 14.25/8.08 thf(fact_4204_type__definition_ORep__range,axiom,
% 14.25/8.08 ! [Rep2: code_natural > nat,Abs2: nat > code_natural,A: set_nat] :
% 14.25/8.08 ( ( type_d4410041424927559462al_nat @ Rep2 @ Abs2 @ A )
% 14.25/8.08 => ( ( image_6215854944615398399al_nat @ Rep2 @ top_to9103716463852441742atural )
% 14.25/8.08 = A ) ) ).
% 14.25/8.08
% 14.25/8.08 thf(fact_4205_refl__Id,axiom,
% 14.25/8.08 refl_on_Product_unit @ top_to1996260823553986621t_unit @ id_Product_unit ).
% 14.25/8.08
% 14.25/8.08 thf(fact_4206_refl__Id,axiom,
% 14.25/8.08 refl_on_o @ top_top_set_o @ id_o2 ).
% 14.25/8.08
% 14.25/8.08 thf(fact_4207_refl__Id,axiom,
% 14.25/8.08 refl_on_nat @ top_top_set_nat @ id_nat ).
% 14.25/8.08
% 14.25/8.08 thf(fact_4208_refl__Id,axiom,
% 14.25/8.08 refl_on_int @ top_top_set_int @ id_int ).
% 14.25/8.08
% 14.25/8.08 thf(fact_4209_boolean__algebra_Ocomplement__unique,axiom,
% 14.25/8.08 ! [A2: product_unit,X: product_unit,Y: product_unit] :
% 14.25/8.08 ( ( ( inf_inf_Product_unit @ A2 @ X )
% 14.25/8.08 = bot_bot_Product_unit )
% 14.25/8.08 => ( ( ( sup_sup_Product_unit @ A2 @ X )
% 14.25/8.08 = top_top_Product_unit )
% 14.25/8.08 => ( ( ( inf_inf_Product_unit @ A2 @ Y )
% 14.25/8.08 = bot_bot_Product_unit )
% 14.25/8.08 => ( ( ( sup_sup_Product_unit @ A2 @ Y )
% 14.25/8.08 = top_top_Product_unit )
% 14.25/8.08 => ( X = Y ) ) ) ) ) ).
% 14.25/8.08
% 14.25/8.08 thf(fact_4210_boolean__algebra_Ocomplement__unique,axiom,
% 14.25/8.08 ! [A2: set_Pr4532377907799695533_nat_o,X: set_Pr4532377907799695533_nat_o,Y: set_Pr4532377907799695533_nat_o] :
% 14.25/8.08 ( ( ( inf_in1906310914598751387_nat_o @ A2 @ X )
% 14.25/8.08 = bot_bo7824918357723582233_nat_o )
% 14.25/8.08 => ( ( ( sup_su5209123915105501825_nat_o @ A2 @ X )
% 14.25/8.08 = top_to4869887016220417533_nat_o )
% 14.25/8.08 => ( ( ( inf_in1906310914598751387_nat_o @ A2 @ Y )
% 14.25/8.08 = bot_bo7824918357723582233_nat_o )
% 14.25/8.08 => ( ( ( sup_su5209123915105501825_nat_o @ A2 @ Y )
% 14.25/8.08 = top_to4869887016220417533_nat_o )
% 14.25/8.08 => ( X = Y ) ) ) ) ) ).
% 14.25/8.08
% 14.25/8.08 thf(fact_4211_boolean__algebra_Ocomplement__unique,axiom,
% 14.25/8.08 ! [A2: assn,X: assn,Y: assn] :
% 14.25/8.08 ( ( ( inf_inf_assn @ A2 @ X )
% 14.25/8.08 = bot_bot_assn )
% 14.25/8.08 => ( ( ( sup_sup_assn @ A2 @ X )
% 14.25/8.08 = top_top_assn )
% 14.25/8.08 => ( ( ( inf_inf_assn @ A2 @ Y )
% 14.25/8.08 = bot_bot_assn )
% 14.25/8.08 => ( ( ( sup_sup_assn @ A2 @ Y )
% 14.25/8.08 = top_top_assn )
% 14.25/8.08 => ( X = Y ) ) ) ) ) ).
% 14.25/8.08
% 14.25/8.08 thf(fact_4212_boolean__algebra_Ocomplement__unique,axiom,
% 14.25/8.08 ! [A2: set_Product_unit,X: set_Product_unit,Y: set_Product_unit] :
% 14.25/8.08 ( ( ( inf_in4660618365625256667t_unit @ A2 @ X )
% 14.25/8.08 = bot_bo3957492148770167129t_unit )
% 14.25/8.08 => ( ( ( sup_su793286257634532545t_unit @ A2 @ X )
% 14.25/8.08 = top_to1996260823553986621t_unit )
% 14.25/8.08 => ( ( ( inf_in4660618365625256667t_unit @ A2 @ Y )
% 14.25/8.08 = bot_bo3957492148770167129t_unit )
% 14.25/8.08 => ( ( ( sup_su793286257634532545t_unit @ A2 @ Y )
% 14.25/8.08 = top_to1996260823553986621t_unit )
% 14.25/8.08 => ( X = Y ) ) ) ) ) ).
% 14.25/8.08
% 14.25/8.08 thf(fact_4213_boolean__algebra_Ocomplement__unique,axiom,
% 14.25/8.08 ! [A2: set_o,X: set_o,Y: set_o] :
% 14.25/8.08 ( ( ( inf_inf_set_o @ A2 @ X )
% 14.25/8.08 = bot_bot_set_o )
% 14.25/8.08 => ( ( ( sup_sup_set_o @ A2 @ X )
% 14.25/8.08 = top_top_set_o )
% 14.25/8.08 => ( ( ( inf_inf_set_o @ A2 @ Y )
% 14.25/8.08 = bot_bot_set_o )
% 14.25/8.08 => ( ( ( sup_sup_set_o @ A2 @ Y )
% 14.25/8.08 = top_top_set_o )
% 14.25/8.08 => ( X = Y ) ) ) ) ) ).
% 14.25/8.08
% 14.25/8.08 thf(fact_4214_boolean__algebra_Ocomplement__unique,axiom,
% 14.25/8.08 ! [A2: set_nat,X: set_nat,Y: set_nat] :
% 14.25/8.08 ( ( ( inf_inf_set_nat @ A2 @ X )
% 14.25/8.08 = bot_bot_set_nat )
% 14.25/8.08 => ( ( ( sup_sup_set_nat @ A2 @ X )
% 14.25/8.08 = top_top_set_nat )
% 14.25/8.08 => ( ( ( inf_inf_set_nat @ A2 @ Y )
% 14.25/8.08 = bot_bot_set_nat )
% 14.25/8.08 => ( ( ( sup_sup_set_nat @ A2 @ Y )
% 14.25/8.08 = top_top_set_nat )
% 14.25/8.08 => ( X = Y ) ) ) ) ) ).
% 14.25/8.08
% 14.25/8.08 thf(fact_4215_boolean__algebra_Ocomplement__unique,axiom,
% 14.25/8.08 ! [A2: set_int,X: set_int,Y: set_int] :
% 14.25/8.08 ( ( ( inf_inf_set_int @ A2 @ X )
% 14.25/8.08 = bot_bot_set_int )
% 14.25/8.08 => ( ( ( sup_sup_set_int @ A2 @ X )
% 14.25/8.08 = top_top_set_int )
% 14.25/8.08 => ( ( ( inf_inf_set_int @ A2 @ Y )
% 14.25/8.08 = bot_bot_set_int )
% 14.25/8.08 => ( ( ( sup_sup_set_int @ A2 @ Y )
% 14.25/8.08 = top_top_set_int )
% 14.25/8.08 => ( X = Y ) ) ) ) ) ).
% 14.25/8.08
% 14.25/8.08 thf(fact_4216_union__fold__insert,axiom,
% 14.25/8.08 ! [A: set_int,B4: set_int] :
% 14.25/8.08 ( ( finite_finite_int @ A )
% 14.25/8.08 => ( ( sup_sup_set_int @ A @ B4 )
% 14.25/8.08 = ( finite1723285575846220376et_int @ insert_int @ B4 @ A ) ) ) ).
% 14.25/8.08
% 14.25/8.08 thf(fact_4217_union__fold__insert,axiom,
% 14.25/8.08 ! [A: set_o,B4: set_o] :
% 14.25/8.08 ( ( finite_finite_o @ A )
% 14.25/8.08 => ( ( sup_sup_set_o @ A @ B4 )
% 14.25/8.08 = ( finite_fold_o_set_o @ insert_o @ B4 @ A ) ) ) ).
% 14.25/8.08
% 14.25/8.08 thf(fact_4218_union__fold__insert,axiom,
% 14.25/8.08 ! [A: set_Product_unit,B4: set_Product_unit] :
% 14.25/8.08 ( ( finite4290736615968046902t_unit @ A )
% 14.25/8.08 => ( ( sup_su793286257634532545t_unit @ A @ B4 )
% 14.25/8.08 = ( finite2740360166580943592t_unit @ insert_Product_unit @ B4 @ A ) ) ) ).
% 14.25/8.08
% 14.25/8.08 thf(fact_4219_union__fold__insert,axiom,
% 14.25/8.08 ! [A: set_Code_integer,B4: set_Code_integer] :
% 14.25/8.08 ( ( finite6017078050557962740nteger @ A )
% 14.25/8.08 => ( ( sup_su848401254843788991nteger @ A @ B4 )
% 14.25/8.08 = ( finite7486036041278456378nteger @ insert_Code_integer @ B4 @ A ) ) ) ).
% 14.25/8.08
% 14.25/8.08 thf(fact_4220_union__fold__insert,axiom,
% 14.25/8.08 ! [A: set_nat,B4: set_nat] :
% 14.25/8.08 ( ( finite_finite_nat @ A )
% 14.25/8.08 => ( ( sup_sup_set_nat @ A @ B4 )
% 14.25/8.08 = ( finite5529483035118572448et_nat @ insert_nat @ B4 @ A ) ) ) ).
% 14.25/8.08
% 14.25/8.08 thf(fact_4221_finite__linorder__min__induct,axiom,
% 14.25/8.08 ! [A: set_Product_unit,P: set_Product_unit > $o] :
% 14.25/8.08 ( ( finite4290736615968046902t_unit @ A )
% 14.25/8.08 => ( ( P @ bot_bo3957492148770167129t_unit )
% 14.25/8.08 => ( ! [B2: product_unit,A9: set_Product_unit] :
% 14.25/8.08 ( ( finite4290736615968046902t_unit @ A9 )
% 14.25/8.08 => ( ! [X6: product_unit] :
% 14.25/8.08 ( ( member_Product_unit @ X6 @ A9 )
% 14.25/8.08 => ( ord_le361264281704409273t_unit @ B2 @ X6 ) )
% 14.25/8.08 => ( ( P @ A9 )
% 14.25/8.08 => ( P @ ( insert_Product_unit @ B2 @ A9 ) ) ) ) )
% 14.25/8.08 => ( P @ A ) ) ) ) ).
% 14.25/8.08
% 14.25/8.08 thf(fact_4222_finite__linorder__min__induct,axiom,
% 14.25/8.08 ! [A: set_o,P: set_o > $o] :
% 14.25/8.08 ( ( finite_finite_o @ A )
% 14.25/8.08 => ( ( P @ bot_bot_set_o )
% 14.25/8.08 => ( ! [B2: $o,A9: set_o] :
% 14.25/8.08 ( ( finite_finite_o @ A9 )
% 14.25/8.08 => ( ! [X6: $o] :
% 14.25/8.08 ( ( member_o @ X6 @ A9 )
% 14.25/8.08 => ( ord_less_o @ B2 @ X6 ) )
% 14.25/8.08 => ( ( P @ A9 )
% 14.25/8.08 => ( P @ ( insert_o @ B2 @ A9 ) ) ) ) )
% 14.25/8.08 => ( P @ A ) ) ) ) ).
% 14.25/8.08
% 14.25/8.08 thf(fact_4223_finite__linorder__min__induct,axiom,
% 14.25/8.08 ! [A: set_nat,P: set_nat > $o] :
% 14.25/8.08 ( ( finite_finite_nat @ A )
% 14.25/8.08 => ( ( P @ bot_bot_set_nat )
% 14.25/8.08 => ( ! [B2: nat,A9: set_nat] :
% 14.25/8.08 ( ( finite_finite_nat @ A9 )
% 14.25/8.08 => ( ! [X6: nat] :
% 14.25/8.08 ( ( member_nat @ X6 @ A9 )
% 14.25/8.08 => ( ord_less_nat @ B2 @ X6 ) )
% 14.25/8.08 => ( ( P @ A9 )
% 14.25/8.08 => ( P @ ( insert_nat @ B2 @ A9 ) ) ) ) )
% 14.25/8.08 => ( P @ A ) ) ) ) ).
% 14.25/8.08
% 14.25/8.08 thf(fact_4224_finite__linorder__min__induct,axiom,
% 14.25/8.08 ! [A: set_int,P: set_int > $o] :
% 14.25/8.08 ( ( finite_finite_int @ A )
% 14.25/8.08 => ( ( P @ bot_bot_set_int )
% 14.25/8.08 => ( ! [B2: int,A9: set_int] :
% 14.25/8.08 ( ( finite_finite_int @ A9 )
% 14.25/8.08 => ( ! [X6: int] :
% 14.25/8.08 ( ( member_int @ X6 @ A9 )
% 14.25/8.08 => ( ord_less_int @ B2 @ X6 ) )
% 14.25/8.08 => ( ( P @ A9 )
% 14.25/8.08 => ( P @ ( insert_int @ B2 @ A9 ) ) ) ) )
% 14.25/8.08 => ( P @ A ) ) ) ) ).
% 14.25/8.08
% 14.25/8.08 thf(fact_4225_finite__linorder__min__induct,axiom,
% 14.25/8.08 ! [A: set_Code_integer,P: set_Code_integer > $o] :
% 14.25/8.08 ( ( finite6017078050557962740nteger @ A )
% 14.25/8.08 => ( ( P @ bot_bo3990330152332043303nteger )
% 14.25/8.08 => ( ! [B2: code_integer,A9: set_Code_integer] :
% 14.25/8.08 ( ( finite6017078050557962740nteger @ A9 )
% 14.25/8.08 => ( ! [X6: code_integer] :
% 14.25/8.08 ( ( member_Code_integer @ X6 @ A9 )
% 14.25/8.08 => ( ord_le6747313008572928689nteger @ B2 @ X6 ) )
% 14.25/8.08 => ( ( P @ A9 )
% 14.25/8.08 => ( P @ ( insert_Code_integer @ B2 @ A9 ) ) ) ) )
% 14.25/8.08 => ( P @ A ) ) ) ) ).
% 14.25/8.08
% 14.25/8.08 thf(fact_4226_finite__linorder__min__induct,axiom,
% 14.25/8.08 ! [A: set_rat,P: set_rat > $o] :
% 14.25/8.08 ( ( finite_finite_rat @ A )
% 14.25/8.08 => ( ( P @ bot_bot_set_rat )
% 14.25/8.08 => ( ! [B2: rat,A9: set_rat] :
% 14.25/8.08 ( ( finite_finite_rat @ A9 )
% 14.25/8.08 => ( ! [X6: rat] :
% 14.25/8.08 ( ( member_rat @ X6 @ A9 )
% 14.25/8.08 => ( ord_less_rat @ B2 @ X6 ) )
% 14.25/8.08 => ( ( P @ A9 )
% 14.25/8.08 => ( P @ ( insert_rat @ B2 @ A9 ) ) ) ) )
% 14.25/8.08 => ( P @ A ) ) ) ) ).
% 14.25/8.08
% 14.25/8.08 thf(fact_4227_finite__linorder__max__induct,axiom,
% 14.25/8.08 ! [A: set_Product_unit,P: set_Product_unit > $o] :
% 14.25/8.08 ( ( finite4290736615968046902t_unit @ A )
% 14.25/8.08 => ( ( P @ bot_bo3957492148770167129t_unit )
% 14.25/8.08 => ( ! [B2: product_unit,A9: set_Product_unit] :
% 14.25/8.08 ( ( finite4290736615968046902t_unit @ A9 )
% 14.25/8.08 => ( ! [X6: product_unit] :
% 14.25/8.08 ( ( member_Product_unit @ X6 @ A9 )
% 14.25/8.08 => ( ord_le361264281704409273t_unit @ X6 @ B2 ) )
% 14.25/8.08 => ( ( P @ A9 )
% 14.25/8.08 => ( P @ ( insert_Product_unit @ B2 @ A9 ) ) ) ) )
% 14.25/8.08 => ( P @ A ) ) ) ) ).
% 14.25/8.08
% 14.25/8.08 thf(fact_4228_finite__linorder__max__induct,axiom,
% 14.25/8.08 ! [A: set_o,P: set_o > $o] :
% 14.25/8.08 ( ( finite_finite_o @ A )
% 14.25/8.08 => ( ( P @ bot_bot_set_o )
% 14.25/8.08 => ( ! [B2: $o,A9: set_o] :
% 14.25/8.08 ( ( finite_finite_o @ A9 )
% 14.25/8.08 => ( ! [X6: $o] :
% 14.25/8.08 ( ( member_o @ X6 @ A9 )
% 14.25/8.08 => ( ord_less_o @ X6 @ B2 ) )
% 14.25/8.08 => ( ( P @ A9 )
% 14.25/8.08 => ( P @ ( insert_o @ B2 @ A9 ) ) ) ) )
% 14.25/8.08 => ( P @ A ) ) ) ) ).
% 14.25/8.08
% 14.25/8.08 thf(fact_4229_finite__linorder__max__induct,axiom,
% 14.25/8.08 ! [A: set_nat,P: set_nat > $o] :
% 14.25/8.08 ( ( finite_finite_nat @ A )
% 14.25/8.08 => ( ( P @ bot_bot_set_nat )
% 14.25/8.08 => ( ! [B2: nat,A9: set_nat] :
% 14.25/8.08 ( ( finite_finite_nat @ A9 )
% 14.25/8.08 => ( ! [X6: nat] :
% 14.25/8.08 ( ( member_nat @ X6 @ A9 )
% 14.25/8.08 => ( ord_less_nat @ X6 @ B2 ) )
% 14.25/8.08 => ( ( P @ A9 )
% 14.25/8.08 => ( P @ ( insert_nat @ B2 @ A9 ) ) ) ) )
% 14.25/8.08 => ( P @ A ) ) ) ) ).
% 14.25/8.08
% 14.25/8.08 thf(fact_4230_finite__linorder__max__induct,axiom,
% 14.25/8.08 ! [A: set_int,P: set_int > $o] :
% 14.25/8.08 ( ( finite_finite_int @ A )
% 14.25/8.08 => ( ( P @ bot_bot_set_int )
% 14.25/8.08 => ( ! [B2: int,A9: set_int] :
% 14.25/8.08 ( ( finite_finite_int @ A9 )
% 14.25/8.08 => ( ! [X6: int] :
% 14.25/8.08 ( ( member_int @ X6 @ A9 )
% 14.25/8.08 => ( ord_less_int @ X6 @ B2 ) )
% 14.25/8.08 => ( ( P @ A9 )
% 14.25/8.08 => ( P @ ( insert_int @ B2 @ A9 ) ) ) ) )
% 14.25/8.08 => ( P @ A ) ) ) ) ).
% 14.25/8.08
% 14.25/8.08 thf(fact_4231_finite__linorder__max__induct,axiom,
% 14.25/8.08 ! [A: set_Code_integer,P: set_Code_integer > $o] :
% 14.25/8.08 ( ( finite6017078050557962740nteger @ A )
% 14.25/8.08 => ( ( P @ bot_bo3990330152332043303nteger )
% 14.25/8.08 => ( ! [B2: code_integer,A9: set_Code_integer] :
% 14.25/8.08 ( ( finite6017078050557962740nteger @ A9 )
% 14.25/8.08 => ( ! [X6: code_integer] :
% 14.25/8.08 ( ( member_Code_integer @ X6 @ A9 )
% 14.25/8.08 => ( ord_le6747313008572928689nteger @ X6 @ B2 ) )
% 14.25/8.08 => ( ( P @ A9 )
% 14.25/8.08 => ( P @ ( insert_Code_integer @ B2 @ A9 ) ) ) ) )
% 14.25/8.08 => ( P @ A ) ) ) ) ).
% 14.25/8.08
% 14.25/8.08 thf(fact_4232_finite__linorder__max__induct,axiom,
% 14.25/8.08 ! [A: set_rat,P: set_rat > $o] :
% 14.25/8.08 ( ( finite_finite_rat @ A )
% 14.25/8.08 => ( ( P @ bot_bot_set_rat )
% 14.25/8.08 => ( ! [B2: rat,A9: set_rat] :
% 14.25/8.08 ( ( finite_finite_rat @ A9 )
% 14.25/8.08 => ( ! [X6: rat] :
% 14.25/8.08 ( ( member_rat @ X6 @ A9 )
% 14.25/8.08 => ( ord_less_rat @ X6 @ B2 ) )
% 14.25/8.08 => ( ( P @ A9 )
% 14.25/8.08 => ( P @ ( insert_rat @ B2 @ A9 ) ) ) ) )
% 14.25/8.08 => ( P @ A ) ) ) ) ).
% 14.25/8.08
% 14.25/8.08 thf(fact_4233_finite__Sup__less__iff,axiom,
% 14.25/8.08 ! [X4: set_int,A2: int] :
% 14.25/8.08 ( ( finite_finite_int @ X4 )
% 14.25/8.08 => ( ( X4 != bot_bot_set_int )
% 14.25/8.08 => ( ( ord_less_int @ ( complete_Sup_Sup_int @ X4 ) @ A2 )
% 14.25/8.08 = ( ! [X2: int] :
% 14.25/8.08 ( ( member_int @ X2 @ X4 )
% 14.25/8.08 => ( ord_less_int @ X2 @ A2 ) ) ) ) ) ) ).
% 14.25/8.08
% 14.25/8.08 thf(fact_4234_finite__Sup__less__iff,axiom,
% 14.25/8.08 ! [X4: set_nat,A2: nat] :
% 14.25/8.08 ( ( finite_finite_nat @ X4 )
% 14.25/8.08 => ( ( X4 != bot_bot_set_nat )
% 14.25/8.08 => ( ( ord_less_nat @ ( complete_Sup_Sup_nat @ X4 ) @ A2 )
% 14.25/8.08 = ( ! [X2: nat] :
% 14.25/8.08 ( ( member_nat @ X2 @ X4 )
% 14.25/8.08 => ( ord_less_nat @ X2 @ A2 ) ) ) ) ) ) ).
% 14.25/8.08
% 14.25/8.08 thf(fact_4235_finite__Sup__less__iff,axiom,
% 14.25/8.08 ! [X4: set_Product_unit,A2: product_unit] :
% 14.25/8.08 ( ( finite4290736615968046902t_unit @ X4 )
% 14.25/8.08 => ( ( X4 != bot_bo3957492148770167129t_unit )
% 14.25/8.08 => ( ( ord_le361264281704409273t_unit @ ( comple4687483117567863418t_unit @ X4 ) @ A2 )
% 14.25/8.08 = ( ! [X2: product_unit] :
% 14.25/8.08 ( ( member_Product_unit @ X2 @ X4 )
% 14.25/8.08 => ( ord_le361264281704409273t_unit @ X2 @ A2 ) ) ) ) ) ) ).
% 14.25/8.08
% 14.25/8.08 thf(fact_4236_Max__eq__iff,axiom,
% 14.25/8.08 ! [A: set_Code_integer,M4: code_integer] :
% 14.25/8.08 ( ( finite6017078050557962740nteger @ A )
% 14.25/8.08 => ( ( A != bot_bo3990330152332043303nteger )
% 14.25/8.08 => ( ( ( lattic4901227151466704046nteger @ A )
% 14.25/8.08 = M4 )
% 14.25/8.08 = ( ( member_Code_integer @ M4 @ A )
% 14.25/8.08 & ! [X2: code_integer] :
% 14.25/8.08 ( ( member_Code_integer @ X2 @ A )
% 14.25/8.08 => ( ord_le3102999989581377725nteger @ X2 @ M4 ) ) ) ) ) ) ).
% 14.25/8.08
% 14.25/8.08 thf(fact_4237_Max__eq__iff,axiom,
% 14.25/8.08 ! [A: set_Product_unit,M4: product_unit] :
% 14.25/8.08 ( ( finite4290736615968046902t_unit @ A )
% 14.25/8.08 => ( ( A != bot_bo3957492148770167129t_unit )
% 14.25/8.08 => ( ( ( lattic1210392890313158908t_unit @ A )
% 14.25/8.08 = M4 )
% 14.25/8.08 = ( ( member_Product_unit @ M4 @ A )
% 14.25/8.08 & ! [X2: product_unit] :
% 14.25/8.08 ( ( member_Product_unit @ X2 @ A )
% 14.25/8.08 => ( ord_le3221252021190050221t_unit @ X2 @ M4 ) ) ) ) ) ) ).
% 14.25/8.08
% 14.25/8.08 thf(fact_4238_Max__eq__iff,axiom,
% 14.25/8.08 ! [A: set_o,M4: $o] :
% 14.25/8.08 ( ( finite_finite_o @ A )
% 14.25/8.08 => ( ( A != bot_bot_set_o )
% 14.25/8.08 => ( ( ( lattic1921953407002678535_Max_o @ A )
% 14.25/8.08 = M4 )
% 14.25/8.08 = ( ( member_o @ M4 @ A )
% 14.25/8.08 & ! [X2: $o] :
% 14.25/8.08 ( ( member_o @ X2 @ A )
% 14.25/8.08 => ( ord_less_eq_o @ X2 @ M4 ) ) ) ) ) ) ).
% 14.25/8.08
% 14.25/8.08 thf(fact_4239_Max__eq__iff,axiom,
% 14.25/8.08 ! [A: set_rat,M4: rat] :
% 14.25/8.08 ( ( finite_finite_rat @ A )
% 14.25/8.08 => ( ( A != bot_bot_set_rat )
% 14.25/8.08 => ( ( ( lattic7630753665789217321ax_rat @ A )
% 14.25/8.08 = M4 )
% 14.25/8.08 = ( ( member_rat @ M4 @ A )
% 14.25/8.08 & ! [X2: rat] :
% 14.25/8.08 ( ( member_rat @ X2 @ A )
% 14.25/8.08 => ( ord_less_eq_rat @ X2 @ M4 ) ) ) ) ) ) ).
% 14.25/8.08
% 14.25/8.08 thf(fact_4240_Max__eq__iff,axiom,
% 14.25/8.08 ! [A: set_int,M4: int] :
% 14.25/8.08 ( ( finite_finite_int @ A )
% 14.25/8.08 => ( ( A != bot_bot_set_int )
% 14.25/8.08 => ( ( ( lattic8263393255366662781ax_int @ A )
% 14.25/8.08 = M4 )
% 14.25/8.08 = ( ( member_int @ M4 @ A )
% 14.25/8.08 & ! [X2: int] :
% 14.25/8.08 ( ( member_int @ X2 @ A )
% 14.25/8.08 => ( ord_less_eq_int @ X2 @ M4 ) ) ) ) ) ) ).
% 14.25/8.08
% 14.25/8.08 thf(fact_4241_Max__eq__iff,axiom,
% 14.25/8.08 ! [A: set_nat,M4: nat] :
% 14.25/8.08 ( ( finite_finite_nat @ A )
% 14.25/8.08 => ( ( A != bot_bot_set_nat )
% 14.25/8.08 => ( ( ( lattic8265883725875713057ax_nat @ A )
% 14.25/8.08 = M4 )
% 14.25/8.08 = ( ( member_nat @ M4 @ A )
% 14.25/8.08 & ! [X2: nat] :
% 14.25/8.08 ( ( member_nat @ X2 @ A )
% 14.25/8.08 => ( ord_less_eq_nat @ X2 @ M4 ) ) ) ) ) ) ).
% 14.25/8.08
% 14.25/8.08 thf(fact_4242_Max__ge__iff,axiom,
% 14.25/8.08 ! [A: set_Code_integer,X: code_integer] :
% 14.25/8.08 ( ( finite6017078050557962740nteger @ A )
% 14.25/8.08 => ( ( A != bot_bo3990330152332043303nteger )
% 14.25/8.08 => ( ( ord_le3102999989581377725nteger @ X @ ( lattic4901227151466704046nteger @ A ) )
% 14.25/8.08 = ( ? [X2: code_integer] :
% 14.25/8.08 ( ( member_Code_integer @ X2 @ A )
% 14.25/8.08 & ( ord_le3102999989581377725nteger @ X @ X2 ) ) ) ) ) ) ).
% 14.25/8.08
% 14.25/8.08 thf(fact_4243_Max__ge__iff,axiom,
% 14.25/8.08 ! [A: set_Product_unit,X: product_unit] :
% 14.25/8.08 ( ( finite4290736615968046902t_unit @ A )
% 14.25/8.08 => ( ( A != bot_bo3957492148770167129t_unit )
% 14.25/8.08 => ( ( ord_le3221252021190050221t_unit @ X @ ( lattic1210392890313158908t_unit @ A ) )
% 14.25/8.08 = ( ? [X2: product_unit] :
% 14.25/8.08 ( ( member_Product_unit @ X2 @ A )
% 14.25/8.08 & ( ord_le3221252021190050221t_unit @ X @ X2 ) ) ) ) ) ) ).
% 14.25/8.08
% 14.25/8.08 thf(fact_4244_Max__ge__iff,axiom,
% 14.25/8.08 ! [A: set_o,X: $o] :
% 14.25/8.08 ( ( finite_finite_o @ A )
% 14.25/8.08 => ( ( A != bot_bot_set_o )
% 14.25/8.08 => ( ( ord_less_eq_o @ X @ ( lattic1921953407002678535_Max_o @ A ) )
% 14.25/8.08 = ( ? [X2: $o] :
% 14.25/8.08 ( ( member_o @ X2 @ A )
% 14.25/8.08 & ( ord_less_eq_o @ X @ X2 ) ) ) ) ) ) ).
% 14.25/8.08
% 14.25/8.08 thf(fact_4245_Max__ge__iff,axiom,
% 14.25/8.08 ! [A: set_rat,X: rat] :
% 14.25/8.08 ( ( finite_finite_rat @ A )
% 14.25/8.08 => ( ( A != bot_bot_set_rat )
% 14.25/8.08 => ( ( ord_less_eq_rat @ X @ ( lattic7630753665789217321ax_rat @ A ) )
% 14.25/8.08 = ( ? [X2: rat] :
% 14.25/8.08 ( ( member_rat @ X2 @ A )
% 14.25/8.08 & ( ord_less_eq_rat @ X @ X2 ) ) ) ) ) ) ).
% 14.25/8.08
% 14.25/8.08 thf(fact_4246_Max__ge__iff,axiom,
% 14.25/8.08 ! [A: set_int,X: int] :
% 14.25/8.08 ( ( finite_finite_int @ A )
% 14.25/8.08 => ( ( A != bot_bot_set_int )
% 14.25/8.08 => ( ( ord_less_eq_int @ X @ ( lattic8263393255366662781ax_int @ A ) )
% 14.25/8.08 = ( ? [X2: int] :
% 14.25/8.08 ( ( member_int @ X2 @ A )
% 14.25/8.08 & ( ord_less_eq_int @ X @ X2 ) ) ) ) ) ) ).
% 14.25/8.08
% 14.25/8.08 thf(fact_4247_Max__ge__iff,axiom,
% 14.25/8.08 ! [A: set_nat,X: nat] :
% 14.25/8.08 ( ( finite_finite_nat @ A )
% 14.25/8.08 => ( ( A != bot_bot_set_nat )
% 14.25/8.08 => ( ( ord_less_eq_nat @ X @ ( lattic8265883725875713057ax_nat @ A ) )
% 14.25/8.08 = ( ? [X2: nat] :
% 14.25/8.08 ( ( member_nat @ X2 @ A )
% 14.25/8.08 & ( ord_less_eq_nat @ X @ X2 ) ) ) ) ) ) ).
% 14.25/8.08
% 14.25/8.08 thf(fact_4248_eq__Max__iff,axiom,
% 14.25/8.08 ! [A: set_Code_integer,M4: code_integer] :
% 14.25/8.08 ( ( finite6017078050557962740nteger @ A )
% 14.25/8.08 => ( ( A != bot_bo3990330152332043303nteger )
% 14.25/8.08 => ( ( M4
% 14.25/8.08 = ( lattic4901227151466704046nteger @ A ) )
% 14.25/8.08 = ( ( member_Code_integer @ M4 @ A )
% 14.25/8.08 & ! [X2: code_integer] :
% 14.25/8.08 ( ( member_Code_integer @ X2 @ A )
% 14.25/8.08 => ( ord_le3102999989581377725nteger @ X2 @ M4 ) ) ) ) ) ) ).
% 14.25/8.08
% 14.25/8.08 thf(fact_4249_eq__Max__iff,axiom,
% 14.25/8.08 ! [A: set_Product_unit,M4: product_unit] :
% 14.25/8.08 ( ( finite4290736615968046902t_unit @ A )
% 14.25/8.08 => ( ( A != bot_bo3957492148770167129t_unit )
% 14.25/8.08 => ( ( M4
% 14.25/8.08 = ( lattic1210392890313158908t_unit @ A ) )
% 14.25/8.08 = ( ( member_Product_unit @ M4 @ A )
% 14.25/8.08 & ! [X2: product_unit] :
% 14.25/8.08 ( ( member_Product_unit @ X2 @ A )
% 14.25/8.08 => ( ord_le3221252021190050221t_unit @ X2 @ M4 ) ) ) ) ) ) ).
% 14.25/8.08
% 14.25/8.08 thf(fact_4250_eq__Max__iff,axiom,
% 14.25/8.08 ! [A: set_o,M4: $o] :
% 14.25/8.08 ( ( finite_finite_o @ A )
% 14.25/8.08 => ( ( A != bot_bot_set_o )
% 14.25/8.08 => ( ( M4
% 14.25/8.08 = ( lattic1921953407002678535_Max_o @ A ) )
% 14.25/8.08 = ( ( member_o @ M4 @ A )
% 14.25/8.08 & ! [X2: $o] :
% 14.25/8.08 ( ( member_o @ X2 @ A )
% 14.25/8.08 => ( ord_less_eq_o @ X2 @ M4 ) ) ) ) ) ) ).
% 14.25/8.08
% 14.25/8.08 thf(fact_4251_eq__Max__iff,axiom,
% 14.25/8.08 ! [A: set_rat,M4: rat] :
% 14.25/8.08 ( ( finite_finite_rat @ A )
% 14.25/8.08 => ( ( A != bot_bot_set_rat )
% 14.25/8.08 => ( ( M4
% 14.25/8.08 = ( lattic7630753665789217321ax_rat @ A ) )
% 14.25/8.08 = ( ( member_rat @ M4 @ A )
% 14.25/8.08 & ! [X2: rat] :
% 14.25/8.08 ( ( member_rat @ X2 @ A )
% 14.25/8.08 => ( ord_less_eq_rat @ X2 @ M4 ) ) ) ) ) ) ).
% 14.25/8.08
% 14.25/8.08 thf(fact_4252_eq__Max__iff,axiom,
% 14.25/8.08 ! [A: set_int,M4: int] :
% 14.25/8.08 ( ( finite_finite_int @ A )
% 14.25/8.08 => ( ( A != bot_bot_set_int )
% 14.25/8.08 => ( ( M4
% 14.25/8.08 = ( lattic8263393255366662781ax_int @ A ) )
% 14.25/8.08 = ( ( member_int @ M4 @ A )
% 14.25/8.08 & ! [X2: int] :
% 14.25/8.08 ( ( member_int @ X2 @ A )
% 14.25/8.08 => ( ord_less_eq_int @ X2 @ M4 ) ) ) ) ) ) ).
% 14.25/8.08
% 14.25/8.08 thf(fact_4253_eq__Max__iff,axiom,
% 14.25/8.08 ! [A: set_nat,M4: nat] :
% 14.25/8.08 ( ( finite_finite_nat @ A )
% 14.25/8.08 => ( ( A != bot_bot_set_nat )
% 14.25/8.08 => ( ( M4
% 14.25/8.08 = ( lattic8265883725875713057ax_nat @ A ) )
% 14.25/8.08 = ( ( member_nat @ M4 @ A )
% 14.25/8.08 & ! [X2: nat] :
% 14.25/8.08 ( ( member_nat @ X2 @ A )
% 14.25/8.08 => ( ord_less_eq_nat @ X2 @ M4 ) ) ) ) ) ) ).
% 14.25/8.08
% 14.25/8.08 thf(fact_4254_Max_OboundedE,axiom,
% 14.25/8.08 ! [A: set_Code_integer,X: code_integer] :
% 14.25/8.08 ( ( finite6017078050557962740nteger @ A )
% 14.25/8.08 => ( ( A != bot_bo3990330152332043303nteger )
% 14.25/8.08 => ( ( ord_le3102999989581377725nteger @ ( lattic4901227151466704046nteger @ A ) @ X )
% 14.25/8.08 => ! [A10: code_integer] :
% 14.25/8.08 ( ( member_Code_integer @ A10 @ A )
% 14.25/8.08 => ( ord_le3102999989581377725nteger @ A10 @ X ) ) ) ) ) ).
% 14.25/8.08
% 14.25/8.08 thf(fact_4255_Max_OboundedE,axiom,
% 14.25/8.08 ! [A: set_Product_unit,X: product_unit] :
% 14.25/8.08 ( ( finite4290736615968046902t_unit @ A )
% 14.25/8.08 => ( ( A != bot_bo3957492148770167129t_unit )
% 14.25/8.08 => ( ( ord_le3221252021190050221t_unit @ ( lattic1210392890313158908t_unit @ A ) @ X )
% 14.25/8.08 => ! [A10: product_unit] :
% 14.25/8.08 ( ( member_Product_unit @ A10 @ A )
% 14.25/8.08 => ( ord_le3221252021190050221t_unit @ A10 @ X ) ) ) ) ) ).
% 14.25/8.08
% 14.25/8.08 thf(fact_4256_Max_OboundedE,axiom,
% 14.25/8.08 ! [A: set_o,X: $o] :
% 14.25/8.08 ( ( finite_finite_o @ A )
% 14.25/8.08 => ( ( A != bot_bot_set_o )
% 14.25/8.08 => ( ( ord_less_eq_o @ ( lattic1921953407002678535_Max_o @ A ) @ X )
% 14.25/8.08 => ! [A10: $o] :
% 14.25/8.08 ( ( member_o @ A10 @ A )
% 14.25/8.08 => ( ord_less_eq_o @ A10 @ X ) ) ) ) ) ).
% 14.25/8.08
% 14.25/8.08 thf(fact_4257_Max_OboundedE,axiom,
% 14.25/8.08 ! [A: set_rat,X: rat] :
% 14.25/8.08 ( ( finite_finite_rat @ A )
% 14.25/8.08 => ( ( A != bot_bot_set_rat )
% 14.25/8.08 => ( ( ord_less_eq_rat @ ( lattic7630753665789217321ax_rat @ A ) @ X )
% 14.25/8.08 => ! [A10: rat] :
% 14.25/8.08 ( ( member_rat @ A10 @ A )
% 14.25/8.08 => ( ord_less_eq_rat @ A10 @ X ) ) ) ) ) ).
% 14.25/8.08
% 14.25/8.08 thf(fact_4258_Max_OboundedE,axiom,
% 14.25/8.08 ! [A: set_int,X: int] :
% 14.25/8.08 ( ( finite_finite_int @ A )
% 14.25/8.08 => ( ( A != bot_bot_set_int )
% 14.25/8.08 => ( ( ord_less_eq_int @ ( lattic8263393255366662781ax_int @ A ) @ X )
% 14.25/8.08 => ! [A10: int] :
% 14.25/8.08 ( ( member_int @ A10 @ A )
% 14.25/8.08 => ( ord_less_eq_int @ A10 @ X ) ) ) ) ) ).
% 14.25/8.08
% 14.25/8.08 thf(fact_4259_Max_OboundedE,axiom,
% 14.25/8.08 ! [A: set_nat,X: nat] :
% 14.25/8.08 ( ( finite_finite_nat @ A )
% 14.25/8.08 => ( ( A != bot_bot_set_nat )
% 14.25/8.08 => ( ( ord_less_eq_nat @ ( lattic8265883725875713057ax_nat @ A ) @ X )
% 14.25/8.08 => ! [A10: nat] :
% 14.25/8.08 ( ( member_nat @ A10 @ A )
% 14.25/8.08 => ( ord_less_eq_nat @ A10 @ X ) ) ) ) ) ).
% 14.25/8.08
% 14.25/8.08 thf(fact_4260_Max_OboundedI,axiom,
% 14.25/8.08 ! [A: set_Code_integer,X: code_integer] :
% 14.25/8.08 ( ( finite6017078050557962740nteger @ A )
% 14.25/8.08 => ( ( A != bot_bo3990330152332043303nteger )
% 14.25/8.08 => ( ! [A4: code_integer] :
% 14.25/8.08 ( ( member_Code_integer @ A4 @ A )
% 14.25/8.08 => ( ord_le3102999989581377725nteger @ A4 @ X ) )
% 14.25/8.08 => ( ord_le3102999989581377725nteger @ ( lattic4901227151466704046nteger @ A ) @ X ) ) ) ) ).
% 14.25/8.08
% 14.25/8.08 thf(fact_4261_Max_OboundedI,axiom,
% 14.25/8.08 ! [A: set_Product_unit,X: product_unit] :
% 14.25/8.08 ( ( finite4290736615968046902t_unit @ A )
% 14.25/8.08 => ( ( A != bot_bo3957492148770167129t_unit )
% 14.25/8.08 => ( ! [A4: product_unit] :
% 14.25/8.08 ( ( member_Product_unit @ A4 @ A )
% 14.25/8.08 => ( ord_le3221252021190050221t_unit @ A4 @ X ) )
% 14.25/8.08 => ( ord_le3221252021190050221t_unit @ ( lattic1210392890313158908t_unit @ A ) @ X ) ) ) ) ).
% 14.25/8.08
% 14.25/8.08 thf(fact_4262_Max_OboundedI,axiom,
% 14.25/8.08 ! [A: set_o,X: $o] :
% 14.25/8.08 ( ( finite_finite_o @ A )
% 14.25/8.08 => ( ( A != bot_bot_set_o )
% 14.25/8.08 => ( ! [A4: $o] :
% 14.25/8.08 ( ( member_o @ A4 @ A )
% 14.25/8.08 => ( ord_less_eq_o @ A4 @ X ) )
% 14.25/8.08 => ( ord_less_eq_o @ ( lattic1921953407002678535_Max_o @ A ) @ X ) ) ) ) ).
% 14.25/8.08
% 14.25/8.08 thf(fact_4263_Max_OboundedI,axiom,
% 14.25/8.08 ! [A: set_rat,X: rat] :
% 14.25/8.08 ( ( finite_finite_rat @ A )
% 14.25/8.08 => ( ( A != bot_bot_set_rat )
% 14.25/8.08 => ( ! [A4: rat] :
% 14.25/8.08 ( ( member_rat @ A4 @ A )
% 14.25/8.08 => ( ord_less_eq_rat @ A4 @ X ) )
% 14.25/8.08 => ( ord_less_eq_rat @ ( lattic7630753665789217321ax_rat @ A ) @ X ) ) ) ) ).
% 14.25/8.08
% 14.25/8.08 thf(fact_4264_Max_OboundedI,axiom,
% 14.25/8.08 ! [A: set_int,X: int] :
% 14.25/8.08 ( ( finite_finite_int @ A )
% 14.25/8.08 => ( ( A != bot_bot_set_int )
% 14.25/8.08 => ( ! [A4: int] :
% 14.25/8.08 ( ( member_int @ A4 @ A )
% 14.25/8.08 => ( ord_less_eq_int @ A4 @ X ) )
% 14.25/8.08 => ( ord_less_eq_int @ ( lattic8263393255366662781ax_int @ A ) @ X ) ) ) ) ).
% 14.25/8.08
% 14.25/8.08 thf(fact_4265_Max_OboundedI,axiom,
% 14.25/8.08 ! [A: set_nat,X: nat] :
% 14.25/8.08 ( ( finite_finite_nat @ A )
% 14.25/8.08 => ( ( A != bot_bot_set_nat )
% 14.25/8.08 => ( ! [A4: nat] :
% 14.25/8.08 ( ( member_nat @ A4 @ A )
% 14.25/8.08 => ( ord_less_eq_nat @ A4 @ X ) )
% 14.25/8.08 => ( ord_less_eq_nat @ ( lattic8265883725875713057ax_nat @ A ) @ X ) ) ) ) ).
% 14.25/8.08
% 14.25/8.08 thf(fact_4266_Max__insert2,axiom,
% 14.25/8.08 ! [A: set_o,A2: $o] :
% 14.25/8.08 ( ( finite_finite_o @ A )
% 14.25/8.08 => ( ! [B2: $o] :
% 14.25/8.08 ( ( member_o @ B2 @ A )
% 14.25/8.08 => ( ord_less_eq_o @ B2 @ A2 ) )
% 14.25/8.08 => ( ( lattic1921953407002678535_Max_o @ ( insert_o @ A2 @ A ) )
% 14.25/8.08 = A2 ) ) ) ).
% 14.25/8.08
% 14.25/8.08 thf(fact_4267_Max__insert2,axiom,
% 14.25/8.08 ! [A: set_Product_unit,A2: product_unit] :
% 14.25/8.08 ( ( finite4290736615968046902t_unit @ A )
% 14.25/8.08 => ( ! [B2: product_unit] :
% 14.25/8.08 ( ( member_Product_unit @ B2 @ A )
% 14.25/8.08 => ( ord_le3221252021190050221t_unit @ B2 @ A2 ) )
% 14.25/8.08 => ( ( lattic1210392890313158908t_unit @ ( insert_Product_unit @ A2 @ A ) )
% 14.25/8.08 = A2 ) ) ) ).
% 14.25/8.08
% 14.25/8.08 thf(fact_4268_Max__insert2,axiom,
% 14.25/8.08 ! [A: set_Code_integer,A2: code_integer] :
% 14.25/8.08 ( ( finite6017078050557962740nteger @ A )
% 14.25/8.08 => ( ! [B2: code_integer] :
% 14.25/8.08 ( ( member_Code_integer @ B2 @ A )
% 14.25/8.08 => ( ord_le3102999989581377725nteger @ B2 @ A2 ) )
% 14.25/8.08 => ( ( lattic4901227151466704046nteger @ ( insert_Code_integer @ A2 @ A ) )
% 14.25/8.08 = A2 ) ) ) ).
% 14.25/8.08
% 14.25/8.08 thf(fact_4269_Max__insert2,axiom,
% 14.25/8.08 ! [A: set_rat,A2: rat] :
% 14.25/8.08 ( ( finite_finite_rat @ A )
% 14.25/8.08 => ( ! [B2: rat] :
% 14.25/8.08 ( ( member_rat @ B2 @ A )
% 14.25/8.08 => ( ord_less_eq_rat @ B2 @ A2 ) )
% 14.25/8.08 => ( ( lattic7630753665789217321ax_rat @ ( insert_rat @ A2 @ A ) )
% 14.25/8.08 = A2 ) ) ) ).
% 14.25/8.08
% 14.25/8.08 thf(fact_4270_Max__insert2,axiom,
% 14.25/8.08 ! [A: set_int,A2: int] :
% 14.25/8.08 ( ( finite_finite_int @ A )
% 14.25/8.08 => ( ! [B2: int] :
% 14.25/8.08 ( ( member_int @ B2 @ A )
% 14.25/8.08 => ( ord_less_eq_int @ B2 @ A2 ) )
% 14.25/8.08 => ( ( lattic8263393255366662781ax_int @ ( insert_int @ A2 @ A ) )
% 14.25/8.08 = A2 ) ) ) ).
% 14.25/8.08
% 14.25/8.08 thf(fact_4271_Max__insert2,axiom,
% 14.25/8.08 ! [A: set_nat,A2: nat] :
% 14.25/8.08 ( ( finite_finite_nat @ A )
% 14.25/8.08 => ( ! [B2: nat] :
% 14.25/8.08 ( ( member_nat @ B2 @ A )
% 14.25/8.08 => ( ord_less_eq_nat @ B2 @ A2 ) )
% 14.25/8.08 => ( ( lattic8265883725875713057ax_nat @ ( insert_nat @ A2 @ A ) )
% 14.25/8.08 = A2 ) ) ) ).
% 14.25/8.08
% 14.25/8.08 thf(fact_4272_remove__subset,axiom,
% 14.25/8.08 ! [X: product_unit,S: set_Product_unit] :
% 14.25/8.08 ( ( member_Product_unit @ X @ S )
% 14.25/8.08 => ( ord_le8056459307392131481t_unit @ ( minus_6452836326544984404t_unit @ S @ ( insert_Product_unit @ X @ bot_bo3957492148770167129t_unit ) ) @ S ) ) ).
% 14.25/8.08
% 14.25/8.08 thf(fact_4273_remove__subset,axiom,
% 14.25/8.08 ! [X: $o,S: set_o] :
% 14.25/8.08 ( ( member_o @ X @ S )
% 14.25/8.08 => ( ord_less_set_o @ ( minus_minus_set_o @ S @ ( insert_o @ X @ bot_bot_set_o ) ) @ S ) ) ).
% 14.25/8.08
% 14.25/8.08 thf(fact_4274_remove__subset,axiom,
% 14.25/8.08 ! [X: int,S: set_int] :
% 14.25/8.08 ( ( member_int @ X @ S )
% 14.25/8.08 => ( ord_less_set_int @ ( minus_minus_set_int @ S @ ( insert_int @ X @ bot_bot_set_int ) ) @ S ) ) ).
% 14.25/8.08
% 14.25/8.08 thf(fact_4275_remove__subset,axiom,
% 14.25/8.08 ! [X: produc3658429121746597890et_nat > $o,S: set_Pr4532377907799695533_nat_o] :
% 14.25/8.08 ( ( member6576561426505652726_nat_o @ X @ S )
% 14.25/8.08 => ( ord_le2453136405763929_nat_o @ ( minus_1801376950450012436_nat_o @ S @ ( insert5175938949040314269_nat_o @ X @ bot_bo7824918357723582233_nat_o ) ) @ S ) ) ).
% 14.25/8.08
% 14.25/8.08 thf(fact_4276_remove__subset,axiom,
% 14.25/8.08 ! [X: nat,S: set_nat] :
% 14.25/8.08 ( ( member_nat @ X @ S )
% 14.25/8.08 => ( ord_less_set_nat @ ( minus_minus_set_nat @ S @ ( insert_nat @ X @ bot_bot_set_nat ) ) @ S ) ) ).
% 14.25/8.08
% 14.25/8.08 thf(fact_4277_cSup__eq__Max,axiom,
% 14.25/8.08 ! [X4: set_int] :
% 14.25/8.08 ( ( finite_finite_int @ X4 )
% 14.25/8.08 => ( ( X4 != bot_bot_set_int )
% 14.25/8.08 => ( ( complete_Sup_Sup_int @ X4 )
% 14.25/8.08 = ( lattic8263393255366662781ax_int @ X4 ) ) ) ) ).
% 14.25/8.08
% 14.25/8.08 thf(fact_4278_cSup__eq__Max,axiom,
% 14.25/8.08 ! [X4: set_Product_unit] :
% 14.25/8.08 ( ( finite4290736615968046902t_unit @ X4 )
% 14.25/8.08 => ( ( X4 != bot_bo3957492148770167129t_unit )
% 14.25/8.08 => ( ( comple4687483117567863418t_unit @ X4 )
% 14.25/8.08 = ( lattic1210392890313158908t_unit @ X4 ) ) ) ) ).
% 14.25/8.08
% 14.25/8.08 thf(fact_4279_cSup__eq__Max,axiom,
% 14.25/8.08 ! [X4: set_nat] :
% 14.25/8.08 ( ( finite_finite_nat @ X4 )
% 14.25/8.08 => ( ( X4 != bot_bot_set_nat )
% 14.25/8.08 => ( ( complete_Sup_Sup_nat @ X4 )
% 14.25/8.08 = ( lattic8265883725875713057ax_nat @ X4 ) ) ) ) ).
% 14.25/8.08
% 14.25/8.08 thf(fact_4280_Max__Sup,axiom,
% 14.25/8.08 ! [A: set_Product_unit] :
% 14.25/8.08 ( ( finite4290736615968046902t_unit @ A )
% 14.25/8.08 => ( ( A != bot_bo3957492148770167129t_unit )
% 14.25/8.08 => ( ( lattic1210392890313158908t_unit @ A )
% 14.25/8.08 = ( comple4687483117567863418t_unit @ A ) ) ) ) ).
% 14.25/8.08
% 14.25/8.08 thf(fact_4281_disjoint__alt__simp3,axiom,
% 14.25/8.08 ! [A: set_Product_unit,B4: set_Product_unit] :
% 14.25/8.08 ( ( ord_le8056459307392131481t_unit @ ( minus_6452836326544984404t_unit @ A @ B4 ) @ A )
% 14.25/8.08 = ( ( inf_in4660618365625256667t_unit @ A @ B4 )
% 14.25/8.08 != bot_bo3957492148770167129t_unit ) ) ).
% 14.25/8.08
% 14.25/8.08 thf(fact_4282_disjoint__alt__simp3,axiom,
% 14.25/8.08 ! [A: set_o,B4: set_o] :
% 14.25/8.08 ( ( ord_less_set_o @ ( minus_minus_set_o @ A @ B4 ) @ A )
% 14.25/8.08 = ( ( inf_inf_set_o @ A @ B4 )
% 14.25/8.08 != bot_bot_set_o ) ) ).
% 14.25/8.08
% 14.25/8.08 thf(fact_4283_disjoint__alt__simp3,axiom,
% 14.25/8.08 ! [A: set_int,B4: set_int] :
% 14.25/8.08 ( ( ord_less_set_int @ ( minus_minus_set_int @ A @ B4 ) @ A )
% 14.25/8.08 = ( ( inf_inf_set_int @ A @ B4 )
% 14.25/8.08 != bot_bot_set_int ) ) ).
% 14.25/8.08
% 14.25/8.08 thf(fact_4284_disjoint__alt__simp3,axiom,
% 14.25/8.08 ! [A: set_Pr4532377907799695533_nat_o,B4: set_Pr4532377907799695533_nat_o] :
% 14.25/8.08 ( ( ord_le2453136405763929_nat_o @ ( minus_1801376950450012436_nat_o @ A @ B4 ) @ A )
% 14.25/8.08 = ( ( inf_in1906310914598751387_nat_o @ A @ B4 )
% 14.25/8.08 != bot_bo7824918357723582233_nat_o ) ) ).
% 14.25/8.08
% 14.25/8.08 thf(fact_4285_disjoint__alt__simp3,axiom,
% 14.25/8.08 ! [A: set_nat,B4: set_nat] :
% 14.25/8.08 ( ( ord_less_set_nat @ ( minus_minus_set_nat @ A @ B4 ) @ A )
% 14.25/8.08 = ( ( inf_inf_set_nat @ A @ B4 )
% 14.25/8.08 != bot_bot_set_nat ) ) ).
% 14.25/8.08
% 14.25/8.08 thf(fact_4286_range__eq__singletonD,axiom,
% 14.25/8.08 ! [F: code_integer > code_integer,A2: code_integer,X: code_integer] :
% 14.25/8.08 ( ( ( image_4470545334726330049nteger @ F @ top_to4645266643341252675nteger )
% 14.25/8.08 = ( insert_Code_integer @ A2 @ bot_bo3990330152332043303nteger ) )
% 14.25/8.08 => ( ( F @ X )
% 14.25/8.08 = A2 ) ) ).
% 14.25/8.08
% 14.25/8.08 thf(fact_4287_range__eq__singletonD,axiom,
% 14.25/8.08 ! [F: product_unit > product_unit,A2: product_unit,X: product_unit] :
% 14.25/8.08 ( ( ( image_405062704495631173t_unit @ F @ top_to1996260823553986621t_unit )
% 14.25/8.08 = ( insert_Product_unit @ A2 @ bot_bo3957492148770167129t_unit ) )
% 14.25/8.08 => ( ( F @ X )
% 14.25/8.08 = A2 ) ) ).
% 14.25/8.08
% 14.25/8.08 thf(fact_4288_range__eq__singletonD,axiom,
% 14.25/8.08 ! [F: product_unit > $o,A2: $o,X: product_unit] :
% 14.25/8.08 ( ( ( image_Product_unit_o2 @ F @ top_to1996260823553986621t_unit )
% 14.25/8.08 = ( insert_o @ A2 @ bot_bot_set_o ) )
% 14.25/8.08 => ( ( F @ X )
% 14.25/8.08 = A2 ) ) ).
% 14.25/8.08
% 14.25/8.08 thf(fact_4289_range__eq__singletonD,axiom,
% 14.25/8.08 ! [F: product_unit > nat,A2: nat,X: product_unit] :
% 14.25/8.08 ( ( ( image_875570014554754200it_nat @ F @ top_to1996260823553986621t_unit )
% 14.25/8.08 = ( insert_nat @ A2 @ bot_bot_set_nat ) )
% 14.25/8.08 => ( ( F @ X )
% 14.25/8.08 = A2 ) ) ).
% 14.25/8.08
% 14.25/8.08 thf(fact_4290_range__eq__singletonD,axiom,
% 14.25/8.08 ! [F: product_unit > int,A2: int,X: product_unit] :
% 14.25/8.08 ( ( ( image_873079544045703924it_int @ F @ top_to1996260823553986621t_unit )
% 14.25/8.08 = ( insert_int @ A2 @ bot_bot_set_int ) )
% 14.25/8.08 => ( ( F @ X )
% 14.25/8.08 = A2 ) ) ).
% 14.25/8.08
% 14.25/8.08 thf(fact_4291_range__eq__singletonD,axiom,
% 14.25/8.08 ! [F: $o > product_unit,A2: product_unit,X: $o] :
% 14.25/8.08 ( ( ( image_o_Product_unit2 @ F @ top_top_set_o )
% 14.25/8.08 = ( insert_Product_unit @ A2 @ bot_bo3957492148770167129t_unit ) )
% 14.25/8.08 => ( ( F @ X )
% 14.25/8.08 = A2 ) ) ).
% 14.25/8.08
% 14.25/8.08 thf(fact_4292_range__eq__singletonD,axiom,
% 14.25/8.08 ! [F: $o > $o,A2: $o,X: $o] :
% 14.25/8.08 ( ( ( image_o_o2 @ F @ top_top_set_o )
% 14.25/8.08 = ( insert_o @ A2 @ bot_bot_set_o ) )
% 14.25/8.08 => ( ( F @ X )
% 14.25/8.08 = A2 ) ) ).
% 14.25/8.08
% 14.25/8.08 thf(fact_4293_range__eq__singletonD,axiom,
% 14.25/8.08 ! [F: $o > nat,A2: nat,X: $o] :
% 14.25/8.08 ( ( ( image_o_nat2 @ F @ top_top_set_o )
% 14.25/8.08 = ( insert_nat @ A2 @ bot_bot_set_nat ) )
% 14.25/8.08 => ( ( F @ X )
% 14.25/8.08 = A2 ) ) ).
% 14.25/8.08
% 14.25/8.08 thf(fact_4294_range__eq__singletonD,axiom,
% 14.25/8.08 ! [F: $o > int,A2: int,X: $o] :
% 14.25/8.08 ( ( ( image_o_int2 @ F @ top_top_set_o )
% 14.25/8.08 = ( insert_int @ A2 @ bot_bot_set_int ) )
% 14.25/8.08 => ( ( F @ X )
% 14.25/8.08 = A2 ) ) ).
% 14.25/8.08
% 14.25/8.08 thf(fact_4295_range__eq__singletonD,axiom,
% 14.25/8.08 ! [F: nat > code_integer,A2: code_integer,X: nat] :
% 14.25/8.08 ( ( ( image_1215581382706833972nteger @ F @ top_top_set_nat )
% 14.25/8.08 = ( insert_Code_integer @ A2 @ bot_bo3990330152332043303nteger ) )
% 14.25/8.08 => ( ( F @ X )
% 14.25/8.08 = A2 ) ) ).
% 14.25/8.08
% 14.25/8.08 thf(fact_4296_less__cSup__iff,axiom,
% 14.25/8.08 ! [X4: set_int,Y: int] :
% 14.25/8.08 ( ( X4 != bot_bot_set_int )
% 14.25/8.08 => ( ( condit2212336002400062152ve_int @ X4 )
% 14.25/8.08 => ( ( ord_less_int @ Y @ ( complete_Sup_Sup_int @ X4 ) )
% 14.25/8.08 = ( ? [X2: int] :
% 14.25/8.08 ( ( member_int @ X2 @ X4 )
% 14.25/8.08 & ( ord_less_int @ Y @ X2 ) ) ) ) ) ) ).
% 14.25/8.08
% 14.25/8.08 thf(fact_4297_less__cSup__iff,axiom,
% 14.25/8.08 ! [X4: set_nat,Y: nat] :
% 14.25/8.08 ( ( X4 != bot_bot_set_nat )
% 14.25/8.08 => ( ( condit2214826472909112428ve_nat @ X4 )
% 14.25/8.08 => ( ( ord_less_nat @ Y @ ( complete_Sup_Sup_nat @ X4 ) )
% 14.25/8.08 = ( ? [X2: nat] :
% 14.25/8.08 ( ( member_nat @ X2 @ X4 )
% 14.25/8.08 & ( ord_less_nat @ Y @ X2 ) ) ) ) ) ) ).
% 14.25/8.08
% 14.25/8.08 thf(fact_4298_less__cSup__iff,axiom,
% 14.25/8.08 ! [X4: set_Product_unit,Y: product_unit] :
% 14.25/8.08 ( ( X4 != bot_bo3957492148770167129t_unit )
% 14.25/8.08 => ( ( condit2080475194861589617t_unit @ X4 )
% 14.25/8.08 => ( ( ord_le361264281704409273t_unit @ Y @ ( comple4687483117567863418t_unit @ X4 ) )
% 14.25/8.08 = ( ? [X2: product_unit] :
% 14.25/8.08 ( ( member_Product_unit @ X2 @ X4 )
% 14.25/8.08 & ( ord_le361264281704409273t_unit @ Y @ X2 ) ) ) ) ) ) ).
% 14.25/8.08
% 14.25/8.08 thf(fact_4299_underS__Field3,axiom,
% 14.25/8.08 ! [R2: set_Pr5094982260447487303t_unit,A2: product_unit] :
% 14.25/8.08 ( ( ( field_Product_unit @ R2 )
% 14.25/8.08 != bot_bo3957492148770167129t_unit )
% 14.25/8.08 => ( ord_le8056459307392131481t_unit @ ( order_5809128134054296019t_unit @ R2 @ A2 ) @ ( field_Product_unit @ R2 ) ) ) ).
% 14.25/8.08
% 14.25/8.08 thf(fact_4300_underS__Field3,axiom,
% 14.25/8.08 ! [R2: set_Product_prod_o_o,A2: $o] :
% 14.25/8.08 ( ( ( field_o @ R2 )
% 14.25/8.08 != bot_bot_set_o )
% 14.25/8.08 => ( ord_less_set_o @ ( order_underS_o @ R2 @ A2 ) @ ( field_o @ R2 ) ) ) ).
% 14.25/8.08
% 14.25/8.08 thf(fact_4301_underS__Field3,axiom,
% 14.25/8.08 ! [R2: set_Pr1261947904930325089at_nat,A2: nat] :
% 14.25/8.08 ( ( ( field_nat @ R2 )
% 14.25/8.08 != bot_bot_set_nat )
% 14.25/8.08 => ( ord_less_set_nat @ ( order_underS_nat @ R2 @ A2 ) @ ( field_nat @ R2 ) ) ) ).
% 14.25/8.08
% 14.25/8.08 thf(fact_4302_underS__Field3,axiom,
% 14.25/8.08 ! [R2: set_Pr958786334691620121nt_int,A2: int] :
% 14.25/8.08 ( ( ( field_int @ R2 )
% 14.25/8.08 != bot_bot_set_int )
% 14.25/8.08 => ( ord_less_set_int @ ( order_underS_int @ R2 @ A2 ) @ ( field_int @ R2 ) ) ) ).
% 14.25/8.08
% 14.25/8.08 thf(fact_4303_underS__Field3,axiom,
% 14.25/8.08 ! [R2: set_Pr2161125870931222855_nat_o,A2: produc3658429121746597890et_nat > $o] :
% 14.25/8.08 ( ( ( field_81821745747897882_nat_o @ R2 )
% 14.25/8.08 != bot_bo7824918357723582233_nat_o )
% 14.25/8.08 => ( ord_le2453136405763929_nat_o @ ( order_7262550146012401043_nat_o @ R2 @ A2 ) @ ( field_81821745747897882_nat_o @ R2 ) ) ) ).
% 14.25/8.08
% 14.25/8.08 thf(fact_4304_arg__min__if__finite_I2_J,axiom,
% 14.25/8.08 ! [S: set_Code_integer,F: code_integer > assn] :
% 14.25/8.08 ( ( finite6017078050557962740nteger @ S )
% 14.25/8.08 => ( ( S != bot_bo3990330152332043303nteger )
% 14.25/8.08 => ~ ? [X6: code_integer] :
% 14.25/8.08 ( ( member_Code_integer @ X6 @ S )
% 14.25/8.08 & ( ord_less_assn @ ( F @ X6 ) @ ( F @ ( lattic4682378690128378784r_assn @ F @ S ) ) ) ) ) ) ).
% 14.25/8.08
% 14.25/8.08 thf(fact_4305_arg__min__if__finite_I2_J,axiom,
% 14.25/8.08 ! [S: set_Product_unit,F: product_unit > assn] :
% 14.25/8.08 ( ( finite4290736615968046902t_unit @ S )
% 14.25/8.08 => ( ( S != bot_bo3957492148770167129t_unit )
% 14.25/8.08 => ~ ? [X6: product_unit] :
% 14.25/8.08 ( ( member_Product_unit @ X6 @ S )
% 14.25/8.08 & ( ord_less_assn @ ( F @ X6 ) @ ( F @ ( lattic4376901495776413404t_assn @ F @ S ) ) ) ) ) ) ).
% 14.25/8.08
% 14.25/8.08 thf(fact_4306_arg__min__if__finite_I2_J,axiom,
% 14.25/8.08 ! [S: set_o,F: $o > assn] :
% 14.25/8.08 ( ( finite_finite_o @ S )
% 14.25/8.08 => ( ( S != bot_bot_set_o )
% 14.25/8.08 => ~ ? [X6: $o] :
% 14.25/8.08 ( ( member_o @ X6 @ S )
% 14.25/8.08 & ( ord_less_assn @ ( F @ X6 ) @ ( F @ ( lattic7708394068118247271o_assn @ F @ S ) ) ) ) ) ) ).
% 14.25/8.08
% 14.25/8.08 thf(fact_4307_arg__min__if__finite_I2_J,axiom,
% 14.25/8.08 ! [S: set_nat,F: nat > assn] :
% 14.25/8.08 ( ( finite_finite_nat @ S )
% 14.25/8.08 => ( ( S != bot_bot_set_nat )
% 14.25/8.08 => ~ ? [X6: nat] :
% 14.25/8.08 ( ( member_nat @ X6 @ S )
% 14.25/8.08 & ( ord_less_assn @ ( F @ X6 ) @ ( F @ ( lattic2187264146484958483t_assn @ F @ S ) ) ) ) ) ) ).
% 14.25/8.08
% 14.25/8.08 thf(fact_4308_arg__min__if__finite_I2_J,axiom,
% 14.25/8.08 ! [S: set_int,F: int > assn] :
% 14.25/8.08 ( ( finite_finite_int @ S )
% 14.25/8.08 => ( ( S != bot_bot_set_int )
% 14.25/8.08 => ~ ? [X6: int] :
% 14.25/8.08 ( ( member_int @ X6 @ S )
% 14.25/8.08 & ( ord_less_assn @ ( F @ X6 ) @ ( F @ ( lattic3162799611691135855t_assn @ F @ S ) ) ) ) ) ) ).
% 14.25/8.08
% 14.25/8.08 thf(fact_4309_arg__min__if__finite_I2_J,axiom,
% 14.25/8.08 ! [S: set_Code_integer,F: code_integer > nat] :
% 14.25/8.08 ( ( finite6017078050557962740nteger @ S )
% 14.25/8.08 => ( ( S != bot_bo3990330152332043303nteger )
% 14.25/8.08 => ~ ? [X6: code_integer] :
% 14.25/8.08 ( ( member_Code_integer @ X6 @ S )
% 14.25/8.08 & ( ord_less_nat @ ( F @ X6 ) @ ( F @ ( lattic6074936555552774728er_nat @ F @ S ) ) ) ) ) ) ).
% 14.25/8.08
% 14.25/8.08 thf(fact_4310_arg__min__if__finite_I2_J,axiom,
% 14.25/8.08 ! [S: set_Product_unit,F: product_unit > nat] :
% 14.25/8.08 ( ( finite4290736615968046902t_unit @ S )
% 14.25/8.08 => ( ( S != bot_bo3957492148770167129t_unit )
% 14.25/8.08 => ~ ? [X6: product_unit] :
% 14.25/8.08 ( ( member_Product_unit @ X6 @ S )
% 14.25/8.08 & ( ord_less_nat @ ( F @ X6 ) @ ( F @ ( lattic1360832392136924548it_nat @ F @ S ) ) ) ) ) ) ).
% 14.25/8.08
% 14.25/8.08 thf(fact_4311_arg__min__if__finite_I2_J,axiom,
% 14.25/8.08 ! [S: set_o,F: $o > nat] :
% 14.25/8.08 ( ( finite_finite_o @ S )
% 14.25/8.08 => ( ( S != bot_bot_set_o )
% 14.25/8.08 => ~ ? [X6: $o] :
% 14.25/8.08 ( ( member_o @ X6 @ S )
% 14.25/8.08 & ( ord_less_nat @ ( F @ X6 ) @ ( F @ ( lattic2775856028456453135_o_nat @ F @ S ) ) ) ) ) ) ).
% 14.25/8.08
% 14.25/8.08 thf(fact_4312_arg__min__if__finite_I2_J,axiom,
% 14.25/8.08 ! [S: set_nat,F: nat > nat] :
% 14.25/8.08 ( ( finite_finite_nat @ S )
% 14.25/8.08 => ( ( S != bot_bot_set_nat )
% 14.25/8.08 => ~ ? [X6: nat] :
% 14.25/8.08 ( ( member_nat @ X6 @ S )
% 14.25/8.08 & ( ord_less_nat @ ( F @ X6 ) @ ( F @ ( lattic7446932960582359483at_nat @ F @ S ) ) ) ) ) ) ).
% 14.25/8.08
% 14.25/8.08 thf(fact_4313_arg__min__if__finite_I2_J,axiom,
% 14.25/8.08 ! [S: set_int,F: int > nat] :
% 14.25/8.08 ( ( finite_finite_int @ S )
% 14.25/8.08 => ( ( S != bot_bot_set_int )
% 14.25/8.08 => ~ ? [X6: int] :
% 14.25/8.08 ( ( member_int @ X6 @ S )
% 14.25/8.08 & ( ord_less_nat @ ( F @ X6 ) @ ( F @ ( lattic8446286672483414039nt_nat @ F @ S ) ) ) ) ) ) ).
% 14.25/8.08
% 14.25/8.08 thf(fact_4314_Sup__fold__sup,axiom,
% 14.25/8.08 ! [A: set_set_Product_unit] :
% 14.25/8.08 ( ( finite1772178364199683094t_unit @ A )
% 14.25/8.08 => ( ( comple3349607820504768346t_unit @ A )
% 14.25/8.08 = ( finite4243028863286625800t_unit @ sup_su793286257634532545t_unit @ bot_bo3957492148770167129t_unit @ A ) ) ) ).
% 14.25/8.08
% 14.25/8.08 thf(fact_4315_Sup__fold__sup,axiom,
% 14.25/8.08 ! [A: set_set_o] :
% 14.25/8.08 ( ( finite_finite_set_o @ A )
% 14.25/8.08 => ( ( comple90263536869209701_set_o @ A )
% 14.25/8.08 = ( finite4337638375924247368_set_o @ sup_sup_set_o @ bot_bot_set_o @ A ) ) ) ).
% 14.25/8.08
% 14.25/8.08 thf(fact_4316_Sup__fold__sup,axiom,
% 14.25/8.08 ! [A: set_set_nat] :
% 14.25/8.08 ( ( finite1152437895449049373et_nat @ A )
% 14.25/8.08 => ( ( comple7399068483239264473et_nat @ A )
% 14.25/8.08 = ( finite677925301803182934et_nat @ sup_sup_set_nat @ bot_bot_set_nat @ A ) ) ) ).
% 14.25/8.08
% 14.25/8.08 thf(fact_4317_Sup__fold__sup,axiom,
% 14.25/8.08 ! [A: set_set_int] :
% 14.25/8.08 ( ( finite6197958912794628473et_int @ A )
% 14.25/8.08 => ( ( comple3221217463730067765et_int @ A )
% 14.25/8.08 = ( finite2508883933289561102et_int @ sup_sup_set_int @ bot_bot_set_int @ A ) ) ) ).
% 14.25/8.08
% 14.25/8.08 thf(fact_4318_Sup__fold__sup,axiom,
% 14.25/8.08 ! [A: set_se1353816476878259085_nat_o] :
% 14.25/8.08 ( ( finite6069017635265841110_nat_o @ A )
% 14.25/8.08 => ( ( comple3382087578465172250_nat_o @ A )
% 14.25/8.08 = ( finite8456856254070787592_nat_o @ sup_su5209123915105501825_nat_o @ bot_bo7824918357723582233_nat_o @ A ) ) ) ).
% 14.25/8.08
% 14.25/8.08 thf(fact_4319_Sup__fold__sup,axiom,
% 14.25/8.08 ! [A: set_Product_unit] :
% 14.25/8.08 ( ( finite4290736615968046902t_unit @ A )
% 14.25/8.08 => ( ( comple4687483117567863418t_unit @ A )
% 14.25/8.08 = ( finite8834431295148555784t_unit @ sup_sup_Product_unit @ bot_bot_Product_unit @ A ) ) ) ).
% 14.25/8.08
% 14.25/8.08 thf(fact_4320_SUP__fold__sup,axiom,
% 14.25/8.08 ! [A: set_nat,F: nat > product_unit] :
% 14.25/8.08 ( ( finite_finite_nat @ A )
% 14.25/8.08 => ( ( comple4687483117567863418t_unit @ ( image_8730104196221521654t_unit @ F @ A ) )
% 14.25/8.08 = ( finite6602901031542678131t_unit @ ( comp_P6204500334276302049it_nat @ sup_sup_Product_unit @ F ) @ bot_bot_Product_unit @ A ) ) ) ).
% 14.25/8.08
% 14.25/8.08 thf(fact_4321_SUP__fold__sup,axiom,
% 14.25/8.08 ! [A: set_Code_integer,F: code_integer > product_unit] :
% 14.25/8.08 ( ( finite6017078050557962740nteger @ A )
% 14.25/8.08 => ( ( comple4687483117567863418t_unit @ ( image_7182656606676592809t_unit @ F @ A ) )
% 14.25/8.08 = ( finite2281032298429728806t_unit @ ( comp_P4558236313141964142nteger @ sup_sup_Product_unit @ F ) @ bot_bot_Product_unit @ A ) ) ) ).
% 14.25/8.08
% 14.25/8.08 thf(fact_4322_SUP__fold__sup,axiom,
% 14.25/8.08 ! [A: set_nat,F: nat > set_Product_unit] :
% 14.25/8.08 ( ( finite_finite_nat @ A )
% 14.25/8.08 => ( ( comple3349607820504768346t_unit @ ( image_4824033694279209942t_unit @ F @ A ) )
% 14.25/8.08 = ( finite9145781422887121747t_unit @ ( comp_s6943286182646368705it_nat @ sup_su793286257634532545t_unit @ F ) @ bot_bo3957492148770167129t_unit @ A ) ) ) ).
% 14.25/8.08
% 14.25/8.08 thf(fact_4323_SUP__fold__sup,axiom,
% 14.25/8.08 ! [A: set_Code_integer,F: code_integer > set_Product_unit] :
% 14.25/8.08 ( ( finite6017078050557962740nteger @ A )
% 14.25/8.08 => ( ( comple3349607820504768346t_unit @ ( image_5870443399019143561t_unit @ F @ A ) )
% 14.25/8.08 = ( finite3687622995246107910t_unit @ ( comp_s3199726385246003790nteger @ sup_su793286257634532545t_unit @ F ) @ bot_bo3957492148770167129t_unit @ A ) ) ) ).
% 14.25/8.08
% 14.25/8.08 thf(fact_4324_SUP__fold__sup,axiom,
% 14.25/8.08 ! [A: set_nat,F: nat > set_o] :
% 14.25/8.08 ( ( finite_finite_nat @ A )
% 14.25/8.08 => ( ( comple90263536869209701_set_o @ ( image_nat_set_o @ F @ A ) )
% 14.25/8.08 = ( finite3217087857726763998_set_o @ ( comp_s4424444925706540684_o_nat @ sup_sup_set_o @ F ) @ bot_bot_set_o @ A ) ) ) ).
% 14.25/8.08
% 14.25/8.08 thf(fact_4325_SUP__fold__sup,axiom,
% 14.25/8.08 ! [A: set_Code_integer,F: code_integer > set_o] :
% 14.25/8.08 ( ( finite6017078050557962740nteger @ A )
% 14.25/8.08 => ( ( comple90263536869209701_set_o @ ( image_1867029960091478804_set_o @ F @ A ) )
% 14.25/8.08 = ( finite1928688952044918289_set_o @ ( comp_s90868338361139993nteger @ sup_sup_set_o @ F ) @ bot_bot_set_o @ A ) ) ) ).
% 14.25/8.08
% 14.25/8.08 thf(fact_4326_SUP__fold__sup,axiom,
% 14.25/8.08 ! [A: set_nat,F: nat > set_nat] :
% 14.25/8.08 ( ( finite_finite_nat @ A )
% 14.25/8.08 => ( ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ F @ A ) )
% 14.25/8.08 = ( finite5529483035118572448et_nat @ ( comp_s2351873599094224870at_nat @ sup_sup_set_nat @ F ) @ bot_bot_set_nat @ A ) ) ) ).
% 14.25/8.08
% 14.25/8.08 thf(fact_4327_SUP__fold__sup,axiom,
% 14.25/8.08 ! [A: set_Code_integer,F: code_integer > set_nat] :
% 14.25/8.08 ( ( finite6017078050557962740nteger @ A )
% 14.25/8.08 => ( ( comple7399068483239264473et_nat @ ( image_8829277216472192490et_nat @ F @ A ) )
% 14.25/8.08 = ( finite1590314701948642349et_nat @ ( comp_s3169385134670675059nteger @ sup_sup_set_nat @ F ) @ bot_bot_set_nat @ A ) ) ) ).
% 14.25/8.08
% 14.25/8.08 thf(fact_4328_SUP__fold__sup,axiom,
% 14.25/8.08 ! [A: set_nat,F: nat > set_int] :
% 14.25/8.08 ( ( finite_finite_nat @ A )
% 14.25/8.08 => ( ( comple3221217463730067765et_int @ ( image_nat_set_int @ F @ A ) )
% 14.25/8.08 = ( finite1351632015609375740et_int @ ( comp_s6689493721604790522nt_nat @ sup_sup_set_int @ F ) @ bot_bot_set_int @ A ) ) ) ).
% 14.25/8.08
% 14.25/8.08 thf(fact_4329_SUP__fold__sup,axiom,
% 14.25/8.08 ! [A: set_Code_integer,F: code_integer > set_int] :
% 14.25/8.08 ( ( finite6017078050557962740nteger @ A )
% 14.25/8.08 => ( ( comple3221217463730067765et_int @ ( image_4651426196962995782et_int @ F @ A ) )
% 14.25/8.08 = ( finite6635835719294221449et_int @ ( comp_s3634918272309710215nteger @ sup_sup_set_int @ F ) @ bot_bot_set_int @ A ) ) ) ).
% 14.25/8.08
% 14.25/8.08 thf(fact_4330_finite__subset__Union__chain,axiom,
% 14.25/8.08 ! [A: set_Code_integer,B9: set_set_Code_integer,A11: set_set_Code_integer] :
% 14.25/8.08 ( ( finite6017078050557962740nteger @ A )
% 14.25/8.08 => ( ( ord_le7084787975880047091nteger @ A @ ( comple739944243200306918nteger @ B9 ) )
% 14.25/8.08 => ( ( B9 != bot_bo9204850693367297629nteger )
% 14.25/8.08 => ( ( pred_c5891848916688490280nteger @ A11 @ ord_le1307284697595431911nteger @ B9 )
% 14.25/8.08 => ~ ! [B5: set_Code_integer] :
% 14.25/8.08 ( ( member1834298987299537450nteger @ B5 @ B9 )
% 14.25/8.08 => ~ ( ord_le7084787975880047091nteger @ A @ B5 ) ) ) ) ) ) ).
% 14.25/8.08
% 14.25/8.08 thf(fact_4331_finite__subset__Union__chain,axiom,
% 14.25/8.08 ! [A: set_nat,B9: set_set_nat,A11: set_set_nat] :
% 14.25/8.08 ( ( finite_finite_nat @ A )
% 14.25/8.08 => ( ( ord_less_eq_set_nat @ A @ ( comple7399068483239264473et_nat @ B9 ) )
% 14.25/8.08 => ( ( B9 != bot_bot_set_set_nat )
% 14.25/8.08 => ( ( pred_chain_set_nat @ A11 @ ord_less_set_nat @ B9 )
% 14.25/8.08 => ~ ! [B5: set_nat] :
% 14.25/8.08 ( ( member_set_nat @ B5 @ B9 )
% 14.25/8.08 => ~ ( ord_less_eq_set_nat @ A @ B5 ) ) ) ) ) ) ).
% 14.25/8.08
% 14.25/8.08 thf(fact_4332_merge__true__star,axiom,
% 14.25/8.08 ( ( times_times_assn @ top_top_assn @ top_top_assn )
% 14.25/8.08 = top_top_assn ) ).
% 14.25/8.08
% 14.25/8.08 thf(fact_4333_assn__basic__inequalities_I5_J,axiom,
% 14.25/8.08 top_top_assn != bot_bot_assn ).
% 14.25/8.08
% 14.25/8.08 thf(fact_4334_assn__basic__inequalities_I1_J,axiom,
% 14.25/8.08 top_top_assn != one_one_assn ).
% 14.25/8.08
% 14.25/8.08 thf(fact_4335_mod__true,axiom,
% 14.25/8.08 ! [H2: produc3658429121746597890et_nat] :
% 14.25/8.08 ( ( rep_assn @ top_top_assn @ H2 )
% 14.25/8.08 = ( in_range @ H2 ) ) ).
% 14.25/8.08
% 14.25/8.08 thf(fact_4336_fun__comp__eq__conv,axiom,
% 14.25/8.08 ! [F: code_integer > produc8923325533196201883nteger > produc8923325533196201883nteger,G: code_integer > code_integer,Fg: code_integer > produc8923325533196201883nteger > produc8923325533196201883nteger] :
% 14.25/8.08 ( ( ( comp_C1593894019821074884nteger @ F @ G )
% 14.25/8.08 = Fg )
% 14.25/8.08 = ( ! [X2: code_integer] :
% 14.25/8.08 ( ( F @ ( G @ X2 ) )
% 14.25/8.08 = ( Fg @ X2 ) ) ) ) ).
% 14.25/8.08
% 14.25/8.08 thf(fact_4337_fun__comp__eq__conv,axiom,
% 14.25/8.08 ! [F: ( code_integer > code_integer ) > produc8923325533196201883nteger > produc8923325533196201883nteger,G: code_integer > code_integer > code_integer,Fg: code_integer > produc8923325533196201883nteger > produc8923325533196201883nteger] :
% 14.25/8.08 ( ( ( comp_C8797469213163452608nteger @ F @ G )
% 14.25/8.08 = Fg )
% 14.25/8.08 = ( ! [X2: code_integer] :
% 14.25/8.08 ( ( F @ ( G @ X2 ) )
% 14.25/8.08 = ( Fg @ X2 ) ) ) ) ).
% 14.25/8.08
% 14.25/8.08 thf(fact_4338_comp__cong__right,axiom,
% 14.25/8.08 ! [X: code_integer > code_integer,Y: code_integer > code_integer,F: code_integer > produc8923325533196201883nteger > produc8923325533196201883nteger] :
% 14.25/8.08 ( ( X = Y )
% 14.25/8.08 => ( ( comp_C1593894019821074884nteger @ F @ X )
% 14.25/8.08 = ( comp_C1593894019821074884nteger @ F @ Y ) ) ) ).
% 14.25/8.08
% 14.25/8.08 thf(fact_4339_comp__cong__right,axiom,
% 14.25/8.08 ! [X: code_integer > code_integer > code_integer,Y: code_integer > code_integer > code_integer,F: ( code_integer > code_integer ) > produc8923325533196201883nteger > produc8923325533196201883nteger] :
% 14.25/8.08 ( ( X = Y )
% 14.25/8.08 => ( ( comp_C8797469213163452608nteger @ F @ X )
% 14.25/8.08 = ( comp_C8797469213163452608nteger @ F @ Y ) ) ) ).
% 14.25/8.08
% 14.25/8.08 thf(fact_4340_comp__cong__left,axiom,
% 14.25/8.08 ! [X: code_integer > produc8923325533196201883nteger > produc8923325533196201883nteger,Y: code_integer > produc8923325533196201883nteger > produc8923325533196201883nteger,F: code_integer > code_integer] :
% 14.25/8.08 ( ( X = Y )
% 14.25/8.08 => ( ( comp_C1593894019821074884nteger @ X @ F )
% 14.25/8.08 = ( comp_C1593894019821074884nteger @ Y @ F ) ) ) ).
% 14.25/8.08
% 14.25/8.08 thf(fact_4341_comp__cong__left,axiom,
% 14.25/8.08 ! [X: ( code_integer > code_integer ) > produc8923325533196201883nteger > produc8923325533196201883nteger,Y: ( code_integer > code_integer ) > produc8923325533196201883nteger > produc8923325533196201883nteger,F: code_integer > code_integer > code_integer] :
% 14.25/8.08 ( ( X = Y )
% 14.25/8.08 => ( ( comp_C8797469213163452608nteger @ X @ F )
% 14.25/8.08 = ( comp_C8797469213163452608nteger @ Y @ F ) ) ) ).
% 14.25/8.08
% 14.25/8.08 thf(fact_4342_comp__apply__eq,axiom,
% 14.25/8.08 ! [F: code_integer > produc8923325533196201883nteger > produc8923325533196201883nteger,G: code_integer > code_integer,X: code_integer,H2: code_integer > produc8923325533196201883nteger > produc8923325533196201883nteger,K3: code_integer > code_integer] :
% 14.25/8.08 ( ( ( F @ ( G @ X ) )
% 14.25/8.08 = ( H2 @ ( K3 @ X ) ) )
% 14.25/8.08 => ( ( comp_C1593894019821074884nteger @ F @ G @ X )
% 14.25/8.08 = ( comp_C1593894019821074884nteger @ H2 @ K3 @ X ) ) ) ).
% 14.25/8.08
% 14.25/8.08 thf(fact_4343_comp__apply__eq,axiom,
% 14.25/8.08 ! [F: code_integer > produc8923325533196201883nteger > produc8923325533196201883nteger,G: code_integer > code_integer,X: code_integer,H2: ( code_integer > code_integer ) > produc8923325533196201883nteger > produc8923325533196201883nteger,K3: code_integer > code_integer > code_integer] :
% 14.25/8.08 ( ( ( F @ ( G @ X ) )
% 14.25/8.08 = ( H2 @ ( K3 @ X ) ) )
% 14.25/8.08 => ( ( comp_C1593894019821074884nteger @ F @ G @ X )
% 14.25/8.08 = ( comp_C8797469213163452608nteger @ H2 @ K3 @ X ) ) ) ).
% 14.25/8.08
% 14.25/8.08 thf(fact_4344_comp__apply__eq,axiom,
% 14.25/8.08 ! [F: ( code_integer > code_integer ) > produc8923325533196201883nteger > produc8923325533196201883nteger,G: code_integer > code_integer > code_integer,X: code_integer,H2: code_integer > produc8923325533196201883nteger > produc8923325533196201883nteger,K3: code_integer > code_integer] :
% 14.25/8.08 ( ( ( F @ ( G @ X ) )
% 14.25/8.08 = ( H2 @ ( K3 @ X ) ) )
% 14.25/8.08 => ( ( comp_C8797469213163452608nteger @ F @ G @ X )
% 14.25/8.08 = ( comp_C1593894019821074884nteger @ H2 @ K3 @ X ) ) ) ).
% 14.25/8.08
% 14.25/8.08 thf(fact_4345_comp__apply__eq,axiom,
% 14.25/8.08 ! [F: ( code_integer > code_integer ) > produc8923325533196201883nteger > produc8923325533196201883nteger,G: code_integer > code_integer > code_integer,X: code_integer,H2: ( code_integer > code_integer ) > produc8923325533196201883nteger > produc8923325533196201883nteger,K3: code_integer > code_integer > code_integer] :
% 14.25/8.08 ( ( ( F @ ( G @ X ) )
% 14.25/8.08 = ( H2 @ ( K3 @ X ) ) )
% 14.25/8.08 => ( ( comp_C8797469213163452608nteger @ F @ G @ X )
% 14.25/8.08 = ( comp_C8797469213163452608nteger @ H2 @ K3 @ X ) ) ) ).
% 14.25/8.08
% 14.25/8.08 thf(fact_4346_type__copy__set__map0,axiom,
% 14.25/8.08 ! [Rep2: ( produc8923325533196201883nteger > produc8923325533196201883nteger ) > code_integer,Abs2: code_integer > produc8923325533196201883nteger > produc8923325533196201883nteger,S: code_integer > set_nat,M: code_integer > code_integer,F: nat > nat,S4: code_integer > set_nat,G: code_integer > code_integer] :
% 14.25/8.08 ( ( type_d6544512618915850416nteger @ Rep2 @ Abs2 @ top_to4645266643341252675nteger )
% 14.25/8.08 => ( ( ( comp_C6537567860429791781nteger @ S @ M )
% 14.25/8.08 = ( comp_s4605850120322998606nteger @ ( image_nat_nat2 @ F ) @ S4 ) )
% 14.25/8.08 => ( ( comp_P8262219321096090349nteger @ ( comp_C3184339336801759213nteger @ S @ Rep2 ) @ ( comp_C1593894019821074884nteger @ ( comp_C1593894019821074884nteger @ Abs2 @ M ) @ G ) )
% 14.25/8.08 = ( comp_s4605850120322998606nteger @ ( image_nat_nat2 @ F ) @ ( comp_C6537567860429791781nteger @ S4 @ G ) ) ) ) ) ).
% 14.25/8.08
% 14.25/8.08 thf(fact_4347_type__copy__set__map0,axiom,
% 14.25/8.08 ! [Rep2: ( produc8923325533196201883nteger > produc8923325533196201883nteger ) > code_integer,Abs2: code_integer > produc8923325533196201883nteger > produc8923325533196201883nteger,S: code_integer > set_Code_integer,M: code_integer > code_integer,F: code_integer > code_integer,S4: code_integer > set_Code_integer,G: code_integer > code_integer] :
% 14.25/8.08 ( ( type_d6544512618915850416nteger @ Rep2 @ Abs2 @ top_to4645266643341252675nteger )
% 14.25/8.08 => ( ( ( comp_C8186931911283765298nteger @ S @ M )
% 14.25/8.08 = ( comp_s956550612328592744nteger @ ( image_4470545334726330049nteger @ F ) @ S4 ) )
% 14.25/8.08 => ( ( comp_P4885729792186999034nteger @ ( comp_C6344129587964912378nteger @ S @ Rep2 ) @ ( comp_C1593894019821074884nteger @ ( comp_C1593894019821074884nteger @ Abs2 @ M ) @ G ) )
% 14.25/8.08 = ( comp_s956550612328592744nteger @ ( image_4470545334726330049nteger @ F ) @ ( comp_C8186931911283765298nteger @ S4 @ G ) ) ) ) ) ).
% 14.25/8.08
% 14.25/8.08 thf(fact_4348_type__copy__set__map0,axiom,
% 14.25/8.08 ! [Rep2: ( produc8923325533196201883nteger > produc8923325533196201883nteger ) > code_integer,Abs2: code_integer > produc8923325533196201883nteger > produc8923325533196201883nteger,S: code_integer > set_Code_integer,M: code_integer > code_integer,F: nat > code_integer,S4: code_integer > set_nat,G: code_integer > code_integer] :
% 14.25/8.08 ( ( type_d6544512618915850416nteger @ Rep2 @ Abs2 @ top_to4645266643341252675nteger )
% 14.25/8.08 => ( ( ( comp_C8186931911283765298nteger @ S @ M )
% 14.25/8.08 = ( comp_s8292494440407567195nteger @ ( image_1215581382706833972nteger @ F ) @ S4 ) )
% 14.25/8.08 => ( ( comp_P4885729792186999034nteger @ ( comp_C6344129587964912378nteger @ S @ Rep2 ) @ ( comp_C1593894019821074884nteger @ ( comp_C1593894019821074884nteger @ Abs2 @ M ) @ G ) )
% 14.25/8.08 = ( comp_s8292494440407567195nteger @ ( image_1215581382706833972nteger @ F ) @ ( comp_C6537567860429791781nteger @ S4 @ G ) ) ) ) ) ).
% 14.25/8.08
% 14.25/8.08 thf(fact_4349_type__copy__set__map0,axiom,
% 14.25/8.08 ! [Rep2: ( produc8923325533196201883nteger > produc8923325533196201883nteger ) > product_unit,Abs2: product_unit > produc8923325533196201883nteger > produc8923325533196201883nteger,S: product_unit > set_nat,M: code_integer > product_unit,F: nat > nat,S4: code_integer > set_nat,G: code_integer > code_integer] :
% 14.25/8.08 ( ( type_d6996789741066924922t_unit @ Rep2 @ Abs2 @ top_to1996260823553986621t_unit )
% 14.25/8.08 => ( ( ( comp_P5878433733084393551nteger @ S @ M )
% 14.25/8.08 = ( comp_s4605850120322998606nteger @ ( image_nat_nat2 @ F ) @ S4 ) )
% 14.25/8.08 => ( ( comp_P8262219321096090349nteger @ ( comp_P6101996899264665623nteger @ S @ Rep2 ) @ ( comp_C1593894019821074884nteger @ ( comp_P4511551582283981294nteger @ Abs2 @ M ) @ G ) )
% 14.25/8.08 = ( comp_s4605850120322998606nteger @ ( image_nat_nat2 @ F ) @ ( comp_C6537567860429791781nteger @ S4 @ G ) ) ) ) ) ).
% 14.25/8.08
% 14.25/8.08 thf(fact_4350_type__copy__set__map0,axiom,
% 14.25/8.08 ! [Rep2: ( produc8923325533196201883nteger > produc8923325533196201883nteger ) > product_unit,Abs2: product_unit > produc8923325533196201883nteger > produc8923325533196201883nteger,S: product_unit > set_Code_integer,M: code_integer > product_unit,F: code_integer > code_integer,S4: code_integer > set_Code_integer,G: code_integer > code_integer] :
% 14.25/8.08 ( ( type_d6996789741066924922t_unit @ Rep2 @ Abs2 @ top_to1996260823553986621t_unit )
% 14.25/8.08 => ( ( ( comp_P4767411901762164316nteger @ S @ M )
% 14.25/8.08 = ( comp_s956550612328592744nteger @ ( image_4470545334726330049nteger @ F ) @ S4 ) )
% 14.25/8.08 => ( ( comp_P4885729792186999034nteger @ ( comp_P4635212033913086756nteger @ S @ Rep2 ) @ ( comp_C1593894019821074884nteger @ ( comp_P4511551582283981294nteger @ Abs2 @ M ) @ G ) )
% 14.25/8.08 = ( comp_s956550612328592744nteger @ ( image_4470545334726330049nteger @ F ) @ ( comp_C8186931911283765298nteger @ S4 @ G ) ) ) ) ) ).
% 14.25/8.08
% 14.25/8.08 thf(fact_4351_type__copy__set__map0,axiom,
% 14.25/8.08 ! [Rep2: ( produc8923325533196201883nteger > produc8923325533196201883nteger ) > product_unit,Abs2: product_unit > produc8923325533196201883nteger > produc8923325533196201883nteger,S: product_unit > set_Code_integer,M: code_integer > product_unit,F: nat > code_integer,S4: code_integer > set_nat,G: code_integer > code_integer] :
% 14.25/8.08 ( ( type_d6996789741066924922t_unit @ Rep2 @ Abs2 @ top_to1996260823553986621t_unit )
% 14.25/8.08 => ( ( ( comp_P4767411901762164316nteger @ S @ M )
% 14.25/8.08 = ( comp_s8292494440407567195nteger @ ( image_1215581382706833972nteger @ F ) @ S4 ) )
% 14.25/8.08 => ( ( comp_P4885729792186999034nteger @ ( comp_P4635212033913086756nteger @ S @ Rep2 ) @ ( comp_C1593894019821074884nteger @ ( comp_P4511551582283981294nteger @ Abs2 @ M ) @ G ) )
% 14.25/8.08 = ( comp_s8292494440407567195nteger @ ( image_1215581382706833972nteger @ F ) @ ( comp_C6537567860429791781nteger @ S4 @ G ) ) ) ) ) ).
% 14.25/8.08
% 14.25/8.08 thf(fact_4352_type__copy__set__map0,axiom,
% 14.25/8.08 ! [Rep2: ( produc8923325533196201883nteger > produc8923325533196201883nteger ) > $o,Abs2: $o > produc8923325533196201883nteger > produc8923325533196201883nteger,S: $o > set_nat,M: code_integer > $o,F: nat > nat,S4: code_integer > set_nat,G: code_integer > code_integer] :
% 14.25/8.08 ( ( type_d5661033022651575429eger_o @ Rep2 @ Abs2 @ top_top_set_o )
% 14.25/8.08 => ( ( ( comp_o172754640513974618nteger @ S @ M )
% 14.25/8.08 = ( comp_s4605850120322998606nteger @ ( image_nat_nat2 @ F ) @ S4 ) )
% 14.25/8.08 => ( ( comp_P8262219321096090349nteger @ ( comp_o1289984117834210338nteger @ S @ Rep2 ) @ ( comp_C1593894019821074884nteger @ ( comp_o8922910837708301817nteger @ Abs2 @ M ) @ G ) )
% 14.25/8.08 = ( comp_s4605850120322998606nteger @ ( image_nat_nat2 @ F ) @ ( comp_C6537567860429791781nteger @ S4 @ G ) ) ) ) ) ).
% 14.25/8.08
% 14.25/8.08 thf(fact_4353_type__copy__set__map0,axiom,
% 14.25/8.08 ! [Rep2: ( produc8923325533196201883nteger > produc8923325533196201883nteger ) > $o,Abs2: $o > produc8923325533196201883nteger > produc8923325533196201883nteger,S: $o > set_Code_integer,M: code_integer > $o,F: code_integer > code_integer,S4: code_integer > set_Code_integer,G: code_integer > code_integer] :
% 14.25/8.08 ( ( type_d5661033022651575429eger_o @ Rep2 @ Abs2 @ top_top_set_o )
% 14.25/8.08 => ( ( ( comp_o4291842030903409511nteger @ S @ M )
% 14.25/8.08 = ( comp_s956550612328592744nteger @ ( image_4470545334726330049nteger @ F ) @ S4 ) )
% 14.25/8.08 => ( ( comp_P4885729792186999034nteger @ ( comp_o7929101295636871983nteger @ S @ Rep2 ) @ ( comp_C1593894019821074884nteger @ ( comp_o8922910837708301817nteger @ Abs2 @ M ) @ G ) )
% 14.25/8.08 = ( comp_s956550612328592744nteger @ ( image_4470545334726330049nteger @ F ) @ ( comp_C8186931911283765298nteger @ S4 @ G ) ) ) ) ) ).
% 14.25/8.08
% 14.25/8.08 thf(fact_4354_type__copy__set__map0,axiom,
% 14.25/8.08 ! [Rep2: ( produc8923325533196201883nteger > produc8923325533196201883nteger ) > $o,Abs2: $o > produc8923325533196201883nteger > produc8923325533196201883nteger,S: $o > set_Code_integer,M: code_integer > $o,F: nat > code_integer,S4: code_integer > set_nat,G: code_integer > code_integer] :
% 14.25/8.08 ( ( type_d5661033022651575429eger_o @ Rep2 @ Abs2 @ top_top_set_o )
% 14.25/8.08 => ( ( ( comp_o4291842030903409511nteger @ S @ M )
% 14.25/8.08 = ( comp_s8292494440407567195nteger @ ( image_1215581382706833972nteger @ F ) @ S4 ) )
% 14.25/8.08 => ( ( comp_P4885729792186999034nteger @ ( comp_o7929101295636871983nteger @ S @ Rep2 ) @ ( comp_C1593894019821074884nteger @ ( comp_o8922910837708301817nteger @ Abs2 @ M ) @ G ) )
% 14.25/8.08 = ( comp_s8292494440407567195nteger @ ( image_1215581382706833972nteger @ F ) @ ( comp_C6537567860429791781nteger @ S4 @ G ) ) ) ) ) ).
% 14.25/8.08
% 14.25/8.08 thf(fact_4355_type__copy__set__map0,axiom,
% 14.25/8.08 ! [Rep2: ( produc8923325533196201883nteger > produc8923325533196201883nteger ) > nat,Abs2: nat > produc8923325533196201883nteger > produc8923325533196201883nteger,S: nat > set_nat,M: code_integer > nat,F: nat > nat,S4: code_integer > set_nat,G: code_integer > code_integer] :
% 14.25/8.08 ( ( type_d6278479724763174947er_nat @ Rep2 @ Abs2 @ top_top_set_nat )
% 14.25/8.08 => ( ( ( comp_n5963683855531483544nteger @ S @ M )
% 14.25/8.08 = ( comp_s4605850120322998606nteger @ ( image_nat_nat2 @ F ) @ S4 ) )
% 14.25/8.08 => ( ( comp_P8262219321096090349nteger @ ( comp_n3623798160786426464nteger @ S @ Rep2 ) @ ( comp_C1593894019821074884nteger @ ( comp_n2033352843805742135nteger @ Abs2 @ M ) @ G ) )
% 14.25/8.08 = ( comp_s4605850120322998606nteger @ ( image_nat_nat2 @ F ) @ ( comp_C6537567860429791781nteger @ S4 @ G ) ) ) ) ) ).
% 14.25/8.08
% 14.25/8.08 thf(fact_4356_top__empty__eq,axiom,
% 14.25/8.08 ( top_to2462227307520033056at_o_o
% 14.25/8.08 = ( ^ [X2: produc3658429121746597890et_nat > $o] : ( member6576561426505652726_nat_o @ X2 @ top_to4869887016220417533_nat_o ) ) ) ).
% 14.25/8.08
% 14.25/8.08 thf(fact_4357_top__empty__eq,axiom,
% 14.25/8.08 ( top_to2465898995584390880unit_o
% 14.25/8.08 = ( ^ [X2: product_unit] : ( member_Product_unit @ X2 @ top_to1996260823553986621t_unit ) ) ) ).
% 14.25/8.08
% 14.25/8.08 thf(fact_4358_top__empty__eq,axiom,
% 14.25/8.08 ( top_top_o_o
% 14.25/8.08 = ( ^ [X2: $o] : ( member_o @ X2 @ top_top_set_o ) ) ) ).
% 14.25/8.08
% 14.25/8.08 thf(fact_4359_top__empty__eq,axiom,
% 14.25/8.08 ( top_top_nat_o
% 14.25/8.08 = ( ^ [X2: nat] : ( member_nat @ X2 @ top_top_set_nat ) ) ) ).
% 14.25/8.08
% 14.25/8.08 thf(fact_4360_top__empty__eq,axiom,
% 14.25/8.08 ( top_top_int_o
% 14.25/8.08 = ( ^ [X2: int] : ( member_int @ X2 @ top_top_set_int ) ) ) ).
% 14.25/8.08
% 14.25/8.08 thf(fact_4361_top__set__def,axiom,
% 14.25/8.08 ( top_to4869887016220417533_nat_o
% 14.25/8.08 = ( collec939566748876313656_nat_o @ top_to2462227307520033056at_o_o ) ) ).
% 14.25/8.08
% 14.25/8.08 thf(fact_4362_top__set__def,axiom,
% 14.25/8.08 ( top_to1996260823553986621t_unit
% 14.25/8.08 = ( collect_Product_unit @ top_to2465898995584390880unit_o ) ) ).
% 14.25/8.08
% 14.25/8.08 thf(fact_4363_top__set__def,axiom,
% 14.25/8.08 ( top_top_set_o
% 14.25/8.08 = ( collect_o @ top_top_o_o ) ) ).
% 14.25/8.08
% 14.25/8.08 thf(fact_4364_top__set__def,axiom,
% 14.25/8.08 ( top_top_set_nat
% 14.25/8.08 = ( collect_nat @ top_top_nat_o ) ) ).
% 14.25/8.08
% 14.25/8.08 thf(fact_4365_top__set__def,axiom,
% 14.25/8.08 ( top_top_set_int
% 14.25/8.08 = ( collect_int @ top_top_int_o ) ) ).
% 14.25/8.08
% 14.25/8.08 thf(fact_4366_pred__on_Ochain__empty,axiom,
% 14.25/8.08 ! [A: set_Product_unit,P: product_unit > product_unit > $o] : ( pred_c6915491785556283576t_unit @ A @ P @ bot_bo3957492148770167129t_unit ) ).
% 14.25/8.08
% 14.25/8.08 thf(fact_4367_pred__on_Ochain__empty,axiom,
% 14.25/8.08 ! [A: set_o,P: $o > $o > $o] : ( pred_chain_o @ A @ P @ bot_bot_set_o ) ).
% 14.25/8.08
% 14.25/8.08 thf(fact_4368_pred__on_Ochain__empty,axiom,
% 14.25/8.08 ! [A: set_nat,P: nat > nat > $o] : ( pred_chain_nat @ A @ P @ bot_bot_set_nat ) ).
% 14.25/8.08
% 14.25/8.08 thf(fact_4369_pred__on_Ochain__empty,axiom,
% 14.25/8.08 ! [A: set_int,P: int > int > $o] : ( pred_chain_int @ A @ P @ bot_bot_set_int ) ).
% 14.25/8.08
% 14.25/8.08 thf(fact_4370_pred__on_Ochain__empty,axiom,
% 14.25/8.08 ! [A: set_Pr4532377907799695533_nat_o,P: ( produc3658429121746597890et_nat > $o ) > ( produc3658429121746597890et_nat > $o ) > $o] : ( pred_c177362287423414904_nat_o @ A @ P @ bot_bo7824918357723582233_nat_o ) ).
% 14.25/8.08
% 14.25/8.08 thf(fact_4371_surj__fun__eq,axiom,
% 14.25/8.08 ! [F: code_integer > code_integer > code_integer,X4: set_Code_integer,G1: ( code_integer > code_integer ) > produc8923325533196201883nteger > produc8923325533196201883nteger,G2: ( code_integer > code_integer ) > produc8923325533196201883nteger > produc8923325533196201883nteger] :
% 14.25/8.08 ( ( ( image_2177946750938214461nteger @ F @ X4 )
% 14.25/8.08 = top_to3804862627464599871nteger )
% 14.25/8.08 => ( ! [X3: code_integer] :
% 14.25/8.08 ( ( member_Code_integer @ X3 @ X4 )
% 14.25/8.08 => ( ( comp_C8797469213163452608nteger @ G1 @ F @ X3 )
% 14.25/8.08 = ( comp_C8797469213163452608nteger @ G2 @ F @ X3 ) ) )
% 14.25/8.08 => ( G1 = G2 ) ) ) ).
% 14.25/8.08
% 14.25/8.08 thf(fact_4372_surj__fun__eq,axiom,
% 14.25/8.08 ! [F: code_integer > code_integer,X4: set_Code_integer,G1: code_integer > produc8923325533196201883nteger > produc8923325533196201883nteger,G2: code_integer > produc8923325533196201883nteger > produc8923325533196201883nteger] :
% 14.25/8.08 ( ( ( image_4470545334726330049nteger @ F @ X4 )
% 14.25/8.08 = top_to4645266643341252675nteger )
% 14.25/8.08 => ( ! [X3: code_integer] :
% 14.25/8.08 ( ( member_Code_integer @ X3 @ X4 )
% 14.25/8.08 => ( ( comp_C1593894019821074884nteger @ G1 @ F @ X3 )
% 14.25/8.08 = ( comp_C1593894019821074884nteger @ G2 @ F @ X3 ) ) )
% 14.25/8.08 => ( G1 = G2 ) ) ) ).
% 14.25/8.08
% 14.25/8.08 thf(fact_4373_type__copy__map__comp0,axiom,
% 14.25/8.08 ! [Rep2: code_integer > code_integer,Abs2: code_integer > code_integer,M: code_integer > code_integer,M1: code_integer > code_integer,M22: code_integer > code_integer,F: code_integer > produc8923325533196201883nteger > produc8923325533196201883nteger,G: code_integer > code_integer] :
% 14.25/8.08 ( ( type_d4796611474492816872nteger @ Rep2 @ Abs2 @ top_to4645266643341252675nteger )
% 14.25/8.08 => ( ( M
% 14.25/8.08 = ( comp_C7449957260575251196nteger @ M1 @ M22 ) )
% 14.25/8.08 => ( ( comp_C1593894019821074884nteger @ ( comp_C1593894019821074884nteger @ F @ M ) @ G )
% 14.25/8.08 = ( comp_C1593894019821074884nteger @ ( comp_C1593894019821074884nteger @ ( comp_C1593894019821074884nteger @ F @ M1 ) @ Rep2 ) @ ( comp_C7449957260575251196nteger @ ( comp_C7449957260575251196nteger @ Abs2 @ M22 ) @ G ) ) ) ) ) ).
% 14.25/8.08
% 14.25/8.08 thf(fact_4374_type__copy__map__comp0,axiom,
% 14.25/8.08 ! [Rep2: code_integer > product_unit,Abs2: product_unit > code_integer,M: code_integer > code_integer,M1: product_unit > code_integer,M22: code_integer > product_unit,F: code_integer > produc8923325533196201883nteger > produc8923325533196201883nteger,G: code_integer > code_integer] :
% 14.25/8.08 ( ( type_d9006111508436661314t_unit @ Rep2 @ Abs2 @ top_to1996260823553986621t_unit )
% 14.25/8.08 => ( ( M
% 14.25/8.08 = ( comp_P6790823133229852966nteger @ M1 @ M22 ) )
% 14.25/8.08 => ( ( comp_C1593894019821074884nteger @ ( comp_C1593894019821074884nteger @ F @ M ) @ G )
% 14.25/8.08 = ( comp_C1593894019821074884nteger @ ( comp_P4511551582283981294nteger @ ( comp_C5854136495726678886t_unit @ F @ M1 ) @ Rep2 ) @ ( comp_C7449957260575251196nteger @ ( comp_P6790823133229852966nteger @ Abs2 @ M22 ) @ G ) ) ) ) ) ).
% 14.25/8.08
% 14.25/8.08 thf(fact_4375_type__copy__map__comp0,axiom,
% 14.25/8.08 ! [Rep2: code_integer > $o,Abs2: $o > code_integer,M: code_integer > code_integer,M1: $o > code_integer,M22: code_integer > $o,F: code_integer > produc8923325533196201883nteger > produc8923325533196201883nteger,G: code_integer > code_integer] :
% 14.25/8.08 ( ( type_d6812736842639505741eger_o @ Rep2 @ Abs2 @ top_top_set_o )
% 14.25/8.08 => ( ( M
% 14.25/8.08 = ( comp_o1085144040659434033nteger @ M1 @ M22 ) )
% 14.25/8.08 => ( ( comp_C1593894019821074884nteger @ ( comp_C1593894019821074884nteger @ F @ M ) @ G )
% 14.25/8.08 = ( comp_C1593894019821074884nteger @ ( comp_o8922910837708301817nteger @ ( comp_C5733264393235242609eger_o @ F @ M1 ) @ Rep2 ) @ ( comp_C7449957260575251196nteger @ ( comp_o1085144040659434033nteger @ Abs2 @ M22 ) @ G ) ) ) ) ) ).
% 14.25/8.08
% 14.25/8.08 thf(fact_4376_type__copy__map__comp0,axiom,
% 14.25/8.08 ! [Rep2: code_integer > nat,Abs2: nat > code_integer,M: code_integer > code_integer,M1: nat > code_integer,M22: code_integer > nat,F: code_integer > produc8923325533196201883nteger > produc8923325533196201883nteger,G: code_integer > code_integer] :
% 14.25/8.08 ( ( type_d8368584451094728027er_nat @ Rep2 @ Abs2 @ top_top_set_nat )
% 14.25/8.08 => ( ( M
% 14.25/8.08 = ( comp_n6876073255676942959nteger @ M1 @ M22 ) )
% 14.25/8.08 => ( ( comp_C1593894019821074884nteger @ ( comp_C1593894019821074884nteger @ F @ M ) @ G )
% 14.25/8.08 = ( comp_C1593894019821074884nteger @ ( comp_n2033352843805742135nteger @ ( comp_C836733381939181879er_nat @ F @ M1 ) @ Rep2 ) @ ( comp_C7449957260575251196nteger @ ( comp_n6876073255676942959nteger @ Abs2 @ M22 ) @ G ) ) ) ) ) ).
% 14.25/8.08
% 14.25/8.08 thf(fact_4377_type__copy__map__comp0,axiom,
% 14.25/8.08 ! [Rep2: code_integer > int,Abs2: int > code_integer,M: code_integer > code_integer,M1: int > code_integer,M22: code_integer > int,F: code_integer > produc8923325533196201883nteger > produc8923325533196201883nteger,G: code_integer > code_integer] :
% 14.25/8.08 ( ( type_d8366093980585677751er_int @ Rep2 @ Abs2 @ top_top_set_int )
% 14.25/8.08 => ( ( M
% 14.25/8.08 = ( comp_i6014793192326455499nteger @ M1 @ M22 ) )
% 14.25/8.08 => ( ( comp_C1593894019821074884nteger @ ( comp_C1593894019821074884nteger @ F @ M ) @ G )
% 14.25/8.08 = ( comp_C1593894019821074884nteger @ ( comp_i2874689474467862163nteger @ ( comp_C834242911430131603er_int @ F @ M1 ) @ Rep2 ) @ ( comp_C7449957260575251196nteger @ ( comp_i6014793192326455499nteger @ Abs2 @ M22 ) @ G ) ) ) ) ) ).
% 14.25/8.08
% 14.25/8.08 thf(fact_4378_type__copy__map__comp0,axiom,
% 14.25/8.08 ! [Rep2: product_unit > $o,Abs2: $o > product_unit,M: code_integer > code_integer,M1: $o > code_integer,M22: code_integer > $o,F: code_integer > produc8923325533196201883nteger > produc8923325533196201883nteger,G: code_integer > code_integer] :
% 14.25/8.08 ( ( type_d6188575255521822967unit_o @ Rep2 @ Abs2 @ top_top_set_o )
% 14.25/8.08 => ( ( M
% 14.25/8.08 = ( comp_o1085144040659434033nteger @ M1 @ M22 ) )
% 14.25/8.08 => ( ( comp_C1593894019821074884nteger @ ( comp_C1593894019821074884nteger @ F @ M ) @ G )
% 14.25/8.08 = ( comp_P4511551582283981294nteger @ ( comp_o6961622296474427249t_unit @ ( comp_C5733264393235242609eger_o @ F @ M1 ) @ Rep2 ) @ ( comp_C8146212659816229034nteger @ ( comp_o6635994240230588853nteger @ Abs2 @ M22 ) @ G ) ) ) ) ) ).
% 14.25/8.08
% 14.25/8.08 thf(fact_4379_type__copy__map__comp0,axiom,
% 14.25/8.08 ! [Rep2: code_natural > nat,Abs2: nat > code_natural,M: code_integer > code_integer,M1: nat > code_integer,M22: code_integer > nat,F: code_integer > produc8923325533196201883nteger > produc8923325533196201883nteger,G: code_integer > code_integer] :
% 14.25/8.08 ( ( type_d4410041424927559462al_nat @ Rep2 @ Abs2 @ top_top_set_nat )
% 14.25/8.08 => ( ( M
% 14.25/8.08 = ( comp_n6876073255676942959nteger @ M1 @ M22 ) )
% 14.25/8.08 => ( ( comp_C1593894019821074884nteger @ ( comp_C1593894019821074884nteger @ F @ M ) @ G )
% 14.25/8.08 = ( comp_C456341964420721295nteger @ ( comp_n856947995562459650atural @ ( comp_C836733381939181879er_nat @ F @ M1 ) @ Rep2 ) @ ( comp_C1891057325546811591nteger @ ( comp_n1317173320648503354nteger @ Abs2 @ M22 ) @ G ) ) ) ) ) ).
% 14.25/8.08
% 14.25/8.08 thf(fact_4380_type__copy__map__comp0,axiom,
% 14.25/8.08 ! [Rep2: code_integer > code_integer > code_integer,Abs2: ( code_integer > code_integer ) > code_integer,M: code_integer > code_integer,M1: ( code_integer > code_integer ) > code_integer,M22: code_integer > code_integer > code_integer,F: code_integer > produc8923325533196201883nteger > produc8923325533196201883nteger,G: code_integer > code_integer] :
% 14.25/8.08 ( ( type_d669523880323286244nteger @ Rep2 @ Abs2 @ top_to3804862627464599871nteger )
% 14.25/8.08 => ( ( M
% 14.25/8.08 = ( comp_C5076291335850163704nteger @ M1 @ M22 ) )
% 14.25/8.08 => ( ( comp_C1593894019821074884nteger @ ( comp_C1593894019821074884nteger @ F @ M ) @ G )
% 14.25/8.08 = ( comp_C1593894019821074884nteger @ ( comp_C8797469213163452608nteger @ ( comp_C95226695241045696nteger @ F @ M1 ) @ Rep2 ) @ ( comp_C7449957260575251196nteger @ ( comp_C5076291335850163704nteger @ Abs2 @ M22 ) @ G ) ) ) ) ) ).
% 14.25/8.08
% 14.25/8.08 thf(fact_4381_type__copy__map__comp0,axiom,
% 14.25/8.08 ! [Rep2: ( code_integer > code_integer ) > code_integer,Abs2: code_integer > code_integer > code_integer,M: code_integer > code_integer,M1: code_integer > code_integer,M22: code_integer > code_integer,F: code_integer > produc8923325533196201883nteger > produc8923325533196201883nteger,G: code_integer > code_integer] :
% 14.25/8.08 ( ( type_d5515310227294319076nteger @ Rep2 @ Abs2 @ top_to4645266643341252675nteger )
% 14.25/8.08 => ( ( M
% 14.25/8.08 = ( comp_C7449957260575251196nteger @ M1 @ M22 ) )
% 14.25/8.08 => ( ( comp_C1593894019821074884nteger @ ( comp_C1593894019821074884nteger @ F @ M ) @ G )
% 14.25/8.08 = ( comp_C8797469213163452608nteger @ ( comp_C95226695241045696nteger @ ( comp_C1593894019821074884nteger @ F @ M1 ) @ Rep2 ) @ ( comp_C2931745410781319416nteger @ ( comp_C2931745410781319416nteger @ Abs2 @ M22 ) @ G ) ) ) ) ) ).
% 14.25/8.08
% 14.25/8.08 thf(fact_4382_type__copy__map__comp0,axiom,
% 14.25/8.08 ! [Rep2: code_integer > code_integer,Abs2: code_integer > code_integer,M: code_integer > code_integer > code_integer,M1: code_integer > code_integer > code_integer,M22: code_integer > code_integer,F: ( code_integer > code_integer ) > produc8923325533196201883nteger > produc8923325533196201883nteger,G: code_integer > code_integer] :
% 14.25/8.08 ( ( type_d4796611474492816872nteger @ Rep2 @ Abs2 @ top_to4645266643341252675nteger )
% 14.25/8.08 => ( ( M
% 14.25/8.08 = ( comp_C2931745410781319416nteger @ M1 @ M22 ) )
% 14.25/8.08 => ( ( comp_C1593894019821074884nteger @ ( comp_C8797469213163452608nteger @ F @ M ) @ G )
% 14.25/8.08 = ( comp_C1593894019821074884nteger @ ( comp_C1593894019821074884nteger @ ( comp_C8797469213163452608nteger @ F @ M1 ) @ Rep2 ) @ ( comp_C7449957260575251196nteger @ ( comp_C7449957260575251196nteger @ Abs2 @ M22 ) @ G ) ) ) ) ) ).
% 14.25/8.08
% 14.25/8.08 thf(fact_4383_type__copy__map__comp0__undo,axiom,
% 14.25/8.08 ! [Rep2: code_integer > int,Abs2: int > code_integer,Rep3: code_integer > int,Abs3: int > code_integer,Rep4: code_integer > int,Abs4: int > code_integer,M: int > int,M1: int > int,M22: int > int] :
% 14.25/8.08 ( ( type_d8366093980585677751er_int @ Rep2 @ Abs2 @ top_top_set_int )
% 14.25/8.08 => ( ( type_d8366093980585677751er_int @ Rep3 @ Abs3 @ top_top_set_int )
% 14.25/8.08 => ( ( type_d8366093980585677751er_int @ Rep4 @ Abs4 @ top_top_set_int )
% 14.25/8.08 => ( ( ( comp_i6014793192326455499nteger @ ( comp_i1585864551200866970er_int @ Abs3 @ M ) @ Rep4 )
% 14.25/8.08 = ( comp_C7449957260575251196nteger @ ( comp_i6014793192326455499nteger @ ( comp_i1585864551200866970er_int @ Abs3 @ M1 ) @ Rep2 ) @ ( comp_i6014793192326455499nteger @ ( comp_i1585864551200866970er_int @ Abs2 @ M22 ) @ Rep4 ) ) )
% 14.25/8.08 => ( ( comp_int_int_int @ M1 @ M22 )
% 14.25/8.08 = M ) ) ) ) ) ).
% 14.25/8.08
% 14.25/8.08 thf(fact_4384_type__copy__map__comp0__undo,axiom,
% 14.25/8.08 ! [Rep2: code_integer > int,Abs2: int > code_integer,Rep3: code_integer > int,Abs3: int > code_integer,Rep4: product_unit > $o,Abs4: $o > product_unit,M: $o > int,M1: int > int,M22: $o > int] :
% 14.25/8.08 ( ( type_d8366093980585677751er_int @ Rep2 @ Abs2 @ top_top_set_int )
% 14.25/8.08 => ( ( type_d8366093980585677751er_int @ Rep3 @ Abs3 @ top_top_set_int )
% 14.25/8.08 => ( ( type_d6188575255521822967unit_o @ Rep4 @ Abs4 @ top_top_set_o )
% 14.25/8.08 => ( ( ( comp_o780160699970281785t_unit @ ( comp_i5286552343147518890eger_o @ Abs3 @ M ) @ Rep4 )
% 14.25/8.08 = ( comp_C2290379119555921966t_unit @ ( comp_i6014793192326455499nteger @ ( comp_i1585864551200866970er_int @ Abs3 @ M1 ) @ Rep2 ) @ ( comp_o780160699970281785t_unit @ ( comp_i5286552343147518890eger_o @ Abs2 @ M22 ) @ Rep4 ) ) )
% 14.25/8.08 => ( ( comp_int_int_o @ M1 @ M22 )
% 14.25/8.08 = M ) ) ) ) ) ).
% 14.25/8.08
% 14.25/8.08 thf(fact_4385_type__copy__map__comp0__undo,axiom,
% 14.25/8.08 ! [Rep2: code_integer > int,Abs2: int > code_integer,Rep3: code_integer > int,Abs3: int > code_integer,Rep4: code_natural > nat,Abs4: nat > code_natural,M: nat > int,M1: int > int,M22: nat > int] :
% 14.25/8.08 ( ( type_d8366093980585677751er_int @ Rep2 @ Abs2 @ top_top_set_int )
% 14.25/8.08 => ( ( type_d8366093980585677751er_int @ Rep3 @ Abs3 @ top_top_set_int )
% 14.25/8.08 => ( ( type_d4410041424927559462al_nat @ Rep4 @ Abs4 @ top_top_set_nat )
% 14.25/8.08 => ( ( ( comp_n5699668407433660474atural @ ( comp_i1588355021709917246er_nat @ Abs3 @ M ) @ Rep4 )
% 14.25/8.08 = ( comp_C6273552412331968711atural @ ( comp_i6014793192326455499nteger @ ( comp_i1585864551200866970er_int @ Abs3 @ M1 ) @ Rep2 ) @ ( comp_n5699668407433660474atural @ ( comp_i1588355021709917246er_nat @ Abs2 @ M22 ) @ Rep4 ) ) )
% 14.25/8.08 => ( ( comp_int_int_nat @ M1 @ M22 )
% 14.25/8.08 = M ) ) ) ) ) ).
% 14.25/8.08
% 14.25/8.08 thf(fact_4386_type__copy__map__comp0__undo,axiom,
% 14.25/8.08 ! [Rep2: code_integer > int,Abs2: int > code_integer,Rep3: product_unit > $o,Abs3: $o > product_unit,Rep4: code_integer > int,Abs4: int > code_integer,M: int > $o,M1: int > $o,M22: int > int] :
% 14.25/8.08 ( ( type_d8366093980585677751er_int @ Rep2 @ Abs2 @ top_top_set_int )
% 14.25/8.08 => ( ( type_d6188575255521822967unit_o @ Rep3 @ Abs3 @ top_top_set_o )
% 14.25/8.08 => ( ( type_d8366093980585677751er_int @ Rep4 @ Abs4 @ top_top_set_int )
% 14.25/8.08 => ( ( ( comp_i8439899303502915483nteger @ ( comp_o7049022088938261380it_int @ Abs3 @ M ) @ Rep4 )
% 14.25/8.08 = ( comp_C8146212659816229034nteger @ ( comp_i8439899303502915483nteger @ ( comp_o7049022088938261380it_int @ Abs3 @ M1 ) @ Rep2 ) @ ( comp_i6014793192326455499nteger @ ( comp_i1585864551200866970er_int @ Abs2 @ M22 ) @ Rep4 ) ) )
% 14.25/8.08 => ( ( comp_int_o_int @ M1 @ M22 )
% 14.25/8.08 = M ) ) ) ) ) ).
% 14.25/8.08
% 14.25/8.08 thf(fact_4387_type__copy__map__comp0__undo,axiom,
% 14.25/8.08 ! [Rep2: code_integer > int,Abs2: int > code_integer,Rep3: product_unit > $o,Abs3: $o > product_unit,Rep4: product_unit > $o,Abs4: $o > product_unit,M: $o > $o,M1: int > $o,M22: $o > int] :
% 14.25/8.08 ( ( type_d8366093980585677751er_int @ Rep2 @ Abs2 @ top_top_set_int )
% 14.25/8.08 => ( ( type_d6188575255521822967unit_o @ Rep3 @ Abs3 @ top_top_set_o )
% 14.25/8.08 => ( ( type_d6188575255521822967unit_o @ Rep4 @ Abs4 @ top_top_set_o )
% 14.25/8.08 => ( ( ( comp_o7724735231917782709t_unit @ ( comp_o3916225744453444672unit_o @ Abs3 @ M ) @ Rep4 )
% 14.25/8.08 = ( comp_C2408090164605745152t_unit @ ( comp_i8439899303502915483nteger @ ( comp_o7049022088938261380it_int @ Abs3 @ M1 ) @ Rep2 ) @ ( comp_o780160699970281785t_unit @ ( comp_i5286552343147518890eger_o @ Abs2 @ M22 ) @ Rep4 ) ) )
% 14.25/8.08 => ( ( comp_int_o_o @ M1 @ M22 )
% 14.25/8.08 = M ) ) ) ) ) ).
% 14.25/8.08
% 14.25/8.08 thf(fact_4388_type__copy__map__comp0__undo,axiom,
% 14.25/8.08 ! [Rep2: code_integer > int,Abs2: int > code_integer,Rep3: product_unit > $o,Abs3: $o > product_unit,Rep4: code_natural > nat,Abs4: nat > code_natural,M: nat > $o,M1: int > $o,M22: nat > int] :
% 14.25/8.08 ( ( type_d8366093980585677751er_int @ Rep2 @ Abs2 @ top_top_set_int )
% 14.25/8.08 => ( ( type_d6188575255521822967unit_o @ Rep3 @ Abs3 @ top_top_set_o )
% 14.25/8.08 => ( ( type_d4410041424927559462al_nat @ Rep4 @ Abs4 @ top_top_set_nat )
% 14.25/8.08 => ( ( ( comp_n8360577435903631554atural @ ( comp_o7051512559447311656it_nat @ Abs3 @ M ) @ Rep4 )
% 14.25/8.08 = ( comp_C6969807811572946549atural @ ( comp_i8439899303502915483nteger @ ( comp_o7049022088938261380it_int @ Abs3 @ M1 ) @ Rep2 ) @ ( comp_n5699668407433660474atural @ ( comp_i1588355021709917246er_nat @ Abs2 @ M22 ) @ Rep4 ) ) )
% 14.25/8.08 => ( ( comp_int_o_nat @ M1 @ M22 )
% 14.25/8.08 = M ) ) ) ) ) ).
% 14.25/8.08
% 14.25/8.08 thf(fact_4389_type__copy__map__comp0__undo,axiom,
% 14.25/8.08 ! [Rep2: code_integer > int,Abs2: int > code_integer,Rep3: code_natural > nat,Abs3: nat > code_natural,Rep4: code_integer > int,Abs4: int > code_integer,M: int > nat,M1: int > nat,M22: int > int] :
% 14.25/8.08 ( ( type_d8366093980585677751er_int @ Rep2 @ Abs2 @ top_top_set_int )
% 14.25/8.08 => ( ( type_d4410041424927559462al_nat @ Rep3 @ Abs3 @ top_top_set_nat )
% 14.25/8.08 => ( ( type_d8366093980585677751er_int @ Rep4 @ Abs4 @ top_top_set_int )
% 14.25/8.08 => ( ( ( comp_i455893257298015894nteger @ ( comp_n9160511493981425161al_int @ Abs3 @ M ) @ Rep4 )
% 14.25/8.08 = ( comp_C1891057325546811591nteger @ ( comp_i455893257298015894nteger @ ( comp_n9160511493981425161al_int @ Abs3 @ M1 ) @ Rep2 ) @ ( comp_i6014793192326455499nteger @ ( comp_i1585864551200866970er_int @ Abs2 @ M22 ) @ Rep4 ) ) )
% 14.25/8.08 => ( ( comp_int_nat_int @ M1 @ M22 )
% 14.25/8.08 = M ) ) ) ) ) ).
% 14.25/8.08
% 14.25/8.08 thf(fact_4390_type__copy__map__comp0__undo,axiom,
% 14.25/8.08 ! [Rep2: code_integer > int,Abs2: int > code_integer,Rep3: code_natural > nat,Abs3: nat > code_natural,Rep4: product_unit > $o,Abs4: $o > product_unit,M: $o > nat,M1: int > nat,M22: $o > int] :
% 14.25/8.08 ( ( type_d8366093980585677751er_int @ Rep2 @ Abs2 @ top_top_set_int )
% 14.25/8.08 => ( ( type_d4410041424927559462al_nat @ Rep3 @ Abs3 @ top_top_set_nat )
% 14.25/8.08 => ( ( type_d6188575255521822967unit_o @ Rep4 @ Abs4 @ top_top_set_o )
% 14.25/8.08 => ( ( ( comp_o2643362028263920430t_unit @ ( comp_n2857367870785791611ural_o @ Abs3 @ M ) @ Rep4 )
% 14.25/8.08 = ( comp_C4153580447849560611t_unit @ ( comp_i455893257298015894nteger @ ( comp_n9160511493981425161al_int @ Abs3 @ M1 ) @ Rep2 ) @ ( comp_o780160699970281785t_unit @ ( comp_i5286552343147518890eger_o @ Abs2 @ M22 ) @ Rep4 ) ) )
% 14.25/8.08 => ( ( comp_int_nat_o @ M1 @ M22 )
% 14.25/8.08 = M ) ) ) ) ) ).
% 14.25/8.08
% 14.25/8.08 thf(fact_4391_type__copy__map__comp0__undo,axiom,
% 14.25/8.08 ! [Rep2: code_integer > int,Abs2: int > code_integer,Rep3: code_natural > nat,Abs3: nat > code_natural,Rep4: code_natural > nat,Abs4: nat > code_natural,M: nat > nat,M1: int > nat,M22: nat > int] :
% 14.25/8.08 ( ( type_d8366093980585677751er_int @ Rep2 @ Abs2 @ top_top_set_int )
% 14.25/8.08 => ( ( type_d4410041424927559462al_nat @ Rep3 @ Abs3 @ top_top_set_nat )
% 14.25/8.08 => ( ( type_d4410041424927559462al_nat @ Rep4 @ Abs4 @ top_top_set_nat )
% 14.25/8.08 => ( ( ( comp_n140768472405220869atural @ ( comp_n9163001964490475437al_nat @ Abs3 @ M ) @ Rep4 )
% 14.25/8.08 = ( comp_C714652477303529106atural @ ( comp_i455893257298015894nteger @ ( comp_n9160511493981425161al_int @ Abs3 @ M1 ) @ Rep2 ) @ ( comp_n5699668407433660474atural @ ( comp_i1588355021709917246er_nat @ Abs2 @ M22 ) @ Rep4 ) ) )
% 14.25/8.08 => ( ( comp_int_nat_nat @ M1 @ M22 )
% 14.25/8.08 = M ) ) ) ) ) ).
% 14.25/8.08
% 14.25/8.08 thf(fact_4392_type__copy__map__comp0__undo,axiom,
% 14.25/8.08 ! [Rep2: product_unit > $o,Abs2: $o > product_unit,Rep3: code_integer > int,Abs3: int > code_integer,Rep4: code_integer > int,Abs4: int > code_integer,M: int > int,M1: $o > int,M22: int > $o] :
% 14.25/8.08 ( ( type_d6188575255521822967unit_o @ Rep2 @ Abs2 @ top_top_set_o )
% 14.25/8.08 => ( ( type_d8366093980585677751er_int @ Rep3 @ Abs3 @ top_top_set_int )
% 14.25/8.08 => ( ( type_d8366093980585677751er_int @ Rep4 @ Abs4 @ top_top_set_int )
% 14.25/8.08 => ( ( ( comp_i6014793192326455499nteger @ ( comp_i1585864551200866970er_int @ Abs3 @ M ) @ Rep4 )
% 14.25/8.08 = ( comp_P6790823133229852966nteger @ ( comp_o780160699970281785t_unit @ ( comp_i5286552343147518890eger_o @ Abs3 @ M1 ) @ Rep2 ) @ ( comp_i8439899303502915483nteger @ ( comp_o7049022088938261380it_int @ Abs2 @ M22 ) @ Rep4 ) ) )
% 14.25/8.08 => ( ( comp_o_int_int @ M1 @ M22 )
% 14.25/8.08 = M ) ) ) ) ) ).
% 14.25/8.08
% 14.25/8.08 thf(fact_4393_wf__bounded__measure,axiom,
% 14.25/8.08 ! [R2: set_Pr1261947904930325089at_nat,Ub: nat > nat,F: nat > nat] :
% 14.25/8.08 ( ! [A4: nat,B2: nat] :
% 14.25/8.08 ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ B2 @ A4 ) @ R2 )
% 14.25/8.08 => ( ( ord_less_eq_nat @ ( Ub @ B2 ) @ ( Ub @ A4 ) )
% 14.25/8.08 & ( ord_less_eq_nat @ ( F @ B2 ) @ ( Ub @ A4 ) )
% 14.25/8.08 & ( ord_less_nat @ ( F @ A4 ) @ ( F @ B2 ) ) ) )
% 14.25/8.08 => ( wf_nat @ R2 ) ) ).
% 14.25/8.08
% 14.25/8.08 thf(fact_4394_wf__bounded__measure,axiom,
% 14.25/8.08 ! [R2: set_Pr958786334691620121nt_int,Ub: int > nat,F: int > nat] :
% 14.25/8.08 ( ! [A4: int,B2: int] :
% 14.25/8.08 ( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ B2 @ A4 ) @ R2 )
% 14.25/8.08 => ( ( ord_less_eq_nat @ ( Ub @ B2 ) @ ( Ub @ A4 ) )
% 14.25/8.08 & ( ord_less_eq_nat @ ( F @ B2 ) @ ( Ub @ A4 ) )
% 14.25/8.08 & ( ord_less_nat @ ( F @ A4 ) @ ( F @ B2 ) ) ) )
% 14.25/8.08 => ( wf_int @ R2 ) ) ).
% 14.25/8.08
% 14.25/8.08 thf(fact_4395_less__assn__def,axiom,
% 14.25/8.08 ( ord_less_assn
% 14.25/8.08 = ( ^ [A6: assn,B6: assn] :
% 14.25/8.08 ( ( ord_less_eq_assn @ A6 @ B6 )
% 14.25/8.08 & ( A6 != B6 ) ) ) ) ).
% 14.25/8.08
% 14.25/8.08 thf(fact_4396_top__assn__def,axiom,
% 14.25/8.08 ( top_top_assn
% 14.25/8.08 = ( abs_assn @ in_range ) ) ).
% 14.25/8.08
% 14.25/8.08 thf(fact_4397_pred__on_Ochain__extend,axiom,
% 14.25/8.08 ! [A: set_Product_unit,P: product_unit > product_unit > $o,C5: set_Product_unit,Z: product_unit] :
% 14.25/8.08 ( ( pred_c6915491785556283576t_unit @ A @ P @ C5 )
% 14.25/8.08 => ( ( member_Product_unit @ Z @ A )
% 14.25/8.08 => ( ! [X3: product_unit] :
% 14.25/8.08 ( ( member_Product_unit @ X3 @ C5 )
% 14.25/8.08 => ( sup_su3411595859437804162unit_o @ P
% 14.25/8.08 @ ^ [Y4: product_unit,Z4: product_unit] : ( Y4 = Z4 )
% 14.25/8.08 @ X3
% 14.25/8.08 @ Z ) )
% 14.25/8.08 => ( pred_c6915491785556283576t_unit @ A @ P @ ( sup_su793286257634532545t_unit @ ( insert_Product_unit @ Z @ bot_bo3957492148770167129t_unit ) @ C5 ) ) ) ) ) ).
% 14.25/8.08
% 14.25/8.08 thf(fact_4398_pred__on_Ochain__extend,axiom,
% 14.25/8.08 ! [A: set_o,P: $o > $o > $o,C5: set_o,Z: $o] :
% 14.25/8.08 ( ( pred_chain_o @ A @ P @ C5 )
% 14.25/8.08 => ( ( member_o @ Z @ A )
% 14.25/8.08 => ( ! [X3: $o] :
% 14.25/8.08 ( ( member_o @ X3 @ C5 )
% 14.25/8.08 => ( sup_sup_o_o_o @ P
% 14.25/8.08 @ ^ [Y4: $o,Z4: $o] : ( Y4 = Z4 )
% 14.25/8.08 @ X3
% 14.25/8.08 @ Z ) )
% 14.25/8.08 => ( pred_chain_o @ A @ P @ ( sup_sup_set_o @ ( insert_o @ Z @ bot_bot_set_o ) @ C5 ) ) ) ) ) ).
% 14.25/8.08
% 14.25/8.08 thf(fact_4399_pred__on_Ochain__extend,axiom,
% 14.25/8.08 ! [A: set_nat,P: nat > nat > $o,C5: set_nat,Z: nat] :
% 14.25/8.08 ( ( pred_chain_nat @ A @ P @ C5 )
% 14.25/8.08 => ( ( member_nat @ Z @ A )
% 14.25/8.08 => ( ! [X3: nat] :
% 14.25/8.08 ( ( member_nat @ X3 @ C5 )
% 14.25/8.08 => ( sup_sup_nat_nat_o @ P
% 14.25/8.08 @ ^ [Y4: nat,Z4: nat] : ( Y4 = Z4 )
% 14.25/8.08 @ X3
% 14.25/8.08 @ Z ) )
% 14.25/8.08 => ( pred_chain_nat @ A @ P @ ( sup_sup_set_nat @ ( insert_nat @ Z @ bot_bot_set_nat ) @ C5 ) ) ) ) ) ).
% 14.25/8.08
% 14.25/8.08 thf(fact_4400_pred__on_Ochain__extend,axiom,
% 14.25/8.08 ! [A: set_int,P: int > int > $o,C5: set_int,Z: int] :
% 14.25/8.08 ( ( pred_chain_int @ A @ P @ C5 )
% 14.25/8.08 => ( ( member_int @ Z @ A )
% 14.25/8.08 => ( ! [X3: int] :
% 14.25/8.08 ( ( member_int @ X3 @ C5 )
% 14.25/8.08 => ( sup_sup_int_int_o @ P
% 14.25/8.08 @ ^ [Y4: int,Z4: int] : ( Y4 = Z4 )
% 14.25/8.08 @ X3
% 14.25/8.08 @ Z ) )
% 14.25/8.08 => ( pred_chain_int @ A @ P @ ( sup_sup_set_int @ ( insert_int @ Z @ bot_bot_set_int ) @ C5 ) ) ) ) ) ).
% 14.25/8.08
% 14.25/8.08 thf(fact_4401_pred__on_Ochain__extend,axiom,
% 14.25/8.08 ! [A: set_Pr4532377907799695533_nat_o,P: ( produc3658429121746597890et_nat > $o ) > ( produc3658429121746597890et_nat > $o ) > $o,C5: set_Pr4532377907799695533_nat_o,Z: produc3658429121746597890et_nat > $o] :
% 14.25/8.08 ( ( pred_c177362287423414904_nat_o @ A @ P @ C5 )
% 14.25/8.08 => ( ( member6576561426505652726_nat_o @ Z @ A )
% 14.25/8.08 => ( ! [X3: produc3658429121746597890et_nat > $o] :
% 14.25/8.08 ( ( member6576561426505652726_nat_o @ X3 @ C5 )
% 14.25/8.08 => ( sup_su1942872429929737346at_o_o @ P
% 14.25/8.08 @ ^ [Y4: produc3658429121746597890et_nat > $o,Z4: produc3658429121746597890et_nat > $o] : ( Y4 = Z4 )
% 14.25/8.08 @ X3
% 14.25/8.08 @ Z ) )
% 14.25/8.08 => ( pred_c177362287423414904_nat_o @ A @ P @ ( sup_su5209123915105501825_nat_o @ ( insert5175938949040314269_nat_o @ Z @ bot_bo7824918357723582233_nat_o ) @ C5 ) ) ) ) ) ).
% 14.25/8.08
% 14.25/8.08 thf(fact_4402_subset__Zorn__nonempty,axiom,
% 14.25/8.08 ! [A11: set_set_nat] :
% 14.25/8.08 ( ( A11 != bot_bot_set_set_nat )
% 14.25/8.08 => ( ! [C7: set_set_nat] :
% 14.25/8.08 ( ( C7 != bot_bot_set_set_nat )
% 14.25/8.08 => ( ( pred_chain_set_nat @ A11 @ ord_less_set_nat @ C7 )
% 14.25/8.08 => ( member_set_nat @ ( comple7399068483239264473et_nat @ C7 ) @ A11 ) ) )
% 14.25/8.08 => ? [X3: set_nat] :
% 14.25/8.08 ( ( member_set_nat @ X3 @ A11 )
% 14.25/8.08 & ! [Xa2: set_nat] :
% 14.25/8.08 ( ( member_set_nat @ Xa2 @ A11 )
% 14.25/8.08 => ( ( ord_less_eq_set_nat @ X3 @ Xa2 )
% 14.25/8.08 => ( Xa2 = X3 ) ) ) ) ) ) ).
% 14.25/8.08
% 14.25/8.08 thf(fact_4403_in__measure,axiom,
% 14.25/8.08 ! [X: nat,Y: nat,F: nat > nat] :
% 14.25/8.08 ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ Y ) @ ( measure_nat @ F ) )
% 14.25/8.08 = ( ord_less_nat @ ( F @ X ) @ ( F @ Y ) ) ) ).
% 14.25/8.08
% 14.25/8.08 thf(fact_4404_in__measure,axiom,
% 14.25/8.08 ! [X: int,Y: int,F: int > nat] :
% 14.25/8.08 ( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ X @ Y ) @ ( measure_int @ F ) )
% 14.25/8.08 = ( ord_less_nat @ ( F @ X ) @ ( F @ Y ) ) ) ).
% 14.25/8.08
% 14.25/8.08 thf(fact_4405_top_Oordering__top__axioms,axiom,
% 14.25/8.08 orderi2935705153647960990t_unit @ ord_le3507040750410214029t_unit @ ord_le8056459307392131481t_unit @ top_to1996260823553986621t_unit ).
% 14.25/8.08
% 14.25/8.08 thf(fact_4406_top_Oordering__top__axioms,axiom,
% 14.25/8.08 ordering_top_set_o @ ord_less_eq_set_o @ ord_less_set_o @ top_top_set_o ).
% 14.25/8.08
% 14.25/8.08 thf(fact_4407_top_Oordering__top__axioms,axiom,
% 14.25/8.08 ordering_top_set_int @ ord_less_eq_set_int @ ord_less_set_int @ top_top_set_int ).
% 14.25/8.08
% 14.25/8.08 thf(fact_4408_top_Oordering__top__axioms,axiom,
% 14.25/8.08 ordering_top_assn @ ord_less_eq_assn @ ord_less_assn @ top_top_assn ).
% 14.25/8.08
% 14.25/8.08 thf(fact_4409_top_Oordering__top__axioms,axiom,
% 14.25/8.08 orderi2807725203697798511er_nat @ ord_le2510731241096832064er_nat @ ord_less_filter_nat @ top_top_filter_nat ).
% 14.25/8.08
% 14.25/8.08 thf(fact_4410_top_Oordering__top__axioms,axiom,
% 14.25/8.08 ordering_top_set_nat @ ord_less_eq_set_nat @ ord_less_set_nat @ top_top_set_nat ).
% 14.25/8.08
% 14.25/8.08 thf(fact_4411_Max_Oinsert__remove,axiom,
% 14.25/8.08 ! [A: set_Code_integer,X: code_integer] :
% 14.25/8.08 ( ( finite6017078050557962740nteger @ A )
% 14.25/8.08 => ( ( ( ( minus_2355218937544613996nteger @ A @ ( insert_Code_integer @ X @ bot_bo3990330152332043303nteger ) )
% 14.25/8.08 = bot_bo3990330152332043303nteger )
% 14.25/8.08 => ( ( lattic4901227151466704046nteger @ ( insert_Code_integer @ X @ A ) )
% 14.25/8.08 = X ) )
% 14.25/8.08 & ( ( ( minus_2355218937544613996nteger @ A @ ( insert_Code_integer @ X @ bot_bo3990330152332043303nteger ) )
% 14.25/8.08 != bot_bo3990330152332043303nteger )
% 14.25/8.08 => ( ( lattic4901227151466704046nteger @ ( insert_Code_integer @ X @ A ) )
% 14.25/8.08 = ( ord_max_Code_integer @ X @ ( lattic4901227151466704046nteger @ ( minus_2355218937544613996nteger @ A @ ( insert_Code_integer @ X @ bot_bo3990330152332043303nteger ) ) ) ) ) ) ) ) ).
% 14.25/8.08
% 14.25/8.08 thf(fact_4412_Max_Oinsert__remove,axiom,
% 14.25/8.08 ! [A: set_Product_unit,X: product_unit] :
% 14.25/8.08 ( ( finite4290736615968046902t_unit @ A )
% 14.25/8.08 => ( ( ( ( minus_6452836326544984404t_unit @ A @ ( insert_Product_unit @ X @ bot_bo3957492148770167129t_unit ) )
% 14.25/8.08 = bot_bo3957492148770167129t_unit )
% 14.25/8.08 => ( ( lattic1210392890313158908t_unit @ ( insert_Product_unit @ X @ A ) )
% 14.25/8.08 = X ) )
% 14.25/8.08 & ( ( ( minus_6452836326544984404t_unit @ A @ ( insert_Product_unit @ X @ bot_bo3957492148770167129t_unit ) )
% 14.25/8.08 != bot_bo3957492148770167129t_unit )
% 14.25/8.08 => ( ( lattic1210392890313158908t_unit @ ( insert_Product_unit @ X @ A ) )
% 14.25/8.08 = ( ord_max_Product_unit @ X @ ( lattic1210392890313158908t_unit @ ( minus_6452836326544984404t_unit @ A @ ( insert_Product_unit @ X @ bot_bo3957492148770167129t_unit ) ) ) ) ) ) ) ) ).
% 14.25/8.08
% 14.25/8.08 thf(fact_4413_Max_Oinsert__remove,axiom,
% 14.25/8.08 ! [A: set_o,X: $o] :
% 14.25/8.08 ( ( finite_finite_o @ A )
% 14.25/8.08 => ( ( lattic1921953407002678535_Max_o @ ( insert_o @ X @ A ) )
% 14.25/8.08 = ( ( ( ( minus_minus_set_o @ A @ ( insert_o @ X @ bot_bot_set_o ) )
% 14.25/8.08 = bot_bot_set_o )
% 14.25/8.08 => X )
% 14.25/8.08 & ( ( ( minus_minus_set_o @ A @ ( insert_o @ X @ bot_bot_set_o ) )
% 14.25/8.08 != bot_bot_set_o )
% 14.25/8.08 => ( ord_max_o @ X @ ( lattic1921953407002678535_Max_o @ ( minus_minus_set_o @ A @ ( insert_o @ X @ bot_bot_set_o ) ) ) ) ) ) ) ) ).
% 14.25/8.08
% 14.25/8.08 thf(fact_4414_Max_Oinsert__remove,axiom,
% 14.25/8.08 ! [A: set_int,X: int] :
% 14.25/8.08 ( ( finite_finite_int @ A )
% 14.25/8.08 => ( ( ( ( minus_minus_set_int @ A @ ( insert_int @ X @ bot_bot_set_int ) )
% 14.25/8.08 = bot_bot_set_int )
% 14.25/8.08 => ( ( lattic8263393255366662781ax_int @ ( insert_int @ X @ A ) )
% 14.25/8.08 = X ) )
% 14.25/8.08 & ( ( ( minus_minus_set_int @ A @ ( insert_int @ X @ bot_bot_set_int ) )
% 14.25/8.08 != bot_bot_set_int )
% 14.25/8.08 => ( ( lattic8263393255366662781ax_int @ ( insert_int @ X @ A ) )
% 14.25/8.08 = ( ord_max_int @ X @ ( lattic8263393255366662781ax_int @ ( minus_minus_set_int @ A @ ( insert_int @ X @ bot_bot_set_int ) ) ) ) ) ) ) ) ).
% 14.25/8.08
% 14.25/8.08 thf(fact_4415_Max_Oinsert__remove,axiom,
% 14.25/8.08 ! [A: set_nat,X: nat] :
% 14.25/8.08 ( ( finite_finite_nat @ A )
% 14.25/8.08 => ( ( ( ( minus_minus_set_nat @ A @ ( insert_nat @ X @ bot_bot_set_nat ) )
% 14.25/8.08 = bot_bot_set_nat )
% 14.25/8.08 => ( ( lattic8265883725875713057ax_nat @ ( insert_nat @ X @ A ) )
% 14.25/8.08 = X ) )
% 14.25/8.08 & ( ( ( minus_minus_set_nat @ A @ ( insert_nat @ X @ bot_bot_set_nat ) )
% 14.25/8.08 != bot_bot_set_nat )
% 14.25/8.08 => ( ( lattic8265883725875713057ax_nat @ ( insert_nat @ X @ A ) )
% 14.25/8.08 = ( ord_max_nat @ X @ ( lattic8265883725875713057ax_nat @ ( minus_minus_set_nat @ A @ ( insert_nat @ X @ bot_bot_set_nat ) ) ) ) ) ) ) ) ).
% 14.25/8.08
% 14.25/8.08 thf(fact_4416_Max_Oremove,axiom,
% 14.25/8.08 ! [A: set_Code_integer,X: code_integer] :
% 14.25/8.08 ( ( finite6017078050557962740nteger @ A )
% 14.25/8.08 => ( ( member_Code_integer @ X @ A )
% 14.25/8.08 => ( ( ( ( minus_2355218937544613996nteger @ A @ ( insert_Code_integer @ X @ bot_bo3990330152332043303nteger ) )
% 14.25/8.08 = bot_bo3990330152332043303nteger )
% 14.25/8.08 => ( ( lattic4901227151466704046nteger @ A )
% 14.25/8.08 = X ) )
% 14.25/8.08 & ( ( ( minus_2355218937544613996nteger @ A @ ( insert_Code_integer @ X @ bot_bo3990330152332043303nteger ) )
% 14.25/8.08 != bot_bo3990330152332043303nteger )
% 14.25/8.08 => ( ( lattic4901227151466704046nteger @ A )
% 14.25/8.08 = ( ord_max_Code_integer @ X @ ( lattic4901227151466704046nteger @ ( minus_2355218937544613996nteger @ A @ ( insert_Code_integer @ X @ bot_bo3990330152332043303nteger ) ) ) ) ) ) ) ) ) ).
% 14.25/8.08
% 14.25/8.08 thf(fact_4417_Max_Oremove,axiom,
% 14.25/8.08 ! [A: set_Product_unit,X: product_unit] :
% 14.25/8.08 ( ( finite4290736615968046902t_unit @ A )
% 14.25/8.08 => ( ( member_Product_unit @ X @ A )
% 14.25/8.08 => ( ( ( ( minus_6452836326544984404t_unit @ A @ ( insert_Product_unit @ X @ bot_bo3957492148770167129t_unit ) )
% 14.25/8.08 = bot_bo3957492148770167129t_unit )
% 14.25/8.08 => ( ( lattic1210392890313158908t_unit @ A )
% 14.25/8.08 = X ) )
% 14.25/8.08 & ( ( ( minus_6452836326544984404t_unit @ A @ ( insert_Product_unit @ X @ bot_bo3957492148770167129t_unit ) )
% 14.25/8.08 != bot_bo3957492148770167129t_unit )
% 14.25/8.08 => ( ( lattic1210392890313158908t_unit @ A )
% 14.25/8.08 = ( ord_max_Product_unit @ X @ ( lattic1210392890313158908t_unit @ ( minus_6452836326544984404t_unit @ A @ ( insert_Product_unit @ X @ bot_bo3957492148770167129t_unit ) ) ) ) ) ) ) ) ) ).
% 14.25/8.08
% 14.25/8.08 thf(fact_4418_Max_Oremove,axiom,
% 14.25/8.08 ! [A: set_o,X: $o] :
% 14.25/8.08 ( ( finite_finite_o @ A )
% 14.25/8.08 => ( ( member_o @ X @ A )
% 14.25/8.08 => ( ( lattic1921953407002678535_Max_o @ A )
% 14.25/8.08 = ( ( ( ( minus_minus_set_o @ A @ ( insert_o @ X @ bot_bot_set_o ) )
% 14.25/8.08 = bot_bot_set_o )
% 14.25/8.08 => X )
% 14.25/8.08 & ( ( ( minus_minus_set_o @ A @ ( insert_o @ X @ bot_bot_set_o ) )
% 14.25/8.08 != bot_bot_set_o )
% 14.25/8.08 => ( ord_max_o @ X @ ( lattic1921953407002678535_Max_o @ ( minus_minus_set_o @ A @ ( insert_o @ X @ bot_bot_set_o ) ) ) ) ) ) ) ) ) ).
% 14.25/8.08
% 14.25/8.08 thf(fact_4419_Max_Oremove,axiom,
% 14.25/8.08 ! [A: set_int,X: int] :
% 14.25/8.08 ( ( finite_finite_int @ A )
% 14.25/8.08 => ( ( member_int @ X @ A )
% 14.25/8.08 => ( ( ( ( minus_minus_set_int @ A @ ( insert_int @ X @ bot_bot_set_int ) )
% 14.25/8.08 = bot_bot_set_int )
% 14.25/8.08 => ( ( lattic8263393255366662781ax_int @ A )
% 14.25/8.08 = X ) )
% 14.25/8.08 & ( ( ( minus_minus_set_int @ A @ ( insert_int @ X @ bot_bot_set_int ) )
% 14.25/8.08 != bot_bot_set_int )
% 14.25/8.08 => ( ( lattic8263393255366662781ax_int @ A )
% 14.25/8.08 = ( ord_max_int @ X @ ( lattic8263393255366662781ax_int @ ( minus_minus_set_int @ A @ ( insert_int @ X @ bot_bot_set_int ) ) ) ) ) ) ) ) ) ).
% 14.25/8.08
% 14.25/8.08 thf(fact_4420_Max_Oremove,axiom,
% 14.25/8.08 ! [A: set_nat,X: nat] :
% 14.25/8.08 ( ( finite_finite_nat @ A )
% 14.25/8.08 => ( ( member_nat @ X @ A )
% 14.25/8.08 => ( ( ( ( minus_minus_set_nat @ A @ ( insert_nat @ X @ bot_bot_set_nat ) )
% 14.25/8.08 = bot_bot_set_nat )
% 14.25/8.08 => ( ( lattic8265883725875713057ax_nat @ A )
% 14.25/8.08 = X ) )
% 14.25/8.08 & ( ( ( minus_minus_set_nat @ A @ ( insert_nat @ X @ bot_bot_set_nat ) )
% 14.25/8.08 != bot_bot_set_nat )
% 14.25/8.08 => ( ( lattic8265883725875713057ax_nat @ A )
% 14.25/8.08 = ( ord_max_nat @ X @ ( lattic8265883725875713057ax_nat @ ( minus_minus_set_nat @ A @ ( insert_nat @ X @ bot_bot_set_nat ) ) ) ) ) ) ) ) ) ).
% 14.25/8.08
% 14.25/8.08 thf(fact_4421_inj__on__Un,axiom,
% 14.25/8.08 ! [F: code_integer > code_integer,A: set_Code_integer,B4: set_Code_integer] :
% 14.25/8.08 ( ( inj_on1158064722259180589nteger @ F @ ( sup_su848401254843788991nteger @ A @ B4 ) )
% 14.25/8.08 = ( ( inj_on1158064722259180589nteger @ F @ A )
% 14.25/8.08 & ( inj_on1158064722259180589nteger @ F @ B4 )
% 14.25/8.08 & ( ( inf_in1364745209274528805nteger @ ( image_4470545334726330049nteger @ F @ ( minus_2355218937544613996nteger @ A @ B4 ) ) @ ( image_4470545334726330049nteger @ F @ ( minus_2355218937544613996nteger @ B4 @ A ) ) )
% 14.25/8.08 = bot_bo3990330152332043303nteger ) ) ) ).
% 14.25/8.08
% 14.25/8.08 thf(fact_4422_inj__on__Un,axiom,
% 14.25/8.08 ! [F: nat > set_nat,A: set_nat,B4: set_nat] :
% 14.25/8.08 ( ( inj_on_nat_set_nat @ F @ ( sup_sup_set_nat @ A @ B4 ) )
% 14.25/8.08 = ( ( inj_on_nat_set_nat @ F @ A )
% 14.25/8.08 & ( inj_on_nat_set_nat @ F @ B4 )
% 14.25/8.08 & ( ( inf_inf_set_set_nat @ ( image_nat_set_nat @ F @ ( minus_minus_set_nat @ A @ B4 ) ) @ ( image_nat_set_nat @ F @ ( minus_minus_set_nat @ B4 @ A ) ) )
% 14.25/8.08 = bot_bot_set_set_nat ) ) ) ).
% 14.25/8.08
% 14.25/8.08 thf(fact_4423_inj__on__Un,axiom,
% 14.25/8.08 ! [F: nat > code_integer,A: set_nat,B4: set_nat] :
% 14.25/8.08 ( ( inj_on2245834603434889120nteger @ F @ ( sup_sup_set_nat @ A @ B4 ) )
% 14.25/8.08 = ( ( inj_on2245834603434889120nteger @ F @ A )
% 14.25/8.08 & ( inj_on2245834603434889120nteger @ F @ B4 )
% 14.25/8.08 & ( ( inf_in1364745209274528805nteger @ ( image_1215581382706833972nteger @ F @ ( minus_minus_set_nat @ A @ B4 ) ) @ ( image_1215581382706833972nteger @ F @ ( minus_minus_set_nat @ B4 @ A ) ) )
% 14.25/8.08 = bot_bo3990330152332043303nteger ) ) ) ).
% 14.25/8.08
% 14.25/8.08 thf(fact_4424_inj__on__Un,axiom,
% 14.25/8.08 ! [F: nat > product_unit,A: set_nat,B4: set_nat] :
% 14.25/8.08 ( ( inj_on7061601236592826506t_unit @ F @ ( sup_sup_set_nat @ A @ B4 ) )
% 14.25/8.08 = ( ( inj_on7061601236592826506t_unit @ F @ A )
% 14.25/8.08 & ( inj_on7061601236592826506t_unit @ F @ B4 )
% 14.25/8.08 & ( ( inf_in4660618365625256667t_unit @ ( image_8730104196221521654t_unit @ F @ ( minus_minus_set_nat @ A @ B4 ) ) @ ( image_8730104196221521654t_unit @ F @ ( minus_minus_set_nat @ B4 @ A ) ) )
% 14.25/8.08 = bot_bo3957492148770167129t_unit ) ) ) ).
% 14.25/8.08
% 14.25/8.08 thf(fact_4425_inj__on__Un,axiom,
% 14.25/8.08 ! [F: nat > $o,A: set_nat,B4: set_nat] :
% 14.25/8.08 ( ( inj_on_nat_o @ F @ ( sup_sup_set_nat @ A @ B4 ) )
% 14.25/8.08 = ( ( inj_on_nat_o @ F @ A )
% 14.25/8.08 & ( inj_on_nat_o @ F @ B4 )
% 14.25/8.08 & ( ( inf_inf_set_o @ ( image_nat_o2 @ F @ ( minus_minus_set_nat @ A @ B4 ) ) @ ( image_nat_o2 @ F @ ( minus_minus_set_nat @ B4 @ A ) ) )
% 14.25/8.08 = bot_bot_set_o ) ) ) ).
% 14.25/8.08
% 14.25/8.08 thf(fact_4426_inj__on__Un,axiom,
% 14.25/8.08 ! [F: nat > nat,A: set_nat,B4: set_nat] :
% 14.25/8.08 ( ( inj_on_nat_nat @ F @ ( sup_sup_set_nat @ A @ B4 ) )
% 14.25/8.08 = ( ( inj_on_nat_nat @ F @ A )
% 14.25/8.08 & ( inj_on_nat_nat @ F @ B4 )
% 14.25/8.08 & ( ( inf_inf_set_nat @ ( image_nat_nat2 @ F @ ( minus_minus_set_nat @ A @ B4 ) ) @ ( image_nat_nat2 @ F @ ( minus_minus_set_nat @ B4 @ A ) ) )
% 14.25/8.08 = bot_bot_set_nat ) ) ) ).
% 14.25/8.08
% 14.25/8.08 thf(fact_4427_inj__on__Un,axiom,
% 14.25/8.08 ! [F: nat > int,A: set_nat,B4: set_nat] :
% 14.25/8.08 ( ( inj_on_nat_int @ F @ ( sup_sup_set_nat @ A @ B4 ) )
% 14.25/8.08 = ( ( inj_on_nat_int @ F @ A )
% 14.25/8.08 & ( inj_on_nat_int @ F @ B4 )
% 14.25/8.08 & ( ( inf_inf_set_int @ ( image_nat_int2 @ F @ ( minus_minus_set_nat @ A @ B4 ) ) @ ( image_nat_int2 @ F @ ( minus_minus_set_nat @ B4 @ A ) ) )
% 14.25/8.08 = bot_bot_set_int ) ) ) ).
% 14.25/8.08
% 14.25/8.08 thf(fact_4428_inj__on__Un,axiom,
% 14.25/8.08 ! [F: nat > produc3658429121746597890et_nat > $o,A: set_nat,B4: set_nat] :
% 14.25/8.08 ( ( inj_on7779444506608719434_nat_o @ F @ ( sup_sup_set_nat @ A @ B4 ) )
% 14.25/8.08 = ( ( inj_on7779444506608719434_nat_o @ F @ A )
% 14.25/8.08 & ( inj_on7779444506608719434_nat_o @ F @ B4 )
% 14.25/8.08 & ( ( inf_in1906310914598751387_nat_o @ ( image_4682239675366380726_nat_o @ F @ ( minus_minus_set_nat @ A @ B4 ) ) @ ( image_4682239675366380726_nat_o @ F @ ( minus_minus_set_nat @ B4 @ A ) ) )
% 14.25/8.08 = bot_bo7824918357723582233_nat_o ) ) ) ).
% 14.25/8.08
% 14.25/8.08 thf(fact_4429_Max__insert,axiom,
% 14.25/8.08 ! [A: set_Code_integer,X: code_integer] :
% 14.25/8.08 ( ( finite6017078050557962740nteger @ A )
% 14.25/8.08 => ( ( A != bot_bo3990330152332043303nteger )
% 14.25/8.08 => ( ( lattic4901227151466704046nteger @ ( insert_Code_integer @ X @ A ) )
% 14.25/8.08 = ( ord_max_Code_integer @ X @ ( lattic4901227151466704046nteger @ A ) ) ) ) ) ).
% 14.25/8.08
% 14.25/8.08 thf(fact_4430_Max__insert,axiom,
% 14.25/8.08 ! [A: set_Product_unit,X: product_unit] :
% 14.25/8.08 ( ( finite4290736615968046902t_unit @ A )
% 14.25/8.08 => ( ( A != bot_bo3957492148770167129t_unit )
% 14.25/8.08 => ( ( lattic1210392890313158908t_unit @ ( insert_Product_unit @ X @ A ) )
% 14.25/8.08 = ( ord_max_Product_unit @ X @ ( lattic1210392890313158908t_unit @ A ) ) ) ) ) ).
% 14.25/8.08
% 14.25/8.08 thf(fact_4431_Max__insert,axiom,
% 14.25/8.08 ! [A: set_o,X: $o] :
% 14.25/8.08 ( ( finite_finite_o @ A )
% 14.25/8.08 => ( ( A != bot_bot_set_o )
% 14.25/8.08 => ( ( lattic1921953407002678535_Max_o @ ( insert_o @ X @ A ) )
% 14.25/8.08 = ( ord_max_o @ X @ ( lattic1921953407002678535_Max_o @ A ) ) ) ) ) ).
% 14.25/8.08
% 14.25/8.08 thf(fact_4432_Max__insert,axiom,
% 14.25/8.08 ! [A: set_int,X: int] :
% 14.25/8.08 ( ( finite_finite_int @ A )
% 14.25/8.08 => ( ( A != bot_bot_set_int )
% 14.25/8.08 => ( ( lattic8263393255366662781ax_int @ ( insert_int @ X @ A ) )
% 14.25/8.08 = ( ord_max_int @ X @ ( lattic8263393255366662781ax_int @ A ) ) ) ) ) ).
% 14.25/8.08
% 14.25/8.08 thf(fact_4433_Max__insert,axiom,
% 14.25/8.08 ! [A: set_nat,X: nat] :
% 14.25/8.08 ( ( finite_finite_nat @ A )
% 14.25/8.08 => ( ( A != bot_bot_set_nat )
% 14.25/8.08 => ( ( lattic8265883725875713057ax_nat @ ( insert_nat @ X @ A ) )
% 14.25/8.08 = ( ord_max_nat @ X @ ( lattic8265883725875713057ax_nat @ A ) ) ) ) ) ).
% 14.25/8.08
% 14.25/8.08 thf(fact_4434_map__prod__surj,axiom,
% 14.25/8.08 ! [F: code_integer > code_integer,G: code_integer > code_integer] :
% 14.25/8.08 ( ( ( image_4470545334726330049nteger @ F @ top_to4645266643341252675nteger )
% 14.25/8.08 = top_to4645266643341252675nteger )
% 14.25/8.08 => ( ( ( image_4470545334726330049nteger @ G @ top_to4645266643341252675nteger )
% 14.25/8.08 = top_to4645266643341252675nteger )
% 14.25/8.08 => ( ( image_7082141833288935621nteger @ ( produc1802105962441033582nteger @ F @ G ) @ top_to2363841527343591755nteger )
% 14.25/8.08 = top_to2363841527343591755nteger ) ) ) ).
% 14.25/8.08
% 14.25/8.08 thf(fact_4435_map__prod__surj,axiom,
% 14.25/8.08 ! [F: code_integer > code_integer,G: product_unit > product_unit] :
% 14.25/8.08 ( ( ( image_4470545334726330049nteger @ F @ top_to4645266643341252675nteger )
% 14.25/8.08 = top_to4645266643341252675nteger )
% 14.25/8.08 => ( ( ( image_405062704495631173t_unit @ G @ top_to1996260823553986621t_unit )
% 14.25/8.08 = top_to1996260823553986621t_unit )
% 14.25/8.08 => ( ( image_3456600171786721233t_unit @ ( produc3346700480963112818t_unit @ F @ G ) @ top_to3244471725241573579t_unit )
% 14.25/8.08 = top_to3244471725241573579t_unit ) ) ) ).
% 14.25/8.08
% 14.25/8.08 thf(fact_4436_map__prod__surj,axiom,
% 14.25/8.08 ! [F: code_integer > code_integer,G: product_unit > $o] :
% 14.25/8.08 ( ( ( image_4470545334726330049nteger @ F @ top_to4645266643341252675nteger )
% 14.25/8.08 = top_to4645266643341252675nteger )
% 14.25/8.08 => ( ( ( image_Product_unit_o2 @ G @ top_to1996260823553986621t_unit )
% 14.25/8.08 = top_top_set_o )
% 14.25/8.08 => ( ( image_7060974388358999388eger_o @ ( produc8116347370398923901unit_o @ F @ G ) @ top_to3244471725241573579t_unit )
% 14.25/8.08 = top_to2658285827425157974eger_o ) ) ) ).
% 14.25/8.08
% 14.25/8.08 thf(fact_4437_map__prod__surj,axiom,
% 14.25/8.08 ! [F: code_integer > code_integer,G: product_unit > nat] :
% 14.25/8.08 ( ( ( image_4470545334726330049nteger @ F @ top_to4645266643341252675nteger )
% 14.25/8.08 = top_to4645266643341252675nteger )
% 14.25/8.08 => ( ( ( image_875570014554754200it_nat @ G @ top_to1996260823553986621t_unit )
% 14.25/8.08 = top_top_set_nat )
% 14.25/8.08 => ( ( image_8236373689398998434er_nat @ ( produc4022400886476617259it_nat @ F @ G ) @ top_to3244471725241573579t_unit )
% 14.25/8.08 = top_to8760411888770933182er_nat ) ) ) ).
% 14.25/8.08
% 14.25/8.08 thf(fact_4438_map__prod__surj,axiom,
% 14.25/8.08 ! [F: code_integer > code_integer,G: product_unit > int] :
% 14.25/8.08 ( ( ( image_4470545334726330049nteger @ F @ top_to4645266643341252675nteger )
% 14.25/8.08 = top_to4645266643341252675nteger )
% 14.25/8.08 => ( ( ( image_873079544045703924it_int @ G @ top_to1996260823553986621t_unit )
% 14.25/8.08 = top_top_set_int )
% 14.25/8.08 => ( ( image_4058522669889801726er_int @ ( produc4019910415967566983it_int @ F @ G ) @ top_to3244471725241573579t_unit )
% 14.25/8.08 = top_to6270328743839206426er_int ) ) ) ).
% 14.25/8.08
% 14.25/8.08 thf(fact_4439_map__prod__surj,axiom,
% 14.25/8.08 ! [F: code_integer > code_integer,G: $o > product_unit] :
% 14.25/8.08 ( ( ( image_4470545334726330049nteger @ F @ top_to4645266643341252675nteger )
% 14.25/8.08 = top_to4645266643341252675nteger )
% 14.25/8.08 => ( ( ( image_o_Product_unit2 @ G @ top_top_set_o )
% 14.25/8.08 = top_to1996260823553986621t_unit )
% 14.25/8.08 => ( ( image_2425804817718344284t_unit @ ( produc3530376873132787239t_unit @ F @ G ) @ top_to2658285827425157974eger_o )
% 14.25/8.08 = top_to3244471725241573579t_unit ) ) ) ).
% 14.25/8.08
% 14.25/8.08 thf(fact_4440_map__prod__surj,axiom,
% 14.25/8.08 ! [F: code_integer > code_integer,G: $o > $o] :
% 14.25/8.08 ( ( ( image_4470545334726330049nteger @ F @ top_to4645266643341252675nteger )
% 14.25/8.08 = top_to4645266643341252675nteger )
% 14.25/8.08 => ( ( ( image_o_o2 @ G @ top_top_set_o )
% 14.25/8.08 = top_top_set_o )
% 14.25/8.08 => ( ( image_227735840630799463eger_o @ ( produc1084426483498010290er_o_o @ F @ G ) @ top_to2658285827425157974eger_o )
% 14.25/8.08 = top_to2658285827425157974eger_o ) ) ) ).
% 14.25/8.08
% 14.25/8.08 thf(fact_4441_map__prod__surj,axiom,
% 14.25/8.08 ! [F: code_integer > code_integer,G: $o > nat] :
% 14.25/8.08 ( ( ( image_4470545334726330049nteger @ F @ top_to4645266643341252675nteger )
% 14.25/8.08 = top_to4645266643341252675nteger )
% 14.25/8.08 => ( ( ( image_o_nat2 @ G @ top_top_set_o )
% 14.25/8.08 = top_top_set_nat )
% 14.25/8.08 => ( ( image_1252617143230709591er_nat @ ( produc7550393335539917622_o_nat @ F @ G ) @ top_to2658285827425157974eger_o )
% 14.25/8.08 = top_to8760411888770933182er_nat ) ) ) ).
% 14.25/8.08
% 14.25/8.08 thf(fact_4442_map__prod__surj,axiom,
% 14.25/8.08 ! [F: code_integer > code_integer,G: $o > int] :
% 14.25/8.08 ( ( ( image_4470545334726330049nteger @ F @ top_to4645266643341252675nteger )
% 14.25/8.08 = top_to4645266643341252675nteger )
% 14.25/8.08 => ( ( ( image_o_int2 @ G @ top_top_set_o )
% 14.25/8.08 = top_top_set_int )
% 14.25/8.08 => ( ( image_6298138160576288691er_int @ ( produc7547902865030867346_o_int @ F @ G ) @ top_to2658285827425157974eger_o )
% 14.25/8.08 = top_to6270328743839206426er_int ) ) ) ).
% 14.25/8.08
% 14.25/8.08 thf(fact_4443_map__prod__surj,axiom,
% 14.25/8.08 ! [F: code_integer > code_integer,G: nat > code_integer] :
% 14.25/8.08 ( ( ( image_4470545334726330049nteger @ F @ top_to4645266643341252675nteger )
% 14.25/8.08 = top_to4645266643341252675nteger )
% 14.25/8.08 => ( ( ( image_1215581382706833972nteger @ G @ top_top_set_nat )
% 14.25/8.08 = top_to4645266643341252675nteger )
% 14.25/8.08 => ( ( image_4921214744877245202nteger @ ( produc1622059938646302433nteger @ F @ G ) @ top_to8760411888770933182er_nat )
% 14.25/8.08 = top_to2363841527343591755nteger ) ) ) ).
% 14.25/8.08
% 14.25/8.08 thf(fact_4444_inj__on__insert,axiom,
% 14.25/8.08 ! [F: code_integer > code_integer,A2: code_integer,A: set_Code_integer] :
% 14.25/8.08 ( ( inj_on1158064722259180589nteger @ F @ ( insert_Code_integer @ A2 @ A ) )
% 14.25/8.08 = ( ( inj_on1158064722259180589nteger @ F @ A )
% 14.25/8.08 & ~ ( member_Code_integer @ ( F @ A2 ) @ ( image_4470545334726330049nteger @ F @ ( minus_2355218937544613996nteger @ A @ ( insert_Code_integer @ A2 @ bot_bo3990330152332043303nteger ) ) ) ) ) ) ).
% 14.25/8.08
% 14.25/8.08 thf(fact_4445_inj__on__insert,axiom,
% 14.25/8.08 ! [F: product_unit > nat,A2: product_unit,A: set_Product_unit] :
% 14.25/8.08 ( ( inj_on8430439091780834860it_nat @ F @ ( insert_Product_unit @ A2 @ A ) )
% 14.25/8.08 = ( ( inj_on8430439091780834860it_nat @ F @ A )
% 14.25/8.08 & ~ ( member_nat @ ( F @ A2 ) @ ( image_875570014554754200it_nat @ F @ ( minus_6452836326544984404t_unit @ A @ ( insert_Product_unit @ A2 @ bot_bo3957492148770167129t_unit ) ) ) ) ) ) ).
% 14.25/8.08
% 14.25/8.08 thf(fact_4446_inj__on__insert,axiom,
% 14.25/8.08 ! [F: product_unit > int,A2: product_unit,A: set_Product_unit] :
% 14.25/8.08 ( ( inj_on8427948621271784584it_int @ F @ ( insert_Product_unit @ A2 @ A ) )
% 14.25/8.08 = ( ( inj_on8427948621271784584it_int @ F @ A )
% 14.25/8.08 & ~ ( member_int @ ( F @ A2 ) @ ( image_873079544045703924it_int @ F @ ( minus_6452836326544984404t_unit @ A @ ( insert_Product_unit @ A2 @ bot_bo3957492148770167129t_unit ) ) ) ) ) ) ).
% 14.25/8.08
% 14.25/8.08 thf(fact_4447_inj__on__insert,axiom,
% 14.25/8.08 ! [F: product_unit > $o,A2: product_unit,A: set_Product_unit] :
% 14.25/8.08 ( ( inj_on8854722734121536060unit_o @ F @ ( insert_Product_unit @ A2 @ A ) )
% 14.25/8.08 = ( ( inj_on8854722734121536060unit_o @ F @ A )
% 14.25/8.08 & ~ ( member_o @ ( F @ A2 ) @ ( image_Product_unit_o2 @ F @ ( minus_6452836326544984404t_unit @ A @ ( insert_Product_unit @ A2 @ bot_bo3957492148770167129t_unit ) ) ) ) ) ) ).
% 14.25/8.08
% 14.25/8.08 thf(fact_4448_inj__on__insert,axiom,
% 14.25/8.08 ! [F: product_unit > product_unit,A2: product_unit,A: set_Product_unit] :
% 14.25/8.08 ( ( inj_on8151373323710067377t_unit @ F @ ( insert_Product_unit @ A2 @ A ) )
% 14.25/8.08 = ( ( inj_on8151373323710067377t_unit @ F @ A )
% 14.25/8.08 & ~ ( member_Product_unit @ ( F @ A2 ) @ ( image_405062704495631173t_unit @ F @ ( minus_6452836326544984404t_unit @ A @ ( insert_Product_unit @ A2 @ bot_bo3957492148770167129t_unit ) ) ) ) ) ) ).
% 14.25/8.08
% 14.25/8.08 thf(fact_4449_inj__on__insert,axiom,
% 14.25/8.08 ! [F: $o > nat,A2: $o,A: set_o] :
% 14.25/8.08 ( ( inj_on_o_nat @ F @ ( insert_o @ A2 @ A ) )
% 14.25/8.08 = ( ( inj_on_o_nat @ F @ A )
% 14.25/8.08 & ~ ( member_nat @ ( F @ A2 ) @ ( image_o_nat2 @ F @ ( minus_minus_set_o @ A @ ( insert_o @ A2 @ bot_bot_set_o ) ) ) ) ) ) ).
% 14.25/8.08
% 14.25/8.08 thf(fact_4450_inj__on__insert,axiom,
% 14.25/8.08 ! [F: $o > int,A2: $o,A: set_o] :
% 14.25/8.08 ( ( inj_on_o_int @ F @ ( insert_o @ A2 @ A ) )
% 14.25/8.08 = ( ( inj_on_o_int @ F @ A )
% 14.25/8.08 & ~ ( member_int @ ( F @ A2 ) @ ( image_o_int2 @ F @ ( minus_minus_set_o @ A @ ( insert_o @ A2 @ bot_bot_set_o ) ) ) ) ) ) ).
% 14.25/8.08
% 14.25/8.08 thf(fact_4451_inj__on__insert,axiom,
% 14.25/8.08 ! [F: $o > $o,A2: $o,A: set_o] :
% 14.25/8.08 ( ( inj_on_o_o @ F @ ( insert_o @ A2 @ A ) )
% 14.25/8.08 = ( ( inj_on_o_o @ F @ A )
% 14.25/8.08 & ~ ( member_o @ ( F @ A2 ) @ ( image_o_o2 @ F @ ( minus_minus_set_o @ A @ ( insert_o @ A2 @ bot_bot_set_o ) ) ) ) ) ) ).
% 14.25/8.08
% 14.25/8.08 thf(fact_4452_inj__on__insert,axiom,
% 14.25/8.08 ! [F: $o > product_unit,A2: $o,A: set_o] :
% 14.25/8.08 ( ( inj_on4268752236855399398t_unit @ F @ ( insert_o @ A2 @ A ) )
% 14.25/8.08 = ( ( inj_on4268752236855399398t_unit @ F @ A )
% 14.25/8.08 & ~ ( member_Product_unit @ ( F @ A2 ) @ ( image_o_Product_unit2 @ F @ ( minus_minus_set_o @ A @ ( insert_o @ A2 @ bot_bot_set_o ) ) ) ) ) ) ).
% 14.25/8.08
% 14.25/8.08 thf(fact_4453_inj__on__insert,axiom,
% 14.25/8.08 ! [F: int > nat,A2: int,A: set_int] :
% 14.25/8.08 ( ( inj_on_int_nat @ F @ ( insert_int @ A2 @ A ) )
% 14.25/8.08 = ( ( inj_on_int_nat @ F @ A )
% 14.25/8.08 & ~ ( member_nat @ ( F @ A2 ) @ ( image_int_nat2 @ F @ ( minus_minus_set_int @ A @ ( insert_int @ A2 @ bot_bot_set_int ) ) ) ) ) ) ).
% 14.25/8.08
% 14.25/8.08 thf(fact_4454_max__bot,axiom,
% 14.25/8.08 ! [X: set_Product_unit] :
% 14.25/8.08 ( ( ord_ma508538259134630082t_unit @ bot_bo3957492148770167129t_unit @ X )
% 14.25/8.08 = X ) ).
% 14.25/8.08
% 14.25/8.08 thf(fact_4455_max__bot,axiom,
% 14.25/8.08 ! [X: set_o] :
% 14.25/8.08 ( ( ord_max_set_o @ bot_bot_set_o @ X )
% 14.25/8.08 = X ) ).
% 14.25/8.08
% 14.25/8.08 thf(fact_4456_max__bot,axiom,
% 14.25/8.08 ! [X: set_nat] :
% 14.25/8.08 ( ( ord_max_set_nat @ bot_bot_set_nat @ X )
% 14.25/8.08 = X ) ).
% 14.25/8.08
% 14.25/8.08 thf(fact_4457_max__bot,axiom,
% 14.25/8.08 ! [X: set_int] :
% 14.25/8.08 ( ( ord_max_set_int @ bot_bot_set_int @ X )
% 14.25/8.08 = X ) ).
% 14.25/8.08
% 14.25/8.08 thf(fact_4458_max__bot,axiom,
% 14.25/8.08 ! [X: set_Pr4532377907799695533_nat_o] :
% 14.25/8.08 ( ( ord_ma8206644353667601538_nat_o @ bot_bo7824918357723582233_nat_o @ X )
% 14.25/8.08 = X ) ).
% 14.25/8.08
% 14.25/8.08 thf(fact_4459_max__bot,axiom,
% 14.25/8.08 ! [X: nat] :
% 14.25/8.08 ( ( ord_max_nat @ bot_bot_nat @ X )
% 14.25/8.08 = X ) ).
% 14.25/8.08
% 14.25/8.08 thf(fact_4460_max__bot2,axiom,
% 14.25/8.08 ! [X: set_Product_unit] :
% 14.25/8.08 ( ( ord_ma508538259134630082t_unit @ X @ bot_bo3957492148770167129t_unit )
% 14.25/8.08 = X ) ).
% 14.25/8.08
% 14.25/8.08 thf(fact_4461_max__bot2,axiom,
% 14.25/8.08 ! [X: set_o] :
% 14.25/8.08 ( ( ord_max_set_o @ X @ bot_bot_set_o )
% 14.25/8.08 = X ) ).
% 14.25/8.08
% 14.25/8.08 thf(fact_4462_max__bot2,axiom,
% 14.25/8.08 ! [X: set_nat] :
% 14.25/8.08 ( ( ord_max_set_nat @ X @ bot_bot_set_nat )
% 14.25/8.08 = X ) ).
% 14.25/8.08
% 14.25/8.08 thf(fact_4463_max__bot2,axiom,
% 14.25/8.08 ! [X: set_int] :
% 14.25/8.08 ( ( ord_max_set_int @ X @ bot_bot_set_int )
% 14.25/8.08 = X ) ).
% 14.25/8.08
% 14.25/8.08 thf(fact_4464_max__bot2,axiom,
% 14.25/8.08 ! [X: set_Pr4532377907799695533_nat_o] :
% 14.25/8.08 ( ( ord_ma8206644353667601538_nat_o @ X @ bot_bo7824918357723582233_nat_o )
% 14.25/8.08 = X ) ).
% 14.25/8.08
% 14.25/8.08 thf(fact_4465_max__bot2,axiom,
% 14.25/8.08 ! [X: nat] :
% 14.25/8.08 ( ( ord_max_nat @ X @ bot_bot_nat )
% 14.25/8.08 = X ) ).
% 14.25/8.08
% 14.25/8.08 thf(fact_4466_max__top,axiom,
% 14.25/8.08 ! [X: assn] :
% 14.25/8.08 ( ( ord_max_assn @ top_top_assn @ X )
% 14.25/8.08 = top_top_assn ) ).
% 14.25/8.08
% 14.25/8.08 thf(fact_4467_max__top,axiom,
% 14.25/8.08 ! [X: set_Product_unit] :
% 14.25/8.08 ( ( ord_ma508538259134630082t_unit @ top_to1996260823553986621t_unit @ X )
% 14.25/8.08 = top_to1996260823553986621t_unit ) ).
% 14.25/8.08
% 14.25/8.08 thf(fact_4468_max__top,axiom,
% 14.25/8.08 ! [X: set_o] :
% 14.25/8.08 ( ( ord_max_set_o @ top_top_set_o @ X )
% 14.25/8.08 = top_top_set_o ) ).
% 14.25/8.08
% 14.25/8.08 thf(fact_4469_max__top,axiom,
% 14.25/8.08 ! [X: set_nat] :
% 14.25/8.08 ( ( ord_max_set_nat @ top_top_set_nat @ X )
% 14.25/8.08 = top_top_set_nat ) ).
% 14.25/8.08
% 14.25/8.08 thf(fact_4470_max__top,axiom,
% 14.25/8.08 ! [X: set_int] :
% 14.25/8.08 ( ( ord_max_set_int @ top_top_set_int @ X )
% 14.25/8.08 = top_top_set_int ) ).
% 14.25/8.08
% 14.25/8.08 thf(fact_4471_max__top2,axiom,
% 14.25/8.08 ! [X: assn] :
% 14.25/8.08 ( ( ord_max_assn @ X @ top_top_assn )
% 14.25/8.08 = top_top_assn ) ).
% 14.25/8.08
% 14.25/8.08 thf(fact_4472_max__top2,axiom,
% 14.25/8.08 ! [X: set_Product_unit] :
% 14.25/8.08 ( ( ord_ma508538259134630082t_unit @ X @ top_to1996260823553986621t_unit )
% 14.25/8.08 = top_to1996260823553986621t_unit ) ).
% 14.25/8.08
% 14.25/8.08 thf(fact_4473_max__top2,axiom,
% 14.25/8.08 ! [X: set_o] :
% 14.25/8.08 ( ( ord_max_set_o @ X @ top_top_set_o )
% 14.25/8.08 = top_top_set_o ) ).
% 14.25/8.08
% 14.25/8.08 thf(fact_4474_max__top2,axiom,
% 14.25/8.08 ! [X: set_nat] :
% 14.25/8.08 ( ( ord_max_set_nat @ X @ top_top_set_nat )
% 14.25/8.08 = top_top_set_nat ) ).
% 14.25/8.08
% 14.25/8.08 thf(fact_4475_max__top2,axiom,
% 14.25/8.08 ! [X: set_int] :
% 14.25/8.08 ( ( ord_max_set_int @ X @ top_top_set_int )
% 14.25/8.08 = top_top_set_int ) ).
% 14.25/8.08
% 14.25/8.08 thf(fact_4476_map__prod__simp,axiom,
% 14.25/8.08 ! [F: nat > nat,G: nat > nat,A2: nat,B: nat] :
% 14.25/8.08 ( ( produc6977886695330630970at_nat @ F @ G @ ( product_Pair_nat_nat @ A2 @ B ) )
% 14.25/8.08 = ( product_Pair_nat_nat @ ( F @ A2 ) @ ( G @ B ) ) ) ).
% 14.25/8.08
% 14.25/8.08 thf(fact_4477_map__prod__simp,axiom,
% 14.25/8.08 ! [F: nat > int,G: nat > int,A2: nat,B: nat] :
% 14.25/8.08 ( ( produc7347049785058425330at_int @ F @ G @ ( product_Pair_nat_nat @ A2 @ B ) )
% 14.25/8.08 = ( product_Pair_int_int @ ( F @ A2 ) @ ( G @ B ) ) ) ).
% 14.25/8.08
% 14.25/8.08 thf(fact_4478_map__prod__simp,axiom,
% 14.25/8.08 ! [F: int > nat,G: int > nat,A2: int,B: int] :
% 14.25/8.08 ( ( produc5667422475138734578nt_nat @ F @ G @ ( product_Pair_int_int @ A2 @ B ) )
% 14.25/8.08 = ( product_Pair_nat_nat @ ( F @ A2 ) @ ( G @ B ) ) ) ).
% 14.25/8.08
% 14.25/8.08 thf(fact_4479_map__prod__simp,axiom,
% 14.25/8.08 ! [F: int > int,G: int > int,A2: int,B: int] :
% 14.25/8.08 ( ( produc6036585564866528938nt_int @ F @ G @ ( product_Pair_int_int @ A2 @ B ) )
% 14.25/8.08 = ( product_Pair_int_int @ ( F @ A2 ) @ ( G @ B ) ) ) ).
% 14.25/8.09
% 14.25/8.09 thf(fact_4480_map__prod__simp,axiom,
% 14.25/8.09 ! [F: heap_e7401611519738050253t_unit > nat,G: set_nat > nat,A2: heap_e7401611519738050253t_unit,B: set_nat] :
% 14.25/8.09 ( ( produc8285742784527409791at_nat @ F @ G @ ( produc7507926704131184380et_nat @ A2 @ B ) )
% 14.25/8.09 = ( product_Pair_nat_nat @ ( F @ A2 ) @ ( G @ B ) ) ) ).
% 14.25/8.09
% 14.25/8.09 thf(fact_4481_map__prod__simp,axiom,
% 14.25/8.09 ! [F: heap_e7401611519738050253t_unit > int,G: set_nat > int,A2: heap_e7401611519738050253t_unit,B: set_nat] :
% 14.25/8.09 ( ( produc5973434381925408567at_int @ F @ G @ ( produc7507926704131184380et_nat @ A2 @ B ) )
% 14.25/8.09 = ( product_Pair_int_int @ ( F @ A2 ) @ ( G @ B ) ) ) ).
% 14.25/8.09
% 14.25/8.09 thf(fact_4482_map__prod__simp,axiom,
% 14.25/8.09 ! [F: nat > heap_e7401611519738050253t_unit,G: nat > set_nat,A2: nat,B: nat] :
% 14.25/8.09 ( ( produc2064153442785501et_nat @ F @ G @ ( product_Pair_nat_nat @ A2 @ B ) )
% 14.25/8.09 = ( produc7507926704131184380et_nat @ ( F @ A2 ) @ ( G @ B ) ) ) ).
% 14.25/8.09
% 14.25/8.09 thf(fact_4483_map__prod__simp,axiom,
% 14.25/8.09 ! [F: int > heap_e7401611519738050253t_unit,G: int > set_nat,A2: int,B: int] :
% 14.25/8.09 ( ( produc3152090231653401565et_nat @ F @ G @ ( product_Pair_int_int @ A2 @ B ) )
% 14.25/8.09 = ( produc7507926704131184380et_nat @ ( F @ A2 ) @ ( G @ B ) ) ) ).
% 14.25/8.09
% 14.25/8.09 thf(fact_4484_map__prod__simp,axiom,
% 14.25/8.09 ! [F: heap_e7401611519738050253t_unit > heap_e7401611519738050253t_unit,G: set_nat > set_nat,A2: heap_e7401611519738050253t_unit,B: set_nat] :
% 14.25/8.09 ( ( produc8234977478398262212et_nat @ F @ G @ ( produc7507926704131184380et_nat @ A2 @ B ) )
% 14.25/8.09 = ( produc7507926704131184380et_nat @ ( F @ A2 ) @ ( G @ B ) ) ) ).
% 14.25/8.09
% 14.25/8.09 thf(fact_4485_map__prod__simp,axiom,
% 14.25/8.09 ! [F: nat > produc3658429121746597890et_nat > $o,G: nat > produc3658429121746597890et_nat,A2: nat,B: nat] :
% 14.25/8.09 ( ( produc79747477514316377et_nat @ F @ G @ ( product_Pair_nat_nat @ A2 @ B ) )
% 14.25/8.09 = ( produc5001842942810119800et_nat @ ( F @ A2 ) @ ( G @ B ) ) ) ).
% 14.25/8.09
% 14.25/8.09 thf(fact_4486_snd__comp__map__prod,axiom,
% 14.25/8.09 ! [F: heap_e7401611519738050253t_unit > heap_e7401611519738050253t_unit,G: set_nat > set_nat] :
% 14.25/8.09 ( ( comp_P2930521774885306735et_nat @ produc8586169260539613262et_nat @ ( produc8234977478398262212et_nat @ F @ G ) )
% 14.25/8.09 = ( comp_s5696514601250091763et_nat @ G @ produc8586169260539613262et_nat ) ) ).
% 14.25/8.09
% 14.25/8.09 thf(fact_4487_snd__comp__map__prod,axiom,
% 14.25/8.09 ! [F: int > heap_e7401611519738050253t_unit,G: int > set_nat] :
% 14.25/8.09 ( ( comp_P8606889026643445772nt_int @ produc8586169260539613262et_nat @ ( produc3152090231653401565et_nat @ F @ G ) )
% 14.25/8.09 = ( comp_i6200299785623974754nt_int @ G @ product_snd_int_int ) ) ).
% 14.25/8.09
% 14.25/8.09 thf(fact_4488_snd__comp__map__prod,axiom,
% 14.25/8.09 ! [F: heap_e7401611519738050253t_unit > int,G: set_nat > int] :
% 14.25/8.09 ( ( comp_P8308861049717174982et_nat @ product_snd_int_int @ ( produc5973434381925408567at_int @ F @ G ) )
% 14.25/8.09 = ( comp_s1601710177542341017et_nat @ G @ produc8586169260539613262et_nat ) ) ).
% 14.25/8.09
% 14.25/8.09 thf(fact_4489_snd__comp__map__prod,axiom,
% 14.25/8.09 ! [F: int > int,G: int > int] :
% 14.25/8.09 ( ( comp_P5356232565778667509nt_int @ product_snd_int_int @ ( produc6036585564866528938nt_int @ F @ G ) )
% 14.25/8.09 = ( comp_i4330622015127193916nt_int @ G @ product_snd_int_int ) ) ).
% 14.25/8.09
% 14.25/8.09 thf(fact_4490_snd__map__prod,axiom,
% 14.25/8.09 ! [F: heap_e7401611519738050253t_unit > heap_e7401611519738050253t_unit,G: set_nat > set_nat,X: produc3658429121746597890et_nat] :
% 14.25/8.09 ( ( produc8586169260539613262et_nat @ ( produc8234977478398262212et_nat @ F @ G @ X ) )
% 14.25/8.09 = ( G @ ( produc8586169260539613262et_nat @ X ) ) ) ).
% 14.25/8.09
% 14.25/8.09 thf(fact_4491_snd__map__prod,axiom,
% 14.25/8.09 ! [F: int > heap_e7401611519738050253t_unit,G: int > set_nat,X: product_prod_int_int] :
% 14.25/8.09 ( ( produc8586169260539613262et_nat @ ( produc3152090231653401565et_nat @ F @ G @ X ) )
% 14.25/8.09 = ( G @ ( product_snd_int_int @ X ) ) ) ).
% 14.25/8.09
% 14.25/8.09 thf(fact_4492_snd__map__prod,axiom,
% 14.25/8.09 ! [F: heap_e7401611519738050253t_unit > int,G: set_nat > int,X: produc3658429121746597890et_nat] :
% 14.25/8.09 ( ( product_snd_int_int @ ( produc5973434381925408567at_int @ F @ G @ X ) )
% 14.25/8.09 = ( G @ ( produc8586169260539613262et_nat @ X ) ) ) ).
% 14.25/8.09
% 14.25/8.09 thf(fact_4493_snd__map__prod,axiom,
% 14.25/8.09 ! [F: int > int,G: int > int,X: product_prod_int_int] :
% 14.25/8.09 ( ( product_snd_int_int @ ( produc6036585564866528938nt_int @ F @ G @ X ) )
% 14.25/8.09 = ( G @ ( product_snd_int_int @ X ) ) ) ).
% 14.25/8.09
% 14.25/8.09 thf(fact_4494_map__prod__imageI,axiom,
% 14.25/8.09 ! [A2: nat,B: nat,R3: set_Pr1261947904930325089at_nat,F: nat > nat,G: nat > nat] :
% 14.25/8.09 ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A2 @ B ) @ R3 )
% 14.25/8.09 => ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ ( F @ A2 ) @ ( G @ B ) ) @ ( image_5168914502847457605at_nat @ ( produc6977886695330630970at_nat @ F @ G ) @ R3 ) ) ) ).
% 14.25/8.09
% 14.25/8.09 thf(fact_4495_map__prod__imageI,axiom,
% 14.25/8.09 ! [A2: nat,B: nat,R3: set_Pr1261947904930325089at_nat,F: nat > int,G: nat > int] :
% 14.25/8.09 ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A2 @ B ) @ R3 )
% 14.25/8.09 => ( member5262025264175285858nt_int @ ( product_Pair_int_int @ ( F @ A2 ) @ ( G @ B ) ) @ ( image_1990417195239315453nt_int @ ( produc7347049785058425330at_int @ F @ G ) @ R3 ) ) ) ).
% 14.25/8.09
% 14.25/8.09 thf(fact_4496_map__prod__imageI,axiom,
% 14.25/8.09 ! [A2: int,B: int,R3: set_Pr958786334691620121nt_int,F: int > nat,G: int > nat] :
% 14.25/8.09 ( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ A2 @ B ) @ R3 )
% 14.25/8.09 => ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ ( F @ A2 ) @ ( G @ B ) ) @ ( image_5831868185956570253at_nat @ ( produc5667422475138734578nt_nat @ F @ G ) @ R3 ) ) ) ).
% 14.25/8.09
% 14.25/8.09 thf(fact_4497_map__prod__imageI,axiom,
% 14.25/8.09 ! [A2: int,B: int,R3: set_Pr958786334691620121nt_int,F: int > int,G: int > int] :
% 14.25/8.09 ( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ A2 @ B ) @ R3 )
% 14.25/8.09 => ( member5262025264175285858nt_int @ ( product_Pair_int_int @ ( F @ A2 ) @ ( G @ B ) ) @ ( image_2653370878348428101nt_int @ ( produc6036585564866528938nt_int @ F @ G ) @ R3 ) ) ) ).
% 14.25/8.09
% 14.25/8.09 thf(fact_4498_map__prod__imageI,axiom,
% 14.25/8.09 ! [A2: heap_e7401611519738050253t_unit,B: set_nat,R3: set_Pr3948176798113811640et_nat,F: heap_e7401611519738050253t_unit > nat,G: set_nat > nat] :
% 14.25/8.09 ( ( member6260224972018164377et_nat @ ( produc7507926704131184380et_nat @ A2 @ B ) @ R3 )
% 14.25/8.09 => ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ ( F @ A2 ) @ ( G @ B ) ) @ ( image_7757775281999735736at_nat @ ( produc8285742784527409791at_nat @ F @ G ) @ R3 ) ) ) ).
% 14.25/8.09
% 14.25/8.09 thf(fact_4499_map__prod__imageI,axiom,
% 14.25/8.09 ! [A2: heap_e7401611519738050253t_unit,B: set_nat,R3: set_Pr3948176798113811640et_nat,F: heap_e7401611519738050253t_unit > int,G: set_nat > int] :
% 14.25/8.09 ( ( member6260224972018164377et_nat @ ( produc7507926704131184380et_nat @ A2 @ B ) @ R3 )
% 14.25/8.09 => ( member5262025264175285858nt_int @ ( product_Pair_int_int @ ( F @ A2 ) @ ( G @ B ) ) @ ( image_4579277974391593584nt_int @ ( produc5973434381925408567at_int @ F @ G ) @ R3 ) ) ) ).
% 14.25/8.09
% 14.25/8.09 thf(fact_4500_map__prod__imageI,axiom,
% 14.25/8.09 ! [A2: nat,B: nat,R3: set_Pr1261947904930325089at_nat,F: nat > heap_e7401611519738050253t_unit,G: nat > set_nat] :
% 14.25/8.09 ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A2 @ B ) @ R3 )
% 14.25/8.09 => ( member6260224972018164377et_nat @ ( produc7507926704131184380et_nat @ ( F @ A2 ) @ ( G @ B ) ) @ ( image_9068448769188195518et_nat @ ( produc2064153442785501et_nat @ F @ G ) @ R3 ) ) ) ).
% 14.25/8.09
% 14.25/8.09 thf(fact_4501_map__prod__imageI,axiom,
% 14.25/8.09 ! [A2: int,B: int,R3: set_Pr958786334691620121nt_int,F: int > heap_e7401611519738050253t_unit,G: int > set_nat] :
% 14.25/8.09 ( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ A2 @ B ) @ R3 )
% 14.25/8.09 => ( member6260224972018164377et_nat @ ( produc7507926704131184380et_nat @ ( F @ A2 ) @ ( G @ B ) ) @ ( image_2127501072796355446et_nat @ ( produc3152090231653401565et_nat @ F @ G ) @ R3 ) ) ) ).
% 14.25/8.09
% 14.25/8.09 thf(fact_4502_map__prod__imageI,axiom,
% 14.25/8.09 ! [A2: heap_e7401611519738050253t_unit,B: set_nat,R3: set_Pr3948176798113811640et_nat,F: heap_e7401611519738050253t_unit > heap_e7401611519738050253t_unit,G: set_nat > set_nat] :
% 14.25/8.09 ( ( member6260224972018164377et_nat @ ( produc7507926704131184380et_nat @ A2 @ B ) @ R3 )
% 14.25/8.09 => ( member6260224972018164377et_nat @ ( produc7507926704131184380et_nat @ ( F @ A2 ) @ ( G @ B ) ) @ ( image_7426123528223433739et_nat @ ( produc8234977478398262212et_nat @ F @ G ) @ R3 ) ) ) ).
% 14.25/8.09
% 14.25/8.09 thf(fact_4503_map__prod__imageI,axiom,
% 14.25/8.09 ! [A2: nat,B: nat,R3: set_Pr1261947904930325089at_nat,F: nat > produc3658429121746597890et_nat > $o,G: nat > produc3658429121746597890et_nat] :
% 14.25/8.09 ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A2 @ B ) @ R3 )
% 14.25/8.09 => ( member1996754912294343701et_nat @ ( produc5001842942810119800et_nat @ ( F @ A2 ) @ ( G @ B ) ) @ ( image_5744643625575355962et_nat @ ( produc79747477514316377et_nat @ F @ G ) @ R3 ) ) ) ).
% 14.25/8.09
% 14.25/8.09 thf(fact_4504_exists__leI,axiom,
% 14.25/8.09 ! [N2: nat,P: nat > $o] :
% 14.25/8.09 ( ( ! [N5: nat] :
% 14.25/8.09 ( ( ord_less_nat @ N5 @ N2 )
% 14.25/8.09 => ~ ( P @ N5 ) )
% 14.25/8.09 => ( P @ N2 ) )
% 14.25/8.09 => ? [N6: nat] :
% 14.25/8.09 ( ( ord_less_eq_nat @ N6 @ N2 )
% 14.25/8.09 & ( P @ N6 ) ) ) ).
% 14.25/8.09
% 14.25/8.09 thf(fact_4505_max__def,axiom,
% 14.25/8.09 ( ord_max_filter_nat
% 14.25/8.09 = ( ^ [A6: filter_nat,B6: filter_nat] : ( if_filter_nat @ ( ord_le2510731241096832064er_nat @ A6 @ B6 ) @ B6 @ A6 ) ) ) ).
% 14.25/8.09
% 14.25/8.09 thf(fact_4506_max__def,axiom,
% 14.25/8.09 ( ord_max_set_nat
% 14.25/8.09 = ( ^ [A6: set_nat,B6: set_nat] : ( if_set_nat @ ( ord_less_eq_set_nat @ A6 @ B6 ) @ B6 @ A6 ) ) ) ).
% 14.25/8.09
% 14.25/8.09 thf(fact_4507_max__def,axiom,
% 14.25/8.09 ( ord_max_rat
% 14.25/8.09 = ( ^ [A6: rat,B6: rat] : ( if_rat @ ( ord_less_eq_rat @ A6 @ B6 ) @ B6 @ A6 ) ) ) ).
% 14.25/8.09
% 14.25/8.09 thf(fact_4508_max__def,axiom,
% 14.25/8.09 ( ord_max_nat
% 14.25/8.09 = ( ^ [A6: nat,B6: nat] : ( if_nat @ ( ord_less_eq_nat @ A6 @ B6 ) @ B6 @ A6 ) ) ) ).
% 14.25/8.09
% 14.25/8.09 thf(fact_4509_max__def,axiom,
% 14.25/8.09 ( ord_max_int
% 14.25/8.09 = ( ^ [A6: int,B6: int] : ( if_int @ ( ord_less_eq_int @ A6 @ B6 ) @ B6 @ A6 ) ) ) ).
% 14.25/8.09
% 14.25/8.09 thf(fact_4510_max__absorb1,axiom,
% 14.25/8.09 ! [Y: filter_nat,X: filter_nat] :
% 14.25/8.09 ( ( ord_le2510731241096832064er_nat @ Y @ X )
% 14.25/8.09 => ( ( ord_max_filter_nat @ X @ Y )
% 14.25/8.09 = X ) ) ).
% 14.25/8.09
% 14.25/8.09 thf(fact_4511_max__absorb1,axiom,
% 14.25/8.09 ! [Y: set_nat,X: set_nat] :
% 14.25/8.09 ( ( ord_less_eq_set_nat @ Y @ X )
% 14.25/8.09 => ( ( ord_max_set_nat @ X @ Y )
% 14.25/8.09 = X ) ) ).
% 14.25/8.09
% 14.25/8.09 thf(fact_4512_max__absorb1,axiom,
% 14.25/8.09 ! [Y: rat,X: rat] :
% 14.25/8.09 ( ( ord_less_eq_rat @ Y @ X )
% 14.25/8.09 => ( ( ord_max_rat @ X @ Y )
% 14.25/8.09 = X ) ) ).
% 14.25/8.09
% 14.25/8.09 thf(fact_4513_max__absorb1,axiom,
% 14.25/8.09 ! [Y: nat,X: nat] :
% 14.25/8.09 ( ( ord_less_eq_nat @ Y @ X )
% 14.25/8.09 => ( ( ord_max_nat @ X @ Y )
% 14.25/8.09 = X ) ) ).
% 14.25/8.09
% 14.25/8.09 thf(fact_4514_max__absorb1,axiom,
% 14.25/8.09 ! [Y: int,X: int] :
% 14.25/8.09 ( ( ord_less_eq_int @ Y @ X )
% 14.25/8.09 => ( ( ord_max_int @ X @ Y )
% 14.25/8.09 = X ) ) ).
% 14.25/8.09
% 14.25/8.09 thf(fact_4515_max__absorb2,axiom,
% 14.25/8.09 ! [X: filter_nat,Y: filter_nat] :
% 14.25/8.09 ( ( ord_le2510731241096832064er_nat @ X @ Y )
% 14.25/8.09 => ( ( ord_max_filter_nat @ X @ Y )
% 14.25/8.09 = Y ) ) ).
% 14.25/8.09
% 14.25/8.09 thf(fact_4516_max__absorb2,axiom,
% 14.25/8.09 ! [X: set_nat,Y: set_nat] :
% 14.25/8.09 ( ( ord_less_eq_set_nat @ X @ Y )
% 14.25/8.09 => ( ( ord_max_set_nat @ X @ Y )
% 14.25/8.09 = Y ) ) ).
% 14.25/8.09
% 14.25/8.09 thf(fact_4517_max__absorb2,axiom,
% 14.25/8.09 ! [X: rat,Y: rat] :
% 14.25/8.09 ( ( ord_less_eq_rat @ X @ Y )
% 14.25/8.09 => ( ( ord_max_rat @ X @ Y )
% 14.25/8.09 = Y ) ) ).
% 14.25/8.09
% 14.25/8.09 thf(fact_4518_max__absorb2,axiom,
% 14.25/8.09 ! [X: nat,Y: nat] :
% 14.25/8.09 ( ( ord_less_eq_nat @ X @ Y )
% 14.25/8.09 => ( ( ord_max_nat @ X @ Y )
% 14.25/8.09 = Y ) ) ).
% 14.25/8.09
% 14.25/8.09 thf(fact_4519_max__absorb2,axiom,
% 14.25/8.09 ! [X: int,Y: int] :
% 14.25/8.09 ( ( ord_less_eq_int @ X @ Y )
% 14.25/8.09 => ( ( ord_max_int @ X @ Y )
% 14.25/8.09 = Y ) ) ).
% 14.25/8.09
% 14.25/8.09 thf(fact_4520_max__diff__distrib__left,axiom,
% 14.25/8.09 ! [X: rat,Y: rat,Z: rat] :
% 14.25/8.09 ( ( minus_minus_rat @ ( ord_max_rat @ X @ Y ) @ Z )
% 14.25/8.09 = ( ord_max_rat @ ( minus_minus_rat @ X @ Z ) @ ( minus_minus_rat @ Y @ Z ) ) ) ).
% 14.25/8.09
% 14.25/8.09 thf(fact_4521_max__diff__distrib__left,axiom,
% 14.25/8.09 ! [X: int,Y: int,Z: int] :
% 14.25/8.09 ( ( minus_minus_int @ ( ord_max_int @ X @ Y ) @ Z )
% 14.25/8.09 = ( ord_max_int @ ( minus_minus_int @ X @ Z ) @ ( minus_minus_int @ Y @ Z ) ) ) ).
% 14.25/8.09
% 14.25/8.09 thf(fact_4522_map__prod_Ocomp,axiom,
% 14.25/8.09 ! [F: code_integer > produc8923325533196201883nteger > produc8923325533196201883nteger,G: code_integer > produc8923325533196201883nteger > produc8923325533196201883nteger,H2: code_integer > code_integer,I2: code_integer > code_integer] :
% 14.25/8.09 ( ( comp_P234064951390806312nteger @ ( produc5081366107664765182nteger @ F @ G ) @ ( produc1802105962441033582nteger @ H2 @ I2 ) )
% 14.25/8.09 = ( produc5081366107664765182nteger @ ( comp_C1593894019821074884nteger @ F @ H2 ) @ ( comp_C1593894019821074884nteger @ G @ I2 ) ) ) ).
% 14.25/8.09
% 14.25/8.09 thf(fact_4523_map__prod_Ocomp,axiom,
% 14.25/8.09 ! [F: code_integer > produc8923325533196201883nteger > produc8923325533196201883nteger,G: ( code_integer > code_integer ) > produc8923325533196201883nteger > produc8923325533196201883nteger,H2: code_integer > code_integer,I2: code_integer > code_integer > code_integer] :
% 14.25/8.09 ( ( comp_P6312383820422015140nteger @ ( produc3090791198996863994nteger @ F @ G ) @ ( produc6023007152014423402nteger @ H2 @ I2 ) )
% 14.25/8.09 = ( produc5081366107664765182nteger @ ( comp_C1593894019821074884nteger @ F @ H2 ) @ ( comp_C8797469213163452608nteger @ G @ I2 ) ) ) ).
% 14.25/8.09
% 14.25/8.09 thf(fact_4524_map__prod_Ocomp,axiom,
% 14.25/8.09 ! [F: ( code_integer > code_integer ) > produc8923325533196201883nteger > produc8923325533196201883nteger,G: code_integer > produc8923325533196201883nteger > produc8923325533196201883nteger,H2: code_integer > code_integer > code_integer,I2: code_integer > code_integer] :
% 14.25/8.09 ( ( comp_P1414880308642389412nteger @ ( produc6158254626097298938nteger @ F @ G ) @ ( produc3789967387199524714nteger @ H2 @ I2 ) )
% 14.25/8.09 = ( produc5081366107664765182nteger @ ( comp_C8797469213163452608nteger @ F @ H2 ) @ ( comp_C1593894019821074884nteger @ G @ I2 ) ) ) ).
% 14.25/8.09
% 14.25/8.09 thf(fact_4525_map__prod_Ocomp,axiom,
% 14.25/8.09 ! [F: ( code_integer > code_integer ) > produc8923325533196201883nteger > produc8923325533196201883nteger,G: ( code_integer > code_integer ) > produc8923325533196201883nteger > produc8923325533196201883nteger,H2: code_integer > code_integer > code_integer,I2: code_integer > code_integer > code_integer] :
% 14.25/8.09 ( ( comp_P5544053077415136032nteger @ ( produc4724772261320961782nteger @ F @ G ) @ ( produc7818937854452556646nteger @ H2 @ I2 ) )
% 14.25/8.09 = ( produc5081366107664765182nteger @ ( comp_C8797469213163452608nteger @ F @ H2 ) @ ( comp_C8797469213163452608nteger @ G @ I2 ) ) ) ).
% 14.25/8.09
% 14.25/8.09 thf(fact_4526_map__prod_Ocompositionality,axiom,
% 14.25/8.09 ! [F: code_integer > produc8923325533196201883nteger > produc8923325533196201883nteger,G: code_integer > produc8923325533196201883nteger > produc8923325533196201883nteger,H2: code_integer > code_integer,I2: code_integer > code_integer,Prod: produc8923325533196201883nteger] :
% 14.25/8.09 ( ( produc5081366107664765182nteger @ F @ G @ ( produc1802105962441033582nteger @ H2 @ I2 @ Prod ) )
% 14.25/8.09 = ( produc5081366107664765182nteger @ ( comp_C1593894019821074884nteger @ F @ H2 ) @ ( comp_C1593894019821074884nteger @ G @ I2 ) @ Prod ) ) ).
% 14.25/8.09
% 14.25/8.09 thf(fact_4527_map__prod_Ocompositionality,axiom,
% 14.25/8.09 ! [F: code_integer > produc8923325533196201883nteger > produc8923325533196201883nteger,G: ( code_integer > code_integer ) > produc8923325533196201883nteger > produc8923325533196201883nteger,H2: code_integer > code_integer,I2: code_integer > code_integer > code_integer,Prod: produc8923325533196201883nteger] :
% 14.25/8.09 ( ( produc3090791198996863994nteger @ F @ G @ ( produc6023007152014423402nteger @ H2 @ I2 @ Prod ) )
% 14.25/8.09 = ( produc5081366107664765182nteger @ ( comp_C1593894019821074884nteger @ F @ H2 ) @ ( comp_C8797469213163452608nteger @ G @ I2 ) @ Prod ) ) ).
% 14.25/8.09
% 14.25/8.09 thf(fact_4528_map__prod_Ocompositionality,axiom,
% 14.25/8.09 ! [F: ( code_integer > code_integer ) > produc8923325533196201883nteger > produc8923325533196201883nteger,G: code_integer > produc8923325533196201883nteger > produc8923325533196201883nteger,H2: code_integer > code_integer > code_integer,I2: code_integer > code_integer,Prod: produc8923325533196201883nteger] :
% 14.25/8.09 ( ( produc6158254626097298938nteger @ F @ G @ ( produc3789967387199524714nteger @ H2 @ I2 @ Prod ) )
% 14.25/8.09 = ( produc5081366107664765182nteger @ ( comp_C8797469213163452608nteger @ F @ H2 ) @ ( comp_C1593894019821074884nteger @ G @ I2 ) @ Prod ) ) ).
% 14.25/8.09
% 14.25/8.09 thf(fact_4529_map__prod_Ocompositionality,axiom,
% 14.25/8.09 ! [F: ( code_integer > code_integer ) > produc8923325533196201883nteger > produc8923325533196201883nteger,G: ( code_integer > code_integer ) > produc8923325533196201883nteger > produc8923325533196201883nteger,H2: code_integer > code_integer > code_integer,I2: code_integer > code_integer > code_integer,Prod: produc8923325533196201883nteger] :
% 14.25/8.09 ( ( produc4724772261320961782nteger @ F @ G @ ( produc7818937854452556646nteger @ H2 @ I2 @ Prod ) )
% 14.25/8.09 = ( produc5081366107664765182nteger @ ( comp_C8797469213163452608nteger @ F @ H2 ) @ ( comp_C8797469213163452608nteger @ G @ I2 ) @ Prod ) ) ).
% 14.25/8.09
% 14.25/8.09 thf(fact_4530_map__prod__compose,axiom,
% 14.25/8.09 ! [F1: code_integer > produc8923325533196201883nteger > produc8923325533196201883nteger,F22: code_integer > code_integer,G1: code_integer > produc8923325533196201883nteger > produc8923325533196201883nteger,G2: code_integer > code_integer] :
% 14.25/8.09 ( ( produc5081366107664765182nteger @ ( comp_C1593894019821074884nteger @ F1 @ F22 ) @ ( comp_C1593894019821074884nteger @ G1 @ G2 ) )
% 14.25/8.09 = ( comp_P234064951390806312nteger @ ( produc5081366107664765182nteger @ F1 @ G1 ) @ ( produc1802105962441033582nteger @ F22 @ G2 ) ) ) ).
% 14.25/8.09
% 14.25/8.09 thf(fact_4531_map__prod__compose,axiom,
% 14.25/8.09 ! [F1: code_integer > produc8923325533196201883nteger > produc8923325533196201883nteger,F22: code_integer > code_integer,G1: ( code_integer > code_integer ) > produc8923325533196201883nteger > produc8923325533196201883nteger,G2: code_integer > code_integer > code_integer] :
% 14.25/8.09 ( ( produc5081366107664765182nteger @ ( comp_C1593894019821074884nteger @ F1 @ F22 ) @ ( comp_C8797469213163452608nteger @ G1 @ G2 ) )
% 14.25/8.09 = ( comp_P6312383820422015140nteger @ ( produc3090791198996863994nteger @ F1 @ G1 ) @ ( produc6023007152014423402nteger @ F22 @ G2 ) ) ) ).
% 14.25/8.09
% 14.25/8.09 thf(fact_4532_map__prod__compose,axiom,
% 14.25/8.09 ! [F1: ( code_integer > code_integer ) > produc8923325533196201883nteger > produc8923325533196201883nteger,F22: code_integer > code_integer > code_integer,G1: code_integer > produc8923325533196201883nteger > produc8923325533196201883nteger,G2: code_integer > code_integer] :
% 14.25/8.09 ( ( produc5081366107664765182nteger @ ( comp_C8797469213163452608nteger @ F1 @ F22 ) @ ( comp_C1593894019821074884nteger @ G1 @ G2 ) )
% 14.25/8.09 = ( comp_P1414880308642389412nteger @ ( produc6158254626097298938nteger @ F1 @ G1 ) @ ( produc3789967387199524714nteger @ F22 @ G2 ) ) ) ).
% 14.25/8.09
% 14.25/8.09 thf(fact_4533_map__prod__compose,axiom,
% 14.25/8.09 ! [F1: ( code_integer > code_integer ) > produc8923325533196201883nteger > produc8923325533196201883nteger,F22: code_integer > code_integer > code_integer,G1: ( code_integer > code_integer ) > produc8923325533196201883nteger > produc8923325533196201883nteger,G2: code_integer > code_integer > code_integer] :
% 14.25/8.09 ( ( produc5081366107664765182nteger @ ( comp_C8797469213163452608nteger @ F1 @ F22 ) @ ( comp_C8797469213163452608nteger @ G1 @ G2 ) )
% 14.25/8.09 = ( comp_P5544053077415136032nteger @ ( produc4724772261320961782nteger @ F1 @ G1 ) @ ( produc7818937854452556646nteger @ F22 @ G2 ) ) ) ).
% 14.25/8.09
% 14.25/8.09 thf(fact_4534_inj__img__insertE,axiom,
% 14.25/8.09 ! [F: code_integer > code_integer,A: set_Code_integer,X: code_integer,B4: set_Code_integer] :
% 14.25/8.09 ( ( inj_on1158064722259180589nteger @ F @ A )
% 14.25/8.09 => ( ~ ( member_Code_integer @ X @ B4 )
% 14.25/8.09 => ( ( ( insert_Code_integer @ X @ B4 )
% 14.25/8.09 = ( image_4470545334726330049nteger @ F @ A ) )
% 14.25/8.09 => ~ ! [X10: code_integer,A7: set_Code_integer] :
% 14.25/8.09 ( ~ ( member_Code_integer @ X10 @ A7 )
% 14.25/8.09 => ( ( A
% 14.25/8.09 = ( insert_Code_integer @ X10 @ A7 ) )
% 14.25/8.09 => ( ( X
% 14.25/8.09 = ( F @ X10 ) )
% 14.25/8.09 => ( B4
% 14.25/8.09 != ( image_4470545334726330049nteger @ F @ A7 ) ) ) ) ) ) ) ) ).
% 14.25/8.09
% 14.25/8.09 thf(fact_4535_inj__img__insertE,axiom,
% 14.25/8.09 ! [F: nat > code_integer,A: set_nat,X: code_integer,B4: set_Code_integer] :
% 14.25/8.09 ( ( inj_on2245834603434889120nteger @ F @ A )
% 14.25/8.09 => ( ~ ( member_Code_integer @ X @ B4 )
% 14.25/8.09 => ( ( ( insert_Code_integer @ X @ B4 )
% 14.25/8.09 = ( image_1215581382706833972nteger @ F @ A ) )
% 14.25/8.09 => ~ ! [X10: nat,A7: set_nat] :
% 14.25/8.09 ( ~ ( member_nat @ X10 @ A7 )
% 14.25/8.09 => ( ( A
% 14.25/8.09 = ( insert_nat @ X10 @ A7 ) )
% 14.25/8.09 => ( ( X
% 14.25/8.09 = ( F @ X10 ) )
% 14.25/8.09 => ( B4
% 14.25/8.09 != ( image_1215581382706833972nteger @ F @ A7 ) ) ) ) ) ) ) ) ).
% 14.25/8.09
% 14.25/8.09 thf(fact_4536_inj__img__insertE,axiom,
% 14.25/8.09 ! [F: nat > nat,A: set_nat,X: nat,B4: set_nat] :
% 14.25/8.09 ( ( inj_on_nat_nat @ F @ A )
% 14.25/8.09 => ( ~ ( member_nat @ X @ B4 )
% 14.25/8.09 => ( ( ( insert_nat @ X @ B4 )
% 14.25/8.09 = ( image_nat_nat2 @ F @ A ) )
% 14.25/8.09 => ~ ! [X10: nat,A7: set_nat] :
% 14.25/8.09 ( ~ ( member_nat @ X10 @ A7 )
% 14.25/8.09 => ( ( A
% 14.25/8.09 = ( insert_nat @ X10 @ A7 ) )
% 14.25/8.09 => ( ( X
% 14.25/8.09 = ( F @ X10 ) )
% 14.25/8.09 => ( B4
% 14.25/8.09 != ( image_nat_nat2 @ F @ A7 ) ) ) ) ) ) ) ) ).
% 14.25/8.09
% 14.25/8.09 thf(fact_4537_inj__img__insertE,axiom,
% 14.25/8.09 ! [F: int > nat,A: set_int,X: nat,B4: set_nat] :
% 14.25/8.09 ( ( inj_on_int_nat @ F @ A )
% 14.25/8.09 => ( ~ ( member_nat @ X @ B4 )
% 14.25/8.09 => ( ( ( insert_nat @ X @ B4 )
% 14.25/8.09 = ( image_int_nat2 @ F @ A ) )
% 14.25/8.09 => ~ ! [X10: int,A7: set_int] :
% 14.25/8.09 ( ~ ( member_int @ X10 @ A7 )
% 14.25/8.09 => ( ( A
% 14.25/8.09 = ( insert_int @ X10 @ A7 ) )
% 14.25/8.09 => ( ( X
% 14.25/8.09 = ( F @ X10 ) )
% 14.25/8.09 => ( B4
% 14.25/8.09 != ( image_int_nat2 @ F @ A7 ) ) ) ) ) ) ) ) ).
% 14.25/8.09
% 14.25/8.09 thf(fact_4538_inj__img__insertE,axiom,
% 14.25/8.09 ! [F: $o > nat,A: set_o,X: nat,B4: set_nat] :
% 14.25/8.09 ( ( inj_on_o_nat @ F @ A )
% 14.25/8.09 => ( ~ ( member_nat @ X @ B4 )
% 14.25/8.09 => ( ( ( insert_nat @ X @ B4 )
% 14.25/8.09 = ( image_o_nat2 @ F @ A ) )
% 14.25/8.09 => ~ ! [X10: $o,A7: set_o] :
% 14.25/8.09 ( ~ ( member_o @ X10 @ A7 )
% 14.25/8.09 => ( ( A
% 14.25/8.09 = ( insert_o @ X10 @ A7 ) )
% 14.25/8.09 => ( ( X
% 14.25/8.09 = ( F @ X10 ) )
% 14.25/8.09 => ( B4
% 14.25/8.09 != ( image_o_nat2 @ F @ A7 ) ) ) ) ) ) ) ) ).
% 14.25/8.09
% 14.25/8.09 thf(fact_4539_inj__img__insertE,axiom,
% 14.25/8.09 ! [F: product_unit > nat,A: set_Product_unit,X: nat,B4: set_nat] :
% 14.25/8.09 ( ( inj_on8430439091780834860it_nat @ F @ A )
% 14.25/8.09 => ( ~ ( member_nat @ X @ B4 )
% 14.25/8.09 => ( ( ( insert_nat @ X @ B4 )
% 14.25/8.09 = ( image_875570014554754200it_nat @ F @ A ) )
% 14.25/8.09 => ~ ! [X10: product_unit,A7: set_Product_unit] :
% 14.25/8.09 ( ~ ( member_Product_unit @ X10 @ A7 )
% 14.25/8.09 => ( ( A
% 14.25/8.09 = ( insert_Product_unit @ X10 @ A7 ) )
% 14.25/8.09 => ( ( X
% 14.25/8.09 = ( F @ X10 ) )
% 14.25/8.09 => ( B4
% 14.25/8.09 != ( image_875570014554754200it_nat @ F @ A7 ) ) ) ) ) ) ) ) ).
% 14.25/8.09
% 14.25/8.09 thf(fact_4540_inj__img__insertE,axiom,
% 14.25/8.09 ! [F: nat > int,A: set_nat,X: int,B4: set_int] :
% 14.25/8.09 ( ( inj_on_nat_int @ F @ A )
% 14.25/8.09 => ( ~ ( member_int @ X @ B4 )
% 14.25/8.09 => ( ( ( insert_int @ X @ B4 )
% 14.25/8.09 = ( image_nat_int2 @ F @ A ) )
% 14.25/8.09 => ~ ! [X10: nat,A7: set_nat] :
% 14.25/8.09 ( ~ ( member_nat @ X10 @ A7 )
% 14.25/8.09 => ( ( A
% 14.25/8.09 = ( insert_nat @ X10 @ A7 ) )
% 14.25/8.09 => ( ( X
% 14.25/8.09 = ( F @ X10 ) )
% 14.25/8.09 => ( B4
% 14.25/8.09 != ( image_nat_int2 @ F @ A7 ) ) ) ) ) ) ) ) ).
% 14.25/8.09
% 14.25/8.09 thf(fact_4541_inj__img__insertE,axiom,
% 14.25/8.09 ! [F: int > int,A: set_int,X: int,B4: set_int] :
% 14.25/8.09 ( ( inj_on_int_int @ F @ A )
% 14.25/8.09 => ( ~ ( member_int @ X @ B4 )
% 14.25/8.09 => ( ( ( insert_int @ X @ B4 )
% 14.25/8.09 = ( image_int_int2 @ F @ A ) )
% 14.25/8.09 => ~ ! [X10: int,A7: set_int] :
% 14.25/8.09 ( ~ ( member_int @ X10 @ A7 )
% 14.25/8.09 => ( ( A
% 14.25/8.09 = ( insert_int @ X10 @ A7 ) )
% 14.25/8.09 => ( ( X
% 14.25/8.09 = ( F @ X10 ) )
% 14.25/8.09 => ( B4
% 14.25/8.09 != ( image_int_int2 @ F @ A7 ) ) ) ) ) ) ) ) ).
% 14.25/8.09
% 14.25/8.09 thf(fact_4542_inj__img__insertE,axiom,
% 14.25/8.09 ! [F: $o > int,A: set_o,X: int,B4: set_int] :
% 14.25/8.09 ( ( inj_on_o_int @ F @ A )
% 14.25/8.09 => ( ~ ( member_int @ X @ B4 )
% 14.25/8.09 => ( ( ( insert_int @ X @ B4 )
% 14.25/8.09 = ( image_o_int2 @ F @ A ) )
% 14.25/8.09 => ~ ! [X10: $o,A7: set_o] :
% 14.25/8.09 ( ~ ( member_o @ X10 @ A7 )
% 14.25/8.09 => ( ( A
% 14.25/8.09 = ( insert_o @ X10 @ A7 ) )
% 14.25/8.09 => ( ( X
% 14.25/8.09 = ( F @ X10 ) )
% 14.25/8.09 => ( B4
% 14.25/8.09 != ( image_o_int2 @ F @ A7 ) ) ) ) ) ) ) ) ).
% 14.25/8.09
% 14.25/8.09 thf(fact_4543_inj__img__insertE,axiom,
% 14.25/8.09 ! [F: product_unit > int,A: set_Product_unit,X: int,B4: set_int] :
% 14.25/8.09 ( ( inj_on8427948621271784584it_int @ F @ A )
% 14.25/8.09 => ( ~ ( member_int @ X @ B4 )
% 14.25/8.09 => ( ( ( insert_int @ X @ B4 )
% 14.25/8.09 = ( image_873079544045703924it_int @ F @ A ) )
% 14.25/8.09 => ~ ! [X10: product_unit,A7: set_Product_unit] :
% 14.25/8.09 ( ~ ( member_Product_unit @ X10 @ A7 )
% 14.25/8.09 => ( ( A
% 14.25/8.09 = ( insert_Product_unit @ X10 @ A7 ) )
% 14.25/8.09 => ( ( X
% 14.25/8.09 = ( F @ X10 ) )
% 14.25/8.09 => ( B4
% 14.25/8.09 != ( image_873079544045703924it_int @ F @ A7 ) ) ) ) ) ) ) ) ).
% 14.25/8.09
% 14.25/8.09 thf(fact_4544_type__copy__wit,axiom,
% 14.25/8.09 ! [Rep2: code_integer > int,Abs2: int > code_integer,X: nat,S: int > set_nat,Y: int] :
% 14.25/8.09 ( ( type_d8366093980585677751er_int @ Rep2 @ Abs2 @ top_top_set_int )
% 14.25/8.09 => ( ( member_nat @ X @ ( comp_i5102403792180996084nteger @ S @ Rep2 @ ( Abs2 @ Y ) ) )
% 14.25/8.09 => ( member_nat @ X @ ( S @ Y ) ) ) ) ).
% 14.25/8.09
% 14.25/8.09 thf(fact_4545_type__copy__wit,axiom,
% 14.25/8.09 ! [Rep2: code_integer > int,Abs2: int > code_integer,X: int,S: int > set_int,Y: int] :
% 14.25/8.09 ( ( type_d8366093980585677751er_int @ Rep2 @ Abs2 @ top_top_set_int )
% 14.25/8.09 => ( ( member_int @ X @ ( comp_i1887841406321795152nteger @ S @ Rep2 @ ( Abs2 @ Y ) ) )
% 14.25/8.09 => ( member_int @ X @ ( S @ Y ) ) ) ) ).
% 14.25/8.09
% 14.25/8.09 thf(fact_4546_type__copy__wit,axiom,
% 14.25/8.09 ! [Rep2: code_integer > int,Abs2: int > code_integer,X: $o,S: int > set_o,Y: int] :
% 14.25/8.09 ( ( type_d8366093980585677751er_int @ Rep2 @ Abs2 @ top_top_set_int )
% 14.25/8.09 => ( ( member_o @ X @ ( comp_i5479505385629679622nteger @ S @ Rep2 @ ( Abs2 @ Y ) ) )
% 14.25/8.09 => ( member_o @ X @ ( S @ Y ) ) ) ) ).
% 14.25/8.09
% 14.25/8.09 thf(fact_4547_type__copy__wit,axiom,
% 14.25/8.09 ! [Rep2: code_integer > int,Abs2: int > code_integer,X: product_unit,S: int > set_Product_unit,Y: int] :
% 14.25/8.09 ( ( type_d8366093980585677751er_int @ Rep2 @ Abs2 @ top_top_set_int )
% 14.25/8.09 => ( ( member_Product_unit @ X @ ( comp_i3319814986458681467nteger @ S @ Rep2 @ ( Abs2 @ Y ) ) )
% 14.25/8.09 => ( member_Product_unit @ X @ ( S @ Y ) ) ) ) ).
% 14.25/8.09
% 14.25/8.09 thf(fact_4548_type__copy__wit,axiom,
% 14.25/8.09 ! [Rep2: product_unit > $o,Abs2: $o > product_unit,X: nat,S: $o > set_nat,Y: $o] :
% 14.25/8.09 ( ( type_d6188575255521822967unit_o @ Rep2 @ Abs2 @ top_top_set_o )
% 14.25/8.09 => ( ( member_nat @ X @ ( comp_o9190160103024335696t_unit @ S @ Rep2 @ ( Abs2 @ Y ) ) )
% 14.25/8.09 => ( member_nat @ X @ ( S @ Y ) ) ) ) ).
% 14.25/8.09
% 14.25/8.09 thf(fact_4549_type__copy__wit,axiom,
% 14.25/8.09 ! [Rep2: product_unit > $o,Abs2: $o > product_unit,X: int,S: $o > set_int,Y: $o] :
% 14.25/8.09 ( ( type_d6188575255521822967unit_o @ Rep2 @ Abs2 @ top_top_set_o )
% 14.25/8.09 => ( ( member_int @ X @ ( comp_o1521574261646630388t_unit @ S @ Rep2 @ ( Abs2 @ Y ) ) )
% 14.25/8.09 => ( member_int @ X @ ( S @ Y ) ) ) ) ).
% 14.25/8.09
% 14.25/8.09 thf(fact_4550_type__copy__wit,axiom,
% 14.25/8.09 ! [Rep2: product_unit > $o,Abs2: $o > product_unit,X: $o,S: $o > set_o,Y: $o] :
% 14.25/8.09 ( ( type_d6188575255521822967unit_o @ Rep2 @ Abs2 @ top_top_set_o )
% 14.25/8.09 => ( ( member_o @ X @ ( comp_o6895741221744486154t_unit @ S @ Rep2 @ ( Abs2 @ Y ) ) )
% 14.25/8.09 => ( member_o @ X @ ( S @ Y ) ) ) ) ).
% 14.25/8.09
% 14.25/8.09 thf(fact_4551_type__copy__wit,axiom,
% 14.25/8.09 ! [Rep2: product_unit > $o,Abs2: $o > product_unit,X: product_unit,S: $o > set_Product_unit,Y: $o] :
% 14.25/8.09 ( ( type_d6188575255521822967unit_o @ Rep2 @ Abs2 @ top_top_set_o )
% 14.25/8.09 => ( ( member_Product_unit @ X @ ( comp_o2086126948152544725t_unit @ S @ Rep2 @ ( Abs2 @ Y ) ) )
% 14.25/8.09 => ( member_Product_unit @ X @ ( S @ Y ) ) ) ) ).
% 14.25/8.09
% 14.25/8.09 thf(fact_4552_type__copy__wit,axiom,
% 14.25/8.09 ! [Rep2: code_natural > nat,Abs2: nat > code_natural,X: nat,S: nat > set_nat,Y: nat] :
% 14.25/8.09 ( ( type_d4410041424927559462al_nat @ Rep2 @ Abs2 @ top_top_set_nat )
% 14.25/8.09 => ( ( member_nat @ X @ ( comp_n4787279007288201059atural @ S @ Rep2 @ ( Abs2 @ Y ) ) )
% 14.25/8.09 => ( member_nat @ X @ ( S @ Y ) ) ) ) ).
% 14.25/8.09
% 14.25/8.09 thf(fact_4553_type__copy__wit,axiom,
% 14.25/8.09 ! [Rep2: code_natural > nat,Abs2: nat > code_natural,X: int,S: nat > set_int,Y: nat] :
% 14.25/8.09 ( ( type_d4410041424927559462al_nat @ Rep2 @ Abs2 @ top_top_set_nat )
% 14.25/8.09 => ( ( member_int @ X @ ( comp_n1572716621429000127atural @ S @ Rep2 @ ( Abs2 @ Y ) ) )
% 14.25/8.09 => ( member_int @ X @ ( S @ Y ) ) ) ) ).
% 14.25/8.09
% 14.25/8.09 thf(fact_4554_prod__fun__imageE,axiom,
% 14.25/8.09 ! [C: product_prod_nat_nat,F: nat > nat,G: nat > nat,R3: set_Pr1261947904930325089at_nat] :
% 14.25/8.09 ( ( member8440522571783428010at_nat @ C @ ( image_5168914502847457605at_nat @ ( produc6977886695330630970at_nat @ F @ G ) @ R3 ) )
% 14.25/8.09 => ~ ! [X3: nat,Y2: nat] :
% 14.25/8.09 ( ( C
% 14.25/8.09 = ( product_Pair_nat_nat @ ( F @ X3 ) @ ( G @ Y2 ) ) )
% 14.25/8.09 => ~ ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X3 @ Y2 ) @ R3 ) ) ) ).
% 14.25/8.09
% 14.25/8.09 thf(fact_4555_prod__fun__imageE,axiom,
% 14.25/8.09 ! [C: product_prod_nat_nat,F: int > nat,G: int > nat,R3: set_Pr958786334691620121nt_int] :
% 14.25/8.09 ( ( member8440522571783428010at_nat @ C @ ( image_5831868185956570253at_nat @ ( produc5667422475138734578nt_nat @ F @ G ) @ R3 ) )
% 14.25/8.09 => ~ ! [X3: int,Y2: int] :
% 14.25/8.09 ( ( C
% 14.25/8.09 = ( product_Pair_nat_nat @ ( F @ X3 ) @ ( G @ Y2 ) ) )
% 14.25/8.09 => ~ ( member5262025264175285858nt_int @ ( product_Pair_int_int @ X3 @ Y2 ) @ R3 ) ) ) ).
% 14.25/8.09
% 14.25/8.09 thf(fact_4556_prod__fun__imageE,axiom,
% 14.25/8.09 ! [C: product_prod_int_int,F: nat > int,G: nat > int,R3: set_Pr1261947904930325089at_nat] :
% 14.25/8.09 ( ( member5262025264175285858nt_int @ C @ ( image_1990417195239315453nt_int @ ( produc7347049785058425330at_int @ F @ G ) @ R3 ) )
% 14.25/8.09 => ~ ! [X3: nat,Y2: nat] :
% 14.25/8.09 ( ( C
% 14.25/8.09 = ( product_Pair_int_int @ ( F @ X3 ) @ ( G @ Y2 ) ) )
% 14.25/8.09 => ~ ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X3 @ Y2 ) @ R3 ) ) ) ).
% 14.25/8.09
% 14.25/8.09 thf(fact_4557_prod__fun__imageE,axiom,
% 14.25/8.09 ! [C: product_prod_int_int,F: int > int,G: int > int,R3: set_Pr958786334691620121nt_int] :
% 14.25/8.09 ( ( member5262025264175285858nt_int @ C @ ( image_2653370878348428101nt_int @ ( produc6036585564866528938nt_int @ F @ G ) @ R3 ) )
% 14.25/8.09 => ~ ! [X3: int,Y2: int] :
% 14.25/8.09 ( ( C
% 14.25/8.09 = ( product_Pair_int_int @ ( F @ X3 ) @ ( G @ Y2 ) ) )
% 14.25/8.09 => ~ ( member5262025264175285858nt_int @ ( product_Pair_int_int @ X3 @ Y2 ) @ R3 ) ) ) ).
% 14.25/8.09
% 14.25/8.09 thf(fact_4558_prod__fun__imageE,axiom,
% 14.25/8.09 ! [C: produc3658429121746597890et_nat,F: nat > heap_e7401611519738050253t_unit,G: nat > set_nat,R3: set_Pr1261947904930325089at_nat] :
% 14.25/8.09 ( ( member6260224972018164377et_nat @ C @ ( image_9068448769188195518et_nat @ ( produc2064153442785501et_nat @ F @ G ) @ R3 ) )
% 14.25/8.09 => ~ ! [X3: nat,Y2: nat] :
% 14.25/8.09 ( ( C
% 14.25/8.09 = ( produc7507926704131184380et_nat @ ( F @ X3 ) @ ( G @ Y2 ) ) )
% 14.25/8.09 => ~ ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X3 @ Y2 ) @ R3 ) ) ) ).
% 14.25/8.09
% 14.25/8.09 thf(fact_4559_prod__fun__imageE,axiom,
% 14.25/8.09 ! [C: produc3658429121746597890et_nat,F: int > heap_e7401611519738050253t_unit,G: int > set_nat,R3: set_Pr958786334691620121nt_int] :
% 14.25/8.09 ( ( member6260224972018164377et_nat @ C @ ( image_2127501072796355446et_nat @ ( produc3152090231653401565et_nat @ F @ G ) @ R3 ) )
% 14.25/8.09 => ~ ! [X3: int,Y2: int] :
% 14.25/8.09 ( ( C
% 14.25/8.09 = ( produc7507926704131184380et_nat @ ( F @ X3 ) @ ( G @ Y2 ) ) )
% 14.25/8.09 => ~ ( member5262025264175285858nt_int @ ( product_Pair_int_int @ X3 @ Y2 ) @ R3 ) ) ) ).
% 14.25/8.09
% 14.25/8.09 thf(fact_4560_prod__fun__imageE,axiom,
% 14.25/8.09 ! [C: product_prod_nat_nat,F: heap_e7401611519738050253t_unit > nat,G: set_nat > nat,R3: set_Pr3948176798113811640et_nat] :
% 14.25/8.09 ( ( member8440522571783428010at_nat @ C @ ( image_7757775281999735736at_nat @ ( produc8285742784527409791at_nat @ F @ G ) @ R3 ) )
% 14.25/8.09 => ~ ! [X3: heap_e7401611519738050253t_unit,Y2: set_nat] :
% 14.25/8.09 ( ( C
% 14.25/8.09 = ( product_Pair_nat_nat @ ( F @ X3 ) @ ( G @ Y2 ) ) )
% 14.25/8.09 => ~ ( member6260224972018164377et_nat @ ( produc7507926704131184380et_nat @ X3 @ Y2 ) @ R3 ) ) ) ).
% 14.25/8.09
% 14.25/8.09 thf(fact_4561_prod__fun__imageE,axiom,
% 14.25/8.09 ! [C: product_prod_int_int,F: heap_e7401611519738050253t_unit > int,G: set_nat > int,R3: set_Pr3948176798113811640et_nat] :
% 14.25/8.09 ( ( member5262025264175285858nt_int @ C @ ( image_4579277974391593584nt_int @ ( produc5973434381925408567at_int @ F @ G ) @ R3 ) )
% 14.25/8.09 => ~ ! [X3: heap_e7401611519738050253t_unit,Y2: set_nat] :
% 14.25/8.09 ( ( C
% 14.25/8.09 = ( product_Pair_int_int @ ( F @ X3 ) @ ( G @ Y2 ) ) )
% 14.25/8.09 => ~ ( member6260224972018164377et_nat @ ( produc7507926704131184380et_nat @ X3 @ Y2 ) @ R3 ) ) ) ).
% 14.25/8.09
% 14.25/8.09 thf(fact_4562_prod__fun__imageE,axiom,
% 14.25/8.09 ! [C: produc3658429121746597890et_nat,F: heap_e7401611519738050253t_unit > heap_e7401611519738050253t_unit,G: set_nat > set_nat,R3: set_Pr3948176798113811640et_nat] :
% 14.25/8.09 ( ( member6260224972018164377et_nat @ C @ ( image_7426123528223433739et_nat @ ( produc8234977478398262212et_nat @ F @ G ) @ R3 ) )
% 14.25/8.09 => ~ ! [X3: heap_e7401611519738050253t_unit,Y2: set_nat] :
% 14.25/8.09 ( ( C
% 14.25/8.09 = ( produc7507926704131184380et_nat @ ( F @ X3 ) @ ( G @ Y2 ) ) )
% 14.25/8.09 => ~ ( member6260224972018164377et_nat @ ( produc7507926704131184380et_nat @ X3 @ Y2 ) @ R3 ) ) ) ).
% 14.25/8.09
% 14.25/8.09 thf(fact_4563_prod__fun__imageE,axiom,
% 14.25/8.09 ! [C: product_prod_nat_nat,F: ( produc3658429121746597890et_nat > $o ) > nat,G: produc3658429121746597890et_nat > nat,R3: set_Pr3286484037609594932et_nat] :
% 14.25/8.09 ( ( member8440522571783428010at_nat @ C @ ( image_7073998564157519676at_nat @ ( produc2784369760062397435at_nat @ F @ G ) @ R3 ) )
% 14.25/8.09 => ~ ! [X3: produc3658429121746597890et_nat > $o,Y2: produc3658429121746597890et_nat] :
% 14.25/8.09 ( ( C
% 14.25/8.09 = ( product_Pair_nat_nat @ ( F @ X3 ) @ ( G @ Y2 ) ) )
% 14.25/8.09 => ~ ( member1996754912294343701et_nat @ ( produc5001842942810119800et_nat @ X3 @ Y2 ) @ R3 ) ) ) ).
% 14.25/8.09
% 14.25/8.09 thf(fact_4564_inj__on__iff__surj,axiom,
% 14.25/8.09 ! [A: set_Code_integer,A8: set_Code_integer] :
% 14.25/8.09 ( ( A != bot_bo3990330152332043303nteger )
% 14.25/8.09 => ( ( ? [F4: code_integer > code_integer] :
% 14.25/8.09 ( ( inj_on1158064722259180589nteger @ F4 @ A )
% 14.25/8.09 & ( ord_le7084787975880047091nteger @ ( image_4470545334726330049nteger @ F4 @ A ) @ A8 ) ) )
% 14.25/8.09 = ( ? [G3: code_integer > code_integer] :
% 14.25/8.09 ( ( image_4470545334726330049nteger @ G3 @ A8 )
% 14.25/8.09 = A ) ) ) ) ).
% 14.25/8.09
% 14.25/8.09 thf(fact_4565_inj__on__iff__surj,axiom,
% 14.25/8.09 ! [A: set_nat,A8: set_set_nat] :
% 14.25/8.09 ( ( A != bot_bot_set_nat )
% 14.25/8.09 => ( ( ? [F4: nat > set_nat] :
% 14.25/8.09 ( ( inj_on_nat_set_nat @ F4 @ A )
% 14.25/8.09 & ( ord_le6893508408891458716et_nat @ ( image_nat_set_nat @ F4 @ A ) @ A8 ) ) )
% 14.25/8.09 = ( ? [G3: set_nat > nat] :
% 14.25/8.09 ( ( image_set_nat_nat @ G3 @ A8 )
% 14.25/8.09 = A ) ) ) ) ).
% 14.25/8.09
% 14.25/8.09 thf(fact_4566_inj__on__iff__surj,axiom,
% 14.25/8.09 ! [A: set_nat,A8: set_Code_integer] :
% 14.25/8.09 ( ( A != bot_bot_set_nat )
% 14.25/8.09 => ( ( ? [F4: nat > code_integer] :
% 14.25/8.09 ( ( inj_on2245834603434889120nteger @ F4 @ A )
% 14.25/8.09 & ( ord_le7084787975880047091nteger @ ( image_1215581382706833972nteger @ F4 @ A ) @ A8 ) ) )
% 14.25/8.09 = ( ? [G3: code_integer > nat] :
% 14.25/8.09 ( ( image_951025933927791156er_nat @ G3 @ A8 )
% 14.25/8.09 = A ) ) ) ) ).
% 14.25/8.09
% 14.25/8.09 thf(fact_4567_inj__on__iff__surj,axiom,
% 14.25/8.09 ! [A: set_set_nat,A8: set_nat] :
% 14.25/8.09 ( ( A != bot_bot_set_set_nat )
% 14.25/8.09 => ( ( ? [F4: set_nat > nat] :
% 14.25/8.09 ( ( inj_on_set_nat_nat @ F4 @ A )
% 14.25/8.09 & ( ord_less_eq_set_nat @ ( image_set_nat_nat @ F4 @ A ) @ A8 ) ) )
% 14.25/8.09 = ( ? [G3: nat > set_nat] :
% 14.25/8.09 ( ( image_nat_set_nat @ G3 @ A8 )
% 14.25/8.09 = A ) ) ) ) ).
% 14.25/8.09
% 14.25/8.09 thf(fact_4568_inj__on__iff__surj,axiom,
% 14.25/8.09 ! [A: set_Code_integer,A8: set_nat] :
% 14.25/8.09 ( ( A != bot_bo3990330152332043303nteger )
% 14.25/8.09 => ( ( ? [F4: code_integer > nat] :
% 14.25/8.09 ( ( inj_on1981279154655846304er_nat @ F4 @ A )
% 14.25/8.09 & ( ord_less_eq_set_nat @ ( image_951025933927791156er_nat @ F4 @ A ) @ A8 ) ) )
% 14.25/8.09 = ( ? [G3: nat > code_integer] :
% 14.25/8.09 ( ( image_1215581382706833972nteger @ G3 @ A8 )
% 14.25/8.09 = A ) ) ) ) ).
% 14.25/8.09
% 14.25/8.09 thf(fact_4569_inj__on__iff__surj,axiom,
% 14.25/8.09 ! [A: set_Product_unit,A8: set_nat] :
% 14.25/8.09 ( ( A != bot_bo3957492148770167129t_unit )
% 14.25/8.09 => ( ( ? [F4: product_unit > nat] :
% 14.25/8.09 ( ( inj_on8430439091780834860it_nat @ F4 @ A )
% 14.25/8.09 & ( ord_less_eq_set_nat @ ( image_875570014554754200it_nat @ F4 @ A ) @ A8 ) ) )
% 14.25/8.09 = ( ? [G3: nat > product_unit] :
% 14.25/8.09 ( ( image_8730104196221521654t_unit @ G3 @ A8 )
% 14.25/8.09 = A ) ) ) ) ).
% 14.25/8.09
% 14.25/8.09 thf(fact_4570_inj__on__iff__surj,axiom,
% 14.25/8.09 ! [A: set_o,A8: set_nat] :
% 14.25/8.09 ( ( A != bot_bot_set_o )
% 14.25/8.09 => ( ( ? [F4: $o > nat] :
% 14.25/8.09 ( ( inj_on_o_nat @ F4 @ A )
% 14.25/8.09 & ( ord_less_eq_set_nat @ ( image_o_nat2 @ F4 @ A ) @ A8 ) ) )
% 14.25/8.09 = ( ? [G3: nat > $o] :
% 14.25/8.09 ( ( image_nat_o2 @ G3 @ A8 )
% 14.25/8.09 = A ) ) ) ) ).
% 14.25/8.09
% 14.25/8.09 thf(fact_4571_inj__on__iff__surj,axiom,
% 14.25/8.09 ! [A: set_nat,A8: set_nat] :
% 14.25/8.09 ( ( A != bot_bot_set_nat )
% 14.25/8.09 => ( ( ? [F4: nat > nat] :
% 14.25/8.09 ( ( inj_on_nat_nat @ F4 @ A )
% 14.25/8.09 & ( ord_less_eq_set_nat @ ( image_nat_nat2 @ F4 @ A ) @ A8 ) ) )
% 14.25/8.09 = ( ? [G3: nat > nat] :
% 14.25/8.09 ( ( image_nat_nat2 @ G3 @ A8 )
% 14.25/8.09 = A ) ) ) ) ).
% 14.25/8.09
% 14.25/8.09 thf(fact_4572_inj__on__iff__surj,axiom,
% 14.25/8.09 ! [A: set_int,A8: set_nat] :
% 14.25/8.09 ( ( A != bot_bot_set_int )
% 14.25/8.09 => ( ( ? [F4: int > nat] :
% 14.25/8.09 ( ( inj_on_int_nat @ F4 @ A )
% 14.25/8.09 & ( ord_less_eq_set_nat @ ( image_int_nat2 @ F4 @ A ) @ A8 ) ) )
% 14.25/8.09 = ( ? [G3: nat > int] :
% 14.25/8.09 ( ( image_nat_int2 @ G3 @ A8 )
% 14.25/8.09 = A ) ) ) ) ).
% 14.25/8.09
% 14.25/8.09 thf(fact_4573_inj__on__iff__surj,axiom,
% 14.25/8.09 ! [A: set_Pr4532377907799695533_nat_o,A8: set_nat] :
% 14.25/8.09 ( ( A != bot_bo7824918357723582233_nat_o )
% 14.25/8.09 => ( ( ? [F4: ( produc3658429121746597890et_nat > $o ) > nat] :
% 14.25/8.09 ( ( inj_on5271586719803267564_o_nat @ F4 @ A )
% 14.25/8.09 & ( ord_less_eq_set_nat @ ( image_2174381888560928856_o_nat @ F4 @ A ) @ A8 ) ) )
% 14.25/8.09 = ( ? [G3: nat > produc3658429121746597890et_nat > $o] :
% 14.25/8.09 ( ( image_4682239675366380726_nat_o @ G3 @ A8 )
% 14.25/8.09 = A ) ) ) ) ).
% 14.25/8.09
% 14.25/8.09 thf(fact_4574_Sup__insert__finite,axiom,
% 14.25/8.09 ! [S: set_int,X: int] :
% 14.25/8.09 ( ( finite_finite_int @ S )
% 14.25/8.09 => ( ( ( S = bot_bot_set_int )
% 14.25/8.09 => ( ( complete_Sup_Sup_int @ ( insert_int @ X @ S ) )
% 14.25/8.09 = X ) )
% 14.25/8.09 & ( ( S != bot_bot_set_int )
% 14.25/8.09 => ( ( complete_Sup_Sup_int @ ( insert_int @ X @ S ) )
% 14.25/8.09 = ( ord_max_int @ X @ ( complete_Sup_Sup_int @ S ) ) ) ) ) ) ).
% 14.25/8.09
% 14.25/8.09 thf(fact_4575_Sup__insert__finite,axiom,
% 14.25/8.09 ! [S: set_nat,X: nat] :
% 14.25/8.09 ( ( finite_finite_nat @ S )
% 14.25/8.09 => ( ( ( S = bot_bot_set_nat )
% 14.25/8.09 => ( ( complete_Sup_Sup_nat @ ( insert_nat @ X @ S ) )
% 14.25/8.09 = X ) )
% 14.25/8.09 & ( ( S != bot_bot_set_nat )
% 14.25/8.09 => ( ( complete_Sup_Sup_nat @ ( insert_nat @ X @ S ) )
% 14.25/8.09 = ( ord_max_nat @ X @ ( complete_Sup_Sup_nat @ S ) ) ) ) ) ) ).
% 14.25/8.09
% 14.25/8.09 thf(fact_4576_Sup__insert__finite,axiom,
% 14.25/8.09 ! [S: set_Product_unit,X: product_unit] :
% 14.25/8.09 ( ( finite4290736615968046902t_unit @ S )
% 14.25/8.09 => ( ( ( S = bot_bo3957492148770167129t_unit )
% 14.25/8.09 => ( ( comple4687483117567863418t_unit @ ( insert_Product_unit @ X @ S ) )
% 14.25/8.09 = X ) )
% 14.25/8.09 & ( ( S != bot_bo3957492148770167129t_unit )
% 14.25/8.09 => ( ( comple4687483117567863418t_unit @ ( insert_Product_unit @ X @ S ) )
% 14.25/8.09 = ( ord_max_Product_unit @ X @ ( comple4687483117567863418t_unit @ S ) ) ) ) ) ) ).
% 14.25/8.09
% 14.25/8.09 thf(fact_4577_hom__Max__commute,axiom,
% 14.25/8.09 ! [H2: code_integer > code_integer,N: set_Code_integer] :
% 14.25/8.09 ( ! [X3: code_integer,Y2: code_integer] :
% 14.25/8.09 ( ( H2 @ ( ord_max_Code_integer @ X3 @ Y2 ) )
% 14.25/8.09 = ( ord_max_Code_integer @ ( H2 @ X3 ) @ ( H2 @ Y2 ) ) )
% 14.25/8.09 => ( ( finite6017078050557962740nteger @ N )
% 14.25/8.09 => ( ( N != bot_bo3990330152332043303nteger )
% 14.25/8.09 => ( ( H2 @ ( lattic4901227151466704046nteger @ N ) )
% 14.25/8.09 = ( lattic4901227151466704046nteger @ ( image_4470545334726330049nteger @ H2 @ N ) ) ) ) ) ) ).
% 14.25/8.09
% 14.25/8.09 thf(fact_4578_hom__Max__commute,axiom,
% 14.25/8.09 ! [H2: product_unit > product_unit,N: set_Product_unit] :
% 14.25/8.09 ( ! [X3: product_unit,Y2: product_unit] :
% 14.25/8.09 ( ( H2 @ ( ord_max_Product_unit @ X3 @ Y2 ) )
% 14.25/8.09 = ( ord_max_Product_unit @ ( H2 @ X3 ) @ ( H2 @ Y2 ) ) )
% 14.25/8.09 => ( ( finite4290736615968046902t_unit @ N )
% 14.25/8.09 => ( ( N != bot_bo3957492148770167129t_unit )
% 14.25/8.09 => ( ( H2 @ ( lattic1210392890313158908t_unit @ N ) )
% 14.25/8.09 = ( lattic1210392890313158908t_unit @ ( image_405062704495631173t_unit @ H2 @ N ) ) ) ) ) ) ).
% 14.25/8.09
% 14.25/8.09 thf(fact_4579_hom__Max__commute,axiom,
% 14.25/8.09 ! [H2: $o > $o,N: set_o] :
% 14.25/8.09 ( ! [X3: $o,Y2: $o] :
% 14.25/8.09 ( ( H2 @ ( ord_max_o @ X3 @ Y2 ) )
% 14.25/8.09 = ( ord_max_o @ ( H2 @ X3 ) @ ( H2 @ Y2 ) ) )
% 14.25/8.09 => ( ( finite_finite_o @ N )
% 14.25/8.09 => ( ( N != bot_bot_set_o )
% 14.25/8.09 => ( ( H2 @ ( lattic1921953407002678535_Max_o @ N ) )
% 14.25/8.09 = ( lattic1921953407002678535_Max_o @ ( image_o_o2 @ H2 @ N ) ) ) ) ) ) ).
% 14.25/8.09
% 14.25/8.09 thf(fact_4580_hom__Max__commute,axiom,
% 14.25/8.09 ! [H2: int > int,N: set_int] :
% 14.25/8.09 ( ! [X3: int,Y2: int] :
% 14.25/8.09 ( ( H2 @ ( ord_max_int @ X3 @ Y2 ) )
% 14.25/8.09 = ( ord_max_int @ ( H2 @ X3 ) @ ( H2 @ Y2 ) ) )
% 14.25/8.09 => ( ( finite_finite_int @ N )
% 14.25/8.09 => ( ( N != bot_bot_set_int )
% 14.25/8.09 => ( ( H2 @ ( lattic8263393255366662781ax_int @ N ) )
% 14.25/8.09 = ( lattic8263393255366662781ax_int @ ( image_int_int2 @ H2 @ N ) ) ) ) ) ) ).
% 14.25/8.09
% 14.25/8.09 thf(fact_4581_hom__Max__commute,axiom,
% 14.25/8.09 ! [H2: nat > nat,N: set_nat] :
% 14.25/8.09 ( ! [X3: nat,Y2: nat] :
% 14.25/8.09 ( ( H2 @ ( ord_max_nat @ X3 @ Y2 ) )
% 14.25/8.09 = ( ord_max_nat @ ( H2 @ X3 ) @ ( H2 @ Y2 ) ) )
% 14.25/8.09 => ( ( finite_finite_nat @ N )
% 14.25/8.09 => ( ( N != bot_bot_set_nat )
% 14.25/8.09 => ( ( H2 @ ( lattic8265883725875713057ax_nat @ N ) )
% 14.25/8.09 = ( lattic8265883725875713057ax_nat @ ( image_nat_nat2 @ H2 @ N ) ) ) ) ) ) ).
% 14.25/8.09
% 14.25/8.09 thf(fact_4582_Max_Osubset,axiom,
% 14.25/8.09 ! [A: set_Code_integer,B4: set_Code_integer] :
% 14.25/8.09 ( ( finite6017078050557962740nteger @ A )
% 14.25/8.09 => ( ( B4 != bot_bo3990330152332043303nteger )
% 14.25/8.09 => ( ( ord_le7084787975880047091nteger @ B4 @ A )
% 14.25/8.09 => ( ( ord_max_Code_integer @ ( lattic4901227151466704046nteger @ B4 ) @ ( lattic4901227151466704046nteger @ A ) )
% 14.25/8.09 = ( lattic4901227151466704046nteger @ A ) ) ) ) ) ).
% 14.25/8.09
% 14.25/8.09 thf(fact_4583_Max_Osubset,axiom,
% 14.25/8.09 ! [A: set_Product_unit,B4: set_Product_unit] :
% 14.25/8.09 ( ( finite4290736615968046902t_unit @ A )
% 14.25/8.09 => ( ( B4 != bot_bo3957492148770167129t_unit )
% 14.25/8.09 => ( ( ord_le3507040750410214029t_unit @ B4 @ A )
% 14.25/8.09 => ( ( ord_max_Product_unit @ ( lattic1210392890313158908t_unit @ B4 ) @ ( lattic1210392890313158908t_unit @ A ) )
% 14.25/8.09 = ( lattic1210392890313158908t_unit @ A ) ) ) ) ) ).
% 14.25/8.09
% 14.25/8.09 thf(fact_4584_Max_Osubset,axiom,
% 14.25/8.09 ! [A: set_o,B4: set_o] :
% 14.25/8.09 ( ( finite_finite_o @ A )
% 14.25/8.09 => ( ( B4 != bot_bot_set_o )
% 14.25/8.09 => ( ( ord_less_eq_set_o @ B4 @ A )
% 14.25/8.09 => ( ( ord_max_o @ ( lattic1921953407002678535_Max_o @ B4 ) @ ( lattic1921953407002678535_Max_o @ A ) )
% 14.25/8.09 = ( lattic1921953407002678535_Max_o @ A ) ) ) ) ) ).
% 14.25/8.09
% 14.25/8.09 thf(fact_4585_Max_Osubset,axiom,
% 14.25/8.09 ! [A: set_int,B4: set_int] :
% 14.25/8.09 ( ( finite_finite_int @ A )
% 14.25/8.09 => ( ( B4 != bot_bot_set_int )
% 14.25/8.09 => ( ( ord_less_eq_set_int @ B4 @ A )
% 14.25/8.09 => ( ( ord_max_int @ ( lattic8263393255366662781ax_int @ B4 ) @ ( lattic8263393255366662781ax_int @ A ) )
% 14.25/8.09 = ( lattic8263393255366662781ax_int @ A ) ) ) ) ) ).
% 14.25/8.09
% 14.25/8.09 thf(fact_4586_Max_Osubset,axiom,
% 14.25/8.09 ! [A: set_nat,B4: set_nat] :
% 14.25/8.09 ( ( finite_finite_nat @ A )
% 14.25/8.09 => ( ( B4 != bot_bot_set_nat )
% 14.25/8.09 => ( ( ord_less_eq_set_nat @ B4 @ A )
% 14.25/8.09 => ( ( ord_max_nat @ ( lattic8265883725875713057ax_nat @ B4 ) @ ( lattic8265883725875713057ax_nat @ A ) )
% 14.25/8.09 = ( lattic8265883725875713057ax_nat @ A ) ) ) ) ) ).
% 14.25/8.09
% 14.25/8.09 thf(fact_4587_Max_Oinsert__not__elem,axiom,
% 14.25/8.09 ! [A: set_Code_integer,X: code_integer] :
% 14.25/8.09 ( ( finite6017078050557962740nteger @ A )
% 14.25/8.09 => ( ~ ( member_Code_integer @ X @ A )
% 14.25/8.09 => ( ( A != bot_bo3990330152332043303nteger )
% 14.25/8.09 => ( ( lattic4901227151466704046nteger @ ( insert_Code_integer @ X @ A ) )
% 14.25/8.09 = ( ord_max_Code_integer @ X @ ( lattic4901227151466704046nteger @ A ) ) ) ) ) ) ).
% 14.25/8.09
% 14.25/8.09 thf(fact_4588_Max_Oinsert__not__elem,axiom,
% 14.25/8.09 ! [A: set_Product_unit,X: product_unit] :
% 14.25/8.09 ( ( finite4290736615968046902t_unit @ A )
% 14.25/8.09 => ( ~ ( member_Product_unit @ X @ A )
% 14.25/8.09 => ( ( A != bot_bo3957492148770167129t_unit )
% 14.25/8.09 => ( ( lattic1210392890313158908t_unit @ ( insert_Product_unit @ X @ A ) )
% 14.25/8.09 = ( ord_max_Product_unit @ X @ ( lattic1210392890313158908t_unit @ A ) ) ) ) ) ) ).
% 14.25/8.09
% 14.25/8.09 thf(fact_4589_Max_Oinsert__not__elem,axiom,
% 14.25/8.09 ! [A: set_o,X: $o] :
% 14.25/8.09 ( ( finite_finite_o @ A )
% 14.25/8.09 => ( ~ ( member_o @ X @ A )
% 14.25/8.09 => ( ( A != bot_bot_set_o )
% 14.25/8.09 => ( ( lattic1921953407002678535_Max_o @ ( insert_o @ X @ A ) )
% 14.25/8.09 = ( ord_max_o @ X @ ( lattic1921953407002678535_Max_o @ A ) ) ) ) ) ) ).
% 14.25/8.09
% 14.25/8.09 thf(fact_4590_Max_Oinsert__not__elem,axiom,
% 14.25/8.09 ! [A: set_int,X: int] :
% 14.25/8.09 ( ( finite_finite_int @ A )
% 14.25/8.09 => ( ~ ( member_int @ X @ A )
% 14.25/8.09 => ( ( A != bot_bot_set_int )
% 14.25/8.09 => ( ( lattic8263393255366662781ax_int @ ( insert_int @ X @ A ) )
% 14.25/8.09 = ( ord_max_int @ X @ ( lattic8263393255366662781ax_int @ A ) ) ) ) ) ) ).
% 14.25/8.09
% 14.25/8.09 thf(fact_4591_Max_Oinsert__not__elem,axiom,
% 14.25/8.09 ! [A: set_nat,X: nat] :
% 14.25/8.09 ( ( finite_finite_nat @ A )
% 14.25/8.09 => ( ~ ( member_nat @ X @ A )
% 14.25/8.09 => ( ( A != bot_bot_set_nat )
% 14.25/8.09 => ( ( lattic8265883725875713057ax_nat @ ( insert_nat @ X @ A ) )
% 14.25/8.09 = ( ord_max_nat @ X @ ( lattic8265883725875713057ax_nat @ A ) ) ) ) ) ) ).
% 14.25/8.09
% 14.25/8.09 thf(fact_4592_Max_Oclosed,axiom,
% 14.25/8.09 ! [A: set_Code_integer] :
% 14.25/8.09 ( ( finite6017078050557962740nteger @ A )
% 14.25/8.09 => ( ( A != bot_bo3990330152332043303nteger )
% 14.25/8.09 => ( ! [X3: code_integer,Y2: code_integer] : ( member_Code_integer @ ( ord_max_Code_integer @ X3 @ Y2 ) @ ( insert_Code_integer @ X3 @ ( insert_Code_integer @ Y2 @ bot_bo3990330152332043303nteger ) ) )
% 14.25/8.09 => ( member_Code_integer @ ( lattic4901227151466704046nteger @ A ) @ A ) ) ) ) ).
% 14.25/8.09
% 14.25/8.09 thf(fact_4593_Max_Oclosed,axiom,
% 14.25/8.09 ! [A: set_Product_unit] :
% 14.25/8.09 ( ( finite4290736615968046902t_unit @ A )
% 14.25/8.09 => ( ( A != bot_bo3957492148770167129t_unit )
% 14.25/8.09 => ( ! [X3: product_unit,Y2: product_unit] : ( member_Product_unit @ ( ord_max_Product_unit @ X3 @ Y2 ) @ ( insert_Product_unit @ X3 @ ( insert_Product_unit @ Y2 @ bot_bo3957492148770167129t_unit ) ) )
% 14.25/8.09 => ( member_Product_unit @ ( lattic1210392890313158908t_unit @ A ) @ A ) ) ) ) ).
% 14.25/8.09
% 14.25/8.09 thf(fact_4594_Max_Oclosed,axiom,
% 14.25/8.09 ! [A: set_o] :
% 14.25/8.09 ( ( finite_finite_o @ A )
% 14.25/8.09 => ( ( A != bot_bot_set_o )
% 14.25/8.09 => ( ! [X3: $o,Y2: $o] : ( member_o @ ( ord_max_o @ X3 @ Y2 ) @ ( insert_o @ X3 @ ( insert_o @ Y2 @ bot_bot_set_o ) ) )
% 14.25/8.09 => ( member_o @ ( lattic1921953407002678535_Max_o @ A ) @ A ) ) ) ) ).
% 14.25/8.09
% 14.25/8.09 thf(fact_4595_Max_Oclosed,axiom,
% 14.25/8.09 ! [A: set_int] :
% 14.25/8.09 ( ( finite_finite_int @ A )
% 14.25/8.09 => ( ( A != bot_bot_set_int )
% 14.25/8.09 => ( ! [X3: int,Y2: int] : ( member_int @ ( ord_max_int @ X3 @ Y2 ) @ ( insert_int @ X3 @ ( insert_int @ Y2 @ bot_bot_set_int ) ) )
% 14.25/8.09 => ( member_int @ ( lattic8263393255366662781ax_int @ A ) @ A ) ) ) ) ).
% 14.25/8.09
% 14.25/8.09 thf(fact_4596_Max_Oclosed,axiom,
% 14.25/8.09 ! [A: set_nat] :
% 14.25/8.09 ( ( finite_finite_nat @ A )
% 14.25/8.09 => ( ( A != bot_bot_set_nat )
% 14.25/8.09 => ( ! [X3: nat,Y2: nat] : ( member_nat @ ( ord_max_nat @ X3 @ Y2 ) @ ( insert_nat @ X3 @ ( insert_nat @ Y2 @ bot_bot_set_nat ) ) )
% 14.25/8.09 => ( member_nat @ ( lattic8265883725875713057ax_nat @ A ) @ A ) ) ) ) ).
% 14.25/8.09
% 14.25/8.09 thf(fact_4597_Max_Ounion,axiom,
% 14.25/8.09 ! [A: set_Code_integer,B4: set_Code_integer] :
% 14.25/8.09 ( ( finite6017078050557962740nteger @ A )
% 14.25/8.09 => ( ( A != bot_bo3990330152332043303nteger )
% 14.25/8.09 => ( ( finite6017078050557962740nteger @ B4 )
% 14.25/8.09 => ( ( B4 != bot_bo3990330152332043303nteger )
% 14.25/8.09 => ( ( lattic4901227151466704046nteger @ ( sup_su848401254843788991nteger @ A @ B4 ) )
% 14.25/8.09 = ( ord_max_Code_integer @ ( lattic4901227151466704046nteger @ A ) @ ( lattic4901227151466704046nteger @ B4 ) ) ) ) ) ) ) ).
% 14.25/8.09
% 14.25/8.09 thf(fact_4598_Max_Ounion,axiom,
% 14.25/8.09 ! [A: set_Product_unit,B4: set_Product_unit] :
% 14.25/8.09 ( ( finite4290736615968046902t_unit @ A )
% 14.25/8.09 => ( ( A != bot_bo3957492148770167129t_unit )
% 14.25/8.09 => ( ( finite4290736615968046902t_unit @ B4 )
% 14.25/8.09 => ( ( B4 != bot_bo3957492148770167129t_unit )
% 14.25/8.09 => ( ( lattic1210392890313158908t_unit @ ( sup_su793286257634532545t_unit @ A @ B4 ) )
% 14.25/8.09 = ( ord_max_Product_unit @ ( lattic1210392890313158908t_unit @ A ) @ ( lattic1210392890313158908t_unit @ B4 ) ) ) ) ) ) ) ).
% 14.25/8.09
% 14.25/8.09 thf(fact_4599_Max_Ounion,axiom,
% 14.25/8.09 ! [A: set_o,B4: set_o] :
% 14.25/8.09 ( ( finite_finite_o @ A )
% 14.25/8.09 => ( ( A != bot_bot_set_o )
% 14.25/8.09 => ( ( finite_finite_o @ B4 )
% 14.25/8.09 => ( ( B4 != bot_bot_set_o )
% 14.25/8.09 => ( ( lattic1921953407002678535_Max_o @ ( sup_sup_set_o @ A @ B4 ) )
% 14.25/8.09 = ( ord_max_o @ ( lattic1921953407002678535_Max_o @ A ) @ ( lattic1921953407002678535_Max_o @ B4 ) ) ) ) ) ) ) ).
% 14.25/8.09
% 14.25/8.09 thf(fact_4600_Max_Ounion,axiom,
% 14.25/8.09 ! [A: set_int,B4: set_int] :
% 14.25/8.09 ( ( finite_finite_int @ A )
% 14.25/8.09 => ( ( A != bot_bot_set_int )
% 14.25/8.09 => ( ( finite_finite_int @ B4 )
% 14.25/8.09 => ( ( B4 != bot_bot_set_int )
% 14.25/8.09 => ( ( lattic8263393255366662781ax_int @ ( sup_sup_set_int @ A @ B4 ) )
% 14.25/8.09 = ( ord_max_int @ ( lattic8263393255366662781ax_int @ A ) @ ( lattic8263393255366662781ax_int @ B4 ) ) ) ) ) ) ) ).
% 14.25/8.09
% 14.25/8.09 thf(fact_4601_Max_Ounion,axiom,
% 14.25/8.09 ! [A: set_nat,B4: set_nat] :
% 14.25/8.09 ( ( finite_finite_nat @ A )
% 14.25/8.09 => ( ( A != bot_bot_set_nat )
% 14.25/8.09 => ( ( finite_finite_nat @ B4 )
% 14.25/8.09 => ( ( B4 != bot_bot_set_nat )
% 14.25/8.09 => ( ( lattic8265883725875713057ax_nat @ ( sup_sup_set_nat @ A @ B4 ) )
% 14.25/8.09 = ( ord_max_nat @ ( lattic8265883725875713057ax_nat @ A ) @ ( lattic8265883725875713057ax_nat @ B4 ) ) ) ) ) ) ) ).
% 14.25/8.09
% 14.25/8.09 thf(fact_4602_Max_Oeq__fold,axiom,
% 14.25/8.09 ! [A: set_o,X: $o] :
% 14.25/8.09 ( ( finite_finite_o @ A )
% 14.25/8.09 => ( ( lattic1921953407002678535_Max_o @ ( insert_o @ X @ A ) )
% 14.25/8.09 = ( finite_fold_o_o @ ord_max_o @ X @ A ) ) ) ).
% 14.25/8.09
% 14.25/8.09 thf(fact_4603_Max_Oeq__fold,axiom,
% 14.25/8.09 ! [A: set_Product_unit,X: product_unit] :
% 14.25/8.09 ( ( finite4290736615968046902t_unit @ A )
% 14.25/8.09 => ( ( lattic1210392890313158908t_unit @ ( insert_Product_unit @ X @ A ) )
% 14.25/8.09 = ( finite8834431295148555784t_unit @ ord_max_Product_unit @ X @ A ) ) ) ).
% 14.25/8.09
% 14.25/8.09 thf(fact_4604_Max_Oeq__fold,axiom,
% 14.25/8.09 ! [A: set_Code_integer,X: code_integer] :
% 14.25/8.09 ( ( finite6017078050557962740nteger @ A )
% 14.25/8.09 => ( ( lattic4901227151466704046nteger @ ( insert_Code_integer @ X @ A ) )
% 14.25/8.09 = ( finite6454954857057555716nteger @ ord_max_Code_integer @ X @ A ) ) ) ).
% 14.25/8.09
% 14.25/8.09 thf(fact_4605_Max_Oeq__fold,axiom,
% 14.25/8.09 ! [A: set_int,X: int] :
% 14.25/8.09 ( ( finite_finite_int @ A )
% 14.25/8.09 => ( ( lattic8263393255366662781ax_int @ ( insert_int @ X @ A ) )
% 14.25/8.09 = ( finite_fold_int_int @ ord_max_int @ X @ A ) ) ) ).
% 14.25/8.09
% 14.25/8.09 thf(fact_4606_Max_Oeq__fold,axiom,
% 14.25/8.09 ! [A: set_nat,X: nat] :
% 14.25/8.09 ( ( finite_finite_nat @ A )
% 14.25/8.09 => ( ( lattic8265883725875713057ax_nat @ ( insert_nat @ X @ A ) )
% 14.25/8.09 = ( finite_fold_nat_nat @ ord_max_nat @ X @ A ) ) ) ).
% 14.25/8.09
% 14.25/8.09 thf(fact_4607_finite__range__prod,axiom,
% 14.25/8.09 ! [F: code_integer > produc2591615631378489102nteger] :
% 14.25/8.09 ( ( finite_finite_nat @ ( image_951025933927791156er_nat @ ( comp_P5725896914063526938nteger @ produc8897218493789310642nteger @ F ) @ top_to4645266643341252675nteger ) )
% 14.25/8.09 => ( ( finite6017078050557962740nteger @ ( image_4470545334726330049nteger @ ( comp_P6122662217367900327nteger @ produc1781891006948753140nteger @ F ) @ top_to4645266643341252675nteger ) )
% 14.25/8.09 => ( finite1154042800025562295nteger @ ( image_3408592110290467754nteger @ F @ top_to4645266643341252675nteger ) ) ) ) ).
% 14.25/8.09
% 14.25/8.09 thf(fact_4608_finite__range__prod,axiom,
% 14.25/8.09 ! [F: code_integer > produc6433947425880249742er_nat] :
% 14.25/8.09 ( ( finite6017078050557962740nteger @ ( image_4470545334726330049nteger @ ( comp_P7101464685127059239nteger @ produc8632663045010267826er_nat @ F ) @ top_to4645266643341252675nteger ) )
% 14.25/8.09 => ( ( finite_finite_nat @ ( image_951025933927791156er_nat @ ( comp_P5964733648194011802nteger @ produc1517335558169710324er_nat @ F ) @ top_to4645266643341252675nteger ) )
% 14.25/8.09 => ( finite4996374594527322935er_nat @ ( image_7250923904792228394er_nat @ F @ top_to4645266643341252675nteger ) ) ) ) ).
% 14.25/8.09
% 14.25/8.09 thf(fact_4609_finite__range__prod,axiom,
% 14.25/8.09 ! [F: code_integer > produc8923325533196201883nteger] :
% 14.25/8.09 ( ( finite6017078050557962740nteger @ ( image_4470545334726330049nteger @ ( comp_P7111796090210262324nteger @ produc8508995932063986495nteger @ F ) @ top_to4645266643341252675nteger ) )
% 14.25/8.09 => ( ( finite6017078050557962740nteger @ ( image_4470545334726330049nteger @ ( comp_P7111796090210262324nteger @ produc6174133586879617921nteger @ F ) @ top_to4645266643341252675nteger ) )
% 14.25/8.09 => ( finite2154694888287295812nteger @ ( image_8049062792287394871nteger @ F @ top_to4645266643341252675nteger ) ) ) ) ).
% 14.25/8.09
% 14.25/8.09 thf(fact_4610_finite__range__prod,axiom,
% 14.25/8.09 ! [F: product_unit > product_prod_nat_nat] :
% 14.25/8.09 ( ( finite_finite_nat @ ( image_875570014554754200it_nat @ ( comp_P2980510830910040157t_unit @ product_fst_nat_nat @ F ) @ top_to1996260823553986621t_unit ) )
% 14.25/8.09 => ( ( finite_finite_nat @ ( image_875570014554754200it_nat @ ( comp_P2980510830910040157t_unit @ product_snd_nat_nat @ F ) @ top_to1996260823553986621t_unit ) )
% 14.25/8.09 => ( finite6177210948735845034at_nat @ ( image_7281959405765017145at_nat @ F @ top_to1996260823553986621t_unit ) ) ) ) ).
% 14.25/8.09
% 14.25/8.09 thf(fact_4611_finite__range__prod,axiom,
% 14.25/8.09 ! [F: product_unit > produc2591615631378489102nteger] :
% 14.25/8.09 ( ( finite_finite_nat @ ( image_875570014554754200it_nat @ ( comp_P7621350044865347728t_unit @ produc8897218493789310642nteger @ F ) @ top_to1996260823553986621t_unit ) )
% 14.25/8.09 => ( ( finite6017078050557962740nteger @ ( image_3815118110082124069nteger @ ( comp_P7320554561093743171t_unit @ produc1781891006948753140nteger @ F ) @ top_to1996260823553986621t_unit ) )
% 14.25/8.09 => ( finite1154042800025562295nteger @ ( image_4946983642536848454nteger @ F @ top_to1996260823553986621t_unit ) ) ) ) ).
% 14.25/8.09
% 14.25/8.09 thf(fact_4612_finite__range__prod,axiom,
% 14.25/8.09 ! [F: product_unit > produc6433947425880249742er_nat] :
% 14.25/8.09 ( ( finite6017078050557962740nteger @ ( image_3815118110082124069nteger @ ( comp_P7581484221039074243t_unit @ produc8632663045010267826er_nat @ F ) @ top_to1996260823553986621t_unit ) )
% 14.25/8.09 => ( ( finite_finite_nat @ ( image_875570014554754200it_nat @ ( comp_P6821003776282389008t_unit @ produc1517335558169710324er_nat @ F ) @ top_to1996260823553986621t_unit ) )
% 14.25/8.09 => ( finite4996374594527322935er_nat @ ( image_8789315437038609094er_nat @ F @ top_to1996260823553986621t_unit ) ) ) ) ).
% 14.25/8.09
% 14.25/8.09 thf(fact_4613_finite__range__prod,axiom,
% 14.25/8.09 ! [F: product_unit > produc8923325533196201883nteger] :
% 14.25/8.09 ( ( finite6017078050557962740nteger @ ( image_3815118110082124069nteger @ ( comp_P659088277599099894t_unit @ produc8508995932063986495nteger @ F ) @ top_to1996260823553986621t_unit ) )
% 14.25/8.09 => ( ( finite6017078050557962740nteger @ ( image_3815118110082124069nteger @ ( comp_P659088277599099894t_unit @ produc6174133586879617921nteger @ F ) @ top_to1996260823553986621t_unit ) )
% 14.25/8.09 => ( finite2154694888287295812nteger @ ( image_8089323762101378259nteger @ F @ top_to1996260823553986621t_unit ) ) ) ) ).
% 14.25/8.09
% 14.25/8.09 thf(fact_4614_finite__range__prod,axiom,
% 14.25/8.09 ! [F: product_unit > product_prod_int_int] :
% 14.25/8.09 ( ( finite_finite_int @ ( image_873079544045703924it_int @ ( comp_P6796073005956588617t_unit @ product_fst_int_int @ F ) @ top_to1996260823553986621t_unit ) )
% 14.25/8.09 => ( ( finite_finite_int @ ( image_873079544045703924it_int @ ( comp_P6796073005956588617t_unit @ product_snd_int_int @ F ) @ top_to1996260823553986621t_unit ) )
% 14.25/8.09 => ( finite2998713641127702882nt_int @ ( image_4103462098156874993nt_int @ F @ top_to1996260823553986621t_unit ) ) ) ) ).
% 14.25/8.09
% 14.25/8.09 thf(fact_4615_finite__range__prod,axiom,
% 14.25/8.09 ! [F: $o > product_prod_nat_nat] :
% 14.25/8.09 ( ( finite_finite_nat @ ( image_o_nat2 @ ( comp_P4394204175247225320_nat_o @ product_fst_nat_nat @ F ) @ top_top_set_o ) )
% 14.25/8.09 => ( ( finite_finite_nat @ ( image_o_nat2 @ ( comp_P4394204175247225320_nat_o @ product_snd_nat_nat @ F ) @ top_top_set_o ) )
% 14.25/8.09 => ( finite6177210948735845034at_nat @ ( image_3855930084881510382at_nat @ F @ top_top_set_o ) ) ) ) ).
% 14.25/8.09
% 14.25/8.09 thf(fact_4616_finite__range__prod,axiom,
% 14.25/8.09 ! [F: $o > produc2591615631378489102nteger] :
% 14.25/8.09 ( ( finite_finite_nat @ ( image_o_nat2 @ ( comp_P7179991235626930843_nat_o @ produc8897218493789310642nteger @ F ) @ top_top_set_o ) )
% 14.25/8.09 => ( ( finite6017078050557962740nteger @ ( image_o_Code_integer @ ( comp_P7177471617286032590eger_o @ produc1781891006948753140nteger @ F ) @ top_top_set_o ) )
% 14.25/8.09 => ( finite1154042800025562295nteger @ ( image_8407096251609565691nteger @ F @ top_top_set_o ) ) ) ) ).
% 14.25/8.09
% 14.25/8.09 thf(fact_4617_inj__apsnd,axiom,
% 14.25/8.09 ! [F: code_integer > code_integer] :
% 14.25/8.09 ( ( inj_on20428373856169009nteger @ ( produc6499014454317279255nteger @ F ) @ top_to2363841527343591755nteger )
% 14.25/8.09 = ( inj_on1158064722259180589nteger @ F @ top_to4645266643341252675nteger ) ) ).
% 14.25/8.09
% 14.25/8.09 thf(fact_4618_total__inv__image,axiom,
% 14.25/8.09 ! [F: product_unit > product_unit,R2: set_Pr5094982260447487303t_unit] :
% 14.25/8.09 ( ( inj_on8151373323710067377t_unit @ F @ top_to1996260823553986621t_unit )
% 14.25/8.09 => ( ( total_5665700600833547665t_unit @ top_to1996260823553986621t_unit @ R2 )
% 14.25/8.09 => ( total_5665700600833547665t_unit @ top_to1996260823553986621t_unit @ ( inv_im8770536620514187154t_unit @ R2 @ F ) ) ) ) ).
% 14.25/8.09
% 14.25/8.09 thf(fact_4619_total__inv__image,axiom,
% 14.25/8.09 ! [F: product_unit > $o,R2: set_Product_prod_o_o] :
% 14.25/8.09 ( ( inj_on8854722734121536060unit_o @ F @ top_to1996260823553986621t_unit )
% 14.25/8.09 => ( ( total_on_o @ top_top_set_o @ R2 )
% 14.25/8.09 => ( total_5665700600833547665t_unit @ top_to1996260823553986621t_unit @ ( inv_im4602722609573551687t_unit @ R2 @ F ) ) ) ) ).
% 14.25/8.09
% 14.25/8.09 thf(fact_4620_total__inv__image,axiom,
% 14.25/8.09 ! [F: product_unit > nat,R2: set_Pr1261947904930325089at_nat] :
% 14.25/8.09 ( ( inj_on8430439091780834860it_nat @ F @ top_to1996260823553986621t_unit )
% 14.25/8.09 => ( ( total_on_nat @ top_top_set_nat @ R2 )
% 14.25/8.09 => ( total_5665700600833547665t_unit @ top_to1996260823553986621t_unit @ ( inv_im7624054837341743209t_unit @ R2 @ F ) ) ) ) ).
% 14.25/8.09
% 14.25/8.09 thf(fact_4621_total__inv__image,axiom,
% 14.25/8.09 ! [F: product_unit > int,R2: set_Pr958786334691620121nt_int] :
% 14.25/8.09 ( ( inj_on8427948621271784584it_int @ F @ top_to1996260823553986621t_unit )
% 14.25/8.09 => ( ( total_on_int @ top_top_set_int @ R2 )
% 14.25/8.09 => ( total_5665700600833547665t_unit @ top_to1996260823553986621t_unit @ ( inv_im4740476784543923469t_unit @ R2 @ F ) ) ) ) ).
% 14.25/8.09
% 14.25/8.09 thf(fact_4622_total__inv__image,axiom,
% 14.25/8.09 ! [F: $o > product_unit,R2: set_Pr5094982260447487303t_unit] :
% 14.25/8.09 ( ( inj_on4268752236855399398t_unit @ F @ top_top_set_o )
% 14.25/8.09 => ( ( total_5665700600833547665t_unit @ top_to1996260823553986621t_unit @ R2 )
% 14.25/8.09 => ( total_on_o @ top_top_set_o @ ( inv_im9188693106839688349unit_o @ R2 @ F ) ) ) ) ).
% 14.25/8.09
% 14.25/8.09 thf(fact_4623_total__inv__image,axiom,
% 14.25/8.09 ! [F: $o > $o,R2: set_Product_prod_o_o] :
% 14.25/8.09 ( ( inj_on_o_o @ F @ top_top_set_o )
% 14.25/8.09 => ( ( total_on_o @ top_top_set_o @ R2 )
% 14.25/8.09 => ( total_on_o @ top_top_set_o @ ( inv_image_o_o @ R2 @ F ) ) ) ) ).
% 14.25/8.09
% 14.25/8.09 thf(fact_4624_total__inv__image,axiom,
% 14.25/8.09 ! [F: $o > nat,R2: set_Pr1261947904930325089at_nat] :
% 14.25/8.09 ( ( inj_on_o_nat @ F @ top_top_set_o )
% 14.25/8.09 => ( ( total_on_nat @ top_top_set_nat @ R2 )
% 14.25/8.09 => ( total_on_o @ top_top_set_o @ ( inv_image_nat_o @ R2 @ F ) ) ) ) ).
% 14.25/8.09
% 14.25/8.09 thf(fact_4625_total__inv__image,axiom,
% 14.25/8.09 ! [F: $o > int,R2: set_Pr958786334691620121nt_int] :
% 14.25/8.09 ( ( inj_on_o_int @ F @ top_top_set_o )
% 14.25/8.09 => ( ( total_on_int @ top_top_set_int @ R2 )
% 14.25/8.09 => ( total_on_o @ top_top_set_o @ ( inv_image_int_o @ R2 @ F ) ) ) ) ).
% 14.25/8.09
% 14.25/8.09 thf(fact_4626_total__inv__image,axiom,
% 14.25/8.09 ! [F: nat > product_unit,R2: set_Pr5094982260447487303t_unit] :
% 14.25/8.09 ( ( inj_on7061601236592826506t_unit @ F @ top_top_set_nat )
% 14.25/8.09 => ( ( total_5665700600833547665t_unit @ top_to1996260823553986621t_unit @ R2 )
% 14.25/8.09 => ( total_on_nat @ top_top_set_nat @ ( inv_im8992892692529751563it_nat @ R2 @ F ) ) ) ) ).
% 14.25/8.09
% 14.25/8.09 thf(fact_4627_total__inv__image,axiom,
% 14.25/8.09 ! [F: nat > $o,R2: set_Product_prod_o_o] :
% 14.25/8.09 ( ( inj_on_nat_o @ F @ top_top_set_nat )
% 14.25/8.09 => ( ( total_on_o @ top_top_set_o @ R2 )
% 14.25/8.09 => ( total_on_nat @ top_top_set_nat @ ( inv_image_o_nat @ R2 @ F ) ) ) ) ).
% 14.25/8.09
% 14.25/8.09 thf(fact_4628_card__Diff1__less,axiom,
% 14.25/8.09 ! [A: set_Code_integer,X: code_integer] :
% 14.25/8.09 ( ( finite6017078050557962740nteger @ A )
% 14.25/8.09 => ( ( member_Code_integer @ X @ A )
% 14.25/8.09 => ( ord_less_nat @ ( finite4902975817058060853nteger @ ( minus_2355218937544613996nteger @ A @ ( insert_Code_integer @ X @ bot_bo3990330152332043303nteger ) ) ) @ ( finite4902975817058060853nteger @ A ) ) ) ) ).
% 14.25/8.09
% 14.25/8.09 thf(fact_4629_card__Diff1__less,axiom,
% 14.25/8.09 ! [A: set_Product_unit,X: product_unit] :
% 14.25/8.09 ( ( finite4290736615968046902t_unit @ A )
% 14.25/8.09 => ( ( member_Product_unit @ X @ A )
% 14.25/8.09 => ( ord_less_nat @ ( finite410649719033368117t_unit @ ( minus_6452836326544984404t_unit @ A @ ( insert_Product_unit @ X @ bot_bo3957492148770167129t_unit ) ) ) @ ( finite410649719033368117t_unit @ A ) ) ) ) ).
% 14.25/8.09
% 14.25/8.09 thf(fact_4630_card__Diff1__less,axiom,
% 14.25/8.09 ! [A: set_o,X: $o] :
% 14.25/8.09 ( ( finite_finite_o @ A )
% 14.25/8.09 => ( ( member_o @ X @ A )
% 14.25/8.09 => ( ord_less_nat @ ( finite_card_o @ ( minus_minus_set_o @ A @ ( insert_o @ X @ bot_bot_set_o ) ) ) @ ( finite_card_o @ A ) ) ) ) ).
% 14.25/8.09
% 14.25/8.09 thf(fact_4631_card__Diff1__less,axiom,
% 14.25/8.09 ! [A: set_int,X: int] :
% 14.25/8.09 ( ( finite_finite_int @ A )
% 14.25/8.09 => ( ( member_int @ X @ A )
% 14.25/8.09 => ( ord_less_nat @ ( finite_card_int @ ( minus_minus_set_int @ A @ ( insert_int @ X @ bot_bot_set_int ) ) ) @ ( finite_card_int @ A ) ) ) ) ).
% 14.25/8.09
% 14.25/8.09 thf(fact_4632_card__Diff1__less,axiom,
% 14.25/8.09 ! [A: set_Pr4532377907799695533_nat_o,X: produc3658429121746597890et_nat > $o] :
% 14.25/8.09 ( ( finite3252695134891459830_nat_o @ A )
% 14.25/8.09 => ( ( member6576561426505652726_nat_o @ X @ A )
% 14.25/8.09 => ( ord_less_nat @ ( finite1363419556375932405_nat_o @ ( minus_1801376950450012436_nat_o @ A @ ( insert5175938949040314269_nat_o @ X @ bot_bo7824918357723582233_nat_o ) ) ) @ ( finite1363419556375932405_nat_o @ A ) ) ) ) ).
% 14.25/8.09
% 14.25/8.09 thf(fact_4633_card__Diff1__less,axiom,
% 14.25/8.09 ! [A: set_nat,X: nat] :
% 14.25/8.09 ( ( finite_finite_nat @ A )
% 14.25/8.09 => ( ( member_nat @ X @ A )
% 14.25/8.09 => ( ord_less_nat @ ( finite_card_nat @ ( minus_minus_set_nat @ A @ ( insert_nat @ X @ bot_bot_set_nat ) ) ) @ ( finite_card_nat @ A ) ) ) ) ).
% 14.25/8.09
% 14.25/8.09 thf(fact_4634_card__Diff2__less,axiom,
% 14.25/8.09 ! [A: set_Code_integer,X: code_integer,Y: code_integer] :
% 14.25/8.09 ( ( finite6017078050557962740nteger @ A )
% 14.25/8.09 => ( ( member_Code_integer @ X @ A )
% 14.25/8.09 => ( ( member_Code_integer @ Y @ A )
% 14.25/8.09 => ( ord_less_nat @ ( finite4902975817058060853nteger @ ( minus_2355218937544613996nteger @ ( minus_2355218937544613996nteger @ A @ ( insert_Code_integer @ X @ bot_bo3990330152332043303nteger ) ) @ ( insert_Code_integer @ Y @ bot_bo3990330152332043303nteger ) ) ) @ ( finite4902975817058060853nteger @ A ) ) ) ) ) ).
% 14.25/8.09
% 14.25/8.09 thf(fact_4635_card__Diff2__less,axiom,
% 14.25/8.09 ! [A: set_Product_unit,X: product_unit,Y: product_unit] :
% 14.25/8.09 ( ( finite4290736615968046902t_unit @ A )
% 14.25/8.09 => ( ( member_Product_unit @ X @ A )
% 14.25/8.09 => ( ( member_Product_unit @ Y @ A )
% 14.25/8.09 => ( ord_less_nat @ ( finite410649719033368117t_unit @ ( minus_6452836326544984404t_unit @ ( minus_6452836326544984404t_unit @ A @ ( insert_Product_unit @ X @ bot_bo3957492148770167129t_unit ) ) @ ( insert_Product_unit @ Y @ bot_bo3957492148770167129t_unit ) ) ) @ ( finite410649719033368117t_unit @ A ) ) ) ) ) ).
% 14.25/8.09
% 14.25/8.09 thf(fact_4636_card__Diff2__less,axiom,
% 14.25/8.09 ! [A: set_o,X: $o,Y: $o] :
% 14.25/8.09 ( ( finite_finite_o @ A )
% 14.25/8.09 => ( ( member_o @ X @ A )
% 14.25/8.09 => ( ( member_o @ Y @ A )
% 14.25/8.09 => ( ord_less_nat @ ( finite_card_o @ ( minus_minus_set_o @ ( minus_minus_set_o @ A @ ( insert_o @ X @ bot_bot_set_o ) ) @ ( insert_o @ Y @ bot_bot_set_o ) ) ) @ ( finite_card_o @ A ) ) ) ) ) ).
% 14.25/8.09
% 14.25/8.09 thf(fact_4637_card__Diff2__less,axiom,
% 14.25/8.09 ! [A: set_int,X: int,Y: int] :
% 14.25/8.09 ( ( finite_finite_int @ A )
% 14.25/8.09 => ( ( member_int @ X @ A )
% 14.25/8.09 => ( ( member_int @ Y @ A )
% 14.25/8.09 => ( ord_less_nat @ ( finite_card_int @ ( minus_minus_set_int @ ( minus_minus_set_int @ A @ ( insert_int @ X @ bot_bot_set_int ) ) @ ( insert_int @ Y @ bot_bot_set_int ) ) ) @ ( finite_card_int @ A ) ) ) ) ) ).
% 14.25/8.09
% 14.25/8.09 thf(fact_4638_card__Diff2__less,axiom,
% 14.25/8.09 ! [A: set_Pr4532377907799695533_nat_o,X: produc3658429121746597890et_nat > $o,Y: produc3658429121746597890et_nat > $o] :
% 14.25/8.09 ( ( finite3252695134891459830_nat_o @ A )
% 14.25/8.09 => ( ( member6576561426505652726_nat_o @ X @ A )
% 14.25/8.09 => ( ( member6576561426505652726_nat_o @ Y @ A )
% 14.25/8.09 => ( ord_less_nat @ ( finite1363419556375932405_nat_o @ ( minus_1801376950450012436_nat_o @ ( minus_1801376950450012436_nat_o @ A @ ( insert5175938949040314269_nat_o @ X @ bot_bo7824918357723582233_nat_o ) ) @ ( insert5175938949040314269_nat_o @ Y @ bot_bo7824918357723582233_nat_o ) ) ) @ ( finite1363419556375932405_nat_o @ A ) ) ) ) ) ).
% 14.25/8.09
% 14.25/8.09 thf(fact_4639_card__Diff2__less,axiom,
% 14.25/8.09 ! [A: set_nat,X: nat,Y: nat] :
% 14.25/8.09 ( ( finite_finite_nat @ A )
% 14.25/8.09 => ( ( member_nat @ X @ A )
% 14.25/8.09 => ( ( member_nat @ Y @ A )
% 14.25/8.09 => ( ord_less_nat @ ( finite_card_nat @ ( minus_minus_set_nat @ ( minus_minus_set_nat @ A @ ( insert_nat @ X @ bot_bot_set_nat ) ) @ ( insert_nat @ Y @ bot_bot_set_nat ) ) ) @ ( finite_card_nat @ A ) ) ) ) ) ).
% 14.25/8.09
% 14.25/8.09 thf(fact_4640_card__Diff1__less__iff,axiom,
% 14.25/8.09 ! [A: set_Code_integer,X: code_integer] :
% 14.25/8.09 ( ( ord_less_nat @ ( finite4902975817058060853nteger @ ( minus_2355218937544613996nteger @ A @ ( insert_Code_integer @ X @ bot_bo3990330152332043303nteger ) ) ) @ ( finite4902975817058060853nteger @ A ) )
% 14.25/8.09 = ( ( finite6017078050557962740nteger @ A )
% 14.25/8.09 & ( member_Code_integer @ X @ A ) ) ) ).
% 14.25/8.09
% 14.25/8.09 thf(fact_4641_card__Diff1__less__iff,axiom,
% 14.25/8.09 ! [A: set_Product_unit,X: product_unit] :
% 14.25/8.09 ( ( ord_less_nat @ ( finite410649719033368117t_unit @ ( minus_6452836326544984404t_unit @ A @ ( insert_Product_unit @ X @ bot_bo3957492148770167129t_unit ) ) ) @ ( finite410649719033368117t_unit @ A ) )
% 14.25/8.09 = ( ( finite4290736615968046902t_unit @ A )
% 14.25/8.09 & ( member_Product_unit @ X @ A ) ) ) ).
% 14.25/8.09
% 14.25/8.09 thf(fact_4642_card__Diff1__less__iff,axiom,
% 14.25/8.09 ! [A: set_o,X: $o] :
% 14.25/8.09 ( ( ord_less_nat @ ( finite_card_o @ ( minus_minus_set_o @ A @ ( insert_o @ X @ bot_bot_set_o ) ) ) @ ( finite_card_o @ A ) )
% 14.25/8.09 = ( ( finite_finite_o @ A )
% 14.25/8.09 & ( member_o @ X @ A ) ) ) ).
% 14.25/8.09
% 14.25/8.09 thf(fact_4643_card__Diff1__less__iff,axiom,
% 14.25/8.09 ! [A: set_int,X: int] :
% 14.25/8.09 ( ( ord_less_nat @ ( finite_card_int @ ( minus_minus_set_int @ A @ ( insert_int @ X @ bot_bot_set_int ) ) ) @ ( finite_card_int @ A ) )
% 14.25/8.09 = ( ( finite_finite_int @ A )
% 14.25/8.09 & ( member_int @ X @ A ) ) ) ).
% 14.25/8.09
% 14.25/8.09 thf(fact_4644_card__Diff1__less__iff,axiom,
% 14.25/8.09 ! [A: set_Pr4532377907799695533_nat_o,X: produc3658429121746597890et_nat > $o] :
% 14.25/8.09 ( ( ord_less_nat @ ( finite1363419556375932405_nat_o @ ( minus_1801376950450012436_nat_o @ A @ ( insert5175938949040314269_nat_o @ X @ bot_bo7824918357723582233_nat_o ) ) ) @ ( finite1363419556375932405_nat_o @ A ) )
% 14.25/8.09 = ( ( finite3252695134891459830_nat_o @ A )
% 14.25/8.09 & ( member6576561426505652726_nat_o @ X @ A ) ) ) ).
% 14.25/8.09
% 14.25/8.09 thf(fact_4645_card__Diff1__less__iff,axiom,
% 14.25/8.09 ! [A: set_nat,X: nat] :
% 14.25/8.09 ( ( ord_less_nat @ ( finite_card_nat @ ( minus_minus_set_nat @ A @ ( insert_nat @ X @ bot_bot_set_nat ) ) ) @ ( finite_card_nat @ A ) )
% 14.25/8.09 = ( ( finite_finite_nat @ A )
% 14.25/8.09 & ( member_nat @ X @ A ) ) ) ).
% 14.25/8.09
% 14.25/8.09 thf(fact_4646_Inter__in__chain,axiom,
% 14.25/8.09 ! [B9: set_set_nat,A11: set_set_nat] :
% 14.25/8.09 ( ( finite1152437895449049373et_nat @ B9 )
% 14.25/8.09 => ( ( B9 != bot_bot_set_set_nat )
% 14.25/8.09 => ( ( pred_chain_set_nat @ A11 @ ord_less_set_nat @ B9 )
% 14.25/8.09 => ( member_set_nat @ ( comple7806235888213564991et_nat @ B9 ) @ B9 ) ) ) ) ).
% 14.25/8.09
% 14.25/8.09 thf(fact_4647_card__eq__UNIV2,axiom,
% 14.25/8.09 ! [S: set_Product_unit] :
% 14.25/8.09 ( ( ( finite410649719033368117t_unit @ top_to1996260823553986621t_unit )
% 14.25/8.09 = ( finite410649719033368117t_unit @ S ) )
% 14.25/8.09 = ( S = top_to1996260823553986621t_unit ) ) ).
% 14.25/8.09
% 14.25/8.09 thf(fact_4648_card__eq__UNIV2,axiom,
% 14.25/8.09 ! [S: set_o] :
% 14.25/8.09 ( ( ( finite_card_o @ top_top_set_o )
% 14.25/8.09 = ( finite_card_o @ S ) )
% 14.25/8.09 = ( S = top_top_set_o ) ) ).
% 14.25/8.09
% 14.25/8.09 thf(fact_4649_card__eq__UNIV,axiom,
% 14.25/8.09 ! [S: set_Product_unit] :
% 14.25/8.09 ( ( ( finite410649719033368117t_unit @ S )
% 14.25/8.09 = ( finite410649719033368117t_unit @ top_to1996260823553986621t_unit ) )
% 14.25/8.09 = ( S = top_to1996260823553986621t_unit ) ) ).
% 14.25/8.09
% 14.25/8.09 thf(fact_4650_card__eq__UNIV,axiom,
% 14.25/8.09 ! [S: set_o] :
% 14.25/8.09 ( ( ( finite_card_o @ S )
% 14.25/8.09 = ( finite_card_o @ top_top_set_o ) )
% 14.25/8.09 = ( S = top_top_set_o ) ) ).
% 14.25/8.09
% 14.25/8.09 thf(fact_4651_fst__map__prod,axiom,
% 14.25/8.09 ! [F: int > int,G: int > int,X: product_prod_int_int] :
% 14.25/8.09 ( ( product_fst_int_int @ ( produc6036585564866528938nt_int @ F @ G @ X ) )
% 14.25/8.09 = ( F @ ( product_fst_int_int @ X ) ) ) ).
% 14.25/8.09
% 14.25/8.09 thf(fact_4652_apfst__conv,axiom,
% 14.25/8.09 ! [F: heap_e7401611519738050253t_unit > heap_e7401611519738050253t_unit,X: heap_e7401611519738050253t_unit,Y: set_nat] :
% 14.25/8.09 ( ( produc4172003873565521026et_nat @ F @ ( produc7507926704131184380et_nat @ X @ Y ) )
% 14.25/8.09 = ( produc7507926704131184380et_nat @ ( F @ X ) @ Y ) ) ).
% 14.25/8.09
% 14.25/8.09 thf(fact_4653_apfst__conv,axiom,
% 14.25/8.09 ! [F: nat > nat,X: nat,Y: nat] :
% 14.25/8.09 ( ( produc986720760941809198at_nat @ F @ ( product_Pair_nat_nat @ X @ Y ) )
% 14.25/8.09 = ( product_Pair_nat_nat @ ( F @ X ) @ Y ) ) ).
% 14.25/8.09
% 14.25/8.09 thf(fact_4654_apfst__conv,axiom,
% 14.25/8.09 ! [F: int > int,X: int,Y: int] :
% 14.25/8.09 ( ( produc2355237562570658114nt_int @ F @ ( product_Pair_int_int @ X @ Y ) )
% 14.25/8.09 = ( product_Pair_int_int @ ( F @ X ) @ Y ) ) ).
% 14.25/8.09
% 14.25/8.09 thf(fact_4655_apfst__conv,axiom,
% 14.25/8.09 ! [F: ( produc3658429121746597890et_nat > $o ) > produc3658429121746597890et_nat > $o,X: produc3658429121746597890et_nat > $o,Y: produc3658429121746597890et_nat] :
% 14.25/8.09 ( ( produc3198230245617193214et_nat @ F @ ( produc5001842942810119800et_nat @ X @ Y ) )
% 14.25/8.09 = ( produc5001842942810119800et_nat @ ( F @ X ) @ Y ) ) ).
% 14.25/8.09
% 14.25/8.09 thf(fact_4656_apfst__conv,axiom,
% 14.25/8.09 ! [F: ( produc3658429121746597890et_nat > $o ) > produc3658429121746597890et_nat > $o,X: produc3658429121746597890et_nat > $o,Y: produc3925858234332021118et_nat] :
% 14.25/8.09 ( ( produc1442628905706551930et_nat @ F @ ( produc2245416461498447860et_nat @ X @ Y ) )
% 14.25/8.09 = ( produc2245416461498447860et_nat @ ( F @ X ) @ Y ) ) ).
% 14.25/8.09
% 14.25/8.09 thf(fact_4657_apsnd__conv,axiom,
% 14.25/8.09 ! [F: set_nat > set_nat,X: heap_e7401611519738050253t_unit,Y: set_nat] :
% 14.25/8.09 ( ( produc3286488329990770625t_unit @ F @ ( produc7507926704131184380et_nat @ X @ Y ) )
% 14.25/8.09 = ( produc7507926704131184380et_nat @ X @ ( F @ Y ) ) ) ).
% 14.25/8.09
% 14.25/8.09 thf(fact_4658_apsnd__conv,axiom,
% 14.25/8.09 ! [F: nat > nat,X: nat,Y: nat] :
% 14.25/8.09 ( ( produc3094765310956027504at_nat @ F @ ( product_Pair_nat_nat @ X @ Y ) )
% 14.25/8.09 = ( product_Pair_nat_nat @ X @ ( F @ Y ) ) ) ).
% 14.25/8.09
% 14.25/8.09 thf(fact_4659_apsnd__conv,axiom,
% 14.25/8.09 ! [F: int > int,X: int,Y: int] :
% 14.25/8.09 ( ( produc4463282112584876420nt_int @ F @ ( product_Pair_int_int @ X @ Y ) )
% 14.25/8.09 = ( product_Pair_int_int @ X @ ( F @ Y ) ) ) ).
% 14.25/8.09
% 14.25/8.09 thf(fact_4660_apsnd__conv,axiom,
% 14.25/8.09 ! [F: produc3658429121746597890et_nat > produc3658429121746597890et_nat,X: produc3658429121746597890et_nat > $o,Y: produc3658429121746597890et_nat] :
% 14.25/8.09 ( ( produc8750854537940449737_nat_o @ F @ ( produc5001842942810119800et_nat @ X @ Y ) )
% 14.25/8.09 = ( produc5001842942810119800et_nat @ X @ ( F @ Y ) ) ) ).
% 14.25/8.09
% 14.25/8.09 thf(fact_4661_apsnd__conv,axiom,
% 14.25/8.09 ! [F: produc3658429121746597890et_nat > produc3925858234332021118et_nat,X: produc3658429121746597890et_nat > $o,Y: produc3658429121746597890et_nat] :
% 14.25/8.09 ( ( produc7714581247149323085_nat_o @ F @ ( produc5001842942810119800et_nat @ X @ Y ) )
% 14.25/8.09 = ( produc2245416461498447860et_nat @ X @ ( F @ Y ) ) ) ).
% 14.25/8.09
% 14.25/8.09 thf(fact_4662_apsnd__conv,axiom,
% 14.25/8.09 ! [F: produc3925858234332021118et_nat > produc3658429121746597890et_nat,X: produc3658429121746597890et_nat > $o,Y: produc3925858234332021118et_nat] :
% 14.25/8.09 ( ( produc1515462096303866701_nat_o @ F @ ( produc2245416461498447860et_nat @ X @ Y ) )
% 14.25/8.09 = ( produc5001842942810119800et_nat @ X @ ( F @ Y ) ) ) ).
% 14.25/8.09
% 14.25/8.09 thf(fact_4663_apsnd__conv,axiom,
% 14.25/8.09 ! [F: produc3925858234332021118et_nat > produc3925858234332021118et_nat,X: produc3658429121746597890et_nat > $o,Y: produc3925858234332021118et_nat] :
% 14.25/8.09 ( ( produc969530845752564945_nat_o @ F @ ( produc2245416461498447860et_nat @ X @ Y ) )
% 14.25/8.09 = ( produc2245416461498447860et_nat @ X @ ( F @ Y ) ) ) ).
% 14.25/8.09
% 14.25/8.09 thf(fact_4664_apsnd__conv,axiom,
% 14.25/8.09 ! [F: code_integer > code_integer,X: code_integer,Y: code_integer] :
% 14.25/8.09 ( ( produc6499014454317279255nteger @ F @ ( produc1086072967326762835nteger @ X @ Y ) )
% 14.25/8.09 = ( produc1086072967326762835nteger @ X @ ( F @ Y ) ) ) ).
% 14.25/8.09
% 14.25/8.09 thf(fact_4665_fst__apfst,axiom,
% 14.25/8.09 ! [F: int > int,X: product_prod_int_int] :
% 14.25/8.09 ( ( product_fst_int_int @ ( produc2355237562570658114nt_int @ F @ X ) )
% 14.25/8.09 = ( F @ ( product_fst_int_int @ X ) ) ) ).
% 14.25/8.09
% 14.25/8.09 thf(fact_4666_snd__apfst,axiom,
% 14.25/8.09 ! [F: heap_e7401611519738050253t_unit > heap_e7401611519738050253t_unit,X: produc3658429121746597890et_nat] :
% 14.25/8.09 ( ( produc8586169260539613262et_nat @ ( produc4172003873565521026et_nat @ F @ X ) )
% 14.25/8.09 = ( produc8586169260539613262et_nat @ X ) ) ).
% 14.25/8.09
% 14.25/8.09 thf(fact_4667_snd__apfst,axiom,
% 14.25/8.09 ! [F: int > int,X: product_prod_int_int] :
% 14.25/8.09 ( ( product_snd_int_int @ ( produc2355237562570658114nt_int @ F @ X ) )
% 14.25/8.09 = ( product_snd_int_int @ X ) ) ).
% 14.25/8.09
% 14.25/8.09 thf(fact_4668_in__inv__image,axiom,
% 14.25/8.09 ! [X: nat,Y: nat,R2: set_Pr1261947904930325089at_nat,F: nat > nat] :
% 14.25/8.09 ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ Y ) @ ( inv_image_nat_nat @ R2 @ F ) )
% 14.25/8.09 = ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ ( F @ X ) @ ( F @ Y ) ) @ R2 ) ) ).
% 14.25/8.09
% 14.25/8.09 thf(fact_4669_in__inv__image,axiom,
% 14.25/8.09 ! [X: nat,Y: nat,R2: set_Pr958786334691620121nt_int,F: nat > int] :
% 14.25/8.09 ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ Y ) @ ( inv_image_int_nat @ R2 @ F ) )
% 14.25/8.09 = ( member5262025264175285858nt_int @ ( product_Pair_int_int @ ( F @ X ) @ ( F @ Y ) ) @ R2 ) ) ).
% 14.25/8.09
% 14.25/8.09 thf(fact_4670_in__inv__image,axiom,
% 14.25/8.09 ! [X: int,Y: int,R2: set_Pr1261947904930325089at_nat,F: int > nat] :
% 14.25/8.09 ( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ X @ Y ) @ ( inv_image_nat_int @ R2 @ F ) )
% 14.25/8.09 = ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ ( F @ X ) @ ( F @ Y ) ) @ R2 ) ) ).
% 14.25/8.09
% 14.25/8.09 thf(fact_4671_in__inv__image,axiom,
% 14.25/8.09 ! [X: int,Y: int,R2: set_Pr958786334691620121nt_int,F: int > int] :
% 14.25/8.09 ( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ X @ Y ) @ ( inv_image_int_int @ R2 @ F ) )
% 14.25/8.09 = ( member5262025264175285858nt_int @ ( product_Pair_int_int @ ( F @ X ) @ ( F @ Y ) ) @ R2 ) ) ).
% 14.25/8.09
% 14.25/8.09 thf(fact_4672_fst__apsnd,axiom,
% 14.25/8.09 ! [F: code_integer > code_integer,X: produc8923325533196201883nteger] :
% 14.25/8.09 ( ( produc8508995932063986495nteger @ ( produc6499014454317279255nteger @ F @ X ) )
% 14.25/8.09 = ( produc8508995932063986495nteger @ X ) ) ).
% 14.25/8.09
% 14.25/8.09 thf(fact_4673_fst__apsnd,axiom,
% 14.25/8.09 ! [F: int > int,X: product_prod_int_int] :
% 14.25/8.09 ( ( product_fst_int_int @ ( produc4463282112584876420nt_int @ F @ X ) )
% 14.25/8.09 = ( product_fst_int_int @ X ) ) ).
% 14.25/8.09
% 14.25/8.09 thf(fact_4674_apsnd__eq__conv,axiom,
% 14.25/8.09 ! [F: code_integer > code_integer,X: produc8923325533196201883nteger,G: code_integer > code_integer] :
% 14.25/8.09 ( ( ( produc6499014454317279255nteger @ F @ X )
% 14.25/8.09 = ( produc6499014454317279255nteger @ G @ X ) )
% 14.25/8.09 = ( ( F @ ( produc6174133586879617921nteger @ X ) )
% 14.25/8.09 = ( G @ ( produc6174133586879617921nteger @ X ) ) ) ) ).
% 14.25/8.09
% 14.25/8.09 thf(fact_4675_snd__apsnd,axiom,
% 14.25/8.09 ! [F: code_integer > code_integer,X: produc8923325533196201883nteger] :
% 14.25/8.09 ( ( produc6174133586879617921nteger @ ( produc6499014454317279255nteger @ F @ X ) )
% 14.25/8.09 = ( F @ ( produc6174133586879617921nteger @ X ) ) ) ).
% 14.25/8.09
% 14.25/8.09 thf(fact_4676_snd__apsnd,axiom,
% 14.25/8.09 ! [F: set_nat > set_nat,X: produc3658429121746597890et_nat] :
% 14.25/8.09 ( ( produc8586169260539613262et_nat @ ( produc3286488329990770625t_unit @ F @ X ) )
% 14.25/8.09 = ( F @ ( produc8586169260539613262et_nat @ X ) ) ) ).
% 14.25/8.09
% 14.25/8.09 thf(fact_4677_snd__apsnd,axiom,
% 14.25/8.09 ! [F: int > int,X: product_prod_int_int] :
% 14.25/8.09 ( ( product_snd_int_int @ ( produc4463282112584876420nt_int @ F @ X ) )
% 14.25/8.09 = ( F @ ( product_snd_int_int @ X ) ) ) ).
% 14.25/8.09
% 14.25/8.09 thf(fact_4678_Inf__eq__bot__iff,axiom,
% 14.25/8.09 ! [A: set_Product_unit] :
% 14.25/8.09 ( ( ( comple2584293577114468500t_unit @ A )
% 14.25/8.09 = bot_bot_Product_unit )
% 14.25/8.09 = ( ! [X2: product_unit] :
% 14.25/8.09 ( ( ord_le361264281704409273t_unit @ bot_bot_Product_unit @ X2 )
% 14.25/8.09 => ? [Y3: product_unit] :
% 14.25/8.09 ( ( member_Product_unit @ Y3 @ A )
% 14.25/8.09 & ( ord_le361264281704409273t_unit @ Y3 @ X2 ) ) ) ) ) ).
% 14.25/8.09
% 14.25/8.09 thf(fact_4679_Inf__empty,axiom,
% 14.25/8.09 ( ( complete_Inf_Inf_o @ bot_bot_set_o )
% 14.25/8.09 = top_top_o ) ).
% 14.25/8.09
% 14.25/8.09 thf(fact_4680_Inf__empty,axiom,
% 14.25/8.09 ( ( comple90455603224345172_nat_o @ bot_bo7824918357723582233_nat_o )
% 14.25/8.09 = top_to5213382387105117981_nat_o ) ).
% 14.25/8.09
% 14.25/8.09 thf(fact_4681_Inf__empty,axiom,
% 14.25/8.09 ( ( comple7216939928495492468t_unit @ bot_bo6284674557190688313t_unit )
% 14.25/8.09 = top_to1996260823553986621t_unit ) ).
% 14.25/8.09
% 14.25/8.09 thf(fact_4682_Inf__empty,axiom,
% 14.25/8.09 ( ( comple3063163877087187839_set_o @ bot_bot_set_set_o )
% 14.25/8.09 = top_top_set_o ) ).
% 14.25/8.09
% 14.25/8.09 thf(fact_4683_Inf__empty,axiom,
% 14.25/8.09 ( ( comple3628384868704368283et_int @ bot_bot_set_set_int )
% 14.25/8.09 = top_top_set_int ) ).
% 14.25/8.09
% 14.25/8.09 thf(fact_4684_Inf__empty,axiom,
% 14.25/8.09 ( ( comple7806235888213564991et_nat @ bot_bot_set_set_nat )
% 14.25/8.09 = top_top_set_nat ) ).
% 14.25/8.09
% 14.25/8.09 thf(fact_4685_Inf__empty,axiom,
% 14.25/8.09 ( ( comple2584293577114468500t_unit @ bot_bo3957492148770167129t_unit )
% 14.25/8.09 = top_top_Product_unit ) ).
% 14.25/8.09
% 14.25/8.09 thf(fact_4686_Inf__UNIV,axiom,
% 14.25/8.09 ( ( comple7216939928495492468t_unit @ top_to1767297665138865437t_unit )
% 14.25/8.09 = bot_bo3957492148770167129t_unit ) ).
% 14.25/8.09
% 14.25/8.09 thf(fact_4687_Inf__UNIV,axiom,
% 14.25/8.09 ( ( comple3063163877087187839_set_o @ top_top_set_set_o )
% 14.25/8.09 = bot_bot_set_o ) ).
% 14.25/8.09
% 14.25/8.09 thf(fact_4688_Inf__UNIV,axiom,
% 14.25/8.09 ( ( comple3628384868704368283et_int @ top_top_set_set_int )
% 14.25/8.09 = bot_bot_set_int ) ).
% 14.25/8.09
% 14.25/8.09 thf(fact_4689_Inf__UNIV,axiom,
% 14.25/8.09 ( ( comple79274577958421812_nat_o @ top_to8515903933492314333_nat_o )
% 14.25/8.09 = bot_bo7824918357723582233_nat_o ) ).
% 14.25/8.09
% 14.25/8.09 thf(fact_4690_Inf__UNIV,axiom,
% 14.25/8.09 ( ( complete_Inf_Inf_o @ top_top_set_o )
% 14.25/8.09 = bot_bot_o ) ).
% 14.25/8.09
% 14.25/8.09 thf(fact_4691_Inf__UNIV,axiom,
% 14.25/8.09 ( ( comple7806235888213564991et_nat @ top_top_set_set_nat )
% 14.25/8.09 = bot_bot_set_nat ) ).
% 14.25/8.09
% 14.25/8.09 thf(fact_4692_Inf__UNIV,axiom,
% 14.25/8.09 ( ( comple2584293577114468500t_unit @ top_to1996260823553986621t_unit )
% 14.25/8.09 = bot_bot_Product_unit ) ).
% 14.25/8.09
% 14.25/8.09 thf(fact_4693_cInf__singleton,axiom,
% 14.25/8.09 ! [X: $o] :
% 14.25/8.09 ( ( complete_Inf_Inf_o @ ( insert_o @ X @ bot_bot_set_o ) )
% 14.25/8.09 = X ) ).
% 14.25/8.09
% 14.25/8.09 thf(fact_4694_cInf__singleton,axiom,
% 14.25/8.09 ! [X: int] :
% 14.25/8.09 ( ( complete_Inf_Inf_int @ ( insert_int @ X @ bot_bot_set_int ) )
% 14.25/8.09 = X ) ).
% 14.25/8.09
% 14.25/8.09 thf(fact_4695_cInf__singleton,axiom,
% 14.25/8.09 ! [X: produc3658429121746597890et_nat > $o] :
% 14.25/8.09 ( ( comple90455603224345172_nat_o @ ( insert5175938949040314269_nat_o @ X @ bot_bo7824918357723582233_nat_o ) )
% 14.25/8.09 = X ) ).
% 14.25/8.09
% 14.25/8.09 thf(fact_4696_cInf__singleton,axiom,
% 14.25/8.09 ! [X: nat] :
% 14.25/8.09 ( ( complete_Inf_Inf_nat @ ( insert_nat @ X @ bot_bot_set_nat ) )
% 14.25/8.09 = X ) ).
% 14.25/8.09
% 14.25/8.09 thf(fact_4697_cInf__singleton,axiom,
% 14.25/8.09 ! [X: set_nat] :
% 14.25/8.09 ( ( comple7806235888213564991et_nat @ ( insert_set_nat @ X @ bot_bot_set_set_nat ) )
% 14.25/8.09 = X ) ).
% 14.25/8.09
% 14.25/8.09 thf(fact_4698_cInf__singleton,axiom,
% 14.25/8.09 ! [X: product_unit] :
% 14.25/8.09 ( ( comple2584293577114468500t_unit @ ( insert_Product_unit @ X @ bot_bo3957492148770167129t_unit ) )
% 14.25/8.09 = X ) ).
% 14.25/8.09
% 14.25/8.09 thf(fact_4699_Inf__insert,axiom,
% 14.25/8.09 ! [A2: $o,A: set_o] :
% 14.25/8.09 ( ( complete_Inf_Inf_o @ ( insert_o @ A2 @ A ) )
% 14.25/8.09 = ( inf_inf_o @ A2 @ ( complete_Inf_Inf_o @ A ) ) ) ).
% 14.25/8.09
% 14.25/8.09 thf(fact_4700_Inf__insert,axiom,
% 14.25/8.09 ! [A2: set_nat,A: set_set_nat] :
% 14.25/8.09 ( ( comple7806235888213564991et_nat @ ( insert_set_nat @ A2 @ A ) )
% 14.25/8.09 = ( inf_inf_set_nat @ A2 @ ( comple7806235888213564991et_nat @ A ) ) ) ).
% 14.25/8.09
% 14.25/8.09 thf(fact_4701_Inf__insert,axiom,
% 14.25/8.09 ! [A2: product_unit,A: set_Product_unit] :
% 14.25/8.09 ( ( comple2584293577114468500t_unit @ ( insert_Product_unit @ A2 @ A ) )
% 14.25/8.09 = ( inf_inf_Product_unit @ A2 @ ( comple2584293577114468500t_unit @ A ) ) ) ).
% 14.25/8.09
% 14.25/8.09 thf(fact_4702_prod_Ocollapse,axiom,
% 14.25/8.09 ! [Prod: product_prod_nat_nat] :
% 14.25/8.09 ( ( product_Pair_nat_nat @ ( product_fst_nat_nat @ Prod ) @ ( product_snd_nat_nat @ Prod ) )
% 14.25/8.09 = Prod ) ).
% 14.25/8.09
% 14.25/8.09 thf(fact_4703_prod_Ocollapse,axiom,
% 14.25/8.09 ! [Prod: produc3925858234332021118et_nat] :
% 14.25/8.09 ( ( produc5001842942810119800et_nat @ ( produc995936583742144908et_nat @ Prod ) @ ( produc4011572625026189258et_nat @ Prod ) )
% 14.25/8.09 = Prod ) ).
% 14.25/8.09
% 14.25/8.09 thf(fact_4704_prod_Ocollapse,axiom,
% 14.25/8.09 ! [Prod: produc2732055786443039994et_nat] :
% 14.25/8.09 ( ( produc2245416461498447860et_nat @ ( produc180342877477747464et_nat @ Prod ) @ ( produc5374455773327741254et_nat @ Prod ) )
% 14.25/8.09 = Prod ) ).
% 14.25/8.09
% 14.25/8.09 thf(fact_4705_prod_Ocollapse,axiom,
% 14.25/8.09 ! [Prod: produc3658429121746597890et_nat] :
% 14.25/8.09 ( ( produc7507926704131184380et_nat @ ( produc1824681642469235216et_nat @ Prod ) @ ( produc8586169260539613262et_nat @ Prod ) )
% 14.25/8.09 = Prod ) ).
% 14.25/8.09
% 14.25/8.09 thf(fact_4706_prod_Ocollapse,axiom,
% 14.25/8.09 ! [Prod: product_prod_int_int] :
% 14.25/8.09 ( ( product_Pair_int_int @ ( product_fst_int_int @ Prod ) @ ( product_snd_int_int @ Prod ) )
% 14.25/8.09 = Prod ) ).
% 14.25/8.09
% 14.25/8.09 thf(fact_4707_card__ge__UNIV,axiom,
% 14.25/8.09 ! [S: set_Product_unit] :
% 14.25/8.09 ( ( ord_less_eq_nat @ ( finite410649719033368117t_unit @ top_to1996260823553986621t_unit ) @ ( finite410649719033368117t_unit @ S ) )
% 14.25/8.09 = ( S = top_to1996260823553986621t_unit ) ) ).
% 14.25/8.09
% 14.25/8.09 thf(fact_4708_card__ge__UNIV,axiom,
% 14.25/8.09 ! [S: set_o] :
% 14.25/8.09 ( ( ord_less_eq_nat @ ( finite_card_o @ top_top_set_o ) @ ( finite_card_o @ S ) )
% 14.25/8.09 = ( S = top_top_set_o ) ) ).
% 14.25/8.09
% 14.25/8.09 thf(fact_4709_card__Diff__insert,axiom,
% 14.25/8.09 ! [A2: produc3658429121746597890et_nat > $o,A: set_Pr4532377907799695533_nat_o,B4: set_Pr4532377907799695533_nat_o] :
% 14.25/8.09 ( ( member6576561426505652726_nat_o @ A2 @ A )
% 14.25/8.09 => ( ~ ( member6576561426505652726_nat_o @ A2 @ B4 )
% 14.25/8.09 => ( ( finite1363419556375932405_nat_o @ ( minus_1801376950450012436_nat_o @ A @ ( insert5175938949040314269_nat_o @ A2 @ B4 ) ) )
% 14.25/8.09 = ( minus_minus_nat @ ( finite1363419556375932405_nat_o @ ( minus_1801376950450012436_nat_o @ A @ B4 ) ) @ one_one_nat ) ) ) ) ).
% 14.25/8.09
% 14.25/8.09 thf(fact_4710_card__Diff__insert,axiom,
% 14.25/8.09 ! [A2: int,A: set_int,B4: set_int] :
% 14.25/8.09 ( ( member_int @ A2 @ A )
% 14.25/8.09 => ( ~ ( member_int @ A2 @ B4 )
% 14.25/8.09 => ( ( finite_card_int @ ( minus_minus_set_int @ A @ ( insert_int @ A2 @ B4 ) ) )
% 14.25/8.09 = ( minus_minus_nat @ ( finite_card_int @ ( minus_minus_set_int @ A @ B4 ) ) @ one_one_nat ) ) ) ) ).
% 14.25/8.09
% 14.25/8.09 thf(fact_4711_card__Diff__insert,axiom,
% 14.25/8.09 ! [A2: $o,A: set_o,B4: set_o] :
% 14.25/8.09 ( ( member_o @ A2 @ A )
% 14.25/8.09 => ( ~ ( member_o @ A2 @ B4 )
% 14.25/8.09 => ( ( finite_card_o @ ( minus_minus_set_o @ A @ ( insert_o @ A2 @ B4 ) ) )
% 14.25/8.09 = ( minus_minus_nat @ ( finite_card_o @ ( minus_minus_set_o @ A @ B4 ) ) @ one_one_nat ) ) ) ) ).
% 14.25/8.09
% 14.25/8.09 thf(fact_4712_card__Diff__insert,axiom,
% 14.25/8.09 ! [A2: product_unit,A: set_Product_unit,B4: set_Product_unit] :
% 14.25/8.09 ( ( member_Product_unit @ A2 @ A )
% 14.25/8.09 => ( ~ ( member_Product_unit @ A2 @ B4 )
% 14.25/8.09 => ( ( finite410649719033368117t_unit @ ( minus_6452836326544984404t_unit @ A @ ( insert_Product_unit @ A2 @ B4 ) ) )
% 14.25/8.09 = ( minus_minus_nat @ ( finite410649719033368117t_unit @ ( minus_6452836326544984404t_unit @ A @ B4 ) ) @ one_one_nat ) ) ) ) ).
% 14.25/8.09
% 14.25/8.09 thf(fact_4713_card__Diff__insert,axiom,
% 14.25/8.09 ! [A2: nat,A: set_nat,B4: set_nat] :
% 14.25/8.09 ( ( member_nat @ A2 @ A )
% 14.25/8.09 => ( ~ ( member_nat @ A2 @ B4 )
% 14.25/8.09 => ( ( finite_card_nat @ ( minus_minus_set_nat @ A @ ( insert_nat @ A2 @ B4 ) ) )
% 14.25/8.09 = ( minus_minus_nat @ ( finite_card_nat @ ( minus_minus_set_nat @ A @ B4 ) ) @ one_one_nat ) ) ) ) ).
% 14.25/8.09
% 14.25/8.09 thf(fact_4714_img__fst,axiom,
% 14.25/8.09 ! [A2: heap_e7401611519738050253t_unit,B: set_nat,S: set_Pr3948176798113811640et_nat] :
% 14.25/8.09 ( ( member6260224972018164377et_nat @ ( produc7507926704131184380et_nat @ A2 @ B ) @ S )
% 14.25/8.09 => ( member242389114213480694t_unit @ A2 @ ( image_6824039242672219652t_unit @ produc1824681642469235216et_nat @ S ) ) ) ).
% 14.25/8.09
% 14.25/8.09 thf(fact_4715_img__fst,axiom,
% 14.25/8.09 ! [A2: nat,B: nat,S: set_Pr1261947904930325089at_nat] :
% 14.25/8.09 ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A2 @ B ) @ S )
% 14.25/8.09 => ( member_nat @ A2 @ ( image_2486076414777270412at_nat @ product_fst_nat_nat @ S ) ) ) ).
% 14.25/8.09
% 14.25/8.09 thf(fact_4716_img__fst,axiom,
% 14.25/8.09 ! [A2: produc3658429121746597890et_nat > $o,B: produc3658429121746597890et_nat,S: set_Pr3286484037609594932et_nat] :
% 14.25/8.09 ( ( member1996754912294343701et_nat @ ( produc5001842942810119800et_nat @ A2 @ B ) @ S )
% 14.25/8.09 => ( member6576561426505652726_nat_o @ A2 @ ( image_6802380912191114376_nat_o @ produc995936583742144908et_nat @ S ) ) ) ).
% 14.25/8.09
% 14.25/8.09 thf(fact_4717_img__fst,axiom,
% 14.25/8.09 ! [A2: produc3658429121746597890et_nat > $o,B: produc3925858234332021118et_nat,S: set_Pr8536935166611901872et_nat] :
% 14.25/8.09 ( ( member6124377750444531601et_nat @ ( produc2245416461498447860et_nat @ A2 @ B ) @ S )
% 14.25/8.09 => ( member6576561426505652726_nat_o @ A2 @ ( image_2517677226577294348_nat_o @ produc180342877477747464et_nat @ S ) ) ) ).
% 14.25/8.09
% 14.25/8.09 thf(fact_4718_img__fst,axiom,
% 14.25/8.09 ! [A2: int,B: int,S: set_Pr958786334691620121nt_int] :
% 14.25/8.09 ( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ A2 @ B ) @ S )
% 14.25/8.09 => ( member_int @ A2 @ ( image_5042161079198086560nt_int @ product_fst_int_int @ S ) ) ) ).
% 14.25/8.09
% 14.25/8.09 thf(fact_4719_fst__swap,axiom,
% 14.25/8.09 ! [X: produc3658429121746597890et_nat] :
% 14.25/8.09 ( ( produc6012101397789842562t_unit @ ( produc6300697153377358172et_nat @ X ) )
% 14.25/8.09 = ( produc8586169260539613262et_nat @ X ) ) ).
% 14.25/8.09
% 14.25/8.09 thf(fact_4720_fst__swap,axiom,
% 14.25/8.09 ! [X: product_prod_int_int] :
% 14.25/8.09 ( ( product_fst_int_int @ ( product_swap_int_int @ X ) )
% 14.25/8.09 = ( product_snd_int_int @ X ) ) ).
% 14.25/8.09
% 14.25/8.09 thf(fact_4721_snd__swap,axiom,
% 14.25/8.09 ! [X: produc7348995623955488188t_unit] :
% 14.25/8.09 ( ( produc8586169260539613262et_nat @ ( produc1264744871843189710t_unit @ X ) )
% 14.25/8.09 = ( produc6012101397789842562t_unit @ X ) ) ).
% 14.25/8.09
% 14.25/8.09 thf(fact_4722_snd__swap,axiom,
% 14.25/8.09 ! [X: product_prod_int_int] :
% 14.25/8.09 ( ( product_snd_int_int @ ( product_swap_int_int @ X ) )
% 14.25/8.09 = ( product_fst_int_int @ X ) ) ).
% 14.25/8.09
% 14.25/8.09 thf(fact_4723_snd__comp__apfst,axiom,
% 14.25/8.09 ! [F: heap_e7401611519738050253t_unit > heap_e7401611519738050253t_unit] :
% 14.25/8.09 ( ( comp_P2930521774885306735et_nat @ produc8586169260539613262et_nat @ ( produc4172003873565521026et_nat @ F ) )
% 14.25/8.09 = produc8586169260539613262et_nat ) ).
% 14.25/8.09
% 14.25/8.09 thf(fact_4724_snd__comp__apfst,axiom,
% 14.25/8.09 ! [F: int > int] :
% 14.25/8.09 ( ( comp_P5356232565778667509nt_int @ product_snd_int_int @ ( produc2355237562570658114nt_int @ F ) )
% 14.25/8.09 = product_snd_int_int ) ).
% 14.25/8.09
% 14.25/8.09 thf(fact_4725_fst__comp__apsnd,axiom,
% 14.25/8.09 ! [F: code_integer > code_integer] :
% 14.25/8.09 ( ( comp_P3428181540658402820nteger @ produc8508995932063986495nteger @ ( produc6499014454317279255nteger @ F ) )
% 14.25/8.09 = produc8508995932063986495nteger ) ).
% 14.25/8.09
% 14.25/8.09 thf(fact_4726_fst__comp__apsnd,axiom,
% 14.25/8.09 ! [F: int > int] :
% 14.25/8.09 ( ( comp_P5356232565778667509nt_int @ product_fst_int_int @ ( produc4463282112584876420nt_int @ F ) )
% 14.25/8.09 = product_fst_int_int ) ).
% 14.25/8.09
% 14.25/8.09 thf(fact_4727_range__fst,axiom,
% 14.25/8.09 ( ( image_5042161079198086560nt_int @ product_fst_int_int @ top_to4366644338036079209nt_int )
% 14.25/8.09 = top_top_set_int ) ).
% 14.25/8.09
% 14.25/8.09 thf(fact_4728_fst__comp__map__prod,axiom,
% 14.25/8.09 ! [F: int > int,G: int > int] :
% 14.25/8.09 ( ( comp_P5356232565778667509nt_int @ product_fst_int_int @ ( produc6036585564866528938nt_int @ F @ G ) )
% 14.25/8.09 = ( comp_i4330622015127193916nt_int @ F @ product_fst_int_int ) ) ).
% 14.25/8.09
% 14.25/8.09 thf(fact_4729_fst__comp__apfst,axiom,
% 14.25/8.09 ! [F: int > int] :
% 14.25/8.09 ( ( comp_P5356232565778667509nt_int @ product_fst_int_int @ ( produc2355237562570658114nt_int @ F ) )
% 14.25/8.09 = ( comp_i4330622015127193916nt_int @ F @ product_fst_int_int ) ) ).
% 14.25/8.09
% 14.25/8.09 thf(fact_4730_snd__comp__apsnd,axiom,
% 14.25/8.09 ! [F: code_integer > code_integer] :
% 14.25/8.09 ( ( comp_P3428181540658402820nteger @ produc6174133586879617921nteger @ ( produc6499014454317279255nteger @ F ) )
% 14.25/8.09 = ( comp_C1043215000170864188nteger @ F @ produc6174133586879617921nteger ) ) ).
% 14.25/8.09
% 14.25/8.09 thf(fact_4731_snd__comp__apsnd,axiom,
% 14.25/8.09 ! [F: set_nat > set_nat] :
% 14.25/8.09 ( ( comp_P2930521774885306735et_nat @ produc8586169260539613262et_nat @ ( produc3286488329990770625t_unit @ F ) )
% 14.25/8.09 = ( comp_s5696514601250091763et_nat @ F @ produc8586169260539613262et_nat ) ) ).
% 14.25/8.09
% 14.25/8.09 thf(fact_4732_snd__comp__apsnd,axiom,
% 14.25/8.09 ! [F: int > int] :
% 14.25/8.09 ( ( comp_P5356232565778667509nt_int @ product_snd_int_int @ ( produc4463282112584876420nt_int @ F ) )
% 14.25/8.09 = ( comp_i4330622015127193916nt_int @ F @ product_snd_int_int ) ) ).
% 14.25/8.09
% 14.25/8.09 thf(fact_4733_apfst__apsnd,axiom,
% 14.25/8.09 ! [F: heap_e7401611519738050253t_unit > heap_e7401611519738050253t_unit,G: set_nat > set_nat,X: produc3658429121746597890et_nat] :
% 14.25/8.09 ( ( produc4172003873565521026et_nat @ F @ ( produc3286488329990770625t_unit @ G @ X ) )
% 14.25/8.09 = ( produc7507926704131184380et_nat @ ( F @ ( produc1824681642469235216et_nat @ X ) ) @ ( G @ ( produc8586169260539613262et_nat @ X ) ) ) ) ).
% 14.25/8.09
% 14.25/8.09 thf(fact_4734_apfst__apsnd,axiom,
% 14.25/8.09 ! [F: heap_e7401611519738050253t_unit > nat,G: set_nat > nat,X: produc3658429121746597890et_nat] :
% 14.25/8.09 ( ( produc8233931525780029747at_nat @ F @ ( produc8442695800884449271t_unit @ G @ X ) )
% 14.25/8.09 = ( product_Pair_nat_nat @ ( F @ ( produc1824681642469235216et_nat @ X ) ) @ ( G @ ( produc8586169260539613262et_nat @ X ) ) ) ) ).
% 14.25/8.09
% 14.25/8.09 thf(fact_4735_apfst__apsnd,axiom,
% 14.25/8.09 ! [F: heap_e7401611519738050253t_unit > int,G: set_nat > int,X: produc3658429121746597890et_nat] :
% 14.25/8.09 ( ( produc7422730317258219nt_int @ F @ ( produc7345612820240450715t_unit @ G @ X ) )
% 14.25/8.09 = ( product_Pair_int_int @ ( F @ ( produc1824681642469235216et_nat @ X ) ) @ ( G @ ( produc8586169260539613262et_nat @ X ) ) ) ) ).
% 14.25/8.09
% 14.25/8.09 thf(fact_4736_apfst__apsnd,axiom,
% 14.25/8.09 ! [F: heap_e7401611519738050253t_unit > produc3658429121746597890et_nat > $o,G: set_nat > produc3658429121746597890et_nat,X: produc3658429121746597890et_nat] :
% 14.25/8.09 ( ( produc6095824665904122878et_nat @ F @ ( produc7441988914112407621t_unit @ G @ X ) )
% 14.25/8.09 = ( produc5001842942810119800et_nat @ ( F @ ( produc1824681642469235216et_nat @ X ) ) @ ( G @ ( produc8586169260539613262et_nat @ X ) ) ) ) ).
% 14.25/8.09
% 14.25/8.09 thf(fact_4737_apfst__apsnd,axiom,
% 14.25/8.09 ! [F: heap_e7401611519738050253t_unit > produc3658429121746597890et_nat > $o,G: set_nat > produc3925858234332021118et_nat,X: produc3658429121746597890et_nat] :
% 14.25/8.09 ( ( produc4273547285574235514et_nat @ F @ ( produc7176435795764513225t_unit @ G @ X ) )
% 14.25/8.09 = ( produc2245416461498447860et_nat @ ( F @ ( produc1824681642469235216et_nat @ X ) ) @ ( G @ ( produc8586169260539613262et_nat @ X ) ) ) ) ).
% 14.25/8.09
% 14.25/8.09 thf(fact_4738_apfst__apsnd,axiom,
% 14.25/8.09 ! [F: int > heap_e7401611519738050253t_unit,G: int > set_nat,X: product_prod_int_int] :
% 14.25/8.09 ( ( produc7086134074927288299et_nat @ F @ ( produc1253519502038700766at_int @ G @ X ) )
% 14.25/8.09 = ( produc7507926704131184380et_nat @ ( F @ ( product_fst_int_int @ X ) ) @ ( G @ ( product_snd_int_int @ X ) ) ) ) ).
% 14.25/8.09
% 14.25/8.09 thf(fact_4739_apfst__apsnd,axiom,
% 14.25/8.09 ! [F: int > nat,G: int > nat,X: product_prod_int_int] :
% 14.25/8.09 ( ( produc1358374321178653834at_nat @ F @ ( produc3463928400683821864at_int @ G @ X ) )
% 14.25/8.09 = ( product_Pair_nat_nat @ ( F @ ( product_fst_int_int @ X ) ) @ ( G @ ( product_snd_int_int @ X ) ) ) ) ).
% 14.25/8.09
% 14.25/8.09 thf(fact_4740_apfst__apsnd,axiom,
% 14.25/8.09 ! [F: int > int,G: int > int,X: product_prod_int_int] :
% 14.25/8.09 ( ( produc2355237562570658114nt_int @ F @ ( produc4463282112584876420nt_int @ G @ X ) )
% 14.25/8.09 = ( product_Pair_int_int @ ( F @ ( product_fst_int_int @ X ) ) @ ( G @ ( product_snd_int_int @ X ) ) ) ) ).
% 14.25/8.09
% 14.25/8.09 thf(fact_4741_apfst__apsnd,axiom,
% 14.25/8.09 ! [F: int > produc3658429121746597890et_nat > $o,G: int > produc3658429121746597890et_nat,X: product_prod_int_int] :
% 14.25/8.09 ( ( produc5259063908904861543et_nat @ F @ ( produc9121118195053341018at_int @ G @ X ) )
% 14.25/8.09 = ( produc5001842942810119800et_nat @ ( F @ ( product_fst_int_int @ X ) ) @ ( G @ ( product_snd_int_int @ X ) ) ) ) ).
% 14.25/8.09
% 14.25/8.09 thf(fact_4742_apfst__apsnd,axiom,
% 14.25/8.09 ! [F: int > produc3658429121746597890et_nat > $o,G: int > produc3925858234332021118et_nat,X: product_prod_int_int] :
% 14.25/8.09 ( ( produc3602029435013415139et_nat @ F @ ( produc2878475440847162070at_int @ G @ X ) )
% 14.25/8.09 = ( produc2245416461498447860et_nat @ ( F @ ( product_fst_int_int @ X ) ) @ ( G @ ( product_snd_int_int @ X ) ) ) ) ).
% 14.25/8.09
% 14.25/8.09 thf(fact_4743_apsnd__apfst,axiom,
% 14.25/8.09 ! [F: set_nat > set_nat,G: heap_e7401611519738050253t_unit > heap_e7401611519738050253t_unit,X: produc3658429121746597890et_nat] :
% 14.25/8.09 ( ( produc3286488329990770625t_unit @ F @ ( produc4172003873565521026et_nat @ G @ X ) )
% 14.25/8.09 = ( produc7507926704131184380et_nat @ ( G @ ( produc1824681642469235216et_nat @ X ) ) @ ( F @ ( produc8586169260539613262et_nat @ X ) ) ) ) ).
% 14.25/8.09
% 14.25/8.09 thf(fact_4744_apsnd__apfst,axiom,
% 14.25/8.09 ! [F: set_nat > nat,G: heap_e7401611519738050253t_unit > nat,X: produc3658429121746597890et_nat] :
% 14.25/8.09 ( ( produc1747169720485888550at_nat @ F @ ( produc2974404638222180585et_nat @ G @ X ) )
% 14.25/8.09 = ( product_Pair_nat_nat @ ( G @ ( produc1824681642469235216et_nat @ X ) ) @ ( F @ ( produc8586169260539613262et_nat @ X ) ) ) ) ).
% 14.25/8.09
% 14.25/8.09 thf(fact_4745_apsnd__apfst,axiom,
% 14.25/8.09 ! [F: set_nat > int,G: heap_e7401611519738050253t_unit > int,X: produc3658429121746597890et_nat] :
% 14.25/8.09 ( ( produc2744032961877892830nt_int @ F @ ( produc3346058198459025221et_nat @ G @ X ) )
% 14.25/8.09 = ( product_Pair_int_int @ ( G @ ( produc1824681642469235216et_nat @ X ) ) @ ( F @ ( produc8586169260539613262et_nat @ X ) ) ) ) ).
% 14.25/8.09
% 14.25/8.09 thf(fact_4746_apsnd__apfst,axiom,
% 14.25/8.09 ! [F: set_nat > produc3658429121746597890et_nat,G: heap_e7401611519738050253t_unit > produc3658429121746597890et_nat > $o,X: produc3658429121746597890et_nat] :
% 14.25/8.09 ( ( produc1718959742329256773_nat_o @ F @ ( produc962520821680634754et_nat @ G @ X ) )
% 14.25/8.09 = ( produc5001842942810119800et_nat @ ( G @ ( produc1824681642469235216et_nat @ X ) ) @ ( F @ ( produc8586169260539613262et_nat @ X ) ) ) ) ).
% 14.25/8.09
% 14.25/8.09 thf(fact_4747_apsnd__apfst,axiom,
% 14.25/8.09 ! [F: set_nat > produc3925858234332021118et_nat,G: heap_e7401611519738050253t_unit > produc3658429121746597890et_nat > $o,X: produc3658429121746597890et_nat] :
% 14.25/8.09 ( ( produc3009152965630989513_nat_o @ F @ ( produc962520821680634754et_nat @ G @ X ) )
% 14.25/8.09 = ( produc2245416461498447860et_nat @ ( G @ ( produc1824681642469235216et_nat @ X ) ) @ ( F @ ( produc8586169260539613262et_nat @ X ) ) ) ) ).
% 14.25/8.09
% 14.25/8.09 thf(fact_4748_apsnd__apfst,axiom,
% 14.25/8.09 ! [F: int > set_nat,G: int > heap_e7401611519738050253t_unit,X: product_prod_int_int] :
% 14.25/8.09 ( ( produc2122153897363312795t_unit @ F @ ( produc1158806734383251089it_int @ G @ X ) )
% 14.25/8.09 = ( produc7507926704131184380et_nat @ ( G @ ( product_fst_int_int @ X ) ) @ ( F @ ( product_snd_int_int @ X ) ) ) ) ).
% 14.25/8.09
% 14.25/8.09 thf(fact_4749_apsnd__apfst,axiom,
% 14.25/8.09 ! [F: int > nat,G: int > nat,X: product_prod_int_int] :
% 14.25/8.09 ( ( produc3466418871192872140at_nat @ F @ ( produc1355883850669603558at_int @ G @ X ) )
% 14.25/8.09 = ( product_Pair_nat_nat @ ( G @ ( product_fst_int_int @ X ) ) @ ( F @ ( product_snd_int_int @ X ) ) ) ) ).
% 14.25/8.09
% 14.25/8.09 thf(fact_4750_apsnd__apfst,axiom,
% 14.25/8.09 ! [F: int > int,G: int > int,X: product_prod_int_int] :
% 14.25/8.09 ( ( produc4463282112584876420nt_int @ F @ ( produc2355237562570658114nt_int @ G @ X ) )
% 14.25/8.09 = ( product_Pair_int_int @ ( G @ ( product_fst_int_int @ X ) ) @ ( F @ ( product_snd_int_int @ X ) ) ) ) ).
% 14.25/8.09
% 14.25/8.09 thf(fact_4751_apsnd__apfst,axiom,
% 14.25/8.09 ! [F: int > produc3658429121746597890et_nat,G: int > produc3658429121746597890et_nat > $o,X: product_prod_int_int] :
% 14.25/8.09 ( ( produc3447581998352395295_nat_o @ F @ ( produc1933289243015553937_o_int @ G @ X ) )
% 14.25/8.09 = ( produc5001842942810119800et_nat @ ( G @ ( product_fst_int_int @ X ) ) @ ( F @ ( product_snd_int_int @ X ) ) ) ) ).
% 14.25/8.09
% 14.25/8.09 thf(fact_4752_apsnd__apfst,axiom,
% 14.25/8.09 ! [F: int > produc3925858234332021118et_nat,G: int > produc3658429121746597890et_nat > $o,X: product_prod_int_int] :
% 14.25/8.09 ( ( produc1783401272518604195_nat_o @ F @ ( produc1933289243015553937_o_int @ G @ X ) )
% 14.25/8.09 = ( produc2245416461498447860et_nat @ ( G @ ( product_fst_int_int @ X ) ) @ ( F @ ( product_snd_int_int @ X ) ) ) ) ).
% 14.25/8.09
% 14.25/8.09 thf(fact_4753_apsnd__apfst__commute,axiom,
% 14.25/8.09 ! [F: code_integer > code_integer,G: code_integer > code_integer,P3: produc8923325533196201883nteger] :
% 14.25/8.09 ( ( produc6499014454317279255nteger @ F @ ( produc3195571438000033749nteger @ G @ P3 ) )
% 14.25/8.09 = ( produc3195571438000033749nteger @ G @ ( produc6499014454317279255nteger @ F @ P3 ) ) ) ).
% 14.25/8.09
% 14.25/8.09 thf(fact_4754_fstI,axiom,
% 14.25/8.09 ! [X: produc3658429121746597890et_nat,Y: heap_e7401611519738050253t_unit,Z: set_nat] :
% 14.25/8.09 ( ( X
% 14.25/8.09 = ( produc7507926704131184380et_nat @ Y @ Z ) )
% 14.25/8.09 => ( ( produc1824681642469235216et_nat @ X )
% 14.25/8.09 = Y ) ) ).
% 14.25/8.09
% 14.25/8.09 thf(fact_4755_fstI,axiom,
% 14.25/8.09 ! [X: product_prod_nat_nat,Y: nat,Z: nat] :
% 14.25/8.09 ( ( X
% 14.25/8.09 = ( product_Pair_nat_nat @ Y @ Z ) )
% 14.25/8.09 => ( ( product_fst_nat_nat @ X )
% 14.25/8.09 = Y ) ) ).
% 14.25/8.09
% 14.25/8.09 thf(fact_4756_fstI,axiom,
% 14.25/8.09 ! [X: produc3925858234332021118et_nat,Y: produc3658429121746597890et_nat > $o,Z: produc3658429121746597890et_nat] :
% 14.25/8.09 ( ( X
% 14.25/8.09 = ( produc5001842942810119800et_nat @ Y @ Z ) )
% 14.25/8.09 => ( ( produc995936583742144908et_nat @ X )
% 14.25/8.09 = Y ) ) ).
% 14.25/8.09
% 14.25/8.09 thf(fact_4757_fstI,axiom,
% 14.25/8.09 ! [X: produc2732055786443039994et_nat,Y: produc3658429121746597890et_nat > $o,Z: produc3925858234332021118et_nat] :
% 14.25/8.09 ( ( X
% 14.25/8.09 = ( produc2245416461498447860et_nat @ Y @ Z ) )
% 14.25/8.09 => ( ( produc180342877477747464et_nat @ X )
% 14.25/8.09 = Y ) ) ).
% 14.25/8.09
% 14.25/8.09 thf(fact_4758_fstI,axiom,
% 14.25/8.09 ! [X: product_prod_int_int,Y: int,Z: int] :
% 14.25/8.09 ( ( X
% 14.25/8.09 = ( product_Pair_int_int @ Y @ Z ) )
% 14.25/8.09 => ( ( product_fst_int_int @ X )
% 14.25/8.09 = Y ) ) ).
% 14.25/8.09
% 14.25/8.09 thf(fact_4759_fst__eqD,axiom,
% 14.25/8.09 ! [X: heap_e7401611519738050253t_unit,Y: set_nat,A2: heap_e7401611519738050253t_unit] :
% 14.25/8.09 ( ( ( produc1824681642469235216et_nat @ ( produc7507926704131184380et_nat @ X @ Y ) )
% 14.25/8.09 = A2 )
% 14.25/8.09 => ( X = A2 ) ) ).
% 14.25/8.09
% 14.25/8.09 thf(fact_4760_fst__eqD,axiom,
% 14.25/8.09 ! [X: nat,Y: nat,A2: nat] :
% 14.25/8.09 ( ( ( product_fst_nat_nat @ ( product_Pair_nat_nat @ X @ Y ) )
% 14.25/8.09 = A2 )
% 14.25/8.09 => ( X = A2 ) ) ).
% 14.25/8.09
% 14.25/8.09 thf(fact_4761_fst__eqD,axiom,
% 14.25/8.09 ! [X: produc3658429121746597890et_nat > $o,Y: produc3658429121746597890et_nat,A2: produc3658429121746597890et_nat > $o] :
% 14.25/8.09 ( ( ( produc995936583742144908et_nat @ ( produc5001842942810119800et_nat @ X @ Y ) )
% 14.25/8.09 = A2 )
% 14.25/8.09 => ( X = A2 ) ) ).
% 14.25/8.09
% 14.25/8.09 thf(fact_4762_fst__eqD,axiom,
% 14.25/8.09 ! [X: produc3658429121746597890et_nat > $o,Y: produc3925858234332021118et_nat,A2: produc3658429121746597890et_nat > $o] :
% 14.25/8.09 ( ( ( produc180342877477747464et_nat @ ( produc2245416461498447860et_nat @ X @ Y ) )
% 14.25/8.09 = A2 )
% 14.25/8.09 => ( X = A2 ) ) ).
% 14.25/8.09
% 14.25/8.09 thf(fact_4763_fst__eqD,axiom,
% 14.25/8.09 ! [X: int,Y: int,A2: int] :
% 14.25/8.09 ( ( ( product_fst_int_int @ ( product_Pair_int_int @ X @ Y ) )
% 14.25/8.09 = A2 )
% 14.25/8.09 => ( X = A2 ) ) ).
% 14.25/8.09
% 14.25/8.09 thf(fact_4764_fst__conv,axiom,
% 14.25/8.09 ! [X1: heap_e7401611519738050253t_unit,X22: set_nat] :
% 14.25/8.09 ( ( produc1824681642469235216et_nat @ ( produc7507926704131184380et_nat @ X1 @ X22 ) )
% 14.25/8.09 = X1 ) ).
% 14.25/8.09
% 14.25/8.09 thf(fact_4765_fst__conv,axiom,
% 14.25/8.09 ! [X1: nat,X22: nat] :
% 14.25/8.09 ( ( product_fst_nat_nat @ ( product_Pair_nat_nat @ X1 @ X22 ) )
% 14.25/8.09 = X1 ) ).
% 14.25/8.09
% 14.25/8.09 thf(fact_4766_fst__conv,axiom,
% 14.25/8.09 ! [X1: produc3658429121746597890et_nat > $o,X22: produc3658429121746597890et_nat] :
% 14.25/8.09 ( ( produc995936583742144908et_nat @ ( produc5001842942810119800et_nat @ X1 @ X22 ) )
% 14.25/8.09 = X1 ) ).
% 14.25/8.09
% 14.25/8.09 thf(fact_4767_fst__conv,axiom,
% 14.25/8.09 ! [X1: produc3658429121746597890et_nat > $o,X22: produc3925858234332021118et_nat] :
% 14.25/8.09 ( ( produc180342877477747464et_nat @ ( produc2245416461498447860et_nat @ X1 @ X22 ) )
% 14.25/8.09 = X1 ) ).
% 14.25/8.09
% 14.25/8.09 thf(fact_4768_fst__conv,axiom,
% 14.25/8.09 ! [X1: int,X22: int] :
% 14.25/8.09 ( ( product_fst_int_int @ ( product_Pair_int_int @ X1 @ X22 ) )
% 14.25/8.09 = X1 ) ).
% 14.25/8.09
% 14.25/8.09 thf(fact_4769_eq__fst__iff,axiom,
% 14.25/8.09 ! [A2: heap_e7401611519738050253t_unit,P3: produc3658429121746597890et_nat] :
% 14.25/8.09 ( ( A2
% 14.25/8.09 = ( produc1824681642469235216et_nat @ P3 ) )
% 14.25/8.09 = ( ? [B6: set_nat] :
% 14.25/8.09 ( P3
% 14.25/8.09 = ( produc7507926704131184380et_nat @ A2 @ B6 ) ) ) ) ).
% 14.25/8.09
% 14.25/8.09 thf(fact_4770_eq__fst__iff,axiom,
% 14.25/8.09 ! [A2: nat,P3: product_prod_nat_nat] :
% 14.25/8.09 ( ( A2
% 14.25/8.09 = ( product_fst_nat_nat @ P3 ) )
% 14.25/8.09 = ( ? [B6: nat] :
% 14.25/8.09 ( P3
% 14.25/8.09 = ( product_Pair_nat_nat @ A2 @ B6 ) ) ) ) ).
% 14.25/8.09
% 14.25/8.09 thf(fact_4771_eq__fst__iff,axiom,
% 14.25/8.09 ! [A2: produc3658429121746597890et_nat > $o,P3: produc3925858234332021118et_nat] :
% 14.25/8.09 ( ( A2
% 14.25/8.09 = ( produc995936583742144908et_nat @ P3 ) )
% 14.25/8.09 = ( ? [B6: produc3658429121746597890et_nat] :
% 14.25/8.09 ( P3
% 14.25/8.09 = ( produc5001842942810119800et_nat @ A2 @ B6 ) ) ) ) ).
% 14.25/8.09
% 14.25/8.09 thf(fact_4772_eq__fst__iff,axiom,
% 14.25/8.09 ! [A2: produc3658429121746597890et_nat > $o,P3: produc2732055786443039994et_nat] :
% 14.25/8.09 ( ( A2
% 14.25/8.09 = ( produc180342877477747464et_nat @ P3 ) )
% 14.25/8.09 = ( ? [B6: produc3925858234332021118et_nat] :
% 14.25/8.09 ( P3
% 14.25/8.09 = ( produc2245416461498447860et_nat @ A2 @ B6 ) ) ) ) ).
% 14.25/8.09
% 14.25/8.09 thf(fact_4773_eq__fst__iff,axiom,
% 14.25/8.09 ! [A2: int,P3: product_prod_int_int] :
% 14.25/8.09 ( ( A2
% 14.25/8.09 = ( product_fst_int_int @ P3 ) )
% 14.25/8.09 = ( ? [B6: int] :
% 14.25/8.09 ( P3
% 14.25/8.09 = ( product_Pair_int_int @ A2 @ B6 ) ) ) ) ).
% 14.25/8.09
% 14.25/8.09 thf(fact_4774_fstE,axiom,
% 14.25/8.09 ! [X: produc3658429121746597890et_nat,A2: heap_e7401611519738050253t_unit,B: set_nat,P: heap_e7401611519738050253t_unit > $o] :
% 14.25/8.09 ( ( X
% 14.25/8.09 = ( produc7507926704131184380et_nat @ A2 @ B ) )
% 14.25/8.09 => ( ( P @ ( produc1824681642469235216et_nat @ X ) )
% 14.25/8.09 => ( P @ A2 ) ) ) ).
% 14.25/8.09
% 14.25/8.09 thf(fact_4775_fstE,axiom,
% 14.25/8.09 ! [X: product_prod_nat_nat,A2: nat,B: nat,P: nat > $o] :
% 14.25/8.09 ( ( X
% 14.25/8.09 = ( product_Pair_nat_nat @ A2 @ B ) )
% 14.25/8.09 => ( ( P @ ( product_fst_nat_nat @ X ) )
% 14.25/8.09 => ( P @ A2 ) ) ) ).
% 14.25/8.09
% 14.25/8.09 thf(fact_4776_fstE,axiom,
% 14.25/8.09 ! [X: produc3925858234332021118et_nat,A2: produc3658429121746597890et_nat > $o,B: produc3658429121746597890et_nat,P: ( produc3658429121746597890et_nat > $o ) > $o] :
% 14.25/8.09 ( ( X
% 14.25/8.09 = ( produc5001842942810119800et_nat @ A2 @ B ) )
% 14.25/8.09 => ( ( P @ ( produc995936583742144908et_nat @ X ) )
% 14.25/8.09 => ( P @ A2 ) ) ) ).
% 14.25/8.09
% 14.25/8.09 thf(fact_4777_fstE,axiom,
% 14.25/8.09 ! [X: produc2732055786443039994et_nat,A2: produc3658429121746597890et_nat > $o,B: produc3925858234332021118et_nat,P: ( produc3658429121746597890et_nat > $o ) > $o] :
% 14.25/8.09 ( ( X
% 14.25/8.09 = ( produc2245416461498447860et_nat @ A2 @ B ) )
% 14.25/8.09 => ( ( P @ ( produc180342877477747464et_nat @ X ) )
% 14.25/8.09 => ( P @ A2 ) ) ) ).
% 14.25/8.09
% 14.25/8.09 thf(fact_4778_fstE,axiom,
% 14.25/8.09 ! [X: product_prod_int_int,A2: int,B: int,P: int > $o] :
% 14.25/8.09 ( ( X
% 14.25/8.09 = ( product_Pair_int_int @ A2 @ B ) )
% 14.25/8.09 => ( ( P @ ( product_fst_int_int @ X ) )
% 14.25/8.09 => ( P @ A2 ) ) ) ).
% 14.25/8.09
% 14.25/8.09 thf(fact_4779_prod__eq__iff,axiom,
% 14.25/8.09 ( ( ^ [Y4: produc3658429121746597890et_nat,Z4: produc3658429121746597890et_nat] : ( Y4 = Z4 ) )
% 14.25/8.09 = ( ^ [S7: produc3658429121746597890et_nat,T: produc3658429121746597890et_nat] :
% 14.25/8.09 ( ( ( produc1824681642469235216et_nat @ S7 )
% 14.25/8.09 = ( produc1824681642469235216et_nat @ T ) )
% 14.25/8.09 & ( ( produc8586169260539613262et_nat @ S7 )
% 14.25/8.09 = ( produc8586169260539613262et_nat @ T ) ) ) ) ) ).
% 14.25/8.09
% 14.25/8.09 thf(fact_4780_prod__eq__iff,axiom,
% 14.25/8.09 ( ( ^ [Y4: product_prod_int_int,Z4: product_prod_int_int] : ( Y4 = Z4 ) )
% 14.25/8.09 = ( ^ [S7: product_prod_int_int,T: product_prod_int_int] :
% 14.25/8.09 ( ( ( product_fst_int_int @ S7 )
% 14.25/8.09 = ( product_fst_int_int @ T ) )
% 14.25/8.09 & ( ( product_snd_int_int @ S7 )
% 14.25/8.09 = ( product_snd_int_int @ T ) ) ) ) ) ).
% 14.25/8.09
% 14.25/8.09 thf(fact_4781_prod__eqI,axiom,
% 14.25/8.09 ! [P3: produc3658429121746597890et_nat,Q4: produc3658429121746597890et_nat] :
% 14.25/8.09 ( ( ( produc1824681642469235216et_nat @ P3 )
% 14.25/8.09 = ( produc1824681642469235216et_nat @ Q4 ) )
% 14.25/8.09 => ( ( ( produc8586169260539613262et_nat @ P3 )
% 14.25/8.09 = ( produc8586169260539613262et_nat @ Q4 ) )
% 14.25/8.09 => ( P3 = Q4 ) ) ) ).
% 14.25/8.09
% 14.25/8.09 thf(fact_4782_prod__eqI,axiom,
% 14.25/8.09 ! [P3: product_prod_int_int,Q4: product_prod_int_int] :
% 14.25/8.09 ( ( ( product_fst_int_int @ P3 )
% 14.25/8.09 = ( product_fst_int_int @ Q4 ) )
% 14.25/8.09 => ( ( ( product_snd_int_int @ P3 )
% 14.25/8.09 = ( product_snd_int_int @ Q4 ) )
% 14.25/8.09 => ( P3 = Q4 ) ) ) ).
% 14.25/8.09
% 14.25/8.09 thf(fact_4783_prod_Oexpand,axiom,
% 14.25/8.09 ! [Prod: produc3658429121746597890et_nat,Prod2: produc3658429121746597890et_nat] :
% 14.25/8.09 ( ( ( ( produc1824681642469235216et_nat @ Prod )
% 14.25/8.09 = ( produc1824681642469235216et_nat @ Prod2 ) )
% 14.25/8.09 & ( ( produc8586169260539613262et_nat @ Prod )
% 14.25/8.09 = ( produc8586169260539613262et_nat @ Prod2 ) ) )
% 14.25/8.09 => ( Prod = Prod2 ) ) ).
% 14.25/8.09
% 14.25/8.09 thf(fact_4784_prod_Oexpand,axiom,
% 14.25/8.09 ! [Prod: product_prod_int_int,Prod2: product_prod_int_int] :
% 14.25/8.09 ( ( ( ( product_fst_int_int @ Prod )
% 14.25/8.09 = ( product_fst_int_int @ Prod2 ) )
% 14.25/8.09 & ( ( product_snd_int_int @ Prod )
% 14.25/8.09 = ( product_snd_int_int @ Prod2 ) ) )
% 14.25/8.09 => ( Prod = Prod2 ) ) ).
% 14.25/8.09
% 14.25/8.09 thf(fact_4785_All__prod__contract,axiom,
% 14.25/8.09 ! [P: heap_e7401611519738050253t_unit > set_nat > $o] :
% 14.25/8.09 ( ( ! [A6: heap_e7401611519738050253t_unit,X7: set_nat] : ( P @ A6 @ X7 ) )
% 14.25/8.09 = ( ! [Z2: produc3658429121746597890et_nat] : ( P @ ( produc1824681642469235216et_nat @ Z2 ) @ ( produc8586169260539613262et_nat @ Z2 ) ) ) ) ).
% 14.25/8.09
% 14.25/8.09 thf(fact_4786_All__prod__contract,axiom,
% 14.25/8.09 ! [P: int > int > $o] :
% 14.25/8.09 ( ( ! [A6: int,X7: int] : ( P @ A6 @ X7 ) )
% 14.25/8.09 = ( ! [Z2: product_prod_int_int] : ( P @ ( product_fst_int_int @ Z2 ) @ ( product_snd_int_int @ Z2 ) ) ) ) ).
% 14.25/8.09
% 14.25/8.09 thf(fact_4787_Ex__prod__contract,axiom,
% 14.25/8.09 ! [P: heap_e7401611519738050253t_unit > set_nat > $o] :
% 14.25/8.09 ( ( ? [A6: heap_e7401611519738050253t_unit,X7: set_nat] : ( P @ A6 @ X7 ) )
% 14.25/8.09 = ( ? [Z2: produc3658429121746597890et_nat] : ( P @ ( produc1824681642469235216et_nat @ Z2 ) @ ( produc8586169260539613262et_nat @ Z2 ) ) ) ) ).
% 14.25/8.09
% 14.25/8.09 thf(fact_4788_Ex__prod__contract,axiom,
% 14.25/8.09 ! [P: int > int > $o] :
% 14.25/8.09 ( ( ? [A6: int,X7: int] : ( P @ A6 @ X7 ) )
% 14.25/8.09 = ( ? [Z2: product_prod_int_int] : ( P @ ( product_fst_int_int @ Z2 ) @ ( product_snd_int_int @ Z2 ) ) ) ) ).
% 14.25/8.09
% 14.25/8.09 thf(fact_4789_apsnd__compose,axiom,
% 14.25/8.09 ! [F: code_integer > produc8923325533196201883nteger > produc8923325533196201883nteger,G: code_integer > code_integer,X: produc8923325533196201883nteger] :
% 14.25/8.09 ( ( produc9020447175693601247nteger @ F @ ( produc6499014454317279255nteger @ G @ X ) )
% 14.25/8.09 = ( produc9020447175693601247nteger @ ( comp_C1593894019821074884nteger @ F @ G ) @ X ) ) ).
% 14.25/8.09
% 14.25/8.09 thf(fact_4790_apsnd__compose,axiom,
% 14.25/8.09 ! [F: code_integer > code_integer,G: code_integer > code_integer,X: produc8923325533196201883nteger] :
% 14.25/8.09 ( ( produc6499014454317279255nteger @ F @ ( produc6499014454317279255nteger @ G @ X ) )
% 14.25/8.09 = ( produc6499014454317279255nteger @ ( comp_C7449957260575251196nteger @ F @ G ) @ X ) ) ).
% 14.25/8.09
% 14.25/8.09 thf(fact_4791_refl__on__INTER,axiom,
% 14.25/8.09 ! [S: set_nat,A: nat > set_nat,R2: nat > set_Pr1261947904930325089at_nat] :
% 14.25/8.09 ( ! [X3: nat] :
% 14.25/8.09 ( ( member_nat @ X3 @ S )
% 14.25/8.09 => ( refl_on_nat @ ( A @ X3 ) @ ( R2 @ X3 ) ) )
% 14.25/8.09 => ( refl_on_nat @ ( comple7806235888213564991et_nat @ ( image_nat_set_nat @ A @ S ) ) @ ( comple1930127330929219688at_nat @ ( image_7178329752028323786at_nat @ R2 @ S ) ) ) ) ).
% 14.25/8.09
% 14.25/8.09 thf(fact_4792_cInf__eq__non__empty,axiom,
% 14.25/8.09 ! [X4: set_o,A2: $o] :
% 14.25/8.09 ( ( X4 != bot_bot_set_o )
% 14.25/8.09 => ( ! [X3: $o] :
% 14.25/8.09 ( ( member_o @ X3 @ X4 )
% 14.25/8.09 => ( ord_less_eq_o @ A2 @ X3 ) )
% 14.25/8.09 => ( ! [Y2: $o] :
% 14.25/8.09 ( ! [X6: $o] :
% 14.25/8.09 ( ( member_o @ X6 @ X4 )
% 14.25/8.09 => ( ord_less_eq_o @ Y2 @ X6 ) )
% 14.25/8.09 => ( ord_less_eq_o @ Y2 @ A2 ) )
% 14.25/8.09 => ( ( complete_Inf_Inf_o @ X4 )
% 14.25/8.09 = A2 ) ) ) ) ).
% 14.25/8.09
% 14.25/8.09 thf(fact_4793_cInf__eq__non__empty,axiom,
% 14.25/8.09 ! [X4: set_Pr4532377907799695533_nat_o,A2: produc3658429121746597890et_nat > $o] :
% 14.25/8.09 ( ( X4 != bot_bo7824918357723582233_nat_o )
% 14.25/8.09 => ( ! [X3: produc3658429121746597890et_nat > $o] :
% 14.25/8.09 ( ( member6576561426505652726_nat_o @ X3 @ X4 )
% 14.25/8.09 => ( ord_le729326519192465773_nat_o @ A2 @ X3 ) )
% 14.25/8.09 => ( ! [Y2: produc3658429121746597890et_nat > $o] :
% 14.25/8.09 ( ! [X6: produc3658429121746597890et_nat > $o] :
% 14.25/8.09 ( ( member6576561426505652726_nat_o @ X6 @ X4 )
% 14.25/8.09 => ( ord_le729326519192465773_nat_o @ Y2 @ X6 ) )
% 14.25/8.09 => ( ord_le729326519192465773_nat_o @ Y2 @ A2 ) )
% 14.25/8.09 => ( ( comple90455603224345172_nat_o @ X4 )
% 14.25/8.09 = A2 ) ) ) ) ).
% 14.25/8.09
% 14.25/8.09 thf(fact_4794_cInf__eq__non__empty,axiom,
% 14.25/8.09 ! [X4: set_filter_nat,A2: filter_nat] :
% 14.25/8.09 ( ( X4 != bot_bo498966703094740906er_nat )
% 14.25/8.09 => ( ! [X3: filter_nat] :
% 14.25/8.09 ( ( member_filter_nat @ X3 @ X4 )
% 14.25/8.09 => ( ord_le2510731241096832064er_nat @ A2 @ X3 ) )
% 14.25/8.09 => ( ! [Y2: filter_nat] :
% 14.25/8.09 ( ! [X6: filter_nat] :
% 14.25/8.09 ( ( member_filter_nat @ X6 @ X4 )
% 14.25/8.09 => ( ord_le2510731241096832064er_nat @ Y2 @ X6 ) )
% 14.25/8.09 => ( ord_le2510731241096832064er_nat @ Y2 @ A2 ) )
% 14.25/8.09 => ( ( comple6612786467464916249er_nat @ X4 )
% 14.25/8.09 = A2 ) ) ) ) ).
% 14.25/8.09
% 14.25/8.09 thf(fact_4795_cInf__eq__non__empty,axiom,
% 14.25/8.09 ! [X4: set_int,A2: int] :
% 14.25/8.09 ( ( X4 != bot_bot_set_int )
% 14.25/8.09 => ( ! [X3: int] :
% 14.25/8.09 ( ( member_int @ X3 @ X4 )
% 14.25/8.09 => ( ord_less_eq_int @ A2 @ X3 ) )
% 14.25/8.09 => ( ! [Y2: int] :
% 14.25/8.09 ( ! [X6: int] :
% 14.25/8.09 ( ( member_int @ X6 @ X4 )
% 14.25/8.09 => ( ord_less_eq_int @ Y2 @ X6 ) )
% 14.25/8.09 => ( ord_less_eq_int @ Y2 @ A2 ) )
% 14.25/8.09 => ( ( complete_Inf_Inf_int @ X4 )
% 14.25/8.09 = A2 ) ) ) ) ).
% 14.25/8.09
% 14.25/8.09 thf(fact_4796_cInf__eq__non__empty,axiom,
% 14.25/8.09 ! [X4: set_nat,A2: nat] :
% 14.25/8.09 ( ( X4 != bot_bot_set_nat )
% 14.25/8.09 => ( ! [X3: nat] :
% 14.25/8.09 ( ( member_nat @ X3 @ X4 )
% 14.25/8.09 => ( ord_less_eq_nat @ A2 @ X3 ) )
% 14.25/8.09 => ( ! [Y2: nat] :
% 14.25/8.09 ( ! [X6: nat] :
% 14.25/8.09 ( ( member_nat @ X6 @ X4 )
% 14.25/8.09 => ( ord_less_eq_nat @ Y2 @ X6 ) )
% 14.25/8.09 => ( ord_less_eq_nat @ Y2 @ A2 ) )
% 14.25/8.09 => ( ( complete_Inf_Inf_nat @ X4 )
% 14.25/8.09 = A2 ) ) ) ) ).
% 14.25/8.09
% 14.25/8.09 thf(fact_4797_cInf__eq__non__empty,axiom,
% 14.25/8.09 ! [X4: set_set_nat,A2: set_nat] :
% 14.25/8.09 ( ( X4 != bot_bot_set_set_nat )
% 14.25/8.09 => ( ! [X3: set_nat] :
% 14.25/8.09 ( ( member_set_nat @ X3 @ X4 )
% 14.25/8.09 => ( ord_less_eq_set_nat @ A2 @ X3 ) )
% 14.25/8.09 => ( ! [Y2: set_nat] :
% 14.25/8.09 ( ! [X6: set_nat] :
% 14.25/8.09 ( ( member_set_nat @ X6 @ X4 )
% 14.25/8.09 => ( ord_less_eq_set_nat @ Y2 @ X6 ) )
% 14.25/8.09 => ( ord_less_eq_set_nat @ Y2 @ A2 ) )
% 14.25/8.09 => ( ( comple7806235888213564991et_nat @ X4 )
% 14.25/8.09 = A2 ) ) ) ) ).
% 14.25/8.09
% 14.25/8.09 thf(fact_4798_cInf__eq__non__empty,axiom,
% 14.25/8.09 ! [X4: set_Product_unit,A2: product_unit] :
% 14.25/8.09 ( ( X4 != bot_bo3957492148770167129t_unit )
% 14.25/8.09 => ( ! [X3: product_unit] :
% 14.25/8.09 ( ( member_Product_unit @ X3 @ X4 )
% 14.25/8.09 => ( ord_le3221252021190050221t_unit @ A2 @ X3 ) )
% 14.25/8.09 => ( ! [Y2: product_unit] :
% 14.25/8.09 ( ! [X6: product_unit] :
% 14.25/8.09 ( ( member_Product_unit @ X6 @ X4 )
% 14.25/8.09 => ( ord_le3221252021190050221t_unit @ Y2 @ X6 ) )
% 14.25/8.09 => ( ord_le3221252021190050221t_unit @ Y2 @ A2 ) )
% 14.25/8.09 => ( ( comple2584293577114468500t_unit @ X4 )
% 14.25/8.09 = A2 ) ) ) ) ).
% 14.25/8.09
% 14.25/8.09 thf(fact_4799_cInf__greatest,axiom,
% 14.25/8.09 ! [X4: set_o,Z: $o] :
% 14.25/8.09 ( ( X4 != bot_bot_set_o )
% 14.25/8.09 => ( ! [X3: $o] :
% 14.25/8.09 ( ( member_o @ X3 @ X4 )
% 14.25/8.09 => ( ord_less_eq_o @ Z @ X3 ) )
% 14.25/8.09 => ( ord_less_eq_o @ Z @ ( complete_Inf_Inf_o @ X4 ) ) ) ) ).
% 14.25/8.09
% 14.25/8.09 thf(fact_4800_cInf__greatest,axiom,
% 14.25/8.09 ! [X4: set_Pr4532377907799695533_nat_o,Z: produc3658429121746597890et_nat > $o] :
% 14.25/8.09 ( ( X4 != bot_bo7824918357723582233_nat_o )
% 14.25/8.09 => ( ! [X3: produc3658429121746597890et_nat > $o] :
% 14.25/8.09 ( ( member6576561426505652726_nat_o @ X3 @ X4 )
% 14.25/8.09 => ( ord_le729326519192465773_nat_o @ Z @ X3 ) )
% 14.25/8.09 => ( ord_le729326519192465773_nat_o @ Z @ ( comple90455603224345172_nat_o @ X4 ) ) ) ) ).
% 14.25/8.09
% 14.25/8.09 thf(fact_4801_cInf__greatest,axiom,
% 14.25/8.09 ! [X4: set_filter_nat,Z: filter_nat] :
% 14.25/8.09 ( ( X4 != bot_bo498966703094740906er_nat )
% 14.25/8.09 => ( ! [X3: filter_nat] :
% 14.25/8.09 ( ( member_filter_nat @ X3 @ X4 )
% 14.25/8.09 => ( ord_le2510731241096832064er_nat @ Z @ X3 ) )
% 14.25/8.09 => ( ord_le2510731241096832064er_nat @ Z @ ( comple6612786467464916249er_nat @ X4 ) ) ) ) ).
% 14.25/8.09
% 14.25/8.09 thf(fact_4802_cInf__greatest,axiom,
% 14.25/8.09 ! [X4: set_int,Z: int] :
% 14.25/8.09 ( ( X4 != bot_bot_set_int )
% 14.25/8.09 => ( ! [X3: int] :
% 14.25/8.09 ( ( member_int @ X3 @ X4 )
% 14.25/8.09 => ( ord_less_eq_int @ Z @ X3 ) )
% 14.25/8.09 => ( ord_less_eq_int @ Z @ ( complete_Inf_Inf_int @ X4 ) ) ) ) ).
% 14.25/8.09
% 14.25/8.09 thf(fact_4803_cInf__greatest,axiom,
% 14.25/8.09 ! [X4: set_nat,Z: nat] :
% 14.25/8.09 ( ( X4 != bot_bot_set_nat )
% 14.25/8.09 => ( ! [X3: nat] :
% 14.25/8.09 ( ( member_nat @ X3 @ X4 )
% 14.25/8.09 => ( ord_less_eq_nat @ Z @ X3 ) )
% 14.25/8.09 => ( ord_less_eq_nat @ Z @ ( complete_Inf_Inf_nat @ X4 ) ) ) ) ).
% 14.25/8.09
% 14.25/8.09 thf(fact_4804_cInf__greatest,axiom,
% 14.25/8.09 ! [X4: set_set_nat,Z: set_nat] :
% 14.25/8.09 ( ( X4 != bot_bot_set_set_nat )
% 14.25/8.09 => ( ! [X3: set_nat] :
% 14.25/8.09 ( ( member_set_nat @ X3 @ X4 )
% 14.25/8.09 => ( ord_less_eq_set_nat @ Z @ X3 ) )
% 14.25/8.09 => ( ord_less_eq_set_nat @ Z @ ( comple7806235888213564991et_nat @ X4 ) ) ) ) ).
% 14.25/8.09
% 14.25/8.09 thf(fact_4805_cInf__greatest,axiom,
% 14.25/8.09 ! [X4: set_Product_unit,Z: product_unit] :
% 14.25/8.09 ( ( X4 != bot_bo3957492148770167129t_unit )
% 14.25/8.09 => ( ! [X3: product_unit] :
% 14.25/8.09 ( ( member_Product_unit @ X3 @ X4 )
% 14.25/8.09 => ( ord_le3221252021190050221t_unit @ Z @ X3 ) )
% 14.25/8.09 => ( ord_le3221252021190050221t_unit @ Z @ ( comple2584293577114468500t_unit @ X4 ) ) ) ) ).
% 14.25/8.09
% 14.25/8.09 thf(fact_4806_Inf__less__eq,axiom,
% 14.25/8.09 ! [A: set_o,U: $o] :
% 14.25/8.09 ( ! [V3: $o] :
% 14.25/8.09 ( ( member_o @ V3 @ A )
% 14.25/8.09 => ( ord_less_eq_o @ V3 @ U ) )
% 14.25/8.09 => ( ( A != bot_bot_set_o )
% 14.25/8.09 => ( ord_less_eq_o @ ( complete_Inf_Inf_o @ A ) @ U ) ) ) ).
% 14.25/8.09
% 14.25/8.09 thf(fact_4807_Inf__less__eq,axiom,
% 14.25/8.09 ! [A: set_Pr4532377907799695533_nat_o,U: produc3658429121746597890et_nat > $o] :
% 14.25/8.09 ( ! [V3: produc3658429121746597890et_nat > $o] :
% 14.25/8.09 ( ( member6576561426505652726_nat_o @ V3 @ A )
% 14.25/8.09 => ( ord_le729326519192465773_nat_o @ V3 @ U ) )
% 14.25/8.09 => ( ( A != bot_bo7824918357723582233_nat_o )
% 14.25/8.09 => ( ord_le729326519192465773_nat_o @ ( comple90455603224345172_nat_o @ A ) @ U ) ) ) ).
% 14.25/8.09
% 14.25/8.09 thf(fact_4808_Inf__less__eq,axiom,
% 14.25/8.09 ! [A: set_filter_nat,U: filter_nat] :
% 14.25/8.09 ( ! [V3: filter_nat] :
% 14.25/8.09 ( ( member_filter_nat @ V3 @ A )
% 14.25/8.09 => ( ord_le2510731241096832064er_nat @ V3 @ U ) )
% 14.25/8.09 => ( ( A != bot_bo498966703094740906er_nat )
% 14.25/8.09 => ( ord_le2510731241096832064er_nat @ ( comple6612786467464916249er_nat @ A ) @ U ) ) ) ).
% 14.25/8.09
% 14.25/8.09 thf(fact_4809_Inf__less__eq,axiom,
% 14.25/8.09 ! [A: set_set_nat,U: set_nat] :
% 14.25/8.09 ( ! [V3: set_nat] :
% 14.25/8.09 ( ( member_set_nat @ V3 @ A )
% 14.25/8.09 => ( ord_less_eq_set_nat @ V3 @ U ) )
% 14.25/8.09 => ( ( A != bot_bot_set_set_nat )
% 14.25/8.09 => ( ord_less_eq_set_nat @ ( comple7806235888213564991et_nat @ A ) @ U ) ) ) ).
% 14.25/8.09
% 14.25/8.09 thf(fact_4810_Inf__less__eq,axiom,
% 14.25/8.09 ! [A: set_Product_unit,U: product_unit] :
% 14.25/8.09 ( ! [V3: product_unit] :
% 14.25/8.09 ( ( member_Product_unit @ V3 @ A )
% 14.25/8.09 => ( ord_le3221252021190050221t_unit @ V3 @ U ) )
% 14.25/8.09 => ( ( A != bot_bo3957492148770167129t_unit )
% 14.25/8.09 => ( ord_le3221252021190050221t_unit @ ( comple2584293577114468500t_unit @ A ) @ U ) ) ) ).
% 14.25/8.09
% 14.25/8.09 thf(fact_4811_cInf__lessD,axiom,
% 14.25/8.09 ! [X4: set_int,Z: int] :
% 14.25/8.09 ( ( X4 != bot_bot_set_int )
% 14.25/8.09 => ( ( ord_less_int @ ( complete_Inf_Inf_int @ X4 ) @ Z )
% 14.25/8.09 => ? [X3: int] :
% 14.25/8.09 ( ( member_int @ X3 @ X4 )
% 14.25/8.09 & ( ord_less_int @ X3 @ Z ) ) ) ) ).
% 14.25/8.09
% 14.25/8.09 thf(fact_4812_cInf__lessD,axiom,
% 14.25/8.09 ! [X4: set_nat,Z: nat] :
% 14.25/8.09 ( ( X4 != bot_bot_set_nat )
% 14.25/8.09 => ( ( ord_less_nat @ ( complete_Inf_Inf_nat @ X4 ) @ Z )
% 14.25/8.09 => ? [X3: nat] :
% 14.25/8.09 ( ( member_nat @ X3 @ X4 )
% 14.25/8.09 & ( ord_less_nat @ X3 @ Z ) ) ) ) ).
% 14.25/8.09
% 14.25/8.09 thf(fact_4813_cInf__lessD,axiom,
% 14.25/8.09 ! [X4: set_Product_unit,Z: product_unit] :
% 14.25/8.09 ( ( X4 != bot_bo3957492148770167129t_unit )
% 14.25/8.09 => ( ( ord_le361264281704409273t_unit @ ( comple2584293577114468500t_unit @ X4 ) @ Z )
% 14.25/8.09 => ? [X3: product_unit] :
% 14.25/8.09 ( ( member_Product_unit @ X3 @ X4 )
% 14.25/8.09 & ( ord_le361264281704409273t_unit @ X3 @ Z ) ) ) ) ).
% 14.25/8.09
% 14.25/8.09 thf(fact_4814_INF__eq__const,axiom,
% 14.25/8.09 ! [I: set_Product_unit,F: product_unit > set_nat,X: set_nat] :
% 14.25/8.09 ( ( I != bot_bo3957492148770167129t_unit )
% 14.25/8.09 => ( ! [I3: product_unit] :
% 14.25/8.09 ( ( member_Product_unit @ I3 @ I )
% 14.25/8.09 => ( ( F @ I3 )
% 14.25/8.09 = X ) )
% 14.25/8.09 => ( ( comple7806235888213564991et_nat @ ( image_8173849991827986510et_nat @ F @ I ) )
% 14.25/8.09 = X ) ) ) ).
% 14.25/8.09
% 14.25/8.09 thf(fact_4815_INF__eq__const,axiom,
% 14.25/8.09 ! [I: set_o,F: $o > set_nat,X: set_nat] :
% 14.25/8.09 ( ( I != bot_bot_set_o )
% 14.25/8.09 => ( ! [I3: $o] :
% 14.25/8.09 ( ( member_o @ I3 @ I )
% 14.25/8.09 => ( ( F @ I3 )
% 14.25/8.09 = X ) )
% 14.25/8.09 => ( ( comple7806235888213564991et_nat @ ( image_o_set_nat @ F @ I ) )
% 14.25/8.09 = X ) ) ) ).
% 14.25/8.09
% 14.25/8.09 thf(fact_4816_INF__eq__const,axiom,
% 14.25/8.09 ! [I: set_nat,F: nat > set_nat,X: set_nat] :
% 14.25/8.09 ( ( I != bot_bot_set_nat )
% 14.25/8.09 => ( ! [I3: nat] :
% 14.25/8.09 ( ( member_nat @ I3 @ I )
% 14.25/8.09 => ( ( F @ I3 )
% 14.25/8.09 = X ) )
% 14.25/8.09 => ( ( comple7806235888213564991et_nat @ ( image_nat_set_nat @ F @ I ) )
% 14.25/8.09 = X ) ) ) ).
% 14.25/8.09
% 14.25/8.09 thf(fact_4817_INF__eq__const,axiom,
% 14.25/8.09 ! [I: set_int,F: int > set_nat,X: set_nat] :
% 14.25/8.09 ( ( I != bot_bot_set_int )
% 14.25/8.09 => ( ! [I3: int] :
% 14.25/8.09 ( ( member_int @ I3 @ I )
% 14.25/8.09 => ( ( F @ I3 )
% 14.25/8.09 = X ) )
% 14.25/8.09 => ( ( comple7806235888213564991et_nat @ ( image_int_set_nat @ F @ I ) )
% 14.25/8.09 = X ) ) ) ).
% 14.25/8.09
% 14.25/8.09 thf(fact_4818_INF__eq__const,axiom,
% 14.25/8.09 ! [I: set_Pr4532377907799695533_nat_o,F: ( produc3658429121746597890et_nat > $o ) > set_nat,X: set_nat] :
% 14.25/8.09 ( ( I != bot_bo7824918357723582233_nat_o )
% 14.25/8.09 => ( ! [I3: produc3658429121746597890et_nat > $o] :
% 14.25/8.09 ( ( member6576561426505652726_nat_o @ I3 @ I )
% 14.25/8.09 => ( ( F @ I3 )
% 14.25/8.09 = X ) )
% 14.25/8.09 => ( ( comple7806235888213564991et_nat @ ( image_6888641473154130958et_nat @ F @ I ) )
% 14.25/8.09 = X ) ) ) ).
% 14.25/8.09
% 14.25/8.09 thf(fact_4819_INF__eq__const,axiom,
% 14.25/8.09 ! [I: set_Product_unit,F: product_unit > product_unit,X: product_unit] :
% 14.25/8.09 ( ( I != bot_bo3957492148770167129t_unit )
% 14.25/8.09 => ( ! [I3: product_unit] :
% 14.25/8.09 ( ( member_Product_unit @ I3 @ I )
% 14.25/8.09 => ( ( F @ I3 )
% 14.25/8.09 = X ) )
% 14.25/8.09 => ( ( comple2584293577114468500t_unit @ ( image_405062704495631173t_unit @ F @ I ) )
% 14.25/8.09 = X ) ) ) ).
% 14.25/8.09
% 14.25/8.09 thf(fact_4820_INF__eq__const,axiom,
% 14.25/8.09 ! [I: set_o,F: $o > product_unit,X: product_unit] :
% 14.25/8.09 ( ( I != bot_bot_set_o )
% 14.25/8.09 => ( ! [I3: $o] :
% 14.25/8.09 ( ( member_o @ I3 @ I )
% 14.25/8.09 => ( ( F @ I3 )
% 14.25/8.09 = X ) )
% 14.25/8.09 => ( ( comple2584293577114468500t_unit @ ( image_o_Product_unit2 @ F @ I ) )
% 14.25/8.09 = X ) ) ) ).
% 14.25/8.09
% 14.25/8.09 thf(fact_4821_INF__eq__const,axiom,
% 14.25/8.09 ! [I: set_nat,F: nat > product_unit,X: product_unit] :
% 14.25/8.09 ( ( I != bot_bot_set_nat )
% 14.25/8.09 => ( ! [I3: nat] :
% 14.25/8.09 ( ( member_nat @ I3 @ I )
% 14.25/8.09 => ( ( F @ I3 )
% 14.25/8.09 = X ) )
% 14.25/8.09 => ( ( comple2584293577114468500t_unit @ ( image_8730104196221521654t_unit @ F @ I ) )
% 14.25/8.09 = X ) ) ) ).
% 14.25/8.09
% 14.25/8.09 thf(fact_4822_INF__eq__const,axiom,
% 14.25/8.09 ! [I: set_int,F: int > product_unit,X: product_unit] :
% 14.25/8.09 ( ( I != bot_bot_set_int )
% 14.25/8.09 => ( ! [I3: int] :
% 14.25/8.09 ( ( member_int @ I3 @ I )
% 14.25/8.09 => ( ( F @ I3 )
% 14.25/8.09 = X ) )
% 14.25/8.09 => ( ( comple2584293577114468500t_unit @ ( image_5846526143423701914t_unit @ F @ I ) )
% 14.25/8.09 = X ) ) ) ).
% 14.25/8.09
% 14.25/8.09 thf(fact_4823_INF__eq__const,axiom,
% 14.25/8.09 ! [I: set_Pr4532377907799695533_nat_o,F: ( produc3658429121746597890et_nat > $o ) > product_unit,X: product_unit] :
% 14.25/8.09 ( ( I != bot_bo7824918357723582233_nat_o )
% 14.25/8.09 => ( ! [I3: produc3658429121746597890et_nat > $o] :
% 14.25/8.09 ( ( member6576561426505652726_nat_o @ I3 @ I )
% 14.25/8.09 => ( ( F @ I3 )
% 14.25/8.09 = X ) )
% 14.25/8.09 => ( ( comple2584293577114468500t_unit @ ( image_8451000719036923781t_unit @ F @ I ) )
% 14.25/8.09 = X ) ) ) ).
% 14.25/8.09
% 14.25/8.09 thf(fact_4824_card__1__singletonE,axiom,
% 14.25/8.09 ! [A: set_Product_unit] :
% 14.25/8.09 ( ( ( finite410649719033368117t_unit @ A )
% 14.25/8.09 = one_one_nat )
% 14.25/8.09 => ~ ! [X3: product_unit] :
% 14.25/8.09 ( A
% 14.25/8.09 != ( insert_Product_unit @ X3 @ bot_bo3957492148770167129t_unit ) ) ) ).
% 14.25/8.09
% 14.25/8.09 thf(fact_4825_card__1__singletonE,axiom,
% 14.25/8.09 ! [A: set_o] :
% 14.25/8.09 ( ( ( finite_card_o @ A )
% 14.25/8.09 = one_one_nat )
% 14.25/8.09 => ~ ! [X3: $o] :
% 14.25/8.09 ( A
% 14.25/8.09 != ( insert_o @ X3 @ bot_bot_set_o ) ) ) ).
% 14.25/8.09
% 14.25/8.09 thf(fact_4826_card__1__singletonE,axiom,
% 14.25/8.09 ! [A: set_nat] :
% 14.25/8.09 ( ( ( finite_card_nat @ A )
% 14.25/8.09 = one_one_nat )
% 14.25/8.09 => ~ ! [X3: nat] :
% 14.25/8.09 ( A
% 14.25/8.09 != ( insert_nat @ X3 @ bot_bot_set_nat ) ) ) ).
% 14.25/8.09
% 14.25/8.09 thf(fact_4827_card__1__singletonE,axiom,
% 14.25/8.09 ! [A: set_int] :
% 14.25/8.09 ( ( ( finite_card_int @ A )
% 14.25/8.09 = one_one_nat )
% 14.25/8.09 => ~ ! [X3: int] :
% 14.25/8.09 ( A
% 14.25/8.09 != ( insert_int @ X3 @ bot_bot_set_int ) ) ) ).
% 14.25/8.09
% 14.25/8.09 thf(fact_4828_card__1__singletonE,axiom,
% 14.25/8.09 ! [A: set_Pr4532377907799695533_nat_o] :
% 14.25/8.09 ( ( ( finite1363419556375932405_nat_o @ A )
% 14.25/8.09 = one_one_nat )
% 14.25/8.09 => ~ ! [X3: produc3658429121746597890et_nat > $o] :
% 14.25/8.09 ( A
% 14.25/8.09 != ( insert5175938949040314269_nat_o @ X3 @ bot_bo7824918357723582233_nat_o ) ) ) ).
% 14.25/8.09
% 14.25/8.09 thf(fact_4829_BNF__Greatest__Fixpoint_Osubst__Pair,axiom,
% 14.25/8.09 ! [P: nat > nat > $o,X: nat,Y: nat,A2: product_prod_nat_nat] :
% 14.25/8.09 ( ( P @ X @ Y )
% 14.25/8.09 => ( ( A2
% 14.25/8.09 = ( product_Pair_nat_nat @ X @ Y ) )
% 14.25/8.09 => ( P @ ( product_fst_nat_nat @ A2 ) @ ( product_snd_nat_nat @ A2 ) ) ) ) ).
% 14.25/8.09
% 14.25/8.09 thf(fact_4830_BNF__Greatest__Fixpoint_Osubst__Pair,axiom,
% 14.25/8.09 ! [P: ( produc3658429121746597890et_nat > $o ) > produc3658429121746597890et_nat > $o,X: produc3658429121746597890et_nat > $o,Y: produc3658429121746597890et_nat,A2: produc3925858234332021118et_nat] :
% 14.25/8.09 ( ( P @ X @ Y )
% 14.25/8.09 => ( ( A2
% 14.25/8.09 = ( produc5001842942810119800et_nat @ X @ Y ) )
% 14.25/8.09 => ( P @ ( produc995936583742144908et_nat @ A2 ) @ ( produc4011572625026189258et_nat @ A2 ) ) ) ) ).
% 14.25/8.09
% 14.25/8.09 thf(fact_4831_BNF__Greatest__Fixpoint_Osubst__Pair,axiom,
% 14.25/8.09 ! [P: ( produc3658429121746597890et_nat > $o ) > produc3925858234332021118et_nat > $o,X: produc3658429121746597890et_nat > $o,Y: produc3925858234332021118et_nat,A2: produc2732055786443039994et_nat] :
% 14.25/8.09 ( ( P @ X @ Y )
% 14.25/8.09 => ( ( A2
% 14.25/8.09 = ( produc2245416461498447860et_nat @ X @ Y ) )
% 14.25/8.09 => ( P @ ( produc180342877477747464et_nat @ A2 ) @ ( produc5374455773327741254et_nat @ A2 ) ) ) ) ).
% 14.25/8.09
% 14.25/8.09 thf(fact_4832_BNF__Greatest__Fixpoint_Osubst__Pair,axiom,
% 14.25/8.09 ! [P: heap_e7401611519738050253t_unit > set_nat > $o,X: heap_e7401611519738050253t_unit,Y: set_nat,A2: produc3658429121746597890et_nat] :
% 14.25/8.09 ( ( P @ X @ Y )
% 14.25/8.09 => ( ( A2
% 14.25/8.09 = ( produc7507926704131184380et_nat @ X @ Y ) )
% 14.25/8.09 => ( P @ ( produc1824681642469235216et_nat @ A2 ) @ ( produc8586169260539613262et_nat @ A2 ) ) ) ) ).
% 14.25/8.09
% 14.25/8.09 thf(fact_4833_BNF__Greatest__Fixpoint_Osubst__Pair,axiom,
% 14.25/8.09 ! [P: int > int > $o,X: int,Y: int,A2: product_prod_int_int] :
% 14.25/8.09 ( ( P @ X @ Y )
% 14.25/8.09 => ( ( A2
% 14.25/8.09 = ( product_Pair_int_int @ X @ Y ) )
% 14.25/8.09 => ( P @ ( product_fst_int_int @ A2 ) @ ( product_snd_int_int @ A2 ) ) ) ) ).
% 14.25/8.09
% 14.25/8.09 thf(fact_4834_prod_Oexhaust__sel,axiom,
% 14.25/8.09 ! [Prod: product_prod_nat_nat] :
% 14.25/8.09 ( Prod
% 14.25/8.09 = ( product_Pair_nat_nat @ ( product_fst_nat_nat @ Prod ) @ ( product_snd_nat_nat @ Prod ) ) ) ).
% 14.25/8.09
% 14.25/8.09 thf(fact_4835_prod_Oexhaust__sel,axiom,
% 14.25/8.09 ! [Prod: produc3925858234332021118et_nat] :
% 14.25/8.09 ( Prod
% 14.25/8.09 = ( produc5001842942810119800et_nat @ ( produc995936583742144908et_nat @ Prod ) @ ( produc4011572625026189258et_nat @ Prod ) ) ) ).
% 14.25/8.09
% 14.25/8.09 thf(fact_4836_prod_Oexhaust__sel,axiom,
% 14.25/8.09 ! [Prod: produc2732055786443039994et_nat] :
% 14.25/8.09 ( Prod
% 14.25/8.09 = ( produc2245416461498447860et_nat @ ( produc180342877477747464et_nat @ Prod ) @ ( produc5374455773327741254et_nat @ Prod ) ) ) ).
% 14.25/8.09
% 14.25/8.09 thf(fact_4837_prod_Oexhaust__sel,axiom,
% 14.25/8.09 ! [Prod: produc3658429121746597890et_nat] :
% 14.25/8.09 ( Prod
% 14.25/8.09 = ( produc7507926704131184380et_nat @ ( produc1824681642469235216et_nat @ Prod ) @ ( produc8586169260539613262et_nat @ Prod ) ) ) ).
% 14.25/8.09
% 14.25/8.09 thf(fact_4838_prod_Oexhaust__sel,axiom,
% 14.25/8.09 ! [Prod: product_prod_int_int] :
% 14.25/8.09 ( Prod
% 14.25/8.09 = ( product_Pair_int_int @ ( product_fst_int_int @ Prod ) @ ( product_snd_int_int @ Prod ) ) ) ).
% 14.25/8.09
% 14.25/8.09 thf(fact_4839_surjective__pairing,axiom,
% 14.25/8.09 ! [T4: product_prod_nat_nat] :
% 14.25/8.09 ( T4
% 14.25/8.09 = ( product_Pair_nat_nat @ ( product_fst_nat_nat @ T4 ) @ ( product_snd_nat_nat @ T4 ) ) ) ).
% 14.25/8.09
% 14.25/8.09 thf(fact_4840_surjective__pairing,axiom,
% 14.25/8.09 ! [T4: produc3925858234332021118et_nat] :
% 14.25/8.09 ( T4
% 14.25/8.09 = ( produc5001842942810119800et_nat @ ( produc995936583742144908et_nat @ T4 ) @ ( produc4011572625026189258et_nat @ T4 ) ) ) ).
% 14.25/8.09
% 14.25/8.09 thf(fact_4841_surjective__pairing,axiom,
% 14.25/8.09 ! [T4: produc2732055786443039994et_nat] :
% 14.25/8.09 ( T4
% 14.25/8.09 = ( produc2245416461498447860et_nat @ ( produc180342877477747464et_nat @ T4 ) @ ( produc5374455773327741254et_nat @ T4 ) ) ) ).
% 14.25/8.09
% 14.25/8.09 thf(fact_4842_surjective__pairing,axiom,
% 14.25/8.09 ! [T4: produc3658429121746597890et_nat] :
% 14.25/8.09 ( T4
% 14.25/8.09 = ( produc7507926704131184380et_nat @ ( produc1824681642469235216et_nat @ T4 ) @ ( produc8586169260539613262et_nat @ T4 ) ) ) ).
% 14.25/8.09
% 14.25/8.09 thf(fact_4843_surjective__pairing,axiom,
% 14.25/8.09 ! [T4: product_prod_int_int] :
% 14.25/8.09 ( T4
% 14.25/8.09 = ( product_Pair_int_int @ ( product_fst_int_int @ T4 ) @ ( product_snd_int_int @ T4 ) ) ) ).
% 14.25/8.09
% 14.25/8.09 thf(fact_4844_card__insert__le,axiom,
% 14.25/8.09 ! [A: set_nat,X: nat] : ( ord_less_eq_nat @ ( finite_card_nat @ A ) @ ( finite_card_nat @ ( insert_nat @ X @ A ) ) ) ).
% 14.25/8.09
% 14.25/8.09 thf(fact_4845_card__insert__le,axiom,
% 14.25/8.09 ! [A: set_int,X: int] : ( ord_less_eq_nat @ ( finite_card_int @ A ) @ ( finite_card_int @ ( insert_int @ X @ A ) ) ) ).
% 14.25/8.09
% 14.25/8.09 thf(fact_4846_card__insert__le,axiom,
% 14.25/8.09 ! [A: set_o,X: $o] : ( ord_less_eq_nat @ ( finite_card_o @ A ) @ ( finite_card_o @ ( insert_o @ X @ A ) ) ) ).
% 14.25/8.09
% 14.25/8.09 thf(fact_4847_card__insert__le,axiom,
% 14.25/8.09 ! [A: set_Product_unit,X: product_unit] : ( ord_less_eq_nat @ ( finite410649719033368117t_unit @ A ) @ ( finite410649719033368117t_unit @ ( insert_Product_unit @ X @ A ) ) ) ).
% 14.25/8.09
% 14.25/8.09 thf(fact_4848_in__fst__imageE,axiom,
% 14.25/8.09 ! [X: heap_e7401611519738050253t_unit,S: set_Pr3948176798113811640et_nat] :
% 14.25/8.09 ( ( member242389114213480694t_unit @ X @ ( image_6824039242672219652t_unit @ produc1824681642469235216et_nat @ S ) )
% 14.25/8.09 => ~ ! [Y2: set_nat] :
% 14.25/8.09 ~ ( member6260224972018164377et_nat @ ( produc7507926704131184380et_nat @ X @ Y2 ) @ S ) ) ).
% 14.25/8.09
% 14.25/8.09 thf(fact_4849_in__fst__imageE,axiom,
% 14.25/8.09 ! [X: nat,S: set_Pr1261947904930325089at_nat] :
% 14.25/8.09 ( ( member_nat @ X @ ( image_2486076414777270412at_nat @ product_fst_nat_nat @ S ) )
% 14.25/8.09 => ~ ! [Y2: nat] :
% 14.25/8.09 ~ ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ Y2 ) @ S ) ) ).
% 14.25/8.09
% 14.25/8.09 thf(fact_4850_in__fst__imageE,axiom,
% 14.25/8.09 ! [X: produc3658429121746597890et_nat > $o,S: set_Pr3286484037609594932et_nat] :
% 14.25/8.09 ( ( member6576561426505652726_nat_o @ X @ ( image_6802380912191114376_nat_o @ produc995936583742144908et_nat @ S ) )
% 14.25/8.09 => ~ ! [Y2: produc3658429121746597890et_nat] :
% 14.25/8.09 ~ ( member1996754912294343701et_nat @ ( produc5001842942810119800et_nat @ X @ Y2 ) @ S ) ) ).
% 14.25/8.09
% 14.25/8.09 thf(fact_4851_in__fst__imageE,axiom,
% 14.25/8.09 ! [X: produc3658429121746597890et_nat > $o,S: set_Pr8536935166611901872et_nat] :
% 14.25/8.09 ( ( member6576561426505652726_nat_o @ X @ ( image_2517677226577294348_nat_o @ produc180342877477747464et_nat @ S ) )
% 14.25/8.09 => ~ ! [Y2: produc3925858234332021118et_nat] :
% 14.25/8.09 ~ ( member6124377750444531601et_nat @ ( produc2245416461498447860et_nat @ X @ Y2 ) @ S ) ) ).
% 14.25/8.09
% 14.25/8.09 thf(fact_4852_in__fst__imageE,axiom,
% 14.25/8.09 ! [X: int,S: set_Pr958786334691620121nt_int] :
% 14.25/8.09 ( ( member_int @ X @ ( image_5042161079198086560nt_int @ product_fst_int_int @ S ) )
% 14.25/8.09 => ~ ! [Y2: int] :
% 14.25/8.09 ~ ( member5262025264175285858nt_int @ ( product_Pair_int_int @ X @ Y2 ) @ S ) ) ).
% 14.25/8.09
% 14.25/8.09 thf(fact_4853_Inter__subset,axiom,
% 14.25/8.09 ! [A: set_set_nat,B4: set_nat] :
% 14.25/8.09 ( ! [X8: set_nat] :
% 14.25/8.09 ( ( member_set_nat @ X8 @ A )
% 14.25/8.09 => ( ord_less_eq_set_nat @ X8 @ B4 ) )
% 14.25/8.09 => ( ( A != bot_bot_set_set_nat )
% 14.25/8.09 => ( ord_less_eq_set_nat @ ( comple7806235888213564991et_nat @ A ) @ B4 ) ) ) ).
% 14.25/8.09
% 14.25/8.09 thf(fact_4854_Inter__empty,axiom,
% 14.25/8.09 ( ( comple7216939928495492468t_unit @ bot_bo6284674557190688313t_unit )
% 14.25/8.09 = top_to1996260823553986621t_unit ) ).
% 14.25/8.09
% 14.25/8.09 thf(fact_4855_Inter__empty,axiom,
% 14.25/8.09 ( ( comple3063163877087187839_set_o @ bot_bot_set_set_o )
% 14.25/8.09 = top_top_set_o ) ).
% 14.25/8.09
% 14.25/8.09 thf(fact_4856_Inter__empty,axiom,
% 14.25/8.09 ( ( comple3628384868704368283et_int @ bot_bot_set_set_int )
% 14.25/8.09 = top_top_set_int ) ).
% 14.25/8.09
% 14.25/8.09 thf(fact_4857_Inter__empty,axiom,
% 14.25/8.09 ( ( comple7806235888213564991et_nat @ bot_bot_set_set_nat )
% 14.25/8.09 = top_top_set_nat ) ).
% 14.25/8.09
% 14.25/8.09 thf(fact_4858_fst__eq__Domain,axiom,
% 14.25/8.09 ! [R3: set_Pr958786334691620121nt_int] :
% 14.25/8.09 ( ( image_5042161079198086560nt_int @ product_fst_int_int @ R3 )
% 14.25/8.09 = ( domain_int_int @ R3 ) ) ).
% 14.25/8.09
% 14.25/8.09 thf(fact_4859_Domain__fst,axiom,
% 14.25/8.09 ( domain_int_int
% 14.25/8.09 = ( image_5042161079198086560nt_int @ product_fst_int_int ) ) ).
% 14.25/8.09
% 14.25/8.09 thf(fact_4860_card__partition,axiom,
% 14.25/8.09 ! [C5: set_set_Code_integer,K3: nat] :
% 14.25/8.09 ( ( finite6931041176100689706nteger @ C5 )
% 14.25/8.09 => ( ( finite6017078050557962740nteger @ ( comple739944243200306918nteger @ C5 ) )
% 14.25/8.09 => ( ! [C2: set_Code_integer] :
% 14.25/8.09 ( ( member1834298987299537450nteger @ C2 @ C5 )
% 14.25/8.09 => ( ( finite4902975817058060853nteger @ C2 )
% 14.25/8.09 = K3 ) )
% 14.25/8.09 => ( ! [C1: set_Code_integer,C22: set_Code_integer] :
% 14.25/8.09 ( ( member1834298987299537450nteger @ C1 @ C5 )
% 14.25/8.09 => ( ( member1834298987299537450nteger @ C22 @ C5 )
% 14.25/8.09 => ( ( C1 != C22 )
% 14.25/8.09 => ( ( inf_in1364745209274528805nteger @ C1 @ C22 )
% 14.25/8.09 = bot_bo3990330152332043303nteger ) ) ) )
% 14.25/8.09 => ( ( times_times_nat @ K3 @ ( finite1340570857867686763nteger @ C5 ) )
% 14.25/8.09 = ( finite4902975817058060853nteger @ ( comple739944243200306918nteger @ C5 ) ) ) ) ) ) ) ).
% 14.25/8.09
% 14.25/8.09 thf(fact_4861_card__partition,axiom,
% 14.25/8.09 ! [C5: set_set_Product_unit,K3: nat] :
% 14.25/8.09 ( ( finite1772178364199683094t_unit @ C5 )
% 14.25/8.09 => ( ( finite4290736615968046902t_unit @ ( comple3349607820504768346t_unit @ C5 ) )
% 14.25/8.09 => ( ! [C2: set_Product_unit] :
% 14.25/8.09 ( ( member5877623283571906838t_unit @ C2 @ C5 )
% 14.25/8.09 => ( ( finite410649719033368117t_unit @ C2 )
% 14.25/8.09 = K3 ) )
% 14.25/8.09 => ( ! [C1: set_Product_unit,C22: set_Product_unit] :
% 14.25/8.09 ( ( member5877623283571906838t_unit @ C1 @ C5 )
% 14.25/8.09 => ( ( member5877623283571906838t_unit @ C22 @ C5 )
% 14.25/8.09 => ( ( C1 != C22 )
% 14.25/8.09 => ( ( inf_in4660618365625256667t_unit @ C1 @ C22 )
% 14.25/8.09 = bot_bo3957492148770167129t_unit ) ) ) )
% 14.25/8.09 => ( ( times_times_nat @ K3 @ ( finite22775419188893461t_unit @ C5 ) )
% 14.25/8.09 = ( finite410649719033368117t_unit @ ( comple3349607820504768346t_unit @ C5 ) ) ) ) ) ) ) ).
% 14.25/8.09
% 14.25/8.09 thf(fact_4862_card__partition,axiom,
% 14.25/8.09 ! [C5: set_set_o,K3: nat] :
% 14.25/8.09 ( ( finite_finite_set_o @ C5 )
% 14.25/8.09 => ( ( finite_finite_o @ ( comple90263536869209701_set_o @ C5 ) )
% 14.25/8.09 => ( ! [C2: set_o] :
% 14.25/8.09 ( ( member_set_o @ C2 @ C5 )
% 14.25/8.09 => ( ( finite_card_o @ C2 )
% 14.25/8.09 = K3 ) )
% 14.25/8.09 => ( ! [C1: set_o,C22: set_o] :
% 14.25/8.09 ( ( member_set_o @ C1 @ C5 )
% 14.25/8.09 => ( ( member_set_o @ C22 @ C5 )
% 14.25/8.09 => ( ( C1 != C22 )
% 14.25/8.09 => ( ( inf_inf_set_o @ C1 @ C22 )
% 14.25/8.09 = bot_bot_set_o ) ) ) )
% 14.25/8.09 => ( ( times_times_nat @ K3 @ ( finite_card_set_o @ C5 ) )
% 14.25/8.09 = ( finite_card_o @ ( comple90263536869209701_set_o @ C5 ) ) ) ) ) ) ) ).
% 14.25/8.09
% 14.25/8.09 thf(fact_4863_card__partition,axiom,
% 14.25/8.09 ! [C5: set_set_nat,K3: nat] :
% 14.25/8.09 ( ( finite1152437895449049373et_nat @ C5 )
% 14.25/8.09 => ( ( finite_finite_nat @ ( comple7399068483239264473et_nat @ C5 ) )
% 14.25/8.09 => ( ! [C2: set_nat] :
% 14.25/8.09 ( ( member_set_nat @ C2 @ C5 )
% 14.25/8.09 => ( ( finite_card_nat @ C2 )
% 14.25/8.09 = K3 ) )
% 14.25/8.09 => ( ! [C1: set_nat,C22: set_nat] :
% 14.25/8.09 ( ( member_set_nat @ C1 @ C5 )
% 14.25/8.09 => ( ( member_set_nat @ C22 @ C5 )
% 14.25/8.09 => ( ( C1 != C22 )
% 14.25/8.09 => ( ( inf_inf_set_nat @ C1 @ C22 )
% 14.25/8.09 = bot_bot_set_nat ) ) ) )
% 14.25/8.09 => ( ( times_times_nat @ K3 @ ( finite_card_set_nat @ C5 ) )
% 14.25/8.09 = ( finite_card_nat @ ( comple7399068483239264473et_nat @ C5 ) ) ) ) ) ) ) ).
% 14.25/8.09
% 14.25/8.09 thf(fact_4864_card__partition,axiom,
% 14.25/8.09 ! [C5: set_set_int,K3: nat] :
% 14.25/8.09 ( ( finite6197958912794628473et_int @ C5 )
% 14.25/8.09 => ( ( finite_finite_int @ ( comple3221217463730067765et_int @ C5 ) )
% 14.25/8.09 => ( ! [C2: set_int] :
% 14.25/8.09 ( ( member_set_int @ C2 @ C5 )
% 14.25/8.09 => ( ( finite_card_int @ C2 )
% 14.25/8.09 = K3 ) )
% 14.25/8.09 => ( ! [C1: set_int,C22: set_int] :
% 14.25/8.09 ( ( member_set_int @ C1 @ C5 )
% 14.25/8.09 => ( ( member_set_int @ C22 @ C5 )
% 14.25/8.09 => ( ( C1 != C22 )
% 14.25/8.09 => ( ( inf_inf_set_int @ C1 @ C22 )
% 14.25/8.09 = bot_bot_set_int ) ) ) )
% 14.25/8.09 => ( ( times_times_nat @ K3 @ ( finite_card_set_int @ C5 ) )
% 14.25/8.09 = ( finite_card_int @ ( comple3221217463730067765et_int @ C5 ) ) ) ) ) ) ) ).
% 14.25/8.09
% 14.25/8.09 thf(fact_4865_card__partition,axiom,
% 14.25/8.09 ! [C5: set_se1353816476878259085_nat_o,K3: nat] :
% 14.25/8.09 ( ( finite6069017635265841110_nat_o @ C5 )
% 14.25/8.09 => ( ( finite3252695134891459830_nat_o @ ( comple3382087578465172250_nat_o @ C5 ) )
% 14.25/8.09 => ( ! [C2: set_Pr4532377907799695533_nat_o] :
% 14.25/8.09 ( ( member1452989533377600726_nat_o @ C2 @ C5 )
% 14.25/8.09 => ( ( finite1363419556375932405_nat_o @ C2 )
% 14.25/8.09 = K3 ) )
% 14.25/8.09 => ( ! [C1: set_Pr4532377907799695533_nat_o,C22: set_Pr4532377907799695533_nat_o] :
% 14.25/8.09 ( ( member1452989533377600726_nat_o @ C1 @ C5 )
% 14.25/8.09 => ( ( member1452989533377600726_nat_o @ C22 @ C5 )
% 14.25/8.09 => ( ( C1 != C22 )
% 14.25/8.09 => ( ( inf_in1906310914598751387_nat_o @ C1 @ C22 )
% 14.25/8.09 = bot_bo7824918357723582233_nat_o ) ) ) )
% 14.25/8.09 => ( ( times_times_nat @ K3 @ ( finite4731484169140532437_nat_o @ C5 ) )
% 14.25/8.09 = ( finite1363419556375932405_nat_o @ ( comple3382087578465172250_nat_o @ C5 ) ) ) ) ) ) ) ).
% 14.25/8.09
% 14.25/8.09 thf(fact_4866_Inter__UNIV,axiom,
% 14.25/8.09 ( ( comple7216939928495492468t_unit @ top_to1767297665138865437t_unit )
% 14.25/8.09 = bot_bo3957492148770167129t_unit ) ).
% 14.25/8.09
% 14.25/8.09 thf(fact_4867_Inter__UNIV,axiom,
% 14.25/8.09 ( ( comple3063163877087187839_set_o @ top_top_set_set_o )
% 14.25/8.09 = bot_bot_set_o ) ).
% 14.25/8.09
% 14.25/8.09 thf(fact_4868_Inter__UNIV,axiom,
% 14.25/8.09 ( ( comple3628384868704368283et_int @ top_top_set_set_int )
% 14.25/8.09 = bot_bot_set_int ) ).
% 14.25/8.09
% 14.25/8.09 thf(fact_4869_Inter__UNIV,axiom,
% 14.25/8.09 ( ( comple79274577958421812_nat_o @ top_to8515903933492314333_nat_o )
% 14.25/8.09 = bot_bo7824918357723582233_nat_o ) ).
% 14.25/8.09
% 14.25/8.09 thf(fact_4870_Inter__UNIV,axiom,
% 14.25/8.09 ( ( comple7806235888213564991et_nat @ top_top_set_set_nat )
% 14.25/8.09 = bot_bot_set_nat ) ).
% 14.25/8.09
% 14.25/8.09 thf(fact_4871_inv__on__f__range,axiom,
% 14.25/8.09 ! [Y: code_integer,F: code_integer > code_integer,A: set_Code_integer] :
% 14.25/8.09 ( ( member_Code_integer @ Y @ ( image_4470545334726330049nteger @ F @ A ) )
% 14.25/8.09 => ( member_Code_integer @ ( inv_on4869000451854154612nteger @ F @ A @ Y ) @ A ) ) ).
% 14.25/8.09
% 14.25/8.09 thf(fact_4872_inv__on__f__range,axiom,
% 14.25/8.09 ! [Y: code_integer,F: nat > code_integer,A: set_nat] :
% 14.25/8.09 ( ( member_Code_integer @ Y @ ( image_1215581382706833972nteger @ F @ A ) )
% 14.25/8.09 => ( member_nat @ ( inv_on1369624593536199399nteger @ F @ A @ Y ) @ A ) ) ).
% 14.25/8.09
% 14.25/8.09 thf(fact_4873_inv__on__f__range,axiom,
% 14.25/8.09 ! [Y: nat,F: nat > nat,A: set_nat] :
% 14.25/8.09 ( ( member_nat @ Y @ ( image_nat_nat2 @ F @ A ) )
% 14.25/8.09 => ( member_nat @ ( inv_on_nat_nat @ F @ A @ Y ) @ A ) ) ).
% 14.25/8.09
% 14.25/8.09 thf(fact_4874_inv__on__f__range,axiom,
% 14.25/8.09 ! [Y: nat,F: int > nat,A: set_int] :
% 14.25/8.09 ( ( member_nat @ Y @ ( image_int_nat2 @ F @ A ) )
% 14.25/8.09 => ( member_int @ ( inv_on_int_nat @ F @ A @ Y ) @ A ) ) ).
% 14.25/8.09
% 14.25/8.09 thf(fact_4875_inv__on__f__range,axiom,
% 14.25/8.09 ! [Y: nat,F: $o > nat,A: set_o] :
% 14.25/8.09 ( ( member_nat @ Y @ ( image_o_nat2 @ F @ A ) )
% 14.25/8.09 => ( member_o @ ( inv_on_o_nat @ F @ A @ Y ) @ A ) ) ).
% 14.25/8.09
% 14.25/8.09 thf(fact_4876_inv__on__f__range,axiom,
% 14.25/8.09 ! [Y: nat,F: product_unit > nat,A: set_Product_unit] :
% 14.25/8.09 ( ( member_nat @ Y @ ( image_875570014554754200it_nat @ F @ A ) )
% 14.25/8.09 => ( member_Product_unit @ ( inv_on7727615724625388965it_nat @ F @ A @ Y ) @ A ) ) ).
% 14.25/8.09
% 14.25/8.09 thf(fact_4877_inv__on__f__range,axiom,
% 14.25/8.09 ! [Y: int,F: nat > int,A: set_nat] :
% 14.25/8.09 ( ( member_int @ Y @ ( image_nat_int2 @ F @ A ) )
% 14.25/8.09 => ( member_nat @ ( inv_on_nat_int @ F @ A @ Y ) @ A ) ) ).
% 14.25/8.09
% 14.25/8.09 thf(fact_4878_inv__on__f__range,axiom,
% 14.25/8.09 ! [Y: int,F: int > int,A: set_int] :
% 14.25/8.09 ( ( member_int @ Y @ ( image_int_int2 @ F @ A ) )
% 14.25/8.09 => ( member_int @ ( inv_on_int_int @ F @ A @ Y ) @ A ) ) ).
% 14.25/8.09
% 14.25/8.09 thf(fact_4879_inv__on__f__range,axiom,
% 14.25/8.09 ! [Y: int,F: $o > int,A: set_o] :
% 14.25/8.09 ( ( member_int @ Y @ ( image_o_int2 @ F @ A ) )
% 14.25/8.09 => ( member_o @ ( inv_on_o_int @ F @ A @ Y ) @ A ) ) ).
% 14.25/8.09
% 14.25/8.09 thf(fact_4880_inv__on__f__range,axiom,
% 14.25/8.09 ! [Y: int,F: product_unit > int,A: set_Product_unit] :
% 14.25/8.09 ( ( member_int @ Y @ ( image_873079544045703924it_int @ F @ A ) )
% 14.25/8.09 => ( member_Product_unit @ ( inv_on7725125254116338689it_int @ F @ A @ Y ) @ A ) ) ).
% 14.25/8.09
% 14.25/8.09 thf(fact_4881_f__inv__on__f,axiom,
% 14.25/8.09 ! [Y: set_nat,F: nat > set_nat,A: set_nat] :
% 14.25/8.09 ( ( member_set_nat @ Y @ ( image_nat_set_nat @ F @ A ) )
% 14.25/8.09 => ( ( F @ ( inv_on_nat_set_nat @ F @ A @ Y ) )
% 14.25/8.09 = Y ) ) ).
% 14.25/8.09
% 14.25/8.09 thf(fact_4882_f__inv__on__f,axiom,
% 14.25/8.09 ! [Y: code_integer,F: code_integer > code_integer,A: set_Code_integer] :
% 14.25/8.09 ( ( member_Code_integer @ Y @ ( image_4470545334726330049nteger @ F @ A ) )
% 14.25/8.09 => ( ( F @ ( inv_on4869000451854154612nteger @ F @ A @ Y ) )
% 14.25/8.09 = Y ) ) ).
% 14.25/8.09
% 14.25/8.09 thf(fact_4883_f__inv__on__f,axiom,
% 14.25/8.09 ! [Y: code_integer,F: nat > code_integer,A: set_nat] :
% 14.25/8.09 ( ( member_Code_integer @ Y @ ( image_1215581382706833972nteger @ F @ A ) )
% 14.25/8.09 => ( ( F @ ( inv_on1369624593536199399nteger @ F @ A @ Y ) )
% 14.25/8.09 = Y ) ) ).
% 14.25/8.09
% 14.25/8.09 thf(fact_4884_f__inv__on__f,axiom,
% 14.25/8.09 ! [Y: nat,F: nat > nat,A: set_nat] :
% 14.25/8.09 ( ( member_nat @ Y @ ( image_nat_nat2 @ F @ A ) )
% 14.25/8.09 => ( ( F @ ( inv_on_nat_nat @ F @ A @ Y ) )
% 14.25/8.09 = Y ) ) ).
% 14.25/8.09
% 14.25/8.09 thf(fact_4885_cINF__greatest,axiom,
% 14.25/8.09 ! [A: set_Product_unit,M4: int,F: product_unit > int] :
% 14.25/8.09 ( ( A != bot_bo3957492148770167129t_unit )
% 14.25/8.09 => ( ! [X3: product_unit] :
% 14.25/8.09 ( ( member_Product_unit @ X3 @ A )
% 14.25/8.09 => ( ord_less_eq_int @ M4 @ ( F @ X3 ) ) )
% 14.25/8.09 => ( ord_less_eq_int @ M4 @ ( complete_Inf_Inf_int @ ( image_873079544045703924it_int @ F @ A ) ) ) ) ) ).
% 14.25/8.09
% 14.25/8.09 thf(fact_4886_cINF__greatest,axiom,
% 14.25/8.09 ! [A: set_o,M4: int,F: $o > int] :
% 14.25/8.09 ( ( A != bot_bot_set_o )
% 14.25/8.09 => ( ! [X3: $o] :
% 14.25/8.09 ( ( member_o @ X3 @ A )
% 14.25/8.09 => ( ord_less_eq_int @ M4 @ ( F @ X3 ) ) )
% 14.25/8.09 => ( ord_less_eq_int @ M4 @ ( complete_Inf_Inf_int @ ( image_o_int2 @ F @ A ) ) ) ) ) ).
% 14.25/8.09
% 14.25/8.09 thf(fact_4887_cINF__greatest,axiom,
% 14.25/8.09 ! [A: set_nat,M4: int,F: nat > int] :
% 14.25/8.09 ( ( A != bot_bot_set_nat )
% 14.25/8.09 => ( ! [X3: nat] :
% 14.25/8.09 ( ( member_nat @ X3 @ A )
% 14.25/8.09 => ( ord_less_eq_int @ M4 @ ( F @ X3 ) ) )
% 14.25/8.09 => ( ord_less_eq_int @ M4 @ ( complete_Inf_Inf_int @ ( image_nat_int2 @ F @ A ) ) ) ) ) ).
% 14.25/8.09
% 14.25/8.09 thf(fact_4888_cINF__greatest,axiom,
% 14.25/8.09 ! [A: set_int,M4: int,F: int > int] :
% 14.25/8.09 ( ( A != bot_bot_set_int )
% 14.25/8.09 => ( ! [X3: int] :
% 14.25/8.09 ( ( member_int @ X3 @ A )
% 14.25/8.09 => ( ord_less_eq_int @ M4 @ ( F @ X3 ) ) )
% 14.25/8.09 => ( ord_less_eq_int @ M4 @ ( complete_Inf_Inf_int @ ( image_int_int2 @ F @ A ) ) ) ) ) ).
% 14.25/8.09
% 14.25/8.09 thf(fact_4889_cINF__greatest,axiom,
% 14.25/8.09 ! [A: set_Product_unit,M4: nat,F: product_unit > nat] :
% 14.25/8.09 ( ( A != bot_bo3957492148770167129t_unit )
% 14.25/8.09 => ( ! [X3: product_unit] :
% 14.25/8.09 ( ( member_Product_unit @ X3 @ A )
% 14.25/8.09 => ( ord_less_eq_nat @ M4 @ ( F @ X3 ) ) )
% 14.25/8.09 => ( ord_less_eq_nat @ M4 @ ( complete_Inf_Inf_nat @ ( image_875570014554754200it_nat @ F @ A ) ) ) ) ) ).
% 14.25/8.09
% 14.25/8.09 thf(fact_4890_cINF__greatest,axiom,
% 14.25/8.09 ! [A: set_o,M4: nat,F: $o > nat] :
% 14.25/8.09 ( ( A != bot_bot_set_o )
% 14.25/8.09 => ( ! [X3: $o] :
% 14.25/8.09 ( ( member_o @ X3 @ A )
% 14.25/8.09 => ( ord_less_eq_nat @ M4 @ ( F @ X3 ) ) )
% 14.25/8.09 => ( ord_less_eq_nat @ M4 @ ( complete_Inf_Inf_nat @ ( image_o_nat2 @ F @ A ) ) ) ) ) ).
% 14.25/8.09
% 14.25/8.09 thf(fact_4891_cINF__greatest,axiom,
% 14.25/8.09 ! [A: set_nat,M4: nat,F: nat > nat] :
% 14.25/8.09 ( ( A != bot_bot_set_nat )
% 14.25/8.09 => ( ! [X3: nat] :
% 14.25/8.09 ( ( member_nat @ X3 @ A )
% 14.25/8.09 => ( ord_less_eq_nat @ M4 @ ( F @ X3 ) ) )
% 14.25/8.09 => ( ord_less_eq_nat @ M4 @ ( complete_Inf_Inf_nat @ ( image_nat_nat2 @ F @ A ) ) ) ) ) ).
% 14.25/8.09
% 14.25/8.09 thf(fact_4892_cINF__greatest,axiom,
% 14.25/8.09 ! [A: set_int,M4: nat,F: int > nat] :
% 14.25/8.09 ( ( A != bot_bot_set_int )
% 14.25/8.09 => ( ! [X3: int] :
% 14.25/8.09 ( ( member_int @ X3 @ A )
% 14.25/8.09 => ( ord_less_eq_nat @ M4 @ ( F @ X3 ) ) )
% 14.25/8.09 => ( ord_less_eq_nat @ M4 @ ( complete_Inf_Inf_nat @ ( image_int_nat2 @ F @ A ) ) ) ) ) ).
% 14.25/8.09
% 14.25/8.09 thf(fact_4893_cINF__greatest,axiom,
% 14.25/8.09 ! [A: set_Product_unit,M4: product_unit,F: product_unit > product_unit] :
% 14.25/8.09 ( ( A != bot_bo3957492148770167129t_unit )
% 14.25/8.09 => ( ! [X3: product_unit] :
% 14.25/8.09 ( ( member_Product_unit @ X3 @ A )
% 14.25/8.09 => ( ord_le3221252021190050221t_unit @ M4 @ ( F @ X3 ) ) )
% 14.25/8.09 => ( ord_le3221252021190050221t_unit @ M4 @ ( comple2584293577114468500t_unit @ ( image_405062704495631173t_unit @ F @ A ) ) ) ) ) ).
% 14.25/8.09
% 14.25/8.09 thf(fact_4894_cINF__greatest,axiom,
% 14.25/8.09 ! [A: set_o,M4: product_unit,F: $o > product_unit] :
% 14.25/8.09 ( ( A != bot_bot_set_o )
% 14.25/8.09 => ( ! [X3: $o] :
% 14.25/8.09 ( ( member_o @ X3 @ A )
% 14.25/8.09 => ( ord_le3221252021190050221t_unit @ M4 @ ( F @ X3 ) ) )
% 14.25/8.09 => ( ord_le3221252021190050221t_unit @ M4 @ ( comple2584293577114468500t_unit @ ( image_o_Product_unit2 @ F @ A ) ) ) ) ) ).
% 14.25/8.09
% 14.25/8.09 thf(fact_4895_INF__eq__iff,axiom,
% 14.25/8.09 ! [I: set_Product_unit,F: product_unit > product_unit,C: product_unit] :
% 14.25/8.09 ( ( I != bot_bo3957492148770167129t_unit )
% 14.25/8.09 => ( ! [I3: product_unit] :
% 14.25/8.09 ( ( member_Product_unit @ I3 @ I )
% 14.25/8.09 => ( ord_le3221252021190050221t_unit @ ( F @ I3 ) @ C ) )
% 14.25/8.09 => ( ( ( comple2584293577114468500t_unit @ ( image_405062704495631173t_unit @ F @ I ) )
% 14.25/8.09 = C )
% 14.25/8.09 = ( ! [X2: product_unit] :
% 14.25/8.09 ( ( member_Product_unit @ X2 @ I )
% 14.25/8.09 => ( ( F @ X2 )
% 14.25/8.09 = C ) ) ) ) ) ) ).
% 14.25/8.09
% 14.25/8.09 thf(fact_4896_INF__eq__iff,axiom,
% 14.25/8.09 ! [I: set_o,F: $o > product_unit,C: product_unit] :
% 14.25/8.09 ( ( I != bot_bot_set_o )
% 14.25/8.09 => ( ! [I3: $o] :
% 14.25/8.09 ( ( member_o @ I3 @ I )
% 14.25/8.09 => ( ord_le3221252021190050221t_unit @ ( F @ I3 ) @ C ) )
% 14.25/8.09 => ( ( ( comple2584293577114468500t_unit @ ( image_o_Product_unit2 @ F @ I ) )
% 14.25/8.09 = C )
% 14.25/8.09 = ( ! [X2: $o] :
% 14.25/8.09 ( ( member_o @ X2 @ I )
% 14.25/8.09 => ( ( F @ X2 )
% 14.25/8.09 = C ) ) ) ) ) ) ).
% 14.25/8.09
% 14.25/8.09 thf(fact_4897_INF__eq__iff,axiom,
% 14.25/8.09 ! [I: set_nat,F: nat > product_unit,C: product_unit] :
% 14.25/8.09 ( ( I != bot_bot_set_nat )
% 14.25/8.09 => ( ! [I3: nat] :
% 14.25/8.09 ( ( member_nat @ I3 @ I )
% 14.25/8.09 => ( ord_le3221252021190050221t_unit @ ( F @ I3 ) @ C ) )
% 14.25/8.09 => ( ( ( comple2584293577114468500t_unit @ ( image_8730104196221521654t_unit @ F @ I ) )
% 14.25/8.09 = C )
% 14.25/8.09 = ( ! [X2: nat] :
% 14.25/8.09 ( ( member_nat @ X2 @ I )
% 14.25/8.09 => ( ( F @ X2 )
% 14.25/8.09 = C ) ) ) ) ) ) ).
% 14.25/8.09
% 14.25/8.09 thf(fact_4898_INF__eq__iff,axiom,
% 14.25/8.09 ! [I: set_int,F: int > product_unit,C: product_unit] :
% 14.25/8.09 ( ( I != bot_bot_set_int )
% 14.25/8.09 => ( ! [I3: int] :
% 14.25/8.09 ( ( member_int @ I3 @ I )
% 14.25/8.09 => ( ord_le3221252021190050221t_unit @ ( F @ I3 ) @ C ) )
% 14.25/8.09 => ( ( ( comple2584293577114468500t_unit @ ( image_5846526143423701914t_unit @ F @ I ) )
% 14.25/8.09 = C )
% 14.25/8.09 = ( ! [X2: int] :
% 14.25/8.09 ( ( member_int @ X2 @ I )
% 14.25/8.09 => ( ( F @ X2 )
% 14.25/8.09 = C ) ) ) ) ) ) ).
% 14.25/8.09
% 14.25/8.09 thf(fact_4899_INF__eq__iff,axiom,
% 14.25/8.09 ! [I: set_Product_unit,F: product_unit > filter_nat,C: filter_nat] :
% 14.25/8.09 ( ( I != bot_bo3957492148770167129t_unit )
% 14.25/8.09 => ( ! [I3: product_unit] :
% 14.25/8.09 ( ( member_Product_unit @ I3 @ I )
% 14.25/8.09 => ( ord_le2510731241096832064er_nat @ ( F @ I3 ) @ C ) )
% 14.25/8.09 => ( ( ( comple6612786467464916249er_nat @ ( image_1853478764106424488er_nat @ F @ I ) )
% 14.25/8.09 = C )
% 14.25/8.09 = ( ! [X2: product_unit] :
% 14.25/8.09 ( ( member_Product_unit @ X2 @ I )
% 14.25/8.09 => ( ( F @ X2 )
% 14.25/8.09 = C ) ) ) ) ) ) ).
% 14.25/8.09
% 14.25/8.09 thf(fact_4900_INF__eq__iff,axiom,
% 14.25/8.09 ! [I: set_o,F: $o > filter_nat,C: filter_nat] :
% 14.25/8.09 ( ( I != bot_bot_set_o )
% 14.25/8.09 => ( ! [I3: $o] :
% 14.25/8.09 ( ( member_o @ I3 @ I )
% 14.25/8.09 => ( ord_le2510731241096832064er_nat @ ( F @ I3 ) @ C ) )
% 14.25/8.09 => ( ( ( comple6612786467464916249er_nat @ ( image_o_filter_nat @ F @ I ) )
% 14.25/8.09 = C )
% 14.25/8.09 = ( ! [X2: $o] :
% 14.25/8.09 ( ( member_o @ X2 @ I )
% 14.25/8.09 => ( ( F @ X2 )
% 14.25/8.09 = C ) ) ) ) ) ) ).
% 14.25/8.09
% 14.25/8.09 thf(fact_4901_INF__eq__iff,axiom,
% 14.25/8.09 ! [I: set_nat,F: nat > filter_nat,C: filter_nat] :
% 14.25/8.09 ( ( I != bot_bot_set_nat )
% 14.25/8.09 => ( ! [I3: nat] :
% 14.25/8.09 ( ( member_nat @ I3 @ I )
% 14.25/8.09 => ( ord_le2510731241096832064er_nat @ ( F @ I3 ) @ C ) )
% 14.25/8.09 => ( ( ( comple6612786467464916249er_nat @ ( image_nat_filter_nat @ F @ I ) )
% 14.25/8.09 = C )
% 14.25/8.09 = ( ! [X2: nat] :
% 14.25/8.09 ( ( member_nat @ X2 @ I )
% 14.25/8.09 => ( ( F @ X2 )
% 14.25/8.09 = C ) ) ) ) ) ) ).
% 14.25/8.09
% 14.25/8.09 thf(fact_4902_INF__eq__iff,axiom,
% 14.25/8.09 ! [I: set_int,F: int > filter_nat,C: filter_nat] :
% 14.25/8.09 ( ( I != bot_bot_set_int )
% 14.25/8.09 => ( ! [I3: int] :
% 14.25/8.09 ( ( member_int @ I3 @ I )
% 14.25/8.09 => ( ord_le2510731241096832064er_nat @ ( F @ I3 ) @ C ) )
% 14.25/8.09 => ( ( ( comple6612786467464916249er_nat @ ( image_int_filter_nat @ F @ I ) )
% 14.25/8.09 = C )
% 14.25/8.09 = ( ! [X2: int] :
% 14.25/8.09 ( ( member_int @ X2 @ I )
% 14.25/8.09 => ( ( F @ X2 )
% 14.25/8.09 = C ) ) ) ) ) ) ).
% 14.25/8.09
% 14.25/8.09 thf(fact_4903_INF__eq__iff,axiom,
% 14.25/8.09 ! [I: set_Product_unit,F: product_unit > set_nat,C: set_nat] :
% 14.25/8.09 ( ( I != bot_bo3957492148770167129t_unit )
% 14.25/8.09 => ( ! [I3: product_unit] :
% 14.25/8.09 ( ( member_Product_unit @ I3 @ I )
% 14.25/8.09 => ( ord_less_eq_set_nat @ ( F @ I3 ) @ C ) )
% 14.25/8.09 => ( ( ( comple7806235888213564991et_nat @ ( image_8173849991827986510et_nat @ F @ I ) )
% 14.25/8.09 = C )
% 14.25/8.09 = ( ! [X2: product_unit] :
% 14.25/8.09 ( ( member_Product_unit @ X2 @ I )
% 14.25/8.09 => ( ( F @ X2 )
% 14.25/8.09 = C ) ) ) ) ) ) ).
% 14.25/8.09
% 14.25/8.09 thf(fact_4904_INF__eq__iff,axiom,
% 14.25/8.09 ! [I: set_o,F: $o > set_nat,C: set_nat] :
% 14.25/8.09 ( ( I != bot_bot_set_o )
% 14.25/8.09 => ( ! [I3: $o] :
% 14.25/8.09 ( ( member_o @ I3 @ I )
% 14.25/8.09 => ( ord_less_eq_set_nat @ ( F @ I3 ) @ C ) )
% 14.25/8.09 => ( ( ( comple7806235888213564991et_nat @ ( image_o_set_nat @ F @ I ) )
% 14.25/8.09 = C )
% 14.25/8.09 = ( ! [X2: $o] :
% 14.25/8.09 ( ( member_o @ X2 @ I )
% 14.25/8.09 => ( ( F @ X2 )
% 14.25/8.09 = C ) ) ) ) ) ) ).
% 14.25/8.09
% 14.25/8.09 thf(fact_4905_finite__less__Inf__iff,axiom,
% 14.25/8.09 ! [X4: set_int,A2: int] :
% 14.25/8.09 ( ( finite_finite_int @ X4 )
% 14.25/8.09 => ( ( X4 != bot_bot_set_int )
% 14.25/8.09 => ( ( ord_less_int @ A2 @ ( complete_Inf_Inf_int @ X4 ) )
% 14.25/8.09 = ( ! [X2: int] :
% 14.25/8.09 ( ( member_int @ X2 @ X4 )
% 14.25/8.09 => ( ord_less_int @ A2 @ X2 ) ) ) ) ) ) ).
% 14.25/8.09
% 14.25/8.09 thf(fact_4906_finite__less__Inf__iff,axiom,
% 14.25/8.09 ! [X4: set_nat,A2: nat] :
% 14.25/8.09 ( ( finite_finite_nat @ X4 )
% 14.25/8.09 => ( ( X4 != bot_bot_set_nat )
% 14.25/8.09 => ( ( ord_less_nat @ A2 @ ( complete_Inf_Inf_nat @ X4 ) )
% 14.25/8.09 = ( ! [X2: nat] :
% 14.25/8.09 ( ( member_nat @ X2 @ X4 )
% 14.25/8.09 => ( ord_less_nat @ A2 @ X2 ) ) ) ) ) ) ).
% 14.25/8.09
% 14.25/8.09 thf(fact_4907_finite__less__Inf__iff,axiom,
% 14.25/8.09 ! [X4: set_Product_unit,A2: product_unit] :
% 14.25/8.09 ( ( finite4290736615968046902t_unit @ X4 )
% 14.25/8.09 => ( ( X4 != bot_bo3957492148770167129t_unit )
% 14.25/8.09 => ( ( ord_le361264281704409273t_unit @ A2 @ ( comple2584293577114468500t_unit @ X4 ) )
% 14.25/8.09 = ( ! [X2: product_unit] :
% 14.25/8.09 ( ( member_Product_unit @ X2 @ X4 )
% 14.25/8.09 => ( ord_le361264281704409273t_unit @ A2 @ X2 ) ) ) ) ) ) ).
% 14.25/8.09
% 14.25/8.09 thf(fact_4908_Inf__le__Sup,axiom,
% 14.25/8.09 ! [A: set_o] :
% 14.25/8.09 ( ( A != bot_bot_set_o )
% 14.25/8.09 => ( ord_less_eq_o @ ( complete_Inf_Inf_o @ A ) @ ( complete_Sup_Sup_o @ A ) ) ) ).
% 14.25/8.09
% 14.25/8.09 thf(fact_4909_Inf__le__Sup,axiom,
% 14.25/8.09 ! [A: set_Pr4532377907799695533_nat_o] :
% 14.25/8.09 ( ( A != bot_bo7824918357723582233_nat_o )
% 14.25/8.09 => ( ord_le729326519192465773_nat_o @ ( comple90455603224345172_nat_o @ A ) @ ( comple4225350640907012154_nat_o @ A ) ) ) ).
% 14.25/8.09
% 14.25/8.09 thf(fact_4910_Inf__le__Sup,axiom,
% 14.25/8.09 ! [A: set_filter_nat] :
% 14.25/8.09 ( ( A != bot_bo498966703094740906er_nat )
% 14.25/8.09 => ( ord_le2510731241096832064er_nat @ ( comple6612786467464916249er_nat @ A ) @ ( comple2454491259644266675er_nat @ A ) ) ) ).
% 14.25/8.09
% 14.25/8.09 thf(fact_4911_Inf__le__Sup,axiom,
% 14.25/8.09 ! [A: set_set_nat] :
% 14.25/8.09 ( ( A != bot_bot_set_set_nat )
% 14.25/8.09 => ( ord_less_eq_set_nat @ ( comple7806235888213564991et_nat @ A ) @ ( comple7399068483239264473et_nat @ A ) ) ) ).
% 14.25/8.09
% 14.25/8.09 thf(fact_4912_Inf__le__Sup,axiom,
% 14.25/8.09 ! [A: set_Product_unit] :
% 14.25/8.09 ( ( A != bot_bo3957492148770167129t_unit )
% 14.25/8.09 => ( ord_le3221252021190050221t_unit @ ( comple2584293577114468500t_unit @ A ) @ ( comple4687483117567863418t_unit @ A ) ) ) ).
% 14.25/8.09
% 14.25/8.09 thf(fact_4913_finite__Inf__in,axiom,
% 14.25/8.09 ! [A: set_o] :
% 14.25/8.09 ( ( finite_finite_o @ A )
% 14.25/8.09 => ( ( A != bot_bot_set_o )
% 14.25/8.09 => ( ! [X3: $o,Y2: $o] :
% 14.25/8.09 ( ( member_o @ X3 @ A )
% 14.25/8.09 => ( ( member_o @ Y2 @ A )
% 14.25/8.09 => ( member_o @ ( inf_inf_o @ X3 @ Y2 ) @ A ) ) )
% 14.25/8.09 => ( member_o @ ( complete_Inf_Inf_o @ A ) @ A ) ) ) ) ).
% 14.25/8.09
% 14.25/8.09 thf(fact_4914_finite__Inf__in,axiom,
% 14.25/8.09 ! [A: set_Pr4532377907799695533_nat_o] :
% 14.25/8.09 ( ( finite3252695134891459830_nat_o @ A )
% 14.25/8.09 => ( ( A != bot_bo7824918357723582233_nat_o )
% 14.25/8.09 => ( ! [X3: produc3658429121746597890et_nat > $o,Y2: produc3658429121746597890et_nat > $o] :
% 14.25/8.09 ( ( member6576561426505652726_nat_o @ X3 @ A )
% 14.25/8.09 => ( ( member6576561426505652726_nat_o @ Y2 @ A )
% 14.25/8.09 => ( member6576561426505652726_nat_o @ ( inf_in1318976480646536635_nat_o @ X3 @ Y2 ) @ A ) ) )
% 14.25/8.09 => ( member6576561426505652726_nat_o @ ( comple90455603224345172_nat_o @ A ) @ A ) ) ) ) ).
% 14.25/8.09
% 14.25/8.09 thf(fact_4915_finite__Inf__in,axiom,
% 14.25/8.09 ! [A: set_set_nat] :
% 14.25/8.09 ( ( finite1152437895449049373et_nat @ A )
% 14.25/8.09 => ( ( A != bot_bot_set_set_nat )
% 14.25/8.09 => ( ! [X3: set_nat,Y2: set_nat] :
% 14.25/8.09 ( ( member_set_nat @ X3 @ A )
% 14.25/8.09 => ( ( member_set_nat @ Y2 @ A )
% 14.25/8.09 => ( member_set_nat @ ( inf_inf_set_nat @ X3 @ Y2 ) @ A ) ) )
% 14.25/8.10 => ( member_set_nat @ ( comple7806235888213564991et_nat @ A ) @ A ) ) ) ) ).
% 14.25/8.10
% 14.25/8.10 thf(fact_4916_finite__Inf__in,axiom,
% 14.25/8.10 ! [A: set_Product_unit] :
% 14.25/8.10 ( ( finite4290736615968046902t_unit @ A )
% 14.25/8.10 => ( ( A != bot_bo3957492148770167129t_unit )
% 14.25/8.10 => ( ! [X3: product_unit,Y2: product_unit] :
% 14.25/8.10 ( ( member_Product_unit @ X3 @ A )
% 14.25/8.10 => ( ( member_Product_unit @ Y2 @ A )
% 14.25/8.10 => ( member_Product_unit @ ( inf_inf_Product_unit @ X3 @ Y2 ) @ A ) ) )
% 14.25/8.10 => ( member_Product_unit @ ( comple2584293577114468500t_unit @ A ) @ A ) ) ) ) ).
% 14.25/8.10
% 14.25/8.10 thf(fact_4917_card__1__singletonI,axiom,
% 14.25/8.10 ! [S: set_Code_integer,X: code_integer] :
% 14.25/8.10 ( ( finite6017078050557962740nteger @ S )
% 14.25/8.10 => ( ( ( finite4902975817058060853nteger @ S )
% 14.25/8.10 = one_one_nat )
% 14.25/8.10 => ( ( member_Code_integer @ X @ S )
% 14.25/8.10 => ( S
% 14.25/8.10 = ( insert_Code_integer @ X @ bot_bo3990330152332043303nteger ) ) ) ) ) ).
% 14.25/8.10
% 14.25/8.10 thf(fact_4918_card__1__singletonI,axiom,
% 14.25/8.10 ! [S: set_Product_unit,X: product_unit] :
% 14.25/8.10 ( ( finite4290736615968046902t_unit @ S )
% 14.25/8.10 => ( ( ( finite410649719033368117t_unit @ S )
% 14.25/8.10 = one_one_nat )
% 14.25/8.10 => ( ( member_Product_unit @ X @ S )
% 14.25/8.10 => ( S
% 14.25/8.10 = ( insert_Product_unit @ X @ bot_bo3957492148770167129t_unit ) ) ) ) ) ).
% 14.25/8.10
% 14.25/8.10 thf(fact_4919_card__1__singletonI,axiom,
% 14.25/8.10 ! [S: set_o,X: $o] :
% 14.25/8.10 ( ( finite_finite_o @ S )
% 14.25/8.10 => ( ( ( finite_card_o @ S )
% 14.25/8.10 = one_one_nat )
% 14.25/8.10 => ( ( member_o @ X @ S )
% 14.25/8.10 => ( S
% 14.25/8.10 = ( insert_o @ X @ bot_bot_set_o ) ) ) ) ) ).
% 14.25/8.10
% 14.25/8.10 thf(fact_4920_card__1__singletonI,axiom,
% 14.25/8.10 ! [S: set_nat,X: nat] :
% 14.25/8.10 ( ( finite_finite_nat @ S )
% 14.25/8.10 => ( ( ( finite_card_nat @ S )
% 14.25/8.10 = one_one_nat )
% 14.25/8.10 => ( ( member_nat @ X @ S )
% 14.25/8.10 => ( S
% 14.25/8.10 = ( insert_nat @ X @ bot_bot_set_nat ) ) ) ) ) ).
% 14.25/8.10
% 14.25/8.10 thf(fact_4921_card__1__singletonI,axiom,
% 14.25/8.10 ! [S: set_int,X: int] :
% 14.25/8.10 ( ( finite_finite_int @ S )
% 14.25/8.10 => ( ( ( finite_card_int @ S )
% 14.25/8.10 = one_one_nat )
% 14.25/8.10 => ( ( member_int @ X @ S )
% 14.25/8.10 => ( S
% 14.25/8.10 = ( insert_int @ X @ bot_bot_set_int ) ) ) ) ) ).
% 14.25/8.10
% 14.25/8.10 thf(fact_4922_card__1__singletonI,axiom,
% 14.25/8.10 ! [S: set_Pr4532377907799695533_nat_o,X: produc3658429121746597890et_nat > $o] :
% 14.25/8.10 ( ( finite3252695134891459830_nat_o @ S )
% 14.25/8.10 => ( ( ( finite1363419556375932405_nat_o @ S )
% 14.25/8.10 = one_one_nat )
% 14.25/8.10 => ( ( member6576561426505652726_nat_o @ X @ S )
% 14.25/8.10 => ( S
% 14.25/8.10 = ( insert5175938949040314269_nat_o @ X @ bot_bo7824918357723582233_nat_o ) ) ) ) ) ).
% 14.25/8.10
% 14.25/8.10 thf(fact_4923_card__Diff__singleton__if,axiom,
% 14.25/8.10 ! [X: product_unit,A: set_Product_unit] :
% 14.25/8.10 ( ( ( member_Product_unit @ X @ A )
% 14.25/8.10 => ( ( finite410649719033368117t_unit @ ( minus_6452836326544984404t_unit @ A @ ( insert_Product_unit @ X @ bot_bo3957492148770167129t_unit ) ) )
% 14.25/8.10 = ( minus_minus_nat @ ( finite410649719033368117t_unit @ A ) @ one_one_nat ) ) )
% 14.25/8.10 & ( ~ ( member_Product_unit @ X @ A )
% 14.25/8.10 => ( ( finite410649719033368117t_unit @ ( minus_6452836326544984404t_unit @ A @ ( insert_Product_unit @ X @ bot_bo3957492148770167129t_unit ) ) )
% 14.25/8.10 = ( finite410649719033368117t_unit @ A ) ) ) ) ).
% 14.25/8.10
% 14.25/8.10 thf(fact_4924_card__Diff__singleton__if,axiom,
% 14.25/8.10 ! [X: $o,A: set_o] :
% 14.25/8.10 ( ( ( member_o @ X @ A )
% 14.25/8.10 => ( ( finite_card_o @ ( minus_minus_set_o @ A @ ( insert_o @ X @ bot_bot_set_o ) ) )
% 14.25/8.10 = ( minus_minus_nat @ ( finite_card_o @ A ) @ one_one_nat ) ) )
% 14.25/8.10 & ( ~ ( member_o @ X @ A )
% 14.25/8.10 => ( ( finite_card_o @ ( minus_minus_set_o @ A @ ( insert_o @ X @ bot_bot_set_o ) ) )
% 14.25/8.10 = ( finite_card_o @ A ) ) ) ) ).
% 14.25/8.10
% 14.25/8.10 thf(fact_4925_card__Diff__singleton__if,axiom,
% 14.25/8.10 ! [X: int,A: set_int] :
% 14.25/8.10 ( ( ( member_int @ X @ A )
% 14.25/8.10 => ( ( finite_card_int @ ( minus_minus_set_int @ A @ ( insert_int @ X @ bot_bot_set_int ) ) )
% 14.25/8.10 = ( minus_minus_nat @ ( finite_card_int @ A ) @ one_one_nat ) ) )
% 14.25/8.10 & ( ~ ( member_int @ X @ A )
% 14.25/8.10 => ( ( finite_card_int @ ( minus_minus_set_int @ A @ ( insert_int @ X @ bot_bot_set_int ) ) )
% 14.25/8.10 = ( finite_card_int @ A ) ) ) ) ).
% 14.25/8.10
% 14.25/8.10 thf(fact_4926_card__Diff__singleton__if,axiom,
% 14.25/8.10 ! [X: produc3658429121746597890et_nat > $o,A: set_Pr4532377907799695533_nat_o] :
% 14.25/8.10 ( ( ( member6576561426505652726_nat_o @ X @ A )
% 14.25/8.10 => ( ( finite1363419556375932405_nat_o @ ( minus_1801376950450012436_nat_o @ A @ ( insert5175938949040314269_nat_o @ X @ bot_bo7824918357723582233_nat_o ) ) )
% 14.25/8.10 = ( minus_minus_nat @ ( finite1363419556375932405_nat_o @ A ) @ one_one_nat ) ) )
% 14.25/8.10 & ( ~ ( member6576561426505652726_nat_o @ X @ A )
% 14.25/8.10 => ( ( finite1363419556375932405_nat_o @ ( minus_1801376950450012436_nat_o @ A @ ( insert5175938949040314269_nat_o @ X @ bot_bo7824918357723582233_nat_o ) ) )
% 14.25/8.10 = ( finite1363419556375932405_nat_o @ A ) ) ) ) ).
% 14.25/8.10
% 14.25/8.10 thf(fact_4927_card__Diff__singleton__if,axiom,
% 14.25/8.10 ! [X: nat,A: set_nat] :
% 14.25/8.10 ( ( ( member_nat @ X @ A )
% 14.25/8.10 => ( ( finite_card_nat @ ( minus_minus_set_nat @ A @ ( insert_nat @ X @ bot_bot_set_nat ) ) )
% 14.25/8.10 = ( minus_minus_nat @ ( finite_card_nat @ A ) @ one_one_nat ) ) )
% 14.25/8.10 & ( ~ ( member_nat @ X @ A )
% 14.25/8.10 => ( ( finite_card_nat @ ( minus_minus_set_nat @ A @ ( insert_nat @ X @ bot_bot_set_nat ) ) )
% 14.25/8.10 = ( finite_card_nat @ A ) ) ) ) ).
% 14.25/8.10
% 14.25/8.10 thf(fact_4928_card__Diff__singleton,axiom,
% 14.25/8.10 ! [X: product_unit,A: set_Product_unit] :
% 14.25/8.10 ( ( member_Product_unit @ X @ A )
% 14.25/8.10 => ( ( finite410649719033368117t_unit @ ( minus_6452836326544984404t_unit @ A @ ( insert_Product_unit @ X @ bot_bo3957492148770167129t_unit ) ) )
% 14.25/8.10 = ( minus_minus_nat @ ( finite410649719033368117t_unit @ A ) @ one_one_nat ) ) ) ).
% 14.25/8.10
% 14.25/8.10 thf(fact_4929_card__Diff__singleton,axiom,
% 14.25/8.10 ! [X: $o,A: set_o] :
% 14.25/8.10 ( ( member_o @ X @ A )
% 14.25/8.10 => ( ( finite_card_o @ ( minus_minus_set_o @ A @ ( insert_o @ X @ bot_bot_set_o ) ) )
% 14.25/8.10 = ( minus_minus_nat @ ( finite_card_o @ A ) @ one_one_nat ) ) ) ).
% 14.25/8.10
% 14.25/8.10 thf(fact_4930_card__Diff__singleton,axiom,
% 14.25/8.10 ! [X: int,A: set_int] :
% 14.25/8.10 ( ( member_int @ X @ A )
% 14.25/8.10 => ( ( finite_card_int @ ( minus_minus_set_int @ A @ ( insert_int @ X @ bot_bot_set_int ) ) )
% 14.25/8.10 = ( minus_minus_nat @ ( finite_card_int @ A ) @ one_one_nat ) ) ) ).
% 14.25/8.10
% 14.25/8.10 thf(fact_4931_card__Diff__singleton,axiom,
% 14.25/8.10 ! [X: produc3658429121746597890et_nat > $o,A: set_Pr4532377907799695533_nat_o] :
% 14.25/8.10 ( ( member6576561426505652726_nat_o @ X @ A )
% 14.25/8.10 => ( ( finite1363419556375932405_nat_o @ ( minus_1801376950450012436_nat_o @ A @ ( insert5175938949040314269_nat_o @ X @ bot_bo7824918357723582233_nat_o ) ) )
% 14.25/8.10 = ( minus_minus_nat @ ( finite1363419556375932405_nat_o @ A ) @ one_one_nat ) ) ) ).
% 14.25/8.10
% 14.25/8.10 thf(fact_4932_card__Diff__singleton,axiom,
% 14.25/8.10 ! [X: nat,A: set_nat] :
% 14.25/8.10 ( ( member_nat @ X @ A )
% 14.25/8.10 => ( ( finite_card_nat @ ( minus_minus_set_nat @ A @ ( insert_nat @ X @ bot_bot_set_nat ) ) )
% 14.25/8.10 = ( minus_minus_nat @ ( finite_card_nat @ A ) @ one_one_nat ) ) ) ).
% 14.25/8.10
% 14.25/8.10 thf(fact_4933_fst__image__mp,axiom,
% 14.25/8.10 ! [A: set_Pr3948176798113811640et_nat,B4: set_he8644174446132254637t_unit,X: heap_e7401611519738050253t_unit,Y: set_nat] :
% 14.25/8.10 ( ( ord_le4903435234951335757t_unit @ ( image_6824039242672219652t_unit @ produc1824681642469235216et_nat @ A ) @ B4 )
% 14.25/8.10 => ( ( member6260224972018164377et_nat @ ( produc7507926704131184380et_nat @ X @ Y ) @ A )
% 14.25/8.10 => ( member242389114213480694t_unit @ X @ B4 ) ) ) ).
% 14.25/8.10
% 14.25/8.10 thf(fact_4934_fst__image__mp,axiom,
% 14.25/8.10 ! [A: set_Pr3286484037609594932et_nat,B4: set_Pr4532377907799695533_nat_o,X: produc3658429121746597890et_nat > $o,Y: produc3658429121746597890et_nat] :
% 14.25/8.10 ( ( ord_le2965882846123202637_nat_o @ ( image_6802380912191114376_nat_o @ produc995936583742144908et_nat @ A ) @ B4 )
% 14.25/8.10 => ( ( member1996754912294343701et_nat @ ( produc5001842942810119800et_nat @ X @ Y ) @ A )
% 14.25/8.10 => ( member6576561426505652726_nat_o @ X @ B4 ) ) ) ).
% 14.25/8.10
% 14.25/8.10 thf(fact_4935_fst__image__mp,axiom,
% 14.25/8.10 ! [A: set_Pr8536935166611901872et_nat,B4: set_Pr4532377907799695533_nat_o,X: produc3658429121746597890et_nat > $o,Y: produc3925858234332021118et_nat] :
% 14.25/8.10 ( ( ord_le2965882846123202637_nat_o @ ( image_2517677226577294348_nat_o @ produc180342877477747464et_nat @ A ) @ B4 )
% 14.25/8.10 => ( ( member6124377750444531601et_nat @ ( produc2245416461498447860et_nat @ X @ Y ) @ A )
% 14.25/8.10 => ( member6576561426505652726_nat_o @ X @ B4 ) ) ) ).
% 14.25/8.10
% 14.25/8.10 thf(fact_4936_fst__image__mp,axiom,
% 14.25/8.10 ! [A: set_Pr958786334691620121nt_int,B4: set_int,X: int,Y: int] :
% 14.25/8.10 ( ( ord_less_eq_set_int @ ( image_5042161079198086560nt_int @ product_fst_int_int @ A ) @ B4 )
% 14.25/8.10 => ( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ X @ Y ) @ A )
% 14.25/8.10 => ( member_int @ X @ B4 ) ) ) ).
% 14.25/8.10
% 14.25/8.10 thf(fact_4937_fst__image__mp,axiom,
% 14.25/8.10 ! [A: set_Pr1261947904930325089at_nat,B4: set_nat,X: nat,Y: nat] :
% 14.25/8.10 ( ( ord_less_eq_set_nat @ ( image_2486076414777270412at_nat @ product_fst_nat_nat @ A ) @ B4 )
% 14.25/8.10 => ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ Y ) @ A )
% 14.25/8.10 => ( member_nat @ X @ B4 ) ) ) ).
% 14.25/8.10
% 14.25/8.10 thf(fact_4938_fst__in__Field,axiom,
% 14.25/8.10 ! [R3: set_Pr958786334691620121nt_int] : ( ord_less_eq_set_int @ ( image_5042161079198086560nt_int @ product_fst_int_int @ R3 ) @ ( field_int @ R3 ) ) ).
% 14.25/8.10
% 14.25/8.10 thf(fact_4939_fst__in__Field,axiom,
% 14.25/8.10 ! [R3: set_Pr1261947904930325089at_nat] : ( ord_less_eq_set_nat @ ( image_2486076414777270412at_nat @ product_fst_nat_nat @ R3 ) @ ( field_nat @ R3 ) ) ).
% 14.25/8.10
% 14.25/8.10 thf(fact_4940_Inf__fin__Inf,axiom,
% 14.25/8.10 ! [A: set_o] :
% 14.25/8.10 ( ( finite_finite_o @ A )
% 14.25/8.10 => ( ( A != bot_bot_set_o )
% 14.25/8.10 => ( ( lattic4107685809792843317_fin_o @ A )
% 14.25/8.10 = ( complete_Inf_Inf_o @ A ) ) ) ) ).
% 14.25/8.10
% 14.25/8.10 thf(fact_4941_Inf__fin__Inf,axiom,
% 14.25/8.10 ! [A: set_Pr4532377907799695533_nat_o] :
% 14.25/8.10 ( ( finite3252695134891459830_nat_o @ A )
% 14.25/8.10 => ( ( A != bot_bo7824918357723582233_nat_o )
% 14.25/8.10 => ( ( lattic956194824204696298_nat_o @ A )
% 14.25/8.10 = ( comple90455603224345172_nat_o @ A ) ) ) ) ).
% 14.25/8.10
% 14.25/8.10 thf(fact_4942_Inf__fin__Inf,axiom,
% 14.25/8.10 ! [A: set_set_nat] :
% 14.25/8.10 ( ( finite1152437895449049373et_nat @ A )
% 14.25/8.10 => ( ( A != bot_bot_set_set_nat )
% 14.25/8.10 => ( ( lattic3014633134055518761et_nat @ A )
% 14.25/8.10 = ( comple7806235888213564991et_nat @ A ) ) ) ) ).
% 14.25/8.10
% 14.25/8.10 thf(fact_4943_Inf__fin__Inf,axiom,
% 14.25/8.10 ! [A: set_Product_unit] :
% 14.25/8.10 ( ( finite4290736615968046902t_unit @ A )
% 14.25/8.10 => ( ( A != bot_bo3957492148770167129t_unit )
% 14.25/8.10 => ( ( lattic1263872656861969706t_unit @ A )
% 14.25/8.10 = ( comple2584293577114468500t_unit @ A ) ) ) ) ).
% 14.25/8.10
% 14.25/8.10 thf(fact_4944_cInf__eq__Inf__fin,axiom,
% 14.25/8.10 ! [X4: set_o] :
% 14.25/8.10 ( ( finite_finite_o @ X4 )
% 14.25/8.10 => ( ( X4 != bot_bot_set_o )
% 14.25/8.10 => ( ( complete_Inf_Inf_o @ X4 )
% 14.25/8.10 = ( lattic4107685809792843317_fin_o @ X4 ) ) ) ) ).
% 14.25/8.10
% 14.25/8.10 thf(fact_4945_cInf__eq__Inf__fin,axiom,
% 14.25/8.10 ! [X4: set_int] :
% 14.25/8.10 ( ( finite_finite_int @ X4 )
% 14.25/8.10 => ( ( X4 != bot_bot_set_int )
% 14.25/8.10 => ( ( complete_Inf_Inf_int @ X4 )
% 14.25/8.10 = ( lattic5235898064620869839in_int @ X4 ) ) ) ) ).
% 14.25/8.10
% 14.25/8.10 thf(fact_4946_cInf__eq__Inf__fin,axiom,
% 14.25/8.10 ! [X4: set_Pr4532377907799695533_nat_o] :
% 14.25/8.10 ( ( finite3252695134891459830_nat_o @ X4 )
% 14.25/8.10 => ( ( X4 != bot_bo7824918357723582233_nat_o )
% 14.25/8.10 => ( ( comple90455603224345172_nat_o @ X4 )
% 14.25/8.10 = ( lattic956194824204696298_nat_o @ X4 ) ) ) ) ).
% 14.25/8.10
% 14.25/8.10 thf(fact_4947_cInf__eq__Inf__fin,axiom,
% 14.25/8.10 ! [X4: set_nat] :
% 14.25/8.10 ( ( finite_finite_nat @ X4 )
% 14.25/8.10 => ( ( X4 != bot_bot_set_nat )
% 14.25/8.10 => ( ( complete_Inf_Inf_nat @ X4 )
% 14.25/8.10 = ( lattic5238388535129920115in_nat @ X4 ) ) ) ) ).
% 14.25/8.10
% 14.25/8.10 thf(fact_4948_cInf__eq__Inf__fin,axiom,
% 14.25/8.10 ! [X4: set_set_nat] :
% 14.25/8.10 ( ( finite1152437895449049373et_nat @ X4 )
% 14.25/8.10 => ( ( X4 != bot_bot_set_set_nat )
% 14.25/8.10 => ( ( comple7806235888213564991et_nat @ X4 )
% 14.25/8.10 = ( lattic3014633134055518761et_nat @ X4 ) ) ) ) ).
% 14.25/8.10
% 14.25/8.10 thf(fact_4949_cInf__eq__Inf__fin,axiom,
% 14.25/8.10 ! [X4: set_Product_unit] :
% 14.25/8.10 ( ( finite4290736615968046902t_unit @ X4 )
% 14.25/8.10 => ( ( X4 != bot_bo3957492148770167129t_unit )
% 14.25/8.10 => ( ( comple2584293577114468500t_unit @ X4 )
% 14.25/8.10 = ( lattic1263872656861969706t_unit @ X4 ) ) ) ) ).
% 14.25/8.10
% 14.25/8.10 thf(fact_4950_prod__set__defs_I1_J,axiom,
% 14.25/8.10 ( basic_fsts_int_int
% 14.25/8.10 = ( ^ [P5: product_prod_int_int] : ( insert_int @ ( product_fst_int_int @ P5 ) @ bot_bot_set_int ) ) ) ).
% 14.25/8.10
% 14.25/8.10 thf(fact_4951_prod_Oswap__def,axiom,
% 14.25/8.10 ( produc1264744871843189710t_unit
% 14.25/8.10 = ( ^ [P5: produc7348995623955488188t_unit] : ( produc7507926704131184380et_nat @ ( produc3550216979005444800t_unit @ P5 ) @ ( produc6012101397789842562t_unit @ P5 ) ) ) ) ).
% 14.25/8.10
% 14.25/8.10 thf(fact_4952_prod_Oswap__def,axiom,
% 14.25/8.10 ( product_swap_nat_nat
% 14.25/8.10 = ( ^ [P5: product_prod_nat_nat] : ( product_Pair_nat_nat @ ( product_snd_nat_nat @ P5 ) @ ( product_fst_nat_nat @ P5 ) ) ) ) ).
% 14.25/8.10
% 14.25/8.10 thf(fact_4953_prod_Oswap__def,axiom,
% 14.25/8.10 ( produc1078497009909708626_nat_o
% 14.25/8.10 = ( ^ [P5: produc6907119039986491712_nat_o] : ( produc5001842942810119800et_nat @ ( produc1838968430882301508_nat_o @ P5 ) @ ( produc8046704426453032966_nat_o @ P5 ) ) ) ) ).
% 14.25/8.10
% 14.25/8.10 thf(fact_4954_prod_Oswap__def,axiom,
% 14.25/8.10 ( produc2580581461503396054_nat_o
% 14.25/8.10 = ( ^ [P5: produc4962441903510821572_nat_o] : ( produc2245416461498447860et_nat @ ( produc6803565743942592456_nat_o @ P5 ) @ ( produc1609452848092598666_nat_o @ P5 ) ) ) ) ).
% 14.25/8.10
% 14.25/8.10 thf(fact_4955_prod_Oswap__def,axiom,
% 14.25/8.10 ( produc6300697153377358172et_nat
% 14.25/8.10 = ( ^ [P5: produc3658429121746597890et_nat] : ( produc2471974422597015918t_unit @ ( produc8586169260539613262et_nat @ P5 ) @ ( produc1824681642469235216et_nat @ P5 ) ) ) ) ).
% 14.25/8.10
% 14.25/8.10 thf(fact_4956_prod_Oswap__def,axiom,
% 14.25/8.10 ( product_swap_int_int
% 14.25/8.10 = ( ^ [P5: product_prod_int_int] : ( product_Pair_int_int @ ( product_snd_int_int @ P5 ) @ ( product_fst_int_int @ P5 ) ) ) ) ).
% 14.25/8.10
% 14.25/8.10 thf(fact_4957_equiv__proj,axiom,
% 14.25/8.10 ! [A: set_int,R3: set_Pr958786334691620121nt_int,Z: product_prod_int_int] :
% 14.25/8.10 ( ( equiv_equiv_int @ A @ R3 )
% 14.25/8.10 => ( ( member5262025264175285858nt_int @ Z @ R3 )
% 14.25/8.10 => ( ( comp_i5103216804979976198nt_int @ ( equiv_proj_int_int @ R3 ) @ product_fst_int_int @ Z )
% 14.25/8.10 = ( comp_i5103216804979976198nt_int @ ( equiv_proj_int_int @ R3 ) @ product_snd_int_int @ Z ) ) ) ) ).
% 14.25/8.10
% 14.25/8.10 thf(fact_4958_INF__le__SUP,axiom,
% 14.25/8.10 ! [A: set_Product_unit,F: product_unit > product_unit] :
% 14.25/8.10 ( ( A != bot_bo3957492148770167129t_unit )
% 14.25/8.10 => ( ord_le3221252021190050221t_unit @ ( comple2584293577114468500t_unit @ ( image_405062704495631173t_unit @ F @ A ) ) @ ( comple4687483117567863418t_unit @ ( image_405062704495631173t_unit @ F @ A ) ) ) ) ).
% 14.25/8.10
% 14.25/8.10 thf(fact_4959_INF__le__SUP,axiom,
% 14.25/8.10 ! [A: set_o,F: $o > product_unit] :
% 14.25/8.10 ( ( A != bot_bot_set_o )
% 14.25/8.10 => ( ord_le3221252021190050221t_unit @ ( comple2584293577114468500t_unit @ ( image_o_Product_unit2 @ F @ A ) ) @ ( comple4687483117567863418t_unit @ ( image_o_Product_unit2 @ F @ A ) ) ) ) ).
% 14.25/8.10
% 14.25/8.10 thf(fact_4960_INF__le__SUP,axiom,
% 14.25/8.10 ! [A: set_nat,F: nat > product_unit] :
% 14.25/8.10 ( ( A != bot_bot_set_nat )
% 14.25/8.10 => ( ord_le3221252021190050221t_unit @ ( comple2584293577114468500t_unit @ ( image_8730104196221521654t_unit @ F @ A ) ) @ ( comple4687483117567863418t_unit @ ( image_8730104196221521654t_unit @ F @ A ) ) ) ) ).
% 14.25/8.10
% 14.25/8.10 thf(fact_4961_INF__le__SUP,axiom,
% 14.25/8.10 ! [A: set_int,F: int > product_unit] :
% 14.25/8.10 ( ( A != bot_bot_set_int )
% 14.25/8.10 => ( ord_le3221252021190050221t_unit @ ( comple2584293577114468500t_unit @ ( image_5846526143423701914t_unit @ F @ A ) ) @ ( comple4687483117567863418t_unit @ ( image_5846526143423701914t_unit @ F @ A ) ) ) ) ).
% 14.25/8.10
% 14.25/8.10 thf(fact_4962_INF__le__SUP,axiom,
% 14.25/8.10 ! [A: set_Product_unit,F: product_unit > filter_nat] :
% 14.25/8.10 ( ( A != bot_bo3957492148770167129t_unit )
% 14.25/8.10 => ( ord_le2510731241096832064er_nat @ ( comple6612786467464916249er_nat @ ( image_1853478764106424488er_nat @ F @ A ) ) @ ( comple2454491259644266675er_nat @ ( image_1853478764106424488er_nat @ F @ A ) ) ) ) ).
% 14.25/8.10
% 14.25/8.10 thf(fact_4963_INF__le__SUP,axiom,
% 14.25/8.10 ! [A: set_o,F: $o > filter_nat] :
% 14.25/8.10 ( ( A != bot_bot_set_o )
% 14.25/8.10 => ( ord_le2510731241096832064er_nat @ ( comple6612786467464916249er_nat @ ( image_o_filter_nat @ F @ A ) ) @ ( comple2454491259644266675er_nat @ ( image_o_filter_nat @ F @ A ) ) ) ) ).
% 14.25/8.10
% 14.25/8.10 thf(fact_4964_INF__le__SUP,axiom,
% 14.25/8.10 ! [A: set_nat,F: nat > filter_nat] :
% 14.25/8.10 ( ( A != bot_bot_set_nat )
% 14.25/8.10 => ( ord_le2510731241096832064er_nat @ ( comple6612786467464916249er_nat @ ( image_nat_filter_nat @ F @ A ) ) @ ( comple2454491259644266675er_nat @ ( image_nat_filter_nat @ F @ A ) ) ) ) ).
% 14.25/8.10
% 14.25/8.10 thf(fact_4965_INF__le__SUP,axiom,
% 14.25/8.10 ! [A: set_int,F: int > filter_nat] :
% 14.25/8.10 ( ( A != bot_bot_set_int )
% 14.25/8.10 => ( ord_le2510731241096832064er_nat @ ( comple6612786467464916249er_nat @ ( image_int_filter_nat @ F @ A ) ) @ ( comple2454491259644266675er_nat @ ( image_int_filter_nat @ F @ A ) ) ) ) ).
% 14.25/8.10
% 14.25/8.10 thf(fact_4966_INF__le__SUP,axiom,
% 14.25/8.10 ! [A: set_Product_unit,F: product_unit > set_nat] :
% 14.25/8.10 ( ( A != bot_bo3957492148770167129t_unit )
% 14.25/8.10 => ( ord_less_eq_set_nat @ ( comple7806235888213564991et_nat @ ( image_8173849991827986510et_nat @ F @ A ) ) @ ( comple7399068483239264473et_nat @ ( image_8173849991827986510et_nat @ F @ A ) ) ) ) ).
% 14.25/8.10
% 14.25/8.10 thf(fact_4967_INF__le__SUP,axiom,
% 14.25/8.10 ! [A: set_o,F: $o > set_nat] :
% 14.25/8.10 ( ( A != bot_bot_set_o )
% 14.25/8.10 => ( ord_less_eq_set_nat @ ( comple7806235888213564991et_nat @ ( image_o_set_nat @ F @ A ) ) @ ( comple7399068483239264473et_nat @ ( image_o_set_nat @ F @ A ) ) ) ) ).
% 14.25/8.10
% 14.25/8.10 thf(fact_4968_card__Diff1__le,axiom,
% 14.25/8.10 ! [A: set_Product_unit,X: product_unit] : ( ord_less_eq_nat @ ( finite410649719033368117t_unit @ ( minus_6452836326544984404t_unit @ A @ ( insert_Product_unit @ X @ bot_bo3957492148770167129t_unit ) ) ) @ ( finite410649719033368117t_unit @ A ) ) ).
% 14.25/8.10
% 14.25/8.10 thf(fact_4969_card__Diff1__le,axiom,
% 14.25/8.10 ! [A: set_o,X: $o] : ( ord_less_eq_nat @ ( finite_card_o @ ( minus_minus_set_o @ A @ ( insert_o @ X @ bot_bot_set_o ) ) ) @ ( finite_card_o @ A ) ) ).
% 14.25/8.10
% 14.25/8.10 thf(fact_4970_card__Diff1__le,axiom,
% 14.25/8.10 ! [A: set_int,X: int] : ( ord_less_eq_nat @ ( finite_card_int @ ( minus_minus_set_int @ A @ ( insert_int @ X @ bot_bot_set_int ) ) ) @ ( finite_card_int @ A ) ) ).
% 14.25/8.10
% 14.25/8.10 thf(fact_4971_card__Diff1__le,axiom,
% 14.25/8.10 ! [A: set_Pr4532377907799695533_nat_o,X: produc3658429121746597890et_nat > $o] : ( ord_less_eq_nat @ ( finite1363419556375932405_nat_o @ ( minus_1801376950450012436_nat_o @ A @ ( insert5175938949040314269_nat_o @ X @ bot_bo7824918357723582233_nat_o ) ) ) @ ( finite1363419556375932405_nat_o @ A ) ) ).
% 14.25/8.10
% 14.25/8.10 thf(fact_4972_card__Diff1__le,axiom,
% 14.25/8.10 ! [A: set_nat,X: nat] : ( ord_less_eq_nat @ ( finite_card_nat @ ( minus_minus_set_nat @ A @ ( insert_nat @ X @ bot_bot_set_nat ) ) ) @ ( finite_card_nat @ A ) ) ).
% 14.25/8.10
% 14.25/8.10 thf(fact_4973_exI__realizer,axiom,
% 14.25/8.10 ! [P: nat > nat > $o,Y: nat,X: nat] :
% 14.25/8.10 ( ( P @ Y @ X )
% 14.25/8.10 => ( P @ ( product_snd_nat_nat @ ( product_Pair_nat_nat @ X @ Y ) ) @ ( product_fst_nat_nat @ ( product_Pair_nat_nat @ X @ Y ) ) ) ) ).
% 14.25/8.10
% 14.25/8.10 thf(fact_4974_exI__realizer,axiom,
% 14.25/8.10 ! [P: produc3658429121746597890et_nat > ( produc3658429121746597890et_nat > $o ) > $o,Y: produc3658429121746597890et_nat,X: produc3658429121746597890et_nat > $o] :
% 14.25/8.10 ( ( P @ Y @ X )
% 14.25/8.10 => ( P @ ( produc4011572625026189258et_nat @ ( produc5001842942810119800et_nat @ X @ Y ) ) @ ( produc995936583742144908et_nat @ ( produc5001842942810119800et_nat @ X @ Y ) ) ) ) ).
% 14.25/8.10
% 14.25/8.10 thf(fact_4975_exI__realizer,axiom,
% 14.25/8.10 ! [P: produc3925858234332021118et_nat > ( produc3658429121746597890et_nat > $o ) > $o,Y: produc3925858234332021118et_nat,X: produc3658429121746597890et_nat > $o] :
% 14.25/8.10 ( ( P @ Y @ X )
% 14.25/8.10 => ( P @ ( produc5374455773327741254et_nat @ ( produc2245416461498447860et_nat @ X @ Y ) ) @ ( produc180342877477747464et_nat @ ( produc2245416461498447860et_nat @ X @ Y ) ) ) ) ).
% 14.25/8.10
% 14.25/8.10 thf(fact_4976_exI__realizer,axiom,
% 14.25/8.10 ! [P: set_nat > heap_e7401611519738050253t_unit > $o,Y: set_nat,X: heap_e7401611519738050253t_unit] :
% 14.25/8.10 ( ( P @ Y @ X )
% 14.25/8.10 => ( P @ ( produc8586169260539613262et_nat @ ( produc7507926704131184380et_nat @ X @ Y ) ) @ ( produc1824681642469235216et_nat @ ( produc7507926704131184380et_nat @ X @ Y ) ) ) ) ).
% 14.25/8.10
% 14.25/8.10 thf(fact_4977_exI__realizer,axiom,
% 14.25/8.10 ! [P: int > int > $o,Y: int,X: int] :
% 14.25/8.10 ( ( P @ Y @ X )
% 14.25/8.10 => ( P @ ( product_snd_int_int @ ( product_Pair_int_int @ X @ Y ) ) @ ( product_fst_int_int @ ( product_Pair_int_int @ X @ Y ) ) ) ) ).
% 14.25/8.10
% 14.25/8.10 thf(fact_4978_conjI__realizer,axiom,
% 14.25/8.10 ! [P: nat > $o,P3: nat,Q2: nat > $o,Q4: nat] :
% 14.25/8.10 ( ( P @ P3 )
% 14.25/8.10 => ( ( Q2 @ Q4 )
% 14.25/8.10 => ( ( P @ ( product_fst_nat_nat @ ( product_Pair_nat_nat @ P3 @ Q4 ) ) )
% 14.25/8.10 & ( Q2 @ ( product_snd_nat_nat @ ( product_Pair_nat_nat @ P3 @ Q4 ) ) ) ) ) ) ).
% 14.25/8.10
% 14.25/8.10 thf(fact_4979_conjI__realizer,axiom,
% 14.25/8.10 ! [P: ( produc3658429121746597890et_nat > $o ) > $o,P3: produc3658429121746597890et_nat > $o,Q2: produc3658429121746597890et_nat > $o,Q4: produc3658429121746597890et_nat] :
% 14.25/8.10 ( ( P @ P3 )
% 14.25/8.10 => ( ( Q2 @ Q4 )
% 14.25/8.10 => ( ( P @ ( produc995936583742144908et_nat @ ( produc5001842942810119800et_nat @ P3 @ Q4 ) ) )
% 14.25/8.10 & ( Q2 @ ( produc4011572625026189258et_nat @ ( produc5001842942810119800et_nat @ P3 @ Q4 ) ) ) ) ) ) ).
% 14.25/8.10
% 14.25/8.10 thf(fact_4980_conjI__realizer,axiom,
% 14.25/8.10 ! [P: ( produc3658429121746597890et_nat > $o ) > $o,P3: produc3658429121746597890et_nat > $o,Q2: produc3925858234332021118et_nat > $o,Q4: produc3925858234332021118et_nat] :
% 14.25/8.10 ( ( P @ P3 )
% 14.25/8.10 => ( ( Q2 @ Q4 )
% 14.25/8.10 => ( ( P @ ( produc180342877477747464et_nat @ ( produc2245416461498447860et_nat @ P3 @ Q4 ) ) )
% 14.25/8.10 & ( Q2 @ ( produc5374455773327741254et_nat @ ( produc2245416461498447860et_nat @ P3 @ Q4 ) ) ) ) ) ) ).
% 14.25/8.10
% 14.25/8.10 thf(fact_4981_conjI__realizer,axiom,
% 14.25/8.10 ! [P: heap_e7401611519738050253t_unit > $o,P3: heap_e7401611519738050253t_unit,Q2: set_nat > $o,Q4: set_nat] :
% 14.25/8.10 ( ( P @ P3 )
% 14.25/8.10 => ( ( Q2 @ Q4 )
% 14.25/8.10 => ( ( P @ ( produc1824681642469235216et_nat @ ( produc7507926704131184380et_nat @ P3 @ Q4 ) ) )
% 14.25/8.10 & ( Q2 @ ( produc8586169260539613262et_nat @ ( produc7507926704131184380et_nat @ P3 @ Q4 ) ) ) ) ) ) ).
% 14.25/8.10
% 14.25/8.10 thf(fact_4982_conjI__realizer,axiom,
% 14.25/8.10 ! [P: int > $o,P3: int,Q2: int > $o,Q4: int] :
% 14.25/8.10 ( ( P @ P3 )
% 14.25/8.10 => ( ( Q2 @ Q4 )
% 14.25/8.10 => ( ( P @ ( product_fst_int_int @ ( product_Pair_int_int @ P3 @ Q4 ) ) )
% 14.25/8.10 & ( Q2 @ ( product_snd_int_int @ ( product_Pair_int_int @ P3 @ Q4 ) ) ) ) ) ) ).
% 14.25/8.10
% 14.25/8.10 thf(fact_4983_apfst__convE,axiom,
% 14.25/8.10 ! [Q4: produc3658429121746597890et_nat,F: heap_e7401611519738050253t_unit > heap_e7401611519738050253t_unit,P3: produc3658429121746597890et_nat] :
% 14.25/8.10 ( ( Q4
% 14.25/8.10 = ( produc4172003873565521026et_nat @ F @ P3 ) )
% 14.25/8.10 => ~ ! [X3: heap_e7401611519738050253t_unit,Y2: set_nat] :
% 14.25/8.10 ( ( P3
% 14.25/8.10 = ( produc7507926704131184380et_nat @ X3 @ Y2 ) )
% 14.25/8.10 => ( Q4
% 14.25/8.10 != ( produc7507926704131184380et_nat @ ( F @ X3 ) @ Y2 ) ) ) ) ).
% 14.25/8.10
% 14.25/8.10 thf(fact_4984_apfst__convE,axiom,
% 14.25/8.10 ! [Q4: product_prod_nat_nat,F: nat > nat,P3: product_prod_nat_nat] :
% 14.25/8.10 ( ( Q4
% 14.25/8.10 = ( produc986720760941809198at_nat @ F @ P3 ) )
% 14.25/8.10 => ~ ! [X3: nat,Y2: nat] :
% 14.25/8.10 ( ( P3
% 14.25/8.10 = ( product_Pair_nat_nat @ X3 @ Y2 ) )
% 14.25/8.10 => ( Q4
% 14.25/8.10 != ( product_Pair_nat_nat @ ( F @ X3 ) @ Y2 ) ) ) ) ).
% 14.25/8.10
% 14.25/8.10 thf(fact_4985_apfst__convE,axiom,
% 14.25/8.10 ! [Q4: product_prod_int_int,F: int > int,P3: product_prod_int_int] :
% 14.25/8.10 ( ( Q4
% 14.25/8.10 = ( produc2355237562570658114nt_int @ F @ P3 ) )
% 14.25/8.10 => ~ ! [X3: int,Y2: int] :
% 14.25/8.10 ( ( P3
% 14.25/8.10 = ( product_Pair_int_int @ X3 @ Y2 ) )
% 14.25/8.10 => ( Q4
% 14.25/8.10 != ( product_Pair_int_int @ ( F @ X3 ) @ Y2 ) ) ) ) ).
% 14.25/8.10
% 14.25/8.10 thf(fact_4986_apfst__convE,axiom,
% 14.25/8.10 ! [Q4: produc3925858234332021118et_nat,F: ( produc3658429121746597890et_nat > $o ) > produc3658429121746597890et_nat > $o,P3: produc3925858234332021118et_nat] :
% 14.25/8.10 ( ( Q4
% 14.25/8.10 = ( produc3198230245617193214et_nat @ F @ P3 ) )
% 14.25/8.10 => ~ ! [X3: produc3658429121746597890et_nat > $o,Y2: produc3658429121746597890et_nat] :
% 14.25/8.10 ( ( P3
% 14.25/8.10 = ( produc5001842942810119800et_nat @ X3 @ Y2 ) )
% 14.25/8.10 => ( Q4
% 14.25/8.10 != ( produc5001842942810119800et_nat @ ( F @ X3 ) @ Y2 ) ) ) ) ).
% 14.25/8.10
% 14.25/8.10 thf(fact_4987_apfst__convE,axiom,
% 14.25/8.10 ! [Q4: produc2732055786443039994et_nat,F: ( produc3658429121746597890et_nat > $o ) > produc3658429121746597890et_nat > $o,P3: produc2732055786443039994et_nat] :
% 14.25/8.10 ( ( Q4
% 14.25/8.10 = ( produc1442628905706551930et_nat @ F @ P3 ) )
% 14.25/8.10 => ~ ! [X3: produc3658429121746597890et_nat > $o,Y2: produc3925858234332021118et_nat] :
% 14.25/8.10 ( ( P3
% 14.25/8.10 = ( produc2245416461498447860et_nat @ X3 @ Y2 ) )
% 14.25/8.10 => ( Q4
% 14.25/8.10 != ( produc2245416461498447860et_nat @ ( F @ X3 ) @ Y2 ) ) ) ) ).
% 14.25/8.10
% 14.25/8.10 thf(fact_4988_exE__realizer_H,axiom,
% 14.25/8.10 ! [P: set_nat > heap_e7401611519738050253t_unit > $o,P3: produc3658429121746597890et_nat] :
% 14.25/8.10 ( ( P @ ( produc8586169260539613262et_nat @ P3 ) @ ( produc1824681642469235216et_nat @ P3 ) )
% 14.25/8.10 => ~ ! [X3: heap_e7401611519738050253t_unit,Y2: set_nat] :
% 14.25/8.10 ~ ( P @ Y2 @ X3 ) ) ).
% 14.25/8.10
% 14.25/8.10 thf(fact_4989_exE__realizer_H,axiom,
% 14.25/8.10 ! [P: int > int > $o,P3: product_prod_int_int] :
% 14.25/8.10 ( ( P @ ( product_snd_int_int @ P3 ) @ ( product_fst_int_int @ P3 ) )
% 14.25/8.10 => ~ ! [X3: int,Y2: int] :
% 14.25/8.10 ~ ( P @ Y2 @ X3 ) ) ).
% 14.25/8.10
% 14.25/8.10 thf(fact_4990_snd__sndOp,axiom,
% 14.25/8.10 ! [P: heap_e7401611519738050253t_unit > heap_e7401611519738050253t_unit > $o,Q2: heap_e7401611519738050253t_unit > set_nat > $o] :
% 14.25/8.10 ( produc8586169260539613262et_nat
% 14.25/8.10 = ( comp_P2930521774885306735et_nat @ produc8586169260539613262et_nat @ ( bNF_sn4039337771197282234et_nat @ P @ Q2 ) ) ) ).
% 14.25/8.10
% 14.25/8.10 thf(fact_4991_snd__sndOp,axiom,
% 14.25/8.10 ! [P: int > int > $o,Q2: int > int > $o] :
% 14.25/8.10 ( product_snd_int_int
% 14.25/8.10 = ( comp_P5356232565778667509nt_int @ product_snd_int_int @ ( bNF_sn1062102010912252026nt_int @ P @ Q2 ) ) ) ).
% 14.25/8.10
% 14.25/8.10 thf(fact_4992_fst__fstOp,axiom,
% 14.25/8.10 ! [P: int > int > $o,Q2: int > int > $o] :
% 14.25/8.10 ( product_fst_int_int
% 14.25/8.10 = ( comp_P5356232565778667509nt_int @ product_fst_int_int @ ( bNF_fs8167890477030535480nt_int @ P @ Q2 ) ) ) ).
% 14.25/8.10
% 14.25/8.10 thf(fact_4993_card__Un__disjoint,axiom,
% 14.25/8.10 ! [A: set_Code_integer,B4: set_Code_integer] :
% 14.25/8.10 ( ( finite6017078050557962740nteger @ A )
% 14.25/8.10 => ( ( finite6017078050557962740nteger @ B4 )
% 14.25/8.10 => ( ( ( inf_in1364745209274528805nteger @ A @ B4 )
% 14.25/8.10 = bot_bo3990330152332043303nteger )
% 14.25/8.10 => ( ( finite4902975817058060853nteger @ ( sup_su848401254843788991nteger @ A @ B4 ) )
% 14.25/8.10 = ( plus_plus_nat @ ( finite4902975817058060853nteger @ A ) @ ( finite4902975817058060853nteger @ B4 ) ) ) ) ) ) ).
% 14.25/8.10
% 14.25/8.10 thf(fact_4994_card__Un__disjoint,axiom,
% 14.25/8.10 ! [A: set_Product_unit,B4: set_Product_unit] :
% 14.25/8.10 ( ( finite4290736615968046902t_unit @ A )
% 14.25/8.10 => ( ( finite4290736615968046902t_unit @ B4 )
% 14.25/8.10 => ( ( ( inf_in4660618365625256667t_unit @ A @ B4 )
% 14.25/8.10 = bot_bo3957492148770167129t_unit )
% 14.25/8.10 => ( ( finite410649719033368117t_unit @ ( sup_su793286257634532545t_unit @ A @ B4 ) )
% 14.25/8.10 = ( plus_plus_nat @ ( finite410649719033368117t_unit @ A ) @ ( finite410649719033368117t_unit @ B4 ) ) ) ) ) ) ).
% 14.25/8.10
% 14.25/8.10 thf(fact_4995_card__Un__disjoint,axiom,
% 14.25/8.10 ! [A: set_o,B4: set_o] :
% 14.25/8.10 ( ( finite_finite_o @ A )
% 14.25/8.10 => ( ( finite_finite_o @ B4 )
% 14.25/8.10 => ( ( ( inf_inf_set_o @ A @ B4 )
% 14.25/8.10 = bot_bot_set_o )
% 14.25/8.10 => ( ( finite_card_o @ ( sup_sup_set_o @ A @ B4 ) )
% 14.25/8.10 = ( plus_plus_nat @ ( finite_card_o @ A ) @ ( finite_card_o @ B4 ) ) ) ) ) ) ).
% 14.25/8.10
% 14.25/8.10 thf(fact_4996_card__Un__disjoint,axiom,
% 14.25/8.10 ! [A: set_nat,B4: set_nat] :
% 14.25/8.10 ( ( finite_finite_nat @ A )
% 14.25/8.10 => ( ( finite_finite_nat @ B4 )
% 14.25/8.10 => ( ( ( inf_inf_set_nat @ A @ B4 )
% 14.25/8.10 = bot_bot_set_nat )
% 14.25/8.10 => ( ( finite_card_nat @ ( sup_sup_set_nat @ A @ B4 ) )
% 14.25/8.10 = ( plus_plus_nat @ ( finite_card_nat @ A ) @ ( finite_card_nat @ B4 ) ) ) ) ) ) ).
% 14.25/8.10
% 14.25/8.10 thf(fact_4997_card__Un__disjoint,axiom,
% 14.25/8.10 ! [A: set_int,B4: set_int] :
% 14.25/8.10 ( ( finite_finite_int @ A )
% 14.25/8.10 => ( ( finite_finite_int @ B4 )
% 14.25/8.10 => ( ( ( inf_inf_set_int @ A @ B4 )
% 14.25/8.10 = bot_bot_set_int )
% 14.25/8.10 => ( ( finite_card_int @ ( sup_sup_set_int @ A @ B4 ) )
% 14.25/8.10 = ( plus_plus_nat @ ( finite_card_int @ A ) @ ( finite_card_int @ B4 ) ) ) ) ) ) ).
% 14.25/8.10
% 14.25/8.10 thf(fact_4998_card__Un__disjoint,axiom,
% 14.25/8.10 ! [A: set_Pr4532377907799695533_nat_o,B4: set_Pr4532377907799695533_nat_o] :
% 14.25/8.10 ( ( finite3252695134891459830_nat_o @ A )
% 14.25/8.10 => ( ( finite3252695134891459830_nat_o @ B4 )
% 14.25/8.10 => ( ( ( inf_in1906310914598751387_nat_o @ A @ B4 )
% 14.25/8.10 = bot_bo7824918357723582233_nat_o )
% 14.25/8.10 => ( ( finite1363419556375932405_nat_o @ ( sup_su5209123915105501825_nat_o @ A @ B4 ) )
% 14.25/8.10 = ( plus_plus_nat @ ( finite1363419556375932405_nat_o @ A ) @ ( finite1363419556375932405_nat_o @ B4 ) ) ) ) ) ) ).
% 14.25/8.10
% 14.25/8.10 thf(fact_4999_card__Suc__Diff1,axiom,
% 14.25/8.10 ! [A: set_Code_integer,X: code_integer] :
% 14.25/8.10 ( ( finite6017078050557962740nteger @ A )
% 14.25/8.10 => ( ( member_Code_integer @ X @ A )
% 14.25/8.10 => ( ( suc @ ( finite4902975817058060853nteger @ ( minus_2355218937544613996nteger @ A @ ( insert_Code_integer @ X @ bot_bo3990330152332043303nteger ) ) ) )
% 14.25/8.10 = ( finite4902975817058060853nteger @ A ) ) ) ) ).
% 14.25/8.10
% 14.25/8.10 thf(fact_5000_card__Suc__Diff1,axiom,
% 14.25/8.10 ! [A: set_Product_unit,X: product_unit] :
% 14.25/8.10 ( ( finite4290736615968046902t_unit @ A )
% 14.25/8.10 => ( ( member_Product_unit @ X @ A )
% 14.25/8.10 => ( ( suc @ ( finite410649719033368117t_unit @ ( minus_6452836326544984404t_unit @ A @ ( insert_Product_unit @ X @ bot_bo3957492148770167129t_unit ) ) ) )
% 14.25/8.10 = ( finite410649719033368117t_unit @ A ) ) ) ) ).
% 14.25/8.10
% 14.25/8.10 thf(fact_5001_card__Suc__Diff1,axiom,
% 14.25/8.10 ! [A: set_o,X: $o] :
% 14.25/8.10 ( ( finite_finite_o @ A )
% 14.25/8.10 => ( ( member_o @ X @ A )
% 14.25/8.10 => ( ( suc @ ( finite_card_o @ ( minus_minus_set_o @ A @ ( insert_o @ X @ bot_bot_set_o ) ) ) )
% 14.25/8.10 = ( finite_card_o @ A ) ) ) ) ).
% 14.25/8.10
% 14.25/8.10 thf(fact_5002_card__Suc__Diff1,axiom,
% 14.25/8.10 ! [A: set_int,X: int] :
% 14.25/8.10 ( ( finite_finite_int @ A )
% 14.25/8.10 => ( ( member_int @ X @ A )
% 14.25/8.10 => ( ( suc @ ( finite_card_int @ ( minus_minus_set_int @ A @ ( insert_int @ X @ bot_bot_set_int ) ) ) )
% 14.25/8.10 = ( finite_card_int @ A ) ) ) ) ).
% 14.25/8.10
% 14.25/8.10 thf(fact_5003_card__Suc__Diff1,axiom,
% 14.25/8.10 ! [A: set_Pr4532377907799695533_nat_o,X: produc3658429121746597890et_nat > $o] :
% 14.25/8.10 ( ( finite3252695134891459830_nat_o @ A )
% 14.25/8.10 => ( ( member6576561426505652726_nat_o @ X @ A )
% 14.25/8.10 => ( ( suc @ ( finite1363419556375932405_nat_o @ ( minus_1801376950450012436_nat_o @ A @ ( insert5175938949040314269_nat_o @ X @ bot_bo7824918357723582233_nat_o ) ) ) )
% 14.25/8.10 = ( finite1363419556375932405_nat_o @ A ) ) ) ) ).
% 14.25/8.10
% 14.25/8.10 thf(fact_5004_card__Suc__Diff1,axiom,
% 14.25/8.10 ! [A: set_nat,X: nat] :
% 14.25/8.10 ( ( finite_finite_nat @ A )
% 14.25/8.10 => ( ( member_nat @ X @ A )
% 14.25/8.10 => ( ( suc @ ( finite_card_nat @ ( minus_minus_set_nat @ A @ ( insert_nat @ X @ bot_bot_set_nat ) ) ) )
% 14.25/8.10 = ( finite_card_nat @ A ) ) ) ) ).
% 14.25/8.10
% 14.25/8.10 thf(fact_5005_add__left__cancel,axiom,
% 14.25/8.10 ! [A2: code_integer,B: code_integer,C: code_integer] :
% 14.25/8.10 ( ( ( plus_p5714425477246183910nteger @ A2 @ B )
% 14.25/8.10 = ( plus_p5714425477246183910nteger @ A2 @ C ) )
% 14.25/8.10 = ( B = C ) ) ).
% 14.25/8.10
% 14.25/8.10 thf(fact_5006_add__left__cancel,axiom,
% 14.25/8.10 ! [A2: rat,B: rat,C: rat] :
% 14.25/8.10 ( ( ( plus_plus_rat @ A2 @ B )
% 14.25/8.10 = ( plus_plus_rat @ A2 @ C ) )
% 14.25/8.10 = ( B = C ) ) ).
% 14.25/8.10
% 14.25/8.10 thf(fact_5007_add__left__cancel,axiom,
% 14.25/8.10 ! [A2: nat,B: nat,C: nat] :
% 14.25/8.10 ( ( ( plus_plus_nat @ A2 @ B )
% 14.25/8.10 = ( plus_plus_nat @ A2 @ C ) )
% 14.25/8.10 = ( B = C ) ) ).
% 14.25/8.10
% 14.25/8.10 thf(fact_5008_add__left__cancel,axiom,
% 14.25/8.10 ! [A2: int,B: int,C: int] :
% 14.25/8.10 ( ( ( plus_plus_int @ A2 @ B )
% 14.25/8.10 = ( plus_plus_int @ A2 @ C ) )
% 14.25/8.10 = ( B = C ) ) ).
% 14.25/8.10
% 14.25/8.10 thf(fact_5009_add__right__cancel,axiom,
% 14.25/8.10 ! [B: code_integer,A2: code_integer,C: code_integer] :
% 14.25/8.10 ( ( ( plus_p5714425477246183910nteger @ B @ A2 )
% 14.25/8.10 = ( plus_p5714425477246183910nteger @ C @ A2 ) )
% 14.25/8.10 = ( B = C ) ) ).
% 14.25/8.10
% 14.25/8.10 thf(fact_5010_add__right__cancel,axiom,
% 14.25/8.10 ! [B: rat,A2: rat,C: rat] :
% 14.25/8.10 ( ( ( plus_plus_rat @ B @ A2 )
% 14.25/8.10 = ( plus_plus_rat @ C @ A2 ) )
% 14.25/8.10 = ( B = C ) ) ).
% 14.25/8.10
% 14.25/8.10 thf(fact_5011_add__right__cancel,axiom,
% 14.25/8.10 ! [B: nat,A2: nat,C: nat] :
% 14.25/8.10 ( ( ( plus_plus_nat @ B @ A2 )
% 14.25/8.10 = ( plus_plus_nat @ C @ A2 ) )
% 14.25/8.10 = ( B = C ) ) ).
% 14.25/8.10
% 14.25/8.10 thf(fact_5012_add__right__cancel,axiom,
% 14.25/8.10 ! [B: int,A2: int,C: int] :
% 14.25/8.10 ( ( ( plus_plus_int @ B @ A2 )
% 14.25/8.10 = ( plus_plus_int @ C @ A2 ) )
% 14.25/8.10 = ( B = C ) ) ).
% 14.25/8.10
% 14.25/8.10 thf(fact_5013_add__le__cancel__left,axiom,
% 14.25/8.10 ! [C: code_integer,A2: code_integer,B: code_integer] :
% 14.25/8.10 ( ( ord_le3102999989581377725nteger @ ( plus_p5714425477246183910nteger @ C @ A2 ) @ ( plus_p5714425477246183910nteger @ C @ B ) )
% 14.25/8.10 = ( ord_le3102999989581377725nteger @ A2 @ B ) ) ).
% 14.25/8.10
% 14.25/8.10 thf(fact_5014_add__le__cancel__left,axiom,
% 14.25/8.10 ! [C: rat,A2: rat,B: rat] :
% 14.25/8.10 ( ( ord_less_eq_rat @ ( plus_plus_rat @ C @ A2 ) @ ( plus_plus_rat @ C @ B ) )
% 14.25/8.10 = ( ord_less_eq_rat @ A2 @ B ) ) ).
% 14.25/8.10
% 14.25/8.10 thf(fact_5015_add__le__cancel__left,axiom,
% 14.25/8.10 ! [C: nat,A2: nat,B: nat] :
% 14.25/8.10 ( ( ord_less_eq_nat @ ( plus_plus_nat @ C @ A2 ) @ ( plus_plus_nat @ C @ B ) )
% 14.25/8.10 = ( ord_less_eq_nat @ A2 @ B ) ) ).
% 14.25/8.10
% 14.25/8.10 thf(fact_5016_add__le__cancel__left,axiom,
% 14.25/8.10 ! [C: int,A2: int,B: int] :
% 14.25/8.10 ( ( ord_less_eq_int @ ( plus_plus_int @ C @ A2 ) @ ( plus_plus_int @ C @ B ) )
% 14.25/8.10 = ( ord_less_eq_int @ A2 @ B ) ) ).
% 14.25/8.10
% 14.25/8.10 thf(fact_5017_add__le__cancel__right,axiom,
% 14.25/8.10 ! [A2: code_integer,C: code_integer,B: code_integer] :
% 14.25/8.10 ( ( ord_le3102999989581377725nteger @ ( plus_p5714425477246183910nteger @ A2 @ C ) @ ( plus_p5714425477246183910nteger @ B @ C ) )
% 14.25/8.10 = ( ord_le3102999989581377725nteger @ A2 @ B ) ) ).
% 14.25/8.10
% 14.25/8.10 thf(fact_5018_add__le__cancel__right,axiom,
% 14.25/8.10 ! [A2: rat,C: rat,B: rat] :
% 14.25/8.10 ( ( ord_less_eq_rat @ ( plus_plus_rat @ A2 @ C ) @ ( plus_plus_rat @ B @ C ) )
% 14.25/8.10 = ( ord_less_eq_rat @ A2 @ B ) ) ).
% 14.25/8.10
% 14.25/8.10 thf(fact_5019_add__le__cancel__right,axiom,
% 14.25/8.10 ! [A2: nat,C: nat,B: nat] :
% 14.25/8.10 ( ( ord_less_eq_nat @ ( plus_plus_nat @ A2 @ C ) @ ( plus_plus_nat @ B @ C ) )
% 14.25/8.10 = ( ord_less_eq_nat @ A2 @ B ) ) ).
% 14.25/8.10
% 14.25/8.10 thf(fact_5020_add__le__cancel__right,axiom,
% 14.25/8.10 ! [A2: int,C: int,B: int] :
% 14.25/8.10 ( ( ord_less_eq_int @ ( plus_plus_int @ A2 @ C ) @ ( plus_plus_int @ B @ C ) )
% 14.25/8.10 = ( ord_less_eq_int @ A2 @ B ) ) ).
% 14.25/8.10
% 14.25/8.10 thf(fact_5021_add__less__cancel__left,axiom,
% 14.25/8.10 ! [C: nat,A2: nat,B: nat] :
% 14.25/8.10 ( ( ord_less_nat @ ( plus_plus_nat @ C @ A2 ) @ ( plus_plus_nat @ C @ B ) )
% 14.25/8.10 = ( ord_less_nat @ A2 @ B ) ) ).
% 14.25/8.10
% 14.25/8.10 thf(fact_5022_add__less__cancel__left,axiom,
% 14.25/8.10 ! [C: int,A2: int,B: int] :
% 14.25/8.10 ( ( ord_less_int @ ( plus_plus_int @ C @ A2 ) @ ( plus_plus_int @ C @ B ) )
% 14.25/8.10 = ( ord_less_int @ A2 @ B ) ) ).
% 14.25/8.10
% 14.25/8.10 thf(fact_5023_add__less__cancel__left,axiom,
% 14.25/8.10 ! [C: code_integer,A2: code_integer,B: code_integer] :
% 14.25/8.10 ( ( ord_le6747313008572928689nteger @ ( plus_p5714425477246183910nteger @ C @ A2 ) @ ( plus_p5714425477246183910nteger @ C @ B ) )
% 14.25/8.10 = ( ord_le6747313008572928689nteger @ A2 @ B ) ) ).
% 14.25/8.10
% 14.25/8.10 thf(fact_5024_add__less__cancel__left,axiom,
% 14.25/8.10 ! [C: rat,A2: rat,B: rat] :
% 14.25/8.10 ( ( ord_less_rat @ ( plus_plus_rat @ C @ A2 ) @ ( plus_plus_rat @ C @ B ) )
% 14.25/8.10 = ( ord_less_rat @ A2 @ B ) ) ).
% 14.25/8.10
% 14.25/8.10 thf(fact_5025_add__less__cancel__right,axiom,
% 14.25/8.10 ! [A2: nat,C: nat,B: nat] :
% 14.25/8.10 ( ( ord_less_nat @ ( plus_plus_nat @ A2 @ C ) @ ( plus_plus_nat @ B @ C ) )
% 14.25/8.10 = ( ord_less_nat @ A2 @ B ) ) ).
% 14.25/8.10
% 14.25/8.10 thf(fact_5026_add__less__cancel__right,axiom,
% 14.25/8.10 ! [A2: int,C: int,B: int] :
% 14.25/8.10 ( ( ord_less_int @ ( plus_plus_int @ A2 @ C ) @ ( plus_plus_int @ B @ C ) )
% 14.25/8.10 = ( ord_less_int @ A2 @ B ) ) ).
% 14.25/8.10
% 14.25/8.10 thf(fact_5027_add__less__cancel__right,axiom,
% 14.25/8.10 ! [A2: code_integer,C: code_integer,B: code_integer] :
% 14.25/8.10 ( ( ord_le6747313008572928689nteger @ ( plus_p5714425477246183910nteger @ A2 @ C ) @ ( plus_p5714425477246183910nteger @ B @ C ) )
% 14.25/8.10 = ( ord_le6747313008572928689nteger @ A2 @ B ) ) ).
% 14.25/8.10
% 14.25/8.10 thf(fact_5028_add__less__cancel__right,axiom,
% 14.25/8.10 ! [A2: rat,C: rat,B: rat] :
% 14.25/8.10 ( ( ord_less_rat @ ( plus_plus_rat @ A2 @ C ) @ ( plus_plus_rat @ B @ C ) )
% 14.25/8.10 = ( ord_less_rat @ A2 @ B ) ) ).
% 14.25/8.10
% 14.25/8.10 thf(fact_5029_add__diff__cancel,axiom,
% 14.25/8.10 ! [A2: code_integer,B: code_integer] :
% 14.25/8.10 ( ( minus_8373710615458151222nteger @ ( plus_p5714425477246183910nteger @ A2 @ B ) @ B )
% 14.25/8.10 = A2 ) ).
% 14.25/8.10
% 14.25/8.10 thf(fact_5030_add__diff__cancel,axiom,
% 14.25/8.10 ! [A2: rat,B: rat] :
% 14.25/8.10 ( ( minus_minus_rat @ ( plus_plus_rat @ A2 @ B ) @ B )
% 14.25/8.10 = A2 ) ).
% 14.25/8.10
% 14.25/8.10 thf(fact_5031_add__diff__cancel,axiom,
% 14.25/8.10 ! [A2: int,B: int] :
% 14.25/8.10 ( ( minus_minus_int @ ( plus_plus_int @ A2 @ B ) @ B )
% 14.25/8.10 = A2 ) ).
% 14.25/8.10
% 14.25/8.10 thf(fact_5032_diff__add__cancel,axiom,
% 14.25/8.10 ! [A2: code_integer,B: code_integer] :
% 14.25/8.10 ( ( plus_p5714425477246183910nteger @ ( minus_8373710615458151222nteger @ A2 @ B ) @ B )
% 14.25/8.10 = A2 ) ).
% 14.25/8.10
% 14.25/8.10 thf(fact_5033_diff__add__cancel,axiom,
% 14.25/8.10 ! [A2: rat,B: rat] :
% 14.25/8.10 ( ( plus_plus_rat @ ( minus_minus_rat @ A2 @ B ) @ B )
% 14.25/8.10 = A2 ) ).
% 14.25/8.10
% 14.25/8.10 thf(fact_5034_diff__add__cancel,axiom,
% 14.25/8.10 ! [A2: int,B: int] :
% 14.25/8.10 ( ( plus_plus_int @ ( minus_minus_int @ A2 @ B ) @ B )
% 14.25/8.10 = A2 ) ).
% 14.25/8.10
% 14.25/8.10 thf(fact_5035_add__diff__cancel__left,axiom,
% 14.25/8.10 ! [C: code_integer,A2: code_integer,B: code_integer] :
% 14.25/8.10 ( ( minus_8373710615458151222nteger @ ( plus_p5714425477246183910nteger @ C @ A2 ) @ ( plus_p5714425477246183910nteger @ C @ B ) )
% 14.25/8.10 = ( minus_8373710615458151222nteger @ A2 @ B ) ) ).
% 14.25/8.10
% 14.25/8.10 thf(fact_5036_add__diff__cancel__left,axiom,
% 14.25/8.10 ! [C: rat,A2: rat,B: rat] :
% 14.25/8.10 ( ( minus_minus_rat @ ( plus_plus_rat @ C @ A2 ) @ ( plus_plus_rat @ C @ B ) )
% 14.25/8.10 = ( minus_minus_rat @ A2 @ B ) ) ).
% 14.25/8.10
% 14.25/8.10 thf(fact_5037_add__diff__cancel__left,axiom,
% 14.25/8.10 ! [C: nat,A2: nat,B: nat] :
% 14.25/8.10 ( ( minus_minus_nat @ ( plus_plus_nat @ C @ A2 ) @ ( plus_plus_nat @ C @ B ) )
% 14.25/8.10 = ( minus_minus_nat @ A2 @ B ) ) ).
% 14.25/8.10
% 14.25/8.10 thf(fact_5038_add__diff__cancel__left,axiom,
% 14.25/8.10 ! [C: int,A2: int,B: int] :
% 14.25/8.10 ( ( minus_minus_int @ ( plus_plus_int @ C @ A2 ) @ ( plus_plus_int @ C @ B ) )
% 14.25/8.10 = ( minus_minus_int @ A2 @ B ) ) ).
% 14.25/8.10
% 14.25/8.10 thf(fact_5039_add__diff__cancel__left_H,axiom,
% 14.25/8.10 ! [A2: code_integer,B: code_integer] :
% 14.25/8.10 ( ( minus_8373710615458151222nteger @ ( plus_p5714425477246183910nteger @ A2 @ B ) @ A2 )
% 14.25/8.10 = B ) ).
% 14.25/8.10
% 14.25/8.10 thf(fact_5040_add__diff__cancel__left_H,axiom,
% 14.25/8.10 ! [A2: rat,B: rat] :
% 14.25/8.10 ( ( minus_minus_rat @ ( plus_plus_rat @ A2 @ B ) @ A2 )
% 14.25/8.10 = B ) ).
% 14.25/8.10
% 14.25/8.10 thf(fact_5041_add__diff__cancel__left_H,axiom,
% 14.25/8.10 ! [A2: nat,B: nat] :
% 14.25/8.10 ( ( minus_minus_nat @ ( plus_plus_nat @ A2 @ B ) @ A2 )
% 14.25/8.10 = B ) ).
% 14.25/8.10
% 14.25/8.10 thf(fact_5042_add__diff__cancel__left_H,axiom,
% 14.25/8.10 ! [A2: int,B: int] :
% 14.25/8.10 ( ( minus_minus_int @ ( plus_plus_int @ A2 @ B ) @ A2 )
% 14.25/8.10 = B ) ).
% 14.25/8.10
% 14.25/8.10 thf(fact_5043_add__diff__cancel__right,axiom,
% 14.25/8.10 ! [A2: code_integer,C: code_integer,B: code_integer] :
% 14.25/8.10 ( ( minus_8373710615458151222nteger @ ( plus_p5714425477246183910nteger @ A2 @ C ) @ ( plus_p5714425477246183910nteger @ B @ C ) )
% 14.25/8.10 = ( minus_8373710615458151222nteger @ A2 @ B ) ) ).
% 14.25/8.10
% 14.25/8.10 thf(fact_5044_add__diff__cancel__right,axiom,
% 14.25/8.10 ! [A2: rat,C: rat,B: rat] :
% 14.25/8.10 ( ( minus_minus_rat @ ( plus_plus_rat @ A2 @ C ) @ ( plus_plus_rat @ B @ C ) )
% 14.25/8.10 = ( minus_minus_rat @ A2 @ B ) ) ).
% 14.25/8.10
% 14.25/8.10 thf(fact_5045_add__diff__cancel__right,axiom,
% 14.25/8.10 ! [A2: nat,C: nat,B: nat] :
% 14.25/8.10 ( ( minus_minus_nat @ ( plus_plus_nat @ A2 @ C ) @ ( plus_plus_nat @ B @ C ) )
% 14.25/8.10 = ( minus_minus_nat @ A2 @ B ) ) ).
% 14.25/8.10
% 14.25/8.10 thf(fact_5046_add__diff__cancel__right,axiom,
% 14.25/8.10 ! [A2: int,C: int,B: int] :
% 14.25/8.10 ( ( minus_minus_int @ ( plus_plus_int @ A2 @ C ) @ ( plus_plus_int @ B @ C ) )
% 14.25/8.10 = ( minus_minus_int @ A2 @ B ) ) ).
% 14.25/8.10
% 14.25/8.10 thf(fact_5047_add__diff__cancel__right_H,axiom,
% 14.25/8.10 ! [A2: code_integer,B: code_integer] :
% 14.25/8.10 ( ( minus_8373710615458151222nteger @ ( plus_p5714425477246183910nteger @ A2 @ B ) @ B )
% 14.25/8.10 = A2 ) ).
% 14.25/8.10
% 14.25/8.10 thf(fact_5048_add__diff__cancel__right_H,axiom,
% 14.25/8.10 ! [A2: rat,B: rat] :
% 14.25/8.10 ( ( minus_minus_rat @ ( plus_plus_rat @ A2 @ B ) @ B )
% 14.25/8.10 = A2 ) ).
% 14.25/8.10
% 14.25/8.10 thf(fact_5049_add__diff__cancel__right_H,axiom,
% 14.25/8.10 ! [A2: nat,B: nat] :
% 14.25/8.10 ( ( minus_minus_nat @ ( plus_plus_nat @ A2 @ B ) @ B )
% 14.25/8.10 = A2 ) ).
% 14.25/8.10
% 14.25/8.10 thf(fact_5050_add__diff__cancel__right_H,axiom,
% 14.25/8.10 ! [A2: int,B: int] :
% 14.25/8.10 ( ( minus_minus_int @ ( plus_plus_int @ A2 @ B ) @ B )
% 14.25/8.10 = A2 ) ).
% 14.25/8.10
% 14.25/8.10 thf(fact_5051_card__insert__disjoint,axiom,
% 14.25/8.10 ! [A: set_Pr4532377907799695533_nat_o,X: produc3658429121746597890et_nat > $o] :
% 14.25/8.10 ( ( finite3252695134891459830_nat_o @ A )
% 14.25/8.10 => ( ~ ( member6576561426505652726_nat_o @ X @ A )
% 14.25/8.10 => ( ( finite1363419556375932405_nat_o @ ( insert5175938949040314269_nat_o @ X @ A ) )
% 14.25/8.10 = ( suc @ ( finite1363419556375932405_nat_o @ A ) ) ) ) ) ).
% 14.25/8.10
% 14.25/8.10 thf(fact_5052_card__insert__disjoint,axiom,
% 14.25/8.10 ! [A: set_int,X: int] :
% 14.25/8.10 ( ( finite_finite_int @ A )
% 14.25/8.10 => ( ~ ( member_int @ X @ A )
% 14.25/8.10 => ( ( finite_card_int @ ( insert_int @ X @ A ) )
% 14.25/8.10 = ( suc @ ( finite_card_int @ A ) ) ) ) ) ).
% 14.25/8.10
% 14.25/8.10 thf(fact_5053_card__insert__disjoint,axiom,
% 14.25/8.10 ! [A: set_o,X: $o] :
% 14.25/8.10 ( ( finite_finite_o @ A )
% 14.25/8.10 => ( ~ ( member_o @ X @ A )
% 14.25/8.10 => ( ( finite_card_o @ ( insert_o @ X @ A ) )
% 14.25/8.10 = ( suc @ ( finite_card_o @ A ) ) ) ) ) ).
% 14.25/8.10
% 14.25/8.10 thf(fact_5054_card__insert__disjoint,axiom,
% 14.25/8.10 ! [A: set_Product_unit,X: product_unit] :
% 14.25/8.10 ( ( finite4290736615968046902t_unit @ A )
% 14.25/8.10 => ( ~ ( member_Product_unit @ X @ A )
% 14.25/8.10 => ( ( finite410649719033368117t_unit @ ( insert_Product_unit @ X @ A ) )
% 14.25/8.10 = ( suc @ ( finite410649719033368117t_unit @ A ) ) ) ) ) ).
% 14.25/8.10
% 14.25/8.10 thf(fact_5055_card__insert__disjoint,axiom,
% 14.25/8.10 ! [A: set_nat,X: nat] :
% 14.25/8.10 ( ( finite_finite_nat @ A )
% 14.25/8.10 => ( ~ ( member_nat @ X @ A )
% 14.25/8.10 => ( ( finite_card_nat @ ( insert_nat @ X @ A ) )
% 14.25/8.10 = ( suc @ ( finite_card_nat @ A ) ) ) ) ) ).
% 14.25/8.10
% 14.25/8.10 thf(fact_5056_card__insert__disjoint,axiom,
% 14.25/8.10 ! [A: set_Code_integer,X: code_integer] :
% 14.25/8.10 ( ( finite6017078050557962740nteger @ A )
% 14.25/8.10 => ( ~ ( member_Code_integer @ X @ A )
% 14.25/8.10 => ( ( finite4902975817058060853nteger @ ( insert_Code_integer @ X @ A ) )
% 14.25/8.10 = ( suc @ ( finite4902975817058060853nteger @ A ) ) ) ) ) ).
% 14.25/8.10
% 14.25/8.10 thf(fact_5057_Suc__diff,axiom,
% 14.25/8.10 ! [M4: nat,N2: nat] :
% 14.25/8.10 ( ( ord_less_eq_nat @ M4 @ N2 )
% 14.25/8.10 => ( ( ord_less_eq_nat @ one_one_nat @ M4 )
% 14.25/8.10 => ( ( suc @ ( minus_minus_nat @ N2 @ M4 ) )
% 14.25/8.10 = ( minus_minus_nat @ N2 @ ( minus_minus_nat @ M4 @ one_one_nat ) ) ) ) ) ).
% 14.25/8.10
% 14.25/8.10 thf(fact_5058_add__mono__thms__linordered__semiring_I4_J,axiom,
% 14.25/8.10 ! [I2: code_integer,J: code_integer,K3: code_integer,L: code_integer] :
% 14.25/8.10 ( ( ( I2 = J )
% 14.25/8.10 & ( K3 = L ) )
% 14.25/8.10 => ( ( plus_p5714425477246183910nteger @ I2 @ K3 )
% 14.25/8.10 = ( plus_p5714425477246183910nteger @ J @ L ) ) ) ).
% 14.25/8.10
% 14.25/8.10 thf(fact_5059_add__mono__thms__linordered__semiring_I4_J,axiom,
% 14.25/8.10 ! [I2: rat,J: rat,K3: rat,L: rat] :
% 14.25/8.10 ( ( ( I2 = J )
% 14.25/8.10 & ( K3 = L ) )
% 14.25/8.10 => ( ( plus_plus_rat @ I2 @ K3 )
% 14.25/8.10 = ( plus_plus_rat @ J @ L ) ) ) ).
% 14.25/8.10
% 14.25/8.10 thf(fact_5060_add__mono__thms__linordered__semiring_I4_J,axiom,
% 14.25/8.10 ! [I2: nat,J: nat,K3: nat,L: nat] :
% 14.25/8.10 ( ( ( I2 = J )
% 14.25/8.10 & ( K3 = L ) )
% 14.25/8.10 => ( ( plus_plus_nat @ I2 @ K3 )
% 14.25/8.10 = ( plus_plus_nat @ J @ L ) ) ) ).
% 14.25/8.10
% 14.25/8.10 thf(fact_5061_add__mono__thms__linordered__semiring_I4_J,axiom,
% 14.25/8.10 ! [I2: int,J: int,K3: int,L: int] :
% 14.25/8.10 ( ( ( I2 = J )
% 14.25/8.10 & ( K3 = L ) )
% 14.25/8.10 => ( ( plus_plus_int @ I2 @ K3 )
% 14.25/8.10 = ( plus_plus_int @ J @ L ) ) ) ).
% 14.25/8.10
% 14.25/8.10 thf(fact_5062_group__cancel_Oadd1,axiom,
% 14.25/8.10 ! [A: code_integer,K3: code_integer,A2: code_integer,B: code_integer] :
% 14.25/8.10 ( ( A
% 14.25/8.10 = ( plus_p5714425477246183910nteger @ K3 @ A2 ) )
% 14.25/8.10 => ( ( plus_p5714425477246183910nteger @ A @ B )
% 14.25/8.10 = ( plus_p5714425477246183910nteger @ K3 @ ( plus_p5714425477246183910nteger @ A2 @ B ) ) ) ) ).
% 14.25/8.10
% 14.25/8.10 thf(fact_5063_group__cancel_Oadd1,axiom,
% 14.25/8.10 ! [A: rat,K3: rat,A2: rat,B: rat] :
% 14.25/8.10 ( ( A
% 14.25/8.10 = ( plus_plus_rat @ K3 @ A2 ) )
% 14.25/8.10 => ( ( plus_plus_rat @ A @ B )
% 14.25/8.10 = ( plus_plus_rat @ K3 @ ( plus_plus_rat @ A2 @ B ) ) ) ) ).
% 14.25/8.10
% 14.25/8.10 thf(fact_5064_group__cancel_Oadd1,axiom,
% 14.25/8.10 ! [A: nat,K3: nat,A2: nat,B: nat] :
% 14.25/8.10 ( ( A
% 14.25/8.10 = ( plus_plus_nat @ K3 @ A2 ) )
% 14.25/8.10 => ( ( plus_plus_nat @ A @ B )
% 14.25/8.10 = ( plus_plus_nat @ K3 @ ( plus_plus_nat @ A2 @ B ) ) ) ) ).
% 14.25/8.10
% 14.25/8.10 thf(fact_5065_group__cancel_Oadd1,axiom,
% 14.25/8.10 ! [A: int,K3: int,A2: int,B: int] :
% 14.25/8.10 ( ( A
% 14.25/8.10 = ( plus_plus_int @ K3 @ A2 ) )
% 14.25/8.10 => ( ( plus_plus_int @ A @ B )
% 14.25/8.10 = ( plus_plus_int @ K3 @ ( plus_plus_int @ A2 @ B ) ) ) ) ).
% 14.25/8.10
% 14.25/8.10 thf(fact_5066_group__cancel_Oadd2,axiom,
% 14.25/8.10 ! [B4: code_integer,K3: code_integer,B: code_integer,A2: code_integer] :
% 14.25/8.10 ( ( B4
% 14.25/8.10 = ( plus_p5714425477246183910nteger @ K3 @ B ) )
% 14.25/8.10 => ( ( plus_p5714425477246183910nteger @ A2 @ B4 )
% 14.25/8.10 = ( plus_p5714425477246183910nteger @ K3 @ ( plus_p5714425477246183910nteger @ A2 @ B ) ) ) ) ).
% 14.25/8.10
% 14.25/8.10 thf(fact_5067_group__cancel_Oadd2,axiom,
% 14.25/8.10 ! [B4: rat,K3: rat,B: rat,A2: rat] :
% 14.25/8.10 ( ( B4
% 14.25/8.10 = ( plus_plus_rat @ K3 @ B ) )
% 14.25/8.10 => ( ( plus_plus_rat @ A2 @ B4 )
% 14.25/8.10 = ( plus_plus_rat @ K3 @ ( plus_plus_rat @ A2 @ B ) ) ) ) ).
% 14.25/8.10
% 14.25/8.10 thf(fact_5068_group__cancel_Oadd2,axiom,
% 14.25/8.10 ! [B4: nat,K3: nat,B: nat,A2: nat] :
% 14.25/8.10 ( ( B4
% 14.25/8.10 = ( plus_plus_nat @ K3 @ B ) )
% 14.25/8.10 => ( ( plus_plus_nat @ A2 @ B4 )
% 14.25/8.10 = ( plus_plus_nat @ K3 @ ( plus_plus_nat @ A2 @ B ) ) ) ) ).
% 14.25/8.10
% 14.25/8.10 thf(fact_5069_group__cancel_Oadd2,axiom,
% 14.25/8.10 ! [B4: int,K3: int,B: int,A2: int] :
% 14.25/8.10 ( ( B4
% 14.25/8.10 = ( plus_plus_int @ K3 @ B ) )
% 14.25/8.10 => ( ( plus_plus_int @ A2 @ B4 )
% 14.25/8.10 = ( plus_plus_int @ K3 @ ( plus_plus_int @ A2 @ B ) ) ) ) ).
% 14.25/8.10
% 14.25/8.10 thf(fact_5070_add_Oassoc,axiom,
% 14.25/8.10 ! [A2: code_integer,B: code_integer,C: code_integer] :
% 14.25/8.10 ( ( plus_p5714425477246183910nteger @ ( plus_p5714425477246183910nteger @ A2 @ B ) @ C )
% 14.25/8.10 = ( plus_p5714425477246183910nteger @ A2 @ ( plus_p5714425477246183910nteger @ B @ C ) ) ) ).
% 14.25/8.10
% 14.25/8.10 thf(fact_5071_add_Oassoc,axiom,
% 14.25/8.10 ! [A2: rat,B: rat,C: rat] :
% 14.25/8.10 ( ( plus_plus_rat @ ( plus_plus_rat @ A2 @ B ) @ C )
% 14.25/8.10 = ( plus_plus_rat @ A2 @ ( plus_plus_rat @ B @ C ) ) ) ).
% 14.25/8.10
% 14.25/8.10 thf(fact_5072_add_Oassoc,axiom,
% 14.25/8.10 ! [A2: nat,B: nat,C: nat] :
% 14.25/8.10 ( ( plus_plus_nat @ ( plus_plus_nat @ A2 @ B ) @ C )
% 14.25/8.10 = ( plus_plus_nat @ A2 @ ( plus_plus_nat @ B @ C ) ) ) ).
% 14.25/8.10
% 14.25/8.10 thf(fact_5073_add_Oassoc,axiom,
% 14.25/8.10 ! [A2: int,B: int,C: int] :
% 14.25/8.10 ( ( plus_plus_int @ ( plus_plus_int @ A2 @ B ) @ C )
% 14.25/8.10 = ( plus_plus_int @ A2 @ ( plus_plus_int @ B @ C ) ) ) ).
% 14.25/8.10
% 14.25/8.10 thf(fact_5074_add_Oleft__cancel,axiom,
% 14.25/8.10 ! [A2: code_integer,B: code_integer,C: code_integer] :
% 14.25/8.10 ( ( ( plus_p5714425477246183910nteger @ A2 @ B )
% 14.25/8.10 = ( plus_p5714425477246183910nteger @ A2 @ C ) )
% 14.25/8.10 = ( B = C ) ) ).
% 14.25/8.10
% 14.25/8.10 thf(fact_5075_add_Oleft__cancel,axiom,
% 14.25/8.10 ! [A2: rat,B: rat,C: rat] :
% 14.25/8.10 ( ( ( plus_plus_rat @ A2 @ B )
% 14.25/8.10 = ( plus_plus_rat @ A2 @ C ) )
% 14.25/8.10 = ( B = C ) ) ).
% 14.25/8.10
% 14.25/8.10 thf(fact_5076_add_Oleft__cancel,axiom,
% 14.25/8.10 ! [A2: int,B: int,C: int] :
% 14.25/8.10 ( ( ( plus_plus_int @ A2 @ B )
% 14.25/8.10 = ( plus_plus_int @ A2 @ C ) )
% 14.25/8.10 = ( B = C ) ) ).
% 14.25/8.10
% 14.25/8.10 thf(fact_5077_add_Oright__cancel,axiom,
% 14.25/8.10 ! [B: code_integer,A2: code_integer,C: code_integer] :
% 14.25/8.10 ( ( ( plus_p5714425477246183910nteger @ B @ A2 )
% 14.25/8.10 = ( plus_p5714425477246183910nteger @ C @ A2 ) )
% 14.25/8.10 = ( B = C ) ) ).
% 14.25/8.10
% 14.25/8.10 thf(fact_5078_add_Oright__cancel,axiom,
% 14.25/8.10 ! [B: rat,A2: rat,C: rat] :
% 14.25/8.10 ( ( ( plus_plus_rat @ B @ A2 )
% 14.25/8.10 = ( plus_plus_rat @ C @ A2 ) )
% 14.25/8.10 = ( B = C ) ) ).
% 14.25/8.10
% 14.25/8.10 thf(fact_5079_add_Oright__cancel,axiom,
% 14.25/8.10 ! [B: int,A2: int,C: int] :
% 14.25/8.10 ( ( ( plus_plus_int @ B @ A2 )
% 14.25/8.10 = ( plus_plus_int @ C @ A2 ) )
% 14.25/8.10 = ( B = C ) ) ).
% 14.25/8.10
% 14.25/8.10 thf(fact_5080_ab__semigroup__add__class_Oadd_Ocommute,axiom,
% 14.25/8.10 ( plus_p5714425477246183910nteger
% 14.25/8.10 = ( ^ [A6: code_integer,B6: code_integer] : ( plus_p5714425477246183910nteger @ B6 @ A6 ) ) ) ).
% 14.25/8.10
% 14.25/8.10 thf(fact_5081_ab__semigroup__add__class_Oadd_Ocommute,axiom,
% 14.25/8.10 ( plus_plus_rat
% 14.25/8.10 = ( ^ [A6: rat,B6: rat] : ( plus_plus_rat @ B6 @ A6 ) ) ) ).
% 14.25/8.10
% 14.25/8.10 thf(fact_5082_ab__semigroup__add__class_Oadd_Ocommute,axiom,
% 14.25/8.10 ( plus_plus_nat
% 14.25/8.10 = ( ^ [A6: nat,B6: nat] : ( plus_plus_nat @ B6 @ A6 ) ) ) ).
% 14.25/8.10
% 14.25/8.10 thf(fact_5083_ab__semigroup__add__class_Oadd_Ocommute,axiom,
% 14.25/8.10 ( plus_plus_int
% 14.25/8.10 = ( ^ [A6: int,B6: int] : ( plus_plus_int @ B6 @ A6 ) ) ) ).
% 14.25/8.10
% 14.25/8.10 thf(fact_5084_ab__semigroup__add__class_Oadd_Oleft__commute,axiom,
% 14.25/8.10 ! [B: code_integer,A2: code_integer,C: code_integer] :
% 14.25/8.10 ( ( plus_p5714425477246183910nteger @ B @ ( plus_p5714425477246183910nteger @ A2 @ C ) )
% 14.25/8.10 = ( plus_p5714425477246183910nteger @ A2 @ ( plus_p5714425477246183910nteger @ B @ C ) ) ) ).
% 14.25/8.10
% 14.25/8.10 thf(fact_5085_ab__semigroup__add__class_Oadd_Oleft__commute,axiom,
% 14.25/8.10 ! [B: rat,A2: rat,C: rat] :
% 14.25/8.10 ( ( plus_plus_rat @ B @ ( plus_plus_rat @ A2 @ C ) )
% 14.25/8.10 = ( plus_plus_rat @ A2 @ ( plus_plus_rat @ B @ C ) ) ) ).
% 14.25/8.10
% 14.25/8.10 thf(fact_5086_ab__semigroup__add__class_Oadd_Oleft__commute,axiom,
% 14.25/8.10 ! [B: nat,A2: nat,C: nat] :
% 14.25/8.10 ( ( plus_plus_nat @ B @ ( plus_plus_nat @ A2 @ C ) )
% 14.25/8.10 = ( plus_plus_nat @ A2 @ ( plus_plus_nat @ B @ C ) ) ) ).
% 14.25/8.10
% 14.25/8.10 thf(fact_5087_ab__semigroup__add__class_Oadd_Oleft__commute,axiom,
% 14.25/8.10 ! [B: int,A2: int,C: int] :
% 14.25/8.10 ( ( plus_plus_int @ B @ ( plus_plus_int @ A2 @ C ) )
% 14.25/8.10 = ( plus_plus_int @ A2 @ ( plus_plus_int @ B @ C ) ) ) ).
% 14.25/8.10
% 14.25/8.10 thf(fact_5088_add__left__imp__eq,axiom,
% 14.25/8.10 ! [A2: code_integer,B: code_integer,C: code_integer] :
% 14.25/8.10 ( ( ( plus_p5714425477246183910nteger @ A2 @ B )
% 14.25/8.10 = ( plus_p5714425477246183910nteger @ A2 @ C ) )
% 14.25/8.10 => ( B = C ) ) ).
% 14.25/8.10
% 14.25/8.10 thf(fact_5089_add__left__imp__eq,axiom,
% 14.25/8.10 ! [A2: rat,B: rat,C: rat] :
% 14.25/8.10 ( ( ( plus_plus_rat @ A2 @ B )
% 14.25/8.10 = ( plus_plus_rat @ A2 @ C ) )
% 14.25/8.10 => ( B = C ) ) ).
% 14.25/8.10
% 14.25/8.10 thf(fact_5090_add__left__imp__eq,axiom,
% 14.25/8.10 ! [A2: nat,B: nat,C: nat] :
% 14.25/8.10 ( ( ( plus_plus_nat @ A2 @ B )
% 14.25/8.10 = ( plus_plus_nat @ A2 @ C ) )
% 14.25/8.10 => ( B = C ) ) ).
% 14.25/8.10
% 14.25/8.10 thf(fact_5091_add__left__imp__eq,axiom,
% 14.25/8.10 ! [A2: int,B: int,C: int] :
% 14.25/8.10 ( ( ( plus_plus_int @ A2 @ B )
% 14.25/8.10 = ( plus_plus_int @ A2 @ C ) )
% 14.25/8.10 => ( B = C ) ) ).
% 14.25/8.10
% 14.25/8.10 thf(fact_5092_add__right__imp__eq,axiom,
% 14.25/8.10 ! [B: code_integer,A2: code_integer,C: code_integer] :
% 14.25/8.10 ( ( ( plus_p5714425477246183910nteger @ B @ A2 )
% 14.25/8.10 = ( plus_p5714425477246183910nteger @ C @ A2 ) )
% 14.25/8.10 => ( B = C ) ) ).
% 14.25/8.10
% 14.25/8.10 thf(fact_5093_add__right__imp__eq,axiom,
% 14.25/8.10 ! [B: rat,A2: rat,C: rat] :
% 14.25/8.10 ( ( ( plus_plus_rat @ B @ A2 )
% 14.25/8.10 = ( plus_plus_rat @ C @ A2 ) )
% 14.25/8.10 => ( B = C ) ) ).
% 14.25/8.10
% 14.25/8.10 thf(fact_5094_add__right__imp__eq,axiom,
% 14.25/8.10 ! [B: nat,A2: nat,C: nat] :
% 14.25/8.10 ( ( ( plus_plus_nat @ B @ A2 )
% 14.25/8.10 = ( plus_plus_nat @ C @ A2 ) )
% 14.25/8.10 => ( B = C ) ) ).
% 14.25/8.10
% 14.25/8.10 thf(fact_5095_add__right__imp__eq,axiom,
% 14.25/8.10 ! [B: int,A2: int,C: int] :
% 14.25/8.10 ( ( ( plus_plus_int @ B @ A2 )
% 14.25/8.10 = ( plus_plus_int @ C @ A2 ) )
% 14.25/8.10 => ( B = C ) ) ).
% 14.25/8.10
% 14.25/8.10 thf(fact_5096_add__mono__thms__linordered__semiring_I3_J,axiom,
% 14.25/8.10 ! [I2: code_integer,J: code_integer,K3: code_integer,L: code_integer] :
% 14.25/8.10 ( ( ( ord_le3102999989581377725nteger @ I2 @ J )
% 14.25/8.10 & ( K3 = L ) )
% 14.25/8.10 => ( ord_le3102999989581377725nteger @ ( plus_p5714425477246183910nteger @ I2 @ K3 ) @ ( plus_p5714425477246183910nteger @ J @ L ) ) ) ).
% 14.25/8.10
% 14.25/8.10 thf(fact_5097_add__mono__thms__linordered__semiring_I3_J,axiom,
% 14.25/8.10 ! [I2: rat,J: rat,K3: rat,L: rat] :
% 14.25/8.10 ( ( ( ord_less_eq_rat @ I2 @ J )
% 14.25/8.10 & ( K3 = L ) )
% 14.25/8.10 => ( ord_less_eq_rat @ ( plus_plus_rat @ I2 @ K3 ) @ ( plus_plus_rat @ J @ L ) ) ) ).
% 14.25/8.10
% 14.25/8.10 thf(fact_5098_add__mono__thms__linordered__semiring_I3_J,axiom,
% 14.25/8.10 ! [I2: nat,J: nat,K3: nat,L: nat] :
% 14.25/8.10 ( ( ( ord_less_eq_nat @ I2 @ J )
% 14.25/8.10 & ( K3 = L ) )
% 14.25/8.10 => ( ord_less_eq_nat @ ( plus_plus_nat @ I2 @ K3 ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% 14.25/8.10
% 14.25/8.10 thf(fact_5099_add__mono__thms__linordered__semiring_I3_J,axiom,
% 14.25/8.10 ! [I2: int,J: int,K3: int,L: int] :
% 14.25/8.10 ( ( ( ord_less_eq_int @ I2 @ J )
% 14.25/8.10 & ( K3 = L ) )
% 14.25/8.10 => ( ord_less_eq_int @ ( plus_plus_int @ I2 @ K3 ) @ ( plus_plus_int @ J @ L ) ) ) ).
% 14.25/8.10
% 14.25/8.10 thf(fact_5100_add__mono__thms__linordered__semiring_I2_J,axiom,
% 14.25/8.10 ! [I2: code_integer,J: code_integer,K3: code_integer,L: code_integer] :
% 14.25/8.10 ( ( ( I2 = J )
% 14.25/8.10 & ( ord_le3102999989581377725nteger @ K3 @ L ) )
% 14.25/8.10 => ( ord_le3102999989581377725nteger @ ( plus_p5714425477246183910nteger @ I2 @ K3 ) @ ( plus_p5714425477246183910nteger @ J @ L ) ) ) ).
% 14.25/8.10
% 14.25/8.10 thf(fact_5101_add__mono__thms__linordered__semiring_I2_J,axiom,
% 14.25/8.10 ! [I2: rat,J: rat,K3: rat,L: rat] :
% 14.25/8.10 ( ( ( I2 = J )
% 14.25/8.10 & ( ord_less_eq_rat @ K3 @ L ) )
% 14.25/8.10 => ( ord_less_eq_rat @ ( plus_plus_rat @ I2 @ K3 ) @ ( plus_plus_rat @ J @ L ) ) ) ).
% 14.25/8.10
% 14.25/8.10 thf(fact_5102_add__mono__thms__linordered__semiring_I2_J,axiom,
% 14.25/8.10 ! [I2: nat,J: nat,K3: nat,L: nat] :
% 14.25/8.10 ( ( ( I2 = J )
% 14.25/8.10 & ( ord_less_eq_nat @ K3 @ L ) )
% 14.25/8.10 => ( ord_less_eq_nat @ ( plus_plus_nat @ I2 @ K3 ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% 14.25/8.10
% 14.25/8.10 thf(fact_5103_add__mono__thms__linordered__semiring_I2_J,axiom,
% 14.25/8.10 ! [I2: int,J: int,K3: int,L: int] :
% 14.25/8.10 ( ( ( I2 = J )
% 14.25/8.10 & ( ord_less_eq_int @ K3 @ L ) )
% 14.25/8.10 => ( ord_less_eq_int @ ( plus_plus_int @ I2 @ K3 ) @ ( plus_plus_int @ J @ L ) ) ) ).
% 14.25/8.10
% 14.25/8.10 thf(fact_5104_add__mono__thms__linordered__semiring_I1_J,axiom,
% 14.25/8.10 ! [I2: code_integer,J: code_integer,K3: code_integer,L: code_integer] :
% 14.25/8.10 ( ( ( ord_le3102999989581377725nteger @ I2 @ J )
% 14.25/8.10 & ( ord_le3102999989581377725nteger @ K3 @ L ) )
% 14.25/8.10 => ( ord_le3102999989581377725nteger @ ( plus_p5714425477246183910nteger @ I2 @ K3 ) @ ( plus_p5714425477246183910nteger @ J @ L ) ) ) ).
% 14.25/8.10
% 14.25/8.10 thf(fact_5105_add__mono__thms__linordered__semiring_I1_J,axiom,
% 14.25/8.10 ! [I2: rat,J: rat,K3: rat,L: rat] :
% 14.25/8.10 ( ( ( ord_less_eq_rat @ I2 @ J )
% 14.25/8.10 & ( ord_less_eq_rat @ K3 @ L ) )
% 14.25/8.10 => ( ord_less_eq_rat @ ( plus_plus_rat @ I2 @ K3 ) @ ( plus_plus_rat @ J @ L ) ) ) ).
% 14.25/8.10
% 14.25/8.10 thf(fact_5106_add__mono__thms__linordered__semiring_I1_J,axiom,
% 14.25/8.10 ! [I2: nat,J: nat,K3: nat,L: nat] :
% 14.25/8.10 ( ( ( ord_less_eq_nat @ I2 @ J )
% 14.25/8.10 & ( ord_less_eq_nat @ K3 @ L ) )
% 14.25/8.10 => ( ord_less_eq_nat @ ( plus_plus_nat @ I2 @ K3 ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% 14.25/8.10
% 14.25/8.10 thf(fact_5107_add__mono__thms__linordered__semiring_I1_J,axiom,
% 14.25/8.10 ! [I2: int,J: int,K3: int,L: int] :
% 14.25/8.10 ( ( ( ord_less_eq_int @ I2 @ J )
% 14.25/8.10 & ( ord_less_eq_int @ K3 @ L ) )
% 14.25/8.10 => ( ord_less_eq_int @ ( plus_plus_int @ I2 @ K3 ) @ ( plus_plus_int @ J @ L ) ) ) ).
% 14.25/8.10
% 14.25/8.10 thf(fact_5108_add__mono,axiom,
% 14.25/8.10 ! [A2: code_integer,B: code_integer,C: code_integer,D2: code_integer] :
% 14.25/8.10 ( ( ord_le3102999989581377725nteger @ A2 @ B )
% 14.25/8.10 => ( ( ord_le3102999989581377725nteger @ C @ D2 )
% 14.25/8.10 => ( ord_le3102999989581377725nteger @ ( plus_p5714425477246183910nteger @ A2 @ C ) @ ( plus_p5714425477246183910nteger @ B @ D2 ) ) ) ) ).
% 14.25/8.10
% 14.25/8.10 thf(fact_5109_add__mono,axiom,
% 14.25/8.10 ! [A2: rat,B: rat,C: rat,D2: rat] :
% 14.25/8.10 ( ( ord_less_eq_rat @ A2 @ B )
% 14.25/8.10 => ( ( ord_less_eq_rat @ C @ D2 )
% 14.25/8.10 => ( ord_less_eq_rat @ ( plus_plus_rat @ A2 @ C ) @ ( plus_plus_rat @ B @ D2 ) ) ) ) ).
% 14.25/8.10
% 14.25/8.10 thf(fact_5110_add__mono,axiom,
% 14.25/8.10 ! [A2: nat,B: nat,C: nat,D2: nat] :
% 14.25/8.10 ( ( ord_less_eq_nat @ A2 @ B )
% 14.25/8.10 => ( ( ord_less_eq_nat @ C @ D2 )
% 14.25/8.10 => ( ord_less_eq_nat @ ( plus_plus_nat @ A2 @ C ) @ ( plus_plus_nat @ B @ D2 ) ) ) ) ).
% 14.25/8.10
% 14.25/8.10 thf(fact_5111_add__mono,axiom,
% 14.25/8.10 ! [A2: int,B: int,C: int,D2: int] :
% 14.25/8.10 ( ( ord_less_eq_int @ A2 @ B )
% 14.25/8.10 => ( ( ord_less_eq_int @ C @ D2 )
% 14.25/8.10 => ( ord_less_eq_int @ ( plus_plus_int @ A2 @ C ) @ ( plus_plus_int @ B @ D2 ) ) ) ) ).
% 14.25/8.10
% 14.25/8.10 thf(fact_5112_add__left__mono,axiom,
% 14.25/8.10 ! [A2: code_integer,B: code_integer,C: code_integer] :
% 14.25/8.10 ( ( ord_le3102999989581377725nteger @ A2 @ B )
% 14.25/8.10 => ( ord_le3102999989581377725nteger @ ( plus_p5714425477246183910nteger @ C @ A2 ) @ ( plus_p5714425477246183910nteger @ C @ B ) ) ) ).
% 14.25/8.10
% 14.25/8.10 thf(fact_5113_add__left__mono,axiom,
% 14.25/8.10 ! [A2: rat,B: rat,C: rat] :
% 14.25/8.10 ( ( ord_less_eq_rat @ A2 @ B )
% 14.25/8.10 => ( ord_less_eq_rat @ ( plus_plus_rat @ C @ A2 ) @ ( plus_plus_rat @ C @ B ) ) ) ).
% 14.25/8.10
% 14.25/8.10 thf(fact_5114_add__left__mono,axiom,
% 14.25/8.10 ! [A2: nat,B: nat,C: nat] :
% 14.25/8.10 ( ( ord_less_eq_nat @ A2 @ B )
% 14.25/8.10 => ( ord_less_eq_nat @ ( plus_plus_nat @ C @ A2 ) @ ( plus_plus_nat @ C @ B ) ) ) ).
% 14.25/8.10
% 14.25/8.10 thf(fact_5115_add__left__mono,axiom,
% 14.25/8.10 ! [A2: int,B: int,C: int] :
% 14.25/8.10 ( ( ord_less_eq_int @ A2 @ B )
% 14.25/8.10 => ( ord_less_eq_int @ ( plus_plus_int @ C @ A2 ) @ ( plus_plus_int @ C @ B ) ) ) ).
% 14.25/8.10
% 14.25/8.10 thf(fact_5116_less__eqE,axiom,
% 14.25/8.10 ! [A2: nat,B: nat] :
% 14.25/8.10 ( ( ord_less_eq_nat @ A2 @ B )
% 14.25/8.10 => ~ ! [C2: nat] :
% 14.25/8.10 ( B
% 14.25/8.10 != ( plus_plus_nat @ A2 @ C2 ) ) ) ).
% 14.25/8.10
% 14.25/8.10 thf(fact_5117_add__right__mono,axiom,
% 14.25/8.10 ! [A2: code_integer,B: code_integer,C: code_integer] :
% 14.25/8.10 ( ( ord_le3102999989581377725nteger @ A2 @ B )
% 14.25/8.10 => ( ord_le3102999989581377725nteger @ ( plus_p5714425477246183910nteger @ A2 @ C ) @ ( plus_p5714425477246183910nteger @ B @ C ) ) ) ).
% 14.25/8.10
% 14.25/8.10 thf(fact_5118_add__right__mono,axiom,
% 14.25/8.10 ! [A2: rat,B: rat,C: rat] :
% 14.25/8.10 ( ( ord_less_eq_rat @ A2 @ B )
% 14.25/8.10 => ( ord_less_eq_rat @ ( plus_plus_rat @ A2 @ C ) @ ( plus_plus_rat @ B @ C ) ) ) ).
% 14.25/8.10
% 14.25/8.10 thf(fact_5119_add__right__mono,axiom,
% 14.25/8.10 ! [A2: nat,B: nat,C: nat] :
% 14.25/8.10 ( ( ord_less_eq_nat @ A2 @ B )
% 14.25/8.10 => ( ord_less_eq_nat @ ( plus_plus_nat @ A2 @ C ) @ ( plus_plus_nat @ B @ C ) ) ) ).
% 14.25/8.10
% 14.25/8.10 thf(fact_5120_add__right__mono,axiom,
% 14.25/8.10 ! [A2: int,B: int,C: int] :
% 14.25/8.10 ( ( ord_less_eq_int @ A2 @ B )
% 14.25/8.10 => ( ord_less_eq_int @ ( plus_plus_int @ A2 @ C ) @ ( plus_plus_int @ B @ C ) ) ) ).
% 14.25/8.10
% 14.25/8.10 thf(fact_5121_le__iff__add,axiom,
% 14.25/8.10 ( ord_less_eq_nat
% 14.25/8.10 = ( ^ [A6: nat,B6: nat] :
% 14.25/8.10 ? [C4: nat] :
% 14.25/8.10 ( B6
% 14.25/8.10 = ( plus_plus_nat @ A6 @ C4 ) ) ) ) ).
% 14.25/8.10
% 14.25/8.10 thf(fact_5122_add__le__imp__le__left,axiom,
% 14.25/8.10 ! [C: code_integer,A2: code_integer,B: code_integer] :
% 14.25/8.10 ( ( ord_le3102999989581377725nteger @ ( plus_p5714425477246183910nteger @ C @ A2 ) @ ( plus_p5714425477246183910nteger @ C @ B ) )
% 14.25/8.10 => ( ord_le3102999989581377725nteger @ A2 @ B ) ) ).
% 14.25/8.10
% 14.25/8.10 thf(fact_5123_add__le__imp__le__left,axiom,
% 14.25/8.10 ! [C: rat,A2: rat,B: rat] :
% 14.25/8.10 ( ( ord_less_eq_rat @ ( plus_plus_rat @ C @ A2 ) @ ( plus_plus_rat @ C @ B ) )
% 14.25/8.10 => ( ord_less_eq_rat @ A2 @ B ) ) ).
% 14.25/8.10
% 14.25/8.10 thf(fact_5124_add__le__imp__le__left,axiom,
% 14.25/8.10 ! [C: nat,A2: nat,B: nat] :
% 14.25/8.10 ( ( ord_less_eq_nat @ ( plus_plus_nat @ C @ A2 ) @ ( plus_plus_nat @ C @ B ) )
% 14.25/8.10 => ( ord_less_eq_nat @ A2 @ B ) ) ).
% 14.25/8.10
% 14.25/8.10 thf(fact_5125_add__le__imp__le__left,axiom,
% 14.25/8.10 ! [C: int,A2: int,B: int] :
% 14.25/8.10 ( ( ord_less_eq_int @ ( plus_plus_int @ C @ A2 ) @ ( plus_plus_int @ C @ B ) )
% 14.25/8.10 => ( ord_less_eq_int @ A2 @ B ) ) ).
% 14.25/8.10
% 14.25/8.10 thf(fact_5126_add__le__imp__le__right,axiom,
% 14.25/8.10 ! [A2: code_integer,C: code_integer,B: code_integer] :
% 14.25/8.10 ( ( ord_le3102999989581377725nteger @ ( plus_p5714425477246183910nteger @ A2 @ C ) @ ( plus_p5714425477246183910nteger @ B @ C ) )
% 14.25/8.10 => ( ord_le3102999989581377725nteger @ A2 @ B ) ) ).
% 14.25/8.10
% 14.25/8.10 thf(fact_5127_add__le__imp__le__right,axiom,
% 14.25/8.10 ! [A2: rat,C: rat,B: rat] :
% 14.25/8.10 ( ( ord_less_eq_rat @ ( plus_plus_rat @ A2 @ C ) @ ( plus_plus_rat @ B @ C ) )
% 14.25/8.10 => ( ord_less_eq_rat @ A2 @ B ) ) ).
% 14.25/8.10
% 14.25/8.10 thf(fact_5128_add__le__imp__le__right,axiom,
% 14.25/8.10 ! [A2: nat,C: nat,B: nat] :
% 14.25/8.10 ( ( ord_less_eq_nat @ ( plus_plus_nat @ A2 @ C ) @ ( plus_plus_nat @ B @ C ) )
% 14.25/8.10 => ( ord_less_eq_nat @ A2 @ B ) ) ).
% 14.25/8.10
% 14.25/8.10 thf(fact_5129_add__le__imp__le__right,axiom,
% 14.25/8.10 ! [A2: int,C: int,B: int] :
% 14.25/8.10 ( ( ord_less_eq_int @ ( plus_plus_int @ A2 @ C ) @ ( plus_plus_int @ B @ C ) )
% 14.25/8.10 => ( ord_less_eq_int @ A2 @ B ) ) ).
% 14.25/8.10
% 14.25/8.10 thf(fact_5130_add__mono__thms__linordered__field_I5_J,axiom,
% 14.25/8.10 ! [I2: nat,J: nat,K3: nat,L: nat] :
% 14.25/8.10 ( ( ( ord_less_nat @ I2 @ J )
% 14.25/8.10 & ( ord_less_nat @ K3 @ L ) )
% 14.25/8.10 => ( ord_less_nat @ ( plus_plus_nat @ I2 @ K3 ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% 14.25/8.10
% 14.25/8.10 thf(fact_5131_add__mono__thms__linordered__field_I5_J,axiom,
% 14.25/8.10 ! [I2: int,J: int,K3: int,L: int] :
% 14.25/8.10 ( ( ( ord_less_int @ I2 @ J )
% 14.25/8.10 & ( ord_less_int @ K3 @ L ) )
% 14.25/8.10 => ( ord_less_int @ ( plus_plus_int @ I2 @ K3 ) @ ( plus_plus_int @ J @ L ) ) ) ).
% 14.25/8.10
% 14.25/8.10 thf(fact_5132_add__mono__thms__linordered__field_I5_J,axiom,
% 14.25/8.10 ! [I2: code_integer,J: code_integer,K3: code_integer,L: code_integer] :
% 14.25/8.10 ( ( ( ord_le6747313008572928689nteger @ I2 @ J )
% 14.25/8.10 & ( ord_le6747313008572928689nteger @ K3 @ L ) )
% 14.25/8.10 => ( ord_le6747313008572928689nteger @ ( plus_p5714425477246183910nteger @ I2 @ K3 ) @ ( plus_p5714425477246183910nteger @ J @ L ) ) ) ).
% 14.25/8.10
% 14.25/8.10 thf(fact_5133_add__mono__thms__linordered__field_I5_J,axiom,
% 14.25/8.10 ! [I2: rat,J: rat,K3: rat,L: rat] :
% 14.25/8.10 ( ( ( ord_less_rat @ I2 @ J )
% 14.25/8.10 & ( ord_less_rat @ K3 @ L ) )
% 14.25/8.10 => ( ord_less_rat @ ( plus_plus_rat @ I2 @ K3 ) @ ( plus_plus_rat @ J @ L ) ) ) ).
% 14.25/8.10
% 14.25/8.10 thf(fact_5134_add__mono__thms__linordered__field_I2_J,axiom,
% 14.25/8.10 ! [I2: nat,J: nat,K3: nat,L: nat] :
% 14.25/8.10 ( ( ( I2 = J )
% 14.25/8.10 & ( ord_less_nat @ K3 @ L ) )
% 14.25/8.10 => ( ord_less_nat @ ( plus_plus_nat @ I2 @ K3 ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% 14.25/8.10
% 14.25/8.10 thf(fact_5135_add__mono__thms__linordered__field_I2_J,axiom,
% 14.25/8.10 ! [I2: int,J: int,K3: int,L: int] :
% 14.25/8.10 ( ( ( I2 = J )
% 14.25/8.10 & ( ord_less_int @ K3 @ L ) )
% 14.25/8.10 => ( ord_less_int @ ( plus_plus_int @ I2 @ K3 ) @ ( plus_plus_int @ J @ L ) ) ) ).
% 14.25/8.10
% 14.25/8.10 thf(fact_5136_add__mono__thms__linordered__field_I2_J,axiom,
% 14.25/8.10 ! [I2: code_integer,J: code_integer,K3: code_integer,L: code_integer] :
% 14.25/8.10 ( ( ( I2 = J )
% 14.25/8.10 & ( ord_le6747313008572928689nteger @ K3 @ L ) )
% 14.25/8.10 => ( ord_le6747313008572928689nteger @ ( plus_p5714425477246183910nteger @ I2 @ K3 ) @ ( plus_p5714425477246183910nteger @ J @ L ) ) ) ).
% 14.25/8.10
% 14.25/8.10 thf(fact_5137_add__mono__thms__linordered__field_I2_J,axiom,
% 14.25/8.10 ! [I2: rat,J: rat,K3: rat,L: rat] :
% 14.25/8.10 ( ( ( I2 = J )
% 14.25/8.10 & ( ord_less_rat @ K3 @ L ) )
% 14.25/8.10 => ( ord_less_rat @ ( plus_plus_rat @ I2 @ K3 ) @ ( plus_plus_rat @ J @ L ) ) ) ).
% 14.25/8.10
% 14.25/8.10 thf(fact_5138_add__mono__thms__linordered__field_I1_J,axiom,
% 14.25/8.10 ! [I2: nat,J: nat,K3: nat,L: nat] :
% 14.25/8.10 ( ( ( ord_less_nat @ I2 @ J )
% 14.25/8.10 & ( K3 = L ) )
% 14.25/8.10 => ( ord_less_nat @ ( plus_plus_nat @ I2 @ K3 ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% 14.25/8.10
% 14.25/8.10 thf(fact_5139_add__mono__thms__linordered__field_I1_J,axiom,
% 14.25/8.10 ! [I2: int,J: int,K3: int,L: int] :
% 14.25/8.10 ( ( ( ord_less_int @ I2 @ J )
% 14.25/8.10 & ( K3 = L ) )
% 14.25/8.10 => ( ord_less_int @ ( plus_plus_int @ I2 @ K3 ) @ ( plus_plus_int @ J @ L ) ) ) ).
% 14.25/8.10
% 14.25/8.10 thf(fact_5140_add__mono__thms__linordered__field_I1_J,axiom,
% 14.25/8.10 ! [I2: code_integer,J: code_integer,K3: code_integer,L: code_integer] :
% 14.25/8.10 ( ( ( ord_le6747313008572928689nteger @ I2 @ J )
% 14.25/8.10 & ( K3 = L ) )
% 14.25/8.10 => ( ord_le6747313008572928689nteger @ ( plus_p5714425477246183910nteger @ I2 @ K3 ) @ ( plus_p5714425477246183910nteger @ J @ L ) ) ) ).
% 14.25/8.10
% 14.25/8.10 thf(fact_5141_add__mono__thms__linordered__field_I1_J,axiom,
% 14.25/8.10 ! [I2: rat,J: rat,K3: rat,L: rat] :
% 14.25/8.10 ( ( ( ord_less_rat @ I2 @ J )
% 14.25/8.10 & ( K3 = L ) )
% 14.25/8.10 => ( ord_less_rat @ ( plus_plus_rat @ I2 @ K3 ) @ ( plus_plus_rat @ J @ L ) ) ) ).
% 14.25/8.10
% 14.25/8.10 thf(fact_5142_add__strict__mono,axiom,
% 14.25/8.10 ! [A2: nat,B: nat,C: nat,D2: nat] :
% 14.25/8.10 ( ( ord_less_nat @ A2 @ B )
% 14.25/8.10 => ( ( ord_less_nat @ C @ D2 )
% 14.25/8.10 => ( ord_less_nat @ ( plus_plus_nat @ A2 @ C ) @ ( plus_plus_nat @ B @ D2 ) ) ) ) ).
% 14.25/8.10
% 14.25/8.10 thf(fact_5143_add__strict__mono,axiom,
% 14.25/8.10 ! [A2: int,B: int,C: int,D2: int] :
% 14.25/8.10 ( ( ord_less_int @ A2 @ B )
% 14.25/8.10 => ( ( ord_less_int @ C @ D2 )
% 14.25/8.10 => ( ord_less_int @ ( plus_plus_int @ A2 @ C ) @ ( plus_plus_int @ B @ D2 ) ) ) ) ).
% 14.25/8.10
% 14.25/8.10 thf(fact_5144_add__strict__mono,axiom,
% 14.25/8.10 ! [A2: code_integer,B: code_integer,C: code_integer,D2: code_integer] :
% 14.25/8.10 ( ( ord_le6747313008572928689nteger @ A2 @ B )
% 14.25/8.10 => ( ( ord_le6747313008572928689nteger @ C @ D2 )
% 14.25/8.10 => ( ord_le6747313008572928689nteger @ ( plus_p5714425477246183910nteger @ A2 @ C ) @ ( plus_p5714425477246183910nteger @ B @ D2 ) ) ) ) ).
% 14.25/8.10
% 14.25/8.10 thf(fact_5145_add__strict__mono,axiom,
% 14.25/8.10 ! [A2: rat,B: rat,C: rat,D2: rat] :
% 14.25/8.10 ( ( ord_less_rat @ A2 @ B )
% 14.25/8.10 => ( ( ord_less_rat @ C @ D2 )
% 14.25/8.10 => ( ord_less_rat @ ( plus_plus_rat @ A2 @ C ) @ ( plus_plus_rat @ B @ D2 ) ) ) ) ).
% 14.25/8.10
% 14.25/8.10 thf(fact_5146_add__strict__left__mono,axiom,
% 14.25/8.10 ! [A2: nat,B: nat,C: nat] :
% 14.25/8.10 ( ( ord_less_nat @ A2 @ B )
% 14.25/8.10 => ( ord_less_nat @ ( plus_plus_nat @ C @ A2 ) @ ( plus_plus_nat @ C @ B ) ) ) ).
% 14.25/8.10
% 14.25/8.10 thf(fact_5147_add__strict__left__mono,axiom,
% 14.25/8.10 ! [A2: int,B: int,C: int] :
% 14.25/8.10 ( ( ord_less_int @ A2 @ B )
% 14.25/8.10 => ( ord_less_int @ ( plus_plus_int @ C @ A2 ) @ ( plus_plus_int @ C @ B ) ) ) ).
% 14.25/8.10
% 14.25/8.10 thf(fact_5148_add__strict__left__mono,axiom,
% 14.25/8.10 ! [A2: code_integer,B: code_integer,C: code_integer] :
% 14.25/8.10 ( ( ord_le6747313008572928689nteger @ A2 @ B )
% 14.25/8.10 => ( ord_le6747313008572928689nteger @ ( plus_p5714425477246183910nteger @ C @ A2 ) @ ( plus_p5714425477246183910nteger @ C @ B ) ) ) ).
% 14.25/8.10
% 14.25/8.10 thf(fact_5149_add__strict__left__mono,axiom,
% 14.25/8.10 ! [A2: rat,B: rat,C: rat] :
% 14.25/8.10 ( ( ord_less_rat @ A2 @ B )
% 14.25/8.10 => ( ord_less_rat @ ( plus_plus_rat @ C @ A2 ) @ ( plus_plus_rat @ C @ B ) ) ) ).
% 14.25/8.10
% 14.25/8.10 thf(fact_5150_add__strict__right__mono,axiom,
% 14.25/8.10 ! [A2: nat,B: nat,C: nat] :
% 14.25/8.10 ( ( ord_less_nat @ A2 @ B )
% 14.25/8.10 => ( ord_less_nat @ ( plus_plus_nat @ A2 @ C ) @ ( plus_plus_nat @ B @ C ) ) ) ).
% 14.25/8.10
% 14.25/8.10 thf(fact_5151_add__strict__right__mono,axiom,
% 14.25/8.10 ! [A2: int,B: int,C: int] :
% 14.25/8.10 ( ( ord_less_int @ A2 @ B )
% 14.25/8.10 => ( ord_less_int @ ( plus_plus_int @ A2 @ C ) @ ( plus_plus_int @ B @ C ) ) ) ).
% 14.25/8.10
% 14.25/8.10 thf(fact_5152_add__strict__right__mono,axiom,
% 14.25/8.10 ! [A2: code_integer,B: code_integer,C: code_integer] :
% 14.25/8.10 ( ( ord_le6747313008572928689nteger @ A2 @ B )
% 14.25/8.10 => ( ord_le6747313008572928689nteger @ ( plus_p5714425477246183910nteger @ A2 @ C ) @ ( plus_p5714425477246183910nteger @ B @ C ) ) ) ).
% 14.25/8.10
% 14.25/8.10 thf(fact_5153_add__strict__right__mono,axiom,
% 14.25/8.10 ! [A2: rat,B: rat,C: rat] :
% 14.25/8.10 ( ( ord_less_rat @ A2 @ B )
% 14.25/8.10 => ( ord_less_rat @ ( plus_plus_rat @ A2 @ C ) @ ( plus_plus_rat @ B @ C ) ) ) ).
% 14.25/8.10
% 14.25/8.10 thf(fact_5154_add__less__imp__less__left,axiom,
% 14.25/8.10 ! [C: nat,A2: nat,B: nat] :
% 14.25/8.10 ( ( ord_less_nat @ ( plus_plus_nat @ C @ A2 ) @ ( plus_plus_nat @ C @ B ) )
% 14.25/8.10 => ( ord_less_nat @ A2 @ B ) ) ).
% 14.25/8.10
% 14.25/8.10 thf(fact_5155_add__less__imp__less__left,axiom,
% 14.25/8.10 ! [C: int,A2: int,B: int] :
% 14.25/8.10 ( ( ord_less_int @ ( plus_plus_int @ C @ A2 ) @ ( plus_plus_int @ C @ B ) )
% 14.25/8.10 => ( ord_less_int @ A2 @ B ) ) ).
% 14.25/8.10
% 14.25/8.10 thf(fact_5156_add__less__imp__less__left,axiom,
% 14.25/8.10 ! [C: code_integer,A2: code_integer,B: code_integer] :
% 14.25/8.10 ( ( ord_le6747313008572928689nteger @ ( plus_p5714425477246183910nteger @ C @ A2 ) @ ( plus_p5714425477246183910nteger @ C @ B ) )
% 14.25/8.10 => ( ord_le6747313008572928689nteger @ A2 @ B ) ) ).
% 14.25/8.10
% 14.25/8.10 thf(fact_5157_add__less__imp__less__left,axiom,
% 14.25/8.10 ! [C: rat,A2: rat,B: rat] :
% 14.25/8.10 ( ( ord_less_rat @ ( plus_plus_rat @ C @ A2 ) @ ( plus_plus_rat @ C @ B ) )
% 14.25/8.10 => ( ord_less_rat @ A2 @ B ) ) ).
% 14.25/8.10
% 14.25/8.10 thf(fact_5158_add__less__imp__less__right,axiom,
% 14.25/8.10 ! [A2: nat,C: nat,B: nat] :
% 14.25/8.10 ( ( ord_less_nat @ ( plus_plus_nat @ A2 @ C ) @ ( plus_plus_nat @ B @ C ) )
% 14.25/8.10 => ( ord_less_nat @ A2 @ B ) ) ).
% 14.25/8.10
% 14.25/8.10 thf(fact_5159_add__less__imp__less__right,axiom,
% 14.25/8.10 ! [A2: int,C: int,B: int] :
% 14.25/8.10 ( ( ord_less_int @ ( plus_plus_int @ A2 @ C ) @ ( plus_plus_int @ B @ C ) )
% 14.25/8.10 => ( ord_less_int @ A2 @ B ) ) ).
% 14.25/8.10
% 14.25/8.10 thf(fact_5160_add__less__imp__less__right,axiom,
% 14.25/8.10 ! [A2: code_integer,C: code_integer,B: code_integer] :
% 14.25/8.10 ( ( ord_le6747313008572928689nteger @ ( plus_p5714425477246183910nteger @ A2 @ C ) @ ( plus_p5714425477246183910nteger @ B @ C ) )
% 14.25/8.10 => ( ord_le6747313008572928689nteger @ A2 @ B ) ) ).
% 14.25/8.10
% 14.25/8.10 thf(fact_5161_add__less__imp__less__right,axiom,
% 14.25/8.10 ! [A2: rat,C: rat,B: rat] :
% 14.25/8.10 ( ( ord_less_rat @ ( plus_plus_rat @ A2 @ C ) @ ( plus_plus_rat @ B @ C ) )
% 14.25/8.10 => ( ord_less_rat @ A2 @ B ) ) ).
% 14.25/8.10
% 14.25/8.10 thf(fact_5162_combine__common__factor,axiom,
% 14.25/8.10 ! [A2: code_integer,E2: code_integer,B: code_integer,C: code_integer] :
% 14.25/8.10 ( ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ A2 @ E2 ) @ ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ B @ E2 ) @ C ) )
% 14.25/8.10 = ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ ( plus_p5714425477246183910nteger @ A2 @ B ) @ E2 ) @ C ) ) ).
% 14.25/8.10
% 14.25/8.10 thf(fact_5163_combine__common__factor,axiom,
% 14.25/8.10 ! [A2: rat,E2: rat,B: rat,C: rat] :
% 14.25/8.10 ( ( plus_plus_rat @ ( times_times_rat @ A2 @ E2 ) @ ( plus_plus_rat @ ( times_times_rat @ B @ E2 ) @ C ) )
% 14.25/8.10 = ( plus_plus_rat @ ( times_times_rat @ ( plus_plus_rat @ A2 @ B ) @ E2 ) @ C ) ) ).
% 14.25/8.10
% 14.25/8.10 thf(fact_5164_combine__common__factor,axiom,
% 14.25/8.10 ! [A2: nat,E2: nat,B: nat,C: nat] :
% 14.25/8.10 ( ( plus_plus_nat @ ( times_times_nat @ A2 @ E2 ) @ ( plus_plus_nat @ ( times_times_nat @ B @ E2 ) @ C ) )
% 14.25/8.10 = ( plus_plus_nat @ ( times_times_nat @ ( plus_plus_nat @ A2 @ B ) @ E2 ) @ C ) ) ).
% 14.25/8.10
% 14.25/8.10 thf(fact_5165_combine__common__factor,axiom,
% 14.25/8.10 ! [A2: int,E2: int,B: int,C: int] :
% 14.25/8.10 ( ( plus_plus_int @ ( times_times_int @ A2 @ E2 ) @ ( plus_plus_int @ ( times_times_int @ B @ E2 ) @ C ) )
% 14.25/8.10 = ( plus_plus_int @ ( times_times_int @ ( plus_plus_int @ A2 @ B ) @ E2 ) @ C ) ) ).
% 14.25/8.10
% 14.25/8.10 thf(fact_5166_distrib__right,axiom,
% 14.25/8.10 ! [A2: code_integer,B: code_integer,C: code_integer] :
% 14.25/8.10 ( ( times_3573771949741848930nteger @ ( plus_p5714425477246183910nteger @ A2 @ B ) @ C )
% 14.25/8.10 = ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ A2 @ C ) @ ( times_3573771949741848930nteger @ B @ C ) ) ) ).
% 14.25/8.10
% 14.25/8.10 thf(fact_5167_distrib__right,axiom,
% 14.25/8.10 ! [A2: rat,B: rat,C: rat] :
% 14.25/8.10 ( ( times_times_rat @ ( plus_plus_rat @ A2 @ B ) @ C )
% 14.25/8.10 = ( plus_plus_rat @ ( times_times_rat @ A2 @ C ) @ ( times_times_rat @ B @ C ) ) ) ).
% 14.25/8.10
% 14.25/8.10 thf(fact_5168_distrib__right,axiom,
% 14.25/8.10 ! [A2: nat,B: nat,C: nat] :
% 14.25/8.10 ( ( times_times_nat @ ( plus_plus_nat @ A2 @ B ) @ C )
% 14.25/8.10 = ( plus_plus_nat @ ( times_times_nat @ A2 @ C ) @ ( times_times_nat @ B @ C ) ) ) ).
% 14.25/8.10
% 14.25/8.10 thf(fact_5169_distrib__right,axiom,
% 14.25/8.10 ! [A2: int,B: int,C: int] :
% 14.25/8.10 ( ( times_times_int @ ( plus_plus_int @ A2 @ B ) @ C )
% 14.25/8.10 = ( plus_plus_int @ ( times_times_int @ A2 @ C ) @ ( times_times_int @ B @ C ) ) ) ).
% 14.25/8.10
% 14.25/8.10 thf(fact_5170_distrib__left,axiom,
% 14.25/8.10 ! [A2: code_integer,B: code_integer,C: code_integer] :
% 14.25/8.10 ( ( times_3573771949741848930nteger @ A2 @ ( plus_p5714425477246183910nteger @ B @ C ) )
% 14.25/8.10 = ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ A2 @ B ) @ ( times_3573771949741848930nteger @ A2 @ C ) ) ) ).
% 14.25/8.10
% 14.25/8.10 thf(fact_5171_distrib__left,axiom,
% 14.25/8.10 ! [A2: rat,B: rat,C: rat] :
% 14.25/8.10 ( ( times_times_rat @ A2 @ ( plus_plus_rat @ B @ C ) )
% 14.25/8.10 = ( plus_plus_rat @ ( times_times_rat @ A2 @ B ) @ ( times_times_rat @ A2 @ C ) ) ) ).
% 14.25/8.10
% 14.25/8.10 thf(fact_5172_distrib__left,axiom,
% 14.25/8.10 ! [A2: nat,B: nat,C: nat] :
% 14.25/8.10 ( ( times_times_nat @ A2 @ ( plus_plus_nat @ B @ C ) )
% 14.25/8.10 = ( plus_plus_nat @ ( times_times_nat @ A2 @ B ) @ ( times_times_nat @ A2 @ C ) ) ) ).
% 14.25/8.10
% 14.25/8.10 thf(fact_5173_distrib__left,axiom,
% 14.25/8.10 ! [A2: int,B: int,C: int] :
% 14.25/8.10 ( ( times_times_int @ A2 @ ( plus_plus_int @ B @ C ) )
% 14.25/8.10 = ( plus_plus_int @ ( times_times_int @ A2 @ B ) @ ( times_times_int @ A2 @ C ) ) ) ).
% 14.25/8.10
% 14.25/8.10 thf(fact_5174_comm__semiring__class_Odistrib,axiom,
% 14.25/8.10 ! [A2: code_integer,B: code_integer,C: code_integer] :
% 14.25/8.10 ( ( times_3573771949741848930nteger @ ( plus_p5714425477246183910nteger @ A2 @ B ) @ C )
% 14.25/8.10 = ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ A2 @ C ) @ ( times_3573771949741848930nteger @ B @ C ) ) ) ).
% 14.25/8.10
% 14.25/8.10 thf(fact_5175_comm__semiring__class_Odistrib,axiom,
% 14.25/8.10 ! [A2: rat,B: rat,C: rat] :
% 14.25/8.10 ( ( times_times_rat @ ( plus_plus_rat @ A2 @ B ) @ C )
% 14.25/8.10 = ( plus_plus_rat @ ( times_times_rat @ A2 @ C ) @ ( times_times_rat @ B @ C ) ) ) ).
% 14.25/8.10
% 14.25/8.10 thf(fact_5176_comm__semiring__class_Odistrib,axiom,
% 14.25/8.10 ! [A2: nat,B: nat,C: nat] :
% 14.25/8.10 ( ( times_times_nat @ ( plus_plus_nat @ A2 @ B ) @ C )
% 14.25/8.10 = ( plus_plus_nat @ ( times_times_nat @ A2 @ C ) @ ( times_times_nat @ B @ C ) ) ) ).
% 14.25/8.10
% 14.25/8.10 thf(fact_5177_comm__semiring__class_Odistrib,axiom,
% 14.25/8.10 ! [A2: int,B: int,C: int] :
% 14.25/8.10 ( ( times_times_int @ ( plus_plus_int @ A2 @ B ) @ C )
% 14.25/8.10 = ( plus_plus_int @ ( times_times_int @ A2 @ C ) @ ( times_times_int @ B @ C ) ) ) ).
% 14.25/8.10
% 14.25/8.10 thf(fact_5178_ring__class_Oring__distribs_I1_J,axiom,
% 14.25/8.10 ! [A2: code_integer,B: code_integer,C: code_integer] :
% 14.25/8.10 ( ( times_3573771949741848930nteger @ A2 @ ( plus_p5714425477246183910nteger @ B @ C ) )
% 14.25/8.10 = ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ A2 @ B ) @ ( times_3573771949741848930nteger @ A2 @ C ) ) ) ).
% 14.25/8.10
% 14.25/8.10 thf(fact_5179_ring__class_Oring__distribs_I1_J,axiom,
% 14.25/8.10 ! [A2: rat,B: rat,C: rat] :
% 14.25/8.10 ( ( times_times_rat @ A2 @ ( plus_plus_rat @ B @ C ) )
% 14.25/8.10 = ( plus_plus_rat @ ( times_times_rat @ A2 @ B ) @ ( times_times_rat @ A2 @ C ) ) ) ).
% 14.25/8.10
% 14.25/8.10 thf(fact_5180_ring__class_Oring__distribs_I1_J,axiom,
% 14.25/8.10 ! [A2: int,B: int,C: int] :
% 14.25/8.10 ( ( times_times_int @ A2 @ ( plus_plus_int @ B @ C ) )
% 14.25/8.10 = ( plus_plus_int @ ( times_times_int @ A2 @ B ) @ ( times_times_int @ A2 @ C ) ) ) ).
% 14.25/8.10
% 14.25/8.10 thf(fact_5181_ring__class_Oring__distribs_I2_J,axiom,
% 14.25/8.10 ! [A2: code_integer,B: code_integer,C: code_integer] :
% 14.25/8.10 ( ( times_3573771949741848930nteger @ ( plus_p5714425477246183910nteger @ A2 @ B ) @ C )
% 14.25/8.10 = ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ A2 @ C ) @ ( times_3573771949741848930nteger @ B @ C ) ) ) ).
% 14.25/8.10
% 14.25/8.10 thf(fact_5182_ring__class_Oring__distribs_I2_J,axiom,
% 14.25/8.10 ! [A2: rat,B: rat,C: rat] :
% 14.25/8.10 ( ( times_times_rat @ ( plus_plus_rat @ A2 @ B ) @ C )
% 14.25/8.10 = ( plus_plus_rat @ ( times_times_rat @ A2 @ C ) @ ( times_times_rat @ B @ C ) ) ) ).
% 14.25/8.10
% 14.25/8.10 thf(fact_5183_ring__class_Oring__distribs_I2_J,axiom,
% 14.25/8.10 ! [A2: int,B: int,C: int] :
% 14.25/8.10 ( ( times_times_int @ ( plus_plus_int @ A2 @ B ) @ C )
% 14.25/8.10 = ( plus_plus_int @ ( times_times_int @ A2 @ C ) @ ( times_times_int @ B @ C ) ) ) ).
% 14.25/8.10
% 14.25/8.10 thf(fact_5184_group__cancel_Osub1,axiom,
% 14.25/8.10 ! [A: code_integer,K3: code_integer,A2: code_integer,B: code_integer] :
% 14.25/8.10 ( ( A
% 14.25/8.10 = ( plus_p5714425477246183910nteger @ K3 @ A2 ) )
% 14.25/8.10 => ( ( minus_8373710615458151222nteger @ A @ B )
% 14.25/8.10 = ( plus_p5714425477246183910nteger @ K3 @ ( minus_8373710615458151222nteger @ A2 @ B ) ) ) ) ).
% 14.25/8.10
% 14.25/8.10 thf(fact_5185_group__cancel_Osub1,axiom,
% 14.25/8.10 ! [A: rat,K3: rat,A2: rat,B: rat] :
% 14.25/8.10 ( ( A
% 14.25/8.10 = ( plus_plus_rat @ K3 @ A2 ) )
% 14.25/8.10 => ( ( minus_minus_rat @ A @ B )
% 14.25/8.10 = ( plus_plus_rat @ K3 @ ( minus_minus_rat @ A2 @ B ) ) ) ) ).
% 14.25/8.10
% 14.25/8.10 thf(fact_5186_group__cancel_Osub1,axiom,
% 14.25/8.10 ! [A: int,K3: int,A2: int,B: int] :
% 14.25/8.10 ( ( A
% 14.25/8.10 = ( plus_plus_int @ K3 @ A2 ) )
% 14.25/8.10 => ( ( minus_minus_int @ A @ B )
% 14.25/8.10 = ( plus_plus_int @ K3 @ ( minus_minus_int @ A2 @ B ) ) ) ) ).
% 14.25/8.10
% 14.25/8.10 thf(fact_5187_diff__eq__eq,axiom,
% 14.25/8.10 ! [A2: code_integer,B: code_integer,C: code_integer] :
% 14.25/8.10 ( ( ( minus_8373710615458151222nteger @ A2 @ B )
% 14.25/8.10 = C )
% 14.25/8.10 = ( A2
% 14.25/8.10 = ( plus_p5714425477246183910nteger @ C @ B ) ) ) ).
% 14.25/8.10
% 14.25/8.10 thf(fact_5188_diff__eq__eq,axiom,
% 14.25/8.10 ! [A2: rat,B: rat,C: rat] :
% 14.25/8.10 ( ( ( minus_minus_rat @ A2 @ B )
% 14.25/8.10 = C )
% 14.25/8.10 = ( A2
% 14.25/8.10 = ( plus_plus_rat @ C @ B ) ) ) ).
% 14.25/8.10
% 14.25/8.10 thf(fact_5189_diff__eq__eq,axiom,
% 14.25/8.10 ! [A2: int,B: int,C: int] :
% 14.25/8.10 ( ( ( minus_minus_int @ A2 @ B )
% 14.25/8.10 = C )
% 14.25/8.10 = ( A2
% 14.25/8.10 = ( plus_plus_int @ C @ B ) ) ) ).
% 14.25/8.10
% 14.25/8.10 thf(fact_5190_eq__diff__eq,axiom,
% 14.25/8.10 ! [A2: code_integer,C: code_integer,B: code_integer] :
% 14.25/8.10 ( ( A2
% 14.25/8.10 = ( minus_8373710615458151222nteger @ C @ B ) )
% 14.25/8.10 = ( ( plus_p5714425477246183910nteger @ A2 @ B )
% 14.25/8.10 = C ) ) ).
% 14.25/8.10
% 14.25/8.10 thf(fact_5191_eq__diff__eq,axiom,
% 14.25/8.10 ! [A2: rat,C: rat,B: rat] :
% 14.25/8.10 ( ( A2
% 14.25/8.10 = ( minus_minus_rat @ C @ B ) )
% 14.25/8.10 = ( ( plus_plus_rat @ A2 @ B )
% 14.25/8.10 = C ) ) ).
% 14.25/8.10
% 14.25/8.10 thf(fact_5192_eq__diff__eq,axiom,
% 14.25/8.10 ! [A2: int,C: int,B: int] :
% 14.25/8.10 ( ( A2
% 14.25/8.10 = ( minus_minus_int @ C @ B ) )
% 14.25/8.10 = ( ( plus_plus_int @ A2 @ B )
% 14.25/8.10 = C ) ) ).
% 14.25/8.10
% 14.25/8.10 thf(fact_5193_add__diff__eq,axiom,
% 14.25/8.10 ! [A2: code_integer,B: code_integer,C: code_integer] :
% 14.25/8.10 ( ( plus_p5714425477246183910nteger @ A2 @ ( minus_8373710615458151222nteger @ B @ C ) )
% 14.25/8.10 = ( minus_8373710615458151222nteger @ ( plus_p5714425477246183910nteger @ A2 @ B ) @ C ) ) ).
% 14.25/8.10
% 14.25/8.10 thf(fact_5194_add__diff__eq,axiom,
% 14.25/8.10 ! [A2: rat,B: rat,C: rat] :
% 14.25/8.10 ( ( plus_plus_rat @ A2 @ ( minus_minus_rat @ B @ C ) )
% 14.25/8.10 = ( minus_minus_rat @ ( plus_plus_rat @ A2 @ B ) @ C ) ) ).
% 14.25/8.10
% 14.25/8.10 thf(fact_5195_add__diff__eq,axiom,
% 14.25/8.10 ! [A2: int,B: int,C: int] :
% 14.25/8.10 ( ( plus_plus_int @ A2 @ ( minus_minus_int @ B @ C ) )
% 14.25/8.10 = ( minus_minus_int @ ( plus_plus_int @ A2 @ B ) @ C ) ) ).
% 14.25/8.10
% 14.25/8.10 thf(fact_5196_diff__diff__eq2,axiom,
% 14.25/8.10 ! [A2: code_integer,B: code_integer,C: code_integer] :
% 14.25/8.10 ( ( minus_8373710615458151222nteger @ A2 @ ( minus_8373710615458151222nteger @ B @ C ) )
% 14.25/8.10 = ( minus_8373710615458151222nteger @ ( plus_p5714425477246183910nteger @ A2 @ C ) @ B ) ) ).
% 14.25/8.10
% 14.25/8.10 thf(fact_5197_diff__diff__eq2,axiom,
% 14.25/8.10 ! [A2: rat,B: rat,C: rat] :
% 14.25/8.10 ( ( minus_minus_rat @ A2 @ ( minus_minus_rat @ B @ C ) )
% 14.25/8.10 = ( minus_minus_rat @ ( plus_plus_rat @ A2 @ C ) @ B ) ) ).
% 14.25/8.10
% 14.25/8.10 thf(fact_5198_diff__diff__eq2,axiom,
% 14.25/8.10 ! [A2: int,B: int,C: int] :
% 14.25/8.10 ( ( minus_minus_int @ A2 @ ( minus_minus_int @ B @ C ) )
% 14.25/8.10 = ( minus_minus_int @ ( plus_plus_int @ A2 @ C ) @ B ) ) ).
% 14.25/8.10
% 14.25/8.10 thf(fact_5199_diff__add__eq,axiom,
% 14.25/8.10 ! [A2: code_integer,B: code_integer,C: code_integer] :
% 14.25/8.10 ( ( plus_p5714425477246183910nteger @ ( minus_8373710615458151222nteger @ A2 @ B ) @ C )
% 14.25/8.10 = ( minus_8373710615458151222nteger @ ( plus_p5714425477246183910nteger @ A2 @ C ) @ B ) ) ).
% 14.25/8.10
% 14.25/8.10 thf(fact_5200_diff__add__eq,axiom,
% 14.25/8.10 ! [A2: rat,B: rat,C: rat] :
% 14.25/8.10 ( ( plus_plus_rat @ ( minus_minus_rat @ A2 @ B ) @ C )
% 14.25/8.10 = ( minus_minus_rat @ ( plus_plus_rat @ A2 @ C ) @ B ) ) ).
% 14.25/8.10
% 14.25/8.10 thf(fact_5201_diff__add__eq,axiom,
% 14.25/8.10 ! [A2: int,B: int,C: int] :
% 14.25/8.10 ( ( plus_plus_int @ ( minus_minus_int @ A2 @ B ) @ C )
% 14.25/8.10 = ( minus_minus_int @ ( plus_plus_int @ A2 @ C ) @ B ) ) ).
% 14.25/8.10
% 14.25/8.10 thf(fact_5202_diff__add__eq__diff__diff__swap,axiom,
% 14.25/8.10 ! [A2: code_integer,B: code_integer,C: code_integer] :
% 14.25/8.10 ( ( minus_8373710615458151222nteger @ A2 @ ( plus_p5714425477246183910nteger @ B @ C ) )
% 14.25/8.10 = ( minus_8373710615458151222nteger @ ( minus_8373710615458151222nteger @ A2 @ C ) @ B ) ) ).
% 14.25/8.10
% 14.25/8.10 thf(fact_5203_diff__add__eq__diff__diff__swap,axiom,
% 14.25/8.10 ! [A2: rat,B: rat,C: rat] :
% 14.25/8.10 ( ( minus_minus_rat @ A2 @ ( plus_plus_rat @ B @ C ) )
% 14.25/8.10 = ( minus_minus_rat @ ( minus_minus_rat @ A2 @ C ) @ B ) ) ).
% 14.25/8.10
% 14.25/8.10 thf(fact_5204_diff__add__eq__diff__diff__swap,axiom,
% 14.25/8.10 ! [A2: int,B: int,C: int] :
% 14.25/8.10 ( ( minus_minus_int @ A2 @ ( plus_plus_int @ B @ C ) )
% 14.25/8.10 = ( minus_minus_int @ ( minus_minus_int @ A2 @ C ) @ B ) ) ).
% 14.25/8.10
% 14.25/8.10 thf(fact_5205_add__implies__diff,axiom,
% 14.25/8.10 ! [C: code_integer,B: code_integer,A2: code_integer] :
% 14.25/8.10 ( ( ( plus_p5714425477246183910nteger @ C @ B )
% 14.25/8.10 = A2 )
% 14.25/8.10 => ( C
% 14.25/8.10 = ( minus_8373710615458151222nteger @ A2 @ B ) ) ) ).
% 14.25/8.10
% 14.25/8.10 thf(fact_5206_add__implies__diff,axiom,
% 14.25/8.10 ! [C: rat,B: rat,A2: rat] :
% 14.25/8.10 ( ( ( plus_plus_rat @ C @ B )
% 14.25/8.10 = A2 )
% 14.25/8.10 => ( C
% 14.25/8.10 = ( minus_minus_rat @ A2 @ B ) ) ) ).
% 14.25/8.10
% 14.25/8.10 thf(fact_5207_add__implies__diff,axiom,
% 14.25/8.10 ! [C: nat,B: nat,A2: nat] :
% 14.25/8.10 ( ( ( plus_plus_nat @ C @ B )
% 14.25/8.10 = A2 )
% 14.25/8.10 => ( C
% 14.25/8.10 = ( minus_minus_nat @ A2 @ B ) ) ) ).
% 14.25/8.10
% 14.25/8.10 thf(fact_5208_add__implies__diff,axiom,
% 14.25/8.10 ! [C: int,B: int,A2: int] :
% 14.25/8.10 ( ( ( plus_plus_int @ C @ B )
% 14.25/8.10 = A2 )
% 14.25/8.10 => ( C
% 14.25/8.10 = ( minus_minus_int @ A2 @ B ) ) ) ).
% 14.25/8.10
% 14.25/8.10 thf(fact_5209_diff__diff__eq,axiom,
% 14.25/8.10 ! [A2: code_integer,B: code_integer,C: code_integer] :
% 14.25/8.10 ( ( minus_8373710615458151222nteger @ ( minus_8373710615458151222nteger @ A2 @ B ) @ C )
% 14.25/8.10 = ( minus_8373710615458151222nteger @ A2 @ ( plus_p5714425477246183910nteger @ B @ C ) ) ) ).
% 14.25/8.10
% 14.25/8.10 thf(fact_5210_diff__diff__eq,axiom,
% 14.25/8.10 ! [A2: rat,B: rat,C: rat] :
% 14.25/8.10 ( ( minus_minus_rat @ ( minus_minus_rat @ A2 @ B ) @ C )
% 14.25/8.10 = ( minus_minus_rat @ A2 @ ( plus_plus_rat @ B @ C ) ) ) ).
% 14.25/8.10
% 14.25/8.10 thf(fact_5211_diff__diff__eq,axiom,
% 14.25/8.10 ! [A2: nat,B: nat,C: nat] :
% 14.25/8.10 ( ( minus_minus_nat @ ( minus_minus_nat @ A2 @ B ) @ C )
% 14.25/8.10 = ( minus_minus_nat @ A2 @ ( plus_plus_nat @ B @ C ) ) ) ).
% 14.25/8.10
% 14.25/8.10 thf(fact_5212_diff__diff__eq,axiom,
% 14.25/8.10 ! [A2: int,B: int,C: int] :
% 14.25/8.10 ( ( minus_minus_int @ ( minus_minus_int @ A2 @ B ) @ C )
% 14.25/8.10 = ( minus_minus_int @ A2 @ ( plus_plus_int @ B @ C ) ) ) ).
% 14.25/8.10
% 14.25/8.10 thf(fact_5213_mlex__bound,axiom,
% 14.25/8.10 ! [A2: nat,A: nat,B: nat,N: nat] :
% 14.25/8.10 ( ( ord_less_nat @ A2 @ A )
% 14.25/8.10 => ( ( ord_less_nat @ B @ N )
% 14.25/8.10 => ( ord_less_nat @ ( plus_plus_nat @ ( times_times_nat @ A2 @ N ) @ B ) @ ( times_times_nat @ A @ N ) ) ) ) ).
% 14.25/8.10
% 14.25/8.10 thf(fact_5214_mlex__fst__decrI,axiom,
% 14.25/8.10 ! [A2: nat,A5: nat,B: nat,N: nat,B3: nat] :
% 14.25/8.10 ( ( ord_less_nat @ A2 @ A5 )
% 14.25/8.10 => ( ( ord_less_nat @ B @ N )
% 14.25/8.10 => ( ( ord_less_nat @ B3 @ N )
% 14.25/8.10 => ( ord_less_nat @ ( plus_plus_nat @ ( times_times_nat @ A2 @ N ) @ B ) @ ( plus_plus_nat @ ( times_times_nat @ A5 @ N ) @ B3 ) ) ) ) ) ).
% 14.25/8.10
% 14.25/8.10 thf(fact_5215_mlex__snd__decrI,axiom,
% 14.25/8.10 ! [A2: nat,A5: nat,B: nat,B3: nat,N: nat] :
% 14.25/8.10 ( ( A2 = A5 )
% 14.25/8.10 => ( ( ord_less_nat @ B @ B3 )
% 14.25/8.10 => ( ord_less_nat @ ( plus_plus_nat @ ( times_times_nat @ A2 @ N ) @ B ) @ ( plus_plus_nat @ ( times_times_nat @ A5 @ N ) @ B3 ) ) ) ) ).
% 14.25/8.10
% 14.25/8.10 thf(fact_5216_mlex__leI,axiom,
% 14.25/8.10 ! [A2: nat,A5: nat,B: nat,B3: nat,N: nat] :
% 14.25/8.10 ( ( ord_less_eq_nat @ A2 @ A5 )
% 14.25/8.10 => ( ( ord_less_eq_nat @ B @ B3 )
% 14.25/8.10 => ( ord_less_eq_nat @ ( plus_plus_nat @ ( times_times_nat @ A2 @ N ) @ B ) @ ( plus_plus_nat @ ( times_times_nat @ A5 @ N ) @ B3 ) ) ) ) ).
% 14.25/8.10
% 14.25/8.10 thf(fact_5217_Inf__nat__def1,axiom,
% 14.25/8.10 ! [K4: set_nat] :
% 14.25/8.10 ( ( K4 != bot_bot_set_nat )
% 14.25/8.10 => ( member_nat @ ( complete_Inf_Inf_nat @ K4 ) @ K4 ) ) ).
% 14.25/8.10
% 14.25/8.10 thf(fact_5218_max__add__distrib__left,axiom,
% 14.25/8.10 ! [X: code_integer,Y: code_integer,Z: code_integer] :
% 14.25/8.10 ( ( plus_p5714425477246183910nteger @ ( ord_max_Code_integer @ X @ Y ) @ Z )
% 14.25/8.10 = ( ord_max_Code_integer @ ( plus_p5714425477246183910nteger @ X @ Z ) @ ( plus_p5714425477246183910nteger @ Y @ Z ) ) ) ).
% 14.25/8.10
% 14.25/8.10 thf(fact_5219_max__add__distrib__left,axiom,
% 14.25/8.10 ! [X: rat,Y: rat,Z: rat] :
% 14.25/8.10 ( ( plus_plus_rat @ ( ord_max_rat @ X @ Y ) @ Z )
% 14.25/8.10 = ( ord_max_rat @ ( plus_plus_rat @ X @ Z ) @ ( plus_plus_rat @ Y @ Z ) ) ) ).
% 14.25/8.10
% 14.25/8.10 thf(fact_5220_max__add__distrib__left,axiom,
% 14.25/8.10 ! [X: nat,Y: nat,Z: nat] :
% 14.25/8.10 ( ( plus_plus_nat @ ( ord_max_nat @ X @ Y ) @ Z )
% 14.25/8.10 = ( ord_max_nat @ ( plus_plus_nat @ X @ Z ) @ ( plus_plus_nat @ Y @ Z ) ) ) ).
% 14.25/8.10
% 14.25/8.10 thf(fact_5221_max__add__distrib__left,axiom,
% 14.25/8.10 ! [X: int,Y: int,Z: int] :
% 14.25/8.10 ( ( plus_plus_int @ ( ord_max_int @ X @ Y ) @ Z )
% 14.25/8.10 = ( ord_max_int @ ( plus_plus_int @ X @ Z ) @ ( plus_plus_int @ Y @ Z ) ) ) ).
% 14.25/8.10
% 14.25/8.10 thf(fact_5222_max__add__distrib__right,axiom,
% 14.25/8.10 ! [X: code_integer,Y: code_integer,Z: code_integer] :
% 14.25/8.10 ( ( plus_p5714425477246183910nteger @ X @ ( ord_max_Code_integer @ Y @ Z ) )
% 14.25/8.10 = ( ord_max_Code_integer @ ( plus_p5714425477246183910nteger @ X @ Y ) @ ( plus_p5714425477246183910nteger @ X @ Z ) ) ) ).
% 14.25/8.10
% 14.25/8.10 thf(fact_5223_max__add__distrib__right,axiom,
% 14.25/8.10 ! [X: rat,Y: rat,Z: rat] :
% 14.25/8.10 ( ( plus_plus_rat @ X @ ( ord_max_rat @ Y @ Z ) )
% 14.25/8.10 = ( ord_max_rat @ ( plus_plus_rat @ X @ Y ) @ ( plus_plus_rat @ X @ Z ) ) ) ).
% 14.25/8.10
% 14.25/8.10 thf(fact_5224_max__add__distrib__right,axiom,
% 14.25/8.10 ! [X: nat,Y: nat,Z: nat] :
% 14.25/8.10 ( ( plus_plus_nat @ X @ ( ord_max_nat @ Y @ Z ) )
% 14.25/8.10 = ( ord_max_nat @ ( plus_plus_nat @ X @ Y ) @ ( plus_plus_nat @ X @ Z ) ) ) ).
% 14.25/8.10
% 14.25/8.10 thf(fact_5225_max__add__distrib__right,axiom,
% 14.25/8.10 ! [X: int,Y: int,Z: int] :
% 14.25/8.10 ( ( plus_plus_int @ X @ ( ord_max_int @ Y @ Z ) )
% 14.25/8.10 = ( ord_max_int @ ( plus_plus_int @ X @ Y ) @ ( plus_plus_int @ X @ Z ) ) ) ).
% 14.25/8.10
% 14.25/8.10 thf(fact_5226_add__mono__thms__linordered__field_I4_J,axiom,
% 14.25/8.10 ! [I2: code_integer,J: code_integer,K3: code_integer,L: code_integer] :
% 14.25/8.10 ( ( ( ord_le3102999989581377725nteger @ I2 @ J )
% 14.25/8.10 & ( ord_le6747313008572928689nteger @ K3 @ L ) )
% 14.25/8.10 => ( ord_le6747313008572928689nteger @ ( plus_p5714425477246183910nteger @ I2 @ K3 ) @ ( plus_p5714425477246183910nteger @ J @ L ) ) ) ).
% 14.25/8.10
% 14.25/8.10 thf(fact_5227_add__mono__thms__linordered__field_I4_J,axiom,
% 14.25/8.10 ! [I2: rat,J: rat,K3: rat,L: rat] :
% 14.25/8.10 ( ( ( ord_less_eq_rat @ I2 @ J )
% 14.25/8.10 & ( ord_less_rat @ K3 @ L ) )
% 14.25/8.10 => ( ord_less_rat @ ( plus_plus_rat @ I2 @ K3 ) @ ( plus_plus_rat @ J @ L ) ) ) ).
% 14.25/8.10
% 14.25/8.10 thf(fact_5228_add__mono__thms__linordered__field_I4_J,axiom,
% 14.25/8.10 ! [I2: nat,J: nat,K3: nat,L: nat] :
% 14.25/8.10 ( ( ( ord_less_eq_nat @ I2 @ J )
% 14.25/8.10 & ( ord_less_nat @ K3 @ L ) )
% 14.25/8.10 => ( ord_less_nat @ ( plus_plus_nat @ I2 @ K3 ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% 14.25/8.10
% 14.25/8.10 thf(fact_5229_add__mono__thms__linordered__field_I4_J,axiom,
% 14.25/8.10 ! [I2: int,J: int,K3: int,L: int] :
% 14.25/8.10 ( ( ( ord_less_eq_int @ I2 @ J )
% 14.25/8.10 & ( ord_less_int @ K3 @ L ) )
% 14.25/8.10 => ( ord_less_int @ ( plus_plus_int @ I2 @ K3 ) @ ( plus_plus_int @ J @ L ) ) ) ).
% 14.25/8.10
% 14.25/8.10 thf(fact_5230_add__mono__thms__linordered__field_I3_J,axiom,
% 14.25/8.10 ! [I2: code_integer,J: code_integer,K3: code_integer,L: code_integer] :
% 14.25/8.10 ( ( ( ord_le6747313008572928689nteger @ I2 @ J )
% 14.25/8.10 & ( ord_le3102999989581377725nteger @ K3 @ L ) )
% 14.25/8.10 => ( ord_le6747313008572928689nteger @ ( plus_p5714425477246183910nteger @ I2 @ K3 ) @ ( plus_p5714425477246183910nteger @ J @ L ) ) ) ).
% 14.25/8.10
% 14.25/8.10 thf(fact_5231_add__mono__thms__linordered__field_I3_J,axiom,
% 14.25/8.10 ! [I2: rat,J: rat,K3: rat,L: rat] :
% 14.25/8.10 ( ( ( ord_less_rat @ I2 @ J )
% 14.25/8.10 & ( ord_less_eq_rat @ K3 @ L ) )
% 14.25/8.10 => ( ord_less_rat @ ( plus_plus_rat @ I2 @ K3 ) @ ( plus_plus_rat @ J @ L ) ) ) ).
% 14.25/8.10
% 14.25/8.10 thf(fact_5232_add__mono__thms__linordered__field_I3_J,axiom,
% 14.25/8.10 ! [I2: nat,J: nat,K3: nat,L: nat] :
% 14.25/8.10 ( ( ( ord_less_nat @ I2 @ J )
% 14.25/8.10 & ( ord_less_eq_nat @ K3 @ L ) )
% 14.25/8.10 => ( ord_less_nat @ ( plus_plus_nat @ I2 @ K3 ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% 14.25/8.10
% 14.25/8.10 thf(fact_5233_add__mono__thms__linordered__field_I3_J,axiom,
% 14.25/8.10 ! [I2: int,J: int,K3: int,L: int] :
% 14.25/8.10 ( ( ( ord_less_int @ I2 @ J )
% 14.25/8.10 & ( ord_less_eq_int @ K3 @ L ) )
% 14.25/8.10 => ( ord_less_int @ ( plus_plus_int @ I2 @ K3 ) @ ( plus_plus_int @ J @ L ) ) ) ).
% 14.25/8.10
% 14.25/8.10 thf(fact_5234_add__le__less__mono,axiom,
% 14.25/8.10 ! [A2: code_integer,B: code_integer,C: code_integer,D2: code_integer] :
% 14.25/8.10 ( ( ord_le3102999989581377725nteger @ A2 @ B )
% 14.25/8.10 => ( ( ord_le6747313008572928689nteger @ C @ D2 )
% 14.25/8.10 => ( ord_le6747313008572928689nteger @ ( plus_p5714425477246183910nteger @ A2 @ C ) @ ( plus_p5714425477246183910nteger @ B @ D2 ) ) ) ) ).
% 14.25/8.10
% 14.25/8.10 thf(fact_5235_add__le__less__mono,axiom,
% 14.25/8.10 ! [A2: rat,B: rat,C: rat,D2: rat] :
% 14.25/8.10 ( ( ord_less_eq_rat @ A2 @ B )
% 14.25/8.10 => ( ( ord_less_rat @ C @ D2 )
% 14.25/8.10 => ( ord_less_rat @ ( plus_plus_rat @ A2 @ C ) @ ( plus_plus_rat @ B @ D2 ) ) ) ) ).
% 14.25/8.10
% 14.25/8.10 thf(fact_5236_add__le__less__mono,axiom,
% 14.25/8.10 ! [A2: nat,B: nat,C: nat,D2: nat] :
% 14.25/8.10 ( ( ord_less_eq_nat @ A2 @ B )
% 14.25/8.10 => ( ( ord_less_nat @ C @ D2 )
% 14.25/8.10 => ( ord_less_nat @ ( plus_plus_nat @ A2 @ C ) @ ( plus_plus_nat @ B @ D2 ) ) ) ) ).
% 14.25/8.10
% 14.25/8.10 thf(fact_5237_add__le__less__mono,axiom,
% 14.25/8.10 ! [A2: int,B: int,C: int,D2: int] :
% 14.25/8.10 ( ( ord_less_eq_int @ A2 @ B )
% 14.25/8.10 => ( ( ord_less_int @ C @ D2 )
% 14.25/8.10 => ( ord_less_int @ ( plus_plus_int @ A2 @ C ) @ ( plus_plus_int @ B @ D2 ) ) ) ) ).
% 14.25/8.10
% 14.25/8.10 thf(fact_5238_add__less__le__mono,axiom,
% 14.25/8.10 ! [A2: code_integer,B: code_integer,C: code_integer,D2: code_integer] :
% 14.25/8.10 ( ( ord_le6747313008572928689nteger @ A2 @ B )
% 14.25/8.10 => ( ( ord_le3102999989581377725nteger @ C @ D2 )
% 14.25/8.10 => ( ord_le6747313008572928689nteger @ ( plus_p5714425477246183910nteger @ A2 @ C ) @ ( plus_p5714425477246183910nteger @ B @ D2 ) ) ) ) ).
% 14.25/8.10
% 14.25/8.10 thf(fact_5239_add__less__le__mono,axiom,
% 14.25/8.10 ! [A2: rat,B: rat,C: rat,D2: rat] :
% 14.25/8.10 ( ( ord_less_rat @ A2 @ B )
% 14.25/8.10 => ( ( ord_less_eq_rat @ C @ D2 )
% 14.25/8.10 => ( ord_less_rat @ ( plus_plus_rat @ A2 @ C ) @ ( plus_plus_rat @ B @ D2 ) ) ) ) ).
% 14.25/8.10
% 14.25/8.10 thf(fact_5240_add__less__le__mono,axiom,
% 14.25/8.10 ! [A2: nat,B: nat,C: nat,D2: nat] :
% 14.25/8.10 ( ( ord_less_nat @ A2 @ B )
% 14.25/8.10 => ( ( ord_less_eq_nat @ C @ D2 )
% 14.25/8.10 => ( ord_less_nat @ ( plus_plus_nat @ A2 @ C ) @ ( plus_plus_nat @ B @ D2 ) ) ) ) ).
% 14.25/8.10
% 14.25/8.10 thf(fact_5241_add__less__le__mono,axiom,
% 14.25/8.10 ! [A2: int,B: int,C: int,D2: int] :
% 14.25/8.10 ( ( ord_less_int @ A2 @ B )
% 14.25/8.10 => ( ( ord_less_eq_int @ C @ D2 )
% 14.25/8.10 => ( ord_less_int @ ( plus_plus_int @ A2 @ C ) @ ( plus_plus_int @ B @ D2 ) ) ) ) ).
% 14.25/8.10
% 14.25/8.10 thf(fact_5242_ordered__cancel__comm__monoid__diff__class_Ole__imp__diff__is__add,axiom,
% 14.25/8.10 ! [A2: nat,B: nat,C: nat] :
% 14.25/8.10 ( ( ord_less_eq_nat @ A2 @ B )
% 14.25/8.10 => ( ( ord_less_eq_nat @ A2 @ B )
% 14.25/8.10 => ( ( ( minus_minus_nat @ B @ A2 )
% 14.25/8.10 = C )
% 14.25/8.10 = ( B
% 14.25/8.10 = ( plus_plus_nat @ C @ A2 ) ) ) ) ) ).
% 14.25/8.10
% 14.25/8.10 thf(fact_5243_ordered__cancel__comm__monoid__diff__class_Oadd__diff__inverse,axiom,
% 14.25/8.10 ! [A2: nat,B: nat] :
% 14.25/8.10 ( ( ord_less_eq_nat @ A2 @ B )
% 14.25/8.10 => ( ( plus_plus_nat @ A2 @ ( minus_minus_nat @ B @ A2 ) )
% 14.25/8.10 = B ) ) ).
% 14.25/8.10
% 14.25/8.10 thf(fact_5244_ordered__cancel__comm__monoid__diff__class_Odiff__diff__right,axiom,
% 14.25/8.10 ! [A2: nat,B: nat,C: nat] :
% 14.25/8.10 ( ( ord_less_eq_nat @ A2 @ B )
% 14.25/8.10 => ( ( minus_minus_nat @ C @ ( minus_minus_nat @ B @ A2 ) )
% 14.25/8.10 = ( minus_minus_nat @ ( plus_plus_nat @ C @ A2 ) @ B ) ) ) ).
% 14.25/8.10
% 14.25/8.10 thf(fact_5245_ordered__cancel__comm__monoid__diff__class_Odiff__add__assoc2,axiom,
% 14.25/8.10 ! [A2: nat,B: nat,C: nat] :
% 14.25/8.10 ( ( ord_less_eq_nat @ A2 @ B )
% 14.25/8.10 => ( ( minus_minus_nat @ ( plus_plus_nat @ B @ C ) @ A2 )
% 14.25/8.10 = ( plus_plus_nat @ ( minus_minus_nat @ B @ A2 ) @ C ) ) ) ).
% 14.25/8.10
% 14.25/8.10 thf(fact_5246_ordered__cancel__comm__monoid__diff__class_Oadd__diff__assoc2,axiom,
% 14.25/8.10 ! [A2: nat,B: nat,C: nat] :
% 14.25/8.10 ( ( ord_less_eq_nat @ A2 @ B )
% 14.25/8.10 => ( ( plus_plus_nat @ ( minus_minus_nat @ B @ A2 ) @ C )
% 14.25/8.10 = ( minus_minus_nat @ ( plus_plus_nat @ B @ C ) @ A2 ) ) ) ).
% 14.25/8.10
% 14.25/8.10 thf(fact_5247_ordered__cancel__comm__monoid__diff__class_Odiff__add__assoc,axiom,
% 14.25/8.10 ! [A2: nat,B: nat,C: nat] :
% 14.25/8.10 ( ( ord_less_eq_nat @ A2 @ B )
% 14.25/8.10 => ( ( minus_minus_nat @ ( plus_plus_nat @ C @ B ) @ A2 )
% 14.25/8.10 = ( plus_plus_nat @ C @ ( minus_minus_nat @ B @ A2 ) ) ) ) ).
% 14.25/8.10
% 14.25/8.10 thf(fact_5248_ordered__cancel__comm__monoid__diff__class_Oadd__diff__assoc,axiom,
% 14.25/8.10 ! [A2: nat,B: nat,C: nat] :
% 14.25/8.10 ( ( ord_less_eq_nat @ A2 @ B )
% 14.25/8.10 => ( ( plus_plus_nat @ C @ ( minus_minus_nat @ B @ A2 ) )
% 14.25/8.10 = ( minus_minus_nat @ ( plus_plus_nat @ C @ B ) @ A2 ) ) ) ).
% 14.25/8.10
% 14.25/8.10 thf(fact_5249_ordered__cancel__comm__monoid__diff__class_Ole__diff__conv2,axiom,
% 14.25/8.10 ! [A2: nat,B: nat,C: nat] :
% 14.25/8.10 ( ( ord_less_eq_nat @ A2 @ B )
% 14.25/8.10 => ( ( ord_less_eq_nat @ C @ ( minus_minus_nat @ B @ A2 ) )
% 14.25/8.10 = ( ord_less_eq_nat @ ( plus_plus_nat @ C @ A2 ) @ B ) ) ) ).
% 14.25/8.10
% 14.25/8.10 thf(fact_5250_le__add__diff,axiom,
% 14.25/8.10 ! [A2: nat,B: nat,C: nat] :
% 14.25/8.10 ( ( ord_less_eq_nat @ A2 @ B )
% 14.25/8.10 => ( ord_less_eq_nat @ C @ ( minus_minus_nat @ ( plus_plus_nat @ B @ C ) @ A2 ) ) ) ).
% 14.25/8.10
% 14.25/8.10 thf(fact_5251_ordered__cancel__comm__monoid__diff__class_Odiff__add,axiom,
% 14.25/8.10 ! [A2: nat,B: nat] :
% 14.25/8.10 ( ( ord_less_eq_nat @ A2 @ B )
% 14.25/8.10 => ( ( plus_plus_nat @ ( minus_minus_nat @ B @ A2 ) @ A2 )
% 14.25/8.10 = B ) ) ).
% 14.25/8.10
% 14.25/8.10 thf(fact_5252_le__diff__eq,axiom,
% 14.25/8.10 ! [A2: code_integer,C: code_integer,B: code_integer] :
% 14.25/8.10 ( ( ord_le3102999989581377725nteger @ A2 @ ( minus_8373710615458151222nteger @ C @ B ) )
% 14.25/8.10 = ( ord_le3102999989581377725nteger @ ( plus_p5714425477246183910nteger @ A2 @ B ) @ C ) ) ).
% 14.25/8.10
% 14.25/8.10 thf(fact_5253_le__diff__eq,axiom,
% 14.25/8.10 ! [A2: rat,C: rat,B: rat] :
% 14.25/8.10 ( ( ord_less_eq_rat @ A2 @ ( minus_minus_rat @ C @ B ) )
% 14.25/8.10 = ( ord_less_eq_rat @ ( plus_plus_rat @ A2 @ B ) @ C ) ) ).
% 14.25/8.10
% 14.25/8.10 thf(fact_5254_le__diff__eq,axiom,
% 14.25/8.10 ! [A2: int,C: int,B: int] :
% 14.25/8.10 ( ( ord_less_eq_int @ A2 @ ( minus_minus_int @ C @ B ) )
% 14.25/8.10 = ( ord_less_eq_int @ ( plus_plus_int @ A2 @ B ) @ C ) ) ).
% 14.25/8.10
% 14.25/8.10 thf(fact_5255_diff__le__eq,axiom,
% 14.25/8.10 ! [A2: code_integer,B: code_integer,C: code_integer] :
% 14.25/8.10 ( ( ord_le3102999989581377725nteger @ ( minus_8373710615458151222nteger @ A2 @ B ) @ C )
% 14.25/8.10 = ( ord_le3102999989581377725nteger @ A2 @ ( plus_p5714425477246183910nteger @ C @ B ) ) ) ).
% 14.25/8.10
% 14.25/8.10 thf(fact_5256_diff__le__eq,axiom,
% 14.25/8.10 ! [A2: rat,B: rat,C: rat] :
% 14.25/8.10 ( ( ord_less_eq_rat @ ( minus_minus_rat @ A2 @ B ) @ C )
% 14.25/8.10 = ( ord_less_eq_rat @ A2 @ ( plus_plus_rat @ C @ B ) ) ) ).
% 14.25/8.10
% 14.25/8.10 thf(fact_5257_diff__le__eq,axiom,
% 14.25/8.10 ! [A2: int,B: int,C: int] :
% 14.25/8.10 ( ( ord_less_eq_int @ ( minus_minus_int @ A2 @ B ) @ C )
% 14.25/8.10 = ( ord_less_eq_int @ A2 @ ( plus_plus_int @ C @ B ) ) ) ).
% 14.25/8.10
% 14.25/8.10 thf(fact_5258_less__add__one,axiom,
% 14.25/8.10 ! [A2: code_natural] : ( ord_le5570908160329646204atural @ A2 @ ( plus_p4538020629002901425atural @ A2 @ one_one_Code_natural ) ) ).
% 14.25/8.10
% 14.25/8.10 thf(fact_5259_less__add__one,axiom,
% 14.25/8.10 ! [A2: nat] : ( ord_less_nat @ A2 @ ( plus_plus_nat @ A2 @ one_one_nat ) ) ).
% 14.25/8.10
% 14.25/8.10 thf(fact_5260_less__add__one,axiom,
% 14.25/8.10 ! [A2: int] : ( ord_less_int @ A2 @ ( plus_plus_int @ A2 @ one_one_int ) ) ).
% 14.25/8.10
% 14.25/8.10 thf(fact_5261_less__add__one,axiom,
% 14.25/8.10 ! [A2: code_integer] : ( ord_le6747313008572928689nteger @ A2 @ ( plus_p5714425477246183910nteger @ A2 @ one_one_Code_integer ) ) ).
% 14.25/8.10
% 14.25/8.10 thf(fact_5262_less__add__one,axiom,
% 14.25/8.10 ! [A2: rat] : ( ord_less_rat @ A2 @ ( plus_plus_rat @ A2 @ one_one_rat ) ) ).
% 14.25/8.10
% 14.25/8.10 thf(fact_5263_add__mono1,axiom,
% 14.25/8.10 ! [A2: code_natural,B: code_natural] :
% 14.25/8.10 ( ( ord_le5570908160329646204atural @ A2 @ B )
% 14.25/8.10 => ( ord_le5570908160329646204atural @ ( plus_p4538020629002901425atural @ A2 @ one_one_Code_natural ) @ ( plus_p4538020629002901425atural @ B @ one_one_Code_natural ) ) ) ).
% 14.25/8.10
% 14.25/8.10 thf(fact_5264_add__mono1,axiom,
% 14.25/8.10 ! [A2: nat,B: nat] :
% 14.25/8.10 ( ( ord_less_nat @ A2 @ B )
% 14.25/8.10 => ( ord_less_nat @ ( plus_plus_nat @ A2 @ one_one_nat ) @ ( plus_plus_nat @ B @ one_one_nat ) ) ) ).
% 14.25/8.10
% 14.25/8.10 thf(fact_5265_add__mono1,axiom,
% 14.25/8.10 ! [A2: int,B: int] :
% 14.25/8.10 ( ( ord_less_int @ A2 @ B )
% 14.25/8.10 => ( ord_less_int @ ( plus_plus_int @ A2 @ one_one_int ) @ ( plus_plus_int @ B @ one_one_int ) ) ) ).
% 14.25/8.10
% 14.25/8.10 thf(fact_5266_add__mono1,axiom,
% 14.25/8.10 ! [A2: code_integer,B: code_integer] :
% 14.25/8.10 ( ( ord_le6747313008572928689nteger @ A2 @ B )
% 14.25/8.10 => ( ord_le6747313008572928689nteger @ ( plus_p5714425477246183910nteger @ A2 @ one_one_Code_integer ) @ ( plus_p5714425477246183910nteger @ B @ one_one_Code_integer ) ) ) ).
% 14.25/8.10
% 14.25/8.10 thf(fact_5267_add__mono1,axiom,
% 14.25/8.10 ! [A2: rat,B: rat] :
% 14.25/8.10 ( ( ord_less_rat @ A2 @ B )
% 14.25/8.10 => ( ord_less_rat @ ( plus_plus_rat @ A2 @ one_one_rat ) @ ( plus_plus_rat @ B @ one_one_rat ) ) ) ).
% 14.25/8.10
% 14.25/8.10 thf(fact_5268_diff__less__eq,axiom,
% 14.25/8.10 ! [A2: int,B: int,C: int] :
% 14.25/8.10 ( ( ord_less_int @ ( minus_minus_int @ A2 @ B ) @ C )
% 14.25/8.10 = ( ord_less_int @ A2 @ ( plus_plus_int @ C @ B ) ) ) ).
% 14.25/8.10
% 14.25/8.10 thf(fact_5269_diff__less__eq,axiom,
% 14.25/8.10 ! [A2: code_integer,B: code_integer,C: code_integer] :
% 14.25/8.10 ( ( ord_le6747313008572928689nteger @ ( minus_8373710615458151222nteger @ A2 @ B ) @ C )
% 14.25/8.10 = ( ord_le6747313008572928689nteger @ A2 @ ( plus_p5714425477246183910nteger @ C @ B ) ) ) ).
% 14.25/8.10
% 14.25/8.10 thf(fact_5270_diff__less__eq,axiom,
% 14.25/8.10 ! [A2: rat,B: rat,C: rat] :
% 14.25/8.10 ( ( ord_less_rat @ ( minus_minus_rat @ A2 @ B ) @ C )
% 14.25/8.10 = ( ord_less_rat @ A2 @ ( plus_plus_rat @ C @ B ) ) ) ).
% 14.25/8.10
% 14.25/8.10 thf(fact_5271_less__diff__eq,axiom,
% 14.25/8.10 ! [A2: int,C: int,B: int] :
% 14.25/8.10 ( ( ord_less_int @ A2 @ ( minus_minus_int @ C @ B ) )
% 14.25/8.10 = ( ord_less_int @ ( plus_plus_int @ A2 @ B ) @ C ) ) ).
% 14.25/8.10
% 14.25/8.10 thf(fact_5272_less__diff__eq,axiom,
% 14.25/8.10 ! [A2: code_integer,C: code_integer,B: code_integer] :
% 14.25/8.10 ( ( ord_le6747313008572928689nteger @ A2 @ ( minus_8373710615458151222nteger @ C @ B ) )
% 14.25/8.10 = ( ord_le6747313008572928689nteger @ ( plus_p5714425477246183910nteger @ A2 @ B ) @ C ) ) ).
% 14.25/8.10
% 14.25/8.10 thf(fact_5273_less__diff__eq,axiom,
% 14.25/8.10 ! [A2: rat,C: rat,B: rat] :
% 14.25/8.10 ( ( ord_less_rat @ A2 @ ( minus_minus_rat @ C @ B ) )
% 14.25/8.10 = ( ord_less_rat @ ( plus_plus_rat @ A2 @ B ) @ C ) ) ).
% 14.25/8.10
% 14.25/8.10 thf(fact_5274_square__diff__square__factored,axiom,
% 14.25/8.10 ! [X: code_integer,Y: code_integer] :
% 14.25/8.10 ( ( minus_8373710615458151222nteger @ ( times_3573771949741848930nteger @ X @ X ) @ ( times_3573771949741848930nteger @ Y @ Y ) )
% 14.25/8.10 = ( times_3573771949741848930nteger @ ( plus_p5714425477246183910nteger @ X @ Y ) @ ( minus_8373710615458151222nteger @ X @ Y ) ) ) ).
% 14.25/8.10
% 14.25/8.10 thf(fact_5275_square__diff__square__factored,axiom,
% 14.25/8.10 ! [X: rat,Y: rat] :
% 14.25/8.10 ( ( minus_minus_rat @ ( times_times_rat @ X @ X ) @ ( times_times_rat @ Y @ Y ) )
% 14.25/8.10 = ( times_times_rat @ ( plus_plus_rat @ X @ Y ) @ ( minus_minus_rat @ X @ Y ) ) ) ).
% 14.25/8.10
% 14.25/8.10 thf(fact_5276_square__diff__square__factored,axiom,
% 14.25/8.10 ! [X: int,Y: int] :
% 14.25/8.10 ( ( minus_minus_int @ ( times_times_int @ X @ X ) @ ( times_times_int @ Y @ Y ) )
% 14.25/8.10 = ( times_times_int @ ( plus_plus_int @ X @ Y ) @ ( minus_minus_int @ X @ Y ) ) ) ).
% 14.25/8.10
% 14.25/8.10 thf(fact_5277_eq__add__iff2,axiom,
% 14.25/8.10 ! [A2: code_integer,E2: code_integer,C: code_integer,B: code_integer,D2: code_integer] :
% 14.25/8.10 ( ( ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ A2 @ E2 ) @ C )
% 14.25/8.10 = ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ B @ E2 ) @ D2 ) )
% 14.25/8.10 = ( C
% 14.25/8.10 = ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ ( minus_8373710615458151222nteger @ B @ A2 ) @ E2 ) @ D2 ) ) ) ).
% 14.25/8.10
% 14.25/8.10 thf(fact_5278_eq__add__iff2,axiom,
% 14.25/8.10 ! [A2: rat,E2: rat,C: rat,B: rat,D2: rat] :
% 14.25/8.10 ( ( ( plus_plus_rat @ ( times_times_rat @ A2 @ E2 ) @ C )
% 14.25/8.10 = ( plus_plus_rat @ ( times_times_rat @ B @ E2 ) @ D2 ) )
% 14.25/8.10 = ( C
% 14.25/8.10 = ( plus_plus_rat @ ( times_times_rat @ ( minus_minus_rat @ B @ A2 ) @ E2 ) @ D2 ) ) ) ).
% 14.25/8.10
% 14.25/8.10 thf(fact_5279_eq__add__iff2,axiom,
% 14.25/8.10 ! [A2: int,E2: int,C: int,B: int,D2: int] :
% 14.25/8.10 ( ( ( plus_plus_int @ ( times_times_int @ A2 @ E2 ) @ C )
% 14.25/8.10 = ( plus_plus_int @ ( times_times_int @ B @ E2 ) @ D2 ) )
% 14.25/8.10 = ( C
% 14.25/8.10 = ( plus_plus_int @ ( times_times_int @ ( minus_minus_int @ B @ A2 ) @ E2 ) @ D2 ) ) ) ).
% 14.25/8.10
% 14.25/8.10 thf(fact_5280_eq__add__iff1,axiom,
% 14.25/8.10 ! [A2: code_integer,E2: code_integer,C: code_integer,B: code_integer,D2: code_integer] :
% 14.25/8.10 ( ( ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ A2 @ E2 ) @ C )
% 14.25/8.10 = ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ B @ E2 ) @ D2 ) )
% 14.25/8.10 = ( ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ ( minus_8373710615458151222nteger @ A2 @ B ) @ E2 ) @ C )
% 14.25/8.10 = D2 ) ) ).
% 14.25/8.10
% 14.25/8.10 thf(fact_5281_eq__add__iff1,axiom,
% 14.25/8.10 ! [A2: rat,E2: rat,C: rat,B: rat,D2: rat] :
% 14.25/8.10 ( ( ( plus_plus_rat @ ( times_times_rat @ A2 @ E2 ) @ C )
% 14.25/8.10 = ( plus_plus_rat @ ( times_times_rat @ B @ E2 ) @ D2 ) )
% 14.25/8.10 = ( ( plus_plus_rat @ ( times_times_rat @ ( minus_minus_rat @ A2 @ B ) @ E2 ) @ C )
% 14.25/8.10 = D2 ) ) ).
% 14.25/8.10
% 14.25/8.10 thf(fact_5282_eq__add__iff1,axiom,
% 14.25/8.10 ! [A2: int,E2: int,C: int,B: int,D2: int] :
% 14.25/8.10 ( ( ( plus_plus_int @ ( times_times_int @ A2 @ E2 ) @ C )
% 14.25/8.10 = ( plus_plus_int @ ( times_times_int @ B @ E2 ) @ D2 ) )
% 14.25/8.10 = ( ( plus_plus_int @ ( times_times_int @ ( minus_minus_int @ A2 @ B ) @ E2 ) @ C )
% 14.25/8.10 = D2 ) ) ).
% 14.25/8.10
% 14.25/8.10 thf(fact_5283_Suc__n__minus__m__eq,axiom,
% 14.25/8.10 ! [M4: nat,N2: nat] :
% 14.25/8.10 ( ( ord_less_eq_nat @ M4 @ N2 )
% 14.25/8.10 => ( ( ord_less_nat @ one_one_nat @ M4 )
% 14.25/8.10 => ( ( suc @ ( minus_minus_nat @ N2 @ M4 ) )
% 14.25/8.10 = ( minus_minus_nat @ N2 @ ( minus_minus_nat @ M4 @ one_one_nat ) ) ) ) ) ).
% 14.25/8.10
% 14.25/8.10 thf(fact_5284_nat__in__between__eq_I1_J,axiom,
% 14.25/8.10 ! [A2: nat,B: nat] :
% 14.25/8.10 ( ( ( ord_less_nat @ A2 @ B )
% 14.25/8.10 & ( ord_less_eq_nat @ B @ ( suc @ A2 ) ) )
% 14.25/8.10 = ( B
% 14.25/8.10 = ( suc @ A2 ) ) ) ).
% 14.25/8.10
% 14.25/8.10 thf(fact_5285_nat__in__between__eq_I2_J,axiom,
% 14.25/8.10 ! [A2: nat,B: nat] :
% 14.25/8.10 ( ( ( ord_less_eq_nat @ A2 @ B )
% 14.25/8.10 & ( ord_less_nat @ B @ ( suc @ A2 ) ) )
% 14.25/8.10 = ( B = A2 ) ) ).
% 14.25/8.10
% 14.25/8.10 thf(fact_5286_wf__no__infinite__down__chainE,axiom,
% 14.25/8.10 ! [R2: set_Pr1261947904930325089at_nat,F: nat > nat] :
% 14.25/8.10 ( ( wf_nat @ R2 )
% 14.25/8.10 => ~ ! [K: nat] : ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ ( F @ ( suc @ K ) ) @ ( F @ K ) ) @ R2 ) ) ).
% 14.25/8.10
% 14.25/8.10 thf(fact_5287_wf__no__infinite__down__chainE,axiom,
% 14.25/8.10 ! [R2: set_Pr958786334691620121nt_int,F: nat > int] :
% 14.25/8.10 ( ( wf_int @ R2 )
% 14.25/8.10 => ~ ! [K: nat] : ( member5262025264175285858nt_int @ ( product_Pair_int_int @ ( F @ ( suc @ K ) ) @ ( F @ K ) ) @ R2 ) ) ).
% 14.25/8.10
% 14.25/8.10 thf(fact_5288_wf__iff__no__infinite__down__chain,axiom,
% 14.25/8.10 ( wf_nat
% 14.25/8.10 = ( ^ [R4: set_Pr1261947904930325089at_nat] :
% 14.25/8.10 ~ ? [F4: nat > nat] :
% 14.25/8.10 ! [I4: nat] : ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ ( F4 @ ( suc @ I4 ) ) @ ( F4 @ I4 ) ) @ R4 ) ) ) ).
% 14.25/8.10
% 14.25/8.10 thf(fact_5289_wf__iff__no__infinite__down__chain,axiom,
% 14.25/8.10 ( wf_int
% 14.25/8.10 = ( ^ [R4: set_Pr958786334691620121nt_int] :
% 14.25/8.10 ~ ? [F4: nat > int] :
% 14.25/8.10 ! [I4: nat] : ( member5262025264175285858nt_int @ ( product_Pair_int_int @ ( F4 @ ( suc @ I4 ) ) @ ( F4 @ I4 ) ) @ R4 ) ) ) ).
% 14.25/8.10
% 14.25/8.10 thf(fact_5290_Inf__INT__eq,axiom,
% 14.25/8.10 ( comple4605987646318907561at_o_o
% 14.25/8.10 = ( ^ [S6: set_Pr6409761941060870310at_o_o,X2: produc3658429121746597890et_nat > $o] : ( member6576561426505652726_nat_o @ X2 @ ( comple79274577958421812_nat_o @ ( image_8639947063239330806_nat_o @ collec939566748876313656_nat_o @ S6 ) ) ) ) ) ).
% 14.25/8.10
% 14.25/8.10 thf(fact_5291_Inf__INT__eq,axiom,
% 14.25/8.10 ( comple4393433247856107946_int_o
% 14.25/8.10 = ( ^ [S6: set_int_o,X2: int] : ( member_int @ X2 @ ( comple3628384868704368283et_int @ ( image_int_o_set_int @ collect_int @ S6 ) ) ) ) ) ).
% 14.25/8.10
% 14.25/8.10 thf(fact_5292_Inf__INT__eq,axiom,
% 14.25/8.10 ( complete_Inf_Inf_o_o
% 14.25/8.10 = ( ^ [S6: set_o_o,X2: $o] : ( member_o @ X2 @ ( comple3063163877087187839_set_o @ ( image_o_o_set_o @ collect_o @ S6 ) ) ) ) ) ).
% 14.25/8.10
% 14.25/8.10 thf(fact_5293_Inf__INT__eq,axiom,
% 14.25/8.10 ( comple2092403528524206185unit_o
% 14.25/8.10 = ( ^ [S6: set_Product_unit_o,X2: product_unit] : ( member_Product_unit @ X2 @ ( comple7216939928495492468t_unit @ ( image_5138372075374782070t_unit @ collect_Product_unit @ S6 ) ) ) ) ) ).
% 14.25/8.10
% 14.25/8.10 thf(fact_5294_Inf__INT__eq,axiom,
% 14.25/8.10 ( comple6214475593288795910_nat_o
% 14.25/8.10 = ( ^ [S6: set_nat_o,X2: nat] : ( member_nat @ X2 @ ( comple7806235888213564991et_nat @ ( image_nat_o_set_nat @ collect_nat @ S6 ) ) ) ) ) ).
% 14.25/8.10
% 14.25/8.10 thf(fact_5295_le__add__iff2,axiom,
% 14.25/8.10 ! [A2: code_integer,E2: code_integer,C: code_integer,B: code_integer,D2: code_integer] :
% 14.25/8.10 ( ( ord_le3102999989581377725nteger @ ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ A2 @ E2 ) @ C ) @ ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ B @ E2 ) @ D2 ) )
% 14.25/8.10 = ( ord_le3102999989581377725nteger @ C @ ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ ( minus_8373710615458151222nteger @ B @ A2 ) @ E2 ) @ D2 ) ) ) ).
% 14.25/8.10
% 14.25/8.10 thf(fact_5296_le__add__iff2,axiom,
% 14.25/8.10 ! [A2: rat,E2: rat,C: rat,B: rat,D2: rat] :
% 14.25/8.10 ( ( ord_less_eq_rat @ ( plus_plus_rat @ ( times_times_rat @ A2 @ E2 ) @ C ) @ ( plus_plus_rat @ ( times_times_rat @ B @ E2 ) @ D2 ) )
% 14.25/8.10 = ( ord_less_eq_rat @ C @ ( plus_plus_rat @ ( times_times_rat @ ( minus_minus_rat @ B @ A2 ) @ E2 ) @ D2 ) ) ) ).
% 14.25/8.10
% 14.25/8.10 thf(fact_5297_le__add__iff2,axiom,
% 14.25/8.10 ! [A2: int,E2: int,C: int,B: int,D2: int] :
% 14.25/8.10 ( ( ord_less_eq_int @ ( plus_plus_int @ ( times_times_int @ A2 @ E2 ) @ C ) @ ( plus_plus_int @ ( times_times_int @ B @ E2 ) @ D2 ) )
% 14.25/8.10 = ( ord_less_eq_int @ C @ ( plus_plus_int @ ( times_times_int @ ( minus_minus_int @ B @ A2 ) @ E2 ) @ D2 ) ) ) ).
% 14.25/8.10
% 14.25/8.10 thf(fact_5298_le__add__iff1,axiom,
% 14.25/8.10 ! [A2: code_integer,E2: code_integer,C: code_integer,B: code_integer,D2: code_integer] :
% 14.25/8.10 ( ( ord_le3102999989581377725nteger @ ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ A2 @ E2 ) @ C ) @ ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ B @ E2 ) @ D2 ) )
% 14.25/8.10 = ( ord_le3102999989581377725nteger @ ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ ( minus_8373710615458151222nteger @ A2 @ B ) @ E2 ) @ C ) @ D2 ) ) ).
% 14.25/8.10
% 14.25/8.10 thf(fact_5299_le__add__iff1,axiom,
% 14.25/8.10 ! [A2: rat,E2: rat,C: rat,B: rat,D2: rat] :
% 14.25/8.10 ( ( ord_less_eq_rat @ ( plus_plus_rat @ ( times_times_rat @ A2 @ E2 ) @ C ) @ ( plus_plus_rat @ ( times_times_rat @ B @ E2 ) @ D2 ) )
% 14.25/8.10 = ( ord_less_eq_rat @ ( plus_plus_rat @ ( times_times_rat @ ( minus_minus_rat @ A2 @ B ) @ E2 ) @ C ) @ D2 ) ) ).
% 14.25/8.10
% 14.25/8.10 thf(fact_5300_le__add__iff1,axiom,
% 14.25/8.10 ! [A2: int,E2: int,C: int,B: int,D2: int] :
% 14.25/8.10 ( ( ord_less_eq_int @ ( plus_plus_int @ ( times_times_int @ A2 @ E2 ) @ C ) @ ( plus_plus_int @ ( times_times_int @ B @ E2 ) @ D2 ) )
% 14.25/8.10 = ( ord_less_eq_int @ ( plus_plus_int @ ( times_times_int @ ( minus_minus_int @ A2 @ B ) @ E2 ) @ C ) @ D2 ) ) ).
% 14.25/8.10
% 14.25/8.10 thf(fact_5301_less__add__iff2,axiom,
% 14.25/8.10 ! [A2: int,E2: int,C: int,B: int,D2: int] :
% 14.25/8.10 ( ( ord_less_int @ ( plus_plus_int @ ( times_times_int @ A2 @ E2 ) @ C ) @ ( plus_plus_int @ ( times_times_int @ B @ E2 ) @ D2 ) )
% 14.25/8.10 = ( ord_less_int @ C @ ( plus_plus_int @ ( times_times_int @ ( minus_minus_int @ B @ A2 ) @ E2 ) @ D2 ) ) ) ).
% 14.25/8.10
% 14.25/8.10 thf(fact_5302_less__add__iff2,axiom,
% 14.25/8.10 ! [A2: code_integer,E2: code_integer,C: code_integer,B: code_integer,D2: code_integer] :
% 14.25/8.10 ( ( ord_le6747313008572928689nteger @ ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ A2 @ E2 ) @ C ) @ ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ B @ E2 ) @ D2 ) )
% 14.25/8.10 = ( ord_le6747313008572928689nteger @ C @ ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ ( minus_8373710615458151222nteger @ B @ A2 ) @ E2 ) @ D2 ) ) ) ).
% 14.25/8.10
% 14.25/8.10 thf(fact_5303_less__add__iff2,axiom,
% 14.25/8.10 ! [A2: rat,E2: rat,C: rat,B: rat,D2: rat] :
% 14.25/8.10 ( ( ord_less_rat @ ( plus_plus_rat @ ( times_times_rat @ A2 @ E2 ) @ C ) @ ( plus_plus_rat @ ( times_times_rat @ B @ E2 ) @ D2 ) )
% 14.25/8.10 = ( ord_less_rat @ C @ ( plus_plus_rat @ ( times_times_rat @ ( minus_minus_rat @ B @ A2 ) @ E2 ) @ D2 ) ) ) ).
% 14.25/8.10
% 14.25/8.10 thf(fact_5304_less__add__iff1,axiom,
% 14.25/8.10 ! [A2: int,E2: int,C: int,B: int,D2: int] :
% 14.25/8.10 ( ( ord_less_int @ ( plus_plus_int @ ( times_times_int @ A2 @ E2 ) @ C ) @ ( plus_plus_int @ ( times_times_int @ B @ E2 ) @ D2 ) )
% 14.25/8.10 = ( ord_less_int @ ( plus_plus_int @ ( times_times_int @ ( minus_minus_int @ A2 @ B ) @ E2 ) @ C ) @ D2 ) ) ).
% 14.25/8.10
% 14.25/8.10 thf(fact_5305_less__add__iff1,axiom,
% 14.25/8.10 ! [A2: code_integer,E2: code_integer,C: code_integer,B: code_integer,D2: code_integer] :
% 14.25/8.10 ( ( ord_le6747313008572928689nteger @ ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ A2 @ E2 ) @ C ) @ ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ B @ E2 ) @ D2 ) )
% 14.25/8.10 = ( ord_le6747313008572928689nteger @ ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ ( minus_8373710615458151222nteger @ A2 @ B ) @ E2 ) @ C ) @ D2 ) ) ).
% 14.25/8.10
% 14.25/8.10 thf(fact_5306_less__add__iff1,axiom,
% 14.25/8.10 ! [A2: rat,E2: rat,C: rat,B: rat,D2: rat] :
% 14.25/8.10 ( ( ord_less_rat @ ( plus_plus_rat @ ( times_times_rat @ A2 @ E2 ) @ C ) @ ( plus_plus_rat @ ( times_times_rat @ B @ E2 ) @ D2 ) )
% 14.25/8.10 = ( ord_less_rat @ ( plus_plus_rat @ ( times_times_rat @ ( minus_minus_rat @ A2 @ B ) @ E2 ) @ C ) @ D2 ) ) ).
% 14.25/8.10
% 14.25/8.10 thf(fact_5307_square__diff__one__factored,axiom,
% 14.25/8.10 ! [X: code_integer] :
% 14.25/8.10 ( ( minus_8373710615458151222nteger @ ( times_3573771949741848930nteger @ X @ X ) @ one_one_Code_integer )
% 14.25/8.10 = ( times_3573771949741848930nteger @ ( plus_p5714425477246183910nteger @ X @ one_one_Code_integer ) @ ( minus_8373710615458151222nteger @ X @ one_one_Code_integer ) ) ) ).
% 14.25/8.10
% 14.25/8.10 thf(fact_5308_square__diff__one__factored,axiom,
% 14.25/8.10 ! [X: rat] :
% 14.25/8.10 ( ( minus_minus_rat @ ( times_times_rat @ X @ X ) @ one_one_rat )
% 14.25/8.10 = ( times_times_rat @ ( plus_plus_rat @ X @ one_one_rat ) @ ( minus_minus_rat @ X @ one_one_rat ) ) ) ).
% 14.25/8.10
% 14.25/8.10 thf(fact_5309_square__diff__one__factored,axiom,
% 14.25/8.10 ! [X: int] :
% 14.25/8.10 ( ( minus_minus_int @ ( times_times_int @ X @ X ) @ one_one_int )
% 14.25/8.10 = ( times_times_int @ ( plus_plus_int @ X @ one_one_int ) @ ( minus_minus_int @ X @ one_one_int ) ) ) ).
% 14.25/8.10
% 14.25/8.10 thf(fact_5310_card__Suc__eq__finite,axiom,
% 14.25/8.10 ! [A: set_Pr4532377907799695533_nat_o,K3: nat] :
% 14.25/8.10 ( ( ( finite1363419556375932405_nat_o @ A )
% 14.25/8.10 = ( suc @ K3 ) )
% 14.25/8.10 = ( ? [B6: produc3658429121746597890et_nat > $o,B7: set_Pr4532377907799695533_nat_o] :
% 14.25/8.10 ( ( A
% 14.25/8.10 = ( insert5175938949040314269_nat_o @ B6 @ B7 ) )
% 14.25/8.10 & ~ ( member6576561426505652726_nat_o @ B6 @ B7 )
% 14.25/8.10 & ( ( finite1363419556375932405_nat_o @ B7 )
% 14.25/8.10 = K3 )
% 14.25/8.10 & ( finite3252695134891459830_nat_o @ B7 ) ) ) ) ).
% 14.25/8.10
% 14.25/8.10 thf(fact_5311_card__Suc__eq__finite,axiom,
% 14.25/8.10 ! [A: set_int,K3: nat] :
% 14.25/8.10 ( ( ( finite_card_int @ A )
% 14.25/8.10 = ( suc @ K3 ) )
% 14.25/8.10 = ( ? [B6: int,B7: set_int] :
% 14.25/8.10 ( ( A
% 14.25/8.10 = ( insert_int @ B6 @ B7 ) )
% 14.25/8.10 & ~ ( member_int @ B6 @ B7 )
% 14.25/8.10 & ( ( finite_card_int @ B7 )
% 14.25/8.10 = K3 )
% 14.25/8.10 & ( finite_finite_int @ B7 ) ) ) ) ).
% 14.25/8.10
% 14.25/8.10 thf(fact_5312_card__Suc__eq__finite,axiom,
% 14.25/8.10 ! [A: set_o,K3: nat] :
% 14.25/8.10 ( ( ( finite_card_o @ A )
% 14.25/8.10 = ( suc @ K3 ) )
% 14.25/8.10 = ( ? [B6: $o,B7: set_o] :
% 14.25/8.10 ( ( A
% 14.25/8.10 = ( insert_o @ B6 @ B7 ) )
% 14.25/8.10 & ~ ( member_o @ B6 @ B7 )
% 14.25/8.10 & ( ( finite_card_o @ B7 )
% 14.25/8.10 = K3 )
% 14.25/8.10 & ( finite_finite_o @ B7 ) ) ) ) ).
% 14.25/8.10
% 14.25/8.10 thf(fact_5313_card__Suc__eq__finite,axiom,
% 14.25/8.10 ! [A: set_Product_unit,K3: nat] :
% 14.25/8.10 ( ( ( finite410649719033368117t_unit @ A )
% 14.25/8.10 = ( suc @ K3 ) )
% 14.25/8.10 = ( ? [B6: product_unit,B7: set_Product_unit] :
% 14.25/8.10 ( ( A
% 14.25/8.10 = ( insert_Product_unit @ B6 @ B7 ) )
% 14.25/8.10 & ~ ( member_Product_unit @ B6 @ B7 )
% 14.25/8.10 & ( ( finite410649719033368117t_unit @ B7 )
% 14.25/8.10 = K3 )
% 14.25/8.10 & ( finite4290736615968046902t_unit @ B7 ) ) ) ) ).
% 14.25/8.10
% 14.25/8.10 thf(fact_5314_card__Suc__eq__finite,axiom,
% 14.25/8.10 ! [A: set_nat,K3: nat] :
% 14.25/8.10 ( ( ( finite_card_nat @ A )
% 14.25/8.10 = ( suc @ K3 ) )
% 14.25/8.10 = ( ? [B6: nat,B7: set_nat] :
% 14.25/8.10 ( ( A
% 14.25/8.10 = ( insert_nat @ B6 @ B7 ) )
% 14.25/8.10 & ~ ( member_nat @ B6 @ B7 )
% 14.25/8.10 & ( ( finite_card_nat @ B7 )
% 14.25/8.10 = K3 )
% 14.25/8.10 & ( finite_finite_nat @ B7 ) ) ) ) ).
% 14.25/8.10
% 14.25/8.10 thf(fact_5315_card__Suc__eq__finite,axiom,
% 14.25/8.10 ! [A: set_Code_integer,K3: nat] :
% 14.25/8.10 ( ( ( finite4902975817058060853nteger @ A )
% 14.25/8.10 = ( suc @ K3 ) )
% 14.25/8.10 = ( ? [B6: code_integer,B7: set_Code_integer] :
% 14.25/8.10 ( ( A
% 14.25/8.10 = ( insert_Code_integer @ B6 @ B7 ) )
% 14.25/8.10 & ~ ( member_Code_integer @ B6 @ B7 )
% 14.25/8.10 & ( ( finite4902975817058060853nteger @ B7 )
% 14.25/8.10 = K3 )
% 14.25/8.10 & ( finite6017078050557962740nteger @ B7 ) ) ) ) ).
% 14.25/8.10
% 14.25/8.10 thf(fact_5316_card__insert__if,axiom,
% 14.25/8.10 ! [A: set_Pr4532377907799695533_nat_o,X: produc3658429121746597890et_nat > $o] :
% 14.25/8.10 ( ( finite3252695134891459830_nat_o @ A )
% 14.25/8.10 => ( ( ( member6576561426505652726_nat_o @ X @ A )
% 14.25/8.10 => ( ( finite1363419556375932405_nat_o @ ( insert5175938949040314269_nat_o @ X @ A ) )
% 14.25/8.10 = ( finite1363419556375932405_nat_o @ A ) ) )
% 14.25/8.10 & ( ~ ( member6576561426505652726_nat_o @ X @ A )
% 14.25/8.10 => ( ( finite1363419556375932405_nat_o @ ( insert5175938949040314269_nat_o @ X @ A ) )
% 14.25/8.10 = ( suc @ ( finite1363419556375932405_nat_o @ A ) ) ) ) ) ) ).
% 14.25/8.10
% 14.25/8.10 thf(fact_5317_card__insert__if,axiom,
% 14.25/8.10 ! [A: set_int,X: int] :
% 14.25/8.10 ( ( finite_finite_int @ A )
% 14.25/8.10 => ( ( ( member_int @ X @ A )
% 14.25/8.10 => ( ( finite_card_int @ ( insert_int @ X @ A ) )
% 14.25/8.10 = ( finite_card_int @ A ) ) )
% 14.25/8.10 & ( ~ ( member_int @ X @ A )
% 14.25/8.10 => ( ( finite_card_int @ ( insert_int @ X @ A ) )
% 14.25/8.10 = ( suc @ ( finite_card_int @ A ) ) ) ) ) ) ).
% 14.25/8.10
% 14.25/8.10 thf(fact_5318_card__insert__if,axiom,
% 14.25/8.10 ! [A: set_o,X: $o] :
% 14.25/8.10 ( ( finite_finite_o @ A )
% 14.25/8.10 => ( ( ( member_o @ X @ A )
% 14.25/8.10 => ( ( finite_card_o @ ( insert_o @ X @ A ) )
% 14.25/8.10 = ( finite_card_o @ A ) ) )
% 14.25/8.10 & ( ~ ( member_o @ X @ A )
% 14.25/8.10 => ( ( finite_card_o @ ( insert_o @ X @ A ) )
% 14.25/8.10 = ( suc @ ( finite_card_o @ A ) ) ) ) ) ) ).
% 14.25/8.10
% 14.25/8.10 thf(fact_5319_card__insert__if,axiom,
% 14.25/8.10 ! [A: set_Product_unit,X: product_unit] :
% 14.25/8.10 ( ( finite4290736615968046902t_unit @ A )
% 14.25/8.10 => ( ( ( member_Product_unit @ X @ A )
% 14.25/8.10 => ( ( finite410649719033368117t_unit @ ( insert_Product_unit @ X @ A ) )
% 14.25/8.10 = ( finite410649719033368117t_unit @ A ) ) )
% 14.25/8.10 & ( ~ ( member_Product_unit @ X @ A )
% 14.25/8.10 => ( ( finite410649719033368117t_unit @ ( insert_Product_unit @ X @ A ) )
% 14.25/8.10 = ( suc @ ( finite410649719033368117t_unit @ A ) ) ) ) ) ) ).
% 14.25/8.10
% 14.25/8.10 thf(fact_5320_card__insert__if,axiom,
% 14.25/8.10 ! [A: set_nat,X: nat] :
% 14.25/8.10 ( ( finite_finite_nat @ A )
% 14.25/8.10 => ( ( ( member_nat @ X @ A )
% 14.25/8.10 => ( ( finite_card_nat @ ( insert_nat @ X @ A ) )
% 14.25/8.10 = ( finite_card_nat @ A ) ) )
% 14.25/8.10 & ( ~ ( member_nat @ X @ A )
% 14.25/8.10 => ( ( finite_card_nat @ ( insert_nat @ X @ A ) )
% 14.25/8.10 = ( suc @ ( finite_card_nat @ A ) ) ) ) ) ) ).
% 14.25/8.10
% 14.25/8.10 thf(fact_5321_card__insert__if,axiom,
% 14.25/8.10 ! [A: set_Code_integer,X: code_integer] :
% 14.25/8.10 ( ( finite6017078050557962740nteger @ A )
% 14.25/8.10 => ( ( ( member_Code_integer @ X @ A )
% 14.25/8.10 => ( ( finite4902975817058060853nteger @ ( insert_Code_integer @ X @ A ) )
% 14.25/8.10 = ( finite4902975817058060853nteger @ A ) ) )
% 14.25/8.10 & ( ~ ( member_Code_integer @ X @ A )
% 14.25/8.10 => ( ( finite4902975817058060853nteger @ ( insert_Code_integer @ X @ A ) )
% 14.25/8.10 = ( suc @ ( finite4902975817058060853nteger @ A ) ) ) ) ) ) ).
% 14.25/8.10
% 14.25/8.10 thf(fact_5322_card__le__Suc__iff,axiom,
% 14.25/8.10 ! [N2: nat,A: set_Pr4532377907799695533_nat_o] :
% 14.25/8.10 ( ( ord_less_eq_nat @ ( suc @ N2 ) @ ( finite1363419556375932405_nat_o @ A ) )
% 14.25/8.10 = ( ? [A6: produc3658429121746597890et_nat > $o,B7: set_Pr4532377907799695533_nat_o] :
% 14.25/8.10 ( ( A
% 14.25/8.10 = ( insert5175938949040314269_nat_o @ A6 @ B7 ) )
% 14.25/8.10 & ~ ( member6576561426505652726_nat_o @ A6 @ B7 )
% 14.25/8.10 & ( ord_less_eq_nat @ N2 @ ( finite1363419556375932405_nat_o @ B7 ) )
% 14.25/8.10 & ( finite3252695134891459830_nat_o @ B7 ) ) ) ) ).
% 14.25/8.10
% 14.25/8.10 thf(fact_5323_card__le__Suc__iff,axiom,
% 14.25/8.10 ! [N2: nat,A: set_int] :
% 14.25/8.10 ( ( ord_less_eq_nat @ ( suc @ N2 ) @ ( finite_card_int @ A ) )
% 14.25/8.10 = ( ? [A6: int,B7: set_int] :
% 14.25/8.10 ( ( A
% 14.25/8.10 = ( insert_int @ A6 @ B7 ) )
% 14.25/8.10 & ~ ( member_int @ A6 @ B7 )
% 14.25/8.10 & ( ord_less_eq_nat @ N2 @ ( finite_card_int @ B7 ) )
% 14.25/8.10 & ( finite_finite_int @ B7 ) ) ) ) ).
% 14.25/8.10
% 14.25/8.10 thf(fact_5324_card__le__Suc__iff,axiom,
% 14.25/8.10 ! [N2: nat,A: set_o] :
% 14.25/8.10 ( ( ord_less_eq_nat @ ( suc @ N2 ) @ ( finite_card_o @ A ) )
% 14.25/8.10 = ( ? [A6: $o,B7: set_o] :
% 14.25/8.10 ( ( A
% 14.25/8.10 = ( insert_o @ A6 @ B7 ) )
% 14.25/8.10 & ~ ( member_o @ A6 @ B7 )
% 14.25/8.10 & ( ord_less_eq_nat @ N2 @ ( finite_card_o @ B7 ) )
% 14.25/8.10 & ( finite_finite_o @ B7 ) ) ) ) ).
% 14.25/8.10
% 14.25/8.10 thf(fact_5325_card__le__Suc__iff,axiom,
% 14.25/8.10 ! [N2: nat,A: set_Product_unit] :
% 14.25/8.10 ( ( ord_less_eq_nat @ ( suc @ N2 ) @ ( finite410649719033368117t_unit @ A ) )
% 14.25/8.10 = ( ? [A6: product_unit,B7: set_Product_unit] :
% 14.25/8.10 ( ( A
% 14.25/8.10 = ( insert_Product_unit @ A6 @ B7 ) )
% 14.25/8.10 & ~ ( member_Product_unit @ A6 @ B7 )
% 14.25/8.10 & ( ord_less_eq_nat @ N2 @ ( finite410649719033368117t_unit @ B7 ) )
% 14.25/8.10 & ( finite4290736615968046902t_unit @ B7 ) ) ) ) ).
% 14.25/8.10
% 14.25/8.10 thf(fact_5326_card__le__Suc__iff,axiom,
% 14.25/8.10 ! [N2: nat,A: set_nat] :
% 14.25/8.10 ( ( ord_less_eq_nat @ ( suc @ N2 ) @ ( finite_card_nat @ A ) )
% 14.25/8.10 = ( ? [A6: nat,B7: set_nat] :
% 14.25/8.10 ( ( A
% 14.25/8.10 = ( insert_nat @ A6 @ B7 ) )
% 14.25/8.10 & ~ ( member_nat @ A6 @ B7 )
% 14.25/8.10 & ( ord_less_eq_nat @ N2 @ ( finite_card_nat @ B7 ) )
% 14.25/8.10 & ( finite_finite_nat @ B7 ) ) ) ) ).
% 14.25/8.10
% 14.25/8.10 thf(fact_5327_card__le__Suc__iff,axiom,
% 14.25/8.10 ! [N2: nat,A: set_Code_integer] :
% 14.25/8.10 ( ( ord_less_eq_nat @ ( suc @ N2 ) @ ( finite4902975817058060853nteger @ A ) )
% 14.25/8.10 = ( ? [A6: code_integer,B7: set_Code_integer] :
% 14.25/8.10 ( ( A
% 14.25/8.10 = ( insert_Code_integer @ A6 @ B7 ) )
% 14.25/8.10 & ~ ( member_Code_integer @ A6 @ B7 )
% 14.25/8.10 & ( ord_less_eq_nat @ N2 @ ( finite4902975817058060853nteger @ B7 ) )
% 14.25/8.10 & ( finite6017078050557962740nteger @ B7 ) ) ) ) ).
% 14.25/8.10
% 14.25/8.10 thf(fact_5328_card_Oremove,axiom,
% 14.25/8.10 ! [A: set_Code_integer,X: code_integer] :
% 14.25/8.10 ( ( finite6017078050557962740nteger @ A )
% 14.25/8.10 => ( ( member_Code_integer @ X @ A )
% 14.25/8.10 => ( ( finite4902975817058060853nteger @ A )
% 14.25/8.10 = ( suc @ ( finite4902975817058060853nteger @ ( minus_2355218937544613996nteger @ A @ ( insert_Code_integer @ X @ bot_bo3990330152332043303nteger ) ) ) ) ) ) ) ).
% 14.25/8.10
% 14.25/8.10 thf(fact_5329_card_Oremove,axiom,
% 14.25/8.10 ! [A: set_Product_unit,X: product_unit] :
% 14.25/8.10 ( ( finite4290736615968046902t_unit @ A )
% 14.25/8.10 => ( ( member_Product_unit @ X @ A )
% 14.25/8.10 => ( ( finite410649719033368117t_unit @ A )
% 14.25/8.10 = ( suc @ ( finite410649719033368117t_unit @ ( minus_6452836326544984404t_unit @ A @ ( insert_Product_unit @ X @ bot_bo3957492148770167129t_unit ) ) ) ) ) ) ) ).
% 14.25/8.10
% 14.25/8.10 thf(fact_5330_card_Oremove,axiom,
% 14.25/8.10 ! [A: set_o,X: $o] :
% 14.25/8.10 ( ( finite_finite_o @ A )
% 14.25/8.10 => ( ( member_o @ X @ A )
% 14.25/8.10 => ( ( finite_card_o @ A )
% 14.25/8.10 = ( suc @ ( finite_card_o @ ( minus_minus_set_o @ A @ ( insert_o @ X @ bot_bot_set_o ) ) ) ) ) ) ) ).
% 14.25/8.10
% 14.25/8.10 thf(fact_5331_card_Oremove,axiom,
% 14.25/8.10 ! [A: set_int,X: int] :
% 14.25/8.10 ( ( finite_finite_int @ A )
% 14.25/8.10 => ( ( member_int @ X @ A )
% 14.25/8.10 => ( ( finite_card_int @ A )
% 14.25/8.10 = ( suc @ ( finite_card_int @ ( minus_minus_set_int @ A @ ( insert_int @ X @ bot_bot_set_int ) ) ) ) ) ) ) ).
% 14.25/8.10
% 14.25/8.10 thf(fact_5332_card_Oremove,axiom,
% 14.25/8.10 ! [A: set_Pr4532377907799695533_nat_o,X: produc3658429121746597890et_nat > $o] :
% 14.25/8.10 ( ( finite3252695134891459830_nat_o @ A )
% 14.25/8.10 => ( ( member6576561426505652726_nat_o @ X @ A )
% 14.25/8.10 => ( ( finite1363419556375932405_nat_o @ A )
% 14.25/8.10 = ( suc @ ( finite1363419556375932405_nat_o @ ( minus_1801376950450012436_nat_o @ A @ ( insert5175938949040314269_nat_o @ X @ bot_bo7824918357723582233_nat_o ) ) ) ) ) ) ) ).
% 14.25/8.10
% 14.25/8.10 thf(fact_5333_card_Oremove,axiom,
% 14.25/8.10 ! [A: set_nat,X: nat] :
% 14.25/8.10 ( ( finite_finite_nat @ A )
% 14.25/8.10 => ( ( member_nat @ X @ A )
% 14.25/8.10 => ( ( finite_card_nat @ A )
% 14.25/8.10 = ( suc @ ( finite_card_nat @ ( minus_minus_set_nat @ A @ ( insert_nat @ X @ bot_bot_set_nat ) ) ) ) ) ) ) ).
% 14.25/8.10
% 14.25/8.10 thf(fact_5334_card_Oinsert__remove,axiom,
% 14.25/8.10 ! [A: set_Code_integer,X: code_integer] :
% 14.25/8.10 ( ( finite6017078050557962740nteger @ A )
% 14.25/8.10 => ( ( finite4902975817058060853nteger @ ( insert_Code_integer @ X @ A ) )
% 14.25/8.10 = ( suc @ ( finite4902975817058060853nteger @ ( minus_2355218937544613996nteger @ A @ ( insert_Code_integer @ X @ bot_bo3990330152332043303nteger ) ) ) ) ) ) ).
% 14.25/8.10
% 14.25/8.10 thf(fact_5335_card_Oinsert__remove,axiom,
% 14.25/8.10 ! [A: set_Product_unit,X: product_unit] :
% 14.25/8.10 ( ( finite4290736615968046902t_unit @ A )
% 14.25/8.10 => ( ( finite410649719033368117t_unit @ ( insert_Product_unit @ X @ A ) )
% 14.25/8.10 = ( suc @ ( finite410649719033368117t_unit @ ( minus_6452836326544984404t_unit @ A @ ( insert_Product_unit @ X @ bot_bo3957492148770167129t_unit ) ) ) ) ) ) ).
% 14.25/8.10
% 14.25/8.10 thf(fact_5336_card_Oinsert__remove,axiom,
% 14.25/8.10 ! [A: set_o,X: $o] :
% 14.25/8.10 ( ( finite_finite_o @ A )
% 14.25/8.10 => ( ( finite_card_o @ ( insert_o @ X @ A ) )
% 14.25/8.10 = ( suc @ ( finite_card_o @ ( minus_minus_set_o @ A @ ( insert_o @ X @ bot_bot_set_o ) ) ) ) ) ) ).
% 14.25/8.10
% 14.25/8.10 thf(fact_5337_card_Oinsert__remove,axiom,
% 14.25/8.10 ! [A: set_int,X: int] :
% 14.25/8.10 ( ( finite_finite_int @ A )
% 14.25/8.10 => ( ( finite_card_int @ ( insert_int @ X @ A ) )
% 14.25/8.10 = ( suc @ ( finite_card_int @ ( minus_minus_set_int @ A @ ( insert_int @ X @ bot_bot_set_int ) ) ) ) ) ) ).
% 14.25/8.10
% 14.25/8.10 thf(fact_5338_card_Oinsert__remove,axiom,
% 14.25/8.10 ! [A: set_Pr4532377907799695533_nat_o,X: produc3658429121746597890et_nat > $o] :
% 14.25/8.10 ( ( finite3252695134891459830_nat_o @ A )
% 14.25/8.10 => ( ( finite1363419556375932405_nat_o @ ( insert5175938949040314269_nat_o @ X @ A ) )
% 14.25/8.10 = ( suc @ ( finite1363419556375932405_nat_o @ ( minus_1801376950450012436_nat_o @ A @ ( insert5175938949040314269_nat_o @ X @ bot_bo7824918357723582233_nat_o ) ) ) ) ) ) ).
% 14.25/8.10
% 14.25/8.10 thf(fact_5339_card_Oinsert__remove,axiom,
% 14.25/8.10 ! [A: set_nat,X: nat] :
% 14.25/8.10 ( ( finite_finite_nat @ A )
% 14.25/8.10 => ( ( finite_card_nat @ ( insert_nat @ X @ A ) )
% 14.25/8.10 = ( suc @ ( finite_card_nat @ ( minus_minus_set_nat @ A @ ( insert_nat @ X @ bot_bot_set_nat ) ) ) ) ) ) ).
% 14.25/8.10
% 14.25/8.10 thf(fact_5340_mult__Suc__right,axiom,
% 14.25/8.10 ! [M4: nat,N2: nat] :
% 14.25/8.10 ( ( times_times_nat @ M4 @ ( suc @ N2 ) )
% 14.25/8.10 = ( plus_plus_nat @ M4 @ ( times_times_nat @ M4 @ N2 ) ) ) ).
% 14.25/8.10
% 14.25/8.10 thf(fact_5341_nat__mult__eq__1__iff,axiom,
% 14.25/8.10 ! [M4: nat,N2: nat] :
% 14.25/8.10 ( ( ( times_times_nat @ M4 @ N2 )
% 14.25/8.10 = one_one_nat )
% 14.25/8.10 = ( ( M4 = one_one_nat )
% 14.25/8.10 & ( N2 = one_one_nat ) ) ) ).
% 14.25/8.10
% 14.25/8.10 thf(fact_5342_nat__1__eq__mult__iff,axiom,
% 14.25/8.10 ! [M4: nat,N2: nat] :
% 14.25/8.10 ( ( one_one_nat
% 14.25/8.10 = ( times_times_nat @ M4 @ N2 ) )
% 14.25/8.10 = ( ( M4 = one_one_nat )
% 14.25/8.10 & ( N2 = one_one_nat ) ) ) ).
% 14.25/8.10
% 14.25/8.10 thf(fact_5343_nat__less__add__iff1,axiom,
% 14.25/8.10 ! [J: nat,I2: nat,U: nat,M4: nat,N2: nat] :
% 14.25/8.10 ( ( ord_less_eq_nat @ J @ I2 )
% 14.25/8.10 => ( ( ord_less_nat @ ( plus_plus_nat @ ( times_times_nat @ I2 @ U ) @ M4 ) @ ( plus_plus_nat @ ( times_times_nat @ J @ U ) @ N2 ) )
% 14.25/8.10 = ( ord_less_nat @ ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ I2 @ J ) @ U ) @ M4 ) @ N2 ) ) ) ).
% 14.25/8.10
% 14.25/8.10 thf(fact_5344_nat__less__add__iff2,axiom,
% 14.25/8.10 ! [I2: nat,J: nat,U: nat,M4: nat,N2: nat] :
% 14.25/8.10 ( ( ord_less_eq_nat @ I2 @ J )
% 14.25/8.10 => ( ( ord_less_nat @ ( plus_plus_nat @ ( times_times_nat @ I2 @ U ) @ M4 ) @ ( plus_plus_nat @ ( times_times_nat @ J @ U ) @ N2 ) )
% 14.25/8.10 = ( ord_less_nat @ M4 @ ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ J @ I2 ) @ U ) @ N2 ) ) ) ) ).
% 14.25/8.10
% 14.25/8.10 thf(fact_5345_nat__diff__add__eq2,axiom,
% 14.25/8.10 ! [I2: nat,J: nat,U: nat,M4: nat,N2: nat] :
% 14.25/8.10 ( ( ord_less_eq_nat @ I2 @ J )
% 14.25/8.10 => ( ( minus_minus_nat @ ( plus_plus_nat @ ( times_times_nat @ I2 @ U ) @ M4 ) @ ( plus_plus_nat @ ( times_times_nat @ J @ U ) @ N2 ) )
% 14.25/8.10 = ( minus_minus_nat @ M4 @ ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ J @ I2 ) @ U ) @ N2 ) ) ) ) ).
% 14.25/8.10
% 14.25/8.10 thf(fact_5346_nat__diff__add__eq1,axiom,
% 14.25/8.10 ! [J: nat,I2: nat,U: nat,M4: nat,N2: nat] :
% 14.25/8.10 ( ( ord_less_eq_nat @ J @ I2 )
% 14.25/8.10 => ( ( minus_minus_nat @ ( plus_plus_nat @ ( times_times_nat @ I2 @ U ) @ M4 ) @ ( plus_plus_nat @ ( times_times_nat @ J @ U ) @ N2 ) )
% 14.25/8.10 = ( minus_minus_nat @ ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ I2 @ J ) @ U ) @ M4 ) @ N2 ) ) ) ).
% 14.25/8.10
% 14.25/8.10 thf(fact_5347_nat__le__add__iff2,axiom,
% 14.25/8.10 ! [I2: nat,J: nat,U: nat,M4: nat,N2: nat] :
% 14.25/8.10 ( ( ord_less_eq_nat @ I2 @ J )
% 14.25/8.10 => ( ( ord_less_eq_nat @ ( plus_plus_nat @ ( times_times_nat @ I2 @ U ) @ M4 ) @ ( plus_plus_nat @ ( times_times_nat @ J @ U ) @ N2 ) )
% 14.25/8.10 = ( ord_less_eq_nat @ M4 @ ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ J @ I2 ) @ U ) @ N2 ) ) ) ) ).
% 14.25/8.10
% 14.25/8.10 thf(fact_5348_nat__mult__max__right,axiom,
% 14.25/8.10 ! [M4: nat,N2: nat,Q4: nat] :
% 14.25/8.10 ( ( times_times_nat @ M4 @ ( ord_max_nat @ N2 @ Q4 ) )
% 14.25/8.10 = ( ord_max_nat @ ( times_times_nat @ M4 @ N2 ) @ ( times_times_nat @ M4 @ Q4 ) ) ) ).
% 14.25/8.10
% 14.25/8.10 thf(fact_5349_nat__mult__max__left,axiom,
% 14.25/8.10 ! [M4: nat,N2: nat,Q4: nat] :
% 14.25/8.10 ( ( times_times_nat @ ( ord_max_nat @ M4 @ N2 ) @ Q4 )
% 14.25/8.10 = ( ord_max_nat @ ( times_times_nat @ M4 @ Q4 ) @ ( times_times_nat @ N2 @ Q4 ) ) ) ).
% 14.25/8.10
% 14.25/8.10 thf(fact_5350_Suc__mult__cancel1,axiom,
% 14.25/8.10 ! [K3: nat,M4: nat,N2: nat] :
% 14.25/8.10 ( ( ( times_times_nat @ ( suc @ K3 ) @ M4 )
% 14.25/8.10 = ( times_times_nat @ ( suc @ K3 ) @ N2 ) )
% 14.25/8.10 = ( M4 = N2 ) ) ).
% 14.25/8.10
% 14.25/8.10 thf(fact_5351_mult__le__mono2,axiom,
% 14.25/8.10 ! [I2: nat,J: nat,K3: nat] :
% 14.25/8.10 ( ( ord_less_eq_nat @ I2 @ J )
% 14.25/8.10 => ( ord_less_eq_nat @ ( times_times_nat @ K3 @ I2 ) @ ( times_times_nat @ K3 @ J ) ) ) ).
% 14.25/8.10
% 14.25/8.10 thf(fact_5352_mult__le__mono1,axiom,
% 14.25/8.10 ! [I2: nat,J: nat,K3: nat] :
% 14.25/8.10 ( ( ord_less_eq_nat @ I2 @ J )
% 14.25/8.10 => ( ord_less_eq_nat @ ( times_times_nat @ I2 @ K3 ) @ ( times_times_nat @ J @ K3 ) ) ) ).
% 14.25/8.10
% 14.25/8.10 thf(fact_5353_mult__le__mono,axiom,
% 14.25/8.10 ! [I2: nat,J: nat,K3: nat,L: nat] :
% 14.25/8.10 ( ( ord_less_eq_nat @ I2 @ J )
% 14.25/8.10 => ( ( ord_less_eq_nat @ K3 @ L )
% 14.25/8.10 => ( ord_less_eq_nat @ ( times_times_nat @ I2 @ K3 ) @ ( times_times_nat @ J @ L ) ) ) ) ).
% 14.25/8.10
% 14.25/8.10 thf(fact_5354_le__square,axiom,
% 14.25/8.10 ! [M4: nat] : ( ord_less_eq_nat @ M4 @ ( times_times_nat @ M4 @ M4 ) ) ).
% 14.25/8.10
% 14.25/8.10 thf(fact_5355_le__cube,axiom,
% 14.25/8.10 ! [M4: nat] : ( ord_less_eq_nat @ M4 @ ( times_times_nat @ M4 @ ( times_times_nat @ M4 @ M4 ) ) ) ).
% 14.25/8.10
% 14.25/8.10 thf(fact_5356_add__mult__distrib,axiom,
% 14.25/8.10 ! [M4: nat,N2: nat,K3: nat] :
% 14.25/8.10 ( ( times_times_nat @ ( plus_plus_nat @ M4 @ N2 ) @ K3 )
% 14.25/8.10 = ( plus_plus_nat @ ( times_times_nat @ M4 @ K3 ) @ ( times_times_nat @ N2 @ K3 ) ) ) ).
% 14.25/8.10
% 14.25/8.10 thf(fact_5357_add__mult__distrib2,axiom,
% 14.25/8.10 ! [K3: nat,M4: nat,N2: nat] :
% 14.25/8.10 ( ( times_times_nat @ K3 @ ( plus_plus_nat @ M4 @ N2 ) )
% 14.25/8.10 = ( plus_plus_nat @ ( times_times_nat @ K3 @ M4 ) @ ( times_times_nat @ K3 @ N2 ) ) ) ).
% 14.25/8.10
% 14.25/8.10 thf(fact_5358_left__add__mult__distrib,axiom,
% 14.25/8.10 ! [I2: nat,U: nat,J: nat,K3: nat] :
% 14.25/8.10 ( ( plus_plus_nat @ ( times_times_nat @ I2 @ U ) @ ( plus_plus_nat @ ( times_times_nat @ J @ U ) @ K3 ) )
% 14.25/8.10 = ( plus_plus_nat @ ( times_times_nat @ ( plus_plus_nat @ I2 @ J ) @ U ) @ K3 ) ) ).
% 14.25/8.10
% 14.25/8.10 thf(fact_5359_diff__mult__distrib,axiom,
% 14.25/8.10 ! [M4: nat,N2: nat,K3: nat] :
% 14.25/8.10 ( ( times_times_nat @ ( minus_minus_nat @ M4 @ N2 ) @ K3 )
% 14.25/8.10 = ( minus_minus_nat @ ( times_times_nat @ M4 @ K3 ) @ ( times_times_nat @ N2 @ K3 ) ) ) ).
% 14.25/8.10
% 14.25/8.10 thf(fact_5360_diff__mult__distrib2,axiom,
% 14.25/8.10 ! [K3: nat,M4: nat,N2: nat] :
% 14.25/8.10 ( ( times_times_nat @ K3 @ ( minus_minus_nat @ M4 @ N2 ) )
% 14.25/8.10 = ( minus_minus_nat @ ( times_times_nat @ K3 @ M4 ) @ ( times_times_nat @ K3 @ N2 ) ) ) ).
% 14.25/8.10
% 14.25/8.10 thf(fact_5361_nat__mult__1,axiom,
% 14.25/8.10 ! [N2: nat] :
% 14.25/8.10 ( ( times_times_nat @ one_one_nat @ N2 )
% 14.25/8.10 = N2 ) ).
% 14.25/8.10
% 14.25/8.10 thf(fact_5362_nat__mult__1__right,axiom,
% 14.25/8.10 ! [N2: nat] :
% 14.25/8.10 ( ( times_times_nat @ N2 @ one_one_nat )
% 14.25/8.10 = N2 ) ).
% 14.25/8.10
% 14.25/8.10 thf(fact_5363_Suc__mult__less__cancel1,axiom,
% 14.25/8.10 ! [K3: nat,M4: nat,N2: nat] :
% 14.25/8.10 ( ( ord_less_nat @ ( times_times_nat @ ( suc @ K3 ) @ M4 ) @ ( times_times_nat @ ( suc @ K3 ) @ N2 ) )
% 14.25/8.10 = ( ord_less_nat @ M4 @ N2 ) ) ).
% 14.25/8.10
% 14.25/8.10 thf(fact_5364_Suc__mult__le__cancel1,axiom,
% 14.25/8.10 ! [K3: nat,M4: nat,N2: nat] :
% 14.25/8.10 ( ( ord_less_eq_nat @ ( times_times_nat @ ( suc @ K3 ) @ M4 ) @ ( times_times_nat @ ( suc @ K3 ) @ N2 ) )
% 14.25/8.10 = ( ord_less_eq_nat @ M4 @ N2 ) ) ).
% 14.25/8.10
% 14.25/8.10 thf(fact_5365_mult__Suc,axiom,
% 14.25/8.10 ! [M4: nat,N2: nat] :
% 14.25/8.10 ( ( times_times_nat @ ( suc @ M4 ) @ N2 )
% 14.25/8.10 = ( plus_plus_nat @ N2 @ ( times_times_nat @ M4 @ N2 ) ) ) ).
% 14.25/8.10
% 14.25/8.10 thf(fact_5366_nat__eq__add__iff1,axiom,
% 14.25/8.10 ! [J: nat,I2: nat,U: nat,M4: nat,N2: nat] :
% 14.25/8.10 ( ( ord_less_eq_nat @ J @ I2 )
% 14.25/8.10 => ( ( ( plus_plus_nat @ ( times_times_nat @ I2 @ U ) @ M4 )
% 14.25/8.10 = ( plus_plus_nat @ ( times_times_nat @ J @ U ) @ N2 ) )
% 14.25/8.10 = ( ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ I2 @ J ) @ U ) @ M4 )
% 14.25/8.10 = N2 ) ) ) ).
% 14.25/8.10
% 14.25/8.10 thf(fact_5367_nat__eq__add__iff2,axiom,
% 14.25/8.10 ! [I2: nat,J: nat,U: nat,M4: nat,N2: nat] :
% 14.25/8.10 ( ( ord_less_eq_nat @ I2 @ J )
% 14.25/8.10 => ( ( ( plus_plus_nat @ ( times_times_nat @ I2 @ U ) @ M4 )
% 14.25/8.10 = ( plus_plus_nat @ ( times_times_nat @ J @ U ) @ N2 ) )
% 14.25/8.10 = ( M4
% 14.25/8.10 = ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ J @ I2 ) @ U ) @ N2 ) ) ) ) ).
% 14.25/8.10
% 14.25/8.10 thf(fact_5368_nat__le__add__iff1,axiom,
% 14.25/8.10 ! [J: nat,I2: nat,U: nat,M4: nat,N2: nat] :
% 14.25/8.10 ( ( ord_less_eq_nat @ J @ I2 )
% 14.25/8.10 => ( ( ord_less_eq_nat @ ( plus_plus_nat @ ( times_times_nat @ I2 @ U ) @ M4 ) @ ( plus_plus_nat @ ( times_times_nat @ J @ U ) @ N2 ) )
% 14.25/8.10 = ( ord_less_eq_nat @ ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ I2 @ J ) @ U ) @ M4 ) @ N2 ) ) ) ).
% 14.25/8.10
% 14.25/8.10 thf(fact_5369_discrete,axiom,
% 14.25/8.10 ( ord_le6747313008572928689nteger
% 14.25/8.10 = ( ^ [A6: code_integer] : ( ord_le3102999989581377725nteger @ ( plus_p5714425477246183910nteger @ A6 @ one_one_Code_integer ) ) ) ) ).
% 14.25/8.10
% 14.25/8.10 thf(fact_5370_discrete,axiom,
% 14.25/8.10 ( ord_less_nat
% 14.25/8.10 = ( ^ [A6: nat] : ( ord_less_eq_nat @ ( plus_plus_nat @ A6 @ one_one_nat ) ) ) ) ).
% 14.25/8.10
% 14.25/8.10 thf(fact_5371_discrete,axiom,
% 14.25/8.10 ( ord_less_int
% 14.25/8.10 = ( ^ [A6: int] : ( ord_less_eq_int @ ( plus_plus_int @ A6 @ one_one_int ) ) ) ) ).
% 14.25/8.10
% 14.25/8.10 thf(fact_5372_fstOp__def,axiom,
% 14.25/8.10 ( bNF_fs3871268833404974121et_nat
% 14.25/8.10 = ( ^ [P2: heap_e7401611519738050253t_unit > set_nat > $o,Q: set_nat > set_nat > $o,Ac: produc3658429121746597890et_nat] : ( produc7507926704131184380et_nat @ ( produc1824681642469235216et_nat @ Ac ) @ ( bNF_pi4042512755101277815et_nat @ P2 @ Q @ ( produc1824681642469235216et_nat @ Ac ) @ ( produc8586169260539613262et_nat @ Ac ) ) ) ) ) ).
% 14.25/8.10
% 14.25/8.10 thf(fact_5373_fstOp__def,axiom,
% 14.25/8.10 ( bNF_fs8167890477030535480nt_int
% 14.25/8.10 = ( ^ [P2: int > int > $o,Q: int > int > $o,Ac: product_prod_int_int] : ( product_Pair_int_int @ ( product_fst_int_int @ Ac ) @ ( bNF_pi8853482479757052266nt_int @ P2 @ Q @ ( product_fst_int_int @ Ac ) @ ( product_snd_int_int @ Ac ) ) ) ) ) ).
% 14.25/8.10
% 14.25/8.10 thf(fact_5374_sndOp__def,axiom,
% 14.25/8.10 ( bNF_sn4039337771197282234et_nat
% 14.25/8.10 = ( ^ [P2: heap_e7401611519738050253t_unit > heap_e7401611519738050253t_unit > $o,Q: heap_e7401611519738050253t_unit > set_nat > $o,Ac: produc3658429121746597890et_nat] : ( produc7507926704131184380et_nat @ ( bNF_pi3450177011487049898et_nat @ P2 @ Q @ ( produc1824681642469235216et_nat @ Ac ) @ ( produc8586169260539613262et_nat @ Ac ) ) @ ( produc8586169260539613262et_nat @ Ac ) ) ) ) ).
% 14.25/8.10
% 14.25/8.10 thf(fact_5375_sndOp__def,axiom,
% 14.25/8.10 ( bNF_sn1062102010912252026nt_int
% 14.25/8.10 = ( ^ [P2: int > int > $o,Q: int > int > $o,Ac: product_prod_int_int] : ( product_Pair_int_int @ ( bNF_pi8853482479757052266nt_int @ P2 @ Q @ ( product_fst_int_int @ Ac ) @ ( product_snd_int_int @ Ac ) ) @ ( product_snd_int_int @ Ac ) ) ) ) ).
% 14.25/8.10
% 14.25/8.10 thf(fact_5376_card__insert__le__m1,axiom,
% 14.25/8.10 ! [N2: nat,Y: set_nat,X: nat] :
% 14.25/8.10 ( ( ord_less_nat @ zero_zero_nat @ N2 )
% 14.25/8.10 => ( ( ord_less_eq_nat @ ( finite_card_nat @ Y ) @ ( minus_minus_nat @ N2 @ one_one_nat ) )
% 14.25/8.10 => ( ord_less_eq_nat @ ( finite_card_nat @ ( insert_nat @ X @ Y ) ) @ N2 ) ) ) ).
% 14.25/8.10
% 14.25/8.10 thf(fact_5377_card__insert__le__m1,axiom,
% 14.25/8.10 ! [N2: nat,Y: set_int,X: int] :
% 14.25/8.10 ( ( ord_less_nat @ zero_zero_nat @ N2 )
% 14.25/8.10 => ( ( ord_less_eq_nat @ ( finite_card_int @ Y ) @ ( minus_minus_nat @ N2 @ one_one_nat ) )
% 14.25/8.10 => ( ord_less_eq_nat @ ( finite_card_int @ ( insert_int @ X @ Y ) ) @ N2 ) ) ) ).
% 14.25/8.10
% 14.25/8.10 thf(fact_5378_card__insert__le__m1,axiom,
% 14.25/8.10 ! [N2: nat,Y: set_o,X: $o] :
% 14.25/8.10 ( ( ord_less_nat @ zero_zero_nat @ N2 )
% 14.25/8.10 => ( ( ord_less_eq_nat @ ( finite_card_o @ Y ) @ ( minus_minus_nat @ N2 @ one_one_nat ) )
% 14.25/8.10 => ( ord_less_eq_nat @ ( finite_card_o @ ( insert_o @ X @ Y ) ) @ N2 ) ) ) ).
% 14.25/8.10
% 14.25/8.10 thf(fact_5379_card__insert__le__m1,axiom,
% 14.25/8.10 ! [N2: nat,Y: set_Product_unit,X: product_unit] :
% 14.25/8.10 ( ( ord_less_nat @ zero_zero_nat @ N2 )
% 14.25/8.10 => ( ( ord_less_eq_nat @ ( finite410649719033368117t_unit @ Y ) @ ( minus_minus_nat @ N2 @ one_one_nat ) )
% 14.25/8.10 => ( ord_less_eq_nat @ ( finite410649719033368117t_unit @ ( insert_Product_unit @ X @ Y ) ) @ N2 ) ) ) ).
% 14.25/8.10
% 14.25/8.10 thf(fact_5380_crossproduct__noteq,axiom,
% 14.25/8.10 ! [A2: code_integer,B: code_integer,C: code_integer,D2: code_integer] :
% 14.25/8.10 ( ( ( A2 != B )
% 14.25/8.10 & ( C != D2 ) )
% 14.25/8.10 = ( ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ A2 @ C ) @ ( times_3573771949741848930nteger @ B @ D2 ) )
% 14.25/8.10 != ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ A2 @ D2 ) @ ( times_3573771949741848930nteger @ B @ C ) ) ) ) ).
% 14.25/8.10
% 14.25/8.10 thf(fact_5381_crossproduct__noteq,axiom,
% 14.25/8.10 ! [A2: rat,B: rat,C: rat,D2: rat] :
% 14.25/8.10 ( ( ( A2 != B )
% 14.25/8.10 & ( C != D2 ) )
% 14.25/8.10 = ( ( plus_plus_rat @ ( times_times_rat @ A2 @ C ) @ ( times_times_rat @ B @ D2 ) )
% 14.25/8.10 != ( plus_plus_rat @ ( times_times_rat @ A2 @ D2 ) @ ( times_times_rat @ B @ C ) ) ) ) ).
% 14.25/8.10
% 14.25/8.10 thf(fact_5382_crossproduct__noteq,axiom,
% 14.25/8.10 ! [A2: nat,B: nat,C: nat,D2: nat] :
% 14.25/8.10 ( ( ( A2 != B )
% 14.25/8.10 & ( C != D2 ) )
% 14.25/8.10 = ( ( plus_plus_nat @ ( times_times_nat @ A2 @ C ) @ ( times_times_nat @ B @ D2 ) )
% 14.25/8.10 != ( plus_plus_nat @ ( times_times_nat @ A2 @ D2 ) @ ( times_times_nat @ B @ C ) ) ) ) ).
% 14.25/8.10
% 14.25/8.10 thf(fact_5383_crossproduct__noteq,axiom,
% 14.25/8.10 ! [A2: int,B: int,C: int,D2: int] :
% 14.25/8.10 ( ( ( A2 != B )
% 14.25/8.10 & ( C != D2 ) )
% 14.25/8.10 = ( ( plus_plus_int @ ( times_times_int @ A2 @ C ) @ ( times_times_int @ B @ D2 ) )
% 14.25/8.10 != ( plus_plus_int @ ( times_times_int @ A2 @ D2 ) @ ( times_times_int @ B @ C ) ) ) ) ).
% 14.25/8.10
% 14.25/8.10 thf(fact_5384_crossproduct__eq,axiom,
% 14.25/8.10 ! [W: code_integer,Y: code_integer,X: code_integer,Z: code_integer] :
% 14.25/8.10 ( ( ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ W @ Y ) @ ( times_3573771949741848930nteger @ X @ Z ) )
% 14.25/8.10 = ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ W @ Z ) @ ( times_3573771949741848930nteger @ X @ Y ) ) )
% 14.25/8.10 = ( ( W = X )
% 14.25/8.10 | ( Y = Z ) ) ) ).
% 14.25/8.10
% 14.25/8.10 thf(fact_5385_crossproduct__eq,axiom,
% 14.25/8.10 ! [W: rat,Y: rat,X: rat,Z: rat] :
% 14.25/8.10 ( ( ( plus_plus_rat @ ( times_times_rat @ W @ Y ) @ ( times_times_rat @ X @ Z ) )
% 14.25/8.10 = ( plus_plus_rat @ ( times_times_rat @ W @ Z ) @ ( times_times_rat @ X @ Y ) ) )
% 14.25/8.10 = ( ( W = X )
% 14.25/8.10 | ( Y = Z ) ) ) ).
% 14.25/8.10
% 14.25/8.10 thf(fact_5386_crossproduct__eq,axiom,
% 14.25/8.10 ! [W: nat,Y: nat,X: nat,Z: nat] :
% 14.25/8.10 ( ( ( plus_plus_nat @ ( times_times_nat @ W @ Y ) @ ( times_times_nat @ X @ Z ) )
% 14.25/8.10 = ( plus_plus_nat @ ( times_times_nat @ W @ Z ) @ ( times_times_nat @ X @ Y ) ) )
% 14.25/8.10 = ( ( W = X )
% 14.25/8.10 | ( Y = Z ) ) ) ).
% 14.25/8.10
% 14.25/8.10 thf(fact_5387_crossproduct__eq,axiom,
% 14.25/8.10 ! [W: int,Y: int,X: int,Z: int] :
% 14.25/8.10 ( ( ( plus_plus_int @ ( times_times_int @ W @ Y ) @ ( times_times_int @ X @ Z ) )
% 14.25/8.10 = ( plus_plus_int @ ( times_times_int @ W @ Z ) @ ( times_times_int @ X @ Y ) ) )
% 14.25/8.10 = ( ( W = X )
% 14.25/8.10 | ( Y = Z ) ) ) ).
% 14.25/8.10
% 14.25/8.10 thf(fact_5388_dbl__dec__def,axiom,
% 14.25/8.10 ( neg_nu7757733837767384882nteger
% 14.25/8.10 = ( ^ [X2: code_integer] : ( minus_8373710615458151222nteger @ ( plus_p5714425477246183910nteger @ X2 @ X2 ) @ one_one_Code_integer ) ) ) ).
% 14.25/8.10
% 14.25/8.10 thf(fact_5389_dbl__dec__def,axiom,
% 14.25/8.10 ( neg_nu3179335615603231917ec_rat
% 14.25/8.10 = ( ^ [X2: rat] : ( minus_minus_rat @ ( plus_plus_rat @ X2 @ X2 ) @ one_one_rat ) ) ) ).
% 14.25/8.10
% 14.25/8.10 thf(fact_5390_dbl__dec__def,axiom,
% 14.25/8.10 ( neg_nu3811975205180677377ec_int
% 14.25/8.10 = ( ^ [X2: int] : ( minus_minus_int @ ( plus_plus_int @ X2 @ X2 ) @ one_one_int ) ) ) ).
% 14.25/8.10
% 14.25/8.10 thf(fact_5391_card__insert__disjoint_H,axiom,
% 14.25/8.10 ! [A: set_Pr4532377907799695533_nat_o,X: produc3658429121746597890et_nat > $o] :
% 14.25/8.10 ( ( finite3252695134891459830_nat_o @ A )
% 14.25/8.10 => ( ~ ( member6576561426505652726_nat_o @ X @ A )
% 14.25/8.10 => ( ( minus_minus_nat @ ( finite1363419556375932405_nat_o @ ( insert5175938949040314269_nat_o @ X @ A ) ) @ ( suc @ zero_zero_nat ) )
% 14.25/8.10 = ( finite1363419556375932405_nat_o @ A ) ) ) ) ).
% 14.25/8.10
% 14.25/8.10 thf(fact_5392_card__insert__disjoint_H,axiom,
% 14.25/8.10 ! [A: set_int,X: int] :
% 14.25/8.10 ( ( finite_finite_int @ A )
% 14.25/8.10 => ( ~ ( member_int @ X @ A )
% 14.25/8.10 => ( ( minus_minus_nat @ ( finite_card_int @ ( insert_int @ X @ A ) ) @ ( suc @ zero_zero_nat ) )
% 14.25/8.10 = ( finite_card_int @ A ) ) ) ) ).
% 14.25/8.10
% 14.25/8.10 thf(fact_5393_card__insert__disjoint_H,axiom,
% 14.25/8.10 ! [A: set_o,X: $o] :
% 14.25/8.10 ( ( finite_finite_o @ A )
% 14.25/8.10 => ( ~ ( member_o @ X @ A )
% 14.25/8.10 => ( ( minus_minus_nat @ ( finite_card_o @ ( insert_o @ X @ A ) ) @ ( suc @ zero_zero_nat ) )
% 14.25/8.10 = ( finite_card_o @ A ) ) ) ) ).
% 14.25/8.10
% 14.25/8.10 thf(fact_5394_card__insert__disjoint_H,axiom,
% 14.25/8.10 ! [A: set_Product_unit,X: product_unit] :
% 14.25/8.10 ( ( finite4290736615968046902t_unit @ A )
% 14.25/8.10 => ( ~ ( member_Product_unit @ X @ A )
% 14.25/8.10 => ( ( minus_minus_nat @ ( finite410649719033368117t_unit @ ( insert_Product_unit @ X @ A ) ) @ ( suc @ zero_zero_nat ) )
% 14.25/8.10 = ( finite410649719033368117t_unit @ A ) ) ) ) ).
% 14.25/8.10
% 14.25/8.10 thf(fact_5395_card__insert__disjoint_H,axiom,
% 14.25/8.10 ! [A: set_nat,X: nat] :
% 14.25/8.10 ( ( finite_finite_nat @ A )
% 14.25/8.10 => ( ~ ( member_nat @ X @ A )
% 14.25/8.10 => ( ( minus_minus_nat @ ( finite_card_nat @ ( insert_nat @ X @ A ) ) @ ( suc @ zero_zero_nat ) )
% 14.25/8.10 = ( finite_card_nat @ A ) ) ) ) ).
% 14.25/8.10
% 14.25/8.10 thf(fact_5396_card__insert__disjoint_H,axiom,
% 14.25/8.10 ! [A: set_Code_integer,X: code_integer] :
% 14.25/8.10 ( ( finite6017078050557962740nteger @ A )
% 14.25/8.10 => ( ~ ( member_Code_integer @ X @ A )
% 14.25/8.10 => ( ( minus_minus_nat @ ( finite4902975817058060853nteger @ ( insert_Code_integer @ X @ A ) ) @ ( suc @ zero_zero_nat ) )
% 14.25/8.10 = ( finite4902975817058060853nteger @ A ) ) ) ) ).
% 14.25/8.10
% 14.25/8.10 thf(fact_5397_le__zero__eq,axiom,
% 14.25/8.10 ! [N2: nat] :
% 14.25/8.10 ( ( ord_less_eq_nat @ N2 @ zero_zero_nat )
% 14.25/8.10 = ( N2 = zero_zero_nat ) ) ).
% 14.25/8.10
% 14.25/8.10 thf(fact_5398_not__gr__zero,axiom,
% 14.25/8.10 ! [N2: nat] :
% 14.25/8.10 ( ( ~ ( ord_less_nat @ zero_zero_nat @ N2 ) )
% 14.25/8.10 = ( N2 = zero_zero_nat ) ) ).
% 14.25/8.10
% 14.25/8.10 thf(fact_5399_add_Oright__neutral,axiom,
% 14.25/8.10 ! [A2: rat] :
% 14.25/8.10 ( ( plus_plus_rat @ A2 @ zero_zero_rat )
% 14.25/8.10 = A2 ) ).
% 14.25/8.10
% 14.25/8.10 thf(fact_5400_add_Oright__neutral,axiom,
% 14.25/8.10 ! [A2: nat] :
% 14.25/8.10 ( ( plus_plus_nat @ A2 @ zero_zero_nat )
% 14.25/8.10 = A2 ) ).
% 14.25/8.10
% 14.25/8.10 thf(fact_5401_add_Oright__neutral,axiom,
% 14.25/8.10 ! [A2: int] :
% 14.25/8.10 ( ( plus_plus_int @ A2 @ zero_zero_int )
% 14.25/8.10 = A2 ) ).
% 14.25/8.10
% 14.25/8.10 thf(fact_5402_add_Oright__neutral,axiom,
% 14.25/8.10 ! [A2: code_integer] :
% 14.25/8.10 ( ( plus_p5714425477246183910nteger @ A2 @ zero_z3403309356797280102nteger )
% 14.25/8.10 = A2 ) ).
% 14.25/8.10
% 14.25/8.10 thf(fact_5403_double__zero__sym,axiom,
% 14.25/8.10 ! [A2: rat] :
% 14.25/8.10 ( ( zero_zero_rat
% 14.25/8.10 = ( plus_plus_rat @ A2 @ A2 ) )
% 14.25/8.10 = ( A2 = zero_zero_rat ) ) ).
% 14.25/8.10
% 14.25/8.10 thf(fact_5404_double__zero__sym,axiom,
% 14.25/8.10 ! [A2: int] :
% 14.25/8.10 ( ( zero_zero_int
% 14.25/8.10 = ( plus_plus_int @ A2 @ A2 ) )
% 14.25/8.10 = ( A2 = zero_zero_int ) ) ).
% 14.25/8.10
% 14.25/8.10 thf(fact_5405_double__zero__sym,axiom,
% 14.25/8.10 ! [A2: code_integer] :
% 14.25/8.10 ( ( zero_z3403309356797280102nteger
% 14.25/8.10 = ( plus_p5714425477246183910nteger @ A2 @ A2 ) )
% 14.25/8.10 = ( A2 = zero_z3403309356797280102nteger ) ) ).
% 14.25/8.10
% 14.25/8.10 thf(fact_5406_add__cancel__left__left,axiom,
% 14.25/8.10 ! [B: rat,A2: rat] :
% 14.25/8.10 ( ( ( plus_plus_rat @ B @ A2 )
% 14.25/8.10 = A2 )
% 14.25/8.10 = ( B = zero_zero_rat ) ) ).
% 14.25/8.10
% 14.25/8.10 thf(fact_5407_add__cancel__left__left,axiom,
% 14.25/8.10 ! [B: nat,A2: nat] :
% 14.25/8.10 ( ( ( plus_plus_nat @ B @ A2 )
% 14.25/8.10 = A2 )
% 14.25/8.10 = ( B = zero_zero_nat ) ) ).
% 14.25/8.10
% 14.25/8.10 thf(fact_5408_add__cancel__left__left,axiom,
% 14.25/8.10 ! [B: int,A2: int] :
% 14.25/8.10 ( ( ( plus_plus_int @ B @ A2 )
% 14.25/8.10 = A2 )
% 14.25/8.10 = ( B = zero_zero_int ) ) ).
% 14.25/8.10
% 14.25/8.10 thf(fact_5409_add__cancel__left__left,axiom,
% 14.25/8.10 ! [B: code_integer,A2: code_integer] :
% 14.25/8.10 ( ( ( plus_p5714425477246183910nteger @ B @ A2 )
% 14.25/8.10 = A2 )
% 14.25/8.10 = ( B = zero_z3403309356797280102nteger ) ) ).
% 14.25/8.10
% 14.25/8.10 thf(fact_5410_add__cancel__left__right,axiom,
% 14.25/8.10 ! [A2: rat,B: rat] :
% 14.25/8.10 ( ( ( plus_plus_rat @ A2 @ B )
% 14.25/8.10 = A2 )
% 14.25/8.10 = ( B = zero_zero_rat ) ) ).
% 14.25/8.10
% 14.25/8.10 thf(fact_5411_add__cancel__left__right,axiom,
% 14.25/8.10 ! [A2: nat,B: nat] :
% 14.25/8.10 ( ( ( plus_plus_nat @ A2 @ B )
% 14.25/8.10 = A2 )
% 14.25/8.10 = ( B = zero_zero_nat ) ) ).
% 14.25/8.10
% 14.25/8.10 thf(fact_5412_add__cancel__left__right,axiom,
% 14.25/8.10 ! [A2: int,B: int] :
% 14.25/8.10 ( ( ( plus_plus_int @ A2 @ B )
% 14.25/8.10 = A2 )
% 14.25/8.10 = ( B = zero_zero_int ) ) ).
% 14.25/8.10
% 14.25/8.10 thf(fact_5413_add__cancel__left__right,axiom,
% 14.25/8.10 ! [A2: code_integer,B: code_integer] :
% 14.25/8.10 ( ( ( plus_p5714425477246183910nteger @ A2 @ B )
% 14.25/8.10 = A2 )
% 14.25/8.10 = ( B = zero_z3403309356797280102nteger ) ) ).
% 14.25/8.10
% 14.25/8.10 thf(fact_5414_add__cancel__right__left,axiom,
% 14.25/8.10 ! [A2: rat,B: rat] :
% 14.25/8.10 ( ( A2
% 14.25/8.10 = ( plus_plus_rat @ B @ A2 ) )
% 14.25/8.10 = ( B = zero_zero_rat ) ) ).
% 14.25/8.10
% 14.25/8.10 thf(fact_5415_add__cancel__right__left,axiom,
% 14.25/8.10 ! [A2: nat,B: nat] :
% 14.25/8.10 ( ( A2
% 14.25/8.10 = ( plus_plus_nat @ B @ A2 ) )
% 14.25/8.10 = ( B = zero_zero_nat ) ) ).
% 14.25/8.10
% 14.25/8.10 thf(fact_5416_add__cancel__right__left,axiom,
% 14.25/8.10 ! [A2: int,B: int] :
% 14.25/8.10 ( ( A2
% 14.25/8.10 = ( plus_plus_int @ B @ A2 ) )
% 14.25/8.10 = ( B = zero_zero_int ) ) ).
% 14.25/8.10
% 14.25/8.10 thf(fact_5417_add__cancel__right__left,axiom,
% 14.25/8.10 ! [A2: code_integer,B: code_integer] :
% 14.25/8.10 ( ( A2
% 14.25/8.10 = ( plus_p5714425477246183910nteger @ B @ A2 ) )
% 14.25/8.10 = ( B = zero_z3403309356797280102nteger ) ) ).
% 14.25/8.10
% 14.25/8.10 thf(fact_5418_add__cancel__right__right,axiom,
% 14.25/8.10 ! [A2: rat,B: rat] :
% 14.25/8.10 ( ( A2
% 14.25/8.10 = ( plus_plus_rat @ A2 @ B ) )
% 14.25/8.10 = ( B = zero_zero_rat ) ) ).
% 14.25/8.10
% 14.25/8.10 thf(fact_5419_add__cancel__right__right,axiom,
% 14.25/8.10 ! [A2: nat,B: nat] :
% 14.25/8.10 ( ( A2
% 14.25/8.10 = ( plus_plus_nat @ A2 @ B ) )
% 14.25/8.10 = ( B = zero_zero_nat ) ) ).
% 14.25/8.10
% 14.25/8.10 thf(fact_5420_add__cancel__right__right,axiom,
% 14.25/8.10 ! [A2: int,B: int] :
% 14.25/8.10 ( ( A2
% 14.25/8.10 = ( plus_plus_int @ A2 @ B ) )
% 14.25/8.10 = ( B = zero_zero_int ) ) ).
% 14.25/8.10
% 14.25/8.10 thf(fact_5421_add__cancel__right__right,axiom,
% 14.25/8.10 ! [A2: code_integer,B: code_integer] :
% 14.25/8.10 ( ( A2
% 14.25/8.10 = ( plus_p5714425477246183910nteger @ A2 @ B ) )
% 14.25/8.10 = ( B = zero_z3403309356797280102nteger ) ) ).
% 14.25/8.10
% 14.25/8.10 thf(fact_5422_add__eq__0__iff__both__eq__0,axiom,
% 14.25/8.10 ! [X: nat,Y: nat] :
% 14.25/8.10 ( ( ( plus_plus_nat @ X @ Y )
% 14.25/8.10 = zero_zero_nat )
% 14.25/8.10 = ( ( X = zero_zero_nat )
% 14.25/8.10 & ( Y = zero_zero_nat ) ) ) ).
% 14.25/8.10
% 14.25/8.10 thf(fact_5423_zero__eq__add__iff__both__eq__0,axiom,
% 14.25/8.10 ! [X: nat,Y: nat] :
% 14.25/8.10 ( ( zero_zero_nat
% 14.25/8.10 = ( plus_plus_nat @ X @ Y ) )
% 14.25/8.10 = ( ( X = zero_zero_nat )
% 14.25/8.10 & ( Y = zero_zero_nat ) ) ) ).
% 14.25/8.10
% 14.25/8.10 thf(fact_5424_add__0,axiom,
% 14.25/8.10 ! [A2: rat] :
% 14.25/8.10 ( ( plus_plus_rat @ zero_zero_rat @ A2 )
% 14.25/8.10 = A2 ) ).
% 14.25/8.10
% 14.25/8.10 thf(fact_5425_add__0,axiom,
% 14.25/8.10 ! [A2: nat] :
% 14.25/8.10 ( ( plus_plus_nat @ zero_zero_nat @ A2 )
% 14.25/8.10 = A2 ) ).
% 14.25/8.10
% 14.25/8.10 thf(fact_5426_add__0,axiom,
% 14.25/8.10 ! [A2: int] :
% 14.25/8.10 ( ( plus_plus_int @ zero_zero_int @ A2 )
% 14.25/8.10 = A2 ) ).
% 14.25/8.10
% 14.25/8.10 thf(fact_5427_add__0,axiom,
% 14.25/8.10 ! [A2: code_integer] :
% 14.25/8.10 ( ( plus_p5714425477246183910nteger @ zero_z3403309356797280102nteger @ A2 )
% 14.25/8.10 = A2 ) ).
% 14.25/8.10
% 14.25/8.10 thf(fact_5428_mult__cancel__right,axiom,
% 14.25/8.10 ! [A2: code_integer,C: code_integer,B: code_integer] :
% 14.25/8.10 ( ( ( times_3573771949741848930nteger @ A2 @ C )
% 14.25/8.10 = ( times_3573771949741848930nteger @ B @ C ) )
% 14.25/8.10 = ( ( C = zero_z3403309356797280102nteger )
% 14.25/8.10 | ( A2 = B ) ) ) ).
% 14.25/8.10
% 14.25/8.10 thf(fact_5429_mult__cancel__right,axiom,
% 14.25/8.10 ! [A2: rat,C: rat,B: rat] :
% 14.25/8.10 ( ( ( times_times_rat @ A2 @ C )
% 14.25/8.10 = ( times_times_rat @ B @ C ) )
% 14.25/8.10 = ( ( C = zero_zero_rat )
% 14.25/8.11 | ( A2 = B ) ) ) ).
% 14.25/8.11
% 14.25/8.11 thf(fact_5430_mult__cancel__right,axiom,
% 14.25/8.11 ! [A2: nat,C: nat,B: nat] :
% 14.25/8.11 ( ( ( times_times_nat @ A2 @ C )
% 14.25/8.11 = ( times_times_nat @ B @ C ) )
% 14.25/8.11 = ( ( C = zero_zero_nat )
% 14.25/8.11 | ( A2 = B ) ) ) ).
% 14.25/8.11
% 14.25/8.11 thf(fact_5431_mult__cancel__right,axiom,
% 14.25/8.11 ! [A2: int,C: int,B: int] :
% 14.25/8.11 ( ( ( times_times_int @ A2 @ C )
% 14.25/8.11 = ( times_times_int @ B @ C ) )
% 14.25/8.11 = ( ( C = zero_zero_int )
% 14.25/8.11 | ( A2 = B ) ) ) ).
% 14.25/8.11
% 14.25/8.11 thf(fact_5432_mult__cancel__left,axiom,
% 14.25/8.11 ! [C: code_integer,A2: code_integer,B: code_integer] :
% 14.25/8.11 ( ( ( times_3573771949741848930nteger @ C @ A2 )
% 14.25/8.11 = ( times_3573771949741848930nteger @ C @ B ) )
% 14.25/8.11 = ( ( C = zero_z3403309356797280102nteger )
% 14.25/8.11 | ( A2 = B ) ) ) ).
% 14.25/8.11
% 14.25/8.11 thf(fact_5433_mult__cancel__left,axiom,
% 14.25/8.11 ! [C: rat,A2: rat,B: rat] :
% 14.25/8.11 ( ( ( times_times_rat @ C @ A2 )
% 14.25/8.11 = ( times_times_rat @ C @ B ) )
% 14.25/8.11 = ( ( C = zero_zero_rat )
% 14.25/8.11 | ( A2 = B ) ) ) ).
% 14.25/8.11
% 14.25/8.11 thf(fact_5434_mult__cancel__left,axiom,
% 14.25/8.11 ! [C: nat,A2: nat,B: nat] :
% 14.25/8.11 ( ( ( times_times_nat @ C @ A2 )
% 14.25/8.11 = ( times_times_nat @ C @ B ) )
% 14.25/8.11 = ( ( C = zero_zero_nat )
% 14.25/8.11 | ( A2 = B ) ) ) ).
% 14.25/8.11
% 14.25/8.11 thf(fact_5435_mult__cancel__left,axiom,
% 14.25/8.11 ! [C: int,A2: int,B: int] :
% 14.25/8.11 ( ( ( times_times_int @ C @ A2 )
% 14.25/8.11 = ( times_times_int @ C @ B ) )
% 14.25/8.11 = ( ( C = zero_zero_int )
% 14.25/8.11 | ( A2 = B ) ) ) ).
% 14.25/8.11
% 14.25/8.11 thf(fact_5436_mult__eq__0__iff,axiom,
% 14.25/8.11 ! [A2: code_integer,B: code_integer] :
% 14.25/8.11 ( ( ( times_3573771949741848930nteger @ A2 @ B )
% 14.25/8.11 = zero_z3403309356797280102nteger )
% 14.25/8.11 = ( ( A2 = zero_z3403309356797280102nteger )
% 14.25/8.11 | ( B = zero_z3403309356797280102nteger ) ) ) ).
% 14.25/8.11
% 14.25/8.11 thf(fact_5437_mult__eq__0__iff,axiom,
% 14.25/8.11 ! [A2: rat,B: rat] :
% 14.25/8.11 ( ( ( times_times_rat @ A2 @ B )
% 14.25/8.11 = zero_zero_rat )
% 14.25/8.11 = ( ( A2 = zero_zero_rat )
% 14.25/8.11 | ( B = zero_zero_rat ) ) ) ).
% 14.25/8.11
% 14.25/8.11 thf(fact_5438_mult__eq__0__iff,axiom,
% 14.25/8.11 ! [A2: nat,B: nat] :
% 14.25/8.11 ( ( ( times_times_nat @ A2 @ B )
% 14.25/8.11 = zero_zero_nat )
% 14.25/8.11 = ( ( A2 = zero_zero_nat )
% 14.25/8.11 | ( B = zero_zero_nat ) ) ) ).
% 14.25/8.11
% 14.25/8.11 thf(fact_5439_mult__eq__0__iff,axiom,
% 14.25/8.11 ! [A2: int,B: int] :
% 14.25/8.11 ( ( ( times_times_int @ A2 @ B )
% 14.25/8.11 = zero_zero_int )
% 14.25/8.11 = ( ( A2 = zero_zero_int )
% 14.25/8.11 | ( B = zero_zero_int ) ) ) ).
% 14.25/8.11
% 14.25/8.11 thf(fact_5440_mult__zero__right,axiom,
% 14.25/8.11 ! [A2: code_integer] :
% 14.25/8.11 ( ( times_3573771949741848930nteger @ A2 @ zero_z3403309356797280102nteger )
% 14.25/8.11 = zero_z3403309356797280102nteger ) ).
% 14.25/8.11
% 14.25/8.11 thf(fact_5441_mult__zero__right,axiom,
% 14.25/8.11 ! [A2: rat] :
% 14.25/8.11 ( ( times_times_rat @ A2 @ zero_zero_rat )
% 14.25/8.11 = zero_zero_rat ) ).
% 14.25/8.11
% 14.25/8.11 thf(fact_5442_mult__zero__right,axiom,
% 14.25/8.11 ! [A2: nat] :
% 14.25/8.11 ( ( times_times_nat @ A2 @ zero_zero_nat )
% 14.25/8.11 = zero_zero_nat ) ).
% 14.25/8.11
% 14.25/8.11 thf(fact_5443_mult__zero__right,axiom,
% 14.25/8.11 ! [A2: int] :
% 14.25/8.11 ( ( times_times_int @ A2 @ zero_zero_int )
% 14.25/8.11 = zero_zero_int ) ).
% 14.25/8.11
% 14.25/8.11 thf(fact_5444_mult__zero__left,axiom,
% 14.25/8.11 ! [A2: code_integer] :
% 14.25/8.11 ( ( times_3573771949741848930nteger @ zero_z3403309356797280102nteger @ A2 )
% 14.25/8.11 = zero_z3403309356797280102nteger ) ).
% 14.25/8.11
% 14.25/8.11 thf(fact_5445_mult__zero__left,axiom,
% 14.25/8.11 ! [A2: rat] :
% 14.25/8.11 ( ( times_times_rat @ zero_zero_rat @ A2 )
% 14.25/8.11 = zero_zero_rat ) ).
% 14.25/8.11
% 14.25/8.11 thf(fact_5446_mult__zero__left,axiom,
% 14.25/8.11 ! [A2: nat] :
% 14.25/8.11 ( ( times_times_nat @ zero_zero_nat @ A2 )
% 14.25/8.11 = zero_zero_nat ) ).
% 14.25/8.11
% 14.25/8.11 thf(fact_5447_mult__zero__left,axiom,
% 14.25/8.11 ! [A2: int] :
% 14.25/8.11 ( ( times_times_int @ zero_zero_int @ A2 )
% 14.25/8.11 = zero_zero_int ) ).
% 14.25/8.11
% 14.25/8.11 thf(fact_5448_diff__self,axiom,
% 14.25/8.11 ! [A2: code_integer] :
% 14.25/8.11 ( ( minus_8373710615458151222nteger @ A2 @ A2 )
% 14.25/8.11 = zero_z3403309356797280102nteger ) ).
% 14.25/8.11
% 14.25/8.11 thf(fact_5449_diff__self,axiom,
% 14.25/8.11 ! [A2: rat] :
% 14.25/8.11 ( ( minus_minus_rat @ A2 @ A2 )
% 14.25/8.11 = zero_zero_rat ) ).
% 14.25/8.11
% 14.25/8.11 thf(fact_5450_diff__self,axiom,
% 14.25/8.11 ! [A2: int] :
% 14.25/8.11 ( ( minus_minus_int @ A2 @ A2 )
% 14.25/8.11 = zero_zero_int ) ).
% 14.25/8.11
% 14.25/8.11 thf(fact_5451_diff__0__right,axiom,
% 14.25/8.11 ! [A2: code_integer] :
% 14.25/8.11 ( ( minus_8373710615458151222nteger @ A2 @ zero_z3403309356797280102nteger )
% 14.25/8.11 = A2 ) ).
% 14.25/8.11
% 14.25/8.11 thf(fact_5452_diff__0__right,axiom,
% 14.25/8.11 ! [A2: rat] :
% 14.25/8.11 ( ( minus_minus_rat @ A2 @ zero_zero_rat )
% 14.25/8.11 = A2 ) ).
% 14.25/8.11
% 14.25/8.11 thf(fact_5453_diff__0__right,axiom,
% 14.25/8.11 ! [A2: int] :
% 14.25/8.11 ( ( minus_minus_int @ A2 @ zero_zero_int )
% 14.25/8.11 = A2 ) ).
% 14.25/8.11
% 14.25/8.11 thf(fact_5454_zero__diff,axiom,
% 14.25/8.11 ! [A2: nat] :
% 14.25/8.11 ( ( minus_minus_nat @ zero_zero_nat @ A2 )
% 14.25/8.11 = zero_zero_nat ) ).
% 14.25/8.11
% 14.25/8.11 thf(fact_5455_diff__zero,axiom,
% 14.25/8.11 ! [A2: code_integer] :
% 14.25/8.11 ( ( minus_8373710615458151222nteger @ A2 @ zero_z3403309356797280102nteger )
% 14.25/8.11 = A2 ) ).
% 14.25/8.11
% 14.25/8.11 thf(fact_5456_diff__zero,axiom,
% 14.25/8.11 ! [A2: rat] :
% 14.25/8.11 ( ( minus_minus_rat @ A2 @ zero_zero_rat )
% 14.25/8.11 = A2 ) ).
% 14.25/8.11
% 14.25/8.11 thf(fact_5457_diff__zero,axiom,
% 14.25/8.11 ! [A2: nat] :
% 14.25/8.11 ( ( minus_minus_nat @ A2 @ zero_zero_nat )
% 14.25/8.11 = A2 ) ).
% 14.25/8.11
% 14.25/8.11 thf(fact_5458_diff__zero,axiom,
% 14.25/8.11 ! [A2: int] :
% 14.25/8.11 ( ( minus_minus_int @ A2 @ zero_zero_int )
% 14.25/8.11 = A2 ) ).
% 14.25/8.11
% 14.25/8.11 thf(fact_5459_cancel__comm__monoid__add__class_Odiff__cancel,axiom,
% 14.25/8.11 ! [A2: code_integer] :
% 14.25/8.11 ( ( minus_8373710615458151222nteger @ A2 @ A2 )
% 14.25/8.11 = zero_z3403309356797280102nteger ) ).
% 14.25/8.11
% 14.25/8.11 thf(fact_5460_cancel__comm__monoid__add__class_Odiff__cancel,axiom,
% 14.25/8.11 ! [A2: rat] :
% 14.25/8.11 ( ( minus_minus_rat @ A2 @ A2 )
% 14.25/8.11 = zero_zero_rat ) ).
% 14.25/8.11
% 14.25/8.11 thf(fact_5461_cancel__comm__monoid__add__class_Odiff__cancel,axiom,
% 14.25/8.11 ! [A2: nat] :
% 14.25/8.11 ( ( minus_minus_nat @ A2 @ A2 )
% 14.25/8.11 = zero_zero_nat ) ).
% 14.25/8.11
% 14.25/8.11 thf(fact_5462_cancel__comm__monoid__add__class_Odiff__cancel,axiom,
% 14.25/8.11 ! [A2: int] :
% 14.25/8.11 ( ( minus_minus_int @ A2 @ A2 )
% 14.25/8.11 = zero_zero_int ) ).
% 14.25/8.11
% 14.25/8.11 thf(fact_5463_mult__cancel2,axiom,
% 14.25/8.11 ! [M4: nat,K3: nat,N2: nat] :
% 14.25/8.11 ( ( ( times_times_nat @ M4 @ K3 )
% 14.25/8.11 = ( times_times_nat @ N2 @ K3 ) )
% 14.25/8.11 = ( ( M4 = N2 )
% 14.25/8.11 | ( K3 = zero_zero_nat ) ) ) ).
% 14.25/8.11
% 14.25/8.11 thf(fact_5464_mult__cancel1,axiom,
% 14.25/8.11 ! [K3: nat,M4: nat,N2: nat] :
% 14.25/8.11 ( ( ( times_times_nat @ K3 @ M4 )
% 14.25/8.11 = ( times_times_nat @ K3 @ N2 ) )
% 14.25/8.11 = ( ( M4 = N2 )
% 14.25/8.11 | ( K3 = zero_zero_nat ) ) ) ).
% 14.25/8.11
% 14.25/8.11 thf(fact_5465_mult__0__right,axiom,
% 14.25/8.11 ! [M4: nat] :
% 14.25/8.11 ( ( times_times_nat @ M4 @ zero_zero_nat )
% 14.25/8.11 = zero_zero_nat ) ).
% 14.25/8.11
% 14.25/8.11 thf(fact_5466_mult__is__0,axiom,
% 14.25/8.11 ! [M4: nat,N2: nat] :
% 14.25/8.11 ( ( ( times_times_nat @ M4 @ N2 )
% 14.25/8.11 = zero_zero_nat )
% 14.25/8.11 = ( ( M4 = zero_zero_nat )
% 14.25/8.11 | ( N2 = zero_zero_nat ) ) ) ).
% 14.25/8.11
% 14.25/8.11 thf(fact_5467_Sup__nat__empty,axiom,
% 14.25/8.11 ( ( complete_Sup_Sup_nat @ bot_bot_set_nat )
% 14.25/8.11 = zero_zero_nat ) ).
% 14.25/8.11
% 14.25/8.11 thf(fact_5468_dbl__dec__simps_I3_J,axiom,
% 14.25/8.11 ( ( neg_nu3811975205180677377ec_int @ one_one_int )
% 14.25/8.11 = one_one_int ) ).
% 14.25/8.11
% 14.25/8.11 thf(fact_5469_dbl__dec__simps_I3_J,axiom,
% 14.25/8.11 ( ( neg_nu7757733837767384882nteger @ one_one_Code_integer )
% 14.25/8.11 = one_one_Code_integer ) ).
% 14.25/8.11
% 14.25/8.11 thf(fact_5470_zero__le__double__add__iff__zero__le__single__add,axiom,
% 14.25/8.11 ! [A2: code_integer] :
% 14.25/8.11 ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ ( plus_p5714425477246183910nteger @ A2 @ A2 ) )
% 14.25/8.11 = ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ A2 ) ) ).
% 14.25/8.11
% 14.25/8.11 thf(fact_5471_zero__le__double__add__iff__zero__le__single__add,axiom,
% 14.25/8.11 ! [A2: rat] :
% 14.25/8.11 ( ( ord_less_eq_rat @ zero_zero_rat @ ( plus_plus_rat @ A2 @ A2 ) )
% 14.25/8.11 = ( ord_less_eq_rat @ zero_zero_rat @ A2 ) ) ).
% 14.25/8.11
% 14.25/8.11 thf(fact_5472_zero__le__double__add__iff__zero__le__single__add,axiom,
% 14.25/8.11 ! [A2: int] :
% 14.25/8.11 ( ( ord_less_eq_int @ zero_zero_int @ ( plus_plus_int @ A2 @ A2 ) )
% 14.25/8.11 = ( ord_less_eq_int @ zero_zero_int @ A2 ) ) ).
% 14.25/8.11
% 14.25/8.11 thf(fact_5473_double__add__le__zero__iff__single__add__le__zero,axiom,
% 14.25/8.11 ! [A2: code_integer] :
% 14.25/8.11 ( ( ord_le3102999989581377725nteger @ ( plus_p5714425477246183910nteger @ A2 @ A2 ) @ zero_z3403309356797280102nteger )
% 14.25/8.11 = ( ord_le3102999989581377725nteger @ A2 @ zero_z3403309356797280102nteger ) ) ).
% 14.25/8.11
% 14.25/8.11 thf(fact_5474_double__add__le__zero__iff__single__add__le__zero,axiom,
% 14.25/8.11 ! [A2: rat] :
% 14.25/8.11 ( ( ord_less_eq_rat @ ( plus_plus_rat @ A2 @ A2 ) @ zero_zero_rat )
% 14.25/8.11 = ( ord_less_eq_rat @ A2 @ zero_zero_rat ) ) ).
% 14.25/8.11
% 14.25/8.11 thf(fact_5475_double__add__le__zero__iff__single__add__le__zero,axiom,
% 14.25/8.11 ! [A2: int] :
% 14.25/8.11 ( ( ord_less_eq_int @ ( plus_plus_int @ A2 @ A2 ) @ zero_zero_int )
% 14.25/8.11 = ( ord_less_eq_int @ A2 @ zero_zero_int ) ) ).
% 14.25/8.11
% 14.25/8.11 thf(fact_5476_le__add__same__cancel2,axiom,
% 14.25/8.11 ! [A2: code_integer,B: code_integer] :
% 14.25/8.11 ( ( ord_le3102999989581377725nteger @ A2 @ ( plus_p5714425477246183910nteger @ B @ A2 ) )
% 14.25/8.11 = ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ B ) ) ).
% 14.25/8.11
% 14.25/8.11 thf(fact_5477_le__add__same__cancel2,axiom,
% 14.25/8.11 ! [A2: rat,B: rat] :
% 14.25/8.11 ( ( ord_less_eq_rat @ A2 @ ( plus_plus_rat @ B @ A2 ) )
% 14.25/8.11 = ( ord_less_eq_rat @ zero_zero_rat @ B ) ) ).
% 14.25/8.11
% 14.25/8.11 thf(fact_5478_le__add__same__cancel2,axiom,
% 14.25/8.11 ! [A2: nat,B: nat] :
% 14.25/8.11 ( ( ord_less_eq_nat @ A2 @ ( plus_plus_nat @ B @ A2 ) )
% 14.25/8.11 = ( ord_less_eq_nat @ zero_zero_nat @ B ) ) ).
% 14.25/8.11
% 14.25/8.11 thf(fact_5479_le__add__same__cancel2,axiom,
% 14.25/8.11 ! [A2: int,B: int] :
% 14.25/8.11 ( ( ord_less_eq_int @ A2 @ ( plus_plus_int @ B @ A2 ) )
% 14.25/8.11 = ( ord_less_eq_int @ zero_zero_int @ B ) ) ).
% 14.25/8.11
% 14.25/8.11 thf(fact_5480_le__add__same__cancel1,axiom,
% 14.25/8.11 ! [A2: code_integer,B: code_integer] :
% 14.25/8.11 ( ( ord_le3102999989581377725nteger @ A2 @ ( plus_p5714425477246183910nteger @ A2 @ B ) )
% 14.25/8.11 = ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ B ) ) ).
% 14.25/8.11
% 14.25/8.11 thf(fact_5481_le__add__same__cancel1,axiom,
% 14.25/8.11 ! [A2: rat,B: rat] :
% 14.25/8.11 ( ( ord_less_eq_rat @ A2 @ ( plus_plus_rat @ A2 @ B ) )
% 14.25/8.11 = ( ord_less_eq_rat @ zero_zero_rat @ B ) ) ).
% 14.25/8.11
% 14.25/8.11 thf(fact_5482_le__add__same__cancel1,axiom,
% 14.25/8.11 ! [A2: nat,B: nat] :
% 14.25/8.11 ( ( ord_less_eq_nat @ A2 @ ( plus_plus_nat @ A2 @ B ) )
% 14.25/8.11 = ( ord_less_eq_nat @ zero_zero_nat @ B ) ) ).
% 14.25/8.11
% 14.25/8.11 thf(fact_5483_le__add__same__cancel1,axiom,
% 14.25/8.11 ! [A2: int,B: int] :
% 14.25/8.11 ( ( ord_less_eq_int @ A2 @ ( plus_plus_int @ A2 @ B ) )
% 14.25/8.11 = ( ord_less_eq_int @ zero_zero_int @ B ) ) ).
% 14.25/8.11
% 14.25/8.11 thf(fact_5484_add__le__same__cancel2,axiom,
% 14.25/8.11 ! [A2: code_integer,B: code_integer] :
% 14.25/8.11 ( ( ord_le3102999989581377725nteger @ ( plus_p5714425477246183910nteger @ A2 @ B ) @ B )
% 14.25/8.11 = ( ord_le3102999989581377725nteger @ A2 @ zero_z3403309356797280102nteger ) ) ).
% 14.25/8.11
% 14.25/8.11 thf(fact_5485_add__le__same__cancel2,axiom,
% 14.25/8.11 ! [A2: rat,B: rat] :
% 14.25/8.11 ( ( ord_less_eq_rat @ ( plus_plus_rat @ A2 @ B ) @ B )
% 14.25/8.11 = ( ord_less_eq_rat @ A2 @ zero_zero_rat ) ) ).
% 14.25/8.11
% 14.25/8.11 thf(fact_5486_add__le__same__cancel2,axiom,
% 14.25/8.11 ! [A2: nat,B: nat] :
% 14.25/8.11 ( ( ord_less_eq_nat @ ( plus_plus_nat @ A2 @ B ) @ B )
% 14.25/8.11 = ( ord_less_eq_nat @ A2 @ zero_zero_nat ) ) ).
% 14.25/8.11
% 14.25/8.11 thf(fact_5487_add__le__same__cancel2,axiom,
% 14.25/8.11 ! [A2: int,B: int] :
% 14.25/8.11 ( ( ord_less_eq_int @ ( plus_plus_int @ A2 @ B ) @ B )
% 14.25/8.11 = ( ord_less_eq_int @ A2 @ zero_zero_int ) ) ).
% 14.25/8.11
% 14.25/8.11 thf(fact_5488_add__le__same__cancel1,axiom,
% 14.25/8.11 ! [B: code_integer,A2: code_integer] :
% 14.25/8.11 ( ( ord_le3102999989581377725nteger @ ( plus_p5714425477246183910nteger @ B @ A2 ) @ B )
% 14.25/8.11 = ( ord_le3102999989581377725nteger @ A2 @ zero_z3403309356797280102nteger ) ) ).
% 14.25/8.11
% 14.25/8.11 thf(fact_5489_add__le__same__cancel1,axiom,
% 14.25/8.11 ! [B: rat,A2: rat] :
% 14.25/8.11 ( ( ord_less_eq_rat @ ( plus_plus_rat @ B @ A2 ) @ B )
% 14.25/8.11 = ( ord_less_eq_rat @ A2 @ zero_zero_rat ) ) ).
% 14.25/8.11
% 14.25/8.11 thf(fact_5490_add__le__same__cancel1,axiom,
% 14.25/8.11 ! [B: nat,A2: nat] :
% 14.25/8.11 ( ( ord_less_eq_nat @ ( plus_plus_nat @ B @ A2 ) @ B )
% 14.25/8.11 = ( ord_less_eq_nat @ A2 @ zero_zero_nat ) ) ).
% 14.25/8.11
% 14.25/8.11 thf(fact_5491_add__le__same__cancel1,axiom,
% 14.25/8.11 ! [B: int,A2: int] :
% 14.25/8.11 ( ( ord_less_eq_int @ ( plus_plus_int @ B @ A2 ) @ B )
% 14.25/8.11 = ( ord_less_eq_int @ A2 @ zero_zero_int ) ) ).
% 14.25/8.11
% 14.25/8.11 thf(fact_5492_diff__ge__0__iff__ge,axiom,
% 14.25/8.11 ! [A2: code_integer,B: code_integer] :
% 14.25/8.11 ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ ( minus_8373710615458151222nteger @ A2 @ B ) )
% 14.25/8.11 = ( ord_le3102999989581377725nteger @ B @ A2 ) ) ).
% 14.25/8.11
% 14.25/8.11 thf(fact_5493_diff__ge__0__iff__ge,axiom,
% 14.25/8.11 ! [A2: rat,B: rat] :
% 14.25/8.11 ( ( ord_less_eq_rat @ zero_zero_rat @ ( minus_minus_rat @ A2 @ B ) )
% 14.25/8.11 = ( ord_less_eq_rat @ B @ A2 ) ) ).
% 14.25/8.11
% 14.25/8.11 thf(fact_5494_diff__ge__0__iff__ge,axiom,
% 14.25/8.11 ! [A2: int,B: int] :
% 14.25/8.11 ( ( ord_less_eq_int @ zero_zero_int @ ( minus_minus_int @ A2 @ B ) )
% 14.25/8.11 = ( ord_less_eq_int @ B @ A2 ) ) ).
% 14.25/8.11
% 14.25/8.11 thf(fact_5495_zero__comp__diff__simps_I1_J,axiom,
% 14.25/8.11 ! [A2: code_integer,B: code_integer] :
% 14.25/8.11 ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ ( minus_8373710615458151222nteger @ A2 @ B ) )
% 14.25/8.11 = ( ord_le3102999989581377725nteger @ B @ A2 ) ) ).
% 14.25/8.11
% 14.25/8.11 thf(fact_5496_zero__comp__diff__simps_I1_J,axiom,
% 14.25/8.11 ! [A2: rat,B: rat] :
% 14.25/8.11 ( ( ord_less_eq_rat @ zero_zero_rat @ ( minus_minus_rat @ A2 @ B ) )
% 14.25/8.11 = ( ord_less_eq_rat @ B @ A2 ) ) ).
% 14.25/8.11
% 14.25/8.11 thf(fact_5497_zero__comp__diff__simps_I1_J,axiom,
% 14.25/8.11 ! [A2: int,B: int] :
% 14.25/8.11 ( ( ord_less_eq_int @ zero_zero_int @ ( minus_minus_int @ A2 @ B ) )
% 14.25/8.11 = ( ord_less_eq_int @ B @ A2 ) ) ).
% 14.25/8.11
% 14.25/8.11 thf(fact_5498_zero__less__double__add__iff__zero__less__single__add,axiom,
% 14.25/8.11 ! [A2: int] :
% 14.25/8.11 ( ( ord_less_int @ zero_zero_int @ ( plus_plus_int @ A2 @ A2 ) )
% 14.25/8.11 = ( ord_less_int @ zero_zero_int @ A2 ) ) ).
% 14.25/8.11
% 14.25/8.11 thf(fact_5499_zero__less__double__add__iff__zero__less__single__add,axiom,
% 14.25/8.11 ! [A2: code_integer] :
% 14.25/8.11 ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ ( plus_p5714425477246183910nteger @ A2 @ A2 ) )
% 14.25/8.11 = ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ A2 ) ) ).
% 14.25/8.11
% 14.25/8.11 thf(fact_5500_zero__less__double__add__iff__zero__less__single__add,axiom,
% 14.25/8.11 ! [A2: rat] :
% 14.25/8.11 ( ( ord_less_rat @ zero_zero_rat @ ( plus_plus_rat @ A2 @ A2 ) )
% 14.25/8.11 = ( ord_less_rat @ zero_zero_rat @ A2 ) ) ).
% 14.25/8.11
% 14.25/8.11 thf(fact_5501_double__add__less__zero__iff__single__add__less__zero,axiom,
% 14.25/8.11 ! [A2: int] :
% 14.25/8.11 ( ( ord_less_int @ ( plus_plus_int @ A2 @ A2 ) @ zero_zero_int )
% 14.25/8.11 = ( ord_less_int @ A2 @ zero_zero_int ) ) ).
% 14.25/8.11
% 14.25/8.11 thf(fact_5502_double__add__less__zero__iff__single__add__less__zero,axiom,
% 14.25/8.11 ! [A2: code_integer] :
% 14.25/8.11 ( ( ord_le6747313008572928689nteger @ ( plus_p5714425477246183910nteger @ A2 @ A2 ) @ zero_z3403309356797280102nteger )
% 14.25/8.11 = ( ord_le6747313008572928689nteger @ A2 @ zero_z3403309356797280102nteger ) ) ).
% 14.25/8.11
% 14.25/8.11 thf(fact_5503_double__add__less__zero__iff__single__add__less__zero,axiom,
% 14.25/8.11 ! [A2: rat] :
% 14.25/8.11 ( ( ord_less_rat @ ( plus_plus_rat @ A2 @ A2 ) @ zero_zero_rat )
% 14.25/8.11 = ( ord_less_rat @ A2 @ zero_zero_rat ) ) ).
% 14.25/8.11
% 14.25/8.11 thf(fact_5504_less__add__same__cancel2,axiom,
% 14.25/8.11 ! [A2: nat,B: nat] :
% 14.25/8.11 ( ( ord_less_nat @ A2 @ ( plus_plus_nat @ B @ A2 ) )
% 14.25/8.11 = ( ord_less_nat @ zero_zero_nat @ B ) ) ).
% 14.25/8.11
% 14.25/8.11 thf(fact_5505_less__add__same__cancel2,axiom,
% 14.25/8.11 ! [A2: int,B: int] :
% 14.25/8.11 ( ( ord_less_int @ A2 @ ( plus_plus_int @ B @ A2 ) )
% 14.25/8.11 = ( ord_less_int @ zero_zero_int @ B ) ) ).
% 14.25/8.11
% 14.25/8.11 thf(fact_5506_less__add__same__cancel2,axiom,
% 14.25/8.11 ! [A2: code_integer,B: code_integer] :
% 14.25/8.11 ( ( ord_le6747313008572928689nteger @ A2 @ ( plus_p5714425477246183910nteger @ B @ A2 ) )
% 14.25/8.11 = ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ B ) ) ).
% 14.25/8.11
% 14.25/8.11 thf(fact_5507_less__add__same__cancel2,axiom,
% 14.25/8.11 ! [A2: rat,B: rat] :
% 14.25/8.11 ( ( ord_less_rat @ A2 @ ( plus_plus_rat @ B @ A2 ) )
% 14.25/8.11 = ( ord_less_rat @ zero_zero_rat @ B ) ) ).
% 14.25/8.11
% 14.25/8.11 thf(fact_5508_less__add__same__cancel1,axiom,
% 14.25/8.11 ! [A2: nat,B: nat] :
% 14.25/8.11 ( ( ord_less_nat @ A2 @ ( plus_plus_nat @ A2 @ B ) )
% 14.25/8.11 = ( ord_less_nat @ zero_zero_nat @ B ) ) ).
% 14.25/8.11
% 14.25/8.11 thf(fact_5509_less__add__same__cancel1,axiom,
% 14.25/8.11 ! [A2: int,B: int] :
% 14.25/8.11 ( ( ord_less_int @ A2 @ ( plus_plus_int @ A2 @ B ) )
% 14.25/8.11 = ( ord_less_int @ zero_zero_int @ B ) ) ).
% 14.25/8.11
% 14.25/8.11 thf(fact_5510_less__add__same__cancel1,axiom,
% 14.25/8.11 ! [A2: code_integer,B: code_integer] :
% 14.25/8.11 ( ( ord_le6747313008572928689nteger @ A2 @ ( plus_p5714425477246183910nteger @ A2 @ B ) )
% 14.25/8.11 = ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ B ) ) ).
% 14.25/8.11
% 14.25/8.11 thf(fact_5511_less__add__same__cancel1,axiom,
% 14.25/8.11 ! [A2: rat,B: rat] :
% 14.25/8.11 ( ( ord_less_rat @ A2 @ ( plus_plus_rat @ A2 @ B ) )
% 14.25/8.11 = ( ord_less_rat @ zero_zero_rat @ B ) ) ).
% 14.25/8.11
% 14.25/8.11 thf(fact_5512_add__less__same__cancel2,axiom,
% 14.25/8.11 ! [A2: nat,B: nat] :
% 14.25/8.11 ( ( ord_less_nat @ ( plus_plus_nat @ A2 @ B ) @ B )
% 14.25/8.11 = ( ord_less_nat @ A2 @ zero_zero_nat ) ) ).
% 14.25/8.11
% 14.25/8.11 thf(fact_5513_add__less__same__cancel2,axiom,
% 14.25/8.11 ! [A2: int,B: int] :
% 14.25/8.11 ( ( ord_less_int @ ( plus_plus_int @ A2 @ B ) @ B )
% 14.25/8.11 = ( ord_less_int @ A2 @ zero_zero_int ) ) ).
% 14.25/8.11
% 14.25/8.11 thf(fact_5514_add__less__same__cancel2,axiom,
% 14.25/8.11 ! [A2: code_integer,B: code_integer] :
% 14.25/8.11 ( ( ord_le6747313008572928689nteger @ ( plus_p5714425477246183910nteger @ A2 @ B ) @ B )
% 14.25/8.11 = ( ord_le6747313008572928689nteger @ A2 @ zero_z3403309356797280102nteger ) ) ).
% 14.25/8.11
% 14.25/8.11 thf(fact_5515_add__less__same__cancel2,axiom,
% 14.25/8.11 ! [A2: rat,B: rat] :
% 14.25/8.11 ( ( ord_less_rat @ ( plus_plus_rat @ A2 @ B ) @ B )
% 14.25/8.11 = ( ord_less_rat @ A2 @ zero_zero_rat ) ) ).
% 14.25/8.11
% 14.25/8.11 thf(fact_5516_add__less__same__cancel1,axiom,
% 14.25/8.11 ! [B: nat,A2: nat] :
% 14.25/8.11 ( ( ord_less_nat @ ( plus_plus_nat @ B @ A2 ) @ B )
% 14.25/8.11 = ( ord_less_nat @ A2 @ zero_zero_nat ) ) ).
% 14.25/8.11
% 14.25/8.11 thf(fact_5517_add__less__same__cancel1,axiom,
% 14.25/8.11 ! [B: int,A2: int] :
% 14.25/8.11 ( ( ord_less_int @ ( plus_plus_int @ B @ A2 ) @ B )
% 14.25/8.11 = ( ord_less_int @ A2 @ zero_zero_int ) ) ).
% 14.25/8.11
% 14.25/8.11 thf(fact_5518_add__less__same__cancel1,axiom,
% 14.25/8.11 ! [B: code_integer,A2: code_integer] :
% 14.25/8.11 ( ( ord_le6747313008572928689nteger @ ( plus_p5714425477246183910nteger @ B @ A2 ) @ B )
% 14.25/8.11 = ( ord_le6747313008572928689nteger @ A2 @ zero_z3403309356797280102nteger ) ) ).
% 14.25/8.11
% 14.25/8.11 thf(fact_5519_add__less__same__cancel1,axiom,
% 14.25/8.11 ! [B: rat,A2: rat] :
% 14.25/8.11 ( ( ord_less_rat @ ( plus_plus_rat @ B @ A2 ) @ B )
% 14.25/8.11 = ( ord_less_rat @ A2 @ zero_zero_rat ) ) ).
% 14.25/8.11
% 14.25/8.11 thf(fact_5520_zero__comp__diff__simps_I2_J,axiom,
% 14.25/8.11 ! [A2: int,B: int] :
% 14.25/8.11 ( ( ord_less_int @ zero_zero_int @ ( minus_minus_int @ A2 @ B ) )
% 14.25/8.11 = ( ord_less_int @ B @ A2 ) ) ).
% 14.25/8.11
% 14.25/8.11 thf(fact_5521_zero__comp__diff__simps_I2_J,axiom,
% 14.25/8.11 ! [A2: code_integer,B: code_integer] :
% 14.25/8.11 ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ ( minus_8373710615458151222nteger @ A2 @ B ) )
% 14.25/8.11 = ( ord_le6747313008572928689nteger @ B @ A2 ) ) ).
% 14.25/8.11
% 14.25/8.11 thf(fact_5522_zero__comp__diff__simps_I2_J,axiom,
% 14.25/8.11 ! [A2: rat,B: rat] :
% 14.25/8.11 ( ( ord_less_rat @ zero_zero_rat @ ( minus_minus_rat @ A2 @ B ) )
% 14.25/8.11 = ( ord_less_rat @ B @ A2 ) ) ).
% 14.25/8.11
% 14.25/8.11 thf(fact_5523_diff__gt__0__iff__gt,axiom,
% 14.25/8.11 ! [A2: int,B: int] :
% 14.25/8.11 ( ( ord_less_int @ zero_zero_int @ ( minus_minus_int @ A2 @ B ) )
% 14.25/8.11 = ( ord_less_int @ B @ A2 ) ) ).
% 14.25/8.11
% 14.25/8.11 thf(fact_5524_diff__gt__0__iff__gt,axiom,
% 14.25/8.11 ! [A2: code_integer,B: code_integer] :
% 14.25/8.11 ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ ( minus_8373710615458151222nteger @ A2 @ B ) )
% 14.25/8.11 = ( ord_le6747313008572928689nteger @ B @ A2 ) ) ).
% 14.25/8.11
% 14.25/8.11 thf(fact_5525_diff__gt__0__iff__gt,axiom,
% 14.25/8.11 ! [A2: rat,B: rat] :
% 14.25/8.11 ( ( ord_less_rat @ zero_zero_rat @ ( minus_minus_rat @ A2 @ B ) )
% 14.25/8.11 = ( ord_less_rat @ B @ A2 ) ) ).
% 14.25/8.11
% 14.25/8.11 thf(fact_5526_diff__add__zero,axiom,
% 14.25/8.11 ! [A2: nat,B: nat] :
% 14.25/8.11 ( ( minus_minus_nat @ A2 @ ( plus_plus_nat @ A2 @ B ) )
% 14.25/8.11 = zero_zero_nat ) ).
% 14.25/8.11
% 14.25/8.11 thf(fact_5527_mult__cancel__right2,axiom,
% 14.25/8.11 ! [A2: code_integer,C: code_integer] :
% 14.25/8.11 ( ( ( times_3573771949741848930nteger @ A2 @ C )
% 14.25/8.11 = C )
% 14.25/8.11 = ( ( C = zero_z3403309356797280102nteger )
% 14.25/8.11 | ( A2 = one_one_Code_integer ) ) ) ).
% 14.25/8.11
% 14.25/8.11 thf(fact_5528_mult__cancel__right2,axiom,
% 14.25/8.11 ! [A2: rat,C: rat] :
% 14.25/8.11 ( ( ( times_times_rat @ A2 @ C )
% 14.25/8.11 = C )
% 14.25/8.11 = ( ( C = zero_zero_rat )
% 14.25/8.11 | ( A2 = one_one_rat ) ) ) ).
% 14.25/8.11
% 14.25/8.11 thf(fact_5529_mult__cancel__right2,axiom,
% 14.25/8.11 ! [A2: int,C: int] :
% 14.25/8.11 ( ( ( times_times_int @ A2 @ C )
% 14.25/8.11 = C )
% 14.25/8.11 = ( ( C = zero_zero_int )
% 14.25/8.11 | ( A2 = one_one_int ) ) ) ).
% 14.25/8.11
% 14.25/8.11 thf(fact_5530_mult__cancel__right1,axiom,
% 14.25/8.11 ! [C: code_integer,B: code_integer] :
% 14.25/8.11 ( ( C
% 14.25/8.11 = ( times_3573771949741848930nteger @ B @ C ) )
% 14.25/8.11 = ( ( C = zero_z3403309356797280102nteger )
% 14.25/8.11 | ( B = one_one_Code_integer ) ) ) ).
% 14.25/8.11
% 14.25/8.11 thf(fact_5531_mult__cancel__right1,axiom,
% 14.25/8.11 ! [C: rat,B: rat] :
% 14.25/8.11 ( ( C
% 14.25/8.11 = ( times_times_rat @ B @ C ) )
% 14.25/8.11 = ( ( C = zero_zero_rat )
% 14.25/8.11 | ( B = one_one_rat ) ) ) ).
% 14.25/8.11
% 14.25/8.11 thf(fact_5532_mult__cancel__right1,axiom,
% 14.25/8.11 ! [C: int,B: int] :
% 14.25/8.11 ( ( C
% 14.25/8.11 = ( times_times_int @ B @ C ) )
% 14.25/8.11 = ( ( C = zero_zero_int )
% 14.25/8.11 | ( B = one_one_int ) ) ) ).
% 14.25/8.11
% 14.25/8.11 thf(fact_5533_mult__cancel__left2,axiom,
% 14.25/8.11 ! [C: code_integer,A2: code_integer] :
% 14.25/8.11 ( ( ( times_3573771949741848930nteger @ C @ A2 )
% 14.25/8.11 = C )
% 14.25/8.11 = ( ( C = zero_z3403309356797280102nteger )
% 14.25/8.11 | ( A2 = one_one_Code_integer ) ) ) ).
% 14.25/8.11
% 14.25/8.11 thf(fact_5534_mult__cancel__left2,axiom,
% 14.25/8.11 ! [C: rat,A2: rat] :
% 14.25/8.11 ( ( ( times_times_rat @ C @ A2 )
% 14.25/8.11 = C )
% 14.25/8.11 = ( ( C = zero_zero_rat )
% 14.25/8.11 | ( A2 = one_one_rat ) ) ) ).
% 14.25/8.11
% 14.25/8.11 thf(fact_5535_mult__cancel__left2,axiom,
% 14.25/8.11 ! [C: int,A2: int] :
% 14.25/8.11 ( ( ( times_times_int @ C @ A2 )
% 14.25/8.11 = C )
% 14.25/8.11 = ( ( C = zero_zero_int )
% 14.25/8.11 | ( A2 = one_one_int ) ) ) ).
% 14.25/8.11
% 14.25/8.11 thf(fact_5536_mult__cancel__left1,axiom,
% 14.25/8.11 ! [C: code_integer,B: code_integer] :
% 14.25/8.11 ( ( C
% 14.25/8.11 = ( times_3573771949741848930nteger @ C @ B ) )
% 14.25/8.11 = ( ( C = zero_z3403309356797280102nteger )
% 14.25/8.11 | ( B = one_one_Code_integer ) ) ) ).
% 14.25/8.11
% 14.25/8.11 thf(fact_5537_mult__cancel__left1,axiom,
% 14.25/8.11 ! [C: rat,B: rat] :
% 14.25/8.11 ( ( C
% 14.25/8.11 = ( times_times_rat @ C @ B ) )
% 14.25/8.11 = ( ( C = zero_zero_rat )
% 14.25/8.11 | ( B = one_one_rat ) ) ) ).
% 14.25/8.11
% 14.25/8.11 thf(fact_5538_mult__cancel__left1,axiom,
% 14.25/8.11 ! [C: int,B: int] :
% 14.25/8.11 ( ( C
% 14.25/8.11 = ( times_times_int @ C @ B ) )
% 14.25/8.11 = ( ( C = zero_zero_int )
% 14.25/8.11 | ( B = one_one_int ) ) ) ).
% 14.25/8.11
% 14.25/8.11 thf(fact_5539_image__add__0,axiom,
% 14.25/8.11 ! [S: set_rat] :
% 14.25/8.11 ( ( image_rat_rat @ ( plus_plus_rat @ zero_zero_rat ) @ S )
% 14.25/8.11 = S ) ).
% 14.25/8.11
% 14.25/8.11 thf(fact_5540_image__add__0,axiom,
% 14.25/8.11 ! [S: set_nat] :
% 14.25/8.11 ( ( image_nat_nat2 @ ( plus_plus_nat @ zero_zero_nat ) @ S )
% 14.25/8.11 = S ) ).
% 14.25/8.11
% 14.25/8.11 thf(fact_5541_image__add__0,axiom,
% 14.25/8.11 ! [S: set_int] :
% 14.25/8.11 ( ( image_int_int2 @ ( plus_plus_int @ zero_zero_int ) @ S )
% 14.25/8.11 = S ) ).
% 14.25/8.11
% 14.25/8.11 thf(fact_5542_image__add__0,axiom,
% 14.25/8.11 ! [S: set_Code_integer] :
% 14.25/8.11 ( ( image_4470545334726330049nteger @ ( plus_p5714425477246183910nteger @ zero_z3403309356797280102nteger ) @ S )
% 14.25/8.11 = S ) ).
% 14.25/8.11
% 14.25/8.11 thf(fact_5543_diff__numeral__special_I9_J,axiom,
% 14.25/8.11 ( ( minus_8373710615458151222nteger @ one_one_Code_integer @ one_one_Code_integer )
% 14.25/8.11 = zero_z3403309356797280102nteger ) ).
% 14.25/8.11
% 14.25/8.11 thf(fact_5544_diff__numeral__special_I9_J,axiom,
% 14.25/8.11 ( ( minus_minus_rat @ one_one_rat @ one_one_rat )
% 14.25/8.11 = zero_zero_rat ) ).
% 14.25/8.11
% 14.25/8.11 thf(fact_5545_diff__numeral__special_I9_J,axiom,
% 14.25/8.11 ( ( minus_minus_int @ one_one_int @ one_one_int )
% 14.25/8.11 = zero_zero_int ) ).
% 14.25/8.11
% 14.25/8.11 thf(fact_5546_max__0__1_I2_J,axiom,
% 14.25/8.11 ( ( ord_max_Code_natural @ one_one_Code_natural @ zero_z2226904508553997617atural )
% 14.25/8.11 = one_one_Code_natural ) ).
% 14.25/8.11
% 14.25/8.11 thf(fact_5547_max__0__1_I2_J,axiom,
% 14.25/8.11 ( ( ord_max_Code_integer @ one_one_Code_integer @ zero_z3403309356797280102nteger )
% 14.25/8.11 = one_one_Code_integer ) ).
% 14.25/8.11
% 14.25/8.11 thf(fact_5548_max__0__1_I2_J,axiom,
% 14.25/8.11 ( ( ord_max_nat @ one_one_nat @ zero_zero_nat )
% 14.25/8.11 = one_one_nat ) ).
% 14.25/8.11
% 14.25/8.11 thf(fact_5549_max__0__1_I2_J,axiom,
% 14.25/8.11 ( ( ord_max_int @ one_one_int @ zero_zero_int )
% 14.25/8.11 = one_one_int ) ).
% 14.25/8.11
% 14.25/8.11 thf(fact_5550_max__0__1_I1_J,axiom,
% 14.25/8.11 ( ( ord_max_Code_natural @ zero_z2226904508553997617atural @ one_one_Code_natural )
% 14.25/8.11 = one_one_Code_natural ) ).
% 14.25/8.11
% 14.25/8.11 thf(fact_5551_max__0__1_I1_J,axiom,
% 14.25/8.11 ( ( ord_max_Code_integer @ zero_z3403309356797280102nteger @ one_one_Code_integer )
% 14.25/8.11 = one_one_Code_integer ) ).
% 14.25/8.11
% 14.25/8.11 thf(fact_5552_max__0__1_I1_J,axiom,
% 14.25/8.11 ( ( ord_max_nat @ zero_zero_nat @ one_one_nat )
% 14.25/8.11 = one_one_nat ) ).
% 14.25/8.11
% 14.25/8.11 thf(fact_5553_max__0__1_I1_J,axiom,
% 14.25/8.11 ( ( ord_max_int @ zero_zero_int @ one_one_int )
% 14.25/8.11 = one_one_int ) ).
% 14.25/8.11
% 14.25/8.11 thf(fact_5554_card_Oempty,axiom,
% 14.25/8.11 ( ( finite410649719033368117t_unit @ bot_bo3957492148770167129t_unit )
% 14.25/8.11 = zero_zero_nat ) ).
% 14.25/8.11
% 14.25/8.11 thf(fact_5555_card_Oempty,axiom,
% 14.25/8.11 ( ( finite_card_o @ bot_bot_set_o )
% 14.25/8.11 = zero_zero_nat ) ).
% 14.25/8.11
% 14.25/8.11 thf(fact_5556_card_Oempty,axiom,
% 14.25/8.11 ( ( finite_card_nat @ bot_bot_set_nat )
% 14.25/8.11 = zero_zero_nat ) ).
% 14.25/8.11
% 14.25/8.11 thf(fact_5557_card_Oempty,axiom,
% 14.25/8.11 ( ( finite_card_int @ bot_bot_set_int )
% 14.25/8.11 = zero_zero_nat ) ).
% 14.25/8.11
% 14.25/8.11 thf(fact_5558_card_Oempty,axiom,
% 14.25/8.11 ( ( finite1363419556375932405_nat_o @ bot_bo7824918357723582233_nat_o )
% 14.25/8.11 = zero_zero_nat ) ).
% 14.25/8.11
% 14.25/8.11 thf(fact_5559_one__eq__mult__iff,axiom,
% 14.25/8.11 ! [M4: nat,N2: nat] :
% 14.25/8.11 ( ( ( suc @ zero_zero_nat )
% 14.25/8.11 = ( times_times_nat @ M4 @ N2 ) )
% 14.25/8.11 = ( ( M4
% 14.25/8.11 = ( suc @ zero_zero_nat ) )
% 14.25/8.11 & ( N2
% 14.25/8.11 = ( suc @ zero_zero_nat ) ) ) ) ).
% 14.25/8.11
% 14.25/8.11 thf(fact_5560_mult__eq__1__iff,axiom,
% 14.25/8.11 ! [M4: nat,N2: nat] :
% 14.25/8.11 ( ( ( times_times_nat @ M4 @ N2 )
% 14.25/8.11 = ( suc @ zero_zero_nat ) )
% 14.25/8.11 = ( ( M4
% 14.25/8.11 = ( suc @ zero_zero_nat ) )
% 14.25/8.11 & ( N2
% 14.25/8.11 = ( suc @ zero_zero_nat ) ) ) ) ).
% 14.25/8.11
% 14.25/8.11 thf(fact_5561_nat__mult__less__cancel__disj,axiom,
% 14.25/8.11 ! [K3: nat,M4: nat,N2: nat] :
% 14.25/8.11 ( ( ord_less_nat @ ( times_times_nat @ K3 @ M4 ) @ ( times_times_nat @ K3 @ N2 ) )
% 14.25/8.11 = ( ( ord_less_nat @ zero_zero_nat @ K3 )
% 14.25/8.11 & ( ord_less_nat @ M4 @ N2 ) ) ) ).
% 14.25/8.11
% 14.25/8.11 thf(fact_5562_nat__0__less__mult__iff,axiom,
% 14.25/8.11 ! [M4: nat,N2: nat] :
% 14.25/8.11 ( ( ord_less_nat @ zero_zero_nat @ ( times_times_nat @ M4 @ N2 ) )
% 14.25/8.11 = ( ( ord_less_nat @ zero_zero_nat @ M4 )
% 14.25/8.11 & ( ord_less_nat @ zero_zero_nat @ N2 ) ) ) ).
% 14.25/8.11
% 14.25/8.11 thf(fact_5563_mult__less__cancel2,axiom,
% 14.25/8.11 ! [M4: nat,K3: nat,N2: nat] :
% 14.25/8.11 ( ( ord_less_nat @ ( times_times_nat @ M4 @ K3 ) @ ( times_times_nat @ N2 @ K3 ) )
% 14.25/8.11 = ( ( ord_less_nat @ zero_zero_nat @ K3 )
% 14.25/8.11 & ( ord_less_nat @ M4 @ N2 ) ) ) ).
% 14.25/8.11
% 14.25/8.11 thf(fact_5564_card__0__eq,axiom,
% 14.25/8.11 ! [A: set_Code_integer] :
% 14.25/8.11 ( ( finite6017078050557962740nteger @ A )
% 14.25/8.11 => ( ( ( finite4902975817058060853nteger @ A )
% 14.25/8.11 = zero_zero_nat )
% 14.25/8.11 = ( A = bot_bo3990330152332043303nteger ) ) ) ).
% 14.25/8.11
% 14.25/8.11 thf(fact_5565_card__0__eq,axiom,
% 14.25/8.11 ! [A: set_Product_unit] :
% 14.25/8.11 ( ( finite4290736615968046902t_unit @ A )
% 14.25/8.11 => ( ( ( finite410649719033368117t_unit @ A )
% 14.25/8.11 = zero_zero_nat )
% 14.25/8.11 = ( A = bot_bo3957492148770167129t_unit ) ) ) ).
% 14.25/8.11
% 14.25/8.11 thf(fact_5566_card__0__eq,axiom,
% 14.25/8.11 ! [A: set_o] :
% 14.25/8.11 ( ( finite_finite_o @ A )
% 14.25/8.11 => ( ( ( finite_card_o @ A )
% 14.25/8.11 = zero_zero_nat )
% 14.25/8.11 = ( A = bot_bot_set_o ) ) ) ).
% 14.25/8.11
% 14.25/8.11 thf(fact_5567_card__0__eq,axiom,
% 14.25/8.11 ! [A: set_nat] :
% 14.25/8.11 ( ( finite_finite_nat @ A )
% 14.25/8.11 => ( ( ( finite_card_nat @ A )
% 14.25/8.11 = zero_zero_nat )
% 14.25/8.11 = ( A = bot_bot_set_nat ) ) ) ).
% 14.25/8.11
% 14.25/8.11 thf(fact_5568_card__0__eq,axiom,
% 14.25/8.11 ! [A: set_int] :
% 14.25/8.11 ( ( finite_finite_int @ A )
% 14.25/8.11 => ( ( ( finite_card_int @ A )
% 14.25/8.11 = zero_zero_nat )
% 14.25/8.11 = ( A = bot_bot_set_int ) ) ) ).
% 14.25/8.11
% 14.25/8.11 thf(fact_5569_card__0__eq,axiom,
% 14.25/8.11 ! [A: set_Pr4532377907799695533_nat_o] :
% 14.25/8.11 ( ( finite3252695134891459830_nat_o @ A )
% 14.25/8.11 => ( ( ( finite1363419556375932405_nat_o @ A )
% 14.25/8.11 = zero_zero_nat )
% 14.25/8.11 = ( A = bot_bo7824918357723582233_nat_o ) ) ) ).
% 14.25/8.11
% 14.25/8.11 thf(fact_5570_one__le__mult__iff,axiom,
% 14.25/8.11 ! [M4: nat,N2: nat] :
% 14.25/8.11 ( ( ord_less_eq_nat @ ( suc @ zero_zero_nat ) @ ( times_times_nat @ M4 @ N2 ) )
% 14.25/8.11 = ( ( ord_less_eq_nat @ ( suc @ zero_zero_nat ) @ M4 )
% 14.25/8.11 & ( ord_less_eq_nat @ ( suc @ zero_zero_nat ) @ N2 ) ) ) ).
% 14.25/8.11
% 14.25/8.11 thf(fact_5571_mult__le__cancel2,axiom,
% 14.25/8.11 ! [M4: nat,K3: nat,N2: nat] :
% 14.25/8.11 ( ( ord_less_eq_nat @ ( times_times_nat @ M4 @ K3 ) @ ( times_times_nat @ N2 @ K3 ) )
% 14.25/8.11 = ( ( ord_less_nat @ zero_zero_nat @ K3 )
% 14.25/8.11 => ( ord_less_eq_nat @ M4 @ N2 ) ) ) ).
% 14.25/8.11
% 14.25/8.11 thf(fact_5572_nat__mult__le__cancel__disj,axiom,
% 14.25/8.11 ! [K3: nat,M4: nat,N2: nat] :
% 14.25/8.11 ( ( ord_less_eq_nat @ ( times_times_nat @ K3 @ M4 ) @ ( times_times_nat @ K3 @ N2 ) )
% 14.25/8.11 = ( ( ord_less_nat @ zero_zero_nat @ K3 )
% 14.25/8.11 => ( ord_less_eq_nat @ M4 @ N2 ) ) ) ).
% 14.25/8.11
% 14.25/8.11 thf(fact_5573_bot__nat__def,axiom,
% 14.25/8.11 bot_bot_nat = zero_zero_nat ).
% 14.25/8.11
% 14.25/8.11 thf(fact_5574_zero__reorient,axiom,
% 14.25/8.11 ! [X: nat] :
% 14.25/8.11 ( ( zero_zero_nat = X )
% 14.25/8.11 = ( X = zero_zero_nat ) ) ).
% 14.25/8.11
% 14.25/8.11 thf(fact_5575_zero__reorient,axiom,
% 14.25/8.11 ! [X: int] :
% 14.25/8.11 ( ( zero_zero_int = X )
% 14.25/8.11 = ( X = zero_zero_int ) ) ).
% 14.25/8.11
% 14.25/8.11 thf(fact_5576_zero__reorient,axiom,
% 14.25/8.11 ! [X: code_integer] :
% 14.25/8.11 ( ( zero_z3403309356797280102nteger = X )
% 14.25/8.11 = ( X = zero_z3403309356797280102nteger ) ) ).
% 14.25/8.11
% 14.25/8.11 thf(fact_5577_zero__le,axiom,
% 14.25/8.11 ! [X: nat] : ( ord_less_eq_nat @ zero_zero_nat @ X ) ).
% 14.25/8.11
% 14.25/8.11 thf(fact_5578_gr__zeroI,axiom,
% 14.25/8.11 ! [N2: nat] :
% 14.25/8.11 ( ( N2 != zero_zero_nat )
% 14.25/8.11 => ( ord_less_nat @ zero_zero_nat @ N2 ) ) ).
% 14.25/8.11
% 14.25/8.11 thf(fact_5579_not__less__zero,axiom,
% 14.25/8.11 ! [N2: nat] :
% 14.25/8.11 ~ ( ord_less_nat @ N2 @ zero_zero_nat ) ).
% 14.25/8.11
% 14.25/8.11 thf(fact_5580_gr__implies__not__zero,axiom,
% 14.25/8.11 ! [M4: nat,N2: nat] :
% 14.25/8.11 ( ( ord_less_nat @ M4 @ N2 )
% 14.25/8.11 => ( N2 != zero_zero_nat ) ) ).
% 14.25/8.11
% 14.25/8.11 thf(fact_5581_zero__less__iff__neq__zero,axiom,
% 14.25/8.11 ! [N2: nat] :
% 14.25/8.11 ( ( ord_less_nat @ zero_zero_nat @ N2 )
% 14.25/8.11 = ( N2 != zero_zero_nat ) ) ).
% 14.25/8.11
% 14.25/8.11 thf(fact_5582_comm__monoid__add__class_Oadd__0,axiom,
% 14.25/8.11 ! [A2: rat] :
% 14.25/8.11 ( ( plus_plus_rat @ zero_zero_rat @ A2 )
% 14.25/8.11 = A2 ) ).
% 14.25/8.11
% 14.25/8.11 thf(fact_5583_comm__monoid__add__class_Oadd__0,axiom,
% 14.25/8.11 ! [A2: nat] :
% 14.25/8.11 ( ( plus_plus_nat @ zero_zero_nat @ A2 )
% 14.25/8.11 = A2 ) ).
% 14.25/8.11
% 14.25/8.11 thf(fact_5584_comm__monoid__add__class_Oadd__0,axiom,
% 14.25/8.11 ! [A2: int] :
% 14.25/8.11 ( ( plus_plus_int @ zero_zero_int @ A2 )
% 14.25/8.11 = A2 ) ).
% 14.25/8.11
% 14.25/8.11 thf(fact_5585_comm__monoid__add__class_Oadd__0,axiom,
% 14.25/8.11 ! [A2: code_integer] :
% 14.25/8.11 ( ( plus_p5714425477246183910nteger @ zero_z3403309356797280102nteger @ A2 )
% 14.25/8.11 = A2 ) ).
% 14.25/8.11
% 14.25/8.11 thf(fact_5586_add_Ocomm__neutral,axiom,
% 14.25/8.11 ! [A2: rat] :
% 14.25/8.11 ( ( plus_plus_rat @ A2 @ zero_zero_rat )
% 14.25/8.11 = A2 ) ).
% 14.25/8.11
% 14.25/8.11 thf(fact_5587_add_Ocomm__neutral,axiom,
% 14.25/8.11 ! [A2: nat] :
% 14.25/8.11 ( ( plus_plus_nat @ A2 @ zero_zero_nat )
% 14.25/8.11 = A2 ) ).
% 14.25/8.11
% 14.25/8.11 thf(fact_5588_add_Ocomm__neutral,axiom,
% 14.25/8.11 ! [A2: int] :
% 14.25/8.11 ( ( plus_plus_int @ A2 @ zero_zero_int )
% 14.25/8.11 = A2 ) ).
% 14.25/8.11
% 14.25/8.11 thf(fact_5589_add_Ocomm__neutral,axiom,
% 14.25/8.11 ! [A2: code_integer] :
% 14.25/8.11 ( ( plus_p5714425477246183910nteger @ A2 @ zero_z3403309356797280102nteger )
% 14.25/8.11 = A2 ) ).
% 14.25/8.11
% 14.25/8.11 thf(fact_5590_add_Ogroup__left__neutral,axiom,
% 14.25/8.11 ! [A2: rat] :
% 14.25/8.11 ( ( plus_plus_rat @ zero_zero_rat @ A2 )
% 14.25/8.11 = A2 ) ).
% 14.25/8.11
% 14.25/8.11 thf(fact_5591_add_Ogroup__left__neutral,axiom,
% 14.25/8.11 ! [A2: int] :
% 14.25/8.11 ( ( plus_plus_int @ zero_zero_int @ A2 )
% 14.25/8.11 = A2 ) ).
% 14.25/8.11
% 14.25/8.11 thf(fact_5592_add_Ogroup__left__neutral,axiom,
% 14.25/8.11 ! [A2: code_integer] :
% 14.25/8.11 ( ( plus_p5714425477246183910nteger @ zero_z3403309356797280102nteger @ A2 )
% 14.25/8.11 = A2 ) ).
% 14.25/8.11
% 14.25/8.11 thf(fact_5593_mult__right__cancel,axiom,
% 14.25/8.11 ! [C: code_integer,A2: code_integer,B: code_integer] :
% 14.25/8.11 ( ( C != zero_z3403309356797280102nteger )
% 14.25/8.11 => ( ( ( times_3573771949741848930nteger @ A2 @ C )
% 14.25/8.11 = ( times_3573771949741848930nteger @ B @ C ) )
% 14.25/8.11 = ( A2 = B ) ) ) ).
% 14.25/8.11
% 14.25/8.11 thf(fact_5594_mult__right__cancel,axiom,
% 14.25/8.11 ! [C: rat,A2: rat,B: rat] :
% 14.25/8.11 ( ( C != zero_zero_rat )
% 14.25/8.11 => ( ( ( times_times_rat @ A2 @ C )
% 14.25/8.11 = ( times_times_rat @ B @ C ) )
% 14.25/8.11 = ( A2 = B ) ) ) ).
% 14.25/8.11
% 14.25/8.11 thf(fact_5595_mult__right__cancel,axiom,
% 14.25/8.11 ! [C: nat,A2: nat,B: nat] :
% 14.25/8.11 ( ( C != zero_zero_nat )
% 14.25/8.11 => ( ( ( times_times_nat @ A2 @ C )
% 14.25/8.11 = ( times_times_nat @ B @ C ) )
% 14.25/8.11 = ( A2 = B ) ) ) ).
% 14.25/8.11
% 14.25/8.11 thf(fact_5596_mult__right__cancel,axiom,
% 14.25/8.11 ! [C: int,A2: int,B: int] :
% 14.25/8.11 ( ( C != zero_zero_int )
% 14.25/8.11 => ( ( ( times_times_int @ A2 @ C )
% 14.25/8.11 = ( times_times_int @ B @ C ) )
% 14.25/8.11 = ( A2 = B ) ) ) ).
% 14.25/8.11
% 14.25/8.11 thf(fact_5597_mult__left__cancel,axiom,
% 14.25/8.11 ! [C: code_integer,A2: code_integer,B: code_integer] :
% 14.25/8.11 ( ( C != zero_z3403309356797280102nteger )
% 14.25/8.11 => ( ( ( times_3573771949741848930nteger @ C @ A2 )
% 14.25/8.11 = ( times_3573771949741848930nteger @ C @ B ) )
% 14.25/8.11 = ( A2 = B ) ) ) ).
% 14.25/8.11
% 14.25/8.11 thf(fact_5598_mult__left__cancel,axiom,
% 14.25/8.11 ! [C: rat,A2: rat,B: rat] :
% 14.25/8.11 ( ( C != zero_zero_rat )
% 14.25/8.11 => ( ( ( times_times_rat @ C @ A2 )
% 14.25/8.11 = ( times_times_rat @ C @ B ) )
% 14.25/8.11 = ( A2 = B ) ) ) ).
% 14.25/8.11
% 14.25/8.11 thf(fact_5599_mult__left__cancel,axiom,
% 14.25/8.11 ! [C: nat,A2: nat,B: nat] :
% 14.25/8.11 ( ( C != zero_zero_nat )
% 14.25/8.11 => ( ( ( times_times_nat @ C @ A2 )
% 14.25/8.11 = ( times_times_nat @ C @ B ) )
% 14.25/8.11 = ( A2 = B ) ) ) ).
% 14.25/8.11
% 14.25/8.11 thf(fact_5600_mult__left__cancel,axiom,
% 14.25/8.11 ! [C: int,A2: int,B: int] :
% 14.25/8.11 ( ( C != zero_zero_int )
% 14.25/8.11 => ( ( ( times_times_int @ C @ A2 )
% 14.25/8.11 = ( times_times_int @ C @ B ) )
% 14.25/8.11 = ( A2 = B ) ) ) ).
% 14.25/8.11
% 14.25/8.11 thf(fact_5601_no__zero__divisors,axiom,
% 14.25/8.11 ! [A2: code_integer,B: code_integer] :
% 14.25/8.11 ( ( A2 != zero_z3403309356797280102nteger )
% 14.25/8.11 => ( ( B != zero_z3403309356797280102nteger )
% 14.25/8.11 => ( ( times_3573771949741848930nteger @ A2 @ B )
% 14.25/8.11 != zero_z3403309356797280102nteger ) ) ) ).
% 14.25/8.11
% 14.25/8.11 thf(fact_5602_no__zero__divisors,axiom,
% 14.25/8.11 ! [A2: rat,B: rat] :
% 14.25/8.11 ( ( A2 != zero_zero_rat )
% 14.25/8.11 => ( ( B != zero_zero_rat )
% 14.25/8.11 => ( ( times_times_rat @ A2 @ B )
% 14.25/8.11 != zero_zero_rat ) ) ) ).
% 14.25/8.11
% 14.25/8.11 thf(fact_5603_no__zero__divisors,axiom,
% 14.25/8.11 ! [A2: nat,B: nat] :
% 14.25/8.11 ( ( A2 != zero_zero_nat )
% 14.25/8.11 => ( ( B != zero_zero_nat )
% 14.25/8.11 => ( ( times_times_nat @ A2 @ B )
% 14.25/8.11 != zero_zero_nat ) ) ) ).
% 14.25/8.11
% 14.25/8.11 thf(fact_5604_no__zero__divisors,axiom,
% 14.25/8.11 ! [A2: int,B: int] :
% 14.25/8.11 ( ( A2 != zero_zero_int )
% 14.25/8.11 => ( ( B != zero_zero_int )
% 14.25/8.11 => ( ( times_times_int @ A2 @ B )
% 14.25/8.11 != zero_zero_int ) ) ) ).
% 14.25/8.11
% 14.25/8.11 thf(fact_5605_divisors__zero,axiom,
% 14.25/8.11 ! [A2: code_integer,B: code_integer] :
% 14.25/8.11 ( ( ( times_3573771949741848930nteger @ A2 @ B )
% 14.25/8.11 = zero_z3403309356797280102nteger )
% 14.25/8.11 => ( ( A2 = zero_z3403309356797280102nteger )
% 14.25/8.11 | ( B = zero_z3403309356797280102nteger ) ) ) ).
% 14.25/8.11
% 14.25/8.11 thf(fact_5606_divisors__zero,axiom,
% 14.25/8.11 ! [A2: rat,B: rat] :
% 14.25/8.11 ( ( ( times_times_rat @ A2 @ B )
% 14.25/8.11 = zero_zero_rat )
% 14.25/8.11 => ( ( A2 = zero_zero_rat )
% 14.25/8.11 | ( B = zero_zero_rat ) ) ) ).
% 14.25/8.11
% 14.25/8.11 thf(fact_5607_divisors__zero,axiom,
% 14.25/8.11 ! [A2: nat,B: nat] :
% 14.25/8.11 ( ( ( times_times_nat @ A2 @ B )
% 14.25/8.11 = zero_zero_nat )
% 14.25/8.11 => ( ( A2 = zero_zero_nat )
% 14.25/8.11 | ( B = zero_zero_nat ) ) ) ).
% 14.25/8.11
% 14.25/8.11 thf(fact_5608_divisors__zero,axiom,
% 14.25/8.11 ! [A2: int,B: int] :
% 14.25/8.11 ( ( ( times_times_int @ A2 @ B )
% 14.25/8.11 = zero_zero_int )
% 14.25/8.11 => ( ( A2 = zero_zero_int )
% 14.25/8.11 | ( B = zero_zero_int ) ) ) ).
% 14.25/8.11
% 14.25/8.11 thf(fact_5609_mult__not__zero,axiom,
% 14.25/8.11 ! [A2: code_integer,B: code_integer] :
% 14.25/8.11 ( ( ( times_3573771949741848930nteger @ A2 @ B )
% 14.25/8.11 != zero_z3403309356797280102nteger )
% 14.25/8.11 => ( ( A2 != zero_z3403309356797280102nteger )
% 14.25/8.11 & ( B != zero_z3403309356797280102nteger ) ) ) ).
% 14.25/8.11
% 14.25/8.11 thf(fact_5610_mult__not__zero,axiom,
% 14.25/8.11 ! [A2: rat,B: rat] :
% 14.25/8.11 ( ( ( times_times_rat @ A2 @ B )
% 14.25/8.11 != zero_zero_rat )
% 14.25/8.11 => ( ( A2 != zero_zero_rat )
% 14.25/8.11 & ( B != zero_zero_rat ) ) ) ).
% 14.25/8.11
% 14.25/8.11 thf(fact_5611_mult__not__zero,axiom,
% 14.25/8.11 ! [A2: nat,B: nat] :
% 14.25/8.11 ( ( ( times_times_nat @ A2 @ B )
% 14.25/8.11 != zero_zero_nat )
% 14.25/8.11 => ( ( A2 != zero_zero_nat )
% 14.25/8.11 & ( B != zero_zero_nat ) ) ) ).
% 14.25/8.11
% 14.25/8.11 thf(fact_5612_mult__not__zero,axiom,
% 14.25/8.11 ! [A2: int,B: int] :
% 14.25/8.11 ( ( ( times_times_int @ A2 @ B )
% 14.25/8.11 != zero_zero_int )
% 14.25/8.11 => ( ( A2 != zero_zero_int )
% 14.25/8.11 & ( B != zero_zero_int ) ) ) ).
% 14.25/8.11
% 14.25/8.11 thf(fact_5613_zero__neq__one,axiom,
% 14.25/8.11 zero_z2226904508553997617atural != one_one_Code_natural ).
% 14.25/8.11
% 14.25/8.11 thf(fact_5614_zero__neq__one,axiom,
% 14.25/8.11 zero_zero_nat != one_one_nat ).
% 14.25/8.11
% 14.25/8.11 thf(fact_5615_zero__neq__one,axiom,
% 14.25/8.11 zero_zero_int != one_one_int ).
% 14.25/8.11
% 14.25/8.11 thf(fact_5616_zero__neq__one,axiom,
% 14.25/8.11 zero_z3403309356797280102nteger != one_one_Code_integer ).
% 14.25/8.11
% 14.25/8.11 thf(fact_5617_eq__iff__diff__eq__0,axiom,
% 14.25/8.11 ( ( ^ [Y4: code_integer,Z4: code_integer] : ( Y4 = Z4 ) )
% 14.25/8.11 = ( ^ [A6: code_integer,B6: code_integer] :
% 14.25/8.11 ( ( minus_8373710615458151222nteger @ A6 @ B6 )
% 14.25/8.11 = zero_z3403309356797280102nteger ) ) ) ).
% 14.25/8.11
% 14.25/8.11 thf(fact_5618_eq__iff__diff__eq__0,axiom,
% 14.25/8.11 ( ( ^ [Y4: rat,Z4: rat] : ( Y4 = Z4 ) )
% 14.25/8.11 = ( ^ [A6: rat,B6: rat] :
% 14.25/8.11 ( ( minus_minus_rat @ A6 @ B6 )
% 14.25/8.11 = zero_zero_rat ) ) ) ).
% 14.25/8.11
% 14.25/8.11 thf(fact_5619_eq__iff__diff__eq__0,axiom,
% 14.25/8.11 ( ( ^ [Y4: int,Z4: int] : ( Y4 = Z4 ) )
% 14.25/8.11 = ( ^ [A6: int,B6: int] :
% 14.25/8.11 ( ( minus_minus_int @ A6 @ B6 )
% 14.25/8.11 = zero_zero_int ) ) ) ).
% 14.25/8.11
% 14.25/8.11 thf(fact_5620_mult__0,axiom,
% 14.25/8.11 ! [N2: nat] :
% 14.25/8.11 ( ( times_times_nat @ zero_zero_nat @ N2 )
% 14.25/8.11 = zero_zero_nat ) ).
% 14.25/8.11
% 14.25/8.11 thf(fact_5621_nat__mult__eq__cancel__disj,axiom,
% 14.25/8.11 ! [K3: nat,M4: nat,N2: nat] :
% 14.25/8.11 ( ( ( times_times_nat @ K3 @ M4 )
% 14.25/8.11 = ( times_times_nat @ K3 @ N2 ) )
% 14.25/8.11 = ( ( K3 = zero_zero_nat )
% 14.25/8.11 | ( M4 = N2 ) ) ) ).
% 14.25/8.11
% 14.25/8.11 thf(fact_5622_add__scale__eq__noteq,axiom,
% 14.25/8.11 ! [R2: code_integer,A2: code_integer,B: code_integer,C: code_integer,D2: code_integer] :
% 14.25/8.11 ( ( R2 != zero_z3403309356797280102nteger )
% 14.25/8.11 => ( ( ( A2 = B )
% 14.25/8.11 & ( C != D2 ) )
% 14.25/8.11 => ( ( plus_p5714425477246183910nteger @ A2 @ ( times_3573771949741848930nteger @ R2 @ C ) )
% 14.25/8.11 != ( plus_p5714425477246183910nteger @ B @ ( times_3573771949741848930nteger @ R2 @ D2 ) ) ) ) ) ).
% 14.25/8.11
% 14.25/8.11 thf(fact_5623_add__scale__eq__noteq,axiom,
% 14.25/8.11 ! [R2: rat,A2: rat,B: rat,C: rat,D2: rat] :
% 14.25/8.11 ( ( R2 != zero_zero_rat )
% 14.25/8.11 => ( ( ( A2 = B )
% 14.25/8.11 & ( C != D2 ) )
% 14.25/8.11 => ( ( plus_plus_rat @ A2 @ ( times_times_rat @ R2 @ C ) )
% 14.25/8.11 != ( plus_plus_rat @ B @ ( times_times_rat @ R2 @ D2 ) ) ) ) ) ).
% 14.25/8.11
% 14.25/8.11 thf(fact_5624_add__scale__eq__noteq,axiom,
% 14.25/8.11 ! [R2: nat,A2: nat,B: nat,C: nat,D2: nat] :
% 14.25/8.11 ( ( R2 != zero_zero_nat )
% 14.25/8.11 => ( ( ( A2 = B )
% 14.25/8.11 & ( C != D2 ) )
% 14.25/8.11 => ( ( plus_plus_nat @ A2 @ ( times_times_nat @ R2 @ C ) )
% 14.25/8.11 != ( plus_plus_nat @ B @ ( times_times_nat @ R2 @ D2 ) ) ) ) ) ).
% 14.25/8.11
% 14.25/8.11 thf(fact_5625_add__scale__eq__noteq,axiom,
% 14.25/8.11 ! [R2: int,A2: int,B: int,C: int,D2: int] :
% 14.25/8.11 ( ( R2 != zero_zero_int )
% 14.25/8.11 => ( ( ( A2 = B )
% 14.25/8.11 & ( C != D2 ) )
% 14.25/8.11 => ( ( plus_plus_int @ A2 @ ( times_times_int @ R2 @ C ) )
% 14.25/8.11 != ( plus_plus_int @ B @ ( times_times_int @ R2 @ D2 ) ) ) ) ) ).
% 14.25/8.11
% 14.25/8.11 thf(fact_5626_Sup__nat__def,axiom,
% 14.25/8.11 ( complete_Sup_Sup_nat
% 14.25/8.11 = ( ^ [X7: set_nat] : ( if_nat @ ( X7 = bot_bot_set_nat ) @ zero_zero_nat @ ( lattic8265883725875713057ax_nat @ X7 ) ) ) ) ).
% 14.25/8.11
% 14.25/8.11 thf(fact_5627_add__decreasing,axiom,
% 14.25/8.11 ! [A2: code_integer,C: code_integer,B: code_integer] :
% 14.25/8.11 ( ( ord_le3102999989581377725nteger @ A2 @ zero_z3403309356797280102nteger )
% 14.25/8.11 => ( ( ord_le3102999989581377725nteger @ C @ B )
% 14.25/8.11 => ( ord_le3102999989581377725nteger @ ( plus_p5714425477246183910nteger @ A2 @ C ) @ B ) ) ) ).
% 14.25/8.11
% 14.25/8.11 thf(fact_5628_add__decreasing,axiom,
% 14.25/8.11 ! [A2: rat,C: rat,B: rat] :
% 14.25/8.11 ( ( ord_less_eq_rat @ A2 @ zero_zero_rat )
% 14.25/8.11 => ( ( ord_less_eq_rat @ C @ B )
% 14.25/8.11 => ( ord_less_eq_rat @ ( plus_plus_rat @ A2 @ C ) @ B ) ) ) ).
% 14.25/8.11
% 14.25/8.11 thf(fact_5629_add__decreasing,axiom,
% 14.25/8.11 ! [A2: nat,C: nat,B: nat] :
% 14.25/8.11 ( ( ord_less_eq_nat @ A2 @ zero_zero_nat )
% 14.25/8.11 => ( ( ord_less_eq_nat @ C @ B )
% 14.25/8.11 => ( ord_less_eq_nat @ ( plus_plus_nat @ A2 @ C ) @ B ) ) ) ).
% 14.25/8.11
% 14.25/8.11 thf(fact_5630_add__decreasing,axiom,
% 14.25/8.11 ! [A2: int,C: int,B: int] :
% 14.25/8.11 ( ( ord_less_eq_int @ A2 @ zero_zero_int )
% 14.25/8.11 => ( ( ord_less_eq_int @ C @ B )
% 14.25/8.11 => ( ord_less_eq_int @ ( plus_plus_int @ A2 @ C ) @ B ) ) ) ).
% 14.25/8.11
% 14.25/8.11 thf(fact_5631_add__increasing,axiom,
% 14.25/8.11 ! [A2: code_integer,B: code_integer,C: code_integer] :
% 14.25/8.11 ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ A2 )
% 14.25/8.11 => ( ( ord_le3102999989581377725nteger @ B @ C )
% 14.25/8.11 => ( ord_le3102999989581377725nteger @ B @ ( plus_p5714425477246183910nteger @ A2 @ C ) ) ) ) ).
% 14.25/8.11
% 14.25/8.11 thf(fact_5632_add__increasing,axiom,
% 14.25/8.11 ! [A2: rat,B: rat,C: rat] :
% 14.25/8.11 ( ( ord_less_eq_rat @ zero_zero_rat @ A2 )
% 14.25/8.11 => ( ( ord_less_eq_rat @ B @ C )
% 14.25/8.11 => ( ord_less_eq_rat @ B @ ( plus_plus_rat @ A2 @ C ) ) ) ) ).
% 14.25/8.11
% 14.25/8.11 thf(fact_5633_add__increasing,axiom,
% 14.25/8.11 ! [A2: nat,B: nat,C: nat] :
% 14.25/8.11 ( ( ord_less_eq_nat @ zero_zero_nat @ A2 )
% 14.25/8.11 => ( ( ord_less_eq_nat @ B @ C )
% 14.25/8.11 => ( ord_less_eq_nat @ B @ ( plus_plus_nat @ A2 @ C ) ) ) ) ).
% 14.25/8.11
% 14.25/8.11 thf(fact_5634_add__increasing,axiom,
% 14.25/8.11 ! [A2: int,B: int,C: int] :
% 14.25/8.11 ( ( ord_less_eq_int @ zero_zero_int @ A2 )
% 14.25/8.11 => ( ( ord_less_eq_int @ B @ C )
% 14.25/8.11 => ( ord_less_eq_int @ B @ ( plus_plus_int @ A2 @ C ) ) ) ) ).
% 14.25/8.11
% 14.25/8.11 thf(fact_5635_add__decreasing2,axiom,
% 14.25/8.11 ! [C: code_integer,A2: code_integer,B: code_integer] :
% 14.25/8.11 ( ( ord_le3102999989581377725nteger @ C @ zero_z3403309356797280102nteger )
% 14.25/8.11 => ( ( ord_le3102999989581377725nteger @ A2 @ B )
% 14.25/8.11 => ( ord_le3102999989581377725nteger @ ( plus_p5714425477246183910nteger @ A2 @ C ) @ B ) ) ) ).
% 14.25/8.11
% 14.25/8.11 thf(fact_5636_add__decreasing2,axiom,
% 14.25/8.11 ! [C: rat,A2: rat,B: rat] :
% 14.25/8.11 ( ( ord_less_eq_rat @ C @ zero_zero_rat )
% 14.25/8.11 => ( ( ord_less_eq_rat @ A2 @ B )
% 14.25/8.11 => ( ord_less_eq_rat @ ( plus_plus_rat @ A2 @ C ) @ B ) ) ) ).
% 14.25/8.11
% 14.25/8.11 thf(fact_5637_add__decreasing2,axiom,
% 14.25/8.11 ! [C: nat,A2: nat,B: nat] :
% 14.25/8.11 ( ( ord_less_eq_nat @ C @ zero_zero_nat )
% 14.25/8.11 => ( ( ord_less_eq_nat @ A2 @ B )
% 14.25/8.11 => ( ord_less_eq_nat @ ( plus_plus_nat @ A2 @ C ) @ B ) ) ) ).
% 14.25/8.11
% 14.25/8.11 thf(fact_5638_add__decreasing2,axiom,
% 14.25/8.11 ! [C: int,A2: int,B: int] :
% 14.25/8.11 ( ( ord_less_eq_int @ C @ zero_zero_int )
% 14.25/8.11 => ( ( ord_less_eq_int @ A2 @ B )
% 14.25/8.11 => ( ord_less_eq_int @ ( plus_plus_int @ A2 @ C ) @ B ) ) ) ).
% 14.25/8.11
% 14.25/8.11 thf(fact_5639_add__increasing2,axiom,
% 14.25/8.11 ! [C: code_integer,B: code_integer,A2: code_integer] :
% 14.25/8.11 ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ C )
% 14.25/8.11 => ( ( ord_le3102999989581377725nteger @ B @ A2 )
% 14.25/8.11 => ( ord_le3102999989581377725nteger @ B @ ( plus_p5714425477246183910nteger @ A2 @ C ) ) ) ) ).
% 14.25/8.11
% 14.25/8.11 thf(fact_5640_add__increasing2,axiom,
% 14.25/8.11 ! [C: rat,B: rat,A2: rat] :
% 14.25/8.11 ( ( ord_less_eq_rat @ zero_zero_rat @ C )
% 14.25/8.11 => ( ( ord_less_eq_rat @ B @ A2 )
% 14.25/8.11 => ( ord_less_eq_rat @ B @ ( plus_plus_rat @ A2 @ C ) ) ) ) ).
% 14.25/8.11
% 14.25/8.11 thf(fact_5641_add__increasing2,axiom,
% 14.25/8.11 ! [C: nat,B: nat,A2: nat] :
% 14.25/8.11 ( ( ord_less_eq_nat @ zero_zero_nat @ C )
% 14.25/8.11 => ( ( ord_less_eq_nat @ B @ A2 )
% 14.25/8.11 => ( ord_less_eq_nat @ B @ ( plus_plus_nat @ A2 @ C ) ) ) ) ).
% 14.25/8.11
% 14.25/8.11 thf(fact_5642_add__increasing2,axiom,
% 14.25/8.11 ! [C: int,B: int,A2: int] :
% 14.25/8.11 ( ( ord_less_eq_int @ zero_zero_int @ C )
% 14.25/8.11 => ( ( ord_less_eq_int @ B @ A2 )
% 14.25/8.11 => ( ord_less_eq_int @ B @ ( plus_plus_int @ A2 @ C ) ) ) ) ).
% 14.25/8.11
% 14.25/8.11 thf(fact_5643_add__nonneg__nonneg,axiom,
% 14.25/8.11 ! [A2: code_integer,B: code_integer] :
% 14.25/8.11 ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ A2 )
% 14.25/8.11 => ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ B )
% 14.25/8.11 => ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ ( plus_p5714425477246183910nteger @ A2 @ B ) ) ) ) ).
% 14.25/8.11
% 14.25/8.11 thf(fact_5644_add__nonneg__nonneg,axiom,
% 14.25/8.11 ! [A2: rat,B: rat] :
% 14.25/8.11 ( ( ord_less_eq_rat @ zero_zero_rat @ A2 )
% 14.25/8.11 => ( ( ord_less_eq_rat @ zero_zero_rat @ B )
% 14.25/8.11 => ( ord_less_eq_rat @ zero_zero_rat @ ( plus_plus_rat @ A2 @ B ) ) ) ) ).
% 14.25/8.11
% 14.25/8.11 thf(fact_5645_add__nonneg__nonneg,axiom,
% 14.25/8.11 ! [A2: nat,B: nat] :
% 14.25/8.11 ( ( ord_less_eq_nat @ zero_zero_nat @ A2 )
% 14.25/8.11 => ( ( ord_less_eq_nat @ zero_zero_nat @ B )
% 14.25/8.11 => ( ord_less_eq_nat @ zero_zero_nat @ ( plus_plus_nat @ A2 @ B ) ) ) ) ).
% 14.25/8.11
% 14.25/8.11 thf(fact_5646_add__nonneg__nonneg,axiom,
% 14.25/8.11 ! [A2: int,B: int] :
% 14.25/8.11 ( ( ord_less_eq_int @ zero_zero_int @ A2 )
% 14.25/8.11 => ( ( ord_less_eq_int @ zero_zero_int @ B )
% 14.25/8.11 => ( ord_less_eq_int @ zero_zero_int @ ( plus_plus_int @ A2 @ B ) ) ) ) ).
% 14.25/8.11
% 14.25/8.11 thf(fact_5647_add__nonpos__nonpos,axiom,
% 14.25/8.11 ! [A2: code_integer,B: code_integer] :
% 14.25/8.11 ( ( ord_le3102999989581377725nteger @ A2 @ zero_z3403309356797280102nteger )
% 14.25/8.11 => ( ( ord_le3102999989581377725nteger @ B @ zero_z3403309356797280102nteger )
% 14.25/8.11 => ( ord_le3102999989581377725nteger @ ( plus_p5714425477246183910nteger @ A2 @ B ) @ zero_z3403309356797280102nteger ) ) ) ).
% 14.25/8.11
% 14.25/8.11 thf(fact_5648_add__nonpos__nonpos,axiom,
% 14.25/8.11 ! [A2: rat,B: rat] :
% 14.25/8.11 ( ( ord_less_eq_rat @ A2 @ zero_zero_rat )
% 14.25/8.11 => ( ( ord_less_eq_rat @ B @ zero_zero_rat )
% 14.25/8.11 => ( ord_less_eq_rat @ ( plus_plus_rat @ A2 @ B ) @ zero_zero_rat ) ) ) ).
% 14.25/8.11
% 14.25/8.11 thf(fact_5649_add__nonpos__nonpos,axiom,
% 14.25/8.11 ! [A2: nat,B: nat] :
% 14.25/8.11 ( ( ord_less_eq_nat @ A2 @ zero_zero_nat )
% 14.25/8.11 => ( ( ord_less_eq_nat @ B @ zero_zero_nat )
% 14.25/8.11 => ( ord_less_eq_nat @ ( plus_plus_nat @ A2 @ B ) @ zero_zero_nat ) ) ) ).
% 14.25/8.11
% 14.25/8.11 thf(fact_5650_add__nonpos__nonpos,axiom,
% 14.25/8.11 ! [A2: int,B: int] :
% 14.25/8.11 ( ( ord_less_eq_int @ A2 @ zero_zero_int )
% 14.25/8.11 => ( ( ord_less_eq_int @ B @ zero_zero_int )
% 14.25/8.11 => ( ord_less_eq_int @ ( plus_plus_int @ A2 @ B ) @ zero_zero_int ) ) ) ).
% 14.25/8.11
% 14.25/8.11 thf(fact_5651_add__nonneg__eq__0__iff,axiom,
% 14.25/8.11 ! [X: code_integer,Y: code_integer] :
% 14.25/8.11 ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ X )
% 14.25/8.11 => ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ Y )
% 14.25/8.11 => ( ( ( plus_p5714425477246183910nteger @ X @ Y )
% 14.25/8.11 = zero_z3403309356797280102nteger )
% 14.25/8.11 = ( ( X = zero_z3403309356797280102nteger )
% 14.25/8.11 & ( Y = zero_z3403309356797280102nteger ) ) ) ) ) ).
% 14.25/8.11
% 14.25/8.11 thf(fact_5652_add__nonneg__eq__0__iff,axiom,
% 14.25/8.11 ! [X: rat,Y: rat] :
% 14.25/8.11 ( ( ord_less_eq_rat @ zero_zero_rat @ X )
% 14.25/8.11 => ( ( ord_less_eq_rat @ zero_zero_rat @ Y )
% 14.25/8.11 => ( ( ( plus_plus_rat @ X @ Y )
% 14.25/8.11 = zero_zero_rat )
% 14.25/8.11 = ( ( X = zero_zero_rat )
% 14.25/8.11 & ( Y = zero_zero_rat ) ) ) ) ) ).
% 14.25/8.11
% 14.25/8.11 thf(fact_5653_add__nonneg__eq__0__iff,axiom,
% 14.25/8.11 ! [X: nat,Y: nat] :
% 14.25/8.11 ( ( ord_less_eq_nat @ zero_zero_nat @ X )
% 14.25/8.11 => ( ( ord_less_eq_nat @ zero_zero_nat @ Y )
% 14.25/8.11 => ( ( ( plus_plus_nat @ X @ Y )
% 14.25/8.11 = zero_zero_nat )
% 14.25/8.11 = ( ( X = zero_zero_nat )
% 14.25/8.11 & ( Y = zero_zero_nat ) ) ) ) ) ).
% 14.25/8.11
% 14.25/8.11 thf(fact_5654_add__nonneg__eq__0__iff,axiom,
% 14.25/8.11 ! [X: int,Y: int] :
% 14.25/8.11 ( ( ord_less_eq_int @ zero_zero_int @ X )
% 14.25/8.11 => ( ( ord_less_eq_int @ zero_zero_int @ Y )
% 14.25/8.11 => ( ( ( plus_plus_int @ X @ Y )
% 14.25/8.11 = zero_zero_int )
% 14.25/8.11 = ( ( X = zero_zero_int )
% 14.25/8.11 & ( Y = zero_zero_int ) ) ) ) ) ).
% 14.25/8.11
% 14.25/8.11 thf(fact_5655_add__nonpos__eq__0__iff,axiom,
% 14.25/8.11 ! [X: code_integer,Y: code_integer] :
% 14.25/8.11 ( ( ord_le3102999989581377725nteger @ X @ zero_z3403309356797280102nteger )
% 14.25/8.11 => ( ( ord_le3102999989581377725nteger @ Y @ zero_z3403309356797280102nteger )
% 14.25/8.11 => ( ( ( plus_p5714425477246183910nteger @ X @ Y )
% 14.25/8.11 = zero_z3403309356797280102nteger )
% 14.25/8.11 = ( ( X = zero_z3403309356797280102nteger )
% 14.25/8.11 & ( Y = zero_z3403309356797280102nteger ) ) ) ) ) ).
% 14.25/8.11
% 14.25/8.11 thf(fact_5656_add__nonpos__eq__0__iff,axiom,
% 14.25/8.11 ! [X: rat,Y: rat] :
% 14.25/8.11 ( ( ord_less_eq_rat @ X @ zero_zero_rat )
% 14.25/8.11 => ( ( ord_less_eq_rat @ Y @ zero_zero_rat )
% 14.25/8.11 => ( ( ( plus_plus_rat @ X @ Y )
% 14.25/8.11 = zero_zero_rat )
% 14.25/8.11 = ( ( X = zero_zero_rat )
% 14.25/8.11 & ( Y = zero_zero_rat ) ) ) ) ) ).
% 14.25/8.11
% 14.25/8.11 thf(fact_5657_add__nonpos__eq__0__iff,axiom,
% 14.25/8.11 ! [X: nat,Y: nat] :
% 14.25/8.11 ( ( ord_less_eq_nat @ X @ zero_zero_nat )
% 14.25/8.11 => ( ( ord_less_eq_nat @ Y @ zero_zero_nat )
% 14.25/8.11 => ( ( ( plus_plus_nat @ X @ Y )
% 14.25/8.11 = zero_zero_nat )
% 14.25/8.11 = ( ( X = zero_zero_nat )
% 14.25/8.11 & ( Y = zero_zero_nat ) ) ) ) ) ).
% 14.25/8.11
% 14.25/8.11 thf(fact_5658_add__nonpos__eq__0__iff,axiom,
% 14.25/8.11 ! [X: int,Y: int] :
% 14.25/8.11 ( ( ord_less_eq_int @ X @ zero_zero_int )
% 14.25/8.11 => ( ( ord_less_eq_int @ Y @ zero_zero_int )
% 14.25/8.11 => ( ( ( plus_plus_int @ X @ Y )
% 14.25/8.11 = zero_zero_int )
% 14.25/8.11 = ( ( X = zero_zero_int )
% 14.25/8.11 & ( Y = zero_zero_int ) ) ) ) ) ).
% 14.25/8.11
% 14.25/8.11 thf(fact_5659_ordered__comm__semiring__class_Ocomm__mult__left__mono,axiom,
% 14.25/8.11 ! [A2: code_integer,B: code_integer,C: code_integer] :
% 14.25/8.11 ( ( ord_le3102999989581377725nteger @ A2 @ B )
% 14.25/8.11 => ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ C )
% 14.25/8.11 => ( ord_le3102999989581377725nteger @ ( times_3573771949741848930nteger @ C @ A2 ) @ ( times_3573771949741848930nteger @ C @ B ) ) ) ) ).
% 14.25/8.11
% 14.25/8.11 thf(fact_5660_ordered__comm__semiring__class_Ocomm__mult__left__mono,axiom,
% 14.25/8.11 ! [A2: rat,B: rat,C: rat] :
% 14.25/8.11 ( ( ord_less_eq_rat @ A2 @ B )
% 14.25/8.11 => ( ( ord_less_eq_rat @ zero_zero_rat @ C )
% 14.25/8.11 => ( ord_less_eq_rat @ ( times_times_rat @ C @ A2 ) @ ( times_times_rat @ C @ B ) ) ) ) ).
% 14.25/8.11
% 14.25/8.11 thf(fact_5661_ordered__comm__semiring__class_Ocomm__mult__left__mono,axiom,
% 14.25/8.11 ! [A2: nat,B: nat,C: nat] :
% 14.25/8.11 ( ( ord_less_eq_nat @ A2 @ B )
% 14.25/8.11 => ( ( ord_less_eq_nat @ zero_zero_nat @ C )
% 14.25/8.11 => ( ord_less_eq_nat @ ( times_times_nat @ C @ A2 ) @ ( times_times_nat @ C @ B ) ) ) ) ).
% 14.25/8.11
% 14.25/8.11 thf(fact_5662_ordered__comm__semiring__class_Ocomm__mult__left__mono,axiom,
% 14.25/8.11 ! [A2: int,B: int,C: int] :
% 14.25/8.11 ( ( ord_less_eq_int @ A2 @ B )
% 14.25/8.11 => ( ( ord_less_eq_int @ zero_zero_int @ C )
% 14.25/8.11 => ( ord_less_eq_int @ ( times_times_int @ C @ A2 ) @ ( times_times_int @ C @ B ) ) ) ) ).
% 14.25/8.11
% 14.25/8.11 thf(fact_5663_zero__le__mult__iff,axiom,
% 14.25/8.11 ! [A2: code_integer,B: code_integer] :
% 14.25/8.11 ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ ( times_3573771949741848930nteger @ A2 @ B ) )
% 14.25/8.11 = ( ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ A2 )
% 14.25/8.11 & ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ B ) )
% 14.25/8.11 | ( ( ord_le3102999989581377725nteger @ A2 @ zero_z3403309356797280102nteger )
% 14.25/8.11 & ( ord_le3102999989581377725nteger @ B @ zero_z3403309356797280102nteger ) ) ) ) ).
% 14.25/8.11
% 14.25/8.11 thf(fact_5664_zero__le__mult__iff,axiom,
% 14.25/8.11 ! [A2: rat,B: rat] :
% 14.25/8.11 ( ( ord_less_eq_rat @ zero_zero_rat @ ( times_times_rat @ A2 @ B ) )
% 14.25/8.11 = ( ( ( ord_less_eq_rat @ zero_zero_rat @ A2 )
% 14.25/8.11 & ( ord_less_eq_rat @ zero_zero_rat @ B ) )
% 14.25/8.11 | ( ( ord_less_eq_rat @ A2 @ zero_zero_rat )
% 14.25/8.11 & ( ord_less_eq_rat @ B @ zero_zero_rat ) ) ) ) ).
% 14.25/8.11
% 14.25/8.11 thf(fact_5665_zero__le__mult__iff,axiom,
% 14.25/8.11 ! [A2: int,B: int] :
% 14.25/8.11 ( ( ord_less_eq_int @ zero_zero_int @ ( times_times_int @ A2 @ B ) )
% 14.25/8.11 = ( ( ( ord_less_eq_int @ zero_zero_int @ A2 )
% 14.25/8.11 & ( ord_less_eq_int @ zero_zero_int @ B ) )
% 14.25/8.11 | ( ( ord_less_eq_int @ A2 @ zero_zero_int )
% 14.25/8.11 & ( ord_less_eq_int @ B @ zero_zero_int ) ) ) ) ).
% 14.25/8.11
% 14.25/8.11 thf(fact_5666_mult__nonneg__nonpos2,axiom,
% 14.25/8.11 ! [A2: code_integer,B: code_integer] :
% 14.25/8.11 ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ A2 )
% 14.25/8.11 => ( ( ord_le3102999989581377725nteger @ B @ zero_z3403309356797280102nteger )
% 14.25/8.11 => ( ord_le3102999989581377725nteger @ ( times_3573771949741848930nteger @ B @ A2 ) @ zero_z3403309356797280102nteger ) ) ) ).
% 14.25/8.11
% 14.25/8.11 thf(fact_5667_mult__nonneg__nonpos2,axiom,
% 14.25/8.11 ! [A2: rat,B: rat] :
% 14.25/8.11 ( ( ord_less_eq_rat @ zero_zero_rat @ A2 )
% 14.25/8.11 => ( ( ord_less_eq_rat @ B @ zero_zero_rat )
% 14.25/8.11 => ( ord_less_eq_rat @ ( times_times_rat @ B @ A2 ) @ zero_zero_rat ) ) ) ).
% 14.25/8.11
% 14.25/8.11 thf(fact_5668_mult__nonneg__nonpos2,axiom,
% 14.25/8.11 ! [A2: nat,B: nat] :
% 14.25/8.11 ( ( ord_less_eq_nat @ zero_zero_nat @ A2 )
% 14.25/8.11 => ( ( ord_less_eq_nat @ B @ zero_zero_nat )
% 14.25/8.11 => ( ord_less_eq_nat @ ( times_times_nat @ B @ A2 ) @ zero_zero_nat ) ) ) ).
% 14.25/8.11
% 14.25/8.11 thf(fact_5669_mult__nonneg__nonpos2,axiom,
% 14.25/8.11 ! [A2: int,B: int] :
% 14.25/8.11 ( ( ord_less_eq_int @ zero_zero_int @ A2 )
% 14.25/8.11 => ( ( ord_less_eq_int @ B @ zero_zero_int )
% 14.25/8.11 => ( ord_less_eq_int @ ( times_times_int @ B @ A2 ) @ zero_zero_int ) ) ) ).
% 14.25/8.11
% 14.25/8.11 thf(fact_5670_mult__nonpos__nonneg,axiom,
% 14.25/8.11 ! [A2: code_integer,B: code_integer] :
% 14.25/8.11 ( ( ord_le3102999989581377725nteger @ A2 @ zero_z3403309356797280102nteger )
% 14.25/8.11 => ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ B )
% 14.25/8.11 => ( ord_le3102999989581377725nteger @ ( times_3573771949741848930nteger @ A2 @ B ) @ zero_z3403309356797280102nteger ) ) ) ).
% 14.25/8.11
% 14.25/8.11 thf(fact_5671_mult__nonpos__nonneg,axiom,
% 14.25/8.11 ! [A2: rat,B: rat] :
% 14.25/8.11 ( ( ord_less_eq_rat @ A2 @ zero_zero_rat )
% 14.25/8.11 => ( ( ord_less_eq_rat @ zero_zero_rat @ B )
% 14.25/8.11 => ( ord_less_eq_rat @ ( times_times_rat @ A2 @ B ) @ zero_zero_rat ) ) ) ).
% 14.25/8.11
% 14.25/8.11 thf(fact_5672_mult__nonpos__nonneg,axiom,
% 14.25/8.11 ! [A2: nat,B: nat] :
% 14.25/8.11 ( ( ord_less_eq_nat @ A2 @ zero_zero_nat )
% 14.25/8.11 => ( ( ord_less_eq_nat @ zero_zero_nat @ B )
% 14.25/8.11 => ( ord_less_eq_nat @ ( times_times_nat @ A2 @ B ) @ zero_zero_nat ) ) ) ).
% 14.25/8.11
% 14.25/8.11 thf(fact_5673_mult__nonpos__nonneg,axiom,
% 14.25/8.11 ! [A2: int,B: int] :
% 14.25/8.11 ( ( ord_less_eq_int @ A2 @ zero_zero_int )
% 14.25/8.11 => ( ( ord_less_eq_int @ zero_zero_int @ B )
% 14.25/8.11 => ( ord_less_eq_int @ ( times_times_int @ A2 @ B ) @ zero_zero_int ) ) ) ).
% 14.25/8.11
% 14.25/8.11 thf(fact_5674_mult__nonneg__nonpos,axiom,
% 14.25/8.11 ! [A2: code_integer,B: code_integer] :
% 14.25/8.11 ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ A2 )
% 14.25/8.11 => ( ( ord_le3102999989581377725nteger @ B @ zero_z3403309356797280102nteger )
% 14.25/8.11 => ( ord_le3102999989581377725nteger @ ( times_3573771949741848930nteger @ A2 @ B ) @ zero_z3403309356797280102nteger ) ) ) ).
% 14.25/8.11
% 14.25/8.11 thf(fact_5675_mult__nonneg__nonpos,axiom,
% 14.25/8.11 ! [A2: rat,B: rat] :
% 14.25/8.11 ( ( ord_less_eq_rat @ zero_zero_rat @ A2 )
% 14.25/8.11 => ( ( ord_less_eq_rat @ B @ zero_zero_rat )
% 14.25/8.11 => ( ord_less_eq_rat @ ( times_times_rat @ A2 @ B ) @ zero_zero_rat ) ) ) ).
% 14.25/8.11
% 14.25/8.11 thf(fact_5676_mult__nonneg__nonpos,axiom,
% 14.25/8.11 ! [A2: nat,B: nat] :
% 14.25/8.11 ( ( ord_less_eq_nat @ zero_zero_nat @ A2 )
% 14.25/8.11 => ( ( ord_less_eq_nat @ B @ zero_zero_nat )
% 14.25/8.11 => ( ord_less_eq_nat @ ( times_times_nat @ A2 @ B ) @ zero_zero_nat ) ) ) ).
% 14.25/8.11
% 14.25/8.11 thf(fact_5677_mult__nonneg__nonpos,axiom,
% 14.25/8.11 ! [A2: int,B: int] :
% 14.25/8.11 ( ( ord_less_eq_int @ zero_zero_int @ A2 )
% 14.25/8.11 => ( ( ord_less_eq_int @ B @ zero_zero_int )
% 14.25/8.11 => ( ord_less_eq_int @ ( times_times_int @ A2 @ B ) @ zero_zero_int ) ) ) ).
% 14.25/8.11
% 14.25/8.11 thf(fact_5678_mult__nonneg__nonneg,axiom,
% 14.25/8.11 ! [A2: code_integer,B: code_integer] :
% 14.25/8.11 ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ A2 )
% 14.25/8.11 => ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ B )
% 14.25/8.11 => ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ ( times_3573771949741848930nteger @ A2 @ B ) ) ) ) ).
% 14.25/8.11
% 14.25/8.11 thf(fact_5679_mult__nonneg__nonneg,axiom,
% 14.25/8.11 ! [A2: rat,B: rat] :
% 14.25/8.11 ( ( ord_less_eq_rat @ zero_zero_rat @ A2 )
% 14.25/8.11 => ( ( ord_less_eq_rat @ zero_zero_rat @ B )
% 14.25/8.11 => ( ord_less_eq_rat @ zero_zero_rat @ ( times_times_rat @ A2 @ B ) ) ) ) ).
% 14.25/8.11
% 14.25/8.11 thf(fact_5680_mult__nonneg__nonneg,axiom,
% 14.25/8.11 ! [A2: nat,B: nat] :
% 14.25/8.11 ( ( ord_less_eq_nat @ zero_zero_nat @ A2 )
% 14.25/8.11 => ( ( ord_less_eq_nat @ zero_zero_nat @ B )
% 14.25/8.11 => ( ord_less_eq_nat @ zero_zero_nat @ ( times_times_nat @ A2 @ B ) ) ) ) ).
% 14.25/8.11
% 14.25/8.11 thf(fact_5681_mult__nonneg__nonneg,axiom,
% 14.25/8.11 ! [A2: int,B: int] :
% 14.25/8.11 ( ( ord_less_eq_int @ zero_zero_int @ A2 )
% 14.25/8.11 => ( ( ord_less_eq_int @ zero_zero_int @ B )
% 14.25/8.11 => ( ord_less_eq_int @ zero_zero_int @ ( times_times_int @ A2 @ B ) ) ) ) ).
% 14.25/8.11
% 14.25/8.11 thf(fact_5682_split__mult__neg__le,axiom,
% 14.25/8.11 ! [A2: code_integer,B: code_integer] :
% 14.25/8.11 ( ( ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ A2 )
% 14.25/8.11 & ( ord_le3102999989581377725nteger @ B @ zero_z3403309356797280102nteger ) )
% 14.25/8.11 | ( ( ord_le3102999989581377725nteger @ A2 @ zero_z3403309356797280102nteger )
% 14.25/8.11 & ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ B ) ) )
% 14.25/8.11 => ( ord_le3102999989581377725nteger @ ( times_3573771949741848930nteger @ A2 @ B ) @ zero_z3403309356797280102nteger ) ) ).
% 14.25/8.11
% 14.25/8.11 thf(fact_5683_split__mult__neg__le,axiom,
% 14.25/8.11 ! [A2: rat,B: rat] :
% 14.25/8.11 ( ( ( ( ord_less_eq_rat @ zero_zero_rat @ A2 )
% 14.25/8.11 & ( ord_less_eq_rat @ B @ zero_zero_rat ) )
% 14.25/8.11 | ( ( ord_less_eq_rat @ A2 @ zero_zero_rat )
% 14.25/8.11 & ( ord_less_eq_rat @ zero_zero_rat @ B ) ) )
% 14.25/8.11 => ( ord_less_eq_rat @ ( times_times_rat @ A2 @ B ) @ zero_zero_rat ) ) ).
% 14.25/8.11
% 14.25/8.11 thf(fact_5684_split__mult__neg__le,axiom,
% 14.25/8.11 ! [A2: nat,B: nat] :
% 14.25/8.11 ( ( ( ( ord_less_eq_nat @ zero_zero_nat @ A2 )
% 14.25/8.11 & ( ord_less_eq_nat @ B @ zero_zero_nat ) )
% 14.25/8.11 | ( ( ord_less_eq_nat @ A2 @ zero_zero_nat )
% 14.25/8.11 & ( ord_less_eq_nat @ zero_zero_nat @ B ) ) )
% 14.25/8.11 => ( ord_less_eq_nat @ ( times_times_nat @ A2 @ B ) @ zero_zero_nat ) ) ).
% 14.25/8.11
% 14.25/8.11 thf(fact_5685_split__mult__neg__le,axiom,
% 14.25/8.11 ! [A2: int,B: int] :
% 14.25/8.11 ( ( ( ( ord_less_eq_int @ zero_zero_int @ A2 )
% 14.25/8.11 & ( ord_less_eq_int @ B @ zero_zero_int ) )
% 14.25/8.11 | ( ( ord_less_eq_int @ A2 @ zero_zero_int )
% 14.25/8.11 & ( ord_less_eq_int @ zero_zero_int @ B ) ) )
% 14.25/8.11 => ( ord_less_eq_int @ ( times_times_int @ A2 @ B ) @ zero_zero_int ) ) ).
% 14.25/8.11
% 14.25/8.11 thf(fact_5686_mult__le__0__iff,axiom,
% 14.25/8.11 ! [A2: code_integer,B: code_integer] :
% 14.25/8.11 ( ( ord_le3102999989581377725nteger @ ( times_3573771949741848930nteger @ A2 @ B ) @ zero_z3403309356797280102nteger )
% 14.25/8.11 = ( ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ A2 )
% 14.25/8.11 & ( ord_le3102999989581377725nteger @ B @ zero_z3403309356797280102nteger ) )
% 14.25/8.11 | ( ( ord_le3102999989581377725nteger @ A2 @ zero_z3403309356797280102nteger )
% 14.25/8.11 & ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ B ) ) ) ) ).
% 14.25/8.11
% 14.25/8.11 thf(fact_5687_mult__le__0__iff,axiom,
% 14.25/8.11 ! [A2: rat,B: rat] :
% 14.25/8.11 ( ( ord_less_eq_rat @ ( times_times_rat @ A2 @ B ) @ zero_zero_rat )
% 14.25/8.11 = ( ( ( ord_less_eq_rat @ zero_zero_rat @ A2 )
% 14.25/8.11 & ( ord_less_eq_rat @ B @ zero_zero_rat ) )
% 14.25/8.11 | ( ( ord_less_eq_rat @ A2 @ zero_zero_rat )
% 14.25/8.11 & ( ord_less_eq_rat @ zero_zero_rat @ B ) ) ) ) ).
% 14.25/8.11
% 14.25/8.11 thf(fact_5688_mult__le__0__iff,axiom,
% 14.25/8.11 ! [A2: int,B: int] :
% 14.25/8.11 ( ( ord_less_eq_int @ ( times_times_int @ A2 @ B ) @ zero_zero_int )
% 14.25/8.11 = ( ( ( ord_less_eq_int @ zero_zero_int @ A2 )
% 14.25/8.11 & ( ord_less_eq_int @ B @ zero_zero_int ) )
% 14.25/8.11 | ( ( ord_less_eq_int @ A2 @ zero_zero_int )
% 14.25/8.11 & ( ord_less_eq_int @ zero_zero_int @ B ) ) ) ) ).
% 14.25/8.11
% 14.25/8.11 thf(fact_5689_mult__right__mono,axiom,
% 14.25/8.11 ! [A2: code_integer,B: code_integer,C: code_integer] :
% 14.25/8.11 ( ( ord_le3102999989581377725nteger @ A2 @ B )
% 14.25/8.11 => ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ C )
% 14.25/8.11 => ( ord_le3102999989581377725nteger @ ( times_3573771949741848930nteger @ A2 @ C ) @ ( times_3573771949741848930nteger @ B @ C ) ) ) ) ).
% 14.25/8.11
% 14.25/8.11 thf(fact_5690_mult__right__mono,axiom,
% 14.25/8.11 ! [A2: rat,B: rat,C: rat] :
% 14.25/8.11 ( ( ord_less_eq_rat @ A2 @ B )
% 14.25/8.11 => ( ( ord_less_eq_rat @ zero_zero_rat @ C )
% 14.25/8.11 => ( ord_less_eq_rat @ ( times_times_rat @ A2 @ C ) @ ( times_times_rat @ B @ C ) ) ) ) ).
% 14.25/8.11
% 14.25/8.11 thf(fact_5691_mult__right__mono,axiom,
% 14.25/8.11 ! [A2: nat,B: nat,C: nat] :
% 14.25/8.11 ( ( ord_less_eq_nat @ A2 @ B )
% 14.25/8.11 => ( ( ord_less_eq_nat @ zero_zero_nat @ C )
% 14.25/8.11 => ( ord_less_eq_nat @ ( times_times_nat @ A2 @ C ) @ ( times_times_nat @ B @ C ) ) ) ) ).
% 14.25/8.11
% 14.25/8.11 thf(fact_5692_mult__right__mono,axiom,
% 14.25/8.11 ! [A2: int,B: int,C: int] :
% 14.25/8.11 ( ( ord_less_eq_int @ A2 @ B )
% 14.25/8.11 => ( ( ord_less_eq_int @ zero_zero_int @ C )
% 14.25/8.11 => ( ord_less_eq_int @ ( times_times_int @ A2 @ C ) @ ( times_times_int @ B @ C ) ) ) ) ).
% 14.25/8.11
% 14.25/8.11 thf(fact_5693_mult__right__mono__neg,axiom,
% 14.25/8.11 ! [B: code_integer,A2: code_integer,C: code_integer] :
% 14.25/8.11 ( ( ord_le3102999989581377725nteger @ B @ A2 )
% 14.25/8.11 => ( ( ord_le3102999989581377725nteger @ C @ zero_z3403309356797280102nteger )
% 14.25/8.11 => ( ord_le3102999989581377725nteger @ ( times_3573771949741848930nteger @ A2 @ C ) @ ( times_3573771949741848930nteger @ B @ C ) ) ) ) ).
% 14.25/8.11
% 14.25/8.11 thf(fact_5694_mult__right__mono__neg,axiom,
% 14.25/8.11 ! [B: rat,A2: rat,C: rat] :
% 14.25/8.11 ( ( ord_less_eq_rat @ B @ A2 )
% 14.25/8.11 => ( ( ord_less_eq_rat @ C @ zero_zero_rat )
% 14.25/8.11 => ( ord_less_eq_rat @ ( times_times_rat @ A2 @ C ) @ ( times_times_rat @ B @ C ) ) ) ) ).
% 14.25/8.11
% 14.25/8.11 thf(fact_5695_mult__right__mono__neg,axiom,
% 14.25/8.11 ! [B: int,A2: int,C: int] :
% 14.25/8.11 ( ( ord_less_eq_int @ B @ A2 )
% 14.25/8.11 => ( ( ord_less_eq_int @ C @ zero_zero_int )
% 14.25/8.11 => ( ord_less_eq_int @ ( times_times_int @ A2 @ C ) @ ( times_times_int @ B @ C ) ) ) ) ).
% 14.25/8.11
% 14.25/8.11 thf(fact_5696_mult__left__mono,axiom,
% 14.25/8.11 ! [A2: code_integer,B: code_integer,C: code_integer] :
% 14.25/8.11 ( ( ord_le3102999989581377725nteger @ A2 @ B )
% 14.25/8.11 => ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ C )
% 14.25/8.11 => ( ord_le3102999989581377725nteger @ ( times_3573771949741848930nteger @ C @ A2 ) @ ( times_3573771949741848930nteger @ C @ B ) ) ) ) ).
% 14.25/8.11
% 14.25/8.11 thf(fact_5697_mult__left__mono,axiom,
% 14.25/8.11 ! [A2: rat,B: rat,C: rat] :
% 14.25/8.11 ( ( ord_less_eq_rat @ A2 @ B )
% 14.25/8.11 => ( ( ord_less_eq_rat @ zero_zero_rat @ C )
% 14.25/8.11 => ( ord_less_eq_rat @ ( times_times_rat @ C @ A2 ) @ ( times_times_rat @ C @ B ) ) ) ) ).
% 14.25/8.11
% 14.25/8.11 thf(fact_5698_mult__left__mono,axiom,
% 14.25/8.11 ! [A2: nat,B: nat,C: nat] :
% 14.25/8.11 ( ( ord_less_eq_nat @ A2 @ B )
% 14.25/8.11 => ( ( ord_less_eq_nat @ zero_zero_nat @ C )
% 14.25/8.11 => ( ord_less_eq_nat @ ( times_times_nat @ C @ A2 ) @ ( times_times_nat @ C @ B ) ) ) ) ).
% 14.25/8.11
% 14.25/8.11 thf(fact_5699_mult__left__mono,axiom,
% 14.25/8.11 ! [A2: int,B: int,C: int] :
% 14.25/8.11 ( ( ord_less_eq_int @ A2 @ B )
% 14.25/8.11 => ( ( ord_less_eq_int @ zero_zero_int @ C )
% 14.25/8.11 => ( ord_less_eq_int @ ( times_times_int @ C @ A2 ) @ ( times_times_int @ C @ B ) ) ) ) ).
% 14.25/8.11
% 14.25/8.11 thf(fact_5700_mult__nonpos__nonpos,axiom,
% 14.25/8.11 ! [A2: code_integer,B: code_integer] :
% 14.25/8.11 ( ( ord_le3102999989581377725nteger @ A2 @ zero_z3403309356797280102nteger )
% 14.25/8.11 => ( ( ord_le3102999989581377725nteger @ B @ zero_z3403309356797280102nteger )
% 14.25/8.11 => ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ ( times_3573771949741848930nteger @ A2 @ B ) ) ) ) ).
% 14.25/8.11
% 14.25/8.11 thf(fact_5701_mult__nonpos__nonpos,axiom,
% 14.25/8.11 ! [A2: rat,B: rat] :
% 14.25/8.11 ( ( ord_less_eq_rat @ A2 @ zero_zero_rat )
% 14.25/8.11 => ( ( ord_less_eq_rat @ B @ zero_zero_rat )
% 14.25/8.11 => ( ord_less_eq_rat @ zero_zero_rat @ ( times_times_rat @ A2 @ B ) ) ) ) ).
% 14.25/8.11
% 14.25/8.11 thf(fact_5702_mult__nonpos__nonpos,axiom,
% 14.25/8.11 ! [A2: int,B: int] :
% 14.25/8.11 ( ( ord_less_eq_int @ A2 @ zero_zero_int )
% 14.25/8.11 => ( ( ord_less_eq_int @ B @ zero_zero_int )
% 14.25/8.11 => ( ord_less_eq_int @ zero_zero_int @ ( times_times_int @ A2 @ B ) ) ) ) ).
% 14.25/8.11
% 14.25/8.11 thf(fact_5703_mult__left__mono__neg,axiom,
% 14.25/8.11 ! [B: code_integer,A2: code_integer,C: code_integer] :
% 14.25/8.11 ( ( ord_le3102999989581377725nteger @ B @ A2 )
% 14.25/8.11 => ( ( ord_le3102999989581377725nteger @ C @ zero_z3403309356797280102nteger )
% 14.25/8.11 => ( ord_le3102999989581377725nteger @ ( times_3573771949741848930nteger @ C @ A2 ) @ ( times_3573771949741848930nteger @ C @ B ) ) ) ) ).
% 14.25/8.11
% 14.25/8.11 thf(fact_5704_mult__left__mono__neg,axiom,
% 14.25/8.11 ! [B: rat,A2: rat,C: rat] :
% 14.25/8.11 ( ( ord_less_eq_rat @ B @ A2 )
% 14.25/8.11 => ( ( ord_less_eq_rat @ C @ zero_zero_rat )
% 14.25/8.11 => ( ord_less_eq_rat @ ( times_times_rat @ C @ A2 ) @ ( times_times_rat @ C @ B ) ) ) ) ).
% 14.25/8.11
% 14.25/8.11 thf(fact_5705_mult__left__mono__neg,axiom,
% 14.25/8.11 ! [B: int,A2: int,C: int] :
% 14.25/8.11 ( ( ord_less_eq_int @ B @ A2 )
% 14.25/8.11 => ( ( ord_less_eq_int @ C @ zero_zero_int )
% 14.25/8.11 => ( ord_less_eq_int @ ( times_times_int @ C @ A2 ) @ ( times_times_int @ C @ B ) ) ) ) ).
% 14.25/8.11
% 14.25/8.11 thf(fact_5706_split__mult__pos__le,axiom,
% 14.25/8.11 ! [A2: code_integer,B: code_integer] :
% 14.25/8.11 ( ( ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ A2 )
% 14.25/8.11 & ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ B ) )
% 14.25/8.11 | ( ( ord_le3102999989581377725nteger @ A2 @ zero_z3403309356797280102nteger )
% 14.25/8.11 & ( ord_le3102999989581377725nteger @ B @ zero_z3403309356797280102nteger ) ) )
% 14.25/8.11 => ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ ( times_3573771949741848930nteger @ A2 @ B ) ) ) ).
% 14.25/8.11
% 14.25/8.11 thf(fact_5707_split__mult__pos__le,axiom,
% 14.25/8.11 ! [A2: rat,B: rat] :
% 14.25/8.11 ( ( ( ( ord_less_eq_rat @ zero_zero_rat @ A2 )
% 14.25/8.11 & ( ord_less_eq_rat @ zero_zero_rat @ B ) )
% 14.25/8.11 | ( ( ord_less_eq_rat @ A2 @ zero_zero_rat )
% 14.25/8.11 & ( ord_less_eq_rat @ B @ zero_zero_rat ) ) )
% 14.25/8.11 => ( ord_less_eq_rat @ zero_zero_rat @ ( times_times_rat @ A2 @ B ) ) ) ).
% 14.25/8.11
% 14.25/8.11 thf(fact_5708_split__mult__pos__le,axiom,
% 14.25/8.11 ! [A2: int,B: int] :
% 14.25/8.11 ( ( ( ( ord_less_eq_int @ zero_zero_int @ A2 )
% 14.25/8.11 & ( ord_less_eq_int @ zero_zero_int @ B ) )
% 14.25/8.11 | ( ( ord_less_eq_int @ A2 @ zero_zero_int )
% 14.25/8.11 & ( ord_less_eq_int @ B @ zero_zero_int ) ) )
% 14.25/8.11 => ( ord_less_eq_int @ zero_zero_int @ ( times_times_int @ A2 @ B ) ) ) ).
% 14.25/8.11
% 14.25/8.11 thf(fact_5709_zero__le__square,axiom,
% 14.25/8.11 ! [A2: code_integer] : ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ ( times_3573771949741848930nteger @ A2 @ A2 ) ) ).
% 14.25/8.11
% 14.25/8.11 thf(fact_5710_zero__le__square,axiom,
% 14.25/8.11 ! [A2: rat] : ( ord_less_eq_rat @ zero_zero_rat @ ( times_times_rat @ A2 @ A2 ) ) ).
% 14.25/8.11
% 14.25/8.11 thf(fact_5711_zero__le__square,axiom,
% 14.25/8.11 ! [A2: int] : ( ord_less_eq_int @ zero_zero_int @ ( times_times_int @ A2 @ A2 ) ) ).
% 14.25/8.11
% 14.25/8.11 thf(fact_5712_mult__mono_H,axiom,
% 14.25/8.11 ! [A2: code_integer,B: code_integer,C: code_integer,D2: code_integer] :
% 14.25/8.11 ( ( ord_le3102999989581377725nteger @ A2 @ B )
% 14.25/8.11 => ( ( ord_le3102999989581377725nteger @ C @ D2 )
% 14.25/8.11 => ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ A2 )
% 14.25/8.11 => ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ C )
% 14.25/8.11 => ( ord_le3102999989581377725nteger @ ( times_3573771949741848930nteger @ A2 @ C ) @ ( times_3573771949741848930nteger @ B @ D2 ) ) ) ) ) ) ).
% 14.25/8.11
% 14.25/8.11 thf(fact_5713_mult__mono_H,axiom,
% 14.25/8.11 ! [A2: rat,B: rat,C: rat,D2: rat] :
% 14.25/8.11 ( ( ord_less_eq_rat @ A2 @ B )
% 14.25/8.11 => ( ( ord_less_eq_rat @ C @ D2 )
% 14.25/8.11 => ( ( ord_less_eq_rat @ zero_zero_rat @ A2 )
% 14.25/8.11 => ( ( ord_less_eq_rat @ zero_zero_rat @ C )
% 14.25/8.11 => ( ord_less_eq_rat @ ( times_times_rat @ A2 @ C ) @ ( times_times_rat @ B @ D2 ) ) ) ) ) ) ).
% 14.25/8.11
% 14.25/8.11 thf(fact_5714_mult__mono_H,axiom,
% 14.25/8.11 ! [A2: nat,B: nat,C: nat,D2: nat] :
% 14.25/8.11 ( ( ord_less_eq_nat @ A2 @ B )
% 14.25/8.11 => ( ( ord_less_eq_nat @ C @ D2 )
% 14.25/8.11 => ( ( ord_less_eq_nat @ zero_zero_nat @ A2 )
% 14.25/8.11 => ( ( ord_less_eq_nat @ zero_zero_nat @ C )
% 14.25/8.11 => ( ord_less_eq_nat @ ( times_times_nat @ A2 @ C ) @ ( times_times_nat @ B @ D2 ) ) ) ) ) ) ).
% 14.25/8.11
% 14.25/8.11 thf(fact_5715_mult__mono_H,axiom,
% 14.25/8.11 ! [A2: int,B: int,C: int,D2: int] :
% 14.25/8.11 ( ( ord_less_eq_int @ A2 @ B )
% 14.25/8.11 => ( ( ord_less_eq_int @ C @ D2 )
% 14.25/8.11 => ( ( ord_less_eq_int @ zero_zero_int @ A2 )
% 14.25/8.11 => ( ( ord_less_eq_int @ zero_zero_int @ C )
% 14.25/8.11 => ( ord_less_eq_int @ ( times_times_int @ A2 @ C ) @ ( times_times_int @ B @ D2 ) ) ) ) ) ) ).
% 14.25/8.11
% 14.25/8.11 thf(fact_5716_mult__mono,axiom,
% 14.25/8.11 ! [A2: code_integer,B: code_integer,C: code_integer,D2: code_integer] :
% 14.25/8.11 ( ( ord_le3102999989581377725nteger @ A2 @ B )
% 14.25/8.11 => ( ( ord_le3102999989581377725nteger @ C @ D2 )
% 14.25/8.11 => ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ B )
% 14.25/8.11 => ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ C )
% 14.25/8.11 => ( ord_le3102999989581377725nteger @ ( times_3573771949741848930nteger @ A2 @ C ) @ ( times_3573771949741848930nteger @ B @ D2 ) ) ) ) ) ) ).
% 14.25/8.11
% 14.25/8.11 thf(fact_5717_mult__mono,axiom,
% 14.25/8.11 ! [A2: rat,B: rat,C: rat,D2: rat] :
% 14.25/8.11 ( ( ord_less_eq_rat @ A2 @ B )
% 14.25/8.11 => ( ( ord_less_eq_rat @ C @ D2 )
% 14.25/8.11 => ( ( ord_less_eq_rat @ zero_zero_rat @ B )
% 14.25/8.11 => ( ( ord_less_eq_rat @ zero_zero_rat @ C )
% 14.25/8.11 => ( ord_less_eq_rat @ ( times_times_rat @ A2 @ C ) @ ( times_times_rat @ B @ D2 ) ) ) ) ) ) ).
% 14.25/8.11
% 14.25/8.11 thf(fact_5718_mult__mono,axiom,
% 14.25/8.11 ! [A2: nat,B: nat,C: nat,D2: nat] :
% 14.25/8.11 ( ( ord_less_eq_nat @ A2 @ B )
% 14.25/8.11 => ( ( ord_less_eq_nat @ C @ D2 )
% 14.25/8.11 => ( ( ord_less_eq_nat @ zero_zero_nat @ B )
% 14.25/8.11 => ( ( ord_less_eq_nat @ zero_zero_nat @ C )
% 14.25/8.11 => ( ord_less_eq_nat @ ( times_times_nat @ A2 @ C ) @ ( times_times_nat @ B @ D2 ) ) ) ) ) ) ).
% 14.25/8.11
% 14.25/8.11 thf(fact_5719_mult__mono,axiom,
% 14.25/8.11 ! [A2: int,B: int,C: int,D2: int] :
% 14.25/8.11 ( ( ord_less_eq_int @ A2 @ B )
% 14.25/8.11 => ( ( ord_less_eq_int @ C @ D2 )
% 14.25/8.11 => ( ( ord_less_eq_int @ zero_zero_int @ B )
% 14.25/8.11 => ( ( ord_less_eq_int @ zero_zero_int @ C )
% 14.25/8.11 => ( ord_less_eq_int @ ( times_times_int @ A2 @ C ) @ ( times_times_int @ B @ D2 ) ) ) ) ) ) ).
% 14.25/8.11
% 14.25/8.11 thf(fact_5720_not__one__le__zero,axiom,
% 14.25/8.11 ~ ( ord_le1926595141338095240atural @ one_one_Code_natural @ zero_z2226904508553997617atural ) ).
% 14.25/8.11
% 14.25/8.11 thf(fact_5721_not__one__le__zero,axiom,
% 14.25/8.11 ~ ( ord_le3102999989581377725nteger @ one_one_Code_integer @ zero_z3403309356797280102nteger ) ).
% 14.25/8.11
% 14.25/8.11 thf(fact_5722_not__one__le__zero,axiom,
% 14.25/8.11 ~ ( ord_less_eq_rat @ one_one_rat @ zero_zero_rat ) ).
% 14.25/8.11
% 14.25/8.11 thf(fact_5723_not__one__le__zero,axiom,
% 14.25/8.11 ~ ( ord_less_eq_nat @ one_one_nat @ zero_zero_nat ) ).
% 14.25/8.11
% 14.25/8.11 thf(fact_5724_not__one__le__zero,axiom,
% 14.25/8.11 ~ ( ord_less_eq_int @ one_one_int @ zero_zero_int ) ).
% 14.25/8.11
% 14.25/8.11 thf(fact_5725_linordered__nonzero__semiring__class_Ozero__le__one,axiom,
% 14.25/8.11 ord_le1926595141338095240atural @ zero_z2226904508553997617atural @ one_one_Code_natural ).
% 14.25/8.11
% 14.25/8.11 thf(fact_5726_linordered__nonzero__semiring__class_Ozero__le__one,axiom,
% 14.25/8.11 ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ one_one_Code_integer ).
% 14.25/8.11
% 14.25/8.11 thf(fact_5727_linordered__nonzero__semiring__class_Ozero__le__one,axiom,
% 14.25/8.11 ord_less_eq_rat @ zero_zero_rat @ one_one_rat ).
% 14.25/8.11
% 14.25/8.11 thf(fact_5728_linordered__nonzero__semiring__class_Ozero__le__one,axiom,
% 14.25/8.11 ord_less_eq_nat @ zero_zero_nat @ one_one_nat ).
% 14.25/8.11
% 14.25/8.11 thf(fact_5729_linordered__nonzero__semiring__class_Ozero__le__one,axiom,
% 14.25/8.11 ord_less_eq_int @ zero_zero_int @ one_one_int ).
% 14.25/8.11
% 14.25/8.11 thf(fact_5730_zero__less__one__class_Ozero__le__one,axiom,
% 14.25/8.11 ord_le1926595141338095240atural @ zero_z2226904508553997617atural @ one_one_Code_natural ).
% 14.25/8.11
% 14.25/8.11 thf(fact_5731_zero__less__one__class_Ozero__le__one,axiom,
% 14.25/8.11 ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ one_one_Code_integer ).
% 14.25/8.11
% 14.25/8.11 thf(fact_5732_zero__less__one__class_Ozero__le__one,axiom,
% 14.25/8.11 ord_less_eq_rat @ zero_zero_rat @ one_one_rat ).
% 14.25/8.11
% 14.25/8.11 thf(fact_5733_zero__less__one__class_Ozero__le__one,axiom,
% 14.25/8.11 ord_less_eq_nat @ zero_zero_nat @ one_one_nat ).
% 14.25/8.11
% 14.25/8.11 thf(fact_5734_zero__less__one__class_Ozero__le__one,axiom,
% 14.25/8.11 ord_less_eq_int @ zero_zero_int @ one_one_int ).
% 14.25/8.11
% 14.25/8.11 thf(fact_5735_le__iff__diff__le__0,axiom,
% 14.25/8.11 ( ord_le3102999989581377725nteger
% 14.25/8.11 = ( ^ [A6: code_integer,B6: code_integer] : ( ord_le3102999989581377725nteger @ ( minus_8373710615458151222nteger @ A6 @ B6 ) @ zero_z3403309356797280102nteger ) ) ) ).
% 14.25/8.11
% 14.25/8.11 thf(fact_5736_le__iff__diff__le__0,axiom,
% 14.25/8.11 ( ord_less_eq_rat
% 14.25/8.11 = ( ^ [A6: rat,B6: rat] : ( ord_less_eq_rat @ ( minus_minus_rat @ A6 @ B6 ) @ zero_zero_rat ) ) ) ).
% 14.25/8.11
% 14.25/8.11 thf(fact_5737_le__iff__diff__le__0,axiom,
% 14.25/8.11 ( ord_less_eq_int
% 14.25/8.11 = ( ^ [A6: int,B6: int] : ( ord_less_eq_int @ ( minus_minus_int @ A6 @ B6 ) @ zero_zero_int ) ) ) ).
% 14.25/8.11
% 14.25/8.11 thf(fact_5738_pos__add__strict,axiom,
% 14.25/8.11 ! [A2: nat,B: nat,C: nat] :
% 14.25/8.11 ( ( ord_less_nat @ zero_zero_nat @ A2 )
% 14.25/8.11 => ( ( ord_less_nat @ B @ C )
% 14.25/8.11 => ( ord_less_nat @ B @ ( plus_plus_nat @ A2 @ C ) ) ) ) ).
% 14.25/8.11
% 14.25/8.11 thf(fact_5739_pos__add__strict,axiom,
% 14.25/8.11 ! [A2: int,B: int,C: int] :
% 14.25/8.11 ( ( ord_less_int @ zero_zero_int @ A2 )
% 14.25/8.11 => ( ( ord_less_int @ B @ C )
% 14.25/8.11 => ( ord_less_int @ B @ ( plus_plus_int @ A2 @ C ) ) ) ) ).
% 14.25/8.11
% 14.25/8.11 thf(fact_5740_pos__add__strict,axiom,
% 14.25/8.11 ! [A2: code_integer,B: code_integer,C: code_integer] :
% 14.25/8.11 ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ A2 )
% 14.25/8.11 => ( ( ord_le6747313008572928689nteger @ B @ C )
% 14.25/8.11 => ( ord_le6747313008572928689nteger @ B @ ( plus_p5714425477246183910nteger @ A2 @ C ) ) ) ) ).
% 14.25/8.11
% 14.25/8.11 thf(fact_5741_pos__add__strict,axiom,
% 14.25/8.11 ! [A2: rat,B: rat,C: rat] :
% 14.25/8.11 ( ( ord_less_rat @ zero_zero_rat @ A2 )
% 14.25/8.11 => ( ( ord_less_rat @ B @ C )
% 14.25/8.11 => ( ord_less_rat @ B @ ( plus_plus_rat @ A2 @ C ) ) ) ) ).
% 14.25/8.11
% 14.25/8.11 thf(fact_5742_canonically__ordered__monoid__add__class_OlessE,axiom,
% 14.25/8.11 ! [A2: nat,B: nat] :
% 14.25/8.11 ( ( ord_less_nat @ A2 @ B )
% 14.25/8.11 => ~ ! [C2: nat] :
% 14.25/8.11 ( ( B
% 14.25/8.11 = ( plus_plus_nat @ A2 @ C2 ) )
% 14.25/8.11 => ( C2 = zero_zero_nat ) ) ) ).
% 14.25/8.11
% 14.25/8.11 thf(fact_5743_add__pos__pos,axiom,
% 14.25/8.11 ! [A2: nat,B: nat] :
% 14.25/8.11 ( ( ord_less_nat @ zero_zero_nat @ A2 )
% 14.25/8.11 => ( ( ord_less_nat @ zero_zero_nat @ B )
% 14.25/8.11 => ( ord_less_nat @ zero_zero_nat @ ( plus_plus_nat @ A2 @ B ) ) ) ) ).
% 14.25/8.11
% 14.25/8.11 thf(fact_5744_add__pos__pos,axiom,
% 14.25/8.11 ! [A2: int,B: int] :
% 14.25/8.11 ( ( ord_less_int @ zero_zero_int @ A2 )
% 14.25/8.11 => ( ( ord_less_int @ zero_zero_int @ B )
% 14.25/8.11 => ( ord_less_int @ zero_zero_int @ ( plus_plus_int @ A2 @ B ) ) ) ) ).
% 14.25/8.11
% 14.25/8.11 thf(fact_5745_add__pos__pos,axiom,
% 14.25/8.11 ! [A2: code_integer,B: code_integer] :
% 14.25/8.11 ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ A2 )
% 14.25/8.11 => ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ B )
% 14.25/8.11 => ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ ( plus_p5714425477246183910nteger @ A2 @ B ) ) ) ) ).
% 14.25/8.11
% 14.25/8.11 thf(fact_5746_add__pos__pos,axiom,
% 14.25/8.11 ! [A2: rat,B: rat] :
% 14.25/8.11 ( ( ord_less_rat @ zero_zero_rat @ A2 )
% 14.25/8.11 => ( ( ord_less_rat @ zero_zero_rat @ B )
% 14.25/8.11 => ( ord_less_rat @ zero_zero_rat @ ( plus_plus_rat @ A2 @ B ) ) ) ) ).
% 14.25/8.11
% 14.25/8.11 thf(fact_5747_add__neg__neg,axiom,
% 14.25/8.11 ! [A2: nat,B: nat] :
% 14.25/8.11 ( ( ord_less_nat @ A2 @ zero_zero_nat )
% 14.25/8.11 => ( ( ord_less_nat @ B @ zero_zero_nat )
% 14.25/8.11 => ( ord_less_nat @ ( plus_plus_nat @ A2 @ B ) @ zero_zero_nat ) ) ) ).
% 14.25/8.11
% 14.25/8.11 thf(fact_5748_add__neg__neg,axiom,
% 14.25/8.11 ! [A2: int,B: int] :
% 14.25/8.11 ( ( ord_less_int @ A2 @ zero_zero_int )
% 14.25/8.11 => ( ( ord_less_int @ B @ zero_zero_int )
% 14.25/8.11 => ( ord_less_int @ ( plus_plus_int @ A2 @ B ) @ zero_zero_int ) ) ) ).
% 14.25/8.11
% 14.25/8.11 thf(fact_5749_add__neg__neg,axiom,
% 14.25/8.11 ! [A2: code_integer,B: code_integer] :
% 14.25/8.11 ( ( ord_le6747313008572928689nteger @ A2 @ zero_z3403309356797280102nteger )
% 14.25/8.11 => ( ( ord_le6747313008572928689nteger @ B @ zero_z3403309356797280102nteger )
% 14.25/8.11 => ( ord_le6747313008572928689nteger @ ( plus_p5714425477246183910nteger @ A2 @ B ) @ zero_z3403309356797280102nteger ) ) ) ).
% 14.25/8.11
% 14.25/8.11 thf(fact_5750_add__neg__neg,axiom,
% 14.25/8.11 ! [A2: rat,B: rat] :
% 14.25/8.11 ( ( ord_less_rat @ A2 @ zero_zero_rat )
% 14.25/8.11 => ( ( ord_less_rat @ B @ zero_zero_rat )
% 14.25/8.11 => ( ord_less_rat @ ( plus_plus_rat @ A2 @ B ) @ zero_zero_rat ) ) ) ).
% 14.25/8.11
% 14.25/8.11 thf(fact_5751_mult__neg__neg,axiom,
% 14.25/8.11 ! [A2: int,B: int] :
% 14.25/8.11 ( ( ord_less_int @ A2 @ zero_zero_int )
% 14.25/8.11 => ( ( ord_less_int @ B @ zero_zero_int )
% 14.25/8.11 => ( ord_less_int @ zero_zero_int @ ( times_times_int @ A2 @ B ) ) ) ) ).
% 14.25/8.11
% 14.25/8.11 thf(fact_5752_mult__neg__neg,axiom,
% 14.25/8.11 ! [A2: code_integer,B: code_integer] :
% 14.25/8.11 ( ( ord_le6747313008572928689nteger @ A2 @ zero_z3403309356797280102nteger )
% 14.25/8.11 => ( ( ord_le6747313008572928689nteger @ B @ zero_z3403309356797280102nteger )
% 14.25/8.11 => ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ ( times_3573771949741848930nteger @ A2 @ B ) ) ) ) ).
% 14.25/8.11
% 14.25/8.11 thf(fact_5753_mult__neg__neg,axiom,
% 14.25/8.11 ! [A2: rat,B: rat] :
% 14.25/8.11 ( ( ord_less_rat @ A2 @ zero_zero_rat )
% 14.25/8.11 => ( ( ord_less_rat @ B @ zero_zero_rat )
% 14.25/8.11 => ( ord_less_rat @ zero_zero_rat @ ( times_times_rat @ A2 @ B ) ) ) ) ).
% 14.25/8.11
% 14.25/8.11 thf(fact_5754_not__square__less__zero,axiom,
% 14.25/8.11 ! [A2: int] :
% 14.25/8.11 ~ ( ord_less_int @ ( times_times_int @ A2 @ A2 ) @ zero_zero_int ) ).
% 14.25/8.11
% 14.25/8.11 thf(fact_5755_not__square__less__zero,axiom,
% 14.25/8.11 ! [A2: code_integer] :
% 14.25/8.11 ~ ( ord_le6747313008572928689nteger @ ( times_3573771949741848930nteger @ A2 @ A2 ) @ zero_z3403309356797280102nteger ) ).
% 14.25/8.11
% 14.25/8.11 thf(fact_5756_not__square__less__zero,axiom,
% 14.25/8.11 ! [A2: rat] :
% 14.25/8.11 ~ ( ord_less_rat @ ( times_times_rat @ A2 @ A2 ) @ zero_zero_rat ) ).
% 14.25/8.11
% 14.25/8.11 thf(fact_5757_mult__less__0__iff,axiom,
% 14.25/8.11 ! [A2: int,B: int] :
% 14.25/8.11 ( ( ord_less_int @ ( times_times_int @ A2 @ B ) @ zero_zero_int )
% 14.25/8.11 = ( ( ( ord_less_int @ zero_zero_int @ A2 )
% 14.25/8.11 & ( ord_less_int @ B @ zero_zero_int ) )
% 14.25/8.11 | ( ( ord_less_int @ A2 @ zero_zero_int )
% 14.25/8.11 & ( ord_less_int @ zero_zero_int @ B ) ) ) ) ).
% 14.25/8.11
% 14.25/8.11 thf(fact_5758_mult__less__0__iff,axiom,
% 14.25/8.11 ! [A2: code_integer,B: code_integer] :
% 14.25/8.11 ( ( ord_le6747313008572928689nteger @ ( times_3573771949741848930nteger @ A2 @ B ) @ zero_z3403309356797280102nteger )
% 14.25/8.11 = ( ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ A2 )
% 14.25/8.11 & ( ord_le6747313008572928689nteger @ B @ zero_z3403309356797280102nteger ) )
% 14.25/8.11 | ( ( ord_le6747313008572928689nteger @ A2 @ zero_z3403309356797280102nteger )
% 14.25/8.11 & ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ B ) ) ) ) ).
% 14.25/8.11
% 14.25/8.11 thf(fact_5759_mult__less__0__iff,axiom,
% 14.25/8.11 ! [A2: rat,B: rat] :
% 14.25/8.11 ( ( ord_less_rat @ ( times_times_rat @ A2 @ B ) @ zero_zero_rat )
% 14.25/8.11 = ( ( ( ord_less_rat @ zero_zero_rat @ A2 )
% 14.25/8.11 & ( ord_less_rat @ B @ zero_zero_rat ) )
% 14.25/8.11 | ( ( ord_less_rat @ A2 @ zero_zero_rat )
% 14.25/8.11 & ( ord_less_rat @ zero_zero_rat @ B ) ) ) ) ).
% 14.25/8.11
% 14.25/8.11 thf(fact_5760_mult__neg__pos,axiom,
% 14.25/8.11 ! [A2: nat,B: nat] :
% 14.25/8.11 ( ( ord_less_nat @ A2 @ zero_zero_nat )
% 14.25/8.11 => ( ( ord_less_nat @ zero_zero_nat @ B )
% 14.25/8.11 => ( ord_less_nat @ ( times_times_nat @ A2 @ B ) @ zero_zero_nat ) ) ) ).
% 14.25/8.11
% 14.25/8.11 thf(fact_5761_mult__neg__pos,axiom,
% 14.25/8.11 ! [A2: int,B: int] :
% 14.25/8.11 ( ( ord_less_int @ A2 @ zero_zero_int )
% 14.25/8.11 => ( ( ord_less_int @ zero_zero_int @ B )
% 14.25/8.11 => ( ord_less_int @ ( times_times_int @ A2 @ B ) @ zero_zero_int ) ) ) ).
% 14.25/8.11
% 14.25/8.11 thf(fact_5762_mult__neg__pos,axiom,
% 14.25/8.11 ! [A2: code_integer,B: code_integer] :
% 14.25/8.11 ( ( ord_le6747313008572928689nteger @ A2 @ zero_z3403309356797280102nteger )
% 14.25/8.11 => ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ B )
% 14.25/8.11 => ( ord_le6747313008572928689nteger @ ( times_3573771949741848930nteger @ A2 @ B ) @ zero_z3403309356797280102nteger ) ) ) ).
% 14.25/8.11
% 14.25/8.11 thf(fact_5763_mult__neg__pos,axiom,
% 14.25/8.11 ! [A2: rat,B: rat] :
% 14.25/8.11 ( ( ord_less_rat @ A2 @ zero_zero_rat )
% 14.25/8.11 => ( ( ord_less_rat @ zero_zero_rat @ B )
% 14.25/8.11 => ( ord_less_rat @ ( times_times_rat @ A2 @ B ) @ zero_zero_rat ) ) ) ).
% 14.25/8.11
% 14.25/8.11 thf(fact_5764_mult__pos__neg,axiom,
% 14.25/8.11 ! [A2: nat,B: nat] :
% 14.25/8.11 ( ( ord_less_nat @ zero_zero_nat @ A2 )
% 14.25/8.11 => ( ( ord_less_nat @ B @ zero_zero_nat )
% 14.25/8.11 => ( ord_less_nat @ ( times_times_nat @ A2 @ B ) @ zero_zero_nat ) ) ) ).
% 14.25/8.11
% 14.25/8.11 thf(fact_5765_mult__pos__neg,axiom,
% 14.25/8.11 ! [A2: int,B: int] :
% 14.25/8.11 ( ( ord_less_int @ zero_zero_int @ A2 )
% 14.25/8.11 => ( ( ord_less_int @ B @ zero_zero_int )
% 14.25/8.11 => ( ord_less_int @ ( times_times_int @ A2 @ B ) @ zero_zero_int ) ) ) ).
% 14.25/8.11
% 14.25/8.11 thf(fact_5766_mult__pos__neg,axiom,
% 14.25/8.11 ! [A2: code_integer,B: code_integer] :
% 14.25/8.11 ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ A2 )
% 14.25/8.11 => ( ( ord_le6747313008572928689nteger @ B @ zero_z3403309356797280102nteger )
% 14.25/8.11 => ( ord_le6747313008572928689nteger @ ( times_3573771949741848930nteger @ A2 @ B ) @ zero_z3403309356797280102nteger ) ) ) ).
% 14.25/8.11
% 14.25/8.11 thf(fact_5767_mult__pos__neg,axiom,
% 14.25/8.11 ! [A2: rat,B: rat] :
% 14.25/8.11 ( ( ord_less_rat @ zero_zero_rat @ A2 )
% 14.25/8.11 => ( ( ord_less_rat @ B @ zero_zero_rat )
% 14.25/8.11 => ( ord_less_rat @ ( times_times_rat @ A2 @ B ) @ zero_zero_rat ) ) ) ).
% 14.25/8.11
% 14.25/8.11 thf(fact_5768_mult__pos__pos,axiom,
% 14.25/8.11 ! [A2: nat,B: nat] :
% 14.25/8.11 ( ( ord_less_nat @ zero_zero_nat @ A2 )
% 14.25/8.11 => ( ( ord_less_nat @ zero_zero_nat @ B )
% 14.25/8.11 => ( ord_less_nat @ zero_zero_nat @ ( times_times_nat @ A2 @ B ) ) ) ) ).
% 14.25/8.11
% 14.25/8.11 thf(fact_5769_mult__pos__pos,axiom,
% 14.25/8.11 ! [A2: int,B: int] :
% 14.25/8.11 ( ( ord_less_int @ zero_zero_int @ A2 )
% 14.25/8.11 => ( ( ord_less_int @ zero_zero_int @ B )
% 14.25/8.11 => ( ord_less_int @ zero_zero_int @ ( times_times_int @ A2 @ B ) ) ) ) ).
% 14.25/8.11
% 14.25/8.11 thf(fact_5770_mult__pos__pos,axiom,
% 14.25/8.11 ! [A2: code_integer,B: code_integer] :
% 14.25/8.11 ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ A2 )
% 14.25/8.11 => ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ B )
% 14.25/8.11 => ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ ( times_3573771949741848930nteger @ A2 @ B ) ) ) ) ).
% 14.25/8.11
% 14.25/8.11 thf(fact_5771_mult__pos__pos,axiom,
% 14.25/8.11 ! [A2: rat,B: rat] :
% 14.25/8.11 ( ( ord_less_rat @ zero_zero_rat @ A2 )
% 14.25/8.11 => ( ( ord_less_rat @ zero_zero_rat @ B )
% 14.25/8.11 => ( ord_less_rat @ zero_zero_rat @ ( times_times_rat @ A2 @ B ) ) ) ) ).
% 14.25/8.11
% 14.25/8.11 thf(fact_5772_mult__pos__neg2,axiom,
% 14.25/8.11 ! [A2: nat,B: nat] :
% 14.25/8.11 ( ( ord_less_nat @ zero_zero_nat @ A2 )
% 14.25/8.11 => ( ( ord_less_nat @ B @ zero_zero_nat )
% 14.25/8.11 => ( ord_less_nat @ ( times_times_nat @ B @ A2 ) @ zero_zero_nat ) ) ) ).
% 14.25/8.11
% 14.25/8.11 thf(fact_5773_mult__pos__neg2,axiom,
% 14.25/8.11 ! [A2: int,B: int] :
% 14.25/8.11 ( ( ord_less_int @ zero_zero_int @ A2 )
% 14.25/8.11 => ( ( ord_less_int @ B @ zero_zero_int )
% 14.25/8.11 => ( ord_less_int @ ( times_times_int @ B @ A2 ) @ zero_zero_int ) ) ) ).
% 14.25/8.11
% 14.25/8.11 thf(fact_5774_mult__pos__neg2,axiom,
% 14.25/8.11 ! [A2: code_integer,B: code_integer] :
% 14.25/8.11 ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ A2 )
% 14.25/8.11 => ( ( ord_le6747313008572928689nteger @ B @ zero_z3403309356797280102nteger )
% 14.25/8.11 => ( ord_le6747313008572928689nteger @ ( times_3573771949741848930nteger @ B @ A2 ) @ zero_z3403309356797280102nteger ) ) ) ).
% 14.25/8.11
% 14.25/8.11 thf(fact_5775_mult__pos__neg2,axiom,
% 14.25/8.11 ! [A2: rat,B: rat] :
% 14.25/8.11 ( ( ord_less_rat @ zero_zero_rat @ A2 )
% 14.25/8.11 => ( ( ord_less_rat @ B @ zero_zero_rat )
% 14.25/8.11 => ( ord_less_rat @ ( times_times_rat @ B @ A2 ) @ zero_zero_rat ) ) ) ).
% 14.25/8.11
% 14.25/8.11 thf(fact_5776_zero__less__mult__iff,axiom,
% 14.25/8.11 ! [A2: int,B: int] :
% 14.25/8.11 ( ( ord_less_int @ zero_zero_int @ ( times_times_int @ A2 @ B ) )
% 14.25/8.11 = ( ( ( ord_less_int @ zero_zero_int @ A2 )
% 14.25/8.11 & ( ord_less_int @ zero_zero_int @ B ) )
% 14.25/8.11 | ( ( ord_less_int @ A2 @ zero_zero_int )
% 14.25/8.11 & ( ord_less_int @ B @ zero_zero_int ) ) ) ) ).
% 14.25/8.11
% 14.25/8.11 thf(fact_5777_zero__less__mult__iff,axiom,
% 14.25/8.11 ! [A2: code_integer,B: code_integer] :
% 14.25/8.11 ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ ( times_3573771949741848930nteger @ A2 @ B ) )
% 14.25/8.11 = ( ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ A2 )
% 14.25/8.11 & ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ B ) )
% 14.25/8.11 | ( ( ord_le6747313008572928689nteger @ A2 @ zero_z3403309356797280102nteger )
% 14.25/8.11 & ( ord_le6747313008572928689nteger @ B @ zero_z3403309356797280102nteger ) ) ) ) ).
% 14.25/8.11
% 14.25/8.11 thf(fact_5778_zero__less__mult__iff,axiom,
% 14.25/8.11 ! [A2: rat,B: rat] :
% 14.25/8.11 ( ( ord_less_rat @ zero_zero_rat @ ( times_times_rat @ A2 @ B ) )
% 14.25/8.11 = ( ( ( ord_less_rat @ zero_zero_rat @ A2 )
% 14.25/8.11 & ( ord_less_rat @ zero_zero_rat @ B ) )
% 14.25/8.11 | ( ( ord_less_rat @ A2 @ zero_zero_rat )
% 14.25/8.11 & ( ord_less_rat @ B @ zero_zero_rat ) ) ) ) ).
% 14.25/8.11
% 14.25/8.11 thf(fact_5779_zero__less__mult__pos,axiom,
% 14.25/8.11 ! [A2: nat,B: nat] :
% 14.25/8.11 ( ( ord_less_nat @ zero_zero_nat @ ( times_times_nat @ A2 @ B ) )
% 14.25/8.11 => ( ( ord_less_nat @ zero_zero_nat @ A2 )
% 14.25/8.11 => ( ord_less_nat @ zero_zero_nat @ B ) ) ) ).
% 14.25/8.11
% 14.25/8.11 thf(fact_5780_zero__less__mult__pos,axiom,
% 14.25/8.11 ! [A2: int,B: int] :
% 14.25/8.11 ( ( ord_less_int @ zero_zero_int @ ( times_times_int @ A2 @ B ) )
% 14.25/8.11 => ( ( ord_less_int @ zero_zero_int @ A2 )
% 14.25/8.11 => ( ord_less_int @ zero_zero_int @ B ) ) ) ).
% 14.25/8.11
% 14.25/8.11 thf(fact_5781_zero__less__mult__pos,axiom,
% 14.25/8.11 ! [A2: code_integer,B: code_integer] :
% 14.25/8.11 ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ ( times_3573771949741848930nteger @ A2 @ B ) )
% 14.25/8.11 => ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ A2 )
% 14.25/8.11 => ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ B ) ) ) ).
% 14.25/8.11
% 14.25/8.11 thf(fact_5782_zero__less__mult__pos,axiom,
% 14.25/8.11 ! [A2: rat,B: rat] :
% 14.25/8.11 ( ( ord_less_rat @ zero_zero_rat @ ( times_times_rat @ A2 @ B ) )
% 14.25/8.11 => ( ( ord_less_rat @ zero_zero_rat @ A2 )
% 14.25/8.11 => ( ord_less_rat @ zero_zero_rat @ B ) ) ) ).
% 14.25/8.11
% 14.25/8.11 thf(fact_5783_zero__less__mult__pos2,axiom,
% 14.25/8.11 ! [B: nat,A2: nat] :
% 14.25/8.11 ( ( ord_less_nat @ zero_zero_nat @ ( times_times_nat @ B @ A2 ) )
% 14.25/8.11 => ( ( ord_less_nat @ zero_zero_nat @ A2 )
% 14.25/8.11 => ( ord_less_nat @ zero_zero_nat @ B ) ) ) ).
% 14.25/8.11
% 14.25/8.11 thf(fact_5784_zero__less__mult__pos2,axiom,
% 14.25/8.11 ! [B: int,A2: int] :
% 14.25/8.11 ( ( ord_less_int @ zero_zero_int @ ( times_times_int @ B @ A2 ) )
% 14.25/8.11 => ( ( ord_less_int @ zero_zero_int @ A2 )
% 14.25/8.11 => ( ord_less_int @ zero_zero_int @ B ) ) ) ).
% 14.25/8.11
% 14.25/8.11 thf(fact_5785_zero__less__mult__pos2,axiom,
% 14.25/8.11 ! [B: code_integer,A2: code_integer] :
% 14.25/8.11 ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ ( times_3573771949741848930nteger @ B @ A2 ) )
% 14.25/8.11 => ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ A2 )
% 14.25/8.11 => ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ B ) ) ) ).
% 14.25/8.11
% 14.25/8.11 thf(fact_5786_zero__less__mult__pos2,axiom,
% 14.25/8.11 ! [B: rat,A2: rat] :
% 14.25/8.11 ( ( ord_less_rat @ zero_zero_rat @ ( times_times_rat @ B @ A2 ) )
% 14.25/8.11 => ( ( ord_less_rat @ zero_zero_rat @ A2 )
% 14.25/8.11 => ( ord_less_rat @ zero_zero_rat @ B ) ) ) ).
% 14.25/8.11
% 14.25/8.11 thf(fact_5787_mult__less__cancel__left__neg,axiom,
% 14.25/8.11 ! [C: int,A2: int,B: int] :
% 14.25/8.11 ( ( ord_less_int @ C @ zero_zero_int )
% 14.25/8.11 => ( ( ord_less_int @ ( times_times_int @ C @ A2 ) @ ( times_times_int @ C @ B ) )
% 14.25/8.11 = ( ord_less_int @ B @ A2 ) ) ) ).
% 14.25/8.11
% 14.25/8.11 thf(fact_5788_mult__less__cancel__left__neg,axiom,
% 14.25/8.11 ! [C: code_integer,A2: code_integer,B: code_integer] :
% 14.25/8.11 ( ( ord_le6747313008572928689nteger @ C @ zero_z3403309356797280102nteger )
% 14.25/8.11 => ( ( ord_le6747313008572928689nteger @ ( times_3573771949741848930nteger @ C @ A2 ) @ ( times_3573771949741848930nteger @ C @ B ) )
% 14.25/8.11 = ( ord_le6747313008572928689nteger @ B @ A2 ) ) ) ).
% 14.25/8.11
% 14.25/8.11 thf(fact_5789_mult__less__cancel__left__neg,axiom,
% 14.25/8.11 ! [C: rat,A2: rat,B: rat] :
% 14.25/8.11 ( ( ord_less_rat @ C @ zero_zero_rat )
% 14.25/8.11 => ( ( ord_less_rat @ ( times_times_rat @ C @ A2 ) @ ( times_times_rat @ C @ B ) )
% 14.25/8.11 = ( ord_less_rat @ B @ A2 ) ) ) ).
% 14.25/8.11
% 14.25/8.11 thf(fact_5790_mult__less__cancel__left__pos,axiom,
% 14.25/8.11 ! [C: int,A2: int,B: int] :
% 14.25/8.11 ( ( ord_less_int @ zero_zero_int @ C )
% 14.25/8.11 => ( ( ord_less_int @ ( times_times_int @ C @ A2 ) @ ( times_times_int @ C @ B ) )
% 14.25/8.11 = ( ord_less_int @ A2 @ B ) ) ) ).
% 14.25/8.11
% 14.25/8.11 thf(fact_5791_mult__less__cancel__left__pos,axiom,
% 14.25/8.11 ! [C: code_integer,A2: code_integer,B: code_integer] :
% 14.25/8.11 ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ C )
% 14.25/8.11 => ( ( ord_le6747313008572928689nteger @ ( times_3573771949741848930nteger @ C @ A2 ) @ ( times_3573771949741848930nteger @ C @ B ) )
% 14.25/8.11 = ( ord_le6747313008572928689nteger @ A2 @ B ) ) ) ).
% 14.25/8.11
% 14.25/8.11 thf(fact_5792_mult__less__cancel__left__pos,axiom,
% 14.25/8.11 ! [C: rat,A2: rat,B: rat] :
% 14.25/8.11 ( ( ord_less_rat @ zero_zero_rat @ C )
% 14.25/8.11 => ( ( ord_less_rat @ ( times_times_rat @ C @ A2 ) @ ( times_times_rat @ C @ B ) )
% 14.25/8.11 = ( ord_less_rat @ A2 @ B ) ) ) ).
% 14.25/8.11
% 14.25/8.11 thf(fact_5793_mult__strict__left__mono__neg,axiom,
% 14.25/8.11 ! [B: int,A2: int,C: int] :
% 14.25/8.11 ( ( ord_less_int @ B @ A2 )
% 14.25/8.11 => ( ( ord_less_int @ C @ zero_zero_int )
% 14.25/8.11 => ( ord_less_int @ ( times_times_int @ C @ A2 ) @ ( times_times_int @ C @ B ) ) ) ) ).
% 14.25/8.11
% 14.25/8.11 thf(fact_5794_mult__strict__left__mono__neg,axiom,
% 14.25/8.11 ! [B: code_integer,A2: code_integer,C: code_integer] :
% 14.25/8.11 ( ( ord_le6747313008572928689nteger @ B @ A2 )
% 14.25/8.11 => ( ( ord_le6747313008572928689nteger @ C @ zero_z3403309356797280102nteger )
% 14.25/8.11 => ( ord_le6747313008572928689nteger @ ( times_3573771949741848930nteger @ C @ A2 ) @ ( times_3573771949741848930nteger @ C @ B ) ) ) ) ).
% 14.25/8.11
% 14.25/8.11 thf(fact_5795_mult__strict__left__mono__neg,axiom,
% 14.25/8.11 ! [B: rat,A2: rat,C: rat] :
% 14.25/8.11 ( ( ord_less_rat @ B @ A2 )
% 14.25/8.11 => ( ( ord_less_rat @ C @ zero_zero_rat )
% 14.25/8.11 => ( ord_less_rat @ ( times_times_rat @ C @ A2 ) @ ( times_times_rat @ C @ B ) ) ) ) ).
% 14.25/8.11
% 14.25/8.11 thf(fact_5796_mult__strict__left__mono,axiom,
% 14.25/8.11 ! [A2: nat,B: nat,C: nat] :
% 14.25/8.11 ( ( ord_less_nat @ A2 @ B )
% 14.25/8.11 => ( ( ord_less_nat @ zero_zero_nat @ C )
% 14.25/8.11 => ( ord_less_nat @ ( times_times_nat @ C @ A2 ) @ ( times_times_nat @ C @ B ) ) ) ) ).
% 14.25/8.11
% 14.25/8.11 thf(fact_5797_mult__strict__left__mono,axiom,
% 14.25/8.11 ! [A2: int,B: int,C: int] :
% 14.25/8.11 ( ( ord_less_int @ A2 @ B )
% 14.25/8.11 => ( ( ord_less_int @ zero_zero_int @ C )
% 14.25/8.11 => ( ord_less_int @ ( times_times_int @ C @ A2 ) @ ( times_times_int @ C @ B ) ) ) ) ).
% 14.25/8.11
% 14.25/8.11 thf(fact_5798_mult__strict__left__mono,axiom,
% 14.25/8.11 ! [A2: code_integer,B: code_integer,C: code_integer] :
% 14.25/8.11 ( ( ord_le6747313008572928689nteger @ A2 @ B )
% 14.25/8.11 => ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ C )
% 14.25/8.11 => ( ord_le6747313008572928689nteger @ ( times_3573771949741848930nteger @ C @ A2 ) @ ( times_3573771949741848930nteger @ C @ B ) ) ) ) ).
% 14.25/8.11
% 14.25/8.11 thf(fact_5799_mult__strict__left__mono,axiom,
% 14.25/8.11 ! [A2: rat,B: rat,C: rat] :
% 14.25/8.11 ( ( ord_less_rat @ A2 @ B )
% 14.25/8.11 => ( ( ord_less_rat @ zero_zero_rat @ C )
% 14.25/8.11 => ( ord_less_rat @ ( times_times_rat @ C @ A2 ) @ ( times_times_rat @ C @ B ) ) ) ) ).
% 14.25/8.11
% 14.25/8.11 thf(fact_5800_mult__less__cancel__left__disj,axiom,
% 14.25/8.11 ! [C: int,A2: int,B: int] :
% 14.25/8.11 ( ( ord_less_int @ ( times_times_int @ C @ A2 ) @ ( times_times_int @ C @ B ) )
% 14.25/8.11 = ( ( ( ord_less_int @ zero_zero_int @ C )
% 14.25/8.11 & ( ord_less_int @ A2 @ B ) )
% 14.25/8.11 | ( ( ord_less_int @ C @ zero_zero_int )
% 14.25/8.11 & ( ord_less_int @ B @ A2 ) ) ) ) ).
% 14.25/8.11
% 14.25/8.11 thf(fact_5801_mult__less__cancel__left__disj,axiom,
% 14.25/8.11 ! [C: code_integer,A2: code_integer,B: code_integer] :
% 14.25/8.11 ( ( ord_le6747313008572928689nteger @ ( times_3573771949741848930nteger @ C @ A2 ) @ ( times_3573771949741848930nteger @ C @ B ) )
% 14.25/8.11 = ( ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ C )
% 14.25/8.11 & ( ord_le6747313008572928689nteger @ A2 @ B ) )
% 14.25/8.11 | ( ( ord_le6747313008572928689nteger @ C @ zero_z3403309356797280102nteger )
% 14.25/8.11 & ( ord_le6747313008572928689nteger @ B @ A2 ) ) ) ) ).
% 14.25/8.11
% 14.25/8.11 thf(fact_5802_mult__less__cancel__left__disj,axiom,
% 14.25/8.11 ! [C: rat,A2: rat,B: rat] :
% 14.25/8.11 ( ( ord_less_rat @ ( times_times_rat @ C @ A2 ) @ ( times_times_rat @ C @ B ) )
% 14.25/8.11 = ( ( ( ord_less_rat @ zero_zero_rat @ C )
% 14.25/8.11 & ( ord_less_rat @ A2 @ B ) )
% 14.25/8.11 | ( ( ord_less_rat @ C @ zero_zero_rat )
% 14.25/8.11 & ( ord_less_rat @ B @ A2 ) ) ) ) ).
% 14.25/8.11
% 14.25/8.11 thf(fact_5803_mult__strict__right__mono__neg,axiom,
% 14.25/8.11 ! [B: int,A2: int,C: int] :
% 14.25/8.11 ( ( ord_less_int @ B @ A2 )
% 14.25/8.11 => ( ( ord_less_int @ C @ zero_zero_int )
% 14.25/8.11 => ( ord_less_int @ ( times_times_int @ A2 @ C ) @ ( times_times_int @ B @ C ) ) ) ) ).
% 14.25/8.11
% 14.25/8.11 thf(fact_5804_mult__strict__right__mono__neg,axiom,
% 14.25/8.11 ! [B: code_integer,A2: code_integer,C: code_integer] :
% 14.25/8.11 ( ( ord_le6747313008572928689nteger @ B @ A2 )
% 14.25/8.11 => ( ( ord_le6747313008572928689nteger @ C @ zero_z3403309356797280102nteger )
% 14.25/8.11 => ( ord_le6747313008572928689nteger @ ( times_3573771949741848930nteger @ A2 @ C ) @ ( times_3573771949741848930nteger @ B @ C ) ) ) ) ).
% 14.25/8.11
% 14.25/8.11 thf(fact_5805_mult__strict__right__mono__neg,axiom,
% 14.25/8.11 ! [B: rat,A2: rat,C: rat] :
% 14.25/8.11 ( ( ord_less_rat @ B @ A2 )
% 14.25/8.11 => ( ( ord_less_rat @ C @ zero_zero_rat )
% 14.25/8.11 => ( ord_less_rat @ ( times_times_rat @ A2 @ C ) @ ( times_times_rat @ B @ C ) ) ) ) ).
% 14.25/8.11
% 14.25/8.11 thf(fact_5806_mult__strict__right__mono,axiom,
% 14.25/8.11 ! [A2: nat,B: nat,C: nat] :
% 14.25/8.11 ( ( ord_less_nat @ A2 @ B )
% 14.25/8.11 => ( ( ord_less_nat @ zero_zero_nat @ C )
% 14.25/8.11 => ( ord_less_nat @ ( times_times_nat @ A2 @ C ) @ ( times_times_nat @ B @ C ) ) ) ) ).
% 14.25/8.11
% 14.25/8.11 thf(fact_5807_mult__strict__right__mono,axiom,
% 14.25/8.11 ! [A2: int,B: int,C: int] :
% 14.25/8.11 ( ( ord_less_int @ A2 @ B )
% 14.25/8.11 => ( ( ord_less_int @ zero_zero_int @ C )
% 14.25/8.11 => ( ord_less_int @ ( times_times_int @ A2 @ C ) @ ( times_times_int @ B @ C ) ) ) ) ).
% 14.25/8.11
% 14.25/8.11 thf(fact_5808_mult__strict__right__mono,axiom,
% 14.25/8.11 ! [A2: code_integer,B: code_integer,C: code_integer] :
% 14.25/8.11 ( ( ord_le6747313008572928689nteger @ A2 @ B )
% 14.25/8.11 => ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ C )
% 14.25/8.11 => ( ord_le6747313008572928689nteger @ ( times_3573771949741848930nteger @ A2 @ C ) @ ( times_3573771949741848930nteger @ B @ C ) ) ) ) ).
% 14.25/8.11
% 14.25/8.11 thf(fact_5809_mult__strict__right__mono,axiom,
% 14.25/8.11 ! [A2: rat,B: rat,C: rat] :
% 14.25/8.11 ( ( ord_less_rat @ A2 @ B )
% 14.25/8.11 => ( ( ord_less_rat @ zero_zero_rat @ C )
% 14.25/8.11 => ( ord_less_rat @ ( times_times_rat @ A2 @ C ) @ ( times_times_rat @ B @ C ) ) ) ) ).
% 14.25/8.11
% 14.25/8.11 thf(fact_5810_mult__less__cancel__right__disj,axiom,
% 14.25/8.11 ! [A2: int,C: int,B: int] :
% 14.25/8.11 ( ( ord_less_int @ ( times_times_int @ A2 @ C ) @ ( times_times_int @ B @ C ) )
% 14.25/8.11 = ( ( ( ord_less_int @ zero_zero_int @ C )
% 14.25/8.11 & ( ord_less_int @ A2 @ B ) )
% 14.25/8.11 | ( ( ord_less_int @ C @ zero_zero_int )
% 14.25/8.11 & ( ord_less_int @ B @ A2 ) ) ) ) ).
% 14.25/8.11
% 14.25/8.11 thf(fact_5811_mult__less__cancel__right__disj,axiom,
% 14.25/8.11 ! [A2: code_integer,C: code_integer,B: code_integer] :
% 14.25/8.11 ( ( ord_le6747313008572928689nteger @ ( times_3573771949741848930nteger @ A2 @ C ) @ ( times_3573771949741848930nteger @ B @ C ) )
% 14.25/8.11 = ( ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ C )
% 14.25/8.11 & ( ord_le6747313008572928689nteger @ A2 @ B ) )
% 14.25/8.11 | ( ( ord_le6747313008572928689nteger @ C @ zero_z3403309356797280102nteger )
% 14.25/8.11 & ( ord_le6747313008572928689nteger @ B @ A2 ) ) ) ) ).
% 14.25/8.11
% 14.25/8.11 thf(fact_5812_mult__less__cancel__right__disj,axiom,
% 14.25/8.11 ! [A2: rat,C: rat,B: rat] :
% 14.25/8.11 ( ( ord_less_rat @ ( times_times_rat @ A2 @ C ) @ ( times_times_rat @ B @ C ) )
% 14.25/8.11 = ( ( ( ord_less_rat @ zero_zero_rat @ C )
% 14.25/8.11 & ( ord_less_rat @ A2 @ B ) )
% 14.25/8.11 | ( ( ord_less_rat @ C @ zero_zero_rat )
% 14.25/8.11 & ( ord_less_rat @ B @ A2 ) ) ) ) ).
% 14.25/8.11
% 14.25/8.11 thf(fact_5813_linordered__comm__semiring__strict__class_Ocomm__mult__strict__left__mono,axiom,
% 14.25/8.11 ! [A2: nat,B: nat,C: nat] :
% 14.25/8.11 ( ( ord_less_nat @ A2 @ B )
% 14.25/8.11 => ( ( ord_less_nat @ zero_zero_nat @ C )
% 14.25/8.11 => ( ord_less_nat @ ( times_times_nat @ C @ A2 ) @ ( times_times_nat @ C @ B ) ) ) ) ).
% 14.25/8.11
% 14.25/8.11 thf(fact_5814_linordered__comm__semiring__strict__class_Ocomm__mult__strict__left__mono,axiom,
% 14.25/8.11 ! [A2: int,B: int,C: int] :
% 14.25/8.11 ( ( ord_less_int @ A2 @ B )
% 14.25/8.11 => ( ( ord_less_int @ zero_zero_int @ C )
% 14.25/8.11 => ( ord_less_int @ ( times_times_int @ C @ A2 ) @ ( times_times_int @ C @ B ) ) ) ) ).
% 14.25/8.11
% 14.25/8.11 thf(fact_5815_linordered__comm__semiring__strict__class_Ocomm__mult__strict__left__mono,axiom,
% 14.25/8.11 ! [A2: code_integer,B: code_integer,C: code_integer] :
% 14.25/8.11 ( ( ord_le6747313008572928689nteger @ A2 @ B )
% 14.25/8.11 => ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ C )
% 14.25/8.11 => ( ord_le6747313008572928689nteger @ ( times_3573771949741848930nteger @ C @ A2 ) @ ( times_3573771949741848930nteger @ C @ B ) ) ) ) ).
% 14.25/8.11
% 14.25/8.11 thf(fact_5816_linordered__comm__semiring__strict__class_Ocomm__mult__strict__left__mono,axiom,
% 14.25/8.11 ! [A2: rat,B: rat,C: rat] :
% 14.25/8.11 ( ( ord_less_rat @ A2 @ B )
% 14.25/8.11 => ( ( ord_less_rat @ zero_zero_rat @ C )
% 14.25/8.11 => ( ord_less_rat @ ( times_times_rat @ C @ A2 ) @ ( times_times_rat @ C @ B ) ) ) ) ).
% 14.25/8.11
% 14.25/8.11 thf(fact_5817_less__numeral__extra_I1_J,axiom,
% 14.25/8.11 ord_le5570908160329646204atural @ zero_z2226904508553997617atural @ one_one_Code_natural ).
% 14.25/8.11
% 14.25/8.11 thf(fact_5818_less__numeral__extra_I1_J,axiom,
% 14.25/8.11 ord_less_nat @ zero_zero_nat @ one_one_nat ).
% 14.25/8.11
% 14.25/8.11 thf(fact_5819_less__numeral__extra_I1_J,axiom,
% 14.25/8.11 ord_less_int @ zero_zero_int @ one_one_int ).
% 14.25/8.11
% 14.25/8.11 thf(fact_5820_less__numeral__extra_I1_J,axiom,
% 14.25/8.11 ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ one_one_Code_integer ).
% 14.25/8.11
% 14.25/8.11 thf(fact_5821_less__numeral__extra_I1_J,axiom,
% 14.25/8.11 ord_less_rat @ zero_zero_rat @ one_one_rat ).
% 14.25/8.11
% 14.25/8.11 thf(fact_5822_zero__less__one,axiom,
% 14.25/8.11 ord_le5570908160329646204atural @ zero_z2226904508553997617atural @ one_one_Code_natural ).
% 14.25/8.11
% 14.25/8.11 thf(fact_5823_zero__less__one,axiom,
% 14.25/8.11 ord_less_nat @ zero_zero_nat @ one_one_nat ).
% 14.25/8.11
% 14.25/8.11 thf(fact_5824_zero__less__one,axiom,
% 14.25/8.11 ord_less_int @ zero_zero_int @ one_one_int ).
% 14.25/8.11
% 14.25/8.11 thf(fact_5825_zero__less__one,axiom,
% 14.25/8.11 ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ one_one_Code_integer ).
% 14.25/8.11
% 14.25/8.11 thf(fact_5826_zero__less__one,axiom,
% 14.25/8.11 ord_less_rat @ zero_zero_rat @ one_one_rat ).
% 14.25/8.11
% 14.25/8.11 thf(fact_5827_not__one__less__zero,axiom,
% 14.25/8.11 ~ ( ord_le5570908160329646204atural @ one_one_Code_natural @ zero_z2226904508553997617atural ) ).
% 14.25/8.11
% 14.25/8.11 thf(fact_5828_not__one__less__zero,axiom,
% 14.25/8.11 ~ ( ord_less_nat @ one_one_nat @ zero_zero_nat ) ).
% 14.25/8.11
% 14.25/8.11 thf(fact_5829_not__one__less__zero,axiom,
% 14.25/8.11 ~ ( ord_less_int @ one_one_int @ zero_zero_int ) ).
% 14.25/8.11
% 14.25/8.11 thf(fact_5830_not__one__less__zero,axiom,
% 14.25/8.11 ~ ( ord_le6747313008572928689nteger @ one_one_Code_integer @ zero_z3403309356797280102nteger ) ).
% 14.25/8.11
% 14.25/8.11 thf(fact_5831_not__one__less__zero,axiom,
% 14.25/8.11 ~ ( ord_less_rat @ one_one_rat @ zero_zero_rat ) ).
% 14.25/8.11
% 14.25/8.11 thf(fact_5832_less__iff__diff__less__0,axiom,
% 14.25/8.11 ( ord_less_int
% 14.25/8.11 = ( ^ [A6: int,B6: int] : ( ord_less_int @ ( minus_minus_int @ A6 @ B6 ) @ zero_zero_int ) ) ) ).
% 14.25/8.11
% 14.25/8.11 thf(fact_5833_less__iff__diff__less__0,axiom,
% 14.25/8.11 ( ord_le6747313008572928689nteger
% 14.25/8.11 = ( ^ [A6: code_integer,B6: code_integer] : ( ord_le6747313008572928689nteger @ ( minus_8373710615458151222nteger @ A6 @ B6 ) @ zero_z3403309356797280102nteger ) ) ) ).
% 14.25/8.11
% 14.25/8.11 thf(fact_5834_less__iff__diff__less__0,axiom,
% 14.25/8.11 ( ord_less_rat
% 14.25/8.11 = ( ^ [A6: rat,B6: rat] : ( ord_less_rat @ ( minus_minus_rat @ A6 @ B6 ) @ zero_zero_rat ) ) ) ).
% 14.25/8.11
% 14.25/8.11 thf(fact_5835_inj__on__mult,axiom,
% 14.25/8.11 ! [A2: code_integer,A: set_Code_integer] :
% 14.25/8.11 ( ( A2 != zero_z3403309356797280102nteger )
% 14.25/8.11 => ( inj_on1158064722259180589nteger @ ( times_3573771949741848930nteger @ A2 ) @ A ) ) ).
% 14.25/8.11
% 14.25/8.11 thf(fact_5836_inj__on__mult,axiom,
% 14.25/8.11 ! [A2: rat,A: set_rat] :
% 14.25/8.11 ( ( A2 != zero_zero_rat )
% 14.25/8.11 => ( inj_on_rat_rat @ ( times_times_rat @ A2 ) @ A ) ) ).
% 14.25/8.11
% 14.25/8.11 thf(fact_5837_inj__on__mult,axiom,
% 14.25/8.11 ! [A2: nat,A: set_nat] :
% 14.25/8.11 ( ( A2 != zero_zero_nat )
% 14.25/8.11 => ( inj_on_nat_nat @ ( times_times_nat @ A2 ) @ A ) ) ).
% 14.25/8.11
% 14.25/8.11 thf(fact_5838_inj__on__mult,axiom,
% 14.25/8.11 ! [A2: int,A: set_int] :
% 14.25/8.11 ( ( A2 != zero_zero_int )
% 14.25/8.11 => ( inj_on_int_int @ ( times_times_int @ A2 ) @ A ) ) ).
% 14.25/8.11
% 14.25/8.11 thf(fact_5839_nat__compl__induct,axiom,
% 14.25/8.11 ! [P: nat > $o,N2: nat] :
% 14.25/8.11 ( ( P @ zero_zero_nat )
% 14.25/8.11 => ( ! [N3: nat] :
% 14.25/8.11 ( ! [Nn: nat] :
% 14.25/8.11 ( ( ord_less_eq_nat @ Nn @ N3 )
% 14.25/8.11 => ( P @ Nn ) )
% 14.25/8.11 => ( P @ ( suc @ N3 ) ) )
% 14.25/8.11 => ( P @ N2 ) ) ) ).
% 14.25/8.11
% 14.25/8.11 thf(fact_5840_nat__compl__induct_H,axiom,
% 14.25/8.11 ! [P: nat > $o,N2: nat] :
% 14.25/8.11 ( ( P @ zero_zero_nat )
% 14.25/8.11 => ( ! [N3: nat] :
% 14.25/8.11 ( ! [Nn: nat] :
% 14.25/8.11 ( ( ord_less_eq_nat @ Nn @ N3 )
% 14.25/8.11 => ( P @ Nn ) )
% 14.25/8.11 => ( P @ ( suc @ N3 ) ) )
% 14.25/8.11 => ( P @ N2 ) ) ) ).
% 14.25/8.11
% 14.25/8.11 thf(fact_5841_Suc__to__right,axiom,
% 14.25/8.11 ! [N2: nat,M4: nat] :
% 14.25/8.11 ( ( ( suc @ N2 )
% 14.25/8.11 = M4 )
% 14.25/8.11 => ( N2
% 14.25/8.11 = ( minus_minus_nat @ M4 @ ( suc @ zero_zero_nat ) ) ) ) ).
% 14.25/8.11
% 14.25/8.11 thf(fact_5842_mult__less__mono1,axiom,
% 14.25/8.11 ! [I2: nat,J: nat,K3: nat] :
% 14.25/8.11 ( ( ord_less_nat @ I2 @ J )
% 14.25/8.11 => ( ( ord_less_nat @ zero_zero_nat @ K3 )
% 14.25/8.11 => ( ord_less_nat @ ( times_times_nat @ I2 @ K3 ) @ ( times_times_nat @ J @ K3 ) ) ) ) ).
% 14.25/8.11
% 14.25/8.11 thf(fact_5843_mult__less__mono2,axiom,
% 14.25/8.11 ! [I2: nat,J: nat,K3: nat] :
% 14.25/8.11 ( ( ord_less_nat @ I2 @ J )
% 14.25/8.11 => ( ( ord_less_nat @ zero_zero_nat @ K3 )
% 14.25/8.11 => ( ord_less_nat @ ( times_times_nat @ K3 @ I2 ) @ ( times_times_nat @ K3 @ J ) ) ) ) ).
% 14.25/8.11
% 14.25/8.11 thf(fact_5844_nat__mult__eq__cancel1,axiom,
% 14.25/8.11 ! [K3: nat,M4: nat,N2: nat] :
% 14.25/8.11 ( ( ord_less_nat @ zero_zero_nat @ K3 )
% 14.25/8.11 => ( ( ( times_times_nat @ K3 @ M4 )
% 14.25/8.11 = ( times_times_nat @ K3 @ N2 ) )
% 14.25/8.11 = ( M4 = N2 ) ) ) ).
% 14.25/8.11
% 14.25/8.11 thf(fact_5845_nat__mult__less__cancel1,axiom,
% 14.25/8.11 ! [K3: nat,M4: nat,N2: nat] :
% 14.25/8.11 ( ( ord_less_nat @ zero_zero_nat @ K3 )
% 14.25/8.11 => ( ( ord_less_nat @ ( times_times_nat @ K3 @ M4 ) @ ( times_times_nat @ K3 @ N2 ) )
% 14.25/8.11 = ( ord_less_nat @ M4 @ N2 ) ) ) ).
% 14.25/8.11
% 14.25/8.11 thf(fact_5846_nat__geq__1__eq__neqz,axiom,
% 14.25/8.11 ! [X: nat] :
% 14.25/8.11 ( ( ord_less_eq_nat @ one_one_nat @ X )
% 14.25/8.11 = ( X != zero_zero_nat ) ) ).
% 14.25/8.11
% 14.25/8.11 thf(fact_5847_mult__eq__self__implies__10,axiom,
% 14.25/8.11 ! [M4: nat,N2: nat] :
% 14.25/8.11 ( ( M4
% 14.25/8.11 = ( times_times_nat @ M4 @ N2 ) )
% 14.25/8.11 => ( ( N2 = one_one_nat )
% 14.25/8.11 | ( M4 = zero_zero_nat ) ) ) ).
% 14.25/8.11
% 14.25/8.11 thf(fact_5848_add__strict__increasing2,axiom,
% 14.25/8.11 ! [A2: code_integer,B: code_integer,C: code_integer] :
% 14.25/8.11 ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ A2 )
% 14.25/8.11 => ( ( ord_le6747313008572928689nteger @ B @ C )
% 14.25/8.11 => ( ord_le6747313008572928689nteger @ B @ ( plus_p5714425477246183910nteger @ A2 @ C ) ) ) ) ).
% 14.25/8.11
% 14.25/8.11 thf(fact_5849_add__strict__increasing2,axiom,
% 14.25/8.11 ! [A2: rat,B: rat,C: rat] :
% 14.25/8.11 ( ( ord_less_eq_rat @ zero_zero_rat @ A2 )
% 14.25/8.11 => ( ( ord_less_rat @ B @ C )
% 14.25/8.11 => ( ord_less_rat @ B @ ( plus_plus_rat @ A2 @ C ) ) ) ) ).
% 14.25/8.11
% 14.25/8.11 thf(fact_5850_add__strict__increasing2,axiom,
% 14.25/8.11 ! [A2: nat,B: nat,C: nat] :
% 14.25/8.11 ( ( ord_less_eq_nat @ zero_zero_nat @ A2 )
% 14.25/8.11 => ( ( ord_less_nat @ B @ C )
% 14.25/8.11 => ( ord_less_nat @ B @ ( plus_plus_nat @ A2 @ C ) ) ) ) ).
% 14.25/8.11
% 14.25/8.11 thf(fact_5851_add__strict__increasing2,axiom,
% 14.25/8.11 ! [A2: int,B: int,C: int] :
% 14.25/8.11 ( ( ord_less_eq_int @ zero_zero_int @ A2 )
% 14.25/8.11 => ( ( ord_less_int @ B @ C )
% 14.25/8.11 => ( ord_less_int @ B @ ( plus_plus_int @ A2 @ C ) ) ) ) ).
% 14.25/8.11
% 14.25/8.11 thf(fact_5852_add__strict__increasing,axiom,
% 14.25/8.11 ! [A2: code_integer,B: code_integer,C: code_integer] :
% 14.25/8.11 ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ A2 )
% 14.25/8.11 => ( ( ord_le3102999989581377725nteger @ B @ C )
% 14.25/8.11 => ( ord_le6747313008572928689nteger @ B @ ( plus_p5714425477246183910nteger @ A2 @ C ) ) ) ) ).
% 14.25/8.11
% 14.25/8.11 thf(fact_5853_add__strict__increasing,axiom,
% 14.25/8.11 ! [A2: rat,B: rat,C: rat] :
% 14.25/8.11 ( ( ord_less_rat @ zero_zero_rat @ A2 )
% 14.25/8.11 => ( ( ord_less_eq_rat @ B @ C )
% 14.25/8.11 => ( ord_less_rat @ B @ ( plus_plus_rat @ A2 @ C ) ) ) ) ).
% 14.25/8.11
% 14.25/8.11 thf(fact_5854_add__strict__increasing,axiom,
% 14.25/8.11 ! [A2: nat,B: nat,C: nat] :
% 14.25/8.11 ( ( ord_less_nat @ zero_zero_nat @ A2 )
% 14.25/8.11 => ( ( ord_less_eq_nat @ B @ C )
% 14.25/8.11 => ( ord_less_nat @ B @ ( plus_plus_nat @ A2 @ C ) ) ) ) ).
% 14.25/8.11
% 14.25/8.11 thf(fact_5855_add__strict__increasing,axiom,
% 14.25/8.11 ! [A2: int,B: int,C: int] :
% 14.25/8.11 ( ( ord_less_int @ zero_zero_int @ A2 )
% 14.25/8.11 => ( ( ord_less_eq_int @ B @ C )
% 14.25/8.11 => ( ord_less_int @ B @ ( plus_plus_int @ A2 @ C ) ) ) ) ).
% 14.25/8.11
% 14.25/8.11 thf(fact_5856_add__pos__nonneg,axiom,
% 14.25/8.11 ! [A2: code_integer,B: code_integer] :
% 14.25/8.11 ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ A2 )
% 14.25/8.11 => ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ B )
% 14.25/8.11 => ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ ( plus_p5714425477246183910nteger @ A2 @ B ) ) ) ) ).
% 14.25/8.11
% 14.25/8.11 thf(fact_5857_add__pos__nonneg,axiom,
% 14.25/8.11 ! [A2: rat,B: rat] :
% 14.25/8.11 ( ( ord_less_rat @ zero_zero_rat @ A2 )
% 14.25/8.11 => ( ( ord_less_eq_rat @ zero_zero_rat @ B )
% 14.25/8.11 => ( ord_less_rat @ zero_zero_rat @ ( plus_plus_rat @ A2 @ B ) ) ) ) ).
% 14.25/8.11
% 14.25/8.11 thf(fact_5858_add__pos__nonneg,axiom,
% 14.25/8.11 ! [A2: nat,B: nat] :
% 14.25/8.11 ( ( ord_less_nat @ zero_zero_nat @ A2 )
% 14.25/8.11 => ( ( ord_less_eq_nat @ zero_zero_nat @ B )
% 14.25/8.11 => ( ord_less_nat @ zero_zero_nat @ ( plus_plus_nat @ A2 @ B ) ) ) ) ).
% 14.25/8.11
% 14.25/8.11 thf(fact_5859_add__pos__nonneg,axiom,
% 14.25/8.11 ! [A2: int,B: int] :
% 14.25/8.11 ( ( ord_less_int @ zero_zero_int @ A2 )
% 14.25/8.11 => ( ( ord_less_eq_int @ zero_zero_int @ B )
% 14.25/8.11 => ( ord_less_int @ zero_zero_int @ ( plus_plus_int @ A2 @ B ) ) ) ) ).
% 14.25/8.11
% 14.25/8.11 thf(fact_5860_add__nonpos__neg,axiom,
% 14.25/8.11 ! [A2: code_integer,B: code_integer] :
% 14.25/8.11 ( ( ord_le3102999989581377725nteger @ A2 @ zero_z3403309356797280102nteger )
% 14.25/8.11 => ( ( ord_le6747313008572928689nteger @ B @ zero_z3403309356797280102nteger )
% 14.25/8.11 => ( ord_le6747313008572928689nteger @ ( plus_p5714425477246183910nteger @ A2 @ B ) @ zero_z3403309356797280102nteger ) ) ) ).
% 14.25/8.11
% 14.25/8.11 thf(fact_5861_add__nonpos__neg,axiom,
% 14.25/8.11 ! [A2: rat,B: rat] :
% 14.25/8.11 ( ( ord_less_eq_rat @ A2 @ zero_zero_rat )
% 14.25/8.11 => ( ( ord_less_rat @ B @ zero_zero_rat )
% 14.25/8.11 => ( ord_less_rat @ ( plus_plus_rat @ A2 @ B ) @ zero_zero_rat ) ) ) ).
% 14.25/8.11
% 14.25/8.11 thf(fact_5862_add__nonpos__neg,axiom,
% 14.25/8.11 ! [A2: nat,B: nat] :
% 14.25/8.11 ( ( ord_less_eq_nat @ A2 @ zero_zero_nat )
% 14.25/8.11 => ( ( ord_less_nat @ B @ zero_zero_nat )
% 14.25/8.11 => ( ord_less_nat @ ( plus_plus_nat @ A2 @ B ) @ zero_zero_nat ) ) ) ).
% 14.25/8.11
% 14.25/8.11 thf(fact_5863_add__nonpos__neg,axiom,
% 14.25/8.11 ! [A2: int,B: int] :
% 14.25/8.11 ( ( ord_less_eq_int @ A2 @ zero_zero_int )
% 14.25/8.11 => ( ( ord_less_int @ B @ zero_zero_int )
% 14.25/8.11 => ( ord_less_int @ ( plus_plus_int @ A2 @ B ) @ zero_zero_int ) ) ) ).
% 14.25/8.11
% 14.25/8.11 thf(fact_5864_add__nonneg__pos,axiom,
% 14.25/8.11 ! [A2: code_integer,B: code_integer] :
% 14.25/8.11 ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ A2 )
% 14.25/8.11 => ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ B )
% 14.25/8.11 => ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ ( plus_p5714425477246183910nteger @ A2 @ B ) ) ) ) ).
% 14.25/8.11
% 14.25/8.11 thf(fact_5865_add__nonneg__pos,axiom,
% 14.25/8.11 ! [A2: rat,B: rat] :
% 14.25/8.11 ( ( ord_less_eq_rat @ zero_zero_rat @ A2 )
% 14.25/8.11 => ( ( ord_less_rat @ zero_zero_rat @ B )
% 14.25/8.11 => ( ord_less_rat @ zero_zero_rat @ ( plus_plus_rat @ A2 @ B ) ) ) ) ).
% 14.25/8.11
% 14.25/8.11 thf(fact_5866_add__nonneg__pos,axiom,
% 14.25/8.11 ! [A2: nat,B: nat] :
% 14.25/8.11 ( ( ord_less_eq_nat @ zero_zero_nat @ A2 )
% 14.25/8.11 => ( ( ord_less_nat @ zero_zero_nat @ B )
% 14.25/8.11 => ( ord_less_nat @ zero_zero_nat @ ( plus_plus_nat @ A2 @ B ) ) ) ) ).
% 14.25/8.11
% 14.25/8.11 thf(fact_5867_add__nonneg__pos,axiom,
% 14.25/8.11 ! [A2: int,B: int] :
% 14.25/8.11 ( ( ord_less_eq_int @ zero_zero_int @ A2 )
% 14.25/8.11 => ( ( ord_less_int @ zero_zero_int @ B )
% 14.25/8.11 => ( ord_less_int @ zero_zero_int @ ( plus_plus_int @ A2 @ B ) ) ) ) ).
% 14.25/8.11
% 14.25/8.11 thf(fact_5868_add__neg__nonpos,axiom,
% 14.25/8.11 ! [A2: code_integer,B: code_integer] :
% 14.25/8.11 ( ( ord_le6747313008572928689nteger @ A2 @ zero_z3403309356797280102nteger )
% 14.25/8.11 => ( ( ord_le3102999989581377725nteger @ B @ zero_z3403309356797280102nteger )
% 14.25/8.11 => ( ord_le6747313008572928689nteger @ ( plus_p5714425477246183910nteger @ A2 @ B ) @ zero_z3403309356797280102nteger ) ) ) ).
% 14.25/8.11
% 14.25/8.11 thf(fact_5869_add__neg__nonpos,axiom,
% 14.25/8.11 ! [A2: rat,B: rat] :
% 14.25/8.11 ( ( ord_less_rat @ A2 @ zero_zero_rat )
% 14.25/8.11 => ( ( ord_less_eq_rat @ B @ zero_zero_rat )
% 14.25/8.11 => ( ord_less_rat @ ( plus_plus_rat @ A2 @ B ) @ zero_zero_rat ) ) ) ).
% 14.25/8.11
% 14.25/8.11 thf(fact_5870_add__neg__nonpos,axiom,
% 14.25/8.11 ! [A2: nat,B: nat] :
% 14.25/8.11 ( ( ord_less_nat @ A2 @ zero_zero_nat )
% 14.25/8.11 => ( ( ord_less_eq_nat @ B @ zero_zero_nat )
% 14.25/8.11 => ( ord_less_nat @ ( plus_plus_nat @ A2 @ B ) @ zero_zero_nat ) ) ) ).
% 14.25/8.11
% 14.25/8.11 thf(fact_5871_add__neg__nonpos,axiom,
% 14.25/8.11 ! [A2: int,B: int] :
% 14.25/8.11 ( ( ord_less_int @ A2 @ zero_zero_int )
% 14.25/8.11 => ( ( ord_less_eq_int @ B @ zero_zero_int )
% 14.25/8.11 => ( ord_less_int @ ( plus_plus_int @ A2 @ B ) @ zero_zero_int ) ) ) ).
% 14.25/8.11
% 14.25/8.11 thf(fact_5872_mult__less__le__imp__less,axiom,
% 14.25/8.11 ! [A2: code_integer,B: code_integer,C: code_integer,D2: code_integer] :
% 14.25/8.11 ( ( ord_le6747313008572928689nteger @ A2 @ B )
% 14.25/8.11 => ( ( ord_le3102999989581377725nteger @ C @ D2 )
% 14.25/8.11 => ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ A2 )
% 14.25/8.11 => ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ C )
% 14.25/8.11 => ( ord_le6747313008572928689nteger @ ( times_3573771949741848930nteger @ A2 @ C ) @ ( times_3573771949741848930nteger @ B @ D2 ) ) ) ) ) ) ).
% 14.25/8.11
% 14.25/8.11 thf(fact_5873_mult__less__le__imp__less,axiom,
% 14.25/8.11 ! [A2: rat,B: rat,C: rat,D2: rat] :
% 14.25/8.11 ( ( ord_less_rat @ A2 @ B )
% 14.25/8.11 => ( ( ord_less_eq_rat @ C @ D2 )
% 14.25/8.11 => ( ( ord_less_eq_rat @ zero_zero_rat @ A2 )
% 14.25/8.11 => ( ( ord_less_rat @ zero_zero_rat @ C )
% 14.25/8.11 => ( ord_less_rat @ ( times_times_rat @ A2 @ C ) @ ( times_times_rat @ B @ D2 ) ) ) ) ) ) ).
% 14.25/8.11
% 14.25/8.11 thf(fact_5874_mult__less__le__imp__less,axiom,
% 14.25/8.11 ! [A2: nat,B: nat,C: nat,D2: nat] :
% 14.25/8.11 ( ( ord_less_nat @ A2 @ B )
% 14.25/8.11 => ( ( ord_less_eq_nat @ C @ D2 )
% 14.25/8.11 => ( ( ord_less_eq_nat @ zero_zero_nat @ A2 )
% 14.25/8.11 => ( ( ord_less_nat @ zero_zero_nat @ C )
% 14.25/8.11 => ( ord_less_nat @ ( times_times_nat @ A2 @ C ) @ ( times_times_nat @ B @ D2 ) ) ) ) ) ) ).
% 14.25/8.11
% 14.25/8.11 thf(fact_5875_mult__less__le__imp__less,axiom,
% 14.25/8.11 ! [A2: int,B: int,C: int,D2: int] :
% 14.25/8.11 ( ( ord_less_int @ A2 @ B )
% 14.25/8.11 => ( ( ord_less_eq_int @ C @ D2 )
% 14.25/8.11 => ( ( ord_less_eq_int @ zero_zero_int @ A2 )
% 14.25/8.11 => ( ( ord_less_int @ zero_zero_int @ C )
% 14.25/8.11 => ( ord_less_int @ ( times_times_int @ A2 @ C ) @ ( times_times_int @ B @ D2 ) ) ) ) ) ) ).
% 14.25/8.11
% 14.25/8.11 thf(fact_5876_mult__le__less__imp__less,axiom,
% 14.25/8.11 ! [A2: code_integer,B: code_integer,C: code_integer,D2: code_integer] :
% 14.25/8.11 ( ( ord_le3102999989581377725nteger @ A2 @ B )
% 14.25/8.11 => ( ( ord_le6747313008572928689nteger @ C @ D2 )
% 14.25/8.11 => ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ A2 )
% 14.25/8.11 => ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ C )
% 14.25/8.11 => ( ord_le6747313008572928689nteger @ ( times_3573771949741848930nteger @ A2 @ C ) @ ( times_3573771949741848930nteger @ B @ D2 ) ) ) ) ) ) ).
% 14.25/8.11
% 14.25/8.11 thf(fact_5877_mult__le__less__imp__less,axiom,
% 14.25/8.11 ! [A2: rat,B: rat,C: rat,D2: rat] :
% 14.25/8.11 ( ( ord_less_eq_rat @ A2 @ B )
% 14.25/8.11 => ( ( ord_less_rat @ C @ D2 )
% 14.25/8.11 => ( ( ord_less_rat @ zero_zero_rat @ A2 )
% 14.25/8.11 => ( ( ord_less_eq_rat @ zero_zero_rat @ C )
% 14.25/8.11 => ( ord_less_rat @ ( times_times_rat @ A2 @ C ) @ ( times_times_rat @ B @ D2 ) ) ) ) ) ) ).
% 14.25/8.11
% 14.25/8.11 thf(fact_5878_mult__le__less__imp__less,axiom,
% 14.25/8.11 ! [A2: nat,B: nat,C: nat,D2: nat] :
% 14.25/8.11 ( ( ord_less_eq_nat @ A2 @ B )
% 14.25/8.11 => ( ( ord_less_nat @ C @ D2 )
% 14.25/8.11 => ( ( ord_less_nat @ zero_zero_nat @ A2 )
% 14.25/8.11 => ( ( ord_less_eq_nat @ zero_zero_nat @ C )
% 14.25/8.11 => ( ord_less_nat @ ( times_times_nat @ A2 @ C ) @ ( times_times_nat @ B @ D2 ) ) ) ) ) ) ).
% 14.25/8.11
% 14.25/8.11 thf(fact_5879_mult__le__less__imp__less,axiom,
% 14.25/8.11 ! [A2: int,B: int,C: int,D2: int] :
% 14.25/8.11 ( ( ord_less_eq_int @ A2 @ B )
% 14.25/8.11 => ( ( ord_less_int @ C @ D2 )
% 14.25/8.11 => ( ( ord_less_int @ zero_zero_int @ A2 )
% 14.25/8.11 => ( ( ord_less_eq_int @ zero_zero_int @ C )
% 14.25/8.11 => ( ord_less_int @ ( times_times_int @ A2 @ C ) @ ( times_times_int @ B @ D2 ) ) ) ) ) ) ).
% 14.25/8.11
% 14.25/8.11 thf(fact_5880_mult__right__le__imp__le,axiom,
% 14.25/8.11 ! [A2: code_integer,C: code_integer,B: code_integer] :
% 14.25/8.11 ( ( ord_le3102999989581377725nteger @ ( times_3573771949741848930nteger @ A2 @ C ) @ ( times_3573771949741848930nteger @ B @ C ) )
% 14.25/8.11 => ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ C )
% 14.25/8.11 => ( ord_le3102999989581377725nteger @ A2 @ B ) ) ) ).
% 14.25/8.11
% 14.25/8.11 thf(fact_5881_mult__right__le__imp__le,axiom,
% 14.25/8.11 ! [A2: rat,C: rat,B: rat] :
% 14.25/8.11 ( ( ord_less_eq_rat @ ( times_times_rat @ A2 @ C ) @ ( times_times_rat @ B @ C ) )
% 14.25/8.11 => ( ( ord_less_rat @ zero_zero_rat @ C )
% 14.25/8.11 => ( ord_less_eq_rat @ A2 @ B ) ) ) ).
% 14.25/8.11
% 14.25/8.11 thf(fact_5882_mult__right__le__imp__le,axiom,
% 14.25/8.11 ! [A2: nat,C: nat,B: nat] :
% 14.25/8.11 ( ( ord_less_eq_nat @ ( times_times_nat @ A2 @ C ) @ ( times_times_nat @ B @ C ) )
% 14.25/8.11 => ( ( ord_less_nat @ zero_zero_nat @ C )
% 14.25/8.11 => ( ord_less_eq_nat @ A2 @ B ) ) ) ).
% 14.25/8.11
% 14.25/8.11 thf(fact_5883_mult__right__le__imp__le,axiom,
% 14.25/8.11 ! [A2: int,C: int,B: int] :
% 14.25/8.11 ( ( ord_less_eq_int @ ( times_times_int @ A2 @ C ) @ ( times_times_int @ B @ C ) )
% 14.25/8.11 => ( ( ord_less_int @ zero_zero_int @ C )
% 14.25/8.11 => ( ord_less_eq_int @ A2 @ B ) ) ) ).
% 14.25/8.11
% 14.25/8.11 thf(fact_5884_mult__left__le__imp__le,axiom,
% 14.25/8.11 ! [C: code_integer,A2: code_integer,B: code_integer] :
% 14.25/8.11 ( ( ord_le3102999989581377725nteger @ ( times_3573771949741848930nteger @ C @ A2 ) @ ( times_3573771949741848930nteger @ C @ B ) )
% 14.25/8.11 => ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ C )
% 14.25/8.11 => ( ord_le3102999989581377725nteger @ A2 @ B ) ) ) ).
% 14.25/8.11
% 14.25/8.11 thf(fact_5885_mult__left__le__imp__le,axiom,
% 14.25/8.11 ! [C: rat,A2: rat,B: rat] :
% 14.25/8.11 ( ( ord_less_eq_rat @ ( times_times_rat @ C @ A2 ) @ ( times_times_rat @ C @ B ) )
% 14.25/8.11 => ( ( ord_less_rat @ zero_zero_rat @ C )
% 14.25/8.11 => ( ord_less_eq_rat @ A2 @ B ) ) ) ).
% 14.25/8.11
% 14.25/8.11 thf(fact_5886_mult__left__le__imp__le,axiom,
% 14.25/8.11 ! [C: nat,A2: nat,B: nat] :
% 14.25/8.11 ( ( ord_less_eq_nat @ ( times_times_nat @ C @ A2 ) @ ( times_times_nat @ C @ B ) )
% 14.25/8.11 => ( ( ord_less_nat @ zero_zero_nat @ C )
% 14.25/8.11 => ( ord_less_eq_nat @ A2 @ B ) ) ) ).
% 14.25/8.11
% 14.25/8.11 thf(fact_5887_mult__left__le__imp__le,axiom,
% 14.25/8.11 ! [C: int,A2: int,B: int] :
% 14.25/8.11 ( ( ord_less_eq_int @ ( times_times_int @ C @ A2 ) @ ( times_times_int @ C @ B ) )
% 14.25/8.11 => ( ( ord_less_int @ zero_zero_int @ C )
% 14.25/8.11 => ( ord_less_eq_int @ A2 @ B ) ) ) ).
% 14.25/8.11
% 14.25/8.11 thf(fact_5888_mult__le__cancel__left__pos,axiom,
% 14.25/8.11 ! [C: code_integer,A2: code_integer,B: code_integer] :
% 14.25/8.11 ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ C )
% 14.25/8.11 => ( ( ord_le3102999989581377725nteger @ ( times_3573771949741848930nteger @ C @ A2 ) @ ( times_3573771949741848930nteger @ C @ B ) )
% 14.25/8.11 = ( ord_le3102999989581377725nteger @ A2 @ B ) ) ) ).
% 14.25/8.11
% 14.25/8.11 thf(fact_5889_mult__le__cancel__left__pos,axiom,
% 14.25/8.11 ! [C: rat,A2: rat,B: rat] :
% 14.25/8.11 ( ( ord_less_rat @ zero_zero_rat @ C )
% 14.25/8.11 => ( ( ord_less_eq_rat @ ( times_times_rat @ C @ A2 ) @ ( times_times_rat @ C @ B ) )
% 14.25/8.11 = ( ord_less_eq_rat @ A2 @ B ) ) ) ).
% 14.25/8.11
% 14.25/8.11 thf(fact_5890_mult__le__cancel__left__pos,axiom,
% 14.25/8.11 ! [C: int,A2: int,B: int] :
% 14.25/8.11 ( ( ord_less_int @ zero_zero_int @ C )
% 14.25/8.11 => ( ( ord_less_eq_int @ ( times_times_int @ C @ A2 ) @ ( times_times_int @ C @ B ) )
% 14.25/8.11 = ( ord_less_eq_int @ A2 @ B ) ) ) ).
% 14.25/8.11
% 14.25/8.11 thf(fact_5891_mult__le__cancel__left__neg,axiom,
% 14.25/8.11 ! [C: code_integer,A2: code_integer,B: code_integer] :
% 14.25/8.11 ( ( ord_le6747313008572928689nteger @ C @ zero_z3403309356797280102nteger )
% 14.25/8.11 => ( ( ord_le3102999989581377725nteger @ ( times_3573771949741848930nteger @ C @ A2 ) @ ( times_3573771949741848930nteger @ C @ B ) )
% 14.25/8.11 = ( ord_le3102999989581377725nteger @ B @ A2 ) ) ) ).
% 14.25/8.11
% 14.25/8.11 thf(fact_5892_mult__le__cancel__left__neg,axiom,
% 14.25/8.11 ! [C: rat,A2: rat,B: rat] :
% 14.25/8.11 ( ( ord_less_rat @ C @ zero_zero_rat )
% 14.25/8.11 => ( ( ord_less_eq_rat @ ( times_times_rat @ C @ A2 ) @ ( times_times_rat @ C @ B ) )
% 14.25/8.11 = ( ord_less_eq_rat @ B @ A2 ) ) ) ).
% 14.25/8.11
% 14.25/8.11 thf(fact_5893_mult__le__cancel__left__neg,axiom,
% 14.25/8.11 ! [C: int,A2: int,B: int] :
% 14.25/8.11 ( ( ord_less_int @ C @ zero_zero_int )
% 14.25/8.11 => ( ( ord_less_eq_int @ ( times_times_int @ C @ A2 ) @ ( times_times_int @ C @ B ) )
% 14.25/8.11 = ( ord_less_eq_int @ B @ A2 ) ) ) ).
% 14.25/8.11
% 14.25/8.11 thf(fact_5894_mult__less__cancel__right,axiom,
% 14.25/8.11 ! [A2: code_integer,C: code_integer,B: code_integer] :
% 14.25/8.11 ( ( ord_le6747313008572928689nteger @ ( times_3573771949741848930nteger @ A2 @ C ) @ ( times_3573771949741848930nteger @ B @ C ) )
% 14.25/8.11 = ( ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ C )
% 14.25/8.11 => ( ord_le6747313008572928689nteger @ A2 @ B ) )
% 14.25/8.11 & ( ( ord_le3102999989581377725nteger @ C @ zero_z3403309356797280102nteger )
% 14.25/8.11 => ( ord_le6747313008572928689nteger @ B @ A2 ) ) ) ) ).
% 14.25/8.11
% 14.25/8.11 thf(fact_5895_mult__less__cancel__right,axiom,
% 14.25/8.11 ! [A2: rat,C: rat,B: rat] :
% 14.25/8.11 ( ( ord_less_rat @ ( times_times_rat @ A2 @ C ) @ ( times_times_rat @ B @ C ) )
% 14.25/8.11 = ( ( ( ord_less_eq_rat @ zero_zero_rat @ C )
% 14.25/8.11 => ( ord_less_rat @ A2 @ B ) )
% 14.25/8.11 & ( ( ord_less_eq_rat @ C @ zero_zero_rat )
% 14.25/8.11 => ( ord_less_rat @ B @ A2 ) ) ) ) ).
% 14.25/8.11
% 14.25/8.11 thf(fact_5896_mult__less__cancel__right,axiom,
% 14.25/8.11 ! [A2: int,C: int,B: int] :
% 14.25/8.11 ( ( ord_less_int @ ( times_times_int @ A2 @ C ) @ ( times_times_int @ B @ C ) )
% 14.25/8.11 = ( ( ( ord_less_eq_int @ zero_zero_int @ C )
% 14.25/8.11 => ( ord_less_int @ A2 @ B ) )
% 14.25/8.11 & ( ( ord_less_eq_int @ C @ zero_zero_int )
% 14.25/8.11 => ( ord_less_int @ B @ A2 ) ) ) ) ).
% 14.25/8.11
% 14.25/8.11 thf(fact_5897_mult__strict__mono_H,axiom,
% 14.25/8.11 ! [A2: code_integer,B: code_integer,C: code_integer,D2: code_integer] :
% 14.25/8.11 ( ( ord_le6747313008572928689nteger @ A2 @ B )
% 14.25/8.11 => ( ( ord_le6747313008572928689nteger @ C @ D2 )
% 14.25/8.11 => ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ A2 )
% 14.25/8.11 => ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ C )
% 14.25/8.11 => ( ord_le6747313008572928689nteger @ ( times_3573771949741848930nteger @ A2 @ C ) @ ( times_3573771949741848930nteger @ B @ D2 ) ) ) ) ) ) ).
% 14.25/8.11
% 14.25/8.11 thf(fact_5898_mult__strict__mono_H,axiom,
% 14.25/8.11 ! [A2: rat,B: rat,C: rat,D2: rat] :
% 14.25/8.11 ( ( ord_less_rat @ A2 @ B )
% 14.25/8.11 => ( ( ord_less_rat @ C @ D2 )
% 14.25/8.11 => ( ( ord_less_eq_rat @ zero_zero_rat @ A2 )
% 14.25/8.11 => ( ( ord_less_eq_rat @ zero_zero_rat @ C )
% 14.25/8.11 => ( ord_less_rat @ ( times_times_rat @ A2 @ C ) @ ( times_times_rat @ B @ D2 ) ) ) ) ) ) ).
% 14.25/8.11
% 14.25/8.11 thf(fact_5899_mult__strict__mono_H,axiom,
% 14.25/8.11 ! [A2: nat,B: nat,C: nat,D2: nat] :
% 14.25/8.11 ( ( ord_less_nat @ A2 @ B )
% 14.25/8.11 => ( ( ord_less_nat @ C @ D2 )
% 14.25/8.11 => ( ( ord_less_eq_nat @ zero_zero_nat @ A2 )
% 14.25/8.11 => ( ( ord_less_eq_nat @ zero_zero_nat @ C )
% 14.25/8.11 => ( ord_less_nat @ ( times_times_nat @ A2 @ C ) @ ( times_times_nat @ B @ D2 ) ) ) ) ) ) ).
% 14.25/8.11
% 14.25/8.11 thf(fact_5900_mult__strict__mono_H,axiom,
% 14.25/8.11 ! [A2: int,B: int,C: int,D2: int] :
% 14.25/8.11 ( ( ord_less_int @ A2 @ B )
% 14.25/8.11 => ( ( ord_less_int @ C @ D2 )
% 14.25/8.11 => ( ( ord_less_eq_int @ zero_zero_int @ A2 )
% 14.25/8.11 => ( ( ord_less_eq_int @ zero_zero_int @ C )
% 14.25/8.11 => ( ord_less_int @ ( times_times_int @ A2 @ C ) @ ( times_times_int @ B @ D2 ) ) ) ) ) ) ).
% 14.25/8.11
% 14.25/8.11 thf(fact_5901_mult__right__less__imp__less,axiom,
% 14.25/8.11 ! [A2: code_integer,C: code_integer,B: code_integer] :
% 14.25/8.11 ( ( ord_le6747313008572928689nteger @ ( times_3573771949741848930nteger @ A2 @ C ) @ ( times_3573771949741848930nteger @ B @ C ) )
% 14.25/8.11 => ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ C )
% 14.25/8.11 => ( ord_le6747313008572928689nteger @ A2 @ B ) ) ) ).
% 14.25/8.11
% 14.25/8.11 thf(fact_5902_mult__right__less__imp__less,axiom,
% 14.25/8.11 ! [A2: rat,C: rat,B: rat] :
% 14.25/8.11 ( ( ord_less_rat @ ( times_times_rat @ A2 @ C ) @ ( times_times_rat @ B @ C ) )
% 14.25/8.11 => ( ( ord_less_eq_rat @ zero_zero_rat @ C )
% 14.25/8.11 => ( ord_less_rat @ A2 @ B ) ) ) ).
% 14.25/8.11
% 14.25/8.11 thf(fact_5903_mult__right__less__imp__less,axiom,
% 14.25/8.11 ! [A2: nat,C: nat,B: nat] :
% 14.25/8.11 ( ( ord_less_nat @ ( times_times_nat @ A2 @ C ) @ ( times_times_nat @ B @ C ) )
% 14.25/8.11 => ( ( ord_less_eq_nat @ zero_zero_nat @ C )
% 14.25/8.11 => ( ord_less_nat @ A2 @ B ) ) ) ).
% 14.25/8.11
% 14.25/8.11 thf(fact_5904_mult__right__less__imp__less,axiom,
% 14.25/8.11 ! [A2: int,C: int,B: int] :
% 14.25/8.11 ( ( ord_less_int @ ( times_times_int @ A2 @ C ) @ ( times_times_int @ B @ C ) )
% 14.25/8.11 => ( ( ord_less_eq_int @ zero_zero_int @ C )
% 14.25/8.11 => ( ord_less_int @ A2 @ B ) ) ) ).
% 14.25/8.11
% 14.25/8.11 thf(fact_5905_mult__less__cancel__left,axiom,
% 14.25/8.11 ! [C: code_integer,A2: code_integer,B: code_integer] :
% 14.25/8.11 ( ( ord_le6747313008572928689nteger @ ( times_3573771949741848930nteger @ C @ A2 ) @ ( times_3573771949741848930nteger @ C @ B ) )
% 14.25/8.11 = ( ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ C )
% 14.25/8.11 => ( ord_le6747313008572928689nteger @ A2 @ B ) )
% 14.25/8.11 & ( ( ord_le3102999989581377725nteger @ C @ zero_z3403309356797280102nteger )
% 14.25/8.11 => ( ord_le6747313008572928689nteger @ B @ A2 ) ) ) ) ).
% 14.25/8.11
% 14.25/8.11 thf(fact_5906_mult__less__cancel__left,axiom,
% 14.25/8.11 ! [C: rat,A2: rat,B: rat] :
% 14.25/8.11 ( ( ord_less_rat @ ( times_times_rat @ C @ A2 ) @ ( times_times_rat @ C @ B ) )
% 14.25/8.11 = ( ( ( ord_less_eq_rat @ zero_zero_rat @ C )
% 14.25/8.11 => ( ord_less_rat @ A2 @ B ) )
% 14.25/8.11 & ( ( ord_less_eq_rat @ C @ zero_zero_rat )
% 14.25/8.11 => ( ord_less_rat @ B @ A2 ) ) ) ) ).
% 14.25/8.11
% 14.25/8.11 thf(fact_5907_mult__less__cancel__left,axiom,
% 14.25/8.11 ! [C: int,A2: int,B: int] :
% 14.25/8.11 ( ( ord_less_int @ ( times_times_int @ C @ A2 ) @ ( times_times_int @ C @ B ) )
% 14.25/8.11 = ( ( ( ord_less_eq_int @ zero_zero_int @ C )
% 14.25/8.11 => ( ord_less_int @ A2 @ B ) )
% 14.25/8.11 & ( ( ord_less_eq_int @ C @ zero_zero_int )
% 14.25/8.11 => ( ord_less_int @ B @ A2 ) ) ) ) ).
% 14.25/8.11
% 14.25/8.11 thf(fact_5908_mult__strict__mono,axiom,
% 14.25/8.11 ! [A2: code_integer,B: code_integer,C: code_integer,D2: code_integer] :
% 14.25/8.11 ( ( ord_le6747313008572928689nteger @ A2 @ B )
% 14.25/8.11 => ( ( ord_le6747313008572928689nteger @ C @ D2 )
% 14.25/8.11 => ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ B )
% 14.25/8.11 => ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ C )
% 14.25/8.11 => ( ord_le6747313008572928689nteger @ ( times_3573771949741848930nteger @ A2 @ C ) @ ( times_3573771949741848930nteger @ B @ D2 ) ) ) ) ) ) ).
% 14.25/8.11
% 14.25/8.11 thf(fact_5909_mult__strict__mono,axiom,
% 14.25/8.11 ! [A2: rat,B: rat,C: rat,D2: rat] :
% 14.25/8.11 ( ( ord_less_rat @ A2 @ B )
% 14.25/8.11 => ( ( ord_less_rat @ C @ D2 )
% 14.25/8.11 => ( ( ord_less_rat @ zero_zero_rat @ B )
% 14.25/8.11 => ( ( ord_less_eq_rat @ zero_zero_rat @ C )
% 14.25/8.11 => ( ord_less_rat @ ( times_times_rat @ A2 @ C ) @ ( times_times_rat @ B @ D2 ) ) ) ) ) ) ).
% 14.25/8.11
% 14.25/8.11 thf(fact_5910_mult__strict__mono,axiom,
% 14.25/8.11 ! [A2: nat,B: nat,C: nat,D2: nat] :
% 14.25/8.11 ( ( ord_less_nat @ A2 @ B )
% 14.25/8.11 => ( ( ord_less_nat @ C @ D2 )
% 14.25/8.11 => ( ( ord_less_nat @ zero_zero_nat @ B )
% 14.25/8.11 => ( ( ord_less_eq_nat @ zero_zero_nat @ C )
% 14.25/8.11 => ( ord_less_nat @ ( times_times_nat @ A2 @ C ) @ ( times_times_nat @ B @ D2 ) ) ) ) ) ) ).
% 14.25/8.11
% 14.25/8.11 thf(fact_5911_mult__strict__mono,axiom,
% 14.25/8.11 ! [A2: int,B: int,C: int,D2: int] :
% 14.25/8.11 ( ( ord_less_int @ A2 @ B )
% 14.25/8.11 => ( ( ord_less_int @ C @ D2 )
% 14.25/8.11 => ( ( ord_less_int @ zero_zero_int @ B )
% 14.25/8.11 => ( ( ord_less_eq_int @ zero_zero_int @ C )
% 14.25/8.11 => ( ord_less_int @ ( times_times_int @ A2 @ C ) @ ( times_times_int @ B @ D2 ) ) ) ) ) ) ).
% 14.25/8.11
% 14.25/8.11 thf(fact_5912_mult__left__less__imp__less,axiom,
% 14.25/8.11 ! [C: code_integer,A2: code_integer,B: code_integer] :
% 14.25/8.11 ( ( ord_le6747313008572928689nteger @ ( times_3573771949741848930nteger @ C @ A2 ) @ ( times_3573771949741848930nteger @ C @ B ) )
% 14.25/8.11 => ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ C )
% 14.25/8.11 => ( ord_le6747313008572928689nteger @ A2 @ B ) ) ) ).
% 14.25/8.11
% 14.25/8.11 thf(fact_5913_mult__left__less__imp__less,axiom,
% 14.25/8.11 ! [C: rat,A2: rat,B: rat] :
% 14.25/8.11 ( ( ord_less_rat @ ( times_times_rat @ C @ A2 ) @ ( times_times_rat @ C @ B ) )
% 14.25/8.11 => ( ( ord_less_eq_rat @ zero_zero_rat @ C )
% 14.25/8.11 => ( ord_less_rat @ A2 @ B ) ) ) ).
% 14.25/8.11
% 14.25/8.11 thf(fact_5914_mult__left__less__imp__less,axiom,
% 14.25/8.11 ! [C: nat,A2: nat,B: nat] :
% 14.25/8.11 ( ( ord_less_nat @ ( times_times_nat @ C @ A2 ) @ ( times_times_nat @ C @ B ) )
% 14.25/8.11 => ( ( ord_less_eq_nat @ zero_zero_nat @ C )
% 14.25/8.11 => ( ord_less_nat @ A2 @ B ) ) ) ).
% 14.25/8.11
% 14.25/8.11 thf(fact_5915_mult__left__less__imp__less,axiom,
% 14.25/8.11 ! [C: int,A2: int,B: int] :
% 14.25/8.11 ( ( ord_less_int @ ( times_times_int @ C @ A2 ) @ ( times_times_int @ C @ B ) )
% 14.25/8.11 => ( ( ord_less_eq_int @ zero_zero_int @ C )
% 14.25/8.11 => ( ord_less_int @ A2 @ B ) ) ) ).
% 14.25/8.11
% 14.25/8.11 thf(fact_5916_mult__le__cancel__right,axiom,
% 14.25/8.11 ! [A2: code_integer,C: code_integer,B: code_integer] :
% 14.25/8.11 ( ( ord_le3102999989581377725nteger @ ( times_3573771949741848930nteger @ A2 @ C ) @ ( times_3573771949741848930nteger @ B @ C ) )
% 14.25/8.11 = ( ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ C )
% 14.25/8.11 => ( ord_le3102999989581377725nteger @ A2 @ B ) )
% 14.25/8.11 & ( ( ord_le6747313008572928689nteger @ C @ zero_z3403309356797280102nteger )
% 14.25/8.11 => ( ord_le3102999989581377725nteger @ B @ A2 ) ) ) ) ).
% 14.25/8.11
% 14.25/8.11 thf(fact_5917_mult__le__cancel__right,axiom,
% 14.25/8.11 ! [A2: rat,C: rat,B: rat] :
% 14.25/8.11 ( ( ord_less_eq_rat @ ( times_times_rat @ A2 @ C ) @ ( times_times_rat @ B @ C ) )
% 14.25/8.11 = ( ( ( ord_less_rat @ zero_zero_rat @ C )
% 14.25/8.11 => ( ord_less_eq_rat @ A2 @ B ) )
% 14.25/8.11 & ( ( ord_less_rat @ C @ zero_zero_rat )
% 14.25/8.11 => ( ord_less_eq_rat @ B @ A2 ) ) ) ) ).
% 14.25/8.11
% 14.25/8.11 thf(fact_5918_mult__le__cancel__right,axiom,
% 14.25/8.11 ! [A2: int,C: int,B: int] :
% 14.25/8.11 ( ( ord_less_eq_int @ ( times_times_int @ A2 @ C ) @ ( times_times_int @ B @ C ) )
% 14.25/8.11 = ( ( ( ord_less_int @ zero_zero_int @ C )
% 14.25/8.11 => ( ord_less_eq_int @ A2 @ B ) )
% 14.25/8.11 & ( ( ord_less_int @ C @ zero_zero_int )
% 14.25/8.11 => ( ord_less_eq_int @ B @ A2 ) ) ) ) ).
% 14.25/8.11
% 14.25/8.11 thf(fact_5919_mult__le__cancel__left,axiom,
% 14.25/8.11 ! [C: code_integer,A2: code_integer,B: code_integer] :
% 14.25/8.11 ( ( ord_le3102999989581377725nteger @ ( times_3573771949741848930nteger @ C @ A2 ) @ ( times_3573771949741848930nteger @ C @ B ) )
% 14.25/8.11 = ( ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ C )
% 14.25/8.11 => ( ord_le3102999989581377725nteger @ A2 @ B ) )
% 14.25/8.11 & ( ( ord_le6747313008572928689nteger @ C @ zero_z3403309356797280102nteger )
% 14.25/8.11 => ( ord_le3102999989581377725nteger @ B @ A2 ) ) ) ) ).
% 14.25/8.11
% 14.25/8.11 thf(fact_5920_mult__le__cancel__left,axiom,
% 14.25/8.11 ! [C: rat,A2: rat,B: rat] :
% 14.25/8.11 ( ( ord_less_eq_rat @ ( times_times_rat @ C @ A2 ) @ ( times_times_rat @ C @ B ) )
% 14.25/8.11 = ( ( ( ord_less_rat @ zero_zero_rat @ C )
% 14.25/8.11 => ( ord_less_eq_rat @ A2 @ B ) )
% 14.25/8.11 & ( ( ord_less_rat @ C @ zero_zero_rat )
% 14.25/8.11 => ( ord_less_eq_rat @ B @ A2 ) ) ) ) ).
% 14.25/8.11
% 14.25/8.11 thf(fact_5921_mult__le__cancel__left,axiom,
% 14.25/8.11 ! [C: int,A2: int,B: int] :
% 14.25/8.11 ( ( ord_less_eq_int @ ( times_times_int @ C @ A2 ) @ ( times_times_int @ C @ B ) )
% 14.25/8.11 = ( ( ( ord_less_int @ zero_zero_int @ C )
% 14.25/8.11 => ( ord_less_eq_int @ A2 @ B ) )
% 14.25/8.11 & ( ( ord_less_int @ C @ zero_zero_int )
% 14.25/8.11 => ( ord_less_eq_int @ B @ A2 ) ) ) ) ).
% 14.25/8.11
% 14.25/8.11 thf(fact_5922_sum__squares__ge__zero,axiom,
% 14.25/8.11 ! [X: code_integer,Y: code_integer] : ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ X @ X ) @ ( times_3573771949741848930nteger @ Y @ Y ) ) ) ).
% 14.25/8.11
% 14.25/8.11 thf(fact_5923_sum__squares__ge__zero,axiom,
% 14.25/8.11 ! [X: rat,Y: rat] : ( ord_less_eq_rat @ zero_zero_rat @ ( plus_plus_rat @ ( times_times_rat @ X @ X ) @ ( times_times_rat @ Y @ Y ) ) ) ).
% 14.25/8.11
% 14.25/8.11 thf(fact_5924_sum__squares__ge__zero,axiom,
% 14.25/8.11 ! [X: int,Y: int] : ( ord_less_eq_int @ zero_zero_int @ ( plus_plus_int @ ( times_times_int @ X @ X ) @ ( times_times_int @ Y @ Y ) ) ) ).
% 14.25/8.11
% 14.25/8.11 thf(fact_5925_mult__left__le,axiom,
% 14.25/8.11 ! [C: code_natural,A2: code_natural] :
% 14.25/8.11 ( ( ord_le1926595141338095240atural @ C @ one_one_Code_natural )
% 14.25/8.11 => ( ( ord_le1926595141338095240atural @ zero_z2226904508553997617atural @ A2 )
% 14.25/8.11 => ( ord_le1926595141338095240atural @ ( times_2397367101498566445atural @ A2 @ C ) @ A2 ) ) ) ).
% 14.25/8.11
% 14.25/8.11 thf(fact_5926_mult__left__le,axiom,
% 14.25/8.11 ! [C: code_integer,A2: code_integer] :
% 14.25/8.11 ( ( ord_le3102999989581377725nteger @ C @ one_one_Code_integer )
% 14.25/8.11 => ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ A2 )
% 14.25/8.11 => ( ord_le3102999989581377725nteger @ ( times_3573771949741848930nteger @ A2 @ C ) @ A2 ) ) ) ).
% 14.25/8.11
% 14.25/8.11 thf(fact_5927_mult__left__le,axiom,
% 14.25/8.11 ! [C: rat,A2: rat] :
% 14.25/8.11 ( ( ord_less_eq_rat @ C @ one_one_rat )
% 14.25/8.11 => ( ( ord_less_eq_rat @ zero_zero_rat @ A2 )
% 14.25/8.11 => ( ord_less_eq_rat @ ( times_times_rat @ A2 @ C ) @ A2 ) ) ) ).
% 14.25/8.11
% 14.25/8.11 thf(fact_5928_mult__left__le,axiom,
% 14.25/8.11 ! [C: nat,A2: nat] :
% 14.25/8.11 ( ( ord_less_eq_nat @ C @ one_one_nat )
% 14.25/8.11 => ( ( ord_less_eq_nat @ zero_zero_nat @ A2 )
% 14.25/8.11 => ( ord_less_eq_nat @ ( times_times_nat @ A2 @ C ) @ A2 ) ) ) ).
% 14.25/8.11
% 14.25/8.11 thf(fact_5929_mult__left__le,axiom,
% 14.25/8.11 ! [C: int,A2: int] :
% 14.25/8.11 ( ( ord_less_eq_int @ C @ one_one_int )
% 14.25/8.11 => ( ( ord_less_eq_int @ zero_zero_int @ A2 )
% 14.25/8.11 => ( ord_less_eq_int @ ( times_times_int @ A2 @ C ) @ A2 ) ) ) ).
% 14.25/8.11
% 14.25/8.11 thf(fact_5930_mult__le__one,axiom,
% 14.25/8.11 ! [A2: code_natural,B: code_natural] :
% 14.25/8.11 ( ( ord_le1926595141338095240atural @ A2 @ one_one_Code_natural )
% 14.25/8.11 => ( ( ord_le1926595141338095240atural @ zero_z2226904508553997617atural @ B )
% 14.25/8.11 => ( ( ord_le1926595141338095240atural @ B @ one_one_Code_natural )
% 14.25/8.11 => ( ord_le1926595141338095240atural @ ( times_2397367101498566445atural @ A2 @ B ) @ one_one_Code_natural ) ) ) ) ).
% 14.25/8.11
% 14.25/8.11 thf(fact_5931_mult__le__one,axiom,
% 14.25/8.11 ! [A2: code_integer,B: code_integer] :
% 14.25/8.11 ( ( ord_le3102999989581377725nteger @ A2 @ one_one_Code_integer )
% 14.25/8.11 => ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ B )
% 14.25/8.11 => ( ( ord_le3102999989581377725nteger @ B @ one_one_Code_integer )
% 14.25/8.11 => ( ord_le3102999989581377725nteger @ ( times_3573771949741848930nteger @ A2 @ B ) @ one_one_Code_integer ) ) ) ) ).
% 14.25/8.11
% 14.25/8.11 thf(fact_5932_mult__le__one,axiom,
% 14.25/8.11 ! [A2: rat,B: rat] :
% 14.25/8.11 ( ( ord_less_eq_rat @ A2 @ one_one_rat )
% 14.25/8.11 => ( ( ord_less_eq_rat @ zero_zero_rat @ B )
% 14.25/8.11 => ( ( ord_less_eq_rat @ B @ one_one_rat )
% 14.25/8.11 => ( ord_less_eq_rat @ ( times_times_rat @ A2 @ B ) @ one_one_rat ) ) ) ) ).
% 14.25/8.11
% 14.25/8.11 thf(fact_5933_mult__le__one,axiom,
% 14.25/8.11 ! [A2: nat,B: nat] :
% 14.25/8.11 ( ( ord_less_eq_nat @ A2 @ one_one_nat )
% 14.25/8.11 => ( ( ord_less_eq_nat @ zero_zero_nat @ B )
% 14.25/8.11 => ( ( ord_less_eq_nat @ B @ one_one_nat )
% 14.25/8.11 => ( ord_less_eq_nat @ ( times_times_nat @ A2 @ B ) @ one_one_nat ) ) ) ) ).
% 14.25/8.11
% 14.25/8.11 thf(fact_5934_mult__le__one,axiom,
% 14.25/8.11 ! [A2: int,B: int] :
% 14.25/8.11 ( ( ord_less_eq_int @ A2 @ one_one_int )
% 14.25/8.11 => ( ( ord_less_eq_int @ zero_zero_int @ B )
% 14.25/8.11 => ( ( ord_less_eq_int @ B @ one_one_int )
% 14.25/8.11 => ( ord_less_eq_int @ ( times_times_int @ A2 @ B ) @ one_one_int ) ) ) ) ).
% 14.25/8.11
% 14.25/8.11 thf(fact_5935_mult__right__le__one__le,axiom,
% 14.25/8.11 ! [X: code_integer,Y: code_integer] :
% 14.25/8.11 ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ X )
% 14.25/8.11 => ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ Y )
% 14.25/8.11 => ( ( ord_le3102999989581377725nteger @ Y @ one_one_Code_integer )
% 14.25/8.11 => ( ord_le3102999989581377725nteger @ ( times_3573771949741848930nteger @ X @ Y ) @ X ) ) ) ) ).
% 14.25/8.11
% 14.25/8.11 thf(fact_5936_mult__right__le__one__le,axiom,
% 14.25/8.11 ! [X: rat,Y: rat] :
% 14.25/8.11 ( ( ord_less_eq_rat @ zero_zero_rat @ X )
% 14.25/8.11 => ( ( ord_less_eq_rat @ zero_zero_rat @ Y )
% 14.25/8.11 => ( ( ord_less_eq_rat @ Y @ one_one_rat )
% 14.25/8.11 => ( ord_less_eq_rat @ ( times_times_rat @ X @ Y ) @ X ) ) ) ) ).
% 14.25/8.11
% 14.25/8.11 thf(fact_5937_mult__right__le__one__le,axiom,
% 14.25/8.11 ! [X: int,Y: int] :
% 14.25/8.11 ( ( ord_less_eq_int @ zero_zero_int @ X )
% 14.25/8.11 => ( ( ord_less_eq_int @ zero_zero_int @ Y )
% 14.25/8.11 => ( ( ord_less_eq_int @ Y @ one_one_int )
% 14.25/8.11 => ( ord_less_eq_int @ ( times_times_int @ X @ Y ) @ X ) ) ) ) ).
% 14.25/8.11
% 14.25/8.11 thf(fact_5938_mult__left__le__one__le,axiom,
% 14.25/8.11 ! [X: code_integer,Y: code_integer] :
% 14.25/8.11 ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ X )
% 14.25/8.11 => ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ Y )
% 14.25/8.11 => ( ( ord_le3102999989581377725nteger @ Y @ one_one_Code_integer )
% 14.25/8.11 => ( ord_le3102999989581377725nteger @ ( times_3573771949741848930nteger @ Y @ X ) @ X ) ) ) ) ).
% 14.25/8.11
% 14.25/8.11 thf(fact_5939_mult__left__le__one__le,axiom,
% 14.25/8.11 ! [X: rat,Y: rat] :
% 14.25/8.11 ( ( ord_less_eq_rat @ zero_zero_rat @ X )
% 14.25/8.11 => ( ( ord_less_eq_rat @ zero_zero_rat @ Y )
% 14.25/8.11 => ( ( ord_less_eq_rat @ Y @ one_one_rat )
% 14.25/8.11 => ( ord_less_eq_rat @ ( times_times_rat @ Y @ X ) @ X ) ) ) ) ).
% 14.25/8.11
% 14.25/8.11 thf(fact_5940_mult__left__le__one__le,axiom,
% 14.25/8.11 ! [X: int,Y: int] :
% 14.25/8.11 ( ( ord_less_eq_int @ zero_zero_int @ X )
% 14.25/8.11 => ( ( ord_less_eq_int @ zero_zero_int @ Y )
% 14.25/8.11 => ( ( ord_less_eq_int @ Y @ one_one_int )
% 14.25/8.11 => ( ord_less_eq_int @ ( times_times_int @ Y @ X ) @ X ) ) ) ) ).
% 14.25/8.11
% 14.25/8.11 thf(fact_5941_not__sum__squares__lt__zero,axiom,
% 14.25/8.11 ! [X: int,Y: int] :
% 14.25/8.11 ~ ( ord_less_int @ ( plus_plus_int @ ( times_times_int @ X @ X ) @ ( times_times_int @ Y @ Y ) ) @ zero_zero_int ) ).
% 14.25/8.11
% 14.25/8.11 thf(fact_5942_not__sum__squares__lt__zero,axiom,
% 14.25/8.11 ! [X: code_integer,Y: code_integer] :
% 14.25/8.11 ~ ( ord_le6747313008572928689nteger @ ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ X @ X ) @ ( times_3573771949741848930nteger @ Y @ Y ) ) @ zero_z3403309356797280102nteger ) ).
% 14.25/8.11
% 14.25/8.11 thf(fact_5943_not__sum__squares__lt__zero,axiom,
% 14.25/8.11 ! [X: rat,Y: rat] :
% 14.25/8.11 ~ ( ord_less_rat @ ( plus_plus_rat @ ( times_times_rat @ X @ X ) @ ( times_times_rat @ Y @ Y ) ) @ zero_zero_rat ) ).
% 14.25/8.11
% 14.25/8.11 thf(fact_5944_zero__less__two,axiom,
% 14.25/8.11 ord_le5570908160329646204atural @ zero_z2226904508553997617atural @ ( plus_p4538020629002901425atural @ one_one_Code_natural @ one_one_Code_natural ) ).
% 14.25/8.11
% 14.25/8.11 thf(fact_5945_zero__less__two,axiom,
% 14.25/8.11 ord_less_nat @ zero_zero_nat @ ( plus_plus_nat @ one_one_nat @ one_one_nat ) ).
% 14.25/8.12
% 14.25/8.12 thf(fact_5946_zero__less__two,axiom,
% 14.25/8.12 ord_less_int @ zero_zero_int @ ( plus_plus_int @ one_one_int @ one_one_int ) ).
% 14.25/8.12
% 14.25/8.12 thf(fact_5947_zero__less__two,axiom,
% 14.25/8.12 ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ ( plus_p5714425477246183910nteger @ one_one_Code_integer @ one_one_Code_integer ) ).
% 14.25/8.12
% 14.25/8.12 thf(fact_5948_zero__less__two,axiom,
% 14.25/8.12 ord_less_rat @ zero_zero_rat @ ( plus_plus_rat @ one_one_rat @ one_one_rat ) ).
% 14.25/8.12
% 14.25/8.12 thf(fact_5949_card__eq__0__iff,axiom,
% 14.25/8.12 ! [A: set_Code_integer] :
% 14.25/8.12 ( ( ( finite4902975817058060853nteger @ A )
% 14.25/8.12 = zero_zero_nat )
% 14.25/8.12 = ( ( A = bot_bo3990330152332043303nteger )
% 14.25/8.12 | ~ ( finite6017078050557962740nteger @ A ) ) ) ).
% 14.25/8.12
% 14.25/8.12 thf(fact_5950_card__eq__0__iff,axiom,
% 14.25/8.12 ! [A: set_Product_unit] :
% 14.25/8.12 ( ( ( finite410649719033368117t_unit @ A )
% 14.25/8.12 = zero_zero_nat )
% 14.25/8.12 = ( ( A = bot_bo3957492148770167129t_unit )
% 14.25/8.12 | ~ ( finite4290736615968046902t_unit @ A ) ) ) ).
% 14.25/8.12
% 14.25/8.12 thf(fact_5951_card__eq__0__iff,axiom,
% 14.25/8.12 ! [A: set_o] :
% 14.25/8.12 ( ( ( finite_card_o @ A )
% 14.25/8.12 = zero_zero_nat )
% 14.25/8.12 = ( ( A = bot_bot_set_o )
% 14.25/8.12 | ~ ( finite_finite_o @ A ) ) ) ).
% 14.25/8.12
% 14.25/8.12 thf(fact_5952_card__eq__0__iff,axiom,
% 14.25/8.12 ! [A: set_nat] :
% 14.25/8.12 ( ( ( finite_card_nat @ A )
% 14.25/8.12 = zero_zero_nat )
% 14.25/8.12 = ( ( A = bot_bot_set_nat )
% 14.25/8.12 | ~ ( finite_finite_nat @ A ) ) ) ).
% 14.25/8.12
% 14.25/8.12 thf(fact_5953_card__eq__0__iff,axiom,
% 14.25/8.12 ! [A: set_int] :
% 14.25/8.12 ( ( ( finite_card_int @ A )
% 14.25/8.12 = zero_zero_nat )
% 14.25/8.12 = ( ( A = bot_bot_set_int )
% 14.25/8.12 | ~ ( finite_finite_int @ A ) ) ) ).
% 14.25/8.12
% 14.25/8.12 thf(fact_5954_card__eq__0__iff,axiom,
% 14.25/8.12 ! [A: set_Pr4532377907799695533_nat_o] :
% 14.25/8.12 ( ( ( finite1363419556375932405_nat_o @ A )
% 14.25/8.12 = zero_zero_nat )
% 14.25/8.12 = ( ( A = bot_bo7824918357723582233_nat_o )
% 14.25/8.12 | ~ ( finite3252695134891459830_nat_o @ A ) ) ) ).
% 14.25/8.12
% 14.25/8.12 thf(fact_5955_one__less__mult,axiom,
% 14.25/8.12 ! [N2: nat,M4: nat] :
% 14.25/8.12 ( ( ord_less_nat @ ( suc @ zero_zero_nat ) @ N2 )
% 14.25/8.12 => ( ( ord_less_nat @ ( suc @ zero_zero_nat ) @ M4 )
% 14.25/8.12 => ( ord_less_nat @ ( suc @ zero_zero_nat ) @ ( times_times_nat @ M4 @ N2 ) ) ) ) ).
% 14.25/8.12
% 14.25/8.12 thf(fact_5956_n__less__m__mult__n,axiom,
% 14.25/8.12 ! [N2: nat,M4: nat] :
% 14.25/8.12 ( ( ord_less_nat @ zero_zero_nat @ N2 )
% 14.25/8.12 => ( ( ord_less_nat @ ( suc @ zero_zero_nat ) @ M4 )
% 14.25/8.12 => ( ord_less_nat @ N2 @ ( times_times_nat @ M4 @ N2 ) ) ) ) ).
% 14.25/8.12
% 14.25/8.12 thf(fact_5957_n__less__n__mult__m,axiom,
% 14.25/8.12 ! [N2: nat,M4: nat] :
% 14.25/8.12 ( ( ord_less_nat @ zero_zero_nat @ N2 )
% 14.25/8.12 => ( ( ord_less_nat @ ( suc @ zero_zero_nat ) @ M4 )
% 14.25/8.12 => ( ord_less_nat @ N2 @ ( times_times_nat @ N2 @ M4 ) ) ) ) ).
% 14.25/8.12
% 14.25/8.12 thf(fact_5958_nat__mult__le__cancel1,axiom,
% 14.25/8.12 ! [K3: nat,M4: nat,N2: nat] :
% 14.25/8.12 ( ( ord_less_nat @ zero_zero_nat @ K3 )
% 14.25/8.12 => ( ( ord_less_eq_nat @ ( times_times_nat @ K3 @ M4 ) @ ( times_times_nat @ K3 @ N2 ) )
% 14.25/8.12 = ( ord_less_eq_nat @ M4 @ N2 ) ) ) ).
% 14.25/8.12
% 14.25/8.12 thf(fact_5959_mult__le__cancel__left1,axiom,
% 14.25/8.12 ! [C: code_integer,B: code_integer] :
% 14.25/8.12 ( ( ord_le3102999989581377725nteger @ C @ ( times_3573771949741848930nteger @ C @ B ) )
% 14.25/8.12 = ( ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ C )
% 14.25/8.12 => ( ord_le3102999989581377725nteger @ one_one_Code_integer @ B ) )
% 14.25/8.12 & ( ( ord_le6747313008572928689nteger @ C @ zero_z3403309356797280102nteger )
% 14.25/8.12 => ( ord_le3102999989581377725nteger @ B @ one_one_Code_integer ) ) ) ) ).
% 14.25/8.12
% 14.25/8.12 thf(fact_5960_mult__le__cancel__left1,axiom,
% 14.25/8.12 ! [C: rat,B: rat] :
% 14.25/8.12 ( ( ord_less_eq_rat @ C @ ( times_times_rat @ C @ B ) )
% 14.25/8.12 = ( ( ( ord_less_rat @ zero_zero_rat @ C )
% 14.25/8.12 => ( ord_less_eq_rat @ one_one_rat @ B ) )
% 14.25/8.12 & ( ( ord_less_rat @ C @ zero_zero_rat )
% 14.25/8.12 => ( ord_less_eq_rat @ B @ one_one_rat ) ) ) ) ).
% 14.25/8.12
% 14.25/8.12 thf(fact_5961_mult__le__cancel__left1,axiom,
% 14.25/8.12 ! [C: int,B: int] :
% 14.25/8.12 ( ( ord_less_eq_int @ C @ ( times_times_int @ C @ B ) )
% 14.25/8.12 = ( ( ( ord_less_int @ zero_zero_int @ C )
% 14.25/8.12 => ( ord_less_eq_int @ one_one_int @ B ) )
% 14.25/8.12 & ( ( ord_less_int @ C @ zero_zero_int )
% 14.25/8.12 => ( ord_less_eq_int @ B @ one_one_int ) ) ) ) ).
% 14.25/8.12
% 14.25/8.12 thf(fact_5962_mult__le__cancel__left2,axiom,
% 14.25/8.12 ! [C: code_integer,A2: code_integer] :
% 14.25/8.12 ( ( ord_le3102999989581377725nteger @ ( times_3573771949741848930nteger @ C @ A2 ) @ C )
% 14.25/8.12 = ( ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ C )
% 14.25/8.12 => ( ord_le3102999989581377725nteger @ A2 @ one_one_Code_integer ) )
% 14.25/8.12 & ( ( ord_le6747313008572928689nteger @ C @ zero_z3403309356797280102nteger )
% 14.25/8.12 => ( ord_le3102999989581377725nteger @ one_one_Code_integer @ A2 ) ) ) ) ).
% 14.25/8.12
% 14.25/8.12 thf(fact_5963_mult__le__cancel__left2,axiom,
% 14.25/8.12 ! [C: rat,A2: rat] :
% 14.25/8.12 ( ( ord_less_eq_rat @ ( times_times_rat @ C @ A2 ) @ C )
% 14.25/8.12 = ( ( ( ord_less_rat @ zero_zero_rat @ C )
% 14.25/8.12 => ( ord_less_eq_rat @ A2 @ one_one_rat ) )
% 14.25/8.12 & ( ( ord_less_rat @ C @ zero_zero_rat )
% 14.25/8.12 => ( ord_less_eq_rat @ one_one_rat @ A2 ) ) ) ) ).
% 14.25/8.12
% 14.25/8.12 thf(fact_5964_mult__le__cancel__left2,axiom,
% 14.25/8.12 ! [C: int,A2: int] :
% 14.25/8.12 ( ( ord_less_eq_int @ ( times_times_int @ C @ A2 ) @ C )
% 14.25/8.12 = ( ( ( ord_less_int @ zero_zero_int @ C )
% 14.25/8.12 => ( ord_less_eq_int @ A2 @ one_one_int ) )
% 14.25/8.12 & ( ( ord_less_int @ C @ zero_zero_int )
% 14.25/8.12 => ( ord_less_eq_int @ one_one_int @ A2 ) ) ) ) ).
% 14.25/8.12
% 14.25/8.12 thf(fact_5965_mult__le__cancel__right1,axiom,
% 14.25/8.12 ! [C: code_integer,B: code_integer] :
% 14.25/8.12 ( ( ord_le3102999989581377725nteger @ C @ ( times_3573771949741848930nteger @ B @ C ) )
% 14.25/8.12 = ( ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ C )
% 14.25/8.12 => ( ord_le3102999989581377725nteger @ one_one_Code_integer @ B ) )
% 14.25/8.12 & ( ( ord_le6747313008572928689nteger @ C @ zero_z3403309356797280102nteger )
% 14.25/8.12 => ( ord_le3102999989581377725nteger @ B @ one_one_Code_integer ) ) ) ) ).
% 14.25/8.12
% 14.25/8.12 thf(fact_5966_mult__le__cancel__right1,axiom,
% 14.25/8.12 ! [C: rat,B: rat] :
% 14.25/8.12 ( ( ord_less_eq_rat @ C @ ( times_times_rat @ B @ C ) )
% 14.25/8.12 = ( ( ( ord_less_rat @ zero_zero_rat @ C )
% 14.25/8.12 => ( ord_less_eq_rat @ one_one_rat @ B ) )
% 14.25/8.12 & ( ( ord_less_rat @ C @ zero_zero_rat )
% 14.25/8.12 => ( ord_less_eq_rat @ B @ one_one_rat ) ) ) ) ).
% 14.25/8.12
% 14.25/8.12 thf(fact_5967_mult__le__cancel__right1,axiom,
% 14.25/8.12 ! [C: int,B: int] :
% 14.25/8.12 ( ( ord_less_eq_int @ C @ ( times_times_int @ B @ C ) )
% 14.25/8.12 = ( ( ( ord_less_int @ zero_zero_int @ C )
% 14.25/8.12 => ( ord_less_eq_int @ one_one_int @ B ) )
% 14.25/8.12 & ( ( ord_less_int @ C @ zero_zero_int )
% 14.25/8.12 => ( ord_less_eq_int @ B @ one_one_int ) ) ) ) ).
% 14.25/8.12
% 14.25/8.12 thf(fact_5968_mult__le__cancel__right2,axiom,
% 14.25/8.12 ! [A2: code_integer,C: code_integer] :
% 14.25/8.12 ( ( ord_le3102999989581377725nteger @ ( times_3573771949741848930nteger @ A2 @ C ) @ C )
% 14.25/8.12 = ( ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ C )
% 14.25/8.12 => ( ord_le3102999989581377725nteger @ A2 @ one_one_Code_integer ) )
% 14.25/8.12 & ( ( ord_le6747313008572928689nteger @ C @ zero_z3403309356797280102nteger )
% 14.25/8.12 => ( ord_le3102999989581377725nteger @ one_one_Code_integer @ A2 ) ) ) ) ).
% 14.25/8.12
% 14.25/8.12 thf(fact_5969_mult__le__cancel__right2,axiom,
% 14.25/8.12 ! [A2: rat,C: rat] :
% 14.25/8.12 ( ( ord_less_eq_rat @ ( times_times_rat @ A2 @ C ) @ C )
% 14.25/8.12 = ( ( ( ord_less_rat @ zero_zero_rat @ C )
% 14.25/8.12 => ( ord_less_eq_rat @ A2 @ one_one_rat ) )
% 14.25/8.12 & ( ( ord_less_rat @ C @ zero_zero_rat )
% 14.25/8.12 => ( ord_less_eq_rat @ one_one_rat @ A2 ) ) ) ) ).
% 14.25/8.12
% 14.25/8.12 thf(fact_5970_mult__le__cancel__right2,axiom,
% 14.25/8.12 ! [A2: int,C: int] :
% 14.25/8.12 ( ( ord_less_eq_int @ ( times_times_int @ A2 @ C ) @ C )
% 14.25/8.12 = ( ( ( ord_less_int @ zero_zero_int @ C )
% 14.25/8.12 => ( ord_less_eq_int @ A2 @ one_one_int ) )
% 14.25/8.12 & ( ( ord_less_int @ C @ zero_zero_int )
% 14.25/8.12 => ( ord_less_eq_int @ one_one_int @ A2 ) ) ) ) ).
% 14.25/8.12
% 14.25/8.12 thf(fact_5971_mult__less__cancel__left1,axiom,
% 14.25/8.12 ! [C: code_integer,B: code_integer] :
% 14.25/8.12 ( ( ord_le6747313008572928689nteger @ C @ ( times_3573771949741848930nteger @ C @ B ) )
% 14.25/8.12 = ( ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ C )
% 14.25/8.12 => ( ord_le6747313008572928689nteger @ one_one_Code_integer @ B ) )
% 14.25/8.12 & ( ( ord_le3102999989581377725nteger @ C @ zero_z3403309356797280102nteger )
% 14.25/8.12 => ( ord_le6747313008572928689nteger @ B @ one_one_Code_integer ) ) ) ) ).
% 14.25/8.12
% 14.25/8.12 thf(fact_5972_mult__less__cancel__left1,axiom,
% 14.25/8.12 ! [C: rat,B: rat] :
% 14.25/8.12 ( ( ord_less_rat @ C @ ( times_times_rat @ C @ B ) )
% 14.25/8.12 = ( ( ( ord_less_eq_rat @ zero_zero_rat @ C )
% 14.25/8.12 => ( ord_less_rat @ one_one_rat @ B ) )
% 14.25/8.12 & ( ( ord_less_eq_rat @ C @ zero_zero_rat )
% 14.25/8.12 => ( ord_less_rat @ B @ one_one_rat ) ) ) ) ).
% 14.25/8.12
% 14.25/8.12 thf(fact_5973_mult__less__cancel__left1,axiom,
% 14.25/8.12 ! [C: int,B: int] :
% 14.25/8.12 ( ( ord_less_int @ C @ ( times_times_int @ C @ B ) )
% 14.25/8.12 = ( ( ( ord_less_eq_int @ zero_zero_int @ C )
% 14.25/8.12 => ( ord_less_int @ one_one_int @ B ) )
% 14.25/8.12 & ( ( ord_less_eq_int @ C @ zero_zero_int )
% 14.25/8.12 => ( ord_less_int @ B @ one_one_int ) ) ) ) ).
% 14.25/8.12
% 14.25/8.12 thf(fact_5974_mult__less__cancel__left2,axiom,
% 14.25/8.12 ! [C: code_integer,A2: code_integer] :
% 14.25/8.12 ( ( ord_le6747313008572928689nteger @ ( times_3573771949741848930nteger @ C @ A2 ) @ C )
% 14.25/8.12 = ( ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ C )
% 14.25/8.12 => ( ord_le6747313008572928689nteger @ A2 @ one_one_Code_integer ) )
% 14.25/8.12 & ( ( ord_le3102999989581377725nteger @ C @ zero_z3403309356797280102nteger )
% 14.25/8.12 => ( ord_le6747313008572928689nteger @ one_one_Code_integer @ A2 ) ) ) ) ).
% 14.25/8.12
% 14.25/8.12 thf(fact_5975_mult__less__cancel__left2,axiom,
% 14.25/8.12 ! [C: rat,A2: rat] :
% 14.25/8.12 ( ( ord_less_rat @ ( times_times_rat @ C @ A2 ) @ C )
% 14.25/8.12 = ( ( ( ord_less_eq_rat @ zero_zero_rat @ C )
% 14.25/8.12 => ( ord_less_rat @ A2 @ one_one_rat ) )
% 14.25/8.12 & ( ( ord_less_eq_rat @ C @ zero_zero_rat )
% 14.25/8.12 => ( ord_less_rat @ one_one_rat @ A2 ) ) ) ) ).
% 14.25/8.12
% 14.25/8.12 thf(fact_5976_mult__less__cancel__left2,axiom,
% 14.25/8.12 ! [C: int,A2: int] :
% 14.25/8.12 ( ( ord_less_int @ ( times_times_int @ C @ A2 ) @ C )
% 14.25/8.12 = ( ( ( ord_less_eq_int @ zero_zero_int @ C )
% 14.25/8.12 => ( ord_less_int @ A2 @ one_one_int ) )
% 14.25/8.12 & ( ( ord_less_eq_int @ C @ zero_zero_int )
% 14.25/8.12 => ( ord_less_int @ one_one_int @ A2 ) ) ) ) ).
% 14.25/8.12
% 14.25/8.12 thf(fact_5977_mult__less__cancel__right1,axiom,
% 14.25/8.12 ! [C: code_integer,B: code_integer] :
% 14.25/8.12 ( ( ord_le6747313008572928689nteger @ C @ ( times_3573771949741848930nteger @ B @ C ) )
% 14.25/8.12 = ( ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ C )
% 14.25/8.12 => ( ord_le6747313008572928689nteger @ one_one_Code_integer @ B ) )
% 14.25/8.12 & ( ( ord_le3102999989581377725nteger @ C @ zero_z3403309356797280102nteger )
% 14.25/8.12 => ( ord_le6747313008572928689nteger @ B @ one_one_Code_integer ) ) ) ) ).
% 14.25/8.12
% 14.25/8.12 thf(fact_5978_mult__less__cancel__right1,axiom,
% 14.25/8.12 ! [C: rat,B: rat] :
% 14.25/8.12 ( ( ord_less_rat @ C @ ( times_times_rat @ B @ C ) )
% 14.25/8.12 = ( ( ( ord_less_eq_rat @ zero_zero_rat @ C )
% 14.25/8.12 => ( ord_less_rat @ one_one_rat @ B ) )
% 14.25/8.12 & ( ( ord_less_eq_rat @ C @ zero_zero_rat )
% 14.25/8.12 => ( ord_less_rat @ B @ one_one_rat ) ) ) ) ).
% 14.25/8.12
% 14.25/8.12 thf(fact_5979_mult__less__cancel__right1,axiom,
% 14.25/8.12 ! [C: int,B: int] :
% 14.25/8.12 ( ( ord_less_int @ C @ ( times_times_int @ B @ C ) )
% 14.25/8.12 = ( ( ( ord_less_eq_int @ zero_zero_int @ C )
% 14.25/8.12 => ( ord_less_int @ one_one_int @ B ) )
% 14.25/8.12 & ( ( ord_less_eq_int @ C @ zero_zero_int )
% 14.25/8.12 => ( ord_less_int @ B @ one_one_int ) ) ) ) ).
% 14.25/8.12
% 14.25/8.12 thf(fact_5980_mult__less__cancel__right2,axiom,
% 14.25/8.12 ! [A2: code_integer,C: code_integer] :
% 14.25/8.12 ( ( ord_le6747313008572928689nteger @ ( times_3573771949741848930nteger @ A2 @ C ) @ C )
% 14.25/8.12 = ( ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ C )
% 14.25/8.12 => ( ord_le6747313008572928689nteger @ A2 @ one_one_Code_integer ) )
% 14.25/8.12 & ( ( ord_le3102999989581377725nteger @ C @ zero_z3403309356797280102nteger )
% 14.25/8.12 => ( ord_le6747313008572928689nteger @ one_one_Code_integer @ A2 ) ) ) ) ).
% 14.25/8.12
% 14.25/8.12 thf(fact_5981_mult__less__cancel__right2,axiom,
% 14.25/8.12 ! [A2: rat,C: rat] :
% 14.25/8.12 ( ( ord_less_rat @ ( times_times_rat @ A2 @ C ) @ C )
% 14.25/8.12 = ( ( ( ord_less_eq_rat @ zero_zero_rat @ C )
% 14.25/8.12 => ( ord_less_rat @ A2 @ one_one_rat ) )
% 14.25/8.12 & ( ( ord_less_eq_rat @ C @ zero_zero_rat )
% 14.25/8.12 => ( ord_less_rat @ one_one_rat @ A2 ) ) ) ) ).
% 14.25/8.12
% 14.25/8.12 thf(fact_5982_mult__less__cancel__right2,axiom,
% 14.25/8.12 ! [A2: int,C: int] :
% 14.25/8.12 ( ( ord_less_int @ ( times_times_int @ A2 @ C ) @ C )
% 14.25/8.12 = ( ( ( ord_less_eq_int @ zero_zero_int @ C )
% 14.25/8.12 => ( ord_less_int @ A2 @ one_one_int ) )
% 14.25/8.12 & ( ( ord_less_eq_int @ C @ zero_zero_int )
% 14.25/8.12 => ( ord_less_int @ one_one_int @ A2 ) ) ) ) ).
% 14.25/8.12
% 14.25/8.12 thf(fact_5983_convex__bound__le,axiom,
% 14.25/8.12 ! [X: code_integer,A2: code_integer,Y: code_integer,U: code_integer,V: code_integer] :
% 14.25/8.12 ( ( ord_le3102999989581377725nteger @ X @ A2 )
% 14.25/8.12 => ( ( ord_le3102999989581377725nteger @ Y @ A2 )
% 14.25/8.12 => ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ U )
% 14.25/8.12 => ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ V )
% 14.25/8.12 => ( ( ( plus_p5714425477246183910nteger @ U @ V )
% 14.25/8.12 = one_one_Code_integer )
% 14.25/8.12 => ( ord_le3102999989581377725nteger @ ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ U @ X ) @ ( times_3573771949741848930nteger @ V @ Y ) ) @ A2 ) ) ) ) ) ) ).
% 14.25/8.12
% 14.25/8.12 thf(fact_5984_convex__bound__le,axiom,
% 14.25/8.12 ! [X: rat,A2: rat,Y: rat,U: rat,V: rat] :
% 14.25/8.12 ( ( ord_less_eq_rat @ X @ A2 )
% 14.25/8.12 => ( ( ord_less_eq_rat @ Y @ A2 )
% 14.25/8.12 => ( ( ord_less_eq_rat @ zero_zero_rat @ U )
% 14.25/8.12 => ( ( ord_less_eq_rat @ zero_zero_rat @ V )
% 14.25/8.12 => ( ( ( plus_plus_rat @ U @ V )
% 14.25/8.12 = one_one_rat )
% 14.25/8.12 => ( ord_less_eq_rat @ ( plus_plus_rat @ ( times_times_rat @ U @ X ) @ ( times_times_rat @ V @ Y ) ) @ A2 ) ) ) ) ) ) ).
% 14.25/8.12
% 14.25/8.12 thf(fact_5985_convex__bound__le,axiom,
% 14.25/8.12 ! [X: int,A2: int,Y: int,U: int,V: int] :
% 14.25/8.12 ( ( ord_less_eq_int @ X @ A2 )
% 14.25/8.12 => ( ( ord_less_eq_int @ Y @ A2 )
% 14.25/8.12 => ( ( ord_less_eq_int @ zero_zero_int @ U )
% 14.25/8.12 => ( ( ord_less_eq_int @ zero_zero_int @ V )
% 14.25/8.12 => ( ( ( plus_plus_int @ U @ V )
% 14.25/8.12 = one_one_int )
% 14.25/8.12 => ( ord_less_eq_int @ ( plus_plus_int @ ( times_times_int @ U @ X ) @ ( times_times_int @ V @ Y ) ) @ A2 ) ) ) ) ) ) ).
% 14.25/8.12
% 14.25/8.12 thf(fact_5986_card__gt__0__iff,axiom,
% 14.25/8.12 ! [A: set_Code_integer] :
% 14.25/8.12 ( ( ord_less_nat @ zero_zero_nat @ ( finite4902975817058060853nteger @ A ) )
% 14.25/8.12 = ( ( A != bot_bo3990330152332043303nteger )
% 14.25/8.12 & ( finite6017078050557962740nteger @ A ) ) ) ).
% 14.25/8.12
% 14.25/8.12 thf(fact_5987_card__gt__0__iff,axiom,
% 14.25/8.12 ! [A: set_Product_unit] :
% 14.25/8.12 ( ( ord_less_nat @ zero_zero_nat @ ( finite410649719033368117t_unit @ A ) )
% 14.25/8.12 = ( ( A != bot_bo3957492148770167129t_unit )
% 14.25/8.12 & ( finite4290736615968046902t_unit @ A ) ) ) ).
% 14.25/8.12
% 14.25/8.12 thf(fact_5988_card__gt__0__iff,axiom,
% 14.25/8.12 ! [A: set_o] :
% 14.25/8.12 ( ( ord_less_nat @ zero_zero_nat @ ( finite_card_o @ A ) )
% 14.25/8.12 = ( ( A != bot_bot_set_o )
% 14.25/8.12 & ( finite_finite_o @ A ) ) ) ).
% 14.25/8.12
% 14.25/8.12 thf(fact_5989_card__gt__0__iff,axiom,
% 14.25/8.12 ! [A: set_nat] :
% 14.25/8.12 ( ( ord_less_nat @ zero_zero_nat @ ( finite_card_nat @ A ) )
% 14.25/8.12 = ( ( A != bot_bot_set_nat )
% 14.25/8.12 & ( finite_finite_nat @ A ) ) ) ).
% 14.25/8.12
% 14.25/8.12 thf(fact_5990_card__gt__0__iff,axiom,
% 14.25/8.12 ! [A: set_int] :
% 14.25/8.12 ( ( ord_less_nat @ zero_zero_nat @ ( finite_card_int @ A ) )
% 14.25/8.12 = ( ( A != bot_bot_set_int )
% 14.25/8.12 & ( finite_finite_int @ A ) ) ) ).
% 14.25/8.12
% 14.25/8.12 thf(fact_5991_card__gt__0__iff,axiom,
% 14.25/8.12 ! [A: set_Pr4532377907799695533_nat_o] :
% 14.25/8.12 ( ( ord_less_nat @ zero_zero_nat @ ( finite1363419556375932405_nat_o @ A ) )
% 14.25/8.12 = ( ( A != bot_bo7824918357723582233_nat_o )
% 14.25/8.12 & ( finite3252695134891459830_nat_o @ A ) ) ) ).
% 14.25/8.12
% 14.25/8.12 thf(fact_5992_card__1__singleton__iff,axiom,
% 14.25/8.12 ! [A: set_Product_unit] :
% 14.25/8.12 ( ( ( finite410649719033368117t_unit @ A )
% 14.25/8.12 = ( suc @ zero_zero_nat ) )
% 14.25/8.12 = ( ? [X2: product_unit] :
% 14.25/8.12 ( A
% 14.25/8.12 = ( insert_Product_unit @ X2 @ bot_bo3957492148770167129t_unit ) ) ) ) ).
% 14.25/8.12
% 14.25/8.12 thf(fact_5993_card__1__singleton__iff,axiom,
% 14.25/8.12 ! [A: set_o] :
% 14.25/8.12 ( ( ( finite_card_o @ A )
% 14.25/8.12 = ( suc @ zero_zero_nat ) )
% 14.25/8.12 = ( ? [X2: $o] :
% 14.25/8.12 ( A
% 14.25/8.12 = ( insert_o @ X2 @ bot_bot_set_o ) ) ) ) ).
% 14.25/8.12
% 14.25/8.12 thf(fact_5994_card__1__singleton__iff,axiom,
% 14.25/8.12 ! [A: set_nat] :
% 14.25/8.12 ( ( ( finite_card_nat @ A )
% 14.25/8.12 = ( suc @ zero_zero_nat ) )
% 14.25/8.12 = ( ? [X2: nat] :
% 14.25/8.12 ( A
% 14.25/8.12 = ( insert_nat @ X2 @ bot_bot_set_nat ) ) ) ) ).
% 14.25/8.12
% 14.25/8.12 thf(fact_5995_card__1__singleton__iff,axiom,
% 14.25/8.12 ! [A: set_int] :
% 14.25/8.12 ( ( ( finite_card_int @ A )
% 14.25/8.12 = ( suc @ zero_zero_nat ) )
% 14.25/8.12 = ( ? [X2: int] :
% 14.25/8.12 ( A
% 14.25/8.12 = ( insert_int @ X2 @ bot_bot_set_int ) ) ) ) ).
% 14.25/8.12
% 14.25/8.12 thf(fact_5996_card__1__singleton__iff,axiom,
% 14.25/8.12 ! [A: set_Pr4532377907799695533_nat_o] :
% 14.25/8.12 ( ( ( finite1363419556375932405_nat_o @ A )
% 14.25/8.12 = ( suc @ zero_zero_nat ) )
% 14.25/8.12 = ( ? [X2: produc3658429121746597890et_nat > $o] :
% 14.25/8.12 ( A
% 14.25/8.12 = ( insert5175938949040314269_nat_o @ X2 @ bot_bo7824918357723582233_nat_o ) ) ) ) ).
% 14.25/8.12
% 14.25/8.12 thf(fact_5997_card__eq__SucD,axiom,
% 14.25/8.12 ! [A: set_Product_unit,K3: nat] :
% 14.25/8.12 ( ( ( finite410649719033368117t_unit @ A )
% 14.25/8.12 = ( suc @ K3 ) )
% 14.25/8.12 => ? [B2: product_unit,B5: set_Product_unit] :
% 14.25/8.12 ( ( A
% 14.25/8.12 = ( insert_Product_unit @ B2 @ B5 ) )
% 14.25/8.12 & ~ ( member_Product_unit @ B2 @ B5 )
% 14.25/8.12 & ( ( finite410649719033368117t_unit @ B5 )
% 14.25/8.12 = K3 )
% 14.25/8.12 & ( ( K3 = zero_zero_nat )
% 14.25/8.12 => ( B5 = bot_bo3957492148770167129t_unit ) ) ) ) ).
% 14.25/8.12
% 14.25/8.12 thf(fact_5998_card__eq__SucD,axiom,
% 14.25/8.12 ! [A: set_o,K3: nat] :
% 14.25/8.12 ( ( ( finite_card_o @ A )
% 14.25/8.12 = ( suc @ K3 ) )
% 14.25/8.12 => ? [B2: $o,B5: set_o] :
% 14.25/8.12 ( ( A
% 14.25/8.12 = ( insert_o @ B2 @ B5 ) )
% 14.25/8.12 & ~ ( member_o @ B2 @ B5 )
% 14.25/8.12 & ( ( finite_card_o @ B5 )
% 14.25/8.12 = K3 )
% 14.25/8.12 & ( ( K3 = zero_zero_nat )
% 14.25/8.12 => ( B5 = bot_bot_set_o ) ) ) ) ).
% 14.25/8.12
% 14.25/8.12 thf(fact_5999_card__eq__SucD,axiom,
% 14.25/8.12 ! [A: set_nat,K3: nat] :
% 14.25/8.12 ( ( ( finite_card_nat @ A )
% 14.25/8.12 = ( suc @ K3 ) )
% 14.25/8.12 => ? [B2: nat,B5: set_nat] :
% 14.25/8.12 ( ( A
% 14.25/8.12 = ( insert_nat @ B2 @ B5 ) )
% 14.25/8.12 & ~ ( member_nat @ B2 @ B5 )
% 14.25/8.12 & ( ( finite_card_nat @ B5 )
% 14.25/8.12 = K3 )
% 14.25/8.12 & ( ( K3 = zero_zero_nat )
% 14.25/8.12 => ( B5 = bot_bot_set_nat ) ) ) ) ).
% 14.25/8.12
% 14.25/8.12 thf(fact_6000_card__eq__SucD,axiom,
% 14.25/8.12 ! [A: set_int,K3: nat] :
% 14.25/8.12 ( ( ( finite_card_int @ A )
% 14.25/8.12 = ( suc @ K3 ) )
% 14.25/8.12 => ? [B2: int,B5: set_int] :
% 14.25/8.12 ( ( A
% 14.25/8.12 = ( insert_int @ B2 @ B5 ) )
% 14.25/8.12 & ~ ( member_int @ B2 @ B5 )
% 14.25/8.12 & ( ( finite_card_int @ B5 )
% 14.25/8.12 = K3 )
% 14.25/8.12 & ( ( K3 = zero_zero_nat )
% 14.25/8.12 => ( B5 = bot_bot_set_int ) ) ) ) ).
% 14.25/8.12
% 14.25/8.12 thf(fact_6001_card__eq__SucD,axiom,
% 14.25/8.12 ! [A: set_Pr4532377907799695533_nat_o,K3: nat] :
% 14.25/8.12 ( ( ( finite1363419556375932405_nat_o @ A )
% 14.25/8.12 = ( suc @ K3 ) )
% 14.25/8.12 => ? [B2: produc3658429121746597890et_nat > $o,B5: set_Pr4532377907799695533_nat_o] :
% 14.25/8.12 ( ( A
% 14.25/8.12 = ( insert5175938949040314269_nat_o @ B2 @ B5 ) )
% 14.25/8.12 & ~ ( member6576561426505652726_nat_o @ B2 @ B5 )
% 14.25/8.12 & ( ( finite1363419556375932405_nat_o @ B5 )
% 14.25/8.12 = K3 )
% 14.25/8.12 & ( ( K3 = zero_zero_nat )
% 14.25/8.12 => ( B5 = bot_bo7824918357723582233_nat_o ) ) ) ) ).
% 14.25/8.12
% 14.25/8.12 thf(fact_6002_card__Suc__eq,axiom,
% 14.25/8.12 ! [A: set_Product_unit,K3: nat] :
% 14.25/8.12 ( ( ( finite410649719033368117t_unit @ A )
% 14.25/8.12 = ( suc @ K3 ) )
% 14.25/8.12 = ( ? [B6: product_unit,B7: set_Product_unit] :
% 14.25/8.12 ( ( A
% 14.25/8.12 = ( insert_Product_unit @ B6 @ B7 ) )
% 14.25/8.12 & ~ ( member_Product_unit @ B6 @ B7 )
% 14.25/8.12 & ( ( finite410649719033368117t_unit @ B7 )
% 14.25/8.12 = K3 )
% 14.25/8.12 & ( ( K3 = zero_zero_nat )
% 14.25/8.12 => ( B7 = bot_bo3957492148770167129t_unit ) ) ) ) ) ).
% 14.25/8.12
% 14.25/8.12 thf(fact_6003_card__Suc__eq,axiom,
% 14.25/8.12 ! [A: set_o,K3: nat] :
% 14.25/8.12 ( ( ( finite_card_o @ A )
% 14.25/8.12 = ( suc @ K3 ) )
% 14.25/8.12 = ( ? [B6: $o,B7: set_o] :
% 14.25/8.12 ( ( A
% 14.25/8.12 = ( insert_o @ B6 @ B7 ) )
% 14.25/8.12 & ~ ( member_o @ B6 @ B7 )
% 14.25/8.12 & ( ( finite_card_o @ B7 )
% 14.25/8.12 = K3 )
% 14.25/8.12 & ( ( K3 = zero_zero_nat )
% 14.25/8.12 => ( B7 = bot_bot_set_o ) ) ) ) ) ).
% 14.25/8.12
% 14.25/8.12 thf(fact_6004_card__Suc__eq,axiom,
% 14.25/8.12 ! [A: set_nat,K3: nat] :
% 14.25/8.12 ( ( ( finite_card_nat @ A )
% 14.25/8.12 = ( suc @ K3 ) )
% 14.25/8.12 = ( ? [B6: nat,B7: set_nat] :
% 14.25/8.12 ( ( A
% 14.25/8.12 = ( insert_nat @ B6 @ B7 ) )
% 14.25/8.12 & ~ ( member_nat @ B6 @ B7 )
% 14.25/8.12 & ( ( finite_card_nat @ B7 )
% 14.25/8.12 = K3 )
% 14.25/8.12 & ( ( K3 = zero_zero_nat )
% 14.25/8.12 => ( B7 = bot_bot_set_nat ) ) ) ) ) ).
% 14.25/8.12
% 14.25/8.12 thf(fact_6005_card__Suc__eq,axiom,
% 14.25/8.12 ! [A: set_int,K3: nat] :
% 14.25/8.12 ( ( ( finite_card_int @ A )
% 14.25/8.12 = ( suc @ K3 ) )
% 14.25/8.12 = ( ? [B6: int,B7: set_int] :
% 14.25/8.12 ( ( A
% 14.25/8.12 = ( insert_int @ B6 @ B7 ) )
% 14.25/8.12 & ~ ( member_int @ B6 @ B7 )
% 14.25/8.12 & ( ( finite_card_int @ B7 )
% 14.25/8.12 = K3 )
% 14.25/8.12 & ( ( K3 = zero_zero_nat )
% 14.25/8.12 => ( B7 = bot_bot_set_int ) ) ) ) ) ).
% 14.25/8.12
% 14.25/8.12 thf(fact_6006_card__Suc__eq,axiom,
% 14.25/8.12 ! [A: set_Pr4532377907799695533_nat_o,K3: nat] :
% 14.25/8.12 ( ( ( finite1363419556375932405_nat_o @ A )
% 14.25/8.12 = ( suc @ K3 ) )
% 14.25/8.12 = ( ? [B6: produc3658429121746597890et_nat > $o,B7: set_Pr4532377907799695533_nat_o] :
% 14.25/8.12 ( ( A
% 14.25/8.12 = ( insert5175938949040314269_nat_o @ B6 @ B7 ) )
% 14.25/8.12 & ~ ( member6576561426505652726_nat_o @ B6 @ B7 )
% 14.25/8.12 & ( ( finite1363419556375932405_nat_o @ B7 )
% 14.25/8.12 = K3 )
% 14.25/8.12 & ( ( K3 = zero_zero_nat )
% 14.25/8.12 => ( B7 = bot_bo7824918357723582233_nat_o ) ) ) ) ) ).
% 14.25/8.12
% 14.25/8.12 thf(fact_6007_nz__le__conv__less,axiom,
% 14.25/8.12 ! [K3: nat,M4: nat] :
% 14.25/8.12 ( ( ord_less_nat @ zero_zero_nat @ K3 )
% 14.25/8.12 => ( ( ord_less_eq_nat @ K3 @ M4 )
% 14.25/8.12 => ( ord_less_nat @ ( minus_minus_nat @ K3 @ ( suc @ zero_zero_nat ) ) @ M4 ) ) ) ).
% 14.25/8.12
% 14.25/8.12 thf(fact_6008_mult__eq__if,axiom,
% 14.25/8.12 ( times_times_nat
% 14.25/8.12 = ( ^ [M3: nat,N4: nat] : ( if_nat @ ( M3 = zero_zero_nat ) @ zero_zero_nat @ ( plus_plus_nat @ N4 @ ( times_times_nat @ ( minus_minus_nat @ M3 @ one_one_nat ) @ N4 ) ) ) ) ) ).
% 14.25/8.12
% 14.25/8.12 thf(fact_6009_convex__bound__lt,axiom,
% 14.25/8.12 ! [X: code_integer,A2: code_integer,Y: code_integer,U: code_integer,V: code_integer] :
% 14.25/8.12 ( ( ord_le6747313008572928689nteger @ X @ A2 )
% 14.25/8.12 => ( ( ord_le6747313008572928689nteger @ Y @ A2 )
% 14.25/8.12 => ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ U )
% 14.25/8.12 => ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ V )
% 14.25/8.12 => ( ( ( plus_p5714425477246183910nteger @ U @ V )
% 14.25/8.12 = one_one_Code_integer )
% 14.25/8.12 => ( ord_le6747313008572928689nteger @ ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ U @ X ) @ ( times_3573771949741848930nteger @ V @ Y ) ) @ A2 ) ) ) ) ) ) ).
% 14.25/8.12
% 14.25/8.12 thf(fact_6010_convex__bound__lt,axiom,
% 14.25/8.12 ! [X: rat,A2: rat,Y: rat,U: rat,V: rat] :
% 14.25/8.12 ( ( ord_less_rat @ X @ A2 )
% 14.25/8.12 => ( ( ord_less_rat @ Y @ A2 )
% 14.25/8.12 => ( ( ord_less_eq_rat @ zero_zero_rat @ U )
% 14.25/8.12 => ( ( ord_less_eq_rat @ zero_zero_rat @ V )
% 14.25/8.12 => ( ( ( plus_plus_rat @ U @ V )
% 14.25/8.12 = one_one_rat )
% 14.25/8.12 => ( ord_less_rat @ ( plus_plus_rat @ ( times_times_rat @ U @ X ) @ ( times_times_rat @ V @ Y ) ) @ A2 ) ) ) ) ) ) ).
% 14.25/8.12
% 14.25/8.12 thf(fact_6011_convex__bound__lt,axiom,
% 14.25/8.12 ! [X: int,A2: int,Y: int,U: int,V: int] :
% 14.25/8.12 ( ( ord_less_int @ X @ A2 )
% 14.25/8.12 => ( ( ord_less_int @ Y @ A2 )
% 14.25/8.12 => ( ( ord_less_eq_int @ zero_zero_int @ U )
% 14.25/8.12 => ( ( ord_less_eq_int @ zero_zero_int @ V )
% 14.25/8.12 => ( ( ( plus_plus_int @ U @ V )
% 14.25/8.12 = one_one_int )
% 14.25/8.12 => ( ord_less_int @ ( plus_plus_int @ ( times_times_int @ U @ X ) @ ( times_times_int @ V @ Y ) ) @ A2 ) ) ) ) ) ) ).
% 14.25/8.12
% 14.25/8.12 thf(fact_6012_inj__mult__left,axiom,
% 14.25/8.12 ! [A2: code_integer] :
% 14.25/8.12 ( ( inj_on1158064722259180589nteger @ ( times_3573771949741848930nteger @ A2 ) @ top_to4645266643341252675nteger )
% 14.25/8.12 = ( A2 != zero_z3403309356797280102nteger ) ) ).
% 14.25/8.12
% 14.25/8.12 thf(fact_6013_inj__mult__left,axiom,
% 14.25/8.12 ! [A2: rat] :
% 14.25/8.12 ( ( inj_on_rat_rat @ ( times_times_rat @ A2 ) @ top_top_set_rat )
% 14.25/8.12 = ( A2 != zero_zero_rat ) ) ).
% 14.25/8.12
% 14.25/8.12 thf(fact_6014_inj__mult__left,axiom,
% 14.25/8.12 ! [A2: int] :
% 14.25/8.12 ( ( inj_on_int_int @ ( times_times_int @ A2 ) @ top_top_set_int )
% 14.25/8.12 = ( A2 != zero_zero_int ) ) ).
% 14.25/8.12
% 14.25/8.12 thf(fact_6015_field__le__mult__one__interval,axiom,
% 14.25/8.12 ! [X: rat,Y: rat] :
% 14.25/8.12 ( ! [Z3: rat] :
% 14.25/8.12 ( ( ord_less_rat @ zero_zero_rat @ Z3 )
% 14.25/8.12 => ( ( ord_less_rat @ Z3 @ one_one_rat )
% 14.25/8.12 => ( ord_less_eq_rat @ ( times_times_rat @ Z3 @ X ) @ Y ) ) )
% 14.25/8.12 => ( ord_less_eq_rat @ X @ Y ) ) ).
% 14.25/8.12
% 14.25/8.12 thf(fact_6016_size__prod__simp,axiom,
% 14.25/8.12 ( basic_5567952622663070111et_nat
% 14.25/8.12 = ( ^ [F4: heap_e7401611519738050253t_unit > nat,G3: set_nat > nat,P5: produc3658429121746597890et_nat] : ( plus_plus_nat @ ( plus_plus_nat @ ( F4 @ ( produc1824681642469235216et_nat @ P5 ) ) @ ( G3 @ ( produc8586169260539613262et_nat @ P5 ) ) ) @ ( suc @ zero_zero_nat ) ) ) ) ).
% 14.25/8.12
% 14.25/8.12 thf(fact_6017_size__prod__simp,axiom,
% 14.25/8.12 ( basic_1872990034501187214nt_int
% 14.25/8.12 = ( ^ [F4: int > nat,G3: int > nat,P5: product_prod_int_int] : ( plus_plus_nat @ ( plus_plus_nat @ ( F4 @ ( product_fst_int_int @ P5 ) ) @ ( G3 @ ( product_snd_int_int @ P5 ) ) ) @ ( suc @ zero_zero_nat ) ) ) ) ).
% 14.25/8.12
% 14.25/8.12 thf(fact_6018_sum__squares__gt__zero__iff,axiom,
% 14.25/8.12 ! [X: int,Y: int] :
% 14.25/8.12 ( ( ord_less_int @ zero_zero_int @ ( plus_plus_int @ ( times_times_int @ X @ X ) @ ( times_times_int @ Y @ Y ) ) )
% 14.25/8.12 = ( ( X != zero_zero_int )
% 14.25/8.12 | ( Y != zero_zero_int ) ) ) ).
% 14.25/8.12
% 14.25/8.12 thf(fact_6019_sum__squares__gt__zero__iff,axiom,
% 14.25/8.12 ! [X: code_integer,Y: code_integer] :
% 14.25/8.12 ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ X @ X ) @ ( times_3573771949741848930nteger @ Y @ Y ) ) )
% 14.25/8.12 = ( ( X != zero_z3403309356797280102nteger )
% 14.25/8.12 | ( Y != zero_z3403309356797280102nteger ) ) ) ).
% 14.25/8.12
% 14.25/8.12 thf(fact_6020_sum__squares__gt__zero__iff,axiom,
% 14.25/8.12 ! [X: rat,Y: rat] :
% 14.25/8.12 ( ( ord_less_rat @ zero_zero_rat @ ( plus_plus_rat @ ( times_times_rat @ X @ X ) @ ( times_times_rat @ Y @ Y ) ) )
% 14.25/8.12 = ( ( X != zero_zero_rat )
% 14.25/8.12 | ( Y != zero_zero_rat ) ) ) ).
% 14.25/8.12
% 14.25/8.12 thf(fact_6021_sum__squares__le__zero__iff,axiom,
% 14.25/8.12 ! [X: code_integer,Y: code_integer] :
% 14.25/8.12 ( ( ord_le3102999989581377725nteger @ ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ X @ X ) @ ( times_3573771949741848930nteger @ Y @ Y ) ) @ zero_z3403309356797280102nteger )
% 14.25/8.12 = ( ( X = zero_z3403309356797280102nteger )
% 14.25/8.12 & ( Y = zero_z3403309356797280102nteger ) ) ) ).
% 14.25/8.12
% 14.25/8.12 thf(fact_6022_sum__squares__le__zero__iff,axiom,
% 14.25/8.12 ! [X: rat,Y: rat] :
% 14.25/8.12 ( ( ord_less_eq_rat @ ( plus_plus_rat @ ( times_times_rat @ X @ X ) @ ( times_times_rat @ Y @ Y ) ) @ zero_zero_rat )
% 14.25/8.12 = ( ( X = zero_zero_rat )
% 14.25/8.12 & ( Y = zero_zero_rat ) ) ) ).
% 14.25/8.12
% 14.25/8.12 thf(fact_6023_sum__squares__le__zero__iff,axiom,
% 14.25/8.12 ! [X: int,Y: int] :
% 14.25/8.12 ( ( ord_less_eq_int @ ( plus_plus_int @ ( times_times_int @ X @ X ) @ ( times_times_int @ Y @ Y ) ) @ zero_zero_int )
% 14.25/8.12 = ( ( X = zero_zero_int )
% 14.25/8.12 & ( Y = zero_zero_int ) ) ) ).
% 14.25/8.12
% 14.25/8.12 thf(fact_6024_mult__le__cancel__iff2,axiom,
% 14.25/8.12 ! [Z: code_integer,X: code_integer,Y: code_integer] :
% 14.25/8.12 ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ Z )
% 14.25/8.12 => ( ( ord_le3102999989581377725nteger @ ( times_3573771949741848930nteger @ Z @ X ) @ ( times_3573771949741848930nteger @ Z @ Y ) )
% 14.25/8.12 = ( ord_le3102999989581377725nteger @ X @ Y ) ) ) ).
% 14.25/8.12
% 14.25/8.12 thf(fact_6025_mult__le__cancel__iff2,axiom,
% 14.25/8.12 ! [Z: rat,X: rat,Y: rat] :
% 14.25/8.12 ( ( ord_less_rat @ zero_zero_rat @ Z )
% 14.25/8.12 => ( ( ord_less_eq_rat @ ( times_times_rat @ Z @ X ) @ ( times_times_rat @ Z @ Y ) )
% 14.25/8.12 = ( ord_less_eq_rat @ X @ Y ) ) ) ).
% 14.25/8.12
% 14.25/8.12 thf(fact_6026_mult__le__cancel__iff2,axiom,
% 14.25/8.12 ! [Z: int,X: int,Y: int] :
% 14.25/8.12 ( ( ord_less_int @ zero_zero_int @ Z )
% 14.25/8.12 => ( ( ord_less_eq_int @ ( times_times_int @ Z @ X ) @ ( times_times_int @ Z @ Y ) )
% 14.25/8.12 = ( ord_less_eq_int @ X @ Y ) ) ) ).
% 14.25/8.12
% 14.25/8.12 thf(fact_6027_mult__le__cancel__iff1,axiom,
% 14.25/8.12 ! [Z: code_integer,X: code_integer,Y: code_integer] :
% 14.25/8.12 ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ Z )
% 14.25/8.12 => ( ( ord_le3102999989581377725nteger @ ( times_3573771949741848930nteger @ X @ Z ) @ ( times_3573771949741848930nteger @ Y @ Z ) )
% 14.25/8.12 = ( ord_le3102999989581377725nteger @ X @ Y ) ) ) ).
% 14.25/8.12
% 14.25/8.12 thf(fact_6028_mult__le__cancel__iff1,axiom,
% 14.25/8.12 ! [Z: rat,X: rat,Y: rat] :
% 14.25/8.12 ( ( ord_less_rat @ zero_zero_rat @ Z )
% 14.25/8.12 => ( ( ord_less_eq_rat @ ( times_times_rat @ X @ Z ) @ ( times_times_rat @ Y @ Z ) )
% 14.25/8.12 = ( ord_less_eq_rat @ X @ Y ) ) ) ).
% 14.25/8.12
% 14.25/8.12 thf(fact_6029_mult__le__cancel__iff1,axiom,
% 14.25/8.12 ! [Z: int,X: int,Y: int] :
% 14.25/8.12 ( ( ord_less_int @ zero_zero_int @ Z )
% 14.25/8.12 => ( ( ord_less_eq_int @ ( times_times_int @ X @ Z ) @ ( times_times_int @ Y @ Z ) )
% 14.25/8.12 = ( ord_less_eq_int @ X @ Y ) ) ) ).
% 14.25/8.12
% 14.25/8.12 thf(fact_6030_divides__aux__eq,axiom,
% 14.25/8.12 ! [Q4: nat,R2: nat] :
% 14.25/8.12 ( ( unique6322359934112328802ux_nat @ ( product_Pair_nat_nat @ Q4 @ R2 ) )
% 14.25/8.12 = ( R2 = zero_zero_nat ) ) ).
% 14.25/8.12
% 14.25/8.12 thf(fact_6031_divides__aux__eq,axiom,
% 14.25/8.12 ! [Q4: int,R2: int] :
% 14.25/8.12 ( ( unique6319869463603278526ux_int @ ( product_Pair_int_int @ Q4 @ R2 ) )
% 14.25/8.12 = ( R2 = zero_zero_int ) ) ).
% 14.25/8.12
% 14.25/8.12 thf(fact_6032_divides__aux__eq,axiom,
% 14.25/8.12 ! [Q4: code_integer,R2: code_integer] :
% 14.25/8.12 ( ( unique5706413561485394159nteger @ ( produc1086072967326762835nteger @ Q4 @ R2 ) )
% 14.25/8.12 = ( R2 = zero_z3403309356797280102nteger ) ) ).
% 14.25/8.12
% 14.25/8.12 thf(fact_6033_divides__aux__def,axiom,
% 14.25/8.12 ( unique6322359934112328802ux_nat
% 14.25/8.12 = ( ^ [Qr: product_prod_nat_nat] :
% 14.25/8.12 ( ( product_snd_nat_nat @ Qr )
% 14.25/8.12 = zero_zero_nat ) ) ) ).
% 14.25/8.12
% 14.25/8.12 thf(fact_6034_divides__aux__def,axiom,
% 14.25/8.12 ( unique6319869463603278526ux_int
% 14.25/8.12 = ( ^ [Qr: product_prod_int_int] :
% 14.25/8.12 ( ( product_snd_int_int @ Qr )
% 14.25/8.12 = zero_zero_int ) ) ) ).
% 14.25/8.12
% 14.25/8.12 thf(fact_6035_divides__aux__def,axiom,
% 14.25/8.12 ( unique5706413561485394159nteger
% 14.25/8.12 = ( ^ [Qr: produc8923325533196201883nteger] :
% 14.25/8.12 ( ( produc6174133586879617921nteger @ Qr )
% 14.25/8.12 = zero_z3403309356797280102nteger ) ) ) ).
% 14.25/8.12
% 14.25/8.12 thf(fact_6036_mult__less__iff1,axiom,
% 14.25/8.12 ! [Z: int,X: int,Y: int] :
% 14.25/8.12 ( ( ord_less_int @ zero_zero_int @ Z )
% 14.25/8.12 => ( ( ord_less_int @ ( times_times_int @ X @ Z ) @ ( times_times_int @ Y @ Z ) )
% 14.25/8.12 = ( ord_less_int @ X @ Y ) ) ) ).
% 14.25/8.12
% 14.25/8.12 thf(fact_6037_mult__less__iff1,axiom,
% 14.25/8.12 ! [Z: code_integer,X: code_integer,Y: code_integer] :
% 14.25/8.12 ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ Z )
% 14.25/8.12 => ( ( ord_le6747313008572928689nteger @ ( times_3573771949741848930nteger @ X @ Z ) @ ( times_3573771949741848930nteger @ Y @ Z ) )
% 14.25/8.12 = ( ord_le6747313008572928689nteger @ X @ Y ) ) ) ).
% 14.25/8.12
% 14.25/8.12 thf(fact_6038_mult__less__iff1,axiom,
% 14.25/8.12 ! [Z: rat,X: rat,Y: rat] :
% 14.25/8.12 ( ( ord_less_rat @ zero_zero_rat @ Z )
% 14.25/8.12 => ( ( ord_less_rat @ ( times_times_rat @ X @ Z ) @ ( times_times_rat @ Y @ Z ) )
% 14.25/8.12 = ( ord_less_rat @ X @ Y ) ) ) ).
% 14.25/8.12
% 14.25/8.12 thf(fact_6039_sum__squares__eq__zero__iff,axiom,
% 14.25/8.12 ! [X: code_integer,Y: code_integer] :
% 14.25/8.12 ( ( ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ X @ X ) @ ( times_3573771949741848930nteger @ Y @ Y ) )
% 14.25/8.12 = zero_z3403309356797280102nteger )
% 14.25/8.12 = ( ( X = zero_z3403309356797280102nteger )
% 14.25/8.12 & ( Y = zero_z3403309356797280102nteger ) ) ) ).
% 14.25/8.12
% 14.25/8.12 thf(fact_6040_sum__squares__eq__zero__iff,axiom,
% 14.25/8.12 ! [X: rat,Y: rat] :
% 14.25/8.12 ( ( ( plus_plus_rat @ ( times_times_rat @ X @ X ) @ ( times_times_rat @ Y @ Y ) )
% 14.25/8.12 = zero_zero_rat )
% 14.25/8.12 = ( ( X = zero_zero_rat )
% 14.25/8.12 & ( Y = zero_zero_rat ) ) ) ).
% 14.25/8.12
% 14.25/8.12 thf(fact_6041_sum__squares__eq__zero__iff,axiom,
% 14.25/8.12 ! [X: int,Y: int] :
% 14.25/8.12 ( ( ( plus_plus_int @ ( times_times_int @ X @ X ) @ ( times_times_int @ Y @ Y ) )
% 14.25/8.12 = zero_zero_int )
% 14.25/8.12 = ( ( X = zero_zero_int )
% 14.25/8.12 & ( Y = zero_zero_int ) ) ) ).
% 14.25/8.12
% 14.25/8.12 thf(fact_6042_dbl__inc__simps_I2_J,axiom,
% 14.25/8.12 ( ( neg_nu5851722552734809277nc_int @ zero_zero_int )
% 14.25/8.12 = one_one_int ) ).
% 14.25/8.12
% 14.25/8.12 thf(fact_6043_dbl__inc__simps_I2_J,axiom,
% 14.25/8.12 ( ( neg_nu5831290666863070958nteger @ zero_z3403309356797280102nteger )
% 14.25/8.12 = one_one_Code_integer ) ).
% 14.25/8.12
% 14.25/8.12 thf(fact_6044_enumerate__Suc_H,axiom,
% 14.25/8.12 ! [S: set_Product_unit,N2: nat] :
% 14.25/8.12 ( ( infini7930543730640340914t_unit @ S @ ( suc @ N2 ) )
% 14.25/8.12 = ( infini7930543730640340914t_unit @ ( minus_6452836326544984404t_unit @ S @ ( insert_Product_unit @ ( infini7930543730640340914t_unit @ S @ zero_zero_nat ) @ bot_bo3957492148770167129t_unit ) ) @ N2 ) ) ).
% 14.25/8.12
% 14.25/8.12 thf(fact_6045_enumerate__Suc_H,axiom,
% 14.25/8.12 ! [S: set_nat,N2: nat] :
% 14.25/8.12 ( ( infini8530281810654367211te_nat @ S @ ( suc @ N2 ) )
% 14.25/8.12 = ( infini8530281810654367211te_nat @ ( minus_minus_set_nat @ S @ ( insert_nat @ ( infini8530281810654367211te_nat @ S @ zero_zero_nat ) @ bot_bot_set_nat ) ) @ N2 ) ) ).
% 14.25/8.12
% 14.25/8.12 thf(fact_6046_power__decreasing__iff,axiom,
% 14.25/8.12 ! [B: code_natural,M4: nat,N2: nat] :
% 14.25/8.12 ( ( ord_le5570908160329646204atural @ zero_z2226904508553997617atural @ B )
% 14.25/8.12 => ( ( ord_le5570908160329646204atural @ B @ one_one_Code_natural )
% 14.25/8.12 => ( ( ord_le1926595141338095240atural @ ( power_7079662738309270450atural @ B @ M4 ) @ ( power_7079662738309270450atural @ B @ N2 ) )
% 14.25/8.12 = ( ord_less_eq_nat @ N2 @ M4 ) ) ) ) ).
% 14.25/8.12
% 14.25/8.12 thf(fact_6047_power__decreasing__iff,axiom,
% 14.25/8.12 ! [B: code_integer,M4: nat,N2: nat] :
% 14.25/8.12 ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ B )
% 14.25/8.12 => ( ( ord_le6747313008572928689nteger @ B @ one_one_Code_integer )
% 14.25/8.12 => ( ( ord_le3102999989581377725nteger @ ( power_8256067586552552935nteger @ B @ M4 ) @ ( power_8256067586552552935nteger @ B @ N2 ) )
% 14.25/8.12 = ( ord_less_eq_nat @ N2 @ M4 ) ) ) ) ).
% 14.25/8.12
% 14.25/8.12 thf(fact_6048_power__decreasing__iff,axiom,
% 14.25/8.12 ! [B: rat,M4: nat,N2: nat] :
% 14.25/8.12 ( ( ord_less_rat @ zero_zero_rat @ B )
% 14.25/8.12 => ( ( ord_less_rat @ B @ one_one_rat )
% 14.25/8.12 => ( ( ord_less_eq_rat @ ( power_power_rat @ B @ M4 ) @ ( power_power_rat @ B @ N2 ) )
% 14.25/8.12 = ( ord_less_eq_nat @ N2 @ M4 ) ) ) ) ).
% 14.25/8.12
% 14.25/8.12 thf(fact_6049_power__decreasing__iff,axiom,
% 14.25/8.12 ! [B: nat,M4: nat,N2: nat] :
% 14.25/8.12 ( ( ord_less_nat @ zero_zero_nat @ B )
% 14.25/8.12 => ( ( ord_less_nat @ B @ one_one_nat )
% 14.25/8.12 => ( ( ord_less_eq_nat @ ( power_power_nat @ B @ M4 ) @ ( power_power_nat @ B @ N2 ) )
% 14.25/8.12 = ( ord_less_eq_nat @ N2 @ M4 ) ) ) ) ).
% 14.25/8.12
% 14.25/8.12 thf(fact_6050_power__decreasing__iff,axiom,
% 14.25/8.12 ! [B: int,M4: nat,N2: nat] :
% 14.25/8.12 ( ( ord_less_int @ zero_zero_int @ B )
% 14.25/8.12 => ( ( ord_less_int @ B @ one_one_int )
% 14.25/8.12 => ( ( ord_less_eq_int @ ( power_power_int @ B @ M4 ) @ ( power_power_int @ B @ N2 ) )
% 14.25/8.12 = ( ord_less_eq_nat @ N2 @ M4 ) ) ) ) ).
% 14.25/8.12
% 14.25/8.12 thf(fact_6051_dvd__partition,axiom,
% 14.25/8.12 ! [C5: set_set_Code_integer,K3: nat] :
% 14.25/8.12 ( ( finite6017078050557962740nteger @ ( comple739944243200306918nteger @ C5 ) )
% 14.25/8.12 => ( ! [X3: set_Code_integer] :
% 14.25/8.12 ( ( member1834298987299537450nteger @ X3 @ C5 )
% 14.25/8.12 => ( dvd_dvd_nat @ K3 @ ( finite4902975817058060853nteger @ X3 ) ) )
% 14.25/8.12 => ( ! [X3: set_Code_integer] :
% 14.25/8.12 ( ( member1834298987299537450nteger @ X3 @ C5 )
% 14.25/8.12 => ! [Xa3: set_Code_integer] :
% 14.25/8.12 ( ( member1834298987299537450nteger @ Xa3 @ C5 )
% 14.25/8.12 => ( ( X3 != Xa3 )
% 14.25/8.12 => ( ( inf_in1364745209274528805nteger @ X3 @ Xa3 )
% 14.25/8.12 = bot_bo3990330152332043303nteger ) ) ) )
% 14.25/8.12 => ( dvd_dvd_nat @ K3 @ ( finite4902975817058060853nteger @ ( comple739944243200306918nteger @ C5 ) ) ) ) ) ) ).
% 14.25/8.12
% 14.25/8.12 thf(fact_6052_dvd__partition,axiom,
% 14.25/8.12 ! [C5: set_set_Product_unit,K3: nat] :
% 14.25/8.12 ( ( finite4290736615968046902t_unit @ ( comple3349607820504768346t_unit @ C5 ) )
% 14.25/8.12 => ( ! [X3: set_Product_unit] :
% 14.25/8.12 ( ( member5877623283571906838t_unit @ X3 @ C5 )
% 14.25/8.12 => ( dvd_dvd_nat @ K3 @ ( finite410649719033368117t_unit @ X3 ) ) )
% 14.25/8.12 => ( ! [X3: set_Product_unit] :
% 14.25/8.12 ( ( member5877623283571906838t_unit @ X3 @ C5 )
% 14.25/8.12 => ! [Xa3: set_Product_unit] :
% 14.25/8.12 ( ( member5877623283571906838t_unit @ Xa3 @ C5 )
% 14.25/8.12 => ( ( X3 != Xa3 )
% 14.25/8.12 => ( ( inf_in4660618365625256667t_unit @ X3 @ Xa3 )
% 14.25/8.12 = bot_bo3957492148770167129t_unit ) ) ) )
% 14.25/8.12 => ( dvd_dvd_nat @ K3 @ ( finite410649719033368117t_unit @ ( comple3349607820504768346t_unit @ C5 ) ) ) ) ) ) ).
% 14.25/8.12
% 14.25/8.12 thf(fact_6053_dvd__partition,axiom,
% 14.25/8.12 ! [C5: set_set_o,K3: nat] :
% 14.25/8.12 ( ( finite_finite_o @ ( comple90263536869209701_set_o @ C5 ) )
% 14.25/8.12 => ( ! [X3: set_o] :
% 14.25/8.12 ( ( member_set_o @ X3 @ C5 )
% 14.25/8.12 => ( dvd_dvd_nat @ K3 @ ( finite_card_o @ X3 ) ) )
% 14.25/8.12 => ( ! [X3: set_o] :
% 14.25/8.12 ( ( member_set_o @ X3 @ C5 )
% 14.25/8.12 => ! [Xa3: set_o] :
% 14.25/8.12 ( ( member_set_o @ Xa3 @ C5 )
% 14.25/8.12 => ( ( X3 != Xa3 )
% 14.25/8.12 => ( ( inf_inf_set_o @ X3 @ Xa3 )
% 14.25/8.12 = bot_bot_set_o ) ) ) )
% 14.25/8.12 => ( dvd_dvd_nat @ K3 @ ( finite_card_o @ ( comple90263536869209701_set_o @ C5 ) ) ) ) ) ) ).
% 14.25/8.12
% 14.25/8.12 thf(fact_6054_dvd__partition,axiom,
% 14.25/8.12 ! [C5: set_set_nat,K3: nat] :
% 14.25/8.12 ( ( finite_finite_nat @ ( comple7399068483239264473et_nat @ C5 ) )
% 14.25/8.12 => ( ! [X3: set_nat] :
% 14.25/8.12 ( ( member_set_nat @ X3 @ C5 )
% 14.25/8.12 => ( dvd_dvd_nat @ K3 @ ( finite_card_nat @ X3 ) ) )
% 14.25/8.12 => ( ! [X3: set_nat] :
% 14.25/8.12 ( ( member_set_nat @ X3 @ C5 )
% 14.25/8.12 => ! [Xa3: set_nat] :
% 14.25/8.12 ( ( member_set_nat @ Xa3 @ C5 )
% 14.25/8.12 => ( ( X3 != Xa3 )
% 14.25/8.12 => ( ( inf_inf_set_nat @ X3 @ Xa3 )
% 14.25/8.12 = bot_bot_set_nat ) ) ) )
% 14.25/8.12 => ( dvd_dvd_nat @ K3 @ ( finite_card_nat @ ( comple7399068483239264473et_nat @ C5 ) ) ) ) ) ) ).
% 14.25/8.12
% 14.25/8.12 thf(fact_6055_dvd__partition,axiom,
% 14.25/8.12 ! [C5: set_set_int,K3: nat] :
% 14.25/8.12 ( ( finite_finite_int @ ( comple3221217463730067765et_int @ C5 ) )
% 14.25/8.12 => ( ! [X3: set_int] :
% 14.25/8.12 ( ( member_set_int @ X3 @ C5 )
% 14.25/8.12 => ( dvd_dvd_nat @ K3 @ ( finite_card_int @ X3 ) ) )
% 14.25/8.12 => ( ! [X3: set_int] :
% 14.25/8.12 ( ( member_set_int @ X3 @ C5 )
% 14.25/8.12 => ! [Xa3: set_int] :
% 14.25/8.12 ( ( member_set_int @ Xa3 @ C5 )
% 14.25/8.12 => ( ( X3 != Xa3 )
% 14.25/8.12 => ( ( inf_inf_set_int @ X3 @ Xa3 )
% 14.25/8.12 = bot_bot_set_int ) ) ) )
% 14.25/8.12 => ( dvd_dvd_nat @ K3 @ ( finite_card_int @ ( comple3221217463730067765et_int @ C5 ) ) ) ) ) ) ).
% 14.25/8.12
% 14.25/8.12 thf(fact_6056_dvd__partition,axiom,
% 14.25/8.12 ! [C5: set_se1353816476878259085_nat_o,K3: nat] :
% 14.25/8.12 ( ( finite3252695134891459830_nat_o @ ( comple3382087578465172250_nat_o @ C5 ) )
% 14.25/8.12 => ( ! [X3: set_Pr4532377907799695533_nat_o] :
% 14.25/8.12 ( ( member1452989533377600726_nat_o @ X3 @ C5 )
% 14.25/8.12 => ( dvd_dvd_nat @ K3 @ ( finite1363419556375932405_nat_o @ X3 ) ) )
% 14.25/8.12 => ( ! [X3: set_Pr4532377907799695533_nat_o] :
% 14.25/8.12 ( ( member1452989533377600726_nat_o @ X3 @ C5 )
% 14.25/8.12 => ! [Xa3: set_Pr4532377907799695533_nat_o] :
% 14.25/8.12 ( ( member1452989533377600726_nat_o @ Xa3 @ C5 )
% 14.25/8.12 => ( ( X3 != Xa3 )
% 14.25/8.12 => ( ( inf_in1906310914598751387_nat_o @ X3 @ Xa3 )
% 14.25/8.12 = bot_bo7824918357723582233_nat_o ) ) ) )
% 14.25/8.12 => ( dvd_dvd_nat @ K3 @ ( finite1363419556375932405_nat_o @ ( comple3382087578465172250_nat_o @ C5 ) ) ) ) ) ) ).
% 14.25/8.12
% 14.25/8.12 thf(fact_6057_cInf__le__cSup,axiom,
% 14.25/8.12 ! [A: set_o] :
% 14.25/8.12 ( ( A != bot_bot_set_o )
% 14.25/8.12 => ( ( condit5488710616941104124bove_o @ A )
% 14.25/8.12 => ( ( condit5413489452508810728elow_o @ A )
% 14.25/8.12 => ( ord_less_eq_o @ ( complete_Inf_Inf_o @ A ) @ ( complete_Sup_Sup_o @ A ) ) ) ) ) ).
% 14.25/8.12
% 14.25/8.12 thf(fact_6058_cInf__le__cSup,axiom,
% 14.25/8.12 ! [A: set_Pr4532377907799695533_nat_o] :
% 14.25/8.12 ( ( A != bot_bo7824918357723582233_nat_o )
% 14.25/8.12 => ( ( condit4352657709669162033_nat_o @ A )
% 14.25/8.12 => ( ( condit8399017556247434269_nat_o @ A )
% 14.25/8.12 => ( ord_le729326519192465773_nat_o @ ( comple90455603224345172_nat_o @ A ) @ ( comple4225350640907012154_nat_o @ A ) ) ) ) ) ).
% 14.25/8.12
% 14.25/8.12 thf(fact_6059_cInf__le__cSup,axiom,
% 14.25/8.12 ! [A: set_filter_nat] :
% 14.25/8.12 ( ( A != bot_bo498966703094740906er_nat )
% 14.25/8.12 => ( ( condit4029664723990530684er_nat @ A )
% 14.25/8.12 => ( ( condit3133331395780078992er_nat @ A )
% 14.25/8.12 => ( ord_le2510731241096832064er_nat @ ( comple6612786467464916249er_nat @ A ) @ ( comple2454491259644266675er_nat @ A ) ) ) ) ) ).
% 14.25/8.12
% 14.25/8.12 thf(fact_6060_cInf__le__cSup,axiom,
% 14.25/8.12 ! [A: set_int] :
% 14.25/8.12 ( ( A != bot_bot_set_int )
% 14.25/8.12 => ( ( condit2212336002400062152ve_int @ A )
% 14.25/8.12 => ( ( condit1735850657277959132ow_int @ A )
% 14.25/8.12 => ( ord_less_eq_int @ ( complete_Inf_Inf_int @ A ) @ ( complete_Sup_Sup_int @ A ) ) ) ) ) ).
% 14.25/8.12
% 14.25/8.12 thf(fact_6061_cInf__le__cSup,axiom,
% 14.25/8.12 ! [A: set_set_nat] :
% 14.25/8.12 ( ( A != bot_bot_set_set_nat )
% 14.25/8.12 => ( ( condit5477540289124974626et_nat @ A )
% 14.25/8.12 => ( ( condit68592940725977398et_nat @ A )
% 14.25/8.12 => ( ord_less_eq_set_nat @ ( comple7806235888213564991et_nat @ A ) @ ( comple7399068483239264473et_nat @ A ) ) ) ) ) ).
% 14.25/8.12
% 14.25/8.12 thf(fact_6062_cInf__le__cSup,axiom,
% 14.25/8.12 ! [A: set_nat] :
% 14.25/8.12 ( ( A != bot_bot_set_nat )
% 14.25/8.12 => ( ( condit2214826472909112428ve_nat @ A )
% 14.25/8.12 => ( ( condit1738341127787009408ow_nat @ A )
% 14.25/8.12 => ( ord_less_eq_nat @ ( complete_Inf_Inf_nat @ A ) @ ( complete_Sup_Sup_nat @ A ) ) ) ) ) ).
% 14.25/8.12
% 14.25/8.12 thf(fact_6063_cInf__le__cSup,axiom,
% 14.25/8.12 ! [A: set_Product_unit] :
% 14.25/8.12 ( ( A != bot_bo3957492148770167129t_unit )
% 14.25/8.12 => ( ( condit2080475194861589617t_unit @ A )
% 14.25/8.12 => ( ( condit9161856309968323677t_unit @ A )
% 14.25/8.12 => ( ord_le3221252021190050221t_unit @ ( comple2584293577114468500t_unit @ A ) @ ( comple4687483117567863418t_unit @ A ) ) ) ) ) ).
% 14.25/8.12
% 14.25/8.12 thf(fact_6064_sum__le__card__Max,axiom,
% 14.25/8.12 ! [A: set_Code_integer,F: code_integer > nat] :
% 14.25/8.12 ( ( finite6017078050557962740nteger @ A )
% 14.25/8.12 => ( ord_less_eq_nat @ ( groups7237345082560585321er_nat @ F @ A ) @ ( times_times_nat @ ( finite4902975817058060853nteger @ A ) @ ( lattic8265883725875713057ax_nat @ ( image_951025933927791156er_nat @ F @ A ) ) ) ) ) ).
% 14.25/8.12
% 14.25/8.12 thf(fact_6065_sum__le__card__Max,axiom,
% 14.25/8.12 ! [A: set_nat,F: nat > nat] :
% 14.25/8.12 ( ( finite_finite_nat @ A )
% 14.25/8.12 => ( ord_less_eq_nat @ ( groups3542108847815614940at_nat @ F @ A ) @ ( times_times_nat @ ( finite_card_nat @ A ) @ ( lattic8265883725875713057ax_nat @ ( image_nat_nat2 @ F @ A ) ) ) ) ) ).
% 14.25/8.12
% 14.25/8.12 thf(fact_6066_gbinomial__reduce__nat,axiom,
% 14.25/8.12 ! [K3: nat,A2: rat] :
% 14.25/8.12 ( ( ord_less_nat @ zero_zero_nat @ K3 )
% 14.25/8.12 => ( ( gbinomial_rat @ A2 @ K3 )
% 14.25/8.12 = ( plus_plus_rat @ ( gbinomial_rat @ ( minus_minus_rat @ A2 @ one_one_rat ) @ ( minus_minus_nat @ K3 @ one_one_nat ) ) @ ( gbinomial_rat @ ( minus_minus_rat @ A2 @ one_one_rat ) @ K3 ) ) ) ) ).
% 14.25/8.12
% 14.25/8.12 thf(fact_6067_power__one,axiom,
% 14.25/8.12 ! [N2: nat] :
% 14.25/8.12 ( ( power_power_assn @ one_one_assn @ N2 )
% 14.25/8.12 = one_one_assn ) ).
% 14.25/8.12
% 14.25/8.12 thf(fact_6068_power__one,axiom,
% 14.25/8.12 ! [N2: nat] :
% 14.25/8.12 ( ( power_8256067586552552935nteger @ one_one_Code_integer @ N2 )
% 14.25/8.12 = one_one_Code_integer ) ).
% 14.25/8.12
% 14.25/8.12 thf(fact_6069_power__one,axiom,
% 14.25/8.12 ! [N2: nat] :
% 14.25/8.12 ( ( power_7079662738309270450atural @ one_one_Code_natural @ N2 )
% 14.25/8.12 = one_one_Code_natural ) ).
% 14.25/8.12
% 14.25/8.12 thf(fact_6070_power__one,axiom,
% 14.25/8.12 ! [N2: nat] :
% 14.25/8.12 ( ( power_power_int @ one_one_int @ N2 )
% 14.25/8.12 = one_one_int ) ).
% 14.25/8.12
% 14.25/8.12 thf(fact_6071_power__one,axiom,
% 14.25/8.12 ! [N2: nat] :
% 14.25/8.12 ( ( power_power_nat @ one_one_nat @ N2 )
% 14.25/8.12 = one_one_nat ) ).
% 14.25/8.12
% 14.25/8.12 thf(fact_6072_bdd__below__empty,axiom,
% 14.25/8.12 condit9161856309968323677t_unit @ bot_bo3957492148770167129t_unit ).
% 14.25/8.12
% 14.25/8.12 thf(fact_6073_bdd__below__empty,axiom,
% 14.25/8.12 condit5413489452508810728elow_o @ bot_bot_set_o ).
% 14.25/8.12
% 14.25/8.12 thf(fact_6074_bdd__below__empty,axiom,
% 14.25/8.12 condit1738341127787009408ow_nat @ bot_bot_set_nat ).
% 14.25/8.12
% 14.25/8.12 thf(fact_6075_bdd__below__empty,axiom,
% 14.25/8.12 condit1735850657277959132ow_int @ bot_bot_set_int ).
% 14.25/8.12
% 14.25/8.12 thf(fact_6076_bdd__below__empty,axiom,
% 14.25/8.12 condit8399017556247434269_nat_o @ bot_bo7824918357723582233_nat_o ).
% 14.25/8.12
% 14.25/8.12 thf(fact_6077_bdd__below__insert,axiom,
% 14.25/8.12 ! [A2: nat,A: set_nat] :
% 14.25/8.12 ( ( condit1738341127787009408ow_nat @ ( insert_nat @ A2 @ A ) )
% 14.25/8.12 = ( condit1738341127787009408ow_nat @ A ) ) ).
% 14.25/8.12
% 14.25/8.12 thf(fact_6078_bdd__below__insert,axiom,
% 14.25/8.12 ! [A2: int,A: set_int] :
% 14.25/8.12 ( ( condit1735850657277959132ow_int @ ( insert_int @ A2 @ A ) )
% 14.25/8.12 = ( condit1735850657277959132ow_int @ A ) ) ).
% 14.25/8.12
% 14.25/8.12 thf(fact_6079_bdd__below__insert,axiom,
% 14.25/8.12 ! [A2: $o,A: set_o] :
% 14.25/8.12 ( ( condit5413489452508810728elow_o @ ( insert_o @ A2 @ A ) )
% 14.25/8.12 = ( condit5413489452508810728elow_o @ A ) ) ).
% 14.25/8.12
% 14.25/8.12 thf(fact_6080_bdd__below__insert,axiom,
% 14.25/8.12 ! [A2: product_unit,A: set_Product_unit] :
% 14.25/8.12 ( ( condit9161856309968323677t_unit @ ( insert_Product_unit @ A2 @ A ) )
% 14.25/8.12 = ( condit9161856309968323677t_unit @ A ) ) ).
% 14.25/8.12
% 14.25/8.12 thf(fact_6081_dvd__times__right__cancel__iff,axiom,
% 14.25/8.12 ! [A2: code_integer,B: code_integer,C: code_integer] :
% 14.25/8.12 ( ( A2 != zero_z3403309356797280102nteger )
% 14.25/8.12 => ( ( dvd_dvd_Code_integer @ ( times_3573771949741848930nteger @ B @ A2 ) @ ( times_3573771949741848930nteger @ C @ A2 ) )
% 14.25/8.12 = ( dvd_dvd_Code_integer @ B @ C ) ) ) ).
% 14.25/8.12
% 14.25/8.12 thf(fact_6082_dvd__times__right__cancel__iff,axiom,
% 14.25/8.12 ! [A2: nat,B: nat,C: nat] :
% 14.25/8.12 ( ( A2 != zero_zero_nat )
% 14.25/8.12 => ( ( dvd_dvd_nat @ ( times_times_nat @ B @ A2 ) @ ( times_times_nat @ C @ A2 ) )
% 14.25/8.12 = ( dvd_dvd_nat @ B @ C ) ) ) ).
% 14.25/8.12
% 14.25/8.12 thf(fact_6083_dvd__times__right__cancel__iff,axiom,
% 14.25/8.12 ! [A2: int,B: int,C: int] :
% 14.25/8.12 ( ( A2 != zero_zero_int )
% 14.25/8.12 => ( ( dvd_dvd_int @ ( times_times_int @ B @ A2 ) @ ( times_times_int @ C @ A2 ) )
% 14.25/8.12 = ( dvd_dvd_int @ B @ C ) ) ) ).
% 14.25/8.12
% 14.25/8.12 thf(fact_6084_dvd__times__left__cancel__iff,axiom,
% 14.25/8.12 ! [A2: code_integer,B: code_integer,C: code_integer] :
% 14.25/8.12 ( ( A2 != zero_z3403309356797280102nteger )
% 14.25/8.12 => ( ( dvd_dvd_Code_integer @ ( times_3573771949741848930nteger @ A2 @ B ) @ ( times_3573771949741848930nteger @ A2 @ C ) )
% 14.25/8.12 = ( dvd_dvd_Code_integer @ B @ C ) ) ) ).
% 14.25/8.12
% 14.25/8.12 thf(fact_6085_dvd__times__left__cancel__iff,axiom,
% 14.25/8.12 ! [A2: nat,B: nat,C: nat] :
% 14.25/8.12 ( ( A2 != zero_zero_nat )
% 14.25/8.12 => ( ( dvd_dvd_nat @ ( times_times_nat @ A2 @ B ) @ ( times_times_nat @ A2 @ C ) )
% 14.25/8.12 = ( dvd_dvd_nat @ B @ C ) ) ) ).
% 14.25/8.12
% 14.25/8.12 thf(fact_6086_dvd__times__left__cancel__iff,axiom,
% 14.25/8.12 ! [A2: int,B: int,C: int] :
% 14.25/8.12 ( ( A2 != zero_zero_int )
% 14.25/8.12 => ( ( dvd_dvd_int @ ( times_times_int @ A2 @ B ) @ ( times_times_int @ A2 @ C ) )
% 14.25/8.12 = ( dvd_dvd_int @ B @ C ) ) ) ).
% 14.25/8.12
% 14.25/8.12 thf(fact_6087_dvd__mult__cancel__right,axiom,
% 14.25/8.12 ! [A2: code_integer,C: code_integer,B: code_integer] :
% 14.25/8.12 ( ( dvd_dvd_Code_integer @ ( times_3573771949741848930nteger @ A2 @ C ) @ ( times_3573771949741848930nteger @ B @ C ) )
% 14.25/8.12 = ( ( C = zero_z3403309356797280102nteger )
% 14.25/8.12 | ( dvd_dvd_Code_integer @ A2 @ B ) ) ) ).
% 14.25/8.12
% 14.25/8.12 thf(fact_6088_dvd__mult__cancel__right,axiom,
% 14.25/8.12 ! [A2: rat,C: rat,B: rat] :
% 14.25/8.12 ( ( dvd_dvd_rat @ ( times_times_rat @ A2 @ C ) @ ( times_times_rat @ B @ C ) )
% 14.25/8.12 = ( ( C = zero_zero_rat )
% 14.25/8.12 | ( dvd_dvd_rat @ A2 @ B ) ) ) ).
% 14.25/8.12
% 14.25/8.12 thf(fact_6089_dvd__mult__cancel__right,axiom,
% 14.25/8.12 ! [A2: int,C: int,B: int] :
% 14.25/8.12 ( ( dvd_dvd_int @ ( times_times_int @ A2 @ C ) @ ( times_times_int @ B @ C ) )
% 14.25/8.12 = ( ( C = zero_zero_int )
% 14.25/8.12 | ( dvd_dvd_int @ A2 @ B ) ) ) ).
% 14.25/8.12
% 14.25/8.12 thf(fact_6090_dvd__mult__cancel__left,axiom,
% 14.25/8.12 ! [C: code_integer,A2: code_integer,B: code_integer] :
% 14.25/8.12 ( ( dvd_dvd_Code_integer @ ( times_3573771949741848930nteger @ C @ A2 ) @ ( times_3573771949741848930nteger @ C @ B ) )
% 14.25/8.12 = ( ( C = zero_z3403309356797280102nteger )
% 14.25/8.12 | ( dvd_dvd_Code_integer @ A2 @ B ) ) ) ).
% 14.25/8.12
% 14.25/8.12 thf(fact_6091_dvd__mult__cancel__left,axiom,
% 14.25/8.12 ! [C: rat,A2: rat,B: rat] :
% 14.25/8.12 ( ( dvd_dvd_rat @ ( times_times_rat @ C @ A2 ) @ ( times_times_rat @ C @ B ) )
% 14.25/8.12 = ( ( C = zero_zero_rat )
% 14.25/8.12 | ( dvd_dvd_rat @ A2 @ B ) ) ) ).
% 14.25/8.12
% 14.25/8.12 thf(fact_6092_dvd__mult__cancel__left,axiom,
% 14.25/8.12 ! [C: int,A2: int,B: int] :
% 14.25/8.12 ( ( dvd_dvd_int @ ( times_times_int @ C @ A2 ) @ ( times_times_int @ C @ B ) )
% 14.25/8.12 = ( ( C = zero_zero_int )
% 14.25/8.12 | ( dvd_dvd_int @ A2 @ B ) ) ) ).
% 14.25/8.12
% 14.25/8.12 thf(fact_6093_dvd__add__times__triv__left__iff,axiom,
% 14.25/8.12 ! [A2: code_integer,C: code_integer,B: code_integer] :
% 14.25/8.12 ( ( dvd_dvd_Code_integer @ A2 @ ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ C @ A2 ) @ B ) )
% 14.25/8.12 = ( dvd_dvd_Code_integer @ A2 @ B ) ) ).
% 14.25/8.12
% 14.25/8.12 thf(fact_6094_dvd__add__times__triv__left__iff,axiom,
% 14.25/8.12 ! [A2: rat,C: rat,B: rat] :
% 14.25/8.12 ( ( dvd_dvd_rat @ A2 @ ( plus_plus_rat @ ( times_times_rat @ C @ A2 ) @ B ) )
% 14.25/8.12 = ( dvd_dvd_rat @ A2 @ B ) ) ).
% 14.25/8.12
% 14.25/8.12 thf(fact_6095_dvd__add__times__triv__left__iff,axiom,
% 14.25/8.12 ! [A2: nat,C: nat,B: nat] :
% 14.25/8.12 ( ( dvd_dvd_nat @ A2 @ ( plus_plus_nat @ ( times_times_nat @ C @ A2 ) @ B ) )
% 14.25/8.12 = ( dvd_dvd_nat @ A2 @ B ) ) ).
% 14.25/8.12
% 14.25/8.12 thf(fact_6096_dvd__add__times__triv__left__iff,axiom,
% 14.25/8.12 ! [A2: int,C: int,B: int] :
% 14.25/8.12 ( ( dvd_dvd_int @ A2 @ ( plus_plus_int @ ( times_times_int @ C @ A2 ) @ B ) )
% 14.25/8.12 = ( dvd_dvd_int @ A2 @ B ) ) ).
% 14.25/8.12
% 14.25/8.12 thf(fact_6097_dvd__add__times__triv__right__iff,axiom,
% 14.25/8.12 ! [A2: code_integer,B: code_integer,C: code_integer] :
% 14.25/8.12 ( ( dvd_dvd_Code_integer @ A2 @ ( plus_p5714425477246183910nteger @ B @ ( times_3573771949741848930nteger @ C @ A2 ) ) )
% 14.25/8.12 = ( dvd_dvd_Code_integer @ A2 @ B ) ) ).
% 14.25/8.12
% 14.25/8.12 thf(fact_6098_dvd__add__times__triv__right__iff,axiom,
% 14.25/8.12 ! [A2: rat,B: rat,C: rat] :
% 14.25/8.12 ( ( dvd_dvd_rat @ A2 @ ( plus_plus_rat @ B @ ( times_times_rat @ C @ A2 ) ) )
% 14.25/8.12 = ( dvd_dvd_rat @ A2 @ B ) ) ).
% 14.25/8.12
% 14.25/8.12 thf(fact_6099_dvd__add__times__triv__right__iff,axiom,
% 14.25/8.12 ! [A2: nat,B: nat,C: nat] :
% 14.25/8.12 ( ( dvd_dvd_nat @ A2 @ ( plus_plus_nat @ B @ ( times_times_nat @ C @ A2 ) ) )
% 14.25/8.12 = ( dvd_dvd_nat @ A2 @ B ) ) ).
% 14.25/8.12
% 14.25/8.12 thf(fact_6100_dvd__add__times__triv__right__iff,axiom,
% 14.25/8.12 ! [A2: int,B: int,C: int] :
% 14.25/8.12 ( ( dvd_dvd_int @ A2 @ ( plus_plus_int @ B @ ( times_times_int @ C @ A2 ) ) )
% 14.25/8.12 = ( dvd_dvd_int @ A2 @ B ) ) ).
% 14.25/8.12
% 14.25/8.12 thf(fact_6101_unit__prod,axiom,
% 14.25/8.12 ! [A2: code_integer,B: code_integer] :
% 14.25/8.12 ( ( dvd_dvd_Code_integer @ A2 @ one_one_Code_integer )
% 14.25/8.12 => ( ( dvd_dvd_Code_integer @ B @ one_one_Code_integer )
% 14.25/8.12 => ( dvd_dvd_Code_integer @ ( times_3573771949741848930nteger @ A2 @ B ) @ one_one_Code_integer ) ) ) ).
% 14.25/8.12
% 14.25/8.12 thf(fact_6102_unit__prod,axiom,
% 14.25/8.12 ! [A2: code_natural,B: code_natural] :
% 14.25/8.12 ( ( dvd_dvd_Code_natural @ A2 @ one_one_Code_natural )
% 14.25/8.12 => ( ( dvd_dvd_Code_natural @ B @ one_one_Code_natural )
% 14.25/8.12 => ( dvd_dvd_Code_natural @ ( times_2397367101498566445atural @ A2 @ B ) @ one_one_Code_natural ) ) ) ).
% 14.25/8.12
% 14.25/8.12 thf(fact_6103_unit__prod,axiom,
% 14.25/8.12 ! [A2: nat,B: nat] :
% 14.25/8.12 ( ( dvd_dvd_nat @ A2 @ one_one_nat )
% 14.25/8.12 => ( ( dvd_dvd_nat @ B @ one_one_nat )
% 14.25/8.12 => ( dvd_dvd_nat @ ( times_times_nat @ A2 @ B ) @ one_one_nat ) ) ) ).
% 14.25/8.12
% 14.25/8.12 thf(fact_6104_unit__prod,axiom,
% 14.25/8.12 ! [A2: int,B: int] :
% 14.25/8.12 ( ( dvd_dvd_int @ A2 @ one_one_int )
% 14.25/8.12 => ( ( dvd_dvd_int @ B @ one_one_int )
% 14.25/8.12 => ( dvd_dvd_int @ ( times_times_int @ A2 @ B ) @ one_one_int ) ) ) ).
% 14.25/8.12
% 14.25/8.12 thf(fact_6105_power__inject__exp,axiom,
% 14.25/8.12 ! [A2: code_natural,M4: nat,N2: nat] :
% 14.25/8.12 ( ( ord_le5570908160329646204atural @ one_one_Code_natural @ A2 )
% 14.25/8.12 => ( ( ( power_7079662738309270450atural @ A2 @ M4 )
% 14.25/8.12 = ( power_7079662738309270450atural @ A2 @ N2 ) )
% 14.25/8.12 = ( M4 = N2 ) ) ) ).
% 14.25/8.12
% 14.25/8.12 thf(fact_6106_power__inject__exp,axiom,
% 14.25/8.12 ! [A2: nat,M4: nat,N2: nat] :
% 14.25/8.12 ( ( ord_less_nat @ one_one_nat @ A2 )
% 14.25/8.12 => ( ( ( power_power_nat @ A2 @ M4 )
% 14.25/8.12 = ( power_power_nat @ A2 @ N2 ) )
% 14.25/8.12 = ( M4 = N2 ) ) ) ).
% 14.25/8.12
% 14.25/8.12 thf(fact_6107_power__inject__exp,axiom,
% 14.25/8.12 ! [A2: int,M4: nat,N2: nat] :
% 14.25/8.12 ( ( ord_less_int @ one_one_int @ A2 )
% 14.25/8.12 => ( ( ( power_power_int @ A2 @ M4 )
% 14.25/8.12 = ( power_power_int @ A2 @ N2 ) )
% 14.25/8.12 = ( M4 = N2 ) ) ) ).
% 14.25/8.12
% 14.25/8.12 thf(fact_6108_power__inject__exp,axiom,
% 14.25/8.12 ! [A2: code_integer,M4: nat,N2: nat] :
% 14.25/8.12 ( ( ord_le6747313008572928689nteger @ one_one_Code_integer @ A2 )
% 14.25/8.12 => ( ( ( power_8256067586552552935nteger @ A2 @ M4 )
% 14.25/8.12 = ( power_8256067586552552935nteger @ A2 @ N2 ) )
% 14.25/8.12 = ( M4 = N2 ) ) ) ).
% 14.25/8.12
% 14.25/8.12 thf(fact_6109_power__inject__exp,axiom,
% 14.25/8.12 ! [A2: rat,M4: nat,N2: nat] :
% 14.25/8.12 ( ( ord_less_rat @ one_one_rat @ A2 )
% 14.25/8.12 => ( ( ( power_power_rat @ A2 @ M4 )
% 14.25/8.12 = ( power_power_rat @ A2 @ N2 ) )
% 14.25/8.12 = ( M4 = N2 ) ) ) ).
% 14.25/8.12
% 14.25/8.12 thf(fact_6110_nat__mult__dvd__cancel__disj,axiom,
% 14.25/8.12 ! [K3: nat,M4: nat,N2: nat] :
% 14.25/8.12 ( ( dvd_dvd_nat @ ( times_times_nat @ K3 @ M4 ) @ ( times_times_nat @ K3 @ N2 ) )
% 14.25/8.12 = ( ( K3 = zero_zero_nat )
% 14.25/8.12 | ( dvd_dvd_nat @ M4 @ N2 ) ) ) ).
% 14.25/8.12
% 14.25/8.12 thf(fact_6111_gbinomial__0_I1_J,axiom,
% 14.25/8.12 ! [A2: nat] :
% 14.25/8.12 ( ( gbinomial_nat @ A2 @ zero_zero_nat )
% 14.25/8.12 = one_one_nat ) ).
% 14.25/8.12
% 14.25/8.12 thf(fact_6112_gbinomial__0_I1_J,axiom,
% 14.25/8.12 ! [A2: int] :
% 14.25/8.12 ( ( gbinomial_int @ A2 @ zero_zero_nat )
% 14.25/8.12 = one_one_int ) ).
% 14.25/8.12
% 14.25/8.12 thf(fact_6113_gbinomial__0_I1_J,axiom,
% 14.25/8.12 ! [A2: code_integer] :
% 14.25/8.12 ( ( gbinom8545251970709558553nteger @ A2 @ zero_zero_nat )
% 14.25/8.12 = one_one_Code_integer ) ).
% 14.25/8.12
% 14.25/8.12 thf(fact_6114_gbinomial__0_I1_J,axiom,
% 14.25/8.12 ! [A2: code_natural] :
% 14.25/8.12 ( ( gbinom7368847122466276068atural @ A2 @ zero_zero_nat )
% 14.25/8.12 = one_one_Code_natural ) ).
% 14.25/8.12
% 14.25/8.12 thf(fact_6115_power__strict__increasing__iff,axiom,
% 14.25/8.12 ! [B: code_natural,X: nat,Y: nat] :
% 14.25/8.12 ( ( ord_le5570908160329646204atural @ one_one_Code_natural @ B )
% 14.25/8.12 => ( ( ord_le5570908160329646204atural @ ( power_7079662738309270450atural @ B @ X ) @ ( power_7079662738309270450atural @ B @ Y ) )
% 14.25/8.12 = ( ord_less_nat @ X @ Y ) ) ) ).
% 14.25/8.12
% 14.25/8.12 thf(fact_6116_power__strict__increasing__iff,axiom,
% 14.25/8.12 ! [B: nat,X: nat,Y: nat] :
% 14.25/8.12 ( ( ord_less_nat @ one_one_nat @ B )
% 14.25/8.12 => ( ( ord_less_nat @ ( power_power_nat @ B @ X ) @ ( power_power_nat @ B @ Y ) )
% 14.25/8.12 = ( ord_less_nat @ X @ Y ) ) ) ).
% 14.25/8.12
% 14.25/8.12 thf(fact_6117_power__strict__increasing__iff,axiom,
% 14.25/8.12 ! [B: int,X: nat,Y: nat] :
% 14.25/8.12 ( ( ord_less_int @ one_one_int @ B )
% 14.25/8.12 => ( ( ord_less_int @ ( power_power_int @ B @ X ) @ ( power_power_int @ B @ Y ) )
% 14.25/8.12 = ( ord_less_nat @ X @ Y ) ) ) ).
% 14.25/8.12
% 14.25/8.12 thf(fact_6118_power__strict__increasing__iff,axiom,
% 14.25/8.12 ! [B: code_integer,X: nat,Y: nat] :
% 14.25/8.12 ( ( ord_le6747313008572928689nteger @ one_one_Code_integer @ B )
% 14.25/8.12 => ( ( ord_le6747313008572928689nteger @ ( power_8256067586552552935nteger @ B @ X ) @ ( power_8256067586552552935nteger @ B @ Y ) )
% 14.25/8.12 = ( ord_less_nat @ X @ Y ) ) ) ).
% 14.25/8.12
% 14.25/8.12 thf(fact_6119_power__strict__increasing__iff,axiom,
% 14.25/8.12 ! [B: rat,X: nat,Y: nat] :
% 14.25/8.12 ( ( ord_less_rat @ one_one_rat @ B )
% 14.25/8.12 => ( ( ord_less_rat @ ( power_power_rat @ B @ X ) @ ( power_power_rat @ B @ Y ) )
% 14.25/8.12 = ( ord_less_nat @ X @ Y ) ) ) ).
% 14.25/8.12
% 14.25/8.12 thf(fact_6120_power__strict__decreasing__iff,axiom,
% 14.25/8.12 ! [B: code_natural,M4: nat,N2: nat] :
% 14.25/8.12 ( ( ord_le5570908160329646204atural @ zero_z2226904508553997617atural @ B )
% 14.25/8.12 => ( ( ord_le5570908160329646204atural @ B @ one_one_Code_natural )
% 14.25/8.12 => ( ( ord_le5570908160329646204atural @ ( power_7079662738309270450atural @ B @ M4 ) @ ( power_7079662738309270450atural @ B @ N2 ) )
% 14.25/8.12 = ( ord_less_nat @ N2 @ M4 ) ) ) ) ).
% 14.25/8.12
% 14.25/8.12 thf(fact_6121_power__strict__decreasing__iff,axiom,
% 14.25/8.12 ! [B: nat,M4: nat,N2: nat] :
% 14.25/8.12 ( ( ord_less_nat @ zero_zero_nat @ B )
% 14.25/8.12 => ( ( ord_less_nat @ B @ one_one_nat )
% 14.25/8.12 => ( ( ord_less_nat @ ( power_power_nat @ B @ M4 ) @ ( power_power_nat @ B @ N2 ) )
% 14.25/8.12 = ( ord_less_nat @ N2 @ M4 ) ) ) ) ).
% 14.25/8.12
% 14.25/8.12 thf(fact_6122_power__strict__decreasing__iff,axiom,
% 14.25/8.12 ! [B: int,M4: nat,N2: nat] :
% 14.25/8.12 ( ( ord_less_int @ zero_zero_int @ B )
% 14.25/8.12 => ( ( ord_less_int @ B @ one_one_int )
% 14.25/8.12 => ( ( ord_less_int @ ( power_power_int @ B @ M4 ) @ ( power_power_int @ B @ N2 ) )
% 14.25/8.12 = ( ord_less_nat @ N2 @ M4 ) ) ) ) ).
% 14.25/8.12
% 14.25/8.12 thf(fact_6123_power__strict__decreasing__iff,axiom,
% 14.25/8.12 ! [B: code_integer,M4: nat,N2: nat] :
% 14.25/8.12 ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ B )
% 14.25/8.12 => ( ( ord_le6747313008572928689nteger @ B @ one_one_Code_integer )
% 14.25/8.12 => ( ( ord_le6747313008572928689nteger @ ( power_8256067586552552935nteger @ B @ M4 ) @ ( power_8256067586552552935nteger @ B @ N2 ) )
% 14.25/8.12 = ( ord_less_nat @ N2 @ M4 ) ) ) ) ).
% 14.25/8.12
% 14.25/8.12 thf(fact_6124_power__strict__decreasing__iff,axiom,
% 14.25/8.12 ! [B: rat,M4: nat,N2: nat] :
% 14.25/8.12 ( ( ord_less_rat @ zero_zero_rat @ B )
% 14.25/8.12 => ( ( ord_less_rat @ B @ one_one_rat )
% 14.25/8.12 => ( ( ord_less_rat @ ( power_power_rat @ B @ M4 ) @ ( power_power_rat @ B @ N2 ) )
% 14.25/8.12 = ( ord_less_nat @ N2 @ M4 ) ) ) ) ).
% 14.25/8.12
% 14.25/8.12 thf(fact_6125_power__increasing__iff,axiom,
% 14.25/8.12 ! [B: code_natural,X: nat,Y: nat] :
% 14.25/8.12 ( ( ord_le5570908160329646204atural @ one_one_Code_natural @ B )
% 14.25/8.12 => ( ( ord_le1926595141338095240atural @ ( power_7079662738309270450atural @ B @ X ) @ ( power_7079662738309270450atural @ B @ Y ) )
% 14.25/8.12 = ( ord_less_eq_nat @ X @ Y ) ) ) ).
% 14.25/8.12
% 14.25/8.12 thf(fact_6126_power__increasing__iff,axiom,
% 14.25/8.12 ! [B: code_integer,X: nat,Y: nat] :
% 14.25/8.12 ( ( ord_le6747313008572928689nteger @ one_one_Code_integer @ B )
% 14.25/8.12 => ( ( ord_le3102999989581377725nteger @ ( power_8256067586552552935nteger @ B @ X ) @ ( power_8256067586552552935nteger @ B @ Y ) )
% 14.25/8.12 = ( ord_less_eq_nat @ X @ Y ) ) ) ).
% 14.25/8.12
% 14.25/8.12 thf(fact_6127_power__increasing__iff,axiom,
% 14.25/8.12 ! [B: rat,X: nat,Y: nat] :
% 14.25/8.12 ( ( ord_less_rat @ one_one_rat @ B )
% 14.25/8.12 => ( ( ord_less_eq_rat @ ( power_power_rat @ B @ X ) @ ( power_power_rat @ B @ Y ) )
% 14.25/8.12 = ( ord_less_eq_nat @ X @ Y ) ) ) ).
% 14.25/8.12
% 14.25/8.12 thf(fact_6128_power__increasing__iff,axiom,
% 14.25/8.12 ! [B: nat,X: nat,Y: nat] :
% 14.25/8.12 ( ( ord_less_nat @ one_one_nat @ B )
% 14.25/8.12 => ( ( ord_less_eq_nat @ ( power_power_nat @ B @ X ) @ ( power_power_nat @ B @ Y ) )
% 14.25/8.12 = ( ord_less_eq_nat @ X @ Y ) ) ) ).
% 14.25/8.12
% 14.25/8.12 thf(fact_6129_power__increasing__iff,axiom,
% 14.25/8.12 ! [B: int,X: nat,Y: nat] :
% 14.25/8.12 ( ( ord_less_int @ one_one_int @ B )
% 14.25/8.12 => ( ( ord_less_eq_int @ ( power_power_int @ B @ X ) @ ( power_power_int @ B @ Y ) )
% 14.25/8.12 = ( ord_less_eq_nat @ X @ Y ) ) ) ).
% 14.25/8.12
% 14.25/8.12 thf(fact_6130_is__unit__power__iff,axiom,
% 14.25/8.12 ! [A2: code_integer,N2: nat] :
% 14.25/8.12 ( ( dvd_dvd_Code_integer @ ( power_8256067586552552935nteger @ A2 @ N2 ) @ one_one_Code_integer )
% 14.25/8.12 = ( ( dvd_dvd_Code_integer @ A2 @ one_one_Code_integer )
% 14.25/8.12 | ( N2 = zero_zero_nat ) ) ) ).
% 14.25/8.12
% 14.25/8.12 thf(fact_6131_is__unit__power__iff,axiom,
% 14.25/8.12 ! [A2: code_natural,N2: nat] :
% 14.25/8.12 ( ( dvd_dvd_Code_natural @ ( power_7079662738309270450atural @ A2 @ N2 ) @ one_one_Code_natural )
% 14.25/8.12 = ( ( dvd_dvd_Code_natural @ A2 @ one_one_Code_natural )
% 14.25/8.12 | ( N2 = zero_zero_nat ) ) ) ).
% 14.25/8.12
% 14.25/8.12 thf(fact_6132_is__unit__power__iff,axiom,
% 14.25/8.12 ! [A2: int,N2: nat] :
% 14.25/8.12 ( ( dvd_dvd_int @ ( power_power_int @ A2 @ N2 ) @ one_one_int )
% 14.25/8.12 = ( ( dvd_dvd_int @ A2 @ one_one_int )
% 14.25/8.12 | ( N2 = zero_zero_nat ) ) ) ).
% 14.25/8.12
% 14.25/8.12 thf(fact_6133_is__unit__power__iff,axiom,
% 14.25/8.12 ! [A2: nat,N2: nat] :
% 14.25/8.12 ( ( dvd_dvd_nat @ ( power_power_nat @ A2 @ N2 ) @ one_one_nat )
% 14.25/8.12 = ( ( dvd_dvd_nat @ A2 @ one_one_nat )
% 14.25/8.12 | ( N2 = zero_zero_nat ) ) ) ).
% 14.25/8.12
% 14.25/8.12 thf(fact_6134_power__commuting__commutes,axiom,
% 14.25/8.12 ! [X: assn,Y: assn,N2: nat] :
% 14.25/8.12 ( ( ( times_times_assn @ X @ Y )
% 14.25/8.12 = ( times_times_assn @ Y @ X ) )
% 14.25/8.12 => ( ( times_times_assn @ ( power_power_assn @ X @ N2 ) @ Y )
% 14.25/8.12 = ( times_times_assn @ Y @ ( power_power_assn @ X @ N2 ) ) ) ) ).
% 14.25/8.12
% 14.25/8.12 thf(fact_6135_power__commuting__commutes,axiom,
% 14.25/8.12 ! [X: rat,Y: rat,N2: nat] :
% 14.25/8.12 ( ( ( times_times_rat @ X @ Y )
% 14.25/8.12 = ( times_times_rat @ Y @ X ) )
% 14.25/8.12 => ( ( times_times_rat @ ( power_power_rat @ X @ N2 ) @ Y )
% 14.25/8.12 = ( times_times_rat @ Y @ ( power_power_rat @ X @ N2 ) ) ) ) ).
% 14.25/8.12
% 14.25/8.12 thf(fact_6136_power__commuting__commutes,axiom,
% 14.25/8.12 ! [X: nat,Y: nat,N2: nat] :
% 14.25/8.12 ( ( ( times_times_nat @ X @ Y )
% 14.25/8.12 = ( times_times_nat @ Y @ X ) )
% 14.25/8.12 => ( ( times_times_nat @ ( power_power_nat @ X @ N2 ) @ Y )
% 14.25/8.12 = ( times_times_nat @ Y @ ( power_power_nat @ X @ N2 ) ) ) ) ).
% 14.25/8.12
% 14.25/8.12 thf(fact_6137_power__commuting__commutes,axiom,
% 14.25/8.12 ! [X: int,Y: int,N2: nat] :
% 14.25/8.12 ( ( ( times_times_int @ X @ Y )
% 14.25/8.12 = ( times_times_int @ Y @ X ) )
% 14.25/8.12 => ( ( times_times_int @ ( power_power_int @ X @ N2 ) @ Y )
% 14.25/8.12 = ( times_times_int @ Y @ ( power_power_int @ X @ N2 ) ) ) ) ).
% 14.25/8.12
% 14.25/8.12 thf(fact_6138_power__mult__distrib,axiom,
% 14.25/8.12 ! [A2: assn,B: assn,N2: nat] :
% 14.25/8.12 ( ( power_power_assn @ ( times_times_assn @ A2 @ B ) @ N2 )
% 14.25/8.12 = ( times_times_assn @ ( power_power_assn @ A2 @ N2 ) @ ( power_power_assn @ B @ N2 ) ) ) ).
% 14.25/8.12
% 14.25/8.12 thf(fact_6139_power__mult__distrib,axiom,
% 14.25/8.12 ! [A2: rat,B: rat,N2: nat] :
% 14.25/8.12 ( ( power_power_rat @ ( times_times_rat @ A2 @ B ) @ N2 )
% 14.25/8.12 = ( times_times_rat @ ( power_power_rat @ A2 @ N2 ) @ ( power_power_rat @ B @ N2 ) ) ) ).
% 14.25/8.12
% 14.25/8.12 thf(fact_6140_power__mult__distrib,axiom,
% 14.25/8.12 ! [A2: nat,B: nat,N2: nat] :
% 14.25/8.12 ( ( power_power_nat @ ( times_times_nat @ A2 @ B ) @ N2 )
% 14.25/8.12 = ( times_times_nat @ ( power_power_nat @ A2 @ N2 ) @ ( power_power_nat @ B @ N2 ) ) ) ).
% 14.25/8.12
% 14.25/8.12 thf(fact_6141_power__mult__distrib,axiom,
% 14.25/8.12 ! [A2: int,B: int,N2: nat] :
% 14.25/8.12 ( ( power_power_int @ ( times_times_int @ A2 @ B ) @ N2 )
% 14.25/8.12 = ( times_times_int @ ( power_power_int @ A2 @ N2 ) @ ( power_power_int @ B @ N2 ) ) ) ).
% 14.25/8.12
% 14.25/8.12 thf(fact_6142_power__commutes,axiom,
% 14.25/8.12 ! [A2: assn,N2: nat] :
% 14.25/8.12 ( ( times_times_assn @ ( power_power_assn @ A2 @ N2 ) @ A2 )
% 14.25/8.12 = ( times_times_assn @ A2 @ ( power_power_assn @ A2 @ N2 ) ) ) ).
% 14.25/8.12
% 14.25/8.12 thf(fact_6143_power__commutes,axiom,
% 14.25/8.12 ! [A2: rat,N2: nat] :
% 14.25/8.12 ( ( times_times_rat @ ( power_power_rat @ A2 @ N2 ) @ A2 )
% 14.25/8.12 = ( times_times_rat @ A2 @ ( power_power_rat @ A2 @ N2 ) ) ) ).
% 14.25/8.12
% 14.25/8.12 thf(fact_6144_power__commutes,axiom,
% 14.25/8.12 ! [A2: nat,N2: nat] :
% 14.25/8.12 ( ( times_times_nat @ ( power_power_nat @ A2 @ N2 ) @ A2 )
% 14.25/8.12 = ( times_times_nat @ A2 @ ( power_power_nat @ A2 @ N2 ) ) ) ).
% 14.25/8.12
% 14.25/8.12 thf(fact_6145_power__commutes,axiom,
% 14.25/8.12 ! [A2: int,N2: nat] :
% 14.25/8.12 ( ( times_times_int @ ( power_power_int @ A2 @ N2 ) @ A2 )
% 14.25/8.12 = ( times_times_int @ A2 @ ( power_power_int @ A2 @ N2 ) ) ) ).
% 14.25/8.12
% 14.25/8.12 thf(fact_6146_dvd__triv__right,axiom,
% 14.25/8.12 ! [A2: assn,B: assn] : ( dvd_dvd_assn @ A2 @ ( times_times_assn @ B @ A2 ) ) ).
% 14.25/8.12
% 14.25/8.12 thf(fact_6147_dvd__triv__right,axiom,
% 14.25/8.12 ! [A2: rat,B: rat] : ( dvd_dvd_rat @ A2 @ ( times_times_rat @ B @ A2 ) ) ).
% 14.25/8.12
% 14.25/8.12 thf(fact_6148_dvd__triv__right,axiom,
% 14.25/8.12 ! [A2: nat,B: nat] : ( dvd_dvd_nat @ A2 @ ( times_times_nat @ B @ A2 ) ) ).
% 14.25/8.12
% 14.25/8.12 thf(fact_6149_dvd__triv__right,axiom,
% 14.25/8.12 ! [A2: int,B: int] : ( dvd_dvd_int @ A2 @ ( times_times_int @ B @ A2 ) ) ).
% 14.25/8.12
% 14.25/8.12 thf(fact_6150_dvd__mult__right,axiom,
% 14.25/8.12 ! [A2: assn,B: assn,C: assn] :
% 14.25/8.12 ( ( dvd_dvd_assn @ ( times_times_assn @ A2 @ B ) @ C )
% 14.25/8.12 => ( dvd_dvd_assn @ B @ C ) ) ).
% 14.25/8.12
% 14.25/8.12 thf(fact_6151_dvd__mult__right,axiom,
% 14.25/8.12 ! [A2: rat,B: rat,C: rat] :
% 14.25/8.12 ( ( dvd_dvd_rat @ ( times_times_rat @ A2 @ B ) @ C )
% 14.25/8.12 => ( dvd_dvd_rat @ B @ C ) ) ).
% 14.25/8.12
% 14.25/8.12 thf(fact_6152_dvd__mult__right,axiom,
% 14.25/8.12 ! [A2: nat,B: nat,C: nat] :
% 14.25/8.12 ( ( dvd_dvd_nat @ ( times_times_nat @ A2 @ B ) @ C )
% 14.25/8.12 => ( dvd_dvd_nat @ B @ C ) ) ).
% 14.25/8.12
% 14.25/8.12 thf(fact_6153_dvd__mult__right,axiom,
% 14.25/8.12 ! [A2: int,B: int,C: int] :
% 14.25/8.12 ( ( dvd_dvd_int @ ( times_times_int @ A2 @ B ) @ C )
% 14.25/8.12 => ( dvd_dvd_int @ B @ C ) ) ).
% 14.25/8.12
% 14.25/8.12 thf(fact_6154_mult__dvd__mono,axiom,
% 14.25/8.12 ! [A2: assn,B: assn,C: assn,D2: assn] :
% 14.25/8.12 ( ( dvd_dvd_assn @ A2 @ B )
% 14.25/8.12 => ( ( dvd_dvd_assn @ C @ D2 )
% 14.25/8.12 => ( dvd_dvd_assn @ ( times_times_assn @ A2 @ C ) @ ( times_times_assn @ B @ D2 ) ) ) ) ).
% 14.25/8.12
% 14.25/8.12 thf(fact_6155_mult__dvd__mono,axiom,
% 14.25/8.12 ! [A2: rat,B: rat,C: rat,D2: rat] :
% 14.25/8.12 ( ( dvd_dvd_rat @ A2 @ B )
% 14.25/8.12 => ( ( dvd_dvd_rat @ C @ D2 )
% 14.25/8.12 => ( dvd_dvd_rat @ ( times_times_rat @ A2 @ C ) @ ( times_times_rat @ B @ D2 ) ) ) ) ).
% 14.25/8.12
% 14.25/8.12 thf(fact_6156_mult__dvd__mono,axiom,
% 14.25/8.12 ! [A2: nat,B: nat,C: nat,D2: nat] :
% 14.25/8.12 ( ( dvd_dvd_nat @ A2 @ B )
% 14.25/8.12 => ( ( dvd_dvd_nat @ C @ D2 )
% 14.25/8.12 => ( dvd_dvd_nat @ ( times_times_nat @ A2 @ C ) @ ( times_times_nat @ B @ D2 ) ) ) ) ).
% 14.25/8.12
% 14.25/8.12 thf(fact_6157_mult__dvd__mono,axiom,
% 14.25/8.12 ! [A2: int,B: int,C: int,D2: int] :
% 14.25/8.12 ( ( dvd_dvd_int @ A2 @ B )
% 14.25/8.12 => ( ( dvd_dvd_int @ C @ D2 )
% 14.25/8.12 => ( dvd_dvd_int @ ( times_times_int @ A2 @ C ) @ ( times_times_int @ B @ D2 ) ) ) ) ).
% 14.25/8.12
% 14.25/8.12 thf(fact_6158_dvd__triv__left,axiom,
% 14.25/8.12 ! [A2: assn,B: assn] : ( dvd_dvd_assn @ A2 @ ( times_times_assn @ A2 @ B ) ) ).
% 14.25/8.12
% 14.25/8.12 thf(fact_6159_dvd__triv__left,axiom,
% 14.25/8.12 ! [A2: rat,B: rat] : ( dvd_dvd_rat @ A2 @ ( times_times_rat @ A2 @ B ) ) ).
% 14.25/8.12
% 14.25/8.12 thf(fact_6160_dvd__triv__left,axiom,
% 14.25/8.12 ! [A2: nat,B: nat] : ( dvd_dvd_nat @ A2 @ ( times_times_nat @ A2 @ B ) ) ).
% 14.25/8.12
% 14.25/8.12 thf(fact_6161_dvd__triv__left,axiom,
% 14.25/8.12 ! [A2: int,B: int] : ( dvd_dvd_int @ A2 @ ( times_times_int @ A2 @ B ) ) ).
% 14.25/8.12
% 14.25/8.12 thf(fact_6162_dvd__mult__left,axiom,
% 14.25/8.12 ! [A2: assn,B: assn,C: assn] :
% 14.25/8.12 ( ( dvd_dvd_assn @ ( times_times_assn @ A2 @ B ) @ C )
% 14.25/8.12 => ( dvd_dvd_assn @ A2 @ C ) ) ).
% 14.25/8.12
% 14.25/8.12 thf(fact_6163_dvd__mult__left,axiom,
% 14.25/8.12 ! [A2: rat,B: rat,C: rat] :
% 14.25/8.12 ( ( dvd_dvd_rat @ ( times_times_rat @ A2 @ B ) @ C )
% 14.25/8.12 => ( dvd_dvd_rat @ A2 @ C ) ) ).
% 14.25/8.12
% 14.25/8.12 thf(fact_6164_dvd__mult__left,axiom,
% 14.25/8.12 ! [A2: nat,B: nat,C: nat] :
% 14.25/8.12 ( ( dvd_dvd_nat @ ( times_times_nat @ A2 @ B ) @ C )
% 14.25/8.12 => ( dvd_dvd_nat @ A2 @ C ) ) ).
% 14.25/8.12
% 14.25/8.12 thf(fact_6165_dvd__mult__left,axiom,
% 14.25/8.12 ! [A2: int,B: int,C: int] :
% 14.25/8.12 ( ( dvd_dvd_int @ ( times_times_int @ A2 @ B ) @ C )
% 14.25/8.12 => ( dvd_dvd_int @ A2 @ C ) ) ).
% 14.25/8.12
% 14.25/8.12 thf(fact_6166_dvd__mult2,axiom,
% 14.25/8.12 ! [A2: assn,B: assn,C: assn] :
% 14.25/8.12 ( ( dvd_dvd_assn @ A2 @ B )
% 14.25/8.12 => ( dvd_dvd_assn @ A2 @ ( times_times_assn @ B @ C ) ) ) ).
% 14.25/8.12
% 14.25/8.12 thf(fact_6167_dvd__mult2,axiom,
% 14.25/8.12 ! [A2: rat,B: rat,C: rat] :
% 14.25/8.12 ( ( dvd_dvd_rat @ A2 @ B )
% 14.25/8.12 => ( dvd_dvd_rat @ A2 @ ( times_times_rat @ B @ C ) ) ) ).
% 14.25/8.12
% 14.25/8.12 thf(fact_6168_dvd__mult2,axiom,
% 14.25/8.12 ! [A2: nat,B: nat,C: nat] :
% 14.25/8.12 ( ( dvd_dvd_nat @ A2 @ B )
% 14.25/8.12 => ( dvd_dvd_nat @ A2 @ ( times_times_nat @ B @ C ) ) ) ).
% 14.25/8.12
% 14.25/8.12 thf(fact_6169_dvd__mult2,axiom,
% 14.25/8.12 ! [A2: int,B: int,C: int] :
% 14.25/8.12 ( ( dvd_dvd_int @ A2 @ B )
% 14.25/8.12 => ( dvd_dvd_int @ A2 @ ( times_times_int @ B @ C ) ) ) ).
% 14.25/8.12
% 14.25/8.12 thf(fact_6170_dvd__mult,axiom,
% 14.25/8.12 ! [A2: assn,C: assn,B: assn] :
% 14.25/8.12 ( ( dvd_dvd_assn @ A2 @ C )
% 14.25/8.12 => ( dvd_dvd_assn @ A2 @ ( times_times_assn @ B @ C ) ) ) ).
% 14.25/8.12
% 14.25/8.12 thf(fact_6171_dvd__mult,axiom,
% 14.25/8.12 ! [A2: rat,C: rat,B: rat] :
% 14.25/8.12 ( ( dvd_dvd_rat @ A2 @ C )
% 14.25/8.12 => ( dvd_dvd_rat @ A2 @ ( times_times_rat @ B @ C ) ) ) ).
% 14.25/8.12
% 14.25/8.12 thf(fact_6172_dvd__mult,axiom,
% 14.25/8.12 ! [A2: nat,C: nat,B: nat] :
% 14.25/8.12 ( ( dvd_dvd_nat @ A2 @ C )
% 14.25/8.12 => ( dvd_dvd_nat @ A2 @ ( times_times_nat @ B @ C ) ) ) ).
% 14.25/8.12
% 14.25/8.12 thf(fact_6173_dvd__mult,axiom,
% 14.25/8.12 ! [A2: int,C: int,B: int] :
% 14.25/8.12 ( ( dvd_dvd_int @ A2 @ C )
% 14.25/8.12 => ( dvd_dvd_int @ A2 @ ( times_times_int @ B @ C ) ) ) ).
% 14.25/8.12
% 14.25/8.12 thf(fact_6174_dvd__def,axiom,
% 14.25/8.12 ( dvd_dvd_assn
% 14.25/8.12 = ( ^ [B6: assn,A6: assn] :
% 14.25/8.12 ? [K5: assn] :
% 14.25/8.12 ( A6
% 14.25/8.12 = ( times_times_assn @ B6 @ K5 ) ) ) ) ).
% 14.25/8.12
% 14.25/8.12 thf(fact_6175_dvd__def,axiom,
% 14.25/8.12 ( dvd_dvd_rat
% 14.25/8.12 = ( ^ [B6: rat,A6: rat] :
% 14.25/8.12 ? [K5: rat] :
% 14.25/8.12 ( A6
% 14.25/8.12 = ( times_times_rat @ B6 @ K5 ) ) ) ) ).
% 14.25/8.12
% 14.25/8.12 thf(fact_6176_dvd__def,axiom,
% 14.25/8.12 ( dvd_dvd_nat
% 14.25/8.12 = ( ^ [B6: nat,A6: nat] :
% 14.25/8.12 ? [K5: nat] :
% 14.25/8.12 ( A6
% 14.25/8.12 = ( times_times_nat @ B6 @ K5 ) ) ) ) ).
% 14.25/8.12
% 14.25/8.12 thf(fact_6177_dvd__def,axiom,
% 14.25/8.12 ( dvd_dvd_int
% 14.25/8.12 = ( ^ [B6: int,A6: int] :
% 14.25/8.12 ? [K5: int] :
% 14.25/8.12 ( A6
% 14.25/8.12 = ( times_times_int @ B6 @ K5 ) ) ) ) ).
% 14.25/8.12
% 14.25/8.12 thf(fact_6178_dvdI,axiom,
% 14.25/8.12 ! [A2: assn,B: assn,K3: assn] :
% 14.25/8.12 ( ( A2
% 14.25/8.12 = ( times_times_assn @ B @ K3 ) )
% 14.25/8.12 => ( dvd_dvd_assn @ B @ A2 ) ) ).
% 14.25/8.12
% 14.25/8.12 thf(fact_6179_dvdI,axiom,
% 14.25/8.12 ! [A2: rat,B: rat,K3: rat] :
% 14.25/8.12 ( ( A2
% 14.25/8.12 = ( times_times_rat @ B @ K3 ) )
% 14.25/8.12 => ( dvd_dvd_rat @ B @ A2 ) ) ).
% 14.25/8.12
% 14.25/8.12 thf(fact_6180_dvdI,axiom,
% 14.25/8.12 ! [A2: nat,B: nat,K3: nat] :
% 14.25/8.12 ( ( A2
% 14.25/8.12 = ( times_times_nat @ B @ K3 ) )
% 14.25/8.12 => ( dvd_dvd_nat @ B @ A2 ) ) ).
% 14.25/8.12
% 14.25/8.12 thf(fact_6181_dvdI,axiom,
% 14.25/8.12 ! [A2: int,B: int,K3: int] :
% 14.25/8.12 ( ( A2
% 14.25/8.12 = ( times_times_int @ B @ K3 ) )
% 14.25/8.12 => ( dvd_dvd_int @ B @ A2 ) ) ).
% 14.25/8.12
% 14.25/8.12 thf(fact_6182_dvdE,axiom,
% 14.25/8.12 ! [B: assn,A2: assn] :
% 14.25/8.12 ( ( dvd_dvd_assn @ B @ A2 )
% 14.25/8.12 => ~ ! [K: assn] :
% 14.25/8.12 ( A2
% 14.25/8.12 != ( times_times_assn @ B @ K ) ) ) ).
% 14.25/8.12
% 14.25/8.12 thf(fact_6183_dvdE,axiom,
% 14.25/8.12 ! [B: rat,A2: rat] :
% 14.25/8.12 ( ( dvd_dvd_rat @ B @ A2 )
% 14.25/8.12 => ~ ! [K: rat] :
% 14.25/8.12 ( A2
% 14.25/8.12 != ( times_times_rat @ B @ K ) ) ) ).
% 14.25/8.12
% 14.25/8.12 thf(fact_6184_dvdE,axiom,
% 14.25/8.12 ! [B: nat,A2: nat] :
% 14.25/8.12 ( ( dvd_dvd_nat @ B @ A2 )
% 14.25/8.12 => ~ ! [K: nat] :
% 14.25/8.12 ( A2
% 14.25/8.12 != ( times_times_nat @ B @ K ) ) ) ).
% 14.25/8.12
% 14.25/8.12 thf(fact_6185_dvdE,axiom,
% 14.25/8.12 ! [B: int,A2: int] :
% 14.25/8.12 ( ( dvd_dvd_int @ B @ A2 )
% 14.25/8.12 => ~ ! [K: int] :
% 14.25/8.12 ( A2
% 14.25/8.12 != ( times_times_int @ B @ K ) ) ) ).
% 14.25/8.12
% 14.25/8.12 thf(fact_6186_dvd__unit__imp__unit,axiom,
% 14.25/8.12 ! [A2: nat,B: nat] :
% 14.25/8.12 ( ( dvd_dvd_nat @ A2 @ B )
% 14.25/8.12 => ( ( dvd_dvd_nat @ B @ one_one_nat )
% 14.25/8.12 => ( dvd_dvd_nat @ A2 @ one_one_nat ) ) ) ).
% 14.25/8.12
% 14.25/8.12 thf(fact_6187_dvd__unit__imp__unit,axiom,
% 14.25/8.12 ! [A2: int,B: int] :
% 14.25/8.12 ( ( dvd_dvd_int @ A2 @ B )
% 14.25/8.12 => ( ( dvd_dvd_int @ B @ one_one_int )
% 14.25/8.12 => ( dvd_dvd_int @ A2 @ one_one_int ) ) ) ).
% 14.25/8.12
% 14.25/8.12 thf(fact_6188_dvd__unit__imp__unit,axiom,
% 14.25/8.12 ! [A2: code_integer,B: code_integer] :
% 14.25/8.12 ( ( dvd_dvd_Code_integer @ A2 @ B )
% 14.25/8.12 => ( ( dvd_dvd_Code_integer @ B @ one_one_Code_integer )
% 14.25/8.12 => ( dvd_dvd_Code_integer @ A2 @ one_one_Code_integer ) ) ) ).
% 14.25/8.12
% 14.25/8.12 thf(fact_6189_dvd__unit__imp__unit,axiom,
% 14.25/8.12 ! [A2: code_natural,B: code_natural] :
% 14.25/8.12 ( ( dvd_dvd_Code_natural @ A2 @ B )
% 14.25/8.12 => ( ( dvd_dvd_Code_natural @ B @ one_one_Code_natural )
% 14.25/8.12 => ( dvd_dvd_Code_natural @ A2 @ one_one_Code_natural ) ) ) ).
% 14.25/8.12
% 14.25/8.12 thf(fact_6190_unit__imp__dvd,axiom,
% 14.25/8.12 ! [B: nat,A2: nat] :
% 14.25/8.12 ( ( dvd_dvd_nat @ B @ one_one_nat )
% 14.25/8.12 => ( dvd_dvd_nat @ B @ A2 ) ) ).
% 14.25/8.12
% 14.25/8.12 thf(fact_6191_unit__imp__dvd,axiom,
% 14.25/8.12 ! [B: int,A2: int] :
% 14.25/8.12 ( ( dvd_dvd_int @ B @ one_one_int )
% 14.25/8.12 => ( dvd_dvd_int @ B @ A2 ) ) ).
% 14.25/8.12
% 14.25/8.12 thf(fact_6192_unit__imp__dvd,axiom,
% 14.25/8.12 ! [B: code_integer,A2: code_integer] :
% 14.25/8.12 ( ( dvd_dvd_Code_integer @ B @ one_one_Code_integer )
% 14.25/8.12 => ( dvd_dvd_Code_integer @ B @ A2 ) ) ).
% 14.25/8.12
% 14.25/8.12 thf(fact_6193_unit__imp__dvd,axiom,
% 14.25/8.12 ! [B: code_natural,A2: code_natural] :
% 14.25/8.12 ( ( dvd_dvd_Code_natural @ B @ one_one_Code_natural )
% 14.25/8.12 => ( dvd_dvd_Code_natural @ B @ A2 ) ) ).
% 14.25/8.12
% 14.25/8.12 thf(fact_6194_one__dvd,axiom,
% 14.25/8.12 ! [A2: assn] : ( dvd_dvd_assn @ one_one_assn @ A2 ) ).
% 14.25/8.12
% 14.25/8.12 thf(fact_6195_one__dvd,axiom,
% 14.25/8.12 ! [A2: nat] : ( dvd_dvd_nat @ one_one_nat @ A2 ) ).
% 14.25/8.12
% 14.25/8.12 thf(fact_6196_one__dvd,axiom,
% 14.25/8.12 ! [A2: int] : ( dvd_dvd_int @ one_one_int @ A2 ) ).
% 14.25/8.12
% 14.25/8.12 thf(fact_6197_one__dvd,axiom,
% 14.25/8.12 ! [A2: code_integer] : ( dvd_dvd_Code_integer @ one_one_Code_integer @ A2 ) ).
% 14.25/8.12
% 14.25/8.12 thf(fact_6198_one__dvd,axiom,
% 14.25/8.12 ! [A2: code_natural] : ( dvd_dvd_Code_natural @ one_one_Code_natural @ A2 ) ).
% 14.25/8.12
% 14.25/8.12 thf(fact_6199_power__mult,axiom,
% 14.25/8.12 ! [A2: int,M4: nat,N2: nat] :
% 14.25/8.12 ( ( power_power_int @ A2 @ ( times_times_nat @ M4 @ N2 ) )
% 14.25/8.12 = ( power_power_int @ ( power_power_int @ A2 @ M4 ) @ N2 ) ) ).
% 14.25/8.12
% 14.25/8.12 thf(fact_6200_power__mult,axiom,
% 14.25/8.12 ! [A2: nat,M4: nat,N2: nat] :
% 14.25/8.12 ( ( power_power_nat @ A2 @ ( times_times_nat @ M4 @ N2 ) )
% 14.25/8.12 = ( power_power_nat @ ( power_power_nat @ A2 @ M4 ) @ N2 ) ) ).
% 14.25/8.12
% 14.25/8.12 thf(fact_6201_dvd__power__iff,axiom,
% 14.25/8.12 ! [X: code_natural,M4: nat,N2: nat] :
% 14.25/8.12 ( ( X != zero_z2226904508553997617atural )
% 14.25/8.12 => ( ( dvd_dvd_Code_natural @ ( power_7079662738309270450atural @ X @ M4 ) @ ( power_7079662738309270450atural @ X @ N2 ) )
% 14.25/8.12 = ( ( dvd_dvd_Code_natural @ X @ one_one_Code_natural )
% 14.25/8.12 | ( ord_less_eq_nat @ M4 @ N2 ) ) ) ) ).
% 14.25/8.12
% 14.25/8.12 thf(fact_6202_dvd__power__iff,axiom,
% 14.25/8.12 ! [X: code_integer,M4: nat,N2: nat] :
% 14.25/8.12 ( ( X != zero_z3403309356797280102nteger )
% 14.25/8.12 => ( ( dvd_dvd_Code_integer @ ( power_8256067586552552935nteger @ X @ M4 ) @ ( power_8256067586552552935nteger @ X @ N2 ) )
% 14.25/8.12 = ( ( dvd_dvd_Code_integer @ X @ one_one_Code_integer )
% 14.25/8.12 | ( ord_less_eq_nat @ M4 @ N2 ) ) ) ) ).
% 14.25/8.12
% 14.25/8.12 thf(fact_6203_dvd__power__iff,axiom,
% 14.25/8.12 ! [X: int,M4: nat,N2: nat] :
% 14.25/8.12 ( ( X != zero_zero_int )
% 14.25/8.12 => ( ( dvd_dvd_int @ ( power_power_int @ X @ M4 ) @ ( power_power_int @ X @ N2 ) )
% 14.25/8.12 = ( ( dvd_dvd_int @ X @ one_one_int )
% 14.25/8.12 | ( ord_less_eq_nat @ M4 @ N2 ) ) ) ) ).
% 14.25/8.12
% 14.25/8.12 thf(fact_6204_dvd__power__iff,axiom,
% 14.25/8.12 ! [X: nat,M4: nat,N2: nat] :
% 14.25/8.12 ( ( X != zero_zero_nat )
% 14.25/8.12 => ( ( dvd_dvd_nat @ ( power_power_nat @ X @ M4 ) @ ( power_power_nat @ X @ N2 ) )
% 14.25/8.12 = ( ( dvd_dvd_nat @ X @ one_one_nat )
% 14.25/8.12 | ( ord_less_eq_nat @ M4 @ N2 ) ) ) ) ).
% 14.25/8.12
% 14.25/8.12 thf(fact_6205_dvd__power,axiom,
% 14.25/8.12 ! [N2: nat,X: code_integer] :
% 14.25/8.12 ( ( ( ord_less_nat @ zero_zero_nat @ N2 )
% 14.25/8.12 | ( X = one_one_Code_integer ) )
% 14.25/8.12 => ( dvd_dvd_Code_integer @ X @ ( power_8256067586552552935nteger @ X @ N2 ) ) ) ).
% 14.25/8.12
% 14.25/8.12 thf(fact_6206_dvd__power,axiom,
% 14.25/8.12 ! [N2: nat,X: code_natural] :
% 14.25/8.12 ( ( ( ord_less_nat @ zero_zero_nat @ N2 )
% 14.25/8.12 | ( X = one_one_Code_natural ) )
% 14.25/8.12 => ( dvd_dvd_Code_natural @ X @ ( power_7079662738309270450atural @ X @ N2 ) ) ) ).
% 14.25/8.12
% 14.25/8.12 thf(fact_6207_dvd__power,axiom,
% 14.25/8.12 ! [N2: nat,X: int] :
% 14.25/8.12 ( ( ( ord_less_nat @ zero_zero_nat @ N2 )
% 14.25/8.12 | ( X = one_one_int ) )
% 14.25/8.12 => ( dvd_dvd_int @ X @ ( power_power_int @ X @ N2 ) ) ) ).
% 14.25/8.12
% 14.25/8.12 thf(fact_6208_dvd__power,axiom,
% 14.25/8.12 ! [N2: nat,X: nat] :
% 14.25/8.12 ( ( ( ord_less_nat @ zero_zero_nat @ N2 )
% 14.25/8.12 | ( X = one_one_nat ) )
% 14.25/8.12 => ( dvd_dvd_nat @ X @ ( power_power_nat @ X @ N2 ) ) ) ).
% 14.25/8.12
% 14.25/8.12 thf(fact_6209_gbinomial__Suc__Suc,axiom,
% 14.25/8.12 ! [A2: rat,K3: nat] :
% 14.25/8.12 ( ( gbinomial_rat @ ( plus_plus_rat @ A2 @ one_one_rat ) @ ( suc @ K3 ) )
% 14.25/8.12 = ( plus_plus_rat @ ( gbinomial_rat @ A2 @ K3 ) @ ( gbinomial_rat @ A2 @ ( suc @ K3 ) ) ) ) ).
% 14.25/8.12
% 14.25/8.12 thf(fact_6210_one__le__power,axiom,
% 14.25/8.12 ! [A2: code_integer,N2: nat] :
% 14.25/8.12 ( ( ord_le3102999989581377725nteger @ one_one_Code_integer @ A2 )
% 14.25/8.12 => ( ord_le3102999989581377725nteger @ one_one_Code_integer @ ( power_8256067586552552935nteger @ A2 @ N2 ) ) ) ).
% 14.25/8.12
% 14.25/8.12 thf(fact_6211_one__le__power,axiom,
% 14.25/8.12 ! [A2: code_natural,N2: nat] :
% 14.25/8.12 ( ( ord_le1926595141338095240atural @ one_one_Code_natural @ A2 )
% 14.25/8.12 => ( ord_le1926595141338095240atural @ one_one_Code_natural @ ( power_7079662738309270450atural @ A2 @ N2 ) ) ) ).
% 14.25/8.12
% 14.25/8.12 thf(fact_6212_one__le__power,axiom,
% 14.25/8.12 ! [A2: rat,N2: nat] :
% 14.25/8.12 ( ( ord_less_eq_rat @ one_one_rat @ A2 )
% 14.25/8.12 => ( ord_less_eq_rat @ one_one_rat @ ( power_power_rat @ A2 @ N2 ) ) ) ).
% 14.25/8.12
% 14.25/8.12 thf(fact_6213_one__le__power,axiom,
% 14.25/8.12 ! [A2: nat,N2: nat] :
% 14.25/8.12 ( ( ord_less_eq_nat @ one_one_nat @ A2 )
% 14.25/8.12 => ( ord_less_eq_nat @ one_one_nat @ ( power_power_nat @ A2 @ N2 ) ) ) ).
% 14.25/8.12
% 14.25/8.12 thf(fact_6214_one__le__power,axiom,
% 14.25/8.12 ! [A2: int,N2: nat] :
% 14.25/8.12 ( ( ord_less_eq_int @ one_one_int @ A2 )
% 14.25/8.12 => ( ord_less_eq_int @ one_one_int @ ( power_power_int @ A2 @ N2 ) ) ) ).
% 14.25/8.12
% 14.25/8.12 thf(fact_6215_not__is__unit__0,axiom,
% 14.25/8.12 ~ ( dvd_dvd_Code_natural @ zero_z2226904508553997617atural @ one_one_Code_natural ) ).
% 14.25/8.12
% 14.25/8.12 thf(fact_6216_not__is__unit__0,axiom,
% 14.25/8.12 ~ ( dvd_dvd_nat @ zero_zero_nat @ one_one_nat ) ).
% 14.25/8.12
% 14.25/8.12 thf(fact_6217_not__is__unit__0,axiom,
% 14.25/8.12 ~ ( dvd_dvd_int @ zero_zero_int @ one_one_int ) ).
% 14.25/8.12
% 14.25/8.12 thf(fact_6218_not__is__unit__0,axiom,
% 14.25/8.12 ~ ( dvd_dvd_Code_integer @ zero_z3403309356797280102nteger @ one_one_Code_integer ) ).
% 14.25/8.12
% 14.25/8.12 thf(fact_6219_left__right__inverse__power,axiom,
% 14.25/8.12 ! [X: code_integer,Y: code_integer,N2: nat] :
% 14.25/8.12 ( ( ( times_3573771949741848930nteger @ X @ Y )
% 14.25/8.12 = one_one_Code_integer )
% 14.25/8.12 => ( ( times_3573771949741848930nteger @ ( power_8256067586552552935nteger @ X @ N2 ) @ ( power_8256067586552552935nteger @ Y @ N2 ) )
% 14.25/8.12 = one_one_Code_integer ) ) ).
% 14.25/8.12
% 14.25/8.12 thf(fact_6220_left__right__inverse__power,axiom,
% 14.25/8.12 ! [X: code_natural,Y: code_natural,N2: nat] :
% 14.25/8.12 ( ( ( times_2397367101498566445atural @ X @ Y )
% 14.25/8.12 = one_one_Code_natural )
% 14.25/8.12 => ( ( times_2397367101498566445atural @ ( power_7079662738309270450atural @ X @ N2 ) @ ( power_7079662738309270450atural @ Y @ N2 ) )
% 14.25/8.12 = one_one_Code_natural ) ) ).
% 14.25/8.12
% 14.25/8.12 thf(fact_6221_left__right__inverse__power,axiom,
% 14.25/8.12 ! [X: assn,Y: assn,N2: nat] :
% 14.25/8.12 ( ( ( times_times_assn @ X @ Y )
% 14.25/8.12 = one_one_assn )
% 14.25/8.12 => ( ( times_times_assn @ ( power_power_assn @ X @ N2 ) @ ( power_power_assn @ Y @ N2 ) )
% 14.25/8.12 = one_one_assn ) ) ).
% 14.25/8.12
% 14.25/8.12 thf(fact_6222_left__right__inverse__power,axiom,
% 14.25/8.12 ! [X: rat,Y: rat,N2: nat] :
% 14.25/8.12 ( ( ( times_times_rat @ X @ Y )
% 14.25/8.12 = one_one_rat )
% 14.25/8.12 => ( ( times_times_rat @ ( power_power_rat @ X @ N2 ) @ ( power_power_rat @ Y @ N2 ) )
% 14.25/8.12 = one_one_rat ) ) ).
% 14.25/8.12
% 14.25/8.12 thf(fact_6223_left__right__inverse__power,axiom,
% 14.25/8.12 ! [X: nat,Y: nat,N2: nat] :
% 14.25/8.12 ( ( ( times_times_nat @ X @ Y )
% 14.25/8.12 = one_one_nat )
% 14.25/8.12 => ( ( times_times_nat @ ( power_power_nat @ X @ N2 ) @ ( power_power_nat @ Y @ N2 ) )
% 14.25/8.12 = one_one_nat ) ) ).
% 14.25/8.12
% 14.25/8.12 thf(fact_6224_left__right__inverse__power,axiom,
% 14.25/8.12 ! [X: int,Y: int,N2: nat] :
% 14.25/8.12 ( ( ( times_times_int @ X @ Y )
% 14.25/8.12 = one_one_int )
% 14.25/8.12 => ( ( times_times_int @ ( power_power_int @ X @ N2 ) @ ( power_power_int @ Y @ N2 ) )
% 14.25/8.12 = one_one_int ) ) ).
% 14.25/8.12
% 14.25/8.12 thf(fact_6225_power__Suc2,axiom,
% 14.25/8.12 ! [A2: assn,N2: nat] :
% 14.25/8.12 ( ( power_power_assn @ A2 @ ( suc @ N2 ) )
% 14.25/8.12 = ( times_times_assn @ ( power_power_assn @ A2 @ N2 ) @ A2 ) ) ).
% 14.25/8.12
% 14.25/8.12 thf(fact_6226_power__Suc2,axiom,
% 14.25/8.12 ! [A2: rat,N2: nat] :
% 14.25/8.12 ( ( power_power_rat @ A2 @ ( suc @ N2 ) )
% 14.25/8.12 = ( times_times_rat @ ( power_power_rat @ A2 @ N2 ) @ A2 ) ) ).
% 14.25/8.12
% 14.25/8.12 thf(fact_6227_power__Suc2,axiom,
% 14.25/8.12 ! [A2: nat,N2: nat] :
% 14.25/8.12 ( ( power_power_nat @ A2 @ ( suc @ N2 ) )
% 14.25/8.12 = ( times_times_nat @ ( power_power_nat @ A2 @ N2 ) @ A2 ) ) ).
% 14.25/8.12
% 14.25/8.12 thf(fact_6228_power__Suc2,axiom,
% 14.25/8.12 ! [A2: int,N2: nat] :
% 14.25/8.12 ( ( power_power_int @ A2 @ ( suc @ N2 ) )
% 14.25/8.12 = ( times_times_int @ ( power_power_int @ A2 @ N2 ) @ A2 ) ) ).
% 14.25/8.12
% 14.25/8.12 thf(fact_6229_power__Suc,axiom,
% 14.25/8.12 ! [A2: assn,N2: nat] :
% 14.25/8.12 ( ( power_power_assn @ A2 @ ( suc @ N2 ) )
% 14.25/8.12 = ( times_times_assn @ A2 @ ( power_power_assn @ A2 @ N2 ) ) ) ).
% 14.25/8.12
% 14.25/8.12 thf(fact_6230_power__Suc,axiom,
% 14.25/8.12 ! [A2: rat,N2: nat] :
% 14.25/8.12 ( ( power_power_rat @ A2 @ ( suc @ N2 ) )
% 14.25/8.12 = ( times_times_rat @ A2 @ ( power_power_rat @ A2 @ N2 ) ) ) ).
% 14.25/8.12
% 14.25/8.12 thf(fact_6231_power__Suc,axiom,
% 14.25/8.12 ! [A2: nat,N2: nat] :
% 14.25/8.12 ( ( power_power_nat @ A2 @ ( suc @ N2 ) )
% 14.25/8.12 = ( times_times_nat @ A2 @ ( power_power_nat @ A2 @ N2 ) ) ) ).
% 14.25/8.12
% 14.25/8.12 thf(fact_6232_power__Suc,axiom,
% 14.25/8.12 ! [A2: int,N2: nat] :
% 14.25/8.12 ( ( power_power_int @ A2 @ ( suc @ N2 ) )
% 14.25/8.12 = ( times_times_int @ A2 @ ( power_power_int @ A2 @ N2 ) ) ) ).
% 14.25/8.12
% 14.25/8.12 thf(fact_6233_unit__mult__right__cancel,axiom,
% 14.25/8.12 ! [A2: code_integer,B: code_integer,C: code_integer] :
% 14.25/8.12 ( ( dvd_dvd_Code_integer @ A2 @ one_one_Code_integer )
% 14.25/8.12 => ( ( ( times_3573771949741848930nteger @ B @ A2 )
% 14.25/8.12 = ( times_3573771949741848930nteger @ C @ A2 ) )
% 14.25/8.12 = ( B = C ) ) ) ).
% 14.25/8.12
% 14.25/8.12 thf(fact_6234_unit__mult__right__cancel,axiom,
% 14.25/8.12 ! [A2: code_natural,B: code_natural,C: code_natural] :
% 14.25/8.12 ( ( dvd_dvd_Code_natural @ A2 @ one_one_Code_natural )
% 14.25/8.12 => ( ( ( times_2397367101498566445atural @ B @ A2 )
% 14.25/8.12 = ( times_2397367101498566445atural @ C @ A2 ) )
% 14.25/8.12 = ( B = C ) ) ) ).
% 14.25/8.12
% 14.25/8.12 thf(fact_6235_unit__mult__right__cancel,axiom,
% 14.25/8.12 ! [A2: nat,B: nat,C: nat] :
% 14.25/8.12 ( ( dvd_dvd_nat @ A2 @ one_one_nat )
% 14.25/8.12 => ( ( ( times_times_nat @ B @ A2 )
% 14.25/8.12 = ( times_times_nat @ C @ A2 ) )
% 14.25/8.12 = ( B = C ) ) ) ).
% 14.25/8.12
% 14.25/8.12 thf(fact_6236_unit__mult__right__cancel,axiom,
% 14.25/8.12 ! [A2: int,B: int,C: int] :
% 14.25/8.12 ( ( dvd_dvd_int @ A2 @ one_one_int )
% 14.25/8.12 => ( ( ( times_times_int @ B @ A2 )
% 14.25/8.12 = ( times_times_int @ C @ A2 ) )
% 14.25/8.12 = ( B = C ) ) ) ).
% 14.25/8.12
% 14.25/8.12 thf(fact_6237_unit__mult__left__cancel,axiom,
% 14.25/8.12 ! [A2: code_integer,B: code_integer,C: code_integer] :
% 14.25/8.12 ( ( dvd_dvd_Code_integer @ A2 @ one_one_Code_integer )
% 14.25/8.12 => ( ( ( times_3573771949741848930nteger @ A2 @ B )
% 14.25/8.12 = ( times_3573771949741848930nteger @ A2 @ C ) )
% 14.25/8.12 = ( B = C ) ) ) ).
% 14.25/8.12
% 14.25/8.12 thf(fact_6238_unit__mult__left__cancel,axiom,
% 14.25/8.12 ! [A2: code_natural,B: code_natural,C: code_natural] :
% 14.25/8.12 ( ( dvd_dvd_Code_natural @ A2 @ one_one_Code_natural )
% 14.25/8.12 => ( ( ( times_2397367101498566445atural @ A2 @ B )
% 14.25/8.12 = ( times_2397367101498566445atural @ A2 @ C ) )
% 14.25/8.12 = ( B = C ) ) ) ).
% 14.25/8.12
% 14.25/8.12 thf(fact_6239_unit__mult__left__cancel,axiom,
% 14.25/8.12 ! [A2: nat,B: nat,C: nat] :
% 14.25/8.12 ( ( dvd_dvd_nat @ A2 @ one_one_nat )
% 14.25/8.12 => ( ( ( times_times_nat @ A2 @ B )
% 14.25/8.12 = ( times_times_nat @ A2 @ C ) )
% 14.25/8.12 = ( B = C ) ) ) ).
% 14.25/8.12
% 14.25/8.12 thf(fact_6240_unit__mult__left__cancel,axiom,
% 14.25/8.12 ! [A2: int,B: int,C: int] :
% 14.25/8.12 ( ( dvd_dvd_int @ A2 @ one_one_int )
% 14.25/8.12 => ( ( ( times_times_int @ A2 @ B )
% 14.25/8.12 = ( times_times_int @ A2 @ C ) )
% 14.25/8.12 = ( B = C ) ) ) ).
% 14.25/8.12
% 14.25/8.12 thf(fact_6241_mult__unit__dvd__iff_H,axiom,
% 14.25/8.12 ! [A2: code_integer,B: code_integer,C: code_integer] :
% 14.25/8.12 ( ( dvd_dvd_Code_integer @ A2 @ one_one_Code_integer )
% 14.25/8.12 => ( ( dvd_dvd_Code_integer @ ( times_3573771949741848930nteger @ A2 @ B ) @ C )
% 14.25/8.12 = ( dvd_dvd_Code_integer @ B @ C ) ) ) ).
% 14.25/8.12
% 14.25/8.12 thf(fact_6242_mult__unit__dvd__iff_H,axiom,
% 14.25/8.12 ! [A2: code_natural,B: code_natural,C: code_natural] :
% 14.25/8.12 ( ( dvd_dvd_Code_natural @ A2 @ one_one_Code_natural )
% 14.25/8.12 => ( ( dvd_dvd_Code_natural @ ( times_2397367101498566445atural @ A2 @ B ) @ C )
% 14.25/8.12 = ( dvd_dvd_Code_natural @ B @ C ) ) ) ).
% 14.25/8.12
% 14.25/8.12 thf(fact_6243_mult__unit__dvd__iff_H,axiom,
% 14.25/8.12 ! [A2: nat,B: nat,C: nat] :
% 14.25/8.12 ( ( dvd_dvd_nat @ A2 @ one_one_nat )
% 14.25/8.12 => ( ( dvd_dvd_nat @ ( times_times_nat @ A2 @ B ) @ C )
% 14.25/8.12 = ( dvd_dvd_nat @ B @ C ) ) ) ).
% 14.25/8.12
% 14.25/8.12 thf(fact_6244_mult__unit__dvd__iff_H,axiom,
% 14.25/8.12 ! [A2: int,B: int,C: int] :
% 14.25/8.12 ( ( dvd_dvd_int @ A2 @ one_one_int )
% 14.25/8.12 => ( ( dvd_dvd_int @ ( times_times_int @ A2 @ B ) @ C )
% 14.25/8.12 = ( dvd_dvd_int @ B @ C ) ) ) ).
% 14.25/8.12
% 14.25/8.12 thf(fact_6245_dvd__mult__unit__iff_H,axiom,
% 14.25/8.12 ! [B: code_integer,A2: code_integer,C: code_integer] :
% 14.25/8.12 ( ( dvd_dvd_Code_integer @ B @ one_one_Code_integer )
% 14.25/8.12 => ( ( dvd_dvd_Code_integer @ A2 @ ( times_3573771949741848930nteger @ B @ C ) )
% 14.25/8.12 = ( dvd_dvd_Code_integer @ A2 @ C ) ) ) ).
% 14.25/8.12
% 14.25/8.12 thf(fact_6246_dvd__mult__unit__iff_H,axiom,
% 14.25/8.12 ! [B: code_natural,A2: code_natural,C: code_natural] :
% 14.25/8.12 ( ( dvd_dvd_Code_natural @ B @ one_one_Code_natural )
% 14.25/8.12 => ( ( dvd_dvd_Code_natural @ A2 @ ( times_2397367101498566445atural @ B @ C ) )
% 14.25/8.12 = ( dvd_dvd_Code_natural @ A2 @ C ) ) ) ).
% 14.25/8.12
% 14.25/8.12 thf(fact_6247_dvd__mult__unit__iff_H,axiom,
% 14.25/8.12 ! [B: nat,A2: nat,C: nat] :
% 14.25/8.12 ( ( dvd_dvd_nat @ B @ one_one_nat )
% 14.25/8.12 => ( ( dvd_dvd_nat @ A2 @ ( times_times_nat @ B @ C ) )
% 14.25/8.12 = ( dvd_dvd_nat @ A2 @ C ) ) ) ).
% 14.25/8.12
% 14.25/8.12 thf(fact_6248_dvd__mult__unit__iff_H,axiom,
% 14.25/8.12 ! [B: int,A2: int,C: int] :
% 14.25/8.12 ( ( dvd_dvd_int @ B @ one_one_int )
% 14.25/8.12 => ( ( dvd_dvd_int @ A2 @ ( times_times_int @ B @ C ) )
% 14.25/8.12 = ( dvd_dvd_int @ A2 @ C ) ) ) ).
% 14.25/8.12
% 14.25/8.12 thf(fact_6249_mult__unit__dvd__iff,axiom,
% 14.25/8.12 ! [B: code_integer,A2: code_integer,C: code_integer] :
% 14.25/8.12 ( ( dvd_dvd_Code_integer @ B @ one_one_Code_integer )
% 14.25/8.12 => ( ( dvd_dvd_Code_integer @ ( times_3573771949741848930nteger @ A2 @ B ) @ C )
% 14.25/8.12 = ( dvd_dvd_Code_integer @ A2 @ C ) ) ) ).
% 14.25/8.12
% 14.25/8.12 thf(fact_6250_mult__unit__dvd__iff,axiom,
% 14.25/8.12 ! [B: code_natural,A2: code_natural,C: code_natural] :
% 14.25/8.12 ( ( dvd_dvd_Code_natural @ B @ one_one_Code_natural )
% 14.25/8.12 => ( ( dvd_dvd_Code_natural @ ( times_2397367101498566445atural @ A2 @ B ) @ C )
% 14.25/8.12 = ( dvd_dvd_Code_natural @ A2 @ C ) ) ) ).
% 14.25/8.12
% 14.25/8.12 thf(fact_6251_mult__unit__dvd__iff,axiom,
% 14.25/8.12 ! [B: nat,A2: nat,C: nat] :
% 14.25/8.12 ( ( dvd_dvd_nat @ B @ one_one_nat )
% 14.25/8.12 => ( ( dvd_dvd_nat @ ( times_times_nat @ A2 @ B ) @ C )
% 14.25/8.12 = ( dvd_dvd_nat @ A2 @ C ) ) ) ).
% 14.25/8.12
% 14.25/8.12 thf(fact_6252_mult__unit__dvd__iff,axiom,
% 14.25/8.12 ! [B: int,A2: int,C: int] :
% 14.25/8.12 ( ( dvd_dvd_int @ B @ one_one_int )
% 14.25/8.12 => ( ( dvd_dvd_int @ ( times_times_int @ A2 @ B ) @ C )
% 14.25/8.12 = ( dvd_dvd_int @ A2 @ C ) ) ) ).
% 14.25/8.12
% 14.25/8.12 thf(fact_6253_dvd__mult__unit__iff,axiom,
% 14.25/8.12 ! [B: code_integer,A2: code_integer,C: code_integer] :
% 14.25/8.12 ( ( dvd_dvd_Code_integer @ B @ one_one_Code_integer )
% 14.25/8.12 => ( ( dvd_dvd_Code_integer @ A2 @ ( times_3573771949741848930nteger @ C @ B ) )
% 14.25/8.12 = ( dvd_dvd_Code_integer @ A2 @ C ) ) ) ).
% 14.25/8.12
% 14.25/8.12 thf(fact_6254_dvd__mult__unit__iff,axiom,
% 14.25/8.12 ! [B: code_natural,A2: code_natural,C: code_natural] :
% 14.25/8.12 ( ( dvd_dvd_Code_natural @ B @ one_one_Code_natural )
% 14.25/8.12 => ( ( dvd_dvd_Code_natural @ A2 @ ( times_2397367101498566445atural @ C @ B ) )
% 14.25/8.12 = ( dvd_dvd_Code_natural @ A2 @ C ) ) ) ).
% 14.25/8.12
% 14.25/8.12 thf(fact_6255_dvd__mult__unit__iff,axiom,
% 14.25/8.12 ! [B: nat,A2: nat,C: nat] :
% 14.25/8.12 ( ( dvd_dvd_nat @ B @ one_one_nat )
% 14.25/8.12 => ( ( dvd_dvd_nat @ A2 @ ( times_times_nat @ C @ B ) )
% 14.25/8.12 = ( dvd_dvd_nat @ A2 @ C ) ) ) ).
% 14.25/8.12
% 14.25/8.12 thf(fact_6256_dvd__mult__unit__iff,axiom,
% 14.25/8.12 ! [B: int,A2: int,C: int] :
% 14.25/8.12 ( ( dvd_dvd_int @ B @ one_one_int )
% 14.25/8.12 => ( ( dvd_dvd_int @ A2 @ ( times_times_int @ C @ B ) )
% 14.25/8.12 = ( dvd_dvd_int @ A2 @ C ) ) ) ).
% 14.25/8.12
% 14.25/8.12 thf(fact_6257_is__unit__mult__iff,axiom,
% 14.25/8.12 ! [A2: code_integer,B: code_integer] :
% 14.25/8.12 ( ( dvd_dvd_Code_integer @ ( times_3573771949741848930nteger @ A2 @ B ) @ one_one_Code_integer )
% 14.25/8.12 = ( ( dvd_dvd_Code_integer @ A2 @ one_one_Code_integer )
% 14.25/8.12 & ( dvd_dvd_Code_integer @ B @ one_one_Code_integer ) ) ) ).
% 14.25/8.12
% 14.25/8.12 thf(fact_6258_is__unit__mult__iff,axiom,
% 14.25/8.12 ! [A2: code_natural,B: code_natural] :
% 14.25/8.12 ( ( dvd_dvd_Code_natural @ ( times_2397367101498566445atural @ A2 @ B ) @ one_one_Code_natural )
% 14.25/8.12 = ( ( dvd_dvd_Code_natural @ A2 @ one_one_Code_natural )
% 14.25/8.12 & ( dvd_dvd_Code_natural @ B @ one_one_Code_natural ) ) ) ).
% 14.25/8.12
% 14.25/8.12 thf(fact_6259_is__unit__mult__iff,axiom,
% 14.25/8.12 ! [A2: nat,B: nat] :
% 14.25/8.12 ( ( dvd_dvd_nat @ ( times_times_nat @ A2 @ B ) @ one_one_nat )
% 14.25/8.12 = ( ( dvd_dvd_nat @ A2 @ one_one_nat )
% 14.25/8.12 & ( dvd_dvd_nat @ B @ one_one_nat ) ) ) ).
% 14.25/8.12
% 14.25/8.12 thf(fact_6260_is__unit__mult__iff,axiom,
% 14.25/8.12 ! [A2: int,B: int] :
% 14.25/8.12 ( ( dvd_dvd_int @ ( times_times_int @ A2 @ B ) @ one_one_int )
% 14.25/8.12 = ( ( dvd_dvd_int @ A2 @ one_one_int )
% 14.25/8.12 & ( dvd_dvd_int @ B @ one_one_int ) ) ) ).
% 14.25/8.12
% 14.25/8.12 thf(fact_6261_power__0,axiom,
% 14.25/8.12 ! [A2: assn] :
% 14.25/8.12 ( ( power_power_assn @ A2 @ zero_zero_nat )
% 14.25/8.12 = one_one_assn ) ).
% 14.25/8.12
% 14.25/8.12 thf(fact_6262_power__0,axiom,
% 14.25/8.12 ! [A2: code_integer] :
% 14.25/8.12 ( ( power_8256067586552552935nteger @ A2 @ zero_zero_nat )
% 14.25/8.12 = one_one_Code_integer ) ).
% 14.25/8.12
% 14.25/8.12 thf(fact_6263_power__0,axiom,
% 14.25/8.12 ! [A2: code_natural] :
% 14.25/8.12 ( ( power_7079662738309270450atural @ A2 @ zero_zero_nat )
% 14.25/8.12 = one_one_Code_natural ) ).
% 14.25/8.12
% 14.25/8.12 thf(fact_6264_power__0,axiom,
% 14.25/8.12 ! [A2: int] :
% 14.25/8.12 ( ( power_power_int @ A2 @ zero_zero_nat )
% 14.25/8.12 = one_one_int ) ).
% 14.25/8.12
% 14.25/8.12 thf(fact_6265_power__0,axiom,
% 14.25/8.12 ! [A2: nat] :
% 14.25/8.12 ( ( power_power_nat @ A2 @ zero_zero_nat )
% 14.25/8.12 = one_one_nat ) ).
% 14.25/8.12
% 14.25/8.12 thf(fact_6266_power__add,axiom,
% 14.25/8.12 ! [A2: assn,M4: nat,N2: nat] :
% 14.25/8.12 ( ( power_power_assn @ A2 @ ( plus_plus_nat @ M4 @ N2 ) )
% 14.25/8.12 = ( times_times_assn @ ( power_power_assn @ A2 @ M4 ) @ ( power_power_assn @ A2 @ N2 ) ) ) ).
% 14.25/8.12
% 14.25/8.12 thf(fact_6267_power__add,axiom,
% 14.25/8.12 ! [A2: rat,M4: nat,N2: nat] :
% 14.25/8.12 ( ( power_power_rat @ A2 @ ( plus_plus_nat @ M4 @ N2 ) )
% 14.25/8.12 = ( times_times_rat @ ( power_power_rat @ A2 @ M4 ) @ ( power_power_rat @ A2 @ N2 ) ) ) ).
% 14.25/8.12
% 14.25/8.12 thf(fact_6268_power__add,axiom,
% 14.25/8.12 ! [A2: nat,M4: nat,N2: nat] :
% 14.25/8.12 ( ( power_power_nat @ A2 @ ( plus_plus_nat @ M4 @ N2 ) )
% 14.25/8.12 = ( times_times_nat @ ( power_power_nat @ A2 @ M4 ) @ ( power_power_nat @ A2 @ N2 ) ) ) ).
% 14.25/8.12
% 14.25/8.12 thf(fact_6269_power__add,axiom,
% 14.25/8.12 ! [A2: int,M4: nat,N2: nat] :
% 14.25/8.12 ( ( power_power_int @ A2 @ ( plus_plus_nat @ M4 @ N2 ) )
% 14.25/8.12 = ( times_times_int @ ( power_power_int @ A2 @ M4 ) @ ( power_power_int @ A2 @ N2 ) ) ) ).
% 14.25/8.12
% 14.25/8.12 thf(fact_6270_gbinomial__addition__formula,axiom,
% 14.25/8.12 ! [A2: rat,K3: nat] :
% 14.25/8.12 ( ( gbinomial_rat @ A2 @ ( suc @ K3 ) )
% 14.25/8.12 = ( plus_plus_rat @ ( gbinomial_rat @ ( minus_minus_rat @ A2 @ one_one_rat ) @ ( suc @ K3 ) ) @ ( gbinomial_rat @ ( minus_minus_rat @ A2 @ one_one_rat ) @ K3 ) ) ) ).
% 14.25/8.12
% 14.25/8.12 thf(fact_6271_power__le__one,axiom,
% 14.25/8.12 ! [A2: code_natural,N2: nat] :
% 14.25/8.12 ( ( ord_le1926595141338095240atural @ zero_z2226904508553997617atural @ A2 )
% 14.25/8.12 => ( ( ord_le1926595141338095240atural @ A2 @ one_one_Code_natural )
% 14.25/8.12 => ( ord_le1926595141338095240atural @ ( power_7079662738309270450atural @ A2 @ N2 ) @ one_one_Code_natural ) ) ) ).
% 14.25/8.12
% 14.25/8.12 thf(fact_6272_power__le__one,axiom,
% 14.25/8.12 ! [A2: code_integer,N2: nat] :
% 14.25/8.12 ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ A2 )
% 14.25/8.12 => ( ( ord_le3102999989581377725nteger @ A2 @ one_one_Code_integer )
% 14.25/8.12 => ( ord_le3102999989581377725nteger @ ( power_8256067586552552935nteger @ A2 @ N2 ) @ one_one_Code_integer ) ) ) ).
% 14.25/8.12
% 14.25/8.12 thf(fact_6273_power__le__one,axiom,
% 14.25/8.12 ! [A2: rat,N2: nat] :
% 14.25/8.12 ( ( ord_less_eq_rat @ zero_zero_rat @ A2 )
% 14.25/8.12 => ( ( ord_less_eq_rat @ A2 @ one_one_rat )
% 14.25/8.12 => ( ord_less_eq_rat @ ( power_power_rat @ A2 @ N2 ) @ one_one_rat ) ) ) ).
% 14.25/8.12
% 14.25/8.12 thf(fact_6274_power__le__one,axiom,
% 14.25/8.12 ! [A2: nat,N2: nat] :
% 14.25/8.12 ( ( ord_less_eq_nat @ zero_zero_nat @ A2 )
% 14.25/8.12 => ( ( ord_less_eq_nat @ A2 @ one_one_nat )
% 14.25/8.12 => ( ord_less_eq_nat @ ( power_power_nat @ A2 @ N2 ) @ one_one_nat ) ) ) ).
% 14.25/8.12
% 14.25/8.12 thf(fact_6275_power__le__one,axiom,
% 14.25/8.12 ! [A2: int,N2: nat] :
% 14.25/8.12 ( ( ord_less_eq_int @ zero_zero_int @ A2 )
% 14.25/8.12 => ( ( ord_less_eq_int @ A2 @ one_one_int )
% 14.25/8.12 => ( ord_less_eq_int @ ( power_power_int @ A2 @ N2 ) @ one_one_int ) ) ) ).
% 14.25/8.12
% 14.25/8.12 thf(fact_6276_power__less__power__Suc,axiom,
% 14.25/8.12 ! [A2: code_natural,N2: nat] :
% 14.25/8.12 ( ( ord_le5570908160329646204atural @ one_one_Code_natural @ A2 )
% 14.25/8.12 => ( ord_le5570908160329646204atural @ ( power_7079662738309270450atural @ A2 @ N2 ) @ ( times_2397367101498566445atural @ A2 @ ( power_7079662738309270450atural @ A2 @ N2 ) ) ) ) ).
% 14.25/8.12
% 14.25/8.12 thf(fact_6277_power__less__power__Suc,axiom,
% 14.25/8.12 ! [A2: nat,N2: nat] :
% 14.25/8.12 ( ( ord_less_nat @ one_one_nat @ A2 )
% 14.25/8.12 => ( ord_less_nat @ ( power_power_nat @ A2 @ N2 ) @ ( times_times_nat @ A2 @ ( power_power_nat @ A2 @ N2 ) ) ) ) ).
% 14.25/8.12
% 14.25/8.12 thf(fact_6278_power__less__power__Suc,axiom,
% 14.25/8.12 ! [A2: int,N2: nat] :
% 14.25/8.12 ( ( ord_less_int @ one_one_int @ A2 )
% 14.25/8.12 => ( ord_less_int @ ( power_power_int @ A2 @ N2 ) @ ( times_times_int @ A2 @ ( power_power_int @ A2 @ N2 ) ) ) ) ).
% 14.25/8.12
% 14.25/8.12 thf(fact_6279_power__less__power__Suc,axiom,
% 14.25/8.12 ! [A2: code_integer,N2: nat] :
% 14.25/8.12 ( ( ord_le6747313008572928689nteger @ one_one_Code_integer @ A2 )
% 14.25/8.12 => ( ord_le6747313008572928689nteger @ ( power_8256067586552552935nteger @ A2 @ N2 ) @ ( times_3573771949741848930nteger @ A2 @ ( power_8256067586552552935nteger @ A2 @ N2 ) ) ) ) ).
% 14.25/8.12
% 14.25/8.12 thf(fact_6280_power__less__power__Suc,axiom,
% 14.25/8.12 ! [A2: rat,N2: nat] :
% 14.25/8.12 ( ( ord_less_rat @ one_one_rat @ A2 )
% 14.25/8.12 => ( ord_less_rat @ ( power_power_rat @ A2 @ N2 ) @ ( times_times_rat @ A2 @ ( power_power_rat @ A2 @ N2 ) ) ) ) ).
% 14.25/8.12
% 14.25/8.12 thf(fact_6281_power__gt1__lemma,axiom,
% 14.25/8.12 ! [A2: code_natural,N2: nat] :
% 14.25/8.12 ( ( ord_le5570908160329646204atural @ one_one_Code_natural @ A2 )
% 14.25/8.12 => ( ord_le5570908160329646204atural @ one_one_Code_natural @ ( times_2397367101498566445atural @ A2 @ ( power_7079662738309270450atural @ A2 @ N2 ) ) ) ) ).
% 14.25/8.12
% 14.25/8.12 thf(fact_6282_power__gt1__lemma,axiom,
% 14.25/8.12 ! [A2: nat,N2: nat] :
% 14.25/8.12 ( ( ord_less_nat @ one_one_nat @ A2 )
% 14.25/8.12 => ( ord_less_nat @ one_one_nat @ ( times_times_nat @ A2 @ ( power_power_nat @ A2 @ N2 ) ) ) ) ).
% 14.25/8.12
% 14.25/8.12 thf(fact_6283_power__gt1__lemma,axiom,
% 14.25/8.12 ! [A2: int,N2: nat] :
% 14.25/8.12 ( ( ord_less_int @ one_one_int @ A2 )
% 14.25/8.12 => ( ord_less_int @ one_one_int @ ( times_times_int @ A2 @ ( power_power_int @ A2 @ N2 ) ) ) ) ).
% 14.25/8.12
% 14.25/8.12 thf(fact_6284_power__gt1__lemma,axiom,
% 14.25/8.12 ! [A2: code_integer,N2: nat] :
% 14.25/8.12 ( ( ord_le6747313008572928689nteger @ one_one_Code_integer @ A2 )
% 14.25/8.12 => ( ord_le6747313008572928689nteger @ one_one_Code_integer @ ( times_3573771949741848930nteger @ A2 @ ( power_8256067586552552935nteger @ A2 @ N2 ) ) ) ) ).
% 14.25/8.12
% 14.25/8.12 thf(fact_6285_power__gt1__lemma,axiom,
% 14.25/8.12 ! [A2: rat,N2: nat] :
% 14.25/8.12 ( ( ord_less_rat @ one_one_rat @ A2 )
% 14.25/8.12 => ( ord_less_rat @ one_one_rat @ ( times_times_rat @ A2 @ ( power_power_rat @ A2 @ N2 ) ) ) ) ).
% 14.25/8.12
% 14.25/8.12 thf(fact_6286_unity__coeff__ex,axiom,
% 14.25/8.12 ! [P: code_integer > $o,L: code_integer] :
% 14.25/8.12 ( ( ? [X2: code_integer] : ( P @ ( times_3573771949741848930nteger @ L @ X2 ) ) )
% 14.25/8.12 = ( ? [X2: code_integer] :
% 14.25/8.12 ( ( dvd_dvd_Code_integer @ L @ ( plus_p5714425477246183910nteger @ X2 @ zero_z3403309356797280102nteger ) )
% 14.25/8.12 & ( P @ X2 ) ) ) ) ).
% 14.25/8.12
% 14.25/8.12 thf(fact_6287_unity__coeff__ex,axiom,
% 14.25/8.12 ! [P: rat > $o,L: rat] :
% 14.25/8.12 ( ( ? [X2: rat] : ( P @ ( times_times_rat @ L @ X2 ) ) )
% 14.25/8.12 = ( ? [X2: rat] :
% 14.25/8.12 ( ( dvd_dvd_rat @ L @ ( plus_plus_rat @ X2 @ zero_zero_rat ) )
% 14.25/8.12 & ( P @ X2 ) ) ) ) ).
% 14.25/8.12
% 14.25/8.12 thf(fact_6288_unity__coeff__ex,axiom,
% 14.25/8.12 ! [P: nat > $o,L: nat] :
% 14.25/8.12 ( ( ? [X2: nat] : ( P @ ( times_times_nat @ L @ X2 ) ) )
% 14.25/8.12 = ( ? [X2: nat] :
% 14.25/8.12 ( ( dvd_dvd_nat @ L @ ( plus_plus_nat @ X2 @ zero_zero_nat ) )
% 14.25/8.12 & ( P @ X2 ) ) ) ) ).
% 14.25/8.12
% 14.25/8.12 thf(fact_6289_unity__coeff__ex,axiom,
% 14.25/8.12 ! [P: int > $o,L: int] :
% 14.25/8.12 ( ( ? [X2: int] : ( P @ ( times_times_int @ L @ X2 ) ) )
% 14.25/8.12 = ( ? [X2: int] :
% 14.25/8.12 ( ( dvd_dvd_int @ L @ ( plus_plus_int @ X2 @ zero_zero_int ) )
% 14.25/8.12 & ( P @ X2 ) ) ) ) ).
% 14.25/8.12
% 14.25/8.12 thf(fact_6290_power__0__left,axiom,
% 14.25/8.12 ! [N2: nat] :
% 14.25/8.12 ( ( ( N2 = zero_zero_nat )
% 14.25/8.12 => ( ( power_7079662738309270450atural @ zero_z2226904508553997617atural @ N2 )
% 14.25/8.12 = one_one_Code_natural ) )
% 14.25/8.12 & ( ( N2 != zero_zero_nat )
% 14.25/8.12 => ( ( power_7079662738309270450atural @ zero_z2226904508553997617atural @ N2 )
% 14.25/8.12 = zero_z2226904508553997617atural ) ) ) ).
% 14.25/8.12
% 14.25/8.12 thf(fact_6291_power__0__left,axiom,
% 14.25/8.12 ! [N2: nat] :
% 14.25/8.12 ( ( ( N2 = zero_zero_nat )
% 14.25/8.12 => ( ( power_8256067586552552935nteger @ zero_z3403309356797280102nteger @ N2 )
% 14.25/8.12 = one_one_Code_integer ) )
% 14.25/8.12 & ( ( N2 != zero_zero_nat )
% 14.25/8.12 => ( ( power_8256067586552552935nteger @ zero_z3403309356797280102nteger @ N2 )
% 14.25/8.12 = zero_z3403309356797280102nteger ) ) ) ).
% 14.25/8.12
% 14.25/8.12 thf(fact_6292_power__0__left,axiom,
% 14.25/8.12 ! [N2: nat] :
% 14.25/8.12 ( ( ( N2 = zero_zero_nat )
% 14.25/8.12 => ( ( power_power_int @ zero_zero_int @ N2 )
% 14.25/8.12 = one_one_int ) )
% 14.25/8.12 & ( ( N2 != zero_zero_nat )
% 14.25/8.12 => ( ( power_power_int @ zero_zero_int @ N2 )
% 14.25/8.12 = zero_zero_int ) ) ) ).
% 14.25/8.12
% 14.25/8.12 thf(fact_6293_power__0__left,axiom,
% 14.25/8.12 ! [N2: nat] :
% 14.25/8.12 ( ( ( N2 = zero_zero_nat )
% 14.25/8.12 => ( ( power_power_nat @ zero_zero_nat @ N2 )
% 14.25/8.12 = one_one_nat ) )
% 14.25/8.12 & ( ( N2 != zero_zero_nat )
% 14.25/8.12 => ( ( power_power_nat @ zero_zero_nat @ N2 )
% 14.25/8.12 = zero_zero_nat ) ) ) ).
% 14.25/8.12
% 14.25/8.12 thf(fact_6294_unit__dvdE,axiom,
% 14.25/8.12 ! [A2: code_natural,B: code_natural] :
% 14.25/8.12 ( ( dvd_dvd_Code_natural @ A2 @ one_one_Code_natural )
% 14.25/8.12 => ~ ( ( A2 != zero_z2226904508553997617atural )
% 14.25/8.12 => ! [C2: code_natural] :
% 14.25/8.12 ( B
% 14.25/8.12 != ( times_2397367101498566445atural @ A2 @ C2 ) ) ) ) ).
% 14.25/8.12
% 14.25/8.12 thf(fact_6295_unit__dvdE,axiom,
% 14.25/8.12 ! [A2: code_integer,B: code_integer] :
% 14.25/8.12 ( ( dvd_dvd_Code_integer @ A2 @ one_one_Code_integer )
% 14.25/8.12 => ~ ( ( A2 != zero_z3403309356797280102nteger )
% 14.25/8.12 => ! [C2: code_integer] :
% 14.25/8.12 ( B
% 14.25/8.12 != ( times_3573771949741848930nteger @ A2 @ C2 ) ) ) ) ).
% 14.25/8.12
% 14.25/8.12 thf(fact_6296_unit__dvdE,axiom,
% 14.25/8.12 ! [A2: nat,B: nat] :
% 14.25/8.12 ( ( dvd_dvd_nat @ A2 @ one_one_nat )
% 14.25/8.12 => ~ ( ( A2 != zero_zero_nat )
% 14.25/8.12 => ! [C2: nat] :
% 14.25/8.12 ( B
% 14.25/8.12 != ( times_times_nat @ A2 @ C2 ) ) ) ) ).
% 14.25/8.12
% 14.25/8.12 thf(fact_6297_unit__dvdE,axiom,
% 14.25/8.12 ! [A2: int,B: int] :
% 14.25/8.12 ( ( dvd_dvd_int @ A2 @ one_one_int )
% 14.25/8.12 => ~ ( ( A2 != zero_zero_int )
% 14.25/8.12 => ! [C2: int] :
% 14.25/8.12 ( B
% 14.25/8.12 != ( times_times_int @ A2 @ C2 ) ) ) ) ).
% 14.25/8.12
% 14.25/8.12 thf(fact_6298_power__gt1,axiom,
% 14.25/8.12 ! [A2: code_natural,N2: nat] :
% 14.25/8.12 ( ( ord_le5570908160329646204atural @ one_one_Code_natural @ A2 )
% 14.25/8.12 => ( ord_le5570908160329646204atural @ one_one_Code_natural @ ( power_7079662738309270450atural @ A2 @ ( suc @ N2 ) ) ) ) ).
% 14.25/8.12
% 14.25/8.12 thf(fact_6299_power__gt1,axiom,
% 14.25/8.12 ! [A2: nat,N2: nat] :
% 14.25/8.12 ( ( ord_less_nat @ one_one_nat @ A2 )
% 14.25/8.12 => ( ord_less_nat @ one_one_nat @ ( power_power_nat @ A2 @ ( suc @ N2 ) ) ) ) ).
% 14.25/8.12
% 14.25/8.12 thf(fact_6300_power__gt1,axiom,
% 14.25/8.12 ! [A2: int,N2: nat] :
% 14.25/8.12 ( ( ord_less_int @ one_one_int @ A2 )
% 14.25/8.12 => ( ord_less_int @ one_one_int @ ( power_power_int @ A2 @ ( suc @ N2 ) ) ) ) ).
% 14.25/8.12
% 14.25/8.12 thf(fact_6301_power__gt1,axiom,
% 14.25/8.12 ! [A2: code_integer,N2: nat] :
% 14.25/8.12 ( ( ord_le6747313008572928689nteger @ one_one_Code_integer @ A2 )
% 14.25/8.12 => ( ord_le6747313008572928689nteger @ one_one_Code_integer @ ( power_8256067586552552935nteger @ A2 @ ( suc @ N2 ) ) ) ) ).
% 14.25/8.12
% 14.25/8.12 thf(fact_6302_power__gt1,axiom,
% 14.25/8.12 ! [A2: rat,N2: nat] :
% 14.25/8.12 ( ( ord_less_rat @ one_one_rat @ A2 )
% 14.25/8.12 => ( ord_less_rat @ one_one_rat @ ( power_power_rat @ A2 @ ( suc @ N2 ) ) ) ) ).
% 14.25/8.12
% 14.25/8.12 thf(fact_6303_power__strict__increasing,axiom,
% 14.25/8.12 ! [N2: nat,N: nat,A2: code_natural] :
% 14.25/8.12 ( ( ord_less_nat @ N2 @ N )
% 14.25/8.12 => ( ( ord_le5570908160329646204atural @ one_one_Code_natural @ A2 )
% 14.25/8.12 => ( ord_le5570908160329646204atural @ ( power_7079662738309270450atural @ A2 @ N2 ) @ ( power_7079662738309270450atural @ A2 @ N ) ) ) ) ).
% 14.25/8.12
% 14.25/8.12 thf(fact_6304_power__strict__increasing,axiom,
% 14.25/8.12 ! [N2: nat,N: nat,A2: nat] :
% 14.25/8.12 ( ( ord_less_nat @ N2 @ N )
% 14.25/8.12 => ( ( ord_less_nat @ one_one_nat @ A2 )
% 14.25/8.12 => ( ord_less_nat @ ( power_power_nat @ A2 @ N2 ) @ ( power_power_nat @ A2 @ N ) ) ) ) ).
% 14.25/8.12
% 14.25/8.12 thf(fact_6305_power__strict__increasing,axiom,
% 14.25/8.12 ! [N2: nat,N: nat,A2: int] :
% 14.25/8.12 ( ( ord_less_nat @ N2 @ N )
% 14.25/8.12 => ( ( ord_less_int @ one_one_int @ A2 )
% 14.25/8.12 => ( ord_less_int @ ( power_power_int @ A2 @ N2 ) @ ( power_power_int @ A2 @ N ) ) ) ) ).
% 14.25/8.12
% 14.25/8.12 thf(fact_6306_power__strict__increasing,axiom,
% 14.25/8.12 ! [N2: nat,N: nat,A2: code_integer] :
% 14.25/8.12 ( ( ord_less_nat @ N2 @ N )
% 14.25/8.12 => ( ( ord_le6747313008572928689nteger @ one_one_Code_integer @ A2 )
% 14.25/8.12 => ( ord_le6747313008572928689nteger @ ( power_8256067586552552935nteger @ A2 @ N2 ) @ ( power_8256067586552552935nteger @ A2 @ N ) ) ) ) ).
% 14.25/8.12
% 14.25/8.12 thf(fact_6307_power__strict__increasing,axiom,
% 14.25/8.12 ! [N2: nat,N: nat,A2: rat] :
% 14.25/8.12 ( ( ord_less_nat @ N2 @ N )
% 14.25/8.12 => ( ( ord_less_rat @ one_one_rat @ A2 )
% 14.25/8.12 => ( ord_less_rat @ ( power_power_rat @ A2 @ N2 ) @ ( power_power_rat @ A2 @ N ) ) ) ) ).
% 14.25/8.12
% 14.25/8.12 thf(fact_6308_power__less__imp__less__exp,axiom,
% 14.25/8.12 ! [A2: code_natural,M4: nat,N2: nat] :
% 14.25/8.12 ( ( ord_le5570908160329646204atural @ one_one_Code_natural @ A2 )
% 14.25/8.12 => ( ( ord_le5570908160329646204atural @ ( power_7079662738309270450atural @ A2 @ M4 ) @ ( power_7079662738309270450atural @ A2 @ N2 ) )
% 14.25/8.12 => ( ord_less_nat @ M4 @ N2 ) ) ) ).
% 14.25/8.12
% 14.25/8.12 thf(fact_6309_power__less__imp__less__exp,axiom,
% 14.25/8.12 ! [A2: nat,M4: nat,N2: nat] :
% 14.25/8.12 ( ( ord_less_nat @ one_one_nat @ A2 )
% 14.25/8.12 => ( ( ord_less_nat @ ( power_power_nat @ A2 @ M4 ) @ ( power_power_nat @ A2 @ N2 ) )
% 14.25/8.12 => ( ord_less_nat @ M4 @ N2 ) ) ) ).
% 14.25/8.12
% 14.25/8.12 thf(fact_6310_power__less__imp__less__exp,axiom,
% 14.25/8.12 ! [A2: int,M4: nat,N2: nat] :
% 14.25/8.12 ( ( ord_less_int @ one_one_int @ A2 )
% 14.25/8.12 => ( ( ord_less_int @ ( power_power_int @ A2 @ M4 ) @ ( power_power_int @ A2 @ N2 ) )
% 14.25/8.12 => ( ord_less_nat @ M4 @ N2 ) ) ) ).
% 14.25/8.12
% 14.25/8.12 thf(fact_6311_power__less__imp__less__exp,axiom,
% 14.25/8.12 ! [A2: code_integer,M4: nat,N2: nat] :
% 14.25/8.12 ( ( ord_le6747313008572928689nteger @ one_one_Code_integer @ A2 )
% 14.25/8.12 => ( ( ord_le6747313008572928689nteger @ ( power_8256067586552552935nteger @ A2 @ M4 ) @ ( power_8256067586552552935nteger @ A2 @ N2 ) )
% 14.25/8.12 => ( ord_less_nat @ M4 @ N2 ) ) ) ).
% 14.25/8.12
% 14.25/8.12 thf(fact_6312_power__less__imp__less__exp,axiom,
% 14.25/8.12 ! [A2: rat,M4: nat,N2: nat] :
% 14.25/8.12 ( ( ord_less_rat @ one_one_rat @ A2 )
% 14.25/8.12 => ( ( ord_less_rat @ ( power_power_rat @ A2 @ M4 ) @ ( power_power_rat @ A2 @ N2 ) )
% 14.25/8.12 => ( ord_less_nat @ M4 @ N2 ) ) ) ).
% 14.25/8.12
% 14.25/8.12 thf(fact_6313_power__increasing,axiom,
% 14.25/8.12 ! [N2: nat,N: nat,A2: code_integer] :
% 14.25/8.12 ( ( ord_less_eq_nat @ N2 @ N )
% 14.25/8.12 => ( ( ord_le3102999989581377725nteger @ one_one_Code_integer @ A2 )
% 14.25/8.12 => ( ord_le3102999989581377725nteger @ ( power_8256067586552552935nteger @ A2 @ N2 ) @ ( power_8256067586552552935nteger @ A2 @ N ) ) ) ) ).
% 14.25/8.12
% 14.25/8.12 thf(fact_6314_power__increasing,axiom,
% 14.25/8.12 ! [N2: nat,N: nat,A2: code_natural] :
% 14.25/8.12 ( ( ord_less_eq_nat @ N2 @ N )
% 14.25/8.12 => ( ( ord_le1926595141338095240atural @ one_one_Code_natural @ A2 )
% 14.25/8.12 => ( ord_le1926595141338095240atural @ ( power_7079662738309270450atural @ A2 @ N2 ) @ ( power_7079662738309270450atural @ A2 @ N ) ) ) ) ).
% 14.25/8.12
% 14.25/8.12 thf(fact_6315_power__increasing,axiom,
% 14.25/8.12 ! [N2: nat,N: nat,A2: rat] :
% 14.25/8.12 ( ( ord_less_eq_nat @ N2 @ N )
% 14.25/8.12 => ( ( ord_less_eq_rat @ one_one_rat @ A2 )
% 14.25/8.12 => ( ord_less_eq_rat @ ( power_power_rat @ A2 @ N2 ) @ ( power_power_rat @ A2 @ N ) ) ) ) ).
% 14.25/8.12
% 14.25/8.12 thf(fact_6316_power__increasing,axiom,
% 14.25/8.12 ! [N2: nat,N: nat,A2: nat] :
% 14.25/8.12 ( ( ord_less_eq_nat @ N2 @ N )
% 14.25/8.12 => ( ( ord_less_eq_nat @ one_one_nat @ A2 )
% 14.25/8.12 => ( ord_less_eq_nat @ ( power_power_nat @ A2 @ N2 ) @ ( power_power_nat @ A2 @ N ) ) ) ) ).
% 14.25/8.12
% 14.25/8.12 thf(fact_6317_power__increasing,axiom,
% 14.25/8.12 ! [N2: nat,N: nat,A2: int] :
% 14.25/8.12 ( ( ord_less_eq_nat @ N2 @ N )
% 14.25/8.12 => ( ( ord_less_eq_int @ one_one_int @ A2 )
% 14.25/8.12 => ( ord_less_eq_int @ ( power_power_int @ A2 @ N2 ) @ ( power_power_int @ A2 @ N ) ) ) ) ).
% 14.25/8.12
% 14.25/8.12 thf(fact_6318_inf__period_I3_J,axiom,
% 14.25/8.12 ! [D2: code_integer,D3: code_integer,T4: code_integer] :
% 14.25/8.12 ( ( dvd_dvd_Code_integer @ D2 @ D3 )
% 14.25/8.12 => ! [X6: code_integer,K2: code_integer] :
% 14.25/8.12 ( ( dvd_dvd_Code_integer @ D2 @ ( plus_p5714425477246183910nteger @ X6 @ T4 ) )
% 14.25/8.12 = ( dvd_dvd_Code_integer @ D2 @ ( plus_p5714425477246183910nteger @ ( minus_8373710615458151222nteger @ X6 @ ( times_3573771949741848930nteger @ K2 @ D3 ) ) @ T4 ) ) ) ) ).
% 14.25/8.12
% 14.25/8.12 thf(fact_6319_inf__period_I3_J,axiom,
% 14.25/8.12 ! [D2: rat,D3: rat,T4: rat] :
% 14.25/8.12 ( ( dvd_dvd_rat @ D2 @ D3 )
% 14.25/8.12 => ! [X6: rat,K2: rat] :
% 14.25/8.12 ( ( dvd_dvd_rat @ D2 @ ( plus_plus_rat @ X6 @ T4 ) )
% 14.25/8.12 = ( dvd_dvd_rat @ D2 @ ( plus_plus_rat @ ( minus_minus_rat @ X6 @ ( times_times_rat @ K2 @ D3 ) ) @ T4 ) ) ) ) ).
% 14.25/8.12
% 14.25/8.12 thf(fact_6320_inf__period_I3_J,axiom,
% 14.25/8.12 ! [D2: int,D3: int,T4: int] :
% 14.25/8.12 ( ( dvd_dvd_int @ D2 @ D3 )
% 14.25/8.12 => ! [X6: int,K2: int] :
% 14.25/8.12 ( ( dvd_dvd_int @ D2 @ ( plus_plus_int @ X6 @ T4 ) )
% 14.25/8.12 = ( dvd_dvd_int @ D2 @ ( plus_plus_int @ ( minus_minus_int @ X6 @ ( times_times_int @ K2 @ D3 ) ) @ T4 ) ) ) ) ).
% 14.25/8.12
% 14.25/8.12 thf(fact_6321_inf__period_I4_J,axiom,
% 14.25/8.12 ! [D2: code_integer,D3: code_integer,T4: code_integer] :
% 14.25/8.12 ( ( dvd_dvd_Code_integer @ D2 @ D3 )
% 14.25/8.12 => ! [X6: code_integer,K2: code_integer] :
% 14.25/8.12 ( ( ~ ( dvd_dvd_Code_integer @ D2 @ ( plus_p5714425477246183910nteger @ X6 @ T4 ) ) )
% 14.25/8.12 = ( ~ ( dvd_dvd_Code_integer @ D2 @ ( plus_p5714425477246183910nteger @ ( minus_8373710615458151222nteger @ X6 @ ( times_3573771949741848930nteger @ K2 @ D3 ) ) @ T4 ) ) ) ) ) ).
% 14.25/8.12
% 14.25/8.12 thf(fact_6322_inf__period_I4_J,axiom,
% 14.25/8.12 ! [D2: rat,D3: rat,T4: rat] :
% 14.25/8.12 ( ( dvd_dvd_rat @ D2 @ D3 )
% 14.25/8.12 => ! [X6: rat,K2: rat] :
% 14.25/8.12 ( ( ~ ( dvd_dvd_rat @ D2 @ ( plus_plus_rat @ X6 @ T4 ) ) )
% 14.25/8.12 = ( ~ ( dvd_dvd_rat @ D2 @ ( plus_plus_rat @ ( minus_minus_rat @ X6 @ ( times_times_rat @ K2 @ D3 ) ) @ T4 ) ) ) ) ) ).
% 14.25/8.12
% 14.25/8.12 thf(fact_6323_inf__period_I4_J,axiom,
% 14.25/8.12 ! [D2: int,D3: int,T4: int] :
% 14.25/8.12 ( ( dvd_dvd_int @ D2 @ D3 )
% 14.25/8.12 => ! [X6: int,K2: int] :
% 14.25/8.12 ( ( ~ ( dvd_dvd_int @ D2 @ ( plus_plus_int @ X6 @ T4 ) ) )
% 14.25/8.12 = ( ~ ( dvd_dvd_int @ D2 @ ( plus_plus_int @ ( minus_minus_int @ X6 @ ( times_times_int @ K2 @ D3 ) ) @ T4 ) ) ) ) ) ).
% 14.25/8.12
% 14.25/8.12 thf(fact_6324_cInf__mono,axiom,
% 14.25/8.12 ! [B4: set_o,A: set_o] :
% 14.25/8.12 ( ( B4 != bot_bot_set_o )
% 14.25/8.12 => ( ( condit5413489452508810728elow_o @ A )
% 14.25/8.12 => ( ! [B2: $o] :
% 14.25/8.12 ( ( member_o @ B2 @ B4 )
% 14.25/8.12 => ? [X6: $o] :
% 14.25/8.12 ( ( member_o @ X6 @ A )
% 14.25/8.12 & ( ord_less_eq_o @ X6 @ B2 ) ) )
% 14.25/8.12 => ( ord_less_eq_o @ ( complete_Inf_Inf_o @ A ) @ ( complete_Inf_Inf_o @ B4 ) ) ) ) ) ).
% 14.25/8.12
% 14.25/8.12 thf(fact_6325_cInf__mono,axiom,
% 14.25/8.12 ! [B4: set_Pr4532377907799695533_nat_o,A: set_Pr4532377907799695533_nat_o] :
% 14.25/8.12 ( ( B4 != bot_bo7824918357723582233_nat_o )
% 14.25/8.12 => ( ( condit8399017556247434269_nat_o @ A )
% 14.25/8.12 => ( ! [B2: produc3658429121746597890et_nat > $o] :
% 14.25/8.12 ( ( member6576561426505652726_nat_o @ B2 @ B4 )
% 14.25/8.12 => ? [X6: produc3658429121746597890et_nat > $o] :
% 14.25/8.12 ( ( member6576561426505652726_nat_o @ X6 @ A )
% 14.25/8.12 & ( ord_le729326519192465773_nat_o @ X6 @ B2 ) ) )
% 14.25/8.12 => ( ord_le729326519192465773_nat_o @ ( comple90455603224345172_nat_o @ A ) @ ( comple90455603224345172_nat_o @ B4 ) ) ) ) ) ).
% 14.25/8.12
% 14.25/8.12 thf(fact_6326_cInf__mono,axiom,
% 14.25/8.12 ! [B4: set_filter_nat,A: set_filter_nat] :
% 14.25/8.12 ( ( B4 != bot_bo498966703094740906er_nat )
% 14.25/8.12 => ( ( condit3133331395780078992er_nat @ A )
% 14.25/8.12 => ( ! [B2: filter_nat] :
% 14.25/8.12 ( ( member_filter_nat @ B2 @ B4 )
% 14.25/8.12 => ? [X6: filter_nat] :
% 14.25/8.12 ( ( member_filter_nat @ X6 @ A )
% 14.25/8.12 & ( ord_le2510731241096832064er_nat @ X6 @ B2 ) ) )
% 14.25/8.12 => ( ord_le2510731241096832064er_nat @ ( comple6612786467464916249er_nat @ A ) @ ( comple6612786467464916249er_nat @ B4 ) ) ) ) ) ).
% 14.25/8.12
% 14.25/8.12 thf(fact_6327_cInf__mono,axiom,
% 14.25/8.12 ! [B4: set_int,A: set_int] :
% 14.25/8.12 ( ( B4 != bot_bot_set_int )
% 14.25/8.12 => ( ( condit1735850657277959132ow_int @ A )
% 14.25/8.12 => ( ! [B2: int] :
% 14.25/8.12 ( ( member_int @ B2 @ B4 )
% 14.25/8.12 => ? [X6: int] :
% 14.25/8.12 ( ( member_int @ X6 @ A )
% 14.25/8.12 & ( ord_less_eq_int @ X6 @ B2 ) ) )
% 14.25/8.12 => ( ord_less_eq_int @ ( complete_Inf_Inf_int @ A ) @ ( complete_Inf_Inf_int @ B4 ) ) ) ) ) ).
% 14.25/8.12
% 14.25/8.12 thf(fact_6328_cInf__mono,axiom,
% 14.25/8.12 ! [B4: set_nat,A: set_nat] :
% 14.25/8.12 ( ( B4 != bot_bot_set_nat )
% 14.25/8.12 => ( ( condit1738341127787009408ow_nat @ A )
% 14.25/8.12 => ( ! [B2: nat] :
% 14.25/8.12 ( ( member_nat @ B2 @ B4 )
% 14.25/8.12 => ? [X6: nat] :
% 14.25/8.12 ( ( member_nat @ X6 @ A )
% 14.25/8.12 & ( ord_less_eq_nat @ X6 @ B2 ) ) )
% 14.25/8.12 => ( ord_less_eq_nat @ ( complete_Inf_Inf_nat @ A ) @ ( complete_Inf_Inf_nat @ B4 ) ) ) ) ) ).
% 14.25/8.12
% 14.25/8.12 thf(fact_6329_cInf__mono,axiom,
% 14.25/8.12 ! [B4: set_set_nat,A: set_set_nat] :
% 14.25/8.12 ( ( B4 != bot_bot_set_set_nat )
% 14.25/8.12 => ( ( condit68592940725977398et_nat @ A )
% 14.25/8.12 => ( ! [B2: set_nat] :
% 14.25/8.12 ( ( member_set_nat @ B2 @ B4 )
% 14.25/8.12 => ? [X6: set_nat] :
% 14.25/8.12 ( ( member_set_nat @ X6 @ A )
% 14.25/8.12 & ( ord_less_eq_set_nat @ X6 @ B2 ) ) )
% 14.25/8.12 => ( ord_less_eq_set_nat @ ( comple7806235888213564991et_nat @ A ) @ ( comple7806235888213564991et_nat @ B4 ) ) ) ) ) ).
% 14.25/8.12
% 14.25/8.12 thf(fact_6330_cInf__mono,axiom,
% 14.25/8.12 ! [B4: set_Product_unit,A: set_Product_unit] :
% 14.25/8.12 ( ( B4 != bot_bo3957492148770167129t_unit )
% 14.25/8.12 => ( ( condit9161856309968323677t_unit @ A )
% 14.25/8.12 => ( ! [B2: product_unit] :
% 14.25/8.12 ( ( member_Product_unit @ B2 @ B4 )
% 14.25/8.12 => ? [X6: product_unit] :
% 14.25/8.12 ( ( member_Product_unit @ X6 @ A )
% 14.25/8.12 & ( ord_le3221252021190050221t_unit @ X6 @ B2 ) ) )
% 14.25/8.12 => ( ord_le3221252021190050221t_unit @ ( comple2584293577114468500t_unit @ A ) @ ( comple2584293577114468500t_unit @ B4 ) ) ) ) ) ).
% 14.25/8.12
% 14.25/8.12 thf(fact_6331_le__cInf__iff,axiom,
% 14.25/8.12 ! [S: set_o,A2: $o] :
% 14.25/8.12 ( ( S != bot_bot_set_o )
% 14.25/8.12 => ( ( condit5413489452508810728elow_o @ S )
% 14.25/8.12 => ( ( ord_less_eq_o @ A2 @ ( complete_Inf_Inf_o @ S ) )
% 14.25/8.12 = ( ! [X2: $o] :
% 14.25/8.12 ( ( member_o @ X2 @ S )
% 14.25/8.12 => ( ord_less_eq_o @ A2 @ X2 ) ) ) ) ) ) ).
% 14.25/8.12
% 14.25/8.12 thf(fact_6332_le__cInf__iff,axiom,
% 14.25/8.12 ! [S: set_Pr4532377907799695533_nat_o,A2: produc3658429121746597890et_nat > $o] :
% 14.25/8.12 ( ( S != bot_bo7824918357723582233_nat_o )
% 14.25/8.12 => ( ( condit8399017556247434269_nat_o @ S )
% 14.25/8.12 => ( ( ord_le729326519192465773_nat_o @ A2 @ ( comple90455603224345172_nat_o @ S ) )
% 14.25/8.12 = ( ! [X2: produc3658429121746597890et_nat > $o] :
% 14.25/8.12 ( ( member6576561426505652726_nat_o @ X2 @ S )
% 14.25/8.12 => ( ord_le729326519192465773_nat_o @ A2 @ X2 ) ) ) ) ) ) ).
% 14.25/8.12
% 14.25/8.12 thf(fact_6333_le__cInf__iff,axiom,
% 14.25/8.12 ! [S: set_filter_nat,A2: filter_nat] :
% 14.25/8.12 ( ( S != bot_bo498966703094740906er_nat )
% 14.25/8.12 => ( ( condit3133331395780078992er_nat @ S )
% 14.25/8.12 => ( ( ord_le2510731241096832064er_nat @ A2 @ ( comple6612786467464916249er_nat @ S ) )
% 14.25/8.12 = ( ! [X2: filter_nat] :
% 14.25/8.12 ( ( member_filter_nat @ X2 @ S )
% 14.25/8.12 => ( ord_le2510731241096832064er_nat @ A2 @ X2 ) ) ) ) ) ) ).
% 14.25/8.12
% 14.25/8.12 thf(fact_6334_le__cInf__iff,axiom,
% 14.25/8.12 ! [S: set_int,A2: int] :
% 14.25/8.12 ( ( S != bot_bot_set_int )
% 14.25/8.12 => ( ( condit1735850657277959132ow_int @ S )
% 14.25/8.12 => ( ( ord_less_eq_int @ A2 @ ( complete_Inf_Inf_int @ S ) )
% 14.25/8.12 = ( ! [X2: int] :
% 14.25/8.12 ( ( member_int @ X2 @ S )
% 14.25/8.12 => ( ord_less_eq_int @ A2 @ X2 ) ) ) ) ) ) ).
% 14.25/8.12
% 14.25/8.12 thf(fact_6335_le__cInf__iff,axiom,
% 14.25/8.12 ! [S: set_nat,A2: nat] :
% 14.25/8.12 ( ( S != bot_bot_set_nat )
% 14.25/8.12 => ( ( condit1738341127787009408ow_nat @ S )
% 14.25/8.12 => ( ( ord_less_eq_nat @ A2 @ ( complete_Inf_Inf_nat @ S ) )
% 14.25/8.12 = ( ! [X2: nat] :
% 14.25/8.12 ( ( member_nat @ X2 @ S )
% 14.25/8.12 => ( ord_less_eq_nat @ A2 @ X2 ) ) ) ) ) ) ).
% 14.25/8.12
% 14.25/8.12 thf(fact_6336_le__cInf__iff,axiom,
% 14.25/8.12 ! [S: set_set_nat,A2: set_nat] :
% 14.25/8.12 ( ( S != bot_bot_set_set_nat )
% 14.25/8.12 => ( ( condit68592940725977398et_nat @ S )
% 14.25/8.12 => ( ( ord_less_eq_set_nat @ A2 @ ( comple7806235888213564991et_nat @ S ) )
% 14.25/8.12 = ( ! [X2: set_nat] :
% 14.25/8.12 ( ( member_set_nat @ X2 @ S )
% 14.25/8.12 => ( ord_less_eq_set_nat @ A2 @ X2 ) ) ) ) ) ) ).
% 14.25/8.12
% 14.25/8.12 thf(fact_6337_le__cInf__iff,axiom,
% 14.25/8.12 ! [S: set_Product_unit,A2: product_unit] :
% 14.25/8.12 ( ( S != bot_bo3957492148770167129t_unit )
% 14.25/8.12 => ( ( condit9161856309968323677t_unit @ S )
% 14.25/8.12 => ( ( ord_le3221252021190050221t_unit @ A2 @ ( comple2584293577114468500t_unit @ S ) )
% 14.25/8.12 = ( ! [X2: product_unit] :
% 14.25/8.12 ( ( member_Product_unit @ X2 @ S )
% 14.25/8.12 => ( ord_le3221252021190050221t_unit @ A2 @ X2 ) ) ) ) ) ) ).
% 14.25/8.12
% 14.25/8.12 thf(fact_6338_cInf__less__iff,axiom,
% 14.25/8.12 ! [X4: set_int,Y: int] :
% 14.25/8.12 ( ( X4 != bot_bot_set_int )
% 14.25/8.12 => ( ( condit1735850657277959132ow_int @ X4 )
% 14.25/8.12 => ( ( ord_less_int @ ( complete_Inf_Inf_int @ X4 ) @ Y )
% 14.25/8.12 = ( ? [X2: int] :
% 14.25/8.12 ( ( member_int @ X2 @ X4 )
% 14.25/8.12 & ( ord_less_int @ X2 @ Y ) ) ) ) ) ) ).
% 14.25/8.12
% 14.25/8.12 thf(fact_6339_cInf__less__iff,axiom,
% 14.25/8.12 ! [X4: set_nat,Y: nat] :
% 14.25/8.12 ( ( X4 != bot_bot_set_nat )
% 14.25/8.12 => ( ( condit1738341127787009408ow_nat @ X4 )
% 14.25/8.12 => ( ( ord_less_nat @ ( complete_Inf_Inf_nat @ X4 ) @ Y )
% 14.25/8.12 = ( ? [X2: nat] :
% 14.25/8.12 ( ( member_nat @ X2 @ X4 )
% 14.25/8.12 & ( ord_less_nat @ X2 @ Y ) ) ) ) ) ) ).
% 14.25/8.12
% 14.25/8.12 thf(fact_6340_cInf__less__iff,axiom,
% 14.25/8.12 ! [X4: set_Product_unit,Y: product_unit] :
% 14.25/8.12 ( ( X4 != bot_bo3957492148770167129t_unit )
% 14.25/8.12 => ( ( condit9161856309968323677t_unit @ X4 )
% 14.25/8.12 => ( ( ord_le361264281704409273t_unit @ ( comple2584293577114468500t_unit @ X4 ) @ Y )
% 14.25/8.12 = ( ? [X2: product_unit] :
% 14.25/8.12 ( ( member_Product_unit @ X2 @ X4 )
% 14.25/8.12 & ( ord_le361264281704409273t_unit @ X2 @ Y ) ) ) ) ) ) ).
% 14.25/8.12
% 14.25/8.12 thf(fact_6341_dvd__mult__cancel,axiom,
% 14.25/8.12 ! [K3: nat,M4: nat,N2: nat] :
% 14.25/8.12 ( ( dvd_dvd_nat @ ( times_times_nat @ K3 @ M4 ) @ ( times_times_nat @ K3 @ N2 ) )
% 14.25/8.12 => ( ( ord_less_nat @ zero_zero_nat @ K3 )
% 14.25/8.12 => ( dvd_dvd_nat @ M4 @ N2 ) ) ) ).
% 14.25/8.12
% 14.25/8.12 thf(fact_6342_nat__mult__dvd__cancel1,axiom,
% 14.25/8.12 ! [K3: nat,M4: nat,N2: nat] :
% 14.25/8.12 ( ( ord_less_nat @ zero_zero_nat @ K3 )
% 14.25/8.12 => ( ( dvd_dvd_nat @ ( times_times_nat @ K3 @ M4 ) @ ( times_times_nat @ K3 @ N2 ) )
% 14.25/8.12 = ( dvd_dvd_nat @ M4 @ N2 ) ) ) ).
% 14.25/8.12
% 14.25/8.12 thf(fact_6343_power__Suc__less,axiom,
% 14.25/8.12 ! [A2: code_natural,N2: nat] :
% 14.25/8.12 ( ( ord_le5570908160329646204atural @ zero_z2226904508553997617atural @ A2 )
% 14.25/8.12 => ( ( ord_le5570908160329646204atural @ A2 @ one_one_Code_natural )
% 14.25/8.12 => ( ord_le5570908160329646204atural @ ( times_2397367101498566445atural @ A2 @ ( power_7079662738309270450atural @ A2 @ N2 ) ) @ ( power_7079662738309270450atural @ A2 @ N2 ) ) ) ) ).
% 14.25/8.12
% 14.25/8.12 thf(fact_6344_power__Suc__less,axiom,
% 14.25/8.12 ! [A2: nat,N2: nat] :
% 14.25/8.12 ( ( ord_less_nat @ zero_zero_nat @ A2 )
% 14.25/8.12 => ( ( ord_less_nat @ A2 @ one_one_nat )
% 14.25/8.12 => ( ord_less_nat @ ( times_times_nat @ A2 @ ( power_power_nat @ A2 @ N2 ) ) @ ( power_power_nat @ A2 @ N2 ) ) ) ) ).
% 14.25/8.12
% 14.25/8.12 thf(fact_6345_power__Suc__less,axiom,
% 14.25/8.12 ! [A2: int,N2: nat] :
% 14.25/8.12 ( ( ord_less_int @ zero_zero_int @ A2 )
% 14.25/8.12 => ( ( ord_less_int @ A2 @ one_one_int )
% 14.25/8.12 => ( ord_less_int @ ( times_times_int @ A2 @ ( power_power_int @ A2 @ N2 ) ) @ ( power_power_int @ A2 @ N2 ) ) ) ) ).
% 14.25/8.12
% 14.25/8.12 thf(fact_6346_power__Suc__less,axiom,
% 14.25/8.12 ! [A2: code_integer,N2: nat] :
% 14.25/8.12 ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ A2 )
% 14.25/8.12 => ( ( ord_le6747313008572928689nteger @ A2 @ one_one_Code_integer )
% 14.25/8.12 => ( ord_le6747313008572928689nteger @ ( times_3573771949741848930nteger @ A2 @ ( power_8256067586552552935nteger @ A2 @ N2 ) ) @ ( power_8256067586552552935nteger @ A2 @ N2 ) ) ) ) ).
% 14.25/8.12
% 14.25/8.12 thf(fact_6347_power__Suc__less,axiom,
% 14.25/8.12 ! [A2: rat,N2: nat] :
% 14.25/8.12 ( ( ord_less_rat @ zero_zero_rat @ A2 )
% 14.25/8.12 => ( ( ord_less_rat @ A2 @ one_one_rat )
% 14.25/8.12 => ( ord_less_rat @ ( times_times_rat @ A2 @ ( power_power_rat @ A2 @ N2 ) ) @ ( power_power_rat @ A2 @ N2 ) ) ) ) ).
% 14.25/8.12
% 14.25/8.12 thf(fact_6348_power__Suc__le__self,axiom,
% 14.25/8.12 ! [A2: code_natural,N2: nat] :
% 14.25/8.12 ( ( ord_le1926595141338095240atural @ zero_z2226904508553997617atural @ A2 )
% 14.25/8.12 => ( ( ord_le1926595141338095240atural @ A2 @ one_one_Code_natural )
% 14.25/8.12 => ( ord_le1926595141338095240atural @ ( power_7079662738309270450atural @ A2 @ ( suc @ N2 ) ) @ A2 ) ) ) ).
% 14.25/8.12
% 14.25/8.12 thf(fact_6349_power__Suc__le__self,axiom,
% 14.25/8.12 ! [A2: code_integer,N2: nat] :
% 14.25/8.12 ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ A2 )
% 14.25/8.12 => ( ( ord_le3102999989581377725nteger @ A2 @ one_one_Code_integer )
% 14.25/8.12 => ( ord_le3102999989581377725nteger @ ( power_8256067586552552935nteger @ A2 @ ( suc @ N2 ) ) @ A2 ) ) ) ).
% 14.25/8.12
% 14.25/8.12 thf(fact_6350_power__Suc__le__self,axiom,
% 14.25/8.12 ! [A2: rat,N2: nat] :
% 14.25/8.12 ( ( ord_less_eq_rat @ zero_zero_rat @ A2 )
% 14.25/8.12 => ( ( ord_less_eq_rat @ A2 @ one_one_rat )
% 14.25/8.12 => ( ord_less_eq_rat @ ( power_power_rat @ A2 @ ( suc @ N2 ) ) @ A2 ) ) ) ).
% 14.25/8.12
% 14.25/8.12 thf(fact_6351_power__Suc__le__self,axiom,
% 14.25/8.12 ! [A2: nat,N2: nat] :
% 14.25/8.12 ( ( ord_less_eq_nat @ zero_zero_nat @ A2 )
% 14.25/8.12 => ( ( ord_less_eq_nat @ A2 @ one_one_nat )
% 14.25/8.12 => ( ord_less_eq_nat @ ( power_power_nat @ A2 @ ( suc @ N2 ) ) @ A2 ) ) ) ).
% 14.25/8.12
% 14.25/8.12 thf(fact_6352_power__Suc__le__self,axiom,
% 14.25/8.12 ! [A2: int,N2: nat] :
% 14.25/8.12 ( ( ord_less_eq_int @ zero_zero_int @ A2 )
% 14.25/8.12 => ( ( ord_less_eq_int @ A2 @ one_one_int )
% 14.25/8.12 => ( ord_less_eq_int @ ( power_power_int @ A2 @ ( suc @ N2 ) ) @ A2 ) ) ) ).
% 14.25/8.12
% 14.25/8.12 thf(fact_6353_power__Suc__less__one,axiom,
% 14.25/8.12 ! [A2: code_natural,N2: nat] :
% 14.25/8.12 ( ( ord_le5570908160329646204atural @ zero_z2226904508553997617atural @ A2 )
% 14.25/8.12 => ( ( ord_le5570908160329646204atural @ A2 @ one_one_Code_natural )
% 14.25/8.12 => ( ord_le5570908160329646204atural @ ( power_7079662738309270450atural @ A2 @ ( suc @ N2 ) ) @ one_one_Code_natural ) ) ) ).
% 14.25/8.12
% 14.25/8.12 thf(fact_6354_power__Suc__less__one,axiom,
% 14.25/8.12 ! [A2: nat,N2: nat] :
% 14.25/8.12 ( ( ord_less_nat @ zero_zero_nat @ A2 )
% 14.25/8.12 => ( ( ord_less_nat @ A2 @ one_one_nat )
% 14.25/8.12 => ( ord_less_nat @ ( power_power_nat @ A2 @ ( suc @ N2 ) ) @ one_one_nat ) ) ) ).
% 14.25/8.12
% 14.25/8.12 thf(fact_6355_power__Suc__less__one,axiom,
% 14.25/8.12 ! [A2: int,N2: nat] :
% 14.25/8.12 ( ( ord_less_int @ zero_zero_int @ A2 )
% 14.25/8.12 => ( ( ord_less_int @ A2 @ one_one_int )
% 14.25/8.12 => ( ord_less_int @ ( power_power_int @ A2 @ ( suc @ N2 ) ) @ one_one_int ) ) ) ).
% 14.25/8.12
% 14.25/8.12 thf(fact_6356_power__Suc__less__one,axiom,
% 14.25/8.12 ! [A2: code_integer,N2: nat] :
% 14.25/8.12 ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ A2 )
% 14.25/8.12 => ( ( ord_le6747313008572928689nteger @ A2 @ one_one_Code_integer )
% 14.25/8.12 => ( ord_le6747313008572928689nteger @ ( power_8256067586552552935nteger @ A2 @ ( suc @ N2 ) ) @ one_one_Code_integer ) ) ) ).
% 14.25/8.12
% 14.25/8.12 thf(fact_6357_power__Suc__less__one,axiom,
% 14.25/8.12 ! [A2: rat,N2: nat] :
% 14.25/8.12 ( ( ord_less_rat @ zero_zero_rat @ A2 )
% 14.25/8.12 => ( ( ord_less_rat @ A2 @ one_one_rat )
% 14.25/8.12 => ( ord_less_rat @ ( power_power_rat @ A2 @ ( suc @ N2 ) ) @ one_one_rat ) ) ) ).
% 14.25/8.12
% 14.25/8.12 thf(fact_6358_power__strict__decreasing,axiom,
% 14.25/8.12 ! [N2: nat,N: nat,A2: code_natural] :
% 14.25/8.12 ( ( ord_less_nat @ N2 @ N )
% 14.25/8.12 => ( ( ord_le5570908160329646204atural @ zero_z2226904508553997617atural @ A2 )
% 14.25/8.12 => ( ( ord_le5570908160329646204atural @ A2 @ one_one_Code_natural )
% 14.25/8.12 => ( ord_le5570908160329646204atural @ ( power_7079662738309270450atural @ A2 @ N ) @ ( power_7079662738309270450atural @ A2 @ N2 ) ) ) ) ) ).
% 14.25/8.12
% 14.25/8.12 thf(fact_6359_power__strict__decreasing,axiom,
% 14.25/8.12 ! [N2: nat,N: nat,A2: nat] :
% 14.25/8.12 ( ( ord_less_nat @ N2 @ N )
% 14.25/8.12 => ( ( ord_less_nat @ zero_zero_nat @ A2 )
% 14.25/8.12 => ( ( ord_less_nat @ A2 @ one_one_nat )
% 14.25/8.12 => ( ord_less_nat @ ( power_power_nat @ A2 @ N ) @ ( power_power_nat @ A2 @ N2 ) ) ) ) ) ).
% 14.25/8.12
% 14.25/8.12 thf(fact_6360_power__strict__decreasing,axiom,
% 14.25/8.12 ! [N2: nat,N: nat,A2: int] :
% 14.25/8.12 ( ( ord_less_nat @ N2 @ N )
% 14.25/8.12 => ( ( ord_less_int @ zero_zero_int @ A2 )
% 14.25/8.12 => ( ( ord_less_int @ A2 @ one_one_int )
% 14.25/8.12 => ( ord_less_int @ ( power_power_int @ A2 @ N ) @ ( power_power_int @ A2 @ N2 ) ) ) ) ) ).
% 14.25/8.12
% 14.25/8.12 thf(fact_6361_power__strict__decreasing,axiom,
% 14.25/8.12 ! [N2: nat,N: nat,A2: code_integer] :
% 14.25/8.12 ( ( ord_less_nat @ N2 @ N )
% 14.25/8.12 => ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ A2 )
% 14.25/8.12 => ( ( ord_le6747313008572928689nteger @ A2 @ one_one_Code_integer )
% 14.25/8.12 => ( ord_le6747313008572928689nteger @ ( power_8256067586552552935nteger @ A2 @ N ) @ ( power_8256067586552552935nteger @ A2 @ N2 ) ) ) ) ) ).
% 14.25/8.12
% 14.25/8.12 thf(fact_6362_power__strict__decreasing,axiom,
% 14.25/8.12 ! [N2: nat,N: nat,A2: rat] :
% 14.25/8.12 ( ( ord_less_nat @ N2 @ N )
% 14.25/8.12 => ( ( ord_less_rat @ zero_zero_rat @ A2 )
% 14.25/8.12 => ( ( ord_less_rat @ A2 @ one_one_rat )
% 14.25/8.12 => ( ord_less_rat @ ( power_power_rat @ A2 @ N ) @ ( power_power_rat @ A2 @ N2 ) ) ) ) ) ).
% 14.25/8.12
% 14.25/8.12 thf(fact_6363_power__decreasing,axiom,
% 14.25/8.12 ! [N2: nat,N: nat,A2: code_natural] :
% 14.25/8.12 ( ( ord_less_eq_nat @ N2 @ N )
% 14.25/8.12 => ( ( ord_le1926595141338095240atural @ zero_z2226904508553997617atural @ A2 )
% 14.25/8.12 => ( ( ord_le1926595141338095240atural @ A2 @ one_one_Code_natural )
% 14.25/8.12 => ( ord_le1926595141338095240atural @ ( power_7079662738309270450atural @ A2 @ N ) @ ( power_7079662738309270450atural @ A2 @ N2 ) ) ) ) ) ).
% 14.25/8.12
% 14.25/8.12 thf(fact_6364_power__decreasing,axiom,
% 14.25/8.12 ! [N2: nat,N: nat,A2: code_integer] :
% 14.25/8.12 ( ( ord_less_eq_nat @ N2 @ N )
% 14.25/8.12 => ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ A2 )
% 14.25/8.12 => ( ( ord_le3102999989581377725nteger @ A2 @ one_one_Code_integer )
% 14.25/8.12 => ( ord_le3102999989581377725nteger @ ( power_8256067586552552935nteger @ A2 @ N ) @ ( power_8256067586552552935nteger @ A2 @ N2 ) ) ) ) ) ).
% 14.25/8.12
% 14.25/8.12 thf(fact_6365_power__decreasing,axiom,
% 14.25/8.12 ! [N2: nat,N: nat,A2: rat] :
% 14.25/8.12 ( ( ord_less_eq_nat @ N2 @ N )
% 14.25/8.12 => ( ( ord_less_eq_rat @ zero_zero_rat @ A2 )
% 14.25/8.12 => ( ( ord_less_eq_rat @ A2 @ one_one_rat )
% 14.25/8.12 => ( ord_less_eq_rat @ ( power_power_rat @ A2 @ N ) @ ( power_power_rat @ A2 @ N2 ) ) ) ) ) ).
% 14.25/8.12
% 14.25/8.12 thf(fact_6366_power__decreasing,axiom,
% 14.25/8.12 ! [N2: nat,N: nat,A2: nat] :
% 14.25/8.12 ( ( ord_less_eq_nat @ N2 @ N )
% 14.25/8.12 => ( ( ord_less_eq_nat @ zero_zero_nat @ A2 )
% 14.25/8.12 => ( ( ord_less_eq_nat @ A2 @ one_one_nat )
% 14.25/8.12 => ( ord_less_eq_nat @ ( power_power_nat @ A2 @ N ) @ ( power_power_nat @ A2 @ N2 ) ) ) ) ) ).
% 14.25/8.12
% 14.25/8.12 thf(fact_6367_power__decreasing,axiom,
% 14.25/8.12 ! [N2: nat,N: nat,A2: int] :
% 14.25/8.12 ( ( ord_less_eq_nat @ N2 @ N )
% 14.25/8.12 => ( ( ord_less_eq_int @ zero_zero_int @ A2 )
% 14.25/8.12 => ( ( ord_less_eq_int @ A2 @ one_one_int )
% 14.25/8.12 => ( ord_less_eq_int @ ( power_power_int @ A2 @ N ) @ ( power_power_int @ A2 @ N2 ) ) ) ) ) ).
% 14.25/8.12
% 14.25/8.12 thf(fact_6368_power__le__imp__le__exp,axiom,
% 14.25/8.12 ! [A2: code_natural,M4: nat,N2: nat] :
% 14.25/8.12 ( ( ord_le5570908160329646204atural @ one_one_Code_natural @ A2 )
% 14.25/8.12 => ( ( ord_le1926595141338095240atural @ ( power_7079662738309270450atural @ A2 @ M4 ) @ ( power_7079662738309270450atural @ A2 @ N2 ) )
% 14.25/8.12 => ( ord_less_eq_nat @ M4 @ N2 ) ) ) ).
% 14.25/8.12
% 14.25/8.12 thf(fact_6369_power__le__imp__le__exp,axiom,
% 14.25/8.12 ! [A2: code_integer,M4: nat,N2: nat] :
% 14.25/8.12 ( ( ord_le6747313008572928689nteger @ one_one_Code_integer @ A2 )
% 14.25/8.12 => ( ( ord_le3102999989581377725nteger @ ( power_8256067586552552935nteger @ A2 @ M4 ) @ ( power_8256067586552552935nteger @ A2 @ N2 ) )
% 14.25/8.12 => ( ord_less_eq_nat @ M4 @ N2 ) ) ) ).
% 14.25/8.12
% 14.25/8.12 thf(fact_6370_power__le__imp__le__exp,axiom,
% 14.25/8.12 ! [A2: rat,M4: nat,N2: nat] :
% 14.25/8.12 ( ( ord_less_rat @ one_one_rat @ A2 )
% 14.25/8.12 => ( ( ord_less_eq_rat @ ( power_power_rat @ A2 @ M4 ) @ ( power_power_rat @ A2 @ N2 ) )
% 14.25/8.12 => ( ord_less_eq_nat @ M4 @ N2 ) ) ) ).
% 14.25/8.12
% 14.25/8.12 thf(fact_6371_power__le__imp__le__exp,axiom,
% 14.25/8.12 ! [A2: nat,M4: nat,N2: nat] :
% 14.25/8.12 ( ( ord_less_nat @ one_one_nat @ A2 )
% 14.25/8.12 => ( ( ord_less_eq_nat @ ( power_power_nat @ A2 @ M4 ) @ ( power_power_nat @ A2 @ N2 ) )
% 14.25/8.12 => ( ord_less_eq_nat @ M4 @ N2 ) ) ) ).
% 14.25/8.12
% 14.25/8.12 thf(fact_6372_power__le__imp__le__exp,axiom,
% 14.25/8.12 ! [A2: int,M4: nat,N2: nat] :
% 14.25/8.12 ( ( ord_less_int @ one_one_int @ A2 )
% 14.25/8.12 => ( ( ord_less_eq_int @ ( power_power_int @ A2 @ M4 ) @ ( power_power_int @ A2 @ N2 ) )
% 14.25/8.12 => ( ord_less_eq_nat @ M4 @ N2 ) ) ) ).
% 14.25/8.12
% 14.25/8.12 thf(fact_6373_self__le__power,axiom,
% 14.25/8.12 ! [A2: code_integer,N2: nat] :
% 14.25/8.12 ( ( ord_le3102999989581377725nteger @ one_one_Code_integer @ A2 )
% 14.25/8.12 => ( ( ord_less_nat @ zero_zero_nat @ N2 )
% 14.25/8.12 => ( ord_le3102999989581377725nteger @ A2 @ ( power_8256067586552552935nteger @ A2 @ N2 ) ) ) ) ).
% 14.25/8.12
% 14.25/8.12 thf(fact_6374_self__le__power,axiom,
% 14.25/8.12 ! [A2: code_natural,N2: nat] :
% 14.25/8.12 ( ( ord_le1926595141338095240atural @ one_one_Code_natural @ A2 )
% 14.25/8.12 => ( ( ord_less_nat @ zero_zero_nat @ N2 )
% 14.25/8.12 => ( ord_le1926595141338095240atural @ A2 @ ( power_7079662738309270450atural @ A2 @ N2 ) ) ) ) ).
% 14.25/8.12
% 14.25/8.12 thf(fact_6375_self__le__power,axiom,
% 14.25/8.12 ! [A2: rat,N2: nat] :
% 14.25/8.12 ( ( ord_less_eq_rat @ one_one_rat @ A2 )
% 14.25/8.12 => ( ( ord_less_nat @ zero_zero_nat @ N2 )
% 14.25/8.12 => ( ord_less_eq_rat @ A2 @ ( power_power_rat @ A2 @ N2 ) ) ) ) ).
% 14.25/8.12
% 14.25/8.12 thf(fact_6376_self__le__power,axiom,
% 14.25/8.12 ! [A2: nat,N2: nat] :
% 14.25/8.12 ( ( ord_less_eq_nat @ one_one_nat @ A2 )
% 14.25/8.12 => ( ( ord_less_nat @ zero_zero_nat @ N2 )
% 14.25/8.12 => ( ord_less_eq_nat @ A2 @ ( power_power_nat @ A2 @ N2 ) ) ) ) ).
% 14.25/8.12
% 14.25/8.12 thf(fact_6377_self__le__power,axiom,
% 14.25/8.12 ! [A2: int,N2: nat] :
% 14.25/8.12 ( ( ord_less_eq_int @ one_one_int @ A2 )
% 14.25/8.12 => ( ( ord_less_nat @ zero_zero_nat @ N2 )
% 14.25/8.12 => ( ord_less_eq_int @ A2 @ ( power_power_int @ A2 @ N2 ) ) ) ) ).
% 14.25/8.12
% 14.25/8.12 thf(fact_6378_one__less__power,axiom,
% 14.25/8.12 ! [A2: code_natural,N2: nat] :
% 14.25/8.12 ( ( ord_le5570908160329646204atural @ one_one_Code_natural @ A2 )
% 14.25/8.12 => ( ( ord_less_nat @ zero_zero_nat @ N2 )
% 14.25/8.12 => ( ord_le5570908160329646204atural @ one_one_Code_natural @ ( power_7079662738309270450atural @ A2 @ N2 ) ) ) ) ).
% 14.25/8.12
% 14.25/8.12 thf(fact_6379_one__less__power,axiom,
% 14.25/8.12 ! [A2: nat,N2: nat] :
% 14.25/8.12 ( ( ord_less_nat @ one_one_nat @ A2 )
% 14.25/8.12 => ( ( ord_less_nat @ zero_zero_nat @ N2 )
% 14.25/8.12 => ( ord_less_nat @ one_one_nat @ ( power_power_nat @ A2 @ N2 ) ) ) ) ).
% 14.25/8.12
% 14.25/8.12 thf(fact_6380_one__less__power,axiom,
% 14.25/8.12 ! [A2: int,N2: nat] :
% 14.25/8.12 ( ( ord_less_int @ one_one_int @ A2 )
% 14.25/8.12 => ( ( ord_less_nat @ zero_zero_nat @ N2 )
% 14.25/8.12 => ( ord_less_int @ one_one_int @ ( power_power_int @ A2 @ N2 ) ) ) ) ).
% 14.25/8.12
% 14.25/8.12 thf(fact_6381_one__less__power,axiom,
% 14.25/8.12 ! [A2: code_integer,N2: nat] :
% 14.25/8.12 ( ( ord_le6747313008572928689nteger @ one_one_Code_integer @ A2 )
% 14.25/8.12 => ( ( ord_less_nat @ zero_zero_nat @ N2 )
% 14.25/8.12 => ( ord_le6747313008572928689nteger @ one_one_Code_integer @ ( power_8256067586552552935nteger @ A2 @ N2 ) ) ) ) ).
% 14.25/8.12
% 14.25/8.12 thf(fact_6382_one__less__power,axiom,
% 14.25/8.12 ! [A2: rat,N2: nat] :
% 14.25/8.12 ( ( ord_less_rat @ one_one_rat @ A2 )
% 14.25/8.12 => ( ( ord_less_nat @ zero_zero_nat @ N2 )
% 14.25/8.12 => ( ord_less_rat @ one_one_rat @ ( power_power_rat @ A2 @ N2 ) ) ) ) ).
% 14.25/8.12
% 14.25/8.12 thf(fact_6383_cINF__mono,axiom,
% 14.25/8.12 ! [B4: set_Product_unit,F: nat > nat,A: set_nat,G: product_unit > nat] :
% 14.25/8.12 ( ( B4 != bot_bo3957492148770167129t_unit )
% 14.25/8.12 => ( ( condit1738341127787009408ow_nat @ ( image_nat_nat2 @ F @ A ) )
% 14.25/8.12 => ( ! [M2: product_unit] :
% 14.25/8.12 ( ( member_Product_unit @ M2 @ B4 )
% 14.25/8.12 => ? [X6: nat] :
% 14.25/8.12 ( ( member_nat @ X6 @ A )
% 14.25/8.12 & ( ord_less_eq_nat @ ( F @ X6 ) @ ( G @ M2 ) ) ) )
% 14.25/8.12 => ( ord_less_eq_nat @ ( complete_Inf_Inf_nat @ ( image_nat_nat2 @ F @ A ) ) @ ( complete_Inf_Inf_nat @ ( image_875570014554754200it_nat @ G @ B4 ) ) ) ) ) ) ).
% 14.25/8.12
% 14.25/8.12 thf(fact_6384_cINF__mono,axiom,
% 14.25/8.12 ! [B4: set_o,F: nat > nat,A: set_nat,G: $o > nat] :
% 14.25/8.12 ( ( B4 != bot_bot_set_o )
% 14.25/8.12 => ( ( condit1738341127787009408ow_nat @ ( image_nat_nat2 @ F @ A ) )
% 14.25/8.12 => ( ! [M2: $o] :
% 14.25/8.12 ( ( member_o @ M2 @ B4 )
% 14.25/8.12 => ? [X6: nat] :
% 14.25/8.12 ( ( member_nat @ X6 @ A )
% 14.25/8.12 & ( ord_less_eq_nat @ ( F @ X6 ) @ ( G @ M2 ) ) ) )
% 14.25/8.12 => ( ord_less_eq_nat @ ( complete_Inf_Inf_nat @ ( image_nat_nat2 @ F @ A ) ) @ ( complete_Inf_Inf_nat @ ( image_o_nat2 @ G @ B4 ) ) ) ) ) ) ).
% 14.25/8.12
% 14.25/8.12 thf(fact_6385_cINF__mono,axiom,
% 14.25/8.12 ! [B4: set_nat,F: nat > nat,A: set_nat,G: nat > nat] :
% 14.25/8.12 ( ( B4 != bot_bot_set_nat )
% 14.25/8.12 => ( ( condit1738341127787009408ow_nat @ ( image_nat_nat2 @ F @ A ) )
% 14.25/8.12 => ( ! [M2: nat] :
% 14.25/8.12 ( ( member_nat @ M2 @ B4 )
% 14.25/8.12 => ? [X6: nat] :
% 14.25/8.12 ( ( member_nat @ X6 @ A )
% 14.25/8.12 & ( ord_less_eq_nat @ ( F @ X6 ) @ ( G @ M2 ) ) ) )
% 14.25/8.12 => ( ord_less_eq_nat @ ( complete_Inf_Inf_nat @ ( image_nat_nat2 @ F @ A ) ) @ ( complete_Inf_Inf_nat @ ( image_nat_nat2 @ G @ B4 ) ) ) ) ) ) ).
% 14.25/8.12
% 14.25/8.12 thf(fact_6386_cINF__mono,axiom,
% 14.25/8.12 ! [B4: set_int,F: nat > nat,A: set_nat,G: int > nat] :
% 14.25/8.12 ( ( B4 != bot_bot_set_int )
% 14.25/8.12 => ( ( condit1738341127787009408ow_nat @ ( image_nat_nat2 @ F @ A ) )
% 14.25/8.12 => ( ! [M2: int] :
% 14.25/8.12 ( ( member_int @ M2 @ B4 )
% 14.25/8.12 => ? [X6: nat] :
% 14.25/8.12 ( ( member_nat @ X6 @ A )
% 14.25/8.12 & ( ord_less_eq_nat @ ( F @ X6 ) @ ( G @ M2 ) ) ) )
% 14.25/8.12 => ( ord_less_eq_nat @ ( complete_Inf_Inf_nat @ ( image_nat_nat2 @ F @ A ) ) @ ( complete_Inf_Inf_nat @ ( image_int_nat2 @ G @ B4 ) ) ) ) ) ) ).
% 14.25/8.12
% 14.25/8.12 thf(fact_6387_cINF__mono,axiom,
% 14.25/8.12 ! [B4: set_Pr4532377907799695533_nat_o,F: nat > nat,A: set_nat,G: ( produc3658429121746597890et_nat > $o ) > nat] :
% 14.25/8.12 ( ( B4 != bot_bo7824918357723582233_nat_o )
% 14.25/8.12 => ( ( condit1738341127787009408ow_nat @ ( image_nat_nat2 @ F @ A ) )
% 14.25/8.12 => ( ! [M2: produc3658429121746597890et_nat > $o] :
% 14.25/8.12 ( ( member6576561426505652726_nat_o @ M2 @ B4 )
% 14.25/8.12 => ? [X6: nat] :
% 14.25/8.12 ( ( member_nat @ X6 @ A )
% 14.25/8.12 & ( ord_less_eq_nat @ ( F @ X6 ) @ ( G @ M2 ) ) ) )
% 14.25/8.12 => ( ord_less_eq_nat @ ( complete_Inf_Inf_nat @ ( image_nat_nat2 @ F @ A ) ) @ ( complete_Inf_Inf_nat @ ( image_2174381888560928856_o_nat @ G @ B4 ) ) ) ) ) ) ).
% 14.25/8.12
% 14.25/8.12 thf(fact_6388_cINF__mono,axiom,
% 14.25/8.12 ! [B4: set_Product_unit,F: nat > set_nat,A: set_nat,G: product_unit > set_nat] :
% 14.25/8.12 ( ( B4 != bot_bo3957492148770167129t_unit )
% 14.25/8.12 => ( ( condit68592940725977398et_nat @ ( image_nat_set_nat @ F @ A ) )
% 14.25/8.12 => ( ! [M2: product_unit] :
% 14.25/8.12 ( ( member_Product_unit @ M2 @ B4 )
% 14.25/8.12 => ? [X6: nat] :
% 14.25/8.12 ( ( member_nat @ X6 @ A )
% 14.25/8.12 & ( ord_less_eq_set_nat @ ( F @ X6 ) @ ( G @ M2 ) ) ) )
% 14.25/8.12 => ( ord_less_eq_set_nat @ ( comple7806235888213564991et_nat @ ( image_nat_set_nat @ F @ A ) ) @ ( comple7806235888213564991et_nat @ ( image_8173849991827986510et_nat @ G @ B4 ) ) ) ) ) ) ).
% 14.25/8.12
% 14.25/8.12 thf(fact_6389_cINF__mono,axiom,
% 14.25/8.12 ! [B4: set_o,F: nat > set_nat,A: set_nat,G: $o > set_nat] :
% 14.25/8.12 ( ( B4 != bot_bot_set_o )
% 14.25/8.12 => ( ( condit68592940725977398et_nat @ ( image_nat_set_nat @ F @ A ) )
% 14.25/8.12 => ( ! [M2: $o] :
% 14.25/8.12 ( ( member_o @ M2 @ B4 )
% 14.25/8.12 => ? [X6: nat] :
% 14.25/8.12 ( ( member_nat @ X6 @ A )
% 14.25/8.12 & ( ord_less_eq_set_nat @ ( F @ X6 ) @ ( G @ M2 ) ) ) )
% 14.25/8.12 => ( ord_less_eq_set_nat @ ( comple7806235888213564991et_nat @ ( image_nat_set_nat @ F @ A ) ) @ ( comple7806235888213564991et_nat @ ( image_o_set_nat @ G @ B4 ) ) ) ) ) ) ).
% 14.25/8.12
% 14.25/8.12 thf(fact_6390_cINF__mono,axiom,
% 14.25/8.12 ! [B4: set_nat,F: nat > set_nat,A: set_nat,G: nat > set_nat] :
% 14.25/8.12 ( ( B4 != bot_bot_set_nat )
% 14.25/8.12 => ( ( condit68592940725977398et_nat @ ( image_nat_set_nat @ F @ A ) )
% 14.25/8.12 => ( ! [M2: nat] :
% 14.25/8.12 ( ( member_nat @ M2 @ B4 )
% 14.25/8.12 => ? [X6: nat] :
% 14.25/8.12 ( ( member_nat @ X6 @ A )
% 14.25/8.12 & ( ord_less_eq_set_nat @ ( F @ X6 ) @ ( G @ M2 ) ) ) )
% 14.25/8.12 => ( ord_less_eq_set_nat @ ( comple7806235888213564991et_nat @ ( image_nat_set_nat @ F @ A ) ) @ ( comple7806235888213564991et_nat @ ( image_nat_set_nat @ G @ B4 ) ) ) ) ) ) ).
% 14.25/8.12
% 14.25/8.12 thf(fact_6391_cINF__mono,axiom,
% 14.25/8.12 ! [B4: set_int,F: nat > set_nat,A: set_nat,G: int > set_nat] :
% 14.25/8.12 ( ( B4 != bot_bot_set_int )
% 14.25/8.12 => ( ( condit68592940725977398et_nat @ ( image_nat_set_nat @ F @ A ) )
% 14.25/8.12 => ( ! [M2: int] :
% 14.25/8.12 ( ( member_int @ M2 @ B4 )
% 14.25/8.12 => ? [X6: nat] :
% 14.25/8.12 ( ( member_nat @ X6 @ A )
% 14.25/8.12 & ( ord_less_eq_set_nat @ ( F @ X6 ) @ ( G @ M2 ) ) ) )
% 14.25/8.12 => ( ord_less_eq_set_nat @ ( comple7806235888213564991et_nat @ ( image_nat_set_nat @ F @ A ) ) @ ( comple7806235888213564991et_nat @ ( image_int_set_nat @ G @ B4 ) ) ) ) ) ) ).
% 14.25/8.12
% 14.25/8.12 thf(fact_6392_cINF__mono,axiom,
% 14.25/8.12 ! [B4: set_Pr4532377907799695533_nat_o,F: nat > set_nat,A: set_nat,G: ( produc3658429121746597890et_nat > $o ) > set_nat] :
% 14.25/8.12 ( ( B4 != bot_bo7824918357723582233_nat_o )
% 14.25/8.12 => ( ( condit68592940725977398et_nat @ ( image_nat_set_nat @ F @ A ) )
% 14.25/8.12 => ( ! [M2: produc3658429121746597890et_nat > $o] :
% 14.25/8.12 ( ( member6576561426505652726_nat_o @ M2 @ B4 )
% 14.25/8.12 => ? [X6: nat] :
% 14.25/8.12 ( ( member_nat @ X6 @ A )
% 14.25/8.12 & ( ord_less_eq_set_nat @ ( F @ X6 ) @ ( G @ M2 ) ) ) )
% 14.25/8.12 => ( ord_less_eq_set_nat @ ( comple7806235888213564991et_nat @ ( image_nat_set_nat @ F @ A ) ) @ ( comple7806235888213564991et_nat @ ( image_6888641473154130958et_nat @ G @ B4 ) ) ) ) ) ) ).
% 14.25/8.12
% 14.25/8.12 thf(fact_6393_le__cINF__iff,axiom,
% 14.25/8.12 ! [A: set_Product_unit,F: product_unit > int,U: int] :
% 14.25/8.12 ( ( A != bot_bo3957492148770167129t_unit )
% 14.25/8.12 => ( ( condit1735850657277959132ow_int @ ( image_873079544045703924it_int @ F @ A ) )
% 14.25/8.12 => ( ( ord_less_eq_int @ U @ ( complete_Inf_Inf_int @ ( image_873079544045703924it_int @ F @ A ) ) )
% 14.25/8.12 = ( ! [X2: product_unit] :
% 14.25/8.12 ( ( member_Product_unit @ X2 @ A )
% 14.25/8.12 => ( ord_less_eq_int @ U @ ( F @ X2 ) ) ) ) ) ) ) ).
% 14.25/8.12
% 14.25/8.12 thf(fact_6394_le__cINF__iff,axiom,
% 14.25/8.12 ! [A: set_o,F: $o > int,U: int] :
% 14.25/8.12 ( ( A != bot_bot_set_o )
% 14.25/8.12 => ( ( condit1735850657277959132ow_int @ ( image_o_int2 @ F @ A ) )
% 14.25/8.12 => ( ( ord_less_eq_int @ U @ ( complete_Inf_Inf_int @ ( image_o_int2 @ F @ A ) ) )
% 14.25/8.12 = ( ! [X2: $o] :
% 14.25/8.12 ( ( member_o @ X2 @ A )
% 14.25/8.12 => ( ord_less_eq_int @ U @ ( F @ X2 ) ) ) ) ) ) ) ).
% 14.25/8.12
% 14.25/8.12 thf(fact_6395_le__cINF__iff,axiom,
% 14.25/8.12 ! [A: set_nat,F: nat > int,U: int] :
% 14.25/8.12 ( ( A != bot_bot_set_nat )
% 14.25/8.12 => ( ( condit1735850657277959132ow_int @ ( image_nat_int2 @ F @ A ) )
% 14.25/8.12 => ( ( ord_less_eq_int @ U @ ( complete_Inf_Inf_int @ ( image_nat_int2 @ F @ A ) ) )
% 14.25/8.12 = ( ! [X2: nat] :
% 14.25/8.12 ( ( member_nat @ X2 @ A )
% 14.25/8.12 => ( ord_less_eq_int @ U @ ( F @ X2 ) ) ) ) ) ) ) ).
% 14.25/8.12
% 14.25/8.12 thf(fact_6396_le__cINF__iff,axiom,
% 14.25/8.12 ! [A: set_int,F: int > int,U: int] :
% 14.25/8.12 ( ( A != bot_bot_set_int )
% 14.25/8.12 => ( ( condit1735850657277959132ow_int @ ( image_int_int2 @ F @ A ) )
% 14.25/8.12 => ( ( ord_less_eq_int @ U @ ( complete_Inf_Inf_int @ ( image_int_int2 @ F @ A ) ) )
% 14.25/8.12 = ( ! [X2: int] :
% 14.25/8.12 ( ( member_int @ X2 @ A )
% 14.25/8.12 => ( ord_less_eq_int @ U @ ( F @ X2 ) ) ) ) ) ) ) ).
% 14.25/8.12
% 14.25/8.12 thf(fact_6397_le__cINF__iff,axiom,
% 14.25/8.12 ! [A: set_Product_unit,F: product_unit > nat,U: nat] :
% 14.25/8.12 ( ( A != bot_bo3957492148770167129t_unit )
% 14.25/8.12 => ( ( condit1738341127787009408ow_nat @ ( image_875570014554754200it_nat @ F @ A ) )
% 14.25/8.12 => ( ( ord_less_eq_nat @ U @ ( complete_Inf_Inf_nat @ ( image_875570014554754200it_nat @ F @ A ) ) )
% 14.25/8.12 = ( ! [X2: product_unit] :
% 14.25/8.12 ( ( member_Product_unit @ X2 @ A )
% 14.25/8.12 => ( ord_less_eq_nat @ U @ ( F @ X2 ) ) ) ) ) ) ) ).
% 14.25/8.12
% 14.25/8.12 thf(fact_6398_le__cINF__iff,axiom,
% 14.25/8.12 ! [A: set_o,F: $o > nat,U: nat] :
% 14.25/8.12 ( ( A != bot_bot_set_o )
% 14.25/8.12 => ( ( condit1738341127787009408ow_nat @ ( image_o_nat2 @ F @ A ) )
% 14.25/8.12 => ( ( ord_less_eq_nat @ U @ ( complete_Inf_Inf_nat @ ( image_o_nat2 @ F @ A ) ) )
% 14.25/8.12 = ( ! [X2: $o] :
% 14.25/8.12 ( ( member_o @ X2 @ A )
% 14.25/8.12 => ( ord_less_eq_nat @ U @ ( F @ X2 ) ) ) ) ) ) ) ).
% 14.25/8.12
% 14.25/8.12 thf(fact_6399_le__cINF__iff,axiom,
% 14.25/8.12 ! [A: set_nat,F: nat > nat,U: nat] :
% 14.25/8.12 ( ( A != bot_bot_set_nat )
% 14.25/8.12 => ( ( condit1738341127787009408ow_nat @ ( image_nat_nat2 @ F @ A ) )
% 14.25/8.12 => ( ( ord_less_eq_nat @ U @ ( complete_Inf_Inf_nat @ ( image_nat_nat2 @ F @ A ) ) )
% 14.25/8.12 = ( ! [X2: nat] :
% 14.25/8.12 ( ( member_nat @ X2 @ A )
% 14.25/8.12 => ( ord_less_eq_nat @ U @ ( F @ X2 ) ) ) ) ) ) ) ).
% 14.25/8.12
% 14.25/8.12 thf(fact_6400_le__cINF__iff,axiom,
% 14.25/8.12 ! [A: set_int,F: int > nat,U: nat] :
% 14.25/8.12 ( ( A != bot_bot_set_int )
% 14.25/8.12 => ( ( condit1738341127787009408ow_nat @ ( image_int_nat2 @ F @ A ) )
% 14.25/8.12 => ( ( ord_less_eq_nat @ U @ ( complete_Inf_Inf_nat @ ( image_int_nat2 @ F @ A ) ) )
% 14.25/8.12 = ( ! [X2: int] :
% 14.25/8.12 ( ( member_int @ X2 @ A )
% 14.25/8.12 => ( ord_less_eq_nat @ U @ ( F @ X2 ) ) ) ) ) ) ) ).
% 14.25/8.12
% 14.25/8.12 thf(fact_6401_le__cINF__iff,axiom,
% 14.25/8.12 ! [A: set_Product_unit,F: product_unit > product_unit,U: product_unit] :
% 14.25/8.12 ( ( A != bot_bo3957492148770167129t_unit )
% 14.25/8.12 => ( ( condit9161856309968323677t_unit @ ( image_405062704495631173t_unit @ F @ A ) )
% 14.25/8.12 => ( ( ord_le3221252021190050221t_unit @ U @ ( comple2584293577114468500t_unit @ ( image_405062704495631173t_unit @ F @ A ) ) )
% 14.25/8.12 = ( ! [X2: product_unit] :
% 14.25/8.12 ( ( member_Product_unit @ X2 @ A )
% 14.25/8.12 => ( ord_le3221252021190050221t_unit @ U @ ( F @ X2 ) ) ) ) ) ) ) ).
% 14.25/8.12
% 14.25/8.12 thf(fact_6402_le__cINF__iff,axiom,
% 14.25/8.12 ! [A: set_o,F: $o > product_unit,U: product_unit] :
% 14.25/8.12 ( ( A != bot_bot_set_o )
% 14.25/8.12 => ( ( condit9161856309968323677t_unit @ ( image_o_Product_unit2 @ F @ A ) )
% 14.25/8.12 => ( ( ord_le3221252021190050221t_unit @ U @ ( comple2584293577114468500t_unit @ ( image_o_Product_unit2 @ F @ A ) ) )
% 14.25/8.12 = ( ! [X2: $o] :
% 14.25/8.12 ( ( member_o @ X2 @ A )
% 14.25/8.12 => ( ord_le3221252021190050221t_unit @ U @ ( F @ X2 ) ) ) ) ) ) ) ).
% 14.25/8.12
% 14.25/8.12 thf(fact_6403_cInf__superset__mono,axiom,
% 14.25/8.12 ! [A: set_o,B4: set_o] :
% 14.25/8.12 ( ( A != bot_bot_set_o )
% 14.25/8.12 => ( ( condit5413489452508810728elow_o @ B4 )
% 14.25/8.12 => ( ( ord_less_eq_set_o @ A @ B4 )
% 14.25/8.12 => ( ord_less_eq_o @ ( complete_Inf_Inf_o @ B4 ) @ ( complete_Inf_Inf_o @ A ) ) ) ) ) ).
% 14.25/8.12
% 14.25/8.12 thf(fact_6404_cInf__superset__mono,axiom,
% 14.25/8.12 ! [A: set_Pr4532377907799695533_nat_o,B4: set_Pr4532377907799695533_nat_o] :
% 14.25/8.12 ( ( A != bot_bo7824918357723582233_nat_o )
% 14.25/8.12 => ( ( condit8399017556247434269_nat_o @ B4 )
% 14.25/8.12 => ( ( ord_le2965882846123202637_nat_o @ A @ B4 )
% 14.25/8.12 => ( ord_le729326519192465773_nat_o @ ( comple90455603224345172_nat_o @ B4 ) @ ( comple90455603224345172_nat_o @ A ) ) ) ) ) ).
% 14.25/8.12
% 14.25/8.12 thf(fact_6405_cInf__superset__mono,axiom,
% 14.25/8.12 ! [A: set_filter_nat,B4: set_filter_nat] :
% 14.25/8.12 ( ( A != bot_bo498966703094740906er_nat )
% 14.25/8.12 => ( ( condit3133331395780078992er_nat @ B4 )
% 14.25/8.12 => ( ( ord_le2426478655948331894er_nat @ A @ B4 )
% 14.25/8.12 => ( ord_le2510731241096832064er_nat @ ( comple6612786467464916249er_nat @ B4 ) @ ( comple6612786467464916249er_nat @ A ) ) ) ) ) ).
% 14.25/8.12
% 14.25/8.12 thf(fact_6406_cInf__superset__mono,axiom,
% 14.25/8.12 ! [A: set_int,B4: set_int] :
% 14.25/8.12 ( ( A != bot_bot_set_int )
% 14.25/8.12 => ( ( condit1735850657277959132ow_int @ B4 )
% 14.25/8.12 => ( ( ord_less_eq_set_int @ A @ B4 )
% 14.25/8.12 => ( ord_less_eq_int @ ( complete_Inf_Inf_int @ B4 ) @ ( complete_Inf_Inf_int @ A ) ) ) ) ) ).
% 14.25/8.13
% 14.25/8.13 thf(fact_6407_cInf__superset__mono,axiom,
% 14.25/8.13 ! [A: set_nat,B4: set_nat] :
% 14.25/8.13 ( ( A != bot_bot_set_nat )
% 14.25/8.13 => ( ( condit1738341127787009408ow_nat @ B4 )
% 14.25/8.13 => ( ( ord_less_eq_set_nat @ A @ B4 )
% 14.25/8.13 => ( ord_less_eq_nat @ ( complete_Inf_Inf_nat @ B4 ) @ ( complete_Inf_Inf_nat @ A ) ) ) ) ) ).
% 14.25/8.13
% 14.25/8.13 thf(fact_6408_cInf__superset__mono,axiom,
% 14.25/8.13 ! [A: set_set_nat,B4: set_set_nat] :
% 14.25/8.13 ( ( A != bot_bot_set_set_nat )
% 14.25/8.13 => ( ( condit68592940725977398et_nat @ B4 )
% 14.25/8.13 => ( ( ord_le6893508408891458716et_nat @ A @ B4 )
% 14.25/8.13 => ( ord_less_eq_set_nat @ ( comple7806235888213564991et_nat @ B4 ) @ ( comple7806235888213564991et_nat @ A ) ) ) ) ) ).
% 14.25/8.13
% 14.25/8.13 thf(fact_6409_cInf__superset__mono,axiom,
% 14.25/8.13 ! [A: set_Product_unit,B4: set_Product_unit] :
% 14.25/8.13 ( ( A != bot_bo3957492148770167129t_unit )
% 14.25/8.13 => ( ( condit9161856309968323677t_unit @ B4 )
% 14.25/8.13 => ( ( ord_le3507040750410214029t_unit @ A @ B4 )
% 14.25/8.13 => ( ord_le3221252021190050221t_unit @ ( comple2584293577114468500t_unit @ B4 ) @ ( comple2584293577114468500t_unit @ A ) ) ) ) ) ).
% 14.25/8.13
% 14.25/8.13 thf(fact_6410_cInf__insert,axiom,
% 14.25/8.13 ! [X4: set_o,A2: $o] :
% 14.25/8.13 ( ( X4 != bot_bot_set_o )
% 14.25/8.13 => ( ( condit5413489452508810728elow_o @ X4 )
% 14.25/8.13 => ( ( complete_Inf_Inf_o @ ( insert_o @ A2 @ X4 ) )
% 14.25/8.13 = ( inf_inf_o @ A2 @ ( complete_Inf_Inf_o @ X4 ) ) ) ) ) ).
% 14.25/8.13
% 14.25/8.13 thf(fact_6411_cInf__insert,axiom,
% 14.25/8.13 ! [X4: set_int,A2: int] :
% 14.25/8.13 ( ( X4 != bot_bot_set_int )
% 14.25/8.13 => ( ( condit1735850657277959132ow_int @ X4 )
% 14.25/8.13 => ( ( complete_Inf_Inf_int @ ( insert_int @ A2 @ X4 ) )
% 14.25/8.13 = ( inf_inf_int @ A2 @ ( complete_Inf_Inf_int @ X4 ) ) ) ) ) ).
% 14.25/8.13
% 14.25/8.13 thf(fact_6412_cInf__insert,axiom,
% 14.25/8.13 ! [X4: set_Pr4532377907799695533_nat_o,A2: produc3658429121746597890et_nat > $o] :
% 14.25/8.13 ( ( X4 != bot_bo7824918357723582233_nat_o )
% 14.25/8.13 => ( ( condit8399017556247434269_nat_o @ X4 )
% 14.25/8.13 => ( ( comple90455603224345172_nat_o @ ( insert5175938949040314269_nat_o @ A2 @ X4 ) )
% 14.25/8.13 = ( inf_in1318976480646536635_nat_o @ A2 @ ( comple90455603224345172_nat_o @ X4 ) ) ) ) ) ).
% 14.25/8.13
% 14.25/8.13 thf(fact_6413_cInf__insert,axiom,
% 14.25/8.13 ! [X4: set_nat,A2: nat] :
% 14.25/8.13 ( ( X4 != bot_bot_set_nat )
% 14.25/8.13 => ( ( condit1738341127787009408ow_nat @ X4 )
% 14.25/8.13 => ( ( complete_Inf_Inf_nat @ ( insert_nat @ A2 @ X4 ) )
% 14.25/8.13 = ( inf_inf_nat @ A2 @ ( complete_Inf_Inf_nat @ X4 ) ) ) ) ) ).
% 14.25/8.13
% 14.25/8.13 thf(fact_6414_cInf__insert,axiom,
% 14.25/8.13 ! [X4: set_set_nat,A2: set_nat] :
% 14.25/8.13 ( ( X4 != bot_bot_set_set_nat )
% 14.25/8.13 => ( ( condit68592940725977398et_nat @ X4 )
% 14.25/8.13 => ( ( comple7806235888213564991et_nat @ ( insert_set_nat @ A2 @ X4 ) )
% 14.25/8.13 = ( inf_inf_set_nat @ A2 @ ( comple7806235888213564991et_nat @ X4 ) ) ) ) ) ).
% 14.25/8.13
% 14.25/8.13 thf(fact_6415_cInf__insert,axiom,
% 14.25/8.13 ! [X4: set_Product_unit,A2: product_unit] :
% 14.25/8.13 ( ( X4 != bot_bo3957492148770167129t_unit )
% 14.25/8.13 => ( ( condit9161856309968323677t_unit @ X4 )
% 14.25/8.13 => ( ( comple2584293577114468500t_unit @ ( insert_Product_unit @ A2 @ X4 ) )
% 14.25/8.13 = ( inf_inf_Product_unit @ A2 @ ( comple2584293577114468500t_unit @ X4 ) ) ) ) ) ).
% 14.25/8.13
% 14.25/8.13 thf(fact_6416_cInf__insert__If,axiom,
% 14.25/8.13 ! [X4: set_o,A2: $o] :
% 14.25/8.13 ( ( condit5413489452508810728elow_o @ X4 )
% 14.25/8.13 => ( ( complete_Inf_Inf_o @ ( insert_o @ A2 @ X4 ) )
% 14.25/8.13 = ( ( ( X4 = bot_bot_set_o )
% 14.25/8.13 => A2 )
% 14.25/8.13 & ( ( X4 != bot_bot_set_o )
% 14.25/8.13 => ( inf_inf_o @ A2 @ ( complete_Inf_Inf_o @ X4 ) ) ) ) ) ) ).
% 14.25/8.13
% 14.25/8.13 thf(fact_6417_cInf__insert__If,axiom,
% 14.25/8.13 ! [X4: set_int,A2: int] :
% 14.25/8.13 ( ( condit1735850657277959132ow_int @ X4 )
% 14.25/8.13 => ( ( ( X4 = bot_bot_set_int )
% 14.25/8.13 => ( ( complete_Inf_Inf_int @ ( insert_int @ A2 @ X4 ) )
% 14.25/8.13 = A2 ) )
% 14.25/8.13 & ( ( X4 != bot_bot_set_int )
% 14.25/8.13 => ( ( complete_Inf_Inf_int @ ( insert_int @ A2 @ X4 ) )
% 14.25/8.13 = ( inf_inf_int @ A2 @ ( complete_Inf_Inf_int @ X4 ) ) ) ) ) ) ).
% 14.25/8.13
% 14.25/8.13 thf(fact_6418_cInf__insert__If,axiom,
% 14.25/8.13 ! [X4: set_Pr4532377907799695533_nat_o,A2: produc3658429121746597890et_nat > $o] :
% 14.25/8.13 ( ( condit8399017556247434269_nat_o @ X4 )
% 14.25/8.13 => ( ( ( X4 = bot_bo7824918357723582233_nat_o )
% 14.25/8.13 => ( ( comple90455603224345172_nat_o @ ( insert5175938949040314269_nat_o @ A2 @ X4 ) )
% 14.25/8.13 = A2 ) )
% 14.25/8.13 & ( ( X4 != bot_bo7824918357723582233_nat_o )
% 14.25/8.13 => ( ( comple90455603224345172_nat_o @ ( insert5175938949040314269_nat_o @ A2 @ X4 ) )
% 14.25/8.13 = ( inf_in1318976480646536635_nat_o @ A2 @ ( comple90455603224345172_nat_o @ X4 ) ) ) ) ) ) ).
% 14.25/8.13
% 14.25/8.13 thf(fact_6419_cInf__insert__If,axiom,
% 14.25/8.13 ! [X4: set_nat,A2: nat] :
% 14.25/8.13 ( ( condit1738341127787009408ow_nat @ X4 )
% 14.25/8.13 => ( ( ( X4 = bot_bot_set_nat )
% 14.25/8.13 => ( ( complete_Inf_Inf_nat @ ( insert_nat @ A2 @ X4 ) )
% 14.25/8.13 = A2 ) )
% 14.25/8.13 & ( ( X4 != bot_bot_set_nat )
% 14.25/8.13 => ( ( complete_Inf_Inf_nat @ ( insert_nat @ A2 @ X4 ) )
% 14.25/8.13 = ( inf_inf_nat @ A2 @ ( complete_Inf_Inf_nat @ X4 ) ) ) ) ) ) ).
% 14.25/8.13
% 14.25/8.13 thf(fact_6420_cInf__insert__If,axiom,
% 14.25/8.13 ! [X4: set_set_nat,A2: set_nat] :
% 14.25/8.13 ( ( condit68592940725977398et_nat @ X4 )
% 14.25/8.13 => ( ( ( X4 = bot_bot_set_set_nat )
% 14.25/8.13 => ( ( comple7806235888213564991et_nat @ ( insert_set_nat @ A2 @ X4 ) )
% 14.25/8.13 = A2 ) )
% 14.25/8.13 & ( ( X4 != bot_bot_set_set_nat )
% 14.25/8.13 => ( ( comple7806235888213564991et_nat @ ( insert_set_nat @ A2 @ X4 ) )
% 14.25/8.13 = ( inf_inf_set_nat @ A2 @ ( comple7806235888213564991et_nat @ X4 ) ) ) ) ) ) ).
% 14.25/8.13
% 14.25/8.13 thf(fact_6421_cInf__insert__If,axiom,
% 14.25/8.13 ! [X4: set_Product_unit,A2: product_unit] :
% 14.25/8.13 ( ( condit9161856309968323677t_unit @ X4 )
% 14.25/8.13 => ( ( ( X4 = bot_bo3957492148770167129t_unit )
% 14.25/8.13 => ( ( comple2584293577114468500t_unit @ ( insert_Product_unit @ A2 @ X4 ) )
% 14.25/8.13 = A2 ) )
% 14.25/8.13 & ( ( X4 != bot_bo3957492148770167129t_unit )
% 14.25/8.13 => ( ( comple2584293577114468500t_unit @ ( insert_Product_unit @ A2 @ X4 ) )
% 14.25/8.13 = ( inf_inf_Product_unit @ A2 @ ( comple2584293577114468500t_unit @ X4 ) ) ) ) ) ) ).
% 14.25/8.13
% 14.25/8.13 thf(fact_6422_cInf__union__distrib,axiom,
% 14.25/8.13 ! [A: set_o,B4: set_o] :
% 14.25/8.13 ( ( A != bot_bot_set_o )
% 14.25/8.13 => ( ( condit5413489452508810728elow_o @ A )
% 14.25/8.13 => ( ( B4 != bot_bot_set_o )
% 14.25/8.13 => ( ( condit5413489452508810728elow_o @ B4 )
% 14.25/8.13 => ( ( complete_Inf_Inf_o @ ( sup_sup_set_o @ A @ B4 ) )
% 14.25/8.13 = ( inf_inf_o @ ( complete_Inf_Inf_o @ A ) @ ( complete_Inf_Inf_o @ B4 ) ) ) ) ) ) ) ).
% 14.25/8.13
% 14.25/8.13 thf(fact_6423_cInf__union__distrib,axiom,
% 14.25/8.13 ! [A: set_int,B4: set_int] :
% 14.25/8.13 ( ( A != bot_bot_set_int )
% 14.25/8.13 => ( ( condit1735850657277959132ow_int @ A )
% 14.25/8.13 => ( ( B4 != bot_bot_set_int )
% 14.25/8.13 => ( ( condit1735850657277959132ow_int @ B4 )
% 14.25/8.13 => ( ( complete_Inf_Inf_int @ ( sup_sup_set_int @ A @ B4 ) )
% 14.25/8.13 = ( inf_inf_int @ ( complete_Inf_Inf_int @ A ) @ ( complete_Inf_Inf_int @ B4 ) ) ) ) ) ) ) ).
% 14.25/8.13
% 14.25/8.13 thf(fact_6424_cInf__union__distrib,axiom,
% 14.25/8.13 ! [A: set_Pr4532377907799695533_nat_o,B4: set_Pr4532377907799695533_nat_o] :
% 14.25/8.13 ( ( A != bot_bo7824918357723582233_nat_o )
% 14.25/8.13 => ( ( condit8399017556247434269_nat_o @ A )
% 14.25/8.13 => ( ( B4 != bot_bo7824918357723582233_nat_o )
% 14.25/8.13 => ( ( condit8399017556247434269_nat_o @ B4 )
% 14.25/8.13 => ( ( comple90455603224345172_nat_o @ ( sup_su5209123915105501825_nat_o @ A @ B4 ) )
% 14.25/8.13 = ( inf_in1318976480646536635_nat_o @ ( comple90455603224345172_nat_o @ A ) @ ( comple90455603224345172_nat_o @ B4 ) ) ) ) ) ) ) ).
% 14.25/8.13
% 14.25/8.13 thf(fact_6425_cInf__union__distrib,axiom,
% 14.25/8.13 ! [A: set_nat,B4: set_nat] :
% 14.25/8.13 ( ( A != bot_bot_set_nat )
% 14.25/8.13 => ( ( condit1738341127787009408ow_nat @ A )
% 14.25/8.13 => ( ( B4 != bot_bot_set_nat )
% 14.25/8.13 => ( ( condit1738341127787009408ow_nat @ B4 )
% 14.25/8.13 => ( ( complete_Inf_Inf_nat @ ( sup_sup_set_nat @ A @ B4 ) )
% 14.25/8.13 = ( inf_inf_nat @ ( complete_Inf_Inf_nat @ A ) @ ( complete_Inf_Inf_nat @ B4 ) ) ) ) ) ) ) ).
% 14.25/8.13
% 14.25/8.13 thf(fact_6426_cInf__union__distrib,axiom,
% 14.25/8.13 ! [A: set_set_nat,B4: set_set_nat] :
% 14.25/8.13 ( ( A != bot_bot_set_set_nat )
% 14.25/8.13 => ( ( condit68592940725977398et_nat @ A )
% 14.25/8.13 => ( ( B4 != bot_bot_set_set_nat )
% 14.25/8.13 => ( ( condit68592940725977398et_nat @ B4 )
% 14.25/8.13 => ( ( comple7806235888213564991et_nat @ ( sup_sup_set_set_nat @ A @ B4 ) )
% 14.25/8.13 = ( inf_inf_set_nat @ ( comple7806235888213564991et_nat @ A ) @ ( comple7806235888213564991et_nat @ B4 ) ) ) ) ) ) ) ).
% 14.25/8.13
% 14.25/8.13 thf(fact_6427_cInf__union__distrib,axiom,
% 14.25/8.13 ! [A: set_Product_unit,B4: set_Product_unit] :
% 14.25/8.13 ( ( A != bot_bo3957492148770167129t_unit )
% 14.25/8.13 => ( ( condit9161856309968323677t_unit @ A )
% 14.25/8.13 => ( ( B4 != bot_bo3957492148770167129t_unit )
% 14.25/8.13 => ( ( condit9161856309968323677t_unit @ B4 )
% 14.25/8.13 => ( ( comple2584293577114468500t_unit @ ( sup_su793286257634532545t_unit @ A @ B4 ) )
% 14.25/8.13 = ( inf_inf_Product_unit @ ( comple2584293577114468500t_unit @ A ) @ ( comple2584293577114468500t_unit @ B4 ) ) ) ) ) ) ) ).
% 14.25/8.13
% 14.25/8.13 thf(fact_6428_dbl__inc__def,axiom,
% 14.25/8.13 ( neg_nu5831290666863070958nteger
% 14.25/8.13 = ( ^ [X2: code_integer] : ( plus_p5714425477246183910nteger @ ( plus_p5714425477246183910nteger @ X2 @ X2 ) @ one_one_Code_integer ) ) ) ).
% 14.25/8.13
% 14.25/8.13 thf(fact_6429_dbl__inc__def,axiom,
% 14.25/8.13 ( neg_nu5219082963157363817nc_rat
% 14.25/8.13 = ( ^ [X2: rat] : ( plus_plus_rat @ ( plus_plus_rat @ X2 @ X2 ) @ one_one_rat ) ) ) ).
% 14.25/8.13
% 14.25/8.13 thf(fact_6430_dbl__inc__def,axiom,
% 14.25/8.13 ( neg_nu5851722552734809277nc_int
% 14.25/8.13 = ( ^ [X2: int] : ( plus_plus_int @ ( plus_plus_int @ X2 @ X2 ) @ one_one_int ) ) ) ).
% 14.25/8.13
% 14.25/8.13 thf(fact_6431_dvd__mult__cancel1,axiom,
% 14.25/8.13 ! [M4: nat,N2: nat] :
% 14.25/8.13 ( ( ord_less_nat @ zero_zero_nat @ M4 )
% 14.25/8.13 => ( ( dvd_dvd_nat @ ( times_times_nat @ M4 @ N2 ) @ M4 )
% 14.25/8.13 = ( N2 = one_one_nat ) ) ) ).
% 14.25/8.13
% 14.25/8.13 thf(fact_6432_dvd__mult__cancel2,axiom,
% 14.25/8.13 ! [M4: nat,N2: nat] :
% 14.25/8.13 ( ( ord_less_nat @ zero_zero_nat @ M4 )
% 14.25/8.13 => ( ( dvd_dvd_nat @ ( times_times_nat @ N2 @ M4 ) @ M4 )
% 14.25/8.13 = ( N2 = one_one_nat ) ) ) ).
% 14.25/8.13
% 14.25/8.13 thf(fact_6433_dvd__minus__add,axiom,
% 14.25/8.13 ! [Q4: nat,N2: nat,R2: nat,M4: nat] :
% 14.25/8.13 ( ( ord_less_eq_nat @ Q4 @ N2 )
% 14.25/8.13 => ( ( ord_less_eq_nat @ Q4 @ ( times_times_nat @ R2 @ M4 ) )
% 14.25/8.13 => ( ( dvd_dvd_nat @ M4 @ ( minus_minus_nat @ N2 @ Q4 ) )
% 14.25/8.13 = ( dvd_dvd_nat @ M4 @ ( plus_plus_nat @ N2 @ ( minus_minus_nat @ ( times_times_nat @ R2 @ M4 ) @ Q4 ) ) ) ) ) ) ).
% 14.25/8.13
% 14.25/8.13 thf(fact_6434_cINF__superset__mono,axiom,
% 14.25/8.13 ! [A: set_Product_unit,G: product_unit > int,B4: set_Product_unit,F: product_unit > int] :
% 14.25/8.13 ( ( A != bot_bo3957492148770167129t_unit )
% 14.25/8.13 => ( ( condit1735850657277959132ow_int @ ( image_873079544045703924it_int @ G @ B4 ) )
% 14.25/8.13 => ( ( ord_le3507040750410214029t_unit @ A @ B4 )
% 14.25/8.13 => ( ! [X3: product_unit] :
% 14.25/8.13 ( ( member_Product_unit @ X3 @ B4 )
% 14.25/8.13 => ( ord_less_eq_int @ ( G @ X3 ) @ ( F @ X3 ) ) )
% 14.25/8.13 => ( ord_less_eq_int @ ( complete_Inf_Inf_int @ ( image_873079544045703924it_int @ G @ B4 ) ) @ ( complete_Inf_Inf_int @ ( image_873079544045703924it_int @ F @ A ) ) ) ) ) ) ) ).
% 14.25/8.13
% 14.25/8.13 thf(fact_6435_cINF__superset__mono,axiom,
% 14.25/8.13 ! [A: set_o,G: $o > int,B4: set_o,F: $o > int] :
% 14.25/8.13 ( ( A != bot_bot_set_o )
% 14.25/8.13 => ( ( condit1735850657277959132ow_int @ ( image_o_int2 @ G @ B4 ) )
% 14.25/8.13 => ( ( ord_less_eq_set_o @ A @ B4 )
% 14.25/8.13 => ( ! [X3: $o] :
% 14.25/8.13 ( ( member_o @ X3 @ B4 )
% 14.25/8.13 => ( ord_less_eq_int @ ( G @ X3 ) @ ( F @ X3 ) ) )
% 14.25/8.13 => ( ord_less_eq_int @ ( complete_Inf_Inf_int @ ( image_o_int2 @ G @ B4 ) ) @ ( complete_Inf_Inf_int @ ( image_o_int2 @ F @ A ) ) ) ) ) ) ) ).
% 14.25/8.13
% 14.25/8.13 thf(fact_6436_cINF__superset__mono,axiom,
% 14.25/8.13 ! [A: set_int,G: int > int,B4: set_int,F: int > int] :
% 14.25/8.13 ( ( A != bot_bot_set_int )
% 14.25/8.13 => ( ( condit1735850657277959132ow_int @ ( image_int_int2 @ G @ B4 ) )
% 14.25/8.13 => ( ( ord_less_eq_set_int @ A @ B4 )
% 14.25/8.13 => ( ! [X3: int] :
% 14.25/8.13 ( ( member_int @ X3 @ B4 )
% 14.25/8.13 => ( ord_less_eq_int @ ( G @ X3 ) @ ( F @ X3 ) ) )
% 14.25/8.13 => ( ord_less_eq_int @ ( complete_Inf_Inf_int @ ( image_int_int2 @ G @ B4 ) ) @ ( complete_Inf_Inf_int @ ( image_int_int2 @ F @ A ) ) ) ) ) ) ) ).
% 14.25/8.13
% 14.25/8.13 thf(fact_6437_cINF__superset__mono,axiom,
% 14.25/8.13 ! [A: set_nat,G: nat > int,B4: set_nat,F: nat > int] :
% 14.25/8.13 ( ( A != bot_bot_set_nat )
% 14.25/8.13 => ( ( condit1735850657277959132ow_int @ ( image_nat_int2 @ G @ B4 ) )
% 14.25/8.13 => ( ( ord_less_eq_set_nat @ A @ B4 )
% 14.25/8.13 => ( ! [X3: nat] :
% 14.25/8.13 ( ( member_nat @ X3 @ B4 )
% 14.25/8.13 => ( ord_less_eq_int @ ( G @ X3 ) @ ( F @ X3 ) ) )
% 14.25/8.13 => ( ord_less_eq_int @ ( complete_Inf_Inf_int @ ( image_nat_int2 @ G @ B4 ) ) @ ( complete_Inf_Inf_int @ ( image_nat_int2 @ F @ A ) ) ) ) ) ) ) ).
% 14.25/8.13
% 14.25/8.13 thf(fact_6438_cINF__superset__mono,axiom,
% 14.25/8.13 ! [A: set_Product_unit,G: product_unit > nat,B4: set_Product_unit,F: product_unit > nat] :
% 14.25/8.13 ( ( A != bot_bo3957492148770167129t_unit )
% 14.25/8.13 => ( ( condit1738341127787009408ow_nat @ ( image_875570014554754200it_nat @ G @ B4 ) )
% 14.25/8.13 => ( ( ord_le3507040750410214029t_unit @ A @ B4 )
% 14.25/8.13 => ( ! [X3: product_unit] :
% 14.25/8.13 ( ( member_Product_unit @ X3 @ B4 )
% 14.25/8.13 => ( ord_less_eq_nat @ ( G @ X3 ) @ ( F @ X3 ) ) )
% 14.25/8.13 => ( ord_less_eq_nat @ ( complete_Inf_Inf_nat @ ( image_875570014554754200it_nat @ G @ B4 ) ) @ ( complete_Inf_Inf_nat @ ( image_875570014554754200it_nat @ F @ A ) ) ) ) ) ) ) ).
% 14.25/8.13
% 14.25/8.13 thf(fact_6439_cINF__superset__mono,axiom,
% 14.25/8.13 ! [A: set_o,G: $o > nat,B4: set_o,F: $o > nat] :
% 14.25/8.13 ( ( A != bot_bot_set_o )
% 14.25/8.13 => ( ( condit1738341127787009408ow_nat @ ( image_o_nat2 @ G @ B4 ) )
% 14.25/8.13 => ( ( ord_less_eq_set_o @ A @ B4 )
% 14.25/8.13 => ( ! [X3: $o] :
% 14.25/8.13 ( ( member_o @ X3 @ B4 )
% 14.25/8.13 => ( ord_less_eq_nat @ ( G @ X3 ) @ ( F @ X3 ) ) )
% 14.25/8.13 => ( ord_less_eq_nat @ ( complete_Inf_Inf_nat @ ( image_o_nat2 @ G @ B4 ) ) @ ( complete_Inf_Inf_nat @ ( image_o_nat2 @ F @ A ) ) ) ) ) ) ) ).
% 14.25/8.13
% 14.25/8.13 thf(fact_6440_cINF__superset__mono,axiom,
% 14.25/8.13 ! [A: set_int,G: int > nat,B4: set_int,F: int > nat] :
% 14.25/8.13 ( ( A != bot_bot_set_int )
% 14.25/8.13 => ( ( condit1738341127787009408ow_nat @ ( image_int_nat2 @ G @ B4 ) )
% 14.25/8.13 => ( ( ord_less_eq_set_int @ A @ B4 )
% 14.25/8.13 => ( ! [X3: int] :
% 14.25/8.13 ( ( member_int @ X3 @ B4 )
% 14.25/8.13 => ( ord_less_eq_nat @ ( G @ X3 ) @ ( F @ X3 ) ) )
% 14.25/8.13 => ( ord_less_eq_nat @ ( complete_Inf_Inf_nat @ ( image_int_nat2 @ G @ B4 ) ) @ ( complete_Inf_Inf_nat @ ( image_int_nat2 @ F @ A ) ) ) ) ) ) ) ).
% 14.25/8.13
% 14.25/8.13 thf(fact_6441_cINF__superset__mono,axiom,
% 14.25/8.13 ! [A: set_nat,G: nat > nat,B4: set_nat,F: nat > nat] :
% 14.25/8.13 ( ( A != bot_bot_set_nat )
% 14.25/8.13 => ( ( condit1738341127787009408ow_nat @ ( image_nat_nat2 @ G @ B4 ) )
% 14.25/8.13 => ( ( ord_less_eq_set_nat @ A @ B4 )
% 14.25/8.13 => ( ! [X3: nat] :
% 14.25/8.13 ( ( member_nat @ X3 @ B4 )
% 14.25/8.13 => ( ord_less_eq_nat @ ( G @ X3 ) @ ( F @ X3 ) ) )
% 14.25/8.13 => ( ord_less_eq_nat @ ( complete_Inf_Inf_nat @ ( image_nat_nat2 @ G @ B4 ) ) @ ( complete_Inf_Inf_nat @ ( image_nat_nat2 @ F @ A ) ) ) ) ) ) ) ).
% 14.25/8.13
% 14.25/8.13 thf(fact_6442_cINF__superset__mono,axiom,
% 14.25/8.13 ! [A: set_Product_unit,G: product_unit > product_unit,B4: set_Product_unit,F: product_unit > product_unit] :
% 14.25/8.13 ( ( A != bot_bo3957492148770167129t_unit )
% 14.25/8.13 => ( ( condit9161856309968323677t_unit @ ( image_405062704495631173t_unit @ G @ B4 ) )
% 14.25/8.13 => ( ( ord_le3507040750410214029t_unit @ A @ B4 )
% 14.25/8.13 => ( ! [X3: product_unit] :
% 14.25/8.13 ( ( member_Product_unit @ X3 @ B4 )
% 14.25/8.13 => ( ord_le3221252021190050221t_unit @ ( G @ X3 ) @ ( F @ X3 ) ) )
% 14.25/8.13 => ( ord_le3221252021190050221t_unit @ ( comple2584293577114468500t_unit @ ( image_405062704495631173t_unit @ G @ B4 ) ) @ ( comple2584293577114468500t_unit @ ( image_405062704495631173t_unit @ F @ A ) ) ) ) ) ) ) ).
% 14.25/8.13
% 14.25/8.13 thf(fact_6443_cINF__superset__mono,axiom,
% 14.25/8.13 ! [A: set_o,G: $o > product_unit,B4: set_o,F: $o > product_unit] :
% 14.25/8.13 ( ( A != bot_bot_set_o )
% 14.25/8.13 => ( ( condit9161856309968323677t_unit @ ( image_o_Product_unit2 @ G @ B4 ) )
% 14.25/8.13 => ( ( ord_less_eq_set_o @ A @ B4 )
% 14.25/8.13 => ( ! [X3: $o] :
% 14.25/8.13 ( ( member_o @ X3 @ B4 )
% 14.25/8.13 => ( ord_le3221252021190050221t_unit @ ( G @ X3 ) @ ( F @ X3 ) ) )
% 14.25/8.13 => ( ord_le3221252021190050221t_unit @ ( comple2584293577114468500t_unit @ ( image_o_Product_unit2 @ G @ B4 ) ) @ ( comple2584293577114468500t_unit @ ( image_o_Product_unit2 @ F @ A ) ) ) ) ) ) ) ).
% 14.25/8.13
% 14.25/8.13 thf(fact_6444_less__eq__cInf__inter,axiom,
% 14.25/8.13 ! [A: set_o,B4: set_o] :
% 14.25/8.13 ( ( condit5413489452508810728elow_o @ A )
% 14.25/8.13 => ( ( condit5413489452508810728elow_o @ B4 )
% 14.25/8.13 => ( ( ( inf_inf_set_o @ A @ B4 )
% 14.25/8.13 != bot_bot_set_o )
% 14.25/8.13 => ( ord_less_eq_o @ ( inf_inf_o @ ( complete_Inf_Inf_o @ A ) @ ( complete_Inf_Inf_o @ B4 ) ) @ ( complete_Inf_Inf_o @ ( inf_inf_set_o @ A @ B4 ) ) ) ) ) ) ).
% 14.25/8.13
% 14.25/8.13 thf(fact_6445_less__eq__cInf__inter,axiom,
% 14.25/8.13 ! [A: set_Pr4532377907799695533_nat_o,B4: set_Pr4532377907799695533_nat_o] :
% 14.25/8.13 ( ( condit8399017556247434269_nat_o @ A )
% 14.25/8.13 => ( ( condit8399017556247434269_nat_o @ B4 )
% 14.25/8.13 => ( ( ( inf_in1906310914598751387_nat_o @ A @ B4 )
% 14.25/8.13 != bot_bo7824918357723582233_nat_o )
% 14.25/8.13 => ( ord_le729326519192465773_nat_o @ ( inf_in1318976480646536635_nat_o @ ( comple90455603224345172_nat_o @ A ) @ ( comple90455603224345172_nat_o @ B4 ) ) @ ( comple90455603224345172_nat_o @ ( inf_in1906310914598751387_nat_o @ A @ B4 ) ) ) ) ) ) ).
% 14.25/8.13
% 14.25/8.13 thf(fact_6446_less__eq__cInf__inter,axiom,
% 14.25/8.13 ! [A: set_filter_nat,B4: set_filter_nat] :
% 14.25/8.13 ( ( condit3133331395780078992er_nat @ A )
% 14.25/8.13 => ( ( condit3133331395780078992er_nat @ B4 )
% 14.25/8.13 => ( ( ( inf_in7395780696554941096er_nat @ A @ B4 )
% 14.25/8.13 != bot_bo498966703094740906er_nat )
% 14.25/8.13 => ( ord_le2510731241096832064er_nat @ ( inf_inf_filter_nat @ ( comple6612786467464916249er_nat @ A ) @ ( comple6612786467464916249er_nat @ B4 ) ) @ ( comple6612786467464916249er_nat @ ( inf_in7395780696554941096er_nat @ A @ B4 ) ) ) ) ) ) ).
% 14.25/8.13
% 14.25/8.13 thf(fact_6447_less__eq__cInf__inter,axiom,
% 14.25/8.13 ! [A: set_int,B4: set_int] :
% 14.25/8.13 ( ( condit1735850657277959132ow_int @ A )
% 14.25/8.13 => ( ( condit1735850657277959132ow_int @ B4 )
% 14.25/8.13 => ( ( ( inf_inf_set_int @ A @ B4 )
% 14.25/8.13 != bot_bot_set_int )
% 14.25/8.13 => ( ord_less_eq_int @ ( inf_inf_int @ ( complete_Inf_Inf_int @ A ) @ ( complete_Inf_Inf_int @ B4 ) ) @ ( complete_Inf_Inf_int @ ( inf_inf_set_int @ A @ B4 ) ) ) ) ) ) ).
% 14.25/8.13
% 14.25/8.13 thf(fact_6448_less__eq__cInf__inter,axiom,
% 14.25/8.13 ! [A: set_nat,B4: set_nat] :
% 14.25/8.13 ( ( condit1738341127787009408ow_nat @ A )
% 14.25/8.13 => ( ( condit1738341127787009408ow_nat @ B4 )
% 14.25/8.13 => ( ( ( inf_inf_set_nat @ A @ B4 )
% 14.25/8.13 != bot_bot_set_nat )
% 14.25/8.13 => ( ord_less_eq_nat @ ( inf_inf_nat @ ( complete_Inf_Inf_nat @ A ) @ ( complete_Inf_Inf_nat @ B4 ) ) @ ( complete_Inf_Inf_nat @ ( inf_inf_set_nat @ A @ B4 ) ) ) ) ) ) ).
% 14.25/8.13
% 14.25/8.13 thf(fact_6449_less__eq__cInf__inter,axiom,
% 14.25/8.13 ! [A: set_set_nat,B4: set_set_nat] :
% 14.25/8.13 ( ( condit68592940725977398et_nat @ A )
% 14.25/8.13 => ( ( condit68592940725977398et_nat @ B4 )
% 14.25/8.13 => ( ( ( inf_inf_set_set_nat @ A @ B4 )
% 14.25/8.13 != bot_bot_set_set_nat )
% 14.25/8.13 => ( ord_less_eq_set_nat @ ( inf_inf_set_nat @ ( comple7806235888213564991et_nat @ A ) @ ( comple7806235888213564991et_nat @ B4 ) ) @ ( comple7806235888213564991et_nat @ ( inf_inf_set_set_nat @ A @ B4 ) ) ) ) ) ) ).
% 14.25/8.13
% 14.25/8.13 thf(fact_6450_less__eq__cInf__inter,axiom,
% 14.25/8.13 ! [A: set_Product_unit,B4: set_Product_unit] :
% 14.25/8.13 ( ( condit9161856309968323677t_unit @ A )
% 14.25/8.13 => ( ( condit9161856309968323677t_unit @ B4 )
% 14.25/8.13 => ( ( ( inf_in4660618365625256667t_unit @ A @ B4 )
% 14.25/8.13 != bot_bo3957492148770167129t_unit )
% 14.25/8.13 => ( ord_le3221252021190050221t_unit @ ( inf_inf_Product_unit @ ( comple2584293577114468500t_unit @ A ) @ ( comple2584293577114468500t_unit @ B4 ) ) @ ( comple2584293577114468500t_unit @ ( inf_in4660618365625256667t_unit @ A @ B4 ) ) ) ) ) ) ).
% 14.25/8.13
% 14.25/8.13 thf(fact_6451_cINF__insert,axiom,
% 14.25/8.13 ! [A: set_Product_unit,F: product_unit > nat,A2: product_unit] :
% 14.25/8.13 ( ( A != bot_bo3957492148770167129t_unit )
% 14.25/8.13 => ( ( condit1738341127787009408ow_nat @ ( image_875570014554754200it_nat @ F @ A ) )
% 14.25/8.13 => ( ( complete_Inf_Inf_nat @ ( image_875570014554754200it_nat @ F @ ( insert_Product_unit @ A2 @ A ) ) )
% 14.25/8.13 = ( inf_inf_nat @ ( F @ A2 ) @ ( complete_Inf_Inf_nat @ ( image_875570014554754200it_nat @ F @ A ) ) ) ) ) ) ).
% 14.25/8.13
% 14.25/8.13 thf(fact_6452_cINF__insert,axiom,
% 14.25/8.13 ! [A: set_o,F: $o > nat,A2: $o] :
% 14.25/8.13 ( ( A != bot_bot_set_o )
% 14.25/8.13 => ( ( condit1738341127787009408ow_nat @ ( image_o_nat2 @ F @ A ) )
% 14.25/8.13 => ( ( complete_Inf_Inf_nat @ ( image_o_nat2 @ F @ ( insert_o @ A2 @ A ) ) )
% 14.25/8.13 = ( inf_inf_nat @ ( F @ A2 ) @ ( complete_Inf_Inf_nat @ ( image_o_nat2 @ F @ A ) ) ) ) ) ) ).
% 14.25/8.13
% 14.25/8.13 thf(fact_6453_cINF__insert,axiom,
% 14.25/8.13 ! [A: set_nat,F: nat > nat,A2: nat] :
% 14.25/8.13 ( ( A != bot_bot_set_nat )
% 14.25/8.13 => ( ( condit1738341127787009408ow_nat @ ( image_nat_nat2 @ F @ A ) )
% 14.25/8.13 => ( ( complete_Inf_Inf_nat @ ( image_nat_nat2 @ F @ ( insert_nat @ A2 @ A ) ) )
% 14.25/8.13 = ( inf_inf_nat @ ( F @ A2 ) @ ( complete_Inf_Inf_nat @ ( image_nat_nat2 @ F @ A ) ) ) ) ) ) ).
% 14.25/8.13
% 14.25/8.13 thf(fact_6454_cINF__insert,axiom,
% 14.25/8.13 ! [A: set_int,F: int > nat,A2: int] :
% 14.25/8.13 ( ( A != bot_bot_set_int )
% 14.25/8.13 => ( ( condit1738341127787009408ow_nat @ ( image_int_nat2 @ F @ A ) )
% 14.25/8.13 => ( ( complete_Inf_Inf_nat @ ( image_int_nat2 @ F @ ( insert_int @ A2 @ A ) ) )
% 14.25/8.13 = ( inf_inf_nat @ ( F @ A2 ) @ ( complete_Inf_Inf_nat @ ( image_int_nat2 @ F @ A ) ) ) ) ) ) ).
% 14.25/8.13
% 14.25/8.13 thf(fact_6455_cINF__insert,axiom,
% 14.25/8.13 ! [A: set_Product_unit,F: product_unit > product_unit,A2: product_unit] :
% 14.25/8.13 ( ( A != bot_bo3957492148770167129t_unit )
% 14.25/8.13 => ( ( condit9161856309968323677t_unit @ ( image_405062704495631173t_unit @ F @ A ) )
% 14.25/8.13 => ( ( comple2584293577114468500t_unit @ ( image_405062704495631173t_unit @ F @ ( insert_Product_unit @ A2 @ A ) ) )
% 14.25/8.13 = ( inf_inf_Product_unit @ ( F @ A2 ) @ ( comple2584293577114468500t_unit @ ( image_405062704495631173t_unit @ F @ A ) ) ) ) ) ) ).
% 14.25/8.13
% 14.25/8.13 thf(fact_6456_cINF__insert,axiom,
% 14.25/8.13 ! [A: set_o,F: $o > product_unit,A2: $o] :
% 14.25/8.13 ( ( A != bot_bot_set_o )
% 14.25/8.13 => ( ( condit9161856309968323677t_unit @ ( image_o_Product_unit2 @ F @ A ) )
% 14.25/8.13 => ( ( comple2584293577114468500t_unit @ ( image_o_Product_unit2 @ F @ ( insert_o @ A2 @ A ) ) )
% 14.25/8.13 = ( inf_inf_Product_unit @ ( F @ A2 ) @ ( comple2584293577114468500t_unit @ ( image_o_Product_unit2 @ F @ A ) ) ) ) ) ) ).
% 14.25/8.13
% 14.25/8.13 thf(fact_6457_cINF__insert,axiom,
% 14.25/8.13 ! [A: set_nat,F: nat > product_unit,A2: nat] :
% 14.25/8.13 ( ( A != bot_bot_set_nat )
% 14.25/8.13 => ( ( condit9161856309968323677t_unit @ ( image_8730104196221521654t_unit @ F @ A ) )
% 14.25/8.13 => ( ( comple2584293577114468500t_unit @ ( image_8730104196221521654t_unit @ F @ ( insert_nat @ A2 @ A ) ) )
% 14.25/8.13 = ( inf_inf_Product_unit @ ( F @ A2 ) @ ( comple2584293577114468500t_unit @ ( image_8730104196221521654t_unit @ F @ A ) ) ) ) ) ) ).
% 14.25/8.13
% 14.25/8.13 thf(fact_6458_cINF__insert,axiom,
% 14.25/8.13 ! [A: set_int,F: int > product_unit,A2: int] :
% 14.25/8.13 ( ( A != bot_bot_set_int )
% 14.25/8.13 => ( ( condit9161856309968323677t_unit @ ( image_5846526143423701914t_unit @ F @ A ) )
% 14.25/8.13 => ( ( comple2584293577114468500t_unit @ ( image_5846526143423701914t_unit @ F @ ( insert_int @ A2 @ A ) ) )
% 14.25/8.13 = ( inf_inf_Product_unit @ ( F @ A2 ) @ ( comple2584293577114468500t_unit @ ( image_5846526143423701914t_unit @ F @ A ) ) ) ) ) ) ).
% 14.25/8.13
% 14.25/8.13 thf(fact_6459_cINF__insert,axiom,
% 14.25/8.13 ! [A: set_Product_unit,F: product_unit > set_nat,A2: product_unit] :
% 14.25/8.13 ( ( A != bot_bo3957492148770167129t_unit )
% 14.25/8.13 => ( ( condit68592940725977398et_nat @ ( image_8173849991827986510et_nat @ F @ A ) )
% 14.25/8.13 => ( ( comple7806235888213564991et_nat @ ( image_8173849991827986510et_nat @ F @ ( insert_Product_unit @ A2 @ A ) ) )
% 14.25/8.13 = ( inf_inf_set_nat @ ( F @ A2 ) @ ( comple7806235888213564991et_nat @ ( image_8173849991827986510et_nat @ F @ A ) ) ) ) ) ) ).
% 14.25/8.13
% 14.25/8.13 thf(fact_6460_cINF__insert,axiom,
% 14.25/8.13 ! [A: set_o,F: $o > set_nat,A2: $o] :
% 14.25/8.13 ( ( A != bot_bot_set_o )
% 14.25/8.13 => ( ( condit68592940725977398et_nat @ ( image_o_set_nat @ F @ A ) )
% 14.25/8.13 => ( ( comple7806235888213564991et_nat @ ( image_o_set_nat @ F @ ( insert_o @ A2 @ A ) ) )
% 14.25/8.13 = ( inf_inf_set_nat @ ( F @ A2 ) @ ( comple7806235888213564991et_nat @ ( image_o_set_nat @ F @ A ) ) ) ) ) ) ).
% 14.25/8.13
% 14.25/8.13 thf(fact_6461_power__eq__if,axiom,
% 14.25/8.13 ( power_8256067586552552935nteger
% 14.25/8.13 = ( ^ [P5: code_integer,M3: nat] : ( if_Code_integer @ ( M3 = zero_zero_nat ) @ one_one_Code_integer @ ( times_3573771949741848930nteger @ P5 @ ( power_8256067586552552935nteger @ P5 @ ( minus_minus_nat @ M3 @ one_one_nat ) ) ) ) ) ) ).
% 14.25/8.13
% 14.25/8.13 thf(fact_6462_power__eq__if,axiom,
% 14.25/8.13 ( power_7079662738309270450atural
% 14.25/8.13 = ( ^ [P5: code_natural,M3: nat] : ( if_Code_natural @ ( M3 = zero_zero_nat ) @ one_one_Code_natural @ ( times_2397367101498566445atural @ P5 @ ( power_7079662738309270450atural @ P5 @ ( minus_minus_nat @ M3 @ one_one_nat ) ) ) ) ) ) ).
% 14.25/8.13
% 14.25/8.13 thf(fact_6463_power__eq__if,axiom,
% 14.25/8.13 ( power_power_assn
% 14.25/8.13 = ( ^ [P5: assn,M3: nat] : ( if_assn @ ( M3 = zero_zero_nat ) @ one_one_assn @ ( times_times_assn @ P5 @ ( power_power_assn @ P5 @ ( minus_minus_nat @ M3 @ one_one_nat ) ) ) ) ) ) ).
% 14.25/8.13
% 14.25/8.13 thf(fact_6464_power__eq__if,axiom,
% 14.25/8.13 ( power_power_rat
% 14.25/8.13 = ( ^ [P5: rat,M3: nat] : ( if_rat @ ( M3 = zero_zero_nat ) @ one_one_rat @ ( times_times_rat @ P5 @ ( power_power_rat @ P5 @ ( minus_minus_nat @ M3 @ one_one_nat ) ) ) ) ) ) ).
% 14.25/8.13
% 14.25/8.13 thf(fact_6465_power__eq__if,axiom,
% 14.25/8.13 ( power_power_nat
% 14.25/8.13 = ( ^ [P5: nat,M3: nat] : ( if_nat @ ( M3 = zero_zero_nat ) @ one_one_nat @ ( times_times_nat @ P5 @ ( power_power_nat @ P5 @ ( minus_minus_nat @ M3 @ one_one_nat ) ) ) ) ) ) ).
% 14.25/8.13
% 14.25/8.13 thf(fact_6466_power__eq__if,axiom,
% 14.25/8.13 ( power_power_int
% 14.25/8.13 = ( ^ [P5: int,M3: nat] : ( if_int @ ( M3 = zero_zero_nat ) @ one_one_int @ ( times_times_int @ P5 @ ( power_power_int @ P5 @ ( minus_minus_nat @ M3 @ one_one_nat ) ) ) ) ) ) ).
% 14.25/8.13
% 14.25/8.13 thf(fact_6467_power__minus__mult,axiom,
% 14.25/8.13 ! [N2: nat,A2: assn] :
% 14.25/8.13 ( ( ord_less_nat @ zero_zero_nat @ N2 )
% 14.25/8.13 => ( ( times_times_assn @ ( power_power_assn @ A2 @ ( minus_minus_nat @ N2 @ one_one_nat ) ) @ A2 )
% 14.25/8.13 = ( power_power_assn @ A2 @ N2 ) ) ) ).
% 14.25/8.13
% 14.25/8.13 thf(fact_6468_power__minus__mult,axiom,
% 14.25/8.13 ! [N2: nat,A2: rat] :
% 14.25/8.13 ( ( ord_less_nat @ zero_zero_nat @ N2 )
% 14.25/8.13 => ( ( times_times_rat @ ( power_power_rat @ A2 @ ( minus_minus_nat @ N2 @ one_one_nat ) ) @ A2 )
% 14.25/8.13 = ( power_power_rat @ A2 @ N2 ) ) ) ).
% 14.25/8.13
% 14.25/8.13 thf(fact_6469_power__minus__mult,axiom,
% 14.25/8.13 ! [N2: nat,A2: nat] :
% 14.25/8.13 ( ( ord_less_nat @ zero_zero_nat @ N2 )
% 14.25/8.13 => ( ( times_times_nat @ ( power_power_nat @ A2 @ ( minus_minus_nat @ N2 @ one_one_nat ) ) @ A2 )
% 14.25/8.13 = ( power_power_nat @ A2 @ N2 ) ) ) ).
% 14.25/8.13
% 14.25/8.13 thf(fact_6470_power__minus__mult,axiom,
% 14.25/8.13 ! [N2: nat,A2: int] :
% 14.25/8.13 ( ( ord_less_nat @ zero_zero_nat @ N2 )
% 14.25/8.13 => ( ( times_times_int @ ( power_power_int @ A2 @ ( minus_minus_nat @ N2 @ one_one_nat ) ) @ A2 )
% 14.25/8.13 = ( power_power_int @ A2 @ N2 ) ) ) ).
% 14.25/8.13
% 14.25/8.13 thf(fact_6471_cINF__union,axiom,
% 14.25/8.13 ! [A: set_Product_unit,F: product_unit > nat,B4: set_Product_unit] :
% 14.25/8.13 ( ( A != bot_bo3957492148770167129t_unit )
% 14.25/8.13 => ( ( condit1738341127787009408ow_nat @ ( image_875570014554754200it_nat @ F @ A ) )
% 14.25/8.13 => ( ( B4 != bot_bo3957492148770167129t_unit )
% 14.25/8.13 => ( ( condit1738341127787009408ow_nat @ ( image_875570014554754200it_nat @ F @ B4 ) )
% 14.25/8.13 => ( ( complete_Inf_Inf_nat @ ( image_875570014554754200it_nat @ F @ ( sup_su793286257634532545t_unit @ A @ B4 ) ) )
% 14.25/8.13 = ( inf_inf_nat @ ( complete_Inf_Inf_nat @ ( image_875570014554754200it_nat @ F @ A ) ) @ ( complete_Inf_Inf_nat @ ( image_875570014554754200it_nat @ F @ B4 ) ) ) ) ) ) ) ) ).
% 14.25/8.13
% 14.25/8.13 thf(fact_6472_cINF__union,axiom,
% 14.25/8.13 ! [A: set_o,F: $o > nat,B4: set_o] :
% 14.25/8.13 ( ( A != bot_bot_set_o )
% 14.25/8.13 => ( ( condit1738341127787009408ow_nat @ ( image_o_nat2 @ F @ A ) )
% 14.25/8.13 => ( ( B4 != bot_bot_set_o )
% 14.25/8.13 => ( ( condit1738341127787009408ow_nat @ ( image_o_nat2 @ F @ B4 ) )
% 14.25/8.13 => ( ( complete_Inf_Inf_nat @ ( image_o_nat2 @ F @ ( sup_sup_set_o @ A @ B4 ) ) )
% 14.25/8.13 = ( inf_inf_nat @ ( complete_Inf_Inf_nat @ ( image_o_nat2 @ F @ A ) ) @ ( complete_Inf_Inf_nat @ ( image_o_nat2 @ F @ B4 ) ) ) ) ) ) ) ) ).
% 14.25/8.13
% 14.25/8.13 thf(fact_6473_cINF__union,axiom,
% 14.25/8.13 ! [A: set_nat,F: nat > nat,B4: set_nat] :
% 14.25/8.13 ( ( A != bot_bot_set_nat )
% 14.25/8.13 => ( ( condit1738341127787009408ow_nat @ ( image_nat_nat2 @ F @ A ) )
% 14.25/8.13 => ( ( B4 != bot_bot_set_nat )
% 14.25/8.13 => ( ( condit1738341127787009408ow_nat @ ( image_nat_nat2 @ F @ B4 ) )
% 14.25/8.13 => ( ( complete_Inf_Inf_nat @ ( image_nat_nat2 @ F @ ( sup_sup_set_nat @ A @ B4 ) ) )
% 14.25/8.13 = ( inf_inf_nat @ ( complete_Inf_Inf_nat @ ( image_nat_nat2 @ F @ A ) ) @ ( complete_Inf_Inf_nat @ ( image_nat_nat2 @ F @ B4 ) ) ) ) ) ) ) ) ).
% 14.25/8.13
% 14.25/8.13 thf(fact_6474_cINF__union,axiom,
% 14.25/8.13 ! [A: set_int,F: int > nat,B4: set_int] :
% 14.25/8.13 ( ( A != bot_bot_set_int )
% 14.25/8.13 => ( ( condit1738341127787009408ow_nat @ ( image_int_nat2 @ F @ A ) )
% 14.25/8.13 => ( ( B4 != bot_bot_set_int )
% 14.25/8.13 => ( ( condit1738341127787009408ow_nat @ ( image_int_nat2 @ F @ B4 ) )
% 14.25/8.13 => ( ( complete_Inf_Inf_nat @ ( image_int_nat2 @ F @ ( sup_sup_set_int @ A @ B4 ) ) )
% 14.25/8.13 = ( inf_inf_nat @ ( complete_Inf_Inf_nat @ ( image_int_nat2 @ F @ A ) ) @ ( complete_Inf_Inf_nat @ ( image_int_nat2 @ F @ B4 ) ) ) ) ) ) ) ) ).
% 14.25/8.13
% 14.25/8.13 thf(fact_6475_cINF__union,axiom,
% 14.25/8.13 ! [A: set_Product_unit,F: product_unit > product_unit,B4: set_Product_unit] :
% 14.25/8.13 ( ( A != bot_bo3957492148770167129t_unit )
% 14.25/8.13 => ( ( condit9161856309968323677t_unit @ ( image_405062704495631173t_unit @ F @ A ) )
% 14.25/8.13 => ( ( B4 != bot_bo3957492148770167129t_unit )
% 14.25/8.13 => ( ( condit9161856309968323677t_unit @ ( image_405062704495631173t_unit @ F @ B4 ) )
% 14.25/8.13 => ( ( comple2584293577114468500t_unit @ ( image_405062704495631173t_unit @ F @ ( sup_su793286257634532545t_unit @ A @ B4 ) ) )
% 14.25/8.13 = ( inf_inf_Product_unit @ ( comple2584293577114468500t_unit @ ( image_405062704495631173t_unit @ F @ A ) ) @ ( comple2584293577114468500t_unit @ ( image_405062704495631173t_unit @ F @ B4 ) ) ) ) ) ) ) ) ).
% 14.25/8.13
% 14.25/8.13 thf(fact_6476_cINF__union,axiom,
% 14.25/8.13 ! [A: set_o,F: $o > product_unit,B4: set_o] :
% 14.25/8.13 ( ( A != bot_bot_set_o )
% 14.25/8.13 => ( ( condit9161856309968323677t_unit @ ( image_o_Product_unit2 @ F @ A ) )
% 14.25/8.13 => ( ( B4 != bot_bot_set_o )
% 14.25/8.13 => ( ( condit9161856309968323677t_unit @ ( image_o_Product_unit2 @ F @ B4 ) )
% 14.25/8.13 => ( ( comple2584293577114468500t_unit @ ( image_o_Product_unit2 @ F @ ( sup_sup_set_o @ A @ B4 ) ) )
% 14.25/8.13 = ( inf_inf_Product_unit @ ( comple2584293577114468500t_unit @ ( image_o_Product_unit2 @ F @ A ) ) @ ( comple2584293577114468500t_unit @ ( image_o_Product_unit2 @ F @ B4 ) ) ) ) ) ) ) ) ).
% 14.25/8.13
% 14.25/8.13 thf(fact_6477_cINF__union,axiom,
% 14.25/8.13 ! [A: set_nat,F: nat > product_unit,B4: set_nat] :
% 14.25/8.13 ( ( A != bot_bot_set_nat )
% 14.25/8.13 => ( ( condit9161856309968323677t_unit @ ( image_8730104196221521654t_unit @ F @ A ) )
% 14.25/8.13 => ( ( B4 != bot_bot_set_nat )
% 14.25/8.13 => ( ( condit9161856309968323677t_unit @ ( image_8730104196221521654t_unit @ F @ B4 ) )
% 14.25/8.13 => ( ( comple2584293577114468500t_unit @ ( image_8730104196221521654t_unit @ F @ ( sup_sup_set_nat @ A @ B4 ) ) )
% 14.25/8.13 = ( inf_inf_Product_unit @ ( comple2584293577114468500t_unit @ ( image_8730104196221521654t_unit @ F @ A ) ) @ ( comple2584293577114468500t_unit @ ( image_8730104196221521654t_unit @ F @ B4 ) ) ) ) ) ) ) ) ).
% 14.25/8.13
% 14.25/8.13 thf(fact_6478_cINF__union,axiom,
% 14.25/8.13 ! [A: set_int,F: int > product_unit,B4: set_int] :
% 14.25/8.13 ( ( A != bot_bot_set_int )
% 14.25/8.13 => ( ( condit9161856309968323677t_unit @ ( image_5846526143423701914t_unit @ F @ A ) )
% 14.25/8.13 => ( ( B4 != bot_bot_set_int )
% 14.25/8.13 => ( ( condit9161856309968323677t_unit @ ( image_5846526143423701914t_unit @ F @ B4 ) )
% 14.25/8.13 => ( ( comple2584293577114468500t_unit @ ( image_5846526143423701914t_unit @ F @ ( sup_sup_set_int @ A @ B4 ) ) )
% 14.25/8.13 = ( inf_inf_Product_unit @ ( comple2584293577114468500t_unit @ ( image_5846526143423701914t_unit @ F @ A ) ) @ ( comple2584293577114468500t_unit @ ( image_5846526143423701914t_unit @ F @ B4 ) ) ) ) ) ) ) ) ).
% 14.25/8.13
% 14.25/8.13 thf(fact_6479_cINF__union,axiom,
% 14.25/8.13 ! [A: set_Product_unit,F: product_unit > set_nat,B4: set_Product_unit] :
% 14.25/8.13 ( ( A != bot_bo3957492148770167129t_unit )
% 14.25/8.13 => ( ( condit68592940725977398et_nat @ ( image_8173849991827986510et_nat @ F @ A ) )
% 14.25/8.13 => ( ( B4 != bot_bo3957492148770167129t_unit )
% 14.25/8.13 => ( ( condit68592940725977398et_nat @ ( image_8173849991827986510et_nat @ F @ B4 ) )
% 14.25/8.13 => ( ( comple7806235888213564991et_nat @ ( image_8173849991827986510et_nat @ F @ ( sup_su793286257634532545t_unit @ A @ B4 ) ) )
% 14.25/8.13 = ( inf_inf_set_nat @ ( comple7806235888213564991et_nat @ ( image_8173849991827986510et_nat @ F @ A ) ) @ ( comple7806235888213564991et_nat @ ( image_8173849991827986510et_nat @ F @ B4 ) ) ) ) ) ) ) ) ).
% 14.25/8.13
% 14.25/8.13 thf(fact_6480_cINF__union,axiom,
% 14.25/8.13 ! [A: set_o,F: $o > set_nat,B4: set_o] :
% 14.25/8.13 ( ( A != bot_bot_set_o )
% 14.25/8.13 => ( ( condit68592940725977398et_nat @ ( image_o_set_nat @ F @ A ) )
% 14.25/8.13 => ( ( B4 != bot_bot_set_o )
% 14.25/8.13 => ( ( condit68592940725977398et_nat @ ( image_o_set_nat @ F @ B4 ) )
% 14.25/8.13 => ( ( comple7806235888213564991et_nat @ ( image_o_set_nat @ F @ ( sup_sup_set_o @ A @ B4 ) ) )
% 14.25/8.13 = ( inf_inf_set_nat @ ( comple7806235888213564991et_nat @ ( image_o_set_nat @ F @ A ) ) @ ( comple7806235888213564991et_nat @ ( image_o_set_nat @ F @ B4 ) ) ) ) ) ) ) ) ).
% 14.25/8.13
% 14.25/8.13 thf(fact_6481_sum_Oinsert,axiom,
% 14.25/8.13 ! [A: set_int,X: int,G: int > code_integer] :
% 14.25/8.13 ( ( finite_finite_int @ A )
% 14.25/8.13 => ( ~ ( member_int @ X @ A )
% 14.25/8.13 => ( ( groups7873554091576472773nteger @ G @ ( insert_int @ X @ A ) )
% 14.25/8.13 = ( plus_p5714425477246183910nteger @ ( G @ X ) @ ( groups7873554091576472773nteger @ G @ A ) ) ) ) ) ).
% 14.25/8.13
% 14.25/8.13 thf(fact_6482_sum_Oinsert,axiom,
% 14.25/8.13 ! [A: set_o,X: $o,G: $o > code_integer] :
% 14.25/8.13 ( ( finite_finite_o @ A )
% 14.25/8.13 => ( ~ ( member_o @ X @ A )
% 14.25/8.13 => ( ( groups4406642042086082107nteger @ G @ ( insert_o @ X @ A ) )
% 14.25/8.13 = ( plus_p5714425477246183910nteger @ ( G @ X ) @ ( groups4406642042086082107nteger @ G @ A ) ) ) ) ) ).
% 14.25/8.13
% 14.25/8.13 thf(fact_6483_sum_Oinsert,axiom,
% 14.25/8.13 ! [A: set_Product_unit,X: product_unit,G: product_unit > code_integer] :
% 14.25/8.13 ( ( finite4290736615968046902t_unit @ A )
% 14.25/8.13 => ( ~ ( member_Product_unit @ X @ A )
% 14.25/8.13 => ( ( groups5393439102872367408nteger @ G @ ( insert_Product_unit @ X @ A ) )
% 14.25/8.13 = ( plus_p5714425477246183910nteger @ ( G @ X ) @ ( groups5393439102872367408nteger @ G @ A ) ) ) ) ) ).
% 14.25/8.13
% 14.25/8.13 thf(fact_6484_sum_Oinsert,axiom,
% 14.25/8.13 ! [A: set_nat,X: nat,G: nat > code_integer] :
% 14.25/8.13 ( ( finite_finite_nat @ A )
% 14.25/8.13 => ( ~ ( member_nat @ X @ A )
% 14.25/8.13 => ( ( groups7501900531339628137nteger @ G @ ( insert_nat @ X @ A ) )
% 14.25/8.13 = ( plus_p5714425477246183910nteger @ ( G @ X ) @ ( groups7501900531339628137nteger @ G @ A ) ) ) ) ) ).
% 14.25/8.13
% 14.25/8.13 thf(fact_6485_sum_Oinsert,axiom,
% 14.25/8.13 ! [A: set_Code_integer,X: code_integer,G: code_integer > code_integer] :
% 14.25/8.13 ( ( finite6017078050557962740nteger @ A )
% 14.25/8.13 => ( ~ ( member_Code_integer @ X @ A )
% 14.25/8.13 => ( ( groups879477027807139574nteger @ G @ ( insert_Code_integer @ X @ A ) )
% 14.25/8.13 = ( plus_p5714425477246183910nteger @ ( G @ X ) @ ( groups879477027807139574nteger @ G @ A ) ) ) ) ) ).
% 14.25/8.13
% 14.25/8.13 thf(fact_6486_sum_Oinsert,axiom,
% 14.25/8.13 ! [A: set_int,X: int,G: int > rat] :
% 14.25/8.13 ( ( finite_finite_int @ A )
% 14.25/8.13 => ( ~ ( member_int @ X @ A )
% 14.25/8.13 => ( ( groups3906332499630173760nt_rat @ G @ ( insert_int @ X @ A ) )
% 14.25/8.13 = ( plus_plus_rat @ ( G @ X ) @ ( groups3906332499630173760nt_rat @ G @ A ) ) ) ) ) ).
% 14.25/8.13
% 14.25/8.13 thf(fact_6487_sum_Oinsert,axiom,
% 14.25/8.13 ! [A: set_o,X: $o,G: $o > rat] :
% 14.25/8.13 ( ( finite_finite_o @ A )
% 14.25/8.13 => ( ~ ( member_o @ X @ A )
% 14.25/8.13 => ( ( groups7872700643590313910_o_rat @ G @ ( insert_o @ X @ A ) )
% 14.25/8.13 = ( plus_plus_rat @ ( G @ X ) @ ( groups7872700643590313910_o_rat @ G @ A ) ) ) ) ) ).
% 14.25/8.13
% 14.25/8.13 thf(fact_6488_sum_Oinsert,axiom,
% 14.25/8.13 ! [A: set_Product_unit,X: product_unit,G: product_unit > rat] :
% 14.25/8.13 ( ( finite4290736615968046902t_unit @ A )
% 14.25/8.13 => ( ~ ( member_Product_unit @ X @ A )
% 14.25/8.13 => ( ( groups8131667117488461483it_rat @ G @ ( insert_Product_unit @ X @ A ) )
% 14.25/8.13 = ( plus_plus_rat @ ( G @ X ) @ ( groups8131667117488461483it_rat @ G @ A ) ) ) ) ) ).
% 14.25/8.13
% 14.25/8.13 thf(fact_6489_sum_Oinsert,axiom,
% 14.25/8.13 ! [A: set_nat,X: nat,G: nat > rat] :
% 14.25/8.13 ( ( finite_finite_nat @ A )
% 14.25/8.13 => ( ~ ( member_nat @ X @ A )
% 14.25/8.13 => ( ( groups2906978787729119204at_rat @ G @ ( insert_nat @ X @ A ) )
% 14.25/8.13 = ( plus_plus_rat @ ( G @ X ) @ ( groups2906978787729119204at_rat @ G @ A ) ) ) ) ) ).
% 14.25/8.13
% 14.25/8.13 thf(fact_6490_sum_Oinsert,axiom,
% 14.25/8.13 ! [A: set_Code_integer,X: code_integer,G: code_integer > rat] :
% 14.25/8.13 ( ( finite6017078050557962740nteger @ A )
% 14.25/8.13 => ( ~ ( member_Code_integer @ X @ A )
% 14.25/8.13 => ( ( groups6602215022474089585er_rat @ G @ ( insert_Code_integer @ X @ A ) )
% 14.25/8.13 = ( plus_plus_rat @ ( G @ X ) @ ( groups6602215022474089585er_rat @ G @ A ) ) ) ) ) ).
% 14.25/8.13
% 14.25/8.13 thf(fact_6491_sum_Oempty,axiom,
% 14.25/8.13 ! [G: product_unit > nat] :
% 14.25/8.13 ( ( groups8766797177574957219it_nat @ G @ bot_bo3957492148770167129t_unit )
% 14.25/8.13 = zero_zero_nat ) ).
% 14.25/8.13
% 14.25/8.13 thf(fact_6492_sum_Oempty,axiom,
% 14.25/8.13 ! [G: product_unit > int] :
% 14.25/8.13 ( ( groups8764306707065906943it_int @ G @ bot_bo3957492148770167129t_unit )
% 14.25/8.13 = zero_zero_int ) ).
% 14.25/8.13
% 14.25/8.13 thf(fact_6493_sum_Oempty,axiom,
% 14.25/8.13 ! [G: product_unit > code_integer] :
% 14.25/8.13 ( ( groups5393439102872367408nteger @ G @ bot_bo3957492148770167129t_unit )
% 14.25/8.13 = zero_z3403309356797280102nteger ) ).
% 14.25/8.13
% 14.25/8.13 thf(fact_6494_sum_Oempty,axiom,
% 14.25/8.13 ! [G: $o > nat] :
% 14.25/8.13 ( ( groups8507830703676809646_o_nat @ G @ bot_bot_set_o )
% 14.25/8.13 = zero_zero_nat ) ).
% 14.25/8.13
% 14.25/8.13 thf(fact_6495_sum_Oempty,axiom,
% 14.25/8.13 ! [G: $o > int] :
% 14.25/8.13 ( ( groups8505340233167759370_o_int @ G @ bot_bot_set_o )
% 14.25/8.13 = zero_zero_int ) ).
% 14.25/8.13
% 14.25/8.13 thf(fact_6496_sum_Oempty,axiom,
% 14.25/8.13 ! [G: $o > code_integer] :
% 14.25/8.13 ( ( groups4406642042086082107nteger @ G @ bot_bot_set_o )
% 14.25/8.13 = zero_z3403309356797280102nteger ) ).
% 14.25/8.13
% 14.25/8.13 thf(fact_6497_sum_Oempty,axiom,
% 14.25/8.13 ! [G: nat > int] :
% 14.25/8.13 ( ( groups3539618377306564664at_int @ G @ bot_bot_set_nat )
% 14.25/8.13 = zero_zero_int ) ).
% 14.25/8.13
% 14.25/8.13 thf(fact_6498_sum_Oempty,axiom,
% 14.25/8.13 ! [G: nat > code_integer] :
% 14.25/8.13 ( ( groups7501900531339628137nteger @ G @ bot_bot_set_nat )
% 14.25/8.13 = zero_z3403309356797280102nteger ) ).
% 14.25/8.13
% 14.25/8.13 thf(fact_6499_sum_Oempty,axiom,
% 14.25/8.13 ! [G: int > nat] :
% 14.25/8.13 ( ( groups4541462559716669496nt_nat @ G @ bot_bot_set_int )
% 14.25/8.13 = zero_zero_nat ) ).
% 14.25/8.13
% 14.25/8.13 thf(fact_6500_sum_Oempty,axiom,
% 14.25/8.13 ! [G: int > code_integer] :
% 14.25/8.13 ( ( groups7873554091576472773nteger @ G @ bot_bot_set_int )
% 14.25/8.13 = zero_z3403309356797280102nteger ) ).
% 14.25/8.13
% 14.25/8.13 thf(fact_6501_member__le__sum,axiom,
% 14.25/8.13 ! [I2: code_integer,A: set_Code_integer,F: code_integer > code_integer] :
% 14.25/8.13 ( ( member_Code_integer @ I2 @ A )
% 14.25/8.13 => ( ! [X3: code_integer] :
% 14.25/8.13 ( ( member_Code_integer @ X3 @ ( minus_2355218937544613996nteger @ A @ ( insert_Code_integer @ I2 @ bot_bo3990330152332043303nteger ) ) )
% 14.25/8.13 => ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ ( F @ X3 ) ) )
% 14.25/8.13 => ( ( finite6017078050557962740nteger @ A )
% 14.25/8.13 => ( ord_le3102999989581377725nteger @ ( F @ I2 ) @ ( groups879477027807139574nteger @ F @ A ) ) ) ) ) ).
% 14.25/8.13
% 14.25/8.13 thf(fact_6502_member__le__sum,axiom,
% 14.25/8.13 ! [I2: product_unit,A: set_Product_unit,F: product_unit > code_integer] :
% 14.25/8.13 ( ( member_Product_unit @ I2 @ A )
% 14.25/8.13 => ( ! [X3: product_unit] :
% 14.25/8.13 ( ( member_Product_unit @ X3 @ ( minus_6452836326544984404t_unit @ A @ ( insert_Product_unit @ I2 @ bot_bo3957492148770167129t_unit ) ) )
% 14.25/8.13 => ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ ( F @ X3 ) ) )
% 14.25/8.13 => ( ( finite4290736615968046902t_unit @ A )
% 14.25/8.13 => ( ord_le3102999989581377725nteger @ ( F @ I2 ) @ ( groups5393439102872367408nteger @ F @ A ) ) ) ) ) ).
% 14.25/8.13
% 14.25/8.13 thf(fact_6503_member__le__sum,axiom,
% 14.25/8.13 ! [I2: $o,A: set_o,F: $o > code_integer] :
% 14.25/8.13 ( ( member_o @ I2 @ A )
% 14.25/8.13 => ( ! [X3: $o] :
% 14.25/8.13 ( ( member_o @ X3 @ ( minus_minus_set_o @ A @ ( insert_o @ I2 @ bot_bot_set_o ) ) )
% 14.25/8.13 => ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ ( F @ X3 ) ) )
% 14.25/8.13 => ( ( finite_finite_o @ A )
% 14.25/8.13 => ( ord_le3102999989581377725nteger @ ( F @ I2 ) @ ( groups4406642042086082107nteger @ F @ A ) ) ) ) ) ).
% 14.25/8.13
% 14.25/8.13 thf(fact_6504_member__le__sum,axiom,
% 14.25/8.13 ! [I2: int,A: set_int,F: int > code_integer] :
% 14.25/8.13 ( ( member_int @ I2 @ A )
% 14.25/8.13 => ( ! [X3: int] :
% 14.25/8.13 ( ( member_int @ X3 @ ( minus_minus_set_int @ A @ ( insert_int @ I2 @ bot_bot_set_int ) ) )
% 14.25/8.13 => ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ ( F @ X3 ) ) )
% 14.25/8.13 => ( ( finite_finite_int @ A )
% 14.25/8.13 => ( ord_le3102999989581377725nteger @ ( F @ I2 ) @ ( groups7873554091576472773nteger @ F @ A ) ) ) ) ) ).
% 14.25/8.13
% 14.25/8.13 thf(fact_6505_member__le__sum,axiom,
% 14.25/8.13 ! [I2: nat,A: set_nat,F: nat > code_integer] :
% 14.25/8.13 ( ( member_nat @ I2 @ A )
% 14.25/8.13 => ( ! [X3: nat] :
% 14.25/8.13 ( ( member_nat @ X3 @ ( minus_minus_set_nat @ A @ ( insert_nat @ I2 @ bot_bot_set_nat ) ) )
% 14.25/8.13 => ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ ( F @ X3 ) ) )
% 14.25/8.13 => ( ( finite_finite_nat @ A )
% 14.25/8.13 => ( ord_le3102999989581377725nteger @ ( F @ I2 ) @ ( groups7501900531339628137nteger @ F @ A ) ) ) ) ) ).
% 14.25/8.13
% 14.25/8.13 thf(fact_6506_member__le__sum,axiom,
% 14.25/8.13 ! [I2: code_integer,A: set_Code_integer,F: code_integer > rat] :
% 14.25/8.13 ( ( member_Code_integer @ I2 @ A )
% 14.25/8.13 => ( ! [X3: code_integer] :
% 14.25/8.13 ( ( member_Code_integer @ X3 @ ( minus_2355218937544613996nteger @ A @ ( insert_Code_integer @ I2 @ bot_bo3990330152332043303nteger ) ) )
% 14.25/8.13 => ( ord_less_eq_rat @ zero_zero_rat @ ( F @ X3 ) ) )
% 14.25/8.13 => ( ( finite6017078050557962740nteger @ A )
% 14.25/8.13 => ( ord_less_eq_rat @ ( F @ I2 ) @ ( groups6602215022474089585er_rat @ F @ A ) ) ) ) ) ).
% 14.25/8.13
% 14.25/8.13 thf(fact_6507_member__le__sum,axiom,
% 14.25/8.13 ! [I2: product_unit,A: set_Product_unit,F: product_unit > rat] :
% 14.25/8.13 ( ( member_Product_unit @ I2 @ A )
% 14.25/8.13 => ( ! [X3: product_unit] :
% 14.25/8.13 ( ( member_Product_unit @ X3 @ ( minus_6452836326544984404t_unit @ A @ ( insert_Product_unit @ I2 @ bot_bo3957492148770167129t_unit ) ) )
% 14.25/8.13 => ( ord_less_eq_rat @ zero_zero_rat @ ( F @ X3 ) ) )
% 14.25/8.13 => ( ( finite4290736615968046902t_unit @ A )
% 14.25/8.13 => ( ord_less_eq_rat @ ( F @ I2 ) @ ( groups8131667117488461483it_rat @ F @ A ) ) ) ) ) ).
% 14.25/8.13
% 14.25/8.13 thf(fact_6508_member__le__sum,axiom,
% 14.25/8.13 ! [I2: $o,A: set_o,F: $o > rat] :
% 14.25/8.13 ( ( member_o @ I2 @ A )
% 14.25/8.13 => ( ! [X3: $o] :
% 14.25/8.13 ( ( member_o @ X3 @ ( minus_minus_set_o @ A @ ( insert_o @ I2 @ bot_bot_set_o ) ) )
% 14.25/8.13 => ( ord_less_eq_rat @ zero_zero_rat @ ( F @ X3 ) ) )
% 14.25/8.13 => ( ( finite_finite_o @ A )
% 14.25/8.13 => ( ord_less_eq_rat @ ( F @ I2 ) @ ( groups7872700643590313910_o_rat @ F @ A ) ) ) ) ) ).
% 14.25/8.13
% 14.25/8.13 thf(fact_6509_member__le__sum,axiom,
% 14.25/8.13 ! [I2: int,A: set_int,F: int > rat] :
% 14.25/8.13 ( ( member_int @ I2 @ A )
% 14.25/8.13 => ( ! [X3: int] :
% 14.25/8.13 ( ( member_int @ X3 @ ( minus_minus_set_int @ A @ ( insert_int @ I2 @ bot_bot_set_int ) ) )
% 14.25/8.13 => ( ord_less_eq_rat @ zero_zero_rat @ ( F @ X3 ) ) )
% 14.25/8.13 => ( ( finite_finite_int @ A )
% 14.25/8.13 => ( ord_less_eq_rat @ ( F @ I2 ) @ ( groups3906332499630173760nt_rat @ F @ A ) ) ) ) ) ).
% 14.25/8.13
% 14.25/8.13 thf(fact_6510_member__le__sum,axiom,
% 14.25/8.13 ! [I2: nat,A: set_nat,F: nat > rat] :
% 14.25/8.13 ( ( member_nat @ I2 @ A )
% 14.25/8.13 => ( ! [X3: nat] :
% 14.25/8.13 ( ( member_nat @ X3 @ ( minus_minus_set_nat @ A @ ( insert_nat @ I2 @ bot_bot_set_nat ) ) )
% 14.25/8.13 => ( ord_less_eq_rat @ zero_zero_rat @ ( F @ X3 ) ) )
% 14.25/8.13 => ( ( finite_finite_nat @ A )
% 14.25/8.13 => ( ord_less_eq_rat @ ( F @ I2 ) @ ( groups2906978787729119204at_rat @ F @ A ) ) ) ) ) ).
% 14.25/8.13
% 14.25/8.13 thf(fact_6511_sum_Ounion__disjoint,axiom,
% 14.25/8.13 ! [A: set_Code_integer,B4: set_Code_integer,G: code_integer > code_integer] :
% 14.25/8.13 ( ( finite6017078050557962740nteger @ A )
% 14.25/8.13 => ( ( finite6017078050557962740nteger @ B4 )
% 14.25/8.13 => ( ( ( inf_in1364745209274528805nteger @ A @ B4 )
% 14.25/8.13 = bot_bo3990330152332043303nteger )
% 14.25/8.13 => ( ( groups879477027807139574nteger @ G @ ( sup_su848401254843788991nteger @ A @ B4 ) )
% 14.25/8.13 = ( plus_p5714425477246183910nteger @ ( groups879477027807139574nteger @ G @ A ) @ ( groups879477027807139574nteger @ G @ B4 ) ) ) ) ) ) ).
% 14.25/8.13
% 14.25/8.13 thf(fact_6512_sum_Ounion__disjoint,axiom,
% 14.25/8.13 ! [A: set_Code_integer,B4: set_Code_integer,G: code_integer > rat] :
% 14.25/8.13 ( ( finite6017078050557962740nteger @ A )
% 14.25/8.13 => ( ( finite6017078050557962740nteger @ B4 )
% 14.25/8.13 => ( ( ( inf_in1364745209274528805nteger @ A @ B4 )
% 14.25/8.13 = bot_bo3990330152332043303nteger )
% 14.25/8.13 => ( ( groups6602215022474089585er_rat @ G @ ( sup_su848401254843788991nteger @ A @ B4 ) )
% 14.25/8.13 = ( plus_plus_rat @ ( groups6602215022474089585er_rat @ G @ A ) @ ( groups6602215022474089585er_rat @ G @ B4 ) ) ) ) ) ) ).
% 14.25/8.13
% 14.25/8.13 thf(fact_6513_sum_Ounion__disjoint,axiom,
% 14.25/8.13 ! [A: set_Code_integer,B4: set_Code_integer,G: code_integer > nat] :
% 14.25/8.13 ( ( finite6017078050557962740nteger @ A )
% 14.25/8.13 => ( ( finite6017078050557962740nteger @ B4 )
% 14.25/8.13 => ( ( ( inf_in1364745209274528805nteger @ A @ B4 )
% 14.25/8.13 = bot_bo3990330152332043303nteger )
% 14.25/8.13 => ( ( groups7237345082560585321er_nat @ G @ ( sup_su848401254843788991nteger @ A @ B4 ) )
% 14.25/8.13 = ( plus_plus_nat @ ( groups7237345082560585321er_nat @ G @ A ) @ ( groups7237345082560585321er_nat @ G @ B4 ) ) ) ) ) ) ).
% 14.25/8.13
% 14.25/8.13 thf(fact_6514_sum_Ounion__disjoint,axiom,
% 14.25/8.13 ! [A: set_Code_integer,B4: set_Code_integer,G: code_integer > int] :
% 14.25/8.13 ( ( finite6017078050557962740nteger @ A )
% 14.25/8.13 => ( ( finite6017078050557962740nteger @ B4 )
% 14.25/8.13 => ( ( ( inf_in1364745209274528805nteger @ A @ B4 )
% 14.25/8.13 = bot_bo3990330152332043303nteger )
% 14.25/8.13 => ( ( groups7234854612051535045er_int @ G @ ( sup_su848401254843788991nteger @ A @ B4 ) )
% 14.25/8.13 = ( plus_plus_int @ ( groups7234854612051535045er_int @ G @ A ) @ ( groups7234854612051535045er_int @ G @ B4 ) ) ) ) ) ) ).
% 14.25/8.13
% 14.25/8.13 thf(fact_6515_sum_Ounion__disjoint,axiom,
% 14.25/8.13 ! [A: set_Product_unit,B4: set_Product_unit,G: product_unit > code_integer] :
% 14.25/8.13 ( ( finite4290736615968046902t_unit @ A )
% 14.25/8.13 => ( ( finite4290736615968046902t_unit @ B4 )
% 14.25/8.13 => ( ( ( inf_in4660618365625256667t_unit @ A @ B4 )
% 14.25/8.13 = bot_bo3957492148770167129t_unit )
% 14.25/8.13 => ( ( groups5393439102872367408nteger @ G @ ( sup_su793286257634532545t_unit @ A @ B4 ) )
% 14.25/8.13 = ( plus_p5714425477246183910nteger @ ( groups5393439102872367408nteger @ G @ A ) @ ( groups5393439102872367408nteger @ G @ B4 ) ) ) ) ) ) ).
% 14.25/8.13
% 14.25/8.13 thf(fact_6516_sum_Ounion__disjoint,axiom,
% 14.25/8.13 ! [A: set_Product_unit,B4: set_Product_unit,G: product_unit > rat] :
% 14.25/8.13 ( ( finite4290736615968046902t_unit @ A )
% 14.25/8.13 => ( ( finite4290736615968046902t_unit @ B4 )
% 14.25/8.13 => ( ( ( inf_in4660618365625256667t_unit @ A @ B4 )
% 14.25/8.13 = bot_bo3957492148770167129t_unit )
% 14.25/8.13 => ( ( groups8131667117488461483it_rat @ G @ ( sup_su793286257634532545t_unit @ A @ B4 ) )
% 14.25/8.13 = ( plus_plus_rat @ ( groups8131667117488461483it_rat @ G @ A ) @ ( groups8131667117488461483it_rat @ G @ B4 ) ) ) ) ) ) ).
% 14.25/8.13
% 14.25/8.13 thf(fact_6517_sum_Ounion__disjoint,axiom,
% 14.25/8.13 ! [A: set_Product_unit,B4: set_Product_unit,G: product_unit > nat] :
% 14.25/8.13 ( ( finite4290736615968046902t_unit @ A )
% 14.25/8.13 => ( ( finite4290736615968046902t_unit @ B4 )
% 14.25/8.13 => ( ( ( inf_in4660618365625256667t_unit @ A @ B4 )
% 14.25/8.13 = bot_bo3957492148770167129t_unit )
% 14.25/8.13 => ( ( groups8766797177574957219it_nat @ G @ ( sup_su793286257634532545t_unit @ A @ B4 ) )
% 14.25/8.13 = ( plus_plus_nat @ ( groups8766797177574957219it_nat @ G @ A ) @ ( groups8766797177574957219it_nat @ G @ B4 ) ) ) ) ) ) ).
% 14.25/8.13
% 14.25/8.13 thf(fact_6518_sum_Ounion__disjoint,axiom,
% 14.25/8.13 ! [A: set_Product_unit,B4: set_Product_unit,G: product_unit > int] :
% 14.25/8.13 ( ( finite4290736615968046902t_unit @ A )
% 14.25/8.13 => ( ( finite4290736615968046902t_unit @ B4 )
% 14.25/8.13 => ( ( ( inf_in4660618365625256667t_unit @ A @ B4 )
% 14.25/8.13 = bot_bo3957492148770167129t_unit )
% 14.25/8.13 => ( ( groups8764306707065906943it_int @ G @ ( sup_su793286257634532545t_unit @ A @ B4 ) )
% 14.25/8.13 = ( plus_plus_int @ ( groups8764306707065906943it_int @ G @ A ) @ ( groups8764306707065906943it_int @ G @ B4 ) ) ) ) ) ) ).
% 14.25/8.13
% 14.25/8.13 thf(fact_6519_sum_Ounion__disjoint,axiom,
% 14.25/8.13 ! [A: set_o,B4: set_o,G: $o > code_integer] :
% 14.25/8.13 ( ( finite_finite_o @ A )
% 14.25/8.13 => ( ( finite_finite_o @ B4 )
% 14.25/8.13 => ( ( ( inf_inf_set_o @ A @ B4 )
% 14.25/8.13 = bot_bot_set_o )
% 14.25/8.13 => ( ( groups4406642042086082107nteger @ G @ ( sup_sup_set_o @ A @ B4 ) )
% 14.25/8.13 = ( plus_p5714425477246183910nteger @ ( groups4406642042086082107nteger @ G @ A ) @ ( groups4406642042086082107nteger @ G @ B4 ) ) ) ) ) ) ).
% 14.25/8.13
% 14.25/8.13 thf(fact_6520_sum_Ounion__disjoint,axiom,
% 14.25/8.13 ! [A: set_o,B4: set_o,G: $o > rat] :
% 14.25/8.13 ( ( finite_finite_o @ A )
% 14.25/8.13 => ( ( finite_finite_o @ B4 )
% 14.25/8.13 => ( ( ( inf_inf_set_o @ A @ B4 )
% 14.25/8.13 = bot_bot_set_o )
% 14.25/8.13 => ( ( groups7872700643590313910_o_rat @ G @ ( sup_sup_set_o @ A @ B4 ) )
% 14.25/8.13 = ( plus_plus_rat @ ( groups7872700643590313910_o_rat @ G @ A ) @ ( groups7872700643590313910_o_rat @ G @ B4 ) ) ) ) ) ) ).
% 14.25/8.13
% 14.25/8.13 thf(fact_6521_sum__diff1,axiom,
% 14.25/8.13 ! [A: set_Code_integer,A2: code_integer,F: code_integer > rat] :
% 14.25/8.13 ( ( finite6017078050557962740nteger @ A )
% 14.25/8.13 => ( ( ( member_Code_integer @ A2 @ A )
% 14.25/8.13 => ( ( groups6602215022474089585er_rat @ F @ ( minus_2355218937544613996nteger @ A @ ( insert_Code_integer @ A2 @ bot_bo3990330152332043303nteger ) ) )
% 14.25/8.13 = ( minus_minus_rat @ ( groups6602215022474089585er_rat @ F @ A ) @ ( F @ A2 ) ) ) )
% 14.25/8.13 & ( ~ ( member_Code_integer @ A2 @ A )
% 14.25/8.13 => ( ( groups6602215022474089585er_rat @ F @ ( minus_2355218937544613996nteger @ A @ ( insert_Code_integer @ A2 @ bot_bo3990330152332043303nteger ) ) )
% 14.25/8.13 = ( groups6602215022474089585er_rat @ F @ A ) ) ) ) ) ).
% 14.25/8.13
% 14.25/8.13 thf(fact_6522_sum__diff1,axiom,
% 14.25/8.13 ! [A: set_Product_unit,A2: product_unit,F: product_unit > rat] :
% 14.25/8.13 ( ( finite4290736615968046902t_unit @ A )
% 14.25/8.13 => ( ( ( member_Product_unit @ A2 @ A )
% 14.25/8.13 => ( ( groups8131667117488461483it_rat @ F @ ( minus_6452836326544984404t_unit @ A @ ( insert_Product_unit @ A2 @ bot_bo3957492148770167129t_unit ) ) )
% 14.25/8.13 = ( minus_minus_rat @ ( groups8131667117488461483it_rat @ F @ A ) @ ( F @ A2 ) ) ) )
% 14.25/8.13 & ( ~ ( member_Product_unit @ A2 @ A )
% 14.25/8.13 => ( ( groups8131667117488461483it_rat @ F @ ( minus_6452836326544984404t_unit @ A @ ( insert_Product_unit @ A2 @ bot_bo3957492148770167129t_unit ) ) )
% 14.25/8.13 = ( groups8131667117488461483it_rat @ F @ A ) ) ) ) ) ).
% 14.25/8.13
% 14.25/8.13 thf(fact_6523_sum__diff1,axiom,
% 14.25/8.13 ! [A: set_o,A2: $o,F: $o > rat] :
% 14.25/8.13 ( ( finite_finite_o @ A )
% 14.25/8.13 => ( ( ( member_o @ A2 @ A )
% 14.25/8.13 => ( ( groups7872700643590313910_o_rat @ F @ ( minus_minus_set_o @ A @ ( insert_o @ A2 @ bot_bot_set_o ) ) )
% 14.25/8.13 = ( minus_minus_rat @ ( groups7872700643590313910_o_rat @ F @ A ) @ ( F @ A2 ) ) ) )
% 14.25/8.13 & ( ~ ( member_o @ A2 @ A )
% 14.25/8.13 => ( ( groups7872700643590313910_o_rat @ F @ ( minus_minus_set_o @ A @ ( insert_o @ A2 @ bot_bot_set_o ) ) )
% 14.25/8.13 = ( groups7872700643590313910_o_rat @ F @ A ) ) ) ) ) ).
% 14.25/8.13
% 14.25/8.13 thf(fact_6524_sum__diff1,axiom,
% 14.25/8.13 ! [A: set_int,A2: int,F: int > rat] :
% 14.25/8.13 ( ( finite_finite_int @ A )
% 14.25/8.13 => ( ( ( member_int @ A2 @ A )
% 14.25/8.13 => ( ( groups3906332499630173760nt_rat @ F @ ( minus_minus_set_int @ A @ ( insert_int @ A2 @ bot_bot_set_int ) ) )
% 14.25/8.13 = ( minus_minus_rat @ ( groups3906332499630173760nt_rat @ F @ A ) @ ( F @ A2 ) ) ) )
% 14.25/8.13 & ( ~ ( member_int @ A2 @ A )
% 14.25/8.13 => ( ( groups3906332499630173760nt_rat @ F @ ( minus_minus_set_int @ A @ ( insert_int @ A2 @ bot_bot_set_int ) ) )
% 14.25/8.13 = ( groups3906332499630173760nt_rat @ F @ A ) ) ) ) ) ).
% 14.25/8.13
% 14.25/8.13 thf(fact_6525_sum__diff1,axiom,
% 14.25/8.13 ! [A: set_Code_integer,A2: code_integer,F: code_integer > int] :
% 14.25/8.13 ( ( finite6017078050557962740nteger @ A )
% 14.25/8.13 => ( ( ( member_Code_integer @ A2 @ A )
% 14.25/8.13 => ( ( groups7234854612051535045er_int @ F @ ( minus_2355218937544613996nteger @ A @ ( insert_Code_integer @ A2 @ bot_bo3990330152332043303nteger ) ) )
% 14.25/8.13 = ( minus_minus_int @ ( groups7234854612051535045er_int @ F @ A ) @ ( F @ A2 ) ) ) )
% 14.25/8.13 & ( ~ ( member_Code_integer @ A2 @ A )
% 14.25/8.13 => ( ( groups7234854612051535045er_int @ F @ ( minus_2355218937544613996nteger @ A @ ( insert_Code_integer @ A2 @ bot_bo3990330152332043303nteger ) ) )
% 14.25/8.13 = ( groups7234854612051535045er_int @ F @ A ) ) ) ) ) ).
% 14.25/8.13
% 14.25/8.13 thf(fact_6526_sum__diff1,axiom,
% 14.25/8.13 ! [A: set_Product_unit,A2: product_unit,F: product_unit > int] :
% 14.25/8.13 ( ( finite4290736615968046902t_unit @ A )
% 14.25/8.13 => ( ( ( member_Product_unit @ A2 @ A )
% 14.25/8.13 => ( ( groups8764306707065906943it_int @ F @ ( minus_6452836326544984404t_unit @ A @ ( insert_Product_unit @ A2 @ bot_bo3957492148770167129t_unit ) ) )
% 14.25/8.13 = ( minus_minus_int @ ( groups8764306707065906943it_int @ F @ A ) @ ( F @ A2 ) ) ) )
% 14.25/8.13 & ( ~ ( member_Product_unit @ A2 @ A )
% 14.25/8.13 => ( ( groups8764306707065906943it_int @ F @ ( minus_6452836326544984404t_unit @ A @ ( insert_Product_unit @ A2 @ bot_bo3957492148770167129t_unit ) ) )
% 14.25/8.13 = ( groups8764306707065906943it_int @ F @ A ) ) ) ) ) ).
% 14.25/8.13
% 14.25/8.13 thf(fact_6527_sum__diff1,axiom,
% 14.25/8.13 ! [A: set_o,A2: $o,F: $o > int] :
% 14.25/8.13 ( ( finite_finite_o @ A )
% 14.25/8.13 => ( ( ( member_o @ A2 @ A )
% 14.25/8.13 => ( ( groups8505340233167759370_o_int @ F @ ( minus_minus_set_o @ A @ ( insert_o @ A2 @ bot_bot_set_o ) ) )
% 14.25/8.13 = ( minus_minus_int @ ( groups8505340233167759370_o_int @ F @ A ) @ ( F @ A2 ) ) ) )
% 14.25/8.13 & ( ~ ( member_o @ A2 @ A )
% 14.25/8.13 => ( ( groups8505340233167759370_o_int @ F @ ( minus_minus_set_o @ A @ ( insert_o @ A2 @ bot_bot_set_o ) ) )
% 14.25/8.13 = ( groups8505340233167759370_o_int @ F @ A ) ) ) ) ) ).
% 14.25/8.13
% 14.25/8.13 thf(fact_6528_sum__diff1,axiom,
% 14.25/8.13 ! [A: set_nat,A2: nat,F: nat > rat] :
% 14.25/8.13 ( ( finite_finite_nat @ A )
% 14.25/8.13 => ( ( ( member_nat @ A2 @ A )
% 14.25/8.13 => ( ( groups2906978787729119204at_rat @ F @ ( minus_minus_set_nat @ A @ ( insert_nat @ A2 @ bot_bot_set_nat ) ) )
% 14.25/8.13 = ( minus_minus_rat @ ( groups2906978787729119204at_rat @ F @ A ) @ ( F @ A2 ) ) ) )
% 14.25/8.13 & ( ~ ( member_nat @ A2 @ A )
% 14.25/8.13 => ( ( groups2906978787729119204at_rat @ F @ ( minus_minus_set_nat @ A @ ( insert_nat @ A2 @ bot_bot_set_nat ) ) )
% 14.25/8.13 = ( groups2906978787729119204at_rat @ F @ A ) ) ) ) ) ).
% 14.25/8.13
% 14.25/8.13 thf(fact_6529_sum__diff1,axiom,
% 14.25/8.13 ! [A: set_nat,A2: nat,F: nat > int] :
% 14.25/8.13 ( ( finite_finite_nat @ A )
% 14.25/8.13 => ( ( ( member_nat @ A2 @ A )
% 14.25/8.13 => ( ( groups3539618377306564664at_int @ F @ ( minus_minus_set_nat @ A @ ( insert_nat @ A2 @ bot_bot_set_nat ) ) )
% 14.25/8.13 = ( minus_minus_int @ ( groups3539618377306564664at_int @ F @ A ) @ ( F @ A2 ) ) ) )
% 14.25/8.13 & ( ~ ( member_nat @ A2 @ A )
% 14.25/8.13 => ( ( groups3539618377306564664at_int @ F @ ( minus_minus_set_nat @ A @ ( insert_nat @ A2 @ bot_bot_set_nat ) ) )
% 14.25/8.13 = ( groups3539618377306564664at_int @ F @ A ) ) ) ) ) ).
% 14.25/8.13
% 14.25/8.13 thf(fact_6530_sum__diff1,axiom,
% 14.25/8.13 ! [A: set_int,A2: int,F: int > int] :
% 14.25/8.13 ( ( finite_finite_int @ A )
% 14.25/8.13 => ( ( ( member_int @ A2 @ A )
% 14.25/8.13 => ( ( groups4538972089207619220nt_int @ F @ ( minus_minus_set_int @ A @ ( insert_int @ A2 @ bot_bot_set_int ) ) )
% 14.25/8.13 = ( minus_minus_int @ ( groups4538972089207619220nt_int @ F @ A ) @ ( F @ A2 ) ) ) )
% 14.25/8.13 & ( ~ ( member_int @ A2 @ A )
% 14.25/8.13 => ( ( groups4538972089207619220nt_int @ F @ ( minus_minus_set_int @ A @ ( insert_int @ A2 @ bot_bot_set_int ) ) )
% 14.25/8.13 = ( groups4538972089207619220nt_int @ F @ A ) ) ) ) ) ).
% 14.25/8.13
% 14.25/8.13 thf(fact_6531_sum_Oremove,axiom,
% 14.25/8.13 ! [A: set_Code_integer,X: code_integer,G: code_integer > code_integer] :
% 14.25/8.13 ( ( finite6017078050557962740nteger @ A )
% 14.25/8.13 => ( ( member_Code_integer @ X @ A )
% 14.25/8.13 => ( ( groups879477027807139574nteger @ G @ A )
% 14.25/8.13 = ( plus_p5714425477246183910nteger @ ( G @ X ) @ ( groups879477027807139574nteger @ G @ ( minus_2355218937544613996nteger @ A @ ( insert_Code_integer @ X @ bot_bo3990330152332043303nteger ) ) ) ) ) ) ) ).
% 14.25/8.13
% 14.25/8.13 thf(fact_6532_sum_Oremove,axiom,
% 14.25/8.13 ! [A: set_Code_integer,X: code_integer,G: code_integer > rat] :
% 14.25/8.13 ( ( finite6017078050557962740nteger @ A )
% 14.25/8.13 => ( ( member_Code_integer @ X @ A )
% 14.25/8.13 => ( ( groups6602215022474089585er_rat @ G @ A )
% 14.25/8.13 = ( plus_plus_rat @ ( G @ X ) @ ( groups6602215022474089585er_rat @ G @ ( minus_2355218937544613996nteger @ A @ ( insert_Code_integer @ X @ bot_bo3990330152332043303nteger ) ) ) ) ) ) ) ).
% 14.25/8.13
% 14.25/8.13 thf(fact_6533_sum_Oremove,axiom,
% 14.25/8.13 ! [A: set_Code_integer,X: code_integer,G: code_integer > nat] :
% 14.25/8.13 ( ( finite6017078050557962740nteger @ A )
% 14.25/8.13 => ( ( member_Code_integer @ X @ A )
% 14.25/8.13 => ( ( groups7237345082560585321er_nat @ G @ A )
% 14.25/8.13 = ( plus_plus_nat @ ( G @ X ) @ ( groups7237345082560585321er_nat @ G @ ( minus_2355218937544613996nteger @ A @ ( insert_Code_integer @ X @ bot_bo3990330152332043303nteger ) ) ) ) ) ) ) ).
% 14.25/8.13
% 14.25/8.13 thf(fact_6534_sum_Oremove,axiom,
% 14.25/8.13 ! [A: set_Code_integer,X: code_integer,G: code_integer > int] :
% 14.25/8.13 ( ( finite6017078050557962740nteger @ A )
% 14.25/8.13 => ( ( member_Code_integer @ X @ A )
% 14.25/8.13 => ( ( groups7234854612051535045er_int @ G @ A )
% 14.25/8.13 = ( plus_plus_int @ ( G @ X ) @ ( groups7234854612051535045er_int @ G @ ( minus_2355218937544613996nteger @ A @ ( insert_Code_integer @ X @ bot_bo3990330152332043303nteger ) ) ) ) ) ) ) ).
% 14.25/8.13
% 14.25/8.13 thf(fact_6535_sum_Oremove,axiom,
% 14.25/8.13 ! [A: set_Product_unit,X: product_unit,G: product_unit > code_integer] :
% 14.25/8.13 ( ( finite4290736615968046902t_unit @ A )
% 14.25/8.13 => ( ( member_Product_unit @ X @ A )
% 14.25/8.13 => ( ( groups5393439102872367408nteger @ G @ A )
% 14.25/8.13 = ( plus_p5714425477246183910nteger @ ( G @ X ) @ ( groups5393439102872367408nteger @ G @ ( minus_6452836326544984404t_unit @ A @ ( insert_Product_unit @ X @ bot_bo3957492148770167129t_unit ) ) ) ) ) ) ) ).
% 14.25/8.13
% 14.25/8.13 thf(fact_6536_sum_Oremove,axiom,
% 14.25/8.13 ! [A: set_Product_unit,X: product_unit,G: product_unit > rat] :
% 14.25/8.13 ( ( finite4290736615968046902t_unit @ A )
% 14.25/8.13 => ( ( member_Product_unit @ X @ A )
% 14.25/8.13 => ( ( groups8131667117488461483it_rat @ G @ A )
% 14.25/8.13 = ( plus_plus_rat @ ( G @ X ) @ ( groups8131667117488461483it_rat @ G @ ( minus_6452836326544984404t_unit @ A @ ( insert_Product_unit @ X @ bot_bo3957492148770167129t_unit ) ) ) ) ) ) ) ).
% 14.25/8.13
% 14.25/8.13 thf(fact_6537_sum_Oremove,axiom,
% 14.25/8.13 ! [A: set_Product_unit,X: product_unit,G: product_unit > nat] :
% 14.25/8.13 ( ( finite4290736615968046902t_unit @ A )
% 14.25/8.13 => ( ( member_Product_unit @ X @ A )
% 14.25/8.13 => ( ( groups8766797177574957219it_nat @ G @ A )
% 14.25/8.13 = ( plus_plus_nat @ ( G @ X ) @ ( groups8766797177574957219it_nat @ G @ ( minus_6452836326544984404t_unit @ A @ ( insert_Product_unit @ X @ bot_bo3957492148770167129t_unit ) ) ) ) ) ) ) ).
% 14.25/8.13
% 14.25/8.13 thf(fact_6538_sum_Oremove,axiom,
% 14.25/8.13 ! [A: set_Product_unit,X: product_unit,G: product_unit > int] :
% 14.25/8.13 ( ( finite4290736615968046902t_unit @ A )
% 14.25/8.13 => ( ( member_Product_unit @ X @ A )
% 14.25/8.13 => ( ( groups8764306707065906943it_int @ G @ A )
% 14.25/8.13 = ( plus_plus_int @ ( G @ X ) @ ( groups8764306707065906943it_int @ G @ ( minus_6452836326544984404t_unit @ A @ ( insert_Product_unit @ X @ bot_bo3957492148770167129t_unit ) ) ) ) ) ) ) ).
% 14.25/8.13
% 14.25/8.13 thf(fact_6539_sum_Oremove,axiom,
% 14.25/8.13 ! [A: set_o,X: $o,G: $o > code_integer] :
% 14.25/8.13 ( ( finite_finite_o @ A )
% 14.25/8.13 => ( ( member_o @ X @ A )
% 14.25/8.13 => ( ( groups4406642042086082107nteger @ G @ A )
% 14.25/8.13 = ( plus_p5714425477246183910nteger @ ( G @ X ) @ ( groups4406642042086082107nteger @ G @ ( minus_minus_set_o @ A @ ( insert_o @ X @ bot_bot_set_o ) ) ) ) ) ) ) ).
% 14.25/8.13
% 14.25/8.13 thf(fact_6540_sum_Oremove,axiom,
% 14.25/8.13 ! [A: set_o,X: $o,G: $o > rat] :
% 14.25/8.13 ( ( finite_finite_o @ A )
% 14.25/8.13 => ( ( member_o @ X @ A )
% 14.25/8.13 => ( ( groups7872700643590313910_o_rat @ G @ A )
% 14.25/8.13 = ( plus_plus_rat @ ( G @ X ) @ ( groups7872700643590313910_o_rat @ G @ ( minus_minus_set_o @ A @ ( insert_o @ X @ bot_bot_set_o ) ) ) ) ) ) ) ).
% 14.25/8.13
% 14.25/8.13 thf(fact_6541_sum_Oinsert__remove,axiom,
% 14.25/8.13 ! [A: set_Code_integer,G: code_integer > code_integer,X: code_integer] :
% 14.25/8.13 ( ( finite6017078050557962740nteger @ A )
% 14.25/8.13 => ( ( groups879477027807139574nteger @ G @ ( insert_Code_integer @ X @ A ) )
% 14.25/8.13 = ( plus_p5714425477246183910nteger @ ( G @ X ) @ ( groups879477027807139574nteger @ G @ ( minus_2355218937544613996nteger @ A @ ( insert_Code_integer @ X @ bot_bo3990330152332043303nteger ) ) ) ) ) ) ).
% 14.25/8.13
% 14.25/8.13 thf(fact_6542_sum_Oinsert__remove,axiom,
% 14.25/8.13 ! [A: set_Code_integer,G: code_integer > rat,X: code_integer] :
% 14.25/8.13 ( ( finite6017078050557962740nteger @ A )
% 14.25/8.13 => ( ( groups6602215022474089585er_rat @ G @ ( insert_Code_integer @ X @ A ) )
% 14.25/8.13 = ( plus_plus_rat @ ( G @ X ) @ ( groups6602215022474089585er_rat @ G @ ( minus_2355218937544613996nteger @ A @ ( insert_Code_integer @ X @ bot_bo3990330152332043303nteger ) ) ) ) ) ) ).
% 14.25/8.13
% 14.25/8.13 thf(fact_6543_sum_Oinsert__remove,axiom,
% 14.25/8.13 ! [A: set_Code_integer,G: code_integer > nat,X: code_integer] :
% 14.25/8.13 ( ( finite6017078050557962740nteger @ A )
% 14.25/8.13 => ( ( groups7237345082560585321er_nat @ G @ ( insert_Code_integer @ X @ A ) )
% 14.25/8.13 = ( plus_plus_nat @ ( G @ X ) @ ( groups7237345082560585321er_nat @ G @ ( minus_2355218937544613996nteger @ A @ ( insert_Code_integer @ X @ bot_bo3990330152332043303nteger ) ) ) ) ) ) ).
% 14.25/8.13
% 14.25/8.13 thf(fact_6544_sum_Oinsert__remove,axiom,
% 14.25/8.13 ! [A: set_Code_integer,G: code_integer > int,X: code_integer] :
% 14.25/8.13 ( ( finite6017078050557962740nteger @ A )
% 14.25/8.13 => ( ( groups7234854612051535045er_int @ G @ ( insert_Code_integer @ X @ A ) )
% 14.25/8.13 = ( plus_plus_int @ ( G @ X ) @ ( groups7234854612051535045er_int @ G @ ( minus_2355218937544613996nteger @ A @ ( insert_Code_integer @ X @ bot_bo3990330152332043303nteger ) ) ) ) ) ) ).
% 14.25/8.13
% 14.25/8.13 thf(fact_6545_sum_Oinsert__remove,axiom,
% 14.25/8.13 ! [A: set_Product_unit,G: product_unit > code_integer,X: product_unit] :
% 14.25/8.13 ( ( finite4290736615968046902t_unit @ A )
% 14.25/8.13 => ( ( groups5393439102872367408nteger @ G @ ( insert_Product_unit @ X @ A ) )
% 14.25/8.13 = ( plus_p5714425477246183910nteger @ ( G @ X ) @ ( groups5393439102872367408nteger @ G @ ( minus_6452836326544984404t_unit @ A @ ( insert_Product_unit @ X @ bot_bo3957492148770167129t_unit ) ) ) ) ) ) ).
% 14.25/8.13
% 14.25/8.13 thf(fact_6546_sum_Oinsert__remove,axiom,
% 14.25/8.13 ! [A: set_Product_unit,G: product_unit > rat,X: product_unit] :
% 14.25/8.13 ( ( finite4290736615968046902t_unit @ A )
% 14.25/8.13 => ( ( groups8131667117488461483it_rat @ G @ ( insert_Product_unit @ X @ A ) )
% 14.25/8.13 = ( plus_plus_rat @ ( G @ X ) @ ( groups8131667117488461483it_rat @ G @ ( minus_6452836326544984404t_unit @ A @ ( insert_Product_unit @ X @ bot_bo3957492148770167129t_unit ) ) ) ) ) ) ).
% 14.25/8.13
% 14.25/8.13 thf(fact_6547_sum_Oinsert__remove,axiom,
% 14.25/8.13 ! [A: set_Product_unit,G: product_unit > nat,X: product_unit] :
% 14.25/8.13 ( ( finite4290736615968046902t_unit @ A )
% 14.25/8.13 => ( ( groups8766797177574957219it_nat @ G @ ( insert_Product_unit @ X @ A ) )
% 14.25/8.13 = ( plus_plus_nat @ ( G @ X ) @ ( groups8766797177574957219it_nat @ G @ ( minus_6452836326544984404t_unit @ A @ ( insert_Product_unit @ X @ bot_bo3957492148770167129t_unit ) ) ) ) ) ) ).
% 14.25/8.13
% 14.25/8.13 thf(fact_6548_sum_Oinsert__remove,axiom,
% 14.25/8.13 ! [A: set_Product_unit,G: product_unit > int,X: product_unit] :
% 14.25/8.13 ( ( finite4290736615968046902t_unit @ A )
% 14.25/8.13 => ( ( groups8764306707065906943it_int @ G @ ( insert_Product_unit @ X @ A ) )
% 14.25/8.13 = ( plus_plus_int @ ( G @ X ) @ ( groups8764306707065906943it_int @ G @ ( minus_6452836326544984404t_unit @ A @ ( insert_Product_unit @ X @ bot_bo3957492148770167129t_unit ) ) ) ) ) ) ).
% 14.25/8.13
% 14.25/8.13 thf(fact_6549_sum_Oinsert__remove,axiom,
% 14.25/8.13 ! [A: set_o,G: $o > code_integer,X: $o] :
% 14.25/8.13 ( ( finite_finite_o @ A )
% 14.25/8.13 => ( ( groups4406642042086082107nteger @ G @ ( insert_o @ X @ A ) )
% 14.25/8.13 = ( plus_p5714425477246183910nteger @ ( G @ X ) @ ( groups4406642042086082107nteger @ G @ ( minus_minus_set_o @ A @ ( insert_o @ X @ bot_bot_set_o ) ) ) ) ) ) ).
% 14.25/8.13
% 14.25/8.13 thf(fact_6550_sum_Oinsert__remove,axiom,
% 14.25/8.13 ! [A: set_o,G: $o > rat,X: $o] :
% 14.25/8.13 ( ( finite_finite_o @ A )
% 14.25/8.13 => ( ( groups7872700643590313910_o_rat @ G @ ( insert_o @ X @ A ) )
% 14.25/8.13 = ( plus_plus_rat @ ( G @ X ) @ ( groups7872700643590313910_o_rat @ G @ ( minus_minus_set_o @ A @ ( insert_o @ X @ bot_bot_set_o ) ) ) ) ) ) ).
% 14.25/8.13
% 14.25/8.13 thf(fact_6551_sum__strict__mono,axiom,
% 14.25/8.13 ! [A: set_Code_integer,F: code_integer > nat,G: code_integer > nat] :
% 14.25/8.13 ( ( finite6017078050557962740nteger @ A )
% 14.25/8.13 => ( ( A != bot_bo3990330152332043303nteger )
% 14.25/8.13 => ( ! [X3: code_integer] :
% 14.25/8.13 ( ( member_Code_integer @ X3 @ A )
% 14.25/8.13 => ( ord_less_nat @ ( F @ X3 ) @ ( G @ X3 ) ) )
% 14.25/8.13 => ( ord_less_nat @ ( groups7237345082560585321er_nat @ F @ A ) @ ( groups7237345082560585321er_nat @ G @ A ) ) ) ) ) ).
% 14.25/8.13
% 14.25/8.13 thf(fact_6552_sum__strict__mono,axiom,
% 14.25/8.13 ! [A: set_Product_unit,F: product_unit > nat,G: product_unit > nat] :
% 14.25/8.13 ( ( finite4290736615968046902t_unit @ A )
% 14.25/8.13 => ( ( A != bot_bo3957492148770167129t_unit )
% 14.25/8.13 => ( ! [X3: product_unit] :
% 14.25/8.13 ( ( member_Product_unit @ X3 @ A )
% 14.25/8.13 => ( ord_less_nat @ ( F @ X3 ) @ ( G @ X3 ) ) )
% 14.25/8.13 => ( ord_less_nat @ ( groups8766797177574957219it_nat @ F @ A ) @ ( groups8766797177574957219it_nat @ G @ A ) ) ) ) ) ).
% 14.25/8.13
% 14.25/8.13 thf(fact_6553_sum__strict__mono,axiom,
% 14.25/8.13 ! [A: set_o,F: $o > nat,G: $o > nat] :
% 14.25/8.13 ( ( finite_finite_o @ A )
% 14.25/8.13 => ( ( A != bot_bot_set_o )
% 14.25/8.13 => ( ! [X3: $o] :
% 14.25/8.13 ( ( member_o @ X3 @ A )
% 14.25/8.13 => ( ord_less_nat @ ( F @ X3 ) @ ( G @ X3 ) ) )
% 14.25/8.13 => ( ord_less_nat @ ( groups8507830703676809646_o_nat @ F @ A ) @ ( groups8507830703676809646_o_nat @ G @ A ) ) ) ) ) ).
% 14.25/8.13
% 14.25/8.13 thf(fact_6554_sum__strict__mono,axiom,
% 14.25/8.13 ! [A: set_int,F: int > nat,G: int > nat] :
% 14.25/8.13 ( ( finite_finite_int @ A )
% 14.25/8.13 => ( ( A != bot_bot_set_int )
% 14.25/8.13 => ( ! [X3: int] :
% 14.25/8.13 ( ( member_int @ X3 @ A )
% 14.25/8.13 => ( ord_less_nat @ ( F @ X3 ) @ ( G @ X3 ) ) )
% 14.25/8.13 => ( ord_less_nat @ ( groups4541462559716669496nt_nat @ F @ A ) @ ( groups4541462559716669496nt_nat @ G @ A ) ) ) ) ) ).
% 14.25/8.13
% 14.25/8.13 thf(fact_6555_sum__strict__mono,axiom,
% 14.25/8.13 ! [A: set_Code_integer,F: code_integer > int,G: code_integer > int] :
% 14.25/8.13 ( ( finite6017078050557962740nteger @ A )
% 14.25/8.13 => ( ( A != bot_bo3990330152332043303nteger )
% 14.25/8.13 => ( ! [X3: code_integer] :
% 14.25/8.13 ( ( member_Code_integer @ X3 @ A )
% 14.25/8.13 => ( ord_less_int @ ( F @ X3 ) @ ( G @ X3 ) ) )
% 14.25/8.13 => ( ord_less_int @ ( groups7234854612051535045er_int @ F @ A ) @ ( groups7234854612051535045er_int @ G @ A ) ) ) ) ) ).
% 14.25/8.13
% 14.25/8.13 thf(fact_6556_sum__strict__mono,axiom,
% 14.25/8.13 ! [A: set_Product_unit,F: product_unit > int,G: product_unit > int] :
% 14.25/8.13 ( ( finite4290736615968046902t_unit @ A )
% 14.25/8.13 => ( ( A != bot_bo3957492148770167129t_unit )
% 14.25/8.13 => ( ! [X3: product_unit] :
% 14.25/8.13 ( ( member_Product_unit @ X3 @ A )
% 14.25/8.13 => ( ord_less_int @ ( F @ X3 ) @ ( G @ X3 ) ) )
% 14.25/8.13 => ( ord_less_int @ ( groups8764306707065906943it_int @ F @ A ) @ ( groups8764306707065906943it_int @ G @ A ) ) ) ) ) ).
% 14.25/8.13
% 14.25/8.13 thf(fact_6557_sum__strict__mono,axiom,
% 14.25/8.13 ! [A: set_o,F: $o > int,G: $o > int] :
% 14.25/8.13 ( ( finite_finite_o @ A )
% 14.25/8.13 => ( ( A != bot_bot_set_o )
% 14.25/8.13 => ( ! [X3: $o] :
% 14.25/8.13 ( ( member_o @ X3 @ A )
% 14.25/8.13 => ( ord_less_int @ ( F @ X3 ) @ ( G @ X3 ) ) )
% 14.25/8.13 => ( ord_less_int @ ( groups8505340233167759370_o_int @ F @ A ) @ ( groups8505340233167759370_o_int @ G @ A ) ) ) ) ) ).
% 14.25/8.13
% 14.25/8.13 thf(fact_6558_sum__strict__mono,axiom,
% 14.25/8.13 ! [A: set_nat,F: nat > int,G: nat > int] :
% 14.25/8.13 ( ( finite_finite_nat @ A )
% 14.25/8.13 => ( ( A != bot_bot_set_nat )
% 14.25/8.13 => ( ! [X3: nat] :
% 14.25/8.13 ( ( member_nat @ X3 @ A )
% 14.25/8.13 => ( ord_less_int @ ( F @ X3 ) @ ( G @ X3 ) ) )
% 14.25/8.13 => ( ord_less_int @ ( groups3539618377306564664at_int @ F @ A ) @ ( groups3539618377306564664at_int @ G @ A ) ) ) ) ) ).
% 14.25/8.13
% 14.25/8.13 thf(fact_6559_sum__strict__mono,axiom,
% 14.25/8.13 ! [A: set_Code_integer,F: code_integer > code_integer,G: code_integer > code_integer] :
% 14.25/8.13 ( ( finite6017078050557962740nteger @ A )
% 14.25/8.13 => ( ( A != bot_bo3990330152332043303nteger )
% 14.25/8.13 => ( ! [X3: code_integer] :
% 14.25/8.13 ( ( member_Code_integer @ X3 @ A )
% 14.25/8.13 => ( ord_le6747313008572928689nteger @ ( F @ X3 ) @ ( G @ X3 ) ) )
% 14.25/8.13 => ( ord_le6747313008572928689nteger @ ( groups879477027807139574nteger @ F @ A ) @ ( groups879477027807139574nteger @ G @ A ) ) ) ) ) ).
% 14.25/8.13
% 14.25/8.13 thf(fact_6560_sum__strict__mono,axiom,
% 14.25/8.13 ! [A: set_Product_unit,F: product_unit > code_integer,G: product_unit > code_integer] :
% 14.25/8.13 ( ( finite4290736615968046902t_unit @ A )
% 14.25/8.13 => ( ( A != bot_bo3957492148770167129t_unit )
% 14.25/8.13 => ( ! [X3: product_unit] :
% 14.25/8.13 ( ( member_Product_unit @ X3 @ A )
% 14.25/8.13 => ( ord_le6747313008572928689nteger @ ( F @ X3 ) @ ( G @ X3 ) ) )
% 14.25/8.13 => ( ord_le6747313008572928689nteger @ ( groups5393439102872367408nteger @ F @ A ) @ ( groups5393439102872367408nteger @ G @ A ) ) ) ) ) ).
% 14.25/8.13
% 14.25/8.13 thf(fact_6561_sum_Oinsert__if,axiom,
% 14.25/8.13 ! [A: set_int,X: int,G: int > code_integer] :
% 14.25/8.13 ( ( finite_finite_int @ A )
% 14.25/8.13 => ( ( ( member_int @ X @ A )
% 14.25/8.13 => ( ( groups7873554091576472773nteger @ G @ ( insert_int @ X @ A ) )
% 14.25/8.13 = ( groups7873554091576472773nteger @ G @ A ) ) )
% 14.25/8.13 & ( ~ ( member_int @ X @ A )
% 14.25/8.13 => ( ( groups7873554091576472773nteger @ G @ ( insert_int @ X @ A ) )
% 14.25/8.13 = ( plus_p5714425477246183910nteger @ ( G @ X ) @ ( groups7873554091576472773nteger @ G @ A ) ) ) ) ) ) ).
% 14.25/8.13
% 14.25/8.13 thf(fact_6562_sum_Oinsert__if,axiom,
% 14.25/8.13 ! [A: set_o,X: $o,G: $o > code_integer] :
% 14.25/8.13 ( ( finite_finite_o @ A )
% 14.25/8.13 => ( ( ( member_o @ X @ A )
% 14.25/8.13 => ( ( groups4406642042086082107nteger @ G @ ( insert_o @ X @ A ) )
% 14.25/8.13 = ( groups4406642042086082107nteger @ G @ A ) ) )
% 14.25/8.13 & ( ~ ( member_o @ X @ A )
% 14.25/8.13 => ( ( groups4406642042086082107nteger @ G @ ( insert_o @ X @ A ) )
% 14.25/8.13 = ( plus_p5714425477246183910nteger @ ( G @ X ) @ ( groups4406642042086082107nteger @ G @ A ) ) ) ) ) ) ).
% 14.25/8.13
% 14.25/8.13 thf(fact_6563_sum_Oinsert__if,axiom,
% 14.25/8.13 ! [A: set_Product_unit,X: product_unit,G: product_unit > code_integer] :
% 14.25/8.13 ( ( finite4290736615968046902t_unit @ A )
% 14.25/8.13 => ( ( ( member_Product_unit @ X @ A )
% 14.25/8.13 => ( ( groups5393439102872367408nteger @ G @ ( insert_Product_unit @ X @ A ) )
% 14.25/8.13 = ( groups5393439102872367408nteger @ G @ A ) ) )
% 14.25/8.13 & ( ~ ( member_Product_unit @ X @ A )
% 14.25/8.13 => ( ( groups5393439102872367408nteger @ G @ ( insert_Product_unit @ X @ A ) )
% 14.25/8.13 = ( plus_p5714425477246183910nteger @ ( G @ X ) @ ( groups5393439102872367408nteger @ G @ A ) ) ) ) ) ) ).
% 14.25/8.13
% 14.25/8.13 thf(fact_6564_sum_Oinsert__if,axiom,
% 14.25/8.13 ! [A: set_nat,X: nat,G: nat > code_integer] :
% 14.25/8.13 ( ( finite_finite_nat @ A )
% 14.25/8.13 => ( ( ( member_nat @ X @ A )
% 14.25/8.13 => ( ( groups7501900531339628137nteger @ G @ ( insert_nat @ X @ A ) )
% 14.25/8.13 = ( groups7501900531339628137nteger @ G @ A ) ) )
% 14.25/8.13 & ( ~ ( member_nat @ X @ A )
% 14.25/8.13 => ( ( groups7501900531339628137nteger @ G @ ( insert_nat @ X @ A ) )
% 14.25/8.13 = ( plus_p5714425477246183910nteger @ ( G @ X ) @ ( groups7501900531339628137nteger @ G @ A ) ) ) ) ) ) ).
% 14.25/8.13
% 14.25/8.13 thf(fact_6565_sum_Oinsert__if,axiom,
% 14.25/8.13 ! [A: set_Code_integer,X: code_integer,G: code_integer > code_integer] :
% 14.25/8.13 ( ( finite6017078050557962740nteger @ A )
% 14.25/8.13 => ( ( ( member_Code_integer @ X @ A )
% 14.25/8.13 => ( ( groups879477027807139574nteger @ G @ ( insert_Code_integer @ X @ A ) )
% 14.25/8.13 = ( groups879477027807139574nteger @ G @ A ) ) )
% 14.25/8.13 & ( ~ ( member_Code_integer @ X @ A )
% 14.25/8.13 => ( ( groups879477027807139574nteger @ G @ ( insert_Code_integer @ X @ A ) )
% 14.25/8.13 = ( plus_p5714425477246183910nteger @ ( G @ X ) @ ( groups879477027807139574nteger @ G @ A ) ) ) ) ) ) ).
% 14.25/8.13
% 14.25/8.13 thf(fact_6566_sum_Oinsert__if,axiom,
% 14.25/8.13 ! [A: set_int,X: int,G: int > rat] :
% 14.25/8.13 ( ( finite_finite_int @ A )
% 14.25/8.13 => ( ( ( member_int @ X @ A )
% 14.25/8.13 => ( ( groups3906332499630173760nt_rat @ G @ ( insert_int @ X @ A ) )
% 14.25/8.13 = ( groups3906332499630173760nt_rat @ G @ A ) ) )
% 14.25/8.13 & ( ~ ( member_int @ X @ A )
% 14.25/8.13 => ( ( groups3906332499630173760nt_rat @ G @ ( insert_int @ X @ A ) )
% 14.25/8.13 = ( plus_plus_rat @ ( G @ X ) @ ( groups3906332499630173760nt_rat @ G @ A ) ) ) ) ) ) ).
% 14.25/8.13
% 14.25/8.13 thf(fact_6567_sum_Oinsert__if,axiom,
% 14.25/8.13 ! [A: set_o,X: $o,G: $o > rat] :
% 14.25/8.13 ( ( finite_finite_o @ A )
% 14.25/8.13 => ( ( ( member_o @ X @ A )
% 14.25/8.13 => ( ( groups7872700643590313910_o_rat @ G @ ( insert_o @ X @ A ) )
% 14.25/8.13 = ( groups7872700643590313910_o_rat @ G @ A ) ) )
% 14.25/8.13 & ( ~ ( member_o @ X @ A )
% 14.25/8.13 => ( ( groups7872700643590313910_o_rat @ G @ ( insert_o @ X @ A ) )
% 14.25/8.13 = ( plus_plus_rat @ ( G @ X ) @ ( groups7872700643590313910_o_rat @ G @ A ) ) ) ) ) ) ).
% 14.25/8.13
% 14.25/8.13 thf(fact_6568_sum_Oinsert__if,axiom,
% 14.25/8.13 ! [A: set_Product_unit,X: product_unit,G: product_unit > rat] :
% 14.25/8.13 ( ( finite4290736615968046902t_unit @ A )
% 14.25/8.13 => ( ( ( member_Product_unit @ X @ A )
% 14.25/8.13 => ( ( groups8131667117488461483it_rat @ G @ ( insert_Product_unit @ X @ A ) )
% 14.25/8.13 = ( groups8131667117488461483it_rat @ G @ A ) ) )
% 14.25/8.13 & ( ~ ( member_Product_unit @ X @ A )
% 14.25/8.13 => ( ( groups8131667117488461483it_rat @ G @ ( insert_Product_unit @ X @ A ) )
% 14.25/8.13 = ( plus_plus_rat @ ( G @ X ) @ ( groups8131667117488461483it_rat @ G @ A ) ) ) ) ) ) ).
% 14.25/8.13
% 14.25/8.13 thf(fact_6569_sum_Oinsert__if,axiom,
% 14.25/8.13 ! [A: set_nat,X: nat,G: nat > rat] :
% 14.25/8.13 ( ( finite_finite_nat @ A )
% 14.25/8.13 => ( ( ( member_nat @ X @ A )
% 14.25/8.13 => ( ( groups2906978787729119204at_rat @ G @ ( insert_nat @ X @ A ) )
% 14.25/8.13 = ( groups2906978787729119204at_rat @ G @ A ) ) )
% 14.25/8.13 & ( ~ ( member_nat @ X @ A )
% 14.25/8.13 => ( ( groups2906978787729119204at_rat @ G @ ( insert_nat @ X @ A ) )
% 14.25/8.13 = ( plus_plus_rat @ ( G @ X ) @ ( groups2906978787729119204at_rat @ G @ A ) ) ) ) ) ) ).
% 14.25/8.13
% 14.25/8.13 thf(fact_6570_sum_Oinsert__if,axiom,
% 14.25/8.13 ! [A: set_Code_integer,X: code_integer,G: code_integer > rat] :
% 14.25/8.13 ( ( finite6017078050557962740nteger @ A )
% 14.25/8.13 => ( ( ( member_Code_integer @ X @ A )
% 14.25/8.13 => ( ( groups6602215022474089585er_rat @ G @ ( insert_Code_integer @ X @ A ) )
% 14.25/8.13 = ( groups6602215022474089585er_rat @ G @ A ) ) )
% 14.25/8.13 & ( ~ ( member_Code_integer @ X @ A )
% 14.25/8.13 => ( ( groups6602215022474089585er_rat @ G @ ( insert_Code_integer @ X @ A ) )
% 14.25/8.13 = ( plus_plus_rat @ ( G @ X ) @ ( groups6602215022474089585er_rat @ G @ A ) ) ) ) ) ) ).
% 14.25/8.13
% 14.25/8.13 thf(fact_6571_sum__pos,axiom,
% 14.25/8.13 ! [I: set_Code_integer,F: code_integer > nat] :
% 14.25/8.13 ( ( finite6017078050557962740nteger @ I )
% 14.25/8.13 => ( ( I != bot_bo3990330152332043303nteger )
% 14.25/8.13 => ( ! [I3: code_integer] :
% 14.25/8.13 ( ( member_Code_integer @ I3 @ I )
% 14.25/8.13 => ( ord_less_nat @ zero_zero_nat @ ( F @ I3 ) ) )
% 14.25/8.13 => ( ord_less_nat @ zero_zero_nat @ ( groups7237345082560585321er_nat @ F @ I ) ) ) ) ) ).
% 14.25/8.13
% 14.25/8.13 thf(fact_6572_sum__pos,axiom,
% 14.25/8.13 ! [I: set_Product_unit,F: product_unit > nat] :
% 14.25/8.13 ( ( finite4290736615968046902t_unit @ I )
% 14.25/8.13 => ( ( I != bot_bo3957492148770167129t_unit )
% 14.25/8.13 => ( ! [I3: product_unit] :
% 14.25/8.13 ( ( member_Product_unit @ I3 @ I )
% 14.25/8.13 => ( ord_less_nat @ zero_zero_nat @ ( F @ I3 ) ) )
% 14.25/8.13 => ( ord_less_nat @ zero_zero_nat @ ( groups8766797177574957219it_nat @ F @ I ) ) ) ) ) ).
% 14.25/8.13
% 14.25/8.13 thf(fact_6573_sum__pos,axiom,
% 14.25/8.13 ! [I: set_o,F: $o > nat] :
% 14.25/8.13 ( ( finite_finite_o @ I )
% 14.25/8.13 => ( ( I != bot_bot_set_o )
% 14.25/8.13 => ( ! [I3: $o] :
% 14.25/8.13 ( ( member_o @ I3 @ I )
% 14.25/8.13 => ( ord_less_nat @ zero_zero_nat @ ( F @ I3 ) ) )
% 14.25/8.13 => ( ord_less_nat @ zero_zero_nat @ ( groups8507830703676809646_o_nat @ F @ I ) ) ) ) ) ).
% 14.25/8.13
% 14.25/8.13 thf(fact_6574_sum__pos,axiom,
% 14.25/8.13 ! [I: set_int,F: int > nat] :
% 14.25/8.13 ( ( finite_finite_int @ I )
% 14.25/8.13 => ( ( I != bot_bot_set_int )
% 14.25/8.13 => ( ! [I3: int] :
% 14.25/8.13 ( ( member_int @ I3 @ I )
% 14.25/8.13 => ( ord_less_nat @ zero_zero_nat @ ( F @ I3 ) ) )
% 14.25/8.13 => ( ord_less_nat @ zero_zero_nat @ ( groups4541462559716669496nt_nat @ F @ I ) ) ) ) ) ).
% 14.25/8.13
% 14.25/8.13 thf(fact_6575_sum__pos,axiom,
% 14.25/8.13 ! [I: set_Code_integer,F: code_integer > int] :
% 14.25/8.13 ( ( finite6017078050557962740nteger @ I )
% 14.25/8.13 => ( ( I != bot_bo3990330152332043303nteger )
% 14.25/8.13 => ( ! [I3: code_integer] :
% 14.25/8.13 ( ( member_Code_integer @ I3 @ I )
% 14.25/8.13 => ( ord_less_int @ zero_zero_int @ ( F @ I3 ) ) )
% 14.25/8.13 => ( ord_less_int @ zero_zero_int @ ( groups7234854612051535045er_int @ F @ I ) ) ) ) ) ).
% 14.25/8.13
% 14.25/8.13 thf(fact_6576_sum__pos,axiom,
% 14.25/8.13 ! [I: set_Product_unit,F: product_unit > int] :
% 14.25/8.13 ( ( finite4290736615968046902t_unit @ I )
% 14.25/8.13 => ( ( I != bot_bo3957492148770167129t_unit )
% 14.25/8.13 => ( ! [I3: product_unit] :
% 14.25/8.13 ( ( member_Product_unit @ I3 @ I )
% 14.25/8.13 => ( ord_less_int @ zero_zero_int @ ( F @ I3 ) ) )
% 14.25/8.13 => ( ord_less_int @ zero_zero_int @ ( groups8764306707065906943it_int @ F @ I ) ) ) ) ) ).
% 14.25/8.13
% 14.25/8.13 thf(fact_6577_sum__pos,axiom,
% 14.25/8.13 ! [I: set_o,F: $o > int] :
% 14.25/8.13 ( ( finite_finite_o @ I )
% 14.25/8.13 => ( ( I != bot_bot_set_o )
% 14.25/8.13 => ( ! [I3: $o] :
% 14.25/8.13 ( ( member_o @ I3 @ I )
% 14.25/8.13 => ( ord_less_int @ zero_zero_int @ ( F @ I3 ) ) )
% 14.25/8.13 => ( ord_less_int @ zero_zero_int @ ( groups8505340233167759370_o_int @ F @ I ) ) ) ) ) ).
% 14.25/8.13
% 14.25/8.13 thf(fact_6578_sum__pos,axiom,
% 14.25/8.13 ! [I: set_nat,F: nat > int] :
% 14.25/8.13 ( ( finite_finite_nat @ I )
% 14.25/8.13 => ( ( I != bot_bot_set_nat )
% 14.25/8.13 => ( ! [I3: nat] :
% 14.25/8.13 ( ( member_nat @ I3 @ I )
% 14.25/8.13 => ( ord_less_int @ zero_zero_int @ ( F @ I3 ) ) )
% 14.25/8.13 => ( ord_less_int @ zero_zero_int @ ( groups3539618377306564664at_int @ F @ I ) ) ) ) ) ).
% 14.25/8.13
% 14.25/8.13 thf(fact_6579_sum__pos,axiom,
% 14.25/8.13 ! [I: set_Code_integer,F: code_integer > code_integer] :
% 14.25/8.13 ( ( finite6017078050557962740nteger @ I )
% 14.25/8.13 => ( ( I != bot_bo3990330152332043303nteger )
% 14.25/8.13 => ( ! [I3: code_integer] :
% 14.25/8.13 ( ( member_Code_integer @ I3 @ I )
% 14.25/8.13 => ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ ( F @ I3 ) ) )
% 14.25/8.13 => ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ ( groups879477027807139574nteger @ F @ I ) ) ) ) ) ).
% 14.25/8.13
% 14.25/8.13 thf(fact_6580_sum__pos,axiom,
% 14.25/8.13 ! [I: set_Product_unit,F: product_unit > code_integer] :
% 14.25/8.13 ( ( finite4290736615968046902t_unit @ I )
% 14.25/8.13 => ( ( I != bot_bo3957492148770167129t_unit )
% 14.25/8.13 => ( ! [I3: product_unit] :
% 14.25/8.13 ( ( member_Product_unit @ I3 @ I )
% 14.25/8.13 => ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ ( F @ I3 ) ) )
% 14.25/8.13 => ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ ( groups5393439102872367408nteger @ F @ I ) ) ) ) ) ).
% 14.25/8.13
% 14.25/8.13 thf(fact_6581_sum_OUnion__disjoint,axiom,
% 14.25/8.13 ! [C5: set_set_nat,G: nat > nat] :
% 14.25/8.13 ( ! [X3: set_nat] :
% 14.25/8.13 ( ( member_set_nat @ X3 @ C5 )
% 14.25/8.13 => ( finite_finite_nat @ X3 ) )
% 14.25/8.13 => ( ! [X3: set_nat] :
% 14.25/8.13 ( ( member_set_nat @ X3 @ C5 )
% 14.25/8.13 => ! [Xa3: set_nat] :
% 14.25/8.13 ( ( member_set_nat @ Xa3 @ C5 )
% 14.25/8.13 => ( ( X3 != Xa3 )
% 14.25/8.13 => ( ( inf_inf_set_nat @ X3 @ Xa3 )
% 14.25/8.13 = bot_bot_set_nat ) ) ) )
% 14.25/8.13 => ( ( groups3542108847815614940at_nat @ G @ ( comple7399068483239264473et_nat @ C5 ) )
% 14.25/8.13 = ( comp_s5456809700045850692at_nat @ groups8294997508430121362at_nat @ groups3542108847815614940at_nat @ G @ C5 ) ) ) ) ).
% 14.25/8.13
% 14.25/8.13 thf(fact_6582_sum_OUnion__disjoint,axiom,
% 14.25/8.13 ! [C5: set_set_int,G: int > int] :
% 14.25/8.13 ( ! [X3: set_int] :
% 14.25/8.13 ( ( member_set_int @ X3 @ C5 )
% 14.25/8.13 => ( finite_finite_int @ X3 ) )
% 14.25/8.13 => ( ! [X3: set_int] :
% 14.25/8.13 ( ( member_set_int @ X3 @ C5 )
% 14.25/8.13 => ! [Xa3: set_int] :
% 14.25/8.13 ( ( member_set_int @ Xa3 @ C5 )
% 14.25/8.13 => ( ( X3 != Xa3 )
% 14.25/8.13 => ( ( inf_inf_set_int @ X3 @ Xa3 )
% 14.25/8.13 = bot_bot_set_int ) ) ) )
% 14.25/8.13 => ( ( groups4538972089207619220nt_int @ G @ ( comple3221217463730067765et_int @ C5 ) )
% 14.25/8.13 = ( comp_s9098231955081793132nt_int @ groups1256056575759317066nt_int @ groups4538972089207619220nt_int @ G @ C5 ) ) ) ) ).
% 14.25/8.13
% 14.25/8.13 thf(fact_6583_sum__diff1__nat,axiom,
% 14.25/8.13 ! [A2: product_unit,A: set_Product_unit,F: product_unit > nat] :
% 14.25/8.13 ( ( ( member_Product_unit @ A2 @ A )
% 14.25/8.13 => ( ( groups8766797177574957219it_nat @ F @ ( minus_6452836326544984404t_unit @ A @ ( insert_Product_unit @ A2 @ bot_bo3957492148770167129t_unit ) ) )
% 14.25/8.13 = ( minus_minus_nat @ ( groups8766797177574957219it_nat @ F @ A ) @ ( F @ A2 ) ) ) )
% 14.25/8.13 & ( ~ ( member_Product_unit @ A2 @ A )
% 14.25/8.13 => ( ( groups8766797177574957219it_nat @ F @ ( minus_6452836326544984404t_unit @ A @ ( insert_Product_unit @ A2 @ bot_bo3957492148770167129t_unit ) ) )
% 14.25/8.13 = ( groups8766797177574957219it_nat @ F @ A ) ) ) ) ).
% 14.25/8.13
% 14.25/8.13 thf(fact_6584_sum__diff1__nat,axiom,
% 14.25/8.13 ! [A2: $o,A: set_o,F: $o > nat] :
% 14.25/8.13 ( ( ( member_o @ A2 @ A )
% 14.25/8.13 => ( ( groups8507830703676809646_o_nat @ F @ ( minus_minus_set_o @ A @ ( insert_o @ A2 @ bot_bot_set_o ) ) )
% 14.25/8.13 = ( minus_minus_nat @ ( groups8507830703676809646_o_nat @ F @ A ) @ ( F @ A2 ) ) ) )
% 14.25/8.13 & ( ~ ( member_o @ A2 @ A )
% 14.25/8.13 => ( ( groups8507830703676809646_o_nat @ F @ ( minus_minus_set_o @ A @ ( insert_o @ A2 @ bot_bot_set_o ) ) )
% 14.25/8.13 = ( groups8507830703676809646_o_nat @ F @ A ) ) ) ) ).
% 14.25/8.13
% 14.25/8.13 thf(fact_6585_sum__diff1__nat,axiom,
% 14.25/8.13 ! [A2: int,A: set_int,F: int > nat] :
% 14.25/8.13 ( ( ( member_int @ A2 @ A )
% 14.25/8.13 => ( ( groups4541462559716669496nt_nat @ F @ ( minus_minus_set_int @ A @ ( insert_int @ A2 @ bot_bot_set_int ) ) )
% 14.25/8.13 = ( minus_minus_nat @ ( groups4541462559716669496nt_nat @ F @ A ) @ ( F @ A2 ) ) ) )
% 14.25/8.13 & ( ~ ( member_int @ A2 @ A )
% 14.25/8.13 => ( ( groups4541462559716669496nt_nat @ F @ ( minus_minus_set_int @ A @ ( insert_int @ A2 @ bot_bot_set_int ) ) )
% 14.25/8.13 = ( groups4541462559716669496nt_nat @ F @ A ) ) ) ) ).
% 14.25/8.13
% 14.25/8.13 thf(fact_6586_sum__diff1__nat,axiom,
% 14.25/8.13 ! [A2: produc3658429121746597890et_nat > $o,A: set_Pr4532377907799695533_nat_o,F: ( produc3658429121746597890et_nat > $o ) > nat] :
% 14.25/8.13 ( ( ( member6576561426505652726_nat_o @ A2 @ A )
% 14.25/8.13 => ( ( groups6742167275697360483_o_nat @ F @ ( minus_1801376950450012436_nat_o @ A @ ( insert5175938949040314269_nat_o @ A2 @ bot_bo7824918357723582233_nat_o ) ) )
% 14.25/8.13 = ( minus_minus_nat @ ( groups6742167275697360483_o_nat @ F @ A ) @ ( F @ A2 ) ) ) )
% 14.25/8.13 & ( ~ ( member6576561426505652726_nat_o @ A2 @ A )
% 14.25/8.13 => ( ( groups6742167275697360483_o_nat @ F @ ( minus_1801376950450012436_nat_o @ A @ ( insert5175938949040314269_nat_o @ A2 @ bot_bo7824918357723582233_nat_o ) ) )
% 14.25/8.13 = ( groups6742167275697360483_o_nat @ F @ A ) ) ) ) ).
% 14.25/8.13
% 14.25/8.13 thf(fact_6587_sum__diff1__nat,axiom,
% 14.25/8.13 ! [A2: nat,A: set_nat,F: nat > nat] :
% 14.25/8.13 ( ( ( member_nat @ A2 @ A )
% 14.25/8.13 => ( ( groups3542108847815614940at_nat @ F @ ( minus_minus_set_nat @ A @ ( insert_nat @ A2 @ bot_bot_set_nat ) ) )
% 14.25/8.13 = ( minus_minus_nat @ ( groups3542108847815614940at_nat @ F @ A ) @ ( F @ A2 ) ) ) )
% 14.25/8.13 & ( ~ ( member_nat @ A2 @ A )
% 14.25/8.13 => ( ( groups3542108847815614940at_nat @ F @ ( minus_minus_set_nat @ A @ ( insert_nat @ A2 @ bot_bot_set_nat ) ) )
% 14.25/8.13 = ( groups3542108847815614940at_nat @ F @ A ) ) ) ) ).
% 14.25/8.13
% 14.25/8.13 thf(fact_6588_bezout__add__strong__nat,axiom,
% 14.25/8.13 ! [A2: nat,B: nat] :
% 14.25/8.13 ( ( A2 != zero_zero_nat )
% 14.25/8.13 => ? [D: nat,X3: nat,Y2: nat] :
% 14.25/8.13 ( ( dvd_dvd_nat @ D @ A2 )
% 14.25/8.13 & ( dvd_dvd_nat @ D @ B )
% 14.25/8.13 & ( ( times_times_nat @ A2 @ X3 )
% 14.25/8.13 = ( plus_plus_nat @ ( times_times_nat @ B @ Y2 ) @ D ) ) ) ) ).
% 14.25/8.13
% 14.25/8.13 thf(fact_6589_bezout1__nat,axiom,
% 14.25/8.13 ! [A2: nat,B: nat] :
% 14.25/8.13 ? [D: nat,X3: nat,Y2: nat] :
% 14.25/8.13 ( ( dvd_dvd_nat @ D @ A2 )
% 14.25/8.13 & ( dvd_dvd_nat @ D @ B )
% 14.25/8.13 & ( ( ( minus_minus_nat @ ( times_times_nat @ A2 @ X3 ) @ ( times_times_nat @ B @ Y2 ) )
% 14.25/8.13 = D )
% 14.25/8.13 | ( ( minus_minus_nat @ ( times_times_nat @ B @ X3 ) @ ( times_times_nat @ A2 @ Y2 ) )
% 14.25/8.13 = D ) ) ) ).
% 14.25/8.13
% 14.25/8.13 thf(fact_6590_bezout__add__nat,axiom,
% 14.25/8.13 ! [A2: nat,B: nat] :
% 14.25/8.13 ? [D: nat,X3: nat,Y2: nat] :
% 14.25/8.13 ( ( dvd_dvd_nat @ D @ A2 )
% 14.25/8.13 & ( dvd_dvd_nat @ D @ B )
% 14.25/8.13 & ( ( ( times_times_nat @ A2 @ X3 )
% 14.25/8.13 = ( plus_plus_nat @ ( times_times_nat @ B @ Y2 ) @ D ) )
% 14.25/8.13 | ( ( times_times_nat @ B @ X3 )
% 14.25/8.13 = ( plus_plus_nat @ ( times_times_nat @ A2 @ Y2 ) @ D ) ) ) ) ).
% 14.25/8.13
% 14.25/8.13 thf(fact_6591_bezout__lemma__nat,axiom,
% 14.25/8.13 ! [D2: nat,A2: nat,B: nat,X: nat,Y: nat] :
% 14.25/8.13 ( ( dvd_dvd_nat @ D2 @ A2 )
% 14.25/8.13 => ( ( dvd_dvd_nat @ D2 @ B )
% 14.25/8.13 => ( ( ( ( times_times_nat @ A2 @ X )
% 14.25/8.13 = ( plus_plus_nat @ ( times_times_nat @ B @ Y ) @ D2 ) )
% 14.25/8.13 | ( ( times_times_nat @ B @ X )
% 14.25/8.13 = ( plus_plus_nat @ ( times_times_nat @ A2 @ Y ) @ D2 ) ) )
% 14.25/8.13 => ? [X3: nat,Y2: nat] :
% 14.25/8.13 ( ( dvd_dvd_nat @ D2 @ A2 )
% 14.25/8.13 & ( dvd_dvd_nat @ D2 @ ( plus_plus_nat @ A2 @ B ) )
% 14.25/8.13 & ( ( ( times_times_nat @ A2 @ X3 )
% 14.25/8.13 = ( plus_plus_nat @ ( times_times_nat @ ( plus_plus_nat @ A2 @ B ) @ Y2 ) @ D2 ) )
% 14.25/8.13 | ( ( times_times_nat @ ( plus_plus_nat @ A2 @ B ) @ X3 )
% 14.25/8.13 = ( plus_plus_nat @ ( times_times_nat @ A2 @ Y2 ) @ D2 ) ) ) ) ) ) ) ).
% 14.25/8.13
% 14.25/8.13 thf(fact_6592_dvd__productE,axiom,
% 14.25/8.13 ! [P3: nat,A2: nat,B: nat] :
% 14.25/8.13 ( ( dvd_dvd_nat @ P3 @ ( times_times_nat @ A2 @ B ) )
% 14.25/8.13 => ~ ! [X3: nat,Y2: nat] :
% 14.25/8.13 ( ( P3
% 14.25/8.13 = ( times_times_nat @ X3 @ Y2 ) )
% 14.25/8.13 => ( ( dvd_dvd_nat @ X3 @ A2 )
% 14.25/8.13 => ~ ( dvd_dvd_nat @ Y2 @ B ) ) ) ) ).
% 14.25/8.13
% 14.25/8.13 thf(fact_6593_dvd__productE,axiom,
% 14.25/8.13 ! [P3: int,A2: int,B: int] :
% 14.25/8.13 ( ( dvd_dvd_int @ P3 @ ( times_times_int @ A2 @ B ) )
% 14.25/8.13 => ~ ! [X3: int,Y2: int] :
% 14.25/8.13 ( ( P3
% 14.25/8.13 = ( times_times_int @ X3 @ Y2 ) )
% 14.25/8.13 => ( ( dvd_dvd_int @ X3 @ A2 )
% 14.25/8.13 => ~ ( dvd_dvd_int @ Y2 @ B ) ) ) ) ).
% 14.25/8.13
% 14.25/8.13 thf(fact_6594_division__decomp,axiom,
% 14.25/8.13 ! [A2: nat,B: nat,C: nat] :
% 14.25/8.13 ( ( dvd_dvd_nat @ A2 @ ( times_times_nat @ B @ C ) )
% 14.25/8.13 => ? [B10: nat,C8: nat] :
% 14.25/8.13 ( ( A2
% 14.25/8.13 = ( times_times_nat @ B10 @ C8 ) )
% 14.25/8.13 & ( dvd_dvd_nat @ B10 @ B )
% 14.25/8.13 & ( dvd_dvd_nat @ C8 @ C ) ) ) ).
% 14.25/8.13
% 14.25/8.13 thf(fact_6595_division__decomp,axiom,
% 14.25/8.13 ! [A2: int,B: int,C: int] :
% 14.25/8.13 ( ( dvd_dvd_int @ A2 @ ( times_times_int @ B @ C ) )
% 14.25/8.13 => ? [B10: int,C8: int] :
% 14.25/8.13 ( ( A2
% 14.25/8.13 = ( times_times_int @ B10 @ C8 ) )
% 14.25/8.13 & ( dvd_dvd_int @ B10 @ B )
% 14.25/8.13 & ( dvd_dvd_int @ C8 @ C ) ) ) ).
% 14.25/8.13
% 14.25/8.13 thf(fact_6596_card__Min__le__sum,axiom,
% 14.25/8.13 ! [A: set_Code_integer,F: code_integer > nat] :
% 14.25/8.13 ( ( finite6017078050557962740nteger @ A )
% 14.25/8.13 => ( ord_less_eq_nat @ ( times_times_nat @ ( finite4902975817058060853nteger @ A ) @ ( lattic8721135487736765967in_nat @ ( image_951025933927791156er_nat @ F @ A ) ) ) @ ( groups7237345082560585321er_nat @ F @ A ) ) ) ).
% 14.25/8.13
% 14.25/8.13 thf(fact_6597_card__Min__le__sum,axiom,
% 14.25/8.13 ! [A: set_nat,F: nat > nat] :
% 14.25/8.13 ( ( finite_finite_nat @ A )
% 14.25/8.13 => ( ord_less_eq_nat @ ( times_times_nat @ ( finite_card_nat @ A ) @ ( lattic8721135487736765967in_nat @ ( image_nat_nat2 @ F @ A ) ) ) @ ( groups3542108847815614940at_nat @ F @ A ) ) ) ).
% 14.25/8.13
% 14.25/8.13 thf(fact_6598_Min__singleton,axiom,
% 14.25/8.13 ! [X: product_unit] :
% 14.25/8.13 ( ( lattic8554435388794137742t_unit @ ( insert_Product_unit @ X @ bot_bo3957492148770167129t_unit ) )
% 14.25/8.13 = X ) ).
% 14.25/8.13
% 14.25/8.13 thf(fact_6599_Min__singleton,axiom,
% 14.25/8.13 ! [X: $o] :
% 14.25/8.13 ( ( lattic1973801136483472281_Min_o @ ( insert_o @ X @ bot_bot_set_o ) )
% 14.25/8.13 = X ) ).
% 14.25/8.13
% 14.25/8.13 thf(fact_6600_Min__singleton,axiom,
% 14.25/8.13 ! [X: nat] :
% 14.25/8.13 ( ( lattic8721135487736765967in_nat @ ( insert_nat @ X @ bot_bot_set_nat ) )
% 14.25/8.13 = X ) ).
% 14.25/8.13
% 14.25/8.13 thf(fact_6601_Min__singleton,axiom,
% 14.25/8.13 ! [X: int] :
% 14.25/8.13 ( ( lattic8718645017227715691in_int @ ( insert_int @ X @ bot_bot_set_int ) )
% 14.25/8.13 = X ) ).
% 14.25/8.13
% 14.25/8.13 thf(fact_6602_Min_Obounded__iff,axiom,
% 14.25/8.13 ! [A: set_Code_integer,X: code_integer] :
% 14.25/8.13 ( ( finite6017078050557962740nteger @ A )
% 14.25/8.13 => ( ( A != bot_bo3990330152332043303nteger )
% 14.25/8.13 => ( ( ord_le3102999989581377725nteger @ X @ ( lattic1063845414844153500nteger @ A ) )
% 14.25/8.13 = ( ! [X2: code_integer] :
% 14.25/8.13 ( ( member_Code_integer @ X2 @ A )
% 14.25/8.13 => ( ord_le3102999989581377725nteger @ X @ X2 ) ) ) ) ) ) ).
% 14.25/8.13
% 14.25/8.13 thf(fact_6603_Min_Obounded__iff,axiom,
% 14.25/8.13 ! [A: set_Product_unit,X: product_unit] :
% 14.25/8.13 ( ( finite4290736615968046902t_unit @ A )
% 14.25/8.13 => ( ( A != bot_bo3957492148770167129t_unit )
% 14.25/8.13 => ( ( ord_le3221252021190050221t_unit @ X @ ( lattic8554435388794137742t_unit @ A ) )
% 14.25/8.13 = ( ! [X2: product_unit] :
% 14.25/8.13 ( ( member_Product_unit @ X2 @ A )
% 14.25/8.13 => ( ord_le3221252021190050221t_unit @ X @ X2 ) ) ) ) ) ) ).
% 14.25/8.13
% 14.25/8.13 thf(fact_6604_Min_Obounded__iff,axiom,
% 14.25/8.13 ! [A: set_o,X: $o] :
% 14.25/8.13 ( ( finite_finite_o @ A )
% 14.25/8.13 => ( ( A != bot_bot_set_o )
% 14.25/8.13 => ( ( ord_less_eq_o @ X @ ( lattic1973801136483472281_Min_o @ A ) )
% 14.25/8.13 = ( ! [X2: $o] :
% 14.25/8.13 ( ( member_o @ X2 @ A )
% 14.25/8.13 => ( ord_less_eq_o @ X @ X2 ) ) ) ) ) ) ).
% 14.25/8.13
% 14.25/8.13 thf(fact_6605_Min_Obounded__iff,axiom,
% 14.25/8.13 ! [A: set_rat,X: rat] :
% 14.25/8.13 ( ( finite_finite_rat @ A )
% 14.25/8.13 => ( ( A != bot_bot_set_rat )
% 14.25/8.13 => ( ( ord_less_eq_rat @ X @ ( lattic8086005427650270231in_rat @ A ) )
% 14.25/8.13 = ( ! [X2: rat] :
% 14.25/8.13 ( ( member_rat @ X2 @ A )
% 14.25/8.13 => ( ord_less_eq_rat @ X @ X2 ) ) ) ) ) ) ).
% 14.25/8.13
% 14.25/8.13 thf(fact_6606_Min_Obounded__iff,axiom,
% 14.25/8.13 ! [A: set_nat,X: nat] :
% 14.25/8.13 ( ( finite_finite_nat @ A )
% 14.25/8.13 => ( ( A != bot_bot_set_nat )
% 14.25/8.13 => ( ( ord_less_eq_nat @ X @ ( lattic8721135487736765967in_nat @ A ) )
% 14.25/8.13 = ( ! [X2: nat] :
% 14.25/8.13 ( ( member_nat @ X2 @ A )
% 14.25/8.13 => ( ord_less_eq_nat @ X @ X2 ) ) ) ) ) ) ).
% 14.25/8.13
% 14.25/8.13 thf(fact_6607_Min_Obounded__iff,axiom,
% 14.25/8.13 ! [A: set_int,X: int] :
% 14.25/8.13 ( ( finite_finite_int @ A )
% 14.25/8.13 => ( ( A != bot_bot_set_int )
% 14.25/8.13 => ( ( ord_less_eq_int @ X @ ( lattic8718645017227715691in_int @ A ) )
% 14.25/8.13 = ( ! [X2: int] :
% 14.25/8.13 ( ( member_int @ X2 @ A )
% 14.25/8.13 => ( ord_less_eq_int @ X @ X2 ) ) ) ) ) ) ).
% 14.25/8.13
% 14.25/8.13 thf(fact_6608_Min__gr__iff,axiom,
% 14.25/8.13 ! [A: set_Product_unit,X: product_unit] :
% 14.25/8.13 ( ( finite4290736615968046902t_unit @ A )
% 14.25/8.13 => ( ( A != bot_bo3957492148770167129t_unit )
% 14.25/8.13 => ( ( ord_le361264281704409273t_unit @ X @ ( lattic8554435388794137742t_unit @ A ) )
% 14.25/8.13 = ( ! [X2: product_unit] :
% 14.25/8.13 ( ( member_Product_unit @ X2 @ A )
% 14.25/8.13 => ( ord_le361264281704409273t_unit @ X @ X2 ) ) ) ) ) ) ).
% 14.25/8.13
% 14.25/8.13 thf(fact_6609_Min__gr__iff,axiom,
% 14.25/8.13 ! [A: set_o,X: $o] :
% 14.25/8.13 ( ( finite_finite_o @ A )
% 14.25/8.13 => ( ( A != bot_bot_set_o )
% 14.25/8.13 => ( ( ord_less_o @ X @ ( lattic1973801136483472281_Min_o @ A ) )
% 14.25/8.13 = ( ! [X2: $o] :
% 14.25/8.13 ( ( member_o @ X2 @ A )
% 14.25/8.13 => ( ord_less_o @ X @ X2 ) ) ) ) ) ) ).
% 14.25/8.13
% 14.25/8.13 thf(fact_6610_Min__gr__iff,axiom,
% 14.25/8.13 ! [A: set_nat,X: nat] :
% 14.25/8.13 ( ( finite_finite_nat @ A )
% 14.25/8.13 => ( ( A != bot_bot_set_nat )
% 14.25/8.13 => ( ( ord_less_nat @ X @ ( lattic8721135487736765967in_nat @ A ) )
% 14.25/8.13 = ( ! [X2: nat] :
% 14.25/8.13 ( ( member_nat @ X2 @ A )
% 14.25/8.13 => ( ord_less_nat @ X @ X2 ) ) ) ) ) ) ).
% 14.25/8.13
% 14.25/8.13 thf(fact_6611_Min__gr__iff,axiom,
% 14.25/8.13 ! [A: set_int,X: int] :
% 14.25/8.13 ( ( finite_finite_int @ A )
% 14.25/8.13 => ( ( A != bot_bot_set_int )
% 14.25/8.13 => ( ( ord_less_int @ X @ ( lattic8718645017227715691in_int @ A ) )
% 14.25/8.13 = ( ! [X2: int] :
% 14.25/8.13 ( ( member_int @ X2 @ A )
% 14.25/8.13 => ( ord_less_int @ X @ X2 ) ) ) ) ) ) ).
% 14.25/8.13
% 14.25/8.13 thf(fact_6612_Min__gr__iff,axiom,
% 14.25/8.13 ! [A: set_Code_integer,X: code_integer] :
% 14.25/8.13 ( ( finite6017078050557962740nteger @ A )
% 14.25/8.13 => ( ( A != bot_bo3990330152332043303nteger )
% 14.25/8.13 => ( ( ord_le6747313008572928689nteger @ X @ ( lattic1063845414844153500nteger @ A ) )
% 14.25/8.13 = ( ! [X2: code_integer] :
% 14.25/8.13 ( ( member_Code_integer @ X2 @ A )
% 14.25/8.13 => ( ord_le6747313008572928689nteger @ X @ X2 ) ) ) ) ) ) ).
% 14.25/8.13
% 14.25/8.13 thf(fact_6613_Min__gr__iff,axiom,
% 14.25/8.13 ! [A: set_rat,X: rat] :
% 14.25/8.13 ( ( finite_finite_rat @ A )
% 14.25/8.13 => ( ( A != bot_bot_set_rat )
% 14.25/8.13 => ( ( ord_less_rat @ X @ ( lattic8086005427650270231in_rat @ A ) )
% 14.25/8.13 = ( ! [X2: rat] :
% 14.25/8.13 ( ( member_rat @ X2 @ A )
% 14.25/8.13 => ( ord_less_rat @ X @ X2 ) ) ) ) ) ) ).
% 14.25/8.13
% 14.25/8.13 thf(fact_6614_Min__in,axiom,
% 14.25/8.13 ! [A: set_Code_integer] :
% 14.25/8.13 ( ( finite6017078050557962740nteger @ A )
% 14.25/8.13 => ( ( A != bot_bo3990330152332043303nteger )
% 14.25/8.13 => ( member_Code_integer @ ( lattic1063845414844153500nteger @ A ) @ A ) ) ) ).
% 14.25/8.13
% 14.25/8.13 thf(fact_6615_Min__in,axiom,
% 14.25/8.13 ! [A: set_Product_unit] :
% 14.25/8.13 ( ( finite4290736615968046902t_unit @ A )
% 14.25/8.13 => ( ( A != bot_bo3957492148770167129t_unit )
% 14.25/8.13 => ( member_Product_unit @ ( lattic8554435388794137742t_unit @ A ) @ A ) ) ) ).
% 14.25/8.13
% 14.25/8.13 thf(fact_6616_Min__in,axiom,
% 14.25/8.13 ! [A: set_o] :
% 14.25/8.13 ( ( finite_finite_o @ A )
% 14.25/8.13 => ( ( A != bot_bot_set_o )
% 14.25/8.13 => ( member_o @ ( lattic1973801136483472281_Min_o @ A ) @ A ) ) ) ).
% 14.25/8.13
% 14.25/8.13 thf(fact_6617_Min__in,axiom,
% 14.25/8.13 ! [A: set_nat] :
% 14.25/8.13 ( ( finite_finite_nat @ A )
% 14.25/8.13 => ( ( A != bot_bot_set_nat )
% 14.25/8.13 => ( member_nat @ ( lattic8721135487736765967in_nat @ A ) @ A ) ) ) ).
% 14.25/8.13
% 14.25/8.13 thf(fact_6618_Min__in,axiom,
% 14.25/8.13 ! [A: set_int] :
% 14.25/8.13 ( ( finite_finite_int @ A )
% 14.25/8.13 => ( ( A != bot_bot_set_int )
% 14.25/8.13 => ( member_int @ ( lattic8718645017227715691in_int @ A ) @ A ) ) ) ).
% 14.25/8.13
% 14.25/8.13 thf(fact_6619_Min__eq__iff,axiom,
% 14.25/8.13 ! [A: set_Code_integer,M4: code_integer] :
% 14.25/8.13 ( ( finite6017078050557962740nteger @ A )
% 14.25/8.13 => ( ( A != bot_bo3990330152332043303nteger )
% 14.25/8.13 => ( ( ( lattic1063845414844153500nteger @ A )
% 14.25/8.13 = M4 )
% 14.25/8.13 = ( ( member_Code_integer @ M4 @ A )
% 14.25/8.13 & ! [X2: code_integer] :
% 14.25/8.13 ( ( member_Code_integer @ X2 @ A )
% 14.25/8.13 => ( ord_le3102999989581377725nteger @ M4 @ X2 ) ) ) ) ) ) ).
% 14.25/8.13
% 14.25/8.13 thf(fact_6620_Min__eq__iff,axiom,
% 14.25/8.13 ! [A: set_Product_unit,M4: product_unit] :
% 14.25/8.13 ( ( finite4290736615968046902t_unit @ A )
% 14.25/8.13 => ( ( A != bot_bo3957492148770167129t_unit )
% 14.25/8.13 => ( ( ( lattic8554435388794137742t_unit @ A )
% 14.25/8.13 = M4 )
% 14.25/8.13 = ( ( member_Product_unit @ M4 @ A )
% 14.25/8.13 & ! [X2: product_unit] :
% 14.25/8.13 ( ( member_Product_unit @ X2 @ A )
% 14.25/8.13 => ( ord_le3221252021190050221t_unit @ M4 @ X2 ) ) ) ) ) ) ).
% 14.25/8.13
% 14.25/8.13 thf(fact_6621_Min__eq__iff,axiom,
% 14.25/8.13 ! [A: set_o,M4: $o] :
% 14.25/8.13 ( ( finite_finite_o @ A )
% 14.25/8.13 => ( ( A != bot_bot_set_o )
% 14.25/8.13 => ( ( ( lattic1973801136483472281_Min_o @ A )
% 14.25/8.13 = M4 )
% 14.25/8.13 = ( ( member_o @ M4 @ A )
% 14.25/8.13 & ! [X2: $o] :
% 14.25/8.13 ( ( member_o @ X2 @ A )
% 14.25/8.13 => ( ord_less_eq_o @ M4 @ X2 ) ) ) ) ) ) ).
% 14.25/8.13
% 14.25/8.13 thf(fact_6622_Min__eq__iff,axiom,
% 14.25/8.13 ! [A: set_rat,M4: rat] :
% 14.25/8.13 ( ( finite_finite_rat @ A )
% 14.25/8.13 => ( ( A != bot_bot_set_rat )
% 14.25/8.13 => ( ( ( lattic8086005427650270231in_rat @ A )
% 14.25/8.13 = M4 )
% 14.25/8.13 = ( ( member_rat @ M4 @ A )
% 14.25/8.13 & ! [X2: rat] :
% 14.25/8.13 ( ( member_rat @ X2 @ A )
% 14.25/8.13 => ( ord_less_eq_rat @ M4 @ X2 ) ) ) ) ) ) ).
% 14.25/8.13
% 14.25/8.13 thf(fact_6623_Min__eq__iff,axiom,
% 14.25/8.13 ! [A: set_nat,M4: nat] :
% 14.25/8.13 ( ( finite_finite_nat @ A )
% 14.25/8.13 => ( ( A != bot_bot_set_nat )
% 14.25/8.13 => ( ( ( lattic8721135487736765967in_nat @ A )
% 14.25/8.13 = M4 )
% 14.25/8.13 = ( ( member_nat @ M4 @ A )
% 14.25/8.13 & ! [X2: nat] :
% 14.25/8.13 ( ( member_nat @ X2 @ A )
% 14.25/8.13 => ( ord_less_eq_nat @ M4 @ X2 ) ) ) ) ) ) ).
% 14.25/8.13
% 14.25/8.13 thf(fact_6624_Min__eq__iff,axiom,
% 14.25/8.13 ! [A: set_int,M4: int] :
% 14.25/8.13 ( ( finite_finite_int @ A )
% 14.25/8.13 => ( ( A != bot_bot_set_int )
% 14.25/8.13 => ( ( ( lattic8718645017227715691in_int @ A )
% 14.25/8.13 = M4 )
% 14.25/8.13 = ( ( member_int @ M4 @ A )
% 14.25/8.13 & ! [X2: int] :
% 14.25/8.13 ( ( member_int @ X2 @ A )
% 14.25/8.13 => ( ord_less_eq_int @ M4 @ X2 ) ) ) ) ) ) ).
% 14.25/8.13
% 14.25/8.13 thf(fact_6625_Min__le__iff,axiom,
% 14.25/8.13 ! [A: set_Code_integer,X: code_integer] :
% 14.25/8.13 ( ( finite6017078050557962740nteger @ A )
% 14.25/8.13 => ( ( A != bot_bo3990330152332043303nteger )
% 14.25/8.13 => ( ( ord_le3102999989581377725nteger @ ( lattic1063845414844153500nteger @ A ) @ X )
% 14.25/8.13 = ( ? [X2: code_integer] :
% 14.25/8.13 ( ( member_Code_integer @ X2 @ A )
% 14.25/8.13 & ( ord_le3102999989581377725nteger @ X2 @ X ) ) ) ) ) ) ).
% 14.25/8.13
% 14.25/8.13 thf(fact_6626_Min__le__iff,axiom,
% 14.25/8.13 ! [A: set_Product_unit,X: product_unit] :
% 14.25/8.13 ( ( finite4290736615968046902t_unit @ A )
% 14.25/8.13 => ( ( A != bot_bo3957492148770167129t_unit )
% 14.25/8.13 => ( ( ord_le3221252021190050221t_unit @ ( lattic8554435388794137742t_unit @ A ) @ X )
% 14.25/8.13 = ( ? [X2: product_unit] :
% 14.25/8.13 ( ( member_Product_unit @ X2 @ A )
% 14.25/8.13 & ( ord_le3221252021190050221t_unit @ X2 @ X ) ) ) ) ) ) ).
% 14.25/8.13
% 14.25/8.13 thf(fact_6627_Min__le__iff,axiom,
% 14.25/8.13 ! [A: set_o,X: $o] :
% 14.25/8.13 ( ( finite_finite_o @ A )
% 14.25/8.13 => ( ( A != bot_bot_set_o )
% 14.25/8.13 => ( ( ord_less_eq_o @ ( lattic1973801136483472281_Min_o @ A ) @ X )
% 14.25/8.13 = ( ? [X2: $o] :
% 14.25/8.13 ( ( member_o @ X2 @ A )
% 14.25/8.13 & ( ord_less_eq_o @ X2 @ X ) ) ) ) ) ) ).
% 14.25/8.13
% 14.25/8.13 thf(fact_6628_Min__le__iff,axiom,
% 14.25/8.13 ! [A: set_rat,X: rat] :
% 14.25/8.13 ( ( finite_finite_rat @ A )
% 14.25/8.13 => ( ( A != bot_bot_set_rat )
% 14.25/8.13 => ( ( ord_less_eq_rat @ ( lattic8086005427650270231in_rat @ A ) @ X )
% 14.25/8.13 = ( ? [X2: rat] :
% 14.25/8.13 ( ( member_rat @ X2 @ A )
% 14.25/8.13 & ( ord_less_eq_rat @ X2 @ X ) ) ) ) ) ) ).
% 14.25/8.13
% 14.25/8.13 thf(fact_6629_Min__le__iff,axiom,
% 14.25/8.13 ! [A: set_nat,X: nat] :
% 14.25/8.13 ( ( finite_finite_nat @ A )
% 14.25/8.13 => ( ( A != bot_bot_set_nat )
% 14.25/8.13 => ( ( ord_less_eq_nat @ ( lattic8721135487736765967in_nat @ A ) @ X )
% 14.25/8.13 = ( ? [X2: nat] :
% 14.25/8.13 ( ( member_nat @ X2 @ A )
% 14.25/8.13 & ( ord_less_eq_nat @ X2 @ X ) ) ) ) ) ) ).
% 14.25/8.13
% 14.25/8.13 thf(fact_6630_Min__le__iff,axiom,
% 14.25/8.13 ! [A: set_int,X: int] :
% 14.25/8.13 ( ( finite_finite_int @ A )
% 14.25/8.13 => ( ( A != bot_bot_set_int )
% 14.25/8.13 => ( ( ord_less_eq_int @ ( lattic8718645017227715691in_int @ A ) @ X )
% 14.25/8.13 = ( ? [X2: int] :
% 14.25/8.13 ( ( member_int @ X2 @ A )
% 14.25/8.13 & ( ord_less_eq_int @ X2 @ X ) ) ) ) ) ) ).
% 14.25/8.13
% 14.25/8.13 thf(fact_6631_eq__Min__iff,axiom,
% 14.25/8.13 ! [A: set_Code_integer,M4: code_integer] :
% 14.25/8.13 ( ( finite6017078050557962740nteger @ A )
% 14.25/8.13 => ( ( A != bot_bo3990330152332043303nteger )
% 14.25/8.13 => ( ( M4
% 14.25/8.13 = ( lattic1063845414844153500nteger @ A ) )
% 14.25/8.13 = ( ( member_Code_integer @ M4 @ A )
% 14.25/8.13 & ! [X2: code_integer] :
% 14.25/8.13 ( ( member_Code_integer @ X2 @ A )
% 14.25/8.13 => ( ord_le3102999989581377725nteger @ M4 @ X2 ) ) ) ) ) ) ).
% 14.25/8.13
% 14.25/8.13 thf(fact_6632_eq__Min__iff,axiom,
% 14.25/8.13 ! [A: set_Product_unit,M4: product_unit] :
% 14.25/8.13 ( ( finite4290736615968046902t_unit @ A )
% 14.25/8.13 => ( ( A != bot_bo3957492148770167129t_unit )
% 14.25/8.13 => ( ( M4
% 14.25/8.13 = ( lattic8554435388794137742t_unit @ A ) )
% 14.25/8.13 = ( ( member_Product_unit @ M4 @ A )
% 14.25/8.13 & ! [X2: product_unit] :
% 14.25/8.13 ( ( member_Product_unit @ X2 @ A )
% 14.25/8.13 => ( ord_le3221252021190050221t_unit @ M4 @ X2 ) ) ) ) ) ) ).
% 14.25/8.13
% 14.25/8.13 thf(fact_6633_eq__Min__iff,axiom,
% 14.25/8.13 ! [A: set_o,M4: $o] :
% 14.25/8.13 ( ( finite_finite_o @ A )
% 14.25/8.13 => ( ( A != bot_bot_set_o )
% 14.25/8.13 => ( ( M4
% 14.25/8.13 = ( lattic1973801136483472281_Min_o @ A ) )
% 14.25/8.13 = ( ( member_o @ M4 @ A )
% 14.25/8.13 & ! [X2: $o] :
% 14.25/8.13 ( ( member_o @ X2 @ A )
% 14.25/8.13 => ( ord_less_eq_o @ M4 @ X2 ) ) ) ) ) ) ).
% 14.25/8.13
% 14.25/8.13 thf(fact_6634_eq__Min__iff,axiom,
% 14.25/8.13 ! [A: set_rat,M4: rat] :
% 14.25/8.13 ( ( finite_finite_rat @ A )
% 14.25/8.13 => ( ( A != bot_bot_set_rat )
% 14.25/8.13 => ( ( M4
% 14.25/8.13 = ( lattic8086005427650270231in_rat @ A ) )
% 14.25/8.13 = ( ( member_rat @ M4 @ A )
% 14.25/8.13 & ! [X2: rat] :
% 14.25/8.13 ( ( member_rat @ X2 @ A )
% 14.25/8.13 => ( ord_less_eq_rat @ M4 @ X2 ) ) ) ) ) ) ).
% 14.25/8.13
% 14.25/8.13 thf(fact_6635_eq__Min__iff,axiom,
% 14.25/8.13 ! [A: set_nat,M4: nat] :
% 14.25/8.13 ( ( finite_finite_nat @ A )
% 14.25/8.13 => ( ( A != bot_bot_set_nat )
% 14.25/8.13 => ( ( M4
% 14.25/8.13 = ( lattic8721135487736765967in_nat @ A ) )
% 14.25/8.13 = ( ( member_nat @ M4 @ A )
% 14.25/8.13 & ! [X2: nat] :
% 14.25/8.13 ( ( member_nat @ X2 @ A )
% 14.25/8.13 => ( ord_less_eq_nat @ M4 @ X2 ) ) ) ) ) ) ).
% 14.25/8.13
% 14.25/8.13 thf(fact_6636_eq__Min__iff,axiom,
% 14.25/8.13 ! [A: set_int,M4: int] :
% 14.25/8.13 ( ( finite_finite_int @ A )
% 14.25/8.13 => ( ( A != bot_bot_set_int )
% 14.25/8.13 => ( ( M4
% 14.25/8.13 = ( lattic8718645017227715691in_int @ A ) )
% 14.25/8.13 = ( ( member_int @ M4 @ A )
% 14.25/8.13 & ! [X2: int] :
% 14.25/8.13 ( ( member_int @ X2 @ A )
% 14.25/8.13 => ( ord_less_eq_int @ M4 @ X2 ) ) ) ) ) ) ).
% 14.25/8.13
% 14.25/8.13 thf(fact_6637_Min_OboundedE,axiom,
% 14.25/8.13 ! [A: set_Code_integer,X: code_integer] :
% 14.25/8.13 ( ( finite6017078050557962740nteger @ A )
% 14.25/8.13 => ( ( A != bot_bo3990330152332043303nteger )
% 14.25/8.13 => ( ( ord_le3102999989581377725nteger @ X @ ( lattic1063845414844153500nteger @ A ) )
% 14.25/8.13 => ! [A10: code_integer] :
% 14.25/8.13 ( ( member_Code_integer @ A10 @ A )
% 14.25/8.13 => ( ord_le3102999989581377725nteger @ X @ A10 ) ) ) ) ) ).
% 14.25/8.13
% 14.25/8.13 thf(fact_6638_Min_OboundedE,axiom,
% 14.25/8.13 ! [A: set_Product_unit,X: product_unit] :
% 14.25/8.13 ( ( finite4290736615968046902t_unit @ A )
% 14.25/8.13 => ( ( A != bot_bo3957492148770167129t_unit )
% 14.25/8.13 => ( ( ord_le3221252021190050221t_unit @ X @ ( lattic8554435388794137742t_unit @ A ) )
% 14.25/8.13 => ! [A10: product_unit] :
% 14.25/8.13 ( ( member_Product_unit @ A10 @ A )
% 14.25/8.13 => ( ord_le3221252021190050221t_unit @ X @ A10 ) ) ) ) ) ).
% 14.25/8.13
% 14.25/8.13 thf(fact_6639_Min_OboundedE,axiom,
% 14.25/8.13 ! [A: set_o,X: $o] :
% 14.25/8.13 ( ( finite_finite_o @ A )
% 14.25/8.13 => ( ( A != bot_bot_set_o )
% 14.25/8.13 => ( ( ord_less_eq_o @ X @ ( lattic1973801136483472281_Min_o @ A ) )
% 14.25/8.13 => ! [A10: $o] :
% 14.25/8.13 ( ( member_o @ A10 @ A )
% 14.25/8.13 => ( ord_less_eq_o @ X @ A10 ) ) ) ) ) ).
% 14.25/8.13
% 14.25/8.13 thf(fact_6640_Min_OboundedE,axiom,
% 14.25/8.13 ! [A: set_rat,X: rat] :
% 14.25/8.13 ( ( finite_finite_rat @ A )
% 14.25/8.13 => ( ( A != bot_bot_set_rat )
% 14.25/8.13 => ( ( ord_less_eq_rat @ X @ ( lattic8086005427650270231in_rat @ A ) )
% 14.25/8.13 => ! [A10: rat] :
% 14.25/8.13 ( ( member_rat @ A10 @ A )
% 14.25/8.13 => ( ord_less_eq_rat @ X @ A10 ) ) ) ) ) ).
% 14.25/8.13
% 14.25/8.13 thf(fact_6641_Min_OboundedE,axiom,
% 14.25/8.13 ! [A: set_nat,X: nat] :
% 14.25/8.13 ( ( finite_finite_nat @ A )
% 14.25/8.13 => ( ( A != bot_bot_set_nat )
% 14.25/8.13 => ( ( ord_less_eq_nat @ X @ ( lattic8721135487736765967in_nat @ A ) )
% 14.25/8.13 => ! [A10: nat] :
% 14.25/8.13 ( ( member_nat @ A10 @ A )
% 14.25/8.13 => ( ord_less_eq_nat @ X @ A10 ) ) ) ) ) ).
% 14.25/8.13
% 14.25/8.13 thf(fact_6642_Min_OboundedE,axiom,
% 14.25/8.13 ! [A: set_int,X: int] :
% 14.25/8.13 ( ( finite_finite_int @ A )
% 14.25/8.13 => ( ( A != bot_bot_set_int )
% 14.25/8.13 => ( ( ord_less_eq_int @ X @ ( lattic8718645017227715691in_int @ A ) )
% 14.25/8.13 => ! [A10: int] :
% 14.25/8.13 ( ( member_int @ A10 @ A )
% 14.25/8.13 => ( ord_less_eq_int @ X @ A10 ) ) ) ) ) ).
% 14.25/8.13
% 14.25/8.13 thf(fact_6643_Min_OboundedI,axiom,
% 14.25/8.13 ! [A: set_Code_integer,X: code_integer] :
% 14.25/8.13 ( ( finite6017078050557962740nteger @ A )
% 14.25/8.13 => ( ( A != bot_bo3990330152332043303nteger )
% 14.25/8.13 => ( ! [A4: code_integer] :
% 14.25/8.13 ( ( member_Code_integer @ A4 @ A )
% 14.25/8.13 => ( ord_le3102999989581377725nteger @ X @ A4 ) )
% 14.25/8.13 => ( ord_le3102999989581377725nteger @ X @ ( lattic1063845414844153500nteger @ A ) ) ) ) ) ).
% 14.25/8.13
% 14.25/8.13 thf(fact_6644_Min_OboundedI,axiom,
% 14.25/8.13 ! [A: set_Product_unit,X: product_unit] :
% 14.25/8.13 ( ( finite4290736615968046902t_unit @ A )
% 14.25/8.13 => ( ( A != bot_bo3957492148770167129t_unit )
% 14.25/8.13 => ( ! [A4: product_unit] :
% 14.25/8.13 ( ( member_Product_unit @ A4 @ A )
% 14.25/8.13 => ( ord_le3221252021190050221t_unit @ X @ A4 ) )
% 14.25/8.13 => ( ord_le3221252021190050221t_unit @ X @ ( lattic8554435388794137742t_unit @ A ) ) ) ) ) ).
% 14.25/8.13
% 14.25/8.13 thf(fact_6645_Min_OboundedI,axiom,
% 14.25/8.13 ! [A: set_o,X: $o] :
% 14.25/8.13 ( ( finite_finite_o @ A )
% 14.25/8.13 => ( ( A != bot_bot_set_o )
% 14.25/8.13 => ( ! [A4: $o] :
% 14.25/8.13 ( ( member_o @ A4 @ A )
% 14.25/8.13 => ( ord_less_eq_o @ X @ A4 ) )
% 14.25/8.13 => ( ord_less_eq_o @ X @ ( lattic1973801136483472281_Min_o @ A ) ) ) ) ) ).
% 14.25/8.13
% 14.25/8.13 thf(fact_6646_Min_OboundedI,axiom,
% 14.25/8.13 ! [A: set_rat,X: rat] :
% 14.25/8.13 ( ( finite_finite_rat @ A )
% 14.25/8.13 => ( ( A != bot_bot_set_rat )
% 14.25/8.13 => ( ! [A4: rat] :
% 14.25/8.13 ( ( member_rat @ A4 @ A )
% 14.25/8.13 => ( ord_less_eq_rat @ X @ A4 ) )
% 14.25/8.13 => ( ord_less_eq_rat @ X @ ( lattic8086005427650270231in_rat @ A ) ) ) ) ) ).
% 14.25/8.13
% 14.25/8.13 thf(fact_6647_Min_OboundedI,axiom,
% 14.25/8.13 ! [A: set_nat,X: nat] :
% 14.25/8.13 ( ( finite_finite_nat @ A )
% 14.25/8.13 => ( ( A != bot_bot_set_nat )
% 14.25/8.13 => ( ! [A4: nat] :
% 14.25/8.13 ( ( member_nat @ A4 @ A )
% 14.25/8.13 => ( ord_less_eq_nat @ X @ A4 ) )
% 14.25/8.13 => ( ord_less_eq_nat @ X @ ( lattic8721135487736765967in_nat @ A ) ) ) ) ) ).
% 14.25/8.13
% 14.25/8.13 thf(fact_6648_Min_OboundedI,axiom,
% 14.25/8.13 ! [A: set_int,X: int] :
% 14.25/8.13 ( ( finite_finite_int @ A )
% 14.25/8.13 => ( ( A != bot_bot_set_int )
% 14.25/8.13 => ( ! [A4: int] :
% 14.25/8.13 ( ( member_int @ A4 @ A )
% 14.25/8.13 => ( ord_less_eq_int @ X @ A4 ) )
% 14.25/8.13 => ( ord_less_eq_int @ X @ ( lattic8718645017227715691in_int @ A ) ) ) ) ) ).
% 14.25/8.13
% 14.25/8.13 thf(fact_6649_Min__less__iff,axiom,
% 14.25/8.13 ! [A: set_Product_unit,X: product_unit] :
% 14.25/8.13 ( ( finite4290736615968046902t_unit @ A )
% 14.25/8.13 => ( ( A != bot_bo3957492148770167129t_unit )
% 14.25/8.13 => ( ( ord_le361264281704409273t_unit @ ( lattic8554435388794137742t_unit @ A ) @ X )
% 14.25/8.13 = ( ? [X2: product_unit] :
% 14.25/8.13 ( ( member_Product_unit @ X2 @ A )
% 14.25/8.13 & ( ord_le361264281704409273t_unit @ X2 @ X ) ) ) ) ) ) ).
% 14.25/8.13
% 14.25/8.13 thf(fact_6650_Min__less__iff,axiom,
% 14.25/8.13 ! [A: set_o,X: $o] :
% 14.25/8.13 ( ( finite_finite_o @ A )
% 14.25/8.13 => ( ( A != bot_bot_set_o )
% 14.25/8.13 => ( ( ord_less_o @ ( lattic1973801136483472281_Min_o @ A ) @ X )
% 14.25/8.13 = ( ? [X2: $o] :
% 14.25/8.13 ( ( member_o @ X2 @ A )
% 14.25/8.13 & ( ord_less_o @ X2 @ X ) ) ) ) ) ) ).
% 14.25/8.13
% 14.25/8.13 thf(fact_6651_Min__less__iff,axiom,
% 14.25/8.13 ! [A: set_nat,X: nat] :
% 14.25/8.13 ( ( finite_finite_nat @ A )
% 14.25/8.13 => ( ( A != bot_bot_set_nat )
% 14.25/8.13 => ( ( ord_less_nat @ ( lattic8721135487736765967in_nat @ A ) @ X )
% 14.25/8.13 = ( ? [X2: nat] :
% 14.25/8.13 ( ( member_nat @ X2 @ A )
% 14.25/8.13 & ( ord_less_nat @ X2 @ X ) ) ) ) ) ) ).
% 14.25/8.13
% 14.25/8.13 thf(fact_6652_Min__less__iff,axiom,
% 14.25/8.13 ! [A: set_int,X: int] :
% 14.25/8.13 ( ( finite_finite_int @ A )
% 14.25/8.13 => ( ( A != bot_bot_set_int )
% 14.25/8.13 => ( ( ord_less_int @ ( lattic8718645017227715691in_int @ A ) @ X )
% 14.25/8.13 = ( ? [X2: int] :
% 14.25/8.13 ( ( member_int @ X2 @ A )
% 14.25/8.13 & ( ord_less_int @ X2 @ X ) ) ) ) ) ) ).
% 14.25/8.13
% 14.25/8.13 thf(fact_6653_Min__less__iff,axiom,
% 14.25/8.13 ! [A: set_Code_integer,X: code_integer] :
% 14.25/8.13 ( ( finite6017078050557962740nteger @ A )
% 14.25/8.13 => ( ( A != bot_bo3990330152332043303nteger )
% 14.25/8.13 => ( ( ord_le6747313008572928689nteger @ ( lattic1063845414844153500nteger @ A ) @ X )
% 14.25/8.13 = ( ? [X2: code_integer] :
% 14.25/8.13 ( ( member_Code_integer @ X2 @ A )
% 14.25/8.13 & ( ord_le6747313008572928689nteger @ X2 @ X ) ) ) ) ) ) ).
% 14.25/8.13
% 14.25/8.13 thf(fact_6654_Min__less__iff,axiom,
% 14.25/8.13 ! [A: set_rat,X: rat] :
% 14.25/8.13 ( ( finite_finite_rat @ A )
% 14.25/8.13 => ( ( A != bot_bot_set_rat )
% 14.25/8.13 => ( ( ord_less_rat @ ( lattic8086005427650270231in_rat @ A ) @ X )
% 14.25/8.13 = ( ? [X2: rat] :
% 14.25/8.13 ( ( member_rat @ X2 @ A )
% 14.25/8.13 & ( ord_less_rat @ X2 @ X ) ) ) ) ) ) ).
% 14.25/8.13
% 14.25/8.13 thf(fact_6655_Min__insert2,axiom,
% 14.25/8.13 ! [A: set_o,A2: $o] :
% 14.25/8.13 ( ( finite_finite_o @ A )
% 14.25/8.13 => ( ! [B2: $o] :
% 14.25/8.13 ( ( member_o @ B2 @ A )
% 14.25/8.13 => ( ord_less_eq_o @ A2 @ B2 ) )
% 14.25/8.13 => ( ( lattic1973801136483472281_Min_o @ ( insert_o @ A2 @ A ) )
% 14.25/8.13 = A2 ) ) ) ).
% 14.25/8.13
% 14.25/8.13 thf(fact_6656_Min__insert2,axiom,
% 14.25/8.13 ! [A: set_Product_unit,A2: product_unit] :
% 14.25/8.13 ( ( finite4290736615968046902t_unit @ A )
% 14.25/8.13 => ( ! [B2: product_unit] :
% 14.25/8.13 ( ( member_Product_unit @ B2 @ A )
% 14.25/8.13 => ( ord_le3221252021190050221t_unit @ A2 @ B2 ) )
% 14.25/8.13 => ( ( lattic8554435388794137742t_unit @ ( insert_Product_unit @ A2 @ A ) )
% 14.25/8.13 = A2 ) ) ) ).
% 14.25/8.13
% 14.25/8.13 thf(fact_6657_Min__insert2,axiom,
% 14.25/8.13 ! [A: set_Code_integer,A2: code_integer] :
% 14.25/8.13 ( ( finite6017078050557962740nteger @ A )
% 14.25/8.13 => ( ! [B2: code_integer] :
% 14.25/8.13 ( ( member_Code_integer @ B2 @ A )
% 14.25/8.13 => ( ord_le3102999989581377725nteger @ A2 @ B2 ) )
% 14.25/8.13 => ( ( lattic1063845414844153500nteger @ ( insert_Code_integer @ A2 @ A ) )
% 14.25/8.13 = A2 ) ) ) ).
% 14.25/8.13
% 14.25/8.13 thf(fact_6658_Min__insert2,axiom,
% 14.25/8.13 ! [A: set_rat,A2: rat] :
% 14.25/8.13 ( ( finite_finite_rat @ A )
% 14.25/8.13 => ( ! [B2: rat] :
% 14.25/8.13 ( ( member_rat @ B2 @ A )
% 14.25/8.13 => ( ord_less_eq_rat @ A2 @ B2 ) )
% 14.25/8.13 => ( ( lattic8086005427650270231in_rat @ ( insert_rat @ A2 @ A ) )
% 14.25/8.13 = A2 ) ) ) ).
% 14.25/8.13
% 14.25/8.13 thf(fact_6659_Min__insert2,axiom,
% 14.25/8.13 ! [A: set_nat,A2: nat] :
% 14.25/8.13 ( ( finite_finite_nat @ A )
% 14.25/8.13 => ( ! [B2: nat] :
% 14.25/8.13 ( ( member_nat @ B2 @ A )
% 14.25/8.13 => ( ord_less_eq_nat @ A2 @ B2 ) )
% 14.25/8.13 => ( ( lattic8721135487736765967in_nat @ ( insert_nat @ A2 @ A ) )
% 14.25/8.13 = A2 ) ) ) ).
% 14.25/8.13
% 14.25/8.13 thf(fact_6660_Min__insert2,axiom,
% 14.25/8.13 ! [A: set_int,A2: int] :
% 14.25/8.13 ( ( finite_finite_int @ A )
% 14.25/8.13 => ( ! [B2: int] :
% 14.25/8.13 ( ( member_int @ B2 @ A )
% 14.25/8.13 => ( ord_less_eq_int @ A2 @ B2 ) )
% 14.25/8.13 => ( ( lattic8718645017227715691in_int @ ( insert_int @ A2 @ A ) )
% 14.25/8.13 = A2 ) ) ) ).
% 14.25/8.13
% 14.25/8.13 thf(fact_6661_cInf__eq__Min,axiom,
% 14.25/8.13 ! [X4: set_int] :
% 14.25/8.13 ( ( finite_finite_int @ X4 )
% 14.25/8.13 => ( ( X4 != bot_bot_set_int )
% 14.25/8.13 => ( ( complete_Inf_Inf_int @ X4 )
% 14.25/8.13 = ( lattic8718645017227715691in_int @ X4 ) ) ) ) ).
% 14.25/8.13
% 14.25/8.13 thf(fact_6662_cInf__eq__Min,axiom,
% 14.25/8.13 ! [X4: set_nat] :
% 14.25/8.13 ( ( finite_finite_nat @ X4 )
% 14.25/8.13 => ( ( X4 != bot_bot_set_nat )
% 14.25/8.13 => ( ( complete_Inf_Inf_nat @ X4 )
% 14.25/8.13 = ( lattic8721135487736765967in_nat @ X4 ) ) ) ) ).
% 14.25/8.13
% 14.25/8.13 thf(fact_6663_cInf__eq__Min,axiom,
% 14.25/8.13 ! [X4: set_Product_unit] :
% 14.25/8.13 ( ( finite4290736615968046902t_unit @ X4 )
% 14.25/8.13 => ( ( X4 != bot_bo3957492148770167129t_unit )
% 14.25/8.13 => ( ( comple2584293577114468500t_unit @ X4 )
% 14.25/8.13 = ( lattic8554435388794137742t_unit @ X4 ) ) ) ) ).
% 14.25/8.13
% 14.25/8.13 thf(fact_6664_Min__Inf,axiom,
% 14.25/8.13 ! [A: set_Product_unit] :
% 14.25/8.13 ( ( finite4290736615968046902t_unit @ A )
% 14.25/8.13 => ( ( A != bot_bo3957492148770167129t_unit )
% 14.25/8.13 => ( ( lattic8554435388794137742t_unit @ A )
% 14.25/8.13 = ( comple2584293577114468500t_unit @ A ) ) ) ) ).
% 14.25/8.13
% 14.25/8.13 thf(fact_6665_Min__antimono,axiom,
% 14.25/8.13 ! [M: set_Code_integer,N: set_Code_integer] :
% 14.25/8.13 ( ( ord_le7084787975880047091nteger @ M @ N )
% 14.25/8.13 => ( ( M != bot_bo3990330152332043303nteger )
% 14.25/8.13 => ( ( finite6017078050557962740nteger @ N )
% 14.25/8.13 => ( ord_le3102999989581377725nteger @ ( lattic1063845414844153500nteger @ N ) @ ( lattic1063845414844153500nteger @ M ) ) ) ) ) ).
% 14.25/8.13
% 14.25/8.13 thf(fact_6666_Min__antimono,axiom,
% 14.25/8.13 ! [M: set_Product_unit,N: set_Product_unit] :
% 14.25/8.13 ( ( ord_le3507040750410214029t_unit @ M @ N )
% 14.25/8.13 => ( ( M != bot_bo3957492148770167129t_unit )
% 14.25/8.13 => ( ( finite4290736615968046902t_unit @ N )
% 14.25/8.13 => ( ord_le3221252021190050221t_unit @ ( lattic8554435388794137742t_unit @ N ) @ ( lattic8554435388794137742t_unit @ M ) ) ) ) ) ).
% 14.25/8.13
% 14.25/8.13 thf(fact_6667_Min__antimono,axiom,
% 14.25/8.13 ! [M: set_o,N: set_o] :
% 14.25/8.13 ( ( ord_less_eq_set_o @ M @ N )
% 14.25/8.13 => ( ( M != bot_bot_set_o )
% 14.25/8.13 => ( ( finite_finite_o @ N )
% 14.25/8.13 => ( ord_less_eq_o @ ( lattic1973801136483472281_Min_o @ N ) @ ( lattic1973801136483472281_Min_o @ M ) ) ) ) ) ).
% 14.25/8.13
% 14.25/8.13 thf(fact_6668_Min__antimono,axiom,
% 14.25/8.13 ! [M: set_rat,N: set_rat] :
% 14.25/8.13 ( ( ord_less_eq_set_rat @ M @ N )
% 14.25/8.13 => ( ( M != bot_bot_set_rat )
% 14.25/8.13 => ( ( finite_finite_rat @ N )
% 14.25/8.13 => ( ord_less_eq_rat @ ( lattic8086005427650270231in_rat @ N ) @ ( lattic8086005427650270231in_rat @ M ) ) ) ) ) ).
% 14.25/8.13
% 14.25/8.13 thf(fact_6669_Min__antimono,axiom,
% 14.25/8.13 ! [M: set_int,N: set_int] :
% 14.25/8.13 ( ( ord_less_eq_set_int @ M @ N )
% 14.25/8.13 => ( ( M != bot_bot_set_int )
% 14.25/8.13 => ( ( finite_finite_int @ N )
% 14.25/8.13 => ( ord_less_eq_int @ ( lattic8718645017227715691in_int @ N ) @ ( lattic8718645017227715691in_int @ M ) ) ) ) ) ).
% 14.25/8.13
% 14.25/8.13 thf(fact_6670_Min__antimono,axiom,
% 14.25/8.13 ! [M: set_nat,N: set_nat] :
% 14.25/8.13 ( ( ord_less_eq_set_nat @ M @ N )
% 14.25/8.13 => ( ( M != bot_bot_set_nat )
% 14.25/8.13 => ( ( finite_finite_nat @ N )
% 14.25/8.13 => ( ord_less_eq_nat @ ( lattic8721135487736765967in_nat @ N ) @ ( lattic8721135487736765967in_nat @ M ) ) ) ) ) ).
% 14.25/8.13
% 14.25/8.13 thf(fact_6671_Min_Osubset__imp,axiom,
% 14.25/8.13 ! [A: set_Code_integer,B4: set_Code_integer] :
% 14.25/8.13 ( ( ord_le7084787975880047091nteger @ A @ B4 )
% 14.25/8.13 => ( ( A != bot_bo3990330152332043303nteger )
% 14.25/8.13 => ( ( finite6017078050557962740nteger @ B4 )
% 14.25/8.13 => ( ord_le3102999989581377725nteger @ ( lattic1063845414844153500nteger @ B4 ) @ ( lattic1063845414844153500nteger @ A ) ) ) ) ) ).
% 14.25/8.13
% 14.25/8.13 thf(fact_6672_Min_Osubset__imp,axiom,
% 14.25/8.13 ! [A: set_Product_unit,B4: set_Product_unit] :
% 14.25/8.13 ( ( ord_le3507040750410214029t_unit @ A @ B4 )
% 14.25/8.13 => ( ( A != bot_bo3957492148770167129t_unit )
% 14.25/8.13 => ( ( finite4290736615968046902t_unit @ B4 )
% 14.25/8.13 => ( ord_le3221252021190050221t_unit @ ( lattic8554435388794137742t_unit @ B4 ) @ ( lattic8554435388794137742t_unit @ A ) ) ) ) ) ).
% 14.25/8.13
% 14.25/8.13 thf(fact_6673_Min_Osubset__imp,axiom,
% 14.25/8.13 ! [A: set_o,B4: set_o] :
% 14.25/8.13 ( ( ord_less_eq_set_o @ A @ B4 )
% 14.25/8.13 => ( ( A != bot_bot_set_o )
% 14.25/8.13 => ( ( finite_finite_o @ B4 )
% 14.25/8.13 => ( ord_less_eq_o @ ( lattic1973801136483472281_Min_o @ B4 ) @ ( lattic1973801136483472281_Min_o @ A ) ) ) ) ) ).
% 14.25/8.13
% 14.25/8.13 thf(fact_6674_Min_Osubset__imp,axiom,
% 14.25/8.13 ! [A: set_rat,B4: set_rat] :
% 14.25/8.13 ( ( ord_less_eq_set_rat @ A @ B4 )
% 14.25/8.13 => ( ( A != bot_bot_set_rat )
% 14.25/8.13 => ( ( finite_finite_rat @ B4 )
% 14.25/8.13 => ( ord_less_eq_rat @ ( lattic8086005427650270231in_rat @ B4 ) @ ( lattic8086005427650270231in_rat @ A ) ) ) ) ) ).
% 14.25/8.13
% 14.25/8.13 thf(fact_6675_Min_Osubset__imp,axiom,
% 14.25/8.13 ! [A: set_int,B4: set_int] :
% 14.25/8.13 ( ( ord_less_eq_set_int @ A @ B4 )
% 14.25/8.13 => ( ( A != bot_bot_set_int )
% 14.25/8.13 => ( ( finite_finite_int @ B4 )
% 14.25/8.13 => ( ord_less_eq_int @ ( lattic8718645017227715691in_int @ B4 ) @ ( lattic8718645017227715691in_int @ A ) ) ) ) ) ).
% 14.25/8.13
% 14.25/8.13 thf(fact_6676_Min_Osubset__imp,axiom,
% 14.25/8.13 ! [A: set_nat,B4: set_nat] :
% 14.25/8.13 ( ( ord_less_eq_set_nat @ A @ B4 )
% 14.25/8.13 => ( ( A != bot_bot_set_nat )
% 14.25/8.13 => ( ( finite_finite_nat @ B4 )
% 14.25/8.13 => ( ord_less_eq_nat @ ( lattic8721135487736765967in_nat @ B4 ) @ ( lattic8721135487736765967in_nat @ A ) ) ) ) ) ).
% 14.25/8.13
% 14.25/8.13 thf(fact_6677_minus__Min__eq__Max,axiom,
% 14.25/8.13 ! [S: set_int] :
% 14.25/8.13 ( ( finite_finite_int @ S )
% 14.25/8.13 => ( ( S != bot_bot_set_int )
% 14.25/8.13 => ( ( uminus_uminus_int @ ( lattic8718645017227715691in_int @ S ) )
% 14.25/8.13 = ( lattic8263393255366662781ax_int @ ( image_int_int2 @ uminus_uminus_int @ S ) ) ) ) ) ).
% 14.25/8.13
% 14.25/8.13 thf(fact_6678_minus__Min__eq__Max,axiom,
% 14.25/8.13 ! [S: set_Code_integer] :
% 14.25/8.13 ( ( finite6017078050557962740nteger @ S )
% 14.25/8.13 => ( ( S != bot_bo3990330152332043303nteger )
% 14.25/8.13 => ( ( uminus1351360451143612070nteger @ ( lattic1063845414844153500nteger @ S ) )
% 14.25/8.13 = ( lattic4901227151466704046nteger @ ( image_4470545334726330049nteger @ uminus1351360451143612070nteger @ S ) ) ) ) ) ).
% 14.25/8.13
% 14.25/8.13 thf(fact_6679_minus__Max__eq__Min,axiom,
% 14.25/8.13 ! [S: set_int] :
% 14.25/8.13 ( ( finite_finite_int @ S )
% 14.25/8.13 => ( ( S != bot_bot_set_int )
% 14.25/8.13 => ( ( uminus_uminus_int @ ( lattic8263393255366662781ax_int @ S ) )
% 14.25/8.13 = ( lattic8718645017227715691in_int @ ( image_int_int2 @ uminus_uminus_int @ S ) ) ) ) ) ).
% 14.25/8.13
% 14.25/8.13 thf(fact_6680_minus__Max__eq__Min,axiom,
% 14.25/8.13 ! [S: set_Code_integer] :
% 14.25/8.13 ( ( finite6017078050557962740nteger @ S )
% 14.25/8.13 => ( ( S != bot_bo3990330152332043303nteger )
% 14.25/8.13 => ( ( uminus1351360451143612070nteger @ ( lattic4901227151466704046nteger @ S ) )
% 14.25/8.13 = ( lattic1063845414844153500nteger @ ( image_4470545334726330049nteger @ uminus1351360451143612070nteger @ S ) ) ) ) ) ).
% 14.25/8.13
% 14.25/8.13 thf(fact_6681_Gcd__0__iff,axiom,
% 14.25/8.13 ! [A: set_int] :
% 14.25/8.13 ( ( ( gcd_Gcd_int @ A )
% 14.25/8.13 = zero_zero_int )
% 14.25/8.13 = ( ord_less_eq_set_int @ A @ ( insert_int @ zero_zero_int @ bot_bot_set_int ) ) ) ).
% 14.25/8.13
% 14.25/8.13 thf(fact_6682_Gcd__0__iff,axiom,
% 14.25/8.13 ! [A: set_nat] :
% 14.25/8.13 ( ( ( gcd_Gcd_nat @ A )
% 14.25/8.13 = zero_zero_nat )
% 14.25/8.13 = ( ord_less_eq_set_nat @ A @ ( insert_nat @ zero_zero_nat @ bot_bot_set_nat ) ) ) ).
% 14.25/8.13
% 14.25/8.13 thf(fact_6683_Gcd__fin__0__iff,axiom,
% 14.25/8.13 ! [A: set_int] :
% 14.25/8.13 ( ( ( semiri4256215615220890538in_int @ A )
% 14.25/8.13 = zero_zero_int )
% 14.25/8.13 = ( ( ord_less_eq_set_int @ A @ ( insert_int @ zero_zero_int @ bot_bot_set_int ) )
% 14.25/8.13 & ( finite_finite_int @ A ) ) ) ).
% 14.25/8.13
% 14.25/8.13 thf(fact_6684_Gcd__fin__0__iff,axiom,
% 14.25/8.13 ! [A: set_nat] :
% 14.25/8.13 ( ( ( semiri4258706085729940814in_nat @ A )
% 14.25/8.13 = zero_zero_nat )
% 14.25/8.13 = ( ( ord_less_eq_set_nat @ A @ ( insert_nat @ zero_zero_nat @ bot_bot_set_nat ) )
% 14.25/8.13 & ( finite_finite_nat @ A ) ) ) ).
% 14.25/8.13
% 14.25/8.13 thf(fact_6685_sum__bounded__above__strict,axiom,
% 14.25/8.13 ! [A: set_nat,F: nat > rat,K4: rat] :
% 14.25/8.13 ( ! [I3: nat] :
% 14.25/8.13 ( ( member_nat @ I3 @ A )
% 14.25/8.13 => ( ord_less_rat @ ( F @ I3 ) @ K4 ) )
% 14.25/8.13 => ( ( ord_less_nat @ zero_zero_nat @ ( finite_card_nat @ A ) )
% 14.25/8.13 => ( ord_less_rat @ ( groups2906978787729119204at_rat @ F @ A ) @ ( times_times_rat @ ( semiri681578069525770553at_rat @ ( finite_card_nat @ A ) ) @ K4 ) ) ) ) ).
% 14.25/8.13
% 14.25/8.13 thf(fact_6686_sum__bounded__above__strict,axiom,
% 14.25/8.13 ! [A: set_int,F: int > rat,K4: rat] :
% 14.25/8.13 ( ! [I3: int] :
% 14.25/8.13 ( ( member_int @ I3 @ A )
% 14.25/8.13 => ( ord_less_rat @ ( F @ I3 ) @ K4 ) )
% 14.25/8.13 => ( ( ord_less_nat @ zero_zero_nat @ ( finite_card_int @ A ) )
% 14.25/8.13 => ( ord_less_rat @ ( groups3906332499630173760nt_rat @ F @ A ) @ ( times_times_rat @ ( semiri681578069525770553at_rat @ ( finite_card_int @ A ) ) @ K4 ) ) ) ) ).
% 14.25/8.13
% 14.25/8.13 thf(fact_6687_sum__bounded__above__strict,axiom,
% 14.25/8.13 ! [A: set_o,F: $o > rat,K4: rat] :
% 14.25/8.13 ( ! [I3: $o] :
% 14.25/8.13 ( ( member_o @ I3 @ A )
% 14.25/8.13 => ( ord_less_rat @ ( F @ I3 ) @ K4 ) )
% 14.25/8.13 => ( ( ord_less_nat @ zero_zero_nat @ ( finite_card_o @ A ) )
% 14.25/8.13 => ( ord_less_rat @ ( groups7872700643590313910_o_rat @ F @ A ) @ ( times_times_rat @ ( semiri681578069525770553at_rat @ ( finite_card_o @ A ) ) @ K4 ) ) ) ) ).
% 14.25/8.13
% 14.25/8.13 thf(fact_6688_sum__bounded__above__strict,axiom,
% 14.25/8.13 ! [A: set_Product_unit,F: product_unit > rat,K4: rat] :
% 14.25/8.13 ( ! [I3: product_unit] :
% 14.25/8.13 ( ( member_Product_unit @ I3 @ A )
% 14.25/8.13 => ( ord_less_rat @ ( F @ I3 ) @ K4 ) )
% 14.25/8.13 => ( ( ord_less_nat @ zero_zero_nat @ ( finite410649719033368117t_unit @ A ) )
% 14.25/8.13 => ( ord_less_rat @ ( groups8131667117488461483it_rat @ F @ A ) @ ( times_times_rat @ ( semiri681578069525770553at_rat @ ( finite410649719033368117t_unit @ A ) ) @ K4 ) ) ) ) ).
% 14.25/8.13
% 14.25/8.13 thf(fact_6689_sum__bounded__above__strict,axiom,
% 14.25/8.13 ! [A: set_nat,F: nat > int,K4: int] :
% 14.25/8.13 ( ! [I3: nat] :
% 14.25/8.13 ( ( member_nat @ I3 @ A )
% 14.25/8.13 => ( ord_less_int @ ( F @ I3 ) @ K4 ) )
% 14.25/8.13 => ( ( ord_less_nat @ zero_zero_nat @ ( finite_card_nat @ A ) )
% 14.25/8.13 => ( ord_less_int @ ( groups3539618377306564664at_int @ F @ A ) @ ( times_times_int @ ( semiri1314217659103216013at_int @ ( finite_card_nat @ A ) ) @ K4 ) ) ) ) ).
% 14.25/8.13
% 14.25/8.13 thf(fact_6690_sum__bounded__above__strict,axiom,
% 14.25/8.13 ! [A: set_o,F: $o > int,K4: int] :
% 14.25/8.13 ( ! [I3: $o] :
% 14.25/8.13 ( ( member_o @ I3 @ A )
% 14.25/8.13 => ( ord_less_int @ ( F @ I3 ) @ K4 ) )
% 14.25/8.13 => ( ( ord_less_nat @ zero_zero_nat @ ( finite_card_o @ A ) )
% 14.25/8.13 => ( ord_less_int @ ( groups8505340233167759370_o_int @ F @ A ) @ ( times_times_int @ ( semiri1314217659103216013at_int @ ( finite_card_o @ A ) ) @ K4 ) ) ) ) ).
% 14.25/8.13
% 14.25/8.13 thf(fact_6691_sum__bounded__above__strict,axiom,
% 14.25/8.13 ! [A: set_Product_unit,F: product_unit > int,K4: int] :
% 14.25/8.13 ( ! [I3: product_unit] :
% 14.25/8.13 ( ( member_Product_unit @ I3 @ A )
% 14.25/8.13 => ( ord_less_int @ ( F @ I3 ) @ K4 ) )
% 14.25/8.13 => ( ( ord_less_nat @ zero_zero_nat @ ( finite410649719033368117t_unit @ A ) )
% 14.25/8.13 => ( ord_less_int @ ( groups8764306707065906943it_int @ F @ A ) @ ( times_times_int @ ( semiri1314217659103216013at_int @ ( finite410649719033368117t_unit @ A ) ) @ K4 ) ) ) ) ).
% 14.25/8.13
% 14.25/8.13 thf(fact_6692_sum__bounded__above__strict,axiom,
% 14.25/8.13 ! [A: set_int,F: int > nat,K4: nat] :
% 14.25/8.13 ( ! [I3: int] :
% 14.25/8.13 ( ( member_int @ I3 @ A )
% 14.25/8.13 => ( ord_less_nat @ ( F @ I3 ) @ K4 ) )
% 14.25/8.13 => ( ( ord_less_nat @ zero_zero_nat @ ( finite_card_int @ A ) )
% 14.25/8.13 => ( ord_less_nat @ ( groups4541462559716669496nt_nat @ F @ A ) @ ( times_times_nat @ ( semiri1316708129612266289at_nat @ ( finite_card_int @ A ) ) @ K4 ) ) ) ) ).
% 14.25/8.13
% 14.25/8.13 thf(fact_6693_sum__bounded__above__strict,axiom,
% 14.25/8.13 ! [A: set_o,F: $o > nat,K4: nat] :
% 14.25/8.13 ( ! [I3: $o] :
% 14.25/8.13 ( ( member_o @ I3 @ A )
% 14.25/8.13 => ( ord_less_nat @ ( F @ I3 ) @ K4 ) )
% 14.25/8.13 => ( ( ord_less_nat @ zero_zero_nat @ ( finite_card_o @ A ) )
% 14.25/8.13 => ( ord_less_nat @ ( groups8507830703676809646_o_nat @ F @ A ) @ ( times_times_nat @ ( semiri1316708129612266289at_nat @ ( finite_card_o @ A ) ) @ K4 ) ) ) ) ).
% 14.25/8.13
% 14.25/8.13 thf(fact_6694_sum__bounded__above__strict,axiom,
% 14.25/8.13 ! [A: set_Product_unit,F: product_unit > nat,K4: nat] :
% 14.25/8.13 ( ! [I3: product_unit] :
% 14.25/8.13 ( ( member_Product_unit @ I3 @ A )
% 14.25/8.13 => ( ord_less_nat @ ( F @ I3 ) @ K4 ) )
% 14.25/8.13 => ( ( ord_less_nat @ zero_zero_nat @ ( finite410649719033368117t_unit @ A ) )
% 14.25/8.13 => ( ord_less_nat @ ( groups8766797177574957219it_nat @ F @ A ) @ ( times_times_nat @ ( semiri1316708129612266289at_nat @ ( finite410649719033368117t_unit @ A ) ) @ K4 ) ) ) ) ).
% 14.25/8.13
% 14.25/8.13 thf(fact_6695_prod__le__power,axiom,
% 14.25/8.13 ! [A: set_nat,F: nat > code_natural,N2: code_natural,K3: nat] :
% 14.25/8.13 ( ! [I3: nat] :
% 14.25/8.13 ( ( member_nat @ I3 @ A )
% 14.25/8.13 => ( ( ord_le1926595141338095240atural @ zero_z2226904508553997617atural @ ( F @ I3 ) )
% 14.25/8.13 & ( ord_le1926595141338095240atural @ ( F @ I3 ) @ N2 ) ) )
% 14.25/8.13 => ( ( ord_less_eq_nat @ ( finite_card_nat @ A ) @ K3 )
% 14.25/8.13 => ( ( ord_le1926595141338095240atural @ one_one_Code_natural @ N2 )
% 14.25/8.13 => ( ord_le1926595141338095240atural @ ( groups2279045934846249631atural @ F @ A ) @ ( power_7079662738309270450atural @ N2 @ K3 ) ) ) ) ) ).
% 14.25/8.13
% 14.25/8.13 thf(fact_6696_prod__le__power,axiom,
% 14.25/8.13 ! [A: set_int,F: int > code_natural,N2: code_natural,K3: nat] :
% 14.25/8.13 ( ! [I3: int] :
% 14.25/8.13 ( ( member_int @ I3 @ A )
% 14.25/8.13 => ( ( ord_le1926595141338095240atural @ zero_z2226904508553997617atural @ ( F @ I3 ) )
% 14.25/8.13 & ( ord_le1926595141338095240atural @ ( F @ I3 ) @ N2 ) ) )
% 14.25/8.13 => ( ( ord_less_eq_nat @ ( finite_card_int @ A ) @ K3 )
% 14.25/8.13 => ( ( ord_le1926595141338095240atural @ one_one_Code_natural @ N2 )
% 14.25/8.13 => ( ord_le1926595141338095240atural @ ( groups2650699495083094267atural @ F @ A ) @ ( power_7079662738309270450atural @ N2 @ K3 ) ) ) ) ) ).
% 14.25/8.13
% 14.25/8.13 thf(fact_6697_prod__le__power,axiom,
% 14.25/8.13 ! [A: set_o,F: $o > code_natural,N2: code_natural,K3: nat] :
% 14.25/8.13 ( ! [I3: $o] :
% 14.25/8.13 ( ( member_o @ I3 @ A )
% 14.25/8.13 => ( ( ord_le1926595141338095240atural @ zero_z2226904508553997617atural @ ( F @ I3 ) )
% 14.25/8.13 & ( ord_le1926595141338095240atural @ ( F @ I3 ) @ N2 ) ) )
% 14.25/8.13 => ( ( ord_less_eq_nat @ ( finite_card_o @ A ) @ K3 )
% 14.25/8.13 => ( ( ord_le1926595141338095240atural @ one_one_Code_natural @ N2 )
% 14.25/8.13 => ( ord_le1926595141338095240atural @ ( groups6518289543945209051atural @ F @ A ) @ ( power_7079662738309270450atural @ N2 @ K3 ) ) ) ) ) ).
% 14.25/8.13
% 14.25/8.13 thf(fact_6698_prod__le__power,axiom,
% 14.25/8.13 ! [A: set_Product_unit,F: product_unit > code_natural,N2: code_natural,K3: nat] :
% 14.25/8.13 ( ! [I3: product_unit] :
% 14.25/8.13 ( ( member_Product_unit @ I3 @ A )
% 14.25/8.13 => ( ( ord_le1926595141338095240atural @ zero_z2226904508553997617atural @ ( F @ I3 ) )
% 14.25/8.13 & ( ord_le1926595141338095240atural @ ( F @ I3 ) @ N2 ) ) )
% 14.25/8.13 => ( ( ord_less_eq_nat @ ( finite410649719033368117t_unit @ A ) @ K3 )
% 14.25/8.13 => ( ( ord_le1926595141338095240atural @ one_one_Code_natural @ N2 )
% 14.25/8.13 => ( ord_le1926595141338095240atural @ ( groups7008954963700796496atural @ F @ A ) @ ( power_7079662738309270450atural @ N2 @ K3 ) ) ) ) ) ).
% 14.25/8.13
% 14.25/8.13 thf(fact_6699_prod__le__power,axiom,
% 14.25/8.13 ! [A: set_nat,F: nat > code_integer,N2: code_integer,K3: nat] :
% 14.25/8.13 ( ! [I3: nat] :
% 14.25/8.13 ( ( member_nat @ I3 @ A )
% 14.25/8.13 => ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ ( F @ I3 ) )
% 14.25/8.13 & ( ord_le3102999989581377725nteger @ ( F @ I3 ) @ N2 ) ) )
% 14.25/8.13 => ( ( ord_less_eq_nat @ ( finite_card_nat @ A ) @ K3 )
% 14.25/8.13 => ( ( ord_le3102999989581377725nteger @ one_one_Code_integer @ N2 )
% 14.25/8.13 => ( ord_le3102999989581377725nteger @ ( groups3455450783089532116nteger @ F @ A ) @ ( power_8256067586552552935nteger @ N2 @ K3 ) ) ) ) ) ).
% 14.25/8.13
% 14.25/8.13 thf(fact_6700_prod__le__power,axiom,
% 14.25/8.13 ! [A: set_int,F: int > code_integer,N2: code_integer,K3: nat] :
% 14.25/8.13 ( ! [I3: int] :
% 14.25/8.13 ( ( member_int @ I3 @ A )
% 14.25/8.13 => ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ ( F @ I3 ) )
% 14.25/8.13 & ( ord_le3102999989581377725nteger @ ( F @ I3 ) @ N2 ) ) )
% 14.25/8.13 => ( ( ord_less_eq_nat @ ( finite_card_int @ A ) @ K3 )
% 14.25/8.13 => ( ( ord_le3102999989581377725nteger @ one_one_Code_integer @ N2 )
% 14.25/8.13 => ( ord_le3102999989581377725nteger @ ( groups3827104343326376752nteger @ F @ A ) @ ( power_8256067586552552935nteger @ N2 @ K3 ) ) ) ) ) ).
% 14.25/8.13
% 14.25/8.13 thf(fact_6701_prod__le__power,axiom,
% 14.25/8.13 ! [A: set_o,F: $o > code_integer,N2: code_integer,K3: nat] :
% 14.25/8.13 ( ! [I3: $o] :
% 14.25/8.13 ( ( member_o @ I3 @ A )
% 14.25/8.13 => ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ ( F @ I3 ) )
% 14.25/8.13 & ( ord_le3102999989581377725nteger @ ( F @ I3 ) @ N2 ) ) )
% 14.25/8.13 => ( ( ord_less_eq_nat @ ( finite_card_o @ A ) @ K3 )
% 14.25/8.13 => ( ( ord_le3102999989581377725nteger @ one_one_Code_integer @ N2 )
% 14.25/8.13 => ( ord_le3102999989581377725nteger @ ( groups7694694392188491536nteger @ F @ A ) @ ( power_8256067586552552935nteger @ N2 @ K3 ) ) ) ) ) ).
% 14.25/8.13
% 14.25/8.13 thf(fact_6702_prod__le__power,axiom,
% 14.25/8.13 ! [A: set_Product_unit,F: product_unit > code_integer,N2: code_integer,K3: nat] :
% 14.25/8.13 ( ! [I3: product_unit] :
% 14.25/8.13 ( ( member_Product_unit @ I3 @ A )
% 14.25/8.13 => ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ ( F @ I3 ) )
% 14.25/8.13 & ( ord_le3102999989581377725nteger @ ( F @ I3 ) @ N2 ) ) )
% 14.25/8.13 => ( ( ord_less_eq_nat @ ( finite410649719033368117t_unit @ A ) @ K3 )
% 14.25/8.13 => ( ( ord_le3102999989581377725nteger @ one_one_Code_integer @ N2 )
% 14.25/8.13 => ( ord_le3102999989581377725nteger @ ( groups8185359811944078981nteger @ F @ A ) @ ( power_8256067586552552935nteger @ N2 @ K3 ) ) ) ) ) ).
% 14.25/8.13
% 14.25/8.13 thf(fact_6703_prod__le__power,axiom,
% 14.25/8.13 ! [A: set_nat,F: nat > rat,N2: rat,K3: nat] :
% 14.25/8.13 ( ! [I3: nat] :
% 14.25/8.13 ( ( member_nat @ I3 @ A )
% 14.25/8.13 => ( ( ord_less_eq_rat @ zero_zero_rat @ ( F @ I3 ) )
% 14.25/8.13 & ( ord_less_eq_rat @ ( F @ I3 ) @ N2 ) ) )
% 14.25/8.13 => ( ( ord_less_eq_nat @ ( finite_card_nat @ A ) @ K3 )
% 14.25/8.13 => ( ( ord_less_eq_rat @ one_one_rat @ N2 )
% 14.25/8.13 => ( ord_less_eq_rat @ ( groups73079841787564623at_rat @ F @ A ) @ ( power_power_rat @ N2 @ K3 ) ) ) ) ) ).
% 14.25/8.13
% 14.25/8.13 thf(fact_6704_prod__le__power,axiom,
% 14.25/8.13 ! [A: set_int,F: int > rat,N2: rat,K3: nat] :
% 14.25/8.13 ( ! [I3: int] :
% 14.25/8.13 ( ( member_int @ I3 @ A )
% 14.25/8.13 => ( ( ord_less_eq_rat @ zero_zero_rat @ ( F @ I3 ) )
% 14.25/8.13 & ( ord_less_eq_rat @ ( F @ I3 ) @ N2 ) ) )
% 14.25/8.13 => ( ( ord_less_eq_nat @ ( finite_card_int @ A ) @ K3 )
% 14.25/8.13 => ( ( ord_less_eq_rat @ one_one_rat @ N2 )
% 14.25/8.13 => ( ord_less_eq_rat @ ( groups1072433553688619179nt_rat @ F @ A ) @ ( power_power_rat @ N2 @ K3 ) ) ) ) ) ).
% 14.25/8.13
% 14.25/8.13 thf(fact_6705_prod__mono2,axiom,
% 14.25/8.13 ! [B4: set_int,A: set_int,F: int > code_integer] :
% 14.25/8.13 ( ( finite_finite_int @ B4 )
% 14.25/8.13 => ( ( ord_less_eq_set_int @ A @ B4 )
% 14.25/8.13 => ( ! [B2: int] :
% 14.25/8.13 ( ( member_int @ B2 @ ( minus_minus_set_int @ B4 @ A ) )
% 14.25/8.13 => ( ord_le3102999989581377725nteger @ one_one_Code_integer @ ( F @ B2 ) ) )
% 14.25/8.13 => ( ! [A4: int] :
% 14.25/8.13 ( ( member_int @ A4 @ A )
% 14.25/8.13 => ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ ( F @ A4 ) ) )
% 14.25/8.13 => ( ord_le3102999989581377725nteger @ ( groups3827104343326376752nteger @ F @ A ) @ ( groups3827104343326376752nteger @ F @ B4 ) ) ) ) ) ) ).
% 14.25/8.13
% 14.25/8.13 thf(fact_6706_prod__mono2,axiom,
% 14.25/8.13 ! [B4: set_o,A: set_o,F: $o > code_integer] :
% 14.25/8.13 ( ( finite_finite_o @ B4 )
% 14.25/8.13 => ( ( ord_less_eq_set_o @ A @ B4 )
% 14.25/8.13 => ( ! [B2: $o] :
% 14.25/8.13 ( ( member_o @ B2 @ ( minus_minus_set_o @ B4 @ A ) )
% 14.25/8.13 => ( ord_le3102999989581377725nteger @ one_one_Code_integer @ ( F @ B2 ) ) )
% 14.25/8.13 => ( ! [A4: $o] :
% 14.25/8.13 ( ( member_o @ A4 @ A )
% 14.25/8.13 => ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ ( F @ A4 ) ) )
% 14.25/8.13 => ( ord_le3102999989581377725nteger @ ( groups7694694392188491536nteger @ F @ A ) @ ( groups7694694392188491536nteger @ F @ B4 ) ) ) ) ) ) ).
% 14.25/8.13
% 14.25/8.13 thf(fact_6707_prod__mono2,axiom,
% 14.25/8.13 ! [B4: set_Product_unit,A: set_Product_unit,F: product_unit > code_integer] :
% 14.25/8.13 ( ( finite4290736615968046902t_unit @ B4 )
% 14.25/8.13 => ( ( ord_le3507040750410214029t_unit @ A @ B4 )
% 14.25/8.13 => ( ! [B2: product_unit] :
% 14.25/8.13 ( ( member_Product_unit @ B2 @ ( minus_6452836326544984404t_unit @ B4 @ A ) )
% 14.25/8.13 => ( ord_le3102999989581377725nteger @ one_one_Code_integer @ ( F @ B2 ) ) )
% 14.25/8.13 => ( ! [A4: product_unit] :
% 14.25/8.13 ( ( member_Product_unit @ A4 @ A )
% 14.25/8.13 => ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ ( F @ A4 ) ) )
% 14.25/8.13 => ( ord_le3102999989581377725nteger @ ( groups8185359811944078981nteger @ F @ A ) @ ( groups8185359811944078981nteger @ F @ B4 ) ) ) ) ) ) ).
% 14.25/8.13
% 14.25/8.13 thf(fact_6708_prod__mono2,axiom,
% 14.25/8.13 ! [B4: set_Code_integer,A: set_Code_integer,F: code_integer > code_integer] :
% 14.25/8.13 ( ( finite6017078050557962740nteger @ B4 )
% 14.25/8.13 => ( ( ord_le7084787975880047091nteger @ A @ B4 )
% 14.25/8.13 => ( ! [B2: code_integer] :
% 14.25/8.13 ( ( member_Code_integer @ B2 @ ( minus_2355218937544613996nteger @ B4 @ A ) )
% 14.25/8.13 => ( ord_le3102999989581377725nteger @ one_one_Code_integer @ ( F @ B2 ) ) )
% 14.25/8.13 => ( ! [A4: code_integer] :
% 14.25/8.13 ( ( member_Code_integer @ A4 @ A )
% 14.25/8.13 => ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ ( F @ A4 ) ) )
% 14.25/8.13 => ( ord_le3102999989581377725nteger @ ( groups3674199335183972705nteger @ F @ A ) @ ( groups3674199335183972705nteger @ F @ B4 ) ) ) ) ) ) ).
% 14.25/8.13
% 14.25/8.13 thf(fact_6709_prod__mono2,axiom,
% 14.25/8.13 ! [B4: set_int,A: set_int,F: int > rat] :
% 14.25/8.13 ( ( finite_finite_int @ B4 )
% 14.25/8.13 => ( ( ord_less_eq_set_int @ A @ B4 )
% 14.25/8.13 => ( ! [B2: int] :
% 14.25/8.13 ( ( member_int @ B2 @ ( minus_minus_set_int @ B4 @ A ) )
% 14.25/8.13 => ( ord_less_eq_rat @ one_one_rat @ ( F @ B2 ) ) )
% 14.25/8.13 => ( ! [A4: int] :
% 14.25/8.13 ( ( member_int @ A4 @ A )
% 14.25/8.13 => ( ord_less_eq_rat @ zero_zero_rat @ ( F @ A4 ) ) )
% 14.25/8.13 => ( ord_less_eq_rat @ ( groups1072433553688619179nt_rat @ F @ A ) @ ( groups1072433553688619179nt_rat @ F @ B4 ) ) ) ) ) ) ).
% 14.25/8.13
% 14.25/8.13 thf(fact_6710_prod__mono2,axiom,
% 14.25/8.13 ! [B4: set_o,A: set_o,F: $o > rat] :
% 14.25/8.13 ( ( finite_finite_o @ B4 )
% 14.25/8.13 => ( ( ord_less_eq_set_o @ A @ B4 )
% 14.25/8.13 => ( ! [B2: $o] :
% 14.25/8.13 ( ( member_o @ B2 @ ( minus_minus_set_o @ B4 @ A ) )
% 14.25/8.13 => ( ord_less_eq_rat @ one_one_rat @ ( F @ B2 ) ) )
% 14.25/8.13 => ( ! [A4: $o] :
% 14.25/8.13 ( ( member_o @ A4 @ A )
% 14.25/8.13 => ( ord_less_eq_rat @ zero_zero_rat @ ( F @ A4 ) ) )
% 14.25/8.13 => ( ord_less_eq_rat @ ( groups2869687844427037835_o_rat @ F @ A ) @ ( groups2869687844427037835_o_rat @ F @ B4 ) ) ) ) ) ) ).
% 14.25/8.13
% 14.25/8.13 thf(fact_6711_prod__mono2,axiom,
% 14.25/8.13 ! [B4: set_Product_unit,A: set_Product_unit,F: product_unit > rat] :
% 14.25/8.13 ( ( finite4290736615968046902t_unit @ B4 )
% 14.25/8.13 => ( ( ord_le3507040750410214029t_unit @ A @ B4 )
% 14.25/8.13 => ( ! [B2: product_unit] :
% 14.25/8.13 ( ( member_Product_unit @ B2 @ ( minus_6452836326544984404t_unit @ B4 @ A ) )
% 14.25/8.13 => ( ord_less_eq_rat @ one_one_rat @ ( F @ B2 ) ) )
% 14.25/8.13 => ( ! [A4: product_unit] :
% 14.25/8.13 ( ( member_Product_unit @ A4 @ A )
% 14.25/8.13 => ( ord_less_eq_rat @ zero_zero_rat @ ( F @ A4 ) ) )
% 14.25/8.13 => ( ord_less_eq_rat @ ( groups8984663225794451456it_rat @ F @ A ) @ ( groups8984663225794451456it_rat @ F @ B4 ) ) ) ) ) ) ).
% 14.25/8.13
% 14.25/8.13 thf(fact_6712_prod__mono2,axiom,
% 14.25/8.13 ! [B4: set_Code_integer,A: set_Code_integer,F: code_integer > rat] :
% 14.25/8.13 ( ( finite6017078050557962740nteger @ B4 )
% 14.25/8.13 => ( ( ord_le7084787975880047091nteger @ A @ B4 )
% 14.25/8.13 => ( ! [B2: code_integer] :
% 14.25/8.13 ( ( member_Code_integer @ B2 @ ( minus_2355218937544613996nteger @ B4 @ A ) )
% 14.25/8.13 => ( ord_less_eq_rat @ one_one_rat @ ( F @ B2 ) ) )
% 14.25/8.13 => ( ! [A4: code_integer] :
% 14.25/8.13 ( ( member_Code_integer @ A4 @ A )
% 14.25/8.13 => ( ord_less_eq_rat @ zero_zero_rat @ ( F @ A4 ) ) )
% 14.25/8.13 => ( ord_less_eq_rat @ ( groups2555765274223993564er_rat @ F @ A ) @ ( groups2555765274223993564er_rat @ F @ B4 ) ) ) ) ) ) ).
% 14.25/8.13
% 14.25/8.13 thf(fact_6713_prod__mono2,axiom,
% 14.25/8.13 ! [B4: set_int,A: set_int,F: int > int] :
% 14.25/8.13 ( ( finite_finite_int @ B4 )
% 14.25/8.13 => ( ( ord_less_eq_set_int @ A @ B4 )
% 14.25/8.13 => ( ! [B2: int] :
% 14.25/8.13 ( ( member_int @ B2 @ ( minus_minus_set_int @ B4 @ A ) )
% 14.25/8.13 => ( ord_less_eq_int @ one_one_int @ ( F @ B2 ) ) )
% 14.25/8.13 => ( ! [A4: int] :
% 14.25/8.13 ( ( member_int @ A4 @ A )
% 14.25/8.13 => ( ord_less_eq_int @ zero_zero_int @ ( F @ A4 ) ) )
% 14.25/8.13 => ( ord_less_eq_int @ ( groups1705073143266064639nt_int @ F @ A ) @ ( groups1705073143266064639nt_int @ F @ B4 ) ) ) ) ) ) ).
% 14.25/8.13
% 14.25/8.13 thf(fact_6714_prod__mono2,axiom,
% 14.25/8.13 ! [B4: set_o,A: set_o,F: $o > int] :
% 14.25/8.13 ( ( finite_finite_o @ B4 )
% 14.25/8.13 => ( ( ord_less_eq_set_o @ A @ B4 )
% 14.25/8.13 => ( ! [B2: $o] :
% 14.25/8.13 ( ( member_o @ B2 @ ( minus_minus_set_o @ B4 @ A ) )
% 14.25/8.13 => ( ord_less_eq_int @ one_one_int @ ( F @ B2 ) ) )
% 14.25/8.13 => ( ! [A4: $o] :
% 14.25/8.13 ( ( member_o @ A4 @ A )
% 14.25/8.13 => ( ord_less_eq_int @ zero_zero_int @ ( F @ A4 ) ) )
% 14.25/8.13 => ( ord_less_eq_int @ ( groups3502327434004483295_o_int @ F @ A ) @ ( groups3502327434004483295_o_int @ F @ B4 ) ) ) ) ) ) ).
% 14.25/8.13
% 14.25/8.13 thf(fact_6715_neg__equal__iff__equal,axiom,
% 14.25/8.13 ! [A2: int,B: int] :
% 14.25/8.13 ( ( ( uminus_uminus_int @ A2 )
% 14.25/8.13 = ( uminus_uminus_int @ B ) )
% 14.25/8.13 = ( A2 = B ) ) ).
% 14.25/8.13
% 14.25/8.13 thf(fact_6716_neg__equal__iff__equal,axiom,
% 14.25/8.13 ! [A2: code_integer,B: code_integer] :
% 14.25/8.13 ( ( ( uminus1351360451143612070nteger @ A2 )
% 14.25/8.13 = ( uminus1351360451143612070nteger @ B ) )
% 14.25/8.13 = ( A2 = B ) ) ).
% 14.25/8.13
% 14.25/8.13 thf(fact_6717_add_Oinverse__inverse,axiom,
% 14.25/8.13 ! [A2: int] :
% 14.25/8.13 ( ( uminus_uminus_int @ ( uminus_uminus_int @ A2 ) )
% 14.25/8.13 = A2 ) ).
% 14.25/8.13
% 14.25/8.13 thf(fact_6718_add_Oinverse__inverse,axiom,
% 14.25/8.13 ! [A2: code_integer] :
% 14.25/8.13 ( ( uminus1351360451143612070nteger @ ( uminus1351360451143612070nteger @ A2 ) )
% 14.25/8.13 = A2 ) ).
% 14.25/8.13
% 14.25/8.13 thf(fact_6719_Compl__subset__Compl__iff,axiom,
% 14.25/8.13 ! [A: set_nat,B4: set_nat] :
% 14.25/8.13 ( ( ord_less_eq_set_nat @ ( uminus5710092332889474511et_nat @ A ) @ ( uminus5710092332889474511et_nat @ B4 ) )
% 14.25/8.13 = ( ord_less_eq_set_nat @ B4 @ A ) ) ).
% 14.25/8.13
% 14.25/8.13 thf(fact_6720_Compl__anti__mono,axiom,
% 14.25/8.13 ! [A: set_nat,B4: set_nat] :
% 14.25/8.13 ( ( ord_less_eq_set_nat @ A @ B4 )
% 14.25/8.13 => ( ord_less_eq_set_nat @ ( uminus5710092332889474511et_nat @ B4 ) @ ( uminus5710092332889474511et_nat @ A ) ) ) ).
% 14.25/8.13
% 14.25/8.13 thf(fact_6721_neg__le__iff__le,axiom,
% 14.25/8.13 ! [B: code_integer,A2: code_integer] :
% 14.25/8.13 ( ( ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ B ) @ ( uminus1351360451143612070nteger @ A2 ) )
% 14.25/8.13 = ( ord_le3102999989581377725nteger @ A2 @ B ) ) ).
% 14.25/8.13
% 14.25/8.13 thf(fact_6722_neg__le__iff__le,axiom,
% 14.25/8.13 ! [B: rat,A2: rat] :
% 14.25/8.13 ( ( ord_less_eq_rat @ ( uminus_uminus_rat @ B ) @ ( uminus_uminus_rat @ A2 ) )
% 14.25/8.13 = ( ord_less_eq_rat @ A2 @ B ) ) ).
% 14.25/8.13
% 14.25/8.13 thf(fact_6723_neg__le__iff__le,axiom,
% 14.25/8.13 ! [B: int,A2: int] :
% 14.25/8.13 ( ( ord_less_eq_int @ ( uminus_uminus_int @ B ) @ ( uminus_uminus_int @ A2 ) )
% 14.25/8.13 = ( ord_less_eq_int @ A2 @ B ) ) ).
% 14.25/8.13
% 14.25/8.13 thf(fact_6724_neg__equal__zero,axiom,
% 14.25/8.13 ! [A2: int] :
% 14.25/8.13 ( ( ( uminus_uminus_int @ A2 )
% 14.25/8.13 = A2 )
% 14.25/8.13 = ( A2 = zero_zero_int ) ) ).
% 14.25/8.13
% 14.25/8.13 thf(fact_6725_neg__equal__zero,axiom,
% 14.25/8.13 ! [A2: code_integer] :
% 14.25/8.13 ( ( ( uminus1351360451143612070nteger @ A2 )
% 14.25/8.13 = A2 )
% 14.25/8.13 = ( A2 = zero_z3403309356797280102nteger ) ) ).
% 14.25/8.13
% 14.25/8.13 thf(fact_6726_equal__neg__zero,axiom,
% 14.25/8.13 ! [A2: int] :
% 14.25/8.13 ( ( A2
% 14.25/8.13 = ( uminus_uminus_int @ A2 ) )
% 14.25/8.13 = ( A2 = zero_zero_int ) ) ).
% 14.25/8.13
% 14.25/8.13 thf(fact_6727_equal__neg__zero,axiom,
% 14.25/8.13 ! [A2: code_integer] :
% 14.25/8.13 ( ( A2
% 14.25/8.13 = ( uminus1351360451143612070nteger @ A2 ) )
% 14.25/8.13 = ( A2 = zero_z3403309356797280102nteger ) ) ).
% 14.25/8.13
% 14.25/8.13 thf(fact_6728_neg__equal__0__iff__equal,axiom,
% 14.25/8.13 ! [A2: int] :
% 14.25/8.13 ( ( ( uminus_uminus_int @ A2 )
% 14.25/8.13 = zero_zero_int )
% 14.25/8.13 = ( A2 = zero_zero_int ) ) ).
% 14.25/8.13
% 14.25/8.13 thf(fact_6729_neg__equal__0__iff__equal,axiom,
% 14.25/8.13 ! [A2: code_integer] :
% 14.25/8.13 ( ( ( uminus1351360451143612070nteger @ A2 )
% 14.25/8.13 = zero_z3403309356797280102nteger )
% 14.25/8.13 = ( A2 = zero_z3403309356797280102nteger ) ) ).
% 14.25/8.13
% 14.25/8.13 thf(fact_6730_neg__0__equal__iff__equal,axiom,
% 14.25/8.13 ! [A2: int] :
% 14.25/8.13 ( ( zero_zero_int
% 14.25/8.13 = ( uminus_uminus_int @ A2 ) )
% 14.25/8.13 = ( zero_zero_int = A2 ) ) ).
% 14.25/8.13
% 14.25/8.13 thf(fact_6731_neg__0__equal__iff__equal,axiom,
% 14.25/8.13 ! [A2: code_integer] :
% 14.25/8.13 ( ( zero_z3403309356797280102nteger
% 14.25/8.13 = ( uminus1351360451143612070nteger @ A2 ) )
% 14.25/8.13 = ( zero_z3403309356797280102nteger = A2 ) ) ).
% 14.25/8.13
% 14.25/8.13 thf(fact_6732_add_Oinverse__neutral,axiom,
% 14.25/8.13 ( ( uminus_uminus_int @ zero_zero_int )
% 14.25/8.13 = zero_zero_int ) ).
% 14.25/8.13
% 14.25/8.13 thf(fact_6733_add_Oinverse__neutral,axiom,
% 14.25/8.13 ( ( uminus1351360451143612070nteger @ zero_z3403309356797280102nteger )
% 14.25/8.13 = zero_z3403309356797280102nteger ) ).
% 14.25/8.13
% 14.25/8.13 thf(fact_6734_neg__less__iff__less,axiom,
% 14.25/8.13 ! [B: rat,A2: rat] :
% 14.25/8.13 ( ( ord_less_rat @ ( uminus_uminus_rat @ B ) @ ( uminus_uminus_rat @ A2 ) )
% 14.25/8.13 = ( ord_less_rat @ A2 @ B ) ) ).
% 14.25/8.13
% 14.25/8.13 thf(fact_6735_neg__less__iff__less,axiom,
% 14.25/8.13 ! [B: int,A2: int] :
% 14.25/8.13 ( ( ord_less_int @ ( uminus_uminus_int @ B ) @ ( uminus_uminus_int @ A2 ) )
% 14.25/8.13 = ( ord_less_int @ A2 @ B ) ) ).
% 14.25/8.13
% 14.25/8.13 thf(fact_6736_neg__less__iff__less,axiom,
% 14.25/8.13 ! [B: code_integer,A2: code_integer] :
% 14.25/8.13 ( ( ord_le6747313008572928689nteger @ ( uminus1351360451143612070nteger @ B ) @ ( uminus1351360451143612070nteger @ A2 ) )
% 14.25/8.13 = ( ord_le6747313008572928689nteger @ A2 @ B ) ) ).
% 14.25/8.13
% 14.25/8.13 thf(fact_6737_minus__add__distrib,axiom,
% 14.25/8.13 ! [A2: rat,B: rat] :
% 14.25/8.13 ( ( uminus_uminus_rat @ ( plus_plus_rat @ A2 @ B ) )
% 14.25/8.13 = ( plus_plus_rat @ ( uminus_uminus_rat @ A2 ) @ ( uminus_uminus_rat @ B ) ) ) ).
% 14.25/8.13
% 14.25/8.13 thf(fact_6738_minus__add__distrib,axiom,
% 14.25/8.13 ! [A2: int,B: int] :
% 14.25/8.13 ( ( uminus_uminus_int @ ( plus_plus_int @ A2 @ B ) )
% 14.25/8.13 = ( plus_plus_int @ ( uminus_uminus_int @ A2 ) @ ( uminus_uminus_int @ B ) ) ) ).
% 14.25/8.13
% 14.25/8.13 thf(fact_6739_minus__add__distrib,axiom,
% 14.25/8.13 ! [A2: code_integer,B: code_integer] :
% 14.25/8.13 ( ( uminus1351360451143612070nteger @ ( plus_p5714425477246183910nteger @ A2 @ B ) )
% 14.25/8.13 = ( plus_p5714425477246183910nteger @ ( uminus1351360451143612070nteger @ A2 ) @ ( uminus1351360451143612070nteger @ B ) ) ) ).
% 14.25/8.13
% 14.25/8.13 thf(fact_6740_minus__add__cancel,axiom,
% 14.25/8.13 ! [A2: rat,B: rat] :
% 14.25/8.13 ( ( plus_plus_rat @ ( uminus_uminus_rat @ A2 ) @ ( plus_plus_rat @ A2 @ B ) )
% 14.25/8.13 = B ) ).
% 14.25/8.13
% 14.25/8.13 thf(fact_6741_minus__add__cancel,axiom,
% 14.25/8.13 ! [A2: int,B: int] :
% 14.25/8.13 ( ( plus_plus_int @ ( uminus_uminus_int @ A2 ) @ ( plus_plus_int @ A2 @ B ) )
% 14.25/8.13 = B ) ).
% 14.25/8.13
% 14.25/8.13 thf(fact_6742_minus__add__cancel,axiom,
% 14.25/8.13 ! [A2: code_integer,B: code_integer] :
% 14.25/8.13 ( ( plus_p5714425477246183910nteger @ ( uminus1351360451143612070nteger @ A2 ) @ ( plus_p5714425477246183910nteger @ A2 @ B ) )
% 14.25/8.13 = B ) ).
% 14.25/8.13
% 14.25/8.13 thf(fact_6743_add__minus__cancel,axiom,
% 14.25/8.13 ! [A2: rat,B: rat] :
% 14.25/8.13 ( ( plus_plus_rat @ A2 @ ( plus_plus_rat @ ( uminus_uminus_rat @ A2 ) @ B ) )
% 14.25/8.13 = B ) ).
% 14.25/8.13
% 14.25/8.13 thf(fact_6744_add__minus__cancel,axiom,
% 14.25/8.13 ! [A2: int,B: int] :
% 14.25/8.13 ( ( plus_plus_int @ A2 @ ( plus_plus_int @ ( uminus_uminus_int @ A2 ) @ B ) )
% 14.25/8.13 = B ) ).
% 14.25/8.13
% 14.25/8.13 thf(fact_6745_add__minus__cancel,axiom,
% 14.25/8.13 ! [A2: code_integer,B: code_integer] :
% 14.25/8.13 ( ( plus_p5714425477246183910nteger @ A2 @ ( plus_p5714425477246183910nteger @ ( uminus1351360451143612070nteger @ A2 ) @ B ) )
% 14.25/8.13 = B ) ).
% 14.25/8.13
% 14.25/8.13 thf(fact_6746_mult__minus__left,axiom,
% 14.25/8.13 ! [A2: rat,B: rat] :
% 14.25/8.13 ( ( times_times_rat @ ( uminus_uminus_rat @ A2 ) @ B )
% 14.25/8.13 = ( uminus_uminus_rat @ ( times_times_rat @ A2 @ B ) ) ) ).
% 14.25/8.13
% 14.25/8.13 thf(fact_6747_mult__minus__left,axiom,
% 14.25/8.13 ! [A2: int,B: int] :
% 14.25/8.13 ( ( times_times_int @ ( uminus_uminus_int @ A2 ) @ B )
% 14.25/8.13 = ( uminus_uminus_int @ ( times_times_int @ A2 @ B ) ) ) ).
% 14.25/8.13
% 14.25/8.13 thf(fact_6748_mult__minus__left,axiom,
% 14.25/8.13 ! [A2: code_integer,B: code_integer] :
% 14.25/8.13 ( ( times_3573771949741848930nteger @ ( uminus1351360451143612070nteger @ A2 ) @ B )
% 14.25/8.13 = ( uminus1351360451143612070nteger @ ( times_3573771949741848930nteger @ A2 @ B ) ) ) ).
% 14.25/8.13
% 14.25/8.13 thf(fact_6749_minus__mult__minus,axiom,
% 14.25/8.13 ! [A2: rat,B: rat] :
% 14.25/8.13 ( ( times_times_rat @ ( uminus_uminus_rat @ A2 ) @ ( uminus_uminus_rat @ B ) )
% 14.25/8.13 = ( times_times_rat @ A2 @ B ) ) ).
% 14.25/8.13
% 14.25/8.13 thf(fact_6750_minus__mult__minus,axiom,
% 14.25/8.13 ! [A2: int,B: int] :
% 14.25/8.13 ( ( times_times_int @ ( uminus_uminus_int @ A2 ) @ ( uminus_uminus_int @ B ) )
% 14.25/8.13 = ( times_times_int @ A2 @ B ) ) ).
% 14.25/8.13
% 14.25/8.13 thf(fact_6751_minus__mult__minus,axiom,
% 14.25/8.13 ! [A2: code_integer,B: code_integer] :
% 14.25/8.13 ( ( times_3573771949741848930nteger @ ( uminus1351360451143612070nteger @ A2 ) @ ( uminus1351360451143612070nteger @ B ) )
% 14.25/8.13 = ( times_3573771949741848930nteger @ A2 @ B ) ) ).
% 14.25/8.13
% 14.25/8.13 thf(fact_6752_mult__minus__right,axiom,
% 14.25/8.13 ! [A2: rat,B: rat] :
% 14.25/8.13 ( ( times_times_rat @ A2 @ ( uminus_uminus_rat @ B ) )
% 14.25/8.13 = ( uminus_uminus_rat @ ( times_times_rat @ A2 @ B ) ) ) ).
% 14.25/8.13
% 14.25/8.13 thf(fact_6753_mult__minus__right,axiom,
% 14.25/8.13 ! [A2: int,B: int] :
% 14.25/8.13 ( ( times_times_int @ A2 @ ( uminus_uminus_int @ B ) )
% 14.25/8.13 = ( uminus_uminus_int @ ( times_times_int @ A2 @ B ) ) ) ).
% 14.25/8.13
% 14.25/8.13 thf(fact_6754_mult__minus__right,axiom,
% 14.25/8.13 ! [A2: code_integer,B: code_integer] :
% 14.25/8.13 ( ( times_3573771949741848930nteger @ A2 @ ( uminus1351360451143612070nteger @ B ) )
% 14.25/8.13 = ( uminus1351360451143612070nteger @ ( times_3573771949741848930nteger @ A2 @ B ) ) ) ).
% 14.25/8.13
% 14.25/8.13 thf(fact_6755_minus__diff__eq,axiom,
% 14.25/8.13 ! [A2: rat,B: rat] :
% 14.25/8.13 ( ( uminus_uminus_rat @ ( minus_minus_rat @ A2 @ B ) )
% 14.25/8.13 = ( minus_minus_rat @ B @ A2 ) ) ).
% 14.25/8.13
% 14.25/8.13 thf(fact_6756_minus__diff__eq,axiom,
% 14.25/8.13 ! [A2: int,B: int] :
% 14.25/8.13 ( ( uminus_uminus_int @ ( minus_minus_int @ A2 @ B ) )
% 14.25/8.13 = ( minus_minus_int @ B @ A2 ) ) ).
% 14.25/8.13
% 14.25/8.13 thf(fact_6757_minus__diff__eq,axiom,
% 14.25/8.13 ! [A2: code_integer,B: code_integer] :
% 14.25/8.13 ( ( uminus1351360451143612070nteger @ ( minus_8373710615458151222nteger @ A2 @ B ) )
% 14.25/8.13 = ( minus_8373710615458151222nteger @ B @ A2 ) ) ).
% 14.25/8.13
% 14.25/8.13 thf(fact_6758_Compl__disjoint,axiom,
% 14.25/8.13 ! [A: set_Product_unit] :
% 14.25/8.13 ( ( inf_in4660618365625256667t_unit @ A @ ( uminus5944136376168626660t_unit @ A ) )
% 14.25/8.13 = bot_bo3957492148770167129t_unit ) ).
% 14.25/8.13
% 14.25/8.13 thf(fact_6759_Compl__disjoint,axiom,
% 14.25/8.13 ! [A: set_o] :
% 14.25/8.13 ( ( inf_inf_set_o @ A @ ( uminus_uminus_set_o @ A ) )
% 14.25/8.13 = bot_bot_set_o ) ).
% 14.25/8.13
% 14.25/8.13 thf(fact_6760_Compl__disjoint,axiom,
% 14.25/8.13 ! [A: set_nat] :
% 14.25/8.13 ( ( inf_inf_set_nat @ A @ ( uminus5710092332889474511et_nat @ A ) )
% 14.25/8.13 = bot_bot_set_nat ) ).
% 14.25/8.13
% 14.25/8.13 thf(fact_6761_Compl__disjoint,axiom,
% 14.25/8.13 ! [A: set_int] :
% 14.25/8.13 ( ( inf_inf_set_int @ A @ ( uminus1532241313380277803et_int @ A ) )
% 14.25/8.13 = bot_bot_set_int ) ).
% 14.25/8.13
% 14.25/8.13 thf(fact_6762_Compl__disjoint,axiom,
% 14.25/8.13 ! [A: set_Pr4532377907799695533_nat_o] :
% 14.25/8.13 ( ( inf_in1906310914598751387_nat_o @ A @ ( uminus5254974436814262692_nat_o @ A ) )
% 14.25/8.13 = bot_bo7824918357723582233_nat_o ) ).
% 14.25/8.13
% 14.25/8.13 thf(fact_6763_Compl__disjoint2,axiom,
% 14.25/8.13 ! [A: set_Product_unit] :
% 14.25/8.13 ( ( inf_in4660618365625256667t_unit @ ( uminus5944136376168626660t_unit @ A ) @ A )
% 14.25/8.13 = bot_bo3957492148770167129t_unit ) ).
% 14.25/8.13
% 14.25/8.13 thf(fact_6764_Compl__disjoint2,axiom,
% 14.25/8.13 ! [A: set_o] :
% 14.25/8.13 ( ( inf_inf_set_o @ ( uminus_uminus_set_o @ A ) @ A )
% 14.25/8.13 = bot_bot_set_o ) ).
% 14.25/8.13
% 14.25/8.13 thf(fact_6765_Compl__disjoint2,axiom,
% 14.25/8.13 ! [A: set_nat] :
% 14.25/8.13 ( ( inf_inf_set_nat @ ( uminus5710092332889474511et_nat @ A ) @ A )
% 14.25/8.13 = bot_bot_set_nat ) ).
% 14.25/8.13
% 14.25/8.13 thf(fact_6766_Compl__disjoint2,axiom,
% 14.25/8.13 ! [A: set_int] :
% 14.25/8.13 ( ( inf_inf_set_int @ ( uminus1532241313380277803et_int @ A ) @ A )
% 14.25/8.13 = bot_bot_set_int ) ).
% 14.25/8.13
% 14.25/8.13 thf(fact_6767_Compl__disjoint2,axiom,
% 14.25/8.13 ! [A: set_Pr4532377907799695533_nat_o] :
% 14.25/8.13 ( ( inf_in1906310914598751387_nat_o @ ( uminus5254974436814262692_nat_o @ A ) @ A )
% 14.25/8.13 = bot_bo7824918357723582233_nat_o ) ).
% 14.25/8.13
% 14.25/8.13 thf(fact_6768_inter__compl__diff__conv,axiom,
% 14.25/8.13 ! [A: set_nat,B4: set_nat] :
% 14.25/8.13 ( ( inf_inf_set_nat @ A @ ( uminus5710092332889474511et_nat @ B4 ) )
% 14.25/8.13 = ( minus_minus_set_nat @ A @ B4 ) ) ).
% 14.25/8.13
% 14.25/8.13 thf(fact_6769_Diff__Compl,axiom,
% 14.25/8.13 ! [A: set_nat,B4: set_nat] :
% 14.25/8.13 ( ( minus_minus_set_nat @ A @ ( uminus5710092332889474511et_nat @ B4 ) )
% 14.25/8.13 = ( inf_inf_set_nat @ A @ B4 ) ) ).
% 14.25/8.13
% 14.25/8.13 thf(fact_6770_Compl__Diff__eq,axiom,
% 14.25/8.13 ! [A: set_nat,B4: set_nat] :
% 14.25/8.13 ( ( uminus5710092332889474511et_nat @ ( minus_minus_set_nat @ A @ B4 ) )
% 14.25/8.13 = ( sup_sup_set_nat @ ( uminus5710092332889474511et_nat @ A ) @ B4 ) ) ).
% 14.25/8.13
% 14.25/8.13 thf(fact_6771_neg__0__le__iff__le,axiom,
% 14.25/8.13 ! [A2: code_integer] :
% 14.25/8.13 ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ ( uminus1351360451143612070nteger @ A2 ) )
% 14.25/8.13 = ( ord_le3102999989581377725nteger @ A2 @ zero_z3403309356797280102nteger ) ) ).
% 14.25/8.13
% 14.25/8.13 thf(fact_6772_neg__0__le__iff__le,axiom,
% 14.25/8.13 ! [A2: rat] :
% 14.25/8.13 ( ( ord_less_eq_rat @ zero_zero_rat @ ( uminus_uminus_rat @ A2 ) )
% 14.25/8.13 = ( ord_less_eq_rat @ A2 @ zero_zero_rat ) ) ).
% 14.25/8.13
% 14.25/8.13 thf(fact_6773_neg__0__le__iff__le,axiom,
% 14.25/8.13 ! [A2: int] :
% 14.25/8.13 ( ( ord_less_eq_int @ zero_zero_int @ ( uminus_uminus_int @ A2 ) )
% 14.25/8.13 = ( ord_less_eq_int @ A2 @ zero_zero_int ) ) ).
% 14.25/8.13
% 14.25/8.13 thf(fact_6774_neg__le__0__iff__le,axiom,
% 14.25/8.13 ! [A2: code_integer] :
% 14.25/8.13 ( ( ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ A2 ) @ zero_z3403309356797280102nteger )
% 14.25/8.13 = ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ A2 ) ) ).
% 14.25/8.13
% 14.25/8.13 thf(fact_6775_neg__le__0__iff__le,axiom,
% 14.25/8.13 ! [A2: rat] :
% 14.25/8.13 ( ( ord_less_eq_rat @ ( uminus_uminus_rat @ A2 ) @ zero_zero_rat )
% 14.25/8.13 = ( ord_less_eq_rat @ zero_zero_rat @ A2 ) ) ).
% 14.25/8.13
% 14.25/8.13 thf(fact_6776_neg__le__0__iff__le,axiom,
% 14.25/8.13 ! [A2: int] :
% 14.25/8.13 ( ( ord_less_eq_int @ ( uminus_uminus_int @ A2 ) @ zero_zero_int )
% 14.25/8.13 = ( ord_less_eq_int @ zero_zero_int @ A2 ) ) ).
% 14.25/8.13
% 14.25/8.13 thf(fact_6777_less__eq__neg__nonpos,axiom,
% 14.25/8.13 ! [A2: code_integer] :
% 14.25/8.13 ( ( ord_le3102999989581377725nteger @ A2 @ ( uminus1351360451143612070nteger @ A2 ) )
% 14.25/8.13 = ( ord_le3102999989581377725nteger @ A2 @ zero_z3403309356797280102nteger ) ) ).
% 14.25/8.13
% 14.25/8.13 thf(fact_6778_less__eq__neg__nonpos,axiom,
% 14.25/8.13 ! [A2: rat] :
% 14.25/8.13 ( ( ord_less_eq_rat @ A2 @ ( uminus_uminus_rat @ A2 ) )
% 14.25/8.13 = ( ord_less_eq_rat @ A2 @ zero_zero_rat ) ) ).
% 14.25/8.13
% 14.25/8.13 thf(fact_6779_less__eq__neg__nonpos,axiom,
% 14.25/8.13 ! [A2: int] :
% 14.25/8.13 ( ( ord_less_eq_int @ A2 @ ( uminus_uminus_int @ A2 ) )
% 14.25/8.13 = ( ord_less_eq_int @ A2 @ zero_zero_int ) ) ).
% 14.25/8.13
% 14.25/8.13 thf(fact_6780_neg__less__eq__nonneg,axiom,
% 14.25/8.13 ! [A2: code_integer] :
% 14.25/8.13 ( ( ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ A2 ) @ A2 )
% 14.25/8.13 = ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ A2 ) ) ).
% 14.25/8.13
% 14.25/8.13 thf(fact_6781_neg__less__eq__nonneg,axiom,
% 14.25/8.13 ! [A2: rat] :
% 14.25/8.13 ( ( ord_less_eq_rat @ ( uminus_uminus_rat @ A2 ) @ A2 )
% 14.25/8.13 = ( ord_less_eq_rat @ zero_zero_rat @ A2 ) ) ).
% 14.25/8.13
% 14.25/8.13 thf(fact_6782_neg__less__eq__nonneg,axiom,
% 14.25/8.13 ! [A2: int] :
% 14.25/8.13 ( ( ord_less_eq_int @ ( uminus_uminus_int @ A2 ) @ A2 )
% 14.25/8.13 = ( ord_less_eq_int @ zero_zero_int @ A2 ) ) ).
% 14.25/8.13
% 14.25/8.13 thf(fact_6783_less__neg__neg,axiom,
% 14.25/8.13 ! [A2: rat] :
% 14.25/8.13 ( ( ord_less_rat @ A2 @ ( uminus_uminus_rat @ A2 ) )
% 14.25/8.13 = ( ord_less_rat @ A2 @ zero_zero_rat ) ) ).
% 14.25/8.13
% 14.25/8.13 thf(fact_6784_less__neg__neg,axiom,
% 14.25/8.13 ! [A2: int] :
% 14.25/8.13 ( ( ord_less_int @ A2 @ ( uminus_uminus_int @ A2 ) )
% 14.25/8.13 = ( ord_less_int @ A2 @ zero_zero_int ) ) ).
% 14.25/8.13
% 14.25/8.13 thf(fact_6785_less__neg__neg,axiom,
% 14.25/8.13 ! [A2: code_integer] :
% 14.25/8.13 ( ( ord_le6747313008572928689nteger @ A2 @ ( uminus1351360451143612070nteger @ A2 ) )
% 14.25/8.13 = ( ord_le6747313008572928689nteger @ A2 @ zero_z3403309356797280102nteger ) ) ).
% 14.25/8.13
% 14.25/8.13 thf(fact_6786_neg__less__pos,axiom,
% 14.25/8.13 ! [A2: rat] :
% 14.25/8.13 ( ( ord_less_rat @ ( uminus_uminus_rat @ A2 ) @ A2 )
% 14.25/8.13 = ( ord_less_rat @ zero_zero_rat @ A2 ) ) ).
% 14.25/8.13
% 14.25/8.13 thf(fact_6787_neg__less__pos,axiom,
% 14.25/8.13 ! [A2: int] :
% 14.25/8.13 ( ( ord_less_int @ ( uminus_uminus_int @ A2 ) @ A2 )
% 14.25/8.13 = ( ord_less_int @ zero_zero_int @ A2 ) ) ).
% 14.25/8.13
% 14.25/8.13 thf(fact_6788_neg__less__pos,axiom,
% 14.25/8.13 ! [A2: code_integer] :
% 14.25/8.13 ( ( ord_le6747313008572928689nteger @ ( uminus1351360451143612070nteger @ A2 ) @ A2 )
% 14.25/8.13 = ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ A2 ) ) ).
% 14.25/8.13
% 14.25/8.13 thf(fact_6789_neg__0__less__iff__less,axiom,
% 14.25/8.13 ! [A2: rat] :
% 14.25/8.13 ( ( ord_less_rat @ zero_zero_rat @ ( uminus_uminus_rat @ A2 ) )
% 14.25/8.13 = ( ord_less_rat @ A2 @ zero_zero_rat ) ) ).
% 14.25/8.13
% 14.25/8.13 thf(fact_6790_neg__0__less__iff__less,axiom,
% 14.25/8.13 ! [A2: int] :
% 14.25/8.13 ( ( ord_less_int @ zero_zero_int @ ( uminus_uminus_int @ A2 ) )
% 14.25/8.13 = ( ord_less_int @ A2 @ zero_zero_int ) ) ).
% 14.25/8.13
% 14.25/8.13 thf(fact_6791_neg__0__less__iff__less,axiom,
% 14.25/8.13 ! [A2: code_integer] :
% 14.25/8.13 ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ ( uminus1351360451143612070nteger @ A2 ) )
% 14.25/8.13 = ( ord_le6747313008572928689nteger @ A2 @ zero_z3403309356797280102nteger ) ) ).
% 14.25/8.13
% 14.25/8.13 thf(fact_6792_neg__less__0__iff__less,axiom,
% 14.25/8.13 ! [A2: rat] :
% 14.25/8.13 ( ( ord_less_rat @ ( uminus_uminus_rat @ A2 ) @ zero_zero_rat )
% 14.25/8.14 = ( ord_less_rat @ zero_zero_rat @ A2 ) ) ).
% 14.25/8.14
% 14.25/8.14 thf(fact_6793_neg__less__0__iff__less,axiom,
% 14.25/8.14 ! [A2: int] :
% 14.25/8.14 ( ( ord_less_int @ ( uminus_uminus_int @ A2 ) @ zero_zero_int )
% 14.25/8.14 = ( ord_less_int @ zero_zero_int @ A2 ) ) ).
% 14.25/8.14
% 14.25/8.14 thf(fact_6794_neg__less__0__iff__less,axiom,
% 14.25/8.14 ! [A2: code_integer] :
% 14.25/8.14 ( ( ord_le6747313008572928689nteger @ ( uminus1351360451143612070nteger @ A2 ) @ zero_z3403309356797280102nteger )
% 14.25/8.14 = ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ A2 ) ) ).
% 14.25/8.14
% 14.25/8.14 thf(fact_6795_ab__left__minus,axiom,
% 14.25/8.14 ! [A2: rat] :
% 14.25/8.14 ( ( plus_plus_rat @ ( uminus_uminus_rat @ A2 ) @ A2 )
% 14.25/8.14 = zero_zero_rat ) ).
% 14.25/8.14
% 14.25/8.14 thf(fact_6796_ab__left__minus,axiom,
% 14.25/8.14 ! [A2: int] :
% 14.25/8.14 ( ( plus_plus_int @ ( uminus_uminus_int @ A2 ) @ A2 )
% 14.25/8.14 = zero_zero_int ) ).
% 14.25/8.14
% 14.25/8.14 thf(fact_6797_ab__left__minus,axiom,
% 14.25/8.14 ! [A2: code_integer] :
% 14.25/8.14 ( ( plus_p5714425477246183910nteger @ ( uminus1351360451143612070nteger @ A2 ) @ A2 )
% 14.25/8.14 = zero_z3403309356797280102nteger ) ).
% 14.25/8.14
% 14.25/8.14 thf(fact_6798_add_Oright__inverse,axiom,
% 14.25/8.14 ! [A2: rat] :
% 14.25/8.14 ( ( plus_plus_rat @ A2 @ ( uminus_uminus_rat @ A2 ) )
% 14.25/8.14 = zero_zero_rat ) ).
% 14.25/8.14
% 14.25/8.14 thf(fact_6799_add_Oright__inverse,axiom,
% 14.25/8.14 ! [A2: int] :
% 14.25/8.14 ( ( plus_plus_int @ A2 @ ( uminus_uminus_int @ A2 ) )
% 14.25/8.14 = zero_zero_int ) ).
% 14.25/8.14
% 14.25/8.14 thf(fact_6800_add_Oright__inverse,axiom,
% 14.25/8.14 ! [A2: code_integer] :
% 14.25/8.14 ( ( plus_p5714425477246183910nteger @ A2 @ ( uminus1351360451143612070nteger @ A2 ) )
% 14.25/8.14 = zero_z3403309356797280102nteger ) ).
% 14.25/8.14
% 14.25/8.14 thf(fact_6801_diff__0,axiom,
% 14.25/8.14 ! [A2: rat] :
% 14.25/8.14 ( ( minus_minus_rat @ zero_zero_rat @ A2 )
% 14.25/8.14 = ( uminus_uminus_rat @ A2 ) ) ).
% 14.25/8.14
% 14.25/8.14 thf(fact_6802_diff__0,axiom,
% 14.25/8.14 ! [A2: int] :
% 14.25/8.14 ( ( minus_minus_int @ zero_zero_int @ A2 )
% 14.25/8.14 = ( uminus_uminus_int @ A2 ) ) ).
% 14.25/8.14
% 14.25/8.14 thf(fact_6803_diff__0,axiom,
% 14.25/8.14 ! [A2: code_integer] :
% 14.25/8.14 ( ( minus_8373710615458151222nteger @ zero_z3403309356797280102nteger @ A2 )
% 14.25/8.14 = ( uminus1351360451143612070nteger @ A2 ) ) ).
% 14.25/8.14
% 14.25/8.14 thf(fact_6804_diff__minus__eq__add,axiom,
% 14.25/8.14 ! [A2: rat,B: rat] :
% 14.25/8.14 ( ( minus_minus_rat @ A2 @ ( uminus_uminus_rat @ B ) )
% 14.25/8.14 = ( plus_plus_rat @ A2 @ B ) ) ).
% 14.25/8.14
% 14.25/8.14 thf(fact_6805_diff__minus__eq__add,axiom,
% 14.25/8.14 ! [A2: int,B: int] :
% 14.25/8.14 ( ( minus_minus_int @ A2 @ ( uminus_uminus_int @ B ) )
% 14.25/8.14 = ( plus_plus_int @ A2 @ B ) ) ).
% 14.25/8.14
% 14.25/8.14 thf(fact_6806_diff__minus__eq__add,axiom,
% 14.25/8.14 ! [A2: code_integer,B: code_integer] :
% 14.25/8.14 ( ( minus_8373710615458151222nteger @ A2 @ ( uminus1351360451143612070nteger @ B ) )
% 14.25/8.14 = ( plus_p5714425477246183910nteger @ A2 @ B ) ) ).
% 14.25/8.14
% 14.25/8.14 thf(fact_6807_uminus__add__conv__diff,axiom,
% 14.25/8.14 ! [A2: rat,B: rat] :
% 14.25/8.14 ( ( plus_plus_rat @ ( uminus_uminus_rat @ A2 ) @ B )
% 14.25/8.14 = ( minus_minus_rat @ B @ A2 ) ) ).
% 14.25/8.14
% 14.25/8.14 thf(fact_6808_uminus__add__conv__diff,axiom,
% 14.25/8.14 ! [A2: int,B: int] :
% 14.25/8.14 ( ( plus_plus_int @ ( uminus_uminus_int @ A2 ) @ B )
% 14.25/8.14 = ( minus_minus_int @ B @ A2 ) ) ).
% 14.25/8.14
% 14.25/8.14 thf(fact_6809_uminus__add__conv__diff,axiom,
% 14.25/8.14 ! [A2: code_integer,B: code_integer] :
% 14.25/8.14 ( ( plus_p5714425477246183910nteger @ ( uminus1351360451143612070nteger @ A2 ) @ B )
% 14.25/8.14 = ( minus_8373710615458151222nteger @ B @ A2 ) ) ).
% 14.25/8.14
% 14.25/8.14 thf(fact_6810_mult__minus1,axiom,
% 14.25/8.14 ! [Z: rat] :
% 14.25/8.14 ( ( times_times_rat @ ( uminus_uminus_rat @ one_one_rat ) @ Z )
% 14.25/8.14 = ( uminus_uminus_rat @ Z ) ) ).
% 14.25/8.14
% 14.25/8.14 thf(fact_6811_mult__minus1,axiom,
% 14.25/8.14 ! [Z: int] :
% 14.25/8.14 ( ( times_times_int @ ( uminus_uminus_int @ one_one_int ) @ Z )
% 14.25/8.14 = ( uminus_uminus_int @ Z ) ) ).
% 14.25/8.14
% 14.25/8.14 thf(fact_6812_mult__minus1,axiom,
% 14.25/8.14 ! [Z: code_integer] :
% 14.25/8.14 ( ( times_3573771949741848930nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ Z )
% 14.25/8.14 = ( uminus1351360451143612070nteger @ Z ) ) ).
% 14.25/8.14
% 14.25/8.14 thf(fact_6813_mult__minus1__right,axiom,
% 14.25/8.14 ! [Z: rat] :
% 14.25/8.14 ( ( times_times_rat @ Z @ ( uminus_uminus_rat @ one_one_rat ) )
% 14.25/8.14 = ( uminus_uminus_rat @ Z ) ) ).
% 14.25/8.14
% 14.25/8.14 thf(fact_6814_mult__minus1__right,axiom,
% 14.25/8.14 ! [Z: int] :
% 14.25/8.14 ( ( times_times_int @ Z @ ( uminus_uminus_int @ one_one_int ) )
% 14.25/8.14 = ( uminus_uminus_int @ Z ) ) ).
% 14.25/8.14
% 14.25/8.14 thf(fact_6815_mult__minus1__right,axiom,
% 14.25/8.14 ! [Z: code_integer] :
% 14.25/8.14 ( ( times_3573771949741848930nteger @ Z @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) )
% 14.25/8.14 = ( uminus1351360451143612070nteger @ Z ) ) ).
% 14.25/8.14
% 14.25/8.14 thf(fact_6816_boolean__algebra_Ocompl__one,axiom,
% 14.25/8.14 ( ( uminus5254974436814262692_nat_o @ top_to4869887016220417533_nat_o )
% 14.25/8.14 = bot_bo7824918357723582233_nat_o ) ).
% 14.25/8.14
% 14.25/8.14 thf(fact_6817_boolean__algebra_Ocompl__one,axiom,
% 14.25/8.14 ( ( uminus5944136376168626660t_unit @ top_to1996260823553986621t_unit )
% 14.25/8.14 = bot_bo3957492148770167129t_unit ) ).
% 14.25/8.14
% 14.25/8.14 thf(fact_6818_boolean__algebra_Ocompl__one,axiom,
% 14.25/8.14 ( ( uminus_uminus_set_o @ top_top_set_o )
% 14.25/8.14 = bot_bot_set_o ) ).
% 14.25/8.14
% 14.25/8.14 thf(fact_6819_boolean__algebra_Ocompl__one,axiom,
% 14.25/8.14 ( ( uminus5710092332889474511et_nat @ top_top_set_nat )
% 14.25/8.14 = bot_bot_set_nat ) ).
% 14.25/8.14
% 14.25/8.14 thf(fact_6820_boolean__algebra_Ocompl__one,axiom,
% 14.25/8.14 ( ( uminus1532241313380277803et_int @ top_top_set_int )
% 14.25/8.14 = bot_bot_set_int ) ).
% 14.25/8.14
% 14.25/8.14 thf(fact_6821_boolean__algebra_Ocompl__one,axiom,
% 14.25/8.14 ( ( uminus_uminus_assn @ top_top_assn )
% 14.25/8.14 = bot_bot_assn ) ).
% 14.25/8.14
% 14.25/8.14 thf(fact_6822_boolean__algebra_Ocompl__one,axiom,
% 14.25/8.14 ( ( uminus2952777764628376836t_unit @ top_top_Product_unit )
% 14.25/8.14 = bot_bot_Product_unit ) ).
% 14.25/8.14
% 14.25/8.14 thf(fact_6823_boolean__algebra_Ocompl__zero,axiom,
% 14.25/8.14 ( ( uminus5254974436814262692_nat_o @ bot_bo7824918357723582233_nat_o )
% 14.25/8.14 = top_to4869887016220417533_nat_o ) ).
% 14.25/8.14
% 14.25/8.14 thf(fact_6824_boolean__algebra_Ocompl__zero,axiom,
% 14.25/8.14 ( ( uminus5944136376168626660t_unit @ bot_bo3957492148770167129t_unit )
% 14.25/8.14 = top_to1996260823553986621t_unit ) ).
% 14.25/8.14
% 14.25/8.14 thf(fact_6825_boolean__algebra_Ocompl__zero,axiom,
% 14.25/8.14 ( ( uminus_uminus_set_o @ bot_bot_set_o )
% 14.25/8.14 = top_top_set_o ) ).
% 14.25/8.14
% 14.25/8.14 thf(fact_6826_boolean__algebra_Ocompl__zero,axiom,
% 14.25/8.14 ( ( uminus5710092332889474511et_nat @ bot_bot_set_nat )
% 14.25/8.14 = top_top_set_nat ) ).
% 14.25/8.14
% 14.25/8.14 thf(fact_6827_boolean__algebra_Ocompl__zero,axiom,
% 14.25/8.14 ( ( uminus1532241313380277803et_int @ bot_bot_set_int )
% 14.25/8.14 = top_top_set_int ) ).
% 14.25/8.14
% 14.25/8.14 thf(fact_6828_boolean__algebra_Ocompl__zero,axiom,
% 14.25/8.14 ( ( uminus_uminus_assn @ bot_bot_assn )
% 14.25/8.14 = top_top_assn ) ).
% 14.25/8.14
% 14.25/8.14 thf(fact_6829_boolean__algebra_Ocompl__zero,axiom,
% 14.25/8.14 ( ( uminus2952777764628376836t_unit @ bot_bot_Product_unit )
% 14.25/8.14 = top_top_Product_unit ) ).
% 14.25/8.14
% 14.25/8.14 thf(fact_6830_boolean__algebra_Oconj__cancel__right,axiom,
% 14.25/8.14 ! [X: set_Product_unit] :
% 14.25/8.14 ( ( inf_in4660618365625256667t_unit @ X @ ( uminus5944136376168626660t_unit @ X ) )
% 14.25/8.14 = bot_bo3957492148770167129t_unit ) ).
% 14.25/8.14
% 14.25/8.14 thf(fact_6831_boolean__algebra_Oconj__cancel__right,axiom,
% 14.25/8.14 ! [X: set_o] :
% 14.25/8.14 ( ( inf_inf_set_o @ X @ ( uminus_uminus_set_o @ X ) )
% 14.25/8.14 = bot_bot_set_o ) ).
% 14.25/8.14
% 14.25/8.14 thf(fact_6832_boolean__algebra_Oconj__cancel__right,axiom,
% 14.25/8.14 ! [X: set_nat] :
% 14.25/8.14 ( ( inf_inf_set_nat @ X @ ( uminus5710092332889474511et_nat @ X ) )
% 14.25/8.14 = bot_bot_set_nat ) ).
% 14.25/8.14
% 14.25/8.14 thf(fact_6833_boolean__algebra_Oconj__cancel__right,axiom,
% 14.25/8.14 ! [X: set_int] :
% 14.25/8.14 ( ( inf_inf_set_int @ X @ ( uminus1532241313380277803et_int @ X ) )
% 14.25/8.14 = bot_bot_set_int ) ).
% 14.25/8.14
% 14.25/8.14 thf(fact_6834_boolean__algebra_Oconj__cancel__right,axiom,
% 14.25/8.14 ! [X: set_Pr4532377907799695533_nat_o] :
% 14.25/8.14 ( ( inf_in1906310914598751387_nat_o @ X @ ( uminus5254974436814262692_nat_o @ X ) )
% 14.25/8.14 = bot_bo7824918357723582233_nat_o ) ).
% 14.25/8.14
% 14.25/8.14 thf(fact_6835_boolean__algebra_Oconj__cancel__right,axiom,
% 14.25/8.14 ! [X: assn] :
% 14.25/8.14 ( ( inf_inf_assn @ X @ ( uminus_uminus_assn @ X ) )
% 14.25/8.14 = bot_bot_assn ) ).
% 14.25/8.14
% 14.25/8.14 thf(fact_6836_boolean__algebra_Oconj__cancel__right,axiom,
% 14.25/8.14 ! [X: product_unit] :
% 14.25/8.14 ( ( inf_inf_Product_unit @ X @ ( uminus2952777764628376836t_unit @ X ) )
% 14.25/8.14 = bot_bot_Product_unit ) ).
% 14.25/8.14
% 14.25/8.14 thf(fact_6837_boolean__algebra_Oconj__cancel__left,axiom,
% 14.25/8.14 ! [X: set_Product_unit] :
% 14.25/8.14 ( ( inf_in4660618365625256667t_unit @ ( uminus5944136376168626660t_unit @ X ) @ X )
% 14.25/8.14 = bot_bo3957492148770167129t_unit ) ).
% 14.25/8.14
% 14.25/8.14 thf(fact_6838_boolean__algebra_Oconj__cancel__left,axiom,
% 14.25/8.14 ! [X: set_o] :
% 14.25/8.14 ( ( inf_inf_set_o @ ( uminus_uminus_set_o @ X ) @ X )
% 14.25/8.14 = bot_bot_set_o ) ).
% 14.25/8.14
% 14.25/8.14 thf(fact_6839_boolean__algebra_Oconj__cancel__left,axiom,
% 14.25/8.14 ! [X: set_nat] :
% 14.25/8.14 ( ( inf_inf_set_nat @ ( uminus5710092332889474511et_nat @ X ) @ X )
% 14.25/8.14 = bot_bot_set_nat ) ).
% 14.25/8.14
% 14.25/8.14 thf(fact_6840_boolean__algebra_Oconj__cancel__left,axiom,
% 14.25/8.14 ! [X: set_int] :
% 14.25/8.14 ( ( inf_inf_set_int @ ( uminus1532241313380277803et_int @ X ) @ X )
% 14.25/8.14 = bot_bot_set_int ) ).
% 14.25/8.14
% 14.25/8.14 thf(fact_6841_boolean__algebra_Oconj__cancel__left,axiom,
% 14.25/8.14 ! [X: set_Pr4532377907799695533_nat_o] :
% 14.25/8.14 ( ( inf_in1906310914598751387_nat_o @ ( uminus5254974436814262692_nat_o @ X ) @ X )
% 14.25/8.14 = bot_bo7824918357723582233_nat_o ) ).
% 14.25/8.14
% 14.25/8.14 thf(fact_6842_boolean__algebra_Oconj__cancel__left,axiom,
% 14.25/8.14 ! [X: assn] :
% 14.25/8.14 ( ( inf_inf_assn @ ( uminus_uminus_assn @ X ) @ X )
% 14.25/8.14 = bot_bot_assn ) ).
% 14.25/8.14
% 14.25/8.14 thf(fact_6843_boolean__algebra_Oconj__cancel__left,axiom,
% 14.25/8.14 ! [X: product_unit] :
% 14.25/8.14 ( ( inf_inf_Product_unit @ ( uminus2952777764628376836t_unit @ X ) @ X )
% 14.25/8.14 = bot_bot_Product_unit ) ).
% 14.25/8.14
% 14.25/8.14 thf(fact_6844_inf__compl__bot__right,axiom,
% 14.25/8.14 ! [X: set_Product_unit,Y: set_Product_unit] :
% 14.25/8.14 ( ( inf_in4660618365625256667t_unit @ X @ ( inf_in4660618365625256667t_unit @ Y @ ( uminus5944136376168626660t_unit @ X ) ) )
% 14.25/8.14 = bot_bo3957492148770167129t_unit ) ).
% 14.25/8.14
% 14.25/8.14 thf(fact_6845_inf__compl__bot__right,axiom,
% 14.25/8.14 ! [X: set_o,Y: set_o] :
% 14.25/8.14 ( ( inf_inf_set_o @ X @ ( inf_inf_set_o @ Y @ ( uminus_uminus_set_o @ X ) ) )
% 14.25/8.14 = bot_bot_set_o ) ).
% 14.25/8.14
% 14.25/8.14 thf(fact_6846_inf__compl__bot__right,axiom,
% 14.25/8.14 ! [X: set_nat,Y: set_nat] :
% 14.25/8.14 ( ( inf_inf_set_nat @ X @ ( inf_inf_set_nat @ Y @ ( uminus5710092332889474511et_nat @ X ) ) )
% 14.25/8.14 = bot_bot_set_nat ) ).
% 14.25/8.14
% 14.25/8.14 thf(fact_6847_inf__compl__bot__right,axiom,
% 14.25/8.14 ! [X: set_int,Y: set_int] :
% 14.25/8.14 ( ( inf_inf_set_int @ X @ ( inf_inf_set_int @ Y @ ( uminus1532241313380277803et_int @ X ) ) )
% 14.25/8.14 = bot_bot_set_int ) ).
% 14.25/8.14
% 14.25/8.14 thf(fact_6848_inf__compl__bot__right,axiom,
% 14.25/8.14 ! [X: set_Pr4532377907799695533_nat_o,Y: set_Pr4532377907799695533_nat_o] :
% 14.25/8.14 ( ( inf_in1906310914598751387_nat_o @ X @ ( inf_in1906310914598751387_nat_o @ Y @ ( uminus5254974436814262692_nat_o @ X ) ) )
% 14.25/8.14 = bot_bo7824918357723582233_nat_o ) ).
% 14.25/8.14
% 14.25/8.14 thf(fact_6849_inf__compl__bot__right,axiom,
% 14.25/8.14 ! [X: assn,Y: assn] :
% 14.25/8.14 ( ( inf_inf_assn @ X @ ( inf_inf_assn @ Y @ ( uminus_uminus_assn @ X ) ) )
% 14.25/8.14 = bot_bot_assn ) ).
% 14.25/8.14
% 14.25/8.14 thf(fact_6850_inf__compl__bot__right,axiom,
% 14.25/8.14 ! [X: product_unit,Y: product_unit] :
% 14.25/8.14 ( ( inf_inf_Product_unit @ X @ ( inf_inf_Product_unit @ Y @ ( uminus2952777764628376836t_unit @ X ) ) )
% 14.25/8.14 = bot_bot_Product_unit ) ).
% 14.25/8.14
% 14.25/8.14 thf(fact_6851_inf__compl__bot__left2,axiom,
% 14.25/8.14 ! [X: set_Product_unit,Y: set_Product_unit] :
% 14.25/8.14 ( ( inf_in4660618365625256667t_unit @ X @ ( inf_in4660618365625256667t_unit @ ( uminus5944136376168626660t_unit @ X ) @ Y ) )
% 14.25/8.14 = bot_bo3957492148770167129t_unit ) ).
% 14.25/8.14
% 14.25/8.14 thf(fact_6852_inf__compl__bot__left2,axiom,
% 14.25/8.14 ! [X: set_o,Y: set_o] :
% 14.25/8.14 ( ( inf_inf_set_o @ X @ ( inf_inf_set_o @ ( uminus_uminus_set_o @ X ) @ Y ) )
% 14.25/8.14 = bot_bot_set_o ) ).
% 14.25/8.14
% 14.25/8.14 thf(fact_6853_inf__compl__bot__left2,axiom,
% 14.25/8.14 ! [X: set_nat,Y: set_nat] :
% 14.25/8.14 ( ( inf_inf_set_nat @ X @ ( inf_inf_set_nat @ ( uminus5710092332889474511et_nat @ X ) @ Y ) )
% 14.25/8.14 = bot_bot_set_nat ) ).
% 14.25/8.14
% 14.25/8.14 thf(fact_6854_inf__compl__bot__left2,axiom,
% 14.25/8.14 ! [X: set_int,Y: set_int] :
% 14.25/8.14 ( ( inf_inf_set_int @ X @ ( inf_inf_set_int @ ( uminus1532241313380277803et_int @ X ) @ Y ) )
% 14.25/8.14 = bot_bot_set_int ) ).
% 14.25/8.14
% 14.25/8.14 thf(fact_6855_inf__compl__bot__left2,axiom,
% 14.25/8.14 ! [X: set_Pr4532377907799695533_nat_o,Y: set_Pr4532377907799695533_nat_o] :
% 14.25/8.14 ( ( inf_in1906310914598751387_nat_o @ X @ ( inf_in1906310914598751387_nat_o @ ( uminus5254974436814262692_nat_o @ X ) @ Y ) )
% 14.25/8.14 = bot_bo7824918357723582233_nat_o ) ).
% 14.25/8.14
% 14.25/8.14 thf(fact_6856_inf__compl__bot__left2,axiom,
% 14.25/8.14 ! [X: assn,Y: assn] :
% 14.25/8.14 ( ( inf_inf_assn @ X @ ( inf_inf_assn @ ( uminus_uminus_assn @ X ) @ Y ) )
% 14.25/8.14 = bot_bot_assn ) ).
% 14.25/8.14
% 14.25/8.14 thf(fact_6857_inf__compl__bot__left2,axiom,
% 14.25/8.14 ! [X: product_unit,Y: product_unit] :
% 14.25/8.14 ( ( inf_inf_Product_unit @ X @ ( inf_inf_Product_unit @ ( uminus2952777764628376836t_unit @ X ) @ Y ) )
% 14.25/8.14 = bot_bot_Product_unit ) ).
% 14.25/8.14
% 14.25/8.14 thf(fact_6858_inf__compl__bot__left1,axiom,
% 14.25/8.14 ! [X: set_Product_unit,Y: set_Product_unit] :
% 14.25/8.14 ( ( inf_in4660618365625256667t_unit @ ( uminus5944136376168626660t_unit @ X ) @ ( inf_in4660618365625256667t_unit @ X @ Y ) )
% 14.25/8.14 = bot_bo3957492148770167129t_unit ) ).
% 14.25/8.14
% 14.25/8.14 thf(fact_6859_inf__compl__bot__left1,axiom,
% 14.25/8.14 ! [X: set_o,Y: set_o] :
% 14.25/8.14 ( ( inf_inf_set_o @ ( uminus_uminus_set_o @ X ) @ ( inf_inf_set_o @ X @ Y ) )
% 14.25/8.14 = bot_bot_set_o ) ).
% 14.25/8.14
% 14.25/8.14 thf(fact_6860_inf__compl__bot__left1,axiom,
% 14.25/8.14 ! [X: set_nat,Y: set_nat] :
% 14.25/8.14 ( ( inf_inf_set_nat @ ( uminus5710092332889474511et_nat @ X ) @ ( inf_inf_set_nat @ X @ Y ) )
% 14.25/8.14 = bot_bot_set_nat ) ).
% 14.25/8.14
% 14.25/8.14 thf(fact_6861_inf__compl__bot__left1,axiom,
% 14.25/8.14 ! [X: set_int,Y: set_int] :
% 14.25/8.14 ( ( inf_inf_set_int @ ( uminus1532241313380277803et_int @ X ) @ ( inf_inf_set_int @ X @ Y ) )
% 14.25/8.14 = bot_bot_set_int ) ).
% 14.25/8.14
% 14.25/8.14 thf(fact_6862_inf__compl__bot__left1,axiom,
% 14.25/8.14 ! [X: set_Pr4532377907799695533_nat_o,Y: set_Pr4532377907799695533_nat_o] :
% 14.25/8.14 ( ( inf_in1906310914598751387_nat_o @ ( uminus5254974436814262692_nat_o @ X ) @ ( inf_in1906310914598751387_nat_o @ X @ Y ) )
% 14.25/8.14 = bot_bo7824918357723582233_nat_o ) ).
% 14.25/8.14
% 14.25/8.14 thf(fact_6863_inf__compl__bot__left1,axiom,
% 14.25/8.14 ! [X: assn,Y: assn] :
% 14.25/8.14 ( ( inf_inf_assn @ ( uminus_uminus_assn @ X ) @ ( inf_inf_assn @ X @ Y ) )
% 14.25/8.14 = bot_bot_assn ) ).
% 14.25/8.14
% 14.25/8.14 thf(fact_6864_inf__compl__bot__left1,axiom,
% 14.25/8.14 ! [X: product_unit,Y: product_unit] :
% 14.25/8.14 ( ( inf_inf_Product_unit @ ( uminus2952777764628376836t_unit @ X ) @ ( inf_inf_Product_unit @ X @ Y ) )
% 14.25/8.14 = bot_bot_Product_unit ) ).
% 14.25/8.14
% 14.25/8.14 thf(fact_6865_subset__Compl__singleton,axiom,
% 14.25/8.14 ! [A: set_Product_unit,B: product_unit] :
% 14.25/8.14 ( ( ord_le3507040750410214029t_unit @ A @ ( uminus5944136376168626660t_unit @ ( insert_Product_unit @ B @ bot_bo3957492148770167129t_unit ) ) )
% 14.25/8.14 = ( ~ ( member_Product_unit @ B @ A ) ) ) ).
% 14.25/8.14
% 14.25/8.14 thf(fact_6866_subset__Compl__singleton,axiom,
% 14.25/8.14 ! [A: set_o,B: $o] :
% 14.25/8.14 ( ( ord_less_eq_set_o @ A @ ( uminus_uminus_set_o @ ( insert_o @ B @ bot_bot_set_o ) ) )
% 14.25/8.14 = ( ~ ( member_o @ B @ A ) ) ) ).
% 14.25/8.14
% 14.25/8.14 thf(fact_6867_subset__Compl__singleton,axiom,
% 14.25/8.14 ! [A: set_int,B: int] :
% 14.25/8.14 ( ( ord_less_eq_set_int @ A @ ( uminus1532241313380277803et_int @ ( insert_int @ B @ bot_bot_set_int ) ) )
% 14.25/8.14 = ( ~ ( member_int @ B @ A ) ) ) ).
% 14.25/8.14
% 14.25/8.14 thf(fact_6868_subset__Compl__singleton,axiom,
% 14.25/8.14 ! [A: set_Pr4532377907799695533_nat_o,B: produc3658429121746597890et_nat > $o] :
% 14.25/8.14 ( ( ord_le2965882846123202637_nat_o @ A @ ( uminus5254974436814262692_nat_o @ ( insert5175938949040314269_nat_o @ B @ bot_bo7824918357723582233_nat_o ) ) )
% 14.25/8.14 = ( ~ ( member6576561426505652726_nat_o @ B @ A ) ) ) ).
% 14.25/8.14
% 14.25/8.14 thf(fact_6869_subset__Compl__singleton,axiom,
% 14.25/8.14 ! [A: set_nat,B: nat] :
% 14.25/8.14 ( ( ord_less_eq_set_nat @ A @ ( uminus5710092332889474511et_nat @ ( insert_nat @ B @ bot_bot_set_nat ) ) )
% 14.25/8.14 = ( ~ ( member_nat @ B @ A ) ) ) ).
% 14.25/8.14
% 14.25/8.14 thf(fact_6870_prod_Oempty,axiom,
% 14.25/8.14 ! [G: product_unit > assn] :
% 14.25/8.14 ( ( groups5360233303959380816t_assn @ G @ bot_bo3957492148770167129t_unit )
% 14.25/8.14 = one_one_assn ) ).
% 14.25/8.14
% 14.25/8.14 thf(fact_6871_prod_Oempty,axiom,
% 14.25/8.14 ! [G: product_unit > nat] :
% 14.25/8.14 ( ( groups396421249026171384it_nat @ G @ bot_bo3957492148770167129t_unit )
% 14.25/8.14 = one_one_nat ) ).
% 14.25/8.14
% 14.25/8.14 thf(fact_6872_prod_Oempty,axiom,
% 14.25/8.14 ! [G: product_unit > int] :
% 14.25/8.14 ( ( groups393930778517121108it_int @ G @ bot_bo3957492148770167129t_unit )
% 14.25/8.14 = one_one_int ) ).
% 14.25/8.14
% 14.25/8.14 thf(fact_6873_prod_Oempty,axiom,
% 14.25/8.14 ! [G: product_unit > code_integer] :
% 14.25/8.14 ( ( groups8185359811944078981nteger @ G @ bot_bo3957492148770167129t_unit )
% 14.25/8.14 = one_one_Code_integer ) ).
% 14.25/8.14
% 14.25/8.14 thf(fact_6874_prod_Oempty,axiom,
% 14.25/8.14 ! [G: product_unit > code_natural] :
% 14.25/8.14 ( ( groups7008954963700796496atural @ G @ bot_bo3957492148770167129t_unit )
% 14.25/8.14 = one_one_Code_natural ) ).
% 14.25/8.14
% 14.25/8.14 thf(fact_6875_prod_Oempty,axiom,
% 14.25/8.14 ! [G: $o > assn] :
% 14.25/8.14 ( ( groups5301882518646026715o_assn @ G @ bot_bot_set_o )
% 14.25/8.14 = one_one_assn ) ).
% 14.25/8.14
% 14.25/8.14 thf(fact_6876_prod_Oempty,axiom,
% 14.25/8.14 ! [G: $o > nat] :
% 14.25/8.14 ( ( groups3504817904513533571_o_nat @ G @ bot_bot_set_o )
% 14.25/8.14 = one_one_nat ) ).
% 14.25/8.14
% 14.25/8.14 thf(fact_6877_prod_Oempty,axiom,
% 14.25/8.14 ! [G: $o > int] :
% 14.25/8.14 ( ( groups3502327434004483295_o_int @ G @ bot_bot_set_o )
% 14.25/8.14 = one_one_int ) ).
% 14.25/8.14
% 14.25/8.14 thf(fact_6878_prod_Oempty,axiom,
% 14.25/8.14 ! [G: $o > code_integer] :
% 14.25/8.14 ( ( groups7694694392188491536nteger @ G @ bot_bot_set_o )
% 14.25/8.14 = one_one_Code_integer ) ).
% 14.25/8.14
% 14.25/8.14 thf(fact_6879_prod_Oempty,axiom,
% 14.25/8.14 ! [G: $o > code_natural] :
% 14.25/8.14 ( ( groups6518289543945209051atural @ G @ bot_bot_set_o )
% 14.25/8.14 = one_one_Code_natural ) ).
% 14.25/8.14
% 14.25/8.14 thf(fact_6880_prod_Oinfinite,axiom,
% 14.25/8.14 ! [A: set_nat,G: nat > assn] :
% 14.25/8.14 ( ~ ( finite_finite_nat @ A )
% 14.25/8.14 => ( ( groups6906906614972039071t_assn @ G @ A )
% 14.25/8.14 = one_one_assn ) ) ).
% 14.25/8.14
% 14.25/8.14 thf(fact_6881_prod_Oinfinite,axiom,
% 14.25/8.14 ! [A: set_Code_integer,G: code_integer > assn] :
% 14.25/8.14 ( ~ ( finite6017078050557962740nteger @ A )
% 14.25/8.14 => ( ( groups1304777262505850412r_assn @ G @ A )
% 14.25/8.14 = one_one_assn ) ) ).
% 14.25/8.14
% 14.25/8.14 thf(fact_6882_prod_Oinfinite,axiom,
% 14.25/8.14 ! [A: set_Code_integer,G: code_integer > nat] :
% 14.25/8.14 ( ~ ( finite6017078050557962740nteger @ A )
% 14.25/8.14 => ( ( groups3190895334310489300er_nat @ G @ A )
% 14.25/8.14 = one_one_nat ) ) ).
% 14.25/8.14
% 14.25/8.14 thf(fact_6883_prod_Oinfinite,axiom,
% 14.25/8.14 ! [A: set_nat,G: nat > int] :
% 14.25/8.14 ( ~ ( finite_finite_nat @ A )
% 14.25/8.14 => ( ( groups705719431365010083at_int @ G @ A )
% 14.25/8.14 = one_one_int ) ) ).
% 14.25/8.14
% 14.25/8.14 thf(fact_6884_prod_Oinfinite,axiom,
% 14.25/8.14 ! [A: set_Code_integer,G: code_integer > int] :
% 14.25/8.14 ( ~ ( finite6017078050557962740nteger @ A )
% 14.25/8.14 => ( ( groups3188404863801439024er_int @ G @ A )
% 14.25/8.14 = one_one_int ) ) ).
% 14.25/8.14
% 14.25/8.14 thf(fact_6885_prod_Oinfinite,axiom,
% 14.25/8.14 ! [A: set_nat,G: nat > code_integer] :
% 14.25/8.14 ( ~ ( finite_finite_nat @ A )
% 14.25/8.14 => ( ( groups3455450783089532116nteger @ G @ A )
% 14.25/8.14 = one_one_Code_integer ) ) ).
% 14.25/8.14
% 14.25/8.14 thf(fact_6886_prod_Oinfinite,axiom,
% 14.25/8.14 ! [A: set_Code_integer,G: code_integer > code_integer] :
% 14.25/8.14 ( ~ ( finite6017078050557962740nteger @ A )
% 14.25/8.14 => ( ( groups3674199335183972705nteger @ G @ A )
% 14.25/8.14 = one_one_Code_integer ) ) ).
% 14.25/8.14
% 14.25/8.14 thf(fact_6887_prod_Oinfinite,axiom,
% 14.25/8.14 ! [A: set_nat,G: nat > code_natural] :
% 14.25/8.14 ( ~ ( finite_finite_nat @ A )
% 14.25/8.14 => ( ( groups2279045934846249631atural @ G @ A )
% 14.25/8.14 = one_one_Code_natural ) ) ).
% 14.25/8.14
% 14.25/8.14 thf(fact_6888_prod_Oinfinite,axiom,
% 14.25/8.14 ! [A: set_Code_integer,G: code_integer > code_natural] :
% 14.25/8.14 ( ~ ( finite6017078050557962740nteger @ A )
% 14.25/8.14 => ( ( groups2497794486940690220atural @ G @ A )
% 14.25/8.14 = one_one_Code_natural ) ) ).
% 14.25/8.14
% 14.25/8.14 thf(fact_6889_prod_Oinfinite,axiom,
% 14.25/8.14 ! [A: set_nat,G: nat > nat] :
% 14.25/8.14 ( ~ ( finite_finite_nat @ A )
% 14.25/8.14 => ( ( groups708209901874060359at_nat @ G @ A )
% 14.25/8.14 = one_one_nat ) ) ).
% 14.25/8.14
% 14.25/8.14 thf(fact_6890_of__nat__mult,axiom,
% 14.25/8.14 ! [M4: nat,N2: nat] :
% 14.25/8.14 ( ( semiri681578069525770553at_rat @ ( times_times_nat @ M4 @ N2 ) )
% 14.25/8.14 = ( times_times_rat @ ( semiri681578069525770553at_rat @ M4 ) @ ( semiri681578069525770553at_rat @ N2 ) ) ) ).
% 14.25/8.14
% 14.25/8.14 thf(fact_6891_of__nat__mult,axiom,
% 14.25/8.14 ! [M4: nat,N2: nat] :
% 14.25/8.14 ( ( semiri1314217659103216013at_int @ ( times_times_nat @ M4 @ N2 ) )
% 14.25/8.14 = ( times_times_int @ ( semiri1314217659103216013at_int @ M4 ) @ ( semiri1314217659103216013at_int @ N2 ) ) ) ).
% 14.25/8.14
% 14.25/8.14 thf(fact_6892_of__nat__mult,axiom,
% 14.25/8.14 ! [M4: nat,N2: nat] :
% 14.25/8.14 ( ( semiri1316708129612266289at_nat @ ( times_times_nat @ M4 @ N2 ) )
% 14.25/8.14 = ( times_times_nat @ ( semiri1316708129612266289at_nat @ M4 ) @ ( semiri1316708129612266289at_nat @ N2 ) ) ) ).
% 14.25/8.14
% 14.25/8.14 thf(fact_6893_of__nat__mult,axiom,
% 14.25/8.14 ! [M4: nat,N2: nat] :
% 14.25/8.14 ( ( semiri4939895301339042750nteger @ ( times_times_nat @ M4 @ N2 ) )
% 14.25/8.14 = ( times_3573771949741848930nteger @ ( semiri4939895301339042750nteger @ M4 ) @ ( semiri4939895301339042750nteger @ N2 ) ) ) ).
% 14.25/8.14
% 14.25/8.14 thf(fact_6894_of__nat__1,axiom,
% 14.25/8.14 ( ( semiri3763490453095760265atural @ one_one_nat )
% 14.25/8.14 = one_one_Code_natural ) ).
% 14.25/8.14
% 14.25/8.14 thf(fact_6895_of__nat__1,axiom,
% 14.25/8.14 ( ( semiri1314217659103216013at_int @ one_one_nat )
% 14.25/8.14 = one_one_int ) ).
% 14.25/8.14
% 14.25/8.14 thf(fact_6896_of__nat__1,axiom,
% 14.25/8.14 ( ( semiri1316708129612266289at_nat @ one_one_nat )
% 14.25/8.14 = one_one_nat ) ).
% 14.25/8.14
% 14.25/8.14 thf(fact_6897_of__nat__1,axiom,
% 14.25/8.14 ( ( semiri4939895301339042750nteger @ one_one_nat )
% 14.25/8.14 = one_one_Code_integer ) ).
% 14.25/8.14
% 14.25/8.14 thf(fact_6898_of__nat__1__eq__iff,axiom,
% 14.25/8.14 ! [N2: nat] :
% 14.25/8.14 ( ( one_one_Code_natural
% 14.25/8.14 = ( semiri3763490453095760265atural @ N2 ) )
% 14.25/8.14 = ( N2 = one_one_nat ) ) ).
% 14.25/8.14
% 14.25/8.14 thf(fact_6899_of__nat__1__eq__iff,axiom,
% 14.25/8.14 ! [N2: nat] :
% 14.25/8.14 ( ( one_one_int
% 14.25/8.14 = ( semiri1314217659103216013at_int @ N2 ) )
% 14.25/8.14 = ( N2 = one_one_nat ) ) ).
% 14.25/8.14
% 14.25/8.14 thf(fact_6900_of__nat__1__eq__iff,axiom,
% 14.25/8.14 ! [N2: nat] :
% 14.25/8.14 ( ( one_one_nat
% 14.25/8.14 = ( semiri1316708129612266289at_nat @ N2 ) )
% 14.25/8.14 = ( N2 = one_one_nat ) ) ).
% 14.25/8.14
% 14.25/8.14 thf(fact_6901_of__nat__1__eq__iff,axiom,
% 14.25/8.14 ! [N2: nat] :
% 14.25/8.14 ( ( one_one_Code_integer
% 14.25/8.14 = ( semiri4939895301339042750nteger @ N2 ) )
% 14.25/8.14 = ( N2 = one_one_nat ) ) ).
% 14.25/8.14
% 14.25/8.14 thf(fact_6902_of__nat__eq__1__iff,axiom,
% 14.25/8.14 ! [N2: nat] :
% 14.25/8.14 ( ( ( semiri3763490453095760265atural @ N2 )
% 14.25/8.14 = one_one_Code_natural )
% 14.25/8.14 = ( N2 = one_one_nat ) ) ).
% 14.25/8.14
% 14.25/8.14 thf(fact_6903_of__nat__eq__1__iff,axiom,
% 14.25/8.14 ! [N2: nat] :
% 14.25/8.14 ( ( ( semiri1314217659103216013at_int @ N2 )
% 14.25/8.14 = one_one_int )
% 14.25/8.14 = ( N2 = one_one_nat ) ) ).
% 14.25/8.14
% 14.25/8.14 thf(fact_6904_of__nat__eq__1__iff,axiom,
% 14.25/8.14 ! [N2: nat] :
% 14.25/8.14 ( ( ( semiri1316708129612266289at_nat @ N2 )
% 14.25/8.14 = one_one_nat )
% 14.25/8.14 = ( N2 = one_one_nat ) ) ).
% 14.25/8.14
% 14.25/8.14 thf(fact_6905_of__nat__eq__1__iff,axiom,
% 14.25/8.14 ! [N2: nat] :
% 14.25/8.14 ( ( ( semiri4939895301339042750nteger @ N2 )
% 14.25/8.14 = one_one_Code_integer )
% 14.25/8.14 = ( N2 = one_one_nat ) ) ).
% 14.25/8.14
% 14.25/8.14 thf(fact_6906_Gcd__empty,axiom,
% 14.25/8.14 ( ( gcd_Gcd_nat @ bot_bot_set_nat )
% 14.25/8.14 = zero_zero_nat ) ).
% 14.25/8.14
% 14.25/8.14 thf(fact_6907_Gcd__empty,axiom,
% 14.25/8.14 ( ( gcd_Gcd_int @ bot_bot_set_int )
% 14.25/8.14 = zero_zero_int ) ).
% 14.25/8.14
% 14.25/8.14 thf(fact_6908_Gcd__UNIV,axiom,
% 14.25/8.14 ( ( gcd_Gcd_nat @ top_top_set_nat )
% 14.25/8.14 = one_one_nat ) ).
% 14.25/8.14
% 14.25/8.14 thf(fact_6909_Gcd__UNIV,axiom,
% 14.25/8.14 ( ( gcd_Gcd_int @ top_top_set_int )
% 14.25/8.14 = one_one_int ) ).
% 14.25/8.14
% 14.25/8.14 thf(fact_6910_Gcd__fin_Oempty,axiom,
% 14.25/8.14 ( ( semiri4258706085729940814in_nat @ bot_bot_set_nat )
% 14.25/8.14 = zero_zero_nat ) ).
% 14.25/8.14
% 14.25/8.14 thf(fact_6911_Gcd__fin_Oempty,axiom,
% 14.25/8.14 ( ( semiri4256215615220890538in_int @ bot_bot_set_int )
% 14.25/8.14 = zero_zero_int ) ).
% 14.25/8.14
% 14.25/8.14 thf(fact_6912_Gcd__fin_Oinfinite,axiom,
% 14.25/8.14 ! [A: set_nat] :
% 14.25/8.14 ( ~ ( finite_finite_nat @ A )
% 14.25/8.14 => ( ( semiri4258706085729940814in_nat @ A )
% 14.25/8.14 = one_one_nat ) ) ).
% 14.25/8.14
% 14.25/8.14 thf(fact_6913_Gcd__fin_Oinfinite,axiom,
% 14.25/8.14 ! [A: set_int] :
% 14.25/8.14 ( ~ ( finite_finite_int @ A )
% 14.25/8.14 => ( ( semiri4256215615220890538in_int @ A )
% 14.25/8.14 = one_one_int ) ) ).
% 14.25/8.14
% 14.25/8.14 thf(fact_6914_is__unit__Gcd__fin__iff,axiom,
% 14.25/8.14 ! [A: set_nat] :
% 14.25/8.14 ( ( dvd_dvd_nat @ ( semiri4258706085729940814in_nat @ A ) @ one_one_nat )
% 14.25/8.14 = ( ( semiri4258706085729940814in_nat @ A )
% 14.25/8.14 = one_one_nat ) ) ).
% 14.25/8.14
% 14.25/8.14 thf(fact_6915_is__unit__Gcd__fin__iff,axiom,
% 14.25/8.14 ! [A: set_int] :
% 14.25/8.14 ( ( dvd_dvd_int @ ( semiri4256215615220890538in_int @ A ) @ one_one_int )
% 14.25/8.14 = ( ( semiri4256215615220890538in_int @ A )
% 14.25/8.14 = one_one_int ) ) ).
% 14.25/8.14
% 14.25/8.14 thf(fact_6916_dbl__inc__simps_I4_J,axiom,
% 14.25/8.14 ( ( neg_nu5851722552734809277nc_int @ ( uminus_uminus_int @ one_one_int ) )
% 14.25/8.14 = ( uminus_uminus_int @ one_one_int ) ) ).
% 14.25/8.14
% 14.25/8.14 thf(fact_6917_dbl__inc__simps_I4_J,axiom,
% 14.25/8.14 ( ( neg_nu5831290666863070958nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) )
% 14.25/8.14 = ( uminus1351360451143612070nteger @ one_one_Code_integer ) ) ).
% 14.25/8.14
% 14.25/8.14 thf(fact_6918_add__neg__numeral__special_I7_J,axiom,
% 14.25/8.14 ( ( plus_plus_rat @ one_one_rat @ ( uminus_uminus_rat @ one_one_rat ) )
% 14.25/8.14 = zero_zero_rat ) ).
% 14.25/8.14
% 14.25/8.14 thf(fact_6919_add__neg__numeral__special_I7_J,axiom,
% 14.25/8.14 ( ( plus_plus_int @ one_one_int @ ( uminus_uminus_int @ one_one_int ) )
% 14.25/8.14 = zero_zero_int ) ).
% 14.25/8.14
% 14.25/8.14 thf(fact_6920_add__neg__numeral__special_I7_J,axiom,
% 14.25/8.14 ( ( plus_p5714425477246183910nteger @ one_one_Code_integer @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) )
% 14.25/8.14 = zero_z3403309356797280102nteger ) ).
% 14.25/8.14
% 14.25/8.14 thf(fact_6921_add__neg__numeral__special_I8_J,axiom,
% 14.25/8.14 ( ( plus_plus_rat @ ( uminus_uminus_rat @ one_one_rat ) @ one_one_rat )
% 14.25/8.14 = zero_zero_rat ) ).
% 14.25/8.14
% 14.25/8.14 thf(fact_6922_add__neg__numeral__special_I8_J,axiom,
% 14.25/8.14 ( ( plus_plus_int @ ( uminus_uminus_int @ one_one_int ) @ one_one_int )
% 14.25/8.14 = zero_zero_int ) ).
% 14.25/8.14
% 14.25/8.14 thf(fact_6923_add__neg__numeral__special_I8_J,axiom,
% 14.25/8.14 ( ( plus_p5714425477246183910nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ one_one_Code_integer )
% 14.25/8.14 = zero_z3403309356797280102nteger ) ).
% 14.25/8.14
% 14.25/8.14 thf(fact_6924_diff__numeral__special_I12_J,axiom,
% 14.25/8.14 ( ( minus_minus_rat @ ( uminus_uminus_rat @ one_one_rat ) @ ( uminus_uminus_rat @ one_one_rat ) )
% 14.25/8.14 = zero_zero_rat ) ).
% 14.25/8.14
% 14.25/8.14 thf(fact_6925_diff__numeral__special_I12_J,axiom,
% 14.25/8.14 ( ( minus_minus_int @ ( uminus_uminus_int @ one_one_int ) @ ( uminus_uminus_int @ one_one_int ) )
% 14.25/8.14 = zero_zero_int ) ).
% 14.25/8.14
% 14.25/8.14 thf(fact_6926_diff__numeral__special_I12_J,axiom,
% 14.25/8.14 ( ( minus_8373710615458151222nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) )
% 14.25/8.14 = zero_z3403309356797280102nteger ) ).
% 14.25/8.14
% 14.25/8.14 thf(fact_6927_minus__one__mult__self,axiom,
% 14.25/8.14 ! [N2: nat] :
% 14.25/8.14 ( ( times_times_rat @ ( power_power_rat @ ( uminus_uminus_rat @ one_one_rat ) @ N2 ) @ ( power_power_rat @ ( uminus_uminus_rat @ one_one_rat ) @ N2 ) )
% 14.25/8.14 = one_one_rat ) ).
% 14.25/8.14
% 14.25/8.14 thf(fact_6928_minus__one__mult__self,axiom,
% 14.25/8.14 ! [N2: nat] :
% 14.25/8.14 ( ( times_times_int @ ( power_power_int @ ( uminus_uminus_int @ one_one_int ) @ N2 ) @ ( power_power_int @ ( uminus_uminus_int @ one_one_int ) @ N2 ) )
% 14.25/8.14 = one_one_int ) ).
% 14.25/8.14
% 14.25/8.14 thf(fact_6929_minus__one__mult__self,axiom,
% 14.25/8.14 ! [N2: nat] :
% 14.25/8.14 ( ( times_3573771949741848930nteger @ ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ N2 ) @ ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ N2 ) )
% 14.25/8.14 = one_one_Code_integer ) ).
% 14.25/8.14
% 14.25/8.14 thf(fact_6930_left__minus__one__mult__self,axiom,
% 14.25/8.14 ! [N2: nat,A2: rat] :
% 14.25/8.14 ( ( times_times_rat @ ( power_power_rat @ ( uminus_uminus_rat @ one_one_rat ) @ N2 ) @ ( times_times_rat @ ( power_power_rat @ ( uminus_uminus_rat @ one_one_rat ) @ N2 ) @ A2 ) )
% 14.25/8.14 = A2 ) ).
% 14.25/8.14
% 14.25/8.14 thf(fact_6931_left__minus__one__mult__self,axiom,
% 14.25/8.14 ! [N2: nat,A2: int] :
% 14.25/8.14 ( ( times_times_int @ ( power_power_int @ ( uminus_uminus_int @ one_one_int ) @ N2 ) @ ( times_times_int @ ( power_power_int @ ( uminus_uminus_int @ one_one_int ) @ N2 ) @ A2 ) )
% 14.25/8.14 = A2 ) ).
% 14.25/8.14
% 14.25/8.14 thf(fact_6932_left__minus__one__mult__self,axiom,
% 14.25/8.14 ! [N2: nat,A2: code_integer] :
% 14.25/8.14 ( ( times_3573771949741848930nteger @ ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ N2 ) @ ( times_3573771949741848930nteger @ ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ N2 ) @ A2 ) )
% 14.25/8.14 = A2 ) ).
% 14.25/8.14
% 14.25/8.14 thf(fact_6933_of__nat__Suc,axiom,
% 14.25/8.14 ! [M4: nat] :
% 14.25/8.14 ( ( semiri3763490453095760265atural @ ( suc @ M4 ) )
% 14.25/8.14 = ( plus_p4538020629002901425atural @ one_one_Code_natural @ ( semiri3763490453095760265atural @ M4 ) ) ) ).
% 14.25/8.14
% 14.25/8.14 thf(fact_6934_of__nat__Suc,axiom,
% 14.25/8.14 ! [M4: nat] :
% 14.25/8.14 ( ( semiri681578069525770553at_rat @ ( suc @ M4 ) )
% 14.25/8.14 = ( plus_plus_rat @ one_one_rat @ ( semiri681578069525770553at_rat @ M4 ) ) ) ).
% 14.25/8.14
% 14.25/8.14 thf(fact_6935_of__nat__Suc,axiom,
% 14.25/8.14 ! [M4: nat] :
% 14.25/8.14 ( ( semiri1314217659103216013at_int @ ( suc @ M4 ) )
% 14.25/8.14 = ( plus_plus_int @ one_one_int @ ( semiri1314217659103216013at_int @ M4 ) ) ) ).
% 14.25/8.14
% 14.25/8.14 thf(fact_6936_of__nat__Suc,axiom,
% 14.25/8.14 ! [M4: nat] :
% 14.25/8.14 ( ( semiri1316708129612266289at_nat @ ( suc @ M4 ) )
% 14.25/8.14 = ( plus_plus_nat @ one_one_nat @ ( semiri1316708129612266289at_nat @ M4 ) ) ) ).
% 14.25/8.14
% 14.25/8.14 thf(fact_6937_of__nat__Suc,axiom,
% 14.25/8.14 ! [M4: nat] :
% 14.25/8.14 ( ( semiri4939895301339042750nteger @ ( suc @ M4 ) )
% 14.25/8.14 = ( plus_p5714425477246183910nteger @ one_one_Code_integer @ ( semiri4939895301339042750nteger @ M4 ) ) ) ).
% 14.25/8.14
% 14.25/8.14 thf(fact_6938_prod_Oinsert,axiom,
% 14.25/8.14 ! [A: set_int,X: int,G: int > assn] :
% 14.25/8.14 ( ( finite_finite_int @ A )
% 14.25/8.14 => ( ~ ( member_int @ X @ A )
% 14.25/8.14 => ( ( groups7882442080178216443t_assn @ G @ ( insert_int @ X @ A ) )
% 14.25/8.14 = ( times_times_assn @ ( G @ X ) @ ( groups7882442080178216443t_assn @ G @ A ) ) ) ) ) ).
% 14.25/8.14
% 14.25/8.14 thf(fact_6939_prod_Oinsert,axiom,
% 14.25/8.14 ! [A: set_o,X: $o,G: $o > assn] :
% 14.25/8.14 ( ( finite_finite_o @ A )
% 14.25/8.14 => ( ~ ( member_o @ X @ A )
% 14.25/8.14 => ( ( groups5301882518646026715o_assn @ G @ ( insert_o @ X @ A ) )
% 14.25/8.14 = ( times_times_assn @ ( G @ X ) @ ( groups5301882518646026715o_assn @ G @ A ) ) ) ) ) ).
% 14.25/8.14
% 14.25/8.14 thf(fact_6940_prod_Oinsert,axiom,
% 14.25/8.14 ! [A: set_Product_unit,X: product_unit,G: product_unit > assn] :
% 14.25/8.14 ( ( finite4290736615968046902t_unit @ A )
% 14.25/8.14 => ( ~ ( member_Product_unit @ X @ A )
% 14.25/8.14 => ( ( groups5360233303959380816t_assn @ G @ ( insert_Product_unit @ X @ A ) )
% 14.25/8.14 = ( times_times_assn @ ( G @ X ) @ ( groups5360233303959380816t_assn @ G @ A ) ) ) ) ) ).
% 14.25/8.14
% 14.25/8.14 thf(fact_6941_prod_Oinsert,axiom,
% 14.25/8.14 ! [A: set_nat,X: nat,G: nat > assn] :
% 14.25/8.14 ( ( finite_finite_nat @ A )
% 14.25/8.14 => ( ~ ( member_nat @ X @ A )
% 14.25/8.14 => ( ( groups6906906614972039071t_assn @ G @ ( insert_nat @ X @ A ) )
% 14.25/8.14 = ( times_times_assn @ ( G @ X ) @ ( groups6906906614972039071t_assn @ G @ A ) ) ) ) ) ).
% 14.25/8.14
% 14.25/8.14 thf(fact_6942_prod_Oinsert,axiom,
% 14.25/8.14 ! [A: set_Code_integer,X: code_integer,G: code_integer > assn] :
% 14.25/8.14 ( ( finite6017078050557962740nteger @ A )
% 14.25/8.14 => ( ~ ( member_Code_integer @ X @ A )
% 14.25/8.14 => ( ( groups1304777262505850412r_assn @ G @ ( insert_Code_integer @ X @ A ) )
% 14.25/8.14 = ( times_times_assn @ ( G @ X ) @ ( groups1304777262505850412r_assn @ G @ A ) ) ) ) ) ).
% 14.25/8.14
% 14.25/8.14 thf(fact_6943_prod_Oinsert,axiom,
% 14.25/8.14 ! [A: set_int,X: int,G: int > rat] :
% 14.25/8.14 ( ( finite_finite_int @ A )
% 14.25/8.14 => ( ~ ( member_int @ X @ A )
% 14.25/8.14 => ( ( groups1072433553688619179nt_rat @ G @ ( insert_int @ X @ A ) )
% 14.25/8.14 = ( times_times_rat @ ( G @ X ) @ ( groups1072433553688619179nt_rat @ G @ A ) ) ) ) ) ).
% 14.25/8.14
% 14.25/8.14 thf(fact_6944_prod_Oinsert,axiom,
% 14.25/8.14 ! [A: set_o,X: $o,G: $o > rat] :
% 14.25/8.14 ( ( finite_finite_o @ A )
% 14.25/8.14 => ( ~ ( member_o @ X @ A )
% 14.25/8.14 => ( ( groups2869687844427037835_o_rat @ G @ ( insert_o @ X @ A ) )
% 14.25/8.14 = ( times_times_rat @ ( G @ X ) @ ( groups2869687844427037835_o_rat @ G @ A ) ) ) ) ) ).
% 14.25/8.14
% 14.25/8.14 thf(fact_6945_prod_Oinsert,axiom,
% 14.25/8.14 ! [A: set_Product_unit,X: product_unit,G: product_unit > rat] :
% 14.25/8.14 ( ( finite4290736615968046902t_unit @ A )
% 14.25/8.14 => ( ~ ( member_Product_unit @ X @ A )
% 14.25/8.14 => ( ( groups8984663225794451456it_rat @ G @ ( insert_Product_unit @ X @ A ) )
% 14.25/8.14 = ( times_times_rat @ ( G @ X ) @ ( groups8984663225794451456it_rat @ G @ A ) ) ) ) ) ).
% 14.25/8.14
% 14.25/8.14 thf(fact_6946_prod_Oinsert,axiom,
% 14.25/8.14 ! [A: set_nat,X: nat,G: nat > rat] :
% 14.25/8.14 ( ( finite_finite_nat @ A )
% 14.25/8.14 => ( ~ ( member_nat @ X @ A )
% 14.25/8.14 => ( ( groups73079841787564623at_rat @ G @ ( insert_nat @ X @ A ) )
% 14.25/8.14 = ( times_times_rat @ ( G @ X ) @ ( groups73079841787564623at_rat @ G @ A ) ) ) ) ) ).
% 14.25/8.14
% 14.25/8.14 thf(fact_6947_prod_Oinsert,axiom,
% 14.25/8.14 ! [A: set_Code_integer,X: code_integer,G: code_integer > rat] :
% 14.25/8.14 ( ( finite6017078050557962740nteger @ A )
% 14.25/8.14 => ( ~ ( member_Code_integer @ X @ A )
% 14.25/8.14 => ( ( groups2555765274223993564er_rat @ G @ ( insert_Code_integer @ X @ A ) )
% 14.25/8.14 = ( times_times_rat @ ( G @ X ) @ ( groups2555765274223993564er_rat @ G @ A ) ) ) ) ) ).
% 14.25/8.14
% 14.25/8.14 thf(fact_6948_dbl__dec__simps_I2_J,axiom,
% 14.25/8.14 ( ( neg_nu3811975205180677377ec_int @ zero_zero_int )
% 14.25/8.14 = ( uminus_uminus_int @ one_one_int ) ) ).
% 14.25/8.14
% 14.25/8.14 thf(fact_6949_dbl__dec__simps_I2_J,axiom,
% 14.25/8.14 ( ( neg_nu7757733837767384882nteger @ zero_z3403309356797280102nteger )
% 14.25/8.14 = ( uminus1351360451143612070nteger @ one_one_Code_integer ) ) ).
% 14.25/8.14
% 14.25/8.14 thf(fact_6950_minus__equation__iff,axiom,
% 14.25/8.14 ! [A2: int,B: int] :
% 14.25/8.14 ( ( ( uminus_uminus_int @ A2 )
% 14.25/8.14 = B )
% 14.25/8.14 = ( ( uminus_uminus_int @ B )
% 14.25/8.14 = A2 ) ) ).
% 14.25/8.14
% 14.25/8.14 thf(fact_6951_minus__equation__iff,axiom,
% 14.25/8.14 ! [A2: code_integer,B: code_integer] :
% 14.25/8.14 ( ( ( uminus1351360451143612070nteger @ A2 )
% 14.25/8.14 = B )
% 14.25/8.14 = ( ( uminus1351360451143612070nteger @ B )
% 14.25/8.14 = A2 ) ) ).
% 14.25/8.14
% 14.25/8.14 thf(fact_6952_equation__minus__iff,axiom,
% 14.25/8.14 ! [A2: int,B: int] :
% 14.25/8.14 ( ( A2
% 14.25/8.14 = ( uminus_uminus_int @ B ) )
% 14.25/8.14 = ( B
% 14.25/8.14 = ( uminus_uminus_int @ A2 ) ) ) ).
% 14.25/8.14
% 14.25/8.14 thf(fact_6953_equation__minus__iff,axiom,
% 14.25/8.14 ! [A2: code_integer,B: code_integer] :
% 14.25/8.14 ( ( A2
% 14.25/8.14 = ( uminus1351360451143612070nteger @ B ) )
% 14.25/8.14 = ( B
% 14.25/8.14 = ( uminus1351360451143612070nteger @ A2 ) ) ) ).
% 14.25/8.14
% 14.25/8.14 thf(fact_6954_prod_Oneutral,axiom,
% 14.25/8.14 ! [A: set_nat,G: nat > nat] :
% 14.25/8.14 ( ! [X3: nat] :
% 14.25/8.14 ( ( member_nat @ X3 @ A )
% 14.25/8.14 => ( ( G @ X3 )
% 14.25/8.14 = one_one_nat ) )
% 14.25/8.14 => ( ( groups708209901874060359at_nat @ G @ A )
% 14.25/8.14 = one_one_nat ) ) ).
% 14.25/8.14
% 14.25/8.14 thf(fact_6955_prod_Onot__neutral__contains__not__neutral,axiom,
% 14.25/8.14 ! [G: nat > assn,A: set_nat] :
% 14.25/8.14 ( ( ( groups6906906614972039071t_assn @ G @ A )
% 14.25/8.14 != one_one_assn )
% 14.25/8.14 => ~ ! [A4: nat] :
% 14.25/8.14 ( ( member_nat @ A4 @ A )
% 14.25/8.14 => ( ( G @ A4 )
% 14.25/8.14 = one_one_assn ) ) ) ).
% 14.25/8.14
% 14.25/8.14 thf(fact_6956_prod_Onot__neutral__contains__not__neutral,axiom,
% 14.25/8.14 ! [G: int > assn,A: set_int] :
% 14.25/8.14 ( ( ( groups7882442080178216443t_assn @ G @ A )
% 14.25/8.14 != one_one_assn )
% 14.25/8.14 => ~ ! [A4: int] :
% 14.25/8.14 ( ( member_int @ A4 @ A )
% 14.25/8.14 => ( ( G @ A4 )
% 14.25/8.14 = one_one_assn ) ) ) ).
% 14.25/8.14
% 14.25/8.14 thf(fact_6957_prod_Onot__neutral__contains__not__neutral,axiom,
% 14.25/8.14 ! [G: $o > assn,A: set_o] :
% 14.25/8.14 ( ( ( groups5301882518646026715o_assn @ G @ A )
% 14.25/8.14 != one_one_assn )
% 14.25/8.14 => ~ ! [A4: $o] :
% 14.25/8.14 ( ( member_o @ A4 @ A )
% 14.25/8.14 => ( ( G @ A4 )
% 14.25/8.14 = one_one_assn ) ) ) ).
% 14.25/8.14
% 14.25/8.14 thf(fact_6958_prod_Onot__neutral__contains__not__neutral,axiom,
% 14.25/8.14 ! [G: product_unit > assn,A: set_Product_unit] :
% 14.25/8.14 ( ( ( groups5360233303959380816t_assn @ G @ A )
% 14.25/8.14 != one_one_assn )
% 14.25/8.14 => ~ ! [A4: product_unit] :
% 14.25/8.14 ( ( member_Product_unit @ A4 @ A )
% 14.25/8.14 => ( ( G @ A4 )
% 14.25/8.14 = one_one_assn ) ) ) ).
% 14.25/8.14
% 14.25/8.14 thf(fact_6959_prod_Onot__neutral__contains__not__neutral,axiom,
% 14.25/8.14 ! [G: int > nat,A: set_int] :
% 14.25/8.14 ( ( ( groups1707563613775114915nt_nat @ G @ A )
% 14.25/8.14 != one_one_nat )
% 14.25/8.14 => ~ ! [A4: int] :
% 14.25/8.14 ( ( member_int @ A4 @ A )
% 14.25/8.14 => ( ( G @ A4 )
% 14.25/8.14 = one_one_nat ) ) ) ).
% 14.25/8.14
% 14.25/8.14 thf(fact_6960_prod_Onot__neutral__contains__not__neutral,axiom,
% 14.25/8.14 ! [G: $o > nat,A: set_o] :
% 14.25/8.14 ( ( ( groups3504817904513533571_o_nat @ G @ A )
% 14.25/8.14 != one_one_nat )
% 14.25/8.14 => ~ ! [A4: $o] :
% 14.25/8.14 ( ( member_o @ A4 @ A )
% 14.25/8.14 => ( ( G @ A4 )
% 14.25/8.14 = one_one_nat ) ) ) ).
% 14.25/8.14
% 14.25/8.14 thf(fact_6961_prod_Onot__neutral__contains__not__neutral,axiom,
% 14.25/8.14 ! [G: product_unit > nat,A: set_Product_unit] :
% 14.25/8.14 ( ( ( groups396421249026171384it_nat @ G @ A )
% 14.25/8.14 != one_one_nat )
% 14.25/8.14 => ~ ! [A4: product_unit] :
% 14.25/8.14 ( ( member_Product_unit @ A4 @ A )
% 14.25/8.14 => ( ( G @ A4 )
% 14.25/8.14 = one_one_nat ) ) ) ).
% 14.25/8.14
% 14.25/8.14 thf(fact_6962_prod_Onot__neutral__contains__not__neutral,axiom,
% 14.25/8.14 ! [G: nat > int,A: set_nat] :
% 14.25/8.14 ( ( ( groups705719431365010083at_int @ G @ A )
% 14.25/8.14 != one_one_int )
% 14.25/8.14 => ~ ! [A4: nat] :
% 14.25/8.14 ( ( member_nat @ A4 @ A )
% 14.25/8.14 => ( ( G @ A4 )
% 14.25/8.14 = one_one_int ) ) ) ).
% 14.25/8.14
% 14.25/8.14 thf(fact_6963_prod_Onot__neutral__contains__not__neutral,axiom,
% 14.25/8.14 ! [G: int > int,A: set_int] :
% 14.25/8.14 ( ( ( groups1705073143266064639nt_int @ G @ A )
% 14.25/8.14 != one_one_int )
% 14.25/8.14 => ~ ! [A4: int] :
% 14.25/8.14 ( ( member_int @ A4 @ A )
% 14.25/8.14 => ( ( G @ A4 )
% 14.25/8.14 = one_one_int ) ) ) ).
% 14.25/8.14
% 14.25/8.14 thf(fact_6964_prod_Onot__neutral__contains__not__neutral,axiom,
% 14.25/8.14 ! [G: $o > int,A: set_o] :
% 14.25/8.14 ( ( ( groups3502327434004483295_o_int @ G @ A )
% 14.25/8.14 != one_one_int )
% 14.25/8.14 => ~ ! [A4: $o] :
% 14.25/8.14 ( ( member_o @ A4 @ A )
% 14.25/8.14 => ( ( G @ A4 )
% 14.25/8.14 = one_one_int ) ) ) ).
% 14.25/8.14
% 14.25/8.14 thf(fact_6965_mult__of__nat__commute,axiom,
% 14.25/8.14 ! [X: nat,Y: rat] :
% 14.25/8.14 ( ( times_times_rat @ ( semiri681578069525770553at_rat @ X ) @ Y )
% 14.25/8.14 = ( times_times_rat @ Y @ ( semiri681578069525770553at_rat @ X ) ) ) ).
% 14.25/8.14
% 14.25/8.14 thf(fact_6966_mult__of__nat__commute,axiom,
% 14.25/8.14 ! [X: nat,Y: int] :
% 14.25/8.14 ( ( times_times_int @ ( semiri1314217659103216013at_int @ X ) @ Y )
% 14.25/8.14 = ( times_times_int @ Y @ ( semiri1314217659103216013at_int @ X ) ) ) ).
% 14.25/8.14
% 14.25/8.14 thf(fact_6967_mult__of__nat__commute,axiom,
% 14.25/8.14 ! [X: nat,Y: nat] :
% 14.25/8.14 ( ( times_times_nat @ ( semiri1316708129612266289at_nat @ X ) @ Y )
% 14.25/8.14 = ( times_times_nat @ Y @ ( semiri1316708129612266289at_nat @ X ) ) ) ).
% 14.25/8.14
% 14.25/8.14 thf(fact_6968_mult__of__nat__commute,axiom,
% 14.25/8.14 ! [X: nat,Y: code_integer] :
% 14.25/8.14 ( ( times_3573771949741848930nteger @ ( semiri4939895301339042750nteger @ X ) @ Y )
% 14.25/8.14 = ( times_3573771949741848930nteger @ Y @ ( semiri4939895301339042750nteger @ X ) ) ) ).
% 14.25/8.14
% 14.25/8.14 thf(fact_6969_le__imp__neg__le,axiom,
% 14.25/8.14 ! [A2: code_integer,B: code_integer] :
% 14.25/8.14 ( ( ord_le3102999989581377725nteger @ A2 @ B )
% 14.25/8.14 => ( ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ B ) @ ( uminus1351360451143612070nteger @ A2 ) ) ) ).
% 14.25/8.14
% 14.25/8.14 thf(fact_6970_le__imp__neg__le,axiom,
% 14.25/8.14 ! [A2: rat,B: rat] :
% 14.25/8.14 ( ( ord_less_eq_rat @ A2 @ B )
% 14.25/8.14 => ( ord_less_eq_rat @ ( uminus_uminus_rat @ B ) @ ( uminus_uminus_rat @ A2 ) ) ) ).
% 14.25/8.14
% 14.25/8.14 thf(fact_6971_le__imp__neg__le,axiom,
% 14.25/8.14 ! [A2: int,B: int] :
% 14.25/8.14 ( ( ord_less_eq_int @ A2 @ B )
% 14.25/8.14 => ( ord_less_eq_int @ ( uminus_uminus_int @ B ) @ ( uminus_uminus_int @ A2 ) ) ) ).
% 14.25/8.14
% 14.25/8.14 thf(fact_6972_minus__le__iff,axiom,
% 14.25/8.14 ! [A2: code_integer,B: code_integer] :
% 14.25/8.14 ( ( ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ A2 ) @ B )
% 14.25/8.14 = ( ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ B ) @ A2 ) ) ).
% 14.25/8.14
% 14.25/8.14 thf(fact_6973_minus__le__iff,axiom,
% 14.25/8.14 ! [A2: rat,B: rat] :
% 14.25/8.14 ( ( ord_less_eq_rat @ ( uminus_uminus_rat @ A2 ) @ B )
% 14.25/8.14 = ( ord_less_eq_rat @ ( uminus_uminus_rat @ B ) @ A2 ) ) ).
% 14.25/8.14
% 14.25/8.14 thf(fact_6974_minus__le__iff,axiom,
% 14.25/8.14 ! [A2: int,B: int] :
% 14.25/8.14 ( ( ord_less_eq_int @ ( uminus_uminus_int @ A2 ) @ B )
% 14.25/8.14 = ( ord_less_eq_int @ ( uminus_uminus_int @ B ) @ A2 ) ) ).
% 14.25/8.14
% 14.25/8.14 thf(fact_6975_le__minus__iff,axiom,
% 14.25/8.14 ! [A2: code_integer,B: code_integer] :
% 14.25/8.14 ( ( ord_le3102999989581377725nteger @ A2 @ ( uminus1351360451143612070nteger @ B ) )
% 14.25/8.14 = ( ord_le3102999989581377725nteger @ B @ ( uminus1351360451143612070nteger @ A2 ) ) ) ).
% 14.25/8.14
% 14.25/8.14 thf(fact_6976_le__minus__iff,axiom,
% 14.25/8.14 ! [A2: rat,B: rat] :
% 14.25/8.14 ( ( ord_less_eq_rat @ A2 @ ( uminus_uminus_rat @ B ) )
% 14.25/8.14 = ( ord_less_eq_rat @ B @ ( uminus_uminus_rat @ A2 ) ) ) ).
% 14.25/8.14
% 14.25/8.14 thf(fact_6977_le__minus__iff,axiom,
% 14.25/8.14 ! [A2: int,B: int] :
% 14.25/8.14 ( ( ord_less_eq_int @ A2 @ ( uminus_uminus_int @ B ) )
% 14.25/8.14 = ( ord_less_eq_int @ B @ ( uminus_uminus_int @ A2 ) ) ) ).
% 14.25/8.14
% 14.25/8.14 thf(fact_6978_minus__less__iff,axiom,
% 14.25/8.14 ! [A2: rat,B: rat] :
% 14.25/8.14 ( ( ord_less_rat @ ( uminus_uminus_rat @ A2 ) @ B )
% 14.25/8.14 = ( ord_less_rat @ ( uminus_uminus_rat @ B ) @ A2 ) ) ).
% 14.25/8.14
% 14.25/8.14 thf(fact_6979_minus__less__iff,axiom,
% 14.25/8.14 ! [A2: int,B: int] :
% 14.25/8.14 ( ( ord_less_int @ ( uminus_uminus_int @ A2 ) @ B )
% 14.25/8.14 = ( ord_less_int @ ( uminus_uminus_int @ B ) @ A2 ) ) ).
% 14.25/8.14
% 14.25/8.14 thf(fact_6980_minus__less__iff,axiom,
% 14.25/8.14 ! [A2: code_integer,B: code_integer] :
% 14.25/8.14 ( ( ord_le6747313008572928689nteger @ ( uminus1351360451143612070nteger @ A2 ) @ B )
% 14.25/8.14 = ( ord_le6747313008572928689nteger @ ( uminus1351360451143612070nteger @ B ) @ A2 ) ) ).
% 14.25/8.14
% 14.25/8.14 thf(fact_6981_less__minus__iff,axiom,
% 14.25/8.14 ! [A2: rat,B: rat] :
% 14.25/8.14 ( ( ord_less_rat @ A2 @ ( uminus_uminus_rat @ B ) )
% 14.25/8.14 = ( ord_less_rat @ B @ ( uminus_uminus_rat @ A2 ) ) ) ).
% 14.25/8.14
% 14.25/8.14 thf(fact_6982_less__minus__iff,axiom,
% 14.25/8.14 ! [A2: int,B: int] :
% 14.25/8.14 ( ( ord_less_int @ A2 @ ( uminus_uminus_int @ B ) )
% 14.25/8.14 = ( ord_less_int @ B @ ( uminus_uminus_int @ A2 ) ) ) ).
% 14.25/8.14
% 14.25/8.14 thf(fact_6983_less__minus__iff,axiom,
% 14.25/8.14 ! [A2: code_integer,B: code_integer] :
% 14.25/8.14 ( ( ord_le6747313008572928689nteger @ A2 @ ( uminus1351360451143612070nteger @ B ) )
% 14.25/8.14 = ( ord_le6747313008572928689nteger @ B @ ( uminus1351360451143612070nteger @ A2 ) ) ) ).
% 14.25/8.14
% 14.25/8.14 thf(fact_6984_add_Oinverse__distrib__swap,axiom,
% 14.25/8.14 ! [A2: rat,B: rat] :
% 14.25/8.14 ( ( uminus_uminus_rat @ ( plus_plus_rat @ A2 @ B ) )
% 14.25/8.14 = ( plus_plus_rat @ ( uminus_uminus_rat @ B ) @ ( uminus_uminus_rat @ A2 ) ) ) ).
% 14.25/8.14
% 14.25/8.14 thf(fact_6985_add_Oinverse__distrib__swap,axiom,
% 14.25/8.14 ! [A2: int,B: int] :
% 14.25/8.14 ( ( uminus_uminus_int @ ( plus_plus_int @ A2 @ B ) )
% 14.25/8.14 = ( plus_plus_int @ ( uminus_uminus_int @ B ) @ ( uminus_uminus_int @ A2 ) ) ) ).
% 14.25/8.14
% 14.25/8.14 thf(fact_6986_add_Oinverse__distrib__swap,axiom,
% 14.25/8.14 ! [A2: code_integer,B: code_integer] :
% 14.25/8.14 ( ( uminus1351360451143612070nteger @ ( plus_p5714425477246183910nteger @ A2 @ B ) )
% 14.25/8.14 = ( plus_p5714425477246183910nteger @ ( uminus1351360451143612070nteger @ B ) @ ( uminus1351360451143612070nteger @ A2 ) ) ) ).
% 14.25/8.14
% 14.25/8.14 thf(fact_6987_group__cancel_Oneg1,axiom,
% 14.25/8.14 ! [A: rat,K3: rat,A2: rat] :
% 14.25/8.14 ( ( A
% 14.25/8.14 = ( plus_plus_rat @ K3 @ A2 ) )
% 14.25/8.14 => ( ( uminus_uminus_rat @ A )
% 14.25/8.14 = ( plus_plus_rat @ ( uminus_uminus_rat @ K3 ) @ ( uminus_uminus_rat @ A2 ) ) ) ) ).
% 14.25/8.14
% 14.25/8.14 thf(fact_6988_group__cancel_Oneg1,axiom,
% 14.25/8.14 ! [A: int,K3: int,A2: int] :
% 14.25/8.14 ( ( A
% 14.25/8.14 = ( plus_plus_int @ K3 @ A2 ) )
% 14.25/8.14 => ( ( uminus_uminus_int @ A )
% 14.25/8.14 = ( plus_plus_int @ ( uminus_uminus_int @ K3 ) @ ( uminus_uminus_int @ A2 ) ) ) ) ).
% 14.25/8.14
% 14.25/8.14 thf(fact_6989_group__cancel_Oneg1,axiom,
% 14.25/8.14 ! [A: code_integer,K3: code_integer,A2: code_integer] :
% 14.25/8.14 ( ( A
% 14.25/8.14 = ( plus_p5714425477246183910nteger @ K3 @ A2 ) )
% 14.25/8.14 => ( ( uminus1351360451143612070nteger @ A )
% 14.25/8.14 = ( plus_p5714425477246183910nteger @ ( uminus1351360451143612070nteger @ K3 ) @ ( uminus1351360451143612070nteger @ A2 ) ) ) ) ).
% 14.25/8.14
% 14.25/8.14 thf(fact_6990_square__eq__iff,axiom,
% 14.25/8.14 ! [A2: rat,B: rat] :
% 14.25/8.14 ( ( ( times_times_rat @ A2 @ A2 )
% 14.25/8.14 = ( times_times_rat @ B @ B ) )
% 14.25/8.14 = ( ( A2 = B )
% 14.25/8.14 | ( A2
% 14.25/8.14 = ( uminus_uminus_rat @ B ) ) ) ) ).
% 14.25/8.14
% 14.25/8.14 thf(fact_6991_square__eq__iff,axiom,
% 14.25/8.14 ! [A2: int,B: int] :
% 14.25/8.14 ( ( ( times_times_int @ A2 @ A2 )
% 14.25/8.14 = ( times_times_int @ B @ B ) )
% 14.25/8.14 = ( ( A2 = B )
% 14.25/8.14 | ( A2
% 14.25/8.14 = ( uminus_uminus_int @ B ) ) ) ) ).
% 14.25/8.14
% 14.25/8.14 thf(fact_6992_square__eq__iff,axiom,
% 14.25/8.14 ! [A2: code_integer,B: code_integer] :
% 14.25/8.14 ( ( ( times_3573771949741848930nteger @ A2 @ A2 )
% 14.25/8.14 = ( times_3573771949741848930nteger @ B @ B ) )
% 14.25/8.14 = ( ( A2 = B )
% 14.25/8.14 | ( A2
% 14.25/8.14 = ( uminus1351360451143612070nteger @ B ) ) ) ) ).
% 14.25/8.14
% 14.25/8.14 thf(fact_6993_minus__mult__commute,axiom,
% 14.25/8.14 ! [A2: rat,B: rat] :
% 14.25/8.14 ( ( times_times_rat @ ( uminus_uminus_rat @ A2 ) @ B )
% 14.25/8.14 = ( times_times_rat @ A2 @ ( uminus_uminus_rat @ B ) ) ) ).
% 14.25/8.14
% 14.25/8.14 thf(fact_6994_minus__mult__commute,axiom,
% 14.25/8.14 ! [A2: int,B: int] :
% 14.25/8.14 ( ( times_times_int @ ( uminus_uminus_int @ A2 ) @ B )
% 14.25/8.14 = ( times_times_int @ A2 @ ( uminus_uminus_int @ B ) ) ) ).
% 14.25/8.14
% 14.25/8.14 thf(fact_6995_minus__mult__commute,axiom,
% 14.25/8.14 ! [A2: code_integer,B: code_integer] :
% 14.25/8.14 ( ( times_3573771949741848930nteger @ ( uminus1351360451143612070nteger @ A2 ) @ B )
% 14.25/8.14 = ( times_3573771949741848930nteger @ A2 @ ( uminus1351360451143612070nteger @ B ) ) ) ).
% 14.25/8.14
% 14.25/8.14 thf(fact_6996_one__neq__neg__one,axiom,
% 14.25/8.14 ( one_one_int
% 14.25/8.14 != ( uminus_uminus_int @ one_one_int ) ) ).
% 14.25/8.14
% 14.25/8.14 thf(fact_6997_one__neq__neg__one,axiom,
% 14.25/8.14 ( one_one_Code_integer
% 14.25/8.14 != ( uminus1351360451143612070nteger @ one_one_Code_integer ) ) ).
% 14.25/8.14
% 14.25/8.14 thf(fact_6998_minus__diff__commute,axiom,
% 14.25/8.14 ! [B: rat,A2: rat] :
% 14.25/8.14 ( ( minus_minus_rat @ ( uminus_uminus_rat @ B ) @ A2 )
% 14.25/8.14 = ( minus_minus_rat @ ( uminus_uminus_rat @ A2 ) @ B ) ) ).
% 14.25/8.14
% 14.25/8.14 thf(fact_6999_minus__diff__commute,axiom,
% 14.25/8.14 ! [B: int,A2: int] :
% 14.25/8.14 ( ( minus_minus_int @ ( uminus_uminus_int @ B ) @ A2 )
% 14.25/8.14 = ( minus_minus_int @ ( uminus_uminus_int @ A2 ) @ B ) ) ).
% 14.25/8.14
% 14.25/8.14 thf(fact_7000_minus__diff__commute,axiom,
% 14.25/8.14 ! [B: code_integer,A2: code_integer] :
% 14.25/8.14 ( ( minus_8373710615458151222nteger @ ( uminus1351360451143612070nteger @ B ) @ A2 )
% 14.25/8.14 = ( minus_8373710615458151222nteger @ ( uminus1351360451143612070nteger @ A2 ) @ B ) ) ).
% 14.25/8.14
% 14.25/8.14 thf(fact_7001_Gcd__1,axiom,
% 14.25/8.14 ! [A: set_nat] :
% 14.25/8.14 ( ( member_nat @ one_one_nat @ A )
% 14.25/8.14 => ( ( gcd_Gcd_nat @ A )
% 14.25/8.14 = one_one_nat ) ) ).
% 14.25/8.14
% 14.25/8.14 thf(fact_7002_Gcd__1,axiom,
% 14.25/8.14 ! [A: set_int] :
% 14.25/8.14 ( ( member_int @ one_one_int @ A )
% 14.25/8.14 => ( ( gcd_Gcd_int @ A )
% 14.25/8.14 = one_one_int ) ) ).
% 14.25/8.14
% 14.25/8.14 thf(fact_7003_minus__assn__def,axiom,
% 14.25/8.14 ( minus_minus_assn
% 14.25/8.14 = ( ^ [A6: assn,B6: assn] : ( inf_inf_assn @ A6 @ ( uminus_uminus_assn @ B6 ) ) ) ) ).
% 14.25/8.14
% 14.25/8.14 thf(fact_7004_prod__ge__1,axiom,
% 14.25/8.14 ! [A: set_nat,F: nat > code_integer] :
% 14.25/8.14 ( ! [X3: nat] :
% 14.25/8.14 ( ( member_nat @ X3 @ A )
% 14.25/8.14 => ( ord_le3102999989581377725nteger @ one_one_Code_integer @ ( F @ X3 ) ) )
% 14.25/8.14 => ( ord_le3102999989581377725nteger @ one_one_Code_integer @ ( groups3455450783089532116nteger @ F @ A ) ) ) ).
% 14.25/8.14
% 14.25/8.14 thf(fact_7005_prod__ge__1,axiom,
% 14.25/8.14 ! [A: set_int,F: int > code_integer] :
% 14.25/8.14 ( ! [X3: int] :
% 14.25/8.14 ( ( member_int @ X3 @ A )
% 14.25/8.14 => ( ord_le3102999989581377725nteger @ one_one_Code_integer @ ( F @ X3 ) ) )
% 14.25/8.14 => ( ord_le3102999989581377725nteger @ one_one_Code_integer @ ( groups3827104343326376752nteger @ F @ A ) ) ) ).
% 14.25/8.14
% 14.25/8.14 thf(fact_7006_prod__ge__1,axiom,
% 14.25/8.14 ! [A: set_o,F: $o > code_integer] :
% 14.25/8.14 ( ! [X3: $o] :
% 14.25/8.14 ( ( member_o @ X3 @ A )
% 14.25/8.14 => ( ord_le3102999989581377725nteger @ one_one_Code_integer @ ( F @ X3 ) ) )
% 14.25/8.14 => ( ord_le3102999989581377725nteger @ one_one_Code_integer @ ( groups7694694392188491536nteger @ F @ A ) ) ) ).
% 14.25/8.14
% 14.25/8.14 thf(fact_7007_prod__ge__1,axiom,
% 14.25/8.14 ! [A: set_Product_unit,F: product_unit > code_integer] :
% 14.25/8.14 ( ! [X3: product_unit] :
% 14.25/8.14 ( ( member_Product_unit @ X3 @ A )
% 14.25/8.14 => ( ord_le3102999989581377725nteger @ one_one_Code_integer @ ( F @ X3 ) ) )
% 14.25/8.14 => ( ord_le3102999989581377725nteger @ one_one_Code_integer @ ( groups8185359811944078981nteger @ F @ A ) ) ) ).
% 14.25/8.14
% 14.25/8.14 thf(fact_7008_prod__ge__1,axiom,
% 14.25/8.14 ! [A: set_nat,F: nat > code_natural] :
% 14.25/8.14 ( ! [X3: nat] :
% 14.25/8.14 ( ( member_nat @ X3 @ A )
% 14.25/8.14 => ( ord_le1926595141338095240atural @ one_one_Code_natural @ ( F @ X3 ) ) )
% 14.25/8.14 => ( ord_le1926595141338095240atural @ one_one_Code_natural @ ( groups2279045934846249631atural @ F @ A ) ) ) ).
% 14.25/8.14
% 14.25/8.14 thf(fact_7009_prod__ge__1,axiom,
% 14.25/8.14 ! [A: set_int,F: int > code_natural] :
% 14.25/8.14 ( ! [X3: int] :
% 14.25/8.14 ( ( member_int @ X3 @ A )
% 14.25/8.14 => ( ord_le1926595141338095240atural @ one_one_Code_natural @ ( F @ X3 ) ) )
% 14.25/8.14 => ( ord_le1926595141338095240atural @ one_one_Code_natural @ ( groups2650699495083094267atural @ F @ A ) ) ) ).
% 14.25/8.14
% 14.25/8.14 thf(fact_7010_prod__ge__1,axiom,
% 14.25/8.14 ! [A: set_o,F: $o > code_natural] :
% 14.25/8.14 ( ! [X3: $o] :
% 14.25/8.14 ( ( member_o @ X3 @ A )
% 14.25/8.14 => ( ord_le1926595141338095240atural @ one_one_Code_natural @ ( F @ X3 ) ) )
% 14.25/8.14 => ( ord_le1926595141338095240atural @ one_one_Code_natural @ ( groups6518289543945209051atural @ F @ A ) ) ) ).
% 14.25/8.14
% 14.25/8.14 thf(fact_7011_prod__ge__1,axiom,
% 14.25/8.14 ! [A: set_Product_unit,F: product_unit > code_natural] :
% 14.25/8.14 ( ! [X3: product_unit] :
% 14.25/8.14 ( ( member_Product_unit @ X3 @ A )
% 14.25/8.14 => ( ord_le1926595141338095240atural @ one_one_Code_natural @ ( F @ X3 ) ) )
% 14.25/8.14 => ( ord_le1926595141338095240atural @ one_one_Code_natural @ ( groups7008954963700796496atural @ F @ A ) ) ) ).
% 14.25/8.14
% 14.25/8.14 thf(fact_7012_prod__ge__1,axiom,
% 14.25/8.14 ! [A: set_nat,F: nat > rat] :
% 14.25/8.14 ( ! [X3: nat] :
% 14.25/8.14 ( ( member_nat @ X3 @ A )
% 14.25/8.14 => ( ord_less_eq_rat @ one_one_rat @ ( F @ X3 ) ) )
% 14.25/8.14 => ( ord_less_eq_rat @ one_one_rat @ ( groups73079841787564623at_rat @ F @ A ) ) ) ).
% 14.25/8.14
% 14.25/8.14 thf(fact_7013_prod__ge__1,axiom,
% 14.25/8.14 ! [A: set_int,F: int > rat] :
% 14.25/8.14 ( ! [X3: int] :
% 14.25/8.14 ( ( member_int @ X3 @ A )
% 14.25/8.14 => ( ord_less_eq_rat @ one_one_rat @ ( F @ X3 ) ) )
% 14.25/8.14 => ( ord_less_eq_rat @ one_one_rat @ ( groups1072433553688619179nt_rat @ F @ A ) ) ) ).
% 14.25/8.14
% 14.25/8.14 thf(fact_7014_prod_OUnion__comp,axiom,
% 14.25/8.14 ! [B4: set_set_int,G: int > assn] :
% 14.25/8.14 ( ! [X3: set_int] :
% 14.25/8.14 ( ( member_set_int @ X3 @ B4 )
% 14.25/8.14 => ( finite_finite_int @ X3 ) )
% 14.25/8.14 => ( ! [A12: set_int] :
% 14.25/8.14 ( ( member_set_int @ A12 @ B4 )
% 14.25/8.14 => ! [A23: set_int] :
% 14.25/8.14 ( ( member_set_int @ A23 @ B4 )
% 14.25/8.14 => ( ( A12 != A23 )
% 14.25/8.14 => ! [X3: int] :
% 14.25/8.14 ( ( member_int @ X3 @ A12 )
% 14.25/8.14 => ( ( member_int @ X3 @ A23 )
% 14.25/8.14 => ( ( G @ X3 )
% 14.25/8.14 = one_one_assn ) ) ) ) ) )
% 14.25/8.14 => ( ( groups7882442080178216443t_assn @ G @ ( comple3221217463730067765et_int @ B4 ) )
% 14.25/8.14 = ( comp_s4595925057662634336t_assn @ groups5950967880310095793t_assn @ groups7882442080178216443t_assn @ G @ B4 ) ) ) ) ).
% 14.25/8.14
% 14.25/8.14 thf(fact_7015_prod_OUnion__comp,axiom,
% 14.25/8.14 ! [B4: set_set_o,G: $o > assn] :
% 14.25/8.14 ( ! [X3: set_o] :
% 14.25/8.14 ( ( member_set_o @ X3 @ B4 )
% 14.25/8.14 => ( finite_finite_o @ X3 ) )
% 14.25/8.14 => ( ! [A12: set_o] :
% 14.25/8.14 ( ( member_set_o @ A12 @ B4 )
% 14.25/8.14 => ! [A23: set_o] :
% 14.25/8.14 ( ( member_set_o @ A23 @ B4 )
% 14.25/8.14 => ( ( A12 != A23 )
% 14.25/8.14 => ! [X3: $o] :
% 14.25/8.14 ( ( member_o @ X3 @ A12 )
% 14.25/8.14 => ( ( member_o @ X3 @ A23 )
% 14.25/8.14 => ( ( G @ X3 )
% 14.25/8.14 = one_one_assn ) ) ) ) ) )
% 14.25/8.14 => ( ( groups5301882518646026715o_assn @ G @ ( comple90263536869209701_set_o @ B4 ) )
% 14.25/8.14 = ( comp_s518380907012905326o_assn @ groups6963733262650904763o_assn @ groups5301882518646026715o_assn @ G @ B4 ) ) ) ) ).
% 14.25/8.14
% 14.25/8.14 thf(fact_7016_prod_OUnion__comp,axiom,
% 14.25/8.14 ! [B4: set_set_Product_unit,G: product_unit > assn] :
% 14.25/8.14 ( ! [X3: set_Product_unit] :
% 14.25/8.14 ( ( member5877623283571906838t_unit @ X3 @ B4 )
% 14.25/8.14 => ( finite4290736615968046902t_unit @ X3 ) )
% 14.25/8.14 => ( ! [A12: set_Product_unit] :
% 14.25/8.14 ( ( member5877623283571906838t_unit @ A12 @ B4 )
% 14.25/8.14 => ! [A23: set_Product_unit] :
% 14.25/8.14 ( ( member5877623283571906838t_unit @ A23 @ B4 )
% 14.25/8.14 => ( ( A12 != A23 )
% 14.25/8.14 => ! [X3: product_unit] :
% 14.25/8.14 ( ( member_Product_unit @ X3 @ A12 )
% 14.25/8.14 => ( ( member_Product_unit @ X3 @ A23 )
% 14.25/8.14 => ( ( G @ X3 )
% 14.25/8.14 = one_one_assn ) ) ) ) ) )
% 14.25/8.14 => ( ( groups5360233303959380816t_assn @ G @ ( comple3349607820504768346t_unit @ B4 ) )
% 14.25/8.14 = ( comp_s8247530669214396195t_assn @ groups2392624107559085104t_assn @ groups5360233303959380816t_assn @ G @ B4 ) ) ) ) ).
% 14.25/8.14
% 14.25/8.14 thf(fact_7017_prod_OUnion__comp,axiom,
% 14.25/8.14 ! [B4: set_set_nat,G: nat > assn] :
% 14.25/8.14 ( ! [X3: set_nat] :
% 14.25/8.14 ( ( member_set_nat @ X3 @ B4 )
% 14.25/8.14 => ( finite_finite_nat @ X3 ) )
% 14.25/8.14 => ( ! [A12: set_nat] :
% 14.25/8.14 ( ( member_set_nat @ A12 @ B4 )
% 14.25/8.14 => ! [A23: set_nat] :
% 14.25/8.14 ( ( member_set_nat @ A23 @ B4 )
% 14.25/8.14 => ( ( A12 != A23 )
% 14.25/8.14 => ! [X3: nat] :
% 14.25/8.14 ( ( member_nat @ X3 @ A12 )
% 14.25/8.14 => ( ( member_nat @ X3 @ A23 )
% 14.25/8.14 => ( ( G @ X3 )
% 14.25/8.14 = one_one_assn ) ) ) ) ) )
% 14.25/8.14 => ( ( groups6906906614972039071t_assn @ G @ ( comple7399068483239264473et_nat @ B4 ) )
% 14.25/8.14 = ( comp_s2451231738238893644t_assn @ groups1535173586962943829t_assn @ groups6906906614972039071t_assn @ G @ B4 ) ) ) ) ).
% 14.25/8.14
% 14.25/8.14 thf(fact_7018_prod_OUnion__comp,axiom,
% 14.25/8.14 ! [B4: set_set_Code_integer,G: code_integer > assn] :
% 14.25/8.14 ( ! [X3: set_Code_integer] :
% 14.25/8.14 ( ( member1834298987299537450nteger @ X3 @ B4 )
% 14.25/8.14 => ( finite6017078050557962740nteger @ X3 ) )
% 14.25/8.14 => ( ! [A12: set_Code_integer] :
% 14.25/8.14 ( ( member1834298987299537450nteger @ A12 @ B4 )
% 14.25/8.14 => ! [A23: set_Code_integer] :
% 14.25/8.14 ( ( member1834298987299537450nteger @ A23 @ B4 )
% 14.25/8.14 => ( ( A12 != A23 )
% 14.25/8.14 => ! [X3: code_integer] :
% 14.25/8.14 ( ( member_Code_integer @ X3 @ A12 )
% 14.25/8.14 => ( ( member_Code_integer @ X3 @ A23 )
% 14.25/8.14 => ( ( G @ X3 )
% 14.25/8.14 = one_one_assn ) ) ) ) ) )
% 14.25/8.14 => ( ( groups1304777262505850412r_assn @ G @ ( comple739944243200306918nteger @ B4 ) )
% 14.25/8.14 = ( comp_s3531641609786562035r_assn @ groups4245288774754739042r_assn @ groups1304777262505850412r_assn @ G @ B4 ) ) ) ) ).
% 14.25/8.14
% 14.25/8.14 thf(fact_7019_prod_OUnion__comp,axiom,
% 14.25/8.14 ! [B4: set_set_int,G: int > nat] :
% 14.25/8.14 ( ! [X3: set_int] :
% 14.25/8.14 ( ( member_set_int @ X3 @ B4 )
% 14.25/8.14 => ( finite_finite_int @ X3 ) )
% 14.25/8.14 => ( ! [A12: set_int] :
% 14.25/8.14 ( ( member_set_int @ A12 @ B4 )
% 14.25/8.14 => ! [A23: set_int] :
% 14.25/8.14 ( ( member_set_int @ A23 @ B4 )
% 14.25/8.14 => ( ( A12 != A23 )
% 14.25/8.14 => ! [X3: int] :
% 14.25/8.14 ( ( member_int @ X3 @ A12 )
% 14.25/8.14 => ( ( member_int @ X3 @ A23 )
% 14.25/8.14 => ( ( G @ X3 )
% 14.25/8.14 = one_one_nat ) ) ) ) ) )
% 14.25/8.14 => ( ( groups1707563613775114915nt_nat @ G @ ( comple3221217463730067765et_int @ B4 ) )
% 14.25/8.14 = ( comp_s3237642481825588056nt_nat @ groups6435469334873047129nt_nat @ groups1707563613775114915nt_nat @ G @ B4 ) ) ) ) ).
% 14.25/8.14
% 14.25/8.14 thf(fact_7020_prod_OUnion__comp,axiom,
% 14.25/8.14 ! [B4: set_set_o,G: $o > nat] :
% 14.25/8.14 ( ! [X3: set_o] :
% 14.25/8.14 ( ( member_set_o @ X3 @ B4 )
% 14.25/8.14 => ( finite_finite_o @ X3 ) )
% 14.25/8.14 => ( ! [A12: set_o] :
% 14.25/8.14 ( ( member_set_o @ A12 @ B4 )
% 14.25/8.14 => ! [A23: set_o] :
% 14.25/8.14 ( ( member_set_o @ A23 @ B4 )
% 14.25/8.14 => ( ( A12 != A23 )
% 14.25/8.14 => ! [X3: $o] :
% 14.25/8.14 ( ( member_o @ X3 @ A12 )
% 14.25/8.14 => ( ( member_o @ X3 @ A23 )
% 14.25/8.14 => ( ( G @ X3 )
% 14.25/8.14 = one_one_nat ) ) ) ) ) )
% 14.25/8.14 => ( ( groups3504817904513533571_o_nat @ G @ ( comple90263536869209701_set_o @ B4 ) )
% 14.25/8.14 = ( comp_s6084864194472624150_o_nat @ groups4868773677410035043_o_nat @ groups3504817904513533571_o_nat @ G @ B4 ) ) ) ) ).
% 14.25/8.14
% 14.25/8.14 thf(fact_7021_prod_OUnion__comp,axiom,
% 14.25/8.14 ! [B4: set_set_Product_unit,G: product_unit > nat] :
% 14.25/8.14 ( ! [X3: set_Product_unit] :
% 14.25/8.14 ( ( member5877623283571906838t_unit @ X3 @ B4 )
% 14.25/8.14 => ( finite4290736615968046902t_unit @ X3 ) )
% 14.25/8.14 => ( ! [A12: set_Product_unit] :
% 14.25/8.14 ( ( member5877623283571906838t_unit @ A12 @ B4 )
% 14.25/8.14 => ! [A23: set_Product_unit] :
% 14.25/8.14 ( ( member5877623283571906838t_unit @ A23 @ B4 )
% 14.25/8.14 => ( ( A12 != A23 )
% 14.25/8.14 => ! [X3: product_unit] :
% 14.25/8.14 ( ( member_Product_unit @ X3 @ A12 )
% 14.25/8.14 => ( ( member_Product_unit @ X3 @ A23 )
% 14.25/8.14 => ( ( G @ X3 )
% 14.25/8.14 = one_one_nat ) ) ) ) ) )
% 14.25/8.14 => ( ( groups396421249026171384it_nat @ G @ ( comple3349607820504768346t_unit @ B4 ) )
% 14.25/8.14 = ( comp_s8928436226432208331it_nat @ groups8454329047868613336it_nat @ groups396421249026171384it_nat @ G @ B4 ) ) ) ) ).
% 14.25/8.14
% 14.25/8.14 thf(fact_7022_prod_OUnion__comp,axiom,
% 14.25/8.14 ! [B4: set_set_Code_integer,G: code_integer > nat] :
% 14.25/8.14 ( ! [X3: set_Code_integer] :
% 14.25/8.14 ( ( member1834298987299537450nteger @ X3 @ B4 )
% 14.25/8.14 => ( finite6017078050557962740nteger @ X3 ) )
% 14.25/8.14 => ( ! [A12: set_Code_integer] :
% 14.25/8.14 ( ( member1834298987299537450nteger @ A12 @ B4 )
% 14.25/8.14 => ! [A23: set_Code_integer] :
% 14.25/8.14 ( ( member1834298987299537450nteger @ A23 @ B4 )
% 14.25/8.14 => ( ( A12 != A23 )
% 14.25/8.14 => ! [X3: code_integer] :
% 14.25/8.14 ( ( member_Code_integer @ X3 @ A12 )
% 14.25/8.14 => ( ( member_Code_integer @ X3 @ A23 )
% 14.25/8.14 => ( ( G @ X3 )
% 14.25/8.14 = one_one_nat ) ) ) ) ) )
% 14.25/8.14 => ( ( groups3190895334310489300er_nat @ G @ ( comple739944243200306918nteger @ B4 ) )
% 14.25/8.14 = ( comp_s8759123648413916139er_nat @ groups6044384099095135754er_nat @ groups3190895334310489300er_nat @ G @ B4 ) ) ) ) ).
% 14.25/8.14
% 14.25/8.14 thf(fact_7023_prod_OUnion__comp,axiom,
% 14.25/8.14 ! [B4: set_set_int,G: int > int] :
% 14.25/8.14 ( ! [X3: set_int] :
% 14.25/8.14 ( ( member_set_int @ X3 @ B4 )
% 14.25/8.14 => ( finite_finite_int @ X3 ) )
% 14.25/8.14 => ( ! [A12: set_int] :
% 14.25/8.14 ( ( member_set_int @ A12 @ B4 )
% 14.25/8.14 => ! [A23: set_int] :
% 14.25/8.14 ( ( member_set_int @ A23 @ B4 )
% 14.25/8.14 => ( ( A12 != A23 )
% 14.25/8.14 => ! [X3: int] :
% 14.25/8.14 ( ( member_int @ X3 @ A12 )
% 14.25/8.14 => ( ( member_int @ X3 @ A23 )
% 14.25/8.14 => ( ( G @ X3 )
% 14.25/8.14 = one_one_int ) ) ) ) ) )
% 14.25/8.14 => ( ( groups1705073143266064639nt_int @ G @ ( comple3221217463730067765et_int @ B4 ) )
% 14.25/8.14 = ( comp_s9098231955081793132nt_int @ groups6432978864363996853nt_int @ groups1705073143266064639nt_int @ G @ B4 ) ) ) ) ).
% 14.25/8.14
% 14.25/8.14 thf(fact_7024_neg__eq__iff__add__eq__0,axiom,
% 14.25/8.14 ! [A2: rat,B: rat] :
% 14.25/8.14 ( ( ( uminus_uminus_rat @ A2 )
% 14.25/8.14 = B )
% 14.25/8.14 = ( ( plus_plus_rat @ A2 @ B )
% 14.25/8.14 = zero_zero_rat ) ) ).
% 14.25/8.14
% 14.25/8.14 thf(fact_7025_neg__eq__iff__add__eq__0,axiom,
% 14.25/8.14 ! [A2: int,B: int] :
% 14.25/8.14 ( ( ( uminus_uminus_int @ A2 )
% 14.25/8.14 = B )
% 14.25/8.14 = ( ( plus_plus_int @ A2 @ B )
% 14.25/8.14 = zero_zero_int ) ) ).
% 14.25/8.14
% 14.25/8.14 thf(fact_7026_neg__eq__iff__add__eq__0,axiom,
% 14.25/8.14 ! [A2: code_integer,B: code_integer] :
% 14.25/8.14 ( ( ( uminus1351360451143612070nteger @ A2 )
% 14.25/8.14 = B )
% 14.25/8.14 = ( ( plus_p5714425477246183910nteger @ A2 @ B )
% 14.25/8.14 = zero_z3403309356797280102nteger ) ) ).
% 14.25/8.14
% 14.25/8.14 thf(fact_7027_eq__neg__iff__add__eq__0,axiom,
% 14.25/8.14 ! [A2: rat,B: rat] :
% 14.25/8.14 ( ( A2
% 14.25/8.14 = ( uminus_uminus_rat @ B ) )
% 14.25/8.14 = ( ( plus_plus_rat @ A2 @ B )
% 14.25/8.14 = zero_zero_rat ) ) ).
% 14.25/8.14
% 14.25/8.14 thf(fact_7028_eq__neg__iff__add__eq__0,axiom,
% 14.25/8.14 ! [A2: int,B: int] :
% 14.25/8.14 ( ( A2
% 14.25/8.14 = ( uminus_uminus_int @ B ) )
% 14.25/8.14 = ( ( plus_plus_int @ A2 @ B )
% 14.25/8.14 = zero_zero_int ) ) ).
% 14.25/8.14
% 14.25/8.14 thf(fact_7029_eq__neg__iff__add__eq__0,axiom,
% 14.25/8.14 ! [A2: code_integer,B: code_integer] :
% 14.25/8.14 ( ( A2
% 14.25/8.14 = ( uminus1351360451143612070nteger @ B ) )
% 14.25/8.14 = ( ( plus_p5714425477246183910nteger @ A2 @ B )
% 14.25/8.14 = zero_z3403309356797280102nteger ) ) ).
% 14.25/8.14
% 14.25/8.14 thf(fact_7030_add_Oinverse__unique,axiom,
% 14.25/8.14 ! [A2: rat,B: rat] :
% 14.25/8.14 ( ( ( plus_plus_rat @ A2 @ B )
% 14.25/8.14 = zero_zero_rat )
% 14.25/8.14 => ( ( uminus_uminus_rat @ A2 )
% 14.25/8.14 = B ) ) ).
% 14.25/8.14
% 14.25/8.14 thf(fact_7031_add_Oinverse__unique,axiom,
% 14.25/8.14 ! [A2: int,B: int] :
% 14.25/8.14 ( ( ( plus_plus_int @ A2 @ B )
% 14.25/8.14 = zero_zero_int )
% 14.25/8.14 => ( ( uminus_uminus_int @ A2 )
% 14.25/8.14 = B ) ) ).
% 14.25/8.14
% 14.25/8.14 thf(fact_7032_add_Oinverse__unique,axiom,
% 14.25/8.14 ! [A2: code_integer,B: code_integer] :
% 14.25/8.14 ( ( ( plus_p5714425477246183910nteger @ A2 @ B )
% 14.25/8.14 = zero_z3403309356797280102nteger )
% 14.25/8.14 => ( ( uminus1351360451143612070nteger @ A2 )
% 14.25/8.14 = B ) ) ).
% 14.25/8.14
% 14.25/8.14 thf(fact_7033_ab__group__add__class_Oab__left__minus,axiom,
% 14.25/8.14 ! [A2: rat] :
% 14.25/8.14 ( ( plus_plus_rat @ ( uminus_uminus_rat @ A2 ) @ A2 )
% 14.25/8.14 = zero_zero_rat ) ).
% 14.25/8.14
% 14.25/8.14 thf(fact_7034_ab__group__add__class_Oab__left__minus,axiom,
% 14.25/8.14 ! [A2: int] :
% 14.25/8.14 ( ( plus_plus_int @ ( uminus_uminus_int @ A2 ) @ A2 )
% 14.25/8.14 = zero_zero_int ) ).
% 14.25/8.14
% 14.25/8.14 thf(fact_7035_ab__group__add__class_Oab__left__minus,axiom,
% 14.25/8.14 ! [A2: code_integer] :
% 14.25/8.14 ( ( plus_p5714425477246183910nteger @ ( uminus1351360451143612070nteger @ A2 ) @ A2 )
% 14.25/8.14 = zero_z3403309356797280102nteger ) ).
% 14.25/8.14
% 14.25/8.14 thf(fact_7036_add__eq__0__iff,axiom,
% 14.25/8.14 ! [A2: rat,B: rat] :
% 14.25/8.14 ( ( ( plus_plus_rat @ A2 @ B )
% 14.25/8.14 = zero_zero_rat )
% 14.25/8.14 = ( B
% 14.25/8.14 = ( uminus_uminus_rat @ A2 ) ) ) ).
% 14.25/8.14
% 14.25/8.14 thf(fact_7037_add__eq__0__iff,axiom,
% 14.25/8.14 ! [A2: int,B: int] :
% 14.25/8.14 ( ( ( plus_plus_int @ A2 @ B )
% 14.25/8.14 = zero_zero_int )
% 14.25/8.14 = ( B
% 14.25/8.14 = ( uminus_uminus_int @ A2 ) ) ) ).
% 14.25/8.14
% 14.25/8.14 thf(fact_7038_add__eq__0__iff,axiom,
% 14.25/8.14 ! [A2: code_integer,B: code_integer] :
% 14.25/8.14 ( ( ( plus_p5714425477246183910nteger @ A2 @ B )
% 14.25/8.14 = zero_z3403309356797280102nteger )
% 14.25/8.14 = ( B
% 14.25/8.14 = ( uminus1351360451143612070nteger @ A2 ) ) ) ).
% 14.25/8.14
% 14.25/8.14 thf(fact_7039_le__minus__one__simps_I4_J,axiom,
% 14.25/8.14 ~ ( ord_le3102999989581377725nteger @ one_one_Code_integer @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) ) ).
% 14.25/8.14
% 14.25/8.14 thf(fact_7040_le__minus__one__simps_I4_J,axiom,
% 14.25/8.14 ~ ( ord_less_eq_rat @ one_one_rat @ ( uminus_uminus_rat @ one_one_rat ) ) ).
% 14.25/8.14
% 14.25/8.14 thf(fact_7041_le__minus__one__simps_I4_J,axiom,
% 14.25/8.14 ~ ( ord_less_eq_int @ one_one_int @ ( uminus_uminus_int @ one_one_int ) ) ).
% 14.25/8.14
% 14.25/8.14 thf(fact_7042_le__minus__one__simps_I2_J,axiom,
% 14.25/8.14 ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ one_one_Code_integer ).
% 14.25/8.14
% 14.25/8.14 thf(fact_7043_le__minus__one__simps_I2_J,axiom,
% 14.25/8.14 ord_less_eq_rat @ ( uminus_uminus_rat @ one_one_rat ) @ one_one_rat ).
% 14.25/8.14
% 14.25/8.14 thf(fact_7044_le__minus__one__simps_I2_J,axiom,
% 14.25/8.14 ord_less_eq_int @ ( uminus_uminus_int @ one_one_int ) @ one_one_int ).
% 14.25/8.14
% 14.25/8.14 thf(fact_7045_zero__neq__neg__one,axiom,
% 14.25/8.14 ( zero_zero_int
% 14.25/8.14 != ( uminus_uminus_int @ one_one_int ) ) ).
% 14.25/8.14
% 14.25/8.14 thf(fact_7046_zero__neq__neg__one,axiom,
% 14.25/8.14 ( zero_z3403309356797280102nteger
% 14.25/8.14 != ( uminus1351360451143612070nteger @ one_one_Code_integer ) ) ).
% 14.25/8.14
% 14.25/8.14 thf(fact_7047_less__minus__one__simps_I2_J,axiom,
% 14.25/8.14 ord_less_rat @ ( uminus_uminus_rat @ one_one_rat ) @ one_one_rat ).
% 14.25/8.14
% 14.25/8.14 thf(fact_7048_less__minus__one__simps_I2_J,axiom,
% 14.25/8.14 ord_less_int @ ( uminus_uminus_int @ one_one_int ) @ one_one_int ).
% 14.25/8.14
% 14.25/8.14 thf(fact_7049_less__minus__one__simps_I2_J,axiom,
% 14.25/8.14 ord_le6747313008572928689nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ one_one_Code_integer ).
% 14.25/8.14
% 14.25/8.14 thf(fact_7050_less__minus__one__simps_I4_J,axiom,
% 14.25/8.14 ~ ( ord_less_rat @ one_one_rat @ ( uminus_uminus_rat @ one_one_rat ) ) ).
% 14.25/8.14
% 14.25/8.14 thf(fact_7051_less__minus__one__simps_I4_J,axiom,
% 14.25/8.14 ~ ( ord_less_int @ one_one_int @ ( uminus_uminus_int @ one_one_int ) ) ).
% 14.25/8.14
% 14.25/8.14 thf(fact_7052_less__minus__one__simps_I4_J,axiom,
% 14.25/8.14 ~ ( ord_le6747313008572928689nteger @ one_one_Code_integer @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) ) ).
% 14.25/8.14
% 14.25/8.14 thf(fact_7053_ab__group__add__class_Oab__diff__conv__add__uminus,axiom,
% 14.25/8.14 ( minus_minus_rat
% 14.25/8.14 = ( ^ [A6: rat,B6: rat] : ( plus_plus_rat @ A6 @ ( uminus_uminus_rat @ B6 ) ) ) ) ).
% 14.25/8.14
% 14.25/8.14 thf(fact_7054_ab__group__add__class_Oab__diff__conv__add__uminus,axiom,
% 14.25/8.14 ( minus_minus_int
% 14.25/8.14 = ( ^ [A6: int,B6: int] : ( plus_plus_int @ A6 @ ( uminus_uminus_int @ B6 ) ) ) ) ).
% 14.25/8.14
% 14.25/8.14 thf(fact_7055_ab__group__add__class_Oab__diff__conv__add__uminus,axiom,
% 14.25/8.14 ( minus_8373710615458151222nteger
% 14.25/8.14 = ( ^ [A6: code_integer,B6: code_integer] : ( plus_p5714425477246183910nteger @ A6 @ ( uminus1351360451143612070nteger @ B6 ) ) ) ) ).
% 14.25/8.14
% 14.25/8.14 thf(fact_7056_diff__conv__add__uminus,axiom,
% 14.25/8.14 ( minus_minus_rat
% 14.25/8.14 = ( ^ [A6: rat,B6: rat] : ( plus_plus_rat @ A6 @ ( uminus_uminus_rat @ B6 ) ) ) ) ).
% 14.25/8.14
% 14.25/8.14 thf(fact_7057_diff__conv__add__uminus,axiom,
% 14.25/8.14 ( minus_minus_int
% 14.25/8.14 = ( ^ [A6: int,B6: int] : ( plus_plus_int @ A6 @ ( uminus_uminus_int @ B6 ) ) ) ) ).
% 14.25/8.14
% 14.25/8.14 thf(fact_7058_diff__conv__add__uminus,axiom,
% 14.25/8.14 ( minus_8373710615458151222nteger
% 14.25/8.14 = ( ^ [A6: code_integer,B6: code_integer] : ( plus_p5714425477246183910nteger @ A6 @ ( uminus1351360451143612070nteger @ B6 ) ) ) ) ).
% 14.25/8.14
% 14.25/8.14 thf(fact_7059_group__cancel_Osub2,axiom,
% 14.25/8.14 ! [B4: rat,K3: rat,B: rat,A2: rat] :
% 14.25/8.14 ( ( B4
% 14.25/8.14 = ( plus_plus_rat @ K3 @ B ) )
% 14.25/8.14 => ( ( minus_minus_rat @ A2 @ B4 )
% 14.25/8.14 = ( plus_plus_rat @ ( uminus_uminus_rat @ K3 ) @ ( minus_minus_rat @ A2 @ B ) ) ) ) ).
% 14.25/8.14
% 14.25/8.14 thf(fact_7060_group__cancel_Osub2,axiom,
% 14.25/8.14 ! [B4: int,K3: int,B: int,A2: int] :
% 14.25/8.14 ( ( B4
% 14.25/8.14 = ( plus_plus_int @ K3 @ B ) )
% 14.25/8.14 => ( ( minus_minus_int @ A2 @ B4 )
% 14.25/8.14 = ( plus_plus_int @ ( uminus_uminus_int @ K3 ) @ ( minus_minus_int @ A2 @ B ) ) ) ) ).
% 14.25/8.14
% 14.25/8.14 thf(fact_7061_group__cancel_Osub2,axiom,
% 14.25/8.14 ! [B4: code_integer,K3: code_integer,B: code_integer,A2: code_integer] :
% 14.25/8.14 ( ( B4
% 14.25/8.14 = ( plus_p5714425477246183910nteger @ K3 @ B ) )
% 14.25/8.14 => ( ( minus_8373710615458151222nteger @ A2 @ B4 )
% 14.25/8.14 = ( plus_p5714425477246183910nteger @ ( uminus1351360451143612070nteger @ K3 ) @ ( minus_8373710615458151222nteger @ A2 @ B ) ) ) ) ).
% 14.25/8.14
% 14.25/8.14 thf(fact_7062_square__eq__1__iff,axiom,
% 14.25/8.14 ! [X: rat] :
% 14.25/8.14 ( ( ( times_times_rat @ X @ X )
% 14.25/8.14 = one_one_rat )
% 14.25/8.14 = ( ( X = one_one_rat )
% 14.25/8.14 | ( X
% 14.25/8.14 = ( uminus_uminus_rat @ one_one_rat ) ) ) ) ).
% 14.25/8.14
% 14.25/8.14 thf(fact_7063_square__eq__1__iff,axiom,
% 14.25/8.14 ! [X: int] :
% 14.25/8.14 ( ( ( times_times_int @ X @ X )
% 14.25/8.14 = one_one_int )
% 14.25/8.14 = ( ( X = one_one_int )
% 14.25/8.14 | ( X
% 14.25/8.14 = ( uminus_uminus_int @ one_one_int ) ) ) ) ).
% 14.25/8.14
% 14.25/8.14 thf(fact_7064_square__eq__1__iff,axiom,
% 14.25/8.14 ! [X: code_integer] :
% 14.25/8.14 ( ( ( times_3573771949741848930nteger @ X @ X )
% 14.25/8.14 = one_one_Code_integer )
% 14.25/8.14 = ( ( X = one_one_Code_integer )
% 14.25/8.14 | ( X
% 14.25/8.14 = ( uminus1351360451143612070nteger @ one_one_Code_integer ) ) ) ) ).
% 14.25/8.14
% 14.25/8.14 thf(fact_7065_inf__cancel__left2,axiom,
% 14.25/8.14 ! [X: set_Product_unit,A2: set_Product_unit,B: set_Product_unit] :
% 14.25/8.14 ( ( inf_in4660618365625256667t_unit @ ( inf_in4660618365625256667t_unit @ ( uminus5944136376168626660t_unit @ X ) @ A2 ) @ ( inf_in4660618365625256667t_unit @ X @ B ) )
% 14.25/8.14 = bot_bo3957492148770167129t_unit ) ).
% 14.25/8.14
% 14.25/8.14 thf(fact_7066_inf__cancel__left2,axiom,
% 14.25/8.14 ! [X: set_o,A2: set_o,B: set_o] :
% 14.25/8.14 ( ( inf_inf_set_o @ ( inf_inf_set_o @ ( uminus_uminus_set_o @ X ) @ A2 ) @ ( inf_inf_set_o @ X @ B ) )
% 14.25/8.14 = bot_bot_set_o ) ).
% 14.25/8.14
% 14.25/8.14 thf(fact_7067_inf__cancel__left2,axiom,
% 14.25/8.14 ! [X: set_nat,A2: set_nat,B: set_nat] :
% 14.25/8.14 ( ( inf_inf_set_nat @ ( inf_inf_set_nat @ ( uminus5710092332889474511et_nat @ X ) @ A2 ) @ ( inf_inf_set_nat @ X @ B ) )
% 14.25/8.14 = bot_bot_set_nat ) ).
% 14.25/8.14
% 14.25/8.14 thf(fact_7068_inf__cancel__left2,axiom,
% 14.25/8.14 ! [X: set_int,A2: set_int,B: set_int] :
% 14.25/8.14 ( ( inf_inf_set_int @ ( inf_inf_set_int @ ( uminus1532241313380277803et_int @ X ) @ A2 ) @ ( inf_inf_set_int @ X @ B ) )
% 14.25/8.14 = bot_bot_set_int ) ).
% 14.25/8.14
% 14.25/8.14 thf(fact_7069_inf__cancel__left2,axiom,
% 14.25/8.14 ! [X: set_Pr4532377907799695533_nat_o,A2: set_Pr4532377907799695533_nat_o,B: set_Pr4532377907799695533_nat_o] :
% 14.25/8.14 ( ( inf_in1906310914598751387_nat_o @ ( inf_in1906310914598751387_nat_o @ ( uminus5254974436814262692_nat_o @ X ) @ A2 ) @ ( inf_in1906310914598751387_nat_o @ X @ B ) )
% 14.25/8.14 = bot_bo7824918357723582233_nat_o ) ).
% 14.25/8.14
% 14.25/8.14 thf(fact_7070_inf__cancel__left2,axiom,
% 14.25/8.14 ! [X: assn,A2: assn,B: assn] :
% 14.25/8.14 ( ( inf_inf_assn @ ( inf_inf_assn @ ( uminus_uminus_assn @ X ) @ A2 ) @ ( inf_inf_assn @ X @ B ) )
% 14.25/8.14 = bot_bot_assn ) ).
% 14.25/8.14
% 14.25/8.14 thf(fact_7071_inf__cancel__left2,axiom,
% 14.25/8.14 ! [X: product_unit,A2: product_unit,B: product_unit] :
% 14.25/8.14 ( ( inf_inf_Product_unit @ ( inf_inf_Product_unit @ ( uminus2952777764628376836t_unit @ X ) @ A2 ) @ ( inf_inf_Product_unit @ X @ B ) )
% 14.25/8.14 = bot_bot_Product_unit ) ).
% 14.25/8.14
% 14.25/8.14 thf(fact_7072_inf__cancel__left1,axiom,
% 14.25/8.14 ! [X: set_Product_unit,A2: set_Product_unit,B: set_Product_unit] :
% 14.25/8.14 ( ( inf_in4660618365625256667t_unit @ ( inf_in4660618365625256667t_unit @ X @ A2 ) @ ( inf_in4660618365625256667t_unit @ ( uminus5944136376168626660t_unit @ X ) @ B ) )
% 14.25/8.14 = bot_bo3957492148770167129t_unit ) ).
% 14.25/8.14
% 14.25/8.14 thf(fact_7073_inf__cancel__left1,axiom,
% 14.25/8.14 ! [X: set_o,A2: set_o,B: set_o] :
% 14.25/8.14 ( ( inf_inf_set_o @ ( inf_inf_set_o @ X @ A2 ) @ ( inf_inf_set_o @ ( uminus_uminus_set_o @ X ) @ B ) )
% 14.25/8.14 = bot_bot_set_o ) ).
% 14.25/8.14
% 14.25/8.14 thf(fact_7074_inf__cancel__left1,axiom,
% 14.25/8.14 ! [X: set_nat,A2: set_nat,B: set_nat] :
% 14.25/8.14 ( ( inf_inf_set_nat @ ( inf_inf_set_nat @ X @ A2 ) @ ( inf_inf_set_nat @ ( uminus5710092332889474511et_nat @ X ) @ B ) )
% 14.25/8.14 = bot_bot_set_nat ) ).
% 14.25/8.14
% 14.25/8.14 thf(fact_7075_inf__cancel__left1,axiom,
% 14.25/8.14 ! [X: set_int,A2: set_int,B: set_int] :
% 14.25/8.14 ( ( inf_inf_set_int @ ( inf_inf_set_int @ X @ A2 ) @ ( inf_inf_set_int @ ( uminus1532241313380277803et_int @ X ) @ B ) )
% 14.25/8.14 = bot_bot_set_int ) ).
% 14.25/8.14
% 14.25/8.14 thf(fact_7076_inf__cancel__left1,axiom,
% 14.25/8.14 ! [X: set_Pr4532377907799695533_nat_o,A2: set_Pr4532377907799695533_nat_o,B: set_Pr4532377907799695533_nat_o] :
% 14.25/8.14 ( ( inf_in1906310914598751387_nat_o @ ( inf_in1906310914598751387_nat_o @ X @ A2 ) @ ( inf_in1906310914598751387_nat_o @ ( uminus5254974436814262692_nat_o @ X ) @ B ) )
% 14.25/8.14 = bot_bo7824918357723582233_nat_o ) ).
% 14.25/8.14
% 14.25/8.14 thf(fact_7077_inf__cancel__left1,axiom,
% 14.25/8.14 ! [X: assn,A2: assn,B: assn] :
% 14.25/8.14 ( ( inf_inf_assn @ ( inf_inf_assn @ X @ A2 ) @ ( inf_inf_assn @ ( uminus_uminus_assn @ X ) @ B ) )
% 14.25/8.14 = bot_bot_assn ) ).
% 14.25/8.14
% 14.25/8.14 thf(fact_7078_inf__cancel__left1,axiom,
% 14.25/8.14 ! [X: product_unit,A2: product_unit,B: product_unit] :
% 14.25/8.14 ( ( inf_inf_Product_unit @ ( inf_inf_Product_unit @ X @ A2 ) @ ( inf_inf_Product_unit @ ( uminus2952777764628376836t_unit @ X ) @ B ) )
% 14.25/8.14 = bot_bot_Product_unit ) ).
% 14.25/8.14
% 14.25/8.14 thf(fact_7079_subset__Compl__self__eq,axiom,
% 14.25/8.14 ! [A: set_Product_unit] :
% 14.25/8.14 ( ( ord_le3507040750410214029t_unit @ A @ ( uminus5944136376168626660t_unit @ A ) )
% 14.25/8.14 = ( A = bot_bo3957492148770167129t_unit ) ) ).
% 14.25/8.14
% 14.25/8.14 thf(fact_7080_subset__Compl__self__eq,axiom,
% 14.25/8.14 ! [A: set_o] :
% 14.25/8.14 ( ( ord_less_eq_set_o @ A @ ( uminus_uminus_set_o @ A ) )
% 14.25/8.14 = ( A = bot_bot_set_o ) ) ).
% 14.25/8.14
% 14.25/8.14 thf(fact_7081_subset__Compl__self__eq,axiom,
% 14.25/8.14 ! [A: set_int] :
% 14.25/8.14 ( ( ord_less_eq_set_int @ A @ ( uminus1532241313380277803et_int @ A ) )
% 14.25/8.14 = ( A = bot_bot_set_int ) ) ).
% 14.25/8.14
% 14.25/8.14 thf(fact_7082_subset__Compl__self__eq,axiom,
% 14.25/8.14 ! [A: set_Pr4532377907799695533_nat_o] :
% 14.25/8.14 ( ( ord_le2965882846123202637_nat_o @ A @ ( uminus5254974436814262692_nat_o @ A ) )
% 14.25/8.14 = ( A = bot_bo7824918357723582233_nat_o ) ) ).
% 14.25/8.14
% 14.25/8.14 thf(fact_7083_subset__Compl__self__eq,axiom,
% 14.25/8.14 ! [A: set_nat] :
% 14.25/8.14 ( ( ord_less_eq_set_nat @ A @ ( uminus5710092332889474511et_nat @ A ) )
% 14.25/8.14 = ( A = bot_bot_set_nat ) ) ).
% 14.25/8.14
% 14.25/8.14 thf(fact_7084_Compl__UNIV__eq,axiom,
% 14.25/8.14 ( ( uminus5254974436814262692_nat_o @ top_to4869887016220417533_nat_o )
% 14.25/8.14 = bot_bo7824918357723582233_nat_o ) ).
% 14.25/8.14
% 14.25/8.14 thf(fact_7085_Compl__UNIV__eq,axiom,
% 14.25/8.14 ( ( uminus5944136376168626660t_unit @ top_to1996260823553986621t_unit )
% 14.25/8.14 = bot_bo3957492148770167129t_unit ) ).
% 14.25/8.14
% 14.25/8.14 thf(fact_7086_Compl__UNIV__eq,axiom,
% 14.25/8.14 ( ( uminus_uminus_set_o @ top_top_set_o )
% 14.25/8.14 = bot_bot_set_o ) ).
% 14.25/8.14
% 14.25/8.14 thf(fact_7087_Compl__UNIV__eq,axiom,
% 14.25/8.14 ( ( uminus5710092332889474511et_nat @ top_top_set_nat )
% 14.25/8.14 = bot_bot_set_nat ) ).
% 14.25/8.14
% 14.25/8.14 thf(fact_7088_Compl__UNIV__eq,axiom,
% 14.25/8.14 ( ( uminus1532241313380277803et_int @ top_top_set_int )
% 14.25/8.14 = bot_bot_set_int ) ).
% 14.25/8.14
% 14.25/8.14 thf(fact_7089_Compl__empty__eq,axiom,
% 14.25/8.14 ( ( uminus5254974436814262692_nat_o @ bot_bo7824918357723582233_nat_o )
% 14.25/8.14 = top_to4869887016220417533_nat_o ) ).
% 14.25/8.14
% 14.25/8.14 thf(fact_7090_Compl__empty__eq,axiom,
% 14.25/8.14 ( ( uminus5944136376168626660t_unit @ bot_bo3957492148770167129t_unit )
% 14.25/8.14 = top_to1996260823553986621t_unit ) ).
% 14.25/8.14
% 14.25/8.14 thf(fact_7091_Compl__empty__eq,axiom,
% 14.25/8.14 ( ( uminus_uminus_set_o @ bot_bot_set_o )
% 14.25/8.14 = top_top_set_o ) ).
% 14.25/8.14
% 14.25/8.14 thf(fact_7092_Compl__empty__eq,axiom,
% 14.25/8.14 ( ( uminus5710092332889474511et_nat @ bot_bot_set_nat )
% 14.25/8.14 = top_top_set_nat ) ).
% 14.25/8.14
% 14.25/8.14 thf(fact_7093_Compl__empty__eq,axiom,
% 14.25/8.14 ( ( uminus1532241313380277803et_int @ bot_bot_set_int )
% 14.25/8.14 = top_top_set_int ) ).
% 14.25/8.14
% 14.25/8.14 thf(fact_7094_Compl__partition2,axiom,
% 14.25/8.14 ! [A: set_Product_unit] :
% 14.25/8.14 ( ( sup_su793286257634532545t_unit @ ( uminus5944136376168626660t_unit @ A ) @ A )
% 14.25/8.14 = top_to1996260823553986621t_unit ) ).
% 14.25/8.14
% 14.25/8.14 thf(fact_7095_Compl__partition2,axiom,
% 14.25/8.14 ! [A: set_o] :
% 14.25/8.14 ( ( sup_sup_set_o @ ( uminus_uminus_set_o @ A ) @ A )
% 14.25/8.14 = top_top_set_o ) ).
% 14.25/8.14
% 14.25/8.14 thf(fact_7096_Compl__partition2,axiom,
% 14.25/8.14 ! [A: set_nat] :
% 14.25/8.14 ( ( sup_sup_set_nat @ ( uminus5710092332889474511et_nat @ A ) @ A )
% 14.25/8.14 = top_top_set_nat ) ).
% 14.25/8.14
% 14.25/8.14 thf(fact_7097_Compl__partition2,axiom,
% 14.25/8.14 ! [A: set_int] :
% 14.25/8.14 ( ( sup_sup_set_int @ ( uminus1532241313380277803et_int @ A ) @ A )
% 14.25/8.14 = top_top_set_int ) ).
% 14.25/8.14
% 14.25/8.14 thf(fact_7098_Compl__partition,axiom,
% 14.25/8.14 ! [A: set_Product_unit] :
% 14.25/8.14 ( ( sup_su793286257634532545t_unit @ A @ ( uminus5944136376168626660t_unit @ A ) )
% 14.25/8.14 = top_to1996260823553986621t_unit ) ).
% 14.25/8.14
% 14.25/8.14 thf(fact_7099_Compl__partition,axiom,
% 14.25/8.14 ! [A: set_o] :
% 14.25/8.14 ( ( sup_sup_set_o @ A @ ( uminus_uminus_set_o @ A ) )
% 14.25/8.14 = top_top_set_o ) ).
% 14.25/8.14
% 14.25/8.14 thf(fact_7100_Compl__partition,axiom,
% 14.25/8.14 ! [A: set_nat] :
% 14.25/8.14 ( ( sup_sup_set_nat @ A @ ( uminus5710092332889474511et_nat @ A ) )
% 14.25/8.14 = top_top_set_nat ) ).
% 14.25/8.14
% 14.25/8.14 thf(fact_7101_Compl__partition,axiom,
% 14.25/8.14 ! [A: set_int] :
% 14.25/8.14 ( ( sup_sup_set_int @ A @ ( uminus1532241313380277803et_int @ A ) )
% 14.25/8.14 = top_top_set_int ) ).
% 14.25/8.14
% 14.25/8.14 thf(fact_7102_Compl__eq__Diff__UNIV,axiom,
% 14.25/8.14 ( uminus5944136376168626660t_unit
% 14.25/8.14 = ( minus_6452836326544984404t_unit @ top_to1996260823553986621t_unit ) ) ).
% 14.25/8.14
% 14.25/8.14 thf(fact_7103_Compl__eq__Diff__UNIV,axiom,
% 14.25/8.14 ( uminus_uminus_set_o
% 14.25/8.14 = ( minus_minus_set_o @ top_top_set_o ) ) ).
% 14.25/8.14
% 14.25/8.14 thf(fact_7104_Compl__eq__Diff__UNIV,axiom,
% 14.25/8.14 ( uminus1532241313380277803et_int
% 14.25/8.14 = ( minus_minus_set_int @ top_top_set_int ) ) ).
% 14.25/8.14
% 14.25/8.14 thf(fact_7105_Compl__eq__Diff__UNIV,axiom,
% 14.25/8.14 ( uminus5710092332889474511et_nat
% 14.25/8.14 = ( minus_minus_set_nat @ top_top_set_nat ) ) ).
% 14.25/8.14
% 14.25/8.14 thf(fact_7106_Compl__Int,axiom,
% 14.25/8.14 ! [A: set_nat,B4: set_nat] :
% 14.25/8.14 ( ( uminus5710092332889474511et_nat @ ( inf_inf_set_nat @ A @ B4 ) )
% 14.25/8.14 = ( sup_sup_set_nat @ ( uminus5710092332889474511et_nat @ A ) @ ( uminus5710092332889474511et_nat @ B4 ) ) ) ).
% 14.25/8.14
% 14.25/8.14 thf(fact_7107_Compl__Un,axiom,
% 14.25/8.14 ! [A: set_nat,B4: set_nat] :
% 14.25/8.14 ( ( uminus5710092332889474511et_nat @ ( sup_sup_set_nat @ A @ B4 ) )
% 14.25/8.14 = ( inf_inf_set_nat @ ( uminus5710092332889474511et_nat @ A ) @ ( uminus5710092332889474511et_nat @ B4 ) ) ) ).
% 14.25/8.14
% 14.25/8.14 thf(fact_7108_gbinomial__negated__upper,axiom,
% 14.25/8.14 ( gbinomial_rat
% 14.25/8.14 = ( ^ [A6: rat,K5: nat] : ( times_times_rat @ ( power_power_rat @ ( uminus_uminus_rat @ one_one_rat ) @ K5 ) @ ( gbinomial_rat @ ( minus_minus_rat @ ( minus_minus_rat @ ( semiri681578069525770553at_rat @ K5 ) @ A6 ) @ one_one_rat ) @ K5 ) ) ) ) ).
% 14.25/8.14
% 14.25/8.14 thf(fact_7109_gbinomial__index__swap,axiom,
% 14.25/8.14 ! [K3: nat,N2: nat] :
% 14.25/8.14 ( ( times_times_rat @ ( power_power_rat @ ( uminus_uminus_rat @ one_one_rat ) @ K3 ) @ ( gbinomial_rat @ ( minus_minus_rat @ ( uminus_uminus_rat @ ( semiri681578069525770553at_rat @ N2 ) ) @ one_one_rat ) @ K3 ) )
% 14.25/8.14 = ( times_times_rat @ ( power_power_rat @ ( uminus_uminus_rat @ one_one_rat ) @ N2 ) @ ( gbinomial_rat @ ( minus_minus_rat @ ( uminus_uminus_rat @ ( semiri681578069525770553at_rat @ K3 ) ) @ one_one_rat ) @ N2 ) ) ) ).
% 14.25/8.14
% 14.25/8.14 thf(fact_7110_prod_OUnion__disjoint,axiom,
% 14.25/8.14 ! [C5: set_set_nat,G: nat > nat] :
% 14.25/8.14 ( ! [X3: set_nat] :
% 14.25/8.14 ( ( member_set_nat @ X3 @ C5 )
% 14.25/8.14 => ( finite_finite_nat @ X3 ) )
% 14.25/8.14 => ( ! [X3: set_nat] :
% 14.25/8.14 ( ( member_set_nat @ X3 @ C5 )
% 14.25/8.14 => ! [Xa3: set_nat] :
% 14.25/8.14 ( ( member_set_nat @ Xa3 @ C5 )
% 14.25/8.14 => ( ( X3 != Xa3 )
% 14.25/8.14 => ( ( inf_inf_set_nat @ X3 @ Xa3 )
% 14.25/8.14 = bot_bot_set_nat ) ) ) )
% 14.25/8.14 => ( ( groups708209901874060359at_nat @ G @ ( comple7399068483239264473et_nat @ C5 ) )
% 14.25/8.14 = ( comp_s5456809700045850692at_nat @ groups4248547760180025341at_nat @ groups708209901874060359at_nat @ G @ C5 ) ) ) ) ).
% 14.25/8.14
% 14.25/8.14 thf(fact_7111_Gcd__remove0__nat,axiom,
% 14.25/8.14 ! [M: set_nat] :
% 14.25/8.14 ( ( finite_finite_nat @ M )
% 14.25/8.14 => ( ( gcd_Gcd_nat @ M )
% 14.25/8.14 = ( gcd_Gcd_nat @ ( minus_minus_set_nat @ M @ ( insert_nat @ zero_zero_nat @ bot_bot_set_nat ) ) ) ) ) ).
% 14.25/8.14
% 14.25/8.14 thf(fact_7112_Gcd__eq__1__I,axiom,
% 14.25/8.14 ! [A2: nat,A: set_nat] :
% 14.25/8.14 ( ( dvd_dvd_nat @ A2 @ one_one_nat )
% 14.25/8.14 => ( ( member_nat @ A2 @ A )
% 14.25/8.14 => ( ( gcd_Gcd_nat @ A )
% 14.25/8.14 = one_one_nat ) ) ) ).
% 14.25/8.14
% 14.25/8.14 thf(fact_7113_Gcd__eq__1__I,axiom,
% 14.25/8.14 ! [A2: int,A: set_int] :
% 14.25/8.14 ( ( dvd_dvd_int @ A2 @ one_one_int )
% 14.25/8.14 => ( ( member_int @ A2 @ A )
% 14.25/8.14 => ( ( gcd_Gcd_int @ A )
% 14.25/8.14 = one_one_int ) ) ) ).
% 14.25/8.14
% 14.25/8.14 thf(fact_7114_prod__le__1,axiom,
% 14.25/8.14 ! [A: set_nat,F: nat > code_natural] :
% 14.25/8.14 ( ! [X3: nat] :
% 14.25/8.14 ( ( member_nat @ X3 @ A )
% 14.25/8.14 => ( ( ord_le1926595141338095240atural @ zero_z2226904508553997617atural @ ( F @ X3 ) )
% 14.25/8.14 & ( ord_le1926595141338095240atural @ ( F @ X3 ) @ one_one_Code_natural ) ) )
% 14.25/8.14 => ( ord_le1926595141338095240atural @ ( groups2279045934846249631atural @ F @ A ) @ one_one_Code_natural ) ) ).
% 14.25/8.14
% 14.25/8.14 thf(fact_7115_prod__le__1,axiom,
% 14.25/8.14 ! [A: set_int,F: int > code_natural] :
% 14.25/8.14 ( ! [X3: int] :
% 14.25/8.14 ( ( member_int @ X3 @ A )
% 14.25/8.14 => ( ( ord_le1926595141338095240atural @ zero_z2226904508553997617atural @ ( F @ X3 ) )
% 14.25/8.14 & ( ord_le1926595141338095240atural @ ( F @ X3 ) @ one_one_Code_natural ) ) )
% 14.25/8.14 => ( ord_le1926595141338095240atural @ ( groups2650699495083094267atural @ F @ A ) @ one_one_Code_natural ) ) ).
% 14.25/8.14
% 14.25/8.14 thf(fact_7116_prod__le__1,axiom,
% 14.25/8.14 ! [A: set_o,F: $o > code_natural] :
% 14.25/8.14 ( ! [X3: $o] :
% 14.25/8.14 ( ( member_o @ X3 @ A )
% 14.25/8.14 => ( ( ord_le1926595141338095240atural @ zero_z2226904508553997617atural @ ( F @ X3 ) )
% 14.25/8.14 & ( ord_le1926595141338095240atural @ ( F @ X3 ) @ one_one_Code_natural ) ) )
% 14.25/8.14 => ( ord_le1926595141338095240atural @ ( groups6518289543945209051atural @ F @ A ) @ one_one_Code_natural ) ) ).
% 14.25/8.14
% 14.25/8.14 thf(fact_7117_prod__le__1,axiom,
% 14.25/8.14 ! [A: set_Product_unit,F: product_unit > code_natural] :
% 14.25/8.14 ( ! [X3: product_unit] :
% 14.25/8.14 ( ( member_Product_unit @ X3 @ A )
% 14.25/8.14 => ( ( ord_le1926595141338095240atural @ zero_z2226904508553997617atural @ ( F @ X3 ) )
% 14.25/8.14 & ( ord_le1926595141338095240atural @ ( F @ X3 ) @ one_one_Code_natural ) ) )
% 14.25/8.14 => ( ord_le1926595141338095240atural @ ( groups7008954963700796496atural @ F @ A ) @ one_one_Code_natural ) ) ).
% 14.25/8.14
% 14.25/8.14 thf(fact_7118_prod__le__1,axiom,
% 14.25/8.14 ! [A: set_nat,F: nat > code_integer] :
% 14.25/8.14 ( ! [X3: nat] :
% 14.25/8.14 ( ( member_nat @ X3 @ A )
% 14.25/8.14 => ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ ( F @ X3 ) )
% 14.25/8.14 & ( ord_le3102999989581377725nteger @ ( F @ X3 ) @ one_one_Code_integer ) ) )
% 14.25/8.14 => ( ord_le3102999989581377725nteger @ ( groups3455450783089532116nteger @ F @ A ) @ one_one_Code_integer ) ) ).
% 14.25/8.14
% 14.25/8.14 thf(fact_7119_prod__le__1,axiom,
% 14.25/8.14 ! [A: set_int,F: int > code_integer] :
% 14.25/8.14 ( ! [X3: int] :
% 14.25/8.14 ( ( member_int @ X3 @ A )
% 14.25/8.14 => ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ ( F @ X3 ) )
% 14.25/8.14 & ( ord_le3102999989581377725nteger @ ( F @ X3 ) @ one_one_Code_integer ) ) )
% 14.25/8.14 => ( ord_le3102999989581377725nteger @ ( groups3827104343326376752nteger @ F @ A ) @ one_one_Code_integer ) ) ).
% 14.25/8.14
% 14.25/8.14 thf(fact_7120_prod__le__1,axiom,
% 14.25/8.14 ! [A: set_o,F: $o > code_integer] :
% 14.25/8.14 ( ! [X3: $o] :
% 14.25/8.14 ( ( member_o @ X3 @ A )
% 14.25/8.14 => ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ ( F @ X3 ) )
% 14.25/8.14 & ( ord_le3102999989581377725nteger @ ( F @ X3 ) @ one_one_Code_integer ) ) )
% 14.25/8.14 => ( ord_le3102999989581377725nteger @ ( groups7694694392188491536nteger @ F @ A ) @ one_one_Code_integer ) ) ).
% 14.25/8.14
% 14.25/8.14 thf(fact_7121_prod__le__1,axiom,
% 14.25/8.14 ! [A: set_Product_unit,F: product_unit > code_integer] :
% 14.25/8.14 ( ! [X3: product_unit] :
% 14.25/8.14 ( ( member_Product_unit @ X3 @ A )
% 14.25/8.14 => ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ ( F @ X3 ) )
% 14.25/8.14 & ( ord_le3102999989581377725nteger @ ( F @ X3 ) @ one_one_Code_integer ) ) )
% 14.25/8.14 => ( ord_le3102999989581377725nteger @ ( groups8185359811944078981nteger @ F @ A ) @ one_one_Code_integer ) ) ).
% 14.25/8.14
% 14.25/8.14 thf(fact_7122_prod__le__1,axiom,
% 14.25/8.14 ! [A: set_nat,F: nat > rat] :
% 14.25/8.14 ( ! [X3: nat] :
% 14.25/8.14 ( ( member_nat @ X3 @ A )
% 14.25/8.14 => ( ( ord_less_eq_rat @ zero_zero_rat @ ( F @ X3 ) )
% 14.25/8.14 & ( ord_less_eq_rat @ ( F @ X3 ) @ one_one_rat ) ) )
% 14.25/8.14 => ( ord_less_eq_rat @ ( groups73079841787564623at_rat @ F @ A ) @ one_one_rat ) ) ).
% 14.25/8.14
% 14.25/8.14 thf(fact_7123_prod__le__1,axiom,
% 14.25/8.14 ! [A: set_int,F: int > rat] :
% 14.25/8.14 ( ! [X3: int] :
% 14.25/8.14 ( ( member_int @ X3 @ A )
% 14.25/8.14 => ( ( ord_less_eq_rat @ zero_zero_rat @ ( F @ X3 ) )
% 14.25/8.14 & ( ord_less_eq_rat @ ( F @ X3 ) @ one_one_rat ) ) )
% 14.25/8.14 => ( ord_less_eq_rat @ ( groups1072433553688619179nt_rat @ F @ A ) @ one_one_rat ) ) ).
% 14.25/8.14
% 14.25/8.14 thf(fact_7124_prod_Orelated,axiom,
% 14.25/8.14 ! [R3: code_integer > code_integer > $o,S: set_nat,H2: nat > code_integer,G: nat > code_integer] :
% 14.25/8.14 ( ( R3 @ one_one_Code_integer @ one_one_Code_integer )
% 14.25/8.14 => ( ! [X12: code_integer,Y12: code_integer,X23: code_integer,Y23: code_integer] :
% 14.25/8.14 ( ( ( R3 @ X12 @ X23 )
% 14.25/8.14 & ( R3 @ Y12 @ Y23 ) )
% 14.25/8.14 => ( R3 @ ( times_3573771949741848930nteger @ X12 @ Y12 ) @ ( times_3573771949741848930nteger @ X23 @ Y23 ) ) )
% 14.25/8.14 => ( ( finite_finite_nat @ S )
% 14.25/8.14 => ( ! [X3: nat] :
% 14.25/8.14 ( ( member_nat @ X3 @ S )
% 14.25/8.14 => ( R3 @ ( H2 @ X3 ) @ ( G @ X3 ) ) )
% 14.25/8.14 => ( R3 @ ( groups3455450783089532116nteger @ H2 @ S ) @ ( groups3455450783089532116nteger @ G @ S ) ) ) ) ) ) ).
% 14.25/8.14
% 14.25/8.14 thf(fact_7125_prod_Orelated,axiom,
% 14.25/8.14 ! [R3: code_integer > code_integer > $o,S: set_Code_integer,H2: code_integer > code_integer,G: code_integer > code_integer] :
% 14.25/8.14 ( ( R3 @ one_one_Code_integer @ one_one_Code_integer )
% 14.25/8.14 => ( ! [X12: code_integer,Y12: code_integer,X23: code_integer,Y23: code_integer] :
% 14.25/8.14 ( ( ( R3 @ X12 @ X23 )
% 14.25/8.14 & ( R3 @ Y12 @ Y23 ) )
% 14.25/8.14 => ( R3 @ ( times_3573771949741848930nteger @ X12 @ Y12 ) @ ( times_3573771949741848930nteger @ X23 @ Y23 ) ) )
% 14.25/8.14 => ( ( finite6017078050557962740nteger @ S )
% 14.25/8.14 => ( ! [X3: code_integer] :
% 14.25/8.14 ( ( member_Code_integer @ X3 @ S )
% 14.25/8.14 => ( R3 @ ( H2 @ X3 ) @ ( G @ X3 ) ) )
% 14.25/8.14 => ( R3 @ ( groups3674199335183972705nteger @ H2 @ S ) @ ( groups3674199335183972705nteger @ G @ S ) ) ) ) ) ) ).
% 14.25/8.14
% 14.25/8.14 thf(fact_7126_prod_Orelated,axiom,
% 14.25/8.14 ! [R3: code_natural > code_natural > $o,S: set_nat,H2: nat > code_natural,G: nat > code_natural] :
% 14.25/8.14 ( ( R3 @ one_one_Code_natural @ one_one_Code_natural )
% 14.25/8.14 => ( ! [X12: code_natural,Y12: code_natural,X23: code_natural,Y23: code_natural] :
% 14.25/8.14 ( ( ( R3 @ X12 @ X23 )
% 14.25/8.14 & ( R3 @ Y12 @ Y23 ) )
% 14.25/8.14 => ( R3 @ ( times_2397367101498566445atural @ X12 @ Y12 ) @ ( times_2397367101498566445atural @ X23 @ Y23 ) ) )
% 14.25/8.14 => ( ( finite_finite_nat @ S )
% 14.25/8.14 => ( ! [X3: nat] :
% 14.25/8.14 ( ( member_nat @ X3 @ S )
% 14.25/8.14 => ( R3 @ ( H2 @ X3 ) @ ( G @ X3 ) ) )
% 14.25/8.14 => ( R3 @ ( groups2279045934846249631atural @ H2 @ S ) @ ( groups2279045934846249631atural @ G @ S ) ) ) ) ) ) ).
% 14.25/8.14
% 14.25/8.14 thf(fact_7127_prod_Orelated,axiom,
% 14.25/8.14 ! [R3: code_natural > code_natural > $o,S: set_Code_integer,H2: code_integer > code_natural,G: code_integer > code_natural] :
% 14.25/8.14 ( ( R3 @ one_one_Code_natural @ one_one_Code_natural )
% 14.25/8.14 => ( ! [X12: code_natural,Y12: code_natural,X23: code_natural,Y23: code_natural] :
% 14.25/8.14 ( ( ( R3 @ X12 @ X23 )
% 14.25/8.14 & ( R3 @ Y12 @ Y23 ) )
% 14.25/8.14 => ( R3 @ ( times_2397367101498566445atural @ X12 @ Y12 ) @ ( times_2397367101498566445atural @ X23 @ Y23 ) ) )
% 14.25/8.14 => ( ( finite6017078050557962740nteger @ S )
% 14.25/8.14 => ( ! [X3: code_integer] :
% 14.25/8.14 ( ( member_Code_integer @ X3 @ S )
% 14.25/8.14 => ( R3 @ ( H2 @ X3 ) @ ( G @ X3 ) ) )
% 14.25/8.14 => ( R3 @ ( groups2497794486940690220atural @ H2 @ S ) @ ( groups2497794486940690220atural @ G @ S ) ) ) ) ) ) ).
% 14.25/8.14
% 14.25/8.14 thf(fact_7128_prod_Orelated,axiom,
% 14.25/8.14 ! [R3: assn > assn > $o,S: set_nat,H2: nat > assn,G: nat > assn] :
% 14.25/8.14 ( ( R3 @ one_one_assn @ one_one_assn )
% 14.25/8.14 => ( ! [X12: assn,Y12: assn,X23: assn,Y23: assn] :
% 14.25/8.14 ( ( ( R3 @ X12 @ X23 )
% 14.25/8.14 & ( R3 @ Y12 @ Y23 ) )
% 14.25/8.14 => ( R3 @ ( times_times_assn @ X12 @ Y12 ) @ ( times_times_assn @ X23 @ Y23 ) ) )
% 14.25/8.14 => ( ( finite_finite_nat @ S )
% 14.25/8.14 => ( ! [X3: nat] :
% 14.25/8.14 ( ( member_nat @ X3 @ S )
% 14.25/8.14 => ( R3 @ ( H2 @ X3 ) @ ( G @ X3 ) ) )
% 14.25/8.14 => ( R3 @ ( groups6906906614972039071t_assn @ H2 @ S ) @ ( groups6906906614972039071t_assn @ G @ S ) ) ) ) ) ) ).
% 14.25/8.14
% 14.25/8.14 thf(fact_7129_prod_Orelated,axiom,
% 14.25/8.14 ! [R3: assn > assn > $o,S: set_Code_integer,H2: code_integer > assn,G: code_integer > assn] :
% 14.25/8.14 ( ( R3 @ one_one_assn @ one_one_assn )
% 14.25/8.14 => ( ! [X12: assn,Y12: assn,X23: assn,Y23: assn] :
% 14.25/8.14 ( ( ( R3 @ X12 @ X23 )
% 14.25/8.14 & ( R3 @ Y12 @ Y23 ) )
% 14.25/8.14 => ( R3 @ ( times_times_assn @ X12 @ Y12 ) @ ( times_times_assn @ X23 @ Y23 ) ) )
% 14.25/8.14 => ( ( finite6017078050557962740nteger @ S )
% 14.25/8.14 => ( ! [X3: code_integer] :
% 14.25/8.14 ( ( member_Code_integer @ X3 @ S )
% 14.25/8.14 => ( R3 @ ( H2 @ X3 ) @ ( G @ X3 ) ) )
% 14.25/8.14 => ( R3 @ ( groups1304777262505850412r_assn @ H2 @ S ) @ ( groups1304777262505850412r_assn @ G @ S ) ) ) ) ) ) ).
% 14.25/8.14
% 14.25/8.14 thf(fact_7130_prod_Orelated,axiom,
% 14.25/8.14 ! [R3: rat > rat > $o,S: set_nat,H2: nat > rat,G: nat > rat] :
% 14.25/8.14 ( ( R3 @ one_one_rat @ one_one_rat )
% 14.25/8.14 => ( ! [X12: rat,Y12: rat,X23: rat,Y23: rat] :
% 14.25/8.14 ( ( ( R3 @ X12 @ X23 )
% 14.25/8.14 & ( R3 @ Y12 @ Y23 ) )
% 14.25/8.14 => ( R3 @ ( times_times_rat @ X12 @ Y12 ) @ ( times_times_rat @ X23 @ Y23 ) ) )
% 14.25/8.14 => ( ( finite_finite_nat @ S )
% 14.25/8.14 => ( ! [X3: nat] :
% 14.25/8.14 ( ( member_nat @ X3 @ S )
% 14.25/8.14 => ( R3 @ ( H2 @ X3 ) @ ( G @ X3 ) ) )
% 14.25/8.14 => ( R3 @ ( groups73079841787564623at_rat @ H2 @ S ) @ ( groups73079841787564623at_rat @ G @ S ) ) ) ) ) ) ).
% 14.25/8.14
% 14.25/8.14 thf(fact_7131_prod_Orelated,axiom,
% 14.25/8.14 ! [R3: rat > rat > $o,S: set_Code_integer,H2: code_integer > rat,G: code_integer > rat] :
% 14.25/8.14 ( ( R3 @ one_one_rat @ one_one_rat )
% 14.25/8.14 => ( ! [X12: rat,Y12: rat,X23: rat,Y23: rat] :
% 14.25/8.14 ( ( ( R3 @ X12 @ X23 )
% 14.25/8.14 & ( R3 @ Y12 @ Y23 ) )
% 14.25/8.14 => ( R3 @ ( times_times_rat @ X12 @ Y12 ) @ ( times_times_rat @ X23 @ Y23 ) ) )
% 14.25/8.14 => ( ( finite6017078050557962740nteger @ S )
% 14.25/8.14 => ( ! [X3: code_integer] :
% 14.25/8.14 ( ( member_Code_integer @ X3 @ S )
% 14.25/8.14 => ( R3 @ ( H2 @ X3 ) @ ( G @ X3 ) ) )
% 14.25/8.14 => ( R3 @ ( groups2555765274223993564er_rat @ H2 @ S ) @ ( groups2555765274223993564er_rat @ G @ S ) ) ) ) ) ) ).
% 14.25/8.14
% 14.25/8.14 thf(fact_7132_prod_Orelated,axiom,
% 14.25/8.14 ! [R3: nat > nat > $o,S: set_Code_integer,H2: code_integer > nat,G: code_integer > nat] :
% 14.25/8.14 ( ( R3 @ one_one_nat @ one_one_nat )
% 14.25/8.14 => ( ! [X12: nat,Y12: nat,X23: nat,Y23: nat] :
% 14.25/8.14 ( ( ( R3 @ X12 @ X23 )
% 14.25/8.14 & ( R3 @ Y12 @ Y23 ) )
% 14.25/8.14 => ( R3 @ ( times_times_nat @ X12 @ Y12 ) @ ( times_times_nat @ X23 @ Y23 ) ) )
% 14.25/8.14 => ( ( finite6017078050557962740nteger @ S )
% 14.25/8.14 => ( ! [X3: code_integer] :
% 14.25/8.14 ( ( member_Code_integer @ X3 @ S )
% 14.25/8.14 => ( R3 @ ( H2 @ X3 ) @ ( G @ X3 ) ) )
% 14.25/8.14 => ( R3 @ ( groups3190895334310489300er_nat @ H2 @ S ) @ ( groups3190895334310489300er_nat @ G @ S ) ) ) ) ) ) ).
% 14.25/8.14
% 14.25/8.14 thf(fact_7133_prod_Orelated,axiom,
% 14.25/8.14 ! [R3: int > int > $o,S: set_nat,H2: nat > int,G: nat > int] :
% 14.25/8.14 ( ( R3 @ one_one_int @ one_one_int )
% 14.25/8.14 => ( ! [X12: int,Y12: int,X23: int,Y23: int] :
% 14.25/8.14 ( ( ( R3 @ X12 @ X23 )
% 14.25/8.14 & ( R3 @ Y12 @ Y23 ) )
% 14.25/8.14 => ( R3 @ ( times_times_int @ X12 @ Y12 ) @ ( times_times_int @ X23 @ Y23 ) ) )
% 14.25/8.14 => ( ( finite_finite_nat @ S )
% 14.25/8.14 => ( ! [X3: nat] :
% 14.25/8.14 ( ( member_nat @ X3 @ S )
% 14.25/8.14 => ( R3 @ ( H2 @ X3 ) @ ( G @ X3 ) ) )
% 14.25/8.14 => ( R3 @ ( groups705719431365010083at_int @ H2 @ S ) @ ( groups705719431365010083at_int @ G @ S ) ) ) ) ) ) ).
% 14.25/8.14
% 14.25/8.14 thf(fact_7134_prod_Oinsert__if,axiom,
% 14.25/8.14 ! [A: set_int,X: int,G: int > assn] :
% 14.25/8.14 ( ( finite_finite_int @ A )
% 14.25/8.14 => ( ( ( member_int @ X @ A )
% 14.25/8.14 => ( ( groups7882442080178216443t_assn @ G @ ( insert_int @ X @ A ) )
% 14.25/8.14 = ( groups7882442080178216443t_assn @ G @ A ) ) )
% 14.25/8.14 & ( ~ ( member_int @ X @ A )
% 14.25/8.14 => ( ( groups7882442080178216443t_assn @ G @ ( insert_int @ X @ A ) )
% 14.25/8.14 = ( times_times_assn @ ( G @ X ) @ ( groups7882442080178216443t_assn @ G @ A ) ) ) ) ) ) ).
% 14.25/8.14
% 14.25/8.14 thf(fact_7135_prod_Oinsert__if,axiom,
% 14.25/8.14 ! [A: set_o,X: $o,G: $o > assn] :
% 14.25/8.14 ( ( finite_finite_o @ A )
% 14.25/8.14 => ( ( ( member_o @ X @ A )
% 14.25/8.14 => ( ( groups5301882518646026715o_assn @ G @ ( insert_o @ X @ A ) )
% 14.25/8.14 = ( groups5301882518646026715o_assn @ G @ A ) ) )
% 14.25/8.14 & ( ~ ( member_o @ X @ A )
% 14.25/8.14 => ( ( groups5301882518646026715o_assn @ G @ ( insert_o @ X @ A ) )
% 14.25/8.14 = ( times_times_assn @ ( G @ X ) @ ( groups5301882518646026715o_assn @ G @ A ) ) ) ) ) ) ).
% 14.25/8.14
% 14.25/8.14 thf(fact_7136_prod_Oinsert__if,axiom,
% 14.25/8.14 ! [A: set_Product_unit,X: product_unit,G: product_unit > assn] :
% 14.25/8.14 ( ( finite4290736615968046902t_unit @ A )
% 14.25/8.14 => ( ( ( member_Product_unit @ X @ A )
% 14.25/8.14 => ( ( groups5360233303959380816t_assn @ G @ ( insert_Product_unit @ X @ A ) )
% 14.25/8.14 = ( groups5360233303959380816t_assn @ G @ A ) ) )
% 14.25/8.14 & ( ~ ( member_Product_unit @ X @ A )
% 14.25/8.14 => ( ( groups5360233303959380816t_assn @ G @ ( insert_Product_unit @ X @ A ) )
% 14.25/8.14 = ( times_times_assn @ ( G @ X ) @ ( groups5360233303959380816t_assn @ G @ A ) ) ) ) ) ) ).
% 14.25/8.14
% 14.25/8.14 thf(fact_7137_prod_Oinsert__if,axiom,
% 14.25/8.14 ! [A: set_nat,X: nat,G: nat > assn] :
% 14.25/8.14 ( ( finite_finite_nat @ A )
% 14.25/8.14 => ( ( ( member_nat @ X @ A )
% 14.25/8.14 => ( ( groups6906906614972039071t_assn @ G @ ( insert_nat @ X @ A ) )
% 14.25/8.14 = ( groups6906906614972039071t_assn @ G @ A ) ) )
% 14.25/8.14 & ( ~ ( member_nat @ X @ A )
% 14.25/8.14 => ( ( groups6906906614972039071t_assn @ G @ ( insert_nat @ X @ A ) )
% 14.25/8.14 = ( times_times_assn @ ( G @ X ) @ ( groups6906906614972039071t_assn @ G @ A ) ) ) ) ) ) ).
% 14.25/8.14
% 14.25/8.14 thf(fact_7138_prod_Oinsert__if,axiom,
% 14.25/8.14 ! [A: set_Code_integer,X: code_integer,G: code_integer > assn] :
% 14.25/8.14 ( ( finite6017078050557962740nteger @ A )
% 14.25/8.14 => ( ( ( member_Code_integer @ X @ A )
% 14.25/8.14 => ( ( groups1304777262505850412r_assn @ G @ ( insert_Code_integer @ X @ A ) )
% 14.25/8.14 = ( groups1304777262505850412r_assn @ G @ A ) ) )
% 14.25/8.14 & ( ~ ( member_Code_integer @ X @ A )
% 14.25/8.14 => ( ( groups1304777262505850412r_assn @ G @ ( insert_Code_integer @ X @ A ) )
% 14.25/8.14 = ( times_times_assn @ ( G @ X ) @ ( groups1304777262505850412r_assn @ G @ A ) ) ) ) ) ) ).
% 14.25/8.14
% 14.25/8.14 thf(fact_7139_prod_Oinsert__if,axiom,
% 14.25/8.14 ! [A: set_int,X: int,G: int > rat] :
% 14.25/8.14 ( ( finite_finite_int @ A )
% 14.25/8.14 => ( ( ( member_int @ X @ A )
% 14.25/8.14 => ( ( groups1072433553688619179nt_rat @ G @ ( insert_int @ X @ A ) )
% 14.25/8.14 = ( groups1072433553688619179nt_rat @ G @ A ) ) )
% 14.25/8.14 & ( ~ ( member_int @ X @ A )
% 14.25/8.14 => ( ( groups1072433553688619179nt_rat @ G @ ( insert_int @ X @ A ) )
% 14.25/8.14 = ( times_times_rat @ ( G @ X ) @ ( groups1072433553688619179nt_rat @ G @ A ) ) ) ) ) ) ).
% 14.25/8.14
% 14.25/8.14 thf(fact_7140_prod_Oinsert__if,axiom,
% 14.25/8.14 ! [A: set_o,X: $o,G: $o > rat] :
% 14.25/8.14 ( ( finite_finite_o @ A )
% 14.25/8.14 => ( ( ( member_o @ X @ A )
% 14.25/8.14 => ( ( groups2869687844427037835_o_rat @ G @ ( insert_o @ X @ A ) )
% 14.25/8.14 = ( groups2869687844427037835_o_rat @ G @ A ) ) )
% 14.25/8.14 & ( ~ ( member_o @ X @ A )
% 14.25/8.14 => ( ( groups2869687844427037835_o_rat @ G @ ( insert_o @ X @ A ) )
% 14.25/8.14 = ( times_times_rat @ ( G @ X ) @ ( groups2869687844427037835_o_rat @ G @ A ) ) ) ) ) ) ).
% 14.25/8.14
% 14.25/8.14 thf(fact_7141_prod_Oinsert__if,axiom,
% 14.25/8.14 ! [A: set_Product_unit,X: product_unit,G: product_unit > rat] :
% 14.25/8.14 ( ( finite4290736615968046902t_unit @ A )
% 14.25/8.14 => ( ( ( member_Product_unit @ X @ A )
% 14.25/8.14 => ( ( groups8984663225794451456it_rat @ G @ ( insert_Product_unit @ X @ A ) )
% 14.25/8.14 = ( groups8984663225794451456it_rat @ G @ A ) ) )
% 14.25/8.14 & ( ~ ( member_Product_unit @ X @ A )
% 14.25/8.14 => ( ( groups8984663225794451456it_rat @ G @ ( insert_Product_unit @ X @ A ) )
% 14.25/8.14 = ( times_times_rat @ ( G @ X ) @ ( groups8984663225794451456it_rat @ G @ A ) ) ) ) ) ) ).
% 14.25/8.14
% 14.25/8.14 thf(fact_7142_prod_Oinsert__if,axiom,
% 14.25/8.14 ! [A: set_nat,X: nat,G: nat > rat] :
% 14.25/8.14 ( ( finite_finite_nat @ A )
% 14.25/8.14 => ( ( ( member_nat @ X @ A )
% 14.25/8.14 => ( ( groups73079841787564623at_rat @ G @ ( insert_nat @ X @ A ) )
% 14.25/8.14 = ( groups73079841787564623at_rat @ G @ A ) ) )
% 14.25/8.14 & ( ~ ( member_nat @ X @ A )
% 14.25/8.14 => ( ( groups73079841787564623at_rat @ G @ ( insert_nat @ X @ A ) )
% 14.25/8.14 = ( times_times_rat @ ( G @ X ) @ ( groups73079841787564623at_rat @ G @ A ) ) ) ) ) ) ).
% 14.25/8.14
% 14.25/8.14 thf(fact_7143_prod_Oinsert__if,axiom,
% 14.25/8.14 ! [A: set_Code_integer,X: code_integer,G: code_integer > rat] :
% 14.25/8.14 ( ( finite6017078050557962740nteger @ A )
% 14.25/8.14 => ( ( ( member_Code_integer @ X @ A )
% 14.25/8.14 => ( ( groups2555765274223993564er_rat @ G @ ( insert_Code_integer @ X @ A ) )
% 14.25/8.14 = ( groups2555765274223993564er_rat @ G @ A ) ) )
% 14.25/8.14 & ( ~ ( member_Code_integer @ X @ A )
% 14.25/8.14 => ( ( groups2555765274223993564er_rat @ G @ ( insert_Code_integer @ X @ A ) )
% 14.25/8.14 = ( times_times_rat @ ( G @ X ) @ ( groups2555765274223993564er_rat @ G @ A ) ) ) ) ) ) ).
% 14.25/8.14
% 14.25/8.14 thf(fact_7144_prod_Oreindex__bij__witness__not__neutral,axiom,
% 14.25/8.14 ! [S4: set_int,T5: set_int,S: set_int,I2: int > int,J: int > int,T2: set_int,G: int > assn,H2: int > assn] :
% 14.25/8.14 ( ( finite_finite_int @ S4 )
% 14.25/8.14 => ( ( finite_finite_int @ T5 )
% 14.25/8.14 => ( ! [A4: int] :
% 14.25/8.14 ( ( member_int @ A4 @ ( minus_minus_set_int @ S @ S4 ) )
% 14.25/8.14 => ( ( I2 @ ( J @ A4 ) )
% 14.25/8.14 = A4 ) )
% 14.25/8.14 => ( ! [A4: int] :
% 14.25/8.14 ( ( member_int @ A4 @ ( minus_minus_set_int @ S @ S4 ) )
% 14.25/8.14 => ( member_int @ ( J @ A4 ) @ ( minus_minus_set_int @ T2 @ T5 ) ) )
% 14.25/8.14 => ( ! [B2: int] :
% 14.25/8.14 ( ( member_int @ B2 @ ( minus_minus_set_int @ T2 @ T5 ) )
% 14.25/8.14 => ( ( J @ ( I2 @ B2 ) )
% 14.25/8.14 = B2 ) )
% 14.25/8.14 => ( ! [B2: int] :
% 14.25/8.14 ( ( member_int @ B2 @ ( minus_minus_set_int @ T2 @ T5 ) )
% 14.25/8.14 => ( member_int @ ( I2 @ B2 ) @ ( minus_minus_set_int @ S @ S4 ) ) )
% 14.25/8.14 => ( ! [A4: int] :
% 14.25/8.14 ( ( member_int @ A4 @ S4 )
% 14.25/8.14 => ( ( G @ A4 )
% 14.25/8.14 = one_one_assn ) )
% 14.25/8.14 => ( ! [B2: int] :
% 14.25/8.14 ( ( member_int @ B2 @ T5 )
% 14.25/8.14 => ( ( H2 @ B2 )
% 14.25/8.14 = one_one_assn ) )
% 14.25/8.14 => ( ! [A4: int] :
% 14.25/8.14 ( ( member_int @ A4 @ S )
% 14.25/8.14 => ( ( H2 @ ( J @ A4 ) )
% 14.25/8.14 = ( G @ A4 ) ) )
% 14.25/8.14 => ( ( groups7882442080178216443t_assn @ G @ S )
% 14.25/8.14 = ( groups7882442080178216443t_assn @ H2 @ T2 ) ) ) ) ) ) ) ) ) ) ) ).
% 14.25/8.14
% 14.25/8.14 thf(fact_7145_prod_Oreindex__bij__witness__not__neutral,axiom,
% 14.25/8.14 ! [S4: set_int,T5: set_o,S: set_int,I2: $o > int,J: int > $o,T2: set_o,G: int > assn,H2: $o > assn] :
% 14.25/8.14 ( ( finite_finite_int @ S4 )
% 14.25/8.14 => ( ( finite_finite_o @ T5 )
% 14.25/8.14 => ( ! [A4: int] :
% 14.25/8.14 ( ( member_int @ A4 @ ( minus_minus_set_int @ S @ S4 ) )
% 14.25/8.14 => ( ( I2 @ ( J @ A4 ) )
% 14.25/8.14 = A4 ) )
% 14.25/8.14 => ( ! [A4: int] :
% 14.25/8.14 ( ( member_int @ A4 @ ( minus_minus_set_int @ S @ S4 ) )
% 14.25/8.14 => ( member_o @ ( J @ A4 ) @ ( minus_minus_set_o @ T2 @ T5 ) ) )
% 14.25/8.14 => ( ! [B2: $o] :
% 14.25/8.14 ( ( member_o @ B2 @ ( minus_minus_set_o @ T2 @ T5 ) )
% 14.25/8.14 => ( ( J @ ( I2 @ B2 ) )
% 14.25/8.14 = B2 ) )
% 14.25/8.14 => ( ! [B2: $o] :
% 14.25/8.14 ( ( member_o @ B2 @ ( minus_minus_set_o @ T2 @ T5 ) )
% 14.25/8.14 => ( member_int @ ( I2 @ B2 ) @ ( minus_minus_set_int @ S @ S4 ) ) )
% 14.25/8.14 => ( ! [A4: int] :
% 14.25/8.14 ( ( member_int @ A4 @ S4 )
% 14.25/8.14 => ( ( G @ A4 )
% 14.25/8.14 = one_one_assn ) )
% 14.25/8.14 => ( ! [B2: $o] :
% 14.25/8.14 ( ( member_o @ B2 @ T5 )
% 14.25/8.14 => ( ( H2 @ B2 )
% 14.25/8.14 = one_one_assn ) )
% 14.25/8.14 => ( ! [A4: int] :
% 14.25/8.14 ( ( member_int @ A4 @ S )
% 14.25/8.14 => ( ( H2 @ ( J @ A4 ) )
% 14.25/8.14 = ( G @ A4 ) ) )
% 14.25/8.14 => ( ( groups7882442080178216443t_assn @ G @ S )
% 14.25/8.14 = ( groups5301882518646026715o_assn @ H2 @ T2 ) ) ) ) ) ) ) ) ) ) ) ).
% 14.25/8.14
% 14.25/8.14 thf(fact_7146_prod_Oreindex__bij__witness__not__neutral,axiom,
% 14.25/8.14 ! [S4: set_int,T5: set_Product_unit,S: set_int,I2: product_unit > int,J: int > product_unit,T2: set_Product_unit,G: int > assn,H2: product_unit > assn] :
% 14.25/8.14 ( ( finite_finite_int @ S4 )
% 14.25/8.14 => ( ( finite4290736615968046902t_unit @ T5 )
% 14.25/8.14 => ( ! [A4: int] :
% 14.25/8.14 ( ( member_int @ A4 @ ( minus_minus_set_int @ S @ S4 ) )
% 14.25/8.14 => ( ( I2 @ ( J @ A4 ) )
% 14.25/8.14 = A4 ) )
% 14.25/8.14 => ( ! [A4: int] :
% 14.25/8.14 ( ( member_int @ A4 @ ( minus_minus_set_int @ S @ S4 ) )
% 14.25/8.14 => ( member_Product_unit @ ( J @ A4 ) @ ( minus_6452836326544984404t_unit @ T2 @ T5 ) ) )
% 14.25/8.14 => ( ! [B2: product_unit] :
% 14.25/8.14 ( ( member_Product_unit @ B2 @ ( minus_6452836326544984404t_unit @ T2 @ T5 ) )
% 14.25/8.14 => ( ( J @ ( I2 @ B2 ) )
% 14.25/8.14 = B2 ) )
% 14.25/8.14 => ( ! [B2: product_unit] :
% 14.25/8.14 ( ( member_Product_unit @ B2 @ ( minus_6452836326544984404t_unit @ T2 @ T5 ) )
% 14.25/8.14 => ( member_int @ ( I2 @ B2 ) @ ( minus_minus_set_int @ S @ S4 ) ) )
% 14.25/8.14 => ( ! [A4: int] :
% 14.25/8.14 ( ( member_int @ A4 @ S4 )
% 14.25/8.14 => ( ( G @ A4 )
% 14.25/8.14 = one_one_assn ) )
% 14.25/8.14 => ( ! [B2: product_unit] :
% 14.25/8.14 ( ( member_Product_unit @ B2 @ T5 )
% 14.25/8.14 => ( ( H2 @ B2 )
% 14.25/8.14 = one_one_assn ) )
% 14.25/8.14 => ( ! [A4: int] :
% 14.25/8.14 ( ( member_int @ A4 @ S )
% 14.25/8.14 => ( ( H2 @ ( J @ A4 ) )
% 14.25/8.14 = ( G @ A4 ) ) )
% 14.25/8.14 => ( ( groups7882442080178216443t_assn @ G @ S )
% 14.25/8.14 = ( groups5360233303959380816t_assn @ H2 @ T2 ) ) ) ) ) ) ) ) ) ) ) ).
% 14.25/8.14
% 14.25/8.14 thf(fact_7147_prod_Oreindex__bij__witness__not__neutral,axiom,
% 14.25/8.14 ! [S4: set_o,T5: set_int,S: set_o,I2: int > $o,J: $o > int,T2: set_int,G: $o > assn,H2: int > assn] :
% 14.25/8.14 ( ( finite_finite_o @ S4 )
% 14.25/8.14 => ( ( finite_finite_int @ T5 )
% 14.25/8.14 => ( ! [A4: $o] :
% 14.25/8.14 ( ( member_o @ A4 @ ( minus_minus_set_o @ S @ S4 ) )
% 14.25/8.14 => ( ( I2 @ ( J @ A4 ) )
% 14.25/8.14 = A4 ) )
% 14.25/8.14 => ( ! [A4: $o] :
% 14.25/8.14 ( ( member_o @ A4 @ ( minus_minus_set_o @ S @ S4 ) )
% 14.25/8.14 => ( member_int @ ( J @ A4 ) @ ( minus_minus_set_int @ T2 @ T5 ) ) )
% 14.25/8.14 => ( ! [B2: int] :
% 14.25/8.14 ( ( member_int @ B2 @ ( minus_minus_set_int @ T2 @ T5 ) )
% 14.25/8.14 => ( ( J @ ( I2 @ B2 ) )
% 14.25/8.14 = B2 ) )
% 14.25/8.14 => ( ! [B2: int] :
% 14.25/8.14 ( ( member_int @ B2 @ ( minus_minus_set_int @ T2 @ T5 ) )
% 14.25/8.14 => ( member_o @ ( I2 @ B2 ) @ ( minus_minus_set_o @ S @ S4 ) ) )
% 14.25/8.14 => ( ! [A4: $o] :
% 14.25/8.14 ( ( member_o @ A4 @ S4 )
% 14.25/8.14 => ( ( G @ A4 )
% 14.25/8.14 = one_one_assn ) )
% 14.25/8.14 => ( ! [B2: int] :
% 14.25/8.14 ( ( member_int @ B2 @ T5 )
% 14.25/8.14 => ( ( H2 @ B2 )
% 14.25/8.14 = one_one_assn ) )
% 14.25/8.14 => ( ! [A4: $o] :
% 14.25/8.14 ( ( member_o @ A4 @ S )
% 14.25/8.14 => ( ( H2 @ ( J @ A4 ) )
% 14.25/8.14 = ( G @ A4 ) ) )
% 14.25/8.14 => ( ( groups5301882518646026715o_assn @ G @ S )
% 14.25/8.14 = ( groups7882442080178216443t_assn @ H2 @ T2 ) ) ) ) ) ) ) ) ) ) ) ).
% 14.25/8.14
% 14.25/8.14 thf(fact_7148_prod_Oreindex__bij__witness__not__neutral,axiom,
% 14.25/8.14 ! [S4: set_o,T5: set_o,S: set_o,I2: $o > $o,J: $o > $o,T2: set_o,G: $o > assn,H2: $o > assn] :
% 14.25/8.14 ( ( finite_finite_o @ S4 )
% 14.25/8.14 => ( ( finite_finite_o @ T5 )
% 14.25/8.14 => ( ! [A4: $o] :
% 14.25/8.14 ( ( member_o @ A4 @ ( minus_minus_set_o @ S @ S4 ) )
% 14.25/8.14 => ( ( I2 @ ( J @ A4 ) )
% 14.25/8.14 = A4 ) )
% 14.25/8.14 => ( ! [A4: $o] :
% 14.25/8.14 ( ( member_o @ A4 @ ( minus_minus_set_o @ S @ S4 ) )
% 14.25/8.14 => ( member_o @ ( J @ A4 ) @ ( minus_minus_set_o @ T2 @ T5 ) ) )
% 14.25/8.14 => ( ! [B2: $o] :
% 14.25/8.14 ( ( member_o @ B2 @ ( minus_minus_set_o @ T2 @ T5 ) )
% 14.25/8.14 => ( ( J @ ( I2 @ B2 ) )
% 14.25/8.14 = B2 ) )
% 14.25/8.14 => ( ! [B2: $o] :
% 14.25/8.14 ( ( member_o @ B2 @ ( minus_minus_set_o @ T2 @ T5 ) )
% 14.25/8.14 => ( member_o @ ( I2 @ B2 ) @ ( minus_minus_set_o @ S @ S4 ) ) )
% 14.25/8.14 => ( ! [A4: $o] :
% 14.25/8.14 ( ( member_o @ A4 @ S4 )
% 14.25/8.14 => ( ( G @ A4 )
% 14.25/8.14 = one_one_assn ) )
% 14.25/8.14 => ( ! [B2: $o] :
% 14.25/8.14 ( ( member_o @ B2 @ T5 )
% 14.25/8.14 => ( ( H2 @ B2 )
% 14.25/8.14 = one_one_assn ) )
% 14.25/8.14 => ( ! [A4: $o] :
% 14.25/8.14 ( ( member_o @ A4 @ S )
% 14.25/8.14 => ( ( H2 @ ( J @ A4 ) )
% 14.25/8.14 = ( G @ A4 ) ) )
% 14.25/8.14 => ( ( groups5301882518646026715o_assn @ G @ S )
% 14.25/8.14 = ( groups5301882518646026715o_assn @ H2 @ T2 ) ) ) ) ) ) ) ) ) ) ) ).
% 14.25/8.14
% 14.25/8.14 thf(fact_7149_prod_Oreindex__bij__witness__not__neutral,axiom,
% 14.25/8.14 ! [S4: set_o,T5: set_Product_unit,S: set_o,I2: product_unit > $o,J: $o > product_unit,T2: set_Product_unit,G: $o > assn,H2: product_unit > assn] :
% 14.25/8.14 ( ( finite_finite_o @ S4 )
% 14.25/8.14 => ( ( finite4290736615968046902t_unit @ T5 )
% 14.25/8.14 => ( ! [A4: $o] :
% 14.25/8.14 ( ( member_o @ A4 @ ( minus_minus_set_o @ S @ S4 ) )
% 14.25/8.14 => ( ( I2 @ ( J @ A4 ) )
% 14.25/8.14 = A4 ) )
% 14.25/8.14 => ( ! [A4: $o] :
% 14.25/8.14 ( ( member_o @ A4 @ ( minus_minus_set_o @ S @ S4 ) )
% 14.25/8.14 => ( member_Product_unit @ ( J @ A4 ) @ ( minus_6452836326544984404t_unit @ T2 @ T5 ) ) )
% 14.25/8.14 => ( ! [B2: product_unit] :
% 14.25/8.14 ( ( member_Product_unit @ B2 @ ( minus_6452836326544984404t_unit @ T2 @ T5 ) )
% 14.25/8.14 => ( ( J @ ( I2 @ B2 ) )
% 14.25/8.14 = B2 ) )
% 14.25/8.14 => ( ! [B2: product_unit] :
% 14.25/8.14 ( ( member_Product_unit @ B2 @ ( minus_6452836326544984404t_unit @ T2 @ T5 ) )
% 14.25/8.14 => ( member_o @ ( I2 @ B2 ) @ ( minus_minus_set_o @ S @ S4 ) ) )
% 14.25/8.14 => ( ! [A4: $o] :
% 14.25/8.14 ( ( member_o @ A4 @ S4 )
% 14.25/8.14 => ( ( G @ A4 )
% 14.25/8.14 = one_one_assn ) )
% 14.25/8.14 => ( ! [B2: product_unit] :
% 14.25/8.14 ( ( member_Product_unit @ B2 @ T5 )
% 14.25/8.14 => ( ( H2 @ B2 )
% 14.25/8.14 = one_one_assn ) )
% 14.25/8.14 => ( ! [A4: $o] :
% 14.25/8.14 ( ( member_o @ A4 @ S )
% 14.25/8.14 => ( ( H2 @ ( J @ A4 ) )
% 14.25/8.14 = ( G @ A4 ) ) )
% 14.25/8.14 => ( ( groups5301882518646026715o_assn @ G @ S )
% 14.25/8.14 = ( groups5360233303959380816t_assn @ H2 @ T2 ) ) ) ) ) ) ) ) ) ) ) ).
% 14.25/8.14
% 14.25/8.14 thf(fact_7150_prod_Oreindex__bij__witness__not__neutral,axiom,
% 14.25/8.14 ! [S4: set_Product_unit,T5: set_int,S: set_Product_unit,I2: int > product_unit,J: product_unit > int,T2: set_int,G: product_unit > assn,H2: int > assn] :
% 14.25/8.14 ( ( finite4290736615968046902t_unit @ S4 )
% 14.25/8.14 => ( ( finite_finite_int @ T5 )
% 14.25/8.14 => ( ! [A4: product_unit] :
% 14.25/8.14 ( ( member_Product_unit @ A4 @ ( minus_6452836326544984404t_unit @ S @ S4 ) )
% 14.25/8.14 => ( ( I2 @ ( J @ A4 ) )
% 14.25/8.14 = A4 ) )
% 14.25/8.14 => ( ! [A4: product_unit] :
% 14.25/8.14 ( ( member_Product_unit @ A4 @ ( minus_6452836326544984404t_unit @ S @ S4 ) )
% 14.25/8.14 => ( member_int @ ( J @ A4 ) @ ( minus_minus_set_int @ T2 @ T5 ) ) )
% 14.25/8.14 => ( ! [B2: int] :
% 14.25/8.14 ( ( member_int @ B2 @ ( minus_minus_set_int @ T2 @ T5 ) )
% 14.25/8.14 => ( ( J @ ( I2 @ B2 ) )
% 14.25/8.14 = B2 ) )
% 14.25/8.14 => ( ! [B2: int] :
% 14.25/8.14 ( ( member_int @ B2 @ ( minus_minus_set_int @ T2 @ T5 ) )
% 14.25/8.14 => ( member_Product_unit @ ( I2 @ B2 ) @ ( minus_6452836326544984404t_unit @ S @ S4 ) ) )
% 14.25/8.14 => ( ! [A4: product_unit] :
% 14.25/8.14 ( ( member_Product_unit @ A4 @ S4 )
% 14.25/8.14 => ( ( G @ A4 )
% 14.25/8.14 = one_one_assn ) )
% 14.25/8.14 => ( ! [B2: int] :
% 14.25/8.14 ( ( member_int @ B2 @ T5 )
% 14.25/8.14 => ( ( H2 @ B2 )
% 14.25/8.14 = one_one_assn ) )
% 14.25/8.14 => ( ! [A4: product_unit] :
% 14.25/8.14 ( ( member_Product_unit @ A4 @ S )
% 14.25/8.14 => ( ( H2 @ ( J @ A4 ) )
% 14.25/8.14 = ( G @ A4 ) ) )
% 14.25/8.14 => ( ( groups5360233303959380816t_assn @ G @ S )
% 14.25/8.14 = ( groups7882442080178216443t_assn @ H2 @ T2 ) ) ) ) ) ) ) ) ) ) ) ).
% 14.25/8.14
% 14.25/8.14 thf(fact_7151_prod_Oreindex__bij__witness__not__neutral,axiom,
% 14.25/8.14 ! [S4: set_Product_unit,T5: set_o,S: set_Product_unit,I2: $o > product_unit,J: product_unit > $o,T2: set_o,G: product_unit > assn,H2: $o > assn] :
% 14.25/8.14 ( ( finite4290736615968046902t_unit @ S4 )
% 14.25/8.14 => ( ( finite_finite_o @ T5 )
% 14.25/8.14 => ( ! [A4: product_unit] :
% 14.25/8.14 ( ( member_Product_unit @ A4 @ ( minus_6452836326544984404t_unit @ S @ S4 ) )
% 14.25/8.14 => ( ( I2 @ ( J @ A4 ) )
% 14.25/8.14 = A4 ) )
% 14.25/8.14 => ( ! [A4: product_unit] :
% 14.25/8.14 ( ( member_Product_unit @ A4 @ ( minus_6452836326544984404t_unit @ S @ S4 ) )
% 14.25/8.14 => ( member_o @ ( J @ A4 ) @ ( minus_minus_set_o @ T2 @ T5 ) ) )
% 14.25/8.14 => ( ! [B2: $o] :
% 14.25/8.14 ( ( member_o @ B2 @ ( minus_minus_set_o @ T2 @ T5 ) )
% 14.25/8.14 => ( ( J @ ( I2 @ B2 ) )
% 14.25/8.14 = B2 ) )
% 14.25/8.14 => ( ! [B2: $o] :
% 14.25/8.14 ( ( member_o @ B2 @ ( minus_minus_set_o @ T2 @ T5 ) )
% 14.25/8.14 => ( member_Product_unit @ ( I2 @ B2 ) @ ( minus_6452836326544984404t_unit @ S @ S4 ) ) )
% 14.25/8.14 => ( ! [A4: product_unit] :
% 14.25/8.14 ( ( member_Product_unit @ A4 @ S4 )
% 14.25/8.14 => ( ( G @ A4 )
% 14.25/8.14 = one_one_assn ) )
% 14.25/8.14 => ( ! [B2: $o] :
% 14.25/8.14 ( ( member_o @ B2 @ T5 )
% 14.25/8.14 => ( ( H2 @ B2 )
% 14.25/8.14 = one_one_assn ) )
% 14.25/8.14 => ( ! [A4: product_unit] :
% 14.25/8.14 ( ( member_Product_unit @ A4 @ S )
% 14.25/8.14 => ( ( H2 @ ( J @ A4 ) )
% 14.25/8.14 = ( G @ A4 ) ) )
% 14.25/8.14 => ( ( groups5360233303959380816t_assn @ G @ S )
% 14.25/8.14 = ( groups5301882518646026715o_assn @ H2 @ T2 ) ) ) ) ) ) ) ) ) ) ) ).
% 14.25/8.14
% 14.25/8.14 thf(fact_7152_prod_Oreindex__bij__witness__not__neutral,axiom,
% 14.25/8.14 ! [S4: set_Product_unit,T5: set_Product_unit,S: set_Product_unit,I2: product_unit > product_unit,J: product_unit > product_unit,T2: set_Product_unit,G: product_unit > assn,H2: product_unit > assn] :
% 14.25/8.14 ( ( finite4290736615968046902t_unit @ S4 )
% 14.25/8.14 => ( ( finite4290736615968046902t_unit @ T5 )
% 14.25/8.14 => ( ! [A4: product_unit] :
% 14.25/8.14 ( ( member_Product_unit @ A4 @ ( minus_6452836326544984404t_unit @ S @ S4 ) )
% 14.25/8.14 => ( ( I2 @ ( J @ A4 ) )
% 14.25/8.14 = A4 ) )
% 14.25/8.14 => ( ! [A4: product_unit] :
% 14.25/8.14 ( ( member_Product_unit @ A4 @ ( minus_6452836326544984404t_unit @ S @ S4 ) )
% 14.25/8.14 => ( member_Product_unit @ ( J @ A4 ) @ ( minus_6452836326544984404t_unit @ T2 @ T5 ) ) )
% 14.25/8.14 => ( ! [B2: product_unit] :
% 14.25/8.14 ( ( member_Product_unit @ B2 @ ( minus_6452836326544984404t_unit @ T2 @ T5 ) )
% 14.25/8.14 => ( ( J @ ( I2 @ B2 ) )
% 14.25/8.14 = B2 ) )
% 14.25/8.14 => ( ! [B2: product_unit] :
% 14.25/8.14 ( ( member_Product_unit @ B2 @ ( minus_6452836326544984404t_unit @ T2 @ T5 ) )
% 14.25/8.14 => ( member_Product_unit @ ( I2 @ B2 ) @ ( minus_6452836326544984404t_unit @ S @ S4 ) ) )
% 14.25/8.14 => ( ! [A4: product_unit] :
% 14.25/8.14 ( ( member_Product_unit @ A4 @ S4 )
% 14.25/8.14 => ( ( G @ A4 )
% 14.25/8.14 = one_one_assn ) )
% 14.25/8.14 => ( ! [B2: product_unit] :
% 14.25/8.14 ( ( member_Product_unit @ B2 @ T5 )
% 14.25/8.14 => ( ( H2 @ B2 )
% 14.25/8.14 = one_one_assn ) )
% 14.25/8.14 => ( ! [A4: product_unit] :
% 14.25/8.14 ( ( member_Product_unit @ A4 @ S )
% 14.25/8.14 => ( ( H2 @ ( J @ A4 ) )
% 14.25/8.14 = ( G @ A4 ) ) )
% 14.25/8.14 => ( ( groups5360233303959380816t_assn @ G @ S )
% 14.25/8.14 = ( groups5360233303959380816t_assn @ H2 @ T2 ) ) ) ) ) ) ) ) ) ) ) ).
% 14.25/8.14
% 14.25/8.14 thf(fact_7153_prod_Oreindex__bij__witness__not__neutral,axiom,
% 14.25/8.14 ! [S4: set_int,T5: set_Code_integer,S: set_int,I2: code_integer > int,J: int > code_integer,T2: set_Code_integer,G: int > assn,H2: code_integer > assn] :
% 14.25/8.14 ( ( finite_finite_int @ S4 )
% 14.25/8.14 => ( ( finite6017078050557962740nteger @ T5 )
% 14.25/8.14 => ( ! [A4: int] :
% 14.25/8.14 ( ( member_int @ A4 @ ( minus_minus_set_int @ S @ S4 ) )
% 14.25/8.14 => ( ( I2 @ ( J @ A4 ) )
% 14.25/8.14 = A4 ) )
% 14.25/8.14 => ( ! [A4: int] :
% 14.25/8.14 ( ( member_int @ A4 @ ( minus_minus_set_int @ S @ S4 ) )
% 14.25/8.14 => ( member_Code_integer @ ( J @ A4 ) @ ( minus_2355218937544613996nteger @ T2 @ T5 ) ) )
% 14.25/8.14 => ( ! [B2: code_integer] :
% 14.25/8.14 ( ( member_Code_integer @ B2 @ ( minus_2355218937544613996nteger @ T2 @ T5 ) )
% 14.25/8.14 => ( ( J @ ( I2 @ B2 ) )
% 14.25/8.14 = B2 ) )
% 14.25/8.14 => ( ! [B2: code_integer] :
% 14.25/8.14 ( ( member_Code_integer @ B2 @ ( minus_2355218937544613996nteger @ T2 @ T5 ) )
% 14.25/8.14 => ( member_int @ ( I2 @ B2 ) @ ( minus_minus_set_int @ S @ S4 ) ) )
% 14.25/8.14 => ( ! [A4: int] :
% 14.25/8.14 ( ( member_int @ A4 @ S4 )
% 14.25/8.14 => ( ( G @ A4 )
% 14.25/8.14 = one_one_assn ) )
% 14.25/8.14 => ( ! [B2: code_integer] :
% 14.25/8.14 ( ( member_Code_integer @ B2 @ T5 )
% 14.25/8.14 => ( ( H2 @ B2 )
% 14.25/8.14 = one_one_assn ) )
% 14.25/8.14 => ( ! [A4: int] :
% 14.25/8.14 ( ( member_int @ A4 @ S )
% 14.25/8.14 => ( ( H2 @ ( J @ A4 ) )
% 14.25/8.14 = ( G @ A4 ) ) )
% 14.25/8.14 => ( ( groups7882442080178216443t_assn @ G @ S )
% 14.25/8.14 = ( groups1304777262505850412r_assn @ H2 @ T2 ) ) ) ) ) ) ) ) ) ) ) ).
% 14.25/8.14
% 14.25/8.14 thf(fact_7154_gbinomial__minus,axiom,
% 14.25/8.14 ! [A2: rat,K3: nat] :
% 14.25/8.14 ( ( gbinomial_rat @ ( uminus_uminus_rat @ A2 ) @ K3 )
% 14.25/8.14 = ( times_times_rat @ ( power_power_rat @ ( uminus_uminus_rat @ one_one_rat ) @ K3 ) @ ( gbinomial_rat @ ( minus_minus_rat @ ( plus_plus_rat @ A2 @ ( semiri681578069525770553at_rat @ K3 ) ) @ one_one_rat ) @ K3 ) ) ) ).
% 14.25/8.14
% 14.25/8.14 thf(fact_7155_le__minus__one__simps_I3_J,axiom,
% 14.25/8.14 ~ ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) ) ).
% 14.25/8.14
% 14.25/8.14 thf(fact_7156_le__minus__one__simps_I3_J,axiom,
% 14.25/8.14 ~ ( ord_less_eq_rat @ zero_zero_rat @ ( uminus_uminus_rat @ one_one_rat ) ) ).
% 14.25/8.14
% 14.25/8.14 thf(fact_7157_le__minus__one__simps_I3_J,axiom,
% 14.25/8.14 ~ ( ord_less_eq_int @ zero_zero_int @ ( uminus_uminus_int @ one_one_int ) ) ).
% 14.25/8.14
% 14.25/8.14 thf(fact_7158_le__minus__one__simps_I1_J,axiom,
% 14.25/8.14 ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ zero_z3403309356797280102nteger ).
% 14.25/8.14
% 14.25/8.14 thf(fact_7159_le__minus__one__simps_I1_J,axiom,
% 14.25/8.14 ord_less_eq_rat @ ( uminus_uminus_rat @ one_one_rat ) @ zero_zero_rat ).
% 14.25/8.14
% 14.25/8.14 thf(fact_7160_le__minus__one__simps_I1_J,axiom,
% 14.25/8.14 ord_less_eq_int @ ( uminus_uminus_int @ one_one_int ) @ zero_zero_int ).
% 14.25/8.14
% 14.25/8.14 thf(fact_7161_less__minus__one__simps_I1_J,axiom,
% 14.25/8.14 ord_less_rat @ ( uminus_uminus_rat @ one_one_rat ) @ zero_zero_rat ).
% 14.25/8.14
% 14.25/8.14 thf(fact_7162_less__minus__one__simps_I1_J,axiom,
% 14.25/8.14 ord_less_int @ ( uminus_uminus_int @ one_one_int ) @ zero_zero_int ).
% 14.25/8.14
% 14.25/8.14 thf(fact_7163_less__minus__one__simps_I1_J,axiom,
% 14.25/8.14 ord_le6747313008572928689nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ zero_z3403309356797280102nteger ).
% 14.25/8.14
% 14.25/8.14 thf(fact_7164_less__minus__one__simps_I3_J,axiom,
% 14.25/8.14 ~ ( ord_less_rat @ zero_zero_rat @ ( uminus_uminus_rat @ one_one_rat ) ) ).
% 14.25/8.14
% 14.25/8.14 thf(fact_7165_less__minus__one__simps_I3_J,axiom,
% 14.25/8.14 ~ ( ord_less_int @ zero_zero_int @ ( uminus_uminus_int @ one_one_int ) ) ).
% 14.25/8.14
% 14.25/8.14 thf(fact_7166_less__minus__one__simps_I3_J,axiom,
% 14.25/8.14 ~ ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) ) ).
% 14.25/8.14
% 14.25/8.14 thf(fact_7167_inf__shunt,axiom,
% 14.25/8.14 ! [X: set_Product_unit,Y: set_Product_unit] :
% 14.25/8.14 ( ( ( inf_in4660618365625256667t_unit @ X @ Y )
% 14.25/8.14 = bot_bo3957492148770167129t_unit )
% 14.25/8.14 = ( ord_le3507040750410214029t_unit @ X @ ( uminus5944136376168626660t_unit @ Y ) ) ) ).
% 14.25/8.14
% 14.25/8.14 thf(fact_7168_inf__shunt,axiom,
% 14.25/8.14 ! [X: set_o,Y: set_o] :
% 14.25/8.14 ( ( ( inf_inf_set_o @ X @ Y )
% 14.25/8.14 = bot_bot_set_o )
% 14.25/8.14 = ( ord_less_eq_set_o @ X @ ( uminus_uminus_set_o @ Y ) ) ) ).
% 14.25/8.14
% 14.25/8.14 thf(fact_7169_inf__shunt,axiom,
% 14.25/8.14 ! [X: set_int,Y: set_int] :
% 14.25/8.14 ( ( ( inf_inf_set_int @ X @ Y )
% 14.25/8.14 = bot_bot_set_int )
% 14.25/8.14 = ( ord_less_eq_set_int @ X @ ( uminus1532241313380277803et_int @ Y ) ) ) ).
% 14.25/8.14
% 14.25/8.14 thf(fact_7170_inf__shunt,axiom,
% 14.25/8.14 ! [X: set_Pr4532377907799695533_nat_o,Y: set_Pr4532377907799695533_nat_o] :
% 14.25/8.14 ( ( ( inf_in1906310914598751387_nat_o @ X @ Y )
% 14.25/8.14 = bot_bo7824918357723582233_nat_o )
% 14.25/8.14 = ( ord_le2965882846123202637_nat_o @ X @ ( uminus5254974436814262692_nat_o @ Y ) ) ) ).
% 14.25/8.14
% 14.25/8.14 thf(fact_7171_inf__shunt,axiom,
% 14.25/8.14 ! [X: assn,Y: assn] :
% 14.25/8.14 ( ( ( inf_inf_assn @ X @ Y )
% 14.25/8.14 = bot_bot_assn )
% 14.25/8.14 = ( ord_less_eq_assn @ X @ ( uminus_uminus_assn @ Y ) ) ) ).
% 14.25/8.14
% 14.25/8.14 thf(fact_7172_inf__shunt,axiom,
% 14.25/8.14 ! [X: product_unit,Y: product_unit] :
% 14.25/8.14 ( ( ( inf_inf_Product_unit @ X @ Y )
% 14.25/8.14 = bot_bot_Product_unit )
% 14.25/8.14 = ( ord_le3221252021190050221t_unit @ X @ ( uminus2952777764628376836t_unit @ Y ) ) ) ).
% 14.25/8.14
% 14.25/8.14 thf(fact_7173_inf__shunt,axiom,
% 14.25/8.14 ! [X: set_nat,Y: set_nat] :
% 14.25/8.14 ( ( ( inf_inf_set_nat @ X @ Y )
% 14.25/8.14 = bot_bot_set_nat )
% 14.25/8.14 = ( ord_less_eq_set_nat @ X @ ( uminus5710092332889474511et_nat @ Y ) ) ) ).
% 14.25/8.14
% 14.25/8.14 thf(fact_7174_power__minus,axiom,
% 14.25/8.14 ! [A2: rat,N2: nat] :
% 14.25/8.14 ( ( power_power_rat @ ( uminus_uminus_rat @ A2 ) @ N2 )
% 14.25/8.14 = ( times_times_rat @ ( power_power_rat @ ( uminus_uminus_rat @ one_one_rat ) @ N2 ) @ ( power_power_rat @ A2 @ N2 ) ) ) ).
% 14.25/8.14
% 14.25/8.14 thf(fact_7175_power__minus,axiom,
% 14.25/8.14 ! [A2: int,N2: nat] :
% 14.25/8.14 ( ( power_power_int @ ( uminus_uminus_int @ A2 ) @ N2 )
% 14.25/8.14 = ( times_times_int @ ( power_power_int @ ( uminus_uminus_int @ one_one_int ) @ N2 ) @ ( power_power_int @ A2 @ N2 ) ) ) ).
% 14.25/8.14
% 14.25/8.14 thf(fact_7176_power__minus,axiom,
% 14.25/8.14 ! [A2: code_integer,N2: nat] :
% 14.25/8.14 ( ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ A2 ) @ N2 )
% 14.25/8.14 = ( times_3573771949741848930nteger @ ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ N2 ) @ ( power_8256067586552552935nteger @ A2 @ N2 ) ) ) ).
% 14.25/8.14
% 14.25/8.14 thf(fact_7177_disjoint__eq__subset__Compl,axiom,
% 14.25/8.14 ! [A: set_Product_unit,B4: set_Product_unit] :
% 14.25/8.14 ( ( ( inf_in4660618365625256667t_unit @ A @ B4 )
% 14.25/8.14 = bot_bo3957492148770167129t_unit )
% 14.25/8.14 = ( ord_le3507040750410214029t_unit @ A @ ( uminus5944136376168626660t_unit @ B4 ) ) ) ).
% 14.25/8.14
% 14.25/8.14 thf(fact_7178_disjoint__eq__subset__Compl,axiom,
% 14.25/8.14 ! [A: set_o,B4: set_o] :
% 14.25/8.14 ( ( ( inf_inf_set_o @ A @ B4 )
% 14.25/8.14 = bot_bot_set_o )
% 14.25/8.14 = ( ord_less_eq_set_o @ A @ ( uminus_uminus_set_o @ B4 ) ) ) ).
% 14.25/8.14
% 14.25/8.14 thf(fact_7179_disjoint__eq__subset__Compl,axiom,
% 14.25/8.14 ! [A: set_int,B4: set_int] :
% 14.25/8.14 ( ( ( inf_inf_set_int @ A @ B4 )
% 14.25/8.14 = bot_bot_set_int )
% 14.25/8.14 = ( ord_less_eq_set_int @ A @ ( uminus1532241313380277803et_int @ B4 ) ) ) ).
% 14.25/8.14
% 14.25/8.14 thf(fact_7180_disjoint__eq__subset__Compl,axiom,
% 14.25/8.14 ! [A: set_Pr4532377907799695533_nat_o,B4: set_Pr4532377907799695533_nat_o] :
% 14.25/8.14 ( ( ( inf_in1906310914598751387_nat_o @ A @ B4 )
% 14.25/8.14 = bot_bo7824918357723582233_nat_o )
% 14.25/8.14 = ( ord_le2965882846123202637_nat_o @ A @ ( uminus5254974436814262692_nat_o @ B4 ) ) ) ).
% 14.25/8.14
% 14.25/8.14 thf(fact_7181_disjoint__eq__subset__Compl,axiom,
% 14.25/8.14 ! [A: set_nat,B4: set_nat] :
% 14.25/8.14 ( ( ( inf_inf_set_nat @ A @ B4 )
% 14.25/8.14 = bot_bot_set_nat )
% 14.25/8.14 = ( ord_less_eq_set_nat @ A @ ( uminus5710092332889474511et_nat @ B4 ) ) ) ).
% 14.25/8.14
% 14.25/8.14 thf(fact_7182_Compl__insert,axiom,
% 14.25/8.14 ! [X: product_unit,A: set_Product_unit] :
% 14.25/8.14 ( ( uminus5944136376168626660t_unit @ ( insert_Product_unit @ X @ A ) )
% 14.25/8.14 = ( minus_6452836326544984404t_unit @ ( uminus5944136376168626660t_unit @ A ) @ ( insert_Product_unit @ X @ bot_bo3957492148770167129t_unit ) ) ) ).
% 14.25/8.14
% 14.25/8.14 thf(fact_7183_Compl__insert,axiom,
% 14.25/8.14 ! [X: $o,A: set_o] :
% 14.25/8.14 ( ( uminus_uminus_set_o @ ( insert_o @ X @ A ) )
% 14.25/8.14 = ( minus_minus_set_o @ ( uminus_uminus_set_o @ A ) @ ( insert_o @ X @ bot_bot_set_o ) ) ) ).
% 14.25/8.14
% 14.25/8.14 thf(fact_7184_Compl__insert,axiom,
% 14.25/8.14 ! [X: int,A: set_int] :
% 14.25/8.14 ( ( uminus1532241313380277803et_int @ ( insert_int @ X @ A ) )
% 14.25/8.14 = ( minus_minus_set_int @ ( uminus1532241313380277803et_int @ A ) @ ( insert_int @ X @ bot_bot_set_int ) ) ) ).
% 14.25/8.14
% 14.25/8.14 thf(fact_7185_Compl__insert,axiom,
% 14.25/8.14 ! [X: produc3658429121746597890et_nat > $o,A: set_Pr4532377907799695533_nat_o] :
% 14.25/8.14 ( ( uminus5254974436814262692_nat_o @ ( insert5175938949040314269_nat_o @ X @ A ) )
% 14.25/8.14 = ( minus_1801376950450012436_nat_o @ ( uminus5254974436814262692_nat_o @ A ) @ ( insert5175938949040314269_nat_o @ X @ bot_bo7824918357723582233_nat_o ) ) ) ).
% 14.25/8.14
% 14.25/8.14 thf(fact_7186_Compl__insert,axiom,
% 14.25/8.14 ! [X: nat,A: set_nat] :
% 14.25/8.14 ( ( uminus5710092332889474511et_nat @ ( insert_nat @ X @ A ) )
% 14.25/8.14 = ( minus_minus_set_nat @ ( uminus5710092332889474511et_nat @ A ) @ ( insert_nat @ X @ bot_bot_set_nat ) ) ) ).
% 14.25/8.14
% 14.25/8.14 thf(fact_7187_Image__subset__eq,axiom,
% 14.25/8.14 ! [R2: set_Pr1261947904930325089at_nat,A: set_nat,B4: set_nat] :
% 14.25/8.14 ( ( ord_less_eq_set_nat @ ( image_nat_nat @ R2 @ A ) @ B4 )
% 14.25/8.14 = ( ord_less_eq_set_nat @ A @ ( uminus5710092332889474511et_nat @ ( image_nat_nat @ ( converse_nat_nat @ R2 ) @ ( uminus5710092332889474511et_nat @ B4 ) ) ) ) ) ).
% 14.25/8.14
% 14.25/8.14 thf(fact_7188_less__1__prod2,axiom,
% 14.25/8.14 ! [I: set_int,I2: int,F: int > code_integer] :
% 14.25/8.14 ( ( finite_finite_int @ I )
% 14.25/8.14 => ( ( member_int @ I2 @ I )
% 14.25/8.14 => ( ( ord_le6747313008572928689nteger @ one_one_Code_integer @ ( F @ I2 ) )
% 14.25/8.14 => ( ! [I3: int] :
% 14.25/8.14 ( ( member_int @ I3 @ I )
% 14.25/8.14 => ( ord_le3102999989581377725nteger @ one_one_Code_integer @ ( F @ I3 ) ) )
% 14.25/8.14 => ( ord_le6747313008572928689nteger @ one_one_Code_integer @ ( groups3827104343326376752nteger @ F @ I ) ) ) ) ) ) ).
% 14.25/8.14
% 14.25/8.14 thf(fact_7189_less__1__prod2,axiom,
% 14.25/8.14 ! [I: set_o,I2: $o,F: $o > code_integer] :
% 14.25/8.14 ( ( finite_finite_o @ I )
% 14.25/8.14 => ( ( member_o @ I2 @ I )
% 14.25/8.14 => ( ( ord_le6747313008572928689nteger @ one_one_Code_integer @ ( F @ I2 ) )
% 14.25/8.14 => ( ! [I3: $o] :
% 14.25/8.14 ( ( member_o @ I3 @ I )
% 14.25/8.14 => ( ord_le3102999989581377725nteger @ one_one_Code_integer @ ( F @ I3 ) ) )
% 14.25/8.14 => ( ord_le6747313008572928689nteger @ one_one_Code_integer @ ( groups7694694392188491536nteger @ F @ I ) ) ) ) ) ) ).
% 14.25/8.14
% 14.25/8.14 thf(fact_7190_less__1__prod2,axiom,
% 14.25/8.14 ! [I: set_Product_unit,I2: product_unit,F: product_unit > code_integer] :
% 14.25/8.14 ( ( finite4290736615968046902t_unit @ I )
% 14.25/8.14 => ( ( member_Product_unit @ I2 @ I )
% 14.25/8.14 => ( ( ord_le6747313008572928689nteger @ one_one_Code_integer @ ( F @ I2 ) )
% 14.25/8.14 => ( ! [I3: product_unit] :
% 14.25/8.14 ( ( member_Product_unit @ I3 @ I )
% 14.25/8.14 => ( ord_le3102999989581377725nteger @ one_one_Code_integer @ ( F @ I3 ) ) )
% 14.25/8.14 => ( ord_le6747313008572928689nteger @ one_one_Code_integer @ ( groups8185359811944078981nteger @ F @ I ) ) ) ) ) ) ).
% 14.25/8.14
% 14.25/8.14 thf(fact_7191_less__1__prod2,axiom,
% 14.25/8.14 ! [I: set_nat,I2: nat,F: nat > code_integer] :
% 14.25/8.14 ( ( finite_finite_nat @ I )
% 14.25/8.14 => ( ( member_nat @ I2 @ I )
% 14.25/8.14 => ( ( ord_le6747313008572928689nteger @ one_one_Code_integer @ ( F @ I2 ) )
% 14.25/8.14 => ( ! [I3: nat] :
% 14.25/8.14 ( ( member_nat @ I3 @ I )
% 14.25/8.14 => ( ord_le3102999989581377725nteger @ one_one_Code_integer @ ( F @ I3 ) ) )
% 14.25/8.14 => ( ord_le6747313008572928689nteger @ one_one_Code_integer @ ( groups3455450783089532116nteger @ F @ I ) ) ) ) ) ) ).
% 14.25/8.14
% 14.25/8.14 thf(fact_7192_less__1__prod2,axiom,
% 14.25/8.14 ! [I: set_Code_integer,I2: code_integer,F: code_integer > code_integer] :
% 14.25/8.14 ( ( finite6017078050557962740nteger @ I )
% 14.25/8.14 => ( ( member_Code_integer @ I2 @ I )
% 14.25/8.14 => ( ( ord_le6747313008572928689nteger @ one_one_Code_integer @ ( F @ I2 ) )
% 14.25/8.14 => ( ! [I3: code_integer] :
% 14.25/8.14 ( ( member_Code_integer @ I3 @ I )
% 14.25/8.14 => ( ord_le3102999989581377725nteger @ one_one_Code_integer @ ( F @ I3 ) ) )
% 14.25/8.14 => ( ord_le6747313008572928689nteger @ one_one_Code_integer @ ( groups3674199335183972705nteger @ F @ I ) ) ) ) ) ) ).
% 14.25/8.14
% 14.25/8.14 thf(fact_7193_less__1__prod2,axiom,
% 14.25/8.14 ! [I: set_int,I2: int,F: int > rat] :
% 14.25/8.14 ( ( finite_finite_int @ I )
% 14.25/8.14 => ( ( member_int @ I2 @ I )
% 14.25/8.14 => ( ( ord_less_rat @ one_one_rat @ ( F @ I2 ) )
% 14.25/8.14 => ( ! [I3: int] :
% 14.25/8.14 ( ( member_int @ I3 @ I )
% 14.25/8.14 => ( ord_less_eq_rat @ one_one_rat @ ( F @ I3 ) ) )
% 14.25/8.14 => ( ord_less_rat @ one_one_rat @ ( groups1072433553688619179nt_rat @ F @ I ) ) ) ) ) ) ).
% 14.25/8.14
% 14.25/8.14 thf(fact_7194_less__1__prod2,axiom,
% 14.25/8.14 ! [I: set_o,I2: $o,F: $o > rat] :
% 14.25/8.14 ( ( finite_finite_o @ I )
% 14.25/8.14 => ( ( member_o @ I2 @ I )
% 14.25/8.14 => ( ( ord_less_rat @ one_one_rat @ ( F @ I2 ) )
% 14.25/8.14 => ( ! [I3: $o] :
% 14.25/8.14 ( ( member_o @ I3 @ I )
% 14.25/8.14 => ( ord_less_eq_rat @ one_one_rat @ ( F @ I3 ) ) )
% 14.25/8.14 => ( ord_less_rat @ one_one_rat @ ( groups2869687844427037835_o_rat @ F @ I ) ) ) ) ) ) ).
% 14.25/8.14
% 14.25/8.14 thf(fact_7195_less__1__prod2,axiom,
% 14.25/8.14 ! [I: set_Product_unit,I2: product_unit,F: product_unit > rat] :
% 14.25/8.14 ( ( finite4290736615968046902t_unit @ I )
% 14.25/8.14 => ( ( member_Product_unit @ I2 @ I )
% 14.25/8.14 => ( ( ord_less_rat @ one_one_rat @ ( F @ I2 ) )
% 14.25/8.14 => ( ! [I3: product_unit] :
% 14.25/8.14 ( ( member_Product_unit @ I3 @ I )
% 14.25/8.14 => ( ord_less_eq_rat @ one_one_rat @ ( F @ I3 ) ) )
% 14.25/8.14 => ( ord_less_rat @ one_one_rat @ ( groups8984663225794451456it_rat @ F @ I ) ) ) ) ) ) ).
% 14.25/8.14
% 14.25/8.14 thf(fact_7196_less__1__prod2,axiom,
% 14.25/8.14 ! [I: set_nat,I2: nat,F: nat > rat] :
% 14.25/8.14 ( ( finite_finite_nat @ I )
% 14.25/8.14 => ( ( member_nat @ I2 @ I )
% 14.25/8.14 => ( ( ord_less_rat @ one_one_rat @ ( F @ I2 ) )
% 14.25/8.14 => ( ! [I3: nat] :
% 14.25/8.14 ( ( member_nat @ I3 @ I )
% 14.25/8.14 => ( ord_less_eq_rat @ one_one_rat @ ( F @ I3 ) ) )
% 14.25/8.14 => ( ord_less_rat @ one_one_rat @ ( groups73079841787564623at_rat @ F @ I ) ) ) ) ) ) ).
% 14.25/8.14
% 14.25/8.14 thf(fact_7197_less__1__prod2,axiom,
% 14.25/8.14 ! [I: set_Code_integer,I2: code_integer,F: code_integer > rat] :
% 14.25/8.14 ( ( finite6017078050557962740nteger @ I )
% 14.25/8.14 => ( ( member_Code_integer @ I2 @ I )
% 14.25/8.14 => ( ( ord_less_rat @ one_one_rat @ ( F @ I2 ) )
% 14.25/8.14 => ( ! [I3: code_integer] :
% 14.25/8.14 ( ( member_Code_integer @ I3 @ I )
% 14.25/8.14 => ( ord_less_eq_rat @ one_one_rat @ ( F @ I3 ) ) )
% 14.25/8.14 => ( ord_less_rat @ one_one_rat @ ( groups2555765274223993564er_rat @ F @ I ) ) ) ) ) ) ).
% 14.25/8.14
% 14.25/8.14 thf(fact_7198_less__1__prod,axiom,
% 14.25/8.14 ! [I: set_Code_integer,F: code_integer > int] :
% 14.25/8.14 ( ( finite6017078050557962740nteger @ I )
% 14.25/8.14 => ( ( I != bot_bo3990330152332043303nteger )
% 14.25/8.14 => ( ! [I3: code_integer] :
% 14.25/8.14 ( ( member_Code_integer @ I3 @ I )
% 14.25/8.14 => ( ord_less_int @ one_one_int @ ( F @ I3 ) ) )
% 14.25/8.14 => ( ord_less_int @ one_one_int @ ( groups3188404863801439024er_int @ F @ I ) ) ) ) ) ).
% 14.25/8.14
% 14.25/8.14 thf(fact_7199_less__1__prod,axiom,
% 14.25/8.14 ! [I: set_Product_unit,F: product_unit > int] :
% 14.25/8.14 ( ( finite4290736615968046902t_unit @ I )
% 14.25/8.14 => ( ( I != bot_bo3957492148770167129t_unit )
% 14.25/8.14 => ( ! [I3: product_unit] :
% 14.25/8.14 ( ( member_Product_unit @ I3 @ I )
% 14.25/8.14 => ( ord_less_int @ one_one_int @ ( F @ I3 ) ) )
% 14.25/8.14 => ( ord_less_int @ one_one_int @ ( groups393930778517121108it_int @ F @ I ) ) ) ) ) ).
% 14.25/8.14
% 14.25/8.14 thf(fact_7200_less__1__prod,axiom,
% 14.25/8.14 ! [I: set_o,F: $o > int] :
% 14.25/8.14 ( ( finite_finite_o @ I )
% 14.25/8.14 => ( ( I != bot_bot_set_o )
% 14.25/8.14 => ( ! [I3: $o] :
% 14.25/8.14 ( ( member_o @ I3 @ I )
% 14.25/8.14 => ( ord_less_int @ one_one_int @ ( F @ I3 ) ) )
% 14.25/8.14 => ( ord_less_int @ one_one_int @ ( groups3502327434004483295_o_int @ F @ I ) ) ) ) ) ).
% 14.25/8.14
% 14.25/8.14 thf(fact_7201_less__1__prod,axiom,
% 14.25/8.14 ! [I: set_nat,F: nat > int] :
% 14.25/8.14 ( ( finite_finite_nat @ I )
% 14.25/8.14 => ( ( I != bot_bot_set_nat )
% 14.25/8.14 => ( ! [I3: nat] :
% 14.25/8.14 ( ( member_nat @ I3 @ I )
% 14.25/8.14 => ( ord_less_int @ one_one_int @ ( F @ I3 ) ) )
% 14.25/8.14 => ( ord_less_int @ one_one_int @ ( groups705719431365010083at_int @ F @ I ) ) ) ) ) ).
% 14.25/8.14
% 14.25/8.14 thf(fact_7202_less__1__prod,axiom,
% 14.25/8.14 ! [I: set_int,F: int > int] :
% 14.25/8.14 ( ( finite_finite_int @ I )
% 14.25/8.14 => ( ( I != bot_bot_set_int )
% 14.25/8.14 => ( ! [I3: int] :
% 14.25/8.14 ( ( member_int @ I3 @ I )
% 14.25/8.14 => ( ord_less_int @ one_one_int @ ( F @ I3 ) ) )
% 14.25/8.14 => ( ord_less_int @ one_one_int @ ( groups1705073143266064639nt_int @ F @ I ) ) ) ) ) ).
% 14.25/8.14
% 14.25/8.14 thf(fact_7203_less__1__prod,axiom,
% 14.25/8.14 ! [I: set_Code_integer,F: code_integer > code_integer] :
% 14.25/8.14 ( ( finite6017078050557962740nteger @ I )
% 14.25/8.14 => ( ( I != bot_bo3990330152332043303nteger )
% 14.25/8.14 => ( ! [I3: code_integer] :
% 14.25/8.14 ( ( member_Code_integer @ I3 @ I )
% 14.25/8.14 => ( ord_le6747313008572928689nteger @ one_one_Code_integer @ ( F @ I3 ) ) )
% 14.25/8.14 => ( ord_le6747313008572928689nteger @ one_one_Code_integer @ ( groups3674199335183972705nteger @ F @ I ) ) ) ) ) ).
% 14.25/8.14
% 14.25/8.14 thf(fact_7204_less__1__prod,axiom,
% 14.25/8.14 ! [I: set_Product_unit,F: product_unit > code_integer] :
% 14.25/8.14 ( ( finite4290736615968046902t_unit @ I )
% 14.25/8.14 => ( ( I != bot_bo3957492148770167129t_unit )
% 14.25/8.14 => ( ! [I3: product_unit] :
% 14.25/8.14 ( ( member_Product_unit @ I3 @ I )
% 14.25/8.14 => ( ord_le6747313008572928689nteger @ one_one_Code_integer @ ( F @ I3 ) ) )
% 14.25/8.14 => ( ord_le6747313008572928689nteger @ one_one_Code_integer @ ( groups8185359811944078981nteger @ F @ I ) ) ) ) ) ).
% 14.25/8.14
% 14.25/8.14 thf(fact_7205_less__1__prod,axiom,
% 14.25/8.14 ! [I: set_o,F: $o > code_integer] :
% 14.25/8.14 ( ( finite_finite_o @ I )
% 14.25/8.14 => ( ( I != bot_bot_set_o )
% 14.25/8.14 => ( ! [I3: $o] :
% 14.25/8.14 ( ( member_o @ I3 @ I )
% 14.25/8.14 => ( ord_le6747313008572928689nteger @ one_one_Code_integer @ ( F @ I3 ) ) )
% 14.25/8.14 => ( ord_le6747313008572928689nteger @ one_one_Code_integer @ ( groups7694694392188491536nteger @ F @ I ) ) ) ) ) ).
% 14.25/8.14
% 14.25/8.14 thf(fact_7206_less__1__prod,axiom,
% 14.25/8.14 ! [I: set_nat,F: nat > code_integer] :
% 14.25/8.14 ( ( finite_finite_nat @ I )
% 14.25/8.14 => ( ( I != bot_bot_set_nat )
% 14.25/8.14 => ( ! [I3: nat] :
% 14.25/8.14 ( ( member_nat @ I3 @ I )
% 14.25/8.14 => ( ord_le6747313008572928689nteger @ one_one_Code_integer @ ( F @ I3 ) ) )
% 14.25/8.14 => ( ord_le6747313008572928689nteger @ one_one_Code_integer @ ( groups3455450783089532116nteger @ F @ I ) ) ) ) ) ).
% 14.25/8.14
% 14.25/8.14 thf(fact_7207_less__1__prod,axiom,
% 14.25/8.14 ! [I: set_int,F: int > code_integer] :
% 14.25/8.14 ( ( finite_finite_int @ I )
% 14.25/8.14 => ( ( I != bot_bot_set_int )
% 14.25/8.14 => ( ! [I3: int] :
% 14.25/8.14 ( ( member_int @ I3 @ I )
% 14.25/8.14 => ( ord_le6747313008572928689nteger @ one_one_Code_integer @ ( F @ I3 ) ) )
% 14.25/8.14 => ( ord_le6747313008572928689nteger @ one_one_Code_integer @ ( groups3827104343326376752nteger @ F @ I ) ) ) ) ) ).
% 14.25/8.14
% 14.25/8.14 thf(fact_7208_prod_Oreindex__nontrivial,axiom,
% 14.25/8.14 ! [A: set_nat,H2: nat > nat,G: nat > assn] :
% 14.25/8.14 ( ( finite_finite_nat @ A )
% 14.25/8.14 => ( ! [X3: nat,Y2: nat] :
% 14.25/8.14 ( ( member_nat @ X3 @ A )
% 14.25/8.14 => ( ( member_nat @ Y2 @ A )
% 14.25/8.14 => ( ( X3 != Y2 )
% 14.25/8.14 => ( ( ( H2 @ X3 )
% 14.25/8.14 = ( H2 @ Y2 ) )
% 14.25/8.14 => ( ( G @ ( H2 @ X3 ) )
% 14.25/8.14 = one_one_assn ) ) ) ) )
% 14.25/8.14 => ( ( groups6906906614972039071t_assn @ G @ ( image_nat_nat2 @ H2 @ A ) )
% 14.25/8.14 = ( groups6906906614972039071t_assn @ ( comp_nat_assn_nat @ G @ H2 ) @ A ) ) ) ) ).
% 14.25/8.14
% 14.25/8.14 thf(fact_7209_prod_Oreindex__nontrivial,axiom,
% 14.25/8.14 ! [A: set_nat,H2: nat > code_integer,G: code_integer > assn] :
% 14.25/8.14 ( ( finite_finite_nat @ A )
% 14.25/8.14 => ( ! [X3: nat,Y2: nat] :
% 14.25/8.14 ( ( member_nat @ X3 @ A )
% 14.25/8.14 => ( ( member_nat @ Y2 @ A )
% 14.25/8.14 => ( ( X3 != Y2 )
% 14.25/8.14 => ( ( ( H2 @ X3 )
% 14.25/8.14 = ( H2 @ Y2 ) )
% 14.25/8.14 => ( ( G @ ( H2 @ X3 ) )
% 14.25/8.14 = one_one_assn ) ) ) ) )
% 14.25/8.14 => ( ( groups1304777262505850412r_assn @ G @ ( image_1215581382706833972nteger @ H2 @ A ) )
% 14.25/8.14 = ( groups6906906614972039071t_assn @ ( comp_C8462447471145125178sn_nat @ G @ H2 ) @ A ) ) ) ) ).
% 14.25/8.14
% 14.25/8.14 thf(fact_7210_prod_Oreindex__nontrivial,axiom,
% 14.25/8.14 ! [A: set_Code_integer,H2: code_integer > code_integer,G: code_integer > assn] :
% 14.25/8.14 ( ( finite6017078050557962740nteger @ A )
% 14.25/8.14 => ( ! [X3: code_integer,Y2: code_integer] :
% 14.25/8.14 ( ( member_Code_integer @ X3 @ A )
% 14.25/8.14 => ( ( member_Code_integer @ Y2 @ A )
% 14.25/8.14 => ( ( X3 != Y2 )
% 14.25/8.14 => ( ( ( H2 @ X3 )
% 14.25/8.14 = ( H2 @ Y2 ) )
% 14.25/8.14 => ( ( G @ ( H2 @ X3 ) )
% 14.25/8.14 = one_one_assn ) ) ) ) )
% 14.25/8.14 => ( ( groups1304777262505850412r_assn @ G @ ( image_4470545334726330049nteger @ H2 @ A ) )
% 14.25/8.14 = ( groups1304777262505850412r_assn @ ( comp_C3026223900822616519nteger @ G @ H2 ) @ A ) ) ) ) ).
% 14.25/8.14
% 14.25/8.14 thf(fact_7211_prod_Oreindex__nontrivial,axiom,
% 14.25/8.14 ! [A: set_Code_integer,H2: code_integer > code_integer,G: code_integer > nat] :
% 14.25/8.14 ( ( finite6017078050557962740nteger @ A )
% 14.25/8.14 => ( ! [X3: code_integer,Y2: code_integer] :
% 14.25/8.14 ( ( member_Code_integer @ X3 @ A )
% 14.25/8.14 => ( ( member_Code_integer @ Y2 @ A )
% 14.25/8.14 => ( ( X3 != Y2 )
% 14.25/8.14 => ( ( ( H2 @ X3 )
% 14.25/8.14 = ( H2 @ Y2 ) )
% 14.25/8.14 => ( ( G @ ( H2 @ X3 ) )
% 14.25/8.14 = one_one_nat ) ) ) ) )
% 14.25/8.14 => ( ( groups3190895334310489300er_nat @ G @ ( image_4470545334726330049nteger @ H2 @ A ) )
% 14.25/8.14 = ( groups3190895334310489300er_nat @ ( comp_C7238605732685392495nteger @ G @ H2 ) @ A ) ) ) ) ).
% 14.25/8.14
% 14.25/8.14 thf(fact_7212_prod_Oreindex__nontrivial,axiom,
% 14.25/8.14 ! [A: set_nat,H2: nat > nat,G: nat > int] :
% 14.25/8.14 ( ( finite_finite_nat @ A )
% 14.25/8.14 => ( ! [X3: nat,Y2: nat] :
% 14.25/8.14 ( ( member_nat @ X3 @ A )
% 14.25/8.14 => ( ( member_nat @ Y2 @ A )
% 14.25/8.14 => ( ( X3 != Y2 )
% 14.25/8.14 => ( ( ( H2 @ X3 )
% 14.25/8.14 = ( H2 @ Y2 ) )
% 14.25/8.14 => ( ( G @ ( H2 @ X3 ) )
% 14.25/8.14 = one_one_int ) ) ) ) )
% 14.25/8.14 => ( ( groups705719431365010083at_int @ G @ ( image_nat_nat2 @ H2 @ A ) )
% 14.25/8.14 = ( groups705719431365010083at_int @ ( comp_nat_int_nat @ G @ H2 ) @ A ) ) ) ) ).
% 14.25/8.14
% 14.25/8.14 thf(fact_7213_prod_Oreindex__nontrivial,axiom,
% 14.25/8.14 ! [A: set_nat,H2: nat > code_integer,G: code_integer > int] :
% 14.25/8.14 ( ( finite_finite_nat @ A )
% 14.25/8.14 => ( ! [X3: nat,Y2: nat] :
% 14.25/8.14 ( ( member_nat @ X3 @ A )
% 14.25/8.14 => ( ( member_nat @ Y2 @ A )
% 14.25/8.14 => ( ( X3 != Y2 )
% 14.25/8.14 => ( ( ( H2 @ X3 )
% 14.25/8.14 = ( H2 @ Y2 ) )
% 14.25/8.14 => ( ( G @ ( H2 @ X3 ) )
% 14.25/8.14 = one_one_int ) ) ) ) )
% 14.25/8.14 => ( ( groups3188404863801439024er_int @ G @ ( image_1215581382706833972nteger @ H2 @ A ) )
% 14.25/8.14 = ( groups705719431365010083at_int @ ( comp_C5048910307564104766nt_nat @ G @ H2 ) @ A ) ) ) ) ).
% 14.25/8.14
% 14.25/8.14 thf(fact_7214_prod_Oreindex__nontrivial,axiom,
% 14.25/8.14 ! [A: set_Code_integer,H2: code_integer > code_integer,G: code_integer > int] :
% 14.25/8.14 ( ( finite6017078050557962740nteger @ A )
% 14.25/8.14 => ( ! [X3: code_integer,Y2: code_integer] :
% 14.25/8.14 ( ( member_Code_integer @ X3 @ A )
% 14.25/8.14 => ( ( member_Code_integer @ Y2 @ A )
% 14.25/8.14 => ( ( X3 != Y2 )
% 14.25/8.14 => ( ( ( H2 @ X3 )
% 14.25/8.14 = ( H2 @ Y2 ) )
% 14.25/8.14 => ( ( G @ ( H2 @ X3 ) )
% 14.25/8.14 = one_one_int ) ) ) ) )
% 14.25/8.14 => ( ( groups3188404863801439024er_int @ G @ ( image_4470545334726330049nteger @ H2 @ A ) )
% 14.25/8.14 = ( groups3188404863801439024er_int @ ( comp_C7610259292922237131nteger @ G @ H2 ) @ A ) ) ) ) ).
% 14.25/8.14
% 14.25/8.14 thf(fact_7215_prod_Oreindex__nontrivial,axiom,
% 14.25/8.14 ! [A: set_nat,H2: nat > nat,G: nat > code_integer] :
% 14.25/8.14 ( ( finite_finite_nat @ A )
% 14.25/8.14 => ( ! [X3: nat,Y2: nat] :
% 14.25/8.14 ( ( member_nat @ X3 @ A )
% 14.25/8.14 => ( ( member_nat @ Y2 @ A )
% 14.25/8.14 => ( ( X3 != Y2 )
% 14.25/8.14 => ( ( ( H2 @ X3 )
% 14.25/8.14 = ( H2 @ Y2 ) )
% 14.25/8.14 => ( ( G @ ( H2 @ X3 ) )
% 14.25/8.14 = one_one_Code_integer ) ) ) ) )
% 14.25/8.14 => ( ( groups3455450783089532116nteger @ G @ ( image_nat_nat2 @ H2 @ A ) )
% 14.25/8.14 = ( groups3455450783089532116nteger @ ( comp_n3898172953802868194er_nat @ G @ H2 ) @ A ) ) ) ) ).
% 14.25/8.14
% 14.25/8.14 thf(fact_7216_prod_Oreindex__nontrivial,axiom,
% 14.25/8.14 ! [A: set_nat,H2: nat > code_integer,G: code_integer > code_integer] :
% 14.25/8.14 ( ( finite_finite_nat @ A )
% 14.25/8.14 => ( ! [X3: nat,Y2: nat] :
% 14.25/8.14 ( ( member_nat @ X3 @ A )
% 14.25/8.14 => ( ( member_nat @ Y2 @ A )
% 14.25/8.14 => ( ( X3 != Y2 )
% 14.25/8.14 => ( ( ( H2 @ X3 )
% 14.25/8.14 = ( H2 @ Y2 ) )
% 14.25/8.14 => ( ( G @ ( H2 @ X3 ) )
% 14.25/8.14 = one_one_Code_integer ) ) ) ) )
% 14.25/8.14 => ( ( groups3674199335183972705nteger @ G @ ( image_1215581382706833972nteger @ H2 @ A ) )
% 14.25/8.14 = ( groups3455450783089532116nteger @ ( comp_C6974050283906349679er_nat @ G @ H2 ) @ A ) ) ) ) ).
% 14.25/8.14
% 14.25/8.14 thf(fact_7217_prod_Oreindex__nontrivial,axiom,
% 14.25/8.14 ! [A: set_Code_integer,H2: code_integer > code_integer,G: code_integer > code_integer] :
% 14.25/8.14 ( ( finite6017078050557962740nteger @ A )
% 14.25/8.14 => ( ! [X3: code_integer,Y2: code_integer] :
% 14.25/8.14 ( ( member_Code_integer @ X3 @ A )
% 14.25/8.14 => ( ( member_Code_integer @ Y2 @ A )
% 14.25/8.14 => ( ( X3 != Y2 )
% 14.25/8.14 => ( ( ( H2 @ X3 )
% 14.25/8.14 = ( H2 @ Y2 ) )
% 14.25/8.14 => ( ( G @ ( H2 @ X3 ) )
% 14.25/8.14 = one_one_Code_integer ) ) ) ) )
% 14.25/8.14 => ( ( groups3674199335183972705nteger @ G @ ( image_4470545334726330049nteger @ H2 @ A ) )
% 14.25/8.14 = ( groups3674199335183972705nteger @ ( comp_C7449957260575251196nteger @ G @ H2 ) @ A ) ) ) ) ).
% 14.25/8.14
% 14.25/8.14 thf(fact_7218_prod_Osubset__diff,axiom,
% 14.25/8.14 ! [B4: set_Code_integer,A: set_Code_integer,G: code_integer > assn] :
% 14.25/8.14 ( ( ord_le7084787975880047091nteger @ B4 @ A )
% 14.25/8.14 => ( ( finite6017078050557962740nteger @ A )
% 14.25/8.14 => ( ( groups1304777262505850412r_assn @ G @ A )
% 14.25/8.14 = ( times_times_assn @ ( groups1304777262505850412r_assn @ G @ ( minus_2355218937544613996nteger @ A @ B4 ) ) @ ( groups1304777262505850412r_assn @ G @ B4 ) ) ) ) ) ).
% 14.25/8.14
% 14.25/8.14 thf(fact_7219_prod_Osubset__diff,axiom,
% 14.25/8.14 ! [B4: set_Code_integer,A: set_Code_integer,G: code_integer > rat] :
% 14.25/8.14 ( ( ord_le7084787975880047091nteger @ B4 @ A )
% 14.25/8.14 => ( ( finite6017078050557962740nteger @ A )
% 14.25/8.14 => ( ( groups2555765274223993564er_rat @ G @ A )
% 14.25/8.14 = ( times_times_rat @ ( groups2555765274223993564er_rat @ G @ ( minus_2355218937544613996nteger @ A @ B4 ) ) @ ( groups2555765274223993564er_rat @ G @ B4 ) ) ) ) ) ).
% 14.25/8.14
% 14.25/8.14 thf(fact_7220_prod_Osubset__diff,axiom,
% 14.25/8.14 ! [B4: set_Code_integer,A: set_Code_integer,G: code_integer > nat] :
% 14.25/8.14 ( ( ord_le7084787975880047091nteger @ B4 @ A )
% 14.25/8.14 => ( ( finite6017078050557962740nteger @ A )
% 14.25/8.14 => ( ( groups3190895334310489300er_nat @ G @ A )
% 14.25/8.14 = ( times_times_nat @ ( groups3190895334310489300er_nat @ G @ ( minus_2355218937544613996nteger @ A @ B4 ) ) @ ( groups3190895334310489300er_nat @ G @ B4 ) ) ) ) ) ).
% 14.25/8.14
% 14.25/8.14 thf(fact_7221_prod_Osubset__diff,axiom,
% 14.25/8.14 ! [B4: set_Code_integer,A: set_Code_integer,G: code_integer > int] :
% 14.25/8.14 ( ( ord_le7084787975880047091nteger @ B4 @ A )
% 14.25/8.14 => ( ( finite6017078050557962740nteger @ A )
% 14.25/8.14 => ( ( groups3188404863801439024er_int @ G @ A )
% 14.25/8.14 = ( times_times_int @ ( groups3188404863801439024er_int @ G @ ( minus_2355218937544613996nteger @ A @ B4 ) ) @ ( groups3188404863801439024er_int @ G @ B4 ) ) ) ) ) ).
% 14.25/8.14
% 14.25/8.14 thf(fact_7222_prod_Osubset__diff,axiom,
% 14.25/8.14 ! [B4: set_nat,A: set_nat,G: nat > assn] :
% 14.25/8.14 ( ( ord_less_eq_set_nat @ B4 @ A )
% 14.25/8.14 => ( ( finite_finite_nat @ A )
% 14.25/8.14 => ( ( groups6906906614972039071t_assn @ G @ A )
% 14.25/8.14 = ( times_times_assn @ ( groups6906906614972039071t_assn @ G @ ( minus_minus_set_nat @ A @ B4 ) ) @ ( groups6906906614972039071t_assn @ G @ B4 ) ) ) ) ) ).
% 14.25/8.14
% 14.25/8.14 thf(fact_7223_prod_Osubset__diff,axiom,
% 14.25/8.14 ! [B4: set_nat,A: set_nat,G: nat > rat] :
% 14.25/8.14 ( ( ord_less_eq_set_nat @ B4 @ A )
% 14.25/8.14 => ( ( finite_finite_nat @ A )
% 14.25/8.14 => ( ( groups73079841787564623at_rat @ G @ A )
% 14.25/8.14 = ( times_times_rat @ ( groups73079841787564623at_rat @ G @ ( minus_minus_set_nat @ A @ B4 ) ) @ ( groups73079841787564623at_rat @ G @ B4 ) ) ) ) ) ).
% 14.25/8.14
% 14.25/8.14 thf(fact_7224_prod_Osubset__diff,axiom,
% 14.25/8.14 ! [B4: set_nat,A: set_nat,G: nat > int] :
% 14.25/8.14 ( ( ord_less_eq_set_nat @ B4 @ A )
% 14.25/8.14 => ( ( finite_finite_nat @ A )
% 14.25/8.14 => ( ( groups705719431365010083at_int @ G @ A )
% 14.25/8.14 = ( times_times_int @ ( groups705719431365010083at_int @ G @ ( minus_minus_set_nat @ A @ B4 ) ) @ ( groups705719431365010083at_int @ G @ B4 ) ) ) ) ) ).
% 14.25/8.14
% 14.25/8.14 thf(fact_7225_prod_Osubset__diff,axiom,
% 14.25/8.14 ! [B4: set_nat,A: set_nat,G: nat > nat] :
% 14.25/8.14 ( ( ord_less_eq_set_nat @ B4 @ A )
% 14.25/8.14 => ( ( finite_finite_nat @ A )
% 14.25/8.14 => ( ( groups708209901874060359at_nat @ G @ A )
% 14.25/8.14 = ( times_times_nat @ ( groups708209901874060359at_nat @ G @ ( minus_minus_set_nat @ A @ B4 ) ) @ ( groups708209901874060359at_nat @ G @ B4 ) ) ) ) ) ).
% 14.25/8.14
% 14.25/8.14 thf(fact_7226_prod_Osame__carrier,axiom,
% 14.25/8.14 ! [C5: set_int,A: set_int,B4: set_int,G: int > assn,H2: int > assn] :
% 14.25/8.14 ( ( finite_finite_int @ C5 )
% 14.25/8.14 => ( ( ord_less_eq_set_int @ A @ C5 )
% 14.25/8.14 => ( ( ord_less_eq_set_int @ B4 @ C5 )
% 14.25/8.14 => ( ! [A4: int] :
% 14.25/8.14 ( ( member_int @ A4 @ ( minus_minus_set_int @ C5 @ A ) )
% 14.25/8.14 => ( ( G @ A4 )
% 14.25/8.14 = one_one_assn ) )
% 14.25/8.14 => ( ! [B2: int] :
% 14.25/8.14 ( ( member_int @ B2 @ ( minus_minus_set_int @ C5 @ B4 ) )
% 14.25/8.14 => ( ( H2 @ B2 )
% 14.25/8.14 = one_one_assn ) )
% 14.25/8.14 => ( ( ( groups7882442080178216443t_assn @ G @ A )
% 14.25/8.14 = ( groups7882442080178216443t_assn @ H2 @ B4 ) )
% 14.25/8.14 = ( ( groups7882442080178216443t_assn @ G @ C5 )
% 14.25/8.14 = ( groups7882442080178216443t_assn @ H2 @ C5 ) ) ) ) ) ) ) ) ).
% 14.25/8.14
% 14.25/8.14 thf(fact_7227_prod_Osame__carrier,axiom,
% 14.25/8.14 ! [C5: set_o,A: set_o,B4: set_o,G: $o > assn,H2: $o > assn] :
% 14.25/8.14 ( ( finite_finite_o @ C5 )
% 14.25/8.14 => ( ( ord_less_eq_set_o @ A @ C5 )
% 14.25/8.14 => ( ( ord_less_eq_set_o @ B4 @ C5 )
% 14.25/8.14 => ( ! [A4: $o] :
% 14.25/8.14 ( ( member_o @ A4 @ ( minus_minus_set_o @ C5 @ A ) )
% 14.25/8.14 => ( ( G @ A4 )
% 14.25/8.14 = one_one_assn ) )
% 14.25/8.14 => ( ! [B2: $o] :
% 14.25/8.14 ( ( member_o @ B2 @ ( minus_minus_set_o @ C5 @ B4 ) )
% 14.25/8.14 => ( ( H2 @ B2 )
% 14.25/8.14 = one_one_assn ) )
% 14.25/8.14 => ( ( ( groups5301882518646026715o_assn @ G @ A )
% 14.25/8.14 = ( groups5301882518646026715o_assn @ H2 @ B4 ) )
% 14.25/8.14 = ( ( groups5301882518646026715o_assn @ G @ C5 )
% 14.25/8.14 = ( groups5301882518646026715o_assn @ H2 @ C5 ) ) ) ) ) ) ) ) ).
% 14.25/8.14
% 14.25/8.14 thf(fact_7228_prod_Osame__carrier,axiom,
% 14.25/8.14 ! [C5: set_Product_unit,A: set_Product_unit,B4: set_Product_unit,G: product_unit > assn,H2: product_unit > assn] :
% 14.25/8.14 ( ( finite4290736615968046902t_unit @ C5 )
% 14.25/8.14 => ( ( ord_le3507040750410214029t_unit @ A @ C5 )
% 14.25/8.14 => ( ( ord_le3507040750410214029t_unit @ B4 @ C5 )
% 14.25/8.14 => ( ! [A4: product_unit] :
% 14.25/8.14 ( ( member_Product_unit @ A4 @ ( minus_6452836326544984404t_unit @ C5 @ A ) )
% 14.25/8.14 => ( ( G @ A4 )
% 14.25/8.14 = one_one_assn ) )
% 14.25/8.14 => ( ! [B2: product_unit] :
% 14.25/8.14 ( ( member_Product_unit @ B2 @ ( minus_6452836326544984404t_unit @ C5 @ B4 ) )
% 14.25/8.14 => ( ( H2 @ B2 )
% 14.25/8.14 = one_one_assn ) )
% 14.25/8.14 => ( ( ( groups5360233303959380816t_assn @ G @ A )
% 14.25/8.14 = ( groups5360233303959380816t_assn @ H2 @ B4 ) )
% 14.25/8.14 = ( ( groups5360233303959380816t_assn @ G @ C5 )
% 14.25/8.14 = ( groups5360233303959380816t_assn @ H2 @ C5 ) ) ) ) ) ) ) ) ).
% 14.25/8.14
% 14.25/8.14 thf(fact_7229_prod_Osame__carrier,axiom,
% 14.25/8.14 ! [C5: set_Code_integer,A: set_Code_integer,B4: set_Code_integer,G: code_integer > assn,H2: code_integer > assn] :
% 14.25/8.14 ( ( finite6017078050557962740nteger @ C5 )
% 14.25/8.14 => ( ( ord_le7084787975880047091nteger @ A @ C5 )
% 14.25/8.14 => ( ( ord_le7084787975880047091nteger @ B4 @ C5 )
% 14.25/8.14 => ( ! [A4: code_integer] :
% 14.25/8.14 ( ( member_Code_integer @ A4 @ ( minus_2355218937544613996nteger @ C5 @ A ) )
% 14.25/8.14 => ( ( G @ A4 )
% 14.25/8.14 = one_one_assn ) )
% 14.25/8.14 => ( ! [B2: code_integer] :
% 14.25/8.14 ( ( member_Code_integer @ B2 @ ( minus_2355218937544613996nteger @ C5 @ B4 ) )
% 14.25/8.14 => ( ( H2 @ B2 )
% 14.25/8.14 = one_one_assn ) )
% 14.25/8.14 => ( ( ( groups1304777262505850412r_assn @ G @ A )
% 14.25/8.14 = ( groups1304777262505850412r_assn @ H2 @ B4 ) )
% 14.25/8.14 = ( ( groups1304777262505850412r_assn @ G @ C5 )
% 14.25/8.14 = ( groups1304777262505850412r_assn @ H2 @ C5 ) ) ) ) ) ) ) ) ).
% 14.25/8.14
% 14.25/8.14 thf(fact_7230_prod_Osame__carrier,axiom,
% 14.25/8.14 ! [C5: set_int,A: set_int,B4: set_int,G: int > nat,H2: int > nat] :
% 14.25/8.14 ( ( finite_finite_int @ C5 )
% 14.25/8.14 => ( ( ord_less_eq_set_int @ A @ C5 )
% 14.25/8.14 => ( ( ord_less_eq_set_int @ B4 @ C5 )
% 14.25/8.14 => ( ! [A4: int] :
% 14.25/8.14 ( ( member_int @ A4 @ ( minus_minus_set_int @ C5 @ A ) )
% 14.25/8.14 => ( ( G @ A4 )
% 14.25/8.14 = one_one_nat ) )
% 14.25/8.14 => ( ! [B2: int] :
% 14.25/8.14 ( ( member_int @ B2 @ ( minus_minus_set_int @ C5 @ B4 ) )
% 14.25/8.14 => ( ( H2 @ B2 )
% 14.25/8.14 = one_one_nat ) )
% 14.25/8.14 => ( ( ( groups1707563613775114915nt_nat @ G @ A )
% 14.25/8.14 = ( groups1707563613775114915nt_nat @ H2 @ B4 ) )
% 14.25/8.14 = ( ( groups1707563613775114915nt_nat @ G @ C5 )
% 14.25/8.14 = ( groups1707563613775114915nt_nat @ H2 @ C5 ) ) ) ) ) ) ) ) ).
% 14.25/8.14
% 14.25/8.14 thf(fact_7231_prod_Osame__carrier,axiom,
% 14.25/8.14 ! [C5: set_o,A: set_o,B4: set_o,G: $o > nat,H2: $o > nat] :
% 14.25/8.14 ( ( finite_finite_o @ C5 )
% 14.25/8.14 => ( ( ord_less_eq_set_o @ A @ C5 )
% 14.25/8.14 => ( ( ord_less_eq_set_o @ B4 @ C5 )
% 14.25/8.14 => ( ! [A4: $o] :
% 14.25/8.14 ( ( member_o @ A4 @ ( minus_minus_set_o @ C5 @ A ) )
% 14.25/8.14 => ( ( G @ A4 )
% 14.25/8.14 = one_one_nat ) )
% 14.25/8.14 => ( ! [B2: $o] :
% 14.25/8.14 ( ( member_o @ B2 @ ( minus_minus_set_o @ C5 @ B4 ) )
% 14.25/8.14 => ( ( H2 @ B2 )
% 14.25/8.14 = one_one_nat ) )
% 14.25/8.14 => ( ( ( groups3504817904513533571_o_nat @ G @ A )
% 14.25/8.14 = ( groups3504817904513533571_o_nat @ H2 @ B4 ) )
% 14.25/8.14 = ( ( groups3504817904513533571_o_nat @ G @ C5 )
% 14.25/8.14 = ( groups3504817904513533571_o_nat @ H2 @ C5 ) ) ) ) ) ) ) ) ).
% 14.25/8.14
% 14.25/8.14 thf(fact_7232_prod_Osame__carrier,axiom,
% 14.25/8.14 ! [C5: set_Product_unit,A: set_Product_unit,B4: set_Product_unit,G: product_unit > nat,H2: product_unit > nat] :
% 14.25/8.14 ( ( finite4290736615968046902t_unit @ C5 )
% 14.25/8.14 => ( ( ord_le3507040750410214029t_unit @ A @ C5 )
% 14.25/8.14 => ( ( ord_le3507040750410214029t_unit @ B4 @ C5 )
% 14.25/8.14 => ( ! [A4: product_unit] :
% 14.25/8.14 ( ( member_Product_unit @ A4 @ ( minus_6452836326544984404t_unit @ C5 @ A ) )
% 14.25/8.14 => ( ( G @ A4 )
% 14.25/8.14 = one_one_nat ) )
% 14.25/8.14 => ( ! [B2: product_unit] :
% 14.25/8.14 ( ( member_Product_unit @ B2 @ ( minus_6452836326544984404t_unit @ C5 @ B4 ) )
% 14.25/8.14 => ( ( H2 @ B2 )
% 14.25/8.14 = one_one_nat ) )
% 14.25/8.14 => ( ( ( groups396421249026171384it_nat @ G @ A )
% 14.25/8.14 = ( groups396421249026171384it_nat @ H2 @ B4 ) )
% 14.25/8.14 = ( ( groups396421249026171384it_nat @ G @ C5 )
% 14.25/8.14 = ( groups396421249026171384it_nat @ H2 @ C5 ) ) ) ) ) ) ) ) ).
% 14.25/8.14
% 14.25/8.14 thf(fact_7233_prod_Osame__carrier,axiom,
% 14.25/8.14 ! [C5: set_Code_integer,A: set_Code_integer,B4: set_Code_integer,G: code_integer > nat,H2: code_integer > nat] :
% 14.25/8.14 ( ( finite6017078050557962740nteger @ C5 )
% 14.25/8.14 => ( ( ord_le7084787975880047091nteger @ A @ C5 )
% 14.25/8.14 => ( ( ord_le7084787975880047091nteger @ B4 @ C5 )
% 14.25/8.14 => ( ! [A4: code_integer] :
% 14.25/8.14 ( ( member_Code_integer @ A4 @ ( minus_2355218937544613996nteger @ C5 @ A ) )
% 14.25/8.14 => ( ( G @ A4 )
% 14.25/8.14 = one_one_nat ) )
% 14.25/8.14 => ( ! [B2: code_integer] :
% 14.25/8.14 ( ( member_Code_integer @ B2 @ ( minus_2355218937544613996nteger @ C5 @ B4 ) )
% 14.25/8.14 => ( ( H2 @ B2 )
% 14.25/8.14 = one_one_nat ) )
% 14.25/8.14 => ( ( ( groups3190895334310489300er_nat @ G @ A )
% 14.25/8.14 = ( groups3190895334310489300er_nat @ H2 @ B4 ) )
% 14.25/8.14 = ( ( groups3190895334310489300er_nat @ G @ C5 )
% 14.25/8.14 = ( groups3190895334310489300er_nat @ H2 @ C5 ) ) ) ) ) ) ) ) ).
% 14.25/8.14
% 14.25/8.14 thf(fact_7234_prod_Osame__carrier,axiom,
% 14.25/8.14 ! [C5: set_int,A: set_int,B4: set_int,G: int > int,H2: int > int] :
% 14.25/8.14 ( ( finite_finite_int @ C5 )
% 14.25/8.14 => ( ( ord_less_eq_set_int @ A @ C5 )
% 14.25/8.14 => ( ( ord_less_eq_set_int @ B4 @ C5 )
% 14.25/8.14 => ( ! [A4: int] :
% 14.25/8.14 ( ( member_int @ A4 @ ( minus_minus_set_int @ C5 @ A ) )
% 14.25/8.14 => ( ( G @ A4 )
% 14.25/8.14 = one_one_int ) )
% 14.25/8.14 => ( ! [B2: int] :
% 14.25/8.14 ( ( member_int @ B2 @ ( minus_minus_set_int @ C5 @ B4 ) )
% 14.25/8.14 => ( ( H2 @ B2 )
% 14.25/8.14 = one_one_int ) )
% 14.25/8.14 => ( ( ( groups1705073143266064639nt_int @ G @ A )
% 14.25/8.14 = ( groups1705073143266064639nt_int @ H2 @ B4 ) )
% 14.25/8.14 = ( ( groups1705073143266064639nt_int @ G @ C5 )
% 14.25/8.14 = ( groups1705073143266064639nt_int @ H2 @ C5 ) ) ) ) ) ) ) ) ).
% 14.25/8.14
% 14.25/8.14 thf(fact_7235_prod_Osame__carrier,axiom,
% 14.25/8.14 ! [C5: set_o,A: set_o,B4: set_o,G: $o > int,H2: $o > int] :
% 14.25/8.14 ( ( finite_finite_o @ C5 )
% 14.25/8.14 => ( ( ord_less_eq_set_o @ A @ C5 )
% 14.25/8.14 => ( ( ord_less_eq_set_o @ B4 @ C5 )
% 14.25/8.14 => ( ! [A4: $o] :
% 14.25/8.14 ( ( member_o @ A4 @ ( minus_minus_set_o @ C5 @ A ) )
% 14.25/8.14 => ( ( G @ A4 )
% 14.25/8.14 = one_one_int ) )
% 14.25/8.14 => ( ! [B2: $o] :
% 14.25/8.14 ( ( member_o @ B2 @ ( minus_minus_set_o @ C5 @ B4 ) )
% 14.25/8.14 => ( ( H2 @ B2 )
% 14.25/8.14 = one_one_int ) )
% 14.25/8.14 => ( ( ( groups3502327434004483295_o_int @ G @ A )
% 14.25/8.14 = ( groups3502327434004483295_o_int @ H2 @ B4 ) )
% 14.25/8.14 = ( ( groups3502327434004483295_o_int @ G @ C5 )
% 14.25/8.14 = ( groups3502327434004483295_o_int @ H2 @ C5 ) ) ) ) ) ) ) ) ).
% 14.25/8.14
% 14.25/8.14 thf(fact_7236_prod_Osame__carrierI,axiom,
% 14.25/8.14 ! [C5: set_int,A: set_int,B4: set_int,G: int > assn,H2: int > assn] :
% 14.25/8.14 ( ( finite_finite_int @ C5 )
% 14.25/8.14 => ( ( ord_less_eq_set_int @ A @ C5 )
% 14.25/8.14 => ( ( ord_less_eq_set_int @ B4 @ C5 )
% 14.25/8.14 => ( ! [A4: int] :
% 14.25/8.14 ( ( member_int @ A4 @ ( minus_minus_set_int @ C5 @ A ) )
% 14.25/8.14 => ( ( G @ A4 )
% 14.25/8.14 = one_one_assn ) )
% 14.25/8.14 => ( ! [B2: int] :
% 14.25/8.14 ( ( member_int @ B2 @ ( minus_minus_set_int @ C5 @ B4 ) )
% 14.25/8.14 => ( ( H2 @ B2 )
% 14.25/8.14 = one_one_assn ) )
% 14.25/8.14 => ( ( ( groups7882442080178216443t_assn @ G @ C5 )
% 14.25/8.14 = ( groups7882442080178216443t_assn @ H2 @ C5 ) )
% 14.25/8.14 => ( ( groups7882442080178216443t_assn @ G @ A )
% 14.25/8.14 = ( groups7882442080178216443t_assn @ H2 @ B4 ) ) ) ) ) ) ) ) ).
% 14.25/8.14
% 14.25/8.14 thf(fact_7237_prod_Osame__carrierI,axiom,
% 14.25/8.14 ! [C5: set_o,A: set_o,B4: set_o,G: $o > assn,H2: $o > assn] :
% 14.25/8.14 ( ( finite_finite_o @ C5 )
% 14.25/8.14 => ( ( ord_less_eq_set_o @ A @ C5 )
% 14.25/8.14 => ( ( ord_less_eq_set_o @ B4 @ C5 )
% 14.25/8.14 => ( ! [A4: $o] :
% 14.25/8.14 ( ( member_o @ A4 @ ( minus_minus_set_o @ C5 @ A ) )
% 14.25/8.14 => ( ( G @ A4 )
% 14.25/8.14 = one_one_assn ) )
% 14.25/8.14 => ( ! [B2: $o] :
% 14.25/8.14 ( ( member_o @ B2 @ ( minus_minus_set_o @ C5 @ B4 ) )
% 14.25/8.14 => ( ( H2 @ B2 )
% 14.25/8.14 = one_one_assn ) )
% 14.25/8.14 => ( ( ( groups5301882518646026715o_assn @ G @ C5 )
% 14.25/8.14 = ( groups5301882518646026715o_assn @ H2 @ C5 ) )
% 14.25/8.14 => ( ( groups5301882518646026715o_assn @ G @ A )
% 14.25/8.14 = ( groups5301882518646026715o_assn @ H2 @ B4 ) ) ) ) ) ) ) ) ).
% 14.25/8.14
% 14.25/8.14 thf(fact_7238_prod_Osame__carrierI,axiom,
% 14.25/8.14 ! [C5: set_Product_unit,A: set_Product_unit,B4: set_Product_unit,G: product_unit > assn,H2: product_unit > assn] :
% 14.25/8.14 ( ( finite4290736615968046902t_unit @ C5 )
% 14.25/8.14 => ( ( ord_le3507040750410214029t_unit @ A @ C5 )
% 14.25/8.14 => ( ( ord_le3507040750410214029t_unit @ B4 @ C5 )
% 14.25/8.14 => ( ! [A4: product_unit] :
% 14.25/8.14 ( ( member_Product_unit @ A4 @ ( minus_6452836326544984404t_unit @ C5 @ A ) )
% 14.25/8.14 => ( ( G @ A4 )
% 14.25/8.14 = one_one_assn ) )
% 14.25/8.14 => ( ! [B2: product_unit] :
% 14.25/8.14 ( ( member_Product_unit @ B2 @ ( minus_6452836326544984404t_unit @ C5 @ B4 ) )
% 14.25/8.14 => ( ( H2 @ B2 )
% 14.25/8.14 = one_one_assn ) )
% 14.25/8.14 => ( ( ( groups5360233303959380816t_assn @ G @ C5 )
% 14.25/8.14 = ( groups5360233303959380816t_assn @ H2 @ C5 ) )
% 14.25/8.14 => ( ( groups5360233303959380816t_assn @ G @ A )
% 14.25/8.14 = ( groups5360233303959380816t_assn @ H2 @ B4 ) ) ) ) ) ) ) ) ).
% 14.25/8.14
% 14.25/8.14 thf(fact_7239_prod_Osame__carrierI,axiom,
% 14.25/8.14 ! [C5: set_Code_integer,A: set_Code_integer,B4: set_Code_integer,G: code_integer > assn,H2: code_integer > assn] :
% 14.25/8.14 ( ( finite6017078050557962740nteger @ C5 )
% 14.25/8.14 => ( ( ord_le7084787975880047091nteger @ A @ C5 )
% 14.25/8.14 => ( ( ord_le7084787975880047091nteger @ B4 @ C5 )
% 14.25/8.14 => ( ! [A4: code_integer] :
% 14.25/8.14 ( ( member_Code_integer @ A4 @ ( minus_2355218937544613996nteger @ C5 @ A ) )
% 14.25/8.14 => ( ( G @ A4 )
% 14.25/8.14 = one_one_assn ) )
% 14.25/8.14 => ( ! [B2: code_integer] :
% 14.25/8.14 ( ( member_Code_integer @ B2 @ ( minus_2355218937544613996nteger @ C5 @ B4 ) )
% 14.25/8.14 => ( ( H2 @ B2 )
% 14.25/8.14 = one_one_assn ) )
% 14.25/8.14 => ( ( ( groups1304777262505850412r_assn @ G @ C5 )
% 14.25/8.14 = ( groups1304777262505850412r_assn @ H2 @ C5 ) )
% 14.25/8.14 => ( ( groups1304777262505850412r_assn @ G @ A )
% 14.25/8.14 = ( groups1304777262505850412r_assn @ H2 @ B4 ) ) ) ) ) ) ) ) ).
% 14.25/8.14
% 14.25/8.14 thf(fact_7240_prod_Osame__carrierI,axiom,
% 14.25/8.14 ! [C5: set_int,A: set_int,B4: set_int,G: int > nat,H2: int > nat] :
% 14.25/8.14 ( ( finite_finite_int @ C5 )
% 14.25/8.14 => ( ( ord_less_eq_set_int @ A @ C5 )
% 14.25/8.14 => ( ( ord_less_eq_set_int @ B4 @ C5 )
% 14.25/8.14 => ( ! [A4: int] :
% 14.25/8.14 ( ( member_int @ A4 @ ( minus_minus_set_int @ C5 @ A ) )
% 14.25/8.14 => ( ( G @ A4 )
% 14.25/8.14 = one_one_nat ) )
% 14.25/8.14 => ( ! [B2: int] :
% 14.25/8.14 ( ( member_int @ B2 @ ( minus_minus_set_int @ C5 @ B4 ) )
% 14.25/8.14 => ( ( H2 @ B2 )
% 14.25/8.14 = one_one_nat ) )
% 14.25/8.14 => ( ( ( groups1707563613775114915nt_nat @ G @ C5 )
% 14.25/8.14 = ( groups1707563613775114915nt_nat @ H2 @ C5 ) )
% 14.25/8.14 => ( ( groups1707563613775114915nt_nat @ G @ A )
% 14.25/8.14 = ( groups1707563613775114915nt_nat @ H2 @ B4 ) ) ) ) ) ) ) ) ).
% 14.25/8.14
% 14.25/8.14 thf(fact_7241_prod_Osame__carrierI,axiom,
% 14.25/8.14 ! [C5: set_o,A: set_o,B4: set_o,G: $o > nat,H2: $o > nat] :
% 14.25/8.14 ( ( finite_finite_o @ C5 )
% 14.25/8.14 => ( ( ord_less_eq_set_o @ A @ C5 )
% 14.25/8.14 => ( ( ord_less_eq_set_o @ B4 @ C5 )
% 14.25/8.14 => ( ! [A4: $o] :
% 14.25/8.14 ( ( member_o @ A4 @ ( minus_minus_set_o @ C5 @ A ) )
% 14.25/8.14 => ( ( G @ A4 )
% 14.25/8.14 = one_one_nat ) )
% 14.25/8.14 => ( ! [B2: $o] :
% 14.25/8.14 ( ( member_o @ B2 @ ( minus_minus_set_o @ C5 @ B4 ) )
% 14.25/8.14 => ( ( H2 @ B2 )
% 14.25/8.14 = one_one_nat ) )
% 14.25/8.14 => ( ( ( groups3504817904513533571_o_nat @ G @ C5 )
% 14.25/8.14 = ( groups3504817904513533571_o_nat @ H2 @ C5 ) )
% 14.25/8.14 => ( ( groups3504817904513533571_o_nat @ G @ A )
% 14.25/8.14 = ( groups3504817904513533571_o_nat @ H2 @ B4 ) ) ) ) ) ) ) ) ).
% 14.25/8.14
% 14.25/8.14 thf(fact_7242_prod_Osame__carrierI,axiom,
% 14.25/8.14 ! [C5: set_Product_unit,A: set_Product_unit,B4: set_Product_unit,G: product_unit > nat,H2: product_unit > nat] :
% 14.25/8.14 ( ( finite4290736615968046902t_unit @ C5 )
% 14.25/8.14 => ( ( ord_le3507040750410214029t_unit @ A @ C5 )
% 14.25/8.14 => ( ( ord_le3507040750410214029t_unit @ B4 @ C5 )
% 14.25/8.14 => ( ! [A4: product_unit] :
% 14.25/8.14 ( ( member_Product_unit @ A4 @ ( minus_6452836326544984404t_unit @ C5 @ A ) )
% 14.25/8.14 => ( ( G @ A4 )
% 14.25/8.14 = one_one_nat ) )
% 14.25/8.14 => ( ! [B2: product_unit] :
% 14.25/8.14 ( ( member_Product_unit @ B2 @ ( minus_6452836326544984404t_unit @ C5 @ B4 ) )
% 14.25/8.14 => ( ( H2 @ B2 )
% 14.25/8.14 = one_one_nat ) )
% 14.25/8.14 => ( ( ( groups396421249026171384it_nat @ G @ C5 )
% 14.25/8.14 = ( groups396421249026171384it_nat @ H2 @ C5 ) )
% 14.25/8.14 => ( ( groups396421249026171384it_nat @ G @ A )
% 14.25/8.14 = ( groups396421249026171384it_nat @ H2 @ B4 ) ) ) ) ) ) ) ) ).
% 14.25/8.14
% 14.25/8.14 thf(fact_7243_prod_Osame__carrierI,axiom,
% 14.25/8.14 ! [C5: set_Code_integer,A: set_Code_integer,B4: set_Code_integer,G: code_integer > nat,H2: code_integer > nat] :
% 14.25/8.14 ( ( finite6017078050557962740nteger @ C5 )
% 14.25/8.14 => ( ( ord_le7084787975880047091nteger @ A @ C5 )
% 14.25/8.14 => ( ( ord_le7084787975880047091nteger @ B4 @ C5 )
% 14.25/8.14 => ( ! [A4: code_integer] :
% 14.25/8.14 ( ( member_Code_integer @ A4 @ ( minus_2355218937544613996nteger @ C5 @ A ) )
% 14.25/8.14 => ( ( G @ A4 )
% 14.25/8.14 = one_one_nat ) )
% 14.25/8.14 => ( ! [B2: code_integer] :
% 14.25/8.14 ( ( member_Code_integer @ B2 @ ( minus_2355218937544613996nteger @ C5 @ B4 ) )
% 14.25/8.14 => ( ( H2 @ B2 )
% 14.25/8.14 = one_one_nat ) )
% 14.25/8.14 => ( ( ( groups3190895334310489300er_nat @ G @ C5 )
% 14.25/8.14 = ( groups3190895334310489300er_nat @ H2 @ C5 ) )
% 14.25/8.14 => ( ( groups3190895334310489300er_nat @ G @ A )
% 14.25/8.14 = ( groups3190895334310489300er_nat @ H2 @ B4 ) ) ) ) ) ) ) ) ).
% 14.25/8.14
% 14.25/8.14 thf(fact_7244_prod_Osame__carrierI,axiom,
% 14.25/8.14 ! [C5: set_int,A: set_int,B4: set_int,G: int > int,H2: int > int] :
% 14.25/8.14 ( ( finite_finite_int @ C5 )
% 14.25/8.14 => ( ( ord_less_eq_set_int @ A @ C5 )
% 14.25/8.14 => ( ( ord_less_eq_set_int @ B4 @ C5 )
% 14.25/8.14 => ( ! [A4: int] :
% 14.25/8.14 ( ( member_int @ A4 @ ( minus_minus_set_int @ C5 @ A ) )
% 14.25/8.14 => ( ( G @ A4 )
% 14.25/8.14 = one_one_int ) )
% 14.25/8.14 => ( ! [B2: int] :
% 14.25/8.14 ( ( member_int @ B2 @ ( minus_minus_set_int @ C5 @ B4 ) )
% 14.25/8.14 => ( ( H2 @ B2 )
% 14.25/8.14 = one_one_int ) )
% 14.25/8.14 => ( ( ( groups1705073143266064639nt_int @ G @ C5 )
% 14.25/8.14 = ( groups1705073143266064639nt_int @ H2 @ C5 ) )
% 14.25/8.14 => ( ( groups1705073143266064639nt_int @ G @ A )
% 14.25/8.14 = ( groups1705073143266064639nt_int @ H2 @ B4 ) ) ) ) ) ) ) ) ).
% 14.25/8.14
% 14.25/8.14 thf(fact_7245_prod_Osame__carrierI,axiom,
% 14.25/8.14 ! [C5: set_o,A: set_o,B4: set_o,G: $o > int,H2: $o > int] :
% 14.25/8.14 ( ( finite_finite_o @ C5 )
% 14.25/8.14 => ( ( ord_less_eq_set_o @ A @ C5 )
% 14.25/8.14 => ( ( ord_less_eq_set_o @ B4 @ C5 )
% 14.25/8.14 => ( ! [A4: $o] :
% 14.25/8.14 ( ( member_o @ A4 @ ( minus_minus_set_o @ C5 @ A ) )
% 14.25/8.14 => ( ( G @ A4 )
% 14.25/8.14 = one_one_int ) )
% 14.25/8.14 => ( ! [B2: $o] :
% 14.25/8.14 ( ( member_o @ B2 @ ( minus_minus_set_o @ C5 @ B4 ) )
% 14.25/8.14 => ( ( H2 @ B2 )
% 14.25/8.14 = one_one_int ) )
% 14.25/8.14 => ( ( ( groups3502327434004483295_o_int @ G @ C5 )
% 14.25/8.14 = ( groups3502327434004483295_o_int @ H2 @ C5 ) )
% 14.25/8.14 => ( ( groups3502327434004483295_o_int @ G @ A )
% 14.25/8.14 = ( groups3502327434004483295_o_int @ H2 @ B4 ) ) ) ) ) ) ) ) ).
% 14.25/8.14
% 14.25/8.14 thf(fact_7246_prod_Omono__neutral__left,axiom,
% 14.25/8.14 ! [T2: set_Code_integer,S: set_Code_integer,G: code_integer > assn] :
% 14.25/8.14 ( ( finite6017078050557962740nteger @ T2 )
% 14.25/8.14 => ( ( ord_le7084787975880047091nteger @ S @ T2 )
% 14.25/8.14 => ( ! [X3: code_integer] :
% 14.25/8.14 ( ( member_Code_integer @ X3 @ ( minus_2355218937544613996nteger @ T2 @ S ) )
% 14.25/8.14 => ( ( G @ X3 )
% 14.25/8.14 = one_one_assn ) )
% 14.25/8.14 => ( ( groups1304777262505850412r_assn @ G @ S )
% 14.25/8.14 = ( groups1304777262505850412r_assn @ G @ T2 ) ) ) ) ) ).
% 14.25/8.14
% 14.25/8.14 thf(fact_7247_prod_Omono__neutral__left,axiom,
% 14.25/8.14 ! [T2: set_Code_integer,S: set_Code_integer,G: code_integer > nat] :
% 14.25/8.14 ( ( finite6017078050557962740nteger @ T2 )
% 14.25/8.14 => ( ( ord_le7084787975880047091nteger @ S @ T2 )
% 14.25/8.14 => ( ! [X3: code_integer] :
% 14.25/8.14 ( ( member_Code_integer @ X3 @ ( minus_2355218937544613996nteger @ T2 @ S ) )
% 14.25/8.14 => ( ( G @ X3 )
% 14.25/8.14 = one_one_nat ) )
% 14.25/8.14 => ( ( groups3190895334310489300er_nat @ G @ S )
% 14.25/8.14 = ( groups3190895334310489300er_nat @ G @ T2 ) ) ) ) ) ).
% 14.25/8.14
% 14.25/8.14 thf(fact_7248_prod_Omono__neutral__left,axiom,
% 14.25/8.14 ! [T2: set_Code_integer,S: set_Code_integer,G: code_integer > int] :
% 14.25/8.14 ( ( finite6017078050557962740nteger @ T2 )
% 14.25/8.14 => ( ( ord_le7084787975880047091nteger @ S @ T2 )
% 14.25/8.14 => ( ! [X3: code_integer] :
% 14.25/8.14 ( ( member_Code_integer @ X3 @ ( minus_2355218937544613996nteger @ T2 @ S ) )
% 14.25/8.14 => ( ( G @ X3 )
% 14.25/8.14 = one_one_int ) )
% 14.25/8.14 => ( ( groups3188404863801439024er_int @ G @ S )
% 14.25/8.14 = ( groups3188404863801439024er_int @ G @ T2 ) ) ) ) ) ).
% 14.25/8.14
% 14.25/8.14 thf(fact_7249_prod_Omono__neutral__left,axiom,
% 14.25/8.14 ! [T2: set_Code_integer,S: set_Code_integer,G: code_integer > code_integer] :
% 14.25/8.14 ( ( finite6017078050557962740nteger @ T2 )
% 14.25/8.14 => ( ( ord_le7084787975880047091nteger @ S @ T2 )
% 14.25/8.14 => ( ! [X3: code_integer] :
% 14.25/8.14 ( ( member_Code_integer @ X3 @ ( minus_2355218937544613996nteger @ T2 @ S ) )
% 14.25/8.14 => ( ( G @ X3 )
% 14.25/8.14 = one_one_Code_integer ) )
% 14.25/8.14 => ( ( groups3674199335183972705nteger @ G @ S )
% 14.25/8.14 = ( groups3674199335183972705nteger @ G @ T2 ) ) ) ) ) ).
% 14.25/8.14
% 14.25/8.14 thf(fact_7250_prod_Omono__neutral__left,axiom,
% 14.25/8.14 ! [T2: set_Code_integer,S: set_Code_integer,G: code_integer > code_natural] :
% 14.25/8.14 ( ( finite6017078050557962740nteger @ T2 )
% 14.25/8.14 => ( ( ord_le7084787975880047091nteger @ S @ T2 )
% 14.25/8.14 => ( ! [X3: code_integer] :
% 14.25/8.14 ( ( member_Code_integer @ X3 @ ( minus_2355218937544613996nteger @ T2 @ S ) )
% 14.25/8.14 => ( ( G @ X3 )
% 14.25/8.14 = one_one_Code_natural ) )
% 14.25/8.14 => ( ( groups2497794486940690220atural @ G @ S )
% 14.25/8.14 = ( groups2497794486940690220atural @ G @ T2 ) ) ) ) ) ).
% 14.25/8.14
% 14.25/8.14 thf(fact_7251_prod_Omono__neutral__left,axiom,
% 14.25/8.14 ! [T2: set_nat,S: set_nat,G: nat > assn] :
% 14.25/8.14 ( ( finite_finite_nat @ T2 )
% 14.25/8.14 => ( ( ord_less_eq_set_nat @ S @ T2 )
% 14.25/8.14 => ( ! [X3: nat] :
% 14.25/8.14 ( ( member_nat @ X3 @ ( minus_minus_set_nat @ T2 @ S ) )
% 14.25/8.14 => ( ( G @ X3 )
% 14.25/8.14 = one_one_assn ) )
% 14.25/8.14 => ( ( groups6906906614972039071t_assn @ G @ S )
% 14.25/8.14 = ( groups6906906614972039071t_assn @ G @ T2 ) ) ) ) ) ).
% 14.25/8.14
% 14.25/8.14 thf(fact_7252_prod_Omono__neutral__left,axiom,
% 14.25/8.14 ! [T2: set_nat,S: set_nat,G: nat > int] :
% 14.25/8.14 ( ( finite_finite_nat @ T2 )
% 14.25/8.14 => ( ( ord_less_eq_set_nat @ S @ T2 )
% 14.25/8.14 => ( ! [X3: nat] :
% 14.25/8.14 ( ( member_nat @ X3 @ ( minus_minus_set_nat @ T2 @ S ) )
% 14.25/8.14 => ( ( G @ X3 )
% 14.25/8.14 = one_one_int ) )
% 14.25/8.14 => ( ( groups705719431365010083at_int @ G @ S )
% 14.25/8.14 = ( groups705719431365010083at_int @ G @ T2 ) ) ) ) ) ).
% 14.25/8.14
% 14.25/8.14 thf(fact_7253_prod_Omono__neutral__left,axiom,
% 14.25/8.14 ! [T2: set_nat,S: set_nat,G: nat > code_integer] :
% 14.25/8.14 ( ( finite_finite_nat @ T2 )
% 14.25/8.14 => ( ( ord_less_eq_set_nat @ S @ T2 )
% 14.25/8.14 => ( ! [X3: nat] :
% 14.25/8.14 ( ( member_nat @ X3 @ ( minus_minus_set_nat @ T2 @ S ) )
% 14.25/8.14 => ( ( G @ X3 )
% 14.25/8.14 = one_one_Code_integer ) )
% 14.25/8.14 => ( ( groups3455450783089532116nteger @ G @ S )
% 14.25/8.14 = ( groups3455450783089532116nteger @ G @ T2 ) ) ) ) ) ).
% 14.25/8.14
% 14.25/8.14 thf(fact_7254_prod_Omono__neutral__left,axiom,
% 14.25/8.14 ! [T2: set_nat,S: set_nat,G: nat > code_natural] :
% 14.25/8.14 ( ( finite_finite_nat @ T2 )
% 14.25/8.14 => ( ( ord_less_eq_set_nat @ S @ T2 )
% 14.25/8.14 => ( ! [X3: nat] :
% 14.25/8.14 ( ( member_nat @ X3 @ ( minus_minus_set_nat @ T2 @ S ) )
% 14.25/8.14 => ( ( G @ X3 )
% 14.25/8.14 = one_one_Code_natural ) )
% 14.25/8.14 => ( ( groups2279045934846249631atural @ G @ S )
% 14.25/8.14 = ( groups2279045934846249631atural @ G @ T2 ) ) ) ) ) ).
% 14.25/8.14
% 14.25/8.14 thf(fact_7255_prod_Omono__neutral__left,axiom,
% 14.25/8.14 ! [T2: set_nat,S: set_nat,G: nat > nat] :
% 14.25/8.14 ( ( finite_finite_nat @ T2 )
% 14.25/8.14 => ( ( ord_less_eq_set_nat @ S @ T2 )
% 14.25/8.14 => ( ! [X3: nat] :
% 14.25/8.14 ( ( member_nat @ X3 @ ( minus_minus_set_nat @ T2 @ S ) )
% 14.25/8.14 => ( ( G @ X3 )
% 14.25/8.14 = one_one_nat ) )
% 14.25/8.14 => ( ( groups708209901874060359at_nat @ G @ S )
% 14.25/8.14 = ( groups708209901874060359at_nat @ G @ T2 ) ) ) ) ) ).
% 14.25/8.14
% 14.25/8.14 thf(fact_7256_prod_Omono__neutral__right,axiom,
% 14.25/8.14 ! [T2: set_Code_integer,S: set_Code_integer,G: code_integer > assn] :
% 14.25/8.14 ( ( finite6017078050557962740nteger @ T2 )
% 14.25/8.14 => ( ( ord_le7084787975880047091nteger @ S @ T2 )
% 14.25/8.14 => ( ! [X3: code_integer] :
% 14.25/8.14 ( ( member_Code_integer @ X3 @ ( minus_2355218937544613996nteger @ T2 @ S ) )
% 14.25/8.14 => ( ( G @ X3 )
% 14.25/8.14 = one_one_assn ) )
% 14.25/8.14 => ( ( groups1304777262505850412r_assn @ G @ T2 )
% 14.25/8.14 = ( groups1304777262505850412r_assn @ G @ S ) ) ) ) ) ).
% 14.25/8.14
% 14.25/8.14 thf(fact_7257_prod_Omono__neutral__right,axiom,
% 14.25/8.14 ! [T2: set_Code_integer,S: set_Code_integer,G: code_integer > nat] :
% 14.25/8.14 ( ( finite6017078050557962740nteger @ T2 )
% 14.25/8.14 => ( ( ord_le7084787975880047091nteger @ S @ T2 )
% 14.25/8.14 => ( ! [X3: code_integer] :
% 14.25/8.14 ( ( member_Code_integer @ X3 @ ( minus_2355218937544613996nteger @ T2 @ S ) )
% 14.25/8.14 => ( ( G @ X3 )
% 14.25/8.14 = one_one_nat ) )
% 14.25/8.14 => ( ( groups3190895334310489300er_nat @ G @ T2 )
% 14.25/8.14 = ( groups3190895334310489300er_nat @ G @ S ) ) ) ) ) ).
% 14.25/8.14
% 14.25/8.14 thf(fact_7258_prod_Omono__neutral__right,axiom,
% 14.25/8.14 ! [T2: set_Code_integer,S: set_Code_integer,G: code_integer > int] :
% 14.25/8.14 ( ( finite6017078050557962740nteger @ T2 )
% 14.25/8.14 => ( ( ord_le7084787975880047091nteger @ S @ T2 )
% 14.25/8.14 => ( ! [X3: code_integer] :
% 14.25/8.14 ( ( member_Code_integer @ X3 @ ( minus_2355218937544613996nteger @ T2 @ S ) )
% 14.25/8.14 => ( ( G @ X3 )
% 14.25/8.14 = one_one_int ) )
% 14.25/8.14 => ( ( groups3188404863801439024er_int @ G @ T2 )
% 14.25/8.14 = ( groups3188404863801439024er_int @ G @ S ) ) ) ) ) ).
% 14.25/8.14
% 14.25/8.14 thf(fact_7259_prod_Omono__neutral__right,axiom,
% 14.25/8.14 ! [T2: set_Code_integer,S: set_Code_integer,G: code_integer > code_integer] :
% 14.25/8.14 ( ( finite6017078050557962740nteger @ T2 )
% 14.25/8.14 => ( ( ord_le7084787975880047091nteger @ S @ T2 )
% 14.25/8.14 => ( ! [X3: code_integer] :
% 14.25/8.14 ( ( member_Code_integer @ X3 @ ( minus_2355218937544613996nteger @ T2 @ S ) )
% 14.25/8.14 => ( ( G @ X3 )
% 14.25/8.14 = one_one_Code_integer ) )
% 14.25/8.14 => ( ( groups3674199335183972705nteger @ G @ T2 )
% 14.25/8.14 = ( groups3674199335183972705nteger @ G @ S ) ) ) ) ) ).
% 14.25/8.14
% 14.25/8.14 thf(fact_7260_prod_Omono__neutral__right,axiom,
% 14.25/8.14 ! [T2: set_Code_integer,S: set_Code_integer,G: code_integer > code_natural] :
% 14.25/8.14 ( ( finite6017078050557962740nteger @ T2 )
% 14.25/8.14 => ( ( ord_le7084787975880047091nteger @ S @ T2 )
% 14.25/8.14 => ( ! [X3: code_integer] :
% 14.25/8.14 ( ( member_Code_integer @ X3 @ ( minus_2355218937544613996nteger @ T2 @ S ) )
% 14.25/8.14 => ( ( G @ X3 )
% 14.25/8.14 = one_one_Code_natural ) )
% 14.25/8.14 => ( ( groups2497794486940690220atural @ G @ T2 )
% 14.25/8.14 = ( groups2497794486940690220atural @ G @ S ) ) ) ) ) ).
% 14.25/8.14
% 14.25/8.14 thf(fact_7261_prod_Omono__neutral__right,axiom,
% 14.25/8.14 ! [T2: set_nat,S: set_nat,G: nat > assn] :
% 14.25/8.14 ( ( finite_finite_nat @ T2 )
% 14.25/8.14 => ( ( ord_less_eq_set_nat @ S @ T2 )
% 14.25/8.14 => ( ! [X3: nat] :
% 14.25/8.14 ( ( member_nat @ X3 @ ( minus_minus_set_nat @ T2 @ S ) )
% 14.25/8.14 => ( ( G @ X3 )
% 14.25/8.14 = one_one_assn ) )
% 14.25/8.14 => ( ( groups6906906614972039071t_assn @ G @ T2 )
% 14.25/8.14 = ( groups6906906614972039071t_assn @ G @ S ) ) ) ) ) ).
% 14.25/8.14
% 14.25/8.14 thf(fact_7262_prod_Omono__neutral__right,axiom,
% 14.25/8.14 ! [T2: set_nat,S: set_nat,G: nat > int] :
% 14.25/8.14 ( ( finite_finite_nat @ T2 )
% 14.25/8.14 => ( ( ord_less_eq_set_nat @ S @ T2 )
% 14.25/8.14 => ( ! [X3: nat] :
% 14.25/8.14 ( ( member_nat @ X3 @ ( minus_minus_set_nat @ T2 @ S ) )
% 14.25/8.14 => ( ( G @ X3 )
% 14.25/8.14 = one_one_int ) )
% 14.25/8.14 => ( ( groups705719431365010083at_int @ G @ T2 )
% 14.25/8.15 = ( groups705719431365010083at_int @ G @ S ) ) ) ) ) ).
% 14.25/8.15
% 14.25/8.15 thf(fact_7263_prod_Omono__neutral__right,axiom,
% 14.25/8.15 ! [T2: set_nat,S: set_nat,G: nat > code_integer] :
% 14.25/8.15 ( ( finite_finite_nat @ T2 )
% 14.25/8.15 => ( ( ord_less_eq_set_nat @ S @ T2 )
% 14.25/8.15 => ( ! [X3: nat] :
% 14.25/8.15 ( ( member_nat @ X3 @ ( minus_minus_set_nat @ T2 @ S ) )
% 14.25/8.15 => ( ( G @ X3 )
% 14.25/8.15 = one_one_Code_integer ) )
% 14.25/8.15 => ( ( groups3455450783089532116nteger @ G @ T2 )
% 14.25/8.15 = ( groups3455450783089532116nteger @ G @ S ) ) ) ) ) ).
% 14.25/8.15
% 14.25/8.15 thf(fact_7264_prod_Omono__neutral__right,axiom,
% 14.25/8.15 ! [T2: set_nat,S: set_nat,G: nat > code_natural] :
% 14.25/8.15 ( ( finite_finite_nat @ T2 )
% 14.25/8.15 => ( ( ord_less_eq_set_nat @ S @ T2 )
% 14.25/8.15 => ( ! [X3: nat] :
% 14.25/8.15 ( ( member_nat @ X3 @ ( minus_minus_set_nat @ T2 @ S ) )
% 14.25/8.15 => ( ( G @ X3 )
% 14.25/8.15 = one_one_Code_natural ) )
% 14.25/8.15 => ( ( groups2279045934846249631atural @ G @ T2 )
% 14.25/8.15 = ( groups2279045934846249631atural @ G @ S ) ) ) ) ) ).
% 14.25/8.15
% 14.25/8.15 thf(fact_7265_prod_Omono__neutral__right,axiom,
% 14.25/8.15 ! [T2: set_nat,S: set_nat,G: nat > nat] :
% 14.25/8.15 ( ( finite_finite_nat @ T2 )
% 14.25/8.15 => ( ( ord_less_eq_set_nat @ S @ T2 )
% 14.25/8.15 => ( ! [X3: nat] :
% 14.25/8.15 ( ( member_nat @ X3 @ ( minus_minus_set_nat @ T2 @ S ) )
% 14.25/8.15 => ( ( G @ X3 )
% 14.25/8.15 = one_one_nat ) )
% 14.25/8.15 => ( ( groups708209901874060359at_nat @ G @ T2 )
% 14.25/8.15 = ( groups708209901874060359at_nat @ G @ S ) ) ) ) ) ).
% 14.25/8.15
% 14.25/8.15 thf(fact_7266_prod_Omono__neutral__cong__left,axiom,
% 14.25/8.15 ! [T2: set_int,S: set_int,H2: int > assn,G: int > assn] :
% 14.25/8.15 ( ( finite_finite_int @ T2 )
% 14.25/8.15 => ( ( ord_less_eq_set_int @ S @ T2 )
% 14.25/8.15 => ( ! [X3: int] :
% 14.25/8.15 ( ( member_int @ X3 @ ( minus_minus_set_int @ T2 @ S ) )
% 14.25/8.15 => ( ( H2 @ X3 )
% 14.25/8.15 = one_one_assn ) )
% 14.25/8.15 => ( ! [X3: int] :
% 14.25/8.15 ( ( member_int @ X3 @ S )
% 14.25/8.15 => ( ( G @ X3 )
% 14.25/8.15 = ( H2 @ X3 ) ) )
% 14.25/8.15 => ( ( groups7882442080178216443t_assn @ G @ S )
% 14.25/8.15 = ( groups7882442080178216443t_assn @ H2 @ T2 ) ) ) ) ) ) ).
% 14.25/8.15
% 14.25/8.15 thf(fact_7267_prod_Omono__neutral__cong__left,axiom,
% 14.25/8.15 ! [T2: set_o,S: set_o,H2: $o > assn,G: $o > assn] :
% 14.25/8.15 ( ( finite_finite_o @ T2 )
% 14.25/8.15 => ( ( ord_less_eq_set_o @ S @ T2 )
% 14.25/8.15 => ( ! [X3: $o] :
% 14.25/8.15 ( ( member_o @ X3 @ ( minus_minus_set_o @ T2 @ S ) )
% 14.25/8.15 => ( ( H2 @ X3 )
% 14.25/8.15 = one_one_assn ) )
% 14.25/8.15 => ( ! [X3: $o] :
% 14.25/8.15 ( ( member_o @ X3 @ S )
% 14.25/8.15 => ( ( G @ X3 )
% 14.25/8.15 = ( H2 @ X3 ) ) )
% 14.25/8.15 => ( ( groups5301882518646026715o_assn @ G @ S )
% 14.25/8.15 = ( groups5301882518646026715o_assn @ H2 @ T2 ) ) ) ) ) ) ).
% 14.25/8.15
% 14.25/8.15 thf(fact_7268_prod_Omono__neutral__cong__left,axiom,
% 14.25/8.15 ! [T2: set_Product_unit,S: set_Product_unit,H2: product_unit > assn,G: product_unit > assn] :
% 14.25/8.15 ( ( finite4290736615968046902t_unit @ T2 )
% 14.25/8.15 => ( ( ord_le3507040750410214029t_unit @ S @ T2 )
% 14.25/8.15 => ( ! [X3: product_unit] :
% 14.25/8.15 ( ( member_Product_unit @ X3 @ ( minus_6452836326544984404t_unit @ T2 @ S ) )
% 14.25/8.15 => ( ( H2 @ X3 )
% 14.25/8.15 = one_one_assn ) )
% 14.25/8.15 => ( ! [X3: product_unit] :
% 14.25/8.15 ( ( member_Product_unit @ X3 @ S )
% 14.25/8.15 => ( ( G @ X3 )
% 14.25/8.15 = ( H2 @ X3 ) ) )
% 14.25/8.15 => ( ( groups5360233303959380816t_assn @ G @ S )
% 14.25/8.15 = ( groups5360233303959380816t_assn @ H2 @ T2 ) ) ) ) ) ) ).
% 14.25/8.15
% 14.25/8.15 thf(fact_7269_prod_Omono__neutral__cong__left,axiom,
% 14.25/8.15 ! [T2: set_Code_integer,S: set_Code_integer,H2: code_integer > assn,G: code_integer > assn] :
% 14.25/8.15 ( ( finite6017078050557962740nteger @ T2 )
% 14.25/8.15 => ( ( ord_le7084787975880047091nteger @ S @ T2 )
% 14.25/8.15 => ( ! [X3: code_integer] :
% 14.25/8.15 ( ( member_Code_integer @ X3 @ ( minus_2355218937544613996nteger @ T2 @ S ) )
% 14.25/8.15 => ( ( H2 @ X3 )
% 14.25/8.15 = one_one_assn ) )
% 14.25/8.15 => ( ! [X3: code_integer] :
% 14.25/8.15 ( ( member_Code_integer @ X3 @ S )
% 14.25/8.15 => ( ( G @ X3 )
% 14.25/8.15 = ( H2 @ X3 ) ) )
% 14.25/8.15 => ( ( groups1304777262505850412r_assn @ G @ S )
% 14.25/8.15 = ( groups1304777262505850412r_assn @ H2 @ T2 ) ) ) ) ) ) ).
% 14.25/8.15
% 14.25/8.15 thf(fact_7270_prod_Omono__neutral__cong__left,axiom,
% 14.25/8.15 ! [T2: set_int,S: set_int,H2: int > nat,G: int > nat] :
% 14.25/8.15 ( ( finite_finite_int @ T2 )
% 14.25/8.15 => ( ( ord_less_eq_set_int @ S @ T2 )
% 14.25/8.15 => ( ! [X3: int] :
% 14.25/8.15 ( ( member_int @ X3 @ ( minus_minus_set_int @ T2 @ S ) )
% 14.25/8.15 => ( ( H2 @ X3 )
% 14.25/8.15 = one_one_nat ) )
% 14.25/8.15 => ( ! [X3: int] :
% 14.25/8.15 ( ( member_int @ X3 @ S )
% 14.25/8.15 => ( ( G @ X3 )
% 14.25/8.15 = ( H2 @ X3 ) ) )
% 14.25/8.15 => ( ( groups1707563613775114915nt_nat @ G @ S )
% 14.25/8.15 = ( groups1707563613775114915nt_nat @ H2 @ T2 ) ) ) ) ) ) ).
% 14.25/8.15
% 14.25/8.15 thf(fact_7271_prod_Omono__neutral__cong__left,axiom,
% 14.25/8.15 ! [T2: set_o,S: set_o,H2: $o > nat,G: $o > nat] :
% 14.25/8.15 ( ( finite_finite_o @ T2 )
% 14.25/8.15 => ( ( ord_less_eq_set_o @ S @ T2 )
% 14.25/8.15 => ( ! [X3: $o] :
% 14.25/8.15 ( ( member_o @ X3 @ ( minus_minus_set_o @ T2 @ S ) )
% 14.25/8.15 => ( ( H2 @ X3 )
% 14.25/8.15 = one_one_nat ) )
% 14.25/8.15 => ( ! [X3: $o] :
% 14.25/8.15 ( ( member_o @ X3 @ S )
% 14.25/8.15 => ( ( G @ X3 )
% 14.25/8.15 = ( H2 @ X3 ) ) )
% 14.25/8.15 => ( ( groups3504817904513533571_o_nat @ G @ S )
% 14.25/8.15 = ( groups3504817904513533571_o_nat @ H2 @ T2 ) ) ) ) ) ) ).
% 14.25/8.15
% 14.25/8.15 thf(fact_7272_prod_Omono__neutral__cong__left,axiom,
% 14.25/8.15 ! [T2: set_Product_unit,S: set_Product_unit,H2: product_unit > nat,G: product_unit > nat] :
% 14.25/8.15 ( ( finite4290736615968046902t_unit @ T2 )
% 14.25/8.15 => ( ( ord_le3507040750410214029t_unit @ S @ T2 )
% 14.25/8.15 => ( ! [X3: product_unit] :
% 14.25/8.15 ( ( member_Product_unit @ X3 @ ( minus_6452836326544984404t_unit @ T2 @ S ) )
% 14.25/8.15 => ( ( H2 @ X3 )
% 14.25/8.15 = one_one_nat ) )
% 14.25/8.15 => ( ! [X3: product_unit] :
% 14.25/8.15 ( ( member_Product_unit @ X3 @ S )
% 14.25/8.15 => ( ( G @ X3 )
% 14.25/8.15 = ( H2 @ X3 ) ) )
% 14.25/8.15 => ( ( groups396421249026171384it_nat @ G @ S )
% 14.25/8.15 = ( groups396421249026171384it_nat @ H2 @ T2 ) ) ) ) ) ) ).
% 14.25/8.15
% 14.25/8.15 thf(fact_7273_prod_Omono__neutral__cong__left,axiom,
% 14.25/8.15 ! [T2: set_Code_integer,S: set_Code_integer,H2: code_integer > nat,G: code_integer > nat] :
% 14.25/8.15 ( ( finite6017078050557962740nteger @ T2 )
% 14.25/8.15 => ( ( ord_le7084787975880047091nteger @ S @ T2 )
% 14.25/8.15 => ( ! [X3: code_integer] :
% 14.25/8.15 ( ( member_Code_integer @ X3 @ ( minus_2355218937544613996nteger @ T2 @ S ) )
% 14.25/8.15 => ( ( H2 @ X3 )
% 14.25/8.15 = one_one_nat ) )
% 14.25/8.15 => ( ! [X3: code_integer] :
% 14.25/8.15 ( ( member_Code_integer @ X3 @ S )
% 14.25/8.15 => ( ( G @ X3 )
% 14.25/8.15 = ( H2 @ X3 ) ) )
% 14.25/8.15 => ( ( groups3190895334310489300er_nat @ G @ S )
% 14.25/8.15 = ( groups3190895334310489300er_nat @ H2 @ T2 ) ) ) ) ) ) ).
% 14.25/8.15
% 14.25/8.15 thf(fact_7274_prod_Omono__neutral__cong__left,axiom,
% 14.25/8.15 ! [T2: set_int,S: set_int,H2: int > int,G: int > int] :
% 14.25/8.15 ( ( finite_finite_int @ T2 )
% 14.25/8.15 => ( ( ord_less_eq_set_int @ S @ T2 )
% 14.25/8.15 => ( ! [X3: int] :
% 14.25/8.15 ( ( member_int @ X3 @ ( minus_minus_set_int @ T2 @ S ) )
% 14.25/8.15 => ( ( H2 @ X3 )
% 14.25/8.15 = one_one_int ) )
% 14.25/8.15 => ( ! [X3: int] :
% 14.25/8.15 ( ( member_int @ X3 @ S )
% 14.25/8.15 => ( ( G @ X3 )
% 14.25/8.15 = ( H2 @ X3 ) ) )
% 14.25/8.15 => ( ( groups1705073143266064639nt_int @ G @ S )
% 14.25/8.15 = ( groups1705073143266064639nt_int @ H2 @ T2 ) ) ) ) ) ) ).
% 14.25/8.15
% 14.25/8.15 thf(fact_7275_prod_Omono__neutral__cong__left,axiom,
% 14.25/8.15 ! [T2: set_o,S: set_o,H2: $o > int,G: $o > int] :
% 14.25/8.15 ( ( finite_finite_o @ T2 )
% 14.25/8.15 => ( ( ord_less_eq_set_o @ S @ T2 )
% 14.25/8.15 => ( ! [X3: $o] :
% 14.25/8.15 ( ( member_o @ X3 @ ( minus_minus_set_o @ T2 @ S ) )
% 14.25/8.15 => ( ( H2 @ X3 )
% 14.25/8.15 = one_one_int ) )
% 14.25/8.15 => ( ! [X3: $o] :
% 14.25/8.15 ( ( member_o @ X3 @ S )
% 14.25/8.15 => ( ( G @ X3 )
% 14.25/8.15 = ( H2 @ X3 ) ) )
% 14.25/8.15 => ( ( groups3502327434004483295_o_int @ G @ S )
% 14.25/8.15 = ( groups3502327434004483295_o_int @ H2 @ T2 ) ) ) ) ) ) ).
% 14.25/8.15
% 14.25/8.15 thf(fact_7276_prod_Omono__neutral__cong__right,axiom,
% 14.25/8.15 ! [T2: set_int,S: set_int,G: int > assn,H2: int > assn] :
% 14.25/8.15 ( ( finite_finite_int @ T2 )
% 14.25/8.15 => ( ( ord_less_eq_set_int @ S @ T2 )
% 14.25/8.15 => ( ! [X3: int] :
% 14.25/8.15 ( ( member_int @ X3 @ ( minus_minus_set_int @ T2 @ S ) )
% 14.25/8.15 => ( ( G @ X3 )
% 14.25/8.15 = one_one_assn ) )
% 14.25/8.15 => ( ! [X3: int] :
% 14.25/8.15 ( ( member_int @ X3 @ S )
% 14.25/8.15 => ( ( G @ X3 )
% 14.25/8.15 = ( H2 @ X3 ) ) )
% 14.25/8.15 => ( ( groups7882442080178216443t_assn @ G @ T2 )
% 14.25/8.15 = ( groups7882442080178216443t_assn @ H2 @ S ) ) ) ) ) ) ).
% 14.25/8.15
% 14.25/8.15 thf(fact_7277_prod_Omono__neutral__cong__right,axiom,
% 14.25/8.15 ! [T2: set_o,S: set_o,G: $o > assn,H2: $o > assn] :
% 14.25/8.15 ( ( finite_finite_o @ T2 )
% 14.25/8.15 => ( ( ord_less_eq_set_o @ S @ T2 )
% 14.25/8.15 => ( ! [X3: $o] :
% 14.25/8.15 ( ( member_o @ X3 @ ( minus_minus_set_o @ T2 @ S ) )
% 14.25/8.15 => ( ( G @ X3 )
% 14.25/8.15 = one_one_assn ) )
% 14.25/8.15 => ( ! [X3: $o] :
% 14.25/8.15 ( ( member_o @ X3 @ S )
% 14.25/8.15 => ( ( G @ X3 )
% 14.25/8.15 = ( H2 @ X3 ) ) )
% 14.25/8.15 => ( ( groups5301882518646026715o_assn @ G @ T2 )
% 14.25/8.15 = ( groups5301882518646026715o_assn @ H2 @ S ) ) ) ) ) ) ).
% 14.25/8.15
% 14.25/8.15 thf(fact_7278_prod_Omono__neutral__cong__right,axiom,
% 14.25/8.15 ! [T2: set_Product_unit,S: set_Product_unit,G: product_unit > assn,H2: product_unit > assn] :
% 14.25/8.15 ( ( finite4290736615968046902t_unit @ T2 )
% 14.25/8.15 => ( ( ord_le3507040750410214029t_unit @ S @ T2 )
% 14.25/8.15 => ( ! [X3: product_unit] :
% 14.25/8.15 ( ( member_Product_unit @ X3 @ ( minus_6452836326544984404t_unit @ T2 @ S ) )
% 14.25/8.15 => ( ( G @ X3 )
% 14.25/8.15 = one_one_assn ) )
% 14.25/8.15 => ( ! [X3: product_unit] :
% 14.25/8.15 ( ( member_Product_unit @ X3 @ S )
% 14.25/8.15 => ( ( G @ X3 )
% 14.25/8.15 = ( H2 @ X3 ) ) )
% 14.25/8.15 => ( ( groups5360233303959380816t_assn @ G @ T2 )
% 14.25/8.15 = ( groups5360233303959380816t_assn @ H2 @ S ) ) ) ) ) ) ).
% 14.25/8.15
% 14.25/8.15 thf(fact_7279_prod_Omono__neutral__cong__right,axiom,
% 14.25/8.15 ! [T2: set_Code_integer,S: set_Code_integer,G: code_integer > assn,H2: code_integer > assn] :
% 14.25/8.15 ( ( finite6017078050557962740nteger @ T2 )
% 14.25/8.15 => ( ( ord_le7084787975880047091nteger @ S @ T2 )
% 14.25/8.15 => ( ! [X3: code_integer] :
% 14.25/8.15 ( ( member_Code_integer @ X3 @ ( minus_2355218937544613996nteger @ T2 @ S ) )
% 14.25/8.15 => ( ( G @ X3 )
% 14.25/8.15 = one_one_assn ) )
% 14.25/8.15 => ( ! [X3: code_integer] :
% 14.25/8.15 ( ( member_Code_integer @ X3 @ S )
% 14.25/8.15 => ( ( G @ X3 )
% 14.25/8.15 = ( H2 @ X3 ) ) )
% 14.25/8.15 => ( ( groups1304777262505850412r_assn @ G @ T2 )
% 14.25/8.15 = ( groups1304777262505850412r_assn @ H2 @ S ) ) ) ) ) ) ).
% 14.25/8.15
% 14.25/8.15 thf(fact_7280_prod_Omono__neutral__cong__right,axiom,
% 14.25/8.15 ! [T2: set_int,S: set_int,G: int > nat,H2: int > nat] :
% 14.25/8.15 ( ( finite_finite_int @ T2 )
% 14.25/8.15 => ( ( ord_less_eq_set_int @ S @ T2 )
% 14.25/8.15 => ( ! [X3: int] :
% 14.25/8.15 ( ( member_int @ X3 @ ( minus_minus_set_int @ T2 @ S ) )
% 14.25/8.15 => ( ( G @ X3 )
% 14.25/8.15 = one_one_nat ) )
% 14.25/8.15 => ( ! [X3: int] :
% 14.25/8.15 ( ( member_int @ X3 @ S )
% 14.25/8.15 => ( ( G @ X3 )
% 14.25/8.15 = ( H2 @ X3 ) ) )
% 14.25/8.15 => ( ( groups1707563613775114915nt_nat @ G @ T2 )
% 14.25/8.15 = ( groups1707563613775114915nt_nat @ H2 @ S ) ) ) ) ) ) ).
% 14.25/8.15
% 14.25/8.15 thf(fact_7281_prod_Omono__neutral__cong__right,axiom,
% 14.25/8.15 ! [T2: set_o,S: set_o,G: $o > nat,H2: $o > nat] :
% 14.25/8.15 ( ( finite_finite_o @ T2 )
% 14.25/8.15 => ( ( ord_less_eq_set_o @ S @ T2 )
% 14.25/8.15 => ( ! [X3: $o] :
% 14.25/8.15 ( ( member_o @ X3 @ ( minus_minus_set_o @ T2 @ S ) )
% 14.25/8.15 => ( ( G @ X3 )
% 14.25/8.15 = one_one_nat ) )
% 14.25/8.15 => ( ! [X3: $o] :
% 14.25/8.15 ( ( member_o @ X3 @ S )
% 14.25/8.15 => ( ( G @ X3 )
% 14.25/8.15 = ( H2 @ X3 ) ) )
% 14.25/8.15 => ( ( groups3504817904513533571_o_nat @ G @ T2 )
% 14.25/8.15 = ( groups3504817904513533571_o_nat @ H2 @ S ) ) ) ) ) ) ).
% 14.25/8.15
% 14.25/8.15 thf(fact_7282_prod_Omono__neutral__cong__right,axiom,
% 14.25/8.15 ! [T2: set_Product_unit,S: set_Product_unit,G: product_unit > nat,H2: product_unit > nat] :
% 14.25/8.15 ( ( finite4290736615968046902t_unit @ T2 )
% 14.25/8.15 => ( ( ord_le3507040750410214029t_unit @ S @ T2 )
% 14.25/8.15 => ( ! [X3: product_unit] :
% 14.25/8.15 ( ( member_Product_unit @ X3 @ ( minus_6452836326544984404t_unit @ T2 @ S ) )
% 14.25/8.15 => ( ( G @ X3 )
% 14.25/8.15 = one_one_nat ) )
% 14.25/8.15 => ( ! [X3: product_unit] :
% 14.25/8.15 ( ( member_Product_unit @ X3 @ S )
% 14.25/8.15 => ( ( G @ X3 )
% 14.25/8.15 = ( H2 @ X3 ) ) )
% 14.25/8.15 => ( ( groups396421249026171384it_nat @ G @ T2 )
% 14.25/8.15 = ( groups396421249026171384it_nat @ H2 @ S ) ) ) ) ) ) ).
% 14.25/8.15
% 14.25/8.15 thf(fact_7283_prod_Omono__neutral__cong__right,axiom,
% 14.25/8.15 ! [T2: set_Code_integer,S: set_Code_integer,G: code_integer > nat,H2: code_integer > nat] :
% 14.25/8.15 ( ( finite6017078050557962740nteger @ T2 )
% 14.25/8.15 => ( ( ord_le7084787975880047091nteger @ S @ T2 )
% 14.25/8.15 => ( ! [X3: code_integer] :
% 14.25/8.15 ( ( member_Code_integer @ X3 @ ( minus_2355218937544613996nteger @ T2 @ S ) )
% 14.25/8.15 => ( ( G @ X3 )
% 14.25/8.15 = one_one_nat ) )
% 14.25/8.15 => ( ! [X3: code_integer] :
% 14.25/8.15 ( ( member_Code_integer @ X3 @ S )
% 14.25/8.15 => ( ( G @ X3 )
% 14.25/8.15 = ( H2 @ X3 ) ) )
% 14.25/8.15 => ( ( groups3190895334310489300er_nat @ G @ T2 )
% 14.25/8.15 = ( groups3190895334310489300er_nat @ H2 @ S ) ) ) ) ) ) ).
% 14.25/8.15
% 14.25/8.15 thf(fact_7284_prod_Omono__neutral__cong__right,axiom,
% 14.25/8.15 ! [T2: set_int,S: set_int,G: int > int,H2: int > int] :
% 14.25/8.15 ( ( finite_finite_int @ T2 )
% 14.25/8.15 => ( ( ord_less_eq_set_int @ S @ T2 )
% 14.25/8.15 => ( ! [X3: int] :
% 14.25/8.15 ( ( member_int @ X3 @ ( minus_minus_set_int @ T2 @ S ) )
% 14.25/8.15 => ( ( G @ X3 )
% 14.25/8.15 = one_one_int ) )
% 14.25/8.15 => ( ! [X3: int] :
% 14.25/8.15 ( ( member_int @ X3 @ S )
% 14.25/8.15 => ( ( G @ X3 )
% 14.25/8.15 = ( H2 @ X3 ) ) )
% 14.25/8.15 => ( ( groups1705073143266064639nt_int @ G @ T2 )
% 14.25/8.15 = ( groups1705073143266064639nt_int @ H2 @ S ) ) ) ) ) ) ).
% 14.25/8.15
% 14.25/8.15 thf(fact_7285_prod_Omono__neutral__cong__right,axiom,
% 14.25/8.15 ! [T2: set_o,S: set_o,G: $o > int,H2: $o > int] :
% 14.25/8.15 ( ( finite_finite_o @ T2 )
% 14.25/8.15 => ( ( ord_less_eq_set_o @ S @ T2 )
% 14.25/8.15 => ( ! [X3: $o] :
% 14.25/8.15 ( ( member_o @ X3 @ ( minus_minus_set_o @ T2 @ S ) )
% 14.25/8.15 => ( ( G @ X3 )
% 14.25/8.15 = one_one_int ) )
% 14.25/8.15 => ( ! [X3: $o] :
% 14.25/8.15 ( ( member_o @ X3 @ S )
% 14.25/8.15 => ( ( G @ X3 )
% 14.25/8.15 = ( H2 @ X3 ) ) )
% 14.25/8.15 => ( ( groups3502327434004483295_o_int @ G @ T2 )
% 14.25/8.15 = ( groups3502327434004483295_o_int @ H2 @ S ) ) ) ) ) ) ).
% 14.25/8.15
% 14.25/8.15 thf(fact_7286_prod_Ounion__inter,axiom,
% 14.25/8.15 ! [A: set_Code_integer,B4: set_Code_integer,G: code_integer > assn] :
% 14.25/8.15 ( ( finite6017078050557962740nteger @ A )
% 14.25/8.15 => ( ( finite6017078050557962740nteger @ B4 )
% 14.25/8.15 => ( ( times_times_assn @ ( groups1304777262505850412r_assn @ G @ ( sup_su848401254843788991nteger @ A @ B4 ) ) @ ( groups1304777262505850412r_assn @ G @ ( inf_in1364745209274528805nteger @ A @ B4 ) ) )
% 14.25/8.15 = ( times_times_assn @ ( groups1304777262505850412r_assn @ G @ A ) @ ( groups1304777262505850412r_assn @ G @ B4 ) ) ) ) ) ).
% 14.25/8.15
% 14.25/8.15 thf(fact_7287_prod_Ounion__inter,axiom,
% 14.25/8.15 ! [A: set_nat,B4: set_nat,G: nat > assn] :
% 14.25/8.15 ( ( finite_finite_nat @ A )
% 14.25/8.15 => ( ( finite_finite_nat @ B4 )
% 14.25/8.15 => ( ( times_times_assn @ ( groups6906906614972039071t_assn @ G @ ( sup_sup_set_nat @ A @ B4 ) ) @ ( groups6906906614972039071t_assn @ G @ ( inf_inf_set_nat @ A @ B4 ) ) )
% 14.25/8.15 = ( times_times_assn @ ( groups6906906614972039071t_assn @ G @ A ) @ ( groups6906906614972039071t_assn @ G @ B4 ) ) ) ) ) ).
% 14.25/8.15
% 14.25/8.15 thf(fact_7288_prod_Ounion__inter,axiom,
% 14.25/8.15 ! [A: set_Code_integer,B4: set_Code_integer,G: code_integer > rat] :
% 14.25/8.15 ( ( finite6017078050557962740nteger @ A )
% 14.25/8.15 => ( ( finite6017078050557962740nteger @ B4 )
% 14.25/8.15 => ( ( times_times_rat @ ( groups2555765274223993564er_rat @ G @ ( sup_su848401254843788991nteger @ A @ B4 ) ) @ ( groups2555765274223993564er_rat @ G @ ( inf_in1364745209274528805nteger @ A @ B4 ) ) )
% 14.25/8.15 = ( times_times_rat @ ( groups2555765274223993564er_rat @ G @ A ) @ ( groups2555765274223993564er_rat @ G @ B4 ) ) ) ) ) ).
% 14.25/8.15
% 14.25/8.15 thf(fact_7289_prod_Ounion__inter,axiom,
% 14.25/8.15 ! [A: set_nat,B4: set_nat,G: nat > rat] :
% 14.25/8.15 ( ( finite_finite_nat @ A )
% 14.25/8.15 => ( ( finite_finite_nat @ B4 )
% 14.25/8.15 => ( ( times_times_rat @ ( groups73079841787564623at_rat @ G @ ( sup_sup_set_nat @ A @ B4 ) ) @ ( groups73079841787564623at_rat @ G @ ( inf_inf_set_nat @ A @ B4 ) ) )
% 14.25/8.15 = ( times_times_rat @ ( groups73079841787564623at_rat @ G @ A ) @ ( groups73079841787564623at_rat @ G @ B4 ) ) ) ) ) ).
% 14.25/8.15
% 14.25/8.15 thf(fact_7290_prod_Ounion__inter,axiom,
% 14.25/8.15 ! [A: set_Code_integer,B4: set_Code_integer,G: code_integer > nat] :
% 14.25/8.15 ( ( finite6017078050557962740nteger @ A )
% 14.25/8.15 => ( ( finite6017078050557962740nteger @ B4 )
% 14.25/8.15 => ( ( times_times_nat @ ( groups3190895334310489300er_nat @ G @ ( sup_su848401254843788991nteger @ A @ B4 ) ) @ ( groups3190895334310489300er_nat @ G @ ( inf_in1364745209274528805nteger @ A @ B4 ) ) )
% 14.25/8.15 = ( times_times_nat @ ( groups3190895334310489300er_nat @ G @ A ) @ ( groups3190895334310489300er_nat @ G @ B4 ) ) ) ) ) ).
% 14.25/8.15
% 14.25/8.15 thf(fact_7291_prod_Ounion__inter,axiom,
% 14.25/8.15 ! [A: set_Code_integer,B4: set_Code_integer,G: code_integer > int] :
% 14.25/8.15 ( ( finite6017078050557962740nteger @ A )
% 14.25/8.15 => ( ( finite6017078050557962740nteger @ B4 )
% 14.25/8.15 => ( ( times_times_int @ ( groups3188404863801439024er_int @ G @ ( sup_su848401254843788991nteger @ A @ B4 ) ) @ ( groups3188404863801439024er_int @ G @ ( inf_in1364745209274528805nteger @ A @ B4 ) ) )
% 14.25/8.15 = ( times_times_int @ ( groups3188404863801439024er_int @ G @ A ) @ ( groups3188404863801439024er_int @ G @ B4 ) ) ) ) ) ).
% 14.25/8.15
% 14.25/8.15 thf(fact_7292_prod_Ounion__inter,axiom,
% 14.25/8.15 ! [A: set_nat,B4: set_nat,G: nat > int] :
% 14.25/8.15 ( ( finite_finite_nat @ A )
% 14.25/8.15 => ( ( finite_finite_nat @ B4 )
% 14.25/8.15 => ( ( times_times_int @ ( groups705719431365010083at_int @ G @ ( sup_sup_set_nat @ A @ B4 ) ) @ ( groups705719431365010083at_int @ G @ ( inf_inf_set_nat @ A @ B4 ) ) )
% 14.25/8.15 = ( times_times_int @ ( groups705719431365010083at_int @ G @ A ) @ ( groups705719431365010083at_int @ G @ B4 ) ) ) ) ) ).
% 14.25/8.15
% 14.25/8.15 thf(fact_7293_prod_Ounion__inter,axiom,
% 14.25/8.15 ! [A: set_nat,B4: set_nat,G: nat > nat] :
% 14.25/8.15 ( ( finite_finite_nat @ A )
% 14.25/8.15 => ( ( finite_finite_nat @ B4 )
% 14.25/8.15 => ( ( times_times_nat @ ( groups708209901874060359at_nat @ G @ ( sup_sup_set_nat @ A @ B4 ) ) @ ( groups708209901874060359at_nat @ G @ ( inf_inf_set_nat @ A @ B4 ) ) )
% 14.25/8.15 = ( times_times_nat @ ( groups708209901874060359at_nat @ G @ A ) @ ( groups708209901874060359at_nat @ G @ B4 ) ) ) ) ) ).
% 14.25/8.15
% 14.25/8.15 thf(fact_7294_prod_OInt__Diff,axiom,
% 14.25/8.15 ! [A: set_Code_integer,G: code_integer > assn,B4: set_Code_integer] :
% 14.25/8.15 ( ( finite6017078050557962740nteger @ A )
% 14.25/8.15 => ( ( groups1304777262505850412r_assn @ G @ A )
% 14.25/8.15 = ( times_times_assn @ ( groups1304777262505850412r_assn @ G @ ( inf_in1364745209274528805nteger @ A @ B4 ) ) @ ( groups1304777262505850412r_assn @ G @ ( minus_2355218937544613996nteger @ A @ B4 ) ) ) ) ) ).
% 14.25/8.15
% 14.25/8.15 thf(fact_7295_prod_OInt__Diff,axiom,
% 14.25/8.15 ! [A: set_nat,G: nat > assn,B4: set_nat] :
% 14.25/8.15 ( ( finite_finite_nat @ A )
% 14.25/8.15 => ( ( groups6906906614972039071t_assn @ G @ A )
% 14.25/8.15 = ( times_times_assn @ ( groups6906906614972039071t_assn @ G @ ( inf_inf_set_nat @ A @ B4 ) ) @ ( groups6906906614972039071t_assn @ G @ ( minus_minus_set_nat @ A @ B4 ) ) ) ) ) ).
% 14.25/8.15
% 14.25/8.15 thf(fact_7296_prod_OInt__Diff,axiom,
% 14.25/8.15 ! [A: set_Code_integer,G: code_integer > rat,B4: set_Code_integer] :
% 14.25/8.15 ( ( finite6017078050557962740nteger @ A )
% 14.25/8.15 => ( ( groups2555765274223993564er_rat @ G @ A )
% 14.25/8.15 = ( times_times_rat @ ( groups2555765274223993564er_rat @ G @ ( inf_in1364745209274528805nteger @ A @ B4 ) ) @ ( groups2555765274223993564er_rat @ G @ ( minus_2355218937544613996nteger @ A @ B4 ) ) ) ) ) ).
% 14.25/8.15
% 14.25/8.15 thf(fact_7297_prod_OInt__Diff,axiom,
% 14.25/8.15 ! [A: set_nat,G: nat > rat,B4: set_nat] :
% 14.25/8.15 ( ( finite_finite_nat @ A )
% 14.25/8.15 => ( ( groups73079841787564623at_rat @ G @ A )
% 14.25/8.15 = ( times_times_rat @ ( groups73079841787564623at_rat @ G @ ( inf_inf_set_nat @ A @ B4 ) ) @ ( groups73079841787564623at_rat @ G @ ( minus_minus_set_nat @ A @ B4 ) ) ) ) ) ).
% 14.25/8.15
% 14.25/8.15 thf(fact_7298_prod_OInt__Diff,axiom,
% 14.25/8.15 ! [A: set_Code_integer,G: code_integer > nat,B4: set_Code_integer] :
% 14.25/8.15 ( ( finite6017078050557962740nteger @ A )
% 14.25/8.15 => ( ( groups3190895334310489300er_nat @ G @ A )
% 14.25/8.15 = ( times_times_nat @ ( groups3190895334310489300er_nat @ G @ ( inf_in1364745209274528805nteger @ A @ B4 ) ) @ ( groups3190895334310489300er_nat @ G @ ( minus_2355218937544613996nteger @ A @ B4 ) ) ) ) ) ).
% 14.25/8.15
% 14.25/8.15 thf(fact_7299_prod_OInt__Diff,axiom,
% 14.25/8.15 ! [A: set_Code_integer,G: code_integer > int,B4: set_Code_integer] :
% 14.25/8.15 ( ( finite6017078050557962740nteger @ A )
% 14.25/8.15 => ( ( groups3188404863801439024er_int @ G @ A )
% 14.25/8.15 = ( times_times_int @ ( groups3188404863801439024er_int @ G @ ( inf_in1364745209274528805nteger @ A @ B4 ) ) @ ( groups3188404863801439024er_int @ G @ ( minus_2355218937544613996nteger @ A @ B4 ) ) ) ) ) ).
% 14.25/8.15
% 14.25/8.15 thf(fact_7300_prod_OInt__Diff,axiom,
% 14.25/8.15 ! [A: set_nat,G: nat > int,B4: set_nat] :
% 14.25/8.15 ( ( finite_finite_nat @ A )
% 14.25/8.15 => ( ( groups705719431365010083at_int @ G @ A )
% 14.25/8.15 = ( times_times_int @ ( groups705719431365010083at_int @ G @ ( inf_inf_set_nat @ A @ B4 ) ) @ ( groups705719431365010083at_int @ G @ ( minus_minus_set_nat @ A @ B4 ) ) ) ) ) ).
% 14.25/8.15
% 14.25/8.15 thf(fact_7301_prod_OInt__Diff,axiom,
% 14.25/8.15 ! [A: set_nat,G: nat > nat,B4: set_nat] :
% 14.25/8.15 ( ( finite_finite_nat @ A )
% 14.25/8.15 => ( ( groups708209901874060359at_nat @ G @ A )
% 14.25/8.15 = ( times_times_nat @ ( groups708209901874060359at_nat @ G @ ( inf_inf_set_nat @ A @ B4 ) ) @ ( groups708209901874060359at_nat @ G @ ( minus_minus_set_nat @ A @ B4 ) ) ) ) ) ).
% 14.25/8.15
% 14.25/8.15 thf(fact_7302_prod_Omono__neutral__cong,axiom,
% 14.25/8.15 ! [T2: set_int,S: set_int,H2: int > assn,G: int > assn] :
% 14.25/8.15 ( ( finite_finite_int @ T2 )
% 14.25/8.15 => ( ( finite_finite_int @ S )
% 14.25/8.15 => ( ! [I3: int] :
% 14.25/8.15 ( ( member_int @ I3 @ ( minus_minus_set_int @ T2 @ S ) )
% 14.25/8.15 => ( ( H2 @ I3 )
% 14.25/8.15 = one_one_assn ) )
% 14.25/8.15 => ( ! [I3: int] :
% 14.25/8.15 ( ( member_int @ I3 @ ( minus_minus_set_int @ S @ T2 ) )
% 14.25/8.15 => ( ( G @ I3 )
% 14.25/8.15 = one_one_assn ) )
% 14.25/8.15 => ( ! [X3: int] :
% 14.25/8.15 ( ( member_int @ X3 @ ( inf_inf_set_int @ S @ T2 ) )
% 14.25/8.15 => ( ( G @ X3 )
% 14.25/8.15 = ( H2 @ X3 ) ) )
% 14.25/8.15 => ( ( groups7882442080178216443t_assn @ G @ S )
% 14.25/8.15 = ( groups7882442080178216443t_assn @ H2 @ T2 ) ) ) ) ) ) ) ).
% 14.25/8.15
% 14.25/8.15 thf(fact_7303_prod_Omono__neutral__cong,axiom,
% 14.25/8.15 ! [T2: set_o,S: set_o,H2: $o > assn,G: $o > assn] :
% 14.25/8.15 ( ( finite_finite_o @ T2 )
% 14.25/8.15 => ( ( finite_finite_o @ S )
% 14.25/8.15 => ( ! [I3: $o] :
% 14.25/8.15 ( ( member_o @ I3 @ ( minus_minus_set_o @ T2 @ S ) )
% 14.25/8.15 => ( ( H2 @ I3 )
% 14.25/8.15 = one_one_assn ) )
% 14.25/8.15 => ( ! [I3: $o] :
% 14.25/8.15 ( ( member_o @ I3 @ ( minus_minus_set_o @ S @ T2 ) )
% 14.25/8.15 => ( ( G @ I3 )
% 14.25/8.15 = one_one_assn ) )
% 14.25/8.15 => ( ! [X3: $o] :
% 14.25/8.15 ( ( member_o @ X3 @ ( inf_inf_set_o @ S @ T2 ) )
% 14.25/8.15 => ( ( G @ X3 )
% 14.25/8.15 = ( H2 @ X3 ) ) )
% 14.25/8.15 => ( ( groups5301882518646026715o_assn @ G @ S )
% 14.25/8.15 = ( groups5301882518646026715o_assn @ H2 @ T2 ) ) ) ) ) ) ) ).
% 14.25/8.15
% 14.25/8.15 thf(fact_7304_prod_Omono__neutral__cong,axiom,
% 14.25/8.15 ! [T2: set_Product_unit,S: set_Product_unit,H2: product_unit > assn,G: product_unit > assn] :
% 14.25/8.15 ( ( finite4290736615968046902t_unit @ T2 )
% 14.25/8.15 => ( ( finite4290736615968046902t_unit @ S )
% 14.25/8.15 => ( ! [I3: product_unit] :
% 14.25/8.15 ( ( member_Product_unit @ I3 @ ( minus_6452836326544984404t_unit @ T2 @ S ) )
% 14.25/8.15 => ( ( H2 @ I3 )
% 14.25/8.15 = one_one_assn ) )
% 14.25/8.15 => ( ! [I3: product_unit] :
% 14.25/8.15 ( ( member_Product_unit @ I3 @ ( minus_6452836326544984404t_unit @ S @ T2 ) )
% 14.25/8.15 => ( ( G @ I3 )
% 14.25/8.15 = one_one_assn ) )
% 14.25/8.15 => ( ! [X3: product_unit] :
% 14.25/8.15 ( ( member_Product_unit @ X3 @ ( inf_in4660618365625256667t_unit @ S @ T2 ) )
% 14.25/8.15 => ( ( G @ X3 )
% 14.25/8.15 = ( H2 @ X3 ) ) )
% 14.25/8.15 => ( ( groups5360233303959380816t_assn @ G @ S )
% 14.25/8.15 = ( groups5360233303959380816t_assn @ H2 @ T2 ) ) ) ) ) ) ) ).
% 14.25/8.15
% 14.25/8.15 thf(fact_7305_prod_Omono__neutral__cong,axiom,
% 14.25/8.15 ! [T2: set_Code_integer,S: set_Code_integer,H2: code_integer > assn,G: code_integer > assn] :
% 14.25/8.15 ( ( finite6017078050557962740nteger @ T2 )
% 14.25/8.15 => ( ( finite6017078050557962740nteger @ S )
% 14.25/8.15 => ( ! [I3: code_integer] :
% 14.25/8.15 ( ( member_Code_integer @ I3 @ ( minus_2355218937544613996nteger @ T2 @ S ) )
% 14.25/8.15 => ( ( H2 @ I3 )
% 14.25/8.15 = one_one_assn ) )
% 14.25/8.15 => ( ! [I3: code_integer] :
% 14.25/8.15 ( ( member_Code_integer @ I3 @ ( minus_2355218937544613996nteger @ S @ T2 ) )
% 14.25/8.15 => ( ( G @ I3 )
% 14.25/8.15 = one_one_assn ) )
% 14.25/8.15 => ( ! [X3: code_integer] :
% 14.25/8.15 ( ( member_Code_integer @ X3 @ ( inf_in1364745209274528805nteger @ S @ T2 ) )
% 14.25/8.15 => ( ( G @ X3 )
% 14.25/8.15 = ( H2 @ X3 ) ) )
% 14.25/8.15 => ( ( groups1304777262505850412r_assn @ G @ S )
% 14.25/8.15 = ( groups1304777262505850412r_assn @ H2 @ T2 ) ) ) ) ) ) ) ).
% 14.25/8.15
% 14.25/8.15 thf(fact_7306_prod_Omono__neutral__cong,axiom,
% 14.25/8.15 ! [T2: set_int,S: set_int,H2: int > nat,G: int > nat] :
% 14.25/8.15 ( ( finite_finite_int @ T2 )
% 14.25/8.15 => ( ( finite_finite_int @ S )
% 14.25/8.15 => ( ! [I3: int] :
% 14.25/8.15 ( ( member_int @ I3 @ ( minus_minus_set_int @ T2 @ S ) )
% 14.25/8.15 => ( ( H2 @ I3 )
% 14.25/8.15 = one_one_nat ) )
% 14.25/8.15 => ( ! [I3: int] :
% 14.25/8.15 ( ( member_int @ I3 @ ( minus_minus_set_int @ S @ T2 ) )
% 14.25/8.15 => ( ( G @ I3 )
% 14.25/8.15 = one_one_nat ) )
% 14.25/8.15 => ( ! [X3: int] :
% 14.25/8.15 ( ( member_int @ X3 @ ( inf_inf_set_int @ S @ T2 ) )
% 14.25/8.15 => ( ( G @ X3 )
% 14.25/8.15 = ( H2 @ X3 ) ) )
% 14.25/8.15 => ( ( groups1707563613775114915nt_nat @ G @ S )
% 14.25/8.15 = ( groups1707563613775114915nt_nat @ H2 @ T2 ) ) ) ) ) ) ) ).
% 14.25/8.15
% 14.25/8.15 thf(fact_7307_prod_Omono__neutral__cong,axiom,
% 14.25/8.15 ! [T2: set_o,S: set_o,H2: $o > nat,G: $o > nat] :
% 14.25/8.15 ( ( finite_finite_o @ T2 )
% 14.25/8.15 => ( ( finite_finite_o @ S )
% 14.25/8.15 => ( ! [I3: $o] :
% 14.25/8.15 ( ( member_o @ I3 @ ( minus_minus_set_o @ T2 @ S ) )
% 14.25/8.15 => ( ( H2 @ I3 )
% 14.25/8.15 = one_one_nat ) )
% 14.25/8.15 => ( ! [I3: $o] :
% 14.25/8.15 ( ( member_o @ I3 @ ( minus_minus_set_o @ S @ T2 ) )
% 14.25/8.15 => ( ( G @ I3 )
% 14.25/8.15 = one_one_nat ) )
% 14.25/8.15 => ( ! [X3: $o] :
% 14.25/8.15 ( ( member_o @ X3 @ ( inf_inf_set_o @ S @ T2 ) )
% 14.25/8.15 => ( ( G @ X3 )
% 14.25/8.15 = ( H2 @ X3 ) ) )
% 14.25/8.15 => ( ( groups3504817904513533571_o_nat @ G @ S )
% 14.25/8.15 = ( groups3504817904513533571_o_nat @ H2 @ T2 ) ) ) ) ) ) ) ).
% 14.25/8.15
% 14.25/8.15 thf(fact_7308_prod_Omono__neutral__cong,axiom,
% 14.25/8.15 ! [T2: set_Product_unit,S: set_Product_unit,H2: product_unit > nat,G: product_unit > nat] :
% 14.25/8.15 ( ( finite4290736615968046902t_unit @ T2 )
% 14.25/8.15 => ( ( finite4290736615968046902t_unit @ S )
% 14.25/8.15 => ( ! [I3: product_unit] :
% 14.25/8.15 ( ( member_Product_unit @ I3 @ ( minus_6452836326544984404t_unit @ T2 @ S ) )
% 14.25/8.15 => ( ( H2 @ I3 )
% 14.25/8.15 = one_one_nat ) )
% 14.25/8.15 => ( ! [I3: product_unit] :
% 14.25/8.15 ( ( member_Product_unit @ I3 @ ( minus_6452836326544984404t_unit @ S @ T2 ) )
% 14.25/8.15 => ( ( G @ I3 )
% 14.25/8.15 = one_one_nat ) )
% 14.25/8.15 => ( ! [X3: product_unit] :
% 14.25/8.15 ( ( member_Product_unit @ X3 @ ( inf_in4660618365625256667t_unit @ S @ T2 ) )
% 14.25/8.15 => ( ( G @ X3 )
% 14.25/8.15 = ( H2 @ X3 ) ) )
% 14.25/8.15 => ( ( groups396421249026171384it_nat @ G @ S )
% 14.25/8.15 = ( groups396421249026171384it_nat @ H2 @ T2 ) ) ) ) ) ) ) ).
% 14.25/8.15
% 14.25/8.15 thf(fact_7309_prod_Omono__neutral__cong,axiom,
% 14.25/8.15 ! [T2: set_Code_integer,S: set_Code_integer,H2: code_integer > nat,G: code_integer > nat] :
% 14.25/8.15 ( ( finite6017078050557962740nteger @ T2 )
% 14.25/8.15 => ( ( finite6017078050557962740nteger @ S )
% 14.25/8.15 => ( ! [I3: code_integer] :
% 14.25/8.15 ( ( member_Code_integer @ I3 @ ( minus_2355218937544613996nteger @ T2 @ S ) )
% 14.25/8.15 => ( ( H2 @ I3 )
% 14.25/8.15 = one_one_nat ) )
% 14.25/8.15 => ( ! [I3: code_integer] :
% 14.25/8.15 ( ( member_Code_integer @ I3 @ ( minus_2355218937544613996nteger @ S @ T2 ) )
% 14.25/8.15 => ( ( G @ I3 )
% 14.25/8.15 = one_one_nat ) )
% 14.25/8.15 => ( ! [X3: code_integer] :
% 14.25/8.15 ( ( member_Code_integer @ X3 @ ( inf_in1364745209274528805nteger @ S @ T2 ) )
% 14.25/8.15 => ( ( G @ X3 )
% 14.25/8.15 = ( H2 @ X3 ) ) )
% 14.25/8.15 => ( ( groups3190895334310489300er_nat @ G @ S )
% 14.25/8.15 = ( groups3190895334310489300er_nat @ H2 @ T2 ) ) ) ) ) ) ) ).
% 14.25/8.15
% 14.25/8.15 thf(fact_7310_prod_Omono__neutral__cong,axiom,
% 14.25/8.15 ! [T2: set_int,S: set_int,H2: int > int,G: int > int] :
% 14.25/8.15 ( ( finite_finite_int @ T2 )
% 14.25/8.15 => ( ( finite_finite_int @ S )
% 14.25/8.15 => ( ! [I3: int] :
% 14.25/8.15 ( ( member_int @ I3 @ ( minus_minus_set_int @ T2 @ S ) )
% 14.25/8.15 => ( ( H2 @ I3 )
% 14.25/8.15 = one_one_int ) )
% 14.25/8.15 => ( ! [I3: int] :
% 14.25/8.15 ( ( member_int @ I3 @ ( minus_minus_set_int @ S @ T2 ) )
% 14.25/8.15 => ( ( G @ I3 )
% 14.25/8.15 = one_one_int ) )
% 14.25/8.15 => ( ! [X3: int] :
% 14.25/8.15 ( ( member_int @ X3 @ ( inf_inf_set_int @ S @ T2 ) )
% 14.25/8.15 => ( ( G @ X3 )
% 14.25/8.15 = ( H2 @ X3 ) ) )
% 14.25/8.15 => ( ( groups1705073143266064639nt_int @ G @ S )
% 14.25/8.15 = ( groups1705073143266064639nt_int @ H2 @ T2 ) ) ) ) ) ) ) ).
% 14.25/8.15
% 14.25/8.15 thf(fact_7311_prod_Omono__neutral__cong,axiom,
% 14.25/8.15 ! [T2: set_o,S: set_o,H2: $o > int,G: $o > int] :
% 14.25/8.15 ( ( finite_finite_o @ T2 )
% 14.25/8.15 => ( ( finite_finite_o @ S )
% 14.25/8.15 => ( ! [I3: $o] :
% 14.25/8.15 ( ( member_o @ I3 @ ( minus_minus_set_o @ T2 @ S ) )
% 14.25/8.15 => ( ( H2 @ I3 )
% 14.25/8.15 = one_one_int ) )
% 14.25/8.15 => ( ! [I3: $o] :
% 14.25/8.15 ( ( member_o @ I3 @ ( minus_minus_set_o @ S @ T2 ) )
% 14.25/8.15 => ( ( G @ I3 )
% 14.25/8.15 = one_one_int ) )
% 14.25/8.15 => ( ! [X3: $o] :
% 14.25/8.15 ( ( member_o @ X3 @ ( inf_inf_set_o @ S @ T2 ) )
% 14.25/8.15 => ( ( G @ X3 )
% 14.25/8.15 = ( H2 @ X3 ) ) )
% 14.25/8.15 => ( ( groups3502327434004483295_o_int @ G @ S )
% 14.25/8.15 = ( groups3502327434004483295_o_int @ H2 @ T2 ) ) ) ) ) ) ) ).
% 14.25/8.15
% 14.25/8.15 thf(fact_7312_prod_Oeq__fold,axiom,
% 14.25/8.15 ( groups708209901874060359at_nat
% 14.25/8.15 = ( ^ [G3: nat > nat] : ( finite_fold_nat_nat @ ( comp_nat_nat_nat_nat @ times_times_nat @ G3 ) @ one_one_nat ) ) ) ).
% 14.25/8.15
% 14.25/8.15 thf(fact_7313_sum__bounded__above,axiom,
% 14.25/8.15 ! [A: set_nat,F: nat > code_integer,K4: code_integer] :
% 14.25/8.15 ( ! [I3: nat] :
% 14.25/8.15 ( ( member_nat @ I3 @ A )
% 14.25/8.15 => ( ord_le3102999989581377725nteger @ ( F @ I3 ) @ K4 ) )
% 14.25/8.15 => ( ord_le3102999989581377725nteger @ ( groups7501900531339628137nteger @ F @ A ) @ ( times_3573771949741848930nteger @ ( semiri4939895301339042750nteger @ ( finite_card_nat @ A ) ) @ K4 ) ) ) ).
% 14.25/8.15
% 14.25/8.15 thf(fact_7314_sum__bounded__above,axiom,
% 14.25/8.15 ! [A: set_int,F: int > code_integer,K4: code_integer] :
% 14.25/8.15 ( ! [I3: int] :
% 14.25/8.15 ( ( member_int @ I3 @ A )
% 14.25/8.15 => ( ord_le3102999989581377725nteger @ ( F @ I3 ) @ K4 ) )
% 14.25/8.15 => ( ord_le3102999989581377725nteger @ ( groups7873554091576472773nteger @ F @ A ) @ ( times_3573771949741848930nteger @ ( semiri4939895301339042750nteger @ ( finite_card_int @ A ) ) @ K4 ) ) ) ).
% 14.25/8.15
% 14.25/8.15 thf(fact_7315_sum__bounded__above,axiom,
% 14.25/8.15 ! [A: set_o,F: $o > code_integer,K4: code_integer] :
% 14.25/8.15 ( ! [I3: $o] :
% 14.25/8.15 ( ( member_o @ I3 @ A )
% 14.25/8.15 => ( ord_le3102999989581377725nteger @ ( F @ I3 ) @ K4 ) )
% 14.25/8.15 => ( ord_le3102999989581377725nteger @ ( groups4406642042086082107nteger @ F @ A ) @ ( times_3573771949741848930nteger @ ( semiri4939895301339042750nteger @ ( finite_card_o @ A ) ) @ K4 ) ) ) ).
% 14.25/8.15
% 14.25/8.15 thf(fact_7316_sum__bounded__above,axiom,
% 14.25/8.15 ! [A: set_Product_unit,F: product_unit > code_integer,K4: code_integer] :
% 14.25/8.15 ( ! [I3: product_unit] :
% 14.25/8.15 ( ( member_Product_unit @ I3 @ A )
% 14.25/8.15 => ( ord_le3102999989581377725nteger @ ( F @ I3 ) @ K4 ) )
% 14.25/8.15 => ( ord_le3102999989581377725nteger @ ( groups5393439102872367408nteger @ F @ A ) @ ( times_3573771949741848930nteger @ ( semiri4939895301339042750nteger @ ( finite410649719033368117t_unit @ A ) ) @ K4 ) ) ) ).
% 14.25/8.15
% 14.25/8.15 thf(fact_7317_sum__bounded__above,axiom,
% 14.25/8.15 ! [A: set_nat,F: nat > rat,K4: rat] :
% 14.25/8.15 ( ! [I3: nat] :
% 14.25/8.15 ( ( member_nat @ I3 @ A )
% 14.25/8.15 => ( ord_less_eq_rat @ ( F @ I3 ) @ K4 ) )
% 14.25/8.15 => ( ord_less_eq_rat @ ( groups2906978787729119204at_rat @ F @ A ) @ ( times_times_rat @ ( semiri681578069525770553at_rat @ ( finite_card_nat @ A ) ) @ K4 ) ) ) ).
% 14.25/8.15
% 14.25/8.15 thf(fact_7318_sum__bounded__above,axiom,
% 14.25/8.15 ! [A: set_int,F: int > rat,K4: rat] :
% 14.25/8.15 ( ! [I3: int] :
% 14.25/8.15 ( ( member_int @ I3 @ A )
% 14.25/8.15 => ( ord_less_eq_rat @ ( F @ I3 ) @ K4 ) )
% 14.25/8.15 => ( ord_less_eq_rat @ ( groups3906332499630173760nt_rat @ F @ A ) @ ( times_times_rat @ ( semiri681578069525770553at_rat @ ( finite_card_int @ A ) ) @ K4 ) ) ) ).
% 14.25/8.15
% 14.25/8.15 thf(fact_7319_sum__bounded__above,axiom,
% 14.25/8.15 ! [A: set_o,F: $o > rat,K4: rat] :
% 14.25/8.15 ( ! [I3: $o] :
% 14.25/8.15 ( ( member_o @ I3 @ A )
% 14.25/8.15 => ( ord_less_eq_rat @ ( F @ I3 ) @ K4 ) )
% 14.25/8.15 => ( ord_less_eq_rat @ ( groups7872700643590313910_o_rat @ F @ A ) @ ( times_times_rat @ ( semiri681578069525770553at_rat @ ( finite_card_o @ A ) ) @ K4 ) ) ) ).
% 14.25/8.15
% 14.25/8.15 thf(fact_7320_sum__bounded__above,axiom,
% 14.25/8.15 ! [A: set_Product_unit,F: product_unit > rat,K4: rat] :
% 14.25/8.15 ( ! [I3: product_unit] :
% 14.25/8.15 ( ( member_Product_unit @ I3 @ A )
% 14.25/8.15 => ( ord_less_eq_rat @ ( F @ I3 ) @ K4 ) )
% 14.25/8.15 => ( ord_less_eq_rat @ ( groups8131667117488461483it_rat @ F @ A ) @ ( times_times_rat @ ( semiri681578069525770553at_rat @ ( finite410649719033368117t_unit @ A ) ) @ K4 ) ) ) ).
% 14.25/8.15
% 14.25/8.15 thf(fact_7321_sum__bounded__above,axiom,
% 14.25/8.15 ! [A: set_int,F: int > nat,K4: nat] :
% 14.25/8.15 ( ! [I3: int] :
% 14.25/8.15 ( ( member_int @ I3 @ A )
% 14.25/8.15 => ( ord_less_eq_nat @ ( F @ I3 ) @ K4 ) )
% 14.25/8.15 => ( ord_less_eq_nat @ ( groups4541462559716669496nt_nat @ F @ A ) @ ( times_times_nat @ ( semiri1316708129612266289at_nat @ ( finite_card_int @ A ) ) @ K4 ) ) ) ).
% 14.25/8.15
% 14.25/8.15 thf(fact_7322_sum__bounded__above,axiom,
% 14.25/8.15 ! [A: set_o,F: $o > nat,K4: nat] :
% 14.25/8.15 ( ! [I3: $o] :
% 14.25/8.15 ( ( member_o @ I3 @ A )
% 14.25/8.15 => ( ord_less_eq_nat @ ( F @ I3 ) @ K4 ) )
% 14.25/8.15 => ( ord_less_eq_nat @ ( groups8507830703676809646_o_nat @ F @ A ) @ ( times_times_nat @ ( semiri1316708129612266289at_nat @ ( finite_card_o @ A ) ) @ K4 ) ) ) ).
% 14.25/8.15
% 14.25/8.15 thf(fact_7323_sum__bounded__below,axiom,
% 14.25/8.15 ! [A: set_nat,K4: code_integer,F: nat > code_integer] :
% 14.25/8.15 ( ! [I3: nat] :
% 14.25/8.15 ( ( member_nat @ I3 @ A )
% 14.25/8.15 => ( ord_le3102999989581377725nteger @ K4 @ ( F @ I3 ) ) )
% 14.25/8.15 => ( ord_le3102999989581377725nteger @ ( times_3573771949741848930nteger @ ( semiri4939895301339042750nteger @ ( finite_card_nat @ A ) ) @ K4 ) @ ( groups7501900531339628137nteger @ F @ A ) ) ) ).
% 14.25/8.15
% 14.25/8.15 thf(fact_7324_sum__bounded__below,axiom,
% 14.25/8.15 ! [A: set_int,K4: code_integer,F: int > code_integer] :
% 14.25/8.15 ( ! [I3: int] :
% 14.25/8.15 ( ( member_int @ I3 @ A )
% 14.25/8.15 => ( ord_le3102999989581377725nteger @ K4 @ ( F @ I3 ) ) )
% 14.25/8.15 => ( ord_le3102999989581377725nteger @ ( times_3573771949741848930nteger @ ( semiri4939895301339042750nteger @ ( finite_card_int @ A ) ) @ K4 ) @ ( groups7873554091576472773nteger @ F @ A ) ) ) ).
% 14.25/8.15
% 14.25/8.15 thf(fact_7325_sum__bounded__below,axiom,
% 14.25/8.15 ! [A: set_o,K4: code_integer,F: $o > code_integer] :
% 14.25/8.15 ( ! [I3: $o] :
% 14.25/8.15 ( ( member_o @ I3 @ A )
% 14.25/8.15 => ( ord_le3102999989581377725nteger @ K4 @ ( F @ I3 ) ) )
% 14.25/8.15 => ( ord_le3102999989581377725nteger @ ( times_3573771949741848930nteger @ ( semiri4939895301339042750nteger @ ( finite_card_o @ A ) ) @ K4 ) @ ( groups4406642042086082107nteger @ F @ A ) ) ) ).
% 14.25/8.15
% 14.25/8.15 thf(fact_7326_sum__bounded__below,axiom,
% 14.25/8.15 ! [A: set_Product_unit,K4: code_integer,F: product_unit > code_integer] :
% 14.25/8.15 ( ! [I3: product_unit] :
% 14.25/8.15 ( ( member_Product_unit @ I3 @ A )
% 14.25/8.15 => ( ord_le3102999989581377725nteger @ K4 @ ( F @ I3 ) ) )
% 14.25/8.15 => ( ord_le3102999989581377725nteger @ ( times_3573771949741848930nteger @ ( semiri4939895301339042750nteger @ ( finite410649719033368117t_unit @ A ) ) @ K4 ) @ ( groups5393439102872367408nteger @ F @ A ) ) ) ).
% 14.25/8.15
% 14.25/8.15 thf(fact_7327_sum__bounded__below,axiom,
% 14.25/8.15 ! [A: set_nat,K4: rat,F: nat > rat] :
% 14.25/8.15 ( ! [I3: nat] :
% 14.25/8.15 ( ( member_nat @ I3 @ A )
% 14.25/8.15 => ( ord_less_eq_rat @ K4 @ ( F @ I3 ) ) )
% 14.25/8.15 => ( ord_less_eq_rat @ ( times_times_rat @ ( semiri681578069525770553at_rat @ ( finite_card_nat @ A ) ) @ K4 ) @ ( groups2906978787729119204at_rat @ F @ A ) ) ) ).
% 14.25/8.15
% 14.25/8.15 thf(fact_7328_sum__bounded__below,axiom,
% 14.25/8.15 ! [A: set_int,K4: rat,F: int > rat] :
% 14.25/8.15 ( ! [I3: int] :
% 14.25/8.15 ( ( member_int @ I3 @ A )
% 14.25/8.15 => ( ord_less_eq_rat @ K4 @ ( F @ I3 ) ) )
% 14.25/8.15 => ( ord_less_eq_rat @ ( times_times_rat @ ( semiri681578069525770553at_rat @ ( finite_card_int @ A ) ) @ K4 ) @ ( groups3906332499630173760nt_rat @ F @ A ) ) ) ).
% 14.25/8.15
% 14.25/8.15 thf(fact_7329_sum__bounded__below,axiom,
% 14.25/8.15 ! [A: set_o,K4: rat,F: $o > rat] :
% 14.25/8.15 ( ! [I3: $o] :
% 14.25/8.15 ( ( member_o @ I3 @ A )
% 14.25/8.15 => ( ord_less_eq_rat @ K4 @ ( F @ I3 ) ) )
% 14.25/8.15 => ( ord_less_eq_rat @ ( times_times_rat @ ( semiri681578069525770553at_rat @ ( finite_card_o @ A ) ) @ K4 ) @ ( groups7872700643590313910_o_rat @ F @ A ) ) ) ).
% 14.25/8.15
% 14.25/8.15 thf(fact_7330_sum__bounded__below,axiom,
% 14.25/8.15 ! [A: set_Product_unit,K4: rat,F: product_unit > rat] :
% 14.25/8.15 ( ! [I3: product_unit] :
% 14.25/8.15 ( ( member_Product_unit @ I3 @ A )
% 14.25/8.15 => ( ord_less_eq_rat @ K4 @ ( F @ I3 ) ) )
% 14.25/8.15 => ( ord_less_eq_rat @ ( times_times_rat @ ( semiri681578069525770553at_rat @ ( finite410649719033368117t_unit @ A ) ) @ K4 ) @ ( groups8131667117488461483it_rat @ F @ A ) ) ) ).
% 14.25/8.15
% 14.25/8.15 thf(fact_7331_sum__bounded__below,axiom,
% 14.25/8.15 ! [A: set_int,K4: nat,F: int > nat] :
% 14.25/8.15 ( ! [I3: int] :
% 14.25/8.15 ( ( member_int @ I3 @ A )
% 14.25/8.15 => ( ord_less_eq_nat @ K4 @ ( F @ I3 ) ) )
% 14.25/8.15 => ( ord_less_eq_nat @ ( times_times_nat @ ( semiri1316708129612266289at_nat @ ( finite_card_int @ A ) ) @ K4 ) @ ( groups4541462559716669496nt_nat @ F @ A ) ) ) ).
% 14.25/8.15
% 14.25/8.15 thf(fact_7332_sum__bounded__below,axiom,
% 14.25/8.15 ! [A: set_o,K4: nat,F: $o > nat] :
% 14.25/8.15 ( ! [I3: $o] :
% 14.25/8.15 ( ( member_o @ I3 @ A )
% 14.25/8.15 => ( ord_less_eq_nat @ K4 @ ( F @ I3 ) ) )
% 14.25/8.15 => ( ord_less_eq_nat @ ( times_times_nat @ ( semiri1316708129612266289at_nat @ ( finite_card_o @ A ) ) @ K4 ) @ ( groups8507830703676809646_o_nat @ F @ A ) ) ) ).
% 14.25/8.15
% 14.25/8.15 thf(fact_7333_gbinomial__absorb__comp,axiom,
% 14.25/8.15 ! [A2: rat,K3: nat] :
% 14.25/8.15 ( ( times_times_rat @ ( minus_minus_rat @ A2 @ ( semiri681578069525770553at_rat @ K3 ) ) @ ( gbinomial_rat @ A2 @ K3 ) )
% 14.25/8.15 = ( times_times_rat @ A2 @ ( gbinomial_rat @ ( minus_minus_rat @ A2 @ one_one_rat ) @ K3 ) ) ) ).
% 14.25/8.15
% 14.25/8.15 thf(fact_7334_gbinomial__mult__1,axiom,
% 14.25/8.15 ! [A2: rat,K3: nat] :
% 14.25/8.15 ( ( times_times_rat @ A2 @ ( gbinomial_rat @ A2 @ K3 ) )
% 14.25/8.15 = ( plus_plus_rat @ ( times_times_rat @ ( semiri681578069525770553at_rat @ K3 ) @ ( gbinomial_rat @ A2 @ K3 ) ) @ ( times_times_rat @ ( semiri681578069525770553at_rat @ ( suc @ K3 ) ) @ ( gbinomial_rat @ A2 @ ( suc @ K3 ) ) ) ) ) ).
% 14.25/8.15
% 14.25/8.15 thf(fact_7335_gbinomial__mult__1_H,axiom,
% 14.25/8.15 ! [A2: rat,K3: nat] :
% 14.25/8.15 ( ( times_times_rat @ ( gbinomial_rat @ A2 @ K3 ) @ A2 )
% 14.25/8.15 = ( plus_plus_rat @ ( times_times_rat @ ( semiri681578069525770553at_rat @ K3 ) @ ( gbinomial_rat @ A2 @ K3 ) ) @ ( times_times_rat @ ( semiri681578069525770553at_rat @ ( suc @ K3 ) ) @ ( gbinomial_rat @ A2 @ ( suc @ K3 ) ) ) ) ) ).
% 14.25/8.15
% 14.25/8.15 thf(fact_7336_boolean__algebra__class_Oboolean__algebra_Ocompl__unique,axiom,
% 14.25/8.15 ! [X: set_Pr4532377907799695533_nat_o,Y: set_Pr4532377907799695533_nat_o] :
% 14.25/8.15 ( ( ( inf_in1906310914598751387_nat_o @ X @ Y )
% 14.25/8.15 = bot_bo7824918357723582233_nat_o )
% 14.25/8.15 => ( ( ( sup_su5209123915105501825_nat_o @ X @ Y )
% 14.25/8.15 = top_to4869887016220417533_nat_o )
% 14.25/8.15 => ( ( uminus5254974436814262692_nat_o @ X )
% 14.25/8.15 = Y ) ) ) ).
% 14.25/8.15
% 14.25/8.15 thf(fact_7337_boolean__algebra__class_Oboolean__algebra_Ocompl__unique,axiom,
% 14.25/8.15 ! [X: set_Product_unit,Y: set_Product_unit] :
% 14.25/8.15 ( ( ( inf_in4660618365625256667t_unit @ X @ Y )
% 14.25/8.15 = bot_bo3957492148770167129t_unit )
% 14.25/8.15 => ( ( ( sup_su793286257634532545t_unit @ X @ Y )
% 14.25/8.15 = top_to1996260823553986621t_unit )
% 14.25/8.15 => ( ( uminus5944136376168626660t_unit @ X )
% 14.25/8.15 = Y ) ) ) ).
% 14.25/8.15
% 14.25/8.15 thf(fact_7338_boolean__algebra__class_Oboolean__algebra_Ocompl__unique,axiom,
% 14.25/8.15 ! [X: set_o,Y: set_o] :
% 14.25/8.15 ( ( ( inf_inf_set_o @ X @ Y )
% 14.25/8.15 = bot_bot_set_o )
% 14.25/8.15 => ( ( ( sup_sup_set_o @ X @ Y )
% 14.25/8.15 = top_top_set_o )
% 14.25/8.15 => ( ( uminus_uminus_set_o @ X )
% 14.25/8.15 = Y ) ) ) ).
% 14.25/8.15
% 14.25/8.15 thf(fact_7339_boolean__algebra__class_Oboolean__algebra_Ocompl__unique,axiom,
% 14.25/8.15 ! [X: set_nat,Y: set_nat] :
% 14.25/8.15 ( ( ( inf_inf_set_nat @ X @ Y )
% 14.25/8.15 = bot_bot_set_nat )
% 14.25/8.15 => ( ( ( sup_sup_set_nat @ X @ Y )
% 14.25/8.15 = top_top_set_nat )
% 14.25/8.15 => ( ( uminus5710092332889474511et_nat @ X )
% 14.25/8.15 = Y ) ) ) ).
% 14.25/8.15
% 14.25/8.15 thf(fact_7340_boolean__algebra__class_Oboolean__algebra_Ocompl__unique,axiom,
% 14.25/8.15 ! [X: set_int,Y: set_int] :
% 14.25/8.15 ( ( ( inf_inf_set_int @ X @ Y )
% 14.25/8.15 = bot_bot_set_int )
% 14.25/8.15 => ( ( ( sup_sup_set_int @ X @ Y )
% 14.25/8.15 = top_top_set_int )
% 14.25/8.15 => ( ( uminus1532241313380277803et_int @ X )
% 14.25/8.15 = Y ) ) ) ).
% 14.25/8.15
% 14.25/8.15 thf(fact_7341_boolean__algebra__class_Oboolean__algebra_Ocompl__unique,axiom,
% 14.25/8.15 ! [X: assn,Y: assn] :
% 14.25/8.15 ( ( ( inf_inf_assn @ X @ Y )
% 14.25/8.15 = bot_bot_assn )
% 14.25/8.15 => ( ( ( sup_sup_assn @ X @ Y )
% 14.25/8.15 = top_top_assn )
% 14.25/8.15 => ( ( uminus_uminus_assn @ X )
% 14.25/8.15 = Y ) ) ) ).
% 14.25/8.15
% 14.25/8.15 thf(fact_7342_boolean__algebra__class_Oboolean__algebra_Ocompl__unique,axiom,
% 14.25/8.15 ! [X: product_unit,Y: product_unit] :
% 14.25/8.15 ( ( ( inf_inf_Product_unit @ X @ Y )
% 14.25/8.15 = bot_bot_Product_unit )
% 14.25/8.15 => ( ( ( sup_sup_Product_unit @ X @ Y )
% 14.25/8.15 = top_top_Product_unit )
% 14.25/8.15 => ( ( uminus2952777764628376836t_unit @ X )
% 14.25/8.15 = Y ) ) ) ).
% 14.25/8.15
% 14.25/8.15 thf(fact_7343_prod__mono__strict,axiom,
% 14.25/8.15 ! [A: set_Code_integer,F: code_integer > code_integer,G: code_integer > code_integer] :
% 14.25/8.15 ( ( finite6017078050557962740nteger @ A )
% 14.25/8.15 => ( ! [I3: code_integer] :
% 14.25/8.15 ( ( member_Code_integer @ I3 @ A )
% 14.25/8.15 => ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ ( F @ I3 ) )
% 14.25/8.15 & ( ord_le6747313008572928689nteger @ ( F @ I3 ) @ ( G @ I3 ) ) ) )
% 14.25/8.15 => ( ( A != bot_bo3990330152332043303nteger )
% 14.25/8.15 => ( ord_le6747313008572928689nteger @ ( groups3674199335183972705nteger @ F @ A ) @ ( groups3674199335183972705nteger @ G @ A ) ) ) ) ) ).
% 14.25/8.15
% 14.25/8.15 thf(fact_7344_prod__mono__strict,axiom,
% 14.25/8.15 ! [A: set_Product_unit,F: product_unit > code_integer,G: product_unit > code_integer] :
% 14.25/8.15 ( ( finite4290736615968046902t_unit @ A )
% 14.25/8.15 => ( ! [I3: product_unit] :
% 14.25/8.15 ( ( member_Product_unit @ I3 @ A )
% 14.25/8.15 => ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ ( F @ I3 ) )
% 14.25/8.15 & ( ord_le6747313008572928689nteger @ ( F @ I3 ) @ ( G @ I3 ) ) ) )
% 14.25/8.15 => ( ( A != bot_bo3957492148770167129t_unit )
% 14.25/8.15 => ( ord_le6747313008572928689nteger @ ( groups8185359811944078981nteger @ F @ A ) @ ( groups8185359811944078981nteger @ G @ A ) ) ) ) ) ).
% 14.25/8.15
% 14.25/8.15 thf(fact_7345_prod__mono__strict,axiom,
% 14.25/8.15 ! [A: set_o,F: $o > code_integer,G: $o > code_integer] :
% 14.25/8.15 ( ( finite_finite_o @ A )
% 14.25/8.15 => ( ! [I3: $o] :
% 14.25/8.15 ( ( member_o @ I3 @ A )
% 14.25/8.15 => ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ ( F @ I3 ) )
% 14.25/8.15 & ( ord_le6747313008572928689nteger @ ( F @ I3 ) @ ( G @ I3 ) ) ) )
% 14.25/8.15 => ( ( A != bot_bot_set_o )
% 14.25/8.15 => ( ord_le6747313008572928689nteger @ ( groups7694694392188491536nteger @ F @ A ) @ ( groups7694694392188491536nteger @ G @ A ) ) ) ) ) ).
% 14.25/8.15
% 14.25/8.15 thf(fact_7346_prod__mono__strict,axiom,
% 14.25/8.15 ! [A: set_nat,F: nat > code_integer,G: nat > code_integer] :
% 14.25/8.15 ( ( finite_finite_nat @ A )
% 14.25/8.15 => ( ! [I3: nat] :
% 14.25/8.15 ( ( member_nat @ I3 @ A )
% 14.25/8.15 => ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ ( F @ I3 ) )
% 14.25/8.15 & ( ord_le6747313008572928689nteger @ ( F @ I3 ) @ ( G @ I3 ) ) ) )
% 14.25/8.15 => ( ( A != bot_bot_set_nat )
% 14.25/8.15 => ( ord_le6747313008572928689nteger @ ( groups3455450783089532116nteger @ F @ A ) @ ( groups3455450783089532116nteger @ G @ A ) ) ) ) ) ).
% 14.25/8.15
% 14.25/8.15 thf(fact_7347_prod__mono__strict,axiom,
% 14.25/8.15 ! [A: set_int,F: int > code_integer,G: int > code_integer] :
% 14.25/8.15 ( ( finite_finite_int @ A )
% 14.25/8.15 => ( ! [I3: int] :
% 14.25/8.15 ( ( member_int @ I3 @ A )
% 14.25/8.15 => ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ ( F @ I3 ) )
% 14.25/8.15 & ( ord_le6747313008572928689nteger @ ( F @ I3 ) @ ( G @ I3 ) ) ) )
% 14.25/8.15 => ( ( A != bot_bot_set_int )
% 14.25/8.15 => ( ord_le6747313008572928689nteger @ ( groups3827104343326376752nteger @ F @ A ) @ ( groups3827104343326376752nteger @ G @ A ) ) ) ) ) ).
% 14.25/8.15
% 14.25/8.15 thf(fact_7348_prod__mono__strict,axiom,
% 14.25/8.15 ! [A: set_Code_integer,F: code_integer > rat,G: code_integer > rat] :
% 14.25/8.15 ( ( finite6017078050557962740nteger @ A )
% 14.25/8.15 => ( ! [I3: code_integer] :
% 14.25/8.15 ( ( member_Code_integer @ I3 @ A )
% 14.25/8.15 => ( ( ord_less_eq_rat @ zero_zero_rat @ ( F @ I3 ) )
% 14.25/8.15 & ( ord_less_rat @ ( F @ I3 ) @ ( G @ I3 ) ) ) )
% 14.25/8.15 => ( ( A != bot_bo3990330152332043303nteger )
% 14.25/8.15 => ( ord_less_rat @ ( groups2555765274223993564er_rat @ F @ A ) @ ( groups2555765274223993564er_rat @ G @ A ) ) ) ) ) ).
% 14.25/8.15
% 14.25/8.15 thf(fact_7349_prod__mono__strict,axiom,
% 14.25/8.15 ! [A: set_Product_unit,F: product_unit > rat,G: product_unit > rat] :
% 14.25/8.15 ( ( finite4290736615968046902t_unit @ A )
% 14.25/8.15 => ( ! [I3: product_unit] :
% 14.25/8.15 ( ( member_Product_unit @ I3 @ A )
% 14.25/8.15 => ( ( ord_less_eq_rat @ zero_zero_rat @ ( F @ I3 ) )
% 14.25/8.15 & ( ord_less_rat @ ( F @ I3 ) @ ( G @ I3 ) ) ) )
% 14.25/8.15 => ( ( A != bot_bo3957492148770167129t_unit )
% 14.25/8.15 => ( ord_less_rat @ ( groups8984663225794451456it_rat @ F @ A ) @ ( groups8984663225794451456it_rat @ G @ A ) ) ) ) ) ).
% 14.25/8.15
% 14.25/8.15 thf(fact_7350_prod__mono__strict,axiom,
% 14.25/8.15 ! [A: set_o,F: $o > rat,G: $o > rat] :
% 14.25/8.15 ( ( finite_finite_o @ A )
% 14.25/8.15 => ( ! [I3: $o] :
% 14.25/8.15 ( ( member_o @ I3 @ A )
% 14.25/8.15 => ( ( ord_less_eq_rat @ zero_zero_rat @ ( F @ I3 ) )
% 14.25/8.15 & ( ord_less_rat @ ( F @ I3 ) @ ( G @ I3 ) ) ) )
% 14.25/8.15 => ( ( A != bot_bot_set_o )
% 14.25/8.15 => ( ord_less_rat @ ( groups2869687844427037835_o_rat @ F @ A ) @ ( groups2869687844427037835_o_rat @ G @ A ) ) ) ) ) ).
% 14.25/8.15
% 14.25/8.15 thf(fact_7351_prod__mono__strict,axiom,
% 14.25/8.15 ! [A: set_nat,F: nat > rat,G: nat > rat] :
% 14.25/8.15 ( ( finite_finite_nat @ A )
% 14.25/8.15 => ( ! [I3: nat] :
% 14.25/8.15 ( ( member_nat @ I3 @ A )
% 14.25/8.15 => ( ( ord_less_eq_rat @ zero_zero_rat @ ( F @ I3 ) )
% 14.25/8.15 & ( ord_less_rat @ ( F @ I3 ) @ ( G @ I3 ) ) ) )
% 14.25/8.15 => ( ( A != bot_bot_set_nat )
% 14.25/8.15 => ( ord_less_rat @ ( groups73079841787564623at_rat @ F @ A ) @ ( groups73079841787564623at_rat @ G @ A ) ) ) ) ) ).
% 14.25/8.15
% 14.25/8.15 thf(fact_7352_prod__mono__strict,axiom,
% 14.25/8.15 ! [A: set_int,F: int > rat,G: int > rat] :
% 14.25/8.15 ( ( finite_finite_int @ A )
% 14.25/8.15 => ( ! [I3: int] :
% 14.25/8.15 ( ( member_int @ I3 @ A )
% 14.25/8.15 => ( ( ord_less_eq_rat @ zero_zero_rat @ ( F @ I3 ) )
% 14.25/8.15 & ( ord_less_rat @ ( F @ I3 ) @ ( G @ I3 ) ) ) )
% 14.25/8.15 => ( ( A != bot_bot_set_int )
% 14.25/8.15 => ( ord_less_rat @ ( groups1072433553688619179nt_rat @ F @ A ) @ ( groups1072433553688619179nt_rat @ G @ A ) ) ) ) ) ).
% 14.25/8.15
% 14.25/8.15 thf(fact_7353_prod_Oinsert__remove,axiom,
% 14.25/8.15 ! [A: set_Code_integer,G: code_integer > assn,X: code_integer] :
% 14.25/8.15 ( ( finite6017078050557962740nteger @ A )
% 14.25/8.15 => ( ( groups1304777262505850412r_assn @ G @ ( insert_Code_integer @ X @ A ) )
% 14.25/8.15 = ( times_times_assn @ ( G @ X ) @ ( groups1304777262505850412r_assn @ G @ ( minus_2355218937544613996nteger @ A @ ( insert_Code_integer @ X @ bot_bo3990330152332043303nteger ) ) ) ) ) ) ).
% 14.25/8.15
% 14.25/8.15 thf(fact_7354_prod_Oinsert__remove,axiom,
% 14.25/8.15 ! [A: set_Product_unit,G: product_unit > assn,X: product_unit] :
% 14.25/8.15 ( ( finite4290736615968046902t_unit @ A )
% 14.25/8.15 => ( ( groups5360233303959380816t_assn @ G @ ( insert_Product_unit @ X @ A ) )
% 14.25/8.15 = ( times_times_assn @ ( G @ X ) @ ( groups5360233303959380816t_assn @ G @ ( minus_6452836326544984404t_unit @ A @ ( insert_Product_unit @ X @ bot_bo3957492148770167129t_unit ) ) ) ) ) ) ).
% 14.25/8.15
% 14.25/8.15 thf(fact_7355_prod_Oinsert__remove,axiom,
% 14.25/8.15 ! [A: set_o,G: $o > assn,X: $o] :
% 14.25/8.15 ( ( finite_finite_o @ A )
% 14.25/8.15 => ( ( groups5301882518646026715o_assn @ G @ ( insert_o @ X @ A ) )
% 14.25/8.15 = ( times_times_assn @ ( G @ X ) @ ( groups5301882518646026715o_assn @ G @ ( minus_minus_set_o @ A @ ( insert_o @ X @ bot_bot_set_o ) ) ) ) ) ) ).
% 14.25/8.15
% 14.25/8.15 thf(fact_7356_prod_Oinsert__remove,axiom,
% 14.25/8.15 ! [A: set_int,G: int > assn,X: int] :
% 14.25/8.15 ( ( finite_finite_int @ A )
% 14.25/8.15 => ( ( groups7882442080178216443t_assn @ G @ ( insert_int @ X @ A ) )
% 14.25/8.15 = ( times_times_assn @ ( G @ X ) @ ( groups7882442080178216443t_assn @ G @ ( minus_minus_set_int @ A @ ( insert_int @ X @ bot_bot_set_int ) ) ) ) ) ) ).
% 14.25/8.15
% 14.25/8.15 thf(fact_7357_prod_Oinsert__remove,axiom,
% 14.25/8.15 ! [A: set_nat,G: nat > assn,X: nat] :
% 14.25/8.15 ( ( finite_finite_nat @ A )
% 14.25/8.15 => ( ( groups6906906614972039071t_assn @ G @ ( insert_nat @ X @ A ) )
% 14.25/8.15 = ( times_times_assn @ ( G @ X ) @ ( groups6906906614972039071t_assn @ G @ ( minus_minus_set_nat @ A @ ( insert_nat @ X @ bot_bot_set_nat ) ) ) ) ) ) ).
% 14.25/8.15
% 14.25/8.15 thf(fact_7358_prod_Oinsert__remove,axiom,
% 14.25/8.15 ! [A: set_Code_integer,G: code_integer > rat,X: code_integer] :
% 14.25/8.15 ( ( finite6017078050557962740nteger @ A )
% 14.25/8.15 => ( ( groups2555765274223993564er_rat @ G @ ( insert_Code_integer @ X @ A ) )
% 14.25/8.15 = ( times_times_rat @ ( G @ X ) @ ( groups2555765274223993564er_rat @ G @ ( minus_2355218937544613996nteger @ A @ ( insert_Code_integer @ X @ bot_bo3990330152332043303nteger ) ) ) ) ) ) ).
% 14.25/8.15
% 14.25/8.15 thf(fact_7359_prod_Oinsert__remove,axiom,
% 14.25/8.15 ! [A: set_Product_unit,G: product_unit > rat,X: product_unit] :
% 14.25/8.15 ( ( finite4290736615968046902t_unit @ A )
% 14.25/8.15 => ( ( groups8984663225794451456it_rat @ G @ ( insert_Product_unit @ X @ A ) )
% 14.25/8.15 = ( times_times_rat @ ( G @ X ) @ ( groups8984663225794451456it_rat @ G @ ( minus_6452836326544984404t_unit @ A @ ( insert_Product_unit @ X @ bot_bo3957492148770167129t_unit ) ) ) ) ) ) ).
% 14.25/8.15
% 14.25/8.15 thf(fact_7360_prod_Oinsert__remove,axiom,
% 14.25/8.15 ! [A: set_o,G: $o > rat,X: $o] :
% 14.25/8.15 ( ( finite_finite_o @ A )
% 14.25/8.15 => ( ( groups2869687844427037835_o_rat @ G @ ( insert_o @ X @ A ) )
% 14.25/8.15 = ( times_times_rat @ ( G @ X ) @ ( groups2869687844427037835_o_rat @ G @ ( minus_minus_set_o @ A @ ( insert_o @ X @ bot_bot_set_o ) ) ) ) ) ) ).
% 14.25/8.15
% 14.25/8.15 thf(fact_7361_prod_Oinsert__remove,axiom,
% 14.25/8.15 ! [A: set_int,G: int > rat,X: int] :
% 14.25/8.15 ( ( finite_finite_int @ A )
% 14.25/8.15 => ( ( groups1072433553688619179nt_rat @ G @ ( insert_int @ X @ A ) )
% 14.25/8.15 = ( times_times_rat @ ( G @ X ) @ ( groups1072433553688619179nt_rat @ G @ ( minus_minus_set_int @ A @ ( insert_int @ X @ bot_bot_set_int ) ) ) ) ) ) ).
% 14.25/8.15
% 14.25/8.15 thf(fact_7362_prod_Oinsert__remove,axiom,
% 14.25/8.15 ! [A: set_nat,G: nat > rat,X: nat] :
% 14.25/8.15 ( ( finite_finite_nat @ A )
% 14.25/8.15 => ( ( groups73079841787564623at_rat @ G @ ( insert_nat @ X @ A ) )
% 14.25/8.15 = ( times_times_rat @ ( G @ X ) @ ( groups73079841787564623at_rat @ G @ ( minus_minus_set_nat @ A @ ( insert_nat @ X @ bot_bot_set_nat ) ) ) ) ) ) ).
% 14.25/8.15
% 14.25/8.15 thf(fact_7363_prod_Oremove,axiom,
% 14.25/8.15 ! [A: set_Code_integer,X: code_integer,G: code_integer > assn] :
% 14.25/8.15 ( ( finite6017078050557962740nteger @ A )
% 14.25/8.15 => ( ( member_Code_integer @ X @ A )
% 14.25/8.15 => ( ( groups1304777262505850412r_assn @ G @ A )
% 14.25/8.15 = ( times_times_assn @ ( G @ X ) @ ( groups1304777262505850412r_assn @ G @ ( minus_2355218937544613996nteger @ A @ ( insert_Code_integer @ X @ bot_bo3990330152332043303nteger ) ) ) ) ) ) ) ).
% 14.25/8.15
% 14.25/8.15 thf(fact_7364_prod_Oremove,axiom,
% 14.25/8.15 ! [A: set_Product_unit,X: product_unit,G: product_unit > assn] :
% 14.25/8.15 ( ( finite4290736615968046902t_unit @ A )
% 14.25/8.15 => ( ( member_Product_unit @ X @ A )
% 14.25/8.15 => ( ( groups5360233303959380816t_assn @ G @ A )
% 14.25/8.15 = ( times_times_assn @ ( G @ X ) @ ( groups5360233303959380816t_assn @ G @ ( minus_6452836326544984404t_unit @ A @ ( insert_Product_unit @ X @ bot_bo3957492148770167129t_unit ) ) ) ) ) ) ) ).
% 14.25/8.15
% 14.25/8.15 thf(fact_7365_prod_Oremove,axiom,
% 14.25/8.15 ! [A: set_o,X: $o,G: $o > assn] :
% 14.25/8.15 ( ( finite_finite_o @ A )
% 14.25/8.15 => ( ( member_o @ X @ A )
% 14.25/8.15 => ( ( groups5301882518646026715o_assn @ G @ A )
% 14.25/8.15 = ( times_times_assn @ ( G @ X ) @ ( groups5301882518646026715o_assn @ G @ ( minus_minus_set_o @ A @ ( insert_o @ X @ bot_bot_set_o ) ) ) ) ) ) ) ).
% 14.25/8.15
% 14.25/8.15 thf(fact_7366_prod_Oremove,axiom,
% 14.25/8.15 ! [A: set_int,X: int,G: int > assn] :
% 14.25/8.15 ( ( finite_finite_int @ A )
% 14.25/8.15 => ( ( member_int @ X @ A )
% 14.25/8.15 => ( ( groups7882442080178216443t_assn @ G @ A )
% 14.25/8.15 = ( times_times_assn @ ( G @ X ) @ ( groups7882442080178216443t_assn @ G @ ( minus_minus_set_int @ A @ ( insert_int @ X @ bot_bot_set_int ) ) ) ) ) ) ) ).
% 14.25/8.15
% 14.25/8.15 thf(fact_7367_prod_Oremove,axiom,
% 14.25/8.15 ! [A: set_nat,X: nat,G: nat > assn] :
% 14.25/8.15 ( ( finite_finite_nat @ A )
% 14.25/8.15 => ( ( member_nat @ X @ A )
% 14.25/8.15 => ( ( groups6906906614972039071t_assn @ G @ A )
% 14.25/8.15 = ( times_times_assn @ ( G @ X ) @ ( groups6906906614972039071t_assn @ G @ ( minus_minus_set_nat @ A @ ( insert_nat @ X @ bot_bot_set_nat ) ) ) ) ) ) ) ).
% 14.25/8.15
% 14.25/8.15 thf(fact_7368_prod_Oremove,axiom,
% 14.25/8.15 ! [A: set_Code_integer,X: code_integer,G: code_integer > rat] :
% 14.25/8.15 ( ( finite6017078050557962740nteger @ A )
% 14.25/8.15 => ( ( member_Code_integer @ X @ A )
% 14.25/8.15 => ( ( groups2555765274223993564er_rat @ G @ A )
% 14.25/8.15 = ( times_times_rat @ ( G @ X ) @ ( groups2555765274223993564er_rat @ G @ ( minus_2355218937544613996nteger @ A @ ( insert_Code_integer @ X @ bot_bo3990330152332043303nteger ) ) ) ) ) ) ) ).
% 14.25/8.15
% 14.25/8.15 thf(fact_7369_prod_Oremove,axiom,
% 14.25/8.15 ! [A: set_Product_unit,X: product_unit,G: product_unit > rat] :
% 14.25/8.15 ( ( finite4290736615968046902t_unit @ A )
% 14.25/8.15 => ( ( member_Product_unit @ X @ A )
% 14.25/8.15 => ( ( groups8984663225794451456it_rat @ G @ A )
% 14.25/8.15 = ( times_times_rat @ ( G @ X ) @ ( groups8984663225794451456it_rat @ G @ ( minus_6452836326544984404t_unit @ A @ ( insert_Product_unit @ X @ bot_bo3957492148770167129t_unit ) ) ) ) ) ) ) ).
% 14.25/8.15
% 14.25/8.15 thf(fact_7370_prod_Oremove,axiom,
% 14.25/8.15 ! [A: set_o,X: $o,G: $o > rat] :
% 14.25/8.15 ( ( finite_finite_o @ A )
% 14.25/8.15 => ( ( member_o @ X @ A )
% 14.25/8.15 => ( ( groups2869687844427037835_o_rat @ G @ A )
% 14.25/8.15 = ( times_times_rat @ ( G @ X ) @ ( groups2869687844427037835_o_rat @ G @ ( minus_minus_set_o @ A @ ( insert_o @ X @ bot_bot_set_o ) ) ) ) ) ) ) ).
% 14.25/8.15
% 14.25/8.15 thf(fact_7371_prod_Oremove,axiom,
% 14.25/8.15 ! [A: set_int,X: int,G: int > rat] :
% 14.25/8.15 ( ( finite_finite_int @ A )
% 14.25/8.15 => ( ( member_int @ X @ A )
% 14.25/8.15 => ( ( groups1072433553688619179nt_rat @ G @ A )
% 14.25/8.15 = ( times_times_rat @ ( G @ X ) @ ( groups1072433553688619179nt_rat @ G @ ( minus_minus_set_int @ A @ ( insert_int @ X @ bot_bot_set_int ) ) ) ) ) ) ) ).
% 14.25/8.15
% 14.25/8.15 thf(fact_7372_prod_Oremove,axiom,
% 14.25/8.15 ! [A: set_nat,X: nat,G: nat > rat] :
% 14.25/8.15 ( ( finite_finite_nat @ A )
% 14.25/8.15 => ( ( member_nat @ X @ A )
% 14.25/8.15 => ( ( groups73079841787564623at_rat @ G @ A )
% 14.25/8.15 = ( times_times_rat @ ( G @ X ) @ ( groups73079841787564623at_rat @ G @ ( minus_minus_set_nat @ A @ ( insert_nat @ X @ bot_bot_set_nat ) ) ) ) ) ) ) ).
% 14.25/8.15
% 14.25/8.15 thf(fact_7373_prod_Ounion__inter__neutral,axiom,
% 14.25/8.15 ! [A: set_Code_integer,B4: set_Code_integer,G: code_integer > code_integer] :
% 14.25/8.15 ( ( finite6017078050557962740nteger @ A )
% 14.25/8.15 => ( ( finite6017078050557962740nteger @ B4 )
% 14.25/8.15 => ( ! [X3: code_integer] :
% 14.25/8.15 ( ( member_Code_integer @ X3 @ ( inf_in1364745209274528805nteger @ A @ B4 ) )
% 14.25/8.15 => ( ( G @ X3 )
% 14.25/8.15 = one_one_Code_integer ) )
% 14.25/8.15 => ( ( groups3674199335183972705nteger @ G @ ( sup_su848401254843788991nteger @ A @ B4 ) )
% 14.25/8.15 = ( times_3573771949741848930nteger @ ( groups3674199335183972705nteger @ G @ A ) @ ( groups3674199335183972705nteger @ G @ B4 ) ) ) ) ) ) ).
% 14.25/8.15
% 14.25/8.15 thf(fact_7374_prod_Ounion__inter__neutral,axiom,
% 14.25/8.15 ! [A: set_Code_integer,B4: set_Code_integer,G: code_integer > code_natural] :
% 14.25/8.15 ( ( finite6017078050557962740nteger @ A )
% 14.25/8.15 => ( ( finite6017078050557962740nteger @ B4 )
% 14.25/8.15 => ( ! [X3: code_integer] :
% 14.25/8.15 ( ( member_Code_integer @ X3 @ ( inf_in1364745209274528805nteger @ A @ B4 ) )
% 14.25/8.15 => ( ( G @ X3 )
% 14.25/8.15 = one_one_Code_natural ) )
% 14.25/8.15 => ( ( groups2497794486940690220atural @ G @ ( sup_su848401254843788991nteger @ A @ B4 ) )
% 14.25/8.15 = ( times_2397367101498566445atural @ ( groups2497794486940690220atural @ G @ A ) @ ( groups2497794486940690220atural @ G @ B4 ) ) ) ) ) ) ).
% 14.25/8.15
% 14.25/8.15 thf(fact_7375_prod_Ounion__inter__neutral,axiom,
% 14.25/8.15 ! [A: set_nat,B4: set_nat,G: nat > code_integer] :
% 14.25/8.15 ( ( finite_finite_nat @ A )
% 14.25/8.15 => ( ( finite_finite_nat @ B4 )
% 14.25/8.15 => ( ! [X3: nat] :
% 14.25/8.15 ( ( member_nat @ X3 @ ( inf_inf_set_nat @ A @ B4 ) )
% 14.25/8.15 => ( ( G @ X3 )
% 14.25/8.15 = one_one_Code_integer ) )
% 14.25/8.15 => ( ( groups3455450783089532116nteger @ G @ ( sup_sup_set_nat @ A @ B4 ) )
% 14.25/8.15 = ( times_3573771949741848930nteger @ ( groups3455450783089532116nteger @ G @ A ) @ ( groups3455450783089532116nteger @ G @ B4 ) ) ) ) ) ) ).
% 14.25/8.15
% 14.25/8.15 thf(fact_7376_prod_Ounion__inter__neutral,axiom,
% 14.25/8.15 ! [A: set_nat,B4: set_nat,G: nat > code_natural] :
% 14.25/8.15 ( ( finite_finite_nat @ A )
% 14.25/8.15 => ( ( finite_finite_nat @ B4 )
% 14.25/8.15 => ( ! [X3: nat] :
% 14.25/8.15 ( ( member_nat @ X3 @ ( inf_inf_set_nat @ A @ B4 ) )
% 14.25/8.15 => ( ( G @ X3 )
% 14.25/8.15 = one_one_Code_natural ) )
% 14.25/8.15 => ( ( groups2279045934846249631atural @ G @ ( sup_sup_set_nat @ A @ B4 ) )
% 14.25/8.15 = ( times_2397367101498566445atural @ ( groups2279045934846249631atural @ G @ A ) @ ( groups2279045934846249631atural @ G @ B4 ) ) ) ) ) ) ).
% 14.25/8.15
% 14.25/8.15 thf(fact_7377_prod_Ounion__inter__neutral,axiom,
% 14.25/8.15 ! [A: set_Code_integer,B4: set_Code_integer,G: code_integer > assn] :
% 14.25/8.15 ( ( finite6017078050557962740nteger @ A )
% 14.25/8.15 => ( ( finite6017078050557962740nteger @ B4 )
% 14.25/8.15 => ( ! [X3: code_integer] :
% 14.25/8.15 ( ( member_Code_integer @ X3 @ ( inf_in1364745209274528805nteger @ A @ B4 ) )
% 14.25/8.15 => ( ( G @ X3 )
% 14.25/8.15 = one_one_assn ) )
% 14.25/8.15 => ( ( groups1304777262505850412r_assn @ G @ ( sup_su848401254843788991nteger @ A @ B4 ) )
% 14.25/8.15 = ( times_times_assn @ ( groups1304777262505850412r_assn @ G @ A ) @ ( groups1304777262505850412r_assn @ G @ B4 ) ) ) ) ) ) ).
% 14.25/8.15
% 14.25/8.15 thf(fact_7378_prod_Ounion__inter__neutral,axiom,
% 14.25/8.15 ! [A: set_nat,B4: set_nat,G: nat > assn] :
% 14.25/8.15 ( ( finite_finite_nat @ A )
% 14.25/8.15 => ( ( finite_finite_nat @ B4 )
% 14.25/8.15 => ( ! [X3: nat] :
% 14.25/8.15 ( ( member_nat @ X3 @ ( inf_inf_set_nat @ A @ B4 ) )
% 14.25/8.15 => ( ( G @ X3 )
% 14.25/8.15 = one_one_assn ) )
% 14.25/8.15 => ( ( groups6906906614972039071t_assn @ G @ ( sup_sup_set_nat @ A @ B4 ) )
% 14.25/8.15 = ( times_times_assn @ ( groups6906906614972039071t_assn @ G @ A ) @ ( groups6906906614972039071t_assn @ G @ B4 ) ) ) ) ) ) ).
% 14.25/8.15
% 14.25/8.15 thf(fact_7379_prod_Ounion__inter__neutral,axiom,
% 14.25/8.15 ! [A: set_Code_integer,B4: set_Code_integer,G: code_integer > rat] :
% 14.25/8.15 ( ( finite6017078050557962740nteger @ A )
% 14.25/8.15 => ( ( finite6017078050557962740nteger @ B4 )
% 14.25/8.15 => ( ! [X3: code_integer] :
% 14.25/8.15 ( ( member_Code_integer @ X3 @ ( inf_in1364745209274528805nteger @ A @ B4 ) )
% 14.25/8.15 => ( ( G @ X3 )
% 14.25/8.15 = one_one_rat ) )
% 14.25/8.15 => ( ( groups2555765274223993564er_rat @ G @ ( sup_su848401254843788991nteger @ A @ B4 ) )
% 14.25/8.15 = ( times_times_rat @ ( groups2555765274223993564er_rat @ G @ A ) @ ( groups2555765274223993564er_rat @ G @ B4 ) ) ) ) ) ) ).
% 14.25/8.15
% 14.25/8.15 thf(fact_7380_prod_Ounion__inter__neutral,axiom,
% 14.25/8.15 ! [A: set_nat,B4: set_nat,G: nat > rat] :
% 14.25/8.15 ( ( finite_finite_nat @ A )
% 14.25/8.15 => ( ( finite_finite_nat @ B4 )
% 14.25/8.15 => ( ! [X3: nat] :
% 14.25/8.15 ( ( member_nat @ X3 @ ( inf_inf_set_nat @ A @ B4 ) )
% 14.25/8.15 => ( ( G @ X3 )
% 14.25/8.15 = one_one_rat ) )
% 14.25/8.15 => ( ( groups73079841787564623at_rat @ G @ ( sup_sup_set_nat @ A @ B4 ) )
% 14.25/8.15 = ( times_times_rat @ ( groups73079841787564623at_rat @ G @ A ) @ ( groups73079841787564623at_rat @ G @ B4 ) ) ) ) ) ) ).
% 14.25/8.15
% 14.25/8.15 thf(fact_7381_prod_Ounion__inter__neutral,axiom,
% 14.25/8.15 ! [A: set_Code_integer,B4: set_Code_integer,G: code_integer > nat] :
% 14.25/8.15 ( ( finite6017078050557962740nteger @ A )
% 14.25/8.15 => ( ( finite6017078050557962740nteger @ B4 )
% 14.25/8.15 => ( ! [X3: code_integer] :
% 14.25/8.15 ( ( member_Code_integer @ X3 @ ( inf_in1364745209274528805nteger @ A @ B4 ) )
% 14.25/8.15 => ( ( G @ X3 )
% 14.25/8.15 = one_one_nat ) )
% 14.25/8.15 => ( ( groups3190895334310489300er_nat @ G @ ( sup_su848401254843788991nteger @ A @ B4 ) )
% 14.25/8.15 = ( times_times_nat @ ( groups3190895334310489300er_nat @ G @ A ) @ ( groups3190895334310489300er_nat @ G @ B4 ) ) ) ) ) ) ).
% 14.25/8.15
% 14.25/8.15 thf(fact_7382_prod_Ounion__inter__neutral,axiom,
% 14.25/8.15 ! [A: set_Code_integer,B4: set_Code_integer,G: code_integer > int] :
% 14.25/8.15 ( ( finite6017078050557962740nteger @ A )
% 14.25/8.15 => ( ( finite6017078050557962740nteger @ B4 )
% 14.25/8.15 => ( ! [X3: code_integer] :
% 14.25/8.15 ( ( member_Code_integer @ X3 @ ( inf_in1364745209274528805nteger @ A @ B4 ) )
% 14.25/8.15 => ( ( G @ X3 )
% 14.25/8.15 = one_one_int ) )
% 14.25/8.15 => ( ( groups3188404863801439024er_int @ G @ ( sup_su848401254843788991nteger @ A @ B4 ) )
% 14.25/8.15 = ( times_times_int @ ( groups3188404863801439024er_int @ G @ A ) @ ( groups3188404863801439024er_int @ G @ B4 ) ) ) ) ) ) ).
% 14.25/8.15
% 14.25/8.15 thf(fact_7383_prod_Ounion__disjoint,axiom,
% 14.25/8.15 ! [A: set_Code_integer,B4: set_Code_integer,G: code_integer > assn] :
% 14.25/8.15 ( ( finite6017078050557962740nteger @ A )
% 14.25/8.15 => ( ( finite6017078050557962740nteger @ B4 )
% 14.25/8.15 => ( ( ( inf_in1364745209274528805nteger @ A @ B4 )
% 14.25/8.15 = bot_bo3990330152332043303nteger )
% 14.25/8.15 => ( ( groups1304777262505850412r_assn @ G @ ( sup_su848401254843788991nteger @ A @ B4 ) )
% 14.25/8.15 = ( times_times_assn @ ( groups1304777262505850412r_assn @ G @ A ) @ ( groups1304777262505850412r_assn @ G @ B4 ) ) ) ) ) ) ).
% 14.25/8.15
% 14.25/8.15 thf(fact_7384_prod_Ounion__disjoint,axiom,
% 14.25/8.15 ! [A: set_Product_unit,B4: set_Product_unit,G: product_unit > assn] :
% 14.25/8.15 ( ( finite4290736615968046902t_unit @ A )
% 14.25/8.15 => ( ( finite4290736615968046902t_unit @ B4 )
% 14.25/8.15 => ( ( ( inf_in4660618365625256667t_unit @ A @ B4 )
% 14.25/8.15 = bot_bo3957492148770167129t_unit )
% 14.25/8.15 => ( ( groups5360233303959380816t_assn @ G @ ( sup_su793286257634532545t_unit @ A @ B4 ) )
% 14.25/8.15 = ( times_times_assn @ ( groups5360233303959380816t_assn @ G @ A ) @ ( groups5360233303959380816t_assn @ G @ B4 ) ) ) ) ) ) ).
% 14.25/8.15
% 14.25/8.15 thf(fact_7385_prod_Ounion__disjoint,axiom,
% 14.25/8.15 ! [A: set_o,B4: set_o,G: $o > assn] :
% 14.25/8.15 ( ( finite_finite_o @ A )
% 14.25/8.15 => ( ( finite_finite_o @ B4 )
% 14.25/8.15 => ( ( ( inf_inf_set_o @ A @ B4 )
% 14.25/8.15 = bot_bot_set_o )
% 14.25/8.15 => ( ( groups5301882518646026715o_assn @ G @ ( sup_sup_set_o @ A @ B4 ) )
% 14.25/8.15 = ( times_times_assn @ ( groups5301882518646026715o_assn @ G @ A ) @ ( groups5301882518646026715o_assn @ G @ B4 ) ) ) ) ) ) ).
% 14.25/8.15
% 14.25/8.15 thf(fact_7386_prod_Ounion__disjoint,axiom,
% 14.25/8.15 ! [A: set_nat,B4: set_nat,G: nat > assn] :
% 14.25/8.15 ( ( finite_finite_nat @ A )
% 14.25/8.15 => ( ( finite_finite_nat @ B4 )
% 14.25/8.15 => ( ( ( inf_inf_set_nat @ A @ B4 )
% 14.25/8.15 = bot_bot_set_nat )
% 14.25/8.15 => ( ( groups6906906614972039071t_assn @ G @ ( sup_sup_set_nat @ A @ B4 ) )
% 14.25/8.15 = ( times_times_assn @ ( groups6906906614972039071t_assn @ G @ A ) @ ( groups6906906614972039071t_assn @ G @ B4 ) ) ) ) ) ) ).
% 14.25/8.15
% 14.25/8.15 thf(fact_7387_prod_Ounion__disjoint,axiom,
% 14.25/8.15 ! [A: set_int,B4: set_int,G: int > assn] :
% 14.25/8.15 ( ( finite_finite_int @ A )
% 14.25/8.15 => ( ( finite_finite_int @ B4 )
% 14.25/8.15 => ( ( ( inf_inf_set_int @ A @ B4 )
% 14.25/8.15 = bot_bot_set_int )
% 14.25/8.15 => ( ( groups7882442080178216443t_assn @ G @ ( sup_sup_set_int @ A @ B4 ) )
% 14.25/8.15 = ( times_times_assn @ ( groups7882442080178216443t_assn @ G @ A ) @ ( groups7882442080178216443t_assn @ G @ B4 ) ) ) ) ) ) ).
% 14.25/8.15
% 14.25/8.15 thf(fact_7388_prod_Ounion__disjoint,axiom,
% 14.25/8.15 ! [A: set_Code_integer,B4: set_Code_integer,G: code_integer > rat] :
% 14.25/8.15 ( ( finite6017078050557962740nteger @ A )
% 14.25/8.15 => ( ( finite6017078050557962740nteger @ B4 )
% 14.25/8.15 => ( ( ( inf_in1364745209274528805nteger @ A @ B4 )
% 14.25/8.15 = bot_bo3990330152332043303nteger )
% 14.25/8.15 => ( ( groups2555765274223993564er_rat @ G @ ( sup_su848401254843788991nteger @ A @ B4 ) )
% 14.25/8.15 = ( times_times_rat @ ( groups2555765274223993564er_rat @ G @ A ) @ ( groups2555765274223993564er_rat @ G @ B4 ) ) ) ) ) ) ).
% 14.25/8.15
% 14.25/8.15 thf(fact_7389_prod_Ounion__disjoint,axiom,
% 14.25/8.15 ! [A: set_Product_unit,B4: set_Product_unit,G: product_unit > rat] :
% 14.25/8.15 ( ( finite4290736615968046902t_unit @ A )
% 14.25/8.15 => ( ( finite4290736615968046902t_unit @ B4 )
% 14.25/8.15 => ( ( ( inf_in4660618365625256667t_unit @ A @ B4 )
% 14.25/8.15 = bot_bo3957492148770167129t_unit )
% 14.25/8.15 => ( ( groups8984663225794451456it_rat @ G @ ( sup_su793286257634532545t_unit @ A @ B4 ) )
% 14.25/8.15 = ( times_times_rat @ ( groups8984663225794451456it_rat @ G @ A ) @ ( groups8984663225794451456it_rat @ G @ B4 ) ) ) ) ) ) ).
% 14.25/8.15
% 14.25/8.15 thf(fact_7390_prod_Ounion__disjoint,axiom,
% 14.25/8.15 ! [A: set_o,B4: set_o,G: $o > rat] :
% 14.25/8.15 ( ( finite_finite_o @ A )
% 14.25/8.15 => ( ( finite_finite_o @ B4 )
% 14.25/8.15 => ( ( ( inf_inf_set_o @ A @ B4 )
% 14.25/8.15 = bot_bot_set_o )
% 14.25/8.15 => ( ( groups2869687844427037835_o_rat @ G @ ( sup_sup_set_o @ A @ B4 ) )
% 14.25/8.15 = ( times_times_rat @ ( groups2869687844427037835_o_rat @ G @ A ) @ ( groups2869687844427037835_o_rat @ G @ B4 ) ) ) ) ) ) ).
% 14.25/8.15
% 14.25/8.15 thf(fact_7391_prod_Ounion__disjoint,axiom,
% 14.25/8.15 ! [A: set_nat,B4: set_nat,G: nat > rat] :
% 14.25/8.15 ( ( finite_finite_nat @ A )
% 14.25/8.15 => ( ( finite_finite_nat @ B4 )
% 14.25/8.15 => ( ( ( inf_inf_set_nat @ A @ B4 )
% 14.25/8.15 = bot_bot_set_nat )
% 14.25/8.15 => ( ( groups73079841787564623at_rat @ G @ ( sup_sup_set_nat @ A @ B4 ) )
% 14.25/8.15 = ( times_times_rat @ ( groups73079841787564623at_rat @ G @ A ) @ ( groups73079841787564623at_rat @ G @ B4 ) ) ) ) ) ) ).
% 14.25/8.15
% 14.25/8.15 thf(fact_7392_prod_Ounion__disjoint,axiom,
% 14.25/8.15 ! [A: set_int,B4: set_int,G: int > rat] :
% 14.25/8.15 ( ( finite_finite_int @ A )
% 14.25/8.15 => ( ( finite_finite_int @ B4 )
% 14.25/8.15 => ( ( ( inf_inf_set_int @ A @ B4 )
% 14.25/8.15 = bot_bot_set_int )
% 14.25/8.15 => ( ( groups1072433553688619179nt_rat @ G @ ( sup_sup_set_int @ A @ B4 ) )
% 14.25/8.15 = ( times_times_rat @ ( groups1072433553688619179nt_rat @ G @ A ) @ ( groups1072433553688619179nt_rat @ G @ B4 ) ) ) ) ) ) ).
% 14.25/8.15
% 14.25/8.15 thf(fact_7393_prod_Ounion__diff2,axiom,
% 14.25/8.15 ! [A: set_Code_integer,B4: set_Code_integer,G: code_integer > assn] :
% 14.25/8.15 ( ( finite6017078050557962740nteger @ A )
% 14.25/8.15 => ( ( finite6017078050557962740nteger @ B4 )
% 14.25/8.15 => ( ( groups1304777262505850412r_assn @ G @ ( sup_su848401254843788991nteger @ A @ B4 ) )
% 14.25/8.15 = ( times_times_assn @ ( times_times_assn @ ( groups1304777262505850412r_assn @ G @ ( minus_2355218937544613996nteger @ A @ B4 ) ) @ ( groups1304777262505850412r_assn @ G @ ( minus_2355218937544613996nteger @ B4 @ A ) ) ) @ ( groups1304777262505850412r_assn @ G @ ( inf_in1364745209274528805nteger @ A @ B4 ) ) ) ) ) ) ).
% 14.25/8.15
% 14.25/8.15 thf(fact_7394_prod_Ounion__diff2,axiom,
% 14.25/8.15 ! [A: set_nat,B4: set_nat,G: nat > assn] :
% 14.25/8.15 ( ( finite_finite_nat @ A )
% 14.25/8.15 => ( ( finite_finite_nat @ B4 )
% 14.25/8.15 => ( ( groups6906906614972039071t_assn @ G @ ( sup_sup_set_nat @ A @ B4 ) )
% 14.25/8.15 = ( times_times_assn @ ( times_times_assn @ ( groups6906906614972039071t_assn @ G @ ( minus_minus_set_nat @ A @ B4 ) ) @ ( groups6906906614972039071t_assn @ G @ ( minus_minus_set_nat @ B4 @ A ) ) ) @ ( groups6906906614972039071t_assn @ G @ ( inf_inf_set_nat @ A @ B4 ) ) ) ) ) ) ).
% 14.25/8.15
% 14.25/8.15 thf(fact_7395_prod_Ounion__diff2,axiom,
% 14.25/8.15 ! [A: set_Code_integer,B4: set_Code_integer,G: code_integer > rat] :
% 14.25/8.15 ( ( finite6017078050557962740nteger @ A )
% 14.25/8.15 => ( ( finite6017078050557962740nteger @ B4 )
% 14.25/8.15 => ( ( groups2555765274223993564er_rat @ G @ ( sup_su848401254843788991nteger @ A @ B4 ) )
% 14.25/8.15 = ( times_times_rat @ ( times_times_rat @ ( groups2555765274223993564er_rat @ G @ ( minus_2355218937544613996nteger @ A @ B4 ) ) @ ( groups2555765274223993564er_rat @ G @ ( minus_2355218937544613996nteger @ B4 @ A ) ) ) @ ( groups2555765274223993564er_rat @ G @ ( inf_in1364745209274528805nteger @ A @ B4 ) ) ) ) ) ) ).
% 14.25/8.15
% 14.25/8.15 thf(fact_7396_prod_Ounion__diff2,axiom,
% 14.25/8.15 ! [A: set_nat,B4: set_nat,G: nat > rat] :
% 14.25/8.15 ( ( finite_finite_nat @ A )
% 14.25/8.15 => ( ( finite_finite_nat @ B4 )
% 14.25/8.15 => ( ( groups73079841787564623at_rat @ G @ ( sup_sup_set_nat @ A @ B4 ) )
% 14.25/8.15 = ( times_times_rat @ ( times_times_rat @ ( groups73079841787564623at_rat @ G @ ( minus_minus_set_nat @ A @ B4 ) ) @ ( groups73079841787564623at_rat @ G @ ( minus_minus_set_nat @ B4 @ A ) ) ) @ ( groups73079841787564623at_rat @ G @ ( inf_inf_set_nat @ A @ B4 ) ) ) ) ) ) ).
% 14.25/8.15
% 14.25/8.15 thf(fact_7397_prod_Ounion__diff2,axiom,
% 14.25/8.15 ! [A: set_Code_integer,B4: set_Code_integer,G: code_integer > nat] :
% 14.25/8.15 ( ( finite6017078050557962740nteger @ A )
% 14.25/8.15 => ( ( finite6017078050557962740nteger @ B4 )
% 14.25/8.15 => ( ( groups3190895334310489300er_nat @ G @ ( sup_su848401254843788991nteger @ A @ B4 ) )
% 14.25/8.15 = ( times_times_nat @ ( times_times_nat @ ( groups3190895334310489300er_nat @ G @ ( minus_2355218937544613996nteger @ A @ B4 ) ) @ ( groups3190895334310489300er_nat @ G @ ( minus_2355218937544613996nteger @ B4 @ A ) ) ) @ ( groups3190895334310489300er_nat @ G @ ( inf_in1364745209274528805nteger @ A @ B4 ) ) ) ) ) ) ).
% 14.25/8.15
% 14.25/8.15 thf(fact_7398_prod_Ounion__diff2,axiom,
% 14.25/8.15 ! [A: set_Code_integer,B4: set_Code_integer,G: code_integer > int] :
% 14.25/8.15 ( ( finite6017078050557962740nteger @ A )
% 14.25/8.15 => ( ( finite6017078050557962740nteger @ B4 )
% 14.25/8.15 => ( ( groups3188404863801439024er_int @ G @ ( sup_su848401254843788991nteger @ A @ B4 ) )
% 14.25/8.15 = ( times_times_int @ ( times_times_int @ ( groups3188404863801439024er_int @ G @ ( minus_2355218937544613996nteger @ A @ B4 ) ) @ ( groups3188404863801439024er_int @ G @ ( minus_2355218937544613996nteger @ B4 @ A ) ) ) @ ( groups3188404863801439024er_int @ G @ ( inf_in1364745209274528805nteger @ A @ B4 ) ) ) ) ) ) ).
% 14.25/8.15
% 14.25/8.15 thf(fact_7399_prod_Ounion__diff2,axiom,
% 14.25/8.15 ! [A: set_nat,B4: set_nat,G: nat > int] :
% 14.25/8.15 ( ( finite_finite_nat @ A )
% 14.25/8.15 => ( ( finite_finite_nat @ B4 )
% 14.25/8.15 => ( ( groups705719431365010083at_int @ G @ ( sup_sup_set_nat @ A @ B4 ) )
% 14.25/8.15 = ( times_times_int @ ( times_times_int @ ( groups705719431365010083at_int @ G @ ( minus_minus_set_nat @ A @ B4 ) ) @ ( groups705719431365010083at_int @ G @ ( minus_minus_set_nat @ B4 @ A ) ) ) @ ( groups705719431365010083at_int @ G @ ( inf_inf_set_nat @ A @ B4 ) ) ) ) ) ) ).
% 14.25/8.15
% 14.25/8.15 thf(fact_7400_prod_Ounion__diff2,axiom,
% 14.25/8.15 ! [A: set_nat,B4: set_nat,G: nat > nat] :
% 14.25/8.15 ( ( finite_finite_nat @ A )
% 14.25/8.15 => ( ( finite_finite_nat @ B4 )
% 14.25/8.15 => ( ( groups708209901874060359at_nat @ G @ ( sup_sup_set_nat @ A @ B4 ) )
% 14.25/8.15 = ( times_times_nat @ ( times_times_nat @ ( groups708209901874060359at_nat @ G @ ( minus_minus_set_nat @ A @ B4 ) ) @ ( groups708209901874060359at_nat @ G @ ( minus_minus_set_nat @ B4 @ A ) ) ) @ ( groups708209901874060359at_nat @ G @ ( inf_inf_set_nat @ A @ B4 ) ) ) ) ) ) ).
% 14.25/8.15
% 14.25/8.15 thf(fact_7401_Suc__times__gbinomial,axiom,
% 14.25/8.15 ! [K3: nat,A2: rat] :
% 14.25/8.15 ( ( times_times_rat @ ( semiri681578069525770553at_rat @ ( suc @ K3 ) ) @ ( gbinomial_rat @ ( plus_plus_rat @ A2 @ one_one_rat ) @ ( suc @ K3 ) ) )
% 14.25/8.15 = ( times_times_rat @ ( plus_plus_rat @ A2 @ one_one_rat ) @ ( gbinomial_rat @ A2 @ K3 ) ) ) ).
% 14.25/8.15
% 14.25/8.15 thf(fact_7402_gbinomial__absorption,axiom,
% 14.25/8.15 ! [K3: nat,A2: rat] :
% 14.25/8.15 ( ( times_times_rat @ ( semiri681578069525770553at_rat @ ( suc @ K3 ) ) @ ( gbinomial_rat @ A2 @ ( suc @ K3 ) ) )
% 14.25/8.15 = ( times_times_rat @ A2 @ ( gbinomial_rat @ ( minus_minus_rat @ A2 @ one_one_rat ) @ K3 ) ) ) ).
% 14.25/8.15
% 14.25/8.15 thf(fact_7403_gbinomial__trinomial__revision,axiom,
% 14.25/8.15 ! [K3: nat,M4: nat,A2: rat] :
% 14.25/8.15 ( ( ord_less_eq_nat @ K3 @ M4 )
% 14.25/8.15 => ( ( times_times_rat @ ( gbinomial_rat @ A2 @ M4 ) @ ( gbinomial_rat @ ( semiri681578069525770553at_rat @ M4 ) @ K3 ) )
% 14.25/8.15 = ( times_times_rat @ ( gbinomial_rat @ A2 @ K3 ) @ ( gbinomial_rat @ ( minus_minus_rat @ A2 @ ( semiri681578069525770553at_rat @ K3 ) ) @ ( minus_minus_nat @ M4 @ K3 ) ) ) ) ) ).
% 14.25/8.15
% 14.25/8.15 thf(fact_7404_neg__one__power__add__eq__neg__one__power__diff,axiom,
% 14.25/8.15 ! [K3: nat,N2: nat] :
% 14.25/8.15 ( ( ord_less_eq_nat @ K3 @ N2 )
% 14.25/8.15 => ( ( power_power_int @ ( uminus_uminus_int @ one_one_int ) @ ( plus_plus_nat @ N2 @ K3 ) )
% 14.25/8.15 = ( power_power_int @ ( uminus_uminus_int @ one_one_int ) @ ( minus_minus_nat @ N2 @ K3 ) ) ) ) ).
% 14.25/8.15
% 14.25/8.15 thf(fact_7405_neg__one__power__add__eq__neg__one__power__diff,axiom,
% 14.25/8.15 ! [K3: nat,N2: nat] :
% 14.25/8.15 ( ( ord_less_eq_nat @ K3 @ N2 )
% 14.25/8.15 => ( ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ ( plus_plus_nat @ N2 @ K3 ) )
% 14.25/8.15 = ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ ( minus_minus_nat @ N2 @ K3 ) ) ) ) ).
% 14.25/8.15
% 14.25/8.15 thf(fact_7406_ex__less__of__nat__mult,axiom,
% 14.25/8.15 ! [X: rat,Y: rat] :
% 14.25/8.15 ( ( ord_less_rat @ zero_zero_rat @ X )
% 14.25/8.15 => ? [N3: nat] : ( ord_less_rat @ Y @ ( times_times_rat @ ( semiri681578069525770553at_rat @ N3 ) @ X ) ) ) ).
% 14.25/8.15
% 14.25/8.15 thf(fact_7407_gbinomial__absorption_H,axiom,
% 14.25/8.15 ! [K3: nat,A2: rat] :
% 14.25/8.15 ( ( ord_less_nat @ zero_zero_nat @ K3 )
% 14.25/8.15 => ( ( gbinomial_rat @ A2 @ K3 )
% 14.25/8.15 = ( times_times_rat @ ( divide_divide_rat @ A2 @ ( semiri681578069525770553at_rat @ K3 ) ) @ ( gbinomial_rat @ ( minus_minus_rat @ A2 @ one_one_rat ) @ ( minus_minus_nat @ K3 @ one_one_nat ) ) ) ) ) ).
% 14.25/8.15
% 14.25/8.15 thf(fact_7408_pochhammer__minus,axiom,
% 14.25/8.15 ! [B: rat,K3: nat] :
% 14.25/8.15 ( ( comm_s4028243227959126397er_rat @ ( uminus_uminus_rat @ B ) @ K3 )
% 14.25/8.15 = ( times_times_rat @ ( power_power_rat @ ( uminus_uminus_rat @ one_one_rat ) @ K3 ) @ ( comm_s4028243227959126397er_rat @ ( plus_plus_rat @ ( minus_minus_rat @ B @ ( semiri681578069525770553at_rat @ K3 ) ) @ one_one_rat ) @ K3 ) ) ) ).
% 14.25/8.15
% 14.25/8.15 thf(fact_7409_pochhammer__minus,axiom,
% 14.25/8.15 ! [B: int,K3: nat] :
% 14.25/8.15 ( ( comm_s4660882817536571857er_int @ ( uminus_uminus_int @ B ) @ K3 )
% 14.25/8.15 = ( times_times_int @ ( power_power_int @ ( uminus_uminus_int @ one_one_int ) @ K3 ) @ ( comm_s4660882817536571857er_int @ ( plus_plus_int @ ( minus_minus_int @ B @ ( semiri1314217659103216013at_int @ K3 ) ) @ one_one_int ) @ K3 ) ) ) ).
% 14.25/8.15
% 14.25/8.15 thf(fact_7410_pochhammer__minus,axiom,
% 14.25/8.15 ! [B: code_integer,K3: nat] :
% 14.25/8.15 ( ( comm_s8582702949713902594nteger @ ( uminus1351360451143612070nteger @ B ) @ K3 )
% 14.25/8.15 = ( times_3573771949741848930nteger @ ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ K3 ) @ ( comm_s8582702949713902594nteger @ ( plus_p5714425477246183910nteger @ ( minus_8373710615458151222nteger @ B @ ( semiri4939895301339042750nteger @ K3 ) ) @ one_one_Code_integer ) @ K3 ) ) ) ).
% 14.25/8.15
% 14.25/8.15 thf(fact_7411_pochhammer__minus_H,axiom,
% 14.25/8.15 ! [B: rat,K3: nat] :
% 14.25/8.15 ( ( comm_s4028243227959126397er_rat @ ( plus_plus_rat @ ( minus_minus_rat @ B @ ( semiri681578069525770553at_rat @ K3 ) ) @ one_one_rat ) @ K3 )
% 14.25/8.15 = ( times_times_rat @ ( power_power_rat @ ( uminus_uminus_rat @ one_one_rat ) @ K3 ) @ ( comm_s4028243227959126397er_rat @ ( uminus_uminus_rat @ B ) @ K3 ) ) ) ).
% 14.25/8.15
% 14.25/8.15 thf(fact_7412_pochhammer__minus_H,axiom,
% 14.25/8.15 ! [B: int,K3: nat] :
% 14.25/8.15 ( ( comm_s4660882817536571857er_int @ ( plus_plus_int @ ( minus_minus_int @ B @ ( semiri1314217659103216013at_int @ K3 ) ) @ one_one_int ) @ K3 )
% 14.25/8.15 = ( times_times_int @ ( power_power_int @ ( uminus_uminus_int @ one_one_int ) @ K3 ) @ ( comm_s4660882817536571857er_int @ ( uminus_uminus_int @ B ) @ K3 ) ) ) ).
% 14.25/8.15
% 14.25/8.15 thf(fact_7413_pochhammer__minus_H,axiom,
% 14.25/8.15 ! [B: code_integer,K3: nat] :
% 14.25/8.15 ( ( comm_s8582702949713902594nteger @ ( plus_p5714425477246183910nteger @ ( minus_8373710615458151222nteger @ B @ ( semiri4939895301339042750nteger @ K3 ) ) @ one_one_Code_integer ) @ K3 )
% 14.25/8.15 = ( times_3573771949741848930nteger @ ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ K3 ) @ ( comm_s8582702949713902594nteger @ ( uminus1351360451143612070nteger @ B ) @ K3 ) ) ) ).
% 14.25/8.15
% 14.25/8.15 thf(fact_7414_zmult__zless__mono2__lemma,axiom,
% 14.25/8.15 ! [I2: int,J: int,K3: nat] :
% 14.25/8.15 ( ( ord_less_int @ I2 @ J )
% 14.25/8.15 => ( ( ord_less_nat @ zero_zero_nat @ K3 )
% 14.25/8.15 => ( ord_less_int @ ( times_times_int @ ( semiri1314217659103216013at_int @ K3 ) @ I2 ) @ ( times_times_int @ ( semiri1314217659103216013at_int @ K3 ) @ J ) ) ) ) ).
% 14.25/8.15
% 14.25/8.15 thf(fact_7415_nat__mult__div__cancel__disj,axiom,
% 14.25/8.15 ! [K3: nat,M4: nat,N2: nat] :
% 14.25/8.15 ( ( ( K3 = zero_zero_nat )
% 14.25/8.15 => ( ( divide_divide_nat @ ( times_times_nat @ K3 @ M4 ) @ ( times_times_nat @ K3 @ N2 ) )
% 14.25/8.15 = zero_zero_nat ) )
% 14.25/8.15 & ( ( K3 != zero_zero_nat )
% 14.25/8.15 => ( ( divide_divide_nat @ ( times_times_nat @ K3 @ M4 ) @ ( times_times_nat @ K3 @ N2 ) )
% 14.25/8.15 = ( divide_divide_nat @ M4 @ N2 ) ) ) ) ).
% 14.25/8.15
% 14.25/8.15 thf(fact_7416_pos__zmult__eq__1__iff__lemma,axiom,
% 14.25/8.15 ! [M4: int,N2: int] :
% 14.25/8.15 ( ( ( times_times_int @ M4 @ N2 )
% 14.25/8.15 = one_one_int )
% 14.25/8.15 => ( ( M4 = one_one_int )
% 14.25/8.15 | ( M4
% 14.25/8.15 = ( uminus_uminus_int @ one_one_int ) ) ) ) ).
% 14.25/8.15
% 14.25/8.15 thf(fact_7417_zmult__eq__1__iff,axiom,
% 14.25/8.15 ! [M4: int,N2: int] :
% 14.25/8.15 ( ( ( times_times_int @ M4 @ N2 )
% 14.25/8.15 = one_one_int )
% 14.25/8.15 = ( ( ( M4 = one_one_int )
% 14.25/8.15 & ( N2 = one_one_int ) )
% 14.25/8.15 | ( ( M4
% 14.25/8.15 = ( uminus_uminus_int @ one_one_int ) )
% 14.25/8.15 & ( N2
% 14.25/8.15 = ( uminus_uminus_int @ one_one_int ) ) ) ) ) ).
% 14.25/8.15
% 14.25/8.15 thf(fact_7418_zdvd__mono,axiom,
% 14.25/8.15 ! [K3: int,M4: int,T4: int] :
% 14.25/8.15 ( ( K3 != zero_zero_int )
% 14.25/8.15 => ( ( dvd_dvd_int @ M4 @ T4 )
% 14.25/8.15 = ( dvd_dvd_int @ ( times_times_int @ K3 @ M4 ) @ ( times_times_int @ K3 @ T4 ) ) ) ) ).
% 14.25/8.15
% 14.25/8.15 thf(fact_7419_zdvd__mult__cancel,axiom,
% 14.25/8.15 ! [K3: int,M4: int,N2: int] :
% 14.25/8.15 ( ( dvd_dvd_int @ ( times_times_int @ K3 @ M4 ) @ ( times_times_int @ K3 @ N2 ) )
% 14.25/8.15 => ( ( K3 != zero_zero_int )
% 14.25/8.15 => ( dvd_dvd_int @ M4 @ N2 ) ) ) ).
% 14.25/8.15
% 14.25/8.15 thf(fact_7420_zdvd__reduce,axiom,
% 14.25/8.15 ! [K3: int,N2: int,M4: int] :
% 14.25/8.15 ( ( dvd_dvd_int @ K3 @ ( plus_plus_int @ N2 @ ( times_times_int @ K3 @ M4 ) ) )
% 14.25/8.15 = ( dvd_dvd_int @ K3 @ N2 ) ) ).
% 14.25/8.15
% 14.25/8.15 thf(fact_7421_zdvd__period,axiom,
% 14.25/8.15 ! [A2: int,D2: int,X: int,T4: int,C: int] :
% 14.25/8.15 ( ( dvd_dvd_int @ A2 @ D2 )
% 14.25/8.15 => ( ( dvd_dvd_int @ A2 @ ( plus_plus_int @ X @ T4 ) )
% 14.25/8.15 = ( dvd_dvd_int @ A2 @ ( plus_plus_int @ ( plus_plus_int @ X @ ( times_times_int @ C @ D2 ) ) @ T4 ) ) ) ) ).
% 14.25/8.15
% 14.25/8.15 thf(fact_7422_unique__quotient__lemma__neg,axiom,
% 14.25/8.15 ! [B: int,Q5: int,R6: int,Q4: int,R2: int] :
% 14.25/8.15 ( ( ord_less_eq_int @ ( plus_plus_int @ ( times_times_int @ B @ Q5 ) @ R6 ) @ ( plus_plus_int @ ( times_times_int @ B @ Q4 ) @ R2 ) )
% 14.25/8.15 => ( ( ord_less_eq_int @ R2 @ zero_zero_int )
% 14.25/8.15 => ( ( ord_less_int @ B @ R2 )
% 14.25/8.15 => ( ( ord_less_int @ B @ R6 )
% 14.25/8.15 => ( ord_less_eq_int @ Q4 @ Q5 ) ) ) ) ) ).
% 14.25/8.15
% 14.25/8.15 thf(fact_7423_unique__quotient__lemma,axiom,
% 14.25/8.15 ! [B: int,Q5: int,R6: int,Q4: int,R2: int] :
% 14.25/8.15 ( ( ord_less_eq_int @ ( plus_plus_int @ ( times_times_int @ B @ Q5 ) @ R6 ) @ ( plus_plus_int @ ( times_times_int @ B @ Q4 ) @ R2 ) )
% 14.25/8.15 => ( ( ord_less_eq_int @ zero_zero_int @ R6 )
% 14.25/8.15 => ( ( ord_less_int @ R6 @ B )
% 14.25/8.15 => ( ( ord_less_int @ R2 @ B )
% 14.25/8.15 => ( ord_less_eq_int @ Q5 @ Q4 ) ) ) ) ) ).
% 14.25/8.15
% 14.25/8.15 thf(fact_7424_zdiv__mono2__neg__lemma,axiom,
% 14.25/8.15 ! [B: int,Q4: int,R2: int,B3: int,Q5: int,R6: int] :
% 14.25/8.15 ( ( ( plus_plus_int @ ( times_times_int @ B @ Q4 ) @ R2 )
% 14.25/8.15 = ( plus_plus_int @ ( times_times_int @ B3 @ Q5 ) @ R6 ) )
% 14.25/8.15 => ( ( ord_less_int @ ( plus_plus_int @ ( times_times_int @ B3 @ Q5 ) @ R6 ) @ zero_zero_int )
% 14.25/8.15 => ( ( ord_less_int @ R2 @ B )
% 14.25/8.15 => ( ( ord_less_eq_int @ zero_zero_int @ R6 )
% 14.25/8.15 => ( ( ord_less_int @ zero_zero_int @ B3 )
% 14.25/8.15 => ( ( ord_less_eq_int @ B3 @ B )
% 14.25/8.15 => ( ord_less_eq_int @ Q5 @ Q4 ) ) ) ) ) ) ) ).
% 14.25/8.15
% 14.25/8.15 thf(fact_7425_zdiv__mono2__lemma,axiom,
% 14.25/8.15 ! [B: int,Q4: int,R2: int,B3: int,Q5: int,R6: int] :
% 14.25/8.15 ( ( ( plus_plus_int @ ( times_times_int @ B @ Q4 ) @ R2 )
% 14.25/8.15 = ( plus_plus_int @ ( times_times_int @ B3 @ Q5 ) @ R6 ) )
% 14.25/8.15 => ( ( ord_less_eq_int @ zero_zero_int @ ( plus_plus_int @ ( times_times_int @ B3 @ Q5 ) @ R6 ) )
% 14.25/8.15 => ( ( ord_less_int @ R6 @ B3 )
% 14.25/8.15 => ( ( ord_less_eq_int @ zero_zero_int @ R2 )
% 14.25/8.15 => ( ( ord_less_int @ zero_zero_int @ B3 )
% 14.25/8.15 => ( ( ord_less_eq_int @ B3 @ B )
% 14.25/8.15 => ( ord_less_eq_int @ Q4 @ Q5 ) ) ) ) ) ) ) ).
% 14.25/8.15
% 14.25/8.15 thf(fact_7426_int__div__pos__eq,axiom,
% 14.25/8.15 ! [A2: int,B: int,Q4: int,R2: int] :
% 14.25/8.15 ( ( A2
% 14.25/8.15 = ( plus_plus_int @ ( times_times_int @ B @ Q4 ) @ R2 ) )
% 14.25/8.15 => ( ( ord_less_eq_int @ zero_zero_int @ R2 )
% 14.25/8.15 => ( ( ord_less_int @ R2 @ B )
% 14.25/8.15 => ( ( divide_divide_int @ A2 @ B )
% 14.25/8.15 = Q4 ) ) ) ) ).
% 14.25/8.15
% 14.25/8.15 thf(fact_7427_int__div__neg__eq,axiom,
% 14.25/8.15 ! [A2: int,B: int,Q4: int,R2: int] :
% 14.25/8.15 ( ( A2
% 14.25/8.15 = ( plus_plus_int @ ( times_times_int @ B @ Q4 ) @ R2 ) )
% 14.25/8.15 => ( ( ord_less_eq_int @ R2 @ zero_zero_int )
% 14.25/8.15 => ( ( ord_less_int @ B @ R2 )
% 14.25/8.15 => ( ( divide_divide_int @ A2 @ B )
% 14.25/8.15 = Q4 ) ) ) ) ).
% 14.25/8.15
% 14.25/8.15 thf(fact_7428_q__pos__lemma,axiom,
% 14.25/8.15 ! [B3: int,Q5: int,R6: int] :
% 14.25/8.15 ( ( ord_less_eq_int @ zero_zero_int @ ( plus_plus_int @ ( times_times_int @ B3 @ Q5 ) @ R6 ) )
% 14.25/8.15 => ( ( ord_less_int @ R6 @ B3 )
% 14.25/8.15 => ( ( ord_less_int @ zero_zero_int @ B3 )
% 14.25/8.15 => ( ord_less_eq_int @ zero_zero_int @ Q5 ) ) ) ) ).
% 14.25/8.15
% 14.25/8.15 thf(fact_7429_split__zdiv,axiom,
% 14.25/8.15 ! [P: int > $o,N2: int,K3: int] :
% 14.25/8.15 ( ( P @ ( divide_divide_int @ N2 @ K3 ) )
% 14.25/8.15 = ( ( ( K3 = zero_zero_int )
% 14.25/8.15 => ( P @ zero_zero_int ) )
% 14.25/8.15 & ( ( ord_less_int @ zero_zero_int @ K3 )
% 14.25/8.15 => ! [I4: int,J3: int] :
% 14.25/8.15 ( ( ( ord_less_eq_int @ zero_zero_int @ J3 )
% 14.25/8.15 & ( ord_less_int @ J3 @ K3 )
% 14.25/8.15 & ( N2
% 14.25/8.15 = ( plus_plus_int @ ( times_times_int @ K3 @ I4 ) @ J3 ) ) )
% 14.25/8.15 => ( P @ I4 ) ) )
% 14.25/8.15 & ( ( ord_less_int @ K3 @ zero_zero_int )
% 14.25/8.15 => ! [I4: int,J3: int] :
% 14.25/8.15 ( ( ( ord_less_int @ K3 @ J3 )
% 14.25/8.15 & ( ord_less_eq_int @ J3 @ zero_zero_int )
% 14.25/8.15 & ( N2
% 14.25/8.15 = ( plus_plus_int @ ( times_times_int @ K3 @ I4 ) @ J3 ) ) )
% 14.25/8.15 => ( P @ I4 ) ) ) ) ) ).
% 14.25/8.15
% 14.25/8.15 thf(fact_7430_incr__mult__lemma,axiom,
% 14.25/8.15 ! [D2: int,P: int > $o,K3: int] :
% 14.25/8.15 ( ( ord_less_int @ zero_zero_int @ D2 )
% 14.25/8.15 => ( ! [X3: int] :
% 14.25/8.15 ( ( P @ X3 )
% 14.25/8.15 => ( P @ ( plus_plus_int @ X3 @ D2 ) ) )
% 14.25/8.15 => ( ( ord_less_eq_int @ zero_zero_int @ K3 )
% 14.25/8.15 => ! [X6: int] :
% 14.25/8.15 ( ( P @ X6 )
% 14.25/8.15 => ( P @ ( plus_plus_int @ X6 @ ( times_times_int @ K3 @ D2 ) ) ) ) ) ) ) ).
% 14.25/8.15
% 14.25/8.15 thf(fact_7431_pos__zmult__eq__1__iff,axiom,
% 14.25/8.15 ! [M4: int,N2: int] :
% 14.25/8.15 ( ( ord_less_int @ zero_zero_int @ M4 )
% 14.25/8.15 => ( ( ( times_times_int @ M4 @ N2 )
% 14.25/8.15 = one_one_int )
% 14.25/8.15 = ( ( M4 = one_one_int )
% 14.25/8.15 & ( N2 = one_one_int ) ) ) ) ).
% 14.25/8.15
% 14.25/8.15 thf(fact_7432_zmult__zless__mono2,axiom,
% 14.25/8.15 ! [I2: int,J: int,K3: int] :
% 14.25/8.15 ( ( ord_less_int @ I2 @ J )
% 14.25/8.15 => ( ( ord_less_int @ zero_zero_int @ K3 )
% 14.25/8.15 => ( ord_less_int @ ( times_times_int @ K3 @ I2 ) @ ( times_times_int @ K3 @ J ) ) ) ) ).
% 14.25/8.15
% 14.25/8.15 thf(fact_7433_minusinfinity,axiom,
% 14.25/8.15 ! [D2: int,P1: int > $o,P: int > $o] :
% 14.25/8.15 ( ( ord_less_int @ zero_zero_int @ D2 )
% 14.25/8.15 => ( ! [X3: int,K: int] :
% 14.25/8.15 ( ( P1 @ X3 )
% 14.25/8.15 = ( P1 @ ( minus_minus_int @ X3 @ ( times_times_int @ K @ D2 ) ) ) )
% 14.25/8.15 => ( ? [Z6: int] :
% 14.25/8.15 ! [X3: int] :
% 14.25/8.15 ( ( ord_less_int @ X3 @ Z6 )
% 14.25/8.15 => ( ( P @ X3 )
% 14.25/8.15 = ( P1 @ X3 ) ) )
% 14.25/8.15 => ( ? [X_12: int] : ( P1 @ X_12 )
% 14.25/8.15 => ? [X_1: int] : ( P @ X_1 ) ) ) ) ) ).
% 14.25/8.15
% 14.25/8.15 thf(fact_7434_plusinfinity,axiom,
% 14.25/8.15 ! [D2: int,P7: int > $o,P: int > $o] :
% 14.25/8.15 ( ( ord_less_int @ zero_zero_int @ D2 )
% 14.25/8.15 => ( ! [X3: int,K: int] :
% 14.25/8.15 ( ( P7 @ X3 )
% 14.25/8.15 = ( P7 @ ( minus_minus_int @ X3 @ ( times_times_int @ K @ D2 ) ) ) )
% 14.25/8.15 => ( ? [Z6: int] :
% 14.25/8.15 ! [X3: int] :
% 14.25/8.15 ( ( ord_less_int @ Z6 @ X3 )
% 14.25/8.15 => ( ( P @ X3 )
% 14.25/8.15 = ( P7 @ X3 ) ) )
% 14.25/8.15 => ( ? [X_12: int] : ( P7 @ X_12 )
% 14.25/8.15 => ? [X_1: int] : ( P @ X_1 ) ) ) ) ) ).
% 14.25/8.15
% 14.25/8.15 thf(fact_7435_decr__mult__lemma,axiom,
% 14.25/8.15 ! [D2: int,P: int > $o,K3: int] :
% 14.25/8.15 ( ( ord_less_int @ zero_zero_int @ D2 )
% 14.25/8.15 => ( ! [X3: int] :
% 14.25/8.15 ( ( P @ X3 )
% 14.25/8.15 => ( P @ ( minus_minus_int @ X3 @ D2 ) ) )
% 14.25/8.15 => ( ( ord_less_eq_int @ zero_zero_int @ K3 )
% 14.25/8.15 => ! [X6: int] :
% 14.25/8.15 ( ( P @ X6 )
% 14.25/8.15 => ( P @ ( minus_minus_int @ X6 @ ( times_times_int @ K3 @ D2 ) ) ) ) ) ) ) ).
% 14.25/8.15
% 14.25/8.15 thf(fact_7436_times__int__code_I2_J,axiom,
% 14.25/8.15 ! [L: int] :
% 14.25/8.15 ( ( times_times_int @ zero_zero_int @ L )
% 14.25/8.15 = zero_zero_int ) ).
% 14.25/8.15
% 14.25/8.15 thf(fact_7437_times__int__code_I1_J,axiom,
% 14.25/8.15 ! [K3: int] :
% 14.25/8.15 ( ( times_times_int @ K3 @ zero_zero_int )
% 14.25/8.15 = zero_zero_int ) ).
% 14.25/8.15
% 14.25/8.15 thf(fact_7438_int__distrib_I1_J,axiom,
% 14.25/8.15 ! [Z1: int,Z22: int,W: int] :
% 14.25/8.15 ( ( times_times_int @ ( plus_plus_int @ Z1 @ Z22 ) @ W )
% 14.25/8.15 = ( plus_plus_int @ ( times_times_int @ Z1 @ W ) @ ( times_times_int @ Z22 @ W ) ) ) ).
% 14.25/8.15
% 14.25/8.15 thf(fact_7439_int__distrib_I2_J,axiom,
% 14.25/8.15 ! [W: int,Z1: int,Z22: int] :
% 14.25/8.15 ( ( times_times_int @ W @ ( plus_plus_int @ Z1 @ Z22 ) )
% 14.25/8.15 = ( plus_plus_int @ ( times_times_int @ W @ Z1 ) @ ( times_times_int @ W @ Z22 ) ) ) ).
% 14.25/8.15
% 14.25/8.15 thf(fact_7440_int__distrib_I4_J,axiom,
% 14.25/8.15 ! [W: int,Z1: int,Z22: int] :
% 14.25/8.15 ( ( times_times_int @ W @ ( minus_minus_int @ Z1 @ Z22 ) )
% 14.25/8.15 = ( minus_minus_int @ ( times_times_int @ W @ Z1 ) @ ( times_times_int @ W @ Z22 ) ) ) ).
% 14.25/8.15
% 14.25/8.15 thf(fact_7441_int__distrib_I3_J,axiom,
% 14.25/8.15 ! [Z1: int,Z22: int,W: int] :
% 14.25/8.15 ( ( times_times_int @ ( minus_minus_int @ Z1 @ Z22 ) @ W )
% 14.25/8.15 = ( minus_minus_int @ ( times_times_int @ Z1 @ W ) @ ( times_times_int @ Z22 @ W ) ) ) ).
% 14.25/8.15
% 14.25/8.15 thf(fact_7442_nat__mult__div__cancel1,axiom,
% 14.25/8.15 ! [K3: nat,M4: nat,N2: nat] :
% 14.25/8.15 ( ( ord_less_nat @ zero_zero_nat @ K3 )
% 14.25/8.15 => ( ( divide_divide_nat @ ( times_times_nat @ K3 @ M4 ) @ ( times_times_nat @ K3 @ N2 ) )
% 14.25/8.15 = ( divide_divide_nat @ M4 @ N2 ) ) ) ).
% 14.25/8.15
% 14.25/8.15 thf(fact_7443_div__mult__self__is__m,axiom,
% 14.25/8.15 ! [N2: nat,M4: nat] :
% 14.25/8.15 ( ( ord_less_nat @ zero_zero_nat @ N2 )
% 14.25/8.15 => ( ( divide_divide_nat @ ( times_times_nat @ M4 @ N2 ) @ N2 )
% 14.25/8.15 = M4 ) ) ).
% 14.25/8.15
% 14.25/8.15 thf(fact_7444_div__mult__self1__is__m,axiom,
% 14.25/8.15 ! [N2: nat,M4: nat] :
% 14.25/8.15 ( ( ord_less_nat @ zero_zero_nat @ N2 )
% 14.25/8.15 => ( ( divide_divide_nat @ ( times_times_nat @ N2 @ M4 ) @ N2 )
% 14.25/8.15 = M4 ) ) ).
% 14.25/8.15
% 14.25/8.15 thf(fact_7445_div__mult2__eq,axiom,
% 14.25/8.15 ! [M4: nat,N2: nat,Q4: nat] :
% 14.25/8.15 ( ( divide_divide_nat @ M4 @ ( times_times_nat @ N2 @ Q4 ) )
% 14.25/8.15 = ( divide_divide_nat @ ( divide_divide_nat @ M4 @ N2 ) @ Q4 ) ) ).
% 14.25/8.15
% 14.25/8.15 thf(fact_7446_zdiv__zmult2__eq,axiom,
% 14.25/8.15 ! [C: int,A2: int,B: int] :
% 14.25/8.15 ( ( ord_less_eq_int @ zero_zero_int @ C )
% 14.25/8.15 => ( ( divide_divide_int @ A2 @ ( times_times_int @ B @ C ) )
% 14.25/8.15 = ( divide_divide_int @ ( divide_divide_int @ A2 @ B ) @ C ) ) ) ).
% 14.25/8.15
% 14.25/8.15 thf(fact_7447_less__mult__imp__div__less,axiom,
% 14.25/8.15 ! [M4: nat,I2: nat,N2: nat] :
% 14.25/8.15 ( ( ord_less_nat @ M4 @ ( times_times_nat @ I2 @ N2 ) )
% 14.25/8.15 => ( ord_less_nat @ ( divide_divide_nat @ M4 @ N2 ) @ I2 ) ) ).
% 14.25/8.15
% 14.25/8.15 thf(fact_7448_div__times__less__eq__dividend,axiom,
% 14.25/8.15 ! [M4: nat,N2: nat] : ( ord_less_eq_nat @ ( times_times_nat @ ( divide_divide_nat @ M4 @ N2 ) @ N2 ) @ M4 ) ).
% 14.25/8.15
% 14.25/8.15 thf(fact_7449_times__div__less__eq__dividend,axiom,
% 14.25/8.15 ! [N2: nat,M4: nat] : ( ord_less_eq_nat @ ( times_times_nat @ N2 @ ( divide_divide_nat @ M4 @ N2 ) ) @ M4 ) ).
% 14.25/8.15
% 14.25/8.15 thf(fact_7450_div__less__iff__less__mult,axiom,
% 14.25/8.15 ! [Q4: nat,M4: nat,N2: nat] :
% 14.25/8.15 ( ( ord_less_nat @ zero_zero_nat @ Q4 )
% 14.25/8.15 => ( ( ord_less_nat @ ( divide_divide_nat @ M4 @ Q4 ) @ N2 )
% 14.25/8.15 = ( ord_less_nat @ M4 @ ( times_times_nat @ N2 @ Q4 ) ) ) ) ).
% 14.25/8.15
% 14.25/8.15 thf(fact_7451_less__eq__div__iff__mult__less__eq,axiom,
% 14.25/8.15 ! [Q4: nat,M4: nat,N2: nat] :
% 14.25/8.15 ( ( ord_less_nat @ zero_zero_nat @ Q4 )
% 14.25/8.15 => ( ( ord_less_eq_nat @ M4 @ ( divide_divide_nat @ N2 @ Q4 ) )
% 14.25/8.15 = ( ord_less_eq_nat @ ( times_times_nat @ M4 @ Q4 ) @ N2 ) ) ) ).
% 14.25/8.15
% 14.25/8.15 thf(fact_7452_div__nat__eqI,axiom,
% 14.25/8.15 ! [N2: nat,Q4: nat,M4: nat] :
% 14.25/8.15 ( ( ord_less_eq_nat @ ( times_times_nat @ N2 @ Q4 ) @ M4 )
% 14.25/8.15 => ( ( ord_less_nat @ M4 @ ( times_times_nat @ N2 @ ( suc @ Q4 ) ) )
% 14.25/8.15 => ( ( divide_divide_nat @ M4 @ N2 )
% 14.25/8.15 = Q4 ) ) ) ).
% 14.25/8.15
% 14.25/8.15 thf(fact_7453_dividend__less__times__div,axiom,
% 14.25/8.15 ! [N2: nat,M4: nat] :
% 14.25/8.15 ( ( ord_less_nat @ zero_zero_nat @ N2 )
% 14.25/8.15 => ( ord_less_nat @ M4 @ ( plus_plus_nat @ N2 @ ( times_times_nat @ N2 @ ( divide_divide_nat @ M4 @ N2 ) ) ) ) ) ).
% 14.25/8.15
% 14.25/8.15 thf(fact_7454_dividend__less__div__times,axiom,
% 14.25/8.15 ! [N2: nat,M4: nat] :
% 14.25/8.15 ( ( ord_less_nat @ zero_zero_nat @ N2 )
% 14.25/8.15 => ( ord_less_nat @ M4 @ ( plus_plus_nat @ N2 @ ( times_times_nat @ ( divide_divide_nat @ M4 @ N2 ) @ N2 ) ) ) ) ).
% 14.25/8.15
% 14.25/8.15 thf(fact_7455_split__div,axiom,
% 14.25/8.15 ! [P: nat > $o,M4: nat,N2: nat] :
% 14.25/8.15 ( ( P @ ( divide_divide_nat @ M4 @ N2 ) )
% 14.25/8.15 = ( ( ( N2 = zero_zero_nat )
% 14.25/8.15 => ( P @ zero_zero_nat ) )
% 14.25/8.15 & ( ( N2 != zero_zero_nat )
% 14.25/8.15 => ! [I4: nat,J3: nat] :
% 14.25/8.15 ( ( ord_less_nat @ J3 @ N2 )
% 14.25/8.15 => ( ( M4
% 14.25/8.15 = ( plus_plus_nat @ ( times_times_nat @ N2 @ I4 ) @ J3 ) )
% 14.25/8.15 => ( P @ I4 ) ) ) ) ) ) ).
% 14.25/8.15
% 14.25/8.15 thf(fact_7456_split__div_H,axiom,
% 14.25/8.15 ! [P: nat > $o,M4: nat,N2: nat] :
% 14.25/8.15 ( ( P @ ( divide_divide_nat @ M4 @ N2 ) )
% 14.25/8.15 = ( ( ( N2 = zero_zero_nat )
% 14.25/8.15 & ( P @ zero_zero_nat ) )
% 14.25/8.15 | ? [Q6: nat] :
% 14.25/8.15 ( ( ord_less_eq_nat @ ( times_times_nat @ N2 @ Q6 ) @ M4 )
% 14.25/8.15 & ( ord_less_nat @ M4 @ ( times_times_nat @ N2 @ ( suc @ Q6 ) ) )
% 14.25/8.15 & ( P @ Q6 ) ) ) ) ).
% 14.25/8.15
% 14.25/8.15 thf(fact_7457_int__ops_I7_J,axiom,
% 14.25/8.15 ! [A2: nat,B: nat] :
% 14.25/8.15 ( ( semiri1314217659103216013at_int @ ( times_times_nat @ A2 @ B ) )
% 14.25/8.15 = ( times_times_int @ ( semiri1314217659103216013at_int @ A2 ) @ ( semiri1314217659103216013at_int @ B ) ) ) ).
% 14.25/8.15
% 14.25/8.15 thf(fact_7458_eucl__rel__int__iff,axiom,
% 14.25/8.15 ! [K3: int,L: int,Q4: int,R2: int] :
% 14.25/8.15 ( ( eucl_rel_int @ K3 @ L @ ( product_Pair_int_int @ Q4 @ R2 ) )
% 14.25/8.15 = ( ( K3
% 14.25/8.15 = ( plus_plus_int @ ( times_times_int @ L @ Q4 ) @ R2 ) )
% 14.25/8.15 & ( ( ord_less_int @ zero_zero_int @ L )
% 14.25/8.15 => ( ( ord_less_eq_int @ zero_zero_int @ R2 )
% 14.25/8.15 & ( ord_less_int @ R2 @ L ) ) )
% 14.25/8.15 & ( ~ ( ord_less_int @ zero_zero_int @ L )
% 14.25/8.15 => ( ( ( ord_less_int @ L @ zero_zero_int )
% 14.25/8.15 => ( ( ord_less_int @ L @ R2 )
% 14.25/8.15 & ( ord_less_eq_int @ R2 @ zero_zero_int ) ) )
% 14.25/8.15 & ( ~ ( ord_less_int @ L @ zero_zero_int )
% 14.25/8.15 => ( Q4 = zero_zero_int ) ) ) ) ) ) ).
% 14.25/8.15
% 14.25/8.15 thf(fact_7459_eucl__rel__int__dividesI,axiom,
% 14.25/8.15 ! [L: int,K3: int,Q4: int] :
% 14.25/8.15 ( ( L != zero_zero_int )
% 14.25/8.15 => ( ( K3
% 14.25/8.15 = ( times_times_int @ Q4 @ L ) )
% 14.25/8.15 => ( eucl_rel_int @ K3 @ L @ ( product_Pair_int_int @ Q4 @ zero_zero_int ) ) ) ) ).
% 14.25/8.15
% 14.25/8.15 thf(fact_7460_split__pos__lemma,axiom,
% 14.25/8.15 ! [K3: int,P: int > int > $o,N2: int] :
% 14.25/8.15 ( ( ord_less_int @ zero_zero_int @ K3 )
% 14.25/8.15 => ( ( P @ ( divide_divide_int @ N2 @ K3 ) @ ( modulo_modulo_int @ N2 @ K3 ) )
% 14.25/8.15 = ( ! [I4: int,J3: int] :
% 14.25/8.15 ( ( ( ord_less_eq_int @ zero_zero_int @ J3 )
% 14.25/8.15 & ( ord_less_int @ J3 @ K3 )
% 14.25/8.15 & ( N2
% 14.25/8.15 = ( plus_plus_int @ ( times_times_int @ K3 @ I4 ) @ J3 ) ) )
% 14.25/8.15 => ( P @ I4 @ J3 ) ) ) ) ) ).
% 14.25/8.15
% 14.25/8.15 thf(fact_7461_split__neg__lemma,axiom,
% 14.25/8.15 ! [K3: int,P: int > int > $o,N2: int] :
% 14.25/8.15 ( ( ord_less_int @ K3 @ zero_zero_int )
% 14.25/8.15 => ( ( P @ ( divide_divide_int @ N2 @ K3 ) @ ( modulo_modulo_int @ N2 @ K3 ) )
% 14.25/8.15 = ( ! [I4: int,J3: int] :
% 14.25/8.15 ( ( ( ord_less_int @ K3 @ J3 )
% 14.25/8.15 & ( ord_less_eq_int @ J3 @ zero_zero_int )
% 14.25/8.15 & ( N2
% 14.25/8.15 = ( plus_plus_int @ ( times_times_int @ K3 @ I4 ) @ J3 ) ) )
% 14.25/8.15 => ( P @ I4 @ J3 ) ) ) ) ) ).
% 14.25/8.15
% 14.25/8.15 thf(fact_7462_decr__lemma,axiom,
% 14.25/8.15 ! [D2: int,X: int,Z: int] :
% 14.25/8.15 ( ( ord_less_int @ zero_zero_int @ D2 )
% 14.25/8.15 => ( ord_less_int @ ( minus_minus_int @ X @ ( times_times_int @ ( plus_plus_int @ ( abs_abs_int @ ( minus_minus_int @ X @ Z ) ) @ one_one_int ) @ D2 ) ) @ Z ) ) ).
% 14.25/8.15
% 14.25/8.15 thf(fact_7463_incr__lemma,axiom,
% 14.25/8.15 ! [D2: int,Z: int,X: int] :
% 14.25/8.15 ( ( ord_less_int @ zero_zero_int @ D2 )
% 14.25/8.15 => ( ord_less_int @ Z @ ( plus_plus_int @ X @ ( times_times_int @ ( plus_plus_int @ ( abs_abs_int @ ( minus_minus_int @ X @ Z ) ) @ one_one_int ) @ D2 ) ) ) ) ).
% 14.25/8.15
% 14.25/8.15 thf(fact_7464_zmod__eq__0__iff,axiom,
% 14.25/8.15 ! [M4: int,D2: int] :
% 14.25/8.15 ( ( ( modulo_modulo_int @ M4 @ D2 )
% 14.25/8.15 = zero_zero_int )
% 14.25/8.15 = ( ? [Q6: int] :
% 14.25/8.15 ( M4
% 14.25/8.15 = ( times_times_int @ D2 @ Q6 ) ) ) ) ).
% 14.25/8.15
% 14.25/8.15 thf(fact_7465_zmod__eq__0D,axiom,
% 14.25/8.15 ! [M4: int,D2: int] :
% 14.25/8.15 ( ( ( modulo_modulo_int @ M4 @ D2 )
% 14.25/8.15 = zero_zero_int )
% 14.25/8.15 => ? [Q7: int] :
% 14.25/8.15 ( M4
% 14.25/8.15 = ( times_times_int @ D2 @ Q7 ) ) ) ).
% 14.25/8.15
% 14.25/8.15 thf(fact_7466_abs__zmult__eq__1,axiom,
% 14.25/8.15 ! [M4: int,N2: int] :
% 14.25/8.15 ( ( ( abs_abs_int @ ( times_times_int @ M4 @ N2 ) )
% 14.25/8.15 = one_one_int )
% 14.25/8.15 => ( ( abs_abs_int @ M4 )
% 14.25/8.15 = one_one_int ) ) ).
% 14.25/8.15
% 14.25/8.15 thf(fact_7467_div__mod__decomp__int,axiom,
% 14.25/8.15 ! [A: int,N2: int] :
% 14.25/8.15 ( A
% 14.25/8.15 = ( plus_plus_int @ ( times_times_int @ ( divide_divide_int @ A @ N2 ) @ N2 ) @ ( modulo_modulo_int @ A @ N2 ) ) ) ).
% 14.25/8.15
% 14.25/8.15 thf(fact_7468_zdvd__mult__cancel1,axiom,
% 14.25/8.15 ! [M4: int,N2: int] :
% 14.25/8.15 ( ( M4 != zero_zero_int )
% 14.25/8.15 => ( ( dvd_dvd_int @ ( times_times_int @ M4 @ N2 ) @ M4 )
% 14.25/8.15 = ( ( abs_abs_int @ N2 )
% 14.25/8.15 = one_one_int ) ) ) ).
% 14.25/8.15
% 14.25/8.15 thf(fact_7469_split__zmod,axiom,
% 14.25/8.15 ! [P: int > $o,N2: int,K3: int] :
% 14.25/8.15 ( ( P @ ( modulo_modulo_int @ N2 @ K3 ) )
% 14.25/8.15 = ( ( ( K3 = zero_zero_int )
% 14.25/8.15 => ( P @ N2 ) )
% 14.25/8.15 & ( ( ord_less_int @ zero_zero_int @ K3 )
% 14.25/8.15 => ! [I4: int,J3: int] :
% 14.25/8.15 ( ( ( ord_less_eq_int @ zero_zero_int @ J3 )
% 14.25/8.15 & ( ord_less_int @ J3 @ K3 )
% 14.25/8.15 & ( N2
% 14.25/8.15 = ( plus_plus_int @ ( times_times_int @ K3 @ I4 ) @ J3 ) ) )
% 14.25/8.15 => ( P @ J3 ) ) )
% 14.25/8.15 & ( ( ord_less_int @ K3 @ zero_zero_int )
% 14.25/8.15 => ! [I4: int,J3: int] :
% 14.25/8.15 ( ( ( ord_less_int @ K3 @ J3 )
% 14.25/8.15 & ( ord_less_eq_int @ J3 @ zero_zero_int )
% 14.25/8.15 & ( N2
% 14.25/8.15 = ( plus_plus_int @ ( times_times_int @ K3 @ I4 ) @ J3 ) ) )
% 14.25/8.15 => ( P @ J3 ) ) ) ) ) ).
% 14.25/8.15
% 14.25/8.15 thf(fact_7470_int__mod__neg__eq,axiom,
% 14.25/8.15 ! [A2: int,B: int,Q4: int,R2: int] :
% 14.25/8.15 ( ( A2
% 14.25/8.15 = ( plus_plus_int @ ( times_times_int @ B @ Q4 ) @ R2 ) )
% 14.25/8.15 => ( ( ord_less_eq_int @ R2 @ zero_zero_int )
% 14.25/8.15 => ( ( ord_less_int @ B @ R2 )
% 14.25/8.15 => ( ( modulo_modulo_int @ A2 @ B )
% 14.25/8.15 = R2 ) ) ) ) ).
% 14.25/8.15
% 14.25/8.15 thf(fact_7471_int__mod__pos__eq,axiom,
% 14.25/8.15 ! [A2: int,B: int,Q4: int,R2: int] :
% 14.25/8.15 ( ( A2
% 14.25/8.15 = ( plus_plus_int @ ( times_times_int @ B @ Q4 ) @ R2 ) )
% 14.25/8.15 => ( ( ord_less_eq_int @ zero_zero_int @ R2 )
% 14.25/8.15 => ( ( ord_less_int @ R2 @ B )
% 14.25/8.15 => ( ( modulo_modulo_int @ A2 @ B )
% 14.25/8.15 = R2 ) ) ) ) ).
% 14.25/8.15
% 14.25/8.15 thf(fact_7472_zmod__zmult2__eq,axiom,
% 14.25/8.15 ! [C: int,A2: int,B: int] :
% 14.25/8.15 ( ( ord_less_eq_int @ zero_zero_int @ C )
% 14.25/8.15 => ( ( modulo_modulo_int @ A2 @ ( times_times_int @ B @ C ) )
% 14.25/8.15 = ( plus_plus_int @ ( times_times_int @ B @ ( modulo_modulo_int @ ( divide_divide_int @ A2 @ B ) @ C ) ) @ ( modulo_modulo_int @ A2 @ B ) ) ) ) ).
% 14.25/8.15
% 14.25/8.15 thf(fact_7473_eucl__rel__int_Osimps,axiom,
% 14.25/8.15 ( eucl_rel_int
% 14.25/8.15 = ( ^ [A13: int,A24: int,A32: product_prod_int_int] :
% 14.25/8.15 ( ? [K5: int] :
% 14.25/8.15 ( ( A13 = K5 )
% 14.25/8.15 & ( A24 = zero_zero_int )
% 14.25/8.15 & ( A32
% 14.25/8.15 = ( product_Pair_int_int @ zero_zero_int @ K5 ) ) )
% 14.25/8.15 | ? [L2: int,K5: int,Q6: int] :
% 14.25/8.15 ( ( A13 = K5 )
% 14.25/8.15 & ( A24 = L2 )
% 14.25/8.15 & ( A32
% 14.25/8.15 = ( product_Pair_int_int @ Q6 @ zero_zero_int ) )
% 14.25/8.15 & ( L2 != zero_zero_int )
% 14.25/8.15 & ( K5
% 14.25/8.15 = ( times_times_int @ Q6 @ L2 ) ) )
% 14.25/8.15 | ? [R4: int,L2: int,K5: int,Q6: int] :
% 14.25/8.15 ( ( A13 = K5 )
% 14.25/8.15 & ( A24 = L2 )
% 14.25/8.15 & ( A32
% 14.25/8.15 = ( product_Pair_int_int @ Q6 @ R4 ) )
% 14.25/8.15 & ( ( sgn_sgn_int @ R4 )
% 14.25/8.15 = ( sgn_sgn_int @ L2 ) )
% 14.25/8.15 & ( ord_less_int @ ( abs_abs_int @ R4 ) @ ( abs_abs_int @ L2 ) )
% 14.25/8.15 & ( K5
% 14.25/8.15 = ( plus_plus_int @ ( times_times_int @ Q6 @ L2 ) @ R4 ) ) ) ) ) ) ).
% 14.25/8.15
% 14.25/8.15 thf(fact_7474_Suc__mod__mult__self4,axiom,
% 14.25/8.15 ! [N2: nat,K3: nat,M4: nat] :
% 14.25/8.15 ( ( modulo_modulo_nat @ ( suc @ ( plus_plus_nat @ ( times_times_nat @ N2 @ K3 ) @ M4 ) ) @ N2 )
% 14.25/8.15 = ( modulo_modulo_nat @ ( suc @ M4 ) @ N2 ) ) ).
% 14.25/8.15
% 14.25/8.15 thf(fact_7475_Suc__mod__mult__self3,axiom,
% 14.25/8.15 ! [K3: nat,N2: nat,M4: nat] :
% 14.25/8.15 ( ( modulo_modulo_nat @ ( suc @ ( plus_plus_nat @ ( times_times_nat @ K3 @ N2 ) @ M4 ) ) @ N2 )
% 14.25/8.15 = ( modulo_modulo_nat @ ( suc @ M4 ) @ N2 ) ) ).
% 14.25/8.15
% 14.25/8.15 thf(fact_7476_Suc__mod__mult__self2,axiom,
% 14.25/8.15 ! [M4: nat,N2: nat,K3: nat] :
% 14.25/8.15 ( ( modulo_modulo_nat @ ( suc @ ( plus_plus_nat @ M4 @ ( times_times_nat @ N2 @ K3 ) ) ) @ N2 )
% 14.25/8.15 = ( modulo_modulo_nat @ ( suc @ M4 ) @ N2 ) ) ).
% 14.25/8.15
% 14.25/8.15 thf(fact_7477_Suc__mod__mult__self1,axiom,
% 14.25/8.15 ! [M4: nat,K3: nat,N2: nat] :
% 14.25/8.15 ( ( modulo_modulo_nat @ ( suc @ ( plus_plus_nat @ M4 @ ( times_times_nat @ K3 @ N2 ) ) ) @ N2 )
% 14.25/8.15 = ( modulo_modulo_nat @ ( suc @ M4 ) @ N2 ) ) ).
% 14.25/8.15
% 14.25/8.15 thf(fact_7478_Suc__times__numeral__mod__eq,axiom,
% 14.25/8.15 ! [K3: num,N2: nat] :
% 14.25/8.15 ( ( ( numeral_numeral_nat @ K3 )
% 14.25/8.15 != one_one_nat )
% 14.25/8.15 => ( ( modulo_modulo_nat @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ K3 ) @ N2 ) ) @ ( numeral_numeral_nat @ K3 ) )
% 14.25/8.15 = one_one_nat ) ) ).
% 14.25/8.15
% 14.25/8.15 thf(fact_7479_dvd__mult__sgn__iff,axiom,
% 14.25/8.15 ! [L: int,K3: int,R2: int] :
% 14.25/8.15 ( ( dvd_dvd_int @ L @ ( times_times_int @ K3 @ ( sgn_sgn_int @ R2 ) ) )
% 14.25/8.15 = ( ( dvd_dvd_int @ L @ K3 )
% 14.25/8.15 | ( R2 = zero_zero_int ) ) ) ).
% 14.25/8.15
% 14.25/8.15 thf(fact_7480_dvd__sgn__mult__iff,axiom,
% 14.25/8.15 ! [L: int,R2: int,K3: int] :
% 14.25/8.15 ( ( dvd_dvd_int @ L @ ( times_times_int @ ( sgn_sgn_int @ R2 ) @ K3 ) )
% 14.25/8.15 = ( ( dvd_dvd_int @ L @ K3 )
% 14.25/8.15 | ( R2 = zero_zero_int ) ) ) ).
% 14.25/8.15
% 14.25/8.15 thf(fact_7481_mult__sgn__dvd__iff,axiom,
% 14.25/8.15 ! [L: int,R2: int,K3: int] :
% 14.25/8.15 ( ( dvd_dvd_int @ ( times_times_int @ L @ ( sgn_sgn_int @ R2 ) ) @ K3 )
% 14.25/8.15 = ( ( dvd_dvd_int @ L @ K3 )
% 14.25/8.15 & ( ( R2 = zero_zero_int )
% 14.25/8.15 => ( K3 = zero_zero_int ) ) ) ) ).
% 14.25/8.15
% 14.25/8.15 thf(fact_7482_sgn__mult__dvd__iff,axiom,
% 14.25/8.15 ! [R2: int,L: int,K3: int] :
% 14.25/8.15 ( ( dvd_dvd_int @ ( times_times_int @ ( sgn_sgn_int @ R2 ) @ L ) @ K3 )
% 14.25/8.15 = ( ( dvd_dvd_int @ L @ K3 )
% 14.25/8.15 & ( ( R2 = zero_zero_int )
% 14.25/8.15 => ( K3 = zero_zero_int ) ) ) ) ).
% 14.25/8.15
% 14.25/8.15 thf(fact_7483_int__sgnE,axiom,
% 14.25/8.15 ! [K3: int] :
% 14.25/8.15 ~ ! [N3: nat,L3: int] :
% 14.25/8.15 ( K3
% 14.25/8.15 != ( times_times_int @ ( sgn_sgn_int @ L3 ) @ ( semiri1314217659103216013at_int @ N3 ) ) ) ).
% 14.25/8.15
% 14.25/8.15 thf(fact_7484_mod__eq__0D,axiom,
% 14.25/8.15 ! [M4: nat,D2: nat] :
% 14.25/8.15 ( ( ( modulo_modulo_nat @ M4 @ D2 )
% 14.25/8.15 = zero_zero_nat )
% 14.25/8.15 => ? [Q7: nat] :
% 14.25/8.15 ( M4
% 14.25/8.15 = ( times_times_nat @ D2 @ Q7 ) ) ) ).
% 14.25/8.15
% 14.25/8.15 thf(fact_7485_nat__mod__eq__iff,axiom,
% 14.25/8.15 ! [X: nat,N2: nat,Y: nat] :
% 14.25/8.15 ( ( ( modulo_modulo_nat @ X @ N2 )
% 14.25/8.15 = ( modulo_modulo_nat @ Y @ N2 ) )
% 14.25/8.15 = ( ? [Q1: nat,Q22: nat] :
% 14.25/8.15 ( ( plus_plus_nat @ X @ ( times_times_nat @ N2 @ Q1 ) )
% 14.25/8.15 = ( plus_plus_nat @ Y @ ( times_times_nat @ N2 @ Q22 ) ) ) ) ) ).
% 14.25/8.15
% 14.25/8.15 thf(fact_7486_nat__mod__eq__lemma,axiom,
% 14.25/8.15 ! [X: nat,N2: nat,Y: nat] :
% 14.25/8.15 ( ( ( modulo_modulo_nat @ X @ N2 )
% 14.25/8.15 = ( modulo_modulo_nat @ Y @ N2 ) )
% 14.25/8.15 => ( ( ord_less_eq_nat @ Y @ X )
% 14.25/8.15 => ? [Q7: nat] :
% 14.25/8.15 ( X
% 14.25/8.15 = ( plus_plus_nat @ Y @ ( times_times_nat @ N2 @ Q7 ) ) ) ) ) ).
% 14.25/8.15
% 14.25/8.15 thf(fact_7487_mod__eq__nat2E,axiom,
% 14.25/8.15 ! [M4: nat,Q4: nat,N2: nat] :
% 14.25/8.15 ( ( ( modulo_modulo_nat @ M4 @ Q4 )
% 14.25/8.15 = ( modulo_modulo_nat @ N2 @ Q4 ) )
% 14.25/8.15 => ( ( ord_less_eq_nat @ M4 @ N2 )
% 14.25/8.15 => ~ ! [S8: nat] :
% 14.25/8.15 ( N2
% 14.25/8.15 != ( plus_plus_nat @ M4 @ ( times_times_nat @ Q4 @ S8 ) ) ) ) ) ).
% 14.25/8.15
% 14.25/8.15 thf(fact_7488_mod__eq__nat1E,axiom,
% 14.25/8.15 ! [M4: nat,Q4: nat,N2: nat] :
% 14.25/8.15 ( ( ( modulo_modulo_nat @ M4 @ Q4 )
% 14.25/8.15 = ( modulo_modulo_nat @ N2 @ Q4 ) )
% 14.25/8.15 => ( ( ord_less_eq_nat @ N2 @ M4 )
% 14.25/8.15 => ~ ! [S8: nat] :
% 14.25/8.15 ( M4
% 14.25/8.15 != ( plus_plus_nat @ N2 @ ( times_times_nat @ Q4 @ S8 ) ) ) ) ) ).
% 14.25/8.15
% 14.25/8.15 thf(fact_7489_div__mod__decomp,axiom,
% 14.25/8.15 ! [A: nat,N2: nat] :
% 14.25/8.15 ( A
% 14.25/8.15 = ( plus_plus_nat @ ( times_times_nat @ ( divide_divide_nat @ A @ N2 ) @ N2 ) @ ( modulo_modulo_nat @ A @ N2 ) ) ) ).
% 14.25/8.15
% 14.25/8.15 thf(fact_7490_mod__mult2__eq,axiom,
% 14.25/8.15 ! [M4: nat,N2: nat,Q4: nat] :
% 14.25/8.15 ( ( modulo_modulo_nat @ M4 @ ( times_times_nat @ N2 @ Q4 ) )
% 14.25/8.15 = ( plus_plus_nat @ ( times_times_nat @ N2 @ ( modulo_modulo_nat @ ( divide_divide_nat @ M4 @ N2 ) @ Q4 ) ) @ ( modulo_modulo_nat @ M4 @ N2 ) ) ) ).
% 14.25/8.15
% 14.25/8.15 thf(fact_7491_modulo__nat__def,axiom,
% 14.25/8.15 ( modulo_modulo_nat
% 14.25/8.15 = ( ^ [M3: nat,N4: nat] : ( minus_minus_nat @ M3 @ ( times_times_nat @ ( divide_divide_nat @ M3 @ N4 ) @ N4 ) ) ) ) ).
% 14.25/8.15
% 14.25/8.15 thf(fact_7492_div__sgn__abs__cancel,axiom,
% 14.25/8.15 ! [V: int,K3: int,L: int] :
% 14.25/8.15 ( ( V != zero_zero_int )
% 14.25/8.15 => ( ( divide_divide_int @ ( times_times_int @ ( sgn_sgn_int @ V ) @ ( abs_abs_int @ K3 ) ) @ ( times_times_int @ ( sgn_sgn_int @ V ) @ ( abs_abs_int @ L ) ) )
% 14.25/8.15 = ( divide_divide_int @ ( abs_abs_int @ K3 ) @ ( abs_abs_int @ L ) ) ) ) ).
% 14.25/8.15
% 14.25/8.15 thf(fact_7493_div__dvd__sgn__abs,axiom,
% 14.25/8.15 ! [L: int,K3: int] :
% 14.25/8.15 ( ( dvd_dvd_int @ L @ K3 )
% 14.25/8.15 => ( ( divide_divide_int @ K3 @ L )
% 14.25/8.15 = ( times_times_int @ ( times_times_int @ ( sgn_sgn_int @ K3 ) @ ( sgn_sgn_int @ L ) ) @ ( divide_divide_int @ ( abs_abs_int @ K3 ) @ ( abs_abs_int @ L ) ) ) ) ) ).
% 14.25/8.15
% 14.25/8.15 thf(fact_7494_split__mod,axiom,
% 14.25/8.15 ! [P: nat > $o,M4: nat,N2: nat] :
% 14.25/8.15 ( ( P @ ( modulo_modulo_nat @ M4 @ N2 ) )
% 14.25/8.15 = ( ( ( N2 = zero_zero_nat )
% 14.25/8.15 => ( P @ M4 ) )
% 14.25/8.15 & ( ( N2 != zero_zero_nat )
% 14.25/8.15 => ! [I4: nat,J3: nat] :
% 14.25/8.15 ( ( ord_less_nat @ J3 @ N2 )
% 14.25/8.15 => ( ( M4
% 14.25/8.15 = ( plus_plus_nat @ ( times_times_nat @ N2 @ I4 ) @ J3 ) )
% 14.25/8.15 => ( P @ J3 ) ) ) ) ) ) ).
% 14.25/8.15
% 14.25/8.15 thf(fact_7495_Suc__times__mod__eq,axiom,
% 14.25/8.15 ! [M4: nat,N2: nat] :
% 14.25/8.15 ( ( ord_less_nat @ ( suc @ zero_zero_nat ) @ M4 )
% 14.25/8.15 => ( ( modulo_modulo_nat @ ( suc @ ( times_times_nat @ M4 @ N2 ) ) @ M4 )
% 14.25/8.15 = one_one_nat ) ) ).
% 14.25/8.15
% 14.25/8.15 thf(fact_7496_eucl__rel__int__remainderI,axiom,
% 14.25/8.15 ! [R2: int,L: int,K3: int,Q4: int] :
% 14.25/8.15 ( ( ( sgn_sgn_int @ R2 )
% 14.25/8.15 = ( sgn_sgn_int @ L ) )
% 14.25/8.15 => ( ( ord_less_int @ ( abs_abs_int @ R2 ) @ ( abs_abs_int @ L ) )
% 14.25/8.15 => ( ( K3
% 14.25/8.15 = ( plus_plus_int @ ( times_times_int @ Q4 @ L ) @ R2 ) )
% 14.25/8.15 => ( eucl_rel_int @ K3 @ L @ ( product_Pair_int_int @ Q4 @ R2 ) ) ) ) ) ).
% 14.25/8.15
% 14.25/8.15 thf(fact_7497_eucl__rel__int_Ocases,axiom,
% 14.25/8.15 ! [A1: int,A22: int,A33: product_prod_int_int] :
% 14.25/8.15 ( ( eucl_rel_int @ A1 @ A22 @ A33 )
% 14.25/8.15 => ( ( ( A22 = zero_zero_int )
% 14.25/8.15 => ( A33
% 14.25/8.15 != ( product_Pair_int_int @ zero_zero_int @ A1 ) ) )
% 14.25/8.15 => ( ! [Q7: int] :
% 14.25/8.15 ( ( A33
% 14.25/8.15 = ( product_Pair_int_int @ Q7 @ zero_zero_int ) )
% 14.25/8.15 => ( ( A22 != zero_zero_int )
% 14.25/8.15 => ( A1
% 14.25/8.15 != ( times_times_int @ Q7 @ A22 ) ) ) )
% 14.25/8.15 => ~ ! [R7: int,Q7: int] :
% 14.25/8.15 ( ( A33
% 14.25/8.15 = ( product_Pair_int_int @ Q7 @ R7 ) )
% 14.25/8.15 => ( ( ( sgn_sgn_int @ R7 )
% 14.25/8.15 = ( sgn_sgn_int @ A22 ) )
% 14.25/8.15 => ( ( ord_less_int @ ( abs_abs_int @ R7 ) @ ( abs_abs_int @ A22 ) )
% 14.25/8.15 => ( A1
% 14.25/8.15 != ( plus_plus_int @ ( times_times_int @ Q7 @ A22 ) @ R7 ) ) ) ) ) ) ) ) ).
% 14.25/8.15
% 14.25/8.15 thf(fact_7498_bezw_Oelims,axiom,
% 14.25/8.15 ! [X: nat,Xa: nat,Y: product_prod_int_int] :
% 14.25/8.15 ( ( ( bezw @ X @ Xa )
% 14.25/8.15 = Y )
% 14.25/8.15 => ( ( ( Xa = zero_zero_nat )
% 14.25/8.15 => ( Y
% 14.25/8.15 = ( product_Pair_int_int @ one_one_int @ zero_zero_int ) ) )
% 14.25/8.15 & ( ( Xa != zero_zero_nat )
% 14.25/8.15 => ( Y
% 14.25/8.15 = ( product_Pair_int_int @ ( product_snd_int_int @ ( bezw @ Xa @ ( modulo_modulo_nat @ X @ Xa ) ) ) @ ( minus_minus_int @ ( product_fst_int_int @ ( bezw @ Xa @ ( modulo_modulo_nat @ X @ Xa ) ) ) @ ( times_times_int @ ( product_snd_int_int @ ( bezw @ Xa @ ( modulo_modulo_nat @ X @ Xa ) ) ) @ ( semiri1314217659103216013at_int @ ( divide_divide_nat @ X @ Xa ) ) ) ) ) ) ) ) ) ).
% 14.25/8.15
% 14.25/8.15 thf(fact_7499_bezw_Osimps,axiom,
% 14.25/8.15 ( bezw
% 14.25/8.15 = ( ^ [X2: nat,Y3: nat] : ( if_Pro3027730157355071871nt_int @ ( Y3 = zero_zero_nat ) @ ( product_Pair_int_int @ one_one_int @ zero_zero_int ) @ ( product_Pair_int_int @ ( product_snd_int_int @ ( bezw @ Y3 @ ( modulo_modulo_nat @ X2 @ Y3 ) ) ) @ ( minus_minus_int @ ( product_fst_int_int @ ( bezw @ Y3 @ ( modulo_modulo_nat @ X2 @ Y3 ) ) ) @ ( times_times_int @ ( product_snd_int_int @ ( bezw @ Y3 @ ( modulo_modulo_nat @ X2 @ Y3 ) ) ) @ ( semiri1314217659103216013at_int @ ( divide_divide_nat @ X2 @ Y3 ) ) ) ) ) ) ) ) ).
% 14.25/8.15
% 14.25/8.15 thf(fact_7500_bezw__non__0,axiom,
% 14.25/8.15 ! [Y: nat,X: nat] :
% 14.25/8.15 ( ( ord_less_nat @ zero_zero_nat @ Y )
% 14.25/8.15 => ( ( bezw @ X @ Y )
% 14.25/8.15 = ( product_Pair_int_int @ ( product_snd_int_int @ ( bezw @ Y @ ( modulo_modulo_nat @ X @ Y ) ) ) @ ( minus_minus_int @ ( product_fst_int_int @ ( bezw @ Y @ ( modulo_modulo_nat @ X @ Y ) ) ) @ ( times_times_int @ ( product_snd_int_int @ ( bezw @ Y @ ( modulo_modulo_nat @ X @ Y ) ) ) @ ( semiri1314217659103216013at_int @ ( divide_divide_nat @ X @ Y ) ) ) ) ) ) ) ).
% 14.25/8.15
% 14.25/8.15 thf(fact_7501_divide__int__unfold,axiom,
% 14.25/8.15 ! [L: int,K3: int,N2: nat,M4: nat] :
% 14.25/8.15 ( ( ( ( ( sgn_sgn_int @ L )
% 14.25/8.15 = zero_zero_int )
% 14.25/8.15 | ( ( sgn_sgn_int @ K3 )
% 14.25/8.15 = zero_zero_int )
% 14.25/8.15 | ( N2 = zero_zero_nat ) )
% 14.25/8.15 => ( ( divide_divide_int @ ( times_times_int @ ( sgn_sgn_int @ K3 ) @ ( semiri1314217659103216013at_int @ M4 ) ) @ ( times_times_int @ ( sgn_sgn_int @ L ) @ ( semiri1314217659103216013at_int @ N2 ) ) )
% 14.25/8.15 = zero_zero_int ) )
% 14.25/8.15 & ( ~ ( ( ( sgn_sgn_int @ L )
% 14.25/8.15 = zero_zero_int )
% 14.25/8.15 | ( ( sgn_sgn_int @ K3 )
% 14.25/8.15 = zero_zero_int )
% 14.25/8.15 | ( N2 = zero_zero_nat ) )
% 14.25/8.15 => ( ( ( ( sgn_sgn_int @ K3 )
% 14.25/8.15 = ( sgn_sgn_int @ L ) )
% 14.25/8.15 => ( ( divide_divide_int @ ( times_times_int @ ( sgn_sgn_int @ K3 ) @ ( semiri1314217659103216013at_int @ M4 ) ) @ ( times_times_int @ ( sgn_sgn_int @ L ) @ ( semiri1314217659103216013at_int @ N2 ) ) )
% 14.25/8.15 = ( semiri1314217659103216013at_int @ ( divide_divide_nat @ M4 @ N2 ) ) ) )
% 14.25/8.15 & ( ( ( sgn_sgn_int @ K3 )
% 14.25/8.15 != ( sgn_sgn_int @ L ) )
% 14.25/8.15 => ( ( divide_divide_int @ ( times_times_int @ ( sgn_sgn_int @ K3 ) @ ( semiri1314217659103216013at_int @ M4 ) ) @ ( times_times_int @ ( sgn_sgn_int @ L ) @ ( semiri1314217659103216013at_int @ N2 ) ) )
% 14.25/8.15 = ( uminus_uminus_int
% 14.25/8.15 @ ( semiri1314217659103216013at_int
% 14.25/8.15 @ ( plus_plus_nat @ ( divide_divide_nat @ M4 @ N2 )
% 14.25/8.15 @ ( zero_n2687167440665602831ol_nat
% 14.25/8.15 @ ~ ( dvd_dvd_nat @ N2 @ M4 ) ) ) ) ) ) ) ) ) ) ).
% 14.25/8.15
% 14.25/8.15 thf(fact_7502_modulo__int__unfold,axiom,
% 14.25/8.15 ! [L: int,K3: int,N2: nat,M4: nat] :
% 14.25/8.15 ( ( ( ( ( sgn_sgn_int @ L )
% 14.25/8.15 = zero_zero_int )
% 14.25/8.15 | ( ( sgn_sgn_int @ K3 )
% 14.25/8.15 = zero_zero_int )
% 14.25/8.15 | ( N2 = zero_zero_nat ) )
% 14.25/8.15 => ( ( modulo_modulo_int @ ( times_times_int @ ( sgn_sgn_int @ K3 ) @ ( semiri1314217659103216013at_int @ M4 ) ) @ ( times_times_int @ ( sgn_sgn_int @ L ) @ ( semiri1314217659103216013at_int @ N2 ) ) )
% 14.25/8.15 = ( times_times_int @ ( sgn_sgn_int @ K3 ) @ ( semiri1314217659103216013at_int @ M4 ) ) ) )
% 14.25/8.15 & ( ~ ( ( ( sgn_sgn_int @ L )
% 14.25/8.15 = zero_zero_int )
% 14.25/8.15 | ( ( sgn_sgn_int @ K3 )
% 14.25/8.15 = zero_zero_int )
% 14.25/8.15 | ( N2 = zero_zero_nat ) )
% 14.25/8.15 => ( ( ( ( sgn_sgn_int @ K3 )
% 14.25/8.15 = ( sgn_sgn_int @ L ) )
% 14.25/8.15 => ( ( modulo_modulo_int @ ( times_times_int @ ( sgn_sgn_int @ K3 ) @ ( semiri1314217659103216013at_int @ M4 ) ) @ ( times_times_int @ ( sgn_sgn_int @ L ) @ ( semiri1314217659103216013at_int @ N2 ) ) )
% 14.25/8.15 = ( times_times_int @ ( sgn_sgn_int @ L ) @ ( semiri1314217659103216013at_int @ ( modulo_modulo_nat @ M4 @ N2 ) ) ) ) )
% 14.25/8.15 & ( ( ( sgn_sgn_int @ K3 )
% 14.25/8.15 != ( sgn_sgn_int @ L ) )
% 14.25/8.15 => ( ( modulo_modulo_int @ ( times_times_int @ ( sgn_sgn_int @ K3 ) @ ( semiri1314217659103216013at_int @ M4 ) ) @ ( times_times_int @ ( sgn_sgn_int @ L ) @ ( semiri1314217659103216013at_int @ N2 ) ) )
% 14.25/8.15 = ( times_times_int @ ( sgn_sgn_int @ L )
% 14.25/8.15 @ ( minus_minus_int
% 14.25/8.15 @ ( semiri1314217659103216013at_int
% 14.25/8.15 @ ( times_times_nat @ N2
% 14.25/8.15 @ ( zero_n2687167440665602831ol_nat
% 14.25/8.15 @ ~ ( dvd_dvd_nat @ N2 @ M4 ) ) ) )
% 14.25/8.15 @ ( semiri1314217659103216013at_int @ ( modulo_modulo_nat @ M4 @ N2 ) ) ) ) ) ) ) ) ) ).
% 14.25/8.15
% 14.25/8.15 thf(fact_7503_bezw_Opelims,axiom,
% 14.25/8.15 ! [X: nat,Xa: nat,Y: product_prod_int_int] :
% 14.25/8.15 ( ( ( bezw @ X @ Xa )
% 14.25/8.15 = Y )
% 14.25/8.15 => ( ( accp_P4275260045618599050at_nat @ bezw_rel @ ( product_Pair_nat_nat @ X @ Xa ) )
% 14.25/8.15 => ~ ( ( ( ( Xa = zero_zero_nat )
% 14.25/8.15 => ( Y
% 14.25/8.15 = ( product_Pair_int_int @ one_one_int @ zero_zero_int ) ) )
% 14.25/8.15 & ( ( Xa != zero_zero_nat )
% 14.25/8.15 => ( Y
% 14.25/8.15 = ( product_Pair_int_int @ ( product_snd_int_int @ ( bezw @ Xa @ ( modulo_modulo_nat @ X @ Xa ) ) ) @ ( minus_minus_int @ ( product_fst_int_int @ ( bezw @ Xa @ ( modulo_modulo_nat @ X @ Xa ) ) ) @ ( times_times_int @ ( product_snd_int_int @ ( bezw @ Xa @ ( modulo_modulo_nat @ X @ Xa ) ) ) @ ( semiri1314217659103216013at_int @ ( divide_divide_nat @ X @ Xa ) ) ) ) ) ) ) )
% 14.25/8.15 => ~ ( accp_P4275260045618599050at_nat @ bezw_rel @ ( product_Pair_nat_nat @ X @ Xa ) ) ) ) ) ).
% 14.25/8.15
% 14.25/8.15 thf(fact_7504_semiring__norm_I12_J,axiom,
% 14.25/8.15 ! [N2: num] :
% 14.25/8.15 ( ( times_times_num @ one @ N2 )
% 14.25/8.15 = N2 ) ).
% 14.25/8.15
% 14.25/8.15 thf(fact_7505_semiring__norm_I11_J,axiom,
% 14.25/8.15 ! [M4: num] :
% 14.25/8.15 ( ( times_times_num @ M4 @ one )
% 14.25/8.15 = M4 ) ).
% 14.25/8.15
% 14.25/8.15 thf(fact_7506_neg__eucl__rel__int__mult__2,axiom,
% 14.25/8.15 ! [B: int,A2: int,Q4: int,R2: int] :
% 14.25/8.15 ( ( ord_less_eq_int @ B @ zero_zero_int )
% 14.25/8.15 => ( ( eucl_rel_int @ ( plus_plus_int @ A2 @ one_one_int ) @ B @ ( product_Pair_int_int @ Q4 @ R2 ) )
% 14.25/8.15 => ( eucl_rel_int @ ( plus_plus_int @ one_one_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A2 ) ) @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B ) @ ( product_Pair_int_int @ Q4 @ ( minus_minus_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ R2 ) @ one_one_int ) ) ) ) ) ).
% 14.25/8.15
% 14.25/8.15 thf(fact_7507_semiring__norm_I13_J,axiom,
% 14.25/8.15 ! [M4: num,N2: num] :
% 14.25/8.15 ( ( times_times_num @ ( bit0 @ M4 ) @ ( bit0 @ N2 ) )
% 14.25/8.15 = ( bit0 @ ( bit0 @ ( times_times_num @ M4 @ N2 ) ) ) ) ).
% 14.25/8.15
% 14.25/8.15 thf(fact_7508_num__double,axiom,
% 14.25/8.15 ! [N2: num] :
% 14.25/8.15 ( ( times_times_num @ ( bit0 @ one ) @ N2 )
% 14.25/8.15 = ( bit0 @ N2 ) ) ).
% 14.25/8.15
% 14.25/8.15 thf(fact_7509_zmod__numeral__Bit0,axiom,
% 14.25/8.15 ! [V: num,W: num] :
% 14.25/8.15 ( ( modulo_modulo_int @ ( numeral_numeral_int @ ( bit0 @ V ) ) @ ( numeral_numeral_int @ ( bit0 @ W ) ) )
% 14.25/8.15 = ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( modulo_modulo_int @ ( numeral_numeral_int @ V ) @ ( numeral_numeral_int @ W ) ) ) ) ).
% 14.25/8.15
% 14.25/8.15 thf(fact_7510_odd__two__times__div__two__nat,axiom,
% 14.25/8.15 ! [N2: nat] :
% 14.25/8.15 ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
% 14.25/8.15 => ( ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ N2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 14.25/8.15 = ( minus_minus_nat @ N2 @ one_one_nat ) ) ) ).
% 14.25/8.15
% 14.25/8.15 thf(fact_7511_Suc__double__not__eq__double,axiom,
% 14.25/8.15 ! [M4: nat,N2: nat] :
% 14.25/8.15 ( ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M4 ) )
% 14.25/8.15 != ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) ).
% 14.25/8.15
% 14.25/8.15 thf(fact_7512_double__not__eq__Suc__double,axiom,
% 14.25/8.15 ! [M4: nat,N2: nat] :
% 14.25/8.15 ( ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M4 )
% 14.25/8.15 != ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) ) ).
% 14.25/8.15
% 14.25/8.15 thf(fact_7513_nat__bit__induct,axiom,
% 14.25/8.15 ! [P: nat > $o,N2: nat] :
% 14.25/8.15 ( ( P @ zero_zero_nat )
% 14.25/8.15 => ( ! [N3: nat] :
% 14.25/8.15 ( ( P @ N3 )
% 14.25/8.15 => ( ( ord_less_nat @ zero_zero_nat @ N3 )
% 14.25/8.15 => ( P @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 ) ) ) )
% 14.25/8.15 => ( ! [N3: nat] :
% 14.25/8.15 ( ( P @ N3 )
% 14.25/8.15 => ( P @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 ) ) ) )
% 14.25/8.15 => ( P @ N2 ) ) ) ) ).
% 14.25/8.15
% 14.25/8.15 thf(fact_7514_int__bit__induct,axiom,
% 14.25/8.15 ! [P: int > $o,K3: int] :
% 14.25/8.15 ( ( P @ zero_zero_int )
% 14.25/8.15 => ( ( P @ ( uminus_uminus_int @ one_one_int ) )
% 14.25/8.15 => ( ! [K: int] :
% 14.25/8.15 ( ( P @ K )
% 14.25/8.15 => ( ( K != zero_zero_int )
% 14.25/8.15 => ( P @ ( times_times_int @ K @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) )
% 14.25/8.15 => ( ! [K: int] :
% 14.25/8.15 ( ( P @ K )
% 14.25/8.15 => ( ( K
% 14.25/8.15 != ( uminus_uminus_int @ one_one_int ) )
% 14.25/8.15 => ( P @ ( plus_plus_int @ one_one_int @ ( times_times_int @ K @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ) )
% 14.25/8.15 => ( P @ K3 ) ) ) ) ) ).
% 14.25/8.15
% 14.25/8.15 thf(fact_7515_pos__zdiv__mult__2,axiom,
% 14.25/8.15 ! [A2: int,B: int] :
% 14.25/8.15 ( ( ord_less_eq_int @ zero_zero_int @ A2 )
% 14.25/8.15 => ( ( divide_divide_int @ ( plus_plus_int @ one_one_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B ) ) @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A2 ) )
% 14.25/8.15 = ( divide_divide_int @ B @ A2 ) ) ) ).
% 14.25/8.15
% 14.25/8.15 thf(fact_7516_neg__zdiv__mult__2,axiom,
% 14.25/8.15 ! [A2: int,B: int] :
% 14.25/8.15 ( ( ord_less_eq_int @ A2 @ zero_zero_int )
% 14.25/8.15 => ( ( divide_divide_int @ ( plus_plus_int @ one_one_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B ) ) @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A2 ) )
% 14.25/8.15 = ( divide_divide_int @ ( plus_plus_int @ B @ one_one_int ) @ A2 ) ) ) ).
% 14.25/8.15
% 14.25/8.15 thf(fact_7517_pos__zmod__mult__2,axiom,
% 14.25/8.15 ! [A2: int,B: int] :
% 14.25/8.15 ( ( ord_less_eq_int @ zero_zero_int @ A2 )
% 14.25/8.15 => ( ( modulo_modulo_int @ ( plus_plus_int @ one_one_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B ) ) @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A2 ) )
% 14.25/8.15 = ( plus_plus_int @ one_one_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( modulo_modulo_int @ B @ A2 ) ) ) ) ) ).
% 14.25/8.15
% 14.25/8.15 thf(fact_7518_neg__zmod__mult__2,axiom,
% 14.25/8.15 ! [A2: int,B: int] :
% 14.25/8.15 ( ( ord_less_eq_int @ A2 @ zero_zero_int )
% 14.25/8.15 => ( ( modulo_modulo_int @ ( plus_plus_int @ one_one_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B ) ) @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A2 ) )
% 14.25/8.15 = ( minus_minus_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( modulo_modulo_int @ ( plus_plus_int @ B @ one_one_int ) @ A2 ) ) @ one_one_int ) ) ) ).
% 14.25/8.15
% 14.25/8.15 thf(fact_7519_pos__eucl__rel__int__mult__2,axiom,
% 14.25/8.15 ! [B: int,A2: int,Q4: int,R2: int] :
% 14.25/8.15 ( ( ord_less_eq_int @ zero_zero_int @ B )
% 14.25/8.15 => ( ( eucl_rel_int @ A2 @ B @ ( product_Pair_int_int @ Q4 @ R2 ) )
% 14.25/8.15 => ( eucl_rel_int @ ( plus_plus_int @ one_one_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A2 ) ) @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B ) @ ( product_Pair_int_int @ Q4 @ ( plus_plus_int @ one_one_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ R2 ) ) ) ) ) ) ).
% 14.25/8.15
% 14.25/8.15 thf(fact_7520_signed__take__bit__Suc__bit0,axiom,
% 14.25/8.15 ! [N2: nat,K3: num] :
% 14.25/8.15 ( ( bit_ri631733984087533419it_int @ ( suc @ N2 ) @ ( numeral_numeral_int @ ( bit0 @ K3 ) ) )
% 14.25/8.15 = ( times_times_int @ ( bit_ri631733984087533419it_int @ N2 @ ( numeral_numeral_int @ K3 ) ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ).
% 14.25/8.15
% 14.25/8.15 thf(fact_7521_signed__take__bit__Suc__minus__bit0,axiom,
% 14.25/8.15 ! [N2: nat,K3: num] :
% 14.25/8.15 ( ( bit_ri631733984087533419it_int @ ( suc @ N2 ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit0 @ K3 ) ) ) )
% 14.25/8.15 = ( times_times_int @ ( bit_ri631733984087533419it_int @ N2 @ ( uminus_uminus_int @ ( numeral_numeral_int @ K3 ) ) ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ).
% 14.25/8.15
% 14.25/8.15 thf(fact_7522_signed__take__bit__mult,axiom,
% 14.25/8.15 ! [N2: nat,K3: int,L: int] :
% 14.25/8.15 ( ( bit_ri631733984087533419it_int @ N2 @ ( times_times_int @ ( bit_ri631733984087533419it_int @ N2 @ K3 ) @ ( bit_ri631733984087533419it_int @ N2 @ L ) ) )
% 14.25/8.15 = ( bit_ri631733984087533419it_int @ N2 @ ( times_times_int @ K3 @ L ) ) ) ).
% 14.25/8.15
% 14.25/8.15 thf(fact_7523_and__int_Oelims,axiom,
% 14.25/8.15 ! [X: int,Xa: int,Y: int] :
% 14.25/8.15 ( ( ( bit_se725231765392027082nd_int @ X @ Xa )
% 14.25/8.15 = Y )
% 14.25/8.15 => ( ( ( ( member_int @ X @ ( insert_int @ zero_zero_int @ ( insert_int @ ( uminus_uminus_int @ one_one_int ) @ bot_bot_set_int ) ) )
% 14.25/8.15 & ( member_int @ Xa @ ( insert_int @ zero_zero_int @ ( insert_int @ ( uminus_uminus_int @ one_one_int ) @ bot_bot_set_int ) ) ) )
% 14.25/8.15 => ( Y
% 14.25/8.15 = ( uminus_uminus_int
% 14.25/8.15 @ ( zero_n2684676970156552555ol_int
% 14.25/8.15 @ ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ X )
% 14.25/8.15 & ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ Xa ) ) ) ) ) )
% 14.25/8.15 & ( ~ ( ( member_int @ X @ ( insert_int @ zero_zero_int @ ( insert_int @ ( uminus_uminus_int @ one_one_int ) @ bot_bot_set_int ) ) )
% 14.25/8.15 & ( member_int @ Xa @ ( insert_int @ zero_zero_int @ ( insert_int @ ( uminus_uminus_int @ one_one_int ) @ bot_bot_set_int ) ) ) )
% 14.25/8.15 => ( Y
% 14.25/8.15 = ( plus_plus_int
% 14.25/8.15 @ ( zero_n2684676970156552555ol_int
% 14.25/8.15 @ ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ X )
% 14.25/8.15 & ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ Xa ) ) )
% 14.25/8.15 @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se725231765392027082nd_int @ ( divide_divide_int @ X @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ ( divide_divide_int @ Xa @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) ).
% 14.25/8.15
% 14.25/8.15 thf(fact_7524_and__int_Osimps,axiom,
% 14.25/8.15 ( bit_se725231765392027082nd_int
% 14.25/8.15 = ( ^ [K5: int,L2: int] :
% 14.25/8.15 ( if_int
% 14.25/8.15 @ ( ( member_int @ K5 @ ( insert_int @ zero_zero_int @ ( insert_int @ ( uminus_uminus_int @ one_one_int ) @ bot_bot_set_int ) ) )
% 14.25/8.15 & ( member_int @ L2 @ ( insert_int @ zero_zero_int @ ( insert_int @ ( uminus_uminus_int @ one_one_int ) @ bot_bot_set_int ) ) ) )
% 14.25/8.15 @ ( uminus_uminus_int
% 14.25/8.15 @ ( zero_n2684676970156552555ol_int
% 14.25/8.15 @ ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ K5 )
% 14.25/8.15 & ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ L2 ) ) ) )
% 14.25/8.15 @ ( plus_plus_int
% 14.25/8.15 @ ( zero_n2684676970156552555ol_int
% 14.25/8.15 @ ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ K5 )
% 14.25/8.15 & ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ L2 ) ) )
% 14.25/8.15 @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se725231765392027082nd_int @ ( divide_divide_int @ K5 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ ( divide_divide_int @ L2 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ).
% 14.25/8.15
% 14.25/8.15 thf(fact_7525_signed__take__bit__Suc__minus__bit1,axiom,
% 14.25/8.15 ! [N2: nat,K3: num] :
% 14.25/8.15 ( ( bit_ri631733984087533419it_int @ ( suc @ N2 ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit1 @ K3 ) ) ) )
% 14.25/8.15 = ( plus_plus_int @ ( times_times_int @ ( bit_ri631733984087533419it_int @ N2 @ ( minus_minus_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ K3 ) ) @ one_one_int ) ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ one_one_int ) ) ).
% 14.25/8.15
% 14.25/8.15 thf(fact_7526_concat__bit__Suc,axiom,
% 14.25/8.15 ! [N2: nat,K3: int,L: int] :
% 14.25/8.15 ( ( bit_concat_bit @ ( suc @ N2 ) @ K3 @ L )
% 14.25/8.15 = ( plus_plus_int @ ( modulo_modulo_int @ K3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_concat_bit @ N2 @ ( divide_divide_int @ K3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ L ) ) ) ) ).
% 14.25/8.15
% 14.25/8.15 thf(fact_7527_semiring__norm_I14_J,axiom,
% 14.25/8.15 ! [M4: num,N2: num] :
% 14.25/8.15 ( ( times_times_num @ ( bit0 @ M4 ) @ ( bit1 @ N2 ) )
% 14.25/8.15 = ( bit0 @ ( times_times_num @ M4 @ ( bit1 @ N2 ) ) ) ) ).
% 14.25/8.15
% 14.25/8.15 thf(fact_7528_semiring__norm_I15_J,axiom,
% 14.25/8.15 ! [M4: num,N2: num] :
% 14.25/8.15 ( ( times_times_num @ ( bit1 @ M4 ) @ ( bit0 @ N2 ) )
% 14.25/8.15 = ( bit0 @ ( times_times_num @ ( bit1 @ M4 ) @ N2 ) ) ) ).
% 14.25/8.15
% 14.25/8.15 thf(fact_7529_semiring__norm_I16_J,axiom,
% 14.25/8.15 ! [M4: num,N2: num] :
% 14.25/8.15 ( ( times_times_num @ ( bit1 @ M4 ) @ ( bit1 @ N2 ) )
% 14.25/8.15 = ( bit1 @ ( plus_plus_num @ ( plus_plus_num @ M4 @ N2 ) @ ( bit0 @ ( times_times_num @ M4 @ N2 ) ) ) ) ) ).
% 14.25/8.15
% 14.25/8.15 thf(fact_7530_zmod__numeral__Bit1,axiom,
% 14.25/8.15 ! [V: num,W: num] :
% 14.25/8.15 ( ( modulo_modulo_int @ ( numeral_numeral_int @ ( bit1 @ V ) ) @ ( numeral_numeral_int @ ( bit0 @ W ) ) )
% 14.25/8.15 = ( plus_plus_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( modulo_modulo_int @ ( numeral_numeral_int @ V ) @ ( numeral_numeral_int @ W ) ) ) @ one_one_int ) ) ).
% 14.25/8.15
% 14.25/8.15 thf(fact_7531_signed__take__bit__Suc__bit1,axiom,
% 14.25/8.15 ! [N2: nat,K3: num] :
% 14.25/8.15 ( ( bit_ri631733984087533419it_int @ ( suc @ N2 ) @ ( numeral_numeral_int @ ( bit1 @ K3 ) ) )
% 14.25/8.15 = ( plus_plus_int @ ( times_times_int @ ( bit_ri631733984087533419it_int @ N2 @ ( numeral_numeral_int @ K3 ) ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ one_one_int ) ) ).
% 14.25/8.15
% 14.25/8.15 thf(fact_7532_take__bit__mult,axiom,
% 14.25/8.15 ! [N2: nat,K3: int,L: int] :
% 14.25/8.15 ( ( bit_se2923211474154528505it_int @ N2 @ ( times_times_int @ ( bit_se2923211474154528505it_int @ N2 @ K3 ) @ ( bit_se2923211474154528505it_int @ N2 @ L ) ) )
% 14.25/8.15 = ( bit_se2923211474154528505it_int @ N2 @ ( times_times_int @ K3 @ L ) ) ) ).
% 14.25/8.15
% 14.25/8.15 thf(fact_7533_take__bit__Suc__minus__bit0,axiom,
% 14.25/8.15 ! [N2: nat,K3: num] :
% 14.25/8.15 ( ( bit_se2923211474154528505it_int @ ( suc @ N2 ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit0 @ K3 ) ) ) )
% 14.25/8.15 = ( times_times_int @ ( bit_se2923211474154528505it_int @ N2 @ ( uminus_uminus_int @ ( numeral_numeral_int @ K3 ) ) ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ).
% 14.25/8.15
% 14.25/8.15 thf(fact_7534_and__int__rec,axiom,
% 14.25/8.15 ( bit_se725231765392027082nd_int
% 14.25/8.15 = ( ^ [K5: int,L2: int] :
% 14.25/8.15 ( plus_plus_int
% 14.25/8.15 @ ( zero_n2684676970156552555ol_int
% 14.25/8.15 @ ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ K5 )
% 14.25/8.15 & ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ L2 ) ) )
% 14.25/8.15 @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se725231765392027082nd_int @ ( divide_divide_int @ K5 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ ( divide_divide_int @ L2 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ) ) ) ).
% 14.25/8.15
% 14.25/8.15 thf(fact_7535_and__int__unfold,axiom,
% 14.25/8.15 ( bit_se725231765392027082nd_int
% 14.25/8.15 = ( ^ [K5: int,L2: int] :
% 14.25/8.15 ( if_int
% 14.25/8.15 @ ( ( K5 = zero_zero_int )
% 14.25/8.15 | ( L2 = zero_zero_int ) )
% 14.25/8.15 @ zero_zero_int
% 14.25/8.15 @ ( if_int
% 14.25/8.15 @ ( K5
% 14.25/8.15 = ( uminus_uminus_int @ one_one_int ) )
% 14.25/8.15 @ L2
% 14.25/8.15 @ ( if_int
% 14.25/8.15 @ ( L2
% 14.25/8.15 = ( uminus_uminus_int @ one_one_int ) )
% 14.25/8.15 @ K5
% 14.25/8.15 @ ( plus_plus_int @ ( times_times_int @ ( modulo_modulo_int @ K5 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ ( modulo_modulo_int @ L2 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se725231765392027082nd_int @ ( divide_divide_int @ K5 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ ( divide_divide_int @ L2 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) ) ).
% 14.25/8.15
% 14.25/8.15 thf(fact_7536_signed__take__bit__numeral__minus__bit1,axiom,
% 14.25/8.15 ! [L: num,K3: num] :
% 14.25/8.15 ( ( bit_ri631733984087533419it_int @ ( numeral_numeral_nat @ L ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit1 @ K3 ) ) ) )
% 14.25/8.15 = ( plus_plus_int @ ( times_times_int @ ( bit_ri631733984087533419it_int @ ( pred_numeral @ L ) @ ( minus_minus_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ K3 ) ) @ one_one_int ) ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ one_one_int ) ) ).
% 14.25/8.15
% 14.25/8.15 thf(fact_7537_signed__take__bit__numeral__bit1,axiom,
% 14.25/8.15 ! [L: num,K3: num] :
% 14.25/8.15 ( ( bit_ri631733984087533419it_int @ ( numeral_numeral_nat @ L ) @ ( numeral_numeral_int @ ( bit1 @ K3 ) ) )
% 14.25/8.15 = ( plus_plus_int @ ( times_times_int @ ( bit_ri631733984087533419it_int @ ( pred_numeral @ L ) @ ( numeral_numeral_int @ K3 ) ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ one_one_int ) ) ).
% 14.25/8.15
% 14.25/8.15 thf(fact_7538_and__int_Opsimps,axiom,
% 14.25/8.15 ! [K3: int,L: int] :
% 14.25/8.15 ( ( accp_P1096762738010456898nt_int @ bit_and_int_rel @ ( product_Pair_int_int @ K3 @ L ) )
% 14.25/8.15 => ( ( ( ( member_int @ K3 @ ( insert_int @ zero_zero_int @ ( insert_int @ ( uminus_uminus_int @ one_one_int ) @ bot_bot_set_int ) ) )
% 14.25/8.15 & ( member_int @ L @ ( insert_int @ zero_zero_int @ ( insert_int @ ( uminus_uminus_int @ one_one_int ) @ bot_bot_set_int ) ) ) )
% 14.25/8.15 => ( ( bit_se725231765392027082nd_int @ K3 @ L )
% 14.25/8.15 = ( uminus_uminus_int
% 14.25/8.15 @ ( zero_n2684676970156552555ol_int
% 14.25/8.15 @ ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ K3 )
% 14.25/8.15 & ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ L ) ) ) ) ) )
% 14.25/8.15 & ( ~ ( ( member_int @ K3 @ ( insert_int @ zero_zero_int @ ( insert_int @ ( uminus_uminus_int @ one_one_int ) @ bot_bot_set_int ) ) )
% 14.25/8.15 & ( member_int @ L @ ( insert_int @ zero_zero_int @ ( insert_int @ ( uminus_uminus_int @ one_one_int ) @ bot_bot_set_int ) ) ) )
% 14.25/8.15 => ( ( bit_se725231765392027082nd_int @ K3 @ L )
% 14.25/8.15 = ( plus_plus_int
% 14.25/8.15 @ ( zero_n2684676970156552555ol_int
% 14.25/8.15 @ ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ K3 )
% 14.25/8.15 & ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ L ) ) )
% 14.25/8.15 @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se725231765392027082nd_int @ ( divide_divide_int @ K3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ ( divide_divide_int @ L @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) ).
% 14.25/8.15
% 14.25/8.15 thf(fact_7539_and__int_Opelims,axiom,
% 14.25/8.15 ! [X: int,Xa: int,Y: int] :
% 14.25/8.15 ( ( ( bit_se725231765392027082nd_int @ X @ Xa )
% 14.25/8.15 = Y )
% 14.25/8.15 => ( ( accp_P1096762738010456898nt_int @ bit_and_int_rel @ ( product_Pair_int_int @ X @ Xa ) )
% 14.25/8.15 => ~ ( ( ( ( ( member_int @ X @ ( insert_int @ zero_zero_int @ ( insert_int @ ( uminus_uminus_int @ one_one_int ) @ bot_bot_set_int ) ) )
% 14.25/8.15 & ( member_int @ Xa @ ( insert_int @ zero_zero_int @ ( insert_int @ ( uminus_uminus_int @ one_one_int ) @ bot_bot_set_int ) ) ) )
% 14.25/8.15 => ( Y
% 14.25/8.15 = ( uminus_uminus_int
% 14.25/8.15 @ ( zero_n2684676970156552555ol_int
% 14.25/8.15 @ ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ X )
% 14.25/8.15 & ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ Xa ) ) ) ) ) )
% 14.25/8.15 & ( ~ ( ( member_int @ X @ ( insert_int @ zero_zero_int @ ( insert_int @ ( uminus_uminus_int @ one_one_int ) @ bot_bot_set_int ) ) )
% 14.25/8.15 & ( member_int @ Xa @ ( insert_int @ zero_zero_int @ ( insert_int @ ( uminus_uminus_int @ one_one_int ) @ bot_bot_set_int ) ) ) )
% 14.25/8.15 => ( Y
% 14.25/8.15 = ( plus_plus_int
% 14.25/8.15 @ ( zero_n2684676970156552555ol_int
% 14.25/8.15 @ ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ X )
% 14.25/8.15 & ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ Xa ) ) )
% 14.25/8.15 @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se725231765392027082nd_int @ ( divide_divide_int @ X @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ ( divide_divide_int @ Xa @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ) ) ) )
% 14.25/8.15 => ~ ( accp_P1096762738010456898nt_int @ bit_and_int_rel @ ( product_Pair_int_int @ X @ Xa ) ) ) ) ) ).
% 14.25/8.15
% 14.25/8.15 thf(fact_7540_signed__take__bit__numeral__bit0,axiom,
% 14.25/8.15 ! [L: num,K3: num] :
% 14.25/8.15 ( ( bit_ri631733984087533419it_int @ ( numeral_numeral_nat @ L ) @ ( numeral_numeral_int @ ( bit0 @ K3 ) ) )
% 14.25/8.15 = ( times_times_int @ ( bit_ri631733984087533419it_int @ ( pred_numeral @ L ) @ ( numeral_numeral_int @ K3 ) ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ).
% 14.25/8.15
% 14.25/8.15 thf(fact_7541_signed__take__bit__numeral__minus__bit0,axiom,
% 14.25/8.15 ! [L: num,K3: num] :
% 14.25/8.15 ( ( bit_ri631733984087533419it_int @ ( numeral_numeral_nat @ L ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit0 @ K3 ) ) ) )
% 14.25/8.15 = ( times_times_int @ ( bit_ri631733984087533419it_int @ ( pred_numeral @ L ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ K3 ) ) ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ).
% 14.25/8.15
% 14.25/8.15 thf(fact_7542_take__bit__numeral__minus__bit0,axiom,
% 14.25/8.15 ! [L: num,K3: num] :
% 14.25/8.15 ( ( bit_se2923211474154528505it_int @ ( numeral_numeral_nat @ L ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit0 @ K3 ) ) ) )
% 14.25/8.15 = ( times_times_int @ ( bit_se2923211474154528505it_int @ ( pred_numeral @ L ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ K3 ) ) ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ).
% 14.25/8.15
% 14.25/8.15 thf(fact_7543_and__nat__unfold,axiom,
% 14.25/8.15 ( bit_se727722235901077358nd_nat
% 14.25/8.15 = ( ^ [M3: nat,N4: nat] :
% 14.25/8.15 ( if_nat
% 14.25/8.15 @ ( ( M3 = zero_zero_nat )
% 14.25/8.15 | ( N4 = zero_zero_nat ) )
% 14.25/8.15 @ zero_zero_nat
% 14.25/8.15 @ ( plus_plus_nat @ ( times_times_nat @ ( modulo_modulo_nat @ M3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( modulo_modulo_nat @ N4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( bit_se727722235901077358nd_nat @ ( divide_divide_nat @ M3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( divide_divide_nat @ N4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ).
% 14.25/8.15
% 14.25/8.15 thf(fact_7544_and__nat__rec,axiom,
% 14.25/8.15 ( bit_se727722235901077358nd_nat
% 14.25/8.15 = ( ^ [M3: nat,N4: nat] :
% 14.25/8.15 ( plus_plus_nat
% 14.25/8.15 @ ( zero_n2687167440665602831ol_nat
% 14.25/8.15 @ ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M3 )
% 14.25/8.15 & ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N4 ) ) )
% 14.25/8.15 @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( bit_se727722235901077358nd_nat @ ( divide_divide_nat @ M3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( divide_divide_nat @ N4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ).
% 14.25/8.15
% 14.25/8.15 thf(fact_7545_and__int_Opinduct,axiom,
% 14.25/8.15 ! [A0: int,A1: int,P: int > int > $o] :
% 14.25/8.15 ( ( accp_P1096762738010456898nt_int @ bit_and_int_rel @ ( product_Pair_int_int @ A0 @ A1 ) )
% 14.25/8.15 => ( ! [K: int,L3: int] :
% 14.25/8.15 ( ( accp_P1096762738010456898nt_int @ bit_and_int_rel @ ( product_Pair_int_int @ K @ L3 ) )
% 14.25/8.15 => ( ( ~ ( ( member_int @ K @ ( insert_int @ zero_zero_int @ ( insert_int @ ( uminus_uminus_int @ one_one_int ) @ bot_bot_set_int ) ) )
% 14.25/8.15 & ( member_int @ L3 @ ( insert_int @ zero_zero_int @ ( insert_int @ ( uminus_uminus_int @ one_one_int ) @ bot_bot_set_int ) ) ) )
% 14.25/8.15 => ( P @ ( divide_divide_int @ K @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ ( divide_divide_int @ L3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) )
% 14.25/8.15 => ( P @ K @ L3 ) ) )
% 14.25/8.15 => ( P @ A0 @ A1 ) ) ) ).
% 14.25/8.15
% 14.25/8.15 thf(fact_7546_take__bit__numeral__minus__bit1,axiom,
% 14.25/8.15 ! [L: num,K3: num] :
% 14.25/8.15 ( ( bit_se2923211474154528505it_int @ ( numeral_numeral_nat @ L ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit1 @ K3 ) ) ) )
% 14.25/8.15 = ( plus_plus_int @ ( times_times_int @ ( bit_se2923211474154528505it_int @ ( pred_numeral @ L ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( inc @ K3 ) ) ) ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ one_one_int ) ) ).
% 14.25/8.15
% 14.25/8.15 thf(fact_7547_take__bit__Suc__minus__bit1,axiom,
% 14.25/8.15 ! [N2: nat,K3: num] :
% 14.25/8.15 ( ( bit_se2923211474154528505it_int @ ( suc @ N2 ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit1 @ K3 ) ) ) )
% 14.25/8.15 = ( plus_plus_int @ ( times_times_int @ ( bit_se2923211474154528505it_int @ N2 @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( inc @ K3 ) ) ) ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ one_one_int ) ) ).
% 14.25/8.15
% 14.25/8.15 thf(fact_7548_binomial__code,axiom,
% 14.25/8.15 ( binomial
% 14.25/8.15 = ( ^ [N4: nat,K5: nat] : ( if_nat @ ( ord_less_nat @ N4 @ K5 ) @ zero_zero_nat @ ( if_nat @ ( ord_less_nat @ N4 @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ K5 ) ) @ ( binomial @ N4 @ ( minus_minus_nat @ N4 @ K5 ) ) @ ( divide_divide_nat @ ( set_fo2584398358068434914at_nat @ times_times_nat @ ( plus_plus_nat @ ( minus_minus_nat @ N4 @ K5 ) @ one_one_nat ) @ N4 @ one_one_nat ) @ ( semiri1408675320244567234ct_nat @ K5 ) ) ) ) ) ) ).
% 14.25/8.15
% 14.25/8.15 thf(fact_7549_modulo__int__def,axiom,
% 14.25/8.15 ( modulo_modulo_int
% 14.25/8.15 = ( ^ [K5: int,L2: int] :
% 14.25/8.15 ( if_int @ ( L2 = zero_zero_int ) @ K5
% 14.25/8.15 @ ( if_int
% 14.25/8.15 @ ( ( sgn_sgn_int @ K5 )
% 14.25/8.15 = ( sgn_sgn_int @ L2 ) )
% 14.25/8.15 @ ( times_times_int @ ( sgn_sgn_int @ L2 ) @ ( semiri1314217659103216013at_int @ ( modulo_modulo_nat @ ( nat2 @ ( abs_abs_int @ K5 ) ) @ ( nat2 @ ( abs_abs_int @ L2 ) ) ) ) )
% 14.25/8.15 @ ( times_times_int @ ( sgn_sgn_int @ L2 )
% 14.25/8.15 @ ( minus_minus_int
% 14.25/8.15 @ ( times_times_int @ ( abs_abs_int @ L2 )
% 14.25/8.15 @ ( zero_n2684676970156552555ol_int
% 14.25/8.15 @ ~ ( dvd_dvd_int @ L2 @ K5 ) ) )
% 14.25/8.15 @ ( semiri1314217659103216013at_int @ ( modulo_modulo_nat @ ( nat2 @ ( abs_abs_int @ K5 ) ) @ ( nat2 @ ( abs_abs_int @ L2 ) ) ) ) ) ) ) ) ) ) ).
% 14.25/8.15
% 14.25/8.15 thf(fact_7550_or__int__unfold,axiom,
% 14.25/8.15 ( bit_se1409905431419307370or_int
% 14.25/8.15 = ( ^ [K5: int,L2: int] :
% 14.25/8.15 ( if_int
% 14.25/8.15 @ ( ( K5
% 14.25/8.15 = ( uminus_uminus_int @ one_one_int ) )
% 14.25/8.15 | ( L2
% 14.25/8.15 = ( uminus_uminus_int @ one_one_int ) ) )
% 14.25/8.15 @ ( uminus_uminus_int @ one_one_int )
% 14.25/8.15 @ ( if_int @ ( K5 = zero_zero_int ) @ L2 @ ( if_int @ ( L2 = zero_zero_int ) @ K5 @ ( plus_plus_int @ ( ord_max_int @ ( modulo_modulo_int @ K5 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ ( modulo_modulo_int @ L2 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se1409905431419307370or_int @ ( divide_divide_int @ K5 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ ( divide_divide_int @ L2 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) ) ).
% 14.25/8.15
% 14.25/8.15 thf(fact_7551_Suc__times__binomial__eq,axiom,
% 14.25/8.15 ! [N2: nat,K3: nat] :
% 14.25/8.15 ( ( times_times_nat @ ( suc @ N2 ) @ ( binomial @ N2 @ K3 ) )
% 14.25/8.15 = ( times_times_nat @ ( binomial @ ( suc @ N2 ) @ ( suc @ K3 ) ) @ ( suc @ K3 ) ) ) ).
% 14.25/8.15
% 14.25/8.15 thf(fact_7552_Suc__times__binomial,axiom,
% 14.25/8.15 ! [K3: nat,N2: nat] :
% 14.25/8.15 ( ( times_times_nat @ ( suc @ K3 ) @ ( binomial @ ( suc @ N2 ) @ ( suc @ K3 ) ) )
% 14.25/8.15 = ( times_times_nat @ ( suc @ N2 ) @ ( binomial @ N2 @ K3 ) ) ) ).
% 14.25/8.15
% 14.25/8.15 thf(fact_7553_choose__mult__lemma,axiom,
% 14.25/8.15 ! [M4: nat,R2: nat,K3: nat] :
% 14.25/8.15 ( ( times_times_nat @ ( binomial @ ( plus_plus_nat @ ( plus_plus_nat @ M4 @ R2 ) @ K3 ) @ ( plus_plus_nat @ M4 @ K3 ) ) @ ( binomial @ ( plus_plus_nat @ M4 @ K3 ) @ K3 ) )
% 14.25/8.15 = ( times_times_nat @ ( binomial @ ( plus_plus_nat @ ( plus_plus_nat @ M4 @ R2 ) @ K3 ) @ K3 ) @ ( binomial @ ( plus_plus_nat @ M4 @ R2 ) @ M4 ) ) ) ).
% 14.25/8.15
% 14.25/8.15 thf(fact_7554_Suc__times__binomial__add,axiom,
% 14.25/8.15 ! [A2: nat,B: nat] :
% 14.25/8.15 ( ( times_times_nat @ ( suc @ A2 ) @ ( binomial @ ( suc @ ( plus_plus_nat @ A2 @ B ) ) @ ( suc @ A2 ) ) )
% 14.25/8.15 = ( times_times_nat @ ( suc @ B ) @ ( binomial @ ( suc @ ( plus_plus_nat @ A2 @ B ) ) @ A2 ) ) ) ).
% 14.25/8.15
% 14.25/8.15 thf(fact_7555_binomial__fact__lemma,axiom,
% 14.25/8.15 ! [K3: nat,N2: nat] :
% 14.25/8.15 ( ( ord_less_eq_nat @ K3 @ N2 )
% 14.25/8.15 => ( ( times_times_nat @ ( times_times_nat @ ( semiri1408675320244567234ct_nat @ K3 ) @ ( semiri1408675320244567234ct_nat @ ( minus_minus_nat @ N2 @ K3 ) ) ) @ ( binomial @ N2 @ K3 ) )
% 14.25/8.15 = ( semiri1408675320244567234ct_nat @ N2 ) ) ) ).
% 14.25/8.15
% 14.25/8.15 thf(fact_7556_choose__mult,axiom,
% 14.25/8.15 ! [K3: nat,M4: nat,N2: nat] :
% 14.25/8.15 ( ( ord_less_eq_nat @ K3 @ M4 )
% 14.25/8.15 => ( ( ord_less_eq_nat @ M4 @ N2 )
% 14.25/8.15 => ( ( times_times_nat @ ( binomial @ N2 @ M4 ) @ ( binomial @ M4 @ K3 ) )
% 14.25/8.15 = ( times_times_nat @ ( binomial @ N2 @ K3 ) @ ( binomial @ ( minus_minus_nat @ N2 @ K3 ) @ ( minus_minus_nat @ M4 @ K3 ) ) ) ) ) ) ).
% 14.25/8.15
% 14.25/8.15 thf(fact_7557_binomial__absorb__comp,axiom,
% 14.25/8.15 ! [N2: nat,K3: nat] :
% 14.25/8.15 ( ( times_times_nat @ ( minus_minus_nat @ N2 @ K3 ) @ ( binomial @ N2 @ K3 ) )
% 14.25/8.15 = ( times_times_nat @ N2 @ ( binomial @ ( minus_minus_nat @ N2 @ one_one_nat ) @ K3 ) ) ) ).
% 14.25/8.15
% 14.25/8.15 thf(fact_7558_binomial__Suc__Suc__eq__times,axiom,
% 14.25/8.15 ! [N2: nat,K3: nat] :
% 14.25/8.15 ( ( binomial @ ( suc @ N2 ) @ ( suc @ K3 ) )
% 14.25/8.15 = ( divide_divide_nat @ ( times_times_nat @ ( suc @ N2 ) @ ( binomial @ N2 @ K3 ) ) @ ( suc @ K3 ) ) ) ).
% 14.25/8.15
% 14.25/8.15 thf(fact_7559_nat__abs__mult__distrib,axiom,
% 14.25/8.15 ! [W: int,Z: int] :
% 14.25/8.15 ( ( nat2 @ ( abs_abs_int @ ( times_times_int @ W @ Z ) ) )
% 14.25/8.15 = ( times_times_nat @ ( nat2 @ ( abs_abs_int @ W ) ) @ ( nat2 @ ( abs_abs_int @ Z ) ) ) ) ).
% 14.25/8.15
% 14.25/8.15 thf(fact_7560_mult__inc,axiom,
% 14.25/8.15 ! [X: num,Y: num] :
% 14.25/8.15 ( ( times_times_num @ X @ ( inc @ Y ) )
% 14.25/8.15 = ( plus_plus_num @ ( times_times_num @ X @ Y ) @ X ) ) ).
% 14.25/8.15
% 14.25/8.15 thf(fact_7561_binomial__absorption,axiom,
% 14.25/8.15 ! [K3: nat,N2: nat] :
% 14.25/8.15 ( ( times_times_nat @ ( suc @ K3 ) @ ( binomial @ N2 @ ( suc @ K3 ) ) )
% 14.25/8.15 = ( times_times_nat @ N2 @ ( binomial @ ( minus_minus_nat @ N2 @ one_one_nat ) @ K3 ) ) ) ).
% 14.25/8.15
% 14.25/8.15 thf(fact_7562_nat__mult__distrib,axiom,
% 14.25/8.15 ! [Z: int,Z7: int] :
% 14.25/8.15 ( ( ord_less_eq_int @ zero_zero_int @ Z )
% 14.25/8.15 => ( ( nat2 @ ( times_times_int @ Z @ Z7 ) )
% 14.25/8.15 = ( times_times_nat @ ( nat2 @ Z ) @ ( nat2 @ Z7 ) ) ) ) ).
% 14.25/8.15
% 14.25/8.15 thf(fact_7563_binomial__altdef__nat,axiom,
% 14.25/8.15 ! [K3: nat,N2: nat] :
% 14.25/8.15 ( ( ord_less_eq_nat @ K3 @ N2 )
% 14.25/8.15 => ( ( binomial @ N2 @ K3 )
% 14.25/8.15 = ( divide_divide_nat @ ( semiri1408675320244567234ct_nat @ N2 ) @ ( times_times_nat @ ( semiri1408675320244567234ct_nat @ K3 ) @ ( semiri1408675320244567234ct_nat @ ( minus_minus_nat @ N2 @ K3 ) ) ) ) ) ) ).
% 14.25/8.15
% 14.25/8.15 thf(fact_7564_times__binomial__minus1__eq,axiom,
% 14.25/8.15 ! [K3: nat,N2: nat] :
% 14.25/8.15 ( ( ord_less_nat @ zero_zero_nat @ K3 )
% 14.25/8.15 => ( ( times_times_nat @ K3 @ ( binomial @ N2 @ K3 ) )
% 14.25/8.15 = ( times_times_nat @ N2 @ ( binomial @ ( minus_minus_nat @ N2 @ one_one_nat ) @ ( minus_minus_nat @ K3 @ one_one_nat ) ) ) ) ) ).
% 14.25/8.15
% 14.25/8.15 thf(fact_7565_nat__mult__distrib__neg,axiom,
% 14.25/8.15 ! [Z: int,Z7: int] :
% 14.25/8.15 ( ( ord_less_eq_int @ Z @ zero_zero_int )
% 14.25/8.15 => ( ( nat2 @ ( times_times_int @ Z @ Z7 ) )
% 14.25/8.15 = ( times_times_nat @ ( nat2 @ ( uminus_uminus_int @ Z ) ) @ ( nat2 @ ( uminus_uminus_int @ Z7 ) ) ) ) ) ).
% 14.25/8.15
% 14.25/8.15 thf(fact_7566_choose__two,axiom,
% 14.25/8.15 ! [N2: nat] :
% 14.25/8.15 ( ( binomial @ N2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 14.25/8.15 = ( divide_divide_nat @ ( times_times_nat @ N2 @ ( minus_minus_nat @ N2 @ one_one_nat ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% 14.25/8.15
% 14.25/8.15 thf(fact_7567_or__int__rec,axiom,
% 14.25/8.15 ( bit_se1409905431419307370or_int
% 14.25/8.15 = ( ^ [K5: int,L2: int] :
% 14.25/8.15 ( plus_plus_int
% 14.25/8.15 @ ( zero_n2684676970156552555ol_int
% 14.25/8.15 @ ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ K5 )
% 14.25/8.15 | ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ L2 ) ) )
% 14.25/8.15 @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se1409905431419307370or_int @ ( divide_divide_int @ K5 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ ( divide_divide_int @ L2 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ) ) ) ).
% 14.25/8.15
% 14.25/8.15 thf(fact_7568_signed__take__bit__eq__take__bit__minus,axiom,
% 14.25/8.15 ( bit_ri631733984087533419it_int
% 14.25/8.15 = ( ^ [N4: nat,K5: int] : ( minus_minus_int @ ( bit_se2923211474154528505it_int @ ( suc @ N4 ) @ K5 ) @ ( times_times_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( suc @ N4 ) ) @ ( zero_n2684676970156552555ol_int @ ( bit_se1146084159140164899it_int @ K5 @ N4 ) ) ) ) ) ) ).
% 14.25/8.15
% 14.25/8.15 thf(fact_7569_or__nat__unfold,axiom,
% 14.25/8.15 ( bit_se1412395901928357646or_nat
% 14.25/8.15 = ( ^ [M3: nat,N4: nat] : ( if_nat @ ( M3 = zero_zero_nat ) @ N4 @ ( if_nat @ ( N4 = zero_zero_nat ) @ M3 @ ( plus_plus_nat @ ( ord_max_nat @ ( modulo_modulo_nat @ M3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( modulo_modulo_nat @ N4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( bit_se1412395901928357646or_nat @ ( divide_divide_nat @ M3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( divide_divide_nat @ N4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) ).
% 14.25/8.15
% 14.25/8.15 thf(fact_7570_push__bit__int__def,axiom,
% 14.25/8.15 ( bit_se545348938243370406it_int
% 14.25/8.15 = ( ^ [N4: nat,K5: int] : ( times_times_int @ K5 @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N4 ) ) ) ) ).
% 14.25/8.15
% 14.25/8.15 thf(fact_7571_push__bit__nat__def,axiom,
% 14.25/8.15 ( bit_se547839408752420682it_nat
% 14.25/8.15 = ( ^ [N4: nat,M3: nat] : ( times_times_nat @ M3 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N4 ) ) ) ) ).
% 14.25/8.15
% 14.25/8.15 thf(fact_7572_set__bit__eq,axiom,
% 14.25/8.15 ( bit_se7879613467334960850it_int
% 14.25/8.15 = ( ^ [N4: nat,K5: int] :
% 14.25/8.15 ( plus_plus_int @ K5
% 14.25/8.15 @ ( times_times_int
% 14.25/8.15 @ ( zero_n2684676970156552555ol_int
% 14.25/8.15 @ ~ ( bit_se1146084159140164899it_int @ K5 @ N4 ) )
% 14.25/8.15 @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N4 ) ) ) ) ) ).
% 14.25/8.15
% 14.25/8.15 thf(fact_7573_unset__bit__eq,axiom,
% 14.25/8.15 ( bit_se4203085406695923979it_int
% 14.25/8.15 = ( ^ [N4: nat,K5: int] : ( minus_minus_int @ K5 @ ( times_times_int @ ( zero_n2684676970156552555ol_int @ ( bit_se1146084159140164899it_int @ K5 @ N4 ) ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N4 ) ) ) ) ) ).
% 14.25/8.15
% 14.25/8.15 thf(fact_7574_or__nat__rec,axiom,
% 14.25/8.15 ( bit_se1412395901928357646or_nat
% 14.25/8.15 = ( ^ [M3: nat,N4: nat] :
% 14.25/8.15 ( plus_plus_nat
% 14.25/8.15 @ ( zero_n2687167440665602831ol_nat
% 14.25/8.15 @ ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M3 )
% 14.25/8.15 | ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N4 ) ) )
% 14.25/8.15 @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( bit_se1412395901928357646or_nat @ ( divide_divide_nat @ M3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( divide_divide_nat @ N4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ).
% 14.25/8.15
% 14.25/8.15 thf(fact_7575_xor__nat__unfold,axiom,
% 14.25/8.15 ( bit_se6528837805403552850or_nat
% 14.25/8.15 = ( ^ [M3: nat,N4: nat] : ( if_nat @ ( M3 = zero_zero_nat ) @ N4 @ ( if_nat @ ( N4 = zero_zero_nat ) @ M3 @ ( plus_plus_nat @ ( modulo_modulo_nat @ ( plus_plus_nat @ ( modulo_modulo_nat @ M3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( modulo_modulo_nat @ N4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( bit_se6528837805403552850or_nat @ ( divide_divide_nat @ M3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( divide_divide_nat @ N4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) ).
% 14.25/8.15
% 14.25/8.15 thf(fact_7576_take__bit__Suc__from__most,axiom,
% 14.25/8.15 ! [N2: nat,K3: int] :
% 14.25/8.15 ( ( bit_se2923211474154528505it_int @ ( suc @ N2 ) @ K3 )
% 14.25/8.15 = ( plus_plus_int @ ( times_times_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N2 ) @ ( zero_n2684676970156552555ol_int @ ( bit_se1146084159140164899it_int @ K3 @ N2 ) ) ) @ ( bit_se2923211474154528505it_int @ N2 @ K3 ) ) ) ).
% 14.25/8.15
% 14.25/8.15 thf(fact_7577_xor__int__rec,axiom,
% 14.25/8.15 ( bit_se6526347334894502574or_int
% 14.25/8.15 = ( ^ [K5: int,L2: int] :
% 14.25/8.15 ( plus_plus_int
% 14.25/8.15 @ ( zero_n2684676970156552555ol_int
% 14.25/8.15 @ ( ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ K5 ) )
% 14.25/8.15 != ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ L2 ) ) ) )
% 14.25/8.15 @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se6526347334894502574or_int @ ( divide_divide_int @ K5 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ ( divide_divide_int @ L2 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ) ) ) ).
% 14.25/8.15
% 14.25/8.15 thf(fact_7578_xor__nat__rec,axiom,
% 14.25/8.15 ( bit_se6528837805403552850or_nat
% 14.25/8.15 = ( ^ [M3: nat,N4: nat] :
% 14.25/8.15 ( plus_plus_nat
% 14.25/8.15 @ ( zero_n2687167440665602831ol_nat
% 14.25/8.15 @ ( ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M3 ) )
% 14.25/8.15 != ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N4 ) ) ) )
% 14.25/8.15 @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( bit_se6528837805403552850or_nat @ ( divide_divide_nat @ M3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( divide_divide_nat @ N4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ).
% 14.25/8.15
% 14.25/8.15 thf(fact_7579_xor__int__unfold,axiom,
% 14.25/8.15 ( bit_se6526347334894502574or_int
% 14.25/8.15 = ( ^ [K5: int,L2: int] :
% 14.25/8.15 ( if_int
% 14.25/8.15 @ ( K5
% 14.25/8.15 = ( uminus_uminus_int @ one_one_int ) )
% 14.25/8.15 @ ( bit_ri7919022796975470100ot_int @ L2 )
% 14.25/8.15 @ ( if_int
% 14.25/8.15 @ ( L2
% 14.25/8.15 = ( uminus_uminus_int @ one_one_int ) )
% 14.25/8.15 @ ( bit_ri7919022796975470100ot_int @ K5 )
% 14.25/8.15 @ ( if_int @ ( K5 = zero_zero_int ) @ L2 @ ( if_int @ ( L2 = zero_zero_int ) @ K5 @ ( plus_plus_int @ ( abs_abs_int @ ( minus_minus_int @ ( modulo_modulo_int @ K5 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ ( modulo_modulo_int @ L2 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se6526347334894502574or_int @ ( divide_divide_int @ K5 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ ( divide_divide_int @ L2 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) ) ) ).
% 14.25/8.15
% 14.25/8.15 thf(fact_7580_not__int__rec,axiom,
% 14.25/8.15 ( bit_ri7919022796975470100ot_int
% 14.25/8.15 = ( ^ [K5: int] : ( plus_plus_int @ ( zero_n2684676970156552555ol_int @ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ K5 ) ) @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_ri7919022796975470100ot_int @ ( divide_divide_int @ K5 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ) ) ) ).
% 14.25/8.15
% 14.25/8.15 thf(fact_7581_or__not__numerals_I8_J,axiom,
% 14.25/8.15 ! [M4: num,N2: num] :
% 14.25/8.15 ( ( bit_se1409905431419307370or_int @ ( numeral_numeral_int @ ( bit1 @ M4 ) ) @ ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ ( bit0 @ N2 ) ) ) )
% 14.25/8.15 = ( plus_plus_int @ one_one_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se1409905431419307370or_int @ ( numeral_numeral_int @ M4 ) @ ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ N2 ) ) ) ) ) ) ).
% 14.25/8.15
% 14.25/8.15 thf(fact_7582_or__not__numerals_I9_J,axiom,
% 14.25/8.15 ! [M4: num,N2: num] :
% 14.25/8.15 ( ( bit_se1409905431419307370or_int @ ( numeral_numeral_int @ ( bit1 @ M4 ) ) @ ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ ( bit1 @ N2 ) ) ) )
% 14.25/8.15 = ( plus_plus_int @ one_one_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se1409905431419307370or_int @ ( numeral_numeral_int @ M4 ) @ ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ N2 ) ) ) ) ) ) ).
% 14.25/8.15
% 14.25/8.15 thf(fact_7583_and__not__numerals_I8_J,axiom,
% 14.25/8.15 ! [M4: num,N2: num] :
% 14.25/8.15 ( ( bit_se725231765392027082nd_int @ ( numeral_numeral_int @ ( bit1 @ M4 ) ) @ ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ ( bit0 @ N2 ) ) ) )
% 14.25/8.15 = ( plus_plus_int @ one_one_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se725231765392027082nd_int @ ( numeral_numeral_int @ M4 ) @ ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ N2 ) ) ) ) ) ) ).
% 14.25/8.15
% 14.25/8.15 thf(fact_7584_bool__assn__proper_I4_J,axiom,
% 14.25/8.15 ! [P: produc3658429121746597890et_nat > $o,Q2: produc3658429121746597890et_nat > $o] :
% 14.25/8.15 ( ( proper @ P )
% 14.25/8.15 => ( ( proper @ Q2 )
% 14.25/8.15 => ( proper
% 14.25/8.15 @ ^ [H4: produc3658429121746597890et_nat] :
% 14.25/8.15 ( ( P @ H4 )
% 14.25/8.15 & ( Q2 @ H4 ) ) ) ) ) ).
% 14.25/8.15
% 14.25/8.15 thf(fact_7585_bool__assn__proper_I3_J,axiom,
% 14.25/8.15 ! [P: produc3658429121746597890et_nat > $o,Q2: produc3658429121746597890et_nat > $o] :
% 14.25/8.15 ( ( proper @ P )
% 14.25/8.15 => ( ( proper @ Q2 )
% 14.25/8.15 => ( proper
% 14.25/8.15 @ ^ [H4: produc3658429121746597890et_nat] :
% 14.25/8.15 ( ( P @ H4 )
% 14.25/8.15 | ( Q2 @ H4 ) ) ) ) ) ).
% 14.25/8.15
% 14.25/8.15 thf(fact_7586_bool__assn__proper_I2_J,axiom,
% 14.25/8.15 ( proper
% 14.25/8.15 @ ^ [Uu: produc3658429121746597890et_nat] : $false ) ).
% 14.25/8.15
% 14.25/8.15 thf(fact_7587_bool__assn__proper_I5_J,axiom,
% 14.25/8.15 ! [P: produc3658429121746597890et_nat > $o] :
% 14.25/8.15 ( ( proper @ P )
% 14.25/8.15 => ( proper
% 14.25/8.15 @ ^ [H4: produc3658429121746597890et_nat] :
% 14.25/8.15 ( ( in_range @ H4 )
% 14.25/8.15 & ~ ( P @ H4 ) ) ) ) ).
% 14.25/8.15
% 14.25/8.15 thf(fact_7588_bot__assn__def,axiom,
% 14.25/8.15 ( bot_bot_assn
% 14.25/8.15 = ( abs_assn
% 14.25/8.15 @ ^ [Uu: produc3658429121746597890et_nat] : $false ) ) ).
% 14.25/8.15
% 14.25/8.15 thf(fact_7589_sup__assn__def,axiom,
% 14.25/8.15 ( sup_sup_assn
% 14.25/8.15 = ( ^ [P2: assn,Q: assn] :
% 14.25/8.15 ( abs_assn
% 14.25/8.15 @ ^ [H4: produc3658429121746597890et_nat] :
% 14.25/8.15 ( ( rep_assn @ P2 @ H4 )
% 14.25/8.15 | ( rep_assn @ Q @ H4 ) ) ) ) ) ).
% 14.25/8.15
% 14.25/8.15 thf(fact_7590_inf__assn__def,axiom,
% 14.25/8.15 ( inf_inf_assn
% 14.25/8.15 = ( ^ [P2: assn,Q: assn] :
% 14.25/8.15 ( abs_assn
% 14.25/8.15 @ ^ [H4: produc3658429121746597890et_nat] :
% 14.25/8.15 ( ( rep_assn @ P2 @ H4 )
% 14.25/8.15 & ( rep_assn @ Q @ H4 ) ) ) ) ) ).
% 14.25/8.15
% 14.25/8.15 thf(fact_7591_nat__times__as__int,axiom,
% 14.25/8.15 ( times_times_nat
% 14.25/8.15 = ( ^ [A6: nat,B6: nat] : ( nat2 @ ( times_times_int @ ( semiri1314217659103216013at_int @ A6 ) @ ( semiri1314217659103216013at_int @ B6 ) ) ) ) ) ).
% 14.25/8.15
% 14.25/8.15 thf(fact_7592_divide__nat__def,axiom,
% 14.25/8.15 ( divide_divide_nat
% 14.25/8.15 = ( ^ [M3: nat,N4: nat] :
% 14.25/8.15 ( if_nat @ ( N4 = zero_zero_nat ) @ zero_zero_nat
% 14.25/8.15 @ ( lattic8265883725875713057ax_nat
% 14.25/8.15 @ ( collect_nat
% 14.25/8.15 @ ^ [K5: nat] : ( ord_less_eq_nat @ ( times_times_nat @ K5 @ N4 ) @ M3 ) ) ) ) ) ) ).
% 14.25/8.15
% 14.25/8.15 thf(fact_7593_uminus__assn__def,axiom,
% 14.25/8.15 ( uminus_uminus_assn
% 14.25/8.15 = ( ^ [P2: assn] :
% 14.25/8.15 ( abs_assn
% 14.25/8.15 @ ^ [H4: produc3658429121746597890et_nat] :
% 14.25/8.15 ( ( in_range @ H4 )
% 14.25/8.15 & ~ ( rep_assn @ P2 @ H4 ) ) ) ) ) ).
% 14.25/8.15
% 14.25/8.15 thf(fact_7594_Gcd__eq__Max,axiom,
% 14.25/8.15 ! [M: set_nat] :
% 14.25/8.15 ( ( finite_finite_nat @ M )
% 14.25/8.15 => ( ( M != bot_bot_set_nat )
% 14.25/8.15 => ( ~ ( member_nat @ zero_zero_nat @ M )
% 14.25/8.15 => ( ( gcd_Gcd_nat @ M )
% 14.25/8.15 = ( lattic8265883725875713057ax_nat
% 14.25/8.15 @ ( comple7806235888213564991et_nat
% 14.25/8.15 @ ( image_nat_set_nat
% 14.25/8.15 @ ^ [M3: nat] :
% 14.25/8.15 ( collect_nat
% 14.25/8.15 @ ^ [D4: nat] : ( dvd_dvd_nat @ D4 @ M3 ) )
% 14.25/8.15 @ M ) ) ) ) ) ) ) ).
% 14.25/8.15
% 14.25/8.15 thf(fact_7595_and__not__numerals_I5_J,axiom,
% 14.25/8.15 ! [M4: num,N2: num] :
% 14.25/8.15 ( ( bit_se725231765392027082nd_int @ ( numeral_numeral_int @ ( bit0 @ M4 ) ) @ ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ ( bit0 @ N2 ) ) ) )
% 14.25/8.15 = ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se725231765392027082nd_int @ ( numeral_numeral_int @ M4 ) @ ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ N2 ) ) ) ) ) ).
% 14.25/8.15
% 14.25/8.15 thf(fact_7596_and__not__numerals_I6_J,axiom,
% 14.25/8.15 ! [M4: num,N2: num] :
% 14.25/8.15 ( ( bit_se725231765392027082nd_int @ ( numeral_numeral_int @ ( bit0 @ M4 ) ) @ ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ ( bit1 @ N2 ) ) ) )
% 14.25/8.15 = ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se725231765392027082nd_int @ ( numeral_numeral_int @ M4 ) @ ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ N2 ) ) ) ) ) ).
% 14.25/8.15
% 14.25/8.15 thf(fact_7597_and__not__numerals_I9_J,axiom,
% 14.25/8.15 ! [M4: num,N2: num] :
% 14.25/8.15 ( ( bit_se725231765392027082nd_int @ ( numeral_numeral_int @ ( bit1 @ M4 ) ) @ ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ ( bit1 @ N2 ) ) ) )
% 14.25/8.15 = ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se725231765392027082nd_int @ ( numeral_numeral_int @ M4 ) @ ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ N2 ) ) ) ) ) ).
% 14.25/8.15
% 14.25/8.15 thf(fact_7598_or__not__numerals_I6_J,axiom,
% 14.25/8.15 ! [M4: num,N2: num] :
% 14.25/8.15 ( ( bit_se1409905431419307370or_int @ ( numeral_numeral_int @ ( bit0 @ M4 ) ) @ ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ ( bit1 @ N2 ) ) ) )
% 14.25/8.15 = ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se1409905431419307370or_int @ ( numeral_numeral_int @ M4 ) @ ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ N2 ) ) ) ) ) ).
% 14.25/8.15
% 14.25/8.15 thf(fact_7599_or__not__numerals_I5_J,axiom,
% 14.25/8.15 ! [M4: num,N2: num] :
% 14.25/8.15 ( ( bit_se1409905431419307370or_int @ ( numeral_numeral_int @ ( bit0 @ M4 ) ) @ ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ ( bit0 @ N2 ) ) ) )
% 14.25/8.15 = ( plus_plus_int @ one_one_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se1409905431419307370or_int @ ( numeral_numeral_int @ M4 ) @ ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ N2 ) ) ) ) ) ) ).
% 14.25/8.15
% 14.25/8.15 thf(fact_7600_atMost__0,axiom,
% 14.25/8.15 ( ( set_ord_atMost_nat @ zero_zero_nat )
% 14.25/8.15 = ( insert_nat @ zero_zero_nat @ bot_bot_set_nat ) ) ).
% 14.25/8.15
% 14.25/8.15 thf(fact_7601_atLeast1__atMost__eq__remove0,axiom,
% 14.25/8.15 ! [N2: nat] :
% 14.25/8.15 ( ( set_or1269000886237332187st_nat @ ( suc @ zero_zero_nat ) @ N2 )
% 14.25/8.15 = ( minus_minus_set_nat @ ( set_ord_atMost_nat @ N2 ) @ ( insert_nat @ zero_zero_nat @ bot_bot_set_nat ) ) ) ).
% 14.25/8.15
% 14.25/8.15 thf(fact_7602_vandermonde,axiom,
% 14.25/8.15 ! [M4: nat,N2: nat,R2: nat] :
% 14.25/8.15 ( ( groups3542108847815614940at_nat
% 14.25/8.15 @ ^ [K5: nat] : ( times_times_nat @ ( binomial @ M4 @ K5 ) @ ( binomial @ N2 @ ( minus_minus_nat @ R2 @ K5 ) ) )
% 14.25/8.15 @ ( set_ord_atMost_nat @ R2 ) )
% 14.25/8.15 = ( binomial @ ( plus_plus_nat @ M4 @ N2 ) @ R2 ) ) ).
% 14.25/8.15
% 14.25/8.15 thf(fact_7603_fact__eq__fact__times,axiom,
% 14.25/8.15 ! [N2: nat,M4: nat] :
% 14.25/8.15 ( ( ord_less_eq_nat @ N2 @ M4 )
% 14.25/8.15 => ( ( semiri1408675320244567234ct_nat @ M4 )
% 14.25/8.15 = ( times_times_nat @ ( semiri1408675320244567234ct_nat @ N2 )
% 14.25/8.15 @ ( groups708209901874060359at_nat
% 14.25/8.15 @ ^ [X2: nat] : X2
% 14.25/8.15 @ ( set_or1269000886237332187st_nat @ ( suc @ N2 ) @ M4 ) ) ) ) ) ).
% 14.25/8.15
% 14.25/8.15 thf(fact_7604_binomial,axiom,
% 14.25/8.15 ! [A2: nat,B: nat,N2: nat] :
% 14.25/8.15 ( ( power_power_nat @ ( plus_plus_nat @ A2 @ B ) @ N2 )
% 14.25/8.15 = ( groups3542108847815614940at_nat
% 14.25/8.15 @ ^ [K5: nat] : ( times_times_nat @ ( times_times_nat @ ( semiri1316708129612266289at_nat @ ( binomial @ N2 @ K5 ) ) @ ( power_power_nat @ A2 @ K5 ) ) @ ( power_power_nat @ B @ ( minus_minus_nat @ N2 @ K5 ) ) )
% 14.25/8.15 @ ( set_ord_atMost_nat @ N2 ) ) ) ).
% 14.25/8.15
% 14.25/8.15 thf(fact_7605_choose__square__sum,axiom,
% 14.25/8.15 ! [N2: nat] :
% 14.25/8.15 ( ( groups3542108847815614940at_nat
% 14.25/8.15 @ ^ [K5: nat] : ( power_power_nat @ ( binomial @ N2 @ K5 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 14.25/8.15 @ ( set_ord_atMost_nat @ N2 ) )
% 14.25/8.15 = ( binomial @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) @ N2 ) ) ).
% 14.25/8.15
% 14.25/8.15 thf(fact_7606_gauss__sum__nat,axiom,
% 14.25/8.15 ! [N2: nat] :
% 14.25/8.15 ( ( groups3542108847815614940at_nat
% 14.25/8.15 @ ^ [X2: nat] : X2
% 14.25/8.15 @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N2 ) )
% 14.25/8.15 = ( divide_divide_nat @ ( times_times_nat @ N2 @ ( suc @ N2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% 14.25/8.15
% 14.25/8.15 thf(fact_7607_binomial__r__part__sum,axiom,
% 14.25/8.15 ! [M4: nat] :
% 14.25/8.15 ( ( groups3542108847815614940at_nat @ ( binomial @ ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M4 ) @ one_one_nat ) ) @ ( set_ord_atMost_nat @ M4 ) )
% 14.25/8.15 = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M4 ) ) ) ).
% 14.25/8.15
% 14.25/8.15 thf(fact_7608_choose__linear__sum,axiom,
% 14.25/8.15 ! [N2: nat] :
% 14.25/8.15 ( ( groups3542108847815614940at_nat
% 14.25/8.15 @ ^ [I4: nat] : ( times_times_nat @ I4 @ ( binomial @ N2 @ I4 ) )
% 14.25/8.15 @ ( set_ord_atMost_nat @ N2 ) )
% 14.25/8.15 = ( times_times_nat @ N2 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( minus_minus_nat @ N2 @ one_one_nat ) ) ) ) ).
% 14.25/8.15
% 14.25/8.15 thf(fact_7609_arith__series__nat,axiom,
% 14.25/8.15 ! [A2: nat,D2: nat,N2: nat] :
% 14.25/8.15 ( ( groups3542108847815614940at_nat
% 14.25/8.15 @ ^ [I4: nat] : ( plus_plus_nat @ A2 @ ( times_times_nat @ I4 @ D2 ) )
% 14.25/8.15 @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N2 ) )
% 14.25/8.15 = ( divide_divide_nat @ ( times_times_nat @ ( suc @ N2 ) @ ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A2 ) @ ( times_times_nat @ N2 @ D2 ) ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% 14.25/8.15
% 14.25/8.15 thf(fact_7610_Sum__Icc__nat,axiom,
% 14.25/8.15 ! [M4: nat,N2: nat] :
% 14.25/8.15 ( ( groups3542108847815614940at_nat
% 14.25/8.15 @ ^ [X2: nat] : X2
% 14.25/8.15 @ ( set_or1269000886237332187st_nat @ M4 @ N2 ) )
% 14.25/8.15 = ( divide_divide_nat @ ( minus_minus_nat @ ( times_times_nat @ N2 @ ( plus_plus_nat @ N2 @ one_one_nat ) ) @ ( times_times_nat @ M4 @ ( minus_minus_nat @ M4 @ one_one_nat ) ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% 14.25/8.15
% 14.25/8.15 thf(fact_7611_periodic__finite__ex,axiom,
% 14.25/8.15 ! [D2: int,P: int > $o] :
% 14.25/8.15 ( ( ord_less_int @ zero_zero_int @ D2 )
% 14.25/8.15 => ( ! [X3: int,K: int] :
% 14.25/8.15 ( ( P @ X3 )
% 14.25/8.15 = ( P @ ( minus_minus_int @ X3 @ ( times_times_int @ K @ D2 ) ) ) )
% 14.25/8.15 => ( ( ? [X7: int] : ( P @ X7 ) )
% 14.25/8.15 = ( ? [X2: int] :
% 14.25/8.15 ( ( member_int @ X2 @ ( set_or1266510415728281911st_int @ one_one_int @ D2 ) )
% 14.25/8.15 & ( P @ X2 ) ) ) ) ) ) ).
% 14.25/8.15
% 14.25/8.15 thf(fact_7612_simp__from__to,axiom,
% 14.25/8.15 ( set_or1266510415728281911st_int
% 14.25/8.15 = ( ^ [I4: int,J3: int] : ( if_set_int @ ( ord_less_int @ J3 @ I4 ) @ bot_bot_set_int @ ( insert_int @ I4 @ ( set_or1266510415728281911st_int @ ( plus_plus_int @ I4 @ one_one_int ) @ J3 ) ) ) ) ) ).
% 14.25/8.15
% 14.25/8.15 thf(fact_7613_cpmi,axiom,
% 14.25/8.15 ! [D3: int,P: int > $o,P7: int > $o,B4: set_int] :
% 14.25/8.15 ( ( ord_less_int @ zero_zero_int @ D3 )
% 14.25/8.15 => ( ? [Z6: int] :
% 14.25/8.15 ! [X3: int] :
% 14.25/8.15 ( ( ord_less_int @ X3 @ Z6 )
% 14.25/8.15 => ( ( P @ X3 )
% 14.25/8.15 = ( P7 @ X3 ) ) )
% 14.25/8.15 => ( ! [X3: int] :
% 14.25/8.15 ( ! [Xa2: int] :
% 14.25/8.15 ( ( member_int @ Xa2 @ ( set_or1266510415728281911st_int @ one_one_int @ D3 ) )
% 14.25/8.15 => ! [Xb2: int] :
% 14.25/8.15 ( ( member_int @ Xb2 @ B4 )
% 14.25/8.15 => ( X3
% 14.25/8.15 != ( plus_plus_int @ Xb2 @ Xa2 ) ) ) )
% 14.25/8.15 => ( ( P @ X3 )
% 14.25/8.15 => ( P @ ( minus_minus_int @ X3 @ D3 ) ) ) )
% 14.25/8.15 => ( ! [X3: int,K: int] :
% 14.25/8.15 ( ( P7 @ X3 )
% 14.25/8.15 = ( P7 @ ( minus_minus_int @ X3 @ ( times_times_int @ K @ D3 ) ) ) )
% 14.25/8.15 => ( ( ? [X7: int] : ( P @ X7 ) )
% 14.25/8.15 = ( ? [X2: int] :
% 14.25/8.15 ( ( member_int @ X2 @ ( set_or1266510415728281911st_int @ one_one_int @ D3 ) )
% 14.25/8.15 & ( P7 @ X2 ) )
% 14.25/8.15 | ? [X2: int] :
% 14.25/8.15 ( ( member_int @ X2 @ ( set_or1266510415728281911st_int @ one_one_int @ D3 ) )
% 14.25/8.15 & ? [Y3: int] :
% 14.25/8.15 ( ( member_int @ Y3 @ B4 )
% 14.25/8.15 & ( P @ ( plus_plus_int @ Y3 @ X2 ) ) ) ) ) ) ) ) ) ) ).
% 14.25/8.15
% 14.25/8.15 thf(fact_7614_cppi,axiom,
% 14.25/8.15 ! [D3: int,P: int > $o,P7: int > $o,A: set_int] :
% 14.25/8.15 ( ( ord_less_int @ zero_zero_int @ D3 )
% 14.25/8.15 => ( ? [Z6: int] :
% 14.25/8.15 ! [X3: int] :
% 14.25/8.15 ( ( ord_less_int @ Z6 @ X3 )
% 14.25/8.15 => ( ( P @ X3 )
% 14.25/8.15 = ( P7 @ X3 ) ) )
% 14.25/8.15 => ( ! [X3: int] :
% 14.25/8.15 ( ! [Xa2: int] :
% 14.25/8.15 ( ( member_int @ Xa2 @ ( set_or1266510415728281911st_int @ one_one_int @ D3 ) )
% 14.25/8.15 => ! [Xb2: int] :
% 14.25/8.15 ( ( member_int @ Xb2 @ A )
% 14.25/8.15 => ( X3
% 14.25/8.15 != ( minus_minus_int @ Xb2 @ Xa2 ) ) ) )
% 14.25/8.15 => ( ( P @ X3 )
% 14.25/8.15 => ( P @ ( plus_plus_int @ X3 @ D3 ) ) ) )
% 14.25/8.15 => ( ! [X3: int,K: int] :
% 14.25/8.15 ( ( P7 @ X3 )
% 14.25/8.15 = ( P7 @ ( minus_minus_int @ X3 @ ( times_times_int @ K @ D3 ) ) ) )
% 14.25/8.15 => ( ( ? [X7: int] : ( P @ X7 ) )
% 14.25/8.15 = ( ? [X2: int] :
% 14.25/8.15 ( ( member_int @ X2 @ ( set_or1266510415728281911st_int @ one_one_int @ D3 ) )
% 14.25/8.15 & ( P7 @ X2 ) )
% 14.25/8.15 | ? [X2: int] :
% 14.25/8.15 ( ( member_int @ X2 @ ( set_or1266510415728281911st_int @ one_one_int @ D3 ) )
% 14.25/8.15 & ? [Y3: int] :
% 14.25/8.15 ( ( member_int @ Y3 @ A )
% 14.25/8.15 & ( P @ ( minus_minus_int @ Y3 @ X2 ) ) ) ) ) ) ) ) ) ) ).
% 14.25/8.15
% 14.25/8.15 thf(fact_7615_divmod__step__nat__def,axiom,
% 14.25/8.15 ( unique5026877609467782581ep_nat
% 14.25/8.15 = ( ^ [L2: num] :
% 14.25/8.15 ( produc2626176000494625587at_nat
% 14.25/8.15 @ ^ [Q6: nat,R4: nat] : ( if_Pro6206227464963214023at_nat @ ( ord_less_eq_nat @ ( numeral_numeral_nat @ L2 ) @ R4 ) @ ( product_Pair_nat_nat @ ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Q6 ) @ one_one_nat ) @ ( minus_minus_nat @ R4 @ ( numeral_numeral_nat @ L2 ) ) ) @ ( product_Pair_nat_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Q6 ) @ R4 ) ) ) ) ) ).
% 14.25/8.15
% 14.25/8.15 thf(fact_7616_divmod__step__int__def,axiom,
% 14.25/8.15 ( unique5024387138958732305ep_int
% 14.25/8.15 = ( ^ [L2: num] :
% 14.25/8.15 ( produc4245557441103728435nt_int
% 14.25/8.15 @ ^ [Q6: int,R4: int] : ( if_Pro3027730157355071871nt_int @ ( ord_less_eq_int @ ( numeral_numeral_int @ L2 ) @ R4 ) @ ( product_Pair_int_int @ ( plus_plus_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ Q6 ) @ one_one_int ) @ ( minus_minus_int @ R4 @ ( numeral_numeral_int @ L2 ) ) ) @ ( product_Pair_int_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ Q6 ) @ R4 ) ) ) ) ) ).
% 14.25/8.15
% 14.25/8.15 thf(fact_7617_Sum__Icc__int,axiom,
% 14.25/8.15 ! [M4: int,N2: int] :
% 14.25/8.15 ( ( ord_less_eq_int @ M4 @ N2 )
% 14.25/8.15 => ( ( groups4538972089207619220nt_int
% 14.25/8.15 @ ^ [X2: int] : X2
% 14.25/8.15 @ ( set_or1266510415728281911st_int @ M4 @ N2 ) )
% 14.25/8.15 = ( divide_divide_int @ ( minus_minus_int @ ( times_times_int @ N2 @ ( plus_plus_int @ N2 @ one_one_int ) ) @ ( times_times_int @ M4 @ ( minus_minus_int @ M4 @ one_one_int ) ) ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ).
% 14.25/8.15
% 14.25/8.15 thf(fact_7618_Sum__Ico__nat,axiom,
% 14.25/8.15 ! [M4: nat,N2: nat] :
% 14.25/8.15 ( ( groups3542108847815614940at_nat
% 14.25/8.15 @ ^ [X2: nat] : X2
% 14.25/8.15 @ ( set_or4665077453230672383an_nat @ M4 @ N2 ) )
% 14.25/8.15 = ( divide_divide_nat @ ( minus_minus_nat @ ( times_times_nat @ N2 @ ( minus_minus_nat @ N2 @ one_one_nat ) ) @ ( times_times_nat @ M4 @ ( minus_minus_nat @ M4 @ one_one_nat ) ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% 14.25/8.15
% 14.25/8.15 thf(fact_7619_atLeastLessThan__singleton,axiom,
% 14.25/8.15 ! [M4: nat] :
% 14.25/8.15 ( ( set_or4665077453230672383an_nat @ M4 @ ( suc @ M4 ) )
% 14.25/8.15 = ( insert_nat @ M4 @ bot_bot_set_nat ) ) ).
% 14.25/8.15
% 14.25/8.15 thf(fact_7620_atLeastLessThan0,axiom,
% 14.25/8.15 ! [M4: nat] :
% 14.25/8.15 ( ( set_or4665077453230672383an_nat @ M4 @ zero_zero_nat )
% 14.25/8.15 = bot_bot_set_nat ) ).
% 14.25/8.15
% 14.25/8.15 thf(fact_7621_atLeastLessThanSuc,axiom,
% 14.25/8.15 ! [M4: nat,N2: nat] :
% 14.25/8.15 ( ( ( ord_less_eq_nat @ M4 @ N2 )
% 14.25/8.15 => ( ( set_or4665077453230672383an_nat @ M4 @ ( suc @ N2 ) )
% 14.25/8.15 = ( insert_nat @ N2 @ ( set_or4665077453230672383an_nat @ M4 @ N2 ) ) ) )
% 14.25/8.15 & ( ~ ( ord_less_eq_nat @ M4 @ N2 )
% 14.25/8.15 => ( ( set_or4665077453230672383an_nat @ M4 @ ( suc @ N2 ) )
% 14.25/8.15 = bot_bot_set_nat ) ) ) ).
% 14.25/8.15
% 14.25/8.15 thf(fact_7622_atLeastLessThan__nat__numeral,axiom,
% 14.25/8.15 ! [M4: nat,K3: num] :
% 14.25/8.15 ( ( ( ord_less_eq_nat @ M4 @ ( pred_numeral @ K3 ) )
% 14.25/8.15 => ( ( set_or4665077453230672383an_nat @ M4 @ ( numeral_numeral_nat @ K3 ) )
% 14.25/8.15 = ( insert_nat @ ( pred_numeral @ K3 ) @ ( set_or4665077453230672383an_nat @ M4 @ ( pred_numeral @ K3 ) ) ) ) )
% 14.25/8.15 & ( ~ ( ord_less_eq_nat @ M4 @ ( pred_numeral @ K3 ) )
% 14.25/8.15 => ( ( set_or4665077453230672383an_nat @ M4 @ ( numeral_numeral_nat @ K3 ) )
% 14.25/8.15 = bot_bot_set_nat ) ) ) ).
% 14.25/8.15
% 14.25/8.15 thf(fact_7623_image__minus__const__atLeastLessThan__nat,axiom,
% 14.25/8.15 ! [C: nat,Y: nat,X: nat] :
% 14.25/8.15 ( ( ( ord_less_nat @ C @ Y )
% 14.25/8.15 => ( ( image_nat_nat2
% 14.25/8.15 @ ^ [I4: nat] : ( minus_minus_nat @ I4 @ C )
% 14.25/8.15 @ ( set_or4665077453230672383an_nat @ X @ Y ) )
% 14.25/8.15 = ( set_or4665077453230672383an_nat @ ( minus_minus_nat @ X @ C ) @ ( minus_minus_nat @ Y @ C ) ) ) )
% 14.25/8.15 & ( ~ ( ord_less_nat @ C @ Y )
% 14.25/8.15 => ( ( ( ord_less_nat @ X @ Y )
% 14.25/8.15 => ( ( image_nat_nat2
% 14.25/8.15 @ ^ [I4: nat] : ( minus_minus_nat @ I4 @ C )
% 14.25/8.15 @ ( set_or4665077453230672383an_nat @ X @ Y ) )
% 14.25/8.15 = ( insert_nat @ zero_zero_nat @ bot_bot_set_nat ) ) )
% 14.25/8.15 & ( ~ ( ord_less_nat @ X @ Y )
% 14.25/8.15 => ( ( image_nat_nat2
% 14.25/8.15 @ ^ [I4: nat] : ( minus_minus_nat @ I4 @ C )
% 14.25/8.15 @ ( set_or4665077453230672383an_nat @ X @ Y ) )
% 14.25/8.15 = bot_bot_set_nat ) ) ) ) ) ).
% 14.25/8.15
% 14.25/8.15 thf(fact_7624_divmod__step__integer__def,axiom,
% 14.25/8.15 ( unique4921790084139445826nteger
% 14.25/8.15 = ( ^ [L2: num] :
% 14.25/8.15 ( produc6916734918728496179nteger
% 14.25/8.15 @ ^ [Q6: code_integer,R4: code_integer] : ( if_Pro6119634080678213985nteger @ ( ord_le3102999989581377725nteger @ ( numera6620942414471956472nteger @ L2 ) @ R4 ) @ ( produc1086072967326762835nteger @ ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ Q6 ) @ one_one_Code_integer ) @ ( minus_8373710615458151222nteger @ R4 @ ( numera6620942414471956472nteger @ L2 ) ) ) @ ( produc1086072967326762835nteger @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ Q6 ) @ R4 ) ) ) ) ) ).
% 14.25/8.15
% 14.25/8.15 thf(fact_7625_triangle__def,axiom,
% 14.25/8.15 ( nat_triangle
% 14.25/8.15 = ( ^ [N4: nat] : ( divide_divide_nat @ ( times_times_nat @ N4 @ ( suc @ N4 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% 14.25/8.15
% 14.25/8.15 thf(fact_7626_image__add__integer__atLeastLessThan,axiom,
% 14.25/8.15 ! [L: code_integer,U: code_integer] :
% 14.25/8.15 ( ( image_4470545334726330049nteger
% 14.25/8.15 @ ^ [X2: code_integer] : ( plus_p5714425477246183910nteger @ X2 @ L )
% 14.25/8.15 @ ( set_or8404916559141939852nteger @ zero_z3403309356797280102nteger @ ( minus_8373710615458151222nteger @ U @ L ) ) )
% 14.25/8.15 = ( set_or8404916559141939852nteger @ L @ U ) ) ).
% 14.25/8.15
% 14.25/8.15 thf(fact_7627_atLeastLessThanPlusOne__atLeastAtMost__integer,axiom,
% 14.25/8.15 ! [L: code_integer,U: code_integer] :
% 14.25/8.15 ( ( set_or8404916559141939852nteger @ L @ ( plus_p5714425477246183910nteger @ U @ one_one_Code_integer ) )
% 14.25/8.15 = ( set_or189985376899183464nteger @ L @ U ) ) ).
% 14.25/8.15
% 14.25/8.15 thf(fact_7628_times__integer__code_I2_J,axiom,
% 14.25/8.15 ! [L: code_integer] :
% 14.25/8.15 ( ( times_3573771949741848930nteger @ zero_z3403309356797280102nteger @ L )
% 14.25/8.15 = zero_z3403309356797280102nteger ) ).
% 14.25/8.15
% 14.25/8.15 thf(fact_7629_times__integer__code_I1_J,axiom,
% 14.25/8.15 ! [K3: code_integer] :
% 14.25/8.15 ( ( times_3573771949741848930nteger @ K3 @ zero_z3403309356797280102nteger )
% 14.25/8.15 = zero_z3403309356797280102nteger ) ).
% 14.25/8.15
% 14.25/8.15 thf(fact_7630_integer__of__int__code,axiom,
% 14.25/8.15 ( code_integer_of_int
% 14.25/8.15 = ( ^ [K5: int] :
% 14.25/8.15 ( if_Code_integer @ ( ord_less_int @ K5 @ zero_zero_int ) @ ( uminus1351360451143612070nteger @ ( code_integer_of_int @ ( uminus_uminus_int @ K5 ) ) )
% 14.25/8.15 @ ( if_Code_integer @ ( K5 = zero_zero_int ) @ zero_z3403309356797280102nteger
% 14.25/8.15 @ ( if_Code_integer
% 14.25/8.15 @ ( ( modulo_modulo_int @ K5 @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
% 14.25/8.15 = zero_zero_int )
% 14.25/8.15 @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( code_integer_of_int @ ( divide_divide_int @ K5 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) )
% 14.25/8.15 @ ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( code_integer_of_int @ ( divide_divide_int @ K5 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) @ one_one_Code_integer ) ) ) ) ) ) ).
% 14.25/8.15
% 14.25/8.15 thf(fact_7631_finite__atLeastLessThan__integer,axiom,
% 14.25/8.15 ! [L: code_integer,U: code_integer] : ( finite6017078050557962740nteger @ ( set_or8404916559141939852nteger @ L @ U ) ) ).
% 14.25/8.15
% 14.25/8.15 thf(fact_7632_finite__atLeastAtMost__integer,axiom,
% 14.25/8.15 ! [L: code_integer,U: code_integer] : ( finite6017078050557962740nteger @ ( set_or189985376899183464nteger @ L @ U ) ) ).
% 14.25/8.15
% 14.25/8.15 thf(fact_7633_finite__atLeastZeroLessThan__integer,axiom,
% 14.25/8.15 ! [U: code_integer] : ( finite6017078050557962740nteger @ ( set_or8404916559141939852nteger @ zero_z3403309356797280102nteger @ U ) ) ).
% 14.25/8.15
% 14.25/8.15 thf(fact_7634_times__integer_Orsp,axiom,
% 14.25/8.15 ( bNF_re711492959462206631nt_int
% 14.25/8.15 @ ^ [Y4: int,Z4: int] : ( Y4 = Z4 )
% 14.25/8.15 @ ( bNF_re4712519889275205905nt_int
% 14.25/8.15 @ ^ [Y4: int,Z4: int] : ( Y4 = Z4 )
% 14.25/8.15 @ ^ [Y4: int,Z4: int] : ( Y4 = Z4 ) )
% 14.25/8.15 @ times_times_int
% 14.25/8.15 @ times_times_int ) ).
% 14.25/8.15
% 14.25/8.15 thf(fact_7635_times__natural_Orsp,axiom,
% 14.25/8.15 ( bNF_re1345281282404953727at_nat
% 14.25/8.15 @ ^ [Y4: nat,Z4: nat] : ( Y4 = Z4 )
% 14.25/8.15 @ ( bNF_re5653821019739307937at_nat
% 14.25/8.15 @ ^ [Y4: nat,Z4: nat] : ( Y4 = Z4 )
% 14.25/8.15 @ ^ [Y4: nat,Z4: nat] : ( Y4 = Z4 ) )
% 14.25/8.15 @ times_times_nat
% 14.25/8.15 @ times_times_nat ) ).
% 14.25/8.15
% 14.25/8.15 thf(fact_7636_times__integer_Oabs__eq,axiom,
% 14.25/8.15 ! [Xa: int,X: int] :
% 14.25/8.15 ( ( times_3573771949741848930nteger @ ( code_integer_of_int @ Xa ) @ ( code_integer_of_int @ X ) )
% 14.25/8.15 = ( code_integer_of_int @ ( times_times_int @ Xa @ X ) ) ) ).
% 14.25/8.15
% 14.25/8.15 thf(fact_7637_divmod__integer__eq__cases,axiom,
% 14.25/8.15 ( code_divmod_integer
% 14.25/8.15 = ( ^ [K5: code_integer,L2: code_integer] :
% 14.25/8.15 ( if_Pro6119634080678213985nteger @ ( K5 = zero_z3403309356797280102nteger ) @ ( produc1086072967326762835nteger @ zero_z3403309356797280102nteger @ zero_z3403309356797280102nteger )
% 14.25/8.15 @ ( if_Pro6119634080678213985nteger @ ( L2 = zero_z3403309356797280102nteger ) @ ( produc1086072967326762835nteger @ zero_z3403309356797280102nteger @ K5 )
% 14.25/8.15 @ ( comp_C1593894019821074884nteger @ ( comp_C8797469213163452608nteger @ produc6499014454317279255nteger @ times_3573771949741848930nteger ) @ sgn_sgn_Code_integer @ L2
% 14.25/8.15 @ ( if_Pro6119634080678213985nteger
% 14.25/8.15 @ ( ( sgn_sgn_Code_integer @ K5 )
% 14.25/8.15 = ( sgn_sgn_Code_integer @ L2 ) )
% 14.25/8.15 @ ( code_divmod_abs @ K5 @ L2 )
% 14.25/8.15 @ ( produc6916734918728496179nteger
% 14.25/8.15 @ ^ [R4: code_integer,S7: code_integer] : ( if_Pro6119634080678213985nteger @ ( S7 = zero_z3403309356797280102nteger ) @ ( produc1086072967326762835nteger @ ( uminus1351360451143612070nteger @ R4 ) @ zero_z3403309356797280102nteger ) @ ( produc1086072967326762835nteger @ ( minus_8373710615458151222nteger @ ( uminus1351360451143612070nteger @ R4 ) @ one_one_Code_integer ) @ ( minus_8373710615458151222nteger @ ( abs_abs_Code_integer @ L2 ) @ S7 ) ) )
% 14.25/8.15 @ ( code_divmod_abs @ K5 @ L2 ) ) ) ) ) ) ) ) ).
% 14.25/8.15
% 14.25/8.15 thf(fact_7638_plus__rat_Otransfer,axiom,
% 14.25/8.15 ( bNF_re7627151682743391978at_rat @ pcr_rat @ ( bNF_re8279943556446156061nt_rat @ pcr_rat @ pcr_rat )
% 14.25/8.15 @ ^ [X2: product_prod_int_int,Y3: product_prod_int_int] : ( product_Pair_int_int @ ( plus_plus_int @ ( times_times_int @ ( product_fst_int_int @ X2 ) @ ( product_snd_int_int @ Y3 ) ) @ ( times_times_int @ ( product_fst_int_int @ Y3 ) @ ( product_snd_int_int @ X2 ) ) ) @ ( times_times_int @ ( product_snd_int_int @ X2 ) @ ( product_snd_int_int @ Y3 ) ) )
% 14.25/8.15 @ plus_plus_rat ) ).
% 14.25/8.15
% 14.25/8.15 thf(fact_7639_times__rat_Otransfer,axiom,
% 14.25/8.15 ( bNF_re7627151682743391978at_rat @ pcr_rat @ ( bNF_re8279943556446156061nt_rat @ pcr_rat @ pcr_rat )
% 14.25/8.15 @ ^ [X2: product_prod_int_int,Y3: product_prod_int_int] : ( product_Pair_int_int @ ( times_times_int @ ( product_fst_int_int @ X2 ) @ ( product_fst_int_int @ Y3 ) ) @ ( times_times_int @ ( product_snd_int_int @ X2 ) @ ( product_snd_int_int @ Y3 ) ) )
% 14.25/8.15 @ times_times_rat ) ).
% 14.25/8.15
% 14.25/8.15 thf(fact_7640_int__of__integer__code,axiom,
% 14.25/8.15 ( code_int_of_integer
% 14.25/8.15 = ( ^ [K5: code_integer] :
% 14.25/8.15 ( if_int @ ( ord_le6747313008572928689nteger @ K5 @ zero_z3403309356797280102nteger ) @ ( uminus_uminus_int @ ( code_int_of_integer @ ( uminus1351360451143612070nteger @ K5 ) ) )
% 14.25/8.15 @ ( if_int @ ( K5 = zero_z3403309356797280102nteger ) @ zero_zero_int
% 14.25/8.15 @ ( produc1553301316500091796er_int
% 14.25/8.15 @ ^ [L2: code_integer,J3: code_integer] : ( if_int @ ( J3 = zero_z3403309356797280102nteger ) @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( code_int_of_integer @ L2 ) ) @ ( plus_plus_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( code_int_of_integer @ L2 ) ) @ one_one_int ) )
% 14.25/8.15 @ ( code_divmod_integer @ K5 @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) ) ) ) ) ).
% 14.25/8.15
% 14.25/8.15 thf(fact_7641_positive_Otransfer,axiom,
% 14.25/8.15 ( bNF_re1494630372529172596at_o_o @ pcr_rat
% 14.25/8.15 @ ^ [Y4: $o,Z4: $o] : ( Y4 = Z4 )
% 14.25/8.15 @ ^ [X2: product_prod_int_int] : ( ord_less_int @ zero_zero_int @ ( times_times_int @ ( product_fst_int_int @ X2 ) @ ( product_snd_int_int @ X2 ) ) )
% 14.25/8.15 @ positive ) ).
% 14.25/8.15
% 14.25/8.15 thf(fact_7642_times__int_Otransfer,axiom,
% 14.25/8.15 ( bNF_re7408651293131936558nt_int @ pcr_int @ ( bNF_re7400052026677387805at_int @ pcr_int @ pcr_int )
% 14.25/8.15 @ ( produc27273713700761075at_nat
% 14.25/8.15 @ ^ [X2: nat,Y3: nat] :
% 14.25/8.15 ( produc2626176000494625587at_nat
% 14.25/8.15 @ ^ [U3: nat,V4: nat] : ( product_Pair_nat_nat @ ( plus_plus_nat @ ( times_times_nat @ X2 @ U3 ) @ ( times_times_nat @ Y3 @ V4 ) ) @ ( plus_plus_nat @ ( times_times_nat @ X2 @ V4 ) @ ( times_times_nat @ Y3 @ U3 ) ) ) ) )
% 14.25/8.15 @ times_times_int ) ).
% 14.25/8.15
% 14.25/8.15 thf(fact_7643_times__int_Oabs__eq,axiom,
% 14.25/8.15 ! [Xa: product_prod_nat_nat,X: product_prod_nat_nat] :
% 14.25/8.15 ( ( times_times_int @ ( abs_Integ @ Xa ) @ ( abs_Integ @ X ) )
% 14.25/8.15 = ( abs_Integ
% 14.25/8.15 @ ( produc27273713700761075at_nat
% 14.25/8.15 @ ^ [X2: nat,Y3: nat] :
% 14.25/8.15 ( produc2626176000494625587at_nat
% 14.25/8.15 @ ^ [U3: nat,V4: nat] : ( product_Pair_nat_nat @ ( plus_plus_nat @ ( times_times_nat @ X2 @ U3 ) @ ( times_times_nat @ Y3 @ V4 ) ) @ ( plus_plus_nat @ ( times_times_nat @ X2 @ V4 ) @ ( times_times_nat @ Y3 @ U3 ) ) ) )
% 14.25/8.15 @ Xa
% 14.25/8.15 @ X ) ) ) ).
% 14.25/8.15
% 14.25/8.15 thf(fact_7644_times__integer_Orep__eq,axiom,
% 14.25/8.15 ! [X: code_integer,Xa: code_integer] :
% 14.25/8.15 ( ( code_int_of_integer @ ( times_3573771949741848930nteger @ X @ Xa ) )
% 14.25/8.15 = ( times_times_int @ ( code_int_of_integer @ X ) @ ( code_int_of_integer @ Xa ) ) ) ).
% 14.25/8.15
% 14.25/8.15 thf(fact_7645_positive__mult,axiom,
% 14.25/8.15 ! [X: rat,Y: rat] :
% 14.25/8.15 ( ( positive @ X )
% 14.25/8.15 => ( ( positive @ Y )
% 14.25/8.15 => ( positive @ ( times_times_rat @ X @ Y ) ) ) ) ).
% 14.25/8.15
% 14.25/8.15 thf(fact_7646_int__of__integer__less__iff,axiom,
% 14.25/8.15 ! [X: code_integer,Y: code_integer] :
% 14.25/8.15 ( ( ord_less_int @ ( code_int_of_integer @ X ) @ ( code_int_of_integer @ Y ) )
% 14.25/8.15 = ( ord_le6747313008572928689nteger @ X @ Y ) ) ).
% 14.25/8.15
% 14.25/8.15 thf(fact_7647_type__definition__integer,axiom,
% 14.25/8.15 type_d8366093980585677751er_int @ code_int_of_integer @ code_integer_of_int @ top_top_set_int ).
% 14.25/8.15
% 14.25/8.15 thf(fact_7648_positive_Orep__eq,axiom,
% 14.25/8.15 ( positive
% 14.25/8.15 = ( ^ [X2: rat] : ( ord_less_int @ zero_zero_int @ ( times_times_int @ ( product_fst_int_int @ ( rep_Rat @ X2 ) ) @ ( product_snd_int_int @ ( rep_Rat @ X2 ) ) ) ) ) ) ).
% 14.25/8.15
% 14.25/8.15 thf(fact_7649_nat__of__integer__less__iff,axiom,
% 14.25/8.15 ! [X: code_integer,Y: code_integer] :
% 14.25/8.15 ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ X )
% 14.25/8.15 => ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ Y )
% 14.25/8.15 => ( ( ord_less_nat @ ( code_nat_of_integer @ X ) @ ( code_nat_of_integer @ Y ) )
% 14.25/8.15 = ( ord_le6747313008572928689nteger @ X @ Y ) ) ) ) ).
% 14.25/8.15
% 14.25/8.15 thf(fact_7650_times__int_Orsp,axiom,
% 14.25/8.15 ( bNF_re3099431351363272937at_nat @ intrel @ ( bNF_re2241393799969408733at_nat @ intrel @ intrel )
% 14.25/8.15 @ ( produc27273713700761075at_nat
% 14.25/8.15 @ ^ [X2: nat,Y3: nat] :
% 14.25/8.15 ( produc2626176000494625587at_nat
% 14.25/8.15 @ ^ [U3: nat,V4: nat] : ( product_Pair_nat_nat @ ( plus_plus_nat @ ( times_times_nat @ X2 @ U3 ) @ ( times_times_nat @ Y3 @ V4 ) ) @ ( plus_plus_nat @ ( times_times_nat @ X2 @ V4 ) @ ( times_times_nat @ Y3 @ U3 ) ) ) ) )
% 14.25/8.15 @ ( produc27273713700761075at_nat
% 14.25/8.15 @ ^ [X2: nat,Y3: nat] :
% 14.25/8.15 ( produc2626176000494625587at_nat
% 14.25/8.15 @ ^ [U3: nat,V4: nat] : ( product_Pair_nat_nat @ ( plus_plus_nat @ ( times_times_nat @ X2 @ U3 ) @ ( times_times_nat @ Y3 @ V4 ) ) @ ( plus_plus_nat @ ( times_times_nat @ X2 @ V4 ) @ ( times_times_nat @ Y3 @ U3 ) ) ) ) ) ) ).
% 14.25/8.15
% 14.25/8.15 thf(fact_7651_atLeast1__lessThan__eq__remove0,axiom,
% 14.25/8.15 ! [N2: nat] :
% 14.25/8.15 ( ( set_or4665077453230672383an_nat @ ( suc @ zero_zero_nat ) @ N2 )
% 14.25/8.15 = ( minus_minus_set_nat @ ( set_ord_lessThan_nat @ N2 ) @ ( insert_nat @ zero_zero_nat @ bot_bot_set_nat ) ) ) ).
% 14.25/8.15
% 14.25/8.15 thf(fact_7652_min__Suc__gt_I1_J,axiom,
% 14.25/8.15 ! [A2: nat,B: nat] :
% 14.25/8.15 ( ( ord_less_nat @ A2 @ B )
% 14.25/8.15 => ( ( ord_min_nat @ ( suc @ A2 ) @ B )
% 14.25/8.15 = ( suc @ A2 ) ) ) ).
% 14.25/8.15
% 14.25/8.15 thf(fact_7653_min__Suc__gt_I2_J,axiom,
% 14.25/8.15 ! [A2: nat,B: nat] :
% 14.25/8.15 ( ( ord_less_nat @ A2 @ B )
% 14.25/8.15 => ( ( ord_min_nat @ B @ ( suc @ A2 ) )
% 14.25/8.15 = ( suc @ A2 ) ) ) ).
% 14.25/8.15
% 14.25/8.15 thf(fact_7654_lessThan__0,axiom,
% 14.25/8.15 ( ( set_ord_lessThan_nat @ zero_zero_nat )
% 14.25/8.15 = bot_bot_set_nat ) ).
% 14.25/8.15
% 14.25/8.15 thf(fact_7655_nat__mult__min__right,axiom,
% 14.25/8.15 ! [M4: nat,N2: nat,Q4: nat] :
% 14.25/8.15 ( ( times_times_nat @ M4 @ ( ord_min_nat @ N2 @ Q4 ) )
% 14.25/8.15 = ( ord_min_nat @ ( times_times_nat @ M4 @ N2 ) @ ( times_times_nat @ M4 @ Q4 ) ) ) ).
% 14.25/8.15
% 14.25/8.15 thf(fact_7656_nat__mult__min__left,axiom,
% 14.25/8.15 ! [M4: nat,N2: nat,Q4: nat] :
% 14.25/8.15 ( ( times_times_nat @ ( ord_min_nat @ M4 @ N2 ) @ Q4 )
% 14.25/8.15 = ( ord_min_nat @ ( times_times_nat @ M4 @ Q4 ) @ ( times_times_nat @ N2 @ Q4 ) ) ) ).
% 14.25/8.15
% 14.25/8.15 thf(fact_7657_lessThan__empty__iff,axiom,
% 14.25/8.15 ! [N2: nat] :
% 14.25/8.15 ( ( ( set_ord_lessThan_nat @ N2 )
% 14.25/8.15 = bot_bot_set_nat )
% 14.25/8.15 = ( N2 = zero_zero_nat ) ) ).
% 14.25/8.15
% 14.25/8.15 thf(fact_7658_image__atLeastZeroLessThan__integer,axiom,
% 14.25/8.15 ! [U: code_integer] :
% 14.25/8.15 ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ U )
% 14.25/8.15 => ( ( set_or8404916559141939852nteger @ zero_z3403309356797280102nteger @ U )
% 14.25/8.15 = ( image_1215581382706833972nteger @ semiri4939895301339042750nteger @ ( set_ord_lessThan_nat @ ( code_nat_of_integer @ U ) ) ) ) ) ).
% 14.25/8.15
% 14.25/8.15 thf(fact_7659_bezw__aux,axiom,
% 14.25/8.15 ! [X: nat,Y: nat] :
% 14.25/8.15 ( ( semiri1314217659103216013at_int @ ( gcd_gcd_nat @ X @ Y ) )
% 14.25/8.15 = ( plus_plus_int @ ( times_times_int @ ( product_fst_int_int @ ( bezw @ X @ Y ) ) @ ( semiri1314217659103216013at_int @ X ) ) @ ( times_times_int @ ( product_snd_int_int @ ( bezw @ X @ Y ) ) @ ( semiri1314217659103216013at_int @ Y ) ) ) ) ).
% 14.25/8.15
% 14.25/8.15 thf(fact_7660_INT__greaterThan__UNIV,axiom,
% 14.25/8.15 ( ( comple7806235888213564991et_nat @ ( image_nat_set_nat @ set_or1210151606488870762an_nat @ top_top_set_nat ) )
% 14.25/8.15 = bot_bot_set_nat ) ).
% 14.25/8.15
% 14.25/8.15 thf(fact_7661_gcd__mult__distrib__nat,axiom,
% 14.25/8.15 ! [K3: nat,M4: nat,N2: nat] :
% 14.25/8.15 ( ( times_times_nat @ K3 @ ( gcd_gcd_nat @ M4 @ N2 ) )
% 14.25/8.15 = ( gcd_gcd_nat @ ( times_times_nat @ K3 @ M4 ) @ ( times_times_nat @ K3 @ N2 ) ) ) ).
% 14.25/8.15
% 14.25/8.15 thf(fact_7662_Gcd__in,axiom,
% 14.25/8.15 ! [A: set_nat] :
% 14.25/8.15 ( ! [A4: nat,B2: nat] :
% 14.25/8.15 ( ( member_nat @ A4 @ A )
% 14.25/8.15 => ( ( member_nat @ B2 @ A )
% 14.25/8.15 => ( member_nat @ ( gcd_gcd_nat @ A4 @ B2 ) @ A ) ) )
% 14.25/8.15 => ( ( A != bot_bot_set_nat )
% 14.25/8.15 => ( member_nat @ ( gcd_Gcd_nat @ A ) @ A ) ) ) ).
% 14.25/8.15
% 14.25/8.15 thf(fact_7663_bezout__nat,axiom,
% 14.25/8.15 ! [A2: nat,B: nat] :
% 14.25/8.15 ( ( A2 != zero_zero_nat )
% 14.25/8.15 => ? [X3: nat,Y2: nat] :
% 14.25/8.15 ( ( times_times_nat @ A2 @ X3 )
% 14.25/8.15 = ( plus_plus_nat @ ( times_times_nat @ B @ Y2 ) @ ( gcd_gcd_nat @ A2 @ B ) ) ) ) ).
% 14.25/8.15
% 14.25/8.15 thf(fact_7664_bezout__gcd__nat_H,axiom,
% 14.25/8.15 ! [B: nat,A2: nat] :
% 14.25/8.15 ? [X3: nat,Y2: nat] :
% 14.25/8.15 ( ( ( ord_less_eq_nat @ ( times_times_nat @ B @ Y2 ) @ ( times_times_nat @ A2 @ X3 ) )
% 14.25/8.15 & ( ( minus_minus_nat @ ( times_times_nat @ A2 @ X3 ) @ ( times_times_nat @ B @ Y2 ) )
% 14.25/8.15 = ( gcd_gcd_nat @ A2 @ B ) ) )
% 14.25/8.15 | ( ( ord_less_eq_nat @ ( times_times_nat @ A2 @ Y2 ) @ ( times_times_nat @ B @ X3 ) )
% 14.25/8.15 & ( ( minus_minus_nat @ ( times_times_nat @ B @ X3 ) @ ( times_times_nat @ A2 @ Y2 ) )
% 14.25/8.15 = ( gcd_gcd_nat @ A2 @ B ) ) ) ) ).
% 14.25/8.15
% 14.25/8.15 thf(fact_7665_greaterThan__Suc,axiom,
% 14.25/8.15 ! [K3: nat] :
% 14.25/8.15 ( ( set_or1210151606488870762an_nat @ ( suc @ K3 ) )
% 14.25/8.15 = ( minus_minus_set_nat @ ( set_or1210151606488870762an_nat @ K3 ) @ ( insert_nat @ ( suc @ K3 ) @ bot_bot_set_nat ) ) ) ).
% 14.25/8.15
% 14.25/8.15 thf(fact_7666_plus__rat_Orsp,axiom,
% 14.25/8.15 ( bNF_re5228765855967844073nt_int @ ratrel @ ( bNF_re7145576690424134365nt_int @ ratrel @ ratrel )
% 14.25/8.15 @ ^ [X2: product_prod_int_int,Y3: product_prod_int_int] : ( product_Pair_int_int @ ( plus_plus_int @ ( times_times_int @ ( product_fst_int_int @ X2 ) @ ( product_snd_int_int @ Y3 ) ) @ ( times_times_int @ ( product_fst_int_int @ Y3 ) @ ( product_snd_int_int @ X2 ) ) ) @ ( times_times_int @ ( product_snd_int_int @ X2 ) @ ( product_snd_int_int @ Y3 ) ) )
% 14.25/8.15 @ ^ [X2: product_prod_int_int,Y3: product_prod_int_int] : ( product_Pair_int_int @ ( plus_plus_int @ ( times_times_int @ ( product_fst_int_int @ X2 ) @ ( product_snd_int_int @ Y3 ) ) @ ( times_times_int @ ( product_fst_int_int @ Y3 ) @ ( product_snd_int_int @ X2 ) ) ) @ ( times_times_int @ ( product_snd_int_int @ X2 ) @ ( product_snd_int_int @ Y3 ) ) ) ) ).
% 14.25/8.15
% 14.25/8.15 thf(fact_7667_positive_Orsp,axiom,
% 14.25/8.15 ( bNF_re8699439704749558557nt_o_o @ ratrel
% 14.25/8.15 @ ^ [Y4: $o,Z4: $o] : ( Y4 = Z4 )
% 14.25/8.15 @ ^ [X2: product_prod_int_int] : ( ord_less_int @ zero_zero_int @ ( times_times_int @ ( product_fst_int_int @ X2 ) @ ( product_snd_int_int @ X2 ) ) )
% 14.25/8.15 @ ^ [X2: product_prod_int_int] : ( ord_less_int @ zero_zero_int @ ( times_times_int @ ( product_fst_int_int @ X2 ) @ ( product_snd_int_int @ X2 ) ) ) ) ).
% 14.25/8.15
% 14.25/8.15 thf(fact_7668_ratrel__iff,axiom,
% 14.25/8.15 ( ratrel
% 14.25/8.15 = ( ^ [X2: product_prod_int_int,Y3: product_prod_int_int] :
% 14.25/8.15 ( ( ( product_snd_int_int @ X2 )
% 14.25/8.15 != zero_zero_int )
% 14.25/8.15 & ( ( product_snd_int_int @ Y3 )
% 14.25/8.15 != zero_zero_int )
% 14.25/8.15 & ( ( times_times_int @ ( product_fst_int_int @ X2 ) @ ( product_snd_int_int @ Y3 ) )
% 14.25/8.15 = ( times_times_int @ ( product_fst_int_int @ Y3 ) @ ( product_snd_int_int @ X2 ) ) ) ) ) ) ).
% 14.25/8.15
% 14.25/8.15 thf(fact_7669_bezout__int,axiom,
% 14.25/8.15 ! [X: int,Y: int] :
% 14.25/8.15 ? [U4: int,V3: int] :
% 14.25/8.15 ( ( plus_plus_int @ ( times_times_int @ U4 @ X ) @ ( times_times_int @ V3 @ Y ) )
% 14.25/8.15 = ( gcd_gcd_int @ X @ Y ) ) ).
% 14.25/8.15
% 14.25/8.15 thf(fact_7670_gcd__mult__distrib__int,axiom,
% 14.25/8.15 ! [K3: int,M4: int,N2: int] :
% 14.25/8.15 ( ( times_times_int @ ( abs_abs_int @ K3 ) @ ( gcd_gcd_int @ M4 @ N2 ) )
% 14.25/8.15 = ( gcd_gcd_int @ ( times_times_int @ K3 @ M4 ) @ ( times_times_int @ K3 @ N2 ) ) ) ).
% 14.25/8.15
% 14.25/8.15 thf(fact_7671_ratrel__def,axiom,
% 14.25/8.15 ( ratrel
% 14.25/8.15 = ( ^ [X2: product_prod_int_int,Y3: product_prod_int_int] :
% 14.25/8.15 ( ( ( product_snd_int_int @ X2 )
% 14.25/8.15 != zero_zero_int )
% 14.25/8.15 & ( ( product_snd_int_int @ Y3 )
% 14.25/8.15 != zero_zero_int )
% 14.25/8.15 & ( ( times_times_int @ ( product_fst_int_int @ X2 ) @ ( product_snd_int_int @ Y3 ) )
% 14.25/8.15 = ( times_times_int @ ( product_fst_int_int @ Y3 ) @ ( product_snd_int_int @ X2 ) ) ) ) ) ) ).
% 14.25/8.15
% 14.25/8.15 thf(fact_7672_times__rat_Orsp,axiom,
% 14.25/8.15 ( bNF_re5228765855967844073nt_int @ ratrel @ ( bNF_re7145576690424134365nt_int @ ratrel @ ratrel )
% 14.25/8.15 @ ^ [X2: product_prod_int_int,Y3: product_prod_int_int] : ( product_Pair_int_int @ ( times_times_int @ ( product_fst_int_int @ X2 ) @ ( product_fst_int_int @ Y3 ) ) @ ( times_times_int @ ( product_snd_int_int @ X2 ) @ ( product_snd_int_int @ Y3 ) ) )
% 14.25/8.15 @ ^ [X2: product_prod_int_int,Y3: product_prod_int_int] : ( product_Pair_int_int @ ( times_times_int @ ( product_fst_int_int @ X2 ) @ ( product_fst_int_int @ Y3 ) ) @ ( times_times_int @ ( product_snd_int_int @ X2 ) @ ( product_snd_int_int @ Y3 ) ) ) ) ).
% 14.25/8.15
% 14.25/8.15 thf(fact_7673_plus__rat_Oabs__eq,axiom,
% 14.25/8.15 ! [Xa: product_prod_int_int,X: product_prod_int_int] :
% 14.25/8.15 ( ( ratrel @ Xa @ Xa )
% 14.25/8.15 => ( ( ratrel @ X @ X )
% 14.25/8.15 => ( ( plus_plus_rat @ ( abs_Rat @ Xa ) @ ( abs_Rat @ X ) )
% 14.25/8.15 = ( abs_Rat @ ( product_Pair_int_int @ ( plus_plus_int @ ( times_times_int @ ( product_fst_int_int @ Xa ) @ ( product_snd_int_int @ X ) ) @ ( times_times_int @ ( product_fst_int_int @ X ) @ ( product_snd_int_int @ Xa ) ) ) @ ( times_times_int @ ( product_snd_int_int @ Xa ) @ ( product_snd_int_int @ X ) ) ) ) ) ) ) ).
% 14.25/8.15
% 14.25/8.15 thf(fact_7674_positive_Oabs__eq,axiom,
% 14.25/8.15 ! [X: product_prod_int_int] :
% 14.25/8.15 ( ( ratrel @ X @ X )
% 14.25/8.15 => ( ( positive @ ( abs_Rat @ X ) )
% 14.25/8.15 = ( ord_less_int @ zero_zero_int @ ( times_times_int @ ( product_fst_int_int @ X ) @ ( product_snd_int_int @ X ) ) ) ) ) ).
% 14.25/8.15
% 14.25/8.15 thf(fact_7675_times__rat_Oabs__eq,axiom,
% 14.25/8.15 ! [Xa: product_prod_int_int,X: product_prod_int_int] :
% 14.25/8.15 ( ( ratrel @ Xa @ Xa )
% 14.25/8.15 => ( ( ratrel @ X @ X )
% 14.25/8.15 => ( ( times_times_rat @ ( abs_Rat @ Xa ) @ ( abs_Rat @ X ) )
% 14.25/8.15 = ( abs_Rat @ ( product_Pair_int_int @ ( times_times_int @ ( product_fst_int_int @ Xa ) @ ( product_fst_int_int @ X ) ) @ ( times_times_int @ ( product_snd_int_int @ Xa ) @ ( product_snd_int_int @ X ) ) ) ) ) ) ) ).
% 14.25/8.15
% 14.25/8.15 thf(fact_7676_finite__greaterThanLessThan__integer,axiom,
% 14.25/8.15 ! [L: code_integer,U: code_integer] : ( finite6017078050557962740nteger @ ( set_or4266950643985792945nteger @ L @ U ) ) ).
% 14.25/8.15
% 14.25/8.15 thf(fact_7677_atLeastPlusOneLessThan__greaterThanLessThan__integer,axiom,
% 14.25/8.15 ! [L: code_integer,U: code_integer] :
% 14.25/8.15 ( ( set_or8404916559141939852nteger @ ( plus_p5714425477246183910nteger @ L @ one_one_Code_integer ) @ U )
% 14.25/8.15 = ( set_or4266950643985792945nteger @ L @ U ) ) ).
% 14.25/8.15
% 14.25/8.15 thf(fact_7678_positive__def,axiom,
% 14.25/8.15 ( positive
% 14.25/8.15 = ( map_fu898904425404107465nt_o_o @ rep_Rat @ id_o
% 14.25/8.15 @ ^ [X2: product_prod_int_int] : ( ord_less_int @ zero_zero_int @ ( times_times_int @ ( product_fst_int_int @ X2 ) @ ( product_snd_int_int @ X2 ) ) ) ) ) ).
% 14.25/8.15
% 14.25/8.15 thf(fact_7679_finite__greaterThanAtMost__integer,axiom,
% 14.25/8.15 ! [L: code_integer,U: code_integer] : ( finite6017078050557962740nteger @ ( set_or2715278749043346189nteger @ L @ U ) ) ).
% 14.25/8.15
% 14.25/8.15 thf(fact_7680_atLeastPlusOneAtMost__greaterThanAtMost__integer,axiom,
% 14.25/8.15 ! [L: code_integer,U: code_integer] :
% 14.25/8.15 ( ( set_or189985376899183464nteger @ ( plus_p5714425477246183910nteger @ L @ one_one_Code_integer ) @ U )
% 14.25/8.15 = ( set_or2715278749043346189nteger @ L @ U ) ) ).
% 14.25/8.15
% 14.25/8.15 thf(fact_7681_atLeast__Suc,axiom,
% 14.25/8.15 ! [K3: nat] :
% 14.25/8.15 ( ( set_ord_atLeast_nat @ ( suc @ K3 ) )
% 14.25/8.15 = ( minus_minus_set_nat @ ( set_ord_atLeast_nat @ K3 ) @ ( insert_nat @ K3 @ bot_bot_set_nat ) ) ) ).
% 14.25/8.15
% 14.25/8.15 thf(fact_7682_eventually__sequentially__Suc,axiom,
% 14.25/8.15 ! [P: nat > $o] :
% 14.25/8.15 ( ( eventually_nat
% 14.25/8.15 @ ^ [I4: nat] : ( P @ ( suc @ I4 ) )
% 14.25/8.15 @ at_top_nat )
% 14.25/8.15 = ( eventually_nat @ P @ at_top_nat ) ) ).
% 14.25/8.15
% 14.25/8.15 thf(fact_7683_eventually__sequentially__seg,axiom,
% 14.25/8.15 ! [P: nat > $o,K3: nat] :
% 14.25/8.15 ( ( eventually_nat
% 14.25/8.15 @ ^ [N4: nat] : ( P @ ( plus_plus_nat @ N4 @ K3 ) )
% 14.25/8.15 @ at_top_nat )
% 14.25/8.15 = ( eventually_nat @ P @ at_top_nat ) ) ).
% 14.25/8.15
% 14.25/8.15 thf(fact_7684_filterlim__Suc,axiom,
% 14.25/8.15 filterlim_nat_nat @ suc @ at_top_nat @ at_top_nat ).
% 14.25/8.15
% 14.25/8.15 thf(fact_7685_eventually__sequentiallyI,axiom,
% 14.25/8.15 ! [C: nat,P: nat > $o] :
% 14.25/8.15 ( ! [X3: nat] :
% 14.25/8.15 ( ( ord_less_eq_nat @ C @ X3 )
% 14.25/8.15 => ( P @ X3 ) )
% 14.25/8.15 => ( eventually_nat @ P @ at_top_nat ) ) ).
% 14.25/8.15
% 14.25/8.15 thf(fact_7686_eventually__sequentially,axiom,
% 14.25/8.15 ! [P: nat > $o] :
% 14.25/8.15 ( ( eventually_nat @ P @ at_top_nat )
% 14.25/8.15 = ( ? [N7: nat] :
% 14.25/8.15 ! [N4: nat] :
% 14.25/8.15 ( ( ord_less_eq_nat @ N7 @ N4 )
% 14.25/8.15 => ( P @ N4 ) ) ) ) ).
% 14.25/8.15
% 14.25/8.15 thf(fact_7687_le__sequentially,axiom,
% 14.25/8.15 ! [F2: filter_nat] :
% 14.25/8.15 ( ( ord_le2510731241096832064er_nat @ F2 @ at_top_nat )
% 14.25/8.15 = ( ! [N7: nat] : ( eventually_nat @ ( ord_less_eq_nat @ N7 ) @ F2 ) ) ) ).
% 14.25/8.15
% 14.25/8.15 thf(fact_7688_filterlim__atMost__at__top,axiom,
% 14.25/8.15 filter3212408913953519116et_nat @ set_ord_atMost_nat @ ( finite3254316476582989077op_nat @ top_top_set_nat ) @ at_top_nat ).
% 14.25/8.15
% 14.25/8.15 thf(fact_7689_filterlim__lessThan__at__top,axiom,
% 14.25/8.15 filter3212408913953519116et_nat @ set_ord_lessThan_nat @ ( finite3254316476582989077op_nat @ top_top_set_nat ) @ at_top_nat ).
% 14.25/8.15
% 14.25/8.15 thf(fact_7690_trivial__limit__sequentially,axiom,
% 14.25/8.15 at_top_nat != bot_bot_filter_nat ).
% 14.25/8.15
% 14.25/8.15 thf(fact_7691_eventually__False__sequentially,axiom,
% 14.25/8.15 ~ ( eventually_nat
% 14.25/8.15 @ ^ [N4: nat] : $false
% 14.25/8.15 @ at_top_nat ) ).
% 14.25/8.15
% 14.25/8.15 thf(fact_7692_plus__rat__def,axiom,
% 14.25/8.15 ( plus_plus_rat
% 14.25/8.15 = ( map_fu4333342158222067775at_rat @ rep_Rat @ ( map_fu5673905371560938248nt_rat @ rep_Rat @ abs_Rat )
% 14.25/8.15 @ ^ [X2: product_prod_int_int,Y3: product_prod_int_int] : ( product_Pair_int_int @ ( plus_plus_int @ ( times_times_int @ ( product_fst_int_int @ X2 ) @ ( product_snd_int_int @ Y3 ) ) @ ( times_times_int @ ( product_fst_int_int @ Y3 ) @ ( product_snd_int_int @ X2 ) ) ) @ ( times_times_int @ ( product_snd_int_int @ X2 ) @ ( product_snd_int_int @ Y3 ) ) ) ) ) ).
% 14.25/8.15
% 14.25/8.15 thf(fact_7693_eventually__prod__sequentially,axiom,
% 14.25/8.15 ! [P: product_prod_nat_nat > $o] :
% 14.25/8.15 ( ( eventu1038000079068216329at_nat @ P @ ( prod_filter_nat_nat @ at_top_nat @ at_top_nat ) )
% 14.25/8.15 = ( ? [N7: nat] :
% 14.25/8.15 ! [M3: nat] :
% 14.25/8.15 ( ( ord_less_eq_nat @ N7 @ M3 )
% 14.25/8.15 => ! [N4: nat] :
% 14.25/8.15 ( ( ord_less_eq_nat @ N7 @ N4 )
% 14.25/8.15 => ( P @ ( product_Pair_nat_nat @ N4 @ M3 ) ) ) ) ) ) ).
% 14.25/8.15
% 14.25/8.15 thf(fact_7694_times__rat__def,axiom,
% 14.25/8.15 ( times_times_rat
% 14.25/8.15 = ( map_fu4333342158222067775at_rat @ rep_Rat @ ( map_fu5673905371560938248nt_rat @ rep_Rat @ abs_Rat )
% 14.25/8.15 @ ^ [X2: product_prod_int_int,Y3: product_prod_int_int] : ( product_Pair_int_int @ ( times_times_int @ ( product_fst_int_int @ X2 ) @ ( product_fst_int_int @ Y3 ) ) @ ( times_times_int @ ( product_snd_int_int @ X2 ) @ ( product_snd_int_int @ Y3 ) ) ) ) ) ).
% 14.25/8.15
% 14.25/8.15 thf(fact_7695_add__rat,axiom,
% 14.25/8.15 ! [B: int,D2: int,A2: int,C: int] :
% 14.25/8.15 ( ( B != zero_zero_int )
% 14.25/8.15 => ( ( D2 != zero_zero_int )
% 14.25/8.15 => ( ( plus_plus_rat @ ( fract @ A2 @ B ) @ ( fract @ C @ D2 ) )
% 14.25/8.15 = ( fract @ ( plus_plus_int @ ( times_times_int @ A2 @ D2 ) @ ( times_times_int @ C @ B ) ) @ ( times_times_int @ B @ D2 ) ) ) ) ) ).
% 14.25/8.15
% 14.25/8.15 thf(fact_7696_divide__rat,axiom,
% 14.25/8.15 ! [A2: int,B: int,C: int,D2: int] :
% 14.25/8.15 ( ( divide_divide_rat @ ( fract @ A2 @ B ) @ ( fract @ C @ D2 ) )
% 14.25/8.15 = ( fract @ ( times_times_int @ A2 @ D2 ) @ ( times_times_int @ B @ C ) ) ) ).
% 14.25/8.15
% 14.25/8.15 thf(fact_7697_sgn__rat,axiom,
% 14.25/8.15 ! [A2: int,B: int] :
% 14.25/8.15 ( ( sgn_sgn_rat @ ( fract @ A2 @ B ) )
% 14.25/8.15 = ( ring_1_of_int_rat @ ( times_times_int @ ( sgn_sgn_int @ A2 ) @ ( sgn_sgn_int @ B ) ) ) ) ).
% 14.25/8.15
% 14.25/8.15 thf(fact_7698_mult__rat,axiom,
% 14.25/8.15 ! [A2: int,B: int,C: int,D2: int] :
% 14.25/8.15 ( ( times_times_rat @ ( fract @ A2 @ B ) @ ( fract @ C @ D2 ) )
% 14.25/8.15 = ( fract @ ( times_times_int @ A2 @ C ) @ ( times_times_int @ B @ D2 ) ) ) ).
% 14.25/8.15
% 14.25/8.15 thf(fact_7699_le__rat,axiom,
% 14.25/8.15 ! [B: int,D2: int,A2: int,C: int] :
% 14.25/8.15 ( ( B != zero_zero_int )
% 14.25/8.15 => ( ( D2 != zero_zero_int )
% 14.25/8.15 => ( ( ord_less_eq_rat @ ( fract @ A2 @ B ) @ ( fract @ C @ D2 ) )
% 14.25/8.15 = ( ord_less_eq_int @ ( times_times_int @ ( times_times_int @ A2 @ D2 ) @ ( times_times_int @ B @ D2 ) ) @ ( times_times_int @ ( times_times_int @ C @ B ) @ ( times_times_int @ B @ D2 ) ) ) ) ) ) ).
% 14.25/8.15
% 14.25/8.15 thf(fact_7700_less__rat,axiom,
% 14.25/8.15 ! [B: int,D2: int,A2: int,C: int] :
% 14.25/8.15 ( ( B != zero_zero_int )
% 14.25/8.15 => ( ( D2 != zero_zero_int )
% 14.25/8.15 => ( ( ord_less_rat @ ( fract @ A2 @ B ) @ ( fract @ C @ D2 ) )
% 14.25/8.15 = ( ord_less_int @ ( times_times_int @ ( times_times_int @ A2 @ D2 ) @ ( times_times_int @ B @ D2 ) ) @ ( times_times_int @ ( times_times_int @ C @ B ) @ ( times_times_int @ B @ D2 ) ) ) ) ) ) ).
% 14.25/8.15
% 14.25/8.15 thf(fact_7701_diff__rat,axiom,
% 14.25/8.15 ! [B: int,D2: int,A2: int,C: int] :
% 14.25/8.15 ( ( B != zero_zero_int )
% 14.25/8.15 => ( ( D2 != zero_zero_int )
% 14.25/8.15 => ( ( minus_minus_rat @ ( fract @ A2 @ B ) @ ( fract @ C @ D2 ) )
% 14.25/8.15 = ( fract @ ( minus_minus_int @ ( times_times_int @ A2 @ D2 ) @ ( times_times_int @ C @ B ) ) @ ( times_times_int @ B @ D2 ) ) ) ) ) ).
% 14.25/8.15
% 14.25/8.15 thf(fact_7702_eq__rat_I1_J,axiom,
% 14.25/8.15 ! [B: int,D2: int,A2: int,C: int] :
% 14.25/8.15 ( ( B != zero_zero_int )
% 14.25/8.15 => ( ( D2 != zero_zero_int )
% 14.25/8.15 => ( ( ( fract @ A2 @ B )
% 14.25/8.15 = ( fract @ C @ D2 ) )
% 14.25/8.15 = ( ( times_times_int @ A2 @ D2 )
% 14.25/8.15 = ( times_times_int @ C @ B ) ) ) ) ) ).
% 14.25/8.15
% 14.25/8.15 thf(fact_7703_mult__rat__cancel,axiom,
% 14.25/8.15 ! [C: int,A2: int,B: int] :
% 14.25/8.15 ( ( C != zero_zero_int )
% 14.25/8.15 => ( ( fract @ ( times_times_int @ C @ A2 ) @ ( times_times_int @ C @ B ) )
% 14.25/8.15 = ( fract @ A2 @ B ) ) ) ).
% 14.25/8.15
% 14.25/8.15 thf(fact_7704_mono__times__nat,axiom,
% 14.25/8.15 ! [N2: nat] :
% 14.25/8.15 ( ( ord_less_nat @ zero_zero_nat @ N2 )
% 14.25/8.15 => ( order_mono_nat_nat @ ( times_times_nat @ N2 ) ) ) ).
% 14.25/8.15
% 14.25/8.15 thf(fact_7705_positive__rat,axiom,
% 14.25/8.15 ! [A2: int,B: int] :
% 14.25/8.15 ( ( positive @ ( fract @ A2 @ B ) )
% 14.25/8.15 = ( ord_less_int @ zero_zero_int @ ( times_times_int @ A2 @ B ) ) ) ).
% 14.25/8.15
% 14.25/8.15 thf(fact_7706_set__encode__empty,axiom,
% 14.25/8.15 ( ( nat_set_encode @ bot_bot_set_nat )
% 14.25/8.15 = zero_zero_nat ) ).
% 14.25/8.15
% 14.25/8.15 thf(fact_7707_normalize__crossproduct,axiom,
% 14.25/8.15 ! [Q4: int,S2: int,P3: int,R2: int] :
% 14.25/8.15 ( ( Q4 != zero_zero_int )
% 14.25/8.15 => ( ( S2 != zero_zero_int )
% 14.25/8.15 => ( ( ( normalize @ ( product_Pair_int_int @ P3 @ Q4 ) )
% 14.25/8.15 = ( normalize @ ( product_Pair_int_int @ R2 @ S2 ) ) )
% 14.25/8.15 => ( ( times_times_int @ P3 @ S2 )
% 14.25/8.15 = ( times_times_int @ R2 @ Q4 ) ) ) ) ) ).
% 14.25/8.15
% 14.25/8.15 thf(fact_7708_rat__minus__code,axiom,
% 14.25/8.15 ! [P3: rat,Q4: rat] :
% 14.25/8.15 ( ( quotient_of @ ( minus_minus_rat @ P3 @ Q4 ) )
% 14.25/8.15 = ( produc4245557441103728435nt_int
% 14.25/8.15 @ ^ [A6: int,C4: int] :
% 14.25/8.15 ( produc4245557441103728435nt_int
% 14.25/8.15 @ ^ [B6: int,D4: int] : ( normalize @ ( product_Pair_int_int @ ( minus_minus_int @ ( times_times_int @ A6 @ D4 ) @ ( times_times_int @ B6 @ C4 ) ) @ ( times_times_int @ C4 @ D4 ) ) )
% 14.25/8.15 @ ( quotient_of @ Q4 ) )
% 14.25/8.15 @ ( quotient_of @ P3 ) ) ) ).
% 14.25/8.15
% 14.25/8.15 thf(fact_7709_rat__plus__code,axiom,
% 14.25/8.15 ! [P3: rat,Q4: rat] :
% 14.25/8.15 ( ( quotient_of @ ( plus_plus_rat @ P3 @ Q4 ) )
% 14.25/8.15 = ( produc4245557441103728435nt_int
% 14.25/8.15 @ ^ [A6: int,C4: int] :
% 14.25/8.15 ( produc4245557441103728435nt_int
% 14.25/8.15 @ ^ [B6: int,D4: int] : ( normalize @ ( product_Pair_int_int @ ( plus_plus_int @ ( times_times_int @ A6 @ D4 ) @ ( times_times_int @ B6 @ C4 ) ) @ ( times_times_int @ C4 @ D4 ) ) )
% 14.25/8.15 @ ( quotient_of @ Q4 ) )
% 14.25/8.15 @ ( quotient_of @ P3 ) ) ) ).
% 14.25/8.15
% 14.25/8.15 thf(fact_7710_rat__times__code,axiom,
% 14.25/8.15 ! [P3: rat,Q4: rat] :
% 14.25/8.15 ( ( quotient_of @ ( times_times_rat @ P3 @ Q4 ) )
% 14.25/8.15 = ( produc4245557441103728435nt_int
% 14.25/8.15 @ ^ [A6: int,C4: int] :
% 14.25/8.15 ( produc4245557441103728435nt_int
% 14.25/8.15 @ ^ [B6: int,D4: int] : ( normalize @ ( product_Pair_int_int @ ( times_times_int @ A6 @ B6 ) @ ( times_times_int @ C4 @ D4 ) ) )
% 14.25/8.15 @ ( quotient_of @ Q4 ) )
% 14.25/8.15 @ ( quotient_of @ P3 ) ) ) ).
% 14.25/8.15
% 14.25/8.15 thf(fact_7711_rat__less__code,axiom,
% 14.25/8.15 ( ord_less_rat
% 14.25/8.15 = ( ^ [P5: rat,Q6: rat] :
% 14.25/8.15 ( produc4947309494688390418_int_o
% 14.25/8.15 @ ^ [A6: int,C4: int] :
% 14.25/8.15 ( produc4947309494688390418_int_o
% 14.25/8.15 @ ^ [B6: int,D4: int] : ( ord_less_int @ ( times_times_int @ A6 @ D4 ) @ ( times_times_int @ C4 @ B6 ) )
% 14.25/8.15 @ ( quotient_of @ Q6 ) )
% 14.25/8.15 @ ( quotient_of @ P5 ) ) ) ) ).
% 14.25/8.15
% 14.25/8.15 thf(fact_7712_rat__less__eq__code,axiom,
% 14.25/8.15 ( ord_less_eq_rat
% 14.25/8.15 = ( ^ [P5: rat,Q6: rat] :
% 14.25/8.15 ( produc4947309494688390418_int_o
% 14.25/8.15 @ ^ [A6: int,C4: int] :
% 14.25/8.15 ( produc4947309494688390418_int_o
% 14.25/8.15 @ ^ [B6: int,D4: int] : ( ord_less_eq_int @ ( times_times_int @ A6 @ D4 ) @ ( times_times_int @ C4 @ B6 ) )
% 14.25/8.15 @ ( quotient_of @ Q6 ) )
% 14.25/8.15 @ ( quotient_of @ P5 ) ) ) ) ).
% 14.25/8.15
% 14.25/8.15 thf(fact_7713_rat__divide__code,axiom,
% 14.25/8.15 ! [P3: rat,Q4: rat] :
% 14.25/8.15 ( ( quotient_of @ ( divide_divide_rat @ P3 @ Q4 ) )
% 14.25/8.15 = ( produc4245557441103728435nt_int
% 14.25/8.15 @ ^ [A6: int,C4: int] :
% 14.25/8.15 ( produc4245557441103728435nt_int
% 14.25/8.15 @ ^ [B6: int,D4: int] : ( normalize @ ( product_Pair_int_int @ ( times_times_int @ A6 @ D4 ) @ ( times_times_int @ C4 @ B6 ) ) )
% 14.25/8.15 @ ( quotient_of @ Q4 ) )
% 14.25/8.15 @ ( quotient_of @ P3 ) ) ) ).
% 14.25/8.15
% 14.25/8.15 thf(fact_7714_rat__inverse__code,axiom,
% 14.25/8.15 ! [P3: rat] :
% 14.25/8.15 ( ( quotient_of @ ( inverse_inverse_rat @ P3 ) )
% 14.25/8.15 = ( produc4245557441103728435nt_int
% 14.25/8.15 @ ^ [A6: int,B6: int] : ( if_Pro3027730157355071871nt_int @ ( A6 = zero_zero_int ) @ ( product_Pair_int_int @ zero_zero_int @ one_one_int ) @ ( product_Pair_int_int @ ( times_times_int @ ( sgn_sgn_int @ A6 ) @ B6 ) @ ( abs_abs_int @ A6 ) ) )
% 14.25/8.15 @ ( quotient_of @ P3 ) ) ) ).
% 14.25/8.15
% 14.25/8.15 thf(fact_7715_divide__rat__def,axiom,
% 14.25/8.15 ( divide_divide_rat
% 14.25/8.15 = ( ^ [Q6: rat,R4: rat] : ( times_times_rat @ Q6 @ ( inverse_inverse_rat @ R4 ) ) ) ) ).
% 14.25/8.15
% 14.25/8.15 thf(fact_7716_sub__BitM__One__eq,axiom,
% 14.25/8.15 ! [N2: num] :
% 14.25/8.15 ( ( neg_numeral_sub_int @ ( bitM @ N2 ) @ one )
% 14.25/8.15 = ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( neg_numeral_sub_int @ N2 @ one ) ) ) ).
% 14.25/8.15
% 14.25/8.15 thf(fact_7717_set__decode__zero,axiom,
% 14.25/8.15 ( ( nat_set_decode @ zero_zero_nat )
% 14.25/8.15 = bot_bot_set_nat ) ).
% 14.25/8.15
% 14.25/8.15 thf(fact_7718_cofinite__eq__sequentially,axiom,
% 14.25/8.15 cofinite_nat = at_top_nat ).
% 14.25/8.15
% 14.25/8.15 thf(fact_7719_INFM__nat__inductI,axiom,
% 14.25/8.15 ! [P: nat > $o,Q2: nat > $o] :
% 14.25/8.15 ( ( P @ zero_zero_nat )
% 14.25/8.15 => ( ! [I3: nat] :
% 14.25/8.15 ( ( P @ I3 )
% 14.25/8.15 => ? [J4: nat] :
% 14.25/8.15 ( ( ord_less_nat @ I3 @ J4 )
% 14.25/8.15 & ( P @ J4 )
% 14.25/8.15 & ( Q2 @ J4 ) ) )
% 14.25/8.15 => ( frequently_nat @ Q2 @ cofinite_nat ) ) ) ).
% 14.25/8.15
% 14.25/8.15 thf(fact_7720_lcm__altdef__int,axiom,
% 14.25/8.15 ( gcd_lcm_int
% 14.25/8.15 = ( ^ [A6: int,B6: int] : ( divide_divide_int @ ( times_times_int @ ( abs_abs_int @ A6 ) @ ( abs_abs_int @ B6 ) ) @ ( gcd_gcd_int @ A6 @ B6 ) ) ) ) ).
% 14.25/8.15
% 14.25/8.15 thf(fact_7721_times__num__def,axiom,
% 14.25/8.15 ( times_times_num
% 14.25/8.15 = ( ^ [M3: num,N4: num] : ( num_of_nat @ ( times_times_nat @ ( nat_of_num @ M3 ) @ ( nat_of_num @ N4 ) ) ) ) ) ).
% 14.25/8.15
% 14.25/8.15 thf(fact_7722_nat__of__num__mult,axiom,
% 14.25/8.15 ! [X: num,Y: num] :
% 14.25/8.15 ( ( nat_of_num @ ( times_times_num @ X @ Y ) )
% 14.25/8.15 = ( times_times_nat @ ( nat_of_num @ X ) @ ( nat_of_num @ Y ) ) ) ).
% 14.25/8.15
% 14.25/8.15 thf(fact_7723_prod__gcd__lcm__int,axiom,
% 14.25/8.15 ! [M4: int,N2: int] :
% 14.25/8.15 ( ( times_times_int @ ( abs_abs_int @ M4 ) @ ( abs_abs_int @ N2 ) )
% 14.25/8.15 = ( times_times_int @ ( gcd_gcd_int @ M4 @ N2 ) @ ( gcd_lcm_int @ M4 @ N2 ) ) ) ).
% 14.25/8.15
% 14.25/8.15 thf(fact_7724_prod__gcd__lcm__nat,axiom,
% 14.25/8.15 ( times_times_nat
% 14.25/8.15 = ( ^ [M3: nat,N4: nat] : ( times_times_nat @ ( gcd_gcd_nat @ M3 @ N4 ) @ ( gcd_lcm_nat @ M3 @ N4 ) ) ) ) ).
% 14.25/8.15
% 14.25/8.15 thf(fact_7725_lcm__code__integer,axiom,
% 14.25/8.15 ( gcd_lcm_Code_integer
% 14.25/8.15 = ( ^ [A6: code_integer,B6: code_integer] : ( divide6298287555418463151nteger @ ( times_3573771949741848930nteger @ ( abs_abs_Code_integer @ A6 ) @ ( abs_abs_Code_integer @ B6 ) ) @ ( gcd_gcd_Code_integer @ A6 @ B6 ) ) ) ) ).
% 14.25/8.15
% 14.25/8.15 thf(fact_7726_lcm__nat__def,axiom,
% 14.25/8.15 ( gcd_lcm_nat
% 14.25/8.15 = ( ^ [X2: nat,Y3: nat] : ( divide_divide_nat @ ( times_times_nat @ X2 @ Y3 ) @ ( gcd_gcd_nat @ X2 @ Y3 ) ) ) ) ).
% 14.25/8.15
% 14.25/8.15 thf(fact_7727_Lcm__eq__Max__nat,axiom,
% 14.25/8.15 ! [M: set_nat] :
% 14.25/8.15 ( ( finite_finite_nat @ M )
% 14.25/8.15 => ( ( M != bot_bot_set_nat )
% 14.25/8.15 => ( ~ ( member_nat @ zero_zero_nat @ M )
% 14.25/8.15 => ( ! [M2: nat,N3: nat] :
% 14.25/8.15 ( ( member_nat @ M2 @ M )
% 14.25/8.15 => ( ( member_nat @ N3 @ M )
% 14.25/8.15 => ( member_nat @ ( gcd_lcm_nat @ M2 @ N3 ) @ M ) ) )
% 14.25/8.15 => ( ( gcd_Lcm_nat @ M )
% 14.25/8.15 = ( lattic8265883725875713057ax_nat @ M ) ) ) ) ) ) ).
% 14.25/8.15
% 14.25/8.15 thf(fact_7728_nat__of__num__sqr,axiom,
% 14.25/8.15 ! [X: num] :
% 14.25/8.15 ( ( nat_of_num @ ( sqr @ X ) )
% 14.25/8.15 = ( times_times_nat @ ( nat_of_num @ X ) @ ( nat_of_num @ X ) ) ) ).
% 14.25/8.15
% 14.25/8.15 thf(fact_7729_Lcm__nat__empty,axiom,
% 14.25/8.15 ( ( gcd_Lcm_nat @ bot_bot_set_nat )
% 14.25/8.15 = one_one_nat ) ).
% 14.25/8.15
% 14.25/8.15 thf(fact_7730_sqr__conv__mult,axiom,
% 14.25/8.15 ( sqr
% 14.25/8.15 = ( ^ [X2: num] : ( times_times_num @ X2 @ X2 ) ) ) ).
% 14.25/8.15
% 14.25/8.15 thf(fact_7731_Lcm__in__lcm__closed__set__nat,axiom,
% 14.25/8.15 ! [M: set_nat] :
% 14.25/8.15 ( ( finite_finite_nat @ M )
% 14.25/8.15 => ( ( M != bot_bot_set_nat )
% 14.25/8.15 => ( ! [M2: nat,N3: nat] :
% 14.25/8.15 ( ( member_nat @ M2 @ M )
% 14.25/8.15 => ( ( member_nat @ N3 @ M )
% 14.25/8.15 => ( member_nat @ ( gcd_lcm_nat @ M2 @ N3 ) @ M ) ) )
% 14.25/8.15 => ( member_nat @ ( gcd_Lcm_nat @ M ) @ M ) ) ) ) ).
% 14.25/8.15
% 14.25/8.15 thf(fact_7732_pow_Osimps_I3_J,axiom,
% 14.25/8.15 ! [X: num,Y: num] :
% 14.25/8.15 ( ( pow @ X @ ( bit1 @ Y ) )
% 14.25/8.15 = ( times_times_num @ ( sqr @ ( pow @ X @ Y ) ) @ X ) ) ).
% 14.25/8.15
% 14.25/8.15 thf(fact_7733_coprime__crossproduct__nat,axiom,
% 14.25/8.15 ! [A2: nat,D2: nat,B: nat,C: nat] :
% 14.25/8.15 ( ( algebr934650988132801477me_nat @ A2 @ D2 )
% 14.25/8.15 => ( ( algebr934650988132801477me_nat @ B @ C )
% 14.25/8.15 => ( ( ( times_times_nat @ A2 @ C )
% 14.25/8.15 = ( times_times_nat @ B @ D2 ) )
% 14.25/8.15 = ( ( A2 = B )
% 14.25/8.15 & ( C = D2 ) ) ) ) ) ).
% 14.25/8.15
% 14.25/8.15 thf(fact_7734_coprime__crossproduct__int,axiom,
% 14.25/8.15 ! [A2: int,D2: int,B: int,C: int] :
% 14.25/8.15 ( ( algebr932160517623751201me_int @ A2 @ D2 )
% 14.25/8.15 => ( ( algebr932160517623751201me_int @ B @ C )
% 14.25/8.15 => ( ( ( times_times_int @ ( abs_abs_int @ A2 ) @ ( abs_abs_int @ C ) )
% 14.25/8.15 = ( times_times_int @ ( abs_abs_int @ B ) @ ( abs_abs_int @ D2 ) ) )
% 14.25/8.15 = ( ( ( abs_abs_int @ A2 )
% 14.25/8.15 = ( abs_abs_int @ B ) )
% 14.25/8.15 & ( ( abs_abs_int @ C )
% 14.25/8.15 = ( abs_abs_int @ D2 ) ) ) ) ) ) ).
% 14.25/8.15
% 14.25/8.15 thf(fact_7735_in__range_Oelims_I3_J,axiom,
% 14.25/8.15 ! [X: produc3658429121746597890et_nat] :
% 14.25/8.15 ( ~ ( in_range @ X )
% 14.25/8.15 => ~ ! [H: heap_e7401611519738050253t_unit,As2: set_nat] :
% 14.25/8.15 ( ( X
% 14.25/8.15 = ( produc7507926704131184380et_nat @ H @ As2 ) )
% 14.25/8.15 => ! [X3: nat] :
% 14.25/8.15 ( ( member_nat @ X3 @ As2 )
% 14.25/8.15 => ( ord_less_nat @ X3 @ ( lim_Product_unit @ H ) ) ) ) ) ).
% 14.25/8.15
% 14.25/8.15 thf(fact_7736_in__range_Osimps,axiom,
% 14.25/8.15 ! [H2: heap_e7401611519738050253t_unit,As: set_nat] :
% 14.25/8.15 ( ( in_range @ ( produc7507926704131184380et_nat @ H2 @ As ) )
% 14.25/8.15 = ( ! [X2: nat] :
% 14.25/8.15 ( ( member_nat @ X2 @ As )
% 14.25/8.15 => ( ord_less_nat @ X2 @ ( lim_Product_unit @ H2 ) ) ) ) ) ).
% 14.25/8.15
% 14.25/8.15 thf(fact_7737_in__range_Oelims_I1_J,axiom,
% 14.25/8.15 ! [X: produc3658429121746597890et_nat,Y: $o] :
% 14.25/8.15 ( ( ( in_range @ X )
% 14.25/8.15 = Y )
% 14.25/8.15 => ~ ! [H: heap_e7401611519738050253t_unit,As2: set_nat] :
% 14.25/8.16 ( ( X
% 14.25/8.16 = ( produc7507926704131184380et_nat @ H @ As2 ) )
% 14.25/8.16 => ( Y
% 14.25/8.16 = ( ~ ! [X2: nat] :
% 14.25/8.16 ( ( member_nat @ X2 @ As2 )
% 14.25/8.16 => ( ord_less_nat @ X2 @ ( lim_Product_unit @ H ) ) ) ) ) ) ) ).
% 14.25/8.16
% 14.25/8.16 thf(fact_7738_in__range_Oelims_I2_J,axiom,
% 14.25/8.16 ! [X: produc3658429121746597890et_nat] :
% 14.25/8.16 ( ( in_range @ X )
% 14.25/8.16 => ~ ! [H: heap_e7401611519738050253t_unit,As2: set_nat] :
% 14.25/8.16 ( ( X
% 14.25/8.16 = ( produc7507926704131184380et_nat @ H @ As2 ) )
% 14.25/8.16 => ~ ! [X6: nat] :
% 14.25/8.16 ( ( member_nat @ X6 @ As2 )
% 14.25/8.16 => ( ord_less_nat @ X6 @ ( lim_Product_unit @ H ) ) ) ) ) ).
% 14.25/8.16
% 14.25/8.16 thf(fact_7739_UNIV__bool,axiom,
% 14.25/8.16 ( top_top_set_o
% 14.25/8.16 = ( insert_o @ $false @ ( insert_o @ $true @ bot_bot_set_o ) ) ) ).
% 14.25/8.16
% 14.25/8.16 thf(fact_7740_Rep__unit__induct,axiom,
% 14.25/8.16 ! [Y: $o,P: $o > $o] :
% 14.25/8.16 ( ( member_o @ Y @ ( insert_o @ $true @ bot_bot_set_o ) )
% 14.25/8.16 => ( ! [X3: product_unit] : ( P @ ( product_Rep_unit @ X3 ) )
% 14.25/8.16 => ( P @ Y ) ) ) ).
% 14.25/8.16
% 14.25/8.16 thf(fact_7741_Abs__unit__inject,axiom,
% 14.25/8.16 ! [X: $o,Y: $o] :
% 14.25/8.16 ( ( member_o @ X @ ( insert_o @ $true @ bot_bot_set_o ) )
% 14.25/8.16 => ( ( member_o @ Y @ ( insert_o @ $true @ bot_bot_set_o ) )
% 14.25/8.16 => ( ( ( product_Abs_unit @ X )
% 14.25/8.16 = ( product_Abs_unit @ Y ) )
% 14.25/8.16 = ( X = Y ) ) ) ) ).
% 14.25/8.16
% 14.25/8.16 thf(fact_7742_Rep__unit__inject,axiom,
% 14.25/8.16 ! [X: product_unit,Y: product_unit] :
% 14.25/8.16 ( ( ( product_Rep_unit @ X )
% 14.25/8.16 = ( product_Rep_unit @ Y ) )
% 14.25/8.16 = ( X = Y ) ) ).
% 14.25/8.16
% 14.25/8.16 thf(fact_7743_Rep__unit__inverse,axiom,
% 14.25/8.16 ! [X: product_unit] :
% 14.25/8.16 ( ( product_Abs_unit @ ( product_Rep_unit @ X ) )
% 14.25/8.16 = X ) ).
% 14.25/8.16
% 14.25/8.16 thf(fact_7744_Abs__unit__inverse,axiom,
% 14.25/8.16 ! [Y: $o] :
% 14.25/8.16 ( ( member_o @ Y @ ( insert_o @ $true @ bot_bot_set_o ) )
% 14.25/8.16 => ( ( product_Rep_unit @ ( product_Abs_unit @ Y ) )
% 14.25/8.16 = Y ) ) ).
% 14.25/8.16
% 14.25/8.16 thf(fact_7745_type__definition__unit,axiom,
% 14.25/8.16 type_d6188575255521822967unit_o @ product_Rep_unit @ product_Abs_unit @ ( insert_o @ $true @ bot_bot_set_o ) ).
% 14.25/8.16
% 14.25/8.16 thf(fact_7746_Rep__unit,axiom,
% 14.25/8.16 ! [X: product_unit] : ( member_o @ ( product_Rep_unit @ X ) @ ( insert_o @ $true @ bot_bot_set_o ) ) ).
% 14.25/8.16
% 14.25/8.16 thf(fact_7747_Abs__unit__cases,axiom,
% 14.25/8.16 ! [X: product_unit] :
% 14.25/8.16 ~ ! [Y2: $o] :
% 14.25/8.16 ( ( X
% 14.25/8.16 = ( product_Abs_unit @ Y2 ) )
% 14.25/8.16 => ~ ( member_o @ Y2 @ ( insert_o @ $true @ bot_bot_set_o ) ) ) ).
% 14.25/8.16
% 14.25/8.16 thf(fact_7748_Rep__unit__cases,axiom,
% 14.25/8.16 ! [Y: $o] :
% 14.25/8.16 ( ( member_o @ Y @ ( insert_o @ $true @ bot_bot_set_o ) )
% 14.25/8.16 => ~ ! [X3: product_unit] :
% 14.25/8.16 ( Y
% 14.25/8.16 = ( ~ ( product_Rep_unit @ X3 ) ) ) ) ).
% 14.25/8.16
% 14.25/8.16 thf(fact_7749_Abs__unit__induct,axiom,
% 14.25/8.16 ! [P: product_unit > $o,X: product_unit] :
% 14.25/8.16 ( ! [Y2: $o] :
% 14.25/8.16 ( ( member_o @ Y2 @ ( insert_o @ $true @ bot_bot_set_o ) )
% 14.25/8.16 => ( P @ ( product_Abs_unit @ Y2 ) ) )
% 14.25/8.16 => ( P @ X ) ) ).
% 14.25/8.16
% 14.25/8.16 thf(fact_7750_integer__of__char__code,axiom,
% 14.25/8.16 ! [B0: $o,B1: $o,B22: $o,B32: $o,B42: $o,B52: $o,B62: $o,B72: $o] :
% 14.25/8.16 ( ( integer_of_char @ ( char2 @ B0 @ B1 @ B22 @ B32 @ B42 @ B52 @ B62 @ B72 ) )
% 14.25/8.16 = ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ ( zero_n356916108424825756nteger @ B72 ) @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) @ ( zero_n356916108424825756nteger @ B62 ) ) @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) @ ( zero_n356916108424825756nteger @ B52 ) ) @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) @ ( zero_n356916108424825756nteger @ B42 ) ) @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) @ ( zero_n356916108424825756nteger @ B32 ) ) @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) @ ( zero_n356916108424825756nteger @ B22 ) ) @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) @ ( zero_n356916108424825756nteger @ B1 ) ) @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) @ ( zero_n356916108424825756nteger @ B0 ) ) ) ).
% 14.25/8.16
% 14.25/8.16 thf(fact_7751_sum__encode__def,axiom,
% 14.25/8.16 ( nat_sum_encode
% 14.25/8.16 = ( sum_ca6763686470577984908at_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 14.25/8.16 @ ^ [B6: nat] : ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B6 ) ) ) ) ).
% 14.25/8.16
% 14.25/8.16 thf(fact_7752_upt__0__eq__Nil__conv,axiom,
% 14.25/8.16 ! [J: nat] :
% 14.25/8.16 ( ( ( upt @ zero_zero_nat @ J )
% 14.25/8.16 = nil_nat )
% 14.25/8.16 = ( J = zero_zero_nat ) ) ).
% 14.25/8.16
% 14.25/8.16 thf(fact_7753_upt__merge,axiom,
% 14.25/8.16 ! [I2: nat,J: nat,K3: nat] :
% 14.25/8.16 ( ( ( ord_less_eq_nat @ I2 @ J )
% 14.25/8.16 & ( ord_less_eq_nat @ J @ K3 ) )
% 14.25/8.16 => ( ( append_nat @ ( upt @ I2 @ J ) @ ( upt @ J @ K3 ) )
% 14.25/8.16 = ( upt @ I2 @ K3 ) ) ) ).
% 14.25/8.16
% 14.25/8.16 thf(fact_7754_map__add__upt_H,axiom,
% 14.25/8.16 ! [Ofs: nat,A2: nat,B: nat] :
% 14.25/8.16 ( ( map_nat_nat
% 14.25/8.16 @ ^ [I4: nat] : ( plus_plus_nat @ I4 @ Ofs )
% 14.25/8.16 @ ( upt @ A2 @ B ) )
% 14.25/8.16 = ( upt @ ( plus_plus_nat @ A2 @ Ofs ) @ ( plus_plus_nat @ B @ Ofs ) ) ) ).
% 14.25/8.16
% 14.25/8.16 thf(fact_7755_upt__eq__append__conv,axiom,
% 14.25/8.16 ! [I2: nat,J: nat,Xs: list_nat,Ys: list_nat] :
% 14.25/8.16 ( ( ord_less_eq_nat @ I2 @ J )
% 14.25/8.16 => ( ( ( upt @ I2 @ J )
% 14.25/8.16 = ( append_nat @ Xs @ Ys ) )
% 14.25/8.16 = ( ? [K5: nat] :
% 14.25/8.16 ( ( ord_less_eq_nat @ I2 @ K5 )
% 14.25/8.16 & ( ord_less_eq_nat @ K5 @ J )
% 14.25/8.16 & ( ( upt @ I2 @ K5 )
% 14.25/8.16 = Xs )
% 14.25/8.16 & ( ( upt @ K5 @ J )
% 14.25/8.16 = Ys ) ) ) ) ) ).
% 14.25/8.16
% 14.25/8.16 thf(fact_7756_upt__append,axiom,
% 14.25/8.16 ! [I2: nat,J: nat] :
% 14.25/8.16 ( ( ord_less_nat @ I2 @ J )
% 14.25/8.16 => ( ( append_nat @ ( upt @ zero_zero_nat @ I2 ) @ ( upt @ I2 @ J ) )
% 14.25/8.16 = ( upt @ zero_zero_nat @ J ) ) ) ).
% 14.25/8.16
% 14.25/8.16 thf(fact_7757_butlast__upt,axiom,
% 14.25/8.16 ! [M4: nat,N2: nat] :
% 14.25/8.16 ( ( butlast_nat @ ( upt @ M4 @ N2 ) )
% 14.25/8.16 = ( upt @ M4 @ ( minus_minus_nat @ N2 @ one_one_nat ) ) ) ).
% 14.25/8.16
% 14.25/8.16 thf(fact_7758_upt__eq__lel__conv,axiom,
% 14.25/8.16 ! [L: nat,H2: nat,Is1: list_nat,I2: nat,Is2: list_nat] :
% 14.25/8.16 ( ( ( upt @ L @ H2 )
% 14.25/8.16 = ( append_nat @ Is1 @ ( cons_nat @ I2 @ Is2 ) ) )
% 14.25/8.16 = ( ( Is1
% 14.25/8.16 = ( upt @ L @ I2 ) )
% 14.25/8.16 & ( Is2
% 14.25/8.16 = ( upt @ ( suc @ I2 ) @ H2 ) )
% 14.25/8.16 & ( ord_less_eq_nat @ L @ I2 )
% 14.25/8.16 & ( ord_less_nat @ I2 @ H2 ) ) ) ).
% 14.25/8.16
% 14.25/8.16 thf(fact_7759_upt__filter__extend,axiom,
% 14.25/8.16 ! [U: nat,U5: nat,P: nat > $o] :
% 14.25/8.16 ( ( ord_less_eq_nat @ U @ U5 )
% 14.25/8.16 => ( ! [I3: nat] :
% 14.25/8.16 ( ( ( ord_less_eq_nat @ U @ I3 )
% 14.25/8.16 & ( ord_less_nat @ I3 @ U5 ) )
% 14.25/8.16 => ~ ( P @ I3 ) )
% 14.25/8.16 => ( ( filter_nat2 @ P @ ( upt @ zero_zero_nat @ U ) )
% 14.25/8.16 = ( filter_nat2 @ P @ ( upt @ zero_zero_nat @ U5 ) ) ) ) ) ).
% 14.25/8.16
% 14.25/8.16 thf(fact_7760_in__range_Opelims_I3_J,axiom,
% 14.25/8.16 ! [X: produc3658429121746597890et_nat] :
% 14.25/8.16 ( ~ ( in_range @ X )
% 14.25/8.16 => ( ( accp_P5801069581201407417et_nat @ in_range_rel @ X )
% 14.25/8.16 => ~ ! [H: heap_e7401611519738050253t_unit,As2: set_nat] :
% 14.25/8.16 ( ( X
% 14.25/8.16 = ( produc7507926704131184380et_nat @ H @ As2 ) )
% 14.25/8.16 => ( ( accp_P5801069581201407417et_nat @ in_range_rel @ ( produc7507926704131184380et_nat @ H @ As2 ) )
% 14.25/8.16 => ! [X3: nat] :
% 14.25/8.16 ( ( member_nat @ X3 @ As2 )
% 14.25/8.16 => ( ord_less_nat @ X3 @ ( lim_Product_unit @ H ) ) ) ) ) ) ) ).
% 14.25/8.16
% 14.25/8.16 thf(fact_7761_in__range_Opelims_I1_J,axiom,
% 14.25/8.16 ! [X: produc3658429121746597890et_nat,Y: $o] :
% 14.25/8.16 ( ( ( in_range @ X )
% 14.25/8.16 = Y )
% 14.25/8.16 => ( ( accp_P5801069581201407417et_nat @ in_range_rel @ X )
% 14.25/8.16 => ~ ! [H: heap_e7401611519738050253t_unit,As2: set_nat] :
% 14.25/8.16 ( ( X
% 14.25/8.16 = ( produc7507926704131184380et_nat @ H @ As2 ) )
% 14.25/8.16 => ( ( Y
% 14.25/8.16 = ( ! [X2: nat] :
% 14.25/8.16 ( ( member_nat @ X2 @ As2 )
% 14.25/8.16 => ( ord_less_nat @ X2 @ ( lim_Product_unit @ H ) ) ) ) )
% 14.25/8.16 => ~ ( accp_P5801069581201407417et_nat @ in_range_rel @ ( produc7507926704131184380et_nat @ H @ As2 ) ) ) ) ) ) ).
% 14.25/8.16
% 14.25/8.16 thf(fact_7762_in__range_Opelims_I2_J,axiom,
% 14.25/8.16 ! [X: produc3658429121746597890et_nat] :
% 14.25/8.16 ( ( in_range @ X )
% 14.25/8.16 => ( ( accp_P5801069581201407417et_nat @ in_range_rel @ X )
% 14.25/8.16 => ~ ! [H: heap_e7401611519738050253t_unit,As2: set_nat] :
% 14.25/8.16 ( ( X
% 14.25/8.16 = ( produc7507926704131184380et_nat @ H @ As2 ) )
% 14.25/8.16 => ( ( accp_P5801069581201407417et_nat @ in_range_rel @ ( produc7507926704131184380et_nat @ H @ As2 ) )
% 14.25/8.16 => ~ ! [X6: nat] :
% 14.25/8.16 ( ( member_nat @ X6 @ As2 )
% 14.25/8.16 => ( ord_less_nat @ X6 @ ( lim_Product_unit @ H ) ) ) ) ) ) ) ).
% 14.25/8.16
% 14.25/8.16 thf(fact_7763_one__assn__raw_Opelims_I3_J,axiom,
% 14.25/8.16 ! [X: produc3658429121746597890et_nat] :
% 14.25/8.16 ( ~ ( one_assn_raw @ X )
% 14.25/8.16 => ( ( accp_P5801069581201407417et_nat @ one_assn_raw_rel @ X )
% 14.25/8.16 => ~ ! [H: heap_e7401611519738050253t_unit,As2: set_nat] :
% 14.25/8.16 ( ( X
% 14.25/8.16 = ( produc7507926704131184380et_nat @ H @ As2 ) )
% 14.25/8.16 => ( ( accp_P5801069581201407417et_nat @ one_assn_raw_rel @ ( produc7507926704131184380et_nat @ H @ As2 ) )
% 14.25/8.16 => ( As2 = bot_bot_set_nat ) ) ) ) ) ).
% 14.25/8.16
% 14.25/8.16 thf(fact_7764_one__assn__raw_Opelims_I2_J,axiom,
% 14.25/8.16 ! [X: produc3658429121746597890et_nat] :
% 14.25/8.16 ( ( one_assn_raw @ X )
% 14.25/8.16 => ( ( accp_P5801069581201407417et_nat @ one_assn_raw_rel @ X )
% 14.25/8.16 => ~ ! [H: heap_e7401611519738050253t_unit,As2: set_nat] :
% 14.25/8.16 ( ( X
% 14.25/8.16 = ( produc7507926704131184380et_nat @ H @ As2 ) )
% 14.25/8.16 => ( ( accp_P5801069581201407417et_nat @ one_assn_raw_rel @ ( produc7507926704131184380et_nat @ H @ As2 ) )
% 14.25/8.16 => ( As2 != bot_bot_set_nat ) ) ) ) ) ).
% 14.25/8.16
% 14.25/8.16 thf(fact_7765_one__assn__raw_Opelims_I1_J,axiom,
% 14.25/8.16 ! [X: produc3658429121746597890et_nat,Y: $o] :
% 14.25/8.16 ( ( ( one_assn_raw @ X )
% 14.25/8.16 = Y )
% 14.25/8.16 => ( ( accp_P5801069581201407417et_nat @ one_assn_raw_rel @ X )
% 14.25/8.16 => ~ ! [H: heap_e7401611519738050253t_unit,As2: set_nat] :
% 14.25/8.16 ( ( X
% 14.25/8.16 = ( produc7507926704131184380et_nat @ H @ As2 ) )
% 14.25/8.16 => ( ( Y
% 14.25/8.16 = ( As2 = bot_bot_set_nat ) )
% 14.25/8.16 => ~ ( accp_P5801069581201407417et_nat @ one_assn_raw_rel @ ( produc7507926704131184380et_nat @ H @ As2 ) ) ) ) ) ) ).
% 14.25/8.16
% 14.25/8.16 thf(fact_7766_times__int__def,axiom,
% 14.25/8.16 ( times_times_int
% 14.25/8.16 = ( map_fu4960017516451851995nt_int @ rep_Integ @ ( map_fu3667384564859982768at_int @ rep_Integ @ abs_Integ )
% 14.25/8.16 @ ( produc27273713700761075at_nat
% 14.25/8.16 @ ^ [X2: nat,Y3: nat] :
% 14.25/8.16 ( produc2626176000494625587at_nat
% 14.25/8.16 @ ^ [U3: nat,V4: nat] : ( product_Pair_nat_nat @ ( plus_plus_nat @ ( times_times_nat @ X2 @ U3 ) @ ( times_times_nat @ Y3 @ V4 ) ) @ ( plus_plus_nat @ ( times_times_nat @ X2 @ V4 ) @ ( times_times_nat @ Y3 @ U3 ) ) ) ) ) ) ) ).
% 14.25/8.16
% 14.25/8.16 thf(fact_7767_next_Osimps,axiom,
% 14.25/8.16 ! [V: code_natural,W: code_natural] :
% 14.25/8.16 ( ( next @ ( produc3574140220909816553atural @ V @ W ) )
% 14.25/8.16 = ( produc6639722614265839536atural @ ( plus_p4538020629002901425atural @ ( minus_shift @ ( numera5444537566228673987atural @ ( bit0 @ ( bit1 @ ( bit0 @ ( bit1 @ ( bit0 @ ( bit1 @ ( bit0 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ one ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) @ ( minus_shift @ ( numera5444537566228673987atural @ ( bit1 @ ( bit1 @ ( bit0 @ ( bit1 @ ( bit0 @ ( bit1 @ ( bit0 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ one ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) @ ( times_2397367101498566445atural @ ( modulo8411746178871703098atural @ V @ ( numera5444537566228673987atural @ ( bit0 @ ( bit0 @ ( bit1 @ ( bit0 @ ( bit0 @ ( bit1 @ ( bit0 @ ( bit1 @ ( bit1 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit1 @ ( bit0 @ ( bit1 @ one ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) @ ( numera5444537566228673987atural @ ( bit0 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit0 @ ( bit0 @ ( bit1 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit0 @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) @ ( times_2397367101498566445atural @ ( divide5121882707175180666atural @ V @ ( numera5444537566228673987atural @ ( bit0 @ ( bit0 @ ( bit1 @ ( bit0 @ ( bit0 @ ( bit1 @ ( bit0 @ ( bit1 @ ( bit1 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit1 @ ( bit0 @ ( bit1 @ one ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) @ ( numera5444537566228673987atural @ ( bit1 @ ( bit1 @ ( bit0 @ ( bit0 @ ( bit1 @ ( bit1 @ ( bit0 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) @ ( plus_p4538020629002901425atural @ ( minus_shift @ ( numera5444537566228673987atural @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ one ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) @ ( times_2397367101498566445atural @ ( modulo8411746178871703098atural @ W @ ( numera5444537566228673987atural @ ( bit0 @ ( bit1 @ ( bit1 @ ( bit0 @ ( bit0 @ ( bit1 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit0 @ ( bit0 @ ( bit1 @ one ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) @ ( numera5444537566228673987atural @ ( bit0 @ ( bit0 @ ( bit1 @ ( bit0 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit0 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit0 @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) @ ( times_2397367101498566445atural @ ( divide5121882707175180666atural @ W @ ( numera5444537566228673987atural @ ( bit0 @ ( bit1 @ ( bit1 @ ( bit0 @ ( bit0 @ ( bit1 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit0 @ ( bit0 @ ( bit1 @ one ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) @ ( numera5444537566228673987atural @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit0 @ ( bit0 @ ( bit1 @ ( bit1 @ ( bit0 @ ( bit1 @ ( bit1 @ one ) ) ) ) ) ) ) ) ) ) ) ) ) ) @ one_one_Code_natural ) ) @ one_one_Code_natural ) @ ( produc3574140220909816553atural @ ( minus_shift @ ( numera5444537566228673987atural @ ( bit1 @ ( bit1 @ ( bit0 @ ( bit1 @ ( bit0 @ ( bit1 @ ( bit0 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ one ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) @ ( times_2397367101498566445atural @ ( modulo8411746178871703098atural @ V @ ( numera5444537566228673987atural @ ( bit0 @ ( bit0 @ ( bit1 @ ( bit0 @ ( bit0 @ ( bit1 @ ( bit0 @ ( bit1 @ ( bit1 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit1 @ ( bit0 @ ( bit1 @ one ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) @ ( numera5444537566228673987atural @ ( bit0 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit0 @ ( bit0 @ ( bit1 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit0 @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) @ ( times_2397367101498566445atural @ ( divide5121882707175180666atural @ V @ ( numera5444537566228673987atural @ ( bit0 @ ( bit0 @ ( bit1 @ ( bit0 @ ( bit0 @ ( bit1 @ ( bit0 @ ( bit1 @ ( bit1 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit1 @ ( bit0 @ ( bit1 @ one ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) @ ( numera5444537566228673987atural @ ( bit1 @ ( bit1 @ ( bit0 @ ( bit0 @ ( bit1 @ ( bit1 @ ( bit0 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) @ ( minus_shift @ ( numera5444537566228673987atural @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ one ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) @ ( times_2397367101498566445atural @ ( modulo8411746178871703098atural @ W @ ( numera5444537566228673987atural @ ( bit0 @ ( bit1 @ ( bit1 @ ( bit0 @ ( bit0 @ ( bit1 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit0 @ ( bit0 @ ( bit1 @ one ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) @ ( numera5444537566228673987atural @ ( bit0 @ ( bit0 @ ( bit1 @ ( bit0 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit0 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit0 @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) @ ( times_2397367101498566445atural @ ( divide5121882707175180666atural @ W @ ( numera5444537566228673987atural @ ( bit0 @ ( bit1 @ ( bit1 @ ( bit0 @ ( bit0 @ ( bit1 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit0 @ ( bit0 @ ( bit1 @ one ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) @ ( numera5444537566228673987atural @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit0 @ ( bit0 @ ( bit1 @ ( bit1 @ ( bit0 @ ( bit1 @ ( bit1 @ one ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% 14.25/8.16
% 14.25/8.16 thf(fact_7768_Random_Orange__def,axiom,
% 14.25/8.16 ( range
% 14.25/8.16 = ( ^ [K5: code_natural] :
% 14.25/8.16 ( produc5538323210962509403atural
% 14.25/8.16 @ ( iterat8892046348760725948atural @ ( log @ ( numera5444537566228673987atural @ ( bit1 @ ( bit0 @ ( bit0 @ ( bit1 @ ( bit0 @ ( bit1 @ ( bit0 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ one ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) @ K5 )
% 14.25/8.16 @ ^ [L2: code_natural] :
% 14.25/8.16 ( produc5538323210962509403atural @ next
% 14.25/8.16 @ ^ [V4: code_natural] : ( produc6639722614265839536atural @ ( plus_p4538020629002901425atural @ V4 @ ( times_2397367101498566445atural @ L2 @ ( numera5444537566228673987atural @ ( bit1 @ ( bit0 @ ( bit0 @ ( bit1 @ ( bit0 @ ( bit1 @ ( bit0 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ one ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) )
% 14.25/8.16 @ one_one_Code_natural )
% 14.25/8.16 @ ^ [V4: code_natural] : ( produc6639722614265839536atural @ ( modulo8411746178871703098atural @ V4 @ K5 ) ) ) ) ) ).
% 14.25/8.16
% 14.25/8.16 thf(fact_7769_bot__unit__def,axiom,
% 14.25/8.16 bot_bot_Product_unit = product_Unity ).
% 14.25/8.16
% 14.25/8.16 thf(fact_7770_sup__unit__def,axiom,
% 14.25/8.16 ( sup_sup_Product_unit
% 14.25/8.16 = ( ^ [Uu2: product_unit,Uv: product_unit] : product_Unity ) ) ).
% 14.25/8.16
% 14.25/8.16 thf(fact_7771_old_Ounit_Oexhaust,axiom,
% 14.25/8.16 ! [Y: product_unit] : ( Y = product_Unity ) ).
% 14.25/8.16
% 14.25/8.16 thf(fact_7772_Inf__unit__def,axiom,
% 14.25/8.16 ( comple2584293577114468500t_unit
% 14.25/8.16 = ( ^ [Uu2: set_Product_unit] : product_Unity ) ) ).
% 14.25/8.16
% 14.25/8.16 thf(fact_7773_uminus__unit__def,axiom,
% 14.25/8.16 ( uminus2952777764628376836t_unit
% 14.25/8.16 = ( ^ [Uu2: product_unit] : product_Unity ) ) ).
% 14.25/8.16
% 14.25/8.16 thf(fact_7774_top__unit__def,axiom,
% 14.25/8.16 top_top_Product_unit = product_Unity ).
% 14.25/8.16
% 14.25/8.16 thf(fact_7775_Unity__def,axiom,
% 14.25/8.16 ( product_Unity
% 14.25/8.16 = ( product_Abs_unit @ $true ) ) ).
% 14.25/8.16
% 14.25/8.16 thf(fact_7776_Sup__unit__def,axiom,
% 14.25/8.16 ( comple4687483117567863418t_unit
% 14.25/8.16 = ( ^ [Uu2: set_Product_unit] : product_Unity ) ) ).
% 14.25/8.16
% 14.25/8.16 thf(fact_7777_inf__unit__def,axiom,
% 14.25/8.16 ( inf_inf_Product_unit
% 14.25/8.16 = ( ^ [Uu2: product_unit,Uv: product_unit] : product_Unity ) ) ).
% 14.25/8.16
% 14.25/8.16 thf(fact_7778_UNIV__unit,axiom,
% 14.25/8.16 ( top_to1996260823553986621t_unit
% 14.25/8.16 = ( insert_Product_unit @ product_Unity @ bot_bo3957492148770167129t_unit ) ) ).
% 14.25/8.16
% 14.25/8.16 thf(fact_7779_default__unit__def,axiom,
% 14.25/8.16 defaul566961228789861419t_unit = product_Unity ).
% 14.25/8.16
% 14.25/8.16 thf(fact_7780_cone__def,axiom,
% 14.25/8.16 ( bNF_Cardinal_cone
% 14.25/8.16 = ( bNF_Ca7083678892733797993t_unit @ ( insert_Product_unit @ product_Unity @ bot_bo3957492148770167129t_unit ) ) ) ).
% 14.25/8.16
% 14.25/8.16 thf(fact_7781_times__integer__code_I3_J,axiom,
% 14.25/8.16 ! [M4: num,N2: num] :
% 14.25/8.16 ( ( times_3573771949741848930nteger @ ( code_Pos @ M4 ) @ ( code_Pos @ N2 ) )
% 14.25/8.16 = ( code_Pos @ ( times_times_num @ M4 @ N2 ) ) ) ).
% 14.25/8.16
% 14.25/8.16 thf(fact_7782_times__integer__code_I6_J,axiom,
% 14.25/8.16 ! [M4: num,N2: num] :
% 14.25/8.16 ( ( times_3573771949741848930nteger @ ( code_Neg @ M4 ) @ ( code_Neg @ N2 ) )
% 14.25/8.16 = ( code_Pos @ ( times_times_num @ M4 @ N2 ) ) ) ).
% 14.25/8.16
% 14.25/8.16 thf(fact_7783_times__integer__code_I5_J,axiom,
% 14.25/8.16 ! [M4: num,N2: num] :
% 14.25/8.16 ( ( times_3573771949741848930nteger @ ( code_Neg @ M4 ) @ ( code_Pos @ N2 ) )
% 14.25/8.16 = ( code_Neg @ ( times_times_num @ M4 @ N2 ) ) ) ).
% 14.25/8.16
% 14.25/8.16 thf(fact_7784_times__integer__code_I4_J,axiom,
% 14.25/8.16 ! [M4: num,N2: num] :
% 14.25/8.16 ( ( times_3573771949741848930nteger @ ( code_Pos @ M4 ) @ ( code_Neg @ N2 ) )
% 14.25/8.16 = ( code_Neg @ ( times_times_num @ M4 @ N2 ) ) ) ).
% 14.25/8.16
% 14.25/8.16 thf(fact_7785_times__natural_Orep__eq,axiom,
% 14.25/8.16 ! [X: code_natural,Xa: code_natural] :
% 14.25/8.16 ( ( code_nat_of_natural @ ( times_2397367101498566445atural @ X @ Xa ) )
% 14.25/8.16 = ( times_times_nat @ ( code_nat_of_natural @ X ) @ ( code_nat_of_natural @ Xa ) ) ) ).
% 14.25/8.16
% 14.25/8.16 thf(fact_7786_times__int__code_I3_J,axiom,
% 14.25/8.16 ! [M4: num,N2: num] :
% 14.25/8.16 ( ( times_times_int @ ( pos @ M4 ) @ ( pos @ N2 ) )
% 14.25/8.16 = ( pos @ ( times_times_num @ M4 @ N2 ) ) ) ).
% 14.25/8.16
% 14.25/8.16 thf(fact_7787_times__int__code_I4_J,axiom,
% 14.25/8.16 ! [M4: num,N2: num] :
% 14.25/8.16 ( ( times_times_int @ ( pos @ M4 ) @ ( neg @ N2 ) )
% 14.25/8.16 = ( neg @ ( times_times_num @ M4 @ N2 ) ) ) ).
% 14.25/8.16
% 14.25/8.16 thf(fact_7788_times__int__code_I5_J,axiom,
% 14.25/8.16 ! [M4: num,N2: num] :
% 14.25/8.16 ( ( times_times_int @ ( neg @ M4 ) @ ( pos @ N2 ) )
% 14.25/8.16 = ( neg @ ( times_times_num @ M4 @ N2 ) ) ) ).
% 14.25/8.16
% 14.25/8.16 thf(fact_7789_times__int__code_I6_J,axiom,
% 14.25/8.16 ! [M4: num,N2: num] :
% 14.25/8.16 ( ( times_times_int @ ( neg @ M4 ) @ ( neg @ N2 ) )
% 14.25/8.16 = ( pos @ ( times_times_num @ M4 @ N2 ) ) ) ).
% 14.25/8.16
% 14.25/8.16 thf(fact_7790_times__natural_Otransfer,axiom,
% 14.25/8.16 bNF_re88643428490162567atural @ code_pcr_natural @ ( bNF_re3704215830270325841atural @ code_pcr_natural @ code_pcr_natural ) @ times_times_nat @ times_2397367101498566445atural ).
% 14.25/8.16
% 14.25/8.16 thf(fact_7791_times__natural_Oabs__eq,axiom,
% 14.25/8.16 ! [Xa: nat,X: nat] :
% 14.25/8.16 ( ( times_2397367101498566445atural @ ( code_natural_of_nat @ Xa ) @ ( code_natural_of_nat @ X ) )
% 14.25/8.16 = ( code_natural_of_nat @ ( times_times_nat @ Xa @ X ) ) ) ).
% 14.25/8.16
% 14.25/8.16 thf(fact_7792_type__definition__natural,axiom,
% 14.25/8.16 type_d4410041424927559462al_nat @ code_nat_of_natural @ code_natural_of_nat @ top_top_set_nat ).
% 14.25/8.16
% 14.25/8.16 thf(fact_7793_times__natural__def,axiom,
% 14.25/8.16 ( times_2397367101498566445atural
% 14.25/8.16 = ( map_fu6549440983881763648atural @ code_nat_of_natural @ ( map_fu1239815594074539274atural @ code_nat_of_natural @ code_natural_of_nat ) @ times_times_nat ) ) ).
% 14.25/8.16
% 14.25/8.16 thf(fact_7794_times__integer_Otransfer,axiom,
% 14.25/8.16 bNF_re398004352372739002nteger @ code_pcr_integer @ ( bNF_re3379532845092657523nteger @ code_pcr_integer @ code_pcr_integer ) @ times_times_int @ times_3573771949741848930nteger ).
% 14.25/8.16
% 14.25/8.16 thf(fact_7795_times__integer__def,axiom,
% 14.25/8.16 ( times_3573771949741848930nteger
% 14.25/8.16 = ( map_fu8272188784021352819nteger @ code_int_of_integer @ ( map_fu2599414010547811884nteger @ code_int_of_integer @ code_integer_of_int ) @ times_times_int ) ) ).
% 14.25/8.16
% 14.25/8.16 thf(help_If_2_1_If_001t__Int__Oint_T,axiom,
% 14.25/8.16 ! [X: int,Y: int] :
% 14.25/8.16 ( ( if_int @ $false @ X @ Y )
% 14.25/8.16 = Y ) ).
% 14.25/8.16
% 14.25/8.16 thf(help_If_1_1_If_001t__Int__Oint_T,axiom,
% 14.25/8.16 ! [X: int,Y: int] :
% 14.25/8.16 ( ( if_int @ $true @ X @ Y )
% 14.25/8.16 = X ) ).
% 14.25/8.16
% 14.25/8.16 thf(help_If_2_1_If_001t__Nat__Onat_T,axiom,
% 14.25/8.16 ! [X: nat,Y: nat] :
% 14.25/8.16 ( ( if_nat @ $false @ X @ Y )
% 14.25/8.16 = Y ) ).
% 14.25/8.16
% 14.25/8.16 thf(help_If_1_1_If_001t__Nat__Onat_T,axiom,
% 14.25/8.16 ! [X: nat,Y: nat] :
% 14.25/8.16 ( ( if_nat @ $true @ X @ Y )
% 14.25/8.16 = X ) ).
% 14.25/8.16
% 14.25/8.16 thf(help_If_2_1_If_001t__Rat__Orat_T,axiom,
% 14.25/8.16 ! [X: rat,Y: rat] :
% 14.25/8.16 ( ( if_rat @ $false @ X @ Y )
% 14.25/8.16 = Y ) ).
% 14.25/8.16
% 14.25/8.16 thf(help_If_1_1_If_001t__Rat__Orat_T,axiom,
% 14.25/8.16 ! [X: rat,Y: rat] :
% 14.25/8.16 ( ( if_rat @ $true @ X @ Y )
% 14.25/8.16 = X ) ).
% 14.25/8.16
% 14.25/8.16 thf(help_If_2_1_If_001t__Set__Oset_I_Eo_J_T,axiom,
% 14.25/8.16 ! [X: set_o,Y: set_o] :
% 14.25/8.16 ( ( if_set_o @ $false @ X @ Y )
% 14.25/8.16 = Y ) ).
% 14.25/8.16
% 14.25/8.16 thf(help_If_1_1_If_001t__Set__Oset_I_Eo_J_T,axiom,
% 14.25/8.16 ! [X: set_o,Y: set_o] :
% 14.25/8.16 ( ( if_set_o @ $true @ X @ Y )
% 14.25/8.16 = X ) ).
% 14.25/8.16
% 14.25/8.16 thf(help_If_2_1_If_001t__Assertions__Oassn_T,axiom,
% 14.25/8.16 ! [X: assn,Y: assn] :
% 14.25/8.16 ( ( if_assn @ $false @ X @ Y )
% 14.25/8.16 = Y ) ).
% 14.25/8.16
% 14.25/8.16 thf(help_If_1_1_If_001t__Assertions__Oassn_T,axiom,
% 14.25/8.16 ! [X: assn,Y: assn] :
% 14.25/8.16 ( ( if_assn @ $true @ X @ Y )
% 14.25/8.16 = X ) ).
% 14.25/8.16
% 14.25/8.16 thf(help_If_2_1_If_001t__Code____Numeral__Ointeger_T,axiom,
% 14.25/8.16 ! [X: code_integer,Y: code_integer] :
% 14.25/8.16 ( ( if_Code_integer @ $false @ X @ Y )
% 14.25/8.16 = Y ) ).
% 14.25/8.16
% 14.25/8.16 thf(help_If_1_1_If_001t__Code____Numeral__Ointeger_T,axiom,
% 14.25/8.16 ! [X: code_integer,Y: code_integer] :
% 14.25/8.16 ( ( if_Code_integer @ $true @ X @ Y )
% 14.25/8.16 = X ) ).
% 14.25/8.16
% 14.25/8.16 thf(help_If_2_1_If_001t__Code____Numeral__Onatural_T,axiom,
% 14.25/8.16 ! [X: code_natural,Y: code_natural] :
% 14.25/8.16 ( ( if_Code_natural @ $false @ X @ Y )
% 14.25/8.16 = Y ) ).
% 14.25/8.16
% 14.25/8.16 thf(help_If_1_1_If_001t__Code____Numeral__Onatural_T,axiom,
% 14.25/8.16 ! [X: code_natural,Y: code_natural] :
% 14.25/8.16 ( ( if_Code_natural @ $true @ X @ Y )
% 14.25/8.16 = X ) ).
% 14.25/8.16
% 14.25/8.16 thf(help_If_2_1_If_001t__Set__Oset_It__Int__Oint_J_T,axiom,
% 14.25/8.16 ! [X: set_int,Y: set_int] :
% 14.25/8.16 ( ( if_set_int @ $false @ X @ Y )
% 14.25/8.16 = Y ) ).
% 14.25/8.16
% 14.25/8.16 thf(help_If_1_1_If_001t__Set__Oset_It__Int__Oint_J_T,axiom,
% 14.25/8.16 ! [X: set_int,Y: set_int] :
% 14.25/8.16 ( ( if_set_int @ $true @ X @ Y )
% 14.25/8.16 = X ) ).
% 14.25/8.16
% 14.25/8.16 thf(help_If_2_1_If_001t__Set__Oset_It__Nat__Onat_J_T,axiom,
% 14.25/8.16 ! [X: set_nat,Y: set_nat] :
% 14.25/8.16 ( ( if_set_nat @ $false @ X @ Y )
% 14.25/8.16 = Y ) ).
% 14.25/8.16
% 14.25/8.16 thf(help_If_1_1_If_001t__Set__Oset_It__Nat__Onat_J_T,axiom,
% 14.25/8.16 ! [X: set_nat,Y: set_nat] :
% 14.25/8.16 ( ( if_set_nat @ $true @ X @ Y )
% 14.25/8.16 = X ) ).
% 14.25/8.16
% 14.25/8.16 thf(help_If_2_1_If_001t__Filter__Ofilter_It__Nat__Onat_J_T,axiom,
% 14.25/8.16 ! [X: filter_nat,Y: filter_nat] :
% 14.25/8.16 ( ( if_filter_nat @ $false @ X @ Y )
% 14.25/8.16 = Y ) ).
% 14.25/8.16
% 14.25/8.16 thf(help_If_1_1_If_001t__Filter__Ofilter_It__Nat__Onat_J_T,axiom,
% 14.25/8.16 ! [X: filter_nat,Y: filter_nat] :
% 14.25/8.16 ( ( if_filter_nat @ $true @ X @ Y )
% 14.25/8.16 = X ) ).
% 14.25/8.16
% 14.25/8.16 thf(help_If_2_1_If_001t__Set__Oset_It__Product____Type__Ounit_J_T,axiom,
% 14.25/8.16 ! [X: set_Product_unit,Y: set_Product_unit] :
% 14.25/8.16 ( ( if_set_Product_unit @ $false @ X @ Y )
% 14.25/8.16 = Y ) ).
% 14.25/8.16
% 14.25/8.16 thf(help_If_1_1_If_001t__Set__Oset_It__Product____Type__Ounit_J_T,axiom,
% 14.25/8.16 ! [X: set_Product_unit,Y: set_Product_unit] :
% 14.25/8.16 ( ( if_set_Product_unit @ $true @ X @ Y )
% 14.25/8.16 = X ) ).
% 14.25/8.16
% 14.25/8.16 thf(help_If_2_1_If_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_T,axiom,
% 14.25/8.16 ! [X: product_prod_int_int,Y: product_prod_int_int] :
% 14.25/8.16 ( ( if_Pro3027730157355071871nt_int @ $false @ X @ Y )
% 14.25/8.16 = Y ) ).
% 14.25/8.16
% 14.25/8.16 thf(help_If_1_1_If_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_T,axiom,
% 14.25/8.16 ! [X: product_prod_int_int,Y: product_prod_int_int] :
% 14.25/8.16 ( ( if_Pro3027730157355071871nt_int @ $true @ X @ Y )
% 14.25/8.16 = X ) ).
% 14.25/8.16
% 14.25/8.16 thf(help_If_2_1_If_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_T,axiom,
% 14.25/8.16 ! [X: product_prod_nat_nat,Y: product_prod_nat_nat] :
% 14.25/8.16 ( ( if_Pro6206227464963214023at_nat @ $false @ X @ Y )
% 14.25/8.16 = Y ) ).
% 14.25/8.16
% 14.25/8.16 thf(help_If_1_1_If_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_T,axiom,
% 14.25/8.16 ! [X: product_prod_nat_nat,Y: product_prod_nat_nat] :
% 14.25/8.16 ( ( if_Pro6206227464963214023at_nat @ $true @ X @ Y )
% 14.25/8.16 = X ) ).
% 14.25/8.16
% 14.25/8.16 thf(help_If_3_1_If_001t__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J_T,axiom,
% 14.25/8.16 ! [P: $o] :
% 14.25/8.16 ( ( P = $true )
% 14.25/8.16 | ( P = $false ) ) ).
% 14.25/8.16
% 14.25/8.16 thf(help_If_2_1_If_001t__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J_T,axiom,
% 14.25/8.16 ! [X: produc8923325533196201883nteger,Y: produc8923325533196201883nteger] :
% 14.25/8.16 ( ( if_Pro6119634080678213985nteger @ $false @ X @ Y )
% 14.25/8.16 = Y ) ).
% 14.25/8.16
% 14.25/8.16 thf(help_If_1_1_If_001t__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J_T,axiom,
% 14.25/8.16 ! [X: produc8923325533196201883nteger,Y: produc8923325533196201883nteger] :
% 14.25/8.16 ( ( if_Pro6119634080678213985nteger @ $true @ X @ Y )
% 14.25/8.16 = X ) ).
% 14.25/8.16
% 14.25/8.16 thf(conj_0,conjecture,
% 14.25/8.16 ( ( rep_assn @ ( abs_assn @ one_assn_raw ) @ h )
% 14.25/8.16 = ( ( produc8586169260539613262et_nat @ h )
% 14.25/8.16 = bot_bot_set_nat ) ) ).
% 14.25/8.16 ------------------------
% 14.26/8.87 % (22824)lrs+1002_1:128_aac=none:au=on:cnfonf=lazy_not_gen_be_off:sos=all:i=2:si=on:rtra=on_0 on DTF2THF_22696 for (2993ds/2Mi)
% 14.26/8.87 % (22823)lrs+10_1:1_au=on:inj=on:i=2:si=on:rtra=on_0 on DTF2THF_22696 for (2993ds/2Mi)
% 14.26/8.87 % (22822)dis+1010_1:1_au=on:cbe=off:chr=on:fsr=off:hfsq=on:nm=64:sos=theory:sp=weighted_frequency:i=27:si=on:rtra=on_0 on DTF2THF_22696 for (2993ds/27Mi)
% 14.26/8.87 % (22820)lrs+1002_1:8_bd=off:fd=off:hud=10:tnu=1:i=183:si=on:rtra=on_0 on DTF2THF_22696 for (2993ds/183Mi)
% 14.26/8.88 % (22823)Instruction limit reached!
% 14.26/8.88 % (22823)------------------------------
% 14.26/8.88 % (22823)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 14.26/8.88 % (22823)Termination reason: Unknown
% 14.26/8.88 % (22823)Termination phase: shuffling
% 14.26/8.88
% 14.26/8.88 % (22823)Memory used [KB]: 11513
% 14.26/8.88 % (22823)Time elapsed: 0.003 s
% 14.26/8.88 % (22823)Instructions burned: 2 (million)
% 14.26/8.88 % (22823)------------------------------
% 14.26/8.88 % (22823)------------------------------
% 14.26/8.88 % (22824)Instruction limit reached!
% 14.26/8.88 % (22824)------------------------------
% 14.26/8.88 % (22824)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 14.26/8.88 % (22824)Termination reason: Unknown
% 14.26/8.88 % (22824)Termination phase: shuffling
% 14.26/8.88
% 14.26/8.88 % (22824)Memory used [KB]: 11513
% 14.26/8.88 % (22824)Time elapsed: 0.004 s
% 14.26/8.88 % (22824)Instructions burned: 2 (million)
% 14.26/8.88 % (22824)------------------------------
% 14.26/8.88 % (22824)------------------------------
% 14.26/8.88 % (22825)lrs+1002_1:1_au=on:bd=off:e2e=on:sd=2:sos=on:ss=axioms:i=275:si=on:rtra=on_0 on DTF2THF_22696 for (2993ds/275Mi)
% 14.26/8.88 % (22821)lrs+10_1:1_c=on:cnfonf=conj_eager:fd=off:fe=off:kws=frequency:spb=intro:i=4:si=on:rtra=on_0 on DTF2THF_22696 for (2993ds/4Mi)
% 14.26/8.88 % (22826)lrs+1004_1:128_cond=on:e2e=on:sp=weighted_frequency:i=18:si=on:rtra=on_0 on DTF2THF_22696 for (2993ds/18Mi)
% 14.26/8.88 % (22821)Instruction limit reached!
% 14.26/8.88 % (22821)------------------------------
% 14.26/8.88 % (22821)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 14.26/8.88 % (22821)Termination reason: Unknown
% 14.26/8.88 % (22821)Termination phase: shuffling
% 14.26/8.88
% 14.26/8.88 % (22821)Memory used [KB]: 11641
% 14.26/8.88 % (22821)Time elapsed: 0.006 s
% 14.26/8.88 % (22821)Instructions burned: 4 (million)
% 14.26/8.88 % (22821)------------------------------
% 14.26/8.88 % (22821)------------------------------
% 14.26/8.89 % (22827)lrs+10_1:1_bet=on:cnfonf=off:fd=off:hud=5:inj=on:i=3:si=on:rtra=on_0 on DTF2THF_22696 for (2992ds/3Mi)
% 14.26/8.90 % (22822)Instruction limit reached!
% 14.26/8.90 % (22822)------------------------------
% 14.26/8.90 % (22822)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 14.26/8.90 % (22822)Termination reason: Unknown
% 14.26/8.90 % (22822)Termination phase: shuffling
% 14.26/8.90
% 14.26/8.90 % (22822)Memory used [KB]: 13560
% 14.26/8.90 % (22822)Time elapsed: 0.024 s
% 14.26/8.90 % (22822)Instructions burned: 27 (million)
% 14.26/8.90 % (22822)------------------------------
% 14.26/8.90 % (22822)------------------------------
% 14.26/8.90 % (22826)Instruction limit reached!
% 14.26/8.90 % (22826)------------------------------
% 14.26/8.90 % (22826)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 14.26/8.90 % (22826)Termination reason: Unknown
% 14.26/8.90 % (22826)Termination phase: shuffling
% 14.26/8.90
% 14.26/8.90 % (22826)Memory used [KB]: 12409
% 14.26/8.90 % (22826)Time elapsed: 0.021 s
% 14.26/8.90 % (22826)Instructions burned: 18 (million)
% 14.26/8.90 % (22826)------------------------------
% 14.26/8.90 % (22826)------------------------------
% 14.26/8.90 % (22827)Instruction limit reached!
% 14.26/8.90 % (22827)------------------------------
% 14.26/8.90 % (22827)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 14.26/8.90 % (22827)Termination reason: Unknown
% 14.26/8.90 % (22827)Termination phase: shuffling
% 14.26/8.90
% 14.26/8.90 % (22827)Memory used [KB]: 11641
% 14.26/8.90 % (22827)Time elapsed: 0.005 s
% 14.26/8.90 % (22827)Instructions burned: 3 (million)
% 14.26/8.90 % (22827)------------------------------
% 14.26/8.90 % (22827)------------------------------
% 14.26/8.90 % (22828)lrs+1002_1:1_cnfonf=lazy_not_be_gen:hud=14:prag=on:sp=weighted_frequency:tnu=1:i=37:si=on:rtra=on_0 on DTF2THF_22696 for (2992ds/37Mi)
% 14.26/8.91 % (22829)lrs+2_16:1_acc=model:au=on:bd=off:c=on:e2e=on:nm=2:sos=all:i=15:si=on:rtra=on_0 on DTF2THF_22696 for (2992ds/15Mi)
% 14.26/8.92 % (22830)dis+21_1:1_cbe=off:cnfonf=off:fs=off:fsr=off:hud=1:inj=on:i=3:si=on:rtra=on_0 on DTF2THF_22696 for (2992ds/3Mi)
% 14.26/8.92 % (22832)lrs+10_1:1_av=off:chr=on:plsq=on:slsq=on:i=7:si=on:rtra=on_0 on DTF2THF_22696 for (2992ds/7Mi)
% 14.26/8.92 % (22830)Instruction limit reached!
% 14.26/8.92 % (22830)------------------------------
% 14.26/8.92 % (22830)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 14.26/8.92 % (22830)Termination reason: Unknown
% 14.26/8.92 % (22830)Termination phase: shuffling
% 14.26/8.92
% 14.26/8.92 % (22830)Memory used [KB]: 11513
% 14.26/8.92 % (22830)Time elapsed: 0.005 s
% 14.26/8.92 % (22830)Instructions burned: 3 (million)
% 14.26/8.92 % (22830)------------------------------
% 14.26/8.92 % (22830)------------------------------
% 14.26/8.92 % (22831)lrs+1002_1:1_aac=none:au=on:cnfonf=lazy_gen:plsq=on:plsqc=1:plsqr=4203469,65536:i=1041:si=on:rtra=on_0 on DTF2THF_22696 for (2992ds/1041Mi)
% 14.26/8.92 % (22829)Instruction limit reached!
% 14.26/8.92 % (22829)------------------------------
% 14.26/8.92 % (22829)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 14.26/8.92 % (22829)Termination reason: Unknown
% 14.26/8.92 % (22829)Termination phase: shuffling
% 14.26/8.92
% 14.26/8.92 % (22829)Memory used [KB]: 12281
% 14.26/8.92 % (22829)Time elapsed: 0.019 s
% 14.26/8.92 % (22829)Instructions burned: 15 (million)
% 14.26/8.92 % (22829)------------------------------
% 14.26/8.92 % (22829)------------------------------
% 14.26/8.92 % (22832)Instruction limit reached!
% 14.26/8.92 % (22832)------------------------------
% 14.26/8.92 % (22832)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 14.26/8.92 % (22832)Termination reason: Unknown
% 14.26/8.92 % (22832)Termination phase: shuffling
% 14.26/8.92
% 14.26/8.92 % (22832)Memory used [KB]: 11897
% 14.26/8.92 % (22832)Time elapsed: 0.010 s
% 14.26/8.92 % (22832)Instructions burned: 7 (million)
% 14.26/8.92 % (22832)------------------------------
% 14.26/8.92 % (22832)------------------------------
% 14.26/8.93 % (22828)Instruction limit reached!
% 14.26/8.93 % (22828)------------------------------
% 14.26/8.93 % (22828)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 14.26/8.93 % (22828)Termination reason: Unknown
% 14.26/8.93 % (22828)Termination phase: shuffling
% 14.26/8.93
% 14.26/8.93 % (22828)Memory used [KB]: 13688
% 14.26/8.93 % (22828)Time elapsed: 0.033 s
% 14.26/8.93 % (22828)Instructions burned: 38 (million)
% 14.26/8.93 % (22828)------------------------------
% 14.26/8.93 % (22828)------------------------------
% 14.26/8.94 % (22833)lrs+10_1:1_acc=on:amm=sco:cs=on:tgt=full:i=16:si=on:rtra=on_0 on DTF2THF_22696 for (2992ds/16Mi)
% 14.26/8.94 % (22835)lrs+2_1:1_apa=on:au=on:bd=preordered:cnfonf=off:cs=on:ixr=off:sos=on:i=3:si=on:rtra=on_0 on DTF2THF_22696 for (2992ds/3Mi)
% 14.26/8.94 % (22835)Instruction limit reached!
% 14.26/8.94 % (22835)------------------------------
% 14.26/8.94 % (22835)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 14.26/8.94 % (22835)Termination reason: Unknown
% 14.26/8.94 % (22835)Termination phase: shuffling
% 14.26/8.94
% 14.26/8.94 % (22835)Memory used [KB]: 11641
% 14.26/8.94 % (22835)Time elapsed: 0.005 s
% 14.26/8.94 % (22835)Instructions burned: 3 (million)
% 14.26/8.94 % (22835)------------------------------
% 14.26/8.94 % (22835)------------------------------
% 14.26/8.95 % (22834)lrs+21_1:1_au=on:cnfonf=off:fd=preordered:fe=off:fsr=off:hud=11:inj=on:kws=precedence:s2pl=no:sp=weighted_frequency:tgt=full:i=3:si=on:rtra=on_0 on DTF2THF_22696 for (2992ds/3Mi)
% 14.26/8.95 % (22834)Instruction limit reached!
% 14.26/8.95 % (22834)------------------------------
% 14.26/8.95 % (22834)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 14.26/8.95 % (22834)Termination reason: Unknown
% 14.26/8.95 % (22834)Termination phase: shuffling
% 14.26/8.95
% 14.26/8.95 % (22834)Memory used [KB]: 11513
% 14.26/8.95 % (22834)Time elapsed: 0.005 s
% 14.26/8.95 % (22834)Instructions burned: 3 (million)
% 14.26/8.95 % (22834)------------------------------
% 14.26/8.95 % (22834)------------------------------
% 14.26/8.95 % (22836)lrs+10_1:1_cnfonf=off:cs=on:hud=3:prag=on:sup=off:i=7:si=on:rtra=on_0 on DTF2THF_22696 for (2992ds/7Mi)
% 14.26/8.95 % (22833)Instruction limit reached!
% 14.26/8.95 % (22833)------------------------------
% 14.26/8.95 % (22833)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 14.26/8.95 % (22833)Termination reason: Unknown
% 14.26/8.95 % (22833)Termination phase: shuffling
% 14.26/8.95
% 14.26/8.95 % (22833)Memory used [KB]: 12281
% 14.26/8.95 % (22833)Time elapsed: 0.018 s
% 14.26/8.95 % (22833)Instructions burned: 16 (million)
% 14.26/8.95 % (22833)------------------------------
% 14.26/8.95 % (22833)------------------------------
% 14.26/8.96 % (22836)Instruction limit reached!
% 14.26/8.96 % (22836)------------------------------
% 14.26/8.96 % (22836)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 14.26/8.96 % (22836)Termination reason: Unknown
% 14.26/8.96 % (22836)Termination phase: shuffling
% 14.26/8.96
% 14.26/8.96 % (22836)Memory used [KB]: 11897
% 14.26/8.96 % (22836)Time elapsed: 0.010 s
% 14.26/8.96 % (22836)Instructions burned: 7 (million)
% 14.26/8.96 % (22836)------------------------------
% 14.26/8.96 % (22836)------------------------------
% 14.26/8.96 % (22837)dis+1002_1:1_add=large:cnfonf=lazy_pi_sigma_gen:fe=off:prag=on:i=3:si=on:rtra=on_0 on DTF2THF_22696 for (2992ds/3Mi)
% 14.26/8.97 % (22837)Instruction limit reached!
% 14.26/8.97 % (22837)------------------------------
% 14.26/8.97 % (22837)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 14.26/8.97 % (22837)Termination reason: Unknown
% 14.26/8.97 % (22837)Termination phase: shuffling
% 14.26/8.97
% 14.26/8.97 % (22837)Memory used [KB]: 11513
% 14.26/8.97 % (22837)Time elapsed: 0.004 s
% 14.26/8.97 % (22837)Instructions burned: 3 (million)
% 14.26/8.97 % (22837)------------------------------
% 14.26/8.97 % (22837)------------------------------
% 14.26/8.97 % (22838)dis+1004_1:1_cha=on:cs=on:fe=off:hud=1:i=4:si=on:rtra=on_0 on DTF2THF_22696 for (2992ds/4Mi)
% 14.26/8.97 % (22838)Instruction limit reached!
% 14.26/8.97 % (22838)------------------------------
% 14.26/8.97 % (22838)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 14.26/8.97 % (22838)Termination reason: Unknown
% 14.26/8.97 % (22838)Termination phase: shuffling
% 14.26/8.97
% 14.26/8.97 % (22838)Memory used [KB]: 11641
% 14.26/8.97 % (22838)Time elapsed: 0.006 s
% 14.26/8.97 % (22838)Instructions burned: 4 (million)
% 14.26/8.97 % (22838)------------------------------
% 14.26/8.97 % (22838)------------------------------
% 14.26/8.98 % (22839)lrs+1002_1:1_anc=all_dependent:au=on:cbe=off:fde=unused:ntd=on:i=18:si=on:rtra=on_0 on DTF2THF_22696 for (2992ds/18Mi)
% 14.26/8.98 % (22840)lrs+10_1:1_e2e=on:sd=1:sgt=8:ss=axioms:i=710:si=on:rtra=on_0 on DTF2THF_22696 for (2992ds/710Mi)
% 14.26/8.98 % (22841)lrs+1004_1:1_chr=on:prag=on:i=6:si=on:rtra=on_0 on DTF2THF_22696 for (2992ds/6Mi)
% 14.26/8.99 % (22820)Instruction limit reached!
% 14.26/8.99 % (22820)------------------------------
% 14.26/8.99 % (22820)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 14.26/8.99 % (22820)Termination reason: Unknown
% 14.26/8.99 % (22820)Termination phase: shuffling
% 14.26/8.99
% 14.26/8.99 % (22820)Memory used [KB]: 15863
% 14.26/8.99 % (22820)Time elapsed: 0.115 s
% 14.26/8.99 % (22820)Instructions burned: 183 (million)
% 14.26/8.99 % (22820)------------------------------
% 14.26/8.99 % (22820)------------------------------
% 14.26/8.99 % (22841)Instruction limit reached!
% 14.26/8.99 % (22841)------------------------------
% 14.26/8.99 % (22841)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 14.26/8.99 % (22841)Termination reason: Unknown
% 14.26/8.99 % (22841)Termination phase: shuffling
% 14.26/8.99
% 14.26/8.99 % (22841)Memory used [KB]: 11897
% 14.26/8.99 % (22841)Time elapsed: 0.009 s
% 14.26/8.99 % (22841)Instructions burned: 6 (million)
% 14.26/8.99 % (22841)------------------------------
% 14.26/8.99 % (22841)------------------------------
% 14.26/8.99 % (22842)dis+1002_5:1_au=on:bd=off:e2e=on:fde=none:fs=off:fsr=off:sos=on:i=902:si=on:rtra=on_0 on DTF2THF_22696 for (2991ds/902Mi)
% 14.26/8.99 % (22839)Instruction limit reached!
% 14.26/8.99 % (22839)------------------------------
% 14.26/8.99 % (22839)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 14.26/8.99 % (22839)Termination reason: Unknown
% 14.26/8.99 % (22839)Termination phase: shuffling
% 14.26/8.99
% 14.26/8.99 % (22839)Memory used [KB]: 12409
% 14.26/8.99 % (22839)Time elapsed: 0.019 s
% 14.26/8.99 % (22839)Instructions burned: 18 (million)
% 14.26/8.99 % (22839)------------------------------
% 14.26/8.99 % (22839)------------------------------
% 14.26/9.01 % (22844)dis+10_1:1_cnfonf=lazy_gen:fe=off:i=5:si=on:rtra=on_0 on DTF2THF_22696 for (2991ds/5Mi)
% 14.26/9.01 % (22843)dis+21_1:8_apa=on:cnfonf=off:fd=off:fsr=off:hud=0:ins=1:kws=inv_frequency:nwc=10.0:ss=axioms:st=5.0:i=21:si=on:rtra=on_0 on DTF2THF_22696 for (2991ds/21Mi)
% 14.26/9.01 % (22844)Instruction limit reached!
% 14.26/9.01 % (22844)------------------------------
% 14.26/9.01 % (22844)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 14.26/9.01 % (22844)Termination reason: Unknown
% 14.26/9.01 % (22844)Termination phase: shuffling
% 14.26/9.01
% 14.26/9.01 % (22844)Memory used [KB]: 11769
% 14.26/9.01 % (22844)Time elapsed: 0.008 s
% 14.26/9.01 % (22844)Instructions burned: 5 (million)
% 14.26/9.01 % (22844)------------------------------
% 14.26/9.01 % (22844)------------------------------
% 14.26/9.02 % (22845)lrs+2_1:1_cnfonf=lazy_not_gen_be_off:cs=on:fe=off:hud=10:inj=on:ins=3:plsq=on:plsqc=1:sd=10:ss=axioms:tnu=1:i=6:si=on:rtra=on_0 on DTF2THF_22696 for (2991ds/6Mi)
% 14.26/9.02 % (22845)Instruction limit reached!
% 14.26/9.02 % (22845)------------------------------
% 14.26/9.02 % (22845)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 14.26/9.02 % (22845)Termination reason: Unknown
% 14.26/9.02 % (22845)Termination phase: shuffling
% 14.26/9.02
% 14.26/9.02 % (22845)Memory used [KB]: 11769
% 14.26/9.02 % (22845)Time elapsed: 0.010 s
% 14.26/9.02 % (22845)Instructions burned: 6 (million)
% 14.26/9.02 % (22845)------------------------------
% 14.26/9.02 % (22845)------------------------------
% 16.01/9.03 % (22843)Instruction limit reached!
% 16.01/9.03 % (22843)------------------------------
% 16.01/9.03 % (22843)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 16.01/9.03 % (22843)Termination reason: Unknown
% 16.01/9.03 % (22843)Termination phase: shuffling
% 16.01/9.03
% 16.01/9.03 % (22843)Memory used [KB]: 12409
% 16.01/9.03 % (22843)Time elapsed: 0.024 s
% 16.01/9.03 % (22843)Instructions burned: 22 (million)
% 16.01/9.03 % (22843)------------------------------
% 16.01/9.03 % (22843)------------------------------
% 16.01/9.03 % (22846)lrs+1002_1:128_au=on:c=on:fsr=off:piset=equals:i=377:si=on:rtra=on_0 on DTF2THF_22696 for (2991ds/377Mi)
% 16.01/9.05 % (22847)dis+1010_1:4_atotf=0.2:c=on:cbe=off:cnfonf=lazy_simp:fe=off:ins=2:ntd=on:s2a=on:s2at=5.0:sgt=5:ss=axioms:st=1.5:i=779:si=on:rtra=on_0 on DTF2THF_22696 for (2991ds/779Mi)
% 16.01/9.05 % (22825)Instruction limit reached!
% 16.01/9.05 % (22825)------------------------------
% 16.01/9.05 % (22825)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 16.01/9.06 % (22825)Termination reason: Unknown
% 16.01/9.06 % (22825)Termination phase: shuffling
% 16.01/9.06
% 16.01/9.06 % (22825)Memory used [KB]: 17270
% 16.01/9.06 % (22825)Time elapsed: 0.179 s
% 16.01/9.06 % (22825)Instructions burned: 275 (million)
% 16.01/9.06 % (22825)------------------------------
% 16.01/9.06 % (22825)------------------------------
% 16.01/9.06 % (22848)lrs+10_1:1_cnfonf=lazy_not_be_gen:ntd=on:sp=const_min:ss=axioms:sup=off:i=19:si=on:rtra=on_0 on DTF2THF_22696 for (2991ds/19Mi)
% 16.01/9.07 % (22848)Instruction limit reached!
% 16.01/9.07 % (22848)------------------------------
% 16.01/9.07 % (22848)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 16.01/9.07 % (22848)Termination reason: Unknown
% 16.01/9.07 % (22848)Termination phase: shuffling
% 16.01/9.07
% 16.01/9.07 % (22848)Memory used [KB]: 13304
% 16.01/9.07 % (22848)Time elapsed: 0.020 s
% 16.01/9.07 % (22848)Instructions burned: 19 (million)
% 16.01/9.07 % (22848)------------------------------
% 16.01/9.07 % (22848)------------------------------
% 16.01/9.08 % (22849)lrs+1010_1:1_au=on:s2a=on:sd=1:sgt=50:ss=axioms:i=879:si=on:rtra=on_0 on DTF2THF_22696 for (2991ds/879Mi)
% 16.01/9.10 % (22850)dis+1002_1:128_acc=on:er=filter:i=17:si=on:rtra=on_0 on DTF2THF_22696 for (2990ds/17Mi)
% 16.01/9.11 % (22850)Instruction limit reached!
% 16.01/9.11 % (22850)------------------------------
% 16.01/9.11 % (22850)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 16.01/9.11 % (22850)Termination reason: Unknown
% 16.01/9.11 % (22850)Termination phase: shuffling
% 16.01/9.11
% 16.01/9.11 % (22850)Memory used [KB]: 12281
% 16.01/9.11 % (22850)Time elapsed: 0.018 s
% 16.01/9.11 % (22850)Instructions burned: 17 (million)
% 16.01/9.11 % (22850)------------------------------
% 16.01/9.11 % (22850)------------------------------
% 16.51/9.14 % (22851)ott+21_1:1_apa=on:au=on:cnfonf=off:sos=on:i=3:si=on:rtra=on_0 on DTF2THF_22696 for (2990ds/3Mi)
% 16.51/9.14 % (22851)Instruction limit reached!
% 16.51/9.14 % (22851)------------------------------
% 16.51/9.14 % (22851)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 16.51/9.14 % (22851)Termination reason: Unknown
% 16.51/9.14 % (22851)Termination phase: shuffling
% 16.51/9.14
% 16.51/9.14 % (22851)Memory used [KB]: 11513
% 16.51/9.14 % (22851)Time elapsed: 0.005 s
% 16.51/9.14 % (22851)Instructions burned: 3 (million)
% 16.51/9.14 % (22851)------------------------------
% 16.51/9.14 % (22851)------------------------------
% 16.51/9.16 % (22852)lrs+1010_1:8_cnfonf=off:hud=1:inj=on:tnu=5:i=30:si=on:rtra=on_0 on DTF2THF_22696 for (2990ds/30Mi)
% 16.51/9.18 % (22852)Instruction limit reached!
% 16.51/9.18 % (22852)------------------------------
% 16.51/9.18 % (22852)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 16.51/9.18 % (22852)Termination reason: Unknown
% 16.51/9.18 % (22852)Termination phase: shuffling
% 16.51/9.18
% 16.51/9.18 % (22852)Memory used [KB]: 13560
% 16.51/9.18 % (22852)Time elapsed: 0.027 s
% 16.51/9.18 % (22852)Instructions burned: 30 (million)
% 16.51/9.18 % (22852)------------------------------
% 16.51/9.18 % (22852)------------------------------
% 16.79/9.21 % (22853)dis+10_1:1_ixr=off:plsq=on:plsqc=1:plsqr=32,1:s2a=on:i=127:si=on:rtra=on_0 on DTF2THF_22696 for (2989ds/127Mi)
% 16.79/9.24 % (22846)Instruction limit reached!
% 16.79/9.24 % (22846)------------------------------
% 16.79/9.24 % (22846)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 16.79/9.24 % (22846)Termination reason: Unknown
% 16.79/9.24 % (22846)Termination phase: shuffling
% 16.79/9.24
% 16.79/9.24 % (22846)Memory used [KB]: 18677
% 16.79/9.24 % (22846)Time elapsed: 0.212 s
% 16.79/9.24 % (22846)Instructions burned: 378 (million)
% 16.79/9.24 % (22846)------------------------------
% 16.79/9.24 % (22846)------------------------------
% 16.79/9.27 % (22854)lrs+1002_1:1_au=on:cbe=off:cnfonf=conj_eager:cond=on:hi=on:i=100:si=on:rtra=on_0 on DTF2THF_22696 for (2989ds/100Mi)
% 16.79/9.30 % (22853)Instruction limit reached!
% 16.79/9.30 % (22853)------------------------------
% 16.79/9.30 % (22853)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 16.79/9.30 % (22853)Termination reason: Unknown
% 16.79/9.30 % (22853)Termination phase: shuffling
% 16.79/9.30
% 16.79/9.30 % (22853)Memory used [KB]: 14967
% 16.79/9.30 % (22853)Time elapsed: 0.087 s
% 16.79/9.30 % (22853)Instructions burned: 127 (million)
% 16.79/9.30 % (22853)------------------------------
% 16.79/9.30 % (22853)------------------------------
% 17.30/9.32 % (22855)dis+10_1:1_anc=none:cnfonf=lazy_gen:fd=preordered:fe=off:hud=10:ins=3:ixr=off:nwc=5.0:plsq=on:plsqc=1:plsqr=32,1:sp=const_frequency:uhcvi=on:i=3:si=on:rtra=on_0 on DTF2THF_22696 for (2988ds/3Mi)
% 17.30/9.32 % (22855)Instruction limit reached!
% 17.30/9.32 % (22855)------------------------------
% 17.30/9.32 % (22855)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 17.30/9.32 % (22855)Termination reason: Unknown
% 17.30/9.32 % (22855)Termination phase: shuffling
% 17.30/9.32
% 17.30/9.32 % (22855)Memory used [KB]: 11641
% 17.30/9.32 % (22855)Time elapsed: 0.005 s
% 17.30/9.32 % (22855)Instructions burned: 3 (million)
% 17.30/9.32 % (22855)------------------------------
% 17.30/9.32 % (22855)------------------------------
% 17.30/9.32 % (22854)Instruction limit reached!
% 17.30/9.32 % (22854)------------------------------
% 17.30/9.32 % (22854)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 17.30/9.32 % (22854)Termination reason: Unknown
% 17.30/9.32 % (22854)Termination phase: shuffling
% 17.30/9.32
% 17.30/9.32 % (22854)Memory used [KB]: 14711
% 17.30/9.32 % (22854)Time elapsed: 0.058 s
% 17.30/9.32 % (22854)Instructions burned: 102 (million)
% 17.30/9.32 % (22854)------------------------------
% 17.30/9.32 % (22854)------------------------------
% 17.30/9.34 % (22856)lrs+10_8:1_au=on:avsq=on:e2e=on:ins=3:s2a=on:s2at=3.0:ss=axioms:i=20:si=on:rtra=on_0 on DTF2THF_22696 for (2988ds/20Mi)
% 17.30/9.34 % (22857)dis+1002_1:1_cbe=off:hud=5:nm=4:plsq=on:plsqr=7,1:prag=on:sp=const_max:tnu=1:i=86:si=on:rtra=on_0 on DTF2THF_22696 for (2988ds/86Mi)
% 17.30/9.36 % (22856)Instruction limit reached!
% 17.30/9.36 % (22856)------------------------------
% 17.30/9.36 % (22856)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 17.30/9.36 % (22856)Termination reason: Unknown
% 17.30/9.36 % (22856)Termination phase: shuffling
% 17.30/9.36
% 17.30/9.36 % (22856)Memory used [KB]: 12409
% 17.30/9.36 % (22856)Time elapsed: 0.021 s
% 17.30/9.36 % (22856)Instructions burned: 21 (million)
% 17.30/9.36 % (22856)------------------------------
% 17.30/9.36 % (22856)------------------------------
% 17.30/9.36 % (22840)Instruction limit reached!
% 17.30/9.36 % (22840)------------------------------
% 17.30/9.36 % (22840)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 17.30/9.36 % (22840)Termination reason: Unknown
% 17.30/9.36 % (22840)Termination phase: Property scanning
% 17.30/9.36
% 17.30/9.36 % (22840)Memory used [KB]: 22899
% 17.30/9.36 % (22840)Time elapsed: 0.389 s
% 17.30/9.36 % (22840)Instructions burned: 710 (million)
% 17.30/9.36 % (22840)------------------------------
% 17.30/9.36 % (22840)------------------------------
% 17.30/9.38 % (22858)lrs+1010_1:1_au=on:cbe=off:nm=2:ntd=on:sd=2:ss=axioms:st=5.0:i=107:si=on:rtra=on_0 on DTF2THF_22696 for (2988ds/107Mi)
% 17.30/9.39 % (22859)lrs+2_1:1024_cnfonf=lazy_gen:fe=off:hud=15:plsq=on:plsqc=1:plsqr=32,1:i=39:si=on:rtra=on_0 on DTF2THF_22696 for (2987ds/39Mi)
% 17.30/9.39 % (22857)Instruction limit reached!
% 17.30/9.39 % (22857)------------------------------
% 17.30/9.39 % (22857)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 17.30/9.39 % (22857)Termination reason: Unknown
% 17.30/9.39 % (22857)Termination phase: shuffling
% 17.30/9.39
% 17.30/9.39 % (22857)Memory used [KB]: 14583
% 17.30/9.39 % (22857)Time elapsed: 0.056 s
% 17.30/9.39 % (22857)Instructions burned: 86 (million)
% 17.30/9.39 % (22857)------------------------------
% 17.30/9.39 % (22857)------------------------------
% 17.92/9.41 % (22860)dis+10_1:1_cnfonf=lazy_not_gen:fsr=off:kws=precedence:nwc=5.0:s2a=on:ss=axioms:st=1.5:i=448:si=on:rtra=on_0 on DTF2THF_22696 for (2987ds/448Mi)
% 17.92/9.42 % (22859)Instruction limit reached!
% 17.92/9.42 % (22859)------------------------------
% 17.92/9.42 % (22859)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 17.92/9.42 % (22859)Termination reason: Unknown
% 17.92/9.42 % (22859)Termination phase: shuffling
% 17.92/9.42
% 17.92/9.42 % (22859)Memory used [KB]: 13688
% 17.92/9.42 % (22859)Time elapsed: 0.030 s
% 17.92/9.42 % (22859)Instructions burned: 40 (million)
% 17.92/9.42 % (22859)------------------------------
% 17.92/9.42 % (22859)------------------------------
% 17.92/9.44 % (22861)lrs+10_1:512_au=on:fde=unused:lma=on:nm=32:plsq=on:plsqc=1:plsqr=16121663,131072:sfv=off:sp=const_max:ss=axioms:st=3.0:tgt=full:i=46:si=on:rtra=on_0 on DTF2THF_22696 for (2987ds/46Mi)
% 17.92/9.44 % (22858)Instruction limit reached!
% 17.92/9.44 % (22858)------------------------------
% 17.92/9.44 % (22858)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 17.92/9.44 % (22858)Termination reason: Unknown
% 17.92/9.44 % (22858)Termination phase: shuffling
% 17.92/9.44
% 17.92/9.44 % (22858)Memory used [KB]: 14711
% 17.92/9.44 % (22858)Time elapsed: 0.065 s
% 17.92/9.44 % (22858)Instructions burned: 107 (million)
% 17.92/9.44 % (22858)------------------------------
% 17.92/9.44 % (22858)------------------------------
% 17.92/9.45 % (22847)Instruction limit reached!
% 17.92/9.45 % (22847)------------------------------
% 17.92/9.45 % (22847)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 17.92/9.45 % (22847)Termination reason: Unknown
% 17.92/9.45 % (22847)Termination phase: shuffling
% 17.92/9.45
% 17.92/9.45 % (22847)Memory used [KB]: 26097
% 17.92/9.45 % (22847)Time elapsed: 0.400 s
% 17.92/9.45 % (22847)Instructions burned: 779 (million)
% 17.92/9.45 % (22847)------------------------------
% 17.92/9.45 % (22847)------------------------------
% 17.92/9.46 % (22862)lrs+10_1:10_au=on:av=off:cbe=off:cnfonf=lazy_pi_sigma_gen:ntd=on:plsq=on:plsqc=1:plsqr=32,1:i=98:si=on:rtra=on_0 on DTF2THF_22696 for (2987ds/98Mi)
% 17.92/9.47 % (22842)Instruction limit reached!
% 17.92/9.47 % (22842)------------------------------
% 17.92/9.47 % (22842)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 17.92/9.47 % (22842)Termination reason: Unknown
% 17.92/9.47 % (22842)Termination phase: Property scanning
% 17.92/9.47
% 17.92/9.47 % (22842)Memory used [KB]: 22899
% 17.92/9.47 % (22842)Time elapsed: 0.478 s
% 17.92/9.47 % (22842)Instructions burned: 902 (million)
% 17.92/9.47 % (22842)------------------------------
% 17.92/9.47 % (22842)------------------------------
% 17.92/9.47 % (22863)ott+1002_1:1_apa=on:au=on:bd=off:cnfonf=off:fd=off:sos=on:sp=weighted_frequency:i=507:si=on:rtra=on_0 on DTF2THF_22696 for (2987ds/507Mi)
% 17.92/9.47 % (22861)Instruction limit reached!
% 17.92/9.47 % (22861)------------------------------
% 17.92/9.47 % (22861)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 17.92/9.47 % (22861)Termination reason: Unknown
% 17.92/9.47 % (22861)Termination phase: shuffling
% 17.92/9.47
% 17.92/9.47 % (22861)Memory used [KB]: 13816
% 17.92/9.47 % (22861)Time elapsed: 0.034 s
% 17.92/9.47 % (22861)Instructions burned: 46 (million)
% 17.92/9.47 % (22861)------------------------------
% 17.92/9.47 % (22861)------------------------------
% 17.92/9.49 % (22864)dis+1010_2:3_amm=off:fd=preordered:ixr=off:nm=0:pe=on:piset=equals:prag=on:sac=on:tgt=ground:i=149:si=on:rtra=on_0 on DTF2THF_22696 for (2986ds/149Mi)
% 17.92/9.49 % (22865)lrs+21_1:16_au=on:bd=off:piset=and:i=39:si=on:rtra=on_0 on DTF2THF_22696 for (2986ds/39Mi)
% 18.72/9.52 % (22865)Instruction limit reached!
% 18.72/9.52 % (22865)------------------------------
% 18.72/9.52 % (22865)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 18.72/9.52 % (22865)Termination reason: Unknown
% 18.72/9.52 % (22865)Termination phase: shuffling
% 18.72/9.52
% 18.72/9.52 % (22865)Memory used [KB]: 13688
% 18.72/9.52 % (22865)Time elapsed: 0.030 s
% 18.72/9.52 % (22865)Instructions burned: 39 (million)
% 18.72/9.52 % (22862)Instruction limit reached!
% 18.72/9.52 % (22862)------------------------------
% 18.72/9.52 % (22862)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 18.72/9.52 % (22865)------------------------------
% 18.72/9.52 % (22865)------------------------------
% 18.72/9.52 % (22862)Termination reason: Unknown
% 18.72/9.52 % (22862)Termination phase: shuffling
% 18.72/9.52
% 18.72/9.52 % (22862)Memory used [KB]: 14583
% 18.72/9.52 % (22862)Time elapsed: 0.061 s
% 18.72/9.52 % (22862)Instructions burned: 98 (million)
% 18.72/9.52 % (22862)------------------------------
% 18.72/9.52 % (22862)------------------------------
% 18.72/9.54 % (22866)lrs+1004_1:1_cnfonf=lazy_pi_sigma_gen:fe=off:gs=on:hud=10:prag=on:tnu=1:i=6:si=on:rtra=on_0 on DTF2THF_22696 for (2986ds/6Mi)
% 18.72/9.54 % (22867)dis+1002_1:1_aac=none:au=on:e2e=on:nm=2:nwc=10.0:sd=1:ss=axioms:st=3.0:i=1087:si=on:rtra=on_0 on DTF2THF_22696 for (2986ds/1087Mi)
% 18.72/9.54 % (22866)Instruction limit reached!
% 18.72/9.54 % (22866)------------------------------
% 18.72/9.54 % (22866)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 18.72/9.54 % (22866)Termination reason: Unknown
% 18.72/9.54 % (22866)Termination phase: shuffling
% 18.72/9.54
% 18.72/9.54 % (22866)Memory used [KB]: 11769
% 18.72/9.54 % (22866)Time elapsed: 0.009 s
% 18.72/9.54 % (22866)Instructions burned: 6 (million)
% 18.72/9.54 % (22866)------------------------------
% 18.72/9.54 % (22866)------------------------------
% 18.72/9.54 % (22831)Instruction limit reached!
% 18.72/9.54 % (22831)------------------------------
% 18.72/9.54 % (22831)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 18.72/9.54 % (22831)Termination reason: Unknown
% 18.72/9.54 % (22831)Termination phase: shuffling
% 18.72/9.54
% 18.72/9.54 % (22831)Memory used [KB]: 33773
% 18.72/9.54 % (22831)Time elapsed: 0.626 s
% 18.72/9.54 % (22831)Instructions burned: 1041 (million)
% 18.72/9.54 % (22831)------------------------------
% 18.72/9.54 % (22831)------------------------------
% 18.72/9.56 % (22849)Instruction limit reached!
% 18.72/9.56 % (22849)------------------------------
% 18.72/9.56 % (22849)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 18.72/9.56 % (22849)Termination reason: Unknown
% 18.72/9.56 % (22849)Termination phase: Property scanning
% 18.72/9.56
% 18.72/9.56 % (22849)Memory used [KB]: 25074
% 18.72/9.56 % (22849)Time elapsed: 0.482 s
% 18.72/9.56 % (22849)Instructions burned: 879 (million)
% 18.72/9.56 % (22849)------------------------------
% 18.72/9.56 % (22849)------------------------------
% 18.72/9.56 % (22868)dis+10_1:1_aac=none:cs=on:i=47:si=on:rtra=on_0 on DTF2THF_22696 for (2986ds/47Mi)
% 18.72/9.57 % (22869)lrs+1010_1:128_av=off:cbe=off:cnfonf=lazy_simp:fsr=off:ntd=on:ss=axioms:st=1.5:i=44:si=on:rtra=on_0 on DTF2THF_22696 for (2986ds/44Mi)
% 18.72/9.57 % (22864)Instruction limit reached!
% 18.72/9.57 % (22864)------------------------------
% 18.72/9.57 % (22864)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 18.72/9.57 % (22864)Termination reason: Unknown
% 18.72/9.57 % (22864)Termination phase: shuffling
% 18.72/9.57
% 18.72/9.57 % (22864)Memory used [KB]: 15351
% 18.72/9.57 % (22864)Time elapsed: 0.088 s
% 18.72/9.57 % (22864)Instructions burned: 149 (million)
% 18.72/9.57 % (22864)------------------------------
% 18.72/9.57 % (22864)------------------------------
% 18.72/9.58 % (22870)dis+10_1:1_av=off:bd=off:cnfonf=lazy_gen:fe=off:hud=10:plsq=on:plsqc=1:plsqr=32,1:prag=on:tnu=1:i=432:si=on:rtra=on_0 on DTF2THF_22696 for (2986ds/432Mi)
% 18.72/9.60 % (22871)lrs+10_7:1_cnfonf=lazy_gen:fe=off:nwc=5.0:plsq=on:i=62:si=on:rtra=on_0 on DTF2THF_22696 for (2985ds/62Mi)
% 19.54/9.60 % (22868)Instruction limit reached!
% 19.54/9.60 % (22868)------------------------------
% 19.54/9.60 % (22868)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 19.54/9.60 % (22868)Termination reason: Unknown
% 19.54/9.60 % (22868)Termination phase: shuffling
% 19.54/9.60
% 19.54/9.60 % (22868)Memory used [KB]: 13816
% 19.54/9.60 % (22868)Time elapsed: 0.035 s
% 19.54/9.60 % (22868)Instructions burned: 48 (million)
% 19.54/9.60 % (22868)------------------------------
% 19.54/9.60 % (22868)------------------------------
% 19.54/9.60 % (22869)Instruction limit reached!
% 19.54/9.60 % (22869)------------------------------
% 19.54/9.60 % (22869)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 19.54/9.60 % (22869)Termination reason: Unknown
% 19.54/9.60 % (22869)Termination phase: shuffling
% 19.54/9.60
% 19.54/9.60 % (22869)Memory used [KB]: 13816
% 19.54/9.60 % (22869)Time elapsed: 0.033 s
% 19.54/9.60 % (22869)Instructions burned: 45 (million)
% 19.54/9.60 % (22869)------------------------------
% 19.54/9.60 % (22869)------------------------------
% 19.54/9.62 % (22872)lrs+1010_1:16_add=large:amm=off:anc=all_dependent:chr=on:fd=off:ins=2:sp=const_max:tnu=5:uhcvi=on:i=31:si=on:rtra=on_0 on DTF2THF_22696 for (2985ds/31Mi)
% 19.54/9.63 % (22873)dis+10_1:128_hud=1:i=96:si=on:rtra=on_0 on DTF2THF_22696 for (2985ds/96Mi)
% 19.54/9.64 % (22871)Instruction limit reached!
% 19.54/9.64 % (22871)------------------------------
% 19.54/9.64 % (22871)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 19.54/9.64 % (22871)Termination reason: Unknown
% 19.54/9.64 % (22871)Termination phase: shuffling
% 19.54/9.64
% 19.54/9.64 % (22871)Memory used [KB]: 14072
% 19.54/9.64 % (22871)Time elapsed: 0.042 s
% 19.54/9.64 % (22871)Instructions burned: 64 (million)
% 19.54/9.64 % (22871)------------------------------
% 19.54/9.64 % (22871)------------------------------
% 19.54/9.64 % (22860)Instruction limit reached!
% 19.54/9.64 % (22860)------------------------------
% 19.54/9.64 % (22860)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 19.54/9.64 % (22860)Termination reason: Unknown
% 19.54/9.64 % (22860)Termination phase: shuffling
% 19.54/9.64
% 19.54/9.64 % (22860)Memory used [KB]: 21492
% 19.54/9.64 % (22860)Time elapsed: 0.229 s
% 19.54/9.64 % (22860)Instructions burned: 448 (million)
% 19.54/9.64 % (22860)------------------------------
% 19.54/9.64 % (22860)------------------------------
% 19.54/9.64 % (22872)Instruction limit reached!
% 19.54/9.64 % (22872)------------------------------
% 19.54/9.64 % (22872)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 19.54/9.64 % (22872)Termination reason: Unknown
% 19.54/9.64 % (22872)Termination phase: shuffling
% 19.54/9.64
% 19.54/9.64 % (22872)Memory used [KB]: 13560
% 19.54/9.64 % (22872)Time elapsed: 0.027 s
% 19.54/9.64 % (22872)Instructions burned: 31 (million)
% 19.54/9.64 % (22872)------------------------------
% 19.54/9.64 % (22872)------------------------------
% 19.54/9.65 % (22874)lrs+1002_1:4_au=on:cbe=off:cnfonf=lazy_pi_sigma_gen:hud=15:inj=on:nwc=4.0:s2a=on:sac=on:sp=reverse_arity:i=453:si=on:rtra=on_0 on DTF2THF_22696 for (2985ds/453Mi)
% 19.54/9.66 % (22875)lrs-1004_1:1_au=on:bsd=on:cbe=off:cnfonf=off:fe=axiom:fsd=on:i=71:si=on:rtra=on_0 on DTF2THF_22696 for (2985ds/71Mi)
% 19.54/9.66 % (22876)dis+1002_1:1_bd=off:fsr=off:ntd=on:s2a=on:ss=axioms:i=83:si=on:rtra=on_0 on DTF2THF_22696 for (2985ds/83Mi)
% 19.54/9.70 % (22873)Instruction limit reached!
% 19.54/9.70 % (22873)------------------------------
% 19.54/9.70 % (22873)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 19.54/9.70 % (22873)Termination reason: Unknown
% 19.54/9.70 % (22873)Termination phase: shuffling
% 19.54/9.70
% 19.54/9.70 % (22873)Memory used [KB]: 14711
% 19.54/9.70 % (22873)Time elapsed: 0.071 s
% 19.54/9.70 % (22873)Instructions burned: 97 (million)
% 19.54/9.70 % (22873)------------------------------
% 19.54/9.70 % (22873)------------------------------
% 20.00/9.71 % (22875)Instruction limit reached!
% 20.00/9.71 % (22875)------------------------------
% 20.00/9.71 % (22875)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 20.00/9.71 % (22875)Termination reason: Unknown
% 20.00/9.71 % (22875)Termination phase: shuffling
% 20.00/9.71
% 20.00/9.71 % (22875)Memory used [KB]: 14200
% 20.00/9.71 % (22875)Time elapsed: 0.048 s
% 20.00/9.71 % (22875)Instructions burned: 72 (million)
% 20.00/9.71 % (22875)------------------------------
% 20.00/9.71 % (22875)------------------------------
% 20.00/9.71 % (22876)Instruction limit reached!
% 20.00/9.71 % (22876)------------------------------
% 20.00/9.71 % (22876)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 20.00/9.71 % (22876)Termination reason: Unknown
% 20.00/9.71 % (22876)Termination phase: shuffling
% 20.00/9.71
% 20.00/9.71 % (22876)Memory used [KB]: 14456
% 20.00/9.71 % (22876)Time elapsed: 0.052 s
% 20.00/9.71 % (22876)Instructions burned: 83 (million)
% 20.00/9.71 % (22876)------------------------------
% 20.00/9.71 % (22876)------------------------------
% 20.00/9.72 % (22877)lrs+10_1:128_cnfonf=lazy_not_gen_be_off:hud=1:ins=1:i=11:si=on:rtra=on_0 on DTF2THF_22696 for (2984ds/11Mi)
% 20.00/9.73 % (22878)lrs+1010_1:1_au=on:cbe=off:cnfonf=lazy_not_gen_be_off:fsr=off:spb=non_intro:i=41:si=on:rtra=on_0 on DTF2THF_22696 for (2984ds/41Mi)
% 20.00/9.73 % (22877)Instruction limit reached!
% 20.00/9.73 % (22877)------------------------------
% 20.00/9.73 % (22877)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 20.00/9.73 % (22877)Termination reason: Unknown
% 20.00/9.73 % (22877)Termination phase: shuffling
% 20.00/9.73
% 20.00/9.73 % (22877)Memory used [KB]: 12281
% 20.00/9.73 % (22877)Time elapsed: 0.016 s
% 20.00/9.73 % (22877)Instructions burned: 12 (million)
% 20.00/9.73 % (22877)------------------------------
% 20.00/9.73 % (22877)------------------------------
% 20.00/9.73 % (22879)lrs+1004_1:1_chr=on:prag=on:i=6:si=on:rtra=on_0 on DTF2THF_22696 for (2984ds/6Mi)
% 20.00/9.74 % (22879)Instruction limit reached!
% 20.00/9.74 % (22879)------------------------------
% 20.00/9.74 % (22879)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 20.00/9.74 % (22879)Termination reason: Unknown
% 20.00/9.74 % (22879)Termination phase: shuffling
% 20.00/9.74
% 20.00/9.74 % (22879)Memory used [KB]: 11769
% 20.00/9.74 % (22879)Time elapsed: 0.008 s
% 20.00/9.74 % (22879)Instructions burned: 6 (million)
% 20.00/9.74 % (22879)------------------------------
% 20.00/9.74 % (22879)------------------------------
% 20.00/9.74 % (22863)Instruction limit reached!
% 20.00/9.74 % (22863)------------------------------
% 20.00/9.74 % (22863)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 20.00/9.74 % (22863)Termination reason: Unknown
% 20.00/9.74 % (22863)Termination phase: shuffling
% 20.00/9.74
% 20.00/9.74 % (22863)Memory used [KB]: 20340
% 20.00/9.74 % (22863)Time elapsed: 0.278 s
% 20.00/9.74 % (22863)Instructions burned: 507 (million)
% 20.00/9.74 % (22863)------------------------------
% 20.00/9.74 % (22863)------------------------------
% 20.00/9.75 % (22878)Instruction limit reached!
% 20.00/9.75 % (22878)------------------------------
% 20.00/9.75 % (22878)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 20.00/9.75 % (22878)Termination reason: Unknown
% 20.00/9.75 % (22878)Termination phase: shuffling
% 20.00/9.75
% 20.00/9.75 % (22878)Memory used [KB]: 13816
% 20.00/9.75 % (22878)Time elapsed: 0.023 s
% 20.00/9.75 % (22878)Instructions burned: 43 (million)
% 20.00/9.75 % (22878)------------------------------
% 20.00/9.75 % (22878)------------------------------
% 20.00/9.76 % (22880)dis+10_1:1_avsq=on:cnfonf=off:e2e=on:nm=32:i=150:si=on:rtra=on_0 on DTF2THF_22696 for (2984ds/150Mi)
% 20.00/9.76 % (22881)ott+1010_1:1_au=on:s2a=on:sd=7:ss=axioms:st=1.5:i=343:si=on:rtra=on_0 on DTF2THF_22696 for (2984ds/343Mi)
% 20.00/9.77 % (22882)ott+1010_1:4_cbe=off:chr=on:hfsq=on:hfsqr=1,16:hud=3:sac=on:sp=unary_frequency:ss=axioms:st=5.0:tnu=2:i=109:si=on:rtra=on_0 on DTF2THF_22696 for (2984ds/109Mi)
% 20.00/9.77 % (22883)dis+10_1:1_au=on:c=on:e2e=on:fsr=off:sos=all:sp=const_frequency:ss=axioms:i=253:si=on:rtra=on_0 on DTF2THF_22696 for (2984ds/253Mi)
% 20.62/9.80 % (22870)Instruction limit reached!
% 20.62/9.80 % (22870)------------------------------
% 20.62/9.80 % (22870)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 20.62/9.80 % (22870)Termination reason: Unknown
% 20.62/9.80 % (22870)Termination phase: shuffling
% 20.62/9.80
% 20.62/9.80 % (22870)Memory used [KB]: 21364
% 20.62/9.80 % (22870)Time elapsed: 0.222 s
% 20.62/9.80 % (22870)Instructions burned: 432 (million)
% 20.62/9.80 % (22870)------------------------------
% 20.62/9.80 % (22870)------------------------------
% 20.62/9.82 % (22884)lrs+10_1:1_au=on:cnfonf=off:sd=1:sgt=16:ss=axioms:i=140:si=on:rtra=on_0 on DTF2THF_22696 for (2983ds/140Mi)
% 20.62/9.84 % (22882)Instruction limit reached!
% 20.62/9.84 % (22882)------------------------------
% 20.62/9.84 % (22882)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 20.62/9.84 % (22882)Termination reason: Unknown
% 20.62/9.84 % (22882)Termination phase: shuffling
% 20.62/9.84
% 20.62/9.84 % (22882)Memory used [KB]: 14711
% 20.62/9.84 % (22882)Time elapsed: 0.076 s
% 20.62/9.84 % (22882)Instructions burned: 109 (million)
% 20.62/9.84 % (22882)------------------------------
% 20.62/9.84 % (22882)------------------------------
% 20.62/9.85 % (22880)Instruction limit reached!
% 20.62/9.85 % (22880)------------------------------
% 20.62/9.85 % (22880)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 20.62/9.85 % (22880)Termination reason: Unknown
% 20.62/9.85 % (22880)Termination phase: shuffling
% 20.62/9.85
% 20.62/9.85 % (22880)Memory used [KB]: 15351
% 20.62/9.85 % (22880)Time elapsed: 0.089 s
% 20.62/9.85 % (22880)Instructions burned: 150 (million)
% 20.62/9.85 % (22880)------------------------------
% 20.62/9.85 % (22880)------------------------------
% 20.62/9.86 % (22885)dis+1010_16:1_acc=model:au=on:avsq=on:cond=fast:plsq=on:plsqc=2:plsqr=9267,262144:i=614:si=on:rtra=on_0 on DTF2THF_22696 for (2983ds/614Mi)
% 20.62/9.87 % (22886)lrs+1002_1:1_e2e=on:fde=unused:fe=axiom:prag=on:sd=1:sos=on:ss=axioms:st=2.0:i=202:si=on:rtra=on_0 on DTF2THF_22696 for (2983ds/202Mi)
% 20.62/9.88 % (22874)Instruction limit reached!
% 20.62/9.88 % (22874)------------------------------
% 20.62/9.88 % (22874)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 20.62/9.88 % (22874)Termination reason: Unknown
% 20.62/9.88 % (22874)Termination phase: shuffling
% 20.62/9.88
% 20.62/9.88 % (22874)Memory used [KB]: 21620
% 20.62/9.88 % (22874)Time elapsed: 0.230 s
% 20.62/9.88 % (22874)Instructions burned: 455 (million)
% 20.62/9.88 % (22874)------------------------------
% 20.62/9.88 % (22874)------------------------------
% 21.31/9.90 % (22884)Instruction limit reached!
% 21.31/9.90 % (22884)------------------------------
% 21.31/9.90 % (22884)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 21.31/9.90 % (22884)Termination reason: Unknown
% 21.31/9.90 % (22884)Termination phase: shuffling
% 21.31/9.90
% 21.31/9.90 % (22884)Memory used [KB]: 15223
% 21.31/9.90 % (22884)Time elapsed: 0.084 s
% 21.31/9.90 % (22884)Instructions burned: 141 (million)
% 21.31/9.90 % (22884)------------------------------
% 21.31/9.90 % (22884)------------------------------
% 21.31/9.91 % (22887)dis+1002_1:1_au=on:cs=on:ins=3:kws=inv_frequency:nwc=10.0:ss=axioms:st=3.0:i=339:si=on:rtra=on_0 on DTF2THF_22696 for (2982ds/339Mi)
% 21.31/9.91 % (22883)Instruction limit reached!
% 21.31/9.91 % (22883)------------------------------
% 21.31/9.91 % (22883)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 21.31/9.91 % (22883)Termination reason: Unknown
% 21.31/9.91 % (22883)Termination phase: shuffling
% 21.31/9.91
% 21.31/9.91 % (22883)Memory used [KB]: 17014
% 21.31/9.91 % (22883)Time elapsed: 0.145 s
% 21.31/9.91 % (22883)Instructions burned: 254 (million)
% 21.31/9.91 % (22883)------------------------------
% 21.31/9.91 % (22883)------------------------------
% 21.31/9.92 % (22888)dis+10_1:1_anc=none:cnfonf=lazy_gen:fd=preordered:fe=off:hud=10:ins=3:ixr=off:nwc=5.0:plsq=on:plsqc=1:plsqr=32,1:sp=const_frequency:uhcvi=on:i=283:si=on:rtra=on_0 on DTF2THF_22696 for (2982ds/283Mi)
% 21.31/9.93 % (22889)dis+2_1:1_cha=on:hud=7:nwc=3.0:prag=on:tnu=2:i=134:si=on:rtra=on_0 on DTF2THF_22696 for (2982ds/134Mi)
% 21.31/9.95 % (22881)Instruction limit reached!
% 21.31/9.95 % (22881)------------------------------
% 21.31/9.95 % (22881)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 21.31/9.95 % (22881)Termination reason: Unknown
% 21.31/9.95 % (22881)Termination phase: shuffling
% 21.31/9.95
% 21.31/9.95 % (22881)Memory used [KB]: 18166
% 21.31/9.95 % (22881)Time elapsed: 0.192 s
% 21.31/9.95 % (22881)Instructions burned: 344 (million)
% 21.31/9.95 % (22881)------------------------------
% 21.31/9.95 % (22881)------------------------------
% 21.31/9.97 % (22890)dis+1010_1:64_cnfonf=lazy_simp:fd=off:i=298:si=on:rtra=on_0 on DTF2THF_22696 for (2982ds/298Mi)
% 21.31/10.00 % (22886)Instruction limit reached!
% 21.31/10.00 % (22886)------------------------------
% 21.31/10.00 % (22886)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 21.31/10.00 % (22886)Termination reason: Unknown
% 21.31/10.00 % (22886)Termination phase: shuffling
% 21.31/10.00
% 21.31/10.00 % (22886)Memory used [KB]: 16247
% 21.31/10.00 % (22886)Time elapsed: 0.130 s
% 21.31/10.00 % (22886)Instructions burned: 202 (million)
% 21.31/10.00 % (22886)------------------------------
% 21.31/10.00 % (22886)------------------------------
% 21.31/10.01 % (22889)Instruction limit reached!
% 21.31/10.01 % (22889)------------------------------
% 21.31/10.01 % (22889)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 21.31/10.01 % (22889)Termination reason: Unknown
% 21.31/10.01 % (22889)Termination phase: shuffling
% 21.31/10.01
% 21.31/10.01 % (22889)Memory used [KB]: 15095
% 21.31/10.01 % (22889)Time elapsed: 0.079 s
% 21.31/10.01 % (22889)Instructions burned: 134 (million)
% 21.31/10.01 % (22889)------------------------------
% 21.31/10.01 % (22889)------------------------------
% 21.31/10.02 % (22891)lrs+2_1:8_erd=off:plsq=on:plsqr=32,1:sos=on:i=192:si=on:rtra=on_0 on DTF2THF_22696 for (2981ds/192Mi)
% 21.31/10.03 % (22892)lrs+10_1:1_bd=off:hud=1:nm=0:prag=on:sd=2:sgt=32:sos=all:ss=axioms:i=258:si=on:rtra=on_0 on DTF2THF_22696 for (2981ds/258Mi)
% 23.03/10.07 % (22888)Instruction limit reached!
% 23.03/10.07 % (22888)------------------------------
% 23.03/10.07 % (22888)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 23.03/10.07 % (22888)Termination reason: Unknown
% 23.03/10.07 % (22888)Termination phase: shuffling
% 23.03/10.07
% 23.03/10.07 % (22888)Memory used [KB]: 19317
% 23.03/10.07 % (22888)Time elapsed: 0.147 s
% 23.03/10.07 % (22888)Instructions burned: 284 (million)
% 23.03/10.07 % (22888)------------------------------
% 23.03/10.07 % (22888)------------------------------
% 23.03/10.09 % (22887)Instruction limit reached!
% 23.03/10.09 % (22887)------------------------------
% 23.03/10.09 % (22887)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 23.03/10.09 % (22887)Termination reason: Unknown
% 23.03/10.09 % (22887)Termination phase: shuffling
% 23.03/10.09
% 23.03/10.09 % (22887)Memory used [KB]: 18166
% 23.03/10.09 % (22887)Time elapsed: 0.188 s
% 23.03/10.09 % (22887)Instructions burned: 340 (million)
% 23.03/10.09 % (22887)------------------------------
% 23.03/10.09 % (22887)------------------------------
% 23.03/10.10 % (22893)lrs+1002_1:1024_au=on:cond=fast:ntd=on:piset=and:plsq=on:plsqc=1:plsqr=183491,1048576:sac=on:tgt=ground:i=90:si=on:rtra=on_0 on DTF2THF_22696 for (2980ds/90Mi)
% 23.03/10.11 % (22894)dis+1004_16:1_avsq=on:avsqr=1,16:cnfonf=lazy_gen:cs=on:fe=off:tgt=ground:tnu=1:i=231:si=on:rtra=on_0 on DTF2THF_22696 for (2980ds/231Mi)
% 23.03/10.12 % (22890)Instruction limit reached!
% 23.03/10.12 % (22890)------------------------------
% 23.03/10.12 % (22890)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 23.03/10.12 % (22890)Termination reason: Unknown
% 23.03/10.12 % (22890)Termination phase: shuffling
% 23.03/10.12
% 23.03/10.12 % (22890)Memory used [KB]: 19445
% 23.03/10.12 % (22890)Time elapsed: 0.154 s
% 23.03/10.12 % (22890)Instructions burned: 299 (million)
% 23.03/10.12 % (22890)------------------------------
% 23.03/10.12 % (22890)------------------------------
% 23.03/10.13 % (22891)Instruction limit reached!
% 23.03/10.13 % (22891)------------------------------
% 23.03/10.13 % (22891)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 23.03/10.13 % (22891)Termination reason: Unknown
% 23.03/10.13 % (22891)Termination phase: shuffling
% 23.03/10.13
% 23.03/10.13 % (22891)Memory used [KB]: 15991
% 23.03/10.13 % (22891)Time elapsed: 0.111 s
% 23.03/10.13 % (22891)Instructions burned: 192 (million)
% 23.03/10.13 % (22891)------------------------------
% 23.03/10.13 % (22891)------------------------------
% 23.59/10.14 % (22895)lrs+1002_1:1_au=on:s2a=on:sgt=32:slsq=on:slsqc=1:slsql=off:ss=axioms:i=458:si=on:rtra=on_0 on DTF2THF_22696 for (2980ds/458Mi)
% 23.59/10.15 % (22896)lrs+1010_1:16_add=large:amm=off:anc=all_dependent:chr=on:fd=off:ins=2:sp=const_max:tnu=5:uhcvi=on:i=31:si=on:rtra=on_0 on DTF2THF_22696 for (2980ds/31Mi)
% 23.59/10.16 % (22893)Instruction limit reached!
% 23.59/10.16 % (22893)------------------------------
% 23.59/10.16 % (22893)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 23.59/10.16 % (22893)Termination reason: Unknown
% 23.59/10.16 % (22893)Termination phase: shuffling
% 23.59/10.16
% 23.59/10.16 % (22893)Memory used [KB]: 14583
% 23.59/10.16 % (22893)Time elapsed: 0.069 s
% 23.59/10.16 % (22893)Instructions burned: 90 (million)
% 23.59/10.16 % (22893)------------------------------
% 23.59/10.16 % (22893)------------------------------
% 23.59/10.17 % (22896)Instruction limit reached!
% 23.59/10.17 % (22896)------------------------------
% 23.59/10.17 % (22896)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 23.59/10.17 % (22896)Termination reason: Unknown
% 23.59/10.17 % (22896)Termination phase: shuffling
% 23.59/10.17
% 23.59/10.17 % (22896)Memory used [KB]: 13560
% 23.59/10.17 % (22896)Time elapsed: 0.027 s
% 23.59/10.17 % (22896)Instructions burned: 31 (million)
% 23.59/10.17 % (22896)------------------------------
% 23.59/10.17 % (22896)------------------------------
% 23.59/10.18 % (22892)Instruction limit reached!
% 23.59/10.18 % (22892)------------------------------
% 23.59/10.18 % (22892)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 23.59/10.18 % (22892)Termination reason: Unknown
% 23.59/10.18 % (22892)Termination phase: shuffling
% 23.59/10.18
% 23.59/10.18 % (22892)Memory used [KB]: 17014
% 23.59/10.18 % (22892)Time elapsed: 0.148 s
% 23.59/10.18 % (22892)Instructions burned: 259 (million)
% 23.59/10.18 % (22892)------------------------------
% 23.59/10.18 % (22892)------------------------------
% 23.59/10.18 % (22897)lrs+10_1:1_cnfonf=lazy_gen:kws=inv_precedence:nwc=10.0:s2a=on:sp=weighted_frequency:ss=axioms:i=297:si=on:rtra=on_0 on DTF2THF_22696 for (2980ds/297Mi)
% 23.59/10.19 % (22898)dis+2_1:4_amm=off:au=on:cbe=off:cnfonf=lazy_gen:hfaw=0:hfsq=on:hfsql=off:plsq=on:plsqc=1:plsqr=35,4:sac=on:i=987:si=on:rtra=on_0 on DTF2THF_22696 for (2980ds/987Mi)
% 24.03/10.20 % (22899)lrs+10_1:1024_avsq=on:avsqc=4:avsqr=584411,1048576:bd=off:bet=on:br=off:chr=on:fsd=on:fsdmm=1:hud=10:nm=0:ntd=on:nwc=9.0:piset=or:plsq=on:plsqc=2:plsql=on:plsqr=852219,1048576:rawr=on:sp=const_frequency:spb=units:i=182:si=on:rtra=on_0 on DTF2THF_22696 for (2979ds/182Mi)
% 24.03/10.20 % (22885)Instruction limit reached!
% 24.03/10.20 % (22885)------------------------------
% 24.03/10.20 % (22885)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 24.03/10.20 % (22885)Termination reason: Unknown
% 24.03/10.20 % (22885)Termination phase: shuffling
% 24.03/10.20
% 24.03/10.20 % (22885)Memory used [KB]: 22131
% 24.03/10.20 % (22885)Time elapsed: 0.339 s
% 24.03/10.20 % (22885)Instructions burned: 614 (million)
% 24.03/10.20 % (22885)------------------------------
% 24.03/10.20 % (22885)------------------------------
% 24.03/10.21 % (22867)Instruction limit reached!
% 24.03/10.21 % (22867)------------------------------
% 24.03/10.21 % (22867)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 24.03/10.21 % (22867)Termination reason: Unknown
% 24.03/10.21 % (22867)Termination phase: SInE selection
% 24.03/10.21
% 24.03/10.21 % (22867)Memory used [KB]: 23027
% 24.03/10.21 % (22867)Time elapsed: 0.671 s
% 24.03/10.21 % (22867)Instructions burned: 1087 (million)
% 24.03/10.21 % (22867)------------------------------
% 24.03/10.21 % (22867)------------------------------
% 24.03/10.22 % (22900)dis+10_1:1_au=on:e2e=on:fde=none:nwc=10.0:s2a=on:s2at=2.0:i=182:si=on:rtra=on_0 on DTF2THF_22696 for (2979ds/182Mi)
% 24.03/10.23 % (22894)Instruction limit reached!
% 24.03/10.23 % (22894)------------------------------
% 24.03/10.23 % (22894)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 24.03/10.23 % (22894)Termination reason: Unknown
% 24.03/10.23 % (22894)Termination phase: shuffling
% 24.03/10.23
% 24.03/10.23 % (22894)Memory used [KB]: 16502
% 24.03/10.23 % (22894)Time elapsed: 0.122 s
% 24.03/10.23 % (22894)Instructions burned: 231 (million)
% 24.03/10.23 % (22894)------------------------------
% 24.03/10.23 % (22894)------------------------------
% 24.03/10.23 % (22901)dis+10_1:8_cha=on:cnfonf=lazy_pi_sigma_gen:fe=axiom:hud=5:plsq=on:plsqr=32,1:prag=on:sos=all:i=154:si=on:rtra=on_0 on DTF2THF_22696 for (2979ds/154Mi)
% 24.03/10.25 % (22902)dis+1003_1:64_atotf=0.4:au=on:fsr=off:hflw=5:hfsq=on:hfsqc=20:hfsql=off:plsq=on:plsqc=1:plsqr=32,1:i=79:si=on:rtra=on_0 on DTF2THF_22696 for (2979ds/79Mi)
% 24.21/10.30 % (22899)Instruction limit reached!
% 24.21/10.30 % (22899)------------------------------
% 24.21/10.30 % (22899)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 24.21/10.30 % (22899)Termination reason: Unknown
% 24.21/10.30 % (22899)Termination phase: shuffling
% 24.21/10.30
% 24.21/10.30 % (22899)Memory used [KB]: 15863
% 24.21/10.30 % (22899)Time elapsed: 0.107 s
% 24.21/10.30 % (22899)Instructions burned: 182 (million)
% 24.21/10.30 % (22899)------------------------------
% 24.21/10.30 % (22899)------------------------------
% 24.21/10.30 % (22902)Instruction limit reached!
% 24.21/10.30 % (22902)------------------------------
% 24.21/10.30 % (22902)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 24.21/10.30 % (22902)Termination reason: Unknown
% 24.21/10.30 % (22902)Termination phase: shuffling
% 24.21/10.30
% 24.21/10.30 % (22902)Memory used [KB]: 14328
% 24.21/10.30 % (22902)Time elapsed: 0.053 s
% 24.21/10.30 % (22902)Instructions burned: 80 (million)
% 24.21/10.30 % (22902)------------------------------
% 24.21/10.30 % (22902)------------------------------
% 24.21/10.32 % (22901)Instruction limit reached!
% 24.21/10.32 % (22901)------------------------------
% 24.21/10.32 % (22901)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 24.21/10.32 % (22901)Termination reason: Unknown
% 24.21/10.32 % (22901)Termination phase: shuffling
% 24.21/10.32
% 24.21/10.32 % (22901)Memory used [KB]: 15479
% 24.21/10.32 % (22901)Time elapsed: 0.084 s
% 24.21/10.32 % (22901)Instructions burned: 155 (million)
% 24.21/10.32 % (22901)------------------------------
% 24.21/10.32 % (22901)------------------------------
% 24.21/10.32 % (22904)lrs+1002_1:1_fe=off:hud=10:ins=2:sac=on:sp=const_frequency:i=114:si=on:rtra=on_0 on DTF2THF_22696 for (2978ds/114Mi)
% 24.21/10.33 % (22903)dis+1010_5:2_anc=all_dependent:awrs=converge:bd=off:bet=on:c=on:cha=on:cnfonf=lazy_not_gen_be_off:fe=axiom:hud=5:kws=precedence:piset=and:prag=on:s2a=on:sac=on:slsq=on:slsqc=1:sp=reverse_arity:ss=axioms:st=1.5:i=216:si=on:rtra=on_0 on DTF2THF_22696 for (2978ds/216Mi)
% 24.21/10.33 % (22900)Instruction limit reached!
% 24.21/10.33 % (22900)------------------------------
% 24.21/10.33 % (22900)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 24.21/10.33 % (22900)Termination reason: Unknown
% 24.21/10.33 % (22900)Termination phase: shuffling
% 24.21/10.33
% 24.21/10.33 % (22900)Memory used [KB]: 15863
% 24.21/10.33 % (22900)Time elapsed: 0.109 s
% 24.21/10.33 % (22900)Instructions burned: 183 (million)
% 24.21/10.33 % (22900)------------------------------
% 24.21/10.33 % (22900)------------------------------
% 24.21/10.34 % (22905)dis+1002_1:1_acc=on:au=on:bd=off:e2e=on:sgt=8:ss=axioms:i=922:si=on:rtra=on_0 on DTF2THF_22696 for (2978ds/922Mi)
% 24.21/10.34 % (22897)Instruction limit reached!
% 24.21/10.34 % (22897)------------------------------
% 24.21/10.34 % (22897)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 24.21/10.34 % (22897)Termination reason: Unknown
% 24.21/10.34 % (22897)Termination phase: shuffling
% 24.21/10.34
% 24.21/10.34 % (22897)Memory used [KB]: 19445
% 24.21/10.34 % (22897)Time elapsed: 0.157 s
% 24.21/10.34 % (22897)Instructions burned: 298 (million)
% 24.21/10.34 % (22897)------------------------------
% 24.21/10.34 % (22897)------------------------------
% 24.21/10.35 % (22906)lrs+10_1:128_au=on:bd=off:cnfonf=lazy_not_gen_be_off:nwc=10.0:piset=and:i=182:si=on:rtra=on_0 on DTF2THF_22696 for (2978ds/182Mi)
% 24.21/10.36 % (22907)ott+1010_1:1024_au=on:av=off:c=on:cnfonf=conj_eager:piset=equals:s2a=on:sp=occurrence:spb=units:ss=axioms:i=164:si=on:rtra=on_0 on DTF2THF_22696 for (2978ds/164Mi)
% 24.21/10.39 % (22904)Instruction limit reached!
% 24.21/10.39 % (22904)------------------------------
% 24.21/10.39 % (22904)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 24.21/10.39 % (22904)Termination reason: Unknown
% 24.21/10.39 % (22904)Termination phase: shuffling
% 24.21/10.39
% 24.21/10.39 % (22904)Memory used [KB]: 14711
% 24.21/10.39 % (22904)Time elapsed: 0.071 s
% 24.21/10.39 % (22904)Instructions burned: 114 (million)
% 24.21/10.39 % (22904)------------------------------
% 24.21/10.39 % (22904)------------------------------
% 24.91/10.40 % (22895)Instruction limit reached!
% 24.91/10.40 % (22895)------------------------------
% 24.91/10.40 % (22895)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 24.91/10.40 % (22895)Termination reason: Unknown
% 24.91/10.40 % (22895)Termination phase: shuffling
% 24.91/10.40
% 24.91/10.40 % (22895)Memory used [KB]: 19957
% 24.91/10.40 % (22895)Time elapsed: 0.259 s
% 24.91/10.40 % (22895)Instructions burned: 459 (million)
% 24.91/10.40 % (22895)------------------------------
% 24.91/10.40 % (22895)------------------------------
% 24.91/10.42 % (22908)lrs+1004_1:1_cnfonf=lazy_pi_sigma_gen:fe=off:gs=on:hud=10:prag=on:tnu=1:i=50:si=on:rtra=on_0 on DTF2THF_22696 for (2977ds/50Mi)
% 24.91/10.42 % (22909)dis-30_1:1_acc=on:bd=off:fsr=off:plsq=on:plsqr=32,1:i=368:si=on:rtra=on_0 on DTF2THF_22696 for (2977ds/368Mi)
% 24.91/10.44 % (22903)Instruction limit reached!
% 24.91/10.44 % (22903)------------------------------
% 24.91/10.44 % (22903)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 24.91/10.44 % (22903)Termination reason: Unknown
% 24.91/10.44 % (22903)Termination phase: shuffling
% 24.91/10.44
% 24.91/10.44 % (22903)Memory used [KB]: 16247
% 24.91/10.44 % (22903)Time elapsed: 0.117 s
% 24.91/10.44 % (22903)Instructions burned: 217 (million)
% 24.91/10.44 % (22903)------------------------------
% 24.91/10.44 % (22903)------------------------------
% 24.91/10.45 % (22906)Instruction limit reached!
% 24.91/10.45 % (22906)------------------------------
% 24.91/10.45 % (22906)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 24.91/10.45 % (22906)Termination reason: Unknown
% 24.91/10.45 % (22906)Termination phase: shuffling
% 24.91/10.45
% 24.91/10.45 % (22906)Memory used [KB]: 15863
% 24.91/10.45 % (22906)Time elapsed: 0.100 s
% 24.91/10.45 % (22906)Instructions burned: 183 (million)
% 24.91/10.45 % (22906)------------------------------
% 24.91/10.45 % (22906)------------------------------
% 24.91/10.45 % (22908)Instruction limit reached!
% 24.91/10.45 % (22908)------------------------------
% 24.91/10.45 % (22908)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 24.91/10.45 % (22908)Termination reason: Unknown
% 24.91/10.45 % (22908)Termination phase: shuffling
% 24.91/10.45
% 24.91/10.45 % (22908)Memory used [KB]: 13816
% 24.91/10.45 % (22908)Time elapsed: 0.036 s
% 24.91/10.45 % (22908)Instructions burned: 51 (million)
% 24.91/10.45 % (22908)------------------------------
% 24.91/10.45 % (22908)------------------------------
% 24.91/10.45 % (22907)Instruction limit reached!
% 24.91/10.45 % (22907)------------------------------
% 24.91/10.45 % (22907)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 24.91/10.45 % (22907)Termination reason: Unknown
% 24.91/10.45 % (22907)Termination phase: shuffling
% 24.91/10.45
% 24.91/10.45 % (22907)Memory used [KB]: 15607
% 24.91/10.45 % (22907)Time elapsed: 0.094 s
% 24.91/10.45 % (22907)Instructions burned: 165 (million)
% 24.91/10.45 % (22907)------------------------------
% 24.91/10.45 % (22907)------------------------------
% 24.91/10.47 % (22911)dis+10_1:1_cnfonf=lazy_gen:fe=off:plsq=on:plsqc=1:plsqr=32,1:prag=on:i=267:si=on:rtra=on_0 on DTF2THF_22696 for (2977ds/267Mi)
% 24.91/10.47 % (22910)dis+2_1:1_bd=off:e2e=on:hud=1:nm=3:piset=or:prag=on:s2a=on:sp=const_max:i=1021:si=on:rtra=on_0 on DTF2THF_22696 for (2977ds/1021Mi)
% 24.91/10.47 % (22912)dis+21_1:5_au=on:ntd=on:nwc=5.0:s2a=on:ss=axioms:st=5.0:i=204:si=on:rtra=on_0 on DTF2THF_22696 for (2977ds/204Mi)
% 24.91/10.48 % (22913)dis+2_1:1024_atotf=0.0802881:au=on:bs=unit_only:cnfonf=lazy_not_gen_be_off:fde=none:nwc=10.0:prag=on:i=26:si=on:rtra=on_0 on DTF2THF_22696 for (2977ds/26Mi)
% 25.61/10.50 % (22913)Instruction limit reached!
% 25.61/10.50 % (22913)------------------------------
% 25.61/10.50 % (22913)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 25.61/10.50 % (22913)Termination reason: Unknown
% 25.61/10.50 % (22913)Termination phase: shuffling
% 25.61/10.50
% 25.61/10.50 % (22913)Memory used [KB]: 13560
% 25.61/10.50 % (22913)Time elapsed: 0.027 s
% 25.61/10.50 % (22913)Instructions burned: 27 (million)
% 25.61/10.50 % (22913)------------------------------
% 25.61/10.50 % (22913)------------------------------
% 25.61/10.52 % (22914)dis+1002_1:1_bd=preordered:c=on:cbe=off:cnfonf=lazy_not_gen_be_off:hud=10:ixr=off:prag=on:i=53:si=on:rtra=on_0 on DTF2THF_22696 for (2976ds/53Mi)
% 25.61/10.56 % (22914)Instruction limit reached!
% 25.61/10.56 % (22914)------------------------------
% 25.61/10.56 % (22914)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 25.61/10.56 % (22914)Termination reason: Unknown
% 25.61/10.56 % (22914)Termination phase: shuffling
% 25.61/10.56
% 25.61/10.56 % (22914)Memory used [KB]: 13944
% 25.61/10.56 % (22914)Time elapsed: 0.037 s
% 25.61/10.56 % (22914)Instructions burned: 54 (million)
% 25.61/10.56 % (22914)------------------------------
% 25.61/10.56 % (22914)------------------------------
% 25.61/10.58 % (22915)dis+1002_1:1_aac=none:au=on:e2e=on:nm=2:nwc=10.0:sd=1:ss=axioms:st=3.0:i=4436:si=on:rtra=on_0 on DTF2THF_22696 for (2976ds/4436Mi)
% 25.61/10.59 % (22912)Instruction limit reached!
% 25.61/10.59 % (22912)------------------------------
% 25.61/10.59 % (22912)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 25.61/10.59 % (22912)Termination reason: Unknown
% 25.61/10.59 % (22912)Termination phase: shuffling
% 25.61/10.59
% 25.61/10.59 % (22912)Memory used [KB]: 16119
% 25.61/10.59 % (22912)Time elapsed: 0.120 s
% 25.61/10.59 % (22912)Instructions burned: 204 (million)
% 25.61/10.59 % (22912)------------------------------
% 25.61/10.59 % (22912)------------------------------
% 25.85/10.60 % (22911)Instruction limit reached!
% 25.85/10.60 % (22911)------------------------------
% 25.85/10.60 % (22911)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 25.85/10.60 % (22911)Termination reason: Unknown
% 25.85/10.60 % (22911)Termination phase: shuffling
% 25.85/10.60
% 25.85/10.60 % (22911)Memory used [KB]: 19061
% 25.85/10.60 % (22911)Time elapsed: 0.137 s
% 25.85/10.60 % (22911)Instructions burned: 268 (million)
% 25.85/10.60 % (22911)------------------------------
% 25.85/10.60 % (22911)------------------------------
% 25.85/10.61 % (22916)dis+1010_1:1_acc=model:avsq=on:ixr=off:prag=on:s2a=on:sac=on:i=1539:si=on:rtra=on_0 on DTF2THF_22696 for (2975ds/1539Mi)
% 25.85/10.63 % (22917)lrs+2_1:1_bd=off:e2e=on:fde=none:kws=inv_precedence:plsq=on:plsqr=32,1:sos=on:i=408:si=on:rtra=on_0 on DTF2THF_22696 for (2975ds/408Mi)
% 25.85/10.65 % (22909)Instruction limit reached!
% 25.85/10.65 % (22909)------------------------------
% 25.85/10.65 % (22909)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 25.85/10.65 % (22909)Termination reason: Unknown
% 25.85/10.65 % (22909)Termination phase: shuffling
% 25.85/10.65
% 25.85/10.65 % (22909)Memory used [KB]: 18677
% 25.85/10.65 % (22909)Time elapsed: 0.226 s
% 25.85/10.65 % (22909)Instructions burned: 369 (million)
% 25.85/10.65 % (22909)------------------------------
% 25.85/10.65 % (22909)------------------------------
% 25.85/10.67 % (22918)lrs+1002_1:1_aac=none:au=on:cnfonf=lazy_gen:plsq=on:plsqc=1:plsqr=4203469,65536:i=5205:si=on:rtra=on_0 on DTF2THF_22696 for (2975ds/5205Mi)
% 25.85/10.68 % (22898)Instruction limit reached!
% 25.85/10.68 % (22898)------------------------------
% 25.85/10.68 % (22898)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 25.85/10.68 % (22898)Termination reason: Unknown
% 25.85/10.68 % (22898)Termination phase: shuffling
% 25.85/10.68
% 25.85/10.68 % (22898)Memory used [KB]: 33005
% 25.85/10.68 % (22898)Time elapsed: 0.494 s
% 25.85/10.68 % (22898)Instructions burned: 987 (million)
% 25.85/10.68 % (22898)------------------------------
% 25.85/10.68 % (22898)------------------------------
% 26.76/10.70 % (22919)dis+1010_1:4_au=on:av=off:cbe=off:nwc=10.0:s2a=on:s2at=3.0:sd=2:ss=axioms:st=5.0:i=1913:si=on:rtra=on_0 on DTF2THF_22696 for (2974ds/1913Mi)
% 26.77/10.85 % (22905)Instruction limit reached!
% 26.77/10.85 % (22905)------------------------------
% 26.77/10.85 % (22905)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 26.77/10.85 % (22905)Termination reason: Unknown
% 26.77/10.85 % (22905)Termination phase: Property scanning
% 26.77/10.85
% 26.77/10.85 % (22905)Memory used [KB]: 23027
% 26.77/10.85 % (22905)Time elapsed: 0.513 s
% 26.77/10.85 % (22905)Instructions burned: 923 (million)
% 26.77/10.85 % (22905)------------------------------
% 26.77/10.85 % (22905)------------------------------
% 26.77/10.87 % (22920)lrs+2_1:1_bd=off:cbe=off:piset=equals:sd=1:sgt=20:sos=all:ss=axioms:i=4350:si=on:rtra=on_0 on DTF2THF_22696 for (2973ds/4350Mi)
% 26.77/10.89 % (22917)Instruction limit reached!
% 26.77/10.89 % (22917)------------------------------
% 26.77/10.89 % (22917)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 26.77/10.89 % (22917)Termination reason: Unknown
% 26.77/10.89 % (22917)Termination phase: shuffling
% 26.77/10.89
% 26.77/10.89 % (22917)Memory used [KB]: 19317
% 26.77/10.89 % (22917)Time elapsed: 0.263 s
% 26.77/10.89 % (22917)Instructions burned: 409 (million)
% 26.77/10.89 % (22917)------------------------------
% 26.77/10.89 % (22917)------------------------------
% 27.59/10.91 % (22921)ott+1010_1:4_cbe=off:chr=on:hfsq=on:hfsqr=1,16:hud=3:sac=on:sp=unary_frequency:ss=axioms:st=5.0:tnu=2:i=68:si=on:rtra=on_0 on DTF2THF_22696 for (2972ds/68Mi)
% 27.59/10.95 % (22921)Instruction limit reached!
% 27.59/10.95 % (22921)------------------------------
% 27.59/10.95 % (22921)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 27.59/10.95 % (22921)Termination reason: Unknown
% 27.59/10.95 % (22921)Termination phase: shuffling
% 27.59/10.95
% 27.59/10.95 % (22921)Memory used [KB]: 14200
% 27.59/10.95 % (22921)Time elapsed: 0.046 s
% 27.59/10.95 % (22921)Instructions burned: 69 (million)
% 27.59/10.95 % (22921)------------------------------
% 27.59/10.95 % (22921)------------------------------
% 27.59/10.97 % (22922)dis+10_1:1_au=on:nwc=5.0:s2a=on:s2at=2.0:sac=on:sd=7:ss=axioms:st=2.0:i=358:si=on:rtra=on_0 on DTF2THF_22696 for (2972ds/358Mi)
% 27.59/11.00 % (22910)Instruction limit reached!
% 27.59/11.00 % (22910)------------------------------
% 27.59/11.00 % (22910)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 27.59/11.00 % (22910)Termination reason: Unknown
% 27.59/11.00 % (22910)Termination phase: Property scanning
% 27.59/11.00
% 27.59/11.00 % (22910)Memory used [KB]: 23027
% 27.59/11.00 % (22910)Time elapsed: 0.533 s
% 27.59/11.00 % (22910)Instructions burned: 1022 (million)
% 27.59/11.00 % (22910)------------------------------
% 27.59/11.00 % (22910)------------------------------
% 27.59/11.02 % (22923)lrs+1010_1:16_add=large:amm=off:anc=all_dependent:chr=on:fd=off:ins=2:sp=const_max:tnu=5:uhcvi=on:i=31:si=on:rtra=on_0 on DTF2THF_22696 for (2971ds/31Mi)
% 29.99/11.04 % (22923)Instruction limit reached!
% 29.99/11.04 % (22923)------------------------------
% 29.99/11.04 % (22923)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 29.99/11.04 % (22923)Termination reason: Unknown
% 29.99/11.04 % (22923)Termination phase: shuffling
% 29.99/11.04
% 29.99/11.04 % (22923)Memory used [KB]: 13560
% 29.99/11.04 % (22923)Time elapsed: 0.028 s
% 29.99/11.04 % (22923)Instructions burned: 32 (million)
% 29.99/11.04 % (22923)------------------------------
% 29.99/11.04 % (22923)------------------------------
% 29.99/11.06 % (22924)lrs+10_1:1_cnfonf=lazy_simp:sgt=5:sos=on:ss=axioms:i=279:si=on:rtra=on_0 on DTF2THF_22696 for (2971ds/279Mi)
% 30.65/11.17 % (22922)Instruction limit reached!
% 30.65/11.17 % (22922)------------------------------
% 30.65/11.17 % (22922)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 30.65/11.17 % (22922)Termination reason: Unknown
% 30.65/11.17 % (22922)Termination phase: shuffling
% 30.65/11.17
% 30.65/11.17 % (22922)Memory used [KB]: 18421
% 30.65/11.17 % (22922)Time elapsed: 0.201 s
% 30.65/11.17 % (22922)Instructions burned: 358 (million)
% 30.65/11.17 % (22922)------------------------------
% 30.65/11.17 % (22922)------------------------------
% 30.65/11.19 % (22925)lrs+10_8:1_au=on:fde=none:slsq=on:i=258:si=on:rtra=on_0 on DTF2THF_22696 for (2969ds/258Mi)
% 31.06/11.21 % (22924)Instruction limit reached!
% 31.06/11.21 % (22924)------------------------------
% 31.06/11.21 % (22924)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 31.06/11.21 % (22924)Termination reason: Unknown
% 31.06/11.21 % (22924)Termination phase: shuffling
% 31.06/11.21
% 31.06/11.21 % (22924)Memory used [KB]: 19189
% 31.06/11.21 % (22924)Time elapsed: 0.148 s
% 31.06/11.21 % (22924)Instructions burned: 281 (million)
% 31.06/11.21 % (22924)------------------------------
% 31.06/11.21 % (22924)------------------------------
% 31.06/11.23 % (22926)dis+2_1:1024_atotf=0.0802881:au=on:bs=unit_only:cnfonf=lazy_not_gen_be_off:fde=none:nwc=10.0:prag=on:i=26:si=on:rtra=on_0 on DTF2THF_22696 for (2969ds/26Mi)
% 31.06/11.25 % (22926)Instruction limit reached!
% 31.06/11.25 % (22926)------------------------------
% 31.06/11.25 % (22926)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 31.06/11.25 % (22926)Termination reason: Unknown
% 31.06/11.25 % (22926)Termination phase: shuffling
% 31.06/11.25
% 31.06/11.25 % (22926)Memory used [KB]: 13560
% 31.06/11.25 % (22926)Time elapsed: 0.023 s
% 31.06/11.25 % (22926)Instructions burned: 26 (million)
% 31.06/11.25 % (22926)------------------------------
% 31.06/11.25 % (22926)------------------------------
% 31.06/11.28 % (22927)lrs+1010_1:1_au=on:cbe=off:cnfonf=lazy_not_be_gen:ntd=on:pe=on:i=995:si=on:rtra=on_0 on DTF2THF_22696 for (2968ds/995Mi)
% 31.78/11.34 % (22925)Instruction limit reached!
% 31.78/11.34 % (22925)------------------------------
% 31.78/11.34 % (22925)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 31.78/11.34 % (22925)Termination reason: Unknown
% 31.78/11.34 % (22925)Termination phase: shuffling
% 31.78/11.34
% 31.78/11.34 % (22925)Memory used [KB]: 17014
% 31.78/11.34 % (22925)Time elapsed: 0.149 s
% 31.78/11.34 % (22925)Instructions burned: 259 (million)
% 31.78/11.34 % (22925)------------------------------
% 31.78/11.34 % (22925)------------------------------
% 31.78/11.37 % (22928)dis+10_1:8_amm=sco:cbe=off:hflw=5:hfsq=on:hfsqc=14:hfsql=off:hfsqr=4,1:hud=5:ile=on:lma=on:nm=2:ntd=on:prag=on:sac=on:sp=frequency:i=373:si=on:rtra=on_0 on DTF2THF_22696 for (2968ds/373Mi)
% 32.46/11.49 % (22916)Instruction limit reached!
% 32.46/11.49 % (22916)------------------------------
% 32.46/11.49 % (22916)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 32.46/11.49 % (22916)Termination reason: Unknown
% 32.46/11.49 % (22916)Termination phase: Preprocessing 3
% 32.46/11.49
% 32.46/11.49 % (22916)Memory used [KB]: 31854
% 32.46/11.49 % (22916)Time elapsed: 0.885 s
% 32.46/11.49 % (22916)Instructions burned: 1540 (million)
% 32.46/11.49 % (22916)------------------------------
% 32.46/11.49 % (22916)------------------------------
% 32.96/11.51 % (22929)lrs+1010_1:128_au=on:cnfonf=lazy_pi_sigma_gen:hflw=4:hfsq=on:hfsqc=20:hfsql=off:i=253:si=on:rtra=on_0 on DTF2THF_22696 for (2966ds/253Mi)
% 33.36/11.61 % (22928)Instruction limit reached!
% 33.36/11.61 % (22928)------------------------------
% 33.36/11.61 % (22928)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 33.36/11.61 % (22928)Termination reason: Unknown
% 33.36/11.61 % (22928)Termination phase: shuffling
% 33.36/11.61
% 33.36/11.61 % (22928)Memory used [KB]: 18677
% 33.36/11.61 % (22928)Time elapsed: 0.248 s
% 33.36/11.61 % (22928)Instructions burned: 374 (million)
% 33.36/11.61 % (22928)------------------------------
% 33.36/11.61 % (22928)------------------------------
% 33.36/11.64 % (22930)lrs+1002_1:14_au=on:bd=off:e2e=on:fde=unused:sac=on:sos=on:i=832:si=on:rtra=on_0 on DTF2THF_22696 for (2965ds/832Mi)
% 33.36/11.64 % (22929)Instruction limit reached!
% 33.36/11.64 % (22929)------------------------------
% 33.36/11.64 % (22929)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 33.36/11.64 % (22929)Termination reason: Unknown
% 33.36/11.64 % (22929)Termination phase: shuffling
% 33.36/11.64
% 33.36/11.64 % (22929)Memory used [KB]: 16758
% 33.36/11.64 % (22929)Time elapsed: 0.130 s
% 33.36/11.64 % (22929)Instructions burned: 254 (million)
% 33.36/11.64 % (22929)------------------------------
% 33.36/11.64 % (22929)------------------------------
% 33.36/11.66 % (22931)lrs+1010_1:1_chr=on:cnfonf=lazy_not_be_gen:gs=on:hud=5:plsq=on:plsqc=1:plsqr=5,1:prag=on:rawr=on:i=668:si=on:rtra=on_0 on DTF2THF_22696 for (2965ds/668Mi)
% 34.17/11.72 % (22919)Instruction limit reached!
% 34.17/11.72 % (22919)------------------------------
% 34.17/11.72 % (22919)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 34.17/11.72 % (22919)Termination reason: Unknown
% 34.17/11.72 % (22919)Termination phase: Function definition elimination
% 34.17/11.72
% 34.17/11.72 % (22919)Memory used [KB]: 28400
% 34.17/11.72 % (22919)Time elapsed: 1.020 s
% 34.17/11.72 % (22919)Instructions burned: 1913 (million)
% 34.17/11.72 % (22919)------------------------------
% 34.17/11.72 % (22919)------------------------------
% 34.17/11.75 % (22932)dis+10_1:1_anc=none:cnfonf=lazy_gen:fd=preordered:fe=off:hud=10:ins=3:ixr=off:nwc=5.0:plsq=on:plsqc=1:plsqr=32,1:sp=const_frequency:uhcvi=on:i=40:si=on:rtra=on_0 on DTF2THF_22696 for (2964ds/40Mi)
% 34.17/11.77 % (22927)Instruction limit reached!
% 34.17/11.77 % (22927)------------------------------
% 34.17/11.77 % (22927)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 34.17/11.77 % (22927)Termination reason: Unknown
% 34.17/11.77 % (22927)Termination phase: shuffling
% 34.17/11.77
% 34.17/11.77 % (22927)Memory used [KB]: 33133
% 34.17/11.77 % (22927)Time elapsed: 0.500 s
% 34.17/11.77 % (22927)Instructions burned: 995 (million)
% 34.17/11.77 % (22927)------------------------------
% 34.17/11.77 % (22927)------------------------------
% 34.17/11.78 % (22932)Instruction limit reached!
% 34.17/11.78 % (22932)------------------------------
% 34.17/11.78 % (22932)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 34.17/11.78 % (22932)Termination reason: Unknown
% 34.17/11.78 % (22932)Termination phase: shuffling
% 34.17/11.78
% 34.17/11.78 % (22932)Memory used [KB]: 13816
% 34.17/11.78 % (22932)Time elapsed: 0.035 s
% 34.17/11.78 % (22932)Instructions burned: 41 (million)
% 34.17/11.78 % (22932)------------------------------
% 34.17/11.78 % (22932)------------------------------
% 34.17/11.80 % (22933)lrs+1010_1:1_cnfonf=lazy_pi_sigma_gen:fd=off:ntd=on:sd=1:sgt=32:sos=on:ss=axioms:i=294:si=on:rtra=on_0 on DTF2THF_22696 for (2963ds/294Mi)
% 35.11/11.81 % (22934)dis+1010_1:1_avsq=on:fe=off:fsr=off:hud=10:plsq=on:plsqr=32,1:prag=on:i=187:si=on:rtra=on_0 on DTF2THF_22696 for (2963ds/187Mi)
% 35.41/11.91 % (22934)Instruction limit reached!
% 35.41/11.91 % (22934)------------------------------
% 35.41/11.91 % (22934)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 35.41/11.92 % (22934)Termination reason: Unknown
% 35.41/11.92 % (22934)Termination phase: shuffling
% 35.41/11.92
% 35.41/11.92 % (22934)Memory used [KB]: 15863
% 35.41/11.92 % (22934)Time elapsed: 0.111 s
% 35.41/11.92 % (22934)Instructions burned: 189 (million)
% 35.41/11.92 % (22934)------------------------------
% 35.41/11.92 % (22934)------------------------------
% 35.41/11.94 % (22935)lrs+1010_2:3_au=on:fsr=off:sd=4:ss=axioms:st=1.5:i=371:si=on:rtra=on_0 on DTF2THF_22696 for (2962ds/371Mi)
% 35.41/11.96 % (22933)Instruction limit reached!
% 35.41/11.96 % (22933)------------------------------
% 35.41/11.96 % (22933)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 35.41/11.96 % (22933)Termination reason: Unknown
% 35.41/11.96 % (22933)Termination phase: shuffling
% 35.41/11.96
% 35.41/11.96 % (22933)Memory used [KB]: 19317
% 35.41/11.96 % (22933)Time elapsed: 0.164 s
% 35.41/11.96 % (22933)Instructions burned: 294 (million)
% 35.41/11.96 % (22933)------------------------------
% 35.41/11.96 % (22933)------------------------------
% 35.41/11.98 % (22936)dis+1003_1:1_au=on:bd=preordered:fde=none:ixr=off:nwc=10.0:i=1827:si=on:rtra=on_0 on DTF2THF_22696 for (2962ds/1827Mi)
% 35.41/11.99 % (22931)Instruction limit reached!
% 35.41/11.99 % (22931)------------------------------
% 35.41/11.99 % (22931)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 35.41/11.99 % (22931)Termination reason: Unknown
% 35.41/11.99 % (22931)Termination phase: shuffling
% 35.41/11.99
% 35.41/11.99 % (22931)Memory used [KB]: 24562
% 35.41/11.99 % (22931)Time elapsed: 0.333 s
% 35.41/11.99 % (22931)Instructions burned: 668 (million)
% 35.41/11.99 % (22931)------------------------------
% 35.41/11.99 % (22931)------------------------------
% 35.41/12.02 % (22937)lrs+10_1:1_av=off:bd=off:cnfonf=lazy_gen:fe=off:hud=10:plsq=on:plsqc=1:plsqr=32,1:tnu=1:i=70:si=on:rtra=on_0 on DTF2THF_22696 for (2961ds/70Mi)
% 37.02/12.06 % (22937)Instruction limit reached!
% 37.02/12.06 % (22937)------------------------------
% 37.02/12.06 % (22937)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 37.02/12.06 % (22937)Termination reason: Unknown
% 37.02/12.06 % (22937)Termination phase: shuffling
% 37.02/12.06
% 37.02/12.06 % (22937)Memory used [KB]: 14200
% 37.02/12.06 % (22937)Time elapsed: 0.045 s
% 37.02/12.06 % (22937)Instructions burned: 71 (million)
% 37.02/12.06 % (22937)------------------------------
% 37.02/12.06 % (22937)------------------------------
% 37.02/12.09 % (22938)lrs+10_1:20_bd=off:cnfonf=lazy_not_be_gen:fe=axiom:hud=10:ins=1:nm=20:prag=on:sos=on:i=316:si=on:rtra=on_0 on DTF2THF_22696 for (2961ds/316Mi)
% 37.02/12.09 % (22930)Instruction limit reached!
% 37.02/12.09 % (22930)------------------------------
% 37.02/12.09 % (22930)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 37.02/12.09 % (22930)Termination reason: Unknown
% 37.02/12.09 % (22930)Termination phase: Property scanning
% 37.02/12.09
% 37.02/12.09 % (22930)Memory used [KB]: 23027
% 37.02/12.09 % (22930)Time elapsed: 0.460 s
% 37.02/12.09 % (22930)Instructions burned: 832 (million)
% 37.02/12.09 % (22930)------------------------------
% 37.02/12.09 % (22930)------------------------------
% 37.02/12.12 % (22939)ott+1010_1:1_cnfonf=lazy_gen:fe=off:plsq=on:plsqc=1:plsqr=3548551,65536:s2a=on:i=932:si=on:rtra=on_0 on DTF2THF_22696 for (2960ds/932Mi)
% 37.69/12.14 % (22935)Instruction limit reached!
% 37.69/12.14 % (22935)------------------------------
% 37.69/12.14 % (22935)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 37.69/12.14 % (22935)Termination reason: Unknown
% 37.69/12.14 % (22935)Termination phase: shuffling
% 37.69/12.14
% 37.69/12.14 % (22935)Memory used [KB]: 18677
% 37.69/12.14 % (22935)Time elapsed: 0.208 s
% 37.69/12.14 % (22935)Instructions burned: 372 (million)
% 37.69/12.14 % (22935)------------------------------
% 37.69/12.14 % (22935)------------------------------
% 37.69/12.16 % (22940)lrs+1010_1:128_au=on:bd=preordered:cnfonf=lazy_pi_sigma_gen:plsq=on:plsqc=1:plsqr=32,1:sos=all:sp=occurrence:i=4714:si=on:rtra=on_0 on DTF2THF_22696 for (2960ds/4714Mi)
% 37.96/12.27 % (22938)Instruction limit reached!
% 37.96/12.27 % (22938)------------------------------
% 37.96/12.27 % (22938)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 37.96/12.27 % (22938)Termination reason: Unknown
% 37.96/12.27 % (22938)Termination phase: shuffling
% 37.96/12.27
% 37.96/12.27 % (22938)Memory used [KB]: 19701
% 37.96/12.27 % (22938)Time elapsed: 0.185 s
% 37.96/12.27 % (22938)Instructions burned: 317 (million)
% 37.96/12.27 % (22938)------------------------------
% 37.96/12.27 % (22938)------------------------------
% 37.96/12.28 % (22941)dis+10_1:1_au=on:e2e=on:fde=none:nwc=10.0:s2a=on:s2at=2.0:i=8250:si=on:rtra=on_0 on DTF2THF_22696 for (2958ds/8250Mi)
% 39.38/12.59 % (22939)Instruction limit reached!
% 39.38/12.59 % (22939)------------------------------
% 39.38/12.59 % (22939)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 39.38/12.59 % (22939)Termination reason: Unknown
% 39.38/12.59 % (22939)Termination phase: shuffling
% 39.38/12.59
% 39.38/12.59 % (22939)Memory used [KB]: 32238
% 39.38/12.59 % (22939)Time elapsed: 0.481 s
% 39.38/12.59 % (22939)Instructions burned: 932 (million)
% 39.38/12.59 % (22939)------------------------------
% 39.38/12.59 % (22939)------------------------------
% 39.78/12.62 % (22942)lrs+1002_1:1_apa=on:cnfonf=off:fd=off:flr=on:hud=1:lcm=reverse:prag=on:sd=1:sos=on:ss=axioms:i=770:si=on:rtra=on_0 on DTF2THF_22696 for (2955ds/770Mi)
% 43.96/13.03 % (22936)Instruction limit reached!
% 43.96/13.03 % (22936)------------------------------
% 43.96/13.03 % (22936)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 43.96/13.03 % (22936)Termination reason: Unknown
% 43.96/13.03 % (22936)Termination phase: Property scanning
% 43.96/13.03
% 43.96/13.03 % (22936)Memory used [KB]: 35436
% 43.96/13.03 % (22936)Time elapsed: 1.047 s
% 43.96/13.03 % (22936)Instructions burned: 1828 (million)
% 43.96/13.03 % (22936)------------------------------
% 43.96/13.03 % (22936)------------------------------
% 43.96/13.04 % (22920)Instruction limit reached!
% 43.96/13.04 % (22920)------------------------------
% 43.96/13.04 % (22920)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 43.96/13.04 % (22920)Termination reason: Unknown
% 43.96/13.04 % (22920)Termination phase: Saturation
% 43.96/13.04
% 43.96/13.04 % (22920)Memory used [KB]: 33645
% 43.96/13.04 % (22920)Time elapsed: 2.192 s
% 43.96/13.04 % (22920)Instructions burned: 4350 (million)
% 43.96/13.04 % (22920)------------------------------
% 43.96/13.04 % (22920)------------------------------
% 43.96/13.05 % (22942)Instruction limit reached!
% 43.96/13.05 % (22942)------------------------------
% 43.96/13.05 % (22942)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 43.96/13.05 % (22942)Termination reason: Unknown
% 43.96/13.05 % (22942)Termination phase: shuffling
% 43.96/13.05
% 43.96/13.05 % (22942)Memory used [KB]: 26097
% 43.96/13.05 % (22942)Time elapsed: 0.435 s
% 43.96/13.05 % (22942)Instructions burned: 771 (million)
% 43.96/13.05 % (22942)------------------------------
% 43.96/13.05 % (22942)------------------------------
% 43.96/13.05 % (22943)ott+1010_5:1_av=off:fe=off:ins=1:kws=precedence:sp=reverse_arity:i=794:si=on:rtra=on_0 on DTF2THF_22696 for (2951ds/794Mi)
% 43.96/13.06 % (22944)dis+1002_1:1_au=on:avsq=on:avsqr=1,16:cond=on:sp=frequency:i=464:si=on:rtra=on_0 on DTF2THF_22696 for (2951ds/464Mi)
% 43.96/13.08 % (22945)ott+21_1:3_avsq=on:bd=off:cnfonf=lazy_gen:fe=off:hud=10:ixr=off:plsq=on:plsqc=1:plsqr=32,1:prag=on:tnu=1:i=1376:si=on:rtra=on_0 on DTF2THF_22696 for (2951ds/1376Mi)
% 43.96/13.10 % (22915)Instruction limit reached!
% 43.96/13.10 % (22915)------------------------------
% 43.96/13.10 % (22915)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 43.96/13.10 % (22915)Termination reason: Unknown
% 43.96/13.10 % (22915)Termination phase: Saturation
% 43.96/13.10
% 43.96/13.10 % (22915)Memory used [KB]: 34157
% 43.96/13.10 % (22915)Time elapsed: 2.526 s
% 43.96/13.10 % (22915)Instructions burned: 4437 (million)
% 43.96/13.10 % (22915)------------------------------
% 43.96/13.10 % (22915)------------------------------
% 44.48/13.15 % (22946)dis+10_1:1_etr=on:plsq=on:plsqc=1:plsqr=32,1:i=906:si=on:rtra=on_0 on DTF2THF_22696 for (2950ds/906Mi)
% 44.71/13.21 % (22918)Instruction limit reached!
% 44.71/13.21 % (22918)------------------------------
% 44.71/13.21 % (22918)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 44.71/13.21 % (22918)Termination reason: Unknown
% 44.71/13.21 % (22918)Termination phase: Property scanning
% 44.71/13.21
% 44.71/13.21 % (22918)Memory used [KB]: 44007
% 44.71/13.21 % (22918)Time elapsed: 2.535 s
% 44.71/13.21 % (22918)Instructions burned: 5207 (million)
% 44.71/13.21 % (22918)------------------------------
% 44.71/13.21 % (22918)------------------------------
% 44.71/13.23 % (22947)lrs+1002_1:16_bd=off:e2e=on:fde=unused:kws=precedence:sac=on:sos=on:ss=axioms:i=434:si=on:rtra=on_0 on DTF2THF_22696 for (2949ds/434Mi)
% 44.71/13.29 % (22944)Instruction limit reached!
% 44.71/13.29 % (22944)------------------------------
% 44.71/13.29 % (22944)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 44.71/13.29 % (22944)Termination reason: Unknown
% 44.71/13.29 % (22944)Termination phase: shuffling
% 44.71/13.29
% 44.71/13.29 % (22944)Memory used [KB]: 19957
% 44.71/13.29 % (22944)Time elapsed: 0.232 s
% 44.71/13.29 % (22944)Instructions burned: 465 (million)
% 44.71/13.29 % (22944)------------------------------
% 44.71/13.29 % (22944)------------------------------
% 45.17/13.32 % (22948)lrs+10_5:4_cnfonf=conj_eager:fe=axiom:hflw=5:hfsq=on:hfsqc=10:hfsql=off:hud=10:plsq=on:sgt=5:spb=goal_then_units:ss=axioms:i=865:si=on:rtra=on_0 on DTF2THF_22696 for (2948ds/865Mi)
% 45.37/13.47 % (22947)Instruction limit reached!
% 45.37/13.47 % (22947)------------------------------
% 45.37/13.47 % (22947)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 45.37/13.47 % (22947)Termination reason: Unknown
% 45.37/13.47 % (22947)Termination phase: shuffling
% 45.37/13.47
% 45.37/13.47 % (22947)Memory used [KB]: 19701
% 45.37/13.47 % (22947)Time elapsed: 0.242 s
% 45.37/13.47 % (22947)Instructions burned: 434 (million)
% 45.37/13.47 % (22947)------------------------------
% 45.37/13.47 % (22947)------------------------------
% 45.37/13.49 % (22949)dis+1002_1:1_cnfonf=lazy_simp:hud=5:plsq=on:plsqc=1:plsqr=32,1:prag=on:i=593:si=on:rtra=on_0 on DTF2THF_22696 for (2946ds/593Mi)
% 45.37/13.49 % (22943)Instruction limit reached!
% 45.37/13.49 % (22943)------------------------------
% 45.37/13.49 % (22943)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 45.37/13.49 % (22943)Termination reason: Unknown
% 45.37/13.49 % (22943)Termination phase: Property scanning
% 45.37/13.49
% 45.37/13.49 % (22943)Memory used [KB]: 25074
% 45.37/13.49 % (22943)Time elapsed: 0.443 s
% 45.37/13.49 % (22943)Instructions burned: 794 (million)
% 45.37/13.49 % (22943)------------------------------
% 45.37/13.49 % (22943)------------------------------
% 45.82/13.52 % (22950)dis+10_1:1_sd=2:sgt=20:slsq=on:ss=axioms:st=6.0:i=443:si=on:rtra=on_0 on DTF2THF_22696 for (2946ds/443Mi)
% 46.50/13.65 % (22946)Instruction limit reached!
% 46.50/13.65 % (22946)------------------------------
% 46.50/13.65 % (22946)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 46.50/13.65 % (22946)Termination reason: Unknown
% 46.50/13.65 % (22946)Termination phase: Property scanning
% 46.50/13.65
% 46.50/13.65 % (22946)Memory used [KB]: 25074
% 46.50/13.65 % (22946)Time elapsed: 0.498 s
% 46.50/13.65 % (22946)Instructions burned: 906 (million)
% 46.50/13.65 % (22946)------------------------------
% 46.50/13.65 % (22946)------------------------------
% 46.50/13.67 % (22951)dis+10_1:1_atotf=0.1:cbe=off:sd=3:ss=axioms:st=2.0:i=594:si=on:rtra=on_0 on DTF2THF_22696 for (2945ds/594Mi)
% 47.21/13.76 % (22950)Instruction limit reached!
% 47.21/13.76 % (22950)------------------------------
% 47.21/13.76 % (22950)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 47.21/13.76 % (22950)Termination reason: Unknown
% 47.21/13.76 % (22950)Termination phase: shuffling
% 47.21/13.76
% 47.21/13.76 % (22950)Memory used [KB]: 19701
% 47.21/13.76 % (22950)Time elapsed: 0.248 s
% 47.21/13.76 % (22950)Instructions burned: 443 (million)
% 47.21/13.76 % (22950)------------------------------
% 47.21/13.76 % (22950)------------------------------
% 47.21/13.78 % (22948)Instruction limit reached!
% 47.21/13.78 % (22948)------------------------------
% 47.21/13.78 % (22948)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 47.21/13.78 % (22948)Termination reason: Unknown
% 47.21/13.78 % (22948)Termination phase: shuffling
% 47.21/13.78
% 47.21/13.78 % (22948)Memory used [KB]: 31342
% 47.21/13.78 % (22948)Time elapsed: 0.463 s
% 47.21/13.78 % (22948)Instructions burned: 866 (million)
% 47.21/13.78 % (22948)------------------------------
% 47.21/13.78 % (22948)------------------------------
% 47.21/13.79 % (22952)ott+10_1:1024_anc=none:bd=off:e2e=on:fde=none:fsr=off:hud=10:ixr=off:nm=4:nwc=5.0:spb=goal:i=327:si=on:rtra=on_0 on DTF2THF_22696 for (2944ds/327Mi)
% 48.16/13.81 % (22953)lrs+1002_1:1_au=on:bd=off:e2e=on:sd=2:sos=on:ss=axioms:i=5411:si=on:rtra=on_0 on DTF2THF_22696 for (2943ds/5411Mi)
% 48.16/13.81 % (22949)Instruction limit reached!
% 48.16/13.81 % (22949)------------------------------
% 48.16/13.81 % (22949)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 48.16/13.81 % (22949)Termination reason: Unknown
% 48.16/13.81 % (22949)Termination phase: shuffling
% 48.16/13.81
% 48.16/13.81 % (22949)Memory used [KB]: 23539
% 48.16/13.81 % (22949)Time elapsed: 0.318 s
% 48.16/13.81 % (22949)Instructions burned: 594 (million)
% 48.16/13.81 % (22949)------------------------------
% 48.16/13.81 % (22949)------------------------------
% 48.16/13.81 % (22945)Instruction limit reached!
% 48.16/13.81 % (22945)------------------------------
% 48.16/13.81 % (22945)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 48.16/13.81 % (22945)Termination reason: Unknown
% 48.16/13.81 % (22945)Termination phase: Property scanning
% 48.16/13.81
% 48.16/13.81 % (22945)Memory used [KB]: 37739
% 48.16/13.81 % (22945)Time elapsed: 0.739 s
% 48.16/13.81 % (22945)Instructions burned: 1377 (million)
% 48.16/13.81 % (22945)------------------------------
% 48.16/13.81 % (22945)------------------------------
% 48.16/13.83 % (22954)lrs+10_1:20_au=on:c=on:cbe=off:cnfonf=lazy_gen:ins=1:plsq=on:plsqc=1:plsqr=46,31:sac=on:i=1762:si=on:rtra=on_0 on DTF2THF_22696 for (2943ds/1762Mi)
% 48.16/13.84 % (22955)dis+1003_1:64_cha=on:cnfonf=lazy_gen:hud=9:nwc=8.0:prag=on:i=96:si=on:rtra=on_0 on DTF2THF_22696 for (2943ds/96Mi)
% 48.16/13.89 % (22955)Instruction limit reached!
% 48.16/13.89 % (22955)------------------------------
% 48.16/13.89 % (22955)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 48.16/13.89 % (22955)Termination reason: Unknown
% 48.16/13.89 % (22955)Termination phase: shuffling
% 48.16/13.89
% 48.16/13.89 % (22955)Memory used [KB]: 14583
% 48.16/13.89 % (22955)Time elapsed: 0.059 s
% 48.16/13.89 % (22955)Instructions burned: 96 (million)
% 48.16/13.89 % (22955)------------------------------
% 48.16/13.89 % (22955)------------------------------
% 49.54/13.92 % (22956)lrs+21_1:1_hud=5:kws=inv_frequency:nm=16:nwc=2.0:prag=on:sos=on:ss=axioms:i=503:si=on:rtra=on_0 on DTF2THF_22696 for (2942ds/503Mi)
% 49.54/13.99 % (22952)Instruction limit reached!
% 49.54/13.99 % (22952)------------------------------
% 49.54/13.99 % (22952)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 49.54/13.99 % (22952)Termination reason: Unknown
% 49.54/13.99 % (22952)Termination phase: shuffling
% 49.54/13.99
% 49.54/13.99 % (22952)Memory used [KB]: 18166
% 49.54/13.99 % (22952)Time elapsed: 0.205 s
% 49.54/13.99 % (22952)Instructions burned: 328 (million)
% 49.54/13.99 % (22952)------------------------------
% 49.54/13.99 % (22952)------------------------------
% 49.54/14.02 % (22957)lrs+1002_1:1024_au=on:cond=fast:ntd=on:piset=and:plsq=on:plsqc=1:plsqr=183491,1048576:sac=on:tgt=ground:i=90:si=on:rtra=on_0 on DTF2THF_22696 for (2941ds/90Mi)
% 51.00/14.04 % (22951)Instruction limit reached!
% 51.00/14.04 % (22951)------------------------------
% 51.00/14.04 % (22951)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 51.00/14.04 % (22951)Termination reason: Unknown
% 51.00/14.04 % (22951)Termination phase: shuffling
% 51.00/14.04
% 51.00/14.04 % (22951)Memory used [KB]: 21875
% 51.00/14.04 % (22951)Time elapsed: 0.369 s
% 51.00/14.04 % (22951)Instructions burned: 594 (million)
% 51.00/14.04 % (22951)------------------------------
% 51.00/14.04 % (22951)------------------------------
% 51.00/14.06 % (22958)lrs+10_1:64_hud=5:plsq=on:plsqr=2011171,524288:i=593:si=on:rtra=on_0 on DTF2THF_22696 for (2941ds/593Mi)
% 51.00/14.07 % (22957)Instruction limit reached!
% 51.00/14.07 % (22957)------------------------------
% 51.00/14.07 % (22957)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 51.00/14.07 % (22957)Termination reason: Unknown
% 51.00/14.07 % (22957)Termination phase: shuffling
% 51.00/14.07
% 51.00/14.07 % (22957)Memory used [KB]: 14583
% 51.00/14.07 % (22957)Time elapsed: 0.058 s
% 51.00/14.07 % (22957)Instructions burned: 90 (million)
% 51.00/14.07 % (22957)------------------------------
% 51.00/14.07 % (22957)------------------------------
% 51.00/14.09 % (22959)dis-1002_1:1_alpa=false:apa=on:au=on:bd=off:cnfonf=off:e2e=on:er=filter:fde=unused:plsq=on:plsqr=9699663,1048576:s2a=on:sos=on:sp=unary_first:i=1512:si=on:rtra=on_0 on DTF2THF_22696 for (2940ds/1512Mi)
% 51.91/14.22 % (22956)Instruction limit reached!
% 51.91/14.22 % (22956)------------------------------
% 51.91/14.22 % (22956)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 51.91/14.22 % (22956)Termination reason: Unknown
% 51.91/14.22 % (22956)Termination phase: shuffling
% 51.91/14.22
% 51.91/14.22 % (22956)Memory used [KB]: 20468
% 51.91/14.22 % (22956)Time elapsed: 0.306 s
% 51.91/14.22 % (22956)Instructions burned: 503 (million)
% 51.91/14.22 % (22956)------------------------------
% 51.91/14.22 % (22956)------------------------------
% 51.91/14.24 % (22960)lrs+1002_1:128_cbe=off:cnfonf=lazy_not_gen_be_off:ins=1:i=569:si=on:rtra=on_0 on DTF2THF_22696 for (2939ds/569Mi)
% 52.34/14.38 % (22958)Instruction limit reached!
% 52.34/14.38 % (22958)------------------------------
% 52.34/14.38 % (22958)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 52.34/14.38 % (22958)Termination reason: Unknown
% 52.34/14.38 % (22958)Termination phase: shuffling
% 52.34/14.38
% 52.34/14.38 % (22958)Memory used [KB]: 21748
% 52.34/14.38 % (22958)Time elapsed: 0.319 s
% 52.34/14.38 % (22958)Instructions burned: 593 (million)
% 52.34/14.38 % (22958)------------------------------
% 52.34/14.38 % (22958)------------------------------
% 53.09/14.41 % (22961)ott+1002_1:1_bd=off:cnfonf=off:e2e=on:fd=off:fe=axiom:hud=5:prag=on:sos=on:i=501:si=on:rtra=on_0 on DTF2THF_22696 for (2937ds/501Mi)
% 53.09/14.45 % (22940)Instruction limit reached!
% 53.09/14.45 % (22940)------------------------------
% 53.09/14.45 % (22940)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 53.09/14.45 % (22940)Termination reason: Unknown
% 53.09/14.45 % (22940)Termination phase: Property scanning
% 53.09/14.45
% 53.09/14.45 % (22940)Memory used [KB]: 43496
% 53.09/14.45 % (22940)Time elapsed: 2.285 s
% 53.09/14.45 % (22940)Instructions burned: 4714 (million)
% 53.09/14.45 % (22940)------------------------------
% 53.09/14.45 % (22940)------------------------------
% 53.09/14.47 % (22962)lrs+1010_1:3_cnfonf=conj_eager:fsr=off:prag=on:sd=4:sgt=10:ss=axioms:i=562:si=on:rtra=on_0 on DTF2THF_22696 for (2937ds/562Mi)
% 53.46/14.54 % (22960)Instruction limit reached!
% 53.46/14.54 % (22960)------------------------------
% 53.46/14.54 % (22960)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 53.46/14.54 % (22960)Termination reason: Unknown
% 53.46/14.54 % (22960)Termination phase: shuffling
% 53.46/14.54
% 53.46/14.54 % (22960)Memory used [KB]: 23283
% 53.46/14.54 % (22960)Time elapsed: 0.297 s
% 53.46/14.54 % (22960)Instructions burned: 570 (million)
% 53.46/14.54 % (22960)------------------------------
% 53.46/14.54 % (22960)------------------------------
% 53.46/14.56 % (22963)lrs+10_1:1_au=on:cnfonf=off:fde=unused:kws=precedence:tgt=full:i=406:si=on:rtra=on_0 on DTF2THF_22696 for (2936ds/406Mi)
% 54.08/14.60 % (22953)Refutation not found, incomplete strategy
% 54.08/14.60 % (22953)------------------------------
% 54.08/14.60 % (22953)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 54.08/14.60 % (22953)Termination reason: Refutation not found, incomplete strategy
% 54.08/14.60
% 54.08/14.60
% 54.08/14.60 % (22953)Memory used [KB]: 28528
% 54.08/14.60 % (22953)Time elapsed: 0.817 s
% 54.08/14.60 % (22953)Instructions burned: 1323 (million)
% 54.08/14.60 % (22953)------------------------------
% 54.08/14.60 % (22953)------------------------------
% 54.08/14.63 % (22964)dis+1004_1:128_au=on:cbe=off:e2e=on:sac=on:i=582:si=on:rtra=on_0 on DTF2THF_22696 for (2935ds/582Mi)
% 54.08/14.69 % (22954)Instruction limit reached!
% 54.08/14.69 % (22954)------------------------------
% 54.08/14.69 % (22954)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 54.08/14.69 % (22954)Termination reason: Unknown
% 54.08/14.69 % (22954)Termination phase: Property scanning
% 54.08/14.69
% 54.08/14.69 % (22954)Memory used [KB]: 37739
% 54.08/14.69 % (22954)Time elapsed: 0.863 s
% 54.08/14.69 % (22954)Instructions burned: 1762 (million)
% 54.08/14.69 % (22954)------------------------------
% 54.08/14.69 % (22954)------------------------------
% 54.30/14.70 % (22961)Instruction limit reached!
% 54.30/14.70 % (22961)------------------------------
% 54.30/14.70 % (22961)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 54.30/14.70 % (22961)Termination reason: Unknown
% 54.30/14.70 % (22961)Termination phase: shuffling
% 54.30/14.70
% 54.30/14.70 % (22961)Memory used [KB]: 20596
% 54.30/14.70 % (22961)Time elapsed: 0.301 s
% 54.30/14.70 % (22961)Instructions burned: 502 (million)
% 54.30/14.70 % (22961)------------------------------
% 54.30/14.70 % (22961)------------------------------
% 54.30/14.72 % (22965)lrs+2_1:1_au=on:bd=off:bsr=unit_only:sd=2:sgt=32:sos=all:ss=axioms:i=456:si=on:rtra=on_0 on DTF2THF_22696 for (2934ds/456Mi)
% 54.30/14.73 % (22966)lrs+1010_1:1_au=on:cbe=off:cnfonf=conj_eager:tgt=ground:i=393:si=on:rtra=on_0 on DTF2THF_22696 for (2934ds/393Mi)
% 54.30/14.77 % (22962)Instruction limit reached!
% 54.30/14.77 % (22962)------------------------------
% 54.30/14.77 % (22962)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 54.30/14.77 % (22962)Termination reason: Unknown
% 54.30/14.77 % (22962)Termination phase: shuffling
% 54.30/14.77
% 54.30/14.77 % (22962)Memory used [KB]: 23155
% 54.30/14.77 % (22962)Time elapsed: 0.306 s
% 54.30/14.77 % (22962)Instructions burned: 562 (million)
% 54.30/14.77 % (22962)------------------------------
% 54.30/14.77 % (22962)------------------------------
% 54.30/14.79 % (22963)Instruction limit reached!
% 54.30/14.79 % (22963)------------------------------
% 54.30/14.79 % (22963)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 54.30/14.79 % (22963)Termination reason: Unknown
% 54.30/14.79 % (22963)Termination phase: shuffling
% 54.30/14.79
% 54.30/14.79 % (22963)Memory used [KB]: 18933
% 54.30/14.79 % (22963)Time elapsed: 0.226 s
% 54.30/14.79 % (22963)Instructions burned: 407 (million)
% 54.30/14.79 % (22963)------------------------------
% 54.30/14.79 % (22963)------------------------------
% 55.30/14.80 % (22967)lrs+1010_1:6_au=on:cnfonf=conj_eager:nwc=10.0:pe=on:prag=on:s2a=on:ss=axioms:st=1.5:i=1335:si=on:rtra=on_0 on DTF2THF_22696 for (2933ds/1335Mi)
% 55.30/14.81 % (22968)dis+1010_16:1_acc=model:au=on:avsq=on:cond=fast:plsq=on:plsqc=2:plsqr=9267,262144:i=3373:si=on:rtra=on_0 on DTF2THF_22696 for (2933ds/3373Mi)
% 55.40/14.94 % (22966)Instruction limit reached!
% 55.40/14.94 % (22966)------------------------------
% 55.40/14.94 % (22966)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 55.40/14.94 % (22966)Termination reason: Unknown
% 55.40/14.94 % (22966)Termination phase: shuffling
% 55.40/14.94
% 55.40/14.94 % (22966)Memory used [KB]: 20724
% 55.40/14.94 % (22966)Time elapsed: 0.212 s
% 55.40/14.94 % (22966)Instructions burned: 393 (million)
% 55.40/14.94 % (22966)------------------------------
% 55.40/14.94 % (22966)------------------------------
% 55.40/14.96 % (22969)dis+10_1:1024_aac=none:au=on:cnfonf=lazy_gen:plsq=on:plsqc=1:plsqr=32,1:spb=goal_then_units:i=474:si=on:rtra=on_0 on DTF2THF_22696 for (2932ds/474Mi)
% 55.40/14.96 % (22959)Instruction limit reached!
% 55.40/14.96 % (22959)------------------------------
% 55.40/14.96 % (22959)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 55.40/14.96 % (22959)Termination reason: Unknown
% 55.40/14.96 % (22959)Termination phase: Preprocessing 3
% 55.40/14.96
% 55.40/14.96 % (22959)Memory used [KB]: 31982
% 55.40/14.96 % (22959)Time elapsed: 0.874 s
% 55.40/14.96 % (22959)Instructions burned: 1512 (million)
% 55.40/14.96 % (22959)------------------------------
% 55.40/14.96 % (22959)------------------------------
% 55.40/14.99 % (22970)lrs+1002_1:16_amm=sco:cnfonf=lazy_pi_sigma_gen:fe=off:nm=60:piset=all_but_not_eq:plsq=on:plsqc=1:plsqr=2,1:rawr=on:sp=occurrence:ss=included:st=-1.0:i=157:si=on:rtra=on_0 on DTF2THF_22696 for (2931ds/157Mi)
% 55.40/15.00 % (22965)Instruction limit reached!
% 55.40/15.00 % (22965)------------------------------
% 55.40/15.00 % (22965)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 55.40/15.00 % (22965)Termination reason: Unknown
% 55.40/15.00 % (22965)Termination phase: shuffling
% 55.40/15.00
% 55.40/15.00 % (22965)Memory used [KB]: 19829
% 55.40/15.00 % (22965)Time elapsed: 0.283 s
% 55.40/15.00 % (22965)Instructions burned: 457 (million)
% 55.40/15.00 % (22965)------------------------------
% 55.40/15.00 % (22965)------------------------------
% 55.40/15.00 % (22964)Instruction limit reached!
% 55.40/15.00 % (22964)------------------------------
% 55.40/15.00 % (22964)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 55.40/15.00 % (22964)Termination reason: Unknown
% 55.40/15.00 % (22964)Termination phase: shuffling
% 55.40/15.00
% 55.40/15.00 % (22964)Memory used [KB]: 21875
% 55.40/15.00 % (22964)Time elapsed: 0.375 s
% 55.40/15.00 % (22964)Instructions burned: 582 (million)
% 55.40/15.00 % (22964)------------------------------
% 55.40/15.00 % (22964)------------------------------
% 55.40/15.02 % (22971)dis+1010_1:1_amm=off:cs=on:kws=precedence:nwc=10.0:piset=all:s2a=on:sac=on:sp=unary_frequency:spb=goal:ss=axioms:st=3.0:i=590:si=on:rtra=on_0 on DTF2THF_22696 for (2931ds/590Mi)
% 55.40/15.03 % (22972)dis+10_1:1024_kws=precedence:prag=on:rawr=on:sos=all:sp=unary_first:i=104:si=on:rtra=on_0 on DTF2THF_22696 for (2931ds/104Mi)
% 57.97/15.08 % (22970)Instruction limit reached!
% 57.97/15.08 % (22970)------------------------------
% 57.97/15.08 % (22970)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 57.97/15.08 % (22970)Termination reason: Unknown
% 57.97/15.08 % (22970)Termination phase: shuffling
% 57.97/15.08
% 57.97/15.08 % (22970)Memory used [KB]: 15479
% 57.97/15.08 % (22970)Time elapsed: 0.093 s
% 57.97/15.08 % (22970)Instructions burned: 157 (million)
% 57.97/15.08 % (22970)------------------------------
% 57.97/15.08 % (22970)------------------------------
% 57.97/15.09 % (22972)Instruction limit reached!
% 57.97/15.09 % (22972)------------------------------
% 57.97/15.09 % (22972)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 57.97/15.09 % (22972)Termination reason: Unknown
% 57.97/15.09 % (22972)Termination phase: shuffling
% 57.97/15.09
% 57.97/15.09 % (22972)Memory used [KB]: 14583
% 57.97/15.09 % (22972)Time elapsed: 0.069 s
% 57.97/15.09 % (22972)Instructions burned: 104 (million)
% 57.97/15.09 % (22972)------------------------------
% 57.97/15.09 % (22972)------------------------------
% 57.97/15.10 % (22973)lrs+1010_1:10_au=on:bd=off:cnfonf=lazy_pi_sigma_gen:sp=occurrence:i=414:si=on:rtra=on_0 on DTF2THF_22696 for (2930ds/414Mi)
% 57.97/15.12 % (22974)dis+10_1:1_av=off:bsd=on:bsr=unit_only:fd=off:hud=8:nm=16:prag=on:i=715:si=on:rtra=on_0 on DTF2THF_22696 for (2930ds/715Mi)
% 58.81/15.21 % (22969)Instruction limit reached!
% 58.81/15.21 % (22969)------------------------------
% 58.81/15.21 % (22969)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 58.81/15.21 % (22969)Termination reason: Unknown
% 58.81/15.21 % (22969)Termination phase: shuffling
% 58.81/15.21
% 58.81/15.21 % (22969)Memory used [KB]: 21875
% 58.81/15.21 % (22969)Time elapsed: 0.247 s
% 58.81/15.21 % (22969)Instructions burned: 474 (million)
% 58.81/15.21 % (22969)------------------------------
% 58.81/15.21 % (22969)------------------------------
% 58.81/15.23 % (22975)dis+21_1:5_au=on:ntd=on:nwc=5.0:s2a=on:ss=axioms:st=5.0:i=961:si=on:rtra=on_0 on DTF2THF_22696 for (2929ds/961Mi)
% 59.11/15.33 % (22973)Instruction limit reached!
% 59.11/15.33 % (22973)------------------------------
% 59.11/15.33 % (22973)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 59.11/15.33 % (22973)Termination reason: Unknown
% 59.11/15.33 % (22973)Termination phase: shuffling
% 59.11/15.33
% 59.11/15.33 % (22973)Memory used [KB]: 21108
% 59.11/15.33 % (22973)Time elapsed: 0.231 s
% 59.11/15.33 % (22973)Instructions burned: 416 (million)
% 59.11/15.33 % (22973)------------------------------
% 59.11/15.33 % (22973)------------------------------
% 59.11/15.36 % (22977)lrs-1002_1:128_abs=on:bd=preordered:cbe=off:e2e=on:fde=unused:fe=off:tgt=full:i=1213:si=on:rtra=on_0 on DTF2THF_22696 for (2928ds/1213Mi)
% 59.11/15.37 % (22971)Instruction limit reached!
% 59.11/15.37 % (22971)------------------------------
% 59.11/15.37 % (22971)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 59.11/15.37 % (22971)Termination reason: Unknown
% 59.11/15.37 % (22971)Termination phase: shuffling
% 59.11/15.37
% 59.11/15.37 % (22971)Memory used [KB]: 21875
% 59.11/15.37 % (22971)Time elapsed: 0.352 s
% 59.11/15.37 % (22971)Instructions burned: 591 (million)
% 59.11/15.37 % (22971)------------------------------
% 59.11/15.37 % (22971)------------------------------
% 59.11/15.39 % (22978)lrs+1002_1:12_bd=off:hud=1:ntd=on:piset=equals:prag=on:i=1979:si=on:rtra=on_0 on DTF2THF_22696 for (2928ds/1979Mi)
% 60.34/15.51 % (22967)Instruction limit reached!
% 60.34/15.51 % (22967)------------------------------
% 60.34/15.51 % (22967)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 60.34/15.51 % (22967)Termination reason: Unknown
% 60.34/15.51 % (22967)Termination phase: Property scanning
% 60.34/15.51
% 60.34/15.51 % (22967)Memory used [KB]: 37739
% 60.34/15.51 % (22967)Time elapsed: 0.714 s
% 60.34/15.51 % (22967)Instructions burned: 1335 (million)
% 60.34/15.51 % (22967)------------------------------
% 60.34/15.51 % (22967)------------------------------
% 60.34/15.52 % (22974)Instruction limit reached!
% 60.34/15.52 % (22974)------------------------------
% 60.34/15.52 % (22974)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 60.34/15.52 % (22974)Termination reason: Unknown
% 60.34/15.52 % (22974)Termination phase: Property scanning
% 60.34/15.52
% 60.34/15.52 % (22974)Memory used [KB]: 25074
% 60.34/15.52 % (22974)Time elapsed: 0.406 s
% 60.34/15.52 % (22974)Instructions burned: 715 (million)
% 60.34/15.52 % (22974)------------------------------
% 60.34/15.52 % (22974)------------------------------
% 60.34/15.54 % (22979)dis+1002_1:1_amm=off:au=on:bsd=on:cbe=off:cnfonf=lazy_gen:sp=unary_frequency:i=481:si=on:rtra=on_0 on DTF2THF_22696 for (2926ds/481Mi)
% 60.34/15.54 % (22980)dis+10_8:1_apa=on:cnfonf=off:e2e=on:plsq=on:plsqc=1:plsqr=32,1:ss=axioms:i=783:si=on:rtra=on_0 on DTF2THF_22696 for (2926ds/783Mi)
% 61.20/15.79 % (22979)Instruction limit reached!
% 61.20/15.79 % (22979)------------------------------
% 61.20/15.79 % (22979)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 61.20/15.79 % (22979)Termination reason: Unknown
% 61.20/15.79 % (22979)Termination phase: shuffling
% 61.20/15.79
% 61.20/15.79 % (22979)Memory used [KB]: 22003
% 61.20/15.79 % (22979)Time elapsed: 0.252 s
% 61.20/15.79 % (22979)Instructions burned: 481 (million)
% 61.20/15.79 % (22979)------------------------------
% 61.20/15.79 % (22979)------------------------------
% 61.20/15.79 % (22975)Instruction limit reached!
% 61.20/15.79 % (22975)------------------------------
% 61.20/15.79 % (22975)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 61.20/15.79 % (22975)Termination reason: Unknown
% 61.20/15.79 % (22975)Termination phase: Property scanning
% 61.20/15.79
% 61.20/15.79 % (22975)Memory used [KB]: 25074
% 61.20/15.79 % (22975)Time elapsed: 0.566 s
% 61.20/15.79 % (22975)Instructions burned: 961 (million)
% 61.20/15.79 % (22975)------------------------------
% 61.20/15.79 % (22975)------------------------------
% 61.67/15.81 % (22981)lrs+1010_1:1_avsq=on:avsqr=1,16:kws=inv_arity:nwc=5.0:sd=1:ss=axioms:i=432:si=on:rtra=on_0 on DTF2THF_22696 for (2923ds/432Mi)
% 61.67/15.82 % (22982)lrs+1010_1:1_atotf=0.5:au=on:cbe=off:cnfonf=lazy_not_be_gen:fe=off:nwc=5.0:spb=units:i=1670:si=on:rtra=on_0 on DTF2THF_22696 for (2923ds/1670Mi)
% 62.46/16.00 % (22980)Instruction limit reached!
% 62.46/16.00 % (22980)------------------------------
% 62.46/16.00 % (22980)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 62.46/16.00 % (22980)Termination reason: Unknown
% 62.46/16.00 % (22980)Termination phase: Property scanning
% 62.46/16.00
% 62.46/16.00 % (22980)Memory used [KB]: 24178
% 62.46/16.00 % (22980)Time elapsed: 0.462 s
% 62.46/16.00 % (22980)Instructions burned: 783 (million)
% 62.46/16.00 % (22980)------------------------------
% 62.46/16.00 % (22980)------------------------------
% 64.93/16.03 % (22983)dis+2_1:1_au=on:hfaw=0:hfsq=on:sp=weighted_frequency:i=90:si=on:rtra=on_0 on DTF2THF_22696 for (2921ds/90Mi)
% 64.93/16.06 % (22981)Instruction limit reached!
% 64.93/16.06 % (22981)------------------------------
% 64.93/16.06 % (22981)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 64.93/16.06 % (22981)Termination reason: Unknown
% 64.93/16.06 % (22981)Termination phase: shuffling
% 64.93/16.06
% 64.93/16.06 % (22981)Memory used [KB]: 19573
% 64.93/16.06 % (22981)Time elapsed: 0.252 s
% 64.93/16.06 % (22981)Instructions burned: 433 (million)
% 64.93/16.06 % (22981)------------------------------
% 64.93/16.06 % (22981)------------------------------
% 64.93/16.08 % (22984)dis+10_1:1_au=on:cnfonf=lazy_gen:ixr=off:plsq=on:plsqc=1:plsqr=32,1:i=346:si=on:rtra=on_0 on DTF2THF_22696 for (2920ds/346Mi)
% 64.93/16.09 % (22983)Instruction limit reached!
% 64.93/16.09 % (22983)------------------------------
% 64.93/16.09 % (22983)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 64.93/16.09 % (22983)Termination reason: Unknown
% 64.93/16.09 % (22983)Termination phase: shuffling
% 64.93/16.09
% 64.93/16.09 % (22983)Memory used [KB]: 14583
% 64.93/16.09 % (22983)Time elapsed: 0.060 s
% 64.93/16.09 % (22983)Instructions burned: 90 (million)
% 64.93/16.09 % (22983)------------------------------
% 64.93/16.09 % (22983)------------------------------
% 64.93/16.10 % (22977)Instruction limit reached!
% 64.93/16.10 % (22977)------------------------------
% 64.93/16.10 % (22977)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 64.93/16.10 % (22977)Termination reason: Unknown
% 64.93/16.10 % (22977)Termination phase: Naming
% 64.93/16.10
% 64.93/16.10 % (22977)Memory used [KB]: 27888
% 64.93/16.10 % (22977)Time elapsed: 0.746 s
% 64.93/16.10 % (22977)Instructions burned: 1213 (million)
% 64.93/16.10 % (22977)------------------------------
% 64.93/16.10 % (22977)------------------------------
% 64.93/16.11 % (22985)dis+1002_1:1_cnfonf=lazy_not_gen:hud=10:plsq=on:plsqc=1:plsqr=32,1:prag=on:i=401:si=on:rtra=on_0 on DTF2THF_22696 for (2920ds/401Mi)
% 65.56/16.13 % (22986)lrs+1010_1:1_au=on:s2a=on:sd=1:sgt=50:ss=axioms:i=6454:si=on:rtra=on_0 on DTF2THF_22696 for (2920ds/6454Mi)
% 65.84/16.26 % (22984)Instruction limit reached!
% 65.84/16.26 % (22984)------------------------------
% 65.84/16.26 % (22984)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 65.84/16.26 % (22984)Termination reason: Unknown
% 65.84/16.26 % (22984)Termination phase: shuffling
% 65.84/16.26
% 65.84/16.26 % (22984)Memory used [KB]: 20084
% 65.84/16.26 % (22984)Time elapsed: 0.183 s
% 65.84/16.26 % (22984)Instructions burned: 347 (million)
% 65.84/16.26 % (22984)------------------------------
% 65.84/16.26 % (22984)------------------------------
% 65.84/16.29 % (22987)dis+1010_1:1_avsq=on:fe=off:fsr=off:hud=10:kws=arity:prag=on:i=283:si=on:rtra=on_0 on DTF2THF_22696 for (2918ds/283Mi)
% 66.43/16.33 % (22985)Instruction limit reached!
% 66.43/16.33 % (22985)------------------------------
% 66.43/16.33 % (22985)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 66.43/16.33 % (22985)Termination reason: Unknown
% 66.43/16.33 % (22985)Termination phase: shuffling
% 66.43/16.33
% 66.43/16.33 % (22985)Memory used [KB]: 20852
% 66.43/16.33 % (22985)Time elapsed: 0.221 s
% 66.43/16.33 % (22985)Instructions burned: 403 (million)
% 66.43/16.33 % (22985)------------------------------
% 66.43/16.33 % (22985)------------------------------
% 66.43/16.35 % (22988)ott+10_1:1_aac=none:apa=on:au=on:cnfonf=off:ntd=on:nwc=12.0:sos=on:i=865:si=on:rtra=on_0 on DTF2THF_22696 for (2918ds/865Mi)
% 67.07/16.43 % (22987)Instruction limit reached!
% 67.07/16.43 % (22987)------------------------------
% 67.07/16.43 % (22987)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 67.07/16.43 % (22987)Termination reason: Unknown
% 67.07/16.43 % (22987)Termination phase: shuffling
% 67.07/16.43
% 67.07/16.43 % (22987)Memory used [KB]: 17270
% 67.07/16.43 % (22987)Time elapsed: 0.146 s
% 67.07/16.43 % (22987)Instructions burned: 283 (million)
% 67.07/16.43 % (22987)------------------------------
% 67.07/16.43 % (22987)------------------------------
% 67.07/16.45 % (22989)lrs+1002_3:4_au=on:av=off:cnfonf=off:fd=off:fsr=off:ins=2:sd=1:sos=on:ss=axioms:i=4528:si=on:rtra=on_0 on DTF2THF_22696 for (2917ds/4528Mi)
% 67.64/16.52 % (22978)Instruction limit reached!
% 67.64/16.52 % (22978)------------------------------
% 67.64/16.52 % (22978)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 67.64/16.52 % (22978)Termination reason: Unknown
% 67.64/16.52 % (22978)Termination phase: Property scanning
% 67.64/16.52
% 67.64/16.52 % (22978)Memory used [KB]: 35436
% 67.64/16.52 % (22978)Time elapsed: 1.127 s
% 67.64/16.52 % (22978)Instructions burned: 1980 (million)
% 67.64/16.52 % (22978)------------------------------
% 67.64/16.52 % (22978)------------------------------
% 67.64/16.54 % (22990)dis+1002_1:1_cbe=off:hud=5:nm=4:plsq=on:plsqr=7,1:prag=on:sp=const_max:tnu=1:i=1001:si=on:rtra=on_0 on DTF2THF_22696 for (2916ds/1001Mi)
% 67.97/16.67 % (22968)Instruction limit reached!
% 67.97/16.67 % (22968)------------------------------
% 67.97/16.67 % (22968)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 67.97/16.67 % (22968)Termination reason: Unknown
% 67.97/16.67 % (22968)Termination phase: Property scanning
% 67.97/16.67
% 67.97/16.67 % (22968)Memory used [KB]: 38250
% 67.97/16.67 % (22968)Time elapsed: 1.865 s
% 67.97/16.67 % (22968)Instructions burned: 3373 (million)
% 67.97/16.67 % (22968)------------------------------
% 67.97/16.67 % (22968)------------------------------
% 68.27/16.70 % (22991)lrs+1002_1:1_au=on:cbe=off:cnfonf=conj_eager:cond=on:hi=on:i=102:si=on:rtra=on_0 on DTF2THF_22696 for (2914ds/102Mi)
% 68.27/16.75 % (22982)Instruction limit reached!
% 68.27/16.75 % (22982)------------------------------
% 68.27/16.75 % (22982)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 68.27/16.75 % (22982)Termination reason: Unknown
% 68.27/16.75 % (22982)Termination phase: Property scanning
% 68.27/16.75
% 68.27/16.75 % (22982)Memory used [KB]: 37739
% 68.27/16.75 % (22982)Time elapsed: 0.911 s
% 68.27/16.75 % (22982)Instructions burned: 1671 (million)
% 68.27/16.75 % (22982)------------------------------
% 68.27/16.75 % (22982)------------------------------
% 68.27/16.76 % (22991)Instruction limit reached!
% 68.27/16.76 % (22991)------------------------------
% 68.27/16.76 % (22991)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 68.27/16.76 % (22991)Termination reason: Unknown
% 68.27/16.76 % (22991)Termination phase: shuffling
% 68.27/16.76
% 68.27/16.76 % (22991)Memory used [KB]: 14711
% 68.27/16.76 % (22991)Time elapsed: 0.068 s
% 68.27/16.76 % (22991)Instructions burned: 103 (million)
% 68.27/16.76 % (22991)------------------------------
% 68.27/16.76 % (22991)------------------------------
% 68.27/16.77 % (22992)ott+1010_3:1_nm=20:s2a=on:s2at=3.0:sp=const_frequency:i=952:si=on:rtra=on_0 on DTF2THF_22696 for (2914ds/952Mi)
% 68.27/16.78 % (22993)lrs+1010_1:1024_add=large:c=on:cbe=off:cnfonf=lazy_gen:fde=none:nm=30:piset=and:sac=on:uhcvi=on:i=578:si=on:rtra=on_0 on DTF2THF_22696 for (2914ds/578Mi)
% 68.52/16.87 % (22988)Instruction limit reached!
% 68.52/16.87 % (22988)------------------------------
% 68.52/16.87 % (22988)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 68.52/16.87 % (22988)Termination reason: Unknown
% 68.52/16.87 % (22988)Termination phase: Property scanning
% 68.52/16.87
% 68.52/16.87 % (22988)Memory used [KB]: 26353
% 68.52/16.87 % (22988)Time elapsed: 0.514 s
% 68.52/16.87 % (22988)Instructions burned: 865 (million)
% 68.52/16.87 % (22988)------------------------------
% 68.52/16.87 % (22988)------------------------------
% 68.52/16.89 % (22994)dis+1010_1:4_avsq=on:avsqr=16,1:bet=on:cbe=off:cnfonf=lazy_not_be_gen:hud=5:plsq=on:plsqc=1:plsqr=32,1:prag=on:i=643:si=on:rtra=on_0 on DTF2THF_22696 for (2912ds/643Mi)
% 69.16/16.96 % (22941)Instruction limit reached!
% 69.16/16.96 % (22941)------------------------------
% 69.16/16.96 % (22941)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 69.16/16.96 % (22941)Termination reason: Unknown
% 69.16/16.96 % (22941)Termination phase: Saturation
% 69.16/16.96
% 69.16/16.96 % (22941)Memory used [KB]: 55265
% 69.16/16.96 % (22941)Time elapsed: 4.675 s
% 69.16/16.96 % (22941)Instructions burned: 8250 (million)
% 69.16/16.96 % (22941)------------------------------
% 69.16/16.96 % (22941)------------------------------
% 69.16/16.98 % (22995)lrs+1010_1:1_cnfonf=off:ntd=on:nwc=3.0:s2a=on:sp=const_frequency:i=865:si=on:rtra=on_0 on DTF2THF_22696 for (2912ds/865Mi)
% 71.96/17.09 % (22993)Instruction limit reached!
% 71.96/17.09 % (22993)------------------------------
% 71.96/17.09 % (22993)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 71.96/17.09 % (22993)Termination reason: Unknown
% 71.96/17.09 % (22993)Termination phase: shuffling
% 71.96/17.09
% 71.96/17.09 % (22993)Memory used [KB]: 23411
% 71.96/17.09 % (22993)Time elapsed: 0.311 s
% 71.96/17.09 % (22993)Instructions burned: 578 (million)
% 71.96/17.09 % (22993)------------------------------
% 71.96/17.09 % (22993)------------------------------
% 71.96/17.10 % (22990)Instruction limit reached!
% 71.96/17.10 % (22990)------------------------------
% 71.96/17.10 % (22990)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 71.96/17.10 % (22990)Termination reason: Unknown
% 71.96/17.10 % (22990)Termination phase: Property scanning
% 71.96/17.10
% 71.96/17.10 % (22990)Memory used [KB]: 25074
% 71.96/17.10 % (22990)Time elapsed: 0.556 s
% 71.96/17.10 % (22990)Instructions burned: 1001 (million)
% 71.96/17.10 % (22990)------------------------------
% 71.96/17.10 % (22990)------------------------------
% 71.96/17.12 % (22997)ott+1010_1:32_apa=on:atotf=0.3:au=on:c=on:cnfonf=off:fe=off:fsr=off:nm=6:piset=or:sac=on:sos=on:i=644:si=on:rtra=on_0 on DTF2THF_22696 for (2910ds/644Mi)
% 71.96/17.12 % (22996)lrs+10_1:1_au=on:erd=off:s2a=on:s2at=3.0:sos=all:i=1174:si=on:rtra=on_0 on DTF2THF_22696 for (2910ds/1174Mi)
% 72.97/17.20 % (22989)Refutation not found, incomplete strategy
% 72.97/17.20 % (22989)------------------------------
% 72.97/17.20 % (22989)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 72.97/17.20 % (22989)Termination reason: Refutation not found, incomplete strategy
% 72.97/17.20
% 72.97/17.20
% 72.97/17.20 % (22989)Memory used [KB]: 26737
% 72.97/17.20 % (22989)Time elapsed: 0.752 s
% 72.97/17.20 % (22989)Instructions burned: 1388 (million)
% 72.97/17.20 % (22989)------------------------------
% 72.97/17.20 % (22989)------------------------------
% 72.97/17.23 % (22994)Instruction limit reached!
% 72.97/17.23 % (22994)------------------------------
% 72.97/17.23 % (22994)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 72.97/17.23 % (22994)Termination reason: Unknown
% 72.97/17.23 % (22994)Termination phase: shuffling
% 72.97/17.23
% 72.97/17.23 % (22994)Memory used [KB]: 24306
% 72.97/17.23 % (22994)Time elapsed: 0.338 s
% 72.97/17.23 % (22994)Instructions burned: 644 (million)
% 72.97/17.23 % (22994)------------------------------
% 72.97/17.23 % (22994)------------------------------
% 72.97/17.23 % (22998)lrs+10_1:1_alpa=true:au=on:cs=on:tgt=ground:i=1201:si=on:rtra=on_0 on DTF2THF_22696 for (2909ds/1201Mi)
% 72.97/17.25 % (22999)lrs+1666_1:64_cnfonf=lazy_simp:hud=5:plsq=on:plsql=on:plsqr=152565,524288:tnu=2:i=1046:si=on:rtra=on_0 on DTF2THF_22696 for (2909ds/1046Mi)
% 73.32/17.30 % (22992)Instruction limit reached!
% 73.32/17.30 % (22992)------------------------------
% 73.32/17.30 % (22992)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 73.32/17.30 % (22992)Termination reason: Unknown
% 73.32/17.30 % (22992)Termination phase: Property scanning
% 73.32/17.30
% 73.32/17.30 % (22992)Memory used [KB]: 25074
% 73.32/17.30 % (22992)Time elapsed: 0.532 s
% 73.32/17.30 % (22992)Instructions burned: 953 (million)
% 73.32/17.30 % (22992)------------------------------
% 73.32/17.30 % (22992)------------------------------
% 73.32/17.33 % (23000)lrs+1010_1:1_c=on:cnfonf=lazy_simp:fsr=off:hud=5:kws=arity:nm=32:plsq=on:plsqc=1:plsqr=642,29:prag=on:i=222:si=on:rtra=on_0 on DTF2THF_22696 for (2908ds/222Mi)
% 73.54/17.45 % (22995)Instruction limit reached!
% 73.54/17.45 % (22995)------------------------------
% 73.54/17.45 % (22995)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 73.54/17.45 % (22995)Termination reason: Unknown
% 73.54/17.45 % (22995)Termination phase: Property scanning
% 73.54/17.45
% 73.54/17.45 % (22995)Memory used [KB]: 26353
% 73.54/17.45 % (22995)Time elapsed: 0.469 s
% 73.54/17.45 % (22995)Instructions burned: 866 (million)
% 73.54/17.45 % (22995)------------------------------
% 73.54/17.45 % (22995)------------------------------
% 73.54/17.45 % (23000)Instruction limit reached!
% 73.54/17.45 % (23000)------------------------------
% 73.54/17.45 % (23000)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 73.54/17.45 % (23000)Termination reason: Unknown
% 73.54/17.45 % (23000)Termination phase: shuffling
% 73.54/17.45
% 73.54/17.45 % (23000)Memory used [KB]: 16247
% 73.54/17.45 % (23000)Time elapsed: 0.127 s
% 73.54/17.45 % (23000)Instructions burned: 223 (million)
% 73.54/17.45 % (23000)------------------------------
% 73.54/17.45 % (23000)------------------------------
% 73.54/17.47 % (23001)ott+1010_1:4_cbe=off:chr=on:hfsq=on:hfsqr=1,16:hud=3:sac=on:sp=unary_frequency:ss=axioms:st=5.0:tnu=2:i=68:si=on:rtra=on_0 on DTF2THF_22696 for (2907ds/68Mi)
% 73.54/17.47 % (22997)Instruction limit reached!
% 73.54/17.47 % (22997)------------------------------
% 73.54/17.47 % (22997)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 73.54/17.47 % (22997)Termination reason: Unknown
% 73.54/17.47 % (22997)Termination phase: shuffling
% 73.54/17.47
% 73.54/17.47 % (22997)Memory used [KB]: 22259
% 73.54/17.47 % (22997)Time elapsed: 0.357 s
% 73.54/17.47 % (22997)Instructions burned: 644 (million)
% 73.54/17.47 % (22997)------------------------------
% 73.54/17.47 % (22997)------------------------------
% 73.54/17.48 % (23002)lrs+1010_1:1_chr=on:cnfonf=lazy_not_be_gen:gs=on:hud=5:plsq=on:plsqc=1:plsqr=5,1:prag=on:rawr=on:i=454:si=on:rtra=on_0 on DTF2THF_22696 for (2907ds/454Mi)
% 73.54/17.50 % (23003)dis+1002_1:1_cnfonf=lazy_simp:hud=5:plsq=on:plsqc=1:plsqr=32,1:prag=on:i=360:si=on:rtra=on_0 on DTF2THF_22696 for (2906ds/360Mi)
% 74.24/17.51 % (23001)Instruction limit reached!
% 74.24/17.51 % (23001)------------------------------
% 74.24/17.51 % (23001)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 74.24/17.52 % (23001)Termination reason: Unknown
% 74.24/17.52 % (23001)Termination phase: shuffling
% 74.24/17.52
% 74.24/17.52 % (23001)Memory used [KB]: 14200
% 74.24/17.52 % (23001)Time elapsed: 0.047 s
% 74.24/17.52 % (23001)Instructions burned: 69 (million)
% 74.24/17.52 % (23001)------------------------------
% 74.24/17.52 % (23001)------------------------------
% 74.24/17.54 % (23004)lrs+1002_1:16_amm=sco:cnfonf=lazy_pi_sigma_gen:fe=off:nm=60:piset=all_but_not_eq:plsq=on:plsqc=1:plsqr=2,1:rawr=on:sp=occurrence:ss=included:st=-1.0:i=157:si=on:rtra=on_0 on DTF2THF_22696 for (2906ds/157Mi)
% 74.40/17.62 % (23004)Instruction limit reached!
% 74.40/17.62 % (23004)------------------------------
% 74.40/17.62 % (23004)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 74.40/17.62 % (23004)Termination reason: Unknown
% 74.40/17.62 % (23004)Termination phase: shuffling
% 74.40/17.62
% 74.40/17.62 % (23004)Memory used [KB]: 15479
% 74.40/17.62 % (23004)Time elapsed: 0.085 s
% 74.40/17.62 % (23004)Instructions burned: 158 (million)
% 74.40/17.62 % (23004)------------------------------
% 74.40/17.62 % (23004)------------------------------
% 74.40/17.64 % (23005)ott+1004_1:24_anc=none:cbe=off:cnfonf=lazy_not_gen:fe=off:piset=not:prag=on:rp=on:i=5061:si=on:rtra=on_0 on DTF2THF_22696 for (2905ds/5061Mi)
% 74.60/17.71 % (23003)Instruction limit reached!
% 74.60/17.71 % (23003)------------------------------
% 74.60/17.71 % (23003)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 74.60/17.71 % (23003)Termination reason: Unknown
% 74.60/17.71 % (23003)Termination phase: shuffling
% 74.60/17.71
% 74.60/17.71 % (23003)Memory used [KB]: 20340
% 74.60/17.71 % (23003)Time elapsed: 0.213 s
% 74.60/17.71 % (23003)Instructions burned: 361 (million)
% 74.60/17.71 % (23003)------------------------------
% 74.60/17.71 % (23003)------------------------------
% 74.60/17.72 % (23002)Instruction limit reached!
% 74.60/17.72 % (23002)------------------------------
% 74.60/17.72 % (23002)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 74.60/17.72 % (23002)Termination reason: Unknown
% 74.60/17.72 % (23002)Termination phase: shuffling
% 74.60/17.72
% 74.60/17.72 % (23002)Memory used [KB]: 21620
% 74.60/17.72 % (23002)Time elapsed: 0.244 s
% 74.60/17.72 % (23002)Instructions burned: 454 (million)
% 74.60/17.72 % (23002)------------------------------
% 74.60/17.72 % (23002)------------------------------
% 74.60/17.73 % (23006)lrs+1002_1:8_av=off:fd=off:hud=5:tnu=1:i=2401:si=on:rtra=on_0 on DTF2THF_22696 for (2904ds/2401Mi)
% 74.60/17.74 % (23007)dis+10_1:1_av=off:e2e=on:nm=4:s2a=on:i=5798:si=on:rtra=on_0 on DTF2THF_22696 for (2904ds/5798Mi)
% 74.60/17.76 % (22996)Instruction limit reached!
% 74.60/17.76 % (22996)------------------------------
% 74.60/17.76 % (22996)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 74.60/17.76 % (22996)Termination reason: Unknown
% 74.60/17.76 % (22996)Termination phase: SInE selection
% 74.60/17.76
% 74.60/17.76 % (22996)Memory used [KB]: 25202
% 74.60/17.76 % (22996)Time elapsed: 0.645 s
% 74.60/17.76 % (22996)Instructions burned: 1174 (million)
% 74.60/17.76 % (22996)------------------------------
% 74.60/17.76 % (22996)------------------------------
% 74.60/17.77 % (22999)Instruction limit reached!
% 74.60/17.77 % (22999)------------------------------
% 74.60/17.77 % (22999)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 74.60/17.77 % (22999)Termination reason: Unknown
% 74.60/17.77 % (22999)Termination phase: shuffling
% 74.60/17.77
% 74.60/17.77 % (22999)Memory used [KB]: 33901
% 74.60/17.77 % (22999)Time elapsed: 0.520 s
% 74.60/17.77 % (22999)Instructions burned: 1046 (million)
% 74.60/17.77 % (22999)------------------------------
% 74.60/17.77 % (22999)------------------------------
% 74.60/17.78 % (23008)dis+10_1:1_au=on:cnfonf=lazy_gen:ixr=off:plsq=on:plsqc=1:plsqr=32,1:i=346:si=on:rtra=on_0 on DTF2THF_22696 for (2903ds/346Mi)
% 74.60/17.79 % (23009)lrs+1002_1:1_av=off:bd=off:bet=on:e2e=on:fde=unused:fe=off:plsq=on:prag=on:sos=on:i=1321:si=on:rtra=on_0 on DTF2THF_22696 for (2903ds/1321Mi)
% 76.40/17.87 % (22998)Instruction limit reached!
% 76.40/17.87 % (22998)------------------------------
% 76.40/17.87 % (22998)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 76.40/17.87 % (22998)Termination reason: Unknown
% 76.40/17.87 % (22998)Termination phase: Preprocessing 1
% 76.40/17.87
% 76.40/17.87 % (22998)Memory used [KB]: 25969
% 76.40/17.87 % (22998)Time elapsed: 0.647 s
% 76.40/17.87 % (22998)Instructions burned: 1201 (million)
% 76.40/17.87 % (22998)------------------------------
% 76.40/17.87 % (22998)------------------------------
% 77.17/17.90 % (23010)lrs+10_1:1_cnfonf=lazy_gen:kws=inv_precedence:nwc=10.0:i=1312:si=on:rtra=on_0 on DTF2THF_22696 for (2902ds/1312Mi)
% 77.17/17.97 % (23008)Instruction limit reached!
% 77.17/17.97 % (23008)------------------------------
% 77.17/17.97 % (23008)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 77.17/17.97 % (23008)Termination reason: Unknown
% 77.17/17.97 % (23008)Termination phase: shuffling
% 77.17/17.97
% 77.17/17.97 % (23008)Memory used [KB]: 20084
% 77.17/17.97 % (23008)Time elapsed: 0.185 s
% 77.17/17.97 % (23008)Instructions burned: 346 (million)
% 77.17/17.97 % (23008)------------------------------
% 77.17/17.97 % (23008)------------------------------
% 77.17/17.99 % (23011)ott+2_8:1_add=large:c=on:cnfonf=off:e2e=on:er=filter:hud=5:plsq=on:plsqr=1,1:ss=axioms:i=672:si=on:rtra=on_0 on DTF2THF_22696 for (2901ds/672Mi)
% 80.79/18.38 % (23011)Instruction limit reached!
% 80.79/18.38 % (23011)------------------------------
% 80.79/18.38 % (23011)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 80.79/18.38 % (23011)Termination reason: Unknown
% 80.79/18.38 % (23011)Termination phase: shuffling
% 80.79/18.38
% 80.79/18.38 % (23011)Memory used [KB]: 22899
% 80.79/18.38 % (23011)Time elapsed: 0.375 s
% 80.79/18.38 % (23011)Instructions burned: 672 (million)
% 80.79/18.38 % (23011)------------------------------
% 80.79/18.38 % (23011)------------------------------
% 81.49/18.40 % (23012)dis+10_1:1_au=on:c=on:e2e=on:fsr=off:sos=all:sp=const_frequency:ss=axioms:i=1897:si=on:rtra=on_0 on DTF2THF_22696 for (2897ds/1897Mi)
% 82.08/18.55 % (23009)Instruction limit reached!
% 82.08/18.55 % (23009)------------------------------
% 82.08/18.55 % (23009)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 82.08/18.55 % (23009)Termination reason: Unknown
% 82.08/18.55 % (23009)Termination phase: Preprocessing 3
% 82.08/18.55
% 82.08/18.55 % (23009)Memory used [KB]: 31726
% 82.08/18.55 % (23009)Time elapsed: 0.768 s
% 82.08/18.55 % (23009)Instructions burned: 1322 (million)
% 82.08/18.55 % (23009)------------------------------
% 82.08/18.55 % (23009)------------------------------
% 82.08/18.58 % (23013)lrs+10_1:40_av=off:fde=unused:ntd=on:plsq=on:plsqc=2:plsqr=411915,1048576:sp=unary_frequency:tgt=full:i=1202:si=on:rtra=on_0 on DTF2THF_22696 for (2896ds/1202Mi)
% 82.08/18.58 % (23010)Instruction limit reached!
% 82.08/18.58 % (23010)------------------------------
% 82.08/18.58 % (23010)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 82.08/18.58 % (23010)Termination reason: Unknown
% 82.08/18.58 % (23010)Termination phase: shuffling
% 82.08/18.58
% 82.08/18.58 % (23010)Memory used [KB]: 37483
% 82.08/18.58 % (23010)Time elapsed: 0.690 s
% 82.08/18.58 % (23010)Instructions burned: 1312 (million)
% 82.08/18.58 % (23010)------------------------------
% 82.08/18.58 % (23010)------------------------------
% 82.65/18.61 % (23014)dis+21_1:1_au=on:cnfonf=lazy_pi_sigma_gen:hfsq=on:hfsqc=40:hfsqr=8,1:ntd=on:sos=on:sp=occurrence:ss=axioms:i=1493:si=on:rtra=on_0 on DTF2THF_22696 for (2895ds/1493Mi)
% 83.85/19.03 % (23006)Instruction limit reached!
% 83.85/19.03 % (23006)------------------------------
% 83.85/19.03 % (23006)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 83.85/19.03 % (23006)Termination reason: Unknown
% 83.85/19.03 % (23006)Termination phase: Property scanning
% 83.85/19.03
% 83.85/19.03 % (23006)Memory used [KB]: 38123
% 83.85/19.03 % (23006)Time elapsed: 1.296 s
% 83.85/19.03 % (23006)Instructions burned: 2401 (million)
% 83.85/19.03 % (23006)------------------------------
% 83.85/19.03 % (23006)------------------------------
% 85.89/19.05 % (23015)dis+1010_3:1_av=off:fde=unused:hud=10:nwc=10.0:s2a=on:ss=axioms:i=7679:si=on:rtra=on_0 on DTF2THF_22696 for (2891ds/7679Mi)
% 86.94/19.23 % (23013)Instruction limit reached!
% 86.94/19.23 % (23013)------------------------------
% 86.94/19.23 % (23013)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 86.94/19.23 % (23013)Termination reason: Unknown
% 86.94/19.23 % (23013)Termination phase: Preprocessing 1
% 86.94/19.23
% 86.94/19.23 % (23013)Memory used [KB]: 25969
% 86.94/19.23 % (23013)Time elapsed: 0.650 s
% 86.94/19.23 % (23013)Instructions burned: 1202 (million)
% 86.94/19.23 % (23013)------------------------------
% 86.94/19.23 % (23013)------------------------------
% 86.94/19.25 % (23016)lrs+2_1:1024_anc=all:cnfonf=off:cs=on:e2e=on:nm=20:s2at=1.5:slsq=on:slsqr=1,2:spb=goal:i=340:si=on:rtra=on_0 on DTF2THF_22696 for (2889ds/340Mi)
% 87.97/19.44 % (23016)Instruction limit reached!
% 87.97/19.44 % (23016)------------------------------
% 87.97/19.44 % (23016)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 87.97/19.44 % (23016)Termination reason: Unknown
% 87.97/19.44 % (23016)Termination phase: shuffling
% 87.97/19.44
% 87.97/19.44 % (23016)Memory used [KB]: 18166
% 87.97/19.44 % (23016)Time elapsed: 0.194 s
% 87.97/19.44 % (23016)Instructions burned: 340 (million)
% 87.97/19.44 % (23016)------------------------------
% 87.97/19.44 % (23016)------------------------------
% 87.97/19.44 % (23014)Instruction limit reached!
% 87.97/19.44 % (23014)------------------------------
% 87.97/19.44 % (23014)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 87.97/19.44 % (23014)Termination reason: Unknown
% 87.97/19.45 % (23014)Termination phase: Property scanning
% 87.97/19.45
% 87.97/19.45 % (23014)Memory used [KB]: 37739
% 87.97/19.45 % (23014)Time elapsed: 0.838 s
% 87.97/19.45 % (23014)Instructions burned: 1493 (million)
% 87.97/19.45 % (23014)------------------------------
% 87.97/19.45 % (23014)------------------------------
% 87.97/19.47 % (23017)dis+1002_1:1_acc=model:au=on:nwc=5.0:sd=5:ss=axioms:st=2.5:i=3672:si=on:rtra=on_0 on DTF2THF_22696 for (2887ds/3672Mi)
% 87.97/19.47 % (23018)dis-1002_3:1_acc=model:bd=off:bsd=on:bsdmm=1:cnfonf=lazy_not_be_gen:fd=off:fe=off:ntd=on:nwc=10.0:piset=equals:plsq=on:plsqc=1:plsqr=32,1:sp=const_max:uhcvi=on:i=1535:si=on:rtra=on_0 on DTF2THF_22696 for (2887ds/1535Mi)
% 87.97/19.47 % (23012)Instruction limit reached!
% 87.97/19.47 % (23012)------------------------------
% 87.97/19.47 % (23012)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 87.97/19.47 % (23012)Termination reason: Unknown
% 87.97/19.47 % (23012)Termination phase: Saturation
% 87.97/19.47
% 87.97/19.47 % (23012)Memory used [KB]: 30447
% 87.97/19.47 % (23012)Time elapsed: 1.068 s
% 87.97/19.47 % (23012)Instructions burned: 1897 (million)
% 87.97/19.47 % (23012)------------------------------
% 87.97/19.47 % (23012)------------------------------
% 87.97/19.49 % (23019)dis+10_16:1_hud=1:plsq=on:plsqc=1:plsqr=32,1:prag=on:sd=1:sgt=64:ss=axioms:i=4637:si=on:rtra=on_0 on DTF2THF_22696 for (2886ds/4637Mi)
% 89.23/19.70 % (22986)Refutation not found, non-redundant clauses discarded% (22986)------------------------------
% 89.23/19.70 % (22986)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 89.23/19.70 % (22986)Termination reason: Refutation not found, non-redundant clauses discarded
% 89.23/19.70
% 89.23/19.70 % (22986)Memory used [KB]: 32750
% 89.23/19.70 % (22986)Time elapsed: 3.566 s
% 89.23/19.70 % (22986)Instructions burned: 6427 (million)
% 89.23/19.70 % (22986)------------------------------
% 89.23/19.70 % (22986)------------------------------
% 89.23/19.73 % (23020)lrs+10_1:1_fde=unused:prag=on:sd=2:sgt=32:ss=axioms:i=3174:si=on:rtra=on_0 on DTF2THF_22696 for (2884ds/3174Mi)
% 93.59/20.18 % (23005)Instruction limit reached!
% 93.59/20.18 % (23005)------------------------------
% 93.59/20.18 % (23005)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 93.59/20.18 % (23005)Termination reason: Unknown
% 93.59/20.18 % (23005)Termination phase: Property scanning
% 93.59/20.18
% 93.59/20.18 % (23005)Memory used [KB]: 44007
% 93.59/20.18 % (23005)Time elapsed: 2.547 s
% 93.59/20.18 % (23005)Instructions burned: 5061 (million)
% 93.59/20.18 % (23005)------------------------------
% 93.59/20.18 % (23005)------------------------------
% 94.08/20.21 % (23021)lrs+1010_1:10_au=on:bd=off:cnfonf=lazy_pi_sigma_gen:sp=occurrence:i=307:si=on:rtra=on_0 on DTF2THF_22696 for (2879ds/307Mi)
% 94.08/20.24 % (23018)Instruction limit reached!
% 94.08/20.24 % (23018)------------------------------
% 94.08/20.24 % (23018)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 94.08/20.24 % (23018)Termination reason: Unknown
% 94.08/20.24 % (23018)Termination phase: Property scanning
% 94.08/20.24
% 94.08/20.24 % (23018)Memory used [KB]: 37739
% 94.08/20.24 % (23018)Time elapsed: 0.773 s
% 94.08/20.24 % (23018)Instructions burned: 1536 (million)
% 94.08/20.24 % (23018)------------------------------
% 94.08/20.24 % (23018)------------------------------
% 94.08/20.26 % (23022)dis+1002_5:1_au=on:bd=off:e2e=on:fde=none:fs=off:fsr=off:sos=on:i=3619:si=on:rtra=on_0 on DTF2THF_22696 for (2879ds/3619Mi)
% 94.65/20.38 % (23021)Instruction limit reached!
% 94.65/20.38 % (23021)------------------------------
% 94.65/20.38 % (23021)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 94.65/20.38 % (23021)Termination reason: Unknown
% 94.65/20.38 % (23021)Termination phase: shuffling
% 94.65/20.38
% 94.65/20.38 % (23021)Memory used [KB]: 19573
% 94.65/20.38 % (23021)Time elapsed: 0.167 s
% 94.65/20.38 % (23021)Instructions burned: 307 (million)
% 94.65/20.38 % (23021)------------------------------
% 94.65/20.38 % (23021)------------------------------
% 95.34/20.40 % (23023)lrs+1002_1:1_au=on:cbe=off:fde=none:ins=1:sp=const_max:i=2325:si=on:rtra=on_0 on DTF2THF_22696 for (2877ds/2325Mi)
% 96.83/20.78 % (23007)Instruction limit reached!
% 96.83/20.78 % (23007)------------------------------
% 96.83/20.78 % (23007)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 96.83/20.78 % (23007)Termination reason: Unknown
% 96.83/20.78 % (23007)Termination phase: Saturation
% 96.83/20.78
% 96.83/20.78 % (23007)Memory used [KB]: 52451
% 96.83/20.78 % (23007)Time elapsed: 3.036 s
% 96.83/20.78 % (23007)Instructions burned: 5799 (million)
% 96.83/20.78 % (23007)------------------------------
% 96.83/20.78 % (23007)------------------------------
% 97.33/20.80 % (23024)dis+1002_1:1_au=on:sac=on:sd=5:slsq=on:slsqr=1,4:ss=axioms:st=2.0:i=2341:si=on:rtra=on_0 on DTF2THF_22696 for (2873ds/2341Mi)
% 101.78/21.37 % (23017)Instruction limit reached!
% 101.78/21.37 % (23017)------------------------------
% 101.78/21.37 % (23017)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 101.78/21.37 % (23017)Termination reason: Unknown
% 101.78/21.37 % (23017)Termination phase: Saturation
% 101.78/21.37
% 101.78/21.37 % (23017)Memory used [KB]: 41705
% 101.78/21.37 % (23017)Time elapsed: 1.905 s
% 101.78/21.37 % (23017)Instructions burned: 3673 (million)
% 101.78/21.37 % (23017)------------------------------
% 101.78/21.37 % (23017)------------------------------
% 101.78/21.39 % (23025)lrs+1010_1:16_apa=on:bsd=on:cnfonf=off:hud=10:sac=on:sos=on:ss=axioms:i=7257:si=on:rtra=on_0 on DTF2THF_22696 for (2867ds/7257Mi)
% 103.06/21.53 % (23020)Instruction limit reached!
% 103.06/21.53 % (23020)------------------------------
% 103.06/21.53 % (23020)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 103.06/21.53 % (23020)Termination reason: Unknown
% 103.06/21.53 % (23020)Termination phase: Saturation
% 103.06/21.53
% 103.06/21.53 % (23020)Memory used [KB]: 33133
% 103.06/21.53 % (23020)Time elapsed: 1.805 s
% 103.06/21.53 % (23020)Instructions burned: 3175 (million)
% 103.06/21.53 % (23020)------------------------------
% 103.06/21.53 % (23020)------------------------------
% 103.06/21.55 % (23026)ott+1010_1:1_cnfonf=lazy_gen:fe=off:plsq=on:plsqc=1:plsqr=3548551,65536:s2a=on:i=3710:si=on:rtra=on_0 on DTF2THF_22696 for (2866ds/3710Mi)
% 103.61/21.63 % (23023)Instruction limit reached!
% 103.61/21.63 % (23023)------------------------------
% 103.61/21.63 % (23023)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 103.61/21.63 % (23023)Termination reason: Unknown
% 103.61/21.63 % (23023)Termination phase: Inequality splitting
% 103.61/21.63
% 103.61/21.63 % (23023)Memory used [KB]: 42344
% 103.61/21.63 % (23023)Time elapsed: 1.235 s
% 103.61/21.63 % (23023)Instructions burned: 2325 (million)
% 103.61/21.63 % (23023)------------------------------
% 103.61/21.63 % (23023)------------------------------
% 103.61/21.66 % (23027)lrs+1010_1:1_avsq=on:avsqr=1,16:kws=inv_arity:nwc=5.0:sd=1:ss=axioms:i=326:si=on:rtra=on_0 on DTF2THF_22696 for (2865ds/326Mi)
% 104.44/21.83 % (23027)Instruction limit reached!
% 104.44/21.83 % (23027)------------------------------
% 104.44/21.83 % (23027)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 104.44/21.83 % (23027)Termination reason: Unknown
% 104.44/21.83 % (23027)Termination phase: shuffling
% 104.44/21.83
% 104.44/21.83 % (23027)Memory used [KB]: 18038
% 104.44/21.83 % (23027)Time elapsed: 0.178 s
% 104.44/21.83 % (23027)Instructions burned: 327 (million)
% 104.44/21.83 % (23027)------------------------------
% 104.44/21.83 % (23027)------------------------------
% 104.44/21.84 % (23028)lrs+1010_1:64_au=on:c=on:e2e=on:flr=on:hfsq=on:hfsql=off:hfsqr=4,1:ins=1:ntd=on:sp=occurrence:i=1100:si=on:rtra=on_0 on DTF2THF_22696 for (2863ds/1100Mi)
% 104.44/21.88 % (23024)Instruction limit reached!
% 104.44/21.88 % (23024)------------------------------
% 104.44/21.88 % (23024)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 104.44/21.88 % (23024)Termination reason: Unknown
% 104.44/21.88 % (23024)Termination phase: Saturation
% 104.44/21.88
% 104.44/21.88 % (23024)Memory used [KB]: 34541
% 104.44/21.88 % (23024)Time elapsed: 1.081 s
% 104.44/21.88 % (23024)Instructions burned: 2342 (million)
% 104.44/21.88 % (23024)------------------------------
% 104.44/21.88 % (23024)------------------------------
% 104.44/21.89 % (23029)lrs+10_1:1_au=on:cnfonf=off:fde=unused:kws=precedence:tgt=full:i=406:si=on:rtra=on_0 on DTF2THF_22696 for (2862ds/406Mi)
% 105.04/21.90 % (23019)Instruction limit reached!
% 105.04/21.90 % (23019)------------------------------
% 105.04/21.90 % (23019)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 105.04/21.90 % (23019)Termination reason: Unknown
% 105.04/21.90 % (23019)Termination phase: Saturation
% 105.04/21.90
% 105.04/21.90 % (23019)Memory used [KB]: 34285
% 105.04/21.90 % (23019)Time elapsed: 2.405 s
% 105.04/21.90 % (23019)Instructions burned: 4637 (million)
% 105.04/21.90 % (23019)------------------------------
% 105.04/21.90 % (23019)------------------------------
% 105.04/21.91 % (23030)dis+1010_1:1_aac=none:bd=preordered:e2e=on:fde=none:fe=axiom:nm=16:nwc=10.0:prag=on:i=1871:si=on:rtra=on_0 on DTF2THF_22696 for (2862ds/1871Mi)
% 105.04/21.99 % (23022)Instruction limit reached!
% 105.04/21.99 % (23022)------------------------------
% 105.04/21.99 % (23022)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 105.04/21.99 % (23022)Termination reason: Unknown
% 105.04/21.99 % (23022)Termination phase: Saturation
% 105.04/21.99
% 105.04/21.99 % (23022)Memory used [KB]: 47333
% 105.04/21.99 % (23022)Time elapsed: 1.732 s
% 105.04/21.99 % (23022)Instructions burned: 3620 (million)
% 105.04/21.99 % (23022)------------------------------
% 105.04/21.99 % (23022)------------------------------
% 105.04/22.01 % (23031)dis+2_1:1_bd=off:e2e=on:hud=1:nm=3:piset=or:prag=on:s2a=on:sp=const_max:i=1194:si=on:rtra=on_0 on DTF2THF_22696 for (2861ds/1194Mi)
% 106.75/22.04 % (23029)Instruction limit reached!
% 106.75/22.04 % (23029)------------------------------
% 106.75/22.04 % (23029)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 106.75/22.04 % (23029)Termination reason: Unknown
% 106.75/22.04 % (23029)Termination phase: shuffling
% 106.75/22.04
% 106.75/22.04 % (23029)Memory used [KB]: 19061
% 106.75/22.04 % (23029)Time elapsed: 0.144 s
% 106.75/22.04 % (23029)Instructions burned: 406 (million)
% 106.75/22.04 % (23029)------------------------------
% 106.75/22.04 % (23029)------------------------------
% 106.75/22.05 % (23032)dis+10_1:1024_aac=none:au=on:cnfonf=lazy_gen:plsq=on:plsqc=1:plsqr=32,1:spb=goal_then_units:i=474:si=on:rtra=on_0 on DTF2THF_22696 for (2861ds/474Mi)
% 107.30/22.19 % (23032)Instruction limit reached!
% 107.30/22.19 % (23032)------------------------------
% 107.30/22.19 % (23032)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 107.30/22.19 % (23032)Termination reason: Unknown
% 107.30/22.19 % (23032)Termination phase: shuffling
% 107.30/22.19
% 107.30/22.19 % (23032)Memory used [KB]: 22003
% 107.30/22.19 % (23032)Time elapsed: 0.142 s
% 107.30/22.19 % (23032)Instructions burned: 477 (million)
% 107.30/22.19 % (23032)------------------------------
% 107.30/22.19 % (23032)------------------------------
% 107.90/22.20 % (23028)Instruction limit reached!
% 107.90/22.20 % (23028)------------------------------
% 107.90/22.20 % (23028)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 107.90/22.20 % (23028)Termination reason: Unknown
% 107.90/22.20 % (23028)Termination phase: Preprocessing 1
% 107.90/22.20
% 107.90/22.20 % (23028)Memory used [KB]: 23155
% 107.90/22.20 % (23028)Time elapsed: 0.357 s
% 107.90/22.20 % (23028)Instructions burned: 1100 (million)
% 107.90/22.20 % (23028)------------------------------
% 107.90/22.20 % (23028)------------------------------
% 107.90/22.20 % (23076)dis+1010_1:1_au=on:cnfonf=lazy_gen:ins=3:plsq=on:plsqc=1:plsqr=15142123,262144:sp=reverse_arity:i=3115:si=on:rtra=on_0 on DTF2THF_22696 for (2859ds/3115Mi)
% 107.90/22.21 % (23081)dis+10_1:1024_kws=precedence:prag=on:rawr=on:sos=all:sp=unary_first:i=831:si=on:rtra=on_0 on DTF2THF_22696 for (2859ds/831Mi)
% 108.72/22.40 % (23031)Instruction limit reached!
% 108.72/22.40 % (23031)------------------------------
% 108.72/22.40 % (23031)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 108.72/22.40 % (23031)Termination reason: Unknown
% 108.72/22.40 % (23031)Termination phase: SInE selection
% 108.72/22.40
% 108.72/22.40 % (23031)Memory used [KB]: 23155
% 108.72/22.40 % (23031)Time elapsed: 0.391 s
% 108.72/22.40 % (23031)Instructions burned: 1194 (million)
% 108.72/22.40 % (23031)------------------------------
% 108.72/22.40 % (23031)------------------------------
% 108.72/22.41 % (23165)dis+1666_14:1_cnfonf=lazy_gen:fe=off:hud=10:piset=all_but_not_eq:prag=on:sp=const_max:i=1188:si=on:rtra=on_0 on DTF2THF_22696 for (2857ds/1188Mi)
% 108.72/22.49 % (23081)Instruction limit reached!
% 108.72/22.49 % (23081)------------------------------
% 108.72/22.49 % (23081)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 108.72/22.49 % (23081)Termination reason: Unknown
% 108.72/22.49 % (23081)Termination phase: Property scanning
% 108.72/22.49
% 108.72/22.49 % (23081)Memory used [KB]: 25074
% 108.72/22.49 % (23081)Time elapsed: 0.281 s
% 108.72/22.49 % (23081)Instructions burned: 833 (million)
% 108.72/22.49 % (23081)------------------------------
% 108.72/22.49 % (23081)------------------------------
% 109.80/22.50 % (23030)Instruction limit reached!
% 109.80/22.50 % (23030)------------------------------
% 109.80/22.50 % (23030)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 109.80/22.50 % (23030)Termination reason: Unknown
% 109.80/22.50 % (23030)Termination phase: Property scanning
% 109.80/22.50
% 109.80/22.50 % (23030)Memory used [KB]: 38506
% 109.80/22.50 % (23030)Time elapsed: 0.593 s
% 109.80/22.50 % (23030)Instructions burned: 1873 (million)
% 109.80/22.50 % (23030)------------------------------
% 109.80/22.50 % (23030)------------------------------
% 109.80/22.51 % (23202)ott+1010_1:20_cnfonf=lazy_pi_sigma_gen:fe=axiom:plsq=on:plsqc=2:plsqr=32,1:sac=on:sos=all:i=2868:si=on:rtra=on_0 on DTF2THF_22696 for (2856ds/2868Mi)
% 109.80/22.52 % (23206)dis+10_1:4_atotf=0.1:avsq=on:bsr=unit_only:e2e=on:sd=1:sp=const_max:ss=axioms:i=1516:si=on:rtra=on_0 on DTF2THF_22696 for (2856ds/1516Mi)
% 110.12/22.62 % (23015)Instruction limit reached!
% 110.12/22.62 % (23015)------------------------------
% 110.12/22.62 % (23015)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 110.12/22.62 % (23015)Termination reason: Unknown
% 110.12/22.62 % (23015)Termination phase: Saturation
% 110.12/22.62
% 110.12/22.62 % (23015)Memory used [KB]: 40297
% 110.12/22.62 % (23015)Time elapsed: 3.574 s
% 110.12/22.62 % (23015)Instructions burned: 7680 (million)
% 110.12/22.62 % (23015)------------------------------
% 110.12/22.62 % (23015)------------------------------
% 110.12/22.64 % (23229)lrs+1002_1:20_afp=1000:au=on:bsd=on:cnfonf=lazy_not_gen_be_off:sos=all:sp=const_max:i=2321:si=on:rtra=on_0 on DTF2THF_22696 for (2855ds/2321Mi)
% 110.36/22.73 % (23026)Instruction limit reached!
% 110.36/22.73 % (23026)------------------------------
% 110.36/22.73 % (23026)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 110.36/22.73 % (23026)Termination reason: Unknown
% 110.36/22.73 % (23026)Termination phase: Function definition elimination
% 110.36/22.73
% 110.36/22.73 % (23026)Memory used [KB]: 42728
% 110.36/22.73 % (23026)Time elapsed: 1.185 s
% 110.36/22.73 % (23026)Instructions burned: 3712 (million)
% 110.36/22.73 % (23026)------------------------------
% 110.36/22.73 % (23026)------------------------------
% 110.36/22.75 % (23269)dis+1010_1:1_au=on:bd=off:bsd=on:ixr=off:sp=const_max:i=2812:si=on:rtra=on_0 on DTF2THF_22696 for (2854ds/2812Mi)
% 110.36/22.76 % (23165)Instruction limit reached!
% 110.36/22.76 % (23165)------------------------------
% 110.36/22.76 % (23165)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 110.36/22.76 % (23165)Termination reason: Unknown
% 110.36/22.76 % (23165)Termination phase: shuffling
% 110.36/22.76
% 110.36/22.76 % (23165)Memory used [KB]: 35820
% 110.36/22.76 % (23165)Time elapsed: 0.355 s
% 110.36/22.76 % (23165)Instructions burned: 1190 (million)
% 110.36/22.76 % (23165)------------------------------
% 110.36/22.76 % (23165)------------------------------
% 110.36/22.78 % (23270)dis+1002_8:1_bd=off:sd=1:sgt=16:ss=axioms:i=4604:si=on:rtra=on_0 on DTF2THF_22696 for (2853ds/4604Mi)
% 111.00/23.02 % (23206)Instruction limit reached!
% 111.00/23.02 % (23206)------------------------------
% 111.00/23.02 % (23206)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 111.00/23.02 % (23206)Termination reason: Unknown
% 111.00/23.02 % (23206)Termination phase: Saturation
% 111.00/23.02
% 111.00/23.02 % (23206)Memory used [KB]: 28528
% 111.00/23.02 % (23206)Time elapsed: 0.506 s
% 111.00/23.02 % (23206)Instructions burned: 1518 (million)
% 111.00/23.02 % (23206)------------------------------
% 111.00/23.02 % (23206)------------------------------
% 113.87/23.04 % (23271)ott+1002_1:4_c=on:chr=on:cnfonf=lazy_not_be_gen:hud=10:nm=10:piset=all_but_not_eq:rp=on:spb=goal:uhcvi=on:i=2678:si=on:rtra=on_0 on DTF2THF_22696 for (2851ds/2678Mi)
% 113.87/23.07 % (23076)Instruction limit reached!
% 113.87/23.07 % (23076)------------------------------
% 113.87/23.07 % (23076)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 113.87/23.07 % (23076)Termination reason: Unknown
% 113.87/23.07 % (23076)Termination phase: Function definition elimination
% 113.87/23.07
% 113.87/23.07 % (23076)Memory used [KB]: 39914
% 113.87/23.07 % (23076)Time elapsed: 0.872 s
% 113.87/23.07 % (23076)Instructions burned: 3115 (million)
% 113.87/23.07 % (23076)------------------------------
% 113.87/23.07 % (23076)------------------------------
% 113.87/23.09 % (23272)dis+1010_1:4_avsq=on:avsqr=16,1:bet=on:cbe=off:cnfonf=lazy_not_be_gen:hud=5:plsq=on:plsqc=1:plsqr=32,1:prag=on:i=643:si=on:rtra=on_0 on DTF2THF_22696 for (2850ds/643Mi)
% 114.83/23.28 % (23272)Instruction limit reached!
% 114.83/23.28 % (23272)------------------------------
% 114.83/23.28 % (23272)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 114.83/23.28 % (23272)Termination reason: Unknown
% 114.83/23.28 % (23272)Termination phase: shuffling
% 114.83/23.28
% 114.83/23.28 % (23272)Memory used [KB]: 24306
% 114.83/23.28 % (23272)Time elapsed: 0.189 s
% 114.83/23.28 % (23272)Instructions burned: 644 (million)
% 114.83/23.28 % (23272)------------------------------
% 114.83/23.28 % (23272)------------------------------
% 114.83/23.29 % (23273)lrs+1010_1:1_au=on:cbe=off:cnfonf=lazy_not_be_gen:ntd=on:pe=on:i=458:si=on:rtra=on_0 on DTF2THF_22696 for (2848ds/458Mi)
% 114.83/23.29 % (23229)Instruction limit reached!
% 114.83/23.29 % (23229)------------------------------
% 114.83/23.29 % (23229)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 114.83/23.29 % (23229)Termination reason: Unknown
% 114.83/23.29 % (23229)Termination phase: Property scanning
% 114.83/23.29
% 114.83/23.29 % (23229)Memory used [KB]: 39018
% 114.83/23.29 % (23229)Time elapsed: 0.656 s
% 114.83/23.29 % (23229)Instructions burned: 2322 (million)
% 114.83/23.29 % (23229)------------------------------
% 114.83/23.29 % (23229)------------------------------
% 115.55/23.30 % (23202)Instruction limit reached!
% 115.55/23.30 % (23202)------------------------------
% 115.55/23.30 % (23202)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 115.55/23.30 % (23202)Termination reason: Unknown
% 115.55/23.30 % (23202)Termination phase: Property scanning
% 115.55/23.30
% 115.55/23.30 % (23202)Memory used [KB]: 39018
% 115.55/23.30 % (23202)Time elapsed: 0.797 s
% 115.55/23.30 % (23202)Instructions burned: 2871 (million)
% 115.55/23.30 % (23202)------------------------------
% 115.55/23.30 % (23202)------------------------------
% 115.55/23.31 % (23274)lrs+1010_1:1_anc=all_dependent:apa=on:au=on:bd=off:c=on:cnfonf=off:ntd=on:i=2546:si=on:rtra=on_0 on DTF2THF_22696 for (2848ds/2546Mi)
% 115.55/23.32 % (23275)dis+21_1:1_acc=model:au=on:cnfonf=lazy_gen:ins=2:plsq=on:plsqc=1:plsqr=32,1:i=1432:si=on:rtra=on_0 on DTF2THF_22696 for (2848ds/1432Mi)
% 115.85/23.43 % (23273)Instruction limit reached!
% 115.85/23.43 % (23273)------------------------------
% 115.85/23.43 % (23273)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 115.85/23.43 % (23273)Termination reason: Unknown
% 115.85/23.43 % (23273)Termination phase: shuffling
% 115.85/23.43
% 115.85/23.43 % (23273)Memory used [KB]: 21748
% 115.85/23.43 % (23273)Time elapsed: 0.141 s
% 115.85/23.43 % (23273)Instructions burned: 460 (million)
% 115.85/23.43 % (23273)------------------------------
% 115.85/23.43 % (23273)------------------------------
% 115.85/23.44 % (23276)dis+1002_1:1_cnfonf=lazy_not_gen:hud=10:plsq=on:plsqc=1:plsqr=32,1:prag=on:i=386:si=on:rtra=on_0 on DTF2THF_22696 for (2847ds/386Mi)
% 116.30/23.56 % (23276)Instruction limit reached!
% 116.30/23.56 % (23276)------------------------------
% 116.30/23.56 % (23276)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 116.30/23.56 % (23276)Termination reason: Unknown
% 116.30/23.56 % (23276)Termination phase: shuffling
% 116.30/23.56
% 116.30/23.56 % (23276)Memory used [KB]: 20724
% 116.30/23.56 % (23276)Time elapsed: 0.118 s
% 116.30/23.56 % (23276)Instructions burned: 388 (million)
% 116.30/23.56 % (23276)------------------------------
% 116.30/23.56 % (23276)------------------------------
% 116.30/23.57 % (23277)dis+1002_1:1_cnfonf=lazy_simp:hud=5:plsq=on:plsqc=1:plsqr=32,1:prag=on:i=360:si=on:rtra=on_0 on DTF2THF_22696 for (2846ds/360Mi)
% 116.74/23.60 % (23269)Instruction limit reached!
% 116.74/23.60 % (23269)------------------------------
% 116.74/23.60 % (23269)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 116.74/23.60 % (23269)Termination reason: Unknown
% 116.74/23.60 % (23269)Termination phase: Property scanning
% 116.74/23.60
% 116.74/23.60 % (23269)Memory used [KB]: 38250
% 116.74/23.60 % (23269)Time elapsed: 0.850 s
% 116.74/23.60 % (23269)Instructions burned: 2814 (million)
% 116.74/23.60 % (23269)------------------------------
% 116.74/23.60 % (23269)------------------------------
% 116.74/23.61 % (23278)dis+1010_1:1_au=on:cbe=off:chr=on:fsr=off:hfsq=on:nm=64:sos=theory:sp=weighted_frequency:i=5446:si=on:rtra=on_0 on DTF2THF_22696 for (2845ds/5446Mi)
% 116.74/23.68 % (23277)Instruction limit reached!
% 116.74/23.68 % (23277)------------------------------
% 116.74/23.68 % (23277)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 116.74/23.68 % (23277)Termination reason: Unknown
% 116.74/23.68 % (23277)Termination phase: shuffling
% 116.74/23.68
% 116.74/23.68 % (23277)Memory used [KB]: 20340
% 116.74/23.68 % (23277)Time elapsed: 0.111 s
% 116.74/23.68 % (23277)Instructions burned: 361 (million)
% 116.74/23.68 % (23277)------------------------------
% 116.74/23.68 % (23277)------------------------------
% 116.74/23.69 % (23279)dis+10_1:1_cnfonf=lazy_gen:fe=off:plsq=on:plsqc=1:plsqr=32,1:prag=on:i=2573:si=on:rtra=on_0 on DTF2THF_22696 for (2844ds/2573Mi)
% 117.08/23.73 % (23275)Instruction limit reached!
% 117.08/23.73 % (23275)------------------------------
% 117.08/23.73 % (23275)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 117.08/23.73 % (23275)Termination reason: Unknown
% 117.08/23.73 % (23275)Termination phase: Property scanning
% 117.08/23.73
% 117.08/23.73 % (23275)Memory used [KB]: 37739
% 117.08/23.73 % (23275)Time elapsed: 0.419 s
% 117.08/23.73 % (23275)Instructions burned: 1434 (million)
% 117.08/23.73 % (23275)------------------------------
% 117.08/23.73 % (23275)------------------------------
% 117.08/23.75 % (23280)lrs+10_1:1_av=off:bd=off:cnfonf=lazy_gen:fe=off:hud=10:plsq=on:plsqc=1:plsqr=32,1:tnu=1:i=2655:si=on:rtra=on_0 on DTF2THF_22696 for (2844ds/2655Mi)
% 117.08/23.78 % (23271)Instruction limit reached!
% 117.08/23.78 % (23271)------------------------------
% 117.08/23.78 % (23271)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 117.08/23.78 % (23271)Termination reason: Unknown
% 117.08/23.78 % (23271)Termination phase: Property scanning
% 117.08/23.78
% 117.08/23.78 % (23271)Memory used [KB]: 39018
% 117.08/23.78 % (23271)Time elapsed: 0.744 s
% 117.08/23.78 % (23271)Instructions burned: 2679 (million)
% 117.08/23.78 % (23271)------------------------------
% 117.08/23.78 % (23271)------------------------------
% 117.08/23.79 % (23281)dis+1002_16:1_apa=on:au=on:av=off:cnfonf=off:e2e=on:nm=0:nwc=10.0:i=2781:si=on:rtra=on_0 on DTF2THF_22696 for (2843ds/2781Mi)
% 118.47/23.85 % (23025)Instruction limit reached!
% 118.47/23.85 % (23025)------------------------------
% 118.47/23.85 % (23025)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 118.47/23.85 % (23025)Termination reason: Unknown
% 118.47/23.85 % (23025)Termination phase: Saturation
% 118.47/23.85
% 118.47/23.85 % (23025)Memory used [KB]: 43496
% 118.47/23.85 % (23025)Time elapsed: 2.453 s
% 118.47/23.85 % (23025)Instructions burned: 7259 (million)
% 118.47/23.85 % (23025)------------------------------
% 118.47/23.85 % (23025)------------------------------
% 118.47/23.86 % (23282)lrs+1010_1:10_au=on:bd=off:cnfonf=lazy_pi_sigma_gen:sp=occurrence:i=307:si=on:rtra=on_0 on DTF2THF_22696 for (2843ds/307Mi)
% 118.64/23.95 % (23282)Instruction limit reached!
% 118.64/23.95 % (23282)------------------------------
% 118.64/23.95 % (23282)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 118.64/23.95 % (23282)Termination reason: Unknown
% 118.64/23.95 % (23282)Termination phase: shuffling
% 118.64/23.95
% 118.64/23.95 % (23282)Memory used [KB]: 19573
% 118.64/23.95 % (23282)Time elapsed: 0.094 s
% 118.64/23.95 % (23282)Instructions burned: 307 (million)
% 118.64/23.95 % (23282)------------------------------
% 118.64/23.95 % (23282)------------------------------
% 118.64/23.97 % (23283)lrs+1010_1:1_avsq=on:avsqr=1,16:kws=inv_arity:nwc=5.0:sd=1:ss=axioms:i=1548:si=on:rtra=on_0 on DTF2THF_22696 for (2842ds/1548Mi)
% 120.86/24.09 % (23274)Instruction limit reached!
% 120.86/24.09 % (23274)------------------------------
% 120.86/24.09 % (23274)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 120.86/24.09 % (23274)Termination reason: Unknown
% 120.86/24.09 % (23274)Termination phase: Property scanning
% 120.86/24.09
% 120.86/24.09 % (23274)Memory used [KB]: 38762
% 120.86/24.09 % (23274)Time elapsed: 0.787 s
% 120.86/24.09 % (23274)Instructions burned: 2548 (million)
% 120.86/24.09 % (23274)------------------------------
% 120.86/24.09 % (23274)------------------------------
% 120.86/24.11 % (23284)lrs+10_1:3_alpa=true:hud=3:ins=1:ntd=on:plsq=on:plsqc=1:plsqr=32,1:prag=on:s2a=on:s2agt=10:i=5214:si=on:rtra=on_0 on DTF2THF_22696 for (2840ds/5214Mi)
% 122.03/24.25 % (23270)Instruction limit reached!
% 122.03/24.25 % (23270)------------------------------
% 122.03/24.25 % (23270)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 122.03/24.25 % (23270)Termination reason: Unknown
% 122.03/24.25 % (23270)Termination phase: Saturation
% 122.03/24.25
% 122.03/24.25 % (23270)Memory used [KB]: 33133
% 122.03/24.25 % (23270)Time elapsed: 1.472 s
% 122.03/24.25 % (23270)Instructions burned: 4605 (million)
% 122.03/24.25 % (23270)------------------------------
% 122.03/24.25 % (23270)------------------------------
% 122.03/24.26 % (23285)dis+21_1:1_aac=none:bsr=on:nwc=3.0:s2a=on:i=3201:si=on:rtra=on_0 on DTF2THF_22696 for (2839ds/3201Mi)
% 123.08/24.42 % (23279)Instruction limit reached!
% 123.08/24.42 % (23279)------------------------------
% 123.08/24.42 % (23279)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 123.08/24.42 % (23279)Termination reason: Unknown
% 123.08/24.42 % (23279)Termination phase: Property scanning
% 123.08/24.42
% 123.08/24.42 % (23279)Memory used [KB]: 39018
% 123.08/24.42 % (23279)Time elapsed: 0.722 s
% 123.08/24.42 % (23279)Instructions burned: 2573 (million)
% 123.08/24.42 % (23279)------------------------------
% 123.08/24.42 % (23279)------------------------------
% 123.08/24.43 % (23286)lrs+1010_1:1_chr=on:cnfonf=lazy_not_be_gen:gs=on:hud=5:plsq=on:plsqc=1:plsqr=5,1:prag=on:rawr=on:i=454:si=on:rtra=on_0 on DTF2THF_22696 for (2837ds/454Mi)
% 123.08/24.47 % (23283)Instruction limit reached!
% 123.08/24.47 % (23283)------------------------------
% 123.08/24.47 % (23283)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 123.08/24.47 % (23283)Termination reason: Unknown
% 123.08/24.47 % (23283)Termination phase: Saturation
% 123.08/24.47
% 123.08/24.47 % (23283)Memory used [KB]: 30703
% 123.08/24.47 % (23283)Time elapsed: 0.504 s
% 123.08/24.47 % (23283)Instructions burned: 1551 (million)
% 123.08/24.47 % (23283)------------------------------
% 123.08/24.47 % (23283)------------------------------
% 123.08/24.48 % (23287)lrs+1002_1:32_au=on:ins=2:nm=16:ntd=on:piset=all_but_not_eq:sp=occurrence:i=3049:si=on:rtra=on_0 on DTF2THF_22696 for (2836ds/3049Mi)
% 123.08/24.49 % (23280)Instruction limit reached!
% 123.08/24.49 % (23280)------------------------------
% 123.08/24.49 % (23280)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 123.08/24.49 % (23280)Termination reason: Unknown
% 123.08/24.49 % (23280)Termination phase: Property scanning
% 123.08/24.49
% 123.08/24.49 % (23280)Memory used [KB]: 39018
% 123.08/24.49 % (23280)Time elapsed: 0.747 s
% 123.08/24.49 % (23280)Instructions burned: 2658 (million)
% 123.08/24.49 % (23280)------------------------------
% 123.08/24.49 % (23280)------------------------------
% 122.76/24.51 % (23288)lrs+1010_1:1_bd=preordered:cnfonf=off:e2e=on:kws=inv_precedence:lwlo=on:sd=2:ss=axioms:i=7311:si=on:rtra=on_0 on DTF2THF_22696 for (2836ds/7311Mi)
% 122.76/24.57 % (23286)Instruction limit reached!
% 122.76/24.57 % (23286)------------------------------
% 122.76/24.57 % (23286)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 122.76/24.57 % (23286)Termination reason: Unknown
% 122.76/24.57 % (23286)Termination phase: shuffling
% 122.76/24.57
% 122.76/24.57 % (23286)Memory used [KB]: 21620
% 122.76/24.57 % (23286)Time elapsed: 0.140 s
% 122.76/24.57 % (23286)Instructions burned: 457 (million)
% 122.76/24.57 % (23286)------------------------------
% 122.76/24.57 % (23286)------------------------------
% 122.76/24.58 % (23289)lrs+1002_1:14_au=on:bd=off:e2e=on:fde=unused:sac=on:sos=on:i=6804:si=on:rtra=on_0 on DTF2THF_22696 for (2835ds/6804Mi)
% 123.86/24.63 % (23281)Instruction limit reached!
% 123.86/24.63 % (23281)------------------------------
% 123.86/24.63 % (23281)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 123.86/24.63 % (23281)Termination reason: Unknown
% 123.86/24.63 % (23281)Termination phase: Function definition elimination
% 123.86/24.63
% 123.86/24.63 % (23281)Memory used [KB]: 39786
% 123.86/24.63 % (23281)Time elapsed: 0.834 s
% 123.86/24.63 % (23281)Instructions burned: 2784 (million)
% 123.86/24.63 % (23281)------------------------------
% 123.86/24.63 % (23281)------------------------------
% 123.86/24.64 % (23290)dis+1003_1:1_au=on:bd=preordered:fde=none:ixr=off:nwc=10.0:i=1683:si=on:rtra=on_0 on DTF2THF_22696 for (2835ds/1683Mi)
% 127.90/25.11 % (23278)Instruction limit reached!
% 127.90/25.11 % (23278)------------------------------
% 127.90/25.11 % (23278)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 127.90/25.11 % (23278)Termination reason: Unknown
% 127.90/25.11 % (23278)Termination phase: Saturation
% 127.90/25.11
% 127.90/25.11 % (23278)Memory used [KB]: 43624
% 127.90/25.11 % (23278)Time elapsed: 1.500 s
% 127.90/25.11 % (23278)Instructions burned: 5447 (million)
% 127.90/25.11 % (23278)------------------------------
% 127.90/25.11 % (23278)------------------------------
% 128.61/25.13 % (23291)lrs+10_8:1_au=on:av=off:bd=off:cs=on:e2e=on:hfsq=on:hfsqc=50:hfsql=off:nm=0:i=4688:si=on:rtra=on_0 on DTF2THF_22696 for (2830ds/4688Mi)
% 128.61/25.18 % (23290)Instruction limit reached!
% 128.61/25.18 % (23290)------------------------------
% 128.61/25.18 % (23290)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 128.61/25.18 % (23290)Termination reason: Unknown
% 128.61/25.18 % (23290)Termination phase: Property scanning
% 128.61/25.18
% 128.61/25.18 % (23290)Memory used [KB]: 35436
% 128.61/25.18 % (23290)Time elapsed: 0.535 s
% 128.61/25.18 % (23290)Instructions burned: 1684 (million)
% 128.61/25.18 % (23290)------------------------------
% 128.61/25.18 % (23290)------------------------------
% 128.61/25.19 % (23292)dis-1_1:1_amm=off:au=on:cbe=off:spb=intro:i=2207:si=on:rtra=on_0 on DTF2THF_22696 for (2829ds/2207Mi)
% 129.06/25.21 % (23285)Instruction limit reached!
% 129.06/25.21 % (23285)------------------------------
% 129.06/25.21 % (23285)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 129.06/25.21 % (23285)Termination reason: Unknown
% 129.06/25.21 % (23285)Termination phase: Property scanning
% 129.06/25.21
% 129.06/25.21 % (23285)Memory used [KB]: 38378
% 129.06/25.21 % (23285)Time elapsed: 0.946 s
% 129.06/25.21 % (23285)Instructions burned: 3204 (million)
% 129.06/25.21 % (23285)------------------------------
% 129.06/25.21 % (23285)------------------------------
% 129.06/25.22 % (23293)dis+1010_1:4_atotf=0.2:c=on:cbe=off:cnfonf=lazy_simp:fe=off:ins=2:ntd=on:s2a=on:s2at=5.0:sgt=5:ss=axioms:st=1.5:i=8216:si=on:rtra=on_0 on DTF2THF_22696 for (2829ds/8216Mi)
% 129.49/25.34 % (23288)First to succeed.
% 129.49/25.35 % (23288)Refutation found. Thanks to Tanya!
% 129.49/25.35 % SZS status Theorem for DTF2THF_22696
% 129.49/25.35 % SZS output start Proof for DTF2THF_22696
% 129.49/25.35 thf(type_def_5, type, produc3658429121746597890et_nat: $tType).
% 129.49/25.35 thf(type_def_7, type, assn: $tType).
% 129.49/25.35 thf(type_def_8, type, produc6197004810343482825et_nat: $tType).
% 129.49/25.35 thf(type_def_9, type, set_nat: $tType).
% 129.49/25.35 thf(type_def_10, type, heap_e7401611519738050253t_unit: $tType).
% 129.49/25.35 thf(type_def_11, type, produc2732055786443039994et_nat: $tType).
% 129.49/25.35 thf(type_def_12, type, set_Pr5094982260447487303t_unit: $tType).
% 129.49/25.35 thf(type_def_13, type, set_Product_unit: $tType).
% 129.49/25.35 thf(type_def_14, type, int: $tType).
% 129.49/25.35 thf(type_def_15, type, product_prod_int_int: $tType).
% 129.49/25.35 thf(type_def_16, type, code_integer: $tType).
% 129.49/25.35 thf(type_def_17, type, nat: $tType).
% 129.49/25.35 thf(type_def_18, type, code_natural: $tType).
% 129.49/25.35 thf(type_def_19, type, rat: $tType).
% 129.49/25.35 thf(type_def_20, type, product_prod_nat_nat: $tType).
% 129.49/25.35 thf(type_def_21, type, set_Pr958786334691620121nt_int: $tType).
% 129.49/25.35 thf(type_def_22, type, set_Pr2560585780119916871nt_int: $tType).
% 129.49/25.35 thf(type_def_23, type, set_Pr1261947904930325089at_nat: $tType).
% 129.49/25.35 thf(type_def_24, type, set_Pr8693737435421807431at_nat: $tType).
% 129.49/25.35 thf(type_def_25, type, set_Pr2161125870931222855_nat_o: $tType).
% 129.49/25.35 thf(type_def_26, type, set_Product_prod_o_o: $tType).
% 129.49/25.35 thf(type_def_27, type, set_Pr4532377907799695533_nat_o: $tType).
% 129.49/25.35 thf(type_def_28, type, set_o: $tType).
% 129.49/25.35 thf(type_def_29, type, set_int: $tType).
% 129.49/25.35 thf(type_def_30, type, product_unit: $tType).
% 129.49/25.35 thf(type_def_31, type, produc3925858234332021118et_nat: $tType).
% 129.49/25.35 thf(type_def_32, type, set_he8644174446132254637t_unit: $tType).
% 129.49/25.35 thf(type_def_33, type, set_Pr3286484037609594932et_nat: $tType).
% 129.49/25.35 thf(type_def_34, type, set_Pr3948176798113811640et_nat: $tType).
% 129.49/25.35 thf(type_def_35, type, set_set_nat: $tType).
% 129.49/25.35 thf(type_def_36, type, num: $tType).
% 129.49/25.35 thf(type_def_37, type, produc8923325533196201883nteger: $tType).
% 129.49/25.35 thf(type_def_38, type, set_Pr6409761941060870310at_o_o: $tType).
% 129.49/25.35 thf(type_def_39, type, set_o_o: $tType).
% 129.49/25.35 thf(type_def_40, type, set_int_o: $tType).
% 129.49/25.35 thf(type_def_41, type, set_nat_o: $tType).
% 129.49/25.35 thf(type_def_42, type, set_Product_unit_o: $tType).
% 129.49/25.35 thf(type_def_43, type, set_filter_nat: $tType).
% 129.49/25.35 thf(type_def_44, type, filter_nat: $tType).
% 129.49/25.35 thf(type_def_45, type, set_se1353816476878259085_nat_o: $tType).
% 129.49/25.35 thf(type_def_46, type, set_set_o: $tType).
% 129.49/25.35 thf(type_def_47, type, set_set_int: $tType).
% 129.49/25.35 thf(type_def_48, type, set_se7855581050983116737at_nat: $tType).
% 129.49/25.35 thf(type_def_49, type, set_set_Product_unit: $tType).
% 129.49/25.35 thf(type_def_50, type, set_set_Code_integer: $tType).
% 129.49/25.35 thf(type_def_51, type, set_Code_integer: $tType).
% 129.49/25.35 thf(type_def_52, type, set_se7663487083892625191_nat_o: $tType).
% 129.49/25.35 thf(type_def_53, type, set_se491199571785101031od_o_o: $tType).
% 129.49/25.35 thf(type_def_54, type, set_se6260736226359567993nt_int: $tType).
% 129.49/25.35 thf(type_def_55, type, set_se8994758365886191399t_unit: $tType).
% 129.49/25.35 thf(type_def_56, type, filter1242075044329608583at_nat: $tType).
% 129.49/25.35 thf(type_def_57, type, filter_set_nat: $tType).
% 129.49/25.35 thf(type_def_58, type, set_assn: $tType).
% 129.49/25.35 thf(type_def_59, type, set_Pr4811707699266497531nteger: $tType).
% 129.49/25.35 thf(type_def_60, type, set_Pr8986988648414616686er_nat: $tType).
% 129.49/25.35 thf(type_def_61, type, set_Pr2406931829603198446nteger: $tType).
% 129.49/25.35 thf(type_def_62, type, set_rat: $tType).
% 129.49/25.35 thf(type_def_63, type, produc723880050079548563nteger: $tType).
% 129.49/25.35 thf(type_def_64, type, produc101387210554329899nteger: $tType).
% 129.49/25.35 thf(type_def_65, type, produc2221115339011613079nteger: $tType).
% 129.49/25.35 thf(type_def_66, type, produc1397831737260518551nteger: $tType).
% 129.49/25.35 thf(type_def_67, type, produc6433947425880249742er_nat: $tType).
% 129.49/25.35 thf(type_def_68, type, produc2591615631378489102nteger: $tType).
% 129.49/25.35 thf(type_def_69, type, typerep: $tType).
% 129.49/25.35 thf(type_def_70, type, list_nat: $tType).
% 129.49/25.35 thf(type_def_71, type, set_Pr3813038829465207111_nat_o: $tType).
% 129.49/25.35 thf(type_def_72, type, set_Pr4577759397028426247_set_o: $tType).
% 129.49/25.35 thf(type_def_73, type, set_Pr2522554150109002629et_int: $tType).
% 129.49/25.35 thf(type_def_74, type, set_Pr5488025237498180813et_nat: $tType).
% 129.49/25.35 thf(type_def_75, type, set_Pr3980508049012409671t_unit: $tType).
% 129.49/25.35 thf(type_def_76, type, set_Pr8536935166611901872et_nat: $tType).
% 129.49/25.35 thf(type_def_77, type, produc5030937667867769463nt_int: $tType).
% 129.49/25.35 thf(type_def_78, type, produc6128020648511768019nt_int: $tType).
% 129.49/25.35 thf(type_def_79, type, produc3574062574353363227at_nat: $tType).
% 129.49/25.35 thf(type_def_80, type, produc4671145554997361783at_nat: $tType).
% 129.49/25.35 thf(type_def_81, type, sum_sum_nat_nat: $tType).
% 129.49/25.35 thf(type_def_82, type, set_Pr4769669219625338370_o_nat: $tType).
% 129.49/25.35 thf(type_def_83, type, set_Pr2101469702781467981_o_nat: $tType).
% 129.49/25.35 thf(type_def_84, type, set_Pr3448869479623346877nt_nat: $tType).
% 129.49/25.35 thf(type_def_85, type, set_Pr1763845938948868674it_nat: $tType).
% 129.49/25.35 thf(type_def_86, type, set_set_set_nat: $tType).
% 129.49/25.35 thf(type_def_87, type, set_Co726186806682698383nteger: $tType).
% 129.49/25.35 thf(type_def_88, type, set_Code_natural: $tType).
% 129.49/25.35 thf(type_def_89, type, set_Pr448751882837621926eger_o: $tType).
% 129.49/25.35 thf(type_def_90, type, set_Pr6496905503482889930er_int: $tType).
% 129.49/25.35 thf(type_def_91, type, set_Pr4580871404036792859t_unit: $tType).
% 129.49/25.35 thf(type_def_92, type, produc4928098042776334183_nat_o: $tType).
% 129.49/25.35 thf(type_def_93, type, product_prod_o_o: $tType).
% 129.49/25.35 thf(type_def_94, type, product_prod_o_int: $tType).
% 129.49/25.35 thf(type_def_95, type, product_prod_o_nat: $tType).
% 129.49/25.35 thf(type_def_96, type, produc3788275437394581276t_unit: $tType).
% 129.49/25.35 thf(type_def_97, type, produc7822875418678951345atural: $tType).
% 129.49/25.35 thf(type_def_98, type, produc5835291356934675326atural: $tType).
% 129.49/25.35 thf(type_def_99, type, produc8126358835630522215t_unit: $tType).
% 129.49/25.35 thf(type_def_100, type, product_prod_int_o: $tType).
% 129.49/25.35 thf(type_def_101, type, product_prod_int_nat: $tType).
% 129.49/25.35 thf(type_def_102, type, produc6865911496949736918t_unit: $tType).
% 129.49/25.35 thf(type_def_103, type, product_prod_nat_o: $tType).
% 129.49/25.35 thf(type_def_104, type, product_prod_nat_int: $tType).
% 129.49/25.35 thf(type_def_105, type, produc526117512892780850t_unit: $tType).
% 129.49/25.35 thf(type_def_106, type, produc5657529347773406293et_nat: $tType).
% 129.49/25.35 thf(type_def_107, type, produc4962441903510821572_nat_o: $tType).
% 129.49/25.35 thf(type_def_108, type, produc6830853553727218525et_nat: $tType).
% 129.49/25.35 thf(type_def_109, type, produc4658369026151987169et_nat: $tType).
% 129.49/25.35 thf(type_def_110, type, produc6907119039986491712_nat_o: $tType).
% 129.49/25.35 thf(type_def_111, type, produc7389280635074931169et_nat: $tType).
% 129.49/25.35 thf(type_def_112, type, produc8111630337999740517et_nat: $tType).
% 129.49/25.35 thf(type_def_113, type, produc1219242969750017639nt_int: $tType).
% 129.49/25.35 thf(type_def_114, type, produc4806409095325742951nt_nat: $tType).
% 129.49/25.35 thf(type_def_115, type, produc6495656768158660455at_int: $tType).
% 129.49/25.35 thf(type_def_116, type, produc859450856879609959at_nat: $tType).
% 129.49/25.35 thf(type_def_117, type, produc7485711547916492274unit_o: $tType).
% 129.49/25.35 thf(type_def_118, type, produc5646078835734834920it_int: $tType).
% 129.49/25.35 thf(type_def_119, type, produc600557818389255820it_nat: $tType).
% 129.49/25.35 thf(type_def_120, type, produc8459935480633519975t_unit: $tType).
% 129.49/25.35 thf(type_def_121, type, produc6666532470309664615_nat_o: $tType).
% 129.49/25.35 thf(type_def_122, type, produc7369051934464679207_set_o: $tType).
% 129.49/25.35 thf(type_def_123, type, produc6491284506569428743nteger: $tType).
% 129.49/25.35 thf(type_def_124, type, produc2115011035271226405et_int: $tType).
% 129.49/25.35 thf(type_def_125, type, produc7348995623955488188t_unit: $tType).
% 129.49/25.35 thf(type_def_126, type, produc7819656566062154093et_nat: $tType).
% 129.49/25.35 thf(type_def_127, type, produc1357844946079266663t_unit: $tType).
% 129.49/25.35 thf(type_def_128, type, set_Pr8834758594704517033_o_int: $tType).
% 129.49/25.35 thf(type_def_129, type, set_Pr6671815714118170108t_unit: $tType).
% 129.49/25.35 thf(type_def_130, type, set_Pr903927857289325719_int_o: $tType).
% 129.49/25.35 thf(type_def_131, type, set_Pr5889264611543340172t_unit: $tType).
% 129.49/25.35 thf(type_def_132, type, set_Pr5623950326363488168_nat_o: $tType).
% 129.49/25.35 thf(type_def_133, type, set_Pr3149072824959771635_nat_o: $tType).
% 129.49/25.35 thf(type_def_134, type, set_Pr7995236796853374141at_int: $tType).
% 129.49/25.35 thf(type_def_135, type, set_Pr4334478416066269672t_unit: $tType).
% 129.49/25.35 thf(type_def_136, type, set_Pr5464989305201766866unit_o: $tType).
% 129.49/25.35 thf(type_def_137, type, produc7936767299998605306et_nat: $tType).
% 129.49/25.35 thf(type_def_138, type, produc5697189838369619070et_nat: $tType).
% 129.49/25.35 thf(type_def_139, type, produc6500295350249520386et_nat: $tType).
% 129.49/25.35 thf(type_def_140, type, produc2475246329834808232it_int: $tType).
% 129.49/25.35 thf(type_def_141, type, produc2771989624626745363et_nat: $tType).
% 129.49/25.35 thf(type_def_142, type, produc6653097349344004940it_nat: $tType).
% 129.49/25.35 thf(type_def_143, type, produc2400336064389900727et_nat: $tType).
% 129.49/25.35 thf(type_def_144, type, produc3776562785724475560_o_int: $tType).
% 129.49/25.35 thf(type_def_145, type, produc1049655590333717003et_nat: $tType).
% 129.49/25.35 thf(type_def_146, type, produc8065447618272157327et_nat: $tType).
% 129.49/25.35 thf(type_def_147, type, produc5966032636589952099nteger: $tType).
% 129.49/25.35 thf(type_def_148, type, produc6271795597528267376eger_o: $tType).
% 129.49/25.35 thf(type_def_149, type, produc2256096406371053034er_int: $tType).
% 129.49/25.35 thf(type_def_150, type, produc5496629617739085541t_unit: $tType).
% 129.49/25.35 thf(type_def_151, type, set_Pr8497134830871917726it_int: $tType).
% 129.49/25.35 thf(type_def_152, type, set_Pr7705059561495379578_nat_o: $tType).
% 129.49/25.35 thf(type_def_153, type, set_Pr4208978187861016310_nat_o: $tType).
% 129.49/25.35 thf(type_def_154, type, set_Pr4071100604818519410t_unit: $tType).
% 129.49/25.35 thf(type_def_155, type, set_Pr7928877670098842301et_nat: $tType).
% 129.49/25.35 thf(type_def_156, type, set_Pr3721596426557623617et_nat: $tType).
% 129.49/25.35 thf(type_def_157, type, set_Pr8807100098903490369et_nat: $tType).
% 129.49/25.35 thf(type_def_158, type, set_Pr719794911490849221et_nat: $tType).
% 129.49/25.35 thf(type_def_159, type, set_Pr1776176089890104647t_unit: $tType).
% 129.49/25.35 thf(type_def_160, type, char: $tType).
% 129.49/25.35 thf(type_def_161, type, set_Pr3444600963470892981et_nat: $tType).
% 129.49/25.35 thf(type_def_162, type, set_Pr5696620967126086215nt_nat: $tType).
% 129.49/25.35 thf(type_def_163, type, set_Pr5557702248415638087at_int: $tType).
% 129.49/25.35 thf(type_def_164, type, set_Pr7577011563204128103nteger: $tType).
% 129.49/25.35 thf(func_def_0, type, set_Pr3444600963470892981et_nat: $tType).
% 129.49/25.35 thf(func_def_1, type, produc5657529347773406293et_nat: $tType).
% 129.49/25.35 thf(func_def_2, type, set_Pr7928877670098842301et_nat: $tType).
% 129.49/25.35 thf(func_def_3, type, produc6830853553727218525et_nat: $tType).
% 129.49/25.35 thf(func_def_4, type, set_Pr7705059561495379578_nat_o: $tType).
% 129.49/25.35 thf(func_def_5, type, set_Pr8536935166611901872et_nat: $tType).
% 129.49/25.35 thf(func_def_6, type, set_Pr8807100098903490369et_nat: $tType).
% 129.49/25.35 thf(func_def_7, type, set_Pr3721596426557623617et_nat: $tType).
% 129.49/25.35 thf(func_def_8, type, produc4962441903510821572_nat_o: $tType).
% 129.49/25.35 thf(func_def_9, type, produc2732055786443039994et_nat: $tType).
% 129.49/25.35 thf(func_def_10, type, produc7389280635074931169et_nat: $tType).
% 129.49/25.35 thf(func_def_11, type, produc4658369026151987169et_nat: $tType).
% 129.49/25.35 thf(func_def_12, type, produc101387210554329899nteger: $tType).
% 129.49/25.35 thf(func_def_13, type, produc7936767299998605306et_nat: $tType).
% 129.49/25.35 thf(func_def_14, type, set_Pr3813038829465207111_nat_o: $tType).
% 129.49/25.35 thf(func_def_15, type, set_se7663487083892625191_nat_o: $tType).
% 129.49/25.35 thf(func_def_16, type, produc6666532470309664615_nat_o: $tType).
% 129.49/25.35 thf(func_def_17, type, produc1049655590333717003et_nat: $tType).
% 129.49/25.35 thf(func_def_18, type, set_Pr2161125870931222855_nat_o: $tType).
% 129.49/25.35 thf(func_def_19, type, set_Pr4208978187861016310_nat_o: $tType).
% 129.49/25.35 thf(func_def_20, type, set_Pr3286484037609594932et_nat: $tType).
% 129.49/25.35 thf(func_def_21, type, produc4928098042776334183_nat_o: $tType).
% 129.49/25.35 thf(func_def_22, type, set_Pr719794911490849221et_nat: $tType).
% 129.49/25.35 thf(func_def_23, type, produc6907119039986491712_nat_o: $tType).
% 129.49/25.35 thf(func_def_24, type, produc3925858234332021118et_nat: $tType).
% 129.49/25.35 thf(func_def_25, type, produc8111630337999740517et_nat: $tType).
% 129.49/25.35 thf(func_def_26, type, produc5966032636589952099nteger: $tType).
% 129.49/25.35 thf(func_def_27, type, produc5697189838369619070et_nat: $tType).
% 129.49/25.35 thf(func_def_28, type, set_Pr5623950326363488168_nat_o: $tType).
% 129.49/25.35 thf(func_def_29, type, set_Pr4769669219625338370_o_nat: $tType).
% 129.49/25.35 thf(func_def_30, type, produc6500295350249520386et_nat: $tType).
% 129.49/25.35 thf(func_def_31, type, produc3776562785724475560_o_int: $tType).
% 129.49/25.35 thf(func_def_32, type, produc723880050079548563nteger: $tType).
% 129.49/25.35 thf(func_def_33, type, set_se1353816476878259085_nat_o: $tType).
% 129.49/25.35 thf(func_def_34, type, set_Pr1776176089890104647t_unit: $tType).
% 129.49/25.35 thf(func_def_35, type, produc8065447618272157327et_nat: $tType).
% 129.49/25.35 thf(func_def_36, type, set_Pr6409761941060870310at_o_o: $tType).
% 129.49/25.35 thf(func_def_37, type, set_Pr8693737435421807431at_nat: $tType).
% 129.49/25.35 thf(func_def_38, type, set_Pr5557702248415638087at_int: $tType).
% 129.49/25.35 thf(func_def_39, type, set_Pr5696620967126086215nt_nat: $tType).
% 129.49/25.35 thf(func_def_40, type, set_Pr2560585780119916871nt_int: $tType).
% 129.49/25.35 thf(func_def_41, type, produc6197004810343482825et_nat: $tType).
% 129.49/25.35 thf(func_def_42, type, produc5835291356934675326atural: $tType).
% 129.49/25.35 thf(func_def_43, type, set_Pr4532377907799695533_nat_o: $tType).
% 129.49/25.35 thf(func_def_44, type, produc8126358835630522215t_unit: $tType).
% 129.49/25.35 thf(func_def_45, type, produc859450856879609959at_nat: $tType).
% 129.49/25.35 thf(func_def_46, type, produc6495656768158660455at_int: $tType).
% 129.49/25.35 thf(func_def_47, type, produc4806409095325742951nt_nat: $tType).
% 129.49/25.35 thf(func_def_48, type, produc1219242969750017639nt_int: $tType).
% 129.49/25.35 thf(func_def_49, type, set_Pr7577011563204128103nteger: $tType).
% 129.49/25.35 thf(func_def_50, type, set_Pr4071100604818519410t_unit: $tType).
% 129.49/25.35 thf(func_def_51, type, set_Pr3948176798113811640et_nat: $tType).
% 129.49/25.35 thf(func_def_52, type, set_Pr3980508049012409671t_unit: $tType).
% 129.49/25.35 thf(func_def_53, type, produc1397831737260518551nteger: $tType).
% 129.49/25.35 thf(func_def_54, type, produc2221115339011613079nteger: $tType).
% 129.49/25.35 thf(func_def_55, type, produc4671145554997361783at_nat: $tType).
% 129.49/25.35 thf(func_def_56, type, produc3574062574353363227at_nat: $tType).
% 129.49/25.35 thf(func_def_57, type, produc6128020648511768019nt_int: $tType).
% 129.49/25.35 thf(func_def_58, type, produc5030937667867769463nt_int: $tType).
% 129.49/25.35 thf(func_def_59, type, produc6491284506569428743nteger: $tType).
% 129.49/25.35 thf(func_def_60, type, produc7348995623955488188t_unit: $tType).
% 129.49/25.35 thf(func_def_61, type, produc3658429121746597890et_nat: $tType).
% 129.49/25.35 thf(func_def_62, type, set_se8994758365886191399t_unit: $tType).
% 129.49/25.35 thf(func_def_63, type, produc1357844946079266663t_unit: $tType).
% 129.49/25.35 thf(func_def_64, type, set_Pr5488025237498180813et_nat: $tType).
% 129.49/25.35 thf(func_def_65, type, set_Pr2522554150109002629et_int: $tType).
% 129.49/25.35 thf(func_def_66, type, set_Pr4811707699266497531nteger: $tType).
% 129.49/25.35 thf(func_def_67, type, set_Pr4580871404036792859t_unit: $tType).
% 129.49/25.35 thf(func_def_68, type, produc6653097349344004940it_nat: $tType).
% 129.49/25.35 thf(func_def_69, type, produc2475246329834808232it_int: $tType).
% 129.49/25.35 thf(func_def_70, type, set_Pr5094982260447487303t_unit: $tType).
% 129.49/25.35 thf(func_def_71, type, set_se7855581050983116737at_nat: $tType).
% 129.49/25.35 thf(func_def_72, type, set_se6260736226359567993nt_int: $tType).
% 129.49/25.35 thf(func_def_73, type, set_Pr2406931829603198446nteger: $tType).
% 129.49/25.35 thf(func_def_74, type, set_Pr8986988648414616686er_nat: $tType).
% 129.49/25.35 thf(func_def_75, type, set_Pr6496905503482889930er_int: $tType).
% 129.49/25.35 thf(func_def_76, type, produc7819656566062154093et_nat: $tType).
% 129.49/25.35 thf(func_def_77, type, produc2115011035271226405et_int: $tType).
% 129.49/25.35 thf(func_def_78, type, produc7822875418678951345atural: $tType).
% 129.49/25.35 thf(func_def_79, type, produc8923325533196201883nteger: $tType).
% 129.49/25.35 thf(func_def_80, type, set_Pr4577759397028426247_set_o: $tType).
% 129.49/25.35 thf(func_def_81, type, set_Pr1763845938948868674it_nat: $tType).
% 129.49/25.35 thf(func_def_82, type, set_Pr8497134830871917726it_int: $tType).
% 129.49/25.35 thf(func_def_83, type, set_Pr4334478416066269672t_unit: $tType).
% 129.49/25.35 thf(func_def_84, type, set_Pr5889264611543340172t_unit: $tType).
% 129.49/25.35 thf(func_def_85, type, produc5496629617739085541t_unit: $tType).
% 129.49/25.35 thf(func_def_86, type, set_Co726186806682698383nteger: $tType).
% 129.49/25.35 thf(func_def_87, type, produc8459935480633519975t_unit: $tType).
% 129.49/25.35 thf(func_def_88, type, set_Pr448751882837621926eger_o: $tType).
% 129.49/25.35 thf(func_def_89, type, filter1242075044329608583at_nat: $tType).
% 129.49/25.35 thf(func_def_90, type, set_Pr5464989305201766866unit_o: $tType).
% 129.49/25.35 thf(func_def_91, type, set_Pr6671815714118170108t_unit: $tType).
% 129.49/25.35 thf(func_def_92, type, set_he8644174446132254637t_unit: $tType).
% 129.49/25.35 thf(func_def_93, type, set_Pr1261947904930325089at_nat: $tType).
% 129.49/25.35 thf(func_def_94, type, set_Pr7995236796853374141at_int: $tType).
% 129.49/25.35 thf(func_def_95, type, set_Pr3448869479623346877nt_nat: $tType).
% 129.49/25.35 thf(func_def_96, type, set_Pr958786334691620121nt_int: $tType).
% 129.49/25.35 thf(func_def_97, type, produc2400336064389900727et_nat: $tType).
% 129.49/25.35 thf(func_def_98, type, produc2591615631378489102nteger: $tType).
% 129.49/25.35 thf(func_def_99, type, produc2771989624626745363et_nat: $tType).
% 129.49/25.35 thf(func_def_100, type, produc6433947425880249742er_nat: $tType).
% 129.49/25.35 thf(func_def_101, type, produc2256096406371053034er_int: $tType).
% 129.49/25.35 thf(func_def_102, type, set_se491199571785101031od_o_o: $tType).
% 129.49/25.35 thf(func_def_103, type, produc7369051934464679207_set_o: $tType).
% 129.49/25.35 thf(func_def_104, type, produc600557818389255820it_nat: $tType).
% 129.49/25.35 thf(func_def_105, type, produc5646078835734834920it_int: $tType).
% 129.49/25.35 thf(func_def_106, type, produc526117512892780850t_unit: $tType).
% 129.49/25.35 thf(func_def_107, type, produc6865911496949736918t_unit: $tType).
% 129.49/25.35 thf(func_def_108, type, set_Pr3149072824959771635_nat_o: $tType).
% 129.49/25.35 thf(func_def_109, type, set_Pr903927857289325719_int_o: $tType).
% 129.49/25.35 thf(func_def_110, type, set_Pr2101469702781467981_o_nat: $tType).
% 129.49/25.35 thf(func_def_111, type, set_Pr8834758594704517033_o_int: $tType).
% 129.49/25.35 thf(func_def_112, type, produc6271795597528267376eger_o: $tType).
% 129.49/25.35 thf(func_def_113, type, set_set_set_nat: $tType).
% 129.49/25.35 thf(func_def_114, type, set_set_Code_integer: $tType).
% 129.49/25.35 thf(func_def_115, type, produc7485711547916492274unit_o: $tType).
% 129.49/25.35 thf(func_def_116, type, produc3788275437394581276t_unit: $tType).
% 129.49/25.35 thf(func_def_117, type, set_set_Product_unit: $tType).
% 129.49/25.35 thf(func_def_118, type, heap_e7401611519738050253t_unit: $tType).
% 129.49/25.35 thf(func_def_119, type, product_prod_nat_nat: $tType).
% 129.49/25.35 thf(func_def_120, type, product_prod_nat_int: $tType).
% 129.49/25.35 thf(func_def_121, type, product_prod_int_nat: $tType).
% 129.49/25.35 thf(func_def_122, type, product_prod_int_int: $tType).
% 129.49/25.35 thf(func_def_123, type, set_Product_unit_o: $tType).
% 129.49/25.35 thf(func_def_124, type, set_Product_prod_o_o: $tType).
% 129.49/25.35 thf(func_def_125, type, sum_sum_nat_nat: $tType).
% 129.49/25.35 thf(func_def_126, type, set_filter_nat: $tType).
% 129.49/25.35 thf(func_def_127, type, filter_set_nat: $tType).
% 129.49/25.35 thf(func_def_128, type, product_prod_nat_o: $tType).
% 129.49/25.35 thf(func_def_129, type, product_prod_int_o: $tType).
% 129.49/25.35 thf(func_def_130, type, product_prod_o_nat: $tType).
% 129.49/25.35 thf(func_def_131, type, product_prod_o_int: $tType).
% 129.49/25.35 thf(func_def_132, type, set_set_nat: $tType).
% 129.49/25.35 thf(func_def_133, type, set_set_int: $tType).
% 129.49/25.35 thf(func_def_134, type, set_Code_natural: $tType).
% 129.49/25.35 thf(func_def_135, type, set_Code_integer: $tType).
% 129.49/25.35 thf(func_def_136, type, set_Product_unit: $tType).
% 129.49/25.35 thf(func_def_137, type, set_nat_o: $tType).
% 129.49/25.35 thf(func_def_138, type, set_int_o: $tType).
% 129.49/25.35 thf(func_def_139, type, product_prod_o_o: $tType).
% 129.49/25.35 thf(func_def_140, type, set_assn: $tType).
% 129.49/25.35 thf(func_def_141, type, set_set_o: $tType).
% 129.49/25.35 thf(func_def_142, type, filter_nat: $tType).
% 129.49/25.35 thf(func_def_143, type, set_o_o: $tType).
% 129.49/25.35 thf(func_def_144, type, list_nat: $tType).
% 129.49/25.35 thf(func_def_145, type, set_rat: $tType).
% 129.49/25.35 thf(func_def_146, type, set_nat: $tType).
% 129.49/25.35 thf(func_def_147, type, set_int: $tType).
% 129.49/25.35 thf(func_def_148, type, code_natural: $tType).
% 129.49/25.35 thf(func_def_149, type, code_integer: $tType).
% 129.49/25.35 thf(func_def_150, type, product_unit: $tType).
% 129.49/25.35 thf(func_def_151, type, typerep: $tType).
% 129.49/25.35 thf(func_def_152, type, assn: $tType).
% 129.49/25.35 thf(func_def_153, type, set_o: $tType).
% 129.49/25.35 thf(func_def_154, type, char: $tType).
% 129.49/25.35 thf(func_def_155, type, rat: $tType).
% 129.49/25.35 thf(func_def_156, type, num: $tType).
% 129.49/25.35 thf(func_def_157, type, nat: $tType).
% 129.49/25.35 thf(func_def_158, type, int: $tType).
% 129.49/25.35 thf(func_def_159, type, abs_assn: (produc3658429121746597890et_nat > $o) > assn).
% 129.49/25.35 thf(func_def_160, type, rep_assn: assn > produc3658429121746597890et_nat > $o).
% 129.49/25.35 thf(func_def_161, type, in_range: produc3658429121746597890et_nat > $o).
% 129.49/25.35 thf(func_def_162, type, in_range_rel: produc3658429121746597890et_nat > produc3658429121746597890et_nat > $o).
% 129.49/25.35 thf(func_def_163, type, is_pure_assn: assn > $o).
% 129.49/25.35 thf(func_def_164, type, one_assn_raw: produc3658429121746597890et_nat > $o).
% 129.49/25.35 thf(func_def_165, type, one_assn_raw_rel: produc3658429121746597890et_nat > produc3658429121746597890et_nat > $o).
% 129.49/25.35 thf(func_def_166, type, proper: (produc3658429121746597890et_nat > $o) > $o).
% 129.49/25.35 thf(func_def_167, type, pure_assn: $o > assn).
% 129.49/25.35 thf(func_def_168, type, pure_a825153325127701367it_nat: $o > produc3658429121746597890et_nat > $o).
% 129.49/25.35 thf(func_def_169, type, pure_a6022498039421069578it_nat: produc6197004810343482825et_nat > produc6197004810343482825et_nat > $o).
% 129.49/25.35 thf(func_def_170, type, relH: set_nat > heap_e7401611519738050253t_unit > heap_e7401611519738050253t_unit > $o).
% 129.49/25.35 thf(func_def_171, type, times_assn_raw: (produc3658429121746597890et_nat > $o) > (produc3658429121746597890et_nat > $o) > produc3658429121746597890et_nat > $o).
% 129.49/25.35 thf(func_def_172, type, times_assn_raw_rel: produc2732055786443039994et_nat > produc2732055786443039994et_nat > $o).
% 129.49/25.35 thf(func_def_173, type, wand_assn: assn > assn > assn).
% 129.49/25.35 thf(func_def_174, type, wand_raw: (produc3658429121746597890et_nat > $o) > (produc3658429121746597890et_nat > $o) > produc3658429121746597890et_nat > $o).
% 129.49/25.35 thf(func_def_175, type, wand_raw_rel: produc2732055786443039994et_nat > produc2732055786443039994et_nat > $o).
% 129.49/25.35 thf(func_def_176, type, bNF_Cardinal_cone: set_Pr5094982260447487303t_unit).
% 129.49/25.35 thf(func_def_177, type, bNF_Ca7083678892733797993t_unit: set_Product_unit > set_Pr5094982260447487303t_unit).
% 129.49/25.35 thf(func_def_178, type, bNF_fs3871268833404974121et_nat: (heap_e7401611519738050253t_unit > set_nat > $o) > (set_nat > set_nat > $o) > produc3658429121746597890et_nat > produc3658429121746597890et_nat).
% 129.49/25.35 thf(func_def_179, type, bNF_fs8167890477030535480nt_int: (int > int > $o) > (int > int > $o) > product_prod_int_int > product_prod_int_int).
% 129.49/25.35 thf(func_def_180, type, bNF_pi3450177011487049898et_nat: (heap_e7401611519738050253t_unit > heap_e7401611519738050253t_unit > $o) > (heap_e7401611519738050253t_unit > set_nat > $o) > heap_e7401611519738050253t_unit > set_nat > heap_e7401611519738050253t_unit).
% 129.49/25.35 thf(func_def_181, type, bNF_pi4042512755101277815et_nat: (heap_e7401611519738050253t_unit > set_nat > $o) > (set_nat > set_nat > $o) > heap_e7401611519738050253t_unit > set_nat > set_nat).
% 129.49/25.35 thf(func_def_182, type, bNF_pi8853482479757052266nt_int: (int > int > $o) > (int > int > $o) > int > int > int).
% 129.49/25.35 thf(func_def_183, type, bNF_re398004352372739002nteger: (int > code_integer > $o) > ((int > int) > (code_integer > code_integer) > $o) > (int > int > int) > (code_integer > code_integer > code_integer) > $o).
% 129.49/25.35 thf(func_def_184, type, bNF_re3379532845092657523nteger: (int > code_integer > $o) > (int > code_integer > $o) > (int > int) > (code_integer > code_integer) > $o).
% 129.49/25.35 thf(func_def_185, type, bNF_re711492959462206631nt_int: (int > int > $o) > ((int > int) > (int > int) > $o) > (int > int > int) > (int > int > int) > $o).
% 129.49/25.35 thf(func_def_186, type, bNF_re4712519889275205905nt_int: (int > int > $o) > (int > int > $o) > (int > int) > (int > int) > $o).
% 129.49/25.35 thf(func_def_187, type, bNF_re88643428490162567atural: (nat > code_natural > $o) > ((nat > nat) > (code_natural > code_natural) > $o) > (nat > nat > nat) > (code_natural > code_natural > code_natural) > $o).
% 129.49/25.35 thf(func_def_188, type, bNF_re3704215830270325841atural: (nat > code_natural > $o) > (nat > code_natural > $o) > (nat > nat) > (code_natural > code_natural) > $o).
% 129.49/25.35 thf(func_def_189, type, bNF_re1345281282404953727at_nat: (nat > nat > $o) > ((nat > nat) > (nat > nat) > $o) > (nat > nat > nat) > (nat > nat > nat) > $o).
% 129.49/25.35 thf(func_def_190, type, bNF_re5653821019739307937at_nat: (nat > nat > $o) > (nat > nat > $o) > (nat > nat) > (nat > nat) > $o).
% 129.49/25.35 thf(func_def_191, type, bNF_re5228765855967844073nt_int: (product_prod_int_int > product_prod_int_int > $o) > ((product_prod_int_int > product_prod_int_int) > (product_prod_int_int > product_prod_int_int) > $o) > (product_prod_int_int > product_prod_int_int > product_prod_int_int) > (product_prod_int_int > product_prod_int_int > product_prod_int_int) > $o).
% 129.49/25.35 thf(func_def_192, type, bNF_re8699439704749558557nt_o_o: (product_prod_int_int > product_prod_int_int > $o) > ($o > $o > $o) > (product_prod_int_int > $o) > (product_prod_int_int > $o) > $o).
% 129.49/25.35 thf(func_def_193, type, bNF_re7145576690424134365nt_int: (product_prod_int_int > product_prod_int_int > $o) > (product_prod_int_int > product_prod_int_int > $o) > (product_prod_int_int > product_prod_int_int) > (product_prod_int_int > product_prod_int_int) > $o).
% 129.49/25.35 thf(func_def_194, type, bNF_re7627151682743391978at_rat: (product_prod_int_int > rat > $o) > ((product_prod_int_int > product_prod_int_int) > (rat > rat) > $o) > (product_prod_int_int > product_prod_int_int > product_prod_int_int) > (rat > rat > rat) > $o).
% 129.49/25.35 thf(func_def_195, type, bNF_re1494630372529172596at_o_o: (product_prod_int_int > rat > $o) > ($o > $o > $o) > (product_prod_int_int > $o) > (rat > $o) > $o).
% 129.49/25.35 thf(func_def_196, type, bNF_re8279943556446156061nt_rat: (product_prod_int_int > rat > $o) > (product_prod_int_int > rat > $o) > (product_prod_int_int > product_prod_int_int) > (rat > rat) > $o).
% 129.49/25.35 thf(func_def_197, type, bNF_re7408651293131936558nt_int: (product_prod_nat_nat > int > $o) > ((product_prod_nat_nat > product_prod_nat_nat) > (int > int) > $o) > (product_prod_nat_nat > product_prod_nat_nat > product_prod_nat_nat) > (int > int > int) > $o).
% 129.49/25.35 thf(func_def_198, type, bNF_re7400052026677387805at_int: (product_prod_nat_nat > int > $o) > (product_prod_nat_nat > int > $o) > (product_prod_nat_nat > product_prod_nat_nat) > (int > int) > $o).
% 129.49/25.35 thf(func_def_199, type, bNF_re3099431351363272937at_nat: (product_prod_nat_nat > product_prod_nat_nat > $o) > ((product_prod_nat_nat > product_prod_nat_nat) > (product_prod_nat_nat > product_prod_nat_nat) > $o) > (product_prod_nat_nat > product_prod_nat_nat > product_prod_nat_nat) > (product_prod_nat_nat > product_prod_nat_nat > product_prod_nat_nat) > $o).
% 129.49/25.35 thf(func_def_200, type, bNF_re2241393799969408733at_nat: (product_prod_nat_nat > product_prod_nat_nat > $o) > (product_prod_nat_nat > product_prod_nat_nat > $o) > (product_prod_nat_nat > product_prod_nat_nat) > (product_prod_nat_nat > product_prod_nat_nat) > $o).
% 129.49/25.35 thf(func_def_201, type, bNF_sn4039337771197282234et_nat: (heap_e7401611519738050253t_unit > heap_e7401611519738050253t_unit > $o) > (heap_e7401611519738050253t_unit > set_nat > $o) > produc3658429121746597890et_nat > produc3658429121746597890et_nat).
% 129.49/25.35 thf(func_def_202, type, bNF_sn1062102010912252026nt_int: (int > int > $o) > (int > int > $o) > product_prod_int_int > product_prod_int_int).
% 129.49/25.35 thf(func_def_203, type, bNF_We1730399552001240015qr_int: set_Pr958786334691620121nt_int > set_Pr2560585780119916871nt_int).
% 129.49/25.35 thf(func_def_204, type, bNF_We1732890022510290291qr_nat: set_Pr1261947904930325089at_nat > set_Pr8693737435421807431at_nat).
% 129.49/25.35 thf(func_def_205, type, bNF_We740377018556551609_nat_o: set_Pr2161125870931222855_nat_o > $o).
% 129.49/25.35 thf(func_def_206, type, bNF_We6437023980930612612_rel_o: set_Product_prod_o_o > $o).
% 129.49/25.35 thf(func_def_207, type, bNF_We3815749466139970368el_int: set_Pr958786334691620121nt_int > $o).
% 129.49/25.35 thf(func_def_208, type, bNF_We3818239936649020644el_nat: set_Pr1261947904930325089at_nat > $o).
% 129.49/25.35 thf(func_def_209, type, bNF_We2091578341578594809t_unit: set_Pr5094982260447487303t_unit > $o).
% 129.49/25.35 thf(func_def_210, type, bNF_We5608535359669600131_nat_o: set_Pr2161125870931222855_nat_o > set_Pr4532377907799695533_nat_o > (produc3658429121746597890et_nat > $o) > $o).
% 129.49/25.35 thf(func_def_211, type, bNF_We4647864369647134286inim_o: set_Product_prod_o_o > set_o > $o > $o).
% 129.49/25.35 thf(func_def_212, type, bNF_We6511469253196982518im_int: set_Pr958786334691620121nt_int > set_int > int > $o).
% 129.49/25.35 thf(func_def_213, type, bNF_We6513959723706032794im_nat: set_Pr1261947904930325089at_nat > set_nat > nat > $o).
% 129.49/25.35 thf(func_def_214, type, bNF_We909621271159000003t_unit: set_Pr5094982260447487303t_unit > set_Product_unit > product_unit > $o).
% 129.49/25.35 thf(func_def_215, type, bNF_We6400040844883348763_nat_o: set_Pr2161125870931222855_nat_o > (produc3658429121746597890et_nat > $o) > (produc3658429121746597890et_nat > $o) > produc3658429121746597890et_nat > $o).
% 129.49/25.35 thf(func_def_216, type, bNF_We3104086342435940326max2_o: set_Product_prod_o_o > $o > $o > $o).
% 129.49/25.35 thf(func_def_217, type, bNF_We2378448595015613790x2_int: set_Pr958786334691620121nt_int > int > int > int).
% 129.49/25.35 thf(func_def_218, type, bNF_We2380939065524664066x2_nat: set_Pr1261947904930325089at_nat > nat > nat > nat).
% 129.49/25.35 thf(func_def_219, type, bNF_We7204996500647184731t_unit: set_Pr5094982260447487303t_unit > product_unit > product_unit > product_unit).
% 129.49/25.35 thf(func_def_220, type, bNF_We8236795211600914329_nat_o: set_Pr2161125870931222855_nat_o > set_Pr4532377907799695533_nat_o > produc3658429121746597890et_nat > $o).
% 129.49/25.35 thf(func_def_221, type, bNF_We537455116209922916inim_o: set_Product_prod_o_o > set_o > $o).
% 129.49/25.35 thf(func_def_222, type, bNF_We3272633801743244640im_int: set_Pr958786334691620121nt_int > set_int > int).
% 129.49/25.35 thf(func_def_223, type, bNF_We3275124272252294916im_nat: set_Pr1261947904930325089at_nat > set_nat > nat).
% 129.49/25.35 thf(func_def_224, type, bNF_We7185707985517244377t_unit: set_Pr5094982260447487303t_unit > set_Product_unit > product_unit).
% 129.49/25.35 thf(func_def_225, type, basic_5567952622663070111et_nat: (heap_e7401611519738050253t_unit > nat) > (set_nat > nat) > produc3658429121746597890et_nat > nat).
% 129.49/25.35 thf(func_def_226, type, basic_1872990034501187214nt_int: (int > nat) > (int > nat) > product_prod_int_int > nat).
% 129.49/25.35 thf(func_def_227, type, basic_3966130907827183364et_nat: produc2732055786443039994et_nat > set_Pr4532377907799695533_nat_o).
% 129.49/25.35 thf(func_def_228, type, basic_8654530971723826568et_nat: produc3925858234332021118et_nat > set_Pr4532377907799695533_nat_o).
% 129.49/25.35 thf(func_def_229, type, basic_7833836874476830220et_nat: produc3658429121746597890et_nat > set_he8644174446132254637t_unit).
% 129.49/25.35 thf(func_def_230, type, basic_fsts_int_int: product_prod_int_int > set_int).
% 129.49/25.35 thf(func_def_231, type, basic_fsts_nat_nat: product_prod_nat_nat > set_nat).
% 129.49/25.35 thf(func_def_232, type, basic_2768200282441914438et_nat: produc2732055786443039994et_nat > set_Pr3286484037609594932et_nat).
% 129.49/25.35 thf(func_def_233, type, basic_8479756702517612234et_nat: produc3925858234332021118et_nat > set_Pr3948176798113811640et_nat).
% 129.49/25.35 thf(func_def_234, type, basic_2698842327037663054et_nat: produc3658429121746597890et_nat > set_set_nat).
% 129.49/25.35 thf(func_def_235, type, basic_snds_int_int: product_prod_int_int > set_int).
% 129.49/25.35 thf(func_def_236, type, basic_snds_nat_nat: product_prod_nat_nat > set_nat).
% 129.49/25.35 thf(func_def_237, type, binomial: nat > nat > nat).
% 129.49/25.35 thf(func_def_238, type, gbinom8545251970709558553nteger: code_integer > nat > code_integer).
% 129.49/25.35 thf(func_def_239, type, gbinom7368847122466276068atural: code_natural > nat > code_natural).
% 129.49/25.35 thf(func_def_240, type, gbinomial_int: int > nat > int).
% 129.49/25.35 thf(func_def_241, type, gbinomial_nat: nat > nat > nat).
% 129.49/25.35 thf(func_def_242, type, gbinomial_rat: rat > nat > rat).
% 129.49/25.35 thf(func_def_243, type, bit_and_int_rel: product_prod_int_int > product_prod_int_int > $o).
% 129.49/25.35 thf(func_def_244, type, bit_concat_bit: nat > int > int > int).
% 129.49/25.35 thf(func_def_245, type, bit_ri7919022796975470100ot_int: int > int).
% 129.49/25.35 thf(func_def_246, type, bit_ri631733984087533419it_int: nat > int > int).
% 129.49/25.35 thf(func_def_247, type, bit_se725231765392027082nd_int: int > int > int).
% 129.49/25.35 thf(func_def_248, type, bit_se727722235901077358nd_nat: nat > nat > nat).
% 129.49/25.35 thf(func_def_249, type, bit_se1409905431419307370or_int: int > int > int).
% 129.49/25.35 thf(func_def_250, type, bit_se1412395901928357646or_nat: nat > nat > nat).
% 129.49/25.35 thf(func_def_251, type, bit_se545348938243370406it_int: nat > int > int).
% 129.49/25.35 thf(func_def_252, type, bit_se547839408752420682it_nat: nat > nat > nat).
% 129.49/25.35 thf(func_def_253, type, bit_se7879613467334960850it_int: nat > int > int).
% 129.49/25.35 thf(func_def_254, type, bit_se2923211474154528505it_int: nat > int > int).
% 129.49/25.35 thf(func_def_255, type, bit_se4203085406695923979it_int: nat > int > int).
% 129.49/25.35 thf(func_def_256, type, bit_se6526347334894502574or_int: int > int > int).
% 129.49/25.35 thf(func_def_257, type, bit_se6528837805403552850or_nat: nat > nat > nat).
% 129.49/25.35 thf(func_def_258, type, bit_se1146084159140164899it_int: int > nat > $o).
% 129.49/25.35 thf(func_def_259, type, code_Neg: num > code_integer).
% 129.49/25.35 thf(func_def_260, type, code_Pos: num > code_integer).
% 129.49/25.35 thf(func_def_261, type, code_divmod_abs: code_integer > code_integer > produc8923325533196201883nteger).
% 129.49/25.35 thf(func_def_262, type, code_divmod_integer: code_integer > code_integer > produc8923325533196201883nteger).
% 129.49/25.35 thf(func_def_263, type, code_int_of_integer: code_integer > int).
% 129.49/25.35 thf(func_def_264, type, code_integer_of_int: int > code_integer).
% 129.49/25.35 thf(func_def_265, type, code_nat_of_integer: code_integer > nat).
% 129.49/25.35 thf(func_def_266, type, code_nat_of_natural: code_natural > nat).
% 129.49/25.35 thf(func_def_267, type, code_natural_of_nat: nat > code_natural).
% 129.49/25.35 thf(func_def_268, type, code_pcr_integer: int > code_integer > $o).
% 129.49/25.35 thf(func_def_269, type, code_pcr_natural: nat > code_natural > $o).
% 129.49/25.35 thf(func_def_270, type, comple4605987646318907561at_o_o: set_Pr6409761941060870310at_o_o > (produc3658429121746597890et_nat > $o) > $o).
% 129.49/25.35 thf(func_def_271, type, complete_Inf_Inf_o_o: set_o_o > $o > $o).
% 129.49/25.35 thf(func_def_272, type, comple4393433247856107946_int_o: set_int_o > int > $o).
% 129.49/25.35 thf(func_def_273, type, comple6214475593288795910_nat_o: set_nat_o > nat > $o).
% 129.49/25.35 thf(func_def_274, type, comple90455603224345172_nat_o: set_Pr4532377907799695533_nat_o > produc3658429121746597890et_nat > $o).
% 129.49/25.35 thf(func_def_275, type, comple2092403528524206185unit_o: set_Product_unit_o > product_unit > $o).
% 129.49/25.35 thf(func_def_276, type, complete_Inf_Inf_o: set_o > $o).
% 129.49/25.35 thf(func_def_277, type, comple6612786467464916249er_nat: set_filter_nat > filter_nat).
% 129.49/25.35 thf(func_def_278, type, complete_Inf_Inf_int: set_int > int).
% 129.49/25.35 thf(func_def_279, type, complete_Inf_Inf_nat: set_nat > nat).
% 129.49/25.35 thf(func_def_280, type, comple2584293577114468500t_unit: set_Product_unit > product_unit).
% 129.49/25.35 thf(func_def_281, type, comple79274577958421812_nat_o: set_se1353816476878259085_nat_o > set_Pr4532377907799695533_nat_o).
% 129.49/25.35 thf(func_def_282, type, comple3063163877087187839_set_o: set_set_o > set_o).
% 129.49/25.35 thf(func_def_283, type, comple3628384868704368283et_int: set_set_int > set_int).
% 129.49/25.35 thf(func_def_284, type, comple7806235888213564991et_nat: set_set_nat > set_nat).
% 129.49/25.35 thf(func_def_285, type, comple1930127330929219688at_nat: set_se7855581050983116737at_nat > set_Pr1261947904930325089at_nat).
% 129.49/25.35 thf(func_def_286, type, comple7216939928495492468t_unit: set_set_Product_unit > set_Product_unit).
% 129.49/25.35 thf(func_def_287, type, comple3590721164184202819at_o_o: set_Pr6409761941060870310at_o_o > (produc3658429121746597890et_nat > $o) > $o).
% 129.49/25.35 thf(func_def_288, type, complete_Sup_Sup_o_o: set_o_o > $o > $o).
% 129.49/25.35 thf(func_def_289, type, comple6496622788309502864_int_o: set_int_o > int > $o).
% 129.49/25.35 thf(func_def_290, type, comple8317665133742190828_nat_o: set_nat_o > nat > $o).
% 129.49/25.35 thf(func_def_291, type, comple4225350640907012154_nat_o: set_Pr4532377907799695533_nat_o > produc3658429121746597890et_nat > $o).
% 129.49/25.35 thf(func_def_292, type, comple4773641153059738115unit_o: set_Product_unit_o > product_unit > $o).
% 129.49/25.35 thf(func_def_293, type, complete_Sup_Sup_o: set_o > $o).
% 129.49/25.35 thf(func_def_294, type, comple2454491259644266675er_nat: set_filter_nat > filter_nat).
% 129.49/25.35 thf(func_def_295, type, complete_Sup_Sup_int: set_int > int).
% 129.49/25.35 thf(func_def_296, type, complete_Sup_Sup_nat: set_nat > nat).
% 129.49/25.35 thf(func_def_297, type, comple4687483117567863418t_unit: set_Product_unit > product_unit).
% 129.49/25.35 thf(func_def_298, type, comple3382087578465172250_nat_o: set_se1353816476878259085_nat_o > set_Pr4532377907799695533_nat_o).
% 129.49/25.35 thf(func_def_299, type, comple90263536869209701_set_o: set_set_o > set_o).
% 129.49/25.35 thf(func_def_300, type, comple739944243200306918nteger: set_set_Code_integer > set_Code_integer).
% 129.49/25.35 thf(func_def_301, type, comple3221217463730067765et_int: set_set_int > set_int).
% 129.49/25.35 thf(func_def_302, type, comple7399068483239264473et_nat: set_set_nat > set_nat).
% 129.49/25.35 thf(func_def_303, type, comple8164645552537203636_nat_o: set_se7663487083892625191_nat_o > set_Pr2161125870931222855_nat_o).
% 129.49/25.35 thf(func_def_304, type, comple2567785453016644468od_o_o: set_se491199571785101031od_o_o > set_Product_prod_o_o).
% 129.49/25.35 thf(func_def_305, type, comple5382143125604098054nt_int: set_se6260736226359567993nt_int > set_Pr958786334691620121nt_int).
% 129.49/25.35 thf(func_def_306, type, comple5685304695842803022at_nat: set_se7855581050983116737at_nat > set_Pr1261947904930325089at_nat).
% 129.49/25.35 thf(func_def_307, type, comple2798140142350441396t_unit: set_se8994758365886191399t_unit > set_Pr5094982260447487303t_unit).
% 129.49/25.35 thf(func_def_308, type, comple3349607820504768346t_unit: set_set_Product_unit > set_Product_unit).
% 129.49/25.35 thf(func_def_309, type, condit4352657709669162033_nat_o: set_Pr4532377907799695533_nat_o > $o).
% 129.49/25.35 thf(func_def_310, type, condit5488710616941104124bove_o: set_o > $o).
% 129.49/25.35 thf(func_def_311, type, condit4029664723990530684er_nat: set_filter_nat > $o).
% 129.49/25.35 thf(func_def_312, type, condit2212336002400062152ve_int: set_int > $o).
% 129.49/25.35 thf(func_def_313, type, condit2214826472909112428ve_nat: set_nat > $o).
% 129.49/25.35 thf(func_def_314, type, condit2080475194861589617t_unit: set_Product_unit > $o).
% 129.49/25.35 thf(func_def_315, type, condit5477540289124974626et_nat: set_set_nat > $o).
% 129.49/25.35 thf(func_def_316, type, condit8399017556247434269_nat_o: set_Pr4532377907799695533_nat_o > $o).
% 129.49/25.35 thf(func_def_317, type, condit5413489452508810728elow_o: set_o > $o).
% 129.49/25.35 thf(func_def_318, type, condit3133331395780078992er_nat: set_filter_nat > $o).
% 129.49/25.35 thf(func_def_319, type, condit1735850657277959132ow_int: set_int > $o).
% 129.49/25.35 thf(func_def_320, type, condit1738341127787009408ow_nat: set_nat > $o).
% 129.49/25.35 thf(func_def_321, type, condit9161856309968323677t_unit: set_Product_unit > $o).
% 129.49/25.35 thf(func_def_322, type, condit68592940725977398et_nat: set_set_nat > $o).
% 129.49/25.35 thf(func_def_323, type, eucl_rel_int: int > int > product_prod_int_int > $o).
% 129.49/25.35 thf(func_def_324, type, unique5706413561485394159nteger: produc8923325533196201883nteger > $o).
% 129.49/25.35 thf(func_def_325, type, unique6319869463603278526ux_int: product_prod_int_int > $o).
% 129.49/25.35 thf(func_def_326, type, unique6322359934112328802ux_nat: product_prod_nat_nat > $o).
% 129.49/25.35 thf(func_def_327, type, unique4921790084139445826nteger: num > produc8923325533196201883nteger > produc8923325533196201883nteger).
% 129.49/25.35 thf(func_def_328, type, unique5024387138958732305ep_int: num > product_prod_int_int > product_prod_int_int).
% 129.49/25.35 thf(func_def_329, type, unique5026877609467782581ep_nat: num > product_prod_nat_nat > product_prod_nat_nat).
% 129.49/25.35 thf(func_def_330, type, equiv_6726761681920839555_nat_o: set_Pr4532377907799695533_nat_o > set_Pr2161125870931222855_nat_o > $o).
% 129.49/25.35 thf(func_def_331, type, equiv_equiv_o: set_o > set_Product_prod_o_o > $o).
% 129.49/25.35 thf(func_def_332, type, equiv_equiv_int: set_int > set_Pr958786334691620121nt_int > $o).
% 129.49/25.35 thf(func_def_333, type, equiv_equiv_nat: set_nat > set_Pr1261947904930325089at_nat > $o).
% 129.49/25.35 thf(func_def_334, type, equiv_4189356295871601603t_unit: set_Product_unit > set_Pr5094982260447487303t_unit > $o).
% 129.49/25.35 thf(func_def_335, type, equiv_3235788788526563786_nat_o: set_Pr2161125870931222855_nat_o > (produc3658429121746597890et_nat > $o) > set_Pr4532377907799695533_nat_o).
% 129.49/25.35 thf(func_def_336, type, equiv_proj_o_o: set_Product_prod_o_o > $o > set_o).
% 129.49/25.35 thf(func_def_337, type, equiv_proj_int_int: set_Pr958786334691620121nt_int > int > set_int).
% 129.49/25.35 thf(func_def_338, type, equiv_proj_nat_nat: set_Pr1261947904930325089at_nat > nat > set_nat).
% 129.49/25.35 thf(func_def_339, type, equiv_5763582929720877514t_unit: set_Pr5094982260447487303t_unit > product_unit > set_Product_unit).
% 129.49/25.35 thf(func_def_340, type, equiv_7899955228204526484_nat_o: set_Pr4532377907799695533_nat_o > set_Pr2161125870931222855_nat_o > set_se1353816476878259085_nat_o).
% 129.49/25.35 thf(func_def_341, type, equiv_quotient_o: set_o > set_Product_prod_o_o > set_set_o).
% 129.49/25.35 thf(func_def_342, type, equiv_quotient_int: set_int > set_Pr958786334691620121nt_int > set_set_int).
% 129.49/25.35 thf(func_def_343, type, equiv_quotient_nat: set_nat > set_Pr1261947904930325089at_nat > set_set_nat).
% 129.49/25.35 thf(func_def_344, type, equiv_5172566880711946708t_unit: set_Product_unit > set_Pr5094982260447487303t_unit > set_set_Product_unit).
% 129.49/25.35 thf(func_def_345, type, comm_s8582702949713902594nteger: code_integer > nat > code_integer).
% 129.49/25.35 thf(func_def_346, type, comm_s4660882817536571857er_int: int > nat > int).
% 129.49/25.35 thf(func_def_347, type, comm_s4028243227959126397er_rat: rat > nat > rat).
% 129.49/25.35 thf(func_def_348, type, semiri1408675320244567234ct_nat: nat > nat).
% 129.49/25.35 thf(func_def_349, type, inverse_inverse_rat: rat > rat).
% 129.49/25.35 thf(func_def_350, type, at_top_nat: filter_nat).
% 129.49/25.35 thf(func_def_351, type, cofinite_nat: filter_nat).
% 129.49/25.35 thf(func_def_352, type, eventually_nat: (nat > $o) > filter_nat > $o).
% 129.49/25.35 thf(func_def_353, type, eventu1038000079068216329at_nat: (product_prod_nat_nat > $o) > filter1242075044329608583at_nat > $o).
% 129.49/25.35 thf(func_def_354, type, filterlim_nat_nat: (nat > nat) > filter_nat > filter_nat > $o).
% 129.49/25.35 thf(func_def_355, type, filter3212408913953519116et_nat: (nat > set_nat) > filter_set_nat > filter_nat > $o).
% 129.49/25.35 thf(func_def_356, type, finite3254316476582989077op_nat: set_nat > filter_set_nat).
% 129.49/25.35 thf(func_def_357, type, frequently_nat: (nat > $o) > filter_nat > $o).
% 129.49/25.35 thf(func_def_358, type, prod_filter_nat_nat: filter_nat > filter_nat > filter1242075044329608583at_nat).
% 129.49/25.35 thf(func_def_359, type, finite1363419556375932405_nat_o: set_Pr4532377907799695533_nat_o > nat).
% 129.49/25.35 thf(func_def_360, type, finite_card_o: set_o > nat).
% 129.49/25.35 thf(func_def_361, type, finite4902975817058060853nteger: set_Code_integer > nat).
% 129.49/25.35 thf(func_def_362, type, finite_card_int: set_int > nat).
% 129.49/25.35 thf(func_def_363, type, finite_card_nat: set_nat > nat).
% 129.49/25.35 thf(func_def_364, type, finite410649719033368117t_unit: set_Product_unit > nat).
% 129.49/25.35 thf(func_def_365, type, finite4731484169140532437_nat_o: set_se1353816476878259085_nat_o > nat).
% 129.49/25.35 thf(func_def_366, type, finite_card_set_o: set_set_o > nat).
% 129.49/25.35 thf(func_def_367, type, finite1340570857867686763nteger: set_set_Code_integer > nat).
% 129.49/25.35 thf(func_def_368, type, finite_card_set_int: set_set_int > nat).
% 129.49/25.35 thf(func_def_369, type, finite_card_set_nat: set_set_nat > nat).
% 129.49/25.35 thf(func_def_370, type, finite22775419188893461t_unit: set_set_Product_unit > nat).
% 129.49/25.35 thf(func_def_371, type, finite3252695134891459830_nat_o: set_Pr4532377907799695533_nat_o > $o).
% 129.49/25.35 thf(func_def_372, type, finite_finite_o: set_o > $o).
% 129.49/25.35 thf(func_def_373, type, finite_finite_assn: set_assn > $o).
% 129.49/25.35 thf(func_def_374, type, finite6017078050557962740nteger: set_Code_integer > $o).
% 129.49/25.35 thf(func_def_375, type, finite2119507909894593271er_nat: set_filter_nat > $o).
% 129.49/25.35 thf(func_def_376, type, finite_finite_int: set_int > $o).
% 129.49/25.35 thf(func_def_377, type, finite_finite_nat: set_nat > $o).
% 129.49/25.35 thf(func_def_378, type, finite2154694888287295812nteger: set_Pr4811707699266497531nteger > $o).
% 129.49/25.35 thf(func_def_379, type, finite4996374594527322935er_nat: set_Pr8986988648414616686er_nat > $o).
% 129.49/25.35 thf(func_def_380, type, finite2998713641127702882nt_int: set_Pr958786334691620121nt_int > $o).
% 129.49/25.35 thf(func_def_381, type, finite1154042800025562295nteger: set_Pr2406931829603198446nteger > $o).
% 129.49/25.35 thf(func_def_382, type, finite6177210948735845034at_nat: set_Pr1261947904930325089at_nat > $o).
% 129.49/25.35 thf(func_def_383, type, finite4290736615968046902t_unit: set_Product_unit > $o).
% 129.49/25.35 thf(func_def_384, type, finite_finite_rat: set_rat > $o).
% 129.49/25.35 thf(func_def_385, type, finite6069017635265841110_nat_o: set_se1353816476878259085_nat_o > $o).
% 129.49/25.35 thf(func_def_386, type, finite_finite_set_o: set_set_o > $o).
% 129.49/25.35 thf(func_def_387, type, finite6931041176100689706nteger: set_set_Code_integer > $o).
% 129.49/25.35 thf(func_def_388, type, finite6197958912794628473et_int: set_set_int > $o).
% 129.49/25.35 thf(func_def_389, type, finite1152437895449049373et_nat: set_set_nat > $o).
% 129.49/25.35 thf(func_def_390, type, finite1772178364199683094t_unit: set_set_Product_unit > $o).
% 129.49/25.35 thf(func_def_391, type, finite_fold_o_o: ($o > $o > $o) > $o > set_o > $o).
% 129.49/25.35 thf(func_def_392, type, finite_fold_o_set_o: ($o > set_o > set_o) > set_o > set_o > set_o).
% 129.49/25.35 thf(func_def_393, type, finite1440822074540989594n_assn: (assn > assn > assn) > assn > set_assn > assn).
% 129.49/25.35 thf(func_def_394, type, finite6454954857057555716nteger: (code_integer > code_integer > code_integer) > code_integer > set_Code_integer > code_integer).
% 129.49/25.35 thf(func_def_395, type, finite2281032298429728806t_unit: (code_integer > product_unit > product_unit) > product_unit > set_Code_integer > product_unit).
% 129.49/25.35 thf(func_def_396, type, finite1928688952044918289_set_o: (code_integer > set_o > set_o) > set_o > set_Code_integer > set_o).
% 129.49/25.35 thf(func_def_397, type, finite7486036041278456378nteger: (code_integer > set_Code_integer > set_Code_integer) > set_Code_integer > set_Code_integer > set_Code_integer).
% 129.49/25.35 thf(func_def_398, type, finite6635835719294221449et_int: (code_integer > set_int > set_int) > set_int > set_Code_integer > set_int).
% 129.49/25.35 thf(func_def_399, type, finite1590314701948642349et_nat: (code_integer > set_nat > set_nat) > set_nat > set_Code_integer > set_nat).
% 129.49/25.35 thf(func_def_400, type, finite3687622995246107910t_unit: (code_integer > set_Product_unit > set_Product_unit) > set_Product_unit > set_Code_integer > set_Product_unit).
% 129.49/25.35 thf(func_def_401, type, finite_fold_int_int: (int > int > int) > int > set_int > int).
% 129.49/25.35 thf(func_def_402, type, finite1723285575846220376et_int: (int > set_int > set_int) > set_int > set_int > set_int).
% 129.49/25.35 thf(func_def_403, type, finite_fold_nat_nat: (nat > nat > nat) > nat > set_nat > nat).
% 129.49/25.35 thf(func_def_404, type, finite6602901031542678131t_unit: (nat > product_unit > product_unit) > product_unit > set_nat > product_unit).
% 129.49/25.35 thf(func_def_405, type, finite3217087857726763998_set_o: (nat > set_o > set_o) > set_o > set_nat > set_o).
% 129.49/25.35 thf(func_def_406, type, finite1351632015609375740et_int: (nat > set_int > set_int) > set_int > set_nat > set_int).
% 129.49/25.35 thf(func_def_407, type, finite5529483035118572448et_nat: (nat > set_nat > set_nat) > set_nat > set_nat > set_nat).
% 129.49/25.35 thf(func_def_408, type, finite9145781422887121747t_unit: (nat > set_Product_unit > set_Product_unit) > set_Product_unit > set_nat > set_Product_unit).
% 129.49/25.35 thf(func_def_409, type, finite8834431295148555784t_unit: (product_unit > product_unit > product_unit) > product_unit > set_Product_unit > product_unit).
% 129.49/25.35 thf(func_def_410, type, finite2740360166580943592t_unit: (product_unit > set_Product_unit > set_Product_unit) > set_Product_unit > set_Product_unit > set_Product_unit).
% 129.49/25.35 thf(func_def_411, type, finite8456856254070787592_nat_o: (set_Pr4532377907799695533_nat_o > set_Pr4532377907799695533_nat_o > set_Pr4532377907799695533_nat_o) > set_Pr4532377907799695533_nat_o > set_se1353816476878259085_nat_o > set_Pr4532377907799695533_nat_o).
% 129.49/25.35 thf(func_def_412, type, finite4337638375924247368_set_o: (set_o > set_o > set_o) > set_o > set_set_o > set_o).
% 129.49/25.35 thf(func_def_413, type, finite2508883933289561102et_int: (set_int > set_int > set_int) > set_int > set_set_int > set_int).
% 129.49/25.35 thf(func_def_414, type, finite677925301803182934et_nat: (set_nat > set_nat > set_nat) > set_nat > set_set_nat > set_nat).
% 129.49/25.35 thf(func_def_415, type, finite4243028863286625800t_unit: (set_Product_unit > set_Product_unit > set_Product_unit) > set_Product_unit > set_set_Product_unit > set_Product_unit).
% 129.49/25.35 thf(func_def_416, type, comp_C8797469213163452608nteger: ((code_integer > code_integer) > produc8923325533196201883nteger > produc8923325533196201883nteger) > (code_integer > code_integer > code_integer) > code_integer > produc8923325533196201883nteger > produc8923325533196201883nteger).
% 129.49/25.35 thf(func_def_417, type, comp_C5076291335850163704nteger: ((code_integer > code_integer) > code_integer) > (code_integer > code_integer > code_integer) > code_integer > code_integer).
% 129.49/25.35 thf(func_def_418, type, comp_P4885729792186999034nteger: ((produc8923325533196201883nteger > produc8923325533196201883nteger) > set_Code_integer) > (code_integer > produc8923325533196201883nteger > produc8923325533196201883nteger) > code_integer > set_Code_integer).
% 129.49/25.35 thf(func_def_419, type, comp_P8262219321096090349nteger: ((produc8923325533196201883nteger > produc8923325533196201883nteger) > set_nat) > (code_integer > produc8923325533196201883nteger > produc8923325533196201883nteger) > code_integer > set_nat).
% 129.49/25.35 thf(func_def_420, type, comp_s518380907012905326o_assn: ((set_o > assn) > set_set_o > assn) > (($o > assn) > set_o > assn) > ($o > assn) > set_set_o > assn).
% 129.49/25.35 thf(func_def_421, type, comp_s6084864194472624150_o_nat: ((set_o > nat) > set_set_o > nat) > (($o > nat) > set_o > nat) > ($o > nat) > set_set_o > nat).
% 129.49/25.35 thf(func_def_422, type, comp_s3531641609786562035r_assn: ((set_Code_integer > assn) > set_set_Code_integer > assn) > ((code_integer > assn) > set_Code_integer > assn) > (code_integer > assn) > set_set_Code_integer > assn).
% 129.49/25.35 thf(func_def_423, type, comp_s8759123648413916139er_nat: ((set_Code_integer > nat) > set_set_Code_integer > nat) > ((code_integer > nat) > set_Code_integer > nat) > (code_integer > nat) > set_set_Code_integer > nat).
% 129.49/25.35 thf(func_def_424, type, comp_s4595925057662634336t_assn: ((set_int > assn) > set_set_int > assn) > ((int > assn) > set_int > assn) > (int > assn) > set_set_int > assn).
% 129.49/25.35 thf(func_def_425, type, comp_s9098231955081793132nt_int: ((set_int > int) > set_set_int > int) > ((int > int) > set_int > int) > (int > int) > set_set_int > int).
% 129.49/25.35 thf(func_def_426, type, comp_s3237642481825588056nt_nat: ((set_int > nat) > set_set_int > nat) > ((int > nat) > set_int > nat) > (int > nat) > set_set_int > nat).
% 129.49/25.35 thf(func_def_427, type, comp_s2451231738238893644t_assn: ((set_nat > assn) > set_set_nat > assn) > ((nat > assn) > set_nat > assn) > (nat > assn) > set_set_nat > assn).
% 129.49/25.35 thf(func_def_428, type, comp_s5456809700045850692at_nat: ((set_nat > nat) > set_set_nat > nat) > ((nat > nat) > set_nat > nat) > (nat > nat) > set_set_nat > nat).
% 129.49/25.35 thf(func_def_429, type, comp_s8247530669214396195t_assn: ((set_Product_unit > assn) > set_set_Product_unit > assn) > ((product_unit > assn) > set_Product_unit > assn) > (product_unit > assn) > set_set_Product_unit > assn).
% 129.49/25.35 thf(func_def_430, type, comp_s8928436226432208331it_nat: ((set_Product_unit > nat) > set_set_Product_unit > nat) > ((product_unit > nat) > set_Product_unit > nat) > (product_unit > nat) > set_set_Product_unit > nat).
% 129.49/25.35 thf(func_def_431, type, comp_o8922910837708301817nteger: ($o > produc8923325533196201883nteger > produc8923325533196201883nteger) > (code_integer > $o) > code_integer > produc8923325533196201883nteger > produc8923325533196201883nteger).
% 129.49/25.35 thf(func_def_432, type, comp_o6961622296474427249t_unit: ($o > produc8923325533196201883nteger > produc8923325533196201883nteger) > (product_unit > $o) > product_unit > produc8923325533196201883nteger > produc8923325533196201883nteger).
% 129.49/25.35 thf(func_def_433, type, comp_o1085144040659434033nteger: ($o > code_integer) > (code_integer > $o) > code_integer > code_integer).
% 129.49/25.35 thf(func_def_434, type, comp_o780160699970281785t_unit: ($o > code_integer) > (product_unit > $o) > product_unit > code_integer).
% 129.49/25.35 thf(func_def_435, type, comp_o2643362028263920430t_unit: ($o > code_natural) > (product_unit > $o) > product_unit > code_natural).
% 129.49/25.35 thf(func_def_436, type, comp_o_int_int: ($o > int) > (int > $o) > int > int).
% 129.49/25.35 thf(func_def_437, type, comp_o3916225744453444672unit_o: ($o > product_unit) > ($o > $o) > $o > product_unit).
% 129.49/25.35 thf(func_def_438, type, comp_o6635994240230588853nteger: ($o > product_unit) > (code_integer > $o) > code_integer > product_unit).
% 129.49/25.35 thf(func_def_439, type, comp_o7049022088938261380it_int: ($o > product_unit) > (int > $o) > int > product_unit).
% 129.49/25.35 thf(func_def_440, type, comp_o7051512559447311656it_nat: ($o > product_unit) > (nat > $o) > nat > product_unit).
% 129.49/25.35 thf(func_def_441, type, comp_o7724735231917782709t_unit: ($o > product_unit) > (product_unit > $o) > product_unit > product_unit).
% 129.49/25.35 thf(func_def_442, type, comp_o6895741221744486154t_unit: ($o > set_o) > (product_unit > $o) > product_unit > set_o).
% 129.49/25.35 thf(func_def_443, type, comp_o7929101295636871983nteger: ($o > set_Code_integer) > ((produc8923325533196201883nteger > produc8923325533196201883nteger) > $o) > (produc8923325533196201883nteger > produc8923325533196201883nteger) > set_Code_integer).
% 129.49/25.35 thf(func_def_444, type, comp_o4291842030903409511nteger: ($o > set_Code_integer) > (code_integer > $o) > code_integer > set_Code_integer).
% 129.49/25.35 thf(func_def_445, type, comp_o1521574261646630388t_unit: ($o > set_int) > (product_unit > $o) > product_unit > set_int).
% 129.49/25.35 thf(func_def_446, type, comp_o1289984117834210338nteger: ($o > set_nat) > ((produc8923325533196201883nteger > produc8923325533196201883nteger) > $o) > (produc8923325533196201883nteger > produc8923325533196201883nteger) > set_nat).
% 129.49/25.35 thf(func_def_447, type, comp_o172754640513974618nteger: ($o > set_nat) > (code_integer > $o) > code_integer > set_nat).
% 129.49/25.35 thf(func_def_448, type, comp_o9190160103024335696t_unit: ($o > set_nat) > (product_unit > $o) > product_unit > set_nat).
% 129.49/25.35 thf(func_def_449, type, comp_o2086126948152544725t_unit: ($o > set_Product_unit) > (product_unit > $o) > product_unit > set_Product_unit).
% 129.49/25.35 thf(func_def_450, type, comp_C2931745410781319416nteger: (code_integer > code_integer > code_integer) > (code_integer > code_integer) > code_integer > code_integer > code_integer).
% 129.49/25.35 thf(func_def_451, type, comp_C95226695241045696nteger: (code_integer > produc8923325533196201883nteger > produc8923325533196201883nteger) > ((code_integer > code_integer) > code_integer) > (code_integer > code_integer) > produc8923325533196201883nteger > produc8923325533196201883nteger).
% 129.49/25.35 thf(func_def_452, type, comp_C5733264393235242609eger_o: (code_integer > produc8923325533196201883nteger > produc8923325533196201883nteger) > ($o > code_integer) > $o > produc8923325533196201883nteger > produc8923325533196201883nteger).
% 129.49/25.35 thf(func_def_453, type, comp_C1593894019821074884nteger: (code_integer > produc8923325533196201883nteger > produc8923325533196201883nteger) > (code_integer > code_integer) > code_integer > produc8923325533196201883nteger > produc8923325533196201883nteger).
% 129.49/25.35 thf(func_def_454, type, comp_C834242911430131603er_int: (code_integer > produc8923325533196201883nteger > produc8923325533196201883nteger) > (int > code_integer) > int > produc8923325533196201883nteger > produc8923325533196201883nteger).
% 129.49/25.35 thf(func_def_455, type, comp_C836733381939181879er_nat: (code_integer > produc8923325533196201883nteger > produc8923325533196201883nteger) > (nat > code_integer) > nat > produc8923325533196201883nteger > produc8923325533196201883nteger).
% 129.49/25.35 thf(func_def_456, type, comp_C5854136495726678886t_unit: (code_integer > produc8923325533196201883nteger > produc8923325533196201883nteger) > (product_unit > code_integer) > product_unit > produc8923325533196201883nteger > produc8923325533196201883nteger).
% 129.49/25.35 thf(func_def_457, type, comp_C3026223900822616519nteger: (code_integer > assn) > (code_integer > code_integer) > code_integer > assn).
% 129.49/25.35 thf(func_def_458, type, comp_C8462447471145125178sn_nat: (code_integer > assn) > (nat > code_integer) > nat > assn).
% 129.49/25.35 thf(func_def_459, type, comp_C7449957260575251196nteger: (code_integer > code_integer) > (code_integer > code_integer) > code_integer > code_integer).
% 129.49/25.35 thf(func_def_460, type, comp_C6273552412331968711atural: (code_integer > code_integer) > (code_natural > code_integer) > code_natural > code_integer).
% 129.49/25.35 thf(func_def_461, type, comp_C6974050283906349679er_nat: (code_integer > code_integer) > (nat > code_integer) > nat > code_integer).
% 129.49/25.35 thf(func_def_462, type, comp_C1043215000170864188nteger: (code_integer > code_integer) > (produc8923325533196201883nteger > code_integer) > produc8923325533196201883nteger > code_integer).
% 129.49/25.35 thf(func_def_463, type, comp_C2290379119555921966t_unit: (code_integer > code_integer) > (product_unit > code_integer) > product_unit > code_integer).
% 129.49/25.35 thf(func_def_464, type, comp_C1891057325546811591nteger: (code_integer > code_natural) > (code_integer > code_integer) > code_integer > code_natural).
% 129.49/25.35 thf(func_def_465, type, comp_C714652477303529106atural: (code_integer > code_natural) > (code_natural > code_integer) > code_natural > code_natural).
% 129.49/25.35 thf(func_def_466, type, comp_C4153580447849560611t_unit: (code_integer > code_natural) > (product_unit > code_integer) > product_unit > code_natural).
% 129.49/25.35 thf(func_def_467, type, comp_C7610259292922237131nteger: (code_integer > int) > (code_integer > code_integer) > code_integer > int).
% 129.49/25.35 thf(func_def_468, type, comp_C5048910307564104766nt_nat: (code_integer > int) > (nat > code_integer) > nat > int).
% 129.49/25.35 thf(func_def_469, type, comp_C7238605732685392495nteger: (code_integer > nat) > (code_integer > code_integer) > code_integer > nat).
% 129.49/25.35 thf(func_def_470, type, comp_C8146212659816229034nteger: (code_integer > product_unit) > (code_integer > code_integer) > code_integer > product_unit).
% 129.49/25.35 thf(func_def_471, type, comp_C6969807811572946549atural: (code_integer > product_unit) > (code_natural > code_integer) > code_natural > product_unit).
% 129.49/25.35 thf(func_def_472, type, comp_C2408090164605745152t_unit: (code_integer > product_unit) > (product_unit > code_integer) > product_unit > product_unit).
% 129.49/25.35 thf(func_def_473, type, comp_C6344129587964912378nteger: (code_integer > set_Code_integer) > ((produc8923325533196201883nteger > produc8923325533196201883nteger) > code_integer) > (produc8923325533196201883nteger > produc8923325533196201883nteger) > set_Code_integer).
% 129.49/25.35 thf(func_def_474, type, comp_C8186931911283765298nteger: (code_integer > set_Code_integer) > (code_integer > code_integer) > code_integer > set_Code_integer).
% 129.49/25.35 thf(func_def_475, type, comp_C3184339336801759213nteger: (code_integer > set_nat) > ((produc8923325533196201883nteger > produc8923325533196201883nteger) > code_integer) > (produc8923325533196201883nteger > produc8923325533196201883nteger) > set_nat).
% 129.49/25.35 thf(func_def_476, type, comp_C6537567860429791781nteger: (code_integer > set_nat) > (code_integer > code_integer) > code_integer > set_nat).
% 129.49/25.35 thf(func_def_477, type, comp_C456341964420721295nteger: (code_natural > produc8923325533196201883nteger > produc8923325533196201883nteger) > (code_integer > code_natural) > code_integer > produc8923325533196201883nteger > produc8923325533196201883nteger).
% 129.49/25.35 thf(func_def_478, type, comp_i2874689474467862163nteger: (int > produc8923325533196201883nteger > produc8923325533196201883nteger) > (code_integer > int) > code_integer > produc8923325533196201883nteger > produc8923325533196201883nteger).
% 129.49/25.35 thf(func_def_479, type, comp_int_o_o: (int > $o) > ($o > int) > $o > $o).
% 129.49/25.35 thf(func_def_480, type, comp_int_o_int: (int > $o) > (int > int) > int > $o).
% 129.49/25.35 thf(func_def_481, type, comp_int_o_nat: (int > $o) > (nat > int) > nat > $o).
% 129.49/25.35 thf(func_def_482, type, comp_i5286552343147518890eger_o: (int > code_integer) > ($o > int) > $o > code_integer).
% 129.49/25.35 thf(func_def_483, type, comp_i6014793192326455499nteger: (int > code_integer) > (code_integer > int) > code_integer > code_integer).
% 129.49/25.35 thf(func_def_484, type, comp_i1585864551200866970er_int: (int > code_integer) > (int > int) > int > code_integer).
% 129.49/25.35 thf(func_def_485, type, comp_i1588355021709917246er_nat: (int > code_integer) > (nat > int) > nat > code_integer).
% 129.49/25.35 thf(func_def_486, type, comp_i455893257298015894nteger: (int > code_natural) > (code_integer > int) > code_integer > code_natural).
% 129.49/25.35 thf(func_def_487, type, comp_int_int_o: (int > int) > ($o > int) > $o > int).
% 129.49/25.35 thf(func_def_488, type, comp_int_int_int: (int > int) > (int > int) > int > int).
% 129.49/25.35 thf(func_def_489, type, comp_int_int_nat: (int > int) > (nat > int) > nat > int).
% 129.49/25.35 thf(func_def_490, type, comp_i4330622015127193916nt_int: (int > int) > (product_prod_int_int > int) > product_prod_int_int > int).
% 129.49/25.35 thf(func_def_491, type, comp_int_nat_o: (int > nat) > ($o > int) > $o > nat).
% 129.49/25.35 thf(func_def_492, type, comp_int_nat_int: (int > nat) > (int > int) > int > nat).
% 129.49/25.35 thf(func_def_493, type, comp_int_nat_nat: (int > nat) > (nat > int) > nat > nat).
% 129.49/25.35 thf(func_def_494, type, comp_i8439899303502915483nteger: (int > product_unit) > (code_integer > int) > code_integer > product_unit).
% 129.49/25.35 thf(func_def_495, type, comp_i5479505385629679622nteger: (int > set_o) > (code_integer > int) > code_integer > set_o).
% 129.49/25.35 thf(func_def_496, type, comp_i1887841406321795152nteger: (int > set_int) > (code_integer > int) > code_integer > set_int).
% 129.49/25.35 thf(func_def_497, type, comp_i5103216804979976198nt_int: (int > set_int) > (product_prod_int_int > int) > product_prod_int_int > set_int).
% 129.49/25.35 thf(func_def_498, type, comp_i5102403792180996084nteger: (int > set_nat) > (code_integer > int) > code_integer > set_nat).
% 129.49/25.35 thf(func_def_499, type, comp_i6200299785623974754nt_int: (int > set_nat) > (product_prod_int_int > int) > product_prod_int_int > set_nat).
% 129.49/25.35 thf(func_def_500, type, comp_i3319814986458681467nteger: (int > set_Product_unit) > (code_integer > int) > code_integer > set_Product_unit).
% 129.49/25.35 thf(func_def_501, type, comp_nat_nat_nat_nat: (nat > nat > nat) > (nat > nat) > nat > nat > nat).
% 129.49/25.35 thf(func_def_502, type, comp_n2033352843805742135nteger: (nat > produc8923325533196201883nteger > produc8923325533196201883nteger) > (code_integer > nat) > code_integer > produc8923325533196201883nteger > produc8923325533196201883nteger).
% 129.49/25.35 thf(func_def_503, type, comp_n856947995562459650atural: (nat > produc8923325533196201883nteger > produc8923325533196201883nteger) > (code_natural > nat) > code_natural > produc8923325533196201883nteger > produc8923325533196201883nteger).
% 129.49/25.35 thf(func_def_504, type, comp_nat_assn_nat: (nat > assn) > (nat > nat) > nat > assn).
% 129.49/25.35 thf(func_def_505, type, comp_n6876073255676942959nteger: (nat > code_integer) > (code_integer > nat) > code_integer > code_integer).
% 129.49/25.35 thf(func_def_506, type, comp_n5699668407433660474atural: (nat > code_integer) > (code_natural > nat) > code_natural > code_integer).
% 129.49/25.35 thf(func_def_507, type, comp_n3898172953802868194er_nat: (nat > code_integer) > (nat > nat) > nat > code_integer).
% 129.49/25.35 thf(func_def_508, type, comp_n2857367870785791611ural_o: (nat > code_natural) > ($o > nat) > $o > code_natural).
% 129.49/25.35 thf(func_def_509, type, comp_n1317173320648503354nteger: (nat > code_natural) > (code_integer > nat) > code_integer > code_natural).
% 129.49/25.35 thf(func_def_510, type, comp_n140768472405220869atural: (nat > code_natural) > (code_natural > nat) > code_natural > code_natural).
% 129.49/25.35 thf(func_def_511, type, comp_n9160511493981425161al_int: (nat > code_natural) > (int > nat) > int > code_natural).
% 129.49/25.35 thf(func_def_512, type, comp_n9163001964490475437al_nat: (nat > code_natural) > (nat > nat) > nat > code_natural).
% 129.49/25.35 thf(func_def_513, type, comp_nat_int_nat: (nat > int) > (nat > nat) > nat > int).
% 129.49/25.35 thf(func_def_514, type, comp_n8360577435903631554atural: (nat > product_unit) > (code_natural > nat) > code_natural > product_unit).
% 129.49/25.35 thf(func_def_515, type, comp_n1572716621429000127atural: (nat > set_int) > (code_natural > nat) > code_natural > set_int).
% 129.49/25.35 thf(func_def_516, type, comp_n3623798160786426464nteger: (nat > set_nat) > ((produc8923325533196201883nteger > produc8923325533196201883nteger) > nat) > (produc8923325533196201883nteger > produc8923325533196201883nteger) > set_nat).
% 129.49/25.35 thf(func_def_517, type, comp_n5963683855531483544nteger: (nat > set_nat) > (code_integer > nat) > code_integer > set_nat).
% 129.49/25.35 thf(func_def_518, type, comp_n4787279007288201059atural: (nat > set_nat) > (code_natural > nat) > code_natural > set_nat).
% 129.49/25.35 thf(func_def_519, type, comp_P5544053077415136032nteger: (produc723880050079548563nteger > produc101387210554329899nteger) > (produc8923325533196201883nteger > produc723880050079548563nteger) > produc8923325533196201883nteger > produc101387210554329899nteger).
% 129.49/25.35 thf(func_def_520, type, comp_P1414880308642389412nteger: (produc2221115339011613079nteger > produc101387210554329899nteger) > (produc8923325533196201883nteger > produc2221115339011613079nteger) > produc8923325533196201883nteger > produc101387210554329899nteger).
% 129.49/25.35 thf(func_def_521, type, comp_P6312383820422015140nteger: (produc1397831737260518551nteger > produc101387210554329899nteger) > (produc8923325533196201883nteger > produc1397831737260518551nteger) > produc8923325533196201883nteger > produc101387210554329899nteger).
% 129.49/25.35 thf(func_def_522, type, comp_P7111796090210262324nteger: (produc8923325533196201883nteger > code_integer) > (code_integer > produc8923325533196201883nteger) > code_integer > code_integer).
% 129.49/25.35 thf(func_def_523, type, comp_P3428181540658402820nteger: (produc8923325533196201883nteger > code_integer) > (produc8923325533196201883nteger > produc8923325533196201883nteger) > produc8923325533196201883nteger > code_integer).
% 129.49/25.35 thf(func_def_524, type, comp_P659088277599099894t_unit: (produc8923325533196201883nteger > code_integer) > (product_unit > produc8923325533196201883nteger) > product_unit > code_integer).
% 129.49/25.35 thf(func_def_525, type, comp_P234064951390806312nteger: (produc8923325533196201883nteger > produc101387210554329899nteger) > (produc8923325533196201883nteger > produc8923325533196201883nteger) > produc8923325533196201883nteger > produc101387210554329899nteger).
% 129.49/25.35 thf(func_def_526, type, comp_P7101464685127059239nteger: (produc6433947425880249742er_nat > code_integer) > (code_integer > produc6433947425880249742er_nat) > code_integer > code_integer).
% 129.49/25.35 thf(func_def_527, type, comp_P7581484221039074243t_unit: (produc6433947425880249742er_nat > code_integer) > (product_unit > produc6433947425880249742er_nat) > product_unit > code_integer).
% 129.49/25.35 thf(func_def_528, type, comp_P5964733648194011802nteger: (produc6433947425880249742er_nat > nat) > (code_integer > produc6433947425880249742er_nat) > code_integer > nat).
% 129.49/25.35 thf(func_def_529, type, comp_P6821003776282389008t_unit: (produc6433947425880249742er_nat > nat) > (product_unit > produc6433947425880249742er_nat) > product_unit > nat).
% 129.49/25.35 thf(func_def_530, type, comp_P2930521774885306735et_nat: (produc3658429121746597890et_nat > set_nat) > (produc3658429121746597890et_nat > produc3658429121746597890et_nat) > produc3658429121746597890et_nat > set_nat).
% 129.49/25.35 thf(func_def_531, type, comp_P8606889026643445772nt_int: (produc3658429121746597890et_nat > set_nat) > (product_prod_int_int > produc3658429121746597890et_nat) > product_prod_int_int > set_nat).
% 129.49/25.35 thf(func_def_532, type, comp_P8308861049717174982et_nat: (product_prod_int_int > int) > (produc3658429121746597890et_nat > product_prod_int_int) > produc3658429121746597890et_nat > int).
% 129.49/25.35 thf(func_def_533, type, comp_P5356232565778667509nt_int: (product_prod_int_int > int) > (product_prod_int_int > product_prod_int_int) > product_prod_int_int > int).
% 129.49/25.35 thf(func_def_534, type, comp_P6796073005956588617t_unit: (product_prod_int_int > int) > (product_unit > product_prod_int_int) > product_unit > int).
% 129.49/25.35 thf(func_def_535, type, comp_P7177471617286032590eger_o: (produc2591615631378489102nteger > code_integer) > ($o > produc2591615631378489102nteger) > $o > code_integer).
% 129.49/25.35 thf(func_def_536, type, comp_P6122662217367900327nteger: (produc2591615631378489102nteger > code_integer) > (code_integer > produc2591615631378489102nteger) > code_integer > code_integer).
% 129.49/25.35 thf(func_def_537, type, comp_P7320554561093743171t_unit: (produc2591615631378489102nteger > code_integer) > (product_unit > produc2591615631378489102nteger) > product_unit > code_integer).
% 129.49/25.35 thf(func_def_538, type, comp_P7179991235626930843_nat_o: (produc2591615631378489102nteger > nat) > ($o > produc2591615631378489102nteger) > $o > nat).
% 129.49/25.35 thf(func_def_539, type, comp_P5725896914063526938nteger: (produc2591615631378489102nteger > nat) > (code_integer > produc2591615631378489102nteger) > code_integer > nat).
% 129.49/25.35 thf(func_def_540, type, comp_P7621350044865347728t_unit: (produc2591615631378489102nteger > nat) > (product_unit > produc2591615631378489102nteger) > product_unit > nat).
% 129.49/25.35 thf(func_def_541, type, comp_P4394204175247225320_nat_o: (product_prod_nat_nat > nat) > ($o > product_prod_nat_nat) > $o > nat).
% 129.49/25.35 thf(func_def_542, type, comp_P2980510830910040157t_unit: (product_prod_nat_nat > nat) > (product_unit > product_prod_nat_nat) > product_unit > nat).
% 129.49/25.35 thf(func_def_543, type, comp_P4511551582283981294nteger: (product_unit > produc8923325533196201883nteger > produc8923325533196201883nteger) > (code_integer > product_unit) > code_integer > produc8923325533196201883nteger > produc8923325533196201883nteger).
% 129.49/25.35 thf(func_def_544, type, comp_P4558236313141964142nteger: (product_unit > product_unit > product_unit) > (code_integer > product_unit) > code_integer > product_unit > product_unit).
% 129.49/25.35 thf(func_def_545, type, comp_P6204500334276302049it_nat: (product_unit > product_unit > product_unit) > (nat > product_unit) > nat > product_unit > product_unit).
% 129.49/25.35 thf(func_def_546, type, comp_P6790823133229852966nteger: (product_unit > code_integer) > (code_integer > product_unit) > code_integer > code_integer).
% 129.49/25.35 thf(func_def_547, type, comp_P4635212033913086756nteger: (product_unit > set_Code_integer) > ((produc8923325533196201883nteger > produc8923325533196201883nteger) > product_unit) > (produc8923325533196201883nteger > produc8923325533196201883nteger) > set_Code_integer).
% 129.49/25.35 thf(func_def_548, type, comp_P4767411901762164316nteger: (product_unit > set_Code_integer) > (code_integer > product_unit) > code_integer > set_Code_integer).
% 129.49/25.35 thf(func_def_549, type, comp_P6101996899264665623nteger: (product_unit > set_nat) > ((produc8923325533196201883nteger > produc8923325533196201883nteger) > product_unit) > (produc8923325533196201883nteger > produc8923325533196201883nteger) > set_nat).
% 129.49/25.35 thf(func_def_550, type, comp_P5878433733084393551nteger: (product_unit > set_nat) > (code_integer > product_unit) > code_integer > set_nat).
% 129.49/25.35 thf(func_def_551, type, comp_s90868338361139993nteger: (set_o > set_o > set_o) > (code_integer > set_o) > code_integer > set_o > set_o).
% 129.49/25.35 thf(func_def_552, type, comp_s4424444925706540684_o_nat: (set_o > set_o > set_o) > (nat > set_o) > nat > set_o > set_o).
% 129.49/25.35 thf(func_def_553, type, comp_s956550612328592744nteger: (set_Code_integer > set_Code_integer) > (code_integer > set_Code_integer) > code_integer > set_Code_integer).
% 129.49/25.35 thf(func_def_554, type, comp_s3634918272309710215nteger: (set_int > set_int > set_int) > (code_integer > set_int) > code_integer > set_int > set_int).
% 129.49/25.35 thf(func_def_555, type, comp_s6689493721604790522nt_nat: (set_int > set_int > set_int) > (nat > set_int) > nat > set_int > set_int).
% 129.49/25.35 thf(func_def_556, type, comp_s3169385134670675059nteger: (set_nat > set_nat > set_nat) > (code_integer > set_nat) > code_integer > set_nat > set_nat).
% 129.49/25.35 thf(func_def_557, type, comp_s2351873599094224870at_nat: (set_nat > set_nat > set_nat) > (nat > set_nat) > nat > set_nat > set_nat).
% 129.49/25.35 thf(func_def_558, type, comp_s1601710177542341017et_nat: (set_nat > int) > (produc3658429121746597890et_nat > set_nat) > produc3658429121746597890et_nat > int).
% 129.49/25.35 thf(func_def_559, type, comp_s8292494440407567195nteger: (set_nat > set_Code_integer) > (code_integer > set_nat) > code_integer > set_Code_integer).
% 129.49/25.35 thf(func_def_560, type, comp_s4605850120322998606nteger: (set_nat > set_nat) > (code_integer > set_nat) > code_integer > set_nat).
% 129.49/25.35 thf(func_def_561, type, comp_s5696514601250091763et_nat: (set_nat > set_nat) > (produc3658429121746597890et_nat > set_nat) > produc3658429121746597890et_nat > set_nat).
% 129.49/25.35 thf(func_def_562, type, comp_s3199726385246003790nteger: (set_Product_unit > set_Product_unit > set_Product_unit) > (code_integer > set_Product_unit) > code_integer > set_Product_unit > set_Product_unit).
% 129.49/25.35 thf(func_def_563, type, comp_s6943286182646368705it_nat: (set_Product_unit > set_Product_unit > set_Product_unit) > (nat > set_Product_unit) > nat > set_Product_unit > set_Product_unit).
% 129.49/25.35 thf(func_def_564, type, fun_upd_o_o: ($o > $o) > $o > $o > $o > $o).
% 129.49/25.35 thf(func_def_565, type, fun_upd_o_int: ($o > int) > $o > int > $o > int).
% 129.49/25.35 thf(func_def_566, type, fun_upd_o_nat: ($o > nat) > $o > nat > $o > nat).
% 129.49/25.35 thf(func_def_567, type, fun_up846054379331416818t_unit: ($o > product_unit) > $o > product_unit > $o > product_unit).
% 129.49/25.35 thf(func_def_568, type, fun_upd_o_set_o: ($o > set_o) > $o > set_o > $o > set_o).
% 129.49/25.35 thf(func_def_569, type, fun_upd_o_set_int: ($o > set_int) > $o > set_int > $o > set_int).
% 129.49/25.35 thf(func_def_570, type, fun_upd_o_set_nat: ($o > set_nat) > $o > set_nat > $o > set_nat).
% 129.49/25.35 thf(func_def_571, type, fun_up4718931970374059986t_unit: ($o > set_Product_unit) > $o > set_Product_unit > $o > set_Product_unit).
% 129.49/25.35 thf(func_def_572, type, fun_up8355007323143964473nteger: (code_integer > code_integer) > code_integer > code_integer > code_integer > code_integer).
% 129.49/25.35 thf(func_def_573, type, fun_upd_int_nat: (int > nat) > int > nat > int > nat).
% 129.49/25.35 thf(func_def_574, type, fun_upd_int_set_o: (int > set_o) > int > set_o > int > set_o).
% 129.49/25.35 thf(func_def_575, type, fun_up5054257004045450754t_unit: (int > set_Product_unit) > int > set_Product_unit > int > set_Product_unit).
% 129.49/25.35 thf(func_def_576, type, fun_up3904115110581602988nteger: (nat > code_integer) > nat > code_integer > nat > code_integer).
% 129.49/25.35 thf(func_def_577, type, fun_upd_nat_nat: (nat > nat) > nat > nat > nat > nat).
% 129.49/25.35 thf(func_def_578, type, fun_upd_nat_set_nat: (nat > set_nat) > nat > set_nat > nat > set_nat).
% 129.49/25.35 thf(func_def_579, type, fun_up5432024876597553480unit_o: (product_unit > $o) > product_unit > $o > product_unit > $o).
% 129.49/25.35 thf(func_def_580, type, fun_up6470560416331323516it_int: (product_unit > int) > product_unit > int > product_unit > int).
% 129.49/25.35 thf(func_def_581, type, fun_up6473050886840373792it_nat: (product_unit > nat) > product_unit > nat > product_unit > nat).
% 129.49/25.35 thf(func_def_582, type, fun_up6875000483049295805t_unit: (product_unit > product_unit) > product_unit > product_unit > product_unit > product_unit).
% 129.49/25.35 thf(func_def_583, type, fun_up1762901246852280360_set_o: (product_unit > set_o) > product_unit > set_o > product_unit > set_o).
% 129.49/25.35 thf(func_def_584, type, fun_up391983181201004082et_int: (product_unit > set_int) > product_unit > set_int > product_unit > set_int).
% 129.49/25.35 thf(func_def_585, type, fun_up4569834200710200790et_nat: (product_unit > set_nat) > product_unit > set_nat > product_unit > set_nat).
% 129.49/25.35 thf(func_def_586, type, fun_up488859312337516701t_unit: (product_unit > set_Product_unit) > product_unit > set_Product_unit > product_unit > set_Product_unit).
% 129.49/25.35 thf(func_def_587, type, id_o: $o > $o).
% 129.49/25.35 thf(func_def_588, type, inj_on5271586719803267564_o_nat: ((produc3658429121746597890et_nat > $o) > nat) > set_Pr4532377907799695533_nat_o > $o).
% 129.49/25.35 thf(func_def_589, type, inj_on_o_o: ($o > $o) > set_o > $o).
% 129.49/25.35 thf(func_def_590, type, inj_on_o_int: ($o > int) > set_o > $o).
% 129.49/25.35 thf(func_def_591, type, inj_on_o_nat: ($o > nat) > set_o > $o).
% 129.49/25.35 thf(func_def_592, type, inj_on4268752236855399398t_unit: ($o > product_unit) > set_o > $o).
% 129.49/25.35 thf(func_def_593, type, inj_on1158064722259180589nteger: (code_integer > code_integer) > set_Code_integer > $o).
% 129.49/25.35 thf(func_def_594, type, inj_on1981279154655846304er_nat: (code_integer > nat) > set_Code_integer > $o).
% 129.49/25.35 thf(func_def_595, type, inj_on_int_int: (int > int) > set_int > $o).
% 129.49/25.35 thf(func_def_596, type, inj_on_int_nat: (int > nat) > set_int > $o).
% 129.49/25.35 thf(func_def_597, type, inj_on7779444506608719434_nat_o: (nat > produc3658429121746597890et_nat > $o) > set_nat > $o).
% 129.49/25.35 thf(func_def_598, type, inj_on_nat_o: (nat > $o) > set_nat > $o).
% 129.49/25.35 thf(func_def_599, type, inj_on2245834603434889120nteger: (nat > code_integer) > set_nat > $o).
% 129.49/25.35 thf(func_def_600, type, inj_on_nat_int: (nat > int) > set_nat > $o).
% 129.49/25.35 thf(func_def_601, type, inj_on_nat_nat: (nat > nat) > set_nat > $o).
% 129.49/25.35 thf(func_def_602, type, inj_on7061601236592826506t_unit: (nat > product_unit) > set_nat > $o).
% 129.49/25.35 thf(func_def_603, type, inj_on_nat_set_nat: (nat > set_nat) > set_nat > $o).
% 129.49/25.35 thf(func_def_604, type, inj_on20428373856169009nteger: (produc8923325533196201883nteger > produc8923325533196201883nteger) > set_Pr4811707699266497531nteger > $o).
% 129.49/25.35 thf(func_def_605, type, inj_on8854722734121536060unit_o: (product_unit > $o) > set_Product_unit > $o).
% 129.49/25.35 thf(func_def_606, type, inj_on8427948621271784584it_int: (product_unit > int) > set_Product_unit > $o).
% 129.49/25.35 thf(func_def_607, type, inj_on8430439091780834860it_nat: (product_unit > nat) > set_Product_unit > $o).
% 129.49/25.35 thf(func_def_608, type, inj_on8151373323710067377t_unit: (product_unit > product_unit) > set_Product_unit > $o).
% 129.49/25.35 thf(func_def_609, type, inj_on_rat_rat: (rat > rat) > set_rat > $o).
% 129.49/25.35 thf(func_def_610, type, inj_on_set_nat_nat: (set_nat > nat) > set_set_nat > $o).
% 129.49/25.35 thf(func_def_611, type, map_fu8272188784021352819nteger: (code_integer > int) > ((int > int) > code_integer > code_integer) > (int > int > int) > code_integer > code_integer > code_integer).
% 129.49/25.35 thf(func_def_612, type, map_fu2599414010547811884nteger: (code_integer > int) > (int > code_integer) > (int > int) > code_integer > code_integer).
% 129.49/25.35 thf(func_def_613, type, map_fu6549440983881763648atural: (code_natural > nat) > ((nat > nat) > code_natural > code_natural) > (nat > nat > nat) > code_natural > code_natural > code_natural).
% 129.49/25.35 thf(func_def_614, type, map_fu1239815594074539274atural: (code_natural > nat) > (nat > code_natural) > (nat > nat) > code_natural > code_natural).
% 129.49/25.35 thf(func_def_615, type, map_fu4960017516451851995nt_int: (int > product_prod_nat_nat) > ((product_prod_nat_nat > product_prod_nat_nat) > int > int) > (product_prod_nat_nat > product_prod_nat_nat > product_prod_nat_nat) > int > int > int).
% 129.49/25.35 thf(func_def_616, type, map_fu3667384564859982768at_int: (int > product_prod_nat_nat) > (product_prod_nat_nat > int) > (product_prod_nat_nat > product_prod_nat_nat) > int > int).
% 129.49/25.35 thf(func_def_617, type, map_fu4333342158222067775at_rat: (rat > product_prod_int_int) > ((product_prod_int_int > product_prod_int_int) > rat > rat) > (product_prod_int_int > product_prod_int_int > product_prod_int_int) > rat > rat > rat).
% 129.49/25.35 thf(func_def_618, type, map_fu898904425404107465nt_o_o: (rat > product_prod_int_int) > ($o > $o) > (product_prod_int_int > $o) > rat > $o).
% 129.49/25.35 thf(func_def_619, type, map_fu5673905371560938248nt_rat: (rat > product_prod_int_int) > (product_prod_int_int > rat) > (product_prod_int_int > product_prod_int_int) > rat > rat).
% 129.49/25.35 thf(func_def_620, type, gcd_Gcd_int: set_int > int).
% 129.49/25.35 thf(func_def_621, type, gcd_Gcd_nat: set_nat > nat).
% 129.49/25.35 thf(func_def_622, type, gcd_Lcm_nat: set_nat > nat).
% 129.49/25.35 thf(func_def_623, type, bezw: nat > nat > product_prod_int_int).
% 129.49/25.35 thf(func_def_624, type, bezw_rel: product_prod_nat_nat > product_prod_nat_nat > $o).
% 129.49/25.35 thf(func_def_625, type, gcd_gcd_Code_integer: code_integer > code_integer > code_integer).
% 129.49/25.35 thf(func_def_626, type, gcd_gcd_int: int > int > int).
% 129.49/25.35 thf(func_def_627, type, gcd_gcd_nat: nat > nat > nat).
% 129.49/25.35 thf(func_def_628, type, gcd_lcm_Code_integer: code_integer > code_integer > code_integer).
% 129.49/25.35 thf(func_def_629, type, gcd_lcm_int: int > int > int).
% 129.49/25.35 thf(func_def_630, type, gcd_lcm_nat: nat > nat > nat).
% 129.49/25.35 thf(func_def_631, type, semiri4256215615220890538in_int: set_int > int).
% 129.49/25.35 thf(func_def_632, type, semiri4258706085729940814in_nat: set_nat > nat).
% 129.49/25.35 thf(func_def_633, type, abs_abs_Code_integer: code_integer > code_integer).
% 129.49/25.35 thf(func_def_634, type, abs_abs_int: int > int).
% 129.49/25.35 thf(func_def_635, type, minus_minus_assn: assn > assn > assn).
% 129.49/25.35 thf(func_def_636, type, minus_8373710615458151222nteger: code_integer > code_integer > code_integer).
% 129.49/25.35 thf(func_def_637, type, minus_minus_int: int > int > int).
% 129.49/25.35 thf(func_def_638, type, minus_minus_nat: nat > nat > nat).
% 129.49/25.35 thf(func_def_639, type, minus_minus_rat: rat > rat > rat).
% 129.49/25.35 thf(func_def_640, type, minus_1801376950450012436_nat_o: set_Pr4532377907799695533_nat_o > set_Pr4532377907799695533_nat_o > set_Pr4532377907799695533_nat_o).
% 129.49/25.35 thf(func_def_641, type, minus_minus_set_o: set_o > set_o > set_o).
% 129.49/25.35 thf(func_def_642, type, minus_minus_set_assn: set_assn > set_assn > set_assn).
% 129.49/25.35 thf(func_def_643, type, minus_2355218937544613996nteger: set_Code_integer > set_Code_integer > set_Code_integer).
% 129.49/25.35 thf(func_def_644, type, minus_minus_set_int: set_int > set_int > set_int).
% 129.49/25.35 thf(func_def_645, type, minus_minus_set_nat: set_nat > set_nat > set_nat).
% 129.49/25.35 thf(func_def_646, type, minus_9189903262233110958_nat_o: set_Pr2161125870931222855_nat_o > set_Pr2161125870931222855_nat_o > set_Pr2161125870931222855_nat_o).
% 129.49/25.35 thf(func_def_647, type, minus_7052513330764642158od_o_o: set_Product_prod_o_o > set_Product_prod_o_o > set_Product_prod_o_o).
% 129.49/25.35 thf(func_def_648, type, minus_1052850069191792384nt_int: set_Pr958786334691620121nt_int > set_Pr958786334691620121nt_int > set_Pr958786334691620121nt_int).
% 129.49/25.35 thf(func_def_649, type, minus_1356011639430497352at_nat: set_Pr1261947904930325089at_nat > set_Pr1261947904930325089at_nat > set_Pr1261947904930325089at_nat).
% 129.49/25.35 thf(func_def_650, type, minus_6985556733573110190t_unit: set_Pr5094982260447487303t_unit > set_Pr5094982260447487303t_unit > set_Pr5094982260447487303t_unit).
% 129.49/25.35 thf(func_def_651, type, minus_6452836326544984404t_unit: set_Product_unit > set_Product_unit > set_Product_unit).
% 129.49/25.35 thf(func_def_652, type, minus_2163939370556025621et_nat: set_set_nat > set_set_nat > set_set_nat).
% 129.49/25.35 thf(func_def_653, type, one_one_assn: assn).
% 129.49/25.35 thf(func_def_654, type, one_one_Code_integer: code_integer).
% 129.49/25.35 thf(func_def_655, type, one_one_Code_natural: code_natural).
% 129.49/25.35 thf(func_def_656, type, one_one_int: int).
% 129.49/25.35 thf(func_def_657, type, one_one_nat: nat).
% 129.49/25.35 thf(func_def_658, type, one_one_rat: rat).
% 129.49/25.35 thf(func_def_659, type, plus_p5714425477246183910nteger: code_integer > code_integer > code_integer).
% 129.49/25.35 thf(func_def_660, type, plus_p4538020629002901425atural: code_natural > code_natural > code_natural).
% 129.49/25.35 thf(func_def_661, type, plus_plus_int: int > int > int).
% 129.49/25.35 thf(func_def_662, type, plus_plus_nat: nat > nat > nat).
% 129.49/25.35 thf(func_def_663, type, plus_plus_num: num > num > num).
% 129.49/25.35 thf(func_def_664, type, plus_plus_rat: rat > rat > rat).
% 129.49/25.35 thf(func_def_665, type, sgn_sgn_Code_integer: code_integer > code_integer).
% 129.49/25.35 thf(func_def_666, type, sgn_sgn_int: int > int).
% 129.49/25.35 thf(func_def_667, type, sgn_sgn_rat: rat > rat).
% 129.49/25.35 thf(func_def_668, type, times_times_assn: assn > assn > assn).
% 129.49/25.35 thf(func_def_669, type, times_3573771949741848930nteger: code_integer > code_integer > code_integer).
% 129.49/25.35 thf(func_def_670, type, times_2397367101498566445atural: code_natural > code_natural > code_natural).
% 129.49/25.35 thf(func_def_671, type, times_times_int: int > int > int).
% 129.49/25.35 thf(func_def_672, type, times_times_nat: nat > nat > nat).
% 129.49/25.35 thf(func_def_673, type, times_times_num: num > num > num).
% 129.49/25.35 thf(func_def_674, type, times_times_rat: rat > rat > rat).
% 129.49/25.35 thf(func_def_675, type, uminus_uminus_assn: assn > assn).
% 129.49/25.35 thf(func_def_676, type, uminus1351360451143612070nteger: code_integer > code_integer).
% 129.49/25.35 thf(func_def_677, type, uminus_uminus_int: int > int).
% 129.49/25.35 thf(func_def_678, type, uminus2952777764628376836t_unit: product_unit > product_unit).
% 129.49/25.35 thf(func_def_679, type, uminus_uminus_rat: rat > rat).
% 129.49/25.35 thf(func_def_680, type, uminus5254974436814262692_nat_o: set_Pr4532377907799695533_nat_o > set_Pr4532377907799695533_nat_o).
% 129.49/25.35 thf(func_def_681, type, uminus_uminus_set_o: set_o > set_o).
% 129.49/25.35 thf(func_def_682, type, uminus1532241313380277803et_int: set_int > set_int).
% 129.49/25.35 thf(func_def_683, type, uminus5710092332889474511et_nat: set_nat > set_nat).
% 129.49/25.35 thf(func_def_684, type, uminus5944136376168626660t_unit: set_Product_unit > set_Product_unit).
% 129.49/25.35 thf(func_def_685, type, zero_z3403309356797280102nteger: code_integer).
% 129.49/25.35 thf(func_def_686, type, zero_z2226904508553997617atural: code_natural).
% 129.49/25.35 thf(func_def_687, type, zero_zero_int: int).
% 129.49/25.35 thf(func_def_688, type, zero_zero_nat: nat).
% 129.49/25.35 thf(func_def_689, type, zero_zero_rat: rat).
% 129.49/25.35 thf(func_def_690, type, groups6742167275697360483_o_nat: ((produc3658429121746597890et_nat > $o) > nat) > set_Pr4532377907799695533_nat_o > nat).
% 129.49/25.35 thf(func_def_691, type, groups4406642042086082107nteger: ($o > code_integer) > set_o > code_integer).
% 129.49/25.35 thf(func_def_692, type, groups8505340233167759370_o_int: ($o > int) > set_o > int).
% 129.49/25.35 thf(func_def_693, type, groups8507830703676809646_o_nat: ($o > nat) > set_o > nat).
% 129.49/25.35 thf(func_def_694, type, groups7872700643590313910_o_rat: ($o > rat) > set_o > rat).
% 129.49/25.35 thf(func_def_695, type, groups879477027807139574nteger: (code_integer > code_integer) > set_Code_integer > code_integer).
% 129.49/25.35 thf(func_def_696, type, groups7234854612051535045er_int: (code_integer > int) > set_Code_integer > int).
% 129.49/25.35 thf(func_def_697, type, groups7237345082560585321er_nat: (code_integer > nat) > set_Code_integer > nat).
% 129.49/25.35 thf(func_def_698, type, groups6602215022474089585er_rat: (code_integer > rat) > set_Code_integer > rat).
% 129.49/25.35 thf(func_def_699, type, groups7873554091576472773nteger: (int > code_integer) > set_int > code_integer).
% 129.49/25.35 thf(func_def_700, type, groups4538972089207619220nt_int: (int > int) > set_int > int).
% 129.49/25.35 thf(func_def_701, type, groups4541462559716669496nt_nat: (int > nat) > set_int > nat).
% 129.49/25.35 thf(func_def_702, type, groups3906332499630173760nt_rat: (int > rat) > set_int > rat).
% 129.49/25.35 thf(func_def_703, type, groups7501900531339628137nteger: (nat > code_integer) > set_nat > code_integer).
% 129.49/25.35 thf(func_def_704, type, groups3539618377306564664at_int: (nat > int) > set_nat > int).
% 129.49/25.35 thf(func_def_705, type, groups3542108847815614940at_nat: (nat > nat) > set_nat > nat).
% 129.49/25.35 thf(func_def_706, type, groups2906978787729119204at_rat: (nat > rat) > set_nat > rat).
% 129.49/25.35 thf(func_def_707, type, groups5393439102872367408nteger: (product_unit > code_integer) > set_Product_unit > code_integer).
% 129.49/25.35 thf(func_def_708, type, groups8764306707065906943it_int: (product_unit > int) > set_Product_unit > int).
% 129.49/25.35 thf(func_def_709, type, groups8766797177574957219it_nat: (product_unit > nat) > set_Product_unit > nat).
% 129.49/25.35 thf(func_def_710, type, groups8131667117488461483it_rat: (product_unit > rat) > set_Product_unit > rat).
% 129.49/25.35 thf(func_def_711, type, groups1256056575759317066nt_int: (set_int > int) > set_set_int > int).
% 129.49/25.35 thf(func_def_712, type, groups8294997508430121362at_nat: (set_nat > nat) > set_set_nat > nat).
% 129.49/25.35 thf(func_def_713, type, groups5301882518646026715o_assn: ($o > assn) > set_o > assn).
% 129.49/25.35 thf(func_def_714, type, groups7694694392188491536nteger: ($o > code_integer) > set_o > code_integer).
% 129.49/25.35 thf(func_def_715, type, groups6518289543945209051atural: ($o > code_natural) > set_o > code_natural).
% 129.49/25.35 thf(func_def_716, type, groups3502327434004483295_o_int: ($o > int) > set_o > int).
% 129.49/25.35 thf(func_def_717, type, groups3504817904513533571_o_nat: ($o > nat) > set_o > nat).
% 129.49/25.35 thf(func_def_718, type, groups2869687844427037835_o_rat: ($o > rat) > set_o > rat).
% 129.49/25.35 thf(func_def_719, type, groups1304777262505850412r_assn: (code_integer > assn) > set_Code_integer > assn).
% 129.49/25.35 thf(func_def_720, type, groups3674199335183972705nteger: (code_integer > code_integer) > set_Code_integer > code_integer).
% 129.49/25.35 thf(func_def_721, type, groups2497794486940690220atural: (code_integer > code_natural) > set_Code_integer > code_natural).
% 129.49/25.35 thf(func_def_722, type, groups3188404863801439024er_int: (code_integer > int) > set_Code_integer > int).
% 129.49/25.35 thf(func_def_723, type, groups3190895334310489300er_nat: (code_integer > nat) > set_Code_integer > nat).
% 129.49/25.35 thf(func_def_724, type, groups2555765274223993564er_rat: (code_integer > rat) > set_Code_integer > rat).
% 129.49/25.35 thf(func_def_725, type, groups7882442080178216443t_assn: (int > assn) > set_int > assn).
% 129.49/25.35 thf(func_def_726, type, groups3827104343326376752nteger: (int > code_integer) > set_int > code_integer).
% 129.49/25.35 thf(func_def_727, type, groups2650699495083094267atural: (int > code_natural) > set_int > code_natural).
% 129.49/25.35 thf(func_def_728, type, groups1705073143266064639nt_int: (int > int) > set_int > int).
% 129.49/25.35 thf(func_def_729, type, groups1707563613775114915nt_nat: (int > nat) > set_int > nat).
% 129.49/25.35 thf(func_def_730, type, groups1072433553688619179nt_rat: (int > rat) > set_int > rat).
% 129.49/25.35 thf(func_def_731, type, groups6906906614972039071t_assn: (nat > assn) > set_nat > assn).
% 129.49/25.35 thf(func_def_732, type, groups3455450783089532116nteger: (nat > code_integer) > set_nat > code_integer).
% 129.49/25.35 thf(func_def_733, type, groups2279045934846249631atural: (nat > code_natural) > set_nat > code_natural).
% 129.49/25.35 thf(func_def_734, type, groups705719431365010083at_int: (nat > int) > set_nat > int).
% 129.49/25.35 thf(func_def_735, type, groups708209901874060359at_nat: (nat > nat) > set_nat > nat).
% 129.49/25.35 thf(func_def_736, type, groups73079841787564623at_rat: (nat > rat) > set_nat > rat).
% 129.49/25.35 thf(func_def_737, type, groups5360233303959380816t_assn: (product_unit > assn) > set_Product_unit > assn).
% 129.49/25.35 thf(func_def_738, type, groups8185359811944078981nteger: (product_unit > code_integer) > set_Product_unit > code_integer).
% 129.49/25.35 thf(func_def_739, type, groups7008954963700796496atural: (product_unit > code_natural) > set_Product_unit > code_natural).
% 129.49/25.35 thf(func_def_740, type, groups393930778517121108it_int: (product_unit > int) > set_Product_unit > int).
% 129.49/25.35 thf(func_def_741, type, groups396421249026171384it_nat: (product_unit > nat) > set_Product_unit > nat).
% 129.49/25.35 thf(func_def_742, type, groups8984663225794451456it_rat: (product_unit > rat) > set_Product_unit > rat).
% 129.49/25.35 thf(func_def_743, type, groups6963733262650904763o_assn: (set_o > assn) > set_set_o > assn).
% 129.49/25.35 thf(func_def_744, type, groups4868773677410035043_o_nat: (set_o > nat) > set_set_o > nat).
% 129.49/25.35 thf(func_def_745, type, groups4245288774754739042r_assn: (set_Code_integer > assn) > set_set_Code_integer > assn).
% 129.49/25.35 thf(func_def_746, type, groups6044384099095135754er_nat: (set_Code_integer > nat) > set_set_Code_integer > nat).
% 129.49/25.35 thf(func_def_747, type, groups5950967880310095793t_assn: (set_int > assn) > set_set_int > assn).
% 129.49/25.35 thf(func_def_748, type, groups6432978864363996853nt_int: (set_int > int) > set_set_int > int).
% 129.49/25.35 thf(func_def_749, type, groups6435469334873047129nt_nat: (set_int > nat) > set_set_int > nat).
% 129.49/25.35 thf(func_def_750, type, groups1535173586962943829t_assn: (set_nat > assn) > set_set_nat > assn).
% 129.49/25.35 thf(func_def_751, type, groups4248547760180025341at_nat: (set_nat > nat) > set_set_nat > nat).
% 129.49/25.35 thf(func_def_752, type, groups2392624107559085104t_assn: (set_Product_unit > assn) > set_set_Product_unit > assn).
% 129.49/25.35 thf(func_def_753, type, groups8454329047868613336it_nat: (set_Product_unit > nat) > set_set_Product_unit > nat).
% 129.49/25.35 thf(func_def_754, type, defaul566961228789861419t_unit: product_unit).
% 129.49/25.35 thf(func_def_755, type, arrays_Product_unit: heap_e7401611519738050253t_unit > typerep > nat > list_nat).
% 129.49/25.35 thf(func_def_756, type, lim_Product_unit: heap_e7401611519738050253t_unit > nat).
% 129.49/25.35 thf(func_def_757, type, refs_Product_unit: heap_e7401611519738050253t_unit > typerep > nat > nat).
% 129.49/25.35 thf(func_def_758, type, if_assn: $o > assn > assn > assn).
% 129.49/25.35 thf(func_def_759, type, if_Code_integer: $o > code_integer > code_integer > code_integer).
% 129.49/25.35 thf(func_def_760, type, if_Code_natural: $o > code_natural > code_natural > code_natural).
% 129.49/25.35 thf(func_def_761, type, if_filter_nat: $o > filter_nat > filter_nat > filter_nat).
% 129.49/25.35 thf(func_def_762, type, if_int: $o > int > int > int).
% 129.49/25.35 thf(func_def_763, type, if_nat: $o > nat > nat > nat).
% 129.49/25.35 thf(func_def_764, type, if_Pro6119634080678213985nteger: $o > produc8923325533196201883nteger > produc8923325533196201883nteger > produc8923325533196201883nteger).
% 129.49/25.35 thf(func_def_765, type, if_Pro3027730157355071871nt_int: $o > product_prod_int_int > product_prod_int_int > product_prod_int_int).
% 129.49/25.35 thf(func_def_766, type, if_Pro6206227464963214023at_nat: $o > product_prod_nat_nat > product_prod_nat_nat > product_prod_nat_nat).
% 129.49/25.35 thf(func_def_767, type, if_rat: $o > rat > rat > rat).
% 129.49/25.35 thf(func_def_768, type, if_set_o: $o > set_o > set_o > set_o).
% 129.49/25.35 thf(func_def_769, type, if_set_int: $o > set_int > set_int > set_int).
% 129.49/25.35 thf(func_def_770, type, if_set_nat: $o > set_nat > set_nat > set_nat).
% 129.49/25.35 thf(func_def_771, type, if_set_Product_unit: $o > set_Product_unit > set_Product_unit > set_Product_unit).
% 129.49/25.35 thf(func_def_772, type, infini8530281810654367211te_nat: set_nat > nat > nat).
% 129.49/25.35 thf(func_def_773, type, infini7930543730640340914t_unit: set_Product_unit > nat > product_unit).
% 129.49/25.35 thf(func_def_774, type, abs_Integ: product_prod_nat_nat > int).
% 129.49/25.35 thf(func_def_775, type, neg: num > int).
% 129.49/25.35 thf(func_def_776, type, pos: num > int).
% 129.49/25.35 thf(func_def_777, type, rep_Integ: int > product_prod_nat_nat).
% 129.49/25.35 thf(func_def_778, type, intrel: product_prod_nat_nat > product_prod_nat_nat > $o).
% 129.49/25.35 thf(func_def_779, type, nat2: int > nat).
% 129.49/25.35 thf(func_def_780, type, pcr_int: product_prod_nat_nat > int > $o).
% 129.49/25.35 thf(func_def_781, type, ring_1_of_int_rat: int > rat).
% 129.49/25.35 thf(func_def_782, type, inf_in1318976480646536635_nat_o: (produc3658429121746597890et_nat > $o) > (produc3658429121746597890et_nat > $o) > produc3658429121746597890et_nat > $o).
% 129.49/25.35 thf(func_def_783, type, inf_inf_o: $o > $o > $o).
% 129.49/25.35 thf(func_def_784, type, inf_inf_assn: assn > assn > assn).
% 129.49/25.35 thf(func_def_785, type, inf_inf_filter_nat: filter_nat > filter_nat > filter_nat).
% 129.49/25.35 thf(func_def_786, type, inf_inf_int: int > int > int).
% 129.49/25.35 thf(func_def_787, type, inf_inf_nat: nat > nat > nat).
% 129.49/25.35 thf(func_def_788, type, inf_inf_Product_unit: product_unit > product_unit > product_unit).
% 129.49/25.35 thf(func_def_789, type, inf_in1906310914598751387_nat_o: set_Pr4532377907799695533_nat_o > set_Pr4532377907799695533_nat_o > set_Pr4532377907799695533_nat_o).
% 129.49/25.35 thf(func_def_790, type, inf_inf_set_o: set_o > set_o > set_o).
% 129.49/25.35 thf(func_def_791, type, inf_in1364745209274528805nteger: set_Code_integer > set_Code_integer > set_Code_integer).
% 129.49/25.35 thf(func_def_792, type, inf_in7395780696554941096er_nat: set_filter_nat > set_filter_nat > set_filter_nat).
% 129.49/25.35 thf(func_def_793, type, inf_inf_set_int: set_int > set_int > set_int).
% 129.49/25.35 thf(func_def_794, type, inf_inf_set_nat: set_nat > set_nat > set_nat).
% 129.49/25.35 thf(func_def_795, type, inf_in2269163501485487111nt_int: set_Pr958786334691620121nt_int > set_Pr958786334691620121nt_int > set_Pr958786334691620121nt_int).
% 129.49/25.35 thf(func_def_796, type, inf_in2572325071724192079at_nat: set_Pr1261947904930325089at_nat > set_Pr1261947904930325089at_nat > set_Pr1261947904930325089at_nat).
% 129.49/25.35 thf(func_def_797, type, inf_in4660618365625256667t_unit: set_Product_unit > set_Product_unit > set_Product_unit).
% 129.49/25.35 thf(func_def_798, type, inf_inf_set_set_nat: set_set_nat > set_set_nat > set_set_nat).
% 129.49/25.35 thf(func_def_799, type, sup_su1942872429929737346at_o_o: ((produc3658429121746597890et_nat > $o) > (produc3658429121746597890et_nat > $o) > $o) > ((produc3658429121746597890et_nat > $o) > (produc3658429121746597890et_nat > $o) > $o) > (produc3658429121746597890et_nat > $o) > (produc3658429121746597890et_nat > $o) > $o).
% 129.49/25.35 thf(func_def_800, type, sup_sup_o_o_o: ($o > $o > $o) > ($o > $o > $o) > $o > $o > $o).
% 129.49/25.35 thf(func_def_801, type, sup_sup_int_int_o: (int > int > $o) > (int > int > $o) > int > int > $o).
% 129.49/25.35 thf(func_def_802, type, sup_sup_nat_nat_o: (nat > nat > $o) > (nat > nat > $o) > nat > nat > $o).
% 129.49/25.35 thf(func_def_803, type, sup_su5453871518329203617_nat_o: (produc3658429121746597890et_nat > $o) > (produc3658429121746597890et_nat > $o) > produc3658429121746597890et_nat > $o).
% 129.49/25.35 thf(func_def_804, type, sup_su3411595859437804162unit_o: (product_unit > product_unit > $o) > (product_unit > product_unit > $o) > product_unit > product_unit > $o).
% 129.49/25.35 thf(func_def_805, type, sup_sup_o: $o > $o > $o).
% 129.49/25.35 thf(func_def_806, type, sup_sup_assn: assn > assn > assn).
% 129.49/25.35 thf(func_def_807, type, sup_sup_filter_nat: filter_nat > filter_nat > filter_nat).
% 129.49/25.35 thf(func_def_808, type, sup_sup_int: int > int > int).
% 129.49/25.35 thf(func_def_809, type, sup_sup_nat: nat > nat > nat).
% 129.49/25.35 thf(func_def_810, type, sup_sup_Product_unit: product_unit > product_unit > product_unit).
% 129.49/25.35 thf(func_def_811, type, sup_su5209123915105501825_nat_o: set_Pr4532377907799695533_nat_o > set_Pr4532377907799695533_nat_o > set_Pr4532377907799695533_nat_o).
% 129.49/25.35 thf(func_def_812, type, sup_sup_set_o: set_o > set_o > set_o).
% 129.49/25.35 thf(func_def_813, type, sup_sup_set_assn: set_assn > set_assn > set_assn).
% 129.49/25.35 thf(func_def_814, type, sup_su848401254843788991nteger: set_Code_integer > set_Code_integer > set_Code_integer).
% 129.49/25.35 thf(func_def_815, type, sup_sup_set_int: set_int > set_int > set_int).
% 129.49/25.35 thf(func_def_816, type, sup_sup_set_nat: set_nat > set_nat > set_nat).
% 129.49/25.35 thf(func_def_817, type, sup_su8328026650783933211_nat_o: set_Pr2161125870931222855_nat_o > set_Pr2161125870931222855_nat_o > set_Pr2161125870931222855_nat_o).
% 129.49/25.35 thf(func_def_818, type, sup_su5769328420594410459od_o_o: set_Product_prod_o_o > set_Product_prod_o_o > set_Product_prod_o_o).
% 129.49/25.35 thf(func_def_819, type, sup_su6024340866399070445nt_int: set_Pr958786334691620121nt_int > set_Pr958786334691620121nt_int > set_Pr958786334691620121nt_int).
% 129.49/25.35 thf(func_def_820, type, sup_su6327502436637775413at_nat: set_Pr1261947904930325089at_nat > set_Pr1261947904930325089at_nat > set_Pr1261947904930325089at_nat).
% 129.49/25.35 thf(func_def_821, type, sup_su6481247232241170203t_unit: set_Pr5094982260447487303t_unit > set_Pr5094982260447487303t_unit > set_Pr5094982260447487303t_unit).
% 129.49/25.35 thf(func_def_822, type, sup_su4863159966000489243_nat_o: set_Pr3813038829465207111_nat_o > set_Pr3813038829465207111_nat_o > set_Pr3813038829465207111_nat_o).
% 129.49/25.35 thf(func_def_823, type, sup_su7766833992073560027_set_o: set_Pr4577759397028426247_set_o > set_Pr4577759397028426247_set_o > set_Pr4577759397028426247_set_o).
% 129.49/25.35 thf(func_def_824, type, sup_su1286422330746572633et_int: set_Pr2522554150109002629et_int > set_Pr2522554150109002629et_int > set_Pr2522554150109002629et_int).
% 129.49/25.35 thf(func_def_825, type, sup_su4251893418135750817et_nat: set_Pr5488025237498180813et_nat > set_Pr5488025237498180813et_nat > set_Pr5488025237498180813et_nat).
% 129.49/25.35 thf(func_def_826, type, sup_su3573767163274163995t_unit: set_Pr3980508049012409671t_unit > set_Pr3980508049012409671t_unit > set_Pr3980508049012409671t_unit).
% 129.49/25.35 thf(func_def_827, type, sup_su793286257634532545t_unit: set_Product_unit > set_Product_unit > set_Product_unit).
% 129.49/25.35 thf(func_def_828, type, sup_sup_set_set_o: set_set_o > set_set_o > set_set_o).
% 129.49/25.35 thf(func_def_829, type, sup_sup_set_set_int: set_set_int > set_set_int > set_set_int).
% 129.49/25.35 thf(func_def_830, type, sup_sup_set_set_nat: set_set_nat > set_set_nat > set_set_nat).
% 129.49/25.35 thf(func_def_831, type, sup_su5975557051335651745t_unit: set_set_Product_unit > set_set_Product_unit > set_set_Product_unit).
% 129.49/25.35 thf(func_def_832, type, lattic1921953407002678535_Max_o: set_o > $o).
% 129.49/25.35 thf(func_def_833, type, lattic4901227151466704046nteger: set_Code_integer > code_integer).
% 129.49/25.35 thf(func_def_834, type, lattic8263393255366662781ax_int: set_int > int).
% 129.49/25.35 thf(func_def_835, type, lattic8265883725875713057ax_nat: set_nat > nat).
% 129.49/25.35 thf(func_def_836, type, lattic1210392890313158908t_unit: set_Product_unit > product_unit).
% 129.49/25.35 thf(func_def_837, type, lattic7630753665789217321ax_rat: set_rat > rat).
% 129.49/25.35 thf(func_def_838, type, lattic1973801136483472281_Min_o: set_o > $o).
% 129.49/25.35 thf(func_def_839, type, lattic1063845414844153500nteger: set_Code_integer > code_integer).
% 129.49/25.35 thf(func_def_840, type, lattic8718645017227715691in_int: set_int > int).
% 129.49/25.35 thf(func_def_841, type, lattic8721135487736765967in_nat: set_nat > nat).
% 129.49/25.35 thf(func_def_842, type, lattic8554435388794137742t_unit: set_Product_unit > product_unit).
% 129.49/25.35 thf(func_def_843, type, lattic8086005427650270231in_rat: set_rat > rat).
% 129.49/25.35 thf(func_def_844, type, lattic7708394068118247271o_assn: ($o > assn) > set_o > $o).
% 129.49/25.35 thf(func_def_845, type, lattic2775856028456453135_o_nat: ($o > nat) > set_o > $o).
% 129.49/25.35 thf(func_def_846, type, lattic2140725968369957399_o_rat: ($o > rat) > set_o > $o).
% 129.49/25.35 thf(func_def_847, type, lattic4682378690128378784r_assn: (code_integer > assn) > set_Code_integer > code_integer).
% 129.49/25.35 thf(func_def_848, type, lattic6074936555552774728er_nat: (code_integer > nat) > set_Code_integer > code_integer).
% 129.49/25.35 thf(func_def_849, type, lattic5439806495466278992er_rat: (code_integer > rat) > set_Code_integer > code_integer).
% 129.49/25.35 thf(func_def_850, type, lattic3162799611691135855t_assn: (int > assn) > set_int > int).
% 129.49/25.35 thf(func_def_851, type, lattic8446286672483414039nt_nat: (int > nat) > set_int > int).
% 129.49/25.35 thf(func_def_852, type, lattic7811156612396918303nt_rat: (int > rat) > set_int > int).
% 129.49/25.35 thf(func_def_853, type, lattic2187264146484958483t_assn: (nat > assn) > set_nat > nat).
% 129.49/25.35 thf(func_def_854, type, lattic7446932960582359483at_nat: (nat > nat) > set_nat > nat).
% 129.49/25.35 thf(func_def_855, type, lattic6811802900495863747at_rat: (nat > rat) > set_nat > nat).
% 129.49/25.35 thf(func_def_856, type, lattic4376901495776413404t_assn: (product_unit > assn) > set_Product_unit > product_unit).
% 129.49/25.35 thf(func_def_857, type, lattic1360832392136924548it_nat: (product_unit > nat) > set_Product_unit > product_unit).
% 129.49/25.35 thf(func_def_858, type, lattic725702332050428812it_rat: (product_unit > rat) > set_Product_unit > product_unit).
% 129.49/25.35 thf(func_def_859, type, lattic956194824204696298_nat_o: set_Pr4532377907799695533_nat_o > produc3658429121746597890et_nat > $o).
% 129.49/25.35 thf(func_def_860, type, lattic4107685809792843317_fin_o: set_o > $o).
% 129.49/25.35 thf(func_def_861, type, lattic47131356835913163n_assn: set_assn > assn).
% 129.49/25.35 thf(func_def_862, type, lattic9128708996678779395er_nat: set_filter_nat > filter_nat).
% 129.49/25.35 thf(func_def_863, type, lattic5235898064620869839in_int: set_int > int).
% 129.49/25.35 thf(func_def_864, type, lattic5238388535129920115in_nat: set_nat > nat).
% 129.49/25.35 thf(func_def_865, type, lattic1263872656861969706t_unit: set_Product_unit > product_unit).
% 129.49/25.35 thf(func_def_866, type, lattic4603258475043424379in_rat: set_rat > rat).
% 129.49/25.35 thf(func_def_867, type, lattic3014633134055518761et_nat: set_set_nat > set_nat).
% 129.49/25.35 thf(func_def_868, type, lattic7320199455484906628_nat_o: set_Pr4532377907799695533_nat_o > produc3658429121746597890et_nat > $o).
% 129.49/25.35 thf(func_def_869, type, lattic1508158080041050831_fin_o: set_o > $o).
% 129.49/25.35 thf(func_def_870, type, lattic2150320897289308081n_assn: set_assn > assn).
% 129.49/25.35 thf(func_def_871, type, lattic5930898082463196905er_nat: set_filter_nat > filter_nat).
% 129.49/25.35 thf(func_def_872, type, lattic1091506334969745077in_int: set_int > int).
% 129.49/25.35 thf(func_def_873, type, lattic1093996805478795353in_nat: set_nat > nat).
% 129.49/25.35 thf(func_def_874, type, lattic5294303975357428420t_unit: set_Product_unit > product_unit).
% 129.49/25.35 thf(func_def_875, type, lattic458866745392299617in_rat: set_rat > rat).
% 129.49/25.35 thf(func_def_876, type, lattic3835124923745554447et_nat: set_set_nat > set_nat).
% 129.49/25.35 thf(func_def_877, type, append_nat: list_nat > list_nat > list_nat).
% 129.49/25.35 thf(func_def_878, type, butlast_nat: list_nat > list_nat).
% 129.49/25.35 thf(func_def_879, type, filter_nat2: (nat > $o) > list_nat > list_nat).
% 129.49/25.35 thf(func_def_880, type, cons_nat: nat > list_nat > list_nat).
% 129.49/25.35 thf(func_def_881, type, nil_nat: list_nat).
% 129.49/25.35 thf(func_def_882, type, map_nat_nat: (nat > nat) > list_nat > list_nat).
% 129.49/25.35 thf(func_def_883, type, upt: nat > nat > list_nat).
% 129.49/25.35 thf(func_def_884, type, biject1468766312547416318et_nat: set_Pr8536935166611901872et_nat > $o).
% 129.49/25.35 thf(func_def_885, type, biject2615096655818420098et_nat: set_Pr3286484037609594932et_nat > $o).
% 129.49/25.35 thf(func_def_886, type, biject2219996528949639174et_nat: set_Pr3948176798113811640et_nat > $o).
% 129.49/25.35 thf(func_def_887, type, bijective_int_int: set_Pr958786334691620121nt_int > $o).
% 129.49/25.35 thf(func_def_888, type, bijective_nat_nat: set_Pr1261947904930325089at_nat > $o).
% 129.49/25.35 thf(func_def_889, type, fun_of384177504262425200et_nat: set_Pr8536935166611901872et_nat > (produc3658429121746597890et_nat > $o) > produc3925858234332021118et_nat).
% 129.49/25.35 thf(func_def_890, type, fun_of3673525222065131764et_nat: set_Pr3286484037609594932et_nat > (produc3658429121746597890et_nat > $o) > produc3658429121746597890et_nat).
% 129.49/25.35 thf(func_def_891, type, fun_of5878246503291251064et_nat: set_Pr3948176798113811640et_nat > heap_e7401611519738050253t_unit > set_nat).
% 129.49/25.35 thf(func_def_892, type, fun_of_rel_int_int: set_Pr958786334691620121nt_int > int > int).
% 129.49/25.35 thf(func_def_893, type, fun_of_rel_nat_nat: set_Pr1261947904930325089at_nat > nat > nat).
% 129.49/25.35 thf(func_def_894, type, inv_on_o_int: ($o > int) > set_o > int > $o).
% 129.49/25.35 thf(func_def_895, type, inv_on_o_nat: ($o > nat) > set_o > nat > $o).
% 129.49/25.35 thf(func_def_896, type, inv_on4869000451854154612nteger: (code_integer > code_integer) > set_Code_integer > code_integer > code_integer).
% 129.49/25.35 thf(func_def_897, type, inv_on_int_int: (int > int) > set_int > int > int).
% 129.49/25.35 thf(func_def_898, type, inv_on_int_nat: (int > nat) > set_int > nat > int).
% 129.49/25.35 thf(func_def_899, type, inv_on1369624593536199399nteger: (nat > code_integer) > set_nat > code_integer > nat).
% 129.49/25.35 thf(func_def_900, type, inv_on_nat_int: (nat > int) > set_nat > int > nat).
% 129.49/25.35 thf(func_def_901, type, inv_on_nat_nat: (nat > nat) > set_nat > nat > nat).
% 129.49/25.35 thf(func_def_902, type, inv_on_nat_set_nat: (nat > set_nat) > set_nat > set_nat > nat).
% 129.49/25.35 thf(func_def_903, type, inv_on7725125254116338689it_int: (product_unit > int) > set_Product_unit > int > product_unit).
% 129.49/25.35 thf(func_def_904, type, inv_on7727615724625388965it_nat: (product_unit > nat) > set_Product_unit > nat > product_unit).
% 129.49/25.35 thf(func_def_905, type, pairself_int_int: (int > int) > product_prod_int_int > product_prod_int_int).
% 129.49/25.35 thf(func_def_906, type, pairself_int_nat: (int > nat) > product_prod_int_int > product_prod_nat_nat).
% 129.49/25.35 thf(func_def_907, type, pairself_nat_int: (nat > int) > product_prod_nat_nat > product_prod_int_int).
% 129.49/25.35 thf(func_def_908, type, pairself_nat_nat: (nat > nat) > product_prod_nat_nat > product_prod_nat_nat).
% 129.49/25.35 thf(func_def_909, type, pairself_rel_int_int: produc5030937667867769463nt_int > produc5030937667867769463nt_int > $o).
% 129.49/25.35 thf(func_def_910, type, pairself_rel_int_nat: produc6128020648511768019nt_int > produc6128020648511768019nt_int > $o).
% 129.49/25.35 thf(func_def_911, type, pairself_rel_nat_int: produc3574062574353363227at_nat > produc3574062574353363227at_nat > $o).
% 129.49/25.35 thf(func_def_912, type, pairself_rel_nat_nat: produc4671145554997361783at_nat > produc4671145554997361783at_nat > $o).
% 129.49/25.35 thf(func_def_913, type, rel_re3331361160397388719_nat_o: set_Pr2161125870931222855_nat_o > set_Pr4532377907799695533_nat_o > set_Pr2161125870931222855_nat_o).
% 129.49/25.35 thf(func_def_914, type, rel_restrict_o: set_Product_prod_o_o > set_o > set_Product_prod_o_o).
% 129.49/25.35 thf(func_def_915, type, rel_restrict_int: set_Pr958786334691620121nt_int > set_int > set_Pr958786334691620121nt_int).
% 129.49/25.35 thf(func_def_916, type, rel_restrict_nat: set_Pr1261947904930325089at_nat > set_nat > set_Pr1261947904930325089at_nat).
% 129.49/25.35 thf(func_def_917, type, rel_re9207996470947987439t_unit: set_Pr5094982260447487303t_unit > set_Product_unit > set_Pr5094982260447487303t_unit).
% 129.49/25.35 thf(func_def_918, type, suc: nat > nat).
% 129.49/25.35 thf(func_def_919, type, semiri4939895301339042750nteger: nat > code_integer).
% 129.49/25.35 thf(func_def_920, type, semiri3763490453095760265atural: nat > code_natural).
% 129.49/25.35 thf(func_def_921, type, semiri1314217659103216013at_int: nat > int).
% 129.49/25.35 thf(func_def_922, type, semiri1316708129612266289at_nat: nat > nat).
% 129.49/25.35 thf(func_def_923, type, semiri681578069525770553at_rat: nat > rat).
% 129.49/25.35 thf(func_def_924, type, nat_set_decode: nat > set_nat).
% 129.49/25.35 thf(func_def_925, type, nat_set_encode: set_nat > nat).
% 129.49/25.35 thf(func_def_926, type, nat_sum_encode: sum_sum_nat_nat > nat).
% 129.49/25.35 thf(func_def_927, type, nat_triangle: nat > nat).
% 129.49/25.35 thf(func_def_928, type, bitM: num > num).
% 129.49/25.35 thf(func_def_929, type, inc: num > num).
% 129.49/25.35 thf(func_def_930, type, nat_of_num: num > nat).
% 129.49/25.35 thf(func_def_931, type, neg_nu7757733837767384882nteger: code_integer > code_integer).
% 129.49/25.35 thf(func_def_932, type, neg_nu3811975205180677377ec_int: int > int).
% 129.49/25.35 thf(func_def_933, type, neg_nu3179335615603231917ec_rat: rat > rat).
% 129.49/25.35 thf(func_def_934, type, neg_nu5831290666863070958nteger: code_integer > code_integer).
% 129.49/25.35 thf(func_def_935, type, neg_nu5851722552734809277nc_int: int > int).
% 129.49/25.35 thf(func_def_936, type, neg_nu5219082963157363817nc_rat: rat > rat).
% 129.49/25.35 thf(func_def_937, type, neg_numeral_sub_int: num > num > int).
% 129.49/25.35 thf(func_def_938, type, bit0: num > num).
% 129.49/25.35 thf(func_def_939, type, bit1: num > num).
% 129.49/25.35 thf(func_def_940, type, one: num).
% 129.49/25.35 thf(func_def_941, type, num_of_nat: nat > num).
% 129.49/25.35 thf(func_def_942, type, numera6620942414471956472nteger: num > code_integer).
% 129.49/25.35 thf(func_def_943, type, numera5444537566228673987atural: num > code_natural).
% 129.49/25.35 thf(func_def_944, type, numeral_numeral_int: num > int).
% 129.49/25.35 thf(func_def_945, type, numeral_numeral_nat: num > nat).
% 129.49/25.35 thf(func_def_946, type, pow: num > num > num).
% 129.49/25.35 thf(func_def_947, type, pred_numeral: num > nat).
% 129.49/25.35 thf(func_def_948, type, sqr: num > num).
% 129.49/25.35 thf(func_def_949, type, order_1145710701291701250_nat_o: set_Pr4532377907799695533_nat_o > set_Pr2161125870931222855_nat_o > $o).
% 129.49/25.35 thf(func_def_950, type, order_6238756238976269133r_on_o: set_o > set_Product_prod_o_o > $o).
% 129.49/25.35 thf(func_def_951, type, order_4471489696718655927on_int: set_int > set_Pr958786334691620121nt_int > $o).
% 129.49/25.35 thf(func_def_952, type, order_4473980167227706203on_nat: set_nat > set_Pr1261947904930325089at_nat > $o).
% 129.49/25.35 thf(func_def_953, type, order_888282469287196226t_unit: set_Product_unit > set_Pr5094982260447487303t_unit > $o).
% 129.49/25.35 thf(func_def_954, type, order_3690095022144090562_nat_o: set_Pr4532377907799695533_nat_o > set_Pr2161125870931222855_nat_o > $o).
% 129.49/25.35 thf(func_def_955, type, order_8538205470752950541r_on_o: set_o > set_Product_prod_o_o > $o).
% 129.49/25.35 thf(func_def_956, type, order_5248785102713058295on_int: set_int > set_Pr958786334691620121nt_int > $o).
% 129.49/25.35 thf(func_def_957, type, order_5251275573222108571on_nat: set_nat > set_Pr1261947904930325089at_nat > $o).
% 129.49/25.35 thf(func_def_958, type, order_2505591618835473410t_unit: set_Product_unit > set_Pr5094982260447487303t_unit > $o).
% 129.49/25.35 thf(func_def_959, type, order_174704566861754596_nat_o: set_Pr4532377907799695533_nat_o > set_Pr2161125870931222855_nat_o > $o).
% 129.49/25.35 thf(func_def_960, type, order_preorder_on_o: set_o > set_Product_prod_o_o > $o).
% 129.49/25.35 thf(func_def_961, type, order_4859164337913492053on_int: set_int > set_Pr958786334691620121nt_int > $o).
% 129.49/25.35 thf(func_def_962, type, order_4861654808422542329on_nat: set_nat > set_Pr1261947904930325089at_nat > $o).
% 129.49/25.35 thf(func_def_963, type, order_7534304569044981540t_unit: set_Product_unit > set_Pr5094982260447487303t_unit > $o).
% 129.49/25.35 thf(func_def_964, type, order_7262550146012401043_nat_o: set_Pr2161125870931222855_nat_o > (produc3658429121746597890et_nat > $o) > set_Pr4532377907799695533_nat_o).
% 129.49/25.35 thf(func_def_965, type, order_underS_o: set_Product_prod_o_o > $o > set_o).
% 129.49/25.35 thf(func_def_966, type, order_underS_int: set_Pr958786334691620121nt_int > int > set_int).
% 129.49/25.35 thf(func_def_967, type, order_underS_nat: set_Pr1261947904930325089at_nat > nat > set_nat).
% 129.49/25.35 thf(func_def_968, type, order_5809128134054296019t_unit: set_Pr5094982260447487303t_unit > product_unit > set_Product_unit).
% 129.49/25.35 thf(func_def_969, type, order_2885205318754725938_nat_o: set_Pr2161125870931222855_nat_o > (produc3658429121746597890et_nat > $o) > set_Pr4532377907799695533_nat_o).
% 129.49/25.35 thf(func_def_970, type, order_under_o: set_Product_prod_o_o > $o > set_o).
% 129.49/25.35 thf(func_def_971, type, order_under_int: set_Pr958786334691620121nt_int > int > set_int).
% 129.49/25.35 thf(func_def_972, type, order_under_nat: set_Pr1261947904930325089at_nat > nat > set_nat).
% 129.49/25.35 thf(func_def_973, type, order_6949322460238004850t_unit: set_Pr5094982260447487303t_unit > product_unit > set_Product_unit).
% 129.49/25.35 thf(func_def_974, type, order_2118566763991480939_nat_o: set_Pr4532377907799695533_nat_o > set_Pr2161125870931222855_nat_o > $o).
% 129.49/25.35 thf(func_def_975, type, order_2615775362358406454r_on_o: set_o > set_Product_prod_o_o > $o).
% 129.49/25.35 thf(func_def_976, type, order_2886507596567047182on_int: set_int > set_Pr958786334691620121nt_int > $o).
% 129.49/25.35 thf(func_def_977, type, order_2888998067076097458on_nat: set_nat > set_Pr1261947904930325089at_nat > $o).
% 129.49/25.35 thf(func_def_978, type, order_3164716736300062379t_unit: set_Product_unit > set_Pr5094982260447487303t_unit > $o).
% 129.49/25.35 thf(func_def_979, type, bot_bo7963750851167320836at_o_o: (produc3658429121746597890et_nat > $o) > $o).
% 129.49/25.35 thf(func_def_980, type, bot_bot_o_o: $o > $o).
% 129.49/25.35 thf(func_def_981, type, bot_bot_int_o: int > $o).
% 129.49/25.35 thf(func_def_982, type, bot_bot_nat_o: nat > $o).
% 129.49/25.35 thf(func_def_983, type, bot_bo6993851861578111033_nat_o: produc3658429121746597890et_nat > $o).
% 129.49/25.35 thf(func_def_984, type, bot_bo4642748612307482820unit_o: product_unit > $o).
% 129.49/25.35 thf(func_def_986, type, bot_bot_assn: assn).
% 129.49/25.35 thf(func_def_987, type, bot_bot_filter_nat: filter_nat).
% 129.49/25.35 thf(func_def_988, type, bot_bot_nat: nat).
% 129.49/25.35 thf(func_def_989, type, bot_bot_Product_unit: product_unit).
% 129.49/25.35 thf(func_def_990, type, bot_bo7824918357723582233_nat_o: set_Pr4532377907799695533_nat_o).
% 129.49/25.35 thf(func_def_991, type, bot_bot_set_o: set_o).
% 129.49/25.35 thf(func_def_992, type, bot_bot_set_assn: set_assn).
% 129.49/25.35 thf(func_def_993, type, bot_bo3990330152332043303nteger: set_Code_integer).
% 129.49/25.35 thf(func_def_994, type, bot_bo498966703094740906er_nat: set_filter_nat).
% 129.49/25.35 thf(func_def_995, type, bot_bo3686245532977126937t_unit: set_he8644174446132254637t_unit).
% 129.49/25.35 thf(func_def_996, type, bot_bot_set_int: set_int).
% 129.49/25.35 thf(func_def_997, type, bot_bot_set_nat: set_nat).
% 129.49/25.35 thf(func_def_998, type, bot_bo3657187498359500723_nat_o: set_Pr2161125870931222855_nat_o).
% 129.49/25.35 thf(func_def_999, type, bot_bo1481135142794719944et_nat: set_Pr3286484037609594932et_nat).
% 129.49/25.35 thf(func_def_1000, type, bot_bo7073875226086086771od_o_o: set_Product_prod_o_o).
% 129.49/25.35 thf(func_def_1001, type, bot_bo3771712512152092492et_nat: set_Pr3948176798113811640et_nat).
% 129.49/25.35 thf(func_def_1002, type, bot_bo1796632182523588997nt_int: set_Pr958786334691620121nt_int).
% 129.49/25.35 thf(func_def_1003, type, bot_bo2099793752762293965at_nat: set_Pr1261947904930325089at_nat).
% 129.49/25.35 thf(func_def_1004, type, bot_bo1654022762305692595t_unit: set_Pr5094982260447487303t_unit).
% 129.49/25.35 thf(func_def_1005, type, bot_bo7535334947429593011_nat_o: set_Pr3813038829465207111_nat_o).
% 129.49/25.35 thf(func_def_1006, type, bot_bo8143257439550059635_set_o: set_Pr4577759397028426247_set_o).
% 129.49/25.35 thf(func_def_1007, type, bot_bo1987282549146081265et_int: set_Pr2522554150109002629et_int).
% 129.49/25.35 thf(func_def_1008, type, bot_bo4952753636535259449et_nat: set_Pr5488025237498180813et_nat).
% 129.49/25.35 thf(func_def_1009, type, bot_bo5113747506523071411t_unit: set_Pr3980508049012409671t_unit).
% 129.49/25.35 thf(func_def_1010, type, bot_bo3957492148770167129t_unit: set_Product_unit).
% 129.49/25.35 thf(func_def_1011, type, bot_bot_set_rat: set_rat).
% 129.49/25.35 thf(func_def_1012, type, bot_bo3736544561982865913_nat_o: set_se1353816476878259085_nat_o).
% 129.49/25.35 thf(func_def_1013, type, bot_bot_set_set_o: set_set_o).
% 129.49/25.35 thf(func_def_1014, type, bot_bo9204850693367297629nteger: set_set_Code_integer).
% 129.49/25.35 thf(func_def_1015, type, bot_bot_set_set_int: set_set_int).
% 129.49/25.35 thf(func_def_1016, type, bot_bot_set_set_nat: set_set_nat).
% 129.49/25.35 thf(func_def_1017, type, bot_bo6284674557190688313t_unit: set_set_Product_unit).
% 129.49/25.35 thf(func_def_1018, type, ord_le8176553561544679545_nat_o: (produc3658429121746597890et_nat > $o) > (produc3658429121746597890et_nat > $o) > $o).
% 129.49/25.35 thf(func_def_1019, type, ord_less_o: $o > $o > $o).
% 129.49/25.35 thf(func_def_1020, type, ord_less_assn: assn > assn > $o).
% 129.49/25.35 thf(func_def_1021, type, ord_le6747313008572928689nteger: code_integer > code_integer > $o).
% 129.49/25.35 thf(func_def_1022, type, ord_le5570908160329646204atural: code_natural > code_natural > $o).
% 129.49/25.35 thf(func_def_1023, type, ord_less_filter_nat: filter_nat > filter_nat > $o).
% 129.49/25.35 thf(func_def_1024, type, ord_less_int: int > int > $o).
% 129.49/25.35 thf(func_def_1025, type, ord_less_nat: nat > nat > $o).
% 129.49/25.35 thf(func_def_1026, type, ord_le361264281704409273t_unit: product_unit > product_unit > $o).
% 129.49/25.35 thf(func_def_1027, type, ord_less_rat: rat > rat > $o).
% 129.49/25.35 thf(func_def_1028, type, ord_le2453136405763929_nat_o: set_Pr4532377907799695533_nat_o > set_Pr4532377907799695533_nat_o > $o).
% 129.49/25.35 thf(func_def_1029, type, ord_less_set_o: set_o > set_o > $o).
% 129.49/25.35 thf(func_def_1030, type, ord_le1307284697595431911nteger: set_Code_integer > set_Code_integer > $o).
% 129.49/25.35 thf(func_def_1031, type, ord_less_set_int: set_int > set_int > $o).
% 129.49/25.35 thf(func_def_1032, type, ord_less_set_nat: set_nat > set_nat > $o).
% 129.49/25.35 thf(func_def_1033, type, ord_le8056459307392131481t_unit: set_Product_unit > set_Product_unit > $o).
% 129.49/25.35 thf(func_def_1034, type, ord_le729326519192465773_nat_o: (produc3658429121746597890et_nat > $o) > (produc3658429121746597890et_nat > $o) > $o).
% 129.49/25.35 thf(func_def_1035, type, ord_less_eq_o: $o > $o > $o).
% 129.49/25.35 thf(func_def_1036, type, ord_less_eq_assn: assn > assn > $o).
% 129.49/25.35 thf(func_def_1037, type, ord_le3102999989581377725nteger: code_integer > code_integer > $o).
% 129.49/25.35 thf(func_def_1038, type, ord_le1926595141338095240atural: code_natural > code_natural > $o).
% 129.49/25.35 thf(func_def_1039, type, ord_le2510731241096832064er_nat: filter_nat > filter_nat > $o).
% 129.49/25.35 thf(func_def_1040, type, ord_less_eq_int: int > int > $o).
% 129.49/25.35 thf(func_def_1041, type, ord_less_eq_nat: nat > nat > $o).
% 129.49/25.35 thf(func_def_1042, type, ord_le3221252021190050221t_unit: product_unit > product_unit > $o).
% 129.49/25.35 thf(func_def_1043, type, ord_less_eq_rat: rat > rat > $o).
% 129.49/25.35 thf(func_def_1044, type, ord_le2965882846123202637_nat_o: set_Pr4532377907799695533_nat_o > set_Pr4532377907799695533_nat_o > $o).
% 129.49/25.35 thf(func_def_1045, type, ord_less_eq_set_o: set_o > set_o > $o).
% 129.49/25.35 thf(func_def_1046, type, ord_less_eq_set_assn: set_assn > set_assn > $o).
% 129.49/25.35 thf(func_def_1047, type, ord_le7084787975880047091nteger: set_Code_integer > set_Code_integer > $o).
% 129.49/25.35 thf(func_def_1048, type, ord_le2426478655948331894er_nat: set_filter_nat > set_filter_nat > $o).
% 129.49/25.35 thf(func_def_1049, type, ord_le4903435234951335757t_unit: set_he8644174446132254637t_unit > set_he8644174446132254637t_unit > $o).
% 129.49/25.35 thf(func_def_1050, type, ord_less_eq_set_int: set_int > set_int > $o).
% 129.49/25.35 thf(func_def_1051, type, ord_less_eq_set_nat: set_nat > set_nat > $o).
% 129.49/25.35 thf(func_def_1052, type, ord_le6659818918159358183_nat_o: set_Pr2161125870931222855_nat_o > set_Pr2161125870931222855_nat_o > $o).
% 129.49/25.35 thf(func_def_1053, type, ord_le4460016228128491106_o_nat: set_Pr4769669219625338370_o_nat > set_Pr4769669219625338370_o_nat > $o).
% 129.49/25.35 thf(func_def_1054, type, ord_le4763372923235995152et_nat: set_Pr8536935166611901872et_nat > set_Pr8536935166611901872et_nat > $o).
% 129.49/25.35 thf(func_def_1055, type, ord_le5966269811547037844et_nat: set_Pr3286484037609594932et_nat > set_Pr3286484037609594932et_nat > $o).
% 129.49/25.35 thf(func_def_1056, type, ord_le192195778054999463od_o_o: set_Product_prod_o_o > set_Product_prod_o_o > $o).
% 129.49/25.35 thf(func_def_1057, type, ord_le6472378536984071597_o_nat: set_Pr2101469702781467981_o_nat > set_Pr2101469702781467981_o_nat > $o).
% 129.49/25.35 thf(func_def_1058, type, ord_le8090042543978450200et_nat: set_Pr3948176798113811640et_nat > set_Pr3948176798113811640et_nat > $o).
% 129.49/25.35 thf(func_def_1059, type, ord_le2843351958646193337nt_int: set_Pr958786334691620121nt_int > set_Pr958786334691620121nt_int > $o).
% 129.49/25.35 thf(func_def_1060, type, ord_le5333435103577920093nt_nat: set_Pr3448869479623346877nt_nat > set_Pr3448869479623346877nt_nat > $o).
% 129.49/25.35 thf(func_def_1061, type, ord_le3146513528884898305at_nat: set_Pr1261947904930325089at_nat > set_Pr1261947904930325089at_nat > $o).
% 129.49/25.35 thf(func_def_1062, type, ord_le8121013740822022818it_nat: set_Pr1763845938948868674it_nat > set_Pr1763845938948868674it_nat > $o).
% 129.49/25.35 thf(func_def_1063, type, ord_le930853871177014503t_unit: set_Pr5094982260447487303t_unit > set_Pr5094982260447487303t_unit > $o).
% 129.49/25.35 thf(func_def_1064, type, ord_le8001614231985977575_nat_o: set_Pr3813038829465207111_nat_o > set_Pr3813038829465207111_nat_o > $o).
% 129.49/25.35 thf(func_def_1065, type, ord_le2885120122233735591_set_o: set_Pr4577759397028426247_set_o > set_Pr4577759397028426247_set_o > $o).
% 129.49/25.35 thf(func_def_1066, type, ord_le1436784781161120549et_int: set_Pr2522554150109002629et_int > set_Pr2522554150109002629et_int > $o).
% 129.49/25.35 thf(func_def_1067, type, ord_le4402255868550298733et_nat: set_Pr5488025237498180813et_nat > set_Pr5488025237498180813et_nat > $o).
% 129.49/25.35 thf(func_def_1068, type, ord_le8072594200992201959t_unit: set_Pr3980508049012409671t_unit > set_Pr3980508049012409671t_unit > $o).
% 129.49/25.35 thf(func_def_1069, type, ord_le3507040750410214029t_unit: set_Product_unit > set_Product_unit > $o).
% 129.49/25.35 thf(func_def_1070, type, ord_less_eq_set_rat: set_rat > set_rat > $o).
% 129.49/25.35 thf(func_def_1071, type, ord_le1914454125413604393nteger: set_set_Code_integer > set_set_Code_integer > $o).
% 129.49/25.35 thf(func_def_1072, type, ord_le6893508408891458716et_nat: set_set_nat > set_set_nat > $o).
% 129.49/25.35 thf(func_def_1073, type, ord_le9131159989063066194et_nat: set_set_set_nat > set_set_set_nat > $o).
% 129.49/25.35 thf(func_def_1074, type, ord_max_o: $o > $o > $o).
% 129.49/25.35 thf(func_def_1075, type, ord_max_assn: assn > assn > assn).
% 129.49/25.35 thf(func_def_1076, type, ord_max_Code_integer: code_integer > code_integer > code_integer).
% 129.49/25.35 thf(func_def_1077, type, ord_max_Code_natural: code_natural > code_natural > code_natural).
% 129.49/25.35 thf(func_def_1078, type, ord_max_filter_nat: filter_nat > filter_nat > filter_nat).
% 129.49/25.35 thf(func_def_1079, type, ord_max_int: int > int > int).
% 129.49/25.35 thf(func_def_1080, type, ord_max_nat: nat > nat > nat).
% 129.49/25.35 thf(func_def_1081, type, ord_max_Product_unit: product_unit > product_unit > product_unit).
% 129.49/25.35 thf(func_def_1082, type, ord_max_rat: rat > rat > rat).
% 129.49/25.35 thf(func_def_1083, type, ord_ma8206644353667601538_nat_o: set_Pr4532377907799695533_nat_o > set_Pr4532377907799695533_nat_o > set_Pr4532377907799695533_nat_o).
% 129.49/25.35 thf(func_def_1084, type, ord_max_set_o: set_o > set_o > set_o).
% 129.49/25.35 thf(func_def_1085, type, ord_max_set_int: set_int > set_int > set_int).
% 129.49/25.35 thf(func_def_1086, type, ord_max_set_nat: set_nat > set_nat > set_nat).
% 129.49/25.35 thf(func_def_1087, type, ord_ma508538259134630082t_unit: set_Product_unit > set_Product_unit > set_Product_unit).
% 129.49/25.35 thf(func_def_1088, type, ord_min_nat: nat > nat > nat).
% 129.49/25.35 thf(func_def_1089, type, order_mono_nat_nat: (nat > nat) > $o).
% 129.49/25.35 thf(func_def_1090, type, ordering_top_assn: (assn > assn > $o) > (assn > assn > $o) > assn > $o).
% 129.49/25.35 thf(func_def_1091, type, orderi2807725203697798511er_nat: (filter_nat > filter_nat > $o) > (filter_nat > filter_nat > $o) > filter_nat > $o).
% 129.49/25.35 thf(func_def_1092, type, ordering_top_set_o: (set_o > set_o > $o) > (set_o > set_o > $o) > set_o > $o).
% 129.49/25.35 thf(func_def_1093, type, ordering_top_set_int: (set_int > set_int > $o) > (set_int > set_int > $o) > set_int > $o).
% 129.49/25.35 thf(func_def_1094, type, ordering_top_set_nat: (set_nat > set_nat > $o) > (set_nat > set_nat > $o) > set_nat > $o).
% 129.49/25.35 thf(func_def_1095, type, orderi2935705153647960990t_unit: (set_Product_unit > set_Product_unit > $o) > (set_Product_unit > set_Product_unit > $o) > set_Product_unit > $o).
% 129.49/25.35 thf(func_def_1096, type, top_to2462227307520033056at_o_o: (produc3658429121746597890et_nat > $o) > $o).
% 129.49/25.35 thf(func_def_1097, type, top_top_o_o: $o > $o).
% 129.49/25.35 thf(func_def_1098, type, top_top_int_o: int > $o).
% 129.49/25.35 thf(func_def_1099, type, top_top_nat_o: nat > $o).
% 129.49/25.35 thf(func_def_1100, type, top_to5213382387105117981_nat_o: produc3658429121746597890et_nat > $o).
% 129.49/25.35 thf(func_def_1101, type, top_to2465898995584390880unit_o: product_unit > $o).
% 129.49/25.35 thf(func_def_1103, type, top_top_assn: assn).
% 129.49/25.35 thf(func_def_1104, type, top_top_filter_nat: filter_nat).
% 129.49/25.35 thf(func_def_1105, type, top_top_Product_unit: product_unit).
% 129.49/25.35 thf(func_def_1106, type, top_to3804862627464599871nteger: set_Co726186806682698383nteger).
% 129.49/25.35 thf(func_def_1107, type, top_to4869887016220417533_nat_o: set_Pr4532377907799695533_nat_o).
% 129.49/25.35 thf(func_def_1108, type, top_top_set_o: set_o).
% 129.49/25.35 thf(func_def_1109, type, top_top_set_assn: set_assn).
% 129.49/25.35 thf(func_def_1110, type, top_to4645266643341252675nteger: set_Code_integer).
% 129.49/25.35 thf(func_def_1111, type, top_to9103716463852441742atural: set_Code_natural).
% 129.49/25.35 thf(func_def_1112, type, top_top_set_int: set_int).
% 129.49/25.35 thf(func_def_1113, type, top_top_set_nat: set_nat).
% 129.49/25.35 thf(func_def_1114, type, top_to2658285827425157974eger_o: set_Pr448751882837621926eger_o).
% 129.49/25.35 thf(func_def_1115, type, top_to2363841527343591755nteger: set_Pr4811707699266497531nteger).
% 129.49/25.35 thf(func_def_1116, type, top_to6270328743839206426er_int: set_Pr6496905503482889930er_int).
% 129.49/25.35 thf(func_def_1117, type, top_to8760411888770933182er_nat: set_Pr8986988648414616686er_nat).
% 129.49/25.35 thf(func_def_1118, type, top_to3244471725241573579t_unit: set_Pr4580871404036792859t_unit).
% 129.49/25.35 thf(func_def_1119, type, top_to2524957550908328808et_nat: set_Pr3948176798113811640et_nat).
% 129.49/25.35 thf(func_def_1120, type, top_to4366644338036079209nt_int: set_Pr958786334691620121nt_int).
% 129.49/25.35 thf(func_def_1121, type, top_to1996260823553986621t_unit: set_Product_unit).
% 129.49/25.35 thf(func_def_1122, type, top_top_set_rat: set_rat).
% 129.49/25.35 thf(func_def_1123, type, top_to8515903933492314333_nat_o: set_se1353816476878259085_nat_o).
% 129.49/25.35 thf(func_def_1124, type, top_top_set_set_o: set_set_o).
% 129.49/25.35 thf(func_def_1125, type, top_top_set_set_int: set_set_int).
% 129.49/25.35 thf(func_def_1126, type, top_top_set_set_nat: set_set_nat).
% 129.49/25.35 thf(func_def_1127, type, top_to1767297665138865437t_unit: set_set_Product_unit).
% 129.49/25.35 thf(func_def_1128, type, power_power_assn: assn > nat > assn).
% 129.49/25.35 thf(func_def_1129, type, power_8256067586552552935nteger: code_integer > nat > code_integer).
% 129.49/25.35 thf(func_def_1130, type, power_7079662738309270450atural: code_natural > nat > code_natural).
% 129.49/25.35 thf(func_def_1131, type, power_power_int: int > nat > int).
% 129.49/25.35 thf(func_def_1132, type, power_power_nat: nat > nat > nat).
% 129.49/25.35 thf(func_def_1133, type, power_power_rat: rat > nat > rat).
% 129.49/25.35 thf(func_def_1134, type, produc1721739103881910569nt_int: (int > int) > product_prod_int_int > produc5030937667867769463nt_int).
% 129.49/25.35 thf(func_def_1135, type, produc2818822084525909125nt_int: (int > nat) > product_prod_int_int > produc6128020648511768019nt_int).
% 129.49/25.35 thf(func_def_1136, type, produc3140199747736941517at_nat: (nat > int) > product_prod_nat_nat > produc3574062574353363227at_nat).
% 129.49/25.35 thf(func_def_1137, type, produc4237282728380940073at_nat: (nat > nat) > product_prod_nat_nat > produc4671145554997361783at_nat).
% 129.49/25.35 thf(func_def_1138, type, produc7368190662567826135_nat_o: (produc3658429121746597890et_nat > $o) > (produc3658429121746597890et_nat > $o) > produc4928098042776334183_nat_o).
% 129.49/25.35 thf(func_def_1139, type, produc2245416461498447860et_nat: (produc3658429121746597890et_nat > $o) > produc3925858234332021118et_nat > produc2732055786443039994et_nat).
% 129.49/25.35 thf(func_def_1140, type, produc5001842942810119800et_nat: (produc3658429121746597890et_nat > $o) > produc3658429121746597890et_nat > produc3925858234332021118et_nat).
% 129.49/25.35 thf(func_def_1141, type, product_Pair_o_o: $o > $o > product_prod_o_o).
% 129.49/25.35 thf(func_def_1142, type, product_Pair_o_int: $o > int > product_prod_o_int).
% 129.49/25.35 thf(func_def_1143, type, product_Pair_o_nat: $o > nat > product_prod_o_nat).
% 129.49/25.35 thf(func_def_1144, type, produc3762353314782720579et_nat: $o > produc3658429121746597890et_nat > produc6197004810343482825et_nat).
% 129.49/25.35 thf(func_def_1145, type, produc3901946521647900940t_unit: $o > product_unit > produc3788275437394581276t_unit).
% 129.49/25.35 thf(func_def_1146, type, produc1086072967326762835nteger: code_integer > code_integer > produc8923325533196201883nteger).
% 129.49/25.35 thf(func_def_1147, type, produc3574140220909816553atural: code_natural > code_natural > produc7822875418678951345atural).
% 129.49/25.35 thf(func_def_1148, type, produc6639722614265839536atural: code_natural > produc7822875418678951345atural > produc5835291356934675326atural).
% 129.49/25.35 thf(func_def_1149, type, produc8256076736985659095t_unit: heap_e7401611519738050253t_unit > heap_e7401611519738050253t_unit > produc8126358835630522215t_unit).
% 129.49/25.35 thf(func_def_1150, type, produc7507926704131184380et_nat: heap_e7401611519738050253t_unit > set_nat > produc3658429121746597890et_nat).
% 129.49/25.35 thf(func_def_1151, type, product_Pair_int_o: int > $o > product_prod_int_o).
% 129.49/25.35 thf(func_def_1152, type, product_Pair_int_int: int > int > product_prod_int_int).
% 129.49/25.35 thf(func_def_1153, type, product_Pair_int_nat: int > nat > product_prod_int_nat).
% 129.49/25.35 thf(func_def_1154, type, produc550872421827702152t_unit: int > product_unit > produc6865911496949736918t_unit).
% 129.49/25.35 thf(func_def_1155, type, product_Pair_nat_o: nat > $o > product_prod_nat_o).
% 129.49/25.35 thf(func_def_1156, type, product_Pair_nat_int: nat > int > product_prod_nat_int).
% 129.49/25.35 thf(func_def_1157, type, product_Pair_nat_nat: nat > nat > product_prod_nat_nat).
% 129.49/25.35 thf(func_def_1158, type, produc3434450474625521892t_unit: nat > product_unit > produc526117512892780850t_unit).
% 129.49/25.35 thf(func_def_1159, type, produc1940133919992309389et_nat: produc2732055786443039994et_nat > produc2732055786443039994et_nat > produc5657529347773406293et_nat).
% 129.49/25.35 thf(func_def_1160, type, produc3674526432113299062_nat_o: produc3925858234332021118et_nat > (produc3658429121746597890et_nat > $o) > produc4962441903510821572_nat_o).
% 129.49/25.35 thf(func_def_1161, type, produc8599840265553166229et_nat: produc3925858234332021118et_nat > produc3925858234332021118et_nat > produc6830853553727218525et_nat).
% 129.49/25.35 thf(func_def_1162, type, produc5530789758353788441et_nat: produc3925858234332021118et_nat > produc3658429121746597890et_nat > produc4658369026151987169et_nat).
% 129.49/25.35 thf(func_def_1163, type, produc2829238748666232050_nat_o: produc3658429121746597890et_nat > (produc3658429121746597890et_nat > $o) > produc6907119039986491712_nat_o).
% 129.49/25.35 thf(func_def_1164, type, produc3909112513931304473et_nat: produc3658429121746597890et_nat > produc3925858234332021118et_nat > produc7389280635074931169et_nat).
% 129.49/25.35 thf(func_def_1165, type, produc8199053930788261021et_nat: produc3658429121746597890et_nat > produc3658429121746597890et_nat > produc8111630337999740517et_nat).
% 129.49/25.35 thf(func_def_1166, type, produc3646306378393792727nt_int: product_prod_int_int > product_prod_int_int > produc1219242969750017639nt_int).
% 129.49/25.35 thf(func_def_1167, type, produc8921240378546987991nt_nat: product_prod_int_nat > product_prod_int_nat > produc4806409095325742951nt_nat).
% 129.49/25.35 thf(func_def_1168, type, produc886916002739626967at_int: product_prod_nat_int > product_prod_nat_int > produc6495656768158660455at_int).
% 129.49/25.35 thf(func_def_1169, type, produc6161850002892822231at_nat: product_prod_nat_nat > product_prod_nat_nat > produc859450856879609959at_nat).
% 129.49/25.35 thf(func_def_1170, type, produc8487917018914037602unit_o: product_unit > $o > produc7485711547916492274unit_o).
% 129.49/25.35 thf(func_def_1171, type, produc4800797859304479970it_int: product_unit > int > produc5646078835734834920it_int).
% 129.49/25.35 thf(func_def_1172, type, produc4803288329813530246it_nat: product_unit > nat > produc600557818389255820it_nat).
% 129.49/25.35 thf(func_def_1173, type, produc5692694829384537815t_unit: product_unit > product_unit > produc8459935480633519975t_unit).
% 129.49/25.35 thf(func_def_1174, type, produc5149316574908490455_nat_o: set_Pr4532377907799695533_nat_o > set_Pr4532377907799695533_nat_o > produc6666532470309664615_nat_o).
% 129.49/25.35 thf(func_def_1175, type, produc5838405689764958487_set_o: set_o > set_o > produc7369051934464679207_set_o).
% 129.49/25.35 thf(func_def_1176, type, produc7443773368509356479nteger: set_Code_integer > set_Code_integer > produc6491284506569428743nteger).
% 129.49/25.35 thf(func_def_1177, type, produc6363374080413544029et_int: set_int > set_int > produc2115011035271226405et_int).
% 129.49/25.35 thf(func_def_1178, type, produc2471974422597015918t_unit: set_nat > heap_e7401611519738050253t_unit > produc7348995623955488188t_unit).
% 129.49/25.35 thf(func_def_1179, type, produc4532415448927165861et_nat: set_nat > set_nat > produc7819656566062154093et_nat).
% 129.49/25.35 thf(func_def_1180, type, produc7269801343787132631t_unit: set_Product_unit > set_Product_unit > produc1357844946079266663t_unit).
% 129.49/25.35 thf(func_def_1181, type, produc3142209312014666069_o_nat: set_Pr4532377907799695533_nat_o > ((produc3658429121746597890et_nat > $o) > set_nat) > set_Pr4769669219625338370_o_nat).
% 129.49/25.35 thf(func_def_1182, type, produc7016705354093686019et_nat: set_Pr4532377907799695533_nat_o > ((produc3658429121746597890et_nat > $o) > set_Pr3286484037609594932et_nat) > set_Pr8536935166611901872et_nat).
% 129.49/25.35 thf(func_def_1183, type, produc8052021730408606599et_nat: set_Pr4532377907799695533_nat_o > ((produc3658429121746597890et_nat > $o) > set_Pr3948176798113811640et_nat) > set_Pr3286484037609594932et_nat).
% 129.49/25.35 thf(func_def_1184, type, product_Sigma_o_o: set_o > ($o > set_o) > set_Product_prod_o_o).
% 129.49/25.35 thf(func_def_1185, type, product_Sigma_o_int: set_o > ($o > set_int) > set_Pr8834758594704517033_o_int).
% 129.49/25.35 thf(func_def_1186, type, product_Sigma_o_nat: set_o > ($o > set_nat) > set_Pr2101469702781467981_o_nat).
% 129.49/25.35 thf(func_def_1187, type, produc411009820408249917t_unit: set_o > ($o > set_Product_unit) > set_Pr6671815714118170108t_unit).
% 129.49/25.35 thf(func_def_1188, type, produc5260988393495149579et_nat: set_he8644174446132254637t_unit > (heap_e7401611519738050253t_unit > set_set_nat) > set_Pr3948176798113811640et_nat).
% 129.49/25.35 thf(func_def_1189, type, product_Sigma_int_o: set_int > (int > set_o) > set_Pr903927857289325719_int_o).
% 129.49/25.35 thf(func_def_1190, type, produc1453890548195736866nt_int: set_int > (int > set_int) > set_Pr958786334691620121nt_int).
% 129.49/25.35 thf(func_def_1191, type, produc1456381018704787142nt_nat: set_int > (int > set_nat) > set_Pr3448869479623346877nt_nat).
% 129.49/25.35 thf(func_def_1192, type, produc2622961094310184599t_unit: set_int > (int > set_Product_unit) > set_Pr5889264611543340172t_unit).
% 129.49/25.35 thf(func_def_1193, type, produc5650067098820117939_nat_o: set_nat > (nat > set_Pr4532377907799695533_nat_o) > set_Pr5623950326363488168_nat_o).
% 129.49/25.35 thf(func_def_1194, type, product_Sigma_nat_o: set_nat > (nat > set_o) > set_Pr3149072824959771635_nat_o).
% 129.49/25.35 thf(func_def_1195, type, produc454536836294682310at_int: set_nat > (nat > set_int) > set_Pr7995236796853374141at_int).
% 129.49/25.35 thf(func_def_1196, type, produc457027306803732586at_nat: set_nat > (nat > set_nat) > set_Pr1261947904930325089at_nat).
% 129.49/25.35 thf(func_def_1197, type, produc5506539147108004339t_unit: set_nat > (nat > set_Product_unit) > set_Pr4334478416066269672t_unit).
% 129.49/25.35 thf(func_def_1198, type, produc4996980317674386579unit_o: set_Product_unit > (product_unit > set_o) > set_Pr5464989305201766866unit_o).
% 129.49/25.35 thf(func_def_1199, type, produc6875377002296012693it_nat: set_Product_unit > (product_unit > set_nat) > set_Pr1763845938948868674it_nat).
% 129.49/25.35 thf(func_def_1200, type, produc6121607364029222536t_unit: set_Product_unit > (product_unit > set_Product_unit) > set_Pr5094982260447487303t_unit).
% 129.49/25.35 thf(func_def_1201, type, product_Unity: product_unit).
% 129.49/25.35 thf(func_def_1202, type, produc1442628905706551930et_nat: ((produc3658429121746597890et_nat > $o) > produc3658429121746597890et_nat > $o) > produc2732055786443039994et_nat > produc2732055786443039994et_nat).
% 129.49/25.35 thf(func_def_1203, type, produc3198230245617193214et_nat: ((produc3658429121746597890et_nat > $o) > produc3658429121746597890et_nat > $o) > produc3925858234332021118et_nat > produc3925858234332021118et_nat).
% 129.49/25.35 thf(func_def_1204, type, produc3195571438000033749nteger: (code_integer > code_integer) > produc8923325533196201883nteger > produc8923325533196201883nteger).
% 129.49/25.35 thf(func_def_1205, type, produc4273547285574235514et_nat: (heap_e7401611519738050253t_unit > produc3658429121746597890et_nat > $o) > produc7936767299998605306et_nat > produc2732055786443039994et_nat).
% 129.49/25.35 thf(func_def_1206, type, produc6095824665904122878et_nat: (heap_e7401611519738050253t_unit > produc3658429121746597890et_nat > $o) > produc5697189838369619070et_nat > produc3925858234332021118et_nat).
% 129.49/25.35 thf(func_def_1207, type, produc962520821680634754et_nat: (heap_e7401611519738050253t_unit > produc3658429121746597890et_nat > $o) > produc3658429121746597890et_nat > produc6500295350249520386et_nat).
% 129.49/25.35 thf(func_def_1208, type, produc4172003873565521026et_nat: (heap_e7401611519738050253t_unit > heap_e7401611519738050253t_unit) > produc3658429121746597890et_nat > produc3658429121746597890et_nat).
% 129.49/25.35 thf(func_def_1209, type, produc7422730317258219nt_int: (heap_e7401611519738050253t_unit > int) > produc2475246329834808232it_int > product_prod_int_int).
% 129.49/25.35 thf(func_def_1210, type, produc3346058198459025221et_nat: (heap_e7401611519738050253t_unit > int) > produc3658429121746597890et_nat > produc2771989624626745363et_nat).
% 129.49/25.35 thf(func_def_1211, type, produc8233931525780029747at_nat: (heap_e7401611519738050253t_unit > nat) > produc6653097349344004940it_nat > product_prod_nat_nat).
% 129.49/25.35 thf(func_def_1212, type, produc2974404638222180585et_nat: (heap_e7401611519738050253t_unit > nat) > produc3658429121746597890et_nat > produc2400336064389900727et_nat).
% 129.49/25.35 thf(func_def_1213, type, produc1933289243015553937_o_int: (int > produc3658429121746597890et_nat > $o) > product_prod_int_int > produc3776562785724475560_o_int).
% 129.49/25.35 thf(func_def_1214, type, produc3602029435013415139et_nat: (int > produc3658429121746597890et_nat > $o) > produc1049655590333717003et_nat > produc2732055786443039994et_nat).
% 129.49/25.35 thf(func_def_1215, type, produc5259063908904861543et_nat: (int > produc3658429121746597890et_nat > $o) > produc8065447618272157327et_nat > produc3925858234332021118et_nat).
% 129.49/25.35 thf(func_def_1216, type, produc1158806734383251089it_int: (int > heap_e7401611519738050253t_unit) > product_prod_int_int > produc2475246329834808232it_int).
% 129.49/25.35 thf(func_def_1217, type, produc7086134074927288299et_nat: (int > heap_e7401611519738050253t_unit) > produc2771989624626745363et_nat > produc3658429121746597890et_nat).
% 129.49/25.35 thf(func_def_1218, type, produc2355237562570658114nt_int: (int > int) > product_prod_int_int > product_prod_int_int).
% 129.49/25.35 thf(func_def_1219, type, produc1355883850669603558at_int: (int > nat) > product_prod_int_int > product_prod_nat_int).
% 129.49/25.35 thf(func_def_1220, type, produc1358374321178653834at_nat: (int > nat) > product_prod_int_nat > product_prod_nat_nat).
% 129.49/25.35 thf(func_def_1221, type, produc986720760941809198at_nat: (nat > nat) > product_prod_nat_nat > product_prod_nat_nat).
% 129.49/25.35 thf(func_def_1222, type, produc9020447175693601247nteger: (code_integer > produc8923325533196201883nteger > produc8923325533196201883nteger) > produc8923325533196201883nteger > produc5966032636589952099nteger).
% 129.49/25.35 thf(func_def_1223, type, produc6499014454317279255nteger: (code_integer > code_integer) > produc8923325533196201883nteger > produc8923325533196201883nteger).
% 129.49/25.35 thf(func_def_1224, type, produc4463282112584876420nt_int: (int > int) > product_prod_int_int > product_prod_int_int).
% 129.49/25.35 thf(func_def_1225, type, produc3463928400683821864at_int: (int > nat) > product_prod_int_int > product_prod_int_nat).
% 129.49/25.35 thf(func_def_1226, type, produc3466418871192872140at_nat: (int > nat) > product_prod_nat_int > product_prod_nat_nat).
% 129.49/25.35 thf(func_def_1227, type, produc1783401272518604195_nat_o: (int > produc3925858234332021118et_nat) > produc3776562785724475560_o_int > produc2732055786443039994et_nat).
% 129.49/25.35 thf(func_def_1228, type, produc2878475440847162070at_int: (int > produc3925858234332021118et_nat) > product_prod_int_int > produc1049655590333717003et_nat).
% 129.49/25.35 thf(func_def_1229, type, produc3447581998352395295_nat_o: (int > produc3658429121746597890et_nat) > produc3776562785724475560_o_int > produc3925858234332021118et_nat).
% 129.49/25.35 thf(func_def_1230, type, produc9121118195053341018at_int: (int > produc3658429121746597890et_nat) > product_prod_int_int > produc8065447618272157327et_nat).
% 129.49/25.35 thf(func_def_1231, type, produc2122153897363312795t_unit: (int > set_nat) > produc2475246329834808232it_int > produc3658429121746597890et_nat).
% 129.49/25.35 thf(func_def_1232, type, produc1253519502038700766at_int: (int > set_nat) > product_prod_int_int > produc2771989624626745363et_nat).
% 129.49/25.35 thf(func_def_1233, type, produc3094765310956027504at_nat: (nat > nat) > product_prod_nat_nat > product_prod_nat_nat).
% 129.49/25.35 thf(func_def_1234, type, produc969530845752564945_nat_o: (produc3925858234332021118et_nat > produc3925858234332021118et_nat) > produc2732055786443039994et_nat > produc2732055786443039994et_nat).
% 129.49/25.35 thf(func_def_1235, type, produc1515462096303866701_nat_o: (produc3925858234332021118et_nat > produc3658429121746597890et_nat) > produc2732055786443039994et_nat > produc3925858234332021118et_nat).
% 129.49/25.35 thf(func_def_1236, type, produc7714581247149323085_nat_o: (produc3658429121746597890et_nat > produc3925858234332021118et_nat) > produc3925858234332021118et_nat > produc2732055786443039994et_nat).
% 129.49/25.35 thf(func_def_1237, type, produc8750854537940449737_nat_o: (produc3658429121746597890et_nat > produc3658429121746597890et_nat) > produc3925858234332021118et_nat > produc3925858234332021118et_nat).
% 129.49/25.35 thf(func_def_1238, type, produc7345612820240450715t_unit: (set_nat > int) > produc3658429121746597890et_nat > produc2475246329834808232it_int).
% 129.49/25.35 thf(func_def_1239, type, produc2744032961877892830nt_int: (set_nat > int) > produc2771989624626745363et_nat > product_prod_int_int).
% 129.49/25.35 thf(func_def_1240, type, produc8442695800884449271t_unit: (set_nat > nat) > produc3658429121746597890et_nat > produc6653097349344004940it_nat).
% 129.49/25.35 thf(func_def_1241, type, produc1747169720485888550at_nat: (set_nat > nat) > produc2400336064389900727et_nat > product_prod_nat_nat).
% 129.49/25.35 thf(func_def_1242, type, produc3009152965630989513_nat_o: (set_nat > produc3925858234332021118et_nat) > produc6500295350249520386et_nat > produc2732055786443039994et_nat).
% 129.49/25.35 thf(func_def_1243, type, produc7176435795764513225t_unit: (set_nat > produc3925858234332021118et_nat) > produc3658429121746597890et_nat > produc7936767299998605306et_nat).
% 129.49/25.35 thf(func_def_1244, type, produc1718959742329256773_nat_o: (set_nat > produc3658429121746597890et_nat) > produc6500295350249520386et_nat > produc3925858234332021118et_nat).
% 129.49/25.35 thf(func_def_1245, type, produc7441988914112407621t_unit: (set_nat > produc3658429121746597890et_nat) > produc3658429121746597890et_nat > produc5697189838369619070et_nat).
% 129.49/25.35 thf(func_def_1246, type, produc3286488329990770625t_unit: (set_nat > set_nat) > produc3658429121746597890et_nat > produc3658429121746597890et_nat).
% 129.49/25.35 thf(func_def_1247, type, produc5101573711933517782_nat_o: (produc2732055786443039994et_nat > $o) > (produc3658429121746597890et_nat > $o) > produc3925858234332021118et_nat > $o).
% 129.49/25.35 thf(func_def_1248, type, produc6216949301066131538_nat_o: (produc3925858234332021118et_nat > $o) > (produc3658429121746597890et_nat > $o) > produc3658429121746597890et_nat > $o).
% 129.49/25.35 thf(func_def_1249, type, produc1807123492089482958_nat_o: (produc3658429121746597890et_nat > $o) > heap_e7401611519738050253t_unit > set_nat > $o).
% 129.49/25.35 thf(func_def_1250, type, produc175634133007206835_int_o: (product_prod_int_int > $o) > int > int > $o).
% 129.49/25.35 thf(func_def_1251, type, produc1310100445399344235_nat_o: (product_prod_nat_nat > $o) > nat > nat > $o).
% 129.49/25.35 thf(func_def_1252, type, produc4724772261320961782nteger: ((code_integer > code_integer) > produc8923325533196201883nteger > produc8923325533196201883nteger) > ((code_integer > code_integer) > produc8923325533196201883nteger > produc8923325533196201883nteger) > produc723880050079548563nteger > produc101387210554329899nteger).
% 129.49/25.35 thf(func_def_1253, type, produc6158254626097298938nteger: ((code_integer > code_integer) > produc8923325533196201883nteger > produc8923325533196201883nteger) > (code_integer > produc8923325533196201883nteger > produc8923325533196201883nteger) > produc2221115339011613079nteger > produc101387210554329899nteger).
% 129.49/25.35 thf(func_def_1254, type, produc2784369760062397435at_nat: ((produc3658429121746597890et_nat > $o) > nat) > (produc3658429121746597890et_nat > nat) > produc3925858234332021118et_nat > product_prod_nat_nat).
% 129.49/25.35 thf(func_def_1255, type, produc7818937854452556646nteger: (code_integer > code_integer > code_integer) > (code_integer > code_integer > code_integer) > produc8923325533196201883nteger > produc723880050079548563nteger).
% 129.49/25.35 thf(func_def_1256, type, produc3789967387199524714nteger: (code_integer > code_integer > code_integer) > (code_integer > code_integer) > produc8923325533196201883nteger > produc2221115339011613079nteger).
% 129.49/25.35 thf(func_def_1257, type, produc3090791198996863994nteger: (code_integer > produc8923325533196201883nteger > produc8923325533196201883nteger) > ((code_integer > code_integer) > produc8923325533196201883nteger > produc8923325533196201883nteger) > produc1397831737260518551nteger > produc101387210554329899nteger).
% 129.49/25.35 thf(func_def_1258, type, produc5081366107664765182nteger: (code_integer > produc8923325533196201883nteger > produc8923325533196201883nteger) > (code_integer > produc8923325533196201883nteger > produc8923325533196201883nteger) > produc8923325533196201883nteger > produc101387210554329899nteger).
% 129.49/25.35 thf(func_def_1259, type, produc1084426483498010290er_o_o: (code_integer > code_integer) > ($o > $o) > produc6271795597528267376eger_o > produc6271795597528267376eger_o).
% 129.49/25.35 thf(func_def_1260, type, produc7547902865030867346_o_int: (code_integer > code_integer) > ($o > int) > produc6271795597528267376eger_o > produc2256096406371053034er_int).
% 129.49/25.35 thf(func_def_1261, type, produc7550393335539917622_o_nat: (code_integer > code_integer) > ($o > nat) > produc6271795597528267376eger_o > produc6433947425880249742er_nat).
% 129.49/25.35 thf(func_def_1262, type, produc3530376873132787239t_unit: (code_integer > code_integer) > ($o > product_unit) > produc6271795597528267376eger_o > produc5496629617739085541t_unit).
% 129.49/25.35 thf(func_def_1263, type, produc6023007152014423402nteger: (code_integer > code_integer) > (code_integer > code_integer > code_integer) > produc8923325533196201883nteger > produc1397831737260518551nteger).
% 129.49/25.35 thf(func_def_1264, type, produc1802105962441033582nteger: (code_integer > code_integer) > (code_integer > code_integer) > produc8923325533196201883nteger > produc8923325533196201883nteger).
% 129.49/25.35 thf(func_def_1265, type, produc1622059938646302433nteger: (code_integer > code_integer) > (nat > code_integer) > produc6433947425880249742er_nat > produc8923325533196201883nteger).
% 129.49/25.35 thf(func_def_1266, type, produc8116347370398923901unit_o: (code_integer > code_integer) > (product_unit > $o) > produc5496629617739085541t_unit > produc6271795597528267376eger_o).
% 129.49/25.35 thf(func_def_1267, type, produc4019910415967566983it_int: (code_integer > code_integer) > (product_unit > int) > produc5496629617739085541t_unit > produc2256096406371053034er_int).
% 129.49/25.35 thf(func_def_1268, type, produc4022400886476617259it_nat: (code_integer > code_integer) > (product_unit > nat) > produc5496629617739085541t_unit > produc6433947425880249742er_nat).
% 129.49/25.35 thf(func_def_1269, type, produc3346700480963112818t_unit: (code_integer > code_integer) > (product_unit > product_unit) > produc5496629617739085541t_unit > produc5496629617739085541t_unit).
% 129.49/25.35 thf(func_def_1270, type, produc8234977478398262212et_nat: (heap_e7401611519738050253t_unit > heap_e7401611519738050253t_unit) > (set_nat > set_nat) > produc3658429121746597890et_nat > produc3658429121746597890et_nat).
% 129.49/25.35 thf(func_def_1271, type, produc5973434381925408567at_int: (heap_e7401611519738050253t_unit > int) > (set_nat > int) > produc3658429121746597890et_nat > product_prod_int_int).
% 129.49/25.35 thf(func_def_1272, type, produc8285742784527409791at_nat: (heap_e7401611519738050253t_unit > nat) > (set_nat > nat) > produc3658429121746597890et_nat > product_prod_nat_nat).
% 129.49/25.35 thf(func_def_1273, type, produc3152090231653401565et_nat: (int > heap_e7401611519738050253t_unit) > (int > set_nat) > product_prod_int_int > produc3658429121746597890et_nat).
% 129.49/25.35 thf(func_def_1274, type, produc6036585564866528938nt_int: (int > int) > (int > int) > product_prod_int_int > product_prod_int_int).
% 129.49/25.35 thf(func_def_1275, type, produc5667422475138734578nt_nat: (int > nat) > (int > nat) > product_prod_int_int > product_prod_nat_nat).
% 129.49/25.35 thf(func_def_1276, type, produc79747477514316377et_nat: (nat > produc3658429121746597890et_nat > $o) > (nat > produc3658429121746597890et_nat) > product_prod_nat_nat > produc3925858234332021118et_nat).
% 129.49/25.35 thf(func_def_1277, type, produc2064153442785501et_nat: (nat > heap_e7401611519738050253t_unit) > (nat > set_nat) > product_prod_nat_nat > produc3658429121746597890et_nat).
% 129.49/25.35 thf(func_def_1278, type, produc7347049785058425330at_int: (nat > int) > (nat > int) > product_prod_nat_nat > product_prod_int_int).
% 129.49/25.35 thf(func_def_1279, type, produc6977886695330630970at_nat: (nat > nat) > (nat > nat) > product_prod_nat_nat > product_prod_nat_nat).
% 129.49/25.35 thf(func_def_1280, type, produc1553301316500091796er_int: (code_integer > code_integer > int) > produc8923325533196201883nteger > int).
% 129.49/25.35 thf(func_def_1281, type, produc6916734918728496179nteger: (code_integer > code_integer > produc8923325533196201883nteger) > produc8923325533196201883nteger > produc8923325533196201883nteger).
% 129.49/25.35 thf(func_def_1282, type, produc4947309494688390418_int_o: (int > int > $o) > product_prod_int_int > $o).
% 129.49/25.35 thf(func_def_1283, type, produc4245557441103728435nt_int: (int > int > product_prod_int_int) > product_prod_int_int > product_prod_int_int).
% 129.49/25.35 thf(func_def_1284, type, produc27273713700761075at_nat: (nat > nat > product_prod_nat_nat > product_prod_nat_nat) > product_prod_nat_nat > product_prod_nat_nat > product_prod_nat_nat).
% 129.49/25.35 thf(func_def_1285, type, produc2626176000494625587at_nat: (nat > nat > product_prod_nat_nat) > product_prod_nat_nat > product_prod_nat_nat).
% 129.49/25.35 thf(func_def_1286, type, produc180342877477747464et_nat: produc2732055786443039994et_nat > produc3658429121746597890et_nat > $o).
% 129.49/25.35 thf(func_def_1287, type, produc995936583742144908et_nat: produc3925858234332021118et_nat > produc3658429121746597890et_nat > $o).
% 129.49/25.35 thf(func_def_1288, type, produc8508995932063986495nteger: produc8923325533196201883nteger > code_integer).
% 129.49/25.35 thf(func_def_1289, type, produc8632663045010267826er_nat: produc6433947425880249742er_nat > code_integer).
% 129.49/25.35 thf(func_def_1290, type, produc1824681642469235216et_nat: produc3658429121746597890et_nat > heap_e7401611519738050253t_unit).
% 129.49/25.35 thf(func_def_1291, type, product_fst_int_int: product_prod_int_int > int).
% 129.49/25.35 thf(func_def_1292, type, produc8897218493789310642nteger: produc2591615631378489102nteger > nat).
% 129.49/25.35 thf(func_def_1293, type, product_fst_nat_nat: product_prod_nat_nat > nat).
% 129.49/25.35 thf(func_def_1294, type, produc1609452848092598666_nat_o: produc4962441903510821572_nat_o > produc3925858234332021118et_nat).
% 129.49/25.35 thf(func_def_1295, type, produc8046704426453032966_nat_o: produc6907119039986491712_nat_o > produc3658429121746597890et_nat).
% 129.49/25.35 thf(func_def_1296, type, produc6012101397789842562t_unit: produc7348995623955488188t_unit > set_nat).
% 129.49/25.35 thf(func_def_1297, type, produc5374455773327741254et_nat: produc2732055786443039994et_nat > produc3925858234332021118et_nat).
% 129.49/25.35 thf(func_def_1298, type, produc4011572625026189258et_nat: produc3925858234332021118et_nat > produc3658429121746597890et_nat).
% 129.49/25.35 thf(func_def_1299, type, produc6174133586879617921nteger: produc8923325533196201883nteger > code_integer).
% 129.49/25.35 thf(func_def_1300, type, produc1517335558169710324er_nat: produc6433947425880249742er_nat > nat).
% 129.49/25.35 thf(func_def_1301, type, produc8586169260539613262et_nat: produc3658429121746597890et_nat > set_nat).
% 129.49/25.35 thf(func_def_1302, type, product_snd_int_int: product_prod_int_int > int).
% 129.49/25.35 thf(func_def_1303, type, produc1781891006948753140nteger: produc2591615631378489102nteger > code_integer).
% 129.49/25.35 thf(func_def_1304, type, product_snd_nat_nat: product_prod_nat_nat > nat).
% 129.49/25.35 thf(func_def_1305, type, produc6803565743942592456_nat_o: produc4962441903510821572_nat_o > produc3658429121746597890et_nat > $o).
% 129.49/25.35 thf(func_def_1306, type, produc1838968430882301508_nat_o: produc6907119039986491712_nat_o > produc3658429121746597890et_nat > $o).
% 129.49/25.35 thf(func_def_1307, type, produc3550216979005444800t_unit: produc7348995623955488188t_unit > heap_e7401611519738050253t_unit).
% 129.49/25.35 thf(func_def_1308, type, produc1151471490888544852et_nat: produc2732055786443039994et_nat > produc4962441903510821572_nat_o).
% 129.49/25.35 thf(func_def_1309, type, produc3251101204053596376et_nat: produc3925858234332021118et_nat > produc6907119039986491712_nat_o).
% 129.49/25.35 thf(func_def_1310, type, produc6300697153377358172et_nat: produc3658429121746597890et_nat > produc7348995623955488188t_unit).
% 129.49/25.35 thf(func_def_1311, type, product_swap_int_int: product_prod_int_int > product_prod_int_int).
% 129.49/25.35 thf(func_def_1312, type, product_swap_nat_nat: product_prod_nat_nat > product_prod_nat_nat).
% 129.49/25.35 thf(func_def_1313, type, produc2580581461503396054_nat_o: produc4962441903510821572_nat_o > produc2732055786443039994et_nat).
% 129.49/25.35 thf(func_def_1314, type, produc1078497009909708626_nat_o: produc6907119039986491712_nat_o > produc3925858234332021118et_nat).
% 129.49/25.35 thf(func_def_1315, type, produc1264744871843189710t_unit: produc7348995623955488188t_unit > produc3658429121746597890et_nat).
% 129.49/25.35 thf(func_def_1316, type, produc5538323210962509403atural: (produc7822875418678951345atural > produc5835291356934675326atural) > (code_natural > produc7822875418678951345atural > produc5835291356934675326atural) > produc7822875418678951345atural > produc5835291356934675326atural).
% 129.49/25.35 thf(func_def_1317, type, product_Abs_unit: $o > product_unit).
% 129.49/25.35 thf(func_def_1318, type, product_Rep_unit: product_unit > $o).
% 129.49/25.35 thf(func_def_1319, type, iterat8892046348760725948atural: code_natural > (code_natural > produc7822875418678951345atural > produc5835291356934675326atural) > code_natural > produc7822875418678951345atural > produc5835291356934675326atural).
% 129.49/25.35 thf(func_def_1320, type, log: code_natural > code_natural > code_natural).
% 129.49/25.35 thf(func_def_1321, type, minus_shift: code_natural > code_natural > code_natural > code_natural).
% 129.49/25.35 thf(func_def_1322, type, next: produc7822875418678951345atural > produc5835291356934675326atural).
% 129.49/25.35 thf(func_def_1323, type, range: code_natural > produc7822875418678951345atural > produc5835291356934675326atural).
% 129.49/25.35 thf(func_def_1324, type, abs_Rat: product_prod_int_int > rat).
% 129.49/25.35 thf(func_def_1325, type, fract: int > int > rat).
% 129.49/25.35 thf(func_def_1326, type, rep_Rat: rat > product_prod_int_int).
% 129.49/25.35 thf(func_def_1327, type, normalize: product_prod_int_int > product_prod_int_int).
% 129.49/25.35 thf(func_def_1328, type, pcr_rat: product_prod_int_int > rat > $o).
% 129.49/25.35 thf(func_def_1329, type, positive: rat > $o).
% 129.49/25.35 thf(func_def_1330, type, quotient_of: rat > product_prod_int_int).
% 129.49/25.35 thf(func_def_1331, type, ratrel: product_prod_int_int > product_prod_int_int > $o).
% 129.49/25.35 thf(func_def_1332, type, domain4881241903163524940_nat_o: set_Pr2161125870931222855_nat_o > set_Pr4532377907799695533_nat_o).
% 129.49/25.35 thf(func_def_1333, type, domain3618434998481624895et_nat: set_Pr8536935166611901872et_nat > set_Pr4532377907799695533_nat_o).
% 129.49/25.35 thf(func_def_1334, type, domain2358961613337345475et_nat: set_Pr3286484037609594932et_nat > set_Pr4532377907799695533_nat_o).
% 129.49/25.35 thf(func_def_1335, type, domain_o_o: set_Product_prod_o_o > set_o).
% 129.49/25.35 thf(func_def_1336, type, domain2007548326387822151et_nat: set_Pr3948176798113811640et_nat > set_he8644174446132254637t_unit).
% 129.49/25.35 thf(func_def_1337, type, domain_int_int: set_Pr958786334691620121nt_int > set_int).
% 129.49/25.35 thf(func_def_1338, type, domain_nat_nat: set_Pr1261947904930325089at_nat > set_nat).
% 129.49/25.35 thf(func_def_1339, type, domain4292024440186670924t_unit: set_Pr5094982260447487303t_unit > set_Product_unit).
% 129.49/25.35 thf(func_def_1340, type, field_81821745747897882_nat_o: set_Pr2161125870931222855_nat_o > set_Pr4532377907799695533_nat_o).
% 129.49/25.35 thf(func_def_1341, type, field_o: set_Product_prod_o_o > set_o).
% 129.49/25.35 thf(func_def_1342, type, field_Code_integer: set_Pr4811707699266497531nteger > set_Code_integer).
% 129.49/25.35 thf(func_def_1343, type, field_int: set_Pr958786334691620121nt_int > set_int).
% 129.49/25.35 thf(func_def_1344, type, field_nat: set_Pr1261947904930325089at_nat > set_nat).
% 129.49/25.35 thf(func_def_1345, type, field_Product_unit: set_Pr5094982260447487303t_unit > set_Product_unit).
% 129.49/25.35 thf(func_def_1346, type, id_Pro6775173067513470761_nat_o: set_Pr2161125870931222855_nat_o).
% 129.49/25.35 thf(func_def_1347, type, id_o2: set_Product_prod_o_o).
% 129.49/25.35 thf(func_def_1348, type, id_int: set_Pr958786334691620121nt_int).
% 129.49/25.35 thf(func_def_1349, type, id_nat: set_Pr1261947904930325089at_nat).
% 129.49/25.35 thf(func_def_1350, type, id_Product_unit: set_Pr5094982260447487303t_unit).
% 129.49/25.35 thf(func_def_1351, type, id_on_7438695062414216554_nat_o: set_Pr4532377907799695533_nat_o > set_Pr2161125870931222855_nat_o).
% 129.49/25.35 thf(func_def_1352, type, id_on_o: set_o > set_Product_prod_o_o).
% 129.49/25.35 thf(func_def_1353, type, id_on_int: set_int > set_Pr958786334691620121nt_int).
% 129.49/25.35 thf(func_def_1354, type, id_on_nat: set_nat > set_Pr1261947904930325089at_nat).
% 129.49/25.35 thf(func_def_1355, type, id_on_Product_unit: set_Product_unit > set_Pr5094982260447487303t_unit).
% 129.49/25.35 thf(func_def_1356, type, image_2232984160439939859_nat_o: set_Pr2161125870931222855_nat_o > set_Pr4532377907799695533_nat_o > set_Pr4532377907799695533_nat_o).
% 129.49/25.35 thf(func_def_1357, type, image_7240710759097644216et_nat: set_Pr8536935166611901872et_nat > set_Pr4532377907799695533_nat_o > set_Pr3286484037609594932et_nat).
% 129.49/25.35 thf(func_def_1358, type, image_1814916760386648892et_nat: set_Pr3286484037609594932et_nat > set_Pr4532377907799695533_nat_o > set_Pr3948176798113811640et_nat).
% 129.49/25.35 thf(func_def_1359, type, image_o_o: set_Product_prod_o_o > set_o > set_o).
% 129.49/25.35 thf(func_def_1360, type, image_o_int: set_Pr8834758594704517033_o_int > set_o > set_int).
% 129.49/25.35 thf(func_def_1361, type, image_o_nat: set_Pr2101469702781467981_o_nat > set_o > set_nat).
% 129.49/25.35 thf(func_def_1362, type, image_o_Product_unit: set_Pr6671815714118170108t_unit > set_o > set_Product_unit).
% 129.49/25.35 thf(func_def_1363, type, image_4147621952751189391nteger: set_Pr4811707699266497531nteger > set_Code_integer > set_Code_integer).
% 129.49/25.35 thf(func_def_1364, type, image_3942343239045813184et_nat: set_Pr3948176798113811640et_nat > set_he8644174446132254637t_unit > set_set_nat).
% 129.49/25.35 thf(func_def_1365, type, image_int_o: set_Pr903927857289325719_int_o > set_int > set_o).
% 129.49/25.35 thf(func_def_1366, type, image_int_int: set_Pr958786334691620121nt_int > set_int > set_int).
% 129.49/25.35 thf(func_def_1367, type, image_int_nat: set_Pr3448869479623346877nt_nat > set_int > set_nat).
% 129.49/25.35 thf(func_def_1368, type, image_1405326909533657676t_unit: set_Pr5889264611543340172t_unit > set_int > set_Product_unit).
% 129.49/25.35 thf(func_def_1369, type, image_nat_o: set_Pr3149072824959771635_nat_o > set_nat > set_o).
% 129.49/25.35 thf(func_def_1370, type, image_nat_int: set_Pr7995236796853374141at_int > set_nat > set_int).
% 129.49/25.35 thf(func_def_1371, type, image_nat_nat: set_Pr1261947904930325089at_nat > set_nat > set_nat).
% 129.49/25.35 thf(func_def_1372, type, image_4288904962331477416t_unit: set_Pr4334478416066269672t_unit > set_nat > set_Product_unit).
% 129.49/25.35 thf(func_def_1373, type, image_Product_unit_o: set_Pr5464989305201766866unit_o > set_Product_unit > set_o).
% 129.49/25.35 thf(func_def_1374, type, image_5655252347010435494it_int: set_Pr8497134830871917726it_int > set_Product_unit > set_int).
% 129.49/25.35 thf(func_def_1375, type, image_5657742817519485770it_nat: set_Pr1763845938948868674it_nat > set_Product_unit > set_nat).
% 129.49/25.35 thf(func_def_1376, type, image_8431784415008483091t_unit: set_Pr5094982260447487303t_unit > set_Product_unit > set_Product_unit).
% 129.49/25.35 thf(func_def_1377, type, range_5289318983786108341_nat_o: set_Pr2161125870931222855_nat_o > set_Pr4532377907799695533_nat_o).
% 129.49/25.35 thf(func_def_1378, type, range_7260058354244137878et_nat: set_Pr8536935166611901872et_nat > set_Pr3286484037609594932et_nat).
% 129.49/25.35 thf(func_def_1379, type, range_3398330936903821850et_nat: set_Pr3286484037609594932et_nat > set_Pr3948176798113811640et_nat).
% 129.49/25.35 thf(func_def_1380, type, range_o_o: set_Product_prod_o_o > set_o).
% 129.49/25.35 thf(func_def_1381, type, range_1907724884897625393nteger: set_Pr4811707699266497531nteger > set_Code_integer).
% 129.49/25.35 thf(func_def_1382, type, range_9132311542094983838et_nat: set_Pr3948176798113811640et_nat > set_set_nat).
% 129.49/25.35 thf(func_def_1383, type, range_int_int: set_Pr958786334691620121nt_int > set_int).
% 129.49/25.35 thf(func_def_1384, type, range_nat_nat: set_Pr1261947904930325089at_nat > set_nat).
% 129.49/25.35 thf(func_def_1385, type, range_6617724647562260917t_unit: set_Pr5094982260447487303t_unit > set_Product_unit).
% 129.49/25.35 thf(func_def_1386, type, antisy9190549550370998639_nat_o: set_Pr2161125870931222855_nat_o > $o).
% 129.49/25.35 thf(func_def_1387, type, antisym_o: set_Product_prod_o_o > $o).
% 129.49/25.35 thf(func_def_1388, type, antisym_int: set_Pr958786334691620121nt_int > $o).
% 129.49/25.35 thf(func_def_1389, type, antisym_nat: set_Pr1261947904930325089at_nat > $o).
% 129.49/25.35 thf(func_def_1390, type, antisym_Product_unit: set_Pr5094982260447487303t_unit > $o).
% 129.49/25.35 thf(func_def_1391, type, conver2764731894203590139_nat_o: set_Pr2161125870931222855_nat_o > set_Pr2161125870931222855_nat_o).
% 129.49/25.35 thf(func_def_1392, type, conver3577387295554834384et_nat: set_Pr8536935166611901872et_nat > set_Pr7705059561495379578_nat_o).
% 129.49/25.35 thf(func_def_1393, type, conver4003969491454187092et_nat: set_Pr3286484037609594932et_nat > set_Pr4208978187861016310_nat_o).
% 129.49/25.35 thf(func_def_1394, type, converse_o_o: set_Product_prod_o_o > set_Product_prod_o_o).
% 129.49/25.35 thf(func_def_1395, type, conver7272282955447398104et_nat: set_Pr3948176798113811640et_nat > set_Pr4071100604818519410t_unit).
% 129.49/25.35 thf(func_def_1396, type, converse_int_int: set_Pr958786334691620121nt_int > set_Pr958786334691620121nt_int).
% 129.49/25.35 thf(func_def_1397, type, converse_nat_nat: set_Pr1261947904930325089at_nat > set_Pr1261947904930325089at_nat).
% 129.49/25.35 thf(func_def_1398, type, conver5006497266169685586_nat_o: set_Pr7705059561495379578_nat_o > set_Pr8536935166611901872et_nat).
% 129.49/25.35 thf(func_def_1399, type, conver1831365297310299342_nat_o: set_Pr4208978187861016310_nat_o > set_Pr3286484037609594932et_nat).
% 129.49/25.35 thf(func_def_1400, type, conver2495244158383847931t_unit: set_Pr5094982260447487303t_unit > set_Pr5094982260447487303t_unit).
% 129.49/25.35 thf(func_def_1401, type, conver2236330673913229642t_unit: set_Pr4071100604818519410t_unit > set_Pr3948176798113811640et_nat).
% 129.49/25.35 thf(func_def_1402, type, inv_image_o_o: set_Product_prod_o_o > ($o > $o) > set_Product_prod_o_o).
% 129.49/25.35 thf(func_def_1403, type, inv_image_o_nat: set_Product_prod_o_o > (nat > $o) > set_Pr1261947904930325089at_nat).
% 129.49/25.35 thf(func_def_1404, type, inv_im4602722609573551687t_unit: set_Product_prod_o_o > (product_unit > $o) > set_Pr5094982260447487303t_unit).
% 129.49/25.35 thf(func_def_1405, type, inv_image_int_o: set_Pr958786334691620121nt_int > ($o > int) > set_Product_prod_o_o).
% 129.49/25.35 thf(func_def_1406, type, inv_image_int_int: set_Pr958786334691620121nt_int > (int > int) > set_Pr958786334691620121nt_int).
% 129.49/25.35 thf(func_def_1407, type, inv_image_int_nat: set_Pr958786334691620121nt_int > (nat > int) > set_Pr1261947904930325089at_nat).
% 129.49/25.35 thf(func_def_1408, type, inv_im4740476784543923469t_unit: set_Pr958786334691620121nt_int > (product_unit > int) > set_Pr5094982260447487303t_unit).
% 129.49/25.35 thf(func_def_1409, type, inv_image_nat_o: set_Pr1261947904930325089at_nat > ($o > nat) > set_Product_prod_o_o).
% 129.49/25.35 thf(func_def_1410, type, inv_image_nat_int: set_Pr1261947904930325089at_nat > (int > nat) > set_Pr958786334691620121nt_int).
% 129.49/25.35 thf(func_def_1411, type, inv_image_nat_nat: set_Pr1261947904930325089at_nat > (nat > nat) > set_Pr1261947904930325089at_nat).
% 129.49/25.35 thf(func_def_1412, type, inv_im7624054837341743209t_unit: set_Pr1261947904930325089at_nat > (product_unit > nat) > set_Pr5094982260447487303t_unit).
% 129.49/25.35 thf(func_def_1413, type, inv_im9188693106839688349unit_o: set_Pr5094982260447487303t_unit > ($o > product_unit) > set_Product_prod_o_o).
% 129.49/25.35 thf(func_def_1414, type, inv_im8992892692529751563it_nat: set_Pr5094982260447487303t_unit > (nat > product_unit) > set_Pr1261947904930325089at_nat).
% 129.49/25.35 thf(func_def_1415, type, inv_im8770536620514187154t_unit: set_Pr5094982260447487303t_unit > (product_unit > product_unit) > set_Pr5094982260447487303t_unit).
% 129.49/25.35 thf(func_def_1416, type, refl_o1075864025857821388_nat_o: set_Pr4532377907799695533_nat_o > set_Pr2161125870931222855_nat_o > $o).
% 129.49/25.35 thf(func_def_1417, type, refl_on_o: set_o > set_Product_prod_o_o > $o).
% 129.49/25.35 thf(func_def_1418, type, refl_on_int: set_int > set_Pr958786334691620121nt_int > $o).
% 129.49/25.35 thf(func_def_1419, type, refl_on_nat: set_nat > set_Pr1261947904930325089at_nat > $o).
% 129.49/25.35 thf(func_def_1420, type, refl_on_Product_unit: set_Product_unit > set_Pr5094982260447487303t_unit > $o).
% 129.49/25.35 thf(func_def_1421, type, relcom6251035627269349452_nat_o: set_Pr2161125870931222855_nat_o > set_Pr2161125870931222855_nat_o > set_Pr2161125870931222855_nat_o).
% 129.49/25.35 thf(func_def_1422, type, relcom2608444778321954367et_nat: set_Pr2161125870931222855_nat_o > set_Pr8536935166611901872et_nat > set_Pr8536935166611901872et_nat).
% 129.49/25.35 thf(func_def_1423, type, relcom6619595054755144899et_nat: set_Pr2161125870931222855_nat_o > set_Pr3286484037609594932et_nat > set_Pr3286484037609594932et_nat).
% 129.49/25.35 thf(func_def_1424, type, relcom8303140135939483018et_nat: set_Pr8536935166611901872et_nat > set_Pr7928877670098842301et_nat > set_Pr8536935166611901872et_nat).
% 129.49/25.35 thf(func_def_1425, type, relcom430014394250001422et_nat: set_Pr8536935166611901872et_nat > set_Pr3721596426557623617et_nat > set_Pr3286484037609594932et_nat).
% 129.49/25.35 thf(func_def_1426, type, relcom8031709186682293262et_nat: set_Pr3286484037609594932et_nat > set_Pr8807100098903490369et_nat > set_Pr8536935166611901872et_nat).
% 129.49/25.35 thf(func_def_1427, type, relcom5975814107203797650et_nat: set_Pr3286484037609594932et_nat > set_Pr719794911490849221et_nat > set_Pr3286484037609594932et_nat).
% 129.49/25.35 thf(func_def_1428, type, relcomp_o_o_o: set_Product_prod_o_o > set_Product_prod_o_o > set_Product_prod_o_o).
% 129.49/25.35 thf(func_def_1429, type, relcom4930750947919833671et_nat: set_Pr1776176089890104647t_unit > set_Pr3948176798113811640et_nat > set_Pr3948176798113811640et_nat).
% 129.49/25.35 thf(func_def_1430, type, relcom8881708762719574682et_nat: set_Pr3948176798113811640et_nat > set_Pr5488025237498180813et_nat > set_Pr3948176798113811640et_nat).
% 129.49/25.35 thf(func_def_1431, type, relcomp_int_int_int: set_Pr958786334691620121nt_int > set_Pr958786334691620121nt_int > set_Pr958786334691620121nt_int).
% 129.49/25.35 thf(func_def_1432, type, relcomp_nat_nat_nat: set_Pr1261947904930325089at_nat > set_Pr1261947904930325089at_nat > set_Pr1261947904930325089at_nat).
% 129.49/25.35 thf(func_def_1433, type, relcom848357998638694028t_unit: set_Pr5094982260447487303t_unit > set_Pr5094982260447487303t_unit > set_Pr5094982260447487303t_unit).
% 129.49/25.35 thf(func_def_1434, type, relcom7236094467932586796_nat_o: set_Pr3813038829465207111_nat_o > set_Pr3813038829465207111_nat_o > set_Pr3813038829465207111_nat_o).
% 129.49/25.35 thf(func_def_1435, type, relcom7534720040753365047_set_o: set_Pr4577759397028426247_set_o > set_Pr4577759397028426247_set_o > set_Pr4577759397028426247_set_o).
% 129.49/25.35 thf(func_def_1436, type, relcom4034950705519254697et_int: set_Pr2522554150109002629et_int > set_Pr2522554150109002629et_int > set_Pr2522554150109002629et_int).
% 129.49/25.35 thf(func_def_1437, type, relcom236561585145426837et_nat: set_Pr5488025237498180813et_nat > set_Pr5488025237498180813et_nat > set_Pr5488025237498180813et_nat).
% 129.49/25.35 thf(func_def_1438, type, relcom7108816970757792620t_unit: set_Pr3980508049012409671t_unit > set_Pr3980508049012409671t_unit > set_Pr3980508049012409671t_unit).
% 129.49/25.35 thf(func_def_1439, type, single261872189793494280et_nat: set_Pr8536935166611901872et_nat > $o).
% 129.49/25.35 thf(func_def_1440, type, single7715245794981374860et_nat: set_Pr3286484037609594932et_nat > $o).
% 129.49/25.35 thf(func_def_1441, type, single6620584844202787856et_nat: set_Pr3948176798113811640et_nat > $o).
% 129.49/25.35 thf(func_def_1442, type, single6503947681363884253nt_int: set_Pr958786334691620121nt_int > $o).
% 129.49/25.35 thf(func_def_1443, type, single5507084439971879973at_nat: set_Pr1261947904930325089at_nat > $o).
% 129.49/25.35 thf(func_def_1444, type, total_7429123086324524369_nat_o: set_Pr4532377907799695533_nat_o > set_Pr2161125870931222855_nat_o > $o).
% 129.49/25.35 thf(func_def_1445, type, total_on_o: set_o > set_Product_prod_o_o > $o).
% 129.49/25.35 thf(func_def_1446, type, total_on_int: set_int > set_Pr958786334691620121nt_int > $o).
% 129.49/25.35 thf(func_def_1447, type, total_on_nat: set_nat > set_Pr1261947904930325089at_nat > $o).
% 129.49/25.35 thf(func_def_1448, type, total_5665700600833547665t_unit: set_Product_unit > set_Pr5094982260447487303t_unit > $o).
% 129.49/25.35 thf(func_def_1449, type, trans_int: set_Pr958786334691620121nt_int > $o).
% 129.49/25.35 thf(func_def_1450, type, trans_nat: set_Pr1261947904930325089at_nat > $o).
% 129.49/25.35 thf(func_def_1451, type, algebr932160517623751201me_int: int > int > $o).
% 129.49/25.35 thf(func_def_1452, type, algebr934650988132801477me_nat: nat > nat > $o).
% 129.49/25.35 thf(func_def_1453, type, divide6298287555418463151nteger: code_integer > code_integer > code_integer).
% 129.49/25.35 thf(func_def_1454, type, divide5121882707175180666atural: code_natural > code_natural > code_natural).
% 129.49/25.35 thf(func_def_1455, type, divide_divide_int: int > int > int).
% 129.49/25.35 thf(func_def_1456, type, divide_divide_nat: nat > nat > nat).
% 129.49/25.35 thf(func_def_1457, type, divide_divide_rat: rat > rat > rat).
% 129.49/25.35 thf(func_def_1458, type, dvd_dvd_assn: assn > assn > $o).
% 129.49/25.35 thf(func_def_1459, type, dvd_dvd_Code_integer: code_integer > code_integer > $o).
% 129.49/25.35 thf(func_def_1460, type, dvd_dvd_Code_natural: code_natural > code_natural > $o).
% 129.49/25.35 thf(func_def_1461, type, dvd_dvd_int: int > int > $o).
% 129.49/25.35 thf(func_def_1462, type, dvd_dvd_nat: nat > nat > $o).
% 129.49/25.35 thf(func_def_1463, type, dvd_dvd_rat: rat > rat > $o).
% 129.49/25.35 thf(func_def_1464, type, modulo8411746178871703098atural: code_natural > code_natural > code_natural).
% 129.49/25.35 thf(func_def_1465, type, modulo_modulo_int: int > int > int).
% 129.49/25.35 thf(func_def_1466, type, modulo_modulo_nat: nat > nat > nat).
% 129.49/25.35 thf(func_def_1467, type, zero_n356916108424825756nteger: $o > code_integer).
% 129.49/25.35 thf(func_def_1468, type, zero_n2684676970156552555ol_int: $o > int).
% 129.49/25.35 thf(func_def_1469, type, zero_n2687167440665602831ol_nat: $o > nat).
% 129.49/25.35 thf(func_def_1470, type, collec939566748876313656_nat_o: ((produc3658429121746597890et_nat > $o) > $o) > set_Pr4532377907799695533_nat_o).
% 129.49/25.35 thf(func_def_1471, type, collect_o: ($o > $o) > set_o).
% 129.49/25.35 thf(func_def_1472, type, collect_int: (int > $o) > set_int).
% 129.49/25.35 thf(func_def_1473, type, collect_nat: (nat > $o) > set_nat).
% 129.49/25.35 thf(func_def_1474, type, collect_Product_unit: (product_unit > $o) > set_Product_unit).
% 129.49/25.35 thf(func_def_1475, type, pow_Pr7829143134171777386_nat_o: set_Pr4532377907799695533_nat_o > set_se1353816476878259085_nat_o).
% 129.49/25.35 thf(func_def_1476, type, pow_o: set_o > set_set_o).
% 129.49/25.35 thf(func_def_1477, type, pow_Code_integer: set_Code_integer > set_set_Code_integer).
% 129.49/25.35 thf(func_def_1478, type, pow_int: set_int > set_set_int).
% 129.49/25.35 thf(func_def_1479, type, pow_nat: set_nat > set_set_nat).
% 129.49/25.35 thf(func_def_1480, type, pow_Product_unit: set_Product_unit > set_set_Product_unit).
% 129.49/25.35 thf(func_def_1481, type, pow_set_nat: set_set_nat > set_set_set_nat).
% 129.49/25.35 thf(func_def_1482, type, disjnt7869448817988085652_nat_o: set_Pr4532377907799695533_nat_o > set_Pr4532377907799695533_nat_o > $o).
% 129.49/25.35 thf(func_def_1483, type, disjnt_o: set_o > set_o > $o).
% 129.49/25.35 thf(func_def_1484, type, disjnt_int: set_int > set_int > $o).
% 129.49/25.35 thf(func_def_1485, type, disjnt_nat: set_nat > set_nat > $o).
% 129.49/25.35 thf(func_def_1486, type, disjnt4476176636800211563_nat_o: set_Pr5623950326363488168_nat_o > set_Pr5623950326363488168_nat_o > $o).
% 129.49/25.35 thf(func_def_1487, type, disjnt5652777605541633334_nat_o: set_Pr3149072824959771635_nat_o > set_Pr3149072824959771635_nat_o > $o).
% 129.49/25.35 thf(func_def_1488, type, disjnt7759068009871657124at_int: set_Pr7995236796853374141at_int > set_Pr7995236796853374141at_int > $o).
% 129.49/25.35 thf(func_def_1489, type, disjnt2713546992526078024at_nat: set_Pr1261947904930325089at_nat > set_Pr1261947904930325089at_nat > $o).
% 129.49/25.35 thf(func_def_1490, type, disjnt595863415355979947t_unit: set_Pr4334478416066269672t_unit > set_Pr4334478416066269672t_unit > $o).
% 129.49/25.35 thf(func_def_1491, type, disjnt_Product_unit: set_Product_unit > set_Product_unit > $o).
% 129.49/25.35 thf(func_def_1492, type, image_8639947063239330806_nat_o: (((produc3658429121746597890et_nat > $o) > $o) > set_Pr4532377907799695533_nat_o) > set_Pr6409761941060870310at_o_o > set_se1353816476878259085_nat_o).
% 129.49/25.35 thf(func_def_1493, type, image_o_o_set_o: (($o > $o) > set_o) > set_o_o > set_set_o).
% 129.49/25.35 thf(func_def_1494, type, image_int_o_set_int: ((int > $o) > set_int) > set_int_o > set_set_int).
% 129.49/25.35 thf(func_def_1495, type, image_nat_o_set_nat: ((nat > $o) > set_nat) > set_nat_o > set_set_nat).
% 129.49/25.35 thf(func_def_1496, type, image_3399141053782081349_nat_o: ((produc3658429121746597890et_nat > $o) > produc3658429121746597890et_nat > $o) > set_Pr4532377907799695533_nat_o > set_Pr4532377907799695533_nat_o).
% 129.49/25.35 thf(func_def_1497, type, image_6451031022632655280o_assn: ((produc3658429121746597890et_nat > $o) > assn) > set_Pr4532377907799695533_nat_o > set_assn).
% 129.49/25.35 thf(func_def_1498, type, image_2174381888560928856_o_nat: ((produc3658429121746597890et_nat > $o) > nat) > set_Pr4532377907799695533_nat_o > set_nat).
% 129.49/25.35 thf(func_def_1499, type, image_8451000719036923781t_unit: ((produc3658429121746597890et_nat > $o) > product_unit) > set_Pr4532377907799695533_nat_o > set_Product_unit).
% 129.49/25.35 thf(func_def_1500, type, image_8657708916238040603nteger: ((produc3658429121746597890et_nat > $o) > set_Code_integer) > set_Pr4532377907799695533_nat_o > set_set_Code_integer).
% 129.49/25.35 thf(func_def_1501, type, image_6888641473154130958et_nat: ((produc3658429121746597890et_nat > $o) > set_nat) > set_Pr4532377907799695533_nat_o > set_set_nat).
% 129.49/25.35 thf(func_def_1502, type, image_5138372075374782070t_unit: ((product_unit > $o) > set_Product_unit) > set_Product_unit_o > set_set_Product_unit).
% 129.49/25.35 thf(func_def_1503, type, image_o_o2: ($o > $o) > set_o > set_o).
% 129.49/25.35 thf(func_def_1504, type, image_o_Code_integer: ($o > code_integer) > set_o > set_Code_integer).
% 129.49/25.35 thf(func_def_1505, type, image_o_filter_nat: ($o > filter_nat) > set_o > set_filter_nat).
% 129.49/25.35 thf(func_def_1506, type, image_o_int2: ($o > int) > set_o > set_int).
% 129.49/25.35 thf(func_def_1507, type, image_o_nat2: ($o > nat) > set_o > set_nat).
% 129.49/25.35 thf(func_def_1508, type, image_8407096251609565691nteger: ($o > produc2591615631378489102nteger) > set_o > set_Pr2406931829603198446nteger).
% 129.49/25.35 thf(func_def_1509, type, image_3855930084881510382at_nat: ($o > product_prod_nat_nat) > set_o > set_Pr1261947904930325089at_nat).
% 129.49/25.35 thf(func_def_1510, type, image_o_Product_unit2: ($o > product_unit) > set_o > set_Product_unit).
% 129.49/25.35 thf(func_def_1511, type, image_o_set_o: ($o > set_o) > set_o > set_set_o).
% 129.49/25.35 thf(func_def_1512, type, image_6491638074533506790nteger: ($o > set_Code_integer) > set_o > set_set_Code_integer).
% 129.49/25.35 thf(func_def_1513, type, image_o_set_int: ($o > set_int) > set_o > set_set_int).
% 129.49/25.35 thf(func_def_1514, type, image_o_set_nat: ($o > set_nat) > set_o > set_set_nat).
% 129.49/25.35 thf(func_def_1515, type, image_663195431845683034t_unit: ($o > set_Product_unit) > set_o > set_set_Product_unit).
% 129.49/25.35 thf(func_def_1516, type, image_5465535159827709278_nat_o: (assn > produc3658429121746597890et_nat > $o) > set_assn > set_Pr4532377907799695533_nat_o).
% 129.49/25.35 thf(func_def_1517, type, image_assn_assn: (assn > assn) > set_assn > set_assn).
% 129.49/25.35 thf(func_def_1518, type, image_2177946750938214461nteger: (code_integer > code_integer > code_integer) > set_Code_integer > set_Co726186806682698383nteger).
% 129.49/25.35 thf(func_def_1519, type, image_Code_integer_o: (code_integer > $o) > set_Code_integer > set_o).
% 129.49/25.35 thf(func_def_1520, type, image_4470545334726330049nteger: (code_integer > code_integer) > set_Code_integer > set_Code_integer).
% 129.49/25.35 thf(func_def_1521, type, image_948535463418740880er_int: (code_integer > int) > set_Code_integer > set_int).
% 129.49/25.35 thf(func_def_1522, type, image_951025933927791156er_nat: (code_integer > nat) > set_Code_integer > set_nat).
% 129.49/25.35 thf(func_def_1523, type, image_8049062792287394871nteger: (code_integer > produc8923325533196201883nteger) > set_Code_integer > set_Pr4811707699266497531nteger).
% 129.49/25.35 thf(func_def_1524, type, image_7250923904792228394er_nat: (code_integer > produc6433947425880249742er_nat) > set_Code_integer > set_Pr8986988648414616686er_nat).
% 129.49/25.35 thf(func_def_1525, type, image_3408592110290467754nteger: (code_integer > produc2591615631378489102nteger) > set_Code_integer > set_Pr2406931829603198446nteger).
% 129.49/25.35 thf(func_def_1526, type, image_7182656606676592809t_unit: (code_integer > product_unit) > set_Code_integer > set_Product_unit).
% 129.49/25.35 thf(func_def_1527, type, image_1867029960091478804_set_o: (code_integer > set_o) > set_Code_integer > set_set_o).
% 129.49/25.35 thf(func_def_1528, type, image_4651426196962995782et_int: (code_integer > set_int) > set_Code_integer > set_set_int).
% 129.49/25.35 thf(func_def_1529, type, image_8829277216472192490et_nat: (code_integer > set_nat) > set_Code_integer > set_set_nat).
% 129.49/25.35 thf(func_def_1530, type, image_5870443399019143561t_unit: (code_integer > set_Product_unit) > set_Code_integer > set_set_Product_unit).
% 129.49/25.35 thf(func_def_1531, type, image_6215854944615398399al_nat: (code_natural > nat) > set_Code_natural > set_nat).
% 129.49/25.35 thf(func_def_1532, type, image_int_o2: (int > $o) > set_int > set_o).
% 129.49/25.35 thf(func_def_1533, type, image_1587234942943678608nteger: (int > code_integer) > set_int > set_Code_integer).
% 129.49/25.35 thf(func_def_1534, type, image_int_filter_nat: (int > filter_nat) > set_int > set_filter_nat).
% 129.49/25.35 thf(func_def_1535, type, image_int_int2: (int > int) > set_int > set_int).
% 129.49/25.35 thf(func_def_1536, type, image_int_nat2: (int > nat) > set_int > set_nat).
% 129.49/25.35 thf(func_def_1537, type, image_5846526143423701914t_unit: (int > product_unit) > set_int > set_Product_unit).
% 129.49/25.35 thf(func_def_1538, type, image_int_set_o: (int > set_o) > set_int > set_set_o).
% 129.49/25.35 thf(func_def_1539, type, image_75573793354812870nteger: (int > set_Code_integer) > set_int > set_set_Code_integer).
% 129.49/25.35 thf(func_def_1540, type, image_int_set_nat: (int > set_nat) > set_int > set_set_nat).
% 129.49/25.35 thf(func_def_1541, type, image_8658272795163236474t_unit: (int > set_Product_unit) > set_int > set_set_Product_unit).
% 129.49/25.35 thf(func_def_1542, type, image_4682239675366380726_nat_o: (nat > produc3658429121746597890et_nat > $o) > set_nat > set_Pr4532377907799695533_nat_o).
% 129.49/25.35 thf(func_def_1543, type, image_nat_o2: (nat > $o) > set_nat > set_o).
% 129.49/25.35 thf(func_def_1544, type, image_1215581382706833972nteger: (nat > code_integer) > set_nat > set_Code_integer).
% 129.49/25.35 thf(func_def_1545, type, image_39176534463551487atural: (nat > code_natural) > set_nat > set_Code_natural).
% 129.49/25.35 thf(func_def_1546, type, image_nat_filter_nat: (nat > filter_nat) > set_nat > set_filter_nat).
% 129.49/25.35 thf(func_def_1547, type, image_nat_int2: (nat > int) > set_nat > set_int).
% 129.49/25.35 thf(func_def_1548, type, image_nat_nat2: (nat > nat) > set_nat > set_nat).
% 129.49/25.35 thf(func_def_1549, type, image_8730104196221521654t_unit: (nat > product_unit) > set_nat > set_Product_unit).
% 129.49/25.35 thf(func_def_1550, type, image_nat_set_o: (nat > set_o) > set_nat > set_set_o).
% 129.49/25.35 thf(func_def_1551, type, image_2385391725447763818nteger: (nat > set_Code_integer) > set_nat > set_set_Code_integer).
% 129.49/25.35 thf(func_def_1552, type, image_nat_set_int: (nat > set_int) > set_nat > set_set_int).
% 129.49/25.35 thf(func_def_1553, type, image_nat_set_nat: (nat > set_nat) > set_nat > set_set_nat).
% 129.49/25.35 thf(func_def_1554, type, image_7178329752028323786at_nat: (nat > set_Pr1261947904930325089at_nat) > set_nat > set_se7855581050983116737at_nat).
% 129.49/25.35 thf(func_def_1555, type, image_4824033694279209942t_unit: (nat > set_Product_unit) > set_nat > set_set_Product_unit).
% 129.49/25.35 thf(func_def_1556, type, image_2517677226577294348_nat_o: (produc2732055786443039994et_nat > produc3658429121746597890et_nat > $o) > set_Pr8536935166611901872et_nat > set_Pr4532377907799695533_nat_o).
% 129.49/25.35 thf(func_def_1557, type, image_4557310310409863807et_nat: (produc2732055786443039994et_nat > produc3925858234332021118et_nat) > set_Pr8536935166611901872et_nat > set_Pr3286484037609594932et_nat).
% 129.49/25.35 thf(func_def_1558, type, image_4071802983491761093_nat_o: (produc2732055786443039994et_nat > produc4962441903510821572_nat_o) > set_Pr8536935166611901872et_nat > set_Pr7705059561495379578_nat_o).
% 129.49/25.35 thf(func_def_1559, type, image_6802380912191114376_nat_o: (produc3925858234332021118et_nat > produc3658429121746597890et_nat > $o) > set_Pr3286484037609594932et_nat > set_Pr4532377907799695533_nat_o).
% 129.49/25.35 thf(func_def_1560, type, image_4390311411407041927et_nat: (produc3925858234332021118et_nat > produc3658429121746597890et_nat) > set_Pr3286484037609594932et_nat > set_Pr3948176798113811640et_nat).
% 129.49/25.35 thf(func_def_1561, type, image_7073998564157519676at_nat: (produc3925858234332021118et_nat > product_prod_nat_nat) > set_Pr3286484037609594932et_nat > set_Pr1261947904930325089at_nat).
% 129.49/25.35 thf(func_def_1562, type, image_6229134628169569989_nat_o: (produc3925858234332021118et_nat > produc6907119039986491712_nat_o) > set_Pr3286484037609594932et_nat > set_Pr4208978187861016310_nat_o).
% 129.49/25.35 thf(func_def_1563, type, image_227735840630799463eger_o: (produc6271795597528267376eger_o > produc6271795597528267376eger_o) > set_Pr448751882837621926eger_o > set_Pr448751882837621926eger_o).
% 129.49/25.35 thf(func_def_1564, type, image_6298138160576288691er_int: (produc6271795597528267376eger_o > produc2256096406371053034er_int) > set_Pr448751882837621926eger_o > set_Pr6496905503482889930er_int).
% 129.49/25.35 thf(func_def_1565, type, image_1252617143230709591er_nat: (produc6271795597528267376eger_o > produc6433947425880249742er_nat) > set_Pr448751882837621926eger_o > set_Pr8986988648414616686er_nat).
% 129.49/25.35 thf(func_def_1566, type, image_2425804817718344284t_unit: (produc6271795597528267376eger_o > produc5496629617739085541t_unit) > set_Pr448751882837621926eger_o > set_Pr4580871404036792859t_unit).
% 129.49/25.35 thf(func_def_1567, type, image_7082141833288935621nteger: (produc8923325533196201883nteger > produc8923325533196201883nteger) > set_Pr4811707699266497531nteger > set_Pr4811707699266497531nteger).
% 129.49/25.35 thf(func_def_1568, type, image_4921214744877245202nteger: (produc6433947425880249742er_nat > produc8923325533196201883nteger) > set_Pr8986988648414616686er_nat > set_Pr4811707699266497531nteger).
% 129.49/25.35 thf(func_def_1569, type, image_7060974388358999388eger_o: (produc5496629617739085541t_unit > produc6271795597528267376eger_o) > set_Pr4580871404036792859t_unit > set_Pr448751882837621926eger_o).
% 129.49/25.35 thf(func_def_1570, type, image_4058522669889801726er_int: (produc5496629617739085541t_unit > produc2256096406371053034er_int) > set_Pr4580871404036792859t_unit > set_Pr6496905503482889930er_int).
% 129.49/25.35 thf(func_def_1571, type, image_8236373689398998434er_nat: (produc5496629617739085541t_unit > produc6433947425880249742er_nat) > set_Pr4580871404036792859t_unit > set_Pr8986988648414616686er_nat).
% 129.49/25.35 thf(func_def_1572, type, image_3456600171786721233t_unit: (produc5496629617739085541t_unit > produc5496629617739085541t_unit) > set_Pr4580871404036792859t_unit > set_Pr4580871404036792859t_unit).
% 129.49/25.35 thf(func_def_1573, type, image_6824039242672219652t_unit: (produc3658429121746597890et_nat > heap_e7401611519738050253t_unit) > set_Pr3948176798113811640et_nat > set_he8644174446132254637t_unit).
% 129.49/25.35 thf(func_def_1574, type, image_7426123528223433739et_nat: (produc3658429121746597890et_nat > produc3658429121746597890et_nat) > set_Pr3948176798113811640et_nat > set_Pr3948176798113811640et_nat).
% 129.49/25.35 thf(func_def_1575, type, image_4579277974391593584nt_int: (produc3658429121746597890et_nat > product_prod_int_int) > set_Pr3948176798113811640et_nat > set_Pr958786334691620121nt_int).
% 129.49/25.35 thf(func_def_1576, type, image_7757775281999735736at_nat: (produc3658429121746597890et_nat > product_prod_nat_nat) > set_Pr3948176798113811640et_nat > set_Pr1261947904930325089at_nat).
% 129.49/25.35 thf(func_def_1577, type, image_1893317993577548229t_unit: (produc3658429121746597890et_nat > produc7348995623955488188t_unit) > set_Pr3948176798113811640et_nat > set_Pr4071100604818519410t_unit).
% 129.49/25.35 thf(func_def_1578, type, image_6088997261112373647et_nat: (produc3658429121746597890et_nat > set_nat) > set_Pr3948176798113811640et_nat > set_set_nat).
% 129.49/25.35 thf(func_def_1579, type, image_5042161079198086560nt_int: (product_prod_int_int > int) > set_Pr958786334691620121nt_int > set_int).
% 129.49/25.35 thf(func_def_1580, type, image_2127501072796355446et_nat: (product_prod_int_int > produc3658429121746597890et_nat) > set_Pr958786334691620121nt_int > set_Pr3948176798113811640et_nat).
% 129.49/25.35 thf(func_def_1581, type, image_2653370878348428101nt_int: (product_prod_int_int > product_prod_int_int) > set_Pr958786334691620121nt_int > set_Pr958786334691620121nt_int).
% 129.49/25.35 thf(func_def_1582, type, image_5831868185956570253at_nat: (product_prod_int_int > product_prod_nat_nat) > set_Pr958786334691620121nt_int > set_Pr1261947904930325089at_nat).
% 129.49/25.35 thf(func_def_1583, type, image_2486076414777270412at_nat: (product_prod_nat_nat > nat) > set_Pr1261947904930325089at_nat > set_nat).
% 129.49/25.35 thf(func_def_1584, type, image_5744643625575355962et_nat: (product_prod_nat_nat > produc3925858234332021118et_nat) > set_Pr1261947904930325089at_nat > set_Pr3286484037609594932et_nat).
% 129.49/25.35 thf(func_def_1585, type, image_9068448769188195518et_nat: (product_prod_nat_nat > produc3658429121746597890et_nat) > set_Pr1261947904930325089at_nat > set_Pr3948176798113811640et_nat).
% 129.49/25.35 thf(func_def_1586, type, image_1990417195239315453nt_int: (product_prod_nat_nat > product_prod_int_int) > set_Pr1261947904930325089at_nat > set_Pr958786334691620121nt_int).
% 129.49/25.35 thf(func_def_1587, type, image_5168914502847457605at_nat: (product_prod_nat_nat > product_prod_nat_nat) > set_Pr1261947904930325089at_nat > set_Pr1261947904930325089at_nat).
% 129.49/25.35 thf(func_def_1588, type, image_7255662321850427333et_nat: (produc4962441903510821572_nat_o > produc2732055786443039994et_nat) > set_Pr7705059561495379578_nat_o > set_Pr8536935166611901872et_nat).
% 129.49/25.35 thf(func_def_1589, type, image_4501728140019976901et_nat: (produc6907119039986491712_nat_o > produc3925858234332021118et_nat) > set_Pr4208978187861016310_nat_o > set_Pr3286484037609594932et_nat).
% 129.49/25.35 thf(func_def_1590, type, image_5838825082199883205et_nat: (produc7348995623955488188t_unit > produc3658429121746597890et_nat) > set_Pr4071100604818519410t_unit > set_Pr3948176798113811640et_nat).
% 129.49/25.35 thf(func_def_1591, type, image_Product_unit_o2: (product_unit > $o) > set_Product_unit > set_o).
% 129.49/25.35 thf(func_def_1592, type, image_3815118110082124069nteger: (product_unit > code_integer) > set_Product_unit > set_Code_integer).
% 129.49/25.35 thf(func_def_1593, type, image_1853478764106424488er_nat: (product_unit > filter_nat) > set_Product_unit > set_filter_nat).
% 129.49/25.35 thf(func_def_1594, type, image_873079544045703924it_int: (product_unit > int) > set_Product_unit > set_int).
% 129.49/25.35 thf(func_def_1595, type, image_875570014554754200it_nat: (product_unit > nat) > set_Product_unit > set_nat).
% 129.49/25.35 thf(func_def_1596, type, image_8089323762101378259nteger: (product_unit > produc8923325533196201883nteger) > set_Product_unit > set_Pr4811707699266497531nteger).
% 129.49/25.35 thf(func_def_1597, type, image_8789315437038609094er_nat: (product_unit > produc6433947425880249742er_nat) > set_Product_unit > set_Pr8986988648414616686er_nat).
% 129.49/25.35 thf(func_def_1598, type, image_4103462098156874993nt_int: (product_unit > product_prod_int_int) > set_Product_unit > set_Pr958786334691620121nt_int).
% 129.49/25.35 thf(func_def_1599, type, image_4946983642536848454nteger: (product_unit > produc2591615631378489102nteger) > set_Product_unit > set_Pr2406931829603198446nteger).
% 129.49/25.35 thf(func_def_1600, type, image_7281959405765017145at_nat: (product_unit > product_prod_nat_nat) > set_Product_unit > set_Pr1261947904930325089at_nat).
% 129.49/25.35 thf(func_def_1601, type, image_405062704495631173t_unit: (product_unit > product_unit) > set_Product_unit > set_Product_unit).
% 129.49/25.35 thf(func_def_1602, type, image_6930536745178679216_set_o: (product_unit > set_o) > set_Product_unit > set_set_o).
% 129.49/25.35 thf(func_def_1603, type, image_4571654851662262875nteger: (product_unit > set_Code_integer) > set_Product_unit > set_set_Code_integer).
% 129.49/25.35 thf(func_def_1604, type, image_3995998972318789802et_int: (product_unit > set_int) > set_Product_unit > set_set_int).
% 129.49/25.35 thf(func_def_1605, type, image_8173849991827986510et_nat: (product_unit > set_nat) > set_Product_unit > set_set_nat).
% 129.49/25.35 thf(func_def_1606, type, image_7778121751606633509t_unit: (product_unit > set_Product_unit) > set_Product_unit > set_set_Product_unit).
% 129.49/25.35 thf(func_def_1607, type, image_rat_rat: (rat > rat) > set_rat > set_rat).
% 129.49/25.35 thf(func_def_1608, type, image_8925030434418351941_nat_o: (set_Pr4532377907799695533_nat_o > set_Pr4532377907799695533_nat_o) > set_se1353816476878259085_nat_o > set_se1353816476878259085_nat_o).
% 129.49/25.35 thf(func_def_1609, type, image_set_o_set_o: (set_o > set_o) > set_set_o > set_set_o).
% 129.49/25.35 thf(func_def_1610, type, image_6270750266955655469nteger: (set_Code_integer > set_Code_integer) > set_set_Code_integer > set_set_Code_integer).
% 129.49/25.35 thf(func_def_1611, type, image_524474410958335435et_int: (set_int > set_int) > set_set_int > set_set_int).
% 129.49/25.35 thf(func_def_1612, type, image_set_nat_nat: (set_nat > nat) > set_set_nat > set_nat).
% 129.49/25.35 thf(func_def_1613, type, image_6916300684244607776nteger: (set_nat > set_Code_integer) > set_set_nat > set_set_Code_integer).
% 129.49/25.35 thf(func_def_1614, type, image_7916887816326733075et_nat: (set_nat > set_nat) > set_set_nat > set_set_nat).
% 129.49/25.35 thf(func_def_1615, type, image_6725021117256019401et_nat: (set_nat > set_set_nat) > set_set_nat > set_set_set_nat).
% 129.49/25.35 thf(func_def_1616, type, image_4019945301284682565t_unit: (set_Product_unit > set_Product_unit) > set_set_Product_unit > set_set_Product_unit).
% 129.49/25.35 thf(func_def_1617, type, insert5175938949040314269_nat_o: (produc3658429121746597890et_nat > $o) > set_Pr4532377907799695533_nat_o > set_Pr4532377907799695533_nat_o).
% 129.49/25.35 thf(func_def_1618, type, insert_o: $o > set_o > set_o).
% 129.49/25.35 thf(func_def_1619, type, insert_assn: assn > set_assn > set_assn).
% 129.49/25.35 thf(func_def_1620, type, insert_Code_integer: code_integer > set_Code_integer > set_Code_integer).
% 129.49/25.35 thf(func_def_1621, type, insert8563697753282147485t_unit: heap_e7401611519738050253t_unit > set_he8644174446132254637t_unit > set_he8644174446132254637t_unit).
% 129.49/25.35 thf(func_def_1622, type, insert_int: int > set_int > set_int).
% 129.49/25.35 thf(func_def_1623, type, insert_nat: nat > set_nat > set_nat).
% 129.49/25.35 thf(func_def_1624, type, insert5309882203591314743_nat_o: produc4928098042776334183_nat_o > set_Pr2161125870931222855_nat_o > set_Pr2161125870931222855_nat_o).
% 129.49/25.35 thf(func_def_1625, type, insert8297122186708525866et_nat: produc2732055786443039994et_nat > set_Pr8536935166611901872et_nat > set_Pr8536935166611901872et_nat).
% 129.49/25.35 thf(func_def_1626, type, insert3111075074545270702et_nat: produc3925858234332021118et_nat > set_Pr3286484037609594932et_nat > set_Pr3286484037609594932et_nat).
% 129.49/25.35 thf(func_def_1627, type, insert6201435330877294327od_o_o: product_prod_o_o > set_Product_prod_o_o > set_Product_prod_o_o).
% 129.49/25.35 thf(func_def_1628, type, insert1271470778893645362et_nat: produc3658429121746597890et_nat > set_Pr3948176798113811640et_nat > set_Pr3948176798113811640et_nat).
% 129.49/25.35 thf(func_def_1629, type, insert5033312907999012233nt_int: product_prod_int_int > set_Pr958786334691620121nt_int > set_Pr958786334691620121nt_int).
% 129.49/25.35 thf(func_def_1630, type, insert8211810215607154385at_nat: product_prod_nat_nat > set_Pr1261947904930325089at_nat > set_Pr1261947904930325089at_nat).
% 129.49/25.35 thf(func_def_1631, type, insert4616891024742550839t_unit: produc8459935480633519975t_unit > set_Pr5094982260447487303t_unit > set_Pr5094982260447487303t_unit).
% 129.49/25.35 thf(func_def_1632, type, insert3244108804717745463_nat_o: produc6666532470309664615_nat_o > set_Pr3813038829465207111_nat_o > set_Pr3813038829465207111_nat_o).
% 129.49/25.35 thf(func_def_1633, type, insert1494450971503307511_set_o: produc7369051934464679207_set_o > set_Pr4577759397028426247_set_o > set_Pr4577759397028426247_set_o).
% 129.49/25.35 thf(func_def_1634, type, insert7328952640415156725et_int: produc2115011035271226405et_int > set_Pr2522554150109002629et_int > set_Pr2522554150109002629et_int).
% 129.49/25.35 thf(func_def_1635, type, insert3810226134351308605et_nat: produc7819656566062154093et_nat > set_Pr5488025237498180813et_nat > set_Pr5488025237498180813et_nat).
% 129.49/25.35 thf(func_def_1636, type, insert6793119152790066487t_unit: produc1357844946079266663t_unit > set_Pr3980508049012409671t_unit > set_Pr3980508049012409671t_unit).
% 129.49/25.35 thf(func_def_1637, type, insert_Product_unit: product_unit > set_Product_unit > set_Product_unit).
% 129.49/25.35 thf(func_def_1638, type, insert_rat: rat > set_rat > set_rat).
% 129.49/25.35 thf(func_def_1639, type, insert2507855216428400253_nat_o: set_Pr4532377907799695533_nat_o > set_se1353816476878259085_nat_o > set_se1353816476878259085_nat_o).
% 129.49/25.35 thf(func_def_1640, type, insert_set_o: set_o > set_set_o > set_set_o).
% 129.49/25.35 thf(func_def_1641, type, insert_set_int: set_int > set_set_int > set_set_int).
% 129.49/25.35 thf(func_def_1642, type, insert_set_nat: set_nat > set_set_nat > set_set_nat).
% 129.49/25.35 thf(func_def_1643, type, insert8986028188087627965t_unit: set_Product_unit > set_set_Product_unit > set_set_Product_unit).
% 129.49/25.35 thf(func_def_1644, type, is_emp6650082560925436063_nat_o: set_Pr4532377907799695533_nat_o > $o).
% 129.49/25.35 thf(func_def_1645, type, is_empty_o: set_o > $o).
% 129.49/25.35 thf(func_def_1646, type, is_empty_int: set_int > $o).
% 129.49/25.35 thf(func_def_1647, type, is_empty_nat: set_nat > $o).
% 129.49/25.35 thf(func_def_1648, type, is_emp5529367068699929823t_unit: set_Product_unit > $o).
% 129.49/25.35 thf(func_def_1649, type, is_sin5180296473474724033_nat_o: set_Pr4532377907799695533_nat_o > $o).
% 129.49/25.35 thf(func_def_1650, type, is_singleton_o: set_o > $o).
% 129.49/25.35 thf(func_def_1651, type, is_singleton_int: set_int > $o).
% 129.49/25.35 thf(func_def_1652, type, is_singleton_nat: set_nat > $o).
% 129.49/25.35 thf(func_def_1653, type, is_sin2160648248035936513t_unit: set_Product_unit > $o).
% 129.49/25.35 thf(func_def_1654, type, pairwi2432143822833250616_nat_o: ((produc3658429121746597890et_nat > $o) > (produc3658429121746597890et_nat > $o) > $o) > set_Pr4532377907799695533_nat_o > $o).
% 129.49/25.35 thf(func_def_1655, type, pairwise_o: ($o > $o > $o) > set_o > $o).
% 129.49/25.35 thf(func_def_1656, type, pairwi4741737654671186738nteger: (code_integer > code_integer > $o) > set_Code_integer > $o).
% 129.49/25.35 thf(func_def_1657, type, pairwise_int: (int > int > $o) > set_int > $o).
% 129.49/25.35 thf(func_def_1658, type, pairwise_nat: (nat > nat > $o) > set_nat > $o).
% 129.49/25.35 thf(func_def_1659, type, pairwi7919127943033184120t_unit: (product_unit > product_unit > $o) > set_Product_unit > $o).
% 129.49/25.35 thf(func_def_1660, type, pairwise_set_nat: (set_nat > set_nat > $o) > set_set_nat > $o).
% 129.49/25.35 thf(func_def_1661, type, remove4651630035290841522_nat_o: (produc3658429121746597890et_nat > $o) > set_Pr4532377907799695533_nat_o > set_Pr4532377907799695533_nat_o).
% 129.49/25.35 thf(func_def_1662, type, remove_o: $o > set_o > set_o).
% 129.49/25.35 thf(func_def_1663, type, remove_int: int > set_int > set_int).
% 129.49/25.35 thf(func_def_1664, type, remove_nat: nat > set_nat > set_nat).
% 129.49/25.35 thf(func_def_1665, type, remove_Product_unit: product_unit > set_Product_unit > set_Product_unit).
% 129.49/25.35 thf(func_def_1666, type, the_el5959152443890766018_nat_o: set_Pr4532377907799695533_nat_o > produc3658429121746597890et_nat > $o).
% 129.49/25.35 thf(func_def_1667, type, the_elem_o: set_o > $o).
% 129.49/25.35 thf(func_def_1668, type, the_el4656377885804969768nteger: set_Code_integer > code_integer).
% 129.49/25.35 thf(func_def_1669, type, the_elem_int: set_int > int).
% 129.49/25.35 thf(func_def_1670, type, the_elem_nat: set_nat > nat).
% 129.49/25.35 thf(func_def_1671, type, the_el608902216710161154t_unit: set_Product_unit > product_unit).
% 129.49/25.35 thf(func_def_1672, type, the_elem_set_nat: set_set_nat > set_nat).
% 129.49/25.35 thf(func_def_1673, type, vimage_o_o: ($o > $o) > set_o > set_o).
% 129.49/25.35 thf(func_def_1674, type, vimage_o_int: ($o > int) > set_int > set_o).
% 129.49/25.35 thf(func_def_1675, type, vimage_o_nat: ($o > nat) > set_nat > set_o).
% 129.49/25.35 thf(func_def_1676, type, vimage8945963521958007626od_o_o: ($o > product_prod_o_o) > set_Product_prod_o_o > set_o).
% 129.49/25.35 thf(func_def_1677, type, vimage4304346610107161470_int_o: ($o > product_prod_int_o) > set_Pr903927857289325719_int_o > set_o).
% 129.49/25.35 thf(func_def_1678, type, vimage6125388955539849434_nat_o: ($o > product_prod_nat_o) > set_Pr3149072824959771635_nat_o > set_o).
% 129.49/25.35 thf(func_def_1679, type, vimage9156843197228287893unit_o: ($o > produc7485711547916492274unit_o) > set_Pr5464989305201766866unit_o > set_o).
% 129.49/25.35 thf(func_def_1680, type, vimage4222377557387472176t_unit: ($o > product_unit) > set_Product_unit > set_o).
% 129.49/25.35 thf(func_def_1681, type, vimage3005800680117336574eger_o: (code_integer > $o) > set_o > set_Code_integer).
% 129.49/25.35 thf(func_def_1682, type, vimage1318793397690208119nteger: (code_integer > code_integer) > set_Code_integer > set_Code_integer).
% 129.49/25.35 thf(func_def_1683, type, vimage5837348839036363078er_int: (code_integer > int) > set_int > set_Code_integer).
% 129.49/25.35 thf(func_def_1684, type, vimage5839839309545413354er_nat: (code_integer > nat) > set_nat > set_Code_integer).
% 129.49/25.35 thf(func_def_1685, type, vimage5651688891684837235t_unit: (code_integer > product_unit) > set_Product_unit > set_Code_integer).
% 129.49/25.35 thf(func_def_1686, type, vimage_int_o: (int > $o) > set_o > set_int).
% 129.49/25.35 thf(func_def_1687, type, vimage_int_int: (int > int) > set_int > set_int).
% 129.49/25.35 thf(func_def_1688, type, vimage_int_nat: (int > nat) > set_nat > set_int).
% 129.49/25.35 thf(func_def_1689, type, vimage7872673496510460816nt_int: (int > product_prod_int_int) > set_Pr958786334691620121nt_int > set_int).
% 129.49/25.35 thf(func_def_1690, type, vimage2000912565490760292t_unit: (int > product_unit) > set_Product_unit > set_int).
% 129.49/25.35 thf(func_def_1691, type, vimage8201027140005291904_nat_o: (nat > produc3658429121746597890et_nat > $o) > set_Pr4532377907799695533_nat_o > set_nat).
% 129.49/25.35 thf(func_def_1692, type, vimage_nat_o: (nat > $o) > set_o > set_nat).
% 129.49/25.35 thf(func_def_1693, type, vimage6104394758324456170nteger: (nat > code_integer) > set_Code_integer > set_nat).
% 129.49/25.35 thf(func_def_1694, type, vimage_nat_int: (nat > int) > set_int > set_nat).
% 129.49/25.35 thf(func_def_1695, type, vimage_nat_nat: (nat > nat) > set_nat > set_nat).
% 129.49/25.35 thf(func_def_1696, type, vimage5656379182745576100_o_nat: (nat > product_prod_o_nat) > set_Pr2101469702781467981_o_nat > set_nat).
% 129.49/25.35 thf(func_def_1697, type, vimage9012682431555523728nt_nat: (nat > product_prod_int_nat) > set_Pr3448869479623346877nt_nat > set_nat).
% 129.49/25.35 thf(func_def_1698, type, vimage8013328719654469172at_nat: (nat > product_prod_nat_nat) > set_Pr1261947904930325089at_nat > set_nat).
% 129.49/25.35 thf(func_def_1699, type, vimage4884490618288580032t_unit: (nat > product_unit) > set_Product_unit > set_nat).
% 129.49/25.35 thf(func_def_1700, type, vimage_nat_set_nat: (nat > set_nat) > set_set_nat > set_nat).
% 129.49/25.35 thf(func_def_1701, type, vimage7160910655466636085et_nat: (produc3925858234332021118et_nat > produc2732055786443039994et_nat) > set_Pr8536935166611901872et_nat > set_Pr3286484037609594932et_nat).
% 129.49/25.35 thf(func_def_1702, type, vimage1990136491886319165et_nat: (produc3658429121746597890et_nat > produc3925858234332021118et_nat) > set_Pr3286484037609594932et_nat > set_Pr3948176798113811640et_nat).
% 129.49/25.35 thf(func_def_1703, type, vimage8808348054653608838unit_o: (product_unit > $o) > set_o > set_Product_unit).
% 129.49/25.35 thf(func_def_1704, type, vimage6250838002967538110it_int: (product_unit > int) > set_int > set_Product_unit).
% 129.49/25.35 thf(func_def_1705, type, vimage6253328473476588386it_nat: (product_unit > nat) > set_nat > set_Product_unit).
% 129.49/25.35 thf(func_def_1706, type, vimage508909887551888138t_unit: (product_unit > produc3788275437394581276t_unit) > set_Pr6671815714118170108t_unit > set_Product_unit).
% 129.49/25.35 thf(func_def_1707, type, vimage4422279681566046568t_unit: (product_unit > produc6865911496949736918t_unit) > set_Pr5889264611543340172t_unit > set_Product_unit).
% 129.49/25.35 thf(func_def_1708, type, vimage7305857734363866308t_unit: (product_unit > produc526117512892780850t_unit) > set_Pr4334478416066269672t_unit > set_Product_unit).
% 129.49/25.35 thf(func_def_1709, type, vimage8114940889160087125t_unit: (product_unit > produc8459935480633519975t_unit) > set_Pr5094982260447487303t_unit > set_Product_unit).
% 129.49/25.35 thf(func_def_1710, type, vimage7995052115951654139t_unit: (product_unit > product_unit) > set_Product_unit > set_Product_unit).
% 129.49/25.35 thf(func_def_1711, type, vimage5412618565253405765et_nat: (set_nat > produc3658429121746597890et_nat) > set_Pr3948176798113811640et_nat > set_set_nat).
% 129.49/25.35 thf(func_def_1712, type, set_fo2584398358068434914at_nat: (nat > nat > nat) > nat > nat > nat > nat).
% 129.49/25.35 thf(func_def_1713, type, set_or189985376899183464nteger: code_integer > code_integer > set_Code_integer).
% 129.49/25.35 thf(func_def_1714, type, set_or1266510415728281911st_int: int > int > set_int).
% 129.49/25.35 thf(func_def_1715, type, set_or1269000886237332187st_nat: nat > nat > set_nat).
% 129.49/25.35 thf(func_def_1716, type, set_or8404916559141939852nteger: code_integer > code_integer > set_Code_integer).
% 129.49/25.35 thf(func_def_1717, type, set_or4665077453230672383an_nat: nat > nat > set_nat).
% 129.49/25.35 thf(func_def_1718, type, set_ord_atLeast_nat: nat > set_nat).
% 129.49/25.35 thf(func_def_1719, type, set_ord_atMost_nat: nat > set_nat).
% 129.49/25.35 thf(func_def_1720, type, set_or2715278749043346189nteger: code_integer > code_integer > set_Code_integer).
% 129.49/25.35 thf(func_def_1721, type, set_or4266950643985792945nteger: code_integer > code_integer > set_Code_integer).
% 129.49/25.35 thf(func_def_1722, type, set_or1210151606488870762an_nat: nat > set_nat).
% 129.49/25.35 thf(func_def_1723, type, set_ord_lessThan_nat: nat > set_nat).
% 129.49/25.35 thf(func_def_1724, type, char2: $o > $o > $o > $o > $o > $o > $o > $o > char).
% 129.49/25.35 thf(func_def_1725, type, integer_of_char: char > code_integer).
% 129.49/25.35 thf(func_def_1726, type, sum_ca6763686470577984908at_nat: (nat > nat) > (nat > nat) > sum_sum_nat_nat > nat).
% 129.49/25.35 thf(func_def_1727, type, syntax7398250324933576852n_assn: assn > assn > $o).
% 129.49/25.35 thf(func_def_1728, type, syntax5678989248478167196nt_int: int > int > $o).
% 129.49/25.35 thf(func_def_1729, type, syntax4682126007086162916at_nat: nat > nat > $o).
% 129.49/25.35 thf(func_def_1730, type, syntax3730441303064801268at_rat: rat > rat > $o).
% 129.49/25.35 thf(func_def_1731, type, transi3136567867383547549ic_int: set_Pr958786334691620121nt_int > $o).
% 129.49/25.35 thf(func_def_1732, type, transi3139058337892597825ic_nat: set_Pr1261947904930325089at_nat > $o).
% 129.49/25.35 thf(func_def_1733, type, transi712980445759334096_nat_o: set_Pr2161125870931222855_nat_o > set_Pr2161125870931222855_nat_o).
% 129.49/25.35 thf(func_def_1734, type, transitive_rtrancl_o: set_Product_prod_o_o > set_Product_prod_o_o).
% 129.49/25.35 thf(func_def_1735, type, transi3984306202085910426nteger: set_Pr4811707699266497531nteger > set_Pr4811707699266497531nteger).
% 129.49/25.35 thf(func_def_1736, type, transi2902850859426252137cl_int: set_Pr958786334691620121nt_int > set_Pr958786334691620121nt_int).
% 129.49/25.35 thf(func_def_1737, type, transi2905341329935302413cl_nat: set_Pr1261947904930325089at_nat > set_Pr1261947904930325089at_nat).
% 129.49/25.35 thf(func_def_1738, type, transi344104063936286263et_nat: set_Pr3444600963470892981et_nat > set_Pr3444600963470892981et_nat).
% 129.49/25.35 thf(func_def_1739, type, transi2414916928880341691et_nat: set_Pr7928877670098842301et_nat > set_Pr7928877670098842301et_nat).
% 129.49/25.35 thf(func_def_1740, type, transi6919426408697524543et_nat: set_Pr719794911490849221et_nat > set_Pr719794911490849221et_nat).
% 129.49/25.35 thf(func_def_1741, type, transi5430920176653612092nt_int: set_Pr2560585780119916871nt_int > set_Pr2560585780119916871nt_int).
% 129.49/25.35 thf(func_def_1742, type, transi8609417484261754244at_nat: set_Pr8693737435421807431at_nat > set_Pr8693737435421807431at_nat).
% 129.49/25.35 thf(func_def_1743, type, transi6481887417741755664t_unit: set_Pr5094982260447487303t_unit > set_Pr5094982260447487303t_unit).
% 129.49/25.35 thf(func_def_1744, type, transi4009562729418357614_nat_o: set_Pr2161125870931222855_nat_o > set_Pr2161125870931222855_nat_o).
% 129.49/25.35 thf(func_def_1745, type, transitive_trancl_o: set_Product_prod_o_o > set_Product_prod_o_o).
% 129.49/25.35 thf(func_def_1746, type, transi6261509568448316235cl_int: set_Pr958786334691620121nt_int > set_Pr958786334691620121nt_int).
% 129.49/25.35 thf(func_def_1747, type, transi6264000038957366511cl_nat: set_Pr1261947904930325089at_nat > set_Pr1261947904930325089at_nat).
% 129.49/25.35 thf(func_def_1748, type, transi5221092739591632921et_nat: set_Pr3444600963470892981et_nat > set_Pr3444600963470892981et_nat).
% 129.49/25.35 thf(func_def_1749, type, transi3145040225084697757et_nat: set_Pr7928877670098842301et_nat > set_Pr7928877670098842301et_nat).
% 129.49/25.35 thf(func_def_1750, type, transi4137310414928362273et_nat: set_Pr719794911490849221et_nat > set_Pr719794911490849221et_nat).
% 129.49/25.35 thf(func_def_1751, type, transi6288783178788033498nt_int: set_Pr2560585780119916871nt_int > set_Pr2560585780119916871nt_int).
% 129.49/25.35 thf(func_def_1752, type, transi243908449541399842at_nat: set_Pr8693737435421807431at_nat > set_Pr8693737435421807431at_nat).
% 129.49/25.35 thf(func_def_1753, type, transi430408012122777006t_unit: set_Pr5094982260447487303t_unit > set_Pr5094982260447487303t_unit).
% 129.49/25.35 thf(func_def_1754, type, type_d5515310227294319076nteger: ((code_integer > code_integer) > code_integer) > (code_integer > code_integer > code_integer) > set_Code_integer > $o).
% 129.49/25.35 thf(func_def_1755, type, type_d5661033022651575429eger_o: ((produc8923325533196201883nteger > produc8923325533196201883nteger) > $o) > ($o > produc8923325533196201883nteger > produc8923325533196201883nteger) > set_o > $o).
% 129.49/25.35 thf(func_def_1756, type, type_d6544512618915850416nteger: ((produc8923325533196201883nteger > produc8923325533196201883nteger) > code_integer) > (code_integer > produc8923325533196201883nteger > produc8923325533196201883nteger) > set_Code_integer > $o).
% 129.49/25.35 thf(func_def_1757, type, type_d6278479724763174947er_nat: ((produc8923325533196201883nteger > produc8923325533196201883nteger) > nat) > (nat > produc8923325533196201883nteger > produc8923325533196201883nteger) > set_nat > $o).
% 129.49/25.35 thf(func_def_1758, type, type_d6996789741066924922t_unit: ((produc8923325533196201883nteger > produc8923325533196201883nteger) > product_unit) > (product_unit > produc8923325533196201883nteger > produc8923325533196201883nteger) > set_Product_unit > $o).
% 129.49/25.35 thf(func_def_1759, type, type_d3909072315231072503_nat_o: (assn > produc3658429121746597890et_nat > $o) > ((produc3658429121746597890et_nat > $o) > assn) > set_Pr4532377907799695533_nat_o > $o).
% 129.49/25.35 thf(func_def_1760, type, type_d669523880323286244nteger: (code_integer > code_integer > code_integer) > ((code_integer > code_integer) > code_integer) > set_Co726186806682698383nteger > $o).
% 129.49/25.35 thf(func_def_1761, type, type_d6812736842639505741eger_o: (code_integer > $o) > ($o > code_integer) > set_o > $o).
% 129.49/25.35 thf(func_def_1762, type, type_d4796611474492816872nteger: (code_integer > code_integer) > (code_integer > code_integer) > set_Code_integer > $o).
% 129.49/25.35 thf(func_def_1763, type, type_d8366093980585677751er_int: (code_integer > int) > (int > code_integer) > set_int > $o).
% 129.49/25.35 thf(func_def_1764, type, type_d8368584451094728027er_nat: (code_integer > nat) > (nat > code_integer) > set_nat > $o).
% 129.49/25.35 thf(func_def_1765, type, type_d9006111508436661314t_unit: (code_integer > product_unit) > (product_unit > code_integer) > set_Product_unit > $o).
% 129.49/25.35 thf(func_def_1766, type, type_d4410041424927559462al_nat: (code_natural > nat) > (nat > code_natural) > set_nat > $o).
% 129.49/25.35 thf(func_def_1767, type, type_d8633139899873770843nteger: (nat > code_integer) > (code_integer > nat) > set_Code_integer > $o).
% 129.49/25.35 thf(func_def_1768, type, type_d6250493948777748686at_nat: (nat > nat) > (nat > nat) > set_nat > $o).
% 129.49/25.35 thf(func_def_1769, type, type_d3768499744764857476et_nat: (nat > set_nat) > (set_nat > nat) > set_set_nat > $o).
% 129.49/25.35 thf(func_def_1770, type, type_d6188575255521822967unit_o: (product_unit > $o) > ($o > product_unit) > set_o > $o).
% 129.49/25.35 thf(func_def_1771, type, type_d202864840109488260at_nat: (set_nat > nat) > (nat > set_nat) > set_nat > $o).
% 129.49/25.35 thf(func_def_1772, type, accp_P4507279754199951662nt_int: (produc5030937667867769463nt_int > produc5030937667867769463nt_int > $o) > produc5030937667867769463nt_int > $o).
% 129.49/25.35 thf(func_def_1773, type, accp_P5604362734843950218nt_int: (produc6128020648511768019nt_int > produc6128020648511768019nt_int > $o) > produc6128020648511768019nt_int > $o).
% 129.49/25.35 thf(func_def_1774, type, accp_P3050404660685545426at_nat: (produc3574062574353363227at_nat > produc3574062574353363227at_nat > $o) > produc3574062574353363227at_nat > $o).
% 129.49/25.35 thf(func_def_1775, type, accp_P4147487641329543982at_nat: (produc4671145554997361783at_nat > produc4671145554997361783at_nat > $o) > produc4671145554997361783at_nat > $o).
% 129.49/25.35 thf(func_def_1776, type, accp_P1862375125659990705et_nat: (produc2732055786443039994et_nat > produc2732055786443039994et_nat > $o) > produc2732055786443039994et_nat > $o).
% 129.49/25.35 thf(func_def_1777, type, accp_P8458817951426537472et_nat: (produc6197004810343482825et_nat > produc6197004810343482825et_nat > $o) > produc6197004810343482825et_nat > $o).
% 129.49/25.35 thf(func_def_1778, type, accp_P5801069581201407417et_nat: (produc3658429121746597890et_nat > produc3658429121746597890et_nat > $o) > produc3658429121746597890et_nat > $o).
% 129.49/25.35 thf(func_def_1779, type, accp_P1096762738010456898nt_int: (product_prod_int_int > product_prod_int_int > $o) > product_prod_int_int > $o).
% 129.49/25.35 thf(func_def_1780, type, accp_P4275260045618599050at_nat: (product_prod_nat_nat > product_prod_nat_nat > $o) > product_prod_nat_nat > $o).
% 129.49/25.35 thf(func_def_1781, type, lex_pr4722427456421979612et_nat: set_Pr2161125870931222855_nat_o > set_Pr7928877670098842301et_nat > set_Pr3444600963470892981et_nat).
% 129.49/25.35 thf(func_def_1782, type, lex_pr5340572901959109728et_nat: set_Pr2161125870931222855_nat_o > set_Pr719794911490849221et_nat > set_Pr7928877670098842301et_nat).
% 129.49/25.35 thf(func_def_1783, type, lex_pr538634069021852900et_nat: set_Pr1776176089890104647t_unit > set_Pr5488025237498180813et_nat > set_Pr719794911490849221et_nat).
% 129.49/25.35 thf(func_def_1784, type, lex_prod_int_int: set_Pr958786334691620121nt_int > set_Pr958786334691620121nt_int > set_Pr2560585780119916871nt_int).
% 129.49/25.35 thf(func_def_1785, type, lex_prod_int_nat: set_Pr958786334691620121nt_int > set_Pr1261947904930325089at_nat > set_Pr5696620967126086215nt_nat).
% 129.49/25.35 thf(func_def_1786, type, lex_prod_nat_int: set_Pr1261947904930325089at_nat > set_Pr958786334691620121nt_int > set_Pr5557702248415638087at_int).
% 129.49/25.35 thf(func_def_1787, type, lex_prod_nat_nat: set_Pr1261947904930325089at_nat > set_Pr1261947904930325089at_nat > set_Pr8693737435421807431at_nat).
% 129.49/25.35 thf(func_def_1788, type, max_ex5265246208886859912_nat_o: set_Pr2161125870931222855_nat_o > set_Pr3813038829465207111_nat_o).
% 129.49/25.35 thf(func_def_1789, type, max_ext_o: set_Product_prod_o_o > set_Pr4577759397028426247_set_o).
% 129.49/25.35 thf(func_def_1790, type, max_ext_Code_integer: set_Pr4811707699266497531nteger > set_Pr7577011563204128103nteger).
% 129.49/25.35 thf(func_def_1791, type, max_ext_int: set_Pr958786334691620121nt_int > set_Pr2522554150109002629et_int).
% 129.49/25.35 thf(func_def_1792, type, max_ext_nat: set_Pr1261947904930325089at_nat > set_Pr5488025237498180813et_nat).
% 129.49/25.35 thf(func_def_1793, type, max_ext_Product_unit: set_Pr5094982260447487303t_unit > set_Pr3980508049012409671t_unit).
% 129.49/25.35 thf(func_def_1794, type, measure_int: (int > nat) > set_Pr958786334691620121nt_int).
% 129.49/25.35 thf(func_def_1795, type, measure_nat: (nat > nat) > set_Pr1261947904930325089at_nat).
% 129.49/25.35 thf(func_def_1796, type, min_ex3730736537596016502_nat_o: set_Pr2161125870931222855_nat_o > set_Pr3813038829465207111_nat_o).
% 129.49/25.35 thf(func_def_1797, type, min_ext_o: set_Product_prod_o_o > set_Pr4577759397028426247_set_o).
% 129.49/25.35 thf(func_def_1798, type, min_ext_int: set_Pr958786334691620121nt_int > set_Pr2522554150109002629et_int).
% 129.49/25.35 thf(func_def_1799, type, min_ext_nat: set_Pr1261947904930325089at_nat > set_Pr5488025237498180813et_nat).
% 129.49/25.35 thf(func_def_1800, type, min_ext_Product_unit: set_Pr5094982260447487303t_unit > set_Pr3980508049012409671t_unit).
% 129.49/25.35 thf(func_def_1801, type, wf_Pro8787509627421085110_nat_o: set_Pr2161125870931222855_nat_o > $o).
% 129.49/25.35 thf(func_def_1802, type, wf_o: set_Product_prod_o_o > $o).
% 129.49/25.35 thf(func_def_1803, type, wf_int: set_Pr958786334691620121nt_int > $o).
% 129.49/25.35 thf(func_def_1804, type, wf_nat: set_Pr1261947904930325089at_nat > $o).
% 129.49/25.35 thf(func_def_1805, type, wf_Product_unit: set_Pr5094982260447487303t_unit > $o).
% 129.49/25.35 thf(func_def_1806, type, chains_int: set_Pr958786334691620121nt_int > set_set_int).
% 129.49/25.35 thf(func_def_1807, type, chains_nat: set_Pr1261947904930325089at_nat > set_set_nat).
% 129.49/25.35 thf(func_def_1808, type, chains_nat2: set_set_nat > set_set_set_nat).
% 129.49/25.35 thf(func_def_1809, type, pred_c177362287423414904_nat_o: set_Pr4532377907799695533_nat_o > ((produc3658429121746597890et_nat > $o) > (produc3658429121746597890et_nat > $o) > $o) > set_Pr4532377907799695533_nat_o > $o).
% 129.49/25.35 thf(func_def_1810, type, pred_chain_o: set_o > ($o > $o > $o) > set_o > $o).
% 129.49/25.35 thf(func_def_1811, type, pred_chain_int: set_int > (int > int > $o) > set_int > $o).
% 129.49/25.35 thf(func_def_1812, type, pred_chain_nat: set_nat > (nat > nat > $o) > set_nat > $o).
% 129.49/25.35 thf(func_def_1813, type, pred_c6915491785556283576t_unit: set_Product_unit > (product_unit > product_unit > $o) > set_Product_unit > $o).
% 129.49/25.35 thf(func_def_1814, type, pred_c5891848916688490280nteger: set_set_Code_integer > (set_Code_integer > set_Code_integer > $o) > set_set_Code_integer > $o).
% 129.49/25.35 thf(func_def_1815, type, pred_chain_set_nat: set_set_nat > (set_nat > set_nat > $o) > set_set_nat > $o).
% 129.49/25.35 thf(func_def_1816, type, member6576561426505652726_nat_o: (produc3658429121746597890et_nat > $o) > set_Pr4532377907799695533_nat_o > $o).
% 129.49/25.35 thf(func_def_1817, type, member_o: $o > set_o > $o).
% 129.49/25.35 thf(func_def_1818, type, member_assn: assn > set_assn > $o).
% 129.49/25.35 thf(func_def_1819, type, member_Code_integer: code_integer > set_Code_integer > $o).
% 129.49/25.35 thf(func_def_1820, type, member_filter_nat: filter_nat > set_filter_nat > $o).
% 129.49/25.35 thf(func_def_1821, type, member242389114213480694t_unit: heap_e7401611519738050253t_unit > set_he8644174446132254637t_unit > $o).
% 129.49/25.35 thf(func_def_1822, type, member_int: int > set_int > $o).
% 129.49/25.35 thf(func_def_1823, type, member_nat: nat > set_nat > $o).
% 129.49/25.35 thf(func_def_1824, type, member8781333585448626064_nat_o: produc4928098042776334183_nat_o > set_Pr2161125870931222855_nat_o > $o).
% 129.49/25.35 thf(func_def_1825, type, member6124377750444531601et_nat: produc2732055786443039994et_nat > set_Pr8536935166611901872et_nat > $o).
% 129.49/25.35 thf(func_def_1826, type, member1996754912294343701et_nat: produc3925858234332021118et_nat > set_Pr3286484037609594932et_nat > $o).
% 129.49/25.35 thf(func_def_1827, type, member7466972457876170832od_o_o: product_prod_o_o > set_Product_prod_o_o > $o).
% 129.49/25.35 thf(func_def_1828, type, member7847949116333733898_o_int: product_prod_o_int > set_Pr8834758594704517033_o_int > $o).
% 129.49/25.35 thf(func_def_1829, type, member2802428098988154798_o_nat: product_prod_o_nat > set_Pr2101469702781467981_o_nat > $o).
% 129.49/25.35 thf(func_def_1830, type, member9215336023991391429t_unit: produc3788275437394581276t_unit > set_Pr6671815714118170108t_unit > $o).
% 129.49/25.35 thf(func_def_1831, type, member157494554546826820nteger: produc8923325533196201883nteger > set_Pr4811707699266497531nteger > $o).
% 129.49/25.35 thf(func_def_1832, type, member5046970461208427408t_unit: produc8126358835630522215t_unit > set_Pr1776176089890104647t_unit > $o).
% 129.49/25.35 thf(func_def_1833, type, member6260224972018164377et_nat: produc3658429121746597890et_nat > set_Pr3948176798113811640et_nat > $o).
% 129.49/25.35 thf(func_def_1834, type, member4489920277610959864_int_o: product_prod_int_o > set_Pr903927857289325719_int_o > $o).
% 129.49/25.35 thf(func_def_1835, type, member5262025264175285858nt_int: product_prod_int_int > set_Pr958786334691620121nt_int > $o).
% 129.49/25.35 thf(func_def_1836, type, member216504246829706758nt_nat: product_prod_int_nat > set_Pr3448869479623346877nt_nat > $o).
% 129.49/25.35 thf(func_def_1837, type, member6688357403439981677t_unit: produc6865911496949736918t_unit > set_Pr5889264611543340172t_unit > $o).
% 129.49/25.35 thf(func_def_1838, type, member6310962623043647828_nat_o: product_prod_nat_o > set_Pr3149072824959771635_nat_o > $o).
% 129.49/25.35 thf(func_def_1839, type, member4262671552274231302at_int: product_prod_nat_int > set_Pr7995236796853374141at_int > $o).
% 129.49/25.35 thf(func_def_1840, type, member8440522571783428010at_nat: product_prod_nat_nat > set_Pr1261947904930325089at_nat > $o).
% 129.49/25.35 thf(func_def_1841, type, member348563419383025609t_unit: produc526117512892780850t_unit > set_Pr4334478416066269672t_unit > $o).
% 129.49/25.35 thf(func_def_1842, type, member6341495586645257982et_nat: produc5657529347773406293et_nat > set_Pr3444600963470892981et_nat > $o).
% 129.49/25.35 thf(func_def_1843, type, member8354763867512313179_nat_o: produc4962441903510821572_nat_o > set_Pr7705059561495379578_nat_o > $o).
% 129.49/25.35 thf(func_def_1844, type, member4763271486408492550et_nat: produc6830853553727218525et_nat > set_Pr7928877670098842301et_nat > $o).
% 129.49/25.35 thf(func_def_1845, type, member6613709072892554et_nat: produc4658369026151987169et_nat > set_Pr3721596426557623617et_nat > $o).
% 129.49/25.35 thf(func_def_1846, type, member4978015717948814295_nat_o: produc6907119039986491712_nat_o > set_Pr4208978187861016310_nat_o > $o).
% 129.49/25.35 thf(func_def_1847, type, member2737525317995836554et_nat: produc7389280635074931169et_nat > set_Pr8807100098903490369et_nat > $o).
% 129.49/25.35 thf(func_def_1848, type, member6099555550032318734et_nat: produc8111630337999740517et_nat > set_Pr719794911490849221et_nat > $o).
% 129.49/25.35 thf(func_def_1849, type, member8566619992076573584nt_int: produc1219242969750017639nt_int > set_Pr2560585780119916871nt_int > $o).
% 129.49/25.35 thf(func_def_1850, type, member2930414080797523088nt_nat: produc4806409095325742951nt_nat > set_Pr5696620967126086215nt_nat > $o).
% 129.49/25.35 thf(func_def_1851, type, member4619661753630440592at_int: produc6495656768158660455at_int > set_Pr5557702248415638087at_int > $o).
% 129.49/25.35 thf(func_def_1852, type, member8206827879206165904at_nat: produc859450856879609959at_nat > set_Pr8693737435421807431at_nat > $o).
% 129.49/25.35 thf(func_def_1853, type, member3689400097658526619unit_o: produc7485711547916492274unit_o > set_Pr5464989305201766866unit_o > $o).
% 129.49/25.35 thf(func_def_1854, type, member5468524742225079679it_int: produc5646078835734834920it_int > set_Pr8497134830871917726it_int > $o).
% 129.49/25.35 thf(func_def_1855, type, member423003724879500579it_nat: produc600557818389255820it_nat > set_Pr1763845938948868674it_nat > $o).
% 129.49/25.35 thf(func_def_1856, type, member7821138191923116944t_unit: produc8459935480633519975t_unit > set_Pr5094982260447487303t_unit > $o).
% 129.49/25.35 thf(func_def_1857, type, member6623172407746373520_nat_o: produc6666532470309664615_nat_o > set_Pr3813038829465207111_nat_o > $o).
% 129.49/25.35 thf(func_def_1858, type, member9116954335612470352_set_o: produc7369051934464679207_set_o > set_Pr4577759397028426247_set_o > $o).
% 129.49/25.35 thf(func_def_1859, type, member4307123515891402160nteger: produc6491284506569428743nteger > set_Pr7577011563204128103nteger > $o).
% 129.49/25.35 thf(func_def_1860, type, member2572552093476627150et_int: produc2115011035271226405et_int > set_Pr2522554150109002629et_int > $o).
% 129.49/25.35 thf(func_def_1861, type, member727419437372278867t_unit: produc7348995623955488188t_unit > set_Pr4071100604818519410t_unit > $o).
% 129.49/25.35 thf(func_def_1862, type, member8277197624267554838et_nat: produc7819656566062154093et_nat > set_Pr5488025237498180813et_nat > $o).
% 129.49/25.35 thf(func_def_1863, type, member4189953012975926160t_unit: produc1357844946079266663t_unit > set_Pr3980508049012409671t_unit > $o).
% 129.49/25.35 thf(func_def_1864, type, member_Product_unit: product_unit > set_Product_unit > $o).
% 129.49/25.35 thf(func_def_1865, type, member_rat: rat > set_rat > $o).
% 129.49/25.35 thf(func_def_1866, type, member1452989533377600726_nat_o: set_Pr4532377907799695533_nat_o > set_se1353816476878259085_nat_o > $o).
% 129.49/25.35 thf(func_def_1867, type, member_set_o: set_o > set_set_o > $o).
% 129.49/25.35 thf(func_def_1868, type, member1834298987299537450nteger: set_Code_integer > set_set_Code_integer > $o).
% 129.49/25.35 thf(func_def_1869, type, member_set_int: set_int > set_set_int > $o).
% 129.49/25.35 thf(func_def_1870, type, member_set_nat: set_nat > set_set_nat > $o).
% 129.49/25.35 thf(func_def_1871, type, member5475208728337331568_nat_o: set_Pr2161125870931222855_nat_o > set_se7663487083892625191_nat_o > $o).
% 129.49/25.35 thf(func_def_1872, type, member3157015407008874800od_o_o: set_Product_prod_o_o > set_se491199571785101031od_o_o > $o).
% 129.49/25.35 thf(func_def_1873, type, member2340774599025711042nt_int: set_Pr958786334691620121nt_int > set_se6260736226359567993nt_int > $o).
% 129.49/25.35 thf(func_def_1874, type, member2643936169264416010at_nat: set_Pr1261947904930325089at_nat > set_se7855581050983116737at_nat > $o).
% 129.49/25.35 thf(func_def_1875, type, member1248447801810324848t_unit: set_Pr5094982260447487303t_unit > set_se8994758365886191399t_unit > $o).
% 129.49/25.35 thf(func_def_1876, type, member5877623283571906838t_unit: set_Product_unit > set_set_Product_unit > $o).
% 129.49/25.35 thf(func_def_1877, type, member_set_set_nat: set_set_nat > set_set_set_nat > $o).
% 129.49/25.35 thf(func_def_1878, type, h: produc3658429121746597890et_nat).
% 129.49/25.35 thf(func_def_1897, type, sP0: set_nat > nat > set_nat > $o).
% 129.49/25.35 thf(func_def_1898, type, sP1: set_nat > nat > set_nat > $o).
% 129.49/25.35 thf(func_def_1899, type, sK2: set_nat > set_nat > nat).
% 129.49/25.35 thf(func_def_1900, type, sK3: produc3658429121746597890et_nat > $o > heap_e7401611519738050253t_unit).
% 129.49/25.35 thf(func_def_1901, type, sK4: produc3658429121746597890et_nat > $o > set_nat).
% 129.49/25.35 thf(func_def_1902, type, sK5: assn > produc3658429121746597890et_nat).
% 129.49/25.35 thf(func_def_1903, type, sK6: assn > produc3658429121746597890et_nat).
% 129.49/25.35 thf(func_def_1904, type, sK7: set_nat > nat).
% 129.49/25.35 thf(func_def_1905, type, sK8: set_nat > set_nat > set_nat > set_nat).
% 129.49/25.35 thf(func_def_1906, type, sK9: set_nat > set_nat > set_nat > set_nat).
% 129.49/25.35 thf(func_def_1907, type, sK10: produc3658429121746597890et_nat > set_nat).
% 129.49/25.35 thf(func_def_1908, type, sK11: produc3658429121746597890et_nat > heap_e7401611519738050253t_unit).
% 129.49/25.35 thf(func_def_1909, type, sK12: set_Pr4532377907799695533_nat_o > produc3658429121746597890et_nat > $o).
% 129.49/25.35 thf(func_def_1910, type, sK13: (produc3658429121746597890et_nat > $o) > assn > produc3658429121746597890et_nat).
% 129.49/25.35 thf(func_def_1911, type, sK14: set_nat > nat).
% 129.49/25.35 thf(func_def_1912, type, sK15: set_nat > set_nat > nat).
% 129.49/25.35 thf(func_def_1913, type, sK16: produc3658429121746597890et_nat > heap_e7401611519738050253t_unit).
% 129.49/25.35 thf(func_def_1914, type, sK17: produc3658429121746597890et_nat > set_nat).
% 129.49/25.35 thf(func_def_1915, type, sK18: produc3658429121746597890et_nat > heap_e7401611519738050253t_unit).
% 129.49/25.35 thf(func_def_1916, type, sK19: produc3658429121746597890et_nat > set_nat).
% 129.49/25.35 thf(func_def_1917, type, sK20: ((produc3658429121746597890et_nat > $o) > $o) > ((produc3658429121746597890et_nat > $o) > $o) > produc3658429121746597890et_nat > $o).
% 129.49/25.35 thf(func_def_1918, type, sK21: set_Pr4532377907799695533_nat_o > produc3658429121746597890et_nat > $o).
% 129.49/25.35 thf(func_def_1919, type, sK22: assn > (produc3658429121746597890et_nat > $o) > produc3658429121746597890et_nat).
% 129.49/25.35 thf(func_def_1920, type, sK23: (produc3658429121746597890et_nat > $o) > assn).
% 129.49/25.35 thf(func_def_1921, type, sK24: produc3658429121746597890et_nat > heap_e7401611519738050253t_unit).
% 129.49/25.35 thf(func_def_1922, type, sK25: produc3658429121746597890et_nat > set_nat).
% 129.49/25.35 thf(func_def_1923, type, sK26: ((produc3658429121746597890et_nat > $o) > $o) > assn).
% 129.49/25.35 thf(func_def_1924, type, sK27: set_nat > nat).
% 129.49/25.35 thf(func_def_1925, type, sK28: ((produc3658429121746597890et_nat > $o) > $o) > ((produc3658429121746597890et_nat > $o) > $o) > produc3658429121746597890et_nat > $o).
% 129.49/25.35 thf(func_def_1926, type, sK29: (nat > $o) > set_nat > nat).
% 129.49/25.35 thf(func_def_1927, type, sK30: (nat > $o) > set_nat > nat).
% 129.49/25.35 thf(func_def_1928, type, sK31: (nat > $o) > set_nat > set_nat > nat).
% 129.49/25.35 thf(func_def_1929, type, sK32: assn > produc3658429121746597890et_nat > assn > set_nat).
% 129.49/25.35 thf(func_def_1930, type, sK33: assn > produc3658429121746597890et_nat > assn > heap_e7401611519738050253t_unit).
% 129.49/25.35 thf(func_def_1931, type, sK34: assn > produc3658429121746597890et_nat > assn > set_nat).
% 129.49/25.35 thf(func_def_1932, type, sK35: set_Pr4532377907799695533_nat_o > produc3658429121746597890et_nat > $o).
% 129.49/25.35 thf(func_def_1933, type, sK36: (produc3658429121746597890et_nat > $o) > set_nat).
% 129.49/25.35 thf(func_def_1934, type, sK37: (produc3658429121746597890et_nat > $o) > heap_e7401611519738050253t_unit).
% 129.49/25.35 thf(func_def_1935, type, sK38: set_nat > (nat > $o) > nat).
% 129.49/25.35 thf(func_def_1936, type, sK39: (nat > $o) > set_nat > nat).
% 129.49/25.35 thf(func_def_1937, type, sK40: set_nat > (nat > $o) > set_nat > nat).
% 129.49/25.35 thf(func_def_1938, type, sK41: set_nat > produc3658429121746597890et_nat > heap_e7401611519738050253t_unit).
% 129.49/25.35 thf(func_def_1939, type, sK42: assn > assn > produc3658429121746597890et_nat).
% 129.49/25.35 thf(func_def_1940, type, sK43: assn > assn > produc3658429121746597890et_nat).
% 129.49/25.35 thf(func_def_1941, type, sK44: assn > produc3658429121746597890et_nat > $o).
% 129.49/25.35 thf(func_def_1942, type, sK45: set_nat > set_nat > nat).
% 129.49/25.35 thf(func_def_1943, type, sK46: ((produc3658429121746597890et_nat > $o) > $o) > produc3658429121746597890et_nat > $o).
% 129.49/25.35 thf(func_def_1944, type, sK47: produc3658429121746597890et_nat > set_nat).
% 129.49/25.35 thf(func_def_1945, type, sK48: produc3658429121746597890et_nat > heap_e7401611519738050253t_unit).
% 129.49/25.35 thf(func_def_1946, type, sK49: ((produc3658429121746597890et_nat > $o) > $o) > produc3658429121746597890et_nat > $o).
% 129.49/25.35 thf(func_def_1947, type, sK50: (assn > $o) > produc3658429121746597890et_nat > $o).
% 129.49/25.35 thf(func_def_1948, type, sK51: set_nat > set_nat > nat).
% 129.49/25.35 thf(func_def_1949, type, sK52: set_nat > set_nat > nat).
% 129.49/25.35 thf(func_def_1950, type, sK53: set_Pr4532377907799695533_nat_o > ((produc3658429121746597890et_nat > $o) > $o) > ((produc3658429121746597890et_nat > $o) > $o) > produc3658429121746597890et_nat > $o).
% 129.49/25.35 thf(func_def_1951, type, sK54: set_Pr4532377907799695533_nat_o > produc3658429121746597890et_nat > $o).
% 129.49/25.35 thf(func_def_1952, type, sK55: set_nat > nat).
% 129.49/25.35 thf(func_def_1953, type, sK56: ((produc3658429121746597890et_nat > $o) > $o) > ((produc3658429121746597890et_nat > $o) > $o) > produc3658429121746597890et_nat > $o).
% 129.49/25.35 thf(func_def_1954, type, sK57: set_Pr4532377907799695533_nat_o > ((produc3658429121746597890et_nat > $o) > $o) > produc3658429121746597890et_nat > $o).
% 129.49/25.35 thf(func_def_1956, type, ph59: !>[X0: $tType]:(X0)).
% 129.49/25.35 thf(func_def_1957, type, sK60: produc3658429121746597890et_nat).
% 129.49/25.35 thf(func_def_1958, type, sK61: assn > assn > produc3658429121746597890et_nat).
% 129.49/25.35 thf(func_def_1959, type, sK62: assn > assn > produc3658429121746597890et_nat).
% 129.49/25.35 thf(func_def_1960, type, sK63: assn > assn > produc3658429121746597890et_nat).
% 129.49/25.35 thf(func_def_1961, type, sK64: assn > produc3658429121746597890et_nat).
% 129.49/25.35 thf(func_def_1962, type, sK65: produc3658429121746597890et_nat).
% 129.49/25.35 thf(f26173,plain,(
% 129.49/25.35 $false),
% 129.49/25.35 inference(avatar_sat_refutation,[status(thm)],[f23972,f23975,f25697,f26172])).
% 129.49/25.35 thf(f26172,plain,(
% 129.49/25.35 ~spl58_1 | spl58_2),
% 129.49/25.35 inference(avatar_contradiction_clause,[status(thm)],[f26171])).
% 129.49/25.35 thf(f26171,plain,(
% 129.49/25.35 $false | (~spl58_1 | spl58_2)),
% 129.49/25.35 inference(subsumption_resolution,[status(thm)],[f26170,f26128])).
% 129.49/25.35 thf(f26128,plain,(
% 129.49/25.35 ((one_assn_raw @ h) != $true) | spl58_2),
% 129.49/25.35 inference(superposition,[status(thm)],[f23970,f24618])).
% 129.49/25.35 thf(f24618,plain,(
% 129.49/25.35 ( ! [X1 : produc3658429121746597890et_nat] : (((one_assn_raw @ X1) = (rep_assn @ (abs_assn @ one_assn_raw) @ X1))) )),
% 129.49/25.35 inference(argument_congruence,[status(thm)],[f24167])).
% 129.49/25.35 thf(f24167,plain,(
% 129.49/25.35 (one_assn_raw = (rep_assn @ (abs_assn @ one_assn_raw)))),
% 129.49/25.35 inference(trivial_inequality_removal,[status(thm)],[f24165])).
% 129.49/25.35 thf(f24165,plain,(
% 129.49/25.35 (one_assn_raw = (rep_assn @ (abs_assn @ one_assn_raw))) | ($true != $true)),
% 129.49/25.35 inference(superposition,[status(thm)],[f23977,f23559])).
% 129.49/25.35 thf(f23559,plain,(
% 129.49/25.35 ((proper @ one_assn_raw) = $true)),
% 129.49/25.35 inference(cnf_transformation,[status(thm)],[f10142])).
% 129.49/25.35 thf(f10142,plain,(
% 129.49/25.35 ((proper @ one_assn_raw) = $true)),
% 129.49/25.35 inference(fool_elimination,[status(thm)],[f10141])).
% 129.49/25.35 thf(f10141,plain,(
% 129.49/25.35 (proper @ one_assn_raw)),
% 129.49/25.35 inference(rectify,[status(thm)],[f89])).
% 129.49/25.35 thf(f89,axiom,(
% 129.49/25.35 (proper @ one_assn_raw)),
% 129.49/25.35 file('/export/starexec/sandbox/tmp/tmp.2EMJwA8PHy/DTF2THF_22696.p',fact_88_one__assn__proper)).
% 129.49/25.35 thf(f23977,plain,(
% 129.49/25.35 ( ! [X0 : produc3658429121746597890et_nat > $o] : (((proper @ X0) != $true) | ((rep_assn @ (abs_assn @ X0)) = X0)) )),
% 129.49/25.35 inference(backward_demodulation,[status(thm)],[f23620,f23648])).
% 129.49/25.35 thf(f23648,plain,(
% 129.49/25.35 ( ! [X0 : produc3658429121746597890et_nat > $o,X1 : (produc3658429121746597890et_nat > $o) > $o] : (((X1 @ X0) = (member6576561426505652726_nat_o @ X0 @ (collec939566748876313656_nat_o @ X1)))) )),
% 129.49/25.35 inference(cnf_transformation,[status(thm)],[f12283])).
% 129.49/25.35 thf(f12283,plain,(
% 129.49/25.35 ! [X0 : produc3658429121746597890et_nat > $o,X1 : (produc3658429121746597890et_nat > $o) > $o] : ((X1 @ X0) = (member6576561426505652726_nat_o @ X0 @ (collec939566748876313656_nat_o @ X1)))),
% 129.49/25.35 inference(fool_elimination,[status(thm)],[f12282])).
% 129.49/25.35 thf(f12282,plain,(
% 129.49/25.35 ! [X0 : produc3658429121746597890et_nat > $o,X1 : (produc3658429121746597890et_nat > $o) > $o] : ((X1 @ X0) = (member6576561426505652726_nat_o @ X0 @ (collec939566748876313656_nat_o @ X1)))),
% 129.49/25.35 inference(rectify,[status(thm)],[f103])).
% 129.49/25.35 thf(f103,axiom,(
% 129.49/25.35 ! [X8 : produc3658429121746597890et_nat > $o,X2 : (produc3658429121746597890et_nat > $o) > $o] : ((member6576561426505652726_nat_o @ X8 @ (collec939566748876313656_nat_o @ X2)) = (X2 @ X8))),
% 129.49/25.35 file('/export/starexec/sandbox/tmp/tmp.2EMJwA8PHy/DTF2THF_22696.p',fact_102_mem__Collect__eq)).
% 129.49/25.35 thf(f23620,plain,(
% 129.49/25.35 ( ! [X0 : produc3658429121746597890et_nat > $o] : (((member6576561426505652726_nat_o @ X0 @ (collec939566748876313656_nat_o @ proper)) != $true) | ((rep_assn @ (abs_assn @ X0)) = X0)) )),
% 129.49/25.35 inference(cnf_transformation,[status(thm)],[f23198])).
% 129.49/25.35 thf(f23198,plain,(
% 129.49/25.35 ! [X0 : produc3658429121746597890et_nat > $o] : (((member6576561426505652726_nat_o @ X0 @ (collec939566748876313656_nat_o @ proper)) != $true) | ((rep_assn @ (abs_assn @ X0)) = X0))),
% 129.49/25.35 inference(ennf_transformation,[status(thm)],[f11495])).
% 129.49/25.35 thf(f11495,plain,(
% 129.49/25.35 ! [X0 : produc3658429121746597890et_nat > $o] : (((member6576561426505652726_nat_o @ X0 @ (collec939566748876313656_nat_o @ proper)) = $true) => ((rep_assn @ (abs_assn @ X0)) = X0))),
% 129.49/25.35 inference(fool_elimination,[status(thm)],[f11494])).
% 129.49/25.35 thf(f11494,plain,(
% 129.49/25.35 ! [X0 : produc3658429121746597890et_nat > $o] : ((member6576561426505652726_nat_o @ X0 @ (collec939566748876313656_nat_o @ proper)) => ((rep_assn @ (abs_assn @ X0)) = X0))),
% 129.49/25.35 inference(rectify,[status(thm)],[f78])).
% 129.49/25.35 thf(f78,axiom,(
% 129.49/25.35 ! [X1 : produc3658429121746597890et_nat > $o] : ((member6576561426505652726_nat_o @ X1 @ (collec939566748876313656_nat_o @ proper)) => ((rep_assn @ (abs_assn @ X1)) = X1))),
% 129.49/25.35 file('/export/starexec/sandbox/tmp/tmp.2EMJwA8PHy/DTF2THF_22696.p',fact_77_Abs__assn__inverse)).
% 129.49/25.35 thf(f23970,plain,(
% 129.49/25.35 ((rep_assn @ (abs_assn @ one_assn_raw) @ h) != $true) | spl58_2),
% 129.49/25.35 inference(avatar_component_clause,[status(thm)],[f23969])).
% 129.49/25.35 thf(f23969,definition,(
% 129.49/25.35 spl58_2 <=> ((rep_assn @ (abs_assn @ one_assn_raw) @ h) = $true)),
% 129.49/25.35 introduced(definition,[new_symbols(naming,[spl58_2])],[avatar_definition])).
% 129.49/25.35 thf(f26170,plain,(
% 129.49/25.35 ((one_assn_raw @ h) = $true) | ~spl58_1),
% 129.49/25.35 inference(equality_resolution,[status(thm)],[f25959])).
% 129.49/25.35 thf(f25959,plain,(
% 129.49/25.35 ( ! [X0 : produc3658429121746597890et_nat] : (((sK47 @ h) != (sK47 @ X0)) | ((one_assn_raw @ X0) = $true)) ) | ~spl58_1),
% 129.49/25.35 inference(forward_demodulation,[status(thm)],[f24604,f25886])).
% 129.49/25.35 thf(f25886,plain,(
% 129.49/25.35 (bot_bot_set_nat = (sK47 @ h)) | ~spl58_1),
% 129.49/25.35 inference(forward_demodulation,[status(thm)],[f23967,f24025])).
% 129.49/25.35 thf(f24025,plain,(
% 129.49/25.35 ( ! [X0 : produc3658429121746597890et_nat] : (((produc8586169260539613262et_nat @ X0) = (sK47 @ X0))) )),
% 129.49/25.35 inference(superposition,[status(thm)],[f23857,f23649])).
% 129.49/25.35 thf(f23649,plain,(
% 129.49/25.35 ( ! [X0 : produc3658429121746597890et_nat] : (((produc7507926704131184380et_nat @ (sK48 @ X0) @ (sK47 @ X0)) = X0)) )),
% 129.49/25.35 inference(cnf_transformation,[status(thm)],[f23336])).
% 129.49/25.35 thf(f23336,plain,(
% 129.49/25.35 ! [X0 : produc3658429121746597890et_nat] : ((produc7507926704131184380et_nat @ (sK48 @ X0) @ (sK47 @ X0)) = X0)),
% 129.49/25.35 inference(skolemisation,[status(esa),new_symbols(skolem,[sK47,sK48])],[f23167,f23335])).
% 129.49/25.35 thf(f23335,plain,(
% 129.49/25.35 ! [X0 : produc3658429121746597890et_nat] : (? [X1 : set_nat,X2 : heap_e7401611519738050253t_unit] : ((produc7507926704131184380et_nat @ X2 @ X1) = X0) => ((produc7507926704131184380et_nat @ (sK48 @ X0) @ (sK47 @ X0)) = X0))),
% 129.49/25.35 introduced(definition,[],[choice_axiom])).
% 129.49/25.35 thf(f23167,plain,(
% 129.49/25.35 ! [X0 : produc3658429121746597890et_nat] : ? [X1 : set_nat,X2 : heap_e7401611519738050253t_unit] : ((produc7507926704131184380et_nat @ X2 @ X1) = X0)),
% 129.49/25.35 inference(ennf_transformation,[status(thm)],[f9906])).
% 129.49/25.35 thf(f9906,plain,(
% 129.49/25.35 ! [X0 : produc3658429121746597890et_nat] : ~! [X1 : set_nat,X2 : heap_e7401611519738050253t_unit] : ((produc7507926704131184380et_nat @ X2 @ X1) != X0)),
% 129.49/25.35 inference(fool_elimination,[status(thm)],[f9905])).
% 129.49/25.35 thf(f9905,plain,(
% 129.49/25.35 ! [X0 : produc3658429121746597890et_nat] : ~! [X1 : set_nat,X2 : heap_e7401611519738050253t_unit] : ((produc7507926704131184380et_nat @ X2 @ X1) != X0)),
% 129.49/25.35 inference(rectify,[status(thm)],[f161])).
% 129.49/25.35 thf(f161,axiom,(
% 129.49/25.35 ! [X1 : produc3658429121746597890et_nat] : ~! [X21 : set_nat,X20 : heap_e7401611519738050253t_unit] : ((produc7507926704131184380et_nat @ X20 @ X21) != X1)),
% 129.49/25.35 file('/export/starexec/sandbox/tmp/tmp.2EMJwA8PHy/DTF2THF_22696.p',fact_160_old_Oprod_Oexhaust)).
% 129.49/25.35 thf(f23857,plain,(
% 129.49/25.35 ( ! [X2 : heap_e7401611519738050253t_unit,X1 : set_nat] : (((produc8586169260539613262et_nat @ (produc7507926704131184380et_nat @ X2 @ X1)) = X1)) )),
% 129.49/25.35 inference(equality_resolution,[status(thm)],[f23587])).
% 129.49/25.35 thf(f23587,plain,(
% 129.49/25.35 ( ! [X2 : heap_e7401611519738050253t_unit,X0 : produc3658429121746597890et_nat,X1 : set_nat] : (((produc8586169260539613262et_nat @ X0) = X1) | ((produc7507926704131184380et_nat @ X2 @ X1) != X0)) )),
% 129.49/25.35 inference(cnf_transformation,[status(thm)],[f23314])).
% 129.49/25.35 thf(f23314,plain,(
% 129.49/25.35 ! [X0 : produc3658429121746597890et_nat,X1 : set_nat] : ((((produc8586169260539613262et_nat @ X0) = X1) | ! [X2 : heap_e7401611519738050253t_unit] : ((produc7507926704131184380et_nat @ X2 @ X1) != X0)) & (((produc7507926704131184380et_nat @ (sK41 @ X1 @ X0) @ X1) = X0) | ((produc8586169260539613262et_nat @ X0) != X1)))),
% 129.49/25.35 inference(skolemisation,[status(esa),new_symbols(skolem,[sK41])],[f23312,f23313])).
% 129.49/25.35 thf(f23313,plain,(
% 129.49/25.35 ! [X0 : produc3658429121746597890et_nat,X1 : set_nat] : (? [X3 : heap_e7401611519738050253t_unit] : ((produc7507926704131184380et_nat @ X3 @ X1) = X0) => ((produc7507926704131184380et_nat @ (sK41 @ X1 @ X0) @ X1) = X0))),
% 129.49/25.35 introduced(definition,[],[choice_axiom])).
% 129.49/25.35 thf(f23312,plain,(
% 129.49/25.35 ! [X0 : produc3658429121746597890et_nat,X1 : set_nat] : ((((produc8586169260539613262et_nat @ X0) = X1) | ! [X2 : heap_e7401611519738050253t_unit] : ((produc7507926704131184380et_nat @ X2 @ X1) != X0)) & (? [X3 : heap_e7401611519738050253t_unit] : ((produc7507926704131184380et_nat @ X3 @ X1) = X0) | ((produc8586169260539613262et_nat @ X0) != X1)))),
% 129.49/25.35 inference(rectify,[status(thm)],[f23311])).
% 129.49/25.35 thf(f23311,plain,(
% 129.49/25.35 ! [X0 : produc3658429121746597890et_nat,X1 : set_nat] : ((((produc8586169260539613262et_nat @ X0) = X1) | ! [X2 : heap_e7401611519738050253t_unit] : ((produc7507926704131184380et_nat @ X2 @ X1) != X0)) & (? [X2 : heap_e7401611519738050253t_unit] : ((produc7507926704131184380et_nat @ X2 @ X1) = X0) | ((produc8586169260539613262et_nat @ X0) != X1)))),
% 129.49/25.35 inference(nnf_transformation,[status(thm)],[f15946])).
% 129.49/25.35 thf(f15946,plain,(
% 129.49/25.35 ! [X0 : produc3658429121746597890et_nat,X1 : set_nat] : (((produc8586169260539613262et_nat @ X0) = X1) <=> ? [X2 : heap_e7401611519738050253t_unit] : ((produc7507926704131184380et_nat @ X2 @ X1) = X0))),
% 129.49/25.35 inference(fool_elimination,[status(thm)],[f15945])).
% 129.49/25.35 thf(f15945,plain,(
% 129.49/25.35 ! [X0 : produc3658429121746597890et_nat,X1 : set_nat] : (? [X2 : heap_e7401611519738050253t_unit] : ((produc7507926704131184380et_nat @ X2 @ X1) = X0) = ((produc8586169260539613262et_nat @ X0) = X1))),
% 129.49/25.35 inference(rectify,[status(thm)],[f152])).
% 129.49/25.35 thf(f152,axiom,(
% 129.49/25.35 ! [X30 : produc3658429121746597890et_nat,X19 : set_nat] : (? [X31 : heap_e7401611519738050253t_unit] : ((produc7507926704131184380et_nat @ X31 @ X19) = X30) = ((produc8586169260539613262et_nat @ X30) = X19))),
% 129.49/25.35 file('/export/starexec/sandbox/tmp/tmp.2EMJwA8PHy/DTF2THF_22696.p',fact_151_eq__snd__iff)).
% 129.49/25.35 thf(f23967,plain,(
% 129.49/25.35 (bot_bot_set_nat = (produc8586169260539613262et_nat @ h)) | ~spl58_1),
% 129.49/25.35 inference(avatar_component_clause,[status(thm)],[f23965])).
% 129.49/25.35 thf(f23965,definition,(
% 129.49/25.35 spl58_1 <=> (bot_bot_set_nat = (produc8586169260539613262et_nat @ h))),
% 129.49/25.35 introduced(definition,[new_symbols(naming,[spl58_1])],[avatar_definition])).
% 129.49/25.35 thf(f24604,plain,(
% 129.49/25.35 ( ! [X0 : produc3658429121746597890et_nat] : ((bot_bot_set_nat != (sK47 @ X0)) | ((one_assn_raw @ X0) = $true)) )),
% 129.49/25.35 inference(duplicate_literal_removal,[status(thm)],[f24603])).
% 129.49/25.35 thf(f24603,plain,(
% 129.49/25.35 ( ! [X0 : produc3658429121746597890et_nat] : ((bot_bot_set_nat != (sK47 @ X0)) | ((one_assn_raw @ X0) = $true) | ((one_assn_raw @ X0) = $true)) )),
% 129.49/25.35 inference(superposition,[status(thm)],[f23451,f24197])).
% 129.49/25.35 thf(f24197,plain,(
% 129.49/25.35 ( ! [X0 : produc3658429121746597890et_nat] : (((sK47 @ X0) = (sK19 @ X0)) | ((one_assn_raw @ X0) = $true)) )),
% 129.49/25.35 inference(superposition,[status(thm)],[f24036,f23450])).
% 129.49/25.35 thf(f23450,plain,(
% 129.49/25.35 ( ! [X0 : produc3658429121746597890et_nat] : (((produc7507926704131184380et_nat @ (sK18 @ X0) @ (sK19 @ X0)) = X0) | ((one_assn_raw @ X0) = $true)) )),
% 129.49/25.35 inference(cnf_transformation,[status(thm)],[f23249])).
% 129.49/25.35 thf(f23249,plain,(
% 129.49/25.35 ! [X0 : produc3658429121746597890et_nat] : (((bot_bot_set_nat != (sK19 @ X0)) & ((produc7507926704131184380et_nat @ (sK18 @ X0) @ (sK19 @ X0)) = X0)) | ((one_assn_raw @ X0) = $true))),
% 129.49/25.35 inference(skolemisation,[status(esa),new_symbols(skolem,[sK18,sK19])],[f23203,f23248])).
% 129.49/25.35 thf(f23248,plain,(
% 129.49/25.35 ! [X0 : produc3658429121746597890et_nat] : (? [X1 : heap_e7401611519738050253t_unit,X2 : set_nat] : ((bot_bot_set_nat != X2) & ((produc7507926704131184380et_nat @ X1 @ X2) = X0)) => ((bot_bot_set_nat != (sK19 @ X0)) & ((produc7507926704131184380et_nat @ (sK18 @ X0) @ (sK19 @ X0)) = X0)))),
% 129.49/25.35 introduced(definition,[],[choice_axiom])).
% 129.49/25.35 thf(f23203,plain,(
% 129.49/25.35 ! [X0 : produc3658429121746597890et_nat] : (? [X1 : heap_e7401611519738050253t_unit,X2 : set_nat] : ((bot_bot_set_nat != X2) & ((produc7507926704131184380et_nat @ X1 @ X2) = X0)) | ((one_assn_raw @ X0) = $true))),
% 129.49/25.35 inference(ennf_transformation,[status(thm)],[f23085])).
% 129.49/25.35 thf(f23085,plain,(
% 129.49/25.35 ! [X0 : produc3658429121746597890et_nat] : (((one_assn_raw @ X0) != $true) => ~! [X1 : heap_e7401611519738050253t_unit,X2 : set_nat] : (((produc7507926704131184380et_nat @ X1 @ X2) = X0) => (bot_bot_set_nat = X2)))),
% 129.49/25.35 inference(flattening,[status(thm)],[f13534])).
% 129.49/25.35 thf(f13534,plain,(
% 129.49/25.35 ! [X0 : produc3658429121746597890et_nat] : (~((one_assn_raw @ X0) = $true) => ~! [X1 : heap_e7401611519738050253t_unit,X2 : set_nat] : (((produc7507926704131184380et_nat @ X1 @ X2) = X0) => (bot_bot_set_nat = X2)))),
% 129.49/25.35 inference(fool_elimination,[status(thm)],[f13533])).
% 129.49/25.35 thf(f13533,plain,(
% 129.49/25.35 ! [X0 : produc3658429121746597890et_nat] : (~(one_assn_raw @ X0) => ~! [X1 : heap_e7401611519738050253t_unit,X2 : set_nat] : (((produc7507926704131184380et_nat @ X1 @ X2) = X0) => (bot_bot_set_nat = X2)))),
% 129.49/25.35 inference(rectify,[status(thm)],[f82])).
% 129.49/25.35 thf(f82,axiom,(
% 129.49/25.35 ! [X0 : produc3658429121746597890et_nat] : (~(one_assn_raw @ X0) => ~! [X4 : heap_e7401611519738050253t_unit,X17 : set_nat] : (((produc7507926704131184380et_nat @ X4 @ X17) = X0) => (bot_bot_set_nat = X17)))),
% 129.49/25.35 file('/export/starexec/sandbox/tmp/tmp.2EMJwA8PHy/DTF2THF_22696.p',fact_81_one__assn__raw_Oelims_I3_J)).
% 129.49/25.35 thf(f24036,plain,(
% 129.49/25.35 ( ! [X0 : heap_e7401611519738050253t_unit,X1 : set_nat] : (((sK47 @ (produc7507926704131184380et_nat @ X0 @ X1)) = X1)) )),
% 129.49/25.35 inference(superposition,[status(thm)],[f23857,f24025])).
% 129.49/25.35 thf(f23451,plain,(
% 129.49/25.35 ( ! [X0 : produc3658429121746597890et_nat] : ((bot_bot_set_nat != (sK19 @ X0)) | ((one_assn_raw @ X0) = $true)) )),
% 129.49/25.35 inference(cnf_transformation,[status(thm)],[f23249])).
% 129.49/25.35 thf(f25697,plain,(
% 129.49/25.35 spl58_1 | ~spl58_2),
% 129.49/25.35 inference(avatar_contradiction_clause,[status(thm)],[f25696])).
% 129.49/25.35 thf(f25696,plain,(
% 129.49/25.35 $false | (spl58_1 | ~spl58_2)),
% 129.49/25.35 inference(subsumption_resolution,[status(thm)],[f25695,f24032])).
% 129.49/25.35 thf(f24032,plain,(
% 129.49/25.35 (bot_bot_set_nat != (sK47 @ h)) | spl58_1),
% 129.49/25.35 inference(backward_demodulation,[status(thm)],[f24023,f24026])).
% 129.49/25.35 thf(f24026,plain,(
% 129.49/25.35 ( ! [X0 : produc3658429121746597890et_nat] : (((sK47 @ X0) = (sK25 @ X0))) )),
% 129.49/25.35 inference(backward_demodulation,[status(thm)],[f24018,f24025])).
% 129.49/25.35 thf(f24018,plain,(
% 129.49/25.35 ( ! [X0 : produc3658429121746597890et_nat] : (((produc8586169260539613262et_nat @ X0) = (sK25 @ X0))) )),
% 129.49/25.35 inference(superposition,[status(thm)],[f23857,f23493])).
% 129.49/25.35 thf(f23493,plain,(
% 129.49/25.35 ( ! [X0 : produc3658429121746597890et_nat] : (((produc7507926704131184380et_nat @ (sK24 @ X0) @ (sK25 @ X0)) = X0)) )),
% 129.49/25.35 inference(cnf_transformation,[status(thm)],[f23268])).
% 129.49/25.35 thf(f23268,plain,(
% 129.49/25.35 ! [X0 : produc3658429121746597890et_nat] : ((produc7507926704131184380et_nat @ (sK24 @ X0) @ (sK25 @ X0)) = X0)),
% 129.49/25.35 inference(skolemisation,[status(esa),new_symbols(skolem,[sK24,sK25])],[f23153,f23267])).
% 129.49/25.35 thf(f23267,plain,(
% 129.49/25.35 ! [X0 : produc3658429121746597890et_nat] : (? [X1 : heap_e7401611519738050253t_unit,X2 : set_nat] : ((produc7507926704131184380et_nat @ X1 @ X2) = X0) => ((produc7507926704131184380et_nat @ (sK24 @ X0) @ (sK25 @ X0)) = X0))),
% 129.49/25.35 introduced(definition,[],[choice_axiom])).
% 129.49/25.35 thf(f23153,plain,(
% 129.49/25.35 ! [X0 : produc3658429121746597890et_nat] : ? [X1 : heap_e7401611519738050253t_unit,X2 : set_nat] : ((produc7507926704131184380et_nat @ X1 @ X2) = X0)),
% 129.49/25.35 inference(ennf_transformation,[status(thm)],[f17946])).
% 129.49/25.35 thf(f17946,plain,(
% 129.49/25.35 ! [X0 : produc3658429121746597890et_nat] : ~! [X1 : heap_e7401611519738050253t_unit,X2 : set_nat] : ((produc7507926704131184380et_nat @ X1 @ X2) != X0)),
% 129.49/25.35 inference(fool_elimination,[status(thm)],[f17945])).
% 129.49/25.35 thf(f17945,plain,(
% 129.49/25.35 ! [X0 : produc3658429121746597890et_nat] : ~! [X1 : heap_e7401611519738050253t_unit,X2 : set_nat] : ((produc7507926704131184380et_nat @ X1 @ X2) != X0)),
% 129.49/25.35 inference(rectify,[status(thm)],[f96])).
% 129.49/25.35 thf(f96,axiom,(
% 129.49/25.35 ! [X0 : produc3658429121746597890et_nat] : ~! [X4 : heap_e7401611519738050253t_unit,X17 : set_nat] : ((produc7507926704131184380et_nat @ X4 @ X17) != X0)),
% 129.49/25.35 file('/export/starexec/sandbox/tmp/tmp.2EMJwA8PHy/DTF2THF_22696.p',fact_95_one__assn__raw_Ocases)).
% 129.49/25.35 thf(f24023,plain,(
% 129.49/25.35 (bot_bot_set_nat != (sK25 @ h)) | spl58_1),
% 129.49/25.35 inference(backward_demodulation,[status(thm)],[f24014,f24019])).
% 129.49/25.35 thf(f24019,plain,(
% 129.49/25.35 ( ! [X0 : produc3658429121746597890et_nat] : (((sK25 @ X0) = (sK10 @ X0))) )),
% 129.49/25.35 inference(backward_demodulation,[status(thm)],[f24012,f24018])).
% 129.49/25.35 thf(f24012,plain,(
% 129.49/25.35 ( ! [X0 : produc3658429121746597890et_nat] : (((produc8586169260539613262et_nat @ X0) = (sK10 @ X0))) )),
% 129.49/25.35 inference(superposition,[status(thm)],[f23857,f23416])).
% 129.49/25.35 thf(f23416,plain,(
% 129.49/25.35 ( ! [X0 : produc3658429121746597890et_nat] : (((produc7507926704131184380et_nat @ (sK11 @ X0) @ (sK10 @ X0)) = X0)) )),
% 129.49/25.35 inference(cnf_transformation,[status(thm)],[f23231])).
% 129.49/25.35 thf(f23231,plain,(
% 129.49/25.35 ! [X0 : produc3658429121746597890et_nat] : ((produc7507926704131184380et_nat @ (sK11 @ X0) @ (sK10 @ X0)) = X0)),
% 129.49/25.35 inference(skolemisation,[status(esa),new_symbols(skolem,[sK10,sK11])],[f18413,f23230])).
% 129.49/25.35 thf(f23230,plain,(
% 129.49/25.35 ! [X0 : produc3658429121746597890et_nat] : (? [X1 : set_nat,X2 : heap_e7401611519738050253t_unit] : ((produc7507926704131184380et_nat @ X2 @ X1) = X0) => ((produc7507926704131184380et_nat @ (sK11 @ X0) @ (sK10 @ X0)) = X0))),
% 129.49/25.35 introduced(definition,[],[choice_axiom])).
% 129.49/25.35 thf(f18413,plain,(
% 129.49/25.35 ! [X0 : produc3658429121746597890et_nat] : ? [X1 : set_nat,X2 : heap_e7401611519738050253t_unit] : ((produc7507926704131184380et_nat @ X2 @ X1) = X0)),
% 129.49/25.35 inference(fool_elimination,[status(thm)],[f18412])).
% 129.49/25.35 thf(f18412,plain,(
% 129.49/25.35 ! [X0 : produc3658429121746597890et_nat] : ? [X1 : set_nat,X2 : heap_e7401611519738050253t_unit] : ((produc7507926704131184380et_nat @ X2 @ X1) = X0)),
% 129.49/25.35 inference(rectify,[status(thm)],[f166])).
% 129.49/25.35 thf(f166,axiom,(
% 129.49/25.35 ! [X30 : produc3658429121746597890et_nat] : ? [X9 : set_nat,X11 : heap_e7401611519738050253t_unit] : ((produc7507926704131184380et_nat @ X11 @ X9) = X30)),
% 129.49/25.35 file('/export/starexec/sandbox/tmp/tmp.2EMJwA8PHy/DTF2THF_22696.p',fact_165_surj__pair)).
% 129.49/25.35 thf(f24014,plain,(
% 129.49/25.35 (bot_bot_set_nat != (sK10 @ h)) | spl58_1),
% 129.49/25.35 inference(backward_demodulation,[status(thm)],[f23966,f24012])).
% 129.49/25.35 thf(f23966,plain,(
% 129.49/25.35 (bot_bot_set_nat != (produc8586169260539613262et_nat @ h)) | spl58_1),
% 129.49/25.35 inference(avatar_component_clause,[status(thm)],[f23965])).
% 129.49/25.35 thf(f25695,plain,(
% 129.49/25.35 (bot_bot_set_nat = (sK47 @ h)) | ~spl58_2),
% 129.49/25.35 inference(trivial_inequality_removal,[status(thm)],[f25683])).
% 129.49/25.35 thf(f25683,plain,(
% 129.49/25.35 (bot_bot_set_nat = (sK47 @ h)) | ($true != $true) | ~spl58_2),
% 129.49/25.35 inference(superposition,[status(thm)],[f24030,f25665])).
% 129.49/25.35 thf(f25665,plain,(
% 129.49/25.35 ((one_assn_raw @ h) = $true) | ~spl58_2),
% 129.49/25.35 inference(superposition,[status(thm)],[f23971,f24618])).
% 129.49/25.35 thf(f23971,plain,(
% 129.49/25.35 ((rep_assn @ (abs_assn @ one_assn_raw) @ h) = $true) | ~spl58_2),
% 129.49/25.35 inference(avatar_component_clause,[status(thm)],[f23969])).
% 129.49/25.35 thf(f24030,plain,(
% 129.49/25.35 ( ! [X0 : produc3658429121746597890et_nat] : (((one_assn_raw @ X0) != $true) | (bot_bot_set_nat = (sK47 @ X0))) )),
% 129.49/25.35 inference(backward_demodulation,[status(thm)],[f24017,f24026])).
% 129.49/25.35 thf(f24017,plain,(
% 129.49/25.35 ( ! [X0 : produc3658429121746597890et_nat] : ((bot_bot_set_nat = (sK25 @ X0)) | ((one_assn_raw @ X0) != $true)) )),
% 129.49/25.35 inference(superposition,[status(thm)],[f23721,f23493])).
% 129.49/25.35 thf(f23721,plain,(
% 129.49/25.35 ( ! [X0 : set_nat,X1 : heap_e7401611519738050253t_unit] : (((one_assn_raw @ (produc7507926704131184380et_nat @ X1 @ X0)) != $true) | (bot_bot_set_nat = X0)) )),
% 129.49/25.35 inference(cnf_transformation,[status(thm)],[f23376])).
% 129.49/25.35 thf(f23376,plain,(
% 129.49/25.35 ! [X0 : set_nat,X1 : heap_e7401611519738050253t_unit] : ((((one_assn_raw @ (produc7507926704131184380et_nat @ X1 @ X0)) = $true) | (bot_bot_set_nat != X0)) & ((bot_bot_set_nat = X0) | ((one_assn_raw @ (produc7507926704131184380et_nat @ X1 @ X0)) != $true)))),
% 129.49/25.35 inference(nnf_transformation,[status(thm)],[f13111])).
% 129.49/25.35 thf(f13111,plain,(
% 129.49/25.35 ! [X0 : set_nat,X1 : heap_e7401611519738050253t_unit] : (((one_assn_raw @ (produc7507926704131184380et_nat @ X1 @ X0)) = $true) <=> (bot_bot_set_nat = X0))),
% 129.49/25.35 inference(fool_elimination,[status(thm)],[f13110])).
% 129.49/25.35 thf(f13110,plain,(
% 129.49/25.35 ! [X0 : set_nat,X1 : heap_e7401611519738050253t_unit] : ((bot_bot_set_nat = X0) = (one_assn_raw @ (produc7507926704131184380et_nat @ X1 @ X0)))),
% 129.49/25.35 inference(rectify,[status(thm)],[f79])).
% 129.49/25.35 thf(f79,axiom,(
% 129.49/25.35 ! [X16 : set_nat,X12 : heap_e7401611519738050253t_unit] : ((bot_bot_set_nat = X16) = (one_assn_raw @ (produc7507926704131184380et_nat @ X12 @ X16)))),
% 129.49/25.35 file('/export/starexec/sandbox/tmp/tmp.2EMJwA8PHy/DTF2THF_22696.p',fact_78_one__assn__raw_Osimps)).
% 129.49/25.35 thf(f23975,plain,(
% 129.49/25.35 ~spl58_2 | ~spl58_1),
% 129.49/25.35 inference(avatar_split_clause,[status(thm)],[f23724,f23965,f23969])).
% 129.49/25.35 thf(f23724,plain,(
% 129.49/25.35 ((rep_assn @ (abs_assn @ one_assn_raw) @ h) != $true) | (bot_bot_set_nat != (produc8586169260539613262et_nat @ h))),
% 129.49/25.35 inference(cnf_transformation,[status(thm)],[f23377])).
% 129.49/25.35 thf(f23377,plain,(
% 129.49/25.35 (((rep_assn @ (abs_assn @ one_assn_raw) @ h) != $true) | (bot_bot_set_nat != (produc8586169260539613262et_nat @ h))) & (((rep_assn @ (abs_assn @ one_assn_raw) @ h) = $true) | (bot_bot_set_nat = (produc8586169260539613262et_nat @ h)))),
% 129.49/25.35 inference(nnf_transformation,[status(thm)],[f23199])).
% 129.49/25.35 thf(f23199,plain,(
% 129.49/25.35 (bot_bot_set_nat = (produc8586169260539613262et_nat @ h)) <~> ((rep_assn @ (abs_assn @ one_assn_raw) @ h) = $true)),
% 129.49/25.35 inference(ennf_transformation,[status(thm)],[f15878])).
% 129.49/25.35 thf(f15878,plain,(
% 129.49/25.35 ~((bot_bot_set_nat = (produc8586169260539613262et_nat @ h)) <=> ((rep_assn @ (abs_assn @ one_assn_raw) @ h) = $true))),
% 129.49/25.35 inference(fool_elimination,[status(thm)],[f7827])).
% 129.49/25.35 thf(f7827,negated_conjecture,(
% 129.49/25.35 ~((rep_assn @ (abs_assn @ one_assn_raw) @ h) = (bot_bot_set_nat = (produc8586169260539613262et_nat @ h)))),
% 129.49/25.35 inference(negated_conjecture,[status(cth)],[f7826])).
% 129.49/25.35 thf(f7826,conjecture,(
% 129.49/25.35 ((rep_assn @ (abs_assn @ one_assn_raw) @ h) = (bot_bot_set_nat = (produc8586169260539613262et_nat @ h)))),
% 129.49/25.35 file('/export/starexec/sandbox/tmp/tmp.2EMJwA8PHy/DTF2THF_22696.p',conj_0)).
% 129.49/25.35 thf(f23972,plain,(
% 129.49/25.35 spl58_1 | spl58_2),
% 129.49/25.35 inference(avatar_split_clause,[status(thm)],[f23723,f23969,f23965])).
% 129.49/25.35 thf(f23723,plain,(
% 129.49/25.35 ((rep_assn @ (abs_assn @ one_assn_raw) @ h) = $true) | (bot_bot_set_nat = (produc8586169260539613262et_nat @ h))),
% 129.49/25.35 inference(cnf_transformation,[status(thm)],[f23377])).
% 129.49/25.35 % SZS output end Proof for DTF2THF_22696
% 129.49/25.35 % (23288)------------------------------
% 129.49/25.35 % (23288)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 129.49/25.35 % (23288)Termination reason: Refutation
% 129.49/25.35
% 129.49/25.35 % (23288)Memory used [KB]: 33133
% 129.49/25.35 % (23288)Time elapsed: 0.836 s
% 129.49/25.35 % (23288)Instructions burned: 2580 (million)
% 129.49/25.35 % (23288)------------------------------
% 129.49/25.35 % (23288)------------------------------
% 129.49/25.35 % (22819)Success in time 17.195 s
%------------------------------------------------------------------------------